(* ========================================================================= *) (* Paths, connectedness, homotopy, simple connectedness & contractibility. *) (* *) (* (c) Copyright, John Harrison 1998-2008 *) (* (c) Copyright, Valentina Bruno 2010 *) (* ========================================================================= *) needs "Multivariate/homology.ml";; needs "Multivariate/convex.ml";; (* ------------------------------------------------------------------------- *) (* Paths and arcs. *) (* ------------------------------------------------------------------------- *) let path = new_definition `!g:real^1->real^N. path g <=> g continuous_on interval[vec 0,vec 1]`;; let pathstart = new_definition `pathstart (g:real^1->real^N) = g(vec 0)`;; let pathfinish = new_definition `pathfinish (g:real^1->real^N) = g(vec 1)`;; let path_image = new_definition `path_image (g:real^1->real^N) = IMAGE g (interval[vec 0,vec 1])`;; let reversepath = new_definition `reversepath (g:real^1->real^N) = \x. g(vec 1 - x)`;; let joinpaths = new_definition `(g1 ++ g2) = \x. if drop x <= &1 / &2 then g1(&2 % x) else g2(&2 % x - vec 1)`;; let simple_path = new_definition `simple_path (g:real^1->real^N) <=> path g /\ !x y. x IN interval[vec 0,vec 1] /\ y IN interval[vec 0,vec 1] /\ g x = g y ==> x = y \/ x = vec 0 /\ y = vec 1 \/ x = vec 1 /\ y = vec 0`;; let arc = new_definition `arc (g:real^1->real^N) <=> path g /\ !x y. x IN interval [vec 0,vec 1] /\ y IN interval [vec 0,vec 1] /\ g x = g y ==> x = y`;; (* ------------------------------------------------------------------------- *) (* Relate to topological general case. *) (* ------------------------------------------------------------------------- *) let PATH_IN_EUCLIDEAN = prove (`!s:real^N->bool g. path_in (subtopology euclidean s) g <=> path (g o drop) /\ path_image (g o drop) SUBSET s`, REWRITE_TAC[path_in; path; GSYM CONTINUOUS_MAP_EUCLIDEAN] THEN REWRITE_TAC[path_image; INTERVAL_REAL_INTERVAL; DROP_VEC] THEN REWRITE_TAC[GSYM IMAGE_o; GSYM o_ASSOC] THEN ONCE_REWRITE_TAC[IMAGE_o] THEN REWRITE_TAC[IMAGE_LIFT_DROP; CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN REPEAT GEN_TAC THEN REWRITE_TAC[TOPSPACE_EUCLIDEANREAL_SUBTOPOLOGY] THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THENL [ALL_TAC; SUBGOAL_THEN `g:real->real^N = (g o drop) o lift` SUBST1_TAC THENL [REWRITE_TAC[FUN_EQ_THM; o_THM; LIFT_DROP]; ALL_TAC]] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] CONTINUOUS_MAP_COMPOSE)) THEN REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_LIFT; CONTINUOUS_MAP_DROP] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; SUBSET_REFL; TOPSPACE_EUCLIDEANREAL_SUBTOPOLOGY] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; o_DEF; LIFT_DROP]);; let PATH_EUCLIDEAN = prove (`!s g:real^1->real^N. path g /\ path_image g SUBSET s <=> path_in (subtopology euclidean s) (g o lift)`, REWRITE_TAC[PATH_IN_EUCLIDEAN] THEN REWRITE_TAC[o_DEF; LIFT_DROP; ETA_AX]);; let PATH_PATH_IN = prove (`!g:real^1->real^N. path g <=> path_in euclidean (g o lift)`, GEN_TAC THEN ONCE_REWRITE_TAC[GSYM SUBTOPOLOGY_UNIV] THEN REWRITE_TAC[GSYM PATH_EUCLIDEAN; SUBSET_UNIV]);; (* ------------------------------------------------------------------------- *) (* Invariance theorems. *) (* ------------------------------------------------------------------------- *) let PATH_EQ = prove (`!p q. (!t. t IN interval[vec 0,vec 1] ==> p t = q t) /\ path p ==> path q`, REWRITE_TAC[path; CONTINUOUS_ON_EQ]);; let PATH_CONTINUOUS_IMAGE = prove (`!f:real^M->real^N g. path g /\ f continuous_on path_image g ==> path(f o g)`, REWRITE_TAC[path; path_image; CONTINUOUS_ON_COMPOSE]);; let PATH_TRANSLATION_EQ = prove (`!a g:real^1->real^N. path((\x. a + x) o g) <=> path g`, REPEAT GEN_TAC THEN REWRITE_TAC[path] THEN EQ_TAC THEN DISCH_TAC THENL [SUBGOAL_THEN `(g:real^1->real^N) = (\x. --a + x) o (\x. a + x) o g` SUBST1_TAC THENL [REWRITE_TAC[FUN_EQ_THM; o_THM] THEN VECTOR_ARITH_TAC; ALL_TAC]; ALL_TAC] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST]);; add_translation_invariants [PATH_TRANSLATION_EQ];; let PATH_LINEAR_IMAGE_EQ = prove (`!f:real^M->real^N g. linear f /\ (!x y. f x = f y ==> x = y) ==> (path(f o g) <=> path g)`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(X_CHOOSE_THEN `h:real^N->real^M` STRIP_ASSUME_TAC o MATCH_MP LINEAR_INJECTIVE_LEFT_INVERSE) THEN REWRITE_TAC[path] THEN EQ_TAC THEN DISCH_TAC THENL [SUBGOAL_THEN `g:real^1->real^M = h o (f:real^M->real^N) o g` SUBST1_TAC THENL [ASM_REWRITE_TAC[o_ASSOC; I_O_ID]; ALL_TAC]; ALL_TAC] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[LINEAR_CONTINUOUS_ON]);; add_linear_invariants [PATH_LINEAR_IMAGE_EQ];; let PATHSTART_TRANSLATION = prove (`!a g. pathstart((\x. a + x) o g) = a + pathstart g`, REWRITE_TAC[pathstart; o_THM]);; add_translation_invariants [PATHSTART_TRANSLATION];; let PATHSTART_LINEAR_IMAGE_EQ = prove (`!f g. linear f ==> pathstart(f o g) = f(pathstart g)`, REWRITE_TAC[pathstart; o_THM]);; add_linear_invariants [PATHSTART_LINEAR_IMAGE_EQ];; let PATHFINISH_TRANSLATION = prove (`!a g. pathfinish((\x. a + x) o g) = a + pathfinish g`, REWRITE_TAC[pathfinish; o_THM]);; add_translation_invariants [PATHFINISH_TRANSLATION];; let PATHFINISH_LINEAR_IMAGE = prove (`!f g. linear f ==> pathfinish(f o g) = f(pathfinish g)`, REWRITE_TAC[pathfinish; o_THM]);; add_linear_invariants [PATHFINISH_LINEAR_IMAGE];; let PATH_IMAGE_TRANSLATION = prove (`!a g. path_image((\x. a + x) o g) = IMAGE (\x. a + x) (path_image g)`, REWRITE_TAC[path_image; IMAGE_o]);; add_translation_invariants [PATH_IMAGE_TRANSLATION];; let PATH_IMAGE_LINEAR_IMAGE = prove (`!f g. linear f ==> path_image(f o g) = IMAGE f (path_image g)`, REWRITE_TAC[path_image; IMAGE_o]);; add_linear_invariants [PATH_IMAGE_LINEAR_IMAGE];; let REVERSEPATH_TRANSLATION = prove (`!a g. reversepath((\x. a + x) o g) = (\x. a + x) o reversepath g`, REWRITE_TAC[FUN_EQ_THM; reversepath; o_THM]);; add_translation_invariants [REVERSEPATH_TRANSLATION];; let REVERSEPATH_LINEAR_IMAGE = prove (`!f g. linear f ==> reversepath(f o g) = f o reversepath g`, REWRITE_TAC[FUN_EQ_THM; reversepath; o_THM]);; add_linear_invariants [REVERSEPATH_LINEAR_IMAGE];; let JOINPATHS_TRANSLATION = prove (`!a:real^N g1 g2. ((\x. a + x) o g1) ++ ((\x. a + x) o g2) = (\x. a + x) o (g1 ++ g2)`, REWRITE_TAC[joinpaths; FUN_EQ_THM] THEN REPEAT GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[o_THM]);; add_translation_invariants [JOINPATHS_TRANSLATION];; let JOINPATHS_LINEAR_IMAGE = prove (`!f g1 g2. linear f ==> (f o g1) ++ (f o g2) = f o (g1 ++ g2)`, REWRITE_TAC[joinpaths; FUN_EQ_THM] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[o_THM]);; add_linear_invariants [JOINPATHS_LINEAR_IMAGE];; let SIMPLE_PATH_TRANSLATION_EQ = prove (`!a g:real^1->real^N. simple_path((\x. a + x) o g) <=> simple_path g`, REPEAT GEN_TAC THEN REWRITE_TAC[simple_path; PATH_TRANSLATION_EQ] THEN REWRITE_TAC[o_THM; VECTOR_ARITH `a + x:real^N = a + y <=> x = y`]);; add_translation_invariants [SIMPLE_PATH_TRANSLATION_EQ];; let SIMPLE_PATH_LINEAR_IMAGE_EQ = prove (`!f:real^M->real^N g. linear f /\ (!x y. f x = f y ==> x = y) ==> (simple_path(f o g) <=> simple_path g)`, REPEAT STRIP_TAC THEN REWRITE_TAC[simple_path; PATH_TRANSLATION_EQ] THEN BINOP_TAC THENL [ASM_MESON_TAC[PATH_LINEAR_IMAGE_EQ]; ALL_TAC] THEN REWRITE_TAC[o_THM] THEN ASM_MESON_TAC[]);; add_linear_invariants [SIMPLE_PATH_LINEAR_IMAGE_EQ];; let ARC_TRANSLATION_EQ = prove (`!a g:real^1->real^N. arc((\x. a + x) o g) <=> arc g`, REPEAT GEN_TAC THEN REWRITE_TAC[arc; PATH_TRANSLATION_EQ] THEN REWRITE_TAC[o_THM; VECTOR_ARITH `a + x:real^N = a + y <=> x = y`]);; add_translation_invariants [ARC_TRANSLATION_EQ];; let ARC_LINEAR_IMAGE_EQ = prove (`!f:real^M->real^N g. linear f /\ (!x y. f x = f y ==> x = y) ==> (arc(f o g) <=> arc g)`, REPEAT STRIP_TAC THEN REWRITE_TAC[arc; PATH_TRANSLATION_EQ] THEN BINOP_TAC THENL [ASM_MESON_TAC[PATH_LINEAR_IMAGE_EQ]; ALL_TAC] THEN REWRITE_TAC[o_THM] THEN ASM_MESON_TAC[]);; add_linear_invariants [ARC_LINEAR_IMAGE_EQ];; let SIMPLE_PATH_CONTINUOUS_IMAGE = prove (`!f g. simple_path g /\ f continuous_on path_image g /\ (!x y. x IN path_image g /\ y IN path_image g /\ f x = f y ==> x = y) ==> simple_path(f o g)`, REWRITE_TAC[simple_path; INJECTIVE_ON_ALT] THEN SIMP_TAC[PATH_CONTINUOUS_IMAGE] THEN REWRITE_TAC[path_image; o_THM] THEN SET_TAC[]);; let ARC_CONTINUOUS_IMAGE = prove (`!f g:real^1->real^N. arc g /\ f continuous_on path_image g /\ (!x y. x IN path_image g /\ y IN path_image g /\ f x = f y ==> x = y) ==> arc(f o g)`, REWRITE_TAC[arc; INJECTIVE_ON_ALT] THEN SIMP_TAC[PATH_CONTINUOUS_IMAGE] THEN REWRITE_TAC[path_image; o_THM] THEN SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* Basic lemmas about paths. *) (* ------------------------------------------------------------------------- *) let ARC_IMP_SIMPLE_PATH = prove (`!g. arc g ==> simple_path g`, REWRITE_TAC[arc; simple_path] THEN MESON_TAC[]);; let ARC_IMP_PATH = prove (`!g. arc g ==> path g`, REWRITE_TAC[arc] THEN MESON_TAC[]);; let SIMPLE_PATH_IMP_PATH = prove (`!g. simple_path g ==> path g`, REWRITE_TAC[simple_path] THEN MESON_TAC[]);; let SIMPLE_PATH_CASES = prove (`!g:real^1->real^N. simple_path g ==> arc g \/ pathfinish g = pathstart g`, REWRITE_TAC[simple_path; arc; pathfinish; pathstart] THEN REPEAT STRIP_TAC THEN ASM_CASES_TAC `(g:real^1->real^N) (vec 0) = g(vec 1)` THEN ASM_REWRITE_TAC[] THEN MAP_EVERY X_GEN_TAC [`u:real^1`; `v:real^1`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`u:real^1`; `v:real^1`]) THEN ASM_MESON_TAC[]);; let SIMPLE_PATH_IMP_ARC = prove (`!g:real^1->real^N. simple_path g /\ ~(pathfinish g = pathstart g) ==> arc g`, MESON_TAC[SIMPLE_PATH_CASES]);; let ARC_DISTINCT_ENDS = prove (`!g:real^1->real^N. arc g ==> ~(pathfinish g = pathstart g)`, GEN_TAC THEN REWRITE_TAC[arc; pathfinish; pathstart] THEN ONCE_REWRITE_TAC[TAUT `a /\ b /\ c ==> d <=> a /\ b /\ ~d ==> ~c`] THEN DISCH_THEN(MATCH_MP_TAC o CONJUNCT2) THEN REWRITE_TAC[GSYM DROP_EQ; IN_INTERVAL_1; DROP_VEC] THEN CONV_TAC REAL_RAT_REDUCE_CONV);; let ARC_SIMPLE_PATH = prove (`!g:real^1->real^N. arc g <=> simple_path g /\ ~(pathfinish g = pathstart g)`, MESON_TAC[SIMPLE_PATH_CASES; ARC_IMP_SIMPLE_PATH; ARC_DISTINCT_ENDS]);; let SIMPLE_PATH_EQ_ARC = prove (`!g. ~(pathstart g = pathfinish g) ==> (simple_path g <=> arc g)`, SIMP_TAC[ARC_SIMPLE_PATH]);; let PATH_IMAGE_NONEMPTY = prove (`!g. ~(path_image g = {})`, REWRITE_TAC[path_image; IMAGE_EQ_EMPTY; INTERVAL_EQ_EMPTY] THEN SIMP_TAC[DIMINDEX_1; CONJ_ASSOC; LE_ANTISYM; UNWIND_THM1; VEC_COMPONENT; ARITH; REAL_OF_NUM_LT]);; let PATHSTART_IN_PATH_IMAGE = prove (`!g. (pathstart g) IN path_image g`, GEN_TAC THEN REWRITE_TAC[pathstart; path_image] THEN MATCH_MP_TAC FUN_IN_IMAGE THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; REAL_POS]);; let PATHFINISH_IN_PATH_IMAGE = prove (`!g. (pathfinish g) IN path_image g`, GEN_TAC THEN REWRITE_TAC[pathfinish; path_image] THEN MATCH_MP_TAC FUN_IN_IMAGE THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN REAL_ARITH_TAC);; let CONNECTED_PATH_IMAGE = prove (`!g. path g ==> connected(path_image g)`, REWRITE_TAC[path; path_image] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN ASM_SIMP_TAC[CONVEX_CONNECTED; CONVEX_INTERVAL]);; let COMPACT_PATH_IMAGE = prove (`!g. path g ==> compact(path_image g)`, REWRITE_TAC[path; path_image] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[COMPACT_INTERVAL]);; let BOUNDED_PATH_IMAGE = prove (`!g. path g ==> bounded(path_image g)`, MESON_TAC[COMPACT_PATH_IMAGE; COMPACT_IMP_BOUNDED]);; let CLOSED_PATH_IMAGE = prove (`!g. path g ==> closed(path_image g)`, MESON_TAC[COMPACT_PATH_IMAGE; COMPACT_IMP_CLOSED]);; let CONNECTED_SIMPLE_PATH_IMAGE = prove (`!g. simple_path g ==> connected(path_image g)`, MESON_TAC[CONNECTED_PATH_IMAGE; SIMPLE_PATH_IMP_PATH]);; let COMPACT_SIMPLE_PATH_IMAGE = prove (`!g. simple_path g ==> compact(path_image g)`, MESON_TAC[COMPACT_PATH_IMAGE; SIMPLE_PATH_IMP_PATH]);; let BOUNDED_SIMPLE_PATH_IMAGE = prove (`!g. simple_path g ==> bounded(path_image g)`, MESON_TAC[BOUNDED_PATH_IMAGE; SIMPLE_PATH_IMP_PATH]);; let CLOSED_SIMPLE_PATH_IMAGE = prove (`!g. simple_path g ==> closed(path_image g)`, MESON_TAC[CLOSED_PATH_IMAGE; SIMPLE_PATH_IMP_PATH]);; let CONNECTED_ARC_IMAGE = prove (`!g. arc g ==> connected(path_image g)`, MESON_TAC[CONNECTED_PATH_IMAGE; ARC_IMP_PATH]);; let COMPACT_ARC_IMAGE = prove (`!g. arc g ==> compact(path_image g)`, MESON_TAC[COMPACT_PATH_IMAGE; ARC_IMP_PATH]);; let BOUNDED_ARC_IMAGE = prove (`!g. arc g ==> bounded(path_image g)`, MESON_TAC[BOUNDED_PATH_IMAGE; ARC_IMP_PATH]);; let CLOSED_ARC_IMAGE = prove (`!g. arc g ==> closed(path_image g)`, MESON_TAC[CLOSED_PATH_IMAGE; ARC_IMP_PATH]);; let PATHSTART_COMPOSE = prove (`!f p. pathstart(f o p) = f(pathstart p)`, REWRITE_TAC[pathstart; o_THM]);; let PATHFINISH_COMPOSE = prove (`!f p. pathfinish(f o p) = f(pathfinish p)`, REWRITE_TAC[pathfinish; o_THM]);; let PATH_IMAGE_COMPOSE = prove (`!f p. path_image (f o p) = IMAGE f (path_image p)`, REWRITE_TAC[path_image; IMAGE_o]);; let PATH_COMPOSE_JOIN = prove (`!f p q. f o (p ++ q) = (f o p) ++ (f o q)`, REWRITE_TAC[joinpaths; o_DEF; FUN_EQ_THM] THEN MESON_TAC[]);; let PATH_COMPOSE_REVERSEPATH = prove (`!f p. f o reversepath p = reversepath(f o p)`, REWRITE_TAC[reversepath; o_DEF; FUN_EQ_THM] THEN MESON_TAC[]);; let JOIN_PATHS_EQ = prove (`!p q:real^1->real^N. (!t. t IN interval[vec 0,vec 1] ==> p t = p' t) /\ (!t. t IN interval[vec 0,vec 1] ==> q t = q' t) ==> !t. t IN interval[vec 0,vec 1] ==> (p ++ q) t = (p' ++ q') t`, REWRITE_TAC[joinpaths; IN_INTERVAL_1; DROP_VEC] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[IN_INTERVAL_1; DROP_CMUL; DROP_SUB; DROP_VEC] THEN ASM_REAL_ARITH_TAC);; let CARD_EQ_SIMPLE_PATH_IMAGE = prove (`!g. simple_path g ==> path_image g =_c (:real)`, SIMP_TAC[CONNECTED_CARD_EQ_IFF_NONTRIVIAL; CONNECTED_SIMPLE_PATH_IMAGE] THEN GEN_TAC THEN REWRITE_TAC[simple_path; path_image] THEN MATCH_MP_TAC(SET_RULE `(?u v. u IN s /\ v IN s /\ ~(u = a) /\ ~(v = a) /\ ~(u = v)) ==> P /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y \/ x = a /\ y = b \/ x = b /\ y = a) ==> ~(?c. IMAGE f s SUBSET {c})`) THEN MAP_EVERY EXISTS_TAC [`lift(&1 / &3)`; `lift(&1 / &2)`] THEN REWRITE_TAC[IN_INTERVAL_1; LIFT_DROP; GSYM LIFT_NUM; LIFT_EQ] THEN CONV_TAC REAL_RAT_REDUCE_CONV);; let INFINITE_SIMPLE_PATH_IMAGE = prove (`!g. simple_path g ==> INFINITE(path_image g)`, MESON_TAC[CARD_EQ_SIMPLE_PATH_IMAGE; INFINITE; FINITE_IMP_COUNTABLE; UNCOUNTABLE_REAL; CARD_COUNTABLE_CONG]);; let CARD_EQ_ARC_IMAGE = prove (`!g. arc g ==> path_image g =_c (:real)`, MESON_TAC[ARC_IMP_SIMPLE_PATH; CARD_EQ_SIMPLE_PATH_IMAGE]);; let INFINITE_ARC_IMAGE = prove (`!g. arc g ==> INFINITE(path_image g)`, MESON_TAC[ARC_IMP_SIMPLE_PATH; INFINITE_SIMPLE_PATH_IMAGE]);; (* ------------------------------------------------------------------------- *) (* The operations on paths. *) (* ------------------------------------------------------------------------- *) let JOINPATHS = prove (`!g1 g2. pathfinish g1 = pathstart g2 ==> g1 ++ g2 = \x. if drop x < &1 / &2 then g1(&2 % x) else g2 (&2 % x - vec 1)`, REWRITE_TAC[pathstart; pathfinish] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[joinpaths; FUN_EQ_THM] THEN X_GEN_TAC `x:real^1` THEN ASM_CASES_TAC `drop x = &1 / &2` THENL [FIRST_X_ASSUM(MP_TAC o AP_TERM `lift`) THEN REWRITE_TAC[LIFT_DROP] THEN DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[LIFT_DROP; REAL_LE_REFL; GSYM LIFT_CMUL; REAL_LT_REFL] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[LIFT_NUM; VECTOR_SUB_REFL]; REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN ASM_REAL_ARITH_TAC]);; let REVERSEPATH_REVERSEPATH = prove (`!g:real^1->real^N. reversepath(reversepath g) = g`, REWRITE_TAC[reversepath; ETA_AX; VECTOR_ARITH `vec 1 - (vec 1 - x):real^1 = x`]);; let PATHSTART_REVERSEPATH = prove (`pathstart(reversepath g) = pathfinish g`, REWRITE_TAC[pathstart; reversepath; pathfinish; VECTOR_SUB_RZERO]);; let PATHFINISH_REVERSEPATH = prove (`pathfinish(reversepath g) = pathstart g`, REWRITE_TAC[pathstart; reversepath; pathfinish; VECTOR_SUB_REFL]);; let PATHSTART_JOIN = prove (`!g1 g2. pathstart(g1 ++ g2) = pathstart g1`, REWRITE_TAC[joinpaths; pathstart; pathstart; DROP_VEC; VECTOR_MUL_RZERO] THEN CONV_TAC REAL_RAT_REDUCE_CONV);; let PATHFINISH_JOIN = prove (`!g1 g2. pathfinish(g1 ++ g2) = pathfinish g2`, REPEAT GEN_TAC THEN REWRITE_TAC[joinpaths; pathfinish; DROP_VEC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN AP_TERM_TAC THEN VECTOR_ARITH_TAC);; let PATH_IMAGE_REVERSEPATH = prove (`!g:real^1->real^N. path_image(reversepath g) = path_image g`, SUBGOAL_THEN `!g:real^1->real^N. path_image(reversepath g) SUBSET path_image g` (fun th -> MESON_TAC[th; REVERSEPATH_REVERSEPATH; SUBSET_ANTISYM]) THEN REWRITE_TAC[SUBSET; path_image; FORALL_IN_IMAGE] THEN MAP_EVERY X_GEN_TAC [`g:real^1->real^N`; `x:real^1`] THEN DISCH_TAC THEN REWRITE_TAC[reversepath; IN_IMAGE] THEN EXISTS_TAC `vec 1 - x:real^1` THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[IN_INTERVAL_1; DROP_SUB; DROP_VEC] THEN REAL_ARITH_TAC);; let PATH_REVERSEPATH = prove (`!g:real^1->real^N. path(reversepath g) <=> path g`, SUBGOAL_THEN `!g:real^1->real^N. path g ==> path(reversepath g)` (fun th -> MESON_TAC[th; REVERSEPATH_REVERSEPATH]) THEN GEN_TAC THEN REWRITE_TAC[path; reversepath] THEN STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `interval[vec 0:real^1,vec 1]` THEN ASM_REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTERVAL_1] THEN REWRITE_TAC[DROP_VEC; DROP_SUB] THEN REAL_ARITH_TAC);; let PATH_JOIN = prove (`!g1 g2:real^1->real^N. pathfinish g1 = pathstart g2 ==> (path(g1 ++ g2) <=> path g1 /\ path g2)`, REWRITE_TAC[path; pathfinish; pathstart] THEN REPEAT STRIP_TAC THEN EQ_TAC THENL [STRIP_TAC THEN CONJ_TAC THENL [SUBGOAL_THEN `(g1:real^1->real^N) = (\x. g1 (&2 % x)) o (\x. &1 / &2 % x)` SUBST1_TAC THENL [REWRITE_TAC[FUN_EQ_THM; o_THM] THEN GEN_TAC THEN AP_TERM_TAC THEN VECTOR_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_CMUL; CONTINUOUS_ON_ID] THEN MATCH_MP_TAC CONTINUOUS_ON_EQ THEN EXISTS_TAC `(g1 ++ g2):real^1->real^N` THEN CONJ_TAC THENL [REWRITE_TAC[FORALL_IN_IMAGE; joinpaths; IN_INTERVAL_1; DROP_CMUL] THEN SIMP_TAC[DROP_VEC; REAL_ARITH `&1 / &2 * x <= &1 / &2 <=> x <= &1`]; ALL_TAC] THEN MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `interval[vec 0:real^1,vec 1]` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTERVAL_1; DROP_CMUL] THEN REWRITE_TAC[DROP_VEC] THEN REAL_ARITH_TAC; SUBGOAL_THEN `(g2:real^1->real^N) = (\x. g2 (&2 % x - vec 1)) o (\x. &1 / &2 % (x + vec 1))` SUBST1_TAC THENL [REWRITE_TAC[FUN_EQ_THM; o_THM] THEN GEN_TAC THEN AP_TERM_TAC THEN VECTOR_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_CMUL; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ADD] THEN MATCH_MP_TAC CONTINUOUS_ON_EQ THEN EXISTS_TAC `(g1 ++ g2):real^1->real^N` THEN CONJ_TAC THENL [REWRITE_TAC[FORALL_IN_IMAGE; joinpaths; IN_INTERVAL_1; DROP_CMUL] THEN REWRITE_TAC[DROP_VEC; DROP_ADD; REAL_ARITH `&1 / &2 * (x + &1) <= &1 / &2 <=> x <= &0`] THEN SIMP_TAC[REAL_ARITH `&0 <= x ==> (x:real <= &0 <=> x = &0)`; LIFT_NUM; VECTOR_MUL_ASSOC; GSYM LIFT_EQ; LIFT_DROP; GSYM LIFT_CMUL] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[VECTOR_ADD_LID; VECTOR_MUL_LID] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[VECTOR_ARITH `(x + vec 1):real^N - vec 1 = x`]; ALL_TAC] THEN MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `interval[vec 0:real^1,vec 1]` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTERVAL_1; DROP_CMUL] THEN REWRITE_TAC[DROP_VEC; DROP_ADD] THEN REAL_ARITH_TAC]; STRIP_TAC THEN SUBGOAL_THEN `interval[vec 0,vec 1] = interval[vec 0,lift(&1 / &2)] UNION interval[lift(&1 / &2),vec 1]` SUBST1_TAC THENL [SIMP_TAC[EXTENSION; IN_UNION; IN_INTERVAL_1; LIFT_DROP; DROP_VEC] THEN REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC CONTINUOUS_ON_UNION THEN REWRITE_TAC[CLOSED_INTERVAL] THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_EQ THENL [EXISTS_TAC `\x. (g1:real^1->real^N) (&2 % x)`; EXISTS_TAC `\x. (g2:real^1->real^N) (&2 % x - vec 1)`] THEN REWRITE_TAC[joinpaths] THEN SIMP_TAC[IN_INTERVAL_1; LIFT_DROP] THENL [GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_CMUL; CONTINUOUS_ON_ID] THEN ONCE_REWRITE_TAC[VECTOR_ARITH `&2 % (x:real^1) = &2 % x + vec 0`] THEN REWRITE_TAC[IMAGE_AFFINITY_INTERVAL] THEN REWRITE_TAC[REAL_POS; INTERVAL_EQ_EMPTY_1; LIFT_DROP; DROP_VEC] THEN REWRITE_TAC[GSYM LIFT_CMUL; VECTOR_ADD_RID; VECTOR_MUL_RZERO] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[LIFT_NUM]; ALL_TAC] THEN CONJ_TAC THENL [SIMP_TAC[REAL_ARITH `&1 / &2 <= x ==> (x <= &1 / &2 <=> x = &1 / &2)`; GSYM LIFT_EQ; LIFT_DROP; GSYM LIFT_CMUL] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish]) THEN ASM_REWRITE_TAC[LIFT_NUM] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM LIFT_CMUL] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[LIFT_NUM; VECTOR_SUB_REFL]; ALL_TAC] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_CMUL; CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN ONCE_REWRITE_TAC[VECTOR_ARITH `&2 % x:real^N - vec 1 = &2 % x + --vec 1`] THEN REWRITE_TAC[IMAGE_AFFINITY_INTERVAL] THEN REWRITE_TAC[REAL_POS; INTERVAL_EQ_EMPTY_1; LIFT_DROP; DROP_VEC] THEN REWRITE_TAC[GSYM LIFT_CMUL; VECTOR_ADD_RID; VECTOR_MUL_RZERO] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[LIFT_NUM] THEN ASM_REWRITE_TAC[VECTOR_ARITH `&2 % x + --x:real^N = x /\ x + --x = vec 0`]]);; let PATH_JOIN_IMP = prove (`!g1 g2:real^1->real^N. path g1 /\ path g2 /\ pathfinish g1 = pathstart g2 ==> path(g1 ++ g2)`, MESON_TAC[PATH_JOIN]);; let PATH_IMAGE_JOIN_SUBSET = prove (`!g1 g2:real^1->real^N. path_image(g1 ++ g2) SUBSET (path_image g1 UNION path_image g2)`, REWRITE_TAC[path_image; FORALL_IN_IMAGE; SUBSET] THEN GEN_TAC THEN GEN_TAC THEN X_GEN_TAC `x:real^1` THEN REWRITE_TAC[IN_INTERVAL_1; IN_UNION; IN_IMAGE; DROP_VEC; joinpaths] THEN STRIP_TAC THEN ASM_CASES_TAC `drop x <= &1 / &2` THEN ASM_REWRITE_TAC[] THENL [DISJ1_TAC THEN EXISTS_TAC `&2 % x:real^1` THEN REWRITE_TAC[DROP_CMUL]; DISJ2_TAC THEN EXISTS_TAC `&2 % x - vec 1:real^1` THEN REWRITE_TAC[DROP_CMUL; DROP_SUB; DROP_VEC]] THEN ASM_REAL_ARITH_TAC);; let SUBSET_PATH_IMAGE_JOIN = prove (`!g1 g2:real^1->real^N s. path_image g1 SUBSET s /\ path_image g2 SUBSET s ==> path_image(g1 ++ g2) SUBSET s`, MP_TAC PATH_IMAGE_JOIN_SUBSET THEN REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN SET_TAC[]);; let PATH_IMAGE_JOIN = prove (`!g1 g2. pathfinish g1 = pathstart g2 ==> path_image(g1 ++ g2) = path_image g1 UNION path_image g2`, REWRITE_TAC[pathfinish; pathstart] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[PATH_IMAGE_JOIN_SUBSET] THEN REWRITE_TAC[path_image; SUBSET; FORALL_AND_THM; IN_UNION; TAUT `a \/ b ==> c <=> (a ==> c) /\ (b ==> c)`] THEN REWRITE_TAC[FORALL_IN_IMAGE; joinpaths] THEN REWRITE_TAC[IN_INTERVAL_1; IN_IMAGE; DROP_VEC] THEN CONJ_TAC THEN X_GEN_TAC `x:real^1` THEN REPEAT STRIP_TAC THENL [EXISTS_TAC `(&1 / &2) % x:real^1` THEN ASM_REWRITE_TAC[DROP_CMUL; REAL_ARITH `&1 / &2 * x <= &1 / &2 <=> x <= &1`] THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[VECTOR_MUL_LID]; EXISTS_TAC `(&1 / &2) % (x + vec 1):real^1` THEN ASM_REWRITE_TAC[DROP_CMUL; DROP_ADD; DROP_VEC] THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[VECTOR_MUL_LID; VECTOR_ARITH `(x + vec 1) - vec 1 = x`] THEN ASM_SIMP_TAC[REAL_ARITH `&0 <= x ==> (&1 / &2 * (x + &1) <= &1 / &2 <=> x = &0)`] THEN REWRITE_TAC[GSYM LIFT_EQ; LIFT_DROP; LIFT_NUM] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[VECTOR_ADD_LID; DROP_VEC]] THEN ASM_REAL_ARITH_TAC);; let NOT_IN_PATH_IMAGE_JOIN = prove (`!g1 g2 x. ~(x IN path_image g1) /\ ~(x IN path_image g2) ==> ~(x IN path_image(g1 ++ g2))`, MESON_TAC[PATH_IMAGE_JOIN_SUBSET; SUBSET; IN_UNION]);; let ARC_REVERSEPATH = prove (`!g. arc g ==> arc(reversepath g)`, GEN_TAC THEN SIMP_TAC[arc; PATH_REVERSEPATH] THEN REWRITE_TAC[arc; reversepath] THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`x:real^1`; `y:real^1`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`vec 1 - x:real^1`; `vec 1 - y:real^1`]) THEN ASM_REWRITE_TAC[] THEN REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[IN_INTERVAL_1; GSYM DROP_EQ; DROP_SUB; DROP_VEC] THEN REAL_ARITH_TAC);; let ARC_REVERSEPATH_EQ = prove (`!g:real^1->real^N. arc(reversepath g) <=> arc g`, MESON_TAC[ARC_REVERSEPATH; REVERSEPATH_REVERSEPATH]);; let SIMPLE_PATH_REVERSEPATH = prove (`!g. simple_path g ==> simple_path (reversepath g)`, GEN_TAC THEN SIMP_TAC[simple_path; PATH_REVERSEPATH] THEN REWRITE_TAC[simple_path; reversepath] THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`x:real^1`; `y:real^1`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`vec 1 - x:real^1`; `vec 1 - y:real^1`]) THEN ASM_REWRITE_TAC[] THEN REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[IN_INTERVAL_1; GSYM DROP_EQ; DROP_SUB; DROP_VEC] THEN REAL_ARITH_TAC);; let SIMPLE_PATH_REVERSEPATH_EQ = prove (`!g:real^1->real^N. simple_path(reversepath g) <=> simple_path g`, MESON_TAC[SIMPLE_PATH_REVERSEPATH; REVERSEPATH_REVERSEPATH]);; let SIMPLE_PATH_JOIN_LOOP = prove (`!g1 g2:real^1->real^N. arc g1 /\ arc g2 /\ pathfinish g1 = pathstart g2 /\ pathfinish g2 = pathstart g1 /\ (path_image g1 INTER path_image g2) SUBSET {pathstart g1,pathstart g2} ==> simple_path(g1 ++ g2)`, REPEAT GEN_TAC THEN REWRITE_TAC[arc; simple_path] THEN MATCH_MP_TAC(TAUT `(a /\ b /\ c /\ d ==> f) /\ (a' /\ b' /\ c /\ d /\ e ==> g) ==> (a /\ a') /\ (b /\ b') /\ c /\ d /\ e ==> f /\ g`) THEN CONJ_TAC THENL [MESON_TAC[PATH_JOIN]; ALL_TAC] THEN REWRITE_TAC[arc; simple_path; SUBSET; IN_INTER; pathstart; pathfinish; IN_INTERVAL_1; DROP_VEC; IN_INSERT; NOT_IN_EMPTY] THEN DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "G1") MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "G2") MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "G0")) THEN MATCH_MP_TAC DROP_WLOG_LE THEN CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN REPEAT GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[joinpaths] THEN MAP_EVERY ASM_CASES_TAC [`drop x <= &1 / &2`; `drop y <= &1 / &2`] THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THENL [REMOVE_THEN "G1" (MP_TAC o SPECL [`&2 % x:real^1`; `&2 % y:real^1`]) THEN ASM_REWRITE_TAC[DROP_CMUL; DROP_VEC; DROP_SUB] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(fun th -> DISJ1_TAC THEN MP_TAC th) THEN VECTOR_ARITH_TAC; ALL_TAC; ASM_REAL_ARITH_TAC; REMOVE_THEN "G2" (MP_TAC o SPECL [`&2 % x:real^1 - vec 1`; `&2 % y:real^1 - vec 1`]) THEN ASM_REWRITE_TAC[DROP_CMUL; DROP_VEC; DROP_SUB] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(fun th -> DISJ1_TAC THEN MP_TAC th) THEN VECTOR_ARITH_TAC] THEN REMOVE_THEN "G0" (MP_TAC o SPEC `(g1:real^1->real^N) (&2 % x)`) THEN ANTS_TAC THENL [CONJ_TAC THENL [REWRITE_TAC[path_image; IN_IMAGE] THEN EXISTS_TAC `&2 % x:real^1` THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; DROP_CMUL] THEN ASM_REAL_ARITH_TAC; ASM_REWRITE_TAC[path_image; IN_IMAGE] THEN EXISTS_TAC `&2 % y:real^1 - vec 1` THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; DROP_CMUL; DROP_SUB] THEN ASM_REAL_ARITH_TAC]; ALL_TAC] THEN STRIP_TAC THENL [DISJ2_TAC THEN DISJ1_TAC; DISJ1_TAC THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `&1 / &2 % vec 1:real^1`] THEN MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL [SUBGOAL_THEN `&2 % x:real^1 = vec 0` MP_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN REMOVE_THEN "G1" MATCH_MP_TAC; DISCH_THEN SUBST_ALL_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[VECTOR_MUL_RZERO]) THEN UNDISCH_TAC `T` THEN REWRITE_TAC[] THEN SUBGOAL_THEN `&2 % y:real^1 - vec 1 = vec 1` MP_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN REMOVE_THEN "G2" MATCH_MP_TAC; SUBGOAL_THEN `&2 % x:real^1 = vec 1` MP_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN REMOVE_THEN "G1" MATCH_MP_TAC; DISCH_THEN SUBST_ALL_TAC THEN SUBGOAL_THEN `&2 % y:real^1 - vec 1 = vec 0` MP_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN REMOVE_THEN "G2" MATCH_MP_TAC] THEN (REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN ASM_REWRITE_TAC[DROP_CMUL; DROP_SUB; DROP_VEC] THEN ASM_REAL_ARITH_TAC));; let ARC_JOIN = prove (`!g1 g2:real^1->real^N. arc g1 /\ arc g2 /\ pathfinish g1 = pathstart g2 /\ (path_image g1 INTER path_image g2) SUBSET {pathstart g2} ==> arc(g1 ++ g2)`, REPEAT GEN_TAC THEN REWRITE_TAC[arc; simple_path] THEN MATCH_MP_TAC(TAUT `(a /\ b /\ c /\ d ==> f) /\ (a' /\ b' /\ c /\ d ==> g) ==> (a /\ a') /\ (b /\ b') /\ c /\ d ==> f /\ g`) THEN CONJ_TAC THENL [MESON_TAC[PATH_JOIN]; ALL_TAC] THEN REWRITE_TAC[arc; simple_path; SUBSET; IN_INTER; pathstart; pathfinish; IN_INTERVAL_1; DROP_VEC; IN_INSERT; NOT_IN_EMPTY] THEN DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "G1") MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "G2") MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "G0")) THEN MATCH_MP_TAC DROP_WLOG_LE THEN CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN REPEAT GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[joinpaths] THEN MAP_EVERY ASM_CASES_TAC [`drop x <= &1 / &2`; `drop y <= &1 / &2`] THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THENL [REMOVE_THEN "G1" (MP_TAC o SPECL [`&2 % x:real^1`; `&2 % y:real^1`]) THEN ASM_REWRITE_TAC[DROP_CMUL; DROP_VEC; DROP_SUB] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN VECTOR_ARITH_TAC; ALL_TAC; ASM_REAL_ARITH_TAC; REMOVE_THEN "G2" (MP_TAC o SPECL [`&2 % x:real^1 - vec 1`; `&2 % y:real^1 - vec 1`]) THEN ASM_REWRITE_TAC[DROP_CMUL; DROP_VEC; DROP_SUB] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN VECTOR_ARITH_TAC] THEN REMOVE_THEN "G0" (MP_TAC o SPEC `(g1:real^1->real^N) (&2 % x)`) THEN ANTS_TAC THENL [CONJ_TAC THENL [REWRITE_TAC[path_image; IN_IMAGE] THEN EXISTS_TAC `&2 % x:real^1` THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; DROP_CMUL] THEN ASM_REAL_ARITH_TAC; ASM_REWRITE_TAC[path_image; IN_IMAGE] THEN EXISTS_TAC `&2 % y:real^1 - vec 1` THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; DROP_CMUL; DROP_SUB] THEN ASM_REAL_ARITH_TAC]; ALL_TAC] THEN STRIP_TAC THEN SUBGOAL_THEN `x:real^1 = &1 / &2 % vec 1` SUBST_ALL_TAC THENL [SUBGOAL_THEN `&2 % x:real^1 = vec 1` MP_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN REMOVE_THEN "G1" MATCH_MP_TAC; SUBGOAL_THEN `&2 % y:real^1 - vec 1 = vec 0` MP_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN REMOVE_THEN "G2" MATCH_MP_TAC] THEN (REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN ASM_REWRITE_TAC[DROP_CMUL; DROP_SUB; DROP_VEC] THEN ASM_REAL_ARITH_TAC));; let REVERSEPATH_JOINPATHS = prove (`!g1 g2. pathfinish g1 = pathstart g2 ==> reversepath(g1 ++ g2) = reversepath g2 ++ reversepath g1`, REPEAT GEN_TAC THEN REWRITE_TAC[reversepath; joinpaths; pathfinish; pathstart; FUN_EQ_THM] THEN DISCH_TAC THEN X_GEN_TAC `t:real^1` THEN REWRITE_TAC[DROP_VEC; DROP_SUB; REAL_ARITH `&1 - x <= &1 / &2 <=> &1 / &2 <= x`] THEN ASM_CASES_TAC `t = lift(&1 / &2)` THENL [ASM_REWRITE_TAC[LIFT_DROP; REAL_LE_REFL; GSYM LIFT_NUM; GSYM LIFT_SUB; GSYM LIFT_CMUL] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[LIFT_NUM]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [GSYM DROP_EQ]) THEN REWRITE_TAC[LIFT_DROP] THEN DISCH_TAC THEN ASM_SIMP_TAC[REAL_ARITH `~(x = &1 / &2) ==> (&1 / &2 <= x <=> ~(x <= &1 / &2))`] THEN ASM_CASES_TAC `drop t <= &1 / &2` THEN ASM_REWRITE_TAC[] THEN AP_TERM_TAC THEN REWRITE_TAC[VECTOR_SUB_LDISTRIB] THEN VECTOR_ARITH_TAC]);; (* ------------------------------------------------------------------------- *) (* Some reversed and "if and only if" versions of joining theorems. *) (* ------------------------------------------------------------------------- *) let PATH_JOIN_PATH_ENDS = prove (`!g1 g2:real^1->real^N. path g2 /\ path(g1 ++ g2) ==> pathfinish g1 = pathstart g2`, REPEAT GEN_TAC THEN DISJ_CASES_TAC(NORM_ARITH `pathfinish g1:real^N = pathstart g2 \/ &0 < dist(pathfinish g1,pathstart g2)`) THEN ASM_REWRITE_TAC[path; continuous_on; joinpaths] THEN RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish]) THEN REWRITE_TAC[pathstart; pathfinish] THEN ABBREV_TAC `e = dist((g1:real^1->real^N)(vec 1),g2(vec 0:real^1))` THEN DISCH_THEN(CONJUNCTS_THEN2 (MP_TAC o SPEC `vec 0:real^1`) (MP_TAC o SPEC `lift(&1 / &2)`)) THEN REWRITE_TAC[ENDS_IN_UNIT_INTERVAL; LIFT_DROP; REAL_LE_REFL] THEN REWRITE_TAC[GSYM LIFT_CMUL; IN_INTERVAL_1; DROP_VEC; LIFT_DROP] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[LIFT_NUM] THEN DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN DISCH_THEN(X_CHOOSE_THEN `d1:real` (CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "1"))) THEN DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN DISCH_THEN(X_CHOOSE_THEN `d2:real` (CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "2"))) THEN REMOVE_THEN "2" (MP_TAC o SPEC `lift(min (&1 / &2) (min d1 d2) / &2)`) THEN REWRITE_TAC[LIFT_DROP; DIST_LIFT; DIST_0; NORM_REAL; GSYM drop] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN REMOVE_THEN "1" (MP_TAC o SPEC `lift(&1 / &2 + min (&1 / &2) (min d1 d2) / &4)`) THEN REWRITE_TAC[LIFT_DROP; DIST_LIFT] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN COND_CASES_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[GSYM LIFT_CMUL; LIFT_ADD; REAL_ADD_LDISTRIB] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[LIFT_NUM] THEN REWRITE_TAC[VECTOR_ADD_SUB; REAL_ARITH `&2 * x / &4 = x / &2`] THEN REPEAT(POP_ASSUM MP_TAC) THEN NORM_ARITH_TAC);; let PATH_JOIN_EQ = prove (`!g1 g2:real^1->real^N. path g1 /\ path g2 ==> (path(g1 ++ g2) <=> pathfinish g1 = pathstart g2)`, MESON_TAC[PATH_JOIN_PATH_ENDS; PATH_JOIN_IMP]);; let SIMPLE_PATH_JOIN_IMP = prove (`!g1 g2:real^1->real^N. simple_path(g1 ++ g2) /\ pathfinish g1 = pathstart g2 ==> arc g1 /\ arc g2 /\ path_image g1 INTER path_image g2 SUBSET {pathstart g1, pathstart g2}`, REPEAT GEN_TAC THEN ASM_CASES_TAC `path(g1:real^1->real^N) /\ path(g2:real^1->real^N)` THENL [ALL_TAC; ASM_MESON_TAC[PATH_JOIN; SIMPLE_PATH_IMP_PATH]] THEN REWRITE_TAC[simple_path; pathstart; pathfinish; arc] THEN STRIP_TAC THEN REPEAT CONJ_TAC THEN ASM_REWRITE_TAC[] THENL [MAP_EVERY X_GEN_TAC [`x:real^1`; `y:real^1`] THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`&1 / &2 % x:real^1`; `&1 / &2 % y:real^1`]) THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; joinpaths; DROP_CMUL] THEN REPEAT(COND_CASES_TAC THEN TRY ASM_REAL_ARITH_TAC) THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[GSYM DROP_EQ; DROP_CMUL; VECTOR_MUL_LID; DROP_VEC] THEN ASM_REAL_ARITH_TAC; MAP_EVERY X_GEN_TAC [`x:real^1`; `y:real^1`] THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`&1 / &2 % (x + vec 1):real^1`; `&1 / &2 % (y + vec 1):real^1`]) THEN ASM_SIMP_TAC[JOINPATHS; pathstart; pathfinish] THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; DROP_ADD; DROP_CMUL] THEN REPEAT(COND_CASES_TAC THEN TRY ASM_REAL_ARITH_TAC) THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[VECTOR_MUL_LID; VECTOR_ARITH `(a + b) - b:real^N = a`] THEN ASM_REWRITE_TAC[GSYM DROP_EQ; DROP_CMUL; VECTOR_MUL_LID; DROP_VEC; DROP_ADD] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[SET_RULE `s INTER t SUBSET u <=> !x. x IN s ==> x IN t ==> x IN u`] THEN REWRITE_TAC[path_image; FORALL_IN_IMAGE] THEN X_GEN_TAC `x:real^1` THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN STRIP_TAC THEN REWRITE_TAC[IN_IMAGE; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `y:real^1` THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN STRIP_TAC THEN SUBST1_TAC(SYM(ASSUME `(g1:real^1->real^N)(vec 1) = g2(vec 0:real^1)`)) THEN MATCH_MP_TAC(SET_RULE `x = a \/ x = b ==> f x IN {f a,f b}`) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`&1 / &2 % x:real^1`; `&1 / &2 % (y + vec 1):real^1`]) THEN ANTS_TAC THENL [REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; DROP_CMUL; DROP_ADD] THEN REPEAT(CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]) THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [joinpaths] THEN ASM_SIMP_TAC[JOINPATHS; pathstart; pathfinish] THEN REWRITE_TAC[DROP_ADD; DROP_CMUL; DROP_VEC] THEN REPEAT(COND_CASES_TAC THEN TRY ASM_REAL_ARITH_TAC) THEN REWRITE_TAC[VECTOR_ARITH `&2 % &1 / &2 % x:real^N = x`] THEN ASM_REWRITE_TAC[VECTOR_ARITH `(a + b) - b:real^N = a`]; REWRITE_TAC[GSYM DROP_EQ; DROP_CMUL; DROP_ADD; DROP_VEC] THEN ASM_REAL_ARITH_TAC]]);; let SIMPLE_PATH_JOIN_LOOP_EQ = prove (`!g1 g2:real^1->real^N. pathfinish g2 = pathstart g1 /\ pathfinish g1 = pathstart g2 ==> (simple_path(g1 ++ g2) <=> arc g1 /\ arc g2 /\ path_image g1 INTER path_image g2 SUBSET {pathstart g1, pathstart g2})`, MESON_TAC[SIMPLE_PATH_JOIN_IMP; SIMPLE_PATH_JOIN_LOOP]);; let SIMPLE_PATH_JOIN_LOOP_EQ_ALT = prove (`!g1 g2:real^1->real^N. pathfinish g2 = pathstart g1 /\ pathfinish g1 = pathstart g2 ==> (simple_path(g1 ++ g2) <=> arc g1 /\ arc g2 /\ path_image g1 INTER path_image g2 = {pathstart g1, pathstart g2})`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[SIMPLE_PATH_JOIN_LOOP_EQ] THEN AP_TERM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC(SET_RULE `a IN s /\ b IN s ==> (s SUBSET {a,b} <=> s = {a,b})`) THEN REWRITE_TAC[IN_INTER] THEN ASM_MESON_TAC[PATHSTART_IN_PATH_IMAGE; PATHFINISH_IN_PATH_IMAGE]);; let ARC_JOIN_EQ = prove (`!g1 g2:real^1->real^N. pathfinish g1 = pathstart g2 ==> (arc(g1 ++ g2) <=> arc g1 /\ arc g2 /\ path_image g1 INTER path_image g2 SUBSET {pathstart g2})`, REPEAT STRIP_TAC THEN EQ_TAC THEN ASM_SIMP_TAC[ARC_JOIN] THEN GEN_REWRITE_TAC LAND_CONV [ARC_SIMPLE_PATH] THEN REWRITE_TAC[PATHFINISH_JOIN; PATHSTART_JOIN] THEN STRIP_TAC THEN MP_TAC(ISPECL [`g1:real^1->real^N`; `g2:real^1->real^N`] SIMPLE_PATH_JOIN_IMP) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `~((pathstart g1:real^N) IN path_image g2)` (fun th -> MP_TAC th THEN ASM SET_TAC[]) THEN REWRITE_TAC[path_image; IN_IMAGE; IN_INTERVAL_1; DROP_VEC] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^1` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [simple_path]) THEN DISCH_THEN(MP_TAC o SPECL [`vec 0:real^1`; `lift(&1 / &2) + inv(&2) % u`] o CONJUNCT2) THEN REWRITE_TAC[GSYM DROP_EQ; IN_INTERVAL_1; DROP_ADD; DROP_VEC; DROP_CMUL; LIFT_DROP; joinpaths] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; REAL_LT_IMP_NZ; REAL_ARITH `&0 <= x ==> &0 < &1 / &2 + &1 / &2 * x`] THEN REWRITE_TAC[REAL_ARITH `&1 / &2 + &1 / &2 * u = &1 <=> u = &1`] THEN ASM_SIMP_TAC[REAL_ARITH `&0 <= u ==> (&1 / &2 + &1 / &2 * u <= &1 / &2 <=> u = &0)`] THEN RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish]) THEN ASM_REWRITE_TAC[VECTOR_MUL_RZERO] THEN ASM_SIMP_TAC[REAL_ARITH `u <= &1 ==> &1 / &2 + &1 / &2 * u <= &1`] THEN REWRITE_TAC[GSYM LIFT_EQ; LIFT_NUM; LIFT_DROP] THEN COND_CASES_TAC THENL [ASM_REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_RID; GSYM LIFT_CMUL] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[LIFT_NUM] THEN ASM_REWRITE_TAC[VEC_EQ] THEN ARITH_TAC; REWRITE_TAC[VECTOR_ADD_LDISTRIB; GSYM LIFT_CMUL] THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[LIFT_NUM; VECTOR_MUL_LID; VECTOR_ADD_SUB] THEN ASM_MESON_TAC[]]);; let ARC_JOIN_EQ_ALT = prove (`!g1 g2:real^1->real^N. pathfinish g1 = pathstart g2 ==> (arc(g1 ++ g2) <=> arc g1 /\ arc g2 /\ path_image g1 INTER path_image g2 = {pathstart g2})`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[ARC_JOIN_EQ] THEN MP_TAC(ISPEC `g1:real^1->real^N` PATHFINISH_IN_PATH_IMAGE) THEN MP_TAC(ISPEC `g2:real^1->real^N` PATHSTART_IN_PATH_IMAGE) THEN ASM SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* Reassociating a joined path doesn't matter for various properties. *) (* ------------------------------------------------------------------------- *) let PATH_ASSOC = prove (`!p q r:real^1->real^N. pathfinish p = pathstart q /\ pathfinish q = pathstart r ==> (path(p ++ (q ++ r)) <=> path((p ++ q) ++ r))`, SIMP_TAC[PATH_JOIN; PATHSTART_JOIN; PATHFINISH_JOIN] THEN CONV_TAC TAUT);; let SIMPLE_PATH_ASSOC = prove (`!p q r:real^1->real^N. pathfinish p = pathstart q /\ pathfinish q = pathstart r ==> (simple_path(p ++ (q ++ r)) <=> simple_path((p ++ q) ++ r))`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `pathstart(p:real^1->real^N) = pathfinish r` THENL [ALL_TAC; ASM_SIMP_TAC[SIMPLE_PATH_EQ_ARC; PATHSTART_JOIN; PATHFINISH_JOIN]] THEN ASM_SIMP_TAC[SIMPLE_PATH_JOIN_LOOP_EQ; PATHSTART_JOIN; PATHFINISH_JOIN; ARC_JOIN_EQ; PATH_IMAGE_JOIN] THEN MAP_EVERY ASM_CASES_TAC [`arc(p:real^1->real^N)`; `arc(q:real^1->real^N)`; `arc(r:real^1->real^N)`] THEN ASM_REWRITE_TAC[UNION_OVER_INTER; UNION_SUBSET; ONCE_REWRITE_RULE[INTER_COMM] UNION_OVER_INTER] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP ARC_DISTINCT_ENDS)) THEN MAP_EVERY (fun t -> MP_TAC(ISPEC t PATHSTART_IN_PATH_IMAGE) THEN MP_TAC(ISPEC t PATHFINISH_IN_PATH_IMAGE)) [`p:real^1->real^N`; `q:real^1->real^N`; `r:real^1->real^N`] THEN ASM SET_TAC[]);; let ARC_ASSOC = prove (`!p q r:real^1->real^N. pathfinish p = pathstart q /\ pathfinish q = pathstart r ==> (arc(p ++ (q ++ r)) <=> arc((p ++ q) ++ r))`, SIMP_TAC[ARC_SIMPLE_PATH; SIMPLE_PATH_ASSOC] THEN SIMP_TAC[PATHSTART_JOIN; PATHFINISH_JOIN]);; (* ------------------------------------------------------------------------- *) (* In the case of a loop, neither does symmetry. *) (* ------------------------------------------------------------------------- *) let PATH_SYM = prove (`!p q. pathfinish p = pathstart q /\ pathfinish q = pathstart p ==> (path(p ++ q) <=> path(q ++ p))`, SIMP_TAC[PATH_JOIN; CONJ_ACI]);; let SIMPLE_PATH_SYM = prove (`!p q. pathfinish p = pathstart q /\ pathfinish q = pathstart p ==> (simple_path(p ++ q) <=> simple_path(q ++ p))`, SIMP_TAC[SIMPLE_PATH_JOIN_LOOP_EQ; INTER_ACI; CONJ_ACI; INSERT_AC]);; let PATH_IMAGE_SYM = prove (`!p q. pathfinish p = pathstart q /\ pathfinish q = pathstart p ==> path_image(p ++ q) = path_image(q ++ p)`, SIMP_TAC[PATH_IMAGE_JOIN; UNION_ACI]);; (* ------------------------------------------------------------------------- *) (* Reparametrizing a closed curve to start at some chosen point. *) (* ------------------------------------------------------------------------- *) let shiftpath = new_definition `shiftpath a (f:real^1->real^N) = \x. if drop(a + x) <= &1 then f(a + x) else f(a + x - vec 1)`;; let SHIFTPATH_TRANSLATION = prove (`!a t g. shiftpath t ((\x. a + x) o g) = (\x. a + x) o shiftpath t g`, REWRITE_TAC[FUN_EQ_THM; shiftpath; o_THM] THEN MESON_TAC[]);; add_translation_invariants [SHIFTPATH_TRANSLATION];; let SHIFTPATH_LINEAR_IMAGE = prove (`!f t g. linear f ==> shiftpath t (f o g) = f o shiftpath t g`, REWRITE_TAC[FUN_EQ_THM; shiftpath; o_THM] THEN MESON_TAC[]);; add_linear_invariants [SHIFTPATH_LINEAR_IMAGE];; let PATHSTART_SHIFTPATH = prove (`!a g. drop a <= &1 ==> pathstart(shiftpath a g) = g(a)`, SIMP_TAC[pathstart; shiftpath; VECTOR_ADD_RID]);; let PATHFINISH_SHIFTPATH = prove (`!a g. &0 <= drop a /\ pathfinish g = pathstart g ==> pathfinish(shiftpath a g) = g(a)`, SIMP_TAC[pathfinish; shiftpath; pathstart; DROP_ADD; DROP_VEC] THEN REWRITE_TAC[VECTOR_ARITH `a + vec 1 - vec 1 = a`] THEN ASM_SIMP_TAC[REAL_ARITH `&0 <= x ==> (x + &1 <= &1 <=> x = &0)`] THEN SIMP_TAC[DROP_EQ_0; VECTOR_ADD_LID] THEN MESON_TAC[]);; let ENDPOINTS_SHIFTPATH = prove (`!a g. pathfinish g = pathstart g /\ a IN interval[vec 0,vec 1] ==> pathfinish(shiftpath a g) = g a /\ pathstart(shiftpath a g) = g a`, SIMP_TAC[IN_INTERVAL_1; DROP_VEC; PATHSTART_SHIFTPATH; PATHFINISH_SHIFTPATH]);; let CLOSED_SHIFTPATH = prove (`!a g. pathfinish g = pathstart g /\ a IN interval[vec 0,vec 1] ==> pathfinish(shiftpath a g) = pathstart(shiftpath a g)`, SIMP_TAC[IN_INTERVAL_1; PATHSTART_SHIFTPATH; PATHFINISH_SHIFTPATH; DROP_VEC]);; let PATH_SHIFTPATH = prove (`!g a. path g /\ pathfinish g:real^N = pathstart g /\ a IN interval[vec 0,vec 1] ==> path(shiftpath a g)`, REWRITE_TAC[shiftpath; path] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `interval[vec 0,vec 1] = interval[vec 0,vec 1 - a:real^1] UNION interval[vec 1 - a,vec 1]` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; IN_UNION; IN_INTERVAL_1] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL_1]) THEN REWRITE_TAC[DROP_SUB; DROP_VEC] THEN REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC CONTINUOUS_ON_UNION THEN REWRITE_TAC[CLOSED_INTERVAL] THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_EQ THENL [EXISTS_TAC `(\x. g(a + x)):real^1->real^N` THEN REWRITE_TAC[IN_INTERVAL_1; DROP_ADD; DROP_VEC; DROP_SUB] THEN SIMP_TAC[REAL_ARITH `a + x <= &1 <=> x <= &1 - a`]; EXISTS_TAC `(\x. g(a + x - vec 1)):real^1->real^N` THEN REWRITE_TAC[IN_INTERVAL_1; DROP_ADD; DROP_VEC; DROP_SUB] THEN SIMP_TAC[REAL_ARITH `&1 - a <= x ==> (a + x <= &1 <=> a + x = &1)`] THEN ONCE_REWRITE_TAC[COND_RAND] THEN REWRITE_TAC[VECTOR_ARITH `a + x - vec 1 = (a + x) - vec 1`] THEN RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish]) THEN ASM_SIMP_TAC[GSYM LIFT_EQ; LIFT_ADD; LIFT_NUM; LIFT_DROP] THEN REWRITE_TAC[VECTOR_SUB_REFL; COND_ID]] THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_COMPOSE) THEN SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID; CONTINUOUS_ON_SUB] THEN MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `interval[vec 0:real^1,vec 1]` THEN ASM_REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL_1]) THEN REWRITE_TAC[IN_INTERVAL_1; DROP_SUB; DROP_VEC; DROP_ADD] THEN REAL_ARITH_TAC);; let SHIFTPATH_SHIFTPATH = prove (`!g a x. a IN interval[vec 0,vec 1] /\ pathfinish g = pathstart g /\ x IN interval[vec 0,vec 1] ==> shiftpath (vec 1 - a) (shiftpath a g) x = g x`, REWRITE_TAC[shiftpath; pathfinish; pathstart] THEN REWRITE_TAC[DROP_ADD; DROP_SUB; DROP_VEC] THEN REPEAT STRIP_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL_1])) THEN REWRITE_TAC[DROP_VEC] THEN REPEAT STRIP_TAC THENL [ALL_TAC; AP_TERM_TAC THEN VECTOR_ARITH_TAC; AP_TERM_TAC THEN VECTOR_ARITH_TAC; ASM_REAL_ARITH_TAC] THEN SUBGOAL_THEN `x:real^1 = vec 0` SUBST1_TAC THENL [REWRITE_TAC[GSYM DROP_EQ; DROP_VEC] THEN ASM_REAL_ARITH_TAC; ASM_REWRITE_TAC[VECTOR_ARITH `a + vec 1 - a + vec 0:real^1 = vec 1`]]);; let PATH_IMAGE_SHIFTPATH = prove (`!a g:real^1->real^N. a IN interval[vec 0,vec 1] /\ pathfinish g = pathstart g ==> path_image(shiftpath a g) = path_image g`, REWRITE_TAC[IN_INTERVAL_1; pathfinish; pathstart] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THEN REWRITE_TAC[path_image; shiftpath; FORALL_IN_IMAGE; SUBSET] THEN REWRITE_TAC[IN_IMAGE] THEN REPEAT STRIP_TAC THEN REPEAT COND_CASES_TAC THEN ASM_REWRITE_TAC[IN_IMAGE] THENL [EXISTS_TAC `a + x:real^1`; EXISTS_TAC `a + x - vec 1:real^1`; ALL_TAC] THEN REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; DROP_SUB; DROP_ADD] THEN TRY REAL_ARITH_TAC THEN REPEAT STRIP_TAC THEN ASM_CASES_TAC `drop a <= drop x` THENL [EXISTS_TAC `x - a:real^1` THEN REWRITE_TAC[VECTOR_ARITH `a + x - a:real^1 = x`; DROP_SUB] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC; EXISTS_TAC `vec 1 + x - a:real^1` THEN REWRITE_TAC[VECTOR_ARITH `a + (v + x - a) - v:real^1 = x`] THEN REWRITE_TAC[DROP_ADD; DROP_SUB; DROP_VEC] THEN ASM_CASES_TAC `x:real^1 = vec 0` THEN ASM_REWRITE_TAC[VECTOR_ARITH `a + v + x - a:real^1 = v + x`] THEN ASM_REWRITE_TAC[VECTOR_ADD_RID; DROP_VEC; COND_ID] THEN ASM_REWRITE_TAC[REAL_ARITH `a + &1 + x - a <= &1 <=> x <= &0`] THEN REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[GSYM DROP_EQ; DROP_VEC] THEN TRY(COND_CASES_TAC THEN POP_ASSUM MP_TAC) THEN REWRITE_TAC[] THEN REAL_ARITH_TAC]);; let SIMPLE_PATH_SHIFTPATH = prove (`!g a. simple_path g /\ pathfinish g = pathstart g /\ a IN interval[vec 0,vec 1] ==> simple_path(shiftpath a g)`, REPEAT GEN_TAC THEN REWRITE_TAC[simple_path] THEN MATCH_MP_TAC(TAUT `(a /\ c /\ d ==> e) /\ (b /\ c /\ d ==> f) ==> (a /\ b) /\ c /\ d ==> e /\ f`) THEN CONJ_TAC THENL [MESON_TAC[PATH_SHIFTPATH]; ALL_TAC] THEN REWRITE_TAC[simple_path; shiftpath; IN_INTERVAL_1; DROP_VEC; DROP_ADD; DROP_SUB] THEN REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN ONCE_REWRITE_TAC[TAUT `a /\ b /\ c ==> d <=> c ==> a /\ b ==> d`] THEN STRIP_TAC THEN REPEAT GEN_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN DISCH_THEN(fun th -> FIRST_X_ASSUM(MP_TAC o C MATCH_MP th)) THEN REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[DROP_ADD; DROP_SUB; DROP_VEC; GSYM DROP_EQ] THEN REAL_ARITH_TAC);; (* ------------------------------------------------------------------------- *) (* Choosing a sub-path of an existing path. *) (* ------------------------------------------------------------------------- *) let subpath = new_definition `subpath u v g = \x. g(u + drop(v - u) % x)`;; let SUBPATH_SCALING_LEMMA = prove (`!u v. IMAGE (\x. u + drop(v - u) % x) (interval[vec 0,vec 1]) = segment[u,v]`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[VECTOR_ADD_SYM] THEN REWRITE_TAC[IMAGE_AFFINITY_INTERVAL; SEGMENT_1] THEN REWRITE_TAC[DROP_SUB; REAL_SUB_LE; INTERVAL_EQ_EMPTY_1; DROP_VEC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN BINOP_TAC THEN REWRITE_TAC[GSYM LIFT_EQ_CMUL; VECTOR_MUL_RZERO] THEN REWRITE_TAC[LIFT_DROP; LIFT_SUB] THEN VECTOR_ARITH_TAC);; let PATH_IMAGE_SUBPATH_GEN = prove (`!u v g:real^1->real^N. path_image(subpath u v g) = IMAGE g (segment[u,v])`, REPEAT GEN_TAC THEN REWRITE_TAC[path_image; subpath] THEN ONCE_REWRITE_TAC[GSYM o_DEF] THEN REWRITE_TAC[IMAGE_o; SUBPATH_SCALING_LEMMA]);; let PATH_IMAGE_SUBPATH = prove (`!u v g:real^1->real^N. drop u <= drop v ==> path_image(subpath u v g) = IMAGE g (interval[u,v])`, SIMP_TAC[PATH_IMAGE_SUBPATH_GEN; SEGMENT_1]);; let PATH_IMAGE_SUBPATH_COMBINE = prove (`!g:real^1->real^N u. path g /\ u IN interval[vec 0,vec 1] ==> path_image(subpath (vec 0) u g) UNION path_image(subpath u (vec 1) g) = path_image g`, REWRITE_TAC[IN_INTERVAL_1] THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[PATH_IMAGE_SUBPATH] THEN REWRITE_TAC[path_image; GSYM IMAGE_UNION] THEN AP_TERM_TAC THEN MATCH_MP_TAC UNION_INTERVAL_1 THEN ASM_REWRITE_TAC[IN_INTERVAL_1]);; let PATH_SUBPATH = prove (`!u v g:real^1->real^N. path g /\ u IN interval[vec 0,vec 1] /\ v IN interval[vec 0,vec 1] ==> path(subpath u v g)`, REWRITE_TAC[path; subpath] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_COMPOSE) THEN SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_CMUL; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBPATH_SCALING_LEMMA; SEGMENT_1] THEN COND_CASES_TAC THEN REWRITE_TAC[SUBSET_INTERVAL_1] THEN REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN REAL_ARITH_TAC);; let PATHSTART_SUBPATH = prove (`!u v g:real^1->real^N. pathstart(subpath u v g) = g(u)`, REWRITE_TAC[pathstart; subpath; VECTOR_MUL_RZERO; VECTOR_ADD_RID]);; let PATHFINISH_SUBPATH = prove (`!u v g:real^1->real^N. pathfinish(subpath u v g) = g(v)`, REWRITE_TAC[pathfinish; subpath; GSYM LIFT_EQ_CMUL] THEN REWRITE_TAC[LIFT_DROP; VECTOR_ARITH `u + v - u:real^N = v`]);; let SUBPATH_TRIVIAL = prove (`!g. subpath (vec 0) (vec 1) g = g`, REWRITE_TAC[subpath; VECTOR_SUB_RZERO; DROP_VEC; VECTOR_MUL_LID; VECTOR_ADD_LID; ETA_AX]);; let SUBPATH_REVERSEPATH = prove (`!g. subpath (vec 1) (vec 0) g = reversepath g`, REWRITE_TAC[subpath; reversepath; VECTOR_SUB_LZERO; DROP_NEG; DROP_VEC] THEN REWRITE_TAC[VECTOR_ARITH `a + -- &1 % b:real^N = a - b`]);; let REVERSEPATH_SUBPATH = prove (`!g u v. reversepath(subpath u v g) = subpath v u g`, REWRITE_TAC[reversepath; subpath; FUN_EQ_THM] THEN REPEAT GEN_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[DROP_SUB; VECTOR_SUB_LDISTRIB] THEN REWRITE_TAC[GSYM LIFT_EQ_CMUL; LIFT_SUB; LIFT_DROP] THEN VECTOR_ARITH_TAC);; let SUBPATH_TRANSLATION = prove (`!a g:real^1->real^N u v. subpath u v ((\x. a + x) o g) = (\x. a + x) o subpath u v g`, REWRITE_TAC[FUN_EQ_THM; subpath; o_THM]);; add_translation_invariants [SUBPATH_TRANSLATION];; let SUBPATH_LINEAR_IMAGE = prove (`!f:real^M->real^N g u v. linear f ==> subpath u v (f o g) = f o subpath u v g`, REWRITE_TAC[FUN_EQ_THM; subpath; o_THM]);; add_linear_invariants [SUBPATH_LINEAR_IMAGE];; let SIMPLE_PATH_SUBPATH_EQ = prove (`!g u v. simple_path(subpath u v g) <=> path(subpath u v g) /\ ~(u = v) /\ (!x y. x IN segment[u,v] /\ y IN segment[u,v] /\ g x = g y ==> x = y \/ x = u /\ y = v \/ x = v /\ y = u)`, REPEAT GEN_TAC THEN REWRITE_TAC[simple_path; subpath] THEN AP_TERM_TAC THEN REWRITE_TAC[GSYM SUBPATH_SCALING_LEMMA] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN REWRITE_TAC[VECTOR_ARITH `u + a % x = u <=> a % x = vec 0`; VECTOR_ARITH `a + x:real^N = a + y <=> x = y`] THEN REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_MUL_LCANCEL] THEN REWRITE_TAC[GSYM DROP_EQ; DROP_CMUL; DROP_ADD; DROP_SUB; REAL_RING `u + (v - u) * y = v <=> v = u \/ y = &1`] THEN REWRITE_TAC[REAL_SUB_0; DROP_EQ; GSYM DROP_VEC] THEN ASM_CASES_TAC `v:real^1 = u` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[REAL_SUB_REFL; DROP_VEC; VECTOR_MUL_LZERO] THEN REWRITE_TAC[RIGHT_IMP_FORALL_THM] THEN DISCH_THEN(MP_TAC o SPECL [`lift(&1 / &2)`; `lift(&3 / &4)`]) THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; GSYM DROP_EQ; LIFT_DROP] THEN CONV_TAC REAL_RAT_REDUCE_CONV);; let ARC_SUBPATH_EQ = prove (`!g u v. arc(subpath u v g) <=> path(subpath u v g) /\ ~(u = v) /\ (!x y. x IN segment[u,v] /\ y IN segment[u,v] /\ g x = g y ==> x = y)`, REPEAT GEN_TAC THEN REWRITE_TAC[arc; subpath] THEN AP_TERM_TAC THEN REWRITE_TAC[GSYM SUBPATH_SCALING_LEMMA] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN REWRITE_TAC[VECTOR_ARITH `u + a % x = u + a % y <=> a % (x - y) = vec 0`; VECTOR_MUL_EQ_0; DROP_EQ_0; VECTOR_SUB_EQ] THEN ASM_CASES_TAC `v:real^1 = u` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[VECTOR_SUB_REFL; DROP_VEC; VECTOR_MUL_LZERO] THEN REWRITE_TAC[RIGHT_IMP_FORALL_THM] THEN DISCH_THEN(MP_TAC o SPECL [`lift(&1 / &2)`; `lift(&3 / &4)`]) THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; GSYM DROP_EQ; LIFT_DROP] THEN CONV_TAC REAL_RAT_REDUCE_CONV);; let SIMPLE_PATH_SUBPATH = prove (`!g u v. simple_path g /\ u IN interval[vec 0,vec 1] /\ v IN interval[vec 0,vec 1] /\ ~(u = v) ==> simple_path(subpath u v g)`, SIMP_TAC[SIMPLE_PATH_SUBPATH_EQ; PATH_SUBPATH; SIMPLE_PATH_IMP_PATH] THEN REWRITE_TAC[simple_path] THEN GEN_TAC THEN REWRITE_TAC[FORALL_LIFT] THEN MATCH_MP_TAC REAL_WLOG_LT THEN REWRITE_TAC[FORALL_DROP; LIFT_DROP] THEN CONJ_TAC THENL [MESON_TAC[SEGMENT_SYM]; ALL_TAC] THEN SIMP_TAC[SEGMENT_1; REAL_LT_IMP_LE] THEN MAP_EVERY X_GEN_TAC [`u:real^1`; `v:real^1`] THEN DISCH_TAC THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`x:real^1`; `y:real^1`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^1`; `y:real^1`]) THEN SUBGOAL_THEN `!x:real^1. x IN interval[u,v] ==> x IN interval[vec 0,vec 1]` ASSUME_TAC THENL [REWRITE_TAC[GSYM SUBSET; SUBSET_INTERVAL_1] THEN ASM_MESON_TAC[IN_INTERVAL_1; DROP_VEC; REAL_LE_TRANS]; ASM_SIMP_TAC[]] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL_1])) THEN REWRITE_TAC[DROP_VEC; GSYM DROP_EQ] THEN ASM_REAL_ARITH_TAC);; let ARC_SIMPLE_PATH_SUBPATH = prove (`!g u v. simple_path g /\ u IN interval[vec 0,vec 1] /\ v IN interval[vec 0,vec 1] /\ ~(g u = g v) ==> arc(subpath u v g)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SIMPLE_PATH_IMP_ARC THEN ASM_REWRITE_TAC[PATHSTART_SUBPATH; PATHFINISH_SUBPATH] THEN ASM_MESON_TAC[SIMPLE_PATH_SUBPATH]);; let ARC_SUBPATH_ARC = prove (`!u v g. arc g /\ u IN interval [vec 0,vec 1] /\ v IN interval [vec 0,vec 1] /\ ~(u = v) ==> arc(subpath u v g)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC ARC_SIMPLE_PATH_SUBPATH THEN ASM_MESON_TAC[ARC_IMP_SIMPLE_PATH; arc]);; let ARC_SIMPLE_PATH_SUBPATH_INTERIOR = prove (`!g u v. simple_path g /\ u IN interval[vec 0,vec 1] /\ v IN interval[vec 0,vec 1] /\ ~(u = v) /\ abs(drop u - drop v) < &1 ==> arc(subpath u v g)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC ARC_SIMPLE_PATH_SUBPATH THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [simple_path]) THEN DISCH_THEN(MP_TAC o SPECL [`u:real^1`; `v:real^1`] o CONJUNCT2) THEN ASM_REWRITE_TAC[] THEN POP_ASSUM_LIST(MP_TAC o end_itlist CONJ) THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[] THEN STRIP_TAC THEN ASM_REWRITE_TAC[DROP_VEC] THEN REAL_ARITH_TAC);; let PATH_IMAGE_SUBPATH_SUBSET = prove (`!u v g:real^1->real^N. path g /\ u IN interval[vec 0,vec 1] /\ v IN interval[vec 0,vec 1] ==> path_image(subpath u v g) SUBSET path_image g`, SIMP_TAC[PATH_IMAGE_SUBPATH_GEN] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[path_image] THEN MATCH_MP_TAC IMAGE_SUBSET THEN SIMP_TAC[SEGMENT_CONVEX_HULL; SUBSET_HULL; CONVEX_INTERVAL] THEN ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET]);; let JOIN_SUBPATHS_MIDDLE = prove (`!p:real^1->real^N. subpath (vec 0) (lift(&1 / &2)) p ++ subpath (lift(&1 / &2)) (vec 1) p = p`, REWRITE_TAC[FUN_EQ_THM] THEN REPEAT GEN_TAC THEN REWRITE_TAC[joinpaths; subpath] THEN COND_CASES_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[GSYM DROP_EQ; DROP_ADD; DROP_SUB; DROP_CMUL; LIFT_DROP; DROP_VEC] THEN REAL_ARITH_TAC);; (* ------------------------------------------------------------------------- *) (* Some additional lemmas about choosing sub-paths. *) (* ------------------------------------------------------------------------- *) let EXISTS_SUBPATH_OF_PATH = prove (`!g a b:real^N. path g /\ a IN path_image g /\ b IN path_image g ==> ?h. path h /\ pathstart h = a /\ pathfinish h = b /\ path_image h SUBSET path_image g`, REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; path_image; FORALL_IN_IMAGE] THEN GEN_TAC THEN DISCH_TAC THEN X_GEN_TAC `u:real^1` THEN DISCH_TAC THEN X_GEN_TAC `v:real^1` THEN DISCH_TAC THEN EXISTS_TAC `subpath u v (g:real^1->real^N)` THEN ASM_REWRITE_TAC[GSYM path_image; PATH_IMAGE_SUBPATH_GEN] THEN ASM_SIMP_TAC[PATH_SUBPATH; PATHSTART_SUBPATH; PATHFINISH_SUBPATH] THEN REWRITE_TAC[path_image] THEN MATCH_MP_TAC IMAGE_SUBSET THEN SIMP_TAC[SEGMENT_CONVEX_HULL; SUBSET_HULL; CONVEX_INTERVAL] THEN ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET]);; let EXISTS_SUBPATH_OF_ARC_NOENDS = prove (`!g a b:real^N. arc g /\ a IN path_image g /\ b IN path_image g /\ {a,b} INTER {pathstart g,pathfinish g} = {} ==> ?h. path h /\ pathstart h = a /\ pathfinish h = b /\ path_image h SUBSET (path_image g) DIFF {pathstart g,pathfinish g}`, REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; path_image; FORALL_IN_IMAGE] THEN GEN_TAC THEN DISCH_TAC THEN X_GEN_TAC `u:real^1` THEN DISCH_TAC THEN X_GEN_TAC `v:real^1` THEN DISCH_TAC THEN DISCH_TAC THEN EXISTS_TAC `subpath u v (g:real^1->real^N)` THEN ASM_SIMP_TAC[PATH_SUBPATH; PATHSTART_SUBPATH; PATHFINISH_SUBPATH; ARC_IMP_PATH; GSYM path_image; PATH_IMAGE_SUBPATH_GEN] THEN REWRITE_TAC[path_image; pathstart; pathfinish] THEN REWRITE_TAC[SET_RULE `s SUBSET t DIFF {a,b} <=> s SUBSET t /\ ~(a IN s) /\ ~(b IN s)`] THEN REWRITE_TAC[IN_IMAGE] THEN SUBGOAL_THEN `~(vec 0 IN segment[u:real^1,v]) /\ ~(vec 1 IN segment[u,v])` STRIP_ASSUME_TAC THENL [REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL_1])) THEN REWRITE_TAC[SEGMENT_1] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN SIMP_TAC[REAL_ARITH `a:real <= b ==> (b <= a <=> a = b)`] THEN REWRITE_TAC[GSYM DROP_VEC; DROP_EQ] THEN RULE_ASSUM_TAC(REWRITE_RULE[arc; pathstart; pathfinish]) THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `segment[u:real^1,v] SUBSET interval[vec 0,vec 1]` MP_TAC THENL [SIMP_TAC[SEGMENT_CONVEX_HULL; SUBSET_HULL; CONVEX_INTERVAL] THEN ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET]; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[arc; pathstart; pathfinish]) THEN SUBGOAL_THEN `(vec 0:real^1) IN interval[vec 0,vec 1] /\ (vec 1:real^1) IN interval[vec 0,vec 1]` MP_TAC THENL [REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; REAL_POS; REAL_LE_REFL]; ASM SET_TAC[]]);; let EXISTS_SUBARC_OF_ARC_NOENDS = prove (`!g a b:real^N. arc g /\ a IN path_image g /\ b IN path_image g /\ ~(a = b) /\ {a,b} INTER {pathstart g,pathfinish g} = {} ==> ?h. arc h /\ pathstart h = a /\ pathfinish h = b /\ path_image h SUBSET (path_image g) DIFF {pathstart g,pathfinish g}`, REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; path_image; FORALL_IN_IMAGE] THEN GEN_TAC THEN DISCH_TAC THEN X_GEN_TAC `u:real^1` THEN DISCH_TAC THEN X_GEN_TAC `v:real^1` THEN REPEAT DISCH_TAC THEN EXISTS_TAC `subpath u v (g:real^1->real^N)` THEN ASM_SIMP_TAC[PATH_SUBPATH; PATHSTART_SUBPATH; PATHFINISH_SUBPATH; ARC_IMP_PATH; GSYM path_image; PATH_IMAGE_SUBPATH_GEN] THEN CONJ_TAC THENL [MATCH_MP_TAC ARC_SIMPLE_PATH_SUBPATH THEN ASM_SIMP_TAC[ARC_IMP_SIMPLE_PATH]; ALL_TAC] THEN REWRITE_TAC[path_image; pathstart; pathfinish] THEN REWRITE_TAC[SET_RULE `s SUBSET t DIFF {a,b} <=> s SUBSET t /\ ~(a IN s) /\ ~(b IN s)`] THEN REWRITE_TAC[IN_IMAGE] THEN SUBGOAL_THEN `~(vec 0 IN segment[u:real^1,v]) /\ ~(vec 1 IN segment[u,v])` STRIP_ASSUME_TAC THENL [REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL_1])) THEN REWRITE_TAC[SEGMENT_1] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN SIMP_TAC[REAL_ARITH `a:real <= b ==> (b <= a <=> a = b)`] THEN REWRITE_TAC[GSYM DROP_VEC; DROP_EQ] THEN RULE_ASSUM_TAC(REWRITE_RULE[arc; pathstart; pathfinish]) THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `segment[u:real^1,v] SUBSET interval[vec 0,vec 1]` MP_TAC THENL [SIMP_TAC[SEGMENT_CONVEX_HULL; SUBSET_HULL; CONVEX_INTERVAL] THEN ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET]; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[arc; pathstart; pathfinish]) THEN SUBGOAL_THEN `(vec 0:real^1) IN interval[vec 0,vec 1] /\ (vec 1:real^1) IN interval[vec 0,vec 1]` MP_TAC THENL [REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; REAL_POS; REAL_LE_REFL]; ASM SET_TAC[]]);; let EXISTS_ARC_PSUBSET_SIMPLE_PATH = prove (`!g:real^1->real^N. simple_path g /\ closed s /\ s PSUBSET path_image g ==> ?h. arc h /\ s SUBSET path_image h /\ path_image h SUBSET path_image g`, REPEAT STRIP_TAC THEN FIRST_ASSUM(DISJ_CASES_TAC o MATCH_MP SIMPLE_PATH_CASES) THENL [EXISTS_TAC `g:real^1->real^N` THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [PSUBSET_ALT]) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [path_image] THEN REWRITE_TAC[EXISTS_IN_IMAGE] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^1` STRIP_ASSUME_TAC) THEN ABBREV_TAC `(h:real^1->real^N) = shiftpath u g` THEN SUBGOAL_THEN `simple_path(h:real^1->real^N) /\ pathstart h = (g:real^1->real^N) u /\ pathfinish h = (g:real^1->real^N) u /\ path_image h = path_image g` MP_TAC THENL [EXPAND_TAC "h" THEN ASM_MESON_TAC[SIMPLE_PATH_SHIFTPATH; PATH_IMAGE_SHIFTPATH; PATHSTART_SHIFTPATH; PATHFINISH_SHIFTPATH; IN_INTERVAL_1; DROP_VEC]; REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(SUBST_ALL_TAC o SYM) THEN UNDISCH_THEN `pathstart(h:real^1->real^N) = (g:real^1->real^N) u` (SUBST_ALL_TAC o SYM)] THEN SUBGOAL_THEN `open_in (subtopology euclidean (interval[vec 0,vec 1])) {x:real^1 | x IN interval[vec 0,vec 1] /\ (h x) IN ((:real^N) DIFF s)}` MP_TAC THENL [MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE THEN ASM_SIMP_TAC[GSYM path; GSYM closed; SIMPLE_PATH_IMP_PATH]; REWRITE_TAC[open_in] THEN DISCH_THEN(MP_TAC o CONJUNCT2)] THEN REWRITE_TAC[IN_DIFF; IN_UNIV; IN_ELIM_THM] THEN DISCH_THEN(fun th -> MP_TAC(SPEC `vec 0:real^1` th) THEN MP_TAC(SPEC `vec 1:real^1` th)) THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; REAL_POS; REAL_LE_REFL] THEN REWRITE_TAC[DIST_REAL; VEC_COMPONENT; REAL_SUB_RZERO] THEN SIMP_TAC[GSYM drop] THEN ANTS_TAC THENL [ASM_MESON_TAC[pathfinish]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `d2:real` STRIP_ASSUME_TAC) THEN ANTS_TAC THENL [ASM_MESON_TAC[pathstart]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `d1:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `subpath (lift(min d1 (&1 / &4))) (lift(&1 - min d2 (&1 / &4))) (h:real^1->real^N)` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC ARC_SIMPLE_PATH_SUBPATH_INTERIOR THEN ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; LIFT_DROP; LIFT_EQ] THEN ASM_REAL_ARITH_TAC; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET t ==> t INTER s SUBSET u ==> s SUBSET u`)) THEN REWRITE_TAC[SUBSET; IN_INTER; IMP_CONJ] THEN SIMP_TAC[PATH_IMAGE_SUBPATH; LIFT_DROP; REAL_ARITH `min d1 (&1 / &4) <= &1 - min d2 (&1 / &4)`] THEN REWRITE_TAC[FORALL_IN_IMAGE; path_image; IN_INTERVAL_1; DROP_VEC] THEN X_GEN_TAC `x:real^1` THEN REPEAT STRIP_TAC THEN REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `x:real^1` THEN ASM_REWRITE_TAC[IN_INTERVAL_1; LIFT_DROP] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `x:real^1`)) THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC; MATCH_MP_TAC PATH_IMAGE_SUBPATH_SUBSET THEN ASM_SIMP_TAC[SIMPLE_PATH_IMP_PATH; IN_INTERVAL_1; LIFT_DROP; DROP_VEC] THEN ASM_REAL_ARITH_TAC]);; let EXISTS_DOUBLE_ARC_EXPLICIT = prove (`!g:real^1->real^N a b. simple_path g /\ pathfinish g = pathstart g /\ a IN interval[vec 0,vec 1] /\ b IN interval[vec 0,vec 1] /\ drop a <= drop b /\ ~(g a = g b) ==> ?u d. arc u /\ arc d /\ pathstart u = g a /\ pathfinish u = g b /\ pathstart d = g b /\ pathfinish d = g a /\ path_image u = IMAGE g (interval[a,b]) /\ path_image d = IMAGE g (interval[vec 0,vec 1] DIFF interval(a,b)) /\ (path_image u) INTER (path_image d) = {g a,g b} /\ (path_image u) UNION (path_image d) = path_image g`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `a:real^1 = vec 0` THENL [MAP_EVERY EXISTS_TAC [`subpath (vec 0) b (g:real^1->real^N)`; `subpath b (vec 1) (g:real^1->real^N)`] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC ARC_SIMPLE_PATH_SUBPATH THEN ASM_REWRITE_TAC[ENDS_IN_UNIT_INTERVAL] THEN ASM_MESON_TAC[]; MATCH_MP_TAC ARC_SIMPLE_PATH_SUBPATH THEN ASM_REWRITE_TAC[ENDS_IN_UNIT_INTERVAL] THEN ASM_MESON_TAC[pathfinish; pathstart]; ASM_REWRITE_TAC[PATHSTART_SUBPATH]; ASM_REWRITE_TAC[PATHFINISH_SUBPATH]; ASM_REWRITE_TAC[PATHSTART_SUBPATH]; ASM_REWRITE_TAC[PATHFINISH_SUBPATH] THEN ASM_MESON_TAC[pathfinish; pathstart]; RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_SIMP_TAC[PATH_IMAGE_SUBPATH]; RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN ASM_SIMP_TAC[PATH_IMAGE_SUBPATH; DROP_VEC; GSYM IMAGE_UNION] THEN RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `g u = g l ==> u IN s /\ u IN t /\ (!x. ~(x = l) ==> (x IN s <=> x IN t)) ==> IMAGE g s = IMAGE g t`)) THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; GSYM DROP_EQ; IN_DIFF] THEN ASM_REAL_ARITH_TAC; RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN ASM_SIMP_TAC[PATH_IMAGE_SUBPATH; DROP_VEC; GSYM IMAGE_UNION] THEN RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `g c = g a ==> a IN ab /\ b IN ab /\ b IN b1 /\ c IN b1 /\ (!x y. g x = g y /\ x IN ab /\ y IN b1 ==> x = a \/ x = b) ==> IMAGE g ab INTER IMAGE g b1 = {g a,g b}`)) THEN ASM_REWRITE_TAC[ENDS_IN_INTERVAL; INTERVAL_NE_EMPTY_1; DROP_VEC] THEN MAP_EVERY X_GEN_TAC [`x:real^1`; `y:real^1`] THEN FIRST_X_ASSUM(MP_TAC o CONJUNCT2 o GEN_REWRITE_RULE I [simple_path]) THEN DISCH_THEN(MP_TAC o SPECL [`x:real^1`; `y:real^1`]) THEN ASM_CASES_TAC `(g:real^1->real^N) x = g y` THEN ASM_REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[DROP_VEC]) THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; GSYM DROP_EQ] THEN ASM_REAL_ARITH_TAC; RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish]) THEN ASM_SIMP_TAC[PATH_IMAGE_JOIN; PATHSTART_SUBPATH; PATHFINISH_SUBPATH] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN ASM_SIMP_TAC[PATH_IMAGE_SUBPATH; GSYM IMAGE_UNION; DROP_VEC] THEN REWRITE_TAC[path_image] THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_UNION; IN_INTERVAL_1; DROP_VEC] THEN ASM_REAL_ARITH_TAC]; ALL_TAC] THEN ASM_CASES_TAC `b:real^1 = vec 1` THENL [MAP_EVERY EXISTS_TAC [`subpath a b (g:real^1->real^N)`; `subpath (vec 0) a (g:real^1->real^N)`] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC ARC_SIMPLE_PATH_SUBPATH THEN ASM_REWRITE_TAC[ENDS_IN_UNIT_INTERVAL] THEN ASM_MESON_TAC[]; MATCH_MP_TAC ARC_SIMPLE_PATH_SUBPATH THEN ASM_REWRITE_TAC[ENDS_IN_UNIT_INTERVAL] THEN ASM_MESON_TAC[pathfinish; pathstart]; ASM_REWRITE_TAC[PATHSTART_SUBPATH]; ASM_REWRITE_TAC[PATHFINISH_SUBPATH]; ASM_REWRITE_TAC[PATHSTART_SUBPATH] THEN ASM_MESON_TAC[pathfinish; pathstart]; ASM_REWRITE_TAC[PATHFINISH_SUBPATH]; RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_SIMP_TAC[PATH_IMAGE_SUBPATH]; RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN ASM_SIMP_TAC[PATH_IMAGE_SUBPATH; DROP_VEC; GSYM IMAGE_UNION] THEN RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `g u = g l ==> l IN s /\ u IN t /\ (!x. ~(x = u) ==> (x IN s <=> x IN t)) ==> IMAGE g s = IMAGE g t`)) THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; GSYM DROP_EQ; IN_DIFF] THEN ASM_REAL_ARITH_TAC; RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN ASM_SIMP_TAC[PATH_IMAGE_SUBPATH; DROP_VEC; GSYM IMAGE_UNION] THEN RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `g b = g c ==> a IN a1 /\ b IN a1 /\ a IN a0 /\ c IN a0 /\ (!x y. g x = g y /\ x IN a0 /\ y IN a1 ==> x = a \/ x = c) ==> IMAGE g a1 INTER IMAGE g a0 = {g a,g b}`)) THEN ASM_REWRITE_TAC[ENDS_IN_INTERVAL; INTERVAL_NE_EMPTY_1; DROP_VEC] THEN MAP_EVERY X_GEN_TAC [`x:real^1`; `y:real^1`] THEN FIRST_X_ASSUM(MP_TAC o CONJUNCT2 o GEN_REWRITE_RULE I [simple_path]) THEN DISCH_THEN(MP_TAC o SPECL [`x:real^1`; `y:real^1`]) THEN ASM_CASES_TAC `(g:real^1->real^N) x = g y` THEN ASM_REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[DROP_VEC]) THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; GSYM DROP_EQ] THEN ASM_REAL_ARITH_TAC; RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish]) THEN ASM_SIMP_TAC[PATH_IMAGE_JOIN; PATHSTART_SUBPATH; PATHFINISH_SUBPATH] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN ASM_SIMP_TAC[PATH_IMAGE_SUBPATH; GSYM IMAGE_UNION; DROP_VEC] THEN REWRITE_TAC[path_image] THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_UNION; IN_INTERVAL_1; DROP_VEC] THEN ASM_REAL_ARITH_TAC]; ALL_TAC] THEN MAP_EVERY EXISTS_TAC [`subpath a b (g:real^1->real^N)`; `subpath b (vec 1) (g:real^1->real^N) ++ subpath (vec 0) a g`] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC ARC_SIMPLE_PATH_SUBPATH THEN ASM_REWRITE_TAC[]; MATCH_MP_TAC ARC_JOIN THEN REWRITE_TAC[PATHSTART_SUBPATH; PATHFINISH_SUBPATH] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC ARC_SIMPLE_PATH_SUBPATH THEN ASM_REWRITE_TAC[ENDS_IN_UNIT_INTERVAL] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o CONJUNCT2 o GEN_REWRITE_RULE I [simple_path]) THEN DISCH_THEN(MP_TAC o SPECL [`b:real^1`; `vec 1:real^1`]) THEN ASM_REWRITE_TAC[ENDS_IN_UNIT_INTERVAL] THEN REWRITE_TAC[GSYM DROP_EQ; DROP_VEC] THEN RULE_ASSUM_TAC(REWRITE_RULE[GSYM DROP_EQ; IN_INTERVAL_1; DROP_VEC]) THEN ASM_REAL_ARITH_TAC; MATCH_MP_TAC ARC_SIMPLE_PATH_SUBPATH THEN ASM_REWRITE_TAC[ENDS_IN_UNIT_INTERVAL] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o CONJUNCT2 o GEN_REWRITE_RULE I [simple_path]) THEN DISCH_THEN(MP_TAC o SPECL [`a:real^1`; `vec 0:real^1`]) THEN ASM_REWRITE_TAC[ENDS_IN_UNIT_INTERVAL] THEN REWRITE_TAC[GSYM DROP_EQ; DROP_VEC] THEN RULE_ASSUM_TAC(REWRITE_RULE[GSYM DROP_EQ; IN_INTERVAL_1; DROP_VEC]) THEN ASM_REAL_ARITH_TAC; ASM_MESON_TAC[pathstart; pathfinish]; RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_SIMP_TAC[PATH_IMAGE_SUBPATH; GSYM IMAGE_UNION] THEN RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `g u = g l ==> (!x y. x IN b1 /\ y IN a0 /\ g x = g y ==> x = l \/ x = u) ==> IMAGE g b1 INTER IMAGE g a0 SUBSET {g l}`)) THEN MAP_EVERY X_GEN_TAC [`x:real^1`; `y:real^1`] THEN FIRST_X_ASSUM(MP_TAC o CONJUNCT2 o GEN_REWRITE_RULE I [simple_path]) THEN DISCH_THEN(MP_TAC o SPECL [`x:real^1`; `y:real^1`]) THEN ASM_CASES_TAC `(g:real^1->real^N) x = g y` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `a:real^1 = b` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN ASM_REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[DROP_VEC; GSYM DROP_EQ]) THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; GSYM DROP_EQ] THEN ASM_REAL_ARITH_TAC]; REWRITE_TAC[PATHSTART_SUBPATH]; REWRITE_TAC[PATHFINISH_SUBPATH]; REWRITE_TAC[PATHSTART_JOIN; PATHSTART_SUBPATH]; REWRITE_TAC[PATHFINISH_JOIN; PATHFINISH_SUBPATH]; ASM_SIMP_TAC[PATH_IMAGE_SUBPATH]; RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish]) THEN ASM_SIMP_TAC[PATH_IMAGE_JOIN; PATHSTART_SUBPATH; PATHFINISH_SUBPATH] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_SIMP_TAC[PATH_IMAGE_SUBPATH; GSYM IMAGE_UNION] THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_UNION; IN_DIFF; IN_INTERVAL_1] THEN ASM_REAL_ARITH_TAC; RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish]) THEN ASM_SIMP_TAC[PATH_IMAGE_JOIN; PATHSTART_SUBPATH; PATHFINISH_SUBPATH] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_SIMP_TAC[PATH_IMAGE_SUBPATH; GSYM IMAGE_UNION] THEN MATCH_MP_TAC(SET_RULE `a IN ab /\ b IN ab /\ a IN a0 /\ b IN b1 /\ (!x y. g x = g y /\ x IN ab /\ (y IN b1 \/ y IN a0) ==> x = a \/ x = b) ==> IMAGE g ab INTER IMAGE g (b1 UNION a0) = {g a,g b}`) THEN ASM_REWRITE_TAC[ENDS_IN_INTERVAL; INTERVAL_NE_EMPTY_1] THEN MAP_EVERY X_GEN_TAC [`x:real^1`; `y:real^1`] THEN FIRST_X_ASSUM(MP_TAC o CONJUNCT2 o GEN_REWRITE_RULE I [simple_path]) THEN DISCH_THEN(MP_TAC o SPECL [`x:real^1`; `y:real^1`]) THEN ASM_CASES_TAC `(g:real^1->real^N) x = g y` THEN ASM_REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[DROP_VEC]) THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; GSYM DROP_EQ] THEN ASM_REAL_ARITH_TAC; RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish]) THEN ASM_SIMP_TAC[PATH_IMAGE_JOIN; PATHSTART_SUBPATH; PATHFINISH_SUBPATH] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_SIMP_TAC[PATH_IMAGE_SUBPATH; GSYM IMAGE_UNION] THEN REWRITE_TAC[path_image] THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_UNION; IN_INTERVAL_1] THEN ASM_REAL_ARITH_TAC]);; let EXISTS_DOUBLE_ARC = prove (`!g:real^1->real^N a b. simple_path g /\ pathfinish g = pathstart g /\ a IN path_image g /\ b IN path_image g /\ ~(a = b) ==> ?u d. arc u /\ arc d /\ pathstart u = a /\ pathfinish u = b /\ pathstart d = b /\ pathfinish d = a /\ (path_image u) INTER (path_image d) = {a,b} /\ (path_image u) UNION (path_image d) = path_image g`, REPEAT STRIP_TAC THEN UNDISCH_TAC `(b:real^N) IN path_image g` THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [path_image] THEN UNDISCH_TAC `(a:real^N) IN path_image g` THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [path_image] THEN REWRITE_TAC[IN_IMAGE; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `u:real^1` THEN DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC) THEN X_GEN_TAC `v:real^1` THEN DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC) THEN DISJ_CASES_TAC(REAL_ARITH `drop u <= drop v \/ drop v <= drop u`) THENL [MP_TAC(ISPECL [`g:real^1->real^N`; `u:real^1`; `v:real^1`] EXISTS_DOUBLE_ARC_EXPLICIT) THEN ASM_REWRITE_TAC[] THEN MESON_TAC[]; MP_TAC(ISPECL [`g:real^1->real^N`; `v:real^1`; `u:real^1`] EXISTS_DOUBLE_ARC_EXPLICIT) THEN ASM_REWRITE_TAC[] THEN MESON_TAC[INTER_COMM; UNION_COMM; INSERT_AC]]);; let SUBPATH_TO_FRONTIER_EXPLICIT = prove (`!g:real^1->real^N s. path g /\ pathstart g IN s /\ ~(pathfinish g IN s) ==> ?u. u IN interval[vec 0,vec 1] /\ (!x. &0 <= drop x /\ drop x < drop u ==> g x IN interior s) /\ ~(g u IN interior s) /\ (u = vec 0 \/ g u IN closure s)`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `{u | lift u IN interval[vec 0,vec 1] /\ g(lift u) IN closure((:real^N) DIFF s)}` COMPACT_ATTAINS_INF) THEN SIMP_TAC[LIFT_DROP; SET_RULE `(!x. lift(drop x) = x) ==> IMAGE lift {x | P(lift x)} = {x | P x}`] THEN ANTS_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[path; pathstart; pathfinish; SUBSET; path_image; FORALL_IN_IMAGE]) THEN CONJ_TAC THENL [REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED] THEN CONJ_TAC THENL [MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `interval[vec 0:real^1,vec 1]` THEN REWRITE_TAC[BOUNDED_INTERVAL] THEN SET_TAC[]; MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE THEN ASM_REWRITE_TAC[CLOSED_CLOSURE; CLOSED_INTERVAL]]; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_ELIM_THM] THEN EXISTS_TAC `&1` THEN REWRITE_TAC[LIFT_NUM] THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; REAL_POS; REAL_LE_REFL] THEN MATCH_MP_TAC(REWRITE_RULE[SUBSET] CLOSURE_SUBSET) THEN ASM_REWRITE_TAC[IN_DIFF; IN_UNIV]]; ALL_TAC] THEN REWRITE_TAC[EXISTS_DROP; FORALL_DROP; IN_ELIM_THM; LIFT_DROP] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^1` THEN REWRITE_TAC[CLOSURE_COMPLEMENT; IN_DIFF; IN_UNIV] THEN STRIP_TAC THEN ASM_REWRITE_TAC[subpath; VECTOR_SUB_RZERO; VECTOR_ADD_LID] THEN ASM_REWRITE_TAC[GSYM LIFT_EQ_CMUL; LIFT_DROP] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; GSYM DROP_EQ] THEN MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL [REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE BINDER_CONV [TAUT `a /\ ~b ==> c <=> a /\ ~c ==> b`]) THEN ASM_REAL_ARITH_TAC; FIRST_X_ASSUM(K ALL_TAC o SPEC `x:real^1`) THEN DISCH_TAC] THEN ASM_CASES_TAC `drop u = &0` THEN ASM_REWRITE_TAC[frontier; IN_DIFF; CLOSURE_APPROACHABLE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[path; pathstart; pathfinish]) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [continuous_on]) THEN DISCH_THEN(MP_TAC o SPEC `u:real^1`) THEN ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(MP_TAC o SPEC `lift(max (&0) (drop u - d / &2))`) THEN REWRITE_TAC[LIFT_DROP; DIST_REAL; GSYM drop] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC (MESON[] `P a ==> dist(a,y) < e ==> ?x. P x /\ dist(x,y) < e`) THEN MATCH_MP_TAC(REWRITE_RULE[SUBSET] INTERIOR_SUBSET) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[LIFT_DROP] THEN ASM_ARITH_TAC);; let SUBPATH_TO_FRONTIER_STRONG = prove (`!g:real^1->real^N s. path g /\ pathstart g IN s /\ ~(pathfinish g IN s) ==> ?u. u IN interval[vec 0,vec 1] /\ ~(pathfinish(subpath (vec 0) u g) IN interior s) /\ (u = vec 0 \/ (!x. x IN interval[vec 0,vec 1] /\ ~(x = vec 1) ==> (subpath (vec 0) u g x) IN interior s) /\ pathfinish(subpath (vec 0) u g) IN closure s)`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP SUBPATH_TO_FRONTIER_EXPLICIT) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^1` THEN REWRITE_TAC[subpath; pathfinish; VECTOR_SUB_RZERO; VECTOR_ADD_LID] THEN ASM_CASES_TAC `u:real^1 = vec 0` THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ASM_REWRITE_TAC[DROP_VEC; VECTOR_MUL_LZERO] THEN ASM_REWRITE_TAC[GSYM LIFT_EQ_CMUL; LIFT_DROP] THEN X_GEN_TAC `x:real^1` THEN REWRITE_TAC[IN_INTERVAL_1; GSYM DROP_EQ; DROP_VEC] THEN STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN ASM_SIMP_TAC[DROP_CMUL; REAL_LE_MUL] THEN REWRITE_TAC[REAL_ARITH `u * x < u <=> &0 < u * (&1 - x)`] THEN MATCH_MP_TAC REAL_LT_MUL THEN REWRITE_TAC[REAL_SUB_LT] THEN ASM_REWRITE_TAC[REAL_LT_LE] THEN ASM_REWRITE_TAC[GSYM LIFT_EQ; LIFT_DROP; LIFT_NUM]);; let SUBPATH_TO_FRONTIER = prove (`!g:real^1->real^N s. path g /\ pathstart g IN s /\ ~(pathfinish g IN s) ==> ?u. u IN interval[vec 0,vec 1] /\ pathfinish(subpath (vec 0) u g) IN frontier s /\ (path_image(subpath (vec 0) u g) DELETE pathfinish(subpath (vec 0) u g)) SUBSET interior s`, REPEAT GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[frontier; IN_DIFF] THEN FIRST_ASSUM(MP_TAC o MATCH_MP SUBPATH_TO_FRONTIER_STRONG) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^1` THEN ASM_CASES_TAC `u:real^1 = vec 0` THEN ASM_REWRITE_TAC[] THENL [REWRITE_TAC[PATHSTART_SUBPATH; PATHFINISH_SUBPATH] THEN RULE_ASSUM_TAC(REWRITE_RULE[pathstart]) THEN STRIP_TAC THEN ASM_SIMP_TAC[REWRITE_RULE[SUBSET] CLOSURE_SUBSET] THEN REWRITE_TAC[subpath; path_image; VECTOR_SUB_REFL; DROP_VEC; VECTOR_MUL_LZERO; VECTOR_ADD_LID] THEN SET_TAC[]; STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET; path_image; FORALL_IN_IMAGE; IN_DELETE; IMP_CONJ] THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; pathfinish] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN ASM_MESON_TAC[]]);; let EXISTS_PATH_SUBPATH_TO_FRONTIER = prove (`!g:real^1->real^N s. path g /\ pathstart g IN s /\ ~(pathfinish g IN s) ==> ?h. path h /\ pathstart h = pathstart g /\ (path_image h) SUBSET (path_image g) /\ (path_image h DELETE (pathfinish h)) SUBSET interior s /\ pathfinish h IN frontier s`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP SUBPATH_TO_FRONTIER) THEN EXISTS_TAC `subpath (vec 0) u (g:real^1->real^N)` THEN ASM_SIMP_TAC[PATH_SUBPATH; IN_INTERVAL_1; DROP_VEC; REAL_POS; REAL_LE_REFL; PATHSTART_SUBPATH; PATH_IMAGE_SUBPATH_SUBSET] THEN REWRITE_TAC[pathstart]);; let EXISTS_PATH_SUBPATH_TO_FRONTIER_CLOSED = prove (`!g:real^1->real^N s. closed s /\ path g /\ pathstart g IN s /\ ~(pathfinish g IN s) ==> ?h. path h /\ pathstart h = pathstart g /\ (path_image h) SUBSET (path_image g) INTER s /\ pathfinish h IN frontier s`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o MATCH_MP EXISTS_PATH_SUBPATH_TO_FRONTIER) THEN MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[SUBSET_INTER] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `(pathfinish h:real^N) INSERT (path_image h DELETE pathfinish h)` THEN CONJ_TAC THENL [SET_TAC[]; REWRITE_TAC[INSERT_SUBSET]] THEN CONJ_TAC THENL [ASM_MESON_TAC[frontier; CLOSURE_EQ; IN_DIFF]; ASM_MESON_TAC[SUBSET_TRANS; INTERIOR_SUBSET]]);; let PATH_COMBINE = prove (`!u g:real^1->real^N. u IN interval[vec 0,vec 1] ==> (path g <=> path(subpath (vec 0) u g) /\ path(subpath u (vec 1) g))`, REPEAT STRIP_TAC THEN EQ_TAC THEN ASM_SIMP_TAC[PATH_SUBPATH; ENDS_IN_UNIT_INTERVAL] THEN ASM_CASES_TAC `u:real^1 = vec 0` THEN ASM_SIMP_TAC[SUBPATH_TRIVIAL] THEN ASM_CASES_TAC `u:real^1 = vec 1` THEN ASM_SIMP_TAC[SUBPATH_TRIVIAL] THEN REWRITE_TAC[path; subpath; VECTOR_ADD_LID; VECTOR_SUB_RZERO] THEN STRIP_TAC THEN SUBGOAL_THEN `interval[vec 0:real^1,vec 1] = interval[vec 0,u] UNION interval[u,vec 1]` SUBST1_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL_1]) THEN REWRITE_TAC[EXTENSION; IN_UNION; IN_INTERVAL_1] THEN REAL_ARITH_TAC; MATCH_MP_TAC CONTINUOUS_ON_UNION THEN REWRITE_TAC[CLOSED_INTERVAL] THEN CONJ_TAC THENL [SUBGOAL_THEN `(g:real^1->real^N) = (\x. g(drop u % x)) o (\x. inv(drop u) % x)` SUBST1_TAC THENL [REWRITE_TAC[FUN_EQ_THM; o_THM; VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[REAL_MUL_RINV; VECTOR_MUL_LID; GSYM LIFT_EQ; LIFT_NUM; LIFT_DROP]; MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[LINEAR_SCALING; LINEAR_CONTINUOUS_ON] THEN ONCE_REWRITE_TAC[VECTOR_ARITH `inv u % x:real^N = inv u % x + vec 0`] THEN REWRITE_TAC[IMAGE_AFFINITY_INTERVAL] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; GSYM DROP_EQ; DROP_VEC]) THEN ASM_REWRITE_TAC[INTERVAL_EQ_EMPTY_1; DROP_VEC; GSYM REAL_NOT_LE; REAL_LE_INV_EQ] THEN REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_RID] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN MATCH_MP_TAC(SET_RULE `s = t ==> s SUBSET t`) THEN AP_TERM_TAC THEN REWRITE_TAC[CONS_11; PAIR_EQ] THEN ASM_SIMP_TAC[GSYM DROP_EQ; DROP_CMUL; DROP_VEC; REAL_MUL_LINV]]; SUBGOAL_THEN `(g:real^1->real^N) = (\x. g(u + drop(vec 1 - u) % x)) o (\x. inv(drop(vec 1 - u)) % (x - u))` SUBST1_TAC THENL [REWRITE_TAC[FUN_EQ_THM; o_THM; VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[REAL_MUL_RINV; VECTOR_MUL_LID; VECTOR_SUB_EQ; GSYM LIFT_EQ; LIFT_NUM; LIFT_DROP] THEN REWRITE_TAC[VECTOR_ARITH `u + x - u:real^N = x`]; MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_CMUL; CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN MATCH_MP_TAC(SET_RULE `s = t ==> s SUBSET t`) THEN REWRITE_TAC[VECTOR_ARITH `c % (x - u):real^N = c % x + --(c % u)`] THEN REWRITE_TAC[IMAGE_AFFINITY_INTERVAL] THEN REWRITE_TAC[IMAGE_AFFINITY_INTERVAL] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; GSYM DROP_EQ; DROP_VEC]) THEN ASM_REWRITE_TAC[INTERVAL_EQ_EMPTY_1; DROP_VEC; GSYM REAL_NOT_LE; REAL_LE_INV_EQ; DROP_SUB; REAL_SUB_LE] THEN AP_TERM_TAC THEN REWRITE_TAC[CONS_11; PAIR_EQ] THEN REWRITE_TAC[GSYM DROP_EQ; DROP_CMUL; DROP_VEC; DROP_ADD; DROP_NEG] THEN REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_FIELD]]]);; (* ------------------------------------------------------------------------- *) (* Special case of straight-line paths. *) (* ------------------------------------------------------------------------- *) let linepath = new_definition `linepath(a,b) = \x. (&1 - drop x) % a + drop x % b`;; let LINEPATH_TRANSLATION = prove (`!a b c. linepath(a + b,a + c) = (\x. a + x) o linepath(b,c)`, REWRITE_TAC[linepath; o_THM; FUN_EQ_THM] THEN VECTOR_ARITH_TAC);; add_translation_invariants [LINEPATH_TRANSLATION];; let LINEPATH_LINEAR_IMAGE = prove (`!f. linear f ==> !b c. linepath(f b,f c) = f o linepath(b,c)`, REWRITE_TAC[linepath; o_THM; FUN_EQ_THM] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP LINEAR_ADD) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP LINEAR_CMUL) THEN ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC);; add_linear_invariants [LINEPATH_LINEAR_IMAGE];; let PATHSTART_LINEPATH = prove (`!a b. pathstart(linepath(a,b)) = a`, REWRITE_TAC[linepath; pathstart; DROP_VEC] THEN VECTOR_ARITH_TAC);; let PATHFINISH_LINEPATH = prove (`!a b. pathfinish(linepath(a,b)) = b`, REWRITE_TAC[linepath; pathfinish; DROP_VEC] THEN VECTOR_ARITH_TAC);; let CONTINUOUS_LINEPATH_AT = prove (`!a b x. linepath(a,b) continuous (at x)`, REPEAT GEN_TAC THEN REWRITE_TAC[linepath] THEN REWRITE_TAC[VECTOR_ARITH `(&1 - u) % x + y = x + u % --x + y`] THEN MATCH_MP_TAC CONTINUOUS_ADD THEN REWRITE_TAC[CONTINUOUS_CONST] THEN MATCH_MP_TAC CONTINUOUS_ADD THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_VMUL THEN REWRITE_TAC[o_DEF; LIFT_DROP; CONTINUOUS_AT_ID]);; let CONTINUOUS_ON_LINEPATH = prove (`!a b s. linepath(a,b) continuous_on s`, MESON_TAC[CONTINUOUS_AT_IMP_CONTINUOUS_ON; CONTINUOUS_LINEPATH_AT]);; let PATH_LINEPATH = prove (`!a b. path(linepath(a,b))`, REWRITE_TAC[path; CONTINUOUS_ON_LINEPATH]);; let PATH_IMAGE_LINEPATH = prove (`!a b. path_image(linepath (a,b)) = segment[a,b]`, REWRITE_TAC[segment; path_image; linepath] THEN REWRITE_TAC[EXTENSION; IN_IMAGE; IN_ELIM_THM; IN_INTERVAL] THEN SIMP_TAC[DIMINDEX_1; FORALL_1; VEC_COMPONENT; ARITH] THEN REWRITE_TAC[EXISTS_LIFT; GSYM drop; LIFT_DROP] THEN MESON_TAC[]);; let REVERSEPATH_LINEPATH = prove (`!a b. reversepath(linepath(a,b)) = linepath(b,a)`, REWRITE_TAC[reversepath; linepath; DROP_SUB; DROP_VEC; FUN_EQ_THM] THEN VECTOR_ARITH_TAC);; let ARC_LINEPATH = prove (`!a b. ~(a = b) ==> arc(linepath(a,b))`, REWRITE_TAC[arc; PATH_LINEPATH] THEN REWRITE_TAC[linepath] THEN REWRITE_TAC[VECTOR_ARITH `(&1 - x) % a + x % b:real^N = (&1 - y) % a + y % b <=> (x - y) % (a - b) = vec 0`] THEN SIMP_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ; DROP_EQ; REAL_SUB_0]);; let SIMPLE_PATH_LINEPATH = prove (`!a b. ~(a = b) ==> simple_path(linepath(a,b))`, MESON_TAC[ARC_IMP_SIMPLE_PATH; ARC_LINEPATH]);; let SIMPLE_PATH_LINEPATH_EQ = prove (`!a b:real^N. simple_path(linepath(a,b)) <=> ~(a = b)`, REPEAT GEN_TAC THEN EQ_TAC THEN REWRITE_TAC[SIMPLE_PATH_LINEPATH] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[simple_path] THEN DISCH_THEN SUBST1_TAC THEN DISCH_THEN(MP_TAC o CONJUNCT2) THEN REWRITE_TAC[linepath; GSYM VECTOR_ADD_RDISTRIB] THEN DISCH_THEN(MP_TAC o SPECL [`lift(&0)`; `lift(&1 / &2)`]) THEN REWRITE_TAC[IN_INTERVAL_1; LIFT_DROP; GSYM DROP_EQ; DROP_VEC] THEN CONV_TAC REAL_RAT_REDUCE_CONV);; let ARC_LINEPATH_EQ = prove (`!a b. arc(linepath(a,b)) <=> ~(a = b)`, REPEAT GEN_TAC THEN EQ_TAC THEN REWRITE_TAC[ARC_LINEPATH] THEN MESON_TAC[SIMPLE_PATH_LINEPATH_EQ; ARC_IMP_SIMPLE_PATH]);; let LINEPATH_REFL = prove (`!a. linepath(a,a) = \x. a`, REWRITE_TAC[linepath; VECTOR_ARITH `(&1 - u) % x + u % x:real^N = x`]);; let PATH_IMAGE_CONST = prove (`!a:real^N. path_image (\x. a) = {a}`, REWRITE_TAC[GSYM LINEPATH_REFL; PATH_IMAGE_LINEPATH] THEN REWRITE_TAC[SEGMENT_REFL]);; let SHIFTPATH_TRIVIAL = prove (`!t a. shiftpath t (linepath(a,a)) = linepath(a,a)`, REWRITE_TAC[shiftpath; LINEPATH_REFL; COND_ID]);; let SUBPATH_REFL = prove (`!g a. subpath a a g = linepath(g a,g a)`, REWRITE_TAC[subpath; linepath; VECTOR_SUB_REFL; DROP_VEC; VECTOR_MUL_LZERO; FUN_EQ_THM; VECTOR_ADD_RID] THEN VECTOR_ARITH_TAC);; let SEGMENT_TO_FRONTIER = prove (`!s a b:real^N. a IN interior s /\ ~(b IN interior s) ==> ?c. c IN segment[a,b] /\ ~(c = a) /\ c IN frontier s /\ segment(a,c) SUBSET interior s`, GEOM_ORIGIN_TAC `a:real^N` THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC(MESON[] `(!x. R x ==> Q x) /\ (?x. P x /\ R x /\ S x) ==> ?x. P x /\ Q x /\ R x /\ S x`) THEN CONJ_TAC THENL [ASM_MESON_TAC[frontier; IN_DIFF]; ALL_TAC] THEN MP_TAC(ISPECL [`linepath(vec 0:real^N,b)`; `interior s:real^N->bool`] SUBPATH_TO_FRONTIER) THEN ASM_REWRITE_TAC[PATH_LINEPATH; PATHSTART_LINEPATH; PATHFINISH_LINEPATH] THEN REWRITE_TAC[PATH_IMAGE_LINEPATH; INTERIOR_INTERIOR] THEN REWRITE_TAC[subpath; linepath; VECTOR_ADD_LID; VECTOR_SUB_RZERO; VECTOR_MUL_RZERO; pathstart; pathfinish] THEN REWRITE_TAC[IN_INTERVAL_1; GSYM EXISTS_DROP; DROP_VEC] THEN REWRITE_TAC[DROP_CMUL; path_image; DROP_VEC; REAL_MUL_RID] THEN DISCH_THEN(X_CHOOSE_THEN `u:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `u % b:real^N` THEN REWRITE_TAC[IN_SEGMENT; VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [ASM_MESON_TAC[FRONTIER_INTERIOR_SUBSET; SUBSET]; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS)) THEN ONCE_REWRITE_TAC[segment] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s DIFF {a,b} SUBSET t DELETE b`) THEN REWRITE_TAC[segment; SUBSET; FORALL_IN_GSPEC] THEN X_GEN_TAC `v:real` THEN STRIP_TAC THEN REWRITE_TAC[IN_IMAGE; VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN EXISTS_TAC `lift v` THEN REWRITE_TAC[IN_INTERVAL_1] THEN ASM_REWRITE_TAC[LIFT_DROP; DROP_VEC; VECTOR_MUL_ASSOC] THEN REWRITE_TAC[REAL_MUL_SYM]);; (* ------------------------------------------------------------------------- *) (* Bounding a point away from a path. *) (* ------------------------------------------------------------------------- *) let NOT_ON_PATH_BALL = prove (`!g z:real^N. path g /\ ~(z IN path_image g) ==> ?e. &0 < e /\ ball(z,e) INTER (path_image g) = {}`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`path_image g:real^N->bool`; `z:real^N`] DISTANCE_ATTAINS_INF) THEN REWRITE_TAC[PATH_IMAGE_NONEMPTY] THEN ASM_SIMP_TAC[COMPACT_PATH_IMAGE; COMPACT_IMP_CLOSED] THEN DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `dist(z:real^N,a)` THEN CONJ_TAC THENL [ASM_MESON_TAC[DIST_POS_LT]; ALL_TAC] THEN REWRITE_TAC[EXTENSION; NOT_IN_EMPTY; IN_BALL; IN_INTER] THEN ASM_MESON_TAC[REAL_NOT_LE]);; let NOT_ON_PATH_CBALL = prove (`!g z:real^N. path g /\ ~(z IN path_image g) ==> ?e. &0 < e /\ cball(z,e) INTER (path_image g) = {}`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP NOT_ON_PATH_BALL) THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `e / &2` THEN ASM_REWRITE_TAC[REAL_HALF] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s INTER u = {} ==> t SUBSET s ==> t INTER u = {}`)) THEN REWRITE_TAC[SUBSET; IN_BALL; IN_CBALL] THEN UNDISCH_TAC `&0 < e` THEN REAL_ARITH_TAC);; (* ------------------------------------------------------------------------- *) (* Homeomorphisms of arc images. *) (* ------------------------------------------------------------------------- *) let HOMEOMORPHISM_ARC = prove (`!g:real^1->real^N. arc g ==> ?h. homeomorphism (interval[vec 0,vec 1],path_image g) (g,h)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHISM_COMPACT THEN ASM_REWRITE_TAC[path_image; COMPACT_INTERVAL; GSYM path; GSYM arc]);; let HOMEOMORPHIC_ARC_IMAGE_INTERVAL = prove (`!g:real^1->real^N a b:real^1. arc g /\ drop a < drop b ==> (path_image g) homeomorphic interval[a,b]`, REPEAT STRIP_TAC THEN TRANS_TAC HOMEOMORPHIC_TRANS `interval[vec 0:real^1,vec 1]` THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN REWRITE_TAC[homeomorphic] THEN EXISTS_TAC `g:real^1->real^N` THEN ASM_SIMP_TAC[HOMEOMORPHISM_ARC]; MATCH_MP_TAC HOMEOMORPHIC_CLOSED_INTERVALS THEN ASM_REWRITE_TAC[INTERVAL_NE_EMPTY_1; DROP_VEC; REAL_LT_01]]);; let HOMEOMORPHIC_ARC_IMAGES = prove (`!g:real^1->real^M h:real^1->real^N. arc g /\ arc h ==> (path_image g) homeomorphic (path_image h)`, REPEAT STRIP_TAC THEN TRANS_TAC HOMEOMORPHIC_TRANS `interval[vec 0:real^1,vec 1]` THEN CONJ_TAC THENL [ALL_TAC; ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM]] THEN MATCH_MP_TAC HOMEOMORPHIC_ARC_IMAGE_INTERVAL THEN ASM_REWRITE_TAC[DROP_VEC; REAL_LT_01]);; let HOMEOMORPHIC_ARC_IMAGE_SEGMENT = prove (`!g:real^1->real^N a b:real^M. arc g /\ ~(a = b) ==> (path_image g) homeomorphic segment[a,b]`, REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM PATH_IMAGE_LINEPATH] THEN MATCH_MP_TAC HOMEOMORPHIC_ARC_IMAGES THEN ASM_REWRITE_TAC[ARC_LINEPATH_EQ]);; let HOMEOMORPHIC_ARC_IMAGE_SEGMENT_EQ = prove (`!s:real^N->bool a b:real^M. ~(a = b) ==> (s homeomorphic segment[a,b] <=> ?g. arc g /\ path_image g = s)`, REPEAT STRIP_TAC THEN EQ_TAC THENL [ALL_TAC; ASM_MESON_TAC[HOMEOMORPHIC_ARC_IMAGE_SEGMENT]] THEN REWRITE_TAC[homeomorphic; homeomorphism; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f:real^N->real^M`; `g:real^M->real^N`] THEN STRIP_TAC THEN EXISTS_TAC `(g:real^M->real^N) o linepath(a,b)` THEN ASM_REWRITE_TAC[PATH_IMAGE_COMPOSE; PATH_IMAGE_LINEPATH] THEN MATCH_MP_TAC ARC_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[PATH_IMAGE_LINEPATH; ARC_LINEPATH_EQ] THEN ASM SET_TAC[]);; let CONNECTED_SUBSET_PATH_IMAGE_ARC = prove (`!s g:real^1->real^N. arc g /\ connected s /\ s SUBSET path_image g /\ pathstart g IN s /\ pathfinish g IN s ==> s = path_image g`, REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMEOMORPHISM_ARC) THEN REWRITE_TAC[homeomorphism; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `h:real^N->real^1` THEN STRIP_TAC THEN SUBGOAL_THEN `IMAGE (h:real^N->real^1) (path_image g) SUBSET IMAGE h s` MP_TAC THENL [ASM_REWRITE_TAC[]; ASM SET_TAC[]] THEN W(MP_TAC o PART_MATCH (lhand o rand) INTERVAL_SUBSET_IS_INTERVAL o snd) THEN ANTS_TAC THENL [REWRITE_TAC[IS_INTERVAL_CONNECTED_1] THEN MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; REWRITE_TAC[UNIT_INTERVAL_NONEMPTY] THEN DISCH_THEN SUBST1_TAC] THEN SUBGOAL_THEN `vec 0 IN interval[vec 0:real^1,vec 1] /\ vec 1 IN interval[vec 0:real^1,vec 1]` MP_TAC THENL [REWRITE_TAC[ENDS_IN_UNIT_INTERVAL]; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish; path_image]) THEN ASM SET_TAC[]);; let ARC_IMAGE_UNIQUE = prove (`!g h:real^1->real^N. path g /\ arc h /\ path_image g SUBSET path_image h /\ {pathstart g,pathfinish g} = {pathstart h,pathfinish h} ==> path_image g = path_image h`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE `{a,b} = {c,d} ==> a = c /\ b = d \/ a = d /\ b = c`)) THEN STRIP_TAC THENL [ALL_TAC; GEN_REWRITE_TAC RAND_CONV [GSYM PATH_IMAGE_REVERSEPATH]] THEN MATCH_MP_TAC CONNECTED_SUBSET_PATH_IMAGE_ARC THEN ASM_REWRITE_TAC[ARC_REVERSEPATH_EQ; PATH_IMAGE_REVERSEPATH] THEN REWRITE_TAC[PATHSTART_REVERSEPATH; PATHFINISH_REVERSEPATH] THEN ASM_MESON_TAC[CONNECTED_PATH_IMAGE; ARC_IMP_PATH; PATHSTART_IN_PATH_IMAGE; PATHFINISH_IN_PATH_IMAGE]);; let CONNECTED_SUBSET_ARC_PAIR = prove (`!g h s:real^N->bool. arc g /\ arc h /\ pathstart g = pathstart h /\ pathfinish g = pathfinish h /\ path_image g INTER path_image h = {pathstart g,pathfinish g} /\ connected s /\ s SUBSET path_image g UNION path_image h /\ pathstart g IN s /\ pathfinish g IN s ==> path_image g SUBSET s \/ path_image h SUBSET s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC(SET_RULE `((?x. x IN t /\ ~(x IN s)) /\ (?y. y IN u /\ ~(y IN s)) ==> F) ==> t SUBSET s \/ u SUBSET s`) THEN REWRITE_TAC[path_image; EXISTS_IN_IMAGE] THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `p:real^1` STRIP_ASSUME_TAC) (X_CHOOSE_THEN `q:real^1` STRIP_ASSUME_TAC)) THEN UNDISCH_TAC `connected(s:real^N->bool)` THEN REWRITE_TAC[CONNECTED_OPEN_IN] THEN MAP_EVERY EXISTS_TAC [`s DIFF (path_image (subpath p (vec 1) g) UNION path_image (subpath q (vec 1) h)):real^N->bool`; `s DIFF (path_image (subpath (vec 0) p g) UNION path_image (subpath (vec 0) q h)):real^N->bool`] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_DIFF_CLOSED THEN REWRITE_TAC[OPEN_IN_REFL] THEN MATCH_MP_TAC CLOSED_UNION THEN CONJ_TAC THEN MATCH_MP_TAC CLOSED_PATH_IMAGE THEN MATCH_MP_TAC PATH_SUBPATH THEN ASM_SIMP_TAC[ARC_IMP_PATH; ENDS_IN_UNIT_INTERVAL]; MATCH_MP_TAC OPEN_IN_DIFF_CLOSED THEN REWRITE_TAC[OPEN_IN_REFL] THEN MATCH_MP_TAC CLOSED_UNION THEN CONJ_TAC THEN MATCH_MP_TAC CLOSED_PATH_IMAGE THEN MATCH_MP_TAC PATH_SUBPATH THEN ASM_SIMP_TAC[ARC_IMP_PATH; ENDS_IN_UNIT_INTERVAL]; REWRITE_TAC[SUBSET; IN_UNION; IN_DIFF] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_SIMP_TAC[PATH_IMAGE_SUBPATH; GSYM DE_MORGAN_THM] THEN DISCH_THEN(REPEAT_TCL STRIP_THM_THEN MP_TAC) THEN REWRITE_TAC[IN_IMAGE; LEFT_IMP_EXISTS_THM; NOT_EXISTS_THM] THEN X_GEN_TAC `a:real^1` THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN X_GEN_TAC `b:real^1` THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THENL [UNDISCH_TAC `arc(g:real^1->real^N)` THEN REWRITE_TAC[arc] THEN DISCH_THEN(MP_TAC o SPECL [`a:real^1`; `b:real^1`] o CONJUNCT2) THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN ASM_CASES_TAC `a:real^1 = p` THENL [ASM SET_TAC[]; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[GSYM DROP_EQ]) THEN REWRITE_TAC[GSYM DROP_EQ] THEN ASM_REAL_ARITH_TAC; RULE_ASSUM_TAC(REWRITE_RULE[path_image]) THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE `IMAGE f s INTER IMAGE g s = a ==> !x y. x IN s /\ y IN s /\ f x = g y ==> f(x) IN a`)) THEN DISCH_THEN(MP_TAC o SPECL [`b:real^1`; `a:real^1`]) THEN ASM_REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; DE_MORGAN_THM]] THEN REPEAT STRIP_TAC THENL [UNDISCH_TAC `arc(g:real^1->real^N)` THEN REWRITE_TAC[arc] THEN DISCH_THEN(MP_TAC o SPECL [`b:real^1`; `vec 0:real^1`] o CONJUNCT2); UNDISCH_TAC `arc(h:real^1->real^N)` THEN REWRITE_TAC[arc] THEN DISCH_THEN(MP_TAC o SPECL [`a:real^1`; `vec 1:real^1`] o CONJUNCT2)]; RULE_ASSUM_TAC(REWRITE_RULE[path_image]) THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE `IMAGE f s INTER IMAGE g s = a ==> !x y. x IN s /\ y IN s /\ f x = g y ==> f(x) IN a`)) THEN DISCH_THEN(MP_TAC o SPECL [`a:real^1`; `b:real^1`]) THEN ASM_REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; DE_MORGAN_THM]] THEN REPEAT STRIP_TAC THENL [UNDISCH_TAC `arc(h:real^1->real^N)` THEN REWRITE_TAC[arc] THEN DISCH_THEN(MP_TAC o SPECL [`b:real^1`; `vec 0:real^1`] o CONJUNCT2); UNDISCH_TAC `arc(g:real^1->real^N)` THEN REWRITE_TAC[arc] THEN DISCH_THEN(MP_TAC o SPECL [`a:real^1`; `vec 1:real^1`] o CONJUNCT2)]; UNDISCH_TAC `arc(h:real^1->real^N)` THEN REWRITE_TAC[arc] THEN DISCH_THEN(MP_TAC o SPECL [`a:real^1`; `b:real^1`] o CONJUNCT2) THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN ASM_CASES_TAC `a:real^1 = q` THENL [ASM SET_TAC[]; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[GSYM DROP_EQ]) THEN REWRITE_TAC[GSYM DROP_EQ] THEN ASM_REAL_ARITH_TAC]; MP_TAC(ISPECL [`g:real^1->real^N`; `p:real^1`] PATH_IMAGE_SUBPATH_COMBINE) THEN MP_TAC(ISPECL [`h:real^1->real^N`; `q:real^1`] PATH_IMAGE_SUBPATH_COMBINE) THEN ASM_SIMP_TAC[ARC_IMP_PATH] THEN ASM SET_TAC[]; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `pathstart g:real^N` THEN ASM_REWRITE_TAC[IN_DIFF; IN_UNION; DE_MORGAN_THM] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_SIMP_TAC[PATH_IMAGE_SUBPATH; ARC_IMP_PATH] THEN CONJ_TAC THENL [UNDISCH_TAC `arc(g:real^1->real^N)`; UNDISCH_TAC `arc(h:real^1->real^N)`] THEN REWRITE_TAC[arc; path_image; IN_IMAGE; NOT_EXISTS_THM] THEN DISCH_THEN(MP_TAC o CONJUNCT2) THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `a:real^1` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC(SPEC `vec 0:real^1` th)) THEN ASM_REWRITE_TAC[ENDS_IN_UNIT_INTERVAL; NOT_IMP]; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `pathfinish g:real^N` THEN ASM_REWRITE_TAC[IN_DIFF; IN_UNION; DE_MORGAN_THM] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_SIMP_TAC[PATH_IMAGE_SUBPATH; ARC_IMP_PATH] THEN CONJ_TAC THENL [UNDISCH_TAC `arc(g:real^1->real^N)`; UNDISCH_TAC `arc(h:real^1->real^N)`] THEN REWRITE_TAC[arc; path_image; IN_IMAGE; NOT_EXISTS_THM] THEN DISCH_THEN(MP_TAC o CONJUNCT2) THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `a:real^1` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC(SPEC `vec 1:real^1` th)) THEN ASM_REWRITE_TAC[ENDS_IN_UNIT_INTERVAL; NOT_IMP]] THEN (REWRITE_TAC[ENDS_IN_UNIT_INTERVAL; NOT_IMP] THEN REPEAT CONJ_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN ASM_REAL_ARITH_TAC; ASM_MESON_TAC[pathstart; pathfinish]; DISCH_THEN SUBST_ALL_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN ASM_MESON_TAC[REAL_LE_ANTISYM; LIFT_EQ; LIFT_NUM; LIFT_DROP; pathstart; pathfinish]]));; let HOMEOMORPHIC_SIMPLE_PATH_IMAGES = prove (`!g:real^1->real^M h:real^1->real^N. simple_path g /\ pathfinish g = pathstart g /\ simple_path h /\ pathfinish h = pathstart h ==> (path_image g) homeomorphic (path_image h)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`g:real^1->real^M`; `h:real^1->real^N`; `interval[vec 0:real^1,vec 1]`; `path_image g:real^M->bool`; `path_image h:real^N->bool`] LIFT_TO_QUOTIENT_SPACE_UNIQUE) THEN REWRITE_TAC[path_image; CONJ_ASSOC] THEN DISCH_THEN MATCH_MP_TAC THEN CONJ_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_IMP_QUOTIENT_MAP_EXPLICIT THEN ASM_SIMP_TAC[GSYM path; COMPACT_INTERVAL; SIMPLE_PATH_IMP_PATH]; RULE_ASSUM_TAC(REWRITE_RULE[simple_path; pathstart; pathfinish]) THEN MAP_EVERY X_GEN_TAC [`x:real^1`; `y:real^1`] THEN STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^1`; `y:real^1`] o CONJUNCT2)) THEN ASM_MESON_TAC[]]);; let HOMEOMORPHIC_SIMPLE_PATH_IMAGE_CIRCLE_EQ = prove (`!s:real^N->bool a:real^2 r. &0 < r ==> (s homeomorphic sphere(a,r) <=> ?g. simple_path g /\ pathfinish g = pathstart g /\ path_image g = s)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?p. simple_path p /\ pathfinish p = pathstart p /\ (path_image p:real^2->bool) homeomorphic(sphere(a:real^2,r))` STRIP_ASSUME_TAC THENL [EXISTS_TAC `linepath(vec 0:real^2,basis 1) ++ linepath(basis 1,basis 2) ++ linepath(basis 2,vec 0)` THEN SUBGOAL_THEN `~(basis 2:real^2 = basis 1) /\ ~(basis 1:real^2 = vec 0) /\ ~(basis 2:real^2 = vec 0)` STRIP_ASSUME_TAC THENL [SIMP_TAC[BASIS_INJ_EQ; BASIS_NONZERO; DIMINDEX_2; ARITH]; ALL_TAC] THEN SUBGOAL_THEN `~affine_dependent {vec 0:real^2,basis 1,basis 2}` ASSUME_TAC THENL [MATCH_MP_TAC INDEPENDENT_IMP_AFFINE_DEPENDENT_0 THEN ASM_REWRITE_TAC[independent; DEPENDENT_2] THEN STRIP_TAC THEN FIRST_X_ASSUM(fun th -> MP_TAC(AP_TERM `\x:real^2. x$1` th) THEN MP_TAC(AP_TERM `\x:real^2. x$2` th)) THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN SIMP_TAC[BASIS_COMPONENT; VEC_COMPONENT; DIMINDEX_2; ARITH] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN SIMP_TAC[SIMPLE_PATH_JOIN_LOOP_EQ; ARC_JOIN_EQ; PATHSTART_JOIN; PATHFINISH_JOIN; PATH_IMAGE_JOIN; PATHSTART_LINEPATH; PATHFINISH_LINEPATH] THEN ASM_REWRITE_TAC[ARC_LINEPATH_EQ; PATH_IMAGE_LINEPATH] THEN REWRITE_TAC[UNION_OVER_INTER; UNION_SUBSET; CONJ_ASSOC] THEN CONJ_TAC THENL [GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [SEGMENT_SYM] THEN REPEAT CONJ_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) INTER_SEGMENT o lhand o snd) THEN (ANTS_TAC THENL [DISJ2_TAC; SET_TAC[]]) THEN ASM_REWRITE_TAC[COLLINEAR_3_EQ_AFFINE_DEPENDENT] THEN ASM_MESON_TAC[INSERT_AC]; TRANS_TAC HOMEOMORPHIC_TRANS `relative_frontier(convex hull {vec 0:real^2,basis 1,basis 2})` THEN CONJ_TAC THENL [MATCH_MP_TAC(MESON[HOMEOMORPHIC_REFL] `s = t ==> s homeomorphic t`) THEN ASM_SIMP_TAC[RELATIVE_FRONTIER_OF_CONVEX_HULL] THEN REWRITE_TAC[SET_RULE `{f x | x IN {a,b,c}} = {f a,f b,f c}`] THEN ASM_REWRITE_TAC[DELETE_INSERT; GSYM SEGMENT_CONVEX_HULL; EMPTY_DELETE; SEGMENT_SYM] THEN SET_TAC[]; MP_TAC(ISPECL [`convex hull {vec 0:real^2,basis 1,basis 2}`; `cball(a:real^2,r)`] HOMEOMORPHIC_RELATIVE_FRONTIERS_CONVEX_BOUNDED_SETS) THEN ASM_REWRITE_TAC[CONVEX_CBALL; BOUNDED_CBALL; CONVEX_CONVEX_HULL] THEN ASM_REWRITE_TAC[CONVEX_CONVEX_HULL; BOUNDED_CONVEX_HULL_EQ] THEN REWRITE_TAC[AFF_DIM_CONVEX_HULL; BOUNDED_INSERT; BOUNDED_EMPTY] THEN ASM_SIMP_TAC[RELATIVE_FRONTIER_CBALL; REAL_LT_IMP_NZ] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[AFF_DIM_CBALL; DIMINDEX_2] THEN ASM_SIMP_TAC[AFF_DIM_AFFINE_INDEPENDENT] THEN SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; ARITH] THEN CONV_TAC INT_REDUCE_CONV]]; TRANS_TAC EQ_TRANS `(s:real^N->bool) homeomorphic (path_image p:real^2->bool)` THEN CONJ_TAC THENL [EQ_TAC THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] HOMEOMORPHIC_TRANS) THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN EQ_TAC THENL [REWRITE_TAC[homeomorphic; homeomorphism; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`g:real^N->real^2`; `f:real^2->real^N`] THEN STRIP_TAC THEN EXISTS_TAC `(f:real^2->real^N) o (p:real^1->real^2)` THEN REWRITE_TAC[PATHFINISH_COMPOSE; PATHSTART_COMPOSE] THEN ASM_REWRITE_TAC[PATH_IMAGE_COMPOSE] THEN MATCH_MP_TAC SIMPLE_PATH_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; DISCH_THEN(X_CHOOSE_THEN `g:real^1->real^N` (STRIP_ASSUME_TAC o GSYM)) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC HOMEOMORPHIC_SIMPLE_PATH_IMAGES THEN ASM_REWRITE_TAC[]]]);; let HOMEOMORPHIC_SIMPLE_PATH_IMAGE_CIRCLE = prove (`!g:real^1->real^N a:real^2 r. simple_path g /\ pathfinish g = pathstart g /\ &0 < r ==> (path_image g) homeomorphic sphere(a,r)`, MESON_TAC[HOMEOMORPHIC_SIMPLE_PATH_IMAGE_CIRCLE_EQ]);; (* ------------------------------------------------------------------------- *) (* Path component, considered as a "joinability" relation (from Tom Hales). *) (* ------------------------------------------------------------------------- *) let path_component = new_definition `path_component s x y <=> ?g. path g /\ path_image g SUBSET s /\ pathstart g = x /\ pathfinish g = y`;; let path_components = new_definition `path_components s = {path_component s x | x | x IN s}`;; let PATH_COMPONENT_OF_EUCLIDEAN = prove (`!s:real^N->bool. path_component_of (subtopology euclidean s) = path_component s`, REWRITE_TAC[FUN_EQ_THM; path_component; path_component_of] THEN REWRITE_TAC[PATH_IN_EUCLIDEAN; pathstart; pathfinish; GSYM DROP_VEC] THEN REPEAT GEN_TAC THEN EQ_TAC THENL [MESON_TAC[o_THM]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^1->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(g:real^1->real^N) o lift` THEN ASM_REWRITE_TAC[o_DEF; LIFT_DROP; ETA_AX]);; let PATH_COMPONENTS_OF_EUCLIDEAN = prove (`!s:real^N->bool. path_components_of (subtopology euclidean s) = path_components s`, REWRITE_TAC[path_components_of; path_components] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; PATH_COMPONENT_OF_EUCLIDEAN]);; let PATH_COMPONENT_IN = prove (`!s x y. path_component s x y ==> x IN s /\ y IN s`, REPEAT GEN_TAC THEN REWRITE_TAC[GSYM PATH_COMPONENT_OF_EUCLIDEAN] THEN DISCH_THEN(MP_TAC o MATCH_MP PATH_COMPONENT_IN_TOPSPACE) THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY]);; let PATH_COMPONENT_REFL_EQ = prove (`!s x:real^N. path_component s x x <=> x IN s`, REWRITE_TAC[GSYM PATH_COMPONENT_OF_EUCLIDEAN; PATH_COMPONENT_OF_REFL] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY]);; let PATH_COMPONENT_REFL = prove (`!s x:real^N. x IN s ==> path_component s x x`, REWRITE_TAC[PATH_COMPONENT_REFL_EQ]);; let PATH_COMPONENT_SYM_EQ = prove (`!s x y. path_component s x y <=> path_component s y x`, REWRITE_TAC[GSYM PATH_COMPONENT_OF_EUCLIDEAN] THEN MATCH_ACCEPT_TAC PATH_COMPONENT_OF_SYM);; let PATH_COMPONENT_SYM = prove (`!s x y:real^N. path_component s x y ==> path_component s y x`, MESON_TAC[PATH_COMPONENT_SYM_EQ]);; let PATH_COMPONENT_TRANS = prove (`!s x y:real^N. path_component s x y /\ path_component s y z ==> path_component s x z`, REWRITE_TAC[GSYM PATH_COMPONENT_OF_EUCLIDEAN; PATH_COMPONENT_OF_TRANS]);; let PATH_COMPONENT_OF_SUBSET = prove (`!s t x. s SUBSET t /\ path_component s x y ==> path_component t x y`, REWRITE_TAC[path_component] THEN SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* Can also consider it as a set, as the name suggests. *) (* ------------------------------------------------------------------------- *) let PATH_COMPONENT_SET = prove (`!s x. path_component s x = { y | ?g. path g /\ path_image g SUBSET s /\ pathstart g = x /\ pathfinish g = y }`, REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN REWRITE_TAC[IN; path_component]);; let PATH_COMPONENT_SUBSET = prove (`!s x. (path_component s x) SUBSET s`, REWRITE_TAC[SUBSET; IN] THEN MESON_TAC[PATH_COMPONENT_IN; IN]);; let PATH_COMPONENT_EQ_EMPTY = prove (`!s x. path_component s x = {} <=> ~(x IN s)`, REWRITE_TAC[EXTENSION; NOT_IN_EMPTY] THEN MESON_TAC[IN; PATH_COMPONENT_REFL; PATH_COMPONENT_IN]);; let PATH_COMPONENT_EMPTY = prove (`!x. path_component {} x = {}`, REWRITE_TAC[PATH_COMPONENT_EQ_EMPTY; NOT_IN_EMPTY]);; let UNIONS_PATH_COMPONENT = prove (`!s:real^N->bool. UNIONS {path_component s x |x| x IN s} = s`, GEN_TAC THEN GEN_REWRITE_TAC RAND_CONV [GSYM TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN REWRITE_TAC[GSYM UNIONS_PATH_COMPONENTS_OF] THEN REWRITE_TAC[path_components_of; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN REWRITE_TAC[PATH_COMPONENT_OF_EUCLIDEAN]);; let PATH_COMPONENT_TRANSLATION = prove (`!a s x. path_component (IMAGE (\x. a + x) s) (a + x) = IMAGE (\x. a + x) (path_component s x)`, REWRITE_TAC[PATH_COMPONENT_SET] THEN GEOM_TRANSLATE_TAC[]);; add_translation_invariants [PATH_COMPONENT_TRANSLATION];; let PATH_COMPONENT_LINEAR_IMAGE = prove (`!f s x. linear f /\ (!x y. f x = f y ==> x = y) /\ (!y. ?x. f x = y) ==> path_component (IMAGE f s) (f x) = IMAGE f (path_component s x)`, REWRITE_TAC[PATH_COMPONENT_SET] THEN GEOM_TRANSFORM_TAC[]);; add_linear_invariants [PATH_COMPONENT_LINEAR_IMAGE];; (* ------------------------------------------------------------------------- *) (* Path connectedness of a space. *) (* ------------------------------------------------------------------------- *) let path_connected = new_definition `path_connected s <=> !x y. x IN s /\ y IN s ==> ?g. path g /\ (path_image g) SUBSET s /\ pathstart g = x /\ pathfinish g = y`;; let PATH_CONNECTED_IFF_PATH_COMPONENT = prove (`!s. path_connected s <=> !x y. x IN s /\ y IN s ==> path_component s x y`, REWRITE_TAC[path_connected; path_component]);; let PATH_CONNECTED_IN_EUCLIDEAN = prove (`!s:real^N->bool. path_connected_in euclidean s <=> path_connected s`, REWRITE_TAC[path_connected_in; PATH_CONNECTED_SPACE_IFF_PATH_COMPONENT] THEN REWRITE_TAC[PATH_COMPONENT_OF_EUCLIDEAN; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN REWRITE_TAC[GSYM PATH_CONNECTED_IFF_PATH_COMPONENT] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN; SUBSET_UNIV]);; let PATH_CONNECTED_SPACE_EUCLIDEAN_SUBTOPOLOGY = prove (`!s:real^N->bool. path_connected_space(subtopology euclidean s) <=> path_connected s`, REWRITE_TAC[GSYM PATH_CONNECTED_IN_TOPSPACE] THEN REWRITE_TAC[PATH_CONNECTED_IN_SUBTOPOLOGY] THEN REWRITE_TAC[PATH_CONNECTED_IN_EUCLIDEAN] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; SUBSET_REFL]);; let PATH_CONNECTED_IMP_PATH_COMPONENT = prove (`!s a b:real^N. path_connected s /\ a IN s /\ b IN s ==> path_component s a b`, MESON_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT]);; let PATH_CONNECTED_COMPONENT_SET = prove (`!s. path_connected s <=> !x. x IN s ==> path_component s x = s`, REWRITE_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT; GSYM SUBSET_ANTISYM_EQ] THEN REWRITE_TAC[PATH_COMPONENT_SUBSET] THEN SET_TAC[]);; let PATH_COMPONENT_MONO = prove (`!s t x. s SUBSET t ==> (path_component s x) SUBSET (path_component t x)`, REWRITE_TAC[PATH_COMPONENT_SET] THEN SET_TAC[]);; let PATH_COMPONENT_MAXIMAL = prove (`!s t x. x IN t /\ path_connected t /\ t SUBSET s ==> t SUBSET (path_component s x)`, REWRITE_TAC[path_connected; PATH_COMPONENT_SET; SUBSET; IN_ELIM_THM] THEN MESON_TAC[]);; let PATH_COMPONENT_EQ = prove (`!s x y. y IN path_component s x ==> path_component s y = path_component s x`, REWRITE_TAC[EXTENSION; IN] THEN MESON_TAC[PATH_COMPONENT_SYM; PATH_COMPONENT_TRANS]);; let PATH_CONNECTED_PATH_IMAGE = prove (`!p:real^1->real^N. path p ==> path_connected(path_image p)`, GEN_TAC THEN REWRITE_TAC[PATH_PATH_IN] THEN DISCH_THEN(MP_TAC o MATCH_MP PATH_CONNECTED_IN_PATH_IMAGE) THEN REWRITE_TAC[IMAGE_o; IMAGE_LIFT_REAL_INTERVAL; LIFT_NUM] THEN REWRITE_TAC[PATH_CONNECTED_IN_EUCLIDEAN; path_image]);; let PATH_COMPONENT_PATH_IMAGE_PATHSTART = prove (`!p x:real^N. path p /\ x IN path_image p ==> path_component (path_image p) (pathstart p) x`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP PATH_CONNECTED_PATH_IMAGE) THEN REWRITE_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_SIMP_TAC[PATHSTART_IN_PATH_IMAGE]);; let PATH_CONNECTED_PATH_COMPONENT = prove (`!s x:real^N. path_connected(path_component s x)`, REPEAT GEN_TAC THEN MP_TAC(ISPECL [`subtopology euclidean (s:real^N->bool)`; `x:real^N`] PATH_CONNECTED_IN_PATH_COMPONENT_OF) THEN REWRITE_TAC[PATH_CONNECTED_IN_SUBTOPOLOGY; PATH_CONNECTED_IN_EUCLIDEAN] THEN SIMP_TAC[PATH_COMPONENT_OF_EUCLIDEAN]);; let PATH_COMPONENT = prove (`!s x y:real^N. path_component s x y <=> ?t. path_connected t /\ t SUBSET s /\ x IN t /\ y IN t`, REPEAT GEN_TAC THEN EQ_TAC THEN STRIP_TAC THENL [EXISTS_TAC `path_component s (x:real^N)` THEN REWRITE_TAC[PATH_CONNECTED_PATH_COMPONENT; PATH_COMPONENT_SUBSET] THEN FIRST_ASSUM(MP_TAC o MATCH_MP PATH_COMPONENT_IN) THEN ASM_SIMP_TAC[IN; PATH_COMPONENT_REFL_EQ]; REWRITE_TAC[path_component] THEN ASM_MESON_TAC[path_connected; SUBSET]]);; let PATH_COMPONENT_PATH_COMPONENT = prove (`!s x:real^N. path_component (path_component s x) x = path_component s x`, REPEAT GEN_TAC THEN ASM_CASES_TAC `(x:real^N) IN s` THENL [MATCH_MP_TAC SUBSET_ANTISYM THEN SIMP_TAC[PATH_COMPONENT_MONO; PATH_COMPONENT_SUBSET] THEN MATCH_MP_TAC PATH_COMPONENT_MAXIMAL THEN REWRITE_TAC[SUBSET_REFL; PATH_CONNECTED_PATH_COMPONENT] THEN ASM_REWRITE_TAC[IN; PATH_COMPONENT_REFL_EQ]; MATCH_MP_TAC(SET_RULE `s = {} /\ t = {} ==> s = t`) THEN ASM_REWRITE_TAC[PATH_COMPONENT_EQ_EMPTY] THEN ASM_MESON_TAC[SUBSET; PATH_COMPONENT_SUBSET]]);; let PATH_CONNECTED_LINEPATH = prove (`!s a b:real^N. segment[a,b] SUBSET s ==> path_component s a b`, REPEAT STRIP_TAC THEN REWRITE_TAC[path_component] THEN EXISTS_TAC `linepath(a:real^N,b)` THEN ASM_REWRITE_TAC[PATHSTART_LINEPATH; PATHFINISH_LINEPATH; PATH_LINEPATH] THEN ASM_REWRITE_TAC[PATH_IMAGE_LINEPATH]);; let PATH_COMPONENT_DISJOINT = prove (`!s a b. DISJOINT (path_component s a) (path_component s b) <=> ~(a IN path_component s b)`, REWRITE_TAC[DISJOINT; EXTENSION; IN_INTER; NOT_IN_EMPTY] THEN REWRITE_TAC[IN] THEN MESON_TAC[PATH_COMPONENT_SYM; PATH_COMPONENT_TRANS]);; let PATH_COMPONENT_EQ_EQ = prove (`!s x y:real^N. path_component s x = path_component s y <=> ~(x IN s) /\ ~(y IN s) \/ x IN s /\ y IN s /\ path_component s x y`, REPEAT GEN_TAC THEN ASM_CASES_TAC `(y:real^N) IN s` THENL [ASM_CASES_TAC `(x:real^N) IN s` THEN ASM_REWRITE_TAC[] THENL [REWRITE_TAC[FUN_EQ_THM] THEN ASM_MESON_TAC[PATH_COMPONENT_TRANS; PATH_COMPONENT_REFL; PATH_COMPONENT_SYM]; ASM_MESON_TAC[PATH_COMPONENT_EQ_EMPTY]]; RULE_ASSUM_TAC(REWRITE_RULE[GSYM PATH_COMPONENT_EQ_EMPTY]) THEN ASM_REWRITE_TAC[PATH_COMPONENT_EQ_EMPTY] THEN ONCE_REWRITE_TAC[PATH_COMPONENT_SYM_EQ] THEN ASM_REWRITE_TAC[EMPTY] THEN ASM_MESON_TAC[PATH_COMPONENT_EQ_EMPTY]]);; let PATH_COMPONENT_UNIQUE = prove (`!s c x:real^N. x IN c /\ c SUBSET s /\ path_connected c /\ (!c'. x IN c' /\ c' SUBSET s /\ path_connected c' ==> c' SUBSET c) ==> path_component s x = c`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[PATH_COMPONENT_SUBSET; PATH_CONNECTED_PATH_COMPONENT] THEN REWRITE_TAC[IN] THEN ASM_REWRITE_TAC[PATH_COMPONENT_REFL_EQ] THEN ASM SET_TAC[]; MATCH_MP_TAC PATH_COMPONENT_MAXIMAL THEN ASM_REWRITE_TAC[]]);; let PATH_COMPONENT_INTERMEDIATE_SUBSET = prove (`!t u a:real^N. path_component u a SUBSET t /\ t SUBSET u ==> path_component t a = path_component u a`, REPEAT GEN_TAC THEN ASM_CASES_TAC `(a:real^N) IN u` THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC PATH_COMPONENT_UNIQUE THEN ASM_REWRITE_TAC[PATH_CONNECTED_PATH_COMPONENT] THEN CONJ_TAC THENL [ASM_MESON_TAC[PATH_COMPONENT_REFL; IN]; ALL_TAC] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC PATH_COMPONENT_MAXIMAL THEN ASM SET_TAC[]; ASM_MESON_TAC[PATH_COMPONENT_EQ_EMPTY; SUBSET]]);; let COMPLEMENT_PATH_COMPONENT_UNIONS = prove (`!s x:real^N. s DIFF path_component s x = UNIONS({path_component s y | y | y IN s} DELETE (path_component s x))`, REPEAT GEN_TAC THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM UNIONS_PATH_COMPONENT] THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s DELETE a ==> DISJOINT a x) ==> UNIONS s DIFF a = UNIONS (s DELETE a)`) THEN REWRITE_TAC[IMP_CONJ; FORALL_IN_GSPEC; IN_DELETE] THEN SIMP_TAC[PATH_COMPONENT_DISJOINT; PATH_COMPONENT_EQ_EQ] THEN MESON_TAC[IN; SUBSET; PATH_COMPONENT_SUBSET]);; (* ------------------------------------------------------------------------- *) (* General "locally connected implies connected" type results. *) (* ------------------------------------------------------------------------- *) let OPEN_GENERAL_COMPONENT = prove (`!c. (!s x y. c s x y ==> x IN s /\ y IN s) /\ (!s x y. c s x y ==> c s y x) /\ (!s x y z. c s x y /\ c s y z ==> c s x z) /\ (!s t x y. s SUBSET t /\ c s x y ==> c t x y) /\ (!s x y e. y IN ball(x,e) /\ ball(x,e) SUBSET s ==> c (ball(x,e)) x y) ==> !s x:real^N. open s ==> open(c s x)`, GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "IN") MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "SYM") MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "TRANS") MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "SUBSET") (LABEL_TAC "BALL")) THEN REPEAT GEN_TAC THEN REWRITE_TAC[OPEN_CONTAINS_BALL; SUBSET; IN_BALL] THEN DISCH_TAC THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[SUBSET; IN] THEN STRIP_TAC THEN SUBGOAL_THEN `(x:real^N) IN s /\ y IN s` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o C MATCH_MP (ASSUME `(y:real^N) IN s`)) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `e:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `z:real^N` THEN DISCH_TAC THEN REMOVE_THEN "TRANS" MATCH_MP_TAC THEN EXISTS_TAC `y:real^N` THEN ASM_REWRITE_TAC[] THEN REMOVE_THEN "SUBSET" MATCH_MP_TAC THEN EXISTS_TAC `ball(y:real^N,e)` THEN ASM_REWRITE_TAC[SUBSET; IN_BALL] THEN REMOVE_THEN "BALL" MATCH_MP_TAC THEN REWRITE_TAC[SUBSET; IN_BALL] THEN ASM_MESON_TAC[]);; let OPEN_NON_GENERAL_COMPONENT = prove (`!c. (!s x y. c s x y ==> x IN s /\ y IN s) /\ (!s x y. c s x y ==> c s y x) /\ (!s x y z. c s x y /\ c s y z ==> c s x z) /\ (!s t x y. s SUBSET t /\ c s x y ==> c t x y) /\ (!s x y e. y IN ball(x,e) /\ ball(x,e) SUBSET s ==> c (ball(x,e)) x y) ==> !s x:real^N. open s ==> open(s DIFF c s x)`, GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "IN") MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "SYM") MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "TRANS") MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "SUBSET") (LABEL_TAC "BALL")) THEN REPEAT GEN_TAC THEN REWRITE_TAC[OPEN_CONTAINS_BALL; SUBSET; IN_BALL] THEN DISCH_TAC THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_DIFF] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (ASSUME_TAC o REWRITE_RULE[IN])) THEN FIRST_X_ASSUM(MP_TAC o C MATCH_MP (ASSUME `(y:real^N) IN s`)) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `e:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `z:real^N` THEN DISCH_TAC THEN ASM_SIMP_TAC[] THEN REWRITE_TAC[IN] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o check (is_neg o concl)) THEN REWRITE_TAC[] THEN REMOVE_THEN "TRANS" MATCH_MP_TAC THEN EXISTS_TAC `z:real^N` THEN ASM_REWRITE_TAC[] THEN REMOVE_THEN "SUBSET" MATCH_MP_TAC THEN EXISTS_TAC `ball(y:real^N,e)` THEN ASM_REWRITE_TAC[SUBSET; IN_BALL] THEN REMOVE_THEN "SYM" MATCH_MP_TAC THEN REMOVE_THEN "BALL" MATCH_MP_TAC THEN REWRITE_TAC[SUBSET; IN_BALL] THEN ASM_MESON_TAC[]);; let GENERAL_CONNECTED_OPEN = prove (`!c. (!s x y. c s x y ==> x IN s /\ y IN s) /\ (!s x y. c s x y ==> c s y x) /\ (!s x y z. c s x y /\ c s y z ==> c s x z) /\ (!s t x y. s SUBSET t /\ c s x y ==> c t x y) /\ (!s x y e. y IN ball(x,e) /\ ball(x,e) SUBSET s ==> c (ball(x,e)) x y) ==> !s x y:real^N. open s /\ connected s /\ x IN s /\ y IN s ==> c s x y`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [connected]) THEN REWRITE_TAC[IN] THEN REWRITE_TAC[NOT_EXISTS_THM; LEFT_IMP_FORALL_THM] THEN MAP_EVERY EXISTS_TAC [`c (s:real^N->bool) (x:real^N):real^N->bool`; `s DIFF (c (s:real^N->bool) (x:real^N))`] THEN MATCH_MP_TAC(TAUT `a /\ b /\ c /\ d /\ e /\ (f ==> g) ==> ~(a /\ b /\ c /\ d /\ e /\ ~f) ==> g`) THEN REPEAT CONJ_TAC THENL [MP_TAC(SPEC `c:(real^N->bool)->real^N->real^N->bool` OPEN_GENERAL_COMPONENT) THEN ASM_MESON_TAC[]; MP_TAC(SPEC `c:(real^N->bool)->real^N->real^N->bool` OPEN_NON_GENERAL_COMPONENT) THEN ASM_MESON_TAC[]; SET_TAC[]; SET_TAC[]; ALL_TAC; ASM SET_TAC[]] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_BALL]) THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `x:real^N` THEN ASM_REWRITE_TAC[IN_INTER] THEN REWRITE_TAC[IN] THEN FIRST_ASSUM(MATCH_MP_TAC o SPECL [`ball(x:real^N,e)`; `s:real^N->bool`]) THEN ASM_MESON_TAC[CENTRE_IN_BALL]);; (* ------------------------------------------------------------------------- *) (* Some useful lemmas about path-connectedness. *) (* ------------------------------------------------------------------------- *) let CONVEX_IMP_PATH_CONNECTED = prove (`!s:real^N->bool. convex s ==> path_connected s`, REWRITE_TAC[CONVEX_ALT; path_connected] THEN REPEAT GEN_TAC THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN EXISTS_TAC `\u. (&1 - drop u) % x + drop u % y:real^N` THEN ASM_SIMP_TAC[pathstart; pathfinish; DROP_VEC; path; path_image; SUBSET; FORALL_IN_IMAGE; IN_INTERVAL_1; GSYM FORALL_DROP] THEN CONJ_TAC THENL [ALL_TAC; CONJ_TAC THEN VECTOR_ARITH_TAC] THEN MATCH_MP_TAC CONTINUOUS_ON_ADD THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_VMUL THEN REWRITE_TAC[o_DEF; LIFT_SUB; LIFT_DROP; LIFT_NUM] THEN SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID]);; let PATH_CONNECTED_UNIV = prove (`path_connected(:real^N)`, SIMP_TAC[CONVEX_IMP_PATH_CONNECTED; CONVEX_UNIV]);; let IS_INTERVAL_PATH_CONNECTED = prove (`!s. is_interval s ==> path_connected s`, SIMP_TAC[CONVEX_IMP_PATH_CONNECTED; IS_INTERVAL_CONVEX]);; let PATH_CONNECTED_INTERVAL = prove (`(!a b:real^N. path_connected(interval[a,b])) /\ (!a b:real^N. path_connected(interval(a,b)))`, SIMP_TAC[IS_INTERVAL_PATH_CONNECTED; IS_INTERVAL_INTERVAL]);; let PATH_COMPONENT_UNIV = prove (`!x. path_component(:real^N) x = (:real^N)`, MESON_TAC[PATH_CONNECTED_COMPONENT_SET; PATH_CONNECTED_UNIV; IN_UNIV]);; let PATH_CONNECTED_IMP_CONNECTED = prove (`!s:real^N->bool. path_connected s ==> connected s`, GEN_TAC THEN REWRITE_TAC[path_connected; CONNECTED_IFF_CONNECTED_COMPONENT] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `x:real^N` THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `y:real^N` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^1->real^N` STRIP_ASSUME_TAC) THEN REWRITE_TAC[connected_component] THEN EXISTS_TAC `path_image(g:real^1->real^N)` THEN ASM_MESON_TAC[CONNECTED_PATH_IMAGE; PATHSTART_IN_PATH_IMAGE; PATHFINISH_IN_PATH_IMAGE]);; let OPEN_PATH_COMPONENT = prove (`!s x:real^N. open s ==> open(path_component s x)`, MATCH_MP_TAC OPEN_GENERAL_COMPONENT THEN REWRITE_TAC[PATH_COMPONENT_IN; PATH_COMPONENT_SYM; PATH_COMPONENT_TRANS; PATH_COMPONENT_OF_SUBSET] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC(REWRITE_RULE[PATH_CONNECTED_IFF_PATH_COMPONENT] (MATCH_MP CONVEX_IMP_PATH_CONNECTED (SPEC_ALL CONVEX_BALL))) THEN ASM_MESON_TAC[CENTRE_IN_BALL; BALL_EQ_EMPTY; REAL_NOT_LE; NOT_IN_EMPTY]);; let OPEN_NON_PATH_COMPONENT = prove (`!s x:real^N. open s ==> open(s DIFF path_component s x)`, MATCH_MP_TAC OPEN_NON_GENERAL_COMPONENT THEN REWRITE_TAC[PATH_COMPONENT_IN; PATH_COMPONENT_SYM; PATH_COMPONENT_TRANS; PATH_COMPONENT_OF_SUBSET] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC(REWRITE_RULE[PATH_CONNECTED_IFF_PATH_COMPONENT] (MATCH_MP CONVEX_IMP_PATH_CONNECTED (SPEC_ALL CONVEX_BALL))) THEN ASM_MESON_TAC[CENTRE_IN_BALL; BALL_EQ_EMPTY; REAL_NOT_LE; NOT_IN_EMPTY]);; let PATH_CONNECTED_CONTINUOUS_IMAGE = prove (`!f:real^M->real^N s. f continuous_on s /\ path_connected s ==> path_connected (IMAGE f s)`, REPEAT GEN_TAC THEN REWRITE_TAC[path_connected] THEN STRIP_TAC THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN X_GEN_TAC `y:real^M` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^M`; `y:real^M`]) THEN ASM_REWRITE_TAC[path; path_image; pathstart; pathfinish] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^1->real^M` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(f:real^M->real^N) o (g:real^1->real^M)` THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ASM_REWRITE_TAC[o_DEF] THEN ASM SET_TAC[]]);; let HOMEOMORPHIC_PATH_CONNECTEDNESS = prove (`!s t. s homeomorphic t ==> (path_connected s <=> path_connected t)`, REWRITE_TAC[homeomorphic; homeomorphism] THEN MESON_TAC[PATH_CONNECTED_CONTINUOUS_IMAGE]);; let PATH_CONNECTED_LINEAR_IMAGE = prove (`!f:real^M->real^N s. path_connected s /\ linear f ==> path_connected(IMAGE f s)`, SIMP_TAC[LINEAR_CONTINUOUS_ON; PATH_CONNECTED_CONTINUOUS_IMAGE]);; let PATH_CONNECTED_LINEAR_IMAGE_EQ = prove (`!f s. linear f /\ (!x y. f x = f y ==> x = y) ==> (path_connected (IMAGE f s) <=> path_connected s)`, MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE PATH_CONNECTED_LINEAR_IMAGE));; add_linear_invariants [PATH_CONNECTED_LINEAR_IMAGE_EQ];; let HOMEOMORPHISM_PATH_CONNECTEDNESS = prove (`!f:real^M->real^N g s t k. homeomorphism (s,t) (f,g) /\ k SUBSET s ==> (path_connected(IMAGE f k) <=> path_connected k)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_PATH_CONNECTEDNESS THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN REWRITE_TAC[homeomorphic] THEN MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `g:real^N->real^M`] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]);; let PATH_CONNECTED_EMPTY = prove (`path_connected {}`, REWRITE_TAC[path_connected; NOT_IN_EMPTY]);; let PATH_CONNECTED_SING = prove (`!a:real^N. path_connected {a}`, GEN_TAC THEN REWRITE_TAC[path_connected; IN_SING] THEN REPEAT STRIP_TAC THEN EXISTS_TAC `linepath(a:real^N,a)` THEN ASM_REWRITE_TAC[PATH_LINEPATH; PATHSTART_LINEPATH; PATHFINISH_LINEPATH] THEN REWRITE_TAC[SEGMENT_REFL; PATH_IMAGE_LINEPATH; SUBSET_REFL]);; let PATH_CONNECTED_UNION = prove (`!s t. path_connected s /\ path_connected t /\ ~(s INTER t = {}) ==> path_connected (s UNION t)`, REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; PATH_CONNECTED_IFF_PATH_COMPONENT] THEN REWRITE_TAC[IN_INTER; IN_UNION] THEN MESON_TAC[PATH_COMPONENT_OF_SUBSET; SUBSET_UNION; PATH_COMPONENT_TRANS]);; let PATH_CONNECTED_UNIONS = prove (`!f:(real^N->bool)->bool. (!s. s IN f ==> path_connected s) /\ ~(INTERS f = {}) ==> path_connected(UNIONS f)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN REWRITE_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN MATCH_MP_TAC PATH_COMPONENT_TRANS THEN EXISTS_TAC `a:real^N` THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[PATH_COMPONENT_SYM_EQ] THEN UNDISCH_TAC `(x:real^N) IN UNIONS f`; UNDISCH_TAC `(y:real^N) IN UNIONS f`] THEN REWRITE_TAC[IN_UNIONS; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `c:real^N->bool` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `c:real^N->bool`) THEN ASM_REWRITE_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT] THEN DISCH_THEN(MP_TAC o SPEC `a:real^N`) THENL [DISCH_THEN(MP_TAC o SPEC `x:real^N`); DISCH_THEN(MP_TAC o SPEC `y:real^N`)] THEN (ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s x ==> t x`) THEN REWRITE_TAC[ETA_AX] THEN MATCH_MP_TAC PATH_COMPONENT_MONO THEN ASM SET_TAC[]);; let PATH_CONNECTED_TRANSLATION = prove (`!a s. path_connected s ==> path_connected (IMAGE (\x:real^N. a + x) s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC PATH_CONNECTED_CONTINUOUS_IMAGE THEN ASM_SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST]);; let PATH_CONNECTED_TRANSLATION_EQ = prove (`!a s. path_connected (IMAGE (\x:real^N. a + x) s) <=> path_connected s`, REWRITE_TAC[path_connected] THEN GEOM_TRANSLATE_TAC[]);; add_translation_invariants [PATH_CONNECTED_TRANSLATION_EQ];; let PATH_CONNECTED_PCROSS = prove (`!s:real^M->bool t:real^N->bool. path_connected s /\ path_connected t ==> path_connected (s PCROSS t)`, REPEAT GEN_TAC THEN REWRITE_TAC[PCROSS; path_connected] THEN DISCH_TAC THEN REWRITE_TAC[FORALL_PASTECART; IN_ELIM_PASTECART_THM] THEN MAP_EVERY X_GEN_TAC [`x1:real^M`; `y1:real^N`; `x2:real^M`; `y2:real^N`] THEN STRIP_TAC THEN FIRST_X_ASSUM(CONJUNCTS_THEN2 (MP_TAC o SPECL [`x1:real^M`; `x2:real^M`]) (MP_TAC o SPECL [`y1:real^N`; `y2:real^N`])) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `h:real^1->real^N` THEN STRIP_TAC THEN X_GEN_TAC `g:real^1->real^M` THEN STRIP_TAC THEN EXISTS_TAC `(\t. pastecart (x1:real^M) ((h:real^1->real^N) t)) ++ (\t. pastecart ((g:real^1->real^M) t) (y2:real^N))` THEN RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish; path]) THEN RULE_ASSUM_TAC(REWRITE_RULE[path_image; FORALL_IN_IMAGE; SUBSET]) THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC PATH_JOIN_IMP THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[path] THEN MATCH_MP_TAC CONTINUOUS_ON_PASTECART THEN ASM_REWRITE_TAC[CONTINUOUS_ON_CONST]; REWRITE_TAC[path] THEN MATCH_MP_TAC CONTINUOUS_ON_PASTECART THEN ASM_REWRITE_TAC[CONTINUOUS_ON_CONST]; ASM_REWRITE_TAC[pathstart; pathfinish]]; MATCH_MP_TAC SUBSET_PATH_IMAGE_JOIN THEN ASM_SIMP_TAC[path_image; FORALL_IN_IMAGE; SUBSET; IN_ELIM_PASTECART_THM]; REWRITE_TAC[PATHSTART_JOIN] THEN ASM_REWRITE_TAC[pathstart]; REWRITE_TAC[PATHFINISH_JOIN] THEN ASM_REWRITE_TAC[pathfinish]]);; let PATH_CONNECTED_PCROSS_EQ = prove (`!s:real^M->bool t:real^N->bool. path_connected(s PCROSS t) <=> s = {} \/ t = {} \/ path_connected s /\ path_connected t`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_REWRITE_TAC[PCROSS_EMPTY; PATH_CONNECTED_EMPTY] THEN ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_REWRITE_TAC[PCROSS_EMPTY; PATH_CONNECTED_EMPTY] THEN EQ_TAC THEN REWRITE_TAC[PATH_CONNECTED_PCROSS] THEN REPEAT STRIP_TAC THENL [MP_TAC(ISPECL [`fstcart:real^(M,N)finite_sum->real^M`; `(s:real^M->bool) PCROSS (t:real^N->bool)`] PATH_CONNECTED_LINEAR_IMAGE) THEN ASM_REWRITE_TAC[LINEAR_FSTCART]; MP_TAC(ISPECL [`sndcart:real^(M,N)finite_sum->real^N`; `(s:real^M->bool) PCROSS (t:real^N->bool)`] PATH_CONNECTED_LINEAR_IMAGE) THEN ASM_REWRITE_TAC[LINEAR_SNDCART]] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_IMAGE; EXISTS_PASTECART; PASTECART_IN_PCROSS; FSTCART_PASTECART; SNDCART_PASTECART] THEN ASM SET_TAC[]);; let PATH_COMPONENT_PCROSS = prove (`!s t a:real^M b:real^N. path_component (s PCROSS t) (pastecart a b) = path_component s a PCROSS path_component t b`, REPEAT GEN_TAC THEN ASM_CASES_TAC `(a:real^M) IN s /\ (b:real^N) IN t` THENL [MATCH_MP_TAC PATH_COMPONENT_UNIQUE THEN REWRITE_TAC[PASTECART_IN_PCROSS; SUBSET_PCROSS; PATH_CONNECTED_PCROSS_EQ] THEN REWRITE_TAC[PATH_COMPONENT_SUBSET; PATH_CONNECTED_PATH_COMPONENT] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [IN] THEN ASM_REWRITE_TAC[PATH_COMPONENT_REFL_EQ] THEN X_GEN_TAC `c:real^(M,N)finite_sum->bool` THEN REWRITE_TAC[SUBSET; FORALL_PASTECART; PASTECART_IN_PCROSS] THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`x:real^M`; `y:real^N`] THEN DISCH_TAC THEN REWRITE_TAC[IN] THEN REWRITE_TAC[PATH_COMPONENT] THEN CONJ_TAC THENL [EXISTS_TAC `IMAGE fstcart (c:real^(M,N)finite_sum->bool)`; EXISTS_TAC `IMAGE sndcart (c:real^(M,N)finite_sum->bool)`] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN REWRITE_TAC[FORALL_PASTECART; EXISTS_PASTECART; IN_IMAGE] THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN (CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]]) THEN MATCH_MP_TAC PATH_CONNECTED_CONTINUOUS_IMAGE THEN ASM_SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; LINEAR_SNDCART]; MATCH_MP_TAC(SET_RULE `s = {} /\ t = {} ==> s = t`) THEN REWRITE_TAC[PCROSS_EQ_EMPTY; PATH_COMPONENT_EQ_EMPTY] THEN REWRITE_TAC[PASTECART_IN_PCROSS] THEN ASM_MESON_TAC[]]);; let PATH_CONNECTED_SCALING = prove (`!s:real^N->bool c. path_connected s ==> path_connected (IMAGE (\x. c % x) s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC PATH_CONNECTED_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC LINEAR_CONTINUOUS_AT THEN REWRITE_TAC[linear] THEN CONJ_TAC THEN VECTOR_ARITH_TAC);; let PATH_CONNECTED_SCALING_EQ = prove (`!s:real^N->bool c. path_connected (IMAGE (\x. c % x) s) <=> c = &0 \/ path_connected s`, REPEAT GEN_TAC THEN ASM_CASES_TAC `c = &0` THEN ASM_REWRITE_TAC[] THENL [REWRITE_TAC[IMAGE_CONST; VECTOR_MUL_LZERO] THEN MESON_TAC[PATH_CONNECTED_SING; PATH_CONNECTED_EMPTY]; EQ_TAC THEN REWRITE_TAC[PATH_CONNECTED_SCALING] THEN DISCH_THEN(MP_TAC o SPEC `inv(c):real` o MATCH_MP PATH_CONNECTED_SCALING) THEN REWRITE_TAC[GSYM IMAGE_o; o_DEF; VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[REAL_MUL_LINV; VECTOR_MUL_LID; IMAGE_ID]]);; let PATH_CONNECTED_AFFINITY_EQ = prove (`!s m c:real^N. path_connected (IMAGE (\x. m % x + c) s) <=> m = &0 \/ path_connected s`, REWRITE_TAC[AFFINITY_SCALING_TRANSLATION; PATH_CONNECTED_TRANSLATION_EQ; PATH_CONNECTED_SCALING_EQ; IMAGE_o]);; let PATH_CONNECTED_AFFINITY = prove (`!s m c:real^N. path_connected s ==> path_connected (IMAGE (\x. m % x + c) s)`, SIMP_TAC[PATH_CONNECTED_AFFINITY_EQ]);; let PATH_CONNECTED_NEGATIONS = prove (`!s:real^N->bool. path_connected s ==> path_connected (IMAGE (--) s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC PATH_CONNECTED_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC LINEAR_CONTINUOUS_AT THEN REWRITE_TAC[linear] THEN CONJ_TAC THEN VECTOR_ARITH_TAC);; let PATH_CONNECTED_SUMS = prove (`!s t:real^N->bool. path_connected s /\ path_connected t ==> path_connected {x + y | x IN s /\ y IN t}`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP PATH_CONNECTED_PCROSS) THEN DISCH_THEN(MP_TAC o ISPEC `\z. (fstcart z + sndcart z:real^N)` o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] PATH_CONNECTED_CONTINUOUS_IMAGE)) THEN SIMP_TAC[CONTINUOUS_ON_ADD; LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; LINEAR_SNDCART; PCROSS] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_IMAGE; IN_ELIM_THM; EXISTS_PASTECART] THEN REWRITE_TAC[PASTECART_INJ; FSTCART_PASTECART; SNDCART_PASTECART] THEN MESON_TAC[]);; let IS_INTERVAL_PATH_CONNECTED_1 = prove (`!s:real^1->bool. is_interval s <=> path_connected s`, MESON_TAC[CONVEX_IMP_PATH_CONNECTED; PATH_CONNECTED_IMP_CONNECTED; IS_INTERVAL_CONNECTED_1; IS_INTERVAL_CONVEX_1]);; (* ------------------------------------------------------------------------- *) (* Bounds on components of a continuous image. *) (* ------------------------------------------------------------------------- *) let CARD_LE_PATH_COMPONENTS = prove (`!f:real^M->real^N s. f continuous_on s ==> {path_component (IMAGE f s) y | y | y IN IMAGE f s} <=_c {path_component s x | x | x IN s}`, REPEAT STRIP_TAC THEN REWRITE_TAC[LE_C] THEN SIMP_TAC[FORALL_IN_GSPEC; EXISTS_IN_GSPEC; FORALL_IN_IMAGE] THEN EXISTS_TAC `\c. path_component (IMAGE (f:real^M->real^N) s) (f(@x. x IN c))` THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN EXISTS_TAC `x:real^M` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC PATH_COMPONENT_EQ THEN REWRITE_TAC[IN] THEN ONCE_REWRITE_TAC[PATH_COMPONENT] THEN EXISTS_TAC `IMAGE (f:real^M->real^N) (path_component s x)` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC PATH_CONNECTED_CONTINUOUS_IMAGE THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; PATH_COMPONENT_SUBSET; PATH_CONNECTED_PATH_COMPONENT]; MATCH_MP_TAC IMAGE_SUBSET THEN REWRITE_TAC[PATH_COMPONENT_SUBSET]; ALL_TAC; ALL_TAC] THEN MATCH_MP_TAC FUN_IN_IMAGE THEN REWRITE_TAC[IN] THEN ASM_MESON_TAC[PATH_COMPONENT_REFL_EQ]);; let CARD_LE_CONNECTED_COMPONENTS = prove (`!f:real^M->real^N s. f continuous_on s ==> {connected_component (IMAGE f s) y | y | y IN IMAGE f s} <=_c {connected_component s x | x | x IN s}`, REPEAT STRIP_TAC THEN REWRITE_TAC[LE_C] THEN SIMP_TAC[FORALL_IN_GSPEC; EXISTS_IN_GSPEC; FORALL_IN_IMAGE] THEN EXISTS_TAC `\c. connected_component (IMAGE (f:real^M->real^N) s) (f(@x. x IN c))` THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN EXISTS_TAC `x:real^M` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONNECTED_COMPONENT_EQ THEN REWRITE_TAC[IN] THEN ONCE_REWRITE_TAC[connected_component] THEN EXISTS_TAC `IMAGE (f:real^M->real^N) (connected_component s x)` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; CONNECTED_COMPONENT_SUBSET; CONNECTED_CONNECTED_COMPONENT]; MATCH_MP_TAC IMAGE_SUBSET THEN REWRITE_TAC[CONNECTED_COMPONENT_SUBSET]; ALL_TAC; ALL_TAC] THEN MATCH_MP_TAC FUN_IN_IMAGE THEN REWRITE_TAC[IN] THEN ASM_MESON_TAC[CONNECTED_COMPONENT_REFL_EQ]);; let CARD_LE_COMPONENTS = prove (`!f:real^M->real^N s. f continuous_on s ==> components(IMAGE f s) <=_c components s`, REWRITE_TAC[components; CARD_LE_CONNECTED_COMPONENTS]);; (* ------------------------------------------------------------------------- *) (* More stuff about segments. *) (* ------------------------------------------------------------------------- *) let PATH_CONNECTED_SEGMENT = prove (`(!a b. path_connected(segment[a,b])) /\ (!a b. path_connected(segment(a,b)))`, SIMP_TAC[CONVEX_IMP_PATH_CONNECTED; CONVEX_SEGMENT]);; let PATH_CONNECTED_SEMIOPEN_SEGMENT = prove (`(!a b:real^N. path_connected(segment[a,b] DELETE a)) /\ (!a b:real^N. path_connected(segment[a,b] DELETE b))`, SIMP_TAC[CONVEX_IMP_PATH_CONNECTED; CONVEX_SEMIOPEN_SEGMENT]);; let SUBSET_CONTINUOUS_IMAGE_SEGMENT_1 = prove (`!f:real^N->real^1 a b. f continuous_on segment[a,b] ==> segment[f a,f b] SUBSET IMAGE f (segment[a,b])`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONNECTED_CONTINUOUS_IMAGE)) THEN REWRITE_TAC[CONNECTED_SEGMENT] THEN REWRITE_TAC[GSYM IS_INTERVAL_CONNECTED_1; IS_INTERVAL_CONVEX_1] THEN REWRITE_TAC[CONVEX_CONTAINS_SEGMENT] THEN MESON_TAC[IN_IMAGE; ENDS_IN_SEGMENT]);; let CONTINUOUS_INJECTIVE_IMAGE_SEGMENT_1 = prove (`!f:real^N->real^1 a b. f continuous_on segment[a,b] /\ (!x y. x IN segment[a,b] /\ y IN segment[a,b] /\ f x = f y ==> x = y) ==> IMAGE f (segment[a,b]) = segment[f a,f b]`, let lemma = prove (`!a b c:real^1. ~(a = b) /\ ~(a IN segment(c,b)) /\ ~(b IN segment(a,c)) ==> c IN segment[a,b]`, REWRITE_TAC[FORALL_LIFT; SEGMENT_1; LIFT_DROP] THEN REPEAT GEN_TAC THEN REWRITE_TAC[SEGMENT_1; LIFT_EQ] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL_1; LIFT_DROP]) THEN ASM_REAL_ARITH_TAC) in REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[INJECTIVE_ON_LEFT_INVERSE; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `g:real^1->real^N` THEN DISCH_TAC THEN MP_TAC(ISPECL [`f:real^N->real^1`; `g:real^1->real^N`; `segment[a:real^N,b]`] CONTINUOUS_ON_INVERSE) THEN ASM_REWRITE_TAC[COMPACT_SEGMENT] THEN DISCH_TAC THEN REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN MATCH_MP_TAC(TAUT `q /\ (q ==> p) ==> p /\ q`) THEN CONJ_TAC THENL [ASM_SIMP_TAC[SUBSET_CONTINUOUS_IMAGE_SEGMENT_1]; DISCH_TAC] THEN ASM_CASES_TAC `a:real^N = b` THEN ASM_REWRITE_TAC[SEGMENT_REFL] THENL [SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `c:real^N` THEN DISCH_TAC THEN MATCH_MP_TAC lemma THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [ASM_MESON_TAC[ENDS_IN_SEGMENT]; DISCH_TAC] THEN ONCE_REWRITE_TAC[segment] THEN ASM_REWRITE_TAC[IN_DIFF; IN_INSERT; NOT_IN_EMPTY] THEN REPEAT STRIP_TAC THENL [MP_TAC(ISPECL [`f:real^N->real^1`; `c:real^N`; `b:real^N`] SUBSET_CONTINUOUS_IMAGE_SEGMENT_1) THEN SUBGOAL_THEN `segment[c:real^N,b] SUBSET segment[a,b]` ASSUME_TAC THENL [ASM_REWRITE_TAC[SUBSET_SEGMENT; ENDS_IN_SEGMENT]; ALL_TAC] THEN REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; REWRITE_TAC[SUBSET]] THEN DISCH_THEN(MP_TAC o SPEC `(f:real^N->real^1) a`) THEN ASM_REWRITE_TAC[IN_IMAGE; NOT_EXISTS_THM] THEN X_GEN_TAC `d:real^N` THEN ASM_CASES_TAC `d:real^N = a` THENL [ASM_MESON_TAC[BETWEEN_ANTISYM; BETWEEN_IN_SEGMENT]; ASM_MESON_TAC[ENDS_IN_SEGMENT; SUBSET]]; MP_TAC(ISPECL [`f:real^N->real^1`; `a:real^N`; `c:real^N`] SUBSET_CONTINUOUS_IMAGE_SEGMENT_1) THEN SUBGOAL_THEN `segment[a:real^N,c] SUBSET segment[a,b]` ASSUME_TAC THENL [ASM_REWRITE_TAC[SUBSET_SEGMENT; ENDS_IN_SEGMENT]; ALL_TAC] THEN REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; REWRITE_TAC[SUBSET]] THEN DISCH_THEN(MP_TAC o SPEC `(f:real^N->real^1) b`) THEN ASM_REWRITE_TAC[IN_IMAGE; NOT_EXISTS_THM] THEN X_GEN_TAC `d:real^N` THEN ASM_CASES_TAC `d:real^N = b` THENL [ASM_MESON_TAC[BETWEEN_ANTISYM; BETWEEN_IN_SEGMENT; BETWEEN_SYM]; ASM_MESON_TAC[ENDS_IN_SEGMENT; SUBSET]]]);; let CONTINUOUS_INJECTIVE_IMAGE_OPEN_SEGMENT_1 = prove (`!f:real^N->real^1 a b. f continuous_on segment[a,b] /\ (!x y. x IN segment[a,b] /\ y IN segment[a,b] /\ f x = f y ==> x = y) ==> IMAGE f (segment(a,b)) = segment(f a,f b)`, REPEAT GEN_TAC THEN DISCH_TAC THEN ONCE_REWRITE_TAC[segment] THEN FIRST_ASSUM(MP_TAC o MATCH_MP CONTINUOUS_INJECTIVE_IMAGE_SEGMENT_1) THEN MP_TAC(ISPECL [`a:real^N`; `b:real^N`] ENDS_IN_SEGMENT) THEN MP_TAC(ISPECL [`(f:real^N->real^1) a`; `(f:real^1->real^1) b`] ENDS_IN_SEGMENT) THEN ASM SET_TAC[]);; let CONTINUOUS_IVT_LOCAL_EXTREMUM = prove (`!f:real^N->real^1 a b. f continuous_on segment[a,b] /\ ~(a = b) /\ f(a) = f(b) ==> ?z. z IN segment(a,b) /\ ((!w. w IN segment[a,b] ==> drop(f w) <= drop(f z)) \/ (!w. w IN segment[a,b] ==> drop(f z) <= drop(f w)))`, REPEAT STRIP_TAC THEN MAP_EVERY (MP_TAC o ISPECL [`drop o (f:real^N->real^1)`; `segment[a:real^N,b]`]) [CONTINUOUS_ATTAINS_SUP; CONTINUOUS_ATTAINS_INF] THEN ASM_REWRITE_TAC[o_DEF; LIFT_DROP; ETA_AX] THEN REWRITE_TAC[COMPACT_SEGMENT; SEGMENT_EQ_EMPTY] THEN DISCH_THEN(X_CHOOSE_THEN `d:real^N` STRIP_ASSUME_TAC) THEN ASM_CASES_TAC `(d:real^N) IN segment(a,b)` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `c:real^N` STRIP_ASSUME_TAC) THEN ASM_CASES_TAC `(c:real^N) IN segment(a,b)` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN EXISTS_TAC `midpoint(a:real^N,b)` THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [ASM_REWRITE_TAC[MIDPOINT_IN_SEGMENT]; DISCH_TAC] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [CONJUNCT2 segment]) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE (RAND_CONV o RAND_CONV) [segment])) THEN ASM_REWRITE_TAC[IN_DIFF; IN_INSERT; NOT_IN_EMPTY] THEN REPEAT(DISCH_THEN(DISJ_CASES_THEN SUBST_ALL_TAC)) THEN FIRST_X_ASSUM SUBST_ALL_TAC THEN ASM_MESON_TAC[REAL_LE_ANTISYM; DROP_EQ]);; let FRONTIER_UNIONS_SUBSET_CLOSURE = prove (`!f:(real^N->bool)->bool. frontier(UNIONS f) SUBSET closure(UNIONS {frontier t | t IN f})`, GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [frontier] THEN REWRITE_TAC[SUBSET; IN_DIFF; CLOSURE_APPROACHABLE] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[EXISTS_IN_UNIONS; EXISTS_IN_GSPEC; RIGHT_EXISTS_AND_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real^N->bool` THEN ASM_CASES_TAC `(t:real^N->bool) IN f` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `(x:real^N) IN t` THENL [DISCH_THEN(K ALL_TAC) THEN EXISTS_TAC `x:real^N` THEN ASM_REWRITE_TAC[frontier; DIST_REFL; IN_DIFF] THEN ASM_SIMP_TAC[REWRITE_RULE[SUBSET] CLOSURE_SUBSET] THEN FIRST_X_ASSUM(MP_TAC o check (is_neg o concl)) THEN SPEC_TAC(`x:real^N`,`z:real^N`) THEN REWRITE_TAC[CONTRAPOS_THM; GSYM SUBSET] THEN MATCH_MP_TAC SUBSET_INTERIOR THEN ASM SET_TAC[]; DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`segment[x:real^N,y]`; `t:real^N->bool`] CONNECTED_INTER_FRONTIER) THEN SIMP_TAC[CONNECTED_SEGMENT; GSYM MEMBER_NOT_EMPTY; IN_INTER; IN_DIFF] THEN ANTS_TAC THENL [ASM_MESON_TAC[ENDS_IN_SEGMENT]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `z:real^N` THEN ASM_MESON_TAC[DIST_IN_CLOSED_SEGMENT; DIST_SYM; REAL_LET_TRANS]]);; let FRONTIER_UNIONS_SUBSET = prove (`!f:(real^N->bool)->bool. FINITE f ==> frontier(UNIONS f) SUBSET UNIONS {frontier t | t IN f}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC(MESON[] `s SUBSET closure t /\ closure t = t ==> s SUBSET t`) THEN REWRITE_TAC[FRONTIER_UNIONS_SUBSET_CLOSURE; CLOSURE_EQ] THEN MATCH_MP_TAC CLOSED_UNIONS THEN ASM_SIMP_TAC[SIMPLE_IMAGE; FINITE_IMAGE; FORALL_IN_IMAGE; FRONTIER_CLOSED]);; let CLOSURE_CONVEX_INTER_AFFINE = prove (`!s t:real^N->bool. convex s /\ affine t /\ ~(relative_interior s INTER t = {}) ==> closure(s INTER t) = closure(s) INTER t`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[SUBSET_INTER] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC SUBSET_CLOSURE THEN SET_TAC[]; TRANS_TAC SUBSET_TRANS `closure t:real^N->bool` THEN SIMP_TAC[SUBSET_CLOSURE; INTER_SUBSET] THEN ASM_SIMP_TAC[CLOSURE_CLOSED; CLOSED_AFFINE; SUBSET_REFL]; ALL_TAC] THEN FIRST_X_ASSUM(X_CHOOSE_THEN `a:real^N` MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN POP_ASSUM_LIST(MP_TAC o end_itlist CONJ) THEN GEOM_ORIGIN_TAC `a:real^N` THEN REPEAT GEN_TAC THEN REWRITE_TAC[IN_INTER] THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_SIMP_TAC[AFFINE_EQ_SUBSPACE] THEN STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP(REWRITE_RULE[SUBSET] RELATIVE_INTERIOR_SUBSET)) THEN REWRITE_TAC[SUBSET; IN_INTER] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN ASM_CASES_TAC `x:real^N = vec 0` THENL [MATCH_MP_TAC(REWRITE_RULE[SUBSET] CLOSURE_SUBSET) THEN ASM_REWRITE_TAC[IN_INTER]; ALL_TAC] THEN SUBGOAL_THEN `x IN closure(segment(vec 0:real^N,x))` MP_TAC THENL [ASM_REWRITE_TAC[CLOSURE_SEGMENT; ENDS_IN_SEGMENT]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> x IN s ==> x IN t`) THEN MATCH_MP_TAC SUBSET_CLOSURE THEN REWRITE_TAC[SUBSET_INTER] THEN CONJ_TAC THENL [TRANS_TAC SUBSET_TRANS `relative_interior s:real^N->bool` THEN REWRITE_TAC[RELATIVE_INTERIOR_SUBSET] THEN MATCH_MP_TAC IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT THEN ASM_REWRITE_TAC[]; ASM_SIMP_TAC[SUBSET; IN_SEGMENT; VECTOR_MUL_RZERO; VECTOR_ADD_LID; SUBSPACE_MUL; LEFT_IMP_EXISTS_THM]]);; let RELATIVE_FRONTIER_CONVEX_INTER_AFFINE = prove (`!s t:real^N->bool. convex s /\ affine t /\ ~(interior s INTER t = {}) ==> relative_frontier(s INTER t) = frontier s INTER t`, SIMP_TAC[relative_frontier; RELATIVE_INTERIOR_CONVEX_INTER_AFFINE; frontier] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `~(relative_interior s INTER t:real^N->bool = {})` ASSUME_TAC THENL [MP_TAC(ISPEC `s:real^N->bool` INTERIOR_SUBSET_RELATIVE_INTERIOR) THEN ASM SET_TAC[]; ASM_SIMP_TAC[CLOSURE_CONVEX_INTER_AFFINE] THEN SET_TAC[]]);; let RELATIVE_FRONTIER_CBALL_INTER_AFFINE = prove (`!s a:real^N r. affine s /\ a IN s /\ ~(r = &0) ==> relative_frontier(cball(a,r) INTER s) = sphere(a,r) INTER s`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `r < &0` THENL [ASM_SIMP_TAC[CBALL_EMPTY; SPHERE_EMPTY; INTER_EMPTY] THEN REWRITE_TAC[RELATIVE_FRONTIER_EMPTY]; W(MP_TAC o PART_MATCH (lhand o rand) RELATIVE_FRONTIER_CONVEX_INTER_AFFINE o lhand o snd) THEN REWRITE_TAC[FRONTIER_CBALL] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[CONVEX_CBALL; INTERIOR_CBALL; GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `a:real^N` THEN ASM_REWRITE_TAC[IN_INTER; CENTRE_IN_BALL] THEN ASM_REAL_ARITH_TAC]);; let CONNECTED_COMPONENT_1_GEN = prove (`!s a b:real^N. dimindex(:N) = 1 ==> (connected_component s a b <=> segment[a,b] SUBSET s)`, SIMP_TAC[connected_component; CONNECTED_CONVEX_1_GEN] THEN MESON_TAC[CONVEX_CONTAINS_SEGMENT; SUBSET; CONVEX_SEGMENT; ENDS_IN_SEGMENT]);; let CONNECTED_COMPONENT_1 = prove (`!s a b:real^1. connected_component s a b <=> segment[a,b] SUBSET s`, SIMP_TAC[CONNECTED_COMPONENT_1_GEN; DIMINDEX_1]);; let HOMEOMORPHIC_SEGMENTS = prove (`(!a b:real^M c d:real^N. segment[a,b] homeomorphic segment[c,d] <=> (a = b <=> c = d)) /\ (!a b:real^M c d:real^N. ~(segment[a,b] homeomorphic segment(c,d))) /\ (!a b:real^M c d:real^N. ~(segment(a,b) homeomorphic segment[c,d])) /\ (!a b:real^M c d:real^N. segment(a,b) homeomorphic segment(c,d) <=> (a = b <=> c = d))`, let lemma = prove (`!a b:real^N. (\u:real^1. (&1 - drop u) % a + drop u % b) = (\u. a + u) o (\u. drop u % (b - a))`, REWRITE_TAC[FUN_EQ_THM; o_THM] THEN VECTOR_ARITH_TAC) in ONCE_REWRITE_TAC[TAUT `p /\ q /\ r /\ s <=> (q /\ r) /\ (p /\ s)`] THEN CONJ_TAC THENL [REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMEOMORPHIC_COMPACTNESS) THEN REWRITE_TAC[COMPACT_SEGMENT] THEN POP_ASSUM MP_TAC THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN SIMP_TAC[SEGMENT_REFL; HOMEOMORPHIC_EMPTY; SEGMENT_EQ_EMPTY]; REPEAT STRIP_TAC THEN (EQ_TAC THENL [DISCH_THEN(MP_TAC o MATCH_MP HOMEOMORPHIC_FINITENESS) THEN REWRITE_TAC[FINITE_SEGMENT]; ASM_CASES_TAC `c:real^N = d` THEN ASM_SIMP_TAC[SEGMENT_REFL; HOMEOMORPHIC_SING; HOMEOMORPHIC_EMPTY] THEN DISCH_TAC])] THEN ASM_SIMP_TAC[SEGMENT_IMAGE_INTERVAL] THENL [TRANS_TAC HOMEOMORPHIC_TRANS `interval[vec 0:real^1,vec 1]`; TRANS_TAC HOMEOMORPHIC_TRANS `interval(vec 0:real^1,vec 1)`] THEN (CONJ_TAC THENL [ALL_TAC; ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM]]) THEN REWRITE_TAC[lemma; IMAGE_o; HOMEOMORPHIC_TRANSLATION_LEFT_EQ] THEN W(MP_TAC o PART_MATCH (lhand o rand) HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_LEFT_EQ o snd) THEN REWRITE_TAC[HOMEOMORPHIC_REFL] THEN DISCH_THEN MATCH_MP_TAC THEN SIMP_TAC[LINEAR_VMUL_DROP; LINEAR_ID; VECTOR_MUL_RCANCEL] THEN ASM_REWRITE_TAC[DROP_EQ; VECTOR_SUB_EQ]);; let HOMEOMORPHISM_SEGMENT = prove (`!a b:real^N. ~(a = b) ==> ?h. homeomorphism (interval[vec 0:real^1,vec 1],segment[a,b]) ((\t. (&1 - drop t) % a + drop t % b),h)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHISM_COMPACT THEN REWRITE_TAC[COMPACT_INTERVAL; GSYM SEGMENT_IMAGE_INTERVAL] THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_ADD THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN SIMP_TAC[o_DEF; CONTINUOUS_ON_CONST; LIFT_DROP; CONTINUOUS_ON_ID; LIFT_SUB; CONTINUOUS_ON_SUB]; REWRITE_TAC[VECTOR_ARITH `(&1 - x) % a + x % b:real^N = (&1 - y) % a + y % b <=> (x - y) % (a - b) = vec 0`] THEN ASM_SIMP_TAC[VECTOR_MUL_EQ_0; REAL_SUB_0; VECTOR_SUB_EQ; DROP_EQ]]);; let CONNECTED_SUBSET_SEGMENT = prove (`!s a b:real^N. connected s /\ s SUBSET segment[a,b] /\ a IN s /\ b IN s ==> s = segment[a,b]`, REPEAT GEN_TAC THEN ASM_CASES_TAC `a:real^N = b` THENL [ASM_REWRITE_TAC[SEGMENT_REFL] THEN SET_TAC[]; STRIP_TAC] THEN ASM_REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMEOMORPHISM_SEGMENT) THEN ABBREV_TAC `g = \x. (&1 - drop x) % a + drop x % (b:real^N)` THEN REWRITE_TAC[homeomorphism; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `h:real^N->real^1` THEN SUBGOAL_THEN `(g:real^1->real^N)(vec 0) = a /\ g(vec 1) = b` MP_TAC THENL [EXPAND_TAC "g" THEN REWRITE_TAC[DROP_VEC] THEN CONV_TAC VECTOR_ARITH; FIRST_X_ASSUM(K ALL_TAC o SYM) THEN REPEAT STRIP_TAC] THEN SUBGOAL_THEN `IMAGE (h:real^N->real^1) (segment[a,b]) SUBSET IMAGE h s` MP_TAC THENL [ASM_REWRITE_TAC[]; ASM SET_TAC[]] THEN W(MP_TAC o PART_MATCH (lhand o rand) INTERVAL_SUBSET_IS_INTERVAL o snd) THEN ANTS_TAC THENL [REWRITE_TAC[IS_INTERVAL_CONNECTED_1] THEN MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; REWRITE_TAC[UNIT_INTERVAL_NONEMPTY] THEN DISCH_THEN SUBST1_TAC] THEN SUBGOAL_THEN `vec 0 IN interval[vec 0:real^1,vec 1] /\ vec 1 IN interval[vec 0:real^1,vec 1]` MP_TAC THENL [REWRITE_TAC[ENDS_IN_UNIT_INTERVAL]; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish; path_image]) THEN ASM SET_TAC[]);; let DIAMETER_SEGMENT = prove (`(!a b:real^N. diameter(segment[a,b]) = dist(a,b)) /\ (!a b:real^N. diameter(segment(a,b)) = dist(a,b))`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `a:real^N = b` THEN ASM_REWRITE_TAC[SEGMENT_REFL; DIST_REFL; DIAMETER_SING; DIAMETER_EMPTY] THENL [ALL_TAC; GEN_REWRITE_TAC LAND_CONV [GSYM DIAMETER_CLOSURE] THEN ASM_REWRITE_TAC[CLOSURE_SEGMENT]] THEN (REWRITE_TAC[GSYM REAL_LE_ANTISYM] THEN CONJ_TAC THENL [MATCH_MP_TAC DIAMETER_LE; REWRITE_TAC[dist] THEN MATCH_MP_TAC DIAMETER_BOUNDED_BOUND]) THEN REWRITE_TAC[BOUNDED_SEGMENT; ENDS_IN_SEGMENT; DIST_POS_LE] THEN REWRITE_TAC[GSYM dist; DIST_IN_CLOSED_SEGMENT_2]);; (* ------------------------------------------------------------------------- *) (* Removing points from arcs and simple paths, hence allowing us to *) (* distinguish simple closed curves and arcs topologically. *) (* ------------------------------------------------------------------------- *) let SIMPLE_PATH_ENDLESS = prove (`!c:real^1->real^N. simple_path c ==> path_image c DIFF {pathstart c,pathfinish c} = IMAGE c (interval(vec 0,vec 1))`, REWRITE_TAC[simple_path; path_image; pathstart; pathfinish] THEN REWRITE_TAC[OPEN_CLOSED_INTERVAL_1; path] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC(SET_RULE `(!x y. x IN s /\ y IN s /\ c x = c y ==> x = y \/ x = a /\ y = b \/ x = b /\ y = a) /\ a IN s /\ b IN s ==> IMAGE c s DIFF {c a,c b} = IMAGE c (s DIFF {a,b})`) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; REAL_POS; REAL_LE_REFL]);; let PATH_CONNECTED_SIMPLE_PATH_ENDLESS = prove (`!c:real^1->real^N. simple_path c ==> path_connected(path_image c DIFF {pathstart c,pathfinish c})`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[SIMPLE_PATH_ENDLESS] THEN MATCH_MP_TAC PATH_CONNECTED_CONTINUOUS_IMAGE THEN REWRITE_TAC[GSYM IS_INTERVAL_PATH_CONNECTED_1; IS_INTERVAL_INTERVAL] THEN MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `interval[vec 0:real^1,vec 1]` THEN RULE_ASSUM_TAC(REWRITE_RULE[simple_path; path]) THEN ASM_REWRITE_TAC[INTERVAL_OPEN_SUBSET_CLOSED]);; let CONNECTED_SIMPLE_PATH_ENDLESS = prove (`!c:real^1->real^N. simple_path c ==> connected(path_image c DIFF {pathstart c,pathfinish c})`, SIMP_TAC[PATH_CONNECTED_IMP_CONNECTED; PATH_CONNECTED_SIMPLE_PATH_ENDLESS]);; let NONEMPTY_SIMPLE_PATH_ENDLESS = prove (`!c:real^1->real^N. simple_path c ==> ~(path_image c DIFF {pathstart c,pathfinish c} = {})`, SIMP_TAC[SIMPLE_PATH_ENDLESS; IMAGE_EQ_EMPTY; INTERVAL_EQ_EMPTY_1] THEN REWRITE_TAC[DROP_VEC] THEN REAL_ARITH_TAC);; let CONNECTED_ARC_IMAGE_DELETE = prove (`!g a:real^N. arc g /\ a IN path_image g ==> (connected(path_image g DELETE a) <=> a IN {pathstart g,pathfinish g})`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMEOMORPHISM_ARC) THEN DISCH_THEN(X_CHOOSE_THEN `h:real^N->real^1` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [path_image]) THEN REWRITE_TAC[IN_IMAGE; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `t:real^1` THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN TRANS_TAC EQ_TRANS `connected(IMAGE (h:real^N->real^1) (path_image g DELETE a))` THEN CONJ_TAC THENL [MATCH_MP_TAC HOMEOMORPHIC_CONNECTEDNESS THEN REWRITE_TAC[homeomorphic] THEN MAP_EVERY EXISTS_TAC [`h:real^N->real^1`; `g:real^1->real^N`] THEN RULE_ASSUM_TAC(ONCE_REWRITE_RULE[HOMEOMORPHISM_SYM]) THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]; ALL_TAC] THEN TRANS_TAC EQ_TRANS `connected(interval[vec 0:real^1,vec 1] DELETE t)` THEN CONJ_TAC THENL [AP_TERM_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]; REWRITE_TAC[pathstart; pathfinish]] THEN TRANS_TAC EQ_TRANS `t IN {vec 0:real^1,vec 1}` THEN CONJ_TAC THENL [REWRITE_TAC[GSYM IS_INTERVAL_CONNECTED_1; IS_INTERVAL_1]; EXPAND_TAC "a" THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `x IN s ==> (!x y. x IN s /\ y IN s /\ g x = g y ==> x = y) /\ a IN s /\ b IN s ==> (x IN {a,b} <=> g x IN {g a,g b})`)) THEN REWRITE_TAC[ENDS_IN_UNIT_INTERVAL] THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]] THEN EQ_TAC THENL [DISCH_THEN(MP_TAC o SPECL [`vec 0:real^1`; `vec 1:real^1`; `t:real^1`]) THEN ASM_REWRITE_TAC[GSYM IN_INTERVAL_1; IN_DELETE; ENDS_IN_UNIT_INTERVAL] THEN SET_TAC[]; REWRITE_TAC[IN_DELETE; IN_INSERT; NOT_IN_EMPTY; IN_INTERVAL_1; GSYM DROP_EQ; DROP_VEC] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN ASM_REAL_ARITH_TAC]);; let CONNECTED_SIMPLE_PATH_IMAGE_DELETE = prove (`!g a:real^N. simple_path g /\ pathfinish g = pathstart g ==> connected(path_image g DELETE a)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `(a:real^N) IN path_image g` THEN ASM_SIMP_TAC[SET_RULE `~(a IN s) ==> s DELETE a = s`; CONNECTED_PATH_IMAGE; SIMPLE_PATH_IMP_PATH] THEN RULE_ASSUM_TAC(REWRITE_RULE[path_image]) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_IMAGE]) THEN DISCH_THEN(X_CHOOSE_THEN `t:real^1` (STRIP_ASSUME_TAC o GSYM)) THEN MP_TAC(ISPEC `shiftpath t (g:real^1->real^N)` CONNECTED_SIMPLE_PATH_ENDLESS) THEN ASM_SIMP_TAC[SIMPLE_PATH_SHIFTPATH; PATH_IMAGE_SHIFTPATH] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN ASM_SIMP_TAC[PATHSTART_SHIFTPATH; PATHFINISH_SHIFTPATH] THEN REWRITE_TAC[SET_RULE `s DIFF {a,a} = s DELETE a`]);; let HOMEOMORPHIC_SIMPLE_PATH_ARC = prove (`!g:real^1->real^M h:real^1->real^N. arc g /\ simple_path h /\ (path_image g) homeomorphic (path_image h) ==> arc h`, REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[ARC_SIMPLE_PATH] THEN DISCH_TAC THEN SUBGOAL_THEN `?a:real^M. a IN path_image g /\ ~connected(path_image g DELETE a)` STRIP_ASSUME_TAC THENL [ASM_SIMP_TAC[CONNECTED_ARC_IMAGE_DELETE; TAUT `p /\ ~q <=> ~(p ==> q)`] THEN REWRITE_TAC[path_image; NOT_IMP; EXISTS_IN_IMAGE] THEN EXISTS_TAC `lift(&1 / &2)` THEN REWRITE_TAC[IN_INTERVAL_1; LIFT_DROP; DROP_VEC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[IN_INSERT; pathstart; pathfinish; NOT_IN_EMPTY] THEN REWRITE_TAC[DE_MORGAN_THM] THEN FIRST_ASSUM(MP_TAC o CONJUNCT2 o REWRITE_RULE[arc]) THEN ONCE_REWRITE_TAC[SET_RULE `p /\ q /\ r ==> s <=> p /\ q /\ ~s ==> ~r`] THEN DISCH_TAC THEN CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[IN_INTERVAL_1; GSYM DROP_EQ; DROP_VEC; LIFT_DROP] THEN CONV_TAC REAL_RAT_REDUCE_CONV; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphic]) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`i:real^M->real^N`; `j:real^N->real^M`] THEN DISCH_TAC THEN MP_TAC(ISPECL [`h:real^1->real^N`; `(i:real^M->real^N) a`] CONNECTED_SIMPLE_PATH_IMAGE_DELETE) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o check (is_neg o concl)) THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_CONNECTEDNESS THEN REWRITE_TAC[homeomorphic] THEN MAP_EVERY EXISTS_TAC [`i:real^M->real^N`; `j:real^N->real^M`] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]);; let HOMEOMORPHIC_SIMPLE_PATH_ARC_EQ = prove (`!g:real^1->real^M h:real^1->real^N. simple_path g /\ simple_path h /\ (path_image g) homeomorphic (path_image h) ==> (arc g <=> arc h)`, REPEAT STRIP_TAC THEN EQ_TAC THENL [MP_TAC(ISPECL [`g:real^1->real^M`; `h:real^1->real^N`] HOMEOMORPHIC_SIMPLE_PATH_ARC); MP_TAC(ISPECL [`h:real^1->real^N`; `g:real^1->real^M`] HOMEOMORPHIC_SIMPLE_PATH_ARC)] THEN ASM_MESON_TAC[HOMEOMORPHIC_SYM]);; let ARC_ENDS_UNIQUE = prove (`!g h:real^1->real^N. arc g /\ simple_path h /\ path_image g = path_image h ==> {pathstart g, pathfinish g} = {pathstart h, pathfinish h}`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`g:real^1->real^N`; `h:real^1->real^N`] HOMEOMORPHIC_SIMPLE_PATH_ARC) THEN ASM_REWRITE_TAC[HOMEOMORPHIC_REFL] THEN DISCH_TAC THEN TRANS_TAC EQ_TRANS `{x:real^N | x IN path_image g /\ connected(path_image g DELETE x)}` THEN CONJ_TAC THENL [CONV_TAC SYM_CONV; ASM_REWRITE_TAC[]] THEN MATCH_MP_TAC(SET_RULE `a IN s /\ b IN s /\ (!x. x IN s ==> (P x <=> x IN {a,b})) ==> {x | x IN s /\ P x} = {a,b}`) THEN REWRITE_TAC[PATHSTART_IN_PATH_IMAGE; PATHFINISH_IN_PATH_IMAGE] THEN ASM_SIMP_TAC[CONNECTED_ARC_IMAGE_DELETE]);; let ARC_HOMEOMORPHISM_ENDS = prove (`!g h f f':real^N->real^N. homeomorphism (path_image g,path_image h) (f,f') /\ arc g /\ arc h ==> f(pathstart g) = pathstart h /\ f(pathfinish g) = pathfinish h /\ f'(pathstart h) = pathstart g /\ f'(pathfinish h) = pathfinish g \/ f(pathstart g) = pathfinish h /\ f(pathfinish g) = pathstart h /\ f'(pathstart h) = pathfinish g /\ f'(pathfinish h) = pathstart g`, REWRITE_TAC[homeomorphism] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`(f:real^N->real^N) o (g:real^1->real^N)`; `h:real^1->real^N`] ARC_ENDS_UNIQUE) THEN ASM_REWRITE_TAC[PATH_IMAGE_COMPOSE] THEN ANTS_TAC THENL [ASM_SIMP_TAC[ARC_IMP_SIMPLE_PATH] THEN MATCH_MP_TAC ARC_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; REWRITE_TAC[PATHSTART_COMPOSE; PATHFINISH_COMPOSE; SET_RULE `{a,b} = {a',b'} <=> a = a' /\ b = b' \/ a = b' /\ b = a'`] THEN MATCH_MP_TAC MONO_OR THEN SIMP_TAC[] THEN CONJ_TAC THEN DISCH_THEN(CONJUNCTS_THEN(SUBST1_TAC o SYM)) THEN CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[PATHSTART_IN_PATH_IMAGE; PATHFINISH_IN_PATH_IMAGE]]);; let HOMEOMORPHISM_ARC_IMAGES = prove (`!g:real^1->real^M h:real^1->real^N. arc g /\ arc h ==> ?f f'. homeomorphism (path_image g,path_image h) (f,f') /\ f(pathstart g) = pathstart h /\ f(pathfinish g) = pathfinish h /\ f'(pathstart h) = pathstart g /\ f'(pathfinish h) = pathfinish g`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `h:real^1->real^N` HOMEOMORPHISM_ARC) THEN MP_TAC(ISPEC `g:real^1->real^M` HOMEOMORPHISM_ARC) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `g':real^M->real^1` THEN STRIP_TAC THEN X_GEN_TAC `h':real^N->real^1` THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`(h:real^1->real^N) o (g':real^M->real^1)`; `(g:real^1->real^M) o (h':real^N->real^1)`] THEN CONJ_TAC THENL [MATCH_MP_TAC HOMEOMORPHISM_COMPOSE THEN ASM_MESON_TAC[HOMEOMORPHISM_SYM]; RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM_SIMP_TAC[o_THM; pathstart; pathfinish; ENDS_IN_UNIT_INTERVAL]]);; let COLLINEAR_SIMPLE_PATH_IMAGE = prove (`!g:real^1->real^N. simple_path g /\ collinear(path_image g) ==> path_image g = segment[pathstart g,pathfinish g]`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `path_image g:real^N->bool` COMPACT_CONVEX_COLLINEAR_SEGMENT) THEN ASM_SIMP_TAC[CONVEX_CONNECTED_COLLINEAR; CONNECTED_PATH_IMAGE; COMPACT_PATH_IMAGE; PATH_IMAGE_NONEMPTY; SIMPLE_PATH_IMP_PATH] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN DISCH_TAC THEN ASM_REWRITE_TAC[SEGMENT_EQ] THEN MP_TAC(ISPECL [`linepath(a:real^N,b)`; `g:real^1->real^N`] ARC_ENDS_UNIQUE) THEN ASM_REWRITE_TAC[PATHSTART_LINEPATH; PATHFINISH_LINEPATH] THEN ASM_REWRITE_TAC[PATH_IMAGE_LINEPATH] THEN DISCH_THEN MATCH_MP_TAC THEN MATCH_MP_TAC ARC_LINEPATH THEN DISCH_THEN(SUBST_ALL_TAC o SYM) THEN RULE_ASSUM_TAC(REWRITE_RULE[SEGMENT_REFL]) THEN FIRST_ASSUM(MP_TAC o MATCH_MP NONEMPTY_SIMPLE_PATH_ENDLESS) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s = {a} ==> x IN s ==> {a} DIFF {x,y} = {}`)) THEN REWRITE_TAC[PATHSTART_IN_PATH_IMAGE]);; (* ------------------------------------------------------------------------- *) (* An injective function into R is a homeomorphism and so an open map. *) (* ------------------------------------------------------------------------- *) let INJECTIVE_INTO_1D_EQ_HOMEOMORPHISM = prove (`!f:real^N->real^1 s. f continuous_on s /\ path_connected s ==> ((!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) <=> ?g. homeomorphism (s,IMAGE f s) (f,g))`, REWRITE_TAC[FORALL_LIFT_FUN; EXISTS_FUN_DROP] THEN REWRITE_TAC[GSYM CONTINUOUS_MAP_EUCLIDEAN; GSYM CONTINUOUS_MAP_EQ_LIFT] THEN REPEAT GEN_TAC THEN REWRITE_TAC[o_THM; LIFT_EQ] THEN MATCH_MP_TAC(TAUT `(q ==> (p /\ r <=> s)) ==> p /\ q ==> (r <=> s)`) THEN DISCH_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand o rand) EMBEDDING_MAP_INTO_EUCLIDEANREAL o lhand o lhand o snd) THEN ASM_REWRITE_TAC[PATH_CONNECTED_SPACE_EUCLIDEAN_SUBTOPOLOGY] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[embedding_map; GSYM HOMEOMORPHIC_MAPS_EUCLIDEAN_SUBTOPOLOGY] THEN REWRITE_TAC[HOMEOMORPHIC_MAP_MAPS; homeomorphic_maps] THEN AP_TERM_TAC THEN ABS_TAC THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; FORALL_IN_IMAGE; o_THM; LIFT_DROP; LIFT_EQ; TOPSPACE_EUCLIDEANREAL_SUBTOPOLOGY; CONTINUOUS_MAP_IN_SUBTOPOLOGY; SUBSET] THEN REWRITE_TAC[GSYM CONTINUOUS_MAP_DROP_EQ_GEN; IMAGE_o] THEN REWRITE_TAC[GSYM CONTINUOUS_MAP_EQ_LIFT; LIFT_IN_IMAGE_LIFT]);; let INJECTIVE_INTO_1D_IMP_OPEN_MAP = prove (`!f:real^N->real^1 s t. f continuous_on s /\ path_connected s /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\ open_in (subtopology euclidean s) t ==> open_in (subtopology euclidean (IMAGE f s)) (IMAGE f t)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHISM_IMP_OPEN_MAP THEN ASM_MESON_TAC[INJECTIVE_INTO_1D_EQ_HOMEOMORPHISM]);; let HOMEOMORPHISM_INTO_1D = prove (`!f:real^N->real^1 s t. path_connected s /\ f continuous_on s /\ IMAGE f s = t /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> ?g. homeomorphism(s,t) (f,g)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHISM_INJECTIVE_OPEN_MAP THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC INJECTIVE_INTO_1D_IMP_OPEN_MAP THEN ASM_MESON_TAC[]);; (* ------------------------------------------------------------------------- *) (* Injective function on an interval is strictly increasing or decreasing. *) (* ------------------------------------------------------------------------- *) let CONTINUOUS_INJECTIVE_IFF_MONOTONIC = prove (`!f:real^1->real^1 s. f continuous_on s /\ is_interval s ==> ((!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) <=> (!x y. x IN s /\ y IN s /\ drop x < drop y ==> drop(f x) < drop(f y)) \/ (!x y. x IN s /\ y IN s /\ drop x < drop y ==> drop(f y) < drop(f x)))`, REWRITE_TAC[FORALL_LIFT_IMAGE] THEN REWRITE_TAC[FORALL_LIFT_FUN; FORALL_FUN_DROP] THEN REPEAT GEN_TAC THEN REWRITE_TAC[GSYM IS_REALINTERVAL_IS_INTERVAL] THEN REWRITE_TAC[GSYM CONTINUOUS_MAP_EUCLIDEAN] THEN REWRITE_TAC[GSYM CONTINUOUS_MAP_DROP_EQ_GEN] THEN REWRITE_TAC[GSYM CONTINUOUS_MAP_EQ_LIFT] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN REWRITE_TAC[o_THM; LIFT_DROP; LIFT_EQ] THEN REWRITE_TAC[RIGHT_IMP_FORALL_THM; IMP_IMP] THEN GEN_REWRITE_TAC LAND_CONV [CONJ_SYM] THEN REWRITE_TAC[GSYM CONJ_ASSOC] THEN REWRITE_TAC[INJECTIVE_EQ_MONOTONE_MAP]);; let CONTINUOUS_INJECTIVE_IMP_MONOTONIC = prove (`!f s. f continuous_on s /\ is_interval s /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> (!x y. x IN s /\ y IN s ==> (drop(f x) < drop(f y) <=> drop x < drop y)) \/ (!x y. x IN s /\ y IN s ==> (drop(f x) < drop(f y) <=> drop y < drop x))`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(fun th -> MP_TAC th THEN ASM_SIMP_TAC[CONTINUOUS_INJECTIVE_IFF_MONOTONIC] THEN ASSUME_TAC(REWRITE_RULE[INJECTIVE_ON_ALT] th)) THEN MATCH_MP_TAC MONO_OR THEN CONJ_TAC THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`x:real^1`; `y:real^1`] THEN STRIP_TAC THEN EQ_TAC THEN ASM_SIMP_TAC[] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[REAL_NOT_LT] THEN REWRITE_TAC[REAL_LE_LT] THEN ASM_MESON_TAC[DROP_EQ]);; let HOMEOMORPHISM_1D_IMP_MONOTONIC = prove (`!f g s t. homeomorphism(s,t) (f,g) /\ is_interval s ==> (!x y. x IN s /\ y IN s ==> (drop(f x) < drop(f y) <=> drop x < drop y)) /\ (!x y. x IN t /\ y IN t ==> (drop(g x) < drop(g y) <=> drop x < drop y)) \/ (!x y. x IN s /\ y IN s ==> (drop(f x) < drop(f y) <=> drop y < drop x)) /\ (!x y. x IN t /\ y IN t ==> (drop(g x) < drop(g y) <=> drop y < drop x))`, REWRITE_TAC[homeomorphism] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^1->real^1`; `s:real^1->bool`] CONTINUOUS_INJECTIVE_IMP_MONOTONIC) THEN ASM_REWRITE_TAC[] THEN REPEAT(ANTS_TAC THENL [ASM SET_TAC[]; STRIP_TAC]) THEN ASM_REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[EXTENSION; IN_IMAGE]) THEN ASM_MESON_TAC[REAL_LT_ANTISYM]);; (* ------------------------------------------------------------------------- *) (* Topological rendering of Darboux continuity, proof it implies continuity *) (* for a regulated function from R^1 (having left and right limits). *) (* ------------------------------------------------------------------------- *) let CONVEXITY_PRESERVING = prove (`!f:real^M->real^N s. (!c. c SUBSET s /\ convex c ==> convex(IMAGE f c)) <=> (!a b. segment[a,b] SUBSET s ==> convex(IMAGE f (segment[a,b])))`, REPEAT GEN_TAC THEN EQ_TAC THEN SIMP_TAC[CONVEX_SEGMENT] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[CONVEX_CONTAINS_SEGMENT; FORALL_IN_IMAGE_2] THEN ASM_SIMP_TAC[GSYM CONVEX_CONTAINS_SEGMENT_IMP] THEN MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^M`] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a:real^M`; `b:real^M`]) THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(MP_TAC o MATCH_MP CONVEX_CONTAINS_SEGMENT_IMP) THEN DISCH_THEN(MP_TAC o SPECL [`(f:real^M->real^N) a`; `(f:real^M->real^N) b`]) THEN SIMP_TAC[FUN_IN_IMAGE; ENDS_IN_SEGMENT] THEN ASM SET_TAC[]);; let CONVEXITY_PRESERVING_ALT = prove (`!f:real^M->real^N s. (!c. c SUBSET s /\ convex c ==> convex(IMAGE f c)) <=> (!a b. segment[a,b] SUBSET s ==> segment[f a,f b] SUBSET IMAGE f (segment[a,b]))`, REPEAT STRIP_TAC THEN EQ_TAC THEN REPEAT STRIP_TAC THENL [MATCH_MP_TAC SEGMENT_SUBSET_CONVEX THEN SIMP_TAC[FUN_IN_IMAGE; ENDS_IN_SEGMENT] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[CONVEX_SEGMENT]; REWRITE_TAC[CONVEX_CONTAINS_SEGMENT; FORALL_IN_IMAGE_2] THEN ASM_SIMP_TAC[GSYM CONVEX_CONTAINS_SEGMENT_IMP] THEN MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^M`] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a:real^M`; `b:real^M`]) THEN ASM SET_TAC[]]);; let DARBOUX_AND_REGULATED_IMP_CONTINUOUS = prove (`!f:real^1->real^N s. is_interval s /\ (!c. c SUBSET s /\ connected c ==> connected(IMAGE f c)) /\ (!a. a IN s ==> (?l. (f --> l) (at a within s INTER {x | drop x <= drop a})) /\ (?r. (f --> r) (at a within s INTER {x | drop a <= drop x}))) ==> f continuous_on s`, SUBGOAL_THEN `!f:real^1->real^1 s. is_interval s /\ (!c. c SUBSET s /\ connected c ==> connected(IMAGE f c)) /\ (!a. a IN s ==> (?l. (f --> l) (at a within s INTER {x | drop x <= drop a})) /\ (?r. (f --> r) (at a within s INTER {x | drop a <= drop x}))) ==> f continuous_on s` ASSUME_TAC THENL [ALL_TAC; REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[CONTINUOUS_ON_COMPONENTWISE_LIFT] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [X_GEN_TAC `c:real^1->bool` THEN STRIP_TAC THEN SUBGOAL_THEN `(\x. lift(((f:real^1->real^N) x)$i)) = (\x. lift(x$i)) o f` (fun th -> ONCE_REWRITE_TAC[th]) THENL [REWRITE_TAC[o_DEF]; REWRITE_TAC[IMAGE_o]] THEN MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN ASM_SIMP_TAC[CONTINUOUS_ON_LIFT_COMPONENT]; X_GEN_TAC `a:real^1` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `a:real^1`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THEN DISCH_THEN(X_CHOOSE_TAC `y:real^N`) THEN EXISTS_TAC `lift((y:real^N)$i)` THEN ASM_SIMP_TAC[LIM_COMPONENT]]] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[CONTINUOUS_ON] THEN X_GEN_TAC `a:real^1` THEN DISCH_TAC THEN ONCE_REWRITE_TAC[TWO_SIDED_LIMIT_WITHIN] THEN FIRST_X_ASSUM(MP_TAC o SPEC `a:real^1`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THEN DISCH_THEN(X_CHOOSE_THEN `m:real^1` MP_TAC) THEN MATCH_MP_TAC(MESON[LIM_TRIVIAL] `(~trivial_limit net /\ (f --> l) net ==> m = l) ==> (f --> l) net ==> (f --> m) net`) THEN REWRITE_TAC[TRIVIAL_LIMIT_WITHIN] THEN STRIP_TAC THEN MATCH_MP_TAC(NORM_ARITH `~(&0 < norm(x - y:real^N)) ==> x = y`) THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIM_WITHIN]) THEN DISCH_THEN(MP_TAC o SPEC `norm((f:real^1->real^1) a - m) / &2`) THEN ASM_REWRITE_TAC[REAL_HALF; IN_INTER; IN_ELIM_THM; GSYM DIST_NZ] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIMPT_APPROACHABLE]) THEN DISCH_THEN(MP_TAC o SPEC `d:real`) THEN ASM_REWRITE_TAC[IN_INTER; IN_ELIM_THM; GSYM DIST_NZ] THEN DISCH_THEN(X_CHOOSE_THEN `b:real^1` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`IMAGE (f:real^1->real^1) (segment(a,b))`; `(f:real^1->real^1) a`] CONNECTED_INSERT) THEN ASM_REWRITE_TAC[IMAGE_EQ_EMPTY; SEGMENT_EQ_EMPTY] THEN MATCH_MP_TAC(TAUT `(p /\ q) /\ ~r ==> ~(p ==> (q <=> r))`) THEN (CONJ_TAC THENL [REWRITE_TAC[GSYM(CONJUNCT2 IMAGE_CLAUSES)] THEN CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[CONNECTED_INSERT; CONNECTED_SEGMENT] THEN ASM_REWRITE_TAC[CLOSURE_SEGMENT; ENDS_IN_SEGMENT; INSERT_SUBSET] THEN ASM_MESON_TAC[CONVEX_CONTAINS_OPEN_SEGMENT; IS_INTERVAL_CONVEX_1]; DISCH_THEN(MP_TAC o SPEC `closure(ball(m,norm((f:real^1->real^1) a - m) / &2))` o MATCH_MP(SET_RULE `a IN s ==> !t. s SUBSET t ==> a IN t`)) THEN REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [MATCH_MP_TAC SUBSET_CLOSURE THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `x:real^1` THEN REWRITE_TAC[SEGMENT_1] THEN COND_CASES_TAC THEN TRY(RULE_ASSUM_TAC(REWRITE_RULE[GSYM DROP_EQ]) THEN ASM_REAL_ARITH_TAC) THEN REWRITE_TAC[IN_INTERVAL_1; IN_BALL] THEN STRIP_TAC THEN ONCE_REWRITE_TAC[DIST_SYM] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[GSYM DROP_EQ; DIST_1; GSYM CONJ_ASSOC] THEN RULE_ASSUM_TAC(REWRITE_RULE[DIST_1; IS_INTERVAL_1]) THEN (CONJ_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC]) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[REAL_LT_IMP_LE]; ASM_SIMP_TAC[CLOSURE_BALL; REAL_HALF; IN_CBALL] THEN MATCH_MP_TAC(NORM_ARITH `&0 < norm(y - x) ==> ~(dist(x:real^N,y) <= norm(y - x) / &2)`) THEN ASM_REWRITE_TAC[]]]));; (* ------------------------------------------------------------------------- *) (* Some handy facts about Lipschitz functions. *) (* ------------------------------------------------------------------------- *) let LIPSCHITZ_ON_UNION = prove (`!(f:real^1->real^N) s t l. is_interval s /\ is_interval t /\ ~(s INTER t = {}) /\ (!x y. x IN s /\ y IN s ==> norm(f x - f y) <= l * norm(x - y)) /\ (!x y. x IN t /\ y IN t ==> norm(f x - f y) <= l * norm(x - y)) ==> !x y. x IN s UNION t /\ y IN s UNION t ==> norm(f x - f y) <= l * norm(x - y)`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC(MESON[] `!Q. (!x y. P x y <=> P y x) /\ (!x y. ~Q x /\ ~Q y ==> P x y) /\ (!x y. Q x /\ Q y ==> P x y) /\ (!x y. ~Q x /\ Q y ==> P x y) ==> !x y. P x y`) THEN EXISTS_TAC `\x:real^1. x IN s` THEN ASM_SIMP_TAC[SET_RULE `~(x IN s) ==> (x IN s UNION t <=> x IN t)`] THEN CONJ_TAC THENL [MESON_TAC[NORM_SUB]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`x:real^1`; `y:real^1`] THEN STRIP_TAC THEN DISCH_THEN(ASSUME_TAC o CONJUNCT1) THEN ASM_CASES_TAC `(y:real^1) IN t` THEN ASM_SIMP_TAC[] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN REWRITE_TAC[IN_INTER; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `z:real^1` THEN STRIP_TAC THEN MP_TAC(ISPEC `{z:real^1,x,y}` COLLINEAR_1) THEN REWRITE_TAC[COLLINEAR_BETWEEN_CASES] THEN DISCH_THEN(DISJ_CASES_THEN MP_TAC) THENL [REWRITE_TAC[between; dist] THEN DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[REAL_ADD_LDISTRIB] THEN TRANS_TAC REAL_LE_TRANS `norm((f:real^1->real^N) x - f z) + norm(f z - f y)` THEN CONJ_TAC THENL [CONV_TAC NORM_ARITH; ASM_MESON_TAC[REAL_LE_ADD2]]; RULE_ASSUM_TAC(REWRITE_RULE[IS_INTERVAL_CONVEX_1]) THEN REWRITE_TAC[BETWEEN_IN_SEGMENT] THEN ASM_MESON_TAC[CONVEX_CONTAINS_SEGMENT; SUBSET]]);; let LIPSCHITZ_ON_COMBINE = prove (`!(f:real^1->real^N) a b c l. (!x y. x IN interval[a,b] /\ y IN interval[a,b] ==> norm(f x - f y) <= l * norm(x - y)) /\ (!x y. x IN interval[b,c] /\ y IN interval[b,c] ==> norm(f x - f y) <= l * norm(x - y)) ==> !x y. x IN interval[a,c] /\ y IN interval[a,c] ==> norm(f x - f y) <= l * norm(x - y)`, REPEAT GEN_TAC THEN STRIP_TAC THEN ASM_CASES_TAC `interval[a:real^1,c] = {}` THENL [ASM_MESON_TAC[NOT_IN_EMPTY]; RULE_ASSUM_TAC(REWRITE_RULE[INTERVAL_NE_EMPTY_1])] THEN ASM_CASES_TAC `interval[a,c] SUBSET interval[a,b] \/ interval[a:real^1,c] SUBSET interval[b,c]` THENL [ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN SUBGOAL_THEN `b IN interval[a:real^1,c]` ASSUME_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [DE_MORGAN_THM]) THEN REWRITE_TAC[SUBSET_INTERVAL_1; REAL_LE_REFL; IN_INTERVAL_1] THEN ASM_REAL_ARITH_TAC; FIRST_ASSUM(SUBST1_TAC o SYM o MATCH_MP UNION_INTERVAL_1) THEN MATCH_MP_TAC LIPSCHITZ_ON_UNION THEN ASM_REWRITE_TAC[IS_INTERVAL_INTERVAL] THEN REWRITE_TAC[INTER_INTERVAL_1; INTERVAL_NE_EMPTY_1] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC]);; let LOCALLY_LIPSCHITZ_GEN = prove (`!f:real^M->real^N s b. convex s /\ (!x c. x IN s /\ b < c ==> eventually (\y. norm(f y - f x) <= c * norm(y - x)) (at x within s)) ==> !x y. x IN s /\ y IN s ==> norm(f x - f y) <= b * norm(x - y)`, let lemma = prove (`{x | x IN s /\ !y. P x y} = s INTER INTERS {{x | x IN s /\ P x y} | y IN UNIV}`, REWRITE_TAC[INTERS_GSPEC] THEN SET_TAC[]) in REPEAT STRIP_TAC THEN ASM_CASES_TAC `x:real^M = y` THEN ASM_REWRITE_TAC[VECTOR_SUB_REFL; NORM_0; REAL_MUL_RZERO; REAL_LE_REFL] THEN ASM_SIMP_TAC[GSYM REAL_LE_LDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN GEN_REWRITE_TAC I [REAL_LE_TRANS_LTE] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN X_GEN_TAC `c:real` THEN DISCH_TAC THEN MP_TAC(ISPECL [`x:real^M`; `y:real^M`] (CONJUNCT1 CONNECTED_SEGMENT)) THEN REWRITE_TAC[CONNECTED_CLOPEN] THEN DISCH_THEN(MP_TAC o SPEC `{z | z IN segment[x,y] /\ !t. t IN segment[x,z] ==> norm((f:real^M->real^N) t - f x) <= c * norm(t - x)}`) THEN ANTS_TAC THENL [ALL_TAC; DISCH_THEN(MP_TAC o SPECL [`x:real^M`; `y:real^M`] o MATCH_MP (SET_RULE `!a b. {x | x IN s /\ P x} = {} \/ {x | x IN s /\ P x} = s ==> a IN s /\ b IN s /\ P a ==> P b`)) THEN REWRITE_TAC[ENDS_IN_SEGMENT; DIST_REFL; DIST_POS_LE] THEN REWRITE_TAC[SEGMENT_REFL; FORALL_IN_INSERT; NOT_IN_EMPTY] THEN REWRITE_TAC[VECTOR_SUB_REFL; NORM_0; REAL_MUL_RZERO; REAL_LE_REFL] THEN DISCH_THEN(MP_TAC o SPEC `y:real^M`) THEN REWRITE_TAC[ENDS_IN_SEGMENT; NORM_SUB]] THEN CONJ_TAC THENL [REWRITE_TAC[open_in; SUBSET_RESTRICT; IN_ELIM_THM] THEN X_GEN_TAC `z:real^M` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`z:real^M`; `c:real`]) THEN ASM_REWRITE_TAC[EVENTUALLY_WITHIN] THEN ANTS_TAC THENL [ASM_MESON_TAC[CONVEX_CONTAINS_SEGMENT; SUBSET]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `u:real^M` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `v:real^M` THEN DISCH_TAC THEN ASM_CASES_TAC `v IN segment[x:real^M,z]` THEN ASM_SIMP_TAC[] THEN FIRST_X_ASSUM(MP_TAC o C MATCH_MP (MESON[ENDS_IN_SEGMENT] `z IN segment[x:real^M,z]`)) THEN MATCH_MP_TAC(NORM_ARITH `norm(v - z:real^N) <= d - c ==> norm(z - x) <= c ==> norm(v - x) <= d`) THEN SUBGOAL_THEN `v IN segment[x:real^M,y] /\ ~(u IN segment[x:real^M,z])` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[BETWEEN_TRANS; BETWEEN_IN_SEGMENT; BETWEEN_SYM]; ALL_TAC] THEN SUBGOAL_THEN `u IN segment[z:real^M,y] /\ v IN segment[z,y]` STRIP_ASSUME_TAC THENL [MP_TAC(ISPECL [`x:real^M`; `z:real^M`; `y:real^M`] UNION_SEGMENT) THEN ASM SET_TAC[]; ALL_TAC] THEN TRANS_TAC REAL_LE_TRANS `c * norm(v - z:real^M)` THEN CONJ_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[GSYM DIST_NZ] THEN CONJ_TAC THENL [ASM_MESON_TAC[CONVEX_CONTAINS_SEGMENT; SUBSET]; ALL_TAC] THEN CONJ_TAC THENL [ASM_MESON_TAC[ENDS_IN_SEGMENT]; ALL_TAC] THEN SUBGOAL_THEN `v IN segment[z:real^M,u]` ASSUME_TAC THENL [ALL_TAC; ASM_MESON_TAC[DIST_IN_CLOSED_SEGMENT; REAL_LET_TRANS; DIST_SYM]]; MATCH_MP_TAC REAL_EQ_IMP_LE THEN MATCH_MP_TAC(REAL_RING `z = x + y:real ==> c * x = c * z - c * y`)] THEN RULE_ASSUM_TAC(REWRITE_RULE[GSYM BETWEEN_IN_SEGMENT; between]) THEN REWRITE_TAC[GSYM BETWEEN_IN_SEGMENT; GSYM dist; between] THEN REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[DIST_SYM] THEN REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `{z | z IN segment [x,y] /\ !t. t IN segment[x,z] ==> norm((f:real^M->real^N) t - f x) <= c * norm (t - x)} = {z | z IN segment [x,y] /\ !t. t IN segment(x,z) ==> norm(f t - f x) <= c * norm(t - x)}` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `z:real^M` THEN ASM_CASES_TAC `z IN segment[x:real^M,y]` THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN ASM_CASES_TAC `z:real^M = x` THEN ASM_REWRITE_TAC[SEGMENT_REFL; FORALL_IN_INSERT; NOT_IN_EMPTY] THEN REWRITE_TAC[VECTOR_SUB_REFL; NORM_0; REAL_MUL_RZERO; REAL_LE_REFL] THEN MP_TAC(ISPECL [`\t. lift(c * norm(t - x) - norm((f:real^M->real^N) t - f x))`; `segment(x:real^M,z)`; `{t | &0 <= drop t}`] FORALL_IN_CLOSURE_EQ) THEN ASM_REWRITE_TAC[CLOSURE_SEGMENT; IN_ELIM_THM; LIFT_DROP; REAL_SUB_LE] THEN DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[CLOSED_SING; drop; REWRITE_RULE[real_ge] CLOSED_HALFSPACE_COMPONENT_GE] THEN REWRITE_TAC[LIFT_SUB; LIFT_CMUL] THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN CONJ_TAC THEN TRY(MATCH_MP_TAC CONTINUOUS_ON_CMUL) THEN MATCH_MP_TAC CONTINUOUS_ON_LIFT_NORM_COMPOSE THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN REWRITE_TAC[CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN X_GEN_TAC `w:real^M` THEN DISCH_TAC THEN MATCH_MP_TAC CONTINUOUS_WITHIN_SUBSET THEN EXISTS_TAC `s:real^M->bool` THEN MATCH_MP_TAC(TAUT `q /\ (q ==> p) ==> p /\ q`) THEN CONJ_TAC THENL [TRANS_TAC SUBSET_TRANS `segment[x:real^M,y]` THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[CONVEX_CONTAINS_SEGMENT]] THEN ASM_REWRITE_TAC[SUBSET_SEGMENT; ENDS_IN_SEGMENT]; DISCH_TAC] THEN REWRITE_TAC[continuous_within] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`w:real^M`; `abs b + &1`]) THEN REWRITE_TAC[ARITH_RULE `b < abs b + &1`] THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[EVENTUALLY_WITHIN]] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `min d (e / (abs b + &1))` THEN ASM_SIMP_TAC[REAL_LT_RDIV_EQ; REAL_MUL_LZERO; dist; REAL_LT_MIN; REAL_ARITH `&0 < abs b + &1`] THEN X_GEN_TAC `v:real^M` THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `v:real^M`) THEN ASM_CASES_TAC `v:real^M = w` THEN ASM_REWRITE_TAC[VECTOR_SUB_REFL; NORM_0; GSYM DIST_NZ] THEN ASM_REWRITE_TAC[dist] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[lemma] THEN MATCH_MP_TAC CLOSED_IN_INTER THEN REWRITE_TAC[CLOSED_IN_REFL] THEN MATCH_MP_TAC CLOSED_IN_INTERS THEN REWRITE_TAC[SIMPLE_IMAGE; IMAGE_EQ_EMPTY; SEGMENT_EQ_EMPTY] THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_UNIV; UNIV_NOT_EMPTY; SET_RULE `{x | x IN s /\ (P x ==> Q x)} = {x | x IN s /\ ~P x} UNION {x | x IN s /\ Q x}`] THEN X_GEN_TAC `z:real^M` THEN MATCH_MP_TAC CLOSED_IN_UNION THEN REWRITE_TAC[SET_RULE `{x | x IN s /\ P} = if P then s else {}`] THEN CONJ_TAC THENL [ALL_TAC; MESON_TAC[CLOSED_IN_REFL; CLOSED_IN_EMPTY]] THEN ASM_CASES_TAC `z IN segment[x:real^M,y]` THENL [SUBGOAL_THEN `{w:real^M | w IN segment[x,y] /\ ~(z IN segment (x,w))} = {w | w IN segment[x,y] /\ (z = x \/ z IN segment[w,y])}` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `w:real^M` THEN MP_TAC(ISPECL [`x:real^M`; `w:real^M`; `y:real^M`] UNION_SEGMENT) THEN ASM_CASES_TAC `w IN segment[x:real^M,y]` THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `z:real^M` o REWRITE_RULE[EXTENSION]) THEN ASM_REWRITE_TAC[IN_UNION] THEN MP_TAC(ISPECL [`x:real^M`; `w:real^M`] SEGMENT_CLOSED_OPEN) THEN DISCH_THEN(MP_TAC o SPEC `z:real^M` o REWRITE_RULE[EXTENSION]) THEN ASM_REWRITE_TAC[IN_UNION; IN_INSERT; NOT_IN_EMPTY] THEN MAP_EVERY ASM_CASES_TAC [`z:real^M = x`; `z:real^M = w`] THEN ASM_REWRITE_TAC[ENDS_NOT_IN_SEGMENT; ENDS_IN_SEGMENT] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN MATCH_MP_TAC(TAUT `~(p /\ q) ==> p \/ q ==> (~p <=> q)`) THEN RULE_ASSUM_TAC(REWRITE_RULE[GSYM BETWEEN_IN_SEGMENT; between]) THEN REWRITE_TAC[GSYM BETWEEN_IN_SEGMENT; GSYM dist; between] THEN REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[DIST_SYM] THEN REWRITE_TAC[GSYM DIST_EQ_0; DIST_SYM] THEN REAL_ARITH_TAC; ASM_CASES_TAC `z:real^M = x` THEN ASM_REWRITE_TAC[CLOSED_IN_REFL; SET_RULE `{x | x IN s} = s`] THEN REWRITE_TAC[GSYM BETWEEN_IN_SEGMENT; between] THEN ONCE_REWRITE_TAC[REAL_ARITH `a:real = b /\ c = d <=> a = b /\ d - c = &0`] THEN REWRITE_TAC[GSYM between; BETWEEN_IN_SEGMENT] THEN REWRITE_TAC[GSYM LIFT_EQ; LIFT_NUM] THEN REWRITE_TAC[GSYM IN_SING] THEN MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE THEN REWRITE_TAC[CLOSED_SING; LIFT_SUB; LIFT_ADD] THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN CONJ_TAC THEN TRY(MATCH_MP_TAC CONTINUOUS_ON_ADD) THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN REWRITE_TAC[REWRITE_RULE[o_DEF] CONTINUOUS_ON_LIFT_DIST]]; MATCH_MP_TAC(MESON[CLOSED_IN_REFL] `s = t ==> closed_in (subtopology euclidean t) s`) THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY] THEN X_GEN_TAC `w:real^M` THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[SUBSET] SEGMENT_OPEN_SUBSET_CLOSED)) THEN ASM_MESON_TAC[BETWEEN_TRANS; BETWEEN_IN_SEGMENT; BETWEEN_SYM]]);; let LOCALLY_LIPSCHITZ = prove (`!f:real^M->real^N s b. convex s /\ (!x. x IN s ==> eventually (\y. norm(f y - f x) <= b * norm(y - x)) (at x within s)) ==> !x y. x IN s /\ y IN s ==> norm(f x - f y) <= b * norm(x - y)`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC LOCALLY_LIPSCHITZ_GEN THEN ASM_REWRITE_TAC[] THEN MAP_EVERY X_GEN_TAC [`x:real^M`; `c:real`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN X_GEN_TAC `y:real^M` THEN REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN MATCH_MP_TAC REAL_LE_RMUL THEN ASM_SIMP_TAC[NORM_POS_LE; REAL_LT_IMP_LE]);; (* ------------------------------------------------------------------------- *) (* Some uncountability results for relevant sets. *) (* ------------------------------------------------------------------------- *) let CARD_EQ_SEGMENT = prove (`(!a b:real^N. ~(a = b) ==> segment[a,b] =_c (:real)) /\ (!a b:real^N. ~(a = b) ==> segment(a,b) =_c (:real))`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[SEGMENT_IMAGE_INTERVAL] THENL [TRANS_TAC CARD_EQ_TRANS `interval[vec 0:real^1,vec 1]`; TRANS_TAC CARD_EQ_TRANS `interval(vec 0:real^1,vec 1)`] THEN SIMP_TAC[CARD_EQ_INTERVAL; UNIT_INTERVAL_NONEMPTY] THEN MATCH_MP_TAC CARD_EQ_IMAGE THEN ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ; VECTOR_ARITH `(&1 - x) % a + x % b:real^N = (&1 - y) % a + y % b <=> (x - y) % (a - b) = vec 0`] THEN SIMP_TAC[REAL_SUB_0; DROP_EQ]);; let UNCOUNTABLE_SEGMENT = prove (`(!a b:real^N. ~(a = b) ==> ~COUNTABLE(segment[a,b])) /\ (!a b:real^N. ~(a = b) ==> ~COUNTABLE(segment(a,b)))`, SIMP_TAC[CARD_EQ_REAL_IMP_UNCOUNTABLE; CARD_EQ_SEGMENT]);; let CARD_EQ_PATH_CONNECTED = prove (`!s a b:real^N. path_connected s /\ a IN s /\ b IN s /\ ~(a = b) ==> s =_c (:real)`, MESON_TAC[CARD_EQ_CONNECTED; PATH_CONNECTED_IMP_CONNECTED]);; let UNCOUNTABLE_PATH_CONNECTED = prove (`!s a b:real^N. path_connected s /\ a IN s /\ b IN s /\ ~(a = b) ==> ~COUNTABLE s`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC CARD_EQ_REAL_IMP_UNCOUNTABLE THEN MATCH_MP_TAC CARD_EQ_PATH_CONNECTED THEN ASM_MESON_TAC[]);; let CARD_EQ_CONVEX = prove (`!s a b:real^N. convex s /\ a IN s /\ b IN s /\ ~(a = b) ==> s =_c (:real)`, MESON_TAC[CARD_EQ_PATH_CONNECTED; CONVEX_IMP_PATH_CONNECTED]);; let UNCOUNTABLE_CONVEX = prove (`!s a b:real^N. convex s /\ a IN s /\ b IN s /\ ~(a = b) ==> ~COUNTABLE s`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC CARD_EQ_REAL_IMP_UNCOUNTABLE THEN MATCH_MP_TAC CARD_EQ_CONVEX THEN ASM_MESON_TAC[]);; let CARD_EQ_NONEMPTY_INTERIOR = prove (`!s:real^N->bool. ~(interior s = {}) ==> s =_c (:real)`, REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM CARD_LE_ANTISYM] THEN CONJ_TAC THENL [TRANS_TAC CARD_LE_TRANS `(:real^N)` THEN SIMP_TAC[CARD_LE_UNIV; CARD_EQ_IMP_LE; CARD_EQ_EUCLIDEAN]; TRANS_TAC CARD_LE_TRANS `interior(s:real^N->bool)` THEN SIMP_TAC[CARD_LE_SUBSET; INTERIOR_SUBSET] THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[CARD_EQ_SYM] CARD_EQ_IMP_LE) THEN MATCH_MP_TAC CARD_EQ_OPEN THEN ASM_REWRITE_TAC[OPEN_INTERIOR]]);; let UNCOUNTABLE_NONEMPTY_INTERIOR = prove (`!s:real^N->bool. ~(interior s = {}) ==> ~(COUNTABLE s)`, SIMP_TAC[CARD_EQ_NONEMPTY_INTERIOR; CARD_EQ_REAL_IMP_UNCOUNTABLE]);; let COUNTABLE_EMPTY_INTERIOR = prove (`!s:real^N->bool. COUNTABLE s ==> interior s = {}`, MESON_TAC[UNCOUNTABLE_NONEMPTY_INTERIOR]);; let [CONNECTED_FINITE_IFF_SING; CONNECTED_FINITE_IFF_COUNTABLE; CONNECTED_INFINITE_IFF_CARD_EQ] = (CONJUNCTS o prove) (`(!s:real^N->bool. connected s ==> (FINITE s <=> s = {} \/ ?a. s = {a})) /\ (!s:real^N->bool. connected s ==> (FINITE s <=> COUNTABLE s)) /\ (!s:real^N->bool. connected s ==> (INFINITE s <=> s =_c (:real)))`, REWRITE_TAC[AND_FORALL_THM] THEN GEN_TAC THEN ASM_CASES_TAC `connected(s:real^N->bool)` THEN ASM_REWRITE_TAC[INFINITE] THEN MATCH_MP_TAC(TAUT `(f ==> c) /\ (r ==> ~c) /\ (s ==> f) /\ (~s ==> r) ==> (f <=> s) /\ (f <=> c) /\ (~f <=> r)`) THEN REWRITE_TAC[FINITE_IMP_COUNTABLE] THEN REPEAT CONJ_TAC THEN STRIP_TAC THEN ASM_SIMP_TAC[CARD_EQ_REAL_IMP_UNCOUNTABLE; FINITE_INSERT; FINITE_EMPTY] THEN MATCH_MP_TAC CARD_EQ_CONNECTED THEN ASM SET_TAC[]);; let CONNECTED_FINITE_EQ_LOWDIM = prove (`!s:real^N->bool. connected s ==> (FINITE s <=> aff_dim s <= &0)`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[CONNECTED_FINITE_IFF_SING] THEN REWRITE_TAC[GSYM AFF_DIM_EQ_0; GSYM AFF_DIM_EQ_MINUS1] THEN MP_TAC(ISPEC `s:real^N->bool` AFF_DIM_GE) THEN INT_ARITH_TAC);; let CLOSED_AS_FRONTIER_OF_SUBSET = prove (`!s:real^N->bool. closed s <=> ?t. t SUBSET s /\ s = frontier t`, GEN_TAC THEN EQ_TAC THENL [ALL_TAC; MESON_TAC[FRONTIER_CLOSED]] THEN DISCH_TAC THEN MP_TAC(ISPEC `s:real^N->bool` SEPARABLE) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real^N->bool` THEN SIMP_TAC[frontier] THEN STRIP_TAC THEN MATCH_MP_TAC(SET_RULE `s SUBSET c /\ c SUBSET s /\ i = {} ==> s = c DIFF i`) THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[SUBSET_CLOSURE; CLOSURE_CLOSED]; ASM_MESON_TAC[UNCOUNTABLE_NONEMPTY_INTERIOR]]);; let CLOSED_AS_FRONTIER = prove (`!s:real^N->bool. closed s <=> ?t. s = frontier t`, GEN_TAC THEN EQ_TAC THENL [MESON_TAC[CLOSED_AS_FRONTIER_OF_SUBSET]; MESON_TAC[FRONTIER_CLOSED]]);; let CARD_EQ_PERFECT_SET = prove (`!s:real^N->bool. closed s /\ (!x. x IN s ==> x limit_point_of s) /\ ~(s = {}) ==> s =_c (:real)`, REWRITE_TAC[GSYM CARD_LE_ANTISYM] THEN REPEAT STRIP_TAC THENL [TRANS_TAC CARD_LE_TRANS `(:real^N)` THEN SIMP_TAC[CARD_LE_UNIV; CARD_EQ_IMP_LE; CARD_EQ_EUCLIDEAN]; MATCH_MP_TAC CARD_GE_PERFECT_SET THEN EXISTS_TAC `euclidean:(real^N)topology` THEN ASM_REWRITE_TAC[COMPLETELY_METRIZABLE_SPACE_EUCLIDEAN] THEN REWRITE_TAC[EUCLIDEAN_DERIVED_SET_OF_IFF_LIMIT_POINT_OF] THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN ASM_MESON_TAC[CLOSED_LIMPT]]);; let CARD_EQ_CLOSED = prove (`!s:real^N->bool. closed s ==> s <=_c (:num) \/ s =_c (:real)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP CANTOR_BENDIXSON) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`c:real^N->bool`; `d:real^N->bool`] THEN ASM_CASES_TAC `c:real^N->bool = {}` THEN ASM_SIMP_TAC[UNION_EMPTY; GSYM ge_c; GSYM COUNTABLE] THEN STRIP_TAC THEN DISJ2_TAC THEN TRANS_TAC CARD_EQ_TRANS `c:real^N->bool` THEN ASM_SIMP_TAC[CARD_EQ_PERFECT_SET] THEN MATCH_MP_TAC CARD_UNION_ABSORB_RIGHT THEN MATCH_MP_TAC(TAUT `(p ==> q) /\ p ==> p /\ q`) THEN ASM_SIMP_TAC[CARD_LE_COUNTABLE_INFINITE] THEN REWRITE_TAC[INFINITE_CARD_LE] THEN TRANS_TAC CARD_LE_TRANS `(:real)` THEN SIMP_TAC[CARD_LT_NUM_REAL; CARD_LT_IMP_LE] THEN MATCH_MP_TAC CARD_EQ_IMP_LE THEN ONCE_REWRITE_TAC[CARD_EQ_SYM] THEN ASM_SIMP_TAC[CARD_EQ_PERFECT_SET]);; let CONDENSATION_POINTS_EQ_EMPTY,CARD_EQ_CONDENSATION_POINTS = (CONJ_PAIR o prove) (`(!s:real^N->bool. {x | x condensation_point_of s} = {} <=> COUNTABLE s) /\ (!s:real^N->bool. {x | x condensation_point_of s} =_c (:real) <=> ~(COUNTABLE s))`, REWRITE_TAC[AND_FORALL_THM] THEN GEN_TAC THEN MATCH_MP_TAC(TAUT `(r ==> p) /\ (~r ==> q) /\ (p ==> ~q) ==> (p <=> r) /\ (q <=> ~r)`) THEN REPEAT CONJ_TAC THENL [DISCH_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY] THEN REWRITE_TAC[condensation_point_of] THEN ASM_MESON_TAC[COUNTABLE_SUBSET; INTER_SUBSET; IN_UNIV; OPEN_UNIV]; DISCH_TAC THEN MATCH_MP_TAC(REWRITE_RULE [TAUT `p ==> q \/ r <=> p /\ ~q ==> r`] CARD_EQ_CLOSED) THEN REWRITE_TAC[CLOSED_CONDENSATION_POINTS; GSYM COUNTABLE_ALT] THEN FIRST_ASSUM(MP_TAC o MATCH_MP CARD_EQ_CONDENSATION_POINTS_IN_SET) THEN DISCH_THEN(MP_TAC o MATCH_MP CARD_COUNTABLE_CONG) THEN ASM_REWRITE_TAC[CONTRAPOS_THM] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] COUNTABLE_SUBSET) THEN SET_TAC[]; DISCH_THEN SUBST1_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP CARD_FINITE_CONG) THEN REWRITE_TAC[FINITE_EMPTY; GSYM INFINITE; real_INFINITE]]);; let UNCOUNTABLE_HAS_CONDENSATION_POINT = prove (`!s:real^N->bool. ~COUNTABLE s ==> ?x. x condensation_point_of s`, REWRITE_TAC[GSYM CONDENSATION_POINTS_EQ_EMPTY] THEN SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* Density of sets with small complement, including irrationals. *) (* ------------------------------------------------------------------------- *) let COSMALL_APPROXIMATION = prove (`!s. ((:real) DIFF s) <_c (:real) ==> !x e. &0 < e ==> ?y. y IN s /\ abs(y - x) < e`, let lemma = prove (`!s. ((:real^1) DIFF s) <_c (:real) ==> !x e. &0 < e ==> ?y. y IN s /\ norm(y - x) < e`, REPEAT STRIP_TAC THEN MATCH_MP_TAC(SET_RULE `~({x | P x} SUBSET UNIV DIFF s) ==> ?x. x IN s /\ P x`) THEN MP_TAC(ISPEC `ball(x:real^1,e)` CARD_EQ_OPEN) THEN ASM_REWRITE_TAC[OPEN_BALL; BALL_EQ_EMPTY; REAL_NOT_LE] THEN DISCH_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP CARD_LE_SUBSET) THEN REWRITE_TAC[CARD_NOT_LE] THEN REWRITE_TAC[GSYM(ONCE_REWRITE_RULE[DIST_SYM] dist); GSYM ball] THEN TRANS_TAC CARD_LTE_TRANS `(:real)` THEN ASM_SIMP_TAC[ONCE_REWRITE_RULE[CARD_EQ_SYM] CARD_EQ_IMP_LE]) in REWRITE_TAC[FORALL_DROP_IMAGE; FORALL_DROP; EXISTS_DROP] THEN REWRITE_TAC[GSYM IMAGE_DROP_UNIV; GSYM DROP_SUB; GSYM NORM_1] THEN REWRITE_TAC[DROP_IN_IMAGE_DROP] THEN REWRITE_TAC[GSYM FORALL_DROP] THEN SIMP_TAC[GSYM IMAGE_DIFF_INJ; DROP_EQ] THEN GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC lemma THEN POP_ASSUM MP_TAC THEN MATCH_MP_TAC EQ_IMP THEN MATCH_MP_TAC CARD_LT_CONG THEN REWRITE_TAC[IMAGE_DROP_UNIV; CARD_EQ_REFL] THEN MATCH_MP_TAC CARD_EQ_IMAGE THEN SIMP_TAC[DROP_EQ]);; let COCOUNTABLE_APPROXIMATION = prove (`!s. COUNTABLE((:real) DIFF s) ==> !x e. &0 < e ==> ?y. y IN s /\ abs(y - x) < e`, GEN_TAC THEN REWRITE_TAC[COUNTABLE; ge_c] THEN DISCH_TAC THEN MATCH_MP_TAC COSMALL_APPROXIMATION THEN TRANS_TAC CARD_LET_TRANS `(:num)` THEN ASM_REWRITE_TAC[] THEN TRANS_TAC CARD_LTE_TRANS `(:num->bool)` THEN SIMP_TAC[CANTOR_THM_UNIV] THEN MATCH_MP_TAC CARD_EQ_IMP_LE THEN ONCE_REWRITE_TAC[CARD_EQ_SYM] THEN REWRITE_TAC[CARD_EQ_REAL]);; let OPEN_SET_COSMALL_COORDINATES = prove (`!P. (!i. 1 <= i /\ i <= dimindex(:N) ==> (:real) DIFF {x | P i x} <_c (:real)) ==> !s:real^N->bool. open s /\ ~(s = {}) ==> ?x. x IN s /\ !i. 1 <= i /\ i <= dimindex(:N) ==> P i (x$i)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_CBALL]) THEN DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `!i. 1 <= i /\ i <= dimindex(:N) ==> ?y:real. P i y /\ abs(y - (a:real^N)$i) < d / &(dimindex(:N))` MP_TAC THENL [X_GEN_TAC `i:num` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o MATCH_MP COSMALL_APPROXIMATION) THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; LE_1; DIMINDEX_GE_1]; REWRITE_TAC[LAMBDA_SKOLEM] THEN MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN REWRITE_TAC[IN_CBALL; dist] THEN W(MP_TAC o PART_MATCH lhand NORM_LE_L1 o lhand o snd) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN MATCH_MP_TAC SUM_BOUND_GEN THEN REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; NUMSEG_EMPTY; NOT_LT; DIMINDEX_GE_1] THEN ONCE_REWRITE_TAC[REAL_ABS_SUB] THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; CARD_NUMSEG_1]]);; let OPEN_SET_COCOUNTABLE_COORDINATES = prove (`!P. (!i. 1 <= i /\ i <= dimindex(:N) ==> COUNTABLE((:real) DIFF {x | P i x})) ==> !s:real^N->bool. open s /\ ~(s = {}) ==> ?x. x IN s /\ !i. 1 <= i /\ i <= dimindex(:N) ==> P i (x$i)`, GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC OPEN_SET_COSMALL_COORDINATES THEN REPEAT STRIP_TAC THEN TRANS_TAC CARD_LET_TRANS `(:num)` THEN ASM_SIMP_TAC[GSYM COUNTABLE_ALT] THEN TRANS_TAC CARD_LTE_TRANS `(:num->bool)` THEN SIMP_TAC[CANTOR_THM_UNIV] THEN MATCH_MP_TAC CARD_EQ_IMP_LE THEN ONCE_REWRITE_TAC[CARD_EQ_SYM] THEN REWRITE_TAC[CARD_EQ_REAL]);; let OPEN_SET_IRRATIONAL_COORDINATES = prove (`!s:real^N->bool. open s /\ ~(s = {}) ==> ?x. x IN s /\ !i. 1 <= i /\ i <= dimindex(:N) ==> ~rational(x$i)`, MATCH_MP_TAC OPEN_SET_COCOUNTABLE_COORDINATES THEN REWRITE_TAC[SET_RULE `UNIV DIFF {x | ~P x} = P`; COUNTABLE_RATIONAL]);; let CLOSURE_COSMALL_COORDINATES = prove (`!P. (!i. 1 <= i /\ i <= dimindex(:N) ==> (:real) DIFF {x | P i x} <_c (:real)) ==> closure {x | !i. 1 <= i /\ i <= dimindex (:N) ==> P i (x$i)} = (:real^N)`, GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[CLOSURE_APPROACHABLE; IN_UNIV; EXTENSION; IN_ELIM_THM] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `e:real`] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP OPEN_SET_COSMALL_COORDINATES) THEN DISCH_THEN(MP_TAC o SPEC `ball(x:real^N,e)`) THEN ASM_REWRITE_TAC[OPEN_BALL; BALL_EQ_EMPTY; REAL_NOT_LE; IN_BALL] THEN MESON_TAC[DIST_SYM]);; let CLOSURE_COCOUNTABLE_COORDINATES = prove (`!P. (!i. 1 <= i /\ i <= dimindex(:N) ==> COUNTABLE((:real) DIFF {x | P i x})) ==> closure {x | !i. 1 <= i /\ i <= dimindex (:N) ==> P i (x$i)} = (:real^N)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CLOSURE_COSMALL_COORDINATES THEN REPEAT STRIP_TAC THEN TRANS_TAC CARD_LET_TRANS `(:num)` THEN ASM_SIMP_TAC[GSYM COUNTABLE_ALT] THEN TRANS_TAC CARD_LTE_TRANS `(:num->bool)` THEN SIMP_TAC[CANTOR_THM_UNIV] THEN MATCH_MP_TAC CARD_EQ_IMP_LE THEN ONCE_REWRITE_TAC[CARD_EQ_SYM] THEN REWRITE_TAC[CARD_EQ_REAL]);; let CLOSURE_IRRATIONAL_COORDINATES = prove (`closure {x | !i. 1 <= i /\ i <= dimindex (:N) ==> ~rational(x$i)} = (:real^N)`, MATCH_MP_TAC CLOSURE_COCOUNTABLE_COORDINATES THEN REWRITE_TAC[SET_RULE `UNIV DIFF {x | ~P x} = P`; COUNTABLE_RATIONAL]);; (* ------------------------------------------------------------------------- *) (* Every path between distinct points contains an arc, and hence *) (* that path connection is equivalent to arcwise connection, for distinct *) (* points. The proof is based on Whyburn's "Topological Analysis". *) (* ------------------------------------------------------------------------- *) let HOMEOMORPHIC_MONOTONE_IMAGE_INTERVAL = prove (`!f:real^1->real^N. f continuous_on interval[vec 0,vec 1] /\ (!y. connected {x | x IN interval[vec 0,vec 1] /\ f x = y}) /\ ~(f(vec 1) = f(vec 0)) ==> (IMAGE f (interval[vec 0,vec 1])) homeomorphic (interval[vec 0:real^1,vec 1])`, let closure_dyadic_rationals_in_convex_set_pos_1 = prove (`!s. convex s /\ ~(interior s = {}) /\ (!x. x IN s ==> &0 <= drop x) ==> closure(s INTER { lift(&m / &2 pow n) | m IN (:num) /\ n IN (:num)}) = closure s`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `s:real^1->bool` CLOSURE_DYADIC_RATIONALS_IN_CONVEX_SET) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN AP_TERM_TAC THEN MATCH_MP_TAC(SET_RULE `(!x. x IN t ==> x IN u) /\ (!x. x IN u ==> x IN s ==> x IN t) ==> s INTER t = s INTER u`) THEN REWRITE_TAC[FORALL_IN_GSPEC; IN_UNIV; DIMINDEX_1; FORALL_1] THEN REWRITE_TAC[IN_ELIM_THM; EXISTS_LIFT; GSYM drop; LIFT_DROP] THEN REWRITE_TAC[REAL_ARITH `x / y:real = inv y * x`; LIFT_CMUL] THEN CONJ_TAC THENL [MESON_TAC[INTEGER_CLOSED]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`n:num`; `x:real^1`] THEN REPEAT DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `inv(&2 pow n) % x:real^1`) THEN ASM_SIMP_TAC[DROP_CMUL; REAL_LE_MUL_EQ; REAL_LT_POW2; REAL_LT_INV_EQ] THEN ASM_MESON_TAC[INTEGER_POS; LIFT_DROP]) in let function_on_dyadic_rationals = prove (`!f:num->num->A. (!m n. f (2 * m) (n + 1) = f m n) ==> ?g. !m n. g(&m / &2 pow n) = f m n`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN MP_TAC(ISPECL [`\(m,n). (f:num->num->A) m n`; `\(m,n). &m / &2 pow n`] FUNCTION_FACTORS_LEFT) THEN REWRITE_TAC[FORALL_PAIR_THM; FUN_EQ_THM; o_THM] THEN DISCH_THEN (SUBST1_TAC o SYM) THEN ONCE_REWRITE_TAC[MESON[] `(!a b c d. P a b c d) <=> (!b d a c. P a b c d)`] THEN MATCH_MP_TAC WLOG_LE THEN REPEAT CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN SIMP_TAC[REAL_FIELD `~(y = &0) /\ ~(y' = &0) ==> (x / y = x' / y' <=> y' / y * x = x')`; REAL_POW_EQ_0; REAL_OF_NUM_EQ; REAL_DIV_POW2; ARITH_EQ] THEN SIMP_TAC[LE_EXISTS; LEFT_IMP_EXISTS_THM] THEN SIMP_TAC[ADD_SUB2; REAL_OF_NUM_MUL; REAL_OF_NUM_EQ; REAL_OF_NUM_POW] THEN REWRITE_TAC[MESON[] `(!n n' d. n' = f d n ==> !m m'. g d m = m' ==> P m m' n d) <=> (!d m n. P m (g d m) n d)`] THEN INDUCT_TAC THEN SIMP_TAC[EXP; MULT_CLAUSES; ADD_CLAUSES] THEN REWRITE_TAC[GSYM MULT_ASSOC; ADD1] THEN ASM_MESON_TAC[]) in let recursion_on_dyadic_rationals = prove (`!b:num->A l r. ?f. (!m. f(&m) = b m) /\ (!m n. f(&(4 * m + 1) / &2 pow (n + 1)) = l(f(&(2 * m + 1) / &2 pow n))) /\ (!m n. f(&(4 * m + 3) / &2 pow (n + 1)) = r(f(&(2 * m + 1) / &2 pow n)))`, REPEAT GEN_TAC THEN SUBGOAL_THEN `?f:num->num->A. (!m n. f (2 * m) (n + 1) = f m n) /\ (!m. f m 0 = b m) /\ (!m n. f (4 * m + 1) (n + 1) = l(f (2 * m + 1) n)) /\ (!m n. f (4 * m + 3) (n + 1) = r(f (2 * m + 1) n))` MP_TAC THENL [MP_TAC(prove_recursive_functions_exist num_RECURSION `(!m. f m 0 = (b:num->A) m) /\ (!m n. f m (SUC n) = if EVEN m then f (m DIV 2) n else if EVEN(m DIV 2) then l(f ((m + 1) DIV 2) n) else r(f (m DIV 2) n))`) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:num->num->A` THEN STRIP_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[ADD1]) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[EVEN_MULT; ARITH_EVEN; ARITH_RULE `(2 * m) DIV 2 = m`] THEN REWRITE_TAC[ARITH_RULE `(4 * m + 1) DIV 2 = 2 * m`; ARITH_RULE `(4 * m + 3) DIV 2 = 2 * m + 1`; ARITH_RULE `((4 * m + 1) + 1) DIV 2 = 2 * m + 1`; ARITH_RULE `((4 * m + 3) + 1) DIV 2 = 2 * m + 2`] THEN REWRITE_TAC[EVEN_ADD; EVEN_MULT; EVEN; ARITH_EVEN; SND]; DISCH_THEN(X_CHOOSE_THEN `f:num->num->A` (CONJUNCTS_THEN2 MP_TAC ASSUME_TAC)) THEN DISCH_THEN(MP_TAC o MATCH_MP function_on_dyadic_rationals) THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN DISCH_THEN(fun th -> RULE_ASSUM_TAC(REWRITE_RULE[GSYM th])) THEN RULE_ASSUM_TAC(REWRITE_RULE[REAL_ARITH `x / &2 pow 0 = x`]) THEN ASM_REWRITE_TAC[]]) in let recursion_on_dyadic_rationals_1 = prove (`!b:A l r. ?f. (!m. f(&m / &2) = b) /\ (!m n. 0 < n ==> f(&(4 * m + 1) / &2 pow (n + 1)) = l(f(&(2 * m + 1) / &2 pow n))) /\ (!m n. 0 < n ==> f(&(4 * m + 3) / &2 pow (n + 1)) = r(f(&(2 * m + 1) / &2 pow n)))`, REPEAT GEN_TAC THEN MP_TAC(ISPECL [`(\n. b):num->A`; `l:A->A`; `r:A->A`] recursion_on_dyadic_rationals) THEN REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `f:real->A` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\x. (f:real->A)(&2 * x)` THEN ASM_REWRITE_TAC[REAL_ARITH `&2 * x / &2 = x`] THEN CONJ_TAC THEN GEN_TAC THEN INDUCT_TAC THEN REWRITE_TAC[LT_REFL] THEN ASM_SIMP_TAC[ADD_CLAUSES; real_pow; REAL_POW_EQ_0; REAL_OF_NUM_EQ; ARITH_EQ; REAL_FIELD `~(y = &0) ==> &2 * x / (&2 * y) = x / y`]) in let exists_function_unpair = prove (`(?f:A->B#C. P f) <=> (?f1 f2. P(\x. (f1 x,f2 x)))`, EQ_TAC THENL [ALL_TAC; MESON_TAC[]] THEN STRIP_TAC THEN EXISTS_TAC `\x. FST((f:A->B#C) x)` THEN EXISTS_TAC `\x. SND((f:A->B#C) x)` THEN ASM_REWRITE_TAC[PAIR; ETA_AX]) in let dyadics_in_open_unit_interval = prove (`interval(vec 0,vec 1) INTER {lift(&m / &2 pow n) | m IN (:num) /\ n IN (:num)} = {lift(&m / &2 pow n) | 0 < m /\ m < 2 EXP n}`, MATCH_MP_TAC(SET_RULE `(!m n. (f m n) IN s <=> P m n) ==> s INTER {f m n | m IN UNIV /\ n IN UNIV} = {f m n | P m n}`) THEN REWRITE_TAC[IN_INTERVAL_1; LIFT_DROP; DROP_VEC] THEN SIMP_TAC[REAL_LT_RDIV_EQ; REAL_LT_LDIV_EQ; REAL_LT_POW2] THEN SIMP_TAC[REAL_MUL_LZERO; REAL_MUL_LID; REAL_OF_NUM_POW; REAL_OF_NUM_LT]) in REPEAT STRIP_TAC THEN SUBGOAL_THEN `!a b m. m IN interval[a,b] /\ interval[a,b] SUBSET interval[vec 0,vec 1] ==> ?c d. drop a <= drop c /\ drop c <= drop m /\ drop m <= drop d /\ drop d <= drop b /\ (!x. x IN interval[c,d] ==> f x = f m) /\ (!x. x IN interval[a,c] DELETE c ==> ~(f x = f m)) /\ (!x. x IN interval[d,b] DELETE d ==> ~(f x = f m)) /\ (!x y. x IN interval[a,c] DELETE c /\ y IN interval[d,b] DELETE d ==> ~((f:real^1->real^N) x = f y))` MP_TAC THENL [REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; SUBSET_INTERVAL_1] THEN REPEAT STRIP_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `?c d. {x | x IN interval[a,b] /\ (f:real^1->real^N) x = f m} = interval[c,d]` MP_TAC THENL [SUBGOAL_THEN `{x | x IN interval[a,b] /\ (f:real^1->real^N) x = f m} = interval[a,b] INTER {x | x IN interval[vec 0,vec 1] /\ (f:real^1->real^N) x = f m}` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; IN_INTER; IN_INTERVAL_1; IN_ELIM_THM; DROP_VEC] THEN GEN_TAC THEN EQ_TAC THEN SIMP_TAC[] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `?c d. {x | x IN interval[vec 0,vec 1] /\ (f:real^1->real^N) x = f m} = interval[c,d]` MP_TAC THENL [ASM_REWRITE_TAC[GSYM CONNECTED_COMPACT_INTERVAL_1] THEN ONCE_REWRITE_TAC[SET_RULE `{x | x IN s /\ P x} = s INTER {x | x IN s /\ P x}`] THEN MATCH_MP_TAC COMPACT_INTER_CLOSED THEN REWRITE_TAC[COMPACT_INTERVAL] THEN MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE_CONSTANT THEN ASM_REWRITE_TAC[CLOSED_INTERVAL]; STRIP_TAC THEN ASM_REWRITE_TAC[INTER_INTERVAL_1] THEN MESON_TAC[]]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^1` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real^1` THEN DISCH_TAC THEN SUBGOAL_THEN `m IN interval[c:real^1,d]` MP_TAC THENL [FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [GSYM th]) THEN REWRITE_TAC[IN_ELIM_THM; IN_INTERVAL_1; DROP_VEC] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[IN_INTERVAL_1; IN_DELETE] THEN STRIP_TAC] THEN SUBGOAL_THEN `{c:real^1,d} SUBSET interval[c,d]` MP_TAC THENL [ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; IN_INTERVAL_1] THEN ASM_REAL_ARITH_TAC; FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM th]) THEN REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; IN_ELIM_THM; IN_INTERVAL_1] THEN STRIP_TAC THEN ASM_REWRITE_TAC[]] THEN CONJ_TAC THENL [GEN_TAC THEN REWRITE_TAC[GSYM IN_INTERVAL_1] THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM th]) THEN SIMP_TAC[IN_ELIM_THM]; ALL_TAC] THEN GEN_REWRITE_TAC I [CONJ_ASSOC] THEN CONJ_TAC THENL [CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `{x | x IN s /\ f x = a} = t ==> (!x. P x ==> x IN s) /\ (!x. P x /\ Q x ==> ~(x IN t)) ==> !x. P x /\ Q x ==> ~(f x = a)`)) THEN REWRITE_TAC[IN_INTERVAL_1; GSYM DROP_EQ] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`x:real^1`; `y:real^1`] THEN REWRITE_TAC[GSYM DROP_EQ] THEN STRIP_TAC THEN SUBGOAL_THEN `{x:real^1,y} INTER interval[c,d] = {}` MP_TAC THENL [REWRITE_TAC[SET_RULE `{a,b} INTER s = {} <=> ~(a IN s) /\ ~(b IN s)`; IN_INTERVAL_1] THEN ASM_REAL_ARITH_TAC; FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o LAND_CONV o RAND_CONV) [GSYM th])] THEN REWRITE_TAC[SET_RULE `{a,b} INTER s = {} <=> ~(a IN s) /\ ~(b IN s)`] THEN REWRITE_TAC[IN_ELIM_THM; IN_INTERVAL_1] THEN ASM_CASES_TAC `(f:real^1->real^N) x = f m` THENL [ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_CASES_TAC `(f:real^1->real^N) y = f m` THENL [ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM IS_INTERVAL_CONNECTED_1] o SPEC `(f:real^1->real^N) y`) THEN ASM_REWRITE_TAC[IS_INTERVAL_1] THEN DISCH_THEN(MP_TAC o SPECL [`x:real^1`; `y:real^1`; `m:real^1`]) THEN ASM_REWRITE_TAC[IN_ELIM_THM; IN_INTERVAL_1; DROP_VEC] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[RIGHT_IMP_EXISTS_THM; SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`leftcut:real^1->real^1->real^1->real^1`; `rightcut:real^1->real^1->real^1->real^1`] THEN STRIP_TAC] THEN FIRST_ASSUM(MP_TAC o SPECL [`vec 0:real^1`; `vec 1:real^1`; `vec 0:real^1`]) THEN REWRITE_TAC[SUBSET_REFL; ENDS_IN_UNIT_INTERVAL] THEN ABBREV_TAC `u = (rightcut:real^1->real^1->real^1->real^1) (vec 0) (vec 1) (vec 0)` THEN REWRITE_TAC[CONJ_ASSOC; REAL_LE_ANTISYM; DROP_EQ] THEN REWRITE_TAC[GSYM CONJ_ASSOC] THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[INTERVAL_SING; SET_RULE `~(x IN ({a} DELETE a))`] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPECL [`u:real^1`; `vec 1:real^1`; `vec 1:real^1`]) THEN REWRITE_TAC[ENDS_IN_INTERVAL; SUBSET_INTERVAL_1; INTERVAL_NE_EMPTY_1] THEN ASM_REWRITE_TAC[REAL_LE_REFL] THEN ABBREV_TAC `v = (leftcut:real^1->real^1->real^1->real^1) u (vec 1) (vec 1)` THEN ONCE_REWRITE_TAC[TAUT `a /\ b /\ c /\ d /\ e <=> (c /\ d) /\ a /\ b /\ e`] THEN REWRITE_TAC[REAL_LE_ANTISYM; DROP_EQ] THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[INTERVAL_SING; SET_RULE `~(x IN ({a} DELETE a))`] THEN STRIP_TAC THEN SUBGOAL_THEN `!x. x IN interval[vec 0,v] DELETE v ==> ~((f:real^1->real^N) x = f(vec 1))` ASSUME_TAC THENL [X_GEN_TAC `t:real^1` THEN REWRITE_TAC[IN_DELETE; IN_INTERVAL_1; GSYM DROP_EQ] THEN STRIP_TAC THEN ASM_CASES_TAC `drop t < drop u` THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (MESON[] `~(f1 = f0) ==> ft = f0 ==> ~(ft = f1)`)); ALL_TAC] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL_1; IN_DELETE; GSYM DROP_EQ] THEN ASM_REAL_ARITH_TAC; UNDISCH_THEN `!x. x IN interval[u,v] DELETE v ==> ~((f:real^1->real^N) x = f (vec 1))` (K ALL_TAC)] THEN MP_TAC(ISPECL [`(u:real^1,v:real^1)`; `\(a,b). (a:real^1,leftcut a b (midpoint(a,b)):real^1)`; `\(a,b). (rightcut a b (midpoint(a,b)):real^1,b:real^1)`] recursion_on_dyadic_rationals_1) THEN REWRITE_TAC[exists_function_unpair; PAIR_EQ] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real->real^1`; `b:real->real^1`] THEN ABBREV_TAC `(c:real->real^1) x = midpoint(a x,b x)` THEN REWRITE_TAC[TAUT `a ==> b /\ c <=> (a ==> b) /\ (a ==> c)`] THEN REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC THEN SUBGOAL_THEN `!m n. drop u <= drop(a(&m / &2 pow n)) /\ drop(a(&m / &2 pow n)) <= drop(b(&m / &2 pow n)) /\ drop(b(&m / &2 pow n)) <= drop v` MP_TAC THENL [GEN_REWRITE_TAC I [SWAP_FORALL_THM] THEN MATCH_MP_TAC num_INDUCTION THEN CONJ_TAC THENL [REWRITE_TAC[REAL_ARITH `x / &2 pow 0 = (&2 * x) / &2`] THEN ASM_REWRITE_TAC[REAL_OF_NUM_MUL; REAL_LE_REFL]; X_GEN_TAC `n:num` THEN DISCH_THEN(LABEL_TAC "*")] THEN X_GEN_TAC `p:num` THEN DISJ_CASES_TAC(SPEC `p:num` EVEN_OR_ODD) THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [EVEN_EXISTS]) THEN DISCH_THEN(X_CHOOSE_THEN `m:num` SUBST1_TAC) THEN REWRITE_TAC[GSYM REAL_OF_NUM_MUL; real_pow] THEN ASM_SIMP_TAC[REAL_LT_POW2; REAL_FIELD `&0 < y ==> (&2 * x) / (&2 * y) = x / y`]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [ODD_EXISTS]) THEN DISCH_THEN(X_CHOOSE_THEN `m:num` SUBST1_TAC) THEN DISJ_CASES_TAC(ARITH_RULE `n = 0 \/ 0 < n`) THENL [ASM_REWRITE_TAC[real_pow; REAL_MUL_RID; REAL_LE_REFL]; REWRITE_TAC[ADD1]] THEN DISJ_CASES_TAC(SPEC `m:num` EVEN_OR_ODD) THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [EVEN_EXISTS]) THEN DISCH_THEN(X_CHOOSE_THEN `r:num` SUBST_ALL_TAC) THEN ASM_SIMP_TAC[ARITH_RULE `2 * 2 * r = 4 * r`]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [ODD_EXISTS]) THEN DISCH_THEN(X_CHOOSE_THEN `r:num` SUBST_ALL_TAC) THEN ASM_SIMP_TAC[ARITH_RULE `2 * SUC(2 * r) + 1 = 4 * r + 3`]] THEN (FIRST_X_ASSUM(MP_TAC o SPECL [`a(&(2 * r + 1) / &2 pow n):real^1`; `b(&(2 * r + 1) / &2 pow n):real^1`; `c(&(2 * r + 1) / &2 pow n):real^1`]) THEN ANTS_TAC THENL [FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM th]) THEN REWRITE_TAC[midpoint; IN_INTERVAL_1; SUBSET_INTERVAL_1] THEN REWRITE_TAC[DROP_CMUL; DROP_ADD] THEN UNDISCH_TAC `drop(vec 0) <= drop u` THEN UNDISCH_TAC `drop v <= drop (vec 1)`; ALL_TAC] THEN REMOVE_THEN "*" (MP_TAC o SPEC `2 * r + 1`) THEN REAL_ARITH_TAC); REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC] THEN SUBGOAL_THEN `!m n. drop(vec 0) <= drop(a(&m / &2 pow n))` ASSUME_TAC THENL [ASM_MESON_TAC[REAL_LE_TRANS]; ALL_TAC] THEN SUBGOAL_THEN `!m n. drop(b(&m / &2 pow n)) <= drop(vec 1)` ASSUME_TAC THENL [ASM_MESON_TAC[REAL_LE_TRANS]; ALL_TAC] THEN SUBGOAL_THEN `!m n. drop(a(&m / &2 pow n)) <= drop(c(&m / &2 pow n)) /\ drop(c(&m / &2 pow n)) <= drop(b(&m / &2 pow n))` MP_TAC THENL [UNDISCH_THEN `!x:real. midpoint(a x:real^1,b x) = c x` (fun th -> REWRITE_TAC[GSYM th]) THEN REWRITE_TAC[midpoint; IN_INTERVAL_1; SUBSET_INTERVAL_1] THEN ASM_REWRITE_TAC[DROP_CMUL; DROP_ADD; REAL_ARITH `a <= inv(&2) * (a + b) /\ inv(&2) * (a + b) <= b <=> a <= b`]; REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC] THEN SUBGOAL_THEN `!i m n j. ODD j /\ abs(&i / &2 pow m - &j / &2 pow n) < inv(&2 pow n) ==> drop(a(&j / &2 pow n)) <= drop(c(&i / &2 pow m)) /\ drop(c(&i / &2 pow m)) <= drop(b(&j / &2 pow n))` ASSUME_TAC THENL [REPLICATE_TAC 3 GEN_TAC THEN WF_INDUCT_TAC `m - n:num` THEN DISJ_CASES_TAC(ARITH_RULE `m <= n \/ n:num < m`) THENL [GEN_TAC THEN STRIP_TAC THEN MP_TAC(SPEC `abs(&2 pow n) * abs(&i / &2 pow m - &j / &2 pow n)` REAL_ABS_INTEGER_LEMMA) THEN MATCH_MP_TAC(TAUT `i /\ ~b /\ (n ==> p) ==> (i /\ ~n ==> b) ==> p`) THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[GSYM REAL_ABS_MUL; INTEGER_ABS] THEN REWRITE_TAC[REAL_ARITH `n * (x / m - y / n):real = x * (n / m) - y * (n / n)`] THEN ASM_SIMP_TAC[GSYM REAL_POW_SUB; LE_REFL; REAL_OF_NUM_EQ; ARITH_EQ] THEN MESON_TAC[INTEGER_CLOSED]; SIMP_TAC[REAL_ABS_MUL; REAL_ABS_ABS; REAL_ABS_POW; REAL_ABS_NUM] THEN REWRITE_TAC[REAL_ARITH `~(&1 <= x * y) <=> y * x < &1`] THEN SIMP_TAC[GSYM REAL_LT_RDIV_EQ; REAL_LT_POW2] THEN ASM_REWRITE_TAC[REAL_ARITH `&1 / x = inv x`]; ASM_SIMP_TAC[REAL_ABS_POW; REAL_ABS_NUM; REAL_ENTIRE; REAL_LT_IMP_NZ; REAL_LT_POW2; REAL_ARITH `abs(x - y) = &0 <=> x = y`]]; ALL_TAC] THEN X_GEN_TAC `k:num` THEN REWRITE_TAC[IMP_CONJ; ODD_EXISTS] THEN DISCH_THEN(X_CHOOSE_THEN `j:num` SUBST1_TAC) THEN DISJ_CASES_TAC(ARITH_RULE `n = 0 \/ 0 < n`) THENL [ASM_REWRITE_TAC[REAL_ARITH `x / &2 pow 0 = (&2 * x) / &2`] THEN ASM_REWRITE_TAC[REAL_OF_NUM_MUL] THEN ASM_MESON_TAC[REAL_LE_TRANS]; ALL_TAC] THEN UNDISCH_THEN `n:num < m` (fun th -> let th' = MATCH_MP (ARITH_RULE `n < m ==> m - SUC n < m - n`) th in FIRST_X_ASSUM(MP_TAC o C MATCH_MP th')) THEN REPEAT_TCL DISJ_CASES_THEN ASSUME_TAC (REAL_ARITH `&i / &2 pow m = &(2 * j + 1) / &2 pow n \/ &i / &2 pow m < &(2 * j + 1) / &2 pow n \/ &(2 * j + 1) / &2 pow n < &i / &2 pow m`) THENL [ASM_REWRITE_TAC[ADD1]; DISCH_THEN(MP_TAC o SPEC `4 * j + 1`) THEN REWRITE_TAC[ODD_ADD; ODD_MULT; ARITH] THEN ASM_SIMP_TAC[ADD1] THEN MATCH_MP_TAC MONO_IMP THEN CONJ_TAC THENL [MATCH_MP_TAC(REAL_ARITH `x < i /\ &2 * n1 = n /\ j + n1 = i ==> abs(x - i) < n ==> abs(x - j) < n1`) THEN ASM_REWRITE_TAC[REAL_ARITH `a / b + inv b = (a + &1) / b`] THEN REWRITE_TAC[real_div; REAL_POW_ADD; REAL_INV_MUL] THEN REWRITE_TAC[GSYM REAL_OF_NUM_ADD; GSYM REAL_OF_NUM_MUL] THEN REAL_ARITH_TAC; MATCH_MP_TAC(REAL_ARITH `b' <= b ==> a <= c /\ c <= b' ==> a <= c /\ c <= b`) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a(&(2 * j + 1) / &2 pow n):real^1`; `b(&(2 * j + 1) / &2 pow n):real^1`; `c(&(2 * j + 1) / &2 pow n):real^1`]) THEN ANTS_TAC THENL [ALL_TAC; REAL_ARITH_TAC] THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM th]) THEN REWRITE_TAC[midpoint; IN_INTERVAL_1; SUBSET_INTERVAL_1] THEN REWRITE_TAC[DROP_CMUL; DROP_ADD] THEN ASM_REWRITE_TAC[DROP_CMUL; DROP_ADD; REAL_ARITH `a <= inv(&2) * (a + b) /\ inv(&2) * (a + b) <= b <=> a <= b`]]; DISCH_THEN(MP_TAC o SPEC `4 * j + 3`) THEN REWRITE_TAC[ODD_ADD; ODD_MULT; ARITH] THEN ASM_SIMP_TAC[ADD1] THEN MATCH_MP_TAC MONO_IMP THEN CONJ_TAC THENL [MATCH_MP_TAC(REAL_ARITH `i < x /\ &2 * n1 = n /\ j - n1 = i ==> abs(x - i) < n ==> abs(x - j) < n1`) THEN ASM_REWRITE_TAC[REAL_ARITH `a / b - inv b = (a - &1) / b`] THEN REWRITE_TAC[real_div; REAL_POW_ADD; REAL_INV_MUL] THEN REWRITE_TAC[GSYM REAL_OF_NUM_ADD; GSYM REAL_OF_NUM_MUL] THEN REAL_ARITH_TAC; MATCH_MP_TAC(REAL_ARITH `a <= a' ==> a' <= c /\ c <= b ==> a <= c /\ c <= b`) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a(&(2 * j + 1) / &2 pow n):real^1`; `b(&(2 * j + 1) / &2 pow n):real^1`; `c(&(2 * j + 1) / &2 pow n):real^1`]) THEN ANTS_TAC THENL [ALL_TAC; REAL_ARITH_TAC] THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM th]) THEN REWRITE_TAC[midpoint; IN_INTERVAL_1; SUBSET_INTERVAL_1] THEN REWRITE_TAC[DROP_CMUL; DROP_ADD] THEN ASM_REWRITE_TAC[DROP_CMUL; DROP_ADD; REAL_ARITH `a <= inv(&2) * (a + b) /\ inv(&2) * (a + b) <= b <=> a <= b`]]]; ALL_TAC] THEN SUBGOAL_THEN `!m n. ODD m ==> abs(drop(a(&m / &2 pow n)) - drop(b(&m / &2 pow n))) <= &2 / &2 pow n` ASSUME_TAC THENL [ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN INDUCT_TAC THENL [ASM_REWRITE_TAC[REAL_ARITH `x / &2 pow 0 = (&2 * x) / &2`] THEN ASM_REWRITE_TAC[REAL_OF_NUM_MUL] THEN CONV_TAC NUM_REDUCE_CONV THEN RULE_ASSUM_TAC(REWRITE_RULE[DROP_VEC]) THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN X_GEN_TAC `m:num` THEN REWRITE_TAC[ODD_EXISTS] THEN DISCH_THEN(X_CHOOSE_THEN `k:num` SUBST1_TAC) THEN DISJ_CASES_TAC(ARITH_RULE `n = 0 \/ 0 < n`) THENL [ASM_REWRITE_TAC[ARITH; REAL_POW_1] THEN RULE_ASSUM_TAC(REWRITE_RULE[DROP_VEC]) THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN DISJ_CASES_TAC(SPEC `k:num` EVEN_OR_ODD) THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [EVEN_EXISTS]) THEN DISCH_THEN(X_CHOOSE_THEN `j:num` SUBST1_TAC) THEN REWRITE_TAC[ARITH_RULE `SUC(2 * 2 * j) = 4 * j + 1`] THEN ASM_SIMP_TAC[ADD1] THEN MATCH_MP_TAC(REAL_ARITH `drop c = (drop a + drop b) / &2 /\ abs(drop a - drop b) <= &2 * k /\ drop a <= drop(leftcut a b c) /\ drop(leftcut a b c) <= drop c ==> abs(drop a - drop(leftcut a b c)) <= k`); FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [ODD_EXISTS]) THEN DISCH_THEN(X_CHOOSE_THEN `j:num` SUBST1_TAC) THEN REWRITE_TAC[ARITH_RULE `SUC(2 * SUC(2 * j)) = 4 * j + 3`] THEN ASM_SIMP_TAC[ADD1] THEN MATCH_MP_TAC(REAL_ARITH `drop c = (drop a + drop b) / &2 /\ abs(drop a - drop b) <= &2 * k /\ drop c <= drop(rightcut a b c) /\ drop(rightcut a b c) <= drop b ==> abs(drop(rightcut a b c) - drop b) <= k`)] THEN (CONJ_TAC THENL [UNDISCH_THEN `!x:real. midpoint(a x:real^1,b x) = c x` (fun th -> REWRITE_TAC[GSYM th]) THEN REWRITE_TAC[midpoint; DROP_CMUL; DROP_ADD] THEN REAL_ARITH_TAC; ALL_TAC] THEN CONJ_TAC THENL [REWRITE_TAC[real_div; REAL_POW_ADD; REAL_INV_MUL] THEN REWRITE_TAC[REAL_ARITH `&2 * x * inv y * inv(&2 pow 1) = x / y`] THEN ASM_SIMP_TAC[GSYM real_div; ODD_ADD; ODD_MULT; ARITH]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a(&(2 * j + 1) / &2 pow n):real^1`; `b(&(2 * j + 1) / &2 pow n):real^1`; `c(&(2 * j + 1) / &2 pow n):real^1`]) THEN ANTS_TAC THENL [ALL_TAC; REAL_ARITH_TAC] THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM th]) THEN REWRITE_TAC[midpoint; IN_INTERVAL_1; SUBSET_INTERVAL_1] THEN REWRITE_TAC[DROP_CMUL; DROP_ADD] THEN ASM_REWRITE_TAC[DROP_CMUL; DROP_ADD; REAL_ARITH `a <= inv(&2) * (a + b) /\ inv(&2) * (a + b) <= b <=> a <= b`]); ALL_TAC] THEN SUBGOAL_THEN `!n j. 0 < 2 * j /\ 2 * j < 2 EXP n ==> (f:real^1->real^N)(b(&(2 * j - 1) / &2 pow n)) = f(a(&(2 * j + 1) / &2 pow n))` ASSUME_TAC THENL [MATCH_MP_TAC num_INDUCTION THEN CONJ_TAC THENL [REWRITE_TAC[ARITH_RULE `0 < 2 * j <=> 0 < j`; ARITH_RULE `2 * j < 2 <=> j < 1`] THEN ARITH_TAC; ALL_TAC] THEN X_GEN_TAC `n:num` THEN DISCH_THEN(LABEL_TAC "+") THEN DISJ_CASES_TAC(ARITH_RULE `n = 0 \/ 0 < n`) THENL [ASM_REWRITE_TAC[] THEN CONV_TAC NUM_REDUCE_CONV THEN REWRITE_TAC[ARITH_RULE `0 < 2 * j <=> 0 < j`; ARITH_RULE `2 * j < 2 <=> j < 1`] THEN ARITH_TAC; ALL_TAC] THEN X_GEN_TAC `k:num` THEN DISJ_CASES_TAC(SPEC `k:num` EVEN_OR_ODD) THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [EVEN_EXISTS]) THEN DISCH_THEN(X_CHOOSE_THEN `j:num` SUBST1_TAC) THEN REWRITE_TAC[EXP; ARITH_RULE `0 < 2 * j <=> 0 < j`; LT_MULT_LCANCEL] THEN CONV_TAC NUM_REDUCE_CONV THEN ASM_SIMP_TAC[ARITH_RULE `0 < j ==> 2 * 2 * j - 1 = 4 * (j - 1) + 3`; ADD1; ARITH_RULE `2 * 2 * j + 1 = 4 * j + 1`] THEN SIMP_TAC[ARITH_RULE `0 < j ==> 2 * (j - 1) + 1 = 2 * j - 1`] THEN STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [ODD_EXISTS]) THEN DISCH_THEN(X_CHOOSE_THEN `j:num` SUBST1_TAC) THEN STRIP_TAC THEN ASM_SIMP_TAC[ADD1; ARITH_RULE `2 * SUC(2 * j) - 1 = 4 * j + 1`; ARITH_RULE `2 * SUC(2 * j) + 1 = 4 * j + 3`] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a(&(2 * j + 1) / &2 pow n):real^1`; `b(&(2 * j + 1) / &2 pow n):real^1`; `c(&(2 * j + 1) / &2 pow n):real^1`]) THEN ANTS_TAC THENL [FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM th]) THEN REWRITE_TAC[midpoint; IN_INTERVAL_1; SUBSET_INTERVAL_1] THEN REWRITE_TAC[DROP_CMUL; DROP_ADD] THEN ASM_REWRITE_TAC[DROP_CMUL; DROP_ADD; REAL_ARITH `a <= inv(&2) * (a + b) /\ inv(&2) * (a + b) <= b <=> a <= b`]; REPLICATE_TAC 4 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN MATCH_MP_TAC(MESON[] `a IN s /\ b IN s ==> (!x. x IN s ==> f x = c) ==> f a = f b`) THEN REWRITE_TAC[ENDS_IN_INTERVAL; INTERVAL_NE_EMPTY_1] THEN ASM_MESON_TAC[REAL_LE_TRANS]]]; ALL_TAC] THEN SUBGOAL_THEN `!n j. 0 < j /\ j < 2 EXP n ==> (f:real^1->real^N)(b(&(2 * j - 1) / &2 pow (n + 1))) = f(c(&j / &2 pow n)) /\ f(a(&(2 * j + 1) / &2 pow (n + 1))) = f(c(&j / &2 pow n))` ASSUME_TAC THENL [MATCH_MP_TAC num_INDUCTION THEN REWRITE_TAC[ARITH_RULE `~(0 < j /\ j < 2 EXP 0)`] THEN X_GEN_TAC `n:num` THEN DISCH_THEN(LABEL_TAC "*") THEN X_GEN_TAC `j:num` THEN DISJ_CASES_TAC(SPEC `j:num` EVEN_OR_ODD) THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [EVEN_EXISTS]) THEN DISCH_THEN(X_CHOOSE_THEN `k:num` SUBST1_TAC) THEN REWRITE_TAC[ADD_CLAUSES; EXP; ARITH_RULE `0 < 2 * k <=> 0 < k`; ARITH_RULE `2 * x < 2 * y <=> x < y`] THEN STRIP_TAC THEN REMOVE_THEN "*" (MP_TAC o SPEC `k:num`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(MESON[] `c' = c /\ a' = a /\ b' = b ==> b = c /\ a = c ==> b' = c' /\ a' = c'`) THEN REPEAT CONJ_TAC THEN AP_TERM_TAC THENL [AP_TERM_TAC THEN REWRITE_TAC[real_pow; real_div; REAL_INV_MUL; GSYM REAL_OF_NUM_MUL] THEN REAL_ARITH_TAC; REWRITE_TAC[ADD1; ARITH_RULE `2 * 2 * n = 4 * n`] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ARITH_TAC; SUBGOAL_THEN `k = PRE k + 1` SUBST1_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[ARITH_RULE `2 * (k + 1) - 1 = 2 * k + 1`; ARITH_RULE `2 * 2 * (k + 1) - 1 = 4 * k + 3`] THEN REWRITE_TAC[ADD1] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ARITH_TAC]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [ODD_EXISTS]) THEN DISCH_THEN(X_CHOOSE_THEN `k:num` SUBST1_TAC) THEN REWRITE_TAC[EXP; ARITH_RULE `SUC(2 * k) < 2 * n <=> k < n`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a(&(2 * k + 1) / &2 pow (SUC n)):real^1`; `b(&(2 * k + 1) / &2 pow (SUC n)):real^1`; `c(&(2 * k + 1) / &2 pow (SUC n)):real^1`]) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[midpoint; IN_INTERVAL_1; SUBSET_INTERVAL_1]; REPLICATE_TAC 4 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC)] THEN REWRITE_TAC[ARITH_RULE `SUC(2 * k) = 2 * k + 1`] THEN DISCH_THEN(fun th -> CONJ_TAC THEN MATCH_MP_TAC th) THEN ASM_SIMP_TAC[ARITH_RULE `2 * (2 * k + 1) - 1 = 4 * k + 1`; ADD1; ARITH_RULE `2 * (2 * k + 1) + 1 = 4 * k + 3`; ARITH_RULE `0 < n + 1`] THEN ASM_REWRITE_TAC[IN_INTERVAL_1; GSYM ADD1] THEN ASM_SIMP_TAC[ARITH_RULE `SUC(2 * k) = 2 * k + 1`] THEN ASM_REAL_ARITH_TAC]; ALL_TAC] THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN MATCH_MP_TAC HOMEOMORPHIC_COMPACT THEN REWRITE_TAC[COMPACT_INTERVAL] THEN MP_TAC(ISPECL [`\x. (f:real^1->real^N)(c(drop x))`; `interval(vec 0,vec 1) INTER {lift(&m / &2 pow n) | m IN (:num) /\ n IN (:num)}`] UNIFORMLY_CONTINUOUS_EXTENDS_TO_CLOSURE) THEN SIMP_TAC[closure_dyadic_rationals_in_convex_set_pos_1; CONVEX_INTERVAL; INTERIOR_OPEN; OPEN_INTERVAL; UNIT_INTERVAL_NONEMPTY; IN_INTERVAL_1; REAL_LT_IMP_LE; DROP_VEC; CLOSURE_OPEN_INTERVAL] THEN REWRITE_TAC[dyadics_in_open_unit_interval] THEN ANTS_TAC THENL [REWRITE_TAC[uniformly_continuous_on; FORALL_IN_GSPEC] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN SUBGOAL_THEN `(f:real^1->real^N) uniformly_continuous_on interval[vec 0,vec 1]` MP_TAC THENL [ASM_SIMP_TAC[COMPACT_UNIFORMLY_CONTINUOUS; COMPACT_INTERVAL]; REWRITE_TAC[uniformly_continuous_on]] THEN DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN MP_TAC(SPECL [`inv(&2)`; `min (d:real) (&1 / &4)`] REAL_ARCH_POW_INV) THEN ASM_REWRITE_TAC[REAL_HALF; REAL_POW_INV; REAL_LT_MIN] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN DISCH_THEN(X_CHOOSE_THEN `n:num` MP_TAC) THEN ASM_CASES_TAC `n = 0` THEN ASM_REWRITE_TAC[] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN STRIP_TAC THEN EXISTS_TAC `inv(&2 pow n)` THEN REWRITE_TAC[REAL_LT_POW2; REAL_LT_INV_EQ] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN SUBGOAL_THEN `!i j m. 0 < i /\ i < 2 EXP m /\ 0 < j /\ j < 2 EXP n /\ abs(&i / &2 pow m - &j / &2 pow n) < inv(&2 pow n) ==> norm((f:real^1->real^N)(c(&i / &2 pow m)) - f(c(&j / &2 pow n))) < e / &2` ASSUME_TAC THENL [REPEAT GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(DISJ_CASES_THEN MP_TAC o MATCH_MP (REAL_ARITH `abs(x - a) < e ==> x = a \/ abs(x - (a - e / &2)) < e / &2 \/ abs(x - (a + e / &2)) < e / &2`)) THENL [DISCH_THEN SUBST1_TAC THEN ASM_REWRITE_TAC[VECTOR_SUB_REFL; NORM_0; REAL_HALF]; ALL_TAC] THEN SUBGOAL_THEN `&j / &2 pow n = &(2 * j) / &2 pow (n + 1)` (fun th -> GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [th]) THENL [REWRITE_TAC[REAL_POW_ADD; real_div; REAL_INV_MUL; GSYM REAL_OF_NUM_MUL] THEN REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[real_div; GSYM REAL_INV_MUL] THEN REWRITE_TAC[GSYM real_div; GSYM(ONCE_REWRITE_RULE[REAL_MUL_SYM] (CONJUNCT2 real_pow))] THEN REWRITE_TAC[ADD1; REAL_ARITH `x / n + inv n = (x + &1) / n`; REAL_ARITH `x / n - inv n = (x - &1) / n`] THEN ASM_SIMP_TAC[REAL_OF_NUM_SUB; ARITH_RULE `0 < j ==> 1 <= 2 * j`] THEN REWRITE_TAC[REAL_OF_NUM_ADD] THEN STRIP_TAC THENL [SUBGOAL_THEN `(f:real^1->real^N)(c(&j / &2 pow n)) = f(b (&(2 * j - 1) / &2 pow (n + 1)))` SUBST1_TAC THENL [ASM_SIMP_TAC[]; ALL_TAC]; SUBGOAL_THEN `(f:real^1->real^N)(c(&j / &2 pow n)) = f(a (&(2 * j + 1) / &2 pow (n + 1)))` SUBST1_TAC THENL [ASM_SIMP_TAC[]; ALL_TAC]] THEN REWRITE_TAC[GSYM dist] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[IN_INTERVAL_1] THEN REPEAT(CONJ_TAC THENL [ASM_MESON_TAC[REAL_LE_TRANS]; ALL_TAC]) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`i:num`; `m:num`; `n + 1`]) THENL [DISCH_THEN(MP_TAC o SPEC `2 * j - 1`) THEN REWRITE_TAC[ODD_SUB]; DISCH_THEN(MP_TAC o SPEC `2 * j + 1`) THEN REWRITE_TAC[ODD_ADD]] THEN ASM_REWRITE_TAC[ODD_MULT; ARITH; ARITH_RULE `1 < 2 * j <=> 0 < j`] THEN REWRITE_TAC[DIST_REAL; GSYM drop] THENL [MATCH_MP_TAC(NORM_ARITH `!t. abs(a - b) <= t /\ t < d ==> a <= c /\ c <= b ==> abs(c - b) < d`); MATCH_MP_TAC(NORM_ARITH `!t. abs(a - b) <= t /\ t < d ==> a <= c /\ c <= b ==> abs(c - a) < d`)] THEN EXISTS_TAC `&2 / &2 pow (n + 1)` THEN (CONJ_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[ODD_SUB; ODD_ADD; ODD_MULT; ARITH_ODD] THEN ASM_REWRITE_TAC[ARITH_RULE `1 < 2 * j <=> 0 < j`]; REWRITE_TAC[REAL_POW_ADD; real_div; REAL_INV_MUL] THEN ASM_REAL_ARITH_TAC]); ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`i:num`; `m:num`] THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`k:num`; `p:num`] THEN STRIP_TAC THEN REWRITE_TAC[DIST_LIFT; LIFT_DROP] THEN STRIP_TAC THEN SUBGOAL_THEN `?j. 0 < j /\ j < 2 EXP n /\ abs(&i / &2 pow m - &j / &2 pow n) < inv(&2 pow n) /\ abs(&k / &2 pow p - &j / &2 pow n) < inv(&2 pow n)` STRIP_ASSUME_TAC THENL [MP_TAC(SPEC `max (&2 pow n * &i / &2 pow m) (&2 pow n * &k / &2 pow p)` FLOOR_POS) THEN SIMP_TAC[REAL_LE_MUL; REAL_LE_MAX; REAL_LE_DIV; REAL_POS; REAL_POW_LE] THEN DISCH_THEN(X_CHOOSE_TAC `j:num`) THEN MP_TAC(SPEC `max (&2 pow n * &i / &2 pow m) (&2 pow n * &k / &2 pow p)` FLOOR) THEN ASM_REWRITE_TAC[REAL_LE_MAX; REAL_MAX_LT] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN SIMP_TAC[GSYM REAL_LE_LDIV_EQ; GSYM REAL_LT_RDIV_EQ; REAL_LT_POW2] THEN REWRITE_TAC[REAL_ARITH `(j + &1) / n = j / n + inv n`] THEN ASM_CASES_TAC `j = 0` THENL [ASM_REWRITE_TAC[REAL_ARITH `&0 / x = &0`; REAL_ADD_LID] THEN DISCH_TAC THEN EXISTS_TAC `1` THEN CONV_TAC NUM_REDUCE_CONV THEN REWRITE_TAC[ARITH_RULE `1 < n <=> 2 EXP 1 <= n`] THEN ASM_SIMP_TAC[LE_EXP; LE_1] THEN CONV_TAC NUM_REDUCE_CONV THEN MATCH_MP_TAC(REAL_ARITH `&0 < x /\ x < inv n /\ &0 < y /\ y < inv n ==> abs(x - &1 / n) < inv n /\ abs(y - &1 / n) < inv n`) THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; REAL_LT_POW2]; DISCH_TAC THEN EXISTS_TAC `j:num` THEN ASM_SIMP_TAC[LE_1] THEN REWRITE_TAC[GSYM REAL_OF_NUM_LT; GSYM REAL_OF_NUM_POW] THEN CONJ_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC LAND_CONV [GSYM th]) THEN SIMP_TAC[GSYM REAL_NOT_LE; REAL_LE_FLOOR; INTEGER_CLOSED] THEN REWRITE_TAC[REAL_NOT_LE; REAL_MAX_LT] THEN REWRITE_TAC[REAL_ARITH `n * x < n <=> n * x < n * &1`] THEN SIMP_TAC[REAL_LT_LMUL_EQ; REAL_LT_POW2; REAL_LT_LDIV_EQ] THEN ASM_REWRITE_TAC[REAL_MUL_LID; REAL_OF_NUM_POW; REAL_OF_NUM_LT]]; MATCH_MP_TAC(NORM_ARITH `!u. dist(w:real^N,u) < e / &2 /\ dist(z,u) < e / &2 ==> dist(w,z) < e`) THEN EXISTS_TAC `(f:real^1->real^N)(c(&j / &2 pow n))` THEN REWRITE_TAC[dist] THEN CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[]]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `h:real^1->real^N` THEN REWRITE_TAC[FORALL_IN_GSPEC; LIFT_DROP] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o CONJUNCT1)) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP UNIFORMLY_CONTINUOUS_IMP_CONTINUOUS) THEN ONCE_REWRITE_TAC[MESON[] `h x = f(c(drop x)) <=> f(c(drop x)) = h x`] THEN REWRITE_TAC[IN_INTER; IMP_CONJ_ALT; FORALL_IN_GSPEC] THEN ASM_REWRITE_TAC[IN_UNIV; LIFT_DROP; IMP_IMP; GSYM CONJ_ASSOC] THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; LIFT_DROP] THEN SIMP_TAC[REAL_LT_RDIV_EQ; REAL_LT_LDIV_EQ; REAL_LT_POW2] THEN REWRITE_TAC[REAL_MUL_LZERO; REAL_MUL_LID] THEN REWRITE_TAC[REAL_OF_NUM_POW; REAL_OF_NUM_LT] THEN REWRITE_TAC[GSYM REAL_OF_NUM_POW] THEN DISCH_TAC THEN CONJ_TAC THENL [MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [MP_TAC(ISPEC `interval(vec 0:real^1,vec 1)` closure_dyadic_rationals_in_convex_set_pos_1) THEN SIMP_TAC[CONVEX_INTERVAL; IN_INTERVAL_1; REAL_LT_IMP_LE; DROP_VEC; INTERIOR_OPEN; OPEN_INTERVAL; INTERVAL_NE_EMPTY_1; REAL_LT_01; CLOSURE_OPEN_INTERVAL] THEN DISCH_THEN(fun th -> GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM th]) THEN MATCH_MP_TAC IMAGE_CLOSURE_SUBSET THEN REPEAT CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN REWRITE_TAC[CLOSED_INTERVAL] THEN MATCH_MP_TAC(SET_RULE `s SUBSET u ==> s INTER t SUBSET u`) THEN REWRITE_TAC[INTERVAL_OPEN_SUBSET_CLOSED]; MATCH_MP_TAC COMPACT_IMP_CLOSED THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[COMPACT_INTERVAL]; SIMP_TAC[dyadics_in_open_unit_interval; SUBSET; FORALL_IN_IMAGE] THEN ASM_SIMP_TAC[FORALL_IN_GSPEC] THEN MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN STRIP_TAC THEN MATCH_MP_TAC FUN_IN_IMAGE THEN REWRITE_TAC[IN_INTERVAL_1] THEN ASM_MESON_TAC[REAL_LE_TRANS]]; MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `closure(IMAGE (h:real^1->real^N) (interval (vec 0,vec 1) INTER {lift (&m / &2 pow n) | m IN (:num) /\ n IN (:num)}))` THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC CLOSURE_MINIMAL THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED; COMPACT_INTERVAL; COMPACT_CONTINUOUS_IMAGE] THEN MATCH_MP_TAC IMAGE_SUBSET THEN MATCH_MP_TAC(SET_RULE `s SUBSET u ==> s INTER t SUBSET u`) THEN REWRITE_TAC[INTERVAL_OPEN_SUBSET_CLOSED]] THEN REWRITE_TAC[SUBSET; CLOSURE_APPROACHABLE; FORALL_IN_IMAGE] THEN REWRITE_TAC[dyadics_in_open_unit_interval; EXISTS_IN_IMAGE; EXISTS_IN_GSPEC] THEN X_GEN_TAC `x:real^1` THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN UNDISCH_TAC `(f:real^1->real^N) continuous_on interval [vec 0,vec 1]` THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] COMPACT_UNIFORMLY_CONTINUOUS)) THEN REWRITE_TAC[COMPACT_INTERVAL; uniformly_continuous_on] THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `!n. ~(n = 0) ==> ?m y. ODD m /\ 0 < m /\ m < 2 EXP n /\ y IN interval[a(&m / &2 pow n),b(&m / &2 pow n)] /\ (f:real^1->real^N) y = f x` MP_TAC THENL [ALL_TAC; MP_TAC(SPECL [`inv(&2)`; `min (d / &2) (&1 / &4)`] REAL_ARCH_POW_INV) THEN ASM_REWRITE_TAC[REAL_HALF; REAL_POW_INV; REAL_LT_MIN] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN DISCH_THEN(X_CHOOSE_THEN `n:num` MP_TAC) THEN ASM_CASES_TAC `n = 0` THEN ASM_REWRITE_TAC[] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN STRIP_TAC THEN DISCH_THEN(MP_TAC o SPEC `n:num`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `m:num` THEN DISCH_THEN(X_CHOOSE_THEN `y:real^1` MP_TAC) THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(SUBST1_TAC o SYM) THEN EXISTS_TAC `n:num` THEN ASM_SIMP_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN REWRITE_TAC[DIST_REAL; GSYM drop; IN_INTERVAL_1] THEN REPEAT(CONJ_TAC THENL [ASM_MESON_TAC[REAL_LE_TRANS]; ALL_TAC]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `a <= y /\ y <= b ==> a <= c /\ c <= b /\ abs(a - b) < d ==> abs(c - y) < d`)) THEN REPEAT(CONJ_TAC THENL [ASM_MESON_TAC[REAL_LE_TRANS]; ALL_TAC]) THEN MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `&2 / &2 pow n` THEN ASM_SIMP_TAC[] THEN ASM_REAL_ARITH_TAC] THEN MATCH_MP_TAC num_INDUCTION THEN REWRITE_TAC[NOT_SUC] THEN X_GEN_TAC `n:num` THEN ASM_CASES_TAC `n = 0` THEN ASM_REWRITE_TAC[] THENL [EXISTS_TAC `1` THEN CONV_TAC NUM_REDUCE_CONV THEN ASM_REWRITE_TAC[REAL_POW_1] THEN SUBGOAL_THEN `x IN interval[vec 0:real^1,u] \/ x IN interval[u,v] \/ x IN interval[v,vec 1]` STRIP_ASSUME_TAC THENL [REWRITE_TAC[IN_INTERVAL_1] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC; EXISTS_TAC `u:real^1` THEN ASM_MESON_TAC[ENDS_IN_INTERVAL; INTERVAL_NE_EMPTY_1]; EXISTS_TAC `x:real^1` THEN ASM_MESON_TAC[]; EXISTS_TAC `v:real^1` THEN ASM_MESON_TAC[ENDS_IN_INTERVAL; INTERVAL_NE_EMPTY_1]]; DISCH_THEN(X_CHOOSE_THEN `m:num` (X_CHOOSE_THEN `y:real^1` MP_TAC)) THEN REPLICATE_TAC 3 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC (SUBST1_TAC o SYM)) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [ODD_EXISTS]) THEN DISCH_THEN(X_CHOOSE_THEN `j:num` SUBST_ALL_TAC) THEN REWRITE_TAC[ADD1] THEN DISCH_TAC THEN SUBGOAL_THEN `y IN interval[a(&(2 * j + 1) / &2 pow n):real^1, b(&(4 * j + 1) / &2 pow (n + 1))] \/ y IN interval[b(&(4 * j + 1) / &2 pow (n + 1)), a(&(4 * j + 3) / &2 pow (n + 1))] \/ y IN interval[a(&(4 * j + 3) / &2 pow (n + 1)), b(&(2 * j + 1) / &2 pow n)]` STRIP_ASSUME_TAC THENL [REWRITE_TAC[IN_INTERVAL_1] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC; EXISTS_TAC `4 * j + 1` THEN EXISTS_TAC `y:real^1` THEN REWRITE_TAC[ODD_ADD; ODD_MULT; ARITH; EXP_ADD] THEN REPEAT(CONJ_TAC THENL [ASM_ARITH_TAC; ALL_TAC]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (MESON[] `y IN interval[a,b] ==> a = a' /\ b = b' ==> y IN interval[a',b']`)) THEN ASM_MESON_TAC[LE_1]; EXISTS_TAC `4 * j + 1` THEN EXISTS_TAC `b(&(4 * j + 1) / &2 pow (n + 1)):real^1` THEN REWRITE_TAC[ODD_ADD; ODD_MULT; ARITH; EXP_ADD] THEN REPEAT(CONJ_TAC THENL [ASM_ARITH_TAC; ALL_TAC]) THEN REWRITE_TAC[ENDS_IN_INTERVAL; INTERVAL_NE_EMPTY_1] THEN CONJ_TAC THENL [ASM_MESON_TAC[REAL_LE_TRANS]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a(&(2 * j + 1) / &2 pow n):real^1`; `b(&(2 * j + 1) / &2 pow n):real^1`; `c(&(2 * j + 1) / &2 pow n):real^1`]) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[midpoint; IN_INTERVAL_1; SUBSET_INTERVAL_1]; REPLICATE_TAC 4 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC)] THEN MATCH_MP_TAC(MESON[] `a IN s /\ b IN s ==> (!x. x IN s ==> f x = k) ==> f a = f b`) THEN SUBGOAL_THEN `leftcut (a (&(2 * j + 1) / &2 pow n)) (b (&(2 * j + 1) / &2 pow n)) (c (&(2 * j + 1) / &2 pow n):real^1):real^1 = b(&(4 * j + 1) / &2 pow (n + 1)) /\ rightcut (a (&(2 * j + 1) / &2 pow n)) (b (&(2 * j + 1) / &2 pow n)) (c (&(2 * j + 1) / &2 pow n)):real^1 = a(&(4 * j + 3) / &2 pow (n + 1))` (CONJUNCTS_THEN SUBST_ALL_TAC) THENL [ASM_MESON_TAC[LE_1]; ALL_TAC] THEN REWRITE_TAC[ENDS_IN_INTERVAL; INTERVAL_NE_EMPTY_1] THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (MESON[] `y IN interval[a,b] ==> a = a' /\ b = b' ==> y IN interval[a',b']`)) THEN ASM_MESON_TAC[LE_1]; EXISTS_TAC `4 * j + 3` THEN EXISTS_TAC `y:real^1` THEN REWRITE_TAC[ODD_ADD; ODD_MULT; ARITH; EXP_ADD] THEN REPEAT(CONJ_TAC THENL [ASM_ARITH_TAC; ALL_TAC]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (MESON[] `y IN interval[a,b] ==> a = a' /\ b = b' ==> y IN interval[a',b']`)) THEN ASM_MESON_TAC[LE_1]]]]; ALL_TAC] THEN SUBGOAL_THEN `!n m. drop(a(&m / &2 pow n)) < drop(b(&m / &2 pow n)) /\ (!x. drop(a(&m / &2 pow n)) < drop x /\ drop x <= drop(b(&m / &2 pow n)) ==> ~(f x = f(a(&m / &2 pow n)))) /\ (!x. drop(a(&m / &2 pow n)) <= drop x /\ drop x < drop(b(&m / &2 pow n)) ==> ~(f x :real^N = f(b(&m / &2 pow n))))` ASSUME_TAC THENL [SUBGOAL_THEN `drop u < drop v` ASSUME_TAC THENL [ASM_REWRITE_TAC[REAL_LT_LE; DROP_EQ] THEN DISCH_THEN SUBST_ALL_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE [IN_DELETE; IN_INTERVAL_1; GSYM DROP_EQ; DROP_VEC]) THEN ASM_MESON_TAC[DROP_EQ]; ALL_TAC] THEN SUBGOAL_THEN `(!x. drop u < drop x /\ drop x <= drop v ==> ~((f:real^1->real^N) x = f u)) /\ (!x. drop u <= drop x /\ drop x < drop v ==> ~(f x = f v))` STRIP_ASSUME_TAC THENL [SUBGOAL_THEN `(f:real^1->real^N) u = f(vec 0) /\ (f:real^1->real^N) v = f(vec 1)` (CONJUNCTS_THEN SUBST1_TAC) THENL [CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL_1; REAL_LE_REFL]; ALL_TAC] THEN CONJ_TAC THEN GEN_TAC THEN STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_DELETE; IN_INTERVAL_1; GSYM DROP_EQ] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC num_INDUCTION THEN ASM_REWRITE_TAC[REAL_ARITH `&m / &2 pow 0 = (&2 * &m) / &2`] THEN ASM_REWRITE_TAC[REAL_OF_NUM_MUL] THEN X_GEN_TAC `n:num` THEN DISCH_THEN(LABEL_TAC "*") THEN DISJ_CASES_TAC(ARITH_RULE `n = 0 \/ 0 < n`) THEN ASM_REWRITE_TAC[ARITH; REAL_POW_1] THEN X_GEN_TAC `j:num` THEN DISJ_CASES_TAC(ISPEC `j:num` EVEN_OR_ODD) THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [EVEN_EXISTS]) THEN DISCH_THEN(X_CHOOSE_THEN `k:num` SUBST1_TAC) THEN SIMP_TAC[GSYM REAL_OF_NUM_MUL; real_div; REAL_INV_MUL; real_pow] THEN ASM_REWRITE_TAC[REAL_ARITH `(&2 * p) * inv(&2) * inv q = p / q`]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [ODD_EXISTS]) THEN DISCH_THEN(X_CHOOSE_THEN `k:num` SUBST1_TAC) THEN DISJ_CASES_TAC(ISPEC `k:num` EVEN_OR_ODD) THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [EVEN_EXISTS]) THEN DISCH_THEN(X_CHOOSE_THEN `m:num` SUBST1_TAC) THEN ASM_SIMP_TAC[ARITH_RULE `2 * 2 * m = 4 * m`; ADD1] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a(&(2 * m + 1) / &2 pow n):real^1`; `b(&(2 * m + 1) / &2 pow n):real^1`; `c(&(2 * m + 1) / &2 pow n):real^1`]) THEN ANTS_TAC THENL [REWRITE_TAC[IN_INTERVAL_1; SUBSET_INTERVAL_1] THEN ASM_MESON_TAC[REAL_LE_TRANS]; REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(K ALL_TAC)] THEN SUBGOAL_THEN `(f:real^1->real^N) (leftcut (a (&(2 * m + 1) / &2 pow n):real^1) (b (&(2 * m + 1) / &2 pow n):real^1) (c (&(2 * m + 1) / &2 pow n):real^1)) = (f:real^1->real^N) (c(&(2 * m + 1) / &2 pow n))` ASSUME_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL_1; REAL_LE_REFL] THEN ASM_REAL_ARITH_TAC; ASM_REWRITE_TAC[]] THEN GEN_REWRITE_TAC LAND_CONV [REAL_LT_LE] THEN ASM_REWRITE_TAC[DROP_EQ] THEN REPEAT CONJ_TAC THENL [DISCH_THEN(SUBST_ALL_TAC o SYM) THEN UNDISCH_THEN `(f:real^1->real^N) (a (&(2 * m + 1) / &2 pow n)) = f(c (&(2 * m + 1) / &2 pow n))` (MP_TAC o SYM) THEN REWRITE_TAC[] THEN FIRST_ASSUM(MATCH_MP_TAC o CONJUNCT1 o CONJUNCT2 o SPEC_ALL) THEN REWRITE_TAC[GSYM(ASSUME `!x. midpoint ((a:real->real^1) x,b x) = c x`); midpoint; DROP_CMUL; DROP_ADD] THEN ASM_REWRITE_TAC[REAL_ARITH `a < inv(&2) * (a + b) /\ inv(&2) * (a + b) <= b <=> a < b`]; GEN_TAC THEN STRIP_TAC THEN FIRST_ASSUM(MATCH_MP_TAC o CONJUNCT1 o CONJUNCT2 o SPEC_ALL) THEN ASM_MESON_TAC[REAL_LE_TRANS]; GEN_TAC THEN STRIP_TAC THEN FIRST_X_ASSUM (fun th -> MATCH_MP_TAC th THEN REWRITE_TAC[IN_INTERVAL_1; IN_DELETE; GSYM DROP_EQ] THEN GEN_REWRITE_TAC I [REAL_ARITH `(a <= x /\ x <= b) /\ ~(x = b) <=> a <= x /\ x < b`]) THEN ASM_REWRITE_TAC[]]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [ODD_EXISTS]) THEN DISCH_THEN(X_CHOOSE_THEN `m:num` SUBST1_TAC) THEN ASM_SIMP_TAC[ARITH_RULE `2 * (2 * m + 1) + 1 = 4 * m + 3`; ADD1] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a(&(2 * m + 1) / &2 pow n):real^1`; `b(&(2 * m + 1) / &2 pow n):real^1`; `c(&(2 * m + 1) / &2 pow n):real^1`]) THEN ANTS_TAC THENL [REWRITE_TAC[IN_INTERVAL_1; SUBSET_INTERVAL_1] THEN ASM_MESON_TAC[REAL_LE_TRANS]; REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(K ALL_TAC)] THEN SUBGOAL_THEN `(f:real^1->real^N) (rightcut (a (&(2 * m + 1) / &2 pow n):real^1) (b (&(2 * m + 1) / &2 pow n):real^1) (c (&(2 * m + 1) / &2 pow n):real^1)) = (f:real^1->real^N) (c(&(2 * m + 1) / &2 pow n))` ASSUME_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL_1; REAL_LE_REFL] THEN ASM_REAL_ARITH_TAC; ASM_REWRITE_TAC[]] THEN GEN_REWRITE_TAC LAND_CONV [REAL_LT_LE] THEN ASM_REWRITE_TAC[DROP_EQ] THEN REPEAT CONJ_TAC THENL [DISCH_THEN SUBST_ALL_TAC THEN UNDISCH_THEN `(f:real^1->real^N) (b (&(2 * m + 1) / &2 pow n)) = f(c (&(2 * m + 1) / &2 pow n))` (MP_TAC o SYM) THEN REWRITE_TAC[] THEN FIRST_ASSUM(MATCH_MP_TAC o CONJUNCT2 o CONJUNCT2 o SPEC_ALL) THEN REWRITE_TAC[GSYM(ASSUME `!x. midpoint ((a:real->real^1) x,b x) = c x`); midpoint; DROP_CMUL; DROP_ADD] THEN ASM_REWRITE_TAC[REAL_ARITH `a <= inv(&2) * (a + b) /\ inv(&2) * (a + b) < b <=> a < b`]; GEN_TAC THEN STRIP_TAC THEN FIRST_X_ASSUM (fun th -> MATCH_MP_TAC th THEN REWRITE_TAC[IN_INTERVAL_1; IN_DELETE; GSYM DROP_EQ] THEN GEN_REWRITE_TAC I [REAL_ARITH `(a <= x /\ x <= b) /\ ~(x = a) <=> a < x /\ x <= b`]) THEN ASM_REWRITE_TAC[]; GEN_TAC THEN STRIP_TAC THEN FIRST_ASSUM(MATCH_MP_TAC o CONJUNCT2 o CONJUNCT2 o SPEC_ALL) THEN ASM_MESON_TAC[REAL_LE_TRANS]]]; ALL_TAC] THEN SUBGOAL_THEN `!m i n j. 0 < i /\ i < 2 EXP m /\ 0 < j /\ j < 2 EXP n /\ &i / &2 pow m < &j / &2 pow n ==> drop(c(&i / &2 pow m)) <= drop(c(&j / &2 pow n))` ASSUME_TAC THENL [SUBGOAL_THEN `!N m p i k. 0 < i /\ i < 2 EXP m /\ 0 < k /\ k < 2 EXP p /\ &i / &2 pow m < &k / &2 pow p /\ m + p = N ==> ?j n. ODD(j) /\ ~(n = 0) /\ &i / &2 pow m <= &j / &2 pow n /\ &j / &2 pow n <= &k / &2 pow p /\ abs(&i / &2 pow m - &j / &2 pow n) < inv(&2 pow n) /\ abs(&k / &2 pow p - &j / &2 pow n) < inv(&2 pow n)` MP_TAC THENL [MATCH_MP_TAC num_WF THEN X_GEN_TAC `N:num` THEN DISCH_THEN(LABEL_TAC "I") THEN MAP_EVERY X_GEN_TAC [`m:num`; `p:num`; `i:num`; `k:num`] THEN STRIP_TAC THEN SUBGOAL_THEN `&i / &2 pow m <= &1 / &2 pow 1 /\ &1 / &2 pow 1 <= &k / &2 pow p \/ &k / &2 pow p < &1 / &2 \/ &1 / &2 < &i / &2 pow m` (REPEAT_TCL DISJ_CASES_THEN STRIP_ASSUME_TAC) THENL [ASM_REAL_ARITH_TAC; MAP_EVERY EXISTS_TAC [`1`; `1`] THEN ASM_REWRITE_TAC[ARITH] THEN MATCH_MP_TAC(REAL_ARITH `&0 < i /\ i <= &1 / &2 pow 1 /\ &1 / &2 pow 1 <= k /\ k < &1 ==> abs(i - &1 / &2 pow 1) < inv(&2 pow 1) /\ abs(k - &1 / &2 pow 1) < inv(&2 pow 1)`) THEN ASM_SIMP_TAC[REAL_LT_RDIV_EQ; REAL_LT_LDIV_EQ; REAL_LT_POW2] THEN REWRITE_TAC[MULT_CLAUSES; REAL_OF_NUM_POW; REAL_OF_NUM_MUL] THEN ASM_REWRITE_TAC[REAL_OF_NUM_LT]; REMOVE_THEN "I" MP_TAC THEN POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN SPEC_TAC(`m:num`,`m:num`) THEN INDUCT_TAC THEN REWRITE_TAC[ARITH_RULE `i < 2 EXP 0 <=> ~(0 < i)`] THEN REWRITE_TAC[TAUT `p /\ ~p /\ q <=> F`] THEN POP_ASSUM(K ALL_TAC) THEN SPEC_TAC(`p:num`,`p:num`) THEN INDUCT_TAC THEN REWRITE_TAC[ARITH_RULE `i < 2 EXP 0 <=> ~(0 < i)`] THEN REWRITE_TAC[TAUT `p /\ ~p /\ q <=> F`] THEN POP_ASSUM(K ALL_TAC) THEN STRIP_TAC THEN DISCH_THEN(MP_TAC o SPEC `m + p:num`) THEN ANTS_TAC THENL [EXPAND_TAC "N" THEN ARITH_TAC; ALL_TAC] THEN DISCH_THEN(MP_TAC o SPECL [`m:num`; `p:num`; `i:num`; `k:num`]) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [MAP_EVERY UNDISCH_TAC [`&k / &2 pow SUC p < &1 / &2`; `&i / &2 pow SUC m < &k / &2 pow SUC p`] THEN REWRITE_TAC[real_div; real_pow; REAL_INV_MUL; REAL_ARITH `x * inv(&2) * y = (x * y) * inv(&2)`] THEN SIMP_TAC[GSYM real_div; REAL_LT_DIV2_EQ; REAL_OF_NUM_LT; ARITH] THEN REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `x < y /\ y < &1 ==> x < &1 /\ y < &1`)) THEN SIMP_TAC[REAL_LT_LDIV_EQ; REAL_LT_POW2; REAL_MUL_LID] THEN REWRITE_TAC[REAL_OF_NUM_POW; REAL_OF_NUM_LT]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `j:num` THEN DISCH_THEN(X_CHOOSE_THEN `n:num` STRIP_ASSUME_TAC) THEN EXISTS_TAC `SUC n` THEN ASM_REWRITE_TAC[NOT_SUC] THEN REWRITE_TAC[real_div; real_pow; REAL_INV_MUL; REAL_ARITH `inv(&2) * y = y * inv(&2)`] THEN REWRITE_TAC[GSYM REAL_SUB_RDISTRIB; REAL_MUL_ASSOC; REAL_ABS_MUL; REAL_ABS_INV; REAL_ABS_NUM] THEN REWRITE_TAC[GSYM real_div; REAL_ABS_NUM] THEN ASM_SIMP_TAC[REAL_LT_DIV2_EQ; REAL_LE_DIV2_EQ; REAL_OF_NUM_LT; ARITH]]; REMOVE_THEN "I" MP_TAC THEN POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN SPEC_TAC(`m:num`,`m:num`) THEN INDUCT_TAC THEN REWRITE_TAC[ARITH_RULE `i < 2 EXP 0 <=> ~(0 < i)`] THEN REWRITE_TAC[TAUT `p /\ ~p /\ q <=> F`] THEN POP_ASSUM(K ALL_TAC) THEN SPEC_TAC(`p:num`,`p:num`) THEN INDUCT_TAC THEN REWRITE_TAC[ARITH_RULE `i < 2 EXP 0 <=> ~(0 < i)`] THEN REWRITE_TAC[TAUT `p /\ ~p /\ q <=> F`] THEN POP_ASSUM(K ALL_TAC) THEN STRIP_TAC THEN DISCH_THEN(MP_TAC o SPEC `m + p:num`) THEN ANTS_TAC THENL [EXPAND_TAC "N" THEN ARITH_TAC; ALL_TAC] THEN DISCH_THEN(MP_TAC o SPECL [`m:num`; `p:num`; `i - 2 EXP m`; `k - 2 EXP p`]) THEN ASM_REWRITE_TAC[] THEN MAP_EVERY UNDISCH_TAC [`&1 / &2 < &i / &2 pow SUC m`; `&i / &2 pow SUC m < &k / &2 pow SUC p`] THEN REWRITE_TAC[real_div; real_pow; REAL_INV_MUL; REAL_ARITH `x * inv(&2) * y = (x * y) * inv(&2)`] THEN SIMP_TAC[GSYM real_div; REAL_LT_DIV2_EQ; REAL_OF_NUM_LT; ARITH] THEN GEN_REWRITE_TAC I [IMP_IMP] THEN DISCH_THEN(fun th -> STRIP_ASSUME_TAC th THEN MP_TAC(MATCH_MP (REAL_ARITH `i < k /\ &1 < i ==> &1 < i /\ &1 < k`) th)) THEN SIMP_TAC[REAL_LT_RDIV_EQ; REAL_LT_POW2; REAL_MUL_LID] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [REAL_OF_NUM_POW] THEN SIMP_TAC[REAL_OF_NUM_LT; GSYM REAL_OF_NUM_SUB; LT_IMP_LE] THEN STRIP_TAC THEN REWRITE_TAC[GSYM REAL_OF_NUM_POW] THEN ANTS_TAC THENL [ASM_SIMP_TAC[ARITH_RULE `a < b ==> 0 < b - a`] THEN ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; REAL_LT_POW2] THEN REWRITE_TAC[real_div; REAL_SUB_RDISTRIB] THEN ASM_SIMP_TAC[REAL_MUL_RINV; REAL_LT_IMP_NZ; REAL_LT_POW2] THEN ASM_REWRITE_TAC[REAL_ARITH `u * inv v - &1 < w * inv z - &1 <=> u / v < w / z`] THEN CONJ_TAC THEN MATCH_MP_TAC(ARITH_RULE `i < 2 * m ==> i - m < m`) THEN ASM_REWRITE_TAC[GSYM(CONJUNCT2 EXP)]; REWRITE_TAC[real_div; REAL_SUB_RDISTRIB] THEN ASM_SIMP_TAC[REAL_MUL_RINV; REAL_LT_IMP_NZ; REAL_LT_POW2] THEN REWRITE_TAC[GSYM real_div] THEN DISCH_THEN(X_CHOOSE_THEN `j:num` (X_CHOOSE_THEN `n:num` STRIP_ASSUME_TAC)) THEN EXISTS_TAC `2 EXP n + j` THEN EXISTS_TAC `SUC n` THEN ASM_REWRITE_TAC[NOT_SUC; ODD_ADD; ODD_EXP; ARITH] THEN REWRITE_TAC[GSYM REAL_OF_NUM_ADD; GSYM REAL_OF_NUM_POW] THEN REWRITE_TAC[real_div; real_pow; REAL_INV_MUL; REAL_ARITH `inv(&2) * y = y * inv(&2)`] THEN REWRITE_TAC[GSYM REAL_SUB_RDISTRIB; REAL_MUL_ASSOC; REAL_ABS_MUL; REAL_ABS_INV; REAL_ABS_NUM] THEN REWRITE_TAC[GSYM real_div; REAL_ABS_NUM] THEN ASM_SIMP_TAC[REAL_LT_DIV2_EQ; REAL_LE_DIV2_EQ; REAL_OF_NUM_LT; ARITH] THEN REWRITE_TAC[real_div; REAL_ADD_RDISTRIB] THEN ASM_SIMP_TAC[REAL_MUL_RINV; REAL_LT_IMP_NZ; REAL_LT_POW2] THEN REWRITE_TAC[GSYM real_div] THEN ASM_REAL_ARITH_TAC]]; DISCH_THEN(fun th -> MAP_EVERY X_GEN_TAC [`m:num`; `i:num`; `p:num`; `k:num`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`m + p:num`; `m:num`; `p:num`; `i:num`; `k:num`] th)) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`j:num`; `n:num`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [ODD_EXISTS]) THEN REWRITE_TAC[ADD1; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `q:num` THEN DISCH_THEN SUBST_ALL_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `drop(c(&(2 * q + 1) / &2 pow n))` THEN CONJ_TAC THENL [ASM_CASES_TAC `&i / &2 pow m = &(2 * q + 1) / &2 pow n` THEN ASM_REWRITE_TAC[REAL_LE_REFL] THEN SUBGOAL_THEN `drop(a(&(4 * q + 1) / &2 pow (n + 1))) <= drop(c(&i / &2 pow m)) /\ drop(c(&i / &2 pow m)) <= drop(b(&(4 * q + 1) / &2 pow (n + 1)))` MP_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[ODD_ADD; ODD_MULT; ARITH] THEN SIMP_TAC[real_div; REAL_POW_ADD; REAL_INV_MUL; REAL_MUL_ASSOC] THEN REWRITE_TAC[GSYM real_div; REAL_POW_1] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `abs(i - q) < n ==> i <= q /\ ~(i = q) /\ q = q' + n / &2 ==> abs(i - q') < n / &2`)) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM REAL_OF_NUM_ADD; GSYM REAL_OF_NUM_MUL] THEN REAL_ARITH_TAC; ASM_SIMP_TAC[LE_1] THEN MATCH_MP_TAC(REAL_ARITH `l <= d ==> u <= v /\ c <= l ==> c <= d`) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a(&(2 * q + 1) / &2 pow n):real^1`; `b(&(2 * q + 1) / &2 pow n):real^1`; `c(&(2 * q + 1) / &2 pow n):real^1`]) THEN ANTS_TAC THENL [REWRITE_TAC[IN_INTERVAL_1; SUBSET_INTERVAL_1] THEN ASM_MESON_TAC[REAL_LE_TRANS]; DISCH_THEN(fun th -> REWRITE_TAC[th])]]; ASM_CASES_TAC `&k / &2 pow p = &(2 * q + 1) / &2 pow n` THEN ASM_REWRITE_TAC[REAL_LE_REFL] THEN SUBGOAL_THEN `drop(a(&(4 * q + 3) / &2 pow (n + 1))) <= drop(c(&k / &2 pow p)) /\ drop(c(&k / &2 pow p)) <= drop(b(&(4 * q + 3) / &2 pow (n + 1)))` MP_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[ODD_ADD; ODD_MULT; ARITH] THEN SIMP_TAC[real_div; REAL_POW_ADD; REAL_INV_MUL; REAL_MUL_ASSOC] THEN REWRITE_TAC[GSYM real_div; REAL_POW_1] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `abs(i - q) < n ==> q <= i /\ ~(i = q) /\ q' = q + n / &2 ==> abs(i - q') < n / &2`)) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM REAL_OF_NUM_ADD; GSYM REAL_OF_NUM_MUL] THEN REAL_ARITH_TAC; ASM_SIMP_TAC[LE_1] THEN MATCH_MP_TAC(REAL_ARITH `d <= l ==> l <= c /\ u <= v ==> d <= c`) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a(&(2 * q + 1) / &2 pow n):real^1`; `b(&(2 * q + 1) / &2 pow n):real^1`; `c(&(2 * q + 1) / &2 pow n):real^1`]) THEN ANTS_TAC THENL [REWRITE_TAC[IN_INTERVAL_1; SUBSET_INTERVAL_1] THEN ASM_MESON_TAC[REAL_LE_TRANS]; DISCH_THEN(fun th -> REWRITE_TAC[th])]]]]; ALL_TAC] THEN REWRITE_TAC[FORALL_LIFT] THEN MATCH_MP_TAC REAL_WLOG_LT THEN REWRITE_TAC[] THEN CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN REWRITE_TAC[FORALL_DROP; LIFT_DROP; IN_INTERVAL_1; DROP_VEC] THEN MAP_EVERY X_GEN_TAC [`x1:real^1`; `x2:real^1`] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `?m n. 0 < m /\ m < 2 EXP n /\ drop x1 < &m / &2 pow n /\ &m / &2 pow n < drop x2 /\ ~(h(x1):real^N = h(lift(&m / &2 pow n)))` STRIP_ASSUME_TAC THENL [MP_TAC(ISPEC `interval(vec 0:real^1,vec 1)` closure_dyadic_rationals_in_convex_set_pos_1) THEN SIMP_TAC[CONVEX_INTERVAL; IN_INTERVAL_1; REAL_LT_IMP_LE; DROP_VEC; INTERIOR_OPEN; OPEN_INTERVAL; INTERVAL_NE_EMPTY_1; REAL_LT_01; CLOSURE_OPEN_INTERVAL] THEN REWRITE_TAC[EXTENSION] THEN DISCH_THEN(MP_TAC o SPEC `inv(&2) % (x1 + x2):real^1`) THEN REWRITE_TAC[dyadics_in_open_unit_interval; IN_INTERVAL_1; DROP_VEC] THEN REWRITE_TAC[DROP_CMUL; DROP_ADD] THEN MATCH_MP_TAC(TAUT `p /\ (q ==> r) ==> (q <=> p) ==> r`) THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; REWRITE_TAC[CLOSURE_APPROACHABLE]] THEN DISCH_THEN(MP_TAC o SPEC `(drop x2 - drop x1) / &64`) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; REWRITE_TAC[EXISTS_IN_GSPEC]] THEN REWRITE_TAC[DIST_REAL; GSYM drop; LIFT_DROP; DROP_CMUL; DROP_ADD] THEN DISCH_TAC THEN SUBGOAL_THEN `?m n. (0 < m /\ m < 2 EXP n) /\ abs(&m / &2 pow n - inv (&2) * (drop x1 + drop x2)) < (drop x2 - drop x1) / &64 /\ inv(&2 pow n) < (drop x2 - drop x1) / &128` STRIP_ASSUME_TAC THENL [MP_TAC(ISPECL [`inv(&2)`; `min (&1 / &4) ((drop x2 - drop x1) / &128)`] REAL_ARCH_POW_INV) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `N:num` MP_TAC) THEN ASM_CASES_TAC `N = 0` THENL [ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[REAL_INV_POW; REAL_LT_MIN; EXISTS_IN_GSPEC] THEN STRIP_TAC THEN FIRST_X_ASSUM(X_CHOOSE_THEN `m:num` (X_CHOOSE_THEN `n:num` STRIP_ASSUME_TAC)) THEN EXISTS_TAC `2 EXP N * m` THEN EXISTS_TAC `N + n:num` THEN ASM_SIMP_TAC[EXP_ADD; LT_MULT; EXP_LT_0; LT_MULT_LCANCEL; LE_1; ARITH_EQ] THEN CONJ_TAC THENL [REWRITE_TAC[REAL_POW_ADD; real_div; REAL_INV_MUL] THEN REWRITE_TAC[GSYM REAL_OF_NUM_MUL; GSYM REAL_OF_NUM_POW; REAL_ARITH `(N * n) * inv N * inv m:real = (N / N) * (n / m)`] THEN ASM_SIMP_TAC[REAL_DIV_REFL; REAL_POW_EQ_0; REAL_OF_NUM_EQ; ARITH_EQ; REAL_MUL_LID; GSYM real_div]; MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `inv(&2) pow N` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_POW_MONO_INV THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[LE_ADD]]; REWRITE_TAC[CONJ_ASSOC] THEN MATCH_MP_TAC(MESON[] `!m n m' n'. (P m n /\ P m' n') /\ (P m n /\ P m' n' ==> ~(g m n = g m' n')) ==> (?m n. P m n /\ ~(a = g m n))`) THEN MAP_EVERY EXISTS_TAC [`2 * m + 1`; `n + 1`; `4 * m + 3`; `n + 2`] THEN CONJ_TAC THENL [REWRITE_TAC[EXP_ADD] THEN CONV_TAC NUM_REDUCE_CONV THEN CONJ_TAC THEN (REWRITE_TAC[GSYM CONJ_ASSOC] THEN REPLICATE_TAC 2 (CONJ_TAC THENL [ASM_ARITH_TAC; ALL_TAC])) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `abs(x - inv(&2) * (x1 + x2)) < (x2 - x1) / &64 ==> abs(x - y) < (x2 - x1) / &4 ==> x1 < y /\ y < x2`)) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `n < x / &128 ==> &0 < x /\ y < &4 * n ==> y < x / &4`)) THEN ASM_REWRITE_TAC[REAL_SUB_LT] THEN REWRITE_TAC[GSYM REAL_OF_NUM_ADD; GSYM REAL_OF_NUM_MUL] THEN MATCH_MP_TAC(REAL_ARITH `a / y = x /\ abs(b / y) < z ==> abs(x - (a + b) / y) < z`) THEN ONCE_REWRITE_TAC[ADD_SYM] THEN REWRITE_TAC[REAL_POW_ADD] THEN SIMP_TAC[REAL_ABS_DIV; REAL_ABS_NUM; REAL_ABS_MUL; REAL_ABS_POW] THEN REWRITE_TAC[real_div; REAL_INV_MUL; REAL_MUL_ASSOC] THEN SIMP_TAC[REAL_LT_RMUL_EQ; REAL_EQ_MUL_RCANCEL; REAL_LT_INV_EQ; REAL_LT_POW2; REAL_INV_EQ_0; REAL_POW_EQ_0; ARITH_EQ; REAL_OF_NUM_EQ] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REAL_ARITH_TAC; ASM_SIMP_TAC[] THEN DISCH_THEN(K ALL_TAC) THEN FIRST_X_ASSUM(MP_TAC o CONJUNCT1 o SPECL [`n + 2`; `4 * m + 3`]) THEN UNDISCH_THEN `!x. midpoint ((a:real->real^1) x,b x) = c x` (fun th -> REWRITE_TAC[GSYM th] THEN ASM_SIMP_TAC[ARITH_RULE `n + 2 = (n + 1) + 1 /\ 0 < n + 1`] THEN REWRITE_TAC[th] THEN ASSUME_TAC th) THEN DISCH_TAC THEN CONV_TAC(RAND_CONV SYM_CONV) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a(&(2 * m + 1) / &2 pow (n + 1)):real^1`; `b(&(2 * m + 1) / &2 pow (n + 1)):real^1`; `c(&(2 * m + 1) / &2 pow (n + 1)):real^1`]) THEN ANTS_TAC THENL [REWRITE_TAC[IN_INTERVAL_1; SUBSET_INTERVAL_1] THEN ASM_MESON_TAC[REAL_LE_TRANS]; REPLICATE_TAC 6 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(MATCH_MP_TAC o CONJUNCT1)] THEN REWRITE_TAC[IN_INTERVAL_1; IN_DELETE; GSYM DROP_EQ] THEN REWRITE_TAC[REAL_ARITH `(a <= b /\ b <= c) /\ ~(b = a) <=> a < b /\ b <= c`] THEN REWRITE_TAC[midpoint; DROP_CMUL; DROP_ADD] THEN ASM_REWRITE_TAC[REAL_ARITH `a < inv(&2) * (a + b) /\ inv(&2) * (a + b) <= b <=> a < b`] THEN ASM_REWRITE_TAC[REAL_LT_LE]]]; ALL_TAC] THEN SUBGOAL_THEN `IMAGE h (interval[vec 0,lift(&m / &2 pow n)]) SUBSET IMAGE (f:real^1->real^N) (interval[vec 0,c(&m / &2 pow n)]) /\ IMAGE h (interval[lift(&m / &2 pow n),vec 1]) SUBSET IMAGE (f:real^1->real^N) (interval[c(&m / &2 pow n),vec 1])` MP_TAC THENL [MP_TAC(ISPEC `interval(lift(&m / &2 pow n),vec 1)` closure_dyadic_rationals_in_convex_set_pos_1) THEN MP_TAC(ISPEC `interval(vec 0,lift(&m / &2 pow n))` closure_dyadic_rationals_in_convex_set_pos_1) THEN SUBGOAL_THEN `&0 < &m / &2 pow n /\ &m / &2 pow n < &1` STRIP_ASSUME_TAC THENL [ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_POW2; REAL_OF_NUM_LT; REAL_LT_LDIV_EQ; REAL_OF_NUM_MUL; REAL_OF_NUM_LT; REAL_OF_NUM_POW; MULT_CLAUSES]; ALL_TAC] THEN MATCH_MP_TAC(TAUT `(p1 /\ p2) /\ (q1 ==> r1) /\ (q2 ==> r2) ==> (p1 ==> q1) ==> (p2 ==> q2) ==> r1 /\ r2`) THEN ASM_SIMP_TAC[CONVEX_INTERVAL; IN_INTERVAL_1; REAL_LT_IMP_LE; DROP_VEC; INTERIOR_OPEN; OPEN_INTERVAL; INTERVAL_NE_EMPTY_1; REAL_LT_01; CLOSURE_OPEN_INTERVAL; LIFT_DROP] THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN CONJ_TAC THEN DISCH_THEN(SUBST1_TAC o SYM) THEN (MATCH_MP_TAC IMAGE_CLOSURE_SUBSET THEN REPEAT CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN REWRITE_TAC[CLOSED_INTERVAL] THEN MATCH_MP_TAC(SET_RULE `s SUBSET u ==> s INTER t SUBSET u`) THEN ASM_SIMP_TAC[SUBSET_INTERVAL_1; LIFT_DROP; REAL_LT_IMP_LE; DROP_VEC; REAL_LE_REFL]; MATCH_MP_TAC COMPACT_IMP_CLOSED THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[COMPACT_INTERVAL] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET_INTERVAL_1; REAL_LE_REFL] THEN ASM_MESON_TAC[REAL_LE_TRANS]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN MATCH_MP_TAC(SET_RULE `i SUBSET interval(vec 0,vec 1) /\ (!x. x IN interval(vec 0,vec 1) INTER l ==> x IN i ==> P x) ==> !x. x IN i INTER l ==> P x`) THEN ASM_SIMP_TAC[SUBSET_INTERVAL_1; LIFT_DROP; DROP_VEC; REAL_LT_IMP_LE; REAL_LE_REFL] THEN REWRITE_TAC[dyadics_in_open_unit_interval; FORALL_IN_GSPEC] THEN MAP_EVERY X_GEN_TAC [`k:num`; `p:num`] THEN STRIP_TAC THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; LIFT_DROP] THEN STRIP_TAC THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC FUN_IN_IMAGE THEN REWRITE_TAC[IN_INTERVAL_1] THEN ASM_SIMP_TAC[] THEN ASM_MESON_TAC[REAL_LE_TRANS]]); DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `IMAGE h s SUBSET t /\ IMAGE h s' SUBSET t' ==> !x y. x IN s /\ y IN s' ==> h(x) IN t /\ h(y) IN t'`)) THEN DISCH_THEN(MP_TAC o SPECL [`x1:real^1`; `x2:real^1`]) THEN ASM_SIMP_TAC[IN_INTERVAL_1; LIFT_DROP; DROP_VEC; REAL_LT_IMP_LE] THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `a IN IMAGE f s /\ a IN IMAGE f t ==> ?x y. x IN s /\ y IN t /\ f x = a /\ f y = a`)) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`t1:real^1`; `t2:real^1`] THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `(h:real^1->real^N) x2` o GEN_REWRITE_RULE BINDER_CONV [GSYM IS_INTERVAL_CONNECTED_1]) THEN REWRITE_TAC[IS_INTERVAL_1; IN_ELIM_THM] THEN DISCH_THEN(MP_TAC o SPECL [`t1:real^1`; `t2:real^1`; `c(&m / &2 pow n):real^1`]) THEN UNDISCH_TAC `~(h x1:real^N = h(lift (&m / &2 pow n)))` THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC(TAUT `q ==> p ==> ~q ==> r`) THEN ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN ASM_MESON_TAC[REAL_LE_TRANS]]);; let PATH_CONTAINS_ARC = prove (`!p:real^1->real^N a b. path p /\ pathstart p = a /\ pathfinish p = b /\ ~(a = b) ==> ?q. arc q /\ path_image q SUBSET path_image p /\ pathstart q = a /\ pathfinish q = b`, REWRITE_TAC[pathstart; pathfinish; path] THEN MAP_EVERY X_GEN_TAC [`f:real^1->real^N`; `a:real^N`; `b:real^N`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`\s. s SUBSET interval[vec 0,vec 1] /\ vec 0 IN s /\ vec 1 IN s /\ (!x y. x IN s /\ y IN s /\ segment(x,y) INTER s = {} ==> (f:real^1->real^N)(x) = f(y))`; `interval[vec 0:real^1,vec 1]`] BROUWER_REDUCTION_THEOREM_GEN) THEN ASM_REWRITE_TAC[GSYM path_image; CLOSED_INTERVAL; SUBSET_REFL] THEN ANTS_TAC THENL [CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[ENDS_IN_UNIT_INTERVAL] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE `s INTER i = {} ==> s SUBSET i ==> s = {}`)) THEN REWRITE_TAC[SEGMENT_EQ_EMPTY] THEN ANTS_TAC THENL [ONCE_REWRITE_TAC[segment]; MESON_TAC[]] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s DIFF i SUBSET t`) THEN ASM_MESON_TAC[CONVEX_CONTAINS_SEGMENT; CONVEX_INTERVAL]] THEN X_GEN_TAC `s:num->real^1->bool` THEN REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC THEN CONJ_TAC THENL [REWRITE_TAC[INTERS_GSPEC; SUBSET; IN_ELIM_THM; IN_UNIV] THEN ASM SET_TAC[]; ALL_TAC] THEN REPEAT(CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN REWRITE_TAC[FORALL_LIFT] THEN MATCH_MP_TAC REAL_WLOG_LT THEN REWRITE_TAC[] THEN CONJ_TAC THENL [REWRITE_TAC[SEGMENT_SYM] THEN MESON_TAC[]; REWRITE_TAC[FORALL_DROP; LIFT_DROP]] THEN MAP_EVERY X_GEN_TAC [`x:real^1`; `y:real^1`] THEN REWRITE_TAC[INTERS_GSPEC; IN_UNIV; IN_ELIM_THM] THEN SIMP_TAC[SEGMENT_1; REAL_LT_IMP_LE] THEN DISCH_TAC THEN STRIP_TAC THEN MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] COMPACT_UNIFORMLY_CONTINUOUS)) THEN REWRITE_TAC[COMPACT_INTERVAL; uniformly_continuous_on] THEN DISCH_THEN(MP_TAC o SPEC `norm((f:real^1->real^N) x - f y) / &2`) THEN ASM_REWRITE_TAC[REAL_HALF; NORM_POS_LT; VECTOR_SUB_EQ] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `?u v. u IN interval[vec 0,vec 1] /\ v IN interval[vec 0,vec 1] /\ norm(u - x) < e /\ norm(v - y) < e /\ (f:real^1->real^N) u = f v` STRIP_ASSUME_TAC THENL [ALL_TAC; FIRST_X_ASSUM(fun th -> MP_TAC(ISPECL [`x:real^1`; `u:real^1`] th) THEN MP_TAC(ISPECL [`y:real^1`; `v:real^1`] th)) THEN ASM_REWRITE_TAC[dist] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(TAUT `q /\ (p ==> ~r) ==> p ==> ~(q ==> r)`) THEN CONJ_TAC THENL [ASM SET_TAC[]; CONV_TAC NORM_ARITH]] THEN SUBGOAL_THEN `?w z. w IN interval(x,y) /\ z IN interval(x,y) /\ drop w < drop z /\ norm(w - x) < e /\ norm(z - y) < e` STRIP_ASSUME_TAC THENL [EXISTS_TAC `x + lift(min e (drop y - drop x) / &3)` THEN EXISTS_TAC `y - lift(min e (drop y - drop x) / &3)` THEN REWRITE_TAC[IN_INTERVAL_1; DROP_ADD; DROP_SUB; LIFT_DROP; NORM_REAL; GSYM drop] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN MP_TAC(ISPECL [`interval[w:real^1,z]`; `{s n :real^1->bool | n IN (:num)}`] COMPACT_IMP_FIP) THEN ASM_REWRITE_TAC[COMPACT_INTERVAL; FORALL_IN_GSPEC] THEN MATCH_MP_TAC(TAUT `q /\ (~p ==> r) ==> (p ==> ~q) ==> r`) THEN CONJ_TAC THENL [REWRITE_TAC[INTERS_GSPEC; IN_UNIV] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s INTER u = {} ==> t SUBSET s ==> t INTER u = {}`)) THEN REWRITE_TAC[SUBSET_INTERVAL_1] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[MESON[] `~(!x. P x /\ Q x ==> R x) <=> (?x. P x /\ Q x /\ ~R x)`] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN REWRITE_TAC[EXISTS_FINITE_SUBSET_IMAGE] THEN DISCH_THEN(X_CHOOSE_THEN `k:num->bool` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPEC `\n:num. n` o MATCH_MP UPPER_BOUND_FINITE_SET) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN SUBGOAL_THEN `interval[w,z] INTER (s:num->real^1->bool) n = {}` ASSUME_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `a INTER t = {} ==> s SUBSET t ==> a INTER s = {}`)) THEN REWRITE_TAC[SUBSET; INTERS_IMAGE; IN_ELIM_THM] THEN REWRITE_TAC[SET_RULE `(!x. x IN s n ==> !i. i IN k ==> x IN s i) <=> (!i. i IN k ==> s n SUBSET s i)`] THEN SUBGOAL_THEN `!i n. i <= n ==> (s:num->real^1->bool) n SUBSET s i` (fun th -> ASM_MESON_TAC[th]) THEN MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN ASM_REWRITE_TAC[] THEN SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?u. u IN (s:num->real^1->bool) n /\ u IN interval[x,w] /\ (interval[u,w] DELETE u) INTER (s n) = {}` MP_TAC THENL [ASM_CASES_TAC `w IN (s:num->real^1->bool) n` THENL [EXISTS_TAC `w:real^1` THEN ASM_REWRITE_TAC[ENDS_IN_INTERVAL] THEN REWRITE_TAC[INTERVAL_SING; SET_RULE `{a} DELETE a = {}`] THEN REWRITE_TAC[INTER_EMPTY; INTERVAL_NE_EMPTY_1] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN MP_TAC(ISPECL [`(s:num->real^1->bool) n INTER interval[x,w]`; `w:real^1`] SEGMENT_TO_POINT_EXISTS) THEN ASM_SIMP_TAC[CLOSED_INTER; CLOSED_INTERVAL] THEN ANTS_TAC THENL [REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `x:real^1` THEN ASM_REWRITE_TAC[IN_INTER; IN_INTERVAL_1] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^1` THEN REWRITE_TAC[IN_INTER] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE `s INTER t INTER u = {} ==> s SUBSET u ==> s INTER t = {}`)) THEN REWRITE_TAC[SEGMENT_1] THEN COND_CASES_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_MESON_TAC[DROP_EQ; REAL_LE_ANTISYM]; ANTS_TAC THENL [REWRITE_TAC[SUBSET_INTERVAL_1] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[OPEN_CLOSED_INTERVAL_1] THEN ASM SET_TAC[]]]]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^1` THEN STRIP_TAC THEN SUBGOAL_THEN `?v. v IN (s:num->real^1->bool) n /\ v IN interval[z,y] /\ (interval[z,v] DELETE v) INTER (s n) = {}` MP_TAC THENL [ASM_CASES_TAC `z IN (s:num->real^1->bool) n` THENL [EXISTS_TAC `z:real^1` THEN ASM_REWRITE_TAC[ENDS_IN_INTERVAL] THEN REWRITE_TAC[INTERVAL_SING; SET_RULE `{a} DELETE a = {}`] THEN REWRITE_TAC[INTER_EMPTY; INTERVAL_NE_EMPTY_1] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN MP_TAC(ISPECL [`(s:num->real^1->bool) n INTER interval[z,y]`; `z:real^1`] SEGMENT_TO_POINT_EXISTS) THEN ASM_SIMP_TAC[CLOSED_INTER; CLOSED_INTERVAL] THEN ANTS_TAC THENL [REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `y:real^1` THEN ASM_REWRITE_TAC[IN_INTER; IN_INTERVAL_1] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `v:real^1` THEN REWRITE_TAC[IN_INTER] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE `s INTER t INTER u = {} ==> s SUBSET u ==> s INTER t = {}`)) THEN REWRITE_TAC[SEGMENT_1] THEN COND_CASES_TAC THENL [ANTS_TAC THENL [REWRITE_TAC[SUBSET_INTERVAL_1] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[OPEN_CLOSED_INTERVAL_1] THEN ASM SET_TAC[]]; RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_MESON_TAC[DROP_EQ; REAL_LE_ANTISYM]]]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `v:real^1` THEN STRIP_TAC THEN REPEAT CONJ_TAC THENL [ASM SET_TAC[]; ASM SET_TAC[]; RULE_ASSUM_TAC(REWRITE_RULE[NORM_REAL; GSYM drop; DROP_SUB]) THEN REWRITE_TAC[NORM_REAL; GSYM drop; DROP_SUB] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC; RULE_ASSUM_TAC(REWRITE_RULE[NORM_REAL; GSYM drop; DROP_SUB]) THEN REWRITE_TAC[NORM_REAL; GSYM drop; DROP_SUB] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC; FIRST_X_ASSUM MATCH_MP_TAC THEN EXISTS_TAC `n:num` THEN ASM_REWRITE_TAC[SEGMENT_1] THEN COND_CASES_TAC THENL [MAP_EVERY UNDISCH_TAC [`interval[w,z] INTER (s:num->real^1->bool) n = {}`; `interval[u,w] DELETE u INTER (s:num->real^1->bool) n = {}`; `interval[z,v] DELETE v INTER (s:num->real^1->bool) n = {}`] THEN REWRITE_TAC[IMP_IMP; SET_RULE `s1 INTER t = {} /\ s2 INTER t = {} <=> (s1 UNION s2) INTER t = {}`] THEN MATCH_MP_TAC(SET_RULE `t SUBSET s ==> s INTER u = {} ==> t INTER u = {}`) THEN REWRITE_TAC[SUBSET; IN_UNION; IN_DELETE; GSYM DROP_EQ; IN_INTERVAL_1] THEN ASM_REAL_ARITH_TAC; RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC]]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `t:real^1->bool` STRIP_ASSUME_TAC) THEN ASM_CASES_TAC `t:real^1->bool = {}` THENL [ASM_MESON_TAC[IN_IMAGE; NOT_IN_EMPTY]; ALL_TAC] THEN ABBREV_TAC `h = \x. (f:real^1->real^N)(@y. y IN t /\ segment(x,y) INTER t = {})` THEN SUBGOAL_THEN `!x y. y IN t /\ segment(x,y) INTER t = {} ==> h(x) = (f:real^1->real^N)(y)` ASSUME_TAC THENL [SUBGOAL_THEN `!x y z. y IN t /\ segment(x,y) INTER t = {} /\ z IN t /\ segment(x,z) INTER t = {} ==> (f:real^1->real^N)(y) = f(z)` ASSUME_TAC THENL [REPEAT GEN_TAC THEN ASM_CASES_TAC `(x:real^1) IN t` THENL [ASM_MESON_TAC[]; UNDISCH_TAC `~((x:real^1) IN t)`] THEN ONCE_REWRITE_TAC[TAUT `p ==> a /\ b /\ c /\ d ==> q <=> (a /\ c) ==> p /\ b /\ d ==> q`] THEN STRIP_TAC THEN REWRITE_TAC[SET_RULE `~(x IN t) /\ s INTER t = {} /\ s' INTER t = {} <=> (x INSERT (s UNION s')) INTER t = {}`] THEN DISCH_THEN(fun th -> FIRST_X_ASSUM MATCH_MP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(SET_RULE `s SUBSET s' ==> s' INTER t = {} ==> s INTER t = {}`) THEN REWRITE_TAC[SEGMENT_1; SUBSET; IN_UNION; IN_INSERT; IN_INTERVAL_1] THEN GEN_TAC THEN REWRITE_TAC[GSYM DROP_EQ] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC; REPEAT STRIP_TAC THEN EXPAND_TAC "h" THEN ASM_MESON_TAC[]]; ALL_TAC] THEN SUBGOAL_THEN `!x. x IN t ==> h(x) = (f:real^1->real^N)(x)` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[SEGMENT_REFL; INTER_EMPTY]; ALL_TAC] THEN SUBGOAL_THEN `!x:real^1. ?y. y IN t /\ segment(x,y) INTER t = {}` ASSUME_TAC THENL [X_GEN_TAC `x:real^1` THEN EXISTS_TAC `closest_point t (x:real^1)` THEN ASM_SIMP_TAC[SEGMENT_TO_CLOSEST_POINT; CLOSEST_POINT_EXISTS]; ALL_TAC] THEN SUBGOAL_THEN `!x y. segment(x,y) INTER t = {} ==> (h:real^1->real^N) x = h y` ASSUME_TAC THENL [MAP_EVERY X_GEN_TAC [`x:real^1`; `x':real^1`] THEN ASM_CASES_TAC `(x:real^1) IN t` THENL [ASM_MESON_TAC[SEGMENT_SYM]; ALL_TAC] THEN ASM_CASES_TAC `(x':real^1) IN t` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?y y'. y IN t /\ segment(x,y) INTER t = {} /\ h x = f y /\ y' IN t /\ segment(x',y') INTER t = {} /\ (h:real^1->real^N) x' = f y'` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN MAP_EVERY UNDISCH_TAC [`~((x:real^1) IN t)`; `~((x':real^1) IN t)`; `segment(x:real^1,y) INTER t = {}`; `segment(x':real^1,y') INTER t = {}`; `segment(x:real^1,x') INTER t = {}`] THEN MATCH_MP_TAC(SET_RULE `s SUBSET (x1 INSERT x2 INSERT (s0 UNION s1 UNION s2)) ==> s0 INTER t = {} ==> s1 INTER t = {} ==> s2 INTER t = {} ==> ~(x1 IN t) ==> ~(x2 IN t) ==> s INTER t = {}`) THEN REWRITE_TAC[SEGMENT_1; SUBSET; IN_UNION; IN_INSERT; IN_INTERVAL_1] THEN GEN_TAC THEN REWRITE_TAC[GSYM DROP_EQ] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN MP_TAC(ISPEC `h:real^1->real^N` HOMEOMORPHIC_MONOTONE_IMAGE_INTERVAL) THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [REWRITE_TAC[continuous_on] THEN X_GEN_TAC `u:real^1` THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [continuous_on]) THEN DISCH_THEN(MP_TAC o SPEC `u:real^1`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `v:real^1` THEN STRIP_TAC THEN ASM_CASES_TAC `segment(u:real^1,v) INTER t = {}` THENL [ASM_MESON_TAC[DIST_REFL]; ALL_TAC] THEN SUBGOAL_THEN `(?w:real^1. w IN t /\ w IN segment[u,v] /\ segment(u,w) INTER t = {}) /\ (?z:real^1. z IN t /\ z IN segment[u,v] /\ segment(v,z) INTER t = {})` STRIP_ASSUME_TAC THENL [CONJ_TAC THENL [MP_TAC(ISPECL [`segment[u:real^1,v] INTER t`; `u:real^1`] SEGMENT_TO_POINT_EXISTS); MP_TAC(ISPECL [`segment[u:real^1,v] INTER t`; `v:real^1`] SEGMENT_TO_POINT_EXISTS)] THEN (ASM_SIMP_TAC[CLOSED_INTER; CLOSED_SEGMENT] THEN ANTS_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `~(segment(u,v) INTER t = {}) ==> segment(u,v) SUBSET segment[u,v] ==> ~(segment[u,v] INTER t = {})`)) THEN REWRITE_TAC[SEGMENT_OPEN_SUBSET_CLOSED]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `w:real^1` THEN SIMP_TAC[IN_INTER] THEN MATCH_MP_TAC(SET_RULE `(w IN uv ==> uw SUBSET uv) ==> (w IN uv /\ w IN t) /\ (uw INTER uv INTER t = {}) ==> uw INTER t = {}`) THEN DISCH_TAC THEN REWRITE_TAC[open_segment] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s DIFF u SUBSET t`) THEN REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[GSYM SEGMENT_CONVEX_HULL; CONVEX_SEGMENT] THEN ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; ENDS_IN_SEGMENT]); SUBGOAL_THEN `(h:real^1->real^N) u = (f:real^1->real^N) w /\ (h:real^1->real^N) v = (f:real^1->real^N) z` (fun th -> REWRITE_TAC[th]) THENL [ASM_MESON_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(NORM_ARITH `!u. dist(w:real^N,u) < e / &2 /\ dist(z,u) < e / &2 ==> dist(w,z) < e`) THEN EXISTS_TAC `(f:real^1->real^N) u` THEN CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN (CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `x IN s ==> s SUBSET t ==> x IN t`)) THEN REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN MATCH_MP_TAC HULL_MINIMAL THEN ASM_REWRITE_TAC[CONVEX_INTERVAL; INSERT_SUBSET; EMPTY_SUBSET]; ASM_MESON_TAC[DIST_IN_CLOSED_SEGMENT; REAL_LET_TRANS; DIST_SYM]])]; X_GEN_TAC `z:real^N` THEN REWRITE_TAC[CONNECTED_IFF_CONNECTED_COMPONENT] THEN MAP_EVERY X_GEN_TAC [`u:real^1`; `v:real^1`] THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN REWRITE_TAC[connected_component] THEN EXISTS_TAC `segment[u:real^1,v]` THEN REWRITE_TAC[CONNECTED_SEGMENT; ENDS_IN_SEGMENT] THEN ASM_CASES_TAC `segment(u:real^1,v) INTER t = {}` THENL [REWRITE_TAC[SET_RULE `s SUBSET {x | x IN t /\ P x} <=> s SUBSET t /\ !x. x IN s ==> P x`] THEN CONJ_TAC THENL [ASM_MESON_TAC[CONVEX_CONTAINS_SEGMENT; CONVEX_INTERVAL]; X_GEN_TAC `x:real^1` THEN DISCH_TAC THEN SUBGOAL_THEN `segment(u:real^1,x) INTER t = {}` (fun th -> ASM_MESON_TAC[th]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `uv INTER t = {} ==> ux SUBSET uv ==> ux INTER t = {}`)) THEN UNDISCH_TAC `(x:real^1) IN segment[u,v]` THEN REWRITE_TAC[SEGMENT_1] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL_1; SUBSET_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPEC `t DIFF segment(u:real^1,v)`) THEN ASM_REWRITE_TAC[SET_RULE `t DIFF s PSUBSET t <=> ~(s INTER t = {})`] THEN MATCH_MP_TAC(TAUT `p ==> ~p ==> q`) THEN REPEAT CONJ_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC CLOSED_DIFF THEN ASM_REWRITE_TAC[OPEN_SEGMENT_1]; ASM SET_TAC[]; ASM_REWRITE_TAC[IN_DIFF] THEN MAP_EVERY UNDISCH_TAC [`(u:real^1) IN interval[vec 0,vec 1]`; `(v:real^1) IN interval[vec 0,vec 1]`] THEN REWRITE_TAC[SEGMENT_1] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC; ASM_REWRITE_TAC[IN_DIFF] THEN MAP_EVERY UNDISCH_TAC [`(u:real^1) IN interval[vec 0,vec 1]`; `(v:real^1) IN interval[vec 0,vec 1]`] THEN REWRITE_TAC[SEGMENT_1] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC; MAP_EVERY X_GEN_TAC [`x:real^1`; `y:real^1`] THEN REWRITE_TAC[IN_DIFF] THEN STRIP_TAC THEN ASM_CASES_TAC `segment(x:real^1,y) INTER segment(u,v) = {}` THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(segment(x:real^1,u) SUBSET segment(x,y) DIFF segment(u,v) /\ segment(y:real^1,v) SUBSET segment(x,y) DIFF segment(u,v)) \/ (segment(y:real^1,u) SUBSET segment(x,y) DIFF segment(u,v) /\ segment(x:real^1,v) SUBSET segment(x,y) DIFF segment(u,v))` MP_TAC THENL [MAP_EVERY UNDISCH_TAC [`~(x IN segment(u:real^1,v))`; `~(y IN segment(u:real^1,v))`; `~(segment(x:real^1,y) INTER segment (u,v) = {})`] THEN POP_ASSUM_LIST(K ALL_TAC) THEN MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`v:real^1`; `u:real^1`; `y:real^1`; `x:real^1`] THEN REWRITE_TAC[FORALL_LIFT] THEN MATCH_MP_TAC REAL_WLOG_LE THEN CONJ_TAC THENL [REWRITE_TAC[SEGMENT_SYM] THEN MESON_TAC[]; ALL_TAC] THEN REWRITE_TAC[FORALL_DROP; LIFT_DROP] THEN MAP_EVERY X_GEN_TAC [`x:real^1`; `y:real^1`] THEN DISCH_TAC THEN REWRITE_TAC[FORALL_LIFT] THEN MATCH_MP_TAC REAL_WLOG_LE THEN CONJ_TAC THENL [REWRITE_TAC[SEGMENT_SYM] THEN MESON_TAC[]; ALL_TAC] THEN REWRITE_TAC[FORALL_DROP; LIFT_DROP] THEN MAP_EVERY X_GEN_TAC [`u:real^1`; `v:real^1`] THEN DISCH_TAC THEN ASM_REWRITE_TAC[SEGMENT_1] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN REWRITE_TAC[IN_INTERVAL_1; SUBSET; IN_DIFF; AND_FORALL_THM] THEN ASM_REAL_ARITH_TAC; DISCH_THEN(DISJ_CASES_THEN(CONJUNCTS_THEN (let sl = SET_RULE `i SUBSET xy DIFF uv ==> xy INTER (t DIFF uv) = {} ==> i INTER t = {}` in fun th -> FIRST_ASSUM(MP_TAC o MATCH_MP (MATCH_MP sl th))))) THEN ASM_MESON_TAC[]]]; ASM_MESON_TAC[]]; DISCH_TAC] THEN SUBGOAL_THEN `?q:real^1->real^N. arc q /\ path_image q SUBSET path_image f /\ a IN path_image q /\ b IN path_image q` STRIP_ASSUME_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphic]) THEN REWRITE_TAC[homeomorphism] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `q:real^1->real^N` THEN REWRITE_TAC[arc; path; path_image] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THENL [ASM MESON_TAC[]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; path_image] THEN ASM SET_TAC[]; REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `vec 0:real^1` THEN REWRITE_TAC[ENDS_IN_UNIT_INTERVAL] THEN ASM_MESON_TAC[]; REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `vec 1:real^1` THEN REWRITE_TAC[ENDS_IN_UNIT_INTERVAL] THEN ASM_MESON_TAC[]]; SUBGOAL_THEN `?u v. u IN interval[vec 0,vec 1] /\ a = (q:real^1->real^N) u /\ v IN interval[vec 0,vec 1] /\ b = (q:real^1->real^N) v` STRIP_ASSUME_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[path_image]) THEN ASM SET_TAC[]; ALL_TAC] THEN EXISTS_TAC `subpath u v (q:real^1->real^N)` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC ARC_SIMPLE_PATH_SUBPATH THEN ASM_MESON_TAC[ARC_IMP_SIMPLE_PATH]; ASM_MESON_TAC[SUBSET_TRANS; PATH_IMAGE_SUBPATH_SUBSET; ARC_IMP_PATH]; ASM_MESON_TAC[pathstart; PATHSTART_SUBPATH]; ASM_MESON_TAC[pathfinish; PATHFINISH_SUBPATH]]]);; let PATH_CONNECTED_ARCWISE = prove (`!s:real^N->bool. path_connected s <=> !x y. x IN s /\ y IN s /\ ~(x = y) ==> ?g. arc g /\ path_image g SUBSET s /\ pathstart g = x /\ pathfinish g = y`, GEN_TAC THEN REWRITE_TAC[path_connected] THEN EQ_TAC THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `y:real^N`]) THEN ASM_REWRITE_TAC[] THENL [DISCH_THEN(X_CHOOSE_THEN `g:real^1->real^N` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`g:real^1->real^N`; `x:real^N`; `y:real^N`] PATH_CONTAINS_ARC) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN ASM_MESON_TAC[SUBSET_TRANS]; ASM_CASES_TAC `y:real^N = x` THEN ASM_REWRITE_TAC[] THENL [EXISTS_TAC `linepath(y:real^N,y)` THEN ASM_REWRITE_TAC[PATH_LINEPATH; PATHSTART_LINEPATH; PATHFINISH_LINEPATH; PATH_IMAGE_LINEPATH; SEGMENT_REFL; SING_SUBSET]; MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[ARC_IMP_PATH]]]);; let ARC_CONNECTED_TRANS = prove (`!g h:real^1->real^N. arc g /\ arc h /\ pathfinish g = pathstart h /\ ~(pathstart g = pathfinish h) ==> ?i. arc i /\ path_image i SUBSET (path_image g UNION path_image h) /\ pathstart i = pathstart g /\ pathfinish i = pathfinish h`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`g ++ h:real^1->real^N`; `pathstart(g):real^N`; `pathfinish(h):real^N`] PATH_CONTAINS_ARC) THEN ASM_SIMP_TAC[PATHSTART_JOIN; PATHFINISH_JOIN; PATH_JOIN_EQ; ARC_IMP_PATH; PATH_IMAGE_JOIN]);; (* ------------------------------------------------------------------------- *) (* Local connectedness and local path connectedness. *) (* ------------------------------------------------------------------------- *) let LOCALLY_CONNECTED_SPACE_SUBTOPOLOGY_EUCLIDEAN = prove (`!s:real^N->bool. locally_connected_space (subtopology euclidean s) <=> locally connected s`, GEN_TAC THEN SIMP_TAC[locally_connected_space; NEIGHBOURHOOD_BASE_OF_EUCLIDEAN] THEN GEN_REWRITE_TAC RAND_CONV [LOCALLY_AND_SUBSET] THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN REWRITE_TAC[CONNECTED_IN_SUBTOPOLOGY; CONNECTED_IN_EUCLIDEAN] THEN CONV_TAC TAUT);; let LOCALLY_CONNECTED,LOCALLY_CONNECTED_OPEN_CONNECTED_COMPONENT = (CONJ_PAIR o prove) (`(!s:real^N->bool. locally connected s <=> !v x. open_in (subtopology euclidean s) v /\ x IN v ==> ?u. open_in (subtopology euclidean s) u /\ connected u /\ x IN u /\ u SUBSET v) /\ (!s:real^N->bool. locally connected s <=> !t x. open_in (subtopology euclidean s) t /\ x IN t ==> open_in (subtopology euclidean s) (connected_component t x))`, REWRITE_TAC[AND_FORALL_THM; locally] THEN X_GEN_TAC `s:real^N->bool` THEN MATCH_MP_TAC(TAUT `(q ==> p) /\ (p ==> r) /\ (r ==> q) ==> (p <=> q) /\ (p <=> r)`) THEN REPEAT CONJ_TAC THENL [MESON_TAC[SUBSET_REFL]; DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `y:real^N`] THEN STRIP_TAC THEN ONCE_REWRITE_TAC[OPEN_IN_SUBOPEN] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN FIRST_ASSUM(SUBST1_TAC o SYM o MATCH_MP CONNECTED_COMPONENT_EQ) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`u:real^N->bool`; `x:real^N`]) THEN ANTS_TAC THENL [ASM_MESON_TAC[CONNECTED_COMPONENT_SUBSET; SUBSET]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `v:real^N->bool` (X_CHOOSE_THEN `a:real^N->bool` STRIP_ASSUME_TAC)) THEN EXISTS_TAC `v:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `a:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `x:real^N`] THEN STRIP_TAC THEN EXISTS_TAC `connected_component u (x:real^N)` THEN REWRITE_TAC[CONNECTED_COMPONENT_SUBSET; CONNECTED_CONNECTED_COMPONENT] THEN ASM_SIMP_TAC[IN; CONNECTED_COMPONENT_REFL]]);; let LOCALLY_PATH_CONNECTED_SPACE_SUBTOPOLOGY_EUCLIDEAN = prove (`!s:real^N->bool. locally_path_connected_space (subtopology euclidean s) <=> locally path_connected s`, GEN_TAC THEN SIMP_TAC[locally_path_connected_space; NEIGHBOURHOOD_BASE_OF_EUCLIDEAN] THEN GEN_REWRITE_TAC RAND_CONV [LOCALLY_AND_SUBSET] THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN REWRITE_TAC[PATH_CONNECTED_IN_SUBTOPOLOGY; PATH_CONNECTED_IN_EUCLIDEAN] THEN CONV_TAC TAUT);; let LOCALLY_PATH_CONNECTED,LOCALLY_PATH_CONNECTED_OPEN_PATH_COMPONENT = (CONJ_PAIR o prove) (`(!s:real^N->bool. locally path_connected s <=> !v x. open_in (subtopology euclidean s) v /\ x IN v ==> ?u. open_in (subtopology euclidean s) u /\ path_connected u /\ x IN u /\ u SUBSET v) /\ (!s:real^N->bool. locally path_connected s <=> !t x. open_in (subtopology euclidean s) t /\ x IN t ==> open_in (subtopology euclidean s) (path_component t x))`, REWRITE_TAC[AND_FORALL_THM; locally] THEN X_GEN_TAC `s:real^N->bool` THEN MATCH_MP_TAC(TAUT `(q ==> p) /\ (p ==> r) /\ (r ==> q) ==> (p <=> q) /\ (p <=> r)`) THEN REPEAT CONJ_TAC THENL [MESON_TAC[SUBSET_REFL]; DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `y:real^N`] THEN STRIP_TAC THEN ONCE_REWRITE_TAC[OPEN_IN_SUBOPEN] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN FIRST_ASSUM(SUBST1_TAC o SYM o MATCH_MP PATH_COMPONENT_EQ) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`u:real^N->bool`; `x:real^N`]) THEN ANTS_TAC THENL [ASM_MESON_TAC[PATH_COMPONENT_SUBSET; SUBSET]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `v:real^N->bool` (X_CHOOSE_THEN `a:real^N->bool` STRIP_ASSUME_TAC)) THEN EXISTS_TAC `v:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `a:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC PATH_COMPONENT_MAXIMAL THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `x:real^N`] THEN STRIP_TAC THEN EXISTS_TAC `path_component u (x:real^N)` THEN REWRITE_TAC[PATH_COMPONENT_SUBSET; PATH_CONNECTED_PATH_COMPONENT] THEN ASM_SIMP_TAC[IN; PATH_COMPONENT_REFL]]);; let LOCALLY_CONNECTED_OPEN_COMPONENT = prove (`!s:real^N->bool. locally connected s <=> !t c. open_in (subtopology euclidean s) t /\ c IN components t ==> open_in (subtopology euclidean s) c`, REWRITE_TAC[LOCALLY_CONNECTED_OPEN_CONNECTED_COMPONENT] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; components; FORALL_IN_GSPEC]);; let LOCALLY_CONNECTED_IM_KLEINEN = prove (`!s:real^N->bool. locally connected s <=> !v x. open_in (subtopology euclidean s) v /\ x IN v ==> ?u. open_in (subtopology euclidean s) u /\ x IN u /\ u SUBSET v /\ !y. y IN u ==> ?c. connected c /\ c SUBSET v /\ x IN c /\ y IN c`, GEN_TAC THEN EQ_TAC THENL [REWRITE_TAC[LOCALLY_CONNECTED] THEN MESON_TAC[SUBSET_REFL]; DISCH_TAC] THEN REWRITE_TAC[LOCALLY_CONNECTED_OPEN_COMPONENT] THEN MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `c:real^N->bool`] THEN STRIP_TAC THEN ONCE_REWRITE_TAC[OPEN_IN_SUBOPEN] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`u:real^N->bool`; `x:real^N`]) THEN ANTS_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_SUBSET; SUBSET]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `v:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `k:real^N->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `(k:real^N->bool) SUBSET c` MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC COMPONENTS_MAXIMAL THEN EXISTS_TAC `u:real^N->bool` THEN ASM SET_TAC[]);; let LOCALLY_PATH_CONNECTED_IM_KLEINEN = prove (`!s:real^N->bool. locally path_connected s <=> !v x. open_in (subtopology euclidean s) v /\ x IN v ==> ?u. open_in (subtopology euclidean s) u /\ x IN u /\ u SUBSET v /\ !y. y IN u ==> ?p. path p /\ path_image p SUBSET v /\ pathstart p = x /\ pathfinish p = y`, GEN_TAC THEN EQ_TAC THENL [REWRITE_TAC[LOCALLY_PATH_CONNECTED] THEN REWRITE_TAC[path_connected] THEN REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; REWRITE_TAC[LOCALLY_PATH_CONNECTED_OPEN_PATH_COMPONENT] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `z:real^N`] THEN STRIP_TAC THEN ONCE_REWRITE_TAC[OPEN_IN_SUBOPEN] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`u:real^N->bool`; `x:real^N`]) THEN ANTS_TAC THENL [ASM_MESON_TAC[PATH_COMPONENT_SUBSET; SUBSET]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `v:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `p:real^1->real^N` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `(path_image p) SUBSET path_component u (z:real^N)` MP_TAC THENL [ALL_TAC; ASM_MESON_TAC[PATHFINISH_IN_PATH_IMAGE; SUBSET]] THEN FIRST_ASSUM(SUBST1_TAC o SYM o MATCH_MP PATH_COMPONENT_EQ) THEN MATCH_MP_TAC PATH_COMPONENT_MAXIMAL THEN ASM_SIMP_TAC[PATH_CONNECTED_PATH_IMAGE] THEN ASM_MESON_TAC[PATHSTART_IN_PATH_IMAGE]]);; let LOCALLY_PATH_CONNECTED_IMP_LOCALLY_CONNECTED = prove (`!s:real^N->bool. locally path_connected s ==> locally connected s`, MESON_TAC[LOCALLY_MONO; PATH_CONNECTED_IMP_CONNECTED]);; let LOCALLY_CONNECTED_COMPONENTS = prove (`!s c:real^N->bool. locally connected s /\ c IN components s ==> locally connected c`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] LOCALLY_OPEN_SUBSET)) THEN FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [LOCALLY_CONNECTED_OPEN_COMPONENT]) THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[OPEN_IN_REFL]);; let LOCALLY_CONNECTED_CONNECTED_COMPONENT = prove (`!s x:real^N. locally connected s ==> locally connected (connected_component s x)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `connected_component s (x:real^N) = {}` THEN ASM_REWRITE_TAC[LOCALLY_EMPTY] THEN MATCH_MP_TAC LOCALLY_CONNECTED_COMPONENTS THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[IN_COMPONENTS] THEN ASM_MESON_TAC[CONNECTED_COMPONENT_EQ_EMPTY]);; let LOCALLY_PATH_CONNECTED_COMPONENTS = prove (`!s c:real^N->bool. locally path_connected s /\ c IN components s ==> locally path_connected c`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] LOCALLY_OPEN_SUBSET)) THEN FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [LOCALLY_CONNECTED_OPEN_COMPONENT] o MATCH_MP LOCALLY_PATH_CONNECTED_IMP_LOCALLY_CONNECTED) THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[OPEN_IN_REFL]);; let LOCALLY_PATH_CONNECTED_CONNECTED_COMPONENT = prove (`!s x:real^N. locally path_connected s ==> locally path_connected (connected_component s x)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `connected_component s (x:real^N) = {}` THEN ASM_REWRITE_TAC[LOCALLY_EMPTY] THEN MATCH_MP_TAC LOCALLY_PATH_CONNECTED_COMPONENTS THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[IN_COMPONENTS] THEN ASM_MESON_TAC[CONNECTED_COMPONENT_EQ_EMPTY]);; let OPEN_IMP_LOCALLY_PATH_CONNECTED = prove (`!s:real^N->bool. open s ==> locally path_connected s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC LOCALLY_MONO THEN EXISTS_TAC `convex:(real^N->bool)->bool` THEN REWRITE_TAC[CONVEX_IMP_PATH_CONNECTED] THEN ASM_SIMP_TAC[locally; OPEN_IN_OPEN_EQ] THEN ASM_MESON_TAC[OPEN_CONTAINS_BALL; CENTRE_IN_BALL; OPEN_BALL; CONVEX_BALL; SUBSET]);; let OPEN_IMP_LOCALLY_CONNECTED = prove (`!s:real^N->bool. open s ==> locally connected s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC LOCALLY_MONO THEN EXISTS_TAC `path_connected:(real^N->bool)->bool` THEN ASM_SIMP_TAC[OPEN_IMP_LOCALLY_PATH_CONNECTED; PATH_CONNECTED_IMP_CONNECTED]);; let LOCALLY_PATH_CONNECTED_UNIV = prove (`locally path_connected (:real^N)`, SIMP_TAC[OPEN_IMP_LOCALLY_PATH_CONNECTED; OPEN_UNIV]);; let LOCALLY_CONNECTED_UNIV = prove (`locally connected (:real^N)`, SIMP_TAC[OPEN_IMP_LOCALLY_CONNECTED; OPEN_UNIV]);; let OPEN_IN_CONNECTED_COMPONENT_LOCALLY_CONNECTED = prove (`!s x:real^N. locally connected s ==> open_in (subtopology euclidean s) (connected_component s x)`, REWRITE_TAC[LOCALLY_CONNECTED_OPEN_CONNECTED_COMPONENT] THEN REPEAT STRIP_TAC THEN ASM_CASES_TAC `(x:real^N) IN s` THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[OPEN_IN_SUBTOPOLOGY_REFL; SUBSET_UNIV; TOPSPACE_EUCLIDEAN]; ASM_MESON_TAC[OPEN_IN_EMPTY; CONNECTED_COMPONENT_EQ_EMPTY]]);; let OPEN_IN_COMPONENTS_LOCALLY_CONNECTED = prove (`!s c:real^N->bool. locally connected s /\ c IN components s ==> open_in (subtopology euclidean s) c`, MESON_TAC[LOCALLY_CONNECTED_OPEN_COMPONENT; OPEN_IN_REFL]);; let OPEN_IN_PATH_COMPONENT_LOCALLY_PATH_CONNECTED = prove (`!s x:real^N. locally path_connected s ==> open_in (subtopology euclidean s) (path_component s x)`, REWRITE_TAC[LOCALLY_PATH_CONNECTED_OPEN_PATH_COMPONENT] THEN REPEAT STRIP_TAC THEN ASM_CASES_TAC `(x:real^N) IN s` THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[OPEN_IN_SUBTOPOLOGY_REFL; SUBSET_UNIV; TOPSPACE_EUCLIDEAN]; ASM_MESON_TAC[OPEN_IN_EMPTY; PATH_COMPONENT_EQ_EMPTY]]);; let CLOSED_IN_PATH_COMPONENT_LOCALLY_PATH_CONNECTED = prove (`!s x:real^N. locally path_connected s ==> closed_in (subtopology euclidean s) (path_component s x)`, REWRITE_TAC[closed_in; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; PATH_COMPONENT_SUBSET] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[COMPLEMENT_PATH_COMPONENT_UNIONS] THEN MATCH_MP_TAC OPEN_IN_UNIONS THEN REWRITE_TAC[IMP_CONJ; FORALL_IN_GSPEC; IN_DELETE] THEN ASM_SIMP_TAC[OPEN_IN_PATH_COMPONENT_LOCALLY_PATH_CONNECTED]);; let CONVEX_IMP_LOCALLY_PATH_CONNECTED = prove (`!s:real^N->bool. convex s ==> locally path_connected s`, REPEAT STRIP_TAC THEN REWRITE_TAC[LOCALLY_PATH_CONNECTED] THEN MAP_EVERY X_GEN_TAC [`v:real^N->bool`; `x:real^N`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM SUBST_ALL_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTER]) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_BALL]) THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `s INTER ball(x:real^N,e)` THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[OPEN_IN_OPEN] THEN MESON_TAC[OPEN_BALL]; MATCH_MP_TAC CONVEX_IMP_PATH_CONNECTED THEN ASM_SIMP_TAC[CONVEX_INTER; CONVEX_BALL]; ASM_REWRITE_TAC[IN_INTER; CENTRE_IN_BALL]; ASM SET_TAC[]]);; let OPEN_IN_IMP_LOCALLY_PATH_CONNECTED = prove (`!s:real^N->bool. open_in (subtopology euclidean (affine hull s)) s ==> locally path_connected s`, GEN_TAC THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LOCALLY_OPEN_SUBSET) THEN MATCH_MP_TAC CONVEX_IMP_LOCALLY_PATH_CONNECTED THEN SIMP_TAC[AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL]);; let OPEN_IN_CONNECTED_COMPONENTS = prove (`!s c:real^N->bool. FINITE(components s) /\ c IN components s ==> open_in (subtopology euclidean s) c`, REWRITE_TAC[components; IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_GSPEC] THEN SIMP_TAC[OPEN_IN_CONNECTED_COMPONENT]);; let FINITE_COMPONENTS_MEETING_COMPACT_SUBSET = prove (`!k s:real^N->bool. compact k /\ locally connected s /\ k SUBSET s ==> FINITE {c | c IN components s /\ ~(c INTER k = {})}`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [COMPACT_EQ_HEINE_BOREL_SUBTOPOLOGY]) THEN DISCH_THEN(MP_TAC o SPEC `{k INTER c:real^N->bool |c| c IN {d | d IN components s /\ ~(d INTER k = {})}}`) THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN REWRITE_TAC[GSYM INTER_UNIONS] THEN ANTS_TAC THENL [CONJ_TAC THENL [X_GEN_TAC `c:real^N->bool` THEN STRIP_TAC THEN MATCH_MP_TAC OPEN_IN_SUBTOPOLOGY_INTER_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC OPEN_IN_INTER THEN REWRITE_TAC[OPEN_IN_REFL] THEN ASM_SIMP_TAC[OPEN_IN_COMPONENTS_LOCALLY_CONNECTED]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `k SUBSET s ==> k INTER s SUBSET t ==> k SUBSET k INTER t`)) THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [UNIONS_COMPONENTS] THEN SET_TAC[]]; ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN REWRITE_TAC[SIMPLE_IMAGE; EXISTS_FINITE_SUBSET_IMAGE] THEN REWRITE_TAC[GSYM SIMPLE_IMAGE; GSYM INTER_UNIONS] THEN REWRITE_TAC[SUBSET_INTER; SUBSET_REFL; LEFT_IMP_EXISTS_THM]] THEN X_GEN_TAC `p:(real^N->bool)->bool` THEN STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (MESON[] `FINITE s ==> t = s ==> FINITE t`)) THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `p SUBSET k ==> ~(p PSUBSET k) ==> k = p`)) THEN REWRITE_TAC[PSUBSET_ALT; IN_ELIM_THM] THEN DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` STRIP_ASSUME_TAC o CONJUNCT2) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN REWRITE_TAC[IN_INTER; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `a:real^N` THEN STRIP_TAC THEN SUBGOAL_THEN `(a:real^N) IN UNIONS p` MP_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[IN_UNIONS; NOT_EXISTS_THM]] THEN X_GEN_TAC `d:real^N->bool` THEN STRIP_TAC THEN MP_TAC(ISPEC `s:real^N->bool` PAIRWISE_DISJOINT_COMPONENTS) THEN REWRITE_TAC[pairwise] THEN DISCH_THEN(MP_TAC o SPECL [`c:real^N->bool`; `d:real^N->bool`]) THEN ASM_REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [ASM_MESON_TAC[]; ASM SET_TAC[]]);; let FINITE_COMPONENTS = prove (`!s:real^N->bool. compact s /\ locally connected s ==> FINITE(components s)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `s:real^N->bool`] FINITE_COMPONENTS_MEETING_COMPACT_SUBSET) THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN REWRITE_TAC[SET_RULE `{x | x IN s /\ P x} = s <=> !x. x IN s ==> P x`] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_NONEMPTY) THEN ASM SET_TAC[]);; let FINITE_LOCALLY_CONNECTED_CONNECTED_COMPONENTS = prove (`!s:real^N->bool. compact s /\ locally connected s ==> FINITE {connected_component s x |x| x IN s}`, REWRITE_TAC[GSYM components; FINITE_COMPONENTS]);; let FINITE_LOCALLY_PATH_CONNECTED_PATH_COMPONENTS = prove (`!s:real^N->bool. compact s /\ locally path_connected s ==> FINITE {path_component s x |x| x IN s}`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `{path_component s (x:real^N) |x| x IN s}` o GEN_REWRITE_RULE I [COMPACT_EQ_HEINE_BOREL_SUBTOPOLOGY]) THEN ASM_SIMP_TAC[OPEN_IN_PATH_COMPONENT_LOCALLY_PATH_CONNECTED; FORALL_IN_GSPEC; UNIONS_PATH_COMPONENT; SUBSET_REFL] THEN DISCH_THEN(X_CHOOSE_THEN `cs:(real^N->bool)->bool` MP_TAC) THEN ASM_CASES_TAC `{path_component s (x:real^N) |x| x IN s} = cs` THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC(TAUT `(p ==> ~r) ==> p /\ q /\ r ==> s`) THEN DISCH_TAC THEN SUBGOAL_THEN `?x:real^N. x IN s /\ ~(path_component s x IN cs)` MP_TAC THENL [ASM SET_TAC[]; SIMP_TAC[SUBSET; NOT_FORALL_THM]] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[NOT_IMP] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `?y:real^N. y IN s /\ x IN path_component s y /\ path_component s y IN cs` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o MATCH_MP PATH_COMPONENT_EQ) THEN ASM_MESON_TAC[]);; let CONVEX_IMP_LOCALLY_CONNECTED = prove (`!s:real^N->bool. convex s ==> locally connected s`, MESON_TAC[CONVEX_IMP_LOCALLY_PATH_CONNECTED; LOCALLY_PATH_CONNECTED_IMP_LOCALLY_CONNECTED]);; let HOMEOMORPHIC_LOCAL_CONNECTEDNESS = prove (`!s t. s homeomorphic t ==> (locally connected s <=> locally connected t)`, MATCH_MP_TAC HOMEOMORPHIC_LOCALLY THEN REWRITE_TAC[HOMEOMORPHIC_CONNECTEDNESS]);; let HOMEOMORPHISM_LOCAL_CONNECTEDNESS = prove (`!f:real^M->real^N g s t k. homeomorphism (s,t) (f,g) /\ k SUBSET s ==> (locally connected (IMAGE f k) <=> locally connected k)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_LOCAL_CONNECTEDNESS THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN REWRITE_TAC[homeomorphic] THEN MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `g:real^N->real^M`] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]);; let HOMEOMORPHIC_LOCAL_PATH_CONNECTEDNESS = prove (`!s t. s homeomorphic t ==> (locally path_connected s <=> locally path_connected t)`, MATCH_MP_TAC HOMEOMORPHIC_LOCALLY THEN REWRITE_TAC[HOMEOMORPHIC_PATH_CONNECTEDNESS]);; let HOMEOMORPHISM_LOCAL_PATH_CONNECTEDNESS = prove (`!f:real^M->real^N g s t k. homeomorphism (s,t) (f,g) /\ k SUBSET s ==> (locally path_connected (IMAGE f k) <=> locally path_connected k)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_LOCAL_PATH_CONNECTEDNESS THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN REWRITE_TAC[homeomorphic] THEN MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `g:real^N->real^M`] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]);; let LOCALLY_PATH_CONNECTED_TRANSLATION_EQ = prove (`!a:real^N s. locally path_connected (IMAGE (\x. a + x) s) <=> locally path_connected s`, MATCH_MP_TAC LOCALLY_TRANSLATION THEN REWRITE_TAC[PATH_CONNECTED_TRANSLATION_EQ]);; add_translation_invariants [LOCALLY_PATH_CONNECTED_TRANSLATION_EQ];; let LOCALLY_CONNECTED_TRANSLATION_EQ = prove (`!a:real^N s. locally connected (IMAGE (\x. a + x) s) <=> locally connected s`, MATCH_MP_TAC LOCALLY_TRANSLATION THEN REWRITE_TAC[CONNECTED_TRANSLATION_EQ]);; add_translation_invariants [LOCALLY_CONNECTED_TRANSLATION_EQ];; let LOCALLY_PATH_CONNECTED_LINEAR_IMAGE_EQ = prove (`!f:real^M->real^N s. linear f /\ (!x y. f x = f y ==> x = y) ==> (locally path_connected (IMAGE f s) <=> locally path_connected s)`, MATCH_MP_TAC LOCALLY_INJECTIVE_LINEAR_IMAGE THEN REWRITE_TAC[PATH_CONNECTED_LINEAR_IMAGE_EQ]);; add_linear_invariants [LOCALLY_PATH_CONNECTED_LINEAR_IMAGE_EQ];; let LOCALLY_CONNECTED_LINEAR_IMAGE_EQ = prove (`!f:real^M->real^N s. linear f /\ (!x y. f x = f y ==> x = y) ==> (locally connected (IMAGE f s) <=> locally connected s)`, MATCH_MP_TAC LOCALLY_INJECTIVE_LINEAR_IMAGE THEN REWRITE_TAC[CONNECTED_LINEAR_IMAGE_EQ]);; add_linear_invariants [LOCALLY_CONNECTED_LINEAR_IMAGE_EQ];; let LOCALLY_CONNECTED_QUOTIENT_IMAGE = prove (`!f:real^M->real^N s. (!t. t SUBSET IMAGE f s ==> (open_in (subtopology euclidean s) {x | x IN s /\ f x IN t} <=> open_in (subtopology euclidean (IMAGE f s)) t)) /\ locally connected s ==> locally connected (IMAGE f s)`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[GSYM LOCALLY_CONNECTED_SPACE_SUBTOPOLOGY_EUCLIDEAN] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LOCALLY_CONNECTED_SPACE_QUOTIENT_MAP_IMAGE) THEN EXISTS_TAC `f:real^M->real^N` THEN ASM_REWRITE_TAC[quotient_map; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY]);; let LOCALLY_PATH_CONNECTED_QUOTIENT_IMAGE = prove (`!f:real^M->real^N s. (!t. t SUBSET IMAGE f s ==> (open_in (subtopology euclidean s) {x | x IN s /\ f x IN t} <=> open_in (subtopology euclidean (IMAGE f s)) t)) /\ locally path_connected s ==> locally path_connected (IMAGE f s)`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[GSYM LOCALLY_PATH_CONNECTED_SPACE_SUBTOPOLOGY_EUCLIDEAN] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LOCALLY_PATH_CONNECTED_SPACE_QUOTIENT_MAP_IMAGE) THEN EXISTS_TAC `f:real^M->real^N` THEN ASM_REWRITE_TAC[quotient_map; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY]);; let LOCALLY_CONNECTED_CONTINUOUS_IMAGE_COMPACT = prove (`!f:real^M->real^N s. locally connected s /\ compact s /\ f continuous_on s ==> locally connected (IMAGE f s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC LOCALLY_CONNECTED_QUOTIENT_IMAGE THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CLOSED_MAP_IMP_QUOTIENT_MAP THEN ASM_SIMP_TAC[CLOSED_IN_CLOSED_EQ; COMPACT_IMP_CLOSED; COMPACT_CONTINUOUS_IMAGE; IMAGE_SUBSET] THEN ASM_MESON_TAC[COMPACT_IMP_CLOSED; COMPACT_CONTINUOUS_IMAGE; CONTINUOUS_ON_SUBSET; BOUNDED_SUBSET; COMPACT_EQ_BOUNDED_CLOSED]);; let LOCALLY_PATH_CONNECTED_CONTINUOUS_IMAGE_COMPACT = prove (`!f:real^M->real^N s. locally path_connected s /\ compact s /\ f continuous_on s ==> locally path_connected (IMAGE f s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC LOCALLY_PATH_CONNECTED_QUOTIENT_IMAGE THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CLOSED_MAP_IMP_QUOTIENT_MAP THEN ASM_SIMP_TAC[CLOSED_IN_CLOSED_EQ; COMPACT_IMP_CLOSED; COMPACT_CONTINUOUS_IMAGE; IMAGE_SUBSET] THEN ASM_MESON_TAC[COMPACT_IMP_CLOSED; COMPACT_CONTINUOUS_IMAGE; CONTINUOUS_ON_SUBSET; BOUNDED_SUBSET; COMPACT_EQ_BOUNDED_CLOSED]);; let LOCALLY_PATH_CONNECTED_PATH_IMAGE = prove (`!p:real^1->real^N. path p ==> locally path_connected (path_image p)`, REWRITE_TAC[path; path_image] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC LOCALLY_PATH_CONNECTED_CONTINUOUS_IMAGE_COMPACT THEN ASM_SIMP_TAC[COMPACT_INTERVAL; CONVEX_INTERVAL; CONVEX_IMP_LOCALLY_PATH_CONNECTED]);; let LOCALLY_CONNECTED_PATH_IMAGE = prove (`!p:real^1->real^N. path p ==> locally connected (path_image p)`, SIMP_TAC[LOCALLY_PATH_CONNECTED_PATH_IMAGE; LOCALLY_PATH_CONNECTED_IMP_LOCALLY_CONNECTED]);; let LOCALLY_CONNECTED_LEFT_INVERTIBLE_IMAGE = prove (`!f:real^M->real^N g s. f continuous_on s /\ g continuous_on (IMAGE f s) /\ (!x. x IN s ==> g(f x) = x) /\ locally connected s ==> locally connected (IMAGE f s)`, REPEAT GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LOCALLY_CONNECTED_QUOTIENT_IMAGE) THEN MATCH_MP_TAC CONTINUOUS_LEFT_INVERSE_IMP_QUOTIENT_MAP THEN ASM_MESON_TAC[]);; let LOCALLY_CONNECTED_RIGHT_INVERTIBLE_IMAGE = prove (`!f:real^M->real^N g s. f continuous_on s /\ g continuous_on (IMAGE f s) /\ IMAGE g (IMAGE f s) SUBSET s /\ (!x. x IN IMAGE f s ==> f(g x) = x) /\ locally connected s ==> locally connected (IMAGE f s)`, REPEAT GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LOCALLY_CONNECTED_QUOTIENT_IMAGE) THEN MATCH_MP_TAC CONTINUOUS_RIGHT_INVERSE_IMP_QUOTIENT_MAP THEN EXISTS_TAC `g:real^N->real^M` THEN ASM SET_TAC[]);; let LOCALLY_PATH_CONNECTED_LEFT_INVERTIBLE_IMAGE = prove (`!f:real^M->real^N g s. f continuous_on s /\ g continuous_on (IMAGE f s) /\ (!x. x IN s ==> g(f x) = x) /\ locally path_connected s ==> locally path_connected (IMAGE f s)`, REPEAT GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LOCALLY_PATH_CONNECTED_QUOTIENT_IMAGE) THEN MATCH_MP_TAC CONTINUOUS_LEFT_INVERSE_IMP_QUOTIENT_MAP THEN ASM_MESON_TAC[]);; let LOCALLY_PATH_CONNECTED_RIGHT_INVERTIBLE_IMAGE = prove (`!f:real^M->real^N g s. f continuous_on s /\ g continuous_on (IMAGE f s) /\ IMAGE g (IMAGE f s) SUBSET s /\ (!x. x IN IMAGE f s ==> f(g x) = x) /\ locally path_connected s ==> locally path_connected (IMAGE f s)`, REPEAT GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LOCALLY_PATH_CONNECTED_QUOTIENT_IMAGE) THEN MATCH_MP_TAC CONTINUOUS_RIGHT_INVERSE_IMP_QUOTIENT_MAP THEN EXISTS_TAC `g:real^N->real^M` THEN ASM SET_TAC[]);; let LOCALLY_CONNECTED_PCROSS = prove (`!s:real^M->bool t:real^N->bool. locally connected s /\ locally connected t ==> locally connected (s PCROSS t)`, MATCH_MP_TAC LOCALLY_PCROSS THEN REWRITE_TAC[CONNECTED_PCROSS]);; let LOCALLY_PATH_CONNECTED_PCROSS = prove (`!s:real^M->bool t:real^N->bool. locally path_connected s /\ locally path_connected t ==> locally path_connected (s PCROSS t)`, MATCH_MP_TAC LOCALLY_PCROSS THEN REWRITE_TAC[PATH_CONNECTED_PCROSS]);; let LOCALLY_CONNECTED_PCROSS_EQ = prove (`!s:real^M->bool t:real^N->bool. locally connected (s PCROSS t) <=> s = {} \/ t = {} \/ locally connected s /\ locally connected t`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_REWRITE_TAC[PCROSS_EMPTY; LOCALLY_EMPTY] THEN ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_REWRITE_TAC[PCROSS_EMPTY; LOCALLY_EMPTY] THEN EQ_TAC THEN REWRITE_TAC[LOCALLY_CONNECTED_PCROSS] THEN GEN_REWRITE_TAC LAND_CONV [LOCALLY_CONNECTED] THEN DISCH_TAC THEN REWRITE_TAC[LOCALLY_CONNECTED_IM_KLEINEN] THEN CONJ_TAC THENL [MAP_EVERY X_GEN_TAC [`u:real^M->bool`; `x:real^M`] THEN STRIP_TAC THEN UNDISCH_TAC `~(t:real^N->bool = {})` THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN DISCH_THEN(X_CHOOSE_TAC `y:real^N`) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`(u:real^M->bool) PCROSS (t:real^N->bool)`; `pastecart (x:real^M) (y:real^N)`]); MAP_EVERY X_GEN_TAC [`v:real^N->bool`; `y:real^N`] THEN STRIP_TAC THEN UNDISCH_TAC `~(s:real^M->bool = {})` THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN DISCH_THEN(X_CHOOSE_TAC `x:real^M`) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`(s:real^M->bool) PCROSS (v:real^N->bool)`; `pastecart (x:real^M) (y:real^N)`])] THEN ASM_SIMP_TAC[OPEN_IN_PCROSS_EQ; PASTECART_IN_PCROSS; SUBSET_UNIV; OPEN_IN_SUBTOPOLOGY_REFL; TOPSPACE_EUCLIDEAN; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `w:real^(M,N)finite_sum->bool` THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^M->bool`; `t:real^N->bool`; `w:real^(M,N)finite_sum->bool`; `x:real^M`; `y:real^N`] PASTECART_IN_INTERIOR_SUBTOPOLOGY) THEN ASM_REWRITE_TAC[] THENL [MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u':real^M->bool` THEN DISCH_THEN(X_CHOOSE_THEN `v:real^N->bool` STRIP_ASSUME_TAC) THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; X_GEN_TAC `z:real^M` THEN DISCH_TAC THEN EXISTS_TAC `IMAGE fstcart (w:real^(M,N)finite_sum->bool)` THEN ASM_SIMP_TAC[CONNECTED_LINEAR_IMAGE; LINEAR_FSTCART] THEN REWRITE_TAC[SUBSET; IN_IMAGE; EXISTS_PASTECART; FSTCART_PASTECART]]; DISCH_THEN(X_CHOOSE_THEN `u:real^M->bool` MP_TAC) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `v':real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; X_GEN_TAC `z:real^N` THEN DISCH_TAC THEN EXISTS_TAC `IMAGE sndcart (w:real^(M,N)finite_sum->bool)` THEN ASM_SIMP_TAC[CONNECTED_LINEAR_IMAGE; LINEAR_SNDCART] THEN REWRITE_TAC[SUBSET; IN_IMAGE; EXISTS_PASTECART; SNDCART_PASTECART]]] THEN RULE_ASSUM_TAC(REWRITE_RULE [SUBSET; FORALL_IN_PCROSS; PASTECART_IN_PCROSS; FORALL_PASTECART]) THEN ASM SET_TAC[]);; let LOCALLY_PATH_CONNECTED_PCROSS_EQ = prove (`!s:real^M->bool t:real^N->bool. locally path_connected (s PCROSS t) <=> s = {} \/ t = {} \/ locally path_connected s /\ locally path_connected t`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_REWRITE_TAC[PCROSS_EMPTY; LOCALLY_EMPTY] THEN ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_REWRITE_TAC[PCROSS_EMPTY; LOCALLY_EMPTY] THEN EQ_TAC THEN REWRITE_TAC[LOCALLY_PATH_CONNECTED_PCROSS] THEN GEN_REWRITE_TAC LAND_CONV [LOCALLY_PATH_CONNECTED] THEN DISCH_TAC THEN REWRITE_TAC[LOCALLY_PATH_CONNECTED_IM_KLEINEN] THEN CONJ_TAC THENL [MAP_EVERY X_GEN_TAC [`u:real^M->bool`; `x:real^M`] THEN STRIP_TAC THEN UNDISCH_TAC `~(t:real^N->bool = {})` THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN DISCH_THEN(X_CHOOSE_TAC `y:real^N`) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`(u:real^M->bool) PCROSS (t:real^N->bool)`; `pastecart (x:real^M) (y:real^N)`]); MAP_EVERY X_GEN_TAC [`v:real^N->bool`; `y:real^N`] THEN STRIP_TAC THEN UNDISCH_TAC `~(s:real^M->bool = {})` THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN DISCH_THEN(X_CHOOSE_TAC `x:real^M`) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`(s:real^M->bool) PCROSS (v:real^N->bool)`; `pastecart (x:real^M) (y:real^N)`])] THEN ASM_SIMP_TAC[OPEN_IN_PCROSS_EQ; PASTECART_IN_PCROSS; SUBSET_UNIV; OPEN_IN_SUBTOPOLOGY_REFL; TOPSPACE_EUCLIDEAN; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `w:real^(M,N)finite_sum->bool` THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^M->bool`; `t:real^N->bool`; `w:real^(M,N)finite_sum->bool`; `x:real^M`; `y:real^N`] PASTECART_IN_INTERIOR_SUBTOPOLOGY) THEN ASM_REWRITE_TAC[] THENL [MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u':real^M->bool` THEN DISCH_THEN(X_CHOOSE_THEN `v:real^N->bool` STRIP_ASSUME_TAC) THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; X_GEN_TAC `z:real^M` THEN DISCH_TAC THEN MP_TAC(ISPECL [`fstcart:real^(M,N)finite_sum->real^M`; `w:real^(M,N)finite_sum->bool`] PATH_CONNECTED_LINEAR_IMAGE) THEN ASM_REWRITE_TAC[LINEAR_FSTCART] THEN REWRITE_TAC[path_connected] THEN DISCH_THEN(MP_TAC o SPECL [`x:real^M`; `z:real^M`]) THEN ANTS_TAC THENL [REWRITE_TAC[IN_IMAGE; EXISTS_PASTECART; FSTCART_PASTECART]; MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[SUBSET; IN_IMAGE; EXISTS_PASTECART; FSTCART_PASTECART] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]]]; DISCH_THEN(X_CHOOSE_THEN `u:real^M->bool` MP_TAC) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `v':real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; X_GEN_TAC `z:real^N` THEN DISCH_TAC THEN MP_TAC(ISPECL [`sndcart:real^(M,N)finite_sum->real^N`; `w:real^(M,N)finite_sum->bool`] PATH_CONNECTED_LINEAR_IMAGE) THEN ASM_REWRITE_TAC[LINEAR_SNDCART] THEN REWRITE_TAC[path_connected] THEN DISCH_THEN(MP_TAC o SPECL [`y:real^N`; `z:real^N`]) THEN ANTS_TAC THENL [REWRITE_TAC[IN_IMAGE; EXISTS_PASTECART; SNDCART_PASTECART]; MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[SUBSET; IN_IMAGE; EXISTS_PASTECART; SNDCART_PASTECART] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]]]] THEN RULE_ASSUM_TAC(REWRITE_RULE [SUBSET; FORALL_IN_PCROSS; PASTECART_IN_PCROSS; FORALL_PASTECART]) THEN ASM SET_TAC[]);; let LOCALLY_CONNECTED_SUBREGION = prove (`!s t c:real^N->bool. locally connected s /\ t SUBSET s /\ connected c /\ open_in (subtopology euclidean t) c ==> ?c'. connected c' /\ open_in (subtopology euclidean s) c' /\ c = t INTER c'`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC) THEN ASM_CASES_TAC `s INTER u:real^N->bool = {}` THENL [EXISTS_TAC `{}:real^N->bool` THEN ASM_REWRITE_TAC[CONNECTED_EMPTY; OPEN_IN_EMPTY] THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`s INTER u:real^N->bool`; `t INTER u:real^N->bool`] EXISTS_COMPONENT_SUPERSET) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real^N->bool` THEN REPEAT STRIP_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]; MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `s INTER u:real^N->bool` THEN ASM_SIMP_TAC[OPEN_IN_OPEN_INTER] THEN MATCH_MP_TAC OPEN_IN_COMPONENTS_LOCALLY_CONNECTED THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] LOCALLY_OPEN_SUBSET)) THEN ASM_SIMP_TAC[OPEN_IN_OPEN_INTER]; ASM_REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; SUBSET_INTER; INTER_SUBSET] THEN FIRST_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN SET_TAC[]]);; let CARD_EQ_OPEN_IN = prove (`!u s:real^N->bool. locally connected u /\ open_in (subtopology euclidean u) s /\ (?x. x IN s /\ x limit_point_of u) ==> s =_c (:real)`, REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM CARD_LE_ANTISYM] THEN CONJ_TAC THENL [TRANS_TAC CARD_LE_TRANS `(:real^N)` THEN SIMP_TAC[CARD_EQ_IMP_LE; CARD_EQ_EUCLIDEAN] THEN MATCH_MP_TAC CARD_LE_SUBSET THEN REWRITE_TAC[SUBSET_UNIV]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` STRIP_ASSUME_TAC) THEN UNDISCH_TAC `(x:real^N) IN s` THEN ASM_REWRITE_TAC[IN_INTER] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LOCALLY_CONNECTED]) THEN DISCH_THEN(MP_TAC o SPECL [`u INTER t:real^N->bool`; `x:real^N`]) THEN ASM_SIMP_TAC[OPEN_IN_OPEN_INTER; IN_INTER] THEN REWRITE_TAC[OPEN_IN_OPEN; GSYM CONJ_ASSOC; LEFT_AND_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN REWRITE_TAC[UNWIND_THM2; IN_INTER] THEN DISCH_THEN(X_CHOOSE_THEN `v:real^N->bool` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [limit_point_of]) THEN DISCH_THEN(MP_TAC o SPEC `t INTER v:real^N->bool`) THEN ASM_SIMP_TAC[IN_INTER; OPEN_INTER] THEN DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN TRANS_TAC CARD_LE_TRANS `u INTER v:real^N->bool` THEN ASM_SIMP_TAC[CARD_LE_SUBSET] THEN MATCH_MP_TAC CARD_EQ_IMP_LE THEN ONCE_REWRITE_TAC[CARD_EQ_SYM] THEN MATCH_MP_TAC CARD_EQ_CONNECTED THEN ASM SET_TAC[]);; let CARD_EQ_OPEN_IN_AFFINE = prove (`!u s:real^N->bool. affine u /\ ~(aff_dim u = &0) /\ open_in (subtopology euclidean u) s /\ ~(s = {}) ==> s =_c (:real)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CARD_EQ_OPEN_IN THEN EXISTS_TAC `u:real^N->bool` THEN ASM_SIMP_TAC[CONVEX_IMP_LOCALLY_CONNECTED; AFFINE_IMP_CONVEX] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONNECTED_IMP_PERFECT_AFF_DIM THEN ASM_SIMP_TAC[AFFINE_IMP_CONVEX; CONVEX_CONNECTED] THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ASM SET_TAC[]);; let SEPARATION_BY_CLOSED_INTERMEDIATES = prove (`!u s:real^N->bool. ~connected(u DIFF s) ==> ?t. closed_in (subtopology euclidean u) t /\ t SUBSET s /\ !c. closed_in (subtopology euclidean u) c /\ t SUBSET c /\ c SUBSET s ==> ~connected(u DIFF c)`, REPEAT GEN_TAC THEN MP_TAC(ISPECL [`subtopology euclidean (u:real^N->bool)`; `s:real^N->bool`] SEPARATION_BY_CLOSED_INTERMEDIATES_GEN) THEN REWRITE_TAC[CONNECTED_IN_SUBTOPOLOGY; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN REWRITE_TAC[CONNECTED_IN_EUCLIDEAN; SUBSET_DIFF] THEN SIMP_TAC[METRIZABLE_IMP_HEREDITARILY_NORMAL_SPACE; METRIZABLE_SPACE_SUBTOPOLOGY; METRIZABLE_SPACE_EUCLIDEAN]);; let SEPARATION_BY_CLOSED_INTERMEDIATES_EQ = prove (`!u s:real^N->bool. locally connected u ==> (~connected(u DIFF s) <=> ?t. closed_in (subtopology euclidean u) t /\ t SUBSET s /\ !c. closed_in (subtopology euclidean u) c /\ t SUBSET c /\ c SUBSET s ==> ~connected(u DIFF c))`, REPEAT GEN_TAC THEN MP_TAC(ISPECL [`subtopology euclidean (u:real^N->bool)`; `s:real^N->bool`] SEPARATION_BY_CLOSED_INTERMEDIATES_EQ_GEN) THEN REWRITE_TAC[CONNECTED_IN_SUBTOPOLOGY; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN REWRITE_TAC[CONNECTED_IN_EUCLIDEAN; SUBSET_DIFF] THEN REWRITE_TAC[LOCALLY_CONNECTED_SPACE_SUBTOPOLOGY_EUCLIDEAN] THEN SIMP_TAC[METRIZABLE_IMP_HEREDITARILY_NORMAL_SPACE; METRIZABLE_SPACE_SUBTOPOLOGY; METRIZABLE_SPACE_EUCLIDEAN]);; let LOCALLY_CONNECTED_CLOSED_UNION_GEN = prove (`!s t u:real^N->bool. closed_in (subtopology euclidean u) s /\ closed_in (subtopology euclidean u) t /\ locally connected s /\ locally connected t ==> locally connected (s UNION t)`, REPEAT STRIP_TAC THEN REWRITE_TAC[locally] THEN MAP_EVERY X_GEN_TAC [`v:real^N->bool`; `a:real^N`] THEN STRIP_TAC THEN ASM_CASES_TAC `(a:real^N) IN s` THENL [ALL_TAC; MATCH_MP_TAC(MESON[] `(?x. P x x) ==> (?x y. P x y)`) THEN SUBGOAL_THEN `locally connected(t DIFF s:real^N->bool)` MP_TAC THENL [MATCH_MP_TAC LOCALLY_OPEN_SUBSET THEN EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[SET_RULE `t DIFF s = t DIFF (t INTER s)`] THEN MATCH_MP_TAC OPEN_IN_DIFF THEN REWRITE_TAC[OPEN_IN_REFL] THEN MATCH_MP_TAC CLOSED_IN_SUBTOPOLOGY_INTER_SUBSET THEN EXISTS_TAC `u:real^N->bool` THEN ASM_SIMP_TAC[CLOSED_IN_INTER; CLOSED_IN_REFL] THEN ASM_MESON_TAC[CLOSED_IN_IMP_SUBSET]; REWRITE_TAC[LOCALLY_CONNECTED] THEN DISCH_THEN(MP_TAC o SPECL [`v DIFF s:real^N->bool`; `a:real^N`]) THEN ASM_REWRITE_TAC[IN_DIFF] THEN ANTS_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN REWRITE_TAC[OPEN_IN_OPEN] THEN MATCH_MP_TAC MONO_EXISTS THEN SET_TAC[]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `w:real^N->bool` THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] OPEN_IN_TRANS) THEN ONCE_REWRITE_TAC[SET_RULE `t DIFF s = (s UNION t) DIFF s`] THEN MATCH_MP_TAC OPEN_IN_DIFF THEN REWRITE_TAC[OPEN_IN_REFL] THEN MATCH_MP_TAC CLOSED_IN_SUBSET_TRANS THEN EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET)) THEN SET_TAC[]]]] THEN ASM_CASES_TAC `(a:real^N) IN t` THENL [ALL_TAC; MATCH_MP_TAC(MESON[] `(?x. P x x) ==> (?x y. P x y)`) THEN SUBGOAL_THEN `locally connected(s DIFF t:real^N->bool)` MP_TAC THENL [MATCH_MP_TAC LOCALLY_OPEN_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[SET_RULE `t DIFF s = t DIFF (t INTER s)`] THEN MATCH_MP_TAC OPEN_IN_DIFF THEN REWRITE_TAC[OPEN_IN_REFL] THEN MATCH_MP_TAC CLOSED_IN_SUBTOPOLOGY_INTER_SUBSET THEN EXISTS_TAC `u:real^N->bool` THEN ASM_SIMP_TAC[CLOSED_IN_INTER; CLOSED_IN_REFL] THEN ASM_MESON_TAC[CLOSED_IN_IMP_SUBSET]; REWRITE_TAC[LOCALLY_CONNECTED] THEN DISCH_THEN(MP_TAC o SPECL [`v DIFF t:real^N->bool`; `a:real^N`]) THEN ASM_REWRITE_TAC[IN_DIFF] THEN ANTS_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN REWRITE_TAC[OPEN_IN_OPEN] THEN MATCH_MP_TAC MONO_EXISTS THEN SET_TAC[]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `w:real^N->bool` THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] OPEN_IN_TRANS) THEN ONCE_REWRITE_TAC[SET_RULE `s DIFF t = (s UNION t) DIFF t`] THEN MATCH_MP_TAC OPEN_IN_DIFF THEN REWRITE_TAC[OPEN_IN_REFL] THEN MATCH_MP_TAC CLOSED_IN_SUBSET_TRANS THEN EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET)) THEN SET_TAC[]]]] THEN UNDISCH_TAC `locally connected (t:real^N->bool)` THEN UNDISCH_TAC `locally connected (s:real^N->bool)` THEN REWRITE_TAC[LOCALLY_CONNECTED] THEN DISCH_THEN(MP_TAC o SPECL [`s INTER v:real^N->bool`; `a:real^N`]) THEN ASM_REWRITE_TAC[IN_INTER] THEN ANTS_TAC THENL [MATCH_MP_TAC OPEN_IN_SUBTOPOLOGY_INTER_SUBSET THEN EXISTS_TAC `s UNION t:real^N->bool` THEN ASM_SIMP_TAC[OPEN_IN_INTER; OPEN_IN_REFL; SUBSET_UNION]; GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [OPEN_IN_OPEN] THEN REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN GEN_REWRITE_TAC LAND_CONV [SWAP_EXISTS_THM] THEN ONCE_REWRITE_TAC[TAUT `(p /\ q) /\ r <=> q /\ p /\ r`] THEN REWRITE_TAC[UNWIND_THM2; IN_INTER; SUBSET_INTER] THEN DISCH_THEN(X_CHOOSE_THEN `m:real^N->bool` STRIP_ASSUME_TAC)] THEN DISCH_THEN(MP_TAC o SPECL [`t INTER v:real^N->bool`; `a:real^N`]) THEN ASM_REWRITE_TAC[IN_INTER] THEN ANTS_TAC THENL [MATCH_MP_TAC OPEN_IN_SUBTOPOLOGY_INTER_SUBSET THEN EXISTS_TAC `s UNION t:real^N->bool` THEN ASM_SIMP_TAC[OPEN_IN_INTER; OPEN_IN_REFL; SUBSET_UNION]; GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [OPEN_IN_OPEN] THEN REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN GEN_REWRITE_TAC LAND_CONV [SWAP_EXISTS_THM] THEN ONCE_REWRITE_TAC[TAUT `(p /\ q) /\ r <=> q /\ p /\ r`] THEN REWRITE_TAC[UNWIND_THM2; IN_INTER; SUBSET_INTER] THEN DISCH_THEN(X_CHOOSE_THEN `n:real^N->bool` STRIP_ASSUME_TAC)] THEN EXISTS_TAC `(s UNION t) INTER (m INTER n):real^N->bool` THEN EXISTS_TAC `(s INTER m) UNION (t INTER n):real^N->bool` THEN ASM_SIMP_TAC[OPEN_INTER; OPEN_IN_OPEN_INTER] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CONNECTED_UNION THEN ASM SET_TAC[]);; let LOCALLY_CONNECTED_CLOSED_UNION = prove (`!s t:real^N->bool. locally connected s /\ locally connected t /\ closed s /\ closed t ==> locally connected (s UNION t)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC LOCALLY_CONNECTED_CLOSED_UNION_GEN THEN EXISTS_TAC `(:real^N)` THEN ASM_REWRITE_TAC[SUBTOPOLOGY_UNIV; GSYM CLOSED_IN]);; let LOCALLY_CONNECTED_CLOSED_UNIONS = prove (`!f:(real^N->bool)->bool. FINITE f /\ (!s. s IN f ==> closed s /\ locally connected s) ==> locally connected (UNIONS f)`, REWRITE_TAC[IMP_CONJ] THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN SIMP_TAC[FORALL_IN_INSERT; NOT_IN_EMPTY; UNIONS_0; UNIONS_INSERT] THEN REWRITE_TAC[LOCALLY_EMPTY] THEN ASM_SIMP_TAC[LOCALLY_CONNECTED_CLOSED_UNION; CLOSED_UNIONS]);; let LOCALLY_CONNECTED_FROM_UNION_AND_INTER_GEN = prove (`!s t u:real^N->bool. closed_in (subtopology euclidean u) s /\ closed_in (subtopology euclidean u) t /\ locally connected (s UNION t) /\ locally connected (s INTER t) ==> locally connected s`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `closed_in (subtopology euclidean (s UNION t)) (s:real^N->bool) /\ closed_in (subtopology euclidean (s UNION t)) (t:real^N->bool)` MP_TAC THENL [CONJ_TAC THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] CLOSED_IN_SUBSET_TRANS)) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET)) THEN SET_TAC[]; REPEAT(FIRST_X_ASSUM(K ALL_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET)) THEN REPEAT STRIP_TAC] THEN REWRITE_TAC[LOCALLY_CONNECTED] THEN MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `x:real^N`] THEN STRIP_TAC THEN ASM_CASES_TAC `(x:real^N) IN t` THENL [ALL_TAC; SUBGOAL_THEN `locally connected (s DIFF t:real^N->bool)` MP_TAC THENL [MATCH_MP_TAC LOCALLY_OPEN_SUBSET THEN EXISTS_TAC `s UNION t:real^N->bool` THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[SET_RULE `s DIFF t = (s UNION t) DIFF t`] THEN MATCH_MP_TAC OPEN_IN_DIFF THEN ASM_REWRITE_TAC[OPEN_IN_REFL]; REWRITE_TAC[LOCALLY_CONNECTED] THEN DISCH_THEN(MP_TAC o SPECL [`u DIFF t:real^N->bool`; `x:real^N`]) THEN ASM_REWRITE_TAC[IN_DIFF] THEN ANTS_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN REWRITE_TAC[OPEN_IN_OPEN] THEN MATCH_MP_TAC MONO_EXISTS THEN SET_TAC[]; MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] OPEN_IN_TRANS) THEN MATCH_MP_TAC OPEN_IN_SUBSET_TRANS THEN EXISTS_TAC `s UNION t:real^N->bool` THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN ONCE_REWRITE_TAC[SET_RULE `s DIFF t = (s UNION t) DIFF t`] THEN MATCH_MP_TAC OPEN_IN_DIFF THEN ASM_REWRITE_TAC[OPEN_IN_REFL]]]] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN DISCH_THEN(X_CHOOSE_THEN `g:real^N->bool` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM SUBST_ALL_TAC THEN FIRST_X_ASSUM(CONJUNCTS_THEN ASSUME_TAC o REWRITE_RULE[IN_INTER]) THEN ABBREV_TAC `c = connected_component (s INTER t INTER g) (x:real^N)` THEN MP_TAC(ISPECL [`(s UNION t) INTER g:real^N->bool`; `s INTER t INTER g:real^N->bool`; `c:real^N->bool`] LOCALLY_CONNECTED_SUBREGION) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [MATCH_MP_TAC LOCALLY_OPEN_SUBSET THEN EXISTS_TAC `s UNION t:real^N->bool` THEN ASM_SIMP_TAC[OPEN_IN_OPEN_INTER]; SET_TAC[]; ASM_MESON_TAC[CONNECTED_CONNECTED_COMPONENT]; EXPAND_TAC "c" THEN MATCH_MP_TAC OPEN_IN_CONNECTED_COMPONENT_LOCALLY_CONNECTED THEN MATCH_MP_TAC LOCALLY_OPEN_SUBSET THEN EXISTS_TAC `s INTER t:real^N->bool` THEN ASM_SIMP_TAC[OPEN_IN_OPEN_INTER; GSYM INTER_ASSOC]]; DISCH_THEN(X_CHOOSE_THEN `h:real^N->bool` (STRIP_ASSUME_TAC o GSYM))] THEN EXISTS_TAC `s INTER h:real^N->bool` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_SUBTOPOLOGY_INTER_SUBSET THEN EXISTS_TAC `s UNION t:real^N->bool` THEN REWRITE_TAC[SUBSET_UNION] THEN MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `(s UNION t) INTER g:real^N->bool` THEN ASM_SIMP_TAC[OPEN_IN_OPEN_INTER] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN REWRITE_TAC[OPEN_IN_OPEN] THEN MATCH_MP_TAC MONO_EXISTS THEN SET_TAC[]; ALL_TAC; SUBGOAL_THEN `(x:real^N) IN c` MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN ASM_MESON_TAC[CONNECTED_COMPONENT_REFL_EQ; IN; IN_INTER]; FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ASM SET_TAC[]] THEN MATCH_MP_TAC(TAUT `!q. p /\ q ==> p`) THEN EXISTS_TAC `connected(t INTER h:real^N->bool)` THEN MATCH_MP_TAC CONNECTED_FROM_CLOSED_UNION_AND_INTER_LOCAL THEN EXISTS_TAC `h:real^N->bool` THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [CONJ_TAC THENL [UNDISCH_TAC `closed_in (subtopology euclidean (s UNION t)) (s:real^N->bool)`; UNDISCH_TAC `closed_in (subtopology euclidean (s UNION t)) (t:real^N->bool)`] THEN REWRITE_TAC[CLOSED_IN_CLOSED] THEN MATCH_MP_TAC MONO_EXISTS THEN FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ASM SET_TAC[]; SUBGOAL_THEN `s INTER h UNION t INTER h:real^N->bool = h` SUBST1_TAC THENL [FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ASM SET_TAC[]; ASM_REWRITE_TAC[]] THEN SUBGOAL_THEN `(s INTER h) INTER t INTER h:real^N->bool = c` SUBST1_TAC THENL [FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ASM SET_TAC[]; ASM_MESON_TAC[CONNECTED_CONNECTED_COMPONENT]]]);; let LOCALLY_CONNECTED_FROM_UNION_AND_INTER = prove (`!s t:real^N->bool. closed s /\ closed t /\ locally connected (s UNION t) /\ locally connected (s INTER t) ==> locally connected s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC LOCALLY_CONNECTED_FROM_UNION_AND_INTER_GEN THEN MAP_EVERY EXISTS_TAC [`t:real^N->bool`; `s UNION t:real^N->bool`] THEN ASM_SIMP_TAC[CLOSED_SUBSET; SUBSET_UNION]);; let LOCALLY_CONNECTED_CLOSURE_FROM_FRONTIER = prove (`!s:real^N->bool. locally connected (frontier s) ==> locally connected (closure s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC LOCALLY_CONNECTED_FROM_UNION_AND_INTER THEN EXISTS_TAC `closure((:real^N) DIFF s)` THEN ASM_REWRITE_TAC[GSYM FRONTIER_CLOSURES; CLOSED_CLOSURE] THEN SUBGOAL_THEN `closure s UNION closure ((:real^N) DIFF s) = (:real^N)` (fun th -> REWRITE_TAC[th; LOCALLY_CONNECTED_UNIV]) THEN MATCH_MP_TAC(SET_RULE `s SUBSET closure s /\ (:real^N) DIFF s SUBSET closure((:real^N) DIFF s) ==> closure s UNION closure ((:real^N) DIFF s) = (:real^N)`) THEN REWRITE_TAC[CLOSURE_SUBSET]);; let PATH_CONNECTED_FROM_CLOSED_UNION_AND_INTER_LOCAL, PATH_CONNECTED_FROM_OPEN_UNION_AND_INTER_LOCAL = (CONJ_PAIR o prove) (`(!u s t:real^N->bool. closed_in (subtopology euclidean u) s /\ closed_in (subtopology euclidean u) t /\ path_connected (s UNION t) /\ path_connected (s INTER t) ==> path_connected s /\ path_connected t) /\ (!u s t:real^N->bool. open_in (subtopology euclidean u) s /\ open_in (subtopology euclidean u) t /\ path_connected (s UNION t) /\ path_connected (s INTER t) ==> path_connected s /\ path_connected t)`, let lemma0 = prove (`!g u s:real^N->bool. closed_in (subtopology euclidean u) s /\ path g /\ path_image g SUBSET u /\ ~DISJOINT (path_image g) s ==> ?p. p IN interval[vec 0,vec 1] /\ g p IN s /\ !x. x IN interval[vec 0,vec 1] /\ drop x < drop p ==> ~(g x IN s)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`{x | x IN interval[vec 0,vec 1] /\ (g:real^1->real^N) x IN s}`; `vec 0:real^1`] DISTANCE_ATTAINS_INF) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC CLOSED_IN_CLOSED_TRANS THEN EXISTS_TAC `interval[vec 0:real^1,vec 1]` THEN REWRITE_TAC[CLOSED_INTERVAL] THEN MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE_GEN THEN EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[GSYM path; GSYM path_image] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET)) THEN ASM SET_TAC[]; RULE_ASSUM_TAC(REWRITE_RULE[path_image]) THEN ASM SET_TAC[]]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:real^1` THEN REWRITE_TAC[IN_ELIM_THM; DIST_0; NORM_1; IN_INTERVAL_1; DROP_VEC] THEN DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN ASM_SIMP_TAC[real_abs] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `y:real^1` THEN ASM_CASES_TAC `(g:real^1->real^N) y IN s` THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC]) in let lemma1 = prove (`!g s t u:real^N->bool. (closed_in (subtopology euclidean u) s /\ closed_in (subtopology euclidean u) t \/ open_in (subtopology euclidean u) s /\ open_in (subtopology euclidean u) t) /\ path g /\ pathstart g IN s /\ path_image g SUBSET s UNION t /\ ~(path_image g SUBSET s) ==> ?p. p IN interval[vec 0,vec 1] /\ g p IN t /\ !x. x IN interval[vec 0,p] ==> g x IN s`, REPEAT STRIP_TAC THENL [SUBGOAL_THEN `(s:real^N->bool) SUBSET u /\ (t:real^N->bool) SUBSET u` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[CLOSED_IN_IMP_SUBSET]; ALL_TAC] THEN MP_TAC(ISPECL [`g:real^1->real^N`; `u:real^N->bool`; `t:real^N->bool`] lemma0) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS] THEN X_GEN_TAC `p:real^1` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `q:real^1` THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN STRIP_TAC THEN ASM_CASES_TAC `q:real^1 = vec 0` THENL [ASM_MESON_TAC[pathstart]; ALL_TAC] THEN ASM_CASES_TAC `p:real^1 = vec 0` THENL [RULE_ASSUM_TAC(REWRITE_RULE[GSYM DROP_EQ; DROP_VEC]) THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `&0 < drop p` ASSUME_TAC THENL [ASM_REWRITE_TAC[REAL_LT_LE; GSYM LIFT_EQ; LIFT_NUM; LIFT_DROP] THEN ASM_MESON_TAC[IN_INTERVAL_1; DROP_VEC]; ALL_TAC] THEN ASM_CASES_TAC `q:real^1 = p` THEN ASM_REWRITE_TAC[] THENL [ALL_TAC; RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN FIRST_X_ASSUM(MP_TAC o SPEC `q:real^1`) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[REAL_LT_LE; GSYM LIFT_EQ; LIFT_DROP]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `p SUBSET s UNION t ==> y IN p ==> ~(y IN t) ==> y IN s`)) THEN REWRITE_TAC[path_image; IN_IMAGE] THEN EXISTS_TAC `q:real^1`] THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN ASM_REAL_ARITH_TAC] THEN SUBGOAL_THEN `p IN {x | x IN interval[vec 0,vec 1] /\ (g:real^1->real^N) x IN s}` MP_TAC THENL [ALL_TAC; SET_TAC[]] THEN MATCH_MP_TAC(SET_RULE `!s. x IN closure s /\ closure s SUBSET t ==> x IN t`) THEN EXISTS_TAC `interval(vec 0:real^1,p)` THEN CONJ_TAC THENL [ASM_SIMP_TAC[CLOSURE_OPEN_INTERVAL; INTERVAL_NE_EMPTY_1; DROP_VEC; ENDS_IN_INTERVAL; REAL_LT_IMP_LE]; MATCH_MP_TAC CLOSURE_MINIMAL] THEN CONJ_TAC THENL [REWRITE_TAC[SUBSET; IN_INTERVAL_1; IN_ELIM_THM; DROP_VEC] THEN X_GEN_TAC `r:real^1` THEN STRIP_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPEC `r:real^1`) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `p SUBSET s UNION t ==> y IN p ==> ~(y IN t) ==> y IN s`)) THEN REWRITE_TAC[path_image; IN_IMAGE] THEN EXISTS_TAC `r:real^1` THEN ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN ASM_REAL_ARITH_TAC; MATCH_MP_TAC CLOSED_IN_CLOSED_TRANS THEN EXISTS_TAC `interval[vec 0:real^1,vec 1]` THEN REWRITE_TAC[CLOSED_INTERVAL] THEN MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE_GEN THEN EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[GSYM path; GSYM path_image] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET)) THEN ASM SET_TAC[]]; SUBGOAL_THEN `(s:real^N->bool) SUBSET u /\ (t:real^N->bool) SUBSET u` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[OPEN_IN_IMP_SUBSET]; ALL_TAC] THEN MP_TAC(ISPECL [`g:real^1->real^N`; `u:real^N->bool`; `u DIFF s:real^N->bool`] lemma0) THEN ASM_SIMP_TAC[CLOSED_IN_DIFF; CLOSED_IN_REFL] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `p:real^1` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `(g:real^1->real^N) p IN t` ASSUME_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[path_image]) THEN ASM SET_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `p:real^1 = vec 0` THENL [EXISTS_TAC `vec 0:real^1` THEN REWRITE_TAC[INTERVAL_SING; ENDS_IN_UNIT_INTERVAL] THEN RULE_ASSUM_TAC(REWRITE_RULE[pathstart]) THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `&0 < drop p` ASSUME_TAC THENL [ASM_REWRITE_TAC[REAL_LT_LE; GSYM LIFT_EQ; LIFT_NUM; LIFT_DROP] THEN ASM_MESON_TAC[IN_INTERVAL_1; DROP_VEC]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL_1]) THEN REWRITE_TAC[DROP_VEC] THEN STRIP_TAC THEN MP_TAC(ISPECL [`interval[vec 0:real^1,vec 1]`; `{x | x IN interval[vec 0,vec 1] /\ (g:real^1->real^N) x IN t}`; `interval(vec 0:real^1,p)`] OPEN_IN_INTER_CLOSURE_EQ_EMPTY) THEN ASM_SIMP_TAC[CLOSURE_OPEN_INTERVAL; INTERVAL_NE_EMPTY_1; DROP_VEC] THEN ASM_REWRITE_TAC[SUBSET_INTERVAL_1; DROP_VEC; REAL_LE_REFL] THEN ANTS_TAC THENL [MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE_GEN THEN EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[GSYM path; GSYM path_image] THEN ASM SET_TAC[]; DISCH_THEN(MP_TAC o MATCH_MP (TAUT `(p <=> q) ==> ~p ==> ~q`))] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_ELIM_THM; IN_INTER] THEN ANTS_TAC THENL [EXISTS_TAC `p:real^1` THEN ASM_REWRITE_TAC[ENDS_IN_INTERVAL] THEN REWRITE_TAC[INTERVAL_NE_EMPTY_1; DROP_VEC] THEN ASM_REAL_ARITH_TAC; MATCH_MP_TAC MONO_EXISTS] THEN X_GEN_TAC `q:real^1` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `r:real^1` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `r:real^1`) THEN ANTS_TAC THENL [ALL_TAC; MATCH_MP_TAC(SET_RULE `x IN u ==> ~(x IN u DIFF s) ==> x IN s`) THEN RULE_ASSUM_TAC(REWRITE_RULE[path_image; SUBSET; FORALL_IN_IMAGE]) THEN FIRST_X_ASSUM(MP_TAC o SPEC `r:real^1`) THEN ANTS_TAC THENL [ALL_TAC; ASM SET_TAC[]]] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN ASM_REAL_ARITH_TAC]) in REWRITE_TAC[AND_FORALL_THM] THEN GEN_TAC THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN REWRITE_TAC[TAUT `(p1 /\ q ==> r) /\ (p2 /\ q ==> r) <=> (p1 \/ p2) /\ q ==> r`] THEN MATCH_MP_TAC(MESON[] `(!x y. R x y ==> R y x) /\ (!x y. R x y ==> P x) ==> !x y. R x y ==> P x /\ P y`) THEN CONJ_TAC THENL [REWRITE_TAC[INTER_COMM; UNION_COMM; CONJ_ACI]; ALL_TAC] THEN REPEAT GEN_TAC THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN REWRITE_TAC[path_connected] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN STRIP_TAC THEN UNDISCH_TAC `path_connected (s UNION t:real^N->bool)` THEN REWRITE_TAC[path_connected] THEN DISCH_THEN(MP_TAC o SPECL [`a:real^N`; `b:real^N`]) THEN ASM_REWRITE_TAC[IN_UNION] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^1->real^N` STRIP_ASSUME_TAC) THEN ASM_CASES_TAC `(path_image g:real^N->bool) SUBSET s` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?p q. p IN interval[vec 0,vec 1] /\ q IN interval[vec 0,vec 1] /\ (g:real^1->real^N) p IN s /\ g p IN t /\ g q IN s /\ g q IN t /\ (!x. &0 <= drop x /\ drop x <= &1 /\ (drop x <= drop p \/ drop q <= drop x) ==> g x IN s)` STRIP_ASSUME_TAC THENL [MP_TAC(ISPECL [`g:real^1->real^N`; `s:real^N->bool`; `t:real^N->bool`; `u:real^N->bool`] lemma1) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `p:real^1` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPECL [`reversepath g:real^1->real^N`; `s:real^N->bool`; `t:real^N->bool`; `u:real^N->bool`] lemma1) THEN ASM_REWRITE_TAC[PATH_REVERSEPATH; PATH_IMAGE_REVERSEPATH] THEN ASM_REWRITE_TAC[PATHSTART_REVERSEPATH] THEN REWRITE_TAC[reversepath; o_THM] THEN DISCH_THEN(X_CHOOSE_THEN `q:real^1` STRIP_ASSUME_TAC) THEN EXISTS_TAC `vec 1 - q:real^1` THEN ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_SUB; DROP_VEC] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; REWRITE_TAC[CONJ_ASSOC]] THEN MATCH_MP_TAC(TAUT `(q ==> p) /\ q ==> p /\ q`) THEN CONJ_TAC THENL [DISCH_THEN(fun th -> CONJ_TAC THEN MATCH_MP_TAC th); X_GEN_TAC `x:real^1` THEN STRIP_TAC THEN ASM_SIMP_TAC[] THEN SUBST1_TAC(VECTOR_ARITH `x:real^1 = vec 1 - (vec 1 - x)`) THEN FIRST_X_ASSUM MATCH_MP_TAC] THEN ASM_REWRITE_TAC[DROP_VEC; DROP_SUB] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [path_connected]) THEN DISCH_THEN(MP_TAC o SPECL [`(g:real^1->real^N) p`; `(g:real^1->real^N) q`]) THEN ASM_REWRITE_TAC[IN_INTER; SUBSET_INTER] THEN DISCH_THEN(X_CHOOSE_THEN `h:real^1->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `subpath (vec 0) p g ++ (h:real^1->real^N) ++ subpath q (vec 1) g` THEN ASM_SIMP_TAC[PATHSTART_JOIN; PATHFINISH_JOIN; PATHSTART_SUBPATH; PATHFINISH_SUBPATH; PATH_JOIN; PATH_SUBPATH; IN_INTERVAL_1; DROP_VEC; REAL_POS; REAL_LE_REFL] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[pathstart; pathfinish]] THEN REPEAT(MATCH_MP_TAC SUBSET_PATH_IMAGE_JOIN THEN CONJ_TAC) THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `g SUBSET s UNION t ==> g' SUBSET g /\ (!x. x IN g' ==> x IN s) ==> g' SUBSET s`)) THEN ASM_SIMP_TAC[PATH_IMAGE_SUBPATH_SUBSET; IN_INTERVAL_1; DROP_VEC; REAL_LE_REFL; REAL_POS] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN ASM_SIMP_TAC[PATH_IMAGE_SUBPATH; DROP_VEC; FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REAL_ARITH_TAC);; let PATH_CONNECTED_FROM_CLOSED_UNION_AND_INTER = prove (`!s t:real^N->bool. closed s /\ closed t /\ path_connected (s UNION t) /\ path_connected (s INTER t) ==> path_connected s /\ path_connected t`, REWRITE_TAC[CLOSED_IN] THEN ONCE_REWRITE_TAC[GSYM SUBTOPOLOGY_UNIV] THEN REWRITE_TAC[PATH_CONNECTED_FROM_CLOSED_UNION_AND_INTER_LOCAL]);; let PATH_CONNECTED_CLOSURE_FROM_FRONTIER = prove (`!s:real^N->bool. path_connected(frontier s) ==> path_connected(closure s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC(TAUT `!q. p /\ q ==> p`) THEN EXISTS_TAC `path_connected(closure((:real^N) DIFF s))` THEN MATCH_MP_TAC PATH_CONNECTED_FROM_CLOSED_UNION_AND_INTER THEN ASM_REWRITE_TAC[CLOSED_CLOSURE; GSYM FRONTIER_CLOSURES] THEN SUBGOAL_THEN `closure s UNION closure ((:real^N) DIFF s) = (:real^N)` (fun th -> REWRITE_TAC[th; PATH_CONNECTED_UNIV]) THEN MATCH_MP_TAC(SET_RULE `s SUBSET closure s /\ (:real^N) DIFF s SUBSET closure((:real^N) DIFF s) ==> closure s UNION closure ((:real^N) DIFF s) = (:real^N)`) THEN REWRITE_TAC[CLOSURE_SUBSET]);; let LOCALLY_PATH_CONNECTED_SUBREGION = prove (`!s t c:real^N->bool. locally path_connected s /\ t SUBSET s /\ path_connected c /\ open_in (subtopology euclidean t) c ==> ?c'. path_connected c' /\ open_in (subtopology euclidean s) c' /\ c = t INTER c'`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC) THEN ASM_CASES_TAC `s INTER u:real^N->bool = {}` THENL [EXISTS_TAC `{}:real^N->bool` THEN ASM_REWRITE_TAC[PATH_CONNECTED_EMPTY; OPEN_IN_EMPTY] THEN ASM SET_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `t INTER u:real^N->bool = {}` THENL [EXISTS_TAC `{}:real^N->bool` THEN ASM_REWRITE_TAC[PATH_CONNECTED_EMPTY; OPEN_IN_EMPTY; INTER_EMPTY]; ALL_TAC] THEN SUBGOAL_THEN `?a:real^N. a IN t /\ a IN u` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN EXISTS_TAC `path_component (s INTER u) (a:real^N)` THEN REWRITE_TAC[PATH_CONNECTED_PATH_COMPONENT] THEN CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `s INTER u:real^N->bool` THEN ASM_SIMP_TAC[OPEN_IN_OPEN_INTER] THEN MATCH_MP_TAC OPEN_IN_PATH_COMPONENT_LOCALLY_PATH_CONNECTED THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] LOCALLY_OPEN_SUBSET)) THEN ASM_SIMP_TAC[OPEN_IN_OPEN_INTER]; ASM_REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; SUBSET_INTER; INTER_SUBSET] THEN CONJ_TAC THENL [MATCH_MP_TAC PATH_COMPONENT_MAXIMAL THEN ASM_REWRITE_TAC[IN_INTER] THEN ASM SET_TAC[]; MP_TAC(ISPECL [`s INTER u:real^N->bool`; `a:real^N`] PATH_COMPONENT_SUBSET) THEN ASM SET_TAC[]]]);; let LOCALLY_PATH_CONNECTED_FROM_UNION_AND_INTER_GEN = prove (`!s t u:real^N->bool. closed_in (subtopology euclidean u) s /\ closed_in (subtopology euclidean u) t /\ locally path_connected (s UNION t) /\ locally path_connected (s INTER t) ==> locally path_connected s`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `closed_in (subtopology euclidean (s UNION t)) (s:real^N->bool) /\ closed_in (subtopology euclidean (s UNION t)) (t:real^N->bool)` MP_TAC THENL [CONJ_TAC THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] CLOSED_IN_SUBSET_TRANS)) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET)) THEN SET_TAC[]; REPEAT(FIRST_X_ASSUM(K ALL_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET)) THEN REPEAT STRIP_TAC] THEN REWRITE_TAC[LOCALLY_PATH_CONNECTED] THEN MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `x:real^N`] THEN STRIP_TAC THEN ASM_CASES_TAC `(x:real^N) IN t` THENL [ALL_TAC; SUBGOAL_THEN `locally path_connected (s DIFF t:real^N->bool)` MP_TAC THENL [MATCH_MP_TAC LOCALLY_OPEN_SUBSET THEN EXISTS_TAC `s UNION t:real^N->bool` THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[SET_RULE `s DIFF t = (s UNION t) DIFF t`] THEN MATCH_MP_TAC OPEN_IN_DIFF THEN ASM_REWRITE_TAC[OPEN_IN_REFL]; REWRITE_TAC[LOCALLY_PATH_CONNECTED] THEN DISCH_THEN(MP_TAC o SPECL [`u DIFF t:real^N->bool`; `x:real^N`]) THEN ASM_REWRITE_TAC[IN_DIFF] THEN ANTS_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN REWRITE_TAC[OPEN_IN_OPEN] THEN MATCH_MP_TAC MONO_EXISTS THEN SET_TAC[]; MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] OPEN_IN_TRANS) THEN MATCH_MP_TAC OPEN_IN_SUBSET_TRANS THEN EXISTS_TAC `s UNION t:real^N->bool` THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN ONCE_REWRITE_TAC[SET_RULE `s DIFF t = (s UNION t) DIFF t`] THEN MATCH_MP_TAC OPEN_IN_DIFF THEN ASM_REWRITE_TAC[OPEN_IN_REFL]]]] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN DISCH_THEN(X_CHOOSE_THEN `g:real^N->bool` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM SUBST_ALL_TAC THEN FIRST_X_ASSUM(CONJUNCTS_THEN ASSUME_TAC o REWRITE_RULE[IN_INTER]) THEN ABBREV_TAC `c = path_component (s INTER t INTER g) (x:real^N)` THEN MP_TAC(ISPECL [`(s UNION t) INTER g:real^N->bool`; `s INTER t INTER g:real^N->bool`; `c:real^N->bool`] LOCALLY_PATH_CONNECTED_SUBREGION) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [MATCH_MP_TAC LOCALLY_OPEN_SUBSET THEN EXISTS_TAC `s UNION t:real^N->bool` THEN ASM_SIMP_TAC[OPEN_IN_OPEN_INTER]; SET_TAC[]; ASM_MESON_TAC[PATH_CONNECTED_PATH_COMPONENT]; EXPAND_TAC "c" THEN MATCH_MP_TAC OPEN_IN_PATH_COMPONENT_LOCALLY_PATH_CONNECTED THEN MATCH_MP_TAC LOCALLY_OPEN_SUBSET THEN EXISTS_TAC `s INTER t:real^N->bool` THEN ASM_SIMP_TAC[OPEN_IN_OPEN_INTER; GSYM INTER_ASSOC]]; DISCH_THEN(X_CHOOSE_THEN `h:real^N->bool` (STRIP_ASSUME_TAC o GSYM))] THEN EXISTS_TAC `s INTER h:real^N->bool` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_SUBTOPOLOGY_INTER_SUBSET THEN EXISTS_TAC `s UNION t:real^N->bool` THEN REWRITE_TAC[SUBSET_UNION] THEN MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `(s UNION t) INTER g:real^N->bool` THEN ASM_SIMP_TAC[OPEN_IN_OPEN_INTER] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN REWRITE_TAC[OPEN_IN_OPEN] THEN MATCH_MP_TAC MONO_EXISTS THEN SET_TAC[]; ALL_TAC; SUBGOAL_THEN `(x:real^N) IN c` MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN ASM_MESON_TAC[PATH_COMPONENT_REFL_EQ; IN; IN_INTER]; FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ASM SET_TAC[]] THEN MATCH_MP_TAC(TAUT `!q. p /\ q ==> p`) THEN EXISTS_TAC `path_connected(t INTER h:real^N->bool)` THEN MATCH_MP_TAC PATH_CONNECTED_FROM_CLOSED_UNION_AND_INTER_LOCAL THEN EXISTS_TAC `h:real^N->bool` THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [CONJ_TAC THENL [UNDISCH_TAC `closed_in (subtopology euclidean (s UNION t)) (s:real^N->bool)`; UNDISCH_TAC `closed_in (subtopology euclidean (s UNION t)) (t:real^N->bool)`] THEN REWRITE_TAC[CLOSED_IN_CLOSED] THEN MATCH_MP_TAC MONO_EXISTS THEN FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ASM SET_TAC[]; SUBGOAL_THEN `s INTER h UNION t INTER h:real^N->bool = h` SUBST1_TAC THENL [FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ASM SET_TAC[]; ASM_REWRITE_TAC[]] THEN SUBGOAL_THEN `(s INTER h) INTER t INTER h:real^N->bool = c` SUBST1_TAC THENL [FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ASM SET_TAC[]; ASM_MESON_TAC[PATH_CONNECTED_PATH_COMPONENT]]]);; let LOCALLY_PATH_CONNECTED_FROM_UNION_AND_INTER = prove (`!s t:real^N->bool. closed s /\ closed t /\ locally path_connected (s UNION t) /\ locally path_connected (s INTER t) ==> locally path_connected s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC LOCALLY_PATH_CONNECTED_FROM_UNION_AND_INTER_GEN THEN MAP_EVERY EXISTS_TAC [`t:real^N->bool`; `s UNION t:real^N->bool`] THEN ASM_SIMP_TAC[CLOSED_SUBSET; SUBSET_UNION]);; (* ------------------------------------------------------------------------- *) (* Two uniform variants of local connectedness. ULC is an abbreviation for *) (* "uniformly locally connected"; FCCOVERABLE ("fine connected coverable") *) (* is more usually called "Property S" (Whyburn, Hocking & Young etc.) *) (* ------------------------------------------------------------------------- *) let FCCOVERABLE_IMP_LOCALLY_CONNECTED = prove (`!s:real^N->bool. (!e. &0 < e ==> ?c. FINITE c /\ UNIONS c = s /\ !t. t IN c ==> connected t /\ bounded t /\ diameter t <= e) ==> locally connected s`, GEN_TAC THEN REWRITE_TAC[locally] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `x:real^N`] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN GEN_REWRITE_TAC LAND_CONV [OPEN_IN_CONTAINS_BALL] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `x:real^N`)) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN DISCH_THEN(X_CHOOSE_THEN `c:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN EXISTS_TAC `UNIONS {t | t IN c /\ (x:real^N) IN closure t}` THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r /\ s /\ t <=> q /\ t /\ p /\ r /\ s`] THEN REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_UNIONS_STRONG THEN ASM_SIMP_TAC[IN_ELIM_THM; GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN EXISTS_TAC `x:real^N` THEN SIMP_TAC[INTERS_GSPEC; IN_ELIM_THM; UNIONS_GSPEC; GSYM CONJ_ASSOC] THEN MATCH_MP_TAC(MESON[] `(!t. R t ==> Q t) /\ (?t. P t /\ R t) ==> (?t. P t /\ Q t /\ R t)`) THEN REWRITE_TAC[REWRITE_RULE[SUBSET] CLOSURE_SUBSET] THEN ASM SET_TAC[]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS)) THEN REWRITE_TAC[SUBSET_INTER] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC] THEN X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN TRANS_TAC SUBSET_TRANS `closure t:real^N->bool` THEN REWRITE_TAC[CLOSURE_SUBSET] THEN FIRST_X_ASSUM(MP_TAC o SPEC `t:real^N->bool`) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN REWRITE_TAC[SUBSET; IN_BALL] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `diameter(t:real^N->bool)` THEN CONJ_TAC THENL [REWRITE_TAC[dist]; ASM_REAL_ARITH_TAC] THEN ONCE_REWRITE_TAC[GSYM DIAMETER_CLOSURE] THEN MATCH_MP_TAC DIAMETER_BOUNDED_BOUND THEN ASM_SIMP_TAC[BOUNDED_CLOSURE]; ALL_TAC] THEN EXISTS_TAC `s INTER ball(x:real^N,e) INTER interior ((:real^N) DIFF (s DIFF UNIONS {t | t IN c /\ x IN closure t}))` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_OPEN_INTER THEN SIMP_TAC[OPEN_INTER; OPEN_BALL; OPEN_INTERIOR]; ASM_REWRITE_TAC[IN_INTER; CENTRE_IN_BALL] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[INTERIOR_COMPLEMENT; IN_DIFF; IN_UNIV] THEN DISCH_THEN(MP_TAC o SPEC `closure(UNIONS {t | t IN c /\ ~((x:real^N) IN closure t)})` o MATCH_MP(SET_RULE `x IN s ==> !t. s SUBSET t ==> x IN t`)) THEN REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [MATCH_MP_TAC SUBSET_CLOSURE THEN ASM SET_TAC[]; ASM_SIMP_TAC[CLOSURE_UNIONS; FINITE_RESTRICT] THEN ASM SET_TAC[]]; MATCH_MP_TAC(SET_RULE `interior t SUBSET t /\ s INTER t SUBSET u ==> s INTER b INTER interior t SUBSET u`) THEN REWRITE_TAC[INTERIOR_SUBSET] THEN SET_TAC[]]);; let ULC_IMP_LOCALLY_CONNECTED = prove (`!s:real^N->bool. (!e. &0 < e ==> ?d. &0 < d /\ !x y. x IN s /\ y IN s /\ dist(x,y) < d ==> ?c. x IN c /\ y IN c /\ c SUBSET s /\ connected c /\ bounded c /\ diameter c <= e) ==> locally connected s`, GEN_TAC THEN REWRITE_TAC[LOCALLY_CONNECTED_IM_KLEINEN] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `p:real^N`] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN GEN_REWRITE_TAC LAND_CONV [OPEN_IN_CONTAINS_BALL] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `p:real^N`)) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `s INTER ball(p:real^N,min d e)` THEN ASM_SIMP_TAC[OPEN_IN_OPEN_INTER; OPEN_BALL] THEN ASM_REWRITE_TAC[CENTRE_IN_BALL; REAL_LT_MIN; IN_INTER] THEN REWRITE_TAC[BALL_MIN_INTER; CONJ_ASSOC] THEN CONJ_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[GSYM CONJ_ASSOC]] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_INTER; IN_BALL] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`p:real^N`; `x:real^N`]) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS] THEN X_GEN_TAC `c:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `b INTER s SUBSET u ==> c SUBSET s /\ c SUBSET b ==> c SUBSET u`)) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET; IN_BALL] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `c <= e / &2 ==> &0 < e /\ d <= c ==> d < e`)) THEN ASM_REWRITE_TAC[dist] THEN MATCH_MP_TAC DIAMETER_BOUNDED_BOUND THEN ASM_REWRITE_TAC[]);; let FCCOVERABLE_INTERMEDIATE_CLOSURE = prove (`!s t:real^N->bool. s SUBSET t /\ t SUBSET closure s /\ (!e. &0 < e ==> ?c. FINITE c /\ UNIONS c = s /\ !t. t IN c ==> connected t /\ bounded t /\ diameter t <= e) ==> (!e. &0 < e ==> ?c. FINITE c /\ UNIONS c = t /\ !t. t IN c ==> connected t /\ bounded t /\ diameter t <= e)`, REPEAT GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `e:real` THEN ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `c:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `{t INTER closure k:real^N->bool | k IN c}` THEN ASM_SIMP_TAC[SIMPLE_IMAGE; FINITE_IMAGE; UNIONS_IMAGE; FORALL_IN_IMAGE] THEN CONJ_TAC THENL [MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN SIMP_TAC[SUBSET; IN_INTER; IN_ELIM_THM] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `t SUBSET c ==> (!x. x IN c ==> P x) ==> (!x. x IN t ==> P x)`)) THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN ASM_SIMP_TAC[CLOSURE_UNIONS] THEN SET_TAC[]; X_GEN_TAC `k:real^N->bool` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `k:real^N->bool`) THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THENL [MATCH_MP_TAC CONNECTED_INTERMEDIATE_CLOSURE THEN EXISTS_TAC `k:real^N->bool` THEN ASM_REWRITE_TAC[INTER_SUBSET] THEN REWRITE_TAC[SUBSET_INTER; CLOSURE_SUBSET] THEN ASM SET_TAC[]; ASM_MESON_TAC[BOUNDED_SUBSET; INTER_SUBSET; BOUNDED_CLOSURE]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS)) THEN GEN_REWRITE_TAC RAND_CONV [GSYM DIAMETER_CLOSURE] THEN MATCH_MP_TAC DIAMETER_SUBSET THEN ASM_SIMP_TAC[INTER_SUBSET; BOUNDED_CLOSURE]]]);; let COMPACT_LOCALLY_CONNECTED_IMP_ULC = prove (`!s:real^N->bool. compact s /\ locally connected s ==> (!e. &0 < e ==> ?d. &0 < d /\ !x y. x IN s /\ y IN s /\ dist(x,y) < d ==> ?c. x IN c /\ y IN c /\ c SUBSET s /\ connected c /\ bounded c /\ diameter c <= e)`, GEN_TAC THEN STRIP_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MATCH_MP_TAC(MESON[] `((!x. ~P x) ==> F) ==> ?x. P x`) THEN DISCH_THEN(MP_TAC o GEN `n:num` o SPEC `inv(&2 pow n)`) THEN REWRITE_TAC[REAL_LT_INV_EQ; REAL_LT_POW2; RIGHT_AND_FORALL_THM] THEN GEN_REWRITE_TAC (RAND_CONV o TOP_DEPTH_CONV) [NOT_FORALL_THM] THEN REWRITE_TAC[SKOLEM_THM; NOT_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`x:num->real^N`; `y:num->real^N`] THEN REWRITE_TAC[NOT_IMP; FORALL_AND_THM] THEN STRIP_TAC THEN MP_TAC(ISPEC `(s:real^N->bool) PCROSS s` compact) THEN ASM_REWRITE_TAC[COMPACT_PCROSS_EQ] THEN DISCH_THEN(MP_TAC o SPEC `\n:num. pastecart(x n:real^N) (y n:real^N)`) THEN ASM_REWRITE_TAC[PASTECART_IN_PCROSS; NOT_IMP] THEN ASM_REWRITE_TAC[NOT_EXISTS_THM; FORALL_PASTECART; PASTECART_IN_PCROSS] THEN MAP_EVERY X_GEN_TAC [`w:real^N`; `z:real^N`; `r:num->num`] THEN REWRITE_TAC[o_DEF; LIM_PASTECART_EQ] THEN STRIP_TAC THEN SUBGOAL_THEN `w:real^N = z` SUBST_ALL_TAC THENL [ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THEN EXISTS_TAC `(\n. x((r:num->num) n) - y(r n)):num->real^N` THEN ASM_SIMP_TAC[LIM_SUB; TRIVIAL_LIMIT_SEQUENTIALLY] THEN REWRITE_TAC[LIM_SEQUENTIALLY; DIST_0] THEN X_GEN_TAC `d:real` THEN DISCH_TAC THEN MP_TAC(ISPEC `max (inv d) (inv e)` REAL_ARCH_POW2) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[REAL_MAX_LT] THEN STRIP_TAC THEN X_GEN_TAC `m:num` THEN DISCH_TAC THEN MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `inv(&2 pow ((r:num->num) m))` THEN ASM_SIMP_TAC[GSYM dist; REAL_LT_IMP_LE] THEN GEN_REWRITE_TAC RAND_CONV [GSYM REAL_INV_INV] THEN MATCH_MP_TAC REAL_LT_INV2 THEN ASM_REWRITE_TAC[REAL_LT_INV_EQ] THEN TRANS_TAC REAL_LTE_TRANS `&2 pow n` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_POW_MONO THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN TRANS_TAC LE_TRANS `m:num` THEN ASM_MESON_TAC[MONOTONE_BIGGER]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LOCALLY_CONNECTED]) THEN DISCH_THEN(MP_TAC o SPECL [`s INTER ball(z:real^N,e / &2)`; `z:real^N`]) THEN ASM_SIMP_TAC[OPEN_IN_OPEN_INTER; OPEN_BALL; IN_INTER; CENTRE_IN_BALL; REAL_HALF] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `?d. &0 < d /\ ball(z:real^N,d) INTER s SUBSET u` STRIP_ASSUME_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [open_in]) THEN DISCH_THEN(MP_TAC o SPEC `z:real^N` o CONJUNCT2) THEN ASM_REWRITE_TAC[SUBSET; IN_INTER; IN_BALL] THEN MESON_TAC[DIST_SYM]; ALL_TAC] THEN MAP_EVERY UNDISCH_TAC [`((\m:num. (y:num->real^N) (r m)) --> z) sequentially`; `((\m:num. (x:num->real^N) (r m)) --> z) sequentially`] THEN REWRITE_TAC[TAUT `p ==> ~q <=> ~(p /\ q)`; tendsto; AND_FORALL_THM] THEN DISCH_THEN(MP_TAC o SPEC `min d (e / &2)`) THEN ASM_REWRITE_TAC[REAL_HALF; REAL_LT_MIN; GSYM EVENTUALLY_AND] THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN DISCH_THEN(X_CHOOSE_THEN `n:num` (MP_TAC o SPEC `n:num`)) THEN REWRITE_TAC[LE_REFL] THEN STRIP_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[NOT_EXISTS_THM; RIGHT_AND_FORALL_THM; RIGHT_IMP_FORALL_THM]) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`(r:num->num) n`; `u:real^N->bool`]) THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET]) THEN ASM_REWRITE_TAC[IN_BALL; IN_INTER] THEN ASM_MESON_TAC[DIST_SYM]; FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET]) THEN ASM_REWRITE_TAC[IN_BALL; IN_INTER] THEN ASM_MESON_TAC[DIST_SYM]; ASM SET_TAC[]; MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `ball(z:real^N,e / &2)` THEN REWRITE_TAC[BOUNDED_BALL] THEN ASM SET_TAC[]; TRANS_TAC REAL_LE_TRANS `diameter(ball(z:real^N,e / &2))` THEN CONJ_TAC THENL [MATCH_MP_TAC DIAMETER_SUBSET THEN REWRITE_TAC[BOUNDED_BALL] THEN ASM SET_TAC[]; REWRITE_TAC[DIAMETER_BALL] THEN ASM_REAL_ARITH_TAC]]);; let COMPACT_LOCALLY_CONNECTED_IMP_ULC_ALT = prove (`!s:real^N->bool. compact s /\ locally connected s ==> !e. &0 < e ==> ?d. &0 < d /\ d < e /\ !x y. x IN s /\ y IN s /\ dist(x,y) < d ==> ?c. connected c /\ x IN c /\ y IN c /\ c SUBSET s INTER ball(x,e) INTER ball(y,e)`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `s:real^N->bool` COMPACT_LOCALLY_CONNECTED_IMP_ULC) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `min d e / &2` THEN REPEAT(CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]) THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `y:real^N`]) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[SUBSET_INTER] THEN CONJ_TAC THEN REWRITE_TAC[SUBSET; IN_BALL] THEN X_GEN_TAC `z:real^N` THEN DISCH_TAC THEN TRANS_TAC REAL_LET_TRANS `diameter(c:real^N->bool)` THEN REWRITE_TAC[dist] THEN ASM_SIMP_TAC[DIAMETER_BOUNDED_BOUND] THEN ASM_REAL_ARITH_TAC);; let BOUNDED_ULC_IMP_FCCOVERABLE = prove (`!s:real^N->bool. bounded s /\ (!e. &0 < e ==> ?d. &0 < d /\ !x y. x IN s /\ y IN s /\ dist(x,y) < d ==> ?c. x IN c /\ y IN c /\ c SUBSET s /\ connected c /\ bounded c /\ diameter c <= e) ==> (!e. &0 < e ==> ?c. FINITE c /\ UNIONS c = s /\ !t. t IN c ==> connected t /\ bounded t /\ diameter t <= e)`, REPEAT GEN_TAC THEN STRIP_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN ABBREV_TAC `M = \p. {x | x IN s /\ ?c. (p:real^N) IN c /\ (x:real^N) IN c /\ c SUBSET s /\ connected c /\ bounded c /\ diameter c <= e / &2}` THEN SUBGOAL_THEN `!p:real^N. p IN s ==> ball(p,d) INTER s SUBSET M p` ASSUME_TAC THENL [X_GEN_TAC `p:real^N` THEN DISCH_TAC THEN EXPAND_TAC "M" THEN REWRITE_TAC[SUBSET; IN_INTER; IN_BALL; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[GSYM SUBSET] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`p:real^N`; `x:real^N`]) THEN ASM_MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?k. FINITE k /\ k SUBSET s /\ UNIONS(IMAGE (M:real^N->real^N->bool) k) = s` STRIP_ASSUME_TAC THENL [ONCE_REWRITE_TAC[MESON[] `(?x. P x /\ Q x /\ R x) <=> ~(!x. P x /\ Q x ==> ~R x)`] THEN DISCH_TAC THEN SUBGOAL_THEN `?f:num->real^N. !n. f n IN s DIFF UNIONS(IMAGE ((M:real^N->real^N->bool) o f) {m | m < n})` STRIP_ASSUME_TAC THENL [SUBGOAL_THEN `?f:num->real^N. !n. f n = @x. x IN s DIFF UNIONS(IMAGE ((M:real^N->real^N->bool) o f) {m | m < n})` MP_TAC THENL [MATCH_MP_TAC(MATCH_MP WF_REC WF_num) THEN REPEAT STRIP_TAC THEN REWRITE_TAC[IMAGE_o] THEN AP_TERM_TAC THEN ABS_TAC THEN REPLICATE_TAC 4 AP_TERM_TAC THEN ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:num->real^N` THEN DISCH_TAC THEN MATCH_MP_TAC num_WF THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN ONCE_ASM_REWRITE_TAC[] THEN CONV_TAC SELECT_CONV THEN FIRST_X_ASSUM(MP_TAC o SPEC `IMAGE (f:num->real^N) {m | m < n}`) THEN SIMP_TAC[FINITE_NUMSEG_LT; FINITE_IMAGE] THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[IMAGE_o]] THEN MATCH_MP_TAC(SET_RULE `t SUBSET s ==> ~(t = s) ==> ?x. x IN s DIFF t`) THEN REWRITE_TAC[UNIONS_SUBSET] THEN ONCE_REWRITE_TAC[FORALL_IN_IMAGE] THEN EXPAND_TAC "M" THEN SET_TAC[]]; MP_TAC(ISPECL [`IMAGE (f:num->real^N) (:num)`; `d:real`] DISCRETE_BOUNDED_IMP_FINITE) THEN ASM_REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_UNIV; NOT_IMP] THEN SUBGOAL_THEN `!m n. norm((f:num->real^N) m - f n) < d ==> m = n` ASSUME_TAC THENL [MATCH_MP_TAC WLOG_LT THEN REWRITE_TAC[] THEN CONJ_TAC THENL [MESON_TAC[NORM_SUB]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN REPEAT DISCH_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `n:num`) THEN MATCH_MP_TAC(SET_RULE `x IN t ==> x IN s DIFF t ==> P`) THEN REWRITE_TAC[IN_UNIONS; EXISTS_IN_IMAGE; o_THM; IN_ELIM_THM] THEN EXISTS_TAC `m:num` THEN ASM_REWRITE_TAC[] THEN UNDISCH_THEN `!p:real^N. p IN s ==> ball(p,d) INTER s SUBSET M p` (MP_TAC o SPEC `(f:num->real^N) m`) THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[SUBSET]] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_BALL; IN_INTER; dist] THEN ASM SET_TAC[]; ALL_TAC] THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] BOUNDED_SUBSET)) THEN ASM SET_TAC[]; W(MP_TAC o PART_MATCH (lhand o rand) FINITE_IMAGE_INJ_EQ o rand o snd) THEN REWRITE_TAC[REWRITE_RULE[INFINITE] num_INFINITE] THEN DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[IN_UNIV] THEN ASM_MESON_TAC[VECTOR_SUB_REFL; NORM_0]]]; EXISTS_TAC `IMAGE (M:real^N->real^N->bool) k` THEN ASM_SIMP_TAC[FINITE_IMAGE; FORALL_IN_IMAGE] THEN X_GEN_TAC `p:real^N` THEN EXPAND_TAC "M" THEN REWRITE_TAC[] THEN REPEAT STRIP_TAC THENL [GEN_REWRITE_TAC I [CONNECTED_IFF_CONNECTED_COMPONENT] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(CONJUNCTS_THEN (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` STRIP_ASSUME_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `d:real^N->bool` STRIP_ASSUME_TAC) THEN REWRITE_TAC[connected_component] THEN EXISTS_TAC `c UNION d:real^N->bool` THEN ASM_REWRITE_TAC[IN_UNION; UNION_SUBSET] THEN CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_UNION THEN ASM SET_TAC[]; ASM SET_TAC[]]; MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; MATCH_MP_TAC DIAMETER_LE THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(CONJUNCTS_THEN (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` STRIP_ASSUME_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `d:real^N->bool` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC(NORM_ARITH `!p:real^N. norm(x - p) <= e / &2 /\ norm(y - p) <= e / &2 ==> norm(x - y) <= e`) THEN EXISTS_TAC `p:real^N` THEN CONJ_TAC THENL [TRANS_TAC REAL_LE_TRANS `diameter(d:real^N->bool)`; TRANS_TAC REAL_LE_TRANS `diameter(c:real^N->bool)`] THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC DIAMETER_BOUNDED_BOUND THEN ASM_REWRITE_TAC[]]]);; let COMPACT_LOCALLY_CONNECTED_IMP_FCCOVERABLE = prove (`!s:real^N->bool. compact s /\ locally connected s ==> !e. &0 < e ==> ?c. FINITE c /\ UNIONS c = s /\ !t. t IN c ==> connected t /\ bounded t /\ diameter t <= e`, GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP COMPACT_LOCALLY_CONNECTED_IMP_ULC) THEN FIRST_X_ASSUM STRIP_ASSUME_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP COMPACT_IMP_BOUNDED) THEN GEN_REWRITE_TAC I [IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP BOUNDED_ULC_IMP_FCCOVERABLE) THEN REWRITE_TAC[]);; let COMPACT_LOCALLY_CONNECTED_EQ_FCCCOVERABLE = prove (`!s:real^N->bool. compact s /\ locally connected s <=> !e. &0 < e ==> ?c. FINITE c /\ UNIONS c = s /\ !t. t IN c ==> connected t /\ compact t /\ diameter t <= e`, GEN_TAC THEN EQ_TAC THEN DISCH_TAC THENL [FIRST_ASSUM(MP_TAC o MATCH_MP COMPACT_LOCALLY_CONNECTED_IMP_FCCOVERABLE) THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `e:real` THEN DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `c:(real^N->bool)->bool` THEN STRIP_TAC THEN EXISTS_TAC `IMAGE closure (c:(real^N->bool)->bool)` THEN ASM_SIMP_TAC[FINITE_IMAGE; FORALL_IN_IMAGE; CONNECTED_CLOSURE; COMPACT_CLOSURE; DIAMETER_CLOSURE] THEN ASM_SIMP_TAC[GSYM SIMPLE_IMAGE; GSYM CLOSURE_UNIONS] THEN ASM_SIMP_TAC[CLOSURE_EQ; COMPACT_IMP_CLOSED]; CONJ_TAC THENL [ASM_MESON_TAC[COMPACT_UNIONS; REAL_LT_01]; ALL_TAC] THEN MATCH_MP_TAC FCCOVERABLE_IMP_LOCALLY_CONNECTED THEN ASM_MESON_TAC[COMPACT_IMP_BOUNDED]]);; (* ------------------------------------------------------------------------- *) (* Localization of "property S" *) (* ------------------------------------------------------------------------- *) let LOCALLY_FCCOVERABLE = prove (`!s u a:real^N. (!e. &0 < e ==> ?c. FINITE c /\ UNIONS c = s /\ !t. t IN c ==> connected t /\ bounded t /\ diameter t <= e) /\ open_in (subtopology euclidean s) u /\ a IN u ==> ?v. open_in (subtopology euclidean s) v /\ connected v /\ a IN v /\ v SUBSET u /\ !e. &0 < e ==> ?c. FINITE c /\ UNIONS c = v /\ !t. t IN c ==> connected t /\ bounded t /\ diameter t <= e`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_CONTAINS_CBALL]) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `a:real^N`)) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o MATCH_MP (MESON[REAL_ARITH `&0 < e ==> &0 < e / &2 /\ &2 * e / &2 = e`] `(?e. &0 < e /\ P e) ==> ?r. &0 < r /\ P(&2 * r)`)) THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN ABBREV_TAC `t = \k. {x | ?f. (!i. i <= k ==> connected(f i) /\ f i SUBSET s /\ bounded(f i) /\ diameter(f i) < r / &2 pow i) /\ a IN f 0 /\ (x:real^N) IN f k /\ (!i. i < k ==> ~(f i INTER f(SUC i) = {}))}` THEN EXISTS_TAC `UNIONS {t k | k IN (:num)}:real^N->bool` THEN SUBGOAL_THEN `!k. a IN (t:num->real^N->bool) k` ASSUME_TAC THENL [X_GEN_TAC `k:num` THEN EXPAND_TAC "t" THEN REWRITE_TAC[IN_ELIM_THM] THEN EXISTS_TAC `\i:num. {a:real^N}` THEN ASM_REWRITE_TAC[CONNECTED_SING; SING_SUBSET; BOUNDED_SING] THEN REWRITE_TAC[IN_SING; DIAMETER_SING] THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_POW2] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `!k. (t:num->real^N->bool) k SUBSET s` ASSUME_TAC THENL [X_GEN_TAC `k:num` THEN EXPAND_TAC "t" THEN REWRITE_TAC[IN_ELIM_THM; SUBSET] THEN X_GEN_TAC `x:real^N` THEN DISCH_THEN(CHOOSE_THEN (CONJUNCTS_THEN2 (MP_TAC o SPEC `k:num`) ASSUME_TAC)) THEN REWRITE_TAC[LE_REFL] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `!k. (t:num->real^N->bool) k SUBSET ball(a,&2 * r)` ASSUME_TAC THENL [SUBGOAL_THEN `!k. (t:num->real^N->bool) k SUBSET ball(a,(&2 - inv(&2 pow k)) * r)` MP_TAC THENL [ALL_TAC; MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS) THEN MATCH_MP_TAC SUBSET_BALL THEN ASM_SIMP_TAC[REAL_LE_RMUL_EQ; REAL_ARITH `x - a <= x <=> &0 <= a`] THEN SIMP_TAC[REAL_POW_LE; REAL_POS; REAL_LE_INV_EQ]] THEN MATCH_MP_TAC num_INDUCTION THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN EXPAND_TAC "t" THEN REWRITE_TAC[SUBSET; IN_ELIM_THM; REAL_MUL_LID] THEN CONJ_TAC THENL [REWRITE_TAC[LE; LT; FORALL_UNWIND_THM2] THEN X_GEN_TAC `b:real^N` THEN DISCH_THEN(X_CHOOSE_THEN `f:num->real^N->bool` STRIP_ASSUME_TAC) THEN REWRITE_TAC[IN_BALL] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `x < r / &2 pow 0 ==> a <= x ==> a < r`)) THEN MATCH_MP_TAC DIST_LE_DIAMETER THEN ASM_REWRITE_TAC[]; X_GEN_TAC `k:num` THEN DISCH_TAC THEN X_GEN_TAC `c:real^N` THEN DISCH_THEN(X_CHOOSE_THEN `f:num->real^N->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `~(f k INTER f(SUC k):real^N->bool = {})` MP_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN ARITH_TAC; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER]] THEN DISCH_THEN(X_CHOOSE_THEN `b:real^N` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPEC `b:real^N`) THEN ANTS_TAC THENL [EXISTS_TAC `f:num->real^N->bool` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THEN GEN_TAC THEN DISCH_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; REWRITE_TAC[IN_BALL]] THEN MATCH_MP_TAC(NORM_ARITH `!u. dist(b,c) <= u /\ x + u <= y ==> dist(a:real^N,b) < x ==> dist(a,c) < y`) THEN EXISTS_TAC `diameter((f:num->real^N->bool) (SUC k))` THEN ASM_SIMP_TAC[DIST_LE_DIAMETER; LE_REFL] THEN MATCH_MP_TAC(REAL_ARITH `r * k = r * &2 * k' /\ d < r * k' ==> (&2 - k) * r + d <= (&2 - k') * r`) THEN ASM_SIMP_TAC[GSYM real_div; LE_REFL] THEN REWRITE_TAC[real_pow; real_div; REAL_INV_MUL] THEN REAL_ARITH_TAC]; ALL_TAC] THEN SUBGOAL_THEN `!k. (t:num->real^N->bool) k SUBSET t(SUC k)` ASSUME_TAC THENL [X_GEN_TAC `k:num` THEN EXPAND_TAC "t" THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `b:real^N` THEN DISCH_THEN(X_CHOOSE_THEN `f:num->real^N->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\i. if i <= k then (f:num->real^N->bool) i else {b}` THEN ASM_REWRITE_TAC[LE_0; ARITH_RULE `~(SUC k <= k)`; IN_SING] THEN CONJ_TAC THEN X_GEN_TAC `i:num` THEN ASM_CASES_TAC `i:num <= k` THEN ASM_SIMP_TAC[LT_SUC_LE] THENL [ASM_MESON_TAC[]; REWRITE_TAC[CONNECTED_SING; IN_SING; BOUNDED_SING] THEN DISCH_TAC THEN ASM_REWRITE_TAC[FORALL_UNWIND_THM2; DIAMETER_SING] THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_POW2] THEN ASM_MESON_TAC[LE_REFL]; ASM_SIMP_TAC[ARITH_RULE `i <= k ==> (SUC i <= k <=> ~(i = k))`] THEN REWRITE_TAC[COND_SWAP] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL [ASM SET_TAC[]; FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC]]; ALL_TAC] THEN SUBGOAL_THEN `!k. connected((t:num->real^N->bool) k)` ASSUME_TAC THENL [X_GEN_TAC `k:num` THEN REWRITE_TAC[CONNECTED_IFF_CONNECTED_COMPONENT] THEN MATCH_MP_TAC(MESON[CONNECTED_COMPONENT_SYM; CONNECTED_COMPONENT_TRANS] `!a. (!x. x IN s ==> connected_component s a x) ==> (!x y. x IN s /\ y IN s ==> connected_component s x y)`) THEN EXISTS_TAC `a:real^N` THEN SPEC_TAC(`k:num`,`k:num`) THEN MATCH_MP_TAC num_INDUCTION THEN CONJ_TAC THENL [X_GEN_TAC `x:real^N` THEN EXPAND_TAC "t" THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(X_CHOOSE_THEN `f:num->real^N->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `connected_component (f 0) (a:real^N) x` MP_TAC THENL [REWRITE_TAC[connected_component] THEN EXISTS_TAC `f 0:real^N->bool` THEN ASM_SIMP_TAC[LE_REFL; SUBSET_REFL]; MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s x ==> t x`) THEN REWRITE_TAC[ETA_AX]] THEN MATCH_MP_TAC CONNECTED_COMPONENT_MONO THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `c:real^N` THEN DISCH_TAC THEN EXISTS_TAC `f:num->real^N->bool` THEN ASM_REWRITE_TAC[GSYM SUBSET]; ALL_TAC] THEN X_GEN_TAC `k:num` THEN DISCH_TAC THEN X_GEN_TAC `c:real^N` THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [SYM th]) THEN REWRITE_TAC[IN_ELIM_THM] THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `f:num->real^N->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `~(f k INTER f(SUC k):real^N->bool = {})` MP_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN ARITH_TAC; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER]] THEN DISCH_THEN(X_CHOOSE_THEN `b:real^N` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPEC `b:real^N`) THEN ANTS_TAC THENL [EXPAND_TAC "t" THEN REWRITE_TAC[IN_ELIM_THM] THEN EXISTS_TAC `f:num->real^N->bool` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THEN GEN_TAC THEN DISCH_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; MATCH_MP_TAC(SET_RULE `connected_component k a SUBSET connected_component k' a /\ (connected_component k' a b ==> connected_component k' a c) ==> connected_component k a b ==> connected_component k' a c`)] THEN ASM_SIMP_TAC[CONNECTED_COMPONENT_MONO] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] CONNECTED_COMPONENT_TRANS) THEN REWRITE_TAC[connected_component] THEN EXISTS_TAC `f(SUC k):real^N->bool` THEN ASM_SIMP_TAC[LE_REFL] THEN EXPAND_TAC "t" THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN EXISTS_TAC `f:num->real^N->bool` THEN ASM_REWRITE_TAC[GSYM SUBSET]; ALL_TAC] THEN REPEAT CONJ_TAC THENL [ONCE_REWRITE_TAC[OPEN_IN_SUBOPEN] THEN REWRITE_TAC[FORALL_IN_UNIONS; FORALL_IN_GSPEC; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN X_GEN_TAC `k:num` THEN DISCH_THEN(K ALL_TAC) THEN X_GEN_TAC `x:real^N` THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [SYM th]) THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(X_CHOOSE_THEN `f:num->real^N->bool` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o MATCH_MP FCCOVERABLE_IMP_LOCALLY_CONNECTED) THEN REWRITE_TAC[LOCALLY_CONNECTED] THEN DISCH_THEN(MP_TAC o SPECL [`s INTER ball(x:real^N,r / &2 pow (k + 3))`; `x:real^N`]) THEN SIMP_TAC[OPEN_IN_OPEN_INTER; OPEN_BALL] THEN ASM_SIMP_TAC[IN_INTER; CENTRE_IN_BALL] THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_POW2] THEN ANTS_TAC THENL [ASM_MESON_TAC[SUBSET; LE_REFL]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `v:real^N->bool` THEN REWRITE_TAC[SUBSET_INTER; UNIONS_GSPEC] THEN STRIP_TAC THEN ASM_REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_UNIV] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN EXISTS_TAC `SUC k` THEN EXPAND_TAC "t" THEN REWRITE_TAC[IN_ELIM_THM] THEN EXISTS_TAC `\i. if i <= k then (f:num->real^N->bool) i else v` THEN ASM_REWRITE_TAC[LE_0; ARITH_RULE `~(SUC k <= k)`; IN_SING] THEN CONJ_TAC THEN X_GEN_TAC `i:num` THEN ASM_CASES_TAC `i:num <= k` THEN ASM_SIMP_TAC[LT_SUC_LE] THENL [DISCH_TAC THEN CONJ_TAC THENL [ASM_MESON_TAC[BOUNDED_SUBSET; BOUNDED_BALL]; ALL_TAC] THEN TRANS_TAC REAL_LET_TRANS `r / &2 pow (k + 2)` THEN CONJ_TAC THENL [TRANS_TAC REAL_LE_TRANS `diameter(ball(x:real^N,r / &2 pow (k + 3)))` THEN ASM_SIMP_TAC[DIAMETER_SUBSET; BOUNDED_BALL] THEN ASM_SIMP_TAC[DIAMETER_BALL; REAL_LT_DIV; REAL_LT_POW2; REAL_ARITH `&0 < x ==> ~(x < &0)`] THEN REWRITE_TAC[REAL_POW_ADD; real_div; REAL_INV_MUL] THEN REAL_ARITH_TAC; ASM_SIMP_TAC[real_div; REAL_LT_LMUL_EQ] THEN MATCH_MP_TAC REAL_LT_INV2 THEN REWRITE_TAC[REAL_LT_POW2] THEN MATCH_MP_TAC REAL_POW_MONO_LT THEN REWRITE_TAC[REAL_OF_NUM_LT] THEN ASM_ARITH_TAC]; ASM_SIMP_TAC[ARITH_RULE `i <= k ==> (SUC i <= k <=> ~(i = k))`] THEN REWRITE_TAC[COND_SWAP] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL [ASM SET_TAC[]; FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC]]; MATCH_MP_TAC CONNECTED_UNIONS THEN ASM_REWRITE_TAC[FORALL_IN_GSPEC] THEN ASM SET_TAC[]; ASM SET_TAC[]; REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC; IN_UNIV] THEN X_GEN_TAC `k:num` THEN TRANS_TAC SUBSET_TRANS `s INTER cball(a:real^N,&2 * r)` THEN ASM_REWRITE_TAC[SUBSET_INTER] THEN ASM_MESON_TAC[SUBSET_TRANS; INTER_COMM; BALL_SUBSET_CBALL]; X_GEN_TAC `e:real` THEN DISCH_TAC] THEN SUBGOAL_THEN `?k. r / &2 pow k < e / &4` STRIP_ASSUME_TAC THENL [REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] real_div] THEN ASM_SIMP_TAC[REAL_INV_POW; GSYM REAL_LT_RDIV_EQ] THEN MATCH_MP_TAC REAL_ARCH_POW_INV THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[REAL_ARITH `&0 < (&1 / &4 * x) / y <=> &0 < x / y`] THEN ASM_SIMP_TAC[REAL_LT_DIV]; ALL_TAC] THEN SUBGOAL_THEN `?ws. FINITE ws /\ (t:num->real^N->bool) k SUBSET UNIONS ws /\ !w. w IN ws ==> w SUBSET s /\ ~(t k INTER w = {}) /\ connected w /\ bounded w /\ diameter w < r / &2 pow (k + 1)` STRIP_ASSUME_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPEC `r / &2 pow (k + 2)`) THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_POW2] THEN DISCH_THEN(X_CHOOSE_THEN `ws:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `{w:real^N->bool | w IN ws /\ ~(t(k:num) INTER w = {})}` THEN ASM_SIMP_TAC[FINITE_RESTRICT; FORALL_IN_GSPEC] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN X_GEN_TAC `w:real^N->bool` THEN REPEAT STRIP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN TRANS_TAC REAL_LET_TRANS `r / &2 pow (k + 2)` THEN ASM_SIMP_TAC[] THEN REWRITE_TAC[REAL_POW_ADD; real_div; REAL_INV_MUL] THEN MATCH_MP_TAC(REAL_ARITH `&0 < r / k ==> r * inv k * inv(&2 pow 2) < r * inv k * inv(&2 pow 1)`) THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_POW2]; ALL_TAC] THEN SUBGOAL_THEN `!w:real^N->bool. w IN ws ==> w SUBSET t(SUC k)` ASSUME_TAC THENL [X_GEN_TAC `w:real^N->bool` THEN DISCH_TAC THEN REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `w:real^N->bool`) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN UNDISCH_TAC `~((t:num->real^N->bool) k INTER w = {})` THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `y:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN EXPAND_TAC "t" THEN REWRITE_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `f:num->real^N->bool` THEN STRIP_TAC THEN EXISTS_TAC `\i. if i <= k then (f:num->real^N->bool) i else w` THEN ASM_REWRITE_TAC[LE_0; ARITH_RULE `~(SUC k <= k)`] THEN CONJ_TAC THEN X_GEN_TAC `i:num` THEN ASM_CASES_TAC `i:num <= k` THEN ASM_SIMP_TAC[LT_SUC_LE] THENL [DISCH_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] REAL_LTE_TRANS)) THEN ASM_SIMP_TAC[real_div; REAL_LE_LMUL_EQ] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_LT_POW2] THEN MATCH_MP_TAC REAL_POW_MONO THEN REWRITE_TAC[REAL_OF_NUM_LE] THEN ASM_ARITH_TAC; ASM_SIMP_TAC[ARITH_RULE `i <= k ==> (SUC i <= k <=> ~(i = k))`] THEN REWRITE_TAC[COND_SWAP] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL [ASM SET_TAC[]; FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC]]; ALL_TAC] THEN ABBREV_TAC `q = \w. {x | ?c. connected c /\ c SUBSET UNIONS {t k | k IN (:num)} /\ bounded c /\ diameter c < e / &4 /\ ~(w INTER c = {}) /\ (x:real^N) IN c}` THEN EXISTS_TAC `IMAGE (q:(real^N->bool)->(real^N->bool)) ws` THEN ASM_SIMP_TAC[FINITE_IMAGE; FORALL_IN_IMAGE] THEN CONJ_TAC THENL [ALL_TAC; X_GEN_TAC `w:real^N->bool` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `w:real^N->bool`) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN CONJ_TAC THENL [SUBGOAL_THEN `?b:real^N. b IN w` CHOOSE_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[CONNECTED_IFF_CONNECTED_COMPONENT] THEN MATCH_MP_TAC(MESON[CONNECTED_COMPONENT_SYM; CONNECTED_COMPONENT_TRANS] `!a. (!x. x IN s ==> connected_component s a x) ==> (!x y. x IN s /\ y IN s ==> connected_component s x y)`) THEN EXISTS_TAC `b:real^N` THEN X_GEN_TAC `x:real^N` THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [SYM th]) THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` STRIP_ASSUME_TAC) THEN REWRITE_TAC[connected_component] THEN EXISTS_TAC `w UNION c:real^N->bool` THEN ASM_SIMP_TAC[IN_UNION; CONNECTED_UNION] THEN EXPAND_TAC "q" THEN REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_UNION] THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[GSYM SUBSET] THEN STRIP_TAC THENL [EXISTS_TAC `{y:real^N}` THEN REWRITE_TAC[BOUNDED_SING; IN_SING; CONNECTED_SING] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[DIAMETER_SING] THEN ASM_REAL_ARITH_TAC; EXISTS_TAC `c:real^N->bool` THEN ASM_REWRITE_TAC[]]; ALL_TAC] THEN SUBGOAL_THEN `q w SUBSET {x + y:real^N | x IN w /\ y IN ball(vec 0,e / &4)}` ASSUME_TAC THENL [REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN EXPAND_TAC "q" THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` STRIP_ASSUME_TAC) THEN UNDISCH_TAC `~(w INTER c:real^N->bool = {})` THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `y:real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[IN_BALL_0] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[VECTOR_ARITH `x:real^N = y + z <=> z = x - y`] THEN REWRITE_TAC[UNWIND_THM2; GSYM dist] THEN TRANS_TAC REAL_LET_TRANS `diameter(c:real^N->bool)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC DIST_LE_DIAMETER THEN ASM_REWRITE_TAC[]; CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] BOUNDED_SUBSET)); FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (MESON[DIAMETER_SUBSET; REAL_LE_TRANS] `s SUBSET t ==> bounded t /\ diameter t <= e ==> diameter s <= e`))] THEN ASM_SIMP_TAC[BOUNDED_SUMS; BOUNDED_BALL] THEN W(MP_TAC o PART_MATCH (lhand o rand) DIAMETER_SUMS o lhand o snd) THEN ASM_SIMP_TAC[BOUNDED_BALL; DIAMETER_BALL] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN ASM_SIMP_TAC[REAL_ARITH `&0 < e ==> ~(e / &4 < &0)`] THEN REWRITE_TAC[REAL_ARITH `d + &2 * e / &4 <= e <=> d <= e / &2`] THEN TRANS_TAC REAL_LE_TRANS `r / &2 pow (k + 1)` THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN REWRITE_TAC[real_div; REAL_POW_ADD; REAL_INV_MUL] THEN ASM_REAL_ARITH_TAC]] THEN REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; UNIONS_SUBSET] THEN REWRITE_TAC[FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN CONJ_TAC THENL [X_GEN_TAC `w:real^N->bool` THEN DISCH_TAC THEN REWRITE_TAC[SUBSET; UNIONS_GSPEC; IN_UNION] THEN X_GEN_TAC `x:real^N` THEN EXPAND_TAC "q" THEN REWRITE_TAC[IN_ELIM_THM; IN_UNIV; LEFT_IMP_EXISTS_THM] THEN SET_TAC[]; REWRITE_TAC[IN_UNIV]] THEN X_GEN_TAC `n:num` THEN DISJ_CASES_TAC(ARITH_RULE `n:num <= k \/ k < n`) THENL [TRANS_TAC SUBSET_TRANS `(t:num->real^N->bool) k` THEN CONJ_TAC THENL [UNDISCH_TAC `n:num <= k` THEN MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`k:num`; `n:num`] THEN MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN ASM_REWRITE_TAC[] THEN SET_TAC[]; TRANS_TAC SUBSET_TRANS `UNIONS ws:real^N->bool` THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM IMAGE_ID] THEN MATCH_MP_TAC UNIONS_MONO_IMAGE THEN X_GEN_TAC `w:real^N->bool` THEN DISCH_TAC THEN REWRITE_TAC[SUBSET] THEN EXPAND_TAC "q" THEN REWRITE_TAC[IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN EXISTS_TAC `{x:real^N}` THEN REWRITE_TAC[BOUNDED_SING; CONNECTED_SING; IN_SING] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[DIAMETER_SING] THEN ASM_REAL_ARITH_TAC]; ALL_TAC] THEN REWRITE_TAC[SUBSET] THEN EXPAND_TAC "t" THEN REWRITE_TAC[IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_THEN(X_CHOOSE_THEN `f:num->real^N->bool` STRIP_ASSUME_TAC) THEN REWRITE_TAC[UNIONS_IMAGE; IN_ELIM_THM] THEN EXPAND_TAC "q" THEN REWRITE_TAC[IN_ELIM_THM] THEN SUBGOAL_THEN `?b:real^N. b IN f k /\ b IN f(SUC k)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[IN_INTER; MEMBER_NOT_EMPTY]; ALL_TAC] THEN SUBGOAL_THEN `b IN (t:num->real^N->bool) k` ASSUME_TAC THENL [EXPAND_TAC "t" THEN REWRITE_TAC[IN_ELIM_THM] THEN EXISTS_TAC `f:num->real^N->bool` THEN ASM_MESON_TAC[LT_TRANS; LE_TRANS; LT_IMP_LE]; ALL_TAC] THEN SUBGOAL_THEN `(b:real^N) IN UNIONS ws` MP_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[IN_UNIONS]] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `w:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `UNIONS (IMAGE f (k+1..n)):real^N->bool` THEN REPEAT CONJ_TAC THENL [SUBGOAL_THEN `!i. i <= n ==> connected(UNIONS(IMAGE f (k+1..i)):real^N->bool)` (fun th -> SIMP_TAC[th; LE_REFL]) THEN MATCH_MP_TAC num_INDUCTION THEN SUBGOAL_THEN `k+1..0 = {}` SUBST1_TAC THENL [REWRITE_TAC[NUMSEG_EMPTY] THEN ARITH_TAC; REWRITE_TAC[IMAGE_CLAUSES; UNIONS_0; CONNECTED_EMPTY]] THEN X_GEN_TAC `i:num` THEN DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN ASM_CASES_TAC `i:num = k` THENL [ASM_REWRITE_TAC[ADD1; NUMSEG_SING; IMAGE_CLAUSES; UNIONS_1] THEN ASM_MESON_TAC[ADD1]; REWRITE_TAC[NUMSEG_CLAUSES] THEN COND_CASES_TAC THEN REWRITE_TAC[] THEN DISCH_TAC THEN REWRITE_TAC[IMAGE_CLAUSES; UNIONS_INSERT] THEN MATCH_MP_TAC CONNECTED_UNION THEN ASM_SIMP_TAC[] THEN REWRITE_TAC[INTER_UNIONS; EMPTY_UNIONS; FORALL_IN_GSPEC] THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_NUMSEG] THEN DISCH_THEN(MP_TAC o SPEC `i:num`) THEN REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [ASM_ARITH_TAC; ONCE_REWRITE_TAC[INTER_COMM] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC]]; MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> f x SUBSET g x) /\ s SUBSET t ==> UNIONS(IMAGE f s) SUBSET UNIONS {g x | x IN t}`) THEN REWRITE_TAC[IN_NUMSEG; SUBSET_UNIV] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN REWRITE_TAC[SUBSET] THEN EXPAND_TAC "t" THEN REWRITE_TAC[IN_ELIM_THM] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN EXISTS_TAC `f:num->real^N->bool` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THEN GEN_TAC THEN DISCH_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; MATCH_MP_TAC BOUNDED_UNIONS THEN ASM_SIMP_TAC[FINITE_IMAGE; FINITE_NUMSEG; FORALL_IN_IMAGE; IN_NUMSEG]; ALL_TAC; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `b:real^N` THEN ASM_REWRITE_TAC[UNIONS_IMAGE; IN_ELIM_THM; IN_NUMSEG; IN_INTER] THEN EXISTS_TAC `SUC k` THEN ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC; REWRITE_TAC[UNIONS_IMAGE; IN_ELIM_THM] THEN EXISTS_TAC `n:num` THEN ASM_REWRITE_TAC[IN_NUMSEG] THEN ASM_ARITH_TAC] THEN SUBGOAL_THEN `!d j. j + d = n ==> diameter (UNIONS (IMAGE f (j..n)):real^N->bool) < &2 * r / &2 pow j` (MP_TAC o SPECL [`n - (k + 1)`; `k + 1`]) THENL [ALL_TAC; ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LTE_TRANS) THEN REWRITE_TAC[real_div; REAL_INV_MUL; REAL_POW_ADD] THEN ASM_REAL_ARITH_TAC] THEN MATCH_MP_TAC num_INDUCTION THEN REWRITE_TAC[ADD_CLAUSES; FORALL_UNWIND_THM2] THEN REWRITE_TAC[NUMSEG_SING; UNIONS_1; IMAGE_CLAUSES] THEN ASM_SIMP_TAC[REAL_ARITH `&0 <= x /\ x < n ==> x < &2 * n`; DIAMETER_POS_LE; LE_REFL] THEN X_GEN_TAC `d:num` THEN DISCH_THEN(LABEL_TAC "*") THEN X_GEN_TAC `j:num` THEN DISCH_TAC THEN SUBGOAL_THEN `j:num < n` ASSUME_TAC THENL [ASM_ARITH_TAC; ASM_SIMP_TAC[LT_IMP_LE; GSYM NUMSEG_LREC]] THEN REWRITE_TAC[IMAGE_CLAUSES; UNIONS_INSERT] THEN W(MP_TAC o PART_MATCH (lhand o rand) DIAMETER_UNION_LE o lhand o snd) THEN ANTS_TAC THENL [ASM_SIMP_TAC[BOUNDED_UNIONS; FINITE_IMAGE; FINITE_NUMSEG; FORALL_IN_IMAGE; IN_NUMSEG; LT_IMP_LE] THEN REWRITE_TAC[INTER_UNIONS; EMPTY_UNIONS; FORALL_IN_GSPEC] THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_NUMSEG] THEN DISCH_THEN(MP_TAC o SPEC `SUC j`) THEN REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [ASM_ARITH_TAC; FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC]; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LET_TRANS)] THEN REMOVE_THEN "*" (MP_TAC o SPEC `j + 1`) THEN ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC(REAL_ARITH `d1 < rj /\ &2 * rj' = rj ==> d2 < &2 * rj' ==> d1 + d2 < &2 * rj`) THEN ASM_SIMP_TAC[LT_IMP_LE] THEN REWRITE_TAC[real_div; REAL_POW_ADD; REAL_INV_MUL] THEN REAL_ARITH_TAC);; let LOCALLY_FCCOVERABLE_ALT = prove (`!s u a:real^N. locally compact s /\ locally connected s /\ open_in (subtopology euclidean s) u /\ a IN u ==> ?v. open_in (subtopology euclidean s) v /\ connected v /\ a IN v /\ v SUBSET u /\ !e. &0 < e ==> ?c. FINITE c /\ UNIONS c = v /\ !t. t IN c ==> connected t /\ bounded t /\ diameter t <= e`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?r. &0 < r /\ s INTER cball(a,&2 * r) SUBSET u /\ compact(s INTER cball(a:real^N,&2 * r))` STRIP_ASSUME_TAC THENL [ONCE_REWRITE_TAC[INTER_COMM] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_CONTAINS_CBALL]) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `a:real^N`)) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LOCALLY_COMPACT_INTER_CBALLS]) THEN DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `min d e / &2` THEN ASM_REWRITE_TAC[REAL_HALF; REAL_LT_MIN] THEN CONJ_TAC THENL [REWRITE_TAC[CBALL_MIN_INTER] THEN ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED] THEN CONJ_TAC THENL [MESON_TAC[BOUNDED_SUBSET; INTER_SUBSET; BOUNDED_CBALL]; FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC]; FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET)] THEN ABBREV_TAC `t = \k. {x | ?f. (!i. i <= k ==> connected(f i) /\ f i SUBSET s /\ bounded(f i) /\ diameter(f i) < r / &2 pow i) /\ a IN f 0 /\ (x:real^N) IN f k /\ (!i. i < k ==> ~(f i INTER f(SUC i) = {}))}` THEN EXISTS_TAC `UNIONS {t k | k IN (:num)}:real^N->bool` THEN SUBGOAL_THEN `!k. a IN (t:num->real^N->bool) k` ASSUME_TAC THENL [X_GEN_TAC `k:num` THEN EXPAND_TAC "t" THEN REWRITE_TAC[IN_ELIM_THM] THEN EXISTS_TAC `\i:num. {a:real^N}` THEN ASM_REWRITE_TAC[CONNECTED_SING; SING_SUBSET; BOUNDED_SING] THEN REWRITE_TAC[IN_SING; DIAMETER_SING] THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_POW2] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `!k. (t:num->real^N->bool) k SUBSET s` ASSUME_TAC THENL [X_GEN_TAC `k:num` THEN EXPAND_TAC "t" THEN REWRITE_TAC[IN_ELIM_THM; SUBSET] THEN X_GEN_TAC `x:real^N` THEN DISCH_THEN(CHOOSE_THEN (CONJUNCTS_THEN2 (MP_TAC o SPEC `k:num`) ASSUME_TAC)) THEN REWRITE_TAC[LE_REFL] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `!k. (t:num->real^N->bool) k SUBSET ball(a,&2 * r)` ASSUME_TAC THENL [SUBGOAL_THEN `!k. (t:num->real^N->bool) k SUBSET ball(a,(&2 - inv(&2 pow k)) * r)` MP_TAC THENL [ALL_TAC; MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS) THEN MATCH_MP_TAC SUBSET_BALL THEN ASM_SIMP_TAC[REAL_LE_RMUL_EQ; REAL_ARITH `x - a <= x <=> &0 <= a`] THEN SIMP_TAC[REAL_POW_LE; REAL_POS; REAL_LE_INV_EQ]] THEN MATCH_MP_TAC num_INDUCTION THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN EXPAND_TAC "t" THEN REWRITE_TAC[SUBSET; IN_ELIM_THM; REAL_MUL_LID] THEN CONJ_TAC THENL [REWRITE_TAC[LE; LT; FORALL_UNWIND_THM2] THEN X_GEN_TAC `b:real^N` THEN DISCH_THEN(X_CHOOSE_THEN `f:num->real^N->bool` STRIP_ASSUME_TAC) THEN REWRITE_TAC[IN_BALL] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `x < r / &2 pow 0 ==> a <= x ==> a < r`)) THEN MATCH_MP_TAC DIST_LE_DIAMETER THEN ASM_REWRITE_TAC[]; X_GEN_TAC `k:num` THEN DISCH_TAC THEN X_GEN_TAC `c:real^N` THEN DISCH_THEN(X_CHOOSE_THEN `f:num->real^N->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `~(f k INTER f(SUC k):real^N->bool = {})` MP_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN ARITH_TAC; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER]] THEN DISCH_THEN(X_CHOOSE_THEN `b:real^N` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPEC `b:real^N`) THEN ANTS_TAC THENL [EXISTS_TAC `f:num->real^N->bool` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THEN GEN_TAC THEN DISCH_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; REWRITE_TAC[IN_BALL]] THEN MATCH_MP_TAC(NORM_ARITH `!u. dist(b,c) <= u /\ x + u <= y ==> dist(a:real^N,b) < x ==> dist(a,c) < y`) THEN EXISTS_TAC `diameter((f:num->real^N->bool) (SUC k))` THEN ASM_SIMP_TAC[DIST_LE_DIAMETER; LE_REFL] THEN MATCH_MP_TAC(REAL_ARITH `r * k = r * &2 * k' /\ d < r * k' ==> (&2 - k) * r + d <= (&2 - k') * r`) THEN ASM_SIMP_TAC[GSYM real_div; LE_REFL] THEN REWRITE_TAC[real_pow; real_div; REAL_INV_MUL] THEN REAL_ARITH_TAC]; ALL_TAC] THEN SUBGOAL_THEN `!k. (t:num->real^N->bool) k SUBSET t(SUC k)` ASSUME_TAC THENL [X_GEN_TAC `k:num` THEN EXPAND_TAC "t" THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `b:real^N` THEN DISCH_THEN(X_CHOOSE_THEN `f:num->real^N->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\i. if i <= k then (f:num->real^N->bool) i else {b}` THEN ASM_REWRITE_TAC[LE_0; ARITH_RULE `~(SUC k <= k)`; IN_SING] THEN CONJ_TAC THEN X_GEN_TAC `i:num` THEN ASM_CASES_TAC `i:num <= k` THEN ASM_SIMP_TAC[LT_SUC_LE] THENL [ASM_MESON_TAC[]; REWRITE_TAC[CONNECTED_SING; IN_SING; BOUNDED_SING] THEN DISCH_TAC THEN ASM_REWRITE_TAC[FORALL_UNWIND_THM2; DIAMETER_SING] THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_POW2] THEN ASM_MESON_TAC[LE_REFL]; ASM_SIMP_TAC[ARITH_RULE `i <= k ==> (SUC i <= k <=> ~(i = k))`] THEN REWRITE_TAC[COND_SWAP] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL [ASM SET_TAC[]; FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC]]; ALL_TAC] THEN SUBGOAL_THEN `!k. connected((t:num->real^N->bool) k)` ASSUME_TAC THENL [X_GEN_TAC `k:num` THEN REWRITE_TAC[CONNECTED_IFF_CONNECTED_COMPONENT] THEN MATCH_MP_TAC(MESON[CONNECTED_COMPONENT_SYM; CONNECTED_COMPONENT_TRANS] `!a. (!x. x IN s ==> connected_component s a x) ==> (!x y. x IN s /\ y IN s ==> connected_component s x y)`) THEN EXISTS_TAC `a:real^N` THEN SPEC_TAC(`k:num`,`k:num`) THEN MATCH_MP_TAC num_INDUCTION THEN CONJ_TAC THENL [X_GEN_TAC `x:real^N` THEN EXPAND_TAC "t" THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(X_CHOOSE_THEN `f:num->real^N->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `connected_component (f 0) (a:real^N) x` MP_TAC THENL [REWRITE_TAC[connected_component] THEN EXISTS_TAC `f 0:real^N->bool` THEN ASM_SIMP_TAC[LE_REFL; SUBSET_REFL]; MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s x ==> t x`) THEN REWRITE_TAC[ETA_AX]] THEN MATCH_MP_TAC CONNECTED_COMPONENT_MONO THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `c:real^N` THEN DISCH_TAC THEN EXISTS_TAC `f:num->real^N->bool` THEN ASM_REWRITE_TAC[GSYM SUBSET]; ALL_TAC] THEN X_GEN_TAC `k:num` THEN DISCH_TAC THEN X_GEN_TAC `c:real^N` THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [SYM th]) THEN REWRITE_TAC[IN_ELIM_THM] THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `f:num->real^N->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `~(f k INTER f(SUC k):real^N->bool = {})` MP_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN ARITH_TAC; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER]] THEN DISCH_THEN(X_CHOOSE_THEN `b:real^N` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPEC `b:real^N`) THEN ANTS_TAC THENL [EXPAND_TAC "t" THEN REWRITE_TAC[IN_ELIM_THM] THEN EXISTS_TAC `f:num->real^N->bool` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THEN GEN_TAC THEN DISCH_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; MATCH_MP_TAC(SET_RULE `connected_component k a SUBSET connected_component k' a /\ (connected_component k' a b ==> connected_component k' a c) ==> connected_component k a b ==> connected_component k' a c`)] THEN ASM_SIMP_TAC[CONNECTED_COMPONENT_MONO] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] CONNECTED_COMPONENT_TRANS) THEN REWRITE_TAC[connected_component] THEN EXISTS_TAC `f(SUC k):real^N->bool` THEN ASM_SIMP_TAC[LE_REFL] THEN EXPAND_TAC "t" THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN EXISTS_TAC `f:num->real^N->bool` THEN ASM_REWRITE_TAC[GSYM SUBSET]; ALL_TAC] THEN REPEAT CONJ_TAC THENL [ONCE_REWRITE_TAC[OPEN_IN_SUBOPEN] THEN REWRITE_TAC[FORALL_IN_UNIONS; FORALL_IN_GSPEC; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN X_GEN_TAC `k:num` THEN DISCH_THEN(K ALL_TAC) THEN X_GEN_TAC `x:real^N` THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [SYM th]) THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(X_CHOOSE_THEN `f:num->real^N->bool` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LOCALLY_CONNECTED]) THEN DISCH_THEN(MP_TAC o SPECL [`s INTER ball(x:real^N,r / &2 pow (k + 3))`; `x:real^N`]) THEN SIMP_TAC[OPEN_IN_OPEN_INTER; OPEN_BALL] THEN ASM_SIMP_TAC[IN_INTER; CENTRE_IN_BALL] THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_POW2] THEN ANTS_TAC THENL [ASM_MESON_TAC[SUBSET; LE_REFL]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `v:real^N->bool` THEN REWRITE_TAC[SUBSET_INTER; UNIONS_GSPEC] THEN STRIP_TAC THEN ASM_REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_UNIV] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN EXISTS_TAC `SUC k` THEN EXPAND_TAC "t" THEN REWRITE_TAC[IN_ELIM_THM] THEN EXISTS_TAC `\i. if i <= k then (f:num->real^N->bool) i else v` THEN ASM_REWRITE_TAC[LE_0; ARITH_RULE `~(SUC k <= k)`; IN_SING] THEN CONJ_TAC THEN X_GEN_TAC `i:num` THEN ASM_CASES_TAC `i:num <= k` THEN ASM_SIMP_TAC[LT_SUC_LE] THENL [DISCH_TAC THEN CONJ_TAC THENL [ASM_MESON_TAC[BOUNDED_SUBSET; BOUNDED_BALL]; ALL_TAC] THEN TRANS_TAC REAL_LET_TRANS `r / &2 pow (k + 2)` THEN CONJ_TAC THENL [TRANS_TAC REAL_LE_TRANS `diameter(ball(x:real^N,r / &2 pow (k + 3)))` THEN ASM_SIMP_TAC[DIAMETER_SUBSET; BOUNDED_BALL] THEN ASM_SIMP_TAC[DIAMETER_BALL; REAL_LT_DIV; REAL_LT_POW2; REAL_ARITH `&0 < x ==> ~(x < &0)`] THEN REWRITE_TAC[REAL_POW_ADD; real_div; REAL_INV_MUL] THEN REAL_ARITH_TAC; ASM_SIMP_TAC[real_div; REAL_LT_LMUL_EQ] THEN MATCH_MP_TAC REAL_LT_INV2 THEN REWRITE_TAC[REAL_LT_POW2] THEN MATCH_MP_TAC REAL_POW_MONO_LT THEN REWRITE_TAC[REAL_OF_NUM_LT] THEN ASM_ARITH_TAC]; ASM_SIMP_TAC[ARITH_RULE `i <= k ==> (SUC i <= k <=> ~(i = k))`] THEN REWRITE_TAC[COND_SWAP] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL [ASM SET_TAC[]; FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC]]; MATCH_MP_TAC CONNECTED_UNIONS THEN ASM_REWRITE_TAC[FORALL_IN_GSPEC] THEN ASM SET_TAC[]; ASM SET_TAC[]; REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC; IN_UNIV] THEN X_GEN_TAC `k:num` THEN TRANS_TAC SUBSET_TRANS `s INTER cball(a:real^N,&2 * r)` THEN ASM_REWRITE_TAC[SUBSET_INTER] THEN ASM_MESON_TAC[SUBSET_TRANS; BALL_SUBSET_CBALL]; X_GEN_TAC `e:real` THEN DISCH_TAC] THEN SUBGOAL_THEN `?k. r / &2 pow k < e / &4` STRIP_ASSUME_TAC THENL [REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] real_div] THEN ASM_SIMP_TAC[REAL_INV_POW; GSYM REAL_LT_RDIV_EQ] THEN MATCH_MP_TAC REAL_ARCH_POW_INV THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[REAL_ARITH `&0 < (&1 / &4 * x) / y <=> &0 < x / y`] THEN ASM_SIMP_TAC[REAL_LT_DIV]; ALL_TAC] THEN SUBGOAL_THEN `?ws. FINITE ws /\ (t:num->real^N->bool) k SUBSET UNIONS ws /\ !w. w IN ws ==> w SUBSET s /\ ~(t k INTER w = {}) /\ connected w /\ bounded w /\ diameter w < r / &2 pow (k + 1)` STRIP_ASSUME_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LOCALLY_CONNECTED]) THEN DISCH_THEN(MP_TAC o GEN `x:real^N` o SPECL [`s INTER ball(x:real^N,r / &2 pow (k + 3))`; `x:real^N`]) THEN SIMP_TAC[OPEN_IN_OPEN_INTER; OPEN_BALL] THEN REWRITE_TAC[IN_INTER; CENTRE_IN_BALL] THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_POW2; SUBSET_INTER] THEN GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `uu:real^N->real^N->bool` THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [COMPACT_EQ_HEINE_BOREL_GEN]) THEN DISCH_THEN(MP_TAC o SPECL [`IMAGE (uu:real^N->real^N->bool) s`; `s:real^N->bool`]) THEN ASM_SIMP_TAC[FORALL_IN_IMAGE] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN REWRITE_TAC[EXISTS_FINITE_SUBSET_IMAGE] THEN DISCH_THEN(X_CHOOSE_THEN `ws:real^N->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `{w:real^N->bool | w IN IMAGE (uu:real^N->real^N->bool) ws /\ ~(t(k:num) INTER w = {})}` THEN ASM_SIMP_TAC[FINITE_RESTRICT; FINITE_IMAGE] THEN CONJ_TAC THENL [REWRITE_TAC[SET_RULE `s SUBSET UNIONS {k | k IN f /\ ~(s INTER k = {})} <=> s SUBSET UNIONS f`] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS)) THEN ASM_REWRITE_TAC[SUBSET_INTER] THEN ASM_MESON_TAC[SUBSET_TRANS; BALL_SUBSET_CBALL]; REWRITE_TAC[IN_ELIM_THM; IMP_CONJ; FORALL_IN_IMAGE] THEN X_GEN_TAC `x:real^N` THEN REPEAT DISCH_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ANTS_TAC THENL [ASM SET_TAC[]; STRIP_TAC] THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[BOUNDED_SUBSET; BOUNDED_BALL]; ALL_TAC] THEN TRANS_TAC REAL_LET_TRANS `r / &2 pow (k + 2)` THEN CONJ_TAC THENL [TRANS_TAC REAL_LE_TRANS `diameter(ball(x:real^N,r / &2 pow (k + 3)))` THEN ASM_SIMP_TAC[DIAMETER_SUBSET; BOUNDED_BALL] THEN ASM_SIMP_TAC[DIAMETER_BALL; REAL_LT_DIV; REAL_LT_POW2; REAL_ARITH `&0 < x ==> ~(x < &0)`] THEN REWRITE_TAC[REAL_POW_ADD; real_div; REAL_INV_MUL] THEN REAL_ARITH_TAC; ASM_SIMP_TAC[real_div; REAL_LT_LMUL_EQ] THEN MATCH_MP_TAC REAL_LT_INV2 THEN REWRITE_TAC[REAL_LT_POW2] THEN MATCH_MP_TAC REAL_POW_MONO_LT THEN REWRITE_TAC[REAL_OF_NUM_LT] THEN ASM_ARITH_TAC]]; ALL_TAC] THEN SUBGOAL_THEN `!w:real^N->bool. w IN ws ==> w SUBSET t(SUC k)` ASSUME_TAC THENL [X_GEN_TAC `w:real^N->bool` THEN DISCH_TAC THEN REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `w:real^N->bool`) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN UNDISCH_TAC `~((t:num->real^N->bool) k INTER w = {})` THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `y:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN EXPAND_TAC "t" THEN REWRITE_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `f:num->real^N->bool` THEN STRIP_TAC THEN EXISTS_TAC `\i. if i <= k then (f:num->real^N->bool) i else w` THEN ASM_REWRITE_TAC[LE_0; ARITH_RULE `~(SUC k <= k)`] THEN CONJ_TAC THEN X_GEN_TAC `i:num` THEN ASM_CASES_TAC `i:num <= k` THEN ASM_SIMP_TAC[LT_SUC_LE] THENL [DISCH_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] REAL_LTE_TRANS)) THEN ASM_SIMP_TAC[real_div; REAL_LE_LMUL_EQ] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_LT_POW2] THEN MATCH_MP_TAC REAL_POW_MONO THEN REWRITE_TAC[REAL_OF_NUM_LE] THEN ASM_ARITH_TAC; ASM_SIMP_TAC[ARITH_RULE `i <= k ==> (SUC i <= k <=> ~(i = k))`] THEN REWRITE_TAC[COND_SWAP] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL [ASM SET_TAC[]; FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC]]; ALL_TAC] THEN ABBREV_TAC `q = \w. {x | ?c. connected c /\ c SUBSET UNIONS {t k | k IN (:num)} /\ bounded c /\ diameter c < e / &4 /\ ~(w INTER c = {}) /\ (x:real^N) IN c}` THEN EXISTS_TAC `IMAGE (q:(real^N->bool)->(real^N->bool)) ws` THEN ASM_SIMP_TAC[FINITE_IMAGE; FORALL_IN_IMAGE] THEN CONJ_TAC THENL [ALL_TAC; X_GEN_TAC `w:real^N->bool` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `w:real^N->bool`) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN CONJ_TAC THENL [SUBGOAL_THEN `?b:real^N. b IN w` CHOOSE_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[CONNECTED_IFF_CONNECTED_COMPONENT] THEN MATCH_MP_TAC(MESON[CONNECTED_COMPONENT_SYM; CONNECTED_COMPONENT_TRANS] `!a. (!x. x IN s ==> connected_component s a x) ==> (!x y. x IN s /\ y IN s ==> connected_component s x y)`) THEN EXISTS_TAC `b:real^N` THEN X_GEN_TAC `x:real^N` THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [SYM th]) THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` STRIP_ASSUME_TAC) THEN REWRITE_TAC[connected_component] THEN EXISTS_TAC `w UNION c:real^N->bool` THEN ASM_SIMP_TAC[IN_UNION; CONNECTED_UNION] THEN EXPAND_TAC "q" THEN REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_UNION] THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[GSYM SUBSET] THEN STRIP_TAC THENL [EXISTS_TAC `{y:real^N}` THEN REWRITE_TAC[BOUNDED_SING; IN_SING; CONNECTED_SING] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[DIAMETER_SING] THEN ASM_REAL_ARITH_TAC; EXISTS_TAC `c:real^N->bool` THEN ASM_REWRITE_TAC[]]; ALL_TAC] THEN SUBGOAL_THEN `q w SUBSET {x + y:real^N | x IN w /\ y IN ball(vec 0,e / &4)}` ASSUME_TAC THENL [REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN EXPAND_TAC "q" THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` STRIP_ASSUME_TAC) THEN UNDISCH_TAC `~(w INTER c:real^N->bool = {})` THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `y:real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[IN_BALL_0] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[VECTOR_ARITH `x:real^N = y + z <=> z = x - y`] THEN REWRITE_TAC[UNWIND_THM2; GSYM dist] THEN TRANS_TAC REAL_LET_TRANS `diameter(c:real^N->bool)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC DIST_LE_DIAMETER THEN ASM_REWRITE_TAC[]; CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] BOUNDED_SUBSET)); FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (MESON[DIAMETER_SUBSET; REAL_LE_TRANS] `s SUBSET t ==> bounded t /\ diameter t <= e ==> diameter s <= e`))] THEN ASM_SIMP_TAC[BOUNDED_SUMS; BOUNDED_BALL] THEN W(MP_TAC o PART_MATCH (lhand o rand) DIAMETER_SUMS o lhand o snd) THEN ASM_SIMP_TAC[BOUNDED_BALL; DIAMETER_BALL] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN ASM_SIMP_TAC[REAL_ARITH `&0 < e ==> ~(e / &4 < &0)`] THEN REWRITE_TAC[REAL_ARITH `d + &2 * e / &4 <= e <=> d <= e / &2`] THEN TRANS_TAC REAL_LE_TRANS `r / &2 pow (k + 1)` THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN REWRITE_TAC[real_div; REAL_POW_ADD; REAL_INV_MUL] THEN ASM_REAL_ARITH_TAC]] THEN REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; UNIONS_SUBSET] THEN REWRITE_TAC[FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN CONJ_TAC THENL [X_GEN_TAC `w:real^N->bool` THEN DISCH_TAC THEN REWRITE_TAC[SUBSET; UNIONS_GSPEC; IN_UNION] THEN X_GEN_TAC `x:real^N` THEN EXPAND_TAC "q" THEN REWRITE_TAC[IN_ELIM_THM; IN_UNIV; LEFT_IMP_EXISTS_THM] THEN SET_TAC[]; REWRITE_TAC[IN_UNIV]] THEN X_GEN_TAC `n:num` THEN DISJ_CASES_TAC(ARITH_RULE `n:num <= k \/ k < n`) THENL [TRANS_TAC SUBSET_TRANS `(t:num->real^N->bool) k` THEN CONJ_TAC THENL [UNDISCH_TAC `n:num <= k` THEN MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`k:num`; `n:num`] THEN MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN ASM_REWRITE_TAC[] THEN SET_TAC[]; TRANS_TAC SUBSET_TRANS `UNIONS ws:real^N->bool` THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM IMAGE_ID] THEN MATCH_MP_TAC UNIONS_MONO_IMAGE THEN X_GEN_TAC `w:real^N->bool` THEN DISCH_TAC THEN REWRITE_TAC[SUBSET] THEN EXPAND_TAC "q" THEN REWRITE_TAC[IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN EXISTS_TAC `{x:real^N}` THEN REWRITE_TAC[BOUNDED_SING; CONNECTED_SING; IN_SING] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[DIAMETER_SING] THEN ASM_REAL_ARITH_TAC]; ALL_TAC] THEN REWRITE_TAC[SUBSET] THEN EXPAND_TAC "t" THEN REWRITE_TAC[IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_THEN(X_CHOOSE_THEN `f:num->real^N->bool` STRIP_ASSUME_TAC) THEN REWRITE_TAC[UNIONS_IMAGE; IN_ELIM_THM] THEN EXPAND_TAC "q" THEN REWRITE_TAC[IN_ELIM_THM] THEN SUBGOAL_THEN `?b:real^N. b IN f k /\ b IN f(SUC k)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[IN_INTER; MEMBER_NOT_EMPTY]; ALL_TAC] THEN SUBGOAL_THEN `b IN (t:num->real^N->bool) k` ASSUME_TAC THENL [EXPAND_TAC "t" THEN REWRITE_TAC[IN_ELIM_THM] THEN EXISTS_TAC `f:num->real^N->bool` THEN ASM_MESON_TAC[LT_TRANS; LE_TRANS; LT_IMP_LE]; ALL_TAC] THEN SUBGOAL_THEN `(b:real^N) IN UNIONS ws` MP_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[IN_UNIONS]] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `w:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `UNIONS (IMAGE f (k+1..n)):real^N->bool` THEN REPEAT CONJ_TAC THENL [SUBGOAL_THEN `!i. i <= n ==> connected(UNIONS(IMAGE f (k+1..i)):real^N->bool)` (fun th -> SIMP_TAC[th; LE_REFL]) THEN MATCH_MP_TAC num_INDUCTION THEN SUBGOAL_THEN `k+1..0 = {}` SUBST1_TAC THENL [REWRITE_TAC[NUMSEG_EMPTY] THEN ARITH_TAC; REWRITE_TAC[IMAGE_CLAUSES; UNIONS_0; CONNECTED_EMPTY]] THEN X_GEN_TAC `i:num` THEN DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN ASM_CASES_TAC `i:num = k` THENL [ASM_REWRITE_TAC[ADD1; NUMSEG_SING; IMAGE_CLAUSES; UNIONS_1] THEN ASM_MESON_TAC[ADD1]; REWRITE_TAC[NUMSEG_CLAUSES] THEN COND_CASES_TAC THEN REWRITE_TAC[] THEN DISCH_TAC THEN REWRITE_TAC[IMAGE_CLAUSES; UNIONS_INSERT] THEN MATCH_MP_TAC CONNECTED_UNION THEN ASM_SIMP_TAC[] THEN REWRITE_TAC[INTER_UNIONS; EMPTY_UNIONS; FORALL_IN_GSPEC] THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_NUMSEG] THEN DISCH_THEN(MP_TAC o SPEC `i:num`) THEN REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [ASM_ARITH_TAC; ONCE_REWRITE_TAC[INTER_COMM] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC]]; MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> f x SUBSET g x) /\ s SUBSET t ==> UNIONS(IMAGE f s) SUBSET UNIONS {g x | x IN t}`) THEN REWRITE_TAC[IN_NUMSEG; SUBSET_UNIV] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN REWRITE_TAC[SUBSET] THEN EXPAND_TAC "t" THEN REWRITE_TAC[IN_ELIM_THM] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN EXISTS_TAC `f:num->real^N->bool` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THEN GEN_TAC THEN DISCH_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; MATCH_MP_TAC BOUNDED_UNIONS THEN ASM_SIMP_TAC[FINITE_IMAGE; FINITE_NUMSEG; FORALL_IN_IMAGE; IN_NUMSEG]; ALL_TAC; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `b:real^N` THEN ASM_REWRITE_TAC[UNIONS_IMAGE; IN_ELIM_THM; IN_NUMSEG; IN_INTER] THEN EXISTS_TAC `SUC k` THEN ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC; REWRITE_TAC[UNIONS_IMAGE; IN_ELIM_THM] THEN EXISTS_TAC `n:num` THEN ASM_REWRITE_TAC[IN_NUMSEG] THEN ASM_ARITH_TAC] THEN SUBGOAL_THEN `!d j. j + d = n ==> diameter (UNIONS (IMAGE f (j..n)):real^N->bool) < &2 * r / &2 pow j` (MP_TAC o SPECL [`n - (k + 1)`; `k + 1`]) THENL [ALL_TAC; ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LTE_TRANS) THEN REWRITE_TAC[real_div; REAL_INV_MUL; REAL_POW_ADD] THEN ASM_REAL_ARITH_TAC] THEN MATCH_MP_TAC num_INDUCTION THEN REWRITE_TAC[ADD_CLAUSES; FORALL_UNWIND_THM2] THEN REWRITE_TAC[NUMSEG_SING; UNIONS_1; IMAGE_CLAUSES] THEN ASM_SIMP_TAC[REAL_ARITH `&0 <= x /\ x < n ==> x < &2 * n`; DIAMETER_POS_LE; LE_REFL] THEN X_GEN_TAC `d:num` THEN DISCH_THEN(LABEL_TAC "*") THEN X_GEN_TAC `j:num` THEN DISCH_TAC THEN SUBGOAL_THEN `j:num < n` ASSUME_TAC THENL [ASM_ARITH_TAC; ASM_SIMP_TAC[LT_IMP_LE; GSYM NUMSEG_LREC]] THEN REWRITE_TAC[IMAGE_CLAUSES; UNIONS_INSERT] THEN W(MP_TAC o PART_MATCH (lhand o rand) DIAMETER_UNION_LE o lhand o snd) THEN ANTS_TAC THENL [ASM_SIMP_TAC[BOUNDED_UNIONS; FINITE_IMAGE; FINITE_NUMSEG; FORALL_IN_IMAGE; IN_NUMSEG; LT_IMP_LE] THEN REWRITE_TAC[INTER_UNIONS; EMPTY_UNIONS; FORALL_IN_GSPEC] THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_NUMSEG] THEN DISCH_THEN(MP_TAC o SPEC `SUC j`) THEN REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [ASM_ARITH_TAC; FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC]; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LET_TRANS)] THEN REMOVE_THEN "*" (MP_TAC o SPEC `j + 1`) THEN ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC(REAL_ARITH `d1 < rj /\ &2 * rj' = rj ==> d2 < &2 * rj' ==> d1 + d2 < &2 * rj`) THEN ASM_SIMP_TAC[LT_IMP_LE] THEN REWRITE_TAC[real_div; REAL_POW_ADD; REAL_INV_MUL] THEN REAL_ARITH_TAC);; let LOCALLY_CONNECTED_CONTINUUM = prove (`!s:real^N->bool. locally (\c. compact c /\ connected c /\ locally connected c) s <=> locally compact s /\ locally connected s`, GEN_TAC THEN EQ_TAC THENL [DISCH_THEN(fun th -> CONJ_TAC THEN MP_TAC th) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LOCALLY_MONO) THEN SIMP_TAC[]; STRIP_TAC THEN GEN_REWRITE_TAC I [locally] THEN MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `a:real^N`] THEN STRIP_TAC THEN MP_TAC(ASSUME `locally compact (s:real^N->bool)`) THEN GEN_REWRITE_TAC LAND_CONV [locally] THEN DISCH_THEN(MP_TAC o SPECL [`u:real^N->bool`; `a:real^N`]) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`v:real^N->bool`; `c:real^N->bool`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `v:real^N->bool`; `a:real^N`] LOCALLY_FCCOVERABLE_ALT) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `w:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `closure w:real^N->bool` THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[COMPACT_CLOSURE; COMPACT_IMP_BOUNDED; BOUNDED_SUBSET; SUBSET_TRANS]; ASM_SIMP_TAC[CONNECTED_CLOSURE]; MATCH_MP_TAC FCCOVERABLE_IMP_LOCALLY_CONNECTED THEN MATCH_MP_TAC FCCOVERABLE_INTERMEDIATE_CLOSURE THEN EXISTS_TAC `w:real^N->bool` THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN REWRITE_TAC[CLOSURE_SUBSET]; REWRITE_TAC[CLOSURE_SUBSET]; TRANS_TAC SUBSET_TRANS `c:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED] THEN ASM SET_TAC[]]]);; let COMPACT_LOCALLY_CONNECTED_EQ_FCCCOVERABLE_ALT = prove (`!s:real^N->bool. compact s /\ locally connected s <=> !e. &0 < e ==> ?c. FINITE c /\ UNIONS c = s /\ !t. t IN c ==> connected t /\ compact t /\ locally connected t /\ diameter t <= e`, GEN_TAC THEN EQ_TAC THENL [REPEAT STRIP_TAC THEN REWRITE_TAC[MESON[COMPACT_IMP_BOUNDED] `P c /\ compact c /\ Q c /\ R c <=> (compact c /\ P c /\ Q c) /\ bounded c /\ R c`] THEN MATCH_MP_TAC LOCALLY_FINE_COVERING_COMPACT THEN ASM_REWRITE_TAC[LOCALLY_CONNECTED_CONTINUUM] THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED; CLOSED_IMP_LOCALLY_COMPACT]; REWRITE_TAC[COMPACT_LOCALLY_CONNECTED_EQ_FCCCOVERABLE] THEN MESON_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Sufficient conditions for "semi-local connectedness" *) (* ------------------------------------------------------------------------- *) let SEMI_LOCALLY_CONNECTED = prove (`!s:real^N->bool. connected s /\ locally compact s /\ locally connected s ==> !x v. open_in (subtopology euclidean s) v /\ x IN v ==> ?u. open_in (subtopology euclidean s) u /\ x IN u /\ u SUBSET v /\ FINITE(components(s DIFF u))`, REPEAT STRIP_TAC THEN MP_TAC(ASSUME `locally compact (s:real^N->bool)`) THEN REWRITE_TAC[locally] THEN DISCH_THEN(MP_TAC o SPECL [`v:real^N->bool`; `x:real^N`]) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`d:real^N->bool`; `c:real^N->bool`] THEN STRIP_TAC THEN SUBGOAL_THEN `?u k:real^N->bool. x IN u /\ u SUBSET k /\ k SUBSET d /\ open_in (subtopology euclidean s) u /\ closed_in (subtopology euclidean s) k` STRIP_ASSUME_TAC THENL [UNDISCH_TAC `locally compact (s:real^N->bool)` THEN REWRITE_TAC[locally] THEN DISCH_THEN(MP_TAC o SPECL [`d:real^N->bool`; `x:real^N`]) THEN ASM_REWRITE_TAC[] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[CLOSED_SUBSET; COMPACT_IMP_CLOSED; OPEN_IN_IMP_SUBSET; SUBSET_TRANS]; ALL_TAC] THEN SUBGOAL_THEN `!x:real^N. x IN c DIFF d ==> ?t. open_in (subtopology euclidean s) t /\ connected t /\ x IN t /\ t SUBSET s DIFF k` MP_TAC THENL [X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_DIFF] THEN STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE [LOCALLY_CONNECTED]) THEN ASM_SIMP_TAC[OPEN_IN_DIFF; OPEN_IN_REFL] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET)) THEN ASM SET_TAC[]; ALL_TAC] THEN GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM; SKOLEM_THM] THEN DISCH_THEN(X_CHOOSE_THEN `t:real^N->real^N->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `compact(c DIFF d:real^N->bool)` MP_TAC THENL [UNDISCH_TAC `open_in (subtopology euclidean s) (d:real^N->bool)` THEN REWRITE_TAC[OPEN_IN_OPEN] THEN DISCH_THEN(X_CHOOSE_THEN `w:real^N->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `c DIFF d:real^N->bool = c DIFF w` (fun th -> ASM_SIMP_TAC[th; COMPACT_DIFF]) THEN ASM SET_TAC[]; GEN_REWRITE_TAC LAND_CONV [COMPACT_EQ_HEINE_BOREL_GEN]] THEN DISCH_THEN(MP_TAC o SPECL [`IMAGE (t:real^N->real^N->bool) (c DIFF d)`; `s:real^N->bool`]) THEN ASM_REWRITE_TAC[FORALL_IN_IMAGE] THEN ASM_SIMP_TAC[] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN REWRITE_TAC[EXISTS_FINITE_SUBSET_IMAGE] THEN DISCH_THEN(X_CHOOSE_THEN `q:real^N->bool` STRIP_ASSUME_TAC) THEN ABBREV_TAC `r = (s DIFF d) UNION UNIONS(IMAGE (\x. s INTER closure((t:real^N->real^N->bool) x)) q)` THEN EXISTS_TAC `s DIFF r:real^N->bool` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_DIFF THEN ASM_REWRITE_TAC[OPEN_IN_REFL] THEN EXPAND_TAC "r" THEN MATCH_MP_TAC CLOSED_IN_UNION THEN ASM_SIMP_TAC[CLOSED_IN_DIFF; CLOSED_IN_REFL] THEN MATCH_MP_TAC CLOSED_IN_UNIONS THEN ASM_SIMP_TAC[FINITE_IMAGE] THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN SIMP_TAC[CLOSED_IN_CLOSED_INTER; CLOSED_CLOSURE]; ASM_REWRITE_TAC[IN_DIFF] THEN EXPAND_TAC "r" THEN REWRITE_TAC[IN_UNION; UNIONS_IMAGE; IN_DIFF; IN_ELIM_THM] THEN SUBGOAL_THEN `(x:real^N) IN s` ASSUME_TAC THENL [REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET)) THEN ASM SET_TAC[]; ASM_REWRITE_TAC[IN_INTER; NOT_EXISTS_THM]] THEN X_GEN_TAC `y:real^N` THEN STRIP_TAC THEN SUBGOAL_THEN `s INTER closure((t:real^N->real^N->bool) y) SUBSET s DIFF u` MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CLOSURE_MINIMAL_LOCAL THEN ASM_SIMP_TAC[CLOSED_IN_DIFF; CLOSED_IN_REFL] THEN ASM SET_TAC[]; ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(r:real^N->bool) SUBSET s` ASSUME_TAC THENL [EXPAND_TAC "r" THEN MATCH_MP_TAC(SET_RULE `t SUBSET s ==> (s DIFF d) UNION t SUBSET s`) THEN REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_IMAGE] THEN SET_TAC[]; ASM_SIMP_TAC[SET_RULE `r SUBSET s ==> s DIFF (s DIFF r) = r`]] THEN MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `IMAGE (\x:real^N. connected_component r x) q` THEN ASM_SIMP_TAC[FINITE_IMAGE] THEN REWRITE_TAC[components; SUBSET; FORALL_IN_GSPEC] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN REWRITE_TAC[IN_IMAGE] THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_EQ_EQ] THEN MP_TAC(ASSUME `(y:real^N) IN r`) THEN EXPAND_TAC "r" THEN GEN_REWRITE_TAC LAND_CONV [IN_UNION] THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[IN_DIFF; UNIONS_IMAGE; IN_ELIM_THM] THEN MATCH_MP_TAC(TAUT `(q ==> r) /\ (~q /\ p ==> r) ==> p \/ q ==> r`) THEN CONJ_TAC THENL [MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `z:real^N` THEN REWRITE_TAC[IN_INTER] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[connected_component]] THEN EXISTS_TAC `s INTER closure ((t:real^N->real^N->bool) z)` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_INTERMEDIATE_CLOSURE THEN EXISTS_TAC `(t:real^N->real^N->bool) z` THEN REWRITE_TAC[INTER_SUBSET] THEN CONJ_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[SUBSET_INTER]] THEN REWRITE_TAC[CLOSURE_SUBSET] THEN FIRST_X_ASSUM(MP_TAC o SPEC `z:real^N`) THEN REWRITE_TAC[open_in] THEN ASM SET_TAC[]; EXPAND_TAC "r" THEN REWRITE_TAC[UNIONS_IMAGE] THEN ASM SET_TAC[]; ASM SET_TAC[]; ASM_REWRITE_TAC[IN_INTER] THEN FIRST_X_ASSUM(MP_TAC o SPEC `z:real^N`) THEN REWRITE_TAC[open_in] THEN MP_TAC(ISPEC `(t:real^N->real^N->bool) z` CLOSURE_SUBSET) THEN ASM SET_TAC[]]; ALL_TAC] THEN ASM_CASES_TAC `(y:real^N) IN d` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `(y:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `(y:real^N) IN c` THENL [MATCH_MP_TAC(TAUT `p ==> ~p ==> r`) THEN SUBGOAL_THEN `y IN UNIONS (IMAGE (t:real^N->real^N->bool) q)` MP_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[UNIONS_IMAGE; IN_ELIM_THM]] THEN REWRITE_TAC[IN_INTER] THEN ASM_MESON_TAC[CLOSURE_SUBSET; SUBSET]; DISCH_THEN(K ALL_TAC)] THEN SUBGOAL_THEN `~((s INTER closure(connected_component (s DIFF c) y)) INTER c :real^N->bool = {})` MP_TAC THENL [MATCH_MP_TAC(SET_RULE `~(s INTER l SUBSET s DIFF c) ==> ~((s INTER l) INTER c = {})`) THEN DISCH_TAC THEN MP_TAC(ISPEC `s:real^N->bool` CONNECTED_CLOPEN) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `connected_component (s DIFF c) y:real^N->bool`) THEN ASM_REWRITE_TAC[NOT_IMP; CONNECTED_COMPONENT_EQ_EMPTY; IN_DIFF] THEN REPEAT CONJ_TAC THENL [TRANS_TAC OPEN_IN_TRANS `s DIFF c:real^N->bool` THEN CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_CONNECTED_COMPONENT_LOCALLY_CONNECTED THEN MATCH_MP_TAC LOCALLY_OPEN_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN MATCH_MP_TAC OPEN_IN_DIFF THEN ASM_REWRITE_TAC[OPEN_IN_REFL] THEN ASM_MESON_TAC[CLOSED_SUBSET; COMPACT_IMP_CLOSED; OPEN_IN_IMP_SUBSET; SUBSET_TRANS]; REWRITE_TAC[CLOSED_IN_INTER_CLOSURE] THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL THEN ASM_REWRITE_TAC[IN_INTER] THEN CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `c y /\ c SUBSET closure c ==> y IN closure c`) THEN ASM_REWRITE_TAC[CLOSURE_SUBSET; CONNECTED_COMPONENT_REFL_EQ] THEN ASM_REWRITE_TAC[IN_DIFF]; MATCH_MP_TAC CONNECTED_INTERMEDIATE_CLOSURE THEN EXISTS_TAC `connected_component (s DIFF c) y:real^N->bool` THEN REWRITE_TAC[CONNECTED_CONNECTED_COMPONENT; INTER_SUBSET]]; ALL_TAC] THEN REWRITE_TAC[SUBSET_INTER; CLOSURE_SUBSET] THEN TRANS_TAC SUBSET_TRANS `s DIFF c:real^N->bool` THEN REWRITE_TAC[CONNECTED_COMPONENT_SUBSET] THEN SET_TAC[]; MATCH_MP_TAC(SET_RULE `connected_component (s DIFF c) y SUBSET s DIFF c /\ c SUBSET s /\ ~(c = {}) ==> ~(connected_component (s DIFF c) y = s)`) THEN REWRITE_TAC[CONNECTED_COMPONENT_SUBSET] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET)) THEN ASM SET_TAC[]]; ALL_TAC] THEN REWRITE_TAC[closure] THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `~((s INTER (cl UNION l)) INTER c = {}) ==> cl SUBSET s DIFF c ==> ?x. x IN c /\ x IN s /\ x IN l`)) THEN REWRITE_TAC[CONNECTED_COMPONENT_SUBSET; IN_ELIM_THM] THEN DISCH_THEN(X_CHOOSE_THEN `z:real^N` STRIP_ASSUME_TAC) THEN ASM_CASES_TAC `(z:real^N) IN d` THENL [MP_TAC(ISPECL [`s:real^N->bool`; `connected_component (s DIFF c) (y:real^N)`; `d:real^N->bool`; `z:real^N`] LIMIT_POINT_OF_LOCAL_IMP) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [TRANS_TAC SUBSET_TRANS `s DIFF c:real^N->bool` THEN REWRITE_TAC[CONNECTED_COMPONENT_SUBSET] THEN SET_TAC[]; DISCH_THEN(X_CHOOSE_THEN `w:real^N` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`s DIFF c:real^N->bool`; `y:real^N`] CONNECTED_COMPONENT_SUBSET) THEN ASM SET_TAC[]]; ALL_TAC] THEN SUBGOAL_THEN `z IN UNIONS (IMAGE (t:real^N->real^N->bool) q)` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[UNIONS_IMAGE; IN_ELIM_THM] THEN DISCH_THEN(X_CHOOSE_THEN `w:real^N` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`s:real^N->bool`; `connected_component (s DIFF c) (y:real^N)`; `(t:real^N->real^N->bool) w`; `z:real^N`] LIMIT_POINT_OF_LOCAL_IMP) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN TRANS_TAC SUBSET_TRANS `s DIFF c:real^N->bool` THEN REWRITE_TAC[CONNECTED_COMPONENT_SUBSET] THEN SET_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `t:real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `w:real^N` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[connected_component]] THEN EXISTS_TAC `connected_component (s DIFF c) y UNION (t:real^N->real^N->bool) w` THEN REPEAT STRIP_TAC THENL [MATCH_MP_TAC CONNECTED_UNION THEN REWRITE_TAC[CONNECTED_CONNECTED_COMPONENT] THEN ASM SET_TAC[]; EXPAND_TAC "r" THEN MATCH_MP_TAC(SET_RULE `s SUBSET s' /\ t SUBSET t' ==> s UNION t SUBSET s' UNION t'`) THEN CONJ_TAC THENL [TRANS_TAC SUBSET_TRANS `s DIFF c:real^N->bool` THEN REWRITE_TAC[CONNECTED_COMPONENT_SUBSET] THEN ASM SET_TAC[]; REWRITE_TAC[UNIONS_IMAGE; IN_ELIM_THM; SUBSET] THEN FIRST_X_ASSUM(MP_TAC o SPEC `w:real^N`) THEN REWRITE_TAC[open_in] THEN MP_TAC(ISPEC `(t:real^N->real^N->bool) w` CLOSURE_SUBSET) THEN ASM SET_TAC[]]; MATCH_MP_TAC(SET_RULE `c y ==> y IN c UNION s`) THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ; IN_DIFF]; ASM SET_TAC[]]);; let SEMI_LOCALLY_CONNECTED_GEN = prove (`!s:real^N->bool. FINITE(components s) /\ locally compact s /\ locally connected s ==> !x v. open_in (subtopology euclidean s) v /\ x IN v ==> ?u. open_in (subtopology euclidean s) u /\ x IN u /\ u SUBSET v /\ FINITE(components(s DIFF u))`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o CONJUNCT1 o REWRITE_RULE[open_in]) THEN SUBGOAL_THEN `(x:real^N) IN s` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPEC `connected_component s (x:real^N)` SEMI_LOCALLY_CONNECTED) THEN ANTS_TAC THENL [REWRITE_TAC[CONNECTED_CONNECTED_COMPONENT] THEN ASM_SIMP_TAC[LOCALLY_CONNECTED_CONNECTED_COMPONENT] THEN MATCH_MP_TAC LOCALLY_OPEN_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_SIMP_TAC[OPEN_IN_CONNECTED_COMPONENT_LOCALLY_CONNECTED]; ALL_TAC] THEN DISCH_THEN(MP_TAC o SPECL [`x:real^N`; `connected_component s (x:real^N) INTER v`]) THEN ASM_REWRITE_TAC[IN_INTER] THEN ANTS_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_SUBTOPOLOGY_INTER_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN REWRITE_TAC[CONNECTED_COMPONENT_SUBSET] THEN ASM_SIMP_TAC[OPEN_IN_INTER; OPEN_IN_REFL]; REWRITE_TAC[IN] THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ]]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^N->bool` THEN REWRITE_TAC[SUBSET_INTER] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] OPEN_IN_TRANS)) THEN ASM_SIMP_TAC[OPEN_IN_CONNECTED_COMPONENT_LOCALLY_CONNECTED]; ALL_TAC] THEN MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `components(connected_component s (x:real^N) DIFF u) UNION components s` THEN ASM_REWRITE_TAC[FINITE_UNION] THEN REWRITE_TAC[components; SUBSET; FORALL_IN_GSPEC] THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_DIFF] THEN STRIP_TAC THEN ASM_CASES_TAC `(y:real^N) IN connected_component s x` THEN REWRITE_TAC[IN_UNION] THENL [DISJ1_TAC; DISJ2_TAC] THEN REWRITE_TAC[IN_ELIM_THM] THEN EXISTS_TAC `y:real^N` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `z:real^N` THENL [SUBGOAL_THEN `connected_component s (x:real^N) = connected_component s y` SUBST1_TAC THENL [ASM_REWRITE_TAC[CONNECTED_COMPONENT_EQ_EQ] THEN ASM_MESON_TAC[IN]; ALL_TAC]; ALL_TAC] THEN ONCE_REWRITE_TAC[connected_component] THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `c:real^N->bool` THEN REWRITE_TAC[SET_RULE `s SUBSET t DIFF u <=> s SUBSET t /\ DISJOINT s u`] THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THENL [MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL THEN ASM_REWRITE_TAC[]; ASM_MESON_TAC[SUBSET_TRANS; CONNECTED_COMPONENT_SUBSET]; MP_TAC(ISPECL [`s:real^N->bool`; `y:real^N`; `x:real^N`] CONNECTED_COMPONENT_DISJOINT) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(SET_RULE `s' SUBSET s /\ t' SUBSET t ==> DISJOINT s t ==> DISJOINT s' t'`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL THEN ASM_REWRITE_TAC[]]);; let SEMI_LOCALLY_CONNECTED_COMPACT = prove (`!s:real^N->bool. compact s /\ locally connected s ==> !x v. open_in (subtopology euclidean s) v /\ x IN v ==> ?u. open_in (subtopology euclidean s) u /\ x IN u /\ u SUBSET v /\ FINITE(components(s DIFF u))`, GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC SEMI_LOCALLY_CONNECTED_GEN THEN ASM_SIMP_TAC[FINITE_COMPONENTS; CLOSED_IMP_LOCALLY_COMPACT; COMPACT_IMP_CLOSED]);; (* ------------------------------------------------------------------------- *) (* Locally convex sets. *) (* ------------------------------------------------------------------------- *) let LOCALLY_CONVEX = prove (`!s:real^N->bool. locally convex s <=> !x. x IN s ==> ?u v. x IN u /\ u SUBSET v /\ v SUBSET s /\ open_in (subtopology euclidean s) u /\ convex v`, GEN_TAC THEN REWRITE_TAC[locally] THEN EQ_TAC THEN DISCH_TAC THENL [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM (MP_TAC o SPECL [`s INTER ball(x:real^N,&1)`; `x:real^N`]) THEN ASM_SIMP_TAC[OPEN_IN_OPEN_INTER; OPEN_BALL] THEN ASM_REWRITE_TAC[IN_INTER; CENTRE_IN_BALL; REAL_LT_01] THEN MESON_TAC[SUBSET_INTER]; MAP_EVERY X_GEN_TAC [`w:real^N->bool`; `x:real^N`] THEN REWRITE_TAC[IMP_CONJ] THEN GEN_REWRITE_TAC LAND_CONV [OPEN_IN_OPEN] THEN DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` STRIP_ASSUME_TAC) THEN ASM_REWRITE_TAC[IN_INTER] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `v:real^N->bool`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_CBALL]) THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(s INTER ball(x:real^N,e)) INTER u` THEN EXISTS_TAC `cball(x:real^N,e) INTER v` THEN ASM_SIMP_TAC[OPEN_IN_INTER; OPEN_IN_OPEN_INTER; OPEN_BALL; CENTRE_IN_BALL; CONVEX_INTER; CONVEX_CBALL; IN_INTER] THEN MP_TAC(ISPECL [`x:real^N`; `e:real`] BALL_SUBSET_CBALL) THEN ASM SET_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Various sufficient conditions for continuity. These are mainly from the *) (* papers by Klee & Utz, Pervin & Levine, and Tanaka. *) (* ------------------------------------------------------------------------- *) let PROPER_MAP_TO_COMPACT = prove (`!f:real^M->real^N s t. (!k. k SUBSET t /\ compact k ==> compact {x | x IN s /\ f x IN k}) /\ compact t /\ IMAGE f s SUBSET t ==> f continuous_on s`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`; `t:real^N->bool`] CONTINUOUS_ON_CLOSED_GEN) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN X_GEN_TAC `c:real^N->bool` THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] CLOSED_IN_COMPACT)) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN FIRST_ASSUM(MP_TAC o SPEC `c:real^N->bool`) THEN ASM_SIMP_TAC[CLOSED_SUBSET_EQ; COMPACT_IMP_CLOSED; SUBSET_RESTRICT]);; let CONTINUOUS_WITHIN_SEQUENTIALLY_COMPACT_MAP = prove (`!f:real^M->real^N s x. (!c. c SUBSET s /\ compact c ==> compact(IMAGE f c)) /\ x IN s ==> (f continuous (at x within s) <=> !p y. (!n. p n IN s) /\ (p --> x) sequentially /\ (!n. f(p n) = y) ==> f x = y)`, REPEAT STRIP_TAC THEN EQ_TAC THENL [REWRITE_TAC[CONTINUOUS_WITHIN_SEQUENTIALLY] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `p:num->real^M` THEN DISCH_THEN(fun th -> REPEAT STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THEN EXISTS_TAC `(f:real^M->real^N) o (p:num->real^M)` THEN ASM_REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN ASM_REWRITE_TAC[o_DEF; LIM_CONST]; ALL_TAC] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[CONTINUOUS_WITHIN_SEQUENTIALLY_ALT] THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; IN_DELETE; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`e:real`; `p:num->real^M`] THEN STRIP_TAC THEN ASM_CASES_TAC `?y. INFINITE {n:num | (f:real^M->real^N) (p n) = y}` THENL [FIRST_X_ASSUM(X_CHOOSE_TAC `y:real^N`) THEN FIRST_ASSUM(MP_TAC o MATCH_MP INFINITE_ENUMERATE) THEN DISCH_THEN(X_CHOOSE_THEN `r:num->num` STRIP_ASSUME_TAC) THEN MAP_EVERY EXISTS_TAC [`(p:num->real^M) o (r:num->num)`; `y:real^N`] THEN ASM_SIMP_TAC[o_THM; LIM_SUBSEQUENCE] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN FIRST_ASSUM(UNDISCH_TAC o check is_neg o concl) THEN REWRITE_TAC[CONTRAPOS_THM] THEN DISCH_THEN SUBST1_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (MESON[INFINITE; FINITE_EMPTY] `INFINITE s ==> ~(s = {})`)) THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_ELIM_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN ASM_SIMP_TAC[DIST_REFL]; ALL_TAC] THEN SUBGOAL_THEN `?r. (!n m. m < n ==> (r:num->num) m < r n) /\ (!n m. m < n ==> ~((f:real^M->real^N)(p(r m)) = f(p(r n))))` STRIP_ASSUME_TAC THENL [SUBGOAL_THEN `?r. !n. r n = @y. !m. m < n ==> (r:num->num) m < y /\ ~((f:real^M->real^N)(p(r m)) = f(p y))` MP_TAC THENL [MATCH_MP_TAC(MATCH_MP WF_REC WF_num) THEN REPEAT STRIP_TAC THEN AP_TERM_TAC THEN ABS_TAC THEN ASM_MESON_TAC[]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `r:num->num` THEN REWRITE_TAC[AND_FORALL_THM] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `n:num` THEN DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[TAUT `(p ==> q) /\ (p ==> r) <=> p ==> q /\ r`] THEN CONV_TAC SELECT_CONV THEN MP_TAC(ISPECL [`\i:num. i`; `UNIONS {{i | (f:real^M->real^N)(p i) = f(p(r m:num))} | m | m IN {m:num | m < n}}`] UPPER_BOUND_FINITE_SET) THEN RULE_ASSUM_TAC(REWRITE_RULE[NOT_EXISTS_THM; INFINITE]) THEN ASM_REWRITE_TAC[FINITE_UNIONS; FORALL_IN_GSPEC] THEN ASM_SIMP_TAC[FINITE_IMAGE; SIMPLE_IMAGE; FINITE_NUMSEG_LT] THEN REWRITE_TAC[FORALL_IN_UNIONS; FORALL_IN_IMAGE; IMP_CONJ; RIGHT_FORALL_IMP_THM; IN_ELIM_THM] THEN DISCH_THEN(X_CHOOSE_TAC `N:num`) THEN EXISTS_TAC `N + 1` THEN X_GEN_TAC `m:num` THEN DISCH_THEN(ANTE_RES_THEN MP_TAC) THEN REWRITE_TAC[ARITH_RULE `m < N + 1 <=> m <= N`] THEN MESON_TAC[ARITH_RULE `~(N + 1 <= N)`]]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o SPEC `(x:real^M) INSERT IMAGE (p o (r:num->num)) (:num)`) THEN ANTS_TAC THENL [ASM_SIMP_TAC[COMPACT_SEQUENCE_WITH_LIMIT; LIM_SUBSEQUENCE] THEN REWRITE_TAC[IMAGE_o] THEN ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(MP_TAC o SPEC `ball((f:real^M->real^N) x,e)` o MATCH_MP (REWRITE_RULE[IMP_CONJ] COMPACT_DIFF)) THEN REWRITE_TAC[OPEN_BALL] THEN ASM_SIMP_TAC[CENTRE_IN_BALL; SET_RULE `f x IN b ==> IMAGE f (x INSERT s) DIFF b = IMAGE f s DIFF b`] THEN REWRITE_TAC[IMAGE_o] THEN RULE_ASSUM_TAC(REWRITE_RULE [ONCE_REWRITE_RULE[DIST_SYM] (GSYM IN_BALL); NOT_EXISTS_THM]) THEN ASM_SIMP_TAC[SET_RULE `(!n. ~(f(p n) IN s)) ==> IMAGE f (IMAGE p t) DIFF s = IMAGE f (IMAGE p t)`] THEN GEN_REWRITE_TAC LAND_CONV [COMPACT_EQ_BOLZANO_WEIERSTRASS] THEN SUBGOAL_THEN `!m n. (f:real^M->real^N) (p ((r:num->num) m)) = f (p (r n)) <=> m = n` ASSUME_TAC THENL [MATCH_MP_TAC WLOG_LT THEN ASM_MESON_TAC[]; ALL_TAC] THEN DISCH_THEN(MP_TAC o SPEC `IMAGE (f:real^M->real^N) (IMAGE p (IMAGE (r:num->num) (:num)))`) THEN REWRITE_TAC[SUBSET_REFL] THEN ANTS_TAC THENL [REWRITE_TAC[GSYM IMAGE_o] THEN MATCH_MP_TAC INFINITE_IMAGE THEN REWRITE_TAC[num_INFINITE; IN_UNIV; o_THM] THEN ASM_MESON_TAC[]; ALL_TAC] THEN REWRITE_TAC[EXISTS_IN_IMAGE; IN_UNIV] THEN DISCH_THEN(X_CHOOSE_TAC `i:num`) THEN FIRST_X_ASSUM(MP_TAC o SPEC `(x:real^M) INSERT (IMAGE (p o (r:num->num)) (:num) DELETE p(r i))`) THEN ANTS_TAC THENL [CONJ_TAC THENL [REWRITE_TAC[IMAGE_o] THEN ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] COMPACT_SEQUENCE_WITH_LIMIT_GEN)) THEN REWRITE_TAC[IMAGE_o] THEN ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[IMAGE_CLAUSES] THEN W(MP_TAC o PART_MATCH (lhand o rand) (SET_RULE `(!i. i IN s /\ f i = f a ==> i = a) ==> IMAGE f (s DELETE a) = IMAGE f s DELETE f a`) o rand o rand o lhand o snd) THEN ANTS_TAC THENL [REWRITE_TAC[IMAGE_o] THEN ASM SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN DISCH_THEN(MP_TAC o MATCH_MP COMPACT_IMP_CLOSED) THEN REWRITE_TAC[CLOSED_LIMPT] THEN DISCH_THEN(MP_TAC o SPEC `(f:real^M->real^N) (p((r:num->num) i))`) THEN ASM_SIMP_TAC[LIMPT_INSERT; LIMPT_DELETE; IMAGE_o; IN_DELETE; IN_INSERT] THEN ASM_MESON_TAC[CENTRE_IN_BALL]);; let COMPACT_CLOSED_POINTIMAGES_IMP_CONTINUOUS_ON = prove (`!f:real^M->real^N s. (!c. c SUBSET s /\ compact c ==> compact(IMAGE f c)) /\ (!y. closed_in (subtopology euclidean s) {x | x IN s /\ f x = y}) ==> f continuous_on s`, REPEAT STRIP_TAC THEN REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN ONCE_REWRITE_TAC[TAUT `p <=> ~ ~p`] THEN DISCH_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`; `x:real^M`] CONTINUOUS_WITHIN_SEQUENTIALLY_COMPACT_MAP) THEN ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`p:num->real^M`; `y:real^N`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN REWRITE_TAC[CLOSED_IN_LIMPT] THEN DISCH_THEN(MP_TAC o SPEC `x:real^M` o CONJUNCT2) THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN REWRITE_TAC[LIMPT_SEQUENTIAL] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN DISCH_THEN(MP_TAC o SPEC `p:num->real^M`) THEN ASM_REWRITE_TAC[IN_ELIM_THM; IN_DELETE] THEN ASM_MESON_TAC[]);; let COMPACT_CONTINUOUS_IMAGE_EQ = prove (`!f:real^M->real^N s. (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> (f continuous_on s <=> !t. compact t /\ t SUBSET s ==> compact(IMAGE f t))`, REPEAT STRIP_TAC THEN EQ_TAC THENL [MESON_TAC[COMPACT_CONTINUOUS_IMAGE; CONTINUOUS_ON_SUBSET]; DISCH_TAC] THEN MATCH_MP_TAC COMPACT_CLOSED_POINTIMAGES_IMP_CONTINUOUS_ON THEN CONJ_TAC THENL [ASM_MESON_TAC[]; X_GEN_TAC `y:real^N`] THEN SUBGOAL_THEN `{x | x IN s /\ (f:real^M->real^N) x = y} = {} \/ ?x. x IN s /\ {x | x IN s /\ (f:real^M->real^N) x = y} = {x}` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ASM_REWRITE_TAC[CLOSED_IN_EMPTY]; ASM_REWRITE_TAC[CLOSED_IN_SING]]);; let CONTINUOUS_EQ_COMPACT_CONNECTED_PRESERVING_GEN = prove (`!P f:real^M->real^N s. locally P s /\ (!c. P c ==> connected c) ==> (f continuous_on s <=> (!c. c SUBSET s /\ compact c ==> compact(IMAGE f c)) /\ (!c. c SUBSET s /\ P c ==> connected(IMAGE f c)))`, REPEAT STRIP_TAC THEN EQ_TAC THENL [ASM_MESON_TAC[COMPACT_CONTINUOUS_IMAGE; CONNECTED_CONTINUOUS_IMAGE; CONTINUOUS_ON_SUBSET]; STRIP_TAC] THEN REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN X_GEN_TAC `a:real^M` THEN DISCH_TAC THEN ASM_SIMP_TAC[CONTINUOUS_WITHIN_SEQUENTIALLY_COMPACT_MAP] THEN MAP_EVERY X_GEN_TAC [`p:num->real^M`; `b:real^N`] THEN STRIP_TAC THEN MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THEN SUBGOAL_THEN `!n. ?c x. a IN c /\ x IN c /\ (f:real^M->real^N) x = b /\ P c /\ c SUBSET s /\ c SUBSET ball(a,inv(&n + &1))` MP_TAC THENL [X_GEN_TAC `n:num` THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [locally]) THEN DISCH_THEN(MP_TAC o SPECL [`s INTER ball(a:real^M,inv(&n + &1))`; `a:real^M`]) THEN SIMP_TAC[OPEN_IN_OPEN_INTER; OPEN_BALL; IN_INTER; CENTRE_IN_BALL] THEN ASM_REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN DISCH_THEN(X_CHOOSE_THEN `d:real^M->bool` MP_TAC) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^M->bool` THEN REWRITE_TAC[SUBSET_INTER] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIM_SEQUENTIALLY]) THEN FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [OPEN_IN_CONTAINS_BALL]) THEN FIRST_X_ASSUM(MP_TAC o SPEC `a:real^M`) THEN ASM_REWRITE_TAC[INTER_SUBSET] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `m:num` (MP_TAC o SPEC `m:num`)) THEN REWRITE_TAC[LE_REFL] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN REWRITE_TAC[GSYM IN_BALL] THEN DISCH_TAC THEN EXISTS_TAC `(p:num->real^M) m` THEN ASM SET_TAC[]; PURE_REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM; FORALL_AND_THM] THEN MAP_EVERY X_GEN_TAC [`c:num->real^M->bool`; `x:num->real^M`] THEN STRIP_TAC] THEN SUBGOAL_THEN `!n. ?u. u IN c n /\ (f:real^M->real^N) u IN ball(b,inv(&n + &1)) DELETE b` MP_TAC THENL [X_GEN_TAC `n:num` THEN SUBGOAL_THEN `connected (IMAGE (f:real^M->real^N) (c(n:num)))` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(MP_TAC o SPEC `b:real^N` o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] CONNECTED_IMP_PERFECT)) THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[limit_point_of]] THEN DISCH_THEN(MP_TAC o SPEC `ball(b:real^N,inv(&n + &1))`) THEN ASM_REWRITE_TAC[OPEN_BALL; CENTRE_IN_BALL; REAL_LT_INV_EQ] THEN ANTS_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN REWRITE_TAC[EXISTS_IN_IMAGE] THEN MATCH_MP_TAC MONO_EXISTS THEN ASM SET_TAC[]; PURE_REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM; FORALL_AND_THM] THEN X_GEN_TAC `u:num->real^M` THEN REWRITE_TAC[FORALL_AND_THM; IN_DELETE] THEN STRIP_TAC] THEN SUBGOAL_THEN `compact(IMAGE (f:real^M->real^N) (a INSERT IMAGE u (:num)))` MP_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC COMPACT_SEQUENCE_WITH_LIMIT THEN REWRITE_TAC[LIM_SEQUENTIALLY] THEN MATCH_MP_TAC FORALL_POS_MONO_1 THEN CONJ_TAC THENL [MESON_TAC[REAL_LT_TRANS]; ONCE_REWRITE_TAC[DIST_SYM]] THEN REWRITE_TAC[GSYM IN_BALL] THEN X_GEN_TAC `n:num` THEN EXISTS_TAC `n:num` THEN X_GEN_TAC `m:num` THEN DISCH_TAC THEN SUBGOAL_THEN `ball(a:real^M,inv(&m + &1)) SUBSET ball(a,inv(&n + &1))` (fun th -> ASM SET_TAC[th]) THEN MATCH_MP_TAC SUBSET_BALL THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LT; REAL_OF_NUM_LE] THEN ASM_ARITH_TAC; DISCH_THEN(MP_TAC o MATCH_MP COMPACT_IMP_CLOSED) THEN REWRITE_TAC[CLOSED_LIMPT] THEN DISCH_THEN(MP_TAC o SPEC `b:real^N`) THEN REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[LIMPT_APPROACHABLE; EXISTS_IN_IMAGE] THEN MATCH_MP_TAC FORALL_POS_MONO_1 THEN CONJ_TAC THENL [MESON_TAC[REAL_LT_TRANS]; ONCE_REWRITE_TAC[DIST_SYM]] THEN REWRITE_TAC[GSYM IN_BALL] THEN ASM SET_TAC[]]);; let CONTINUOUS_EQ_COMPACT_CONNECTED_PRESERVING = prove (`!f:real^M->real^N s. locally connected s ==> (f continuous_on s <=> (!c. c SUBSET s /\ compact c ==> compact(IMAGE f c)) /\ (!c. c SUBSET s /\ connected c ==> connected(IMAGE f c)))`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_EQ_COMPACT_CONNECTED_PRESERVING_GEN THEN ASM_REWRITE_TAC[]);; let CONTINUOUS_EQ_COMPACT_PATH_CONNECTED_PRESERVING = prove (`!f:real^M->real^N s. locally path_connected s ==> (f continuous_on s <=> (!c. c SUBSET s /\ compact c ==> compact(IMAGE f c)) /\ (!c. c SUBSET s /\ path_connected c ==> path_connected(IMAGE f c)))`, REPEAT STRIP_TAC THEN EQ_TAC THENL [MESON_TAC[COMPACT_CONTINUOUS_IMAGE; PATH_CONNECTED_CONTINUOUS_IMAGE; CONTINUOUS_ON_SUBSET]; STRIP_TAC THEN FIRST_ASSUM(MP_TAC o ISPEC `f:real^M->real^N` o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] CONTINUOUS_EQ_COMPACT_CONNECTED_PRESERVING_GEN)) THEN ASM_MESON_TAC[PATH_CONNECTED_IMP_CONNECTED]]);; let CONNECTED_CLOSED_POINTIMAGES_IMP_CONTINUOUS_ON = prove (`!f:real^N->real^1 s t. IMAGE f s SUBSET t /\ locally connected s /\ (!c. c SUBSET s /\ connected c ==> connected(IMAGE f c)) /\ (!y. closed_in (subtopology euclidean s) {x | x IN s /\ f x = y}) ==> f continuous_on s`, REPEAT STRIP_TAC THEN REWRITE_TAC[continuous_on] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LOCALLY_CONNECTED]) THEN DISCH_THEN(MP_TAC o SPECL [`s DIFF {y:real^N | y IN s /\ f y IN sphere(f x:real^1,e)}`; `x:real^N`]) THEN ANTS_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_DIFF THEN REWRITE_TAC[OPEN_IN_REFL] THEN ASM_SIMP_TAC[SPHERE_1; REAL_ARITH `&0 < e ==> ~(e < &0)`] THEN ASM_SIMP_TAC[CLOSED_IN_UNION; SET_RULE `{x | x IN s /\ f x IN {a,b}} = {x | x IN s /\ f x = a} UNION {x | x IN s /\ f x = b}`]; ASM_REWRITE_TAC[IN_DIFF; IN_ELIM_THM; IN_SPHERE; DIST_REFL] THEN ASM_SIMP_TAC[REAL_LT_IMP_NZ]]; DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [open_in]) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `x:real^N`)) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `IMAGE (f:real^N->real^1) u SUBSET ball(f x,e)` MP_TAC THENL [MP_TAC(ISPECL [`IMAGE (f:real^N->real^1) u`; `ball((f:real^N->real^1) x,e)`] CONNECTED_INTER_FRONTIER) THEN ASM_SIMP_TAC[FRONTIER_BALL] THEN SUBGOAL_THEN `(f:real^N->real^1) x IN ball(f x,e)` MP_TAC THENL [ASM_REWRITE_TAC[CENTRE_IN_BALL]; ASM SET_TAC[]]; REWRITE_TAC[IN_BALL; FORALL_IN_IMAGE; SUBSET] THEN ASM_MESON_TAC[DIST_SYM]]]);; let CONNECTED_CONNECTED_IMP_CLOSED_POINTIMAGES = prove (`!f:real^M->real^N s. (!c. c SUBSET s /\ connected c ==> connected(IMAGE f c)) /\ (!y. connected {x | x IN s /\ f x = y}) ==> !y. closed_in (subtopology euclidean s) {x | x IN s /\ f x = y}`, REPEAT GEN_TAC THEN STRIP_TAC THEN X_GEN_TAC `b:real^N` THEN ABBREV_TAC `t = {x | x IN s /\ (f:real^M->real^N) x = b}` THEN ASM_CASES_TAC `t:real^M->bool = {}` THEN ASM_SIMP_TAC[CLOSED_IN_EMPTY] THEN REWRITE_TAC[CLOSED_IN_LIMPT] THEN CONJ_TAC THENL [ASM SET_TAC[]; X_GEN_TAC `a:real^M`] THEN FIRST_X_ASSUM(MP_TAC o SPEC `b:real^N`) THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `connected(IMAGE (f:real^M->real^N) (a INSERT t))` MP_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[CONNECTED_INSERT_LIMPT] THEN ASM SET_TAC[]; REWRITE_TAC[CONNECTED_CLOSED_IN; NOT_EXISTS_THM] THEN DISCH_THEN(MP_TAC o SPECL [`{b:real^N}`; `{(f:real^M->real^N) a}`]) THEN REWRITE_TAC[CLOSED_IN_SING] THEN ASM SET_TAC[]]);; let CONNECTED_CONNECTED_POINTIMAGES_IMP_CONTINUOUS_ON = prove (`!f:real^N->real^1 s t. IMAGE f s SUBSET t /\ locally connected s /\ (!c. c SUBSET s /\ connected c ==> connected(IMAGE f c)) /\ (!y. connected {x | x IN s /\ f x = y}) ==> f continuous_on s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CONNECTED_CLOSED_POINTIMAGES_IMP_CONTINUOUS_ON THEN EXISTS_TAC `t:real^1->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONNECTED_CONNECTED_IMP_CLOSED_POINTIMAGES THEN ASM_REWRITE_TAC[]);; let CLOSED_CLOSED_PREIMAGES_IMP_CONTINUOUS_ON = prove (`!f:real^M->real^N s t. compact t /\ (!y. closed_in (subtopology euclidean s) {x | x IN s /\ f x = y}) /\ (!c. closed_in (subtopology euclidean s) c ==> closed_in (subtopology euclidean t) (IMAGE f c)) ==> f continuous_on s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC COMPACT_CLOSED_POINTIMAGES_IMP_CONTINUOUS_ON THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CLOSED_IN_COMPACT THEN EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[CLOSED_SUBSET_EQ; COMPACT_IMP_CLOSED]);; let CLOSED_CONNECTED_PREIMAGES_IMP_CONTINUOUS_ON = prove (`!f:real^M->real^N s t. compact t /\ (!y. connected {x | x IN s /\ f x = y}) /\ (!c. closed_in (subtopology euclidean s) c ==> closed_in (subtopology euclidean t) (IMAGE f c)) /\ (!c. c SUBSET s /\ connected c ==> connected(IMAGE f c)) ==> f continuous_on s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CLOSED_CLOSED_PREIMAGES_IMP_CONTINUOUS_ON THEN EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONNECTED_CONNECTED_IMP_CLOSED_POINTIMAGES THEN ASM_REWRITE_TAC[]);; let BICONNECTED_IMP_CONTINUOUS_ON = prove (`!f:real^M->real^N s t. FINITE (components t) /\ locally compact t /\ locally connected t /\ IMAGE f s = t /\ (!c. c SUBSET s /\ connected c ==> connected(IMAGE f c)) /\ (!c. c SUBSET t /\ connected c ==> connected {x | x IN s /\ f x IN c}) ==> f continuous_on s`, let lemma = prove (`{n | f n IN UNIONS a} = UNIONS {{n | f n IN s} | s IN a}`, REWRITE_TAC[UNIONS_GSPEC] THEN SET_TAC[]) in REPEAT STRIP_TAC THEN REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN X_GEN_TAC `p:real^M` THEN DISCH_TAC THEN REWRITE_TAC[CONTINUOUS_WITHIN_SEQUENTIALLY_INJ] THEN X_GEN_TAC `x:num->real^M` THEN STRIP_TAC THEN REWRITE_TAC[TENDSTO_ALT; EVENTUALLY_SEQUENTIALLY; o_DEF] THEN X_GEN_TAC `v:real^N->bool` THEN STRIP_TAC THEN ASM_CASES_TAC `FINITE {n:num | (f:real^M->real^N) (x n) IN (t DIFF v)}` THENL [FIRST_ASSUM(MP_TAC o ISPEC `\n:num. n` o MATCH_MP UPPER_BOUND_FINITE_SET) THEN REWRITE_TAC[FORALL_IN_GSPEC; IN_DIFF] THEN MATCH_MP_TAC(MESON[] `(!n. P n ==> Q(SUC n)) ==> (?n. P n) ==> (?n. Q n)`) THEN X_GEN_TAC `N:num` THEN DISCH_TAC THEN X_GEN_TAC `n:num` THEN GEN_REWRITE_TAC I [GSYM CONTRAPOS_THM] THEN REWRITE_TAC[ARITH_RULE `~(SUC m <= n) <=> n <= m`] THEN ASM SET_TAC[]; MATCH_MP_TAC(TAUT `F ==> p`)] THEN SUBGOAL_THEN `?u. open_in (subtopology euclidean t) u /\ (f:real^M->real^N) p IN u /\ u SUBSET v /\ INFINITE {n:num | (f:real^M->real^N) (x n) IN t DIFF u} /\ FINITE(components(t DIFF u))` STRIP_ASSUME_TAC THENL [MP_TAC(ISPEC `t:real^N->bool` SEMI_LOCALLY_CONNECTED_GEN) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPECL [`(f:real^M->real^N) p`; `t INTER v:real^N->bool`]) THEN ASM_SIMP_TAC[OPEN_IN_OPEN_INTER] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[GSYM INFINITE]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] INFINITE_SUPERSET)) THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?c. c IN components (t DIFF u) /\ INFINITE {n:num | (f:real^M->real^N)(x n) IN c}` STRIP_ASSUME_TAC THENL [UNDISCH_TAC `INFINITE {n:num | (f:real^M->real^N)(x n) IN t DIFF u}` THEN MP_TAC(ISPEC `t DIFF u:real^N->bool` UNIONS_COMPONENTS) THEN DISCH_THEN(fun th -> GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [th]) THEN REWRITE_TAC[lemma; INFINITE; FINITE_UNIONS] THEN ASM_SIMP_TAC[SIMPLE_IMAGE; FINITE_IMAGE; FORALL_IN_IMAGE] THEN MESON_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPEC `(p:real^M) INSERT {x | x IN s /\ (f:real^M->real^N) x IN c}`) THEN REWRITE_TAC[NOT_IMP] THEN REPEAT CONJ_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC CONNECTED_INSERT_LIMPT THEN CONJ_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN CONJ_TAC THENL [TRANS_TAC SUBSET_TRANS `t DIFF u:real^N->bool` THEN REWRITE_TAC[SUBSET_DIFF] THEN ASM_MESON_TAC[IN_COMPONENTS_SUBSET]; ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]]; FIRST_ASSUM(MP_TAC o MATCH_MP INFINITE_ENUMERATE) THEN DISCH_THEN(X_CHOOSE_THEN `r:num->num` STRIP_ASSUME_TAC) THEN REWRITE_TAC[LIMPT_SEQUENTIAL] THEN EXISTS_TAC `(x:num->real^M) o (r:num->num)` THEN ASM_SIMP_TAC[LIM_SUBSEQUENCE] THEN REWRITE_TAC[o_DEF; IN_ELIM_THM] THEN ASM SET_TAC[]]; SUBGOAL_THEN `IMAGE f (p INSERT {x | x IN s /\ f x IN c}) = (f:real^M->real^N)(p) INSERT c` SUBST1_TAC THENL [MP_TAC(ISPECL [`t DIFF u:real^N->bool`; `c:real^N->bool`] IN_COMPONENTS_SUBSET) THEN ASM SET_TAC[]; ALL_TAC] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_CONNECTED) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_NONEMPTY) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN ASM_SIMP_TAC[CONNECTED_INSERT] THEN REWRITE_TAC[closure; IN_UNION; DE_MORGAN_THM; IN_ELIM_THM] THEN CONJ_TAC THENL [ASM SET_TAC[]; DISCH_TAC] THEN MP_TAC(ISPECL [`t:real^N->bool`; `c:real^N->bool`; `(f:real^M->real^N) p`] LIMIT_POINT_OF_LOCAL) THEN ASM SET_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Topological characterizations of non-strict monotonicity. *) (* ------------------------------------------------------------------------- *) let MONOTONE_TOPOLOGICALLY_IMP = prove (`!f s. (!c. connected c ==> connected {x | x IN s /\ f x IN c}) ==> (!x y. x IN s /\ y IN s /\ drop x <= drop y ==> drop(f x) <= drop(f y)) \/ (!x y. x IN s /\ y IN s /\ drop x <= drop y ==> drop(f y) <= drop(f x))`, REPEAT STRIP_TAC THEN REWRITE_TAC[FORALL_LIFT; REAL_NON_MONOTONE; LIFT_DROP] THEN REWRITE_TAC[NOT_EXISTS_THM; FORALL_DROP; LIFT_DROP] THEN MAP_EVERY X_GEN_TAC [`a:real^1`; `b:real^1`; `c:real^1`] THEN STRIP_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPEC `{y | drop y < drop(f(b:real^1))}`); FIRST_X_ASSUM(MP_TAC o SPEC `{y | drop(f(b:real^1)) < drop y}`)] THEN REWRITE_TAC[NOT_IMP; GSYM IS_INTERVAL_CONNECTED_1] THEN (CONJ_TAC THENL [REWRITE_TAC[IS_INTERVAL_1_CASES] THEN SET_TAC[]; ALL_TAC]) THEN REWRITE_TAC[IS_INTERVAL_1; IN_ELIM_THM] THEN DISCH_THEN(MP_TAC o SPECL [`a:real^1`; `c:real^1`; `b:real^1`]) THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; REAL_LT_REFL]);; let MONOTONE_TOPOLOGICALLY_EQ = prove (`!f s. (!c. connected c ==> connected {x | x IN s /\ f x IN c}) <=> is_interval s /\ ((!x y. x IN s /\ y IN s /\ drop x <= drop y ==> drop(f x) <= drop(f y)) \/ (!x y. x IN s /\ y IN s /\ drop x <= drop y ==> drop(f y) <= drop(f x)))`, REPEAT GEN_TAC THEN EQ_TAC THEN SIMP_TAC[MONOTONE_TOPOLOGICALLY_IMP] THENL [DISCH_THEN(MP_TAC o SPEC `(:real^1)`) THEN REWRITE_TAC[SET_RULE `{x | x IN s /\ f x IN UNIV} = s`] THEN REWRITE_TAC[CONNECTED_UNIV; GSYM IS_INTERVAL_CONNECTED_1]; REWRITE_TAC[GSYM IS_INTERVAL_CONNECTED_1; IS_INTERVAL_1] THEN SET_TAC[]]);; let MONOTONE_TOPOLOGICALLY = prove (`!f s. is_interval s ==> ((!x y. x IN s /\ y IN s /\ drop x <= drop y ==> drop(f x) <= drop(f y)) \/ (!x y. x IN s /\ y IN s /\ drop x <= drop y ==> drop(f y) <= drop(f x)) <=> !c. connected c ==> connected {x | x IN s /\ f x IN c})`, SIMP_TAC[MONOTONE_TOPOLOGICALLY_EQ]);; let MONOTONE_TOPOLOGICALLY_INTO_1D_EQ = prove (`!f:real^N->real^1 s. f continuous_on s ==> ((!k. connected k ==> connected {x | x IN s /\ f x IN k}) <=> connected s /\ (!y. connected {x | x IN s /\ f x = y}))`, REPEAT GEN_TAC THEN REWRITE_TAC[GSYM CONTINUOUS_MAP_EUCLIDEAN] THEN REWRITE_TAC[CONTINUOUS_MAP_EQ_DROP] THEN DISCH_THEN(MP_TAC o MATCH_MP MONOTONE_MAP_INTO_EUCLIDEANREAL_ALT) THEN REWRITE_TAC[MONOTONE_MAP; CONNECTED_SPACE_EUCLIDEAN_SUBTOPOLOGY] THEN REWRITE_TAC[CONNECTED_IN_SUBTOPOLOGY; TOPSPACE_EUCLIDEANREAL; SUBSET_UNIV; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; SUBSET_RESTRICT] THEN REWRITE_TAC[CONNECTED_IN_EUCLIDEAN; o_THM; FORALL_DROP; DROP_EQ] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[FORALL_DROP_IMAGE; DROP_IN_IMAGE_DROP] THEN REWRITE_TAC[IS_REALINTERVAL_IS_INTERVAL] THEN REWRITE_TAC[GSYM IMAGE_o; o_DEF; LIFT_DROP; IMAGE_ID] THEN REWRITE_TAC[IS_INTERVAL_CONNECTED_1]);; let MONOTONE_TOPOLOGICALLY_INTO_1D = prove (`!f:real^N->real^1 s. connected s /\ f continuous_on s /\ (!y. connected {x | x IN s /\ f x = y}) ==> (!k. connected k ==> connected {x | x IN s /\ f x IN k})`, MESON_TAC[MONOTONE_TOPOLOGICALLY_INTO_1D_EQ]);; let MONOTONE_TOPOLOGICALLY_POINTS = prove (`!f:real^1->real^1 s. is_interval s /\ f continuous_on s ==> ((!x y. x IN s /\ y IN s /\ drop x <= drop y ==> drop(f x) <= drop(f y)) \/ (!x y. x IN s /\ y IN s /\ drop x <= drop y ==> drop(f y) <= drop(f x)) <=> !a. connected {x | x IN s /\ f x = a})`, SIMP_TAC[MONOTONE_TOPOLOGICALLY; MONOTONE_TOPOLOGICALLY_INTO_1D_EQ] THEN SIMP_TAC[IS_INTERVAL_CONNECTED]);; let MONOTONE_TOPOLOGICALLY_POINTS_IMP = prove (`!f s. f continuous_on s /\ is_interval s /\ (!y. connected {x | x IN s /\ f x = y}) ==> (!x y. x IN s /\ y IN s /\ drop x <= drop y ==> drop(f x) <= drop(f y)) \/ (!x y. x IN s /\ y IN s /\ drop x <= drop y ==> drop(f y) <= drop(f x))`, SIMP_TAC[MONOTONE_TOPOLOGICALLY_POINTS]);; let MONOTONE_IMP_HOMEOMORPHISM_1D = prove (`!f s t. is_interval s /\ is_interval t /\ IMAGE f s = t /\ ((!x y. x IN s /\ y IN s /\ drop x < drop y ==> drop(f x) < drop(f y)) \/ (!x y. x IN s /\ y IN s /\ drop x < drop y ==> drop(f x) < drop(f y))) ==> ?g. homeomorphism(s,t) (f,g)`, REPEAT GEN_TAC THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN SUBGOAL_THEN `!x y. x IN s /\ y IN s ==> ((f:real^1->real^1) x = f y <=> x = y)` ASSUME_TAC THENL [REWRITE_TAC[GSYM INJECTIVE_ON_ALT] THEN REWRITE_TAC[GSYM DROP_EQ; GSYM REAL_LE_ANTISYM] THEN ASM_MESON_TAC[REAL_NOT_LE]; ALL_TAC] THEN EXPAND_TAC "t" THEN W(MP_TAC o PART_MATCH (rand o rand) INJECTIVE_INTO_1D_EQ_HOMEOMORPHISM o snd) THEN ASM_REWRITE_TAC[INJECTIVE_ON_ALT] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_SIMP_TAC[IS_INTERVAL_PATH_CONNECTED] THEN MATCH_MP_TAC CONNECTED_CONNECTED_POINTIMAGES_IMP_CONTINUOUS_ON THEN EXISTS_TAC `t:real^1->bool` THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN ASM_SIMP_TAC[CONVEX_IMP_LOCALLY_CONNECTED; GSYM IS_INTERVAL_CONVEX_1] THEN CONJ_TAC THENL [MP_TAC(ISPECL [`f:real^1->real^1`; `s:real^1->bool`] INJECTIVE_ON_LEFT_INVERSE) THEN ASM_REWRITE_TAC[INJECTIVE_ON_ALT] THEN DISCH_THEN(X_CHOOSE_TAC `g:real^1->real^1`) THEN MP_TAC(ISPECL [`g:real^1->real^1`; `t:real^1->bool`] MONOTONE_TOPOLOGICALLY) THEN ASM_REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN EXPAND_TAC "t" THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN ASM_SIMP_TAC[GSYM REAL_NOT_LT; CONTRAPOS_THM] THEN DISCH_THEN(MP_TAC o fst o EQ_IMP_RULE) THEN ANTS_TAC THENL [ASM_MESON_TAC[]; MATCH_MP_TAC MONO_FORALL] THEN X_GEN_TAC `c:real^1->bool` THEN DISCH_THEN(fun th -> REPEAT STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN ASM SET_TAC[]; X_GEN_TAC `y:real^1` THEN SUBGOAL_THEN `{x | x IN s /\ (f:real^1->real^1) x = y} = {} \/ ?a. {x | x IN s /\ f x = y} = {a}` MP_TAC THENL [MATCH_MP_TAC(SET_RULE `(!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> {x | x IN s /\ f x = a} = {} \/ ?b. {x | x IN s /\ f x = a} = {b}`) THEN ASM_MESON_TAC[]; STRIP_TAC THEN ASM_REWRITE_TAC[CONNECTED_EMPTY; CONNECTED_SING]]]);; let MONOTONE_CONNECTED_PREIMAGES_IMP_PROPER_MAP = prove (`!f:real^M->real^N s t. IMAGE f s = t /\ locally compact s /\ locally connected t /\ f continuous_on s /\ (!y. compact {x | x IN s /\ f x = y}) /\ (!c. c SUBSET t /\ connected c ==> connected {x | x IN s /\ f x IN c}) ==> (!k. k SUBSET t /\ compact k ==> compact {x | x IN s /\ f x IN k})`, REPEAT GEN_TAC THEN STRIP_TAC THEN ASM_SIMP_TAC[PROPER_MAP; SUBSET_REFL] THEN X_GEN_TAC `k:real^M->bool` THEN STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN REWRITE_TAC[CLOSED_IN_LIMPT] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN X_GEN_TAC `y:real^N` THEN ONCE_REWRITE_TAC[TAUT `p /\ q ==> r <=> q /\ ~r ==> ~p`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^M->bool`; `s DIFF k:real^M->bool`] LOCALLY_COMPACT_OPEN_IN) THEN ASM_SIMP_TAC[OPEN_IN_DIFF; OPEN_IN_REFL] THEN REWRITE_TAC[LOCALLY_COMPACT_COMPACT] THEN DISCH_THEN(MP_TAC o SPEC `{x | x IN s /\ (f:real^M->real^N) x = y}`) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u:real^M->bool`; `v:real^M->bool`] THEN STRIP_TAC THEN ABBREV_TAC `b:real^M->bool = closure u DIFF u` THEN SUBGOAL_THEN `(b:real^M->bool) SUBSET v` ASSUME_TAC THENL [EXPAND_TAC "b" THEN MATCH_MP_TAC(SET_RULE `s SUBSET u ==> s DIFF t SUBSET u`) THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED]; ALL_TAC] THEN SUBGOAL_THEN `compact(b:real^M->bool)` ASSUME_TAC THENL [MATCH_MP_TAC CLOSED_IN_COMPACT THEN EXISTS_TAC `v:real^M->bool` THEN ASM_REWRITE_TAC[] THEN EXPAND_TAC "b" THEN MATCH_MP_TAC CLOSED_IN_DIFF THEN CONJ_TAC THENL [MATCH_MP_TAC CLOSED_SUBSET THEN REWRITE_TAC[CLOSED_CLOSURE] THEN ASM SET_TAC[]; MATCH_MP_TAC OPEN_IN_SUBSET_TRANS THEN EXISTS_TAC `s DIFF k:real^M->bool` THEN ASM SET_TAC[]]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LOCALLY_CONNECTED]) THEN DISCH_THEN(MP_TAC o SPECL [`t DIFF IMAGE (f:real^M->real^N) b`; `y:real^N`]) THEN ANTS_TAC THENL [CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC OPEN_IN_DIFF THEN REWRITE_TAC[OPEN_IN_REFL] THEN MATCH_MP_TAC CLOSED_SUBSET THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC COMPACT_IMP_CLOSED THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; DISCH_THEN(X_CHOOSE_THEN `r:real^N->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`t:real^N->bool`; `IMAGE (f:real^M->real^N) k`; `y:real^N`] LIMIT_POINT_OF_LOCAL) THEN ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN DISCH_THEN(MP_TAC o SPEC `r:real^N->bool`) THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPEC `{x | x IN s /\ (f:real^M->real^N) x IN r}` CONNECTED_OPEN_IN) THEN MATCH_MP_TAC(TAUT `p /\ (r ==> q) ==> (p <=> ~q) ==> ~r`) THEN CONJ_TAC THENL [ASM_MESON_TAC[OPEN_IN_IMP_SUBSET]; DISCH_TAC] THEN MAP_EVERY EXISTS_TAC [`{x | x IN s /\ (f:real^M->real^N) x IN r} INTER u`; `{x | x IN s /\ (f:real^M->real^N) x IN r} DIFF closure u`] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_SUBTOPOLOGY_INTER_SUBSET THEN EXISTS_TAC `s:real^M->bool` THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC OPEN_IN_INTER THEN REWRITE_TAC[OPEN_IN_REFL] THEN MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `s DIFF k:real^M->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC OPEN_IN_DIFF THEN ASM_REWRITE_TAC[OPEN_IN_REFL]; SIMP_TAC[OPEN_IN_DIFF_CLOSED; CLOSED_CLOSURE]; ASM SET_TAC[]; MP_TAC(ISPEC `u:real^M->bool` CLOSURE_SUBSET) THEN ASM SET_TAC[]; ASM SET_TAC[]; ASM SET_TAC[]]]);; let MONOTONE_INTO_1D_IMP_PROPER_MAP = prove (`!f:real^N->real^1 s t. connected s /\ locally compact s /\ f continuous_on s /\ IMAGE f s = t /\ (!y. compact {x | x IN s /\ f x = y}) /\ (!y. connected {x | x IN s /\ f x = y}) ==> (!k. k SUBSET t /\ compact k ==> compact {x | x IN s /\ f x IN k})`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC MONOTONE_CONNECTED_PREIMAGES_IMP_PROPER_MAP THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [MATCH_MP_TAC CONVEX_IMP_LOCALLY_CONNECTED THEN REWRITE_TAC[CONVEX_CONNECTED_1] THEN ASM_MESON_TAC[CONNECTED_CONTINUOUS_IMAGE]; ASM_MESON_TAC[MONOTONE_TOPOLOGICALLY_INTO_1D]]);; let MONOTONE_CONNECTED_PREIMAGES_IMP_PROPER_MAP_GEN = prove (`!f:real^M->real^N s t. IMAGE f s = t /\ locally compact s /\ locally connected t /\ (!c. c SUBSET s /\ compact c ==> compact(IMAGE f c)) /\ (!y. compact {x | x IN s /\ f x = y}) /\ (!c. c SUBSET t /\ connected c ==> connected {x | x IN s /\ f x IN c}) ==> (!k. k SUBSET t /\ compact k ==> compact {x | x IN s /\ f x IN k})`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC MONOTONE_CONNECTED_PREIMAGES_IMP_PROPER_MAP THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC COMPACT_CLOSED_POINTIMAGES_IMP_CONTINUOUS_ON THEN ASM_REWRITE_TAC[] THEN GEN_TAC THEN MATCH_MP_TAC CLOSED_SUBSET THEN REWRITE_TAC[SUBSET_RESTRICT] THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED]);; (* ------------------------------------------------------------------------- *) (* Sura-Bura's results about compact components of sets. *) (* ------------------------------------------------------------------------- *) let SURA_BURA_COMPACT = prove (`!s c:real^N->bool. compact s /\ c IN components s ==> c = INTERS {t | c SUBSET t /\ open_in (subtopology euclidean s) t /\ closed_in (subtopology euclidean s) t}`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`subtopology euclidean (s:real^N->bool)`; `c:real^N->bool`] COMPACT_QUASI_EQ_CONNECTED_COMPONENTS_OF) THEN ASM_REWRITE_TAC[LOCALLY_COMPACT_SPACE_SUBTOPOLOGY_EUCLIDEAN] THEN ASM_REWRITE_TAC[COMPACT_IN_SUBTOPOLOGY; COMPACT_IN_EUCLIDEAN] THEN ASM_SIMP_TAC[EUCLIDEAN_CONNECTED_COMPONENTS_OF; IN_COMPONENTS_SUBSET] THEN ASM_SIMP_TAC[HAUSDORFF_SPACE_EUCLIDEAN; HAUSDORFF_SPACE_SUBTOPOLOGY] THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED; CLOSED_IMP_LOCALLY_COMPACT] THEN ANTS_TAC THENL [ASM_MESON_TAC[COMPACT_COMPONENTS]; ALL_TAC] THEN DISCH_THEN(fun th -> GEN_REWRITE_TAC LAND_CONV [GSYM(MATCH_MP QUASI_COMPONENTS_OF_SET th)]) THEN REWRITE_TAC[CONJ_ACI]);; let SURA_BURA_CLOPEN_SUBSET = prove (`!s c u:real^N->bool. locally compact s /\ c IN components s /\ compact c /\ open u /\ c SUBSET u ==> ?k. open_in (subtopology euclidean s) k /\ compact k /\ c SUBSET k /\ k SUBSET u`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM (MP_TAC o GEN_REWRITE_RULE I [LOCALLY_COMPACT_COMPACT_SUBOPEN]) THEN DISCH_THEN(MP_TAC o SPECL [`c:real^N->bool`; `u:real^N->bool`]) THEN ASM_SIMP_TAC[IN_COMPONENTS_SUBSET; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`v:real^N->bool`; `k:real^N->bool`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`k:real^N->bool`; `c:real^N->bool`] SURA_BURA_COMPACT) THEN ASM_SIMP_TAC[CLOSED_IN_COMPACT_EQ] THEN ANTS_TAC THENL [MATCH_MP_TAC COMPONENTS_INTERMEDIATE_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; DISCH_THEN(ASSUME_TAC o SYM)] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN DISCH_THEN(X_CHOOSE_THEN `w:real^N->bool` (STRIP_ASSUME_TAC o GSYM)) THEN MP_TAC(ISPECL [`(:real^N) DIFF (u INTER w)`; `{t:real^N->bool | c SUBSET t /\ open_in (subtopology euclidean k) t /\ compact t /\ t SUBSET k}`] CLOSED_IMP_FIP_COMPACT) THEN ASM_SIMP_TAC[GSYM OPEN_CLOSED; OPEN_INTER; FORALL_IN_GSPEC] THEN GEN_REWRITE_TAC LAND_CONV [GSYM CONTRAPOS_THM] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_NONEMPTY) THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [SUBSET] THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; IN_ELIM_THM; SET_RULE `(UNIV DIFF u) INTER s = {} <=> s SUBSET u`] THEN DISCH_THEN(X_CHOOSE_THEN `f:(real^N->bool)->bool` MP_TAC) THEN ASM_CASES_TAC `f:(real^N->bool)->bool = {}` THENL [ASM_REWRITE_TAC[NOT_IN_EMPTY; INTERS_0; FINITE_EMPTY] THEN REWRITE_TAC[SET_RULE `UNIV SUBSET s INTER t <=> s = UNIV /\ t = UNIV`] THEN DISCH_THEN(CONJUNCTS_THEN SUBST_ALL_TAC) THEN RULE_ASSUM_TAC(REWRITE_RULE[INTER_UNIV]) THEN UNDISCH_THEN `s:real^N->bool = v` (SUBST_ALL_TAC o SYM) THEN SUBGOAL_THEN `k:real^N->bool = s` SUBST_ALL_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[SUBSET_UNIV]] THEN EXISTS_TAC `s:real^N->bool` THEN ASM_SIMP_TAC[IN_COMPONENTS_SUBSET; OPEN_IN_REFL]; STRIP_TAC THEN EXISTS_TAC `INTERS f:real^N->bool` THEN ASM_SIMP_TAC[COMPACT_INTERS] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `v:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC OPEN_IN_SUBSET_TRANS THEN EXISTS_TAC `k:real^N->bool` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_INTERS THEN ASM_SIMP_TAC[]; EXPAND_TAC "v" THEN REWRITE_TAC[SUBSET_INTER] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC(SET_RULE `(!t. t IN f ==> t SUBSET s) /\ ~(f = {}) ==> INTERS f SUBSET s`) THEN ASM_MESON_TAC[OPEN_IN_IMP_SUBSET; SUBSET_TRANS]]]);; let SURA_BURA_CLOPEN_SUBSET_ALT = prove (`!s c u:real^N->bool. locally compact s /\ c IN components s /\ compact c /\ open_in (subtopology euclidean s) u /\ c SUBSET u ==> ?k. open_in (subtopology euclidean s) k /\ compact k /\ c SUBSET k /\ k SUBSET u`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN DISCH_THEN(X_CHOOSE_THEN `v:real^N->bool` (STRIP_ASSUME_TAC o GSYM)) THEN MP_TAC(ISPECL [`s:real^N->bool`; `c:real^N->bool`; `v:real^N->bool`] SURA_BURA_CLOPEN_SUBSET) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET)) THEN ASM SET_TAC[]);; let SURA_BURA = prove (`!s c:real^N->bool. locally compact s /\ c IN components s /\ compact c ==> c = INTERS {k | c SUBSET k /\ compact k /\ open_in (subtopology euclidean s) k}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^N` THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN MP_TAC(ISPECL [`{x:real^N}`; `c:real^N->bool`] SEPARATION_NORMAL) THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED; CLOSED_SING] THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `v:real^N->bool`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `c:real^N->bool`; `v:real^N->bool`] SURA_BURA_CLOPEN_SUBSET) THEN ASM_REWRITE_TAC[IN_INTERS; NOT_FORALL_THM; IN_ELIM_THM; NOT_IMP] THEN MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN ASM SET_TAC[]);; let COMPONENT_CLOPEN_HAUSDIST_EXPLICIT = prove (`!s c:real^N->bool e. &0 < e /\ locally compact s /\ c IN components s /\ compact c ==> ?k. open_in (subtopology euclidean s) k /\ compact k /\ c SUBSET k /\ k SUBSET {x + d | x IN c /\ d IN ball(vec 0,e)}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SURA_BURA_CLOPEN_SUBSET THEN ASM_SIMP_TAC[OPEN_SUMS; OPEN_BALL] THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MAP_EVERY EXISTS_TAC [`x:real^N`; `vec 0:real^N`] THEN ASM_REWRITE_TAC[CENTRE_IN_BALL; VECTOR_ADD_RID]);; let COMPONENT_CLOPEN_HAUSDIST = prove (`!s c:real^N->bool e. &0 < e /\ locally compact s /\ c IN components s /\ compact c ==> ?k. open_in (subtopology euclidean s) k /\ compact k /\ c SUBSET k /\ hausdist(c,k) < e`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `c:real^N->bool`; `e / &2`] COMPONENT_CLOPEN_HAUSDIST_EXPLICIT) THEN ASM_REWRITE_TAC[REAL_HALF] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MAP_EVERY ASM_CASES_TAC [`c:real^N->bool = {}`; `k:real^N->bool = {}`] THEN ASM_REWRITE_TAC[HAUSDIST_EMPTY] THEN MATCH_MP_TAC(REAL_ARITH `&0 < e /\ x <= e / &2 ==> x < e`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_HAUSDIST_LE_SUMS THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] SUBSET_TRANS)) THENL [REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MAP_EVERY EXISTS_TAC [`x:real^N`; `vec 0:real^N`] THEN ASM_REWRITE_TAC[CENTRE_IN_CBALL; VECTOR_ADD_RID] THEN ASM_REAL_ARITH_TAC; MATCH_MP_TAC(SET_RULE `t SUBSET u ==> {f x y | x IN s /\ y IN t} SUBSET {f x y | x IN s /\ y IN u}`) THEN REWRITE_TAC[BALL_SUBSET_CBALL]]);; let COMPONENT_INTERMEDIATE_CLOPEN = prove (`!s t u:real^N->bool. t IN components s /\ open_in (subtopology euclidean s) u /\ t SUBSET u /\ (dimindex(:N) = 1 \/ (?r:real^1->bool. s homeomorphic r) \/ locally connected s \/ (locally compact s /\ compact t)) ==> ?c. closed_in (subtopology euclidean s) c /\ open_in (subtopology euclidean s) c /\ t SUBSET c /\ c SUBSET u`, let lemma = prove (`!s t u:real^1->bool. bounded s /\ t IN components s /\ open_in (subtopology euclidean s) u /\ t SUBSET u ==> ?c. closed_in (subtopology euclidean s) c /\ open_in (subtopology euclidean s) c /\ t SUBSET c /\ c SUBSET u`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_NONEMPTY) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_COMPONENT) THEN FIRST_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_CONNECTED) THEN REWRITE_TAC[GSYM IS_INTERVAL_CONNECTED_1] THEN DISCH_TAC THEN SUBGOAL_THEN `?a b:real^1. s INTER interval[a,b] = t` STRIP_ASSUME_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CLOSED_IN_CLOSED]) THEN DISCH_THEN(X_CHOOSE_THEN `d:real^1->bool` (STRIP_ASSUME_TAC o GSYM)) THEN MP_TAC(ISPECL [`d:real^1->bool`; `t:real^1->bool`] EXISTS_COMPONENT_SUPERSET) THEN ASM_REWRITE_TAC[GSYM IS_INTERVAL_CONNECTED_1] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `c:real^1->bool` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(X_CHOOSE_TAC `b:real^1` o MATCH_MP BOUNDED_SUBSET_CLOSED_INTERVAL_SYMMETRIC) THEN MP_TAC(ISPEC `c INTER interval[--b:real^1,b]` CONNECTED_COMPACT_INTERVAL_1) THEN MATCH_MP_TAC(TAUT `p /\ (q ==> r) ==> (p <=> q) ==> r`) THEN CONJ_TAC THENL [CONJ_TAC THENL [REWRITE_TAC[GSYM IS_INTERVAL_CONNECTED_1] THEN MATCH_MP_TAC IS_INTERVAL_INTER THEN REWRITE_TAC[IS_INTERVAL_INTERVAL] THEN ASM_MESON_TAC[IS_INTERVAL_CONNECTED_1; IN_COMPONENTS_CONNECTED]; MATCH_MP_TAC CLOSED_INTER_COMPACT THEN ASM_MESON_TAC[CLOSED_COMPONENTS; COMPACT_INTERVAL]]; REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN FIRST_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN ASM SET_TAC[]]; ALL_TAC] THEN SUBGOAL_THEN `drop a <= drop b` ASSUME_TAC THENL [REWRITE_TAC[GSYM INTERVAL_NE_EMPTY_1] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?a'. drop a' <= drop a /\ ~(a' IN s) /\ s INTER interval[a',b] SUBSET u` STRIP_ASSUME_TAC THENL [ASM_CASES_TAC `(a:real^1) IN s` THENL [ALL_TAC; ASM_MESON_TAC[REAL_LE_REFL]] THEN FIRST_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [OPEN_IN_CONTAINS_CBALL]) THEN FIRST_X_ASSUM(MP_TAC o SPEC `a:real^1`) THEN ANTS_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN EXPAND_TAC "t" THEN REWRITE_TAC[IN_INTER] THEN ASM_REWRITE_TAC[ENDS_IN_INTERVAL] THEN ASM SET_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `r:real` THEN STRIP_TAC] THEN SUBGOAL_THEN `~(interval[a - lift r,a] SUBSET s)` MP_TAC THENL [DISCH_TAC THEN MP_TAC(ISPECL [`s:real^1->bool`; `t UNION interval [a - lift r,a]`; `t:real^1->bool`] COMPONENTS_MAXIMAL) THEN ASM_REWRITE_TAC[NOT_IMP] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_UNION THEN ASM_REWRITE_TAC[GSYM IS_INTERVAL_CONNECTED_1; IS_INTERVAL_INTERVAL; GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN EXISTS_TAC `a:real^1` THEN EXPAND_TAC "t" THEN REWRITE_TAC[IN_INTER; ENDS_IN_INTERVAL] THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[INTERVAL_NE_EMPTY_1; DROP_SUB; LIFT_DROP] THEN ASM_REAL_ARITH_TAC; ASM_MESON_TAC[CLOSED_IN_IMP_SUBSET; UNION_SUBSET]; ASM SET_TAC[]; EXPAND_TAC "t" THEN REWRITE_TAC[UNION_SUBSET; SUBSET_INTER] THEN DISCH_THEN(MP_TAC o last o CONJUNCTS) THEN REWRITE_TAC[SUBSET_INTERVAL_1; DROP_SUB; LIFT_DROP] THEN ASM_REAL_ARITH_TAC]; REWRITE_TAC[SUBSET; NOT_FORALL_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a':real^1` THEN REWRITE_TAC[NOT_IMP; IN_INTERVAL_1; DROP_SUB; LIFT_DROP] THEN STRIP_TAC THEN ASM_REWRITE_TAC[GSYM SUBSET] THEN MP_TAC(ISPECL [`a':real^1`; `b:real^1`; `a:real^1`] UNION_INTERVAL_1) THEN ASM_REWRITE_TAC[IN_INTERVAL_1; GSYM INTERVAL_NE_EMPTY_1] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN ASM_REWRITE_TAC[UNION_OVER_INTER; UNION_SUBSET] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `b INTER s SUBSET u ==> i SUBSET b ==> s INTER i SUBSET u`)) THEN REWRITE_TAC[CBALL_INTERVAL; SUBSET_INTERVAL_1] THEN REWRITE_TAC[DROP_ADD; DROP_SUB; LIFT_DROP] THEN ASM_REAL_ARITH_TAC]; ALL_TAC] THEN SUBGOAL_THEN `?b'. drop b <= drop b' /\ ~(b' IN s) /\ s INTER interval[a',b'] SUBSET u` STRIP_ASSUME_TAC THENL [ASM_CASES_TAC `(b:real^1) IN s` THENL [ALL_TAC; ASM_MESON_TAC[REAL_LE_REFL]] THEN FIRST_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [OPEN_IN_CONTAINS_CBALL]) THEN FIRST_X_ASSUM(MP_TAC o SPEC `b:real^1`) THEN ANTS_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN EXPAND_TAC "t" THEN REWRITE_TAC[IN_INTER] THEN ASM_REWRITE_TAC[ENDS_IN_INTERVAL] THEN REWRITE_TAC[INTERVAL_NE_EMPTY_1] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `r:real` THEN STRIP_TAC] THEN SUBGOAL_THEN `~(interval[b,b + lift r] SUBSET s)` MP_TAC THENL [DISCH_TAC THEN MP_TAC(ISPECL [`s:real^1->bool`; `t UNION interval [b,b + lift r]`; `t:real^1->bool`] COMPONENTS_MAXIMAL) THEN ASM_REWRITE_TAC[NOT_IMP] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_UNION THEN ASM_REWRITE_TAC[GSYM IS_INTERVAL_CONNECTED_1; IS_INTERVAL_INTERVAL; GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN EXISTS_TAC `b:real^1` THEN EXPAND_TAC "t" THEN REWRITE_TAC[IN_INTER; ENDS_IN_INTERVAL] THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[INTERVAL_NE_EMPTY_1; DROP_ADD; LIFT_DROP] THEN ASM_REAL_ARITH_TAC; ASM_MESON_TAC[CLOSED_IN_IMP_SUBSET; UNION_SUBSET]; ASM SET_TAC[]; EXPAND_TAC "t" THEN REWRITE_TAC[UNION_SUBSET; SUBSET_INTER] THEN DISCH_THEN(MP_TAC o last o CONJUNCTS) THEN REWRITE_TAC[SUBSET_INTERVAL_1; DROP_ADD; LIFT_DROP] THEN ASM_REAL_ARITH_TAC]; REWRITE_TAC[SUBSET; NOT_FORALL_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `b':real^1` THEN REWRITE_TAC[NOT_IMP; IN_INTERVAL_1; DROP_ADD; LIFT_DROP] THEN STRIP_TAC THEN ASM_REWRITE_TAC[GSYM SUBSET] THEN MP_TAC(ISPECL [`a':real^1`; `b':real^1`; `b:real^1`] UNION_INTERVAL_1) THEN ASM_REWRITE_TAC[IN_INTERVAL_1] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; DISCH_THEN(SUBST1_TAC o SYM)] THEN ASM_REWRITE_TAC[UNION_OVER_INTER; UNION_SUBSET] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `b INTER s SUBSET u ==> i SUBSET b ==> s INTER i SUBSET u`)) THEN REWRITE_TAC[CBALL_INTERVAL; SUBSET_INTERVAL_1] THEN REWRITE_TAC[DROP_ADD; DROP_SUB; LIFT_DROP] THEN ASM_REAL_ARITH_TAC]; ALL_TAC] THEN EXISTS_TAC `s INTER interval[a':real^1,b']` THEN ASM_SIMP_TAC[CLOSED_IN_CLOSED_INTER; CLOSED_INTERVAL] THEN CONJ_TAC THENL [REWRITE_TAC[OPEN_IN_OPEN] THEN EXISTS_TAC `interval(a':real^1,b')` THEN REWRITE_TAC[OPEN_INTERVAL] THEN MP_TAC(ISPECL [`a':real^1`; `b':real^1`] CLOSED_OPEN_INTERVAL_1) THEN ANTS_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN SUBGOAL_THEN `~(interval[a:real^1,b] = {})` MP_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[INTERVAL_EQ_EMPTY_1] THEN ASM_REAL_ARITH_TAC]; EXPAND_TAC "t" THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> a INTER s SUBSET a INTER t`) THEN REWRITE_TAC[SUBSET_INTERVAL_1] THEN ASM_REAL_ARITH_TAC]) in REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[DISJ_ASSOC] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_COMPONENT) THEN DISCH_THEN DISJ_CASES_TAC THENL [SUBGOAL_THEN `?r:real^1->bool. bounded r /\ (s:real^N->bool) homeomorphic r` STRIP_ASSUME_TAC THENL [FIRST_X_ASSUM(MP_TAC o MATCH_MP (MESON[] `p \/ q ==> (p ==> q) ==> q`)) THEN ANTS_TAC THENL [REWRITE_TAC[GSYM DIMINDEX_1; GSYM DIM_UNIV] THEN DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] (REWRITE_RULE[CONJ_ASSOC] HOMEOMORPHIC_SUBSPACES))) THEN REWRITE_TAC[SUBSPACE_UNIV; homeomorphic] THEN GEN_REWRITE_TAC RAND_CONV [SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^N->real^1` THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^1->real^N` THEN DISCH_TAC THEN EXISTS_TAC `IMAGE (f:real^N->real^1) s` THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]; DISCH_THEN(X_CHOOSE_TAC `r:real^1->bool`) THEN SUBGOAL_THEN `?r'. bounded r' /\ (r:real^1->bool) homeomorphic (r':real^1->bool)` MP_TAC THENL [ALL_TAC; MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] HOMEOMORPHIC_TRANS) THEN ASM_REWRITE_TAC[]] THEN MP_TAC(ISPECL [`vec 0:real^1`; `vec 1:real^1`] HOMEOMORPHIC_OPEN_INTERVAL_UNIV) THEN REWRITE_TAC[UNIT_INTERVAL_NONEMPTY] THEN GEN_REWRITE_TAC LAND_CONV [HOMEOMORPHIC_SYM] THEN REWRITE_TAC[homeomorphic; RIGHT_AND_EXISTS_THM] THEN GEN_REWRITE_TAC RAND_CONV [SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^1->real^1` THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^1->real^1` THEN DISCH_TAC THEN EXISTS_TAC `IMAGE (f:real^1->real^1) r` THEN CONJ_TAC THENL [MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `interval(vec 0:real^1,vec 1)` THEN REWRITE_TAC[BOUNDED_INTERVAL]; FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS))] THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphic]) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f:real^N->real^1`; `g:real^1->real^N`] THEN STRIP_TAC THEN FIRST_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [homeomorphism]) THEN MP_TAC(ISPECL [`IMAGE (f:real^N->real^1) s`; `IMAGE (f:real^N->real^1) t`; `IMAGE (f:real^N->real^1) u`] lemma) THEN ASM_SIMP_TAC[IMAGE_SUBSET] THEN ANTS_TAC THENL [CONJ_TAC THENL [FIRST_ASSUM(SUBST1_TAC o MATCH_MP HOMEOMORPHISM_COMPONENTS) THEN MATCH_MP_TAC FUN_IN_IMAGE THEN ASM_REWRITE_TAC[]; EXPAND_TAC "r" THEN FIRST_ASSUM(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OPEN_IN_EQ)) THEN DISCH_THEN(MP_TAC o SPECL [`s:real^N->bool`; `u:real^N->bool`]) THEN ASM_MESON_TAC[SUBSET_REFL; OPEN_IN_IMP_SUBSET]]; DISCH_THEN(X_CHOOSE_THEN `c:real^1->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `IMAGE (g:real^1->real^N) c` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC HOMEOMORPHISM_IMP_CLOSED_MAP; MATCH_MP_TAC HOMEOMORPHISM_IMP_OPEN_MAP; REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET)) THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]; REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET)) THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]] THEN MAP_EVERY EXISTS_TAC [`f:real^N->real^1`; `IMAGE (f:real^N->real^1) s`] THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[HOMEOMORPHISM_SYM] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]]]; FIRST_X_ASSUM(DISJ_CASES_THEN STRIP_ASSUME_TAC) THENL [EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN ASM_MESON_TAC[OPEN_IN_COMPONENTS_LOCALLY_CONNECTED]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN DISCH_THEN(X_CHOOSE_THEN `v:real^N->bool` (STRIP_ASSUME_TAC o GSYM)) THEN MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`; `v:real^N->bool`] SURA_BURA_CLOPEN_SUBSET) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CLOSED_SUBSET THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED]]]);; let COMPONENTS_SUBSETS_CLOPEN_PARTITION = prove (`!u s:real^N->bool. locally compact s /\ FINITE u /\ ~(u = {}) /\ u SUBSET components s /\ (!c. c IN u ==> compact c) ==> ?f. (!c. c IN u ==> open_in (subtopology euclidean s) (f c) /\ closed_in (subtopology euclidean s) (f c) /\ c SUBSET f(c)) /\ pairwise (\c c'. ~(f(c) = f(c'))) u /\ pairwise (\c c'. DISJOINT (f c) (f c')) u /\ UNIONS (IMAGE f u) = s`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?l. !c. c IN u ==> closed_in (subtopology euclidean s) (l c) /\ open_in (subtopology euclidean s) (l c) /\ (c:real^N->bool) SUBSET l c /\ (!c'. c' IN u /\ ~(c' = c) ==> DISJOINT (l c) (l c'))` STRIP_ASSUME_TAC THENL [SUBGOAL_THEN `!c. c IN u ==> ?l. closed_in (subtopology euclidean s) l /\ open_in (subtopology euclidean s) l /\ c SUBSET l /\ (!c':real^N->bool. c' IN u /\ ~(c' = c) ==> DISJOINT c' l)` MP_TAC THENL [X_GEN_TAC `c:real^N->bool` THEN DISCH_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `c:real^N->bool`; `s DIFF UNIONS (u DELETE c):real^N->bool`] COMPONENT_INTERMEDIATE_CLOPEN) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ALL_TAC; MATCH_MP_TAC MONO_EXISTS THEN SET_TAC[]] THEN REPEAT CONJ_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC OPEN_IN_DIFF THEN REWRITE_TAC[OPEN_IN_REFL] THEN MATCH_MP_TAC CLOSED_IN_UNIONS THEN ASM_REWRITE_TAC[FINITE_DELETE] THEN ASM_SIMP_TAC[IN_DELETE; COMPACT_IMP_CLOSED; CLOSED_SUBSET_EQ] THEN ASM_MESON_TAC[IN_COMPONENTS_SUBSET; SUBSET]; MATCH_MP_TAC(SET_RULE `c SUBSET s /\ (!d. d IN u ==> DISJOINT c d) ==> c SUBSET s DIFF UNIONS u`) THEN CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_SUBSET; SUBSET]; REWRITE_TAC[IN_DELETE] THEN ASM_MESON_TAC[REWRITE_RULE[pairwise] PAIRWISE_DISJOINT_COMPONENTS; SUBSET]]; ASM SET_TAC[]]; GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM] THEN DISCH_THEN(X_CHOOSE_TAC `l:(real^N->bool)->(real^N->bool)`) THEN EXISTS_TAC `\c. (l:(real^N->bool)->(real^N->bool)) c DIFF UNIONS (IMAGE l (u DELETE c))` THEN X_GEN_TAC `c:real^N->bool` THEN DISCH_TAC THEN REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CLOSED_IN_DIFF THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC OPEN_IN_UNIONS THEN ASM SET_TAC[]; MATCH_MP_TAC OPEN_IN_DIFF THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC CLOSED_IN_UNIONS THEN ASM_SIMP_TAC[FINITE_DELETE; FINITE_IMAGE; FORALL_IN_IMAGE] THEN ASM SET_TAC[]; MATCH_MP_TAC(SET_RULE `c SUBSET s /\ (!d. d IN u ==> DISJOINT c d) ==> c SUBSET s DIFF UNIONS u`) THEN CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_SUBSET; SUBSET]; REWRITE_TAC[IN_DELETE; FORALL_IN_IMAGE] THEN ASM_MESON_TAC[REWRITE_RULE[pairwise] PAIRWISE_DISJOINT_COMPONENTS; SUBSET]]; SET_TAC[]]]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_TAC `d:real^N->bool`) THEN EXISTS_TAC `\c. if c = d then s DIFF UNIONS (IMAGE l (u DELETE d)) else (l:(real^N->bool)->(real^N->bool)) c` THEN REWRITE_TAC[] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r /\ s <=> p /\ (q /\ r) /\ s`] THEN REWRITE_TAC[PAIRWISE_AND] THEN REPEAT CONJ_TAC THENL [X_GEN_TAC `c:real^N->bool` THEN DISCH_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_DIFF THEN REWRITE_TAC[OPEN_IN_REFL] THEN MATCH_MP_TAC CLOSED_IN_UNIONS THEN ASM_SIMP_TAC[FINITE_DELETE; FINITE_IMAGE; FORALL_IN_IMAGE] THEN ASM SET_TAC[]; MATCH_MP_TAC CLOSED_IN_DIFF THEN REWRITE_TAC[CLOSED_IN_REFL] THEN MATCH_MP_TAC OPEN_IN_UNIONS THEN ASM SET_TAC[]; MATCH_MP_TAC(SET_RULE `c SUBSET s /\ (!d. d IN u ==> DISJOINT c d) ==> c SUBSET s DIFF UNIONS u`) THEN CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_SUBSET; SUBSET]; ALL_TAC] THEN REWRITE_TAC[IN_DELETE; FORALL_IN_IMAGE] THEN ASM_MESON_TAC[SET_RULE `c SUBSET c' /\ DISJOINT c' d ==> DISJOINT c d`]]; FIRST_ASSUM(SUBST1_TAC o MATCH_MP (SET_RULE `a IN s ==> s = a INSERT (s DELETE a)`)) THEN REWRITE_TAC[PAIRWISE_INSERT] THEN ASM_SIMP_TAC[SET_RULE `a IN s ==> a INSERT (s DELETE a) = s`] THEN SUBGOAL_THEN `!c:real^N->bool. c IN u ==> ~(l c:real^N->bool = {})` ASSUME_TAC THENL [ASM_MESON_TAC[SUBSET_EMPTY; IN_COMPONENTS_NONEMPTY; SUBSET]; ALL_TAC] THEN ASM_SIMP_TAC[IN_DELETE; pairwise; SET_RULE `~(c' = {}) ==> ((~(c = c') /\ DISJOINT c c') /\ (~(c' = c) /\ DISJOINT c' c) <=> DISJOINT c c')`] THEN ASM SET_TAC[]; FIRST_ASSUM(SUBST1_TAC o MATCH_MP (SET_RULE `a IN s ==> s = a INSERT (s DELETE a)`)) THEN REWRITE_TAC[IMAGE_CLAUSES; UNIONS_INSERT] THEN ASM_SIMP_TAC[SET_RULE `a IN s ==> a INSERT (s DELETE a) = s`] THEN REWRITE_TAC[SET_RULE `IMAGE (\x. if x = a then f x else g x) (s DELETE a) = IMAGE g (s DELETE a)`] THEN MATCH_MP_TAC(SET_RULE `u SUBSET s ==> (s DIFF u) UNION u = s`) THEN REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_IMAGE; IN_DELETE] THEN ASM_MESON_TAC[OPEN_IN_IMP_SUBSET]]);; (* ------------------------------------------------------------------------- *) (* Relations between components and path components. *) (* ------------------------------------------------------------------------- *) let OPEN_CONNECTED_COMPONENT = prove (`!s x:real^N. open s ==> open(connected_component s x)`, REPEAT GEN_TAC THEN REWRITE_TAC[OPEN_CONTAINS_BALL] THEN DISCH_TAC THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN ANTS_TAC THENL [ASM_MESON_TAC[SUBSET; CONNECTED_COMPONENT_SUBSET]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `e:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `connected_component s (x:real^N) = connected_component s y` SUBST1_TAC THENL [ASM_MESON_TAC[CONNECTED_COMPONENT_EQ]; MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL THEN ASM_REWRITE_TAC[CENTRE_IN_BALL; CONNECTED_BALL]]);; let IN_CLOSURE_CONNECTED_COMPONENT = prove (`!x y:real^N. x IN s /\ open s ==> (x IN closure(connected_component s y) <=> x IN connected_component s y)`, REPEAT STRIP_TAC THEN EQ_TAC THEN REWRITE_TAC[REWRITE_RULE[SUBSET] CLOSURE_SUBSET] THEN DISCH_TAC THEN SUBGOAL_THEN `~((connected_component s (x:real^N)) INTER closure(connected_component s y) = {})` MP_TAC THENL [REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `x:real^N` THEN ASM_REWRITE_TAC[IN_INTER] THEN ASM_REWRITE_TAC[IN; CONNECTED_COMPONENT_REFL_EQ]; ASM_SIMP_TAC[OPEN_INTER_CLOSURE_EQ_EMPTY; OPEN_CONNECTED_COMPONENT] THEN REWRITE_TAC[CONNECTED_COMPONENT_OVERLAP] THEN STRIP_TAC THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN ASM_REWRITE_TAC[IN; CONNECTED_COMPONENT_REFL_EQ]]);; let PATH_COMPONENT_SUBSET_CONNECTED_COMPONENT = prove (`!s x:real^N. (path_component s x) SUBSET (connected_component s x)`, REWRITE_TAC[GSYM PATH_COMPONENT_OF_EUCLIDEAN; GSYM CONNECTED_COMPONENT_OF_EUCLIDEAN] THEN REWRITE_TAC[PATH_COMPONENT_SUBSET_CONNECTED_COMPONENT_OF]);; let PATH_COMPONENT_EQ_CONNECTED_COMPONENT = prove (`!s x:real^N. locally path_connected s ==> (path_component s x = connected_component s x)`, REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM PATH_COMPONENT_OF_EUCLIDEAN; GSYM CONNECTED_COMPONENT_OF_EUCLIDEAN] THEN MATCH_MP_TAC PATH_COMPONENT_EQ_CONNECTED_COMPONENT_OF THEN ASM_REWRITE_TAC[LOCALLY_PATH_CONNECTED_SPACE_SUBTOPOLOGY_EUCLIDEAN]);; let PATH_COMPONENT_IMP_CONNECTED_COMPONENT = prove (`!s a b:real^N. path_component s a b ==> connected_component s a b`, REWRITE_TAC[SET_RULE `(!x. P x ==> Q x) <=> P SUBSET Q`] THEN REWRITE_TAC[PATH_COMPONENT_SUBSET_CONNECTED_COMPONENT; ETA_AX]);; let LOCALLY_PATH_CONNECTED_PATH_COMPONENT = prove (`!s x:real^N. locally path_connected s ==> locally path_connected (path_component s x)`, MESON_TAC[LOCALLY_PATH_CONNECTED_CONNECTED_COMPONENT; PATH_COMPONENT_EQ_CONNECTED_COMPONENT]);; let OPEN_PATH_CONNECTED_COMPONENT = prove (`!s x:real^N. open s ==> path_component s x = connected_component s x`, SIMP_TAC[PATH_COMPONENT_EQ_CONNECTED_COMPONENT; OPEN_IMP_LOCALLY_PATH_CONNECTED]);; let PATH_CONNECTED_EQ_CONNECTED_LPC = prove (`!s. locally path_connected s ==> (path_connected s <=> connected s)`, REWRITE_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT; CONNECTED_IFF_CONNECTED_COMPONENT] THEN SIMP_TAC[PATH_COMPONENT_EQ_CONNECTED_COMPONENT]);; let PATH_CONNECTED_EQ_CONNECTED = prove (`!s. open s ==> (path_connected s <=> connected s)`, SIMP_TAC[PATH_CONNECTED_EQ_CONNECTED_LPC; OPEN_IMP_LOCALLY_PATH_CONNECTED]);; let CONNECTED_OPEN_PATH_CONNECTED = prove (`!s:real^N->bool. open s /\ connected s ==> path_connected s`, SIMP_TAC[PATH_CONNECTED_EQ_CONNECTED]);; let CONNECTED_OPEN_ARC_CONNECTED = prove (`!s:real^N->bool. open s /\ connected s ==> !x y. x IN s /\ y IN s ==> x = y \/ ?g. arc g /\ path_image g SUBSET s /\ pathstart g = x /\ pathfinish g = y`, GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP CONNECTED_OPEN_PATH_CONNECTED) THEN REWRITE_TAC[PATH_CONNECTED_ARCWISE] THEN REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN MESON_TAC[]);; let OPEN_COMPONENTS = prove (`!u:real^N->bool s. open u /\ s IN components u ==> open s`, REPEAT STRIP_TAC THEN STRIP_ASSUME_TAC (MESON[IN_COMPONENTS; ASSUME `s:real^N->bool IN components u`] `?x. s:real^N->bool = connected_component u x`) THEN ASM_SIMP_TAC [OPEN_CONNECTED_COMPONENT]);; let COMPONENTS_OPEN_UNIQUE = prove (`!f:(real^N->bool)->bool s. (!c. c IN f ==> open c /\ connected c /\ ~(c = {})) /\ pairwise DISJOINT f /\ UNIONS f = s ==> components s = f`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CONNECTED_DISJOINT_UNIONS_OPEN_UNIQUE THEN ASM_REWRITE_TAC[GSYM UNIONS_COMPONENTS; PAIRWISE_DISJOINT_COMPONENTS] THEN ASM_MESON_TAC[OPEN_COMPONENTS; IN_COMPONENTS_NONEMPTY; IN_COMPONENTS_CONNECTED; OPEN_UNIONS]);; let COUNTABLE_OPEN_COMPONENTS = prove (`!s:real^N->bool. open s ==> COUNTABLE(components s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC COUNTABLE_DISJOINT_OPEN_SUBSETS THEN REWRITE_TAC[PAIRWISE_DISJOINT_COMPONENTS] THEN ASM_MESON_TAC[OPEN_COMPONENTS]);; let COUNTABLE_OPEN_CONNECTED_COMPONENTS = prove (`!s t:real^N->bool. open s ==> COUNTABLE {connected_component s x | x IN t}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC COUNTABLE_SUBSET THEN EXISTS_TAC `{} INSERT components(s:real^N->bool)` THEN ASM_SIMP_TAC[COUNTABLE_INSERT; COUNTABLE_OPEN_COMPONENTS] THEN REWRITE_TAC[SUBSET; IN_INSERT; components; FORALL_IN_GSPEC] THEN REWRITE_TAC[CONNECTED_COMPONENT_EQ_EMPTY] THEN SET_TAC[]);; let CONTINUOUS_ON_COMPONENTS = prove (`!f:real^M->real^N s. locally connected s /\ (!c. c IN components s ==> f continuous_on c) ==> f continuous_on s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_COMPONENTS_GEN THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[OPEN_IN_COMPONENTS_LOCALLY_CONNECTED]);; let CONTINUOUS_ON_COMPONENTS_EQ = prove (`!f s. locally connected s ==> (f continuous_on s <=> !c. c IN components s ==> f continuous_on c)`, REPEAT STRIP_TAC THEN EQ_TAC THENL [MESON_TAC[CONTINUOUS_ON_SUBSET; IN_COMPONENTS_SUBSET]; ASM_MESON_TAC[CONTINUOUS_ON_COMPONENTS]]);; let CONTINUOUS_ON_COMPONENTS_OPEN = prove (`!f:real^M->real^N s. open s /\ (!c. c IN components s ==> f continuous_on c) ==> f continuous_on s`, ASM_MESON_TAC[CONTINUOUS_ON_COMPONENTS; OPEN_IMP_LOCALLY_CONNECTED]);; let CONTINUOUS_ON_COMPONENTS_OPEN_EQ = prove (`!f s. open s ==> (f continuous_on s <=> !c. c IN components s ==> f continuous_on c)`, REPEAT STRIP_TAC THEN EQ_TAC THENL [MESON_TAC[CONTINUOUS_ON_SUBSET; IN_COMPONENTS_SUBSET]; ASM_MESON_TAC[CONTINUOUS_ON_COMPONENTS_OPEN]]);; let CLOSED_IN_UNION_COMPLEMENT_COMPONENTS = prove (`!u s:real^N->bool c. locally connected u /\ closed_in (subtopology euclidean u) s /\ c SUBSET components(u DIFF s) ==> closed_in (subtopology euclidean u) (s UNION UNIONS c)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `s UNION UNIONS c:real^N->bool = u DIFF (UNIONS(components(u DIFF s) DIFF c))` SUBST1_TAC THENL [MATCH_MP_TAC(SET_RULE `s SUBSET u /\ u DIFF s = c UNION c' /\ DISJOINT c c' ==> s UNION c = u DIFF c'`) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN ASM_SIMP_TAC[GSYM UNIONS_UNION; GSYM UNIONS_COMPONENTS; SET_RULE `s SUBSET t ==> s UNION (t DIFF s) = t`] THEN MATCH_MP_TAC(SET_RULE `(!s t. s IN c /\ t IN c' ==> DISJOINT s t) ==> DISJOINT (UNIONS c) (UNIONS c')`) THEN REWRITE_TAC[IN_DIFF] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPEC `(u:real^N->bool) DIFF s` PAIRWISE_DISJOINT_COMPONENTS) THEN REWRITE_TAC[pairwise] THEN DISCH_THEN MATCH_MP_TAC THEN REPEAT(CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN ASM_MESON_TAC[]; REWRITE_TAC[closed_in; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; SUBSET_DIFF] THEN MATCH_MP_TAC OPEN_IN_DIFF THEN REWRITE_TAC[OPEN_IN_REFL] THEN MATCH_MP_TAC CLOSED_IN_DIFF THEN REWRITE_TAC[CLOSED_IN_REFL] THEN MATCH_MP_TAC OPEN_IN_UNIONS THEN REWRITE_TAC[IN_DIFF] THEN X_GEN_TAC `d:real^N->bool` THEN STRIP_TAC THEN MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `u DIFF s:real^N->bool` THEN CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_COMPONENTS_LOCALLY_CONNECTED THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC LOCALLY_OPEN_SUBSET THEN EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN MATCH_MP_TAC OPEN_IN_DIFF THEN ASM_SIMP_TAC[OPEN_IN_REFL]]);; let CLOSED_UNION_COMPLEMENT_COMPONENTS = prove (`!s c. closed s /\ c SUBSET components((:real^N) DIFF s) ==> closed(s UNION UNIONS c)`, ONCE_REWRITE_TAC[CLOSED_IN] THEN ONCE_REWRITE_TAC[GSYM SUBTOPOLOGY_UNIV] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CLOSED_IN_UNION_COMPLEMENT_COMPONENTS THEN ASM_REWRITE_TAC[LOCALLY_CONNECTED_UNIV]);; let CLOSED_IN_UNION_COMPLEMENT_COMPONENT = prove (`!u s c:real^N->bool. locally connected u /\ closed_in (subtopology euclidean u) s /\ c IN components(u DIFF s) ==> closed_in (subtopology euclidean u) (s UNION c)`, REPEAT STRIP_TAC THEN GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM UNIONS_1] THEN MATCH_MP_TAC CLOSED_IN_UNION_COMPLEMENT_COMPONENTS THEN ASM_REWRITE_TAC[SING_SUBSET]);; let CLOSED_UNION_COMPLEMENT_COMPONENT = prove (`!s c. closed s /\ c IN components((:real^N) DIFF s) ==> closed(s UNION c)`, ONCE_REWRITE_TAC[CLOSED_IN] THEN ONCE_REWRITE_TAC[GSYM SUBTOPOLOGY_UNIV] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CLOSED_IN_UNION_COMPLEMENT_COMPONENT THEN ASM_REWRITE_TAC[LOCALLY_CONNECTED_UNIV]);; let NONSEPARATED_CLOSED_COMPLEMENT_COMPONENTS = prove (`!u s:real^N->bool c. connected u /\ locally connected u /\ closed_in (subtopology euclidean u) s /\ ~(s = {}) /\ c SUBSET components(u DIFF s) /\ ~(c = {}) ==> ~(s INTER closure(UNIONS c) = {})`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`u:real^N->bool`; `s:real^N->bool`; `c:(real^N->bool)->bool`] CLOSED_IN_UNION_COMPLEMENT_COMPONENTS) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM CLOSURE_OF_SUBSET_EQ] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; UNION_SUBSET] THEN DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN REWRITE_TAC[CLOSURE_OF_UNION] THEN MATCH_MP_TAC(SET_RULE `DISJOINT t u /\ ~(t SUBSET v) ==> ~(s UNION t SUBSET u UNION v)`) THEN CONJ_TAC THENL [REWRITE_TAC[CLOSURE_OF_SUBTOPOLOGY; EUCLIDEAN_CLOSURE_OF] THEN ASM_SIMP_TAC[SET_RULE `c SUBSET u ==> u INTER c = c`] THEN ASM SET_TAC[]; W(MP_TAC o PART_MATCH (rand o lhand) CLOSURE_OF_SUBSET_EQ o rand o snd) THEN ASM_REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN DISCH_THEN SUBST1_TAC THEN DISCH_TAC] THEN FIRST_ASSUM(MP_TAC o SPEC `UNIONS c:real^N->bool` o GEN_REWRITE_RULE I [CONNECTED_CLOPEN]) THEN ASM_REWRITE_TAC[NOT_IMP; DE_MORGAN_THM] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_UNIONS THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [LOCALLY_CONNECTED_OPEN_COMPONENT]) THEN EXISTS_TAC `u DIFF (s:real^N->bool)` THEN ASM_SIMP_TAC[OPEN_IN_DIFF; OPEN_IN_REFL] THEN ASM SET_TAC[]; REWRITE_TAC[EMPTY_UNIONS] THEN ASM_MESON_TAC[MEMBER_NOT_EMPTY; IN_COMPONENTS_NONEMPTY; SUBSET]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s INTER closure c = {} ==> c SUBSET closure c /\ s SUBSET u /\ ~(s = {}) ==> ~(c = u)`)) THEN ASM_SIMP_TAC[CLOSURE_SUBSET; CLOSED_IN_SUBSET]]);; let COUNTABLE_CONNECTED_COMPONENTS = prove (`!s:real^N->bool t. locally connected s ==> COUNTABLE {connected_component s x | x IN t}`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`{connected_component s (x:real^N) |x| x IN s}`; `s:real^N->bool`] LINDELOF_OPEN_IN) THEN ASM_SIMP_TAC[FORALL_IN_GSPEC; OPEN_IN_CONNECTED_COMPONENT_LOCALLY_CONNECTED; UNIONS_CONNECTED_COMPONENT] THEN DISCH_THEN(X_CHOOSE_THEN `u:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC COUNTABLE_SUBSET THEN EXISTS_TAC `({}:real^N->bool) INSERT u` THEN ASM_REWRITE_TAC[COUNTABLE_INSERT] THEN REWRITE_TAC[SUBSET; FORALL_IN_GSPEC; IN_INSERT] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[CONNECTED_COMPONENT_EQ_EMPTY] THEN DISCH_TAC THEN ASM_CASES_TAC `(x:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPECL [`s:real^N->bool`; `x:real^N`] COMPLEMENT_CONNECTED_COMPONENT_UNIONS) THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN REWRITE_TAC[EXTENSION] THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[IN_DIFF] THEN ASM_CASES_TAC `(x:real^N) IN connected_component s x` THENL [ALL_TAC; ASM_MESON_TAC[IN; CONNECTED_COMPONENT_REFL]] THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `(x:real^N) IN UNIONS u` MP_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> x IN s ==> x IN t`) THEN MATCH_MP_TAC SUBSET_UNIONS THEN ASM SET_TAC[]);; let COUNTABLE_PATH_COMPONENTS = prove (`!s:real^N->bool t. locally path_connected s ==> COUNTABLE {path_component s x | x IN t}`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`{path_component s (x:real^N) |x| x IN s}`; `s:real^N->bool`] LINDELOF_OPEN_IN) THEN ASM_SIMP_TAC[FORALL_IN_GSPEC; OPEN_IN_PATH_COMPONENT_LOCALLY_PATH_CONNECTED; UNIONS_PATH_COMPONENT] THEN DISCH_THEN(X_CHOOSE_THEN `u:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC COUNTABLE_SUBSET THEN EXISTS_TAC `({}:real^N->bool) INSERT u` THEN ASM_REWRITE_TAC[COUNTABLE_INSERT] THEN REWRITE_TAC[SUBSET; FORALL_IN_GSPEC; IN_INSERT] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[PATH_COMPONENT_EQ_EMPTY] THEN DISCH_TAC THEN ASM_CASES_TAC `(x:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPECL [`s:real^N->bool`; `x:real^N`] COMPLEMENT_PATH_COMPONENT_UNIONS) THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN REWRITE_TAC[EXTENSION] THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[IN_DIFF] THEN ASM_CASES_TAC `(x:real^N) IN path_component s x` THENL [ALL_TAC; ASM_MESON_TAC[IN; PATH_COMPONENT_REFL]] THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `(x:real^N) IN UNIONS u` MP_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> x IN s ==> x IN t`) THEN MATCH_MP_TAC SUBSET_UNIONS THEN ASM SET_TAC[]);; let COUNTABLE_COMPONENTS = prove (`!s:real^N->bool. locally connected s ==> COUNTABLE(components s)`, SIMP_TAC[components; COUNTABLE_CONNECTED_COMPONENTS]);; let FRONTIER_MINIMAL_SEPARATING_CLOSED = prove (`!s c. closed s /\ ~connected((:real^N) DIFF s) /\ (!t. closed t /\ t PSUBSET s ==> connected((:real^N) DIFF t)) /\ c IN components ((:real^N) DIFF s) ==> frontier c = s`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [CONNECTED_EQ_CONNECTED_COMPONENTS_EQ]) THEN DISCH_THEN(MP_TAC o MATCH_MP (MESON[] `~(!x x'. x IN s /\ x' IN s ==> x = x') ==> !x. x IN s ==> ?y. y IN s /\ ~(y = x)`)) THEN DISCH_THEN(MP_TAC o SPEC `c:real^N->bool`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `d:real^N->bool` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPEC `frontier c:real^N->bool`) THEN REWRITE_TAC[SET_RULE `s PSUBSET t <=> s SUBSET t /\ ~(t SUBSET s)`; GSYM SUBSET_ANTISYM_EQ] THEN ASM_SIMP_TAC[FRONTIER_OF_COMPONENTS_CLOSED_COMPLEMENT; FRONTIER_CLOSED] THEN MATCH_MP_TAC(TAUT `~r ==> (~p ==> r) ==> p`) THEN REWRITE_TAC[connected] THEN MAP_EVERY EXISTS_TAC [`c:real^N->bool`; `(:real^N) DIFF closure c`] THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[OPEN_COMPONENTS; closed]; REWRITE_TAC[GSYM closed; CLOSED_CLOSURE]; MP_TAC(ISPEC `c:real^N->bool` INTERIOR_SUBSET) THEN REWRITE_TAC[frontier] THEN SET_TAC[]; MATCH_MP_TAC(SET_RULE `c SUBSET c' ==> c INTER (UNIV DIFF c') INTER s = {}`) THEN REWRITE_TAC[GSYM INTERIOR_COMPLEMENT; CLOSURE_SUBSET]; REWRITE_TAC[frontier] THEN MATCH_MP_TAC(SET_RULE `ci = c /\ ~(c = {}) ==> ~(c INTER (UNIV DIFF (cc DIFF ci)) = {})`) THEN ASM_MESON_TAC[IN_COMPONENTS_NONEMPTY; INTERIOR_OPEN; closed; OPEN_COMPONENTS]; REWRITE_TAC[frontier] THEN MATCH_MP_TAC(SET_RULE `~(UNIV DIFF c = {}) ==> ~((UNIV DIFF c) INTER (UNIV DIFF (c DIFF i)) = {})`) THEN REWRITE_TAC[GSYM INTERIOR_COMPLEMENT] THEN MATCH_MP_TAC(SET_RULE `!t. t SUBSET s /\ ~(t = {}) ==> ~(s = {})`) THEN EXISTS_TAC `d:real^N->bool` THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[IN_COMPONENTS_NONEMPTY]] THEN MATCH_MP_TAC INTERIOR_MAXIMAL THEN REWRITE_TAC[SET_RULE `s SUBSET UNIV DIFF t <=> s INTER t = {}`] THEN ASM_MESON_TAC[COMPONENTS_NONOVERLAP; OPEN_COMPONENTS; GSYM closed]]);; let FRONTIER_MINIMAL_SEPARATING_CLOSED_POINTWISE = prove (`!s a b. closed s /\ ~(a IN s) /\ ~connected_component ((:real^N) DIFF s) a b /\ (!t. closed t /\ t PSUBSET s ==> connected_component((:real^N) DIFF t) a b) ==> frontier(connected_component ((:real^N) DIFF s) a) = s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC(SET_RULE `s SUBSET t /\ ~(s PSUBSET t) ==> s = t`) THEN CONJ_TAC THENL [MATCH_MP_TAC FRONTIER_OF_COMPONENTS_CLOSED_COMPLEMENT THEN ASM_REWRITE_TAC[IN_COMPONENTS; IN_UNIV; IN_DIFF] THEN ASM SET_TAC[]; DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `frontier (connected_component ((:real^N) DIFF s) a)`) THEN ASM_REWRITE_TAC[FRONTIER_CLOSED] THEN GEN_REWRITE_TAC RAND_CONV [connected_component] THEN DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `t SUBSET UNIV DIFF f ==> ~(t INTER f = {}) ==> F`)) THEN MATCH_MP_TAC CONNECTED_INTER_FRONTIER THEN ASM_REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER; IN_DIFF] THEN CONJ_TAC THENL [EXISTS_TAC `a:real^N`; EXISTS_TAC `b:real^N`] THEN ASM_REWRITE_TAC[] THEN ASM_REWRITE_TAC[IN] THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ; IN_UNIV; IN_DIFF]]);; (* ------------------------------------------------------------------------- *) (* "Boundary bumping theorems" and relatives. *) (* ------------------------------------------------------------------------- *) let CONNECTED_COMPONENT_DIFF_NONSEPARATED = prove (`!s t c:real^N->bool. compact s /\ connected s /\ t SUBSET s /\ ~(t = {}) /\ c IN components(s DIFF t) ==> ~(closure(c) INTER closure(t) = {})`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s DIFF {x + d | (x:real^N) IN t /\ d IN ball(vec 0,setdist(c,t) / &2)}`; `c:real^N->bool`; `setdist(c:real^N->bool,t) / &2`] COMPONENT_CLOPEN_HAUSDIST_EXPLICIT) THEN REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC; REAL_HALF] THEN ABBREV_TAC `t' = {x + d | (x:real^N) IN t /\ d IN ball(vec 0,setdist(c,t) / &2)}` THEN SUBGOAL_THEN `open(t':real^N->bool)` ASSUME_TAC THENL [EXPAND_TAC "t'" THEN SIMP_TAC[OPEN_SUMS; OPEN_BALL]; ALL_TAC] THEN SUBGOAL_THEN `compact(s DIFF t':real^N->bool)` ASSUME_TAC THENL [MATCH_MP_TAC COMPACT_DIFF THEN ASM_REWRITE_TAC[]; ASM_SIMP_TAC[COMPACT_IMP_CLOSED; CLOSED_IMP_LOCALLY_COMPACT]] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_NONEMPTY) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [REWRITE_TAC[SETDIST_POS_LT] THEN MP_TAC(ISPECL [`closure c:real^N->bool`; `closure t:real^N->bool`] SETDIST_EQ_0_COMPACT_CLOSED) THEN ASM_REWRITE_TAC[SETDIST_CLOSURE; CLOSED_CLOSURE; CLOSURE_EQ_EMPTY] THEN DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[COMPACT_CLOSURE] THEN MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_SIMP_TAC[COMPACT_IMP_BOUNDED] THEN ASM SET_TAC[]; DISCH_TAC] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] COMPONENTS_INTERMEDIATE_SUBSET)) THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `c SUBSET s DIFF t ==> (!x. x IN t' ==> ~(x IN c)) ==> c SUBSET s DIFF t'`)) THEN EXPAND_TAC "t'" THEN REWRITE_TAC[FORALL_IN_GSPEC; IN_BALL_0] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `d:real^N`] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[REAL_NOT_LT] THEN DISCH_TAC THEN SUBST1_TAC(NORM_ARITH `norm(d:real^N) = dist(x + d,x)`) THEN MATCH_MP_TAC(NORM_ARITH `a <= dist(p:real^N,q) ==> a / &2 <= dist(p,q)`) THEN MATCH_MP_TAC SETDIST_LE_DIST THEN ASM_REWRITE_TAC[]; MATCH_MP_TAC(SET_RULE `t SUBSET t' ==> s DIFF t' SUBSET s DIFF t`) THEN EXPAND_TAC "t'" THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MAP_EVERY EXISTS_TAC [`x:real^N`; `vec 0:real^N`] THEN ASM_REWRITE_TAC[CENTRE_IN_BALL; VECTOR_ADD_RID; REAL_HALF]]; DISCH_TAC] THEN SUBGOAL_THEN `compact(c:real^N->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[COMPACT_COMPONENTS]; ASM_REWRITE_TAC[]] THEN DISCH_THEN(X_CHOOSE_THEN `k:real^N->bool` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONNECTED_CLOPEN]) THEN DISCH_THEN(MP_TAC o SPEC `k:real^N->bool`) THEN ASM_REWRITE_TAC[NOT_IMP; DE_MORGAN_THM] THEN REPEAT CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC CLOSED_SUBSET THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED] THEN FIRST_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN SET_TAC[]; ASM SET_TAC[]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `t SUBSET s ==> ~(t = {}) /\ DISJOINT t k ==> ~(k = s)`)) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `k SUBSET k' ==> (!x. x IN k' ==> ~(x IN t)) ==> DISJOINT t k`)) THEN REWRITE_TAC[FORALL_IN_GSPEC; IN_BALL_0] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `d:real^N`] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[REAL_NOT_LT] THEN DISCH_TAC THEN SUBST1_TAC(NORM_ARITH `norm(d:real^N) = dist(x,x + d)`) THEN MATCH_MP_TAC(NORM_ARITH `a <= dist(p:real^N,q) ==> a / &2 <= dist(p,q)`) THEN MATCH_MP_TAC SETDIST_LE_DIST THEN ASM_REWRITE_TAC[]] THEN MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `s INTER {x + d:real^N | x IN c /\ d IN ball(vec 0,setdist(c,t) / &2)}` THEN ASM_SIMP_TAC[OPEN_IN_OPEN_INTER; OPEN_SUMS; OPEN_BALL] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] OPEN_IN_SUBSET_TRANS)) THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `(!x. x IN u ==> !y. y IN t ==> ~(x = y)) ==> s INTER u SUBSET s DIFF t`) THEN EXPAND_TAC "t'" THEN REWRITE_TAC[FORALL_IN_GSPEC; IN_BALL_0] THEN REWRITE_TAC[RIGHT_IMP_FORALL_THM] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `d:real^N`; `x':real^N`; `d':real^N`] THEN MAP_EVERY ASM_CASES_TAC [`(x:real^N) IN c`; `(x':real^N) IN t`] THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(NORM_ARITH `k <= dist(x:real^N,x') ==> norm d < k / &2 ==> norm d' < k / &2 ==> ~(x + d = x' + d')`) THEN MATCH_MP_TAC SETDIST_LE_DIST THEN ASM_REWRITE_TAC[]);; let CONNECTED_COMPONENT_DIFF_NONSEPARATED_ALT = prove (`!s t c:real^N->bool. compact s /\ connected s /\ t PSUBSET s /\ c IN components t ==> ~(closure(c) INTER closure(s DIFF t) = {})`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC CONNECTED_COMPONENT_DIFF_NONSEPARATED THEN EXISTS_TAC `s:real^N->bool` THEN ASM_SIMP_TAC[SET_RULE `t PSUBSET s ==> s DIFF (s DIFF t) = t`] THEN ASM SET_TAC[]);; let CONNECTED_COMPONENT_DIFF_CLOSED_NONSEPARATED = prove (`!s t c:real^N->bool. compact s /\ connected s /\ closed t /\ t SUBSET s /\ ~(t = {}) /\ c IN components(s DIFF t) ==> ~(closure(c) INTER t = {})`, MESON_TAC[CONNECTED_COMPONENT_DIFF_NONSEPARATED; CLOSURE_CLOSED]);; let NONSEPARATED_CLOSED_COMPLEMENT_COMPONENT = prove (`!u s c:real^N->bool. (compact u \/ locally connected u) /\ connected u /\ closed_in (subtopology euclidean u) s /\ ~(s = {}) /\ c IN components(u DIFF s) ==> ~(s INTER closure c = {})`, REPEAT GEN_TAC THEN STRIP_TAC THENL [ONCE_REWRITE_TAC[INTER_COMM] THEN MATCH_MP_TAC CONNECTED_COMPONENT_DIFF_CLOSED_NONSEPARATED THEN EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[COMPACT_IMP_CLOSED; CLOSED_IN_CLOSED_EQ; CLOSED_IN_SUBSET]; GEN_REWRITE_TAC (RAND_CONV o LAND_CONV o RAND_CONV o RAND_CONV) [GSYM UNIONS_1] THEN MATCH_MP_TAC NONSEPARATED_CLOSED_COMPLEMENT_COMPONENTS THEN EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[NOT_INSERT_EMPTY; SING_SUBSET]]);; let CONNECTED_EQ_NONSEPARATED_CLOSED_COMPLEMENT_COMPONENT = prove (`!u s:real^N->bool. (compact u \/ locally connected u) /\ closed_in (subtopology euclidean u) s /\ connected s /\ ~(s = {}) ==> (connected u <=> !c. c IN components(u DIFF s) ==> ~(s INTER closure c = {}))`, REPEAT GEN_TAC THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN ASM_CASES_TAC `u:real^N->bool = s` THEN ASM_REWRITE_TAC[DIFF_EQ_EMPTY; COMPONENTS_EMPTY; NOT_IN_EMPTY] THEN EQ_TAC THEN DISCH_TAC THENL [ASM_MESON_TAC[NONSEPARATED_CLOSED_COMPLEMENT_COMPONENT]; ALL_TAC] THEN SUBGOAL_THEN `u = UNIONS {c UNION s:real^N->bool |c| c IN components(u DIFF s)}` SUBST1_TAC THENL [FIRST_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN TRANS_TAC EQ_TRANS `UNIONS(components(u DIFF s)) UNION s:real^N->bool` THEN CONJ_TAC THENL [REWRITE_TAC[GSYM UNIONS_COMPONENTS] THEN ASM SET_TAC[]; SUBGOAL_THEN `~(components(u DIFF s:real^N->bool) = {})` MP_TAC THENL [REWRITE_TAC[COMPONENTS_EQ_EMPTY] THEN ASM SET_TAC[]; ALL_TAC] THEN ONCE_REWRITE_TAC[EXTENSION] THEN REWRITE_TAC[NOT_IN_EMPTY; UNIONS_GSPEC] THEN REWRITE_TAC[IN_UNION; IN_UNIONS; IN_ELIM_THM] THEN MESON_TAC[]]; MATCH_MP_TAC CONNECTED_UNIONS THEN CONJ_TAC THENL [REWRITE_TAC[FORALL_IN_GSPEC] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CONNECTED_UNION_STRONG THEN ASM_MESON_TAC[IN_COMPONENTS_CONNECTED; INTER_COMM]; REWRITE_TAC[INTERS_GSPEC] THEN ASM SET_TAC[]]]);; let CONNECTED_EQ_COMPONENT_DIFF_CLOSED_NONSEPARATED = prove (`!s:real^N->bool t. compact s /\ closed t /\ connected t /\ t SUBSET s /\ ~(t = {}) ==> (connected s <=> !c. c IN components (s DIFF t) ==> ~(closure c INTER t = {}))`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[INTER_COMM] THEN MATCH_MP_TAC CONNECTED_EQ_NONSEPARATED_CLOSED_COMPLEMENT_COMPONENT THEN ASM_MESON_TAC[CLOSED_IN_CLOSED_EQ; COMPACT_IMP_CLOSED]);; let CONNECTED_EQ_COMPONENT_DELETE_NONSEPARATED = prove (`!s:real^N->bool a:real^N. (compact s \/ locally connected s \/ FINITE(components(s DELETE a))) /\ a IN s ==> (connected s <=> !c. c IN components (s DELETE a) ==> a IN closure c)`, REWRITE_TAC[DISJ_ASSOC] THEN REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN ASSUME_TAC) THEN EQ_TAC THEN DISCH_TAC THENL [FIRST_X_ASSUM DISJ_CASES_TAC THENL [MP_TAC(ISPECL [`s:real^N->bool`; `{a:real^N}`] CONNECTED_EQ_NONSEPARATED_CLOSED_COMPLEMENT_COMPONENT) THEN ASM_REWRITE_TAC[CLOSED_IN_SING; CONNECTED_SING; NOT_INSERT_EMPTY] THEN REWRITE_TAC[SET_RULE `s DIFF {a} = s DELETE a`] THEN REWRITE_TAC[SET_RULE `~({a} INTER s = {}) <=> a IN s`]; REPEAT STRIP_TAC THEN UNDISCH_TAC `connected(s:real^N->bool)` THEN GEN_REWRITE_TAC I [GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN REWRITE_TAC[GSYM CONNECTED_IN_EUCLIDEAN] THEN REWRITE_TAC[CONNECTED_IN_EQ_NOT_SEPARATED] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN; SUBSET_UNIV] THEN MAP_EVERY EXISTS_TAC [`c:real^N->bool`; `{a:real^N} UNION UNIONS (components(s DELETE a) DELETE c)`] THEN REWRITE_TAC[NOT_INSERT_EMPTY; EMPTY_UNION] THEN REPEAT CONJ_TAC THENL [ASM_SIMP_TAC[GSYM IN_COMPONENTS_UNIONS_COMPLEMENT] THEN FIRST_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN ASM SET_TAC[]; ASM_MESON_TAC[IN_COMPONENTS_NONEMPTY]; REWRITE_TAC[SEPARATED_IN_UNION] THEN ASM_SIMP_TAC[SEPARATED_IN_UNIONS; FINITE_DELETE] THEN ASM_SIMP_TAC[SEPARATED_IN_SING; T1_SPACE_EUCLIDEAN] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN; IN_UNIV; SUBSET_UNIV] THEN ASM_REWRITE_TAC[IN_DELETE; EUCLIDEAN_CLOSURE_OF] THEN MP_TAC(ISPEC `s DELETE (a:real^N)` PAIRWISE_SEPARATED_COMPONENTS) THEN REWRITE_TAC[pairwise] THEN ASM_SIMP_TAC[]]]; SUBGOAL_THEN `s = {a} UNION UNIONS {(a:real^N) INSERT c |c| c IN components(s DELETE a)}` SUBST1_TAC THENL [MP_TAC(ISPEC `s DELETE (a:real^N)` UNIONS_COMPONENTS) THEN REWRITE_TAC[UNIONS_GSPEC] THEN ASM SET_TAC[]; REWRITE_TAC[GSYM UNIONS_INSERT]] THEN MATCH_MP_TAC CONNECTED_UNIONS THEN REWRITE_TAC[FORALL_IN_INSERT; CONNECTED_SING] THEN REWRITE_TAC[FORALL_IN_GSPEC; GSYM MEMBER_NOT_EMPTY] THEN CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED; CONNECTED_INSERT]; EXISTS_TAC `a:real^N` THEN REWRITE_TAC[INTERS_INSERT; IN_INTER; IN_SING] THEN REWRITE_TAC[INTERS_GSPEC; IN_ELIM_THM] THEN ASM SET_TAC[]]]);; let CONNECTED_INSERT_COMPACT = prove (`!s:real^N->bool a:real^N. compact(a INSERT s) ==> (connected(a INSERT s) <=> !c. c IN components s ==> a IN closure c)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `(a:real^N) IN s` THENL [FIRST_ASSUM(ASSUME_TAC o MATCH_MP (SET_RULE `a IN s ==> ~(s = {})`)) THEN ASM_SIMP_TAC[SET_RULE `a IN s ==> a INSERT s = s`] THEN EQ_TAC THENL [DISCH_TAC THEN MP_TAC(ISPEC `s:real^N->bool` COMPONENTS_EQ_SING) THEN ASM_SIMP_TAC[IN_SING; CLOSURE_INC]; DISCH_TAC THEN GEN_REWRITE_TAC RAND_CONV [UNIONS_COMPONENTS] THEN MATCH_MP_TAC CONNECTED_UNIONS THEN REWRITE_TAC[IN_COMPONENTS_CONNECTED] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTERS] THEN EXISTS_TAC `a:real^N` THEN X_GEN_TAC `c:real^N->bool` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `c:real^N->bool`) THEN FIRST_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_COMPONENT) THEN ASM_REWRITE_TAC[CLOSED_IN_INTER_CLOSURE] THEN ASM SET_TAC[]]; MP_TAC(ISPECL [`(a:real^N) INSERT s`; `a:real^N`] CONNECTED_EQ_COMPONENT_DELETE_NONSEPARATED) THEN ASM_REWRITE_TAC[IN_INSERT] THEN ASM_SIMP_TAC[SET_RULE `~(a IN s) ==> (a INSERT s) DELETE a = s`]]);; let BOUNDARY_BUMPING_THEOREM_EUCLIDEAN = prove (`!s t c:real^N->bool. compact s /\ connected s /\ t PSUBSET s /\ c IN components t ==> ~(closure(c) INTER closure(t) INTER closure(s DIFF t) = {})`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC(SET_RULE `~(c INTER s = {}) /\ c SUBSET t ==> ~(c INTER t INTER s = {})`) THEN ASM_SIMP_TAC[CONNECTED_COMPONENT_DIFF_NONSEPARATED_ALT; SUBSET_CLOSURE; IN_COMPONENTS_SUBSET]);; let BOUNDARY_BUMPING_THEOREM_EUCLIDEAN_CLOSED = prove (`!s t c:real^N->bool. compact s /\ connected s /\ closed t /\ t PSUBSET s /\ c IN components t ==> ~(c INTER closure t INTER closure(s DIFF t) = {})`, REPEAT GEN_TAC THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`; `c:real^N->bool`] BOUNDARY_BUMPING_THEOREM_EUCLIDEAN) THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] CLOSED_COMPONENTS)) THEN ASM_SIMP_TAC[CLOSURE_CLOSED]);; let BOUNDARY_BUMPING_THEOREM_EUCLIDEAN_ALT = prove (`!s t c:real^N->bool. compact s /\ connected s /\ open_in (subtopology euclidean s) t /\ t PSUBSET s /\ c IN components(closure t) ==> ~(c INTER (s DIFF t) = {})`, REPEAT GEN_TAC THEN ASM_CASES_TAC `closure t:real^N->bool = s` THENL [REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN MP_TAC(snd(EQ_IMP_RULE(ISPEC `s:real^N->bool` COMPONENTS_EQ_SING))) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN REWRITE_TAC[IN_SING] THEN ASM SET_TAC[]; STRIP_TAC] THEN MP_TAC(ISPECL [`s:real^N->bool`; `closure t:real^N->bool`; `c:real^N->bool`] BOUNDARY_BUMPING_THEOREM_EUCLIDEAN_CLOSED) THEN ASM_REWRITE_TAC[CLOSED_CLOSURE] THEN ANTS_TAC THENL [ASM_REWRITE_TAC[PSUBSET] THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED] THEN ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `s SUBSET u ==> ~(c INTER t INTER s = {}) ==> ~(c INTER u = {})`) THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN ASM_REWRITE_TAC[SET_RULE `k SUBSET s DIFF (s INTER u) <=> k SUBSET s /\ u INTER k = {}`] THEN CONJ_TAC THENL [MATCH_MP_TAC CLOSURE_MINIMAL THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED] THEN SET_TAC[]; ASM_SIMP_TAC[OPEN_INTER_CLOSURE_EQ_EMPTY] THEN MP_TAC(ISPEC `s INTER u:real^N->bool` CLOSURE_SUBSET) THEN SET_TAC[]]);; let BOUNDARY_BUMPING_THEOREM_EUCLIDEAN_OPEN = prove (`!s t c:real^N->bool. (compact s \/ locally connected s) /\ connected s /\ open_in (subtopology euclidean s) t /\ t PSUBSET s /\ c IN components t ==> ~(closure c INTER (s DIFF t) = {})`, REPEAT GEN_TAC THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN ONCE_REWRITE_TAC[INTER_COMM] THEN MATCH_MP_TAC NONSEPARATED_CLOSED_COMPLEMENT_COMPONENT THEN EXISTS_TAC `s:real^N->bool` THEN ASM_SIMP_TAC[ISPEC `euclidean` OPEN_IN_IMP_SUBSET; CLOSED_IN_DIFF; CLOSED_IN_REFL; SET_RULE `t SUBSET s ==> (s DIFF t = {} <=> ~(t PSUBSET s))`; SET_RULE `t SUBSET s ==> s DIFF (s DIFF t) = t`]);; let BOUNDARY_BUMPING_THEOREM_EUCLIDEAN_OPEN_ALT = prove (`!s t c:real^N->bool. (compact s \/ locally connected s) /\ connected s /\ open_in (subtopology euclidean s) t /\ t PSUBSET s /\ c IN components t ==> ~(closure c INTER (closure t DIFF t) = {})`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP BOUNDARY_BUMPING_THEOREM_EUCLIDEAN_OPEN) THEN MATCH_MP_TAC(SET_RULE `c SUBSET t' ==> ~(c INTER (s DIFF t) = {}) ==> ~(c INTER (t' DIFF t) = {})`) THEN ASM_SIMP_TAC[SUBSET_CLOSURE; IN_COMPONENTS_SUBSET]);; let CONTINUUM_UNION_COMPONENTS_INTERMEDIATE_COMPLEMENT = prove (`!s t u c:real^N->bool. compact s /\ connected s /\ compact t /\ s SUBSET t /\ compact u /\ connected u /\ t SUBSET u /\ c IN components(u DIFF t) /\ closure c DIFF c SUBSET s ==> compact(c UNION s) /\ connected(c UNION s)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THENL [ASM_SIMP_TAC[DIFF_EMPTY; SUBSET_EMPTY; UNION_EMPTY] THEN MESON_TAC[COMPACT_COMPONENTS; IN_COMPONENTS_CONNECTED]; STRIP_TAC] THEN MP_TAC(ISPECL [`u:real^N->bool`; `u DIFF t:real^N->bool`; `c:real^N->bool`] BOUNDARY_BUMPING_THEOREM_EUCLIDEAN_OPEN) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC OPEN_IN_DIFF THEN REWRITE_TAC[OPEN_IN_REFL] THEN MATCH_MP_TAC CLOSED_SUBSET THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED] THEN ASM SET_TAC[]; ASM_SIMP_TAC[SET_RULE `t SUBSET u ==> u DIFF (u DIFF t) = t`]] THEN DISCH_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN SUBGOAL_THEN `c UNION s:real^N->bool = closure c UNION s` SUBST1_TAC THENL [MP_TAC(ISPEC `c:real^N->bool` CLOSURE_SUBSET) THEN ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [MATCH_MP_TAC COMPACT_UNION THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[COMPACT_CLOSURE] THEN MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `u:real^N->bool` THEN ASM_SIMP_TAC[COMPACT_IMP_BOUNDED] THEN ASM SET_TAC[]; MATCH_MP_TAC CONNECTED_UNION THEN FIRST_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_CONNECTED) THEN ASM_SIMP_TAC[CONNECTED_CLOSURE] THEN ASM SET_TAC[]]);; let CONTINUUM_UNION_COMPONENTS_COMPLEMENT = prove (`!s u c:real^N->bool. compact s /\ connected s /\ compact u /\ connected u /\ s SUBSET u /\ c IN components(u DIFF s) /\ closure c DIFF c SUBSET s ==> compact(c UNION s) /\ connected(c UNION s)`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC CONTINUUM_UNION_COMPONENTS_INTERMEDIATE_COMPLEMENT THEN MAP_EVERY EXISTS_TAC [`s:real^N->bool`; `u:real^N->bool`] THEN ASM_REWRITE_TAC[SUBSET_REFL]);; (* ------------------------------------------------------------------------- *) (* More compact component properties via the notion of "well-chained". *) (* ------------------------------------------------------------------------- *) let WELLCHAINED_ELEMENTS = prove (`!s:real^N->bool a b e. (?p n. p 0 = a /\ p n = b /\ (!i. i <= n ==> p i IN s) /\ (!i. i < n ==> dist(p i,p(SUC i)) < e)) <=> a IN s /\ b IN s /\ (!c. c SUBSET s /\ a IN c /\ (!x y. x IN c /\ y IN s /\ dist(x,y) < e ==> y IN c) ==> b IN c)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `(a:real^N) IN s` THENL [ALL_TAC; ASM_MESON_TAC[LE_0]] THEN ASM_CASES_TAC `(b:real^N) IN s` THENL [ALL_TAC; ASM_MESON_TAC[LE_REFL]] THEN ASM_REWRITE_TAC[] THEN EQ_TAC THENL [REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`p:num->real^N`; `n:num`] THEN STRIP_TAC THEN X_GEN_TAC `c:real^N->bool` THEN STRIP_TAC THEN SUBGOAL_THEN `!k. k <= n ==> (p:num->real^N) k IN c` (fun th -> ASM_MESON_TAC[th; LE_REFL]) THEN INDUCT_TAC THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN EXISTS_TAC `(p:num->real^N) k` THEN REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; DISCH_THEN(MP_TAC o SPEC `{x:real^N | ?p n. p 0 = a /\ p n = x /\ (!i. i <= n ==> p i IN s) /\ (!i. i < n ==> dist(p i,p(SUC i)) < e)}`) THEN ANTS_TAC THENL [ALL_TAC; REWRITE_TAC[IN_ELIM_THM]] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN ASM_MESON_TAC[LE_REFL]; REWRITE_TAC[IN_ELIM_THM] THEN MAP_EVERY EXISTS_TAC [`(\n. a):num->real^N`; `0`] THEN ASM_REWRITE_TAC[LT]; MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN REWRITE_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`p:num->real^N`; `n:num`] THEN STRIP_TAC THEN EXISTS_TAC `\i. if i <= n then (p:num->real^N) i else y` THEN EXISTS_TAC `SUC n` THEN ASM_REWRITE_TAC[LE_0; ARITH_RULE `~(SUC n <= n)`] THEN REWRITE_TAC[LE; LT; TAUT `p \/ q ==> r <=> (p ==> r) /\ (q ==> r)`; FORALL_AND_THM; FORALL_UNWIND_THM2] THEN REWRITE_TAC[LE_REFL; LE_SUC_LT; LT_REFL] THEN ASM_SIMP_TAC[LT_IMP_LE]]]);; let WELLCHAINED_SETS = prove (`!s:real^N->bool e. (!a b. a IN s /\ b IN s ==> ?p n. p 0 = a /\ p n = b /\ (!i. i <= n ==> p i IN s) /\ (!i. i < n ==> dist(p i,p(SUC i)) < e)) <=> (!c. c SUBSET s /\ ~(c = {}) /\ (!x y. x IN c /\ y IN s /\ dist(x,y) < e ==> y IN c) ==> c = s)`, REPEAT GEN_TAC THEN REWRITE_TAC[WELLCHAINED_ELEMENTS] THEN SIMP_TAC[] THEN REWRITE_TAC[MESON[] `(!a b. P a /\ P b ==> !c. Q a b c ==> R a b c) <=> (!c a b. Q a b c /\ P a /\ P b ==> R a b c)`] THEN AP_TERM_TAC THEN ABS_TAC THEN SIMP_TAC[GSYM MEMBER_NOT_EMPTY; GSYM SUBSET_ANTISYM_EQ] THEN REWRITE_TAC[SUBSET] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN MESON_TAC[]);; let CONNECTED_IMP_WELLCHAINED = prove (`!s e a b:real^N. connected s /\ &0 < e /\ a IN s /\ b IN s ==> ?p n. p 0 = a /\ p n = b /\ (!i. i <= n ==> p i IN s) /\ (!i. i < n ==> dist(p i,p(SUC i)) < e)`, REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REPLICATE_TAC 2 (GEN_TAC THEN DISCH_TAC) THEN REWRITE_TAC[RIGHT_IMP_FORALL_THM; IMP_IMP] THEN REWRITE_TAC[WELLCHAINED_SETS] THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[SET_RULE `c SUBSET s /\ ~(c = {}) ==> (c = s <=> !a b. a IN s /\ b IN s /\ a IN c ==> b IN c)`] THEN MATCH_MP_TAC CONNECTED_INDUCTION_SIMPLE THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN EXISTS_TAC `s INTER ball(a:real^N,e / &2)` THEN ASM_SIMP_TAC[IN_INTER; CENTRE_IN_BALL; REAL_HALF] THEN ASM_SIMP_TAC[OPEN_IN_OPEN_INTER; OPEN_BALL] THEN REWRITE_TAC[IN_BALL] THEN ASM_MESON_TAC[NORM_ARITH `dist(a:real^N,x) < e / &2 /\ dist(a,y) < e / &2 ==> dist(x,y) < e`]);; let CONNECTED_EQ_WELLCHAINED = prove (`!s:real^N->bool. compact s ==> (connected s <=> !e a b. &0 < e /\ a IN s /\ b IN s ==> ?p n. p 0 = a /\ p n = b /\ (!i. i <= n ==> p i IN s) /\ (!i. i < n ==> dist(p i,p(SUC i)) < e))`, REPEAT STRIP_TAC THEN EQ_TAC THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC CONNECTED_IMP_WELLCHAINED THEN ASM_MESON_TAC[]; ONCE_REWRITE_TAC[IMP_CONJ] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM]] THEN REWRITE_TAC[WELLCHAINED_SETS] THEN DISCH_TAC THEN ASM_CASES_TAC `connected(s:real^N->bool)` THENL [ASM_REWRITE_TAC[]; REWRITE_TAC[CONNECTED_CLOSED_IN_EQ]] THEN UNDISCH_TAC `compact(s:real^N->bool)` THEN SIMP_TAC[CLOSED_IN_COMPACT_EQ] THEN DISCH_TAC THEN REWRITE_TAC[NOT_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`k1:real^N->bool`; `k2:real^N->bool`] THEN STRIP_TAC THEN SUBGOAL_THEN `?a:real^N. a IN k1` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPEC `setdist(k1:real^N->bool,k2)`) THEN REWRITE_TAC[NOT_IMP; SETDIST_POS_LT] THEN ASM_SIMP_TAC[SETDIST_EQ_0_COMPACT_CLOSED; COMPACT_IMP_CLOSED] THEN DISCH_THEN(MP_TAC o SPEC `k1:real^N->bool`) THEN ASM_REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r ==> s <=> p /\ q /\ ~s ==> ~r`] THEN REWRITE_TAC[REAL_NOT_LT; GSYM IN_DIFF] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC SETDIST_LE_DIST THEN ASM SET_TAC[]);; let WELLCHAINED_INTERS = prove (`!s:num->(real^N->bool) d e. d < e /\ (!m. compact (s m)) /\ (!m. s(SUC m) SUBSET s m) /\ (!m a b. a IN s m /\ b IN s m ==> ?p n. p 0 = a /\ p n = b /\ (!i. i <= n ==> p i IN s m) /\ (!i. i < n ==> dist(p i,p (SUC i)) < d)) ==> !a b. a IN INTERS {s m | m IN (:num)} /\ b IN INTERS {s m | m IN (:num)} ==> ?p n. p 0 = a /\ p n = b /\ (!i. i <= n ==> p i IN INTERS {s m | m IN (:num)}) /\ (!i. i < n ==> dist(p i,p (SUC i)) < e)`, REWRITE_TAC[WELLCHAINED_SETS] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN ABBREV_TAC `k:real^N->bool = INTERS {s m | m IN (:num)}` THEN ASM_CASES_TAC `k:real^N->bool = {}` THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `compact(k:real^N->bool)` ASSUME_TAC THENL [EXPAND_TAC "k" THEN MATCH_MP_TAC COMPACT_INTERS THEN ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[GSYM WELLCHAINED_SETS] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN STRIP_TAC THEN MP_TAC(ISPEC `s:num->real^N->bool` HAUSDIST_COMPACT_INTERS_LIMIT) THEN ASM_REWRITE_TAC[LIM_SEQUENTIALLY; DIST_0] THEN DISCH_THEN(MP_TAC o SPEC `(e - d) / &2`) THEN ASM_REWRITE_TAC[REAL_SUB_LT; REAL_HALF; NORM_LIFT] THEN DISCH_THEN(X_CHOOSE_THEN `n:num` (MP_TAC o SPEC `n:num`)) THEN REWRITE_TAC[LE_REFL; real_abs; HAUSDIST_POS_LE] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`n:num`; `a:real^N`; `b:real^N`] o GEN_REWRITE_RULE BINDER_CONV [GSYM WELLCHAINED_SETS]) THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `m:num` THEN DISCH_THEN(X_CHOOSE_THEN `p:num->real^N` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `!i. ?y. i <= m ==> y IN k /\ dist((p:num->real^N) i,y) <= (e - d) / &2` MP_TAC THENL [X_GEN_TAC `j:num` THEN REWRITE_TAC[RIGHT_EXISTS_IMP_THM] THEN DISCH_TAC THEN MP_TAC(ISPECL [`(s:num->real^N->bool) n`; `k:real^N->bool`] HAUSDIST_COMPACT_EXISTS) THEN ASM_SIMP_TAC[COMPACT_IMP_BOUNDED] THEN DISCH_THEN(MP_TAC o SPEC `(p:num->real^N) j`) THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM]] THEN X_GEN_TAC `q:num->real^N` THEN DISCH_TAC THEN EXISTS_TAC `\i. if 0 < i /\ i < m then (q:num->real^N) i else p i` THEN ASM_SIMP_TAC[LT_REFL] THEN CONJ_TAC THEN X_GEN_TAC `i:num` THENL [ASM_CASES_TAC `i = 0` THEN ASM_REWRITE_TAC[LT_REFL] THEN ASM_CASES_TAC `i:num = m` THEN ASM_REWRITE_TAC[LT_REFL] THEN REPEAT DISCH_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_SIMP_TAC[]) THEN ASM_ARITH_TAC; ASM_CASES_TAC `i = 0` THEN ASM_SIMP_TAC[LE_1; LT_0; LT_REFL] THEN SIMP_TAC[ARITH_RULE `i < m ==> (SUC i < m <=> ~(SUC i = m))`] THEN REWRITE_TAC[COND_SWAP] THEN DISCH_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL [ASM_MESON_TAC[REAL_LT_TRANS]; MATCH_MP_TAC(NORM_ARITH `dist(a:real^N,p(SUC 0)) < d /\ dist(p(SUC 0),q(SUC 0)) <= (e - d) / &2 ==> dist(a,q(SUC 0)) < e`) THEN ASM_MESON_TAC[ARITH_RULE `0 < m ==> SUC 0 <= m`]; MATCH_MP_TAC(NORM_ARITH `dist((p:num->real^N) i,b) < d /\ dist(p i,q i) <= (e - d) / &2 ==> dist(q i,b) < e`) THEN ASM_MESON_TAC[LT_IMP_LE]; MATCH_MP_TAC(NORM_ARITH `dist(p i:real^N,p(SUC i)) < d /\ dist(p i,q i) <= (e - d) / &2 /\ dist(p(SUC i),q(SUC i)) <= (e - d) / &2 ==> dist(q i,q(SUC i)) < e`) THEN ASM_MESON_TAC[LT_IMP_LE; ARITH_RULE `i < m /\ ~(SUC i = m) ==> SUC i <= m`]]]);; let CONNECTED_COMPONENT_IMP_WELLCHAINED = prove (`!s a b:real^N e. &0 < e /\ connected_component s a b ==> ?p n. p 0 = a /\ p n = b /\ (!i. i <= n ==> p i IN s) /\ (!i. i < n ==> dist(p i,p (SUC i)) < e)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`connected_component s (a:real^N)`; `e:real`; `a:real^N`; `b:real^N`] CONNECTED_IMP_WELLCHAINED) THEN ASM_REWRITE_TAC[CONNECTED_CONNECTED_COMPONENT] THEN ANTS_TAC THENL [REWRITE_TAC[IN] THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ] THEN ASM_MESON_TAC[CONNECTED_COMPONENT_IN]; REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN MP_TAC (ISPECL [`s:real^N->bool`; `a:real^N`] CONNECTED_COMPONENT_SUBSET) THEN ASM SET_TAC[]]);; let CONNECTED_COMPONENT_EQ_WELLCHAINED = prove (`!s a b:real^N. compact s ==> (connected_component s a b <=> a IN s /\ b IN s /\ !e. &0 < e ==> ?p n. p 0 = a /\ p n = b /\ (!i. i <= n ==> p i IN s) /\ (!i. i < n ==> dist(p i,p (SUC i)) < e))`, REPEAT STRIP_TAC THEN EQ_TAC THEN STRIP_TAC THENL [FIRST_ASSUM(ASSUME_TAC o MATCH_MP CONNECTED_COMPONENT_IN) THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CONNECTED_COMPONENT_IMP_WELLCHAINED THEN ASM_MESON_TAC[]; ALL_TAC] THEN ABBREV_TAC `t = \k. {x | (x:real^N) IN s /\ ?p n. p 0 = a /\ p n = x /\ (!i. i <= n ==> p i IN s) /\ (!i. i < n ==> dist(p i,p(SUC i)) < inv(&k + &1))}` THEN REWRITE_TAC[connected_component] THEN EXISTS_TAC `INTERS {t k | k IN (:num)}:real^N->bool` THEN REPEAT CONJ_TAC THENL [ALL_TAC; EXPAND_TAC "t" THEN REWRITE_TAC[INTERS_GSPEC] THEN SET_TAC[]; EXPAND_TAC "t" THEN REWRITE_TAC[INTERS_GSPEC] THEN ASM_REWRITE_TAC[IN_ELIM_THM; IN_UNIV] THEN X_GEN_TAC `j:num` THEN EXISTS_TAC `(\n. a):num->real^N` THEN ASM_REWRITE_TAC[DIST_REFL; REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`]; EXPAND_TAC "t" THEN REWRITE_TAC[INTERS_GSPEC; IN_UNIV; IN_ELIM_THM] THEN X_GEN_TAC `j:num` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[DIST_REFL; REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`]] THEN W(MP_TAC o PART_MATCH (lhand o rand) CONNECTED_EQ_WELLCHAINED o snd) THEN SUBGOAL_THEN `!n. compact((t:num->real^N->bool) n)` ASSUME_TAC THENL [GEN_TAC THEN MATCH_MP_TAC CLOSED_IN_COMPACT THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[closed_in; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN EXPAND_TAC "t" THEN REWRITE_TAC[SUBSET_RESTRICT] THEN REWRITE_TAC[open_in; SET_RULE `s DIFF t SUBSET s`] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[DIFF; IN_ELIM_THM] THEN ASM_CASES_TAC `(x:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN EXISTS_TAC `inv(&n + &1)` THEN REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN X_GEN_TAC `y:real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o check (is_neg o concl)) THEN REWRITE_TAC[CONTRAPOS_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`p:num->real^N`; `m:num`] THEN STRIP_TAC THEN EXISTS_TAC `\j. if j <= m then (p:num->real^N) j else x` THEN EXISTS_TAC `SUC m` THEN ASM_REWRITE_TAC[LE_0; ARITH_RULE `~(SUC m <= m)`] THEN REWRITE_TAC[LE_SUC_LT; LT; LE] THEN CONJ_TAC THEN X_GEN_TAC `j:num` THEN STRIP_TAC THEN ASM_SIMP_TAC[LT_IMP_LE; ARITH_RULE `~(SUC m <= m)`; LE_REFL; LT_REFL]; ALL_TAC] THEN ANTS_TAC THENL [MATCH_MP_TAC COMPACT_INTERS THEN ASM_REWRITE_TAC[FORALL_IN_GSPEC; IN_UNIV] THEN SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN SUBGOAL_THEN `!n. t(SUC n):real^N->bool SUBSET t n` ASSUME_TAC THENL [EXPAND_TAC "t" THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN REPEAT GEN_TAC THEN MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN REPEAT(MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[]) THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LT_TRANS) THEN MATCH_MP_TAC REAL_LT_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LT] THEN ARITH_TAC; ALL_TAC] THEN X_GEN_TAC `e:real` THEN ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[] THEN MP_TAC(SPEC `e / &2` ARCH_EVENTUALLY_INV1) THEN ASM_REWRITE_TAC[REAL_HALF; EVENTUALLY_SEQUENTIALLY] THEN DISCH_THEN(X_CHOOSE_TAC `N:num`) THEN SUBGOAL_THEN `INTERS {t n | n IN (:num)}:real^N->bool = INTERS {t(N + n) | n IN (:num)}` SUBST1_TAC THENL [REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[SUBSET; INTERS_GSPEC; IN_ELIM_THM; IN_UNIV] THEN SUBGOAL_THEN `!m n. m <= n ==> (t:num->real^N->bool) n SUBSET t m` (fun th -> MESON_TAC[th; LE_ADD; ADD_SYM; SUBSET]) THEN MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN ASM_REWRITE_TAC[] THEN SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC WELLCHAINED_INTERS THEN EXISTS_TAC `e / &2` THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ASM_REWRITE_TAC[ADD_CLAUSES]] THEN MAP_EVERY X_GEN_TAC [`m:num`; `x:real^N`; `y:real^N`] THEN EXPAND_TAC "t" THEN REWRITE_TAC[IN_ELIM_THM] THEN MAP_EVERY ASM_CASES_TAC [`(x:real^N) IN s`; `(y:real^N) IN s`] THEN ASM_REWRITE_TAC[IMP_CONJ; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`p1:num->real^N`; `n1:num`] THEN REPEAT DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`p2:num->real^N`; `n2:num`] THEN REPEAT DISCH_TAC THEN EXISTS_TAC `\j. if j <= n1 then (p1:num->real^N) (n1 - j) else p2(j - n1)` THEN EXISTS_TAC `n1 + n2:num` THEN ASM_REWRITE_TAC[LE_0; SUB_0; ADD_SUB2; ARITH_RULE `n - (n + m) = 0`] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[ARITH_RULE `n1 + n2 <= n1 <=> n2 = 0`] THEN ASM_MESON_TAC[]; X_GEN_TAC `i:num` THEN DISCH_TAC THEN ASM_CASES_TAC `(i:num) <= n1` THEN ASM_REWRITE_TAC[] THEN (CONJ_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; ALL_TAC]) THENL [MAP_EVERY EXISTS_TAC [`p1:num->real^N`; `n1 - i:num`]; MAP_EVERY EXISTS_TAC [`p2:num->real^N`; `i - n1:num`]] THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; X_GEN_TAC `i:num` THEN DISCH_TAC THEN ASM_CASES_TAC `SUC i <= n1` THEN ASM_SIMP_TAC[ARITH_RULE `SUC i <= n ==> i <= n`] THENL [ASM_SIMP_TAC[ARITH_RULE `SUC i <= n ==> n - i = SUC(n - SUC i)`] THEN TRANS_TAC REAL_LT_TRANS `inv(&(N + m) + &1)` THEN ASM_SIMP_TAC[LE_ADD] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; ASM_SIMP_TAC[ARITH_RULE `~(SUC i <= n) ==> (i <= n <=> i = n)`] THEN COND_CASES_TAC THENL [ASM_REWRITE_TAC[SUB_REFL; ARITH_RULE `SUC n - n = SUC 0`] THEN SUBGOAL_THEN `a:real^N = p2 0` SUBST1_TAC THENL [ASM_REWRITE_TAC[]; ALL_TAC]; ASM_SIMP_TAC[ARITH_RULE `~(SUC i <= n) ==> SUC i - n = SUC(i - n)`]] THEN (TRANS_TAC REAL_LT_TRANS `inv(&(N + m) + &1)` THEN CONJ_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; ASM_SIMP_TAC[LE_ADD]])]]);; let COMPACT_PARTITION_CONTAINING_CLOSED = prove (`!s t t':real^N->bool. compact s /\ closed t /\ closed t' /\ t SUBSET s /\ t' SUBSET s /\ (!c. c IN components s ==> c INTER t = {} \/ c INTER t' = {}) ==> ?k k'. compact k /\ compact k' /\ t SUBSET k /\ t' SUBSET k' /\ DISJOINT k k' /\ k UNION k' = s`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THENL [MAP_EVERY EXISTS_TAC [`{}:real^N->bool`; `s:real^N->bool`] THEN ASM_REWRITE_TAC[COMPACT_EMPTY] THEN SET_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `t':real^N->bool = {}` THENL [MAP_EVERY EXISTS_TAC [`s:real^N->bool`; `{}:real^N->bool`] THEN ASM_REWRITE_TAC[COMPACT_EMPTY] THEN SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `compact(t:real^N->bool) /\ compact(t':real^N->bool)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[COMPACT_EQ_BOUNDED_CLOSED; BOUNDED_SUBSET]; ALL_TAC] THEN SUBGOAL_THEN `?e. &0 < e /\ !x y. x IN t /\ (y:real^N) IN t' ==> ~(?p n. p 0 = x /\ p n = y /\ (!i. i <= n ==> p i IN s) /\ (!i. i < n ==> dist(p i,p (SUC i)) < e))` STRIP_ASSUME_TAC THENL [ONCE_REWRITE_TAC[MESON[] `(?e. P e /\ Q e) <=> ~(!e. P e ==> ~Q e)`] THEN DISCH_THEN(MP_TAC o GEN `n:num` o SPEC `inv(&n + &1)`) THEN REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN GEN_REWRITE_TAC (RAND_CONV o TOP_DEPTH_CONV) [NOT_FORALL_THM] THEN REWRITE_TAC[SKOLEM_THM] THEN REWRITE_TAC[NOT_EXISTS_THM; NOT_IMP] THEN MAP_EVERY X_GEN_TAC [`x:num->real^N`; `y:num->real^N`] THEN REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC THEN MP_TAC(ISPECL [`(t:real^N->bool) PCROSS (t':real^N->bool)`] compact) THEN ASM_REWRITE_TAC[COMPACT_PCROSS_EQ] THEN DISCH_THEN(MP_TAC o SPEC `\n. pastecart((x:num->real^N) n) (y n:real^N)`) THEN ASM_REWRITE_TAC[PASTECART_IN_PCROSS; o_DEF; EXISTS_PASTECART] THEN REWRITE_TAC[NOT_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`; `r:num->num`] THEN STRIP_TAC THEN SUBGOAL_THEN `(a:real^N) IN s /\ b IN s` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPEC `connected_component s (a:real^N)`) THEN REWRITE_TAC[NOT_IMP; components; IN_ELIM_THM] THEN CONJ_TAC THENL [ASM_MESON_TAC[SUBSET]; REWRITE_TAC[DE_MORGAN_THM]] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN CONJ_TAC THENL [EXISTS_TAC `a:real^N`; EXISTS_TAC `b:real^N`] THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[IN] THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ] THEN ASM_SIMP_TAC[CONNECTED_COMPONENT_EQ_WELLCHAINED] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIM_PASTECART_EQ]) THEN REWRITE_TAC[tendsto; AND_FORALL_THM] THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN SUBGOAL_THEN `eventually ((\n. inv(&n + &1) < e) o r) sequentially` MP_TAC THENL [MATCH_MP_TAC EVENTUALLY_SUBSEQUENCE THEN ASM_REWRITE_TAC[ARCH_EVENTUALLY_INV1]; ASM_REWRITE_TAC[o_DEF; GSYM EVENTUALLY_AND; IMP_IMP]] THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN DISCH_THEN(X_CHOOSE_THEN `NN:num` (MP_TAC o SPEC `NN:num`)) THEN REWRITE_TAC[LE_REFL] THEN ABBREV_TAC `N = (r:num->num) NN` THEN STRIP_TAC THEN FIRST_X_ASSUM(X_CHOOSE_THEN `p:num->real^N` MP_TAC o SPEC `N:num`) THEN DISCH_THEN(X_CHOOSE_THEN `n:num` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\i. if i = 0 then a:real^N else if i <= SUC n then p(i - 1) else b` THEN EXISTS_TAC `n + 2` THEN ASM_REWRITE_TAC[ADD_EQ_0; ARITH_EQ; ARITH_RULE `~(n + 2 <= SUC n)`] THEN MATCH_MP_TAC num_INDUCTION THEN CONV_TAC NUM_REDUCE_CONV THEN ASM_REWRITE_TAC[ARITH_RULE `1 <= SUC n`; NOT_SUC; LE_SUC] THEN CONJ_TAC THENL [ASM_MESON_TAC[DIST_SYM]; ALL_TAC] THEN X_GEN_TAC `i:num` THEN DISCH_THEN(K ALL_TAC) THEN CONJ_TAC THENL [DISCH_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; SIMP_TAC[LE_SUC_LT; ARITH_RULE `SUC i < n + 2 <=> i = n \/ i < n`] THEN STRIP_TAC THEN ASM_SIMP_TAC[LT_IMP_LE; LE_REFL; LT_REFL; SUC_SUB1] THEN TRANS_TAC REAL_LT_TRANS `inv(&N + &1)` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC]; ALL_TAC] THEN EXISTS_TAC `{x | (x:real^N) IN s /\ ?p n. p 0 IN t /\ p n = x /\ (!i. i <= n ==> p i IN s) /\ (!i. i < n ==> dist (p i,p (SUC i)) < e)}` THEN EXISTS_TAC `{x | (x:real^N) IN s /\ ~(?p n. p 0 IN t /\ p n = x /\ (!i. i <= n ==> p i IN s) /\ (!i. i < n ==> dist (p i,p (SUC i)) < e))}` THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN MATCH_MP_TAC(TAUT `q /\ (q ==> p) ==> p /\ q`) THEN CONJ_TAC THENL [ALL_TAC; STRIP_TAC THEN MATCH_MP_TAC COMPACT_IN_SEPARATED_UNION] THEN ASM_SIMP_TAC[SET_RULE `{x | x IN s /\ P x} UNION {x | x IN s /\ ~P x} = s`; SET_RULE `DISJOINT {x | x IN s /\ P x} {x | x IN s /\ ~P x}`] THENL [REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN CONJ_TAC THENL [ALL_TAC; RULE_ASSUM_TAC(REWRITE_RULE[SUBSET]) THEN ASM MESON_TAC[]] THEN X_GEN_TAC `x:real^N` THEN REPEAT STRIP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MAP_EVERY EXISTS_TAC [`(\i. x):num->real^N`; `0`] THEN ASM_REWRITE_TAC[LT; LE] THEN ASM SET_TAC[]; TRANS_TAC REAL_LTE_TRANS `e:real` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_SETDIST THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `!t u. t SUBSET t' /\ u SUBSET u' /\ ~(t = {}) /\ ~(u = {}) ==> ~(t' = {}) /\ ~(u' = {})`) THEN MAP_EVERY EXISTS_TAC [`t:real^N->bool`; `t':real^N->bool`] THEN ASM_REWRITE_TAC[]; MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN REWRITE_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM; IMP_CONJ] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`p:num->real^N`; `n:num`] THEN REPLICATE_TAC 5 DISCH_TAC THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[REAL_NOT_LE] THEN DISCH_TAC THEN MAP_EVERY EXISTS_TAC [`\i. if i <= n then (p:num->real^N) i else y`; `SUC n`] THEN ASM_REWRITE_TAC[LE_0; ARITH_RULE `~(SUC n <= n)`] THEN REWRITE_TAC[LE; LT; TAUT `p \/ q ==> r <=> (p ==> r) /\ (q ==> r)`; FORALL_AND_THM; FORALL_UNWIND_THM2] THEN REWRITE_TAC[LE_REFL; LE_SUC_LT; LT_REFL] THEN ASM_SIMP_TAC[LT_IMP_LE]]]);; let COMPACT_PARTITION_CONTAINING_POINTS = prove (`!s a b:real^N. compact s /\ a IN s /\ b IN s /\ ~(connected_component s a b) ==> ?k k'. compact k /\ compact k' /\ a IN k /\ b IN k' /\ DISJOINT k k' /\ k UNION k' = s`, REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM SING_SUBSET] THEN MATCH_MP_TAC COMPACT_PARTITION_CONTAINING_CLOSED THEN ASM_REWRITE_TAC[SING_SUBSET; CLOSED_SING] THEN X_GEN_TAC `c:real^N->bool` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [connected_component]) THEN REWRITE_TAC[NOT_EXISTS_THM] THEN DISCH_THEN(MP_TAC o SPEC `c:real^N->bool`) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_CONNECTED) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);; let CONNECTED_COMPONENT_LIMIT = prove (`!s x y a b:real^N. compact s /\ (x --> a) sequentially /\ (y --> b) sequentially /\ eventually (\n. connected_component s (x n) (y n)) sequentially ==> connected_component s a b`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_SIMP_TAC[CONNECTED_COMPONENT_EQ_WELLCHAINED] THEN DISCH_TAC THEN REWRITE_TAC[CONJ_ASSOC] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [FIRST_X_ASSUM(CONJUNCTS_THEN STRIP_ASSUME_TAC) THEN CONJ_TAC THEN MATCH_MP_TAC(ISPEC `sequentially` LIM_IN_CLOSED_SET) THENL [EXISTS_TAC `x:num->real^N`; EXISTS_TAC `y:num->real^N`] THEN ASM_SIMP_TAC[TRIVIAL_LIMIT_SEQUENTIALLY; COMPACT_IMP_CLOSED] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] EVENTUALLY_MONO)) THEN SIMP_TAC[]; STRIP_TAC] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o check (is_conj o concl)) THEN REWRITE_TAC[tendsto; CONJ_ASSOC; AND_FORALL_THM] THEN REWRITE_TAC[LEFT_AND_FORALL_THM] THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[GSYM EVENTUALLY_AND; GSYM CONJ_ASSOC] THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN DISCH_THEN(X_CHOOSE_THEN `N:num` (MP_TAC o SPEC `N:num`)) THEN REWRITE_TAC[LE_REFL] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`p:num->real^N`; `n:num`] THEN STRIP_TAC THEN EXISTS_TAC `\i. if i = 0 then a:real^N else if i <= SUC n then p(i - 1) else b` THEN EXISTS_TAC `n + 2` THEN ASM_REWRITE_TAC[ADD_EQ_0; ARITH_EQ; ARITH_RULE `~(n + 2 <= SUC n)`] THEN CONJ_TAC THENL [REPEAT STRIP_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; MATCH_MP_TAC num_INDUCTION THEN CONV_TAC NUM_REDUCE_CONV THEN ASM_REWRITE_TAC[ARITH_RULE `1 <= SUC n`; NOT_SUC; LE_SUC] THEN CONJ_TAC THENL [ASM_MESON_TAC[DIST_SYM]; ALL_TAC] THEN X_GEN_TAC `i:num` THEN DISCH_THEN(K ALL_TAC) THEN SIMP_TAC[LE_SUC_LT; ARITH_RULE `SUC i < n + 2 <=> i = n \/ i < n`] THEN STRIP_TAC THEN ASM_SIMP_TAC[LT_IMP_LE; LE_REFL; LT_REFL; SUC_SUB1]]);; let CLOSED_UNIONS_COMPONENTS_MEETING_CLOSED = prove (`!s t:real^N->bool. compact s /\ closed t ==> closed (UNIONS {c | c IN components s /\ ~(c INTER t = {})})`, REPEAT STRIP_TAC THEN REWRITE_TAC[CLOSED_SEQUENTIAL_LIMITS] THEN MAP_EVERY X_GEN_TAC [`x:num->real^N`; `a:real^N`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE BINDER_CONV [IN_UNIONS]) THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM; IN_ELIM_THM] THEN X_GEN_TAC `c:num->real^N->bool` THEN REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE BINDER_CONV [GSYM MEMBER_NOT_EMPTY]) THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM; IN_INTER] THEN X_GEN_TAC `y:num->real^N` THEN REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC THEN SUBGOAL_THEN `(!n. (x:num->real^N) n IN s) /\ (!n. (y:num->real^N) n IN s)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[SUBSET; IN_COMPONENTS_SUBSET]; ALL_TAC] THEN SUBGOAL_THEN `(a:real^N) IN s` ASSUME_TAC THENL [ASM_MESON_TAC[CLOSED_SEQUENTIAL_LIMITS; COMPACT_IMP_CLOSED]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o SPEC `y:num->real^N` o REWRITE_RULE[compact]) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`b:real^N`; `r:num->num`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `(x:num->real^N) o (r:num->num)`; `(y:num->real^N) o (r:num->num)`; `a:real^N`; `b:real^N`] CONNECTED_COMPONENT_LIMIT) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_SIMP_TAC[LIM_SUBSEQUENCE] THEN MATCH_MP_TAC ALWAYS_EVENTUALLY THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[o_THM] THEN REWRITE_TAC[connected_component] THEN EXISTS_TAC `(c:num->real^N->bool)(r(n:num))` THEN ASM_MESON_TAC[IN_COMPONENTS_CONNECTED; IN_COMPONENTS_SUBSET]; DISCH_TAC THEN REWRITE_TAC[IN_UNIONS; IN_ELIM_THM] THEN EXISTS_TAC `connected_component s (a:real^N)` THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[IN; CONNECTED_COMPONENT_REFL]] THEN FIRST_ASSUM(SUBST1_TAC o SYM o MATCH_MP (REWRITE_RULE[IN] CONNECTED_COMPONENT_EQ)) THEN REWRITE_TAC[components; IN_ELIM_THM] THEN CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN EXISTS_TAC `b:real^N` THEN CONJ_TAC THENL [ASM_MESON_TAC[CONNECTED_COMPONENT_REFL; IN]; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CLOSED_SEQUENTIAL_LIMITS]) THEN EXISTS_TAC `(y:num->real^N) o (r:num->num)` THEN ASM_REWRITE_TAC[o_THM]]);; let ARBITRARILY_SMALL_CONTINUUM = prove (`!s u a:real^N. connected s /\ locally compact s /\ open u /\ {a} PSUBSET s /\ a IN u ==> ?c. {a} PSUBSET c /\ c SUBSET s /\ c SUBSET u /\ compact c /\ connected c`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `(a:real^N) IN s` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?b:real^N. b IN s /\ ~(b = a)` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [locally]) THEN DISCH_THEN(MP_TAC o SPECL [`s INTER (u DELETE (b:real^N))`; `a:real^N`]) THEN ASM_SIMP_TAC[OPEN_IN_OPEN_INTER; OPEN_DELETE; SUBSET_INTER] THEN ASM_REWRITE_TAC[IN_DELETE; IN_INTER; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`v:real^N->bool`; `k:real^N->bool`] THEN STRIP_TAC THEN EXISTS_TAC `connected_component k (a:real^N)` THEN REWRITE_TAC[CONNECTED_CONNECTED_COMPONENT] THEN REPEAT CONJ_TAC THENL [ALL_TAC; TRANS_TAC SUBSET_TRANS `k:real^N->bool` THEN REWRITE_TAC[CONNECTED_COMPONENT_SUBSET] THEN ASM SET_TAC[]; TRANS_TAC SUBSET_TRANS `k:real^N->bool` THEN REWRITE_TAC[CONNECTED_COMPONENT_SUBSET] THEN ASM SET_TAC[]; MATCH_MP_TAC COMPACT_CONNECTED_COMPONENT THEN ASM_REWRITE_TAC[]] THEN MATCH_MP_TAC(SET_RULE `a IN s /\ ~(s = {a}) ==> {a} PSUBSET s`) THEN REWRITE_TAC[IN] THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ] THEN CONJ_TAC THENL [ASM SET_TAC[]; DISCH_TAC] THEN MP_TAC(ISPECL [`k:real^N->bool`; `{a:real^N}`; `v:real^N->bool`] SURA_BURA_CLOPEN_SUBSET_ALT) THEN ASM_REWRITE_TAC[COMPACT_SING; SING_SUBSET; IN_INTER] THEN REWRITE_TAC[NOT_IMP; NOT_EXISTS_THM] THEN REPEAT CONJ_TAC THENL [ASM_SIMP_TAC[CLOSED_IMP_LOCALLY_COMPACT; COMPACT_IMP_CLOSED]; FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN REWRITE_TAC[components; IN_ELIM_THM] THEN EXISTS_TAC `a:real^N` THEN REWRITE_TAC[] THEN ASM SET_TAC[]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] OPEN_IN_SUBSET_TRANS)) THEN ASM SET_TAC[]; ALL_TAC] THEN X_GEN_TAC `c:real^N->bool` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONNECTED_CLOPEN]) THEN DISCH_THEN(MP_TAC o SPEC `c:real^N->bool`) THEN REWRITE_TAC[NOT_IMP; DE_MORGAN_THM; GSYM CONJ_ASSOC] THEN CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `v:real^N->bool` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] OPEN_IN_SUBSET_TRANS)) THEN ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [MATCH_MP_TAC CLOSED_SUBSET THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED]; ALL_TAC] THEN ASM SET_TAC[]);; let BOUNDARY_BUMPING_THEOREM_EUCLIDEAN_INTER = prove (`!s u c:real^N->bool. connected s /\ locally compact s /\ open u /\ ~(s SUBSET u) /\ compact(s INTER closure u) /\ c IN components(s INTER closure u) ==> ~(c INTER frontier u = {})`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s INTER closure u:real^N->bool`; `c:real^N->bool`; `s INTER frontier u:real^N->bool`] COMPACT_PARTITION_CONTAINING_CLOSED) THEN ASM_REWRITE_TAC[NOT_IMP] THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[CLOSED_COMPONENTS; COMPACT_IMP_CLOSED]; SUBGOAL_THEN `s INTER frontier u:real^N->bool = (s INTER closure u) INTER frontier u` (fun th -> ASM_SIMP_TAC[th; CLOSED_INTER; FRONTIER_CLOSED; COMPACT_IMP_CLOSED]) THEN REWRITE_TAC[frontier] THEN SET_TAC[]; ASM_SIMP_TAC[IN_COMPONENTS_SUBSET]; REWRITE_TAC[frontier] THEN SET_TAC[]; MP_TAC(ISPEC `s INTER closure u:real^N->bool` PAIRWISE_DISJOINT_COMPONENTS) THEN REWRITE_TAC[pairwise] THEN DISCH_THEN(MP_TAC o SPEC `c:real^N->bool`) THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `d:real^N->bool` THEN DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN ASM SET_TAC[]; REWRITE_TAC[NOT_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`k:real^N->bool`; `l:real^N->bool`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONNECTED_CLOSED_IN]) THEN REWRITE_TAC[] THEN MAP_EVERY EXISTS_TAC [`k:real^N->bool`; `l UNION (s DIFF closure u):real^N->bool`] THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CLOSED_SUBSET THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED] THEN ASM SET_TAC[]; SUBGOAL_THEN `l UNION s DIFF closure u:real^N->bool = s INTER (l UNION closure(s DIFF closure u))` (fun th -> ASM_SIMP_TAC[th; CLOSED_IN_CLOSED_INTER; CLOSED_UNION; COMPACT_IMP_CLOSED; CLOSED_CLOSURE]) THEN MP_TAC(ISPECL [`u:real^N->bool`; `s DIFF closure u:real^N->bool`] OPEN_INTER_CLOSURE_EQ_EMPTY) THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPEC `u:real^N->bool` CLOSURE_UNION_FRONTIER) THEN MP_TAC (ISPEC `s DIFF closure u:real^N->bool` CLOSURE_UNION_FRONTIER) THEN ASM SET_TAC[]; ASM SET_TAC[]; ASM SET_TAC[]; FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_NONEMPTY) THEN ASM SET_TAC[]; SUBGOAL_THEN `frontier u:real^N->bool = closure u DIFF u` SUBST_ALL_TAC THENL [ASM_SIMP_TAC[frontier; INTERIOR_OPEN]; ALL_TAC] THEN ASM SET_TAC[]]]);; let BOUNDARY_BUMPING_THEOREM_EUCLIDEAN_INTER_ALT = prove (`!s u c:real^N->bool. connected s /\ locally compact s /\ open u /\ ~(s INTER u = {}) /\ ~(s SUBSET u) /\ compact(s INTER closure u) /\ c IN components(s INTER u) ==> ?x. x IN frontier u /\ x limit_point_of c`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `~(closure c INTER frontier u:real^N->bool = {})` MP_TAC THENL [DISCH_TAC; REWRITE_TAC[closure] THEN MATCH_MP_TAC(SET_RULE `s INTER u = {} ==> ~((s UNION {x | P x}) INTER u = {}) ==> ?x. x IN u /\ P x`) THEN ASM_SIMP_TAC[frontier; INTERIOR_OPEN] THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN SET_TAC[]] THEN SUBGOAL_THEN `closed(c:real^N->bool)` ASSUME_TAC THENL [MATCH_MP_TAC CLOSED_IN_CLOSED_TRANS THEN EXISTS_TAC `s INTER closure u:real^N->bool` THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED] THEN FIRST_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_COMPONENT) THEN REWRITE_TAC[CLOSED_IN_LIMPT] THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [REWRITE_TAC[closure] THEN SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE (LAND_CONV o LAND_CONV) [closure]) THEN ASM_SIMP_TAC[frontier; INTERIOR_OPEN] THEN SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`c:real^N->bool`; `(:real^N) DIFF u`] SEPARATION_NORMAL) THEN ASM_REWRITE_TAC[GSYM OPEN_CLOSED; NOT_IMP] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN SET_TAC[]; REWRITE_TAC[NOT_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`h:real^N->bool`; `k:real^N->bool`] THEN STRIP_TAC THEN SUBGOAL_THEN `closure(h:real^N->bool) SUBSET u` ASSUME_TAC THENL [TRANS_TAC SUBSET_TRANS `(:real^N) DIFF k` THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN ASM_REWRITE_TAC[GSYM OPEN_CLOSED] THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`s INTER closure h:real^N->bool`; `c:real^N->bool`] EXISTS_COMPONENT_SUPERSET) THEN REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC] THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [FIRST_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_NONEMPTY) THEN FIRST_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN REWRITE_TAC[closure] THEN ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `c':real^N->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`s:real^N->bool`; `h:real^N->bool`; `c':real^N->bool`] BOUNDARY_BUMPING_THEOREM_EUCLIDEAN_INTER) THEN ASM_REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [SUBGOAL_THEN `s INTER closure(h:real^N->bool) = (s INTER closure u) INTER closure h` (fun th -> ASM_SIMP_TAC[COMPACT_INTER_CLOSED; th; CLOSED_CLOSURE]) THEN MP_TAC(ISPEC `u:real^N->bool` CLOSURE_SUBSET) THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `c':real^N->bool = c` SUBST_ALL_TAC THENL [ASM_REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN MATCH_MP_TAC COMPONENTS_MAXIMAL THEN EXISTS_TAC `s INTER u:real^N->bool` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]; ALL_TAC] THEN CONJ_TAC THENL [TRANS_TAC SUBSET_TRANS `s INTER closure h:real^N->bool` THEN ASM_SIMP_TAC[IN_COMPONENTS_SUBSET] THEN ASM SET_TAC[]; REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_NONEMPTY)) THEN ASM SET_TAC[]]; ASM_SIMP_TAC[frontier; INTERIOR_OPEN] THEN ASM SET_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Equivalence of LC and LPC for locally connected sets. *) (* ------------------------------------------------------------------------- *) let LOCALLY_COMPACT_CONNECTED_IMP_PATH_CONNECTED = prove (`!s:real^N->bool. locally compact s /\ locally connected s /\ connected s ==> path_connected s`, SUBGOAL_THEN `!s:real^N->bool. compact s /\ connected s /\ locally connected s ==> path_connected s` ASSUME_TAC THENL [ALL_TAC; REPEAT STRIP_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) PATH_CONNECTED_EQ_CONNECTED_LPC o snd) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN MATCH_MP_TAC LOCALLY_MONO THEN EXISTS_TAC `\c:real^N->bool. compact c /\ connected c /\ locally connected c` THEN ASM_SIMP_TAC[LOCALLY_CONNECTED_CONTINUUM]] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN STRIP_TAC THEN SUBGOAL_THEN `?f:real^1->real^N. f(vec 0) = a /\ f(vec 1) = b /\ (!x. x IN {lift(&m / &2 pow n) | 0 <= m /\ m <= 2 EXP n} ==> f x IN s) /\ f uniformly_continuous_on {lift(&m / &2 pow n) | 0 <= m /\ m <= 2 EXP n}` STRIP_ASSUME_TAC THENL [ALL_TAC; SUBGOAL_THEN `interval[vec 0:real^1,vec 1] INTER {inv(&2 pow n) % m | n,m | !i. 1 <= i /\ i <= dimindex(:1) ==> integer(m$i)} = {lift (&m / &2 pow n) | 0 <= m /\ m <= 2 EXP n}` ASSUME_TAC THENL [REWRITE_TAC[FORALL_1; DIMINDEX_1; SET_RULE `s INTER t = u <=> (!x. x IN t ==> x IN s ==> x IN u) /\ (!x. x IN u ==> x IN s /\ x IN t)`] THEN REWRITE_TAC[FORALL_IN_GSPEC; IN_INTERVAL_1; DROP_VEC; LIFT_DROP] THEN REWRITE_TAC[GSYM drop; FORALL_LIFT; LIFT_DROP; DROP_CMUL] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ; REAL_LT_POW2] THEN REWRITE_TAC[REAL_MUL_LZERO; REAL_MUL_LID; IMP_CONJ] THEN REWRITE_TAC[REAL_OF_NUM_POW; REAL_OF_NUM_LE; LE_0] THEN REWRITE_TAC[MESON[INTEGER_POS; REAL_POS] `(!m. integer m ==> &0 <= m ==> P m) <=> (!n. P(&n))`] THEN REWRITE_TAC[IN_ELIM_THM; ONCE_REWRITE_RULE[REAL_MUL_SYM] real_div] THEN REWRITE_TAC[REAL_OF_NUM_LE; LIFT_CMUL; EXISTS_LIFT; LIFT_DROP] THEN MESON_TAC[INTEGER_CLOSED]; ALL_TAC] THEN MP_TAC(ISPECL [`f:real^1->real^N`; `interval[vec 0:real^1,vec 1] INTER {inv(&2 pow n) % m | n,m | !i. 1 <= i /\ i <= dimindex(:1) ==> integer(m$i)}`] UNIFORMLY_CONTINUOUS_EXTENDS_TO_CLOSURE) THEN SIMP_TAC[CLOSURE_DYADIC_RATIONALS_IN_CONVEX_SET; CONVEX_INTERVAL; INTERIOR_INTERVAL; UNIT_INTERVAL_NONEMPTY] THEN ASM_REWRITE_TAC[CLOSURE_INTERVAL] THEN REWRITE_TAC[path_component] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^1->real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (ASSUME_TAC o CONJUNCT1)) THEN ASM_SIMP_TAC[path; UNIFORMLY_CONTINUOUS_IMP_CONTINUOUS] THEN CONJ_TAC THENL [MP_TAC(ISPECL [`g:real^1->real^N`; `interval[vec 0:real^1,vec 1] INTER {inv(&2 pow n) % m | n,m | !i. 1 <= i /\ i <= dimindex(:1) ==> integer(m$i)}`; `s:real^N->bool`] FORALL_IN_CLOSURE) THEN SIMP_TAC[CLOSURE_DYADIC_RATIONALS_IN_CONVEX_SET; CONVEX_INTERVAL; INTERIOR_INTERVAL; UNIT_INTERVAL_NONEMPTY] THEN ASM_REWRITE_TAC[CLOSURE_INTERVAL] THEN REWRITE_TAC[path_image; SUBSET; FORALL_IN_IMAGE] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_SIMP_TAC[UNIFORMLY_CONTINUOUS_IMP_CONTINUOUS] THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED] THEN ASM SET_TAC[]; CONJ_TAC THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [SYM th]) THEN REWRITE_TAC[pathstart; pathfinish] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[IN_ELIM_THM] THENL [EXISTS_TAC `0`; EXISTS_TAC `1`] THEN EXISTS_TAC `0` THEN CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[LIFT_NUM]]] THEN SUBGOAL_THEN `?f:real->real^N. f(&0) = a /\ f(&1) = b /\ (!m n. m <= 2 EXP n ==> f(&m / &2 pow n) IN s) /\ (!j. ?d. &0 < d /\ !n m1 m2. m1 <= 2 EXP n /\ m2 <= 2 EXP n /\ abs(&m1 / &2 pow n - &m2 / &2 pow n) < d ==> dist(f(&m1 / &2 pow n),f(&m2 / &2 pow n)) < inv(&2 pow j))` STRIP_ASSUME_TAC THENL [ALL_TAC; EXISTS_TAC `(f:real->real^N) o drop` THEN REWRITE_TAC[GSYM LIFT_NUM; uniformly_continuous_on; FORALL_IN_GSPEC; IMP_CONJ; RIGHT_FORALL_IMP_THM; DIST_LIFT; o_DEF] THEN ASM_REWRITE_TAC[LE_0; LIFT_DROP] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(ISPECL [`inv(&2)`; `e:real`] REAL_ARCH_POW_INV) THEN REWRITE_TAC[REAL_POW_INV] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `j:num` THEN DISCH_TAC THEN FIRST_X_ASSUM(fun th -> MP_TAC(SPEC `j:num` th) THEN MATCH_MP_TAC MONO_EXISTS) THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[RIGHT_IMP_FORALL_THM] THEN ONCE_REWRITE_TAC[MESON[] `(!a b c d. P a b c d) <=> (!b d a c. P a b c d)`] THEN MATCH_MP_TAC WLOG_LE THEN CONJ_TAC THENL [MESON_TAC[DIST_SYM; REAL_ABS_SUB]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`n1:num`; `n2:num`] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`m1:num`; `m2:num`] THEN REPEAT DISCH_TAC THEN TRANS_TAC REAL_LT_TRANS `inv(&2 pow j)` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`n2:num`; `m2:num`; `2 EXP (n2 - n1) * m1`]) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM REAL_OF_NUM_LE; GSYM REAL_OF_NUM_MUL] THEN REWRITE_TAC[GSYM REAL_OF_NUM_POW] THEN ASM_SIMP_TAC[REAL_POW_SUB; REAL_OF_NUM_EQ; ARITH_EQ] THEN SIMP_TAC[REAL_LT_POW2; REAL_FIELD `&0 < n1 /\ &0 < n2 ==> (n2 / n1 * m) / n2 = m / n1`] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[REAL_ARITH `a / b * c <= a <=> a * c / b <= a * &1`] THEN SIMP_TAC[REAL_LE_LMUL_EQ; REAL_LT_POW2; REAL_LE_LDIV_EQ] THEN REWRITE_TAC[REAL_MUL_LID; REAL_OF_NUM_POW] THEN ASM_REWRITE_TAC[REAL_OF_NUM_LE]] THEN SUBGOAL_THEN `?p f. (!n. p n < p(SUC n)) /\ f(&0):real^N = a /\ f(&1) = b /\ (!k1 i1 k2 i2. k1 <= k2 /\ i1 <= 2 EXP (p k1) /\ i2 <= 2 EXP (p k2) /\ abs(&i1 / &2 pow (p k1) - &i2 / &2 pow (p k2)) < inv(&2 pow (p k1)) ==> ?c. connected c /\ c SUBSET s /\ c SUBSET ball(f(&i1 / &2 pow (p k1)),&2 / &2 pow k1) /\ f(&i1 / &2 pow (p k1)) IN c /\ f(&i2 / &2 pow (p k2)) IN c)` MP_TAC THENL [ALL_TAC; DISCH_THEN(X_CHOOSE_THEN `r:num->num` MP_TAC) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real->real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN STRIP_TAC THEN MP_TAC(ISPEC `r:num->num` MONOTONE_BIGGER) THEN ANTS_TAC THENL [MATCH_MP_TAC TRANSITIVE_STEPWISE_LT THEN ASM_REWRITE_TAC[LT_TRANS]; DISCH_THEN(MP_TAC o SPEC `n:num`) THEN DISCH_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`n:num`; `2 EXP (r n - n) * m`; `n:num`; `2 EXP (r n - n) * m`]) THEN REWRITE_TAC[LE_REFL; REAL_SUB_REFL; REAL_ABS_NUM] THEN REWRITE_TAC[CONJ_ASSOC; REAL_LT_INV_EQ; REAL_LT_POW2] THEN REWRITE_TAC[GSYM REAL_OF_NUM_LE; GSYM REAL_OF_NUM_MUL] THEN REWRITE_TAC[GSYM REAL_OF_NUM_POW] THEN ASM_SIMP_TAC[REAL_POW_SUB; REAL_OF_NUM_EQ; ARITH_EQ] THEN REWRITE_TAC[REAL_ARITH `a / b * c <= a <=> a * c / b <= a * &1`] THEN SIMP_TAC[REAL_LE_LMUL_EQ; REAL_LT_POW2; REAL_LE_LDIV_EQ] THEN SIMP_TAC[REAL_LT_POW2; REAL_FIELD `&0 < n1 /\ &0 < n2 ==> (n2 / n1 * m) / n2 = m / n1`] THEN ASM_REWRITE_TAC[REAL_MUL_LID; REAL_OF_NUM_POW; REAL_OF_NUM_LE] THEN SET_TAC[]; X_GEN_TAC `j:num` THEN EXISTS_TAC `inv(&2 pow (r(j + 2)))` THEN REWRITE_TAC[REAL_LT_INV_EQ; REAL_LT_POW2] THEN X_GEN_TAC `n:num` THEN MATCH_MP_TAC WLOG_LT THEN REWRITE_TAC[DIST_REFL; REAL_LT_INV_EQ; REAL_LT_POW2] THEN CONJ_TAC THENL [MESON_TAC[DIST_SYM; REAL_ABS_SUB]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`m1:num`; `m2:num`] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `r(j + 2):num < n` ASSUME_TAC THENL [REWRITE_TAC[GSYM NOT_LE] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `inv(&2 pow n)` o MATCH_MP (REAL_ARITH `a < b ==> !x. x <= a ==> x < b`)) THEN REWRITE_TAC[REAL_NOT_LT; NOT_IMP] THEN ASM_SIMP_TAC[REAL_LE_INV2; REAL_LT_POW2; REAL_POW_MONO; REAL_OF_NUM_LE; ARITH] THEN REWRITE_TAC[real_div; GSYM REAL_SUB_RDISTRIB; REAL_ABS_MUL] THEN REWRITE_TAC[REAL_ABS_INV; REAL_ABS_POW; REAL_ABS_NUM] THEN GEN_REWRITE_TAC LAND_CONV [GSYM REAL_MUL_LID] THEN SIMP_TAC[REAL_LE_RMUL_EQ; REAL_LT_POW2; REAL_LT_INV_EQ] THEN MATCH_MP_TAC REAL_ABS_INTEGER_LEMMA THEN SIMP_TAC[INTEGER_CLOSED; REAL_SUB_0; REAL_OF_NUM_EQ] THEN ASM_ARITH_TAC; ALL_TAC] THEN MP_TAC(ISPEC `r:num->num` MONOTONE_BIGGER) THEN ANTS_TAC THENL [MATCH_MP_TAC TRANSITIVE_STEPWISE_LT THEN ASM_REWRITE_TAC[LT_TRANS]; DISCH_THEN(MP_TAC o SPEC `n:num`) THEN DISCH_TAC] THEN FIRST_X_ASSUM(MP_TAC o ISPECL [`j + 2`; `m2 DIV (2 EXP (n - r(j + 2)))`; `n:num`]) THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM; IMP_CONJ] THEN ANTS_TAC THENL [TRANS_TAC LE_TRANS `r(j + 2):num` THEN ASM_SIMP_TAC[LT_IMP_LE] THEN SPEC_TAC(`j + 2`,`i:num`) THEN MATCH_MP_TAC MONOTONE_BIGGER THEN MATCH_MP_TAC TRANSITIVE_STEPWISE_LT THEN ASM_REWRITE_TAC[LT_TRANS]; ALL_TAC] THEN ANTS_TAC THENL [SIMP_TAC[LE_LDIV_EQ; EXP_EQ_0; ARITH_EQ] THEN MATCH_MP_TAC (ARITH_RULE `~(b = 0) /\ a <= b * c ==> a < b * (c + 1)`) THEN REWRITE_TAC[EXP_EQ_0; ARITH_EQ] THEN ASM_SIMP_TAC[GSYM EXP_ADD; LT_IMP_LE; SUB_ADD]; ALL_TAC] THEN DISCH_THEN(fun th -> MP_TAC(SPEC `2 EXP (r n - n) * m1` th) THEN MP_TAC(SPEC `2 EXP (r n - n) * m2` th)) THEN REWRITE_TAC[GSYM REAL_OF_NUM_LE; GSYM REAL_OF_NUM_MUL] THEN REWRITE_TAC[GSYM REAL_OF_NUM_POW] THEN ASM_SIMP_TAC[REAL_POW_SUB; REAL_OF_NUM_EQ; ARITH_EQ] THEN REWRITE_TAC[REAL_ARITH `a / b * c <= a <=> a * c / b <= a * &1`] THEN SIMP_TAC[REAL_LE_LMUL_EQ; REAL_LT_POW2; REAL_LE_LDIV_EQ] THEN SIMP_TAC[REAL_LT_POW2; REAL_FIELD `&0 < n1 /\ &0 < n2 ==> (n2 / n1 * m) / n2 = m / n1`] THEN ASM_REWRITE_TAC[REAL_MUL_LID; REAL_OF_NUM_POW; REAL_OF_NUM_LE] THEN REWRITE_TAC[GSYM REAL_OF_NUM_POW] THEN MATCH_MP_TAC(TAUT `(q1 /\ q2 ==> r) /\ (p1 /\ p2) ==> (p1 ==> q1) ==> (p2 ==> q2) ==> r`) THEN CONJ_TAC THENL [DISCH_THEN(CONJUNCTS_THEN (MP_TAC o MATCH_MP (SET_RULE `(?c. P c /\ c SUBSET s /\ c SUBSET b /\ x IN c /\ y IN c) ==> y IN b`))) THEN REWRITE_TAC[IN_BALL; IMP_IMP] THEN MATCH_MP_TAC(NORM_ARITH `inv(&2) * j = i ==> dist(x:real^N,a) < i /\ dist(x,b) < i ==> dist(a,b) < j`) THEN REWRITE_TAC[REAL_POW_ADD; real_div; REAL_INV_MUL] THEN REAL_ARITH_TAC; MATCH_MP_TAC(REAL_ARITH `a < b /\ x <= b /\ b - i < x /\ abs(a - b) < i ==> abs(x - b) < i /\ abs(x - a) < i`) THEN ASM_SIMP_TAC[REAL_LT_DIV2_EQ; REAL_LT_POW2; REAL_OF_NUM_LT] THEN SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LT_RDIV_EQ; REAL_LT_POW2] THEN SIMP_TAC[REAL_LT_POW2; REAL_FIELD `&0 < n /\ &0 < j ==> (m / n - inv j) * j = m / (n / j) - &1`] THEN SIMP_TAC[REAL_LT_POW2; REAL_LT_SUB_RADD; REAL_FIELD `&0 < n /\ &0 < j ==> m / n * j = m / (n / j)`] THEN ASM_SIMP_TAC[GSYM REAL_POW_SUB; LT_IMP_LE; REAL_OF_NUM_EQ; ARITH] THEN SIMP_TAC[REAL_LE_RDIV_EQ; REAL_LT_LDIV_EQ; REAL_LT_POW2] THEN REWRITE_TAC[REAL_OF_NUM_POW; REAL_OF_NUM_MUL; REAL_OF_NUM_ADD] THEN REWRITE_TAC[REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ONCE_REWRITE_TAC[MULT_SYM] THEN REWRITE_TAC[DIV_MUL_LE] THEN W(MP_TAC o PART_MATCH (lhand o lhand o rand o lhand o rand) DIVISION o lhand o rand o rand o snd) THEN REWRITE_TAC[EXP_EQ_0; ARITH_EQ] THEN ARITH_TAC]]] THEN SUBGOAL_THEN `?p f. (!n. p n < p(SUC n)) /\ f(&0):real^N = a /\ f(&1) = b /\ (!n m k. m <= 2 EXP (p n) /\ k <= 2 EXP (p(SUC n)) /\ abs(&m / &2 pow (p n) - &k / &2 pow (p(SUC n))) < inv(&2 pow (p n)) ==> ?c. connected c /\ c SUBSET s /\ c SUBSET ball(f(&m / &2 pow (p n)),inv(&2 pow n)) /\ f(&m / &2 pow (p n)) IN c /\ f(&k / &2 pow (p(SUC n))) IN c)` MP_TAC THENL [ALL_TAC; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `r:num->num` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real->real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC BINDER_CONV [SWAP_FORALL_THM] THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN ONCE_REWRITE_TAC[MESON[LE_EXISTS] `(!m n:num. m <= n ==> P m n) <=> !n d. P n (n + d)`] THEN GEN_REWRITE_TAC I [SWAP_FORALL_THM] THEN MATCH_MP_TAC num_INDUCTION THEN CONJ_TAC THENL [MAP_EVERY X_GEN_TAC [`n:num`; `m1:num`; `m2:num`] THEN REWRITE_TAC[ADD_CLAUSES; real_div; GSYM REAL_SUB_RDISTRIB] THEN REWRITE_TAC[REAL_ABS_MUL; REAL_ABS_INV; REAL_ABS_POW; REAL_ABS_NUM] THEN SIMP_TAC[GSYM real_div; REAL_LT_LDIV_EQ; REAL_LT_POW2; REAL_MUL_LINV; REAL_LT_IMP_NZ] THEN SIMP_TAC[GSYM REAL_EQ_INTEGERS; INTEGER_CLOSED] THEN REWRITE_TAC[REAL_OF_NUM_EQ] THEN STRIP_TAC THEN EXISTS_TAC `{f(&m2 / &2 pow r(n:num)):real^N}` THEN ASM_REWRITE_TAC[SING_SUBSET; IN_SING; CENTRE_IN_BALL] THEN REWRITE_TAC[CONNECTED_SING] THEN SIMP_TAC[REAL_LT_DIV; REAL_POW_LT; REAL_OF_NUM_LT; ARITH] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`n:num`; `m2:num`; `2 EXP (r(SUC n) - r n) * m2`]) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ALL_TAC; SET_TAC[]] THEN REWRITE_TAC[GSYM REAL_OF_NUM_LE; GSYM REAL_OF_NUM_MUL] THEN REWRITE_TAC[GSYM REAL_OF_NUM_POW] THEN ASM_SIMP_TAC[REAL_POW_SUB; LT_IMP_LE; REAL_OF_NUM_EQ; ARITH_EQ] THEN SIMP_TAC[REAL_LT_POW2; REAL_FIELD `&0 < r /\ &0 < s ==> (r / s * m) / r = m / s`] THEN REWRITE_TAC[REAL_SUB_REFL; REAL_ABS_NUM; REAL_LT_INV_EQ] THEN REWRITE_TAC[REAL_LT_POW2] THEN ONCE_REWRITE_TAC[REAL_ARITH `a / b * c <= a <=> a * c / b <= a * &1`] THEN SIMP_TAC[REAL_LE_LMUL_EQ; REAL_LT_POW2; REAL_LE_LDIV_EQ] THEN REWRITE_TAC[REAL_MUL_LID] THEN ASM_REWRITE_TAC[REAL_OF_NUM_POW; REAL_OF_NUM_LE]; ALL_TAC] THEN X_GEN_TAC `d:num` THEN DISCH_THEN(LABEL_TAC "*") THEN MAP_EVERY X_GEN_TAC [`n:num`; `m1:num`; `m2:num`] THEN REWRITE_TAC[ADD_CLAUSES] THEN STRIP_TAC THEN SUBGOAL_THEN `?k. k <= 2 EXP r(SUC n) /\ abs(&m1 / &2 pow (r n) - &k / &2 pow r(SUC n)) < inv(&2 pow r n) /\ abs(&k / &2 pow r(SUC n) - &m2 / &2 pow r(SUC(n + d))) < inv (&2 pow r (SUC n))` STRIP_ASSUME_TAC THENL [ALL_TAC; REMOVE_THEN "*" (MP_TAC o SPECL [`SUC n`; `k:num`; `m2:num`]) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`n:num`; `m1:num`; `k:num`]) THEN ASM_REWRITE_TAC[ADD_CLAUSES; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `c1:real^N->bool` THEN STRIP_TAC THEN X_GEN_TAC `c2:real^N->bool` THEN STRIP_TAC THEN EXISTS_TAC `c1 UNION c2:real^N->bool` THEN ASM_REWRITE_TAC[UNION_SUBSET; IN_UNION] THEN CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_UNION THEN ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] SUBSET_TRANS)) THEN REWRITE_TAC[SUBSET_BALLS; DIST_REFL; REAL_ARITH `&0 + inv x <= &2 / x <=> &0 <= inv x`] THEN SIMP_TAC[REAL_LE_INV_EQ; REAL_LT_IMP_LE; REAL_LT_POW2] THEN DISJ1_TAC THEN REWRITE_TAC[real_pow; real_div; REAL_INV_MUL] THEN MATCH_MP_TAC(REAL_ARITH `x < y ==> x + &2 * inv(&2) * y <= &2 * y`) THEN ONCE_REWRITE_TAC[DIST_SYM] THEN REWRITE_TAC[GSYM IN_BALL] THEN REWRITE_TAC[GSYM real_div] THEN ASM SET_TAC[]] THEN SUBGOAL_THEN `!m n. m <= n ==> (r:num->num) m <= r n` ASSUME_TAC THENL [MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN ASM_SIMP_TAC[LT_IMP_LE] THEN ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC(MESON[] `(?k. P k \/ P(k + 1)) ==> ?k. P k`) THEN EXISTS_TAC `m2 DIV 2 EXP (r(SUC(n + d)) - r(SUC n))` THEN REWRITE_TAC[GSYM REAL_OF_NUM_LE; GSYM REAL_OF_NUM_ADD] THEN REWRITE_TAC[GSYM REAL_OF_NUM_POW] THEN ONCE_REWRITE_TAC[REAL_ARITH `x <= y <=> x <= &1 * y`] THEN SIMP_TAC[GSYM REAL_LE_LDIV_EQ; REAL_LT_POW2] THEN REWRITE_TAC[REAL_ARITH `(x + &1) / y = x / y + inv(y)`] THEN MATCH_MP_TAC(REAL_ARITH `x <= b /\ b < x + e /\ abs(a - b) < d /\ e <= d /\ a <= c /\ b <= c ==> x <= c /\ abs(a - x) < d /\ abs(x - b) < e \/ x + e <= c /\ abs(a - (x + e)) < d /\ abs((x + e) - b) < e`) THEN ASM_SIMP_TAC[REAL_LE_INV2; REAL_POW_MONO; REAL_OF_NUM_LE; ARITH; LT_IMP_LE; REAL_LT_POW2] THEN SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LT_POW2; REAL_MUL_LID] THEN ASM_REWRITE_TAC[REAL_OF_NUM_POW; REAL_OF_NUM_LE] THEN REWRITE_TAC[REAL_ARITH `x / y + inv y = (x + &1) / y`] THEN SIMP_TAC[REAL_LT_RDIV_EQ; REAL_LT_POW2; GSYM REAL_OF_NUM_POW] THEN SIMP_TAC[REAL_LT_POW2; REAL_FIELD `&0 < m /\ &0 < n ==> x / m * n = x / (m / n)`] THEN ASM_SIMP_TAC[GSYM REAL_POW_SUB; REAL_OF_NUM_EQ; ARITH_EQ; ARITH_RULE `SUC n <= SUC(n + d)`] THEN SIMP_TAC[REAL_LE_RDIV_EQ; REAL_LT_LDIV_EQ; REAL_LT_POW2] THEN REWRITE_TAC[REAL_OF_NUM_POW; REAL_OF_NUM_MUL; REAL_OF_NUM_ADD] THEN REWRITE_TAC[REAL_OF_NUM_LT; REAL_OF_NUM_LE] THEN W(MP_TAC o PART_MATCH (lhand o lhand o rand o lhand o rand) DIVISION o lhand o lhand o lhand o snd) THEN REWRITE_TAC[EXP_EQ_0; ARITH_EQ] THEN ARITH_TAC] THEN SUBGOAL_THEN `?p f. (!n. p n < p(SUC n)) /\ (!n. f n 0 = (a:real^N)) /\ (!n. f n (2 EXP (p n)) = b) /\ (!n k. k <= 2 EXP (p n) ==> f (SUC n) (2 EXP (p(SUC n) - p n) * k) = f n k) /\ (!n m k. m <= 2 EXP (p n) /\ k <= 2 EXP (p(SUC n)) /\ abs(&m / &2 pow (p n) - &k / &2 pow (p(SUC n))) < inv(&2 pow (p n)) ==> ?c. connected c /\ c SUBSET s /\ c SUBSET ball(f n m,inv(&2 pow n)) /\ f n m IN c /\ f (SUC n) k IN c)` MP_TAC THENL [ALL_TAC; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `r:num->num` THEN DISCH_THEN(X_CHOOSE_THEN `f:num->num->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\x. let t = @t. &(SND t) / &2 pow (r(FST t)) = x in (f:num->num->real^N) (FST t) (SND t)` THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [LET_TAC THEN SUBGOAL_THEN `&(SND t) / &2 pow r(FST t:num) = &0` MP_TAC THENL [EXPAND_TAC "t" THEN CONV_TAC SELECT_CONV THEN EXISTS_TAC `0,0` THEN REWRITE_TAC[real_div; REAL_MUL_LZERO]; REWRITE_TAC[REAL_DIV_EQ_0; REAL_POW_EQ_0] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_SIMP_TAC[REAL_OF_NUM_EQ]]; LET_TAC THEN SUBGOAL_THEN `&(SND t) / &2 pow r(FST t:num) = &1` MP_TAC THENL [EXPAND_TAC "t" THEN CONV_TAC SELECT_CONV THEN EXISTS_TAC `0,2 EXP r 0` THEN SIMP_TAC[GSYM REAL_OF_NUM_POW; REAL_DIV_REFL; REAL_POW_EQ_0; REAL_OF_NUM_EQ; ARITH_EQ]; SIMP_TAC[REAL_EQ_LDIV_EQ; REAL_LT_POW2; REAL_MUL_LID] THEN ASM_SIMP_TAC[REAL_OF_NUM_POW; REAL_OF_NUM_EQ]]; MAP_EVERY X_GEN_TAC [`n:num`; `m:num`; `k:num`] THEN STRIP_TAC THEN ABBREV_TAC `t = @t. &(SND t) / &2 pow r (FST t:num) = &m / &2 pow r n` THEN ABBREV_TAC `u = @t. &(SND t) / &2 pow r (FST t) = &k / &2 pow r(SUC n)` THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN SUBGOAL_THEN `(f:num->num->real^N) (FST t) (SND t) = f n m /\ (f:num->num->real^N) (FST u) (SND u) = f (SUC n) k` (fun th -> ASM_SIMP_TAC[th]) THEN SUBGOAL_THEN `!n n' m m'. &m / &2 pow (r n) = &m' / &2 pow (r n') /\ m' <= 2 EXP (r n') ==> (f:num->num->real^N) n m = f n' m'` (fun th -> CONJ_TAC THEN MATCH_MP_TAC th THEN ASM_REWRITE_TAC[] THEN MAP_EVERY EXPAND_TAC ["t"; "u"] THEN CONV_TAC SELECT_CONV THEN REWRITE_TAC[EXISTS_PAIR_THM] THEN MESON_TAC[]) THEN REWRITE_TAC[GSYM REAL_OF_NUM_LE; GSYM REAL_OF_NUM_POW] THEN ONCE_REWRITE_TAC[REAL_ARITH `x <= y <=> x <= &1 * y`] THEN SIMP_TAC[GSYM REAL_LE_LDIV_EQ; REAL_LT_POW2] THEN MATCH_MP_TAC WLOG_LE THEN CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN REPEAT(FIRST_X_ASSUM(K ALL_TAC o check (fun t -> not(is_forall t)) o concl)) THEN X_GEN_TAC `n:num` THEN MATCH_MP_TAC num_INDUCTION THEN CONJ_TAC THENL [SIMP_TAC[LE] THEN DISCH_THEN SUBST1_TAC THEN SIMP_TAC[REAL_OF_NUM_EQ; REAL_LT_POW2; REAL_FIELD `&0 < z ==> (x / z = y / z <=> x = y)`]; ALL_TAC] THEN X_GEN_TAC `p:num` THEN ASM_CASES_TAC `SUC p = n` THEN ASM_SIMP_TAC[REAL_OF_NUM_EQ; REAL_LT_POW2; REAL_FIELD `&0 < z ==> (x / z = y / z <=> x = y)`] THEN ASM_CASES_TAC `n <= SUC p` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `n:num <= p` THENL [ALL_TAC; ASM_ARITH_TAC] THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(LABEL_TAC "*") THEN MAP_EVERY X_GEN_TAC [`m1:num`; `m2:num`] THEN SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LT_POW2; REAL_MUL_LID] THEN SUBGOAL_THEN `!m n. m <= n ==> (r:num->num) m <= r n` ASSUME_TAC THENL [MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN ASM_SIMP_TAC[LT_IMP_LE] THEN ARITH_TAC; ALL_TAC] THEN SIMP_TAC[REAL_LT_POW2; REAL_FIELD `&0 < m /\ &0 < n ==> (x / m = y / n <=> y = (n / m) * x)`] THEN ASM_SIMP_TAC[GSYM REAL_POW_SUB; REAL_OF_NUM_EQ; ARITH_EQ; LT_IMP_LE] THEN REWRITE_TAC[REAL_OF_NUM_POW; REAL_OF_NUM_MUL; REAL_OF_NUM_EQ] THEN ASM_SIMP_TAC[REAL_OF_NUM_LE] THEN STRIP_TAC THEN SUBGOAL_THEN `r(SUC p) - r n:num = (r(SUC p) - r p) + (r p - r n)` SUBST1_TAC THENL [MATCH_MP_TAC(ARITH_RULE `x <= y /\ y <= z ==> z - x:num = (z - y) + (y - x)`) THEN CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; REWRITE_TAC[EXP_ADD; GSYM MULT_ASSOC]] THEN CONV_TAC SYM_CONV THEN TRANS_TAC EQ_TRANS `(f:num->num->real^N) p (2 EXP (r p - r(n:num)) * m1)` THEN CONJ_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC; CONV_TAC SYM_CONV THEN FIRST_X_ASSUM MATCH_MP_TAC THEN SIMP_TAC[REAL_EQ_RDIV_EQ; REAL_LE_LDIV_EQ; REAL_LT_POW2] THEN ONCE_REWRITE_TAC[REAL_ARITH `m / x * y:real = (y / x) * m`] THEN ASM_SIMP_TAC[GSYM REAL_POW_SUB; REAL_OF_NUM_EQ; ARITH_EQ] THEN REWRITE_TAC[REAL_OF_NUM_MUL; REAL_OF_NUM_POW; MULT_CLAUSES]] THEN UNDISCH_TAC `m2 <= 2 EXP r (SUC p)` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM REAL_OF_NUM_LE; GSYM REAL_OF_NUM_MUL] THEN REWRITE_TAC[GSYM REAL_OF_NUM_POW] THEN ASM_SIMP_TAC[REAL_POW_SUB; REAL_OF_NUM_EQ; ARITH_EQ] THEN REWRITE_TAC[REAL_ARITH `x / y * z <= x <=> x * (z / y) <= x * &1`] THEN SIMP_TAC[REAL_LE_LMUL_EQ; REAL_LT_POW2]]] THEN SUBGOAL_THEN `?t. (!n. SND(t n) 0 = (a:real^N) /\ (!i. 2 EXP (FST(t n)) <= i ==> SND(t n) i = b) /\ !m. m <= 2 EXP (FST(t n)) ==> ?c. connected c /\ c SUBSET s /\ c SUBSET ball(SND(t n) m,inv(&2 pow n)) /\ c SUBSET ball(SND(t n) (SUC m),inv(&2 pow n)) /\ SND(t n) m IN c /\ SND(t n) (SUC m) IN c) /\ (!n. FST(t n) < FST(t(SUC n)) /\ (!k. k <= 2 EXP (FST(t n)) ==> SND(t(SUC n)) (2 EXP (FST(t(SUC n)) - FST(t n)) * k) = SND(t n) k) /\ (!m k. m <= 2 EXP (FST(t n)) /\ k <= 2 EXP (FST(t(SUC n))) /\ abs(&m / &2 pow (FST(t n)) - &k / &2 pow (FST(t(SUC n)))) < inv(&2 pow (FST(t n))) ==> ?c. connected c /\ c SUBSET s /\ c SUBSET ball(SND(t n) m,inv(&2 pow n)) /\ SND(t n) m IN c /\ SND(t(SUC n)) k IN c))` MP_TAC THENL [MATCH_MP_TAC DEPENDENT_CHOICE THEN REWRITE_TAC[EXISTS_PAIR_THM; FORALL_PAIR_THM]; DISCH_THEN(X_CHOOSE_THEN `t:num->num#(num->real^N)` STRIP_ASSUME_TAC) THEN EXISTS_TAC `FST o (t:num->num#(num->real^N))` THEN EXISTS_TAC `SND o (t:num->num#(num->real^N))` THEN ASM_REWRITE_TAC[o_THM] THEN ASM_SIMP_TAC[LE_REFL]] THEN CONJ_TAC THENL [MP_TAC(ISPEC `s:real^N->bool` COMPACT_LOCALLY_CONNECTED_IMP_ULC_ALT) THEN ASM_REWRITE_TAC[] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN DISCH_THEN(MP_TAC o SPEC `&1`) THEN REWRITE_TAC[REAL_LT_01] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`s:real^N->bool`; `d:real`; `a:real^N`; `b:real^N`] CONNECTED_IMP_WELLCHAINED) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `g:num->real^N` MP_TAC) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `l:num` THEN STRIP_TAC THEN EXISTS_TAC `\i. if i <= l then (g:num->real^N) i else b:real^N` THEN ASM_REWRITE_TAC[LE_0] THEN CONJ_TAC THENL [X_GEN_TAC `i:num` THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPEC `l:num` LT_POW2_REFL) THEN ASM_ARITH_TAC; ALL_TAC] THEN X_GEN_TAC `i:num` THEN DISCH_TAC THEN ASM_CASES_TAC `l:num <= i` THENL [ASM_SIMP_TAC[ARITH_RULE `l <= i ==> ~(SUC i <= l)`] THEN ASM_SIMP_TAC[ARITH_RULE `l:num <= i ==> (i <= l <=> i = l)`] THEN EXISTS_TAC `{b:real^N}` THEN REWRITE_TAC[COND_ID; CONNECTED_SING; IN_SING; SING_SUBSET] THEN REWRITE_TAC[CENTRE_IN_BALL; REAL_LT_01] THEN ASM SET_TAC[]; RULE_ASSUM_TAC(REWRITE_RULE[NOT_LE])] THEN ASM_SIMP_TAC[LE_SUC_LT; LT_IMP_LE] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`(g:num->real^N) i`; `g(SUC i):real^N`]) THEN ANTS_TAC THENL [ASM_SIMP_TAC[LE_SUC_LT; LT_IMP_LE]; MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[SUBSET_INTER; CONJ_ACI]]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`n:num`; `m:num`; `f:num->real^N`] THEN STRIP_TAC THEN SUBGOAL_THEN `!r. r <= 2 EXP m ==> ?l g:num->real^N. g 0 = f r /\ (!i. l <= i ==> g i = f (SUC r)) /\ (?c. connected c /\ f r IN c /\ f(SUC r) IN c /\ c SUBSET s /\ c SUBSET ball(f r,inv (&2 pow n)) /\ c SUBSET ball(f(SUC r),inv (&2 pow n)) /\ !i. g i IN c) /\ (!i. ?c. connected c /\ c SUBSET s /\ c SUBSET ball(g i,inv(&2 pow (SUC n))) /\ c SUBSET ball(g(SUC i),inv(&2 pow (SUC n))) /\ g i IN c /\ g(SUC i) IN c)` MP_TAC THENL [X_GEN_TAC `r:num` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `r:num`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPEC `s:real^N->bool` COMPACT_LOCALLY_CONNECTED_IMP_ULC_ALT) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `inv(&2 pow (SUC n))`) THEN REWRITE_TAC[REAL_LT_INV_EQ; REAL_LT_POW2] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`c:real^N->bool`; `d:real`; `(f:num->real^N) r`; `(f:num->real^N) (SUC r)`] CONNECTED_IMP_WELLCHAINED) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `g:num->real^N` MP_TAC) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `l:num` THEN STRIP_TAC THEN EXISTS_TAC `\i. if i <= l then (g:num->real^N) i else f(SUC r)` THEN ASM_REWRITE_TAC[LE_0] THEN CONJ_TAC THENL [ASM_MESON_TAC[LE_ANTISYM]; ALL_TAC] THEN CONJ_TAC THENL [EXISTS_TAC `c:real^N->bool` THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]; ALL_TAC] THEN X_GEN_TAC `i:num` THEN ASM_CASES_TAC `l:num <= i` THENL [ASM_SIMP_TAC[ARITH_RULE `l <= i ==> ~(SUC i <= l)`] THEN ASM_SIMP_TAC[ARITH_RULE `l:num <= i ==> (i <= l <=> i = l)`] THEN EXISTS_TAC `{f(SUC r):real^N}` THEN REWRITE_TAC[COND_ID; CONNECTED_SING; IN_SING; SING_SUBSET] THEN REWRITE_TAC[CENTRE_IN_BALL; REAL_LT_INV_EQ; REAL_LT_POW2] THEN ASM SET_TAC[]; RULE_ASSUM_TAC(REWRITE_RULE[NOT_LE])] THEN ASM_SIMP_TAC[LE_SUC_LT; LT_IMP_LE] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`(g:num->real^N) i`; `g(SUC i):real^N`]) THEN ANTS_TAC THENL [ALL_TAC; REWRITE_TAC[SUBSET_INTER; CONJ_ACI]] THEN REPEAT(FIRST_X_ASSUM(fun th -> MP_TAC(SPEC `i:num` th) THEN MP_TAC(SPEC `SUC i` th))) THEN ASM_SIMP_TAC[LE_SUC_LT; LT_IMP_LE] THEN ASM SET_TAC[]; GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM]] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`l:num->num`; `g:num->num->real^N`] THEN DISCH_THEN(LABEL_TAC "*") THEN MP_TAC(ISPECL [`\n:num. n`; `2 EXP 1 INSERT IMAGE (l:num->num) {r | r <= 2 EXP m}`] UPPER_BOUND_FINITE_SET) THEN SIMP_TAC[FINITE_INSERT; FINITE_IMAGE; FINITE_NUMSEG_LE] THEN DISCH_THEN(X_CHOOSE_THEN `p:num` (MP_TAC o MATCH_MP (MESON[LE_TRANS; LT_LE] `(!x. x IN s ==> x <= p) ==> p < 2 EXP p ==> (!x. x IN s ==> x <= 2 EXP p)`))) THEN ANTS_TAC THEN REWRITE_TAC[LT_POW2_REFL] THEN REWRITE_TAC[FORALL_IN_INSERT; LE_EXP] THEN CONV_TAC NUM_REDUCE_CONV THEN REWRITE_TAC[FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN STRIP_TAC THEN EXISTS_TAC `m + p:num` THEN EXISTS_TAC `\i. if i <= 2 EXP (m + p) then (g:num->num->real^N) (i DIV (2 EXP p)) (i MOD (2 EXP p)) else b` THEN ASM_REWRITE_TAC[ARITH_RULE `m < m + p <=> 1 <= p`] THEN REWRITE_TAC[ADD_SUB2] THEN SIMP_TAC[] THEN SIMP_TAC[DIV_MULT; EXP_EQ_0; ARITH_EQ; EXP_ADD; MOD_MULT; ONCE_REWRITE_RULE[MULT_SYM] DIV_MULT; ONCE_REWRITE_RULE[MULT_SYM] MOD_MULT] THEN REPEAT CONJ_TAC THENL [SIMP_TAC[LE_0; DIV_0; MOD_0; EXP_EQ_0; ARITH_EQ] THEN ASM_SIMP_TAC[LE_0]; SIMP_TAC[ARITH_RULE `m:num <= n ==> (n <= m <=> n = m)`] THEN SIMP_TAC[ONCE_REWRITE_RULE[MULT_SYM] DIV_MULT; ONCE_REWRITE_RULE[MULT_SYM] MOD_MULT; EXP_EQ_0; ARITH_EQ] THEN ASM_SIMP_TAC[LE_REFL; COND_ID]; ALL_TAC; ASM_SIMP_TAC[] THEN ONCE_REWRITE_TAC[ARITH_RULE `p * k:num <= m * p <=> p * k <= p * m`] THEN SIMP_TAC[LE_MULT_LCANCEL]; MAP_EVERY X_GEN_TAC [`r:num`; `k:num`] THEN MP_TAC(ISPECL [`k:num`; `2 EXP p`] DIVISION) THEN MAP_EVERY ABBREV_TAC [`k1 = k DIV 2 EXP p`; `k2 = k MOD 2 EXP p`] THEN REWRITE_TAC[EXP_EQ_0; ARITH_EQ] THEN DISCH_THEN(CONJUNCTS_THEN2 SUBST1_TAC ASSUME_TAC) THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (ARITH_RULE `k * p + k':num <= m ==> k * p <= m`)) THEN REWRITE_TAC[LE_MULT_RCANCEL; EXP_EQ_0; ARITH_EQ] THEN DISCH_TAC THEN REMOVE_THEN "*" (MP_TAC o SPEC `k1:num`) THEN ASM_REWRITE_TAC[] THEN REPLICATE_TAC 2 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(MP_TAC o CONJUNCT1) THEN MATCH_MP_TAC MONO_EXISTS THEN SUBGOAL_THEN `r = k1 \/ r = SUC k1` MP_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE LAND_CONV [REAL_ABS_SUB]) THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM REAL_MUL_LID] THEN REWRITE_TAC[GSYM real_div; GSYM REAL_OF_NUM_ADD; GSYM REAL_OF_NUM_MUL] THEN SIMP_TAC[REAL_LT_RDIV_EQ; REAL_LT_POW2; GSYM REAL_OF_NUM_POW] THEN SIMP_TAC[REAL_POW_ADD; REAL_LT_POW2; REAL_FIELD `&0 < m /\ &0 < p ==> (k1 * p + k2) / (m * p) - r / m = ((k1 - r) * p + k2) / p / m`] THEN REWRITE_TAC[REAL_ABS_DIV; REAL_ABS_POW; REAL_ABS_NUM] THEN SIMP_TAC[REAL_DIV_RMUL; REAL_LT_IMP_NZ; REAL_LT_POW2] THEN SIMP_TAC[REAL_LT_LDIV_EQ; REAL_MUL_LID; REAL_LT_POW2] THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `abs(r + k2) < p ==> &0 <= k2 /\ k2 < p ==> -- &2 * p < r /\ r < &1 * p`)) THEN SIMP_TAC[REAL_LT_RMUL_EQ; REAL_LT_POW2] THEN ASM_REWRITE_TAC[REAL_POS; REAL_OF_NUM_POW; REAL_OF_NUM_LT] THEN SIMP_TAC[REAL_LT_INTEGERS; INTEGER_CLOSED; REAL_POS] THEN REWRITE_TAC[REAL_ARITH `-- &2 + &1:real <= k - r <=> r <= k + &1`; REAL_ARITH `k - r + &1:real <= &1 <=> k <= r`] THEN REWRITE_TAC[REAL_OF_NUM_LE; REAL_OF_NUM_ADD] THEN ARITH_TAC] THEN X_GEN_TAC `k:num` THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL [UNDISCH_TAC `SUC k <= 2 EXP m * 2 EXP p`; DISCH_THEN(MP_TAC o MATCH_MP (ARITH_RULE `k <= n ==> ~(SUC k <= n) ==> k = n`)) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN SIMP_TAC[ONCE_REWRITE_RULE[MULT_SYM] DIV_MULT; ONCE_REWRITE_RULE[MULT_SYM] MOD_MULT; EXP_EQ_0; ARITH_EQ] THEN ASM_SIMP_TAC[LE_REFL] THEN EXISTS_TAC `{b:real^N}` THEN ASM_REWRITE_TAC[CONNECTED_SING; IN_SING; SING_SUBSET] THEN REWRITE_TAC[CENTRE_IN_BALL; REAL_LT_INV_EQ; REAL_LT_POW2]] THEN MP_TAC(ISPECL [`k:num`; `2 EXP p`] DIVISION) THEN MAP_EVERY ABBREV_TAC [`k1 = k DIV 2 EXP p`; `k2 = k MOD 2 EXP p`] THEN REWRITE_TAC[EXP_EQ_0; ARITH_EQ] THEN DISCH_THEN(CONJUNCTS_THEN2 SUBST1_TAC ASSUME_TAC) THEN REWRITE_TAC[LE_SUC_LT] THEN REPEAT DISCH_TAC THEN REWRITE_TAC[ARITH_RULE `SUC(a * b + c) = a * b + SUC c`] THEN SIMP_TAC[DIV_MULT_ADD; MOD_MULT_ADD; EXP_EQ_0; ARITH_EQ] THEN FIRST_ASSUM(MP_TAC o MATCH_MP (ARITH_RULE `a + b:num < c ==> a < c`)) THEN REWRITE_TAC[LT_MULT_RCANCEL; EXP_EQ_0; ARITH_EQ] THEN DISCH_TAC THEN SUBGOAL_THEN `SUC k2 <= 2 EXP p` MP_TAC THENL [ASM_REWRITE_TAC[LE_SUC_LT]; REWRITE_TAC[LE_LT]] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THENL [ASM_SIMP_TAC[DIV_LT; MOD_LT; ADD_CLAUSES] THEN REMOVE_THEN "*" (MP_TAC o SPEC `k1:num`) THEN ASM_SIMP_TAC[LT_IMP_LE] THEN MESON_TAC[]; ASM_SIMP_TAC[DIV_REFL; MOD_REFL; EXP_EQ_0; ARITH_EQ] THEN ASM_SIMP_TAC[ARITH_RULE `a < b ==> a + 1 <= b`] THEN REMOVE_THEN "*" (MP_TAC o SPEC `k1:num`) THEN ASM_SIMP_TAC[LT_IMP_LE] THEN REPLICATE_TAC 2 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(MP_TAC o CONJUNCT2) THEN REWRITE_TAC[GSYM SKOLEM_THM] THEN DISCH_THEN(MP_TAC o SPEC `k2:num`) THEN REWRITE_TAC[GSYM ADD1] THEN SUBGOAL_THEN `(g:num->num->real^N) k1 (SUC k2) = f(SUC k1)` (fun th -> REWRITE_TAC[th]) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[LT_IMP_LE]]);; let LOCALLY_COMPACT_PATH_CONNECTED_EQ_CONNECTED = prove (`!s:real^N->bool. locally compact s /\ locally connected s ==> (path_connected s <=> connected s)`, MESON_TAC[LOCALLY_COMPACT_CONNECTED_IMP_PATH_CONNECTED; PATH_CONNECTED_IMP_CONNECTED]);; let LOCALLY_COMPACT_LOCALLY_CONNECTED_IMP_LOCALLY_PATH_CONNECTED = prove (`!s:real^N->bool. locally compact s /\ locally connected s ==> locally path_connected s`, REPEAT STRIP_TAC THEN MP_TAC(ASSUME `locally connected (s:real^N->bool)`) THEN REWRITE_TAC[LOCALLY_CONNECTED; LOCALLY_PATH_CONNECTED] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `v:real^N->bool` THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `a:real^N` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC LOCALLY_COMPACT_CONNECTED_IMP_PATH_CONNECTED THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THEN MATCH_MP_TAC LOCALLY_OPEN_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[]);; let LOCALLY_COMPACT_LOCALLY_PATH_CONNECTED_EQ_LOCALLY_CONNECTED = prove (`!s:real^N->bool. locally compact s ==> (locally path_connected s <=> locally connected s)`, MESON_TAC[LOCALLY_COMPACT_LOCALLY_CONNECTED_IMP_LOCALLY_PATH_CONNECTED; LOCALLY_PATH_CONNECTED_IMP_LOCALLY_CONNECTED]);; let LOCALLY_PATH_CONNECTED_CLOSURE_FROM_FRONTIER = prove (`!s:real^N->bool. locally connected (frontier s) ==> locally path_connected (closure s)`, SIMP_TAC[LOCALLY_COMPACT_LOCALLY_PATH_CONNECTED_EQ_LOCALLY_CONNECTED; CLOSED_IMP_LOCALLY_COMPACT; CLOSED_CLOSURE; FRONTIER_CLOSED] THEN REWRITE_TAC[LOCALLY_CONNECTED_CLOSURE_FROM_FRONTIER]);; (* ------------------------------------------------------------------------- *) (* If two points are separated by a closed set, there's a minimal one. *) (* ------------------------------------------------------------------------- *) let CLOSED_IRREDUCIBLE_SEPARATOR = prove (`!s a b:real^N. closed s /\ ~connected_component ((:real^N) DIFF s) a b ==> ?t. t SUBSET s /\ closed t /\ ~(t = {}) /\ ~connected_component ((:real^N) DIFF t) a b /\ !u. u PSUBSET t ==> connected_component ((:real^N) DIFF u) a b`, MAP_EVERY X_GEN_TAC [`c:real^N->bool`; `a:real^N`; `b:real^N`] THEN STRIP_TAC THEN ASM_CASES_TAC `(a:real^N) IN c` THENL [EXISTS_TAC `{a:real^N}` THEN ASM_REWRITE_TAC[CLOSED_SING; SING_SUBSET] THEN SIMP_TAC[SET_RULE `s PSUBSET {a} <=> s = {}`; NOT_INSERT_EMPTY] THEN REWRITE_TAC[DIFF_EMPTY; CONNECTED_COMPONENT_UNIV] THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[UNIV]] THEN DISCH_THEN(MP_TAC o MATCH_MP CONNECTED_COMPONENT_IN) THEN SET_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `(b:real^N) IN c` THENL [EXISTS_TAC `{b:real^N}` THEN ASM_REWRITE_TAC[CLOSED_SING; SING_SUBSET] THEN SIMP_TAC[SET_RULE `s PSUBSET {a} <=> s = {}`; NOT_INSERT_EMPTY] THEN REWRITE_TAC[DIFF_EMPTY; CONNECTED_COMPONENT_UNIV] THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[UNIV]] THEN DISCH_THEN(MP_TAC o MATCH_MP CONNECTED_COMPONENT_IN) THEN SET_TAC[]; ALL_TAC] THEN MAP_EVERY ABBREV_TAC [`r = connected_component ((:real^N) DIFF c) a`; `s = connected_component ((:real^N) DIFF closure r) b`] THEN EXISTS_TAC `frontier s:real^N->bool` THEN REWRITE_TAC[FRONTIER_CLOSED] THEN SUBGOAL_THEN `(a:real^N) IN r` ASSUME_TAC THENL [EXPAND_TAC "r" THEN REWRITE_TAC[IN] THEN REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(b:real^N) IN s` ASSUME_TAC THENL [EXPAND_TAC "s" THEN REWRITE_TAC[IN] THEN REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ] THEN ASM_REWRITE_TAC[IN_UNIV; IN_DIFF] THEN REWRITE_TAC[CLOSURE_UNION_FRONTIER; IN_UNION; DE_MORGAN_THM] THEN CONJ_TAC THENL [ASM_REWRITE_TAC[IN]; EXPAND_TAC "r"] THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[SUBSET] FRONTIER_OF_CONNECTED_COMPONENT_SUBSET)) THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `~(b IN s) ==> t SUBSET s ==> b IN t ==> F`)) THEN ASM_REWRITE_TAC[FRONTIER_COMPLEMENT; FRONTIER_SUBSET_EQ]; ALL_TAC] THEN SUBGOAL_THEN `frontier(s:real^N->bool) SUBSET frontier r` ASSUME_TAC THENL [EXPAND_TAC "s" THEN MATCH_MP_TAC(MESON[SUBSET_TRANS; FRONTIER_OF_CONNECTED_COMPONENT_SUBSET] `frontier s SUBSET t ==> frontier(connected_component s a) SUBSET t`) THEN REWRITE_TAC[FRONTIER_COMPLEMENT; FRONTIER_CLOSURE_SUBSET]; ALL_TAC] THEN MATCH_MP_TAC(TAUT `(q ==> r) /\ p /\ ~r /\ s ==> p /\ ~q /\ ~r /\ s`) THEN CONJ_TAC THENL [SIMP_TAC[DIFF_EMPTY; CONNECTED_COMPONENT_UNIV] THEN REWRITE_TAC[UNIV]; ALL_TAC] THEN REPEAT CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] SUBSET_TRANS)) THEN EXPAND_TAC "r" THEN MATCH_MP_TAC(MESON[SUBSET_TRANS; FRONTIER_OF_CONNECTED_COMPONENT_SUBSET] `frontier s SUBSET t ==>frontier (connected_component s a) SUBSET t`) THEN ASM_REWRITE_TAC[FRONTIER_COMPLEMENT; FRONTIER_SUBSET_EQ]; REWRITE_TAC[connected_component; NOT_EXISTS_THM; SET_RULE `s SUBSET UNIV DIFF t <=> s INTER t = {}`] THEN X_GEN_TAC `t:real^N->bool` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC)) THEN REWRITE_TAC[] THEN MATCH_MP_TAC CONNECTED_INTER_FRONTIER THEN ASM_REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN CONJ_TAC THENL [EXISTS_TAC `b:real^N` THEN ASM_REWRITE_TAC[]; EXISTS_TAC `a:real^N`] THEN ASM_REWRITE_TAC[IN_DIFF] THEN EXPAND_TAC "s" THEN REWRITE_TAC[IN] THEN DISCH_THEN(MP_TAC o CONJUNCT2 o MATCH_MP CONNECTED_COMPONENT_IN) THEN REWRITE_TAC[IN_DIFF; IN_UNIV] THEN MATCH_MP_TAC(REWRITE_RULE[SUBSET] CLOSURE_SUBSET) THEN ASM_REWRITE_TAC[]; X_GEN_TAC `u:real^N->bool` THEN REWRITE_TAC[PSUBSET_ALT] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `p:real^N` STRIP_ASSUME_TAC) THEN REWRITE_TAC[connected_component] THEN EXISTS_TAC `(p:real^N) INSERT (s UNION r)` THEN ASM_REWRITE_TAC[IN_INSERT; IN_UNION] THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[SET_RULE `a INSERT (s UNION t) = (a INSERT s) UNION (a INSERT t)`] THEN MATCH_MP_TAC CONNECTED_UNION THEN REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN CONJ_TAC THEN MATCH_MP_TAC CONNECTED_INTERMEDIATE_CLOSURE THENL [EXISTS_TAC `s:real^N->bool`; EXISTS_TAC `r:real^N->bool`] THEN (CONJ_TAC THENL [ASM_MESON_TAC[CONNECTED_CONNECTED_COMPONENT]; ALL_TAC] THEN CONJ_TAC THENL [SET_TAC[]; REWRITE_TAC[INSERT_SUBSET]] THEN REWRITE_TAC[CLOSURE_SUBSET] THEN ASM_REWRITE_TAC[CLOSURE_UNION_FRONTIER; IN_UNION] THEN ASM SET_TAC[]); MATCH_MP_TAC(SET_RULE `s INTER u = {} /\ t INTER u = {} /\ ~(p IN u) ==> p INSERT (s UNION t) SUBSET UNIV DIFF u`) THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `u SUBSET t ==> t INTER s = {} ==> s INTER u = {}`)) THEN REWRITE_TAC[FRONTIER_DISJOINT_EQ] THEN EXPAND_TAC "s"; SUBGOAL_THEN `frontier(r:real^N->bool) INTER r = {}` (fun th -> ASM SET_TAC[th]) THEN REWRITE_TAC[FRONTIER_DISJOINT_EQ] THEN EXPAND_TAC "r"] THEN MATCH_MP_TAC OPEN_CONNECTED_COMPONENT THEN ASM_REWRITE_TAC[GSYM closed; CLOSED_CLOSURE]]]);; (* ------------------------------------------------------------------------- *) (* Lower bound on norms within segment between vectors. *) (* Could have used these for connectedness results below, in fact. *) (* ------------------------------------------------------------------------- *) let NORM_SEGMENT_LOWERBOUND = prove (`!a b x:real^N r d. &0 < r /\ norm(a) = r /\ norm(b) = r /\ x IN segment[a,b] /\ a dot b = d * r pow 2 ==> sqrt((&1 - abs d) / &2) * r <= norm(x)`, REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM real_ge] THEN REWRITE_TAC[NORM_GE_SQUARE] THEN DISJ2_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_SEGMENT]) THEN DISCH_THEN(X_CHOOSE_THEN `u:real` STRIP_ASSUME_TAC) THEN ASM_REWRITE_TAC[real_ge; DOT_LMUL; DOT_RMUL; REAL_MUL_RZERO; VECTOR_ARITH `(a + b) dot (a + b) = a dot a + b dot b + &2 * a dot b`] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `(&1 - u) * (&1 - u) * r pow 2 + u * u * r pow 2 - &2 * (&1 - u) * u * abs d * r pow 2` THEN CONJ_TAC THENL [REWRITE_TAC[REAL_POW_MUL; REAL_MUL_ASSOC] THEN REWRITE_TAC[GSYM REAL_ADD_RDISTRIB; GSYM REAL_SUB_RDISTRIB] THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[REAL_POW_2; REAL_LE_SQUARE] THEN REWRITE_TAC[GSYM REAL_POW_2; REAL_ARITH `(&1 - u) pow 2 + u pow 2 - ((&2 * (&1 - u)) * u) * d = (&1 + d) * (&1 - &2 * u + &2 * u pow 2) - d`] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `(&1 + abs d) * &1 / &2 - abs d` THEN CONJ_TAC THENL [REWRITE_TAC[REAL_ARITH `(&1 + d) * &1 / &2 - d = (&1 - d) / &2`] THEN MATCH_MP_TAC REAL_EQ_IMP_LE THEN MATCH_MP_TAC SQRT_POW_2 THEN MP_TAC(ISPECL [`a:real^N`; `b:real^N`] NORM_CAUCHY_SCHWARZ_ABS) THEN ASM_REWRITE_TAC[REAL_ABS_MUL; REAL_ABS_POW; REAL_POW2_ABS] THEN ASM_REWRITE_TAC[REAL_ARITH `r * r = &1 * r pow 2`] THEN ASM_SIMP_TAC[REAL_LE_RMUL_EQ; REAL_POW_LT] THEN REAL_ARITH_TAC; MATCH_MP_TAC(REAL_ARITH `x <= y ==> x - a <= y - a`) THEN MATCH_MP_TAC REAL_LE_LMUL THEN CONJ_TAC THENL [REAL_ARITH_TAC; MATCH_MP_TAC(REAL_ARITH `&0 <= (u - &1 / &2) * (u - &1 / &2) ==> &1 / &2 <= &1 - &2 * u + &2 * u pow 2`) THEN REWRITE_TAC[REAL_LE_SQUARE]]]; ASM_REWRITE_TAC[GSYM NORM_POW_2; REAL_LE_LADD; real_sub] THEN MATCH_MP_TAC(REAL_ARITH `abs(a) <= --x ==> x <= a`) THEN ASM_REWRITE_TAC[REAL_ABS_MUL; REAL_MUL_LNEG; REAL_NEG_NEG] THEN REWRITE_TAC[REAL_ABS_POW; REAL_POW2_ABS; REAL_ABS_NUM] THEN REWRITE_TAC[REAL_MUL_ASSOC] THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[REAL_POW_2; REAL_LE_SQUARE] THEN ASM_REWRITE_TAC[real_abs; GSYM real_sub; REAL_SUB_LE; REAL_POS] THEN MATCH_MP_TAC REAL_LE_LMUL THEN CONJ_TAC THEN REPEAT(MATCH_MP_TAC REAL_LE_MUL THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC]);; (* ------------------------------------------------------------------------- *) (* Special case of orthogonality (could replace 2 by sqrt(2)). *) (* ------------------------------------------------------------------------- *) let NORM_SEGMENT_ORTHOGONAL_LOWERBOUND = prove (`!a b:real^N x r. r <= norm(a) /\ r <= norm(b) /\ orthogonal a b /\ x IN segment[a,b] ==> r / &2 <= norm(x)`, REPEAT GEN_TAC THEN REWRITE_TAC[GSYM real_ge] THEN REWRITE_TAC[NORM_GE_SQUARE] THEN REWRITE_TAC[real_ge] THEN ASM_CASES_TAC `r <= &0` THEN ASM_REWRITE_TAC[] THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[orthogonal] THEN STRIP_TAC THEN DISJ2_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_SEGMENT]) THEN DISCH_THEN(X_CHOOSE_THEN `u:real` STRIP_ASSUME_TAC) THEN ASM_REWRITE_TAC[DOT_LMUL; DOT_RMUL; REAL_MUL_RZERO; VECTOR_ARITH `(a + b) dot (a + b) = a dot a + b dot b + &2 * a dot b`] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `(&1 - u) * (&1 - u) * r pow 2 + u * u * r pow 2` THEN CONJ_TAC THENL [REWRITE_TAC[REAL_ARITH `(r / &2) pow 2 = &1 / &4 * r pow 2`] THEN REWRITE_TAC[GSYM REAL_ADD_RDISTRIB; REAL_MUL_ASSOC] THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[REAL_POW_2; REAL_LE_SQUARE] THEN MATCH_MP_TAC(REAL_ARITH `&0 <= (u - &1 / &2) * (u - &1 / &2) ==> &1 / &4 <= (&1 - u) * (&1 - u) + u * u`) THEN REWRITE_TAC[REAL_LE_SQUARE]; REWRITE_TAC[REAL_ADD_RID] THEN MATCH_MP_TAC REAL_LE_ADD2 THEN CONJ_TAC THEN REPEAT(MATCH_MP_TAC REAL_LE_LMUL THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]) THEN ASM_REWRITE_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Accessibility of frontier points. *) (* ------------------------------------------------------------------------- *) let DENSE_ACCESSIBLE_FRONTIER_POINTS = prove (`!s:real^N->bool v. open s /\ open_in (subtopology euclidean (frontier s)) v /\ ~(v = {}) ==> ?g. arc g /\ IMAGE g (interval [vec 0,vec 1] DELETE vec 1) SUBSET s /\ pathstart g IN s /\ pathfinish g IN v`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_TAC `z:real^N`) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_CONTAINS_BALL]) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `z:real^N`)) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `r:real` THEN STRIP_TAC THEN SUBGOAL_THEN `(z:real^N) IN frontier s` MP_TAC THENL [ASM SET_TAC[]; DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN REWRITE_TAC[frontier] THEN ASM_SIMP_TAC[IN_DIFF; INTERIOR_OPEN]] THEN REWRITE_TAC[closure; IN_UNION; TAUT `(p \/ q) /\ ~p <=> ~p /\ q`] THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIMPT_INFINITE_BALL]) THEN DISCH_THEN(MP_TAC o SPEC `r:real`) THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `s INTER ball(z:real^N,r) = {}` THENL [ASM_MESON_TAC[INFINITE; FINITE_EMPTY]; DISCH_THEN(K ALL_TAC)] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN REWRITE_TAC[IN_INTER] THEN DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `~((y:real^N) IN frontier s)` ASSUME_TAC THENL [ASM_SIMP_TAC[IN_DIFF; INTERIOR_OPEN; frontier]; ALL_TAC] THEN SUBGOAL_THEN `path_connected(ball(z:real^N,r))` MP_TAC THENL [ASM_SIMP_TAC[CONVEX_BALL; CONVEX_IMP_PATH_CONNECTED]; ALL_TAC] THEN REWRITE_TAC[PATH_CONNECTED_ARCWISE] THEN DISCH_THEN(MP_TAC o SPECL [`y:real^N`; `z:real^N`]) THEN ASM_REWRITE_TAC[CENTRE_IN_BALL] THEN ANTS_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^1->real^N` STRIP_ASSUME_TAC) THEN MP_TAC(ISPEC `IMAGE drop {t | t IN interval[vec 0,vec 1] /\ (g:real^1->real^N) t IN frontier s}` COMPACT_ATTAINS_INF) THEN REWRITE_TAC[EXISTS_IN_IMAGE; IMAGE_EQ_EMPTY; FORALL_IN_IMAGE; IMP_CONJ] THEN REWRITE_TAC[IMP_IMP; FORALL_IN_GSPEC; EXISTS_IN_GSPEC; GSYM IMAGE_o] THEN REWRITE_TAC[o_DEF; LIFT_DROP; IMAGE_ID] THEN ANTS_TAC THENL [CONJ_TAC THENL [REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED] THEN CONJ_TAC THENL [MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `interval[vec 0:real^1,vec 1]` THEN REWRITE_TAC[BOUNDED_INTERVAL; SUBSET_RESTRICT]; MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE THEN REWRITE_TAC[FRONTIER_CLOSED; CLOSED_INTERVAL; GSYM path] THEN ASM_MESON_TAC[arc]]; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `vec 1:real^1` THEN ASM_REWRITE_TAC[IN_ELIM_THM; ENDS_IN_UNIT_INTERVAL] THEN ASM_MESON_TAC[pathfinish; SUBSET]]; DISCH_THEN(X_CHOOSE_THEN `t:real^1` STRIP_ASSUME_TAC) THEN EXISTS_TAC `subpath (vec 0) t (g:real^1->real^N)` THEN ASM_REWRITE_TAC[PATHSTART_SUBPATH; PATHFINISH_SUBPATH] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [MATCH_MP_TAC ARC_SUBPATH_ARC THEN ASM_REWRITE_TAC[ENDS_IN_UNIT_INTERVAL] THEN ASM_MESON_TAC[pathstart]; REWRITE_TAC[arc] THEN STRIP_TAC] THEN GEN_REWRITE_TAC (RAND_CONV o LAND_CONV o LAND_CONV) [GSYM pathstart] THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; RULE_ASSUM_TAC(SIMP_RULE[path_image]) THEN ASM SET_TAC[]] THEN MATCH_MP_TAC(SET_RULE `a IN s /\ IMAGE f s DELETE (f a) SUBSET t /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> IMAGE f (s DELETE a) SUBSET t`) THEN ASM_REWRITE_TAC[ENDS_IN_UNIT_INTERVAL; GSYM path_image] THEN W(MP_TAC o PART_MATCH (lhand o rand) PATH_IMAGE_SUBPATH o lhand o lhand o snd) THEN ANTS_TAC THENL [ASM_MESON_TAC[IN_INTERVAL_1]; DISCH_THEN SUBST1_TAC] THEN REWRITE_TAC[REWRITE_RULE[pathfinish] PATHFINISH_SUBPATH] THEN MATCH_MP_TAC(SET_RULE `IMAGE f (s DELETE a) DIFF t = {} ==> IMAGE f s DELETE f a SUBSET t`) THEN MATCH_MP_TAC(REWRITE_RULE[TAUT `p /\ q /\ ~r ==> ~s <=> p /\ q /\ s ==> r`] CONNECTED_INTER_FRONTIER) THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN CONJ_TAC THENL [FIRST_X_ASSUM(MP_TAC o CONJUNCT1 o GEN_REWRITE_RULE I [arc]) THEN REWRITE_TAC[path] THEN MATCH_MP_TAC (REWRITE_RULE[IMP_CONJ_ALT] CONTINUOUS_ON_SUBSET) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL_1]) THEN REWRITE_TAC[SUBSET; IN_DELETE; GSYM DROP_EQ; IN_INTERVAL_1] THEN REAL_ARITH_TAC; MATCH_MP_TAC CONNECTED_INTERMEDIATE_CLOSURE THEN EXISTS_TAC `interval(vec 0:real^1,t)` THEN REWRITE_TAC[CONNECTED_INTERVAL; CLOSURE_INTERVAL] THEN REWRITE_TAC[INTERVAL_EQ_EMPTY_1] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL_1]) THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[SUBSET; IN_DELETE; GSYM DROP_EQ; IN_INTERVAL_1] THEN REWRITE_TAC[NOT_IN_EMPTY] THEN ASM_REAL_ARITH_TAC]; REWRITE_TAC[SET_RULE `~(IMAGE f s INTER t = {}) <=> ?x. x IN s /\ f x IN t`] THEN EXISTS_TAC `vec 0:real^1` THEN REWRITE_TAC[IN_INTERVAL_1; IN_DELETE; REAL_LE_REFL] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM SET_TAC[pathstart]; REWRITE_TAC[SET_RULE `IMAGE g i INTER s = {} <=> !x. x IN i ==> ~(g x IN s)`] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_DELETE; IN_UNIV; IN_DIFF] THEN X_GEN_TAC `z:real^1` THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[GSYM DROP_EQ; IN_INTERVAL_1] THEN DISCH_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[GSYM REAL_LE_ANTISYM] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL_1] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC]]);; let DENSE_ACCESSIBLE_FRONTIER_POINTS_CONNECTED = prove (`!s:real^N->bool v x. open s /\ connected s /\ x IN s /\ open_in (subtopology euclidean (frontier s)) v /\ ~(v = {}) ==> ?g. arc g /\ IMAGE g (interval [vec 0,vec 1] DELETE vec 1) SUBSET s /\ pathstart g = x /\ pathfinish g IN v`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `v:real^N->bool`] DENSE_ACCESSIBLE_FRONTIER_POINTS) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^1->real^N` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `path_connected(s:real^N->bool)` MP_TAC THENL [ASM_MESON_TAC[CONNECTED_OPEN_PATH_CONNECTED]; ALL_TAC] THEN REWRITE_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT] THEN DISCH_THEN(MP_TAC o SPECL [`x:real^N`; `pathstart g:real^N`]) THEN ASM_REWRITE_TAC[path_component; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `f:real^1->real^N` THEN STRIP_TAC THEN MP_TAC(ISPECL [`f ++ g:real^1->real^N`; `x:real^N`; `pathfinish g:real^N`] PATH_CONTAINS_ARC) THEN ASM_SIMP_TAC[PATH_JOIN_EQ; ARC_IMP_PATH; PATH_IMAGE_JOIN; PATHSTART_JOIN; PATHFINISH_JOIN] THEN FIRST_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN GEN_REWRITE_TAC LAND_CONV [SUBSET] THEN ASM_SIMP_TAC[frontier; INTERIOR_OPEN; IN_DIFF] THEN DISCH_TAC THEN ANTS_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `h:real^1->real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(SET_RULE `a IN s /\ IMAGE f s DELETE (f a) SUBSET t /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> IMAGE f (s DELETE a) SUBSET t`) THEN REWRITE_TAC[ENDS_IN_UNIT_INTERVAL] THEN CONJ_TAC THENL [REWRITE_TAC[GSYM path_image]; ASM_MESON_TAC[arc]] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `h SUBSET f UNION g ==> f SUBSET s /\ g DELETE a SUBSET s ==> h DELETE a SUBSET s`)) THEN ASM_REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[path_image; pathstart; pathfinish]) THEN REWRITE_TAC[path_image] THEN ASM SET_TAC[]);; let DENSE_ACCESSIBLE_FRONTIER_POINT_PAIRS = prove (`!s u v:real^N->bool. open s /\ connected s /\ open_in (subtopology euclidean (frontier s)) u /\ open_in (subtopology euclidean (frontier s)) v /\ ~(u = {}) /\ ~(v = {}) /\ ~(u = v) ==> ?g. arc g /\ pathstart g IN u /\ pathfinish g IN v /\ IMAGE g (interval(vec 0,vec 1)) SUBSET s`, GEN_TAC THEN REWRITE_TAC[CONJ_ASSOC] THEN ONCE_REWRITE_TAC[IMP_CONJ_ALT] THEN GEN_REWRITE_TAC (funpow 2 BINDER_CONV o LAND_CONV o RAND_CONV) [GSYM SUBSET_ANTISYM_EQ] THEN REWRITE_TAC[DE_MORGAN_THM; GSYM CONJ_ASSOC] THEN MATCH_MP_TAC(MESON[] `(!u v. R u v ==> R v u) /\ (!u v. P u v ==> R u v) ==> !u v. P u v \/ P v u ==> R u v`) THEN CONJ_TAC THENL [REPEAT GEN_TAC THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `g:real^1->real^N` THEN STRIP_TAC THEN EXISTS_TAC `reversepath g:real^1->real^N` THEN ASM_SIMP_TAC[ARC_REVERSEPATH; PATHSTART_REVERSEPATH; PATHFINISH_REVERSEPATH] THEN REWRITE_TAC[reversepath] THEN ONCE_REWRITE_TAC[GSYM o_DEF] THEN REWRITE_TAC[IMAGE_o] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE f i SUBSET t ==> IMAGE r i SUBSET i ==> IMAGE f (IMAGE r i) SUBSET t`)) THEN SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTERVAL_1; DROP_SUB; DROP_VEC] THEN REAL_ARITH_TAC; ALL_TAC] THEN REPEAT GEN_TAC THEN DISCH_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[FRONTIER_EMPTY; OPEN_IN_SUBTOPOLOGY_EMPTY] THENL [CONV_TAC TAUT; STRIP_TAC THEN UNDISCH_TAC `~(s:real^N->bool = {})`] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `v:real^N->bool`; `x:real^N`] DENSE_ACCESSIBLE_FRONTIER_POINTS_CONNECTED) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `g:real^1->real^N` THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `(u DELETE pathfinish g):real^N->bool`; `x:real^N`] DENSE_ACCESSIBLE_FRONTIER_POINTS_CONNECTED) THEN ASM_SIMP_TAC[OPEN_IN_DELETE; IN_DELETE; LEFT_IMP_EXISTS_THM] THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN X_GEN_TAC `h:real^1->real^N` THEN STRIP_TAC THEN MP_TAC(ISPECL [`(reversepath h ++ g):real^1->real^N`; `pathfinish h:real^N`; `pathfinish g:real^N`] PATH_CONTAINS_ARC) THEN ASM_SIMP_TAC[PATH_JOIN; PATHSTART_JOIN; PATHFINISH_JOIN; PATHSTART_REVERSEPATH; PATHFINISH_REVERSEPATH; PATH_REVERSEPATH; ARC_IMP_PATH; PATH_IMAGE_JOIN; PATH_IMAGE_REVERSEPATH] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `q:real^1->real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[OPEN_CLOSED_INTERVAL_1] THEN MATCH_MP_TAC(SET_RULE `(!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\ t SUBSET s /\ IMAGE f s SUBSET u UNION IMAGE f t ==> IMAGE f (s DIFF t) SUBSET u`) THEN REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; ENDS_IN_UNIT_INTERVAL] THEN CONJ_TAC THENL [ASM_MESON_TAC[arc]; REWRITE_TAC[GSYM path_image]] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] SUBSET_TRANS)) THEN RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish; path_image]) THEN REWRITE_TAC[path_image] THEN ASM SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* Some simple positive connection theorems. *) (* ------------------------------------------------------------------------- *) let PATH_CONNECTED_CONVEX_DIFF_CARD_LT = prove (`!u s:real^N->bool. convex u /\ ~(collinear u) /\ s <_c (:real) ==> path_connected(u DIFF s)`, REPEAT STRIP_TAC THEN REWRITE_TAC[path_connected; IN_DIFF; IN_UNIV] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN STRIP_TAC THEN ASM_CASES_TAC `a:real^N = b` THENL [EXISTS_TAC `linepath(a:real^N,b)` THEN REWRITE_TAC[PATHSTART_LINEPATH; PATHFINISH_LINEPATH; PATH_LINEPATH] THEN ASM_REWRITE_TAC[PATH_IMAGE_LINEPATH; SEGMENT_REFL] THEN ASM SET_TAC[]; ALL_TAC] THEN ABBREV_TAC `m:real^N = midpoint(a,b)` THEN SUBGOAL_THEN `~(m:real^N = a) /\ ~(m = b)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[MIDPOINT_EQ_ENDPOINT]; ALL_TAC] THEN POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN GEOM_ORIGIN_TAC `m:real^N` THEN REPEAT GEN_TAC THEN GEOM_NORMALIZE_TAC `b:real^N` THEN REWRITE_TAC[] THEN GEN_TAC THEN GEOM_BASIS_MULTIPLE_TAC 1 `b:real^N` THEN X_GEN_TAC `bbb:real` THEN DISCH_TAC THEN SIMP_TAC[NORM_MUL; NORM_BASIS; DIMINDEX_GE_1; LE_REFL] THEN ASM_REWRITE_TAC[real_abs; REAL_MUL_RID] THEN DISCH_THEN SUBST1_TAC THEN POP_ASSUM(K ALL_TAC) THEN REPEAT GEN_TAC THEN REWRITE_TAC[midpoint; VECTOR_MUL_LID] THEN REWRITE_TAC[VECTOR_ARITH `inv(&2) % (a + b):real^N = vec 0 <=> a = --b`] THEN ASM_CASES_TAC `a:real^N = --(basis 1)` THEN ASM_REWRITE_TAC[] THEN POP_ASSUM(K ALL_TAC) THEN REPLICATE_TAC 7 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(K ALL_TAC) THEN SUBGOAL_THEN `segment[--basis 1:real^N,basis 1] SUBSET u` ASSUME_TAC THENL [REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN MATCH_MP_TAC HULL_MINIMAL THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(vec 0:real^N) IN u` ASSUME_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN REWRITE_TAC[IN_SEGMENT] THEN EXISTS_TAC `&1 / &2` THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN VECTOR_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `?c:real^N k. 1 <= k /\ ~(k = 1) /\ k <= dimindex(:N) /\ c IN u /\ ~(c$k = &0)` STRIP_ASSUME_TAC THENL [REWRITE_TAC[GSYM NOT_FORALL_THM; TAUT `a /\ ~b /\ c /\ d /\ ~e <=> ~(d ==> a /\ c ==> ~b ==> e)`] THEN DISCH_TAC THEN UNDISCH_TAC `~collinear(u:real^N->bool)` THEN REWRITE_TAC[COLLINEAR_AFFINE_HULL] THEN MAP_EVERY EXISTS_TAC [`vec 0:real^N`; `basis 1:real^N`] THEN SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC; IN_INSERT; SPAN_INSERT_0] THEN REWRITE_TAC[SPAN_SING; SUBSET; IN_ELIM_THM; IN_UNIV] THEN X_GEN_TAC `c:real^N` THEN DISCH_TAC THEN EXISTS_TAC `(c:real^N)$1` THEN SIMP_TAC[CART_EQ; VECTOR_MUL_COMPONENT; BASIS_COMPONENT] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_MUL_RID; REAL_MUL_RZERO] THEN ASM_MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `~(c:real^N = vec 0)` ASSUME_TAC THENL [ASM_SIMP_TAC[CART_EQ; VEC_COMPONENT] THEN ASM_MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `segment[vec 0:real^N,c] SUBSET u` ASSUME_TAC THENL [REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN MATCH_MP_TAC HULL_MINIMAL THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?z:real^N. z IN segment[vec 0,c] /\ (segment[--basis 1,z] UNION segment[z,basis 1]) INTER s = {}` STRIP_ASSUME_TAC THENL [ALL_TAC; EXISTS_TAC `linepath(--basis 1:real^N,z) ++ linepath(z,basis 1)` THEN ASM_SIMP_TAC[PATH_JOIN; PATHSTART_JOIN; PATHFINISH_JOIN; PATH_LINEPATH; PATHSTART_LINEPATH; PATHFINISH_LINEPATH; PATH_IMAGE_JOIN] THEN REWRITE_TAC[PATH_IMAGE_LINEPATH] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `(t UNION v) INTER s = {} ==> t SUBSET u /\ v SUBSET u ==> (t UNION v) SUBSET u DIFF s`)) THEN REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN CONJ_TAC THEN MATCH_MP_TAC HULL_MINIMAL THEN ASM SET_TAC[]] THEN MATCH_MP_TAC(SET_RULE `~(s SUBSET {z | z IN s /\ ~P z}) ==> ?z. z IN s /\ P z`) THEN DISCH_THEN(MP_TAC o MATCH_MP CARD_LE_SUBSET) THEN REWRITE_TAC[CARD_NOT_LE; SET_RULE `~((b UNION c) INTER s = {}) <=> ~(b INTER s = {}) \/ ~(c INTER s = {})`] THEN REWRITE_TAC[SET_RULE `{x | P x /\ (Q x \/ R x)} = {x | P x /\ Q x} UNION {x | P x /\ R x}`] THEN W(MP_TAC o PART_MATCH lhand UNION_LE_ADD_C o lhand o snd) THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] CARD_LET_TRANS) THEN TRANS_TAC CARD_LTE_TRANS `(:real)` THEN CONJ_TAC THENL [MATCH_MP_TAC CARD_ADD2_ABSORB_LT THEN REWRITE_TAC[real_INFINITE]; MATCH_MP_TAC CARD_EQ_IMP_LE THEN ONCE_REWRITE_TAC[CARD_EQ_SYM] THEN ASM_SIMP_TAC[CARD_EQ_SEGMENT]] THEN REWRITE_TAC[MESON[SEGMENT_SYM] `segment[--a:real^N,b] = segment[b,--a]`] THEN SUBGOAL_THEN `!b:real^N. b IN u /\ ~(b IN s) /\ ~(b = vec 0) /\ b$k = &0 ==> {z | z IN segment[vec 0,c] /\ ~(segment[z,b] INTER s = {})} <_c (:real)` (fun th -> CONJ_TAC THEN MATCH_MP_TAC th THEN REWRITE_TAC[VECTOR_NEG_EQ_0; VECTOR_NEG_COMPONENT] THEN ASM_SIMP_TAC[BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL; BASIS_COMPONENT] THEN REWRITE_TAC[REAL_NEG_0]) THEN REPEAT STRIP_TAC THEN TRANS_TAC CARD_LET_TRANS `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER; RIGHT_AND_EXISTS_THM] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> r /\ p /\ q`] THEN MATCH_MP_TAC CARD_LE_RELATIONAL THEN MAP_EVERY X_GEN_TAC [`w:real^N`; `x1:real^N`; `x2:real^N`] THEN REWRITE_TAC[SEGMENT_SYM] THEN STRIP_TAC THEN ASM_CASES_TAC `x2:real^N = x1` THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPECL [`x1:real^N`; `b:real^N`; `x2:real^N`] INTER_SEGMENT) THEN REWRITE_TAC[NOT_IMP; SEGMENT_SYM] THEN CONJ_TAC THENL [DISJ2_TAC; REWRITE_TAC[SEGMENT_SYM] THEN ASM SET_TAC[]] THEN ONCE_REWRITE_TAC[SET_RULE `{x1,b,x2} = {x1,x2,b}`] THEN ASM_SIMP_TAC[COLLINEAR_3_AFFINE_HULL] THEN STRIP_TAC THEN SUBGOAL_THEN `(b:real^N) IN affine hull {vec 0,c}` MP_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `b IN s ==> s SUBSET t ==> b IN t`)) THEN MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[AFFINE_AFFINE_HULL] THEN MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `segment[c:real^N,vec 0]` THEN CONJ_TAC THENL [ASM SET_TAC[]; ONCE_REWRITE_TAC[SEGMENT_SYM]] THEN REWRITE_TAC[SEGMENT_CONVEX_HULL; CONVEX_HULL_SUBSET_AFFINE_HULL]; REWRITE_TAC[AFFINE_HULL_2_ALT; IN_ELIM_THM; IN_UNIV] THEN REWRITE_TAC[VECTOR_ADD_LID; VECTOR_SUB_RZERO; NOT_EXISTS_THM] THEN X_GEN_TAC `r:real` THEN ASM_CASES_TAC `r = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN CONV_TAC(RAND_CONV SYM_CONV) THEN DISCH_THEN(MP_TAC o AP_TERM `\x:real^N. x$k`) THEN ASM_SIMP_TAC[VECTOR_MUL_COMPONENT; REAL_ENTIRE]]);; let CONNECTED_CONVEX_DIFF_CARD_LT = prove (`!u s. convex u /\ ~collinear u /\ s <_c (:real) ==> connected(u DIFF s)`, SIMP_TAC[PATH_CONNECTED_CONVEX_DIFF_CARD_LT; PATH_CONNECTED_IMP_CONNECTED]);; let PATH_CONNECTED_CONVEX_DIFF_COUNTABLE = prove (`!u s. convex u /\ ~collinear u /\ COUNTABLE s ==> path_connected(u DIFF s)`, MESON_TAC[COUNTABLE_IMP_CARD_LT_REAL; PATH_CONNECTED_CONVEX_DIFF_CARD_LT]);; let CONNECTED_CONVEX_DIFF_COUNTABLE = prove (`!u s. convex u /\ ~collinear u /\ COUNTABLE s ==> connected(u DIFF s)`, MESON_TAC[COUNTABLE_IMP_CARD_LT_REAL; CONNECTED_CONVEX_DIFF_CARD_LT]);; let PATH_CONNECTED_PUNCTURED_CONVEX = prove (`!s a:real^N. convex s /\ ~(aff_dim s = &1) ==> path_connected(s DELETE a)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (INT_ARITH `~(x:int = &1) ==> --(&1) <= x ==> x = -- &1 \/ x = &0 \/ &2 <= x`)) THEN ASM_REWRITE_TAC[AFF_DIM_GE; AFF_DIM_EQ_MINUS1; AFF_DIM_EQ_0] THEN DISCH_THEN(REPEAT_TCL DISJ_CASES_THEN ASSUME_TAC) THEN ASM_REWRITE_TAC[PATH_CONNECTED_EMPTY; SET_RULE `{} DELETE a = {}`] THENL [FIRST_X_ASSUM(X_CHOOSE_THEN `b:real^N` SUBST1_TAC) THEN ASM_CASES_TAC `b:real^N = a` THEN ASM_REWRITE_TAC[PATH_CONNECTED_EMPTY; SET_RULE `{a} DELETE a = {}`] THEN ASM_SIMP_TAC[SET_RULE `~(b = a) ==> {a} DELETE b = {a}`] THEN REWRITE_TAC[PATH_CONNECTED_SING]; REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[SET_RULE `s DELETE a = s DIFF {a}`] THEN MATCH_MP_TAC PATH_CONNECTED_CONVEX_DIFF_COUNTABLE THEN ASM_REWRITE_TAC[COUNTABLE_SING; COLLINEAR_AFF_DIM] THEN ASM_INT_ARITH_TAC]);; let CONNECTED_PUNCTURED_CONVEX = prove (`!s a:real^N. convex s /\ ~(aff_dim s = &1) ==> connected(s DELETE a)`, SIMP_TAC[PATH_CONNECTED_PUNCTURED_CONVEX; PATH_CONNECTED_IMP_CONNECTED]);; let PATH_CONNECTED_COMPLEMENT_CARD_LT = prove (`!s. 2 <= dimindex(:N) /\ s <_c (:real) ==> path_connected((:real^N) DIFF s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC PATH_CONNECTED_CONVEX_DIFF_CARD_LT THEN ASM_REWRITE_TAC[CONVEX_UNIV; COLLINEAR_AFF_DIM; AFF_DIM_UNIV] THEN REWRITE_TAC[INT_OF_NUM_LE] THEN ASM_ARITH_TAC);; let PATH_CONNECTED_CONNECTED_DIFF = prove (`!s t:real^N->bool. connected s /\ s SUBSET closure(s DIFF t) /\ (!x. x IN s ==> ?u. x IN u /\ open_in (subtopology euclidean s) u /\ path_connected(u DIFF t)) ==> path_connected(s DIFF t)`, REPEAT STRIP_TAC THEN REWRITE_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT; IN_DIFF] THEN REWRITE_TAC[TAUT `(p /\ q) /\ (r /\ s) <=> p /\ r /\ q /\ s`] THEN MATCH_MP_TAC CONNECTED_EQUIVALENCE_RELATION_GEN THEN ASM_REWRITE_TAC[PATH_COMPONENT_SYM; PATH_COMPONENT_TRANS] THEN CONJ_TAC THENL [REWRITE_TAC[OPEN_IN_OPEN; LEFT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`v:real^N->bool`; `x:real^N`; `u:real^N->bool`] THEN DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[IN_INTER] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N` o GEN_REWRITE_RULE I [SUBSET]) THEN ASM_REWRITE_TAC[CLOSURE_NONEMPTY_OPEN_INTER] THEN DISCH_THEN(MP_TAC o SPEC `u:real^N->bool`) THEN ASM SET_TAC[]; X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^N->bool` THEN ONCE_REWRITE_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN ASM_REWRITE_TAC[IN_DIFF] THEN REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN MATCH_MP_TAC MONO_IMP THEN SIMP_TAC[] THEN MATCH_MP_TAC(SET_RULE `P SUBSET Q ==> P x ==> Q x`) THEN REWRITE_TAC[ETA_AX] THEN MATCH_MP_TAC PATH_COMPONENT_MONO THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN SET_TAC[]]);; let PATH_CONNECTED_OPEN_IN_DIFF_CARD_LT = prove (`!s t:real^N->bool. connected s /\ open_in (subtopology euclidean (affine hull s)) s /\ ~collinear s /\ t <_c (:real) ==> path_connected(s DIFF t)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC PATH_CONNECTED_CONNECTED_DIFF THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [REWRITE_TAC[SUBSET; CLOSURE_APPROACHABLE; IN_DIFF] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN X_GEN_TAC `d:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(CONJUNCTS_THEN ASSUME_TAC o REWRITE_RULE[OPEN_IN_CONTAINS_BALL]) THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `~((ball(x:real^N,min d e) INTER affine hull s) DIFF t = {})` MP_TAC THENL [REWRITE_TAC[SET_RULE `s DIFF t = {} <=> s SUBSET t`] THEN DISCH_THEN(MP_TAC o MATCH_MP CARD_LE_SUBSET) THEN REWRITE_TAC[CARD_NOT_LE] THEN TRANS_TAC CARD_LTE_TRANS `(:real)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[CARD_EQ_SYM] CARD_EQ_IMP_LE) THEN MATCH_MP_TAC CARD_EQ_CONVEX THEN ASM_SIMP_TAC[CONVEX_BALL; CONVEX_INTER; AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL] THEN EXISTS_TAC `x:real^N` THEN ASM_SIMP_TAC[CENTRE_IN_BALL; REAL_LT_MIN; IN_INTER; HULL_INC] THEN MP_TAC(ISPECL [`s:real^N->bool`; `x:real^N`] CONNECTED_IMP_PERFECT) THEN ANTS_TAC THENL [ASM_MESON_TAC[COLLINEAR_SING]; ALL_TAC] THEN REWRITE_TAC[LIMPT_APPROACHABLE] THEN DISCH_THEN(MP_TAC o SPEC `min d e:real`) THEN ASM_REWRITE_TAC[REAL_LT_MIN; IN_BALL] THEN ASM_MESON_TAC[HULL_INC; DIST_SYM]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER; BALL_MIN_INTER; IN_DIFF; IN_BALL; REAL_LT_MIN] THEN MESON_TAC[DIST_SYM]]; X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(CONJUNCTS_THEN ASSUME_TAC o REWRITE_RULE[OPEN_IN_CONTAINS_BALL]) THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `ball(x:real^N,r) INTER affine hull s` THEN ASM_SIMP_TAC[IN_INTER; CENTRE_IN_BALL; HULL_INC] THEN CONJ_TAC THENL [REWRITE_TAC[OPEN_IN_OPEN] THEN EXISTS_TAC `ball(x:real^N,r)` THEN REWRITE_TAC[OPEN_BALL] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `b INTER t SUBSET s ==> s SUBSET t ==> b INTER t = s INTER b`)) THEN REWRITE_TAC[HULL_SUBSET]; MATCH_MP_TAC PATH_CONNECTED_CONVEX_DIFF_CARD_LT THEN ASM_SIMP_TAC[CONVEX_INTER; AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL; CONVEX_BALL; COLLINEAR_AFF_DIM] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [COLLINEAR_AFF_DIM]) THEN MATCH_MP_TAC(INT_ARITH `x:int = y ==> ~(y <= &1) ==> ~(x <= &1)`) THEN GEN_REWRITE_TAC RAND_CONV [GSYM AFF_DIM_AFFINE_HULL] THEN ONCE_REWRITE_TAC[INTER_COMM] THEN MATCH_MP_TAC AFF_DIM_CONVEX_INTER_OPEN THEN ASM_SIMP_TAC[AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL; OPEN_BALL] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN ASM_MESON_TAC[HULL_INC; CENTRE_IN_BALL]]]);; let CONNECTED_OPEN_IN_DIFF_CARD_LT = prove (`!s t:real^N->bool. connected s /\ open_in (subtopology euclidean (affine hull s)) s /\ ~collinear s /\ t <_c (:real) ==> connected(s DIFF t)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC PATH_CONNECTED_IMP_CONNECTED THEN MATCH_MP_TAC PATH_CONNECTED_OPEN_IN_DIFF_CARD_LT THEN ASM_REWRITE_TAC[]);; let PATH_CONNECTED_OPEN_DIFF_CARD_LT = prove (`!s t:real^N->bool. 2 <= dimindex(:N) /\ open s /\ connected s /\ t <_c (:real) ==> path_connected(s DIFF t)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[EMPTY_DIFF; PATH_CONNECTED_EMPTY] THEN MATCH_MP_TAC PATH_CONNECTED_OPEN_IN_DIFF_CARD_LT THEN ASM_REWRITE_TAC[COLLINEAR_AFF_DIM] THEN ASM_SIMP_TAC[AFFINE_HULL_OPEN; AFF_DIM_OPEN] THEN ASM_REWRITE_TAC[INT_OF_NUM_LE; SUBTOPOLOGY_UNIV; GSYM OPEN_IN] THEN ASM_ARITH_TAC);; let CONNECTED_OPEN_DIFF_CARD_LT = prove (`!s t:real^N->bool. 2 <= dimindex(:N) /\ open s /\ connected s /\ t <_c (:real) ==> connected(s DIFF t)`, SIMP_TAC[PATH_CONNECTED_OPEN_DIFF_CARD_LT; PATH_CONNECTED_IMP_CONNECTED]);; let PATH_CONNECTED_OPEN_DIFF_COUNTABLE = prove (`!s t:real^N->bool. 2 <= dimindex(:N) /\ open s /\ connected s /\ COUNTABLE t ==> path_connected(s DIFF t)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC PATH_CONNECTED_OPEN_DIFF_CARD_LT THEN ASM_REWRITE_TAC[GSYM CARD_NOT_LE] THEN ASM_MESON_TAC[UNCOUNTABLE_REAL; CARD_LE_COUNTABLE]);; let CONNECTED_OPEN_DIFF_COUNTABLE = prove (`!s t:real^N->bool. 2 <= dimindex(:N) /\ open s /\ connected s /\ COUNTABLE t ==> connected(s DIFF t)`, SIMP_TAC[PATH_CONNECTED_OPEN_DIFF_COUNTABLE; PATH_CONNECTED_IMP_CONNECTED]);; let PATH_CONNECTED_OPEN_DELETE = prove (`!s a:real^N. 2 <= dimindex(:N) /\ open s /\ connected s ==> path_connected(s DELETE a)`, REPEAT STRIP_TAC THEN REWRITE_TAC[SET_RULE `s DELETE a = s DIFF {a}`] THEN MATCH_MP_TAC PATH_CONNECTED_OPEN_DIFF_COUNTABLE THEN ASM_REWRITE_TAC[COUNTABLE_SING]);; let CONNECTED_OPEN_DELETE = prove (`!s a:real^N. 2 <= dimindex(:N) /\ open s /\ connected s ==> connected(s DELETE a)`, SIMP_TAC[PATH_CONNECTED_OPEN_DELETE; PATH_CONNECTED_IMP_CONNECTED]);; let PATH_CONNECTED_PUNCTURED_UNIVERSE = prove (`!a. 2 <= dimindex(:N) ==> path_connected((:real^N) DIFF {a})`, REPEAT STRIP_TAC THEN MATCH_MP_TAC PATH_CONNECTED_OPEN_DIFF_COUNTABLE THEN ASM_REWRITE_TAC[OPEN_UNIV; CONNECTED_UNIV; COUNTABLE_SING]);; let CONNECTED_PUNCTURED_UNIVERSE = prove (`!a. 2 <= dimindex(:N) ==> connected((:real^N) DIFF {a})`, SIMP_TAC[PATH_CONNECTED_PUNCTURED_UNIVERSE; PATH_CONNECTED_IMP_CONNECTED]);; let PATH_CONNECTED_PUNCTURED_BALL = prove (`!a:real^N r. 2 <= dimindex(:N) ==> path_connected(ball(a,r) DELETE a)`, SIMP_TAC[PATH_CONNECTED_OPEN_DELETE; OPEN_BALL; CONNECTED_BALL]);; let CONNECTED_PUNCTURED_BALL = prove (`!a:real^N r. 2 <= dimindex(:N) ==> connected(ball(a,r) DELETE a)`, SIMP_TAC[CONNECTED_OPEN_DELETE; OPEN_BALL; CONNECTED_BALL]);; let PATH_CONNECTED_PUNCTURED_CBALL = prove (`!a:real^N r. 2 <= dimindex(:N) ==> path_connected(cball(a,r) DELETE a)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `&0 < r` THENL [MATCH_MP_TAC PATH_CONNECTED_PUNCTURED_CONVEX THEN ASM_REWRITE_TAC[CONVEX_CBALL; AFF_DIM_CBALL; INT_OF_NUM_EQ] THEN ASM_ARITH_TAC; MATCH_MP_TAC(MESON[PATH_CONNECTED_EMPTY] `s = {} ==> path_connected s`) THEN FIRST_X_ASSUM(DISJ_CASES_TAC o MATCH_MP (REAL_ARITH `~(&0 < r) ==> r = &0 \/ r < &0`)) THEN ASM_SIMP_TAC[CBALL_EMPTY; CBALL_SING] THEN ASM SET_TAC[]]);; let CONNECTED_PUNCTURED_CBALL = prove (`!a:real^N r. 2 <= dimindex(:N) ==> connected(cball(a,r) DELETE a)`, SIMP_TAC[PATH_CONNECTED_PUNCTURED_CBALL; PATH_CONNECTED_IMP_CONNECTED]);; let PATH_CONNECTED_SPHERE = prove (`!a:real^N r. 2 <= dimindex(:N) ==> path_connected(sphere(a,r))`, REPEAT GEN_TAC THEN REWRITE_TAC[sphere; dist] THEN ONCE_REWRITE_TAC[NORM_SUB] THEN GEOM_ORIGIN_TAC `a:real^N` THEN GEN_TAC THEN REWRITE_TAC[VECTOR_SUB_RZERO] THEN DISCH_TAC THEN REPEAT_TCL DISJ_CASES_THEN ASSUME_TAC (REAL_ARITH `r < &0 \/ r = &0 \/ &0 < r`) THENL [ASM_SIMP_TAC[NORM_ARITH `r < &0 ==> ~(norm(x:real^N) = r)`] THEN REWRITE_TAC[EMPTY_GSPEC; PATH_CONNECTED_EMPTY]; ASM_REWRITE_TAC[NORM_EQ_0; SING_GSPEC; PATH_CONNECTED_SING]; SUBGOAL_THEN `{x:real^N | norm x = r} = IMAGE (\x. r / norm x % x) ((:real^N) DIFF {vec 0})` SUBST1_TAC THENL [MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_IMAGE; IN_ELIM_THM; IN_DIFF; IN_SING; IN_UNIV] THEN ASM_SIMP_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM; REAL_DIV_RMUL; NORM_EQ_0; REAL_ARITH `&0 < r ==> abs r = r`] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN EXISTS_TAC `x:real^N` THEN ASM_SIMP_TAC[REAL_DIV_REFL; REAL_LT_IMP_NZ; VECTOR_MUL_LID] THEN ASM_MESON_TAC[NORM_0; REAL_LT_IMP_NZ]; MATCH_MP_TAC PATH_CONNECTED_CONTINUOUS_IMAGE THEN ASM_SIMP_TAC[PATH_CONNECTED_PUNCTURED_UNIVERSE] THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN REWRITE_TAC[CONTINUOUS_ON_ID] THEN REWRITE_TAC[o_DEF; CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_DIFF; IN_UNIV; IN_SING] THEN DISCH_TAC THEN REWRITE_TAC[real_div; LIFT_CMUL] THEN MATCH_MP_TAC CONTINUOUS_CMUL THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_AT_WITHIN_INV) THEN ASM_REWRITE_TAC[NORM_EQ_0] THEN MATCH_MP_TAC CONTINUOUS_AT_WITHIN THEN REWRITE_TAC[REWRITE_RULE[o_DEF] CONTINUOUS_AT_LIFT_NORM]]]);; let CONNECTED_SPHERE = prove (`!a:real^N r. 2 <= dimindex(:N) ==> connected(sphere(a,r))`, SIMP_TAC[PATH_CONNECTED_SPHERE; PATH_CONNECTED_IMP_CONNECTED]);; let CONNECTED_SPHERE_EQ = prove (`!a:real^N r. connected(sphere(a,r)) <=> 2 <= dimindex(:N) \/ r <= &0`, let lemma = prove (`!a:real^1 r. &0 < r ==> ?x y. ~(x = y) /\ dist(a,x) = r /\ dist(a,y) = r`, MP_TAC SPHERE_1 THEN REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN COND_CASES_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[EXTENSION; IN_SPHERE; IN_INSERT; NOT_IN_EMPTY] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(MESON[] `~(a = b) ==> ?x y. ~(x = y) /\ (x = a \/ x = b) /\ (y = a \/ y = b)`) THEN REWRITE_TAC[VECTOR_ARITH `a - r:real^1 = a + r <=> r = vec 0`] THEN REWRITE_TAC[GSYM DROP_EQ; DROP_VEC; LIFT_DROP] THEN ASM_REAL_ARITH_TAC) in REPEAT GEN_TAC THEN ASM_CASES_TAC `r < &0` THEN ASM_SIMP_TAC[SPHERE_EMPTY; CONNECTED_EMPTY; REAL_LT_IMP_LE] THEN ASM_CASES_TAC `r = &0` THEN ASM_SIMP_TAC[SPHERE_SING; REAL_LE_REFL; CONNECTED_SING] THEN SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ASM_REWRITE_TAC[GSYM REAL_NOT_LT]] THEN EQ_TAC THEN SIMP_TAC[CONNECTED_SPHERE] THEN DISCH_THEN(MP_TAC o MATCH_MP CONNECTED_FINITE_IFF_SING) THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN SIMP_TAC[DIMINDEX_GE_1; ARITH_RULE `1 <= n ==> (2 <= n <=> ~(n = 1))`] THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM DIMINDEX_1] THEN DISCH_TAC THEN FIRST_ASSUM (fun th -> REWRITE_TAC[GEOM_EQUAL_DIMENSION_RULE th FINITE_SPHERE_1]) THEN REWRITE_TAC[SET_RULE `~(s = {} \/ ?a. s = {a}) <=> ?x y. ~(x = y) /\ x IN s /\ y IN s`] THEN REWRITE_TAC[IN_SPHERE] THEN FIRST_X_ASSUM(MATCH_MP_TAC o C GEOM_EQUAL_DIMENSION_RULE lemma) THEN ASM_REWRITE_TAC[]);; let PATH_CONNECTED_SPHERE_EQ = prove (`!a:real^N r. path_connected(sphere(a,r)) <=> 2 <= dimindex(:N) \/ r <= &0`, REPEAT GEN_TAC THEN EQ_TAC THENL [REWRITE_TAC[GSYM CONNECTED_SPHERE_EQ; PATH_CONNECTED_IMP_CONNECTED]; STRIP_TAC THEN ASM_SIMP_TAC[PATH_CONNECTED_SPHERE]] THEN ASM_CASES_TAC `r < &0` THEN ASM_SIMP_TAC[SPHERE_EMPTY; PATH_CONNECTED_EMPTY] THEN ASM_CASES_TAC `r = &0` THEN ASM_SIMP_TAC[SPHERE_SING; PATH_CONNECTED_SING] THEN ASM_REAL_ARITH_TAC);; let FINITE_SPHERE = prove (`!a:real^N r. FINITE(sphere(a,r)) <=> r <= &0 \/ dimindex(:N) = 1`, REPEAT GEN_TAC THEN ASM_CASES_TAC `dimindex(:N) = 1` THEN ASM_REWRITE_TAC[] THENL [RULE_ASSUM_TAC(REWRITE_RULE[GSYM DIMINDEX_1]) THEN FIRST_ASSUM(MATCH_ACCEPT_TAC o C PROVE_HYP (GEOM_EQUAL_DIMENSION_RULE(ASSUME `dimindex(:N) = dimindex(:1)`) FINITE_SPHERE_1)); ASM_SIMP_TAC[CONNECTED_SPHERE; ARITH_RULE `2 <= n <=> 1 <= n /\ ~(n = 1)`; DIMINDEX_GE_1; CONNECTED_FINITE_IFF_SING] THEN REWRITE_TAC[SET_RULE `(s = {} \/ ?a. s = {a}) <=> (!a b. a IN s /\ b IN s ==> a = b)`] THEN SIMP_TAC[IN_SPHERE] THEN EQ_TAC THENL [ALL_TAC; CONV_TAC NORM_ARITH] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[REAL_NOT_LE] THEN DISCH_TAC THEN MP_TAC(ISPECL [`a:real^N`; `r:real`] VECTOR_CHOOSE_DIST) THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN DISCH_THEN(MP_TAC o SPECL [`x:real^N`; `a - (x - a):real^N`]) THEN FIRST_X_ASSUM(K ALL_TAC o check (is_neg o concl)) THEN REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC NORM_ARITH]);; let LIMIT_POINT_OF_SPHERE = prove (`!a r x:real^N. x limit_point_of sphere(a,r) <=> &0 < r /\ 2 <= dimindex(:N) /\ x IN sphere(a,r)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `FINITE(sphere(a:real^N,r))` THENL [ASM_SIMP_TAC[LIMIT_POINT_FINITE]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o REWRITE_RULE[FINITE_SPHERE]) THEN REWRITE_TAC[DE_MORGAN_THM] THEN STRIP_TAC THEN ASM_REWRITE_TAC[REAL_NOT_LE; ARITH; REAL_NOT_LT] THEN ASM_SIMP_TAC[GSYM REAL_NOT_LE; DIMINDEX_GE_1; ARITH_RULE `1 <= n ==> (2 <= n <=> ~(n = 1))`] THEN EQ_TAC THEN REWRITE_TAC[REWRITE_RULE[CLOSED_LIMPT] CLOSED_SPHERE] THEN DISCH_TAC THEN MATCH_MP_TAC CONNECTED_IMP_PERFECT THEN ASM_SIMP_TAC[CONNECTED_SPHERE_EQ; DIMINDEX_GE_1; ARITH_RULE `1 <= n ==> (2 <= n <=> ~(n = 1))`] THEN ASM_MESON_TAC[FINITE_SING]);; let CARD_EQ_SPHERE = prove (`!a:real^N r. 2 <= dimindex(:N) /\ &0 < r ==> sphere(a,r) =_c (:real)`, SIMP_TAC[CONNECTED_CARD_EQ_IFF_NONTRIVIAL; CONNECTED_SPHERE] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP(REWRITE_RULE[IMP_CONJ_ALT] FINITE_SUBSET)) THEN ASM_REWRITE_TAC[FINITE_SING; FINITE_SPHERE; REAL_NOT_LE; DE_MORGAN_THM] THEN ASM_ARITH_TAC);; let HAS_SIZE_SPHERE_2 = prove (`!a:real^N r. sphere(a,r) HAS_SIZE 2 <=> dimindex(:N) = 1 /\ &0 < r`, REPEAT GEN_TAC THEN ASM_CASES_TAC `r < &0` THEN ASM_SIMP_TAC[HAS_SIZE; SPHERE_EMPTY; CARD_CLAUSES] THENL [CONV_TAC NUM_REDUCE_CONV THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_CASES_TAC `r = &0` THEN ASM_SIMP_TAC[SPHERE_SING; CARD_CLAUSES; FINITE_EMPTY; NOT_IN_EMPTY] THEN REWRITE_TAC[REAL_LT_REFL] THEN CONV_TAC NUM_REDUCE_CONV THEN SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_CASES_TAC `r <= &0` THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_REWRITE_TAC[FINITE_SPHERE] THEN ASM_CASES_TAC `dimindex(:N) = 1` THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `sphere(a:real^N,r) = {a - r % basis 1,a + r % basis 1}` SUBST1_TAC THENL [ASM_REWRITE_TAC[EXTENSION; IN_SPHERE; dist; vector_norm; dot] THEN REWRITE_TAC[SUM_1; GSYM REAL_POW_2; POW_2_SQRT_ABS] THEN ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; CART_EQ; FORALL_1] THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN SIMP_TAC[BASIS_COMPONENT; LE_REFL; DIMINDEX_GE_1] THEN ASM_REAL_ARITH_TAC; SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY; NOT_IN_EMPTY] THEN SIMP_TAC[IN_SING; VECTOR_ARITH `a - r:real^N = a + r <=> r = vec 0`] THEN ASM_SIMP_TAC[VECTOR_MUL_EQ_0; BASIS_NONZERO; LE_REFL; DIMINDEX_GE_1] THEN CONV_TAC NUM_REDUCE_CONV]);; let LOCALLY_PATH_CONNECTED_SPHERE = prove (`!(a:real^N) r. locally path_connected (sphere(a:real^N,r))`, REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `a:real^N` THEN GEN_TAC THEN REWRITE_TAC[GSYM LOCALLY_PATH_CONNECTED_SPACE_SUBTOPOLOGY_EUCLIDEAN] THEN ASM_CASES_TAC `FINITE(sphere(vec 0:real^N,r))` THEN ASM_SIMP_TAC[SUBTOPOLOGY_EUCLIDEAN_EQ_DISCRETE_TOPOLOGY_FINITE; LOCALLY_PATH_CONNECTED_SPACE_DISCRETE_TOPOLOGY] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [FINITE_SPHERE]) THEN REWRITE_TAC[DE_MORGAN_THM; REAL_NOT_LE] THEN STRIP_TAC THEN MATCH_MP_TAC(ISPEC `subtopology euclidean ((:real^N) DELETE vec 0)` LOCALLY_PATH_CONNECTED_SPACE_QUOTIENT_MAP_IMAGE) THEN EXISTS_TAC `\x:real^N. r / norm(x) % x` THEN CONJ_TAC THENL [MATCH_MP_TAC RETRACTION_IMP_QUOTIENT_MAP THEN REWRITE_TAC[retraction_map; retraction_maps] THEN EXISTS_TAC `\x:real^N. x` THEN REWRITE_TAC[CONTINUOUS_MAP_EUCLIDEAN2] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; CONTINUOUS_ON_ID] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_DELETE; IN_SPHERE_0] THEN REWRITE_TAC[IN_UNIV; NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM] THEN ASM_SIMP_TAC[GSYM NORM_EQ_0; REAL_DIV_RMUL; REAL_DIV_REFL; REAL_LT_IMP_NZ; VECTOR_MUL_LID; REAL_ARITH `&0 < r ==> abs r = r`] THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN REWRITE_TAC[CONTINUOUS_ON_ID; o_DEF; real_div; LIFT_CMUL] THEN MATCH_MP_TAC CONTINUOUS_ON_CMUL THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN REWRITE_TAC[NORM_EQ_0; IN_DELETE; IN_UNIV] THEN REWRITE_TAC[REWRITE_RULE[o_DEF] CONTINUOUS_ON_LIFT_NORM]; MATCH_MP_TAC LOCALLY_PATH_CONNECTED_SPACE_OPEN_SUBSET THEN SIMP_TAC[GSYM OPEN_IN; OPEN_DELETE; OPEN_UNIV] THEN GEN_REWRITE_TAC RAND_CONV [GSYM SUBTOPOLOGY_TOPSPACE] THEN REWRITE_TAC[LOCALLY_PATH_CONNECTED_SPACE_SUBTOPOLOGY_EUCLIDEAN] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN; LOCALLY_PATH_CONNECTED_UNIV]]);; let LOCALLY_CONNECTED_SPHERE = prove (`!a:real^N r. locally connected(sphere(a,r))`, SIMP_TAC[LOCALLY_PATH_CONNECTED_SPHERE; LOCALLY_PATH_CONNECTED_IMP_LOCALLY_CONNECTED]);; let FINITE_CIRCLE_INTERSECTION,CARD_CIRCLE_INTERSECTION_LE = (CONJ_PAIR o prove) (`(!a b:real^2 r s. FINITE(sphere(a,r) INTER sphere(b,s)) <=> ~(a = b /\ r = s /\ &0 < r /\ &0 < s)) /\ (!a b:real^2 r s. ~(a = b /\ r = s /\ &0 < r /\ &0 < s) ==> CARD(sphere(a,r) INTER sphere(b,s)) <= 2)`, REWRITE_TAC[AND_FORALL_THM] THEN REPEAT GEN_TAC THEN ASM_CASES_TAC `&0 < r` THEN ASM_REWRITE_TAC[] THENL [ALL_TAC; MATCH_MP_TAC(MESON[FINITE_SUBSET; CARD_SUBSET; LE_TRANS] `!t. s SUBSET t /\ FINITE t /\ CARD t <= n ==> FINITE s /\ CARD s <= n`) THEN EXISTS_TAC `{a:real^2}` THEN REWRITE_TAC[FINITE_SING; CARD_SING; SUBSET; IN_SING; IN_INTER; ARITH] THEN REWRITE_TAC[IN_SPHERE] THEN POP_ASSUM MP_TAC THEN CONV_TAC NORM_ARITH] THEN ASM_CASES_TAC `&0 < s` THEN ASM_REWRITE_TAC[] THENL [ALL_TAC; MATCH_MP_TAC(MESON[FINITE_SUBSET; CARD_SUBSET; LE_TRANS] `!t. s SUBSET t /\ FINITE t /\ CARD t <= n ==> FINITE s /\ CARD s <= n`) THEN EXISTS_TAC `{b:real^2}` THEN REWRITE_TAC[FINITE_SING; CARD_SING; SUBSET; IN_SING; IN_INTER; ARITH] THEN REWRITE_TAC[IN_SPHERE] THEN POP_ASSUM MP_TAC THEN CONV_TAC NORM_ARITH] THEN ASM_CASES_TAC `a:real^2 = b /\ r:real = s` THEN ASM_REWRITE_TAC[INTER_IDEMPOT; FINITE_SPHERE] THEN ASM_REWRITE_TAC[DIMINDEX_2; ARITH_EQ; REAL_NOT_LE] THEN REWRITE_TAC[ARITH_RULE `n <= 2 <=> ~(3 <= n)`] THEN MP_TAC(ISPECL [`3`; `sphere(a:real^2,r) INTER sphere(b,s)`] CHOOSE_SUBSET_STRONG) THEN MATCH_MP_TAC(TAUT `~r ==> ((p ==> q) ==> r) ==> p /\ ~q`) THEN CONV_TAC (ONCE_DEPTH_CONV HAS_SIZE_CONV) THEN DISCH_THEN(CHOOSE_THEN (CONJUNCTS_THEN MP_TAC)) THEN SIMP_TAC[LEFT_IMP_EXISTS_THM] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM SUBST_ALL_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (TAUT `~(p /\ q) ==> p ==> ~q`)) THEN REPEAT(FIRST_X_ASSUM(SUBST1_TAC o MATCH_MP (REAL_ARITH `&0 < r ==> r = abs r`))) THEN REWRITE_TAC[REAL_EQ_SQUARE_ABS] THEN REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; IN_INTER; IN_SPHERE] THEN ONCE_REWRITE_TAC[GSYM DIST_EQ_0] THEN REWRITE_TAC[dist; NORM_EQ_SQUARE; REAL_POS] THEN REWRITE_TAC[DOT_2; VECTOR_SUB_COMPONENT] THEN CONV_TAC REAL_RING);; let INTER_SPHERE_EQ_EMPTY = prove (`!a b:real^N r s. sphere(a,r) INTER sphere(b,s) = {} <=> if dimindex(:N) = 1 then r < &0 \/ s < &0 \/ ~(dist(a,b) = abs(r - s)) /\ ~(dist(a,b) = r + s) else r < &0 \/ s < &0 \/ dist(a,b) < abs(r - s) \/ r + s < dist(a,b)`, REPEAT GEN_TAC THEN COND_CASES_TAC THENL [REWRITE_TAC[EXTENSION; IN_INTER; IN_SPHERE; NOT_IN_EMPTY] THEN REWRITE_TAC[dist; NORM_EQ_SQUARE] THEN ASM_CASES_TAC `&0 <= r` THEN ASM_REWRITE_TAC[GSYM REAL_NOT_LE] THEN ASM_CASES_TAC `&0 <= s` THEN ASM_REWRITE_TAC[] THEN ASM_REWRITE_TAC[dot; SUM_1; VECTOR_SUB_COMPONENT] THEN REWRITE_TAC[GSYM REAL_POW_2; GSYM REAL_EQ_SQUARE_ABS] THEN ASM_SIMP_TAC[REAL_LE_ADD; REAL_ABS_POS; REAL_ABS_ABS] THEN REWRITE_TAC[REAL_ARITH `abs(x - y) = abs r <=> y = x - r \/ y = x + r`] THEN EQ_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN DISCH_THEN(fun th -> MP_TAC(SPEC `(lambda i. (a:real^N)$1 + r):real^N` th) THEN MP_TAC(SPEC `(lambda i. (a:real^N)$1 - r):real^N` th)) THEN ASM_SIMP_TAC[LAMBDA_BETA; ARITH] THEN REAL_ARITH_TAC; EQ_TAC THENL [ALL_TAC; REWRITE_TAC[EXTENSION; IN_INTER; IN_SPHERE; NOT_IN_EMPTY] THEN CONV_TAC NORM_ARITH] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (ARITH_RULE `~(n = 1) ==> 1 <= n ==> 2 <= n`)) THEN REWRITE_TAC[DIMINDEX_GE_1] THEN DISCH_TAC THEN ASM_CASES_TAC `sphere(a:real^N,r) SUBSET cball(b,s)` THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [GSYM SPHERE_UNION_BALL]) THEN ASM_SIMP_TAC[SET_RULE `a INTER b = {} ==> (a SUBSET b UNION c <=> a SUBSET c)`] THEN SIMP_TAC[SPHERE_SUBSET_CONVEX; CONVEX_BALL; SUBSET_BALLS] THEN REAL_ARITH_TAC; ALL_TAC] THEN MP_TAC(ISPECL [`sphere(a:real^N,r)`; `cball(b:real^N,s)`] CONNECTED_INTER_FRONTIER) THEN ASM_SIMP_TAC[CONNECTED_SPHERE; FRONTIER_CBALL; DE_MORGAN_THM] THEN DISCH_THEN DISJ_CASES_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MP_TAC(ISPECL [`cball(b:real^N,s)`; `cball(a:real^N,r)`] CONNECTED_INTER_FRONTIER) THEN REWRITE_TAC[CONNECTED_CBALL; FRONTIER_CBALL] THEN GEN_REWRITE_TAC LAND_CONV [GSYM CONTRAPOS_THM] THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[DE_MORGAN_THM]] THEN REWRITE_TAC[SET_RULE `s DIFF t = {} <=> s SUBSET t`] THEN DISCH_THEN(DISJ_CASES_THEN MP_TAC) THENL [REWRITE_TAC[INTER_BALLS_EQ_EMPTY; DIST_SYM] THEN REAL_ARITH_TAC; GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM SPHERE_UNION_BALL] THEN ASM_SIMP_TAC[SET_RULE `a INTER b = {} ==> (b SUBSET a UNION c <=> b SUBSET c)`] THEN REWRITE_TAC[SUBSET_BALLS; DIST_SYM] THEN REAL_ARITH_TAC]]);; let HAS_SIZE_INTER_SPHERE_1 = prove (`!a b:real^N r s. (sphere(a,r) INTER sphere(b,s)) HAS_SIZE 1 <=> &0 <= r /\ &0 <= s /\ (a = b ==> r = &0 /\ s = &0) /\ (dist(a,b) = r + s \/ dist(a,b) = abs(r - s))`, REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `a:real^N` THEN GEOM_BASIS_MULTIPLE_TAC 1 `b:real^N` THEN SIMP_TAC[DIST_0; NORM_MUL; NORM_BASIS; DIMINDEX_GE_1; LE_REFL] THEN SIMP_TAC[REAL_ARITH `&0 <= b ==> (abs b * &1 = x <=> x = b)`] THEN X_GEN_TAC `b:real` THEN REPEAT STRIP_TAC THEN ASM_CASES_TAC `b = &0` THENL [ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN ASM_CASES_TAC `r:real = s` THEN ASM_REWRITE_TAC[INTER_IDEMPOT; HAS_SIZE_SPHERE_1] THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN EQ_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN CONV_TAC(LAND_CONV HAS_SIZE_CONV) THEN MATCH_MP_TAC(SET_RULE `s = {} ==> (?a. s = {a}) ==> P`) THEN REWRITE_TAC[EXTENSION; IN_INTER; IN_SPHERE_0; NOT_IN_EMPTY] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[CONV_RULE (LAND_CONV SYM_CONV) (SPEC_ALL VECTOR_MUL_EQ_0)] THEN ASM_SIMP_TAC[BASIS_EQ_0; IN_NUMSEG; LE_1; DIMINDEX_GE_1; LE_REFL]] THEN ASM_CASES_TAC `sphere(vec 0:real^N,r) INTER sphere (b % basis 1,s) = {}` THEN ASM_REWRITE_TAC[] THENL [REWRITE_TAC[HAS_SIZE; CARD_CLAUSES; ARITH_EQ] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INTER_SPHERE_EQ_EMPTY]) THEN SIMP_TAC[DIST_0; NORM_MUL; NORM_BASIS; DIMINDEX_GE_1; LE_REFL] THEN ASM_SIMP_TAC[REAL_ARITH `&0 <= b ==> (abs b * &1 = x <=> x = b)`] THEN REAL_ARITH_TAC; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY])] THEN REWRITE_TAC[IN_INTER; IN_SPHERE; DIST_0; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN MATCH_MP_TAC(TAUT `!r. (p ==> r) /\ (r ==> q) /\ (q ==> p) ==> (p <=> q)`) THEN EXISTS_TAC `x:real^N = x$1 % basis 1` THEN REPEAT CONJ_TAC THENL [DISCH_TAC THEN REWRITE_TAC[CART_EQ] THEN SIMP_TAC[VECTOR_MUL_COMPONENT; BASIS_COMPONENT] THEN X_GEN_TAC `k:num` THEN STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_MUL_RID; REAL_MUL_RZERO] THEN FIRST_X_ASSUM(MP_TAC o CONV_RULE HAS_SIZE_CONV) THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `(?a. s = {a}) ==> !x y. x IN s /\ y IN s ==> x = y`)) THEN DISCH_THEN(MP_TAC o SPECL [`x:real^N`; `(lambda i. if i = k then --((x:real^N)$k) else x$i):real^N`]) THEN ANTS_TAC THENL [REWRITE_TAC[IN_INTER; IN_SPHERE] THEN MATCH_MP_TAC(MESON[] `(x = r /\ y = s) /\ (x' = x /\ y' = y) ==> (x = r /\ y = s) /\ (x' = r /\ y' = s)`) THEN CONJ_TAC THENL [ASM_REWRITE_TAC[DIST_0]; ALL_TAC] THEN REWRITE_TAC[dist] THEN CONJ_TAC THEN MATCH_MP_TAC NORM_EQ_COMPONENTWISE THEN SIMP_TAC[VECTOR_SUB_COMPONENT; VEC_COMPONENT; LAMBDA_BETA] THEN SIMP_TAC[VECTOR_MUL_COMPONENT; BASIS_COMPONENT] THEN REPEAT STRIP_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN ASM_REAL_ARITH_TAC; SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN DISCH_THEN(MP_TAC o SPEC `k:num`) THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC]; DISCH_THEN SUBST_ALL_TAC THEN REPEAT(POP_ASSUM MP_TAC) THEN SIMP_TAC[NORM_MUL; NORM_BASIS; DIMINDEX_GE_1; LE_REFL; dist; GSYM VECTOR_SUB_RDISTRIB] THEN REAL_ARITH_TAC; DISCH_TAC THEN CONV_TAC HAS_SIZE_CONV THEN EXISTS_TAC `x:real^N` THEN REWRITE_TAC[EXTENSION; IN_INTER; IN_SPHERE; DIST_0; IN_SING] THEN X_GEN_TAC `y:real^N` THEN EQ_TAC THEN ASM_SIMP_TAC[] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `x:real^N = x$1 % basis 1 /\ y:real^N = y$1 % basis 1` (CONJUNCTS_THEN SUBST_ALL_TAC) THENL [ALL_TAC; AP_THM_TAC THEN AP_TERM_TAC THEN REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[dist; GSYM VECTOR_SUB_RDISTRIB; NORM_MUL] THEN SIMP_TAC[NORM_BASIS; DIMINDEX_GE_1; LE_REFL] THEN REAL_ARITH_TAC] THEN CONJ_TAC THEN MP_TAC(ISPEC `b % basis 1:real^N` COLLINEAR_LEMMA_ALT) THENL [DISCH_THEN(MP_TAC o SPEC `x:real^N`); DISCH_THEN(MP_TAC o SPEC `y:real^N`)] THEN ASM_SIMP_TAC[VECTOR_MUL_EQ_0; BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL] THEN MATCH_MP_TAC(TAUT `p /\ (q ==> r) ==> (p <=> q) ==> r`) THEN SIMP_TAC[LEFT_IMP_EXISTS_THM; VECTOR_MUL_COMPONENT; VECTOR_MUL_ASSOC] THEN SIMP_TAC[BASIS_COMPONENT; DIMINDEX_GE_1; LE_REFL; REAL_MUL_RID] THEN REWRITE_TAC[COLLINEAR_BETWEEN_CASES; between] THEN POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN REWRITE_TAC[DIST_SYM; DIST_0; NORM_MUL] THEN SIMP_TAC[NORM_BASIS; DIMINDEX_GE_1; LE_REFL; real_abs] THEN REAL_ARITH_TAC]);; let PATH_CONNECTED_ANNULUS = prove (`(!a:real^N r1 r2. 2 <= dimindex(:N) ==> path_connected {x | r1 < norm(x - a) /\ norm(x - a) < r2}) /\ (!a:real^N r1 r2. 2 <= dimindex(:N) ==> path_connected {x | r1 < norm(x - a) /\ norm(x - a) <= r2}) /\ (!a:real^N r1 r2. 2 <= dimindex(:N) ==> path_connected {x | r1 <= norm(x - a) /\ norm(x - a) < r2}) /\ (!a:real^N r1 r2. 2 <= dimindex(:N) ==> path_connected {x | r1 <= norm(x - a) /\ norm(x - a) < r2})`, let lemma = prove (`!a:real^N P. 2 <= dimindex(:N) /\ path_connected {lift r | &0 <= r /\ P r} ==> path_connected {x | P(norm(x - a))}`, REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `a:real^N` THEN REWRITE_TAC[VECTOR_SUB_RZERO] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `{x:real^N | P(norm(x))} = IMAGE (\z. drop(fstcart z) % sndcart z) {pastecart x y | x IN {lift x | &0 <= x /\ P x} /\ y IN {y | norm y = &1}}` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; IN_IMAGE] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[EXISTS_IN_GSPEC; FSTCART_PASTECART; SNDCART_PASTECART] THEN X_GEN_TAC `z:real^N` THEN REWRITE_TAC[EXISTS_LIFT; LIFT_DROP] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN REWRITE_TAC[LIFT_IN_IMAGE_LIFT; IMAGE_ID] THEN REWRITE_TAC[IN_ELIM_THM] THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[NORM_MUL; REAL_MUL_RID] THEN ASM_REWRITE_TAC[real_abs] THEN ASM_CASES_TAC `z:real^N = vec 0` THENL [MAP_EVERY EXISTS_TAC [`&0`; `basis 1:real^N`] THEN ASM_SIMP_TAC[NORM_BASIS; DIMINDEX_GE_1; LE_REFL; VECTOR_MUL_LZERO] THEN ASM_MESON_TAC[NORM_0; REAL_ABS_NUM; REAL_LE_REFL]; MAP_EVERY EXISTS_TAC [`norm(z:real^N)`; `inv(norm z) % z:real^N`] THEN ASM_SIMP_TAC[REAL_ABS_NORM; NORM_MUL; VECTOR_MUL_ASSOC; VECTOR_MUL_LID; NORM_POS_LE; REAL_ABS_INV; REAL_MUL_RINV; REAL_MUL_LINV; NORM_EQ_0]]; MATCH_MP_TAC PATH_CONNECTED_CONTINUOUS_IMAGE THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_MUL THEN REWRITE_TAC[o_DEF; LIFT_DROP; ETA_AX] THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; LINEAR_SNDCART]; REWRITE_TAC[GSYM PCROSS] THEN MATCH_MP_TAC PATH_CONNECTED_PCROSS THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[NORM_ARITH `norm y = norm(y - vec 0:real^N)`] THEN ONCE_REWRITE_TAC[NORM_SUB] THEN REWRITE_TAC[REWRITE_RULE[dist] (GSYM sphere)] THEN ASM_SIMP_TAC[PATH_CONNECTED_SPHERE]]]) in REPEAT STRIP_TAC THEN MP_TAC(ISPEC `a:real^N` lemma) THEN DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONVEX_IMP_PATH_CONNECTED THEN MATCH_MP_TAC IS_INTERVAL_CONVEX THEN REWRITE_TAC[is_interval] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN REWRITE_TAC[IN_IMAGE_LIFT_DROP; FORALL_1; DIMINDEX_1] THEN REWRITE_TAC[IN_ELIM_THM; GSYM drop] THEN REAL_ARITH_TAC);; let CONNECTED_ANNULUS = prove (`(!a:real^N r1 r2. 2 <= dimindex(:N) ==> connected {x | r1 < norm(x - a) /\ norm(x - a) < r2}) /\ (!a:real^N r1 r2. 2 <= dimindex(:N) ==> connected {x | r1 < norm(x - a) /\ norm(x - a) <= r2}) /\ (!a:real^N r1 r2. 2 <= dimindex(:N) ==> connected {x | r1 <= norm(x - a) /\ norm(x - a) < r2}) /\ (!a:real^N r1 r2. 2 <= dimindex(:N) ==> connected {x | r1 <= norm(x - a) /\ norm(x - a) < r2})`, REPEAT STRIP_TAC THEN MATCH_MP_TAC PATH_CONNECTED_IMP_CONNECTED THEN ASM_SIMP_TAC[PATH_CONNECTED_ANNULUS]);; let PATH_CONNECTED_COMPLEMENT_BOUNDED_CONVEX = prove (`!s. 2 <= dimindex(:N) /\ bounded s /\ convex s ==> path_connected((:real^N) DIFF s)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_SIMP_TAC[DIFF_EMPTY; CONVEX_IMP_PATH_CONNECTED; CONVEX_UNIV] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN REWRITE_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN REWRITE_TAC[IN_DIFF; IN_UNIV] THEN STRIP_TAC THEN SUBGOAL_THEN `~(x:real^N = a) /\ ~(y = a)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `bounded((x:real^N) INSERT y INSERT s)` MP_TAC THENL [ASM_REWRITE_TAC[BOUNDED_INSERT]; ALL_TAC] THEN DISCH_THEN(MP_TAC o SPEC `a:real^N` o MATCH_MP BOUNDED_SUBSET_BALL) THEN REWRITE_TAC[INSERT_SUBSET] THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC PATH_COMPONENT_TRANS THEN ABBREV_TAC `C = (B / norm(x - a:real^N))` THEN EXISTS_TAC `a + C % (x - a):real^N` THEN CONJ_TAC THENL [MATCH_MP_TAC PATH_CONNECTED_LINEPATH THEN REWRITE_TAC[SUBSET; segment; FORALL_IN_GSPEC; IN_DIFF; IN_UNIV] THEN REWRITE_TAC[VECTOR_ARITH `(&1 - u) % x + u % (a + B % (x - a)):real^N = a + (&1 + (B - &1) * u) % (x - a)`] THEN X_GEN_TAC `u:real` THEN STRIP_TAC THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONVEX_ALT]) THEN DISCH_THEN(MP_TAC o SPECL [`a:real^N`; `a + (&1 + (C - &1) * u) % (x - a):real^N`; `&1 / (&1 + (C - &1) * u)`]) THEN SUBGOAL_THEN `&1 <= &1 + (C - &1) * u` ASSUME_TAC THENL [REWRITE_TAC[REAL_LE_ADDR] THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_REWRITE_TAC[REAL_SUB_LE] THEN EXPAND_TAC "C" THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_BALL; dist]) THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; NORM_ARITH `&1 * norm(x - a) = norm(a - x)`]; FIRST_ASSUM(ASSUME_TAC o MATCH_MP (REAL_ARITH `&1 <= a ==> &0 < a`))] THEN ASM_REWRITE_TAC[NOT_IMP] THEN ASM_SIMP_TAC[REAL_LE_DIV; REAL_POS; REAL_LT_IMP_LE; REAL_LE_LDIV_EQ; REAL_MUL_LID] THEN ASM_SIMP_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC; REAL_DIV_RMUL; REAL_LT_IMP_NZ] THEN UNDISCH_TAC `~((x:real^N) IN s)` THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN VECTOR_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC PATH_COMPONENT_SYM THEN MATCH_MP_TAC PATH_COMPONENT_TRANS THEN ABBREV_TAC `D = (B / norm(y - a:real^N))` THEN EXISTS_TAC `a + D % (y - a):real^N` THEN CONJ_TAC THENL [MATCH_MP_TAC PATH_CONNECTED_LINEPATH THEN REWRITE_TAC[SUBSET; segment; FORALL_IN_GSPEC; IN_DIFF; IN_UNIV] THEN REWRITE_TAC[VECTOR_ARITH `(&1 - u) % y + u % (a + B % (y - a)):real^N = a + (&1 + (B - &1) * u) % (y - a)`] THEN X_GEN_TAC `u:real` THEN STRIP_TAC THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONVEX_ALT]) THEN DISCH_THEN(MP_TAC o SPECL [`a:real^N`; `a + (&1 + (D - &1) * u) % (y - a):real^N`; `&1 / (&1 + (D - &1) * u)`]) THEN SUBGOAL_THEN `&1 <= &1 + (D - &1) * u` ASSUME_TAC THENL [REWRITE_TAC[REAL_LE_ADDR] THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_REWRITE_TAC[REAL_SUB_LE] THEN EXPAND_TAC "D" THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_BALL; dist]) THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; NORM_ARITH `&1 * norm(y - a) = norm(a - y)`]; FIRST_ASSUM(ASSUME_TAC o MATCH_MP (REAL_ARITH `&1 <= a ==> &0 < a`))] THEN ASM_REWRITE_TAC[NOT_IMP] THEN ASM_SIMP_TAC[REAL_LE_DIV; REAL_POS; REAL_LT_IMP_LE; REAL_LE_LDIV_EQ; REAL_MUL_LID] THEN ASM_SIMP_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC; REAL_DIV_RMUL; REAL_LT_IMP_NZ] THEN UNDISCH_TAC `~((y:real^N) IN s)` THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN VECTOR_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC PATH_COMPONENT_OF_SUBSET THEN EXISTS_TAC `{x:real^N | norm(x - a) = B}` THEN CONJ_TAC THENL [UNDISCH_TAC `s SUBSET ball(a:real^N,B)` THEN REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_DIFF; IN_UNIV; IN_BALL; dist] THEN MESON_TAC[NORM_SUB; REAL_LT_REFL]; MP_TAC(ISPECL [`a:real^N`; `B:real`] PATH_CONNECTED_SPHERE) THEN REWRITE_TAC[REWRITE_RULE[ONCE_REWRITE_RULE[DIST_SYM] dist] sphere] THEN ASM_REWRITE_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT] THEN DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[IN_ELIM_THM; VECTOR_ADD_SUB; NORM_MUL] THEN MAP_EVERY EXPAND_TAC ["C"; "D"] THEN REWRITE_TAC[REAL_ABS_DIV; REAL_ABS_NORM] THEN ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_EQ_0; VECTOR_SUB_EQ] THEN ASM_REAL_ARITH_TAC]);; let CONNECTED_COMPLEMENT_BOUNDED_CONVEX = prove (`!s. 2 <= dimindex(:N) /\ bounded s /\ convex s ==> connected((:real^N) DIFF s)`, SIMP_TAC[PATH_CONNECTED_IMP_CONNECTED; PATH_CONNECTED_COMPLEMENT_BOUNDED_CONVEX]);; let CONNECTED_DIFF_BALL = prove (`!s a:real^N r. 2 <= dimindex(:N) /\ connected s /\ cball(a,r) SUBSET s ==> connected(s DIFF ball(a,r))`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CONNECTED_DIFF_OPEN_FROM_CLOSED THEN EXISTS_TAC `cball(a:real^N,r)` THEN ASM_REWRITE_TAC[OPEN_BALL; CLOSED_CBALL; BALL_SUBSET_CBALL] THEN REWRITE_TAC[CBALL_DIFF_BALL] THEN REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] dist] THEN ASM_SIMP_TAC[CONNECTED_SPHERE]);; let PATH_CONNECTED_DIFF_BALL = prove (`!s a:real^N r. 2 <= dimindex(:N) /\ path_connected s /\ cball(a,r) SUBSET s ==> path_connected(s DIFF ball(a,r))`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `ball(a:real^N,r) = {}` THEN ASM_SIMP_TAC[DIFF_EMPTY] THEN RULE_ASSUM_TAC(REWRITE_RULE[BALL_EQ_EMPTY; REAL_NOT_LE]) THEN REWRITE_TAC[path_connected] THEN FIRST_ASSUM(MP_TAC o SPEC `a:real^N` o GEN_REWRITE_RULE I [SUBSET]) THEN ASM_SIMP_TAC[CENTRE_IN_CBALL; REAL_LT_IMP_LE] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN REWRITE_TAC[IN_DIFF] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [path_connected]) THEN DISCH_THEN(fun th -> MP_TAC(SPECL [`x:real^N`; `a:real^N`] th) THEN MP_TAC(SPECL [`y:real^N`; `a:real^N`] th)) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `g2:real^1->real^N` STRIP_ASSUME_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `g1:real^1->real^N` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`g2:real^1->real^N`; `(:real^N) DIFF ball(a,r)`] EXISTS_PATH_SUBPATH_TO_FRONTIER) THEN MP_TAC(ISPECL [`g1:real^1->real^N`; `(:real^N) DIFF ball(a,r)`] EXISTS_PATH_SUBPATH_TO_FRONTIER) THEN ASM_SIMP_TAC[CENTRE_IN_BALL; IN_DIFF; IN_UNIV; LEFT_IMP_EXISTS_THM] THEN ASM_SIMP_TAC[FRONTIER_COMPLEMENT; INTERIOR_COMPLEMENT; CLOSURE_BALL] THEN ASM_SIMP_TAC[FRONTIER_BALL; IN_SPHERE] THEN X_GEN_TAC `h1:real^1->real^N` THEN STRIP_TAC THEN X_GEN_TAC `h2:real^1->real^N` THEN STRIP_TAC THEN MP_TAC(ISPECL [`a:real^N`; `r:real`] PATH_CONNECTED_SPHERE) THEN ASM_REWRITE_TAC[path_connected] THEN DISCH_THEN(MP_TAC o SPECL [`pathfinish h1:real^N`; `pathfinish h2:real^N`]) THEN ASM_SIMP_TAC[IN_SPHERE] THEN DISCH_THEN(X_CHOOSE_THEN `h:real^1->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `h1 ++ h ++ reversepath h2:real^1->real^N` THEN ASM_SIMP_TAC[PATHSTART_JOIN; PATHFINISH_JOIN; PATHSTART_REVERSEPATH; PATHFINISH_REVERSEPATH; PATH_JOIN; PATH_REVERSEPATH; PATH_IMAGE_JOIN; PATH_IMAGE_REVERSEPATH] THEN REWRITE_TAC[UNION_SUBSET] THEN REPEAT CONJ_TAC THENL [ALL_TAC; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] SUBSET_TRANS)) THEN UNDISCH_TAC `cball(a:real^N,r) SUBSET s` THEN SIMP_TAC[SUBSET; IN_CBALL; IN_SPHERE; IN_BALL; IN_DIFF] THEN MESON_TAC[REAL_LE_REFL; REAL_LT_REFL]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t /\ s INTER u = {} ==> s SUBSET t DIFF u`) THEN (CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s DELETE a SUBSET (UNIV DIFF t) ==> ~(a IN u) /\ u SUBSET t ==> s INTER u = {}`)) THEN ASM_REWRITE_TAC[BALL_SUBSET_CBALL; IN_BALL; REAL_LT_REFL]);; let CONNECTED_DELETE_INTERIOR_POINT = prove (`!s a:real^N. 2 <= dimindex(:N) /\ connected s /\ a IN interior s ==> connected(s DELETE a)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERIOR_CBALL]) THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `s DELETE a = (s DIFF ball(a:real^N,r)) UNION (cball(a,r) DELETE a)` SUBST1_TAC THENL [MP_TAC(ISPECL [`a:real^N`; `r:real`] BALL_SUBSET_CBALL) THEN MP_TAC(ISPECL [`a:real^N`; `r:real`] CENTRE_IN_BALL) THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; MATCH_MP_TAC CONNECTED_UNION THEN ASM_SIMP_TAC[CONNECTED_DIFF_BALL; CONNECTED_PUNCTURED_CBALL] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `c SUBSET s ==> ~(c DIFF b = {}) /\ a IN b ==> ~((s DIFF b) INTER (c DELETE a) = {})`)) THEN ASM_REWRITE_TAC[CBALL_DIFF_BALL; CENTRE_IN_BALL; SPHERE_EQ_EMPTY] THEN ASM_REAL_ARITH_TAC]);; let CONNECTED_DELETE_INTERIOR_POINT_EQ = prove (`!s a:real^N. 2 <= dimindex(:N) /\ a IN interior s ==> (connected (s DELETE a) <=> connected s)`, REPEAT STRIP_TAC THEN EQ_TAC THEN DISCH_TAC THEN ASM_SIMP_TAC[CONNECTED_DELETE_INTERIOR_POINT] THEN FIRST_ASSUM (ASSUME_TAC o MATCH_MP (REWRITE_RULE[SUBSET] INTERIOR_SUBSET)) THEN SUBGOAL_THEN `s = (a:real^N) INSERT (s DELETE a)` SUBST1_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC CONNECTED_INSERT_LIMPT] THEN ASM_SIMP_TAC[LIMPT_DELETE; INTERIOR_LIMIT_POINT]);; let CONNECTED_OPEN_DELETE_EQ = prove (`!s a:real^N. 2 <= dimindex(:N) /\ open s ==> (connected(s DELETE a) <=> connected s)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `s DELETE (a:real^N) = s \/ a IN s` STRIP_ASSUME_TAC THENL [SET_TAC[]; ASM_REWRITE_TAC[]; ALL_TAC] THEN MATCH_MP_TAC CONNECTED_DELETE_INTERIOR_POINT_EQ THEN ASM_SIMP_TAC[INTERIOR_OPEN]);; let PATH_CONNECTED_DELETE_INTERIOR_POINT = prove (`!s a:real^N. 2 <= dimindex(:N) /\ path_connected s /\ a IN interior s ==> path_connected(s DELETE a)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERIOR_CBALL]) THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `s DELETE a = (s DIFF ball(a:real^N,r)) UNION (cball(a,r) DELETE a)` SUBST1_TAC THENL [MP_TAC(ISPECL [`a:real^N`; `r:real`] BALL_SUBSET_CBALL) THEN MP_TAC(ISPECL [`a:real^N`; `r:real`] CENTRE_IN_BALL) THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; MATCH_MP_TAC PATH_CONNECTED_UNION THEN ASM_SIMP_TAC[PATH_CONNECTED_DIFF_BALL; PATH_CONNECTED_PUNCTURED_CBALL] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `c SUBSET s ==> ~(c DIFF b = {}) /\ a IN b ==> ~((s DIFF b) INTER (c DELETE a) = {})`)) THEN ASM_REWRITE_TAC[CBALL_DIFF_BALL; CENTRE_IN_BALL; SPHERE_EQ_EMPTY] THEN ASM_REAL_ARITH_TAC]);; let CONNECTED_OPEN_DIFF_CBALL = prove (`!s a:real^N r. 2 <= dimindex (:N) /\ open s /\ connected s /\ cball(a,r) SUBSET s ==> connected(s DIFF cball(a,r))`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `cball(a:real^N,r) = {}` THEN ASM_REWRITE_TAC[DIFF_EMPTY] THEN RULE_ASSUM_TAC(REWRITE_RULE[CBALL_EQ_EMPTY; REAL_NOT_LT]) THEN SUBGOAL_THEN `?r'. r < r' /\ cball(a:real^N,r') SUBSET s` STRIP_ASSUME_TAC THENL [ASM_CASES_TAC `s = (:real^N)` THENL [EXISTS_TAC `r + &1` THEN ASM_SIMP_TAC[SUBSET_UNIV] THEN REAL_ARITH_TAC; ALL_TAC] THEN MP_TAC(ISPECL [`cball(a:real^N,r)`; `(:real^N) DIFF s`] SETDIST_POS_LE) THEN REWRITE_TAC[REAL_ARITH `&0 <= x <=> &0 < x \/ x = &0`] THEN ASM_SIMP_TAC[SETDIST_EQ_0_COMPACT_CLOSED; GSYM OPEN_CLOSED; COMPACT_CBALL; CBALL_EQ_EMPTY] THEN ASM_REWRITE_TAC[SET_RULE `UNIV DIFF s = {} <=> s = UNIV`] THEN ASM_SIMP_TAC[SET_RULE `b INTER (UNIV DIFF s) = {} <=> b SUBSET s`; REAL_ARITH `&0 <= r ==> ~(r < &0)`] THEN STRIP_TAC THEN EXISTS_TAC `r + setdist(cball(a,r),(:real^N) DIFF s) / &2` THEN ASM_REWRITE_TAC[REAL_LT_ADDR; REAL_HALF; SUBSET; IN_CBALL] THEN X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `x:real^N = a` THENL [ASM_MESON_TAC[SUBSET; DIST_REFL; IN_CBALL]; ALL_TAC] THEN ASM_CASES_TAC `(x:real^N) IN s` THEN ASM_REWRITE_TAC[REAL_NOT_LE] THEN MP_TAC(ISPECL [`cball(a:real^N,r)`; `(:real^N) DIFF s`; `a + r / dist(a,x) % (x - a):real^N`; `x:real^N`] SETDIST_LE_DIST) THEN ASM_REWRITE_TAC[IN_DIFF; IN_UNIV; IN_CBALL] THEN REWRITE_TAC[NORM_ARITH `dist(a:real^N,a + x) = norm x`] THEN ASM_SIMP_TAC[NORM_MUL; REAL_ABS_DIV; ONCE_REWRITE_RULE[DIST_SYM] dist; REAL_ABS_NORM; REAL_DIV_RMUL; NORM_EQ_0; VECTOR_SUB_EQ] THEN ASM_REWRITE_TAC[REAL_ARITH `abs r <= r <=> &0 <= r`] THEN REWRITE_TAC[NORM_MUL; VECTOR_ARITH `x - (a + d % (x - a)):real^N = (&1 - d) % (x - a)`] THEN ONCE_REWRITE_TAC[GSYM REAL_ABS_NORM] THEN REWRITE_TAC[GSYM REAL_ABS_MUL] THEN REWRITE_TAC[REAL_ABS_NORM; REAL_SUB_RDISTRIB] THEN ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_EQ_0; VECTOR_SUB_EQ] THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N` o REWRITE_RULE[SUBSET]) THEN ASM_REWRITE_TAC[IN_CBALL; ONCE_REWRITE_RULE[DIST_SYM] dist] THEN REAL_ARITH_TAC; SUBGOAL_THEN `s DIFF cball(a:real^N,r) = s DIFF ball(a,r') UNION {x | r < norm(x - a) /\ norm(x - a) <= r'}` SUBST1_TAC THENL [REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] (GSYM dist)] THEN REWRITE_TAC[GSYM REAL_NOT_LE; GSYM IN_CBALL] THEN MATCH_MP_TAC(SET_RULE `b' SUBSET c' /\ c' SUBSET s /\ c SUBSET b' ==> s DIFF c = (s DIFF b') UNION {x | ~(x IN c) /\ x IN c'}`) THEN ASM_REWRITE_TAC[BALL_SUBSET_CBALL] THEN REWRITE_TAC[SUBSET; IN_BALL; IN_CBALL] THEN ASM_REAL_ARITH_TAC; MATCH_MP_TAC CONNECTED_UNION THEN ASM_SIMP_TAC[CONNECTED_ANNULUS; PATH_CONNECTED_DIFF_BALL; PATH_CONNECTED_IMP_CONNECTED; CONNECTED_OPEN_PATH_CONNECTED] THEN REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] (GSYM dist)] THEN REWRITE_TAC[GSYM REAL_NOT_LE; GSYM IN_CBALL] THEN MATCH_MP_TAC(SET_RULE `c' SUBSET s /\ (?x. x IN c' /\ ~(x IN b') /\ ~(x IN c)) ==> ~((s DIFF b') INTER {x | ~(x IN c) /\ x IN c'} = {})`) THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `a + r' % basis 1:real^N` THEN REWRITE_TAC[IN_BALL; IN_CBALL] THEN REWRITE_TAC[NORM_ARITH `dist(a:real^N,a + x) = norm x`] THEN SIMP_TAC[NORM_MUL; NORM_BASIS; DIMINDEX_GE_1; LE_REFL] THEN ASM_REAL_ARITH_TAC]]);; let PATH_CONNECTED_CONVEX_DIFF_LOWDIM = prove (`!s t:real^N->bool. convex s /\ aff_dim t + &2 <= aff_dim s ==> path_connected(s DIFF t)`, REPEAT STRIP_TAC THEN REWRITE_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN REWRITE_TAC[IN_DIFF] THEN STRIP_TAC THEN REWRITE_TAC[PATH_CONNECTED_COMPONENT_SET] THEN ASM_CASES_TAC `segment[x:real^N,y] INTER t = {}` THENL [MATCH_MP_TAC PATH_CONNECTED_LINEPATH THEN ASM_SIMP_TAC[CONVEX_CONTAINS_SEGMENT_IMP; SET_RULE `s SUBSET t DIFF u <=> s INTER u = {} /\ s SUBSET t`]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(fun th -> REPEAT(POP_ASSUM MP_TAC) THEN X_CHOOSE_THEN `a:real^N` MP_TAC th) THEN GEOM_ORIGIN_TAC `a:real^N` THEN REWRITE_TAC[IN_INTER] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `(vec 0:real^N) IN s` ASSUME_TAC THENL [ASM_MESON_TAC[CONVEX_CONTAINS_SEGMENT; SUBSET]; ALL_TAC] THEN SUBGOAL_THEN `?z:real^N. z IN s /\ ~(z IN span(x INSERT y INSERT span t))` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC(SET_RULE `~(t SUBSET s) ==> ?x. x IN t /\ ~(x IN s)`) THEN DISCH_THEN(MP_TAC o MATCH_MP DIM_SUBSET) THEN UNDISCH_TAC `aff_dim(t:real^N->bool) + &2 <= aff_dim(s:real^N->bool)` THEN ASM_SIMP_TAC[AFF_DIM_DIM_0; HULL_INC; INT_OF_NUM_ADD; INT_OF_NUM_LE] THEN MATCH_MP_TAC(ARITH_RULE `x <= SUC y ==> y + 2 <= s ==> ~(s <= x)`) THEN ONCE_REWRITE_TAC[DIM_INSERT] THEN SUBGOAL_THEN `(x:real^N) IN span(y INSERT span t)` (fun th -> REWRITE_TAC[th; DIM_INSERT; DIM_SPAN] THEN ARITH_TAC) THEN SUBGOAL_THEN `(vec 0:real^N) IN segment(x,y)` MP_TAC THENL [ASM_REWRITE_TAC[open_segment; IN_DIFF; IN_INSERT; NOT_IN_EMPTY] THEN ASM SET_TAC[]; REWRITE_TAC[IN_SEGMENT]] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (X_CHOOSE_THEN `u:real` (STRIP_ASSUME_TAC o GSYM))) THEN FIRST_ASSUM(MP_TAC o AP_TERM `(%) (inv(&1 - u)):real^N->real^N`) THEN REWRITE_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC; VECTOR_MUL_RZERO] THEN ASM_SIMP_TAC[REAL_MUL_LINV; REAL_SUB_0; REAL_LT_IMP_NE] THEN REWRITE_TAC[VECTOR_ARITH `&1 % x + a % y:real^N = vec 0 <=> x = --a % y`] THEN SIMP_TAC[SPAN_MUL; SPAN_SUPERSET; IN_INSERT]; ALL_TAC] THEN MATCH_MP_TAC PATH_COMPONENT_TRANS THEN EXISTS_TAC `z:real^N` THEN SUBGOAL_THEN `~((z:real^N) IN t)` ASSUME_TAC THENL [ASM_MESON_TAC[SPAN_SUPERSET; IN_INSERT]; ALL_TAC] THEN CONJ_TAC THEN MATCH_MP_TAC PATH_CONNECTED_LINEPATH THENL [ALL_TAC; ONCE_REWRITE_TAC[SEGMENT_SYM]] THEN ASM_SIMP_TAC[CONVEX_CONTAINS_SEGMENT_IMP; SET_RULE `s SUBSET t DIFF u <=> s INTER u = {} /\ s SUBSET t`] THEN ASM_REWRITE_TAC[SEGMENT_CLOSED_OPEN; SET_RULE `(s UNION {a,b}) INTER t = {} <=> ~(a IN t) /\ ~(b IN t) /\ s INTER t = {}`] THEN REWRITE_TAC[SET_RULE `s INTER t = {} <=> !x. x IN s ==> ~(x IN t)`] THEN X_GEN_TAC `w:real^N` THEN REWRITE_TAC[IN_SEGMENT] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `u:real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o AP_TERM `(%) (inv u):real^N->real^N`) THEN ASM_SIMP_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC; REAL_MUL_LINV; REAL_LT_IMP_NZ] THEN REWRITE_TAC[VECTOR_ARITH `a:real^N = b + &1 % z <=> z = a - b`] THEN REPEAT DISCH_TAC THEN UNDISCH_TAC `~((z:real^N) IN span (x INSERT y INSERT span t))` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SPAN_SUB THEN CONJ_TAC THEN MATCH_MP_TAC SPAN_MUL THEN ASM_SIMP_TAC[SPAN_SUPERSET; IN_INSERT]);; let PATH_CONNECTED_OPEN_IN_DIFF_LOWDIM = prove (`!s t:real^N->bool. connected s /\ open_in (subtopology euclidean (affine hull s)) s /\ aff_dim t + &2 <= aff_dim s ==> path_connected(s DIFF t)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC PATH_CONNECTED_CONNECTED_DIFF THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [TRANS_TAC SUBSET_TRANS `closure s:real^N->bool` THEN REWRITE_TAC[CLOSURE_SUBSET] THEN MATCH_MP_TAC(SET_RULE `t = s ==> s SUBSET t`) THEN MATCH_MP_TAC DENSE_COMPLEMENT_OPEN_IN_AFFINE_HULL THEN ASM_REWRITE_TAC[] THEN ASM_INT_ARITH_TAC; X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(CONJUNCTS_THEN ASSUME_TAC o REWRITE_RULE[OPEN_IN_CONTAINS_BALL]) THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `ball(x:real^N,r) INTER affine hull s` THEN ASM_SIMP_TAC[IN_INTER; CENTRE_IN_BALL; HULL_INC] THEN CONJ_TAC THENL [REWRITE_TAC[OPEN_IN_OPEN] THEN EXISTS_TAC `ball(x:real^N,r)` THEN REWRITE_TAC[OPEN_BALL] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `b INTER t SUBSET s ==> s SUBSET t ==> b INTER t = s INTER b`)) THEN REWRITE_TAC[HULL_SUBSET]; MATCH_MP_TAC PATH_CONNECTED_CONVEX_DIFF_LOWDIM THEN ASM_SIMP_TAC[CONVEX_INTER; CONVEX_BALL; AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (INT_ARITH `t:int <= s ==> s' = s ==> t <= s'`)) THEN GEN_REWRITE_TAC RAND_CONV [GSYM AFF_DIM_AFFINE_HULL] THEN ONCE_REWRITE_TAC[INTER_COMM] THEN MATCH_MP_TAC AFF_DIM_CONVEX_INTER_OPEN THEN ASM_SIMP_TAC[AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL; OPEN_BALL] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN ASM_MESON_TAC[HULL_INC; CENTRE_IN_BALL]]]);; let PATH_CONNECTED_OPEN_IN_DIFF_UNIONS_LOWDIM = prove (`!(s:real^N->bool) f. connected s /\ open_in (subtopology euclidean (affine hull s)) s /\ FINITE f /\ (!t. t IN f ==> closed t /\ aff_dim t + &2 <= aff_dim s) ==> path_connected(s DIFF UNIONS f)`, GEN_TAC THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REPEAT DISCH_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN CONJ_TAC THENL [DISCH_THEN(K ALL_TAC) THEN REWRITE_TAC[UNIONS_0; DIFF_EMPTY] THEN ASM_MESON_TAC[PATH_CONNECTED_EQ_CONNECTED_LPC; OPEN_IN_IMP_LOCALLY_PATH_CONNECTED]; MAP_EVERY X_GEN_TAC [`t:real^N->bool`; `g:(real^N->bool)->bool`] THEN REWRITE_TAC[FORALL_IN_INSERT; UNIONS_INSERT] THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN REWRITE_TAC[SET_RULE `s DIFF (t UNION u) = (s DIFF u) DIFF t`] THEN ASM_CASES_TAC `s DIFF UNIONS g:real^N->bool = {}` THEN ASM_REWRITE_TAC[PATH_CONNECTED_EMPTY; EMPTY_DIFF] THEN MATCH_MP_TAC PATH_CONNECTED_OPEN_IN_DIFF_LOWDIM THEN ASM_SIMP_TAC[PATH_CONNECTED_IMP_CONNECTED] THEN SUBGOAL_THEN `open_in (subtopology euclidean (affine hull s)) (s DIFF UNIONS g:real^N->bool)` ASSUME_TAC THENL [SUBGOAL_THEN `s DIFF UNIONS g:real^N->bool = s DIFF (affine hull s INTER UNIONS g)` SUBST1_TAC THENL [MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s DIFF u = s DIFF (t INTER u)`) THEN REWRITE_TAC[HULL_SUBSET]; MATCH_MP_TAC OPEN_IN_DIFF THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CLOSED_IN_INTER_CLOSED THEN REWRITE_TAC[CLOSED_IN_REFL] THEN MATCH_MP_TAC CLOSED_UNIONS THEN ASM_SIMP_TAC[]]; CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_SUBSET_TRANS THEN EXISTS_TAC `affine hull s:real^N->bool` THEN ASM_REWRITE_TAC[HULL_SUBSET] THEN MATCH_MP_TAC HULL_MONO THEN SET_TAC[]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (INT_ARITH `t:int <= s ==> u = s ==> t <= u`)) THEN GEN_REWRITE_TAC RAND_CONV [GSYM AFF_DIM_AFFINE_HULL] THEN MATCH_MP_TAC AFF_DIM_OPEN_IN THEN ASM_REWRITE_TAC[AFFINE_AFFINE_HULL]]]]);; let CONNECTED_OPEN_IN_DIFF_UNIONS_LOWDIM = prove (`!f s:real^N->bool. connected s /\ open_in (subtopology euclidean (affine hull s)) s /\ FINITE f /\ (!t. t IN f ==> aff_dim t + &2 <= aff_dim s) ==> connected(s DIFF UNIONS f)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CONNECTED_INTERMEDIATE_CLOSURE THEN EXISTS_TAC `s DIFF UNIONS {closure t:real^N->bool | t IN f}` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC PATH_CONNECTED_IMP_CONNECTED THEN MATCH_MP_TAC PATH_CONNECTED_OPEN_IN_DIFF_UNIONS_LOWDIM THEN ASM_SIMP_TAC[FORALL_IN_GSPEC; CLOSED_CLOSURE; AFF_DIM_CLOSURE] THEN ASM_SIMP_TAC[SIMPLE_IMAGE; FINITE_IMAGE]; MATCH_MP_TAC(SET_RULE `u SUBSET t ==> s DIFF t SUBSET s DIFF u`) THEN MATCH_MP_TAC UNIONS_MONO THEN REWRITE_TAC[EXISTS_IN_GSPEC] THEN MESON_TAC[CLOSURE_SUBSET]; ASM_CASES_TAC `f:(real^N->bool)->bool = {}` THEN ASM_REWRITE_TAC[UNIONS_0; SET_RULE `{f x |x| F} = {}`; NOT_IN_EMPTY; DIFF_EMPTY; CLOSURE_SUBSET] THEN REWRITE_TAC[DIFF_UNIONS] THEN REWRITE_TAC[SET_RULE `{f x | x IN {g y | P y}} = {f(g y) | P y}`] THEN MP_TAC(ISPECL [`s:real^N->bool`; `INTERS {s DIFF closure t:real^N->bool | t IN f}`; `affine hull s:real^N->bool`] CLOSURE_OPEN_IN_INTER_CLOSURE) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [TRANS_TAC SUBSET_TRANS `s:real^N->bool` THEN REWRITE_TAC[HULL_SUBSET; INTERS_GSPEC] THEN ASM SET_TAC[]; DISCH_THEN(SUBST1_TAC o SYM)] THEN MATCH_MP_TAC(SET_RULE `u SUBSET closure u /\ s = u ==> s INTER t SUBSET closure u`) THEN REWRITE_TAC[CLOSURE_SUBSET; SET_RULE `s = s INTER t <=> s SUBSET t`] THEN MATCH_MP_TAC DENSE_OPEN_INTERS THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN ASM_SIMP_TAC[SIMPLE_IMAGE; FINITE_IMAGE] THEN X_GEN_TAC `t:real^N->bool` THEN REPEAT STRIP_TAC THENL [ONCE_REWRITE_TAC[SET_RULE `s DIFF t = s INTER (UNIV DIFF t)`] THEN SIMP_TAC[OPEN_IN_OPEN_INTER; GSYM closed; CLOSED_CLOSURE]; TRANS_TAC SUBSET_TRANS `closure s:real^N->bool` THEN REWRITE_TAC[CLOSURE_SUBSET] THEN MATCH_MP_TAC(SET_RULE `t = s ==> s SUBSET t`) THEN MATCH_MP_TAC DENSE_COMPLEMENT_OPEN_IN_AFFINE_HULL THEN ASM_REWRITE_TAC[AFF_DIM_CLOSURE] THEN FIRST_X_ASSUM(MP_TAC o SPEC `t:real^N->bool`) THEN ASM_REWRITE_TAC[] THEN INT_ARITH_TAC]]);; let BOUNDED_FRONTIER_BOUNDED_OR_COBOUNDED = prove (`!s. 2 <= dimindex(:N) /\ bounded(frontier s) ==> bounded(s) \/ bounded((:real^N) DIFF s)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `vec 0:real^N` o MATCH_MP BOUNDED_SUBSET_BALL) THEN DISCH_THEN(X_CHOOSE_THEN `r:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[DE_MORGAN_THM] THEN STRIP_TAC THEN REWRITE_TAC[SET_RULE `f SUBSET s <=> (UNIV DIFF s) INTER f = {}`] THEN MATCH_MP_TAC CONNECTED_INTER_FRONTIER THEN ASM_SIMP_TAC[CONNECTED_COMPLEMENT_BOUNDED_CONVEX; BOUNDED_BALL; CONVEX_BALL; SET_RULE `UNIV DIFF s DIFF t = {} <=> UNIV DIFF t SUBSET s`; SET_RULE `(UNIV DIFF s) INTER t = {} <=> t SUBSET s`] THEN ASM_MESON_TAC[BOUNDED_SUBSET; BOUNDED_BALL]);; let BOUNDED_COMMON_FRONTIER_DOMAINS = prove (`!s t c:real^N->bool. 2 <= dimindex(:N) /\ bounded c /\ open s /\ connected s /\ open t /\ connected t /\ ~(s = t) /\ frontier s = c /\ frontier t = c ==> bounded s \/ bounded t`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `t:real^N->bool` BOUNDED_FRONTIER_BOUNDED_OR_COBOUNDED) THEN MP_TAC(ISPEC `s:real^N->bool` BOUNDED_FRONTIER_BOUNDED_OR_COBOUNDED) THEN ASM_REWRITE_TAC[] THEN REPEAT(STRIP_TAC THEN ASM_REWRITE_TAC[]) THEN MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`] COMMON_FRONTIER_DOMAINS) THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_THEN(K ALL_TAC) THEN MATCH_MP_TAC(SET_RULE `~((UNIV DIFF s) UNION (UNIV DIFF t) = UNIV) ==> ~DISJOINT s t`) THEN MATCH_MP_TAC(MESON[NOT_BOUNDED_UNIV] `bounded s ==> ~(s = (:real^N))`) THEN ASM_REWRITE_TAC[BOUNDED_UNION]);; let INTERIOR_ARC_IMAGE = prove (`!g:real^1->real^N. 2 <= dimindex(:N) /\ arc g ==> interior(path_image g) = {}`, REPEAT STRIP_TAC THEN SIMP_TAC[path_image; CLOSED_OPEN_INTERVAL_1; DROP_VEC; REAL_POS] THEN REWRITE_TAC[IMAGE_UNION; IMAGE_CLAUSES] THEN SIMP_TAC[INTERIOR_UNION_EQ_EMPTY; CLOSED_INSERT; CLOSED_EMPTY] THEN SIMP_TAC[EMPTY_INTERIOR_FINITE; FINITE_INSERT; FINITE_EMPTY] THEN MATCH_MP_TAC(SET_RULE `(!a. ~(a IN s)) ==> s = {}`) THEN X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] (REWRITE_RULE[CONJ_ASSOC] CONNECTED_DELETE_INTERIOR_POINT))) THEN ASM_REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN REWRITE_TAC[CONNECTED_INTERVAL] THEN MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `interval[vec 0:real^1,vec 1]` THEN ASM_SIMP_TAC[INTERVAL_OPEN_SUBSET_CLOSED; GSYM path; ARC_IMP_PATH]; FIRST_X_ASSUM(MP_TAC o MATCH_MP(REWRITE_RULE[SUBSET] INTERIOR_SUBSET)) THEN SPEC_TAC(`a:real^N`,`a:real^N`) THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN X_GEN_TAC `t:real^1` THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMEOMORPHISM_ARC) THEN DISCH_THEN(X_CHOOSE_THEN `h:real^N->real^1` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o ISPEC `interval(vec 0,vec 1) DELETE (t:real^1)` o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_CONNECTEDNESS)) THEN MATCH_MP_TAC(TAUT `p /\ q /\ ~r ==> (p ==> (q <=> r)) ==> F`) THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s DELETE a SUBSET t`) THEN REWRITE_TAC[INTERVAL_OPEN_SUBSET_CLOSED]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (MESON[] `connected s ==> s = t ==> connected t`)) THEN RULE_ASSUM_TAC(REWRITE_RULE[arc]) THEN MP_TAC(ISPECL [`vec 0:real^1`; `vec 1:real^1`] INTERVAL_OPEN_SUBSET_CLOSED) THEN ASM SET_TAC[]; REWRITE_TAC[GSYM IS_INTERVAL_CONNECTED_1; IS_INTERVAL_1] THEN DISCH_THEN(MP_TAC o SPECL [`midpoint(vec 0:real^1,t)`; `midpoint(vec 1:real^1,t)`; `t:real^1`]) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL_1]) THEN REWRITE_TAC[DROP_VEC; IN_DELETE; IN_INTERVAL_1; GSYM DROP_EQ; midpoint; DROP_ADD; DROP_CMUL] THEN REAL_ARITH_TAC]]);; let INTERIOR_SIMPLE_PATH_IMAGE = prove (`!g:real^1->real^N. 2 <= dimindex(:N) /\ simple_path g ==> interior(path_image g) = {}`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`g:real^1->real^N`; `lift(&1 / &2)`] PATH_IMAGE_SUBPATH_COMBINE) THEN REWRITE_TAC[IN_INTERVAL_1; LIFT_DROP; DROP_VEC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_SIMP_TAC[SIMPLE_PATH_IMP_PATH] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN W(MP_TAC o PART_MATCH (lhs o rand) INTERIOR_UNION_EQ_EMPTY o snd) THEN ANTS_TAC THENL [DISJ1_TAC THEN MATCH_MP_TAC CLOSED_PATH_IMAGE THEN MATCH_MP_TAC PATH_SUBPATH THEN REWRITE_TAC[IN_INTERVAL_1; LIFT_DROP; DROP_VEC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_SIMP_TAC[SIMPLE_PATH_IMP_PATH]; DISCH_THEN SUBST1_TAC THEN CONJ_TAC THEN MATCH_MP_TAC INTERIOR_ARC_IMAGE THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC ARC_SIMPLE_PATH_SUBPATH_INTERIOR THEN ASM_REWRITE_TAC[IN_INTERVAL_1; LIFT_DROP; DROP_VEC; GSYM DROP_EQ] THEN CONV_TAC REAL_RAT_REDUCE_CONV]);; let ENDPOINTS_NOT_IN_INTERIOR_SIMPLE_PATH_IMAGE = prove (`!g:real^1->real^N. simple_path g ==> DISJOINT {pathstart g,pathfinish g} (interior(path_image g))`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `2 <= dimindex(:N)` THENL [ASM_SIMP_TAC[INTERIOR_SIMPLE_PATH_IMAGE] THEN SET_TAC[]; FIRST_ASSUM(MP_TAC o MATCH_MP (ARITH_RULE `~(2 <= p) ==> 1 <= p ==> p = 1`))] THEN REWRITE_TAC[DIMINDEX_GE_1] THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] COLLINEAR_SIMPLE_PATH_IMAGE)) THEN ANTS_TAC THENL [REWRITE_TAC[COLLINEAR_AFF_DIM] THEN ASM_MESON_TAC[AFF_DIM_LE_UNIV]; DISCH_THEN SUBST1_TAC] THEN REWRITE_TAC[DISJOINT_INSERT; DISJOINT_EMPTY; INTERIOR_SEGMENT] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[ENDS_NOT_IN_SEGMENT; NOT_IN_EMPTY]);; (* ------------------------------------------------------------------------- *) (* Existence of unbounded components. *) (* ------------------------------------------------------------------------- *) let COBOUNDED_UNBOUNDED_COMPONENT = prove (`!s. bounded((:real^N) DIFF s) ==> ?x. x IN s /\ ~bounded(connected_component s x)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `vec 0:real^N` o MATCH_MP BOUNDED_SUBSET_BALL) THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `B % basis 1:real^N` THEN CONJ_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPEC `B % basis 1:real^N` o GEN_REWRITE_RULE I [SUBSET]) THEN REWRITE_TAC[IN_UNIV; IN_DIFF; IN_BALL_0] THEN SIMP_TAC[NORM_MUL; NORM_BASIS; DIMINDEX_GE_1; LE_REFL] THEN ASM_SIMP_TAC[REAL_ARITH `&0 < B ==> ~(abs B * &1 < B)`]; MP_TAC(ISPECL [`basis 1:real^N`; `B:real`] BOUNDED_HALFSPACE_GE) THEN SIMP_TAC[BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL; CONTRAPOS_THM] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] BOUNDED_SUBSET) THEN MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL THEN SIMP_TAC[CONVEX_HALFSPACE_GE; CONVEX_CONNECTED] THEN ASM_SIMP_TAC[IN_ELIM_THM; DOT_RMUL; DOT_BASIS_BASIS; DIMINDEX_GE_1; LE_REFL; real_ge; REAL_MUL_RID; REAL_LE_REFL] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `UNIV DIFF s SUBSET b ==> (!x. x IN h ==> ~(x IN b)) ==> h SUBSET s`)) THEN SIMP_TAC[IN_ELIM_THM; DOT_BASIS; IN_BALL_0; DIMINDEX_GE_1; LE_REFL] THEN GEN_TAC THEN REWRITE_TAC[REAL_NOT_LT] THEN MATCH_MP_TAC(REAL_ARITH `abs x <= n ==> b <= x ==> b <= n`) THEN SIMP_TAC[COMPONENT_LE_NORM; DIMINDEX_GE_1; LE_REFL]]);; let COBOUNDED_UNIQUE_UNBOUNDED_COMPONENT = prove (`!s x y:real^N. 2 <= dimindex(:N) /\ bounded((:real^N) DIFF s) /\ ~bounded(connected_component s x) /\ ~bounded(connected_component s y) ==> connected_component s x = connected_component s y`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `vec 0:real^N` o MATCH_MP BOUNDED_SUBSET_BALL) THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN MP_TAC(ISPEC `ball(vec 0:real^N,B)` CONNECTED_COMPLEMENT_BOUNDED_CONVEX) THEN ASM_REWRITE_TAC[BOUNDED_BALL; CONVEX_BALL] THEN DISCH_TAC THEN MAP_EVERY (MP_TAC o SPEC `B:real` o REWRITE_RULE[bounded; NOT_EXISTS_THM] o ASSUME) [`~bounded(connected_component s (y:real^N))`; `~bounded(connected_component s (x:real^N))`] THEN REWRITE_TAC[NOT_FORALL_THM; IN; NOT_IMP] THEN DISCH_THEN(X_CHOOSE_THEN `x':real^N` STRIP_ASSUME_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `y':real^N` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC CONNECTED_COMPONENT_EQ THEN REWRITE_TAC[IN] THEN SUBGOAL_THEN `connected_component s (x':real^N) (y':real^N)` ASSUME_TAC THENL [REWRITE_TAC[connected_component] THEN EXISTS_TAC `(:real^N) DIFF ball (vec 0,B)` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[IN_DIFF; IN_UNIV]] THEN REWRITE_TAC[IN_BALL_0] THEN ASM_MESON_TAC[REAL_LT_IMP_LE]; ASM_MESON_TAC[CONNECTED_COMPONENT_SYM; CONNECTED_COMPONENT_TRANS]]);; let COBOUNDED_UNBOUNDED_COMPONENTS = prove (`!s. bounded ((:real^N) DIFF s) ==> ?c. c IN components s /\ ~bounded c`, REWRITE_TAC[components; EXISTS_IN_GSPEC; COBOUNDED_UNBOUNDED_COMPONENT]);; let COBOUNDED_UNIQUE_UNBOUNDED_COMPONENTS = prove (`!s c c'. 2 <= dimindex(:N) /\ bounded ((:real^N) DIFF s) /\ c IN components s /\ ~bounded c /\ c' IN components s /\ ~bounded c' ==> c' = c`, REWRITE_TAC[components; IN_ELIM_THM] THEN MESON_TAC[COBOUNDED_UNIQUE_UNBOUNDED_COMPONENT]);; let COBOUNDED_HAS_BOUNDED_COMPONENT = prove (`!s. 2 <= dimindex(:N) /\ bounded((:real^N) DIFF s) /\ ~connected s ==> ?c. c IN components s /\ bounded c`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?c c':real^N->bool. c IN components s /\ c' IN components s /\ ~(c = c')` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC(SET_RULE `~(s = {}) /\ ~(?a. s = {a}) ==> ?x y. x IN s /\ y IN s /\ ~(x = y)`) THEN ASM_REWRITE_TAC[COMPONENTS_EQ_SING_EXISTS; COMPONENTS_EQ_EMPTY] THEN ASM_MESON_TAC[DIFF_EMPTY; NOT_BOUNDED_UNIV]; ASM_MESON_TAC[COBOUNDED_UNIQUE_UNBOUNDED_COMPONENTS]]);; (* ------------------------------------------------------------------------- *) (* Self-homeomorphisms shuffling points about in various ways. *) (* ------------------------------------------------------------------------- *) let HOMEOMORPHISM_MOVING_POINT_EXISTS = prove (`!s t a b:real^N. open_in (subtopology euclidean (affine hull s)) s /\ s SUBSET t /\ t SUBSET affine hull s /\ connected s /\ a IN s /\ b IN s ==> ?f g. homeomorphism (t,t) (f,g) /\ f a = b /\ {x | ~(f x = x /\ g x = x)} SUBSET s /\ bounded {x | ~(f x = x /\ g x = x)}`, let lemma1 = prove (`!a t r u:real^N. affine t /\ a IN t /\ u IN ball(a,r) INTER t ==> ?f g. homeomorphism (cball(a,r) INTER t,cball(a,r) INTER t) (f,g) /\ f(a) = u /\ (!x. x IN sphere(a,r) ==> f(x) = x)`, REPEAT STRIP_TAC THEN DISJ_CASES_TAC(REAL_ARITH `r <= &0 \/ &0 < r`) THENL [ASM_MESON_TAC[BALL_EMPTY; INTER_EMPTY; NOT_IN_EMPTY]; ALL_TAC] THEN EXISTS_TAC `\x:real^N. (&1 - norm(x - a) / r) % (u - a) + x` THEN REWRITE_TAC[LEFT_EXISTS_AND_THM] THEN CONJ_TAC THENL [MATCH_MP_TAC HOMEOMORPHISM_COMPACT THEN ASM_SIMP_TAC[COMPACT_INTER_CLOSED; COMPACT_CBALL; CLOSED_AFFINE]; ASM_SIMP_TAC[IN_SPHERE; ONCE_REWRITE_RULE[NORM_SUB] dist; REAL_DIV_REFL; REAL_LT_IMP_NZ; IN_INTER] THEN REWRITE_TAC[real_div; VECTOR_SUB_REFL; NORM_0; REAL_MUL_LZERO] THEN REPEAT STRIP_TAC THEN VECTOR_ARITH_TAC] THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_ADD THEN REWRITE_TAC[CONTINUOUS_ON_ID] THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN REWRITE_TAC[o_DEF; LIFT_SUB] THEN SIMP_TAC[CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST; CONTINUOUS_ON_SUB] THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] real_div; LIFT_CMUL] THEN MATCH_MP_TAC CONTINUOUS_ON_CMUL THEN MATCH_MP_TAC CONTINUOUS_ON_LIFT_NORM_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST; CONTINUOUS_ON_SUB]; ALL_TAC] THEN CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> f x IN s) /\ (!y. y IN s ==> ?x. x IN s /\ f x = y) ==> IMAGE f s = s`) THEN REWRITE_TAC[] THEN ONCE_REWRITE_TAC[VECTOR_ARITH `(&1 - n) % (u - a) + x:real^N = a + (&1 - n) % (u - a) + (x - a)`]; ALL_TAC] THEN REPEAT(POP_ASSUM MP_TAC) THEN GEOM_ORIGIN_TAC `a:real^N` THEN REWRITE_TAC[IN_BALL_0; VECTOR_SUB_RZERO; IN_CBALL_0; IN_INTER] THEN REWRITE_TAC[VECTOR_ADD_LID; VECTOR_ADD_RID; VECTOR_ARITH `a + x:real^N = a + y <=> x = y`; VECTOR_ARITH `(&1 - n) % u + a + x = (&1 - m) % u + a + y <=> (n - m) % u:real^N = x - y`] THEN REWRITE_TAC[REAL_ARITH `x / r - y / r:real = (x - y) / r`] THENL [ALL_TAC; REPEAT GEN_TAC THEN REPEAT DISCH_TAC THEN REPEAT GEN_TAC THEN ASM_CASES_TAC `x:real^N = y` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `norm(x:real^N) = norm(y:real^N)` THEN ASM_REWRITE_TAC[real_div; REAL_SUB_REFL; REAL_MUL_LZERO; VECTOR_MUL_LZERO; VECTOR_ARITH `vec 0:real^N = x - y <=> x = y`] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o AP_TERM `norm:real^N->real`) THEN ASM_SIMP_TAC[NORM_MUL; REAL_ABS_MUL; REAL_ABS_INV] THEN DISCH_THEN(MP_TAC o MATCH_MP (NORM_ARITH `r = norm(x - y:real^N) ==> r < abs(norm x - norm y) * &1 ==> F`)) THEN REWRITE_TAC[GSYM REAL_MUL_ASSOC] THEN MATCH_MP_TAC REAL_LT_LMUL THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ONCE_REWRITE_TAC[REAL_MUL_SYM]] THEN ASM_SIMP_TAC[GSYM real_div; REAL_LT_LDIV_EQ; REAL_ARITH `&0 < r ==> &0 < abs r`] THEN ASM_REAL_ARITH_TAC] THEN REPEAT GEN_TAC THEN ASM_CASES_TAC `subspace(t:real^N->bool)` THENL [ALL_TAC; ASM_MESON_TAC[AFFINE_IMP_SUBSPACE]] THEN ASM_SIMP_TAC[SUBSPACE_ADD; SUBSPACE_MUL] THEN REPEAT STRIP_TAC THENL [MATCH_MP_TAC(NORM_ARITH `norm(x) + norm(y) <= &1 * r ==> norm(x + y:real^N) <= r`) THEN ASM_SIMP_TAC[NORM_MUL; GSYM REAL_LE_LDIV_EQ; REAL_ARITH `(a * u + x) / r:real = a * u / r + x / r`] THEN MATCH_MP_TAC(REAL_ARITH `x <= &1 /\ a <= abs(&1 - x) * &1 ==> a + x <= &1`) THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ] THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[REAL_ABS_POS] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_MUL_LID; REAL_LT_IMP_LE]; ALL_TAC] THEN MP_TAC(ISPECL [`\a. lift((&1 - drop a) * r - norm(y - drop a % u:real^N))`; `vec 0:real^1`; `vec 1:real^1`; `&0`; `1`] IVT_DECREASING_COMPONENT_1) THEN REWRITE_TAC[DIMINDEX_1; GSYM drop; LIFT_DROP; DROP_VEC] THEN REWRITE_TAC[REAL_POS; LE_REFL; REAL_SUB_REFL; VECTOR_MUL_LZERO] THEN REWRITE_TAC[REAL_SUB_RZERO; VECTOR_SUB_RZERO; REAL_MUL_LID] THEN REWRITE_TAC[NORM_ARITH `&0 * r - norm(x:real^N) <= &0`] THEN ASM_REWRITE_TAC[REAL_SUB_LE; GSYM EXISTS_DROP; IN_INTERVAL_1] THEN ANTS_TAC THENL [REPEAT STRIP_TAC THEN REWRITE_TAC[REAL_ARITH `(&1 - x) * r - b:real = r - r * x - b`] THEN REWRITE_TAC[LIFT_SUB; LIFT_CMUL; LIFT_DROP] THEN REPEAT(MATCH_MP_TAC CONTINUOUS_SUB THEN CONJ_TAC THEN REWRITE_TAC[CONTINUOUS_CONST]) THEN SIMP_TAC[CONTINUOUS_CMUL; CONTINUOUS_AT_ID] THEN MATCH_MP_TAC CONTINUOUS_LIFT_NORM_COMPOSE THEN MATCH_MP_TAC CONTINUOUS_SUB THEN REWRITE_TAC[CONTINUOUS_CONST] THEN MATCH_MP_TAC CONTINUOUS_MUL THEN REWRITE_TAC[o_DEF; LIFT_DROP; CONTINUOUS_AT_ID; CONTINUOUS_CONST]; ASM_SIMP_TAC[DROP_VEC; REAL_FIELD `&0 < r ==> ((&1 - x) * r - n = &0 <=> &1 - n / r = x)`] THEN DISCH_THEN(X_CHOOSE_THEN `a:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `y - a % u:real^N` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN ASM_SIMP_TAC[SUBSPACE_SUB; SUBSPACE_MUL] THEN GEN_REWRITE_TAC RAND_CONV [GSYM REAL_MUL_LID] THEN ASM_SIMP_TAC[GSYM REAL_LE_LDIV_EQ] THEN ASM_REAL_ARITH_TAC]) in let lemma2 = prove (`!a t u v:real^N r. affine t /\ a IN t /\ u IN ball(a,r) INTER t /\ v IN ball(a,r) INTER t ==> ?f g. homeomorphism (cball(a,r) INTER t,cball(a,r) INTER t) (f,g) /\ f(u) = v /\ !x. x IN sphere(a,r) /\ x IN t ==> f(x) = x`, REPEAT GEN_TAC THEN DISJ_CASES_TAC(REAL_ARITH `r <= &0 \/ &0 < r`) THENL [ASM_MESON_TAC[BALL_EMPTY; INTER_EMPTY; NOT_IN_EMPTY]; REPLICATE_TAC 2 (DISCH_THEN (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_TAC] THEN MP_TAC(ISPECL [`a:real^N`; `t:real^N->bool`; `r:real`] lemma1) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(fun th -> FIRST_ASSUM(CONJUNCTS_THEN(MP_TAC o MATCH_MP th))) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f1:real^N->real^N`; `g1:real^N->real^N`] THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`f2:real^N->real^N`; `g2:real^N->real^N`] THEN STRIP_TAC THEN EXISTS_TAC `(f1:real^N->real^N) o (g2:real^N->real^N)` THEN EXISTS_TAC `(f2:real^N->real^N) o (g1:real^N->real^N)` THEN REWRITE_TAC[o_THM; SUBSET; IN_ELIM_THM] THEN CONJ_TAC THENL [MATCH_MP_TAC HOMEOMORPHISM_COMPOSE THEN ASM_MESON_TAC[HOMEOMORPHISM_SYM]; RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism; IN_INTER]) THEN CONJ_TAC THENL [MP_TAC(ISPECL [`a:real^N`; `r:real`] CENTRE_IN_CBALL) THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN ASM SET_TAC[]; MP_TAC(ISPECL [`a:real^N`; `r:real`] SPHERE_SUBSET_CBALL) THEN ASM SET_TAC[]]]) in let lemma3 = prove (`!a t u v:real^N r s. affine t /\ a IN t /\ ball(a,r) INTER t SUBSET s /\ s SUBSET t /\ u IN ball(a,r) INTER t /\ v IN ball(a,r) INTER t ==> ?f g. homeomorphism (s,s) (f,g) /\ f(u) = v /\ {x | ~(f x = x /\ g x = x)} SUBSET ball(a,r) INTER t`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`a:real^N`; `t:real^N->bool`; `u:real^N`; `v:real^N`; `r:real`] lemma2) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f:real^N->real^N`; `g:real^N->real^N`] THEN STRIP_TAC THEN EXISTS_TAC `\x:real^N. if x IN ball(a,r) INTER t then f x else x` THEN EXISTS_TAC `\x:real^N. if x IN ball(a,r) INTER t then g x else x` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHISM]) THEN REWRITE_TAC[HOMEOMORPHISM; SUBSET; FORALL_IN_IMAGE] THEN STRIP_TAC THEN SUBGOAL_THEN `(!x:real^N. x IN ball(a,r) INTER t ==> f x IN ball(a,r)) /\ (!x:real^N. x IN ball(a,r) INTER t ==> g x IN ball(a,r))` STRIP_ASSUME_TAC THENL [REWRITE_TAC[GSYM CBALL_DIFF_SPHERE] THEN ASM SET_TAC[]; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[SUBSET]) THEN REWRITE_TAC[IN_INTER] THEN REPEAT CONJ_TAC THEN TRY(X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `x IN ball(a:real^N,r)` THEN ASM_SIMP_TAC[] THEN MP_TAC(ISPECL [`a:real^N`; `r:real`] BALL_SUBSET_CBALL) THEN REPEAT(COND_CASES_TAC THEN ASM_SIMP_TAC[]) THEN ASM SET_TAC[]) THEN MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `(cball(a,r) INTER t) UNION ((t:real^N->bool) DIFF ball(a,r))` THEN (CONJ_TAC THENL [ALL_TAC; MP_TAC(ISPECL [`a:real^N`; `r:real`] BALL_SUBSET_CBALL) THEN ASM SET_TAC[]]) THEN MATCH_MP_TAC CONTINUOUS_ON_CASES THEN ASM_SIMP_TAC[CLOSED_CBALL; CLOSED_DIFF; OPEN_BALL; CONTINUOUS_ON_ID; GSYM IN_DIFF; CBALL_DIFF_BALL; CLOSED_AFFINE; CLOSED_INTER] THEN MP_TAC(ISPECL [`a:real^N`; `r:real`] SPHERE_SUBSET_CBALL) THEN MP_TAC(ISPECL [`a:real^N`; `r:real`] CBALL_DIFF_BALL) THEN ASM SET_TAC[]) in REWRITE_TAC[TAUT `p /\ q /\ r /\ s /\ t ==> u <=> p /\ q /\ r /\ s ==> t ==> u`] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ONCE_REWRITE_TAC[TAUT `p ==> q <=> p ==> p /\ q`] THEN MATCH_MP_TAC CONNECTED_EQUIVALENCE_RELATION THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THEN X_GEN_TAC `a:real^N` THENL [X_GEN_TAC `b:real^N` THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC LAND_CONV [SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^N->real^N` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^N->real^N` THEN REWRITE_TAC[HOMEOMORPHISM] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[TAUT `~(p /\ q) <=> ~(q /\ p)`] THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; MAP_EVERY X_GEN_TAC [`b:real^N`; `c:real^N`] THEN MAP_EVERY (fun t -> ASM_CASES_TAC t THEN ASM_REWRITE_TAC[]) [`(a:real^N) IN s`; `(b:real^N) IN s`; `(c:real^N) IN s`] THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f1:real^N->real^N`; `g1:real^N->real^N`] THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`f2:real^N->real^N`; `g2:real^N->real^N`] THEN STRIP_TAC THEN EXISTS_TAC `(f2:real^N->real^N) o (f1:real^N->real^N)` THEN EXISTS_TAC `(g1:real^N->real^N) o (g2:real^N->real^N)` THEN ASM_REWRITE_TAC[o_THM] THEN CONJ_TAC THENL [ASM_MESON_TAC[HOMEOMORPHISM_COMPOSE]; ALL_TAC] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `{x | ~(f1 x = x /\ g1 x = x)} UNION {x:real^N | ~(f2 x = x /\ g2 x = x)}` THEN ASM_REWRITE_TAC[BOUNDED_UNION] THEN ASM SET_TAC[]; DISCH_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_CONTAINS_BALL]) THEN DISCH_THEN(MP_TAC o SPEC `a:real^N` o CONJUNCT2) THEN ASM_SIMP_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `s INTER ball(a:real^N,r)` THEN ASM_SIMP_TAC[IN_INTER; CENTRE_IN_BALL; OPEN_IN_OPEN_INTER; OPEN_BALL] THEN X_GEN_TAC `b:real^N` THEN STRIP_TAC THEN MP_TAC(ISPECL [`a:real^N`; `affine hull s:real^N->bool`; `a:real^N`; `b:real^N`; `r:real`; `t:real^N->bool`] lemma3) THEN ASM_SIMP_TAC[CENTRE_IN_BALL; AFFINE_AFFINE_HULL; HULL_INC; IN_INTER] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN ASM_MESON_TAC[BOUNDED_SUBSET; BOUNDED_BALL; INTER_SUBSET; SUBSET_TRANS]]);; let HOMEOMORPHISM_MOVING_POINTS_EXISTS_GEN = prove (`!s t x (y:A->real^N) k. &2 <= aff_dim s /\ open_in (subtopology euclidean (affine hull s)) s /\ s SUBSET t /\ t SUBSET affine hull s /\ connected s /\ FINITE k /\ (!i. i IN k ==> x i IN s /\ y i IN s) /\ pairwise (\i j. ~(x i = x j) /\ ~(y i = y j)) k ==> ?f g. homeomorphism (t,t) (f,g) /\ (!i. i IN k ==> f(x i) = y i) /\ {x | ~(f x = x /\ g x = x)} SUBSET s /\ bounded {x | ~(f x = x /\ g x = x)}`, REPEAT GEN_TAC THEN ASM_CASES_TAC `FINITE(k:A->bool)` THEN ASM_REWRITE_TAC[] THEN SPEC_TAC(`s:real^N->bool`,`s:real^N->bool`) THEN POP_ASSUM MP_TAC THEN SPEC_TAC(`k:A->bool`,`k:A->bool`) THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN CONJ_TAC THENL [GEN_TAC THEN STRIP_TAC THEN REPEAT(EXISTS_TAC `I:real^N->real^N`) THEN REWRITE_TAC[HOMEOMORPHISM_I; NOT_IN_EMPTY; I_THM; EMPTY_GSPEC] THEN REWRITE_TAC[EMPTY_SUBSET; BOUNDED_EMPTY]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`i:A`; `k:A->bool`] THEN STRIP_TAC THEN X_GEN_TAC `s:real^N->bool` THEN REWRITE_TAC[PAIRWISE_INSERT; FORALL_IN_INSERT] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `s:real^N->bool`) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f:real^N->real^N`; `g:real^N->real^N`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`s DIFF IMAGE (y:A->real^N) k`; `t:real^N->bool`; `(f:real^N->real^N) ((x:A->real^N) i)`; `(y:A->real^N) i`] HOMEOMORPHISM_MOVING_POINT_EXISTS) THEN SUBGOAL_THEN `affine hull (s DIFF (IMAGE (y:A->real^N) k)) = affine hull s` SUBST1_TAC THENL [MATCH_MP_TAC AFFINE_HULL_OPEN_IN THEN CONJ_TAC THENL [TRANS_TAC OPEN_IN_TRANS `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC OPEN_IN_DIFF THEN REWRITE_TAC[OPEN_IN_REFL] THEN MATCH_MP_TAC FINITE_IMP_CLOSED_IN THEN ASM_SIMP_TAC[FINITE_IMAGE] THEN ASM SET_TAC[]; REWRITE_TAC[SET_RULE `s DIFF t = {} <=> s SUBSET t`] THEN DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] FINITE_SUBSET)) THEN ASM_SIMP_TAC[FINITE_IMAGE; CONNECTED_FINITE_IFF_SING] THEN UNDISCH_TAC `&2 <= aff_dim(s:real^N->bool)` THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[] THEN STRIP_TAC THEN ASM_REWRITE_TAC[AFF_DIM_EMPTY; AFF_DIM_SING] THEN CONV_TAC INT_REDUCE_CONV]; ASM_REWRITE_TAC[]] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_DIFF THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC FINITE_IMP_CLOSED_IN THEN ASM_SIMP_TAC[FINITE_IMAGE] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC HULL_INC THEN ASM SET_TAC[]; ASM SET_TAC[]; MATCH_MP_TAC CONNECTED_OPEN_IN_DIFF_CARD_LT THEN ASM_REWRITE_TAC[COLLINEAR_AFF_DIM; INT_ARITH `~(s:int <= &1) <=> &2 <= s`] THEN MATCH_MP_TAC CARD_LT_FINITE_INFINITE THEN ASM_SIMP_TAC[FINITE_IMAGE; real_INFINITE]; ALL_TAC; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN REWRITE_TAC[IN_DIFF] THEN (CONJ_TAC THENL [ASM SET_TAC[]; ASM_REWRITE_TAC[IN_DIFF]]) THEN SIMP_TAC[SET_RULE `~(y IN IMAGE f s) <=> !x. x IN s ==> ~(f x = y)`] THEN ASM SET_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`h:real^N->real^N`; `k:real^N->real^N`] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`(h:real^N->real^N) o (f:real^N->real^N)`; `(g:real^N->real^N) o (k:real^N->real^N)`] THEN CONJ_TAC THENL [ASM_MESON_TAC[HOMEOMORPHISM_COMPOSE]; ALL_TAC] THEN ASM_SIMP_TAC[o_THM] THEN REPEAT(CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `{x | ~(f x = x /\ g x = x)} UNION {x:real^N | ~(h x = x /\ k x = x)}` THEN ASM_REWRITE_TAC[BOUNDED_UNION] THEN ASM SET_TAC[]]);; let HOMEOMORPHISM_MOVING_POINTS_EXISTS = prove (`!s t x (y:A->real^N) k. 2 <= dimindex(:N) /\ open s /\ connected s /\ s SUBSET t /\ FINITE k /\ (!i. i IN k ==> x i IN s /\ y i IN s) /\ pairwise (\i j. ~(x i = x j) /\ ~(y i = y j)) k ==> ?f g. homeomorphism (t,t) (f,g) /\ (!i. i IN k ==> f(x i) = y i) /\ {x | ~(f x = x /\ g x = x)} SUBSET s /\ bounded {x | ~(f x = x /\ g x = x)}`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [STRIP_TAC THEN REPEAT(EXISTS_TAC `I:real^N->real^N`) THEN REWRITE_TAC[HOMEOMORPHISM_I; NOT_IN_EMPTY; I_THM; EMPTY_GSPEC] THEN REWRITE_TAC[EMPTY_SUBSET; BOUNDED_EMPTY] THEN ASM SET_TAC[]; STRIP_TAC] THEN MATCH_MP_TAC HOMEOMORPHISM_MOVING_POINTS_EXISTS_GEN THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN SUBGOAL_THEN `affine hull s = (:real^N)` SUBST1_TAC THENL [MATCH_MP_TAC AFFINE_HULL_OPEN THEN ASM SET_TAC[]; ASM_REWRITE_TAC[SUBTOPOLOGY_UNIV; GSYM OPEN_IN; AFF_DIM_UNIV] THEN ASM_REWRITE_TAC[INT_OF_NUM_LE; SUBSET_UNIV]]);; let HOMEOMORPHISM_GROUPING_POINTS_EXISTS = prove (`!u s t k:real^N->bool. open u /\ open s /\ connected s /\ ~(u = {}) /\ FINITE k /\ k SUBSET s /\ u SUBSET s /\ s SUBSET t ==> ?f g. homeomorphism (t,t) (f,g) /\ {x | ~(f x = x /\ g x = x)} SUBSET s /\ bounded {x | ~(f x = x /\ g x = x)} /\ !x. x IN k ==> (f x) IN u`, let lemma1 = prove (`!a b:real^1 c d:real^1. drop a < drop b /\ drop c < drop d ==> ?f g. homeomorphism (interval[a,b],interval[c,d]) (f,g) /\ f(a) = c /\ f(b) = d`, REPEAT STRIP_TAC THEN EXISTS_TAC `\x. c + (drop x - drop a) / (drop b - drop a) % (d - c:real^1)` THEN ASM_SIMP_TAC[REAL_DIV_REFL; REAL_SUB_LT; REAL_LT_IMP_NZ; REAL_ARITH `(a - a) / x = &0`; LEFT_EXISTS_AND_THM] THEN CONJ_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN MATCH_MP_TAC HOMEOMORPHISM_COMPACT THEN REWRITE_TAC[COMPACT_INTERVAL] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_ADD THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN MATCH_MP_TAC CONTINUOUS_ON_VMUL THEN REWRITE_TAC[LIFT_CMUL; real_div; o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_VMUL THEN REWRITE_TAC[o_DEF; LIFT_SUB; LIFT_DROP] THEN SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID]; REWRITE_TAC[EXTENSION; IN_INTERVAL_1; IN_IMAGE] THEN ASM_SIMP_TAC[GSYM DROP_EQ; DROP_ADD; DROP_CMUL; DROP_SUB; REAL_FIELD `a < b /\ c < d ==> (x = c + (y - a) / (b - a) * (d - c) <=> a + (x - c) / (d - c) * (b - a) = y)`] THEN REWRITE_TAC[GSYM EXISTS_DROP; UNWIND_THM1] THEN REWRITE_TAC[REAL_ARITH `c <= c + x /\ c + x <= d <=> &0 <= x /\ x <= &1 * (d - c)`] THEN ASM_SIMP_TAC[REAL_LE_MUL_EQ; REAL_LE_RMUL_EQ; REAL_SUB_LT] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ; REAL_SUB_LT] THEN REAL_ARITH_TAC; ASM_SIMP_TAC[VECTOR_ARITH `a + x:real^N = a + y <=> x = y`; REAL_FIELD `a < b ==> (x / (b - a) = y / (b - a) <=> x = y)`; REAL_ARITH `x - a:real = y - a <=> x = y`; VECTOR_MUL_RCANCEL; DROP_EQ; VECTOR_SUB_EQ] THEN ASM_MESON_TAC[REAL_LT_REFL]]) in let lemma2 = prove (`!a b c:real^1 u v w:real^1 f1 g1 f2 g2. homeomorphism (interval[a,b],interval[u,v]) (f1,g1) /\ homeomorphism (interval[b,c],interval[v,w]) (f2,g2) ==> b IN interval[a,c] /\ v IN interval[u,w] /\ f1 a = u /\ f1 b = v /\ f2 b = v /\ f2 c = w ==> ?f g. homeomorphism(interval[a,c],interval[u,w]) (f,g) /\ f a = u /\ f c = w /\ (!x. x IN interval[a,b] ==> f x = f1 x) /\ (!x. x IN interval[b,c] ==> f x = f2 x)`, REWRITE_TAC[IN_INTERVAL_1] THEN REPEAT STRIP_TAC THEN REPEAT(FIRST_X_ASSUM (STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [homeomorphism])) THEN EXISTS_TAC `\x. if drop x <= drop b then (f1:real^1->real^1) x else f2 x` THEN ASM_REWRITE_TAC[LEFT_EXISTS_AND_THM; REAL_LE_REFL] THEN ASM_SIMP_TAC[DROP_EQ; REAL_ARITH `b <= c ==> (c <= b <=> c = b)`] THEN CONJ_TAC THENL [REWRITE_TAC[GSYM CONJ_ASSOC]; ASM_MESON_TAC[]] THEN MATCH_MP_TAC HOMEOMORPHISM_COMPACT THEN REWRITE_TAC[COMPACT_INTERVAL] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_CASES_LE THEN ASM_SIMP_TAC[o_DEF; LIFT_DROP; CONTINUOUS_ON_ID; DROP_EQ] THEN CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN SIMP_TAC[SUBSET; FORALL_DROP; IN_ELIM_THM; IN_INTERVAL_1]; SUBGOAL_THEN `interval[a:real^1,c] = interval[a,b] UNION interval[b,c] /\ interval[u:real^1,w] = interval[u,v] UNION interval[v,w]` (CONJUNCTS_THEN SUBST1_TAC) THENL [REWRITE_TAC[EXTENSION; IN_UNION; IN_INTERVAL_1] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[IMAGE_UNION] THEN BINOP_TAC THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [SYM th]) THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> f x = g x) ==> IMAGE f s = IMAGE g s`) THEN SIMP_TAC[IN_INTERVAL_1; REAL_ARITH `b <= c ==> (c <= b <=> c = b)`] THEN ASM_MESON_TAC[DROP_EQ]]; REWRITE_TAC[FORALL_LIFT] THEN MATCH_MP_TAC REAL_WLOG_LT THEN REWRITE_TAC[] THEN CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN REWRITE_TAC[FORALL_DROP; LIFT_DROP; IN_INTERVAL_1] THEN MAP_EVERY X_GEN_TAC [`x:real^1`; `y:real^1`] THEN DISCH_TAC THEN ASM_CASES_TAC `drop y <= drop b` THEN ASM_REWRITE_TAC[] THENL [COND_CASES_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; REAL_NOT_LE]) THEN ASM_MESON_TAC[]; ALL_TAC] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; REAL_NOT_LE]) THENL [ALL_TAC; ASM_MESON_TAC[REAL_LT_IMP_LE]] THEN STRIP_TAC THEN SUBGOAL_THEN `(f1:real^1->real^1) x IN interval[u,v] INTER interval[v,w]` MP_TAC THENL [REWRITE_TAC[IN_INTER] THEN CONJ_TAC THENL [ALL_TAC; ASM_REWRITE_TAC[]] THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [SYM th]) THEN MATCH_MP_TAC FUN_IN_IMAGE THEN ASM_REWRITE_TAC[IN_INTERVAL_1] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[IN_INTER; IN_INTERVAL_1] THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `(a <= x /\ x <= b) /\ (b <= x /\ x <= c) ==> x = b`)) THEN REWRITE_TAC[DROP_EQ] THEN DISCH_TAC THEN SUBGOAL_THEN `(f1:real^1->real^1) x = f1 b /\ (f2:real^1->real^1) y = f2 b` MP_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(MESON[] `!g1:real^1->real^1 g2:real^1->real^1. g1(f1 x) = x /\ g1(f1 b) = b /\ g2(f2 y) = y /\ g2(f2 b) = b ==> f1 x = f1 b /\ f2 y = f2 b ==> x = y`) THEN MAP_EVERY EXISTS_TAC [`g1:real^1->real^1`; `g2:real^1->real^1`] THEN REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REAL_ARITH_TAC]) in let lemma3 = prove (`!a b c d u v:real^1. interval[c,d] SUBSET interval(a,b) /\ interval[u,v] SUBSET interval(a,b) /\ ~(interval(c,d) = {}) /\ ~(interval(u,v) = {}) ==> ?f g. homeomorphism (interval[a,b],interval[a,b]) (f,g) /\ f a = a /\ f b = b /\ !x. x IN interval[c,d] ==> f(x) IN interval[u,v]`, REPEAT GEN_TAC THEN REWRITE_TAC[SUBSET_INTERVAL_1; INTERVAL_NE_EMPTY_1] THEN ASM_CASES_TAC `drop u < drop v` THEN ASM_SIMP_TAC[REAL_ARITH `u < v ==> ~(v < u)`] THEN ASM_CASES_TAC `interval[c:real^1,d] = {}` THENL [DISCH_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT(EXISTS_TAC `I:real^1->real^1`) THEN REWRITE_TAC[HOMEOMORPHISM_I; NOT_IN_EMPTY; I_THM]; RULE_ASSUM_TAC(REWRITE_RULE[INTERVAL_NE_EMPTY_1]) THEN ASM_SIMP_TAC[REAL_ARITH `c <= d ==> ~(d < c)`] THEN STRIP_TAC] THEN MP_TAC(ISPECL [`d:real^1`; `b:real^1`; `v:real^1`; `b:real^1`] lemma1) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f3:real^1->real^1`; `g3:real^1->real^1`] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`c:real^1`; `d:real^1`; `u:real^1`; `v:real^1`] lemma1) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f2:real^1->real^1`; `g2:real^1->real^1`] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`a:real^1`; `c:real^1`; `a:real^1`; `u:real^1`] lemma1) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f1:real^1->real^1`; `g1:real^1->real^1`] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN GEN_REWRITE_TAC I [IMP_IMP] THEN DISCH_THEN(fun th -> ASSUME_TAC(CONJUNCT2 th) THEN MP_TAC(MATCH_MP lemma2 th)) THEN ASM_SIMP_TAC[IN_INTERVAL_1; REAL_LT_IMP_LE; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f4:real^1->real^1`; `g4:real^1->real^1`] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN GEN_REWRITE_TAC I [IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP lemma2) THEN ASM_SIMP_TAC[IN_INTERVAL_1; REAL_LT_IMP_LE; LEFT_IMP_EXISTS_THM] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN SIMP_TAC[] THEN DISCH_THEN(STRIP_ASSUME_TAC o CONJUNCT2) THEN X_GEN_TAC `x:real^1` THEN STRIP_TAC THEN FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [HOMEOMORPHISM]) THEN RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; FORALL_IN_IMAGE; IN_INTERVAL_1]) THEN SUBGOAL_THEN `drop a <= drop x` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ASM_SIMP_TAC[]]) in let lemma4 = prove (`!s k u t:real^1->bool. open u /\ open s /\ connected s /\ ~(u = {}) /\ FINITE k /\ k SUBSET s /\ u SUBSET s /\ s SUBSET t ==> ?f g. homeomorphism (t,t) (f,g) /\ (!x. x IN k ==> f(x) IN u) /\ {x | ~(f x = x /\ g x = x)} SUBSET s /\ bounded {x | ~(f x = x /\ g x = x)}`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?c d:real^1. ~(interval(c,d) = {}) /\ interval[c,d] SUBSET u` STRIP_ASSUME_TAC THENL [UNDISCH_TAC `open(u:real^1->bool)` THEN REWRITE_TAC[OPEN_CONTAINS_INTERVAL] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_TAC `y:real^1`) THEN DISCH_THEN(MP_TAC o SPEC `y:real^1`) THEN ASM_REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?a b:real^1. ~(interval(a,b) = {}) /\ k SUBSET interval[a,b] /\ interval[a,b] SUBSET s` STRIP_ASSUME_TAC THENL [ASM_CASES_TAC `k:real^1->bool = {}` THENL [ASM_MESON_TAC[SUBSET_TRANS; EMPTY_SUBSET]; ALL_TAC] THEN MP_TAC(SPEC `IMAGE drop k` COMPACT_ATTAINS_SUP) THEN MP_TAC(SPEC `IMAGE drop k` COMPACT_ATTAINS_INF) THEN ASM_SIMP_TAC[GSYM IMAGE_o; o_DEF; LIFT_DROP; IMAGE_EQ_EMPTY; IMAGE_ID; FINITE_IMP_COMPACT; EXISTS_IN_IMAGE; FORALL_IN_IMAGE] THEN DISCH_THEN(X_CHOOSE_THEN `a:real^1` STRIP_ASSUME_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `b:real^1` STRIP_ASSUME_TAC) THEN UNDISCH_TAC `open(s:real^1->bool)` THEN REWRITE_TAC[OPEN_CONTAINS_INTERVAL] THEN DISCH_THEN(MP_TAC o SPEC `b:real^1`) THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`u:real^1`; `v:real^1`] THEN REWRITE_TAC[SUBSET; IN_INTERVAL_1] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`a:real^1`; `v:real^1`] THEN REWRITE_TAC[INTERVAL_NE_EMPTY_1] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM IS_INTERVAL_CONNECTED_1]) THEN REWRITE_TAC[IS_INTERVAL_1] THEN ASM_MESON_TAC[GSYM MEMBER_NOT_EMPTY; REAL_LET_TRANS; REAL_LE_TRANS; REAL_LT_IMP_LE; SUBSET; REAL_LE_TOTAL]; ALL_TAC] THEN SUBGOAL_THEN `?w z:real^1. interval[w,z] SUBSET s /\ interval[a,b] UNION interval[c,d] SUBSET interval(w,z)` STRIP_ASSUME_TAC THENL [SUBGOAL_THEN `?w z:real^1. interval[w,z] SUBSET s /\ interval[a,b] UNION interval[c,d] SUBSET interval[w,z]` STRIP_ASSUME_TAC THENL [EXISTS_TAC `lift(min (drop a) (drop c))` THEN EXISTS_TAC `lift(max (drop b) (drop d))` THEN REWRITE_TAC[UNION_SUBSET; SUBSET_INTERVAL_1; LIFT_DROP] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM IS_INTERVAL_CONNECTED_1]) THEN REWRITE_TAC[IS_INTERVAL_1; SUBSET; IN_INTERVAL_1; LIFT_DROP] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN EXISTS_TAC `lift(min (drop a) (drop c))` THEN EXISTS_TAC `lift(max (drop b) (drop d))` THEN ASM_REWRITE_TAC[LIFT_DROP] THEN REWRITE_TAC[real_min; real_max] THEN CONJ_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[LIFT_DROP] THEN ASM_MESON_TAC[ENDS_IN_INTERVAL; SUBSET; INTERVAL_EQ_EMPTY_1; REAL_LT_IMP_LE]; ASM_REAL_ARITH_TAC]; UNDISCH_TAC `open(s:real^1->bool)` THEN REWRITE_TAC[OPEN_CONTAINS_INTERVAL] THEN DISCH_THEN(fun th -> MP_TAC(SPEC `z:real^1` th) THEN MP_TAC(SPEC `w:real^1` th)) THEN SUBGOAL_THEN `(w:real^1) IN interval[w,z] /\ z IN interval[w,z]` STRIP_ASSUME_TAC THENL [REWRITE_TAC[ENDS_IN_INTERVAL] THEN MP_TAC (ISPECL [`a:real^1`; `b:real^1`] INTERVAL_OPEN_SUBSET_CLOSED) THEN ASM SET_TAC[]; REWRITE_TAC[UNION_SUBSET]] THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`w0:real^1`; `w1:real^1`] THEN REWRITE_TAC[IN_INTERVAL_1; SUBSET] THEN STRIP_TAC THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`z0:real^1`; `z1:real^1`] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`w0:real^1`; `z1:real^1`] THEN RULE_ASSUM_TAC (REWRITE_RULE[ENDS_IN_UNIT_INTERVAL; INTERVAL_NE_EMPTY_1; UNION_SUBSET; SUBSET_INTERVAL_1]) THEN CONJ_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; IN_INTERVAL_1]) THEN X_GEN_TAC `x:real^1` THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `x:real^1`)) THEN ASM_CASES_TAC `(x:real^1) IN s` THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC]; ALL_TAC] THEN FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [UNION_SUBSET]) THEN MP_TAC(ISPECL [`w:real^1`; `z:real^1`; `a:real^1`; `b:real^1`; `c:real^1`; `d:real^1`] lemma3) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f:real^1->real^1`; `g:real^1->real^1`] THEN REWRITE_TAC[homeomorphism] THEN STRIP_TAC THEN EXISTS_TAC `\x:real^1. if x IN interval[w,z] then f x else x` THEN EXISTS_TAC `\x:real^1. if x IN interval[w,z] then g x else x` THEN ASSUME_TAC(ISPECL [`w:real^1`; `z:real^1`]INTERVAL_OPEN_SUBSET_CLOSED) THEN REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [ASM SET_TAC[]; ASM SET_TAC[]; ALL_TAC; ASM SET_TAC[]; ASM SET_TAC[]; ALL_TAC; ASM SET_TAC[]; ASM SET_TAC[]; MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `interval[w:real^1,z]` THEN REWRITE_TAC[BOUNDED_INTERVAL] THEN ASM SET_TAC[]] THEN (SUBGOAL_THEN `t = interval[w:real^1,z] UNION (t DIFF interval(w,z))` (fun th -> SUBST1_TAC th THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL THEN ASSUME_TAC(SYM th)) THENL [ASM SET_TAC[]; ASM_REWRITE_TAC[]] THEN ASM_REWRITE_TAC[CONTINUOUS_ON_ID] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CLOSED_SUBSET THEN REWRITE_TAC[CLOSED_INTERVAL] THEN ASM SET_TAC[]; MATCH_MP_TAC CLOSED_IN_DIFF THEN REWRITE_TAC[CLOSED_IN_REFL] THEN MATCH_MP_TAC OPEN_SUBSET THEN REWRITE_TAC[OPEN_INTERVAL] THEN ASM SET_TAC[]; REWRITE_TAC[CLOSED_DIFF_OPEN_INTERVAL_1; SET_RULE `p /\ ~p \/ x IN t DIFF s /\ x IN u <=> x IN t /\ x IN u DIFF s`] THEN MAP_EVERY (MP_TAC o ISPECL [`w:real^1`; `z:real^1`]) (CONJUNCTS ENDS_IN_INTERVAL) THEN ASM SET_TAC[]])) in REPEAT STRIP_TAC THEN ASM_CASES_TAC `2 <= dimindex(:N)` THENL [MP_TAC(ISPECL [`CARD(k:real^N->bool)`; `u:real^N->bool`] CHOOSE_SUBSET_STRONG) THEN ANTS_TAC THENL [ASM_MESON_TAC[FINITE_IMP_NOT_OPEN]; ALL_TAC] THEN REWRITE_TAC[HAS_SIZE; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `p:real^N->bool` THEN STRIP_TAC THEN MP_TAC(ISPECL [`k:real^N->bool`; `p:real^N->bool`] CARD_EQ_BIJECTION) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `y:real^N->real^N` THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`; `\x:real^N. x`; `y:real^N->real^N`; `k:real^N->bool`] HOMEOMORPHISM_MOVING_POINTS_EXISTS) THEN ASM_REWRITE_TAC[pairwise] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_LE]) THEN SIMP_TAC[DIMINDEX_GE_1; ARITH_RULE `1 <= n ==> (n < 2 <=> n = 1)`] THEN REWRITE_TAC[GSYM DIMINDEX_1] THEN DISCH_THEN(MP_TAC o MATCH_MP ISOMORPHISMS_UNIV_UNIV) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`h:real^N->real^1`; `j:real^1->real^N`] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPECL [`IMAGE (h:real^N->real^1) s`; `IMAGE (h:real^N->real^1) k`; `IMAGE (h:real^N->real^1) u`; `IMAGE (h:real^N->real^1) t`] lemma4) THEN ASM_SIMP_TAC[FINITE_IMAGE; IMAGE_SUBSET; IMAGE_EQ_EMPTY; CONNECTED_CONTINUOUS_IMAGE; LINEAR_CONTINUOUS_ON] THEN ANTS_TAC THENL [ASM_MESON_TAC[OPEN_BIJECTIVE_LINEAR_IMAGE_EQ]; REWRITE_TAC[LEFT_IMP_EXISTS_THM; homeomorphism]] THEN MAP_EVERY X_GEN_TAC [`f:real^1->real^1`; `g:real^1->real^1`] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`(j:real^1->real^N) o (f:real^1->real^1) o (h:real^N->real^1)`; `(j:real^1->real^N) o (g:real^1->real^1) o (h:real^N->real^1)`] THEN ASM_REWRITE_TAC[o_THM; IMAGE_o] THEN ASM_SIMP_TAC[CONTINUOUS_ON_COMPOSE; LINEAR_CONTINUOUS_ON] THEN ASM_REWRITE_TAC[GSYM IMAGE_o; o_DEF; IMAGE_ID] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `{x | ~(j ((f:real^1->real^1) (h x)) = x /\ j (g (h x)) = x)} = IMAGE (j:real^1->real^N) {x | ~(f x = x /\ g x = x)}` SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN ASM_SIMP_TAC[BOUNDED_LINEAR_IMAGE]]);; let HOMEOMORPHISM_GROUPING_POINTS_EXISTS_GEN = prove (`!u s t k:real^N->bool. open_in (subtopology euclidean (affine hull s)) s /\ s SUBSET t /\ t SUBSET affine hull s /\ connected s /\ FINITE k /\ k SUBSET s /\ open_in (subtopology euclidean s) u /\ ~(u = {}) ==> ?f g. homeomorphism (t,t) (f,g) /\ (!x. x IN k ==> f(x) IN u) /\ {x | ~(f x = x /\ g x = x)} SUBSET s /\ bounded {x | ~(f x = x /\ g x = x)}`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `&2 <= aff_dim(s:real^N->bool)` THENL [MP_TAC(ISPECL [`CARD(k:real^N->bool)`; `u:real^N->bool`] CHOOSE_SUBSET_STRONG) THEN ANTS_TAC THENL [MATCH_MP_TAC(TAUT `~p ==> p ==> q`) THEN REWRITE_TAC[GSYM INFINITE] THEN MATCH_MP_TAC INFINITE_OPEN_IN THEN EXISTS_TAC `affine hull s:real^N->bool` THEN CONJ_TAC THENL [ASM_MESON_TAC[OPEN_IN_TRANS]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `y:real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONNECTED_IMP_PERFECT_AFF_DIM THEN ASM_SIMP_TAC[CONVEX_CONNECTED; AFFINE_AFFINE_HULL; AFFINE_IMP_CONVEX; AFF_DIM_AFFINE_HULL] THEN CONJ_TAC THENL [ASM_INT_ARITH_TAC; ALL_TAC] THEN ASM_MESON_TAC[OPEN_IN_IMP_SUBSET; SUBSET]; REWRITE_TAC[HAS_SIZE; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `p:real^N->bool` THEN STRIP_TAC THEN MP_TAC (ISPECL [`k:real^N->bool`; `p:real^N->bool`] CARD_EQ_BIJECTION) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `y:real^N->real^N` THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`; `\x:real^N. x`; `y:real^N->real^N`; `k:real^N->bool`] HOMEOMORPHISM_MOVING_POINTS_EXISTS_GEN) THEN ASM_REWRITE_TAC[pairwise] THEN REPEAT(FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET)) THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INT_NOT_LE])] THEN SIMP_TAC[AFF_DIM_GE; INT_ARITH `--(&1):int <= x ==> (x < &2 <=> x = --(&1) \/ x = &0 \/ x = &1)`] THEN REWRITE_TAC[AFF_DIM_EQ_MINUS1; AFF_DIM_EQ_0] THEN SUBGOAL_THEN `(u:real^N->bool) SUBSET s /\ s SUBSET affine hull s` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[open_in]; ALL_TAC] THEN DISCH_THEN(DISJ_CASES_THEN MP_TAC) THENL [ASM SET_TAC[]; ALL_TAC] THEN STRIP_TAC THENL [REPEAT(EXISTS_TAC `I:real^N->real^N`) THEN REWRITE_TAC[HOMEOMORPHISM_I; I_THM; EMPTY_GSPEC; BOUNDED_EMPTY] THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`affine hull s:real^N->bool`; `(:real^1)`] HOMEOMORPHIC_AFFINE_SETS) THEN ASM_REWRITE_TAC[AFF_DIM_UNIV; AFFINE_AFFINE_HULL; AFFINE_UNIV] THEN ASM_REWRITE_TAC[DIMINDEX_1; AFF_DIM_AFFINE_HULL] THEN REWRITE_TAC[homeomorphic; homeomorphism; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`h:real^N->real^1`; `j:real^1->real^N`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`IMAGE (h:real^N->real^1) u`; `IMAGE (h:real^N->real^1) s`; `IMAGE (h:real^N->real^1) t`; `IMAGE (h:real^N->real^1) k`] HOMEOMORPHISM_GROUPING_POINTS_EXISTS) THEN ASM_SIMP_TAC[FINITE_IMAGE; IMAGE_SUBSET; IMAGE_EQ_EMPTY] THEN ANTS_TAC THENL [MP_TAC(ISPECL [`h:real^N->real^1`; `j:real^1->real^N`; `affine hull s:real^N->bool`; `(:real^1)`] HOMEOMORPHISM_IMP_OPEN_MAP) THEN ASM_SIMP_TAC[homeomorphism; SUBTOPOLOGY_UNIV; GSYM OPEN_IN] THEN REPEAT STRIP_TAC THENL [ASM_MESON_TAC[OPEN_IN_TRANS]; ALL_TAC] THEN MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; REWRITE_TAC[LEFT_IMP_EXISTS_THM; homeomorphism]] THEN MAP_EVERY X_GEN_TAC [`f:real^1->real^1`; `g:real^1->real^1`] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`\x. if x IN affine hull s then ((j:real^1->real^N) o (f:real^1->real^1) o (h:real^N->real^1)) x else x`; `\x. if x IN affine hull s then ((j:real^1->real^N) o (g:real^1->real^1) o (h:real^N->real^1)) x else x`] THEN ASM_SIMP_TAC[o_THM; IMAGE_o] THEN REPEAT CONJ_TAC THENL [ASM SET_TAC[]; ASM_SIMP_TAC[SET_RULE `t SUBSET s ==> IMAGE (\x. if x IN s then f x else x) t = IMAGE f t`] THEN REPLICATE_TAC 3 (ONCE_REWRITE_TAC[GSYM o_DEF]) THEN ASM_REWRITE_TAC[IMAGE_o] THEN ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_ON_EQ THEN EXISTS_TAC `(j:real^1->real^N) o (f:real^1->real^1) o (h:real^N->real^1)` THEN REWRITE_TAC[o_THM] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REPEAT(MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; ASM SET_TAC[]; ASM_SIMP_TAC[SET_RULE `t SUBSET s ==> IMAGE (\x. if x IN s then f x else x) t = IMAGE f t`] THEN REPLICATE_TAC 3 (ONCE_REWRITE_TAC[GSYM o_DEF]) THEN ASM_REWRITE_TAC[IMAGE_o] THEN ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_ON_EQ THEN EXISTS_TAC `(j:real^1->real^N) o (g:real^1->real^1) o (h:real^N->real^1)` THEN REWRITE_TAC[o_THM] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REPEAT(MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; ASM SET_TAC[]; ALL_TAC; ALL_TAC] THEN REWRITE_TAC[MESON[] `(if P then f x else x) = x <=> ~P \/ f x = x`] THEN REWRITE_TAC[DE_MORGAN_THM; GSYM LEFT_OR_DISTRIB] THEN (SUBGOAL_THEN `{x | x IN affine hull s /\ (~(j (f (h x)) = x) \/ ~(j (g (h x)) = x))} = IMAGE (j:real^1->real^N) {x | ~(f x = x /\ g x = x)}` SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THENL [TRANS_TAC SUBSET_TRANS `IMAGE (j:real^1->real^N) (IMAGE (h:real^N->real^1) s)` THEN ASM SET_TAC[]; MATCH_MP_TAC(MESON[CLOSURE_SUBSET; BOUNDED_SUBSET; IMAGE_SUBSET] `bounded (IMAGE f (closure s)) ==> bounded (IMAGE f s)`) THEN MATCH_MP_TAC COMPACT_IMP_BOUNDED THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[COMPACT_CLOSURE] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV]]);; let HOMEOMORPHISM_MOVING_DENSE_COUNTABLE_SUBSETS_EXISTS = prove (`!s:real^M->bool t:real^N->bool. COUNTABLE s /\ closure s = affine hull s /\ COUNTABLE t /\ closure t = affine hull t /\ aff_dim s = aff_dim t ==> ?f g. homeomorphism (affine hull s,affine hull t) (f,g) /\ IMAGE f s = t`, let lemma = prove (`!n s:real^N->bool t:real^N->bool. 1 <= n /\ n <= dimindex(:N) /\ INFINITE s /\ COUNTABLE s /\ closure s = span(IMAGE basis (1..n)) /\ INFINITE t /\ COUNTABLE t /\ closure t = span(IMAGE basis (1..n)) ==> ?f g. homeomorphism (span(IMAGE basis (1..n)),span(IMAGE basis (1..n))) (f,g) /\ IMAGE f s = t`, X_GEN_TAC `n:num` THEN ASM_CASES_TAC `1 <= n` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `n <= dimindex(:N)` THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r /\ s <=> (p /\ q /\ r) /\ s`] THEN MATCH_MP_TAC(METIS[] `!Q. (!s t. P s /\ P t /\ R s t ==> R t s) /\ (!t. P t /\ (!s. P s /\ Q s ==> R s t) ==> (!s. P s ==> R s t)) /\ (!s t. P s /\ Q s /\ P t /\ Q t ==> R s t) ==> !s t. P s /\ P t ==> R s t`) THEN EXISTS_TAC `\s. pairwise (\x y:real^N. !i. 1 <= i /\ i <= n ==> ~(x$i = y$i)) s` THEN REWRITE_TAC[] THEN CONJ_TAC THENL [MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `t:real^N->bool`] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC)) THEN GEN_REWRITE_TAC RAND_CONV [SWAP_EXISTS_THM] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN GEN_REWRITE_TAC (RAND_CONV o LAND_CONV) [HOMEOMORPHISM_SYM] THEN SIMP_TAC[] THEN REWRITE_TAC[homeomorphism] THEN MP_TAC(ISPEC `s:real^N->bool` CLOSURE_SUBSET) THEN ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN X_GEN_TAC `s:real^N->bool` THEN STRIP_TAC THEN MP_TAC(ISPECL [`n:num`; `s:real^N->bool`] ROTATION_TO_GENERAL_POSITION_EXISTS_GEN) THEN ANTS_TAC THENL [ASM_MESON_TAC[CLOSURE_SUBSET]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `f:real^N->real^N` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPEC `IMAGE (f:real^N->real^N) s`) THEN FIRST_ASSUM(MP_TAC o SPECL [`span(IMAGE basis (1..n)):real^N->bool`; `span(IMAGE basis (1..n)):real^N->bool`] o MATCH_MP (REWRITE_RULE[IMP_CONJ] ORTHOGONAL_TRANSFORMATION_IMP_HOMEOMORPHISM)) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `g:real^N->real^N` THEN STRIP_TAC THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [ASM_MESON_TAC[HOMEOMORPHISM_INFINITENESS; CLOSURE_SUBSET]; ASM_MESON_TAC[HOMEOMORPHISM_COUNTABILITY; CLOSURE_SUBSET]; FIRST_ASSUM(MP_TAC o SPEC `s:real^N->bool` o MATCH_MP(REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_CLOSURE_OF)) THEN ASM_SIMP_TAC[CLOSURE_OF_SUBTOPOLOGY; EUCLIDEAN_CLOSURE_OF; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; CLOSURE_MINIMAL; CLOSED_SPAN; SET_RULE `s INTER s = s`; SET_RULE `s SUBSET t ==> t INTER s = s`] THEN ANTS_TAC THENL [ASM_MESON_TAC[CLOSURE_SUBSET]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `c SUBSET d /\ d SUBSET s ==> s = s INTER c ==> d = s`) THEN SIMP_TAC[SUBSET_CLOSURE; INTER_SUBSET] THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN REWRITE_TAC[CLOSED_SPAN] THEN ASM_MESON_TAC[CLOSURE_SUBSET; IMAGE_SUBSET; SUBSET_TRANS]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [pairwise]) THEN REWRITE_TAC[PAIRWISE_IMAGE] THEN REWRITE_TAC[pairwise] THEN MESON_TAC[]]; REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`h:real^N->real^N`; `k:real^N->real^N`] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`(h:real^N->real^N) o (f:real^N->real^N)`; `(g:real^N->real^N) o (k:real^N->real^N)`] THEN ASM_REWRITE_TAC[IMAGE_o] THEN ASM_MESON_TAC[HOMEOMORPHISM_COMPOSE]]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `t:real^N->bool`] THEN STRIP_TAC THEN SUBGOAL_THEN `?f:real^N->real^N. IMAGE f s = t /\ !x y i. x IN s /\ y IN s /\ 1 <= i /\ i <= n ==> real_sgn(f x$i - f y$i) = real_sgn(x$i - y$i)` STRIP_ASSUME_TAC THENL [ALL_TAC; SUBGOAL_THEN `!i. ?g h. 1 <= i /\ i <= n ==> (!x. x IN s ==> (f:real^N->real^N) x$i = drop(g(lift(x$i)))) /\ homeomorphism ((:real^1),(:real^1)) (g,h)` MP_TAC THENL [REWRITE_TAC[RIGHT_EXISTS_IMP_THM] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN SUBGOAL_THEN `?g. !x. x IN s ==> (f:real^N->real^N)(x)$i = g(x$i)` STRIP_ASSUME_TAC THENL [GEN_REWRITE_TAC I [GSYM FUNCTION_FACTORS_LEFT_GEN] THEN RULE_ASSUM_TAC(REWRITE_RULE[pairwise]) THEN ASM_MESON_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`lift o g o drop`; `IMAGE (\x:real^N. lift(x$i)) s`] INCREASING_EXTENDS_FROM_DENSE) THEN ASM_REWRITE_TAC[o_THM; LIFT_DROP] THEN ANTS_TAC THENL [REWRITE_TAC[IMP_CONJ; FORALL_IN_IMAGE; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[LIFT_DROP; CONJ_ASSOC] THEN CONJ_TAC THENL [REWRITE_TAC[GSYM IMAGE_o; o_DEF; LIFT_DROP] THEN SUBGOAL_THEN `IMAGE (\x. lift(g(x$i))) s = IMAGE (\x. lift((f:real^N->real^N) x$i)) s` SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ONCE_REWRITE_TAC[SET_RULE `IMAGE (\x. lift(f x$i)) s = IMAGE (\y. lift(y$i)) (IMAGE f s)`] THEN ASM_REWRITE_TAC[IMAGE_ID] THEN CONJ_TAC THEN W(MP_TAC o PART_MATCH (rand o rand) CLOSURE_LINEAR_IMAGE_SUBSET o lhand o snd) THEN ASM_REWRITE_TAC[LINEAR_LIFT_COMPONENT] THEN MATCH_MP_TAC(SET_RULE `s = UNIV ==> s SUBSET t ==> t = UNIV`) THEN REWRITE_TAC[EXTENSION; IN_IMAGE; IN_SPAN_IMAGE_BASIS; IN_UNIV] THEN X_GEN_TAC `c:real^1` THEN EXISTS_TAC `drop c % basis i:real^N` THEN (SUBGOAL_THEN `i <= dimindex(:N)` ASSUME_TAC THENL [ASM_ARITH_TAC; ALL_TAC]) THEN ASM_SIMP_TAC[VECTOR_MUL_COMPONENT; BASIS_COMPONENT] THEN REWRITE_TAC[IN_NUMSEG; REAL_MUL_RID; LIFT_DROP] THEN GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_MUL_RZERO]; X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL[`x:real^N`; `y:real^N`; `i:num`]) THEN ASM_SIMP_TAC[real_sgn] THEN REAL_ARITH_TAC]; REWRITE_TAC[FORALL_IN_IMAGE; LIFT_DROP] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `h:real^1->real^1` THEN STRIP_TAC THEN REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN ASM_SIMP_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[LIFT_DROP]; ALL_TAC] THEN MATCH_MP_TAC MONOTONE_IMP_HOMEOMORPHISM_1D THEN ASM_REWRITE_TAC[IS_INTERVAL_UNIV; IN_UNIV]]; REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`g:num->real^1->real^1`; `h:num->real^1->real^1`] THEN REWRITE_TAC[HOMEOMORPHISM; SUBSET; FORALL_IN_IMAGE; IN_UNIV] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`(\x. lambda i. if i IN 1..n then drop((g:num->real^1->real^1) i (lift(x$i))) else &0):real^N->real^N`; `(\x. lambda i. if i IN 1..n then drop((h:num->real^1->real^1) i (lift(x$i))) else &0):real^N->real^N`] THEN SIMP_TAC[IN_SPAN_IMAGE_BASIS; LAMBDA_BETA; GSYM CONJ_ASSOC] THEN GEN_REWRITE_TAC I [CONJ_ASSOC] THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[CONTINUOUS_ON_COMPONENTWISE_LIFT] THEN SIMP_TAC[LAMBDA_BETA] THEN CONJ_TAC THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN ASM_CASES_TAC `i IN 1..n` THEN ASM_REWRITE_TAC[CONTINUOUS_ON_CONST; LIFT_DROP] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_SIMP_TAC[CONTINUOUS_ON_LIFT_COMPONENT] THEN MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `(:real^1)` THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_NUMSEG]) THEN ASM_SIMP_TAC[] THEN REWRITE_TAC[SUBSET_UNIV]; ALL_TAC] THEN GEN_REWRITE_TAC I [CONJ_ASSOC] THEN CONJ_TAC THENL [SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN CONJ_TAC THEN X_GEN_TAC `z:real^N` THEN DISCH_TAC THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[LIFT_DROP] THEN REWRITE_TAC[GSYM LIFT_EQ; LIFT_DROP] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_NUMSEG]) THEN ASM_SIMP_TAC[]; EXPAND_TAC "t" THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> f x = g x) ==> IMAGE f s = IMAGE g s`) THEN ASM_SIMP_TAC[CART_EQ; LAMBDA_BETA; IN_NUMSEG] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[] THEN CONV_TAC SYM_CONV THEN SUBGOAL_THEN `(f:real^N->real^N) x IN span(IMAGE basis (1..n))` MP_TAC THENL [ASM_MESON_TAC[CLOSURE_SUBSET; SUBSET; IN_IMAGE]; ALL_TAC] THEN ASM_SIMP_TAC[IN_SPAN_IMAGE_BASIS; IN_NUMSEG]]]] THEN REWRITE_TAC[TAUT `p /\ q /\ r /\ s ==> t <=> p /\ q ==> r /\ s ==> t`] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN MATCH_MP_TAC(MESON[] `(?f. IMAGE f s = t /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\ P f) ==> (?f. IMAGE f s = t /\ P f)`) THEN MATCH_MP_TAC BACK_AND_FORTH_2 THEN ASM_REWRITE_TAC[REAL_SUB_REFL] THEN CONJ_TAC THENL [REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_NEG_SUB] THEN REWRITE_TAC[REAL_SGN_NEG] THEN ASM_MESON_TAC[]; ALL_TAC] THEN POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN ONCE_REWRITE_TAC[TAUT `p ==> q /\ r <=> (p ==> q) /\ (p ==> r)`] THEN REWRITE_TAC[RIGHT_IMP_FORALL_THM] THEN MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`t:real^N->bool`; `s:real^N->bool`] THEN REWRITE_TAC[FORALL_AND_THM] THEN GEN_REWRITE_TAC RAND_CONV [MESON[] `(!s t f s' t' x. P s t f s' t' x) <=> (!s t f t' s' x. P t s f t' s' x)`] THEN REWRITE_TAC[AND_FORALL_THM; IMP_IMP] THEN MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `t:real^N->bool`; `f:real^N->real^N`; `s':real^N->bool`; `t':real^N->bool`; `x:real^N`] THEN MATCH_MP_TAC(TAUT `(q <=> p) /\ p ==> p /\ q`) THEN CONJ_TAC THENL [REWRITE_TAC[EQ_SYM_EQ] THEN REWRITE_TAC[CONJ_ACI]; REWRITE_TAC[IN_DIFF] THEN STRIP_TAC] THEN ABBREV_TAC `u = INTERS {{y:real^N | y$i < (f:real^N->real^N)(z)$i} |i,z| i IN 1..n /\ z IN {z | z IN s' /\ (x:real^N)$i < z$i}} INTER INTERS {{y:real^N | y$i > (f:real^N->real^N)(z)$i} |i,z| i IN 1..n /\ z IN {z | z IN s' /\ (x:real^N)$i > z$i}}` THEN SUBGOAL_THEN `open(u:real^N->bool)` ASSUME_TAC THENL [EXPAND_TAC "u" THEN MATCH_MP_TAC OPEN_INTER THEN CONJ_TAC THEN MATCH_MP_TAC OPEN_INTERS THEN ASM_SIMP_TAC[FINITE_PRODUCT_DEPENDENT; FINITE_NUMSEG; FINITE_RESTRICT; FORALL_IN_GSPEC; OPEN_HALFSPACE_COMPONENT_GT; OPEN_HALFSPACE_COMPONENT_LT]; ALL_TAC] THEN SUBGOAL_THEN `~(u:real^N->bool = {})` ASSUME_TAC THENL [EXPAND_TAC "u" THEN GEN_REWRITE_TAC RAND_CONV [EXTENSION] THEN REWRITE_TAC[INTERS_GSPEC; IN_INTER; NOT_IN_EMPTY; IN_ELIM_THM] THEN REWRITE_TAC[NOT_FORALL_THM; IN_NUMSEG] THEN SUBGOAL_THEN `!i. 1 <= i /\ i <= n <=> (1 <= i /\ i <= dimindex(:N)) /\ i <= n` MP_TAC THENL [ASM_ARITH_TAC; DISCH_THEN(fun th -> REWRITE_TAC[th])] THEN ONCE_REWRITE_TAC[TAUT `(p /\ q) /\ r ==> s <=> p ==> q /\ r ==> s`] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[AND_FORALL_THM] THEN REWRITE_TAC[TAUT `(p ==> q) /\ (p ==> r) <=> p ==> q /\ r`] THEN REWRITE_TAC[GSYM LAMBDA_SKOLEM] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN ASM_CASES_TAC `i:num <= n` THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[SET_RULE `(!x. x IN s /\ P x ==> R y ((f x)$i)) <=> (!x. x IN IMAGE (\x. (f x)$i) {x | x IN s /\ P x} ==> R y x)`] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[real_gt] THEN ASM_SIMP_TAC[REAL_LT_BETWEEN_GEN; FINITE_RESTRICT; FINITE_IMAGE] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `u:real^N` THEN STRIP_TAC THEN X_GEN_TAC `v:real^N` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`u:real^N`; `v:real^N`]) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[REAL_SGN_EQ_INEQ] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `?y:real^N. y IN (t DIFF t') /\ y IN u` MP_TAC THENL [SUBGOAL_THEN `?z:real^N. z IN span(IMAGE basis (1..n)) INTER u` MP_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_TAC `z:real^N`) THEN EXISTS_TAC `(lambda i. if i IN 1..n then (z:real^N)$i else &0):real^N` THEN UNDISCH_TAC `(z:real^N) IN u` THEN EXPAND_TAC "u" THEN REWRITE_TAC[INTERS_GSPEC; IN_INTER; IN_ELIM_THM] THEN SUBGOAL_THEN `!i. i IN 1..n ==> 1 <= i /\ i <= dimindex(:N)` MP_TAC THENL [REWRITE_TAC[IN_NUMSEG] THEN ASM_ARITH_TAC; ALL_TAC] THEN SIMP_TAC[IN_SPAN_IMAGE_BASIS; LAMBDA_BETA]; REWRITE_TAC[IN_INTER] THEN STRIP_TAC] THEN SUBGOAL_THEN `(z:real^N) limit_point_of t` MP_TAC THENL [ONCE_REWRITE_TAC[GSYM LIMPT_OF_CLOSURE] THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONNECTED_IMP_PERFECT_AFF_DIM THEN ASM_SIMP_TAC[CONVEX_SPAN; CONVEX_CONNECTED] THEN SIMP_TAC[AFF_DIM_DIM_SUBSPACE; SUBSPACE_SPAN] THEN REWRITE_TAC[DIM_SPAN; DIM_BASIS_IMAGE] THEN REWRITE_TAC[INTER_NUMSEG; CARD_NUMSEG_1; ARITH_RULE `MAX n n = n`] THEN REWRITE_TAC[INT_OF_NUM_EQ] THEN MATCH_MP_TAC(ARITH_RULE `1 <= m /\ 1 <= n ==> ~(MIN m n = 0)`) THEN ASM_REWRITE_TAC[DIMINDEX_GE_1]; GEN_REWRITE_TAC LAND_CONV [LIMPT_INFINITE_OPEN]] THEN DISCH_THEN(MP_TAC o SPEC `u:real^N->bool`) THEN ASM_REWRITE_TAC[] THEN UNDISCH_TAC `FINITE(t':real^N->bool)` THEN REWRITE_TAC[GSYM IMP_CONJ_ALT] THEN DISCH_THEN(MP_TAC o MATCH_MP INFINITE_DIFF_FINITE) THEN DISCH_THEN(MP_TAC o MATCH_MP INFINITE_NONEMPTY) THEN SET_TAC[]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_DIFF] THEN DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC)] THEN ASM_REWRITE_TAC[] THEN EXPAND_TAC "u" THEN REWRITE_TAC[IN_INTER; INTERS_GSPEC; IN_ELIM_THM] THEN CONV_TAC(RAND_CONV(ONCE_DEPTH_CONV SYM_CONV)) THEN REWRITE_TAC[REAL_SGNS_EQ_ALT; real_gt; REAL_SUB_LT; REAL_SUB_0; REAL_ARITH `x - y < &0 <=> x < y`] THEN SIMP_TAC[IN_NUMSEG] THEN RULE_ASSUM_TAC(REWRITE_RULE[pairwise]) THEN ASM SET_TAC[]) in REPEAT GEN_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THENL [ASM_SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_EQ_MINUS1; AFFINE_HULL_EMPTY] THEN REWRITE_TAC[IMAGE_CLAUSES; GSYM homeomorphic] THEN REWRITE_TAC[HOMEOMORPHIC_EMPTY]; ALL_TAC] THEN ASM_CASES_TAC `aff_dim(t:real^N->bool) = &0` THENL [ASM_REWRITE_TAC[AFF_DIM_EQ_0; CONJ_ASSOC] THEN ONCE_REWRITE_TAC[IMP_CONJ_ALT] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [AFF_DIM_EQ_0]) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `b:real^N` THEN DISCH_THEN SUBST1_TAC THEN X_GEN_TAC `a:real^M` THEN DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[AFFINE_HULL_SING; CLOSURE_SING] THEN DISCH_THEN(K ALL_TAC) THEN MAP_EVERY EXISTS_TAC [`(\x. b):real^M->real^N`; `(\x. a):real^N->real^M`] THEN ASM_REWRITE_TAC[IMAGE_CLAUSES; homeomorphism; CONTINUOUS_ON_CONST] THEN REWRITE_TAC[FORALL_IN_INSERT; NOT_IN_EMPTY]; ALL_TAC] THEN ASM_CASES_TAC `FINITE(s:real^M->bool)` THENL [ASM_SIMP_TAC[CLOSURE_CLOSED; FINITE_IMP_CLOSED] THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN MP_TAC(ISPEC `affine hull s:real^M->bool` CONNECTED_FINITE_IFF_SING) THEN SIMP_TAC[AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL; CONVEX_CONNECTED] THEN ASM_MESON_TAC[AFF_DIM_EQ_0; AFF_DIM_EQ_MINUS1]; ALL_TAC] THEN ASM_CASES_TAC `FINITE(t:real^N->bool)` THENL [ASM_SIMP_TAC[CLOSURE_CLOSED; FINITE_IMP_CLOSED] THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN MP_TAC(ISPEC `affine hull t:real^N->bool` CONNECTED_FINITE_IFF_SING) THEN SIMP_TAC[AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL; CONVEX_CONNECTED] THEN ASM_MESON_TAC[AFF_DIM_EQ_0; AFF_DIM_EQ_MINUS1]; ALL_TAC] THEN STRIP_TAC THEN SUBGOAL_THEN `?n. aff_dim(t:real^N->bool) = &n` (CHOOSE_THEN (fun th -> SUBST_ALL_TAC th THEN ASSUME_TAC th)) THENL [ASM_REWRITE_TAC[INT_OF_NUM_EXISTS; AFF_DIM_POS_LE]; ALL_TAC] THEN SUBGOAL_THEN `&1 <= aff_dim(t:real^N->bool)` MP_TAC THENL [MATCH_MP_TAC(INT_ARITH `-- &1:int <= x /\ ~(x = -- &1) /\ ~(x = &0) ==> &1 <= x`) THEN REWRITE_TAC[AFF_DIM_GE; AFF_DIM_EQ_MINUS1] THEN ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[INT_OF_NUM_LE] THEN DISCH_TAC] THEN MAP_EVERY (C UNDISCH_THEN (K ALL_TAC)) [`~(t:real^N->bool = {})`; `~(&n:int = &0)`] THEN RULE_ASSUM_TAC(REWRITE_RULE[GSYM INFINITE]) THEN MP_TAC(ISPEC `s:real^M->bool` AFF_DIM_LE_UNIV) THEN MP_TAC(ISPEC `t:real^N->bool` AFF_DIM_LE_UNIV) THEN ASM_REWRITE_TAC[INT_OF_NUM_LE] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`affine hull s:real^M->bool`; `span(IMAGE basis (1..n)):real^(M,N)finite_sum->bool`] HOMEOMORPHIC_AFFINE_SETS) THEN SIMP_TAC[AFFINE_AFFINE_HULL; AFFINE_SPAN; AFF_DIM_AFFINE_HULL] THEN SIMP_TAC[AFF_DIM_DIM_SUBSPACE; SUBSPACE_SPAN] THEN SIMP_TAC[DIM_BASIS_IMAGE; DIM_SPAN; INTER_NUMSEG] THEN ASM_SIMP_TAC[ARITH_RULE `MAX x x = x`; CARD_NUMSEG_1] THEN ASM_SIMP_TAC[DIMINDEX_FINITE_SUM; ARITH_RULE `n <= N ==> MIN (M + N) n = n`; homeomorphic; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`h1:real^M->real^(M,N)finite_sum`; `k1:real^(M,N)finite_sum->real^M`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`affine hull t:real^N->bool`; `span(IMAGE basis (1..n)):real^(M,N)finite_sum->bool`] HOMEOMORPHIC_AFFINE_SETS) THEN SIMP_TAC[AFFINE_AFFINE_HULL; AFFINE_SPAN; AFF_DIM_AFFINE_HULL] THEN SIMP_TAC[AFF_DIM_DIM_SUBSPACE; SUBSPACE_SPAN] THEN SIMP_TAC[DIM_BASIS_IMAGE; DIM_SPAN; INTER_NUMSEG] THEN ASM_SIMP_TAC[ARITH_RULE `MAX x x = x`; CARD_NUMSEG_1] THEN ASM_SIMP_TAC[DIMINDEX_FINITE_SUM; ARITH_RULE `n <= N ==> MIN (M + N) n = n`; homeomorphic; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`h2:real^N->real^(M,N)finite_sum`; `k2:real^(M,N)finite_sum->real^N`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`n:num`; `IMAGE (h1:real^M->real^(M,N)finite_sum) s`; `IMAGE (h2:real^N->real^(M,N)finite_sum) t`] lemma) THEN ASM_REWRITE_TAC[DIMINDEX_FINITE_SUM] THEN ANTS_TAC THENL [ALL_TAC; REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f:real^(M,N)finite_sum->real^(M,N)finite_sum`; `g:real^(M,N)finite_sum->real^(M,N)finite_sum`] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`(k2:real^(M,N)finite_sum->real^N) o f o (h1:real^M->real^(M,N)finite_sum)`; `(k1:real^(M,N)finite_sum->real^M) o g o (h2:real^N->real^(M,N)finite_sum)`] THEN CONJ_TAC THENL [GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [o_ASSOC] THEN MATCH_MP_TAC HOMEOMORPHISM_COMPOSE THEN EXISTS_TAC `span(IMAGE basis (1..n)):real^(M,N)finite_sum->bool` THEN GEN_REWRITE_TAC RAND_CONV [HOMEOMORPHISM_SYM] THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC HOMEOMORPHISM_COMPOSE THEN EXISTS_TAC `span(IMAGE basis (1..n)):real^(M,N)finite_sum->bool` THEN ASM_REWRITE_TAC[]; SUBGOAL_THEN `(t:real^N->bool) SUBSET affine hull t` MP_TAC THENL [REWRITE_TAC[HULL_SUBSET]; ASM_REWRITE_TAC[IMAGE_o]] THEN RULE_ASSUM_TAC(REWRITE_RULE[HOMEOMORPHISM]) THEN ASM SET_TAC[]]] THEN CONJ_TAC THENL [ASM_ARITH_TAC; ASM_SIMP_TAC[COUNTABLE_IMAGE]] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r /\ s <=> (p /\ r) /\ (q /\ s)`] THEN CONJ_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC INFINITE_IMAGE THEN ASM_REWRITE_TAC[] THENL [SUBGOAL_THEN `(s:real^M->bool) SUBSET affine hull s` MP_TAC THENL [REWRITE_TAC[HULL_SUBSET]; ALL_TAC]; SUBGOAL_THEN `(t:real^N->bool) SUBSET affine hull t` MP_TAC THENL [REWRITE_TAC[HULL_SUBSET]; ALL_TAC]] THEN RULE_ASSUM_TAC(REWRITE_RULE[HOMEOMORPHISM]) THEN ASM SET_TAC[]; CONJ_TAC THENL [FIRST_ASSUM(MP_TAC o SPEC `s:real^M->bool` o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_CLOSURE_OF)) THEN SUBGOAL_THEN `span(IMAGE basis (1..n)) = IMAGE (h1:real^M->real^(M,N)finite_sum) (affine hull s)` (fun th -> SUBST1_TAC th THEN ASSUME_TAC(SYM th)) THENL [ASM_MESON_TAC[homeomorphism]; ALL_TAC]; FIRST_ASSUM(MP_TAC o SPEC `t:real^N->bool` o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_CLOSURE_OF)) THEN SUBGOAL_THEN `span(IMAGE basis (1..n)) = IMAGE (h2:real^N->real^(M,N)finite_sum) (affine hull t)` (fun th -> SUBST1_TAC th THEN ASSUME_TAC(SYM th)) THENL [ASM_MESON_TAC[homeomorphism]; ALL_TAC]] THEN SIMP_TAC[HULL_SUBSET; CLOSURE_OF_SUBTOPOLOGY; EUCLIDEAN_CLOSURE_OF; IMAGE_SUBSET; SET_RULE `s SUBSET t ==> t INTER s = s`] THEN ASM_REWRITE_TAC[SET_RULE `s INTER s = s`] THEN DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[SET_RULE `c = s INTER c <=> c SUBSET s`] THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN REWRITE_TAC[CLOSED_SPAN] THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [SYM th]) THEN SIMP_TAC[IMAGE_SUBSET; HULL_SUBSET]]);; (* ------------------------------------------------------------------------- *) (* Boring but useful lemmas about the number of unbounded components. *) (* ------------------------------------------------------------------------- *) let HAS_SIZE_UNBOUNDED_COMPONENTS_COMPLEMENT_1 = prove (`!s. bounded s /\ ~(s = {}) ==> {c | c IN components((:real^1) DIFF s) /\ ~bounded c} HAS_SIZE 2`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [BOUNDED_POS]) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; NORM_1] THEN X_GEN_TAC `B:real` THEN STRIP_TAC THEN CONV_TAC HAS_SIZE_CONV THEN MAP_EVERY ABBREV_TAC [`l = connected_component ((:real^1) DIFF s) (--lift(B + &1))`; `r = connected_component ((:real^1) DIFF s) (lift(B + &1))`] THEN SUBGOAL_THEN `l IN components((:real^1) DIFF s) /\ r IN components((:real^1) DIFF s)` STRIP_ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["l"; "r"] THEN REWRITE_TAC[components] THEN REWRITE_TAC[SIMPLE_IMAGE; ETA_AX] THEN CONJ_TAC THEN MATCH_MP_TAC FUN_IN_IMAGE THEN REWRITE_TAC[IN_UNIV; IN_DIFF] THEN FIRST_X_ASSUM(MATCH_MP_TAC o ONCE_REWRITE_RULE[GSYM CONTRAPOS_THM]) THEN REWRITE_TAC[DROP_NEG; LIFT_DROP] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `(!x. drop x < --B ==> x IN l) /\ (!x. B < drop x ==> x IN r)` STRIP_ASSUME_TAC THENL [REPEAT STRIP_TAC THEN MAP_EVERY EXPAND_TAC ["l"; "r"] THEN REWRITE_TAC[IN] THEN REWRITE_TAC[CONNECTED_COMPONENT_1] THEN REWRITE_TAC[SEGMENT_1; SUBSET; IN_UNIV; IN_DIFF; DROP_NEG; LIFT_DROP] THEN GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_NEG; LIFT_DROP] THEN STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o ONCE_REWRITE_RULE[GSYM CONTRAPOS_THM]) THEN REWRITE_TAC[DROP_NEG; LIFT_DROP] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN MAP_EVERY EXISTS_TAC [`l:real^1->bool`; `r:real^1->bool`] THEN ASM_REWRITE_TAC[SET_RULE `{x | P x} = {a,b} <=> P a /\ P b /\ !c. P c ==> c = a \/ c = b`] THEN CONJ_TAC THENL [DISCH_TAC THEN SUBGOAL_THEN `--lift(B + &1) IN l /\ lift(B + &1) IN r` MP_TAC THENL [CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[DROP_NEG; LIFT_DROP] THEN ASM_REAL_ARITH_TAC; ASM_REWRITE_TAC[]] THEN W(MP_TAC o PART_MATCH (rand o rand) CONVEX_CONTAINS_SEGMENT_IMP o rand o snd) THEN REWRITE_TAC[GSYM CONNECTED_CONVEX_1] THEN ANTS_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]; ALL_TAC] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN REWRITE_TAC[SUBSET; NOT_FORALL_THM; SEGMENT_1] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:real^1` THEN DISCH_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `x:real^1`)) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[DROP_NEG; LIFT_DROP] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_NEG; LIFT_DROP] THEN REWRITE_TAC[NOT_IMP] THEN REPEAT STRIP_TAC THEN TRY(FIRST_X_ASSUM MATCH_MP_TAC) THEN TRY ASM_REAL_ARITH_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET)) THEN ASM SET_TAC[]; GEN_REWRITE_TAC I [CONJ_ASSOC]] THEN CONJ_TAC THENL [REWRITE_TAC[bounded; NOT_EXISTS_THM] THEN CONJ_TAC THEN X_GEN_TAC `c:real` THENL [DISCH_THEN(MP_TAC o SPEC `--lift(B + abs c + &1)`); DISCH_THEN(MP_TAC o SPEC `lift(B + abs c + &1)`)] THEN REWRITE_TAC[NOT_IMP; NORM_NEG; NORM_LIFT] THEN (CONJ_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC]) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[DROP_NEG; LIFT_DROP] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN X_GEN_TAC `c:real^1->bool` THEN STRIP_TAC THEN MP_TAC(ISPEC `(:real^1) DIFF s` COMPONENTS_EQ) THEN ASM_SIMP_TAC[] THEN DISCH_THEN(K ALL_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [bounded]) THEN REWRITE_TAC[NOT_EXISTS_THM; NOT_FORALL_THM; NOT_IMP; REAL_NOT_LE] THEN DISCH_THEN(MP_TAC o SPEC `B:real`) THEN REWRITE_TAC[NORM_1; REAL_ARITH `B < abs x <=> x < -- B \/ B < x`] THEN ASM SET_TAC[]);; let HAS_SIZE_UNBOUNDED_COMPONENTS_COMPLEMENT = prove (`!s. bounded s ==> {c | c IN components((:real^N) DIFF s) /\ ~bounded c} HAS_SIZE (if s = {} \/ 2 <= dimindex(:N) then 1 else 2)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[] THENL [ASM_REWRITE_TAC[DIFF_EMPTY; COMPONENTS_UNIV; IN_SING] THEN REWRITE_TAC[MESON[] `c = u /\ ~bounded c <=> c = u /\ ~bounded u`] THEN REWRITE_TAC[NOT_BOUNDED_UNIV; SING_GSPEC] THEN CONV_TAC HAS_SIZE_CONV THEN MESON_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `2 <= dimindex(:N)` THEN ASM_REWRITE_TAC[] THENL [CONV_TAC HAS_SIZE_CONV THEN MP_TAC(ISPEC `(:real^N) DIFF s` COBOUNDED_UNBOUNDED_COMPONENTS) THEN ASM_REWRITE_TAC[COMPL_COMPL] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[SET_RULE `{x | P x} = {a} <=> P a /\ !b. P b ==> b = a`] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC COBOUNDED_UNIQUE_UNBOUNDED_COMPONENTS THEN EXISTS_TAC `(:real^N) DIFF s` THEN ASM_REWRITE_TAC[COMPL_COMPL]; FIRST_ASSUM(MP_TAC o MATCH_MP (ARITH_RULE `~(2 <= n) ==> 1 <= n ==> n = 1`)) THEN REWRITE_TAC[DIMINDEX_GE_1] THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM DIMINDEX_1] THEN DISCH_THEN(MATCH_MP_TAC o C GEOM_EQUAL_DIMENSION_RULE HAS_SIZE_UNBOUNDED_COMPONENTS_COMPLEMENT_1) THEN ASM_REWRITE_TAC[]]);; (* ------------------------------------------------------------------------- *) (* The "inside" and "outside" of a set, i.e. the points respectively in a *) (* bounded or unbounded connected component of the set's complement. *) (* ------------------------------------------------------------------------- *) let inside = new_definition `inside s = {x | ~(x IN s) /\ bounded(connected_component ((:real^N) DIFF s) x)}`;; let outside = new_definition `outside s = {x | ~(x IN s) /\ ~bounded(connected_component ((:real^N) DIFF s) x)}`;; let INSIDE_TRANSLATION = prove (`!a s. inside(IMAGE (\x. a + x) s) = IMAGE (\x. a + x) (inside s)`, REWRITE_TAC[inside] THEN GEOM_TRANSLATE_TAC[]);; let OUTSIDE_TRANSLATION = prove (`!a s. outside(IMAGE (\x. a + x) s) = IMAGE (\x. a + x) (outside s)`, REWRITE_TAC[outside] THEN GEOM_TRANSLATE_TAC[]);; add_translation_invariants [INSIDE_TRANSLATION; OUTSIDE_TRANSLATION];; let INSIDE_LINEAR_IMAGE = prove (`!f s. linear f /\ (!x y. f x = f y ==> x = y) /\ (!y. ?x. f x = y) ==> inside(IMAGE f s) = IMAGE f (inside s)`, REWRITE_TAC[inside] THEN GEOM_TRANSFORM_TAC[]);; let OUTSIDE_LINEAR_IMAGE = prove (`!f s. linear f /\ (!x y. f x = f y ==> x = y) /\ (!y. ?x. f x = y) ==> outside(IMAGE f s) = IMAGE f (outside s)`, REWRITE_TAC[outside] THEN GEOM_TRANSFORM_TAC[]);; add_linear_invariants [INSIDE_LINEAR_IMAGE; OUTSIDE_LINEAR_IMAGE];; let OUTSIDE = prove (`!s. outside s = {x | ~bounded(connected_component((:real^N) DIFF s) x)}`, GEN_TAC THEN REWRITE_TAC[outside; EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `(x:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[BOUNDED_EMPTY; CONNECTED_COMPONENT_EQ_EMPTY; IN_DIFF]);; let INSIDE_NO_OVERLAP = prove (`!s. inside s INTER s = {}`, REWRITE_TAC[inside] THEN SET_TAC[]);; let OUTSIDE_NO_OVERLAP = prove (`!s. outside s INTER s = {}`, REWRITE_TAC[outside] THEN SET_TAC[]);; let INSIDE_INTER_OUTSIDE = prove (`!s. inside s INTER outside s = {}`, REWRITE_TAC[inside; outside] THEN SET_TAC[]);; let INSIDE_UNION_OUTSIDE = prove (`!s. inside s UNION outside s = (:real^N) DIFF s`, REWRITE_TAC[inside; outside] THEN SET_TAC[]);; let INSIDE_EQ_OUTSIDE = prove (`!s. inside s = outside s <=> s = (:real^N)`, REWRITE_TAC[inside; outside] THEN SET_TAC[]);; let INSIDE_OUTSIDE = prove (`!s. inside s = (:real^N) DIFF (s UNION outside s)`, GEN_TAC THEN MAP_EVERY (MP_TAC o ISPEC `s:real^N->bool`) [INSIDE_INTER_OUTSIDE; INSIDE_UNION_OUTSIDE] THEN SET_TAC[]);; let OUTSIDE_INSIDE = prove (`!s. outside s = (:real^N) DIFF (s UNION inside s)`, GEN_TAC THEN MAP_EVERY (MP_TAC o ISPEC `s:real^N->bool`) [INSIDE_INTER_OUTSIDE; INSIDE_UNION_OUTSIDE] THEN SET_TAC[]);; let INSIDE_EMPTY_EQ_NO_BOUNDED_COMPONENT_COMPLEMENT = prove (`!s. inside s = {} <=> !c. c IN components((:real^N) DIFF s) ==> ~bounded c`, REWRITE_TAC[components; FORALL_IN_GSPEC; inside] THEN SET_TAC[]);; let OUTSIDE_EMPTY_EQ_NO_BOUNDED_COMPONENT_COMPLEMENT = prove (`!s. outside s = {} <=> !c. c IN components((:real^N) DIFF s) ==> bounded c`, REWRITE_TAC[components; FORALL_IN_GSPEC; outside] THEN SET_TAC[]);; let INSIDE_SELF_OUTSIDE_EVERSION = prove (`!s t:real^N->bool. s UNION inside s SUBSET inside t <=> t UNION outside t SUBSET outside s`, REPEAT GEN_TAC THEN GEN_REWRITE_TAC RAND_CONV[SET_RULE `s SUBSET t <=> UNIV DIFF t SUBSET UNIV DIFF s`] THEN REWRITE_TAC[GSYM INSIDE_OUTSIDE] THEN REWRITE_TAC[OUTSIDE_INSIDE] THEN ASM SET_TAC[]);; let UNION_WITH_INSIDE = prove (`!s. s UNION inside s = (:real^N) DIFF outside s`, REWRITE_TAC[OUTSIDE_INSIDE] THEN SET_TAC[]);; let UNION_WITH_OUTSIDE = prove (`!s. s UNION outside s = (:real^N) DIFF inside s`, REWRITE_TAC[INSIDE_OUTSIDE] THEN SET_TAC[]);; let OUTSIDE_MONO = prove (`!s t. s SUBSET t ==> outside t SUBSET outside s`, REPEAT GEN_TAC THEN REWRITE_TAC[OUTSIDE; SUBSET; IN_ELIM_THM] THEN DISCH_TAC THEN GEN_TAC THEN REWRITE_TAC[CONTRAPOS_THM] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] BOUNDED_SUBSET) THEN MATCH_MP_TAC CONNECTED_COMPONENT_MONO THEN ASM SET_TAC[]);; let INSIDE_MONO = prove (`!s t. s SUBSET t ==> inside s DIFF t SUBSET inside t`, REPEAT STRIP_TAC THEN SIMP_TAC[SUBSET; IN_DIFF; inside; IN_ELIM_THM] THEN GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) ASSUME_TAC) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] BOUNDED_SUBSET) THEN MATCH_MP_TAC CONNECTED_COMPONENT_MONO THEN ASM SET_TAC[]);; let INSIDE_MONO_ALT = prove (`!s t:real^N->bool. s SUBSET t ==> inside s SUBSET t UNION inside t`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP INSIDE_MONO) THEN SET_TAC[]);; let COBOUNDED_OUTSIDE = prove (`!s:real^N->bool. bounded s ==> bounded((:real^N) DIFF outside s)`, GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[outside] THEN REWRITE_TAC[SET_RULE `UNIV DIFF {x | ~(x IN s) /\ ~P x} = s UNION {x | P x}`] THEN ASM_REWRITE_TAC[BOUNDED_UNION] THEN FIRST_ASSUM(MP_TAC o SPEC `vec 0:real^N` o MATCH_MP BOUNDED_SUBSET_BALL) THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `ball(vec 0:real^N,B)` THEN REWRITE_TAC[BOUNDED_BALL; SUBSET; IN_ELIM_THM; IN_BALL_0] THEN X_GEN_TAC `x:real^N` THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[REAL_NOT_LT] THEN ASM_CASES_TAC `x:real^N = vec 0` THENL [ASM_REWRITE_TAC[NORM_0] THEN ASM_REAL_ARITH_TAC; DISCH_TAC] THEN REWRITE_TAC[BOUNDED_POS] THEN DISCH_THEN(X_CHOOSE_THEN `C:real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPEC `(B + C) / norm(x) % x:real^N`) THEN REWRITE_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM] THEN ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_EQ_0; NOT_IMP] THEN CONJ_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN REWRITE_TAC[IN] THEN REWRITE_TAC[connected_component] THEN EXISTS_TAC `segment[x:real^N,(B + C) / norm(x) % x]` THEN REWRITE_TAC[ENDS_IN_SEGMENT; CONNECTED_SEGMENT] THEN MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `(:real^N) DIFF ball(vec 0,B)` THEN ASM_REWRITE_TAC[SET_RULE `UNIV DIFF s SUBSET UNIV DIFF t <=> t SUBSET s`] THEN REWRITE_TAC[SUBSET; IN_DIFF; IN_UNIV; IN_BALL_0] THEN REWRITE_TAC[segment; FORALL_IN_GSPEC] THEN X_GEN_TAC `u:real` THEN STRIP_TAC THEN REWRITE_TAC[REAL_NOT_LT] THEN REWRITE_TAC[GSYM VECTOR_ADD_RDISTRIB; NORM_MUL; VECTOR_MUL_ASSOC] THEN GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM REAL_ABS_NORM] THEN REWRITE_TAC[GSYM REAL_ABS_MUL] THEN MATCH_MP_TAC(REAL_ARITH `&0 < B /\ B <= x ==> B <= abs x`) THEN ASM_SIMP_TAC[REAL_ADD_RDISTRIB; REAL_DIV_RMUL; NORM_EQ_0; GSYM REAL_MUL_ASSOC] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `(&1 - u) * B + u * (B + C)` THEN ASM_SIMP_TAC[REAL_LE_RADD; REAL_LE_LMUL; REAL_SUB_LE] THEN SIMP_TAC[REAL_ARITH `B <= (&1 - u) * B + u * (B + C) <=> &0 <= u * C`] THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_REAL_ARITH_TAC);; let UNBOUNDED_OUTSIDE = prove (`!s:real^N->bool. bounded s ==> ~bounded(outside s)`, MESON_TAC[COBOUNDED_IMP_UNBOUNDED; COBOUNDED_OUTSIDE]);; let BOUNDED_INSIDE = prove (`!s:real^N->bool. bounded s ==> bounded(inside s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `(:real^N) DIFF outside s` THEN ASM_SIMP_TAC[COBOUNDED_OUTSIDE] THEN MP_TAC(ISPEC `s:real^N->bool` INSIDE_INTER_OUTSIDE) THEN SET_TAC[]);; let CONNECTED_OUTSIDE = prove (`!s:real^N->bool. 2 <= dimindex(:N) /\ bounded s ==> connected(outside s)`, REPEAT STRIP_TAC THEN REWRITE_TAC[CONNECTED_IFF_CONNECTED_COMPONENT] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN REWRITE_TAC[outside; IN_ELIM_THM] THEN STRIP_TAC THEN MATCH_MP_TAC CONNECTED_COMPONENT_OF_SUBSET THEN EXISTS_TAC `connected_component ((:real^N) DIFF s) x` THEN REWRITE_TAC[SUBSET; IN_UNIV; IN_DIFF; IN_ELIM_THM] THEN CONJ_TAC THENL [X_GEN_TAC `z:real^N` THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[SUBSET] CONNECTED_COMPONENT_SUBSET)) THEN REWRITE_TAC[IN_DIFF] THEN ASM_MESON_TAC[CONNECTED_COMPONENT_EQ]; REWRITE_TAC[CONNECTED_COMPONENT_IDEMP] THEN SUBGOAL_THEN `connected_component ((:real^N) DIFF s) x = connected_component ((:real^N) DIFF s) y` SUBST1_TAC THENL [MATCH_MP_TAC COBOUNDED_UNIQUE_UNBOUNDED_COMPONENT THEN ASM_REWRITE_TAC[COMPL_COMPL]; ASM_REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ; IN_DIFF; IN_UNIV]]]);; let OUTSIDE_CONNECTED_COMPONENT_LT = prove (`!s. outside s = {x | !B. ?y. B < norm(y) /\ connected_component((:real^N) DIFF s) x y}`, REWRITE_TAC[OUTSIDE; bounded; EXTENSION; IN_ELIM_THM] THEN REWRITE_TAC[IN] THEN ASM_MESON_TAC[REAL_NOT_LE]);; let OUTSIDE_CONNECTED_COMPONENT_LE = prove (`!s. outside s = {x | !B. ?y. B <= norm(y) /\ connected_component((:real^N) DIFF s) x y}`, GEN_TAC THEN REWRITE_TAC[OUTSIDE_CONNECTED_COMPONENT_LT] THEN GEN_REWRITE_TAC I [EXTENSION] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_ELIM_THM] THEN MESON_TAC[REAL_LT_IMP_LE; REAL_ARITH `B + &1 <= x ==> B < x`]);; let NOT_OUTSIDE_CONNECTED_COMPONENT_LT = prove (`!s. 2 <= dimindex(:N) /\ bounded s ==> (:real^N) DIFF (outside s) = {x | !B. ?y. B < norm(y) /\ ~(connected_component((:real^N) DIFF s) x y)}`, REPEAT STRIP_TAC THEN REWRITE_TAC[OUTSIDE] THEN REWRITE_TAC[EXTENSION; IN_DIFF; IN_UNIV; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[bounded] THEN EQ_TAC THENL [DISCH_THEN(X_CHOOSE_TAC `C:real`) THEN X_GEN_TAC `B:real` THEN EXISTS_TAC `(abs B + abs C + &1) % basis 1:real^N` THEN RULE_ASSUM_TAC(ONCE_REWRITE_RULE[GSYM CONTRAPOS_THM]) THEN RULE_ASSUM_TAC(REWRITE_RULE[IN]) THEN CONJ_TAC THENL [ALL_TAC; FIRST_X_ASSUM MATCH_MP_TAC] THEN SIMP_TAC[NORM_MUL; NORM_BASIS; DIMINDEX_GE_1; LE_REFL] THEN REAL_ARITH_TAC; DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [BOUNDED_POS]) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `B:real` THEN STRIP_TAC THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN] THEN DISCH_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_NOT_LT] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `B:real`) THEN DISCH_THEN (X_CHOOSE_THEN `z:real^N` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN REWRITE_TAC[] THEN MATCH_MP_TAC CONNECTED_COMPONENT_TRANS THEN EXISTS_TAC `y:real^N` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONNECTED_COMPONENT_OF_SUBSET THEN EXISTS_TAC `(:real^N) DIFF cball(vec 0,B)` THEN ASM_REWRITE_TAC[SUBSET; IN_DIFF; IN_CBALL_0; IN_UNIV; CONTRAPOS_THM] THEN REWRITE_TAC[connected_component] THEN EXISTS_TAC `(:real^N) DIFF cball(vec 0,B)` THEN ASM_SIMP_TAC[SUBSET_REFL; IN_DIFF; IN_UNIV; IN_CBALL_0; REAL_NOT_LE] THEN MATCH_MP_TAC CONNECTED_COMPLEMENT_BOUNDED_CONVEX THEN ASM_SIMP_TAC[BOUNDED_CBALL; CONVEX_CBALL]]);; let NOT_OUTSIDE_CONNECTED_COMPONENT_LE = prove (`!s. 2 <= dimindex(:N) /\ bounded s ==> (:real^N) DIFF (outside s) = {x | !B. ?y. B <= norm(y) /\ ~(connected_component((:real^N) DIFF s) x y)}`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[NOT_OUTSIDE_CONNECTED_COMPONENT_LT] THEN GEN_REWRITE_TAC I [EXTENSION] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_ELIM_THM] THEN MESON_TAC[REAL_LT_IMP_LE; REAL_ARITH `B + &1 <= x ==> B < x`]);; let INSIDE_CONNECTED_COMPONENT_LT = prove (`!s. 2 <= dimindex(:N) /\ bounded s ==> inside s = {x:real^N | ~(x IN s) /\ !B. ?y. B < norm(y) /\ ~(connected_component((:real^N) DIFF s) x y)}`, REPEAT STRIP_TAC THEN REWRITE_TAC[INSIDE_OUTSIDE] THEN REWRITE_TAC[SET_RULE `UNIV DIFF (s UNION t) = (UNIV DIFF t) DIFF s`] THEN ASM_SIMP_TAC[NOT_OUTSIDE_CONNECTED_COMPONENT_LT] THEN SET_TAC[]);; let INSIDE_CONNECTED_COMPONENT_LE = prove (`!s. 2 <= dimindex(:N) /\ bounded s ==> inside s = {x:real^N | ~(x IN s) /\ !B. ?y. B <= norm(y) /\ ~(connected_component((:real^N) DIFF s) x y)}`, REPEAT STRIP_TAC THEN REWRITE_TAC[INSIDE_OUTSIDE] THEN REWRITE_TAC[SET_RULE `UNIV DIFF (s UNION t) = (UNIV DIFF t) DIFF s`] THEN ASM_SIMP_TAC[NOT_OUTSIDE_CONNECTED_COMPONENT_LE] THEN SET_TAC[]);; let OUTSIDE_UNION_OUTSIDE_UNION = prove (`!c c1 c2:real^N->bool. c INTER outside(c1 UNION c2) = {} ==> outside(c1 UNION c2) SUBSET outside(c1 UNION c)`, REPEAT STRIP_TAC THEN REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^N` THEN DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN REWRITE_TAC[OUTSIDE_CONNECTED_COMPONENT_LT; IN_ELIM_THM] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `B:real` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `y:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[connected_component] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `t SUBSET outside(c1 UNION c2:real^N->bool)` MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `connected_component((:real^N) DIFF (c1 UNION c2)) x` THEN CONJ_TAC THENL [ASM_MESON_TAC[CONNECTED_COMPONENT_MAXIMAL]; ALL_TAC] THEN UNDISCH_TAC `(x:real^N) IN outside(c1 UNION c2)` THEN REWRITE_TAC[OUTSIDE; IN_ELIM_THM; SUBSET] THEN MESON_TAC[CONNECTED_COMPONENT_EQ]);; let INSIDE_SUBSET = prove (`!s t u. connected u /\ ~bounded u /\ t UNION u = (:real^N) DIFF s ==> inside s SUBSET t`, REPEAT STRIP_TAC THEN REWRITE_TAC[SUBSET; inside; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THEN UNDISCH_TAC `~bounded(u:real^N->bool)` THEN REWRITE_TAC[] THEN MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `connected_component((:real^N) DIFF s) x` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);; let INSIDE_UNIQUE = prove (`!s t u. connected t /\ bounded t /\ connected u /\ ~(bounded u) /\ ~connected((:real^N) DIFF s) /\ t UNION u = (:real^N) DIFF s ==> inside s = t`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [ASM_MESON_TAC[INSIDE_SUBSET]; ALL_TAC] THEN REWRITE_TAC[SUBSET; inside; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(SET_RULE `!s u. c INTER s = {} /\ c INTER u = {} /\ t UNION u = UNIV DIFF s ==> c SUBSET t`) THEN MAP_EVERY EXISTS_TAC [`s:real^N->bool`; `u:real^N->bool`] THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [REWRITE_TAC[SET_RULE `c INTER s = {} <=> c SUBSET (UNIV DIFF s)`] THEN REWRITE_TAC[CONNECTED_COMPONENT_SUBSET]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s /\ x IN t ==> F) ==> s INTER t = {}`) THEN X_GEN_TAC `y:real^N` THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [IN] THEN STRIP_TAC THEN UNDISCH_TAC `~connected((:real^N) DIFF s)` THEN REWRITE_TAC[CONNECTED_IFF_CONNECTED_COMPONENT] THEN MAP_EVERY X_GEN_TAC [`u:real^N`; `v:real^N`] THEN SUBGOAL_THEN `(!w. w IN t ==> connected_component ((:real^N) DIFF s) x w) /\ (!w. w IN u ==> connected_component ((:real^N) DIFF s) y w)` STRIP_ASSUME_TAC THENL [CONJ_TAC THEN GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[connected_component] THENL [EXISTS_TAC `t:real^N->bool`; EXISTS_TAC `u:real^N->bool`] THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; FIRST_ASSUM(SUBST1_TAC o SYM) THEN REWRITE_TAC[IN_UNION] THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[CONNECTED_COMPONENT_TRANS; CONNECTED_COMPONENT_SYM]]);; let INSIDE_OUTSIDE_UNIQUE = prove (`!s t u. connected t /\ bounded t /\ connected u /\ ~(bounded u) /\ ~connected((:real^N) DIFF s) /\ t UNION u = (:real^N) DIFF s ==> inside s = t /\ outside s = u`, REPEAT GEN_TAC THEN STRIP_TAC THEN REWRITE_TAC[OUTSIDE_INSIDE] THEN MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL [ASM_MESON_TAC[INSIDE_UNIQUE]; MP_TAC(ISPEC `(:real^N) DIFF s` INSIDE_NO_OVERLAP) THEN SUBGOAL_THEN `t INTER u:real^N->bool = {}` MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN UNDISCH_TAC `~connected ((:real^N) DIFF s)` THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN DISCH_TAC THEN REWRITE_TAC[] THEN MATCH_MP_TAC CONNECTED_UNION THEN ASM_REWRITE_TAC[]]);; let INTERIOR_INSIDE_FRONTIER = prove (`!s:real^N->bool. bounded s ==> interior s SUBSET inside(frontier s)`, REPEAT STRIP_TAC THEN REWRITE_TAC[inside; SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL [ASM_REWRITE_TAC[frontier; IN_DIFF]; DISCH_TAC] THEN MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[SUBSET] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THEN SUBGOAL_THEN `~(connected_component((:real^N) DIFF frontier s) x INTER frontier s = {})` MP_TAC THENL [MATCH_MP_TAC CONNECTED_INTER_FRONTIER THEN REWRITE_TAC[CONNECTED_CONNECTED_COMPONENT; GSYM MEMBER_NOT_EMPTY] THEN CONJ_TAC THENL [REWRITE_TAC[IN_INTER]; ASM SET_TAC[]] THEN EXISTS_TAC `x:real^N` THEN CONJ_TAC THENL [REWRITE_TAC[IN; CONNECTED_COMPONENT_REFL_EQ] THEN GEN_REWRITE_TAC I [GSYM IN] THEN ASM SET_TAC[]; ASM_MESON_TAC[INTERIOR_SUBSET; SUBSET]]; REWRITE_TAC[SET_RULE `s INTER t = {} <=> s SUBSET (UNIV DIFF t)`] THEN REWRITE_TAC[CONNECTED_COMPONENT_SUBSET]]);; let INSIDE_EMPTY = prove (`inside {} = {}`, REWRITE_TAC[inside; NOT_IN_EMPTY; DIFF_EMPTY; CONNECTED_COMPONENT_UNIV] THEN REWRITE_TAC[NOT_BOUNDED_UNIV; EMPTY_GSPEC]);; let OUTSIDE_EMPTY = prove (`outside {} = (:real^N)`, REWRITE_TAC[OUTSIDE_INSIDE; INSIDE_EMPTY] THEN SET_TAC[]);; let INSIDE_SAME_COMPONENT = prove (`!s x y. connected_component((:real^N) DIFF s) x y /\ x IN inside s ==> y IN inside s`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 (ASSUME_TAC o GEN_REWRITE_RULE I [GSYM IN]) MP_TAC) THEN REWRITE_TAC[inside; IN_ELIM_THM] THEN FIRST_ASSUM(SUBST1_TAC o MATCH_MP CONNECTED_COMPONENT_EQ) THEN RULE_ASSUM_TAC(REWRITE_RULE[IN]) THEN FIRST_ASSUM(MP_TAC o MATCH_MP CONNECTED_COMPONENT_IN) THEN SIMP_TAC[IN_DIFF]);; let OUTSIDE_SAME_COMPONENT = prove (`!s x y. connected_component((:real^N) DIFF s) x y /\ x IN outside s ==> y IN outside s`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 (ASSUME_TAC o GEN_REWRITE_RULE I [GSYM IN]) MP_TAC) THEN REWRITE_TAC[outside; IN_ELIM_THM] THEN FIRST_ASSUM(SUBST1_TAC o MATCH_MP CONNECTED_COMPONENT_EQ) THEN RULE_ASSUM_TAC(REWRITE_RULE[IN]) THEN FIRST_ASSUM(MP_TAC o MATCH_MP CONNECTED_COMPONENT_IN) THEN SIMP_TAC[IN_DIFF]);; let CONNECTED_COMPONENT_INSIDE = prove (`!s a. connected_component (inside s) a = if a IN inside s then connected_component ((:real^N) DIFF s) a else {}`, REPEAT GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_EQ_EMPTY] THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_COMPONENT_MONO THEN REWRITE_TAC[SET_RULE `s SUBSET UNIV DIFF t <=> s INTER t = {}`] THEN REWRITE_TAC[INSIDE_NO_OVERLAP]; GEN_REWRITE_TAC LAND_CONV [GSYM CONNECTED_COMPONENT_IDEMP] THEN MATCH_MP_TAC CONNECTED_COMPONENT_MONO THEN REWRITE_TAC[SUBSET] THEN ASM_MESON_TAC[IN; INSIDE_SAME_COMPONENT]]);; let CONNECTED_COMPONENT_OUTSIDE = prove (`!s a. connected_component (outside s) a = if a IN outside s then connected_component ((:real^N) DIFF s) a else {}`, REPEAT GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_EQ_EMPTY] THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_COMPONENT_MONO THEN REWRITE_TAC[SET_RULE `s SUBSET UNIV DIFF t <=> s INTER t = {}`] THEN REWRITE_TAC[OUTSIDE_NO_OVERLAP]; GEN_REWRITE_TAC LAND_CONV [GSYM CONNECTED_COMPONENT_IDEMP] THEN MATCH_MP_TAC CONNECTED_COMPONENT_MONO THEN REWRITE_TAC[SUBSET] THEN ASM_MESON_TAC[IN; OUTSIDE_SAME_COMPONENT]]);; let BOUNDED_COMPONENTS_INSIDE = prove (`!c:real^N->bool. c IN components(inside s) ==> bounded c`, SIMP_TAC[components; FORALL_IN_GSPEC; CONNECTED_COMPONENT_INSIDE] THEN REWRITE_TAC[inside] THEN SET_TAC[]);; let UNBOUNDED_COMPONENTS_OUTSIDE = prove (`!s c:real^N->bool. c IN components(outside s) ==> ~bounded c`, SIMP_TAC[components; FORALL_IN_GSPEC; CONNECTED_COMPONENT_OUTSIDE] THEN REWRITE_TAC[outside] THEN SET_TAC[]);; let INSIDE_WITH_INSIDE = prove (`!s:real^N->bool. inside(s UNION inside s) = {}`, REPEAT STRIP_TAC THEN REWRITE_TAC[INSIDE_EMPTY_EQ_NO_BOUNDED_COMPONENT_COMPLEMENT] THEN REWRITE_TAC[GSYM OUTSIDE_INSIDE; UNBOUNDED_COMPONENTS_OUTSIDE]);; let OUTSIDE_WITH_OUTSIDE = prove (`!s:real^N->bool. outside(s UNION outside s) = {}`, REPEAT STRIP_TAC THEN REWRITE_TAC[OUTSIDE_EMPTY_EQ_NO_BOUNDED_COMPONENT_COMPLEMENT] THEN REWRITE_TAC[GSYM INSIDE_OUTSIDE] THEN REWRITE_TAC[BOUNDED_COMPONENTS_INSIDE]);; let OUTSIDE_CONVEX = prove (`!s. convex s ==> outside s = (:real^N) DIFF s`, REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; REWRITE_RULE[SET_RULE `t INTER s = {} <=> t SUBSET UNIV DIFF s`] OUTSIDE_NO_OVERLAP] THEN REWRITE_TAC[SUBSET; IN_UNIV; IN_DIFF] THEN MATCH_MP_TAC SET_PROVE_CASES THEN REWRITE_TAC[OUTSIDE_EMPTY; IN_UNIV] THEN X_GEN_TAC `a:real^N` THEN GEOM_ORIGIN_TAC `a:real^N` THEN X_GEN_TAC `t:real^N->bool` THEN DISCH_THEN(K ALL_TAC) THEN MP_TAC(SET_RULE `(vec 0:real^N) IN (vec 0 INSERT t)`) THEN SPEC_TAC(`(vec 0:real^N) INSERT t`,`s:real^N->bool`) THEN GEN_TAC THEN DISCH_TAC THEN DISCH_TAC THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN ASM_REWRITE_TAC[outside; IN_ELIM_THM] THEN SUBGOAL_THEN `~(x:real^N = vec 0)` ASSUME_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN REWRITE_TAC[BOUNDED_POS; NOT_EXISTS_THM] THEN X_GEN_TAC `B:real` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(MP_TAC o SPEC `(max (&2) ((B + &1) / norm(x))) % x:real^N`) THEN REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [REWRITE_TAC[IN] THEN REWRITE_TAC[connected_component] THEN EXISTS_TAC `segment[x:real^N,(max (&2) ((B + &1) / norm(x))) % x]` THEN REWRITE_TAC[ENDS_IN_SEGMENT; CONNECTED_SEGMENT] THEN REWRITE_TAC[segment; SUBSET; FORALL_IN_GSPEC] THEN X_GEN_TAC `u:real` THEN ASM_CASES_TAC `u = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_MUL_LID; REAL_SUB_RZERO; VECTOR_ADD_RID; IN_DIFF; IN_UNIV] THEN DISCH_TAC THEN REWRITE_TAC[VECTOR_ARITH `a % x + b % c % x:real^N = (a + b * c) % x`] THEN ABBREV_TAC `c = &1 - u + u * max (&2) ((B + &1) / norm(x:real^N))` THEN DISCH_TAC THEN SUBGOAL_THEN `&1 < c` ASSUME_TAC THENL [EXPAND_TAC "c" THEN REWRITE_TAC[REAL_ARITH `&1 < &1 - u + u * x <=> &0 < u * (x - &1)`] THEN MATCH_MP_TAC REAL_LT_MUL THEN ASM_REAL_ARITH_TAC; UNDISCH_TAC `~((x:real^N) IN s)` THEN REWRITE_TAC[] THEN SUBGOAL_THEN `x:real^N = (&1 - inv c) % vec 0 + inv c % c % x` SUBST1_TAC THENL [REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID; VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[REAL_MUL_LINV; REAL_ARITH `&1 < x ==> ~(x = &0)`] THEN REWRITE_TAC[VECTOR_MUL_LID]; MATCH_MP_TAC IN_CONVEX_SET THEN ASM_SIMP_TAC[REAL_LE_INV_EQ; REAL_INV_LE_1; REAL_LT_IMP_LE] THEN ASM_REAL_ARITH_TAC]]; ASM_SIMP_TAC[NORM_MUL; REAL_NOT_LE; GSYM REAL_LT_LDIV_EQ; NORM_POS_LT] THEN MATCH_MP_TAC(REAL_ARITH `&0 < b /\ b < c ==> b < abs(max (&2) c)`) THEN ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT; REAL_LT_DIV2_EQ] THEN REAL_ARITH_TAC]);; let INSIDE_CONVEX = prove (`!s. convex s ==> inside s = {}`, SIMP_TAC[INSIDE_OUTSIDE; OUTSIDE_CONVEX] THEN SET_TAC[]);; let OUTSIDE_SUBSET_CONVEX = prove (`!s t. convex t /\ s SUBSET t ==> (:real^N) DIFF t SUBSET outside s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `outside(t:real^N->bool)` THEN ASM_SIMP_TAC[OUTSIDE_MONO] THEN ASM_SIMP_TAC[OUTSIDE_CONVEX; SUBSET_REFL]);; let INSIDE_SUBSET_CONVEX = prove (`!s c:real^N->bool. convex c /\ s SUBSET c ==> inside s SUBSET c`, REPEAT STRIP_TAC THEN REWRITE_TAC[INSIDE_OUTSIDE] THEN MP_TAC(ISPECL [`s:real^N->bool`; `c:real^N->bool`] OUTSIDE_SUBSET_CONVEX) THEN ASM SET_TAC[]);; let INSIDE_SUBSET_CONVEX_HULL = prove (`!s:real^N->bool. inside s SUBSET convex hull s`, SIMP_TAC[INSIDE_SUBSET_CONVEX; CONVEX_CONVEX_HULL; HULL_SUBSET]);; let UNBOUNDED_DISJOINT_IN_OUTSIDE = prove (`!s t x:real^N. connected t /\ ~bounded t /\ x IN t /\ DISJOINT s t ==> x IN outside s`, REPEAT STRIP_TAC THEN REWRITE_TAC[outside; IN_ELIM_THM] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN UNDISCH_TAC `~bounded(t:real^N->bool)` THEN REWRITE_TAC[CONTRAPOS_THM] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] BOUNDED_SUBSET) THEN MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL THEN ASM SET_TAC[]);; let INSIDE_SUBSET_INTERIOR_CONVEX = prove (`!s c:real^N->bool. convex c /\ s SUBSET c ==> inside s SUBSET interior c`, REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM SET_DIFF_FRONTIER] THEN REWRITE_TAC[SET_RULE `s SUBSET t DIFF u <=> s SUBSET t /\ DISJOINT s u`] THEN ASM_SIMP_TAC[INSIDE_SUBSET_CONVEX] THEN REWRITE_TAC[SET_RULE `DISJOINT s t <=> !x. x IN s ==> ~(x IN t)`] THEN X_GEN_TAC `x:real^N` THEN REPEAT DISCH_TAC THEN MP_TAC(ISPECL [`c:real^N->bool`; `x:real^N`] SUPPORTING_HYPERPLANE_FRONTIER) THEN ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN X_GEN_TAC `a:real^N` THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `x INSERT {y:real^N | a dot y < a dot x}`; `x:real^N`] UNBOUNDED_DISJOINT_IN_OUTSIDE) THEN REWRITE_TAC[NOT_IMP] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_INTERMEDIATE_CLOSURE THEN EXISTS_TAC `{y:real^N | a dot y < a dot x}` THEN ASM_SIMP_TAC[CLOSURE_HALFSPACE_LT; CONVEX_CONNECTED; CONVEX_HALFSPACE_LT; INSERT_SUBSET; IN_ELIM_THM; SUBSET; IN_INSERT] THEN REAL_ARITH_TAC; ASM_REWRITE_TAC[BOUNDED_INSERT; BOUNDED_HALFSPACE_LT]; REWRITE_TAC[IN_INSERT]; REWRITE_TAC[SET_RULE `DISJOINT s (x INSERT t) <=> ~(x IN s) /\ (!y. y IN s ==> ~(y IN t))`] THEN CONJ_TAC THENL [MP_TAC(ISPEC `s:real^N->bool` INSIDE_NO_OVERLAP) THEN ASM SET_TAC[]; REWRITE_TAC[IN_ELIM_THM; REAL_NOT_LT] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN MATCH_MP_TAC CLOSURE_INC THEN ASM SET_TAC[]]; MP_TAC(ISPEC `s:real^N->bool` INSIDE_INTER_OUTSIDE) THEN ASM SET_TAC[]]);; let INSIDE_SUBSET_INTERIOR_CONVEX_HULL = prove (`!s:real^N->bool. inside s SUBSET interior(convex hull s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC INSIDE_SUBSET_INTERIOR_CONVEX THEN REWRITE_TAC[CONVEX_CONVEX_HULL; HULL_SUBSET]);; let OUTSIDE_FRONTIER_MISSES_CLOSURE = prove (`!s. bounded s ==> outside(frontier s) SUBSET (:real^N) DIFF closure s`, REPEAT STRIP_TAC THEN REWRITE_TAC[OUTSIDE_INSIDE] THEN SIMP_TAC[SET_RULE `(UNIV DIFF s) SUBSET (UNIV DIFF t) <=> t SUBSET s`] THEN REWRITE_TAC[frontier] THEN MATCH_MP_TAC(SET_RULE `i SUBSET ins ==> c SUBSET (c DIFF i) UNION ins`) THEN ASM_SIMP_TAC[GSYM frontier; INTERIOR_INSIDE_FRONTIER]);; let OUTSIDE_FRONTIER_EQ_COMPLEMENT_CLOSURE = prove (`!s. bounded s /\ convex s ==> outside(frontier s) = (:real^N) DIFF closure s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN ASM_SIMP_TAC[OUTSIDE_FRONTIER_MISSES_CLOSURE] THEN MATCH_MP_TAC OUTSIDE_SUBSET_CONVEX THEN ASM_SIMP_TAC[CONVEX_CLOSURE; frontier] THEN SET_TAC[]);; let INSIDE_FRONTIER_EQ_INTERIOR = prove (`!s:real^N->bool. bounded s /\ convex s ==> inside(frontier s) = interior s`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[INSIDE_OUTSIDE; OUTSIDE_FRONTIER_EQ_COMPLEMENT_CLOSURE] THEN REWRITE_TAC[frontier] THEN MAP_EVERY (MP_TAC o ISPEC `s:real^N->bool`) [CLOSURE_SUBSET; INTERIOR_SUBSET] THEN ASM SET_TAC[]);; let INSIDE_SPHERE = prove (`!a:real^N r. inside(sphere(a,r)) = ball(a,r)`, REWRITE_TAC[GSYM FRONTIER_CBALL] THEN SIMP_TAC[INSIDE_FRONTIER_EQ_INTERIOR; BOUNDED_CBALL; CONVEX_CBALL] THEN REWRITE_TAC[INTERIOR_CBALL]);; let OUTSIDE_SPHERE = prove (`!a r. outside(sphere(a,r)) = (:real^N) DIFF cball(a,r)`, REWRITE_TAC[OUTSIDE_INSIDE; INSIDE_SPHERE; SPHERE_UNION_BALL]);; let OPEN_INSIDE = prove (`!s:real^N->bool. closed s ==> open(inside s)`, REPEAT STRIP_TAC THEN REWRITE_TAC[OPEN_CONTAINS_BALL] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN SUBGOAL_THEN `open(connected_component ((:real^N) DIFF s) x)` MP_TAC THENL [MATCH_MP_TAC OPEN_CONNECTED_COMPONENT THEN ASM_REWRITE_TAC[GSYM closed]; REWRITE_TAC[open_def] THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ANTS_TAC THENL [REWRITE_TAC[IN; CONNECTED_COMPONENT_REFL_EQ] THEN GEN_REWRITE_TAC I [GSYM IN] THEN ASM_REWRITE_TAC[IN_DIFF; IN_UNIV] THEN MP_TAC(ISPEC `s:real^N->bool` INSIDE_NO_OVERLAP) THEN ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `e:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[SUBSET; IN_BALL] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN MATCH_MP_TAC INSIDE_SAME_COMPONENT THEN EXISTS_TAC `x:real^N` THEN ASM_REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN]) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[DIST_SYM]]]);; let OPEN_OUTSIDE = prove (`!s:real^N->bool. closed s ==> open(outside s)`, REPEAT STRIP_TAC THEN REWRITE_TAC[OPEN_CONTAINS_BALL] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN SUBGOAL_THEN `open(connected_component ((:real^N) DIFF s) x)` MP_TAC THENL [MATCH_MP_TAC OPEN_CONNECTED_COMPONENT THEN ASM_REWRITE_TAC[GSYM closed]; REWRITE_TAC[open_def] THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ANTS_TAC THENL [REWRITE_TAC[IN; CONNECTED_COMPONENT_REFL_EQ] THEN GEN_REWRITE_TAC I [GSYM IN] THEN ASM_REWRITE_TAC[IN_DIFF; IN_UNIV] THEN MP_TAC(ISPEC `s:real^N->bool` OUTSIDE_NO_OVERLAP) THEN ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `e:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[SUBSET; IN_BALL] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN MATCH_MP_TAC OUTSIDE_SAME_COMPONENT THEN EXISTS_TAC `x:real^N` THEN ASM_REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN]) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[DIST_SYM]]]);; let CLOSURE_INSIDE_SUBSET = prove (`!s:real^N->bool. closed s ==> closure(inside s) SUBSET s UNION inside s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN ASM_SIMP_TAC[closed; GSYM OUTSIDE_INSIDE; OPEN_OUTSIDE] THEN SET_TAC[]);; let FRONTIER_INSIDE_SUBSET = prove (`!s:real^N->bool. closed s ==> frontier(inside s) SUBSET s`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[frontier; OPEN_INSIDE; INTERIOR_OPEN] THEN FIRST_ASSUM(MP_TAC o MATCH_MP CLOSURE_INSIDE_SUBSET) THEN SET_TAC[]);; let FRONTIER_WITH_INSIDE_SUBSET = prove (`!s:real^N->bool. closed s ==> frontier(s UNION inside s) SUBSET s`, REPEAT STRIP_TAC THEN TRANS_TAC SUBSET_TRANS `frontier s UNION frontier(inside s):real^N->bool` THEN REWRITE_TAC[FRONTIER_UNION_SUBSET; UNION_SUBSET] THEN ASM_SIMP_TAC[FRONTIER_INSIDE_SUBSET; FRONTIER_SUBSET_CLOSED]);; let CLOSURE_OUTSIDE_SUBSET = prove (`!s:real^N->bool. closed s ==> closure(outside s) SUBSET s UNION outside s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN ASM_SIMP_TAC[closed; GSYM INSIDE_OUTSIDE; OPEN_INSIDE] THEN SET_TAC[]);; let FRONTIER_OUTSIDE_SUBSET = prove (`!s:real^N->bool. closed s ==> frontier(outside s) SUBSET s`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[frontier; OPEN_OUTSIDE; INTERIOR_OPEN] THEN FIRST_ASSUM(MP_TAC o MATCH_MP CLOSURE_OUTSIDE_SUBSET) THEN SET_TAC[]);; let FRONTIER_WITH_OUTSIDE_SUBSET = prove (`!s:real^N->bool. closed s ==> frontier(s UNION outside s) SUBSET s`, REPEAT STRIP_TAC THEN TRANS_TAC SUBSET_TRANS `frontier s UNION frontier(outside s):real^N->bool` THEN REWRITE_TAC[FRONTIER_UNION_SUBSET; UNION_SUBSET] THEN ASM_SIMP_TAC[FRONTIER_OUTSIDE_SUBSET; FRONTIER_SUBSET_CLOSED]);; let CLOSED_WITH_INSIDE = prove (`!s:real^N->bool. closed s ==> closed(s UNION inside s)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `s UNION inside s:real^N->bool = s UNION closure(inside s)` SUBST1_TAC THENL [FIRST_ASSUM(MP_TAC o MATCH_MP CLOSURE_INSIDE_SUBSET) THEN MP_TAC(ISPEC `inside s:real^N->bool` CLOSURE_SUBSET) THEN SET_TAC[]; ASM_SIMP_TAC[CLOSED_UNION; CLOSED_CLOSURE]]);; let BOUNDED_WITH_INSIDE = prove (`!s:real^N->bool. bounded s ==> bounded(s UNION inside s)`, SIMP_TAC[BOUNDED_UNION; BOUNDED_INSIDE]);; let COMPACT_WITH_INSIDE = prove (`!s:real^N->bool. compact s ==> compact(s UNION inside s)`, SIMP_TAC[COMPACT_EQ_BOUNDED_CLOSED; BOUNDED_WITH_INSIDE; CLOSED_WITH_INSIDE]);; let INSIDE_COMPLEMENT_UNBOUNDED_CONNECTED_EMPTY = prove (`!s. connected((:real^N) DIFF s) /\ ~bounded((:real^N) DIFF s) ==> inside s = {}`, REWRITE_TAC[inside; CONNECTED_CONNECTED_COMPONENT_SET] THEN REWRITE_TAC[SET_RULE `s = {} <=> !x. x IN s ==> F`] THEN SIMP_TAC[IN_ELIM_THM; IN_DIFF; IN_UNIV; TAUT `~(a /\ b) <=> a ==> ~b`]);; let INSIDE_BOUNDED_COMPLEMENT_CONNECTED_EMPTY = prove (`!s. connected((:real^N) DIFF s) /\ bounded s ==> inside s = {}`, MESON_TAC[INSIDE_COMPLEMENT_UNBOUNDED_CONNECTED_EMPTY; COBOUNDED_IMP_UNBOUNDED]);; let INSIDE_INSIDE = prove (`!s t:real^N->bool. s SUBSET inside t ==> inside s DIFF t SUBSET inside t`, REPEAT STRIP_TAC THEN SIMP_TAC[SUBSET; inside; IN_DIFF; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN ASM_CASES_TAC `s INTER connected_component ((:real^N) DIFF t) x = {}` THENL [MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `connected_component ((:real^N) DIFF s) x` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL THEN REWRITE_TAC[CONNECTED_CONNECTED_COMPONENT; IN] THEN REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ] THEN ASM SET_TAC[]; FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE `~(s INTER t = {}) ==> ?x. x IN s /\ x IN t`)) THEN DISCH_THEN(X_CHOOSE_THEN `y:real^N` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(SUBST_ALL_TAC o SYM o MATCH_MP CONNECTED_COMPONENT_EQ) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN DISCH_THEN(MP_TAC o SPEC `y:real^N`) THEN ASM_SIMP_TAC[inside; IN_ELIM_THM]]);; let INSIDE_INSIDE_SUBSET = prove (`!s:real^N->bool. inside(inside s) SUBSET s`, GEN_TAC THEN MP_TAC (ISPECL [`inside s:real^N->bool`; `s:real^N->bool`] INSIDE_INSIDE) THEN REWRITE_TAC[SUBSET_REFL] THEN MP_TAC(ISPEC `inside s:real^N->bool` INSIDE_NO_OVERLAP) THEN SET_TAC[]);; let INSIDE_OUTSIDE_INTERSECT_CONNECTED = prove (`!s t:real^N->bool. connected t /\ ~(inside s INTER t = {}) /\ ~(outside s INTER t = {}) ==> ~(s INTER t = {})`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN REWRITE_TAC[inside; outside; IN_ELIM_THM] THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `x:real^N` STRIP_ASSUME_TAC) (X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC)) THEN SUBGOAL_THEN `connected_component ((:real^N) DIFF s) y = connected_component ((:real^N) DIFF s) x` (fun th -> ASM_MESON_TAC[th]) THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_EQ_EQ; IN_DIFF; IN_UNIV] THEN REWRITE_TAC[connected_component] THEN EXISTS_TAC `t:real^N->bool` THEN ASM SET_TAC[]);; let OUTSIDE_BOUNDED_NONEMPTY = prove (`!s:real^N->bool. bounded s ==> ~(outside s = {})`, GEN_TAC THEN DISCH_THEN(MP_TAC o SPEC `vec 0:real^N` o MATCH_MP BOUNDED_SUBSET_BALL) THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] OUTSIDE_SUBSET_CONVEX)) THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN SIMP_TAC[CONVEX_BALL; SUBSET_EMPTY] THEN REWRITE_TAC[SET_RULE `s DIFF t = {} <=> s SUBSET t`] THEN MESON_TAC[BOUNDED_BALL; BOUNDED_SUBSET; NOT_BOUNDED_UNIV]);; let OUTSIDE_COMPACT_IN_OPEN = prove (`!s t:real^N->bool. compact s /\ open t /\ s SUBSET t /\ ~(t = {}) ==> ~(outside s INTER t = {})`, REPEAT GEN_TAC THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP OUTSIDE_BOUNDED_NONEMPTY o MATCH_MP COMPACT_IMP_BOUNDED) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM; IN_INTER] THEN X_GEN_TAC `b:real^N` THEN DISCH_TAC THEN X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN ASM_CASES_TAC `(a:real^N) IN t` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`linepath(a:real^N,b)`; `(:real^N) DIFF t`] EXISTS_PATH_SUBPATH_TO_FRONTIER) THEN REWRITE_TAC[PATH_LINEPATH; PATHSTART_LINEPATH; PATHFINISH_LINEPATH] THEN ASM_REWRITE_TAC[IN_DIFF; IN_UNIV; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `g:real^1->real^N` THEN REWRITE_TAC[FRONTIER_COMPLEMENT] THEN REWRITE_TAC[PATH_IMAGE_LINEPATH; INTERIOR_DIFF; INTERIOR_UNIV] THEN ABBREV_TAC `c:real^N = pathfinish g` THEN STRIP_TAC THEN SUBGOAL_THEN `frontier t SUBSET (:real^N) DIFF s` MP_TAC THENL [ONCE_REWRITE_TAC[GSYM FRONTIER_COMPLEMENT] THEN REWRITE_TAC[frontier] THEN ASM_SIMP_TAC[CLOSURE_CLOSED; GSYM OPEN_CLOSED] THEN ASM SET_TAC[]; REWRITE_TAC[SUBSET; IN_DIFF; IN_UNIV]] THEN DISCH_THEN(MP_TAC o SPEC `c:real^N`) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN MP_TAC(ISPEC `(:real^N) DIFF s` OPEN_CONTAINS_CBALL) THEN ASM_SIMP_TAC[GSYM closed; COMPACT_IMP_CLOSED; IN_DIFF; IN_UNIV] THEN DISCH_THEN(MP_TAC o SPEC `c:real^N`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`c:real^N`; `t:real^N->bool`] CLOSURE_APPROACHABLE) THEN RULE_ASSUM_TAC(REWRITE_RULE[frontier; IN_DIFF]) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC OUTSIDE_SAME_COMPONENT THEN EXISTS_TAC `a:real^N` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[connected_component] THEN EXISTS_TAC `path_image(g) UNION segment[c:real^N,d]` THEN REWRITE_TAC[IN_UNION; ENDS_IN_SEGMENT] THEN CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_UNION THEN ASM_SIMP_TAC[CONNECTED_SEGMENT; GSYM MEMBER_NOT_EMPTY; CONNECTED_PATH_IMAGE] THEN EXISTS_TAC `c:real^N` THEN REWRITE_TAC[ENDS_IN_SEGMENT; IN_INTER] THEN ASM_MESON_TAC[PATHFINISH_IN_PATH_IMAGE; SUBSET]; CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[PATHSTART_IN_PATH_IMAGE]] THEN REWRITE_TAC[UNION_SUBSET] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `~(c IN s) ==> (t DELETE c) SUBSET (UNIV DIFF s) ==> t SUBSET (UNIV DIFF s)`)) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] SUBSET_TRANS)) THEN SIMP_TAC[SET_RULE `UNIV DIFF s SUBSET UNIV DIFF t <=> t SUBSET s`] THEN ASM_MESON_TAC[SUBSET_TRANS; CLOSURE_SUBSET]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS)) THEN REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN MATCH_MP_TAC HULL_MINIMAL THEN ASM_SIMP_TAC[CONVEX_CBALL; INSERT_SUBSET; REAL_LT_IMP_LE; EMPTY_SUBSET; CENTRE_IN_CBALL] THEN REWRITE_TAC[IN_CBALL] THEN ASM_MESON_TAC[DIST_SYM; REAL_LT_IMP_LE]]]);; let INSIDE_INSIDE_COMPACT_CONNECTED = prove (`!s t:real^N->bool. closed s /\ compact t /\ s SUBSET inside t /\ connected t ==> inside s SUBSET inside t`, REPEAT GEN_TAC THEN ASM_CASES_TAC `inside t:real^N->bool = {}` THEN ASM_SIMP_TAC[INSIDE_EMPTY; SUBSET_EMPTY; EMPTY_SUBSET] THEN SUBGOAL_THEN `1 <= dimindex(:N)` MP_TAC THENL [REWRITE_TAC[DIMINDEX_GE_1]; REWRITE_TAC[ARITH_RULE `1 <= n <=> n = 1 \/ 2 <= n`]] THEN STRIP_TAC THEN ASM_SIMP_TAC[CONNECTED_CONVEX_1_GEN] THENL [ASM_MESON_TAC[INSIDE_CONVEX]; ALL_TAC] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP INSIDE_INSIDE) THEN MATCH_MP_TAC(SET_RULE `s INTER t = {} ==> s DIFF t SUBSET u ==> s SUBSET u`) THEN SUBGOAL_THEN `compact(s:real^N->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[COMPACT_EQ_BOUNDED_CLOSED; BOUNDED_SUBSET; BOUNDED_INSIDE]; ALL_TAC] THEN MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`] INSIDE_OUTSIDE_INTERSECT_CONNECTED) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(TAUT `r /\ q ==> (~p /\ q ==> ~r) ==> p`) THEN CONJ_TAC THENL [MP_TAC(ISPEC `t:real^N->bool` INSIDE_NO_OVERLAP) THEN ASM SET_TAC[]; ONCE_REWRITE_TAC[INTER_COMM]] THEN MATCH_MP_TAC INSIDE_OUTSIDE_INTERSECT_CONNECTED THEN ASM_SIMP_TAC[CONNECTED_OUTSIDE; COMPACT_IMP_BOUNDED] THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[INTER_COMM] THEN MATCH_MP_TAC OUTSIDE_COMPACT_IN_OPEN THEN ASM_SIMP_TAC[OPEN_INSIDE; COMPACT_IMP_CLOSED]; MP_TAC(ISPECL [`s UNION t:real^N->bool`; `vec 0:real^N`] BOUNDED_SUBSET_BALL) THEN ASM_SIMP_TAC[BOUNDED_UNION; COMPACT_IMP_BOUNDED] THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC(SET_RULE `!u. ~(u = UNIV) /\ UNIV DIFF u SUBSET s /\ UNIV DIFF u SUBSET t ==> ~(s INTER t = {})`) THEN EXISTS_TAC `ball(vec 0:real^N,r)` THEN CONJ_TAC THENL [ASM_MESON_TAC[NOT_BOUNDED_UNIV; BOUNDED_BALL]; ALL_TAC] THEN CONJ_TAC THEN MATCH_MP_TAC OUTSIDE_SUBSET_CONVEX THEN REWRITE_TAC[CONVEX_BALL] THEN ASM SET_TAC[]]);; let INSIDE_SELF_OUTSIDE_COMPACT_CONNECTED = prove (`!s t:real^N->bool. closed s /\ compact t /\ s SUBSET inside t /\ connected t ==> t UNION outside t SUBSET outside s`, REWRITE_TAC[GSYM INSIDE_SELF_OUTSIDE_EVERSION] THEN SIMP_TAC[UNION_SUBSET] THEN REWRITE_TAC[INSIDE_INSIDE_COMPACT_CONNECTED]);; let INSIDE_OUTSIDE_COMPACT_CONNECTED = prove (`!s t:real^N->bool. closed s /\ compact t /\ s SUBSET inside t /\ connected t ==> t SUBSET outside s`, REPEAT STRIP_TAC THEN TRANS_TAC SUBSET_TRANS `t UNION outside t:real^N->bool` THEN ASM_SIMP_TAC[INSIDE_SELF_OUTSIDE_COMPACT_CONNECTED] THEN SET_TAC[]);; let CONNECTED_WITH_INSIDE = prove (`!s:real^N->bool. closed s /\ connected s ==> connected(s UNION inside s)`, GEN_TAC THEN ASM_CASES_TAC `s UNION inside s = (:real^N)` THEN ASM_REWRITE_TAC[CONNECTED_UNIV] THEN REWRITE_TAC[CONNECTED_IFF_CONNECTED_COMPONENT] THEN REWRITE_TAC[CONNECTED_COMPONENT_SET; IN_ELIM_THM] THEN STRIP_TAC THEN SUBGOAL_THEN `!x. x IN (s UNION inside s) ==> ?y:real^N t. y IN s /\ connected t /\ x IN t /\ y IN t /\ t SUBSET (s UNION inside s)` MP_TAC THENL [X_GEN_TAC `a:real^N` THEN REWRITE_TAC[IN_UNION] THEN STRIP_TAC THENL [MAP_EVERY EXISTS_TAC [`a:real^N`; `{a:real^N}`] THEN ASM_REWRITE_TAC[IN_SING; CONNECTED_SING] THEN ASM SET_TAC[]; FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE `~(s UNION t = UNIV) ==> ?b. ~(b IN s) /\ ~(b IN t)`)) THEN DISCH_THEN(X_CHOOSE_THEN `b:real^N` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`linepath(a:real^N,b)`; `inside s:real^N->bool`] EXISTS_PATH_SUBPATH_TO_FRONTIER) THEN ASM_SIMP_TAC[PATH_LINEPATH; PATHSTART_LINEPATH; PATHFINISH_LINEPATH; IN_UNION; OPEN_INSIDE; INTERIOR_OPEN] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^1->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `pathfinish g :real^N` THEN EXISTS_TAC `path_image g :real^N->bool` THEN ASM_SIMP_TAC[PATHFINISH_IN_PATH_IMAGE; CONNECTED_PATH_IMAGE] THEN MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN REPEAT STRIP_TAC THENL [ASM_MESON_TAC[FRONTIER_INSIDE_SUBSET; SUBSET]; ASM_MESON_TAC[PATHSTART_IN_PATH_IMAGE]; ASM SET_TAC[]]]; DISCH_THEN(fun th -> MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN MP_TAC(SPEC `y:real^N` th) THEN MP_TAC(SPEC `x:real^N` th)) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `t:real^N->bool`] THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`b:real^N`; `u:real^N->bool`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a:real^N`; `b:real^N`]) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `v:real^N->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `t UNION v UNION u:real^N->bool` THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REPEAT(MATCH_MP_TAC CONNECTED_UNION THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC) THEN ASM SET_TAC[]]);; let CONNECTED_WITH_OUTSIDE = prove (`!s:real^N->bool. closed s /\ connected s ==> connected(s UNION outside s)`, GEN_TAC THEN ASM_CASES_TAC `s UNION outside s = (:real^N)` THEN ASM_REWRITE_TAC[CONNECTED_UNIV] THEN REWRITE_TAC[CONNECTED_IFF_CONNECTED_COMPONENT] THEN REWRITE_TAC[CONNECTED_COMPONENT_SET; IN_ELIM_THM] THEN STRIP_TAC THEN SUBGOAL_THEN `!x. x IN (s UNION outside s) ==> ?y:real^N t. y IN s /\ connected t /\ x IN t /\ y IN t /\ t SUBSET (s UNION outside s)` MP_TAC THENL [X_GEN_TAC `a:real^N` THEN REWRITE_TAC[IN_UNION] THEN STRIP_TAC THENL [MAP_EVERY EXISTS_TAC [`a:real^N`; `{a:real^N}`] THEN ASM_REWRITE_TAC[IN_SING; CONNECTED_SING] THEN ASM SET_TAC[]; FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE `~(s UNION t = UNIV) ==> ?b. ~(b IN s) /\ ~(b IN t)`)) THEN DISCH_THEN(X_CHOOSE_THEN `b:real^N` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`linepath(a:real^N,b)`; `outside s:real^N->bool`] EXISTS_PATH_SUBPATH_TO_FRONTIER) THEN ASM_SIMP_TAC[PATH_LINEPATH; PATHSTART_LINEPATH; PATHFINISH_LINEPATH; IN_UNION; OPEN_OUTSIDE; INTERIOR_OPEN] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^1->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `pathfinish g :real^N` THEN EXISTS_TAC `path_image g :real^N->bool` THEN ASM_SIMP_TAC[PATHFINISH_IN_PATH_IMAGE; CONNECTED_PATH_IMAGE] THEN MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN REPEAT STRIP_TAC THENL [ASM_MESON_TAC[FRONTIER_OUTSIDE_SUBSET; SUBSET]; ASM_MESON_TAC[PATHSTART_IN_PATH_IMAGE]; ASM SET_TAC[]]]; DISCH_THEN(fun th -> MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN MP_TAC(SPEC `y:real^N` th) THEN MP_TAC(SPEC `x:real^N` th)) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `t:real^N->bool`] THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`b:real^N`; `u:real^N->bool`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a:real^N`; `b:real^N`]) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `v:real^N->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `t UNION v UNION u:real^N->bool` THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REPEAT(MATCH_MP_TAC CONNECTED_UNION THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC) THEN ASM SET_TAC[]]);; let INSIDE_INSIDE_EQ_EMPTY = prove (`!s:real^N->bool. closed s /\ connected s ==> inside(inside s) = {}`, REPEAT STRIP_TAC THEN REWRITE_TAC[EXTENSION; NOT_IN_EMPTY] THEN X_GEN_TAC `x:real^N` THEN ONCE_REWRITE_TAC[inside] THEN REWRITE_TAC[IN_ELIM_THM] THEN ONCE_REWRITE_TAC[INSIDE_OUTSIDE] THEN REWRITE_TAC[COMPL_COMPL] THEN REWRITE_TAC[IN_DIFF; IN_UNIV] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_SIMP_TAC[CONNECTED_COMPONENT_EQ_SELF; CONNECTED_WITH_OUTSIDE] THEN REWRITE_TAC[BOUNDED_UNION] THEN MESON_TAC[UNBOUNDED_OUTSIDE]);; let INSIDE_IN_COMPONENTS = prove (`!s. (inside s) IN components((:real^N) DIFF s) <=> connected(inside s) /\ ~(inside s = {})`, X_GEN_TAC `s:real^N->bool` THEN REWRITE_TAC[IN_COMPONENTS_MAXIMAL] THEN ASM_CASES_TAC `inside s:real^N->bool = {}` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `connected(inside s:real^N->bool)` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SET_RULE `s SUBSET UNIV DIFF t <=> s INTER t = {}`] THEN REWRITE_TAC[INSIDE_NO_OVERLAP] THEN X_GEN_TAC `d:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MATCH_MP_TAC INSIDE_SAME_COMPONENT THEN UNDISCH_TAC `~(inside s:real^N->bool = {})` THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN ASM_REWRITE_TAC[connected_component] THEN EXISTS_TAC `d:real^N->bool` THEN ASM SET_TAC[]);; let OUTSIDE_IN_COMPONENTS = prove (`!s. (outside s) IN components((:real^N) DIFF s) <=> connected(outside s) /\ ~(outside s = {})`, X_GEN_TAC `s:real^N->bool` THEN REWRITE_TAC[IN_COMPONENTS_MAXIMAL] THEN ASM_CASES_TAC `outside s:real^N->bool = {}` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `connected(outside s:real^N->bool)` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SET_RULE `s SUBSET UNIV DIFF t <=> s INTER t = {}`] THEN REWRITE_TAC[OUTSIDE_NO_OVERLAP] THEN X_GEN_TAC `d:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MATCH_MP_TAC OUTSIDE_SAME_COMPONENT THEN UNDISCH_TAC `~(outside s:real^N->bool = {})` THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN ASM_REWRITE_TAC[connected_component] THEN EXISTS_TAC `d:real^N->bool` THEN ASM SET_TAC[]);; let BOUNDED_UNIQUE_OUTSIDE = prove (`!c s. 2 <= dimindex(:N) /\ bounded s ==> (c IN components ((:real^N) DIFF s) /\ ~bounded c <=> c = outside s)`, REPEAT STRIP_TAC THEN EQ_TAC THEN STRIP_TAC THENL [MATCH_MP_TAC COBOUNDED_UNIQUE_UNBOUNDED_COMPONENTS THEN EXISTS_TAC `(:real^N) DIFF s` THEN ASM_REWRITE_TAC[COMPL_COMPL] THEN ASM_REWRITE_TAC[OUTSIDE_IN_COMPONENTS]; ASM_REWRITE_TAC[OUTSIDE_IN_COMPONENTS]] THEN ASM_SIMP_TAC[UNBOUNDED_OUTSIDE; OUTSIDE_BOUNDED_NONEMPTY; CONNECTED_OUTSIDE]);; let EMPTY_INSIDE_PSUBSET_CONVEX_FRONTIER = prove (`!s t:real^N->bool. convex s /\ t PSUBSET frontier s ==> inside t = {}`, REPEAT STRIP_TAC THEN REWRITE_TAC[inside] THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN ASM_CASES_TAC `(x:real^N) IN closure s` THEN ASM_REWRITE_TAC[] THENL [ALL_TAC; SUBGOAL_THEN `x IN UNIONS(components((:real^N) DIFF closure s))` MP_TAC THENL [ASM_REWRITE_TAC[GSYM UNIONS_COMPONENTS; IN_DIFF; IN_UNIV]; REWRITE_TAC[IN_UNIONS] THEN DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] UNBOUNDED_COMPLEMENT_COMPONENT_CONVEX)) THEN ASM_SIMP_TAC[CONVEX_CLOSURE] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] BOUNDED_SUBSET)) THEN MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_CONNECTED) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN RULE_ASSUM_TAC(REWRITE_RULE[frontier]) THEN ASM SET_TAC[]]] THEN SUBGOAL_THEN `?y:real^N. y IN frontier s /\ ~(y IN t) /\ connected_component ((:real^N) DIFF t) x = connected_component ((:real^N) DIFF t) y` STRIP_ASSUME_TAC THENL [ASM_CASES_TAC `(x:real^N) IN frontier s` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o MATCH_MP (SET_RULE `t PSUBSET s ==> ?x. x IN s /\ ~(x IN t)`)) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `y:real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONNECTED_COMPONENT_EQ THEN REWRITE_TAC[IN] THEN REWRITE_TAC[connected_component] THEN EXISTS_TAC `(y:real^N) INSERT interior s` THEN ASM_REWRITE_TAC[IN_INSERT] THEN ASM_SIMP_TAC[IN_INSERT; CONNECTED_INSERT; CONVEX_CONNECTED; CONVEX_INTERIOR; INSERT_SUBSET; IN_DIFF; IN_UNIV] THEN SUBGOAL_THEN `(x:real^N) IN interior s /\ ~(interior s = {})` STRIP_ASSUME_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[frontier]) THEN ASM SET_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[CONVEX_CLOSURE_INTERIOR] THEN RULE_ASSUM_TAC(REWRITE_RULE[frontier]) THEN ASM SET_TAC[]; FIRST_X_ASSUM SUBST_ALL_TAC] THEN MP_TAC(ISPECL [`s:real^N->bool`; `y:real^N`] SUPPORTING_HYPERPLANE_FRONTIER) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o SPEC `y INSERT {u:real^N | a dot u < a dot y}` o MATCH_MP (REWRITE_RULE[IMP_CONJ] BOUNDED_SUBSET)) THEN ASM_REWRITE_TAC[NOT_IMP; BOUNDED_INSERT; BOUNDED_HALFSPACE_LT] THEN MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL THEN ASM_SIMP_TAC[IN_INSERT; CONNECTED_INSERT; CONVEX_CONNECTED; CONVEX_HALFSPACE_LT; CLOSURE_HALFSPACE_LT] THEN REWRITE_TAC[INSERT_SUBSET; IN_ELIM_THM; REAL_LE_REFL] THEN ASM_REWRITE_TAC[IN_DIFF; IN_ELIM_THM; REAL_NOT_LT; IN_UNIV; SET_RULE `s SUBSET UNIV DIFF t <=> !x. x IN t ==> ~(x IN s)`] THEN RULE_ASSUM_TAC(REWRITE_RULE[frontier]) THEN ASM SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* A Euclidean-centric formulation of homotopy. *) (* ------------------------------------------------------------------------- *) let HOMOTOPIC_WITH_EUCLIDEAN = prove (`!P X Y (p:real^M->real^N) q. homotopic_with P (subtopology euclidean X,subtopology euclidean Y) p q <=> ?h:real^(1,M)finite_sum->real^N. h continuous_on (interval[vec 0,vec 1] PCROSS X) /\ IMAGE h (interval[vec 0,vec 1] PCROSS X) SUBSET Y /\ (!x. h(pastecart (vec 0) x) = p x) /\ (!x. h(pastecart (vec 1) x) = q x) /\ (!t. t IN interval[vec 0,vec 1] ==> P(\x. h(pastecart t x)))`, REPEAT GEN_TAC THEN REWRITE_TAC[homotopic_with] THEN REWRITE_TAC[CONJ_ASSOC; GSYM CONTINUOUS_MAP_EUCLIDEAN2] THEN REWRITE_TAC[INTERVAL_REAL_INTERVAL; FORALL_IN_IMAGE; DROP_VEC] THEN EQ_TAC THEN REWRITE_TAC[GSYM CONJ_ASSOC; LEFT_IMP_EXISTS_THM] THENL [X_GEN_TAC `h:real#real^M->real^N` THEN STRIP_TAC THEN EXISTS_TAC `(h:real#real^M->real^N) o (\z. drop(fstcart z),sndcart z)` THEN ASM_REWRITE_TAC[o_THM; FSTCART_PASTECART; SNDCART_PASTECART] THEN ASM_REWRITE_TAC[LIFT_DROP; DROP_VEC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] CONTINUOUS_MAP_COMPOSE)) THEN REWRITE_TAC[CONTINUOUS_MAP_PAIRWISE; o_DEF] THEN REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN REWRITE_TAC[SUBSET; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; FORALL_IN_PCROSS; FORALL_IN_IMAGE; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN SIMP_TAC[FSTCART_PASTECART; SNDCART_PASTECART; LIFT_DROP] THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN REWRITE_TAC[CONTINUOUS_MAP_EUCLIDEAN_EUCLIDEAN] THEN SIMP_TAC[CONTINUOUS_ON_SNDCART; CONTINUOUS_ON_ID] THEN GEN_REWRITE_TAC RAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_MAP_COMPOSE THEN EXISTS_TAC `euclidean:(real^1)topology` THEN REWRITE_TAC[CONTINUOUS_MAP_DROP] THEN GEN_REWRITE_TAC RAND_CONV [GSYM ETA_AX] THEN REWRITE_TAC[CONTINUOUS_MAP_EUCLIDEAN_EUCLIDEAN] THEN SIMP_TAC[CONTINUOUS_ON_FSTCART; CONTINUOUS_ON_ID]; X_GEN_TAC `h:real^(1,M)finite_sum->real^N` THEN STRIP_TAC THEN EXISTS_TAC `(h:real^(1,M)finite_sum->real^N) o (\(x,y). pastecart x y) o (\z. lift(FST z),SND z)` THEN ASM_REWRITE_TAC[o_THM; LIFT_NUM] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] CONTINUOUS_MAP_COMPOSE)) THEN REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; TOPSPACE_PROD_TOPOLOGY; FORALL_PAIR_THM; IN_CROSS; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; PASTECART_IN_PCROSS; TOPSPACE_EUCLIDEANREAL_SUBTOPOLOGY] THEN SIMP_TAC[FUN_IN_IMAGE; o_THM; PASTECART_IN_PCROSS] THEN REWRITE_TAC[GSYM SUBTOPOLOGY_CROSS] THEN MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN MATCH_MP_TAC CONTINUOUS_MAP_COMPOSE THEN EXISTS_TAC `prod_topology (euclidean:(real^1)topology) (euclidean:(real^M)topology)` THEN REWRITE_TAC[CONTINUOUS_MAP_PASTECART] THEN REWRITE_TAC[CONTINUOUS_MAP_PAIRWISE; o_DEF] THEN CONJ_TAC THENL [GEN_REWRITE_TAC RAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_MAP_COMPOSE THEN MESON_TAC[CONTINUOUS_MAP_LIFT; CONTINUOUS_MAP_FST]; REWRITE_TAC[CONTINUOUS_MAP_SND; ETA_AX]]]);; (* ------------------------------------------------------------------------- *) (* We often want to just localize the ending function equality or whatever. *) (* ------------------------------------------------------------------------- *) let HOMOTOPIC_WITH_EUCLIDEAN_ALT = prove (`(!h k. (!x. x IN X ==> h x = k x) ==> (P h <=> P k)) ==> (homotopic_with P (subtopology euclidean X,subtopology euclidean Y) p q <=> ?h:real^(1,M)finite_sum->real^N. h continuous_on (interval[vec 0,vec 1] PCROSS X) /\ IMAGE h (interval[vec 0,vec 1] PCROSS X) SUBSET Y /\ (!x. x IN X ==> h(pastecart (vec 0) x) = p x) /\ (!x. x IN X ==> h(pastecart (vec 1) x) = q x) /\ (!t. t IN interval[vec 0,vec 1] ==> P(\x. h(pastecart t x))))`, REPEAT GEN_TAC THEN DISCH_TAC THEN EQ_TAC THENL [REWRITE_TAC[HOMOTOPIC_WITH_EUCLIDEAN; PCROSS] THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[]; REWRITE_TAC[HOMOTOPIC_WITH_EUCLIDEAN; PCROSS] THEN DISCH_THEN(X_CHOOSE_THEN `h:real^(1,M)finite_sum->real^N` (fun th -> EXISTS_TAC `\y. if sndcart(y) IN X then (h:real^(1,M)finite_sum->real^N) y else if fstcart(y) = vec 0 then p(sndcart y) else q(sndcart y)` THEN MP_TAC th)) THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; VEC_EQ; ARITH_EQ] THEN REPEAT(MATCH_MP_TAC MONO_AND THEN CONJ_TAC) THENL [MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_EQ) THEN SIMP_TAC[FORALL_IN_GSPEC; SNDCART_PASTECART]; SIMP_TAC[FORALL_IN_IMAGE; FORALL_IN_GSPEC; SUBSET] THEN SIMP_TAC[FORALL_IN_GSPEC; SNDCART_PASTECART]; ASM_MESON_TAC[]; ASM_MESON_TAC[]; MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `t:real^1` THEN MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN MATCH_MP_TAC EQ_IMP THEN FIRST_X_ASSUM MATCH_MP_TAC THEN SIMP_TAC[]]]);; (* ------------------------------------------------------------------------- *) (* Trivial properties. *) (* ------------------------------------------------------------------------- *) let HOMOTOPIC_WITH_RESTRICT = prove (`!P s t s' t' f g:real^M->real^N. homotopic_with P (subtopology euclidean s,subtopology euclidean t) f g /\ s' SUBSET s /\ (!h. P h /\ IMAGE h s SUBSET t ==> IMAGE h s' SUBSET t') ==> homotopic_with P (subtopology euclidean s',subtopology euclidean t') f g`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN REWRITE_TAC[HOMOTOPIC_WITH_EUCLIDEAN] THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET_PCROSS] THEN ASM SET_TAC[]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_PCROSS] THEN MAP_EVERY X_GEN_TAC [`a:real^1`; `x:real^M`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `\x. (h:real^(1,M)finite_sum->real^N)(pastecart a x)`) THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE] THEN RULE_ASSUM_TAC(REWRITE_RULE [SUBSET; FORALL_IN_IMAGE; FORALL_IN_PCROSS]) THEN ASM_SIMP_TAC[]]);; let HOMOTOPIC_WITH_IMP_CONTINUOUS = prove (`!P X Y (f:real^M->real^N) g. homotopic_with P (subtopology euclidean X,subtopology euclidean Y) f g ==> f continuous_on X /\ g continuous_on X`, REPEAT GEN_TAC THEN REWRITE_TAC[HOMOTOPIC_WITH_EUCLIDEAN] THEN DISCH_THEN(X_CHOOSE_THEN `h:real^(1,M)finite_sum->real^N` MP_TAC) THEN STRIP_TAC THEN SUBGOAL_THEN `((h:real^(1,M)finite_sum->real^N) o (\x. pastecart (vec 0) x)) continuous_on X /\ ((h:real^(1,M)finite_sum->real^N) o (\x. pastecart (vec 1) x)) continuous_on X` MP_TAC THENL [ALL_TAC; ASM_REWRITE_TAC[o_DEF; ETA_AX]] THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_PASTECART; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; PASTECART_IN_PCROSS] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[GSYM CONJ_ASSOC; FSTCART_PASTECART; SNDCART_PASTECART] THEN SIMP_TAC[RIGHT_EXISTS_AND_THM; UNWIND_THM1; IN_INTERVAL_1] THEN REWRITE_TAC[DROP_VEC; REAL_POS; REAL_LE_REFL]);; let HOMOTOPIC_WITH_IMP_SUBSET = prove (`!P X Y (f:real^M->real^N) g. homotopic_with P (subtopology euclidean X,subtopology euclidean Y) f g ==> IMAGE f X SUBSET Y /\ IMAGE g X SUBSET Y`, REPEAT GEN_TAC THEN REWRITE_TAC[HOMOTOPIC_WITH_EUCLIDEAN] THEN DISCH_THEN(X_CHOOSE_THEN `h:real^(1,M)finite_sum->real^N` MP_TAC) THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN REWRITE_TAC[FORALL_IN_IMAGE; FORALL_IN_PCROSS; SUBSET] THEN DISCH_THEN (fun th -> MP_TAC(SPEC `vec 0:real^1` th) THEN MP_TAC(SPEC `vec 1:real^1` th)) THEN ASM_SIMP_TAC[IN_INTERVAL_1; DROP_VEC; REAL_POS; REAL_LE_REFL]);; let HOMOTOPIC_WITH_MONO = prove (`!P Q X Y f g:real^M->real^N. homotopic_with P (subtopology euclidean X,subtopology euclidean Y) f g /\ (!h. h continuous_on X /\ IMAGE h X SUBSET Y /\ P h ==> Q h) ==> homotopic_with Q (subtopology euclidean X,subtopology euclidean Y) f g`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN REWRITE_TAC[HOMOTOPIC_WITH_EUCLIDEAN; PCROSS] THEN MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[] THEN CONJ_TAC THENL [GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_PASTECART; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; ASM SET_TAC[]]);; let HOMOTOPIC_WITH_SUBSET_LEFT = prove (`!P X Y Z f g. homotopic_with P (subtopology euclidean X,subtopology euclidean Y) f g /\ Z SUBSET X ==> homotopic_with P (subtopology euclidean Z,subtopology euclidean Y) f g`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN REWRITE_TAC[HOMOTOPIC_WITH_EUCLIDEAN; PCROSS] THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; ASM SET_TAC[]]);; let HOMOTOPIC_WITH_SUBSET_RIGHT = prove (`!P X Y Z (f:real^M->real^N) g. homotopic_with P (subtopology euclidean X,subtopology euclidean Y) f g /\ Y SUBSET Z ==> homotopic_with P (subtopology euclidean X,subtopology euclidean Z) f g`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN REWRITE_TAC[HOMOTOPIC_WITH_EUCLIDEAN] THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[] THEN ASM_MESON_TAC[SUBSET_TRANS]);; let HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_RIGHT = prove (`!p f:real^N->real^P g h:real^M->real^N W X Y. homotopic_with (\f. p(f o h)) (subtopology euclidean X,subtopology euclidean Y) f g /\ h continuous_on W /\ IMAGE h W SUBSET X ==> homotopic_with p (subtopology euclidean W,subtopology euclidean Y) (f o h) (g o h)`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[IMP_CONJ_ALT] THEN STRIP_TAC THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_MAP_RIGHT) THEN ASM_REWRITE_TAC[CONTINUOUS_MAP_EUCLIDEAN2]);; let HOMOTOPIC_COMPOSE_CONTINUOUS_RIGHT = prove (`!f:real^N->real^P g h:real^M->real^N W X Y. homotopic_with (\f. T) (subtopology euclidean X,subtopology euclidean Y) f g /\ h continuous_on W /\ IMAGE h W SUBSET X ==> homotopic_with (\f. T) (subtopology euclidean W,subtopology euclidean Y) (f o h) (g o h)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_RIGHT THEN EXISTS_TAC `X:real^N->bool` THEN ASM_REWRITE_TAC[]);; let HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_LEFT = prove (`!p f:real^M->real^N g h:real^N->real^P X Y Z. homotopic_with (\f. p(h o f)) (subtopology euclidean X,subtopology euclidean Y) f g /\ h continuous_on Y /\ IMAGE h Y SUBSET Z ==> homotopic_with p (subtopology euclidean X,subtopology euclidean Z) (h o f) (h o g)`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[IMP_CONJ_ALT] THEN STRIP_TAC THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_MAP_LEFT) THEN ASM_REWRITE_TAC[CONTINUOUS_MAP_EUCLIDEAN2]);; let HOMOTOPIC_COMPOSE_CONTINUOUS_LEFT = prove (`!f:real^M->real^N g h:real^N->real^P X Y Z. homotopic_with (\f. T) (subtopology euclidean X,subtopology euclidean Y) f g /\ h continuous_on Y /\ IMAGE h Y SUBSET Z ==> homotopic_with (\f. T) (subtopology euclidean X,subtopology euclidean Z) (h o f) (h o g)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_LEFT THEN EXISTS_TAC `Y:real^N->bool` THEN ASM_REWRITE_TAC[]);; let HOMOTOPIC_WITH_PCROSS = prove (`!f:real^M->real^N f':real^P->real^Q g g' p p' q s s' t t'. homotopic_with p (subtopology euclidean s,subtopology euclidean t) f g /\ homotopic_with p' (subtopology euclidean s',subtopology euclidean t') f' g' /\ (!f g. p f /\ p' g ==> q(\x. pastecart (f(fstcart x)) (g(sndcart x)))) ==> homotopic_with q (subtopology euclidean (s PCROSS s'), subtopology euclidean (t PCROSS t')) (\z. pastecart (f(fstcart z)) (f'(sndcart z))) (\z. pastecart (g(fstcart z)) (g'(sndcart z)))`, REPEAT GEN_TAC THEN REWRITE_TAC[HOMOTOPIC_WITH_EUCLIDEAN] THEN REWRITE_TAC[CONJ_ASSOC] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN REWRITE_TAC[GSYM CONJ_ASSOC] THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `k:real^(1,M)finite_sum->real^N` STRIP_ASSUME_TAC) (X_CHOOSE_THEN `k':real^(1,P)finite_sum->real^Q` STRIP_ASSUME_TAC)) THEN EXISTS_TAC `\z:real^(1,(M,P)finite_sum)finite_sum. pastecart (k(pastecart (fstcart z) (fstcart(sndcart z))):real^N) (k'(pastecart (fstcart z) (sndcart(sndcart z))):real^Q)` THEN ASM_REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; FORALL_IN_IMAGE; FORALL_IN_PCROSS]) THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_PCROSS; FSTCART_PASTECART; SNDCART_PASTECART; PASTECART_IN_PCROSS; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN MATCH_MP_TAC CONTINUOUS_ON_PASTECART THEN CONJ_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN (CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_PASTECART THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_FSTCART] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; LINEAR_SNDCART]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_PCROSS; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN ASM_SIMP_TAC[FSTCART_PASTECART; SNDCART_PASTECART; PASTECART_IN_PCROSS]]));; (* ------------------------------------------------------------------------- *) (* Homotopy with P is an equivalence relation (on continuous functions *) (* mapping X into Y that satisfy P, though this only affects reflexivity). *) (* ------------------------------------------------------------------------- *) let HOMOTOPIC_WITH_COMPOSE = prove (`!P Q R f f':real^M->real^N g g':real^N->real^P s t u. (!f g. f continuous_on s /\ IMAGE f s SUBSET t /\ P f /\ g continuous_on t /\ IMAGE g t SUBSET u /\ Q g ==> R(g o f)) /\ homotopic_with P (subtopology euclidean s,subtopology euclidean t) f f' /\ homotopic_with Q (subtopology euclidean t,subtopology euclidean u) g g' ==> homotopic_with R (subtopology euclidean s,subtopology euclidean u) (g o f) (g' o f')`, REPEAT STRIP_TAC THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMOTOPIC_WITH_TRANS THEN EXISTS_TAC `(g:real^N->real^P) o (f':real^M->real^N)` THEN CONJ_TAC THENL [MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_LEFT; MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_RIGHT] THEN EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[] THEN (CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMOTOPIC_WITH_MONO)) THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_PROPERTY) THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_CONTINUOUS) THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_SUBSET) THEN ASM_SIMP_TAC[]; ASM_MESON_TAC[HOMOTOPIC_WITH_IMP_CONTINUOUS; HOMOTOPIC_WITH_IMP_SUBSET]]));; let HOMOTOPIC_COMPOSE = prove (`!f f':real^M->real^N g g':real^N->real^P s t u. homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean t) f f' /\ homotopic_with (\x. T) (subtopology euclidean t,subtopology euclidean u) g g' ==> homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean u) (g o f) (g' o f')`, REPEAT GEN_TAC THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] HOMOTOPIC_WITH_COMPOSE) THEN REWRITE_TAC[]);; (* ------------------------------------------------------------------------- *) (* Two characterizations of homotopic triviality, one of which *) (* implicitly incorporates path-connectedness. *) (* ------------------------------------------------------------------------- *) let HOMOTOPIC_TRIVIALITY = prove (`!s:real^M->bool t:real^N->bool. (!f g. f continuous_on s /\ IMAGE f s SUBSET t /\ g continuous_on s /\ IMAGE g s SUBSET t ==> homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean t) f g) <=> (s = {} \/ path_connected t) /\ (!f. f continuous_on s /\ IMAGE f s SUBSET t ==> ?c. homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean t) f (\x. c))`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THENL [ASM_SIMP_TAC[CONTINUOUS_ON_EMPTY; HOMOTOPIC_WITH_EUCLIDEAN_ALT; NOT_IN_EMPTY;PCROSS_EMPTY; IMAGE_CLAUSES; EMPTY_SUBSET]; ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_REWRITE_TAC[SUBSET_EMPTY; IMAGE_EQ_EMPTY; PATH_CONNECTED_EMPTY]] THEN EQ_TAC THEN REPEAT STRIP_TAC THENL [REWRITE_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM PATH_COMPONENT_OF_EUCLIDEAN] THEN MP_TAC(ISPECL [`subtopology euclidean (s:real^M->bool)`; `subtopology euclidean (t:real^N->bool)`] HOMOTOPIC_CONSTANT_MAPS) THEN ASM_REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN DISCH_THEN(fun th -> REWRITE_TAC[GSYM th]) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; CONTINUOUS_ON_CONST] THEN ASM SET_TAC[]; SUBGOAL_THEN `?c:real^N. c IN t` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; CONTINUOUS_ON_CONST]; FIRST_X_ASSUM(fun th -> MP_TAC(ISPEC `g:real^M->real^N` th) THEN MP_TAC(ISPEC `f:real^M->real^N` th)) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `c:real^N` THEN DISCH_TAC THEN X_GEN_TAC `d:real^N` THEN DISCH_TAC THEN TRANS_TAC HOMOTOPIC_WITH_TRANS `(\x. c):real^M->real^N` THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[HOMOTOPIC_WITH_SYM] THEN TRANS_TAC HOMOTOPIC_WITH_TRANS `(\x. d):real^M->real^N` THEN ASM_REWRITE_TAC[HOMOTOPIC_CONSTANT_MAPS; PATH_COMPONENT_OF_EUCLIDEAN; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[PATH_CONNECTED_IFF_PATH_COMPONENT]) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_SUBSET)) THEN ASM SET_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Homotopy on a union of closed-open sets. *) (* ------------------------------------------------------------------------- *) let HOMOTOPIC_ON_CLOPEN_UNIONS = prove (`!f:real^M->real^N g t u. (!s. s IN u ==> closed_in (subtopology euclidean (UNIONS u)) s /\ open_in (subtopology euclidean (UNIONS u)) s /\ homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean t) f g) ==> homotopic_with (\x. T) (subtopology euclidean (UNIONS u),subtopology euclidean t) f g`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?v. v SUBSET u /\ COUNTABLE v /\ UNIONS v :real^M->bool = UNIONS u` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC LINDELOF_OPEN_IN THEN ASM_MESON_TAC[]; FIRST_X_ASSUM(SUBST_ALL_TAC o SYM)] THEN ASM_CASES_TAC `v:(real^M->bool)->bool = {}` THEN ASM_SIMP_TAC[HOMOTOPIC_ON_EMPTY; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; UNIONS_0] THEN MP_TAC(ISPEC `v:(real^M->bool)->bool` COUNTABLE_AS_IMAGE) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `f:num->real^M->bool` THEN DISCH_THEN SUBST_ALL_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN `n:num` o SPEC `(f:num->real^M->bool) n`) THEN DISCH_THEN(MP_TAC o MATCH_MP MONO_FORALL) THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[FORALL_AND_THM]] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [HOMOTOPIC_WITH_EUCLIDEAN] THEN SIMP_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM; HOMOTOPIC_WITH_EUCLIDEAN_ALT] THEN X_GEN_TAC `h:num->real^(1,M)finite_sum->real^N` THEN REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC THEN MP_TAC(ISPECL [`subtopology euclidean ((interval[vec 0,vec 1] PCROSS UNIONS(IMAGE f (:num))) :real^(1,M)finite_sum->bool)`; `euclidean:(real^N)topology`; `h:num->real^(1,M)finite_sum->real^N`; `(\n. interval[vec 0,vec 1] PCROSS (f n DIFF UNIONS {f m | m < n})) :num->real^(1,M)finite_sum->bool`; `(:num)`] PASTING_LEMMA_EXISTS) THEN REWRITE_TAC[CONTINUOUS_MAP_EUCLIDEAN; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; SUBTOPOLOGY_SUBTOPOLOGY] THEN ONCE_REWRITE_TAC[TAUT `open_in a b /\ c <=> ~(open_in a b ==> ~c)`] THEN SIMP_TAC[ISPEC `euclidean` OPEN_IN_IMP_SUBSET; SET_RULE `s SUBSET u ==> u INTER s = s`] THEN REWRITE_TAC[NOT_IMP] THEN REWRITE_TAC[IN_UNIV; FORALL_AND_THM; SUBSET_UNIV; INTER_PCROSS] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [REWRITE_TAC[UNIONS_GSPEC; SUBSET; IN_ELIM_THM; FORALL_PASTECART] THEN REWRITE_TAC[PASTECART_IN_PCROSS; IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_UNIONS; FORALL_IN_IMAGE; IN_UNIV; IMP_CONJ] THEN X_GEN_TAC `t:real^1` THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN X_GEN_TAC `y:real^M` THEN REWRITE_TAC[LEFT_FORALL_IMP_THM; IN_DIFF; IN_ELIM_THM] THEN GEN_REWRITE_TAC LAND_CONV [num_WOP] THEN MESON_TAC[]; X_GEN_TAC `n:num` THEN MATCH_MP_TAC OPEN_IN_PCROSS THEN REWRITE_TAC[OPEN_IN_REFL] THEN MATCH_MP_TAC OPEN_IN_DIFF THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CLOSED_IN_UNIONS THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN SIMP_TAC[FINITE_NUMSEG_LT; FINITE_IMAGE] THEN ASM SET_TAC[]; X_GEN_TAC `n:num` THEN FIRST_X_ASSUM(fun th -> MATCH_MP_TAC(MATCH_MP(REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET) (SPEC `n:num` th))) THEN REWRITE_TAC[SUBSET_PCROSS; SUBSET_REFL; SUBSET_DIFF]; MATCH_MP_TAC WLOG_LT THEN REWRITE_TAC[] THEN CONJ_TAC THENL [REWRITE_TAC[INTER_ACI] THEN MESON_TAC[]; REWRITE_TAC[FORALL_PASTECART; PASTECART_IN_PCROSS] THEN SET_TAC[]]]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^(1,M)finite_sum->real^N` THEN REWRITE_TAC[INTER_ACI; FORALL_PASTECART; PASTECART_IN_PCROSS] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[FORALL_IN_UNIONS; FORALL_IN_IMAGE; IMP_CONJ; SUBSET; RIGHT_FORALL_IMP_THM; IN_UNIV; FORALL_IN_PCROSS] THEN CONJ_TAC THENL [X_GEN_TAC `t:real^1` THEN DISCH_TAC; CONJ_TAC] THEN ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN X_GEN_TAC `y:real^M` THEN REWRITE_TAC[LEFT_FORALL_IMP_THM] THEN GEN_REWRITE_TAC LAND_CONV [num_WOP] THEN DISCH_THEN(X_CHOOSE_THEN `n:num` STRIP_ASSUME_TAC) THENL [FIRST_X_ASSUM(MP_TAC o SPECL [`t:real^1`; `y:real^M`; `n:num`]); FIRST_X_ASSUM(MP_TAC o SPECL [`vec 0:real^1`; `y:real^M`; `n:num`]); FIRST_X_ASSUM(MP_TAC o SPECL [`vec 1:real^1`; `y:real^M`; `n:num`])] THEN ASM_REWRITE_TAC[IN_INTER; UNIONS_IMAGE; IN_UNIV; IN_DIFF; UNIONS_GSPEC; IN_ELIM_THM; ENDS_IN_UNIT_INTERVAL] THEN (ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN SUBST1_TAC]) THEN REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE BINDER_CONV [SUBSET]) THEN REWRITE_TAC[FORALL_IN_IMAGE; FORALL_IN_PCROSS] THEN ASM SET_TAC[]]);; let INESSENTIAL_ON_CLOPEN_UNIONS = prove (`!f:real^M->real^N t u. path_connected t /\ (!s. s IN u ==> closed_in (subtopology euclidean (UNIONS u)) s /\ open_in (subtopology euclidean (UNIONS u)) s /\ ?a. homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean t) f (\x. a)) ==> ?a. homotopic_with (\x. T) (subtopology euclidean (UNIONS u),subtopology euclidean t) f (\x. a)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `UNIONS u:real^M->bool = {}` THEN ASM_SIMP_TAC[UNIONS_0; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; HOMOTOPIC_ON_EMPTY] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [EMPTY_UNIONS]) THEN REWRITE_TAC[NOT_FORALL_THM; LEFT_IMP_EXISTS_THM; NOT_IMP] THEN X_GEN_TAC `c:real^M->bool` THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `c:real^M->bool`) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[]; ALL_TAC] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(X_CHOOSE_THEN `a:real^N` MP_TAC) THEN DISCH_THEN(MP_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_SUBSET) THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE `IMAGE (\x. a) s SUBSET t ==> ~(s = {}) ==> a IN t`)) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN EXISTS_TAC `a:real^N` THEN MATCH_MP_TAC HOMOTOPIC_ON_CLOPEN_UNIONS THEN X_GEN_TAC `s:real^M->bool` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `s:real^M->bool`) THEN ASM_REWRITE_TAC[] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_SIMP_TAC[HOMOTOPIC_ON_EMPTY; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN X_GEN_TAC `b:real^N` THEN DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_WITH_TRANS) THEN REWRITE_TAC[HOMOTOPIC_CONSTANT_MAPS; PATH_COMPONENT_OF_EUCLIDEAN] THEN DISJ2_TAC THEN FIRST_X_ASSUM(STRIP_ASSUME_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_SUBSET) THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE `IMAGE (\x. a) s SUBSET t ==> ~(s = {}) ==> a IN t`)) THEN ASM_MESON_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT]);; (* ------------------------------------------------------------------------- *) (* Homotopy within the space of linear maps. *) (* ------------------------------------------------------------------------- *) let HOMOTOPIC_WITH_REFLECTIONS_ALONG = prove (`!P s t a b:real^N. ~(a = vec 0) /\ ~(b = vec 0) /\ (!c. c IN segment[a,b] ==> P(reflect_along c) /\ IMAGE (reflect_along c) s SUBSET t) ==> homotopic_with P (subtopology euclidean s,subtopology euclidean t) (reflect_along a) (reflect_along b)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `vec 0 IN segment[a:real^N,b]` THENL [SUBGOAL_THEN `reflect_along (b:real^N) = reflect_along a` SUBST1_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM BETWEEN_IN_SEGMENT]) THEN DISCH_THEN(MP_TAC o MATCH_MP BETWEEN_IMP_COLLINEAR) THEN ONCE_REWRITE_TAC[SET_RULE `{a,z,b} = {z,a,b}`] THEN ASM_REWRITE_TAC[COLLINEAR_LEMMA_ALT] THEN DISCH_THEN(X_CHOOSE_THEN `c:real` SUBST_ALL_TAC) THEN REWRITE_TAC[FUN_EQ_THM] THEN GEN_TAC THEN MATCH_MP_TAC REFLECT_ALONG_SCALE THEN ASM_MESON_TAC[VECTOR_MUL_EQ_0]; ASM_SIMP_TAC[HOMOTOPIC_WITH_REFL; ENDS_IN_SEGMENT; LINEAR_CONTINUOUS_ON; CONTINUOUS_MAP_EUCLIDEAN2; LINEAR_REFLECT_ALONG]]; ALL_TAC] THEN REWRITE_TAC[homotopic_with] THEN EXISTS_TAC `(\(c,x). reflect_along (c:real^N) x) o (\(t,x). (&1 - t) % a + t % b,x)` THEN REWRITE_TAC[o_THM] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_MUL_LID] THEN REWRITE_TAC[VECTOR_ADD_LID; VECTOR_ADD_RID; ETA_AX] THEN REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY; TOPSPACE_PROD_TOPOLOGY] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_CROSS; o_THM] THEN REWRITE_TAC[TOPSPACE_EUCLIDEANREAL_SUBTOPOLOGY; IMP_CONJ] THEN REWRITE_TAC[GSYM CONJ_ASSOC] THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_MAP_COMPOSE THEN EXISTS_TAC `prod_topology (subtopology euclidean ((:real^N) DIFF {vec 0})) (euclidean:(real^N)topology)` THEN REWRITE_TAC[CONTINUOUS_MAP_PROD] THEN CONJ_TAC THENL [DISJ2_TAC THEN REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN SIMP_TAC[CONTINUOUS_MAP_ID; CONTINUOUS_MAP_FROM_SUBTOPOLOGY] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; TOPSPACE_EUCLIDEANREAL_SUBTOPOLOGY] THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN REWRITE_TAC[CONTINUOUS_MAP_DROP_EQ; o_DEF; GSYM DROP_VEC] THEN REWRITE_TAC[CONTINUOUS_MAP_EUCLIDEAN_EUCLIDEAN; GSYM DROP_SUB] THEN MATCH_MP_TAC CONTINUOUS_ON_ADD THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_VMUL THEN REWRITE_TAC[o_DEF; LIFT_DROP] THEN SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [IN_SEGMENT]) THEN REWRITE_TAC[IN_REAL_INTERVAL] THEN SET_TAC[]]; REWRITE_TAC[reflect_along; LAMBDA_PAIR] THEN MATCH_MP_TAC CONTINUOUS_MAP_VECTOR_SUB THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM ETA_AX] THEN REWRITE_TAC[CONTINUOUS_MAP_OF_SND; CONTINUOUS_MAP_ID] THEN MATCH_MP_TAC CONTINUOUS_MAP_VECTOR_MUL THEN GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM ETA_AX] THEN REWRITE_TAC[CONTINUOUS_MAP_OF_FST] THEN SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_ID] THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_LMUL THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_DIV THEN REWRITE_TAC[TOPSPACE_PROD_TOPOLOGY; FORALL_IN_CROSS] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; DOT_EQ_0] THEN SIMP_TAC[IN_DIFF; IN_SING] THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_MAP_DOT THEN CONJ_TAC THEN GEN_REWRITE_TAC RAND_CONV [GSYM ETA_AX] THEN REWRITE_TAC[CONTINUOUS_MAP_OF_FST; CONTINUOUS_MAP_OF_SND] THEN SIMP_TAC[CONTINUOUS_MAP_ID; CONTINUOUS_MAP_FROM_SUBTOPOLOGY]]; REWRITE_TAC[AND_FORALL_THM; RIGHT_FORALL_IMP_THM; IN_REAL_INTERVAL; TAUT `(p ==> q) /\ (p ==> r) <=> p ==> q /\ r`] THEN X_GEN_TAC `t:real` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `(&1 - t) % a + t % b:real^N`) THEN REWRITE_TAC[IN_SEGMENT; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN ASM SET_TAC[]]);; let HOMOTOPIC_WITH_ORTHOGONAL_TRANSFORMATIONS = prove (`!f g:real^N->real^N. homotopic_with orthogonal_transformation (subtopology euclidean (sphere(vec 0,&1)), subtopology euclidean (sphere(vec 0,&1))) f g <=> orthogonal_transformation f /\ orthogonal_transformation g /\ det(matrix f) = det(matrix g)`, let lemma = prove (`!f:real^N->real^N. orthogonal_transformation f ==> homotopic_with orthogonal_transformation (subtopology euclidean (sphere(vec 0,&1)), subtopology euclidean (sphere(vec 0,&1))) f (if det(matrix f) = &1 then I else reflect_along(basis 1))`, MATCH_MP_TAC ORTHOGONAL_TRANSFORMATION_REFLECT_INDUCT THEN CONJ_TAC THENL [REWRITE_TAC[MATRIX_I; DET_I; HOMOTOPIC_WITH_REFL] THEN REWRITE_TAC[ORTHOGONAL_TRANSFORMATION_I; CONTINUOUS_MAP_EUCLIDEAN2] THEN REWRITE_TAC[I_DEF; CONTINUOUS_ON_ID; IMAGE_ID; SUBSET_REFL]; MAP_EVERY X_GEN_TAC [`f:real^N->real^N`; `a:real^N`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`orthogonal_transformation:(real^N->real^N)->bool`; `orthogonal_transformation:(real^N->real^N)->bool`; `orthogonal_transformation:(real^N->real^N)->bool`; `f:real^N->real^N`; `if det(matrix f:real^N^N) = &1 then I else reflect_along (basis 1:real^N)`; `reflect_along(a:real^N)`; `reflect_along(basis 1:real^N)`; `sphere(vec 0:real^N,&1)`; `sphere(vec 0:real^N,&1)`; `sphere(vec 0:real^N,&1)`] HOMOTOPIC_WITH_COMPOSE) THEN ASM_SIMP_TAC[ORTHOGONAL_TRANSFORMATION_COMPOSE] THEN ANTS_TAC THENL [MATCH_MP_TAC HOMOTOPIC_WITH_REFLECTIONS_ALONG THEN ASM_SIMP_TAC[BASIS_NONZERO; LE_REFL; DIMINDEX_GE_1] THEN REWRITE_TAC[ORTHOGONAL_TRANSFORMATION_REFLECT_ALONG; SUBSET] THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_SPHERE_0; NORM_REFLECT_ALONG]; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_WITH_TRANS) THEN ASM_SIMP_TAC[MATRIX_COMPOSE; ORTHOGONAL_TRANSFORMATION_LINEAR; DET_MUL; LINEAR_REFLECT_ALONG; DET_MATRIX_REFLECT_ALONG] THEN FIRST_ASSUM(DISJ_CASES_TAC o MATCH_MP DET_ORTHOGONAL_MATRIX o MATCH_MP ORTHOGONAL_MATRIX_MATRIX) THEN ASM_REWRITE_TAC[] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[o_DEF; I_DEF; REFLECT_ALONG_INVOLUTION; ETA_AX] THEN REWRITE_TAC[HOMOTOPIC_WITH_REFL] THEN REWRITE_TAC[CONTINUOUS_MAP_ID; ORTHOGONAL_TRANSFORMATION_ID] THEN REWRITE_TAC[CONTINUOUS_MAP_EUCLIDEAN2; ORTHOGONAL_TRANSFORMATION_REFLECT_ALONG] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_SPHERE_0] THEN REWRITE_TAC[NORM_REFLECT_ALONG] THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_REFLECT_ALONG]]]) in REPEAT GEN_TAC THEN EQ_TAC THEN STRIP_TAC THENL [FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_PROPERTY) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMOTOPIC_WITH_EUCLIDEAN]) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `k:real^(1,N)finite_sum->real^N` THEN STRIP_TAC THEN MP_TAC(ISPECL [`\t. lift(det(matrix((k:real^(1,N)finite_sum->real^N) o pastecart t)))`; `interval[vec 0:real^1,vec 1]`] CONTINUOUS_DISCRETE_RANGE_CONSTANT) THEN REWRITE_TAC[CONNECTED_INTERVAL] THEN ANTS_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_LIFT_DET THEN SIMP_TAC[matrix; LAMBDA_BETA; o_DEF] THEN MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_LIFT_COMPONENT_COMPOSE THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_PASTECART; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE; PASTECART_IN_PCROSS; IN_SPHERE_0; NORM_BASIS]; X_GEN_TAC `t:real^1` THEN DISCH_TAC THEN EXISTS_TAC `&1` THEN REWRITE_TAC[REAL_LT_01] THEN X_GEN_TAC `u:real^1` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[GSYM LIFT_SUB; NORM_LIFT; LIFT_EQ] THEN MATCH_MP_TAC(REAL_ARITH `(a = &1 \/ a = -- &1) /\ (b = &1 \/ b = -- &1) ==> ~(a = b) ==> &1 <= abs(a - b)`) THEN CONJ_TAC THEN MATCH_MP_TAC DET_ORTHOGONAL_MATRIX THEN ASM_SIMP_TAC[ORTHOGONAL_MATRIX_MATRIX; o_DEF]]; DISCH_THEN(MP_TAC o SPECL [`vec 0:real^1`; `vec 1:real^1`] o MATCH_MP (MESON[] `(?y. !x. x IN s ==> f x = y) ==> !x x'. x IN s /\ x' IN s ==> f x = f x'`)) THEN ASM_REWRITE_TAC[ENDS_IN_UNIT_INTERVAL; LIFT_EQ] THEN MATCH_MP_TAC EQ_IMP THEN BINOP_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN ASM_REWRITE_TAC[FUN_EQ_THM; o_THM]]; MP_TAC(SPEC `g:real^N->real^N` lemma) THEN MP_TAC(SPEC `f:real^N->real^N` lemma) THEN ASM_REWRITE_TAC[] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN MESON_TAC[HOMOTOPIC_WITH_SYM; HOMOTOPIC_WITH_TRANS]]);; let HOMOTOPIC_WITH_ORTHOGONAL_TRANSFORMATIONS_GEN = prove (`!P f g:real^N->real^N. (?r. &0 < r /\ P r) ==> (homotopic_with orthogonal_transformation (subtopology euclidean {x | P(norm x)}, subtopology euclidean {x | P(norm x)}) f g <=> orthogonal_transformation f /\ orthogonal_transformation g /\ det(matrix f) = det(matrix g))`, SUBGOAL_THEN `!P f g:real^N->real^N. (?r. &0 < r /\ P r) ==> (homotopic_with orthogonal_transformation (subtopology euclidean {x | P(norm x)}, subtopology euclidean {x | P(norm x)}) f g <=> homotopic_with orthogonal_transformation (subtopology euclidean (:real^N), subtopology euclidean (:real^N)) f g)` ASSUME_TAC THENL [ALL_TAC; ASM_SIMP_TAC[GSYM HOMOTOPIC_WITH_ORTHOGONAL_TRANSFORMATIONS] THEN REPEAT GEN_TAC THEN DISCH_THEN(K ALL_TAC) THEN CONV_TAC SYM_CONV THEN REWRITE_TAC[sphere; DIST_0] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN EXISTS_TAC `&1` THEN CONV_TAC REAL_RAT_REDUCE_CONV] THEN REPEAT STRIP_TAC THEN EQ_TAC THENL [ALL_TAC; DISCH_TAC THEN MATCH_MP_TAC HOMOTOPIC_WITH_RESTRICT THEN REPEAT(EXISTS_TAC `(:real^N)`) THEN ASM_REWRITE_TAC[SUBSET_UNIV] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM] THEN MESON_TAC[ORTHOGONAL_TRANSFORMATION]] THEN REWRITE_TAC[HOMOTOPIC_WITH_EUCLIDEAN] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `h:real^(1,N)finite_sum->real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[SUBSET_UNIV] THEN MATCH_MP_TAC CONTINUOUS_ON_EQ THEN EXISTS_TAC `\z. norm(sndcart z) / r % (h:real^(1,N)finite_sum->real^N) (pastecart (fstcart z) (r / norm(sndcart z) % sndcart z))` THEN REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN; FORALL_IN_PCROSS; IN_UNIV; FSTCART_PASTECART; SNDCART_PASTECART] THEN CONJ_TAC THENL [MAP_EVERY X_GEN_TAC [`a:real^1`; `x:real^N`] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `a:real^1`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o MATCH_MP ORTHOGONAL_TRANSFORMATION_LINEAR) THEN DISCH_THEN(MP_TAC o MATCH_MP LINEAR_CMUL) THEN SIMP_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `&0`) THEN REWRITE_TAC[VECTOR_MUL_LZERO] THEN STRIP_TAC THEN ASM_CASES_TAC `x:real^N = vec 0` THEN ASM_REWRITE_TAC[NORM_0; VECTOR_MUL_LZERO; real_div; REAL_MUL_LZERO] THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; NORM_EQ_0; VECTOR_MUL_LID; REAL_FIELD `~(x = &0) /\ &0 < r ==> (x * inv r) * r * inv x = &1`]; ALL_TAC] THEN REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN MAP_EVERY X_GEN_TAC [`a:real^1`; `x:real^N`] THEN DISCH_TAC THEN ASM_CASES_TAC `x:real^N = vec 0` THENL [ASM_REWRITE_TAC[CONTINUOUS_WITHIN; SNDCART_PASTECART] THEN REWRITE_TAC[NORM_0; real_div; REAL_MUL_LZERO; VECTOR_MUL_LZERO] THEN MATCH_MP_TAC LIM_NULL_COMPARISON THEN EXISTS_TAC `(norm o sndcart):real^(1,N)finite_sum->real` THEN CONJ_TAC THENL [SIMP_TAC[EVENTUALLY_WITHIN; FORALL_PASTECART; PASTECART_IN_PCROSS] THEN EXISTS_TAC `&1` THEN REWRITE_TAC[REAL_LT_01; IN_UNIV] THEN REWRITE_TAC[o_DEF; FSTCART_PASTECART; SNDCART_PASTECART] THEN MAP_EVERY X_GEN_TAC [`b:real^1`; `y:real^N`] THEN STRIP_TAC THEN ASM_CASES_TAC `y:real^N = vec 0` THEN ASM_SIMP_TAC[NORM_0; VECTOR_MUL_LZERO; REAL_MUL_LZERO; REAL_LE_REFL] THEN FIRST_X_ASSUM(MP_TAC o SPEC `b:real^1`) THEN ASM_REWRITE_TAC[ORTHOGONAL_TRANSFORMATION] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN DISCH_THEN(MP_TAC o MATCH_MP LINEAR_CMUL) THEN ASM_SIMP_TAC[NORM_MUL; REAL_ABS_INV; REAL_ABS_MUL; REAL_ABS_NORM] THEN ASM_SIMP_TAC[NORM_EQ_0; REAL_LE_REFL; REAL_FIELD `&0 < r /\ ~(y = &0) ==> (y * inv(abs r)) * (abs r * inv y) * y = y`]; MATCH_MP_TAC(MESON[CONTINUOUS_WITHIN; CONTINUOUS_AT_WITHIN] `f continuous at a /\ f a = l ==> (f --> l) (at a within s)`) THEN REWRITE_TAC[o_DEF; SNDCART_PASTECART; NORM_0; LIFT_NUM] THEN SIMP_TAC[CONTINUOUS_LIFT_NORM_COMPOSE; LINEAR_CONTINUOUS_AT; LINEAR_SNDCART]]; MATCH_MP_TAC CONTINUOUS_MUL THEN REWRITE_TAC[o_DEF; real_div] THEN ONCE_REWRITE_TAC[REAL_ARITH `norm(x:real^N) * inv r = inv r * norm x`] THEN SIMP_TAC[LIFT_CMUL; CONTINUOUS_CMUL; CONTINUOUS_LIFT_NORM_COMPOSE; LINEAR_CONTINUOUS_WITHIN; LINEAR_SNDCART] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_WITHIN_COMPOSE THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_PASTECART THEN SIMP_TAC[LINEAR_CONTINUOUS_WITHIN; LINEAR_FSTCART] THEN MATCH_MP_TAC CONTINUOUS_MUL THEN SIMP_TAC[LINEAR_CONTINUOUS_WITHIN; LINEAR_SNDCART] THEN REWRITE_TAC[o_DEF; LIFT_CMUL] THEN MATCH_MP_TAC CONTINUOUS_CMUL THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_AT_WITHIN_INV) THEN SIMP_TAC[CONTINUOUS_LIFT_NORM_COMPOSE; LINEAR_CONTINUOUS_WITHIN; LINEAR_SNDCART; o_DEF] THEN ASM_REWRITE_TAC[NORM_EQ_0; SNDCART_PASTECART]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN]) THEN REWRITE_TAC[FORALL_IN_PCROSS; FSTCART_PASTECART; SNDCART_PASTECART] THEN DISCH_THEN(MP_TAC o SPECL [`a:real^1`; `r / norm(x) % x:real^N`]) THEN ASM_SIMP_TAC[IN_ELIM_THM; NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM; REAL_MUL_LINV; NORM_EQ_0; real_div; REAL_ABS_INV; REAL_ABS_MUL; REAL_ARITH `&0 < r ==> abs r = r`; REAL_FIELD `&0 < r /\ ~(x = &0) ==> (r * inv x) * x = r`] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] CONTINUOUS_TRANSFORM_WITHIN_SET_IMP) THEN REWRITE_TAC[EVENTUALLY_AT] THEN REWRITE_TAC[GSYM IMP_CONJ_ALT] THEN REWRITE_TAC[IMP_CONJ; FORALL_IN_IMAGE; FORALL_IN_PCROSS; IN_UNIV] THEN EXISTS_TAC `r:real` THEN ASM_REWRITE_TAC[] THEN SIMP_TAC[FSTCART_PASTECART; SNDCART_PASTECART; PASTECART_IN_PCROSS] THEN MAP_EVERY X_GEN_TAC [`b:real^1`; `y:real^N`] THEN STRIP_TAC THEN ASM_SIMP_TAC[IN_ELIM_THM; NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM; REAL_ABS_MUL; REAL_ARITH `&0 < r ==> abs r = r`; REAL_RING `(r * x) * y = r <=> r = &0 \/ x * y = &1`; REAL_LT_IMP_NZ; REAL_FIELD `inv x * x = &1 <=> ~(x = &0)`; NORM_EQ_0] THEN ASM_CASES_TAC `y:real^N = vec 0` THEN ASM_SIMP_TAC[GSYM REAL_MUL_ASSOC; REAL_MUL_LINV; REAL_MUL_RID; NORM_EQ_0] THEN DISCH_THEN(K ALL_TAC) THEN DISCH_THEN(MP_TAC o MATCH_MP (MESON[DIST_LE_PASTECART; REAL_LET_TRANS] `dist(pastecart a b,pastecart c d) < r ==> dist(b,d) < r`)) THEN REWRITE_TAC[DIST_0; VECTOR_MUL_RZERO] THEN REWRITE_TAC[NORM_MUL; REAL_ABS_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN ASM_SIMP_TAC[GSYM REAL_MUL_ASSOC; REAL_MUL_LINV; NORM_EQ_0] THEN ASM_REAL_ARITH_TAC]]);; let HOMOTOPIC_WITH_ORTHOGONAL_TRANSFORMATIONS_ALT = prove (`!f g:real^N->real^N. homotopic_with orthogonal_transformation (subtopology euclidean ((:real^N) DELETE vec 0), subtopology euclidean ((:real^N) DELETE vec 0)) f g <=> orthogonal_transformation f /\ orthogonal_transformation g /\ det(matrix f) = det(matrix g)`, REPEAT GEN_TAC THEN REWRITE_TAC[GSYM NORM_EQ_0; SET_RULE `UNIV DELETE a = {x | ~(x = a)}`] THEN MATCH_MP_TAC HOMOTOPIC_WITH_ORTHOGONAL_TRANSFORMATIONS_GEN THEN EXISTS_TAC `&1` THEN CONV_TAC REAL_RAT_REDUCE_CONV);; let HOMOTOPIC_WITH_ORTHOGONAL_TRANSFORMATIONS_UNIV = prove (`!P f g:real^N->real^N. homotopic_with orthogonal_transformation (subtopology euclidean (:real^N), subtopology euclidean (:real^N)) f g <=> orthogonal_transformation f /\ orthogonal_transformation g /\ det(matrix f) = det(matrix g)`, REPEAT GEN_TAC THEN REWRITE_TAC[GSYM NORM_EQ_0; SET_RULE `UNIV = {x | T}`] THEN MATCH_MP_TAC HOMOTOPIC_WITH_ORTHOGONAL_TRANSFORMATIONS_GEN THEN EXISTS_TAC `&1` THEN CONV_TAC REAL_RAT_REDUCE_CONV);; let HOMOTOPIC_WITH_LINEAR_POSITIVE_DEFINITE_MAPS = prove (`!f g. homotopic_with (\f. linear f /\ positive_definite(matrix f)) (subtopology euclidean ((:real^N) DELETE vec 0), subtopology euclidean ((:real^N) DELETE vec 0)) f g <=> linear f /\ linear g /\ positive_definite(matrix f) /\ positive_definite(matrix g)`, REPEAT GEN_TAC THEN EQ_TAC THEN STRIP_TAC THENL [FIRST_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_PROPERTY) THEN SIMP_TAC[]; REWRITE_TAC[HOMOTOPIC_WITH_EUCLIDEAN]] THEN EXISTS_TAC `\z. (&1 - drop(fstcart z)) % (f:real^N->real^N) (sndcart z) + drop(fstcart z) % (g:real^N->real^N) (sndcart z)` THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; DROP_VEC; SUBSET; FORALL_IN_IMAGE; FORALL_PASTECART; PASTECART_IN_PCROSS; IN_UNIV; IN_DELETE; REAL_SUB_RZERO; REAL_SUB_REFL; VECTOR_MUL_LID; VECTOR_MUL_LZERO; VECTOR_ADD_LID; VECTOR_ADD_RID] THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_ADD THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN SIMP_TAC[o_DEF; LIFT_SUB; LIFT_DROP; CONTINUOUS_ON_SUB; LINEAR_FSTCART; ETA_AX; LINEAR_SNDCART; CONTINUOUS_ON_CONST; LINEAR_CONTINUOUS_ON] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; LINEAR_SNDCART]; ALL_TAC] THEN MATCH_MP_TAC(TAUT `q /\ (q ==> p) ==> p /\ q`) THEN CONJ_TAC THENL [REWRITE_TAC[IN_INTERVAL_1; GSYM FORALL_DROP; DROP_VEC] THEN ASM_SIMP_TAC[LINEAR_COMPOSE_ADD; MATRIX_ADD; LINEAR_COMPOSE_CMUL; MATRIX_CMUL] THEN X_GEN_TAC `t:real` THEN STRIP_TAC THEN ASM_CASES_TAC `t = &0` THEN ASM_REWRITE_TAC[REAL_SUB_RZERO; MATRIX_CMUL_LZERO; MATRIX_ADD_RID; MATRIX_CMUL_LID] THEN ASM_CASES_TAC `t = &1` THEN ASM_REWRITE_TAC[REAL_SUB_REFL; MATRIX_CMUL_LZERO; MATRIX_ADD_LID; MATRIX_CMUL_LID] THEN MATCH_MP_TAC POSITIVE_DEFINITE_ADD THEN CONJ_TAC THEN MATCH_MP_TAC POSITIVE_DEFINITE_CMUL THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[FORALL_LIFT; LIFT_DROP] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`t:real`; `x:real^N`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `t:real`) THEN ASM_REWRITE_TAC [POSITIVE_DEFINITE_EIGENVALUES] THEN DISCH_THEN(MP_TAC o SPECL [`&0`; `x:real^N`] o last o CONJUNCTS) THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO; REAL_LT_REFL; CONTRAPOS_THM] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN MATCH_MP_TAC(REWRITE_RULE[RIGHT_IMP_FORALL_THM] MATRIX_WORKS) THEN ASM_SIMP_TAC[LINEAR_COMPOSE_ADD; LINEAR_COMPOSE_CMUL]]);; let HOMOTOPIC_WITH_LINEAR_MAPS = prove (`!f g:real^N->real^N. homotopic_with linear (subtopology euclidean ((:real^N) DELETE vec 0), subtopology euclidean ((:real^N) DELETE vec 0)) f g <=> linear f /\ linear g /\ &0 < det(matrix f) * det(matrix g)`, let lemma = prove (`!f:real^N->real^N. linear f /\ ~(det(matrix f) = &0) ==> ?f' P. orthogonal_transformation f' /\ positive_definite P /\ f = f' o (\x. P ** x)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM INVERTIBLE_DET_NZ]) THEN REWRITE_TAC[RIGHT_POLAR_DECOMPOSITION_INVERTIBLE] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `S:real^N^N` THEN DISCH_THEN(X_CHOOSE_THEN `P:real^N^N` (STRIP_ASSUME_TAC o GSYM)) THEN EXISTS_TAC `(\x. P ** x):real^N->real^N` THEN ASM_REWRITE_TAC[GSYM ORTHOGONAL_MATRIX_TRANSFORMATION] THEN FIRST_ASSUM(ASSUME_TAC o GSYM o MATCH_MP MATRIX_WORKS) THEN ASM_REWRITE_TAC[FUN_EQ_THM; o_THM; GSYM MATRIX_VECTOR_MUL_ASSOC]) in REPEAT GEN_TAC THEN EQ_TAC THENL [DISCH_TAC THEN FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_PROPERTY) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMOTOPIC_WITH_EUCLIDEAN]) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `h:real^(1,N)finite_sum->real^N` THEN STRIP_TAC THEN SUBGOAL_THEN `(\t. lift(det(matrix((h:real^(1,N)finite_sum->real^N) o pastecart t)))) continuous_on interval[vec 0,vec 1]` MP_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_LIFT_DET THEN SIMP_TAC[matrix; LAMBDA_BETA; o_THM] THEN MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_LIFT_COMPONENT_COMPOSE THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_PASTECART; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE; PASTECART_IN_PCROSS; IN_UNIV; IN_DELETE; BASIS_NONZERO]; DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONNECTED_CONTINUOUS_IMAGE)) THEN REWRITE_TAC[CONNECTED_INTERVAL] THEN REWRITE_TAC[GSYM CONVEX_CONNECTED_1; CONVEX_CONTAINS_SEGMENT] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; o_DEF] THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN DISCH_THEN(MP_TAC o SPEC `vec 0:real^1`) THEN REWRITE_TAC[ENDS_IN_UNIT_INTERVAL] THEN DISCH_THEN(MP_TAC o SPEC `vec 1:real^1`) THEN ASM_REWRITE_TAC[ENDS_IN_UNIT_INTERVAL; ETA_AX] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[REAL_MUL_POS_LT] THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `~(&0 < x /\ &0 < y \/ x < &0 /\ y < &0) ==> abs(x - y) = abs(x - &0) + abs(&0 - y)`)) THEN REWRITE_TAC[GSYM DIST_LIFT; LIFT_NUM; GSYM between] THEN REWRITE_TAC[BETWEEN_IN_SEGMENT] THEN MATCH_MP_TAC(SET_RULE `~(z IN t) ==> z IN s ==> ~(s SUBSET t)`) THEN REWRITE_TAC[IN_IMAGE; GSYM LIFT_NUM; LIFT_EQ] THEN DISCH_THEN(X_CHOOSE_THEN `t:real^1` (STRIP_ASSUME_TAC o GSYM)) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN REWRITE_TAC[FORALL_IN_IMAGE; FORALL_IN_PCROSS] THEN DISCH_THEN(MP_TAC o SPEC `t:real^1`) THEN ASM_REWRITE_TAC[GSYM LIFT_NUM; IN_DELETE; IN_UNIV; CONTRAPOS_THM] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM HOMOGENEOUS_LINEAR_EQUATIONS_DET]) THEN ASM_SIMP_TAC[MATRIX_WORKS; GSYM LIFT_NUM] THEN MESON_TAC[]]; REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN ASM_CASES_TAC `det(matrix f:real^N^N) = &0` THEN ASM_REWRITE_TAC[REAL_MUL_LZERO; REAL_LT_REFL] THEN ASM_CASES_TAC `det(matrix g:real^N^N) = &0` THEN ASM_REWRITE_TAC[REAL_MUL_RZERO; REAL_LT_REFL] THEN MP_TAC(ISPEC `g:real^N->real^N` lemma) THEN MP_TAC(ISPEC `f:real^N->real^N` lemma) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`h:real^N->real^N`; `P:real^N^N`] THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`k:real^N->real^N`; `Q:real^N^N`] THEN STRIP_TAC THEN ASM_SIMP_TAC[MATRIX_COMPOSE; MATRIX_VECTOR_MUL_LINEAR; ORTHOGONAL_TRANSFORMATION_LINEAR; DET_MUL; MATRIX_OF_MATRIX_VECTOR_MUL] THEN ONCE_REWRITE_TAC[REAL_ARITH `(a * b) * (c * d):real = (a * c) * b * d`] THEN ASM_SIMP_TAC[DET_POSITIVE_DEFINITE; REAL_LT_MUL; REAL_LT_MUL_EQ] THEN REWRITE_TAC[REAL_MUL_POS_LT] THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `&0 < x /\ &0 < y \/ x < &0 /\ y < &0 ==> (x = &1 \/ x = -- &1) /\ (y = &1 \/ y = -- &1) ==> x = y`)) THEN ASM_SIMP_TAC[DET_ORTHOGONAL_MATRIX; ORTHOGONAL_MATRIX_MATRIX] THEN DISCH_TAC THEN MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE THEN MAP_EVERY EXISTS_TAC [`\f:real^N->real^N. linear f /\ positive_definite(matrix f)`; `orthogonal_transformation:(real^N->real^N)->bool`; `(:real^N) DELETE vec 0`] THEN ASM_REWRITE_TAC[HOMOTOPIC_WITH_ORTHOGONAL_TRANSFORMATIONS_ALT] THEN SIMP_TAC[LINEAR_COMPOSE; ORTHOGONAL_TRANSFORMATION_LINEAR] THEN REWRITE_TAC[HOMOTOPIC_WITH_LINEAR_POSITIVE_DEFINITE_MAPS] THEN REWRITE_TAC[MATRIX_VECTOR_MUL_LINEAR] THEN ASM_REWRITE_TAC[MATRIX_OF_MATRIX_VECTOR_MUL]]);; (* ------------------------------------------------------------------------- *) (* Homotopy of paths, maintaining the same endpoints. *) (* ------------------------------------------------------------------------- *) let homotopic_paths = new_definition `homotopic_paths s p q = homotopic_with (\r. pathstart r = pathstart p /\ pathfinish r = pathfinish p) (subtopology euclidean (interval[vec 0:real^1,vec 1]), subtopology euclidean s) p q`;; let HOMOTOPIC_PATHS = prove (`!s p q:real^1->real^N. homotopic_paths s p q <=> ?h. h continuous_on interval[vec 0,vec 1] PCROSS interval[vec 0,vec 1] /\ IMAGE h (interval[vec 0,vec 1] PCROSS interval[vec 0,vec 1]) SUBSET s /\ (!x. x IN interval[vec 0,vec 1] ==> h(pastecart (vec 0) x) = p x) /\ (!x. x IN interval[vec 0,vec 1] ==> h(pastecart (vec 1) x) = q x) /\ (!t. t IN interval[vec 0:real^1,vec 1] ==> pathstart(h o pastecart t) = pathstart p /\ pathfinish(h o pastecart t) = pathfinish p)`, REPEAT GEN_TAC THEN REWRITE_TAC[homotopic_paths] THEN W(MP_TAC o PART_MATCH (lhand o rand) HOMOTOPIC_WITH_EUCLIDEAN_ALT o lhand o snd) THEN ANTS_TAC THENL [SIMP_TAC[pathstart; pathfinish; ENDS_IN_UNIT_INTERVAL]; DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[o_DEF]]);; let HOMOTOPIC_PATHS_IMP_PATHSTART = prove (`!s p q. homotopic_paths s p q ==> pathstart p = pathstart q`, REPEAT GEN_TAC THEN REWRITE_TAC[homotopic_paths] THEN DISCH_THEN(MP_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_PROPERTY) THEN SIMP_TAC[]);; let HOMOTOPIC_PATHS_IMP_PATHFINISH = prove (`!s p q. homotopic_paths s p q ==> pathfinish p = pathfinish q`, REPEAT GEN_TAC THEN REWRITE_TAC[homotopic_paths] THEN DISCH_THEN(MP_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_PROPERTY) THEN SIMP_TAC[]);; let HOMOTOPIC_PATHS_IMP_PATH = prove (`!s p q. homotopic_paths s p q ==> path p /\ path q`, REPEAT GEN_TAC THEN REWRITE_TAC[homotopic_paths] THEN DISCH_THEN(MP_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_CONTINUOUS) THEN SIMP_TAC[path]);; let HOMOTOPIC_PATHS_IMP_SUBSET = prove (`!s p q. homotopic_paths s p q ==> path_image p SUBSET s /\ path_image q SUBSET s`, REPEAT GEN_TAC THEN REWRITE_TAC[homotopic_paths] THEN DISCH_THEN(MP_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_SUBSET) THEN SIMP_TAC[path_image]);; let HOMOTOPIC_PATHS_REFL = prove (`!s p. homotopic_paths s p p <=> path p /\ path_image p SUBSET s`, REWRITE_TAC[homotopic_paths; HOMOTOPIC_WITH_REFL; CONTINUOUS_MAP_EUCLIDEAN2; path; path_image]);; let HOMOTOPIC_PATHS_SYM = prove (`!s p q. homotopic_paths s p q <=> homotopic_paths s q p`, REPEAT GEN_TAC THEN EQ_TAC THEN DISCH_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_PATHSTART) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_PATHFINISH) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homotopic_paths]) THEN ONCE_REWRITE_TAC[HOMOTOPIC_WITH_SYM] THEN ASM_SIMP_TAC[homotopic_paths]);; let HOMOTOPIC_PATHS_TRANS = prove (`!s p q r. homotopic_paths s p q /\ homotopic_paths s q r ==> homotopic_paths s p r`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(CONJUNCTS_THEN (fun th -> ASSUME_TAC(MATCH_MP HOMOTOPIC_PATHS_IMP_PATHSTART th) THEN ASSUME_TAC(MATCH_MP HOMOTOPIC_PATHS_IMP_PATHFINISH th))) THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE BINOP_CONV [homotopic_paths]) THEN ASM_REWRITE_TAC[HOMOTOPIC_WITH_TRANS; homotopic_paths]);; let HOMOTOPIC_PATHS_EQ = prove (`!p:real^1->real^N q s. path p /\ path_image p SUBSET s /\ (!t. t IN interval[vec 0,vec 1] ==> p(t) = q(t)) ==> homotopic_paths s p q`, REPEAT STRIP_TAC THEN REWRITE_TAC[homotopic_paths] THEN MATCH_MP_TAC HOMOTOPIC_WITH_EQ THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN REPEAT(EXISTS_TAC `p:real^1->real^N`) THEN ASM_SIMP_TAC[HOMOTOPIC_WITH_REFL; CONTINUOUS_MAP_EUCLIDEAN2] THEN ASM_REWRITE_TAC[GSYM path; GSYM path_image] THEN REWRITE_TAC[pathstart; pathfinish] THEN MESON_TAC[ENDS_IN_UNIT_INTERVAL]);; let HOMOTOPIC_PATHS_REPARAMETRIZE = prove (`!p:real^1->real^N q. path p /\ path_image p SUBSET s /\ (?f. f continuous_on interval[vec 0,vec 1] /\ IMAGE f (interval[vec 0,vec 1]) SUBSET interval[vec 0,vec 1] /\ f(vec 0) = vec 0 /\ f(vec 1) = vec 1 /\ !t. t IN interval[vec 0,vec 1] ==> q(t) = p(f t)) ==> homotopic_paths s p q`, REWRITE_TAC[path; path_image] THEN REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[HOMOTOPIC_PATHS_SYM] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_TRANS THEN EXISTS_TAC `(p:real^1->real^N) o (f:real^1->real^1)` THEN CONJ_TAC THENL [MATCH_MP_TAC HOMOTOPIC_PATHS_EQ THEN ASM_SIMP_TAC[o_THM; pathstart; pathfinish; o_THM; IN_INTERVAL_1; DROP_VEC; REAL_POS; REAL_LE_REFL] THEN REWRITE_TAC[path; path_image] THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_EQ THEN EXISTS_TAC `(p:real^1->real^N) o (f:real^1->real^1)` THEN ASM_SIMP_TAC[o_THM] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ASM SET_TAC[]]; REWRITE_TAC[homotopic_paths; HOMOTOPIC_WITH_EUCLIDEAN; PCROSS] THEN EXISTS_TAC `(p:real^1->real^N) o (\y. (&1 - drop(fstcart y)) % f(sndcart y) + drop(fstcart y) % sndcart y)` THEN ASM_REWRITE_TAC[o_THM; FSTCART_PASTECART; SNDCART_PASTECART; DROP_VEC; pathstart; pathfinish] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_MUL_RZERO; VECTOR_ADD_LID; VECTOR_MUL_LID; VECTOR_ADD_RID] THEN REWRITE_TAC[VECTOR_ARITH `(&1 - u) % x + u % x:real^N = x`] THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_ADD THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN REWRITE_TAC[o_DEF; LIFT_DROP; ETA_AX; LIFT_SUB] THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; CONTINUOUS_ON_CONST; LINEAR_FSTCART; LINEAR_SNDCART; CONTINUOUS_ON_SUB] THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_COMPOSE) THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_SNDCART] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN SIMP_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC; SNDCART_PASTECART]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET))]; ONCE_REWRITE_TAC[IMAGE_o] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE p i SUBSET s ==> IMAGE f x SUBSET i ==> IMAGE p (IMAGE f x) SUBSET s`))] THEN SIMP_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC; SNDCART_PASTECART; FSTCART_PASTECART] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC(REWRITE_RULE[CONVEX_ALT] (CONJUNCT1(SPEC_ALL CONVEX_INTERVAL))) THEN ASM_MESON_TAC[IN_INTERVAL_1; DROP_VEC; SUBSET; IN_IMAGE]]);; let HOMOTOPIC_PATHS_SUBSET = prove (`!s p q. homotopic_paths s p q /\ s SUBSET t ==> homotopic_paths t p q`, REWRITE_TAC[homotopic_paths; HOMOTOPIC_WITH_SUBSET_RIGHT]);; (* ------------------------------------------------------------------------- *) (* A slightly ad-hoc but useful lemma in constructing homotopies. *) (* ------------------------------------------------------------------------- *) let HOMOTOPIC_JOIN_LEMMA = prove (`!p q:real^1->real^1->real^N. (\y. p (fstcart y) (sndcart y)) continuous_on (interval[vec 0,vec 1] PCROSS interval[vec 0,vec 1]) /\ (\y. q (fstcart y) (sndcart y)) continuous_on (interval[vec 0,vec 1] PCROSS interval[vec 0,vec 1]) /\ (!t. t IN interval[vec 0,vec 1] ==> pathfinish(p t) = pathstart(q t)) ==> (\y. (p(fstcart y) ++ q(fstcart y)) (sndcart y)) continuous_on (interval[vec 0,vec 1] PCROSS interval[vec 0,vec 1])`, REWRITE_TAC[joinpaths; PCROSS] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_LE THEN REPEAT CONJ_TAC THENL [SUBGOAL_THEN `(\y. p (fstcart y) (&2 % sndcart y)):real^(1,1)finite_sum->real^N = (\y. p (fstcart y) (sndcart y)) o (\y. pastecart (fstcart y) (&2 % sndcart y))` SUBST1_TAC THENL [REWRITE_TAC[o_DEF; FSTCART_PASTECART; SNDCART_PASTECART]; ALL_TAC]; SUBGOAL_THEN `(\y. q (fstcart y) (&2 % sndcart y - vec 1)):real^(1,1)finite_sum->real^N = (\y. q (fstcart y) (sndcart y)) o (\y. pastecart (fstcart y) (&2 % sndcart y - vec 1))` SUBST1_TAC THENL [REWRITE_TAC[o_DEF; FSTCART_PASTECART; SNDCART_PASTECART]; ALL_TAC]; SIMP_TAC[o_DEF; LIFT_DROP; LINEAR_CONTINUOUS_ON; LINEAR_SNDCART; ETA_AX]; SIMP_TAC[IMP_CONJ; FORALL_IN_GSPEC; FSTCART_PASTECART; SNDCART_PASTECART; GSYM LIFT_EQ; LIFT_DROP; GSYM LIFT_CMUL] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish]) THEN ASM_SIMP_TAC[LIFT_NUM; VECTOR_SUB_REFL]] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN (CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_PASTECART; ALL_TAC]) THEN SIMP_TAC[CONTINUOUS_ON_CMUL; LINEAR_CONTINUOUS_ON; CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; LINEAR_FSTCART; LINEAR_SNDCART] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC; IMP_CONJ] THEN SIMP_TAC[IN_ELIM_PASTECART_THM; FSTCART_PASTECART; SNDCART_PASTECART] THEN REWRITE_TAC[IN_INTERVAL_1; DROP_CMUL; DROP_SUB; DROP_VEC] THEN REAL_ARITH_TAC);; (* ------------------------------------------------------------------------- *) (* Congruence properties of homotopy w.r.t. path-combining operations. *) (* ------------------------------------------------------------------------- *) let HOMOTOPIC_PATHS_REVERSEPATH = prove (`!s p q:real^1->real^N. homotopic_paths s (reversepath p) (reversepath q) <=> homotopic_paths s p q`, GEN_TAC THEN MATCH_MP_TAC(MESON[] `(!p. f(f p) = p) /\ (!a b. homotopic_paths s a b ==> homotopic_paths s (f a) (f b)) ==> !a b. homotopic_paths s (f a) (f b) <=> homotopic_paths s a b`) THEN REWRITE_TAC[REVERSEPATH_REVERSEPATH] THEN REPEAT GEN_TAC THEN REWRITE_TAC[homotopic_paths; HOMOTOPIC_WITH_EUCLIDEAN; PCROSS; o_DEF] THEN DISCH_THEN(X_CHOOSE_THEN `h:real^(1,1)finite_sum->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\y:real^(1,1)finite_sum. (h:real^(1,1)finite_sum->real^N) (pastecart(fstcart y) (vec 1 - sndcart y))` THEN ASM_REWRITE_TAC[o_DEF; FSTCART_PASTECART; SNDCART_PASTECART] THEN RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish]) THEN ASM_SIMP_TAC[reversepath; pathstart; pathfinish; VECTOR_SUB_REFL; VECTOR_SUB_RZERO] THEN CONJ_TAC THENL [GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_PASTECART THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; LINEAR_SNDCART; CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN SIMP_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC; IN_ELIM_PASTECART_THM; FSTCART_PASTECART; SNDCART_PASTECART] THEN REWRITE_TAC[IN_INTERVAL_1; DROP_SUB; DROP_VEC] THEN REAL_ARITH_TAC]; GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM o_DEF] THEN REWRITE_TAC[IMAGE_o] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE h s SUBSET t ==> IMAGE g s SUBSET s ==> IMAGE h (IMAGE g s) SUBSET t`)) THEN SIMP_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC; IN_ELIM_PASTECART_THM; FSTCART_PASTECART; SNDCART_PASTECART] THEN REWRITE_TAC[IN_INTERVAL_1; DROP_SUB; DROP_VEC] THEN REAL_ARITH_TAC]);; let HOMOTOPIC_PATHS_JOIN = prove (`!s p q p' q':real^1->real^N. homotopic_paths s p p' /\ homotopic_paths s q q' /\ pathfinish p = pathstart q ==> homotopic_paths s (p ++ q) (p' ++ q')`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN REWRITE_TAC[homotopic_paths; HOMOTOPIC_WITH_EUCLIDEAN; PCROSS] THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `k1:real^(1,1)finite_sum->real^N` STRIP_ASSUME_TAC) (X_CHOOSE_THEN `k2:real^(1,1)finite_sum->real^N` STRIP_ASSUME_TAC)) THEN EXISTS_TAC `(\y. ((k1 o pastecart (fstcart y)) ++ (k2 o pastecart (fstcart y))) (sndcart y)) :real^(1,1)finite_sum->real^N` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC(REWRITE_RULE[PCROSS] HOMOTOPIC_JOIN_LEMMA) THEN ASM_REWRITE_TAC[o_DEF; PASTECART_FST_SND; ETA_AX] THEN RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish]) THEN ASM_REWRITE_TAC[pathstart; pathfinish] THEN ASM_MESON_TAC[]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[ETA_AX; GSYM path_image; SET_RULE `(!x. x IN i ==> f x IN s) <=> IMAGE f i SUBSET s`] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_PATH_IMAGE_JOIN THEN REWRITE_TAC[path_image; SUBSET; FORALL_IN_IMAGE; o_DEF] THEN ASM SET_TAC[]; ALL_TAC; ALL_TAC; ALL_TAC] THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN ASM_REWRITE_TAC[joinpaths; o_DEF] THEN RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish]) THEN REWRITE_TAC[pathstart; pathfinish; DROP_VEC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_SIMP_TAC[VECTOR_ARITH `&2 % x - x:real^N = x`; VECTOR_MUL_RZERO]);; let HOMOTOPIC_PATHS_CONTINUOUS_IMAGE = prove (`!f:real^1->real^M g h:real^M->real^N s t. homotopic_paths s f g /\ h continuous_on s /\ IMAGE h s SUBSET t ==> homotopic_paths t (h o f) (h o g)`, REWRITE_TAC[homotopic_paths] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_LEFT THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMOTOPIC_WITH_MONO)) THEN SIMP_TAC[pathstart; pathfinish; o_THM]);; (* ------------------------------------------------------------------------- *) (* Group properties for homotopy of paths (so taking equivalence classes *) (* under homotopy would give the fundamental group). *) (* ------------------------------------------------------------------------- *) let HOMOTOPIC_PATHS_RID = prove (`!s p. path p /\ path_image p SUBSET s ==> homotopic_paths s (p ++ linepath(pathfinish p,pathfinish p)) p`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[HOMOTOPIC_PATHS_SYM] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_REPARAMETRIZE THEN ASM_REWRITE_TAC[joinpaths] THEN EXISTS_TAC `\t. if drop t <= &1 / &2 then &2 % t else vec 1` THEN ASM_REWRITE_TAC[DROP_VEC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[VECTOR_MUL_RZERO; linepath; pathfinish; VECTOR_ARITH `(&1 - t) % x + t % x:real^N = x`] THEN REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; MESON_TAC[]] THEN CONJ_TAC THENL [SUBGOAL_THEN `interval[vec 0:real^1,vec 1] = interval[vec 0,lift(&1 / &2)] UNION interval[lift(&1 / &2),vec 1]` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; IN_UNION; IN_INTERVAL_1; LIFT_DROP; DROP_VEC] THEN REAL_ARITH_TAC; MATCH_MP_TAC CONTINUOUS_ON_CASES THEN SIMP_TAC[CLOSED_INTERVAL; CONTINUOUS_ON_CMUL; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST; IN_INTERVAL_1; DROP_VEC; LIFT_DROP; GSYM DROP_EQ; DROP_CMUL] THEN REAL_ARITH_TAC]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTERVAL_1; DROP_VEC] THEN GEN_TAC THEN COND_CASES_TAC THEN REWRITE_TAC[DROP_CMUL; DROP_VEC] THEN ASM_REAL_ARITH_TAC]);; let HOMOTOPIC_PATHS_LID = prove (`!s p:real^1->real^N. path p /\ path_image p SUBSET s ==> homotopic_paths s (linepath(pathstart p,pathstart p) ++ p) p`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM HOMOTOPIC_PATHS_REVERSEPATH] THEN REWRITE_TAC[o_DEF; PATHSTART_REVERSEPATH; PATHFINISH_REVERSEPATH] THEN SIMP_TAC[REVERSEPATH_JOINPATHS; REVERSEPATH_LINEPATH; PATHFINISH_LINEPATH] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN MP_TAC(ISPECL [`s:real^N->bool`; `reversepath p :real^1->real^N`] HOMOTOPIC_PATHS_RID) THEN ASM_SIMP_TAC[PATH_REVERSEPATH; PATH_IMAGE_REVERSEPATH; PATHSTART_REVERSEPATH; PATHFINISH_REVERSEPATH]);; let HOMOTOPIC_PATHS_ASSOC = prove (`!s p q r:real^1->real^N. path p /\ path_image p SUBSET s /\ path q /\ path_image q SUBSET s /\ path r /\ path_image r SUBSET s /\ pathfinish p = pathstart q /\ pathfinish q = pathstart r ==> homotopic_paths s (p ++ (q ++ r)) ((p ++ q) ++ r)`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[HOMOTOPIC_PATHS_SYM] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_REPARAMETRIZE THEN ASM_SIMP_TAC[PATH_JOIN; PATH_IMAGE_JOIN; UNION_SUBSET; PATHSTART_JOIN; PATHFINISH_JOIN] THEN REWRITE_TAC[joinpaths] THEN EXISTS_TAC `\t. if drop t <= &1 / &2 then inv(&2) % t else if drop t <= &3 / &4 then t - lift(&1 / &4) else &2 % t - vec 1` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_CASES_1 THEN SIMP_TAC[CONTINUOUS_ON_CMUL; CONTINUOUS_ON_ID; LIFT_DROP] THEN REWRITE_TAC[GSYM LIFT_SUB; GSYM LIFT_CMUL] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_1 THEN SIMP_TAC[CONTINUOUS_ON_CMUL; CONTINUOUS_ON_SUB; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST] THEN REWRITE_TAC[GSYM LIFT_SUB; GSYM LIFT_CMUL; GSYM LIFT_NUM] THEN CONV_TAC REAL_RAT_REDUCE_CONV; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTERVAL_1; DROP_VEC] THEN REPEAT STRIP_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN REWRITE_TAC[DROP_CMUL; DROP_VEC; LIFT_DROP; DROP_SUB] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[DROP_VEC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[VECTOR_MUL_RZERO]; REWRITE_TAC[DROP_VEC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN VECTOR_ARITH_TAC; X_GEN_TAC `t:real^1` THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN STRIP_TAC THEN ASM_CASES_TAC `drop t <= &1 / &2` THEN ASM_REWRITE_TAC[DROP_CMUL] THEN ASM_REWRITE_TAC[REAL_ARITH `inv(&2) * t <= &1 / &2 <=> t <= &1`] THEN REWRITE_TAC[VECTOR_MUL_ASSOC; REAL_MUL_ASSOC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[REAL_MUL_LID] THEN ASM_CASES_TAC `drop t <= &3 / &4` THEN ASM_REWRITE_TAC[DROP_SUB; DROP_VEC; DROP_CMUL; LIFT_DROP; REAL_ARITH `&2 * (t - &1 / &4) <= &1 / &2 <=> t <= &1 / &2`; REAL_ARITH `&2 * t - &1 <= &1 / &2 <=> t <= &3 / &4`; REAL_ARITH `t - &1 / &4 <= &1 / &2 <=> t <= &3 / &4`] THEN REWRITE_TAC[VECTOR_SUB_LDISTRIB; VECTOR_MUL_ASSOC; GSYM LIFT_CMUL] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[LIFT_NUM] THEN REWRITE_TAC[VECTOR_ARITH `a - b - b:real^N = a - &2 % b`]]);; let HOMOTOPIC_PATHS_RINV = prove (`!s p:real^1->real^N. path p /\ path_image p SUBSET s ==> homotopic_paths s (p ++ reversepath p) (linepath(pathstart p,pathstart p))`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[HOMOTOPIC_PATHS_SYM] THEN REWRITE_TAC[homotopic_paths; HOMOTOPIC_WITH_EUCLIDEAN; PCROSS] THEN EXISTS_TAC `(\y. (subpath (vec 0) (fstcart y) p ++ reversepath(subpath (vec 0) (fstcart y) p)) (sndcart y)) : real^(1,1)finite_sum->real^N` THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; SUBPATH_TRIVIAL] THEN REWRITE_TAC[ETA_AX; PATHSTART_JOIN; PATHFINISH_JOIN] THEN REWRITE_TAC[REVERSEPATH_SUBPATH; PATHSTART_SUBPATH; PATHFINISH_SUBPATH] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[joinpaths] THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_LE THEN RULE_ASSUM_TAC(REWRITE_RULE[path; path_image]) THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[subpath; VECTOR_ADD_LID; VECTOR_SUB_RZERO] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_MUL THEN REWRITE_TAC[o_DEF; LIFT_DROP; ETA_AX] THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; LINEAR_SNDCART; CONTINUOUS_ON_CMUL]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[FORALL_IN_IMAGE; SUBSET; FORALL_IN_GSPEC; IMP_CONJ] THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN REWRITE_TAC[IN_INTERVAL_1; DROP_CMUL; DROP_VEC] THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[REAL_LE_MUL; REAL_POS] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `drop x * &2 * &1 / &2` THEN CONJ_TAC THEN REPEAT(MATCH_MP_TAC REAL_LE_LMUL THEN CONJ_TAC) THEN ASM_REAL_ARITH_TAC]; REWRITE_TAC[subpath; VECTOR_ADD_LID; VECTOR_SUB_RZERO] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_ADD THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_FSTCART] THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN REWRITE_TAC[o_DEF; LIFT_DROP; ETA_AX] THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; LINEAR_SNDCART; CONTINUOUS_ON_CMUL; CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[FORALL_IN_IMAGE; SUBSET; FORALL_IN_GSPEC; IMP_CONJ] THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN REWRITE_TAC[IN_INTERVAL_1; DROP_SUB; DROP_CMUL; DROP_VEC; DROP_ADD; REAL_ARITH `t + (&0 - t) * (&2 * x - &1) = t * &2 * (&1 - x)`] THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[REAL_LE_MUL; REAL_POS; REAL_SUB_LE] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `drop x * &2 * &1 / &2` THEN CONJ_TAC THEN REPEAT(MATCH_MP_TAC REAL_LE_LMUL THEN CONJ_TAC) THEN ASM_REAL_ARITH_TAC]; SIMP_TAC[o_DEF; LIFT_DROP; ETA_AX; LINEAR_CONTINUOUS_ON; LINEAR_SNDCART]; REWRITE_TAC[GSYM LIFT_EQ; LIFT_DROP] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[subpath] THEN AP_TERM_TAC THEN REWRITE_TAC[GSYM DROP_EQ; DROP_SUB; DROP_VEC; DROP_ADD; DROP_CMUL; LIFT_DROP] THEN REAL_ARITH_TAC]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM; IMP_CONJ] THEN X_GEN_TAC `t:real^1` THEN DISCH_TAC THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; ETA_AX; SET_RULE `(!x. x IN s ==> f x IN t) <=> IMAGE f s SUBSET t`] THEN REWRITE_TAC[GSYM path_image] THEN MATCH_MP_TAC SUBSET_PATH_IMAGE_JOIN THEN REWRITE_TAC[PATH_IMAGE_SUBPATH_GEN] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE LAND_CONV [path_image]) THEN MATCH_MP_TAC(SET_RULE `t SUBSET s /\ u SUBSET s ==> IMAGE p s SUBSET v ==> IMAGE p t SUBSET v /\ IMAGE p u SUBSET v`) THEN REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN CONJ_TAC THEN MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[CONVEX_INTERVAL] THEN ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET] THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; REAL_POS; REAL_LE_REFL]; REWRITE_TAC[subpath; linepath; pathstart; joinpaths] THEN REWRITE_TAC[VECTOR_SUB_REFL; DROP_VEC; VECTOR_MUL_LZERO] THEN REWRITE_TAC[VECTOR_ADD_RID; COND_ID] THEN VECTOR_ARITH_TAC; REWRITE_TAC[pathstart; PATHFINISH_LINEPATH; PATHSTART_LINEPATH]]);; let HOMOTOPIC_PATHS_LINV = prove (`!s p:real^1->real^N. path p /\ path_image p SUBSET s ==> homotopic_paths s (reversepath p ++ p) (linepath(pathfinish p,pathfinish p))`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `reversepath p:real^1->real^N`] HOMOTOPIC_PATHS_RINV) THEN ASM_SIMP_TAC[PATH_REVERSEPATH; PATH_IMAGE_REVERSEPATH] THEN REWRITE_TAC[PATHSTART_REVERSEPATH; PATHFINISH_REVERSEPATH; REVERSEPATH_REVERSEPATH]);; let HOMOTOPIC_PATHS_LCANCEL = prove (`!p q r s:real^N->bool. homotopic_paths s (p ++ q) (p ++ r) /\ pathstart q = pathfinish p /\ pathstart r = pathfinish p ==> homotopic_paths s q r`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_PATH) THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_SUBSET) THEN ASM_SIMP_TAC[PATH_JOIN; PATH_IMAGE_JOIN; UNION_SUBSET] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `homotopic_paths (s:real^N->bool) (reversepath p ++ p ++ q) (reversepath p ++ p ++ r)` MP_TAC THENL [MATCH_MP_TAC HOMOTOPIC_PATHS_JOIN THEN ASM_REWRITE_TAC[PATHFINISH_REVERSEPATH; PATHSTART_JOIN] THEN ASM_REWRITE_TAC[HOMOTOPIC_PATHS_REFL; PATH_REVERSEPATH; PATH_IMAGE_REVERSEPATH]; MATCH_MP_TAC(MESON[HOMOTOPIC_PATHS_SYM; HOMOTOPIC_PATHS_TRANS] `homotopic_paths s p p' /\ homotopic_paths s q q' ==> homotopic_paths s p q ==> homotopic_paths s p' q'`) THEN CONJ_TAC THEN W(MP_TAC o PART_MATCH (rator o rand) HOMOTOPIC_PATHS_ASSOC o rator o snd) THEN ASM_REWRITE_TAC[PATH_REVERSEPATH; PATHFINISH_REVERSEPATH; PATH_IMAGE_REVERSEPATH] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_PATHS_TRANS) THEN MP_TAC(ISPEC `s:real^N->bool` HOMOTOPIC_PATHS_LID) THENL [DISCH_THEN(MP_TAC o SPEC `q:real^1->real^N`); DISCH_THEN(MP_TAC o SPEC `r:real^1->real^N`)] THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] HOMOTOPIC_PATHS_TRANS) THEN MATCH_MP_TAC HOMOTOPIC_PATHS_JOIN THEN ASM_REWRITE_TAC[HOMOTOPIC_PATHS_REFL; PATHFINISH_JOIN] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_LINV THEN ASM_REWRITE_TAC[]]);; let HOMOTOPIC_PATHS_LCANCEL_EQ = prove (`!p q r s:real^N->bool. pathstart q = pathfinish p /\ pathstart r = pathfinish p ==> (homotopic_paths s (p ++ q) (p ++ r) <=> path p /\ path_image p SUBSET s /\ homotopic_paths s q r)`, REPEAT(STRIP_TAC ORELSE EQ_TAC) THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_PATH) THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_SUBSET) THEN ASM_SIMP_TAC[PATH_JOIN; PATH_IMAGE_JOIN; UNION_SUBSET] THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[HOMOTOPIC_PATHS_JOIN; HOMOTOPIC_PATHS_REFL] THEN ASM_MESON_TAC[HOMOTOPIC_PATHS_LCANCEL]);; let HOMOTOPIC_PATHS_RCANCEL = prove (`!p q r s:real^N->bool. homotopic_paths s (p ++ r) (q ++ r) /\ pathfinish p = pathstart r /\ pathfinish q = pathstart r ==> homotopic_paths s p q`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_PATH) THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_SUBSET) THEN ASM_SIMP_TAC[PATH_JOIN; PATH_IMAGE_JOIN; UNION_SUBSET] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `homotopic_paths (s:real^N->bool) ((p ++ r) ++ reversepath r) ((q ++ r) ++ reversepath r)` MP_TAC THENL [MATCH_MP_TAC HOMOTOPIC_PATHS_JOIN THEN ASM_REWRITE_TAC[PATHSTART_REVERSEPATH; PATHFINISH_JOIN] THEN ASM_REWRITE_TAC[HOMOTOPIC_PATHS_REFL; PATH_REVERSEPATH; PATH_IMAGE_REVERSEPATH]; MATCH_MP_TAC(MESON[HOMOTOPIC_PATHS_SYM; HOMOTOPIC_PATHS_TRANS] `homotopic_paths s p p' /\ homotopic_paths s q q' ==> homotopic_paths s p q ==> homotopic_paths s p' q'`) THEN CONJ_TAC THEN W(MP_TAC o PART_MATCH (rator o rand) (ONCE_REWRITE_RULE[HOMOTOPIC_PATHS_SYM] HOMOTOPIC_PATHS_ASSOC) o rator o snd) THEN ASM_REWRITE_TAC[PATH_REVERSEPATH; PATHSTART_REVERSEPATH; PATH_IMAGE_REVERSEPATH] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_PATHS_TRANS) THEN MP_TAC(ISPEC `s:real^N->bool` HOMOTOPIC_PATHS_RID) THENL [DISCH_THEN(MP_TAC o SPEC `p:real^1->real^N`); DISCH_THEN(MP_TAC o SPEC `q:real^1->real^N`)] THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] HOMOTOPIC_PATHS_TRANS) THEN MATCH_MP_TAC HOMOTOPIC_PATHS_JOIN THEN ASM_REWRITE_TAC[HOMOTOPIC_PATHS_REFL; PATHSTART_JOIN] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_RINV THEN ASM_REWRITE_TAC[]]);; let HOMOTOPIC_PATHS_RCANCEL_EQ = prove (`!p q r s:real^N->bool. pathfinish p = pathstart r /\ pathfinish q = pathstart r ==> (homotopic_paths s (p ++ r) (q ++ r) <=> homotopic_paths s p q /\ path r /\ path_image r SUBSET s)`, REPEAT(STRIP_TAC ORELSE EQ_TAC) THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_PATH) THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_SUBSET) THEN ASM_SIMP_TAC[PATH_JOIN; PATH_IMAGE_JOIN; UNION_SUBSET] THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[HOMOTOPIC_PATHS_JOIN; HOMOTOPIC_PATHS_REFL] THEN ASM_MESON_TAC[HOMOTOPIC_PATHS_RCANCEL]);; (* ------------------------------------------------------------------------- *) (* Homotopy of loops without requiring preservation of endpoints. *) (* ------------------------------------------------------------------------- *) let homotopic_loops = new_definition `homotopic_loops s p q = homotopic_with (\r. pathfinish r = pathstart r) (subtopology euclidean (interval[vec 0:real^1,vec 1]), subtopology euclidean s) p q`;; let HOMOTOPIC_LOOPS = prove (`!s p q:real^1->real^N. homotopic_loops s p q <=> ?h. h continuous_on interval[vec 0,vec 1] PCROSS interval[vec 0,vec 1] /\ IMAGE h (interval[vec 0,vec 1] PCROSS interval[vec 0,vec 1]) SUBSET s /\ (!x. x IN interval[vec 0,vec 1] ==> h(pastecart (vec 0) x) = p x) /\ (!x. x IN interval[vec 0,vec 1] ==> h(pastecart (vec 1) x) = q x) /\ (!t. t IN interval[vec 0:real^1,vec 1] ==> pathfinish(h o pastecart t) = pathstart(h o pastecart t))`, REPEAT GEN_TAC THEN REWRITE_TAC[homotopic_loops] THEN W(MP_TAC o PART_MATCH (lhand o rand) HOMOTOPIC_WITH_EUCLIDEAN_ALT o lhand o snd) THEN ANTS_TAC THENL [SIMP_TAC[pathstart; pathfinish; ENDS_IN_UNIT_INTERVAL]; DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[o_DEF]]);; let HOMOTOPIC_LOOPS_IMP_LOOP = prove (`!s p q. homotopic_loops s p q ==> pathfinish p = pathstart p /\ pathfinish q = pathstart q`, REPEAT GEN_TAC THEN REWRITE_TAC[homotopic_loops] THEN DISCH_THEN(MP_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_PROPERTY) THEN SIMP_TAC[]);; let HOMOTOPIC_LOOPS_IMP_PATH = prove (`!s p q. homotopic_loops s p q ==> path p /\ path q`, REPEAT GEN_TAC THEN REWRITE_TAC[homotopic_loops] THEN DISCH_THEN(MP_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_CONTINUOUS) THEN SIMP_TAC[path]);; let HOMOTOPIC_LOOPS_IMP_SUBSET = prove (`!s p q. homotopic_loops s p q ==> path_image p SUBSET s /\ path_image q SUBSET s`, REPEAT GEN_TAC THEN REWRITE_TAC[homotopic_loops] THEN DISCH_THEN(MP_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_SUBSET) THEN SIMP_TAC[path_image]);; let HOMOTOPIC_LOOPS_REFL = prove (`!s p. homotopic_loops s p p <=> path p /\ path_image p SUBSET s /\ pathfinish p = pathstart p`, REWRITE_TAC[homotopic_loops; HOMOTOPIC_WITH_REFL; CONTINUOUS_MAP_EUCLIDEAN2; path; path_image] THEN CONV_TAC TAUT);; let HOMOTOPIC_LOOPS_SYM = prove (`!s p q. homotopic_loops s p q <=> homotopic_loops s q p`, REWRITE_TAC[homotopic_loops; HOMOTOPIC_WITH_SYM]);; let HOMOTOPIC_LOOPS_TRANS = prove (`!s p q r. homotopic_loops s p q /\ homotopic_loops s q r ==> homotopic_loops s p r`, REWRITE_TAC[homotopic_loops; HOMOTOPIC_WITH_TRANS]);; let HOMOTOPIC_LOOPS_SUBSET = prove (`!s p q. homotopic_loops s p q /\ s SUBSET t ==> homotopic_loops t p q`, REWRITE_TAC[homotopic_loops; HOMOTOPIC_WITH_SUBSET_RIGHT]);; let HOMOTOPIC_LOOPS_EQ = prove (`!p:real^1->real^N q s. path p /\ path_image p SUBSET s /\ pathfinish p = pathstart p /\ (!t. t IN interval[vec 0,vec 1] ==> p(t) = q(t)) ==> homotopic_loops s p q`, REPEAT STRIP_TAC THEN REWRITE_TAC[homotopic_loops] THEN MATCH_MP_TAC HOMOTOPIC_WITH_EQ THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN REPEAT(EXISTS_TAC `p:real^1->real^N`) THEN ASM_SIMP_TAC[HOMOTOPIC_WITH_REFL; CONTINUOUS_MAP_EUCLIDEAN2] THEN ASM_REWRITE_TAC[GSYM path; GSYM path_image] THEN REWRITE_TAC[pathstart; pathfinish] THEN MESON_TAC[ENDS_IN_UNIT_INTERVAL]);; let HOMOTOPIC_LOOPS_CONTINUOUS_IMAGE = prove (`!f:real^1->real^M g h:real^M->real^N s t. homotopic_loops s f g /\ h continuous_on s /\ IMAGE h s SUBSET t ==> homotopic_loops t (h o f) (h o g)`, REWRITE_TAC[homotopic_loops] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_LEFT THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMOTOPIC_WITH_MONO)) THEN SIMP_TAC[pathstart; pathfinish; o_THM]);; let HOMOTOPIC_LOOPS_SHIFTPATH_SELF = prove (`!p:real^1->real^N t s. path p /\ path_image p SUBSET s /\ pathfinish p = pathstart p /\ t IN interval[vec 0,vec 1] ==> homotopic_loops s p (shiftpath t p)`, REPEAT STRIP_TAC THEN REWRITE_TAC[HOMOTOPIC_LOOPS] THEN EXISTS_TAC `\z. shiftpath (drop t % fstcart z) (p:real^1->real^N) (sndcart z)` THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; o_DEF] THEN REWRITE_TAC[GSYM LIFT_EQ_CMUL; VECTOR_MUL_RZERO; ETA_AX] THEN REPEAT CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_PCROSS] THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN MATCH_MP_TAC(SET_RULE `IMAGE p t SUBSET u /\ (!x. x IN s ==> IMAGE(shiftpath (f x) p) t = IMAGE p t) ==> (!x y. x IN s /\ y IN t ==> shiftpath (f x) p y IN u)`) THEN ASM_REWRITE_TAC[GSYM path_image] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC PATH_IMAGE_SHIFTPATH THEN ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_CMUL; DROP_VEC] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN ASM_SIMP_TAC[REAL_LE_MUL] THEN GEN_REWRITE_TAC RAND_CONV [GSYM REAL_MUL_RID] THEN MATCH_MP_TAC REAL_LE_MUL2 THEN ASM_SIMP_TAC[]; SIMP_TAC[shiftpath; VECTOR_ADD_LID; IN_INTERVAL_1; DROP_VEC]; REWRITE_TAC[LIFT_DROP]; X_GEN_TAC `x:real^1` THEN STRIP_TAC THEN MATCH_MP_TAC CLOSED_SHIFTPATH THEN ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_CMUL; DROP_VEC] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN ASM_SIMP_TAC[REAL_LE_MUL] THEN GEN_REWRITE_TAC RAND_CONV [GSYM REAL_MUL_RID] THEN MATCH_MP_TAC REAL_LE_MUL2 THEN ASM_SIMP_TAC[]] THEN REWRITE_TAC[shiftpath; DROP_ADD; DROP_CMUL] THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_LE THEN REPEAT CONJ_TAC THENL [GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_MUL; o_DEF; LIFT_DROP; LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; LINEAR_SNDCART; CONTINUOUS_ON_CONST] THEN RULE_ASSUM_TAC(REWRITE_RULE[path]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_PASTECART] THEN REWRITE_TAC[IN_ELIM_THM; PASTECART_IN_PCROSS] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN ASM_SIMP_TAC[FSTCART_PASTECART; SNDCART_PASTECART; IN_INTERVAL_1; DROP_ADD; DROP_CMUL; DROP_VEC; REAL_LE_ADD; REAL_LE_MUL]; GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_MUL; o_DEF; LIFT_DROP; LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; LINEAR_SNDCART; CONTINUOUS_ON_CONST; CONTINUOUS_ON_SUB] THEN RULE_ASSUM_TAC(REWRITE_RULE[path]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_PASTECART] THEN REWRITE_TAC[IN_ELIM_THM; PASTECART_IN_PCROSS] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN ASM_SIMP_TAC[FSTCART_PASTECART; SNDCART_PASTECART; IN_INTERVAL_1; DROP_SUB; DROP_ADD; DROP_CMUL; DROP_VEC; REAL_LE_ADD; REAL_LE_MUL] THEN SIMP_TAC[REAL_ARITH `&0 <= x + y - &1 <=> &1 <= x + y`] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC(REAL_ARITH `t * x <= &1 * &1 /\ y <= &1 ==> t * x + y - &1 <= &1`) THEN ASM_SIMP_TAC[REAL_LE_MUL2; REAL_POS]; REWRITE_TAC[o_DEF; LIFT_ADD; LIFT_CMUL; LIFT_DROP] THEN SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_CMUL; LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; LINEAR_SNDCART]; SIMP_TAC[GSYM LIFT_EQ; LIFT_ADD; LIFT_CMUL; LIFT_DROP; LIFT_NUM; VECTOR_ARITH `a + b - c:real^1 = (a + b) - c`] THEN ASM_MESON_TAC[VECTOR_SUB_REFL; pathstart; pathfinish]]);; (* ------------------------------------------------------------------------- *) (* Relations between the two variants of homotopy. *) (* ------------------------------------------------------------------------- *) let HOMOTOPIC_PATHS_IMP_HOMOTOPIC_LOOPS = prove (`!s p q. homotopic_paths s p q /\ pathfinish p = pathstart p /\ pathfinish q = pathstart p ==> homotopic_loops s p q`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN REWRITE_TAC[homotopic_paths; homotopic_loops] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_WITH_MONO) THEN ASM_SIMP_TAC[]);; let HOMOTOPIC_LOOPS_IMP_HOMOTOPIC_PATHS_NULL = prove (`!s p a:real^N. homotopic_loops s p (linepath(a,a)) ==> homotopic_paths s p (linepath(pathstart p,pathstart p))`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o CONJUNCT1 o MATCH_MP HOMOTOPIC_LOOPS_IMP_LOOP) THEN FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP HOMOTOPIC_LOOPS_IMP_PATH) THEN FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP HOMOTOPIC_LOOPS_IMP_SUBSET) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homotopic_loops]) THEN REWRITE_TAC[HOMOTOPIC_WITH_EUCLIDEAN; PCROSS; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `h:real^(1,1)finite_sum->real^N` THEN STRIP_TAC THEN MATCH_MP_TAC HOMOTOPIC_PATHS_TRANS THEN EXISTS_TAC `(p:real^1->real^N) ++ linepath(pathfinish p,pathfinish p)` THEN CONJ_TAC THENL [ASM_MESON_TAC[HOMOTOPIC_PATHS_RID; HOMOTOPIC_PATHS_SYM]; ALL_TAC] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_TRANS THEN EXISTS_TAC `linepath(pathstart p,pathstart p) ++ (p:real^1->real^N) ++ linepath(pathfinish p,pathfinish p)` THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[HOMOTOPIC_PATHS_SYM] THEN MP_TAC(ISPECL [`s:real^N->bool`; `(p:real^1->real^N) ++ linepath(pathfinish p,pathfinish p)`] HOMOTOPIC_PATHS_LID) THEN REWRITE_TAC[PATHSTART_JOIN] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_SIMP_TAC[PATH_JOIN; PATH_LINEPATH; PATHSTART_LINEPATH] THEN MATCH_MP_TAC SUBSET_PATH_IMAGE_JOIN THEN ASM_REWRITE_TAC[PATH_IMAGE_LINEPATH; SEGMENT_REFL] THEN REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET] THEN ASM_MESON_TAC[PATHSTART_IN_PATH_IMAGE; SUBSET]; ALL_TAC] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_TRANS THEN EXISTS_TAC `((\u. (h:real^(1,1)finite_sum->real^N) (pastecart u (vec 0))) ++ linepath(a,a) ++ reversepath(\u. h (pastecart u (vec 0))))` THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC(MESON[HOMOTOPIC_PATHS_LID; HOMOTOPIC_PATHS_JOIN; HOMOTOPIC_PATHS_TRANS; HOMOTOPIC_PATHS_SYM; HOMOTOPIC_PATHS_RINV] `(path p /\ path(reversepath p)) /\ (path_image p SUBSET s /\ path_image(reversepath p) SUBSET s) /\ (pathfinish p = pathstart(linepath(b,b) ++ reversepath p) /\ pathstart(reversepath p) = b) /\ pathstart p = a ==> homotopic_paths s (p ++ linepath(b,b) ++ reversepath p) (linepath(a,a))`) THEN REWRITE_TAC[PATHSTART_REVERSEPATH; PATHSTART_JOIN; PATH_REVERSEPATH; PATH_IMAGE_REVERSEPATH; PATHSTART_LINEPATH] THEN ASM_REWRITE_TAC[path; path_image; pathstart; pathfinish; LINEPATH_REFL] THEN CONJ_TAC THENL [GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_PASTECART; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_PASTECART_THM; ENDS_IN_UNIT_INTERVAL]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS)) THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM o_DEF] THEN REWRITE_TAC[IMAGE_o] THEN MATCH_MP_TAC IMAGE_SUBSET THEN SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_PASTECART_THM; ENDS_IN_UNIT_INTERVAL]]] THEN REWRITE_TAC[homotopic_paths; HOMOTOPIC_WITH_EUCLIDEAN; PCROSS] THEN EXISTS_TAC `\y:real^(1,1)finite_sum. (subpath (vec 0) (fstcart y) (\u. h(pastecart u (vec 0))) ++ (\u. (h:real^(1,1)finite_sum->real^N) (pastecart (fstcart y) u)) ++ subpath (fstcart y) (vec 0) (\u. h(pastecart u (vec 0)))) (sndcart y)` THEN ASM_REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; SUBPATH_TRIVIAL; SUBPATH_REFL; SUBPATH_REVERSEPATH; ETA_AX; PATHSTART_JOIN; PATHFINISH_JOIN; PATHSTART_SUBPATH; PATHFINISH_SUBPATH; PATHSTART_LINEPATH; PATHFINISH_LINEPATH] THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[pathstart]] THEN CONJ_TAC THENL [MATCH_MP_TAC(REWRITE_RULE[PCROSS] HOMOTOPIC_JOIN_LEMMA) THEN REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC(REWRITE_RULE[PCROSS] HOMOTOPIC_JOIN_LEMMA) THEN ASM_REWRITE_TAC[PASTECART_FST_SND; ETA_AX] THEN CONJ_TAC THENL [ALL_TAC; RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish]) THEN REWRITE_TAC[PATHSTART_SUBPATH] THEN ASM_SIMP_TAC[pathstart; pathfinish]]; RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish]) THEN REWRITE_TAC[PATHFINISH_SUBPATH; PATHSTART_JOIN] THEN ASM_SIMP_TAC[pathstart]] THEN REWRITE_TAC[subpath] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN REWRITE_TAC[VECTOR_SUB_RZERO; VECTOR_SUB_LZERO; VECTOR_ADD_LID] THEN (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 5) [CONTINUOUS_ON_PASTECART; CONTINUOUS_ON_ADD; CONTINUOUS_ON_MUL; LIFT_DROP; CONTINUOUS_ON_NEG; DROP_NEG; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID; LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; LINEAR_SNDCART; LIFT_NEG; o_DEF; ETA_AX] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN REWRITE_TAC[IN_ELIM_PASTECART_THM] THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; IN_INTERVAL_1] THEN REWRITE_TAC[DROP_ADD; DROP_NEG; DROP_VEC; DROP_CMUL; REAL_POS] THEN SIMP_TAC[REAL_LE_MUL; REAL_SUB_LE; REAL_ARITH `t + --t * x = t * (&1 - x)`] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC(REAL_ARITH `t * x <= t * &1 /\ &1 * t <= &1 * &1 ==> t * x <= &1`) THEN CONJ_TAC THEN MATCH_MP_TAC REAL_LE_LMUL THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC; IMP_CONJ; RIGHT_FORALL_IMP_THM; FSTCART_PASTECART; SNDCART_PASTECART] THEN X_GEN_TAC `t:real^1` THEN STRIP_TAC THEN REWRITE_TAC[SET_RULE `(!x. x IN s ==> f x IN t) <=> IMAGE f s SUBSET t`] THEN REWRITE_TAC[GSYM path_image; ETA_AX] THEN REPEAT(MATCH_MP_TAC SUBSET_PATH_IMAGE_JOIN THEN CONJ_TAC) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS)) THEN REWRITE_TAC[path_image; subpath] THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM o_DEF] THEN REWRITE_TAC[IMAGE_o] THEN MATCH_MP_TAC IMAGE_SUBSET THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_PASTECART_THM] THEN SIMP_TAC[IN_INTERVAL_1; DROP_SUB; DROP_VEC; DROP_CMUL; DROP_ADD] THEN REWRITE_TAC[REAL_ADD_LID; REAL_SUB_RZERO; REAL_POS] THEN REWRITE_TAC[REAL_ARITH `t + (&0 - t) * x = t * (&1 - x)`] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN ASM_SIMP_TAC[REAL_LE_MUL; REAL_SUB_LE] THEN REPEAT STRIP_TAC THEN GEN_REWRITE_TAC RAND_CONV [GSYM REAL_MUL_RID] THEN MATCH_MP_TAC REAL_LE_MUL2 THEN ASM_REAL_ARITH_TAC]);; let HOMOTOPIC_LOOPS_CONJUGATE = prove (`!p q s:real^N->bool. path p /\ path_image p SUBSET s /\ path q /\ path_image q SUBSET s /\ pathfinish p = pathstart q /\ pathfinish q = pathstart q ==> homotopic_loops s (p ++ q ++ reversepath p) q`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMOTOPIC_LOOPS_TRANS THEN EXISTS_TAC `linepath(pathstart q,pathstart q) ++ (q:real^1->real^N) ++ linepath(pathstart q,pathstart q)` THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC HOMOTOPIC_PATHS_IMP_HOMOTOPIC_LOOPS THEN MP_TAC(ISPECL [`s:real^N->bool`; `(q:real^1->real^N) ++ linepath(pathfinish q,pathfinish q)`] HOMOTOPIC_PATHS_LID) THEN ASM_SIMP_TAC[PATHSTART_JOIN; PATHFINISH_JOIN; UNION_SUBSET; SING_SUBSET; PATHSTART_LINEPATH; PATHFINISH_LINEPATH; PATH_IMAGE_LINEPATH; PATH_JOIN; PATH_IMAGE_JOIN; PATH_LINEPATH; SEGMENT_REFL] THEN ANTS_TAC THENL [ASM_MESON_TAC[PATHSTART_IN_PATH_IMAGE; SUBSET]; ALL_TAC] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_PATHS_TRANS) THEN ASM_MESON_TAC[HOMOTOPIC_PATHS_RID]] THEN REWRITE_TAC[homotopic_loops; HOMOTOPIC_WITH_EUCLIDEAN; PCROSS] THEN EXISTS_TAC `(\y. (subpath (fstcart y) (vec 1) p ++ q ++ subpath (vec 1) (fstcart y) p) (sndcart y)):real^(1,1)finite_sum->real^N` THEN ASM_REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; SUBPATH_TRIVIAL; SUBPATH_REFL; SUBPATH_REVERSEPATH; ETA_AX; PATHSTART_JOIN; PATHFINISH_JOIN; PATHSTART_SUBPATH; PATHFINISH_SUBPATH; PATHSTART_LINEPATH; PATHFINISH_LINEPATH] THEN RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish]) THEN ASM_REWRITE_TAC[pathstart; pathfinish] THEN CONJ_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[path; path_image]) THEN MATCH_MP_TAC(REWRITE_RULE[PCROSS] HOMOTOPIC_JOIN_LEMMA) THEN REPEAT CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC(REWRITE_RULE[PCROSS] HOMOTOPIC_JOIN_LEMMA) THEN REPEAT CONJ_TAC THENL [GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_SNDCART] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN SIMP_TAC[SNDCART_PASTECART]; ALL_TAC; REWRITE_TAC[PATHSTART_SUBPATH] THEN ASM_REWRITE_TAC[pathfinish]]; REWRITE_TAC[PATHSTART_JOIN; PATHFINISH_SUBPATH] THEN ASM_REWRITE_TAC[pathstart]] THEN REWRITE_TAC[subpath] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN (CONJ_TAC THENL [REWRITE_TAC[DROP_SUB] THEN MATCH_MP_TAC CONTINUOUS_ON_ADD THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; CONTINUOUS_ON_CONST; LINEAR_FSTCART] THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_SNDCART] THEN REWRITE_TAC[o_DEF; LIFT_SUB; LIFT_DROP] THEN SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; LINEAR_CONTINUOUS_ON; LINEAR_FSTCART]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; IN_INTERVAL_1] THEN REWRITE_TAC[DROP_ADD; DROP_SUB; DROP_VEC; DROP_CMUL]]) THENL [REPEAT STRIP_TAC THENL [MATCH_MP_TAC REAL_LE_ADD THEN CONJ_TAC THEN TRY(MATCH_MP_TAC REAL_LE_MUL) THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[REAL_ARITH `t + (&1 - t) * x <= &1 <=> (&1 - t) * x <= (&1 - t) * &1`] THEN MATCH_MP_TAC REAL_LE_LMUL THEN ASM_REAL_ARITH_TAC]; REPEAT STRIP_TAC THENL [MATCH_MP_TAC(REAL_ARITH `x * (&1 - t) <= x * &1 /\ x <= &1 ==> &0 <= &1 + (t - &1) * x`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_LMUL THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[REAL_ARITH `a + (t - &1) * x <= a <=> &0 <= (&1 - t) * x`] THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_REAL_ARITH_TAC]]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[ETA_AX; GSYM path_image; SET_RULE `(!x. x IN i ==> f x IN s) <=> IMAGE f i SUBSET s`] THEN REPEAT STRIP_TAC THEN REPEAT(MATCH_MP_TAC SUBSET_PATH_IMAGE_JOIN THEN CONJ_TAC) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `path_image p:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC PATH_IMAGE_SUBPATH_SUBSET THEN ASM_REWRITE_TAC[ENDS_IN_UNIT_INTERVAL]]);; (* ------------------------------------------------------------------------- *) (* Relating homotopy of trivial loops to path-connectedness. *) (* ------------------------------------------------------------------------- *) let PATH_COMPONENT_IMP_HOMOTOPIC_POINTS = prove (`!s a b:real^N. path_component s a b ==> homotopic_loops s (linepath(a,a)) (linepath(b,b))`, REWRITE_TAC[path_component; homotopic_loops; HOMOTOPIC_WITH_EUCLIDEAN; PCROSS] THEN REPEAT GEN_TAC THEN REWRITE_TAC[pathstart; pathfinish; path_image; path] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^1->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\y:real^(1,1)finite_sum. (g(fstcart y):real^N)` THEN ASM_SIMP_TAC[FSTCART_PASTECART; linepath] THEN REWRITE_TAC[VECTOR_ARITH `(&1 - x) % a + x % a:real^N = a`] THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_COMPOSE) THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_FSTCART] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN SIMP_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC; FSTCART_PASTECART]);; let HOMOTOPIC_LOOPS_IMP_PATH_COMPONENT_VALUE = prove (`!s p q:real^1->real^N t. homotopic_loops s p q /\ t IN interval[vec 0,vec 1] ==> path_component s (p t) (q t)`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN REWRITE_TAC[path_component; homotopic_loops; HOMOTOPIC_WITH_EUCLIDEAN; PCROSS] THEN DISCH_THEN(X_CHOOSE_THEN `h:real^(1,1)finite_sum->real^N` MP_TAC) THEN STRIP_TAC THEN EXISTS_TAC `\u. (h:real^(1,1)finite_sum->real^N) (pastecart u t)` THEN ASM_REWRITE_TAC[pathstart; pathfinish] THEN CONJ_TAC THENL [REWRITE_TAC[path] THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_COMPOSE) THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_PASTECART THEN REWRITE_TAC[CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN ASM SET_TAC[]]; REWRITE_TAC[path_image] THEN ASM SET_TAC[]]);; let HOMOTOPIC_POINTS_EQ_PATH_COMPONENT = prove (`!s a b:real^N. homotopic_loops s (linepath(a,a)) (linepath(b,b)) <=> path_component s a b`, REPEAT GEN_TAC THEN EQ_TAC THEN REWRITE_TAC[PATH_COMPONENT_IMP_HOMOTOPIC_POINTS] THEN DISCH_THEN(MP_TAC o SPEC `vec 0:real^1` o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMOTOPIC_LOOPS_IMP_PATH_COMPONENT_VALUE)) THEN REWRITE_TAC[linepath; IN_INTERVAL_1; DROP_VEC; REAL_POS] THEN REWRITE_TAC[VECTOR_ARITH `(&1 - &0) % a + &0 % b:real^N = a`]);; let PATH_CONNECTED_EQ_HOMOTOPIC_POINTS = prove (`!s:real^N->bool. path_connected s <=> !a b. a IN s /\ b IN s ==> homotopic_loops s (linepath(a,a)) (linepath(b,b))`, GEN_TAC THEN REWRITE_TAC[HOMOTOPIC_POINTS_EQ_PATH_COMPONENT] THEN REWRITE_TAC[path_connected; path_component]);; (* ------------------------------------------------------------------------- *) (* Homotopy of "nearby" function, paths and loops. *) (* ------------------------------------------------------------------------- *) let HOMOTOPIC_WITH_LINEAR = prove (`!f g:real^M->real^N s t. f continuous_on s /\ g continuous_on s /\ (!x. x IN s ==> segment[f x,g x] SUBSET t) ==> homotopic_with (\z. T) (subtopology euclidean s,subtopology euclidean t) f g`, REPEAT STRIP_TAC THEN REWRITE_TAC[HOMOTOPIC_WITH_EUCLIDEAN] THEN EXISTS_TAC `\y. ((&1 - drop(fstcart y)) % (f:real^M->real^N)(sndcart y) + drop(fstcart y) % g(sndcart y):real^N)` THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; DROP_VEC] THEN ASM_REWRITE_TAC[REAL_SUB_REFL; REAL_SUB_RZERO] THEN REWRITE_TAC[VECTOR_ARITH `(&1 - t) % a + t % a:real^N = a`] THEN REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_MUL_LID] THEN REWRITE_TAC[VECTOR_ADD_LID; VECTOR_ADD_RID] THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_ADD THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN REWRITE_TAC[o_DEF; LIFT_DROP; LIFT_SUB] THEN SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; ETA_AX] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_SNDCART] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN SIMP_TAC[SNDCART_PASTECART; FORALL_IN_PCROSS]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_PCROSS] THEN MAP_EVERY X_GEN_TAC [`t:real^1`; `u:real^M`] THEN STRIP_TAC THEN SIMP_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; RIGHT_IMP_FORALL_THM; IMP_IMP]) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN EXISTS_TAC `u:real^M` THEN ASM_REWRITE_TAC[IN_SEGMENT] THEN EXISTS_TAC `drop t` THEN ASM_MESON_TAC[IN_INTERVAL_1; DROP_VEC]]);; let HOMOTOPIC_PATHS_LINEAR,HOMOTOPIC_LOOPS_LINEAR = (CONJ_PAIR o prove) (`(!g s:real^N->bool h. path g /\ path h /\ pathstart h = pathstart g /\ pathfinish h = pathfinish g /\ (!t x. t IN interval[vec 0,vec 1] ==> segment[g t,h t] SUBSET s) ==> homotopic_paths s g h) /\ (!g s:real^N->bool h. path g /\ path h /\ pathfinish g = pathstart g /\ pathfinish h = pathstart h /\ (!t x. t IN interval[vec 0,vec 1] ==> segment[g t,h t] SUBSET s) ==> homotopic_loops s g h)`, CONJ_TAC THEN (REWRITE_TAC[pathstart; pathfinish] THEN REWRITE_TAC[SUBSET; RIGHT_IMP_FORALL_THM; IMP_IMP] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[homotopic_paths; homotopic_loops; HOMOTOPIC_WITH_EUCLIDEAN; PCROSS] THEN EXISTS_TAC `\y:real^(1,1)finite_sum. ((&1 - drop(fstcart y)) % g(sndcart y) + drop(fstcart y) % h(sndcart y):real^N)` THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; DROP_VEC] THEN ASM_REWRITE_TAC[pathstart; pathfinish; REAL_SUB_REFL; REAL_SUB_RZERO] THEN REWRITE_TAC[VECTOR_ARITH `(&1 - t) % a + t % a:real^N = a`] THEN REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_MUL_LID] THEN REWRITE_TAC[VECTOR_ADD_LID; VECTOR_ADD_RID] THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_ADD THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN REWRITE_TAC[o_DEF; LIFT_DROP; LIFT_SUB] THEN SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; ETA_AX] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_SNDCART] THEN RULE_ASSUM_TAC(REWRITE_RULE[path]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN SIMP_TAC[SNDCART_PASTECART]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN MAP_EVERY X_GEN_TAC [`t:real^1`; `u:real^1`] THEN STRIP_TAC THEN SIMP_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN EXISTS_TAC `u:real^1` THEN ASM_REWRITE_TAC[IN_SEGMENT] THEN EXISTS_TAC `drop t` THEN ASM_MESON_TAC[IN_INTERVAL_1; DROP_VEC]]));; let HOMOTOPIC_PATHS_NEARBY_EXPLICIT, HOMOTOPIC_LOOPS_NEARBY_EXPLICIT = (CONJ_PAIR o prove) (`(!g s:real^N->bool h. path g /\ path h /\ pathstart h = pathstart g /\ pathfinish h = pathfinish g /\ (!t x. t IN interval[vec 0,vec 1] /\ ~(x IN s) ==> norm(h t - g t) < norm(g t - x)) ==> homotopic_paths s g h) /\ (!g s:real^N->bool h. path g /\ path h /\ pathfinish g = pathstart g /\ pathfinish h = pathstart h /\ (!t x. t IN interval[vec 0,vec 1] /\ ~(x IN s) ==> norm(h t - g t) < norm(g t - x)) ==> homotopic_loops s g h)`, ONCE_REWRITE_TAC[TAUT `p /\ ~q ==> r <=> p /\ ~r ==> q`] THEN REPEAT STRIP_TAC THENL [MATCH_MP_TAC HOMOTOPIC_PATHS_LINEAR; MATCH_MP_TAC HOMOTOPIC_LOOPS_LINEAR] THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET; segment; FORALL_IN_GSPEC] THEN X_GEN_TAC `t:real^1` THEN STRIP_TAC THEN X_GEN_TAC `u:real` THEN STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN EXISTS_TAC `t:real^1` THEN ASM_REWRITE_TAC[REAL_NOT_LT] THEN MP_TAC(ISPECL [`(g:real^1->real^N) t`; `(h:real^1->real^N) t`] DIST_IN_CLOSED_SEGMENT) THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN REWRITE_TAC[segment; FORALL_IN_GSPEC; ONCE_REWRITE_RULE[DIST_SYM] dist] THEN ASM_MESON_TAC[]);; let HOMOTOPIC_NEARBY_PATHS,HOMOTOPIC_NEARBY_LOOPS = (CONJ_PAIR o prove) (`(!g s:real^N->bool. path g /\ open s /\ path_image g SUBSET s ==> ?e. &0 < e /\ !h. path h /\ pathstart h = pathstart g /\ pathfinish h = pathfinish g /\ (!t. t IN interval[vec 0,vec 1] ==> norm(h t - g t) < e) ==> homotopic_paths s g h) /\ (!g s:real^N->bool. path g /\ pathfinish g = pathstart g /\ open s /\ path_image g SUBSET s ==> ?e. &0 < e /\ !h. path h /\ pathfinish h = pathstart h /\ (!t. t IN interval[vec 0,vec 1] ==> norm(h t - g t) < e) ==> homotopic_loops s g h)`, CONJ_TAC THEN REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`path_image g:real^N->bool`; `(:real^N) DIFF s`] SEPARATE_COMPACT_CLOSED) THEN ASM_SIMP_TAC[COMPACT_PATH_IMAGE; GSYM OPEN_CLOSED] THEN (ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[IN_DIFF; IN_UNIV; dist]]) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `e:real` THEN REWRITE_TAC[REAL_NOT_LE] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `h:real^1->real^N` THEN STRIP_TAC THENL [MATCH_MP_TAC HOMOTOPIC_PATHS_NEARBY_EXPLICIT; MATCH_MP_TAC HOMOTOPIC_LOOPS_NEARBY_EXPLICIT] THEN ASM_REWRITE_TAC[] THEN MAP_EVERY X_GEN_TAC [`t:real^1`; `x:real^N`] THEN STRIP_TAC THEN MATCH_MP_TAC REAL_LTE_TRANS THEN EXISTS_TAC `e:real` THEN ASM_SIMP_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[path_image] THEN ASM SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* Homotopy of non-antipodal sphere maps. *) (* ------------------------------------------------------------------------- *) let HOMOTOPIC_NON_MIDPOINT_SPHEREMAPS = prove (`!f g:real^M->real^N s a r. f continuous_on s /\ IMAGE f s SUBSET sphere(a,r) /\ g continuous_on s /\ IMAGE g s SUBSET sphere(a,r) /\ (!x. x IN s ==> ~(midpoint(f x,g x) = a)) ==> homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean (sphere(a,r))) f g`, REPEAT GEN_TAC THEN ASM_CASES_TAC `r <= &0` THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMOTOPIC_WITH_EQ THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN REPEAT(EXISTS_TAC `g:real^M->real^N`) THEN ASM_REWRITE_TAC[HOMOTOPIC_WITH_REFL; CONTINUOUS_MAP_EUCLIDEAN2] THEN SUBGOAL_THEN `?c:real^N. sphere(a,r) SUBSET {c}` MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN ASM_CASES_TAC `r = &0` THEN ASM_SIMP_TAC[SPHERE_SING; SPHERE_EMPTY; REAL_LT_LE] THEN MESON_TAC[SUBSET_REFL; EMPTY_SUBSET]; RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LE]) THEN STRIP_TAC] THEN SUBGOAL_THEN `homotopic_with (\z. T) (subtopology euclidean (s:real^M->bool), subtopology euclidean ((:real^N) DELETE a)) f g` MP_TAC THENL [MATCH_MP_TAC HOMOTOPIC_WITH_LINEAR THEN ASM_REWRITE_TAC[SET_RULE `s SUBSET UNIV DELETE a <=> ~(a IN s)`] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET])) THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_SPHERE; IMP_IMP] THEN REWRITE_TAC[AND_FORALL_THM] THEN DISCH_THEN(MP_TAC o SPEC `x:real^M`) THEN FIRST_X_ASSUM(MP_TAC o GSYM o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[GSYM BETWEEN_IN_SEGMENT; MIDPOINT_BETWEEN] THEN MESON_TAC[DIST_SYM]; ALL_TAC] THEN DISCH_THEN(MP_TAC o ISPECL [`\y:real^N. a + r / norm(y - a) % (y - a)`; `sphere(a:real^N,r)`] o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMOTOPIC_COMPOSE_CONTINUOUS_LEFT)) THEN REWRITE_TAC[o_DEF] THEN ANTS_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_ADD THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN REWRITE_TAC[real_div; o_DEF; LIFT_CMUL] THEN MATCH_MP_TAC CONTINUOUS_ON_CMUL THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN SIMP_TAC[IN_DELETE; NORM_EQ_0; VECTOR_SUB_EQ] THEN MATCH_MP_TAC CONTINUOUS_ON_LIFT_NORM_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID]; SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_UNIV; IN_DELETE; IN_SPHERE] THEN REWRITE_TAC[NORM_ARITH `dist(a:real^N,a + b) = norm b`] THEN SIMP_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM] THEN ASM_SIMP_TAC[real_abs; REAL_LE_RMUL; REAL_DIV_RMUL; NORM_EQ_0; VECTOR_SUB_EQ; REAL_LT_IMP_LE]]; MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_WITH_EQ) THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; FORALL_IN_IMAGE; IN_SPHERE]) THEN ASM_SIMP_TAC[NORM_ARITH `norm(a - b:real^N) = dist(b,a)`] THEN ASM_SIMP_TAC[REAL_DIV_REFL; REAL_LT_IMP_NZ] THEN REPEAT STRIP_TAC THEN VECTOR_ARITH_TAC]);; let HOMOTOPIC_NON_ANTIPODAL_SPHEREMAPS = prove (`!f g:real^M->real^N s r. f continuous_on s /\ IMAGE f s SUBSET sphere(vec 0,r) /\ g continuous_on s /\ IMAGE g s SUBSET sphere(vec 0,r) /\ (!x. x IN s ==> ~(f x = --g x)) ==> homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean (sphere(vec 0,r))) f g`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMOTOPIC_NON_MIDPOINT_SPHEREMAPS THEN ASM_REWRITE_TAC[midpoint; VECTOR_ARITH `inv(&2) % (a + b):real^N = vec 0 <=> a = --b`]);; (* ------------------------------------------------------------------------- *) (* Retracts, in a general sense, preserve (co)homotopic triviality. *) (* ------------------------------------------------------------------------- *) let HOMOTOPICALLY_TRIVIAL_RETRACTION_GEN = prove (`!P Q s:real^M->bool t:real^N->bool u:real^P->bool h k. (h continuous_on s /\ IMAGE h s = t /\ k continuous_on t /\ IMAGE k t SUBSET s /\ (!y. y IN t ==> h(k y) = y) /\ (!f. f continuous_on u /\ IMAGE f u SUBSET t /\ Q f ==> P(k o f)) /\ (!f. f continuous_on u /\ IMAGE f u SUBSET s /\ P f ==> Q(h o f)) /\ (!h k. (!x. x IN u ==> h x = k x) ==> (Q h <=> Q k))) /\ (!f g. f continuous_on u /\ IMAGE f u SUBSET s /\ P f /\ g continuous_on u /\ IMAGE g u SUBSET s /\ P g ==> homotopic_with P (subtopology euclidean u,subtopology euclidean s) f g) ==> (!f g. f continuous_on u /\ IMAGE f u SUBSET t /\ Q f /\ g continuous_on u /\ IMAGE g u SUBSET t /\ Q g ==> homotopic_with Q (subtopology euclidean u,subtopology euclidean t) f g)`, REPEAT GEN_TAC THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`p:real^P->real^N`; `q:real^P->real^N`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`(k:real^N->real^M) o (p:real^P->real^N)`; `(k:real^N->real^M) o (q:real^P->real^N)`]) THEN ANTS_TAC THENL [ASM_SIMP_TAC[IMAGE_o] THEN REPEAT CONJ_TAC THEN TRY(MATCH_MP_TAC CONTINUOUS_ON_COMPOSE) THEN ASM_REWRITE_TAC[] THEN TRY(FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET))) THEN ASM SET_TAC[]; DISCH_TAC] THEN MATCH_MP_TAC HOMOTOPIC_WITH_EQ THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN MAP_EVERY EXISTS_TAC [`(h:real^M->real^N) o (k:real^N->real^M) o (p:real^P->real^N)`; `(h:real^M->real^N) o (k:real^N->real^M) o (q:real^P->real^N)`] THEN ASM_REWRITE_TAC[o_THM] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_LEFT THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMOTOPIC_WITH_MONO)) THEN ASM_SIMP_TAC[]);; let HOMOTOPICALLY_TRIVIAL_RETRACTION_NULL_GEN = prove (`!P Q s:real^M->bool t:real^N->bool u:real^P->bool h k. (h continuous_on s /\ IMAGE h s = t /\ k continuous_on t /\ IMAGE k t SUBSET s /\ (!y. y IN t ==> h(k y) = y) /\ (!f. f continuous_on u /\ IMAGE f u SUBSET t /\ Q f ==> P(k o f)) /\ (!f. f continuous_on u /\ IMAGE f u SUBSET s /\ P f ==> Q(h o f)) /\ (!h k. (!x. x IN u ==> h x = k x) ==> (Q h <=> Q k))) /\ (!f. f continuous_on u /\ IMAGE f u SUBSET s /\ P f ==> ?c. homotopic_with P (subtopology euclidean u,subtopology euclidean s) f (\x. c)) ==> (!f. f continuous_on u /\ IMAGE f u SUBSET t /\ Q f ==> ?c. homotopic_with Q (subtopology euclidean u,subtopology euclidean t) f (\x. c))`, REPEAT GEN_TAC THEN STRIP_TAC THEN X_GEN_TAC `p:real^P->real^N` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `(k:real^N->real^M) o (p:real^P->real^N)`) THEN ANTS_TAC THENL [ASM_SIMP_TAC[IMAGE_o] THEN CONJ_TAC THEN TRY(MATCH_MP_TAC CONTINUOUS_ON_COMPOSE) THEN ASM_REWRITE_TAC[] THEN TRY(FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET))) THEN ASM SET_TAC[]; DISCH_THEN(X_CHOOSE_TAC `c:real^M`)] THEN EXISTS_TAC `(h:real^M->real^N) c` THEN MATCH_MP_TAC HOMOTOPIC_WITH_EQ THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN MAP_EVERY EXISTS_TAC [`(h:real^M->real^N) o (k:real^N->real^M) o (p:real^P->real^N)`; `(h:real^M->real^N) o ((\x. c):real^P->real^M)`] THEN ASM_REWRITE_TAC[o_THM] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_LEFT THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMOTOPIC_WITH_MONO)) THEN ASM_SIMP_TAC[]);; let COHOMOTOPICALLY_TRIVIAL_RETRACTION_GEN = prove (`!P Q s:real^M->bool t:real^N->bool u:real^P->bool h k. (h continuous_on s /\ IMAGE h s = t /\ k continuous_on t /\ IMAGE k t SUBSET s /\ (!y. y IN t ==> h(k y) = y) /\ (!f. f continuous_on t /\ IMAGE f t SUBSET u /\ Q f ==> P(f o h)) /\ (!f. f continuous_on s /\ IMAGE f s SUBSET u /\ P f ==> Q(f o k)) /\ (!h k. (!x. x IN t ==> h x = k x) ==> (Q h <=> Q k))) /\ (!f g. f continuous_on s /\ IMAGE f s SUBSET u /\ P f /\ g continuous_on s /\ IMAGE g s SUBSET u /\ P g ==> homotopic_with P (subtopology euclidean s, subtopology euclidean u) f g) ==> (!f g. f continuous_on t /\ IMAGE f t SUBSET u /\ Q f /\ g continuous_on t /\ IMAGE g t SUBSET u /\ Q g ==> homotopic_with Q (subtopology euclidean t,subtopology euclidean u) f g)`, REPEAT GEN_TAC THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`p:real^N->real^P`; `q:real^N->real^P`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`(p:real^N->real^P) o (h:real^M->real^N)`; `(q:real^N->real^P) o (h:real^M->real^N)`]) THEN ANTS_TAC THENL [ASM_SIMP_TAC[IMAGE_o] THEN REPEAT CONJ_TAC THEN TRY(MATCH_MP_TAC CONTINUOUS_ON_COMPOSE) THEN ASM_REWRITE_TAC[] THEN TRY(FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET))) THEN ASM SET_TAC[]; DISCH_TAC] THEN MATCH_MP_TAC HOMOTOPIC_WITH_EQ THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN MAP_EVERY EXISTS_TAC [`((p:real^N->real^P) o (h:real^M->real^N)) o (k:real^N->real^M)`; `((q:real^N->real^P) o (h:real^M->real^N)) o (k:real^N->real^M)`] THEN ASM_REWRITE_TAC[o_THM] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_RIGHT THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMOTOPIC_WITH_MONO)) THEN ASM_SIMP_TAC[]);; let COHOMOTOPICALLY_TRIVIAL_RETRACTION_NULL_GEN = prove (`!P Q s:real^M->bool t:real^N->bool u:real^P->bool h k. (h continuous_on s /\ IMAGE h s = t /\ k continuous_on t /\ IMAGE k t SUBSET s /\ (!y. y IN t ==> h(k y) = y) /\ (!f. f continuous_on t /\ IMAGE f t SUBSET u /\ Q f ==> P(f o h)) /\ (!f. f continuous_on s /\ IMAGE f s SUBSET u /\ P f ==> Q(f o k)) /\ (!h k. (!x. x IN t ==> h x = k x) ==> (Q h <=> Q k))) /\ (!f. f continuous_on s /\ IMAGE f s SUBSET u /\ P f ==> ?c. homotopic_with P (subtopology euclidean s,subtopology euclidean u) f (\x. c)) ==> (!f. f continuous_on t /\ IMAGE f t SUBSET u /\ Q f ==> ?c. homotopic_with Q (subtopology euclidean t,subtopology euclidean u) f (\x. c))`, REPEAT GEN_TAC THEN STRIP_TAC THEN X_GEN_TAC `p:real^N->real^P` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `(p:real^N->real^P) o (h:real^M->real^N)`) THEN ANTS_TAC THENL [ASM_SIMP_TAC[IMAGE_o] THEN TRY(MATCH_MP_TAC CONTINUOUS_ON_COMPOSE) THEN ASM_REWRITE_TAC[] THEN TRY(FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET))) THEN ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^P` THEN DISCH_TAC] THEN MATCH_MP_TAC HOMOTOPIC_WITH_EQ THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN MAP_EVERY EXISTS_TAC [`((p:real^N->real^P) o (h:real^M->real^N)) o (k:real^N->real^M)`; `((\x. c):real^M->real^P) o (k:real^N->real^M)`] THEN ASM_REWRITE_TAC[o_THM] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_RIGHT THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMOTOPIC_WITH_MONO)) THEN ASM_SIMP_TAC[]);; (* ------------------------------------------------------------------------- *) (* Another useful lemma. *) (* ------------------------------------------------------------------------- *) let HOMOTOPIC_JOIN_SUBPATHS = prove (`!g:real^1->real^N s. path g /\ path_image g SUBSET s /\ u IN interval[vec 0,vec 1] /\ v IN interval[vec 0,vec 1] /\ w IN interval[vec 0,vec 1] ==> homotopic_paths s (subpath u v g ++ subpath v w g) (subpath u w g)`, let lemma1 = prove (`!g:real^1->real^N s. drop u <= drop v /\ drop v <= drop w ==> path g /\ path_image g SUBSET s /\ u IN interval[vec 0,vec 1] /\ v IN interval[vec 0,vec 1] /\ w IN interval[vec 0,vec 1] /\ drop u <= drop v /\ drop v <= drop w ==> homotopic_paths s (subpath u v g ++ subpath v w g) (subpath u w g)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMOTOPIC_PATHS_SUBSET THEN EXISTS_TAC `path_image g:real^N->bool` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `w:real^1 = u` THENL [MP_TAC(ISPECL [`path_image g:real^N->bool`; `subpath u v (g:real^1->real^N)`] HOMOTOPIC_PATHS_RINV) THEN ASM_REWRITE_TAC[REVERSEPATH_SUBPATH; SUBPATH_REFL] THEN REWRITE_TAC[LINEPATH_REFL; PATHSTART_SUBPATH] THEN ASM_SIMP_TAC[PATH_SUBPATH; PATH_IMAGE_SUBPATH_SUBSET]; ALL_TAC] THEN ONCE_REWRITE_TAC[HOMOTOPIC_PATHS_SYM] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_REPARAMETRIZE THEN ASM_SIMP_TAC[PATH_SUBPATH; PATH_IMAGE_SUBPATH_SUBSET] THEN EXISTS_TAC `\t. if drop t <= &1 / &2 then inv(drop(w - u)) % (&2 * drop(v - u)) % t else inv(drop(w - u)) % ((v - u) + drop(w - v) % (&2 % t - vec 1))` THEN REWRITE_TAC[DROP_VEC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[VECTOR_MUL_RZERO] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_CASES_1 THEN REWRITE_TAC[GSYM DROP_EQ; DROP_CMUL; LIFT_DROP; GSYM LIFT_NUM; DROP_ADD; DROP_SUB] THEN (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 5) [CONTINUOUS_ON_MUL; o_DEF; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID; CONTINUOUS_ON_SUB; CONTINUOUS_ON_ADD] THEN REPEAT STRIP_TAC THEN REAL_ARITH_TAC; SUBGOAL_THEN `drop u < drop w` ASSUME_TAC THENL [ASM_SIMP_TAC[REAL_LT_LE; DROP_EQ] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `t:real^1` THEN STRIP_TAC THEN COND_CASES_TAC THEN REWRITE_TAC[IN_INTERVAL_1; DROP_CMUL; DROP_VEC; DROP_ADD; DROP_SUB] THEN ONCE_REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ; REAL_SUB_LT] THEN REWRITE_TAC[REAL_MUL_LZERO; REAL_MUL_LID] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN (CONJ_TAC THENL [REPEAT(MATCH_MP_TAC REAL_LE_ADD THEN CONJ_TAC) THEN REPEAT(MATCH_MP_TAC REAL_LE_MUL THEN CONJ_TAC) THEN ASM_REAL_ARITH_TAC; ALL_TAC]) THEN REWRITE_TAC[REAL_ARITH `v - u + x * t <= w - u <=> x * t <= w - v`; REAL_ARITH `(&2 * x) * t = x * &2 * t`] THEN MATCH_MP_TAC(REAL_ARITH `a * t <= a * &1 /\ a <= b ==> a * t <= b`) THEN (CONJ_TAC THENL [MATCH_MP_TAC REAL_LE_LMUL; ALL_TAC]) THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[GSYM DROP_EQ; DROP_VEC; DROP_ADD; DROP_CMUL; DROP_SUB] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[REAL_ARITH `(v - u) + (w - v) * &1 = w - u`] THEN ASM_SIMP_TAC[REAL_SUB_0; DROP_EQ; REAL_MUL_LINV]; X_GEN_TAC `t:real^1` THEN DISCH_TAC THEN REWRITE_TAC[subpath; joinpaths] THEN COND_CASES_TAC THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_ASSOC] THEN ASM_SIMP_TAC[REAL_MUL_RINV; DROP_EQ_0; VECTOR_SUB_EQ] THEN AP_TERM_TAC THEN REWRITE_TAC[GSYM DROP_EQ; DROP_VEC; DROP_ADD; DROP_CMUL; DROP_SUB] THEN REAL_ARITH_TAC]) in let lemma2 = prove (`path g /\ path_image g SUBSET s /\ u IN interval[vec 0,vec 1] /\ v IN interval[vec 0,vec 1] /\ w IN interval[vec 0,vec 1] /\ homotopic_paths s (subpath u v g ++ subpath v w g) (subpath u w g) ==> homotopic_paths s (subpath w v g ++ subpath v u g) (subpath w u g)`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM HOMOTOPIC_PATHS_REVERSEPATH] THEN SIMP_TAC[REVERSEPATH_JOINPATHS; PATHSTART_SUBPATH; PATHFINISH_SUBPATH] THEN ASM_REWRITE_TAC[REVERSEPATH_SUBPATH]) in let lemma3 = prove (`path (g:real^1->real^N) /\ path_image g SUBSET s /\ u IN interval[vec 0,vec 1] /\ v IN interval[vec 0,vec 1] /\ w IN interval[vec 0,vec 1] /\ homotopic_paths s (subpath u v g ++ subpath v w g) (subpath u w g) ==> homotopic_paths s (subpath v w g ++ subpath w u g) (subpath v u g)`, let tac = ASM_MESON_TAC[PATHSTART_SUBPATH; PATHFINISH_SUBPATH; PATH_SUBPATH; HOMOTOPIC_PATHS_REFL; PATH_IMAGE_SUBPATH_SUBSET; SUBSET_TRANS; PATHSTART_JOIN; PATHFINISH_JOIN] in REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM HOMOTOPIC_PATHS_REVERSEPATH] THEN SIMP_TAC[REVERSEPATH_JOINPATHS; PATHSTART_SUBPATH; PATHFINISH_SUBPATH] THEN ASM_REWRITE_TAC[REVERSEPATH_SUBPATH] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_TRANS THEN EXISTS_TAC `(subpath u v g ++ subpath v w g) ++ subpath w v g:real^1->real^N` THEN CONJ_TAC THENL [MATCH_MP_TAC HOMOTOPIC_PATHS_JOIN THEN ONCE_REWRITE_TAC[HOMOTOPIC_PATHS_SYM] THEN ASM_REWRITE_TAC[HOMOTOPIC_PATHS_REFL] THEN tac; ALL_TAC] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_TRANS THEN EXISTS_TAC `subpath u v g ++ (subpath v w g ++ subpath w v g):real^1->real^N` THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[HOMOTOPIC_PATHS_SYM] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_ASSOC THEN tac; ALL_TAC] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_TRANS THEN EXISTS_TAC `(subpath u v g :real^1->real^N) ++ linepath(pathfinish(subpath u v g),pathfinish(subpath u v g))` THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC HOMOTOPIC_PATHS_RID THEN tac] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_JOIN THEN REPEAT CONJ_TAC THENL [tac; ALL_TAC; tac] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_TRANS THEN EXISTS_TAC `linepath(pathstart(subpath v w g):real^N,pathstart(subpath v w g))` THEN CONJ_TAC THENL [GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM REVERSEPATH_SUBPATH] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_RINV THEN tac; ALL_TAC] THEN REWRITE_TAC[PATHSTART_SUBPATH; PATHFINISH_SUBPATH; HOMOTOPIC_PATHS_REFL; PATH_LINEPATH; PATH_IMAGE_LINEPATH; SEGMENT_REFL; INSERT_SUBSET; EMPTY_SUBSET] THEN ASM_MESON_TAC[path_image; IN_IMAGE; SUBSET]) in REPEAT STRIP_TAC THEN REPEAT_TCL DISJ_CASES_THEN ASSUME_TAC (REAL_ARITH `(drop u <= drop v /\ drop v <= drop w \/ drop w <= drop v /\ drop v <= drop u) \/ (drop u <= drop w /\ drop w <= drop v \/ drop v <= drop w /\ drop w <= drop u) \/ (drop v <= drop u /\ drop u <= drop w \/ drop w <= drop u /\ drop u <= drop v)`) THEN FIRST_ASSUM(MP_TAC o SPECL [`g:real^1->real^N`; `s:real^N->bool`] o MATCH_MP lemma1) THEN ASM_MESON_TAC[lemma2; lemma3]);; let HOMOTOPIC_LOOPS_SHIFTPATH = prove (`!s:real^N->bool p q u. homotopic_loops s p q /\ u IN interval[vec 0,vec 1] ==> homotopic_loops s (shiftpath u p) (shiftpath u q)`, REPEAT GEN_TAC THEN REWRITE_TAC[homotopic_loops; HOMOTOPIC_WITH_EUCLIDEAN; PCROSS] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN DISCH_THEN( (X_CHOOSE_THEN `h:real^(1,1)finite_sum->real^N` STRIP_ASSUME_TAC)) THEN EXISTS_TAC `\z. shiftpath u (\t. (h:real^(1,1)finite_sum->real^N) (pastecart (fstcart z) t)) (sndcart z)` THEN ASM_REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; ETA_AX] THEN ASM_SIMP_TAC[CLOSED_SHIFTPATH] THEN CONJ_TAC THENL [REWRITE_TAC[shiftpath; DROP_ADD; REAL_ARITH `u + z <= &1 <=> z <= &1 - u`] THEN SUBGOAL_THEN `{ pastecart (t:real^1) (x:real^1) | t IN interval[vec 0,vec 1] /\ x IN interval[vec 0,vec 1]} = { pastecart (t:real^1) (x:real^1) | t IN interval[vec 0,vec 1] /\ x IN interval[vec 0,vec 1 - u]} UNION { pastecart (t:real^1) (x:real^1) | t IN interval[vec 0,vec 1] /\ x IN interval[vec 1 - u,vec 1]}` SUBST1_TAC THENL [MATCH_MP_TAC(SET_RULE `s UNION s' = u ==> {f t x | t IN i /\ x IN u} = {f t x | t IN i /\ x IN s} UNION {f t x | t IN i /\ x IN s'}`) THEN UNDISCH_TAC `(u:real^1) IN interval[vec 0,vec 1]` THEN REWRITE_TAC[EXTENSION; IN_INTERVAL_1; IN_UNION; DROP_SUB; DROP_VEC] THEN REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC CONTINUOUS_ON_CASES THEN SIMP_TAC[REWRITE_RULE[PCROSS] CLOSED_PCROSS; CLOSED_INTERVAL] THEN REWRITE_TAC[FORALL_AND_THM; FORALL_IN_GSPEC; TAUT `p /\ q \/ r /\ s ==> t <=> (p ==> q ==> t) /\ (r ==> s ==> t)`] THEN SIMP_TAC[SNDCART_PASTECART; IN_INTERVAL_1; DROP_SUB; DROP_VEC] THEN SIMP_TAC[REAL_ARITH `&1 - u <= x ==> (x <= &1 - u <=> x = &1 - u)`] THEN SIMP_TAC[GSYM LIFT_EQ; LIFT_SUB; LIFT_DROP; LIFT_NUM] THEN REWRITE_TAC[FSTCART_PASTECART; VECTOR_ARITH `u + v - u:real^N = v`; VECTOR_ARITH `u + v - u - v:real^N = vec 0`] THEN RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish]) THEN ASM_SIMP_TAC[GSYM IN_INTERVAL_1; GSYM DROP_VEC] THEN CONJ_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_PASTECART; CONTINUOUS_ON_ADD; CONTINUOUS_ON_CONST; LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; LINEAR_SNDCART; VECTOR_ARITH `u + z - v:real^N = (u - v) + z`] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN UNDISCH_TAC `(u:real^1) IN interval[vec 0,vec 1]` THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; IN_INTERVAL_1; IN_ELIM_PASTECART_THM; DROP_ADD; DROP_SUB; DROP_VEC] THEN REAL_ARITH_TAC; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; SET_RULE `(!t x. t IN i /\ x IN i ==> f t x IN s) <=> (!t. t IN i ==> IMAGE (f t) i SUBSET s)`] THEN X_GEN_TAC `t:real^1` THEN STRIP_TAC THEN REWRITE_TAC[GSYM path_image] THEN ASM_SIMP_TAC[PATH_IMAGE_SHIFTPATH; ETA_AX] THEN REWRITE_TAC[path_image] THEN ASM SET_TAC[]]);; let HOMOTOPIC_PATHS_LOOP_PARTS = prove (`!s p q a:real^N. homotopic_loops s (p ++ reversepath q) (linepath(a,a)) /\ path q ==> homotopic_paths s p q`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_LOOPS_IMP_HOMOTOPIC_PATHS_NULL) THEN REWRITE_TAC[PATHSTART_JOIN] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o CONJUNCT1 o MATCH_MP HOMOTOPIC_PATHS_IMP_PATH) THEN ASM_CASES_TAC `pathfinish p:real^N = pathstart(reversepath q)` THENL [ASM_SIMP_TAC[PATH_JOIN; PATH_REVERSEPATH] THEN STRIP_TAC; ASM_MESON_TAC[PATH_JOIN_PATH_ENDS; PATH_REVERSEPATH]] THEN RULE_ASSUM_TAC(REWRITE_RULE[PATHSTART_REVERSEPATH]) THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_SUBSET) THEN ASM_SIMP_TAC[PATH_IMAGE_JOIN; PATHSTART_JOIN; PATHFINISH_JOIN; PATHSTART_REVERSEPATH; PATHFINISH_REVERSEPATH; UNION_SUBSET; SING_SUBSET; PATH_IMAGE_REVERSEPATH; PATH_IMAGE_LINEPATH; SEGMENT_REFL] THEN STRIP_TAC THEN MATCH_MP_TAC HOMOTOPIC_PATHS_TRANS THEN EXISTS_TAC `p ++ (linepath(pathfinish p:real^N,pathfinish p))` THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[HOMOTOPIC_PATHS_SYM] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_RID THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_TRANS THEN EXISTS_TAC `p ++ (reversepath q ++ q):real^1->real^N` THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[HOMOTOPIC_PATHS_SYM] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_JOIN THEN ASM_SIMP_TAC[HOMOTOPIC_PATHS_LINV; PATHSTART_JOIN; PATHSTART_REVERSEPATH; HOMOTOPIC_PATHS_REFL]; ALL_TAC] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_TRANS THEN EXISTS_TAC `(p ++ reversepath q) ++ q:real^1->real^N` THEN CONJ_TAC THENL [MATCH_MP_TAC HOMOTOPIC_PATHS_ASSOC THEN ASM_REWRITE_TAC[PATHSTART_REVERSEPATH; PATHFINISH_REVERSEPATH; PATH_IMAGE_REVERSEPATH; PATH_REVERSEPATH]; ALL_TAC] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_TRANS THEN EXISTS_TAC `linepath(pathstart p:real^N,pathstart p) ++ q` THEN CONJ_TAC THENL [MATCH_MP_TAC HOMOTOPIC_PATHS_JOIN THEN ASM_REWRITE_TAC[HOMOTOPIC_PATHS_REFL] THEN REWRITE_TAC[PATHFINISH_JOIN; PATHFINISH_REVERSEPATH]; FIRST_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_PATHFINISH) THEN REWRITE_TAC[PATHFINISH_JOIN; PATHFINISH_LINEPATH; PATHFINISH_REVERSEPATH] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN MATCH_MP_TAC HOMOTOPIC_PATHS_LID THEN ASM_REWRITE_TAC[]]);; let HOMOTOPIC_LOOPS_ADD_SYM = prove (`!p q:real^1->real^N. path p /\ path_image p SUBSET s /\ pathfinish p = pathstart p /\ path q /\ path_image q SUBSET s /\ pathfinish q = pathstart q /\ pathstart q = pathstart p ==> homotopic_loops s (p ++ q) (q ++ p)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMOTOPIC_LOOPS_TRANS THEN SUBGOAL_THEN `lift(&1 / &2) IN interval[vec 0,vec 1]` ASSUME_TAC THENL [REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; LIFT_DROP] THEN CONV_TAC REAL_RAT_REDUCE_CONV; ALL_TAC] THEN EXISTS_TAC `shiftpath (lift(&1 / &2)) (p ++ q:real^1->real^N)` THEN CONJ_TAC THENL [MATCH_MP_TAC HOMOTOPIC_LOOPS_SHIFTPATH_SELF; MATCH_MP_TAC HOMOTOPIC_LOOPS_EQ] THEN ASM_SIMP_TAC[PATH_JOIN; PATH_IMAGE_JOIN; PATHSTART_JOIN; PATHFINISH_JOIN; UNION_SUBSET; IN_INTERVAL_1; DROP_VEC; LIFT_DROP; PATH_SHIFTPATH; PATH_IMAGE_SHIFTPATH; CLOSED_SHIFTPATH] THEN SIMP_TAC[shiftpath; joinpaths; LIFT_DROP; DROP_ADD; DROP_SUB; DROP_VEC; REAL_ARITH `&0 <= t ==> (a + t <= a <=> t = &0)`; REAL_ARITH `t <= &1 ==> &1 / &2 + t - &1 <= &1 / &2`; REAL_ARITH `&1 / &2 + t <= &1 <=> t <= &1 / &2`] THEN X_GEN_TAC `t:real^1` THEN STRIP_TAC THEN ASM_CASES_TAC `drop t <= &1 / &2` THEN ASM_REWRITE_TAC[] THENL [REWRITE_TAC[GSYM LIFT_EQ; LIFT_NUM; LIFT_DROP] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[VECTOR_ADD_RID] THENL [REWRITE_TAC[GSYM LIFT_CMUL; VECTOR_MUL_RZERO] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_MESON_TAC[LIFT_NUM; pathstart; pathfinish]; ALL_TAC]; ALL_TAC] THEN AP_TERM_TAC THEN REWRITE_TAC[GSYM DROP_EQ; DROP_SUB; DROP_ADD; DROP_VEC; DROP_CMUL; LIFT_DROP] THEN REAL_ARITH_TAC);; (* ------------------------------------------------------------------------- *) (* Simply connected sets defined as "all loops are homotopic (as loops)". *) (* ------------------------------------------------------------------------- *) let simply_connected = new_definition `simply_connected(s:real^N->bool) <=> !p q. path p /\ pathfinish p = pathstart p /\ path_image p SUBSET s /\ path q /\ pathfinish q = pathstart q /\ path_image q SUBSET s ==> homotopic_loops s p q`;; let SIMPLY_CONNECTED_EMPTY = prove (`simply_connected {}`, REWRITE_TAC[simply_connected; SUBSET_EMPTY] THEN MESON_TAC[PATH_IMAGE_NONEMPTY]);; let SIMPLY_CONNECTED_IMP_PATH_CONNECTED = prove (`!s:real^N->bool. simply_connected s ==> path_connected s`, REWRITE_TAC[simply_connected; PATH_CONNECTED_EQ_HOMOTOPIC_POINTS] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[PATH_LINEPATH; PATHSTART_LINEPATH; PATHFINISH_LINEPATH; PATH_IMAGE_LINEPATH; SEGMENT_REFL] THEN ASM SET_TAC[]);; let SIMPLY_CONNECTED_IMP_CONNECTED = prove (`!s:real^N->bool. simply_connected s ==> connected s`, SIMP_TAC[SIMPLY_CONNECTED_IMP_PATH_CONNECTED; PATH_CONNECTED_IMP_CONNECTED]);; let SIMPLY_CONNECTED_EQ_CONTRACTIBLE_LOOP_ANY = prove (`!s:real^N->bool. simply_connected s <=> !p a. path p /\ path_image p SUBSET s /\ pathfinish p = pathstart p /\ a IN s ==> homotopic_loops s p (linepath(a,a))`, GEN_TAC THEN REWRITE_TAC[simply_connected] THEN EQ_TAC THEN DISCH_TAC THENL [REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[PATH_LINEPATH; PATHSTART_LINEPATH; PATHFINISH_LINEPATH] THEN ASM_REWRITE_TAC[PATH_IMAGE_LINEPATH; SEGMENT_REFL; SING_SUBSET]; MAP_EVERY X_GEN_TAC [`p:real^1->real^N`; `q:real^1->real^N`] THEN STRIP_TAC THEN MATCH_MP_TAC HOMOTOPIC_LOOPS_TRANS THEN EXISTS_TAC `linepath(pathstart p:real^N,pathstart p)` THEN CONJ_TAC THENL [ALL_TAC; ONCE_REWRITE_TAC[HOMOTOPIC_LOOPS_SYM]] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[PATHSTART_IN_PATH_IMAGE; SUBSET]]);; let SIMPLY_CONNECTED_EQ_CONTRACTIBLE_LOOP_SOME = prove (`!s:real^N->bool. simply_connected s <=> path_connected s /\ !p. path p /\ path_image p SUBSET s /\ pathfinish p = pathstart p ==> ?a. a IN s /\ homotopic_loops s p (linepath(a,a))`, GEN_TAC THEN EQ_TAC THEN STRIP_TAC THEN ASM_SIMP_TAC[SIMPLY_CONNECTED_IMP_PATH_CONNECTED] THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SIMPLY_CONNECTED_EQ_CONTRACTIBLE_LOOP_ANY]) THEN MESON_TAC[SUBSET; PATHSTART_IN_PATH_IMAGE]; REWRITE_TAC[SIMPLY_CONNECTED_EQ_CONTRACTIBLE_LOOP_ANY] THEN MAP_EVERY X_GEN_TAC [`p:real^1->real^N`; `a:real^N`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `p:real^1->real^N`) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `b:real^N` THEN STRIP_TAC THEN MATCH_MP_TAC HOMOTOPIC_LOOPS_TRANS THEN EXISTS_TAC `linepath(b:real^N,b)` THEN ASM_REWRITE_TAC[HOMOTOPIC_POINTS_EQ_PATH_COMPONENT] THEN ASM_MESON_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT]]);; let SIMPLY_CONNECTED_EQ_CONTRACTIBLE_LOOP_ALL = prove (`!s:real^N->bool. simply_connected s <=> s = {} \/ ?a. a IN s /\ !p. path p /\ path_image p SUBSET s /\ pathfinish p = pathstart p ==> homotopic_loops s p (linepath(a,a))`, GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[SIMPLY_CONNECTED_EMPTY] THEN REWRITE_TAC[SIMPLY_CONNECTED_EQ_CONTRACTIBLE_LOOP_SOME] THEN EQ_TAC THENL [STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `p:real^1->real^N` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `p:real^1->real^N`) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `b:real^N` THEN STRIP_TAC THEN MATCH_MP_TAC HOMOTOPIC_LOOPS_TRANS THEN EXISTS_TAC `linepath(b:real^N,b)` THEN ASM_REWRITE_TAC[HOMOTOPIC_POINTS_EQ_PATH_COMPONENT] THEN ASM_MESON_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT]; DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN REWRITE_TAC[PATH_CONNECTED_EQ_HOMOTOPIC_POINTS] THEN MAP_EVERY X_GEN_TAC [`b:real^N`; `c:real^N`] THEN STRIP_TAC THEN MATCH_MP_TAC HOMOTOPIC_LOOPS_TRANS THEN EXISTS_TAC `linepath(a:real^N,a)` THEN GEN_REWRITE_TAC RAND_CONV [HOMOTOPIC_LOOPS_SYM] THEN CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[PATH_LINEPATH; PATH_IMAGE_LINEPATH; SEGMENT_REFL; PATHSTART_LINEPATH; PATHFINISH_LINEPATH] THEN ASM SET_TAC[]]);; let SIMPLY_CONNECTED_EQ_CONTRACTIBLE_PATH = prove (`!s:real^N->bool. simply_connected s <=> path_connected s /\ !p. path p /\ path_image p SUBSET s /\ pathfinish p = pathstart p ==> homotopic_paths s p (linepath(pathstart p,pathstart p))`, GEN_TAC THEN EQ_TAC THEN STRIP_TAC THENL [ASM_SIMP_TAC[SIMPLY_CONNECTED_IMP_PATH_CONNECTED] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMOTOPIC_LOOPS_IMP_HOMOTOPIC_PATHS_NULL THEN EXISTS_TAC `pathstart p :real^N` THEN FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SIMPLY_CONNECTED_EQ_CONTRACTIBLE_LOOP_ANY]) THEN ASM_MESON_TAC[PATHSTART_IN_PATH_IMAGE; SUBSET]; REWRITE_TAC[SIMPLY_CONNECTED_EQ_CONTRACTIBLE_LOOP_ANY] THEN MAP_EVERY X_GEN_TAC [`p:real^1->real^N`; `a:real^N`] THEN STRIP_TAC THEN MATCH_MP_TAC HOMOTOPIC_LOOPS_TRANS THEN EXISTS_TAC `linepath(pathstart p:real^N,pathfinish p)` THEN CONJ_TAC THENL [MATCH_MP_TAC HOMOTOPIC_PATHS_IMP_HOMOTOPIC_LOOPS THEN ASM_SIMP_TAC[PATHFINISH_LINEPATH]; ASM_REWRITE_TAC[HOMOTOPIC_POINTS_EQ_PATH_COMPONENT] THEN RULE_ASSUM_TAC(REWRITE_RULE[PATH_CONNECTED_IFF_PATH_COMPONENT]) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[PATHSTART_IN_PATH_IMAGE; SUBSET]]]);; let SIMPLY_CONNECTED_EQ_HOMOTOPIC_PATHS = prove (`!s:real^N->bool. simply_connected s <=> path_connected s /\ !p q. path p /\ path_image p SUBSET s /\ path q /\ path_image q SUBSET s /\ pathstart q = pathstart p /\ pathfinish q = pathfinish p ==> homotopic_paths s p q`, REPEAT GEN_TAC THEN REWRITE_TAC[SIMPLY_CONNECTED_EQ_CONTRACTIBLE_PATH] THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `p:real^1->real^N` THENL [X_GEN_TAC `q:real^1->real^N` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `p ++ reversepath q :real^1->real^N`) THEN ASM_SIMP_TAC[PATH_JOIN; PATHSTART_REVERSEPATH; PATH_REVERSEPATH; PATHSTART_JOIN; PATHFINISH_JOIN; PATHFINISH_REVERSEPATH; PATH_IMAGE_JOIN; UNION_SUBSET; PATH_IMAGE_REVERSEPATH] THEN DISCH_TAC THEN MATCH_MP_TAC HOMOTOPIC_PATHS_TRANS THEN EXISTS_TAC `p ++ linepath(pathfinish p,pathfinish p):real^1->real^N` THEN GEN_REWRITE_TAC LAND_CONV [HOMOTOPIC_PATHS_SYM] THEN ASM_SIMP_TAC[HOMOTOPIC_PATHS_RID] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_TRANS THEN EXISTS_TAC `p ++ (reversepath q ++ q):real^1->real^N` THEN CONJ_TAC THENL [MATCH_MP_TAC HOMOTOPIC_PATHS_JOIN THEN ASM_REWRITE_TAC[HOMOTOPIC_PATHS_REFL; PATHSTART_LINEPATH] THEN ASM_MESON_TAC[HOMOTOPIC_PATHS_LINV; HOMOTOPIC_PATHS_SYM]; ALL_TAC] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_TRANS THEN EXISTS_TAC `(p ++ reversepath q) ++ q:real^1->real^N` THEN CONJ_TAC THENL [MATCH_MP_TAC HOMOTOPIC_PATHS_ASSOC THEN ASM_SIMP_TAC[PATH_REVERSEPATH; PATH_IMAGE_REVERSEPATH] THEN ASM_REWRITE_TAC[PATHSTART_REVERSEPATH; PATHFINISH_REVERSEPATH]; ALL_TAC] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_TRANS THEN EXISTS_TAC `linepath(pathstart q,pathstart q) ++ q:real^1->real^N` THEN CONJ_TAC THENL [MATCH_MP_TAC HOMOTOPIC_PATHS_JOIN THEN ASM_SIMP_TAC[HOMOTOPIC_PATHS_RINV; HOMOTOPIC_PATHS_REFL] THEN ASM_REWRITE_TAC[PATHFINISH_JOIN; PATHFINISH_REVERSEPATH]; ASM_MESON_TAC[HOMOTOPIC_PATHS_LID]]; STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[PATHSTART_LINEPATH; PATHFINISH_LINEPATH; PATH_LINEPATH] THEN REWRITE_TAC[PATH_IMAGE_LINEPATH; SEGMENT_REFL; SING_SUBSET] THEN ASM_MESON_TAC[PATHSTART_IN_PATH_IMAGE; SUBSET]]);; let SIMPLY_CONNECTED_RETRACTION_GEN = prove (`!s:real^M->bool t:real^N->bool h k. h continuous_on s /\ IMAGE h s = t /\ k continuous_on t /\ IMAGE k t SUBSET s /\ (!y. y IN t ==> h(k y) = y) /\ simply_connected s ==> simply_connected t`, REPEAT GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN REWRITE_TAC[simply_connected; path; path_image; homotopic_loops] THEN ONCE_REWRITE_TAC[TAUT `a /\ b /\ c /\ a' /\ b' /\ c' <=> a /\ c /\ b /\ a' /\ c' /\ b'`] THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] HOMOTOPICALLY_TRIVIAL_RETRACTION_GEN) THEN MAP_EVERY EXISTS_TAC [`h:real^M->real^N`; `k:real^N->real^M`] THEN ASM_SIMP_TAC[PATHSTART_COMPOSE; PATHFINISH_COMPOSE] THEN REWRITE_TAC[pathfinish; pathstart] THEN MESON_TAC[ENDS_IN_UNIT_INTERVAL]);; let HOMEOMORPHIC_SIMPLY_CONNECTED = prove (`!s:real^M->bool t:real^N->bool. s homeomorphic t /\ simply_connected s ==> simply_connected t`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] (REWRITE_RULE[CONJ_ASSOC] SIMPLY_CONNECTED_RETRACTION_GEN)) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphic]) THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN SIMP_TAC[homeomorphism; SUBSET_REFL]);; let HOMEOMORPHIC_SIMPLY_CONNECTED_EQ = prove (`!s:real^M->bool t:real^N->bool. s homeomorphic t ==> (simply_connected s <=> simply_connected t)`, REPEAT STRIP_TAC THEN EQ_TAC THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] HOMEOMORPHIC_SIMPLY_CONNECTED) THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN ASM_REWRITE_TAC[]);; let SIMPLY_CONNECTED_TRANSLATION = prove (`!a:real^N s. simply_connected (IMAGE (\x. a + x) s) <=> simply_connected s`, REPEAT GEN_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_SIMPLY_CONNECTED_EQ THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN REWRITE_TAC[HOMEOMORPHIC_TRANSLATION]);; add_translation_invariants [SIMPLY_CONNECTED_TRANSLATION];; let SIMPLY_CONNECTED_INJECTIVE_LINEAR_IMAGE = prove (`!f:real^M->real^N s. linear f /\ (!x y. f x = f y ==> x = y) ==> (simply_connected (IMAGE f s) <=> simply_connected s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_SIMPLY_CONNECTED_EQ THEN ASM_MESON_TAC[HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_LEFT_EQ; HOMEOMORPHIC_REFL]);; add_linear_invariants [SIMPLY_CONNECTED_INJECTIVE_LINEAR_IMAGE];; let HOMEOMORPHISM_SIMPLE_CONNECTEDNESS = prove (`!f:real^M->real^N g s t k. homeomorphism (s,t) (f,g) /\ k SUBSET s ==> (simply_connected(IMAGE f k) <=> simply_connected k)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_SIMPLY_CONNECTED_EQ THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN REWRITE_TAC[homeomorphic] THEN MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `g:real^N->real^M`] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]);; let SIMPLY_CONNECTED_PCROSS = prove (`!s:real^M->bool t:real^N->bool. simply_connected s /\ simply_connected t ==> simply_connected(s PCROSS t)`, REPEAT GEN_TAC THEN REWRITE_TAC[SIMPLY_CONNECTED_EQ_CONTRACTIBLE_LOOP_ANY] THEN REWRITE_TAC[path; path_image; pathstart; pathfinish; FORALL_PASTECART] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`p:real^1->real^(M,N)finite_sum`; `a:real^M`; `b:real^N`] THEN REWRITE_TAC[PASTECART_IN_PCROSS; FORALL_IN_IMAGE; SUBSET] THEN STRIP_TAC THEN FIRST_X_ASSUM(CONJUNCTS_THEN2 (MP_TAC o SPECL [`fstcart o (p:real^1->real^(M,N)finite_sum)`; `a:real^M`]) (MP_TAC o SPECL [`sndcart o (p:real^1->real^(M,N)finite_sum)`; `b:real^N`])) THEN ASM_SIMP_TAC[CONTINUOUS_ON_COMPOSE; LINEAR_FSTCART; LINEAR_SNDCART; LINEAR_CONTINUOUS_ON; homotopic_loops; HOMOTOPIC_WITH_EUCLIDEAN; pathfinish; pathstart; IMAGE_o; o_THM] THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE] THEN ANTS_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[PCROSS; IN_ELIM_THM]) THEN ASM_MESON_TAC[SNDCART_PASTECART]; DISCH_THEN(X_CHOOSE_THEN `k:real^(1,1)finite_sum->real^N` STRIP_ASSUME_TAC)] THEN ANTS_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[PCROSS; IN_ELIM_THM]) THEN ASM_MESON_TAC[FSTCART_PASTECART]; DISCH_THEN(X_CHOOSE_THEN `h:real^(1,1)finite_sum->real^M` STRIP_ASSUME_TAC)] THEN EXISTS_TAC `(\z. pastecart (h z) (k z)) :real^(1,1)finite_sum->real^(M,N)finite_sum` THEN ASM_SIMP_TAC[CONTINUOUS_ON_PASTECART; ETA_AX] THEN REWRITE_TAC[LINEPATH_REFL; PASTECART_FST_SND] THEN ASM_SIMP_TAC[PASTECART_IN_PCROSS]);; let SIMPLY_CONNECTED_PCROSS_EQ = prove (`!s:real^M->bool t:real^N->bool. simply_connected(s PCROSS t) <=> s = {} \/ t = {} \/ simply_connected s /\ simply_connected t`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_REWRITE_TAC[PCROSS_EMPTY; SIMPLY_CONNECTED_EMPTY] THEN ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_REWRITE_TAC[PCROSS_EMPTY; SIMPLY_CONNECTED_EMPTY] THEN EQ_TAC THEN REWRITE_TAC[SIMPLY_CONNECTED_PCROSS] THEN REPEAT STRIP_TAC THENL [REWRITE_TAC[SIMPLY_CONNECTED_EQ_CONTRACTIBLE_LOOP_ANY] THEN MAP_EVERY X_GEN_TAC [`p:real^1->real^M`; `a:real^M`] THEN REWRITE_TAC[path; path_image; pathstart; pathfinish; SUBSET; FORALL_IN_IMAGE] THEN STRIP_TAC THEN UNDISCH_TAC `~(t:real^N->bool = {})` THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN DISCH_THEN(X_CHOOSE_THEN `b:real^N` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SIMPLY_CONNECTED_EQ_CONTRACTIBLE_LOOP_ANY]) THEN DISCH_THEN(MP_TAC o SPECL [`(\t. pastecart (p t) (b)):real^1->real^(M,N)finite_sum`; `pastecart (a:real^M) (b:real^N)`]) THEN ASM_REWRITE_TAC[PASTECART_IN_PCROSS] THEN ASM_SIMP_TAC[path; path_image; pathstart; pathfinish; SUBSET; FORALL_IN_IMAGE; PASTECART_IN_PCROSS; PASTECART_INJ; CONTINUOUS_ON_PASTECART; ETA_AX; CONTINUOUS_ON_CONST] THEN STRIP_TAC THEN MP_TAC(ISPECL [`(\t. pastecart (p t) b):real^1->real^(M,N)finite_sum`; `linepath (pastecart (a:real^M) (b:real^N),pastecart a b)`; `fstcart:real^(M,N)finite_sum->real^M`; `(s:real^M->bool) PCROSS (t:real^N->bool)`; `s:real^M->bool`] HOMOTOPIC_LOOPS_CONTINUOUS_IMAGE) THEN ASM_SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_FSTCART] THEN SIMP_TAC[o_DEF; LINEPATH_REFL; FSTCART_PASTECART; ETA_AX; SUBSET; FORALL_IN_PCROSS; FORALL_IN_IMAGE]; REWRITE_TAC[SIMPLY_CONNECTED_EQ_CONTRACTIBLE_LOOP_ANY] THEN MAP_EVERY X_GEN_TAC [`p:real^1->real^N`; `b:real^N`] THEN REWRITE_TAC[path; path_image; pathstart; pathfinish; SUBSET; FORALL_IN_IMAGE] THEN STRIP_TAC THEN UNDISCH_TAC `~(s:real^M->bool = {})` THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN DISCH_THEN(X_CHOOSE_THEN `a:real^M` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SIMPLY_CONNECTED_EQ_CONTRACTIBLE_LOOP_ANY]) THEN DISCH_THEN(MP_TAC o SPECL [`(\t. pastecart a (p t)):real^1->real^(M,N)finite_sum`; `pastecart (a:real^M) (b:real^N)`]) THEN ASM_REWRITE_TAC[PASTECART_IN_PCROSS] THEN ASM_SIMP_TAC[path; path_image; pathstart; pathfinish; SUBSET; FORALL_IN_IMAGE; PASTECART_IN_PCROSS; PASTECART_INJ; CONTINUOUS_ON_PASTECART; ETA_AX; CONTINUOUS_ON_CONST] THEN STRIP_TAC THEN MP_TAC(ISPECL [`(\t. pastecart a (p t)):real^1->real^(M,N)finite_sum`; `linepath (pastecart (a:real^M) (b:real^N),pastecart a b)`; `sndcart:real^(M,N)finite_sum->real^N`; `(s:real^M->bool) PCROSS (t:real^N->bool)`; `t:real^N->bool`] HOMOTOPIC_LOOPS_CONTINUOUS_IMAGE) THEN ASM_SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_SNDCART] THEN SIMP_TAC[o_DEF; LINEPATH_REFL; SNDCART_PASTECART; ETA_AX; SUBSET; FORALL_IN_PCROSS; FORALL_IN_IMAGE]]);; let SIMPLY_CONNECTED_NESTED_UNIONS = prove (`!f:(real^N->bool)->bool. (!s. s IN f ==> open s /\ simply_connected s) /\ (!s t. s IN f /\ t IN f ==> s SUBSET t \/ t SUBSET s) ==> simply_connected(UNIONS f)`, REPEAT STRIP_TAC THEN REWRITE_TAC[simply_connected] THEN MAP_EVERY X_GEN_TAC [`p:real^1->real^N`; `q:real^1->real^N`] THEN STRIP_TAC THEN MP_TAC(ISPEC `IMAGE (\u:real^N->bool. (path_image p UNION path_image q) DIFF u) f` COMPACT_CHAIN) THEN REWRITE_TAC[FORALL_IN_IMAGE; FORALL_IN_IMAGE_2] THEN MATCH_MP_TAC(TAUT `q /\ r /\ (~p ==> s) ==> (p /\ q ==> ~r) ==> s`) THEN REPEAT CONJ_TAC THENL [ASM SET_TAC[]; MP_TAC(ISPEC `p:real^1->real^N` PATH_IMAGE_NONEMPTY) THEN REWRITE_TAC[INTERS_IMAGE] THEN ASM SET_TAC[]; REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; LEFT_IMP_EXISTS_THM]] THEN X_GEN_TAC `s:real^N->bool` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `s:real^N->bool`)) THEN ASM_REWRITE_TAC[] THEN REPLICATE_TAC 2 STRIP_TAC THEN FIRST_X_ASSUM(K ALL_TAC o check (is_forall o concl)) THEN ASM_SIMP_TAC[COMPACT_DIFF; COMPACT_UNION; COMPACT_PATH_IMAGE] THEN REWRITE_TAC[SET_RULE `(s UNION t) DIFF u = {} <=> s SUBSET u /\ t SUBSET u`] THEN STRIP_TAC THEN MATCH_MP_TAC HOMOTOPIC_LOOPS_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[simply_connected]) THEN ASM_REWRITE_TAC[]);; (* ------------------------------------------------------------------------- *) (* The (carrier of) the fundamental group: homotopy classes of loops at a. *) (* ------------------------------------------------------------------------- *) let fundamental_group = new_definition `fundamental_group(s,a:real^N) = { homotopic_paths s p | p | path p /\ path_image p SUBSET s /\ pathstart p = a /\ pathfinish p = a}`;; let FUNDAMENTAL_GROUP_EQ_EMPTY = prove (`!s a:real^N. fundamental_group (s,a) = {} <=> ~(a IN s)`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[SET_RULE `s = {} <=> !x. x IN s ==> F`] THEN PURE_REWRITE_TAC[fundamental_group; FORALL_IN_GSPEC] THEN EQ_TAC THENL [ALL_TAC; MESON_TAC[SUBSET; PATHSTART_IN_PATH_IMAGE]] THEN DISCH_THEN(MP_TAC o SPEC `linepath(a:real^N,a)`) THEN REWRITE_TAC[PATHSTART_LINEPATH; PATHFINISH_LINEPATH] THEN REWRITE_TAC[PATH_LINEPATH; PATH_IMAGE_LINEPATH; SEGMENT_REFL] THEN SET_TAC[]);; let CARD_EQ_FUNDAMENTAL_GROUPS_BASEPOINTS = prove (`!s a b:real^N. path_connected s /\ a IN s /\ b IN s ==> fundamental_group(s,a) =_c fundamental_group(s,b)`, let lemma = prove (`!g:real^1->real^N. path g /\ path_image g SUBSET s ==> homotopic_paths s g ((@) (homotopic_paths s g)) /\ path ((@) (homotopic_paths s g)) /\ path_image ((@) (homotopic_paths s g)) SUBSET s /\ pathstart ((@) (homotopic_paths s g)) = pathstart g /\ pathfinish ((@) (homotopic_paths s g)) = pathfinish g`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM ETA_AX] THEN CONV_TAC SELECT_CONV THEN ASM_MESON_TAC[HOMOTOPIC_PATHS_REFL]; ASM MESON_TAC[HOMOTOPIC_PATHS_IMP_PATH; HOMOTOPIC_PATHS_IMP_SUBSET; HOMOTOPIC_PATHS_IMP_PATHSTART; HOMOTOPIC_PATHS_IMP_PATHFINISH]]) and tac = ASM_SIMP_TAC[HOMOTOPIC_PATHS_REFL; PATHSTART_JOIN; PATHFINISH_JOIN; PATH_REVERSEPATH; SUBSET_PATH_IMAGE_JOIN; PATHSTART_REVERSEPATH; PATHFINISH_REVERSEPATH; PATH_IMAGE_REVERSEPATH; PATH_JOIN] in REWRITE_TAC[GSYM CARD_LE_ANTISYM; FORALL_AND_THM; TAUT `p ==> q /\ r <=> (p ==> q) /\ (p ==> r)`] THEN MATCH_MP_TAC(TAUT `(a ==> b) /\ a ==> a /\ b`) THEN CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[le_c] THEN SUBGOAL_THEN `?f:real^1->real^N. path f /\ path_image f SUBSET s /\ pathstart f = a /\ pathfinish f = b` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[path_connected]; ALL_TAC] THEN REWRITE_TAC[fundamental_group; FORALL_IN_GSPEC; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN EXISTS_TAC `\g. homotopic_paths (s:real^N->bool) (reversepath f ++ (@) g ++ f)` THEN CONJ_TAC THEN X_GEN_TAC `g:real^1->real^N` THEN REPEAT DISCH_TAC THEN MP_TAC(ISPEC `g:real^1->real^N` lemma) THEN ASM_REWRITE_TAC[] THEN ABBREV_TAC `g':real^1->real^N = (@) (homotopic_paths s g)` THEN STRIP_TAC THENL [MATCH_MP_TAC(SET_RULE `P q ==> homotopic_paths s q IN {homotopic_paths s p | P p}`) THEN tac; X_GEN_TAC `h:real^1->real^N` THEN REPLICATE_TAC 4 DISCH_TAC THEN MP_TAC(ISPEC `h:real^1->real^N` lemma) THEN ASM_REWRITE_TAC[] THEN ABBREV_TAC `h':real^1->real^N = (@) (homotopic_paths s h)` THEN STRIP_TAC THEN DISCH_THEN(MP_TAC o C AP_THM`reversepath f ++ h' ++ f:real^1->real^N`) THEN tac THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMOTOPIC_PATHS_LCANCEL)) THEN tac THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMOTOPIC_PATHS_RCANCEL)) THEN tac THEN REWRITE_TAC[FUN_EQ_THM] THEN ASM_MESON_TAC[HOMOTOPIC_PATHS_SYM; HOMOTOPIC_PATHS_TRANS]]);; let SIMPLY_CONNECTED_FUNDAMENTAL_GROUP = prove (`!s:real^N->bool. simply_connected s <=> path_connected s /\ !a. a IN s ==> fundamental_group(s,a) = {homotopic_paths s (linepath(a,a))}`, GEN_TAC THEN REWRITE_TAC[SIMPLY_CONNECTED_EQ_CONTRACTIBLE_PATH] THEN ASM_CASES_TAC `path_connected(s:real^N->bool)` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SET_RULE `s = {a} <=> a IN s /\ !x. x IN s ==> x = a`] THEN REWRITE_TAC[fundamental_group; FORALL_IN_GSPEC] THEN SIMP_TAC[PATH_LINEPATH; PATHSTART_LINEPATH; PATHFINISH_LINEPATH; PATH_IMAGE_LINEPATH; SEGMENT_REFL; SING_SUBSET; SET_RULE `P q ==> homotopic_paths s q IN {homotopic_paths s p | P p}`] THEN REWRITE_TAC[RIGHT_IMP_FORALL_THM; IMP_IMP; GSYM CONJ_ASSOC] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r /\ s /\ t <=> t /\ q /\ p /\ r /\ s`] THEN ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN REWRITE_TAC[IMP_CONJ; FORALL_UNWIND_THM1] THEN REWRITE_TAC[MESON[PATHFINISH_IN_PATH_IMAGE; SUBSET] `pathfinish p IN s ==> path_image p SUBSET s ==> P <=> path_image p SUBSET s ==> P`] THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN REWRITE_TAC[ISPEC `pathfinish p:real^N` EQ_SYM_EQ] THEN X_GEN_TAC `p:real^1->real^N` THEN REPEAT (MATCH_MP_TAC(TAUT `(p ==> (q <=> r)) ==> ((p ==> q) <=> (p ==> r))`) THEN DISCH_TAC) THEN REPEAT(EQ_TAC ORELSE STRIP_TAC) THEN ASM_REWRITE_TAC[HOMOTOPIC_PATHS_REFL; PATH_LINEPATH] THEN REWRITE_TAC[PATH_IMAGE_LINEPATH; SEGMENT_REFL; SING_SUBSET] THEN ASM_MESON_TAC[HOMOTOPIC_PATHS_TRANS; HOMOTOPIC_PATHS_SYM; PATHFINISH_IN_PATH_IMAGE; SUBSET]);; let FUNDAMENTAL_GROUP_SIMPLY_CONNECTED = prove (`!s a:real^N. simply_connected s /\ a IN s ==> fundamental_group(s,a) = {homotopic_paths s (linepath(a,a))}`, SIMP_TAC[SIMPLY_CONNECTED_FUNDAMENTAL_GROUP]);; (* ------------------------------------------------------------------------- *) (* A mapping out of a sphere is nullhomotopic iff it extends to the ball. *) (* This even works out in the degenerate cases when the radius is <= 0, and *) (* we also don't need to explicitly assume continuity since it's already *) (* implicit in both sides of the equivalence. *) (* ------------------------------------------------------------------------- *) let NULLHOMOTOPIC_FROM_SPHERE_EXTENSION = prove (`!f:real^M->real^N s a r. (?c. homotopic_with (\x. T) (subtopology euclidean (sphere(a,r)), subtopology euclidean s) f (\x. c)) <=> (?g. g continuous_on cball(a,r) /\ IMAGE g (cball(a,r)) SUBSET s /\ !x. x IN sphere(a,r) ==> g x = f x)`, let lemma = prove (`!f:real^M->real^N g a r. (!e. &0 < e ==> ?d. &0 < d /\ !x. ~(x = a) /\ norm(x - a) < d ==> norm(g x - f a) < e) /\ g continuous_on (cball(a,r) DELETE a) /\ (!x. x IN cball(a,r) /\ ~(x = a) ==> f x = g x) ==> f continuous_on cball(a,r)`, REPEAT STRIP_TAC THEN REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_CBALL; dist] THEN STRIP_TAC THEN ASM_CASES_TAC `x:real^M = a` THENL [ASM_REWRITE_TAC[continuous_within; IN_CBALL; dist] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_CBALL; dist]) THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `y:real^M` THEN ASM_CASES_TAC `y:real^M = a` THEN ASM_MESON_TAC[VECTOR_SUB_REFL; NORM_0]; MATCH_MP_TAC CONTINUOUS_TRANSFORM_WITHIN THEN EXISTS_TAC `g:real^M->real^N` THEN EXISTS_TAC `norm(x - a:real^M)` THEN ASM_SIMP_TAC[NORM_POS_LT; VECTOR_SUB_EQ; IN_CBALL; dist] THEN CONJ_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[IN_CBALL; dist]); UNDISCH_TAC `(g:real^M->real^N) continuous_on (cball(a,r) DELETE a)` THEN REWRITE_TAC[continuous_on; continuous_within] THEN DISCH_THEN(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[IN_DELETE; IN_CBALL; dist] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `e:real` THEN ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `min d (norm(x - a:real^M))` THEN ASM_REWRITE_TAC[REAL_LT_MIN; NORM_POS_LT; VECTOR_SUB_EQ]] THEN ASM_MESON_TAC[NORM_SUB; NORM_ARITH `norm(y - x:real^N) < norm(x - a) ==> ~(y = a)`]]) in REWRITE_TAC[sphere; ONCE_REWRITE_RULE[DIST_SYM] dist] THEN REPEAT GEN_TAC THEN REPEAT_TCL DISJ_CASES_THEN ASSUME_TAC (REAL_ARITH `r < &0 \/ r = &0 \/ &0 < r`) THENL [ASM_SIMP_TAC[NORM_ARITH `r < &0 ==> ~(norm x = r)`] THEN FIRST_ASSUM(ASSUME_TAC o GEN_REWRITE_RULE I [GSYM CBALL_EQ_EMPTY]) THEN ASM_SIMP_TAC[HOMOTOPIC_WITH_EUCLIDEAN_ALT; IMAGE_CLAUSES; EMPTY_GSPEC; NOT_IN_EMPTY; PCROSS; SET_RULE `{f t x |x,t| F} = {}`; EMPTY_SUBSET] THEN REWRITE_TAC[CONTINUOUS_ON_EMPTY]; ASM_SIMP_TAC[NORM_EQ_0; VECTOR_SUB_EQ; CBALL_SING] THEN SIMP_TAC[HOMOTOPIC_WITH_EUCLIDEAN_ALT; PCROSS; FORALL_IN_GSPEC; FORALL_UNWIND_THM2] THEN ASM_CASES_TAC `(f:real^M->real^N) a IN s` THENL [MATCH_MP_TAC(TAUT `p /\ q ==> (p <=> q)`) THEN CONJ_TAC THENL [EXISTS_TAC `(f:real^M->real^N) a` THEN EXISTS_TAC `\y:real^(1,M)finite_sum. (f:real^M->real^N) a` THEN ASM_REWRITE_TAC[CONTINUOUS_ON_CONST; SUBSET; FORALL_IN_IMAGE]; EXISTS_TAC `f:real^M->real^N` THEN REWRITE_TAC[CONTINUOUS_ON_SING] THEN ASM SET_TAC[]]; MATCH_MP_TAC(TAUT `~q /\ ~p ==> (p <=> q)`) THEN CONJ_TAC THENL [ASM SET_TAC[]; STRIP_TAC] THEN UNDISCH_TAC `~((f:real^M->real^N) a IN s)` THEN REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE h t SUBSET s ==> (?y. y IN t /\ z = h y) ==> z IN s`)) THEN REWRITE_TAC[EXISTS_IN_GSPEC] THEN EXISTS_TAC `vec 0:real^1` THEN ASM_SIMP_TAC[ENDS_IN_UNIT_INTERVAL] THEN ASM_REWRITE_TAC[EXISTS_IN_GSPEC; UNWIND_THM2]]; ALL_TAC] THEN MATCH_MP_TAC(TAUT `!p. (q ==> p) /\ (r ==> p) /\ (p ==> (q <=> r)) ==> (q <=> r)`) THEN EXISTS_TAC `(f:real^M->real^N) continuous_on {x | norm(x - a) = r} /\ IMAGE f {x | norm(x - a) = r} SUBSET s` THEN REPEAT CONJ_TAC THENL [STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_SUBSET) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_CONTINUOUS) THEN ASM_REWRITE_TAC[]; DISCH_THEN(X_CHOOSE_THEN `g:real^M->real^N` STRIP_ASSUME_TAC) THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_EQ THEN EXISTS_TAC `g:real^M->real^N` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `cball(a:real^M,r)`; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE g t SUBSET s ==> u SUBSET t /\ (!x. x IN u ==> f x = g x) ==> IMAGE f u SUBSET s`)) THEN ASM_SIMP_TAC[]] THEN ASM_SIMP_TAC[SUBSET; IN_CBALL; dist; IN_ELIM_THM] THEN MESON_TAC[REAL_LE_REFL; NORM_SUB]; STRIP_TAC] THEN ONCE_REWRITE_TAC[HOMOTOPIC_WITH_SYM] THEN EQ_TAC THENL [REWRITE_TAC[HOMOTOPIC_WITH_EUCLIDEAN; PCROSS; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`c:real^N`; `h:real^(1,M)finite_sum->real^N`] THEN STRIP_TAC THEN EXISTS_TAC `\x. (h:real^(1,M)finite_sum->real^N) (pastecart (lift(inv(r) * norm(x - a))) (a + (if x = a then r % basis 1 else r / norm(x - a) % (x - a))))` THEN ASM_SIMP_TAC[IN_ELIM_THM; REAL_MUL_LINV; REAL_DIV_REFL; REAL_LT_IMP_NZ; LIFT_NUM; VECTOR_ARITH `a + &1 % (x - a):real^N = x`] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC lemma THEN EXISTS_TAC `\x. (h:real^(1,M)finite_sum->real^N) (pastecart (lift(inv(r) * norm(x - a))) (a + r / norm(x - a) % (x - a)))` THEN SIMP_TAC[] THEN CONJ_TAC THENL [X_GEN_TAC `e:real` THEN DISCH_TAC THEN ASM_REWRITE_TAC[VECTOR_SUB_REFL; NORM_0; REAL_MUL_RZERO; LIFT_NUM] THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] COMPACT_UNIFORMLY_CONTINUOUS)) THEN SIMP_TAC[REWRITE_RULE[PCROSS] COMPACT_PCROSS; REWRITE_RULE[REWRITE_RULE[ONCE_REWRITE_RULE[DIST_SYM] dist] sphere] COMPACT_SPHERE; COMPACT_INTERVAL] THEN REWRITE_TAC[uniformly_continuous_on] THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_GSPEC] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `min r (d * r):real` THEN ASM_SIMP_TAC[REAL_LT_MUL; REAL_LT_MIN] THEN X_GEN_TAC `x:real^M` THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `vec 0:real^1`) THEN REWRITE_TAC[ENDS_IN_UNIT_INTERVAL; RIGHT_IMP_FORALL_THM] THEN ASM_REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC] THEN DISCH_THEN(MP_TAC o MATCH_MP (MESON[] `(!x t y. P x t y) ==> (!t x. P x t x)`)) THEN REWRITE_TAC[dist] THEN DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[IN_INTERVAL_1; LIFT_DROP; DROP_VEC] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_LE_LDIV_EQ] THEN ASM_SIMP_TAC[REAL_MUL_LID; REAL_MUL_LZERO; NORM_POS_LE] THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; CONJ_ASSOC] THEN REWRITE_TAC[VECTOR_ADD_SUB; NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM] THEN ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_EQ_0; VECTOR_SUB_EQ] THEN ASM_SIMP_TAC[REAL_ARITH `&0 < r ==> abs r = r`] THEN REWRITE_TAC[PASTECART_SUB; VECTOR_SUB_REFL; NORM_PASTECART] THEN REWRITE_TAC[NORM_0; VECTOR_SUB_RZERO] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[REAL_ADD_RID] THEN REWRITE_TAC[POW_2_SQRT_ABS; REAL_ABS_NORM; NORM_LIFT] THEN ASM_SIMP_TAC[REAL_ABS_DIV; REAL_LT_LDIV_EQ; REAL_ABS_NORM; REAL_ARITH `&0 < r ==> abs r = r`]; GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_PASTECART THEN SIMP_TAC[CONTINUOUS_ON_CMUL; LIFT_CMUL; CONTINUOUS_ON_SUB; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST; CONTINUOUS_ON_LIFT_NORM_COMPOSE] THEN MATCH_MP_TAC CONTINUOUS_ON_ADD THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST; o_DEF; real_div; LIFT_CMUL] THEN MATCH_MP_TAC CONTINUOUS_ON_CMUL THEN REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN GEN_TAC THEN REWRITE_TAC[IN_DELETE] THEN DISCH_TAC THEN MATCH_MP_TAC CONTINUOUS_AT_WITHIN THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_INV) THEN ASM_SIMP_TAC[NETLIMIT_AT; NORM_EQ_0; VECTOR_SUB_EQ] THEN MATCH_MP_TAC CONTINUOUS_LIFT_NORM_COMPOSE THEN SIMP_TAC[CONTINUOUS_SUB; CONTINUOUS_AT_ID; CONTINUOUS_CONST]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[FORALL_IN_IMAGE; FORALL_IN_GSPEC; SUBSET] THEN REWRITE_TAC[IN_ELIM_PASTECART_THM; IN_DELETE; IN_ELIM_THM] THEN SIMP_TAC[IN_CBALL; NORM_ARITH `dist(a:real^M,a + x) = norm x`] THEN REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] dist] THEN REWRITE_TAC[IN_INTERVAL_1; LIFT_DROP; DROP_VEC] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_LE_LDIV_EQ] THEN ASM_SIMP_TAC[REAL_MUL_LID; REAL_MUL_LZERO; NORM_POS_LE] THEN SIMP_TAC[VECTOR_ADD_SUB; NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM; REAL_DIV_RMUL; NORM_EQ_0; VECTOR_SUB_EQ] THEN ASM_REAL_ARITH_TAC]]; GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM o_DEF] THEN REWRITE_TAC[IMAGE_o] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE g s SUBSET u ==> t SUBSET s ==> IMAGE g t SUBSET u`)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN REWRITE_TAC[IN_ELIM_PASTECART_THM; IN_CBALL; IN_ELIM_THM] THEN X_GEN_TAC `x:real^M` THEN REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] dist] THEN REPEAT STRIP_TAC THENL [REWRITE_TAC[IN_INTERVAL_1; LIFT_DROP; DROP_VEC] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_LE_LDIV_EQ] THEN ASM_REWRITE_TAC[REAL_MUL_LID; REAL_MUL_LZERO; NORM_POS_LE]; REWRITE_TAC[VECTOR_ADD_SUB] THEN COND_CASES_TAC THEN ASM_SIMP_TAC[NORM_MUL; NORM_BASIS; DIMINDEX_GE_1; LE_REFL; REAL_ABS_DIV; REAL_ABS_NORM; REAL_MUL_RID; REAL_ARITH `&0 < r ==> abs r = r`] THEN ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_EQ_0; VECTOR_SUB_EQ]]; GEN_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[VECTOR_SUB_REFL; NORM_0; REAL_LT_IMP_NZ] THEN REWRITE_TAC[VECTOR_ARITH `a + &1 % (x - a):real^N = x`]]; DISCH_THEN(X_CHOOSE_THEN `g:real^M->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(g:real^M->real^N) a` THEN ASM_SIMP_TAC[HOMOTOPIC_WITH_EUCLIDEAN_ALT; PCROSS] THEN EXISTS_TAC `\y:real^(1,M)finite_sum. (g:real^M->real^N) (a + drop(fstcart y) % (sndcart y - a))` THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; DROP_VEC] THEN REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_RID; VECTOR_MUL_LID] THEN ASM_SIMP_TAC[VECTOR_SUB_ADD2] THEN CONJ_TAC THENL [GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_ADD THEN SIMP_TAC[CONTINUOUS_ON_CONST] THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN SIMP_TAC[o_DEF; LIFT_DROP; CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; LINEAR_CONTINUOUS_ON; LINEAR_SNDCART; LINEAR_FSTCART; ETA_AX]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET))]; GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM o_DEF] THEN REWRITE_TAC[IMAGE_o] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE g s SUBSET u ==> t SUBSET s ==> IMAGE g t SUBSET u`))] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; IN_ELIM_THM] THEN REWRITE_TAC[IN_CBALL; NORM_ARITH `dist(a:real^M,a + x) = norm x`] THEN ASM_SIMP_TAC[NORM_MUL; IN_INTERVAL_1; DROP_VEC; REAL_LE_RMUL_EQ; REAL_ARITH `x * r <= r <=> x * r <= &1 * r`] THEN REAL_ARITH_TAC]);; (* ------------------------------------------------------------------------- *) (* Homotopy equivalence. *) (* ------------------------------------------------------------------------- *) parse_as_infix("homotopy_equivalent",(12,"right"));; let homotopy_equivalent = new_definition `(s:real^M->bool) homotopy_equivalent (t:real^N->bool) <=> ?f g. f continuous_on s /\ IMAGE f s SUBSET t /\ g continuous_on t /\ IMAGE g t SUBSET s /\ homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean s) (g o f) I /\ homotopic_with (\x. T) (subtopology euclidean t,subtopology euclidean t) (f o g) I`;; let HOMOTOPY_EQUIVALENT_SPACE_EUCLIDEAN = prove (`!(s:real^M->bool) (t:real^N->bool). (subtopology euclidean s) homotopy_equivalent_space (subtopology euclidean t) <=> s homotopy_equivalent t`, REWRITE_TAC[homotopy_equivalent_space; homotopy_equivalent] THEN REWRITE_TAC[CONTINUOUS_MAP_EUCLIDEAN2] THEN MESON_TAC[]);; let HOMOTOPY_EQUIVALENT = prove (`!s:real^M->bool t:real^N->bool. s homotopy_equivalent t <=> ?f g h. f continuous_on s /\ IMAGE f s SUBSET t /\ g continuous_on t /\ IMAGE g t SUBSET s /\ h continuous_on t /\ IMAGE h t SUBSET s /\ homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean s) (g o f) I /\ homotopic_with (\x. T) (subtopology euclidean t,subtopology euclidean t) (f o h) I`, REPEAT GEN_TAC THEN REWRITE_TAC[homotopy_equivalent] THEN MATCH_MP_TAC(MESON[] `(!x. P x <=> Q x) ==> ((?x. P x) <=> (?x. Q x))`) THEN X_GEN_TAC `f:real^M->real^N` THEN EQ_TAC THENL [MESON_TAC[]; STRIP_TAC] THEN EXISTS_TAC `(g:real^N->real^M) o f o (h:real^N->real^M)` THEN ASM_REWRITE_TAC[IMAGE_o] THEN REPEAT CONJ_TAC THENL [REPEAT(MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC) THEN REWRITE_TAC[IMAGE_o] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; ASM SET_TAC[]; TRANS_TAC HOMOTOPIC_WITH_TRANS `((g:real^N->real^M) o I) o (f:real^M->real^N)` THEN CONJ_TAC THENL [ALL_TAC; ASM_REWRITE_TAC[I_O_ID]] THEN MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_RIGHT THEN EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_LEFT THEN EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[]; TRANS_TAC HOMOTOPIC_WITH_TRANS `(f:real^M->real^N) o I o (h:real^N->real^M)` THEN CONJ_TAC THENL [ALL_TAC; ASM_REWRITE_TAC[I_O_ID]] THEN MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_LEFT THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[o_ASSOC] THEN MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_RIGHT THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[]]);; let HOMEOMORPHIC_IMP_HOMOTOPY_EQUIVALENT = prove (`!s:real^M->bool t:real^N->bool. s homeomorphic t ==> s homotopy_equivalent t`, REWRITE_TAC[GSYM HOMEOMORPHIC_SPACE_EUCLIDEAN; GSYM HOMOTOPY_EQUIVALENT_SPACE_EUCLIDEAN] THEN REWRITE_TAC[HOMEOMORPHIC_IMP_HOMOTOPY_EQUIVALENT_SPACE]);; let HOMOTOPY_EQUIVALENT_REFL = prove (`!s:real^N->bool. s homotopy_equivalent s`, SIMP_TAC[HOMEOMORPHIC_IMP_HOMOTOPY_EQUIVALENT; HOMEOMORPHIC_REFL]);; let HOMOTOPY_EQUIVALENT_SYM = prove (`!s:real^M->bool t:real^N->bool. s homotopy_equivalent t <=> t homotopy_equivalent s`, REWRITE_TAC[GSYM HOMOTOPY_EQUIVALENT_SPACE_EUCLIDEAN] THEN REWRITE_TAC[HOMOTOPY_EQUIVALENT_SPACE_SYM]);; let HOMOTOPY_EQUIVALENT_TRANS = prove (`!s:real^M->bool t:real^N->bool u:real^P->bool. s homotopy_equivalent t /\ t homotopy_equivalent u ==> s homotopy_equivalent u`, REWRITE_TAC[GSYM HOMOTOPY_EQUIVALENT_SPACE_EUCLIDEAN] THEN REWRITE_TAC[HOMOTOPY_EQUIVALENT_SPACE_TRANS]);; let HOMOTOPY_EQUIVALENT_PCROSS = prove (`!s:real^M->bool t:real^N->bool s':real^P->bool t':real^Q->bool. s homotopy_equivalent s' /\ t homotopy_equivalent t' ==> s PCROSS t homotopy_equivalent s' PCROSS t'`, REPEAT GEN_TAC THEN REWRITE_TAC[homotopy_equivalent] THEN REWRITE_TAC[LEFT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f1:real^M->real^P`; `g1:real^P->real^M`; `f2:real^N->real^Q`; `g2:real^Q->real^N`] THEN DISCH_TAC THEN EXISTS_TAC `\z. pastecart ((f1:real^M->real^P) (fstcart z)) ((f2:real^N->real^Q) (sndcart z))` THEN EXISTS_TAC `\z. pastecart ((g1:real^P->real^M) (fstcart z)) ((g2:real^Q->real^N) (sndcart z))` THEN FIRST_X_ASSUM(CONJUNCTS_THEN (CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC o GEN_REWRITE_RULE I [TAUT `p /\ q /\ r /\ s /\ t <=> (p /\ q /\ r /\ s) /\ t`])) THEN ONCE_REWRITE_TAC[TAUT `p /\ q ==> p' /\ q' ==> r <=> (p' /\ p) /\ (q' /\ q) ==> r`] THEN DISCH_THEN(CONJUNCTS_THEN(MP_TAC o MATCH_MP(MESON[] `(!x. p x) ==> (!x. p(\a. T))`) o MATCH_MP (ONCE_REWRITE_RULE[TAUT `p /\ q /\ r ==> s <=> p /\ q ==> r ==> s`] HOMOTOPIC_WITH_PCROSS))) THEN REWRITE_TAC[I_DEF; PASTECART_FST_SND; o_DEF] THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN DISCH_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r /\ s <=> (p /\ r) /\ (q /\ s)`] THEN CONJ_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_PASTECART THEN CONJ_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; LINEAR_SNDCART] THEN REWRITE_TAC[IMAGE_FSTCART_PCROSS; IMAGE_SNDCART_PCROSS] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[CONTINUOUS_ON_EMPTY]; CONJ_TAC THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_PCROSS] THEN REWRITE_TAC[PASTECART_IN_PCROSS; FSTCART_PASTECART; SNDCART_PASTECART] THEN ASM SET_TAC[]]);; let HOMOTOPY_EQUIVALENT_INJECTIVE_LINEAR_IMAGE_SELF = prove (`!f:real^M->real^N s. linear f /\ (!x y. f x = f y ==> x = y) ==> (IMAGE f s) homotopy_equivalent s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_IMP_HOMOTOPY_EQUIVALENT THEN MATCH_MP_TAC HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_SELF THEN ASM_REWRITE_TAC[]);; let HOMOTOPY_EQUIVALENT_INJECTIVE_LINEAR_IMAGE_LEFT_EQ = prove (`!f:real^M->real^N s t. linear f /\ (!x y. f x = f y ==> x = y) ==> ((IMAGE f s) homotopy_equivalent t <=> s homotopy_equivalent t)`, REPEAT GEN_TAC THEN DISCH_THEN(ASSUME_TAC o SPEC `s:real^M->bool` o MATCH_MP HOMOTOPY_EQUIVALENT_INJECTIVE_LINEAR_IMAGE_SELF) THEN EQ_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMOTOPY_EQUIVALENT_SYM]); POP_ASSUM MP_TAC] THEN REWRITE_TAC[IMP_IMP; HOMOTOPY_EQUIVALENT_TRANS]);; let HOMOTOPY_EQUIVALENT_INJECTIVE_LINEAR_IMAGE_RIGHT_EQ = prove (`!f:real^M->real^N s t. linear f /\ (!x y. f x = f y ==> x = y) ==> (s homotopy_equivalent (IMAGE f t) <=> s homotopy_equivalent t)`, ONCE_REWRITE_TAC[HOMOTOPY_EQUIVALENT_SYM] THEN REWRITE_TAC[HOMOTOPY_EQUIVALENT_INJECTIVE_LINEAR_IMAGE_LEFT_EQ]);; add_linear_invariants [HOMOTOPY_EQUIVALENT_INJECTIVE_LINEAR_IMAGE_LEFT_EQ; HOMOTOPY_EQUIVALENT_INJECTIVE_LINEAR_IMAGE_RIGHT_EQ];; let HOMOTOPY_EQUIVALENT_TRANSLATION_SELF = prove (`!a:real^N s. (IMAGE (\x. a + x) s) homotopy_equivalent s`, REPEAT GEN_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_IMP_HOMOTOPY_EQUIVALENT THEN REWRITE_TAC[HOMEOMORPHIC_TRANSLATION_SELF]);; let HOMOTOPY_EQUIVALENT_TRANSLATION_LEFT_EQ = prove (`!a:real^N s t. (IMAGE (\x. a + x) s) homotopy_equivalent t <=> s homotopy_equivalent t`, MESON_TAC[HOMOTOPY_EQUIVALENT_TRANSLATION_SELF; HOMOTOPY_EQUIVALENT_SYM; HOMOTOPY_EQUIVALENT_TRANS]);; let HOMOTOPY_EQUIVALENT_TRANSLATION_RIGHT_EQ = prove (`!a:real^N s t. s homotopy_equivalent (IMAGE (\x. a + x) t) <=> s homotopy_equivalent t`, ONCE_REWRITE_TAC[HOMOTOPY_EQUIVALENT_SYM] THEN REWRITE_TAC[HOMOTOPY_EQUIVALENT_TRANSLATION_LEFT_EQ]);; add_translation_invariants [HOMOTOPY_EQUIVALENT_TRANSLATION_LEFT_EQ; HOMOTOPY_EQUIVALENT_TRANSLATION_RIGHT_EQ];; let HOMOTOPY_EQUIVALENT_HOMOTOPIC_TRIVIALITY = prove (`!s:real^M->bool t:real^N->bool u:real^P->bool. s homotopy_equivalent t ==> ((!f g. f continuous_on u /\ IMAGE f u SUBSET s /\ g continuous_on u /\ IMAGE g u SUBSET s ==> homotopic_with (\x. T) (subtopology euclidean u,subtopology euclidean s) f g) <=> (!f g. f continuous_on u /\ IMAGE f u SUBSET t /\ g continuous_on u /\ IMAGE g u SUBSET t ==> homotopic_with (\x. T) (subtopology euclidean u,subtopology euclidean t) f g))`, let lemma = prove (`!s:real^M->bool t:real^N->bool u:real^P->bool. s homotopy_equivalent t /\ (!f g. f continuous_on u /\ IMAGE f u SUBSET s /\ g continuous_on u /\ IMAGE g u SUBSET s ==> homotopic_with (\x. T) (subtopology euclidean u,subtopology euclidean s) f g) ==> (!f g. f continuous_on u /\ IMAGE f u SUBSET t /\ g continuous_on u /\ IMAGE g u SUBSET t ==> homotopic_with (\x. T) (subtopology euclidean u,subtopology euclidean t) f g)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homotopy_equivalent]) THEN DISCH_THEN(X_CHOOSE_THEN `h:real^M->real^N` (X_CHOOSE_THEN `k:real^N->real^M` STRIP_ASSUME_TAC)) THEN SUBGOAL_THEN `homotopic_with (\x. T) (subtopology euclidean u,subtopology euclidean t) ((h:real^M->real^N) o (k:real^N->real^M) o (f:real^P->real^N)) (h o k o g)` MP_TAC THENL [MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_LEFT THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[IMAGE_o] THEN REPEAT CONJ_TAC THEN TRY(MATCH_MP_TAC CONTINUOUS_ON_COMPOSE) THEN ASM_REWRITE_TAC[] THEN TRY(FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET))) THEN ASM SET_TAC[]; MATCH_MP_TAC(MESON[HOMOTOPIC_WITH_TRANS; HOMOTOPIC_WITH_SYM] `homotopic_with P (subtopology euclidean u,subtopology euclidean t) f f' /\ homotopic_with P (subtopology euclidean u,subtopology euclidean t) g g' ==> homotopic_with P (subtopology euclidean u,subtopology euclidean t) f g ==> homotopic_with P (subtopology euclidean u,subtopology euclidean t) f' g'`) THEN CONJ_TAC THEN GEN_REWRITE_TAC RAND_CONV [GSYM(CONJUNCT1(SPEC_ALL I_O_ID))] THEN REWRITE_TAC[o_ASSOC] THEN MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_RIGHT THEN EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[]]) in REPEAT STRIP_TAC THEN EQ_TAC THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] lemma) THEN ASM_MESON_TAC[HOMOTOPY_EQUIVALENT_SYM]);; let HOMOTOPY_EQUIVALENT_COHOMOTOPIC_TRIVIALITY = prove (`!s:real^M->bool t:real^N->bool u:real^P->bool. s homotopy_equivalent t ==> ((!f g. f continuous_on s /\ IMAGE f s SUBSET u /\ g continuous_on s /\ IMAGE g s SUBSET u ==> homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean u) f g) <=> (!f g. f continuous_on t /\ IMAGE f t SUBSET u /\ g continuous_on t /\ IMAGE g t SUBSET u ==> homotopic_with (\x. T) (subtopology euclidean t,subtopology euclidean u) f g))`, let lemma = prove (`!s:real^M->bool t:real^N->bool u:real^P->bool. s homotopy_equivalent t /\ (!f g. f continuous_on s /\ IMAGE f s SUBSET u /\ g continuous_on s /\ IMAGE g s SUBSET u ==> homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean u) f g) ==> (!f g. f continuous_on t /\ IMAGE f t SUBSET u /\ g continuous_on t /\ IMAGE g t SUBSET u ==> homotopic_with (\x. T) (subtopology euclidean t,subtopology euclidean u) f g)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homotopy_equivalent]) THEN DISCH_THEN(X_CHOOSE_THEN `h:real^M->real^N` (X_CHOOSE_THEN `k:real^N->real^M` STRIP_ASSUME_TAC)) THEN SUBGOAL_THEN `homotopic_with (\x. T) (subtopology euclidean t,subtopology euclidean u) (((f:real^N->real^P) o h) o (k:real^N->real^M)) ((g o h) o k)` MP_TAC THENL [MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_RIGHT THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[IMAGE_o] THEN REPEAT CONJ_TAC THEN TRY(MATCH_MP_TAC CONTINUOUS_ON_COMPOSE) THEN ASM_REWRITE_TAC[] THEN TRY(FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET))) THEN ASM SET_TAC[]; MATCH_MP_TAC(MESON[HOMOTOPIC_WITH_TRANS; HOMOTOPIC_WITH_SYM] `homotopic_with P (subtopology euclidean u,subtopology euclidean t) f f' /\ homotopic_with P (subtopology euclidean u,subtopology euclidean t) g g' ==> homotopic_with P (subtopology euclidean u,subtopology euclidean t) f g ==> homotopic_with P (subtopology euclidean u,subtopology euclidean t) f' g'`) THEN CONJ_TAC THEN GEN_REWRITE_TAC RAND_CONV [GSYM(CONJUNCT2(SPEC_ALL I_O_ID))] THEN REWRITE_TAC[GSYM o_ASSOC] THEN MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_LEFT THEN EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[]]) in REPEAT STRIP_TAC THEN EQ_TAC THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] lemma) THEN ASM_MESON_TAC[HOMOTOPY_EQUIVALENT_SYM]);; let HOMOTOPY_EQUIVALENT_HOMOTOPIC_TRIVIALITY_NULL = prove (`!s:real^M->bool t:real^N->bool u:real^P->bool. s homotopy_equivalent t ==> ((!f. f continuous_on u /\ IMAGE f u SUBSET s ==> ?c. homotopic_with (\x. T) (subtopology euclidean u,subtopology euclidean s) f (\x. c)) <=> (!f. f continuous_on u /\ IMAGE f u SUBSET t ==> ?c. homotopic_with (\x. T) (subtopology euclidean u,subtopology euclidean t) f (\x. c)))`, let lemma = prove (`!s:real^M->bool t:real^N->bool u:real^P->bool. s homotopy_equivalent t /\ (!f. f continuous_on u /\ IMAGE f u SUBSET s ==> ?c. homotopic_with (\x. T) (subtopology euclidean u,subtopology euclidean s) f (\x. c)) ==> (!f. f continuous_on u /\ IMAGE f u SUBSET t ==> ?c. homotopic_with (\x. T) (subtopology euclidean u,subtopology euclidean t) f (\x. c))`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homotopy_equivalent]) THEN DISCH_THEN(X_CHOOSE_THEN `h:real^M->real^N` (X_CHOOSE_THEN `k:real^N->real^M` STRIP_ASSUME_TAC)) THEN FIRST_X_ASSUM(MP_TAC o SPEC `(k:real^N->real^M) o (f:real^P->real^N)`) THEN REWRITE_TAC[IMAGE_o] THEN ANTS_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE; ASM SET_TAC[]] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; DISCH_THEN(X_CHOOSE_TAC `c:real^M`) THEN EXISTS_TAC `(h:real^M->real^N) c`] THEN SUBGOAL_THEN `homotopic_with (\x. T) (subtopology euclidean u,subtopology euclidean t) ((h:real^M->real^N) o (k:real^N->real^M) o (f:real^P->real^N)) (h o (\x. c))` MP_TAC THENL [MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_LEFT THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[]; GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [o_DEF] THEN REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] HOMOTOPIC_WITH_TRANS) THEN GEN_REWRITE_TAC LAND_CONV [GSYM(CONJUNCT1(SPEC_ALL I_O_ID))] THEN REWRITE_TAC[o_ASSOC] THEN ONCE_REWRITE_TAC[HOMOTOPIC_WITH_SYM] THEN MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_RIGHT THEN EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[]]) in REPEAT STRIP_TAC THEN EQ_TAC THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] lemma) THEN ASM_MESON_TAC[HOMOTOPY_EQUIVALENT_SYM]);; let HOMOTOPY_EQUIVALENT_COHOMOTOPIC_TRIVIALITY_NULL = prove (`!s:real^M->bool t:real^N->bool u:real^P->bool. s homotopy_equivalent t ==> ((!f. f continuous_on s /\ IMAGE f s SUBSET u ==> ?c. homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean u) f (\x. c)) <=> (!f. f continuous_on t /\ IMAGE f t SUBSET u ==> ?c. homotopic_with (\x. T) (subtopology euclidean t,subtopology euclidean u) f (\x. c)))`, let lemma = prove (`!s:real^M->bool t:real^N->bool u:real^P->bool. s homotopy_equivalent t /\ (!f. f continuous_on s /\ IMAGE f s SUBSET u ==> ?c. homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean u) f (\x. c)) ==> (!f. f continuous_on t /\ IMAGE f t SUBSET u ==> ?c. homotopic_with (\x. T) (subtopology euclidean t,subtopology euclidean u) f (\x. c))`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homotopy_equivalent]) THEN DISCH_THEN(X_CHOOSE_THEN `h:real^M->real^N` (X_CHOOSE_THEN `k:real^N->real^M` STRIP_ASSUME_TAC)) THEN FIRST_X_ASSUM(MP_TAC o SPEC `(f:real^N->real^P) o (h:real^M->real^N)`) THEN REWRITE_TAC[IMAGE_o] THEN ANTS_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE; ASM SET_TAC[]] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^P` THEN DISCH_TAC] THEN SUBGOAL_THEN `homotopic_with (\x. T) (subtopology euclidean t,subtopology euclidean u) (((f:real^N->real^P) o h) o (k:real^N->real^M)) ((\x. c) o k)` MP_TAC THENL [MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_RIGHT THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[]; GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [o_DEF] THEN REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] HOMOTOPIC_WITH_TRANS) THEN GEN_REWRITE_TAC LAND_CONV [GSYM(CONJUNCT2(SPEC_ALL I_O_ID))] THEN REWRITE_TAC[GSYM o_ASSOC] THEN ONCE_REWRITE_TAC[HOMOTOPIC_WITH_SYM] THEN MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_LEFT THEN EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[]]) in REPEAT STRIP_TAC THEN EQ_TAC THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] lemma) THEN ASM_MESON_TAC[HOMOTOPY_EQUIVALENT_SYM]);; let HOMOTOPIC_WITH_IMP_PATH_COMPONENT = prove (`!f g:real^M->real^N s t a. homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean t) f g /\ a IN s ==> path_component t (f a) (g a)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMOTOPIC_WITH_EUCLIDEAN]) THEN REWRITE_TAC[o_THM; I_THM] THEN DISCH_THEN(X_CHOOSE_THEN `h:real^(1,M)finite_sum->real^N` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC PATH_COMPONENT_OF_SUBSET THEN EXISTS_TAC `IMAGE (h:real^(1,M)finite_sum->real^N) (interval[vec 0,vec 1] PCROSS {a})` THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE f s' SUBSET t ==> s SUBSET s' ==> IMAGE f s SUBSET t`)) THEN REWRITE_TAC[SUBSET_PCROSS] THEN ASM SET_TAC[]; W(MP_TAC o fst o EQ_IMP_RULE o PART_MATCH (rand o lhand) PATH_CONNECTED_IFF_PATH_COMPONENT o lhand o rator o snd) THEN ANTS_TAC THENL [MATCH_MP_TAC PATH_CONNECTED_CONTINUOUS_IMAGE THEN REWRITE_TAC[PATH_CONNECTED_PCROSS_EQ; PATH_CONNECTED_INTERVAL; PATH_CONNECTED_SING] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET_PCROSS] THEN ASM SET_TAC[]; DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[IN_IMAGE; EXISTS_PASTECART; PASTECART_IN_PCROSS] THEN ASM_MESON_TAC[ENDS_IN_UNIT_INTERVAL; IN_SING]]]);; let HOMOTOPY_INVARIANT_CARD_COMPONENTS = prove (`!f:real^M->real^N g s t. f continuous_on s /\ IMAGE f s SUBSET t /\ g continuous_on t /\ IMAGE g t SUBSET s /\ homotopic_with (\x. T) (subtopology euclidean t,subtopology euclidean t) (f o g) I ==> components t <=_c components s`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_SIMP_TAC[SUBSET_EMPTY; IMAGE_EQ_EMPTY; COMPONENTS_EMPTY] THEN REWRITE_TAC[CARD_EMPTY_LE] THEN STRIP_TAC THEN MATCH_MP_TAC CARD_LE_RELATIONAL_FULL THEN EXISTS_TAC `\c d. IMAGE (g:real^N->real^M) d SUBSET c` THEN REWRITE_TAC[] THEN CONJ_TAC THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC EXISTS_COMPONENT_SUPERSET THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN ASM SET_TAC[]; MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN ASM_MESON_TAC[IN_COMPONENTS_SUBSET; CONTINUOUS_ON_SUBSET; IN_COMPONENTS_CONNECTED]]; ALL_TAC] THEN REWRITE_TAC[IMP_CONJ; components; FORALL_IN_GSPEC; RIGHT_FORALL_IMP_THM] THEN SIMP_TAC[CONNECTED_COMPONENT_EQ_EQ] THEN X_GEN_TAC `a:real^M` THEN DISCH_TAC THEN X_GEN_TAC `b:real^N` THEN DISCH_TAC THEN X_GEN_TAC `c:real^N` THEN DISCH_TAC THEN REPEAT DISCH_TAC THEN MATCH_MP_TAC(MESON[CONNECTED_COMPONENT_SYM; CONNECTED_COMPONENT_TRANS] `!f. connected_component s (f b) (f c) /\ connected_component s (f b) b /\ connected_component s (f c) c ==> connected_component s b c`) THEN EXISTS_TAC `(f:real^M->real^N) o (g:real^N->real^M)` THEN CONJ_TAC THENL [ALL_TAC; CONJ_TAC THEN MATCH_MP_TAC PATH_COMPONENT_IMP_CONNECTED_COMPONENT THEN GEN_REWRITE_TAC RAND_CONV [GSYM I_THM] THEN MATCH_MP_TAC HOMOTOPIC_WITH_IMP_PATH_COMPONENT THEN EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[ETA_AX]] THEN REWRITE_TAC[connected_component] THEN EXISTS_TAC `IMAGE (f:real^M->real^N) (connected_component s a)` THEN CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN REWRITE_TAC[CONNECTED_CONNECTED_COMPONENT] THEN ASM_MESON_TAC[CONNECTED_COMPONENT_SUBSET; CONTINUOUS_ON_SUBSET]; ALL_TAC] THEN CONJ_TAC THENL [MP_TAC(ISPECL [`s:real^M->bool`; `a:real^M`] CONNECTED_COMPONENT_SUBSET) THEN ASM SET_TAC[]; REWRITE_TAC[o_THM] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP CONNECTED_COMPONENT_REFL)) THEN ASM SET_TAC[]]);; let HOMOTOPY_INVARIANT_CONNECTEDNESS = prove (`!f:real^M->real^N g s t. f continuous_on s /\ IMAGE f s SUBSET t /\ g continuous_on t /\ IMAGE g t SUBSET s /\ homotopic_with (\x. T) (subtopology euclidean t,subtopology euclidean t) (f o g) I /\ connected s ==> connected t`, REPEAT GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN REWRITE_TAC[CONNECTED_EQ_CARD_COMPONENTS] THEN STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `g:real^N->real^M`; `s:real^M->bool`; `t:real^N->bool`] HOMOTOPY_INVARIANT_CARD_COMPONENTS) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] CARD_LE_FINITE)) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[CARD_LE_CARD_IMP; LE_TRANS]);; let HOMOTOPY_EQUIVALENT_CONNECTEDNESS = prove (`!s:real^M->bool t:real^N->bool. s homotopy_equivalent t ==> (connected s <=> connected t)`, REWRITE_TAC[homotopy_equivalent] THEN REPEAT STRIP_TAC THEN EQ_TAC THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] (REWRITE_RULE[CONJ_ASSOC] HOMOTOPY_INVARIANT_CONNECTEDNESS)) THEN ASM_MESON_TAC[]);; let HOMOTOPY_EQUIVALENT_CARD_EQ_COMPONENTS = prove (`!s:real^M->bool t:real^N->bool. s homotopy_equivalent t ==> components s =_c components t`, REWRITE_TAC[homotopy_equivalent] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM CARD_LE_ANTISYM] THEN CONJ_TAC THEN MATCH_MP_TAC HOMOTOPY_INVARIANT_CARD_COMPONENTS THEN ASM_MESON_TAC[]);; let HOMEOMORPHIC_CARD_EQ_COMPONENTS = prove (`!s:real^M->bool t:real^N->bool. s homeomorphic t ==> components s =_c components t`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMOTOPY_EQUIVALENT_CARD_EQ_COMPONENTS THEN MATCH_MP_TAC HOMEOMORPHIC_IMP_HOMOTOPY_EQUIVALENT THEN ASM_REWRITE_TAC[]);; let HOMOTOPY_INVARIANT_CARD_PATH_COMPONENTS = prove (`!f:real^M->real^N g s t. f continuous_on s /\ IMAGE f s SUBSET t /\ g continuous_on t /\ IMAGE g t SUBSET s /\ homotopic_with (\x. T) (subtopology euclidean t,subtopology euclidean t) (f o g) I ==> {path_component t x | x | x IN t} <=_c {path_component s x | x | x IN s}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CARD_LE_RELATIONAL_FULL THEN EXISTS_TAC `\c d. IMAGE (g:real^N->real^M) d SUBSET c` THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[FORALL_IN_GSPEC; EXISTS_IN_GSPEC] THEN CONJ_TAC THENL [X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN EXISTS_TAC `(g:real^N->real^M) y` THEN CONJ_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC PATH_COMPONENT_MAXIMAL] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC FUN_IN_IMAGE THEN REWRITE_TAC[IN] THEN ASM_SIMP_TAC[PATH_COMPONENT_REFL]; MATCH_MP_TAC PATH_CONNECTED_CONTINUOUS_IMAGE THEN REWRITE_TAC[PATH_CONNECTED_PATH_COMPONENT] THEN ASM_MESON_TAC[PATH_COMPONENT_SUBSET; CONTINUOUS_ON_SUBSET]; MP_TAC(ISPECL [`t:real^N->bool`; `y:real^N`] PATH_COMPONENT_SUBSET) THEN ASM SET_TAC[]]; ALL_TAC] THEN SIMP_TAC[PATH_COMPONENT_EQ_EQ] THEN X_GEN_TAC `a:real^M` THEN DISCH_TAC THEN X_GEN_TAC `b:real^N` THEN DISCH_TAC THEN X_GEN_TAC `c:real^N` THEN DISCH_TAC THEN REPEAT DISCH_TAC THEN MATCH_MP_TAC(MESON[PATH_COMPONENT_SYM; PATH_COMPONENT_TRANS] `!f. path_component s (f b) (f c) /\ path_component s (f b) b /\ path_component s (f c) c ==> path_component s b c`) THEN EXISTS_TAC `(f:real^M->real^N) o (g:real^N->real^M)` THEN CONJ_TAC THENL [ALL_TAC; CONJ_TAC THEN GEN_REWRITE_TAC RAND_CONV [GSYM I_THM] THEN MATCH_MP_TAC HOMOTOPIC_WITH_IMP_PATH_COMPONENT THEN EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[ETA_AX]] THEN REWRITE_TAC[PATH_COMPONENT] THEN EXISTS_TAC `IMAGE (f:real^M->real^N) (path_component s a)` THEN CONJ_TAC THENL [MATCH_MP_TAC PATH_CONNECTED_CONTINUOUS_IMAGE THEN REWRITE_TAC[PATH_CONNECTED_PATH_COMPONENT] THEN ASM_MESON_TAC[PATH_COMPONENT_SUBSET; CONTINUOUS_ON_SUBSET]; ALL_TAC] THEN CONJ_TAC THENL [MP_TAC(ISPECL [`s:real^M->bool`; `a:real^M`] PATH_COMPONENT_SUBSET) THEN ASM SET_TAC[]; REWRITE_TAC[o_THM] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP PATH_COMPONENT_REFL)) THEN ASM SET_TAC[]]);; let HOMOTOPY_INVARIANT_PATH_CONNECTEDNESS = prove (`!f:real^M->real^N g s t. f continuous_on s /\ IMAGE f s SUBSET t /\ g continuous_on t /\ IMAGE g t SUBSET s /\ homotopic_with (\x. T) (subtopology euclidean t,subtopology euclidean t) (f o g) I /\ path_connected s ==> path_connected t`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_SIMP_TAC[SUBSET_EMPTY; IMAGE_EQ_EMPTY; COMPONENTS_EMPTY] THEN REWRITE_TAC[PATH_CONNECTED_EMPTY] THEN STRIP_TAC THEN SUBGOAL_THEN `{path_component s (x:real^M) | x | x IN s} = {s}` ASSUME_TAC THENL [REWRITE_TAC[SET_RULE `s = {a} <=> a IN s /\ !x. x IN s ==> x = a`] THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN REWRITE_TAC[IN_ELIM_THM] THEN MP_TAC(ISPEC `s:real^M->bool` PATH_CONNECTED_COMPONENT_SET) THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`f:real^M->real^N`; `g:real^N->real^M`; `s:real^M->bool`; `t:real^N->bool`] HOMOTOPY_INVARIANT_CARD_PATH_COMPONENTS) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] CARD_LE_FINITE)) THEN REWRITE_TAC[FINITE_SING] THEN DISCH_TAC THEN UNDISCH_TAC `{path_component t (x:real^N) | x | x IN t} <=_c {s:real^M->bool}` THEN ASM_SIMP_TAC[CARD_LE_CARD; FINITE_SING; CARD_SING] THEN UNDISCH_TAC `FINITE {path_component t (x:real^N) | x | x IN t}` THEN REWRITE_TAC[ARITH_RULE `n <= 1 <=> n = 0 \/ n = 1`] THEN REWRITE_TAC[IMP_IMP; LEFT_OR_DISTRIB; GSYM HAS_SIZE] THEN CONV_TAC(ONCE_DEPTH_CONV HAS_SIZE_CONV) THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `s = {} \/ (?a. s = {a}) ==> !x. x IN s ==> !y. y IN s ==> x = y`)) THEN SIMP_TAC[FORALL_IN_GSPEC; PATH_COMPONENT_EQ_EQ] THEN REWRITE_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT] THEN MESON_TAC[]);; let HOMOTOPY_EQUIVALENT_PATH_CONNECTEDNESS = prove (`!s:real^M->bool t:real^N->bool. s homotopy_equivalent t ==> (path_connected s <=> path_connected t)`, REWRITE_TAC[homotopy_equivalent] THEN REPEAT STRIP_TAC THEN EQ_TAC THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] (REWRITE_RULE[CONJ_ASSOC] HOMOTOPY_INVARIANT_PATH_CONNECTEDNESS)) THEN ASM_MESON_TAC[]);; let HOMOTOPY_EQUIVALENT_CARD_EQ_PATH_COMPONENTS = prove (`!s:real^M->bool t:real^N->bool. s homotopy_equivalent t ==> {path_component s x | x | x IN s} =_c {path_component t x | x | x IN t}`, REWRITE_TAC[homotopy_equivalent] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM CARD_LE_ANTISYM] THEN CONJ_TAC THEN MATCH_MP_TAC HOMOTOPY_INVARIANT_CARD_PATH_COMPONENTS THEN ASM_MESON_TAC[]);; let HOMEOMORPHIC_CARD_EQ_PATH_COMPONENTS = prove (`!s:real^M->bool t:real^N->bool. s homeomorphic t ==> {path_component s x | x | x IN s} =_c {path_component t x | x | x IN t}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMOTOPY_EQUIVALENT_CARD_EQ_PATH_COMPONENTS THEN MATCH_MP_TAC HOMEOMORPHIC_IMP_HOMOTOPY_EQUIVALENT THEN ASM_REWRITE_TAC[]);; let FINITE_COMPONENTS_PUNCTURED_CONVEX = prove (`!s a:real^N. convex s ==> FINITE(components(s DELETE a))`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `aff_dim(s:real^N->bool) = &1` THENL [ALL_TAC; ASM_MESON_TAC[CONNECTED_PUNCTURED_CONVEX; CONNECTED_EQ_CARD_COMPONENTS]] THEN ASM_CASES_TAC `(a:real^N) IN s` THENL [ALL_TAC; ASM_SIMP_TAC[SET_RULE `~(a IN s) ==> s DELETE a = s`] THEN MATCH_MP_TAC(MESON[CONNECTED_EQ_CARD_COMPONENTS] `connected(c:real^N->bool) ==> FINITE(components c)`) THEN ASM_SIMP_TAC[CONVEX_CONNECTED]] THEN SUBGOAL_THEN `?t. (s:real^N->bool) homeomorphic (t:real^1->bool)` CHOOSE_TAC THENL [MP_TAC(ISPECL [`affine hull s:real^N->bool`; `(:real^1)`] HOMEOMORPHIC_AFFINE_SETS) THEN ASM_REWRITE_TAC[AFFINE_AFFINE_HULL; AFFINE_UNIV; AFF_DIM_AFFINE_HULL] THEN SIMP_TAC[AFF_DIM_UNIV; DIMINDEX_1; homeomorphic; LEFT_IMP_EXISTS_THM] THEN REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o SPEC `s:real^N->bool` o MATCH_MP (ONCE_REWRITE_RULE [IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN REWRITE_TAC[HULL_SUBSET; SUBSET_UNIV] THEN MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?b. b IN t /\ (s DELETE (a:real^N)) homeomorphic (t DELETE (b:real^1))` (X_CHOOSE_THEN `b:real^1` MP_TAC) THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphic]) THEN REWRITE_TAC[homeomorphic; RIGHT_AND_EXISTS_THM] THEN GEN_REWRITE_TAC RAND_CONV [SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^N->real^1` THEN GEN_REWRITE_TAC RAND_CONV [SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^1->real^N` THEN DISCH_TAC THEN EXISTS_TAC `(f:real^N->real^1) a` THEN CONJ_TAC THENL [ALL_TAC; FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE [IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS))] THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]; DISCH_THEN(CONJUNCTS_THEN MP_TAC) THEN DISCH_THEN(MP_TAC o MATCH_MP HOMEOMORPHIC_CARD_EQ_COMPONENTS) THEN DISCH_THEN(SUBST1_TAC o MATCH_MP CARD_FINITE_CONG) THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMEOMORPHIC_CONNECTEDNESS) THEN ASM_SIMP_TAC[CONVEX_CONNECTED] THEN REWRITE_TAC[GSYM IS_INTERVAL_CONNECTED_1]] THEN POP_ASSUM_LIST(K ALL_TAC) THEN REPEAT STRIP_TAC THEN ASM_CASES_TAC `connected(t DELETE (b:real^1))` THENL [ASM_MESON_TAC[CONNECTED_EQ_CARD_COMPONENTS]; ALL_TAC] THEN MP_TAC(ISPECL [`t DELETE (b:real^1)`; `t INTER {x | drop x < drop b}`; `t INTER {x | drop b < drop x}`] COMPONENTS_UNIQUE_2) THEN ANTS_TAC THENL [ALL_TAC; SIMP_TAC[FINITE_INSERT; FINITE_EMPTY]] THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM IS_INTERVAL_CONNECTED_1] THEN ASM_SIMP_TAC[IS_INTERVAL_INTER; IS_INTERVAL_1_CLAUSES] THEN MATCH_MP_TAC(SET_RULE `(!x. ~P x /\ ~Q x <=> x = b) ==> t INTER {x | P x} UNION t INTER {x | Q x} = t DELETE b`) THEN REWRITE_TAC[GSYM DROP_EQ] THEN REAL_ARITH_TAC);; (* ------------------------------------------------------------------------- *) (* Contractible sets. *) (* ------------------------------------------------------------------------- *) let contractible = new_definition `contractible s <=> ?a. homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean s) (\x. x) (\x. a)`;; let CONTRACTIBLE_SPACE_EUCLIDEAN = prove (`!s:real^N->bool. contractible_space(subtopology euclidean s) <=> contractible s`, REWRITE_TAC[contractible; contractible_space]);; let CONTRACTIBLE_IMP_SIMPLY_CONNECTED = prove (`!s:real^N->bool. contractible s ==> simply_connected s`, GEN_TAC THEN REWRITE_TAC[contractible] THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[SIMPLY_CONNECTED_EMPTY] THEN ASM_REWRITE_TAC[SIMPLY_CONNECTED_EQ_CONTRACTIBLE_LOOP_ALL] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN REWRITE_TAC[homotopic_loops; PCROSS] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_SUBSET) THEN MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL [ASM SET_TAC[]; DISCH_TAC] THEN X_GEN_TAC `p:real^1->real^N` THEN REWRITE_TAC[path; path_image; pathfinish; pathstart] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMOTOPIC_WITH_EUCLIDEAN]) THEN REWRITE_TAC[HOMOTOPIC_WITH_EUCLIDEAN; SUBSET; FORALL_IN_IMAGE; PCROSS] THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN DISCH_THEN(X_CHOOSE_THEN `h:real^(1,N)finite_sum->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(h o (\y. pastecart (fstcart y) (p(sndcart y):real^N))) :real^(1,1)finite_sum->real^N` THEN ASM_SIMP_TAC[FSTCART_PASTECART; SNDCART_PASTECART; linepath; o_THM] THEN CONJ_TAC THENL [ALL_TAC; CONV_TAC VECTOR_ARITH] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_PASTECART THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_FSTCART] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_SNDCART]; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE [IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN ASM_SIMP_TAC[IN_ELIM_PASTECART_THM; FSTCART_PASTECART; SNDCART_PASTECART]);; let CONTRACTIBLE_IMP_CONNECTED = prove (`!s:real^N->bool. contractible s ==> connected s`, SIMP_TAC[CONTRACTIBLE_IMP_SIMPLY_CONNECTED; SIMPLY_CONNECTED_IMP_CONNECTED]);; let CONTRACTIBLE_IMP_PATH_CONNECTED = prove (`!s:real^N->bool. contractible s ==> path_connected s`, SIMP_TAC[CONTRACTIBLE_IMP_SIMPLY_CONNECTED; SIMPLY_CONNECTED_IMP_PATH_CONNECTED]);; let NULLHOMOTOPIC_THROUGH_CONTRACTIBLE = prove (`!f:real^M->real^N g:real^N->real^P s t u. f continuous_on s /\ IMAGE f s SUBSET t /\ g continuous_on t /\ IMAGE g t SUBSET u /\ contractible t ==> ?c. homotopic_with (\h. T) (subtopology euclidean s,subtopology euclidean u) (g o f) (\x. c)`, REWRITE_TAC[GSYM CONTRACTIBLE_SPACE_EUCLIDEAN] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC NULLHOMOTOPIC_THROUGH_CONTRACTIBLE_SPACE THEN EXISTS_TAC `subtopology euclidean (t:real^N->bool)` THEN ASM_REWRITE_TAC[CONTINUOUS_MAP_EUCLIDEAN2]);; let NULLHOMOTOPIC_INTO_CONTRACTIBLE = prove (`!f:real^M->real^N s t. f continuous_on s /\ IMAGE f s SUBSET t /\ contractible t ==> ?c. homotopic_with (\h. T) (subtopology euclidean s,subtopology euclidean t) f (\x. c)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `(f:real^M->real^N) = (\x. x) o f` SUBST1_TAC THENL [REWRITE_TAC[o_THM; FUN_EQ_THM]; MATCH_MP_TAC NULLHOMOTOPIC_THROUGH_CONTRACTIBLE THEN EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[CONTINUOUS_ON_ID] THEN SET_TAC[]]);; let NULLHOMOTOPIC_FROM_CONTRACTIBLE = prove (`!f:real^M->real^N s t. f continuous_on s /\ IMAGE f s SUBSET t /\ contractible s ==> ?c. homotopic_with (\h. T) (subtopology euclidean s,subtopology euclidean t) f (\x. c)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `(f:real^M->real^N) = f o (\x. x)` SUBST1_TAC THENL [REWRITE_TAC[o_THM; FUN_EQ_THM]; MATCH_MP_TAC NULLHOMOTOPIC_THROUGH_CONTRACTIBLE THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[CONTINUOUS_ON_ID] THEN SET_TAC[]]);; let HOMOTOPIC_THROUGH_CONTRACTIBLE = prove (`!f1:real^M->real^N g1:real^N->real^P f2 g2 s t u. f1 continuous_on s /\ IMAGE f1 s SUBSET t /\ g1 continuous_on t /\ IMAGE g1 t SUBSET u /\ f2 continuous_on s /\ IMAGE f2 s SUBSET t /\ g2 continuous_on t /\ IMAGE g2 t SUBSET u /\ contractible t /\ path_connected u ==> homotopic_with (\h. T) (subtopology euclidean s,subtopology euclidean u) (g1 o f1) (g2 o f2)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMOTOPIC_THROUGH_CONTRACTIBLE_SPACE THEN EXISTS_TAC `subtopology euclidean (t:real^N->bool)` THEN ASM_REWRITE_TAC[CONTINUOUS_MAP_EUCLIDEAN2] THEN ASM_REWRITE_TAC[CONTRACTIBLE_SPACE_EUCLIDEAN] THEN ASM_REWRITE_TAC[PATH_CONNECTED_SPACE_EUCLIDEAN_SUBTOPOLOGY]);; let HOMOTOPIC_INTO_CONTRACTIBLE = prove (`!f:real^M->real^N g s t. f continuous_on s /\ IMAGE f s SUBSET t /\ g continuous_on s /\ IMAGE g s SUBSET t /\ contractible t ==> homotopic_with (\h. T) (subtopology euclidean s,subtopology euclidean t) f g`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMOTOPIC_INTO_CONTRACTIBLE_SPACE THEN ASM_REWRITE_TAC[CONTINUOUS_MAP_EUCLIDEAN2; CONTRACTIBLE_SPACE_EUCLIDEAN]);; let HOMOTOPIC_FROM_CONTRACTIBLE = prove (`!f:real^M->real^N g s t. f continuous_on s /\ IMAGE f s SUBSET t /\ g continuous_on s /\ IMAGE g s SUBSET t /\ contractible s /\ path_connected t ==> homotopic_with (\h. T) (subtopology euclidean s,subtopology euclidean t) f g`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMOTOPIC_FROM_CONTRACTIBLE_SPACE THEN ASM_REWRITE_TAC[CONTINUOUS_MAP_EUCLIDEAN2; CONTRACTIBLE_SPACE_EUCLIDEAN; PATH_CONNECTED_SPACE_EUCLIDEAN_SUBTOPOLOGY]);; let HOMOTOPY_EQUIVALENT_CONTRACTIBLE_SETS = prove (`!s:real^M->bool t:real^N->bool. contractible s /\ contractible t /\ (s = {} <=> t = {}) ==> s homotopy_equivalent t`, REPEAT GEN_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_SIMP_TAC[HOMEOMORPHIC_IMP_HOMOTOPY_EQUIVALENT; HOMEOMORPHIC_EMPTY] THEN FIRST_X_ASSUM(X_CHOOSE_TAC `b:real^N` o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN STRIP_TAC THEN REWRITE_TAC[homotopy_equivalent] THEN FIRST_X_ASSUM(X_CHOOSE_TAC `a:real^M` o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN EXISTS_TAC `(\x. b):real^M->real^N` THEN EXISTS_TAC `(\y. a):real^N->real^M` THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN REPLICATE_TAC 2 (CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN CONJ_TAC THEN MATCH_MP_TAC HOMOTOPIC_INTO_CONTRACTIBLE THEN ASM_REWRITE_TAC[o_DEF; IMAGE_ID; I_DEF; SUBSET_REFL; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST] THEN ASM SET_TAC[]);; let STARLIKE_IMP_CONTRACTIBLE_GEN = prove (`!P s. (!a t. a IN s /\ &0 <= t /\ t <= &1 ==> P(\x. (&1 - t) % x + t % a)) /\ starlike s ==> ?a:real^N. homotopic_with P (subtopology euclidean s,subtopology euclidean s) (\x. x) (\x. a)`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[starlike] THEN ONCE_REWRITE_TAC[SEGMENT_SYM] THEN REWRITE_TAC[segment; SUBSET; FORALL_IN_GSPEC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN STRIP_TAC THEN REWRITE_TAC[HOMOTOPIC_WITH_EUCLIDEAN; PCROSS] THEN EXISTS_TAC `\y:real^(1,N)finite_sum. (&1 - drop(fstcart y)) % sndcart y + drop(fstcart y) % a` THEN ASM_SIMP_TAC[FSTCART_PASTECART; SNDCART_PASTECART; DROP_VEC; IN_INTERVAL_1; SUBSET; FORALL_IN_IMAGE; REAL_SUB_RZERO; REAL_SUB_REFL; FORALL_IN_GSPEC; VECTOR_MUL_LZERO; VECTOR_MUL_LID; VECTOR_ADD_LID; VECTOR_ADD_RID] THEN MATCH_MP_TAC CONTINUOUS_ON_ADD THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN SIMP_TAC[o_DEF; LIFT_DROP; ETA_AX; LIFT_SUB; CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; LINEAR_CONTINUOUS_ON; ETA_AX; LINEAR_FSTCART; LINEAR_SNDCART]);; let STARLIKE_IMP_CONTRACTIBLE = prove (`!s:real^N->bool. starlike s ==> contractible s`, SIMP_TAC[contractible; STARLIKE_IMP_CONTRACTIBLE_GEN]);; let CONTRACTIBLE_UNIV = prove (`contractible(:real^N)`, SIMP_TAC[STARLIKE_IMP_CONTRACTIBLE; STARLIKE_UNIV]);; let STARLIKE_IMP_SIMPLY_CONNECTED = prove (`!s:real^N->bool. starlike s ==> simply_connected s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTRACTIBLE_IMP_SIMPLY_CONNECTED THEN MATCH_MP_TAC STARLIKE_IMP_CONTRACTIBLE THEN ASM_REWRITE_TAC[]);; let CONVEX_IMP_SIMPLY_CONNECTED = prove (`!s:real^N->bool. convex s ==> simply_connected s`, MESON_TAC[CONVEX_IMP_STARLIKE; STARLIKE_IMP_SIMPLY_CONNECTED; SIMPLY_CONNECTED_EMPTY]);; let STARLIKE_IMP_PATH_CONNECTED = prove (`!s:real^N->bool. starlike s ==> path_connected s`, MESON_TAC[STARLIKE_IMP_SIMPLY_CONNECTED; SIMPLY_CONNECTED_IMP_PATH_CONNECTED]);; let STARLIKE_IMP_CONNECTED = prove (`!s:real^N->bool. starlike s ==> connected s`, MESON_TAC[STARLIKE_IMP_PATH_CONNECTED; PATH_CONNECTED_IMP_CONNECTED]);; let CONIC_IMP_PATH_CONNECTED = prove (`!s:real^N->bool. conic s ==> path_connected s`, MESON_TAC[STARLIKE_IMP_PATH_CONNECTED; CONIC_IMP_STARLIKE; PATH_CONNECTED_EMPTY]);; let CONIC_IMP_CONNECTED = prove (`!s:real^N->bool. conic s ==> connected s`, MESON_TAC[CONIC_IMP_PATH_CONNECTED; PATH_CONNECTED_IMP_CONNECTED]);; let IS_INTERVAL_SIMPLY_CONNECTED_1 = prove (`!s:real^1->bool. is_interval s <=> simply_connected s`, MESON_TAC[SIMPLY_CONNECTED_IMP_PATH_CONNECTED; IS_INTERVAL_PATH_CONNECTED_1; CONVEX_IMP_SIMPLY_CONNECTED; IS_INTERVAL_CONVEX_1]);; let CONTRACTIBLE_EMPTY = prove (`contractible {}`, SIMP_TAC[contractible; HOMOTOPIC_WITH_EUCLIDEAN_ALT; PCROSS_EMPTY; NOT_IN_EMPTY] THEN REWRITE_TAC[CONTINUOUS_ON_EMPTY] THEN SET_TAC[]);; let CONIC_IMP_CONTRACTIBLE = prove (`!s:real^N->bool. conic s ==> contractible s`, MESON_TAC[CONIC_IMP_STARLIKE; STARLIKE_IMP_CONTRACTIBLE; CONTRACTIBLE_EMPTY]);; let CONIC_IMP_SIMPLY_CONNECTED = prove (`!s:real^N->bool. conic s ==> simply_connected s`, MESON_TAC[CONIC_IMP_CONTRACTIBLE; CONTRACTIBLE_IMP_SIMPLY_CONNECTED]);; let CONTRACTIBLE_CONVEX_TWEAK_BOUNDARY_POINTS = prove (`!s t:real^N->bool. convex s /\ relative_interior s SUBSET t /\ t SUBSET closure s ==> contractible t`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_SIMP_TAC[SUBSET_EMPTY; CLOSURE_EMPTY; CONTRACTIBLE_EMPTY] THEN STRIP_TAC THEN MATCH_MP_TAC STARLIKE_IMP_CONTRACTIBLE THEN MATCH_MP_TAC STARLIKE_CONVEX_TWEAK_BOUNDARY_POINTS THEN ASM_MESON_TAC[]);; let CONVEX_IMP_CONTRACTIBLE = prove (`!s:real^N->bool. convex s ==> contractible s`, MESON_TAC[CONVEX_IMP_STARLIKE; CONTRACTIBLE_EMPTY; STARLIKE_IMP_CONTRACTIBLE]);; let CONTRACTIBLE_SING = prove (`!a:real^N. contractible {a}`, SIMP_TAC[CONVEX_IMP_CONTRACTIBLE; CONVEX_SING]);; let SIMPLY_CONNECTED_SING = prove (`!a:real^N. simply_connected {a}`, SIMP_TAC[CONTRACTIBLE_SING; CONTRACTIBLE_IMP_SIMPLY_CONNECTED]);; let IS_INTERVAL_CONTRACTIBLE_1 = prove (`!s:real^1->bool. is_interval s <=> contractible s`, MESON_TAC[CONTRACTIBLE_IMP_PATH_CONNECTED; IS_INTERVAL_PATH_CONNECTED_1; CONVEX_IMP_CONTRACTIBLE; IS_INTERVAL_CONVEX_1]);; let CONTRACTIBLE_PCROSS = prove (`!s:real^M->bool t:real^N->bool. contractible s /\ contractible t ==> contractible(s PCROSS t)`, REPEAT GEN_TAC THEN REWRITE_TAC[contractible; HOMOTOPIC_WITH_EUCLIDEAN] THEN REWRITE_TAC[IMP_CONJ; LEFT_IMP_EXISTS_THM] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_PCROSS] THEN MAP_EVERY X_GEN_TAC [`a:real^M`; `h:real^(1,M)finite_sum->real^M`] THEN REPEAT DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`b:real^N`; `k:real^(1,N)finite_sum->real^N`] THEN REPEAT DISCH_TAC THEN EXISTS_TAC `pastecart (a:real^M) (b:real^N)` THEN EXISTS_TAC `\z. pastecart ((h:real^(1,M)finite_sum->real^M) (pastecart (fstcart z) (fstcart(sndcart z)))) ((k:real^(1,N)finite_sum->real^N) (pastecart (fstcart z) (sndcart(sndcart z))))` THEN ASM_SIMP_TAC[FORALL_IN_IMAGE; FORALL_PASTECART; PASTECART_IN_PCROSS; FSTCART_PASTECART; SNDCART_PASTECART] THEN MATCH_MP_TAC CONTINUOUS_ON_PASTECART THEN CONJ_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_PASTECART; LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; LINEAR_SNDCART; CONTINUOUS_ON_ID; GSYM o_DEF; CONTINUOUS_ON_COMPOSE] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_PASTECART] THEN SIMP_TAC[PASTECART_IN_PCROSS; FSTCART_PASTECART; SNDCART_PASTECART]);; let CONTRACTIBLE_PCROSS_EQ = prove (`!s:real^M->bool t:real^N->bool. contractible(s PCROSS t) <=> s = {} \/ t = {} \/ contractible s /\ contractible t`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_REWRITE_TAC[PCROSS_EMPTY; CONTRACTIBLE_EMPTY] THEN ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_REWRITE_TAC[PCROSS_EMPTY; CONTRACTIBLE_EMPTY] THEN EQ_TAC THEN REWRITE_TAC[CONTRACTIBLE_PCROSS] THEN REWRITE_TAC[contractible; HOMOTOPIC_WITH_EUCLIDEAN; LEFT_IMP_EXISTS_THM] THEN SIMP_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_PASTECART; PASTECART_IN_PCROSS] THEN MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^N`; `h:real^(1,(M,N)finite_sum)finite_sum->real^(M,N)finite_sum`] THEN STRIP_TAC THEN SUBGOAL_THEN `(a:real^M) IN s /\ (b:real^N) IN t` STRIP_ASSUME_TAC THENL [REWRITE_TAC[GSYM PASTECART_IN_PCROSS] THEN RULE_ASSUM_TAC(REWRITE_RULE[GSYM MEMBER_NOT_EMPTY]) THEN ASM_MESON_TAC[ENDS_IN_UNIT_INTERVAL]; ALL_TAC] THEN CONJ_TAC THENL [EXISTS_TAC `a:real^M` THEN EXISTS_TAC `fstcart o (h:real^(1,(M,N)finite_sum)finite_sum->real^(M,N)finite_sum) o (\z. pastecart (fstcart z) (pastecart (sndcart z) b))`; EXISTS_TAC `b:real^N` THEN EXISTS_TAC `sndcart o (h:real^(1,(M,N)finite_sum)finite_sum->real^(M,N)finite_sum) o (\z. pastecart (fstcart z) (pastecart a (sndcart z)))`] THEN ASM_REWRITE_TAC[o_THM; FSTCART_PASTECART; SNDCART_PASTECART; SUBSET; FORALL_IN_IMAGE; FORALL_IN_PCROSS; o_THM] THEN (CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[PASTECART_FST_SND; PASTECART_IN_PCROSS]]) THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; LINEAR_SNDCART] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_PASTECART; CONTINUOUS_ON_CONST; LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; LINEAR_SNDCART] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_PCROSS] THEN ASM_SIMP_TAC[FSTCART_PASTECART; SNDCART_PASTECART; PASTECART_IN_PCROSS]);; let HOMOTOPY_EQUIVALENT_EMPTY = prove (`(!s. (s:real^M->bool) homotopy_equivalent ({}:real^N->bool) <=> s = {}) /\ (!t. ({}:real^M->bool) homotopy_equivalent (t:real^N->bool) <=> t = {})`, REPEAT STRIP_TAC THEN EQ_TAC THEN SIMP_TAC[HOMOTOPY_EQUIVALENT_CONTRACTIBLE_SETS; CONTRACTIBLE_EMPTY] THEN REWRITE_TAC[homotopy_equivalent] THEN SET_TAC[]);; let HOMOTOPY_EQUIVALENT_CONTRACTIBILITY = prove (`!s:real^M->bool t:real^N->bool. s homotopy_equivalent t ==> (contractible s <=> contractible t)`, REWRITE_TAC[GSYM HOMOTOPY_EQUIVALENT_SPACE_EUCLIDEAN; GSYM CONTRACTIBLE_SPACE_EUCLIDEAN] THEN REWRITE_TAC[HOMOTOPY_EQUIVALENT_SPACE_CONTRACTIBILITY]);; let HOMOTOPY_EQUIVALENT_SING = prove (`!s:real^M->bool a:real^N. s homotopy_equivalent {a} <=> ~(s = {}) /\ contractible s`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_REWRITE_TAC[HOMOTOPY_EQUIVALENT_EMPTY; NOT_INSERT_EMPTY] THEN EQ_TAC THENL [DISCH_THEN(MP_TAC o MATCH_MP HOMOTOPY_EQUIVALENT_CONTRACTIBILITY) THEN REWRITE_TAC[CONTRACTIBLE_SING]; DISCH_TAC THEN MATCH_MP_TAC HOMOTOPY_EQUIVALENT_CONTRACTIBLE_SETS THEN ASM_REWRITE_TAC[CONTRACTIBLE_SING; NOT_INSERT_EMPTY]]);; let HOMEOMORPHIC_CONTRACTIBLE_EQ = prove (`!s:real^M->bool t:real^N->bool. s homeomorphic t ==> (contractible s <=> contractible t)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMOTOPY_EQUIVALENT_CONTRACTIBILITY THEN ASM_SIMP_TAC[HOMEOMORPHIC_IMP_HOMOTOPY_EQUIVALENT]);; let HOMEOMORPHIC_CONTRACTIBLE = prove (`!s:real^M->bool t:real^N->bool. s homeomorphic t /\ contractible s ==> contractible t`, MESON_TAC[HOMEOMORPHIC_CONTRACTIBLE_EQ]);; let CONTRACTIBLE_TRANSLATION = prove (`!a:real^N s. contractible (IMAGE (\x. a + x) s) <=> contractible s`, REPEAT GEN_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_CONTRACTIBLE_EQ THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN REWRITE_TAC[HOMEOMORPHIC_TRANSLATION]);; add_translation_invariants [CONTRACTIBLE_TRANSLATION];; let CONTRACTIBLE_INJECTIVE_LINEAR_IMAGE = prove (`!f:real^M->real^N s. linear f /\ (!x y. f x = f y ==> x = y) ==> (contractible (IMAGE f s) <=> contractible s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_CONTRACTIBLE_EQ THEN ASM_MESON_TAC[HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_LEFT_EQ; HOMEOMORPHIC_REFL]);; add_linear_invariants [CONTRACTIBLE_INJECTIVE_LINEAR_IMAGE];; let HOMEOMORPHISM_CONTRACTIBILITY = prove (`!f:real^M->real^N g s t k. homeomorphism (s,t) (f,g) /\ k SUBSET s ==> (contractible(IMAGE f k) <=> contractible k)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_CONTRACTIBLE_EQ THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN REWRITE_TAC[homeomorphic] THEN MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `g:real^N->real^M`] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* Homeomorphisms between punctured spheres and affine sets. *) (* ------------------------------------------------------------------------- *) let HOMEOMORPHIC_PUNCTURED_AFFINE_SPHERE_AFFINE = prove (`!a r b t:real^N->bool p:real^M->bool. &0 < r /\ b IN sphere(a,r) /\ affine t /\ a IN t /\ b IN t /\ affine p /\ aff_dim t = aff_dim p + &1 ==> ((sphere(a:real^N,r) INTER t) DELETE b) homeomorphic p`, GEOM_ORIGIN_TAC `a:real^N` THEN REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET] THEN REWRITE_TAC[sphere; DIST_0; IN_ELIM_THM] THEN SIMP_TAC[CONJ_ASSOC; NORM_ARITH `&0 < r /\ norm(b:real^N) = r <=> norm(b) = r /\ ~(b = vec 0)`] THEN GEOM_NORMALIZE_TAC `b:real^N` THEN REWRITE_TAC[] THEN GEOM_BASIS_MULTIPLE_TAC 1 `b:real^N` THEN SIMP_TAC[NORM_MUL; real_abs; NORM_BASIS; LE_REFL; DIMINDEX_GE_1] THEN X_GEN_TAC `b:real` THEN REWRITE_TAC[REAL_MUL_RID; VECTOR_MUL_EQ_0] THEN DISCH_THEN(K ALL_TAC) THEN DISCH_THEN SUBST1_TAC THEN REPEAT GEN_TAC THEN REWRITE_TAC[VECTOR_MUL_LID] THEN ASM_CASES_TAC `r = &1` THEN ASM_REWRITE_TAC[] THEN POP_ASSUM(K ALL_TAC) THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN SIMP_TAC[BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL] THEN STRIP_TAC THEN SUBGOAL_THEN `subspace(t:real^N->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[AFFINE_EQ_SUBSPACE]; ALL_TAC] THEN TRANS_TAC HOMEOMORPHIC_TRANS `{x:real^N | x$1 = &0} INTER t` THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC HOMEOMORPHIC_AFFINE_SETS THEN ASM_SIMP_TAC[AFFINE_INTER; AFFINE_STANDARD_HYPERPLANE] THEN ONCE_REWRITE_TAC[INTER_COMM] THEN MP_TAC(ISPECL [`basis 1:real^N`; `&0`; `t:real^N->bool`] AFF_DIM_AFFINE_INTER_HYPERPLANE) THEN ASM_SIMP_TAC[DOT_BASIS; DIMINDEX_GE_1; LE_REFL] THEN DISCH_THEN SUBST1_TAC THEN SUBGOAL_THEN `~(t INTER {x:real^N | x$1 = &0} = {})` ASSUME_TAC THENL [REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER; IN_ELIM_THM] THEN EXISTS_TAC `vec 0:real^N` THEN ASM_REWRITE_TAC[VEC_COMPONENT]; ALL_TAC] THEN SUBGOAL_THEN `~(t SUBSET {v:real^N | v$1 = &0})` ASSUME_TAC THENL [REWRITE_TAC[SUBSET] THEN DISCH_THEN(MP_TAC o SPEC `basis 1:real^N`) THEN ASM_SIMP_TAC[IN_ELIM_THM; BASIS_COMPONENT; DIMINDEX_GE_1; LE_REFL] THEN REAL_ARITH_TAC; ASM_REWRITE_TAC[] THEN INT_ARITH_TAC]] THEN SUBGOAL_THEN `({x:real^N | norm x = &1} INTER t) DELETE (basis 1) = {x | norm x = &1 /\ ~(x$1 = &1)} INTER t` SUBST1_TAC THENL [MATCH_MP_TAC(SET_RULE `s DELETE a = s' ==> (s INTER t) DELETE a = s' INTER t`) THEN MATCH_MP_TAC(SET_RULE `Q a /\ (!x. P x /\ Q x ==> x = a) ==> {x | P x} DELETE a = {x | P x /\ ~Q x}`) THEN SIMP_TAC[BASIS_COMPONENT; CART_EQ; DIMINDEX_GE_1; LE_REFL] THEN REWRITE_TAC[NORM_EQ_SQUARE; REAL_POS; REAL_POW_ONE] THEN X_GEN_TAC `x:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN ASM_SIMP_TAC[dot; SUM_CLAUSES_LEFT; DIMINDEX_GE_1] THEN REWRITE_TAC[REAL_ARITH `&1 * &1 + s = &1 <=> s = &0`] THEN DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] SUM_POS_EQ_0_NUMSEG)) THEN REWRITE_TAC[REAL_LE_SQUARE; REAL_ENTIRE] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[homeomorphic; HOMEOMORPHISM] THEN MAP_EVERY ABBREV_TAC [`f = \x:real^N. &2 % basis 1 + &2 / (&1 - x$1) % (x - basis 1)`; `g = \y:real^N. basis 1 + &4 / (norm y pow 2 + &4) % (y - &2 % basis 1)`] THEN MAP_EVERY EXISTS_TAC [`f:real^N->real^N`; `g:real^N->real^N`] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC(MESON[CONTINUOUS_ON_SUBSET; INTER_SUBSET] `f continuous_on s ==> f continuous_on (s INTER t)`) THEN EXPAND_TAC "f" THEN MATCH_MP_TAC CONTINUOUS_ON_ADD THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN REWRITE_TAC[o_DEF; real_div; LIFT_CMUL] THEN MATCH_MP_TAC CONTINUOUS_ON_CMUL THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN SIMP_TAC[REAL_SUB_0; IN_ELIM_THM] THEN REWRITE_TAC[LIFT_SUB] THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN MATCH_MP_TAC CONTINUOUS_ON_LIFT_COMPONENT THEN REWRITE_TAC[LE_REFL; DIMINDEX_GE_1]; MATCH_MP_TAC(SET_RULE `IMAGE f s SUBSET s' /\ IMAGE f t SUBSET t ==> IMAGE f (s INTER t) SUBSET (s' INTER t)`) THEN EXPAND_TAC "f" THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN ASM_SIMP_TAC[SUBSPACE_ADD; SUBSPACE_MUL; SUBSPACE_SUB] THEN REWRITE_TAC[IN_ELIM_THM; IN_DELETE] THEN SIMP_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; BASIS_COMPONENT; LE_REFL; DIMINDEX_GE_1; VECTOR_SUB_COMPONENT] THEN CONV_TAC REAL_FIELD; MATCH_MP_TAC(MESON[CONTINUOUS_ON_SUBSET; INTER_SUBSET] `f continuous_on s ==> f continuous_on (s INTER t)`) THEN EXPAND_TAC "g" THEN MATCH_MP_TAC CONTINUOUS_ON_ADD THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN REWRITE_TAC[o_DEF; real_div; LIFT_CMUL] THEN MATCH_MP_TAC CONTINUOUS_ON_CMUL THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN SIMP_TAC[LIFT_ADD; REAL_POW_LE; NORM_POS_LE; REAL_ARITH `&0 <= x ==> ~(x + &4 = &0)`] THEN MATCH_MP_TAC CONTINUOUS_ON_ADD THEN REWRITE_TAC[REAL_POW_2; LIFT_CMUL; CONTINUOUS_ON_CONST] THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN REWRITE_TAC[CONTINUOUS_ON_LIFT_NORM; GSYM o_DEF]; MATCH_MP_TAC(SET_RULE `IMAGE f s SUBSET s' /\ IMAGE f t SUBSET t ==> IMAGE f (s INTER t) SUBSET (s' INTER t)`) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM] THEN REWRITE_TAC[NORM_EQ_SQUARE; REAL_POS] THEN EXPAND_TAC "g" THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[SUBSPACE_ADD; SUBSPACE_MUL; SUBSPACE_SUB]] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN REWRITE_TAC[VECTOR_ARITH `b + a % (y - &2 % b):real^N = (&1 - &2 * a) % b + a % y`] THEN REWRITE_TAC[NORM_POW_2; VECTOR_ARITH `(a + b:real^N) dot (a + b) = (a dot a + b dot b) + &2 * a dot b`] THEN ASM_SIMP_TAC[DOT_LMUL; DOT_RMUL; DOT_BASIS; BASIS_COMPONENT; LE_REFL; VECTOR_ADD_COMPONENT; DIMINDEX_GE_1; VECTOR_MUL_COMPONENT] THEN REWRITE_TAC[REAL_MUL_RZERO; REAL_MUL_RID; GSYM REAL_POW_2] THEN SUBGOAL_THEN `~((y:real^N) dot y + &4 = &0)` MP_TAC THENL [MESON_TAC[DOT_POS_LE; REAL_ARITH `&0 <= x ==> ~(x + &4 = &0)`]; CONV_TAC REAL_FIELD]; SUBGOAL_THEN `!x. norm x = &1 /\ ~(x$1 = &1) ==> norm((f:real^N->real^N) x) pow 2 = &4 * (&1 + x$1) / (&1 - x$1)` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN EXPAND_TAC "f" THEN REWRITE_TAC[VECTOR_ARITH `a % b + m % (x - b):real^N = (a - m) % b + m % x`] THEN REWRITE_TAC[NORM_POW_2; VECTOR_ARITH `(a + b:real^N) dot (a + b) = (a dot a + b dot b) + &2 * a dot b`] THEN SIMP_TAC[DOT_LMUL; DOT_RMUL; DOT_BASIS; BASIS_COMPONENT; DIMINDEX_GE_1; LE_REFL; VECTOR_MUL_COMPONENT] THEN ASM_REWRITE_TAC[GSYM NORM_POW_2; GSYM REAL_POW_2; REAL_MUL_RID; REAL_POW_ONE] THEN UNDISCH_TAC `~((x:real^N)$1 = &1)` THEN CONV_TAC REAL_FIELD; ALL_TAC] THEN EXPAND_TAC "g" THEN REWRITE_TAC[IN_INTER; IN_ELIM_THM] THEN ASM_SIMP_TAC[] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN ASM_SIMP_TAC[REAL_FIELD `~(x = &1) ==> &4 * (&1 + x) / (&1 - x) + &4 = &8 / (&1 - x)`] THEN REWRITE_TAC[real_div; REAL_INV_MUL; REAL_INV_INV] THEN REWRITE_TAC[REAL_ARITH `&4 * inv(&8) * x = x / &2`] THEN EXPAND_TAC "f" THEN REWRITE_TAC[VECTOR_ARITH `(a + x) - a:real^N = x`] THEN REWRITE_TAC[VECTOR_MUL_ASSOC; VECTOR_ARITH `b + a % (x - b):real^N = x <=> (&1 - a) % (x - b) = vec 0`] THEN REWRITE_TAC[VECTOR_MUL_EQ_0] THEN DISJ1_TAC THEN UNDISCH_TAC `~((x:real^N)$1 = &1)` THEN CONV_TAC REAL_FIELD; X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_INTER; IN_ELIM_THM] THEN DISCH_TAC THEN SUBGOAL_THEN `~((y:real^N) dot y + &4 = &0)` ASSUME_TAC THENL [MESON_TAC[DOT_POS_LE; REAL_ARITH `&0 <= x ==> ~(x + &4 = &0)`]; ALL_TAC] THEN SUBGOAL_THEN `((g:real^N->real^N) y)$1 = (y dot y - &4) / (y dot y + &4)` ASSUME_TAC THENL [EXPAND_TAC "g" THEN REWRITE_TAC[VECTOR_ADD_COMPONENT] THEN REWRITE_TAC[VECTOR_MUL_COMPONENT; VECTOR_SUB_COMPONENT] THEN ASM_SIMP_TAC[BASIS_COMPONENT; LE_REFL; NORM_POW_2; DIMINDEX_GE_1] THEN UNDISCH_TAC `~((y:real^N) dot y + &4 = &0)` THEN CONV_TAC REAL_FIELD; ALL_TAC] THEN EXPAND_TAC "f" THEN REWRITE_TAC[] THEN ASM_REWRITE_TAC[] THEN EXPAND_TAC "g" THEN SIMP_TAC[VECTOR_ARITH `(a + x) - a:real^N = x`] THEN REWRITE_TAC[VECTOR_MUL_ASSOC; VECTOR_ARITH `b + a % (x - b):real^N = x <=> (&1 - a) % (x - b) = vec 0`] THEN REWRITE_TAC[VECTOR_MUL_EQ_0; NORM_POW_2] THEN DISJ1_TAC THEN UNDISCH_TAC `~((y:real^N) dot y + &4 = &0)` THEN CONV_TAC REAL_FIELD]);; let HOMEOMORPHIC_PUNCTURED_SPHERE_AFFINE_GEN = prove (`!s:real^N->bool t:real^M->bool a. convex s /\ bounded s /\ a IN relative_frontier s /\ affine t /\ aff_dim s = aff_dim t + &1 ==> (relative_frontier s DELETE a) homeomorphic t`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_GE; INT_ARITH `--(&1):int <= s ==> ~(--(&1) = s + &1)`] THEN MP_TAC(ISPECL [`(:real^N)`; `aff_dim(s:real^N->bool)`] CHOOSE_AFFINE_SUBSET) THEN REWRITE_TAC[SUBSET_UNIV] THEN REWRITE_TAC[AFF_DIM_GE; AFF_DIM_LE_UNIV; AFF_DIM_UNIV; AFFINE_UNIV] THEN DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `~(t:real^N->bool = {})` MP_TAC THENL [ASM_MESON_TAC[AFF_DIM_EQ_MINUS1]; ALL_TAC] THEN GEN_REWRITE_TAC LAND_CONV [GSYM MEMBER_NOT_EMPTY] THEN DISCH_THEN(X_CHOOSE_TAC `z:real^N`) THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `ball(z:real^N,&1) INTER t`] HOMEOMORPHIC_RELATIVE_FRONTIERS_CONVEX_BOUNDED_SETS) THEN MP_TAC(ISPECL [`t:real^N->bool`; `ball(z:real^N,&1)`] (ONCE_REWRITE_RULE[INTER_COMM] AFF_DIM_CONVEX_INTER_OPEN)) THEN MP_TAC(ISPECL [`ball(z:real^N,&1)`; `t:real^N->bool`] RELATIVE_FRONTIER_CONVEX_INTER_AFFINE) THEN ASM_SIMP_TAC[CONVEX_INTER; BOUNDED_INTER; BOUNDED_BALL; CONVEX_BALL; AFFINE_IMP_CONVEX; INTERIOR_OPEN; OPEN_BALL; FRONTIER_BALL; REAL_LT_01] THEN SUBGOAL_THEN `~(ball(z:real^N,&1) INTER t = {})` ASSUME_TAC THENL [REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN EXISTS_TAC `z:real^N` THEN ASM_REWRITE_TAC[CENTRE_IN_BALL; REAL_LT_01]; ASM_REWRITE_TAC[] THEN REPEAT(DISCH_THEN SUBST1_TAC) THEN SIMP_TAC[]] THEN REWRITE_TAC[homeomorphic; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`h:real^N->real^N`; `k:real^N->real^N`] THEN STRIP_TAC THEN REWRITE_TAC[GSYM homeomorphic] THEN TRANS_TAC HOMEOMORPHIC_TRANS `(sphere(z,&1) INTER t) DELETE (h:real^N->real^N) a` THEN CONJ_TAC THENL [REWRITE_TAC[homeomorphic] THEN MAP_EVERY EXISTS_TAC [`h:real^N->real^N`; `k:real^N->real^N`] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHISM]) THEN REWRITE_TAC[HOMEOMORPHISM] THEN STRIP_TAC THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; DELETE_SUBSET]; ASM SET_TAC[]; ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; DELETE_SUBSET]; ASM SET_TAC[]; ASM SET_TAC[]; ASM SET_TAC[]]; MATCH_MP_TAC HOMEOMORPHIC_PUNCTURED_AFFINE_SPHERE_AFFINE THEN ASM_REWRITE_TAC[REAL_LT_01; GSYM IN_INTER] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHISM]) THEN ASM SET_TAC[]]);; let HOMEOMORPHIC_PUNCTURED_SPHERE_AFFINE = prove (`!a r b:real^N t:real^M->bool. &0 < r /\ b IN sphere(a,r) /\ affine t /\ aff_dim(t) + &1 = &(dimindex(:N)) ==> (sphere(a:real^N,r) DELETE b) homeomorphic t`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`cball(a:real^N,r)`; `t:real^M->bool`; `b:real^N`] HOMEOMORPHIC_PUNCTURED_SPHERE_AFFINE_GEN) THEN ASM_SIMP_TAC[RELATIVE_FRONTIER_CBALL; REAL_LT_IMP_NZ; AFF_DIM_CBALL; CONVEX_CBALL; BOUNDED_CBALL]);; let HOMEOMORPHIC_PUNCTURED_SPHERE_HYPERPLANE = prove (`!a r b c d. &0 < r /\ b IN sphere(a,r) /\ ~(c = vec 0) ==> (sphere(a:real^N,r) DELETE b) homeomorphic {x:real^N | c dot x = d}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_PUNCTURED_SPHERE_AFFINE THEN ASM_SIMP_TAC[AFFINE_HYPERPLANE; AFF_DIM_HYPERPLANE] THEN INT_ARITH_TAC);; let HOMEOMORPHIC_PUNCTURED_SPHERE_UNIV = prove (`!a r b. &0 < r /\ b IN sphere(a,r) /\ dimindex(:N) = dimindex(:M) + 1 ==> (sphere(a:real^N,r) DELETE b) homeomorphic (:real^M)`, REPEAT STRIP_TAC THEN TRANS_TAC HOMEOMORPHIC_TRANS `{x:real^N | basis 1 dot x = &0}` THEN ASM_SIMP_TAC[HOMEOMORPHIC_HYPERPLANE_UNIV; BASIS_NONZERO; LE_REFL; DIMINDEX_GE_1; HOMEOMORPHIC_PUNCTURED_SPHERE_HYPERPLANE]);; let CONTRACTIBLE_PUNCTURED_SPHERE = prove (`!a r b:real^N. &0 < r /\ b IN sphere(a,r) ==> contractible(sphere(a,r) DELETE b)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `contractible {x:real^N | basis 1 dot x = &0}` MP_TAC THENL [SIMP_TAC[CONVEX_IMP_CONTRACTIBLE; CONVEX_HYPERPLANE]; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] HOMEOMORPHIC_CONTRACTIBLE) THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN MATCH_MP_TAC HOMEOMORPHIC_PUNCTURED_SPHERE_HYPERPLANE THEN ASM_SIMP_TAC[BASIS_NONZERO; LE_REFL; DIMINDEX_GE_1]]);; let CONTRACTIBLE_PUNCTURED_SPHERE_GEN = prove (`!s a:real^N. convex s /\ bounded s /\ a IN relative_frontier s ==> contractible(relative_frontier s DELETE a)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[RELATIVE_FRONTIER_EMPTY; EMPTY_DELETE; CONTRACTIBLE_EMPTY] THEN MP_TAC(SPEC `aff_dim(s:real^N->bool) - &1` AFFINE_EXISTS) THEN REWRITE_TAC[INT_ARITH `--a:int <= s - a <=> &0 <= s`] THEN ASM_SIMP_TAC[AFF_DIM_POS_LE; AFF_DIM_LE_UNIV; LEFT_IMP_EXISTS_THM; INT_ARITH `s:int <= n ==> s - &1 <= n`] THEN X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`; `a:real^N`] HOMEOMORPHIC_PUNCTURED_SPHERE_AFFINE_GEN) THEN ASM_REWRITE_TAC[INT_SUB_ADD] THEN DISCH_THEN(SUBST1_TAC o MATCH_MP HOMEOMORPHIC_CONTRACTIBLE_EQ) THEN ASM_SIMP_TAC[AFFINE_IMP_CONVEX; CONVEX_IMP_CONTRACTIBLE]);; let NULLHOMOTOPIC_NONSURJECTIVE_SPHERE_MAP_GEN = prove (`!f s:real^N->bool. convex s /\ bounded s /\ f continuous_on (relative_frontier s) /\ IMAGE f (relative_frontier s) PSUBSET relative_frontier s ==> ?a. homotopic_with (\x. T) (subtopology euclidean (relative_frontier s), subtopology euclidean (relative_frontier s)) f (\x. a)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC o GEN_REWRITE_RULE I [PSUBSET_ALT]) THEN DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`f:real^N->real^N`; `\x:real^N. x`; `relative_frontier s:real^N->bool`; `relative_frontier s DELETE (a:real^N)`; `relative_frontier s:real^N->bool`] NULLHOMOTOPIC_THROUGH_CONTRACTIBLE) THEN ASM_REWRITE_TAC[IMAGE_ID; SUBSET_DELETE; DELETE_SUBSET; o_DEF; ETA_AX] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_SIMP_TAC[CONTINUOUS_ON_ID; CONTRACTIBLE_PUNCTURED_SPHERE_GEN]);; let CONNECTED_PUNCTURED_SPHERE = prove (`!a r b:real^N. connected(sphere(a,r) DELETE b) <=> (dimindex(:N) = 1 /\ &0 < r ==> b IN sphere(a,r))`, REPEAT GEN_TAC THEN ASM_CASES_TAC `r < &0` THEN ASM_SIMP_TAC[SPHERE_EMPTY; EMPTY_DELETE; NOT_IN_EMPTY; CONNECTED_EMPTY] THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_CASES_TAC `r = &0` THEN ASM_SIMP_TAC[SPHERE_SING; REAL_LT_REFL] THENL [SUBGOAL_THEN `{a:real^N} DELETE b = {a} \/ {a} DELETE b = {}` MP_TAC THENL [SET_TAC[]; DISCH_THEN(DISJ_CASES_THEN SUBST1_TAC)] THEN REWRITE_TAC[CONNECTED_EMPTY; CONNECTED_SING]; ALL_TAC] THEN SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_CASES_TAC `dimindex(:N) = 1` THEN ASM_REWRITE_TAC[] THENL [MP_TAC(ISPECL [`a:real^N`; `r:real`] HAS_SIZE_SPHERE_2) THEN ASM_REWRITE_TAC[] THEN CONV_TAC(LAND_CONV HAS_SIZE_CONV) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u:real^N`; `v:real^N`] THEN STRIP_TAC THEN ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN ASM_CASES_TAC `b:real^N = u` THENL [SUBGOAL_THEN `{u:real^N,v} DELETE b = {v}` SUBST1_TAC THENL [ASM SET_TAC[]; ASM_REWRITE_TAC[CONNECTED_SING]]; ALL_TAC] THEN ASM_CASES_TAC `b:real^N = v` THENL [SUBGOAL_THEN `{u:real^N,v} DELETE b = {u}` SUBST1_TAC THENL [ASM SET_TAC[]; ASM_REWRITE_TAC[CONNECTED_SING]]; ALL_TAC] THEN SUBGOAL_THEN `{u:real^N,v} DELETE b = {u,v}` SUBST1_TAC THENL [ASM SET_TAC[]; ASM_REWRITE_TAC[CONNECTED_2] THEN ASM SET_TAC[]]; ASM_CASES_TAC `b IN sphere(a:real^N,r)` THENL [MP_TAC(ISPECL [`a:real^N`; `r:real`; `b:real^N`; `basis 1:real^N`; `&0`] HOMEOMORPHIC_PUNCTURED_SPHERE_HYPERPLANE) THEN ASM_SIMP_TAC[BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL] THEN DISCH_THEN(MP_TAC o MATCH_MP HOMEOMORPHIC_CONNECTEDNESS) THEN SIMP_TAC[CONVEX_HYPERPLANE; CONVEX_CONNECTED]; ASM_SIMP_TAC[SET_RULE `~(x IN s) ==> s DELETE x = s`] THEN MATCH_MP_TAC CONNECTED_SPHERE THEN MATCH_MP_TAC(ARITH_RULE `1 <= n /\ ~(n = 1) ==> 2 <= n`) THEN ASM_REWRITE_TAC[DIMINDEX_GE_1]]]);; let CONNECTED_IN_SPHERE_DELETE_INTERIOR_POINT_EQ = prove (`!a b r u s:real^N->bool. 3 <= dimindex(:N) /\ open_in (subtopology euclidean (sphere(a,r))) u /\ b IN u /\ u SUBSET s /\ s SUBSET sphere(a,r) ==> (connected(s DELETE b) <=> connected s)`, REPEAT STRIP_TAC THEN STRIP_ASSUME_TAC (SET_RULE `s DELETE (b:real^N) = s \/ b IN s`) THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ASM_CASES_TAC `s = {b:real^N}` THEN ASM_REWRITE_TAC[CONNECTED_SING; CONNECTED_EMPTY; SET_RULE `{b} DELETE b = {}`] THEN ASM_CASES_TAC `r < &0` THENL [MP_TAC(ISPECL [`a:real^N`; `r:real`] SPHERE_EQ_EMPTY) THEN ASM SET_TAC[]; RULE_ASSUM_TAC(REWRITE_RULE[GSYM REAL_NOT_LE])] THEN ASM_CASES_TAC `r = &0` THENL [MP_TAC(ISPECL [`a:real^N`; `r:real`; `a:real^N`] SPHERE_EQ_SING) THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]] THEN ASM_CASES_TAC `s = sphere(a:real^N,r)` THENL [FIRST_X_ASSUM SUBST_ALL_TAC THEN ASM_REWRITE_TAC[CONNECTED_PUNCTURED_SPHERE; CONNECTED_SPHERE_EQ] THEN DISJ1_TAC THEN ASM_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `?c:real^N. c IN sphere(a,r) /\ ~(c IN s) /\ ~(c = b)` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `sphere(a:real^N,r) DELETE c homeomorphic (:real^(N,1)finite_diff)` MP_TAC THENL [MATCH_MP_TAC HOMEOMORPHIC_PUNCTURED_SPHERE_UNIV THEN ASM_REWRITE_TAC[DIMINDEX_FINITE_DIFF; DIMINDEX_1] THEN ASM_ARITH_TAC; REWRITE_TAC[homeomorphic; LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`f:real^N->real^(N,1)finite_diff`; `g:real^(N,1)finite_diff->real^N`] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_CONNECTEDNESS)) THEN DISCH_THEN(fun th -> MP_TAC(SPEC `s DELETE (b:real^N)` th) THEN MP_TAC(SPEC `s:real^N->bool` th)) THEN REPEAT(ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN(SUBST1_TAC o SYM)]) THEN SUBGOAL_THEN `IMAGE (f:real^N->real^(N,1)finite_diff) (s DELETE b) = IMAGE f s DELETE f b` SUBST1_TAC THENL [MATCH_MP_TAC IMAGE_DELETE_INJ_ALT THEN ASM_REWRITE_TAC[INJECTIVE_ON_LEFT_INVERSE] THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism; IN_UNIV]) THEN ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC CONNECTED_DELETE_INTERIOR_POINT_EQ THEN REWRITE_TAC[DIMINDEX_FINITE_DIFF; DIMINDEX_1] THEN CONJ_TAC THENL [ASM_ARITH_TAC; REWRITE_TAC[OPEN_IN]] THEN MATCH_MP_TAC(SET_RULE `!u. b IN u /\ IMAGE f u SUBSET t ==> f b IN t`) THEN EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC INTERIOR_MAXIMAL THEN ASM_SIMP_TAC[IMAGE_SUBSET] THEN REWRITE_TAC[OPEN_IN] THEN ONCE_REWRITE_TAC[GSYM SUBTOPOLOGY_UNIV] THEN SUBGOAL_THEN `open_in (subtopology euclidean (sphere(a:real^N,r) DELETE c)) u` MP_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] OPEN_IN_SUBSET_TRANS)) THEN ASM SET_TAC[]; MATCH_MP_TAC EQ_IMP THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC HOMEOMORPHISM_OPENNESS THEN EXISTS_TAC `g:real^(N,1)finite_diff->real^N` THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]]);; let CONNECTED_OPEN_IN_SPHERE_DELETE_EQ = prove (`!a b r s:real^N->bool. 3 <= dimindex(:N) /\ open_in (subtopology euclidean (sphere(a,r))) s ==> (connected(s DELETE b) <=> connected s)`, REPEAT STRIP_TAC THEN STRIP_ASSUME_TAC (SET_RULE `s DELETE (b:real^N) = s \/ b IN s`) THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN MATCH_MP_TAC CONNECTED_IN_SPHERE_DELETE_INTERIOR_POINT_EQ THEN MAP_EVERY EXISTS_TAC [`a:real^N`; `r:real`; `s:real^N->bool`] THEN ASM_REWRITE_TAC[SUBSET_REFL]);; let FINITE_COMPONENTS_PUNCTURED_CONNECTED_SUBSET_SPHERE = prove (`!r s a b:real^N. connected s /\ s SUBSET sphere(a,r) /\ b IN (subtopology euclidean (sphere (a,r))) interior_of s ==> FINITE(components(s DELETE b))`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `FINITE(sphere(a:real^N,r))` THENL [MATCH_MP_TAC FINITE_COMPONENTS_FINITE THEN ASM_MESON_TAC[FINITE_SUBSET; FINITE_DELETE]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [FINITE_SPHERE])] THEN REWRITE_TAC[DE_MORGAN_THM; REAL_NOT_LE] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (ARITH_RULE `~(n = 1) ==> 1 <= n ==> 3 <= n \/ n = 2`)) THEN REWRITE_TAC[DIMINDEX_GE_1] THEN STRIP_TAC THENL [MP_TAC(ISPECL [`a:real^N`; `b:real^N`; `r:real`; `subtopology euclidean (sphere(a:real^N,r)) interior_of s`; `s:real^N->bool`] CONNECTED_IN_SPHERE_DELETE_INTERIOR_POINT_EQ) THEN ASM_REWRITE_TAC[OPEN_IN_INTERIOR_OF; INTERIOR_OF_SUBSET] THEN SIMP_TAC[CONNECTED_EQ_CARD_COMPONENTS]; FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP (REWRITE_RULE[SUBSET] INTERIOR_OF_SUBSET))] THEN ASM_CASES_TAC `s = sphere(a:real^N,r)` THENL [MATCH_MP_TAC(MESON[CONNECTED_EQ_CARD_COMPONENTS] `connected(c:real^N->bool) ==> FINITE(components c)`) THEN MP_TAC(ISPECL [`a:real^N`; `r:real`; `b:real^N`; `basis 1:real^N`; `&0:real`] HOMEOMORPHIC_PUNCTURED_SPHERE_HYPERPLANE) THEN ASM_SIMP_TAC[BASIS_NONZERO; LE_REFL; DIMINDEX_GE_1] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(SUBST1_TAC o MATCH_MP HOMEOMORPHIC_CONNECTEDNESS) THEN MATCH_MP_TAC CONVEX_CONNECTED THEN REWRITE_TAC[CONVEX_HYPERPLANE]; FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE `~(s = u) ==> s SUBSET u ==> ?z. z IN u /\ ~(z IN s)`))] THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `c:real^N` THEN STRIP_TAC THEN SUBGOAL_THEN `?t. (s:real^N->bool) homeomorphic (t:real^1->bool)` STRIP_ASSUME_TAC THENL [MP_TAC(ISPECL [`a:real^N`; `r:real`; `c:real^N`; `(:real^1)`] HOMEOMORPHIC_PUNCTURED_SPHERE_AFFINE) THEN ASM_REWRITE_TAC[AFFINE_UNIV; AFF_DIM_UNIV; DIMINDEX_1] THEN CONV_TAC INT_REDUCE_CONV THEN REWRITE_TAC[homeomorphic; LEFT_IMP_EXISTS_THM] THEN REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE [IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN REWRITE_TAC[SUBSET_DELETE; SUBSET_UNIV] THEN ASM_MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?d. (s DELETE (b:real^N)) homeomorphic (t DELETE (d:real^1))` (X_CHOOSE_THEN `d:real^1` MP_TAC) THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphic]) THEN REWRITE_TAC[homeomorphic] THEN GEN_REWRITE_TAC RAND_CONV [SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^N->real^1` THEN GEN_REWRITE_TAC RAND_CONV [SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^1->real^N` THEN DISCH_TAC THEN EXISTS_TAC `(f:real^N->real^1) b` THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE [IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]; DISCH_THEN(MP_TAC o MATCH_MP HOMEOMORPHIC_CARD_EQ_COMPONENTS) THEN DISCH_THEN(SUBST1_TAC o MATCH_MP CARD_FINITE_CONG) THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMEOMORPHIC_CONNECTEDNESS) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM CONVEX_CONNECTED_1] THEN REWRITE_TAC[FINITE_COMPONENTS_PUNCTURED_CONVEX]]);; let CARD_EQ_COMPONENTS_IN_COMPACTIFICATION = prove (`!f g (a:real^N) r s z. homeomorphism (sphere(a,r) DELETE z,(:real^M)) (f,g) /\ s SUBSET sphere(a,r) /\ z IN sphere(a,r) /\ ~(z IN closure s) ==> components(sphere(a,r) DIFF s) =_c {1} +_c {c | c IN components((:real^M) DIFF IMAGE f s) /\ bounded c}`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `~((z:real^N) IN s)` ASSUME_TAC THENL [ASM_MESON_TAC[SUBSET; CLOSURE_SUBSET]; ALL_TAC] THEN ABBREV_TAC `c = connected_component (sphere(a:real^N,r) DIFF s) z` THEN SUBGOAL_THEN `c IN components(sphere(a:real^N,r) DIFF s)` ASSUME_TAC THENL [EXPAND_TAC "c" THEN REWRITE_TAC[components; SIMPLE_IMAGE] THEN REWRITE_TAC[ETA_AX] THEN MATCH_MP_TAC FUN_IN_IMAGE THEN ASM_REWRITE_TAC[IN_DIFF]; ALL_TAC] THEN SUBGOAL_THEN `(z:real^N) IN c` ASSUME_TAC THENL [EXPAND_TAC "c" THEN REWRITE_TAC[IN] THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ] THEN ASM_REWRITE_TAC[IN_DIFF]; ALL_TAC] THEN SUBGOAL_THEN `components(sphere(a,r) DIFF s) = {c} UNION (components(sphere(a:real^N,r) DIFF s) DELETE c)` SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhand o rand) CARD_DISJOINT_UNION o lhand o snd) THEN ANTS_TAC THENL [SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] CARD_EQ_TRANS) THEN MATCH_MP_TAC CARD_ADD_CONG THEN SIMP_TAC[CARD_EQ_CARD; CARD_SING; FINITE_SING] THEN REWRITE_TAC[EQ_C_BIJECTIONS] THEN EXISTS_TAC `IMAGE (f:real^N->real^M)` THEN EXISTS_TAC `IMAGE (g:real^M->real^N)` THEN REWRITE_TAC[IN_ELIM_THM] THEN FIRST_ASSUM(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] BOUNDED_IMAGE_IN_COMPACTIFICATION) o ONCE_REWRITE_RULE[SET_RULE `s DELETE z = s DIFF {z}`] o GEN_REWRITE_RULE I [HOMEOMORPHISM_SYM]) THEN REWRITE_TAC[COMPACT_SPHERE; CLOSED_UNIV; SUBSET_UNIV] THEN REWRITE_TAC[SET_RULE `s INTER {z} = {} <=> ~(z IN s)`] THEN DISCH_THEN(fun th -> REWRITE_TAC[th]) THEN CONJ_TAC THENL [X_GEN_TAC `d:real^N->bool` THEN REWRITE_TAC[IN_DELETE] THEN STRIP_TAC THEN SUBGOAL_THEN `~((z:real^N) IN closure d)` ASSUME_TAC THENL [MP_TAC(ISPEC `subtopology euclidean (sphere(a:real^N,r) DIFF s)` PAIRWISE_SEPARATED_CONNECTED_COMPONENTS_OF) THEN REWRITE_TAC[EUCLIDEAN_CONNECTED_COMPONENTS_OF; pairwise] THEN REWRITE_TAC[SEPARATED_IN_SUBTOPOLOGY] THEN REWRITE_TAC[separated_in; EUCLIDEAN_CLOSURE_OF] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `~((z:real^N) IN d)` ASSUME_TAC THENL [ASM_MESON_TAC[REWRITE_RULE[SUBSET] CLOSURE_SUBSET]; ALL_TAC] THEN MATCH_MP_TAC(TAUT `q /\ (q ==> p) ==> p /\ q`) THEN CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `(!x. x IN d ==> g(f x) = x) ==> IMAGE g (IMAGE f d) = d`) THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [homeomorphism]) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[IN_DELETE] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET)) THEN ASM SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[IN_COMPONENTS_MAXIMAL_ALT] THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[IMAGE_EQ_EMPTY; IN_COMPONENTS_NONEMPTY]; REPEAT(FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_SUBSET)) THEN MATCH_MP_TAC(SET_RULE `DISJOINT d s /\ IMAGE f (d INTER s) = IMAGE f d INTER IMAGE f s ==> IMAGE f d SUBSET UNIV DIFF IMAGE f s`) THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [homeomorphism]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `(!x. x IN u ==> g(f x) = x) ==> d SUBSET u /\ s SUBSET u ==> IMAGE f (d INTER s) = IMAGE f d INTER IMAGE f s`)) THEN ASM SET_TAC[]; MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]] THEN FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [homeomorphism]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REPEAT(FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_SUBSET)) THEN ASM SET_TAC[]; X_GEN_TAC `c':real^M->bool` THEN STRIP_TAC THEN SUBGOAL_THEN `IMAGE (g:real^M->real^N) c' SUBSET d` MP_TAC THENL [ALL_TAC; FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [homeomorphism]) THEN ASM SET_TAC[]] THEN MATCH_MP_TAC COMPONENTS_MAXIMAL THEN EXISTS_TAC `sphere (a:real^N,r) DIFF s` THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [homeomorphism]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [homeomorphism]) THEN ASM SET_TAC[]; FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [homeomorphism]) THEN REPEAT(FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_SUBSET)) THEN ASM SET_TAC[]]]; X_GEN_TAC `d:real^M->bool` THEN STRIP_TAC THEN CONJ_TAC THENL [ALL_TAC; FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [homeomorphism]) THEN ASM SET_TAC[]] THEN REWRITE_TAC[IN_DELETE] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[CLOSURE_SUBSET; SUBSET]] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHISM_SYM]) THEN DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN DISCH_THEN(MP_TAC o SPECL [`(:real^M) DIFF IMAGE (f:real^N->real^M) s`; `(sphere(a:real^N,r) DIFF s) DELETE z`]) THEN ANTS_TAC THENL [REPEAT(CONJ_TAC THENL [SET_TAC[]; ALL_TAC]) THEN TRANS_TAC EQ_TRANS `IMAGE (g:real^M->real^N) (:real^M) DIFF IMAGE g (IMAGE (f:real^N->real^M) s)` THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphism]) THEN ASM SET_TAC[]; DISCH_THEN(MP_TAC o MATCH_MP HOMEOMORPHISM_COMPONENTS)] THEN DISCH_THEN(MP_TAC o SPEC `d:real^M->bool` o MATCH_MP (SET_RULE `t = IMAGE f s ==> !d. d IN s ==> f d IN t`)) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[IN_COMPONENTS_MAXIMAL; SUBSET_DELETE] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `c':real^N->bool` THEN ASM_CASES_TAC `(z:real^N) IN c'` THENL [STRIP_TAC; ASM_MESON_TAC[]] THEN SUBGOAL_THEN `IMAGE (g:real^M->real^N) d IN components(c DELETE z)` MP_TAC THENL [ASM_REWRITE_TAC[IN_COMPONENTS_MAXIMAL; SUBSET_DELETE] THEN CONJ_TAC THENL [TRANS_TAC SUBSET_TRANS `c':real^N->bool` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] COMPONENTS_MAXIMAL)) THEN ASM SET_TAC[]; REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET)) THEN ASM SET_TAC[]]; MATCH_MP_TAC(TAUT `~p ==> p ==> q`) THEN DISCH_TAC] THEN MP_TAC(ISPECL [`c:real^N->bool`; `z:real^N`] CONNECTED_EQ_COMPONENT_DELETE_NONSEPARATED) THEN ASM_REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]] THEN REPEAT DISJ2_TAC THEN MATCH_MP_TAC FINITE_COMPONENTS_PUNCTURED_CONNECTED_SUBSET_SPHERE THEN MAP_EVERY EXISTS_TAC [`r:real`; `a:real^N`] THEN CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]; ALL_TAC] THEN CONJ_TAC THENL [REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET)) THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`a:real^N`; `r:real`] LOCALLY_CONNECTED_SPHERE) THEN REWRITE_TAC[LOCALLY_CONNECTED] THEN DISCH_THEN(MP_TAC o SPECL [`sphere(a:real^N,r) DIFF closure s`; `z:real^N`]) THEN SIMP_TAC[OPEN_IN_DIFF_CLOSED; CLOSED_CLOSURE] THEN ASM_REWRITE_TAC[IN_DIFF; interior_of; IN_ELIM_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] COMPONENTS_MAXIMAL)) THEN MP_TAC(ISPEC `s:real^N->bool` CLOSURE_SUBSET) THEN ASM SET_TAC[]]);; let CONNECTED_COMPLEMENT_SUBSET_SIMPLE_PATH_IMAGE = prove (`!g s:real^N->bool. simple_path g /\ pathfinish g = pathstart g /\ s SUBSET path_image g ==> (connected(path_image g DIFF s) <=> connected s)`, SUBGOAL_THEN `!g s:real^N->bool. simple_path g /\ pathfinish g = pathstart g /\ s SUBSET path_image g ==> ~connected s ==> ~connected(path_image g DIFF s)` MP_TAC THENL [ALL_TAC; MESON_TAC[SUBSET_DIFF; SET_RULE `s SUBSET t ==> t DIFF (t DIFF s) = s`]] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [CONNECTED_IFF_CONNECTED_COMPONENT] THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN ASM_CASES_TAC `a:real^N = b` THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ] THEN DISCH_TAC THEN MP_TAC(ISPECL [`g:real^1->real^N`; `a:real^N`; `b:real^N`] EXISTS_DOUBLE_ARC) THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`g1:real^1->real^N`; `g2:real^1->real^N`] THEN STRIP_TAC THEN REWRITE_TAC[CONNECTED_CLOSED] THEN MAP_EVERY EXISTS_TAC [`path_image g1:real^N->bool`;`path_image g2:real^N->bool`] THEN ASM_SIMP_TAC[CLOSED_PATH_IMAGE; ARC_IMP_PATH] THEN REPLICATE_TAC 2 (CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN REPEAT STRIP_TAC THEN UNDISCH_TAC `~connected_component s (a:real^N) b` THEN REWRITE_TAC[connected_component] THENL [EXISTS_TAC `path_image g1:real^N->bool`; EXISTS_TAC `path_image g2:real^N->bool`] THEN ASM_SIMP_TAC[CONNECTED_PATH_IMAGE; ARC_IMP_PATH] THEN (CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN ASM_MESON_TAC[PATHSTART_IN_PATH_IMAGE; PATHFINISH_IN_PATH_IMAGE]);; let CONNECTED_COMPLEMENT_SUBSET_CIRCLE = prove (`!s a:real^N r. dimindex(:N) = 2 /\ s SUBSET sphere(a,r) ==> (connected(sphere(a,r) DIFF s) <=> connected s)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `r < &0` THEN ASM_SIMP_TAC[SPHERE_EMPTY; SUBSET_EMPTY; DIFF_EQ_EMPTY] THEN ASM_CASES_TAC `r = &0` THENL [ASM_SIMP_TAC[SPHERE_SING; SET_RULE `s SUBSET {a} <=> s = {} \/ s = {a}`] THEN STRIP_TAC THEN ASM_SIMP_TAC[DIFF_EMPTY; CONNECTED_EMPTY; CONNECTED_SING; DIFF_EQ_EMPTY]; SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]] THEN STRIP_TAC THEN MP_TAC(ISPECL [`cball(a:real^N,r)`; `cball(vec 0:real^2,r)`] HOMEOMORPHIC_RELATIVE_FRONTIERS_CONVEX_BOUNDED_SETS) THEN REWRITE_TAC[CONVEX_CBALL; BOUNDED_CBALL] THEN ASM_SIMP_TAC[RELATIVE_FRONTIER_CBALL; REAL_LT_IMP_NZ; AFF_DIM_CBALL] THEN ASM_REWRITE_TAC[DIMINDEX_2] THEN ASM_SIMP_TAC[HOMEOMORPHIC_SIMPLE_PATH_IMAGE_CIRCLE_EQ] THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN ASM_MESON_TAC[CONNECTED_COMPLEMENT_SUBSET_SIMPLE_PATH_IMAGE]);; (* ------------------------------------------------------------------------- *) (* When dealing with AR, ANR and ANR later, it's useful to know that any set *) (* at all is homeomorphic to a closed subset of a convex set, and if the *) (* set is locally compact we can take the convex set to be the universe. *) (* ------------------------------------------------------------------------- *) let HOMEOMORPHIC_CLOSED_IN_CONVEX = prove (`!s:real^M->bool. aff_dim s < &(dimindex(:N)) ==> ?u t:real^N->bool. convex u /\ ~(u = {}) /\ closed_in (subtopology euclidean u) t /\ s homeomorphic t`, GEN_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THENL [REPEAT STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`(:real^N)`; `{}:real^N->bool`] THEN REWRITE_TAC[CONVEX_UNIV; UNIV_NOT_EMPTY; CLOSED_IN_EMPTY] THEN ASM_REWRITE_TAC[HOMEOMORPHIC_EMPTY]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY])] THEN DISCH_THEN(X_CHOOSE_THEN `a:real^M` MP_TAC) THEN GEOM_ORIGIN_TAC `a:real^M` THEN SIMP_TAC[AFF_DIM_DIM_0; HULL_INC; INT_OF_NUM_LT] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`{x:real^N | x$1 = &0}`; `dim(s:real^M->bool)`] CHOOSE_SUBSPACE_OF_SUBSPACE) THEN SIMP_TAC[DIM_SPECIAL_HYPERPLANE; DIMINDEX_GE_1; LE_REFL; SUBSET; IN_ELIM_THM; SPAN_OF_SUBSPACE; SUBSPACE_SPECIAL_HYPERPLANE] THEN ANTS_TAC THENL [ASM_ARITH_TAC; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN MP_TAC(ISPECL [`span s:real^M->bool`; `t:real^N->bool`] ISOMETRIES_SUBSPACES) THEN ASM_REWRITE_TAC[SUBSPACE_SPAN; DIM_SPAN; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`h:real^M->real^N`; `k:real^N->real^M`] THEN REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; SUBSET; FORALL_IN_IMAGE] THEN STRIP_TAC THEN MP_TAC(ISPECL [`vec 0:real^N`; `&1`; `basis 1:real^N`; `{x:real^N | basis 1 dot x = &0}`] HOMEOMORPHIC_PUNCTURED_SPHERE_AFFINE) THEN SIMP_TAC[AFFINE_HYPERPLANE; AFF_DIM_HYPERPLANE; BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL; REAL_LT_01; IN_SPHERE_0; NORM_BASIS] THEN ANTS_TAC THENL [INT_ARITH_TAC; ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM]] THEN SIMP_TAC[DOT_BASIS; DIMINDEX_GE_1; LE_REFL; homeomorphic] THEN REWRITE_TAC[HOMEOMORPHISM; LEFT_IMP_EXISTS_THM; IN_ELIM_THM; SUBSET; FORALL_IN_IMAGE; IN_SPHERE_0; IN_DELETE] THEN MAP_EVERY X_GEN_TAC [`f:real^N->real^N`; `g:real^N->real^N`] THEN STRIP_TAC THEN EXISTS_TAC `ball(vec 0,&1) UNION IMAGE ((f:real^N->real^N) o (h:real^M->real^N)) s` THEN EXISTS_TAC `IMAGE ((f:real^N->real^N) o (h:real^M->real^N)) s` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONVEX_INTERMEDIATE_BALL THEN MAP_EVERY EXISTS_TAC [`vec 0:real^N`; `&1`] THEN REWRITE_TAC[SUBSET_UNION; UNION_SUBSET; BALL_SUBSET_CBALL] THEN ASM_REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; o_THM; IN_CBALL_0] THEN ASM_MESON_TAC[SPAN_SUPERSET; REAL_LE_REFL]; REWRITE_TAC[NOT_IN_EMPTY; IMAGE_o] THEN ASM SET_TAC[]; REWRITE_TAC[CLOSED_IN_CLOSED] THEN EXISTS_TAC `sphere(vec 0:real^N,&1)` THEN REWRITE_TAC[CLOSED_SPHERE] THEN MATCH_MP_TAC(SET_RULE `b INTER t = {} /\ s SUBSET t ==> s = (b UNION s) INTER t`) THEN REWRITE_TAC[GSYM CBALL_DIFF_SPHERE] THEN CONJ_TAC THENL [SET_TAC[]; REWRITE_TAC[SUBSET]] THEN REWRITE_TAC[FORALL_IN_IMAGE; o_THM; IN_SPHERE_0] THEN ASM_MESON_TAC[SPAN_SUPERSET]; MAP_EVERY EXISTS_TAC [`(k:real^N->real^M) o (g:real^N->real^N)`; `(f:real^N->real^N) o (h:real^M->real^N)`] THEN REWRITE_TAC[FORALL_IN_IMAGE; o_THM; IMAGE_o] THEN REPEAT CONJ_TAC THEN TRY(MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_SIMP_TAC[LINEAR_CONTINUOUS_ON]) THEN TRY(FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET))) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_SPHERE_0; IN_DELETE] THEN MP_TAC(ISPEC `s:real^M->bool` SPAN_INC) THEN ASM SET_TAC[]]);; let LOCALLY_COMPACT_HOMEOMORPHIC_CLOSED = prove (`!s:real^M->bool. locally compact s /\ dimindex(:M) < dimindex(:N) ==> ?t:real^N->bool. closed t /\ s homeomorphic t`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?t:real^(M,1)finite_sum->bool h. closed t /\ homeomorphism (s,t) (h,fstcart)` STRIP_ASSUME_TAC THENL [ASM_SIMP_TAC[LOCALLY_COMPACT_HOMEOMORPHISM_PROJECTION_CLOSED]; ALL_TAC] THEN ABBREV_TAC `f:real^(M,1)finite_sum->real^N = \x. lambda i. if i <= dimindex(:M) then x$i else x$(dimindex(:M)+1)` THEN ABBREV_TAC `g:real^N->real^(M,1)finite_sum = (\x. lambda i. x$i)` THEN EXISTS_TAC `IMAGE (f:real^(M,1)finite_sum->real^N) t` THEN SUBGOAL_THEN `linear(f:real^(M,1)finite_sum->real^N)` ASSUME_TAC THENL [EXPAND_TAC "f" THEN REWRITE_TAC[linear; CART_EQ] THEN SIMP_TAC[LAMBDA_BETA; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `linear(g:real^N->real^(M,1)finite_sum)` ASSUME_TAC THENL [EXPAND_TAC "g" THEN REWRITE_TAC[linear; CART_EQ] THEN SIMP_TAC[LAMBDA_BETA; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `!x. (g:real^N->real^(M,1)finite_sum)((f:real^(M,1)finite_sum->real^N) x) = x` ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["f"; "g"] THEN FIRST_ASSUM(MP_TAC o MATCH_MP (ARITH_RULE `m < n ==> !i. i <= m + 1 ==> i <= n`)) THEN SIMP_TAC[CART_EQ; LAMBDA_BETA; DIMINDEX_FINITE_SUM; DIMINDEX_1] THEN REWRITE_TAC[ARITH_RULE `i <= n + 1 <=> i <= n \/ i = n + 1`] THEN MESON_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [ASM_MESON_TAC[CLOSED_INJECTIVE_LINEAR_IMAGE]; ALL_TAC] THEN TRANS_TAC HOMEOMORPHIC_TRANS `t:real^(M,1)finite_sum->bool` THEN CONJ_TAC THENL [ASM_MESON_TAC[homeomorphic]; ALL_TAC] THEN REWRITE_TAC[homeomorphic; HOMEOMORPHISM] THEN MAP_EVERY EXISTS_TAC [`f:real^(M,1)finite_sum->real^N`; `g:real^N->real^(M,1)finite_sum`] THEN ASM_SIMP_TAC[LINEAR_CONTINUOUS_ON] THEN ASM SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* Simple connectedness of a union. This is essentially a stripped-down *) (* version of the Seifert - Van Kampen theorem. *) (* ------------------------------------------------------------------------- *) let SIMPLY_CONNECTED_UNION = prove (`!s t:real^N->bool. open_in (subtopology euclidean (s UNION t)) s /\ open_in (subtopology euclidean (s UNION t)) t /\ simply_connected s /\ simply_connected t /\ path_connected (s INTER t) /\ ~(s INTER t = {}) ==> simply_connected (s UNION t)`, REPEAT GEN_TAC THEN REWRITE_TAC[OPEN_IN_OPEN] THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `u:real^N->bool` (STRIP_ASSUME_TAC o GSYM)) MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `v:real^N->bool` (STRIP_ASSUME_TAC o GSYM)) MP_TAC) THEN SIMP_TAC[SIMPLY_CONNECTED_EQ_CONTRACTIBLE_PATH; PATH_CONNECTED_UNION] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `(pathstart p:real^N) IN s UNION t` MP_TAC THENL [ASM_MESON_TAC[PATHSTART_IN_PATH_IMAGE; SUBSET]; REWRITE_TAC[IN_UNION]] THEN POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN ONCE_REWRITE_TAC[TAUT `p ==> q ==> r <=> q ==> p ==> r`] THEN MAP_EVERY (fun s -> let x = mk_var(s,`:real^N->bool`) in SPEC_TAC(x,x)) ["v"; "u"; "t"; "s"] THEN MATCH_MP_TAC(MESON[] `(!s t u v. x IN s ==> P x s t u v) /\ (!x s t u v. P x s t u v ==> P x t s v u) ==> (!s t u v. x IN s \/ x IN t ==> P x s t u v)`) THEN CONJ_TAC THENL [REPEAT STRIP_TAC; REPEAT GEN_TAC THEN REWRITE_TAC[UNION_COMM; INTER_COMM] THEN MATCH_MP_TAC MONO_IMP THEN SIMP_TAC[]] THEN SUBGOAL_THEN `?e. &0 < e /\ !x y. x IN interval[vec 0,vec 1] /\ y IN interval[vec 0,vec 1] /\ norm(x - y) < e ==> path_image(subpath x y p) SUBSET (s:real^N->bool) \/ path_image(subpath x y p) SUBSET t` STRIP_ASSUME_TAC THENL [MP_TAC(ISPEC `path_image(p:real^1->real^N)` HEINE_BOREL_LEMMA) THEN ASM_SIMP_TAC[COMPACT_PATH_IMAGE] THEN DISCH_THEN(MP_TAC o SPEC `{u:real^N->bool,v}`) THEN SIMP_TAC[UNIONS_2; EXISTS_IN_INSERT; FORALL_IN_INSERT; NOT_IN_EMPTY] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`p:real^1->real^N`; `interval[vec 0:real^1,vec 1]`] COMPACT_UNIFORMLY_CONTINUOUS) THEN ASM_REWRITE_TAC[GSYM path; COMPACT_INTERVAL; uniformly_continuous_on] THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[dist] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MAP_EVERY X_GEN_TAC [`x:real^1`; `y:real^1`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `(p:real^1->real^N) x`) THEN ANTS_TAC THENL [REWRITE_TAC[path_image] THEN ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `!p'. p SUBSET b /\ (s UNION t) INTER u = s /\ (s UNION t) INTER v = t /\ p SUBSET p' /\ p' SUBSET s UNION t ==> (b SUBSET u \/ b SUBSET v) ==> p SUBSET s \/ p SUBSET t`) THEN EXISTS_TAC `path_image(p:real^1->real^N)` THEN ASM_SIMP_TAC[PATH_IMAGE_SUBPATH_SUBSET] THEN REWRITE_TAC[PATH_IMAGE_SUBPATH_GEN; SUBSET; FORALL_IN_IMAGE] THEN SUBGOAL_THEN `segment[x,y] SUBSET ball(x:real^1,d)` MP_TAC THENL [REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN MATCH_MP_TAC HULL_MINIMAL THEN ASM_REWRITE_TAC[INSERT_SUBSET; CENTRE_IN_BALL] THEN ASM_REWRITE_TAC[IN_BALL; EMPTY_SUBSET; CONVEX_BALL; dist]; REWRITE_TAC[IN_BALL; dist; SUBSET] THEN STRIP_TAC THEN X_GEN_TAC `z:real^1` THEN DISCH_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [SEGMENT_1]) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL_1])) THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN ASM_REAL_ARITH_TAC]; MP_TAC(SPEC `e:real` REAL_ARCH_INV) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `N:num` THEN STRIP_TAC] THEN SUBGOAL_THEN `!n. n <= N /\ p(lift(&n / &N)) IN s ==> ?q. path(q:real^1->real^N) /\ path_image q SUBSET s /\ homotopic_paths (s UNION t) (subpath (vec 0) (lift(&n / &N)) p) q` MP_TAC THENL [ALL_TAC; DISCH_THEN(MP_TAC o SPEC `N:num`) THEN ASM_SIMP_TAC[REAL_DIV_REFL; REAL_OF_NUM_EQ; LE_REFL; LIFT_NUM] THEN ANTS_TAC THENL [ASM_MESON_TAC[pathfinish]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `q:real^1->real^N` MP_TAC) THEN REWRITE_TAC[SUBPATH_TRIVIAL] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_PATHS_TRANS) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_PATHSTART) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_PATHFINISH) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_MESON_TAC[SUBSET_UNION]] THEN SUBGOAL_THEN `!n. n < N ==> path_image(subpath (lift(&n / &N)) (lift(&(SUC n) / &N)) p) SUBSET (s:real^N->bool) \/ path_image(subpath (lift(&n / &N)) (lift(&(SUC n) / &N)) p) SUBSET t` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[IN_INTERVAL_1; LIFT_DROP; GSYM LIFT_SUB; DROP_VEC; NORM_REAL; GSYM drop; REAL_ARITH `abs(a / c - b / c) = abs((b - a) / c)`] THEN ASM_SIMP_TAC[GSYM REAL_OF_NUM_SUC; REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ; REAL_OF_NUM_LT; LE_1; REAL_ARITH `(x + &1) - x = &1`] THEN ASM_REWRITE_TAC[real_div; REAL_MUL_LID; REAL_MUL_LZERO; REAL_ABS_INV; REAL_ABS_NUM; REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ASM_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC num_WF THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC] THEN STRIP_TAC THEN ASM_CASES_TAC `n = 0` THENL [ASM_REWRITE_TAC[REAL_ARITH `&0 / x = &0`; LIFT_NUM] THEN EXISTS_TAC `linepath((p:real^1->real^N)(vec 0),p(vec 0))` THEN REWRITE_TAC[SUBPATH_REFL; HOMOTOPIC_PATHS_REFL] THEN REWRITE_TAC[PATH_LINEPATH; PATH_IMAGE_LINEPATH; SEGMENT_REFL] THEN UNDISCH_TAC `(pathstart p:real^N) IN s` THEN REWRITE_TAC[pathstart] THEN SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPEC `\m. m < n /\ (p(lift(&m / &N)):real^N) IN s` num_MAX) THEN REWRITE_TAC[] THEN MATCH_MP_TAC(TAUT `p /\ (q ==> r) ==> (p <=> q) ==> r`) THEN CONJ_TAC THENL [CONJ_TAC THENL [EXISTS_TAC `0`; MESON_TAC[LT_IMP_LE]] THEN ASM_SIMP_TAC[REAL_ARITH `&0 / x = &0`; LIFT_NUM; LE_1] THEN ASM_MESON_TAC[pathstart]; DISCH_THEN(X_CHOOSE_THEN `m:num` STRIP_ASSUME_TAC)] THEN SUBGOAL_THEN `?q. path q /\ path_image(q:real^1->real^N) SUBSET s /\ homotopic_paths (s UNION t) (subpath (vec 0) (lift (&m / &N)) p) q` STRIP_ASSUME_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `!i. m < i /\ i <= n ==> path_image(subpath (lift(&m / &N)) (lift(&i / &N)) p) SUBSET s \/ path_image(subpath (lift(&m / &N)) (lift(&i / &N)) p) SUBSET (t:real^N->bool)` MP_TAC THENL [MATCH_MP_TAC num_INDUCTION THEN REWRITE_TAC[CONJUNCT1 LT] THEN X_GEN_TAC `i:num` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_CASES_TAC `i:num = m` THENL [DISCH_THEN(K ALL_TAC) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC]] THEN SUBGOAL_THEN `p(lift(&i / &N)) IN t /\ ~((p(lift(&i / &N)):real^N) IN s)` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC(SET_RULE `x IN s UNION t /\ ~(x IN s) ==> x IN t /\ ~(x IN s)`) THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET t ==> x IN s ==> x IN t`)) THEN REWRITE_TAC[path_image] THEN MATCH_MP_TAC FUN_IN_IMAGE THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; LIFT_DROP] THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_LE_LDIV_EQ; REAL_OF_NUM_LT; LE_1; REAL_MUL_LZERO; REAL_MUL_LID; REAL_OF_NUM_LE] THEN ASM_ARITH_TAC; SUBGOAL_THEN `i < n /\ ~(i:num <= m)` MP_TAC THENL [ASM_ARITH_TAC; ASM_MESON_TAC[]]]; ALL_TAC] THEN SUBGOAL_THEN `path_image(subpath (lift(&i / &N)) (lift (&(SUC i) / &N)) p) SUBSET s \/ path_image(subpath (lift(&i / &N)) (lift (&(SUC i) / &N)) p) SUBSET (t:real^N->bool)` MP_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `~(x IN s) ==> (x IN p /\ x IN q) /\ (q UNION p = r) ==> p SUBSET s \/ p SUBSET t ==> q SUBSET s \/ q SUBSET t ==> r SUBSET s \/ r SUBSET t`)) THEN SIMP_TAC[PATH_IMAGE_SUBPATH_GEN; FUN_IN_IMAGE; ENDS_IN_SEGMENT] THEN REWRITE_TAC[GSYM IMAGE_UNION] THEN AP_TERM_TAC THEN MATCH_MP_TAC UNION_SEGMENT THEN ASM_SIMP_TAC[SEGMENT_1; LIFT_DROP; REAL_LE_DIV2_EQ; REAL_OF_NUM_LT; LE_1; REAL_OF_NUM_LE; LT_IMP_LE; IN_INTERVAL_1] THEN ASM_ARITH_TAC; DISCH_THEN(MP_TAC o SPEC `n:num`) THEN ASM_REWRITE_TAC[LE_REFL]] THEN STRIP_TAC THENL [EXISTS_TAC `(q:real^1->real^N) ++ subpath (lift(&m / &N)) (lift (&n / &N)) p` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC PATH_JOIN_IMP THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_PATHFINISH) THEN ASM_SIMP_TAC[PATHSTART_SUBPATH; PATHFINISH_SUBPATH] THEN DISCH_TAC THEN MATCH_MP_TAC PATH_SUBPATH THEN ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; LIFT_DROP] THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_LE_LDIV_EQ; REAL_OF_NUM_LT; LE_1; REAL_MUL_LZERO; REAL_MUL_LID; REAL_OF_NUM_LE] THEN ASM_ARITH_TAC; MATCH_MP_TAC SUBSET_PATH_IMAGE_JOIN THEN ASM_REWRITE_TAC[]; MATCH_MP_TAC HOMOTOPIC_PATHS_TRANS THEN EXISTS_TAC `subpath (vec 0) (lift(&m / &N)) (p:real^1->real^N) ++ subpath (lift(&m / &N)) (lift(&n / &N)) p` THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[HOMOTOPIC_PATHS_SYM] THEN MATCH_MP_TAC HOMOTOPIC_JOIN_SUBPATHS THEN ASM_REWRITE_TAC[ENDS_IN_UNIT_INTERVAL]; MATCH_MP_TAC HOMOTOPIC_PATHS_JOIN THEN ASM_REWRITE_TAC[PATHSTART_SUBPATH; PATHFINISH_SUBPATH] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[SUBSET_UNION] THEN ASM_REWRITE_TAC[HOMOTOPIC_PATHS_REFL] THEN MATCH_MP_TAC PATH_SUBPATH] THEN ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; LIFT_DROP] THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_LE_LDIV_EQ; REAL_OF_NUM_LT; LE_1; REAL_MUL_LZERO; REAL_MUL_LID; REAL_OF_NUM_LE] THEN ASM_ARITH_TAC]; SUBGOAL_THEN `(p(lift(&m / &N)):real^N) IN t /\ (p(lift(&n / &N)):real^N) IN t` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[PATHSTART_IN_PATH_IMAGE; PATHFINISH_IN_PATH_IMAGE; PATHSTART_SUBPATH; PATHFINISH_SUBPATH; SUBSET]; ALL_TAC] THEN UNDISCH_TAC `path_connected(s INTER t:real^N->bool)` THEN REWRITE_TAC[path_connected] THEN DISCH_THEN(MP_TAC o SPECL [`p(lift(&m / &N)):real^N`; `p(lift(&n / &N)):real^N`]) THEN ASM_REWRITE_TAC[IN_INTER; SUBSET_INTER] THEN DISCH_THEN(X_CHOOSE_THEN `r:real^1->real^N` STRIP_ASSUME_TAC) THEN UNDISCH_THEN `!p. path p /\ path_image p SUBSET t /\ pathfinish p:real^N = pathstart p ==> homotopic_paths t p (linepath (pathstart p,pathstart p))` (MP_TAC o SPEC `subpath (lift(&m / &N)) (lift(&n / &N)) p ++ reversepath(r:real^1->real^N)`) THEN ASM_REWRITE_TAC[PATHSTART_SUBPATH; PATHFINISH_SUBPATH; PATHSTART_JOIN; PATHFINISH_JOIN; PATHFINISH_REVERSEPATH] THEN ANTS_TAC THENL [ASM_SIMP_TAC[SUBSET_PATH_IMAGE_JOIN; PATH_IMAGE_REVERSEPATH] THEN MATCH_MP_TAC PATH_JOIN_IMP THEN ASM_SIMP_TAC[PATH_REVERSEPATH; PATHFINISH_SUBPATH; PATHSTART_REVERSEPATH] THEN MATCH_MP_TAC PATH_SUBPATH THEN ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; LIFT_DROP] THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_LE_LDIV_EQ; REAL_OF_NUM_LT; LE_1; REAL_MUL_LZERO; REAL_MUL_LID; REAL_OF_NUM_LE] THEN ASM_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMOTOPIC_PATHS_IMP_HOMOTOPIC_LOOPS)) THEN ASM_REWRITE_TAC[PATHFINISH_LINEPATH; PATHSTART_SUBPATH; PATHSTART_JOIN; PATHFINISH_JOIN; PATHFINISH_REVERSEPATH] THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMOTOPIC_PATHS_LOOP_PARTS)) THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_PATHSTART) THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_PATHFINISH) THEN REWRITE_TAC[PATHSTART_SUBPATH; PATHFINISH_SUBPATH] THEN REPLICATE_TAC 2 (DISCH_THEN(ASSUME_TAC o SYM)) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN EXISTS_TAC `(q:real^1->real^N) ++ r` THEN ASM_SIMP_TAC[PATH_JOIN; PATH_IMAGE_JOIN; UNION_SUBSET] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_TRANS THEN EXISTS_TAC `subpath (vec 0) (lift(&m / &N)) (p:real^1->real^N) ++ subpath (lift(&m / &N)) (lift(&n / &N)) p` THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[HOMOTOPIC_PATHS_SYM] THEN MATCH_MP_TAC HOMOTOPIC_JOIN_SUBPATHS THEN ASM_REWRITE_TAC[ENDS_IN_UNIT_INTERVAL] THEN ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; LIFT_DROP] THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_LE_LDIV_EQ; REAL_OF_NUM_LT; LE_1; REAL_MUL_LZERO; REAL_MUL_LID; REAL_OF_NUM_LE] THEN ASM_ARITH_TAC; MATCH_MP_TAC HOMOTOPIC_PATHS_JOIN THEN ASM_REWRITE_TAC[PATHSTART_SUBPATH; PATHFINISH_SUBPATH] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_SUBSET THEN EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[SUBSET_UNION]]]);; (* ------------------------------------------------------------------------- *) (* Basic results about topological dimension. *) (* ------------------------------------------------------------------------- *) let dimension = new_definition `(dimension:(real^N->bool)->int) s = if s = {} then --(&1) else &(minimal n. (subtopology euclidean s) dimension_le &n)`;; let DIMENSION_GE = prove (`!s:real^N->bool. -- &1 <= dimension s`, GEN_TAC THEN REWRITE_TAC[dimension; INT_LE_REFL] THEN INT_ARITH_TAC);; let DIMENSION_LE_IMP_GE = prove (`!s:real^N->bool n. dimension s <= n ==> -- &1 <= n`, REWRITE_TAC[GSYM INT_LE_TRANS_LE; DIMENSION_GE]);; let (HOMEOMORPHIC_DIMENSION,DIMENSION_LE_EQ, LOCALLY_DIMENSION_LE, DIMENSION_LE_AFF_DIM,DIMENSION_DIMENSION_LE) = let HOMEOMORPHIC_DIMENSION_LE' = prove (`!s:real^M->bool t:real^N->bool n. s homeomorphic t ==> (subtopology euclidean s dimension_le n <=> subtopology euclidean t dimension_le n)`, REWRITE_TAC[GSYM HOMEOMORPHIC_SPACE_EUCLIDEAN] THEN REWRITE_TAC[HOMEOMORPHIC_SPACE_DIMENSION_LE]) in let DIMENSION_LE_EQ' = prove (`!s:real^N->bool n. (subtopology euclidean s) dimension_le n <=> -- &1 <= n /\ !v a. open v /\ a IN v /\ a IN s ==> ?u. a IN u /\ u SUBSET v /\ open u /\ subtopology euclidean (s INTER frontier u) dimension_le (n - &1)`, REPEAT GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [DIMENSION_LE_EQ_SUBTOPOLOGY] THEN REWRITE_TAC[GSYM OPEN_IN] THEN AP_TERM_TAC THEN GEN_REWRITE_TAC (BINOP_CONV o funpow 2 BINDER_CONV o RAND_CONV o ONCE_DEPTH_CONV) [TAUT `open(u:real^N->bool) /\ p <=> ~(open u ==> ~p)`] THEN SIMP_TAC[frontier; INTERIOR_OPEN; FRONTIER_OF_CLOSURES] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN REWRITE_TAC[SET_RULE `s DIFF s INTER u = s INTER (UNIV DIFF u)`] THEN SIMP_TAC[CLOSURE_OF_CLOSED_IN; CLOSED_IN_CLOSED_INTER; GSYM OPEN_CLOSED] THEN REWRITE_TAC[CLOSURE_OF_SUBTOPOLOGY] THEN REWRITE_TAC[TAUT `~(p ==> ~q) <=> p /\ q`] THEN REWRITE_TAC[SET_RULE `s INTER s INTER u = s INTER u`] THEN REWRITE_TAC[SET_RULE `(s INTER t) INTER (s INTER (UNIV DIFF u)) = (s INTER t) DIFF u`] THEN REWRITE_TAC[EUCLIDEAN_CLOSURE_OF] THEN EQ_TAC THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`v:real^N->bool`; `a:real^N`] THEN STRIP_TAC THENL [ALL_TAC; FIRST_X_ASSUM(MP_TAC o SPECL [`v:real^N->bool`; `a:real^N`]) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^N->bool` THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] DIMENSION_LE_SUBTOPOLOGIES) THEN MATCH_MP_TAC(SET_RULE `t SUBSET t' ==> s INTER t DIFF u SUBSET s INTER (t' DIFF u)`) THEN MATCH_MP_TAC SUBSET_CLOSURE THEN REWRITE_TAC[INTER_SUBSET]] THEN SUBGOAL_THEN `?w:real^N->bool. a IN w /\ open w /\ closure w SUBSET v` STRIP_ASSUME_TAC THENL [MP_TAC(ISPECL [`(:real^N)`; `v:real^N->bool`; `a:real^N`] LOCALLY_CLOSED_IN_EXPLICIT) THEN ASM_SIMP_TAC[GSYM OPEN_IN; SUBTOPOLOGY_UNIV; EUCLIDEAN_CLOSURE_OF] THEN MESON_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`w:real^N->bool`; `a:real^N`]) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`s INTER u:real^N->bool`; `s DIFF closure(s INTER u):real^N->bool`] SEPARATION_CLOSURES) THEN ANTS_TAC THENL [REWRITE_TAC[SEPARATION_OPEN_IN_UNION] THEN CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `s INTER t SUBSET u ==> DISJOINT (s INTER t) (s DIFF u)`) THEN REWRITE_TAC[CLOSURE_SUBSET]; REWRITE_TAC[OPEN_IN_OPEN] THEN CONJ_TAC THENL [EXISTS_TAC `u:real^N->bool`; EXISTS_TAC `(:real^N) DIFF closure(s INTER u)`] THEN ASM_REWRITE_TAC[GSYM closed; CLOSED_CLOSURE] THEN SET_TAC[]]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`t:real^N->bool`; `c:real^N->bool`] THEN STRIP_TAC THEN EXISTS_TAC `t INTER v:real^N->bool` THEN ASM_SIMP_TAC[IN_INTER; INTER_SUBSET; OPEN_INTER] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] DIMENSION_LE_SUBTOPOLOGIES)) THEN MATCH_MP_TAC(SET_RULE `s DIFF d SUBSET s DIFF c /\ s INTER u SUBSET t ==> s INTER (c DIFF t) SUBSET s INTER d DIFF u`) THEN CONJ_TAC THENL [ALL_TAC; MP_TAC(ISPEC `w:real^N->bool` CLOSURE_SUBSET) THEN ASM SET_TAC[]] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s DIFF t SUBSET u ==> s INTER u SUBSET v ==> s DIFF t SUBSET v`)) THEN MATCH_MP_TAC(SET_RULE `d SUBSET UNIV DIFF c ==> s INTER c SUBSET s DIFF d`) THEN ASM_SIMP_TAC[CLOSURE_MINIMAL_EQ; GSYM OPEN_CLOSED] THEN ASM SET_TAC[]) in let LOCALLY_DIMENSION_LE' = prove (`!s:real^N->bool n. (subtopology euclidean s) dimension_le n <=> -- &1 <= n /\ locally (\u. (subtopology euclidean u) dimension_le n) s`, REPEAT GEN_TAC THEN EQ_TAC THENL [DISCH_TAC THEN CONJ_TAC THENL [ASM_MESON_TAC[DIMENSION_LE_BOUND]; ALL_TAC] THEN REWRITE_TAC[locally] THEN MAP_EVERY X_GEN_TAC [`v:real^N->bool`; `a:real^N`] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`v:real^N->bool`; `v:real^N->bool`] THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN FIRST_ASSUM(MP_TAC o SPEC `v:real^N->bool` o MATCH_MP DIMENSION_LE_SUBTOPOLOGY) THEN REWRITE_TAC[SUBTOPOLOGY_SUBTOPOLOGY] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN SET_TAC[]; ALL_TAC] THEN ONCE_REWRITE_TAC[DIMENSION_LE_EQ'] THEN REWRITE_TAC[locally] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MAP_EVERY X_GEN_TAC [`v:real^N->bool`; `a:real^N`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`s INTER v:real^N->bool`; `a:real^N`]) THEN ASM_SIMP_TAC[OPEN_IN_OPEN_INTER; LEFT_IMP_EXISTS_THM; IN_INTER] THEN MAP_EVERY X_GEN_TAC [`w:real^N->bool`; `u:real^N->bool`] THEN STRIP_TAC THEN SUBGOAL_THEN `?b:real^N->bool. a IN b /\ open b /\ s INTER closure b SUBSET w` STRIP_ASSUME_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_CONTAINS_CBALL]) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `a:real^N`)) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `ball(a:real^N,r)` THEN ASM_SIMP_TAC[CLOSURE_BALL; OPEN_BALL; CENTRE_IN_BALL] THEN ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`b INTER v:real^N->bool`; `a:real^N`]) THEN ASM_SIMP_TAC[OPEN_INTER; IN_INTER] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `s INTER frontier c:real^N->bool = u INTER frontier c` (fun th -> ASM_REWRITE_TAC[th]) THEN MP_TAC(ISPECL [`c:real^N->bool`; `b:real^N->bool`] SUBSET_CLOSURE) THEN REWRITE_TAC[frontier] THEN ASM SET_TAC[]) in let DIMENSION_LE_AFF_DIM' = prove (`!s:real^N->bool. (subtopology euclidean s) dimension_le aff_dim s`, SUBGOAL_THEN `!n s:real^N->bool. -- &1 <= n ==> aff_dim s = n ==> (subtopology euclidean s) dimension_le n` MP_TAC THENL [ALL_TAC; MESON_TAC[AFF_DIM_GE]] THEN SIMP_TAC[RIGHT_FORALL_IMP_THM; INT_ARITH `--w:int <= x <=> &0 <= x + w`] THEN REWRITE_TAC[GSYM INT_OF_NUM_EXISTS; LEFT_IMP_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN SIMP_TAC[INT_ARITH `x + &1:int = &n <=> x = &n - &1`] THEN REWRITE_TAC[LEFT_FORALL_IMP_THM; EXISTS_REFL] THEN INDUCT_TAC THEN CONV_TAC INT_REDUCE_CONV THEN SIMP_TAC[AFF_DIM_EQ_MINUS1; DIMENSION_LE_EQ_EMPTY] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; GSYM INT_OF_NUM_SUC] THEN REWRITE_TAC[INT_ARITH `(x + &1) - &1:int = x`] THEN X_GEN_TAC `s:real^N->bool` THEN DISCH_TAC THEN MATCH_MP_TAC DIMENSION_LE_SUBTOPOLOGIES THEN EXISTS_TAC `affine hull s:real^N->bool` THEN REWRITE_TAC[HULL_SUBSET] THEN ONCE_REWRITE_TAC[DIMENSION_LE_EQ'] THEN CONJ_TAC THENL [INT_ARITH_TAC; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`v:real^N->bool`; `a:real^N`] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `a:real^N` o REWRITE_RULE[OPEN_CONTAINS_BALL]) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `r:real` THEN STRIP_TAC THEN EXISTS_TAC `ball(a:real^N,r)` THEN ASM_SIMP_TAC[OPEN_BALL; CENTRE_IN_BALL; FRONTIER_BALL] THEN ONCE_REWRITE_TAC[LOCALLY_DIMENSION_LE'] THEN CONJ_TAC THENL [INT_ARITH_TAC; ALL_TAC] THEN ONCE_REWRITE_TAC[LOCALLY_ON_OPEN_SUBSETS] THEN X_GEN_TAC `b:real^N` THEN REWRITE_TAC[IN_INTER] THEN STRIP_TAC THEN EXISTS_TAC `(affine hull s INTER sphere(a,r)) DELETE (&2 % a - b:real^N)` THEN ASM_SIMP_TAC[OPEN_IN_DELETE; OPEN_IN_REFL; IN_DELETE; IN_INTER] THEN REWRITE_TAC[NORM_ARITH `b:real^N = &2 % a - b <=> dist(a,b) = &0`] THEN CONJ_TAC THENL [ASM_MESON_TAC[IN_SPHERE; REAL_LT_REFL]; ALL_TAC] THEN MATCH_MP_TAC(TAUT `!p. p /\ q ==> q`) THEN EXISTS_TAC `-- &1:int <= &n - &1` THEN REWRITE_TAC[GSYM LOCALLY_DIMENSION_LE'] THEN MATCH_MP_TAC (MESON[HOMEOMORPHIC_DIMENSION_LE'] `!s:real^N->bool t:real^N->bool n. s homeomorphic t /\ subtopology euclidean t dimension_le n ==> subtopology euclidean s dimension_le n`) THEN SUBGOAL_THEN `?s:real^N->bool. affine s /\ aff_dim s = &n - &1` MP_TAC THENL [MATCH_MP_TAC AFFINE_EXISTS THEN MP_TAC(ISPEC `s:real^N->bool` AFF_DIM_LE_UNIV) THEN ASM_REWRITE_TAC[] THEN INT_ARITH_TAC; MATCH_MP_TAC MONO_EXISTS] THEN X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN ASM_SIMP_TAC[] THEN ONCE_REWRITE_TAC[INTER_COMM] THEN MATCH_MP_TAC HOMEOMORPHIC_PUNCTURED_AFFINE_SPHERE_AFFINE THEN ASM_REWRITE_TAC[AFFINE_AFFINE_HULL; AFF_DIM_AFFINE_HULL] THEN SIMP_TAC[IN_SPHERE; NORM_ARITH `dist(a:real^N,&2 % a - b) = dist(a,b)`] THEN ASM_REWRITE_TAC[GSYM IN_SPHERE] THEN CONJ_TAC THENL [ALL_TAC; INT_ARITH_TAC] THEN REWRITE_TAC[VECTOR_ARITH `&2 % a - b:real^N = a - &1 % (b - a)`] THEN MATCH_MP_TAC IN_AFFINE_SUB_MUL_DIFF THEN ASM_REWRITE_TAC[AFFINE_AFFINE_HULL]) in let DIMENSION_DIMENSION_LE = prove (`!s:real^N->bool n. dimension s <= n <=> (subtopology euclidean s) dimension_le n`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[dimension] THENL [EQ_TAC THEN REWRITE_TAC[DIMENSION_LE_BOUND] THEN DISCH_TAC THEN MATCH_MP_TAC DIMENSION_LE_MONO THEN EXISTS_TAC `-- &1:int` THEN ASM_REWRITE_TAC[DIMENSION_LE_EQ_EMPTY; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY]; ALL_TAC] THEN ASM_CASES_TAC `--(&1):int <= n` THENL [ALL_TAC; ASM_MESON_TAC[DIMENSION_LE_BOUND; INT_ARITH `&n:int <= x ==> --(&1):int <= x `]] THEN ASM_CASES_TAC `n:int = --(&1)` THENL [ASM_REWRITE_TAC[DIMENSION_LE_EQ_EMPTY; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN ASM_INT_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `?k. n:int = &k` (CHOOSE_THEN SUBST1_TAC) THENL [REWRITE_TAC[INT_OF_NUM_EXISTS] THEN ASM_INT_ARITH_TAC; ALL_TAC] THEN MP_TAC(fst(EQ_IMP_RULE(ISPEC `\n. subtopology euclidean (s:real^N->bool) dimension_le &n` MINIMAL))) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM AFF_DIM_POS_LE]) THEN REWRITE_TAC[GSYM INT_OF_NUM_EXISTS] THEN MATCH_MP_TAC MONO_EXISTS THEN ASM_MESON_TAC[DIMENSION_LE_AFF_DIM']; ALL_TAC] THEN ABBREV_TAC `d = minimal n. subtopology euclidean (s:real^N->bool) dimension_le &n` THEN REWRITE_TAC[GSYM NOT_LE; CONTRAPOS_THM; INT_OF_NUM_LE] THEN MESON_TAC[INT_OF_NUM_LE; INT_LE_TRANS; DIMENSION_LE_MONO]) in let HOMEOMORPHIC_DIMENSION = prove (`!s:real^M->bool t:real^N->bool. s homeomorphic t ==> dimension s = dimension t`, REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM INT_LE_ANTISYM] THEN ONCE_REWRITE_TAC[INT_LE_TRANS_LE] THEN REWRITE_TAC[DIMENSION_DIMENSION_LE; AND_FORALL_THM] THEN GEN_TAC THEN ONCE_REWRITE_TAC[TAUT `(p ==> q) /\ (q ==> p) <=> (q <=> p)`] THEN MATCH_MP_TAC HOMEOMORPHIC_DIMENSION_LE' THEN ASM_REWRITE_TAC[]) in let DIMENSION_LE_EQ = prove (`!s:real^N->bool n. dimension s <= n <=> -- &1 <= n /\ !v a. open v /\ a IN v /\ a IN s ==> ?u. a IN u /\ u SUBSET v /\ open u /\ dimension(s INTER frontier u) <= n - &1`, REWRITE_TAC[DIMENSION_DIMENSION_LE] THEN MATCH_ACCEPT_TAC DIMENSION_LE_EQ') in let LOCALLY_DIMENSION_LE = prove (`!s:real^N->bool n. dimension s <= n <=> -- &1 <= n /\ locally (\u. dimension u <= n) s`, REWRITE_TAC[DIMENSION_DIMENSION_LE] THEN MATCH_ACCEPT_TAC LOCALLY_DIMENSION_LE') in let DIMENSION_LE_AFF_DIM = prove (`!s:real^N->bool. dimension s <= aff_dim s`, REWRITE_TAC[DIMENSION_DIMENSION_LE] THEN MATCH_ACCEPT_TAC DIMENSION_LE_AFF_DIM') in (HOMEOMORPHIC_DIMENSION,DIMENSION_LE_EQ, LOCALLY_DIMENSION_LE, DIMENSION_LE_AFF_DIM,DIMENSION_DIMENSION_LE);; let DIMENSION_TRANSLATION = prove (`!a:real^N s. dimension(IMAGE (\x. a + x) s) = dimension s`, REPEAT GEN_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_DIMENSION THEN REWRITE_TAC[HOMEOMORPHIC_TRANSLATION_SELF]);; add_translation_invariants [DIMENSION_TRANSLATION];; let DIMENSION_LINEAR_IMAGE = prove (`!f:real^M->real^N. linear f /\ (!x y. f x = f y ==> x = y) ==> dimension(IMAGE f s) = dimension s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_DIMENSION THEN MATCH_MP_TAC HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_SELF THEN ASM_REWRITE_TAC[]);; add_linear_invariants [DIMENSION_LINEAR_IMAGE];; let DIMENSION_LE_DIMINDEX = prove (`!s:real^N->bool. dimension s <= &(dimindex(:N))`, MESON_TAC[INT_LE_TRANS; AFF_DIM_LE_UNIV; DIMENSION_LE_AFF_DIM]);; let DIMENSION_LE_MINUS1 = prove (`!s:real^N->bool. dimension s <= -- &1 <=> s = {}`, REWRITE_TAC[DIMENSION_DIMENSION_LE; DIMENSION_LE_EQ_EMPTY; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY]);; let DIMENSION_EQ_MINUS1 = prove (`!s:real^N->bool. dimension s = -- &1 <=> s = {}`, REWRITE_TAC[GSYM INT_LE_ANTISYM; DIMENSION_GE; DIMENSION_LE_MINUS1]);; let DIMENSION_POS_LE = prove (`!s:real^N->bool. &0 <= dimension s <=> ~(s = {})`, SIMP_TAC[DIMENSION_GE; DIMENSION_EQ_MINUS1; INT_ARITH `-- &1:int <= d ==> (&0 <= d <=> ~(d = -- &1))`]);; let DIMENSION_EMPTY = prove (`dimension {} = -- &1`, REWRITE_TAC[DIMENSION_EQ_MINUS1]);; let DIMENSION_SUBSET = prove (`!s t:real^N->bool. s SUBSET t ==> dimension s <= dimension t`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[INT_LE_TRANS_LE] THEN REWRITE_TAC[DIMENSION_DIMENSION_LE] THEN ASM_MESON_TAC[DIMENSION_LE_SUBTOPOLOGIES]);; let DIMENSION_LE_DISCRETE = prove (`!s:real^N->bool. {x | x limit_point_of s} = {} ==> dimension s <= &0`, GEN_TAC THEN REWRITE_TAC[DIMENSION_DIMENSION_LE] THEN REWRITE_TAC[GSYM EUCLIDEAN_DERIVED_SET_OF_IFF_LIMIT_POINT_OF] THEN SIMP_TAC[SUBTOPOLOGY_EQ_DISCRETE_TOPOLOGY; INTER_EMPTY; TOPSPACE_EUCLIDEAN; SUBSET_UNIV; DIMENSION_LE_DISCRETE_TOPOLOGY]);; let DIMENSION_EQ_ZERO_DISCRETE = prove (`!s:real^N->bool. ~(s = {}) /\ {x | x limit_point_of s} = {} ==> dimension s = &0`, SIMP_TAC[GSYM INT_LE_ANTISYM; DIMENSION_POS_LE; DIMENSION_LE_DISCRETE]);; let DIMENSION_EQ_DISCRETE = prove (`!s:real^N->bool. {x | x limit_point_of s} = {} ==> dimension s = if s = {} then --(&1) else &0`, REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[DIMENSION_EMPTY; DIMENSION_EQ_ZERO_DISCRETE]);; let DIMENSION_LE_EQ_ALT = prove (`!s:real^N->bool n. dimension s <= n <=> -- &1 <= n /\ !v a. open v /\ a IN v /\ a IN s ==> ?u. a IN u /\ u SUBSET v /\ open u /\ dimension(subtopology euclidean s frontier_of (s INTER u)) <= n - &1`, REPEAT GEN_TAC THEN REWRITE_TAC[DIMENSION_DIMENSION_LE] THEN GEN_REWRITE_TAC LAND_CONV [DIMENSION_LE_EQ_SUBTOPOLOGY] THEN REWRITE_TAC[GSYM OPEN_IN]);; let DIMENSION_LE_EQ_LOCAL = prove (`!s:real^N->bool n. dimension s <= n <=> -- &1 <= n /\ !v a. open_in (subtopology euclidean s) v /\ a IN v ==> ?u. a IN u /\ u SUBSET v /\ open_in (subtopology euclidean s) u /\ dimension(subtopology euclidean s frontier_of u) <= n - &1`, REPEAT GEN_TAC THEN REWRITE_TAC[DIMENSION_DIMENSION_LE] THEN GEN_REWRITE_TAC LAND_CONV [DIMENSION_LE_CASES] THEN REWRITE_TAC[SUBTOPOLOGY_SUBTOPOLOGY] THEN REWRITE_TAC[GSYM DIMENSION_DIMENSION_LE] THEN SIMP_TAC[FRONTIER_OF_SUBSET_SUBTOPOLOGY; SET_RULE `t SUBSET s ==> s INTER t = t`]);; let DIMENSION_LE_EQ_GENERAL = prove (`!t s:real^N->bool n. s SUBSET t ==> (dimension s <= n <=> -- &1 <= n /\ !v a. open_in (subtopology euclidean t) v /\ a IN v /\ a IN s ==> ?u. a IN u /\ u SUBSET v /\ open_in (subtopology euclidean t) u /\ dimension(s INTER subtopology euclidean t frontier_of u) <= n - &1)`, REPEAT STRIP_TAC THEN EQ_TAC THENL [GEN_REWRITE_TAC LAND_CONV [DIMENSION_LE_EQ]; GEN_REWRITE_TAC RAND_CONV [DIMENSION_LE_EQ_LOCAL]] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MAP_EVERY X_GEN_TAC [`w:real^N->bool`; `a:real^N`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN DISCH_THEN(X_CHOOSE_THEN `v:real^N->bool` MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC) THEN FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [IN_INTER]) THENL [FIRST_X_ASSUM(MP_TAC o SPECL [`v:real^N->bool`; `a:real^N`]) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `t INTER u:real^N->bool`; SUBGOAL_THEN `(a:real^N) IN t` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`t INTER v:real^N->bool`; `a:real^N`]) THEN ASM_SIMP_TAC[IN_INTER; OPEN_IN_OPEN_INTER] THEN DISCH_THEN(X_CHOOSE_THEN `uu:real^N->bool` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC) THEN FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [IN_INTER]) THEN EXISTS_TAC `s INTER u:real^N->bool`] THEN (ASM_SIMP_TAC[IN_INTER; OPEN_IN_OPEN_INTER] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] INT_LE_TRANS)) THEN MATCH_MP_TAC DIMENSION_SUBSET THEN REWRITE_TAC[FRONTIER_OF_CLOSURES; FRONTIER_CLOSURES] THEN REWRITE_TAC[CLOSURE_OF_SUBTOPOLOGY; EUCLIDEAN_CLOSURE_OF] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY]) THENL [MATCH_MP_TAC(SET_RULE `s SUBSET t /\ c1 SUBSET c1' /\ c2 SUBSET c2' ==> s INTER (t INTER c1) INTER t INTER c2 SUBSET s INTER c1' INTER c2'`); MATCH_MP_TAC(SET_RULE `s SUBSET t /\ c1 SUBSET c1' /\ c2 SUBSET c2' ==> (s INTER c1) INTER s INTER c2 SUBSET s INTER (t INTER c1') INTER (t INTER c2')`)] THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THEN MATCH_MP_TAC SUBSET_CLOSURE THEN ASM SET_TAC[]);; let DIMENSION_LE_EQ_LOCALLY = prove (`!s:real^N->bool n. dimension s <= n <=> --(&1) <= n /\ locally (\u. open_in (subtopology euclidean s) u /\ dimension(subtopology euclidean s frontier_of u) <= n - &1) s`, REPEAT GEN_TAC THEN REWRITE_TAC[locally] THEN GEN_REWRITE_TAC LAND_CONV [DIMENSION_LE_EQ_LOCAL] THEN MESON_TAC[SUBSET]);; let LOCALLY_OPEN_AND_DIMENSION_LE = prove (`!s n. dimension s <= n <=> -- &1 <= n /\ locally (\u. open_in (subtopology euclidean s) u /\ dimension u <= n) s`, REPEAT GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [LOCALLY_DIMENSION_LE] THEN AP_TERM_TAC THEN GEN_REWRITE_TAC RAND_CONV [LOCALLY_AND_OPEN_IN] THEN REWRITE_TAC[locally] THEN MESON_TAC[DIMENSION_SUBSET; INT_LE_TRANS; SUBSET_TRANS]);; let DIMENSION_EQ_ON_NBDS = prove (`!s:real^N->bool n. ~(s = {}) /\ (!x. x IN s ==> ?u v. x IN u /\ open_in (subtopology euclidean s) u /\ u SUBSET v /\ v SUBSET s /\ dimension v = n) ==> dimension s = n`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN FIRST_ASSUM(MP_TAC o SPEC `a:real^N`) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `v:real^N->bool`] THEN STRIP_TAC THEN SUBGOAL_THEN `dimension(v:real^N->bool) <= dimension(s:real^N->bool)` MP_TAC THENL [ASM_SIMP_TAC[DIMENSION_SUBSET]; ASM_REWRITE_TAC[]] THEN SIMP_TAC[GSYM INT_LE_ANTISYM] THEN DISCH_TAC THEN ONCE_REWRITE_TAC[LOCALLY_DIMENSION_LE] THEN CONJ_TAC THENL [ASM_MESON_TAC[DIMENSION_GE]; ALL_TAC] THEN ONCE_REWRITE_TAC[LOCALLY_ON_NBDS] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [GSYM INT_LE_ANTISYM] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [LOCALLY_DIMENSION_LE] THEN SIMP_TAC[]);; let LOCALLY_DIMENSION_EQ = prove (`!s:real^N->bool n. ~(s = {}) /\ locally (\u. dimension u = n) s ==> dimension s = n`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [locally]) THEN DISCH_THEN(MP_TAC o SPEC `s:real^N->bool`) THEN REWRITE_TAC[OPEN_IN_REFL] THEN DISCH_TAC THEN MATCH_MP_TAC DIMENSION_EQ_ON_NBDS THEN ASM_MESON_TAC[]);; let DIMENSION_EQ_ON_OPEN_SUBSETS = prove (`!s:real^N->bool n. ~(s = {}) /\ (!x. x IN s ==> ?u. x IN u /\ open_in (subtopology euclidean s) u /\ dimension u = n) ==> dimension s = n`, REPEAT STRIP_TAC THEN MATCH_MP_TAC DIMENSION_EQ_ON_NBDS THEN ASM_MESON_TAC[SUBSET_REFL; OPEN_IN_IMP_SUBSET]);; let DIMENSION_EQ_LOCALLY_CLOPEN = prove (`!s:real^N->bool. dimension s <= &0 <=> locally (\u. closed_in (subtopology euclidean s) u /\ open_in (subtopology euclidean s) u) s`, REWRITE_TAC[GSYM NEIGHBOURHOOD_BASE_OF_EUCLIDEAN] THEN REWRITE_TAC[DIMENSION_DIMENSION_LE] THEN REWRITE_TAC[DIMENSION_LE_0_NEIGHBOURHOOD_BASE_OF_CLOPEN]);; let SMALL_INDUCTIVE_DIMENSION = prove (`!s:real^N->bool n. dimension s <= n <=> -- &1 <= n /\ !c a. a IN s /\ closed_in (subtopology euclidean s) c /\ ~(a IN c) ==> ?b. closed_in (subtopology euclidean s) b /\ dimension b <= n - &1 /\ ?u v. open_in (subtopology euclidean s) u /\ open_in (subtopology euclidean s) v /\ DISJOINT u v /\ u UNION v = s DIFF b /\ a IN u /\ c SUBSET v`, REPEAT GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [DIMENSION_LE_EQ_LOCAL] THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THENL [MAP_EVERY X_GEN_TAC [`c:real^N->bool`; `a:real^N`] THEN STRIP_TAC THEN SUBGOAL_THEN `?v:real^N->bool. a IN v /\ open_in (subtopology euclidean s) v /\ DISJOINT ((subtopology euclidean s) closure_of v) c` STRIP_ASSUME_TAC THENL [MP_TAC(ISPECL [`s:real^N->bool`; `s DIFF c:real^N->bool`; `a:real^N`] LOCALLY_CLOSED_IN_EXPLICIT) THEN ASM_SIMP_TAC[IN_DIFF; OPEN_IN_DIFF; OPEN_IN_REFL] THEN MATCH_MP_TAC MONO_EXISTS THEN SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`v:real^N->bool`; `a:real^N`]) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `u:real^N->bool` THEN STRIP_TAC THEN EXISTS_TAC `subtopology euclidean s frontier_of u:real^N->bool` THEN ASM_REWRITE_TAC[CLOSED_IN_FRONTIER_OF] THEN MAP_EVERY EXISTS_TAC [`u:real^N->bool`; `s DIFF (subtopology euclidean s closure_of u):real^N->bool`] THEN ASM_SIMP_TAC[OPEN_IN_DIFF; OPEN_IN_REFL; CLOSED_IN_CLOSURE_OF; FRONTIER_OF_CLOSURES; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; CLOSED_IN_DIFF; CLOSED_IN_REFL; CLOSURE_OF_CLOSED_IN] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `u SUBSET t ==> DISJOINT u (s DIFF t)`) THEN MATCH_MP_TAC CLOSURE_OF_SUBSET THEN ASM_MESON_TAC[OPEN_IN_SUBSET]; MATCH_MP_TAC(SET_RULE `u SUBSET u' /\ u' SUBSET s ==> u UNION (s DIFF u') = s DIFF (u' INTER (s DIFF u))`) THEN REWRITE_TAC[CLOSURE_OF_SUBSET_SUBTOPOLOGY] THEN ASM_SIMP_TAC[CLOSURE_OF_SUBSET; OPEN_IN_SUBSET]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `DISJOINT v c ==> c SUBSET s /\ u SUBSET v ==> c SUBSET s DIFF u`)) THEN ASM_MESON_TAC[CLOSURE_OF_MONO; CLOSED_IN_IMP_SUBSET]]; MAP_EVERY X_GEN_TAC [`w:real^N->bool`; `a:real^N`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`s DIFF w:real^N->bool`; `a:real^N`]) THEN ASM_SIMP_TAC[IN_DIFF; CLOSED_IN_DIFF; CLOSED_IN_REFL] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`b:real^N->bool`; `u:real^N->bool`; `v:real^N->bool`] THEN STRIP_TAC THEN EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN TRANS_TAC INT_LE_TRANS `dimension(b:real^N->bool)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC DIMENSION_SUBSET THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `u UNION v = s DIFF b ==> f SUBSET s /\ DISJOINT f u /\ DISJOINT f v ==> f SUBSET b`)) THEN REWRITE_TAC[FRONTIER_OF_SUBSET_SUBTOPOLOGY] THEN ASM_SIMP_TAC[FRONTIER_OF_CLOSURES; CLOSURE_OF_CLOSED_IN; CLOSED_IN_REFL; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; CLOSED_IN_DIFF] THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `v INTER s = {} ==> DISJOINT (s INTER c) v`) THEN ASM_SIMP_TAC[OPEN_IN_INTER_CLOSURE_OF_EQ_EMPTY; OPEN_IN_SUBSET] THEN ASM SET_TAC[]]);; let SMALL_IMP_DIMENSION_LE_0 = prove (`!s:real^N->bool. s <_c (:real) ==> dimension s <= &0`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[DIMENSION_LE_EQ] THEN CONV_TAC INT_REDUCE_CONV THEN SIMP_TAC[DIMENSION_GE; INT_ARITH `a:int <= d ==> (d <= a <=> d = a)`] THEN REWRITE_TAC[DIMENSION_EQ_MINUS1] THEN MAP_EVERY X_GEN_TAC [`v:real^N->bool`; `a:real^N`] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_BALL]) THEN DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `~(interval(vec 0,lift r) SUBSET IMAGE (\x:real^N. lift(dist(a,x))) s)` MP_TAC THENL [DISCH_THEN(MP_TAC o MATCH_MP CARD_LE_SUBSET) THEN W(MP_TAC o PART_MATCH lhand CARD_LE_IMAGE o rand o lhand o snd) THEN PURE_REWRITE_TAC[GSYM IMP_CONJ_ALT] THEN DISCH_THEN(MP_TAC o MATCH_MP CARD_LE_TRANS) THEN REWRITE_TAC[CARD_NOT_LE] THEN TRANS_TAC CARD_LTE_TRANS `(:real)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CARD_EQ_IMP_LE THEN ONCE_REWRITE_TAC[CARD_EQ_SYM] THEN MATCH_MP_TAC(CONJUNCT2 CARD_EQ_INTERVAL) THEN ASM_REWRITE_TAC[INTERVAL_NE_EMPTY_1; DROP_VEC; LIFT_DROP]; REWRITE_TAC[SUBSET; NOT_FORALL_THM; LEFT_IMP_EXISTS_THM; NOT_IMP] THEN REWRITE_TAC[IN_IMAGE; NOT_EXISTS_THM; FORALL_LIFT; IN_INTERVAL_1] THEN X_GEN_TAC `p:real` THEN REWRITE_TAC[LIFT_EQ; LIFT_DROP; DROP_VEC] THEN REWRITE_TAC[TAUT `~(p /\ q) <=> q ==> ~p`] THEN STRIP_TAC THEN EXISTS_TAC `ball(a:real^N,p)` THEN ASM_REWRITE_TAC[OPEN_BALL; CENTRE_IN_BALL; GSYM SUBSET] THEN ASM_SIMP_TAC[FRONTIER_BALL; sphere] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN TRANS_TAC SUBSET_TRANS `ball(a:real^N,r)` THEN ASM_SIMP_TAC[SUBSET_BALL; REAL_LT_IMP_LE]]);; let COUNTABLE_IMP_DIMENSION_LE_0 = prove (`!s:real^N->bool. COUNTABLE s ==> dimension s <= &0`, SIMP_TAC[COUNTABLE_IMP_CARD_LT_REAL; SMALL_IMP_DIMENSION_LE_0]);; let FINITE_IMP_DIMENSION_LE_0 = prove (`!s:real^N->bool. FINITE s ==> dimension s <= &0`, SIMP_TAC[COUNTABLE_IMP_DIMENSION_LE_0; FINITE_IMP_COUNTABLE]);; let DIMENSION_SING = prove (`!a:real^N. dimension {a} = &0`, GEN_TAC THEN MATCH_MP_TAC(INT_ARITH `-- &1:int <= x /\ x <= &0 /\ ~(x = -- &1) ==> x = &0`) THEN REWRITE_TAC[DIMENSION_GE; DIMENSION_EQ_MINUS1; NOT_INSERT_EMPTY] THEN SIMP_TAC[FINITE_IMP_DIMENSION_LE_0; FINITE_SING]);; let CONNECTED_DIMENSION_EQ_SING = prove (`!s:real^N->bool. connected s ==> (dimension s = &0 <=> ?a. s = {a})`, REPEAT GEN_TAC THEN REPEAT(STRIP_TAC ORELSE EQ_TAC) THEN ASM_REWRITE_TAC[DIMENSION_SING] THEN MATCH_MP_TAC(SET_RULE `~(s = {}) /\ (!a b. ~(a = b) /\ a IN s /\ b IN s ==> F) ==> ?a. s = {a}`) THEN ASM_REWRITE_TAC[GSYM DIMENSION_EQ_MINUS1] THEN CONV_TAC INT_REDUCE_CONV THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (INT_ARITH `i:int = &0 ==> i <= &0`)) THEN REWRITE_TAC[] THEN ONCE_REWRITE_TAC[SMALL_INDUCTIVE_DIMENSION] THEN CONV_TAC INT_REDUCE_CONV THEN SIMP_TAC[DIMENSION_GE; INT_ARITH `a:int <= d ==> (d <= a <=> d = a)`] THEN REWRITE_TAC[DIMENSION_EQ_MINUS1] THEN DISCH_THEN(MP_TAC o SPECL [`{b:real^N}`; `a:real^N`]) THEN ASM_REWRITE_TAC[CLOSED_IN_SING; IN_SING; SING_SUBSET] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN REWRITE_TAC[UNWIND_THM2; CLOSED_IN_EMPTY] THEN REWRITE_TAC[NOT_EXISTS_THM; DIFF_EMPTY] THEN MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `v:real^N->bool`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONNECTED_OPEN_IN_EQ]) THEN REWRITE_TAC[] THEN MAP_EVERY EXISTS_TAC [`u:real^N->bool`; `v:real^N->bool`] THEN ASM SET_TAC[]);; let DIMENSION_SUBSET_EXISTS = prove (`!s:real^N->bool n. -- &1 <= n /\ n <= dimension s ==> ?t. closed_in (subtopology euclidean s) t /\ t SUBSET s /\ dimension t = n`, let lemma = prove (`!s:real^N->bool. ~(s = {}) ==> ?t. closed_in (subtopology euclidean s) t /\ dimension t = dimension s - &1`, REPEAT STRIP_TAC THEN MP_TAC(INT_ARITH `dimension(s:real^N->bool) <= dimension(s) /\ ~(dimension s <= dimension s - &1)`) THEN ONCE_REWRITE_TAC[DIMENSION_LE_EQ_ALT] THEN ASM_REWRITE_TAC[DIMENSION_GE; DIMENSION_POS_LE; INT_ARITH `--a:int <= d - a <=> &0 <= d`] THEN DISCH_THEN(MP_TAC o MATCH_MP (MESON[] `(!x y. P x y ==> Q x y) /\ ~(!x y. P x y ==> R x y) ==> (?x y. P x y) ==> ~(!x y. Q x y ==> R x y)`)) THEN ANTS_TAC THENL [ASM_MESON_TAC[OPEN_UNIV; IN_UNIV; MEMBER_NOT_EMPTY]; ALL_TAC] THEN GEN_REWRITE_TAC I [GSYM CONTRAPOS_THM] THEN REWRITE_TAC[NOT_EXISTS_THM; TAUT `~(p /\ q) <=> p ==> ~q`] THEN DISCH_TAC THEN REPEAT GEN_TAC THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN REPEAT(MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[]) THEN MATCH_MP_TAC(INT_ARITH `~(x:int = s - &1) ==> x <= s - &1 ==> x <= s - &1 - &1`) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[CLOSED_IN_FRONTIER_OF]) in REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[INT_ARITH `-- &1:int <= n /\ n <= s <=> &0 <= s - n /\ -- &1 <= s - (s - n)`] THEN ASM_CASES_TAC `!t:real^N->bool. dimension t = n <=> dimension t = dimension(s:real^N->bool) - (dimension s - n)` THENL [POP_ASSUM(fun th -> ONCE_REWRITE_TAC[th]); ASM_INT_ARITH_TAC] THEN SPEC_TAC(`dimension(s:real^N->bool) - n`,`d:int`) THEN REWRITE_TAC[IMP_CONJ; GSYM INT_FORALL_POS] THEN MATCH_MP_TAC num_INDUCTION THEN CONJ_TAC THENL [REWRITE_TAC[INT_SUB_RZERO] THEN MESON_TAC[CLOSED_IN_REFL; SUBSET_REFL]; X_GEN_TAC `d:num`] THEN REWRITE_TAC[GSYM INT_OF_NUM_SUC; INT_ARITH `--a:int <= d - (x + a) <=> x <= d`] THEN DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN FIRST_ASSUM(DISJ_CASES_TAC o MATCH_MP (INT_ARITH `d:int <= s ==> d = s \/ d < s`)) THENL [DISCH_THEN(K ALL_TAC) THEN EXISTS_TAC `{}:real^N->bool` THEN ASM_REWRITE_TAC[CLOSED_IN_EMPTY; EMPTY_SUBSET; DIMENSION_EMPTY] THEN INT_ARITH_TAC; ANTS_TAC THENL [ASM_INT_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPEC `t:real^N->bool` lemma) THEN ASM_REWRITE_TAC[GSYM DIMENSION_EQ_MINUS1] THEN ANTS_TAC THENL [ASM_INT_ARITH_TAC; MATCH_MP_TAC MONO_EXISTS] THEN X_GEN_TAC `u:real^N->bool` THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THENL [ASM_MESON_TAC[CLOSED_IN_TRANS]; ASM_MESON_TAC[CLOSED_IN_IMP_SUBSET; SUBSET_TRANS]; INT_ARITH_TAC]]);; let DIMENSION_UNION_LE_BASIC = prove (`!s t:real^N->bool. dimension(s UNION t) <= dimension s + dimension t + &1`, SUBGOAL_THEN `!n s t. dimension s + dimension t + &2 <= &n ==> dimension(s UNION t:real^N->bool) <= &n - &1` ASSUME_TAC THENL [ALL_TAC; MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `t:real^N->bool`] THEN SUBGOAL_THEN `?n. dimension(s:real^N->bool) + dimension(t:real^N->bool) + &2 = &n` CHOOSE_TAC THENL [REWRITE_TAC[INT_OF_NUM_EXISTS] THEN MATCH_MP_TAC(INT_ARITH `-- &1:int <= x /\ -- &1 <= y ==> &0 <= x + y + &2`) THEN REWRITE_TAC[DIMENSION_GE]; ASM_SIMP_TAC[INT_LE_REFL; INT_ARITH `x:int <= a + b + &1 <=> x <= (a + b + &2) - &1`]]] THEN INDUCT_TAC THEN SIMP_TAC[DIMENSION_GE; DIMENSION_EQ_MINUS1; UNION_EMPTY; DIMENSION_EMPTY; INT_SUB_LZERO; INT_LE_REFL; GSYM INT_OF_NUM_SUC; INT_ARITH `-- &1:int <= x /\ -- &1 <= y ==> (x + y + &2 <= &0 <=> x = -- &1 /\ y = -- &1)`] THEN MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `t:real^N->bool`] THEN REWRITE_TAC[INT_ARITH `(n + &1) - &1:int = n`] THEN REWRITE_TAC[INT_ARITH `s + t + &2:int <= n + &1 <=> s + t <= n - &1`] THEN DISCH_TAC THEN ONCE_REWRITE_TAC[DIMENSION_LE_EQ] THEN CONJ_TAC THENL [INT_ARITH_TAC; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`v:real^N->bool`; `a:real^N`] THEN REWRITE_TAC[SET_RULE `p /\ q /\ x IN s UNION t ==> r <=> (x IN s \/ x IN t ==> p /\ q ==> r)`] THEN UNDISCH_TAC `dimension(s:real^N->bool) + dimension(t:real^N->bool) <= &n - &1` THEN MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`t:real^N->bool`; `s:real^N->bool`] THEN ONCE_REWRITE_TAC[TAUT `p ==> q ==> r <=> q ==> p ==> r`] THEN MATCH_MP_TAC(MESON[] `(!s t. R s t ==> R t s) /\ (!s t. P s ==> R s t) ==> (!s t. P s \/ P t ==> R s t)`) THEN CONJ_TAC THENL [REWRITE_TAC[INT_ADD_SYM; UNION_COMM]; ALL_TAC] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPEC `dimension(s:real^N->bool)` INT_LE_REFL) THEN GEN_REWRITE_TAC LAND_CONV [DIMENSION_LE_EQ] THEN DISCH_THEN(MP_TAC o SPECL [`v:real^N->bool`; `a:real^N`] o CONJUNCT2) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[ONCE_REWRITE_RULE[INTER_COMM] UNION_OVER_INTER] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN MP_TAC(ISPECL [`t INTER frontier u:real^N->bool`; `t:real^N->bool`] DIMENSION_SUBSET) THEN REWRITE_TAC[INTER_SUBSET] THEN ASM_INT_ARITH_TAC);; let DIMENSION_ZERO_REDUCTION_THEOREM = prove (`!s:real^N->bool v. dimension s <= &0 /\ (!n:num. open_in (subtopology euclidean s) (v n)) ==> ?u. (!n. open_in (subtopology euclidean s) (u n)) /\ (!n. u n SUBSET v n) /\ pairwise (\m n. DISJOINT (u m) (u n)) (:num) /\ UNIONS {u n | n IN (:num)} = UNIONS {v n | n IN (:num)}`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `\u:real^N->bool. closed_in (subtopology euclidean s) u /\ open_in (subtopology euclidean s) u` GENERAL_REDUCTION_THEOREM) THEN REWRITE_TAC[OPEN_IN_EMPTY; CLOSED_IN_EMPTY] THEN ANTS_TAC THENL [SIMP_TAC[OPEN_IN_UNION; CLOSED_IN_UNION; CLOSED_IN_DIFF; OPEN_IN_DIFF]; DISCH_THEN(MP_TAC o SPEC `v:num->real^N->bool`)] THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [X_GEN_TAC `n:num` THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [DIMENSION_EQ_LOCALLY_CLOPEN]) THEN DISCH_THEN(MP_TAC o SPEC `(v:num->real^N->bool) n` o MATCH_MP (REWRITE_RULE[IMP_CONJ] LOCALLY_OPEN_SUBSET)) THEN ASM_REWRITE_TAC[LOCALLY_IMP_COUNTABLE_UNION_OF]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:num->real^N->bool` THEN MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[UNION_OF] THEN MESON_TAC[OPEN_IN_UNIONS]]);; let DIMENSION_ZERO_REDUCTION_THEOREM_2 = prove (`!u s t:real^N->bool. dimension u <= &0 /\ open_in (subtopology euclidean u) s /\ open_in (subtopology euclidean u) t ==> ?s' t'. open_in (subtopology euclidean u) s' /\ open_in (subtopology euclidean u) t' /\ s' SUBSET s /\ t' SUBSET t /\ DISJOINT s' t' /\ s' UNION t' = s UNION t`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN ASSUME_TAC) THEN MP_TAC(ISPEC `\v:real^N->bool. closed_in (subtopology euclidean u) v /\ open_in (subtopology euclidean u) v` GENERAL_REDUCTION_THEOREM_2) THEN REWRITE_TAC[OPEN_IN_EMPTY; CLOSED_IN_EMPTY] THEN ANTS_TAC THENL [SIMP_TAC[OPEN_IN_UNION; CLOSED_IN_UNION; CLOSED_IN_DIFF; OPEN_IN_DIFF]; DISCH_THEN(MP_TAC o SPECL [`s:real^N->bool`; `t:real^N->bool`])] THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC LOCALLY_IMP_COUNTABLE_UNION_OF THEN MATCH_MP_TAC LOCALLY_OPEN_SUBSET THEN EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[GSYM DIMENSION_EQ_LOCALLY_CLOPEN]; REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN REPEAT(MATCH_MP_TAC MONO_AND THEN CONJ_TAC) THEN REWRITE_TAC[UNION_OF] THEN MESON_TAC[OPEN_IN_UNIONS]]);; let DIMENSION_ZERO_SEPARATION_THEOREM = prove (`!u s t:real^N->bool. dimension u <= &0 /\ closed_in (subtopology euclidean u) s /\ closed_in (subtopology euclidean u) t /\ DISJOINT s t ==> ?s' t'. closed_in (subtopology euclidean u) s' /\ open_in (subtopology euclidean u) s' /\ closed_in (subtopology euclidean u) t' /\ open_in (subtopology euclidean u) t' /\ s SUBSET s' /\ t SUBSET t' /\ DISJOINT s' t' /\ s' UNION t' = u`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`u:real^N->bool`; `u DIFF s:real^N->bool`; `u DIFF t:real^N->bool`] DIMENSION_ZERO_REDUCTION_THEOREM_2) THEN ASM_SIMP_TAC[OPEN_IN_DIFF; OPEN_IN_REFL; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`s':real^N->bool`; `t':real^N->bool`] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`u DIFF s':real^N->bool`; `u DIFF t':real^N->bool`] THEN ASM_SIMP_TAC[CLOSED_IN_DIFF; CLOSED_IN_REFL] THEN SUBGOAL_THEN `u DIFF s':real^N->bool = t' /\ u DIFF t' = s'` (CONJUNCTS_THEN SUBST1_TAC) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET)) THEN ASM SET_TAC[]);; let DIMENSION_LE_CLOSED_IN_UNIONS,DIMENSION_DECOMPOSITION = let lemma = prove (`!s:real^N->bool n. dimension s <= n /\ &0 <= n ==> ?t. (COUNTABLE UNION_OF (\c. closed_in (subtopology euclidean s) c /\ dimension c <= n - &1)) t /\ dimension(s DIFF t) <= &0`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [DIMENSION_LE_EQ_LOCALLY]) THEN DISCH_THEN(MP_TAC o REWRITE_RULE[LOCALLY_OPEN_BASIS] o CONJUNCT2) THEN DISCH_THEN(X_CHOOSE_THEN `b:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN ABBREV_TAC `t = UNIONS(IMAGE (\c:real^N->bool. subtopology euclidean s frontier_of c) b)` THEN EXISTS_TAC `t:real^N->bool` THEN CONJ_TAC THENL [EXPAND_TAC "t" THEN MATCH_MP_TAC COUNTABLE_UNION_OF_UNIONS THEN ASM_SIMP_TAC[COUNTABLE_IMAGE; FORALL_IN_IMAGE] THEN X_GEN_TAC `c:real^N->bool` THEN DISCH_TAC THEN MATCH_MP_TAC COUNTABLE_UNION_OF_INC THEN ASM_SIMP_TAC[CLOSED_IN_FRONTIER_OF]; GEN_REWRITE_TAC I [DIMENSION_LE_EQ_LOCAL] THEN CONV_TAC INT_REDUCE_CONV THEN REWRITE_TAC[DIMENSION_LE_MINUS1] THEN MAP_EVERY X_GEN_TAC [`v:real^N->bool`; `a:real^N`] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN DISCH_THEN(X_CHOOSE_THEN `w:real^N->bool` (CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC)) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTER]) THEN REWRITE_TAC[IN_DIFF] THEN STRIP_TAC THEN SUBGOAL_THEN `?c. c SUBSET b /\ s INTER w:real^N->bool = UNIONS c` MP_TAC THENL [ASM_SIMP_TAC[OPEN_IN_OPEN_INTER]; ALL_TAC] THEN DISCH_THEN(MP_TAC o SPEC `a:real^N` o MATCH_MP (SET_RULE `(?c. c SUBSET b /\ x = UNIONS c) ==> !a. a IN x ==> ?d. d IN b /\ d SUBSET x /\ a IN d`)) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; IN_INTER; SUBSET_INTER] THEN X_GEN_TAC `d:real^N->bool` THEN STRIP_TAC THEN ANTE_RES_THEN MP_TAC (ASSUME `(d:real^N->bool) IN b`) THEN REPEAT(FIRST_X_ASSUM(K ALL_TAC o check (is_forall o concl))) THEN STRIP_TAC THEN EXISTS_TAC `d DIFF t:real^N->bool` THEN REPLICATE_TAC 2 (CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN CONJ_TAC THENL [UNDISCH_TAC `open_in (subtopology euclidean s) (d:real^N->bool)` THEN REWRITE_TAC[OPEN_IN_OPEN] THEN MATCH_MP_TAC MONO_EXISTS THEN SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[FRONTIER_OF_CLOSURES; CLOSURE_OF_SUBTOPOLOGY; EUCLIDEAN_CLOSURE_OF; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN ASM_SIMP_TAC[SET_RULE `d SUBSET s ==> (s DIFF t) INTER (d DIFF t) = d DIFF t`] THEN ASM_SIMP_TAC[SET_RULE `d SUBSET s ==> (s DIFF t) INTER (s DIFF t DIFF (d DIFF t)) = s DIFF (t UNION d)`] THEN REWRITE_TAC[EXTENSION; IN_INTER; NOT_IN_EMPTY; IN_DIFF] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN MP_TAC(ASSUME `~((x:real^N) IN t)`) THEN EXPAND_TAC "t" THEN REWRITE_TAC[UNIONS_IMAGE; IN_ELIM_THM] THEN EXISTS_TAC `d:real^N->bool` THEN ASM_REWRITE_TAC[CLOSURE_OF_SUBTOPOLOGY; EUCLIDEAN_CLOSURE_OF; FRONTIER_OF_CLOSURES; IN_INTER; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN ASM_SIMP_TAC[SET_RULE `d SUBSET s ==> s INTER d = d /\ s INTER (s DIFF d) = s DIFF d`] THEN CONJ_TAC THENL [UNDISCH_TAC `(x:real^N) IN closure (d DIFF t)`; UNDISCH_TAC `(x:real^N) IN closure (s DIFF (t UNION d))`] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> x IN s ==> x IN t`) THEN MATCH_MP_TAC SUBSET_CLOSURE THEN SET_TAC[]]) and case0 = prove (`!u:real^N->bool c. COUNTABLE c /\ (!s. s IN c ==> closed_in (subtopology euclidean u) s /\ dimension s <= &0) ==> dimension(UNIONS c) <= &0`, MAP_EVERY X_GEN_TAC [`uu:real^N->bool`; `cc:(real^N->bool)->bool`] THEN ASM_CASES_TAC `cc:(real^N->bool)->bool = {}` THEN ASM_REWRITE_TAC[UNIONS_0; DIMENSION_EMPTY] THEN CONV_TAC INT_REDUCE_CONV THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN MP_TAC(ISPEC `cc:(real^N->bool)->bool` COUNTABLE_AS_IMAGE) THEN ASM_REWRITE_TAC[] THEN POP_ASSUM_LIST(K ALL_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `c:num->real^N->bool` SUBST1_TAC) THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_UNIV; FORALL_AND_THM] THEN STRIP_TAC THEN REWRITE_TAC[GSYM SIMPLE_IMAGE] THEN ABBREV_TAC `u:real^N->bool = UNIONS {c n | n IN (:num)}` THEN SUBGOAL_THEN `!n. closed_in (subtopology euclidean u) ((c:num->real^N->bool) n)` ASSUME_TAC THENL [GEN_TAC THEN MATCH_MP_TAC CLOSED_IN_SUBSET_TRANS THEN EXISTS_TAC `uu:real^N->bool` THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(MP_TAC o GEN `n:num` o MATCH_MP CLOSED_IN_IMP_SUBSET o SPEC `n:num`) THEN ASM SET_TAC[]; UNDISCH_THEN `!n. closed_in (subtopology euclidean uu) ((c:num->real^N->bool) n)` (K ALL_TAC)] THEN GEN_REWRITE_TAC I [SMALL_INDUCTIVE_DIMENSION] THEN CONV_TAC INT_REDUCE_CONV THEN MAP_EVERY X_GEN_TAC [`l:real^N->bool`; `aa:real^N`] THEN REWRITE_TAC[SET_RULE `~(a IN l) <=> DISJOINT {a} l`] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [GSYM CLOSED_IN_SING] THEN REWRITE_TAC[GSYM SING_SUBSET] THEN SPEC_TAC(`{aa:real^N}`,`k:real^N->bool`) THEN REPEAT STRIP_TAC THEN REWRITE_TAC[DIMENSION_LE_MINUS1] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN REWRITE_TAC[UNWIND_THM2; CLOSED_IN_EMPTY; DIFF_EMPTY] THEN SUBGOAL_THEN `?g h:num->real^N->bool. (!n. open_in (subtopology euclidean u) (g n) /\ open_in (subtopology euclidean u) (h n) /\ DISJOINT (g n) (h n) /\ DISJOINT ((subtopology euclidean u) closure_of g n) ((subtopology euclidean u) closure_of h n) /\ k SUBSET g n /\ l SUBSET h n /\ c n SUBSET g n UNION h n) /\ (!n. (subtopology euclidean u) closure_of (g n) SUBSET g(SUC n) /\ (subtopology euclidean u) closure_of (h n) SUBSET h(SUC n))` STRIP_ASSUME_TAC THENL [ALL_TAC; MAP_EVERY EXISTS_TAC [`UNIONS {g n | n IN (:num)}:real^N->bool`; `UNIONS {h n | n IN (:num)}:real^N->bool`] THEN ASM_SIMP_TAC[OPEN_IN_UNIONS; FORALL_IN_GSPEC] THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN CONJ_TAC THENL [REWRITE_TAC[SET_RULE `DISJOINT (UNIONS a) (UNIONS b) <=> !s. s IN a ==> !t. t IN b ==> DISJOINT s t`] THEN REWRITE_TAC[FORALL_IN_GSPEC; IN_UNIV] THEN MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN DISJ_CASES_TAC(ARITH_RULE `i:num <= j \/ j <= i`) THENL [MATCH_MP_TAC(SET_RULE `!s'. DISJOINT s' t /\ s SUBSET s' ==> DISJOINT s t`) THEN EXISTS_TAC `(g:num->real^N->bool) j` THEN ASM_REWRITE_TAC[] THEN UNDISCH_TAC `i:num <= j` THEN MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`j:num`; `i:num`]; MATCH_MP_TAC(SET_RULE `!t'. DISJOINT s t' /\ t SUBSET t' ==> DISJOINT s t`) THEN EXISTS_TAC `(h:num->real^N->bool) i` THEN ASM_REWRITE_TAC[] THEN UNDISCH_TAC `j:num <= i` THEN MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`i:num`; `j:num`]] THEN MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN REWRITE_TAC[SUBSET_TRANS; SUBSET_REFL] THEN ASM_MESON_TAC[CLOSURE_OF_SUBSET; SUBSET_TRANS; OPEN_IN_SUBSET]; REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; UNION_SUBSET; UNIONS_SUBSET] THEN REWRITE_TAC[FORALL_IN_GSPEC; IN_UNIV] THEN CONJ_TAC THENL [ASM_MESON_TAC[OPEN_IN_IMP_SUBSET]; ASM SET_TAC[]]]] THEN MATCH_MP_TAC(MESON[] `(?f. R (FST o f) (SND o f)) ==> ?g h. R g h`) THEN REWRITE_TAC[o_THM] THEN MATCH_MP_TAC DEPENDENT_CHOICE THEN REWRITE_TAC[NOT_SUC; EXISTS_PAIR_THM; FORALL_PAIR_THM] THEN CONJ_TAC THENL [MP_TAC(ISPECL [`c 0:real^N->bool`; `c 0 INTER k:real^N->bool`; `c 0 INTER l:real^N->bool`] DIMENSION_ZERO_SEPARATION_THEOREM) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN CONJ_TAC THEN MATCH_MP_TAC CLOSED_IN_SUBSET_TRANS THEN EXISTS_TAC `u:real^N->bool` THEN ASM_SIMP_TAC[CLOSED_IN_INTER] THEN ASM_MESON_TAC[CLOSED_IN_IMP_SUBSET; INTER_SUBSET]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`a:real^N->bool`; `b:real^N->bool`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`k UNION a:real^N->bool`; `l UNION b:real^N->bool`; `u:real^N->bool`] SEPARATION_NORMAL_LOCAL_CLOSURES) THEN ANTS_TAC THENL [ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN CONJ_TAC THEN MATCH_MP_TAC CLOSED_IN_UNION THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CLOSED_IN_TRANS THEN EXISTS_TAC `c 0:real^N->bool` THEN ASM_REWRITE_TAC[]; REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN ASM SET_TAC[]]; MAP_EVERY X_GEN_TAC [`n:num`; `g:real^N->bool`; `h:real^N->bool`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`c (SUC n):real^N->bool`; `c(SUC n) INTER (subtopology euclidean u closure_of g):real^N->bool`; `c(SUC n) INTER (subtopology euclidean u closure_of h):real^N->bool`] DIMENSION_ZERO_SEPARATION_THEOREM) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN CONJ_TAC THEN MATCH_MP_TAC CLOSED_IN_SUBSET_TRANS THEN EXISTS_TAC `u:real^N->bool` THEN ASM_SIMP_TAC[CLOSED_IN_INTER; CLOSED_IN_CLOSURE_OF] THEN ASM_MESON_TAC[CLOSED_IN_IMP_SUBSET; INTER_SUBSET]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`a:real^N->bool`; `b:real^N->bool`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`(subtopology euclidean u closure_of g) UNION a:real^N->bool`; `(subtopology euclidean u closure_of h) UNION b:real^N->bool`; `u:real^N->bool`] SEPARATION_NORMAL_LOCAL_CLOSURES) THEN ANTS_TAC THENL [ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN CONJ_TAC THEN MATCH_MP_TAC CLOSED_IN_UNION THEN REWRITE_TAC[CLOSED_IN_CLOSURE_OF] THEN MATCH_MP_TAC CLOSED_IN_TRANS THEN EXISTS_TAC `c(SUC n):real^N->bool` THEN ASM_REWRITE_TAC[]; REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] SUBSET_TRANS)) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS)) THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s SUBSET t UNION u`) THEN MATCH_MP_TAC CLOSURE_OF_SUBSET THEN ASM_MESON_TAC[OPEN_IN_SUBSET]]]) in let DIMENSION_LE_CLOSED_IN_UNIONS = prove (`!u:real^N->bool c n. -- &1 <= n /\ COUNTABLE c /\ (!s. s IN c ==> closed_in (subtopology euclidean u) s /\ dimension s <= n) ==> dimension(UNIONS c) <= n`, REPEAT GEN_TAC THEN ASM_CASES_TAC `n:int = -- &1` THENL [ASM_REWRITE_TAC[DIMENSION_LE_MINUS1] THEN SET_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `&0:int <= n` THENL [ALL_TAC; ASM_INT_ARITH_TAC] THEN DISCH_THEN(MP_TAC o CONJUNCT2) THEN MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`c:(real^N->bool)->bool`; `u:real^N->bool`] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM INT_OF_NUM_EXISTS]) THEN POP_ASSUM_LIST(K ALL_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `n:num` SUBST1_TAC) THEN SPEC_TAC(`n:num`,`d:num`) THEN INDUCT_TAC THEN REWRITE_TAC[case0] THEN MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `c:(real^N->bool)->bool`] THEN STRIP_TAC THEN SUBGOAL_THEN `!u. u IN c ==> ?t r. DISJOINT t r /\ t UNION r = u /\ dimension(t) <= &d /\ dimension(r:real^N->bool) <= &0 /\ (COUNTABLE UNION_OF closed_in (subtopology euclidean s)) t` MP_TAC THENL [X_GEN_TAC `u:real^N->bool` THEN DISCH_TAC THEN MP_TAC(ISPECL [`u:real^N->bool`; `&(SUC d):int`] lemma) THEN ASM_SIMP_TAC[INT_POS] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN EXISTS_TAC `u DIFF t:real^N->bool` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RATOR_CONV [UNION_OF]) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `b:(real^N->bool)->bool` THEN REWRITE_TAC[GSYM INT_OF_NUM_SUC; INT_ARITH `(x + y) - y:int = x`] THEN REPEAT STRIP_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN MATCH_MP_TAC(SET_RULE `t SUBSET u ==> t UNION (u DIFF t) = u`) THEN EXPAND_TAC "t" THEN REWRITE_TAC[UNIONS_SUBSET] THEN ASM_MESON_TAC[CLOSED_IN_IMP_SUBSET]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] UNION_OF_MONO)) THEN REWRITE_TAC[] THEN ASM_MESON_TAC[CLOSED_IN_TRANS]]; REWRITE_TAC[RIGHT_IMP_EXISTS_THM; SKOLEM_THM; LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`t:(real^N->bool)->(real^N->bool)`; `r:(real^N->bool)->(real^N->bool)`] THEN DISCH_TAC THEN SUBGOAL_THEN `UNIONS c = UNIONS (IMAGE (t:(real^N->bool)->(real^N->bool)) c) UNION (UNIONS (IMAGE r c) DIFF UNIONS (IMAGE t c))` SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[GSYM INT_OF_NUM_SUC] THEN W(MP_TAC o PART_MATCH lhand DIMENSION_UNION_LE_BASIC o lhand o snd) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] INT_LE_TRANS) THEN MATCH_MP_TAC(INT_ARITH `x:int <= d /\ y <= &0 ==> x + y + &1 <= d + &1`) THEN CONJ_TAC THENL [SUBGOAL_THEN `(COUNTABLE UNION_OF (\c. closed_in (subtopology euclidean s) c /\ dimension c <= &d)) (UNIONS (IMAGE (t:(real^N->bool)->(real^N->bool)) c))` MP_TAC THENL [MATCH_MP_TAC COUNTABLE_UNION_OF_UNIONS THEN ASM_SIMP_TAC[COUNTABLE_IMAGE; FORALL_IN_IMAGE] THEN X_GEN_TAC `d:real^N->bool` THEN DISCH_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `d:real^N->bool`)) THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RATOR_CONV [UNION_OF]) THEN REWRITE_TAC[UNION_OF] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `v:(real^N->bool)->bool` THEN STRIP_TAC THEN ASM_SIMP_TAC[] THEN X_GEN_TAC `b:real^N->bool` THEN DISCH_TAC THEN TRANS_TAC INT_LE_TRANS `dimension((t:(real^N->bool)->(real^N->bool)) d)` THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC DIMENSION_SUBSET THEN ASM SET_TAC[]; REWRITE_TAC[UNION_OF] THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[]]; REWRITE_TAC[UNIONS_DIFF] THEN MATCH_MP_TAC case0 THEN EXISTS_TAC `UNIONS (IMAGE (r:(real^N->bool)->(real^N->bool)) c) DIFF UNIONS (IMAGE t c)` THEN ASM_SIMP_TAC[COUNTABLE_IMAGE; SIMPLE_IMAGE; FORALL_IN_IMAGE] THEN X_GEN_TAC `d:real^N->bool` THEN DISCH_TAC THEN CONJ_TAC THENL [SUBGOAL_THEN `r d DIFF UNIONS (IMAGE (t:(real^N->bool)->(real^N->bool)) c) = (UNIONS (IMAGE r c) DIFF UNIONS (IMAGE t c)) INTER d` SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC CLOSED_IN_SUBTOPOLOGY_INTER_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_SIMP_TAC[CLOSED_IN_INTER; CLOSED_IN_REFL] THEN MATCH_MP_TAC(SET_RULE `(!x. x IN c ==> x SUBSET s) ==> UNIONS c DIFF t SUBSET s`) THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN X_GEN_TAC `b:real^N->bool` THEN STRIP_TAC THEN TRANS_TAC SUBSET_TRANS `b:real^N->bool` THEN CONJ_TAC THENL [ASM SET_TAC[]; ASM_MESON_TAC[CLOSED_IN_IMP_SUBSET]]; TRANS_TAC INT_LE_TRANS `dimension((r:(real^N->bool)->(real^N->bool)) d)` THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC DIMENSION_SUBSET THEN SET_TAC[]]]) in let DIMENSION_DECOMPOSITION = prove (`!s:real^N->bool n. &0 <= n ==> (dimension s <= n <=> ?t u. t UNION u = s /\ DISJOINT t u /\ (fsigma relative_to s) t /\ dimension t <= n - &1 /\ dimension u <= &0)`, REPEAT STRIP_TAC THEN EQ_TAC THEN STRIP_TAC THENL [MP_TAC(ISPECL [`s:real^N->bool`; `n:int`] lemma) THEN ASM_REWRITE_TAC[COUNTABLE_UNION_OF_RELATIVE_TO; REWRITE_RULE[GSYM FUN_EQ_THM; ETA_AX] fsigma] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN EXISTS_TAC `s DIFF t:real^N->bool` THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `t SUBSET s ==> t UNION (s DIFF t) = s`) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RATOR_CONV [UNION_OF]) THEN REWRITE_TAC[] THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN ASM_REWRITE_TAC[UNIONS_SUBSET] THEN ASM_MESON_TAC[CLOSED_IN_IMP_SUBSET]; SET_TAC[]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] UNION_OF_MONO)) THEN SIMP_TAC[CLOSED_RELATIVE_TO]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RATOR_CONV [UNION_OF]) THEN REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `u:(real^N->bool)->bool` (STRIP_ASSUME_TAC o GSYM)) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC DIMENSION_LE_CLOSED_IN_UNIONS THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN ASM_INT_ARITH_TAC]; EXPAND_TAC "s" THEN W(MP_TAC o PART_MATCH lhand DIMENSION_UNION_LE_BASIC o lhand o snd) THEN ASM_INT_ARITH_TAC]) in DIMENSION_LE_CLOSED_IN_UNIONS,DIMENSION_DECOMPOSITION;; let DIMENSION_LE_UNIONS_RELATIVE = prove (`!u:real^N->bool c n. -- &1 <= n /\ COUNTABLE c /\ (!s. s IN c ==> (fsigma relative_to u) s /\ dimension s <= n) ==> dimension(UNIONS c) <= n`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `(COUNTABLE UNION_OF (\t. closed_in (subtopology euclidean u) t /\ dimension t <= n)) (UNIONS c:real^N->bool)` MP_TAC THENL [MATCH_MP_TAC COUNTABLE_UNION_OF_UNIONS THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `t:real^N->bool` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `t:real^N->bool`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN REWRITE_TAC[COUNTABLE_UNION_OF_RELATIVE_TO; REWRITE_RULE[GSYM FUN_EQ_THM; ETA_AX] fsigma] THEN REWRITE_TAC[UNION_OF] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:(real^N->bool)->bool` THEN SIMP_TAC[CLOSED_RELATIVE_TO] THEN STRIP_TAC THEN X_GEN_TAC `d:real^N->bool` THEN DISCH_TAC THEN TRANS_TAC INT_LE_TRANS `dimension(UNIONS u:real^N->bool)` THEN CONJ_TAC THENL [MATCH_MP_TAC DIMENSION_SUBSET; ASM_REWRITE_TAC[]] THEN ASM SET_TAC[]; REWRITE_TAC[UNION_OF] THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC DIMENSION_LE_CLOSED_IN_UNIONS THEN EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[]]);; let DIMENSION_LE_UNIONS = prove (`!c:(real^N->bool)->bool n. -- &1 <= n /\ COUNTABLE c /\ (!s. s IN c ==> fsigma s /\ dimension s <= n) ==> dimension(UNIONS c) <= n`, REPEAT STRIP_TAC THEN MATCH_MP_TAC DIMENSION_LE_UNIONS_RELATIVE THEN EXISTS_TAC `(:real^N)` THEN ASM_REWRITE_TAC[RELATIVE_TO_UNIV]);; let DIMENSION_LE_UNION_RELATIVE = prove (`!u s t:real^N->bool n. (fsigma relative_to u) s /\ (fsigma relative_to u) t /\ dimension s <= n /\ dimension t <= n ==> dimension(s UNION t) <= n`, REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM UNIONS_2] THEN MATCH_MP_TAC DIMENSION_LE_UNIONS_RELATIVE THEN EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[COUNTABLE_INSERT; COUNTABLE_EMPTY] THEN ASM_REWRITE_TAC[FORALL_IN_INSERT; NOT_IN_EMPTY] THEN ASM_MESON_TAC[DIMENSION_LE_IMP_GE]);; let DIMENSION_LE_UNION = prove (`!s t:real^N->bool n. fsigma s /\ fsigma t /\ dimension s <= n /\ dimension t <= n ==> dimension(s UNION t) <= n`, REPEAT STRIP_TAC THEN MATCH_MP_TAC DIMENSION_LE_UNION_RELATIVE THEN EXISTS_TAC `(:real^N)` THEN ASM_REWRITE_TAC[RELATIVE_TO_UNIV]);; let DIMENSION_LE_UNION_RELATIVE_GEN = prove (`!u s t:real^N->bool n. ((fsigma relative_to u) s /\ (gdelta relative_to u) s /\ t SUBSET u \/ (fsigma relative_to u) t /\ (gdelta relative_to u) t /\ s SUBSET u) /\ dimension s <= n /\ dimension t <= n ==> dimension(s UNION t) <= n`, ONCE_REWRITE_TAC[MESON[] `(!u s t n. P u s t n) <=> (!u n s t. P u s t n)`] THEN GEN_TAC THEN GEN_TAC THEN MATCH_MP_TAC(MESON[] `(!x y. Q x y ==> Q y x) /\ (!x y. R x y ==> R y x) /\ (!x y. P x y /\ Q x y ==> R x y) ==> !x y. (P x y \/ P y x) /\ Q x y ==> R x y`) THEN SIMP_TAC[] THEN CONJ_TAC THENL [REWRITE_TAC[UNION_COMM]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `t:real^N->bool`] THEN REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[SET_RULE `s UNION t = s UNION (t DIFF s)`] THEN MATCH_MP_TAC DIMENSION_LE_UNION_RELATIVE THEN EXISTS_TAC `s UNION t:real^N->bool` THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC RELATIVE_TO_SUBSET_TRANS THEN EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP RELATIVE_TO_IMP_SUBSET)) THEN ASM SET_TAC[]; ONCE_REWRITE_TAC[SET_RULE `t DIFF s = (s UNION t) DIFF s`] THEN SIMP_TAC[RELATIVE_TO_COMPL; SUBSET_UNION; FSIGMA_COMPLEMENT; ETA_AX] THEN MATCH_MP_TAC RELATIVE_TO_SUBSET_TRANS THEN EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP RELATIVE_TO_IMP_SUBSET)) THEN ASM SET_TAC[]; TRANS_TAC INT_LE_TRANS `dimension(t:real^N->bool)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC DIMENSION_SUBSET THEN SET_TAC[]]);; let DIMENSION_LE_UNION_GEN = prove (`!s t:real^N->bool. (fsigma s /\ gdelta s \/ fsigma t /\ gdelta t) /\ dimension s <= n /\ dimension t <= n ==> dimension(s UNION t) <= n`, REPEAT GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC DIMENSION_LE_UNION_RELATIVE_GEN THEN EXISTS_TAC `(:real^N)` THEN ASM_REWRITE_TAC[RELATIVE_TO_UNIV; SUBSET_UNIV]);; let DIMENSION_LE_UNION_CLOSED_IN = prove (`!u s t:real^N->bool n. (closed_in (subtopology euclidean u) s /\ t SUBSET u \/ closed_in (subtopology euclidean u) t /\ s SUBSET u) /\ dimension s <= n /\ dimension t <= n ==> dimension(s UNION t) <= n`, REWRITE_TAC[GSYM CLOSED_RELATIVE_TO] THEN REPEAT GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC DIMENSION_LE_UNION_RELATIVE_GEN THEN EXISTS_TAC `u:real^N->bool` THEN ASM_MESON_TAC[RELATIVE_TO_MONO; CLOSED_IMP_GDELTA; CLOSED_IMP_FSIGMA]);; let DIMENSION_LE_UNIONS_ZERODIMENSIONAL = prove (`!f:(real^N->bool)->bool. FINITE f /\ (!s. s IN f ==> dimension s <= &0) ==> dimension(UNIONS f) <= &(CARD f) - &1`, REWRITE_TAC[IMP_CONJ] THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN REWRITE_TAC[UNIONS_0; CARD_CLAUSES; DIMENSION_EMPTY] THEN CONV_TAC INT_REDUCE_CONV THEN SIMP_TAC[FORALL_IN_INSERT; UNIONS_INSERT; CARD_CLAUSES] THEN MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `f:(real^N->bool)->bool`] THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[GSYM INT_OF_NUM_SUC] THEN STRIP_TAC THEN TRANS_TAC INT_LE_TRANS `dimension(s:real^N->bool) + dimension(UNIONS f:real^N->bool) + &1` THEN CONJ_TAC THENL [ALL_TAC; ASM_INT_ARITH_TAC] THEN REWRITE_TAC[DIMENSION_UNION_LE_BASIC]);; let DIMENSION_LE_UNIONS_ZERODIMENSIONAL_EQ = prove (`!s:real^N->bool n. dimension s <= n <=> ?f. FINITE f /\ &(CARD f) <= n + &1 /\ (!d. d IN f ==> dimension d <= &0) /\ UNIONS f = s`, REWRITE_TAC[TAUT `(p <=> q) <=> (q ==> p) /\ (p ==> q)`; FORALL_AND_THM] THEN CONJ_TAC THENL [MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `n:int`] THEN DISCH_THEN(X_CHOOSE_THEN `f:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN TRANS_TAC INT_LE_TRANS `&(CARD(f:(real^N->bool)->bool)) - &1:int` THEN CONJ_TAC THENL [ASM_MESON_TAC[DIMENSION_LE_UNIONS_ZERODIMENSIONAL]; ASM_INT_ARITH_TAC]; ONCE_REWRITE_TAC[MESON[DIMENSION_LE_IMP_GE] `dimension(s:real^N->bool) <= n <=> -- &1 <= n /\ dimension s <= n`] THEN GEN_REWRITE_TAC I [SWAP_FORALL_THM] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN ONCE_REWRITE_TAC[INT_ARITH `i:int <= n <=> i + &1 <= n + &1`] THEN X_GEN_TAC `m:int` THEN SPEC_TAC(`m + &1:int`,`n:int`) THEN REWRITE_TAC[INT_ADD_LINV; GSYM INT_FORALL_POS] THEN REWRITE_TAC[INT_LE_RADD; INT_ARITH `d + &1:int <= &1 <=> d <= &0`] THEN CONV_TAC INT_REDUCE_CONV THEN REWRITE_TAC[GSYM INT_LE_SUB_LADD] THEN MATCH_MP_TAC num_INDUCTION THEN CONJ_TAC THENL [CONV_TAC INT_REDUCE_CONV THEN REWRITE_TAC[DIMENSION_LE_MINUS1; FORALL_UNWIND_THM2] THEN EXISTS_TAC `{}:(real^N->bool)->bool` THEN REWRITE_TAC[FINITE_RULES; UNIONS_0; NOT_IN_EMPTY; CARD_CLAUSES] THEN REWRITE_TAC[INT_LE_REFL]; X_GEN_TAC `n:num` THEN DISCH_TAC THEN X_GEN_TAC `s:real^N->bool` THEN REWRITE_TAC[GSYM INT_OF_NUM_SUC] THEN REWRITE_TAC[INT_ARITH `(n + &1) - &1:int = n`] THEN W(MP_TAC o PART_MATCH (lhand o rand) DIMENSION_DECOMPOSITION o lhand o snd) THEN REWRITE_TAC[INT_POS] THEN DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`t:real^N->bool`; `u:real^N->bool`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `t:real^N->bool`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `f:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(u:real^N->bool) INSERT f` THEN ASM_REWRITE_TAC[FORALL_IN_INSERT; FINITE_INSERT] THEN ASM_REWRITE_TAC[UNIONS_INSERT] THEN ONCE_REWRITE_TAC[UNION_COMM] THEN ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[CARD_CLAUSES] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[GSYM INT_OF_NUM_SUC] THEN ASM_INT_ARITH_TAC]]);; let DIMENSION_INSERT = prove (`!s a:real^N. dimension(a INSERT s) = if s = {} then &0 else dimension s`, REPEAT GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[DIMENSION_SING] THEN SIMP_TAC[GSYM INT_LE_ANTISYM; DIMENSION_SUBSET; SET_RULE `s SUBSET a INSERT s`] THEN ONCE_REWRITE_TAC[SET_RULE `a INSERT s = {a} UNION s`] THEN MATCH_MP_TAC DIMENSION_LE_UNION_GEN THEN REWRITE_TAC[FSIGMA_SING; GDELTA_SING; DIMENSION_SING] THEN ASM_REWRITE_TAC[INT_LE_REFL; DIMENSION_POS_LE]);; let DIMENSION_DELETE = prove (`!s a:real^N. dimension(s DELETE a) = if s DELETE a = {} then --(&1) else dimension s`, REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[DIMENSION_EMPTY] THEN SIMP_TAC[GSYM INT_LE_ANTISYM; DIMENSION_SUBSET; DELETE_SUBSET] THEN TRANS_TAC INT_LE_TRANS `dimension((a:real^N) INSERT (s DELETE a))` THEN SIMP_TAC[DIMENSION_SUBSET; SET_RULE `s SUBSET a INSERT (s DELETE a)`] THEN ASM_REWRITE_TAC[DIMENSION_INSERT; INT_LE_REFL]);; let DIMENSION_LE_EQ_GEN = prove (`!s:real^N->bool n. dimension s <= n <=> if s = {} then -- &1 <= n else !v a. open v /\ a IN v ==> ?u. a IN u /\ u SUBSET v /\ open u /\ dimension (s INTER frontier u) <= n - &1`, REPEAT GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[DIMENSION_EMPTY] THEN EQ_TAC THENL [DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`v:real^N->bool`; `a:real^N`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`(a:real^N) INSERT s`; `n:int`] DIMENSION_LE_EQ) THEN ASM_REWRITE_TAC[DIMENSION_INSERT; IN_INSERT] THEN DISCH_THEN(MP_TAC o SPECL [`v:real^N->bool`; `a:real^N`] o CONJUNCT2) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN REPEAT(MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[]) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] INT_LE_TRANS) THEN MATCH_MP_TAC DIMENSION_SUBSET THEN SET_TAC[]; DISCH_TAC THEN GEN_REWRITE_TAC I [DIMENSION_LE_EQ] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPECL [`(:real^N)`; `vec 0:real^N`]) THEN ASM_REWRITE_TAC[OPEN_UNIV; IN_UNIV] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP DIMENSION_LE_IMP_GE) THEN ASM_ARITH_TAC; ASM_MESON_TAC[]]]);; let DIMENSION_PCROSS_LE = prove (`!s:real^M->bool t:real^N->bool. ~(s = {} /\ t = {}) ==> dimension(s PCROSS t) <= dimension s + dimension t`, SUBGOAL_THEN `!n s:real^M->bool t:real^N->bool. dimension s + dimension t <= &n - &1 ==> dimension(s PCROSS t) <= &n - &1` ASSUME_TAC THENL [ALL_TAC; MAP_EVERY X_GEN_TAC [`s:real^M->bool`; `t:real^N->bool`] THEN REWRITE_TAC[GSYM DIMENSION_POS_LE; DE_MORGAN_THM] THEN DISCH_THEN(MP_TAC o MATCH_MP (INT_ARITH `&0:int <= s \/ &0 <= t ==> -- &1 <= s /\ -- &1 <= t ==> &0 <= (s + t) + &1`)) THEN REWRITE_TAC[DIMENSION_GE; GSYM INT_OF_NUM_EXISTS] THEN REWRITE_TAC[INT_ARITH `x + &1:int = n <=> x = n - &1`] THEN ASM_MESON_TAC[INT_LE_REFL]] THEN INDUCT_TAC THENL [REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP (INT_ARITH `s + t:int <= &0 - &1 ==> -- &1 <= s /\ -- &1 <= t ==> s = -- &1 \/ t = -- &1`)) THEN REWRITE_TAC[DIMENSION_GE; DIMENSION_EQ_MINUS1; INT_SUB_LZERO] THEN STRIP_TAC THEN ASM_REWRITE_TAC[PCROSS_EMPTY; DIMENSION_EMPTY; INT_LE_REFL]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`s:real^M->bool`; `t:real^N->bool`] THEN REWRITE_TAC[GSYM INT_OF_NUM_SUC; INT_ARITH `(x + y) - y:int = x`] THEN DISCH_TAC THEN ONCE_REWRITE_TAC[DIMENSION_LE_EQ] THEN CONJ_TAC THENL [INT_ARITH_TAC; ALL_TAC] THEN X_GEN_TAC `w:real^(M,N)finite_sum->bool` THEN SIMP_TAC[FORALL_PASTECART] THEN MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^N`] THEN REWRITE_TAC[PASTECART_IN_PCROSS] THEN STRIP_TAC THEN MP_TAC(ISPECL [ `w:real^(M,N)finite_sum->bool`; `a:real^M`; `b:real^N`] PASTECART_IN_INTERIOR) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`v1:real^M->bool`; `v2:real^N->bool`] THEN STRIP_TAC THEN MP_TAC(ISPEC `dimension(s:real^M->bool)` INT_LE_REFL) THEN GEN_REWRITE_TAC LAND_CONV [DIMENSION_LE_EQ] THEN DISCH_THEN(MP_TAC o SPECL [`v1:real^M->bool`; `a:real^M`] o CONJUNCT2) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `u1:real^M->bool` THEN STRIP_TAC THEN MP_TAC(ISPEC `dimension(t:real^N->bool)` INT_LE_REFL) THEN GEN_REWRITE_TAC LAND_CONV [DIMENSION_LE_EQ] THEN DISCH_THEN(MP_TAC o SPECL [`v2:real^N->bool`; `b:real^N`] o CONJUNCT2) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `u2:real^N->bool` THEN STRIP_TAC THEN EXISTS_TAC `(u1:real^M->bool) PCROSS (u2:real^N->bool)` THEN ASM_REWRITE_TAC[OPEN_PCROSS_EQ; PASTECART_IN_PCROSS] THEN CONJ_TAC THENL [ASM_MESON_TAC[SUBSET_TRANS; SUBSET_PCROSS]; ALL_TAC] THEN REWRITE_TAC[FRONTIER_PCROSS; UNION_OVER_INTER; INTER_PCROSS] THEN MATCH_MP_TAC DIMENSION_LE_UNION_CLOSED_IN THEN EXISTS_TAC `(s:real^M->bool) PCROSS (t:real^N->bool)` THEN CONJ_TAC THENL [DISJ1_TAC THEN REWRITE_TAC[SUBSET_PCROSS; INTER_SUBSET] THEN MATCH_MP_TAC CLOSED_IN_PCROSS THEN SIMP_TAC[CLOSED_IN_CLOSED_INTER; FRONTIER_CLOSED; CLOSED_CLOSURE]; CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN MATCH_MP_TAC INT_LE_TRANS THENL [EXISTS_TAC `(dimension(s:real^M->bool) - &1) + dimension(t:real^N->bool)`; EXISTS_TAC `dimension(s:real^M->bool) + (dimension(t:real^N->bool) - &1)`] THEN (CONJ_TAC THENL [ALL_TAC; ASM_INT_ARITH_TAC]) THEN MATCH_MP_TAC INT_LE_ADD2 THEN ASM_REWRITE_TAC[] THEN SIMP_TAC[DIMENSION_SUBSET; INTER_SUBSET]]);; let DIMENSION_PCROSS_EQ_0 = prove (`!s:real^M->bool t:real^N->bool. dimension(s PCROSS t) = &0 <=> dimension s = &0 /\ dimension t = &0`, REPEAT GEN_TAC THEN MAP_EVERY ASM_CASES_TAC [`s:real^M->bool = {}`; `t:real^N->bool = {}`] THEN ASM_REWRITE_TAC[PCROSS_EMPTY; DIMENSION_EMPTY] THEN CONV_TAC INT_REDUCE_CONV THEN EQ_TAC THEN REPEAT STRIP_TAC THENL [MP_TAC(ASSUME `~(t:real^N->bool = {})`) THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `b:real^N` THEN DISCH_TAC THEN SUBGOAL_THEN `dimension((s:real^M->bool) PCROSS {b:real^N}) = &0` MP_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (INT_ARITH `d2:int = &0 ==> d <= d2 /\ ~(d <= -- &1) ==> d = &0`)) THEN ASM_REWRITE_TAC[DIMENSION_LE_MINUS1; PCROSS_EQ_EMPTY; NOT_INSERT_EMPTY] THEN MATCH_MP_TAC DIMENSION_SUBSET THEN REWRITE_TAC[SUBSET_PCROSS] THEN ASM SET_TAC[]; MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC HOMEOMORPHIC_DIMENSION THEN REWRITE_TAC[HOMEOMORPHIC_PCROSS_SING]]; MP_TAC(ASSUME `~(s:real^M->bool = {})`) THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `a:real^M` THEN DISCH_TAC THEN SUBGOAL_THEN `dimension({a:real^M} PCROSS (t:real^N->bool)) = &0` MP_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (INT_ARITH `d2:int = &0 ==> d <= d2 /\ ~(d <= -- &1) ==> d = &0`)) THEN ASM_REWRITE_TAC[DIMENSION_LE_MINUS1; PCROSS_EQ_EMPTY; NOT_INSERT_EMPTY] THEN MATCH_MP_TAC DIMENSION_SUBSET THEN REWRITE_TAC[SUBSET_PCROSS] THEN ASM SET_TAC[]; MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC HOMEOMORPHIC_DIMENSION THEN REWRITE_TAC[HOMEOMORPHIC_PCROSS_SING]]; MP_TAC(ISPECL [`s:real^M->bool`; `t:real^N->bool`] DIMENSION_PCROSS_LE) THEN MP_TAC(ISPEC `(s:real^M->bool) PCROSS (t:real^N->bool)` DIMENSION_LE_MINUS1) THEN ASM_REWRITE_TAC[PCROSS_EQ_EMPTY] THEN ASM_INT_ARITH_TAC]);; let DIMENSION_SEPARATION_THEOREM = prove (`!t s:real^N->bool n c d. &0 <= n /\ s SUBSET t /\ dimension s <= n /\ closed_in (subtopology euclidean t) c /\ closed_in (subtopology euclidean t) d /\ DISJOINT c d ==> ?b. closed_in (subtopology euclidean t) b /\ dimension(b INTER s) <= n - &1 /\ ?u v. open_in (subtopology euclidean t) u /\ open_in (subtopology euclidean t) v /\ DISJOINT u v /\ u UNION v = t DIFF b /\ c SUBSET u /\ d SUBSET v`, GEN_TAC THEN ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM; GSYM INT_FORALL_POS] THEN MATCH_MP_TAC(MESON[num_CASES] `P 0 /\ (P 0 ==> !n. P(SUC n)) ==> !n. P n`) THEN CONJ_TAC THENL [REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`c:real^N->bool`; `d:real^N->bool`; `t:real^N->bool`] SEPARATION_NORMAL_LOCAL_CLOSURES) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `v:real^N->bool`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `s INTER (subtopology euclidean t closure_of u):real^N->bool`; `s INTER (subtopology euclidean t closure_of v):real^N->bool`] DIMENSION_ZERO_SEPARATION_THEOREM) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN CONJ_TAC THEN MATCH_MP_TAC CLOSED_IN_SUBTOPOLOGY_INTER_SUBSET THEN EXISTS_TAC `t:real^N->bool` THEN ASM_SIMP_TAC[CLOSED_IN_INTER; CLOSED_IN_REFL; CLOSED_IN_CLOSURE_OF]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`c':real^N->bool`; `d':real^N->bool`] THEN STRIP_TAC THEN SUBGOAL_THEN `c SUBSET t /\ d SUBSET t /\ u SUBSET t /\ v SUBSET (t:real^N->bool) /\ c' SUBSET t /\ d' SUBSET t` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[OPEN_IN_IMP_SUBSET; CLOSED_IN_IMP_SUBSET; SUBSET_TRANS]; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE [CLOSURE_OF_SUBTOPOLOGY; EUCLIDEAN_CLOSURE_OF]) THEN SUBGOAL_THEN `t INTER u:real^N->bool = u /\ t INTER v = v` (CONJUNCTS_THEN SUBST_ALL_TAC) THENL [ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`c UNION c':real^N->bool`; `d UNION d':real^N->bool`] SEPARATION_CLOSURES) THEN ANTS_TAC THENL [REWRITE_TAC[CLOSURE_UNION] THEN SUBGOAL_THEN `c' INTER closure v:real^N->bool = {} /\ d' INTER closure u:real^N->bool = {}` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `c' INTER closure d':real^N->bool = {} /\ d' INTER closure c':real^N->bool = {}` STRIP_ASSUME_TAC THENL [ASM_REWRITE_TAC[SEPARATION_CLOSED_IN_UNION]; ALL_TAC] THEN SUBGOAL_THEN `c' INTER closure d:real^N->bool = {} /\ d' INTER closure c:real^N->bool = {}` STRIP_ASSUME_TAC THENL [REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP SUBSET_CLOSURE)) THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `v INTER closure c':real^N->bool = {} /\ u INTER closure d':real^N->bool = {}` STRIP_ASSUME_TAC THENL [MP_TAC(ISPEC `subtopology euclidean (t:real^N->bool)` OPEN_IN_INTER_CLOSURE_OF_EQ_EMPTY) THEN DISCH_THEN(fun th -> MP_TAC(SPECL [`v:real^N->bool`; `c':real^N->bool`] th) THEN MP_TAC(SPECL [`u:real^N->bool`; `d':real^N->bool`] th)) THEN MP_TAC(ISPEC `subtopology euclidean (t:real^N->bool)` CLOSURE_OF_SUBSET) THEN DISCH_THEN(fun th -> MP_TAC(SPEC `u:real^N->bool` th) THEN MP_TAC(SPEC `v:real^N->bool` th)) THEN ASM_REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN ASM_SIMP_TAC[CLOSURE_OF_SUBTOPOLOGY; EUCLIDEAN_CLOSURE_OF; SET_RULE `s SUBSET t ==> t INTER s = s`] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `d INTER closure c':real^N->bool = {} /\ c INTER closure d':real^N->bool = {}` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP CLOSURE_OF_CLOSED_IN)) THEN ASM_SIMP_TAC[CLOSURE_OF_SUBTOPOLOGY; EUCLIDEAN_CLOSURE_OF; SET_RULE `s SUBSET t ==> t INTER s = s`] THEN ASM SET_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`w:real^N->bool`; `z:real^N->bool`] THEN REWRITE_TAC[UNION_SUBSET] THEN STRIP_TAC THEN EXISTS_TAC `t INTER frontier w:real^N->bool` THEN SIMP_TAC[CLOSED_IN_CLOSED_INTER; FRONTIER_CLOSED] THEN CONV_TAC INT_REDUCE_CONV THEN ASM_REWRITE_TAC[DIMENSION_LE_MINUS1] THEN ASM_SIMP_TAC[frontier; INTERIOR_OPEN] THEN CONJ_TAC THENL [MP_TAC(ISPECL [`z:real^N->bool`; `w:real^N->bool`] OPEN_INTER_CLOSURE_EQ_EMPTY) THEN ASM SET_TAC[]; MAP_EVERY EXISTS_TAC [`t INTER w:real^N->bool`; `t INTER ((:real^N) DIFF closure w):real^N->bool`] THEN ASM_SIMP_TAC[OPEN_IN_OPEN_INTER; GSYM closed; CLOSED_CLOSURE] THEN REPEAT CONJ_TAC THENL [MP_TAC(ISPEC `w:real^N->bool` CLOSURE_SUBSET) THEN ASM SET_TAC[]; ASM SET_TAC[]; ASM SET_TAC[]; MP_TAC(ISPECL [`z:real^N->bool`; `w:real^N->bool`] OPEN_INTER_CLOSURE_EQ_EMPTY) THEN ASM SET_TAC[]]]]; REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `&(SUC n):int`] DIMENSION_DECOMPOSITION) THEN ASM_REWRITE_TAC[INT_POS; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`l:real^N->bool`; `z:real^N->bool`] THEN REWRITE_TAC[GSYM INT_OF_NUM_SUC; INT_ARITH `(x + y) - y:int = x`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`z:real^N->bool`; `c:real^N->bool`; `d:real^N->bool`]) THEN ASM_REWRITE_TAC[] THEN CONV_TAC INT_REDUCE_CONV THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[DIMENSION_LE_MINUS1]] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `b:real^N->bool` THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN ASM_REWRITE_TAC[] THEN TRANS_TAC INT_LE_TRANS `dimension(l:real^N->bool)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC DIMENSION_SUBSET THEN ASM SET_TAC[]]);; let LARGE_INDUCTIVE_DIMENSION = prove (`!s:real^N->bool n. dimension s <= n <=> if s = {} then -- &1 <= n else !c d. closed_in (subtopology euclidean s) c /\ closed_in (subtopology euclidean s) d /\ DISJOINT c d ==> ?b. closed_in (subtopology euclidean s) b /\ dimension b <= n - &1 /\ ?u v. open_in (subtopology euclidean s) u /\ open_in (subtopology euclidean s) v /\ DISJOINT u v /\ u UNION v = s DIFF b /\ c SUBSET u /\ d SUBSET v`, REPEAT GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[DIMENSION_EMPTY] THEN EQ_TAC THENL [REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `s:real^N->bool`; `n:int`; `c:real^N->bool`; `d:real^N->bool`] DIMENSION_SEPARATION_THEOREM) THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN ANTS_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[GSYM DIMENSION_POS_LE]) THEN ASM_INT_ARITH_TAC; MESON_TAC[CLOSED_IN_IMP_SUBSET; SET_RULE `b SUBSET s ==> b INTER s = b`]]; STRIP_TAC THEN GEN_REWRITE_TAC I [SMALL_INDUCTIVE_DIMENSION] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPECL [`{}:real^N->bool`; `{}:real^N->bool`]) THEN REWRITE_TAC[CLOSED_IN_EMPTY; DISJOINT_EMPTY] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP DIMENSION_LE_IMP_GE) THEN ASM_ARITH_TAC; ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM SING_SUBSET] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[CLOSED_IN_SING] THEN ASM SET_TAC[]]]);; let TINY_INDUCTIVE_DIMENSION = prove (`!s:real^N->bool n. locally compact s ==> (dimension s <= n <=> if s = {} then -- &1 <= n else &0 <= n /\ !x y. x IN s /\ y IN s /\ ~(x = y) ==> ?b. closed_in (subtopology euclidean s) b /\ dimension b <= n - &1 /\ ?u v. open_in (subtopology euclidean s) u /\ open_in (subtopology euclidean s) v /\ DISJOINT u v /\ u UNION v = s DIFF b /\ x IN u /\ y IN v)`, REPEAT GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[DIMENSION_EMPTY] THEN REPEAT STRIP_TAC THEN EQ_TAC THENL [DISCH_TAC THEN CONJ_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[GSYM DIMENSION_LE_MINUS1]) THEN FIRST_ASSUM(MP_TAC o MATCH_MP DIMENSION_LE_IMP_GE) THEN ASM_INT_ARITH_TAC; MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SMALL_INDUCTIVE_DIMENSION]) THEN DISCH_THEN(MP_TAC o SPECL [`{b:real^N}`; `a:real^N`] o CONJUNCT2) THEN ASM_REWRITE_TAC[IN_SING; CLOSED_IN_SING] THEN REWRITE_TAC[SING_SUBSET; CONJ_ACI]]; STRIP_TAC] THEN ONCE_REWRITE_TAC[LOCALLY_DIMENSION_LE] THEN CONJ_TAC THENL [ASM_INT_ARITH_TAC; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LOCALLY_AND_SUBSET]) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LOCALLY_MONO) THEN X_GEN_TAC `k:real^N->bool` THEN REWRITE_TAC[] THEN STRIP_TAC THEN ONCE_REWRITE_TAC[SMALL_INDUCTIVE_DIMENSION] THEN CONJ_TAC THENL [ASM_INT_ARITH_TAC; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`c:real^N->bool`; `p:real^N`] THEN STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] CLOSED_IN_COMPACT)) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN SUBGOAL_THEN `!q. q IN c ==> ?u. open_in (subtopology euclidean k) u /\ q IN u /\ ~((p:real^N) IN closure u) /\ dimension(subtopology euclidean k frontier_of u) <= n - &1` MP_TAC THENL [X_GEN_TAC `q:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`p:real^N`; `q:real^N`]) THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN X_GEN_TAC `b:real^N->bool` THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `v:real^N->bool`] THEN STRIP_TAC THEN EXISTS_TAC `k INTER v:real^N->bool` THEN CONJ_TAC THENL [ASM_MESON_TAC[OPEN_IN_INTER; OPEN_IN_REFL; OPEN_IN_SUBTOPOLOGY_INTER_SUBSET]; ALL_TAC] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(snd(EQ_IMP_RULE(ISPECL [`k INTER u:real^N->bool`; `k INTER v:real^N->bool`] SEPARATION_OPEN_IN_UNION))) THEN ANTS_TAC THENL [CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [UNDISCH_TAC `open_in (subtopology euclidean s) (u:real^N->bool)`; UNDISCH_TAC `open_in (subtopology euclidean s) (v:real^N->bool)`] THEN REWRITE_TAC[OPEN_IN_OPEN] THEN MATCH_MP_TAC MONO_EXISTS THEN ASM SET_TAC[]; STRIP_TAC] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN TRANS_TAC INT_LE_TRANS `dimension(b:real^N->bool)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC DIMENSION_SUBSET THEN REWRITE_TAC[FRONTIER_OF_CLOSURES; CLOSURE_OF_SUBTOPOLOGY; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; EUCLIDEAN_CLOSURE_OF] THEN REWRITE_TAC[SET_RULE `k INTER k INTER v = k INTER v`; SET_RULE `k INTER (k DIFF k INTER v) = k DIFF v`] THEN SUBGOAL_THEN `closure(k DIFF v):real^N->bool = k DIFF v /\ (k INTER v) SUBSET closure(k INTER v)` MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[CLOSURE_SUBSET; CLOSURE_EQ] THEN MATCH_MP_TAC CLOSED_IN_CLOSED_TRANS THEN EXISTS_TAC `k:real^N->bool` THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED] THEN SUBGOAL_THEN `k DIFF v:real^N->bool = k INTER (s DIFF v)` SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC CLOSED_IN_SUBTOPOLOGY_INTER_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_SIMP_TAC[CLOSED_IN_INTER; CLOSED_IN_DIFF; CLOSED_IN_REFL]; FIRST_X_ASSUM(K ALL_TAC o SPEC `vec 0:real^N`)] THEN GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `u:real^N->real^N->bool` THEN DISCH_TAC THEN MP_TAC(ISPEC `c:real^N->bool` COMPACT_EQ_HEINE_BOREL_GEN) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPECL [`IMAGE (u:real^N->real^N->bool) c`; `k:real^N->bool`]) THEN ASM_SIMP_TAC[FORALL_IN_IMAGE; UNIONS_IMAGE] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN REWRITE_TAC[EXISTS_FINITE_SUBSET_IMAGE] THEN DISCH_THEN(X_CHOOSE_THEN `d:real^N->bool` STRIP_ASSUME_TAC) THEN ABBREV_TAC `v = UNIONS (IMAGE (u:real^N->real^N->bool) d)` THEN EXISTS_TAC `subtopology euclidean k frontier_of v:real^N->bool` THEN REWRITE_TAC[CLOSED_IN_FRONTIER_OF; RIGHT_AND_EXISTS_THM] THEN EXISTS_TAC `k DIFF subtopology euclidean k closure_of v:real^N->bool` THEN EXISTS_TAC `v:real^N->bool` THEN SUBGOAL_THEN `open_in (subtopology euclidean k) (v:real^N->bool)` ASSUME_TAC THENL [EXPAND_TAC "v" THEN MATCH_MP_TAC OPEN_IN_UNIONS THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN ASM SET_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[OPEN_IN_DIFF; OPEN_IN_REFL; CLOSED_IN_CLOSURE_OF] THEN ASM_REWRITE_TAC[IN_DIFF] THEN FIRST_ASSUM(ASSUME_TAC o SYM o MATCH_MP FRONTIER_OF_OPEN_IN) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN MP_TAC(ISPECL [`subtopology euclidean (k:real^N->bool)`; `v:real^N->bool`] CLOSURE_OF_SUBSET) THEN ASM_REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN DISCH_TAC THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r /\ s <=> (q /\ r) /\ p /\ s`] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `subtopology euclidean k frontier_of (v:real^N->bool) SUBSET UNIONS {subtopology euclidean k frontier_of (u q) | (q:real^N) IN d}` ASSUME_TAC THENL [EXPAND_TAC "v" THEN W(MP_TAC o PART_MATCH (lhand o rand) FRONTIER_OF_UNIONS_SUBSET o lhand o snd) THEN ASM_SIMP_TAC[FINITE_IMAGE; SET_RULE `{f x | x IN IMAGE g s} = {f(g x) | x IN s}`]; ALL_TAC] THEN CONJ_TAC THENL [TRANS_TAC INT_LE_TRANS `dimension(UNIONS { subtopology euclidean k frontier_of (u q) | (q:real^N) IN d}:real^N->bool)` THEN ASM_SIMP_TAC[DIMENSION_SUBSET] THEN MATCH_MP_TAC DIMENSION_LE_CLOSED_IN_UNIONS THEN EXISTS_TAC `k:real^N->bool` THEN ASM_SIMP_TAC[SIMPLE_IMAGE; FINITE_IMP_COUNTABLE; COUNTABLE_IMAGE] THEN CONJ_TAC THENL [ASM_INT_ARITH_TAC; REWRITE_TAC[FORALL_IN_IMAGE]] THEN REWRITE_TAC[CLOSED_IN_FRONTIER_OF] THEN ASM SET_TAC[]; FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `c DIFF v = f ==> ~(p IN v) /\ ~(p IN f) ==> ~(p IN c)`)) THEN CONJ_TAC THENL [EXPAND_TAC "v"; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET t ==> ~(p IN t) ==> ~(p IN s)`))] THEN (REWRITE_TAC[IN_UNIONS; EXISTS_IN_IMAGE; EXISTS_IN_GSPEC] THEN DISCH_THEN(X_CHOOSE_THEN `q:real^N` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPEC `q:real^N`) THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THENL [MP_TAC(ISPEC `(u:real^N->real^N->bool) q` CLOSURE_SUBSET) THEN ASM SET_TAC[]; DISCH_THEN(MP_TAC o el 2 o CONJUNCTS) THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [FRONTIER_OF_CLOSURES]) THEN REWRITE_TAC[IN_INTER; CLOSURE_OF_SUBTOPOLOGY; EUCLIDEAN_CLOSURE_OF] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> (x IN k /\ x IN s) /\ P ==> x IN t`) THEN MATCH_MP_TAC SUBSET_CLOSURE THEN REWRITE_TAC[INTER_SUBSET]]]);; let DIMENSION_LE_RATIONAL_COORDINATES = prove (`!n. dimension {x:real^N | {i | i IN 1..dimindex(:N) /\ rational(x$i)} HAS_SIZE n} <= &0`, GEN_TAC THEN MATCH_MP_TAC(MESON[] `(!x. t = x ==> dimension x <= &0) ==> dimension t <= &0`) THEN X_GEN_TAC `s:real^N->bool` THEN DISCH_TAC THEN SUBGOAL_THEN `s = UNIONS {s INTER h | h IN { INTERS {{x:real^N | x$i = q$i} | i IN k} | k IN {l | l SUBSET 1..dimindex(:N) /\ l HAS_SIZE n} /\ q IN {y:real^N | !i. 1 <= i /\ i <= dimindex(:N) ==> rational(y$i)}}}` SUBST1_TAC THENL [REWRITE_TAC[UNIONS_GSPEC; INTERS_GSPEC; EXISTS_IN_GSPEC] THEN EXPAND_TAC "s" THEN GEN_REWRITE_TAC I [EXTENSION] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_ELIM_THM; IN_INTER] THEN ABBREV_TAC `k = {i | i IN 1..dimindex(:N) /\ rational((x:real^N)$i)}` THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `k:num->bool` THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `(lambda i. if i IN k then (x:real^N)$i else &0):real^N` THEN SIMP_TAC[LAMBDA_BETA] THEN EXPAND_TAC "k" THEN REWRITE_TAC[IN_ELIM_THM; SUBSET_RESTRICT] THEN SIMP_TAC[IN_NUMSEG; LAMBDA_BETA] THEN MESON_TAC[RATIONAL_NUM]; ALL_TAC] THEN MATCH_MP_TAC DIMENSION_LE_CLOSED_IN_UNIONS THEN EXISTS_TAC `s:real^N->bool` THEN CONV_TAC INT_REDUCE_CONV THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN MATCH_MP_TAC COUNTABLE_IMAGE THEN MATCH_MP_TAC COUNTABLE_PRODUCT_DEPENDENT THEN REWRITE_TAC[COUNTABLE_RATIONAL_COORDINATES] THEN MATCH_MP_TAC COUNTABLE_SUBSET THEN EXISTS_TAC `{k | k SUBSET (:num) /\ FINITE k}` THEN SIMP_TAC[COUNTABLE_FINITE_SUBSETS; NUM_COUNTABLE] THEN REWRITE_TAC[HAS_SIZE] THEN SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN MAP_EVERY X_GEN_TAC [`k:num->bool`; `q:real^N`] THEN REWRITE_TAC[IN_ELIM_THM] THEN REPEAT STRIP_TAC THENL [MATCH_MP_TAC CLOSED_IN_CLOSED_INTER THEN MATCH_MP_TAC CLOSED_INTERS THEN REWRITE_TAC[FORALL_IN_GSPEC; CLOSED_STANDARD_HYPERPLANE]; ALL_TAC] THEN TRANS_TAC INT_LE_TRANS `dimension {x:real^N | !i. 1 <= i /\ i <= dimindex(:N) ==> (rational(x$i) <=> i IN k)}` THEN CONJ_TAC THENL [MATCH_MP_TAC DIMENSION_SUBSET THEN REWRITE_TAC[SUBSET; INTERS_GSPEC; IN_ELIM_THM; IN_INTER] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN MP_TAC(ISPECL [`k:num->bool`; `{i | i IN 1..dimindex(:N) /\ rational((x:real^N)$i)}`] CARD_SUBSET_LE) THEN ANTS_TAC THENL [ALL_TAC; REWRITE_TAC[IN_NUMSEG] THEN SET_TAC[]] THEN SIMP_TAC[FINITE_RESTRICT; FINITE_NUMSEG] THEN CONJ_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[GSYM IN_NUMSEG]) THEN ASM SET_TAC[]; UNDISCH_TAC `(x:real^N) IN s` THEN EXPAND_TAC "s" THEN UNDISCH_TAC `(k:num->bool) HAS_SIZE n` THEN SIMP_TAC[IN_ELIM_THM; HAS_SIZE; LE_REFL]]; ALL_TAC] THEN GEN_REWRITE_TAC I [DIMENSION_LE_EQ] THEN CONV_TAC INT_REDUCE_CONV THEN MAP_EVERY X_GEN_TAC [`v:real^N->bool`; `a:real^N`] THEN REWRITE_TAC[DIMENSION_LE_MINUS1; IN_ELIM_THM] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `a:real^N` o GEN_REWRITE_RULE I [OPEN_CONTAINS_INTERVAL]) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`l:real^N`; `r:real^N`] THEN REWRITE_TAC[IN_INTERVAL] THEN STRIP_TAC THEN SUBGOAL_THEN `!i. 1 <= i /\ i <= dimindex(:N) ==> ?u v. (rational u <=> ~(i IN k)) /\ (rational v <=> ~(i IN k)) /\ u < (a:real^N)$i /\ a$i < v /\ abs(v - u) < min ((r:real^N)$i - a$i) (a$i - (l:real^N)$i)` MP_TAC THENL [X_GEN_TAC `i:num` THEN STRIP_TAC THEN ASM_CASES_TAC `(i:num) IN k` THEN ASM_REWRITE_TAC[] THENL [MATCH_MP_TAC IRRATIONAL_APPROXIMATION_STRADDLE; MATCH_MP_TAC RATIONAL_APPROXIMATION_STRADDLE] THEN ASM_SIMP_TAC[REAL_LT_MIN; REAL_SUB_LT]; REWRITE_TAC[LAMBDA_SKOLEM; LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN EXISTS_TAC `interval(x:real^N,y)` THEN ASM_SIMP_TAC[IN_INTERVAL; OPEN_INTERVAL] THEN CONJ_TAC THENL [TRANS_TAC SUBSET_TRANS `interval[l:real^N,r]` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET_INTERVAL] THEN DISCH_THEN(K ALL_TAC) THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `i:num`)) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[FRONTIER_OPEN_INTERVAL; INTERVAL_EQ_EMPTY] THEN COND_CASES_TAC THEN REWRITE_TAC[INTER_EMPTY] THEN REWRITE_TAC[EXTENSION; IN_DIFF; IN_INTER; NOT_IN_EMPTY; IN_ELIM_THM] THEN X_GEN_TAC `z:real^N` THEN REWRITE_TAC[IN_INTERVAL] THEN MATCH_MP_TAC(MESON[] `(!x. ~R x ==> ~(P x /\ Q x)) ==> ~((!x. P x) /\ (!x. Q x) /\ ~(!x. R x))`) THEN X_GEN_TAC `i:num` THEN REWRITE_TAC[NOT_IMP] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `i:num`)) THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[REAL_LE_LT]);; let DIMENSION_EXACTLY_RATIONAL_COORDINATES = prove (`!n. 1 <= n /\ n <= dimindex(:N) ==> dimension {x:real^N | {i | i IN 1..dimindex(:N) /\ rational(x$i)} HAS_SIZE n} = &0`, GEN_TAC THEN REWRITE_TAC[GSYM INT_LE_ANTISYM] THEN REWRITE_TAC[DIMENSION_POS_LE; DIMENSION_LE_RATIONAL_COORDINATES] THEN STRIP_TAC THEN SUBGOAL_THEN `?q r. rational q /\ ~rational r` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[RATIONAL_APPROXIMATION; IRRATIONAL_APPROXIMATION; REAL_LT_01]; REWRITE_TAC[EXTENSION; NOT_FORALL_THM; IN_ELIM_THM; NOT_IN_EMPTY] THEN EXISTS_TAC `(lambda i. if i <= n then q else r):real^N` THEN ONCE_REWRITE_TAC[TAUT `p /\ q <=> ~(p ==> ~q)`] THEN SIMP_TAC[IN_NUMSEG; LAMBDA_BETA] THEN ONCE_REWRITE_TAC[COND_RAND] THEN ASM_REWRITE_TAC[TAUT `(if p then T else F) <=> p`; NOT_IMP] THEN ASM_SIMP_TAC[ARITH_RULE `n <= m ==> ((1 <= i /\ i <= m) /\ i <= n <=> 1 <= i /\ i <= n)`] THEN REWRITE_TAC[GSYM numseg; HAS_SIZE_NUMSEG_1]]);; (* ------------------------------------------------------------------------- *) (* Covering spaces and lifting results for them. *) (* ------------------------------------------------------------------------- *) let covering_space = new_definition `covering_space(c,(p:real^M->real^N)) s <=> p continuous_on c /\ IMAGE p c = s /\ !x. x IN s ==> ?t. x IN t /\ open_in (subtopology euclidean s) t /\ ?v. UNIONS v = {x | x IN c /\ p(x) IN t} /\ (!u. u IN v ==> open_in (subtopology euclidean c) u) /\ pairwise DISJOINT v /\ (!u. u IN v ==> ?q. homeomorphism (u,t) (p,q))`;; let COVERING_SPACE_IMP_CONTINUOUS = prove (`!p:real^M->real^N c s. covering_space (c,p) s ==> p continuous_on c`, SIMP_TAC[covering_space]);; let COVERING_SPACE_IMP_SURJECTIVE = prove (`!p:real^M->real^N c s. covering_space (c,p) s ==> IMAGE p c = s`, SIMP_TAC[covering_space]);; let HOMEOMORPHISM_IMP_COVERING_SPACE = prove (`!f:real^M->real^N g s t. homeomorphism (s,t) (f,g) ==> covering_space (s,f) t`, REWRITE_TAC[homeomorphism] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[covering_space] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN EXISTS_TAC `t:real^N->bool` THEN ASM_SIMP_TAC[OPEN_IN_SUBTOPOLOGY_REFL; TOPSPACE_EUCLIDEAN; SUBSET_UNIV] THEN EXISTS_TAC `{s:real^M->bool}` THEN REWRITE_TAC[FORALL_IN_INSERT; NOT_IN_EMPTY; UNIONS_1; PAIRWISE_SING] THEN ASM_SIMP_TAC[OPEN_IN_SUBTOPOLOGY_REFL; TOPSPACE_EUCLIDEAN; SUBSET_UNIV] THEN CONJ_TAC THENL [ASM SET_TAC[]; EXISTS_TAC `g:real^N->real^M`] THEN ASM_REWRITE_TAC[homeomorphism]);; let COVERING_SPACE_LOCAL_HOMEOMORPHISM = prove (`!p:real^M->real^N c s. covering_space (c,p) s ==> !x. x IN c ==> ?t u. x IN t /\ open_in (subtopology euclidean c) t /\ p(x) IN u /\ open_in (subtopology euclidean s) u /\ ?q. homeomorphism (t,u) (p,q)`, REWRITE_TAC[covering_space] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `(p:real^M->real^N) x`) THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` MP_TAC) THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(X_CHOOSE_THEN `v:(real^M->bool)->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `(x:real^M) IN UNIONS v` MP_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[IN_UNIONS]] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^M->bool` THEN STRIP_TAC THEN EXISTS_TAC `t:real^N->bool` THEN ASM_SIMP_TAC[]);; let COVERING_SPACE_LOCAL_HOMEOMORPHISM_ALT = prove (`!p:real^M->real^N c s. covering_space (c,p) s ==> !y. y IN s ==> ?x t u. p(x) = y /\ x IN t /\ open_in (subtopology euclidean c) t /\ y IN u /\ open_in (subtopology euclidean s) u /\ ?q. homeomorphism (t,u) (p,q)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?x. x IN c /\ (p:real^M->real^N) x = y` MP_TAC THENL [FIRST_X_ASSUM(MP_TAC o MATCH_MP COVERING_SPACE_IMP_SURJECTIVE) THEN ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `x:real^M` o MATCH_MP COVERING_SPACE_LOCAL_HOMEOMORPHISM) THEN ASM_MESON_TAC[]]);; let COVERING_SPACE_OPEN_MAP = prove (`!p:real^M->real^N c s t. covering_space (c,p) s /\ open_in (subtopology euclidean c) t ==> open_in (subtopology euclidean s) (IMAGE p t)`, REWRITE_TAC[covering_space] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o CONJUNCT1 o GEN_REWRITE_RULE I [open_in]) THEN ONCE_REWRITE_TAC[OPEN_IN_SUBOPEN] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` MP_TAC) THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(X_CHOOSE_THEN `vs:(real^M->bool)->bool` (STRIP_ASSUME_TAC o GSYM)) THEN SUBGOAL_THEN `?x. x IN {x | x IN c /\ (p:real^M->real^N) x IN u} /\ x IN t /\ p x = y` MP_TAC THENL [ASM SET_TAC[]; ASM_REWRITE_TAC[]] THEN DISCH_THEN(X_CHOOSE_THEN `x:real^M` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_UNIONS]) THEN DISCH_THEN(X_CHOOSE_THEN `v:real^M->bool` STRIP_ASSUME_TAC) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `v:real^M->bool`)) THEN ASM_REWRITE_TAC[homeomorphism] THEN REPEAT DISCH_TAC THEN FIRST_X_ASSUM(X_CHOOSE_THEN `q:real^N->real^M` STRIP_ASSUME_TAC) THEN EXISTS_TAC `IMAGE (p:real^M->real^N) (t INTER v)` THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN SUBGOAL_THEN `IMAGE (p:real^M->real^N) (t INTER v) = {z | z IN u /\ q z IN (t INTER v)}` SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONTINUOUS_ON_OPEN]) THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[INTER_COMM] THEN MATCH_MP_TAC OPEN_IN_SUBTOPOLOGY_INTER_SUBSET THEN EXISTS_TAC `c:real^M->bool` THEN CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_INTER; ASM_MESON_TAC[open_in]] THEN ASM_REWRITE_TAC[OPEN_IN_SUBTOPOLOGY_REFL; TOPSPACE_EUCLIDEAN; SUBSET_UNIV]);; let COVERING_SPACE_QUOTIENT_MAP = prove (`!p:real^M->real^N c s. covering_space (c,p) s ==> !u. u SUBSET s ==> (open_in (subtopology euclidean c) {x | x IN c /\ p x IN u} <=> open_in (subtopology euclidean s) u)`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(SUBST1_TAC o SYM o MATCH_MP COVERING_SPACE_IMP_SURJECTIVE) THEN MATCH_MP_TAC OPEN_MAP_IMP_QUOTIENT_MAP THEN CONJ_TAC THENL [ASM_MESON_TAC[COVERING_SPACE_IMP_CONTINUOUS]; ALL_TAC] THEN FIRST_ASSUM(SUBST1_TAC o MATCH_MP COVERING_SPACE_IMP_SURJECTIVE) THEN ASM_MESON_TAC[COVERING_SPACE_OPEN_MAP]);; let COVERING_SPACE_LOCALIZED_HOMEOMORPHISM = prove (`!p:real^M->real^N c s. covering_space (c,p) s ==> !w x. x IN w /\ open_in (subtopology euclidean c) w ==> ?t u. x IN t /\ open_in (subtopology euclidean c) t /\ p(x) IN u /\ open_in (subtopology euclidean s) u /\ t SUBSET w /\ ?q. homeomorphism (t,u) (p,q)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `x:real^M` o MATCH_MP COVERING_SPACE_LOCAL_HOMEOMORPHISM) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`u:real^M->bool`; `v:real^N->bool`] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(X_CHOOSE_TAC `q:real^N->real^M`) THEN MAP_EVERY EXISTS_TAC [`u INTER w:real^M->bool`; `IMAGE (p:real^M->real^N) (u INTER w)`] THEN ASM_SIMP_TAC[IN_INTER; OPEN_IN_INTER; INTER_SUBSET] THEN REPEAT CONJ_TAC THENL [ASM_SIMP_TAC[FUN_IN_IMAGE; IN_INTER]; ASM_MESON_TAC[COVERING_SPACE_OPEN_MAP; OPEN_IN_INTER]; EXISTS_TAC `q:real^N->real^M` THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]]);; let COVERING_SPACE_LOCALIZED_HOMEOMORPHISM_ALT = prove (`!p:real^M->real^N c s. covering_space (c,p) s ==> !w y. y IN w /\ open_in (subtopology euclidean s) w ==> ?x t u. p(x) = y /\ x IN t /\ open_in (subtopology euclidean c) t /\ y IN u /\ open_in (subtopology euclidean s) u /\ u SUBSET w /\ ?q. homeomorphism (t,u) (p,q)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN SUBGOAL_THEN `?x. x IN c /\ (p:real^M->real^N) x = y` MP_TAC THENL [FIRST_X_ASSUM(MP_TAC o MATCH_MP COVERING_SPACE_IMP_SURJECTIVE) THEN ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPECL [`{x | x IN c /\ (p:real^M->real^N) x IN w}`; `x:real^M`] o MATCH_MP COVERING_SPACE_LOCALIZED_HOMEOMORPHISM) THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN ANTS_TAC THENL [MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE_GEN THEN RULE_ASSUM_TAC(REWRITE_RULE[covering_space]) THEN ASM_MESON_TAC[SUBSET_REFL]; REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN SIMP_TAC[] THEN REWRITE_TAC[homeomorphism] THEN REPEAT STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET)) THEN ASM SET_TAC[]]]);; let COVERING_SPACE_LOCALLY_HOMEOMORPHIC = prove (`!P Q p:real^M->real^N c s. covering_space (c,p) s /\ (!q u v. ~(u = {}) /\ u SUBSET c /\ v SUBSET s /\ homeomorphism (u,v) (p,q) /\ P u ==> Q v) /\ locally P c ==> locally Q s`, REPEAT GEN_TAC THEN REWRITE_TAC[locally] THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`w:real^N->bool`; `y:real^N`] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPECL [`w:real^N->bool`; `y:real^N`] o MATCH_MP COVERING_SPACE_LOCALIZED_HOMEOMORPHISM_ALT) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`x:real^M`; `u:real^M->bool`; `v:real^N->bool`] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(X_CHOOSE_TAC `q:real^N->real^M`) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`u:real^M->bool`; `x:real^M`]) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`n:real^M->bool`; `l:real^M->bool`] THEN STRIP_TAC THEN EXISTS_TAC `IMAGE (p:real^M->real^N) n` THEN EXISTS_TAC `IMAGE (p:real^M->real^N) l` THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[COVERING_SPACE_OPEN_MAP; OPEN_IN_INTER]; FIRST_X_ASSUM MATCH_MP_TAC THEN MAP_EVERY EXISTS_TAC [`q:real^N->real^M`; `l:real^M->bool`] THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS))] THEN RULE_ASSUM_TAC(REWRITE_RULE[covering_space]) THEN RULE_ASSUM_TAC(REWRITE_RULE[HOMEOMORPHISM]) THEN REPEAT(FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET)) THEN ASM SET_TAC[]; ASM SET_TAC[]; ASM SET_TAC[]; RULE_ASSUM_TAC(REWRITE_RULE[HOMEOMORPHISM]) THEN ASM SET_TAC[]]);; let COVERING_SPACE_LOCALLY_HOMEOMORPHIC_EQ = prove (`!P Q p:real^M->real^N c s. covering_space (c,p) s /\ (!q u v. ~(u = {}) /\ u SUBSET c /\ v SUBSET s /\ homeomorphism (u,v) (p,q) ==> (P u <=> Q v)) ==> (locally P c <=> locally Q s)`, REPEAT STRIP_TAC THEN EQ_TAC THENL [ASM_MESON_TAC[COVERING_SPACE_LOCALLY_HOMEOMORPHIC]; ALL_TAC] THEN REWRITE_TAC[locally] THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`w:real^M->bool`; `x:real^M`] THEN STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN FIRST_ASSUM(MP_TAC o SPECL [`w:real^M->bool`; `x:real^M`] o MATCH_MP COVERING_SPACE_LOCALIZED_HOMEOMORPHISM) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u:real^M->bool`; `v:real^N->bool`] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(X_CHOOSE_TAC `q:real^N->real^M`) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`v:real^N->bool`; `(p:real^M->real^N) x`]) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`n:real^N->bool`; `l:real^N->bool`] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`{x | x IN u /\ (p:real^M->real^N) x IN n}`; `{x | x IN u /\ (p:real^M->real^N) x IN l}`] THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `u:real^M->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE_GEN THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN REPEAT(FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET)) THEN RULE_ASSUM_TAC(REWRITE_RULE[covering_space]) THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ASM SET_TAC[]]; CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`q:real^N->real^M`; `{x | x IN u /\ (p:real^M->real^N) x IN l}`; `l:real^N->bool`]) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN REPLICATE_TAC 2 (CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN RULE_ASSUM_TAC(REWRITE_RULE[covering_space]) THEN REPEAT(FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET)) THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN RULE_ASSUM_TAC(REWRITE_RULE[HOMEOMORPHISM]) THEN ASM SET_TAC[]]);; let COVERING_SPACE_LOCALLY = prove (`!P Q p:real^M->real^N c s. covering_space (c,p) s /\ (!t. t SUBSET c /\ P t ==> Q(IMAGE p t)) /\ locally P c ==> locally Q s`, MP_TAC COVERING_SPACE_LOCALLY_HOMEOMORPHIC THEN REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN REWRITE_TAC[homeomorphism] THEN MESON_TAC[]);; let COVERING_SPACE_LOCALLY_EQ = prove (`!P Q p:real^M->real^N c s. covering_space (c,p) s /\ (!t. t SUBSET c /\ P t ==> Q(IMAGE p t)) /\ (!q u. u SUBSET s /\ q continuous_on u /\ Q u ==> P(IMAGE q u)) ==> (locally Q s <=> locally P c)`, MP_TAC COVERING_SPACE_LOCALLY_HOMEOMORPHIC_EQ THEN REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN REWRITE_TAC[homeomorphism] THEN MESON_TAC[]);; let COVERING_SPACE_LOCALLY_COMPACT_EQ = prove (`!p:real^M->real^N c s. covering_space (c,p) s ==> (locally compact s <=> locally compact c)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC COVERING_SPACE_LOCALLY_EQ THEN EXISTS_TAC `p:real^M->real^N` THEN ASM_REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[covering_space]) THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; COMPACT_CONTINUOUS_IMAGE]);; let COVERING_SPACE_LOCALLY_CONNECTED_EQ = prove (`!p:real^M->real^N c s. covering_space (c,p) s ==> (locally connected s <=> locally connected c)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC COVERING_SPACE_LOCALLY_EQ THEN EXISTS_TAC `p:real^M->real^N` THEN ASM_REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[covering_space]) THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; CONNECTED_CONTINUOUS_IMAGE]);; let COVERING_SPACE_LOCALLY_PATH_CONNECTED_EQ = prove (`!p:real^M->real^N c s. covering_space (c,p) s ==> (locally path_connected s <=> locally path_connected c)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC COVERING_SPACE_LOCALLY_EQ THEN EXISTS_TAC `p:real^M->real^N` THEN ASM_REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[covering_space]) THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; PATH_CONNECTED_CONTINUOUS_IMAGE]);; let COVERING_SPACE_LOCALLY_COMPACT = prove (`!p:real^M->real^N c s. covering_space (c,p) s /\ locally compact c ==> locally compact s`, MESON_TAC[COVERING_SPACE_LOCALLY_COMPACT_EQ]);; let COVERING_SPACE_LOCALLY_CONNECTED = prove (`!p:real^M->real^N c s. covering_space (c,p) s /\ locally connected c ==> locally connected s`, MESON_TAC[COVERING_SPACE_LOCALLY_CONNECTED_EQ]);; let COVERING_SPACE_LOCALLY_PATH_CONNECTED = prove (`!p:real^M->real^N c s. covering_space (c,p) s /\ locally path_connected c ==> locally path_connected s`, MESON_TAC[COVERING_SPACE_LOCALLY_PATH_CONNECTED_EQ]);; let COVERING_SPACE_LIFT_UNIQUE_GEN = prove (`!p:real^M->real^N f:real^P->real^N g1 g2 c s t u a x. covering_space (c,p) s /\ f continuous_on t /\ IMAGE f t SUBSET s /\ g1 continuous_on t /\ IMAGE g1 t SUBSET c /\ (!x. x IN t ==> f(x) = p(g1 x)) /\ g2 continuous_on t /\ IMAGE g2 t SUBSET c /\ (!x. x IN t ==> f(x) = p(g2 x)) /\ u IN components t /\ a IN u /\ g1(a) = g2(a) /\ x IN u ==> g1(x) = g2(x)`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN UNDISCH_TAC `(x:real^P) IN u` THEN SPEC_TAC(`x:real^P`,`x:real^P`) THEN MATCH_MP_TAC(SET_RULE `(?a. a IN u /\ g a = z) /\ ({x | x IN u /\ g x = z} = {} \/ {x | x IN u /\ g x = z} = u) ==> !x. x IN u ==> g x = z`) THEN CONJ_TAC THENL [ASM_MESON_TAC[VECTOR_SUB_EQ]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_CONNECTED) THEN REWRITE_TAC[CONNECTED_CLOPEN] THEN DISCH_THEN MATCH_MP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[OPEN_IN_SUBOPEN] THEN REWRITE_TAC[IN_ELIM_THM] THEN X_GEN_TAC `x:real^P` THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `(g1:real^P->real^M) x` o MATCH_MP COVERING_SPACE_LOCAL_HOMEOMORPHISM) THEN ANTS_TAC THENL [ASM SET_TAC[]; SIMP_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`v:real^M->bool`; `w:real^N->bool`] THEN RULE_ASSUM_TAC(REWRITE_RULE[VECTOR_SUB_EQ]) THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN REWRITE_TAC[homeomorphism] THEN DISCH_THEN(X_CHOOSE_THEN `q:real^N->real^M` STRIP_ASSUME_TAC) THEN EXISTS_TAC `{x | x IN u /\ (g1:real^P->real^M) x IN v} INTER {x | x IN u /\ (g2:real^P->real^M) x IN v}` THEN CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_INTER THEN ONCE_REWRITE_TAC[SET_RULE `{x | x IN u /\ g x IN v} = {x | x IN u /\ g x IN (v INTER IMAGE g u)}`] THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_IMP_OPEN_IN THEN (CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ALL_TAC]) THEN UNDISCH_TAC `open_in (subtopology euclidean c) (v:real^M->bool)` THEN REWRITE_TAC[OPEN_IN_OPEN] THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[] THEN ASM SET_TAC[]; REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_INTER; VECTOR_SUB_EQ] THEN ASM SET_TAC[]]; MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE_CONSTANT THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]]);; let COVERING_SPACE_LIFT_UNIQUE = prove (`!p:real^M->real^N f:real^P->real^N g1 g2 c s t a x. covering_space (c,p) s /\ f continuous_on t /\ IMAGE f t SUBSET s /\ g1 continuous_on t /\ IMAGE g1 t SUBSET c /\ (!x. x IN t ==> f(x) = p(g1 x)) /\ g2 continuous_on t /\ IMAGE g2 t SUBSET c /\ (!x. x IN t ==> f(x) = p(g2 x)) /\ connected t /\ a IN t /\ g1(a) = g2(a) /\ x IN t ==> g1(x) = g2(x)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`p:real^M->real^N`; `f:real^P->real^N`; `g1:real^P->real^M`; `g2:real^P->real^M`; `c:real^M->bool`; `s:real^N->bool`; `t:real^P->bool`; `t:real^P->bool`; `a:real^P`; `x:real^P`] COVERING_SPACE_LIFT_UNIQUE_GEN) THEN ASM_REWRITE_TAC[IN_COMPONENTS_SELF] THEN ASM SET_TAC[]);; let COVERING_SPACE_LIFT_UNIQUE_IDENTITY = prove (`!p:real^M->real^N c f s a. covering_space (c,p) s /\ path_connected c /\ f continuous_on c /\ IMAGE f c SUBSET c /\ (!x. x IN c ==> p(f x) = p x) /\ a IN c /\ f(a) = a ==> !x. x IN c ==> f x = x`, REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [path_connected]) THEN DISCH_THEN(MP_TAC o SPECL [`a:real^M`; `x:real^M`]) THEN ASM_REWRITE_TAC[path; path_image; pathstart; pathfinish] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^1->real^M` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`p:real^M->real^N`; `(p:real^M->real^N) o (g:real^1->real^M)`; `(f:real^M->real^M) o (g:real^1->real^M)`; `g:real^1->real^M`; `c:real^M->bool`; `s:real^N->bool`; `interval[vec 0:real^1,vec 1]`; `vec 0:real^1`; `vec 1:real^1`] COVERING_SPACE_LIFT_UNIQUE) THEN ASM_REWRITE_TAC[o_THM; IMAGE_o] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[ENDS_IN_UNIT_INTERVAL; CONNECTED_INTERVAL] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [covering_space]) THEN STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP (SET_RULE `IMAGE p c = s ==> !x. x IN c ==> p(x) IN s`)) THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE] THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM_REWRITE_TAC[SUBSET; FORALL_IN_IMAGE]);; let COVERING_SPACE_LIFT_HOMOTOPY = prove (`!p:real^M->real^N c s (h:real^(1,P)finite_sum->real^N) f u. covering_space (c,p) s /\ h continuous_on (interval[vec 0,vec 1] PCROSS u) /\ IMAGE h (interval[vec 0,vec 1] PCROSS u) SUBSET s /\ (!y. y IN u ==> h (pastecart (vec 0) y) = p(f y)) /\ f continuous_on u /\ IMAGE f u SUBSET c ==> ?k. k continuous_on (interval[vec 0,vec 1] PCROSS u) /\ IMAGE k (interval[vec 0,vec 1] PCROSS u) SUBSET c /\ (!y. y IN u ==> k(pastecart (vec 0) y) = f y) /\ (!z. z IN interval[vec 0,vec 1] PCROSS u ==> h z = p(k z))`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `!y. y IN u ==> ?v. open_in (subtopology euclidean u) v /\ y IN v /\ ?k:real^(1,P)finite_sum->real^M. k continuous_on (interval[vec 0,vec 1] PCROSS v) /\ IMAGE k (interval[vec 0,vec 1] PCROSS v) SUBSET c /\ (!y. y IN v ==> k(pastecart (vec 0) y) = f y) /\ (!z. z IN interval[vec 0,vec 1] PCROSS v ==> h z :real^N = p(k z))` MP_TAC THENL [ALL_TAC; GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM; RIGHT_AND_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`v:real^P->real^P->bool`; `fs:real^P->real^(1,P)finite_sum->real^M`] THEN DISCH_THEN(LABEL_TAC "*") THEN MP_TAC(ISPECL [`subtopology euclidean ((interval[vec 0,vec 1] PCROSS u):real^(1,P)finite_sum->bool)`; `euclidean:(real^M)topology`; `fs:real^P->real^(1,P)finite_sum->real^M`; `(\x. interval[vec 0,vec 1] PCROSS (v x)) :real^P->real^(1,P)finite_sum->bool`; `u:real^P->bool`] PASTING_LEMMA_EXISTS) THEN REWRITE_TAC[CONTINUOUS_MAP_EUCLIDEAN; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; SUBTOPOLOGY_SUBTOPOLOGY] THEN ONCE_REWRITE_TAC[TAUT `open_in a b /\ c <=> ~(open_in a b ==> ~c)`] THEN SIMP_TAC[ISPEC `euclidean` OPEN_IN_IMP_SUBSET; SET_RULE `s SUBSET u ==> u INTER s = s`] THEN REWRITE_TAC[NOT_IMP] THEN ASM_SIMP_TAC[SUBSET_UNIV] THEN ANTS_TAC THENL [ALL_TAC; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:real^(1,P)finite_sum->real^M` THEN STRIP_TAC THEN ASM_REWRITE_TAC[FORALL_IN_IMAGE; FORALL_IN_PCROSS; SUBSET] THEN REPEAT CONJ_TAC THEN TRY(X_GEN_TAC `t:real^1`) THEN X_GEN_TAC `y:real^P` THEN STRIP_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPECL [`pastecart (t:real^1) (y:real^P)`; `y:real^P`]); FIRST_X_ASSUM(MP_TAC o SPECL [`pastecart (vec 0:real^1) (y:real^P)`; `y:real^P`]); FIRST_X_ASSUM(MP_TAC o SPECL [`pastecart (t:real^1) (y:real^P)`; `y:real^P`])] THEN ASM_SIMP_TAC[PASTECART_IN_PCROSS; IN_INTER; ENDS_IN_UNIT_INTERVAL] THEN DISCH_THEN SUBST1_TAC THEN REMOVE_THEN "*" (MP_TAC o SPEC `y:real^P`) THEN ASM_REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[PASTECART_IN_PCROSS]] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[SUBSET; FORALL_IN_PCROSS; UNIONS_GSPEC; IN_ELIM_THM] THEN MAP_EVERY X_GEN_TAC [`t:real^1`; `y:real^P`] THEN STRIP_TAC THEN EXISTS_TAC `y:real^P` THEN ASM_SIMP_TAC[PASTECART_IN_PCROSS]; X_GEN_TAC `y:real^P` THEN DISCH_TAC THEN REMOVE_THEN "*" (MP_TAC o SPEC `y:real^P`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o CONJUNCT1) THEN REWRITE_TAC[OPEN_IN_OPEN] THEN DISCH_THEN(X_CHOOSE_THEN `t:real^P->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(:real^1) PCROSS (t:real^P->bool)` THEN ASM_SIMP_TAC[REWRITE_RULE[GSYM PCROSS] OPEN_PCROSS; OPEN_UNIV] THEN REWRITE_TAC[EXTENSION; FORALL_PASTECART; PASTECART_IN_PCROSS; IN_INTER; IN_UNIV] THEN REPEAT GEN_TAC THEN CONV_TAC TAUT; REWRITE_TAC[FORALL_PASTECART; IN_INTER; PASTECART_IN_PCROSS] THEN MAP_EVERY X_GEN_TAC [`x:real^P`; `z:real^P`; `t:real^1`; `y:real^P`] THEN REWRITE_TAC[CONJ_ACI] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o ISPECL [`h:real^(1,P)finite_sum->real^N`; `(fs:real^P->real^(1,P)finite_sum->real^M) x`; `(fs:real^P->real^(1,P)finite_sum->real^M) z`; `interval[vec 0:real^1,vec 1] PCROSS {y:real^P}`; `pastecart (vec 0:real^1) (y:real^P)`; `pastecart (t:real^1) (y:real^P)`] o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] COVERING_SPACE_LIFT_UNIQUE)) THEN DISCH_THEN MATCH_MP_TAC THEN ASM_SIMP_TAC[PASTECART_IN_PCROSS; IN_SING; ENDS_IN_UNIT_INTERVAL] THEN SIMP_TAC[REWRITE_RULE[GSYM PCROSS] CONNECTED_PCROSS; CONNECTED_INTERVAL; CONNECTED_SING] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[FORALL_PASTECART; SUBSET; PASTECART_IN_PCROSS] THEN ASM_SIMP_TAC[IN_SING]; ALL_TAC] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS)) THEN MATCH_MP_TAC IMAGE_SUBSET THEN REWRITE_TAC[FORALL_PASTECART; SUBSET; PASTECART_IN_PCROSS] THEN ASM_SIMP_TAC[IN_SING]; ALL_TAC] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r /\ s <=> (p /\ q /\ r) /\ s`] THEN CONJ_TAC THENL [REMOVE_THEN "*" (MP_TAC o SPEC `x:real^P`); REMOVE_THEN "*" (MP_TAC o SPEC `z:real^P`)] THEN ASM_REWRITE_TAC[FORALL_IN_GSPEC; SUBSET; FORALL_IN_IMAGE] THEN ASM_SIMP_TAC[FORALL_PASTECART; PASTECART_IN_PCROSS; IN_SING] THEN STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[FORALL_PASTECART; SUBSET; PASTECART_IN_PCROSS] THEN ASM_SIMP_TAC[IN_SING]]] THEN X_GEN_TAC `y:real^P` THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o last o CONJUNCTS o GEN_REWRITE_RULE I [covering_space]) THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `uu:real^N->real^N->bool` THEN DISCH_TAC THEN SUBGOAL_THEN `!t. t IN interval[vec 0,vec 1] ==> ?k n i:real^N. open_in (subtopology euclidean (interval[vec 0,vec 1])) k /\ open_in (subtopology euclidean u) n /\ t IN k /\ y IN n /\ i IN s /\ IMAGE (h:real^(1,P)finite_sum->real^N) (k PCROSS n) SUBSET uu i` MP_TAC THENL [X_GEN_TAC `t:real^1` THEN DISCH_TAC THEN SUBGOAL_THEN `(h:real^(1,P)finite_sum->real^N) (pastecart t y) IN s` ASSUME_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o ONCE_REWRITE_RULE[FORALL_IN_IMAGE] o GEN_REWRITE_RULE I [SUBSET]) THEN ASM_REWRITE_TAC[PASTECART_IN_PCROSS]; ALL_TAC] THEN SUBGOAL_THEN `open_in (subtopology euclidean (interval[vec 0,vec 1] PCROSS u)) {z | z IN (interval[vec 0,vec 1] PCROSS u) /\ (h:real^(1,P)finite_sum->real^N) z IN uu(h(pastecart t y))}` MP_TAC THENL [MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE_GEN THEN EXISTS_TAC `s:real^N->bool` THEN ASM_SIMP_TAC[]; ALL_TAC] THEN DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] PASTECART_IN_INTERIOR_SUBTOPOLOGY)) THEN DISCH_THEN(MP_TAC o SPECL [`t:real^1`; `y:real^P`]) THEN ASM_SIMP_TAC[IN_ELIM_THM; PASTECART_IN_PCROSS] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:real^1->bool` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:real^P->bool` THEN STRIP_TAC THEN EXISTS_TAC `(h:real^(1,P)finite_sum->real^N) (pastecart t y)` THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; ALL_TAC] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [OPEN_IN_OPEN] THEN REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN REWRITE_TAC[MESON[] `(?x y. (P y /\ x = f y) /\ Q x) <=> ?y. P y /\ Q(f y)`] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`kk:real^1->real^1->bool`; `nn:real^1->real^P->bool`; `xx:real^1->real^N`] THEN DISCH_THEN(LABEL_TAC "+") THEN MP_TAC(ISPEC `interval[vec 0:real^1,vec 1] PCROSS {y:real^P}` COMPACT_IMP_HEINE_BOREL) THEN SIMP_TAC[COMPACT_PCROSS; COMPACT_INTERVAL; COMPACT_SING] THEN DISCH_THEN(MP_TAC o SPEC `IMAGE ((\i. kk i PCROSS nn i):real^1->real^(1,P)finite_sum->bool) (interval[vec 0,vec 1])`) THEN ASM_SIMP_TAC[FORALL_IN_IMAGE; OPEN_PCROSS] THEN ANTS_TAC THENL [REWRITE_TAC[SUBSET; FORALL_IN_PCROSS; IN_SING] THEN MAP_EVERY X_GEN_TAC [`t:real^1`; `z:real^P`] THEN STRIP_TAC THEN ASM_REWRITE_TAC[UNIONS_IMAGE; IN_ELIM_THM; PASTECART_IN_PCROSS] THEN ASM_MESON_TAC[IN_INTER]; GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN REWRITE_TAC[EXISTS_FINITE_SUBSET_IMAGE] THEN DISCH_THEN(X_CHOOSE_THEN `tk:real^1->bool` STRIP_ASSUME_TAC)] THEN ABBREV_TAC `n = INTERS (IMAGE (nn:real^1->real^P->bool) tk)` THEN SUBGOAL_THEN `(y:real^P) IN n /\ open n` STRIP_ASSUME_TAC THENL [EXPAND_TAC "n" THEN CONJ_TAC THENL [REWRITE_TAC[INTERS_IMAGE; IN_ELIM_THM]; MATCH_MP_TAC OPEN_INTERS THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN ASM_SIMP_TAC[FINITE_IMAGE]] THEN X_GEN_TAC `t:real^1` THEN DISCH_TAC THEN REMOVE_THEN "+" (MP_TAC o SPEC `t:real^1`) THEN (ANTS_TAC THENL [ASM SET_TAC[]; SIMP_TAC[IN_INTER]]); ALL_TAC] THEN MP_TAC(ISPECL [`interval[vec 0:real^1,vec 1]`; `IMAGE (kk:real^1->real^1->bool) tk`] LEBESGUE_COVERING_LEMMA) THEN REWRITE_TAC[COMPACT_INTERVAL; FORALL_IN_IMAGE; IMAGE_EQ_EMPTY] THEN MATCH_MP_TAC(TAUT `q /\ (p ==> ~q) /\ (q ==> (r ==> s) ==> t) ==> (~p /\ q /\ r ==> s) ==> t`) THEN SIMP_TAC[UNIONS_0; IMAGE_CLAUSES; SUBSET_EMPTY; UNIT_INTERVAL_NONEMPTY] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [UNIONS_IMAGE]) THEN REWRITE_TAC[SUBSET; FORALL_IN_PCROSS; IMP_CONJ; IN_SING] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM; FORALL_UNWIND_THM2] THEN REWRITE_TAC[UNIONS_IMAGE; IN_ELIM_THM; PASTECART_IN_PCROSS] THEN MESON_TAC[]; DISCH_TAC] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN MP_TAC(ISPEC `d:real` REAL_ARCH_INV) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `N:num` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `!n. n <= N ==> ?v k:real^(1,P)finite_sum->real^M. open_in (subtopology euclidean u) v /\ y IN v /\ k continuous_on interval[vec 0,lift(&n / &N)] PCROSS v /\ IMAGE k (interval[vec 0,lift(&n / &N)] PCROSS v) SUBSET c /\ (!y. y IN v ==> k (pastecart (vec 0) y) = f y) /\ (!z. z IN interval[vec 0,lift(&n / &N)] PCROSS v ==> h z:real^N = p (k z))` MP_TAC THENL [ALL_TAC; DISCH_THEN(MP_TAC o SPEC `N:num`) THEN REWRITE_TAC[LE_REFL] THEN ASM_SIMP_TAC[REAL_DIV_REFL; REAL_OF_NUM_EQ; LIFT_NUM]] THEN MATCH_MP_TAC num_INDUCTION THEN CONJ_TAC THENL [DISCH_TAC THEN REWRITE_TAC[real_div; REAL_MUL_LZERO; LIFT_NUM] THEN EXISTS_TAC `u:real^P->bool` THEN EXISTS_TAC `(f o sndcart):real^(1,P)finite_sum->real^M` THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_PCROSS; INTERVAL_SING] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; IN_SING; o_THM] THEN ASM_REWRITE_TAC[FORALL_UNWIND_THM2; SNDCART_PASTECART] THEN REWRITE_TAC[OPEN_IN_SUBTOPOLOGY_REFL; TOPSPACE_EUCLIDEAN; SUBSET_UNIV] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_SNDCART] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_PCROSS] THEN SIMP_TAC[SNDCART_PASTECART]; ALL_TAC] THEN X_GEN_TAC `m:num` THEN ASM_CASES_TAC `SUC m <= N` THEN ASM_SIMP_TAC[ARITH_RULE `SUC m <= N ==> m <= N`; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`v:real^P->bool`; `k:real^(1,P)finite_sum->real^M`] THEN STRIP_TAC THEN FIRST_X_ASSUM (MP_TAC o SPEC `interval[lift(&m / &N),lift(&(SUC m) / &N)]`) THEN ANTS_TAC THENL [REWRITE_TAC[DIAMETER_INTERVAL; SUBSET_INTERVAL_1] THEN REWRITE_TAC[LIFT_DROP; DROP_VEC; INTERVAL_EQ_EMPTY_1; GSYM LIFT_SUB; NORM_LIFT] THEN ASM_SIMP_TAC[REAL_LT_DIV2_EQ; REAL_LE_DIV2_EQ; REAL_OF_NUM_LT; LE_1; REAL_FIELD `&0 < x ==> a / x - b / x = (a - b) / x`] THEN SIMP_TAC[GSYM NOT_LE; ARITH_RULE `m <= SUC m`; REAL_OF_NUM_SUB] THEN ASM_SIMP_TAC[REAL_ABS_DIV; REAL_ABS_NUM; REAL_LE_DIV; REAL_POS; REAL_ABS_NUM; ARITH_RULE `SUC m - m = 1`] THEN ASM_SIMP_TAC[REAL_ARITH `&1 / n = inv(n)`; REAL_LT_IMP_LE] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_OF_NUM_LT; LE_1] THEN ASM_REWRITE_TAC[REAL_MUL_LID; REAL_OF_NUM_LE] THEN ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[EXISTS_IN_IMAGE] THEN DISCH_THEN(X_CHOOSE_THEN `t:real^1` STRIP_ASSUME_TAC) THEN REMOVE_THEN "+" (MP_TAC o SPEC `t:real^1`) THEN ANTS_TAC THENL [ASM SET_TAC[]; STRIP_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPEC `(xx:real^1->real^N) t`) THEN ASM_REWRITE_TAC[] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(X_CHOOSE_THEN `vv:(real^M->bool)->bool` MP_TAC) THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN GEN_REWRITE_TAC LAND_CONV [EXTENSION] THEN DISCH_THEN(MP_TAC o SPEC `(k:real^(1,P)finite_sum->real^M) (pastecart (lift(&m / &N)) y)`) THEN REWRITE_TAC[IN_ELIM_THM] THEN MATCH_MP_TAC(TAUT `q /\ (p ==> r) ==> (p <=> q) ==> r`) THEN REPEAT(FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [IN_INTER])) THEN SUBGOAL_THEN `lift(&m / &N) IN interval[vec 0,lift (&m / &N)] /\ lift(&m / &N) IN interval[lift(&m / &N),lift(&(SUC m) / &N)]` STRIP_ASSUME_TAC THENL [REWRITE_TAC[IN_INTERVAL_1; LIFT_DROP; DROP_VEC] THEN SIMP_TAC[REAL_LE_DIV; REAL_POS; REAL_LE_REFL] THEN ASM_SIMP_TAC[REAL_LE_DIV2_EQ; LE_1; REAL_OF_NUM_LT; REAL_OF_NUM_LE] THEN ARITH_TAC; ALL_TAC] THEN REPEAT CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN MATCH_MP_TAC FUN_IN_IMAGE THEN ASM_REWRITE_TAC[PASTECART_IN_PCROSS]; FIRST_X_ASSUM(MP_TAC o SPEC `pastecart(lift(&m / &N)) (y:real^P)`) THEN ASM_REWRITE_TAC[PASTECART_IN_PCROSS] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE h s SUBSET t ==> x IN s ==> h x IN t`)) THEN ASM_REWRITE_TAC[PASTECART_IN_PCROSS; IN_INTER] THEN ASM_SIMP_TAC[IN_INTERVAL_1; LIFT_DROP; REAL_LE_DIV; REAL_LE_LDIV_EQ; REAL_POS; REAL_OF_NUM_LT; LE_1; DROP_VEC] THEN REWRITE_TAC[REAL_MUL_LID; REAL_OF_NUM_LE] THEN CONJ_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN ASM_REWRITE_TAC[]; GEN_REWRITE_TAC LAND_CONV [IN_UNIONS] THEN DISCH_THEN(X_CHOOSE_THEN `w:real^M->bool` STRIP_ASSUME_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 (MP_TAC o SPEC `w:real^M->bool`) MP_TAC) THEN DISCH_THEN(MP_TAC o SPEC `w:real^M->bool` o CONJUNCT2) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_TAC `p':real^N->real^M`) THEN DISCH_TAC THEN UNDISCH_THEN `(w:real^M->bool) IN vv` (K ALL_TAC)] THEN ABBREV_TAC `w' = (uu:real^N->real^N->bool)(xx(t:real^1))` THEN SUBGOAL_THEN `?n'. open_in (subtopology euclidean u) n' /\ y IN n' /\ IMAGE (k:real^(1,P)finite_sum->real^M) ({lift(&m / &N)} PCROSS n') SUBSET w` STRIP_ASSUME_TAC THENL [EXISTS_TAC `{z | z IN v /\ ((k:real^(1,P)finite_sum->real^M) o pastecart (lift(&m / &N))) z IN w}` THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_PCROSS] THEN ASM_SIMP_TAC[IN_ELIM_THM; IN_SING; o_THM] THEN MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `v:real^P->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE_GEN THEN EXISTS_TAC `c:real^M->bool` THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[GSYM o_DEF] THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_PASTECART; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)); REWRITE_TAC[IMAGE_o] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE k s SUBSET c ==> t SUBSET s ==> IMAGE k t SUBSET c`))] THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE; PASTECART_IN_PCROSS]; ALL_TAC] THEN SUBGOAL_THEN `?q q':real^P->bool. open_in (subtopology euclidean u) q /\ closed_in (subtopology euclidean u) q' /\ y IN q /\ y IN q' /\ q SUBSET q' /\ q SUBSET (u INTER nn(t:real^1)) INTER n' INTER v /\ q' SUBSET (u INTER nn(t:real^1)) INTER n' INTER v` STRIP_ASSUME_TAC THENL [REWRITE_TAC[SET_RULE `y IN q /\ y IN q' /\ q SUBSET q' /\ q SUBSET s /\ q' SUBSET s <=> y IN q /\ q SUBSET q' /\ q' SUBSET s`] THEN UNDISCH_TAC `open_in (subtopology euclidean u) (v:real^P->bool)` THEN UNDISCH_TAC `open_in (subtopology euclidean u) (n':real^P->bool)` THEN REWRITE_TAC[OPEN_IN_OPEN] THEN DISCH_THEN(X_CHOOSE_THEN `vo:real^P->bool` STRIP_ASSUME_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `vx:real^P->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPEC `nn(t:real^1) INTER vo INTER vx:real^P->bool` OPEN_CONTAINS_CBALL) THEN ASM_SIMP_TAC[OPEN_INTER] THEN DISCH_THEN(MP_TAC o SPEC `y:real^P`) THEN ASM_REWRITE_TAC[IN_INTER] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `u INTER ball(y:real^P,e)` THEN EXISTS_TAC `u INTER cball(y:real^P,e)` THEN REWRITE_TAC[CLOSED_IN_CLOSED] THEN CONJ_TAC THENL [MESON_TAC[OPEN_BALL]; ALL_TAC] THEN CONJ_TAC THENL [MESON_TAC[CLOSED_CBALL]; ALL_TAC] THEN ASM_REWRITE_TAC[IN_INTER; CENTRE_IN_BALL] THEN MP_TAC(ISPECL [`y:real^P`; `e:real`] BALL_SUBSET_CBALL) THEN ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [homeomorphism]) THEN EXISTS_TAC `q:real^P->bool` THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPECL [`\x:real^(1,P)finite_sum. x IN interval[vec 0,lift(&m / &N)] PCROSS (q':real^P->bool)`; `k:real^(1,P)finite_sum->real^M`; `(p':real^N->real^M) o (h:real^(1,P)finite_sum->real^N)`; `interval[vec 0,lift(&m / &N)] PCROSS (q':real^P->bool)`; `interval[lift(&m / &N),lift(&(SUC m) / &N)] PCROSS (q':real^P->bool)`] CONTINUOUS_ON_CASES_LOCAL) THEN REWRITE_TAC[TAUT `~(p /\ ~p)`] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [REWRITE_TAC[CLOSED_IN_CLOSED] THEN EXISTS_TAC `interval[vec 0,lift(&m / &N)] PCROSS (:real^P)` THEN SIMP_TAC[CLOSED_PCROSS; CLOSED_INTERVAL; CLOSED_UNIV] THEN REWRITE_TAC[EXTENSION; IN_INTER; IN_UNION; FORALL_PASTECART] THEN REWRITE_TAC[PASTECART_IN_PCROSS; IN_UNIV] THEN CONV_TAC TAUT; REWRITE_TAC[CLOSED_IN_CLOSED] THEN EXISTS_TAC `interval[lift(&m / &N),lift(&(SUC m) / &N)] PCROSS (:real^P)` THEN SIMP_TAC[CLOSED_PCROSS; CLOSED_INTERVAL; CLOSED_UNIV] THEN REWRITE_TAC[EXTENSION; IN_INTER; IN_UNION; FORALL_PASTECART] THEN REWRITE_TAC[PASTECART_IN_PCROSS; IN_UNIV] THEN CONV_TAC TAUT; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_PASTECART; PASTECART_IN_PCROSS] THEN ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THENL [ALL_TAC; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE k s SUBSET c ==> t SUBSET s ==> IMAGE k t SUBSET c`))] THEN MATCH_MP_TAC PCROSS_MONO THEN (CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]]) THEN ASM_REWRITE_TAC[SUBSET_INTERVAL_1; LIFT_DROP; DROP_VEC; SUBSET_INTER] THEN REWRITE_TAC[SUBSET_INTERVAL_1; LIFT_DROP; DROP_VEC] THEN ASM_SIMP_TAC[REAL_LE_MUL; REAL_POS; REAL_LE_DIV2_EQ; REAL_OF_NUM_LT; LE_1] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ; REAL_OF_NUM_LT; LE_1; REAL_MUL_LZERO; REAL_MUL_LID; REAL_OF_NUM_LE] THEN DISJ2_TAC THEN ARITH_TAC; REWRITE_TAC[FORALL_PASTECART; PASTECART_IN_PCROSS] THEN MAP_EVERY X_GEN_TAC [`r:real^1`; `z:real^P`] THEN ASM_CASES_TAC `(z:real^P) IN q'` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `(b <= x /\ x <= c) /\ (a <= x /\ x <= b) ==> x = b`)) THEN REWRITE_TAC[DROP_EQ; o_THM] THEN DISCH_THEN SUBST1_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (MESON[] `(!x. x IN w ==> p' (p x) = x) ==> h z = p(k z) /\ k z IN w ==> k z = p' (h z)`)) THEN CONJ_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[PASTECART_IN_PCROSS] THEN ASM SET_TAC[]; FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN MATCH_MP_TAC FUN_IN_IMAGE THEN REWRITE_TAC[PASTECART_IN_PCROSS; IN_SING] THEN ASM SET_TAC[]]]; SUBGOAL_THEN `interval[vec 0,lift(&m / &N)] UNION interval [lift(&m / &N),lift(&(SUC m) / &N)] = interval[vec 0,lift(&(SUC m) / &N)]` ASSUME_TAC THENL [REWRITE_TAC[EXTENSION; IN_UNION; IN_INTERVAL_1] THEN GEN_TAC THEN MATCH_MP_TAC(REAL_ARITH `a <= b /\ b <= c ==> (a <= x /\ x <= b \/ b <= x /\ x <= c <=> a <= x /\ x <= c)`) THEN SIMP_TAC[LIFT_DROP; DROP_VEC; REAL_LE_DIV; REAL_POS] THEN ASM_SIMP_TAC[REAL_LE_DIV2_EQ; REAL_OF_NUM_LT; REAL_OF_NUM_LE; LE_1] THEN ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `interval[vec 0,lift(&m / &N)] PCROSS (q':real^P->bool) UNION interval [lift(&m / &N),lift(&(SUC m) / &N)] PCROSS q' = interval[vec 0,lift(&(SUC m) / &N)] PCROSS q'` SUBST1_TAC THENL [SIMP_TAC[EXTENSION; IN_UNION; FORALL_PASTECART; PASTECART_IN_PCROSS] THEN ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(MESON[CONTINUOUS_ON_SUBSET] `t SUBSET s /\ (f continuous_on s ==> P f) ==> f continuous_on s ==> ?g. g continuous_on t /\ P g`) THEN ASM_SIMP_TAC[PCROSS_MONO; SUBSET_REFL] THEN DISCH_TAC THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_PCROSS] THEN MAP_EVERY X_GEN_TAC [`r:real^1`; `z:real^P`] THEN STRIP_TAC THEN SUBGOAL_THEN `(z:real^P) IN q'` ASSUME_TAC THENL [ASM SET_TAC[]; ASM_REWRITE_TAC[PASTECART_IN_PCROSS]] THEN COND_CASES_TAC THEN REWRITE_TAC[o_THM] THENL [FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN MATCH_MP_TAC FUN_IN_IMAGE THEN REWRITE_TAC[PASTECART_IN_PCROSS; IN_SING] THEN ASM SET_TAC[]; FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET] o CONJUNCT1 o GEN_REWRITE_RULE I [open_in]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE p w' = w ==> x IN w' ==> p x IN w`))]; X_GEN_TAC `z:real^P` THEN REWRITE_TAC[PASTECART_IN_PCROSS] THEN DISCH_TAC THEN REWRITE_TAC[IN_INTERVAL_1; REAL_LE_REFL] THEN SUBGOAL_THEN `(z:real^P) IN q'` ASSUME_TAC THENL [ASM SET_TAC[]; ASM_REWRITE_TAC[LIFT_DROP; DROP_VEC]] THEN SIMP_TAC[REAL_LE_DIV; REAL_POS] THEN ASM SET_TAC[]; REWRITE_TAC[FORALL_PASTECART; PASTECART_IN_PCROSS] THEN MAP_EVERY X_GEN_TAC [`r:real^1`; `z:real^P`] THEN STRIP_TAC THEN SUBGOAL_THEN `(z:real^P) IN q'` ASSUME_TAC THENL [ASM SET_TAC[]; ASM_REWRITE_TAC[]] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[PASTECART_IN_PCROSS] THEN ASM SET_TAC[]; REWRITE_TAC[o_THM] THEN CONV_TAC SYM_CONV THEN FIRST_X_ASSUM MATCH_MP_TAC]] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE h s SUBSET t ==> x IN s ==> h x IN t`)) THEN ASM_REWRITE_TAC[PASTECART_IN_PCROSS; IN_INTER] THEN REPEAT(CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]]) THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN REWRITE_TAC[IN_INTERVAL_1] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `a <= x /\ x <= b ==> b <= c ==> a <= x /\ x <= c`)) THEN ASM_SIMP_TAC[LIFT_DROP; REAL_LE_LDIV_EQ; REAL_OF_NUM_LT; LE_1] THEN ASM_REWRITE_TAC[DROP_VEC; REAL_MUL_LID; REAL_OF_NUM_LE]]);; let COVERING_SPACE_LIFT_HOMOTOPIC_FUNCTION = prove (`!p:real^M->real^N c s f f' g u:real^P->bool. covering_space (c,p) s /\ g continuous_on u /\ IMAGE g u SUBSET c /\ (!y. y IN u ==> p(g y) = f y) /\ homotopic_with (\x. T) (subtopology euclidean u,subtopology euclidean s) f f' ==> ?g'. g' continuous_on u /\ IMAGE g' u SUBSET c /\ (!y. y IN u ==> p(g' y) = f' y)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(X_CHOOSE_THEN `h:real^(1,P)finite_sum->real^N` STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [HOMOTOPIC_WITH_EUCLIDEAN]) THEN FIRST_ASSUM(MP_TAC o ISPECL [`h:real^(1,P)finite_sum->real^N`; `g:real^P->real^M`; `u:real^P->bool`] o MATCH_MP(ONCE_REWRITE_RULE[IMP_CONJ] COVERING_SPACE_LIFT_HOMOTOPY)) THEN ASM_SIMP_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `k:real^(1,P)finite_sum->real^M` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(k:real^(1,P)finite_sum->real^M) o (\x. pastecart (vec 1) x)` THEN ASM_REWRITE_TAC[IMAGE_o; o_THM] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_PASTECART; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; PASTECART_IN_PCROSS; ENDS_IN_UNIT_INTERVAL]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE k s SUBSET c ==> t SUBSET s ==> IMAGE k t SUBSET c`)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; PASTECART_IN_PCROSS; ENDS_IN_UNIT_INTERVAL]; ASM_MESON_TAC[PASTECART_IN_PCROSS; ENDS_IN_UNIT_INTERVAL]]);; let COVERING_SPACE_LIFT_INESSENTIAL_FUNCTION = prove (`!p:real^M->real^N c s f a u:real^P->bool. covering_space (c,p) s /\ homotopic_with (\x. T) (subtopology euclidean u,subtopology euclidean s) f (\x. a) ==> ?g. g continuous_on u /\ IMAGE g u SUBSET c /\ (!y. y IN u ==> p(g y) = f y)`, ONCE_REWRITE_TAC[HOMOTOPIC_WITH_SYM] THEN REPEAT GEN_TAC THEN DISCH_TAC THEN ASM_CASES_TAC `u:real^P->bool = {}` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY; IMAGE_CLAUSES; EMPTY_SUBSET; CONTINUOUS_ON_EMPTY] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE [TAUT `a /\ b /\ c /\ d /\ e ==> f <=> a /\ e ==> b /\ c /\ d ==> f`] COVERING_SPACE_LIFT_HOMOTOPIC_FUNCTION)) THEN FIRST_X_ASSUM(CONJUNCTS_THEN ASSUME_TAC) THEN FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_SUBSET) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP COVERING_SPACE_IMP_SURJECTIVE) THEN SUBGOAL_THEN `?b. b IN c /\ (p:real^M->real^N) b = a` CHOOSE_TAC THENL [ASM SET_TAC[]; EXISTS_TAC `(\x. b):real^P->real^M`] THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN ASM SET_TAC[]);; let COVERING_SPACE_LIFT_HOMOTOPY_ALT = prove (`!p:real^M->real^N c s (h:real^(P,1)finite_sum->real^N) f u. covering_space (c,p) s /\ h continuous_on (u PCROSS interval[vec 0,vec 1]) /\ IMAGE h (u PCROSS interval[vec 0,vec 1]) SUBSET s /\ (!y. y IN u ==> h (pastecart y (vec 0)) = p(f y)) /\ f continuous_on u /\ IMAGE f u SUBSET c ==> ?k. k continuous_on (u PCROSS interval[vec 0,vec 1]) /\ IMAGE k (u PCROSS interval[vec 0,vec 1]) SUBSET c /\ (!y. y IN u ==> k(pastecart y (vec 0)) = f y) /\ (!z. z IN u PCROSS interval[vec 0,vec 1] ==> h z = p(k z))`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o ISPECL [`(h:real^(P,1)finite_sum->real^N) o (\z. pastecart (sndcart z) (fstcart z))`; `f:real^P->real^M`; `u:real^P->bool`] o MATCH_MP(ONCE_REWRITE_RULE[IMP_CONJ] COVERING_SPACE_LIFT_HOMOTOPY)) THEN ASM_SIMP_TAC[o_THM; FSTCART_PASTECART; SNDCART_PASTECART] THEN ANTS_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_PASTECART; LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; LINEAR_SNDCART] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)); REWRITE_TAC[IMAGE_o] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE k s SUBSET c ==> t SUBSET s ==> IMAGE k t SUBSET c`))] THEN SIMP_TAC[SUBSET; FORALL_IN_IMAGE; PASTECART_IN_PCROSS; FORALL_IN_PCROSS; FSTCART_PASTECART; SNDCART_PASTECART]; DISCH_THEN(X_CHOOSE_THEN `k:real^(1,P)finite_sum->real^M` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(k:real^(1,P)finite_sum->real^M) o (\z. pastecart (sndcart z) (fstcart z))` THEN ASM_SIMP_TAC[o_THM; FSTCART_PASTECART; SNDCART_PASTECART; FORALL_IN_PCROSS; PASTECART_IN_PCROSS] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_PASTECART; LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; LINEAR_SNDCART] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)); REWRITE_TAC[IMAGE_o] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE k s SUBSET c ==> t SUBSET s ==> IMAGE k t SUBSET c`)); MAP_EVERY X_GEN_TAC [`x:real^P`; `t:real^1`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `pastecart (t:real^1) (x:real^P)`)] THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE; PASTECART_IN_PCROSS; FSTCART_PASTECART; SNDCART_PASTECART; FORALL_IN_PCROSS]]);; let COVERING_SPACE_LIFT_PATH_STRONG = prove (`!p:real^M->real^N c s g a. covering_space (c,p) s /\ path g /\ path_image g SUBSET s /\ pathstart g = p(a) /\ a IN c ==> ?h. path h /\ path_image h SUBSET c /\ pathstart h = a /\ !t. t IN interval[vec 0,vec 1] ==> p(h t) = g t`, REWRITE_TAC[path_image; path; pathstart] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o ISPECL [`(g:real^1->real^N) o (fstcart:real^(1,P)finite_sum->real^1)`; `(\y. a):real^P->real^M`; `{arb:real^P}`] o MATCH_MP(ONCE_REWRITE_RULE[IMP_CONJ] COVERING_SPACE_LIFT_HOMOTOPY)) THEN REWRITE_TAC[FORALL_IN_INSERT; NOT_IN_EMPTY; o_THM; FSTCART_PASTECART] THEN ANTS_TAC THENL [ASM_REWRITE_TAC[IMAGE_o; CONTINUOUS_ON_CONST] THEN ASM_REWRITE_TAC[SET_RULE `IMAGE (\y. a) {b} SUBSET s <=> a IN s`] THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[LINEAR_FSTCART; LINEAR_CONTINUOUS_ON] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)); ALL_TAC] THEN ASM_REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_PCROSS] THEN SIMP_TAC[FSTCART_PASTECART] THEN ASM SET_TAC[]; DISCH_THEN(X_CHOOSE_THEN `k:real^(1,P)finite_sum->real^M` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(k:real^(1,P)finite_sum->real^M) o (\t. pastecart t arb)` THEN ASM_REWRITE_TAC[o_THM; IMAGE_o] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_PASTECART; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN SIMP_TAC[SUBSET; FORALL_IN_IMAGE; PASTECART_IN_PCROSS; IN_SING]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE k s SUBSET c ==> t SUBSET s ==> IMAGE k t SUBSET c`)) THEN SIMP_TAC[SUBSET; FORALL_IN_IMAGE; PASTECART_IN_PCROSS; IN_SING]; X_GEN_TAC `t:real^1` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `pastecart (t:real^1) (arb:real^P)`) THEN ASM_SIMP_TAC[PASTECART_IN_PCROSS; FSTCART_PASTECART; IN_SING]]]);; let COVERING_SPACE_LIFT_PATH = prove (`!p:real^M->real^N c s g. covering_space (c,p) s /\ path g /\ path_image g SUBSET s ==> ?h. path h /\ path_image h SUBSET c /\ !t. t IN interval[vec 0,vec 1] ==> p(h t) = g t`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (SET_RULE `IMAGE g i SUBSET s ==> vec 0 IN i ==> g(vec 0) IN s`) o GEN_REWRITE_RULE LAND_CONV [path_image]) THEN REWRITE_TAC[ENDS_IN_UNIT_INTERVAL] THEN FIRST_ASSUM(SUBST1_TAC o SYM o MATCH_MP COVERING_SPACE_IMP_SURJECTIVE) THEN REWRITE_TAC[IN_IMAGE; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `a:real^M` THEN STRIP_TAC THEN MP_TAC(ISPECL [`p:real^M->real^N`; `c:real^M->bool`; `s:real^N->bool`; `g:real^1->real^N`; `a:real^M`] COVERING_SPACE_LIFT_PATH_STRONG) THEN ASM_REWRITE_TAC[pathstart] THEN MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[]);; let COVERING_SPACE_LIFT_HOMOTOPIC_PATHS = prove (`!p:real^M->real^N c s g1 g2 h1 h2. covering_space (c,p) s /\ path g1 /\ path_image g1 SUBSET s /\ path g2 /\ path_image g2 SUBSET s /\ homotopic_paths s g1 g2 /\ path h1 /\ path_image h1 SUBSET c /\ (!t. t IN interval[vec 0,vec 1] ==> p(h1 t) = g1 t) /\ path h2 /\ path_image h2 SUBSET c /\ (!t. t IN interval[vec 0,vec 1] ==> p(h2 t) = g2 t) /\ pathstart h1 = pathstart h2 ==> homotopic_paths c h1 h2`, REPEAT STRIP_TAC THEN REWRITE_TAC[HOMOTOPIC_PATHS] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homotopic_paths]) THEN REWRITE_TAC[HOMOTOPIC_WITH_EUCLIDEAN; pathstart; pathfinish] THEN DISCH_THEN(X_CHOOSE_THEN `h:real^(1,1)finite_sum->real^N` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o ISPECL [`h:real^(1,1)finite_sum->real^N`; `(\x. pathstart h2):real^1->real^M`; `interval[vec 0:real^1,vec 1]`] o MATCH_MP(ONCE_REWRITE_RULE[IMP_CONJ] COVERING_SPACE_LIFT_HOMOTOPY_ALT)) THEN ASM_SIMP_TAC[] THEN ANTS_TAC THENL [REWRITE_TAC[CONTINUOUS_ON_CONST; SUBSET; FORALL_IN_IMAGE] THEN ASM_MESON_TAC[pathstart; ENDS_IN_UNIT_INTERVAL; PATHSTART_IN_PATH_IMAGE; SUBSET]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:real^(1,1)finite_sum->real^M` THEN STRIP_TAC THEN ASM_SIMP_TAC[o_DEF] THEN MATCH_MP_TAC(TAUT `(p /\ q) /\ (p /\ q ==> r) ==> p /\ q /\ r`) THEN CONJ_TAC THENL [CONJ_TAC THEN FIRST_ASSUM(MATCH_MP_TAC o REWRITE_RULE[RIGHT_FORALL_IMP_THM] o ONCE_REWRITE_RULE[IMP_CONJ] o REWRITE_RULE[CONJ_ASSOC] o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] COVERING_SPACE_LIFT_UNIQUE)) THEN REWRITE_TAC[GSYM CONJ_ASSOC] THENL [MAP_EVERY EXISTS_TAC [`g1:real^1->real^N`; `vec 0:real^1`]; MAP_EVERY EXISTS_TAC [`g2:real^1->real^N`; `vec 0:real^1`]] THEN ASM_SIMP_TAC[ENDS_IN_UNIT_INTERVAL; UNIT_INTERVAL_NONEMPTY] THEN RULE_ASSUM_TAC(REWRITE_RULE[path_image; pathstart; pathfinish; path]) THEN ASM_REWRITE_TAC[CONNECTED_INTERVAL; pathstart; pathfinish] THEN REWRITE_TAC[CONJ_ASSOC] THEN (REPEAT CONJ_TAC THENL [GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_PASTECART; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)); GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM o_DEF] THEN REWRITE_TAC[IMAGE_o] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE k s SUBSET c ==> t SUBSET s ==> IMAGE k t SUBSET c`)); ASM_MESON_TAC[PASTECART_IN_PCROSS; ENDS_IN_UNIT_INTERVAL]] THEN SIMP_TAC[SUBSET; FORALL_IN_IMAGE; PASTECART_IN_PCROSS; FORALL_IN_PCROSS; FSTCART_PASTECART; SNDCART_PASTECART; ENDS_IN_UNIT_INTERVAL]); STRIP_TAC THEN REWRITE_TAC[TAUT `p ==> q /\ r <=> (p ==> q) /\ (p ==> r)`] THEN REWRITE_TAC[FORALL_AND_THM] THEN CONJ_TAC THENL [ASM_MESON_TAC[pathstart; ENDS_IN_UNIT_INTERVAL]; ALL_TAC] THEN FIRST_ASSUM(MATCH_MP_TAC o REWRITE_RULE[RIGHT_FORALL_IMP_THM] o ONCE_REWRITE_RULE[IMP_CONJ] o REWRITE_RULE[CONJ_ASSOC] o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] COVERING_SPACE_LIFT_UNIQUE)) THEN MAP_EVERY EXISTS_TAC [`(\x. pathfinish g1):real^1->real^N`; `vec 0:real^1`] THEN ASM_SIMP_TAC[ENDS_IN_UNIT_INTERVAL; CONNECTED_INTERVAL] THEN REWRITE_TAC[CONTINUOUS_ON_CONST; pathfinish] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN ASM_MESON_TAC[SUBSET; pathfinish; PATHFINISH_IN_PATH_IMAGE]; GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_PASTECART; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN SIMP_TAC[SUBSET; FORALL_IN_IMAGE; PASTECART_IN_PCROSS; FORALL_IN_PCROSS; FSTCART_PASTECART; SNDCART_PASTECART; ENDS_IN_UNIT_INTERVAL]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `t:real^1` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `pastecart (t:real^1) (vec 1:real^1)` o REWRITE_RULE[FORALL_IN_IMAGE] o GEN_REWRITE_RULE I [SUBSET]) THEN ASM_REWRITE_TAC[PASTECART_IN_PCROSS; ENDS_IN_UNIT_INTERVAL]; ASM_MESON_TAC[PASTECART_IN_PCROSS; ENDS_IN_UNIT_INTERVAL]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN ASM_MESON_TAC[SUBSET; pathfinish; PATHFINISH_IN_PATH_IMAGE]]]);; let COVERING_SPACE_MONODROMY = prove (`!p:real^M->real^N c s g1 g2 h1 h2. covering_space (c,p) s /\ path g1 /\ path_image g1 SUBSET s /\ path g2 /\ path_image g2 SUBSET s /\ homotopic_paths s g1 g2 /\ path h1 /\ path_image h1 SUBSET c /\ (!t. t IN interval[vec 0,vec 1] ==> p(h1 t) = g1 t) /\ path h2 /\ path_image h2 SUBSET c /\ (!t. t IN interval[vec 0,vec 1] ==> p(h2 t) = g2 t) /\ pathstart h1 = pathstart h2 ==> pathfinish h1 = pathfinish h2`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP COVERING_SPACE_LIFT_HOMOTOPIC_PATHS) THEN REWRITE_TAC[HOMOTOPIC_PATHS_IMP_PATHFINISH]);; let COVERING_SPACE_LIFT_HOMOTOPIC_PATH = prove (`!p:real^M->real^N c s f f' g a b. covering_space (c,p) s /\ homotopic_paths s f f' /\ path g /\ path_image g SUBSET c /\ pathstart g = a /\ pathfinish g = b /\ (!t. t IN interval[vec 0,vec 1] ==> p(g t) = f t) ==> ?g'. path g' /\ path_image g' SUBSET c /\ pathstart g' = a /\ pathfinish g' = b /\ (!t. t IN interval[vec 0,vec 1] ==> p(g' t) = f' t)`, ONCE_REWRITE_TAC[HOMOTOPIC_PATHS_SYM] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_PATH) THEN FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_SUBSET) THEN FIRST_ASSUM(MP_TAC o ISPECL [`f':real^1->real^N`; `a:real^M`] o MATCH_MP(ONCE_REWRITE_RULE[IMP_CONJ] COVERING_SPACE_LIFT_PATH_STRONG)) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[pathstart; ENDS_IN_UNIT_INTERVAL; HOMOTOPIC_PATHS_IMP_PATHSTART]; ASM_MESON_TAC[PATHSTART_IN_PATH_IMAGE; SUBSET]]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g':real^1->real^M` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN SUBST1_TAC(SYM(ASSUME `pathfinish g:real^M = b`)) THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP(ONCE_REWRITE_RULE[IMP_CONJ] COVERING_SPACE_MONODROMY)) THEN MAP_EVERY EXISTS_TAC [`f':real^1->real^N`; `f:real^1->real^N`] THEN ASM_REWRITE_TAC[]]);; let COVERING_SPACE_INESSENTIAL_LOOP_LIFT_IS_LOOP = prove (`!p:real^M->real^N c s g h a. covering_space (c,p) s /\ path g /\ path_image g SUBSET s /\ pathfinish g = pathstart g /\ homotopic_paths s g (linepath(a,a)) /\ path h /\ path_image h SUBSET c /\ (!t. t IN interval[vec 0,vec 1] ==> p(h t) = g t) ==> pathfinish h = pathstart h`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_SUBSET) THEN REWRITE_TAC[PATH_IMAGE_LINEPATH; SEGMENT_REFL; SING_SUBSET] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP COVERING_SPACE_IMP_SURJECTIVE) THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_PATHSTART) THEN REWRITE_TAC[PATHSTART_LINEPATH] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o ISPECL [`g:real^1->real^N`; `linepath(a:real^N,a)`; `h:real^1->real^M`; `linepath(pathstart h:real^M,pathstart h)`] o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] COVERING_SPACE_MONODROMY)) THEN ASM_REWRITE_TAC[PATH_LINEPATH; PATH_IMAGE_LINEPATH; SEGMENT_REFL] THEN ASM_REWRITE_TAC[SING_SUBSET; PATHSTART_LINEPATH; PATHFINISH_LINEPATH] THEN DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[LINEPATH_REFL] THEN CONJ_TAC THENL [ASM_MESON_TAC[PATHSTART_IN_PATH_IMAGE; SUBSET]; REPEAT STRIP_TAC THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [SYM th]) THEN REWRITE_TAC[pathstart] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[ENDS_IN_UNIT_INTERVAL]]);; let COVERING_SPACE_SIMPLY_CONNECTED_LOOP_LIFT_IS_LOOP = prove (`!p:real^M->real^N c s g h. covering_space (c,p) s /\ simply_connected s /\ path g /\ path_image g SUBSET s /\ pathfinish g = pathstart g /\ path h /\ path_image h SUBSET c /\ (!t. t IN interval[vec 0,vec 1] ==> p(h t) = g t) ==> pathfinish h = pathstart h`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] COVERING_SPACE_INESSENTIAL_LOOP_LIFT_IS_LOOP)) THEN EXISTS_TAC `g:real^1->real^N` THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[SIMPLY_CONNECTED_EQ_CONTRACTIBLE_PATH]);; let COVERING_SPACE_HOMOTOPIC_PATHS_CANCEL = prove (`!p:real^M->real^N c s g h. covering_space (c,p) s /\ path g /\ path_image g SUBSET c /\ path h /\ path_image h SUBSET c /\ pathstart g = pathstart h /\ homotopic_paths s (p o g) (p o h) ==> homotopic_paths c g h`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] COVERING_SPACE_LIFT_HOMOTOPIC_PATHS)) THEN MAP_EVERY EXISTS_TAC [`(p:real^M->real^N) o (g:real^1->real^M)`; `(p:real^M->real^N) o (h:real^1->real^M)`] THEN ASM_REWRITE_TAC[o_THM] THEN ASM_MESON_TAC[HOMOTOPIC_PATHS_IMP_PATH; HOMOTOPIC_PATHS_IMP_SUBSET]);; let COVERING_SPACE_HOMOTOPIC_PATHS_CANCEL_EQ = prove (`!p:real^M->real^N c s g h. covering_space (c,p) s /\ path g /\ path_image g SUBSET c /\ path h /\ path_image h SUBSET c /\ pathstart g = pathstart h ==> (homotopic_paths s (p o g) (p o h) <=> homotopic_paths c g h)`, REPEAT STRIP_TAC THEN EQ_TAC THENL [ASM_MESON_TAC[COVERING_SPACE_HOMOTOPIC_PATHS_CANCEL]; DISCH_TAC] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_CONTINUOUS_IMAGE THEN EXISTS_TAC `c:real^M->bool` THEN RULE_ASSUM_TAC(REWRITE_RULE[covering_space]) THEN ASM_REWRITE_TAC[SUBSET_REFL]);; (* ------------------------------------------------------------------------- *) (* Lifting of general functions to covering space *) (* ------------------------------------------------------------------------- *) let COVERING_SPACE_LIFT_GENERAL = prove (`!p:real^M->real^N c s f:real^P->real^N u a z. covering_space (c,p) s /\ a IN c /\ z IN u /\ path_connected u /\ locally path_connected u /\ f continuous_on u /\ IMAGE f u SUBSET s /\ f z = p a /\ (!r. path r /\ path_image r SUBSET u /\ pathstart r = z /\ pathfinish r = z ==> ?q. path q /\ path_image q SUBSET c /\ pathstart q = a /\ pathfinish q = a /\ homotopic_paths s (f o r) (p o q)) ==> ?g. g continuous_on u /\ IMAGE g u SUBSET c /\ g z = a /\ (!y. y IN u ==> p(g y) = f y)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `!y. y IN u ==> ?g h. path g /\ path_image g SUBSET u /\ pathstart g = z /\ pathfinish g = y /\ path h /\ path_image h SUBSET c /\ pathstart h = a /\ (!t. t IN interval[vec 0,vec 1] ==> (p:real^M->real^N)(h t) = (f:real^P->real^N)(g t))` (LABEL_TAC "*") THENL [X_GEN_TAC `y:real^P` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [path_connected]) THEN DISCH_THEN(MP_TAC o SPECL [`z:real^P`; `y:real^P`]) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^1->real^P` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC COVERING_SPACE_LIFT_PATH_STRONG THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[GSYM o_DEF] THEN ASM_REWRITE_TAC[PATH_IMAGE_COMPOSE; PATHSTART_COMPOSE] THEN CONJ_TAC THENL [MATCH_MP_TAC PATH_CONTINUOUS_IMAGE THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ASM SET_TAC[]]; ALL_TAC] THEN SUBGOAL_THEN `?l. !y g h. path g /\ path_image g SUBSET u /\ pathstart g = z /\ pathfinish g = y /\ path h /\ path_image h SUBSET c /\ pathstart h = a /\ (!t. t IN interval[vec 0,vec 1] ==> (p:real^M->real^N)(h t) = (f:real^P->real^N)(g t)) ==> pathfinish h = l y` MP_TAC THENL [REWRITE_TAC[GSYM SKOLEM_THM] THEN X_GEN_TAC `y:real^P` THEN MATCH_MP_TAC(MESON[] `(!g h g' h'. P g h /\ P g' h' ==> f h = f h') ==> ?z. !g h. P g h ==> f h = z`) THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `(g ++ reversepath g'):real^1->real^P`) THEN ASM_SIMP_TAC[PATH_JOIN; PATHSTART_JOIN; PATHFINISH_JOIN; PATH_REVERSEPATH; PATHSTART_REVERSEPATH; PATHFINISH_REVERSEPATH; SUBSET_PATH_IMAGE_JOIN; PATH_IMAGE_REVERSEPATH] THEN DISCH_THEN(X_CHOOSE_THEN `q:real^1->real^M` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o ISPECL [`(p:real^M->real^N) o (q:real^1->real^M)`; `(f:real^P->real^N) o (g ++ reversepath g')`; `q:real^1->real^M`; `pathstart q:real^M`; `pathfinish q:real^M`] o MATCH_MP(ONCE_REWRITE_RULE[IMP_CONJ] (ONCE_REWRITE_RULE[HOMOTOPIC_PATHS_SYM] COVERING_SPACE_LIFT_HOMOTOPIC_PATH))) THEN ASM_REWRITE_TAC[o_THM] THEN DISCH_THEN(X_CHOOSE_THEN `q':real^1->real^M` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `path(h ++ reversepath h':real^1->real^M)` MP_TAC THENL [ALL_TAC; ASM_SIMP_TAC[PATH_JOIN_EQ; PATH_REVERSEPATH; PATHSTART_REVERSEPATH]] THEN MATCH_MP_TAC PATH_EQ THEN EXISTS_TAC `q':real^1->real^M` THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `t:real^1` THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN STRIP_TAC THEN REWRITE_TAC[joinpaths] THEN COND_CASES_TAC THENL [FIRST_ASSUM(MP_TAC o ISPECL [`(f:real^P->real^N) o (g:real^1->real^P) o (\t. &2 % t)`; `q':real^1->real^M`; `(h:real^1->real^M) o (\t. &2 % t)`; `interval[vec 0,lift(&1 / &2)]`; `vec 0:real^1`; `t:real^1`] o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] COVERING_SPACE_LIFT_UNIQUE)) THEN REWRITE_TAC[o_THM] THEN DISCH_THEN MATCH_MP_TAC THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_EQ THEN EXISTS_TAC `(f:real^P->real^N) o (g ++ reversepath g')` THEN CONJ_TAC THENL [SIMP_TAC[IN_INTERVAL_1; LIFT_DROP; joinpaths; o_THM]; ALL_TAC] THEN MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `interval[vec 0:real^1,vec 1]` THEN CONJ_TAC THENL [ASM_MESON_TAC[HOMOTOPIC_PATHS_IMP_PATH; path]; REWRITE_TAC[SUBSET_INTERVAL_1; LIFT_DROP; DROP_VEC] THEN REAL_ARITH_TAC]; MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `path_image ((f:real^P->real^N) o (g ++ reversepath g'))` THEN CONJ_TAC THENL[ALL_TAC; ASM_MESON_TAC[HOMOTOPIC_PATHS_IMP_SUBSET]] THEN REWRITE_TAC[path_image] THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> f x = g x) /\ s SUBSET t ==> IMAGE f s SUBSET IMAGE g t`) THEN REWRITE_TAC[SUBSET_INTERVAL_1; LIFT_DROP; DROP_VEC; IN_INTERVAL_1] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN SIMP_TAC[joinpaths; o_THM]; MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `interval[vec 0:real^1,vec 1]` THEN ASM_REWRITE_TAC[GSYM path] THEN REWRITE_TAC[SUBSET_INTERVAL_1; LIFT_DROP; DROP_VEC] THEN REAL_ARITH_TAC; MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `path_image(q':real^1->real^M)` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[path_image] THEN MATCH_MP_TAC IMAGE_SUBSET THEN REWRITE_TAC[SUBSET_INTERVAL_1; LIFT_DROP; DROP_VEC] THEN REAL_ARITH_TAC; X_GEN_TAC `t':real^1` THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; LIFT_DROP] THEN STRIP_TAC THEN FIRST_X_ASSUM(fun th -> W(MP_TAC o PART_MATCH (lhand o rand) th o rand o snd)) THEN ASM_SIMP_TAC[IN_INTERVAL_1; joinpaths; DROP_VEC] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; SIMP_TAC[]]; MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_CMUL; CONTINUOUS_ON_ID] THEN MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `interval[vec 0:real^1,vec 1]` THEN ASM_SIMP_TAC[GSYM path] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; DROP_CMUL; LIFT_DROP] THEN REAL_ARITH_TAC; MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `path_image(h:real^1->real^M)` THEN CONJ_TAC THENL [ALL_TAC; ASM_SIMP_TAC[]] THEN REWRITE_TAC[path_image; IMAGE_o] THEN MATCH_MP_TAC IMAGE_SUBSET THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTERVAL_1] THEN REWRITE_TAC[DROP_VEC; DROP_CMUL; LIFT_DROP] THEN REAL_ARITH_TAC; X_GEN_TAC `t':real^1` THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; LIFT_DROP] THEN STRIP_TAC THEN CONV_TAC SYM_CONV THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; DROP_CMUL] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[CONNECTED_INTERVAL]; REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; LIFT_DROP] THEN REAL_ARITH_TAC; GEN_REWRITE_TAC LAND_CONV [GSYM pathstart] THEN ASM_REWRITE_TAC[] THEN SUBST1_TAC(SYM(ASSUME `pathstart h:real^M = a`)) THEN REWRITE_TAC[pathstart] THEN AP_TERM_TAC THEN REWRITE_TAC[VECTOR_MUL_RZERO]; REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; LIFT_DROP] THEN ASM_REAL_ARITH_TAC]; FIRST_ASSUM(MP_TAC o ISPECL [`(f:real^P->real^N) o reversepath(g':real^1->real^P) o (\t. &2 % t - vec 1)`; `q':real^1->real^M`; `reversepath(h':real^1->real^M) o (\t. &2 % t - vec 1)`; `{t | &1 / &2 < drop t /\ drop t <= &1}`; `vec 1:real^1`; `t:real^1`] o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] COVERING_SPACE_LIFT_UNIQUE)) THEN REWRITE_TAC[o_THM] THEN DISCH_THEN MATCH_MP_TAC THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_EQ THEN EXISTS_TAC `(f:real^P->real^N) o (g ++ reversepath g')` THEN CONJ_TAC THENL [SIMP_TAC[IN_ELIM_THM; GSYM REAL_NOT_LE; joinpaths; o_THM]; ALL_TAC] THEN MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `interval[vec 0:real^1,vec 1]` THEN CONJ_TAC THENL [ASM_MESON_TAC[HOMOTOPIC_PATHS_IMP_PATH; path]; REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_INTERVAL_1; DROP_VEC] THEN REAL_ARITH_TAC]; MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `path_image ((f:real^P->real^N) o (g ++ reversepath g'))` THEN CONJ_TAC THENL[ALL_TAC; ASM_MESON_TAC[HOMOTOPIC_PATHS_IMP_SUBSET]] THEN REWRITE_TAC[path_image] THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> f x = g x) /\ s SUBSET t ==> IMAGE f s SUBSET IMAGE g t`) THEN SIMP_TAC[IN_ELIM_THM; GSYM REAL_NOT_LE; joinpaths; o_THM] THEN REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_INTERVAL_1; DROP_VEC] THEN REAL_ARITH_TAC; MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `interval[vec 0:real^1,vec 1]` THEN ASM_REWRITE_TAC[GSYM path] THEN REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_INTERVAL_1; DROP_VEC] THEN REAL_ARITH_TAC; MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `path_image(q':real^1->real^M)` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[path_image] THEN MATCH_MP_TAC IMAGE_SUBSET THEN REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_INTERVAL_1; DROP_VEC] THEN REAL_ARITH_TAC; X_GEN_TAC `t':real^1` THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN FIRST_X_ASSUM(fun th -> W(MP_TAC o PART_MATCH (lhand o rand) th o rand o snd)) THEN ASM_SIMP_TAC[IN_INTERVAL_1; joinpaths; DROP_VEC; GSYM REAL_NOT_LT] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; SIMP_TAC[]]; MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_CMUL; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST] THEN MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `interval[vec 0:real^1,vec 1]` THEN ASM_SIMP_TAC[GSYM path; PATH_REVERSEPATH] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM] THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; DROP_CMUL; DROP_SUB] THEN REAL_ARITH_TAC; MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `path_image(reversepath h':real^1->real^M)` THEN CONJ_TAC THENL [ALL_TAC; ASM_SIMP_TAC[PATH_IMAGE_REVERSEPATH]] THEN REWRITE_TAC[path_image; IMAGE_o] THEN MATCH_MP_TAC IMAGE_SUBSET THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM] THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; DROP_CMUL; DROP_SUB] THEN REAL_ARITH_TAC; X_GEN_TAC `t':real^1` THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN REWRITE_TAC[reversepath] THEN CONV_TAC SYM_CONV THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[IN_INTERVAL_1; DROP_SUB; DROP_VEC; DROP_CMUL] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[GSYM IS_INTERVAL_CONNECTED_1; IS_INTERVAL_1] THEN REWRITE_TAC[IN_ELIM_THM] THEN REAL_ARITH_TAC; REWRITE_TAC[IN_ELIM_THM; DROP_VEC] THEN REAL_ARITH_TAC; GEN_REWRITE_TAC LAND_CONV [GSYM pathfinish] THEN ASM_REWRITE_TAC[reversepath] THEN SUBST1_TAC(SYM(ASSUME `pathstart h':real^M = a`)) THEN REWRITE_TAC[pathstart] THEN AP_TERM_TAC THEN REWRITE_TAC[GSYM DROP_EQ; DROP_SUB; DROP_CMUL; DROP_VEC] THEN REAL_ARITH_TAC; REWRITE_TAC[IN_ELIM_THM] THEN ASM_REAL_ARITH_TAC]]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `l:real^P->real^M` THEN DISCH_THEN(LABEL_TAC "+") THEN MATCH_MP_TAC(TAUT `(q ==> p) /\ q ==> p /\ q`) THEN REPEAT CONJ_TAC THENL [STRIP_TAC; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `y:real^P` THEN DISCH_TAC THEN REMOVE_THEN "*" (MP_TAC o SPEC `y:real^P`) THEN ASM_MESON_TAC[PATHFINISH_IN_PATH_IMAGE; SUBSET]; FIRST_ASSUM(MP_TAC o SPECL [`z:real^P`; `linepath(z:real^P,z)`; `linepath(a:real^M,a)`]) THEN REWRITE_TAC[PATH_LINEPATH; PATH_IMAGE_LINEPATH; SEGMENT_REFL] THEN REWRITE_TAC[PATHSTART_LINEPATH; PATHFINISH_LINEPATH] THEN ASM_SIMP_TAC[LINEPATH_REFL; SING_SUBSET]; X_GEN_TAC `y:real^P` THEN DISCH_TAC THEN REMOVE_THEN "*" (MP_TAC o SPEC `y:real^P`) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`g:real^1->real^P`; `h:real^1->real^M`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`y:real^P`; `g:real^1->real^P`; `h:real^1->real^M`]) THEN ASM_MESON_TAC[pathfinish; ENDS_IN_UNIT_INTERVAL]] THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC I [MATCH_MP CONTINUOUS_ON_OPEN_GEN th]) THEN X_GEN_TAC `n:real^M->bool` THEN DISCH_TAC THEN ONCE_REWRITE_TAC[OPEN_IN_SUBOPEN] THEN X_GEN_TAC `y:real^P` THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o CONJUNCT1 o GEN_REWRITE_RULE I [open_in]) THEN FIRST_ASSUM(MP_TAC o SPEC `(f:real^P->real^N) y` o last o CONJUNCTS o GEN_REWRITE_RULE I [covering_space]) THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `w:real^N->bool` MP_TAC) THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(X_CHOOSE_THEN `vv:(real^M->bool)->bool` MP_TAC) THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN GEN_REWRITE_TAC LAND_CONV [EXTENSION] THEN DISCH_THEN(MP_TAC o SPEC `(l:real^P->real^M) y`) THEN MATCH_MP_TAC(TAUT `q /\ (p ==> r) ==> (p <=> q) ==> r`) THEN CONJ_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[IN_UNIONS]] THEN DISCH_THEN(X_CHOOSE_THEN `w':real^M->bool` STRIP_ASSUME_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 (MP_TAC o SPEC `w':real^M->bool`) MP_TAC) THEN DISCH_THEN(MP_TAC o SPEC `w':real^M->bool` o CONJUNCT2) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_TAC `p':real^N->real^M`) THEN DISCH_TAC THEN UNDISCH_THEN `(w':real^M->bool) IN vv` (K ALL_TAC) THEN SUBGOAL_THEN `?v. y IN v /\ y IN u /\ IMAGE (f:real^P->real^N) v SUBSET w /\ v SUBSET u /\ path_connected v /\ open_in (subtopology euclidean u) v` STRIP_ASSUME_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LOCALLY_PATH_CONNECTED]) THEN DISCH_THEN(MP_TAC o SPECL [`{x | x IN u /\ (f:real^P->real^N) x IN w}`; `y:real^P`]) THEN ANTS_TAC THENL [ALL_TAC; MATCH_MP_TAC MONO_EXISTS THEN ASM SET_TAC[]] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE_GEN THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [homeomorphism]) THEN SUBGOAL_THEN `(w':real^M->bool) SUBSET c /\ (w:real^N->bool) SUBSET s` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[open_in]; ALL_TAC] THEN EXISTS_TAC `v INTER {x | x IN u /\ (f:real^P->real^N) x IN {x | x IN w /\ (p':real^N->real^M) x IN w' INTER n}}` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_INTER THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE_GEN THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `w:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE_GEN THEN EXISTS_TAC `w':real^M->bool` THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN UNDISCH_TAC `open_in (subtopology euclidean c) (n:real^M->bool)` THEN REWRITE_TAC[OPEN_IN_OPEN] THEN MATCH_MP_TAC MONO_EXISTS THEN ASM SET_TAC[]; ASM SET_TAC[]; ALL_TAC] THEN SIMP_TAC[SUBSET; IN_INTER; IN_ELIM_THM] THEN X_GEN_TAC `y':real^P` THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [path_connected]) THEN DISCH_THEN(MP_TAC o SPECL [`y:real^P`; `y':real^P`]) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `r:real^1->real^P` STRIP_ASSUME_TAC) THEN REMOVE_THEN "*" (MP_TAC o SPEC `y:real^P`) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`pp:real^1->real^P`; `qq:real^1->real^M`] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPECL [`y':real^P`; `(pp:real^1->real^P) ++ r`; `(qq:real^1->real^M) ++ ((p':real^N->real^M) o (f:real^P->real^N) o (r:real^1->real^P))`]) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`y:real^P`; `pp:real^1->real^P`; `qq:real^1->real^M`]) THEN ASM_SIMP_TAC[o_THM; PATHSTART_JOIN; PATHFINISH_JOIN] THEN DISCH_TAC THEN SUBGOAL_THEN `path_image ((pp:real^1->real^P) ++ r) SUBSET u` ASSUME_TAC THENL [MATCH_MP_TAC SUBSET_PATH_IMAGE_JOIN THEN ASM SET_TAC[]; ALL_TAC] THEN ANTS_TAC THENL [ALL_TAC; ASM_REWRITE_TAC[PATHFINISH_COMPOSE] THEN ASM_MESON_TAC[]] THEN REPEAT CONJ_TAC THENL [ASM_SIMP_TAC[PATH_JOIN]; ASM_SIMP_TAC[SUBSET_PATH_IMAGE_JOIN]; MATCH_MP_TAC PATH_JOIN_IMP THEN ASM_SIMP_TAC[PATHSTART_COMPOSE] THEN CONJ_TAC THENL [REWRITE_TAC[o_ASSOC] THEN MATCH_MP_TAC PATH_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; REWRITE_TAC[pathfinish] THEN ASM SET_TAC[]]; MATCH_MP_TAC SUBSET_PATH_IMAGE_JOIN THEN ASM_SIMP_TAC[] THEN REWRITE_TAC[PATH_IMAGE_COMPOSE] THEN ASM SET_TAC[]; X_GEN_TAC `tt:real^1` THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN STRIP_TAC THEN REWRITE_TAC[joinpaths; o_THM] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL [ABBREV_TAC `t:real^1 = &2 % tt`; ABBREV_TAC `t:real^1 = &2 % tt - vec 1`] THEN (SUBGOAL_THEN `t IN interval[vec 0:real^1,vec 1]` ASSUME_TAC THENL [EXPAND_TAC "t" THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; DROP_CMUL; DROP_SUB] THEN ASM_REAL_ARITH_TAC; ALL_TAC]) THEN ASM_SIMP_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[path_image]) THEN ASM SET_TAC[]]);; let COVERING_SPACE_LIFT_STRONGER = prove (`!p:real^M->real^N c s f:real^P->real^N u a z. covering_space (c,p) s /\ a IN c /\ z IN u /\ path_connected u /\ locally path_connected u /\ f continuous_on u /\ IMAGE f u SUBSET s /\ f z = p a /\ (!r. path r /\ path_image r SUBSET u /\ pathstart r = z /\ pathfinish r = z ==> ?b. homotopic_paths s (f o r) (linepath(b,b))) ==> ?g. g continuous_on u /\ IMAGE g u SUBSET c /\ g z = a /\ (!y. y IN u ==> p(g y) = f y)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] COVERING_SPACE_LIFT_GENERAL)) THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `r:real^1->real^P` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `r:real^1->real^P`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_TAC `b:real^N`) THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_PATHSTART) THEN ASM_REWRITE_TAC[PATHSTART_COMPOSE; PATHSTART_LINEPATH] THEN DISCH_THEN(SUBST_ALL_TAC o SYM) THEN EXISTS_TAC `linepath(a:real^M,a)` THEN REWRITE_TAC[PATH_LINEPATH; PATHSTART_LINEPATH; PATHFINISH_LINEPATH] THEN ASM_REWRITE_TAC[PATH_IMAGE_LINEPATH; SEGMENT_REFL; SING_SUBSET] THEN RULE_ASSUM_TAC(REWRITE_RULE[o_DEF; LINEPATH_REFL]) THEN ASM_REWRITE_TAC[o_DEF; LINEPATH_REFL]);; let COVERING_SPACE_LIFT_STRONG = prove (`!p:real^M->real^N c s f:real^P->real^N u a z. covering_space (c,p) s /\ a IN c /\ z IN u /\ simply_connected u /\ locally path_connected u /\ f continuous_on u /\ IMAGE f u SUBSET s /\ f z = p a ==> ?g. g continuous_on u /\ IMAGE g u SUBSET c /\ g z = a /\ (!y. y IN u ==> p(g y) = f y)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] COVERING_SPACE_LIFT_STRONGER)) THEN ASM_SIMP_TAC[SIMPLY_CONNECTED_IMP_PATH_CONNECTED] THEN X_GEN_TAC `r:real^1->real^P` THEN STRIP_TAC THEN EXISTS_TAC `(f:real^P->real^N) z` THEN SUBGOAL_THEN `linepath(f z,f z) = (f:real^P->real^N) o linepath(z,z)` SUBST1_TAC THENL [REWRITE_TAC[o_DEF; LINEPATH_REFL]; ALL_TAC] THEN MATCH_MP_TAC HOMOTOPIC_PATHS_CONTINUOUS_IMAGE THEN EXISTS_TAC `u:real^P->bool` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o CONJUNCT2 o GEN_REWRITE_RULE I [SIMPLY_CONNECTED_EQ_HOMOTOPIC_PATHS]) THEN ASM_REWRITE_TAC[PATH_LINEPATH; PATHSTART_LINEPATH; PATHFINISH_LINEPATH] THEN ASM_REWRITE_TAC[PATH_IMAGE_LINEPATH; SEGMENT_REFL; SING_SUBSET]);; let COVERING_SPACE_LIFT = prove (`!p:real^M->real^N c s f:real^P->real^N u. covering_space (c,p) s /\ simply_connected u /\ locally path_connected u /\ f continuous_on u /\ IMAGE f u SUBSET s ==> ?g. g continuous_on u /\ IMAGE g u SUBSET c /\ (!y. y IN u ==> p(g y) = f y)`, MP_TAC COVERING_SPACE_LIFT_STRONG THEN REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th THEN ASM_REWRITE_TAC[]) THEN ASM_CASES_TAC `u:real^P->bool = {}` THEN ASM_REWRITE_TAC[CONTINUOUS_ON_EMPTY; IMAGE_CLAUSES; EMPTY_SUBSET; NOT_IN_EMPTY] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_TAC `a:real^P`) THEN FIRST_ASSUM(MP_TAC o MATCH_MP COVERING_SPACE_IMP_SURJECTIVE) THEN GEN_REWRITE_TAC LAND_CONV [EXTENSION] THEN DISCH_THEN(MP_TAC o SPEC `(f:real^P->real^N) a`) THEN MATCH_MP_TAC(TAUT `q /\ (p ==> r) ==> (p <=> q) ==> r`) THEN CONJ_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[IN_IMAGE]] THEN ASM_MESON_TAC[]);; (* ------------------------------------------------------------------------- *) (* Some additional lemmas about covering spaces. *) (* ------------------------------------------------------------------------- *) let CARD_EQ_COVERING_MAP_FIBRES = prove (`!p:real^M->real^N c s a b. covering_space (c,p) s /\ path_connected s /\ a IN s /\ b IN s ==> {x | x IN c /\ p(x) = a} =_c {x | x IN c /\ p(x) = b}`, REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REPEAT GEN_TAC THEN REPEAT DISCH_TAC THEN REWRITE_TAC[GSYM CARD_LE_ANTISYM] THEN REWRITE_TAC[RIGHT_IMP_FORALL_THM; IMP_IMP; FORALL_AND_THM; TAUT `p ==> q /\ r <=> (p ==> q) /\ (p ==> r)`] THEN GEN_REWRITE_TAC (LAND_CONV o funpow 2 BINDER_CONV o LAND_CONV) [CONJ_SYM] THEN MATCH_MP_TAC(MESON[] `(!a b. P a b) ==> (!a b. P a b) /\ (!a b. P b a)`) THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a:real^N`; `b:real^N`] o GEN_REWRITE_RULE I [path_connected]) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `g:real^1->real^N` THEN STRIP_TAC THEN SUBGOAL_THEN `!z. ?h. z IN c /\ p z = a ==> path h /\ path_image h SUBSET c /\ pathstart h = z /\ !t. t IN interval[vec 0,vec 1] ==> (p:real^M->real^N)(h t) = g t` MP_TAC THENL [REWRITE_TAC[RIGHT_EXISTS_IMP_THM] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC COVERING_SPACE_LIFT_PATH_STRONG THEN REWRITE_TAC[ETA_AX] THEN ASM_MESON_TAC[]; REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `h:real^M->real^1->real^M` THEN DISCH_TAC] THEN REWRITE_TAC[le_c; IN_ELIM_THM] THEN EXISTS_TAC `\z. pathfinish((h:real^M->real^1->real^M) z)` THEN ASM_REWRITE_TAC[pathfinish] THEN CONJ_TAC THENL [X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[SUBSET; path_image; pathstart; FORALL_IN_IMAGE] THEN ASM_MESON_TAC[pathfinish; ENDS_IN_UNIT_INTERVAL]; MAP_EVERY X_GEN_TAC [`x:real^M`; `y:real^M`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`p:real^M->real^N`; `c:real^M->bool`; `s:real^N->bool`; `reversepath(g:real^1->real^N)`; `reversepath(g:real^1->real^N)`; `reversepath((h:real^M->real^1->real^M) x)`; `reversepath((h:real^M->real^1->real^M) y)`] COVERING_SPACE_MONODROMY) THEN ASM_SIMP_TAC[PATHSTART_REVERSEPATH; PATHFINISH_REVERSEPATH] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_SIMP_TAC[PATH_REVERSEPATH; PATH_IMAGE_REVERSEPATH; HOMOTOPIC_PATHS_REFL] THEN ASM_REWRITE_TAC[pathfinish; reversepath; IN_INTERVAL_1; DROP_VEC] THEN REPEAT STRIP_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`); FIRST_X_ASSUM(MP_TAC o SPEC `y:real^M`)] THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MATCH_MP_TAC o last o CONJUNCTS) THEN REWRITE_TAC[IN_INTERVAL_1; DROP_SUB; DROP_VEC] THEN ASM_REAL_ARITH_TAC]);; let COVERING_SPACE_INJECTIVE = prove (`!p:real^M->real^N c s. covering_space (c,p) s /\ path_connected c /\ simply_connected s ==> (!x y. x IN c /\ y IN c /\ p x = p y ==> x = y)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP COVERING_SPACE_IMP_SURJECTIVE) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP COVERING_SPACE_IMP_CONTINUOUS) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [path_connected]) THEN DISCH_THEN(MP_TAC o SPECL [`x:real^M`; `y:real^M`]) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^1->real^M` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] COVERING_SPACE_LIFT_PATH_STRONG)) THEN GEN_REWRITE_TAC LAND_CONV [SWAP_FORALL_THM] THEN DISCH_THEN(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(fun th -> MP_TAC(SPEC `(p:real^M->real^N) o (g:real^1->real^M)` th) THEN MP_TAC(SPEC `(p:real^M->real^N) o linepath(x:real^M,x)` th)) THEN SUBGOAL_THEN `(path ((p:real^M->real^N) o linepath(x,x)) /\ path (p o g)) /\ (path_image (p o linepath(x:real^M,x)) SUBSET s /\ path_image (p o g) SUBSET s)` STRIP_ASSUME_TAC THENL [CONJ_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC PATH_CONTINUOUS_IMAGE THEN REWRITE_TAC[PATH_LINEPATH; PATH_IMAGE_LINEPATH] THEN ASM_REWRITE_TAC[CONTINUOUS_ON_SING; SEGMENT_REFL] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; REWRITE_TAC[PATH_IMAGE_COMPOSE; PATH_IMAGE_LINEPATH] THEN REWRITE_TAC[SEGMENT_REFL] THEN ASM SET_TAC[]]; ALL_TAC] THEN ASM_REWRITE_TAC[PATHSTART_COMPOSE; PATHSTART_LINEPATH] THEN DISCH_THEN(X_CHOOSE_THEN `h1:real^1->real^M` STRIP_ASSUME_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `h2:real^1->real^M` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o SPECL [`(p:real^M->real^N) o linepath(x:real^M,x)`; `(p:real^M->real^N) o (g:real^1->real^M)`; `h1:real^1->real^M`; `h2:real^1->real^M`] o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] COVERING_SPACE_MONODROMY)) THEN ASM_SIMP_TAC[] THEN ANTS_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o CONJUNCT2 o GEN_REWRITE_RULE I [SIMPLY_CONNECTED_EQ_HOMOTOPIC_PATHS]) THEN ASM_REWRITE_TAC[PATHSTART_COMPOSE; PATHFINISH_COMPOSE] THEN ASM_REWRITE_TAC[PATHSTART_LINEPATH; PATHFINISH_LINEPATH]; ALL_TAC] THEN MATCH_MP_TAC EQ_IMP THEN BINOP_TAC THENL [MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `pathfinish(linepath(x:real^M,x))` THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[PATHFINISH_LINEPATH]]; FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [SYM th])] THEN REWRITE_TAC[pathfinish] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] COVERING_SPACE_LIFT_UNIQUE)) THENL [EXISTS_TAC `(p:real^M->real^N) o (h1:real^1->real^M)`; EXISTS_TAC `(p:real^M->real^N) o (h2:real^1->real^M)`] THEN MAP_EVERY EXISTS_TAC [`interval[vec 0:real^1,vec 1]`; `vec 0:real^1`] THEN REWRITE_TAC[CONNECTED_INTERVAL; ENDS_IN_UNIT_INTERVAL] THEN ASM_REWRITE_TAC[GSYM path; PATH_LINEPATH; GSYM path_image] THEN RULE_ASSUM_TAC(REWRITE_RULE[o_THM]) THEN ASM_REWRITE_TAC[o_THM] THEN ASM_REWRITE_TAC[PATH_IMAGE_LINEPATH; SEGMENT_REFL; SING_SUBSET] THEN RULE_ASSUM_TAC(REWRITE_RULE[pathstart]) THEN ASM_REWRITE_TAC[LINEPATH_REFL; PATH_IMAGE_COMPOSE] THEN (CONJ_TAC THENL [ASM_MESON_TAC[PATH_CONTINUOUS_IMAGE; CONTINUOUS_ON_SUBSET]; ASM SET_TAC[]]));; let COVERING_SPACE_HOMEOMORPHISM = prove (`!p:real^M->real^N c s. covering_space (c,p) s /\ path_connected c /\ simply_connected s ==> ?q. homeomorphism (c,s) (p,q)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHISM_INJECTIVE_OPEN_MAP THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[COVERING_SPACE_IMP_CONTINUOUS]; ASM_MESON_TAC[COVERING_SPACE_IMP_SURJECTIVE]; ASM_MESON_TAC[COVERING_SPACE_INJECTIVE]; ASM_MESON_TAC[COVERING_SPACE_OPEN_MAP]]);; (* ------------------------------------------------------------------------- *) (* Results on finiteness of the number of sheets in a covering space. *) (* ------------------------------------------------------------------------- *) let COVERING_SPACE_FIBRE_NO_LIMPT = prove (`!p:real^M->real^N c s a b. covering_space (c,p) s /\ a IN c ==> ~(a limit_point_of {x | x IN c /\ p x = b})`, REPEAT STRIP_TAC THEN FIRST_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [covering_space]) THEN FIRST_X_ASSUM(MP_TAC o SPEC `(p:real^M->real^N) a`) THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` MP_TAC) THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(X_CHOOSE_THEN `vv:(real^M->bool)->bool` MP_TAC) THEN GEN_REWRITE_TAC I [IMP_CONJ] THEN REWRITE_TAC[EXTENSION; IN_UNIONS; IN_ELIM_THM] THEN DISCH_THEN(MP_TAC o SPEC `a:real^M`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `t:real^M->bool` STRIP_ASSUME_TAC) THEN STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `t:real^M->bool`)) THEN ASM_REWRITE_TAC[] THEN REPEAT DISCH_TAC THEN FIRST_X_ASSUM(X_CHOOSE_THEN `q:real^N->real^M` MP_TAC) THEN REWRITE_TAC[homeomorphism] THEN STRIP_TAC THEN UNDISCH_TAC `open_in (subtopology euclidean c) (t:real^M->bool)` THEN REWRITE_TAC[OPEN_IN_OPEN] THEN DISCH_THEN(X_CHOOSE_THEN `v:real^M->bool` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPEC `v:real^M->bool` o GEN_REWRITE_RULE I [LIMPT_INFINITE_OPEN]) THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[INFINITE]] THEN MATCH_MP_TAC(MESON[FINITE_SING; FINITE_SUBSET] `(?a. s SUBSET {a}) ==> FINITE s`) THEN ASM SET_TAC[]);; let COVERING_SPACE_COUNTABLE_SHEETS = prove (`!p:real^M->real^N c s b. covering_space (c,p) s ==> COUNTABLE {x | x IN c /\ p x = b}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC(REWRITE_RULE[] (ONCE_REWRITE_RULE[GSYM CONTRAPOS_THM] UNCOUNTABLE_CONTAINS_LIMIT_POINT)) THEN REWRITE_TAC[IN_ELIM_THM] THEN ASM_MESON_TAC[COVERING_SPACE_FIBRE_NO_LIMPT]);; let COVERING_SPACE_FINITE_EQ_COMPACT_FIBRE = prove (`!p:real^M->real^N c s b. covering_space (c,p) s ==> (FINITE {x | x IN c /\ p x = b} <=> compact {x | x IN c /\ p x = b})`, REPEAT STRIP_TAC THEN EQ_TAC THEN SIMP_TAC[FINITE_IMP_COMPACT] THEN DISCH_TAC THEN ASM_CASES_TAC `(b:real^N) IN s` THENL [ONCE_REWRITE_TAC[TAUT `p <=> (~p ==> F)`] THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `{x | x IN c /\ (p:real^M->real^N) x = b}` o GEN_REWRITE_RULE I [COMPACT_EQ_BOLZANO_WEIERSTRASS]) THEN ASM_REWRITE_TAC[INFINITE; SUBSET_REFL; IN_ELIM_THM] THEN DISCH_THEN(X_CHOOSE_THEN `a:real^M` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a:real^M`; `b:real^N`] o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] COVERING_SPACE_FIBRE_NO_LIMPT)) THEN ASM_REWRITE_TAC[]; SUBGOAL_THEN `{x | x IN c /\ (p:real^M->real^N) x = b} = {}` (fun th -> REWRITE_TAC[th; FINITE_EMPTY]) THEN FIRST_ASSUM(MP_TAC o MATCH_MP COVERING_SPACE_IMP_SURJECTIVE) THEN ASM SET_TAC[]]);; let COVERING_SPACE_CLOSED_MAP = prove (`!p:real^M->real^N c s t. covering_space (c,p) s /\ (!b. b IN s ==> FINITE {x | x IN c /\ p x = b}) /\ closed_in (subtopology euclidean c) t ==> closed_in (subtopology euclidean s) (IMAGE p t)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP COVERING_SPACE_IMP_SURJECTIVE) THEN REWRITE_TAC[closed_in; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN CONJ_TAC THENL [ASM SET_TAC[]; ONCE_REWRITE_TAC[OPEN_IN_SUBOPEN]] THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_DIFF] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `y:real^N` o last o CONJUNCTS o GEN_REWRITE_RULE I [covering_space]) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `v:real^N->bool` THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN DISCH_THEN(X_CHOOSE_THEN `uu:(real^M->bool)->bool` STRIP_ASSUME_TAC) THEN ASM_CASES_TAC `uu:(real^M->bool)->bool = {}` THENL [ASM_REWRITE_TAC[UNIONS_0; NOT_IN_EMPTY] THEN ASM SET_TAC[]; ALL_TAC] THEN EXISTS_TAC `INTERS {IMAGE (p:real^M->real^N) (u DIFF t) | u IN uu}` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_INTERS THEN ASM_REWRITE_TAC[SIMPLE_IMAGE; FORALL_IN_IMAGE; IMAGE_EQ_EMPTY] THEN CONJ_TAC THENL [MATCH_MP_TAC FINITE_IMAGE THEN SUBGOAL_THEN `!u. u IN uu ==> ?x. x IN u /\ (p:real^M->real^N) x = y` ASSUME_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `FINITE (IMAGE (\u. @x. x IN u /\ (p:real^M->real^N) x = y) uu)` MP_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] FINITE_SUBSET)) THEN ASM SET_TAC[]; MATCH_MP_TAC EQ_IMP THEN MATCH_MP_TAC FINITE_IMAGE_INJ_EQ THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [pairwise]) THEN REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN ASM SET_TAC[]]; X_GEN_TAC `u:real^M->bool` THEN DISCH_TAC THEN MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `v:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC HOMEOMORPHISM_IMP_OPEN_MAP THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN EXISTS_TAC `u:real^M->bool` THEN ASM_SIMP_TAC[LEFT_EXISTS_AND_THM] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CLOSED_IN_CLOSED]) THEN DISCH_THEN(X_CHOOSE_THEN `k:real^M->bool` STRIP_ASSUME_TAC) THEN ASM_REWRITE_TAC[OPEN_IN_OPEN] THEN EXISTS_TAC `(:real^M) DIFF k` THEN ASM_REWRITE_TAC[GSYM closed] THEN ASM SET_TAC[]]; REWRITE_TAC[IN_INTERS; FORALL_IN_GSPEC] THEN X_GEN_TAC `u:real^M->bool` THEN DISCH_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `u:real^M->bool`)) THEN ASM_REWRITE_TAC[homeomorphism] THEN ASM SET_TAC[]; REWRITE_TAC[SUBSET; INTERS_GSPEC; IN_DIFF; IN_ELIM_THM] THEN X_GEN_TAC `z:real^N` THEN DISCH_TAC THEN CONJ_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[IN_IMAGE]] THEN DISCH_THEN(X_CHOOSE_THEN `w:real^M` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM SUBST_ALL_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [EXTENSION]) THEN DISCH_THEN(MP_TAC o SPEC `w:real^M`) THEN REWRITE_TAC[IN_ELIM_THM] THEN MATCH_MP_TAC(TAUT `q /\ r /\ ~s ==> ~(s <=> q /\ r)`) THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN REPEAT(CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN REWRITE_TAC[IN_UNIONS] THEN ASM SET_TAC[]]);; let COVERING_SPACE_FINITE_SHEETS_EQ_CLOSED_MAP_STRONG = prove (`!p:real^M->real^N c s. covering_space (c,p) s /\ (!b. b IN s ==> b limit_point_of s) ==> ((!b. b IN s ==> FINITE {x | x IN c /\ p x = b}) <=> (!t. closed_in (subtopology euclidean c) t ==> closed_in (subtopology euclidean s) (IMAGE p t)))`, let lemma = prove (`!f:num->real^N. (!n. ~(s = v n) ==> DISJOINT s (v n)) ==> (!n. f n IN v n) /\ (!m n. v m = v n <=> m = n) ==> ?n. IMAGE f (:num) INTER s SUBSET {f n}`, ASM_CASES_TAC `?n. s = (v:num->real^N->bool) n` THENL [REPEAT STRIP_TAC THEN FIRST_X_ASSUM(fun th -> MP_TAC th THEN MATCH_MP_TAC MONO_EXISTS); RULE_ASSUM_TAC(REWRITE_RULE[NOT_EXISTS_THM]) THEN ASM_REWRITE_TAC[]] THEN ASM SET_TAC[]) in REPEAT STRIP_TAC THEN EQ_TAC THEN REPEAT STRIP_TAC THENL [MATCH_MP_TAC COVERING_SPACE_CLOSED_MAP THEN EXISTS_TAC `c:real^M->bool` THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN REWRITE_TAC[MESON[INFINITE] `FINITE s <=> ~INFINITE s`] THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `b:real^N` o last o CONJUNCTS o GEN_REWRITE_RULE I [covering_space]) THEN ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN X_GEN_TAC `t:real^N->bool` THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(X_CHOOSE_THEN `vv:(real^M->bool)->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `(b:real^N) limit_point_of t` MP_TAC THENL [MATCH_MP_TAC LIMPT_OF_OPEN_IN THEN ASM_MESON_TAC[]; PURE_REWRITE_TAC[LIMPT_SEQUENTIAL_INJ]] THEN DISCH_THEN(X_CHOOSE_THEN `y:num->real^N` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `INFINITE(vv:(real^M->bool)->bool)` MP_TAC THENL [FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CARD_LE_INFINITE)) THEN REWRITE_TAC[le_c] THEN SUBGOAL_THEN `!x. ?v. x IN c /\ (p:real^M->real^N) x = b ==> v IN vv /\ x IN v` MP_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[SKOLEM_THM]] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^M->real^M->bool` THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_TAC THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`x:real^M`; `y:real^M`] THEN STRIP_TAC THEN FIRST_X_ASSUM(fun th -> MP_TAC(SPEC `x:real^M` th) THEN MP_TAC(SPEC `y:real^M` th)) THEN ASM_REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[INFINITE_CARD_LE; le_c; INJECTIVE_ON_ALT] THEN REWRITE_TAC[IN_UNIV] THEN DISCH_THEN(X_CHOOSE_THEN `v:num->real^M->bool` STRIP_ASSUME_TAC) THEN UNDISCH_THEN `!u. u IN vv ==> ?q:real^N->real^M. homeomorphism (u,t) (p,q)` (MP_TAC o GEN `n:num` o SPEC `(v:num->real^M->bool) n`) THEN ASM_REWRITE_TAC[SKOLEM_THM; homeomorphism; FORALL_AND_THM] THEN DISCH_THEN(X_CHOOSE_THEN `q:num->real^N->real^M` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `closed_in (subtopology euclidean s) (IMAGE (p:real^M->real^N) (IMAGE (\n. q n (y n:real^N)) (:num)))` MP_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[CLOSED_IN_LIMPT; SUBSET; FORALL_IN_IMAGE] THEN CONJ_TAC THENL [ASM SET_TAC[]; X_GEN_TAC `a:real^M`] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP LIMPT_OF_SEQUENCE_SUBSEQUENCE) THEN DISCH_THEN(X_CHOOSE_THEN `r:num->num` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `(p:real^M->real^N) a = b` ASSUME_TAC THENL [MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THEN EXISTS_TAC `(p:real^M->real^N) o (\n:num. q n (y n :real^N)) o (r:num->num)` THEN REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN CONJ_TAC THENL [MATCH_MP_TAC(GEN_ALL(REWRITE_RULE[RIGHT_IMP_FORALL_THM; IMP_IMP] (fst(EQ_IMP_RULE(SPEC_ALL CONTINUOUS_ON_SEQUENTIALLY))))) THEN EXISTS_TAC `c:real^M->bool` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[COVERING_SPACE_IMP_CONTINUOUS]; REWRITE_TAC[o_DEF] THEN ASM SET_TAC[]]; REWRITE_TAC[o_ASSOC] THEN MATCH_MP_TAC LIM_SUBSEQUENCE THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] LIM_TRANSFORM_EVENTUALLY)) THEN MATCH_MP_TAC ALWAYS_EVENTUALLY THEN REWRITE_TAC[o_DEF] THEN ASM SET_TAC[]]; SUBGOAL_THEN `?u. u IN vv /\ (a:real^M) IN u` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?w:real^M->bool. open w /\ u = c INTER w` (CHOOSE_THEN (CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC)) THENL [ASM_MESON_TAC[OPEN_IN_OPEN]; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTER]) THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIMPT_INFINITE_OPEN]) THEN DISCH_THEN(MP_TAC o SPEC `w:real^M->bool`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o MATCH_MP (MESON[] `INFINITE s ==> !k. s INTER k = s ==> INFINITE(s INTER k)`)) THEN DISCH_THEN(MP_TAC o SPEC `c:real^M->bool`) THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[INTER_ASSOC]] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[INFINITE] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [pairwise]) THEN DISCH_THEN(MP_TAC o SPEC `c INTER w:real^M->bool`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o GEN `n:num` o SPEC `(v:num->real^M->bool) n`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `\n. (q:num->real^N->real^M) n (y n)` o MATCH_MP lemma) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MESON_TAC[FINITE_SUBSET; FINITE_SING; INTER_COMM]]; SUBGOAL_THEN `IMAGE (p:real^M->real^N) (IMAGE (\n. q n (y n:real^N)) (:num)) = IMAGE y (:num)` SUBST1_TAC THENL [REWRITE_TAC[GSYM IMAGE_o; o_DEF] THEN ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[CLOSED_IN_LIMPT] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `b:real^N`)) THEN ASM_REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[LIMPT_SEQUENTIAL_INJ] THEN EXISTS_TAC `y:num->real^N` THEN ASM SET_TAC[]]);; let COVERING_SPACE_FINITE_SHEETS_EQ_CLOSED_MAP = prove (`!p:real^M->real^N c s. covering_space (c,p) s /\ connected s /\ ~(?a. s = {a}) ==> ((!b. b IN s ==> FINITE {x | x IN c /\ p x = b}) <=> (!t. closed_in (subtopology euclidean c) t ==> closed_in (subtopology euclidean s) (IMAGE p t)))`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [SUBGOAL_THEN `c:real^M->bool = {}` ASSUME_TAC THENL [FIRST_ASSUM(MP_TAC o MATCH_MP COVERING_SPACE_IMP_SURJECTIVE) THEN ASM_REWRITE_TAC[IMAGE_EQ_EMPTY]; ASM_REWRITE_TAC[OPEN_IN_SUBTOPOLOGY_EMPTY; CLOSED_IN_SUBTOPOLOGY_EMPTY; IMAGE_EQ_EMPTY; NOT_IN_EMPTY]]; MATCH_MP_TAC COVERING_SPACE_FINITE_SHEETS_EQ_CLOSED_MAP_STRONG THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CONNECTED_IMP_PERFECT THEN ASM SET_TAC[]]);; let COVERING_SPACE_FINITE_SHEETS_EQ_PROPER_MAP = prove (`!p:real^M->real^N c s. covering_space (c,p) s ==> ((!b. b IN s ==> FINITE {x | x IN c /\ p x = b}) <=> (!k. k SUBSET s /\ compact k ==> compact {x | x IN c /\ p(x) IN k}))`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP COVERING_SPACE_IMP_SURJECTIVE) THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `s = t ==> s SUBSET t`)) THEN DISCH_THEN(fun th -> REWRITE_TAC[MATCH_MP PROPER_MAP th]) THEN FIRST_ASSUM(fun th -> REWRITE_TAC [GSYM(MATCH_MP COVERING_SPACE_FINITE_EQ_COMPACT_FIBRE th)]) THEN REWRITE_TAC[TAUT `(p <=> q /\ p) <=> (p ==> q)`] THEN ASM_MESON_TAC[COVERING_SPACE_CLOSED_MAP]);; (* ------------------------------------------------------------------------- *) (* Special cases where one or both of the sets is compact. *) (* ------------------------------------------------------------------------- *) let COVERING_SPACE_FINITE_SHEETS = prove (`!p:real^M->real^N c s b. covering_space (c,p) s /\ compact c ==> FINITE {x | x IN c /\ p x = b}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC BOLZANO_WEIERSTRASS_CONTRAPOS THEN EXISTS_TAC `c:real^M->bool` THEN ASM_REWRITE_TAC[SUBSET_RESTRICT] THEN ASM_MESON_TAC[COVERING_SPACE_FIBRE_NO_LIMPT]);; let COVERING_SPACE_COMPACT = prove (`!p:real^M->real^N c s. covering_space (c,p) s ==> (compact c <=> compact s /\ (!b. b IN s ==> FINITE {x | x IN c /\ p x = b}))`, REPEAT STRIP_TAC THEN EQ_TAC THEN REPEAT STRIP_TAC THENL [ASM_MESON_TAC[covering_space; COMPACT_CONTINUOUS_IMAGE]; MATCH_MP_TAC COVERING_SPACE_FINITE_SHEETS THEN ASM_MESON_TAC[]; FIRST_ASSUM(MP_TAC o MATCH_MP COVERING_SPACE_FINITE_SHEETS_EQ_PROPER_MAP) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `s:real^N->bool`) THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP COVERING_SPACE_IMP_SURJECTIVE) THEN SET_TAC[]]);; (* ------------------------------------------------------------------------- *) (* A proper (or closed) local homeomorphism is in fact a covering map. *) (* ------------------------------------------------------------------------- *) let PROPER_LOCAL_HOMEOMORPHISM_IMP_COVERING_MAP = prove (`!p:real^M->real^N c s. IMAGE p c = s /\ (!k. k SUBSET s /\ compact k ==> compact {x | x IN c /\ p x IN k}) /\ (!x. x IN c ==> ?t u q. x IN t /\ open_in (subtopology euclidean c) t /\ open_in (subtopology euclidean s) u /\ homeomorphism (t,u) (p,q)) ==> covering_space (c,p) s`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `(p:real^M->real^N) continuous_on c` ASSUME_TAC THENL [REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `t:real^M->bool` STRIP_ASSUME_TAC) THEN UNDISCH_TAC `open_in (subtopology euclidean c) (t:real^M->bool)` THEN REWRITE_TAC[CONTINUOUS_WITHIN_OPEN; OPEN_IN_OPEN] THEN DISCH_THEN(X_CHOOSE_THEN `v:real^M->bool` STRIP_ASSUME_TAC) THEN X_GEN_TAC `w:real^N->bool` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o CONJUNCT1 o GEN_REWRITE_RULE I [HOMEOMORPHISM]) THEN REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN REWRITE_TAC[CONTINUOUS_WITHIN_OPEN] THEN DISCH_THEN(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `w:real^N->bool`) THEN ASM_REWRITE_TAC[IN_INTER] THEN DISCH_THEN(X_CHOOSE_THEN `y:real^M->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `y INTER v:real^M->bool` THEN ASM_SIMP_TAC[OPEN_INTER; IN_INTER] THEN ASM SET_TAC[]; ALL_TAC] THEN ASM_REWRITE_TAC[covering_space] THEN SUBGOAL_THEN `!y. y IN s ==> FINITE {x | x IN c /\ (p:real^M->real^N) x = y}` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC BOLZANO_WEIERSTRASS_CONTRAPOS THEN EXISTS_TAC `{x | x IN c /\ (p:real^M->real^N) x = y}` THEN REWRITE_TAC[SUBSET_REFL; FORALL_IN_GSPEC] THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[GSYM IN_SING] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[SING_SUBSET; COMPACT_SING]; X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[limit_point_of; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`t:real^M->bool`; `v:real^N->bool`; `q:real^N->real^M`] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN REWRITE_TAC[OPEN_IN_OPEN] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^M->bool` (CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC)) THEN FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [IN_INTER]) THEN DISCH_THEN(MP_TAC o SPEC `u:real^M->bool`) THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHISM]) THEN ASM SET_TAC[]]; ALL_TAC] THEN SUBGOAL_THEN `!y. y IN s ==> ?v uu. y IN v /\ open_in (subtopology euclidean s) v /\ pairwise DISJOINT (IMAGE uu {x | x IN c /\ p x = y}) /\ !x. x IN c /\ (p:real^M->real^N) x = y ==> x IN uu x /\ open_in (subtopology euclidean c) (uu x) /\ ?q. homeomorphism (uu x,v) (p,q)` ASSUME_TAC THENL [X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN SUBGOAL_THEN `?uu:real^M->real^M->bool vv:real^M->real^N->bool. pairwise DISJOINT (IMAGE uu {x | x IN c /\ p x = y}) /\ !x. x IN c /\ (p:real^M->real^N) x = y ==> x IN uu x /\ open_in (subtopology euclidean c) (uu x) /\ open_in (subtopology euclidean s) (vv x) /\ (?q. homeomorphism (uu x,vv x) (p,q))` STRIP_ASSUME_TAC THENL [MP_TAC(ISPEC `{x | x IN c /\ (p:real^M->real^N) x = y}` FINITE_EQ_BOUNDED_DISCRETE) THEN ASM_SIMP_TAC[IN_ELIM_THM] THEN DISCH_THEN(X_CHOOSE_THEN `r:real` MP_TAC o CONJUNCT2) THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r ==> s <=> p /\ q /\ ~s ==> ~r`] THEN REWRITE_TAC[GSYM CONJ_ASSOC; REAL_NOT_LT] THEN STRIP_TAC THEN SUBGOAL_THEN `!x. x IN c /\ (p:real^M->real^N) x = y ==> ?u v. x IN u /\ open_in (subtopology euclidean c) u /\ open_in (subtopology euclidean s) v /\ ?q. homeomorphism (u,v) (p,q)` MP_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`uu:real^M->real^M->bool`; `vv:real^M->real^N->bool`] THEN DISCH_TAC THEN EXISTS_TAC `\x:real^M. ball(x,r / &2) INTER uu x` THEN EXISTS_TAC `\x. IMAGE (p:real^M->real^N) (ball(x,r / &2) INTER uu x)` THEN ASM_REWRITE_TAC[IN_INTER; CENTRE_IN_BALL; REAL_HALF] THEN CONJ_TAC THENL [REWRITE_TAC[pairwise; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN X_GEN_TAC `x':real^M` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^M`; `x':real^M`]) THEN ASM_CASES_TAC `x':real^M = x` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(SET_RULE `(!y. y IN s /\ y IN t ==> ~P) ==> P ==> Q ==> DISJOINT (s INTER s') (t INTER t')`) THEN REWRITE_TAC[IN_BALL] THEN CONV_TAC NORM_ARITH; ALL_TAC] THEN X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `q:real^N->real^M` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [ASM_SIMP_TAC[OPEN_BALL; ONCE_REWRITE_RULE[INTER_COMM]OPEN_IN_INTER_OPEN]; MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `(vv:real^M->real^N->bool) x` THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_IMP_OPEN_MAP)) THEN ONCE_REWRITE_TAC[INTER_COMM] THEN SIMP_TAC[OPEN_IN_OPEN_INTER; OPEN_BALL]; FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHISM]) THEN SET_TAC[]]; ALL_TAC] THEN ABBREV_TAC `v = INTERS (IMAGE (vv:real^M->real^N->bool) {x | x IN c /\ (p:real^M->real^N) x = y})` THEN EXISTS_TAC `v:real^N->bool` THEN EXISTS_TAC `\x:real^M. {w | w IN uu x /\ (p:real^M->real^N) w IN v}` THEN REWRITE_TAC[] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [EXPAND_TAC "v" THEN REWRITE_TAC[INTERS_IMAGE; IN_ELIM_THM] THEN X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[HOMEOMORPHISM] THEN ASM SET_TAC[]; DISCH_TAC] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [EXPAND_TAC "v" THEN MATCH_MP_TAC OPEN_IN_INTERS THEN ASM_SIMP_TAC[FORALL_IN_IMAGE; IMAGE_EQ_EMPTY; FORALL_IN_GSPEC; FINITE_IMAGE] THEN ASM SET_TAC[]; DISCH_TAC] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [pairwise]) THEN REWRITE_TAC[pairwise; RIGHT_FORALL_IMP_THM; IMP_CONJ] THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN SET_TAC[]; ALL_TAC] THEN X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `q:real^N->real^M` THEN STRIP_TAC THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `(uu:real^M->real^M->bool) x` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE_GEN THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN REPEAT(FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET)) THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)); FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHISM])] THEN ASM SET_TAC[]; FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHISM]) THEN REPEAT STRIP_TAC THENL [ASM SET_TAC[]; ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN EXPAND_TAC "v" THEN REWRITE_TAC[INTERS_IMAGE; IN_ELIM_THM; SUBSET] THEN X_GEN_TAC `z:real^N` THEN DISCH_THEN(fun th -> MP_TAC th THEN MP_TAC(SPEC `x:real^M` th)) THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC(SET_RULE `z IN IMAGE p s /\ P z ==> z IN IMAGE p {x | x IN s /\ P(p x)}`) THEN ASM SET_TAC[]]; ALL_TAC] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`)) THEN ASM_REWRITE_TAC[] THEN REPEAT DISCH_TAC THEN FIRST_X_ASSUM(X_CHOOSE_THEN `v:real^N->bool` MP_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `uu:real^M->real^M->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `?w. y IN w /\ open_in (subtopology euclidean s) w /\ w SUBSET v /\ {x | x IN c /\ (p:real^M->real^N) x IN w} SUBSET UNIONS (IMAGE uu {x | x IN c /\ p x = y})` MP_TAC THENL [SUBGOAL_THEN `?w. y IN w /\ open_in (subtopology euclidean s) w /\ {x | x IN c /\ (p:real^M->real^N) x IN w} SUBSET UNIONS (IMAGE uu {x | x IN c /\ p x = y})` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC(MESON[] `~(!w. P w /\ Q w ==> ~R w) ==> ?w. P w /\ Q w /\ R w`) THEN DISCH_THEN(MP_TAC o GEN `n:num` o SPEC `s INTER ball(y:real^N,inv(&n + &1))`) THEN SIMP_TAC[OPEN_IN_OPEN_INTER; OPEN_BALL; IN_INTER; CENTRE_IN_BALL] THEN ASM_REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN REWRITE_TAC[UNIONS_IMAGE; SKOLEM_THM; SET_RULE `~(s SUBSET t) <=> ?x. x IN s /\ ~(x IN t)`] THEN REWRITE_TAC[IN_ELIM_THM; IN_BALL; FORALL_AND_THM] THEN REWRITE_TAC[NOT_EXISTS_THM; TAUT `~(p /\ q) <=> p ==> ~q`] THEN X_GEN_TAC `z:num->real^M` THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `y INSERT IMAGE ((p:real^M->real^N) o (z:num->real^M)) (:num)`) THEN REWRITE_TAC[NOT_IMP] THEN SUBGOAL_THEN `(((p:real^M->real^N) o (z:num->real^M)) --> y) sequentially` ASSUME_TAC THENL [REWRITE_TAC[LIM_SEQUENTIALLY; o_THM] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(SPEC `e:real` REAL_ARCH_INV) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `N:num` THEN STRIP_TAC THEN EXISTS_TAC `N:num` THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN ONCE_REWRITE_TAC[DIST_SYM] THEN TRANS_TAC REAL_LTE_TRANS `inv(&n + &1)` THEN ASM_REWRITE_TAC[] THEN TRANS_TAC REAL_LE_TRANS `inv(&N)` THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN MATCH_MP_TAC REAL_LE_INV2 THEN ASM_REWRITE_TAC[REAL_OF_NUM_LE; REAL_OF_NUM_LT; REAL_OF_NUM_ADD] THEN ASM_ARITH_TAC; ALL_TAC] THEN CONJ_TAC THENL [ASM_REWRITE_TAC[SUBSET; FORALL_IN_INSERT; FORALL_IN_IMAGE; o_THM] THEN MATCH_MP_TAC COMPACT_SEQUENCE_WITH_LIMIT THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN REWRITE_TAC[compact] THEN DISCH_THEN(MP_TAC o SPEC `z:num->real^M`) THEN ASM_REWRITE_TAC[IN_ELIM_THM; NOT_IMP] THEN CONJ_TAC THENL [REWRITE_TAC[IN_INSERT; IN_IMAGE; o_DEF] THEN SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[NOT_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`x:real^M`; `r:num->num`] THEN STRIP_TAC THEN SUBGOAL_THEN `(p:real^M->real^N) x = y` ASSUME_TAC THENL [MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THEN EXISTS_TAC `(p:real^M->real^N) o (z:num->real^M) o (r:num->num)` THEN REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN]) THEN DISCH_THEN(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[CONTINUOUS_WITHIN_SEQUENTIALLY] THEN DISCH_THEN(MP_TAC o SPEC `(z:num->real^M) o (r:num->num)`) THEN ASM_REWRITE_TAC[o_THM]; REWRITE_TAC[o_ASSOC] THEN MATCH_MP_TAC LIM_SUBSEQUENCE THEN ASM_REWRITE_TAC[]]; FIRST_X_ASSUM(MP_TAC o GEN `n:num` o SPECL [`n:num`; `x:real^M`]) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`)) THEN ASM_REWRITE_TAC[] THEN REPEAT(DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC)) THEN UNDISCH_TAC `open_in (subtopology euclidean c) ((uu:real^M->real^M->bool) x)` THEN REWRITE_TAC[open_in] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `x:real^M`)) THEN ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN X_GEN_TAC `e:real` THEN STRIP_TAC THEN UNDISCH_TAC `(((z:num->real^M) o (r:num->num)) --> x) sequentially` THEN REWRITE_TAC[LIM_SEQUENTIALLY] THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[o_THM] THEN ASM_MESON_TAC[LE_REFL]]; EXISTS_TAC `v INTER w:real^N->bool` THEN ASM_SIMP_TAC[IN_INTER; OPEN_IN_INTER; INTER_SUBSET] THEN ASM SET_TAC[]]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `IMAGE (\x:real^M. {w | w IN uu x /\ (p:real^M->real^N) w IN t}) {x | x IN c /\ p x = y}` THEN REWRITE_TAC[FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[UNIONS_IMAGE; EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `x:real^M` THEN RULE_ASSUM_TAC(REWRITE_RULE[UNIONS_IMAGE]) THEN EQ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN DISCH_THEN(X_CHOOSE_THEN `z:real^M` STRIP_ASSUME_TAC) THEN ASM_REWRITE_TAC[] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `z:real^M`)) THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET)) THEN ASM SET_TAC[]; X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `(uu:real^M->real^M->bool) x` THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE_GEN THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`)) THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET)) THEN REPEAT STRIP_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)); FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHISM])] THEN ASM SET_TAC[]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [pairwise]) THEN REWRITE_TAC[pairwise; RIGHT_FORALL_IMP_THM; IMP_CONJ] THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN SET_TAC[]; X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`)) THEN ASM_REWRITE_TAC[] THEN REPEAT(DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC)) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] (REWRITE_RULE[IMP_IMP] MONO_EXISTS))) THEN GEN_TAC THEN DISCH_THEN(fun th -> MP_TAC th THEN MP_TAC th) THEN GEN_REWRITE_TAC LAND_CONV [HOMEOMORPHISM] THEN STRIP_TAC THEN DISCH_THEN(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN ASM_REWRITE_TAC[SUBSET_RESTRICT] THEN ASM SET_TAC[]]);; let CLOSED_LOCAL_HOMEOMORPHISM_IMP_COVERING_MAP = prove (`!p:real^M->real^N c s. (!x. connected_component c x = {x} ==> c = {x}) /\ IMAGE p c = s /\ (!k. closed_in (subtopology euclidean c) k ==> closed_in (subtopology euclidean s) (IMAGE p k)) /\ (!x. x IN c ==> ?t u q. x IN t /\ open_in (subtopology euclidean c) t /\ open_in (subtopology euclidean s) u /\ homeomorphism (t,u) (p,q)) ==> covering_space (c,p) s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC PROPER_LOCAL_HOMEOMORPHISM_IMP_COVERING_MAP THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CLOSED_LOCAL_HOMEOMORPHISM_IMP_PROPER THEN ASM_REWRITE_TAC[SUBSET_REFL]);; let PROPER_LOCAL_HOMEOMORPHISM_GLOBAL = prove (`!f:real^M->real^N s t. path_connected s /\ simply_connected t /\ (s = {} ==> t = {}) /\ (!k. k SUBSET t /\ compact k ==> compact {x | x IN s /\ f x IN k}) /\ (!x. x IN s ==> ?u v q. x IN u /\ open_in (subtopology euclidean s) u /\ open_in (subtopology euclidean t) v /\ homeomorphism (u,v) (f,q)) ==> ?g. homeomorphism (s,t) (f,g)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_REWRITE_TAC[] THENL [ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN STRIP_TAC THEN ASM_REWRITE_TAC[HOMEOMORPHISM; NOT_IN_EMPTY; CONTINUOUS_ON_EMPTY] THEN SET_TAC[]; STRIP_TAC] THEN MATCH_MP_TAC COVERING_SPACE_HOMEOMORPHISM THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC PROPER_LOCAL_HOMEOMORPHISM_IMP_COVERING_MAP THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(MP_TAC o MATCH_MP SIMPLY_CONNECTED_IMP_CONNECTED) THEN REWRITE_TAC[CONNECTED_CLOPEN] THEN DISCH_THEN(MP_TAC o SPEC `IMAGE (f:real^M->real^N) s`) THEN ASM_REWRITE_TAC[IMAGE_EQ_EMPTY] THEN DISCH_THEN MATCH_MP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP LOCAL_HOMEOMORPHISM_IMP_OPEN_MAP) THEN DISCH_THEN(MP_TAC o SPEC `s:real^M->bool`) THEN REWRITE_TAC[OPEN_IN_REFL] THEN DISCH_THEN(MP_TAC o MATCH_MP PROPER_MAP o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ASM_SIMP_TAC[CLOSED_IN_REFL] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP LOCAL_HOMEOMORPHISM_IMP_OPEN_MAP) THEN REWRITE_TAC[OPEN_IN_REFL]);; let CLOSED_LOCAL_HOMEOMORPHISM_GLOBAL = prove (`!f:real^M->real^N s t. path_connected s /\ simply_connected t /\ (s = {} ==> t = {}) /\ (!c. closed_in (subtopology euclidean s) c ==> closed_in (subtopology euclidean t) (IMAGE f c)) /\ (!x. x IN s ==> ?u v g. x IN u /\ open_in (subtopology euclidean s) u /\ open_in (subtopology euclidean t) v /\ homeomorphism (u,v) (f,g)) ==> ?g. homeomorphism (s,t) (f,g)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THENL [ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN STRIP_TAC THEN ASM_REWRITE_TAC[HOMEOMORPHISM; NOT_IN_EMPTY; CONTINUOUS_ON_EMPTY] THEN SET_TAC[]; STRIP_TAC] THEN MATCH_MP_TAC COVERING_SPACE_HOMEOMORPHISM THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CLOSED_LOCAL_HOMEOMORPHISM_IMP_COVERING_MAP THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [X_GEN_TAC `x:real^M` THEN ASM_CASES_TAC `connected_component s (x:real^M) = {}` THEN ASM_REWRITE_TAC[NOT_INSERT_EMPTY] THEN RULE_ASSUM_TAC(REWRITE_RULE[CONNECTED_COMPONENT_EQ_EMPTY]) THEN ASM_MESON_TAC[CONNECTED_CONNECTED_COMPONENT_SET; PATH_CONNECTED_IMP_CONNECTED]; FIRST_ASSUM(MP_TAC o MATCH_MP SIMPLY_CONNECTED_IMP_CONNECTED) THEN REWRITE_TAC[CONNECTED_CLOPEN] THEN DISCH_THEN(MP_TAC o SPEC `IMAGE (f:real^M->real^N) s`) THEN ASM_REWRITE_TAC[IMAGE_EQ_EMPTY] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_SIMP_TAC[CLOSED_IN_REFL] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP LOCAL_HOMEOMORPHISM_IMP_OPEN_MAP) THEN REWRITE_TAC[OPEN_IN_REFL]]);; let PROPER_LOCALLY_INJECTIVE_OPEN_IMP_COVERING_MAP = prove (`!p:real^M->real^N c s. p continuous_on c /\ IMAGE p c = s /\ (!k. k SUBSET s /\ compact k ==> compact {x | x IN c /\ p x IN k}) /\ (!u. open_in (subtopology euclidean c) u ==> open_in (subtopology euclidean s) (IMAGE p u)) /\ (!x. x IN c ==> ?t. x IN t /\ open_in (subtopology euclidean c) t /\ (!y z. y IN t /\ z IN t /\ p y = p z ==> y = z)) ==> covering_space (c,p) s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC PROPER_LOCAL_HOMEOMORPHISM_IMP_COVERING_MAP THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real^M->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `IMAGE (p:real^M->real^N) t` THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC HOMEOMORPHISM_INJECTIVE_OPEN_MAP THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ASM SET_TAC[]; X_GEN_TAC `u:real^M->bool` THEN DISCH_TAC THEN MATCH_MP_TAC OPEN_IN_SUBSET_TRANS THEN EXISTS_TAC `s:real^N->bool` THEN CONJ_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[OPEN_IN_TRANS]; REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET)) THEN ASM SET_TAC[]]]);; let PROPER_LOCALLY_INJECTIVE_OPEN_IMP_COVERING_MAP_GEN = prove (`!p:real^M->real^N c s. p continuous_on c /\ IMAGE p c = s /\ (!k. k SUBSET s /\ compact k ==> compact {x | x IN c /\ p x IN k}) /\ (!u. open_in (subtopology euclidean c) u ==> open_in (subtopology euclidean c) {x | x IN c /\ p x IN IMAGE p u}) /\ (!x. x IN c ==> ?t. x IN t /\ open_in (subtopology euclidean c) t /\ (!y z. y IN t /\ z IN t /\ p y = p z ==> y = z)) ==> covering_space (c,p) s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC PROPER_LOCALLY_INJECTIVE_OPEN_IMP_COVERING_MAP THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CLOSED_MAP_IMP_OPEN_MAP THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[PROPER_MAP; SUBSET_REFL]);; let PROPER_LOCALLY_INJECTIVE_OPEN_IMP_HOMEOMORPHISM = prove (`!f:real^M->real^N s t. path_connected s /\ simply_connected t /\ f continuous_on s /\ IMAGE f s = t /\ (!k. k SUBSET t /\ compact k ==> compact {x | x IN s /\ f x IN k}) /\ (!u. open_in (subtopology euclidean s) u ==> open_in (subtopology euclidean t) (IMAGE f u)) /\ (!x. x IN s ==> ?u. x IN u /\ open_in (subtopology euclidean s) u /\ !y z. y IN u /\ z IN u /\ f y = f z ==> y = z) ==> ?g. homeomorphism (s,t) (f,g)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC COVERING_SPACE_HOMEOMORPHISM THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC PROPER_LOCALLY_INJECTIVE_OPEN_IMP_COVERING_MAP THEN ASM_REWRITE_TAC[]);; let PROPER_LOCALLY_INJECTIVE_OPEN_IMP_HOMEOMORPHISM_GEN = prove (`!f:real^M->real^N s t. path_connected s /\ simply_connected t /\ f continuous_on s /\ IMAGE f s = t /\ (!k. k SUBSET t /\ compact k ==> compact {x | x IN s /\ f x IN k}) /\ (!u. open_in (subtopology euclidean s) u ==> open_in (subtopology euclidean s) {x | x IN s /\ f x IN IMAGE f u}) /\ (!x. x IN s ==> ?u. x IN u /\ open_in (subtopology euclidean s) u /\ !y z. y IN u /\ z IN u /\ f y = f z ==> y = z) ==> ?g. homeomorphism (s,t) (f,g)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC COVERING_SPACE_HOMEOMORPHISM THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC PROPER_LOCALLY_INJECTIVE_OPEN_IMP_COVERING_MAP_GEN THEN ASM_REWRITE_TAC[]);; (* ------------------------------------------------------------------------- *) (* A simply connected covering space is universal. *) (* ------------------------------------------------------------------------- *) let UNIVERSAL_COVERING_SPACE = prove (`!c p:real^M->real^P c' p':real^N->real^P s. covering_space (c,p) s /\ covering_space (c',p') s /\ locally path_connected c /\ simply_connected c /\ connected c' ==> ?q. covering_space (c,q) c' /\ !x. x IN c ==> p'(q x) = p x`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^P->bool = {}` THENL [ASM_SIMP_TAC[covering_space; IMAGE_EQ_EMPTY; NOT_IN_EMPTY] THEN REWRITE_TAC[CONTINUOUS_ON_EMPTY]; FIRST_X_ASSUM(X_CHOOSE_TAC `a:real^P` o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY])] THEN STRIP_TAC THEN SUBGOAL_THEN `path_connected(c':real^N->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[COVERING_SPACE_LOCALLY_PATH_CONNECTED_EQ; PATH_CONNECTED_EQ_CONNECTED_LPC]; ALL_TAC] THEN SUBGOAL_THEN `locally connected (s:real^P->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[LOCALLY_PATH_CONNECTED_IMP_LOCALLY_CONNECTED; COVERING_SPACE_LOCALLY_PATH_CONNECTED_EQ]; ALL_TAC] THEN SUBGOAL_THEN `?b:real^M b':real^N. b IN c /\ b' IN c' /\ (p:real^M->real^P) b = a /\ p' b' = a` STRIP_ASSUME_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[covering_space]) THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`p':real^N->real^P`; `c':real^N->bool`; `s:real^P->bool`; `p:real^M->real^P`; `c:real^M->bool`; `b':real^N`; `b:real^M`] COVERING_SPACE_LIFT_STRONG) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_MESON_TAC[covering_space; SUBSET_REFL]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `q:real^M->real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[covering_space] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [ASM_REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN REWRITE_TAC[SUBSET; IN_IMAGE] THEN X_GEN_TAC `z:real^N` THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [path_connected]) THEN DISCH_THEN(MP_TAC o SPECL [`b':real^N`; `z:real^N`]) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `g':real^1->real^N` THEN STRIP_TAC THEN MP_TAC(ISPECL [`p:real^M->real^P`; `c:real^M->bool`; `s:real^P->bool`; `(p':real^N->real^P) o (g':real^1->real^N)`; `b:real^M`] COVERING_SPACE_LIFT_PATH_STRONG) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_REWRITE_TAC[PATHSTART_COMPOSE; PATH_IMAGE_COMPOSE] THEN CONJ_TAC THENL [MATCH_MP_TAC PATH_CONTINUOUS_IMAGE; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[covering_space]) THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ASM SET_TAC[]]; REWRITE_TAC[o_THM] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^1->real^M` STRIP_ASSUME_TAC)] THEN MP_TAC(ISPECL [`p':real^N->real^P`; `c':real^N->bool`; `s:real^P->bool`; `(p':real^N->real^P) o (g':real^1->real^N)`; `(p':real^N->real^P) o q o (g:real^1->real^M)`; `g':real^1->real^N`; `(q:real^M->real^N) o (g:real^1->real^M)`] COVERING_SPACE_MONODROMY) THEN ASM_REWRITE_TAC[PATHSTART_COMPOSE; PATHFINISH_COMPOSE] THEN ANTS_TAC THENL [REWRITE_TAC[o_THM; PATH_IMAGE_COMPOSE] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC PATH_CONTINUOUS_IMAGE THEN RULE_ASSUM_TAC(REWRITE_RULE[covering_space]) THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; RULE_ASSUM_TAC(REWRITE_RULE[covering_space]) THEN ASM SET_TAC[]; MATCH_MP_TAC PATH_CONTINUOUS_IMAGE THEN REWRITE_TAC[PATH_IMAGE_COMPOSE] THEN CONJ_TAC THENL [MATCH_MP_TAC PATH_CONTINUOUS_IMAGE THEN RULE_ASSUM_TAC(REWRITE_RULE[covering_space]) THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; REPEAT(FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [covering_space])) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]]; RULE_ASSUM_TAC(REWRITE_RULE[covering_space]) THEN ASM SET_TAC[]; MATCH_MP_TAC HOMOTOPIC_PATHS_EQ THEN ASM_SIMP_TAC[o_THM] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC PATH_CONTINUOUS_IMAGE THEN RULE_ASSUM_TAC(REWRITE_RULE[covering_space]) THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; REWRITE_TAC[PATH_IMAGE_COMPOSE] THEN RULE_ASSUM_TAC(REWRITE_RULE[covering_space]) THEN ASM SET_TAC[]; RULE_ASSUM_TAC(REWRITE_RULE[path_image]) THEN ASM SET_TAC[]]; MATCH_MP_TAC PATH_CONTINUOUS_IMAGE THEN RULE_ASSUM_TAC(REWRITE_RULE[covering_space]) THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; RULE_ASSUM_TAC(REWRITE_RULE[covering_space]) THEN ASM SET_TAC[]]; DISCH_THEN SUBST1_TAC THEN EXISTS_TAC `pathfinish(g:real^1->real^M)` THEN ASM_MESON_TAC[PATHFINISH_IN_PATH_IMAGE; SUBSET]]; DISCH_TAC] THEN X_GEN_TAC `z:real^N` THEN DISCH_TAC THEN SUBGOAL_THEN `?t vv vv'. connected t /\ open_in (subtopology euclidean s) t /\ (p':real^N->real^P) z IN t /\ UNIONS vv = {x:real^M | x IN c /\ p x IN t} /\ (!u. u IN vv ==> open_in (subtopology euclidean c) u) /\ pairwise DISJOINT vv /\ (!u. u IN vv ==> (?r. homeomorphism (u,t) (p,r))) /\ UNIONS vv' = {x:real^N | x IN c' /\ p' x IN t} /\ (!u. u IN vv' ==> open_in (subtopology euclidean c') u) /\ pairwise DISJOINT vv' /\ (!u. u IN vv' ==> (?r'. homeomorphism (u,t) (p',r')))` STRIP_ASSUME_TAC THENL [UNDISCH_TAC `covering_space (c,p:real^M->real^P) s` THEN REWRITE_TAC[covering_space] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(MP_TAC o SPEC `(p':real^N->real^P) z`) THEN ANTS_TAC THENL [FIRST_ASSUM(MP_TAC o MATCH_MP COVERING_SPACE_IMP_SURJECTIVE) THEN ASM SET_TAC[]; DISCH_THEN(X_CHOOSE_THEN `t:real^P->bool` MP_TAC) THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(X_CHOOSE_THEN `vv:(real^M->bool)->bool` STRIP_ASSUME_TAC)] THEN UNDISCH_TAC `covering_space (c',p':real^N->real^P) s` THEN REWRITE_TAC[covering_space] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(MP_TAC o SPEC `(p':real^N->real^P) z`) THEN ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN(X_CHOOSE_THEN `t':real^P->bool` MP_TAC) THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(X_CHOOSE_THEN `vv':(real^N->bool)->bool` STRIP_ASSUME_TAC)] THEN ABBREV_TAC `u = connected_component (t INTER t') ((p':real^N->real^P) z)` THEN SUBGOAL_THEN `(u:real^P->bool) SUBSET t INTER t'` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[CONNECTED_COMPONENT_SUBSET]; ALL_TAC] THEN MAP_EVERY EXISTS_TAC [`u:real^P->bool`; `{{x | x IN v /\ (p:real^M->real^P) x IN u} | v IN vv}`; `{{x | x IN v /\ (p':real^N->real^P) x IN u} | v IN vv'}`] THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[CONNECTED_CONNECTED_COMPONENT]; MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `t INTER t':real^P->bool` THEN ASM_SIMP_TAC[OPEN_IN_INTER] THEN EXPAND_TAC "u" THEN MATCH_MP_TAC OPEN_IN_CONNECTED_COMPONENT_LOCALLY_CONNECTED THEN ASM_MESON_TAC[LOCALLY_INTER_OPEN; LOCALLY_OPEN_SUBSET]; REWRITE_TAC[IN] THEN EXPAND_TAC "u" THEN REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ] THEN ASM_REWRITE_TAC[IN_INTER]; REWRITE_TAC[SET_RULE `{x | x IN v /\ P x} = v INTER {x | P x}`] THEN ASM_REWRITE_TAC[GSYM INTER_UNIONS] THEN ASM SET_TAC[]; REWRITE_TAC[FORALL_IN_GSPEC] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE_SUBSET THEN EXISTS_TAC `s:real^P->bool` THEN ASM_SIMP_TAC[SUBSET_REFL] THEN MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `t INTER t':real^P->bool` THEN ASM_SIMP_TAC[OPEN_IN_INTER] THEN EXPAND_TAC "u" THEN MATCH_MP_TAC OPEN_IN_CONNECTED_COMPONENT_LOCALLY_CONNECTED THEN ASM_MESON_TAC[LOCALLY_INTER_OPEN; LOCALLY_OPEN_SUBSET]; REWRITE_TAC[PAIRWISE_IMAGE; SIMPLE_IMAGE] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] PAIRWISE_IMP)) THEN SET_TAC[]; REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `v:real^M->bool` THEN DISCH_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `v:real^M->bool`)) THEN ASM_REWRITE_TAC[] THEN REPEAT DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o check(is_exists o concl)) THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN DISCH_THEN(fun th -> STRIP_ASSUME_TAC(REWRITE_RULE[homeomorphism] th) THEN MP_TAC th THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] HOMEOMORPHISM_OF_SUBSETS)) THEN ASM SET_TAC[]; REWRITE_TAC[SET_RULE `{x | x IN v /\ P x} = v INTER {x | P x}`] THEN ASM_REWRITE_TAC[GSYM INTER_UNIONS] THEN ASM SET_TAC[]; REWRITE_TAC[FORALL_IN_GSPEC] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE_SUBSET THEN EXISTS_TAC `s:real^P->bool` THEN ASM_SIMP_TAC[SUBSET_REFL] THEN MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `t INTER t':real^P->bool` THEN ASM_SIMP_TAC[OPEN_IN_INTER] THEN EXPAND_TAC "u" THEN MATCH_MP_TAC OPEN_IN_CONNECTED_COMPONENT_LOCALLY_CONNECTED THEN ASM_MESON_TAC[LOCALLY_INTER_OPEN; LOCALLY_OPEN_SUBSET]; REWRITE_TAC[PAIRWISE_IMAGE; SIMPLE_IMAGE] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] PAIRWISE_IMP)) THEN SET_TAC[]; REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `v:real^N->bool` THEN DISCH_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `v:real^N->bool`)) THEN ASM_REWRITE_TAC[] THEN REPEAT DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o check(is_exists o concl)) THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN DISCH_THEN(fun th -> STRIP_ASSUME_TAC(REWRITE_RULE[homeomorphism] th) THEN MP_TAC th THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] HOMEOMORPHISM_OF_SUBSETS)) THEN ASM SET_TAC[]]; ALL_TAC] THEN SUBGOAL_THEN `?v':real^N->bool. v' IN vv' /\ z IN v'` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `v' SUBSET {x | x IN c' /\ (p':real^N->real^P) x IN t}` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `open_in (subtopology euclidean c') (v':real^N->bool) /\ ?r':real^P->real^N. homeomorphism (v',t) (p',r')` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN EXISTS_TAC `v':real^N->bool` THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `!u. u IN vv ==> ?u'. u' IN vv' /\ IMAGE (q:real^M->real^N) u SUBSET u'` ASSUME_TAC THENL [X_GEN_TAC `u:real^M->bool` THEN DISCH_TAC THEN SUBGOAL_THEN `IMAGE (q:real^M->real^N) u SUBSET UNIONS vv'` MP_TAC THENL [ASM_REWRITE_TAC[IN_ELIM_THM; SUBSET; FORALL_IN_IMAGE] THEN ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `~(UNIONS uu = {}) /\ (!u. u IN uu ==> DISJOINT s u \/ DISJOINT s (UNIONS(uu DELETE u))) ==> s SUBSET UNIONS uu ==> ?u. u IN uu /\ s SUBSET u`) THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN X_GEN_TAC `v:real^N->bool` THEN DISCH_TAC THEN SUBGOAL_THEN `connected(IMAGE (q:real^M->real^N) u)` MP_TAC THENL [MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; OPEN_IN_IMP_SUBSET]; ALL_TAC] THEN UNDISCH_TAC `connected(t:real^P->bool)` THEN MATCH_MP_TAC EQ_IMP THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC HOMEOMORPHIC_CONNECTEDNESS THEN REWRITE_TAC[homeomorphic] THEN ASM_MESON_TAC[]; REWRITE_TAC[CONNECTED_OPEN_IN; NOT_EXISTS_THM; TAUT `~(p /\ q /\ r /\ s /\ t) <=> p /\ q /\ r /\ s ==> ~t`] THEN REWRITE_TAC[DE_MORGAN_THM; DISJOINT]] THEN DISCH_THEN MATCH_MP_TAC THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC OPEN_IN_SUBTOPOLOGY_INTER_SUBSET THEN EXISTS_TAC `c':real^N->bool` THEN (CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]]) THEN MATCH_MP_TAC OPEN_IN_INTER THEN ASM_SIMP_TAC[OPEN_IN_REFL] THEN MATCH_MP_TAC OPEN_IN_UNIONS THEN ASM_SIMP_TAC[IN_DELETE]; REWRITE_TAC[GSYM UNION_OVER_INTER] THEN ASM_SIMP_TAC[GSYM UNIONS_INSERT; SET_RULE `a IN s ==> a INSERT (s DELETE a) = s`] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `s INTER t = {} ==> (u INTER s) INTER (u INTER t) = {}`) THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM UNIONS_1] THEN W(MP_TAC o PART_MATCH (lhand o rand) INTER_UNIONS_PAIRWISE_DISJOINT o lhand o snd) THEN ASM_SIMP_TAC[SET_RULE `a IN s ==> {a} UNION (s DELETE a) = s`] THEN SET_TAC[]]; ALL_TAC] THEN EXISTS_TAC `{v:real^M->bool | v IN vv /\ IMAGE (q:real^M->real^N) v SUBSET v'}` THEN ASM_SIMP_TAC[FORALL_IN_GSPEC] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [ASM SET_TAC[]; GEN_REWRITE_TAC I [SUBSET]] THEN REWRITE_TAC[IN_UNIONS; IN_ELIM_THM] THEN X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN SUBGOAL_THEN `(x:real^M) IN UNIONS vv` MP_TAC THENL [ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; REWRITE_TAC[IN_UNIONS]] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^M->bool` THEN RULE_ASSUM_TAC(REWRITE_RULE[pairwise]) THEN ASM SET_TAC[]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] PAIRWISE_MONO)) THEN SET_TAC[]; X_GEN_TAC `v:real^M->bool` THEN STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `v:real^M->bool`)) THEN ASM_REWRITE_TAC[] THEN REPEAT DISCH_TAC THEN FIRST_X_ASSUM(X_CHOOSE_TAC `r:real^P->real^M`) THEN EXISTS_TAC `(r:real^P->real^M) o (p':real^N->real^P)` THEN MATCH_MP_TAC HOMEOMORPHISM_EQ THEN MAP_EVERY EXISTS_TAC [`(r':real^P->real^N) o (p:real^M->real^P)`; `(r:real^P->real^M) o (p':real^N->real^P)`] THEN CONJ_TAC THENL [MATCH_MP_TAC HOMEOMORPHISM_COMPOSE THEN ASM_MESON_TAC[HOMEOMORPHISM_SYM]; RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN REWRITE_TAC[o_DEF] THEN ASM SET_TAC[]]]);; (* ------------------------------------------------------------------------- *) (* Size of fundamental group of a covering space (this could be generalized *) (* with structural properties of the bijections of course). *) (* ------------------------------------------------------------------------- *) let CARD_EQ_FUNDAMENTAL_GROUP_COVERING_SPACE_ALT = prove (`!p:real^M->real^N c s a. covering_space(c,p) s /\ path_connected c /\ a IN c ==> fundamental_group(s,p a) =_c {homotopic_paths c g |g| path g /\ path_image g SUBSET c /\ pathstart g = a /\ p(pathfinish g) = p a}`, let tac = REPEAT (FIRST_ASSUM(ASSUME_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_PATH) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_SUBSET) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_PATHSTART) THEN FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP HOMOTOPIC_PATHS_IMP_PATHFINISH)) THEN ASM_REWRITE_TAC[PATHSTART_COMPOSE; PATHFINISH_COMPOSE] THEN RULE_ASSUM_TAC(REWRITE_RULE[covering_space]) THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; PATH_CONTINUOUS_IMAGE]; ALL_TAC] THEN REWRITE_TAC[PATH_IMAGE_COMPOSE] THEN ASM SET_TAC[] in REPEAT STRIP_TAC THEN REWRITE_TAC[fundamental_group] THEN ONCE_REWRITE_TAC[CARD_EQ_SYM] THEN REWRITE_TAC[eq_c] THEN EXISTS_TAC `homotopic_paths s o ((o) (p:real^M->real^N)) o (@)` THEN REWRITE_TAC[FORALL_IN_GSPEC; EXISTS_IN_GSPEC; EXISTS_UNIQUE_DEF; GSYM CONJ_ASSOC; IMP_CONJ; o_THM; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC] THEN CONJ_TAC THENL [X_GEN_TAC `g:real^1->real^M` THEN STRIP_TAC THEN SUBGOAL_THEN `homotopic_paths (c:real^M->bool) g ((@) (homotopic_paths c g))` MP_TAC THENL [GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM ETA_AX] THEN CONV_TAC SELECT_CONV THEN ASM_MESON_TAC[HOMOTOPIC_PATHS_REFL]; ABBREV_TAC `h:real^1->real^M = (@) (homotopic_paths c g)` THEN ONCE_REWRITE_TAC[HOMOTOPIC_PATHS_SYM] THEN DISCH_TAC] THEN MATCH_MP_TAC(SET_RULE `P q ==> homotopic_paths s q IN {homotopic_paths s p | P p}`) THEN tac; X_GEN_TAC `h:real^1->real^N` THEN STRIP_TAC THEN CONJ_TAC THENL [FIRST_ASSUM(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] COVERING_SPACE_LIFT_PATH_STRONG)) THEN DISCH_THEN(MP_TAC o SPECL [`h:real^1->real^N`; `a:real^M`]) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^1->real^M` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[pathfinish; ENDS_IN_UNIT_INTERVAL]; ALL_TAC] THEN SUBGOAL_THEN `homotopic_paths (c:real^M->bool) g ((@) (homotopic_paths c g))` MP_TAC THENL [GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM ETA_AX] THEN CONV_TAC SELECT_CONV THEN ASM_MESON_TAC[HOMOTOPIC_PATHS_REFL]; ABBREV_TAC `g':real^1->real^M = (@) (homotopic_paths c g)` THEN ONCE_REWRITE_TAC[HOMOTOPIC_PATHS_SYM] THEN DISCH_TAC] THEN ONCE_REWRITE_TAC[FUN_EQ_THM] THEN MATCH_MP_TAC(MESON[HOMOTOPIC_PATHS_SYM; HOMOTOPIC_PATHS_TRANS] `homotopic_paths s p q ==> !r. homotopic_paths s p r <=> homotopic_paths s q r`) THEN TRANS_TAC HOMOTOPIC_PATHS_TRANS `(p:real^M->real^N) o (g:real^1->real^M)` THEN CONJ_TAC THENL [MATCH_MP_TAC HOMOTOPIC_PATHS_CONTINUOUS_IMAGE THEN EXISTS_TAC `c:real^M->bool` THEN ASM_REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[covering_space]) THEN ASM SET_TAC[]; MATCH_MP_TAC HOMOTOPIC_PATHS_EQ THEN ASM_REWRITE_TAC[o_THM] THEN tac]; X_GEN_TAC `g:real^1->real^M` THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(SUBST1_TAC o SYM) THEN X_GEN_TAC `g':real^1->real^M` THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN SUBGOAL_THEN `homotopic_paths (c:real^M->bool) g ((@) (homotopic_paths c g))` MP_TAC THENL [GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM ETA_AX] THEN CONV_TAC SELECT_CONV THEN ASM_MESON_TAC[HOMOTOPIC_PATHS_REFL]; ABBREV_TAC `k:real^1->real^M = (@) (homotopic_paths c g)` THEN ONCE_REWRITE_TAC[HOMOTOPIC_PATHS_SYM] THEN DISCH_TAC] THEN SUBGOAL_THEN `homotopic_paths (c:real^M->bool) g' ((@) (homotopic_paths c g'))` MP_TAC THENL [GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM ETA_AX] THEN CONV_TAC SELECT_CONV THEN ASM_MESON_TAC[HOMOTOPIC_PATHS_REFL]; ABBREV_TAC `k':real^1->real^M = (@) (homotopic_paths c g')` THEN ONCE_REWRITE_TAC[HOMOTOPIC_PATHS_SYM] THEN DISCH_TAC] THEN DISCH_THEN(MP_TAC o C AP_THM `(p:real^M->real^N) o (k:real^1->real^M)`) THEN DISCH_THEN(MP_TAC o snd o EQ_IMP_RULE) THEN REWRITE_TAC[HOMOTOPIC_PATHS_REFL] THEN ANTS_TAC THENL [tac; ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhand o rand) COVERING_SPACE_HOMOTOPIC_PATHS_CANCEL_EQ o lhand o snd) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [tac; DISCH_THEN SUBST1_TAC] THEN REWRITE_TAC[FUN_EQ_THM] THEN ASM_MESON_TAC[HOMOTOPIC_PATHS_SYM; HOMOTOPIC_PATHS_TRANS]]]);; let CARD_EQ_FUNDAMENTAL_GROUP_COVERING_SPACE = prove (`!p:real^M->real^N c s a. covering_space(c,p) s /\ path_connected c /\ a IN c ==> fundamental_group(s,p a) =_c fundamental_group(c,a) *_c {a' | a' IN c /\ p a' = p a}`, let lemma = prove (`!g:real^1->real^N. path g /\ path_image g SUBSET c ==> homotopic_paths c g ((@) (homotopic_paths c g)) /\ path ((@) (homotopic_paths c g)) /\ path_image ((@) (homotopic_paths c g)) SUBSET c /\ pathstart ((@) (homotopic_paths c g)) = pathstart g /\ pathfinish ((@) (homotopic_paths c g)) = pathfinish g`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM ETA_AX] THEN CONV_TAC SELECT_CONV THEN ASM_MESON_TAC[HOMOTOPIC_PATHS_REFL]; ASM MESON_TAC[HOMOTOPIC_PATHS_IMP_PATH; HOMOTOPIC_PATHS_IMP_SUBSET; HOMOTOPIC_PATHS_IMP_PATHSTART; HOMOTOPIC_PATHS_IMP_PATHFINISH]]) in REPEAT STRIP_TAC THEN SUBGOAL_THEN `!b:real^M. b IN c ==> ?g. path g /\ path_image g SUBSET c /\ pathstart g = a /\ pathfinish g = b` MP_TAC THENL [ASM_MESON_TAC[path_connected]; ALL_TAC] THEN REWRITE_TAC[RIGHT_IMP_EXISTS_THM; SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `f:real^M->real^1->real^M` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `a:real^M` o MATCH_MP (REWRITE_RULE[IMP_CONJ] CARD_EQ_FUNDAMENTAL_GROUP_COVERING_SPACE_ALT)) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] CARD_EQ_TRANS) THEN REWRITE_TAC[EQ_C_BIJECTIONS] THEN REWRITE_TAC[mul_c; FORALL_IN_GSPEC] THEN MAP_EVERY EXISTS_TAC [`\g. homotopic_paths c ((@) g ++ reversepath(f(pathfinish((@) g):real^M))),pathfinish((@) g)`; `\(g,b:real^M). homotopic_paths (c:real^M->bool) ((@) g ++ f b)`] THEN REWRITE_TAC[fundamental_group; IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_GSPEC; IN_ELIM_PAIR_THM] THEN CONJ_TAC THEN X_GEN_TAC `g:real^1->real^M` THEN REPEAT DISCH_TAC THEN (SUBGOAL_THEN `(pathfinish g:real^M) IN c` ASSUME_TAC THENL [ASM_MESON_TAC[PATHFINISH_IN_PATH_IMAGE; SUBSET]; ALL_TAC]) THEN MP_TAC(ISPEC `g:real^1->real^M` lemma) THEN ABBREV_TAC `g':real^1->real^M = (@) (homotopic_paths c g)` THEN ASM_REWRITE_TAC[GSYM CONJ_ASSOC] THEN STRIP_TAC THEN TRY(X_GEN_TAC `b:real^M` THEN REPEAT DISCH_TAC) THEN (CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `P q ==> homotopic_paths s q IN {homotopic_paths s p | P p}`) THEN ASM_SIMP_TAC[PATHSTART_JOIN; PATHFINISH_JOIN; PATHFINISH_REVERSEPATH; PATH_JOIN; PATH_REVERSEPATH; SUBSET_PATH_IMAGE_JOIN; PATHSTART_REVERSEPATH; PATH_IMAGE_REVERSEPATH]; ALL_TAC]) THEN ASM_REWRITE_TAC[IN_ELIM_THM] THENL [MP_TAC(ISPEC `g' ++ reversepath(f(pathfinish g')):real^1->real^M` lemma) THEN ASM_SIMP_TAC[PATH_JOIN; SUBSET_PATH_IMAGE_JOIN; PATH_IMAGE_REVERSEPATH; PATH_REVERSEPATH; PATHSTART_REVERSEPATH] THEN ABBREV_TAC `g'':real^1->real^M = (@) (homotopic_paths c (g' ++ reversepath(f(pathfinish(g:real^1->real^M)))))` THEN ASM_SIMP_TAC[PATHSTART_JOIN; PATHFINISH_JOIN; PATHFINISH_REVERSEPATH] THEN STRIP_TAC; MP_TAC(ISPEC `g' ++ f(b:real^M):real^1->real^M` lemma) THEN ASM_SIMP_TAC[PATH_JOIN; SUBSET_PATH_IMAGE_JOIN] THEN ABBREV_TAC `g'':real^1->real^M = (@) (homotopic_paths c (g' ++ f(b:real^M):real^1->real^M))` THEN ASM_SIMP_TAC[PATHSTART_JOIN; PATHFINISH_JOIN; PAIR_EQ] THEN STRIP_TAC] THEN ONCE_REWRITE_TAC[FUN_EQ_THM] THEN MATCH_MP_TAC(MESON[HOMOTOPIC_PATHS_SYM; HOMOTOPIC_PATHS_TRANS] `homotopic_paths s p q ==> !r. homotopic_paths s p r <=> homotopic_paths s q r`) THENL [TRANS_TAC HOMOTOPIC_PATHS_TRANS `(g' ++ reversepath(f(pathfinish g:real^M))) ++ f(pathfinish g):real^1->real^M` THEN CONJ_TAC THENL [MATCH_MP_TAC HOMOTOPIC_PATHS_JOIN THEN ASM_SIMP_TAC[HOMOTOPIC_PATHS_REFL] THEN ASM_MESON_TAC[HOMOTOPIC_PATHS_SYM]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (rator o rand) (ONCE_REWRITE_RULE[HOMOTOPIC_PATHS_SYM] HOMOTOPIC_PATHS_ASSOC) o rator o snd) THEN ASM_SIMP_TAC[PATHSTART_REVERSEPATH; PATHFINISH_REVERSEPATH; PATH_REVERSEPATH; PATH_IMAGE_REVERSEPATH] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_PATHS_TRANS) THEN TRANS_TAC HOMOTOPIC_PATHS_TRANS `g' ++ linepath(pathfinish(f(pathfinish g:real^M)):real^M, pathfinish(f(pathfinish g)))` THEN CONJ_TAC THENL [MATCH_MP_TAC HOMOTOPIC_PATHS_JOIN THEN ASM_SIMP_TAC[HOMOTOPIC_PATHS_LINV; HOMOTOPIC_PATHS_REFL; PATHSTART_REVERSEPATH; PATHSTART_JOIN]; ALL_TAC] THEN TRANS_TAC HOMOTOPIC_PATHS_TRANS `g':real^1->real^M` THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[HOMOTOPIC_PATHS_SYM]] THEN ASM_SIMP_TAC[] THEN SUBST1_TAC(SYM(ASSUME `pathfinish g':real^M = pathfinish g`)) THEN MATCH_MP_TAC HOMOTOPIC_PATHS_RID THEN ASM_REWRITE_TAC[]; TRANS_TAC HOMOTOPIC_PATHS_TRANS `(g' ++ f(b:real^M)) ++ reversepath(f b):real^1->real^M` THEN CONJ_TAC THENL [MATCH_MP_TAC HOMOTOPIC_PATHS_JOIN THEN ASM_SIMP_TAC[HOMOTOPIC_PATHS_REFL; PATH_REVERSEPATH; PATH_IMAGE_REVERSEPATH; PATHSTART_REVERSEPATH] THEN ASM_MESON_TAC[HOMOTOPIC_PATHS_SYM]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (rator o rand) (ONCE_REWRITE_RULE[HOMOTOPIC_PATHS_SYM] HOMOTOPIC_PATHS_ASSOC) o rator o snd) THEN ASM_SIMP_TAC[PATHSTART_REVERSEPATH; PATHFINISH_REVERSEPATH; PATH_REVERSEPATH; PATH_IMAGE_REVERSEPATH] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_PATHS_TRANS) THEN TRANS_TAC HOMOTOPIC_PATHS_TRANS `g' ++ linepath(pathstart(f(b:real^M)),pathstart(f b)) :real^1->real^M` THEN CONJ_TAC THENL [MATCH_MP_TAC HOMOTOPIC_PATHS_JOIN THEN ASM_SIMP_TAC[HOMOTOPIC_PATHS_RINV; HOMOTOPIC_PATHS_REFL; PATHSTART_REVERSEPATH; PATHSTART_JOIN]; ALL_TAC] THEN TRANS_TAC HOMOTOPIC_PATHS_TRANS `g':real^1->real^M` THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[HOMOTOPIC_PATHS_SYM]] THEN ASM_SIMP_TAC[] THEN SUBST1_TAC(SYM(ASSUME `pathfinish g':real^M = a`)) THEN MATCH_MP_TAC HOMOTOPIC_PATHS_RID THEN ASM_REWRITE_TAC[]]);; let COVERING_SPACE_SELF_FINITE_FUNDAMENTAL_GROUP = prove (`!p s a:real^N. covering_space (s,p) s /\ path_connected s /\ a IN s /\ FINITE(fundamental_group(s,a)) ==> ?q. homeomorphism (s,s) (p,q)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHISM_INJECTIVE_OPEN_MAP THEN REPEAT CONJ_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[covering_space]) THEN ASM_MESON_TAC[]; RULE_ASSUM_TAC(REWRITE_RULE[covering_space]) THEN ASM_MESON_TAC[]; ALL_TAC; ASM_MESON_TAC[COVERING_SPACE_OPEN_MAP]] THEN MAP_EVERY X_GEN_TAC [`b:real^N`; `c:real^N`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `a:real^N`; `b:real^N`] CARD_EQ_FUNDAMENTAL_GROUPS_BASEPOINTS) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o MATCH_MP CARD_FINITE_CONG) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `(p:real^N->real^N) b`; `b:real^N`] CARD_EQ_FUNDAMENTAL_GROUPS_BASEPOINTS) THEN ANTS_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[covering_space]) THEN ASM SET_TAC[]; DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP CARD_FINITE_CONG)] THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN MP_TAC(ISPECL [`p:real^N->real^N`; `s:real^N->bool`;` s:real^N->bool`; `b:real^N`] CARD_EQ_FUNDAMENTAL_GROUP_COVERING_SPACE) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(fun th -> MP_TAC(MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] CARD_EQ_CARD_IMP) th) THEN MP_TAC(MATCH_MP CARD_FINITE_CONG th)) THEN ASM_REWRITE_TAC[CARD_MUL_FINITE_EQ; FUNDAMENTAL_GROUP_EQ_EMPTY] THEN DISCH_THEN(ASSUME_TAC o MATCH_MP (MESON[FINITE_EMPTY] `s = {} \/ FINITE s ==> FINITE s`)) THEN ASM_SIMP_TAC[CARD_MUL_C] THEN FIRST_ASSUM (MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] CARD_EQ_CARD_IMP)) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN ASM_REWRITE_TAC[NUM_RING `a = a * b <=> a = 0 \/ b = 1`] THEN ASM_SIMP_TAC[CARD_EQ_0; FUNDAMENTAL_GROUP_EQ_EMPTY] THEN ASM_SIMP_TAC[MESON[HAS_SIZE] `FINITE s ==> (CARD s = n <=> s HAS_SIZE n)`] THEN CONV_TAC(LAND_CONV HAS_SIZE_CONV) THEN ASM SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* Invariance of dimension and domain in setting of R^n. *) (* ------------------------------------------------------------------------- *) let HOMEOMORPHIC_SUBSPACE_EUCLIDEAN_SPACE = prove (`!(s:real^N->bool) n. subspace s ==> (subtopology euclidean s homeomorphic_space euclidean_space n <=> dim s = n)`, REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN GEN_TAC THEN DISCH_TAC THEN SUBGOAL_THEN `subtopology euclidean (s:real^N->bool) homeomorphic_space euclidean_space (dim s)` ASSUME_TAC THENL [TRANS_TAC HOMEOMORPHIC_SPACE_TRANS `subtopology euclidean (span (IMAGE basis (1..dim(s:real^N->bool))):real^N->bool)` THEN CONJ_TAC THENL [REWRITE_TAC[HOMEOMORPHIC_SPACE_EUCLIDEAN] THEN MATCH_MP_TAC HOMEOMORPHIC_SUBSPACES THEN ASM_REWRITE_TAC[SUBSPACE_SPAN; DIM_SPAN; DIM_BASIS_IMAGE] THEN REWRITE_TAC[INTER_NUMSEG; ARITH_RULE `MAX 1 1 = 1`] THEN SIMP_TAC[DIM_SUBSET_UNIV; ARITH_RULE `s <= n ==> MIN n s = s`] THEN REWRITE_TAC[CARD_NUMSEG_1]; ONCE_REWRITE_TAC[HOMEOMORPHIC_SPACE_SYM] THEN MP_TAC(SPEC `dim(s:real^N->bool)` HOMEOMORPHIC_MAPS_EUCLIDEAN_SPACE_EUCLIDEAN_GEN) THEN REWRITE_TAC[homeomorphic_space; DIM_SUBSET_UNIV] THEN MESON_TAC[]]; X_GEN_TAC `n:num` THEN EQ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN REWRITE_TAC[GSYM INVARIANCE_OF_DIMENSION_EUCLIDEAN_SPACE] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] HOMEOMORPHIC_SPACE_TRANS) THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SPACE_SYM] THEN ASM_REWRITE_TAC[]]);; let HOMEOMORPHIC_SUBSPACE_EUCLIDEAN_SPACE_DIM = prove (`!(s:real^N->bool). subspace s ==> subtopology euclidean s homeomorphic_space euclidean_space (dim s)`, SIMP_TAC[HOMEOMORPHIC_SUBSPACE_EUCLIDEAN_SPACE]);; let HOMEOMORPHIC_SUBSPACES_EQ = prove (`!s:real^M->bool t:real^N->bool. subspace s /\ subspace t ==> (s homeomorphic t <=> dim s = dim t)`, REPEAT STRIP_TAC THEN EQ_TAC THENL [DISCH_TAC; ASM_MESON_TAC[HOMEOMORPHIC_SUBSPACES]] THEN REWRITE_TAC[GSYM INVARIANCE_OF_DIMENSION_EUCLIDEAN_SPACE] THEN TRANS_TAC HOMEOMORPHIC_SPACE_TRANS `subtopology euclidean (t:real^N->bool)` THEN ASM_SIMP_TAC[HOMEOMORPHIC_SUBSPACE_EUCLIDEAN_SPACE] THEN TRANS_TAC HOMEOMORPHIC_SPACE_TRANS `subtopology euclidean (s:real^M->bool)` THEN ASM_REWRITE_TAC[HOMEOMORPHIC_SPACE_EUCLIDEAN] THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SPACE_SYM] THEN ASM_SIMP_TAC[HOMEOMORPHIC_SUBSPACE_EUCLIDEAN_SPACE]);; let HOMEOMORPHIC_AFFINE_EUCLIDEAN_SPACE = prove (`!(s:real^N->bool) n. affine s ==> (subtopology euclidean s homeomorphic_space euclidean_space n <=> aff_dim s = &n)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [ASM_SIMP_TAC[HOMEOMORPHIC_EMPTY_SPACE_EQ; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; NONEMPTY_EUCLIDEAN_SPACE; AFF_DIM_EMPTY] THEN INT_ARITH_TAC; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY])] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN TRANS_TAC EQ_TRANS `subtopology euclidean (IMAGE (\x:real^N. --a + x) s) homeomorphic_space euclidean_space n` THEN CONJ_TAC THENL [EQ_TAC THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] HOMEOMORPHIC_SPACE_TRANS) THEN REWRITE_TAC[HOMEOMORPHIC_SPACE_EUCLIDEAN] THEN REWRITE_TAC[HOMEOMORPHIC_TRANSLATION_LEFT_EQ; HOMEOMORPHIC_TRANSLATION_RIGHT_EQ; HOMEOMORPHIC_REFL]; W(MP_TAC o PART_MATCH (lhand o rand) HOMEOMORPHIC_SUBSPACE_EUCLIDEAN_SPACE o lhand o snd) THEN SIMP_TAC[GSYM INT_OF_NUM_EQ; GSYM AFF_DIM_DIM_SUBSPACE] THEN REWRITE_TAC[AFF_DIM_TRANSLATION_EQ] THEN DISCH_THEN MATCH_MP_TAC THEN MATCH_MP_TAC AFFINE_IMP_SUBSPACE THEN ASM_SIMP_TAC[AFFINE_TRANSLATION; GSYM IN_TRANSLATION_GALOIS_ALT] THEN ASM_REWRITE_TAC[VECTOR_ADD_RID]]);; let HOMEOMORPHIC_AFFINE_SETS_EQ = prove (`!s:real^M->bool t:real^N->bool. affine s /\ affine t ==> (s homeomorphic t <=> aff_dim s = aff_dim t)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_REWRITE_TAC[AFF_DIM_EMPTY; AFF_DIM_EQ_MINUS1; HOMEOMORPHIC_EMPTY] THEN POP_ASSUM MP_TAC THEN GEN_REWRITE_TAC (funpow 3 RAND_CONV) [EQ_SYM_EQ] THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_EQ_MINUS1; HOMEOMORPHIC_EMPTY] THEN POP_ASSUM MP_TAC THEN REWRITE_TAC [GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM; RIGHT_IMP_FORALL_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^N`] THEN GEOM_ORIGIN_TAC `a:real^M` THEN GEOM_ORIGIN_TAC `b:real^N` THEN SIMP_TAC[AFFINE_EQ_SUBSPACE; HOMEOMORPHIC_SUBSPACES_EQ; AFF_DIM_DIM_0; HULL_INC; INT_OF_NUM_EQ] THEN MESON_TAC[]);; let INVARIANCE_OF_DOMAIN_SUBSPACES = prove (`!f:real^M->real^N u v s. subspace u /\ subspace v /\ dim v <= dim u /\ f continuous_on s /\ IMAGE f s SUBSET v /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\ open_in (subtopology euclidean u) s ==> open_in (subtopology euclidean v) (IMAGE f s)`, X_GEN_TAC `h:real^M->real^N` THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN MAP_EVERY (MP_TAC o C ISPEC HOMEOMORPHIC_SUBSPACE_EUCLIDEAN_SPACE_DIM) [`v:real^N->bool`; `u:real^M->bool`] THEN ASM_REWRITE_TAC[homeomorphic_space; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f:real^M->num->real`; `f':(num->real)->real^M`] THEN REWRITE_TAC[HOMEOMORPHIC_MAPS_MAP; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`g:real^N->num->real`; `g':(num->real)->real^N`] THEN STRIP_TAC THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC I [MATCH_MP HOMEOMORPHIC_MAP_OPENNESS_EQ th]) THEN CONJ_TAC THENL [ALL_TAC; MP_TAC(ISPECL [`dim(u:real^M->bool)`; `dim(v:real^N->bool)`; `IMAGE (f:real^M->num->real) s`; `(g:real^N->num->real) o h o (f':(num->real)->real^M)`] INVARIANCE_OF_DOMAIN_EUCLIDEAN_SPACE_GEN) THEN ASM_REWRITE_TAC[GSYM IMAGE_o; GSYM o_ASSOC] THEN ANTS_TAC THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[HOMEOMORPHIC_MAP_OPENNESS_EQ]; MATCH_MP_TAC CONTINUOUS_MAP_COMPOSE THEN EXISTS_TAC `subtopology euclidean (v:real^N->bool)` THEN ASM_SIMP_TAC[HOMEOMORPHIC_IMP_CONTINUOUS_MAP] THEN MATCH_MP_TAC CONTINUOUS_MAP_COMPOSE THEN EXISTS_TAC `subtopology euclidean (s:real^M->bool)` THEN ASM_REWRITE_TAC[CONTINUOUS_MAP_EUCLIDEAN2] THEN SUBGOAL_THEN `subtopology euclidean (s:real^M->bool) = subtopology (subtopology euclidean u) (s:real^M->bool)` SUBST1_TAC THENL [REWRITE_TAC[SUBTOPOLOGY_SUBTOPOLOGY] THEN AP_TERM_TAC THEN ASM SET_TAC[]; ONCE_REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN ASM_SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; HOMEOMORPHIC_IMP_CONTINUOUS_MAP]]; REWRITE_TAC[FORALL_IN_IMAGE; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN RULE_ASSUM_TAC(REWRITE_RULE[SUBSET]) THEN ASM_SIMP_TAC[o_THM]; MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN REWRITE_TAC[IMAGE_o] THEN AP_TERM_TAC THEN AP_TERM_TAC]] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP HOMEOMORPHIC_IMP_CONTINUOUS_MAP)) THEN REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[continuous_map; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; TOPSPACE_SUBTOPOLOGY; o_DEF] THEN SET_TAC[]);; let INVARIANCE_OF_DOMAIN = prove (`!f:real^N->real^N s. f continuous_on s /\ open s /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> open(IMAGE f s)`, ONCE_REWRITE_TAC[OPEN_IN] THEN ONCE_REWRITE_TAC[GSYM SUBTOPOLOGY_UNIV] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC INVARIANCE_OF_DOMAIN_SUBSPACES THEN EXISTS_TAC `(:real^N)` THEN ASM_REWRITE_TAC[SUBSPACE_UNIV; SUBSET_UNIV; LE_REFL]);; let INVARIANCE_OF_DIMENSION_SUBSPACES = prove (`!f:real^M->real^N u v s. subspace u /\ subspace v /\ ~(s = {}) /\ open_in (subtopology euclidean u) s /\ f continuous_on s /\ IMAGE f s SUBSET v /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> dim u <= dim v`, REWRITE_TAC[GSYM NOT_LT] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`u:real^M->bool`; `dim(v:real^N->bool)`] CHOOSE_SUBSPACE_OF_SUBSPACE) THEN ASM_SIMP_TAC[SPAN_OF_SUBSPACE; LE_LT] THEN DISCH_THEN(X_CHOOSE_THEN `t:real^M->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`v:real^N->bool`; `t:real^M->bool`] HOMEOMORPHIC_SUBSPACES) THEN FIRST_X_ASSUM(SUBST_ALL_TAC o SYM) THEN ASM_REWRITE_TAC[homeomorphic; homeomorphism; NOT_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`h:real^N->real^M`; `k:real^M->real^N`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`(h:real^N->real^M) o (f:real^M->real^N)`; `u:real^M->bool`; `u:real^M->bool`; `s:real^M->bool`] INVARIANCE_OF_DOMAIN_SUBSPACES) THEN ASM_REWRITE_TAC[LE_LT; NOT_IMP] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; REWRITE_TAC[IMAGE_o] THEN ASM SET_TAC[]; REWRITE_TAC[o_THM] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `IMAGE ((h:real^N->real^M) o (f:real^M->real^N)) s SUBSET t` ASSUME_TAC THENL [REWRITE_TAC[IMAGE_o] THEN ASM SET_TAC[]; ALL_TAC] THEN ABBREV_TAC `w = IMAGE ((h:real^N->real^M) o (f:real^M->real^N)) s` THEN DISCH_TAC THEN UNDISCH_TAC `dim(t:real^M->bool) < dim(u:real^M->bool)` THEN REWRITE_TAC[NOT_LT] THEN MP_TAC(ISPECL [`w:real^M->bool`; `u:real^M->bool`] DIM_OPEN_IN) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_MESON_TAC[IMAGE_EQ_EMPTY]; DISCH_THEN(SUBST1_TAC o SYM)] THEN ASM_SIMP_TAC[DIM_SUBSET]);; let INVARIANCE_OF_DOMAIN_AFFINE_SETS = prove (`!f:real^M->real^N u v s. affine u /\ affine v /\ aff_dim v <= aff_dim u /\ f continuous_on s /\ IMAGE f s SUBSET v /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\ open_in (subtopology euclidean u) s ==> open_in (subtopology euclidean v) (IMAGE f s)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_REWRITE_TAC[IMAGE_CLAUSES; OPEN_IN_EMPTY; INJECTIVE_ON_ALT] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `?a:real^M b:real^N. a IN s /\ a IN u /\ b IN v` STRIP_ASSUME_TAC THENL [FIRST_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`(\x. --b + x) o (f:real^M->real^N) o (\x. a + x)`; `IMAGE (\x:real^M. --a + x) u`; `IMAGE (\x:real^N. --b + x) v`; `IMAGE (\x:real^M. --a + x) s`] INVARIANCE_OF_DOMAIN_SUBSPACES) THEN REWRITE_TAC[IMAGE_o; INJECTIVE_ON_ALT; OPEN_IN_TRANSLATION_EQ] THEN SIMP_TAC[IMP_CONJ; GSYM INT_OF_NUM_LE; GSYM AFF_DIM_DIM_SUBSPACE] THEN ASM_REWRITE_TAC[AFF_DIM_TRANSLATION_EQ; RIGHT_FORALL_IMP_THM] THEN SIMP_TAC[FORALL_IN_IMAGE; o_THM; GSYM IMAGE_o; IMP_IMP; GSYM CONJ_ASSOC] THEN ANTS_TAC THENL [ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC AFFINE_IMP_SUBSPACE THEN ASM_REWRITE_TAC[AFFINE_TRANSLATION_EQ] THEN REWRITE_TAC[IN_IMAGE; VECTOR_ARITH `vec 0:real^N = --a + x <=> x = a`] THEN ASM_MESON_TAC[]; REPEAT CONJ_TAC THENL [REPEAT(MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID]); REWRITE_TAC[GSYM o_ASSOC] THEN REWRITE_TAC[IMAGE_o] THEN MATCH_MP_TAC IMAGE_SUBSET; REWRITE_TAC[VECTOR_ARITH `a + x:real^N = a + y <=> x = y`]]]; ALL_TAC] THEN ASM_SIMP_TAC[VECTOR_ARITH `a + --a + x:real^N = x`; GSYM IMAGE_o; o_DEF; IMAGE_ID; ETA_AX]);; let INVARIANCE_OF_DIMENSION_AFFINE_SETS = prove (`!f:real^M->real^N u v s. affine u /\ affine v /\ ~(s = {}) /\ open_in (subtopology euclidean u) s /\ f continuous_on s /\ IMAGE f s SUBSET v /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> aff_dim u <= aff_dim v`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_REWRITE_TAC[IMAGE_CLAUSES; OPEN_IN_EMPTY; INJECTIVE_ON_ALT] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `?a:real^M b:real^N. a IN s /\ a IN u /\ b IN v` STRIP_ASSUME_TAC THENL [FIRST_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`(\x. --b + x) o (f:real^M->real^N) o (\x. a + x)`; `IMAGE (\x:real^M. --a + x) u`; `IMAGE (\x:real^N. --b + x) v`; `IMAGE (\x:real^M. --a + x) s`] INVARIANCE_OF_DIMENSION_SUBSPACES) THEN REWRITE_TAC[IMAGE_o; INJECTIVE_ON_ALT; OPEN_IN_TRANSLATION_EQ] THEN SIMP_TAC[IMP_CONJ; GSYM INT_OF_NUM_LE; GSYM AFF_DIM_DIM_SUBSPACE] THEN ASM_REWRITE_TAC[AFF_DIM_TRANSLATION_EQ; RIGHT_FORALL_IMP_THM] THEN SIMP_TAC[FORALL_IN_IMAGE; o_THM; GSYM IMAGE_o; IMP_IMP; GSYM CONJ_ASSOC] THEN DISCH_THEN MATCH_MP_TAC THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC AFFINE_IMP_SUBSPACE THEN ASM_REWRITE_TAC[AFFINE_TRANSLATION_EQ] THEN REWRITE_TAC[IN_IMAGE; VECTOR_ARITH `vec 0:real^N = --a + x <=> x = a`] THEN ASM_MESON_TAC[]; ASM_REWRITE_TAC[IMAGE_EQ_EMPTY] THEN REPEAT CONJ_TAC THENL [REPEAT(MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID]); REWRITE_TAC[GSYM o_ASSOC] THEN REWRITE_TAC[IMAGE_o] THEN MATCH_MP_TAC IMAGE_SUBSET; REWRITE_TAC[VECTOR_ARITH `a + x:real^N = a + y <=> x = y`]]] THEN ASM_SIMP_TAC[VECTOR_ARITH `a + --a + x:real^N = x`; GSYM IMAGE_o; o_DEF; IMAGE_ID; ETA_AX]);; let INVARIANCE_OF_DIMENSION = prove (`!f:real^M->real^N s. f continuous_on s /\ open s /\ ~(s = {}) /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> dimindex(:M) <= dimindex(:N)`, REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM DIM_UNIV] THEN MATCH_MP_TAC INVARIANCE_OF_DIMENSION_SUBSPACES THEN MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `s:real^M->bool`] THEN ASM_REWRITE_TAC[SUBSPACE_UNIV; SUBSET_UNIV; SUBTOPOLOGY_UNIV; GSYM OPEN_IN]);; let CONTINUOUS_INJECTIVE_IMAGE_SUBSPACE_DIM_LE = prove (`!f:real^M->real^N s t. subspace s /\ subspace t /\ f continuous_on s /\ IMAGE f s SUBSET t /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> dim(s) <= dim(t)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC INVARIANCE_OF_DIMENSION_SUBSPACES THEN MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `s:real^M->bool`] THEN ASM_REWRITE_TAC[OPEN_IN_REFL] THEN ASM_SIMP_TAC[SUBSPACE_IMP_NONEMPTY]);; let INVARIANCE_OF_DIMENSION_CONVEX_DOMAIN = prove (`!f:real^M->real^N s t. convex s /\ f continuous_on s /\ IMAGE f s SUBSET affine hull t /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> aff_dim(s) <= aff_dim(t)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_REWRITE_TAC[AFF_DIM_EMPTY; AFF_DIM_GE] THEN MP_TAC(ISPECL [`f:real^M->real^N`; `affine hull s:real^M->bool`; `affine hull t:real^N->bool`; `relative_interior s:real^M->bool`] INVARIANCE_OF_DIMENSION_AFFINE_SETS) THEN ASM_REWRITE_TAC[AFFINE_AFFINE_HULL; AFF_DIM_AFFINE_HULL; OPEN_IN_RELATIVE_INTERIOR] THEN DISCH_THEN MATCH_MP_TAC THEN CONJ_TAC THENL [ASM_MESON_TAC[RELATIVE_INTERIOR_EQ_EMPTY]; ALL_TAC] THEN ASSUME_TAC(ISPEC `s:real^M->bool` RELATIVE_INTERIOR_SUBSET) THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ASM SET_TAC[]]);; let HOMEOMORPHIC_CONVEX_SETS = prove (`!s:real^M->bool t:real^N->bool. convex s /\ convex t /\ s homeomorphic t ==> aff_dim s = aff_dim t`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphic]) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; GSYM INT_LE_ANTISYM; homeomorphism] THEN MAP_EVERY X_GEN_TAC [`f:real^M->real^N`; `g:real^N->real^M`] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC INVARIANCE_OF_DIMENSION_CONVEX_DOMAIN THENL [EXISTS_TAC `f:real^M->real^N`; EXISTS_TAC `g:real^N->real^M`] THEN ASM_REWRITE_TAC[HULL_SUBSET] THEN ASM SET_TAC[]);; let HOMEOMORPHIC_CONVEX_COMPACT_SETS_EQ = prove (`!s:real^M->bool t:real^N->bool. convex s /\ compact s /\ convex t /\ compact t ==> (s homeomorphic t <=> aff_dim s = aff_dim t)`, MESON_TAC[HOMEOMORPHIC_CONVEX_SETS; HOMEOMORPHIC_CONVEX_COMPACT_SETS]);; let INVARIANCE_OF_DOMAIN_GEN = prove (`!f:real^M->real^N s. dimindex(:N) <= dimindex(:M) /\ f continuous_on s /\ open s /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> open(IMAGE f s)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `(:real^M)`; `(:real^N)`; `s:real^M->bool`] INVARIANCE_OF_DOMAIN_SUBSPACES) THEN ASM_REWRITE_TAC[SUBTOPOLOGY_UNIV; GSYM OPEN_IN; SUBSPACE_UNIV; DIM_UNIV; SUBSET_UNIV]);; let INJECTIVE_INTO_1D_IMP_OPEN_MAP_UNIV = prove (`!f:real^N->real^1 s t. f continuous_on s /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\ open t /\ t SUBSET s ==> open (IMAGE f t)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC INVARIANCE_OF_DOMAIN_GEN THEN ASM_REWRITE_TAC[DIMINDEX_1; DIMINDEX_GE_1] THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ASM SET_TAC[]]);; let CONTINUOUS_ON_INVERSE_OPEN = prove (`!f:real^M->real^N g s. dimindex(:N) <= dimindex(:M) /\ f continuous_on s /\ open s /\ (!x. x IN s ==> g(f x) = x) ==> g continuous_on IMAGE f s`, REPEAT STRIP_TAC THEN REWRITE_TAC[CONTINUOUS_OPEN_IN_PREIMAGE_EQ] THEN X_GEN_TAC `t:real^M->bool` THEN DISCH_TAC THEN SUBGOAL_THEN `{x | x IN IMAGE f s /\ g x IN t} = IMAGE (f:real^M->real^N) (s INTER t)` SUBST1_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC OPEN_OPEN_IN_TRANS] THEN REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN CONJ_TAC THEN MATCH_MP_TAC INVARIANCE_OF_DOMAIN_GEN THEN ASM_SIMP_TAC[OPEN_INTER; IN_INTER] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; INTER_SUBSET]);; let CONTINUOUS_ON_INVERSE_INTO_1D = prove (`!f:real^N->real^1 g s t. f continuous_on s /\ (path_connected s \/ connected s /\ (locally compact s \/ locally connected s) \/ compact s \/ open s) /\ IMAGE f s = t /\ (!x. x IN s ==> g(f x) = x) ==> g continuous_on t`, REPEAT STRIP_TAC THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_INVERSE_OPEN_MAP THEN MAP_EVERY EXISTS_TAC [`f:real^N->real^1`; `s:real^N->bool`] THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(SUBST1_TAC o SYM) THEN MATCH_MP_TAC INJECTIVE_INTO_1D_IMP_OPEN_MAP THEN ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_INVERSE_INJECTIVE_PROPER_MAP THEN MAP_EVERY EXISTS_TAC [`f:real^N->real^1`; `s:real^N->bool`] THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONOTONE_INTO_1D_IMP_PROPER_MAP THEN ASM_REWRITE_TAC[AND_FORALL_THM] THEN X_GEN_TAC `y:real^1` THEN SUBGOAL_THEN `{x | x IN s /\ (f:real^N->real^1) x = y} = {} \/ ?a. {x | x IN s /\ (f:real^N->real^1) x = y} = {a}` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ASM_REWRITE_TAC[COMPACT_EMPTY; CONNECTED_EMPTY]; ASM_REWRITE_TAC[COMPACT_SING; CONNECTED_SING]]; MATCH_MP_TAC CONTINUOUS_ON_INVERSE_OPEN_MAP THEN MAP_EVERY EXISTS_TAC [`f:real^N->real^1`; `s:real^N->bool`] THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `w:real^N->bool` THEN DISCH_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN REWRITE_TAC[open_in; SUBSET; FORALL_IN_IMAGE] THEN CONJ_TAC THENL [ASM SET_TAC[]; X_GEN_TAC `x:real^N` THEN DISCH_TAC] THEN ABBREV_TAC `u = connected_component w (x:real^N)` THEN SUBGOAL_THEN `connected u /\ (x:real^N) IN u /\ u SUBSET s /\ u SUBSET w /\ open_in (subtopology euclidean s) u` STRIP_ASSUME_TAC THENL [EXPAND_TAC "u" THEN REPEAT CONJ_TAC THENL [MESON_TAC[CONNECTED_CONNECTED_COMPONENT]; ASM_MESON_TAC[IN; CONNECTED_COMPONENT_REFL]; ASM_MESON_TAC[CONNECTED_COMPONENT_SUBSET; SUBSET_TRANS]; ASM_MESON_TAC[CONNECTED_COMPONENT_SUBSET]; ASM_MESON_TAC[LOCALLY_CONNECTED_OPEN_CONNECTED_COMPONENT]]; SUBGOAL_THEN `?e. &0 < e /\ !y. y IN t /\ dist(y,(f:real^N->real^1) x) < e ==> y IN IMAGE f u` ASSUME_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REPEAT(FIRST_X_ASSUM(K ALL_TAC o check (free_in `w:real^N->bool`) o concl))] THEN SUBGOAL_THEN `(?e. &0 < e /\ !y. y IN t /\ drop(f x) <= drop y /\ drop y < drop(f x) + e ==> y IN IMAGE f (u:real^N->bool)) /\ (?e. &0 < e /\ !y. y IN t /\ drop(f x) - e < drop y /\ drop y <= drop(f x) ==> y IN IMAGE f (u:real^N->bool))` MP_TAC THENL [ALL_TAC; DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `d:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) (X_CHOOSE_THEN `e:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC))) THEN REWRITE_TAC[IMP_IMP; AND_FORALL_THM; TAUT `(p ==> r) /\ (q ==> r) <=> (p \/ q ==> r)`] THEN DISCH_TAC THEN EXISTS_TAC `min d e:real` THEN ASM_REWRITE_TAC[REAL_LT_MIN; DIST_1] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC] THEN MP_TAC(ISPECL [`f:real^N->real^1`; `s:real^N->bool`] MONOTONE_TOPOLOGICALLY_INTO_1D) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[] THEN X_GEN_TAC `y:real^1` THEN SUBGOAL_THEN `{x | x IN s /\ (f:real^N->real^1) x = y} = {} \/ ?a. {x | x IN s /\ (f:real^N->real^1) x = y} = {a}` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ASM_REWRITE_TAC[COMPACT_EMPTY; CONNECTED_EMPTY]; ASM_REWRITE_TAC[COMPACT_SING; CONNECTED_SING]]; ALL_TAC] THEN DISCH_THEN(fun th -> CONJ_TAC THEN MP_TAC th) THENL [DISCH_THEN(MP_TAC o SPEC `{y | y IN IMAGE (f:real^N->real^1) s /\ drop(f x) <= drop y}`); DISCH_THEN(MP_TAC o SPEC `{y | y IN IMAGE (f:real^N->real^1) s /\ drop y <= drop(f x)}`)] THEN (ANTS_TAC THENL [REWRITE_TAC[SET_RULE `{x | x IN s /\ P x} = s INTER {x | P x}`] THEN REWRITE_TAC[GSYM CONVEX_CONNECTED_1] THEN MATCH_MP_TAC CONVEX_INTER THEN REWRITE_TAC[drop; CONVEX_HALFSPACE_COMPONENT_LE; REWRITE_RULE[real_ge] CONVEX_HALFSPACE_COMPONENT_GE] THEN REWRITE_TAC[CONVEX_CONNECTED_1] THEN ASM_MESON_TAC[CONNECTED_CONTINUOUS_IMAGE]; ALL_TAC]) THEN REWRITE_TAC[IN_ELIM_THM; SET_RULE `x IN s /\ f x IN IMAGE f s /\ P x <=> x IN s /\ P x`] THEN REWRITE_TAC[CONNECTED_CLOSED_IN; NOT_EXISTS_THM] THEN DISCH_THEN(MP_TAC o SPEC `{x:real^N}`) THEN REWRITE_TAC[CLOSED_IN_SING; IN_ELIM_THM; REAL_LE_REFL] THEN REWRITE_TAC[NOT_INSERT_EMPTY] THENL [DISCH_THEN(MP_TAC o SPEC `{w:real^N | w IN s DIFF u /\ drop(f x) <= drop(f w)}`); DISCH_THEN(MP_TAC o SPEC `{w:real^N | w IN s DIFF u /\ drop(f w) <= drop(f x)}`)] THEN DISCH_THEN(MP_TAC o MATCH_MP (TAUT `~(p /\ q /\ r /\ s /\ ~t) ==> p /\ q /\ s ==> t \/ ~r`)) THEN (ANTS_TAC THENL [CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[SET_RULE `{x | x IN s DIFF u /\ P x} = {x | x IN s /\ P x} INTER (s DIFF u)`] THEN MATCH_MP_TAC CLOSED_IN_SUBTOPOLOGY_INTER_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CLOSED_IN_INTER THEN REWRITE_TAC[CLOSED_IN_REFL] THEN MATCH_MP_TAC CLOSED_IN_DIFF THEN ASM_REWRITE_TAC[CLOSED_IN_REFL]; ALL_TAC]) THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `{x | x IN s DIFF u /\ P x} = {} \/ ~({x | x IN s /\ P x} SUBSET {a} UNION {x | x IN s DIFF u /\ P x}) ==> u SUBSET s ==> (!x. x IN s /\ P x ==> x = a) \/ ?x. x IN u /\ ~(x = a) /\ P x`)) THEN ASM_REWRITE_TAC[] THEN (DISCH_THEN(DISJ_CASES_THEN2 ASSUME_TAC MP_TAC) THENL [EXISTS_TAC `&1` THEN REWRITE_TAC[REAL_LT_01] THEN ASM SET_TAC[]; ALL_TAC]) THEN DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `dist((f:real^N->real^1) x,f y)` THEN ASM_REWRITE_TAC[GSYM DIST_NZ] THEN (CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN X_GEN_TAC `z:real^1` THEN STRIP_TAC THEN (SUBGOAL_THEN `is_interval(IMAGE (f:real^N->real^1) u)` MP_TAC THENL [REWRITE_TAC[IS_INTERVAL_CONNECTED_1] THEN MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; REWRITE_TAC[IS_INTERVAL_1] THEN DISCH_THEN MATCH_MP_TAC]) THENL [ALL_TAC; ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM]] THEN REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN ONCE_REWRITE_TAC[EXISTS_IN_IMAGE] THEN EXISTS_TAC `x:real^N` THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `(f:real^N->real^1) y` THEN ASM_SIMP_TAC[FUN_IN_IMAGE] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM REAL_NOT_LE]) THEN REWRITE_TAC[DIST_1] THEN ASM_REAL_ARITH_TAC; ASM_MESON_TAC[CONTINUOUS_ON_INVERSE]; FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN MATCH_MP_TAC CONTINUOUS_ON_INVERSE_OPEN THEN ASM_REWRITE_TAC[DIMINDEX_1; DIMINDEX_GE_1]]);; let INVARIANCE_OF_DOMAIN_HOMEOMORPHISM = prove (`!f:real^M->real^N s. dimindex(:N) <= dimindex(:M) /\ f continuous_on s /\ open s /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> ?g. homeomorphism (s,IMAGE f s) (f,g)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INJECTIVE_ON_LEFT_INVERSE]) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^N->real^M` THEN DISCH_TAC THEN ASM_REWRITE_TAC[homeomorphism] THEN ASM_SIMP_TAC[CONTINUOUS_ON_INVERSE_OPEN] THEN ASM SET_TAC[]);; let INVARIANCE_OF_DOMAIN_HOMEOMORPHIC = prove (`!f:real^M->real^N s. dimindex(:N) <= dimindex(:M) /\ f continuous_on s /\ open s /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> s homeomorphic (IMAGE f s)`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP INVARIANCE_OF_DOMAIN_HOMEOMORPHISM) THEN REWRITE_TAC[homeomorphic] THEN MESON_TAC[]);; let HOMEOMORPHIC_INTERVALS_EQ = prove (`(!a b:real^M c d:real^N. interval[a,b] homeomorphic interval[c,d] <=> aff_dim(interval[a,b]) = aff_dim(interval[c,d])) /\ (!a b:real^M c d:real^N. interval[a,b] homeomorphic interval(c,d) <=> interval[a,b] = {} /\ interval(c,d) = {}) /\ (!a b:real^M c d:real^N. interval(a,b) homeomorphic interval[c,d] <=> interval(a,b) = {} /\ interval[c,d] = {}) /\ (!a b:real^M c d:real^N. interval(a,b) homeomorphic interval(c,d) <=> interval(a,b) = {} /\ interval(c,d) = {} \/ ~(interval(a,b) = {}) /\ ~(interval(c,d) = {}) /\ dimindex(:M) = dimindex(:N))`, SIMP_TAC[HOMEOMORPHIC_CONVEX_COMPACT_SETS_EQ; CONVEX_INTERVAL; COMPACT_INTERVAL] THEN REPEAT STRIP_TAC THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[HOMEOMORPHIC_EMPTY] THENL [FIRST_ASSUM(MP_TAC o MATCH_MP HOMEOMORPHIC_COMPACTNESS) THEN REWRITE_TAC[COMPACT_INTERVAL_EQ] THEN ASM_MESON_TAC[HOMEOMORPHIC_EMPTY]; FIRST_ASSUM(MP_TAC o MATCH_MP HOMEOMORPHIC_COMPACTNESS) THEN REWRITE_TAC[COMPACT_INTERVAL_EQ] THEN ASM_MESON_TAC[HOMEOMORPHIC_EMPTY]; MATCH_MP_TAC(TAUT `(p <=> q) /\ (~p /\ ~q ==> r) ==> p /\ q \/ ~p /\ ~q /\ r`) THEN CONJ_TAC THENL [ASM_MESON_TAC[HOMEOMORPHIC_EMPTY]; STRIP_TAC] THEN REWRITE_TAC[GSYM LE_ANTISYM] THEN CONJ_TAC THEN MATCH_MP_TAC INVARIANCE_OF_DIMENSION THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphic]) THENL [ALL_TAC; GEN_REWRITE_TAC LAND_CONV [SWAP_EXISTS_THM]] THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN REWRITE_TAC[homeomorphism] THEN STRIP_TAC THENL [EXISTS_TAC `interval(a:real^M,b)`; EXISTS_TAC `interval(c:real^N,d)`] THEN ASM_REWRITE_TAC[OPEN_INTERVAL] THEN ASM SET_TAC[]; TRANS_TAC HOMEOMORPHIC_TRANS `IMAGE ((\x. lambda i. x$i):real^M->real^N) (interval(a,b))` THEN CONJ_TAC THENL [MATCH_MP_TAC INVARIANCE_OF_DOMAIN_HOMEOMORPHIC THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[LE_REFL]; MATCH_MP_TAC LINEAR_CONTINUOUS_ON THEN SIMP_TAC[linear; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; LAMBDA_BETA; CART_EQ]; REWRITE_TAC[OPEN_INTERVAL]; SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN ASM_MESON_TAC[]]; ALL_TAC] THEN SUBGOAL_THEN `IMAGE ((\x. lambda i. x$i):real^M->real^N) (interval(a,b)) = interval((lambda i. a$i),(lambda i. b$i))` SUBST1_TAC THENL [MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN SIMP_TAC[IN_INTERVAL; LAMBDA_BETA] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN EXISTS_TAC `(lambda i. (y:real^N)$i):real^M` THEN SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN FIRST_ASSUM(SUBST1_TAC o SYM) THEN SIMP_TAC[CART_EQ; LAMBDA_BETA]; MATCH_MP_TAC HOMEOMORPHIC_OPEN_INTERVALS THEN GEN_REWRITE_TAC I [TAUT `(p <=> q) <=> (~p <=> ~q)`] THEN SIMP_TAC[INTERVAL_NE_EMPTY; LAMBDA_BETA] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INTERVAL_NE_EMPTY])) THEN ASM_MESON_TAC[]]]);; let CONTINUOUS_IMAGE_SUBSET_INTERIOR = prove (`!f:real^M->real^N s. f continuous_on s /\ dimindex(:N) <= dimindex(:M) /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> IMAGE f (interior s) SUBSET interior(IMAGE f s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC INTERIOR_MAXIMAL THEN SIMP_TAC[IMAGE_SUBSET; INTERIOR_SUBSET] THEN ASM_CASES_TAC `interior s:real^M->bool = {}` THENL [ASM_REWRITE_TAC[INTERIOR_EMPTY; OPEN_EMPTY; IMAGE_CLAUSES]; MATCH_MP_TAC INVARIANCE_OF_DOMAIN_GEN] THEN ASM_REWRITE_TAC[OPEN_INTERIOR] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; INTERIOR_SUBSET; SUBSET]);; let HOMEOMORPHIC_INTERIORS_SAME_DIMENSION = prove (`!s:real^M->bool t:real^N->bool. dimindex(:M) = dimindex(:N) /\ s homeomorphic t ==> (interior s) homeomorphic (interior t)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHIC_MINIMAL]) THEN REWRITE_TAC[HOMEOMORPHIC_MINIMAL] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^M->real^N` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^N->real^M` THEN STRIP_TAC THEN ASM_SIMP_TAC[REWRITE_RULE[SUBSET] INTERIOR_SUBSET] THEN REWRITE_TAC[SET_RULE `(!x. x IN s ==> f x IN t) <=> IMAGE f s SUBSET t`] THEN REPEAT CONJ_TAC THENL [SUBGOAL_THEN `t = IMAGE (f:real^M->real^N) s` SUBST1_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_IMAGE_SUBSET_INTERIOR THEN ASM_MESON_TAC[LE_REFL]]; SUBGOAL_THEN `s = IMAGE (g:real^N->real^M) t` SUBST1_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_IMAGE_SUBSET_INTERIOR THEN ASM_MESON_TAC[LE_REFL]]; ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; INTERIOR_SUBSET]; ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; INTERIOR_SUBSET]]);; let HOMEOMORPHIC_INTERIORS = prove (`!s:real^M->bool t:real^N->bool. s homeomorphic t /\ (interior s = {} <=> interior t = {}) ==> (interior s) homeomorphic (interior t)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `interior t:real^N->bool = {}` THEN ASM_SIMP_TAC[HOMEOMORPHIC_EMPTY] THEN STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_INTERIORS_SAME_DIMENSION THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM (STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [HOMEOMORPHIC_MINIMAL]) THEN REWRITE_TAC[GSYM LE_ANTISYM] THEN CONJ_TAC THEN MATCH_MP_TAC INVARIANCE_OF_DIMENSION THENL [MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `interior s:real^M->bool`]; MAP_EVERY EXISTS_TAC [`g:real^N->real^M`; `interior t:real^N->bool`]] THEN ASM_REWRITE_TAC[OPEN_INTERIOR] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; INTERIOR_SUBSET; SUBSET]);; let HOMEOMORPHIC_FRONTIERS_SAME_DIMENSION = prove (`!s:real^M->bool t:real^N->bool. dimindex(:M) = dimindex(:N) /\ s homeomorphic t /\ closed s /\ closed t ==> (frontier s) homeomorphic (frontier t)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHIC_MINIMAL]) THEN REWRITE_TAC[HOMEOMORPHIC_MINIMAL] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^M->real^N` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^N->real^M` THEN ASM_SIMP_TAC[REWRITE_RULE[SUBSET] FRONTIER_SUBSET_CLOSED] THEN STRIP_TAC THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[FRONTIER_SUBSET_CLOSED; CONTINUOUS_ON_SUBSET]] THEN ASM_SIMP_TAC[frontier; CLOSURE_CLOSED] THEN SUBGOAL_THEN `(!x:real^M. x IN interior s ==> f x IN interior t) /\ (!y:real^N. y IN interior t ==> g y IN interior s)` MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[SET_RULE `(!x. x IN s ==> f x IN t) <=> IMAGE f s SUBSET t`] THEN CONJ_TAC THENL [SUBGOAL_THEN `t = IMAGE (f:real^M->real^N) s` SUBST1_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_IMAGE_SUBSET_INTERIOR THEN ASM_MESON_TAC[LE_REFL]]; SUBGOAL_THEN `s = IMAGE (g:real^N->real^M) t` SUBST1_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_IMAGE_SUBSET_INTERIOR THEN ASM_MESON_TAC[LE_REFL]]]);; let HOMEOMORPHIC_FRONTIERS = prove (`!s:real^M->bool t:real^N->bool. s homeomorphic t /\ closed s /\ closed t /\ (interior s = {} <=> interior t = {}) ==> (frontier s) homeomorphic (frontier t)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `interior t:real^N->bool = {}` THENL [ASM_SIMP_TAC[frontier; CLOSURE_CLOSED; DIFF_EMPTY]; STRIP_TAC] THEN MATCH_MP_TAC HOMEOMORPHIC_FRONTIERS_SAME_DIMENSION THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM (STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [HOMEOMORPHIC_MINIMAL]) THEN REWRITE_TAC[GSYM LE_ANTISYM] THEN CONJ_TAC THEN MATCH_MP_TAC INVARIANCE_OF_DIMENSION THENL [MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `interior s:real^M->bool`]; MAP_EVERY EXISTS_TAC [`g:real^N->real^M`; `interior t:real^N->bool`]] THEN ASM_REWRITE_TAC[OPEN_INTERIOR] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; INTERIOR_SUBSET; SUBSET]);; let CONTINUOUS_IMAGE_SUBSET_RELATIVE_INTERIOR = prove (`!f:real^M->real^N s t. f continuous_on s /\ IMAGE f s SUBSET t /\ aff_dim t <= aff_dim s /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> IMAGE f (relative_interior s) SUBSET relative_interior(IMAGE f s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC RELATIVE_INTERIOR_MAXIMAL THEN SIMP_TAC[IMAGE_SUBSET; RELATIVE_INTERIOR_SUBSET] THEN MATCH_MP_TAC INVARIANCE_OF_DOMAIN_AFFINE_SETS THEN EXISTS_TAC `affine hull s:real^M->bool` THEN ASM_REWRITE_TAC[AFFINE_AFFINE_HULL; AFF_DIM_AFFINE_HULL] THEN REWRITE_TAC[OPEN_IN_RELATIVE_INTERIOR] THEN CONJ_TAC THENL [ASM_MESON_TAC[AFF_DIM_SUBSET; INT_LE_TRANS]; ALL_TAC] THEN ASSUME_TAC(ISPEC `s:real^M->bool` RELATIVE_INTERIOR_SUBSET) THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `IMAGE (f:real^M->real^N) s` THEN SIMP_TAC[IMAGE_SUBSET; RELATIVE_INTERIOR_SUBSET; HULL_SUBSET]);; let HOMEOMORPHIC_RELATIVE_INTERIORS_SAME_DIMENSION = prove (`!s:real^M->bool t:real^N->bool. aff_dim s = aff_dim t /\ s homeomorphic t ==> (relative_interior s) homeomorphic (relative_interior t)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHIC_MINIMAL]) THEN REWRITE_TAC[HOMEOMORPHIC_MINIMAL] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^M->real^N` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^N->real^M` THEN STRIP_TAC THEN ASM_SIMP_TAC[REWRITE_RULE[SUBSET] RELATIVE_INTERIOR_SUBSET] THEN REWRITE_TAC[SET_RULE `(!x. x IN s ==> f x IN t) <=> IMAGE f s SUBSET t`] THEN REPEAT CONJ_TAC THENL [SUBGOAL_THEN `t = IMAGE (f:real^M->real^N) s` SUBST1_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_IMAGE_SUBSET_RELATIVE_INTERIOR THEN EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[INT_LE_REFL] THEN ASM SET_TAC[]]; SUBGOAL_THEN `s = IMAGE (g:real^N->real^M) t` SUBST1_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_IMAGE_SUBSET_RELATIVE_INTERIOR THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[INT_LE_REFL] THEN ASM SET_TAC[]]; ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; RELATIVE_INTERIOR_SUBSET]; ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; RELATIVE_INTERIOR_SUBSET]]);; let HOMEOMORPHIC_RELATIVE_INTERIORS = prove (`!s:real^M->bool t:real^N->bool. s homeomorphic t /\ (relative_interior s = {} <=> relative_interior t = {}) ==> (relative_interior s) homeomorphic (relative_interior t)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `relative_interior t:real^N->bool = {}` THEN ASM_SIMP_TAC[HOMEOMORPHIC_EMPTY] THEN STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_RELATIVE_INTERIORS_SAME_DIMENSION THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM (STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [HOMEOMORPHIC_MINIMAL]) THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN REWRITE_TAC[GSYM INT_LE_ANTISYM] THEN CONJ_TAC THEN MATCH_MP_TAC INVARIANCE_OF_DIMENSION_AFFINE_SETS THENL [MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `relative_interior s:real^M->bool`]; MAP_EVERY EXISTS_TAC [`g:real^N->real^M`; `relative_interior t:real^N->bool`]] THEN ASM_REWRITE_TAC[OPEN_IN_RELATIVE_INTERIOR; AFFINE_AFFINE_HULL] THEN (REPEAT CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; RELATIVE_INTERIOR_SUBSET]; ASM_MESON_TAC[RELATIVE_INTERIOR_SUBSET; HULL_SUBSET; SET_RULE `(!x. x IN s ==> f x IN t) /\ s' SUBSET s /\ t SUBSET t' ==> IMAGE f s' SUBSET t'`]; ASM_MESON_TAC[RELATIVE_INTERIOR_SUBSET; SUBSET]]));; let HOMEOMORPHIC_RELATIVE_BOUNDARIES_SAME_DIMENSION = prove (`!s:real^M->bool t:real^N->bool. aff_dim s = aff_dim t /\ s homeomorphic t ==> (s DIFF relative_interior s) homeomorphic (t DIFF relative_interior t)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHIC_MINIMAL]) THEN REWRITE_TAC[HOMEOMORPHIC_MINIMAL] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^M->real^N` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^N->real^M` THEN STRIP_TAC THEN ASM_SIMP_TAC[IN_DIFF] THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[SUBSET_DIFF; CONTINUOUS_ON_SUBSET]] THEN SUBGOAL_THEN `(!x:real^M. x IN relative_interior s ==> f x IN relative_interior t) /\ (!y:real^N. y IN relative_interior t ==> g y IN relative_interior s)` MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[SET_RULE `(!x. x IN s ==> f x IN t) <=> IMAGE f s SUBSET t`] THEN CONJ_TAC THENL [SUBGOAL_THEN `t = IMAGE (f:real^M->real^N) s` SUBST1_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_IMAGE_SUBSET_RELATIVE_INTERIOR THEN EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[INT_LE_REFL] THEN ASM SET_TAC[]]; SUBGOAL_THEN `s = IMAGE (g:real^N->real^M) t` SUBST1_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_IMAGE_SUBSET_RELATIVE_INTERIOR THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[INT_LE_REFL] THEN ASM SET_TAC[]]]);; let HOMEOMORPHIC_RELATIVE_BOUNDARIES = prove (`!s:real^M->bool t:real^N->bool. s homeomorphic t /\ (relative_interior s = {} <=> relative_interior t = {}) ==> (s DIFF relative_interior s) homeomorphic (t DIFF relative_interior t)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `relative_interior t:real^N->bool = {}` THEN ASM_SIMP_TAC[DIFF_EMPTY] THEN STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_RELATIVE_BOUNDARIES_SAME_DIMENSION THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM (STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [HOMEOMORPHIC_MINIMAL]) THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN REWRITE_TAC[GSYM INT_LE_ANTISYM] THEN CONJ_TAC THEN MATCH_MP_TAC INVARIANCE_OF_DIMENSION_AFFINE_SETS THENL [MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `relative_interior s:real^M->bool`]; MAP_EVERY EXISTS_TAC [`g:real^N->real^M`; `relative_interior t:real^N->bool`]] THEN ASM_REWRITE_TAC[OPEN_IN_RELATIVE_INTERIOR; AFFINE_AFFINE_HULL] THEN (REPEAT CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; RELATIVE_INTERIOR_SUBSET]; ASM_MESON_TAC[RELATIVE_INTERIOR_SUBSET; HULL_SUBSET; SET_RULE `(!x. x IN s ==> f x IN t) /\ s' SUBSET s /\ t SUBSET t' ==> IMAGE f s' SUBSET t'`]; ASM_MESON_TAC[RELATIVE_INTERIOR_SUBSET; SUBSET]]));; let UNIFORMLY_CONTINUOUS_HOMEOMORPHISM_UNIV_TRIVIAL = prove (`!f g s:real^N->bool. homeomorphism (s,(:real^N)) (f,g) /\ f uniformly_continuous_on s ==> s = (:real^N)`, REPEAT GEN_TAC THEN REWRITE_TAC[homeomorphism; IN_UNIV] THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY] THENL [SET_TAC[]; STRIP_TAC] THEN MP_TAC(ISPEC `s:real^N->bool` CLOPEN) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN CONJ_TAC THENL [REWRITE_TAC[GSYM COMPLETE_EQ_CLOSED; complete] THEN X_GEN_TAC `x:num->real^N` THEN STRIP_TAC THEN SUBGOAL_THEN `cauchy ((f:real^N->real^N) o x)` MP_TAC THENL [ASM_MESON_TAC[UNIFORMLY_CONTINUOUS_IMP_CAUCHY_CONTINUOUS]; ALL_TAC] THEN REWRITE_TAC[GSYM CONVERGENT_EQ_CAUCHY] THEN DISCH_THEN(X_CHOOSE_TAC `l:real^N`) THEN EXISTS_TAC `(g:real^N->real^N) l` THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC LIM_TRANSFORM_EVENTUALLY THEN EXISTS_TAC `(g:real^N->real^N) o (f:real^N->real^N) o (x:num->real^N)` THEN REWRITE_TAC[o_DEF] THEN CONJ_TAC THENL [MATCH_MP_TAC ALWAYS_EVENTUALLY THEN ASM SET_TAC[]; MATCH_MP_TAC LIM_CONTINUOUS_FUNCTION THEN ASM_SIMP_TAC[GSYM o_DEF] THEN ASM_MESON_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_AT; OPEN_UNIV; IN_UNIV]]; FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [SYM th]) THEN MATCH_MP_TAC INVARIANCE_OF_DOMAIN THEN ASM_REWRITE_TAC[OPEN_UNIV] THEN ASM SET_TAC[]]);; let INVARIANCE_OF_DOMAIN_SPHERE_AFFINE_SET_GEN = prove (`!f:real^M->real^N u s t. f continuous_on s /\ IMAGE f s SUBSET t /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\ bounded u /\ convex u /\ affine t /\ aff_dim t < aff_dim u /\ open_in (subtopology euclidean (relative_frontier u)) s ==> open_in (subtopology euclidean t) (IMAGE f s)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `relative_frontier u:real^M->bool = {}` THEN ASM_SIMP_TAC[OPEN_IN_SUBTOPOLOGY_EMPTY; IMAGE_CLAUSES; OPEN_IN_EMPTY] THEN STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN SUBGOAL_THEN `?b c:real^M. b IN relative_frontier u /\ c IN relative_frontier u /\ ~(b = c)` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC(SET_RULE `~(s = {} \/ ?x. s = {x}) ==> ?a b. a IN s /\ b IN s /\ ~(a = b)`) THEN ASM_MESON_TAC[RELATIVE_FRONTIER_NOT_SING]; ALL_TAC] THEN MP_TAC(ISPECL [`(:real^M)`; `aff_dim(u:real^M->bool) - &1`] CHOOSE_AFFINE_SUBSET) THEN REWRITE_TAC[SUBSET_UNIV; AFFINE_UNIV] THEN ANTS_TAC THENL [MATCH_MP_TAC(INT_ARITH `&0:int <= t /\ t <= n ==> --a <= t - a /\ t - &1 <= n`) THEN REWRITE_TAC[AFF_DIM_LE_UNIV; AFF_DIM_UNIV; AFF_DIM_POS_LE] THEN ASM_MESON_TAC[RELATIVE_FRONTIER_EMPTY; NOT_IN_EMPTY]; DISCH_THEN(X_CHOOSE_THEN `af:real^M->bool` STRIP_ASSUME_TAC)] THEN MP_TAC(ISPECL [`u:real^M->bool`; `af:real^M->bool`] HOMEOMORPHIC_PUNCTURED_SPHERE_AFFINE_GEN) THEN ASM_REWRITE_TAC[INT_ARITH `x - a + a:int = x`] THEN DISCH_THEN(fun th -> MP_TAC(SPEC `c:real^M` th) THEN MP_TAC(SPEC `b:real^M` th)) THEN ASM_REWRITE_TAC[homeomorphic; homeomorphism; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`g:real^M->real^M`; `h:real^M->real^M`] THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`j:real^M->real^M`; `k:real^M->real^M`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`(f:real^M->real^N) o (k:real^M->real^M)`; `(af:real^M->bool)`; `t:real^N->bool`; `IMAGE (j:real^M->real^M) (s DELETE c)`] INVARIANCE_OF_DOMAIN_AFFINE_SETS) THEN MP_TAC(ISPECL [`(f:real^M->real^N) o (h:real^M->real^M)`; `(af:real^M->bool)`; `t:real^N->bool`; `IMAGE (g:real^M->real^M) (s DELETE b)`] INVARIANCE_OF_DOMAIN_AFFINE_SETS) THEN ASM_REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN ASM_REWRITE_TAC[IMP_IMP; INT_ARITH `x:int <= y - &1 <=> x < y`] THEN MATCH_MP_TAC(TAUT `(p1 /\ p2) /\ (q1 /\ q2 ==> r) ==> (p1 ==> q1) /\ (p2 ==> q2) ==> r`) THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; REWRITE_TAC[IMAGE_o] THEN ASM SET_TAC[]; RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; IN_DELETE]) THEN ASM_SIMP_TAC[o_THM; IN_DELETE; IMP_CONJ] THEN ASM_MESON_TAC[]; MATCH_MP_TAC HOMEOMORPHISM_IMP_OPEN_MAP THEN MAP_EVERY EXISTS_TAC [`h:real^M->real^M`; `relative_frontier u DELETE (b:real^M)`] THEN ASM_SIMP_TAC[homeomorphism; DOT_BASIS; DIMINDEX_GE_1; LE_REFL] THEN ASM_REWRITE_TAC[IN_ELIM_THM; OPEN_IN_OPEN] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN MATCH_MP_TAC MONO_EXISTS THEN SET_TAC[]; MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; REWRITE_TAC[IMAGE_o] THEN ASM SET_TAC[]; RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; IN_DELETE]) THEN ASM_SIMP_TAC[o_THM; IN_DELETE; IMP_CONJ] THEN ASM_MESON_TAC[]; MATCH_MP_TAC HOMEOMORPHISM_IMP_OPEN_MAP THEN MAP_EVERY EXISTS_TAC [`k:real^M->real^M`; `relative_frontier u DELETE (c:real^M)`] THEN ASM_SIMP_TAC[homeomorphism; DOT_BASIS; DIMINDEX_GE_1; LE_REFL] THEN ASM_REWRITE_TAC[IN_ELIM_THM; OPEN_IN_OPEN] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN MATCH_MP_TAC MONO_EXISTS THEN SET_TAC[]; DISCH_THEN(MP_TAC o MATCH_MP OPEN_IN_UNION) THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `IMAGE (f:real^M->real^N) ((s DELETE b) UNION (s DELETE c))` THEN CONJ_TAC THENL [REWRITE_TAC[IMAGE_UNION] THEN BINOP_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[IMAGE_o] THEN AP_TERM_TAC THEN ASM SET_TAC[]]);; let INVARIANCE_OF_DOMAIN_SPHERE_AFFINE_SET = prove (`!f:real^M->real^N a r s t. f continuous_on s /\ IMAGE f s SUBSET t /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\ ~(r = &0) /\ affine t /\ aff_dim t < &(dimindex(:M)) /\ open_in (subtopology euclidean (sphere(a,r))) s ==> open_in (subtopology euclidean t) (IMAGE f s)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `sphere(a:real^M,r) = {}` THEN ASM_SIMP_TAC[OPEN_IN_SUBTOPOLOGY_EMPTY; OPEN_IN_EMPTY; IMAGE_CLAUSES] THEN RULE_ASSUM_TAC(REWRITE_RULE[SPHERE_EQ_EMPTY; REAL_NOT_LT]) THEN STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `cball(a:real^M,r)`; `s:real^M->bool`; `t:real^N->bool`] INVARIANCE_OF_DOMAIN_SPHERE_AFFINE_SET_GEN) THEN ASM_REWRITE_TAC[AFF_DIM_CBALL; RELATIVE_FRONTIER_CBALL; BOUNDED_CBALL; CONVEX_CBALL] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC);; let NO_EMBEDDING_SPHERE_LOWDIM = prove (`!f:real^M->real^N a r. &0 < r /\ f continuous_on sphere(a,r) /\ (!x y. x IN sphere(a,r) /\ y IN sphere(a,r) /\ f x = f y ==> x = y) ==> dimindex(:M) <= dimindex(:N)`, REWRITE_TAC[GSYM NOT_LT] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPEC `IMAGE (f:real^M->real^N) (sphere(a:real^M,r))` COMPACT_OPEN) THEN ASM_SIMP_TAC[COMPACT_CONTINUOUS_IMAGE; IMAGE_EQ_EMPTY; COMPACT_SPHERE; SPHERE_EQ_EMPTY; REAL_ARITH `&0 < r ==> ~(r < &0)`] THEN ONCE_REWRITE_TAC[OPEN_IN] THEN ONCE_REWRITE_TAC[GSYM SUBTOPOLOGY_UNIV] THEN MATCH_MP_TAC INVARIANCE_OF_DOMAIN_SPHERE_AFFINE_SET THEN MAP_EVERY EXISTS_TAC [`a:real^M`; `r:real`] THEN ASM_REWRITE_TAC[AFFINE_UNIV; SUBSET_UNIV; AFF_DIM_UNIV; OPEN_IN_REFL; INT_OF_NUM_LT] THEN ASM_REAL_ARITH_TAC);; let EMPTY_INTERIOR_LOWDIM_GEN = prove (`!s:real^N->bool t:real^M->bool. dimindex(:M) < dimindex(:N) /\ s homeomorphic t ==> interior s = {}`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`(:real^M)`; `(:real^N)`] ISOMETRY_SUBSET_SUBSPACE) THEN ASM_SIMP_TAC[SUBSPACE_UNIV; DIM_UNIV; LT_IMP_LE; IN_UNIV; SUBSET_UNIV] THEN DISCH_THEN(X_CHOOSE_THEN `h:real^M->real^N` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC(MESON[HOMEOMORPHIC_EMPTY] `!t. interior(t:real^N->bool) homeomorphic interior(s:real^N->bool) /\ interior t = {} ==> interior s = {}`) THEN EXISTS_TAC `IMAGE (h:real^M->real^N) t` THEN CONJ_TAC THENL [MATCH_MP_TAC HOMEOMORPHIC_INTERIORS_SAME_DIMENSION THEN REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHIC_SYM]) THEN MATCH_MP_TAC EQ_IMP THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_LEFT_EQ THEN ASM_MESON_TAC[PRESERVES_NORM_INJECTIVE]; MATCH_MP_TAC(SET_RULE `!t. s SUBSET t /\ t = {} ==> s = {}`) THEN EXISTS_TAC `interior(IMAGE (h:real^M->real^N) (:real^M))` THEN SIMP_TAC[SUBSET_INTERIOR; SET_RULE `IMAGE f s SUBSET IMAGE f UNIV`] THEN MATCH_MP_TAC EMPTY_INTERIOR_LOWDIM THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] LET_TRANS)) THEN REWRITE_TAC[GSYM DIM_UNIV] THEN MATCH_MP_TAC EQ_IMP_LE THEN MATCH_MP_TAC DIM_INJECTIVE_LINEAR_IMAGE THEN ASM_MESON_TAC[PRESERVES_NORM_INJECTIVE]]);; let EMPTY_INTERIOR_LOWDIM_GEN_LE = prove (`!s:real^N->bool t:real^M->bool. dimindex(:M) <= dimindex(:N) /\ interior t = {} /\ s homeomorphic t ==> interior s = {}`, REWRITE_TAC[LE_LT] THEN REPEAT STRIP_TAC THENL [ASM_MESON_TAC[EMPTY_INTERIOR_LOWDIM_GEN]; ASM_MESON_TAC[HOMEOMORPHIC_INTERIORS_SAME_DIMENSION; HOMEOMORPHIC_EMPTY]]);; let HOMEOMORPHIC_HYPERPLANES_EQ = prove (`!a:real^M b c:real^N d. ~(a = vec 0) /\ ~(c = vec 0) ==> ({x | a dot x = b} homeomorphic {x | c dot x = d} <=> dimindex(:M) = dimindex(:N))`, SIMP_TAC[HOMEOMORPHIC_AFFINE_SETS_EQ; AFFINE_HYPERPLANE] THEN SIMP_TAC[AFF_DIM_HYPERPLANE; INT_OF_NUM_EQ; INT_ARITH `x - &1:int = y - &1 <=> x = y`]);; let HOMEOMORPHIC_UNIV_UNIV = prove (`(:real^M) homeomorphic (:real^N) <=> dimindex(:M) = dimindex(:N)`, SIMP_TAC[HOMEOMORPHIC_SUBSPACES_EQ; DIM_UNIV; SUBSPACE_UNIV]);; let HOMEOMORPHIC_CBALLS_EQ = prove (`!a:real^M b:real^N r s. cball(a,r) homeomorphic cball(b,s) <=> r < &0 /\ s < &0 \/ r = &0 /\ s = &0 \/ &0 < r /\ &0 < s /\ dimindex(:M) = dimindex(:N)`, let lemma = let d = `dimindex(:M) = dimindex(:N)` and t = `?a:real^M b:real^N. ~(cball(a,r) homeomorphic cball(b,s))` in DISCH d (DISCH t (GEOM_EQUAL_DIMENSION_RULE (ASSUME d) (ASSUME t))) in REPEAT GEN_TAC THEN ASM_CASES_TAC `r < &0` THENL [ASM_SIMP_TAC[CBALL_EMPTY; HOMEOMORPHIC_EMPTY; CBALL_EQ_EMPTY] THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_CASES_TAC `r = &0` THEN ASM_REWRITE_TAC[REAL_LT_REFL] THENL [ASM_SIMP_TAC[CBALL_TRIVIAL; FINITE_SING; HOMEOMORPHIC_FINITE_STRONG] THEN REWRITE_TAC[FINITE_CBALL] THEN ASM_CASES_TAC `s < &0` THEN ASM_SIMP_TAC[CBALL_EMPTY; CARD_CLAUSES; FINITE_EMPTY; NOT_IN_EMPTY; ARITH; REAL_LT_IMP_NE] THEN ASM_CASES_TAC `s = &0` THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN ASM_SIMP_TAC[CBALL_TRIVIAL; CARD_CLAUSES; FINITE_EMPTY; NOT_IN_EMPTY; REAL_LE_REFL; ARITH]; ALL_TAC] THEN SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_CASES_TAC `s <= &0` THEN ASM_SIMP_TAC[HOMEOMORPHIC_FINITE_STRONG; FINITE_CBALL] THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `&0 < s` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_REWRITE_TAC[] THEN EQ_TAC THENL [REWRITE_TAC[homeomorphic; HOMEOMORPHISM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f:real^M->real^N`; `g:real^N->real^M`] THEN STRIP_TAC THEN REWRITE_TAC[GSYM LE_ANTISYM] THEN CONJ_TAC THEN MATCH_MP_TAC INVARIANCE_OF_DIMENSION THENL [MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `ball(a:real^M,r)`] THEN MP_TAC(ISPECL [`a:real^M`; `r:real`] BALL_SUBSET_CBALL); MAP_EVERY EXISTS_TAC [`g:real^N->real^M`; `ball(b:real^N,s)`] THEN MP_TAC(ISPECL [`b:real^N`; `s:real`] BALL_SUBSET_CBALL)] THEN ASM_REWRITE_TAC[BALL_EQ_EMPTY; OPEN_BALL; REAL_NOT_LE] THEN ASM_MESON_TAC[SUBSET; CONTINUOUS_ON_SUBSET]; DISCH_THEN(MP_TAC o MATCH_MP lemma) THEN GEN_REWRITE_TAC LAND_CONV [GSYM CONTRAPOS_THM] THEN REWRITE_TAC[NOT_EXISTS_THM] THEN ASM_SIMP_TAC[HOMEOMORPHIC_CBALLS]]);; let HOMEOMORPHIC_BALLS_EQ = prove (`!a:real^M b:real^N r s. ball(a,r) homeomorphic ball(b,s) <=> r <= &0 /\ s <= &0 \/ &0 < r /\ &0 < s /\ dimindex(:M) = dimindex(:N)`, let lemma = let d = `dimindex(:M) = dimindex(:N)` and t = `?a:real^M b:real^N. ~(ball(a,r) homeomorphic ball(b,s))` in DISCH d (DISCH t (GEOM_EQUAL_DIMENSION_RULE (ASSUME d) (ASSUME t))) in REPEAT GEN_TAC THEN ASM_CASES_TAC `r <= &0` THENL [ASM_SIMP_TAC[BALL_EMPTY; HOMEOMORPHIC_EMPTY; BALL_EQ_EMPTY] THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_CASES_TAC `s <= &0` THENL [ASM_SIMP_TAC[BALL_EMPTY; HOMEOMORPHIC_EMPTY; BALL_EQ_EMPTY] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LE]) THEN ASM_REWRITE_TAC[] THEN EQ_TAC THENL [REWRITE_TAC[homeomorphic; HOMEOMORPHISM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f:real^M->real^N`; `g:real^N->real^M`] THEN STRIP_TAC THEN REWRITE_TAC[GSYM LE_ANTISYM] THEN CONJ_TAC THEN MATCH_MP_TAC INVARIANCE_OF_DIMENSION THENL [MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `ball(a:real^M,r)`]; MAP_EVERY EXISTS_TAC [`g:real^N->real^M`; `ball(b:real^N,s)`]] THEN ASM_REWRITE_TAC[BALL_EQ_EMPTY; OPEN_BALL; REAL_NOT_LE] THEN ASM SET_TAC[]; DISCH_THEN(MP_TAC o MATCH_MP lemma) THEN GEN_REWRITE_TAC LAND_CONV [GSYM CONTRAPOS_THM] THEN REWRITE_TAC[NOT_EXISTS_THM] THEN ASM_SIMP_TAC[HOMEOMORPHIC_BALLS]]);; (* ------------------------------------------------------------------------- *) (* General forms of the Jordan curve theorem in setting of R^n. *) (* ------------------------------------------------------------------------- *) let CARD_EQ_COMPONENTS_COMPLEMENTS = prove (`!s t:real^N->bool. closed s /\ closed t /\ s homeomorphic t ==> components((:real^N) DIFF s) =_c components((:real^N) DIFF t)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?f g. homeomorphic_maps (euclidean_space (dimindex (:N)),euclidean) (f:(num->real)->real^N,g)` STRIP_ASSUME_TAC THENL [MESON_TAC[HOMEOMORPHIC_MAPS_EUCLIDEAN_SPACE_EUCLIDEAN]; ALL_TAC] THEN MP_TAC(ISPECL [`dimindex(:N)`; `IMAGE (g:real^N->num->real) s`; `IMAGE (g:real^N->num->real) t`] CARD_EQ_CONNECTED_COMPONENTS_EUCLIDEAN_COMPLEMENTS) THEN ANTS_TAC THENL [REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [CONJ_TAC THENL [UNDISCH_TAC `closed(s:real^N->bool)`; UNDISCH_TAC `closed(t:real^N->bool)`] THEN REWRITE_TAC[CLOSED_IN] THEN MATCH_MP_TAC EQ_IMP THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC HOMEOMORPHIC_MAP_CLOSEDNESS THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN; SUBSET_UNIV] THEN ASM_MESON_TAC[HOMEOMORPHIC_MAPS_MAP]; TRANS_TAC HOMEOMORPHIC_SPACE_TRANS `subtopology euclidean (s:real^N->bool)` THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[HOMEOMORPHIC_SPACE_SYM]; TRANS_TAC HOMEOMORPHIC_SPACE_TRANS `subtopology euclidean (t:real^N->bool)` THEN ASM_REWRITE_TAC[HOMEOMORPHIC_SPACE_EUCLIDEAN]] THEN REWRITE_TAC[HOMEOMORPHIC_SPACE] THEN EXISTS_TAC `g:real^N->num->real` THEN MATCH_MP_TAC HOMEOMORPHIC_MAP_SUBTOPOLOGIES THEN RULE_ASSUM_TAC(REWRITE_RULE[HOMEOMORPHIC_MAPS_MAP]) THEN ASM_REWRITE_TAC[TOPSPACE_EUCLIDEAN; INTER_UNIV] THEN RULE_ASSUM_TAC(REWRITE_RULE [TOPSPACE_EUCLIDEAN; HOMEOMORPHIC_EQ_EVERYTHING_MAP]) THEN ASM SET_TAC[]]; MATCH_MP_TAC EQ_IMP THEN MATCH_MP_TAC CARD_EQ_CONG] THEN W(fun (asl,w) -> SUBGOAL_THEN(mk_forall(`s:real^N->bool`,lhand w)) (fun th -> REWRITE_TAC[th])) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHIC_MAPS_MAP]) THEN POP_ASSUM_LIST(K ALL_TAC) THEN SPEC_TAC(`dimindex(:N)`,`n:num`) THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN; IN_UNIV] THEN STRIP_TAC THEN X_GEN_TAC `s:real^N->bool` THEN FIRST_ASSUM(MP_TAC o SPECL [`(:real^N) DIFF s`; `topspace (euclidean_space n) DIFF IMAGE g (s:real^N->bool)`] o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMEOMORPHIC_MAP_SUBTOPOLOGIES)) THEN ANTS_TAC THENL [ALL_TAC; DISCH_THEN(SUBST1_TAC o MATCH_MP HOMEOMORPHIC_MAP_CONNECTED_COMPONENTS_OF) THEN REWRITE_TAC[EUCLIDEAN_CONNECTED_COMPONENTS_OF] THEN MATCH_MP_TAC CARD_EQ_IMAGE] THEN REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[HOMEOMORPHIC_EQ_EVERYTHING_MAP; TOPSPACE_EUCLIDEAN] THEN SET_TAC[]);; let JORDAN_CURVE_THEOREM_GEN = prove (`!s:real^N->bool. 2 <= dimindex(:N) /\ s homeomorphic sphere(vec 0:real^N,&1) ==> ?ins out. ~(ins = {}) /\ open ins /\ connected ins /\ ~(out = {}) /\ open out /\ connected out /\ bounded ins /\ ~bounded out /\ ins INTER out = {} /\ ins UNION out = (:real^N) DIFF s /\ frontier ins = s /\ frontier out = s`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `compact(s:real^N->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[HOMEOMORPHIC_COMPACTNESS; COMPACT_SPHERE]; ALL_TAC] THEN MP_TAC(ISPECL [`s:real^N->bool`; `sphere(vec 0:real^N,&1)`] CARD_EQ_COMPONENTS_COMPLEMENTS) THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED; CLOSED_SPHERE] THEN DISCH_THEN(MP_TAC o SPEC `2` o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] CARD_HAS_SIZE_CONG)) THEN DISCH_THEN(MP_TAC o snd o EQ_IMP_RULE) THEN ANTS_TAC THENL [REWRITE_TAC[HAS_SIZE_CONV `s HAS_SIZE 2`] THEN MAP_EVERY EXISTS_TAC [`ball(vec 0:real^N,&1)`; `(:real^N) DIFF cball(vec 0,&1)`] THEN CONJ_TAC THENL [DISCH_THEN(MP_TAC o AP_TERM `bounded:(real^N->bool)->bool`) THEN REWRITE_TAC[BOUNDED_BALL] THEN MATCH_MP_TAC COBOUNDED_IMP_UNBOUNDED THEN REWRITE_TAC[BOUNDED_CBALL; COMPL_COMPL]; MATCH_MP_TAC COMPONENTS_UNIQUE_2 THEN REWRITE_TAC[CONNECTED_BALL] THEN ASM_SIMP_TAC[CONNECTED_COMPLEMENT_BOUNDED_CONVEX; BOUNDED_CBALL; CONVEX_CBALL] THEN REWRITE_TAC[EXTENSION; IN_UNION; IN_DIFF; IN_UNIV] THEN REWRITE_TAC[IN_CBALL_0; IN_BALL_0; IN_SPHERE_0] THEN CONJ_TAC THENL [REAL_ARITH_TAC; DISCH_TAC] THEN MP_TAC(ISPECL [`\x:real^N. lift(norm x)`; `(:real^N) DIFF sphere(vec 0,&1)`] CONNECTED_CONTINUOUS_IMAGE) THEN ASM_SIMP_TAC[CONTINUOUS_ON_LIFT_NORM_COMPOSE; CONTINUOUS_ON_ID] THEN REWRITE_TAC[GSYM IS_INTERVAL_CONNECTED_1; IS_INTERVAL_1] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM; IMP_CONJ; FORALL_IN_IMAGE] THEN DISCH_THEN(MP_TAC o SPEC `vec 0:real^N`) THEN REWRITE_TAC[NORM_0; IN_DIFF; IN_SPHERE_0; IN_UNIV; IN_IMAGE] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN DISCH_THEN(MP_TAC o SPEC `&2 % basis 1:real^N`) THEN REWRITE_TAC[FORALL_LIFT; LIFT_DROP; NORM_MUL; IN_IMAGE] THEN SIMP_TAC[NORM_BASIS; DIMINDEX_GE_1; LE_REFL] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN DISCH_THEN(MP_TAC o SPEC `&1:real`) THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN MESON_TAC[LIFT_EQ]]; DISCH_TAC] THEN SUBGOAL_THEN `~connected((:real^N) DIFF s)` ASSUME_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[HAS_SIZE]) THEN ASM_SIMP_TAC[CONNECTED_EQ_CARD_COMPONENTS] THEN CONV_TAC NUM_REDUCE_CONV; ALL_TAC] THEN MP_TAC(ISPEC `(:real^N) DIFF s` COBOUNDED_HAS_BOUNDED_COMPONENT) THEN ASM_SIMP_TAC[COMPL_COMPL; COMPACT_IMP_BOUNDED] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `ins:real^N->bool` THEN STRIP_TAC THEN MP_TAC(ISPEC `(:real^N) DIFF s` COBOUNDED_UNBOUNDED_COMPONENTS) THEN ASM_SIMP_TAC[COMPL_COMPL; COMPACT_IMP_BOUNDED] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `out:real^N->bool` THEN STRIP_TAC THEN REWRITE_TAC[GSYM CONNECTED_EQ_CARD_COMPONENTS] THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[TAUT `a /\ b /\ c /\ d /\ e /\ f /\ g <=> (a /\ d) /\ (b /\ e) /\ (c /\ f) /\ g`] THEN CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_NONEMPTY]; ALL_TAC] THEN CONJ_TAC THENL [ASM_MESON_TAC[OPEN_COMPONENTS; closed; COMPACT_IMP_CLOSED]; ALL_TAC] THEN CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]; ALL_TAC] THEN CONJ_TAC THENL [ASM_MESON_TAC[COMPONENTS_NONOVERLAP]; ALL_TAC] THEN CONJ_TAC THENL [GEN_REWRITE_TAC RAND_CONV [UNIONS_COMPONENTS] THEN ASM_REWRITE_TAC[GSYM UNIONS_2] THEN AP_TERM_TAC THEN FIRST_X_ASSUM(MP_TAC o CONV_RULE HAS_SIZE_CONV) THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `!t. closed t /\ t PSUBSET s ==> connected ((:real^N) DIFF t)` ASSUME_TAC THENL [ALL_TAC; CONJ_TAC THEN MATCH_MP_TAC FRONTIER_MINIMAL_SEPARATING_CLOSED THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED]] THEN SUBGOAL_THEN `!t. t PSUBSET sphere(vec 0,&1) ==> connected((:real^N) DIFF t)` (LABEL_TAC "*") THENL [REWRITE_TAC[PSUBSET_ALT; SUBSET; IN_SPHERE; GSYM REAL_LE_ANTISYM] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `(:real^N) DIFF t = {x:real^N | dist(vec 0,x) <= &1 /\ ~(x IN t)} UNION {x:real^N | &1 <= dist(vec 0,x) /\ ~(x IN t)}` SUBST1_TAC THENL [SET_TAC[REAL_LE_TOTAL]; MATCH_MP_TAC CONNECTED_UNION] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_INTERMEDIATE_CLOSURE THEN EXISTS_TAC `ball(vec 0:real^N,&1)` THEN ASM_SIMP_TAC[CONNECTED_BALL; CLOSURE_BALL; SUBSET; IN_BALL; IN_CBALL; REAL_LT_01; IN_ELIM_THM] THEN ASM_MESON_TAC[REAL_LT_IMP_LE; REAL_NOT_LE]; MATCH_MP_TAC CONNECTED_INTERMEDIATE_CLOSURE THEN EXISTS_TAC `(:real^N) DIFF cball(vec 0,&1)` THEN REWRITE_TAC[CLOSURE_COMPLEMENT; SUBSET; IN_DIFF; IN_UNIV; IN_BALL; IN_CBALL; IN_ELIM_THM; INTERIOR_CBALL] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[REAL_LT_IMP_LE; REAL_NOT_LE]] THEN MATCH_MP_TAC CONNECTED_OPEN_DIFF_CBALL THEN ASM_REWRITE_TAC[SUBSET_UNIV; CONNECTED_UNIV; OPEN_UNIV]; ASM SET_TAC[]]; X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphic]) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f:real^N->real^N`; `g:real^N->real^N`] THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o SPECL [`t:real^N->bool`; `IMAGE (f:real^N->real^N) t`] o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN ANTS_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]; DISCH_THEN(ASSUME_TAC o MATCH_MP HOMEOMORPHISM_IMP_HOMEOMORPHIC)] THEN FIRST_X_ASSUM(MP_TAC o SPEC `IMAGE (f:real^N->real^N) t`) THEN ANTS_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]; MATCH_MP_TAC EQ_IMP] THEN REWRITE_TAC[CONNECTED_EQ_CARD_COMPONENTS] THEN MATCH_MP_TAC(MESON[HAS_SIZE] `(!n. s HAS_SIZE n <=> t HAS_SIZE n) ==> (FINITE s /\ CARD s <= m <=> FINITE t /\ CARD t <= m)`) THEN GEN_TAC THEN MATCH_MP_TAC CARD_HAS_SIZE_CONG THEN MATCH_MP_TAC CARD_EQ_COMPONENTS_COMPLEMENTS THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[HOMEOMORPHIC_SYM]] THEN MATCH_MP_TAC COMPACT_IMP_CLOSED THEN SUBGOAL_THEN `compact(t:real^N->bool)` MP_TAC THENL [ASM_MESON_TAC[COMPACT_EQ_BOUNDED_CLOSED; BOUNDED_SUBSET; PSUBSET]; MATCH_MP_TAC EQ_IMP] THEN MATCH_MP_TAC HOMEOMORPHIC_COMPACTNESS THEN ASM_REWRITE_TAC[]]);; let JORDAN_INSIDE_OUTSIDE_GEN = prove (`!s:real^N->bool. 2 <= dimindex(:N) /\ s homeomorphic sphere(vec 0:real^N,&1) ==> ~(inside s = {}) /\ open(inside s) /\ connected(inside s) /\ ~(outside s = {}) /\ open(outside s) /\ connected(outside s) /\ bounded(inside s) /\ ~bounded(outside s) /\ inside s INTER outside s = {} /\ inside s UNION outside s = (:real^N) DIFF s /\ frontier(inside s) = s /\ frontier(outside s) = s`, GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP JORDAN_CURVE_THEOREM_GEN) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`ins:real^N->bool`; `out:real^N->bool`] THEN STRIP_TAC THEN SUBGOAL_THEN `inside s :real^N->bool = ins /\ outside s:real^N->bool = out ` (fun th -> ASM_REWRITE_TAC[th]) THEN MATCH_MP_TAC INSIDE_OUTSIDE_UNIQUE THEN SUBGOAL_THEN `compact(s:real^N->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[HOMEOMORPHIC_COMPACTNESS; COMPACT_SPHERE]; ALL_TAC] THEN ASM_SIMP_TAC[CONNECTED_OPEN_SET; GSYM closed; COMPACT_IMP_CLOSED] THEN ASM_MESON_TAC[]);; let JORDAN_BROUWER_FRONTIER = prove (`!s t a:real^N r. 2 <= dimindex(:N) /\ s homeomorphic sphere(a,r) /\ t IN components((:real^N) DIFF s) ==> frontier t = s`, REPEAT GEN_TAC THEN ASM_CASES_TAC `r < &0` THENL [ASM_SIMP_TAC[SPHERE_EMPTY; HOMEOMORPHIC_EMPTY; IMP_CONJ; DIFF_EMPTY] THEN SIMP_TAC[snd(EQ_IMP_RULE(SPEC_ALL COMPONENTS_EQ_SING)); UNIV_NOT_EMPTY; CONNECTED_UNIV; IN_SING; FRONTIER_UNIV]; ALL_TAC] THEN ASM_CASES_TAC `r = &0` THENL [ASM_SIMP_TAC[HOMEOMORPHIC_FINITE_STRONG; SPHERE_SING; FINITE_SING] THEN SIMP_TAC[CARD_CLAUSES; FINITE_EMPTY; GSYM HAS_SIZE; NOT_IN_EMPTY] THEN REWRITE_TAC[HAS_SIZE_CLAUSES; UNWIND_THM2; NOT_IN_EMPTY; IMP_CONJ] THEN SIMP_TAC[LEFT_IMP_EXISTS_THM; CONNECTED_PUNCTURED_UNIVERSE; IN_SING; snd(EQ_IMP_RULE(SPEC_ALL COMPONENTS_EQ_SING)); FRONTIER_SING; SET_RULE `UNIV DIFF s = {} <=> s = UNIV`; FRONTIER_COMPLEMENT; MESON[BOUNDED_SING; NOT_BOUNDED_UNIV] `~((:real^N) = {a})`]; ALL_TAC] THEN SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; STRIP_TAC] THEN MP_TAC(ISPEC `s:real^N->bool` JORDAN_CURVE_THEOREM_GEN) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[] THEN TRANS_TAC HOMEOMORPHIC_TRANS `sphere(a:real^N,r)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC HOMEOMORPHIC_SPHERES THEN ASM_REWRITE_TAC[REAL_LT_01]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`ins:real^N->bool`; `out:real^N->bool`] THEN STRIP_TAC THEN SUBGOAL_THEN `components((:real^N) DIFF s) = {ins,out}` SUBST_ALL_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC COMPONENTS_UNIQUE_2 THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `compact(s:real^N->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[HOMEOMORPHIC_COMPACTNESS; COMPACT_SPHERE]; ALL_TAC] THEN ASM_SIMP_TAC[CONNECTED_OPEN_SET; GSYM closed; COMPACT_IMP_CLOSED] THEN ASM_MESON_TAC[]);; let JORDAN_BROUWER_NONSEPARATION = prove (`!s t a:real^N r. 2 <= dimindex(:N) /\ s homeomorphic sphere(a,r) /\ t PSUBSET s ==> connected((:real^N) DIFF t)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MINIMAL_SEPARATING_COMMON_COMPONENT_FRONTIER THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [DISCH_THEN SUBST_ALL_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMEOMORPHIC_COMPACTNESS) THEN REWRITE_TAC[COMPACT_SPHERE; NOT_COMPACT_UNIV]; REPEAT STRIP_TAC THEN MATCH_MP_TAC JORDAN_BROUWER_FRONTIER THEN ASM_MESON_TAC[]]);; let JORDAN_BROUWER_SEPARATION = prove (`!s a:real^N r. &0 < r /\ s homeomorphic sphere(a,r) ==> ~connected((:real^N) DIFF s)`, REPEAT GEN_TAC THEN STRIP_TAC THEN SUBGOAL_THEN `compact(s:real^N->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[HOMEOMORPHIC_COMPACTNESS; COMPACT_SPHERE]; ALL_TAC] THEN SUBGOAL_THEN `1 <= dimindex(:N)` MP_TAC THENL [REWRITE_TAC[DIMINDEX_GE_1]; REWRITE_TAC[ARITH_RULE `1 <= n <=> n = 1 \/ 2 <= n`]] THEN STRIP_TAC THENL [ASM_SIMP_TAC[BOUNDED_SEPARATION_1D; COMPACT_IMP_BOUNDED] THEN DISCH_THEN SUBST_ALL_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHIC_EMPTY]) THEN REWRITE_TAC[SPHERE_EQ_EMPTY] THEN ASM_REAL_ARITH_TAC; ASM_SIMP_TAC[CONNECTED_OPEN_SET; GSYM closed; COMPACT_IMP_CLOSED] THEN MP_TAC(ISPEC `s:real^N->bool` JORDAN_CURVE_THEOREM_GEN) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[] THEN TRANS_TAC HOMEOMORPHIC_TRANS `sphere(a:real^N,r)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC HOMEOMORPHIC_SPHERES THEN ASM_REWRITE_TAC[REAL_LT_01]; MESON_TAC[]]]);; let JORDAN_CURVE_THEOREM = prove (`!c:real^1->real^2. simple_path c /\ pathfinish c = pathstart c ==> ?ins out. ~(ins = {}) /\ open ins /\ connected ins /\ ~(out = {}) /\ open out /\ connected out /\ bounded ins /\ ~bounded out /\ ins INTER out = {} /\ ins UNION out = (:real^2) DIFF path_image c /\ frontier ins = path_image c /\ frontier out = path_image c`, REPEAT STRIP_TAC THEN MATCH_MP_TAC JORDAN_CURVE_THEOREM_GEN THEN REWRITE_TAC[DIMINDEX_2; LE_REFL] THEN ASM_SIMP_TAC[HOMEOMORPHIC_SIMPLE_PATH_IMAGE_CIRCLE; REAL_LT_01]);; let JORDAN_DISCONNECTED = prove (`!c. simple_path c /\ pathfinish c = pathstart c ==> ~connected((:real^2) DIFF path_image c)`, GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[connected] THEN FIRST_ASSUM(MP_TAC o MATCH_MP JORDAN_CURVE_THEOREM) THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);; let JORDAN_INSIDE_OUTSIDE = prove (`!c:real^1->real^2. simple_path c /\ pathfinish c = pathstart c ==> ~(inside(path_image c) = {}) /\ open(inside(path_image c)) /\ connected(inside(path_image c)) /\ ~(outside(path_image c) = {}) /\ open(outside(path_image c)) /\ connected(outside(path_image c)) /\ bounded(inside(path_image c)) /\ ~bounded(outside(path_image c)) /\ inside(path_image c) INTER outside(path_image c) = {} /\ inside(path_image c) UNION outside(path_image c) = (:real^2) DIFF path_image c /\ frontier(inside(path_image c)) = path_image c /\ frontier(outside(path_image c)) = path_image c`, GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC JORDAN_INSIDE_OUTSIDE_GEN THEN REWRITE_TAC[DIMINDEX_2; LE_REFL] THEN ASM_SIMP_TAC[HOMEOMORPHIC_SIMPLE_PATH_IMAGE_CIRCLE; REAL_LT_01]);; let JORDAN_COMPONENTS = prove (`!g. simple_path g /\ pathfinish g = pathstart g ==> components((:real^2) DIFF path_image g) = {inside(path_image g),outside(path_image g)}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC COMPONENTS_OPEN_UNIQUE THEN REWRITE_TAC[UNIONS_2; PAIRWISE_INSERT; NOT_IN_EMPTY; FORALL_IN_INSERT; IMP_CONJ; PAIRWISE_EMPTY] THEN MP_TAC(ISPEC `g:real^1->real^2` JORDAN_INSIDE_OUTSIDE) THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPEC `path_image g:real^2->bool` INSIDE_INTER_OUTSIDE) THEN REPLICATE_TAC 2 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);; let HOMEOMORPHIC_SEPARATION_SPHERE_CARD_EQ = prove (`!s t:real^N->bool a r. s SUBSET sphere(a,r) /\ t SUBSET sphere(a,r) /\ closed s /\ closed t /\ s homeomorphic t ==> (components(sphere(a,r) DIFF s) =_c components(sphere(a,r) DIFF t))`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `compact(s:real^N->bool) /\ compact(t:real^N->bool)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[COMPACT_EQ_BOUNDED_CLOSED; BOUNDED_SUBSET; BOUNDED_SPHERE]; ALL_TAC] THEN ASM_CASES_TAC `FINITE(sphere(a:real^N,r))` THENL [FIRST_ASSUM(MP_TAC o MATCH_MP HOMEOMORPHIC_IMP_CARD_EQ) THEN SUBGOAL_THEN `FINITE(s:real^N->bool) /\ FINITE(t:real^N->bool)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[FINITE_SUBSET]; ALL_TAC] THEN ASM_SIMP_TAC[COMPONENTS_FINITE; FINITE_DIFF; CARD_EQ_CARD; SIMPLE_IMAGE; FINITE_IMAGE; CARD_IMAGE_INJ; CARD_DIFF; SET_RULE `{a} = {b} <=> a = b`]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [FINITE_SPHERE]) THEN REWRITE_TAC[DE_MORGAN_THM; REAL_NOT_LE] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (ARITH_RULE `~(n = 1) ==> 1 <= n ==> 2 <= n`)) THEN REWRITE_TAC[DIMINDEX_GE_1] THEN DISCH_TAC] THEN SUBGOAL_THEN `s PSUBSET sphere(a:real^N,r) <=> t PSUBSET sphere(a:real^N,r)` ASSUME_TAC THENL [REPEAT(POP_ASSUM MP_TAC) THEN MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`t:real^N->bool`; `s:real^N->bool`] THEN REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC] THEN MATCH_MP_TAC(MESON[] `(!s t. R s t ==> R t s) /\ (!s t. R s t ==> P s ==> P t) ==> !s t. R s t ==> (P s <=> P t)`) THEN CONJ_TAC THENL [REWRITE_TAC[HOMEOMORPHIC_SYM; CONJ_ACI]; ALL_TAC] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[PSUBSET] THEN DISCH_THEN SUBST_ALL_TAC THEN MP_TAC(ISPECL [`sphere(a:real^N,r)`; `s:real^N->bool`; `a:real^N`; `r:real`] JORDAN_BROUWER_NONSEPARATION) THEN ASM_REWRITE_TAC[HOMEOMORPHIC_REFL] THEN MATCH_MP_TAC JORDAN_BROUWER_SEPARATION THEN ASM_MESON_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE `(s PSUBSET u <=> t PSUBSET u) ==> s SUBSET u /\ t SUBSET u ==> s = u /\ t = u \/ s PSUBSET u /\ t PSUBSET u`)) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ASM_REWRITE_TAC[DIFF_EQ_EMPTY; COMPONENTS_EMPTY; CARD_EQ_REFL] THEN SUBGOAL_THEN `?w z:real^N. w IN sphere(a,r) /\ z IN sphere(a,r) /\ ~(w IN s) /\ ~(z IN t)` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(sphere(a:real^N,r) DELETE w) homeomorphic (:real^(N,1)finite_diff)` MP_TAC THENL [MATCH_MP_TAC HOMEOMORPHIC_PUNCTURED_SPHERE_UNIV THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[DIMINDEX_FINITE_DIFF; DIMINDEX_1] THEN ASM_ARITH_TAC; REWRITE_TAC[homeomorphic; LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`f:real^N->real^(N,1)finite_diff`; `g:real^(N,1)finite_diff->real^N`] THEN DISCH_TAC THEN SUBGOAL_THEN `(sphere(a:real^N,r) DELETE z) homeomorphic (:real^(N,1)finite_diff)` MP_TAC THENL [MATCH_MP_TAC HOMEOMORPHIC_PUNCTURED_SPHERE_UNIV THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[DIMINDEX_FINITE_DIFF; DIMINDEX_1] THEN ASM_ARITH_TAC; REWRITE_TAC[homeomorphic; LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`h:real^N->real^(N,1)finite_diff`; `k:real^(N,1)finite_diff->real^N`] THEN DISCH_TAC THEN MP_TAC(ISPECL [`f:real^N->real^(N,1)finite_diff`; `g:real^(N,1)finite_diff->real^N`; `a:real^N`; `r:real`; `s:real^N->bool`; `w:real^N`] CARD_EQ_COMPONENTS_IN_COMPACTIFICATION) THEN ASM_SIMP_TAC[CLOSURE_CLOSED] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] CARD_EQ_TRANS) THEN MP_TAC(ISPECL [`h:real^N->real^(N,1)finite_diff`; `k:real^(N,1)finite_diff->real^N`; `a:real^N`; `r:real`; `t:real^N->bool`; `z:real^N`] CARD_EQ_COMPONENTS_IN_COMPACTIFICATION) THEN ASM_SIMP_TAC[CLOSURE_CLOSED] THEN GEN_REWRITE_TAC LAND_CONV [CARD_EQ_SYM] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] CARD_EQ_TRANS) THEN MATCH_MP_TAC CARD_ADD_CONG THEN REWRITE_TAC[CARD_EQ_REFL] THEN SUBGOAL_THEN `compact(IMAGE (f:real^N->real^(N,1)finite_diff) s) /\ compact(IMAGE (h:real^N->real^(N,1)finite_diff) t)` STRIP_ASSUME_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[] THEN REPEAT(FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [homeomorphism])) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; ALL_TAC] THEN ONCE_REWRITE_TAC[SET_RULE `{x | x IN s /\ P x} = s DIFF {x | x IN s /\ ~P x}`] THEN MATCH_MP_TAC CARD_DIFF_CONG THEN REWRITE_TAC[SUBSET_RESTRICT] THEN CONJ_TAC THENL [MATCH_MP_TAC CARD_EQ_COMPONENTS_COMPLEMENTS THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED] THEN TRANS_TAC HOMEOMORPHIC_TRANS `s:real^N->bool` THEN CONJ_TAC THENL [ALL_TAC; TRANS_TAC HOMEOMORPHIC_TRANS `t:real^N->bool` THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM]] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMEOMORPHIC_SELF_IMAGE)) THEN ASM SET_TAC[]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhand o rand) HAS_SIZE_UNBOUNDED_COMPONENTS_COMPLEMENT o rand o lhand o snd) THEN W(MP_TAC o PART_MATCH (lhand o rand) HAS_SIZE_UNBOUNDED_COMPONENTS_COMPLEMENT o lhand o lhand o rand o snd) THEN ASM_SIMP_TAC[COMPACT_IMP_BOUNDED] THEN SIMP_TAC[CARD_EQ_CARD; HAS_SIZE] THEN FIRST_ASSUM(MP_TAC o MATCH_MP (MESON[HOMEOMORPHIC_EMPTY] `s homeomorphic t ==> (s = {} <=> t = {})`)) THEN SIMP_TAC[IMAGE_EQ_EMPTY; CARD_LT_FINITE_INFINITE]);; let HOMEOMORPHIC_SEPARATION_SPHERE_HAS_SIZE_EQ = prove (`!s t:real^N->bool a r n. s SUBSET sphere(a,r) /\ t SUBSET sphere(a,r) /\ s homeomorphic t ==> (components(sphere(a,r) DIFF s) HAS_SIZE n <=> components(sphere(a,r) DIFF t) HAS_SIZE n)`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `subtopology euclidean (sphere(a:real^N,r))` KURATOWSKI_COMPONENT_NUMBER_INVARIANCE) THEN SIMP_TAC[METRIZABLE_IMP_HEREDITARILY_NORMAL_SPACE; METRIZABLE_SPACE_EUCLIDEAN; METRIZABLE_SPACE_SUBTOPOLOGY; HAUSDORFF_SPACE_SUBTOPOLOGY; HAUSDORFF_SPACE_EUCLIDEAN] THEN REWRITE_TAC[COMPACT_SPACE_EUCLIDEAN_SUBTOPOLOGY; COMPACT_SPHERE] THEN REWRITE_TAC[LOCALLY_CONNECTED_SPACE_SUBTOPOLOGY_EUCLIDEAN; LOCALLY_CONNECTED_SPHERE] THEN SIMP_TAC[CLOSED_IN_CLOSED_EQ; CLOSED_SPHERE] THEN REWRITE_TAC[HOMEOMORPHIC_SPACE_EUCLIDEAN; SUBTOPOLOGY_SUBTOPOLOGY; EUCLIDEAN_CONNECTED_COMPONENTS_OF; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN REWRITE_TAC[SET_RULE `s INTER (s DIFF t) = s DIFF t`] THEN DISCH_THEN(MP_TAC o fst o EQ_IMP_RULE) THEN ANTS_TAC THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC CARD_HAS_SIZE_CONG THEN MATCH_MP_TAC HOMEOMORPHIC_SEPARATION_SPHERE_CARD_EQ; ALL_TAC] THEN ASM_MESON_TAC[SET_RULE `s SUBSET u ==> u INTER s = s`]);; let HOMEOMORPHIC_SEPARATION_SPHERE = prove (`!s t:real^N->bool a r. s SUBSET sphere(a,r) /\ t SUBSET sphere(a,r) /\ s homeomorphic t ==> (connected (sphere(a,r) DIFF s) <=> connected(sphere(a,r) DIFF t))`, REPEAT STRIP_TAC THEN REWRITE_TAC[CONNECTED_EQ_CARD_COMPONENTS] THEN REWRITE_TAC[ARITH_RULE `c <= 1 <=> c = 0 \/ c = 1`; LEFT_OR_DISTRIB] THEN REWRITE_TAC[GSYM HAS_SIZE] THEN BINOP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_SEPARATION_SPHERE_HAS_SIZE_EQ THEN ASM_REWRITE_TAC[]);; let HAS_SIZE_EQ_COMPONENTS_COMPLEMENTS = prove (`!s (t:real^N->bool) n. (closed s /\ closed t \/ bounded s /\ bounded t) /\ s homeomorphic t ==> (components((:real^N) DIFF s) HAS_SIZE n <=> components((:real^N) DIFF t) HAS_SIZE n)`, REPEAT STRIP_TAC THENL [MATCH_MP_TAC CARD_HAS_SIZE_CONG THEN ASM_SIMP_TAC[CARD_EQ_COMPONENTS_COMPLEMENTS]; ALL_TAC] THEN ONCE_REWRITE_TAC[SET_RULE `s = {c | c IN s /\ ~bounded c} UNION {c | c IN s /\ bounded c}`] THEN MP_TAC(ISPEC `t:real^N->bool` HAS_SIZE_UNBOUNDED_COMPONENTS_COMPLEMENT) THEN MP_TAC(ISPEC `s:real^N->bool` HAS_SIZE_UNBOUNDED_COMPONENTS_COMPLEMENT) THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(SUBST1_TAC o MATCH_MP (MESON[HOMEOMORPHIC_EMPTY] `s homeomorphic t ==> (s = {} <=> t = {})`)) THEN ABBREV_TAC `m = if t:real^N->bool = {} \/ 2 <= dimindex (:N) then 1 else 2` THEN REWRITE_TAC[HAS_SIZE; IMP_IMP] THEN GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [TAUT `p /\ q <=> ~(p ==> ~q)`] THEN SIMP_TAC[FINITE_UNION; CARD_UNION; SET_RULE `{x | x IN s /\ ~P x} INTER {x | x IN s /\ P x} = {}`] THEN DISCH_THEN(K ALL_TAC) THEN FIRST_X_ASSUM(K ALL_TAC o SYM) THEN REWRITE_TAC[NOT_IMP] THEN MATCH_MP_TAC(MESON[EQ_ADD_LCANCEL] `(!n. FINITE s /\ CARD s = n <=> FINITE t /\ CARD t = n) ==> (FINITE s /\ m + CARD s = n <=> FINITE t /\ m + CARD t = n)`) THEN X_GEN_TAC `n:num` THEN ABBREV_TAC `z:real^(N,1)finite_sum = basis 1` THEN SUBGOAL_THEN `z IN sphere(vec 0:real^(N,1)finite_sum,&1)` ASSUME_TAC THENL [EXPAND_TAC "z" THEN REWRITE_TAC[IN_SPHERE_0] THEN SIMP_TAC[NORM_BASIS; DIMINDEX_GE_1; LE_REFL]; ALL_TAC] THEN SUBGOAL_THEN `(sphere(vec 0:real^(N,1)finite_sum,&1) DELETE z) homeomorphic (:real^N)` MP_TAC THENL [MATCH_MP_TAC HOMEOMORPHIC_PUNCTURED_SPHERE_UNIV THEN ASM_REWRITE_TAC[REAL_LT_01; DIMINDEX_FINITE_SUM; DIMINDEX_1]; REWRITE_TAC[homeomorphic; LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`f:real^(N,1)finite_sum->real^N`; `g:real^N->real^(N,1)finite_sum`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`IMAGE (g:real^N->real^(N,1)finite_sum) s`; `IMAGE (g:real^N->real^(N,1)finite_sum) t`; `vec 0:real^(N,1)finite_sum`; `&1:real`; `n + 1`] HOMEOMORPHIC_SEPARATION_SPHERE_HAS_SIZE_EQ) THEN SUBGOAL_THEN `IMAGE (g:real^N->real^(N,1)finite_sum) s SUBSET sphere(vec 0,&1) /\ IMAGE (g:real^N->real^(N,1)finite_sum) t SUBSET sphere(vec 0,&1)` STRIP_ASSUME_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]; ASM_REWRITE_TAC[]] THEN ANTS_TAC THENL [TRANS_TAC HOMEOMORPHIC_TRANS `s:real^N->bool` THEN CONJ_TAC THENL [ALL_TAC; TRANS_TAC HOMEOMORPHIC_TRANS `t:real^N->bool` THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM]] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMEOMORPHIC_SELF_IMAGE) o GEN_REWRITE_RULE I [HOMEOMORPHISM_SYM]) THEN ASM SET_TAC[]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] CARD_EQ_COMPONENTS_IN_COMPACTIFICATION)) THEN DISCH_THEN(fun th -> MP_TAC(SPEC `IMAGE (g:real^N->real^(N,1)finite_sum) t` th) THEN MP_TAC(SPEC `IMAGE (g:real^N->real^(N,1)finite_sum) s` th)) THEN FIRST_ASSUM(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] BOUNDED_IMAGE_IN_COMPACTIFICATION) o ONCE_REWRITE_RULE[SET_RULE `s DELETE z = s DIFF {z}`] o GEN_REWRITE_RULE I [HOMEOMORPHISM_SYM]) THEN ASM_REWRITE_TAC[COMPACT_SPHERE; CLOSED_UNIV; SUBSET_UNIV] THEN REWRITE_TAC[SET_RULE `s INTER {z} = {} <=> ~(z IN s)`] THEN DISCH_THEN(ASSUME_TAC o GSYM) THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC I [IMP_IMP] THEN DISCH_THEN(CONJUNCTS_THEN(fun th -> REWRITE_TAC[MATCH_MP CARD_HAS_SIZE_CONG th])) THEN SUBGOAL_THEN `IMAGE f (IMAGE (g:real^N->real^(N,1)finite_sum) s) = s /\ IMAGE f (IMAGE (g:real^N->real^(N,1)finite_sum) t) = t` (CONJUNCTS_THEN SUBST1_TAC) THENL [RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]; REWRITE_TAC[HAS_SIZE]] THEN ONCE_REWRITE_TAC[TAUT `p /\ q <=> ~(p ==> ~q)`] THEN SIMP_TAC[CARD_ADD_C; FINITE_SING; CARD_ADD_FINITE_EQ] THEN REWRITE_TAC[CARD_SING; ARITH_RULE `1 + m = n + 1 <=> m = n`]);; let HOMEOMORPHIC_SEPARATION = prove (`!s t. bounded s /\ bounded t /\ s homeomorphic t ==> (connected((:real^N) DIFF s) <=> connected((:real^N) DIFF t))`, REPEAT STRIP_TAC THEN REWRITE_TAC[CONNECTED_EQ_CARD_COMPONENTS] THEN REWRITE_TAC[ARITH_RULE `c <= 1 <=> c = 0 \/ c = 1`; LEFT_OR_DISTRIB] THEN REWRITE_TAC[GSYM HAS_SIZE] THEN BINOP_TAC THEN MATCH_MP_TAC HAS_SIZE_EQ_COMPONENTS_COMPLEMENTS THEN ASM_REWRITE_TAC[]);;