(* ========================================================================= *) (* Convex sets, functions and related things. *) (* *) (* (c) Copyright, John Harrison 1998-2008 *) (* (c) Copyright, Lars Schewe 2007 *) (* (c) Copyright, Valentina Bruno 2010 *) (* (c) Copyright, Marco Maggesi 2014 *) (* ========================================================================= *) needs "Multivariate/topology.ml";; (* ------------------------------------------------------------------------- *) (* Some miscelleneous things that are convenient to prove here. *) (* ------------------------------------------------------------------------- *) let TRANSLATION_EQ_IMP = prove (`!P:(real^N->bool)->bool. (!a s. P(IMAGE (\x. a + x) s) <=> P s) <=> (!a s. P s ==> P (IMAGE (\x. a + x) s))`, REPEAT GEN_TAC THEN EQ_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `s:real^N->bool`] THEN EQ_TAC THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN FIRST_X_ASSUM (MP_TAC o SPECL [`--a:real^N`; `IMAGE (\x:real^N. a + x) s`]) THEN ASM_REWRITE_TAC[GSYM IMAGE_o; o_DEF; IMAGE_ID; VECTOR_ARITH `--a + a + x:real^N = x`]);; let DIM_HYPERPLANE = prove (`!a:real^N. ~(a = vec 0) ==> dim {x | a dot x = &0} = dimindex(:N) - 1`, GEOM_BASIS_MULTIPLE_TAC 1 `a:real^N` THEN SIMP_TAC[VECTOR_MUL_EQ_0; DE_MORGAN_THM; DOT_LMUL; DOT_BASIS; DIMINDEX_GE_1; LE_REFL; REAL_ENTIRE; DIM_SPECIAL_HYPERPLANE]);; let DIM_EQ_HYPERPLANE = prove (`!s. dim s = dimindex(:N) - 1 <=> ?a:real^N. ~(a = vec 0) /\ span s = {x | a dot x = &0}`, MESON_TAC[DIM_HYPERPLANE; LOWDIM_EQ_HYPERPLANE; DIM_SPAN]);; (* ------------------------------------------------------------------------- *) (* Affine set and affine hull. *) (* ------------------------------------------------------------------------- *) let affine = new_definition `affine s <=> !x y u v. x IN s /\ y IN s /\ (u + v = &1) ==> (u % x + v % y) IN s`;; let AFFINE_ALT = prove (`affine s <=> !x y u. x IN s /\ y IN s ==> ((&1 - u) % x + u % y) IN s`, REWRITE_TAC[affine] THEN MESON_TAC[REAL_ARITH `(u + v = &1) <=> (u = &1 - v)`]);; let AFFINE_SCALING = prove (`!s c. affine s ==> affine (IMAGE (\x. c % x) s)`, REWRITE_TAC[affine; IN_IMAGE] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[VECTOR_ARITH `u % c % x + v % c % y = c % (u % x + v % y)`] THEN ASM_MESON_TAC[]);; let AFFINE_NEGATIONS = prove (`!s. affine s ==> affine (IMAGE (--) s)`, REWRITE_TAC[affine; IN_IMAGE] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[VECTOR_ARITH `u % --x + v % --y = --(u % x + v % y)`] THEN ASM_MESON_TAC[]);; let AFFINE_SUMS = prove (`!s t. affine s /\ affine t ==> affine {x + y | x IN s /\ y IN t}`, REWRITE_TAC[affine; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[VECTOR_ARITH `u % (a + b) + v % (c + d) = (u % a + v % c) + (u % b + v % d)`] THEN ASM_MESON_TAC[]);; let AFFINE_DIFFERENCES = prove (`!s t. affine s /\ affine t ==> affine {x - y | x IN s /\ y IN t}`, REWRITE_TAC[affine; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[VECTOR_ARITH `u % (a - b) + v % (c - d) = (u % a + v % c) - (u % b + v % d)`] THEN ASM_MESON_TAC[]);; let AFFINE_TRANSLATION_EQ = prove (`!a:real^N s. affine (IMAGE (\x. a + x) s) <=> affine s`, REWRITE_TAC[AFFINE_ALT; IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_IMAGE; UNWIND_THM1; VECTOR_ARITH `(&1 - u) % (a + x) + u % (a + y) = a + z <=> (&1 - u) % x + u % y = z`]);; add_translation_invariants [AFFINE_TRANSLATION_EQ];; let AFFINE_TRANSLATION = prove (`!s a:real^N. affine s ==> affine (IMAGE (\x. a + x) s)`, REWRITE_TAC[AFFINE_TRANSLATION_EQ]);; let AFFINE_LINEAR_IMAGE = prove (`!f s. affine s /\ linear f ==> affine(IMAGE f s)`, REWRITE_TAC[affine; FORALL_IN_IMAGE; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[IN_IMAGE; linear] THEN MESON_TAC[]);; let AFFINE_LINEAR_IMAGE_EQ = prove (`!f s. linear f /\ (!x y. f x = f y ==> x = y) ==> (affine (IMAGE f s) <=> affine s)`, MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE AFFINE_LINEAR_IMAGE));; add_linear_invariants [AFFINE_LINEAR_IMAGE_EQ];; let AFFINE_LINEAR_PREIMAGE = prove (`!f:real^M->real^N s. linear f /\ affine s ==> affine {x | f(x) IN s}`, REWRITE_TAC[affine; IN_ELIM_THM] THEN SIMP_TAC[LINEAR_ADD; LINEAR_CMUL]);; let AFFINE_EMPTY = prove (`affine {}`, REWRITE_TAC[affine; NOT_IN_EMPTY]);; let AFFINE_SING = prove (`!x. affine {x}`, SIMP_TAC[AFFINE_ALT; IN_SING] THEN REWRITE_TAC[GSYM VECTOR_ADD_RDISTRIB] THEN REWRITE_TAC[REAL_SUB_ADD; VECTOR_MUL_LID]);; let AFFINE_SCALING_EQ = prove (`!s:real^N->bool c. affine (IMAGE (\x. c % x) s) <=> c = &0 \/ affine 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[AFFINE_SING; AFFINE_EMPTY]; EQ_TAC THEN REWRITE_TAC[AFFINE_SCALING] THEN DISCH_THEN(MP_TAC o SPEC `inv(c):real` o MATCH_MP AFFINE_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 AFFINE_AFFINITY_EQ = prove (`!s m c:real^N. affine (IMAGE (\x. m % x + c) s) <=> m = &0 \/ affine s`, REWRITE_TAC[AFFINITY_SCALING_TRANSLATION; AFFINE_TRANSLATION_EQ; AFFINE_SCALING_EQ; IMAGE_o]);; let AFFINE_AFFINITY = prove (`!s m c:real^N. affine s ==> affine (IMAGE (\x. m % x + c) s)`, SIMP_TAC[AFFINE_AFFINITY_EQ]);; let AFFINE_UNIV = prove (`affine(UNIV:real^N->bool)`, REWRITE_TAC[affine; IN_UNIV]);; let AFFINE_HYPERPLANE = prove (`!a b. affine {x | a dot x = b}`, REWRITE_TAC[affine; IN_ELIM_THM; DOT_RADD; DOT_RMUL] THEN CONV_TAC REAL_RING);; let AFFINE_STANDARD_HYPERPLANE = prove (`!b k. affine {x:real^N | x$k = b}`, REPEAT GEN_TAC THEN SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i` CHOOSE_TAC THENL [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN MP_TAC(ISPECL [`basis i:real^N`; `b:real`] AFFINE_HYPERPLANE) THEN ASM_SIMP_TAC[DOT_BASIS]);; let AFFINE_INTERS = prove (`(!s. s IN f ==> affine s) ==> affine(INTERS f)`, REWRITE_TAC[affine; IN_INTERS] THEN MESON_TAC[]);; let AFFINE_INTER = prove (`!s t. affine s /\ affine t ==> affine(s INTER t)`, REWRITE_TAC[affine; IN_INTER] THEN MESON_TAC[]);; let AFFINE_AFFINE_HULL = prove (`!s. affine(affine hull s)`, SIMP_TAC[P_HULL; AFFINE_INTERS]);; let AFFINE_HULL_EQ = prove (`!s. (affine hull s = s) <=> affine s`, SIMP_TAC[HULL_EQ; AFFINE_INTERS]);; let IS_AFFINE_HULL = prove (`!s. affine s <=> ?t. s = affine hull t`, GEN_TAC THEN MATCH_MP_TAC IS_HULL THEN SIMP_TAC[AFFINE_INTERS]);; let AFFINE_HULL_UNIV = prove (`affine hull (:real^N) = (:real^N)`, REWRITE_TAC[AFFINE_HULL_EQ; AFFINE_UNIV]);; let AFFINE_HULLS_EQ = prove (`!s t. s SUBSET affine hull t /\ t SUBSET affine hull s ==> affine hull s = affine hull t`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HULLS_EQ THEN ASM_SIMP_TAC[AFFINE_INTERS]);; let AFFINE_HULL_TRANSLATION = prove (`!a s. affine hull (IMAGE (\x. a + x) s) = IMAGE (\x. a + x) (affine hull s)`, REWRITE_TAC[hull] THEN GEOM_TRANSLATE_TAC[]);; add_translation_invariants [AFFINE_HULL_TRANSLATION];; let AFFINE_HULL_LINEAR_IMAGE = prove (`!f s. linear f ==> affine hull (IMAGE f s) = IMAGE f (affine hull s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN CONJ_TAC THEN MATCH_MP_TAC HULL_INDUCT THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN SIMP_TAC[FUN_IN_IMAGE; HULL_INC] THEN REWRITE_TAC[affine; IN_ELIM_THM] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THENL [FIRST_ASSUM(fun th -> REWRITE_TAC[GSYM(MATCH_MP LINEAR_CMUL th)]) THEN FIRST_ASSUM(fun th -> REWRITE_TAC[GSYM(MATCH_MP LINEAR_ADD th)]) THEN REWRITE_TAC[IN_IMAGE] THEN MESON_TAC[REWRITE_RULE[affine] AFFINE_AFFINE_HULL]; ASM_SIMP_TAC[LINEAR_ADD; LINEAR_CMUL] THEN MESON_TAC[REWRITE_RULE[affine] AFFINE_AFFINE_HULL]]);; add_linear_invariants [AFFINE_HULL_LINEAR_IMAGE];; let IN_AFFINE_HULL_LINEAR_IMAGE = prove (`!f:real^M->real^N s x. linear f /\ x IN affine hull s ==> (f x) IN affine hull (IMAGE f s)`, SIMP_TAC[AFFINE_HULL_LINEAR_IMAGE] THEN SET_TAC[]);; let SAME_DISTANCES_TO_AFFINE_HULL = prove (`!s a b:real^N. (!x. x IN s ==> dist(x,a) = dist(x,b)) ==> (!x. x IN affine hull s ==> dist(x,a) = dist(x,b))`, REPEAT GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC HULL_INDUCT THEN ASM_REWRITE_TAC[AFFINE_ALT; IN_ELIM_THM] THEN REWRITE_TAC[dist; NORM_EQ_SQUARE; NORM_POS_LE; VECTOR_ARITH `((&1 - u) % x + u % y) - a:real^N = (&1 - u) % (x - a) + u % (y - a)`] THEN REWRITE_TAC[NORM_POW_2; DOT_LMUL; DOT_RMUL; VECTOR_ARITH `(x + y) dot (x + y):real^N = (x dot x + y dot y) + &2 * x dot y`] THEN SIMP_TAC[DOT_LSUB; DOT_RSUB; DOT_SYM] THEN CONV_TAC REAL_RING);; (* ------------------------------------------------------------------------- *) (* Some convenient lemmas about common affine combinations. *) (* ------------------------------------------------------------------------- *) let IN_AFFINE_ADD_MUL = prove (`!s a x:real^N d. affine s /\ a IN s /\ (a + x) IN s ==> (a + d % x) IN s`, REWRITE_TAC[affine] THEN REPEAT STRIP_TAC THEN SUBST1_TAC(VECTOR_ARITH `a + d % x:real^N = (&1 - d) % a + d % (a + x)`) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);; let IN_AFFINE_ADD_MUL_DIFF = prove (`!s a x y z:real^N. affine s /\ x IN s /\ y IN s /\ z IN s ==> (x + a % (y - z)) IN s`, REWRITE_TAC[affine] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[VECTOR_ARITH `x + a % (y - z):real^N = &1 / &2 % ((&1 - &2 * a) % x + (&2 * a) % y) + &1 / &2 % ((&1 + &2 * a) % x + (-- &2 * a) % z)`] THEN FIRST_ASSUM MATCH_MP_TAC THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN CONJ_TAC THEN FIRST_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);; let IN_AFFINE_MUL_DIFF_ADD = prove (`!s a x y z:real^N. affine s /\ x IN s /\ y IN s /\ z IN s ==> a % (x - y) + z IN s`, ONCE_REWRITE_TAC[VECTOR_ADD_SYM] THEN SIMP_TAC[IN_AFFINE_ADD_MUL_DIFF]);; let IN_AFFINE_SUB_MUL_DIFF = prove (`!s a x y z:real^N. affine s /\ x IN s /\ y IN s /\ z IN s ==> x - a % (y - z) IN s`, REWRITE_TAC[VECTOR_ARITH `x - a % (y - z):real^N = x + a % (z - y)`] THEN SIMP_TAC[IN_AFFINE_ADD_MUL_DIFF]);; let AFFINE_DIFFS_SUBSPACE = prove (`!s:real^N->bool a. affine s /\ a IN s ==> subspace {x - a | x IN s}`, REWRITE_TAC[subspace; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN REWRITE_TAC[IN_ELIM_THM] THEN REWRITE_TAC[VECTOR_ARITH `vec 0:real^N = x - a <=> x = a`; VECTOR_ARITH `x - a + y - a:real^N = z - a <=> z = (a + &1 % (x - a)) + &1 % (y - a)`; VECTOR_ARITH `c % (x - a):real^N = y - a <=> y = a + c % (x - a)`] THEN MESON_TAC[IN_AFFINE_ADD_MUL_DIFF]);; (* ------------------------------------------------------------------------- *) (* Explicit formulations for affine combinations. *) (* ------------------------------------------------------------------------- *) let AFFINE_VSUM = prove (`!s k u x:A->real^N. FINITE k /\ affine s /\ sum k u = &1 /\ (!i. i IN k ==> x i IN s) ==> vsum k (\i. u i % x i) IN s`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [ASM_REWRITE_TAC[NOT_IN_EMPTY; GSYM NOT_EXISTS_THM; MEMBER_NOT_EMPTY] THEN ASM_CASES_TAC `k:A->bool = {}` THEN ASM_REWRITE_TAC[SUM_CLAUSES] THEN REAL_ARITH_TAC; ALL_TAC] THEN 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 MP_TAC(ISPECL [`s:real^N->bool`; `a:real^N`] AFFINE_DIFFS_SUBSPACE) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN MP_TAC(ISPECL [`{x - a:real^N | x IN s}`; `(\i. u i % (x i - a)):A->real^N`; `k:A->bool`] SUBSPACE_VSUM) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSPACE_MUL THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; ASM_SIMP_TAC[VSUM_SUB; IN_ELIM_THM; VECTOR_SUB_LDISTRIB; VSUM_RMUL] THEN REWRITE_TAC[VECTOR_ARITH `x - &1 % a:real^N = y - a <=> x = y`] THEN ASM_MESON_TAC[]]);; let AFFINE_VSUM_STRONG = prove (`!s k u x:A->real^N. affine s /\ sum k u = &1 /\ (!i. i IN k ==> u i = &0 \/ x i IN s) ==> vsum k (\i. u i % x i) IN s`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `vsum k (\i. u i % (x:A->real^N) i) = vsum {i | i IN k /\ ~(u i = &0)} (\i. u i % x i)` SUBST1_TAC THENL [MATCH_MP_TAC VSUM_SUPERSET THEN REWRITE_TAC[VECTOR_MUL_EQ_0] THEN SET_TAC[]; MATCH_MP_TAC AFFINE_VSUM THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[SUM_DEGENERATE; REAL_ARITH `~(&1 = &0)`]; FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC SUM_SUPERSET THEN ASM SET_TAC[]; ASM SET_TAC[]]]);; let AFFINE_INDEXED = prove (`!s:real^N->bool. affine s <=> !k u x. (!i:num. 1 <= i /\ i <= k ==> x(i) IN s) /\ (sum (1..k) u = &1) ==> vsum (1..k) (\i. u(i) % x(i)) IN s`, REPEAT GEN_TAC THEN EQ_TAC THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC AFFINE_VSUM THEN ASM_REWRITE_TAC[IN_NUMSEG; FINITE_NUMSEG]; DISCH_TAC THEN REWRITE_TAC[affine] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `u:real`; `v:real`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `2`) THEN DISCH_THEN(MP_TAC o SPEC `\n. if n = 1 then u else v:real`) THEN DISCH_THEN(MP_TAC o SPEC `\n. if n = 1 then x else y:real^N`) THEN REWRITE_TAC[num_CONV `2`; SUM_CLAUSES_NUMSEG; VSUM_CLAUSES_NUMSEG; NUMSEG_SING; VSUM_SING; SUM_SING] THEN REWRITE_TAC[ARITH] THEN ASM_MESON_TAC[]]);; let AFFINE_HULL_INDEXED = prove (`!s. affine hull s = {y:real^N | ?k u x. (!i. 1 <= i /\ i <= k ==> x i IN s) /\ (sum (1..k) u = &1) /\ (vsum (1..k) (\i. u i % x i) = y)}`, GEN_TAC THEN MATCH_MP_TAC HULL_UNIQUE THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MAP_EVERY EXISTS_TAC [`1`; `\i:num. &1`; `\i:num. x:real^N`] THEN ASM_SIMP_TAC[FINITE_RULES; IN_SING; SUM_SING; VECTOR_MUL_LID; VSUM_SING; REAL_POS; NUMSEG_SING]; ALL_TAC; REWRITE_TAC[AFFINE_INDEXED; SUBSET; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN MESON_TAC[]] THEN REWRITE_TAC[affine; IN_ELIM_THM] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `u:real`; `v:real`] THEN REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`k1:num`; `u1:num->real`; `x1:num->real^N`; `k2:num`; `u2:num->real`; `x2:num->real^N`] THEN STRIP_TAC THEN EXISTS_TAC `k1 + k2:num` THEN EXISTS_TAC `\i:num. if i <= k1 then u * u1(i) else v * u2(i - k1):real` THEN EXISTS_TAC `\i:num. if i <= k1 then x1(i) else x2(i - k1):real^N` THEN ASM_SIMP_TAC[NUMSEG_ADD_SPLIT; ARITH_RULE `1 <= x + 1 /\ x < x + 1`; IN_NUMSEG; SUM_UNION; VSUM_UNION; FINITE_NUMSEG; DISJOINT_NUMSEG; ARITH_RULE `k1 + 1 <= i ==> ~(i <= k1)`] THEN REWRITE_TAC[ONCE_REWRITE_RULE[ADD_SYM] NUMSEG_OFFSET_IMAGE] THEN ASM_SIMP_TAC[SUM_IMAGE; VSUM_IMAGE; EQ_ADD_LCANCEL; FINITE_NUMSEG] THEN ASM_SIMP_TAC[o_DEF; ADD_SUB2; SUM_LMUL; VSUM_LMUL; GSYM VECTOR_MUL_ASSOC; FINITE_NUMSEG; REAL_MUL_RID] THEN ASM_MESON_TAC[REAL_LE_MUL; ARITH_RULE `i <= k1 + k2 /\ ~(i <= k1) ==> 1 <= i - k1 /\ i - k1 <= k2`]);; let AFFINE = prove (`!V:real^N->bool. affine V <=> !(s:real^N->bool) (u:real^N->real). FINITE s /\ ~(s = {}) /\ s SUBSET V /\ sum s u = &1 ==> vsum s (\x. u x % x) IN V`, GEN_TAC THEN EQ_TAC THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC AFFINE_VSUM THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; REWRITE_TAC[affine] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `u:real`; `v:real`] THEN STRIP_TAC THEN ASM_CASES_TAC `x:real^N = y` THENL [FIRST_X_ASSUM SUBST_ALL_TAC THEN ASM_REWRITE_TAC[GSYM VECTOR_ADD_RDISTRIB;VECTOR_MUL_LID];ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPEC `{x:real^N,y}`) THEN DISCH_THEN(MP_TAC o SPEC `\w. if w = x:real^N then u else v:real`) THEN ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; FINITE_RULES; NUMSEG_SING; VSUM_SING; SUM_SING;SUBSET;IN_INSERT;NOT_IN_EMPTY] THEN ASM SET_TAC[]]);; let AFFINE_EXPLICIT = prove (`!s:real^N->bool. affine s <=> !t u. FINITE t /\ t SUBSET s /\ sum t u = &1 ==> vsum t (\x. u(x) % x) IN s`, GEN_TAC THEN REWRITE_TAC[AFFINE] THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `t:real^N->bool` THEN REWRITE_TAC[] THEN AP_TERM_TAC THEN ABS_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_REWRITE_TAC[SUM_CLAUSES] THEN CONV_TAC REAL_RAT_REDUCE_CONV);; let AFFINE_HULL_EXPLICIT = prove (`!(p:real^N -> bool). affine hull p = {y | ?s u. FINITE s /\ ~(s = {}) /\ s SUBSET p /\ sum s u = &1 /\ vsum s (\v. u v % v) = y}`, GEN_TAC THEN MATCH_MP_TAC HULL_UNIQUE THEN REPEAT CONJ_TAC 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}`;`\v:real^N. &1:real`] THEN ASM_SIMP_TAC[FINITE_RULES;IN_SING;SUM_SING;VSUM_SING;VECTOR_MUL_LID] THEN SET_TAC[]; REWRITE_TAC[affine;IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN EXISTS_TAC `(s UNION s'):real^N->bool` THEN EXISTS_TAC `\a:real^N. (\b:real^N.if (b IN s) then (u * (u' b)) else &0) a + (\b:real^N.if (b IN s') then v * (u'' b) else &0) a` THEN REPEAT CONJ_TAC THENL [ASM_REWRITE_TAC[FINITE_UNION]; ASM SET_TAC[]; ASM_REWRITE_TAC[UNION_SUBSET]; ASM_SIMP_TAC[REWRITE_RULE[REAL_ARITH `a + b = c + d <=> c = a + b - d`] SUM_INCL_EXCL; GSYM SUM_RESTRICT_SET; SET_RULE `{a | a IN (s:A->bool) /\ a IN s'} = s INTER s'`; SUM_ADD;SUM_LMUL;REAL_MUL_RID; FINITE_INTER;INTER_IDEMPOT] THEN ASM_REWRITE_TAC[SET_RULE `(a INTER b) INTER a = a INTER b`; SET_RULE `(a INTER b) INTER b = a INTER b`; REAL_ARITH `(a + b) + (c + d) - (e + b) = (a + d) + c - e`; REAL_ARITH `a + b - c = a <=> b = c`] THEN AP_TERM_TAC THEN REWRITE_TAC[INTER_COMM]; ASM_SIMP_TAC[REWRITE_RULE [VECTOR_ARITH `(a:real^N) + b = c + d <=> c = a + b - d`] VSUM_INCL_EXCL;GSYM VSUM_RESTRICT_SET; SET_RULE `{a | a IN (s:A->bool) /\ a IN s'} = s INTER s'`; VSUM_ADD;FINITE_INTER;INTER_IDEMPOT;VECTOR_ADD_RDISTRIB; GSYM VECTOR_MUL_ASSOC;VSUM_LMUL; MESON[] `(if P then a else b) % (x:real^N) = (if P then a % x else b % x)`; VECTOR_MUL_LZERO;GSYM VSUM_RESTRICT_SET] THEN ASM_REWRITE_TAC[SET_RULE `(a INTER b) INTER a = a INTER b`; SET_RULE `(a INTER b) INTER b = a INTER b`; VECTOR_ARITH `((a:real^N) + b) + (c + d) - (e + b) = (a + d) + c - e`; VECTOR_ARITH `(a:real^N) + b - c = a <=> b = c`] THEN AP_TERM_TAC THEN REWRITE_TAC[INTER_COMM]]; ASM_CASES_TAC `(p:real^N->bool) = {}` THENL [FIRST_X_ASSUM SUBST_ALL_TAC THEN REWRITE_TAC[SUBSET_EMPTY;EMPTY_SUBSET] THEN ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[AFFINE; SUBSET; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN ASM SET_TAC[]]);; let AFFINE_HULL_EXPLICIT_ALT = prove (`!(p:real^N -> bool). affine hull p = {y | ?s u. FINITE s /\ s SUBSET p /\ sum s u = &1 /\ vsum s (\v. u v % v) = y}`, GEN_TAC THEN REWRITE_TAC[AFFINE_HULL_EXPLICIT] THEN GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[IN_ELIM_THM] THEN GEN_TAC THEN REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN EQ_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN POP_ASSUM MP_TAC THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN SIMP_TAC[SUM_CLAUSES; REAL_OF_NUM_EQ; ARITH_EQ]);; let AFFINE_HULL_FINITE = prove (`!s:real^N->bool. affine hull s = {y | ?u. sum s u = &1 /\ vsum s (\v. u v % v) = y}`, GEN_TAC THEN GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[AFFINE_HULL_EXPLICIT; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN EQ_TAC THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THENL [MAP_EVERY X_GEN_TAC [`t:real^N->bool`; `f:real^N->real`] THEN STRIP_TAC THEN EXISTS_TAC `\x:real^N. if x IN t then f x else &0` THEN REWRITE_TAC[COND_RAND; COND_RATOR; VECTOR_MUL_LZERO] THEN REWRITE_TAC[GSYM SUM_RESTRICT_SET; GSYM VSUM_RESTRICT_SET] THEN ASM_SIMP_TAC[SET_RULE `t SUBSET s ==> {x | x IN s /\ x IN t} = t`]; X_GEN_TAC `f:real^N->real` THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[SUM_CLAUSES; REAL_OF_NUM_EQ; ARITH] THEN STRIP_TAC THEN EXISTS_TAC `support (+) (f:real^N->real) s` THEN EXISTS_TAC `f:real^N->real` THEN MP_TAC(ASSUME `sum s (f:real^N->real) = &1`) THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [sum] THEN REWRITE_TAC[iterate] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[NEUTRAL_REAL_ADD; REAL_OF_NUM_EQ; ARITH] THEN DISCH_THEN(K ALL_TAC) THEN UNDISCH_TAC `sum s (f:real^N->real) = &1` THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [GSYM SUM_SUPPORT] THEN ASM_CASES_TAC `support (+) (f:real^N->real) s = {}` THEN ASM_SIMP_TAC[SUM_CLAUSES; REAL_OF_NUM_EQ; ARITH] THEN DISCH_TAC THEN REWRITE_TAC[SUPPORT_SUBSET] THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [GSYM th]) THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC VSUM_SUPERSET THEN REWRITE_TAC[SUPPORT_SUBSET] THEN REWRITE_TAC[support; IN_ELIM_THM; NEUTRAL_REAL_ADD] THEN MESON_TAC[VECTOR_MUL_LZERO]]);; let AFFINE_HULL_0_EXPLICIT = prove (`!s:real^N->bool. vec 0 IN affine hull s <=> ?t u. FINITE t /\ ~(t = {}) /\ t SUBSET s /\ ~(sum t u = &0) /\ vsum t (\x. u x % x) = vec 0`, GEN_TAC THEN REWRITE_TAC[AFFINE_HULL_EXPLICIT; IN_ELIM_THM] THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `t:real^N->bool` THEN REWRITE_TAC[] THEN EQ_TAC THENL [MESON_TAC[REAL_RAT_REDUCE_CONV `&1 = &0`]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\x:real^N. inv(sum t u) * u x` THEN ASM_REWRITE_TAC[SUM_LMUL; VSUM_LMUL; GSYM VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[REAL_MUL_LINV; VECTOR_MUL_RZERO]);; (* ------------------------------------------------------------------------- *) (* Stepping theorems and hence small special cases. *) (* ------------------------------------------------------------------------- *) let AFFINE_HULL_EMPTY = prove (`affine hull {} = {}`, MATCH_MP_TAC HULL_UNIQUE THEN REWRITE_TAC[SUBSET_REFL; AFFINE_EMPTY; EMPTY_SUBSET]);; let AFFINE_HULL_EQ_EMPTY = prove (`!s. (affine hull s = {}) <=> (s = {})`, GEN_TAC THEN EQ_TAC THEN MESON_TAC[SUBSET_EMPTY; HULL_SUBSET; AFFINE_HULL_EMPTY]);; let AFFINE_HULL_FINITE_STEP_GEN = prove (`!P:real^N->real->bool. ((?u. (!x. x IN {} ==> P x (u x)) /\ sum {} u = w /\ vsum {} (\x. u(x) % x) = y) <=> w = &0 /\ y = vec 0) /\ (FINITE(s:real^N->bool) /\ (!y. a IN s /\ P a y ==> P a (y / &2)) /\ (!x y. a IN s /\ P a x /\ P a y ==> P a (x + y)) ==> ((?u. (!x. x IN (a INSERT s) ==> P x (u x)) /\ sum (a INSERT s) u = w /\ vsum (a INSERT s) (\x. u(x) % x) = y) <=> ?v u. P a v /\ (!x. x IN s ==> P x (u x)) /\ sum s u = w - v /\ vsum s (\x. u(x) % x) = y - v % a))`, GEN_TAC THEN SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; NOT_IN_EMPTY] THEN CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN DISCH_TAC THEN ASM_CASES_TAC `(a:real^N) IN s` THEN ASM_REWRITE_TAC[] THENL [ASM_SIMP_TAC[SET_RULE `a IN s ==> a INSERT s = s`] THEN EQ_TAC THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THENL [X_GEN_TAC `u:real^N->real` THEN STRIP_TAC THEN EXISTS_TAC `(u:real^N->real) a / &2` THEN EXISTS_TAC `\x:real^N. if x = a then u x / &2 else u x`; MAP_EVERY X_GEN_TAC [`v:real`; `u:real^N->real`] THEN STRIP_TAC THEN EXISTS_TAC `\x:real^N. if x = a then u x + v else u x`] THEN ASM_SIMP_TAC[] THEN (CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC]) THEN ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN ASM_SIMP_TAC[VSUM_CASES; SUM_CASES] THEN ASM_SIMP_TAC[GSYM DELETE; SUM_DELETE; VSUM_DELETE] THEN ASM_SIMP_TAC[SET_RULE `a IN s ==> {x | x IN s /\ x = a} = {a}`] THEN REWRITE_TAC[SUM_SING; VSUM_SING] THEN (CONJ_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC]); EQ_TAC THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THENL [X_GEN_TAC `u:real^N->real` THEN STRIP_TAC THEN EXISTS_TAC `(u:real^N->real) a` THEN EXISTS_TAC `u:real^N->real` THEN ASM_SIMP_TAC[IN_INSERT] THEN REPEAT(FIRST_X_ASSUM(SUBST1_TAC o SYM)) THEN CONJ_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC]; MAP_EVERY X_GEN_TAC [`v:real`; `u:real^N->real`] THEN STRIP_TAC THEN EXISTS_TAC `\x:real^N. if x = a then v:real else u x` THEN ASM_SIMP_TAC[IN_INSERT] THEN CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN ASM_SIMP_TAC[VSUM_CASES; SUM_CASES] THEN ASM_SIMP_TAC[GSYM DELETE; SUM_DELETE; VSUM_DELETE] THEN ASM_SIMP_TAC[SET_RULE `~(a IN s) ==> {x | x IN s /\ x = a} = {}`] THEN ASM_SIMP_TAC[SET_RULE `~(a IN s) ==> s DELETE a = s`] THEN REWRITE_TAC[SUM_CLAUSES; VSUM_CLAUSES] THEN CONJ_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC]]]);; let AFFINE_HULL_FINITE_STEP = prove (`((?u. sum {} u = w /\ vsum {} (\x. u(x) % x) = y) <=> w = &0 /\ y = vec 0) /\ (FINITE(s:real^N->bool) ==> ((?u. sum (a INSERT s) u = w /\ vsum (a INSERT s) (\x. u(x) % x) = y) <=> ?v u. sum s u = w - v /\ vsum s (\x. u(x) % x) = y - v % a))`, MATCH_ACCEPT_TAC (REWRITE_RULE[] (ISPEC `\x:real^N y:real. T` AFFINE_HULL_FINITE_STEP_GEN)));; let AFFINE_HULL_2 = prove (`!a b. affine hull {a,b} = {u % a + v % b | u + v = &1}`, SIMP_TAC[AFFINE_HULL_FINITE; FINITE_INSERT; FINITE_RULES] THEN SIMP_TAC[AFFINE_HULL_FINITE_STEP; FINITE_INSERT; FINITE_RULES] THEN REWRITE_TAC[REAL_ARITH `x - y = z:real <=> x = y + z`; VECTOR_ARITH `x - y = z:real^N <=> x = y + z`] THEN REWRITE_TAC[VECTOR_ADD_RID; REAL_ADD_RID] THEN SET_TAC[]);; let AFFINE_HULL_2_ALT = prove (`!a b. affine hull {a,b} = {a + u % (b - a) | u IN (:real)}`, REPEAT GEN_TAC THEN REWRITE_TAC[AFFINE_HULL_2] THEN MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN REWRITE_TAC[IN_ELIM_THM; IN_UNIV; ARITH_RULE `u + v = &1 <=> v = &1 - u`; FORALL_UNWIND_THM2; UNWIND_THM2] THEN CONJ_TAC THEN X_GEN_TAC `u:real` THEN EXISTS_TAC `&1 - u` THEN VECTOR_ARITH_TAC);; let AFFINE_HULL_3 = prove (`affine hull {a,b,c} = { u % a + v % b + w % c | u + v + w = &1}`, SIMP_TAC[AFFINE_HULL_FINITE; FINITE_INSERT; FINITE_RULES] THEN SIMP_TAC[AFFINE_HULL_FINITE_STEP; FINITE_INSERT; FINITE_RULES] THEN REWRITE_TAC[REAL_ARITH `x - y = z:real <=> x = y + z`; VECTOR_ARITH `x - y = z:real^N <=> x = y + z`] THEN REWRITE_TAC[VECTOR_ADD_RID; REAL_ADD_RID] THEN SET_TAC[]);; let AFFINE_HULL_0_2_EXPLICIT = prove (`!x y:real^N. vec 0 IN affine hull {x,y} <=> ?a b. a % x + b % y = vec 0 /\ ~(a + b = &0)`, REPEAT GEN_TAC THEN REWRITE_TAC[AFFINE_HULL_2; IN_ELIM_THM] THEN EQ_TAC THENL [MESON_TAC[REAL_ARITH `~(&1 = &0)`]; ALL_TAC] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real`; `b:real`] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`a / (a + b):real`; `b / (a + b):real`] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] real_div] THEN ASM_REWRITE_TAC[GSYM REAL_ADD_LDISTRIB; GSYM VECTOR_MUL_ASSOC; GSYM VECTOR_ADD_LDISTRIB; VECTOR_MUL_RZERO] THEN ASM_SIMP_TAC[REAL_MUL_LINV]);; let AFFINE_HULL_0_3_EXPLICIT = prove (`!x y z:real^N. vec 0 IN affine hull {x,y,z} <=> ?a b c. a % x + b % y + c % z = vec 0 /\ ~(a + b + c = &0)`, REPEAT GEN_TAC THEN REWRITE_TAC[AFFINE_HULL_3; IN_ELIM_THM] THEN EQ_TAC THENL [MESON_TAC[REAL_ARITH `~(&1 = &0)`]; ALL_TAC] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real`; `b:real`; `c:real`] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`a / (a + b + c):real`; `b / (a + b + c):real`; `c / (a + b + c):real`] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] real_div] THEN ASM_REWRITE_TAC[GSYM REAL_ADD_LDISTRIB; GSYM VECTOR_MUL_ASSOC; GSYM VECTOR_ADD_LDISTRIB; VECTOR_MUL_RZERO] THEN ASM_SIMP_TAC[REAL_MUL_LINV]);; (* ------------------------------------------------------------------------- *) (* Some relations between affine hull and subspaces. *) (* ------------------------------------------------------------------------- *) let AFFINE_HULL_INSERT_SUBSET_SPAN = prove (`!a:real^N s. affine hull (a INSERT s) SUBSET {a + v | v | v IN span {x - a | x IN s}}`, REPEAT GEN_TAC THEN GEN_REWRITE_TAC I [SUBSET] THEN REWRITE_TAC[AFFINE_HULL_EXPLICIT; SPAN_EXPLICIT; IN_ELIM_THM] THEN REWRITE_TAC[SIMPLE_IMAGE; CONJ_ASSOC; FINITE_SUBSET_IMAGE] THEN REWRITE_TAC[MESON[] `(?s u. (?t. P t /\ s = f t) /\ Q s u) <=> (?t u. P t /\ Q (f t) u)`] THEN REWRITE_TAC[MESON[] `(?v. (?s u. P s /\ f s u = v) /\ (x = g a v)) <=> (?s u. ~(P s ==> ~(g a (f s u) = x)))`] THEN SIMP_TAC[VSUM_IMAGE; VECTOR_ARITH `x - a:real^N = y - a <=> x = y`] THEN REWRITE_TAC[o_DEF] THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`t:real^N->bool`; `u:real^N->real`] THEN DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC (SUBST1_TAC o SYM)) THEN MAP_EVERY EXISTS_TAC [`t DELETE (a:real^N)`; `\x. (u:real^N->real)(x + a)`] THEN ASM_SIMP_TAC[FINITE_DELETE; VECTOR_SUB_ADD; SET_RULE `t SUBSET (a INSERT s) ==> t DELETE a SUBSET s`] THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `a + vsum t (\x. u x % (x - a)):real^N` THEN CONJ_TAC THENL [AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC VSUM_SUPERSET THEN REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ] THEN SET_TAC[]; ASM_SIMP_TAC[VECTOR_SUB_LDISTRIB; FINITE_DELETE; VSUM_SUB] THEN ASM_REWRITE_TAC[VSUM_RMUL] THEN REWRITE_TAC[VECTOR_ARITH `a + x - &1 % a:real^N = x`]]);; let AFFINE_HULL_INSERT_SPAN = prove (`!a:real^N s. ~(a IN s) ==> affine hull (a INSERT s) = {a + v | v | v IN span {x - a | x IN s}}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[AFFINE_HULL_INSERT_SUBSET_SPAN] THEN REWRITE_TAC[SUBSET] THEN REWRITE_TAC[AFFINE_HULL_EXPLICIT; SPAN_EXPLICIT; IN_ELIM_THM] THEN REWRITE_TAC[SIMPLE_IMAGE; CONJ_ASSOC; FINITE_SUBSET_IMAGE] THEN REWRITE_TAC[MESON[] `(?s u. (?t. P t /\ s = f t) /\ Q s u) <=> (?t u. P t /\ Q (f t) u)`] THEN REWRITE_TAC[MESON[] `(?v. (?s u. P s /\ f s u = v) /\ (x = g a v)) <=> (?s u. ~(P s ==> ~(g a (f s u) = x)))`] THEN SIMP_TAC[VSUM_IMAGE; VECTOR_ARITH `x - a:real^N = y - a <=> x = y`] THEN REWRITE_TAC[o_DEF] THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[NOT_IMP; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`t:real^N->bool`; `u:real^N->real`] THEN DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC (SUBST1_TAC o SYM)) THEN MAP_EVERY EXISTS_TAC [`(a:real^N) INSERT t`; `\x. if x = a then &1 - sum t (\x. u(x - a)) else (u:real^N->real)(x - a)`] THEN ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES] THEN ASM_CASES_TAC `(a:real^N) IN t` THENL [ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN ASM_SIMP_TAC[FINITE_INSERT; NOT_INSERT_EMPTY; SET_RULE `s SUBSET t ==> (a INSERT s) SUBSET (a INSERT t)`] THEN SUBGOAL_THEN `!x:real^N. x IN t ==> ~(x = a)` MP_TAC THENL [ASM SET_TAC[]; SIMP_TAC[] THEN DISCH_THEN(K ALL_TAC)] THEN CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[VECTOR_SUB_LDISTRIB; FINITE_DELETE; VSUM_SUB] THEN ASM_REWRITE_TAC[VSUM_RMUL] THEN VECTOR_ARITH_TAC);; let AFFINE_HULL_SPAN = prove (`!a:real^N s. a IN s ==> (affine hull s = {a + v | v | v IN span {x - a | x | x IN (s DELETE a)}})`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`a:real^N`; `s DELETE (a:real^N)`] AFFINE_HULL_INSERT_SPAN) THEN ASM_REWRITE_TAC[IN_DELETE] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN AP_TERM_TAC THEN ASM SET_TAC[]);; let DIFFS_AFFINE_HULL_SPAN = prove (`!a:real^N s. a IN s ==> {x - a | x IN affine hull s} = span {x - a | x IN s}`, REPEAT STRIP_TAC THEN FIRST_ASSUM(SUBST1_TAC o MATCH_MP AFFINE_HULL_SPAN) THEN REWRITE_TAC[SIMPLE_IMAGE; GSYM IMAGE_o; o_DEF; VECTOR_ADD_SUB; IMAGE_ID] THEN SIMP_TAC[IMAGE_DELETE_INJ; VECTOR_ARITH `x - a:real^N = y - a <=> x = y`] THEN REWRITE_TAC[VECTOR_SUB_REFL; SPAN_DELETE_0]);; let AFFINE_HULL_SING = prove (`!a. affine hull {a} = {a}`, SIMP_TAC[AFFINE_HULL_INSERT_SPAN; NOT_IN_EMPTY] THEN REWRITE_TAC[SET_RULE `{f x | x | F} = {}`; SPAN_EMPTY] THEN REWRITE_TAC[SET_RULE `{f x | x IN {a}} = {f a}`; VECTOR_ADD_RID]);; let AFFINE_HULL_EQ_SING = prove (`!s a:real^N. affine hull s = {a} <=> s = {a}`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[AFFINE_HULL_EMPTY] THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[AFFINE_HULL_SING] THEN MATCH_MP_TAC(SET_RULE `~(s = {}) /\ s SUBSET {a} ==> s = {a}`) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN REWRITE_TAC[HULL_SUBSET]);; let AFFINE_HULL_SCALING = prove (`!s:real^N->bool c. affine hull (IMAGE (\x. c % x) s) = IMAGE (\x. c % x) (affine hull s)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `c = &0` THENL [ASM_SIMP_TAC[IMAGE_CONST; VECTOR_MUL_LZERO; AFFINE_HULL_EQ_EMPTY] THEN COND_CASES_TAC THEN REWRITE_TAC[AFFINE_HULL_EMPTY; AFFINE_HULL_SING]; ALL_TAC] THEN MATCH_MP_TAC HULL_IMAGE THEN ASM_SIMP_TAC[AFFINE_SCALING_EQ; AFFINE_AFFINE_HULL] THEN REWRITE_TAC[VECTOR_ARITH `c % x = c % y <=> c % (x - y) = vec 0`] THEN ASM_SIMP_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ] THEN X_GEN_TAC `x:real^N` THEN EXISTS_TAC `inv c % x:real^N` THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; VECTOR_MUL_LID]);; let AFFINE_HULL_AFFINITY = prove (`!s a:real^N c. affine hull (IMAGE (\x. c % x + a) s) = IMAGE (\x. c % x + a) (affine hull s)`, REPEAT GEN_TAC THEN REWRITE_TAC[AFFINITY_SCALING_TRANSLATION] THEN ASM_SIMP_TAC[IMAGE_o; AFFINE_HULL_TRANSLATION; AFFINE_HULL_SCALING]);; (* ------------------------------------------------------------------------- *) (* Convexity. *) (* ------------------------------------------------------------------------- *) let convex = new_definition `convex s <=> !x y u v. x IN s /\ y IN s /\ &0 <= u /\ &0 <= v /\ (u + v = &1) ==> (u % x + v % y) IN s`;; let CONVEX_ALT = prove (`convex s <=> !x y u. x IN s /\ y IN s /\ &0 <= u /\ u <= &1 ==> ((&1 - u) % x + u % y) IN s`, REWRITE_TAC[convex] THEN MESON_TAC[REAL_ARITH `&0 <= u /\ &0 <= v /\ (u + v = &1) ==> v <= &1 /\ (u = &1 - v)`; REAL_ARITH `u <= &1 ==> &0 <= &1 - u /\ ((&1 - u) + u = &1)`]);; let IN_CONVEX_SET = prove (`!s a b u. convex s /\ a IN s /\ b IN s /\ &0 <= u /\ u <= &1 ==> ((&1 - u) % a + u % b) IN s`, MESON_TAC[CONVEX_ALT]);; let MIDPOINT_IN_CONVEX = prove (`!s x y:real^N. convex s /\ x IN s /\ y IN s ==> midpoint(x,y) IN s`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `x:real^N`; `y:real^N`; `&1 / &2`] IN_CONVEX_SET) THEN ASM_REWRITE_TAC[midpoint] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN CONV_TAC VECTOR_ARITH);; let CONVEX_CONTAINS_SEGMENT = prove (`!s. convex s <=> !a b. a IN s /\ b IN s ==> segment[a,b] SUBSET s`, REWRITE_TAC[CONVEX_ALT; segment; SUBSET; IN_ELIM_THM] THEN MESON_TAC[]);; let CONVEX_CONTAINS_OPEN_SEGMENT = prove (`!s. convex s <=> !a b. a IN s /\ b IN s ==> segment(a,b) SUBSET s`, ONCE_REWRITE_TAC[segment] THEN REWRITE_TAC[CONVEX_CONTAINS_SEGMENT] THEN SET_TAC[]);; let CONVEX_CONTAINS_SEGMENT_EQ = prove (`!s:real^N->bool. convex s <=> !a b. segment[a,b] SUBSET s <=> a IN s /\ b IN s`, REWRITE_TAC[CONVEX_CONTAINS_SEGMENT; SUBSET] THEN MESON_TAC[ENDS_IN_SEGMENT]);; let CONVEX_CONTAINS_SEGMENT_IMP = prove (`!s a b. convex s ==> (segment[a,b] SUBSET s <=> a IN s /\ b IN s)`, SIMP_TAC[CONVEX_CONTAINS_SEGMENT_EQ]);; let SEGMENT_SUBSET_CONVEX = prove (`!s a b:real^N. convex s /\ a IN s /\ b IN s ==> segment[a,b] SUBSET s`, MESON_TAC[CONVEX_CONTAINS_SEGMENT]);; let CONVEX_EMPTY = prove (`convex {}`, REWRITE_TAC[convex; NOT_IN_EMPTY]);; let CONVEX_SING = prove (`!a. convex {a}`, SIMP_TAC[convex; IN_SING; GSYM VECTOR_ADD_RDISTRIB; VECTOR_MUL_LID]);; let CONVEX_UNIV = prove (`convex(UNIV:real^N->bool)`, REWRITE_TAC[convex; IN_UNIV]);; let CONVEX_INTERS = prove (`(!s. s IN f ==> convex s) ==> convex(INTERS f)`, REWRITE_TAC[convex; IN_INTERS] THEN MESON_TAC[]);; let CONVEX_INTER = prove (`!s t. convex s /\ convex t ==> convex(s INTER t)`, REWRITE_TAC[convex; IN_INTER] THEN MESON_TAC[]);; let CONVEX_HALFSPACE_LE = prove (`!a b. convex {x | a dot x <= b}`, REWRITE_TAC[convex; IN_ELIM_THM; DOT_RADD; DOT_RMUL] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `(u + v) * b` THEN CONJ_TAC THENL [ASM_MESON_TAC[REAL_ADD_RDISTRIB; REAL_LE_ADD2; REAL_LE_LMUL]; ASM_MESON_TAC[REAL_MUL_LID; REAL_LE_REFL]]);; let CONVEX_HALFSPACE_COMPONENT_LE = prove (`!a k. convex {x:real^N | x$k <= a}`, REPEAT GEN_TAC THEN SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i` CHOOSE_TAC THENL [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN MP_TAC(ISPECL [`basis i:real^N`; `a:real`] CONVEX_HALFSPACE_LE) THEN ASM_SIMP_TAC[DOT_BASIS]);; let CONVEX_HALFSPACE_GE = prove (`!a b. convex {x:real^N | a dot x >= b}`, REPEAT GEN_TAC THEN SUBGOAL_THEN `{x:real^N | a dot x >= b} = {x | --a dot x <= --b}` (fun th -> REWRITE_TAC[th; CONVEX_HALFSPACE_LE]) THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM; DOT_LNEG] THEN REAL_ARITH_TAC);; let CONVEX_HALFSPACE_COMPONENT_GE = prove (`!a k. convex {x:real^N | x$k >= a}`, REPEAT GEN_TAC THEN SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i` CHOOSE_TAC THENL [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN MP_TAC(ISPECL [`basis i:real^N`; `a:real`] CONVEX_HALFSPACE_GE) THEN ASM_SIMP_TAC[DOT_BASIS]);; let CONVEX_HYPERPLANE = prove (`!a b. convex {x:real^N | a dot x = b}`, REPEAT GEN_TAC THEN SUBGOAL_THEN `{x:real^N | a dot x = b} = {x | a dot x <= b} INTER {x | a dot x >= b}` (fun th -> SIMP_TAC[th; CONVEX_INTER; CONVEX_HALFSPACE_LE; CONVEX_HALFSPACE_GE]) THEN REWRITE_TAC[EXTENSION; IN_INTER; IN_ELIM_THM] THEN REAL_ARITH_TAC);; let CONVEX_STANDARD_HYPERPLANE = prove (`!k a. convex {x:real^N | x$k = a}`, REPEAT GEN_TAC THEN SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i` CHOOSE_TAC THENL [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN MP_TAC(ISPECL [`basis i:real^N`; `a:real`] CONVEX_HYPERPLANE) THEN ASM_SIMP_TAC[DOT_BASIS]);; let CONVEX_HALFSPACE_LT = prove (`!a b. convex {x | a dot x < b}`, REWRITE_TAC[convex; IN_ELIM_THM; DOT_RADD; DOT_RMUL] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_CONVEX_BOUND_LT THEN ASM_REWRITE_TAC[]);; let CONVEX_HALFSPACE_COMPONENT_LT = prove (`!a k. convex {x:real^N | x$k < a}`, REPEAT GEN_TAC THEN SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i` CHOOSE_TAC THENL [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN MP_TAC(ISPECL [`basis i:real^N`; `a:real`] CONVEX_HALFSPACE_LT) THEN ASM_SIMP_TAC[DOT_BASIS]);; let CONVEX_HALFSPACE_GT = prove (`!a b. convex {x | a dot x > b}`, REWRITE_TAC[REAL_ARITH `ax > b <=> --ax < --b`] THEN REWRITE_TAC[GSYM DOT_LNEG; CONVEX_HALFSPACE_LT]);; let CONVEX_HALFSPACE_COMPONENT_GT = prove (`!a k. convex {x:real^N | x$k > a}`, REPEAT GEN_TAC THEN SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i` CHOOSE_TAC THENL [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN MP_TAC(ISPECL [`basis i:real^N`; `a:real`] CONVEX_HALFSPACE_GT) THEN ASM_SIMP_TAC[DOT_BASIS]);; let CONVEX_STRIP_COMPONENT_LE = prove (`!a k. convex {x:real^N | abs(x$k) <= a}`, REWRITE_TAC[REAL_ARITH `abs(x) <= a <=> x <= a /\ x >= --a`] THEN REWRITE_TAC[SET_RULE `{x | P x /\ Q x} = {x | P x} INTER {x | Q x}`] THEN SIMP_TAC[CONVEX_HALFSPACE_COMPONENT_LE; CONVEX_HALFSPACE_COMPONENT_GE; CONVEX_INTER]);; let CONVEX_STRIP_COMPONENT_LT = prove (`!a k. convex {x:real^N | abs(x$k) < a}`, REWRITE_TAC[REAL_ARITH `abs(x) < a <=> x < a /\ x > --a`] THEN REWRITE_TAC[SET_RULE `{x | P x /\ Q x} = {x | P x} INTER {x | Q x}`] THEN SIMP_TAC[CONVEX_HALFSPACE_COMPONENT_LT; CONVEX_HALFSPACE_COMPONENT_GT; CONVEX_INTER]);; let CONVEX_HALFSPACE_SGN = prove (`!a b. convex {x:real^N | real_sgn(a dot x) = b}`, REPEAT GEN_TAC THEN ASM_CASES_TAC `b = &0` THEN ASM_REWRITE_TAC[CONVEX_HYPERPLANE; REAL_SGN_EQ] THEN ASM_CASES_TAC `b = -- &1` THEN ASM_REWRITE_TAC[CONVEX_HALFSPACE_LT; REAL_SGN_EQ] THEN ASM_CASES_TAC `b = &1` THEN ASM_REWRITE_TAC[CONVEX_HALFSPACE_GT; REAL_SGN_EQ] THEN ASM_SIMP_TAC[CONVEX_EMPTY; MATCH_MP (SET_RULE `(!x. P(real_sgn x)) ==> ~(P b) ==> {x | real_sgn(f x) = b} = {}`) REAL_SGN_CASES]);; let CONVEX_HALFSPACE_COMPONENT_SGN = prove (`!a k. convex {x:real^N | real_sgn(x$k) = a}`, REPEAT GEN_TAC THEN SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i` CHOOSE_TAC THENL [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN MP_TAC(ISPECL [`basis i:real^N`; `a:real`] CONVEX_HALFSPACE_SGN) THEN ASM_SIMP_TAC[DOT_BASIS]);; let CONVEX_POSITIVE_ORTHANT = prove (`convex {x:real^N | !i. 1 <= i /\ i <= dimindex(:N) ==> &0 <= x$i}`, SIMP_TAC[convex; IN_ELIM_THM; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; REAL_LE_MUL; REAL_LE_ADD]);; let LIMPT_OF_CONVEX = prove (`!s x:real^N. convex s /\ x IN s ==> (x limit_point_of s <=> ~(s = {x}))`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s = {x:real^N}` THEN ASM_REWRITE_TAC[LIMPT_SING] THEN SUBGOAL_THEN `?y:real^N. y IN s /\ ~(y = x)` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[LIMPT_APPROACHABLE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN ABBREV_TAC `u = min (&1 / &2) (e / &2 / norm(y - x:real^N))` THEN SUBGOAL_THEN `&0 < u /\ u < &1` STRIP_ASSUME_TAC THENL [EXPAND_TAC "u" THEN REWRITE_TAC[REAL_LT_MIN; REAL_MIN_LT] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_SIMP_TAC[REAL_HALF; REAL_LT_DIV; NORM_POS_LT; VECTOR_SUB_EQ]; ALL_TAC] THEN EXISTS_TAC `(&1 - u) % x + u % y:real^N` THEN REPEAT CONJ_TAC THENL [FIRST_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_ALT]) THEN ASM_SIMP_TAC[REAL_LT_IMP_LE]; ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ; VECTOR_ARITH `(&1 - u) % x + u % y:real^N = x <=> u % (y - x) = vec 0`] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[dist; NORM_MUL; VECTOR_ARITH `((&1 - u) % x + u % y) - x:real^N = u % (y - x)`] THEN ASM_SIMP_TAC[REAL_ARITH `&0 < u ==> abs u = u`] THEN MATCH_MP_TAC(REAL_ARITH `x <= e / &2 /\ &0 < e ==> x < e`) THEN ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN ASM_REAL_ARITH_TAC]);; let TRIVIAL_LIMIT_WITHIN_CONVEX = prove (`!s x:real^N. convex s /\ x IN s ==> (trivial_limit(at x within s) <=> s = {x})`, SIMP_TAC[TRIVIAL_LIMIT_WITHIN; LIMPT_OF_CONVEX]);; (* ------------------------------------------------------------------------- *) (* Some invariance theorems for convex sets. *) (* ------------------------------------------------------------------------- *) let CONVEX_TRANSLATION_EQ = prove (`!a:real^N s. convex (IMAGE (\x. a + x) s) <=> convex s`, REWRITE_TAC[CONVEX_ALT; IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_IMAGE; UNWIND_THM1; VECTOR_ARITH `(&1 - u) % (a + x) + u % (a + y) = a + z <=> (&1 - u) % x + u % y = z`]);; add_translation_invariants [CONVEX_TRANSLATION_EQ];; let CONVEX_TRANSLATION = prove (`!s a:real^N. convex s ==> convex (IMAGE (\x. a + x) s)`, REWRITE_TAC[CONVEX_TRANSLATION_EQ]);; let CONVEX_LINEAR_IMAGE = prove (`!f s. convex s /\ linear f ==> convex(IMAGE f s)`, REWRITE_TAC[convex; FORALL_IN_IMAGE; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[IN_IMAGE; linear] THEN MESON_TAC[]);; let CONVEX_LINEAR_IMAGE_EQ = prove (`!f s. linear f /\ (!x y. f x = f y ==> x = y) ==> (convex (IMAGE f s) <=> convex s)`, MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE CONVEX_LINEAR_IMAGE));; add_linear_invariants [CONVEX_LINEAR_IMAGE_EQ];; (* ------------------------------------------------------------------------- *) (* Explicit expressions for convexity in terms of arbitrary sums. *) (* ------------------------------------------------------------------------- *) let CONVEX_VSUM = prove (`!s k u x:A->real^N. FINITE k /\ convex s /\ sum k u = &1 /\ (!i. i IN k ==> &0 <= u i /\ x i IN s) ==> vsum k (\i. u i % x i) IN s`, GEN_TAC THEN ASM_CASES_TAC `convex(s:real^N->bool)` THEN ASM_REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; FORALL_IN_INSERT] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN MAP_EVERY X_GEN_TAC [`i:A`; `k:A->bool`] THEN GEN_REWRITE_TAC (BINOP_CONV o DEPTH_CONV) [RIGHT_IMP_FORALL_THM] THEN REWRITE_TAC[IMP_IMP] THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`u:A->real`; `x:A->real^N`] THEN ASM_CASES_TAC `(u:A->real) i = &1` THENL [ASM_REWRITE_TAC[REAL_ARITH `&1 + a = &1 <=> a = &0`] THEN STRIP_TAC THEN SUBGOAL_THEN `vsum k (\i:A. u i % x(i):real^N) = vec 0` (fun th -> ASM_SIMP_TAC[th; VECTOR_ADD_RID; VECTOR_MUL_LID]) THEN MATCH_MP_TAC VSUM_EQ_0 THEN REWRITE_TAC[VECTOR_MUL_EQ_0] THEN REPEAT STRIP_TAC THEN DISJ1_TAC THEN ASM_MESON_TAC[SUM_POS_EQ_0]; STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `\j:A. u(j) / (&1 - u(i))`) THEN ASM_REWRITE_TAC[real_div] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[SUM_LMUL; VSUM_LMUL; GSYM VECTOR_MUL_ASSOC] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[GSYM real_div] THEN SUBGOAL_THEN `&0 < &1 - u(i:A)` ASSUME_TAC THENL [ASM_MESON_TAC[SUM_POS_LE; REAL_ADD_SYM; REAL_ARITH `&0 <= a /\ &0 <= b /\ b + a = &1 /\ ~(a = &1) ==> &0 < &1 - a`]; ALL_TAC] THEN ASM_SIMP_TAC[REAL_LE_DIV; REAL_LT_IMP_LE] THEN ASM_SIMP_TAC[REAL_EQ_LDIV_EQ; REAL_MUL_LID; REAL_EQ_SUB_LADD] THEN DISCH_TAC THEN ONCE_REWRITE_TAC[VECTOR_ADD_SYM] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [convex]) THEN DISCH_THEN(MP_TAC o SPECL [`vsum k (\j. (u j / (&1 - u(i:A))) % x(j) :real^N)`; `x(i:A):real^N`; `&1 - u(i:A)`; `u(i:A):real`]) THEN REWRITE_TAC[real_div] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[GSYM VECTOR_MUL_ASSOC; VSUM_LMUL] THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; REAL_LT_IMP_NZ] THEN REWRITE_TAC[VECTOR_MUL_LID] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; VSUM_LMUL] THEN CONJ_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC; REAL_ARITH_TAC] THEN ASM_MESON_TAC[REAL_ADD_SYM]]);; let CONVEX_VSUM_STRONG = prove (`!s k u x:A->real^N. convex s /\ sum k u = &1 /\ (!i. i IN k ==> &0 <= u i /\ (u i = &0 \/ x i IN s)) ==> vsum k (\i. u i % x i) IN s`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `vsum k (\i. u i % (x:A->real^N) i) = vsum {i | i IN k /\ ~(u i = &0)} (\i. u i % x i)` SUBST1_TAC THENL [MATCH_MP_TAC VSUM_SUPERSET THEN REWRITE_TAC[VECTOR_MUL_EQ_0] THEN SET_TAC[]; MATCH_MP_TAC CONVEX_VSUM THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[SUM_DEGENERATE; REAL_ARITH `~(&1 = &0)`]; FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC SUM_SUPERSET THEN ASM SET_TAC[]; ASM SET_TAC[]]]);; let CONVEX_INDEXED = prove (`!s:real^N->bool. convex s <=> !k u x. (!i:num. 1 <= i /\ i <= k ==> &0 <= u(i) /\ x(i) IN s) /\ (sum (1..k) u = &1) ==> vsum (1..k) (\i. u(i) % x(i)) IN s`, REPEAT GEN_TAC THEN EQ_TAC THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC CONVEX_VSUM THEN ASM_REWRITE_TAC[IN_NUMSEG; FINITE_NUMSEG]; DISCH_TAC THEN REWRITE_TAC[convex] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `u:real`; `v:real`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `2`) THEN DISCH_THEN(MP_TAC o SPEC `\n. if n = 1 then u else v:real`) THEN DISCH_THEN(MP_TAC o SPEC `\n. if n = 1 then x else y:real^N`) THEN REWRITE_TAC[num_CONV `2`; SUM_CLAUSES_NUMSEG; VSUM_CLAUSES_NUMSEG; NUMSEG_SING; VSUM_SING; SUM_SING] THEN REWRITE_TAC[ARITH] THEN ASM_MESON_TAC[]]);; let CONVEX_EXPLICIT = prove (`!s:real^N->bool. convex s <=> !t u. FINITE t /\ t SUBSET s /\ (!x. x IN t ==> &0 <= u x) /\ sum t u = &1 ==> vsum t (\x. u(x) % x) IN s`, REPEAT GEN_TAC THEN EQ_TAC THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC CONVEX_VSUM THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; DISCH_TAC THEN REWRITE_TAC[convex] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `u:real`; `v:real`] THEN ASM_CASES_TAC `x:real^N = y` THENL [ASM_SIMP_TAC[GSYM VECTOR_ADD_RDISTRIB; VECTOR_MUL_LID]; ALL_TAC] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `{x:real^N,y}`) THEN DISCH_THEN(MP_TAC o SPEC `\z:real^N. if z = x then u else v:real`) THEN ASM_SIMP_TAC[FINITE_INSERT; FINITE_RULES; SUM_CLAUSES; VSUM_CLAUSES; NOT_IN_EMPTY] THEN ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; REAL_ADD_RID; SUBSET] THEN REWRITE_TAC[VECTOR_ADD_RID] THEN ASM_MESON_TAC[]]);; let CONVEX = prove (`!V:real^N->bool. convex V <=> !(s:real^N->bool) (u:real^N->real). FINITE s /\ ~(s = {}) /\ s SUBSET V /\ (!x. x IN s ==> &0 <= u x) /\ sum s u = &1 ==> vsum s (\x. u x % x) IN V`, GEN_TAC THEN REWRITE_TAC[CONVEX_EXPLICIT] THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `t:real^N->bool` THEN REWRITE_TAC[] THEN AP_TERM_TAC THEN ABS_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_REWRITE_TAC[SUM_CLAUSES] THEN CONV_TAC REAL_RAT_REDUCE_CONV);; let CONVEX_FINITE = prove (`!s:real^N->bool. FINITE s ==> (convex s <=> !u. (!x. x IN s ==> &0 <= u x) /\ sum s u = &1 ==> vsum s (\x. u(x) % x) IN s)`, REPEAT STRIP_TAC THEN REWRITE_TAC[CONVEX_EXPLICIT] THEN EQ_TAC THENL [ASM_MESON_TAC[SUBSET_REFL]; ALL_TAC] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`t:real^N->bool`; `u:real^N->real`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `\x:real^N. if x IN t then u x else &0`) THEN ASM_SIMP_TAC[GSYM SUM_RESTRICT_SET] THEN ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN ASM_SIMP_TAC[VECTOR_MUL_LZERO; REAL_LE_REFL; GSYM VSUM_RESTRICT_SET] THEN ASM_SIMP_TAC[COND_ID; SET_RULE `t SUBSET s ==> {x | x IN s /\ x IN t} = t`]);; let AFFINE_PCROSS = prove (`!s:real^M->bool t:real^N->bool. affine s /\ affine t ==> affine(s PCROSS t)`, REWRITE_TAC[affine; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN SIMP_TAC[FORALL_IN_PCROSS; GSYM PASTECART_CMUL; PASTECART_ADD] THEN SIMP_TAC[PASTECART_IN_PCROSS]);; let AFFINE_PCROSS_EQ = prove (`!s:real^M->bool t:real^N->bool. affine(s PCROSS t) <=> s = {} \/ t = {} \/ affine s /\ affine t`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_REWRITE_TAC[PCROSS_EMPTY; AFFINE_EMPTY] THEN ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_REWRITE_TAC[PCROSS_EMPTY; AFFINE_EMPTY] THEN EQ_TAC THEN REWRITE_TAC[AFFINE_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)`] AFFINE_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)`] AFFINE_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 CONVEX_PCROSS = prove (`!s:real^M->bool t:real^N->bool. convex s /\ convex t ==> convex(s PCROSS t)`, REWRITE_TAC[convex; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN SIMP_TAC[FORALL_IN_PCROSS; GSYM PASTECART_CMUL; PASTECART_ADD] THEN SIMP_TAC[PASTECART_IN_PCROSS]);; let CONVEX_PCROSS_EQ = prove (`!s:real^M->bool t:real^N->bool. convex(s PCROSS t) <=> s = {} \/ t = {} \/ convex s /\ convex t`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_REWRITE_TAC[PCROSS_EMPTY; CONVEX_EMPTY] THEN ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_REWRITE_TAC[PCROSS_EMPTY; CONVEX_EMPTY] THEN EQ_TAC THEN REWRITE_TAC[CONVEX_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)`] CONVEX_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)`] CONVEX_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[]);; (* ------------------------------------------------------------------------- *) (* Conic sets and conic hull. *) (* ------------------------------------------------------------------------- *) let conic = new_definition `conic s <=> !x c. x IN s /\ &0 <= c ==> (c % x) IN s`;; let SUBSPACE_IMP_CONIC = prove (`!s. subspace s ==> conic s`, SIMP_TAC[subspace; conic]);; let CONIC_EMPTY = prove (`conic {}`, REWRITE_TAC[conic; NOT_IN_EMPTY]);; let CONIC_UNIV = prove (`conic (UNIV:real^N->bool)`, REWRITE_TAC[conic; IN_UNIV]);; let CONIC_INTERS = prove (`(!s. s IN f ==> conic s) ==> conic(INTERS f)`, REWRITE_TAC[conic; IN_INTERS] THEN MESON_TAC[]);; let CONIC_LINEAR_IMAGE = prove (`!f s. conic s /\ linear f ==> conic(IMAGE f s)`, REWRITE_TAC[conic; IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_IMAGE] THEN MESON_TAC[LINEAR_CMUL]);; let CONIC_LINEAR_IMAGE_EQ = prove (`!f s. linear f /\ (!x y. f x = f y ==> x = y) ==> (conic (IMAGE f s) <=> conic s)`, MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE CONIC_LINEAR_IMAGE));; add_linear_invariants [CONIC_LINEAR_IMAGE_EQ];; let CONIC_MUL = prove (`!s c x:real^N. conic s /\ x IN s /\ &0 <= c ==> (c % x) IN s`, REWRITE_TAC[conic] THEN MESON_TAC[]);; let CONIC_CONIC_HULL = prove (`!s. conic(conic hull s)`, SIMP_TAC[P_HULL; CONIC_INTERS]);; let CONIC_HULL_EQ = prove (`!s. (conic hull s = s) <=> conic s`, SIMP_TAC[HULL_EQ; CONIC_INTERS]);; let CONIC_HULL_UNIV = prove (`conic hull (:real^N) = (:real^N)`, REWRITE_TAC[HULL_UNIV]);; let CONIC_NEGATIONS = prove (`!s. conic s ==> conic (IMAGE (--) s)`, REWRITE_TAC[conic; RIGHT_FORALL_IMP_THM; IMP_CONJ; FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_IMAGE; VECTOR_MUL_RNEG] THEN MESON_TAC[]);; let CONIC_SPAN = prove (`!s. conic(span s)`, SIMP_TAC[SUBSPACE_IMP_CONIC; SUBSPACE_SPAN]);; let CONIC_HULL_EXPLICIT = prove (`!s:real^N->bool. conic hull s = {c % x | &0 <= c /\ x IN s}`, GEN_TAC THEN MATCH_MP_TAC HULL_UNIQUE THEN REWRITE_TAC[conic; SUBSET; RIGHT_FORALL_IMP_THM; IMP_CONJ] THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN REWRITE_TAC[RIGHT_IMP_FORALL_THM; IMP_IMP; IN_ELIM_THM] THEN REPEAT CONJ_TAC THENL [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MAP_EVERY EXISTS_TAC [`&1`; `x:real^N`] THEN ASM_SIMP_TAC[REAL_POS; VECTOR_MUL_LID]; REWRITE_TAC[VECTOR_MUL_ASSOC] THEN MESON_TAC[REAL_LE_MUL]; MESON_TAC[]]);; let CONIC_HULL_AS_IMAGE = prove (`!s:real^N->bool. conic hull s = IMAGE (\z. drop(fstcart z) % sndcart z) ({t | &0 <= drop t} PCROSS s)`, REWRITE_TAC[EXTENSION; IN_IMAGE; EXISTS_PASTECART; PASTECART_IN_PCROSS] THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; CONIC_HULL_EXPLICIT] THEN REWRITE_TAC[IN_ELIM_THM; GSYM EXISTS_DROP] THEN MESON_TAC[]);; let CONIC_HULL_POINTLESS_AS_IMAGE = prove (`!s:real^N->bool. conic hull s DELETE vec 0 = IMAGE (\z. drop(fstcart z) % sndcart z) ({t | &0 < drop t} PCROSS (s DELETE vec 0))`, GEN_TAC THEN REWRITE_TAC[CONIC_HULL_AS_IMAGE; EXTENSION] THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_IMAGE; IN_DELETE] THEN REWRITE_TAC[EXISTS_PASTECART; FSTCART_PASTECART; SNDCART_PASTECART] THEN REWRITE_TAC[PASTECART_IN_PCROSS; IN_DELETE; IN_ELIM_THM] THEN REWRITE_TAC[GSYM EXISTS_DROP; LEFT_AND_EXISTS_THM] THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `a:real` THEN REWRITE_TAC[] THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `b:real^N` THEN ASM_CASES_TAC `y:real^N = a % b` THEN ASM_REWRITE_TAC[VECTOR_MUL_EQ_0] THEN REWRITE_TAC[REAL_LT_LE] THEN MESON_TAC[]);; let CONIC_HULL_LINEAR_IMAGE = prove (`!f s. linear f ==> conic hull (IMAGE f s) = IMAGE f (conic hull s)`, REPEAT GEN_TAC THEN REWRITE_TAC[CONIC_HULL_EXPLICIT] THEN REWRITE_TAC[SET_RULE `IMAGE f {c % x | P c x} = {f(c % x) | P c x}`] THEN REWRITE_TAC[SET_RULE `{c % x | &0 <= c /\ x IN IMAGE f s} = {c % f(x) | &0 <= c /\ x IN s}`] THEN DISCH_THEN(fun th -> REWRITE_TAC[MATCH_MP LINEAR_CMUL th]));; add_linear_invariants [CONIC_HULL_LINEAR_IMAGE];; let CONIC_HULL_IMAGE_SCALE = prove (`!c s:real^N->bool. (!x. x IN s ==> &0 < c x) ==> conic hull (IMAGE (\x. c x % x) s) = conic hull s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THEN MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[CONIC_CONIC_HULL; SUBSET; FORALL_IN_IMAGE] THEN ASM_SIMP_TAC[HULL_INC; CONIC_MUL; CONIC_CONIC_HULL; REAL_LT_IMP_LE] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN SUBGOAL_THEN `x:real^N = inv(c x) % c x % x` SUBST1_TAC THENL [REWRITE_TAC[VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[REAL_MUL_LINV; VECTOR_MUL_LID; REAL_LT_IMP_NZ]; MATCH_MP_TAC CONIC_MUL THEN ASM_SIMP_TAC[CONIC_CONIC_HULL; REAL_LE_INV_EQ; REAL_LT_IMP_LE] THEN MATCH_MP_TAC HULL_INC THEN ASM SET_TAC[]]);; let CONVEX_CONIC_HULL = prove (`!s:real^N->bool. convex s ==> convex (conic hull s)`, REWRITE_TAC[CONIC_HULL_EXPLICIT] THEN REWRITE_TAC[CONVEX_ALT; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN REWRITE_TAC[IN_ELIM_THM; IMP_IMP] THEN X_GEN_TAC `s:real^N->bool` THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`c:real`; `x:real^N`] THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`d:real`; `y:real^N`] THEN STRIP_TAC THEN X_GEN_TAC `u:real` THEN STRIP_TAC THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN ASM_CASES_TAC `(&1 - u) * c = &0` THENL [ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_LID] THEN ASM_MESON_TAC[REAL_LE_MUL]; ALL_TAC] THEN SUBGOAL_THEN `&0 < (&1 - u) * c + u * d` ASSUME_TAC THENL [MATCH_MP_TAC REAL_LTE_ADD THEN ASM_REWRITE_TAC[REAL_LT_LE] THEN CONJ_TAC THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN EXISTS_TAC `(&1 - u) * c + u * d:real` THEN EXISTS_TAC `((&1 - u) * c) / ((&1 - u) * c + u * d) % x + (u * d) / ((&1 - u) * c + u * d) % y:real^N` THEN REWRITE_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[REAL_DIV_LMUL; REAL_LT_IMP_NZ] THEN ASM_SIMP_TAC[REAL_LE_ADD; REAL_LE_MUL; REAL_SUB_LE] THEN ASM_SIMP_TAC[REAL_FIELD `&0 < u + v ==> u / (u + v) = &1 - (v / (u + v))`] THEN RULE_ASSUM_TAC(REWRITE_RULE[RIGHT_IMP_FORALL_THM; IMP_IMP]) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ] THEN ASM_SIMP_TAC[REAL_MUL_LZERO; REAL_LE_MUL; REAL_MUL_LID; REAL_LE_ADDL; REAL_SUB_LE]);; let CONIC_HALFSPACE_LE = prove (`!a. conic {x | a dot x <= &0}`, REWRITE_TAC[conic; IN_ELIM_THM; DOT_RMUL] THEN REWRITE_TAC[REAL_ARITH `a <= &0 <=> &0 <= --a`] THEN SIMP_TAC[GSYM REAL_MUL_RNEG; REAL_LE_MUL]);; let CONIC_HALFSPACE_GE = prove (`!a. conic {x | a dot x >= &0}`, SIMP_TAC[conic; IN_ELIM_THM; DOT_RMUL; real_ge; REAL_LE_MUL]);; let CONIC_HULL_EMPTY = prove (`conic hull {} = {}`, MATCH_MP_TAC HULL_UNIQUE THEN REWRITE_TAC[SUBSET_REFL; CONIC_EMPTY; EMPTY_SUBSET]);; let CONIC_CONTAINS_0 = prove (`!s:real^N->bool. conic s ==> (vec 0 IN s <=> ~(s = {}))`, REPEAT STRIP_TAC THEN EQ_TAC THENL [SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN DISCH_THEN(X_CHOOSE_TAC `x:real^N`) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [conic]) THEN DISCH_THEN(MP_TAC o SPECL [`x:real^N`; `&0`]) THEN ASM_REWRITE_TAC[REAL_POS; VECTOR_MUL_LZERO]);; let CONIC_HULL_EQ_EMPTY = prove (`!s. (conic hull s = {}) <=> (s = {})`, GEN_TAC THEN EQ_TAC THEN MESON_TAC[SUBSET_EMPTY; HULL_SUBSET; CONIC_HULL_EMPTY]);; let CONIC_SUMS = prove (`!s t. conic s /\ conic t ==> conic {x + y:real^N | x IN s /\ y IN t}`, REWRITE_TAC[conic; IN_ELIM_THM] THEN MESON_TAC[VECTOR_ADD_LDISTRIB]);; let CONIC_PCROSS = prove (`!s:real^M->bool t:real^N->bool. conic s /\ conic t ==> conic(s PCROSS t)`, REWRITE_TAC[conic; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN SIMP_TAC[FORALL_IN_PCROSS; GSYM PASTECART_CMUL; PASTECART_ADD] THEN SIMP_TAC[PASTECART_IN_PCROSS]);; let CONIC_PCROSS_EQ = prove (`!s:real^M->bool t:real^N->bool. conic(s PCROSS t) <=> s = {} \/ t = {} \/ conic s /\ conic t`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_REWRITE_TAC[PCROSS_EMPTY; CONIC_EMPTY] THEN ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_REWRITE_TAC[PCROSS_EMPTY; CONIC_EMPTY] THEN EQ_TAC THEN REWRITE_TAC[CONIC_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)`] CONIC_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)`] CONIC_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 CONIC_POSITIVE_ORTHANT = prove (`conic {x:real^N | !i. 1 <= i /\ i <= dimindex(:N) ==> &0 <= x$i}`, SIMP_TAC[conic; IN_ELIM_THM; REAL_LE_MUL; VECTOR_MUL_COMPONENT]);; let CONIC_HULL_0 = prove (`conic hull {vec 0} = {vec 0}`, REWRITE_TAC[EXTENSION; IN_SING; CONIC_HULL_EXPLICIT; IN_ELIM_THM] THEN MESON_TAC[VECTOR_MUL_RZERO; REAL_POS]);; let CONIC_HULL_CONTAINS_0 = prove (`!s:real^N->bool. vec 0 IN conic hull s <=> ~(s = {})`, SIMP_TAC[CONIC_CONTAINS_0; CONIC_HULL_EQ_EMPTY; CONIC_CONIC_HULL]);; let CONIC_HULL_EQ_SING = prove (`!s x:real^N. conic hull s = {x} <=> s = {vec 0} /\ x = vec 0`, REPEAT GEN_TAC THEN EQ_TAC THEN SIMP_TAC[CONIC_HULL_0] THEN ASM_CASES_TAC `s SUBSET {x:real^N}` THENL [ALL_TAC; ASM_MESON_TAC[HULL_SUBSET]] THEN FIRST_X_ASSUM(STRIP_ASSUME_TAC o MATCH_MP (SET_RULE `s SUBSET {a} ==> s = {} \/ s = {a}`)) THEN ASM_REWRITE_TAC[CONIC_HULL_EMPTY; NOT_INSERT_EMPTY] THEN ASM_CASES_TAC `x:real^N = vec 0` THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN MP_TAC(ISPEC `{x:real^N}` CONIC_HULL_CONTAINS_0) THEN ASM_REWRITE_TAC[IN_SING; NOT_INSERT_EMPTY]);; let CONIC_HULL_INTER_AFFINE_HULL = prove (`!s f:real^N->bool. f SUBSET s /\ ~(vec 0 IN affine hull s) ==> (conic hull f) INTER (affine hull s) = f`, REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; SUBSET_INTER; HULL_SUBSET] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[HULL_MONO; HULL_SUBSET; SUBSET_TRANS]] THEN REWRITE_TAC[SUBSET; IN_INTER; CONIC_HULL_EXPLICIT; IMP_CONJ; FORALL_IN_GSPEC] THEN MAP_EVERY X_GEN_TAC [`c:real`; `x:real^N`] THEN ASM_CASES_TAC `c = &1` THEN ASM_SIMP_TAC[VECTOR_MUL_LID] THEN REPEAT STRIP_TAC THEN UNDISCH_TAC `~((vec 0:real^N) IN affine hull s)` THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_THEN(K ALL_TAC) THEN SUBGOAL_THEN `vec 0:real^N = inv(&1 - c) % c % x + (&1 - inv(&1 - c)) % x` SUBST1_TAC THENL [CONV_TAC SYM_CONV THEN REWRITE_TAC[VECTOR_MUL_ASSOC; GSYM VECTOR_ADD_RDISTRIB] THEN REWRITE_TAC[VECTOR_MUL_EQ_0] THEN DISJ1_TAC THEN UNDISCH_TAC `~(c = &1)` THEN CONV_TAC REAL_FIELD; MP_TAC(ISPEC `affine hull s:real^N->bool` affine) THEN REWRITE_TAC[AFFINE_AFFINE_HULL] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[HULL_INC; SUBSET]; UNDISCH_TAC `~(c = &1)` THEN CONV_TAC REAL_FIELD]]);; let SEPARATE_CLOSED_CONES = prove (`!c d:real^N->bool. conic c /\ closed c /\ conic d /\ closed d /\ c INTER d SUBSET {vec 0} ==> ?e. &0 < e /\ !x y. x IN c /\ y IN d ==> dist(x,y) >= e * max (norm x) (norm y)`, SUBGOAL_THEN `!c d:real^N->bool. conic c /\ closed c /\ conic d /\ closed d /\ c INTER d SUBSET {vec 0} ==> ?e. &0 < e /\ !x y. x IN c /\ y IN d ==> dist(x,y) >= e * norm x` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN REWRITE_TAC[real_ge] THEN MP_TAC(ISPECL [`c INTER sphere(vec 0:real^N,&1)`; `d:real^N->bool`] SEPARATE_COMPACT_CLOSED) THEN ASM_SIMP_TAC[CLOSED_INTER_COMPACT; COMPACT_SPHERE] THEN ANTS_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `c INTER d SUBSET {a} ==> ~(a IN s) ==> (c INTER s) INTER d = {}`)) THEN REWRITE_TAC[IN_SPHERE_0; NORM_0] THEN REAL_ARITH_TAC; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `e:real` THEN REWRITE_TAC[IN_INTER; IN_SPHERE_0] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN ASM_CASES_TAC `x:real^N = vec 0` THEN ASM_REWRITE_TAC[DIST_POS_LE; REAL_MUL_RZERO; NORM_0] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`inv(norm x) % x:real^N`; `inv(norm(x:real^N)) % y:real^N`]) THEN REWRITE_TAC[dist; NORM_MUL; GSYM VECTOR_SUB_LDISTRIB] THEN REWRITE_TAC[REAL_ARITH `abs x * a = a * abs x`] THEN REWRITE_TAC[REAL_ABS_INV; GSYM real_div; REAL_ABS_NORM] THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; NORM_POS_LT] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_SIMP_TAC[REAL_DIV_REFL; NORM_EQ_0] THEN RULE_ASSUM_TAC(REWRITE_RULE[conic]) THEN CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[REAL_LE_INV_EQ; NORM_POS_LE]]; REPEAT STRIP_TAC THEN FIRST_X_ASSUM(fun th -> MP_TAC(SPECL [`c:real^N->bool`; `d:real^N->bool`] th) THEN MP_TAC(SPECL [`d:real^N->bool`; `c:real^N->bool`] th)) THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[INTER_COMM] THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; real_ge] THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN X_GEN_TAC `e:real` THEN STRIP_TAC THEN EXISTS_TAC `min d e:real` THEN ASM_REWRITE_TAC[REAL_LT_MIN] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN REWRITE_TAC[real_max] THEN COND_CASES_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THENL [EXISTS_TAC `d * norm(y:real^N)` THEN ONCE_REWRITE_TAC[DIST_SYM]; EXISTS_TAC `e * norm(x:real^N)`] THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC REAL_LE_RMUL THEN NORM_ARITH_TAC]);; let CONTINUOUS_ON_COMPACT_SURFACE_PROJECTION = prove (`!s:real^N->bool v d:real^N->real. compact s /\ s SUBSET (v DELETE (vec 0)) /\ conic v /\ (!x k. x IN v DELETE (vec 0) ==> (&0 < k /\ (k % x) IN s <=> d x = k)) ==> (\x. d x % x) continuous_on (v DELETE (vec 0))`, let lemma = prove (`!s:real^N->real^N p srf:real^N->bool pnc. compact srf /\ srf SUBSET pnc /\ IMAGE s pnc SUBSET srf /\ (!x. x IN srf ==> s x = x) /\ p continuous_on pnc /\ (!x. x IN pnc ==> s(p x) = s x /\ p(s x) = p x) ==> s continuous_on pnc`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_EQ THEN EXISTS_TAC `(s:real^N->real^N) o (p:real^N->real^N)` THEN CONJ_TAC THENL [ASM_SIMP_TAC[o_DEF]; ALL_TAC] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `IMAGE (p:real^N->real^N) pnc = IMAGE p srf` SUBST1_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_ON_INVERSE THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ASM SET_TAC[]]]) in REWRITE_TAC[conic; IN_DELETE; SUBSET] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC lemma THEN MAP_EVERY EXISTS_TAC [`\x:real^N. inv(norm x) % x`; `s:real^N->bool`] THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_MUL THEN SIMP_TAC[o_DEF; CONTINUOUS_ON_ID] THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN SIMP_TAC[IN_DELETE; NORM_EQ_0; SIMP_RULE[o_DEF] CONTINUOUS_ON_LIFT_NORM]; REWRITE_TAC[IN_UNIV; IN_DELETE]] THEN CONJ_TAC THENL [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `&1`]) THEN ASM_REWRITE_TAC[VECTOR_MUL_LID; REAL_LT_01; IN_DELETE] THEN ASM_MESON_TAC[VECTOR_MUL_LID; SUBSET; IN_DELETE]; ALL_TAC] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN CONJ_TAC THENL [FIRST_ASSUM(MP_TAC o SPECL [`inv(norm x) % x:real^N`; `norm x * (d:real^N->real) x`]) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `(d:real^N->real) x`]) THEN ASM_SIMP_TAC[VECTOR_MUL_EQ_0; REAL_INV_EQ_0; NORM_EQ_0] THEN STRIP_TAC THEN ASM_SIMP_TAC[REAL_LE_INV_EQ; NORM_POS_LE; REAL_LT_MUL; NORM_POS_LT] THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; NORM_EQ_0; REAL_FIELD `~(n = &0) ==> (n * d) * inv n = d`]; FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `(d:real^N->real) x`]) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ASM_SIMP_TAC[NORM_MUL; VECTOR_MUL_ASSOC; REAL_INV_MUL] THEN ASM_SIMP_TAC[real_abs; REAL_LT_IMP_LE] THEN ASM_SIMP_TAC[REAL_FIELD `&0 < x ==> (inv(x) * y) * x = y`]]);; (* ------------------------------------------------------------------------- *) (* Affine dependence and consequential theorems (from Lars Schewe). *) (* ------------------------------------------------------------------------- *) let affine_dependent = new_definition `affine_dependent (s:real^N -> bool) <=> ?x. x IN s /\ x IN (affine hull (s DELETE x))`;; let AFFINE_DEPENDENT_EXPLICIT = prove (`!p. affine_dependent (p:real^N -> bool) <=> (?s u. FINITE s /\ s SUBSET p /\ sum s u = &0 /\ (?v. v IN s /\ ~(u v = &0)) /\ vsum s (\v. u v % v) = (vec 0):real^N)`, X_GEN_TAC `p:real^N->bool` THEN EQ_TAC THENL [REWRITE_TAC[affine_dependent;AFFINE_HULL_EXPLICIT; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN EXISTS_TAC `(x:real^N) INSERT s` THEN EXISTS_TAC `\v:real^N.if v = x then -- &1 else u v` THEN ASM_SIMP_TAC[FINITE_INSERT;SUM_CLAUSES;VSUM_CLAUSES;INSERT_SUBSET] THEN REPEAT CONJ_TAC THENL [ASM SET_TAC[]; COND_CASES_TAC THENL [ASM SET_TAC[];ALL_TAC] THEN ASM_SIMP_TAC[SUM_CASES; SUM_CLAUSES; SET_RULE `~((x:real^N) IN s) ==> {v | v IN s /\ v = x} = {} /\ {v | v IN s /\ ~(v = x)} = s`] THEN REAL_ARITH_TAC; SET_TAC[REAL_ARITH `~(-- &1 = &0)`]; MP_TAC (SET_RULE `s SUBSET p DELETE (x:real^N) ==> ~(x IN s)`) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_SIMP_TAC[VECTOR_ARITH `(-- &1 % (x:real^N)) + a = vec 0 <=> a = x`] THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `vsum s (\v:real^N. u v % v)` THEN CONJ_TAC THENL [ MATCH_MP_TAC VSUM_EQ THEN ASM_SIMP_TAC[] THEN ASM SET_TAC[]; ASM_REWRITE_TAC[]]]; ALL_TAC] THEN REWRITE_TAC[affine_dependent;AFFINE_HULL_EXPLICIT;IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN EXISTS_TAC `v:real^N` THEN CONJ_TAC THENL [ASM SET_TAC[];ALL_TAC] THEN EXISTS_TAC `s DELETE (v:real^N)` THEN EXISTS_TAC `\x:real^N. -- (&1 / (u v)) * u x` THEN ASM_SIMP_TAC[FINITE_DELETE;SUM_DELETE;VSUM_DELETE_CASES] THEN ASM_SIMP_TAC[SUM_LMUL;GSYM VECTOR_MUL_ASSOC;VSUM_LMUL; VECTOR_MUL_RZERO;VECTOR_ARITH `vec 0 - -- a % x = a % x:real^N`; REAL_MUL_RZERO;REAL_ARITH `&0 - -- a * b = a * b`] THEN ASM_SIMP_TAC[REAL_FIELD `~(x = &0) ==> &1 / x * x = &1`; VECTOR_MUL_ASSOC;VECTOR_MUL_LID] THEN CONJ_TAC THENL [ALL_TAC;ASM SET_TAC[]] THEN ASM_SIMP_TAC[SET_RULE `v IN s ==> (s DELETE v = {} <=> s = {v})`] THEN ASM_CASES_TAC `s = {v:real^N}` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM SUBST_ALL_TAC THEN FIND_ASSUM MP_TAC `sum {v:real^N} u = &0` THEN REWRITE_TAC[SUM_SING] THEN ASM_REWRITE_TAC[]);; let AFFINE_DEPENDENT_EXPLICIT_FINITE = prove (`!s. FINITE(s:real^N -> bool) ==> (affine_dependent s <=> ?u. sum s u = &0 /\ (?v. v IN s /\ ~(u v = &0)) /\ vsum s (\v. u v % v) = vec 0)`, REPEAT STRIP_TAC THEN REWRITE_TAC[AFFINE_DEPENDENT_EXPLICIT] THEN EQ_TAC THENL [ALL_TAC; ASM_MESON_TAC[SUBSET_REFL]] THEN DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` (X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC)) THEN EXISTS_TAC `\x:real^N. if x IN t then u(x) else &0` THEN REWRITE_TAC[COND_RAND; COND_RATOR; VECTOR_MUL_LZERO] THEN ASM_SIMP_TAC[GSYM SUM_RESTRICT_SET; GSYM VSUM_RESTRICT_SET] THEN ASM_SIMP_TAC[SET_RULE `t SUBSET s ==> {x | x IN s /\ x IN t} = t`] THEN ASM SET_TAC[]);; let AFFINE_DEPENDENT_TRANSLATION_EQ = prove (`!a s. affine_dependent (IMAGE (\x. a + x) s) <=> affine_dependent s`, REWRITE_TAC[affine_dependent] THEN GEOM_TRANSLATE_TAC[]);; add_translation_invariants [AFFINE_DEPENDENT_TRANSLATION_EQ];; let AFFINE_DEPENDENT_TRANSLATION = prove (`!s a. affine_dependent s ==> affine_dependent (IMAGE (\x. a + x) s)`, REWRITE_TAC[AFFINE_DEPENDENT_TRANSLATION_EQ]);; let AFFINE_DEPENDENT_LINEAR_IMAGE_EQ = prove (`!f:real^M->real^N s. linear f /\ (!x y. f x = f y ==> x = y) ==> (affine_dependent(IMAGE f s) <=> affine_dependent s)`, REWRITE_TAC[affine_dependent] THEN GEOM_TRANSFORM_TAC[]);; add_linear_invariants [AFFINE_DEPENDENT_LINEAR_IMAGE_EQ];; let AFFINE_DEPENDENT_LINEAR_IMAGE = prove (`!f:real^M->real^N s. linear f /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\ affine_dependent(s) ==> affine_dependent(IMAGE f s)`, REPEAT GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN REWRITE_TAC[affine_dependent; EXISTS_IN_IMAGE] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^M` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `IMAGE (f:real^M->real^N) s DELETE f a = IMAGE f (s DELETE a)` (fun t -> ASM_SIMP_TAC[FUN_IN_IMAGE; AFFINE_HULL_LINEAR_IMAGE; t]) THEN ASM SET_TAC[]);; let AFFINE_DEPENDENT_MONO = prove (`!s t:real^N->bool. affine_dependent s /\ s SUBSET t ==> affine_dependent t`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN REWRITE_TAC[affine_dependent] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:real^N` THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP HULL_MONO o SPEC `x:real^N` o MATCH_MP (SET_RULE `!x. s SUBSET t ==> (s DELETE x) SUBSET (t DELETE x)`)) THEN ASM SET_TAC[]);; let AFFINE_INDEPENDENT_EMPTY = prove (`~(affine_dependent {})`, REWRITE_TAC[affine_dependent; NOT_IN_EMPTY]);; let AFFINE_INDEPENDENT_1 = prove (`!a:real^N. ~(affine_dependent {a})`, REWRITE_TAC[affine_dependent; EXISTS_IN_INSERT; NOT_IN_EMPTY] THEN REWRITE_TAC[SET_RULE `{a} DELETE a = {}`; AFFINE_HULL_EMPTY; NOT_IN_EMPTY]);; let AFFINE_INDEPENDENT_2 = prove (`!a b:real^N. ~(affine_dependent {a,b})`, REPEAT GEN_TAC THEN ASM_CASES_TAC `b:real^N = a` THENL [ASM_REWRITE_TAC[INSERT_AC; AFFINE_INDEPENDENT_1]; REWRITE_TAC[affine_dependent; EXISTS_IN_INSERT; NOT_IN_EMPTY] THEN ASM_SIMP_TAC[SET_RULE `~(a = b) ==> {a,b} DELETE a = {b} /\ {a,b} DELETE b = {a}`] THEN ASM_REWRITE_TAC[AFFINE_HULL_SING; IN_SING]]);; let AFFINE_INDEPENDENT_SUBSET = prove (`!s t. ~affine_dependent t /\ s SUBSET t ==> ~affine_dependent s`, REWRITE_TAC[IMP_CONJ_ALT; CONTRAPOS_THM] THEN REWRITE_TAC[GSYM IMP_CONJ_ALT; AFFINE_DEPENDENT_MONO]);; let AFFINE_INDEPENDENT_DELETE = prove (`!s a. ~affine_dependent s ==> ~affine_dependent(s DELETE a)`, REPEAT GEN_TAC THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] AFFINE_INDEPENDENT_SUBSET) THEN SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* Coplanarity, and collinearity in terms of affine hull. *) (* ------------------------------------------------------------------------- *) let coplanar = new_definition `coplanar s <=> ?u v w. s SUBSET affine hull {u,v,w}`;; let COLLINEAR_AFFINE_HULL = prove (`!s:real^N->bool. collinear s <=> ?u v. s SUBSET affine hull {u,v}`, GEN_TAC THEN REWRITE_TAC[collinear; AFFINE_HULL_2] THEN EQ_TAC THEN REWRITE_TAC[SUBSET; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN REWRITE_TAC[REAL_ARITH `u + v = &1 <=> &1 - u = v`; UNWIND_THM1] THENL [X_GEN_TAC `u:real^N` THEN DISCH_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY] 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 DISCH_TAC THEN EXISTS_TAC `x + u:real^N` THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `y:real^N`]) THEN ASM_REWRITE_TAC[VECTOR_ARITH `x - y:real^N = z <=> x = y + z`] THEN DISCH_THEN(X_CHOOSE_THEN `c:real` SUBST1_TAC) THEN EXISTS_TAC `&1 + c` THEN VECTOR_ARITH_TAC; MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN DISCH_TAC THEN EXISTS_TAC `b - a:real^N` THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN FIRST_X_ASSUM(fun th -> MP_TAC(SPEC `y:real^N` th) THEN MP_TAC(SPEC `x:real^N` th)) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `r:real` THEN DISCH_THEN SUBST1_TAC THEN X_GEN_TAC `s:real` THEN DISCH_THEN SUBST1_TAC THEN EXISTS_TAC `s - r:real` THEN VECTOR_ARITH_TAC]);; let COLLINEAR_IMP_COPLANAR = prove (`!s. collinear s ==> coplanar s`, REWRITE_TAC[coplanar; COLLINEAR_AFFINE_HULL] THEN MESON_TAC[INSERT_AC]);; let COPLANAR_SMALL = prove (`!s. FINITE s /\ CARD s <= 3 ==> coplanar s`, GEN_TAC THEN REWRITE_TAC[ARITH_RULE `s <= 3 <=> s <= 2 \/ s = 3`] THEN REWRITE_TAC[LEFT_OR_DISTRIB; GSYM HAS_SIZE] THEN DISCH_THEN(DISJ_CASES_THEN MP_TAC) THEN SIMP_TAC[COLLINEAR_IMP_COPLANAR; COLLINEAR_SMALL] THEN CONV_TAC(LAND_CONV HAS_SIZE_CONV) THEN REWRITE_TAC[coplanar] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN STRIP_TAC THEN ASM_REWRITE_TAC[HULL_INC; SUBSET]);; let COPLANAR_EMPTY = prove (`coplanar {}`, SIMP_TAC[COLLINEAR_IMP_COPLANAR; COLLINEAR_EMPTY]);; let COPLANAR_SING = prove (`!a. coplanar {a}`, SIMP_TAC[COLLINEAR_IMP_COPLANAR; COLLINEAR_SING]);; let COPLANAR_2 = prove (`!a b. coplanar {a,b}`, SIMP_TAC[COLLINEAR_IMP_COPLANAR; COLLINEAR_2]);; let COPLANAR_3 = prove (`!a b c. coplanar {a,b,c}`, REPEAT GEN_TAC THEN MATCH_MP_TAC COPLANAR_SMALL THEN SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_RULES] THEN ARITH_TAC);; let COLLINEAR_AFFINE_HULL_COLLINEAR = prove (`!s. collinear(affine hull s) <=> collinear s`, REWRITE_TAC[COLLINEAR_AFFINE_HULL] THEN MESON_TAC[HULL_HULL; HULL_MONO; HULL_INC; SUBSET]);; let COPLANAR_AFFINE_HULL_COPLANAR = prove (`!s. coplanar(affine hull s) <=> coplanar s`, REWRITE_TAC[coplanar] THEN MESON_TAC[HULL_HULL; HULL_MONO; HULL_INC; SUBSET]);; let COPLANAR_TRANSLATION_EQ = prove (`!a:real^N s. coplanar(IMAGE (\x. a + x) s) <=> coplanar s`, REWRITE_TAC[coplanar] THEN GEOM_TRANSLATE_TAC[]);; let COPLANAR_TRANSLATION = prove (`!a:real^N s. coplanar s ==> coplanar(IMAGE (\x. a + x) s)`, REWRITE_TAC[COPLANAR_TRANSLATION_EQ]);; add_translation_invariants [COPLANAR_TRANSLATION_EQ];; let COPLANAR_LINEAR_IMAGE = prove (`!f:real^M->real^N s. coplanar s /\ linear f ==> coplanar(IMAGE f s)`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN REWRITE_TAC[coplanar; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^M`; `c:real^M`] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`(f:real^M->real^N) a`; `(f:real^M->real^N) b`; `(f:real^M->real^N) c`] THEN REWRITE_TAC[SET_RULE `{f a,f b,f c} = IMAGE f {a,b,c}`] THEN ASM_SIMP_TAC[AFFINE_HULL_LINEAR_IMAGE; IMAGE_SUBSET]);; let COPLANAR_LINEAR_IMAGE_EQ = prove (`!f s. linear f /\ (!x y. f x = f y ==> x = y) ==> (coplanar (IMAGE f s) <=> coplanar s)`, MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE COPLANAR_LINEAR_IMAGE));; add_linear_invariants [COPLANAR_LINEAR_IMAGE_EQ];; let COPLANAR_SUBSET = prove (`!s t. coplanar t /\ s SUBSET t ==> coplanar s`, REWRITE_TAC[coplanar] THEN SET_TAC[]);; let AFFINE_HULL_3_IMP_COLLINEAR = prove (`!a b c. c IN affine hull {a,b} ==> collinear {a,b,c}`, ONCE_REWRITE_TAC[GSYM COLLINEAR_AFFINE_HULL_COLLINEAR] THEN SIMP_TAC[HULL_REDUNDANT_EQ; INSERT_AC] THEN REWRITE_TAC[COLLINEAR_AFFINE_HULL_COLLINEAR; COLLINEAR_2]);; let COLLINEAR_3_AFFINE_HULL = prove (`!a b c:real^N. ~(a = b) ==> (collinear {a,b,c} <=> c IN affine hull {a,b})`, REPEAT STRIP_TAC THEN EQ_TAC THEN SIMP_TAC[AFFINE_HULL_3_IMP_COLLINEAR] THEN REWRITE_TAC[collinear] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(fun th -> MP_TAC(SPECL [`b:real^N`; `a:real^N`] th) THEN MP_TAC(SPECL [`c:real^N`; `a:real^N`] th)) THEN REWRITE_TAC[IN_INSERT; AFFINE_HULL_2; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN REWRITE_TAC[VECTOR_ARITH `a - b:real^N = c <=> a = b + c`] THEN X_GEN_TAC `x:real` THEN DISCH_TAC THEN X_GEN_TAC `y:real` THEN ASM_CASES_TAC `y = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_RID] THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN MAP_EVERY EXISTS_TAC [`&1 - x / y`; `x / y:real`] THEN CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[REAL_DIV_RMUL] THEN VECTOR_ARITH_TAC);; let COLLINEAR_3_EQ_AFFINE_DEPENDENT = prove (`!a b c:real^N. collinear{a,b,c} <=> a = b \/ a = c \/ b = c \/ affine_dependent {a,b,c}`, REPEAT GEN_TAC THEN MAP_EVERY (fun t -> ASM_CASES_TAC t THENL [ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC]) [`a:real^N = b`; `a:real^N = c`; `b:real^N = c`] THEN ASM_REWRITE_TAC[affine_dependent] THEN EQ_TAC THENL [ASM_SIMP_TAC[COLLINEAR_3_AFFINE_HULL] THEN DISCH_TAC THEN EXISTS_TAC `c:real^N` THEN REWRITE_TAC[IN_INSERT]; REWRITE_TAC[EXISTS_IN_INSERT; NOT_IN_EMPTY] THEN STRIP_TAC THENL [ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {b,c,a}`]; ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {c,a,b}`]; ALL_TAC] THEN ASM_SIMP_TAC[COLLINEAR_3_AFFINE_HULL]] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `x IN s ==> s SUBSET t ==> x IN t`)) THEN MATCH_MP_TAC HULL_MONO THEN ASM SET_TAC[]);; let AFFINE_DEPENDENT_IMP_COLLINEAR_3 = prove (`!a b c:real^N. affine_dependent {a,b,c} ==> collinear{a,b,c}`, REPEAT GEN_TAC THEN REWRITE_TAC[affine_dependent] THEN REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; RIGHT_OR_DISTRIB] THEN REWRITE_TAC[EXISTS_OR_THM; UNWIND_THM2; COLLINEAR_AFFINE_HULL] THEN STRIP_TAC THENL [MAP_EVERY EXISTS_TAC [`b:real^N`; `c:real^N`]; MAP_EVERY EXISTS_TAC [`a:real^N`; `c:real^N`]; MAP_EVERY EXISTS_TAC [`a:real^N`; `b:real^N`]] THEN SIMP_TAC[INSERT_SUBSET; EMPTY_SUBSET; HULL_INC; IN_INSERT] THEN POP_ASSUM MP_TAC THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> a IN s ==> a IN t`) THEN MATCH_MP_TAC HULL_MONO THEN SET_TAC[]);; let COLLINEAR_3_IN_AFFINE_HULL = prove (`!v0 v1 x:real^N. ~(v1 = v0) ==> (collinear {v0,v1,x} <=> x IN affine hull {v0,v1})`, REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `v0:real^N` THEN REWRITE_TAC[COLLINEAR_LEMMA; AFFINE_HULL_2] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID; IN_ELIM_THM] THEN ASM_CASES_TAC `x:real^N = vec 0` THEN ASM_REWRITE_TAC[] THENL [MAP_EVERY EXISTS_TAC [`&1`; `&0`] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN VECTOR_ARITH_TAC; MESON_TAC[REAL_ARITH `u + v = &1 <=> u = &1 - v`]]);; let COLLINEAR_3_EXPLICIT = prove (`!x y z:real^N. collinear {x,y,z} <=> ?a b c. a % x + b % y + c % z = vec 0 /\ a + b + c = &0 /\ ~(a = &0 /\ b = &0 /\ c = &0)`, MATCH_MP_TAC(MESON[] `(!x y z. P x y z ==> P y z x) /\ (!x z. P x x z) /\ (!x y z. ~(x = y) /\ ~(x = z) /\ ~(y = z) ==> P x y z) ==> !x y z. P x y z`) THEN CONJ_TAC THENL [REWRITE_TAC[INSERT_AC; REAL_ADD_AC; VECTOR_ADD_AC; CONJ_ACI] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MESON_TAC[REAL_ADD_AC; VECTOR_ADD_AC]; ALL_TAC] THEN CONJ_TAC THENL [REPEAT GEN_TAC THEN REWRITE_TAC[INSERT_AC; COLLINEAR_2] THEN MAP_EVERY EXISTS_TAC [`&1`; `-- &1`; `&0`] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN VECTOR_ARITH_TAC; ALL_TAC] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[COLLINEAR_3_EQ_AFFINE_DEPENDENT] THEN SIMP_TAC[AFFINE_DEPENDENT_EXPLICIT_FINITE; FINITE_INSERT; FINITE_EMPTY; SUM_CLAUSES; VSUM_CLAUSES; EXISTS_IN_INSERT; NOT_IN_EMPTY] THEN ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; VECTOR_ADD_RID; REAL_ADD_RID] THEN EQ_TAC THENL [MESON_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`a:real`; `b:real`; `c:real`] THEN STRIP_TAC THEN EXISTS_TAC `(\w. if w = x then a else if w = y then b else c):real^N->real` THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]);; (* ------------------------------------------------------------------------- *) (* A general lemma. *) (* ------------------------------------------------------------------------- *) let CONVEX_CONNECTED = prove (`!s:real^N->bool. convex s ==> connected s`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[CONNECTED_IFF_CONNECTABLE_POINTS] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN STRIP_TAC THEN EXISTS_TAC `segment[a:real^N,b]` THEN ASM_SIMP_TAC[CONNECTED_SEGMENT; ENDS_IN_SEGMENT; SEGMENT_SUBSET_CONVEX]);; (* ------------------------------------------------------------------------- *) (* Convex functions into the reals. *) (* ------------------------------------------------------------------------- *) parse_as_infix ("convex_on",(12,"right"));; let convex_on = new_definition `f convex_on s <=> !x y u v. x IN s /\ y IN s /\ &0 <= u /\ &0 <= v /\ (u + v = &1) ==> f(u % x + v % y) <= u * f(x) + v * f(y)`;; let CONVEX_ON_EMPTY = prove (`!f:real^N->real. f convex_on {}`, REWRITE_TAC[convex_on; NOT_IN_EMPTY]);; let CONVEX_ON_SUBSET = prove (`!f s t. f convex_on t /\ s SUBSET t ==> f convex_on s`, REWRITE_TAC[convex_on; SUBSET] THEN MESON_TAC[]);; let CONVEX_ON_EQ = prove (`!f g s. convex s /\ (!x. x IN s ==> f x = g x) /\ f convex_on s ==> g convex_on s`, REWRITE_TAC[convex_on; convex] THEN MESON_TAC[]);; let CONVEX_ON_CONST = prove (`!s a. (\x. a) convex_on s`, SIMP_TAC[convex_on; GSYM REAL_ADD_RDISTRIB; REAL_MUL_LID; REAL_LE_REFL]);; let LINEAR_IMP_CONVEX_ON = prove (`!f s:real^N->bool. linear (lift o f) ==> f convex_on s`, REWRITE_TAC[linear; convex_on] THEN SIMP_TAC[GSYM DROP_EQ; DROP_ADD; o_DEF; LIFT_DROP; DROP_CMUL] THEN REWRITE_TAC[REAL_LE_REFL]);; let CONVEX_ON_SING = prove (`!f a:real^N. f convex_on {a}`, REPEAT GEN_TAC THEN MATCH_MP_TAC CONVEX_ON_EQ THEN EXISTS_TAC `\x:real^N. (f:real^N->real) a` THEN SIMP_TAC[IN_SING; CONVEX_SING; CONVEX_ON_CONST]);; let CONVEX_ADD = prove (`!s f g. f convex_on s /\ g convex_on s ==> (\x. f(x) + g(x)) convex_on s`, REWRITE_TAC[convex_on; AND_FORALL_THM] THEN REPEAT(MATCH_MP_TAC MONO_FORALL ORELSE GEN_TAC) THEN MATCH_MP_TAC(TAUT `(b /\ c ==> d) ==> (a ==> b) /\ (a ==> c) ==> a ==> d`) THEN REAL_ARITH_TAC);; let CONVEX_ADD_EQ = prove (`!a f s:real^N->bool. (\x. a + f x) convex_on s <=> f convex_on s`, REPEAT STRIP_TAC THEN EQ_TAC THEN SIMP_TAC[CONVEX_ADD; CONVEX_ON_CONST] THEN DISCH_THEN(MP_TAC o ISPEC `(\x. --a):real^N->real` o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] CONVEX_ADD)) THEN REWRITE_TAC[CONVEX_ON_CONST; ETA_AX; REAL_ARITH `--a + a + x:real = x`]);; let CONVEX_CMUL = prove (`!s c f. &0 <= c /\ f convex_on s ==> (\x. c * f(x)) convex_on s`, SIMP_TAC[convex_on; REAL_LE_LMUL; REAL_ARITH `u * c * fx + v * c * fy = c * (u * fx + v * fy)`]);; let CONVEX_MAX = prove (`!f g s. f convex_on s /\ g convex_on s ==> (\x. max (f x) (g x)) convex_on s`, REWRITE_TAC[convex_on; REAL_MAX_LE] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(fun th -> W(MP_TAC o PART_MATCH (lhand o rand) th o lhand o snd)) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN MATCH_MP_TAC REAL_LE_ADD2 THEN CONJ_TAC THEN MATCH_MP_TAC REAL_LE_LMUL THEN ASM_REAL_ARITH_TAC);; let CONVEX_ON_SUM = prove (`!t f:A->real^N->real s. FINITE s /\ (!a. a IN s ==> f a convex_on t) ==> (\x. sum s (\a. f a x)) convex_on t`, GEN_TAC THEN GEN_TAC THEN REWRITE_TAC[IMP_CONJ] THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN SIMP_TAC[SUM_CLAUSES; CONVEX_ON_CONST; FORALL_IN_INSERT] THEN SIMP_TAC[CONVEX_ADD; ETA_AX]);; let CONVEX_ON_IMP_MIDPOINT_CONVEX = prove (`!f s x y:real^N. f convex_on s /\ x IN s /\ y IN s ==> f(midpoint(x,y)) <= (f x + f y) / &2`, REWRITE_TAC[convex_on] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[midpoint; VECTOR_ADD_LDISTRIB; REAL_ARITH `(x + y) / &2 = inv(&2) * x + inv(&2) * y`] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[]);; let CONVEX_LOWER = prove (`!f s x y:real^N u v. f convex_on s /\ x IN s /\ y IN s /\ &0 <= u /\ &0 <= v /\ (u + v = &1) ==> f(u % x + v % y) <= max (f(x)) (f(y))`, REWRITE_TAC[convex_on] THEN REPEAT STRIP_TAC THEN GEN_REWRITE_TAC RAND_CONV [GSYM REAL_MUL_LID] THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (RAND_CONV o LAND_CONV) [SYM th]) THEN REWRITE_TAC[REAL_ADD_RDISTRIB] THEN MATCH_MP_TAC REAL_LE_TRANS THEN ASM_MESON_TAC[REAL_LE_ADD2; REAL_LE_LMUL; REAL_MAX_MAX]);; let CONVEX_LOWER_SEGMENT = prove (`!f s a b x:real^N. f convex_on s /\ a IN s /\ b IN s /\ x IN segment[a,b] ==> f(x) <= max (f a) (f b)`, REWRITE_TAC[IN_SEGMENT] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONVEX_LOWER THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC);; let CONVEX_LOWER_SEGMENT_LT = prove (`!f s a b x:real^N. f convex_on s /\ a IN s /\ b IN s /\ x IN segment[a,b] /\ ~(x = b) /\ f a < f b ==> f x < f b`, REPEAT STRIP_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 o GSYM)) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a:real^N`; `b:real^N`; `&1 - u:real`; `u:real`] o GEN_REWRITE_RULE I [convex_on]) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LET_TRANS) THEN REWRITE_TAC[REAL_ARITH `a + u * b < b <=> a < (&1 - u) * b`] THEN MATCH_MP_TAC REAL_LT_LMUL THEN ASM_REWRITE_TAC[REAL_SUB_LT] THEN ASM_CASES_TAC `u = &1` THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN UNDISCH_TAC `~(x:real^N = b)` THEN EXPAND_TAC "x" THEN MATCH_MP_TAC(TAUT `~p ==> p ==> q`) THEN REWRITE_TAC[ASSUME `u = &1`] THEN VECTOR_ARITH_TAC);; let CONVEX_LOCAL_GLOBAL_MINIMUM_SEGMENT = prove (`!f s x:real^N. f convex_on s /\ x IN s /\ (!z. z IN s /\ ~(z = x) ==> ?y. y IN segment[x,z] /\ y IN s /\ ~(y = x) /\ f(x) <= f(y)) ==> !z. z IN s ==> f(x) <= f(z)`, REWRITE_TAC[IN_OPEN_SEGMENT] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `z:real^N`) THEN ASM_CASES_TAC `z:real^N = x` THEN ASM_REWRITE_TAC[REAL_LE_REFL] THEN DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`f:real^N->real`; `s:real^N->bool`; `z:real^N`; `x:real^N`; `y:real^N`] CONVEX_LOWER_SEGMENT_LT) THEN ASM_SIMP_TAC[REWRITE_RULE[SUBSET] SEGMENT_OPEN_SUBSET_CLOSED] THEN ONCE_REWRITE_TAC[SEGMENT_SYM] THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC);; let CONVEX_LOCAL_GLOBAL_MINIMUM_GEN = prove (`!f s t x:real^N. f convex_on s /\ x IN t /\ open_in (subtopology euclidean (affine hull s)) t /\ t SUBSET s /\ (!y. y IN t ==> f(x) <= f(y)) ==> !y. y IN s ==> f(x) <= f(y)`, REWRITE_TAC[open_in] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC CONVEX_LOCAL_GLOBAL_MINIMUM_SEGMENT THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN X_GEN_TAC `z:real^N` THEN STRIP_TAC THEN FIRST_X_ASSUM(fun th -> MP_TAC(SPEC `x:real^N` th) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC)) THEN EXISTS_TAC `x + min (&1) (d / &2 / norm(z - x:real^N)) % (z - x)` THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[IN_SEGMENT] THEN EXISTS_TAC `min (&1) (d / &2 / norm(z - x:real^N))` THEN REWRITE_TAC[REAL_MIN_LE; REAL_LE_MIN; REAL_LE_REFL; REAL_POS] THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; REAL_LE_DIV; REAL_POS; NORM_POS_LE] THEN VECTOR_ARITH_TAC; FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]); REWRITE_TAC[VECTOR_ARITH `x + a:real^N = x <=> a = vec 0`] THEN ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ] THEN MATCH_MP_TAC(REAL_ARITH `&0 < x ==> ~(min (&1) x = &0)`) THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_HALF; NORM_POS_LT; VECTOR_SUB_EQ]; FIRST_X_ASSUM MATCH_MP_TAC] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[SUBSET]) THEN ASM_SIMP_TAC[IN_AFFINE_ADD_MUL_DIFF; HULL_INC; AFFINE_AFFINE_HULL] THEN REWRITE_TAC[NORM_ARITH `dist(x + a:real^N,x) = norm a`] THEN ASM_SIMP_TAC[NORM_MUL; GSYM REAL_LT_RDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN MATCH_MP_TAC(REAL_ARITH `&0 < x /\ x < d ==> abs(min (&1) x) < d`) THEN MATCH_MP_TAC(REAL_ARITH `&0 < x / y ==> &0 < x / &2 / y /\ x / &2 / y < x / y`) THEN ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT; VECTOR_SUB_EQ]);; let CONVEX_LOCAL_GLOBAL_MINIMUM = prove (`!f s t x:real^N. f convex_on s /\ x IN t /\ open t /\ t SUBSET s /\ (!y. y IN t ==> f(x) <= f(y)) ==> !y. y IN s ==> f(x) <= f(y)`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC CONVEX_LOCAL_GLOBAL_MINIMUM_GEN THEN EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC OPEN_SUBSET THEN ASM_REWRITE_TAC[] THEN TRANS_TAC SUBSET_TRANS `s:real^N->bool` THEN ASM_REWRITE_TAC[HULL_SUBSET]);; let CONVEX_DISTANCE = prove (`!s a. (\x. dist(a,x)) convex_on s`, REWRITE_TAC[convex_on; dist] THEN REPEAT STRIP_TAC THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV o LAND_CONV) [GSYM VECTOR_MUL_LID] THEN FIRST_ASSUM(SUBST1_TAC o SYM) THEN REWRITE_TAC[VECTOR_ARITH `(u + v) % z - (u % x + v % y) = u % (z - x) + v % (z - y)`] THEN ASM_MESON_TAC[NORM_TRIANGLE; NORM_MUL; REAL_ABS_REFL]);; let CONVEX_NORM = prove (`!s:real^N->bool. norm convex_on s`, GEN_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `vec 0:real^N`] CONVEX_DISTANCE) THEN REWRITE_TAC[DIST_0; ETA_AX]);; let CONVEX_ON_COMPOSE_LINEAR = prove (`!f g:real^M->real^N s. f convex_on (IMAGE g s) /\ linear g ==> (f o g) convex_on s`, REWRITE_TAC[convex_on; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[FORALL_IN_IMAGE; o_THM] THEN REWRITE_TAC[RIGHT_IMP_FORALL_THM; IMP_IMP; GSYM CONJ_ASSOC] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP LINEAR_ADD th]) THEN FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP LINEAR_CMUL th]) THEN ASM_SIMP_TAC[]);; let CONVEX_ON_TRANSLATION = prove (`!f a:real^N. f convex_on (IMAGE (\x. a + x) s) <=> (\x. f(a + x)) convex_on s`, REWRITE_TAC[convex_on; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[FORALL_IN_IMAGE; o_THM] THEN REWRITE_TAC[VECTOR_ARITH `u % (a + x) + v % (a + y):real^N = (u + v) % a + u % x + v % y`] THEN SIMP_TAC[VECTOR_MUL_LID]);; let LINEAR_CONVEX_ON_1 = prove (`!f:real^N->real^1. linear f <=> f(vec 0) = vec 0 /\ (drop o f) convex_on UNIV /\ ((--) o drop o f) convex_on UNIV`, GEN_TAC THEN REWRITE_TAC[convex_on; IN_UNIV; o_THM] THEN REWRITE_TAC[AND_FORALL_THM; TAUT `(p ==> q) /\ (p ==> r) <=> p ==> q /\ r`; REAL_ARITH `--a <= u * --x + v * --y <=> u * x + v * y <= a`] THEN REWRITE_TAC[REAL_LE_ANTISYM] THEN REWRITE_TAC[GSYM DROP_CMUL; GSYM DROP_ADD; DROP_EQ] THEN EQ_TAC THENL [MESON_TAC[LINEAR_ADD; LINEAR_CMUL; LINEAR_0]; STRIP_TAC THEN REWRITE_TAC[linear]] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN FIRST_ASSUM(fun th -> MP_TAC(SPECL[`x:real^N`; `y:real^N`; `&1 / &2`; `&1 / &2`] th) THEN MP_TAC(SPECL[`x + y:real^N`; `vec 0:real^N`; `&1 / &2`; `&1 / &2`] th)) THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_RID] THEN REWRITE_TAC[VECTOR_ADD_LDISTRIB] THEN DISCH_THEN SUBST1_TAC THEN CONV_TAC VECTOR_ARITH; DISCH_TAC] THEN SUBGOAL_THEN `!c x:real^N. &0 <= c /\ c <= &1 ==> (f:real^N->real^1)(c % x) = c % f x` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `vec 0:real^N`; `c:real`; `&1 - c`]) THEN ASM_REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_RID] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `!c x:real^N. &0 <= c ==> (f:real^N->real^1)(c % x) = c % f x` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN ASM_CASES_TAC `c <= &1` THEN ASM_SIMP_TAC[] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`inv c:real`; `c % x:real^N`]) THEN SUBGOAL_THEN `&1 <= c /\ ~(c = &0)` STRIP_ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[REAL_INV_LE_1; REAL_LE_INV_EQ] THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; VECTOR_MUL_LID] THEN ASM_SIMP_TAC[REAL_MUL_RINV; VECTOR_MUL_LID]; ALL_TAC] THEN SUBGOAL_THEN `!x. (f:real^N->real^1) (--x) = --(f x)` ASSUME_TAC THENL [GEN_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `--x:real^N`; `inv(&2)`; `inv(&2)`]) THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[VECTOR_ARITH `a % x + a % --x:real^N = vec 0`] THEN ASM_REWRITE_TAC[] THEN CONV_TAC VECTOR_ARITH; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`c:real`; `x:real^N`] THEN ASM_CASES_TAC `&0 <= c` THEN ASM_SIMP_TAC[] THEN SUBGOAL_THEN `(f:real^N->real^1)(--c % x) = --c % f x` MP_TAC THENL [ASM_SIMP_TAC[REAL_ARITH `~(&0 <= c) ==> &0 <= --c`]; ASM_REWRITE_TAC[VECTOR_MUL_LNEG; VECTOR_EQ_NEG2]]);; let CONVEX_CONCAVE_EQ_AFFINE = prove (`!f:real^N->real. f convex_on UNIV /\ ((--) o f) convex_on UNIV <=> (?a b. f = \x. a dot x + b)`, GEN_TAC THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THENL [MP_TAC(ISPEC `\x. lift(--f(vec 0) + (f:real^N->real) x)` LINEAR_CONVEX_ON_1) THEN REWRITE_TAC[o_DEF; LIFT_DROP; REAL_ADD_LINV; LIFT_NUM] THEN RULE_ASSUM_TAC(REWRITE_RULE[o_DEF]) THEN ASM_SIMP_TAC[CONVEX_ADD; CONVEX_ON_CONST; REAL_NEG_ADD] THEN REWRITE_TAC[LINEAR_TO_1] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN REWRITE_TAC[FUN_EQ_THM; GSYM DROP_EQ; LIFT_DROP] THEN DISCH_THEN(fun th -> REWRITE_TAC[GSYM th]) THEN EXISTS_TAC `(f:real^N->real) (vec 0)` THEN REAL_ARITH_TAC; REWRITE_TAC[o_DEF; REAL_NEG_ADD; GSYM DOT_LNEG] THEN CONJ_TAC THEN MATCH_MP_TAC CONVEX_ADD THEN REWRITE_TAC[CONVEX_ON_CONST] THEN MATCH_MP_TAC LINEAR_IMP_CONVEX_ON THEN REWRITE_TAC[o_DEF; LINEAR_LIFT_DOT]]);; (* ------------------------------------------------------------------------- *) (* Open and closed balls are convex and hence connected. *) (* ------------------------------------------------------------------------- *) let CONVEX_BALL = prove (`!x:real^N e. convex(ball(x,e))`, let lemma = REWRITE_RULE[convex_on; IN_UNIV] (ISPEC `(:real^N)` CONVEX_DISTANCE) in REWRITE_TAC[convex; IN_BALL] THEN REPEAT STRIP_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) lemma o lhand o snd) THEN ASM_MESON_TAC[REAL_LET_TRANS; REAL_CONVEX_BOUND_LT]);; let CONNECTED_BALL = prove (`!x:real^N e. connected(ball(x,e))`, SIMP_TAC[CONVEX_CONNECTED; CONVEX_BALL]);; let CONVEX_CBALL = prove (`!x:real^N e. convex(cball(x,e))`, REWRITE_TAC[convex; IN_CBALL; dist] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `e:real`; `y:real^N`; `z:real^N`; `u:real`; `v:real`] THEN STRIP_TAC THEN ONCE_REWRITE_TAC[VECTOR_ARITH `a - b = &1 % a - b`] THEN FIRST_ASSUM(SUBST1_TAC o SYM) THEN REWRITE_TAC[VECTOR_ARITH `(a + b) % x - (a % y + b % z) = a % (x - y) + b % (x - z)`] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `norm(u % (x - y)) + norm(v % (x - z):real^N)` THEN REWRITE_TAC[NORM_TRIANGLE; NORM_MUL] THEN MATCH_MP_TAC REAL_CONVEX_BOUND_LE THEN ASM_REWRITE_TAC[REAL_ABS_POS] THEN ASM_SIMP_TAC[REAL_ARITH `&0 <= u /\ &0 <= v /\ (u + v = &1) ==> (abs(u) + abs(v) = &1)`]);; let CONNECTED_CBALL = prove (`!x:real^N e. connected(cball(x,e))`, SIMP_TAC[CONVEX_CONNECTED; CONVEX_CBALL]);; let CONVEX_INTERMEDIATE_BALL = prove (`!a:real^N r t. ball(a,r) SUBSET t /\ t SUBSET cball(a,r) ==> convex t`, REPEAT STRIP_TAC THEN REWRITE_TAC[CONVEX_CONTAINS_OPEN_SEGMENT] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS)) THEN REWRITE_TAC[SUBSET; IN_BALL; IN_CBALL] THEN GEN_TAC THEN DISCH_THEN (MP_TAC o SPEC `a:real^N` o MATCH_MP DIST_DECREASES_OPEN_SEGMENT) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN REWRITE_TAC[IN_CBALL] THEN ASM_MESON_TAC[REAL_LTE_TRANS]);; let FRONTIER_OF_CONNECTED_COMPONENT_SUBSET = prove (`!s x:real^N. frontier(connected_component s x) SUBSET frontier s`, REPEAT GEN_TAC THEN REWRITE_TAC[frontier; SUBSET; IN_DIFF] THEN X_GEN_TAC `y:real^N` THEN REPEAT STRIP_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `y IN s ==> s SUBSET t ==> y IN t`)) THEN MATCH_MP_TAC SUBSET_CLOSURE THEN REWRITE_TAC[CONNECTED_COMPONENT_SUBSET]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERIOR]) THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `ball(y:real^N,e) SUBSET connected_component s y` ASSUME_TAC THENL [MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL THEN ASM_REWRITE_TAC[CONNECTED_BALL; CENTRE_IN_BALL]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CLOSURE_APPROACHABLE]) THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] (GSYM IN_BALL)] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o check (is_neg o concl)) THEN REWRITE_TAC[IN_INTERIOR] THEN EXISTS_TAC `e:real` THEN MP_TAC(ISPECL [`s:real^N->bool`; `x:real^N`; `y:real^N`] CONNECTED_COMPONENT_OVERLAP) THEN MATCH_MP_TAC(TAUT `p /\ (q ==> r) ==> (p <=> q) ==> r`) THEN ASM_SIMP_TAC[] THEN ASM SET_TAC[]]]);; let FRONTIER_OF_COMPONENTS_SUBSET = prove (`!s c:real^N->bool. c IN components s ==> frontier c SUBSET frontier s`, SIMP_TAC[components; FORALL_IN_GSPEC; FRONTIER_OF_CONNECTED_COMPONENT_SUBSET]);; let FRONTIER_OF_COMPONENTS_CLOSED_COMPLEMENT = prove (`!s c. closed s /\ c IN components ((:real^N) DIFF s) ==> frontier c SUBSET s`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP FRONTIER_OF_COMPONENTS_SUBSET) THEN REWRITE_TAC[FRONTIER_COMPLEMENT] THEN ASM_MESON_TAC[FRONTIER_SUBSET_EQ; SUBSET_TRANS]);; let CONTAINS_COMPONENT_OF_COMPACT_FRONTIER = prove (`!s:real^N->bool c. compact s /\ c IN components s ==> ?d. d IN components(frontier s) /\ d SUBSET c`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `~(UNIONS(components(frontier s)) INTER c:real^N->bool = {})` MP_TAC THENL [REWRITE_TAC[GSYM UNIONS_COMPONENTS] THEN FIRST_ASSUM(MP_TAC o MATCH_MP FRONTIER_OF_COMPONENTS_SUBSET) THEN MATCH_MP_TAC(SET_RULE `f SUBSET c /\ ~(f = {}) ==> f SUBSET s ==> ~(s INTER c = {})`) THEN REWRITE_TAC[FRONTIER_SUBSET_EQ; FRONTIER_EQ_EMPTY; DE_MORGAN_THM] THEN MP_TAC(ISPECL [`s:real^N->bool`; `c:real^N->bool`] COMPACT_COMPONENTS) THEN ASM_REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THENL [ASM_MESON_TAC[IN_COMPONENTS_NONEMPTY]; ASM_MESON_TAC[NOT_BOUNDED_UNIV]]; REWRITE_TAC[INTER_UNIONS; EMPTY_UNIONS; FORALL_IN_GSPEC] THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC COMPONENTS_MAXIMAL THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[INTER_COMM] THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]; ALL_TAC] THEN TRANS_TAC SUBSET_TRANS `frontier s:real^N->bool` THEN ASM_SIMP_TAC[FRONTIER_SUBSET_EQ; COMPACT_IMP_CLOSED] THEN ASM_MESON_TAC[IN_COMPONENTS_SUBSET]]);; let CARD_LE_COMPONENTS_FRONTIER = prove (`!s:real^N->bool. compact s ==> components s <=_c components(frontier s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CARD_LE_RELATIONAL_FULL THEN EXISTS_TAC `\s t:real^N->bool. s SUBSET t` THEN ASM_SIMP_TAC[CONTAINS_COMPONENT_OF_COMPACT_FRONTIER] THEN MAP_EVERY X_GEN_TAC [`c:real^N->bool`; `d:real^N->bool`; `e:real^N->bool`] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN DISCH_THEN(SUBST1_TAC o MATCH_MP COMPONENTS_EQ) THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_NONEMPTY) THEN SET_TAC[]);; let CONTAINS_COMPONENT_OF_CLOSURE_FRONTIER = prove (`!s:real^N->bool c. bounded s /\ c IN components(closure s) ==> ?d. d IN components(frontier s) /\ d SUBSET c`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`closure s:real^N->bool`; `c:real^N->bool`] CONTAINS_COMPONENT_OF_COMPACT_FRONTIER) THEN ASM_REWRITE_TAC[COMPACT_CLOSURE] THEN DISCH_TAC THEN SUBGOAL_THEN `~(UNIONS(components(frontier s)) INTER c:real^N->bool = {})` MP_TAC THENL [REWRITE_TAC[GSYM UNIONS_COMPONENTS] THEN MP_TAC(ISPEC `s:real^N->bool` FRONTIER_CLOSURE_SUBSET) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `(?d. P d) ==> (!d. P d ==> ~(d = {}) /\ d SUBSET f /\ d SUBSET c) ==> f SUBSET g ==> ~(g INTER c = {})`)) THEN MESON_TAC[IN_COMPONENTS_SUBSET; IN_COMPONENTS_NONEMPTY]; REWRITE_TAC[INTER_UNIONS; EMPTY_UNIONS; FORALL_IN_GSPEC] THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC COMPONENTS_MAXIMAL THEN EXISTS_TAC `closure s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[INTER_COMM] THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]; ALL_TAC] THEN TRANS_TAC SUBSET_TRANS `frontier s:real^N->bool` THEN CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_SUBSET]; REWRITE_TAC[frontier] THEN SET_TAC[]]]);; let CARD_LE_COMPONENTS_CLOSURE_FRONTIER = prove (`!s:real^N->bool. bounded s ==> components(closure s) <=_c components(frontier s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CARD_LE_RELATIONAL_FULL THEN EXISTS_TAC `\s t:real^N->bool. s SUBSET t` THEN ASM_SIMP_TAC[CONTAINS_COMPONENT_OF_CLOSURE_FRONTIER] THEN MAP_EVERY X_GEN_TAC [`c:real^N->bool`; `d:real^N->bool`; `e:real^N->bool`] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN DISCH_THEN(SUBST1_TAC o MATCH_MP COMPONENTS_EQ) THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_NONEMPTY) THEN SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* A couple of lemmas about components (see Newman IV, 3.3 and 3.4). *) (* ------------------------------------------------------------------------- *) let CONNECTED_UNION_CLOPEN_IN_COMPLEMENT = prove (`!s t u:real^N->bool. connected s /\ connected u /\ s SUBSET u /\ open_in (subtopology euclidean (u DIFF s)) t /\ closed_in (subtopology euclidean (u DIFF s)) t ==> connected (s UNION t)`, MAP_EVERY X_GEN_TAC [`c:real^N->bool`; `h:real^N->bool`; `s:real^N->bool`] THEN STRIP_TAC THEN REWRITE_TAC[CONNECTED_CLOSED_IN_EQ; NOT_EXISTS_THM] THEN MATCH_MP_TAC(MESON[] `!Q. (!x y. P x y <=> P y x) /\ (!x y. P x y ==> Q x \/ Q y) /\ (!x y. P x y /\ Q x ==> F) ==> (!x y. ~(P x y))`) THEN EXISTS_TAC `\x:real^N->bool. c SUBSET x` THEN CONJ_TAC THENL [MESON_TAC[INTER_COMM; UNION_COMM]; ALL_TAC] THEN REWRITE_TAC[] THEN CONJ_TAC THEN MAP_EVERY X_GEN_TAC [`h1:real^N->bool`; `h2:real^N->bool`] THENL [STRIP_TAC THEN UNDISCH_TAC `connected(c:real^N->bool)` THEN REWRITE_TAC[CONNECTED_CLOSED_IN; NOT_EXISTS_THM] THEN DISCH_THEN(MP_TAC o SPECL [`c INTER h1:real^N->bool`; `c INTER h2:real^N->bool`]) THEN MATCH_MP_TAC(TAUT `(p /\ q) /\ (~r ==> s) ==> ~(p /\ q /\ r) ==> s`) THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN CONJ_TAC THENL [UNDISCH_TAC `closed_in(subtopology euclidean (c UNION h)) (h1:real^N->bool)`; UNDISCH_TAC `closed_in(subtopology euclidean (c UNION h)) (h2:real^N->bool)`] THEN REWRITE_TAC[CLOSED_IN_CLOSED] THEN MATCH_MP_TAC MONO_EXISTS THEN ASM SET_TAC[]; STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o CONJUNCT1 o GEN_REWRITE_RULE I [open_in]) THEN SUBGOAL_THEN `(h2:real^N->bool) SUBSET h` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN UNDISCH_TAC `connected(s:real^N->bool)` THEN REWRITE_TAC[CONNECTED_CLOPEN] THEN DISCH_THEN(MP_TAC o SPEC `h2:real^N->bool`) THEN REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN SUBGOAL_THEN `s:real^N->bool = (s DIFF c) UNION (c UNION h)` SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_SUBTOPOLOGY_UNION THEN MATCH_MP_TAC(TAUT `q /\ (q ==> p) ==> p /\ q`) THEN CONJ_TAC THENL [REWRITE_TAC[OPEN_IN_CLOSED_IN_EQ; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(c UNION h) DIFF h2:real^N->bool = h1` (fun th -> ASM_REWRITE_TAC[th]) THEN ASM SET_TAC[]; DISCH_TAC THEN MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `h:real^N->bool` THEN ASM_REWRITE_TAC[] THEN UNDISCH_TAC `open_in(subtopology euclidean (c UNION h)) (h2:real^N->bool)` THEN REWRITE_TAC[OPEN_IN_OPEN] THEN MATCH_MP_TAC MONO_EXISTS THEN ASM SET_TAC[]]; MATCH_MP_TAC CLOSED_IN_SUBTOPOLOGY_UNION THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CLOSED_IN_TRANS THEN EXISTS_TAC `h:real^N->bool` THEN ASM_REWRITE_TAC[] THEN UNDISCH_TAC `closed_in(subtopology euclidean (c UNION h)) (h2:real^N->bool)` THEN REWRITE_TAC[CLOSED_IN_CLOSED] THEN MATCH_MP_TAC MONO_EXISTS THEN ASM SET_TAC[]]]);; let COMPONENT_COMPLEMENT_CONNECTED = prove (`!s u c:real^N->bool. connected s /\ connected u /\ s SUBSET u /\ c IN components (u DIFF s) ==> connected(u DIFF c)`, MAP_EVERY X_GEN_TAC [`a:real^N->bool`; `s:real^N->bool`; `c:real^N->bool`] THEN STRIP_TAC THEN UNDISCH_TAC `connected(a:real^N->bool)` THEN REWRITE_TAC[CONNECTED_CLOSED_IN_EQ; NOT_EXISTS_THM] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`h3:real^N->bool`; `h4:real^N->bool`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a INTER h3:real^N->bool`; `a INTER h4:real^N->bool`]) THEN FIRST_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_NONEMPTY) THEN FIRST_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN EVERY_ASSUM(fun th -> try MP_TAC(CONJUNCT1(GEN_REWRITE_RULE I [closed_in] th)) with Failure _ -> ALL_TAC) THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN REPEAT DISCH_TAC THEN REPEAT CONJ_TAC THENL [UNDISCH_TAC `closed_in (subtopology euclidean (s DIFF c)) (h3:real^N->bool)` THEN REWRITE_TAC[CLOSED_IN_CLOSED] THEN MATCH_MP_TAC MONO_EXISTS THEN ASM SET_TAC[]; UNDISCH_TAC `closed_in (subtopology euclidean (s DIFF c)) (h4:real^N->bool)` THEN REWRITE_TAC[CLOSED_IN_CLOSED] THEN MATCH_MP_TAC MONO_EXISTS THEN ASM SET_TAC[]; ASM SET_TAC[]; ASM SET_TAC[]; DISCH_TAC THEN MP_TAC(ISPECL [`s DIFF a:real^N->bool`; `c UNION h3:real^N->bool`; `c:real^N->bool`] COMPONENTS_MAXIMAL) THEN ASM_REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CONNECTED_UNION_CLOPEN_IN_COMPLEMENT THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]; ASM SET_TAC[]; REWRITE_TAC[OPEN_IN_CLOSED_IN_EQ; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `s DIFF c DIFF h3:real^N->bool = h4` SUBST1_TAC THEN ASM SET_TAC[]]; DISCH_TAC THEN MP_TAC(ISPECL [`s DIFF a:real^N->bool`; `c UNION h4:real^N->bool`; `c:real^N->bool`] COMPONENTS_MAXIMAL) THEN ASM_REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CONNECTED_UNION_CLOPEN_IN_COMPLEMENT THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]; ASM SET_TAC[]; REWRITE_TAC[OPEN_IN_CLOSED_IN_EQ; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `s DIFF c DIFF h4:real^N->bool = h3` SUBST1_TAC THEN ASM SET_TAC[]]]);; (* ------------------------------------------------------------------------- *) (* Condition for an open map's image to contain a ball. *) (* ------------------------------------------------------------------------- *) let BALL_SUBSET_OPEN_MAP_IMAGE = prove (`!f:real^M->real^N s a r. bounded s /\ f continuous_on closure s /\ open(IMAGE f (interior s)) /\ a IN s /\ &0 < r /\ (!z. z IN frontier s ==> r <= norm(f z - f a)) ==> ball(f(a),r) SUBSET IMAGE f s`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`ball((f:real^M->real^N) a,r)`; `(:real^N) DIFF IMAGE (f:real^M->real^N) s`] CONNECTED_INTER_FRONTIER) THEN REWRITE_TAC[CONNECTED_BALL] THEN MATCH_MP_TAC(SET_RULE `~(b INTER s = {}) /\ b INTER f = {} ==> (~(b INTER (UNIV DIFF s) = {}) /\ ~(b DIFF (UNIV DIFF s) = {}) ==> ~(b INTER f = {})) ==> b SUBSET s`) THEN REWRITE_TAC[FRONTIER_COMPLEMENT] THEN CONJ_TAC THENL [REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN EXISTS_TAC `(f:real^M->real^N) a` THEN ASM_REWRITE_TAC[CENTRE_IN_BALL] THEN ASM SET_TAC[]; REWRITE_TAC[SET_RULE `s INTER t = {} <=> !x. x IN t ==> ~(x IN s)`] THEN REWRITE_TAC[IN_BALL; REAL_NOT_LT]] THEN MP_TAC(ISPECL[`frontier(IMAGE (f:real^M->real^N) s)`; `(f:real^M->real^N) a`] DISTANCE_ATTAINS_INF) THEN REWRITE_TAC[FRONTIER_CLOSED; FRONTIER_EQ_EMPTY] THEN ANTS_TAC THENL [SIMP_TAC[DE_MORGAN_THM] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(MESON[NOT_BOUNDED_UNIV] `bounded s ==> ~(s = UNIV)`) THEN MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `IMAGE (f:real^M->real^N) (closure s)` THEN SIMP_TAC[IMAGE_SUBSET; CLOSURE_SUBSET] THEN MATCH_MP_TAC COMPACT_IMP_BOUNDED THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[COMPACT_CLOSURE]; DISCH_THEN(X_CHOOSE_THEN `w:real^N` STRIP_ASSUME_TAC)] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [frontier]) THEN REWRITE_TAC[IN_DIFF] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN REWRITE_TAC[CLOSURE_SEQUENTIAL] THEN DISCH_THEN(X_CHOOSE_THEN `y:num->real^N` (CONJUNCTS_THEN2 MP_TAC ASSUME_TAC)) THEN REWRITE_TAC[IN_IMAGE; SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `z:num->real^M` THEN REWRITE_TAC[FORALL_AND_THM] THEN ONCE_REWRITE_TAC[GSYM FUN_EQ_THM] THEN DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM COMPACT_CLOSURE]) THEN REWRITE_TAC[compact] THEN DISCH_THEN(MP_TAC o SPEC `z:num->real^M`) THEN ASM_SIMP_TAC[REWRITE_RULE[SUBSET] CLOSURE_SUBSET; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`y:real^M`; `r:num->num`] THEN STRIP_TAC THEN SUBGOAL_THEN `(((\n. (f:real^M->real^N)(z n)) o (r:num->num)) --> w) sequentially` MP_TAC THENL [MATCH_MP_TAC LIM_SUBSEQUENCE THEN ASM_REWRITE_TAC[]; ONCE_REWRITE_TAC[GSYM o_DEF] THEN REWRITE_TAC[GSYM o_ASSOC]] THEN DISCH_TAC THEN SUBGOAL_THEN `!n. ((z:num->real^M) o (r:num->num)) n IN s` MP_TAC THENL [ASM_REWRITE_TAC[o_THM]; UNDISCH_THEN `((\n. (f:real^M->real^N) ((z:num->real^M) n)) --> w) sequentially` (K ALL_TAC) THEN UNDISCH_THEN `!n. (z:num->real^M) n IN s` (K ALL_TAC)] THEN POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN SPEC_TAC(`(z:num->real^M) o (r:num->num)`, `z:num->real^M`) THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `w = (f:real^M->real^N) y` SUBST_ALL_TAC THENL [MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THEN EXISTS_TAC `(f:real^M->real^N) o (z:num->real^M)` THEN ASM_REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN ASM_MESON_TAC[CONTINUOUS_ON_CLOSURE_SEQUENTIALLY]; ALL_TAC] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `norm(f y - (f:real^M->real^N) a)` THEN CONJ_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC; ASM_MESON_TAC[dist; NORM_SUB]] THEN ASM_REWRITE_TAC[frontier; IN_DIFF] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o check (is_neg o concl)) THEN REWRITE_TAC[interior; IN_ELIM_THM] THEN EXISTS_TAC `IMAGE (f:real^M->real^N) (interior s)` THEN ASM_SIMP_TAC[IMAGE_SUBSET; INTERIOR_SUBSET] THEN ASM SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* Arithmetic operations on sets preserve convexity. *) (* ------------------------------------------------------------------------- *) let CONVEX_SCALING = prove (`!s c. convex s ==> convex (IMAGE (\x. c % x) s)`, REWRITE_TAC[convex; IN_IMAGE] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[VECTOR_ARITH `u % c % x + v % c % y = c % (u % x + v % y)`] THEN ASM_MESON_TAC[]);; let CONVEX_SCALING_EQ = prove (`!s:real^N->bool c. convex (IMAGE (\x. c % x) s) <=> c = &0 \/ convex 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[CONVEX_SING; CONVEX_EMPTY]; EQ_TAC THEN REWRITE_TAC[CONVEX_SCALING] THEN DISCH_THEN(MP_TAC o SPEC `inv(c):real` o MATCH_MP CONVEX_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 CONVEX_NEGATIONS = prove (`!s. convex s ==> convex (IMAGE (--) s)`, REWRITE_TAC[convex; IN_IMAGE] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[VECTOR_ARITH `u % --x + v % --y = --(u % x + v % y)`] THEN ASM_MESON_TAC[]);; let CONVEX_SUMS = prove (`!s t. convex s /\ convex t ==> convex {x + y | x IN s /\ y IN t}`, REWRITE_TAC[convex; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[VECTOR_ARITH `u % (a + b) + v % (c + d) = (u % a + v % c) + (u % b + v % d)`] THEN ASM_MESON_TAC[]);; let CONVEX_DIFFERENCES = prove (`!s t. convex s /\ convex t ==> convex {x - y | x IN s /\ y IN t}`, REWRITE_TAC[convex; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[VECTOR_ARITH `u % (a - b) + v % (c - d) = (u % a + v % c) - (u % b + v % d)`] THEN ASM_MESON_TAC[]);; let CONVEX_AFFINITY_EQ = prove (`!s m c:real^N. convex (IMAGE (\x. m % x + c) s) <=> m = &0 \/ convex s`, REWRITE_TAC[AFFINITY_SCALING_TRANSLATION; CONVEX_TRANSLATION_EQ; CONVEX_SCALING_EQ; IMAGE_o]);; let CONVEX_AFFINITY = prove (`!s m c:real^N. convex s ==> convex (IMAGE (\x. m % x + c) s)`, SIMP_TAC[CONVEX_AFFINITY_EQ]);; let CONVEX_LINEAR_PREIMAGE = prove (`!f:real^M->real^N. linear f /\ convex s ==> convex {x | f(x) IN s}`, REWRITE_TAC[CONVEX_ALT; IN_ELIM_THM] THEN SIMP_TAC[LINEAR_ADD; LINEAR_CMUL]);; let CONVEX_SUMS_MULTIPLES = prove (`!s:real^N->bool c d. convex s /\ &0 <= c /\ &0 <= d ==> {c % x + d % y | x IN s /\ y IN s} = IMAGE (\x. (c + d) % x) s`, REPEAT STRIP_TAC THEN REWRITE_TAC[VECTOR_ADD_RDISTRIB] THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN SUBGOAL_THEN `c = &0 /\ d = &0 \/ &0 < c + d` STRIP_ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_LID] THEN SET_TAC[]; REWRITE_TAC[SUBSET; FORALL_IN_GSPEC; GSYM VECTOR_ADD_RDISTRIB] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `c / (c + d) % x + d / (c + d) % y:real^N` THEN CONJ_TAC THENL [REWRITE_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[REAL_DIV_LMUL; REAL_LT_IMP_NZ]; FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [convex]) THEN ASM_SIMP_TAC[REAL_LE_DIV; REAL_LT_IMP_LE] THEN UNDISCH_TAC `&0 < c + d` THEN CONV_TAC REAL_FIELD]]);; let CONVEX_TRANSLATION_SUBSET_PREIMAGE = prove (`!s t:real^N->bool. convex t ==> convex {a | IMAGE (\x. a + x) s SUBSET t}`, REPEAT GEN_TAC THEN REWRITE_TAC[CONVEX_ALT] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`; `u:real`] THEN REWRITE_TAC[SUBSET; IN_ELIM_THM; FORALL_IN_IMAGE] THEN STRIP_TAC THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN ONCE_REWRITE_TAC[VECTOR_ARITH `((&1 - u) % a + u % b) + x:real^N = (&1 - u) % (a + x) + u % (b + x)`] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[]);; let CONVEX_TRANSLATION_SUPERSET_PREIMAGE = prove (`!s t:real^N->bool. convex t ==> convex {a | s SUBSET IMAGE (\x. a + x) t}`, REWRITE_TAC[TRANSLATION_SUBSET_GALOIS_RIGHT] THEN ASM_SIMP_TAC[VECTOR_NEG_NEG; CONVEX_NEGATIONS; CONVEX_TRANSLATION_SUBSET_PREIMAGE; SET_RULE `(!x:real^N. --(--x) = x) ==> {a:real^N | P(--a)} = IMAGE (--) {a | P a}`]);; (* ------------------------------------------------------------------------- *) (* Some interesting "cancellation" properties for sum-sets. *) (* ------------------------------------------------------------------------- *) let SUBSET_SUMS_LCANCEL = prove (`!s t u:real^N->bool. ~(s = {}) /\ bounded s /\ closed u /\ convex u /\ {x + y | x IN s /\ y IN t} SUBSET {x + z | x IN s /\ z IN u} ==> t SUBSET u`, REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN REWRITE_TAC[IN_ELIM_THM] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN X_GEN_TAC `b:real^N` THEN DISCH_TAC THEN SUBGOAL_THEN `!n. ?w z:real^N. w IN s /\ z IN u /\ (&n + &1) % (b - z) = w - a` MP_TAC THENL [INDUCT_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPECL [`a:real^N`; `b:real^N`]) THEN ASM_REWRITE_TAC[REAL_ADD_LID; VECTOR_MUL_LID] THEN REWRITE_TAC[VECTOR_ARITH `b - z:real^N = w - a <=> a + b = w + z`] THEN MESON_TAC[]; FIRST_X_ASSUM(X_CHOOSE_THEN `a':real^N` (X_CHOOSE_THEN `c':real^N` STRIP_ASSUME_TAC)) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a':real^N`; `b:real^N`]) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a'':real^N`; `c'':real^N`] THEN STRIP_TAC THEN EXISTS_TAC `a'':real^N` THEN EXISTS_TAC `(&1 - &1 / (&n + &2)) % c' + &1 / (&n + &2) % c'':real^N` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[CONVEX_ALT]) THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_LE_LDIV_EQ; REAL_ARITH `&0 < &n + &2`] THEN REAL_ARITH_TAC; FIRST_X_ASSUM(SUBST1_TAC o GEN_REWRITE_RULE I [VECTOR_ARITH `a' + b:real^N = a'' + c <=> a'' = (a' + b) - c`]) THEN REWRITE_TAC[VECTOR_ARITH `(&n + &1) % (b - c):real^N = (a' + b) - c'' - a <=> &n % b - (&n + &1) % c = (a' - c'') - a`] THEN SIMP_TAC[GSYM REAL_OF_NUM_SUC; VECTOR_MUL_ASSOC; VECTOR_ADD_LDISTRIB; REAL_ARITH `(n + &1) + &1 = n + &2`] THEN REWRITE_TAC[VECTOR_MUL_LID; REAL_FIELD `(&n + &2) * (&1 - (&1 / (&n + &2))) = &n + &1 /\ (&n + &2) * &1 / (&n + &2) = &1`] THEN ASM_REWRITE_TAC[VECTOR_ARITH `n % b - (n % c + d):real^N = n % (b - c) - d`] THEN CONV_TAC VECTOR_ARITH]]; FIRST_X_ASSUM(K ALL_TAC o check is_forall o concl) THEN MP_TAC(ISPECL [`s:real^N->bool`; `s:real^N->bool`] BOUNDED_DIFFS) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[BOUNDED_POS; FORALL_IN_GSPEC] THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN DISCH_TAC THEN FIRST_X_ASSUM(fun th -> ONCE_REWRITE_TAC[GSYM(MATCH_MP CLOSED_APPROACHABLE th)]) THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(SPEC `e:real` REAL_ARCH) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `B:real`) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MATCH_MP_TAC num_INDUCTION THEN REWRITE_TAC[REAL_MUL_LZERO] THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; X_GEN_TAC `n:num`] THEN DISCH_THEN(K ALL_TAC) THEN REWRITE_TAC[GSYM REAL_OF_NUM_SUC] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `n:num`) THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^N` THEN DISCH_THEN(X_CHOOSE_THEN `d:real^N` STRIP_ASSUME_TAC) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LT_LCANCEL_IMP THEN EXISTS_TAC `abs(&n + &1)` THEN ONCE_REWRITE_TAC[DIST_SYM] THEN CONJ_TAC THENL [REAL_ARITH_TAC; REWRITE_TAC[dist]] THEN ASM_REWRITE_TAC[GSYM NORM_MUL] THEN REWRITE_TAC[REAL_ARITH `abs(&n + &1) = &n + &1`] THEN ASM_MESON_TAC[REAL_LET_TRANS]]);; let SUBSET_SUMS_RCANCEL = prove (`!s t u:real^N->bool. closed t /\ convex t /\ bounded u /\ ~(u = {}) /\ {x + z | x IN s /\ z IN u} SUBSET {y + z | y IN t /\ z IN u} ==> s SUBSET t`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_SUMS_LCANCEL THEN EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[SUMS_SYM] THEN ASM_REWRITE_TAC[]);; let EQ_SUMS_LCANCEL = prove (`!s t u. ~(s = {}) /\ bounded s /\ closed t /\ convex t /\ closed u /\ convex u /\ {x + y | x IN s /\ y IN t} = {x + z | x IN s /\ z IN u} ==> t = u`, REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; EMPTY_SUBSET] THEN REWRITE_TAC[SUBSET_EMPTY] THEN MESON_TAC[SUBSET_SUMS_LCANCEL]);; let EQ_SUMS_RCANCEL = prove (`!s t u. closed s /\ convex s /\ closed t /\ convex t /\ bounded u /\ ~(u = {}) /\ {x + z | x IN s /\ z IN u} = {y + z | y IN t /\ z IN u} ==> s = t`, REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; EMPTY_SUBSET] THEN REWRITE_TAC[SUBSET_EMPTY] THEN MESON_TAC[SUBSET_SUMS_RCANCEL]);; (* ------------------------------------------------------------------------- *) (* Convex hull. *) (* ------------------------------------------------------------------------- *) let CONVEX_CONVEX_HULL = prove (`!s. convex(convex hull s)`, SIMP_TAC[P_HULL; CONVEX_INTERS]);; let CONVEX_HULL_EQ = prove (`!s. (convex hull s = s) <=> convex s`, SIMP_TAC[HULL_EQ; CONVEX_INTERS]);; let CONVEX_HULLS_EQ = prove (`!s t. s SUBSET convex hull t /\ t SUBSET convex hull s ==> convex hull s = convex hull t`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HULLS_EQ THEN ASM_SIMP_TAC[CONVEX_INTERS]);; let IS_CONVEX_HULL = prove (`!s. convex s <=> ?t. s = convex hull t`, GEN_TAC THEN MATCH_MP_TAC IS_HULL THEN SIMP_TAC[CONVEX_INTERS]);; let MIDPOINTS_IN_CONVEX_HULL = prove (`!x:real^N s. x IN convex hull s /\ y IN convex hull s ==> midpoint(x,y) IN convex hull s`, MESON_TAC[MIDPOINT_IN_CONVEX; CONVEX_CONVEX_HULL]);; let CONVEX_HULL_UNIV = prove (`convex hull (:real^N) = (:real^N)`, REWRITE_TAC[CONVEX_HULL_EQ; CONVEX_UNIV]);; let BOUNDED_CONVEX_HULL = prove (`!s:real^N->bool. bounded s ==> bounded(convex hull s)`, GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [bounded] THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `cball(vec 0:real^N,B)` THEN SIMP_TAC[BOUNDED_CBALL; SUBSET_HULL; CONVEX_CBALL] THEN ASM_REWRITE_TAC[IN_CBALL; SUBSET; dist; VECTOR_SUB_LZERO; NORM_NEG]);; let BOUNDED_CONVEX_HULL_EQ = prove (`!s. bounded(convex hull s) <=> bounded s`, MESON_TAC[BOUNDED_CONVEX_HULL; HULL_SUBSET; BOUNDED_SUBSET]);; let FINITE_IMP_BOUNDED_CONVEX_HULL = prove (`!s. FINITE s ==> bounded(convex hull s)`, SIMP_TAC[BOUNDED_CONVEX_HULL; FINITE_IMP_BOUNDED]);; (* ------------------------------------------------------------------------- *) (* Stepping theorems for convex hulls of finite sets. *) (* ------------------------------------------------------------------------- *) let CONVEX_HULL_EMPTY = prove (`convex hull {} = {}`, MATCH_MP_TAC HULL_UNIQUE THEN REWRITE_TAC[SUBSET_REFL; CONVEX_EMPTY; EMPTY_SUBSET]);; let CONVEX_HULL_EQ_EMPTY = prove (`!s. (convex hull s = {}) <=> (s = {})`, GEN_TAC THEN EQ_TAC THEN MESON_TAC[SUBSET_EMPTY; HULL_SUBSET; CONVEX_HULL_EMPTY]);; let CONVEX_HULL_SING = prove (`!a. convex hull {a} = {a}`, REWRITE_TAC[CONVEX_HULL_EQ; CONVEX_SING]);; let CONVEX_HULL_EQ_SING = prove (`!s a:real^N. convex hull s = {a} <=> s = {a}`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[CONVEX_HULL_EMPTY] THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[CONVEX_HULL_SING] THEN MATCH_MP_TAC(SET_RULE `~(s = {}) /\ s SUBSET {a} ==> s = {a}`) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN REWRITE_TAC[HULL_SUBSET]);; let CONVEX_HULL_INSERT = prove (`!s a. ~(s = {}) ==> convex hull (a INSERT s) = {x:real^N | ?u v b. &0 <= u /\ &0 <= v /\ u + v = &1 /\ b IN (convex hull s) /\ x = u % a + v % b}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [MATCH_MP_TAC HULL_MINIMAL THEN CONJ_TAC THENL [REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_INSERT] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THENL [MAP_EVERY EXISTS_TAC [`&1`; `&0`]; MAP_EVERY EXISTS_TAC [`&0`; `&1`]] THEN ASM_REWRITE_TAC[VECTOR_MUL_LID; VECTOR_MUL_LZERO] THEN ASM_REWRITE_TAC[VECTOR_ADD_LID; VECTOR_ADD_RID] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_MESON_TAC[MEMBER_NOT_EMPTY; HULL_SUBSET; SUBSET]; ALL_TAC]; REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[convex] CONVEX_CONVEX_HULL) THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[HULL_SUBSET; SUBSET; IN_INSERT; HULL_MONO]] THEN REWRITE_TAC[convex; IN_ELIM_THM] 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 [`x:real^N`; `y:real^N`; `u:real`; `v:real`; `u1:real`; `v1:real`; `b1:real^N`; `u2:real`; `v2:real`; `b2:real^N`] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MAP_EVERY EXISTS_TAC [`u * u1 + v * u2`; `u * v1 + v * v2`] THEN REWRITE_TAC[VECTOR_ARITH `u % (u1 % a + v1 % b1) + v % (u2 % a + v2 % b2):real^N = (u * u1 + v * u2) % a + (u * v1) % b1 + (v * v2) % b2`] THEN ASM_SIMP_TAC[REAL_LE_ADD; REAL_LE_MUL] THEN ASM_REWRITE_TAC[REAL_MUL_RID; REAL_ARITH `(u * u1 + v * u2) + (u * v1 + v * v2) = u * (u1 + v1) + v * (u2 + v2)`] THEN ASM_CASES_TAC `u * v1 + v * v2 = &0` THENL [FIRST_X_ASSUM(MP_TAC o MATCH_MP (REAL_ARITH `(a + b = &0) ==> &0 <= a /\ &0 <= b ==> (a = &0) /\ (b = &0)`)) THEN ASM_SIMP_TAC[REAL_LE_MUL; REAL_ADD_LID; VECTOR_MUL_LZERO; VECTOR_ADD_RID] THEN ASM_MESON_TAC[]; ALL_TAC] THEN EXISTS_TAC `(u * v1) / (u * v1 + v * v2) % b1 + (v * v2) / (u * v1 + v * v2) % b2 :real^N` THEN ASM_SIMP_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC; REAL_DIV_LMUL] THEN MATCH_MP_TAC(REWRITE_RULE[convex] CONVEX_CONVEX_HULL) THEN ASM_SIMP_TAC[REAL_LE_DIV; REAL_LE_MUL; REAL_LE_ADD] THEN ASM_SIMP_TAC[real_div; GSYM REAL_ADD_RDISTRIB; REAL_MUL_RINV]);; let CONVEX_HULL_INSERT_ALT = prove (`!s a:real^N. convex hull (a INSERT s) = if s = {} then {a} else {(&1 - u) % a + u % x | &0 <= u /\ u <= &1 /\ x IN convex hull s}`, REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[CONVEX_HULL_SING] THEN ASM_SIMP_TAC[CONVEX_HULL_INSERT] THEN ONCE_REWRITE_TAC[TAUT `a /\ b /\ c /\ d <=> b /\ c /\ a /\ d`] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN REWRITE_TAC[RIGHT_EXISTS_AND_THM; UNWIND_THM2; REAL_SUB_LE; REAL_ARITH `u + v = &1 <=> u = &1 - v`] THEN SET_TAC[]);; let CONVEX_HULL_INSERT_SEGMENTS = prove (`!s a:real^N. convex hull (a INSERT s) = if s = {} then {a} else UNIONS {segment[a,x] | x IN convex hull s}`, REPEAT GEN_TAC THEN REWRITE_TAC[CONVEX_HULL_INSERT_ALT] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[UNIONS_GSPEC; IN_SEGMENT] THEN SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* Explicit expressions for convex hull. *) (* ------------------------------------------------------------------------- *) let CONVEX_HULL_INDEXED = prove (`!s. convex hull s = {y:real^N | ?k u x. (!i. 1 <= i /\ i <= k ==> &0 <= u i /\ x i IN s) /\ (sum (1..k) u = &1) /\ (vsum (1..k) (\i. u i % x i) = y)}`, GEN_TAC THEN MATCH_MP_TAC HULL_UNIQUE THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MAP_EVERY EXISTS_TAC [`1`; `\i:num. &1`; `\i:num. x:real^N`] THEN ASM_SIMP_TAC[FINITE_RULES; IN_SING; SUM_SING; VECTOR_MUL_LID; VSUM_SING; REAL_POS; NUMSEG_SING]; ALL_TAC; REWRITE_TAC[CONVEX_INDEXED; SUBSET; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN MESON_TAC[]] THEN REWRITE_TAC[convex; IN_ELIM_THM] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `u:real`; `v:real`] THEN REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`k1:num`; `u1:num->real`; `x1:num->real^N`; `k2:num`; `u2:num->real`; `x2:num->real^N`] THEN STRIP_TAC THEN EXISTS_TAC `k1 + k2:num` THEN EXISTS_TAC `\i:num. if i <= k1 then u * u1(i) else v * u2(i - k1):real` THEN EXISTS_TAC `\i:num. if i <= k1 then x1(i) else x2(i - k1):real^N` THEN ASM_SIMP_TAC[NUMSEG_ADD_SPLIT; ARITH_RULE `1 <= x + 1 /\ x < x + 1`; IN_NUMSEG; SUM_UNION; VSUM_UNION; FINITE_NUMSEG; DISJOINT_NUMSEG; ARITH_RULE `k1 + 1 <= i ==> ~(i <= k1)`] THEN REWRITE_TAC[ONCE_REWRITE_RULE[ADD_SYM] NUMSEG_OFFSET_IMAGE] THEN ASM_SIMP_TAC[SUM_IMAGE; VSUM_IMAGE; EQ_ADD_LCANCEL; FINITE_NUMSEG] THEN ASM_SIMP_TAC[o_DEF; ADD_SUB2; SUM_LMUL; VSUM_LMUL; GSYM VECTOR_MUL_ASSOC; FINITE_NUMSEG; REAL_MUL_RID] THEN ASM_MESON_TAC[REAL_LE_MUL; ARITH_RULE `i <= k1 + k2 /\ ~(i <= k1) ==> 1 <= i - k1 /\ i - k1 <= k2`]);; let CONVEX_HULL_FINITE_IMAGE_EXPLICIT = prove (`!f:A->real^N k. FINITE k ==> convex hull (IMAGE f k) = {y | ?u. (!a. a IN k ==> &0 <= u a) /\ sum k u = &1 /\ vsum k (\a. u a % f a) = y}`, GEN_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN REWRITE_TAC[SUM_CLAUSES; REAL_OF_NUM_EQ; CONVEX_HULL_EMPTY; ARITH_EQ; IMAGE_CLAUSES; EMPTY_GSPEC] THEN MAP_EVERY X_GEN_TAC [`b:A`; `k:A->bool`] THEN ASM_CASES_TAC `k:A->bool = {}` THENL [ASM_REWRITE_TAC[IMAGE_CLAUSES; SUM_SING; VSUM_SING; CONVEX_HULL_SING] THEN REWRITE_TAC[FORALL_IN_INSERT; NOT_IN_EMPTY] THEN REWRITE_TAC[MESON[] `&0 <= u /\ u = &1 /\ u % x:real^N = y <=> u = &1 /\ &0 <= &1 /\ &1 % x = y`] THEN REWRITE_TAC[REAL_POS; VECTOR_MUL_LID; LEFT_EXISTS_AND_THM] THEN DISCH_THEN(K ALL_TAC) THEN SUBGOAL_THEN `?u:A->real. u b = &1` (fun th -> REWRITE_TAC[th] THEN SET_TAC[]) THEN EXISTS_TAC `\a:A. &1` THEN REWRITE_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[CONVEX_HULL_INSERT; IMAGE_EQ_EMPTY] THEN STRIP_TAC THEN ASM_SIMP_TAC[IN_ELIM_THM; SUM_CLAUSES; VSUM_CLAUSES] THEN GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[FORALL_IN_INSERT] THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_ELIM_THM] THEN REWRITE_TAC[LEFT_AND_EXISTS_THM; RIGHT_AND_EXISTS_THM] THEN EQ_TAC THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THENL [MAP_EVERY X_GEN_TAC [`u:real`; `v:real`; `z:real^N`; `c:A->real`] THEN STRIP_TAC THEN EXISTS_TAC `\a. if a = b then u else v * (c:A->real) a` THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP (SET_RULE `~(b IN k) ==> !a. a IN k ==> ~(a = b)`)) THEN ASM_SIMP_TAC[GSYM VECTOR_MUL_ASSOC; VSUM_LMUL; SUM_LMUL] THEN ASM_SIMP_TAC[REAL_LE_MUL; REAL_MUL_RID]; X_GEN_TAC `c:A->real` THEN STRIP_TAC THEN ASM_CASES_TAC `(c:A->real) b = &1` THENL [UNDISCH_TAC `c(b:A) + sum k c = &1` THEN ASM_REWRITE_TAC[REAL_ARITH `&1 + x = &1 <=> x = &0`] THEN DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] (REWRITE_RULE[CONJ_ASSOC] SUM_POS_EQ_0))) THEN ASM_SIMP_TAC[] THEN DISCH_TAC THEN MAP_EVERY EXISTS_TAC [`&1`; `&0`] THEN EXPAND_TAC "y" THEN REWRITE_TAC[VECTOR_ARITH `c % f + v:real^N = &1 % f + &0 % b <=> v = (&1 - c) % f`] THEN ASM_SIMP_TAC[VECTOR_MUL_LZERO; VSUM_0; REAL_SUB_REFL] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[CONJ_ASSOC] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN REWRITE_TAC[UNWIND_THM1] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_TAC `d:A`) THEN EXISTS_TAC `\a:A. if a = d then &1 else &0` THEN ASM_REWRITE_TAC[SUM_DELTA] THEN MESON_TAC[REAL_POS]; MAP_EVERY EXISTS_TAC [`(c:A->real) b`; `&1 - (c:A->real) b`; `vsum k (\a. (c:A->real) a / (&1 - c b) % f a):real^N`; `\a. (c:A->real) a / (&1 - c b)`] THEN ASM_REWRITE_TAC[REAL_ARITH `x + &1 - x = &1`] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] real_div] THEN REWRITE_TAC[GSYM VECTOR_MUL_ASSOC; SUM_LMUL; VSUM_LMUL] THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; REAL_SUB_0] THEN ASM_REWRITE_TAC[VECTOR_MUL_LID] THEN ASM_SIMP_TAC[REAL_FIELD `~(c = &1) ==> (inv(&1 - c) * b = &1 <=> c + b = &1)`] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN ASM_SIMP_TAC[REAL_LE_MUL; REAL_LE_INV_EQ] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `c + s = &1 ==> &0 <= s ==> &0 <= &1 - c`)) THEN ASM_SIMP_TAC[SUM_POS_LE]]]);; (* ------------------------------------------------------------------------- *) (* Another formulation from Lars Schewe. *) (* ------------------------------------------------------------------------- *) let CONVEX_HULL_EXPLICIT = prove (`!p. convex hull p = {y:real^N | ?s u. FINITE s /\ s SUBSET p /\ (!x. x IN s ==> &0 <= u x) /\ sum s u = &1 /\ vsum s (\v. u v % v) = y}`, REWRITE_TAC[CONVEX_HULL_INDEXED;EXTENSION;IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN EQ_TAC THEN REPEAT STRIP_TAC THENL [MAP_EVERY EXISTS_TAC [`IMAGE (x':num->real^N) (1..k)`; `\v:real^N.sum {i | i IN (1..k) /\ x' i = v} u`] THEN ASM_SIMP_TAC[FINITE_IMAGE;FINITE_NUMSEG;IN_IMAGE] THEN REPEAT STRIP_TAC THENL [REWRITE_TAC[IMAGE;SUBSET;IN_ELIM_THM;IN_NUMSEG] THEN ASM_MESON_TAC[]; MATCH_MP_TAC SUM_POS_LE THEN ASM_SIMP_TAC[FINITE_NUMSEG;FINITE_RESTRICT;IN_ELIM_THM;IN_NUMSEG]; ASM_SIMP_TAC[GSYM SUM_IMAGE_GEN;FINITE_IMAGE;FINITE_NUMSEG]; FIRST_X_ASSUM (fun th -> REWRITE_TAC[GSYM th]) THEN ASM_SIMP_TAC[GSYM VSUM_IMAGE_GEN;FINITE_IMAGE; FINITE_NUMSEG;VSUM_VMUL;FINITE_RESTRICT] THEN MP_TAC (ISPECL [`x':num->real^N`;`\i:num.u i % (x' i):real^N`;`(1..k)`] (GSYM VSUM_IMAGE_GEN)) THEN ASM_SIMP_TAC[FINITE_NUMSEG]];ALL_TAC] THEN STRIP_ASSUME_TAC (ASM_REWRITE_RULE [ASSUME `FINITE (s:real^N->bool)`] (ISPEC `s:real^N->bool` FINITE_INDEX_NUMSEG)) THEN MAP_EVERY EXISTS_TAC [`CARD (s:real^N->bool)`; `(u:real^N->real) o (f:num->real^N)`; `(f:num->real^N)`] THEN REPEAT STRIP_TAC THENL [REWRITE_TAC[o_DEF] THEN FIRST_ASSUM MATCH_MP_TAC THEN FIRST_ASSUM SUBST1_TAC THEN REWRITE_TAC[IN_IMAGE;IN_NUMSEG] THEN ASM_MESON_TAC[]; MATCH_MP_TAC (REWRITE_RULE [SUBSET] (ASSUME `(s:real^N->bool) SUBSET p`)) THEN FIRST_ASSUM SUBST1_TAC THEN REWRITE_TAC[IN_IMAGE;IN_NUMSEG] THEN ASM_MESON_TAC[]; MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `sum (s:real^N->bool) u` THEN CONJ_TAC THENL [ALL_TAC;ASM_REWRITE_TAC[]] THEN GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [ASSUME `(s:real^N->bool) = IMAGE f (1..CARD s)`] THEN MATCH_MP_TAC (GSYM SUM_IMAGE) THEN ASM_MESON_TAC[]; REWRITE_TAC[MESON [o_THM;FUN_EQ_THM] `(\i:num. (u o f) i % f i) = (\v:real^N. u v % v) o f`] THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `vsum (s:real^N->bool) (\v. u v % v)` THEN CONJ_TAC THENL [ALL_TAC;ASM_REWRITE_TAC[]] THEN GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [ASSUME `(s:real^N->bool) = IMAGE f (1..CARD s)`] THEN MATCH_MP_TAC (GSYM VSUM_IMAGE) THEN ASM SET_TAC[FINITE_NUMSEG]]);; let CONVEX_HULL_FINITE = prove (`!s:real^N->bool. convex hull s = {y | ?u. (!x. x IN s ==> &0 <= u x) /\ sum s u = &1 /\ vsum s (\x. u x % x) = y}`, GEN_TAC THEN GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[CONVEX_HULL_EXPLICIT; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN EQ_TAC THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THENL [MAP_EVERY X_GEN_TAC [`t:real^N->bool`; `f:real^N->real`] THEN STRIP_TAC THEN EXISTS_TAC `\x:real^N. if x IN t then f x else &0` THEN REWRITE_TAC[COND_RAND; COND_RATOR; VECTOR_MUL_LZERO] THEN REWRITE_TAC[GSYM SUM_RESTRICT_SET; GSYM VSUM_RESTRICT_SET] THEN ASM_SIMP_TAC[SET_RULE `t SUBSET s ==> {x | x IN s /\ x IN t} = t`] THEN REWRITE_TAC[REAL_LE_REFL; COND_ID]; X_GEN_TAC `f:real^N->real` THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[SUM_CLAUSES; REAL_OF_NUM_EQ; ARITH] THEN STRIP_TAC THEN EXISTS_TAC `support (+) (f:real^N->real) s` THEN EXISTS_TAC `f:real^N->real` THEN MP_TAC(ASSUME `sum s (f:real^N->real) = &1`) THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [sum] THEN REWRITE_TAC[iterate] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[NEUTRAL_REAL_ADD; REAL_OF_NUM_EQ; ARITH] THEN DISCH_THEN(K ALL_TAC) THEN UNDISCH_TAC `sum s (f:real^N->real) = &1` THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [GSYM SUM_SUPPORT] THEN ASM_CASES_TAC `support (+) (f:real^N->real) s = {}` THEN ASM_SIMP_TAC[SUM_CLAUSES; REAL_OF_NUM_EQ; ARITH] THEN DISCH_TAC THEN REWRITE_TAC[SUPPORT_SUBSET] THEN CONJ_TAC THENL [ASM_SIMP_TAC[support; IN_ELIM_THM]; ALL_TAC] THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [GSYM th]) THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC VSUM_SUPERSET THEN REWRITE_TAC[SUPPORT_SUBSET] THEN REWRITE_TAC[support; IN_ELIM_THM; NEUTRAL_REAL_ADD] THEN MESON_TAC[VECTOR_MUL_LZERO]]);; let CONVEX_HULL_IMAGE = prove (`!f:A->real^N k. convex hull (IMAGE f k) = {y | ?c u. FINITE c /\ c SUBSET k /\ (!a. a IN c ==> &0 <= u a) /\ sum c u = &1 /\ vsum c (\a. u a % f a) = y}`, REPEAT GEN_TAC THEN REWRITE_TAC[CONVEX_HULL_EXPLICIT] THEN GEN_REWRITE_TAC I [EXTENSION] THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_ELIM_THM] THEN EQ_TAC THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THENL [ONCE_REWRITE_TAC[TAUT `p /\ q /\ r ==> s <=> p /\ q ==> r ==> s`] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[FORALL_FINITE_SUBSET_IMAGE_INJ] THEN X_GEN_TAC `c:A->bool` THEN STRIP_TAC THEN X_GEN_TAC `u:real^N->real` THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN W(MP_TAC o PART_MATCH (lhand o rand) SUM_IMAGE o lhand o lhand o rand o lhand o snd) THEN ANTS_TAC THENL [ASM_MESON_TAC[]; DISCH_THEN SUBST1_TAC] THEN W(MP_TAC o PART_MATCH (lhand o rand) VSUM_IMAGE o lhand o rand o rand o lhand o snd) THEN ANTS_TAC THENL [ASM_MESON_TAC[]; DISCH_THEN SUBST1_TAC] THEN REWRITE_TAC[o_DEF] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`c:A->bool`; `(u:real^N->real) o (f:A->real^N)`] THEN ASM_SIMP_TAC[GSYM SUM_IMAGE; GSYM VSUM_IMAGE] THEN ASM_SIMP_TAC[o_DEF]; MAP_EVERY X_GEN_TAC [`c:A->bool`; `u:A->real`] THEN STRIP_TAC THEN EXISTS_TAC `IMAGE (f:A->real^N) c` THEN EXISTS_TAC `\y. sum {a | a IN c /\ (f:A->real^N) a = y} u` THEN ASM_SIMP_TAC[GSYM SUM_IMAGE_GEN; FINITE_IMAGE; IMAGE_SUBSET] THEN CONJ_TAC THENL [REWRITE_TAC[FORALL_IN_IMAGE] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC SUM_POS_LE THEN ASM SET_TAC[]; FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [SYM th]) THEN MP_TAC(GEN `g:A->real^N` (ISPECL [`f:A->real^N`; `g:A->real^N`; `c:A->bool`] VSUM_IMAGE_GEN)) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(fun th -> REWRITE_TAC[th]) THEN MATCH_MP_TAC VSUM_EQ THEN SIMP_TAC[FORALL_IN_IMAGE; GSYM VSUM_RMUL] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC VSUM_EQ THEN SET_TAC[]]]);; let CONVEX_HULL_IMAGE_LT = prove (`!f:A->real^N k. convex hull (IMAGE f k) = {y | ?c u. FINITE c /\ c SUBSET k /\ (!a. a IN c ==> &0 < u a) /\ sum c u = &1 /\ vsum c (\a. u a % f a) = y}`, REPEAT GEN_TAC THEN REWRITE_TAC[CONVEX_HULL_IMAGE] THEN GEN_REWRITE_TAC I [EXTENSION] THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_ELIM_THM] THEN EQ_TAC THENL [ONCE_REWRITE_TAC[SWAP_EXISTS_THM]; MESON_TAC[REAL_LT_IMP_LE]] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:A->real` THEN DISCH_THEN(X_CHOOSE_THEN `c:A->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `{a | a IN c /\ &0 < (u:A->real) a}` THEN ASM_SIMP_TAC[FINITE_RESTRICT; IN_ELIM_THM] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [SYM th]) THEN CONV_TAC SYM_CONV THENL [MATCH_MP_TAC SUM_SUPERSET; MATCH_MP_TAC VSUM_SUPERSET] THEN REWRITE_TAC[SUBSET_RESTRICT; IN_ELIM_THM] THEN ASM_SIMP_TAC[IMP_CONJ; REAL_LT_LE] THEN MESON_TAC[VECTOR_MUL_LZERO]);; let CONVEX_HULL_UNION_EXPLICIT = prove (`!s t:real^N->bool. convex s /\ convex t ==> convex hull (s UNION t) = s UNION t UNION {(&1 - u) % x + u % y | x IN s /\ y IN t /\ &0 <= u /\ u <= &1}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [REWRITE_TAC[CONVEX_HULL_EXPLICIT] THEN GEN_REWRITE_TAC I [SUBSET] THEN REWRITE_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`y:real^N`; `u:real^N->bool`; `f:real^N->real`] THEN REPLICATE_TAC 3 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN SUBST1_TAC(SET_RULE `u:real^N->bool = (u INTER s) UNION (u DIFF s)`) THEN ASM_SIMP_TAC[SUM_UNION; VSUM_UNION; FINITE_INTER; FINITE_DIFF; SET_RULE `DISJOINT (u INTER s) (u DIFF s)`] THEN ASM_CASES_TAC `sum (u INTER s) (f:real^N->real) = &0` THENL [SUBGOAL_THEN `!x. x IN (u INTER s) ==> (f:real^N->real) x = &0` ASSUME_TAC THENL [ASM_MESON_TAC[SUM_POS_EQ_0; FINITE_INTER; IN_INTER]; ASM_SIMP_TAC[VECTOR_MUL_LZERO; VSUM_0] THEN REWRITE_TAC[VECTOR_ADD_LID; REAL_ADD_LID] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (SUBST1_TAC o SYM)) THEN REWRITE_TAC[IN_UNION] THEN DISJ2_TAC THEN DISJ1_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_EXPLICIT]) THEN ASM_SIMP_TAC[FINITE_DIFF; IN_DIFF] THEN ASM SET_TAC[]]; ALL_TAC] THEN ASM_CASES_TAC `sum (u DIFF s) (f:real^N->real) = &0` THENL [SUBGOAL_THEN `!x. x IN (u DIFF s) ==> (f:real^N->real) x = &0` ASSUME_TAC THENL [ASM_MESON_TAC[SUM_POS_EQ_0; FINITE_DIFF; IN_DIFF]; ASM_SIMP_TAC[VECTOR_MUL_LZERO; VSUM_0] THEN REWRITE_TAC[VECTOR_ADD_RID; REAL_ADD_RID] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (SUBST1_TAC o SYM)) THEN REWRITE_TAC[IN_UNION] THEN DISJ1_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_EXPLICIT]) THEN ASM_SIMP_TAC[FINITE_INTER; IN_INTER] THEN ASM SET_TAC[]]; ALL_TAC] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (SUBST1_TAC o SYM)) THEN REWRITE_TAC[IN_UNION; IN_ELIM_THM] THEN DISJ2_TAC THEN DISJ2_TAC THEN MAP_EVERY EXISTS_TAC [`vsum(u INTER s) (\v:real^N. (f v / sum(u INTER s) f) % v)`; `sum(u DIFF s) (f:real^N->real)`; `vsum(u DIFF s) (\v:real^N. (f v / sum(u DIFF s) f) % v)`] THEN REPEAT CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_EXPLICIT]) THEN ASM_SIMP_TAC[INTER_SUBSET; FINITE_INTER; SUM_POS_LE; REAL_LE_DIV; IN_INTER; real_div; SUM_RMUL; REAL_MUL_RINV]; FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_EXPLICIT]) THEN ASM_SIMP_TAC[SUBSET_DIFF; FINITE_DIFF; SUM_POS_LE; REAL_LE_DIV; IN_DIFF; real_div; SUM_RMUL; REAL_MUL_RINV] THEN ASM SET_TAC[]; ASM_SIMP_TAC[SUM_POS_LE; IN_DIFF; FINITE_DIFF]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `a + b = &1 ==> &0 <= a ==> b <= &1`)) THEN ASM_SIMP_TAC[SUM_POS_LE; IN_INTER; FINITE_INTER]; ASM_SIMP_TAC[GSYM VSUM_LMUL; FINITE_INTER; FINITE_DIFF] THEN SIMP_TAC[VECTOR_MUL_ASSOC; REAL_ARITH `a * b / c:real = a / c * b`] THEN FIRST_ASSUM(SUBST1_TAC o MATCH_MP (REAL_ARITH `a + b = &1 ==> &1 - b = a`)) THEN ASM_SIMP_TAC[REAL_DIV_REFL; REAL_MUL_LID]]; REWRITE_TAC[GSYM UNION_ASSOC] THEN ONCE_REWRITE_TAC[UNION_SUBSET] THEN REWRITE_TAC[HULL_SUBSET] THEN REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `u:real`; `y:real^N`] THEN STRIP_TAC THEN MATCH_MP_TAC(REWRITE_RULE[CONVEX_ALT] CONVEX_CONVEX_HULL) THEN ASM_SIMP_TAC[HULL_INC; IN_UNION]]);; let CONVEX_HULL_UNION_NONEMPTY_EXPLICIT = prove (`!s t:real^N->bool. convex s /\ ~(s = {}) /\ convex t /\ ~(t = {}) ==> convex hull (s UNION t) = {(&1 - u) % x + u % y | x IN s /\ y IN t /\ &0 <= u /\ u <= &1}`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[CONVEX_HULL_UNION_EXPLICIT] THEN SIMP_TAC[SET_RULE `s UNION t UNION u = u <=> s SUBSET u /\ t SUBSET u`] THEN CONJ_TAC THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `z:real^N` THEN DISCH_TAC THENL [MAP_EVERY EXISTS_TAC [`z:real^N`; `&0`] THEN REWRITE_TAC[REAL_SUB_RZERO; VECTOR_MUL_LID; REAL_POS; VECTOR_MUL_LZERO; VECTOR_ADD_RID] THEN ASM SET_TAC[]; SUBGOAL_THEN `?a:real^N. a IN s` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN DISCH_TAC THEN MAP_EVERY EXISTS_TAC [`&1`; `z:real^N`] THEN ASM_REWRITE_TAC[REAL_POS; REAL_LE_REFL] THEN VECTOR_ARITH_TAC]);; let CONVEX_HULL_UNION_UNIONS = prove (`!f s:real^N->bool. convex(UNIONS f) /\ ~(f = {}) ==> convex hull (s UNION UNIONS f) = UNIONS {convex hull (s UNION t) | t IN f}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC HULL_MONO THEN ASM SET_TAC[]] THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [ASM_SIMP_TAC[UNION_EMPTY; HULL_P; UNIONS_SUBSET] THEN X_GEN_TAC `u:real^N->bool` THEN DISCH_TAC THEN MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `convex hull u:real^N->bool` THEN REWRITE_TAC[HULL_SUBSET] THEN ASM SET_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `UNIONS f :real^N->bool = {}` THENL [ASM_REWRITE_TAC[UNION_EMPTY] THEN SUBGOAL_THEN `?u:real^N->bool. u IN f` CHOOSE_TAC THENL [ASM_REWRITE_TAC[MEMBER_NOT_EMPTY]; ALL_TAC] THEN MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `convex hull (s UNION u:real^N->bool)` THEN ASM_SIMP_TAC[HULL_MONO; SUBSET_UNION] THEN ASM SET_TAC[]; ALL_TAC] THEN GEN_REWRITE_TAC LAND_CONV [HULL_UNION_LEFT] THEN ASM_SIMP_TAC[CONVEX_HULL_UNION_NONEMPTY_EXPLICIT; CONVEX_HULL_EQ_EMPTY; CONVEX_CONVEX_HULL] THEN REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_UNIONS] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`a:real`; `u:real^N->bool`] THEN DISCH_TAC THEN X_GEN_TAC `y:real^N` THEN REPEAT STRIP_TAC THEN REWRITE_TAC[IN_UNIONS; EXISTS_IN_GSPEC] THEN EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[CONVEX_ALT] CONVEX_CONVEX_HULL) THEN ASM_MESON_TAC[HULL_MONO; IN_UNION; SUBSET; HULL_INC]);; (* ------------------------------------------------------------------------- *) (* A stepping theorem for that expansion. *) (* ------------------------------------------------------------------------- *) let CONVEX_HULL_FINITE_STEP = prove (`((?u. (!x. x IN {} ==> &0 <= u x) /\ sum {} u = w /\ vsum {} (\x. u(x) % x) = y) <=> w = &0 /\ y = vec 0) /\ (FINITE(s:real^N->bool) ==> ((?u. (!x. x IN (a INSERT s) ==> &0 <= u x) /\ sum (a INSERT s) u = w /\ vsum (a INSERT s) (\x. u(x) % x) = y) <=> ?v. &0 <= v /\ ?u. (!x. x IN s ==> &0 <= u x) /\ sum s u = w - v /\ vsum s (\x. u(x) % x) = y - v % a))`, MP_TAC(ISPEC `\x:real^N y:real. &0 <= y` AFFINE_HULL_FINITE_STEP_GEN) THEN SIMP_TAC[REAL_ARITH `&0 <= x / &2 <=> &0 <= x`; REAL_LE_ADD] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM]);; (* ------------------------------------------------------------------------- *) (* Hence some special cases. *) (* ------------------------------------------------------------------------- *) let CONVEX_HULL_2 = prove (`!a b. convex hull {a,b} = {u % a + v % b | &0 <= u /\ &0 <= v /\ u + v = &1}`, SIMP_TAC[CONVEX_HULL_FINITE; FINITE_INSERT; FINITE_RULES] THEN SIMP_TAC[CONVEX_HULL_FINITE_STEP; FINITE_INSERT; FINITE_RULES] THEN REWRITE_TAC[REAL_ARITH `x - y = z:real <=> x = y + z`; VECTOR_ARITH `x - y = z:real^N <=> x = y + z`] THEN REWRITE_TAC[VECTOR_ADD_RID; REAL_ADD_RID] THEN SET_TAC[]);; let CONVEX_HULL_2_ALT = prove (`!a b. convex hull {a,b} = {a + u % (b - a) | &0 <= u /\ u <= &1}`, ONCE_REWRITE_TAC[SET_RULE `{a,b} = {b,a}`] THEN REWRITE_TAC[CONVEX_HULL_2; EXTENSION; IN_ELIM_THM] THEN REWRITE_TAC[REAL_ADD_ASSOC; CONJ_ASSOC] THEN REWRITE_TAC[TAUT `(a /\ x + y = &1) /\ b <=> x + y = &1 /\ a /\ b`] THEN REWRITE_TAC[REAL_ARITH `x + y = &1 <=> y = &1 - x`; UNWIND_THM2] THEN REPEAT GEN_TAC THEN REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN BINOP_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC]);; let CONVEX_HULL_3 = prove (`convex hull {a,b,c} = { u % a + v % b + w % c | &0 <= u /\ &0 <= v /\ &0 <= w /\ u + v + w = &1}`, SIMP_TAC[CONVEX_HULL_FINITE; FINITE_INSERT; FINITE_RULES] THEN SIMP_TAC[CONVEX_HULL_FINITE_STEP; FINITE_INSERT; FINITE_RULES] THEN REWRITE_TAC[REAL_ARITH `x - y = z:real <=> x = y + z`; VECTOR_ARITH `x - y = z:real^N <=> x = y + z`] THEN REWRITE_TAC[VECTOR_ADD_RID; REAL_ADD_RID] THEN SET_TAC[]);; let CONVEX_HULL_3_ALT = prove (`!a b c. convex hull {a,b,c} = {a + u % (b - a) + v % (c - a) | &0 <= u /\ &0 <= v /\ u + v <= &1}`, ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {b,c,a}`] THEN REWRITE_TAC[CONVEX_HULL_3; EXTENSION; IN_ELIM_THM] THEN REWRITE_TAC[REAL_ADD_ASSOC; CONJ_ASSOC] THEN REWRITE_TAC[TAUT `(a /\ x + y = &1) /\ b <=> x + y = &1 /\ a /\ b`] THEN REWRITE_TAC[REAL_ARITH `x + y = &1 <=> y = &1 - x`; UNWIND_THM2] THEN REPEAT GEN_TAC THEN REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN BINOP_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC]);; let CONVEX_HULL_SUMS = prove (`!s t:real^N->bool. convex hull {x + y | x IN s /\ y IN t} = {x + y | x IN convex hull s /\ y IN convex hull t}`, REPEAT GEN_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [MATCH_MP_TAC HULL_MINIMAL THEN SIMP_TAC[CONVEX_SUMS; CONVEX_CONVEX_HULL] THEN REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN REWRITE_TAC[IN_ELIM_THM] THEN MESON_TAC[HULL_INC]; REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [CONVEX_HULL_INDEXED] THEN REWRITE_TAC[IN_ELIM_THM; LEFT_AND_EXISTS_THM] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`k1:num`; `u1:num->real`; `x1:num->real^N`; `k2:num`; `u2:num->real`; `x2:num->real^N`] THEN STRIP_TAC THEN SUBGOAL_THEN `x + y:real^N = vsum(1..k1) (\i. vsum(1..k2) (\j. u1 i % u2 j % (x1 i + x2 j)))` SUBST1_TAC THENL [REWRITE_TAC[VECTOR_ADD_LDISTRIB; VSUM_ADD_NUMSEG] THEN ASM_SIMP_TAC[VSUM_LMUL; VSUM_RMUL; VECTOR_MUL_LID]; REWRITE_TAC[VSUM_LMUL] THEN MATCH_MP_TAC CONVEX_VSUM THEN ASM_SIMP_TAC[FINITE_NUMSEG; CONVEX_CONVEX_HULL; IN_NUMSEG] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CONVEX_VSUM THEN ASM_SIMP_TAC[FINITE_NUMSEG; CONVEX_CONVEX_HULL; IN_NUMSEG] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC HULL_INC THEN ASM SET_TAC[]]]);; let AFFINE_HULL_SUMS = prove (`!s t:real^N->bool. affine hull {x + y | x IN s /\ y IN t} = {x + y | x IN affine hull s /\ y IN affine hull t}`, REPEAT GEN_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [MATCH_MP_TAC HULL_MINIMAL THEN SIMP_TAC[AFFINE_SUMS; AFFINE_AFFINE_HULL] THEN REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN REWRITE_TAC[IN_ELIM_THM] THEN MESON_TAC[HULL_INC]; REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [AFFINE_HULL_INDEXED] THEN REWRITE_TAC[IN_ELIM_THM; LEFT_AND_EXISTS_THM] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`k1:num`; `u1:num->real`; `x1:num->real^N`; `k2:num`; `u2:num->real`; `x2:num->real^N`] THEN STRIP_TAC THEN SUBGOAL_THEN `x + y:real^N = vsum(1..k1) (\i. vsum(1..k2) (\j. u1 i % u2 j % (x1 i + x2 j)))` SUBST1_TAC THENL [REWRITE_TAC[VECTOR_ADD_LDISTRIB; VSUM_ADD_NUMSEG] THEN ASM_SIMP_TAC[VSUM_LMUL; VSUM_RMUL; VECTOR_MUL_LID]; REWRITE_TAC[VSUM_LMUL] THEN MATCH_MP_TAC AFFINE_VSUM THEN ASM_SIMP_TAC[FINITE_NUMSEG; AFFINE_AFFINE_HULL; IN_NUMSEG] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC AFFINE_VSUM THEN ASM_SIMP_TAC[FINITE_NUMSEG; AFFINE_AFFINE_HULL; IN_NUMSEG] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC HULL_INC THEN ASM SET_TAC[]]]);; let AFFINE_HULL_PCROSS,CONVEX_HULL_PCROSS = (CONJ_PAIR o prove) (`(!s:real^M->bool t:real^N->bool. affine hull (s PCROSS t) = (affine hull s) PCROSS (affine hull t)) /\ (!s:real^M->bool t:real^N->bool. convex hull (s PCROSS t) = (convex hull s) PCROSS (convex hull t))`, let lemma1 = prove (`!u v x y:real^M z:real^N. u + v = &1 ==> pastecart z (u % x + v % y) = u % pastecart z x + v % pastecart z y /\ pastecart (u % x + v % y) z = u % pastecart x z + v % pastecart y z`, REWRITE_TAC[PASTECART_ADD; GSYM PASTECART_CMUL] THEN SIMP_TAC[GSYM VECTOR_ADD_RDISTRIB; VECTOR_MUL_LID]) and lemma2 = prove (`INTERS {{x | pastecart x y IN u} | y IN t} = {x | !y. y IN t ==> pastecart x y IN u}`, REWRITE_TAC[INTERS_GSPEC; EXTENSION; IN_ELIM_THM] THEN SET_TAC[]) in CONJ_TAC THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [MATCH_MP_TAC HULL_MINIMAL THEN SIMP_TAC[AFFINE_PCROSS; AFFINE_AFFINE_HULL; HULL_SUBSET; PCROSS_MONO]; REWRITE_TAC[SUBSET; FORALL_IN_PCROSS] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN MATCH_MP_TAC HULL_INDUCT THEN CONJ_TAC THENL [X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN MATCH_MP_TAC HULL_INDUCT THEN CONJ_TAC THENL [X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN SUBGOAL_THEN `pastecart (x:real^M) (y:real^N) IN s PCROSS t` MP_TAC THENL [ASM_REWRITE_TAC[PASTECART_IN_PCROSS]; ALL_TAC] THEN REWRITE_TAC[HULL_INC]; ALL_TAC]; REWRITE_TAC[GSYM lemma2] THEN MATCH_MP_TAC AFFINE_INTERS THEN REWRITE_TAC[FORALL_IN_GSPEC]] THEN SIMP_TAC[affine; IN_ELIM_THM; lemma1; ONCE_REWRITE_RULE[affine] AFFINE_AFFINE_HULL]]; REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [MATCH_MP_TAC HULL_MINIMAL THEN SIMP_TAC[CONVEX_PCROSS; CONVEX_CONVEX_HULL; HULL_SUBSET; PCROSS_MONO]; REWRITE_TAC[SUBSET; FORALL_IN_PCROSS] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN MATCH_MP_TAC HULL_INDUCT THEN CONJ_TAC THENL [X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN MATCH_MP_TAC HULL_INDUCT THEN CONJ_TAC THENL [X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN SUBGOAL_THEN `pastecart (x:real^M) (y:real^N) IN s PCROSS t` MP_TAC THENL [ASM_REWRITE_TAC[PASTECART_IN_PCROSS]; ALL_TAC] THEN REWRITE_TAC[HULL_INC]; ALL_TAC]; REWRITE_TAC[GSYM lemma2] THEN MATCH_MP_TAC CONVEX_INTERS THEN REWRITE_TAC[FORALL_IN_GSPEC]] THEN SIMP_TAC[convex; IN_ELIM_THM; lemma1; ONCE_REWRITE_RULE[convex] CONVEX_CONVEX_HULL]]]);; (* ------------------------------------------------------------------------- *) (* Relations among closure notions and corresponding hulls. *) (* ------------------------------------------------------------------------- *) let SUBSPACE_IMP_AFFINE = prove (`!s. subspace s ==> affine s`, REWRITE_TAC[subspace; affine] THEN MESON_TAC[]);; let AFFINE_IMP_CONVEX = prove (`!s. affine s ==> convex s`, REWRITE_TAC[affine; convex] THEN MESON_TAC[]);; let SUBSPACE_IMP_CONVEX = prove (`!s. subspace s ==> convex s`, MESON_TAC[SUBSPACE_IMP_AFFINE; AFFINE_IMP_CONVEX]);; let AFFINE_HULL_SUBSET_SPAN = prove (`!s. (affine hull s) SUBSET (span s)`, GEN_TAC THEN REWRITE_TAC[span] THEN MATCH_MP_TAC HULL_ANTIMONO THEN REWRITE_TAC[SUBSET; IN; SUBSPACE_IMP_AFFINE]);; let CONVEX_HULL_SUBSET_SPAN = prove (`!s. (convex hull s) SUBSET (span s)`, GEN_TAC THEN REWRITE_TAC[span] THEN MATCH_MP_TAC HULL_ANTIMONO THEN REWRITE_TAC[SUBSET; IN; SUBSPACE_IMP_CONVEX]);; let CONVEX_HULL_SUBSET_AFFINE_HULL = prove (`!s. (convex hull s) SUBSET (affine hull s)`, GEN_TAC THEN REWRITE_TAC[span] THEN MATCH_MP_TAC HULL_ANTIMONO THEN REWRITE_TAC[SUBSET; IN; AFFINE_IMP_CONVEX]);; let COLLINEAR_CONVEX_HULL_COLLINEAR = prove (`!s:real^N->bool. collinear(convex hull s) <=> collinear s`, MESON_TAC[COLLINEAR_SUBSET; HULL_SUBSET; SUBSET_TRANS; COLLINEAR_AFFINE_HULL_COLLINEAR; CONVEX_HULL_SUBSET_AFFINE_HULL]);; let AFFINE_SPAN = prove (`!s. affine(span s)`, SIMP_TAC[SUBSPACE_IMP_AFFINE; SUBSPACE_SPAN]);; let CONVEX_SPAN = prove (`!s. convex(span s)`, SIMP_TAC[SUBSPACE_IMP_CONVEX; SUBSPACE_SPAN]);; let SEGMENT_SUBSET_LINE = prove (`(!a b:real^N. segment[a,b] SUBSET affine hull {a,b}) /\ (!a b:real^N. segment(a,b) SUBSET affine hull {a,b})`, REWRITE_TAC[open_segment] THEN MATCH_MP_TAC(TAUT `(p ==> q) /\ p ==> p /\ q`) THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN SIMP_TAC[CONVEX_CONTAINS_SEGMENT_IMP; AFFINE_AFFINE_HULL; AFFINE_IMP_CONVEX; HULL_INC; IN_INSERT]);; let SPAN_CONVEX_HULL = prove (`!s:real^N->bool. span(convex hull s) = span s`, GEN_TAC THEN REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN SIMP_TAC[SPAN_MONO; HULL_SUBSET] THEN MATCH_MP_TAC SPAN_SUBSET_SUBSPACE THEN REWRITE_TAC[SUBSPACE_SPAN] THEN MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[SPAN_INC; CONVEX_SPAN]);; let DIM_CONVEX_HULL = prove (`!s:real^N->bool. dim(convex hull s) = dim s`, MESON_TAC[SPAN_CONVEX_HULL; DIM_SPAN]);; let AFFINE_EQ_SUBSPACE = prove (`!s:real^N->bool. vec 0 IN s ==> (affine s <=> subspace s)`, REPEAT STRIP_TAC THEN EQ_TAC THEN ASM_SIMP_TAC[subspace; affine] THEN DISCH_TAC THEN MATCH_MP_TAC(TAUT `b /\ (b ==> a) ==> a /\ b`) THEN CONJ_TAC THENL [MAP_EVERY X_GEN_TAC [`c:real`; `x:real^N`] THEN STRIP_TAC THEN SUBST1_TAC(VECTOR_ARITH `c % x:real^N = c % x + (&1 - c) % vec 0`) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC; DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN SUBST1_TAC(VECTOR_ARITH `x + y:real^N = &2 % (&1 / &2 % x + &1 / &2 % y)`) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC]);; let AFFINE_IMP_SUBSPACE = prove (`!s. affine s /\ vec 0 IN s ==> subspace s`, SIMP_TAC[GSYM AFFINE_EQ_SUBSPACE]);; let SUBSPACE_EQ_AFFINE = prove (`!s:real^N->bool. subspace s <=> affine s /\ vec 0 IN s`, MESON_TAC[AFFINE_IMP_SUBSPACE; SUBSPACE_IMP_AFFINE; SUBSPACE_0]);; let AFFINE_HULL_EQ_SPAN = prove (`!s:real^N->bool. (vec 0) IN affine hull s ==> affine hull s = span s`, GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[AFFINE_HULL_SUBSET_SPAN] THEN REWRITE_TAC[SUBSET] THEN MATCH_MP_TAC SPAN_INDUCT THEN ASM_REWRITE_TAC[SUBSET; subspace; IN_ELIM_THM; HULL_INC] THEN REPEAT STRIP_TAC THENL [SUBST1_TAC(VECTOR_ARITH `x + y:real^N = &2 % (&1 / &2 % x + &1 / &2 % y) + --(&1) % vec 0`) THEN MATCH_MP_TAC(REWRITE_RULE[affine] AFFINE_AFFINE_HULL) THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[affine] AFFINE_AFFINE_HULL) THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[]; SUBST1_TAC(VECTOR_ARITH `c % x:real^N = c % x + (&1 - c) % vec 0`) THEN MATCH_MP_TAC(REWRITE_RULE[affine] AFFINE_AFFINE_HULL) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC]);; let SPAN_AFFINE_HULL_INSERT = prove (`!s:real^N->bool. span s = affine hull (vec 0 INSERT s)`, SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC; IN_INSERT; SPAN_INSERT_0]);; let CLOSED_AFFINE = prove (`!s:real^N->bool. affine s ==> closed s`, GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[CLOSED_EMPTY] 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 SUBGOAL_THEN `affine (IMAGE (\x:real^N. --a + x) s) ==> closed (IMAGE (\x:real^N. --a + x) s)` MP_TAC THENL [DISCH_THEN(fun th -> MATCH_MP_TAC CLOSED_SUBSPACE THEN MP_TAC th) THEN MATCH_MP_TAC EQ_IMP THEN MATCH_MP_TAC AFFINE_EQ_SUBSPACE THEN REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `a:real^N` THEN ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC; REWRITE_TAC[AFFINE_TRANSLATION_EQ; CLOSED_TRANSLATION_EQ]]);; let CLOSED_AFFINE_HULL = prove (`!s. closed(affine hull s)`, SIMP_TAC[CLOSED_AFFINE; AFFINE_AFFINE_HULL]);; let CLOSURE_SUBSET_AFFINE_HULL = prove (`!s. closure s SUBSET affine hull s`, GEN_TAC THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN REWRITE_TAC[CLOSED_AFFINE_HULL; HULL_SUBSET]);; let AFFINE_HULL_CLOSURE = prove (`!s:real^N->bool. affine hull (closure s) = affine hull s`, GEN_TAC THEN MATCH_MP_TAC HULL_UNIQUE THEN REWRITE_TAC[CLOSURE_SUBSET_AFFINE_HULL; AFFINE_AFFINE_HULL] THEN X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN MATCH_MP_TAC HULL_MINIMAL THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[CLOSURE_SUBSET; SUBSET]);; let AFFINE_HULL_EQ_SPAN_EQ = prove (`!s:real^N->bool. (affine hull s = span s) <=> (vec 0) IN affine hull s`, GEN_TAC THEN EQ_TAC THEN SIMP_TAC[SPAN_0; AFFINE_HULL_EQ_SPAN]);; let AFFINE_DEPENDENT_IMP_DEPENDENT = prove (`!s. affine_dependent s ==> dependent s`, REWRITE_TAC[affine_dependent; dependent] THEN MESON_TAC[SUBSET; AFFINE_HULL_SUBSET_SPAN]);; let DEPENDENT_AFFINE_DEPENDENT_CASES = prove (`!s:real^N->bool. dependent s <=> affine_dependent s \/ (vec 0) IN affine hull s`, REWRITE_TAC[DEPENDENT_EXPLICIT; AFFINE_DEPENDENT_EXPLICIT; AFFINE_HULL_EXPLICIT_ALT; IN_ELIM_THM] THEN GEN_TAC THEN ONCE_REWRITE_TAC[OR_EXISTS_THM] THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `t:real^N->bool` THEN ASM_CASES_TAC `FINITE(t:real^N->bool)` THEN ASM_REWRITE_TAC[] THEN EQ_TAC THEN DISCH_THEN(REPEAT_TCL DISJ_CASES_THEN (X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC)) THENL [ASM_CASES_TAC `sum t (u:real^N->real) = &0` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN DISJ2_TAC THEN EXISTS_TAC `\v:real^N. inv(sum t u) * u v` THEN ASM_SIMP_TAC[SUM_LMUL; VSUM_LMUL; GSYM VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[VECTOR_MUL_RZERO; REAL_MUL_LINV]; EXISTS_TAC `u:real^N->real` THEN ASM_MESON_TAC[]; EXISTS_TAC `u:real^N->real` THEN ASM_REWRITE_TAC[SET_RULE `(?v. v IN t /\ ~p v) <=> ~(!v. v IN t ==> p v)`] THEN DISCH_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `x = &1 ==> x = &0 ==> F`)) THEN ASM_MESON_TAC[SUM_EQ_0]]);; let DEPENDENT_IMP_AFFINE_DEPENDENT = prove (`!a:real^N s. dependent {x - a | x IN s} /\ ~(a IN s) ==> affine_dependent(a INSERT s)`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN REWRITE_TAC[DEPENDENT_EXPLICIT; AFFINE_DEPENDENT_EXPLICIT] THEN REWRITE_TAC[SIMPLE_IMAGE; CONJ_ASSOC; FINITE_SUBSET_IMAGE] THEN REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN REWRITE_TAC[GSYM CONJ_ASSOC] THEN GEN_REWRITE_TAC LAND_CONV [SWAP_EXISTS_THM] THEN GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [SWAP_EXISTS_THM] THEN REWRITE_TAC[TAUT `a /\ x = IMAGE f s /\ b <=> x = IMAGE f s /\ a /\ b`] THEN REWRITE_TAC[UNWIND_THM2; EXISTS_IN_IMAGE] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` (X_CHOOSE_THEN `t:real^N->bool` STRIP_ASSUME_TAC)) THEN FIRST_X_ASSUM(MP_TAC o check (is_eq o concl)) THEN ASM_SIMP_TAC[VSUM_IMAGE; VECTOR_ARITH `x - a:real^N = y - a <=> x = y`] THEN ASM_SIMP_TAC[o_DEF; VECTOR_SUB_LDISTRIB; VSUM_SUB; VSUM_RMUL] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`(a:real^N) INSERT t`; `\x. if x = a then --sum t (\x. u (x - a)) else (u:real^N->real) (x - a)`] THEN ASM_REWRITE_TAC[FINITE_INSERT; SUBSET_REFL] THEN ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL [ASM SET_TAC[]; ALL_TAC] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC(REAL_ARITH `x = y ==> --x + y = &0`) THEN MATCH_MP_TAC SUM_EQ THEN ASM_MESON_TAC[]; EXISTS_TAC `x:real^N` THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]; MATCH_MP_TAC(VECTOR_ARITH `!s. s - t % a = vec 0 /\ s = u ==> --t % a + u = vec 0`) THEN EXISTS_TAC `vsum t (\x:real^N. u(x - a) % x)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC VSUM_EQ THEN REPEAT STRIP_TAC THEN REWRITE_TAC[] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]]);; let AFFINE_DEPENDENT_BIGGERSET = prove (`!s:real^N->bool. (FINITE s ==> CARD s >= dimindex(:N) + 2) ==> affine_dependent s`, GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_SIMP_TAC[CARD_CLAUSES; ARITH_RULE `~(0 >= n + 2)`; FINITE_RULES] 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(SUBST1_TAC o MATCH_MP (SET_RULE `x IN s ==> s = x INSERT (s DELETE x)`)) THEN SIMP_TAC[FINITE_INSERT; CARD_CLAUSES; IN_DELETE] THEN REWRITE_TAC[ARITH_RULE `SUC x >= n + 2 <=> x > n`] THEN DISCH_TAC THEN MATCH_MP_TAC DEPENDENT_IMP_AFFINE_DEPENDENT THEN REWRITE_TAC[IN_DELETE] THEN MATCH_MP_TAC DEPENDENT_BIGGERSET THEN REWRITE_TAC[SET_RULE `{x - a:real^N | x | x IN s /\ ~(x = a)} = IMAGE (\x. x - a) (s DELETE a)`] THEN ASM_SIMP_TAC[FINITE_IMAGE_INJ_EQ; VECTOR_ARITH `x - a = y - a <=> x:real^N = y`; CARD_IMAGE_INJ]);; let AFFINE_DEPENDENT_BIGGERSET_GENERAL = prove (`!s:real^N->bool. (FINITE s ==> CARD s >= dim s + 2) ==> affine_dependent s`, GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_SIMP_TAC[CARD_CLAUSES; ARITH_RULE `~(0 >= n + 2)`; FINITE_RULES] 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(SUBST1_TAC o MATCH_MP (SET_RULE `x IN s ==> s = x INSERT (s DELETE x)`)) THEN SIMP_TAC[FINITE_INSERT; CARD_CLAUSES; IN_DELETE] THEN REWRITE_TAC[ARITH_RULE `SUC x >= n + 2 <=> x > n`] THEN DISCH_TAC THEN MATCH_MP_TAC DEPENDENT_IMP_AFFINE_DEPENDENT THEN REWRITE_TAC[IN_DELETE] THEN MATCH_MP_TAC DEPENDENT_BIGGERSET_GENERAL THEN REWRITE_TAC[SET_RULE `{x - a:real^N | x | x IN s /\ ~(x = a)} = IMAGE (\x. x - a) (s DELETE a)`] THEN ASM_SIMP_TAC[FINITE_IMAGE_INJ_EQ; FINITE_DELETE; VECTOR_ARITH `x - a = y - a <=> x:real^N = y`; CARD_IMAGE_INJ] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o check(is_imp o concl)) THEN ASM_REWRITE_TAC[FINITE_DELETE] THEN MATCH_MP_TAC(ARITH_RULE `c:num <= b ==> (a > b ==> a > c)`) THEN MATCH_MP_TAC SUBSET_LE_DIM THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN SIMP_TAC[SPAN_SUB; SPAN_SUPERSET; IN_INSERT]);; let AFFINE_INDEPENDENT_IMP_FINITE = prove (`!s:real^N->bool. ~(affine_dependent s) ==> FINITE s`, MESON_TAC[AFFINE_DEPENDENT_BIGGERSET]);; let AFFINE_INDEPENDENT_CARD_LE = prove (`!s:real^N->bool. ~(affine_dependent s) ==> CARD s <= dimindex(:N) + 1`, REWRITE_TAC[ARITH_RULE `s <= n + 1 <=> ~(n + 2 <= s)`; CONTRAPOS_THM] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC AFFINE_DEPENDENT_BIGGERSET THEN ASM_REWRITE_TAC[GE]);; let AFFINE_INDEPENDENT_CONVEX_AFFINE_HULL = prove (`!s t:real^N->bool. ~affine_dependent s /\ t SUBSET s ==> convex hull t = affine hull t INTER convex hull s`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN SUBGOAL_THEN `FINITE(t:real^N->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[FINITE_SUBSET]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `ct SUBSET a /\ ct SUBSET cs /\ a INTER cs SUBSET ct ==> ct = a INTER cs`) THEN ASM_SIMP_TAC[HULL_MONO; CONVEX_HULL_SUBSET_AFFINE_HULL] THEN REWRITE_TAC[SUBSET; IN_INTER; CONVEX_HULL_FINITE; AFFINE_HULL_FINITE] THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) (X_CHOOSE_THEN `v:real^N->real` STRIP_ASSUME_TAC)) THEN EXISTS_TAC `u:real^N->real` THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [AFFINE_DEPENDENT_EXPLICIT]) THEN REWRITE_TAC[NOT_EXISTS_THM] THEN DISCH_THEN(MP_TAC o SPECL [`s:real^N->bool`; `\x:real^N. if x IN t then v x - u x:real else v x`]) THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN REWRITE_TAC[MESON[] `(if p then a else b) % x = if p then a % x else b % x`] THEN ASM_SIMP_TAC[VSUM_CASES; SUM_CASES; SET_RULE `t SUBSET s ==> {x | x IN s /\ x IN t} = t`] THEN ASM_SIMP_TAC[GSYM DIFF; SUM_DIFF; VSUM_DIFF; VECTOR_SUB_RDISTRIB; SUM_SUB; VSUM_SUB] THEN REWRITE_TAC[REAL_ARITH `a - b + b - a = &0`; NOT_EXISTS_THM; VECTOR_ARITH `a - b + b - a:real^N = vec 0`] THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[REAL_SUB_0] THEN ASM SET_TAC[]);; let DISJOINT_AFFINE_HULL = prove (`!s t u:real^N->bool. ~affine_dependent s /\ t SUBSET s /\ u SUBSET s /\ DISJOINT t u ==> DISJOINT (affine hull t) (affine hull u)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN SUBGOAL_THEN `FINITE(t:real^N->bool) /\ FINITE (u:real^N->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[FINITE_SUBSET]; ALL_TAC] THEN REWRITE_TAC[IN_DISJOINT; AFFINE_HULL_FINITE; IN_ELIM_THM] THEN DISCH_THEN(X_CHOOSE_THEN `y:real^N` MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `a:real^N->real` STRIP_ASSUME_TAC) (X_CHOOSE_THEN `b:real^N->real` STRIP_ASSUME_TAC)) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [AFFINE_DEPENDENT_EXPLICIT]) THEN REWRITE_TAC[NOT_EXISTS_THM] THEN MAP_EVERY EXISTS_TAC [`s:real^N->bool`; `\x:real^N. if x IN t then a x else if x IN u then --(b x) else &0`] THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN REWRITE_TAC[MESON[] `(if p then a else b) % x = if p then a % x else b % x`] THEN ASM_SIMP_TAC[SUM_CASES; SUBSET_REFL; VSUM_CASES; GSYM DIFF; SUM_DIFF; VSUM_DIFF; SET_RULE `t SUBSET s ==> {x | x IN s /\ x IN t} = t`] THEN ASM_SIMP_TAC[SUM_0; VSUM_0; VECTOR_MUL_LZERO; SUM_NEG; VSUM_NEG; VECTOR_MUL_LNEG; SET_RULE `DISJOINT t u ==> ~(x IN t /\ x IN u)`] THEN REWRITE_TAC[EMPTY_GSPEC; SUM_CLAUSES; VSUM_CLAUSES] THEN CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN CONJ_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN UNDISCH_TAC `sum t (a:real^N->real) = &1` THEN ASM_CASES_TAC `!x:real^N. x IN t ==> a x = &0` THEN ASM_SIMP_TAC[SUM_EQ_0; REAL_OF_NUM_EQ; ARITH_EQ] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_FORALL_THM]) THEN MATCH_MP_TAC MONO_EXISTS THEN ASM SET_TAC[]);; let AFFINE_INDEPENDENT_SPAN_EQ = prove (`!s. ~(affine_dependent s) /\ CARD s = dimindex(:N) + 1 ==> affine hull s = (:real^N)`, MATCH_MP_TAC SET_PROVE_CASES THEN REWRITE_TAC[CARD_CLAUSES; ARITH_RULE `~(0 = n + 1)`] THEN SIMP_TAC[IMP_CONJ; AFFINE_INDEPENDENT_IMP_FINITE; MESON[HAS_SIZE] `FINITE s ==> (CARD s = n <=> s HAS_SIZE n)`] THEN X_GEN_TAC `orig:real^N` THEN GEOM_ORIGIN_TAC `orig:real^N` THEN SIMP_TAC[AFFINE_HULL_EQ_SPAN; IN_INSERT; SPAN_INSERT_0; HULL_INC] THEN SIMP_TAC[HAS_SIZE; CARD_CLAUSES; FINITE_INSERT; IMP_CONJ] THEN REWRITE_TAC[ARITH_RULE `SUC n = m + 1 <=> n = m`; GSYM UNIV_SUBSET] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CARD_GE_DIM_INDEPENDENT THEN ASM_REWRITE_TAC[DIM_UNIV; SUBSET_UNIV; LE_REFL; independent] THEN UNDISCH_TAC `~affine_dependent((vec 0:real^N) INSERT s)` THEN REWRITE_TAC[CONTRAPOS_THM] THEN DISCH_TAC THEN MATCH_MP_TAC DEPENDENT_IMP_AFFINE_DEPENDENT THEN ASM_REWRITE_TAC[VECTOR_SUB_RZERO; SET_RULE `{x | x IN s} = s`]);; let AFFINE_INDEPENDENT_SPAN_GT = prove (`!s:real^N->bool. ~(affine_dependent s) /\ dimindex(:N) < CARD s ==> affine hull s = (:real^N)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC AFFINE_INDEPENDENT_SPAN_EQ THEN ASM_REWRITE_TAC[] THEN MP_TAC(SPEC `s:real^N->bool` AFFINE_DEPENDENT_BIGGERSET) THEN ASM_SIMP_TAC[AFFINE_INDEPENDENT_IMP_FINITE] THEN ASM_ARITH_TAC);; let EMPTY_INTERIOR_AFFINE_HULL = prove (`!s:real^N->bool. FINITE s /\ CARD(s) <= dimindex(:N) ==> interior(affine hull s) = {}`, REWRITE_TAC[IMP_CONJ] THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN REWRITE_TAC[AFFINE_HULL_EMPTY; INTERIOR_EMPTY] THEN SUBGOAL_THEN `!x s:real^N->bool n. ~(x IN s) /\ (x INSERT s) HAS_SIZE n /\ n <= dimindex(:N) ==> interior(affine hull(x INSERT s)) = {}` (fun th -> MESON_TAC[th; HAS_SIZE; FINITE_INSERT]) THEN X_GEN_TAC `orig:real^N` THEN GEOM_ORIGIN_TAC `orig:real^N` THEN SIMP_TAC[AFFINE_HULL_EQ_SPAN; IN_INSERT; SPAN_INSERT_0; HULL_INC] THEN REWRITE_TAC[HAS_SIZE; FINITE_INSERT; IMP_CONJ] THEN SIMP_TAC[CARD_CLAUSES] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC EMPTY_INTERIOR_LOWDIM THEN MATCH_MP_TAC LET_TRANS THEN EXISTS_TAC `CARD(s:real^N->bool)` THEN ASM_SIMP_TAC[DIM_LE_CARD; DIM_SPAN] THEN ASM_ARITH_TAC);; let EMPTY_INTERIOR_CONVEX_HULL = prove (`!s:real^N->bool. FINITE s /\ CARD(s) <= dimindex(:N) ==> interior(convex hull s) = {}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC(SET_RULE `!t. s SUBSET t /\ t = {} ==> s = {}`) THEN EXISTS_TAC `interior(affine hull s):real^N->bool` THEN SIMP_TAC[SUBSET_INTERIOR; CONVEX_HULL_SUBSET_AFFINE_HULL] THEN ASM_SIMP_TAC[EMPTY_INTERIOR_AFFINE_HULL]);; let AFFINE_DEPENDENT_CHOOSE = prove (`!s a:real^N. ~(affine_dependent s) ==> (affine_dependent(a INSERT s) <=> ~(a IN s) /\ a IN affine hull s)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `(a:real^N) IN s` THEN ASM_SIMP_TAC[SET_RULE `a IN s ==> a INSERT s = s`] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN EQ_TAC THENL [UNDISCH_TAC `~(affine_dependent(s:real^N->bool))` THEN ASM_SIMP_TAC[AFFINE_DEPENDENT_EXPLICIT_FINITE; AFFINE_HULL_FINITE; FINITE_INSERT; IN_ELIM_THM; SUM_CLAUSES; VSUM_CLAUSES] THEN DISCH_TAC THEN REWRITE_TAC[EXISTS_IN_INSERT] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` MP_TAC) THEN ASM_CASES_TAC `(u:real^N->real) a = &0` THEN ASM_REWRITE_TAC[] THENL [REWRITE_TAC[REAL_ADD_LID; VECTOR_MUL_LZERO; VECTOR_ADD_LID] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN DISCH_THEN(MP_TAC o SPEC `u:real^N->real`) THEN ASM_REWRITE_TAC[]; ONCE_REWRITE_TAC[REAL_ARITH `ua + sa = &0 <=> sa = --ua`; VECTOR_ARITH `va + sa:real^N = vec 0 <=> sa = --va`] THEN STRIP_TAC THEN EXISTS_TAC `(\x. --(inv(u a)) * u x):real^N->real` THEN ASM_SIMP_TAC[SUM_LMUL; GSYM VECTOR_MUL_ASSOC; VSUM_LMUL] THEN ASM_REWRITE_TAC[VECTOR_MUL_ASSOC; GSYM VECTOR_MUL_LNEG] THEN REWRITE_TAC[REAL_ARITH `--a * --b:real = a * b`] THEN ASM_SIMP_TAC[REAL_MUL_LINV; VECTOR_MUL_LID]]; DISCH_TAC THEN REWRITE_TAC[affine_dependent] THEN EXISTS_TAC `a:real^N` THEN ASM_SIMP_TAC[IN_INSERT; SET_RULE `~(a IN s) ==> (a INSERT s) DELETE a = s`]]);; let AFFINE_INDEPENDENT_INSERT = prove (`!s a:real^N. ~(affine_dependent s) /\ ~(a IN affine hull s) ==> ~(affine_dependent(a INSERT s))`, SIMP_TAC[AFFINE_DEPENDENT_CHOOSE]);; let AFFINE_HULL_EXPLICIT_UNIQUE = prove (`!s:real^N->bool u u'. ~(affine_dependent s) /\ sum s u = &1 /\ sum s u' = &1 /\ vsum s (\x. u x % x) = vsum s (\x. u' x % x) ==> !x. x IN s ==> u x = u' x`, REPEAT GEN_TAC THEN STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN FIRST_ASSUM(MP_TAC o MATCH_MP AFFINE_DEPENDENT_EXPLICIT_FINITE) THEN ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN DISCH_THEN(MP_TAC o SPEC `(\x. u x - u' x):real^N->real`) THEN ASM_SIMP_TAC[VSUM_SUB; SUM_SUB; REAL_SUB_REFL; VECTOR_SUB_RDISTRIB; VECTOR_SUB_REFL; VECTOR_SUB_EQ; REAL_SUB_0] THEN MESON_TAC[]);; let INDEPENDENT_IMP_AFFINE_DEPENDENT_0 = prove (`!s. independent s ==> ~(affine_dependent(vec 0 INSERT s))`, REWRITE_TAC[independent; DEPENDENT_AFFINE_DEPENDENT_CASES] THEN SIMP_TAC[DE_MORGAN_THM; AFFINE_INDEPENDENT_INSERT]);; let AFFINE_INDEPENDENT_STDBASIS = prove (`~(affine_dependent ((vec 0:real^N) INSERT {basis i | 1 <= i /\ i <= dimindex (:N)}))`, SIMP_TAC[INDEPENDENT_IMP_AFFINE_DEPENDENT_0; INDEPENDENT_STDBASIS]);; let SPAN_CONIC_HULL = prove (`!s:real^N->bool. span(conic hull s) = span s`, GEN_TAC THEN REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN SIMP_TAC[HULL_SUBSET; SPAN_MONO] THEN MATCH_MP_TAC SPAN_SUBSET_SUBSPACE THEN REWRITE_TAC[SUBSPACE_SPAN; CONIC_HULL_EXPLICIT] THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN MESON_TAC[SPAN_SUPERSET; SPAN_MUL]);; let CONIC_HULLS_EQ_IMP_SPANS_EQ = prove (`!s t:real^N->bool. conic hull s = conic hull t ==> span s = span t`, MESON_TAC[SPAN_CONIC_HULL]);; let DIM_CONIC_HULL = prove (`!s:real^N->bool. dim(conic hull s) = dim s`, MESON_TAC[DIM_SPAN; SPAN_CONIC_HULL]);; let CONIC_HULL_SUBSET_SPAN = prove (`!s:real^N->bool. conic hull s SUBSET span s`, MESON_TAC[SPAN_CONIC_HULL; SPAN_INC]);; let CONIC_IMAGE_MULTIPLE_EQ = prove (`!s:real^N->bool. conic s <=> !a. &0 <= a ==> IMAGE (\x. a % x) s SUBSET s`, SIMP_TAC[conic; SUBSET; FORALL_IN_IMAGE] THEN MESON_TAC[]);; let CONIC_IMAGE_MULTIPLE = prove (`!s:real^N->bool a. conic s /\ &0 < a ==> IMAGE (\x. a % x) s = s`, REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN CONJ_TAC THENL [ASM_MESON_TAC[REAL_LT_IMP_LE; CONIC_IMAGE_MULTIPLE_EQ]; ALL_TAC] THEN MP_TAC(ISPEC `IMAGE (\x:real^N. a % x) s` CONIC_IMAGE_MULTIPLE_EQ) THEN ASM_SIMP_TAC[CONIC_LINEAR_IMAGE; LINEAR_SCALING] THEN DISCH_THEN(MP_TAC o SPEC `inv a:real`) THEN ASM_SIMP_TAC[REAL_LT_INV_EQ; REAL_LT_IMP_LE; GSYM IMAGE_o] THEN ASM_SIMP_TAC[o_DEF; VECTOR_MUL_ASSOC; REAL_MUL_LINV; REAL_LT_IMP_NZ] THEN REWRITE_TAC[VECTOR_MUL_LID; IMAGE_ID]);; (* ------------------------------------------------------------------------- *) (* Nonempty affine sets are translates of (unique) subspaces. *) (* ------------------------------------------------------------------------- *) let AFFINE_TRANSLATION_SUBSPACE = prove (`!t:real^N->bool. affine t /\ ~(t = {}) <=> ?a s. subspace s /\ t = IMAGE (\x. a + x) s`, GEN_TAC THEN EQ_TAC THEN STRIP_TAC THEN ASM_SIMP_TAC[SUBSPACE_IMP_NONEMPTY; IMAGE_EQ_EMPTY; AFFINE_TRANSLATION; SUBSPACE_IMP_AFFINE] THEN FIRST_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 DISCH_TAC THEN ONCE_REWRITE_TAC[TRANSLATION_GALOIS] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[UNWIND_THM2] THEN MATCH_MP_TAC AFFINE_IMP_SUBSPACE THEN ASM_REWRITE_TAC[AFFINE_TRANSLATION_EQ; IN_IMAGE] THEN EXISTS_TAC `a:real^N` THEN ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC);; let AFFINE_TRANSLATION_UNIQUE_SUBSPACE = prove (`!t:real^N->bool. affine t /\ ~(t = {}) <=> ?!s. ?a. subspace s /\ t = IMAGE (\x. a + x) s`, GEN_TAC THEN REWRITE_TAC[AFFINE_TRANSLATION_SUBSPACE] THEN MATCH_MP_TAC(MESON[] `(!a a' s s'. P s a /\ P s' a' ==> s = s') ==> ((?a s. P s a) <=> (?!s. ?a. P s a))`) THEN REPEAT GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC)) THEN DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[TRANSLATION_GALOIS] THEN DISCH_THEN SUBST1_TAC THEN CONV_TAC SYM_CONV THEN REWRITE_TAC[GSYM IMAGE_o; o_DEF; VECTOR_ADD_ASSOC] THEN MATCH_MP_TAC SUBSPACE_TRANSLATION_SELF THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[VECTOR_ARITH `--a' + a:real^N = --(a' - a)`] THEN MATCH_MP_TAC SUBSPACE_NEG THEN ASM_REWRITE_TAC[] THEN UNDISCH_TAC `t = IMAGE (\x:real^N. a' + x) s'` THEN DISCH_THEN(MP_TAC o AP_TERM `\s. (a':real^N) IN s`) THEN REWRITE_TAC[IN_IMAGE; VECTOR_ARITH `a:real^N = a + x <=> x = vec 0`] THEN ASM_SIMP_TAC[UNWIND_THM2; SUBSPACE_0] THEN REWRITE_TAC[IN_IMAGE; VECTOR_ARITH `a':real^N = a + x <=> x = a' - a`] THEN REWRITE_TAC[UNWIND_THM2]);; let AFFINE_TRANSLATION_SUBSPACE_EXPLICIT = prove (`!t:real^N->bool a. affine t /\ a IN t ==> subspace {x - a | x IN t} /\ t = IMAGE (\x. a + x) {x - a | x IN t}`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[AFFINE_DIFFS_SUBSPACE] THEN ASM_REWRITE_TAC[SIMPLE_IMAGE; GSYM IMAGE_o] THEN REWRITE_TAC[o_DEF; VECTOR_SUB_ADD2; IMAGE_ID]);; (* ------------------------------------------------------------------------- *) (* If we take a slice out of a set, we can do it perpendicularly, *) (* with the normal vector to the slice parallel to the affine hull. *) (* ------------------------------------------------------------------------- *) let AFFINE_PARALLEL_SLICE = prove (`!s a:real^N b. affine s ==> s INTER {x | a dot x <= b} = {} \/ s SUBSET {x | a dot x <= b} \/ ?a' b'. ~(a' = vec 0) /\ s INTER {x | a' dot x <= b'} = s INTER {x | a dot x <= b} /\ s INTER {x | a' dot x = b'} = s INTER {x | a dot x = b} /\ !w. w IN s ==> (w + a') IN s`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s INTER {x:real^N | a dot x = b} = {}` THENL [MATCH_MP_TAC(TAUT `~(~p /\ ~q) ==> p \/ q \/ r`) THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `?u v:real^N. u IN s /\ v IN s /\ a dot u <= b /\ ~(a dot v <= b)` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(a:real^N) dot u < b` ASSUME_TAC THENL [ASM_REWRITE_TAC[REAL_LT_LE] THEN ASM SET_TAC[]; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LE]) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [EXTENSION]) THEN REWRITE_TAC[NOT_IN_EMPTY; IN_INTER; NOT_FORALL_THM; IN_ELIM_THM] THEN EXISTS_TAC `u + (b - a dot u) / (a dot v - a dot u) % (v - u):real^N` THEN ASM_SIMP_TAC[IN_AFFINE_ADD_MUL_DIFF] THEN REWRITE_TAC[DOT_RADD; DOT_RMUL; DOT_RSUB] THEN REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_FIELD; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_THEN `z:real^N` MP_TAC) THEN REWRITE_TAC[IN_INTER; IN_ELIM_THM] THEN POP_ASSUM MP_TAC THEN GEN_GEOM_ORIGIN_TAC `z:real^N` ["a"; "a'"; "b'"; "w"] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN REWRITE_TAC[VECTOR_ADD_RID; FORALL_IN_IMAGE] THEN REWRITE_TAC[DOT_RADD; REAL_ARITH `a + x <= a <=> x <= &0`] THEN SUBGOAL_THEN `subspace(s:real^N->bool) /\ span s = s` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[AFFINE_IMP_SUBSPACE; SPAN_EQ_SELF]; ALL_TAC] THEN MP_TAC(ISPECL [`s:real^N->bool`; `a:real^N`] ORTHOGONAL_SUBSPACE_DECOMP_EXISTS) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; orthogonal] THEN MAP_EVERY X_GEN_TAC [`a':real^N`; `a'':real^N`] THEN ASM_CASES_TAC `a':real^N = vec 0` THENL [ASM_REWRITE_TAC[VECTOR_ADD_LID] THEN ASM_CASES_TAC `a'':real^N = a` THEN ASM_REWRITE_TAC[] THEN SIMP_TAC[SUBSET; IN_ELIM_THM; REAL_LE_REFL]; ALL_TAC] THEN STRIP_TAC THEN REPEAT DISJ2_TAC THEN EXISTS_TAC `a':real^N` THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `(a':real^N) dot z` THEN REPEAT(CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> (p x <=> q x)) ==> s INTER {x | p x} = s INTER {x | q x}`) THEN ASM_SIMP_TAC[DOT_LADD] THEN REAL_ARITH_TAC; ALL_TAC]) THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `x + a':real^N` THEN ASM_SIMP_TAC[SUBSPACE_ADD; VECTOR_ADD_ASSOC]]);; (* ------------------------------------------------------------------------- *) (* Affine dimension. *) (* ------------------------------------------------------------------------- *) let MAXIMAL_AFFINE_INDEPENDENT_SUBSET = prove (`!s b:real^N->bool. b SUBSET s /\ ~(affine_dependent b) /\ (!b'. b SUBSET b' /\ b' SUBSET s /\ ~(affine_dependent b') ==> b' = b) ==> s SUBSET (affine hull b)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC(SET_RULE `(!a. a IN t /\ ~(a IN s) ==> F) ==> t SUBSET s`) THEN X_GEN_TAC `a:real^N` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `(a:real^N) INSERT b`) THEN FIRST_ASSUM(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[GSYM CONTRAPOS_THM] HULL_INC)) THEN ASM_SIMP_TAC[AFFINE_INDEPENDENT_INSERT; INSERT_SUBSET] THEN ASM SET_TAC[]);; let MAXIMAL_AFFINE_INDEPENDENT_SUBSET_AFFINE = prove (`!s b:real^N->bool. affine s /\ b SUBSET s /\ ~(affine_dependent b) /\ (!b'. b SUBSET b' /\ b' SUBSET s /\ ~(affine_dependent b') ==> b' = b) ==> affine hull b = s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [ASM_MESON_TAC[HULL_MONO; HULL_P]; ASM_MESON_TAC[MAXIMAL_AFFINE_INDEPENDENT_SUBSET]]);; let EXTEND_TO_AFFINE_BASIS = prove (`!s u:real^N->bool. ~(affine_dependent s) /\ s SUBSET u ==> ?t. ~(affine_dependent t) /\ s SUBSET t /\ t SUBSET u /\ affine hull t = affine hull u`, REPEAT STRIP_TAC THEN MP_TAC(SPEC `\n. ?t:real^N->bool. ~(affine_dependent t) /\ s SUBSET t /\ t SUBSET u /\ CARD t = n` num_MAX) THEN DISCH_THEN(MP_TAC o fst o EQ_IMP_RULE) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_MESON_TAC[SUBSET_REFL; AFFINE_INDEPENDENT_CARD_LE]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `n:num` (CONJUNCTS_THEN2 MP_TAC ASSUME_TAC)) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [ASM_MESON_TAC[HULL_MONO; HULL_P]; ALL_TAC] THEN MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[AFFINE_AFFINE_HULL] THEN MATCH_MP_TAC MAXIMAL_AFFINE_INDEPENDENT_SUBSET THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `c:real^N->bool` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `CARD(c:real^N->bool)`) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN DISCH_THEN(MP_TAC o SPEC `c:real^N->bool`) THEN ANTS_TAC THENL [ASM SET_TAC[]; DISCH_TAC] THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC CARD_SUBSET_LE THEN ASM_MESON_TAC[AFFINE_INDEPENDENT_IMP_FINITE]);; let AFFINE_BASIS_EXISTS = prove (`!s:real^N->bool. ?b. ~(affine_dependent b) /\ b SUBSET s /\ affine hull b = affine hull s`, GEN_TAC THEN MP_TAC(ISPECL [`{}:real^N->bool`; `s:real^N->bool`] EXTEND_TO_AFFINE_BASIS) THEN REWRITE_TAC[AFFINE_INDEPENDENT_EMPTY; EMPTY_SUBSET]);; let aff_dim = new_definition `aff_dim s = @d:int. ?b. affine hull b = affine hull s /\ ~(affine_dependent b) /\ &(CARD b) = d + &1`;; let AFF_DIM = prove (`!s. ?b. affine hull b = affine hull s /\ ~(affine_dependent b) /\ aff_dim s = &(CARD b) - &1`, GEN_TAC THEN REWRITE_TAC[aff_dim; INT_ARITH `y:int = x + &1 <=> x = y - &1`] THEN CONV_TAC SELECT_CONV THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN REWRITE_TAC[RIGHT_EXISTS_AND_THM; EXISTS_REFL] THEN MESON_TAC[AFFINE_BASIS_EXISTS]);; let AFF_DIM_EMPTY = prove (`aff_dim {} = -- &1`, REWRITE_TAC[aff_dim; AFFINE_HULL_EMPTY; AFFINE_HULL_EQ_EMPTY] THEN REWRITE_TAC[UNWIND_THM2; AFFINE_INDEPENDENT_EMPTY; CARD_CLAUSES] THEN REWRITE_TAC[INT_ARITH `&0 = d + &1 <=> d:int = -- &1`; SELECT_REFL]);; let AFF_DIM_AFFINE_HULL = prove (`!s. aff_dim(affine hull s) = aff_dim s`, REWRITE_TAC[aff_dim; HULL_HULL]);; let AFF_DIM_TRANSLATION_EQ = prove (`!a:real^N s. aff_dim (IMAGE (\x. a + x) s) = aff_dim s`, REWRITE_TAC[aff_dim] THEN GEOM_TRANSLATE_TAC[] THEN ONCE_REWRITE_TAC[TAUT `a /\ b /\ c <=> ~(a /\ b ==> ~c)`] THEN SIMP_TAC[AFFINE_INDEPENDENT_IMP_FINITE; CARD_IMAGE_INJ; VECTOR_ARITH `a + x:real^N = a + y <=> x = y`]);; add_translation_invariants [AFF_DIM_TRANSLATION_EQ];; let AFFINE_HULL_CONIC_HULL = prove (`!s:real^N->bool. affine hull (conic hull s) = if s = {} then {} else affine hull (vec 0 INSERT s)`, GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[CONIC_HULL_EMPTY; AFFINE_HULL_EMPTY] THEN ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; IN_INSERT; CONIC_HULL_CONTAINS_0; HULL_INC; SPAN_INSERT_0; SPAN_CONIC_HULL]);; let AFFINE_INDEPENDENT_CARD_DIM_DIFFS = prove (`!s a:real^N. ~affine_dependent s /\ a IN s ==> CARD s = dim {x - a | x IN s} + 1`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN MATCH_MP_TAC(ARITH_RULE `~(s = 0) /\ v = s - 1 ==> s = v + 1`) THEN ASM_SIMP_TAC[CARD_EQ_0] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC DIM_UNIQUE THEN EXISTS_TAC `{b - a:real^N |b| b IN (s DELETE a)}` THEN REPEAT CONJ_TAC THENL [SET_TAC[]; REWRITE_TAC[SIMPLE_IMAGE; SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN ASM_CASES_TAC `x:real^N = a` THENL [ASM_REWRITE_TAC[VECTOR_SUB_REFL; SPAN_0]; MATCH_MP_TAC SPAN_SUPERSET THEN ASM SET_TAC[]]; UNDISCH_TAC `~affine_dependent(s:real^N->bool)` THEN REWRITE_TAC[independent; CONTRAPOS_THM] THEN DISCH_TAC THEN SUBGOAL_THEN `s = (a:real^N) INSERT (s DELETE a)` SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC DEPENDENT_IMP_AFFINE_DEPENDENT THEN ASM_REWRITE_TAC[IN_DELETE]; REWRITE_TAC[SIMPLE_IMAGE] THEN MATCH_MP_TAC HAS_SIZE_IMAGE_INJ THEN SIMP_TAC[VECTOR_ARITH `x - a:real^N = y - a <=> x = y`] THEN ASM_SIMP_TAC[HAS_SIZE; FINITE_DELETE; CARD_DELETE]]);; let AFF_DIM_DIM_0 = prove (`!s:real^N->bool. vec 0 IN affine hull s ==> aff_dim s = &(dim s)`, let lemma = prove (`!a:real^N s. affine s /\ a IN s ==> aff_dim s = &(dim {x - a | x IN s})`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `s:real^N->bool` AFF_DIM) THEN DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` MP_TAC) THEN ASM_CASES_TAC `b:real^N->bool = {}` THENL [ASM_MESON_TAC[AFFINE_HULL_EQ_EMPTY; NOT_IN_EMPTY]; ALL_TAC] THEN STRIP_TAC THEN ASM_REWRITE_TAC[INT_EQ_SUB_RADD; INT_OF_NUM_ADD; INT_OF_NUM_EQ] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_TAC `c:real^N`) THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `dim {x - c:real^N | x IN b} + 1` THEN CONJ_TAC THENL [MATCH_MP_TAC AFFINE_INDEPENDENT_CARD_DIM_DIFFS THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `dim {x - c:real^N | x IN affine hull b} + 1` THEN CONJ_TAC THENL [ASM_SIMP_TAC[DIFFS_AFFINE_HULL_SPAN; DIM_SPAN]; ALL_TAC] THEN ASM_REWRITE_TAC[] THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN SUBGOAL_THEN `affine hull s:real^N->bool = s` SUBST1_TAC THENL [ASM_MESON_TAC[AFFINE_HULL_EQ]; ALL_TAC] THEN SUBGOAL_THEN `(c:real^N) IN s` ASSUME_TAC THENL [ASM_MESON_TAC[AFFINE_HULL_EQ; HULL_INC]; ALL_TAC] THEN MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN REWRITE_TAC[IN_ELIM_THM] THEN SIMP_TAC[VECTOR_ARITH `x - c:real^N = y - a <=> y = x + &1 % (a - c)`] THEN ASM_MESON_TAC[IN_AFFINE_ADD_MUL_DIFF]) in REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`vec 0:real^N`; `affine hull s:real^N->bool`] lemma) THEN ASM_REWRITE_TAC[AFFINE_AFFINE_HULL; VECTOR_SUB_RZERO] THEN REWRITE_TAC[AFF_DIM_AFFINE_HULL; SET_RULE `{x | x IN s} = s`] THEN ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; DIM_SPAN]);; let AFF_DIM_DIM_SUBSPACE = prove (`!s:real^N->bool. subspace s ==> aff_dim s = &(dim s)`, MESON_TAC[AFF_DIM_DIM_0; SUBSPACE_0; HULL_INC]);; let AFF_DIM_DIM_AFFINE_DIFFS_STRONG = prove (`!a:real^N s. a IN affine hull s ==> aff_dim s = &(dim {x - a | x IN s})`, REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `a:real^N` THEN REWRITE_TAC[VECTOR_SUB_RZERO; SET_RULE `{x | x IN s} = s`] THEN REWRITE_TAC[AFF_DIM_DIM_0]);; let AFF_DIM_DIM_AFFINE_DIFFS = prove (`!a:real^N s. a IN s ==> aff_dim s = &(dim {x - a | x IN s})`, SIMP_TAC[AFF_DIM_DIM_AFFINE_DIFFS_STRONG; HULL_INC]);; let AFF_DIM_LINEAR_IMAGE_LE = prove (`!f:real^M->real^N s. linear f ==> aff_dim(IMAGE f s) <= aff_dim s`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN ASM_SIMP_TAC[AFFINE_HULL_LINEAR_IMAGE] THEN MP_TAC(ISPEC `s:real^M->bool` AFFINE_AFFINE_HULL) THEN SPEC_TAC(`affine hull s:real^M->bool`,`s:real^M->bool`) THEN GEN_TAC THEN DISCH_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_REWRITE_TAC[IMAGE_CLAUSES; AFF_DIM_EMPTY; INT_LE_REFL] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_TAC `a:real^M`) THEN SUBGOAL_THEN `dim {x - f(a) |x| x IN IMAGE (f:real^M->real^N) s} <= dim {x - a | x IN s}` MP_TAC THENL [REWRITE_TAC[SET_RULE `{f x | x IN IMAGE g s} = {f (g x) | x IN s}`] THEN ASM_SIMP_TAC[GSYM LINEAR_SUB] THEN REWRITE_TAC[SIMPLE_IMAGE] THEN ONCE_REWRITE_TAC[GSYM o_DEF] THEN REWRITE_TAC[IMAGE_o] THEN MATCH_MP_TAC DIM_LINEAR_IMAGE_LE THEN ASM_REWRITE_TAC[]; MATCH_MP_TAC EQ_IMP THEN REWRITE_TAC[GSYM INT_OF_NUM_LE] THEN BINOP_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC AFF_DIM_DIM_AFFINE_DIFFS THEN ASM_SIMP_TAC[AFFINE_LINEAR_IMAGE; FUN_IN_IMAGE]]);; let AFF_DIM_INJECTIVE_LINEAR_IMAGE = prove (`!f:real^M->real^N s. linear f /\ (!x y. f x = f y ==> x = y) ==> aff_dim(IMAGE f s) = aff_dim s`, REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM INT_LE_ANTISYM] THEN CONJ_TAC THENL [ASM_MESON_TAC[AFF_DIM_LINEAR_IMAGE_LE]; ALL_TAC] THEN MP_TAC(ISPEC `f:real^M->real^N` LINEAR_INJECTIVE_LEFT_INVERSE) THEN ASM_REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^N->real^M` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC INT_LE_TRANS THEN EXISTS_TAC `aff_dim(IMAGE (g:real^N->real^M) (IMAGE (f:real^M->real^N) s))` THEN CONJ_TAC THENL [ASM_REWRITE_TAC[GSYM IMAGE_o; o_DEF; IMAGE_ID; INT_LE_REFL]; MATCH_MP_TAC AFF_DIM_LINEAR_IMAGE_LE THEN ASM_REWRITE_TAC[]]);; add_linear_invariants [AFF_DIM_INJECTIVE_LINEAR_IMAGE];; let AFF_DIM_AFFINE_INDEPENDENT = prove (`!b:real^N->bool. ~(affine_dependent b) ==> aff_dim b = &(CARD b) - &1`, GEN_TAC THEN ASM_CASES_TAC `b:real^N->bool = {}` THENL [ASM_REWRITE_TAC[CARD_CLAUSES; AFF_DIM_EMPTY] THEN INT_ARITH_TAC; ALL_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 DISCH_TAC THEN MP_TAC(ISPECL [`b:real^N->bool`; `a:real^N`] AFFINE_INDEPENDENT_CARD_DIM_DIFFS) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[GSYM INT_OF_NUM_ADD; INT_ARITH `(a + b) - b:int = a`] THEN MP_TAC(ISPECL [`a:real^N`; `affine hull b:real^N->bool`] AFF_DIM_DIM_AFFINE_DIFFS) THEN ASM_SIMP_TAC[AFFINE_AFFINE_HULL; HULL_INC; AFF_DIM_AFFINE_HULL] THEN DISCH_THEN(K ALL_TAC) THEN AP_TERM_TAC THEN ASM_MESON_TAC[DIFFS_AFFINE_HULL_SPAN; DIM_SPAN]);; let AFF_DIM_UNIQUE = prove (`!s b:real^N->bool. affine hull b = affine hull s /\ ~(affine_dependent b) ==> aff_dim s = &(CARD b) - &1`, MESON_TAC[AFF_DIM_AFFINE_HULL; AFF_DIM_AFFINE_INDEPENDENT]);; let AFF_DIM_SING = prove (`!a:real^N. aff_dim {a} = &0`, GEN_TAC THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `&(CARD {a:real^N}) - &1:int` THEN CONJ_TAC THENL [MATCH_MP_TAC AFF_DIM_AFFINE_INDEPENDENT THEN REWRITE_TAC[AFFINE_INDEPENDENT_1]; SIMP_TAC[CARD_CLAUSES; FINITE_RULES; ARITH; NOT_IN_EMPTY; INT_SUB_REFL]]);; let AFF_DIM_LE_CARD = prove (`!s:real^N->bool. FINITE s ==> aff_dim s <= &(CARD s) - &1`, MATCH_MP_TAC SET_PROVE_CASES THEN SIMP_TAC[AFF_DIM_EMPTY; CARD_CLAUSES] THEN CONV_TAC INT_REDUCE_CONV THEN GEOM_ORIGIN_TAC `a:real^N` THEN SIMP_TAC[AFF_DIM_DIM_0; IN_INSERT; HULL_INC] THEN SIMP_TAC[CARD_IMAGE_INJ; VECTOR_ARITH `a + x:real^N = a + y <=> x = y`] THEN SIMP_TAC[DIM_INSERT_0; INT_LE_SUB_LADD; CARD_CLAUSES; FINITE_INSERT] THEN REWRITE_TAC[INT_OF_NUM_ADD; INT_OF_NUM_LE; ADD1; LE_ADD_RCANCEL] THEN SIMP_TAC[DIM_LE_CARD]);; let AFF_DIM_GE = prove (`!s:real^N->bool. -- &1 <= aff_dim s`, GEN_TAC THEN MP_TAC(ISPEC `s:real^N->bool` AFF_DIM) THEN STRIP_TAC THEN ASM_REWRITE_TAC[INT_LE_SUB_LADD; INT_ADD_LINV; INT_POS]);; let AFF_DIM_SUBSET = prove (`!s t:real^N->bool. s SUBSET t ==> aff_dim s <= aff_dim t`, MATCH_MP_TAC SET_PROVE_CASES THEN REWRITE_TAC[AFF_DIM_GE; AFF_DIM_EMPTY] THEN GEOM_ORIGIN_TAC `a:real^N` THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `(vec 0:real^N) IN t` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[AFF_DIM_DIM_0; IN_INSERT; HULL_INC; INT_OF_NUM_LE; DIM_SUBSET]);; let AFF_DIM_LE_DIM = prove (`!s:real^N->bool. aff_dim s <= &(dim s)`, GEN_TAC THEN ONCE_REWRITE_TAC[GSYM DIM_SPAN] THEN ASM_SIMP_TAC[GSYM AFF_DIM_DIM_SUBSPACE; SUBSPACE_SPAN] THEN MATCH_MP_TAC AFF_DIM_SUBSET THEN REWRITE_TAC[SPAN_INC]);; let AFF_DIM_CONVEX_HULL = prove (`!s:real^N->bool. aff_dim(convex hull s) = aff_dim s`, GEN_TAC THEN MATCH_MP_TAC(INT_ARITH `!c:int. c = a /\ a <= b /\ b <= c ==> b = a`) THEN EXISTS_TAC `aff_dim(affine hull s:real^N->bool)` THEN SIMP_TAC[AFF_DIM_AFFINE_HULL; AFF_DIM_SUBSET; HULL_SUBSET; CONVEX_HULL_SUBSET_AFFINE_HULL]);; let AFF_DIM_CLOSURE = prove (`!s:real^N->bool. aff_dim(closure s) = aff_dim s`, GEN_TAC THEN MATCH_MP_TAC(INT_ARITH `!h. h = s /\ s <= c /\ c <= h ==> c:int = s`) THEN EXISTS_TAC `aff_dim(affine hull s:real^N->bool)` THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[AFF_DIM_AFFINE_HULL]; MATCH_MP_TAC AFF_DIM_SUBSET THEN REWRITE_TAC[CLOSURE_SUBSET]; MATCH_MP_TAC AFF_DIM_SUBSET THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN REWRITE_TAC[CLOSED_AFFINE_HULL; HULL_SUBSET]]);; let AFF_DIM_2 = prove (`!a b:real^N. aff_dim {a,b} = if a = b then &0 else &1`, REPEAT GEN_TAC THEN COND_CASES_TAC THENL [ASM_REWRITE_TAC[INSERT_AC; AFF_DIM_SING]; ALL_TAC] THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `&(CARD {a:real^N,b}) - &1:int` THEN ASM_SIMP_TAC[AFF_DIM_AFFINE_INDEPENDENT; AFFINE_INDEPENDENT_2] THEN ASM_SIMP_TAC[CARD_CLAUSES; FINITE_RULES; IN_INSERT; NOT_IN_EMPTY] THEN CONV_TAC NUM_REDUCE_CONV THEN INT_ARITH_TAC);; let AFF_DIM_EQ_MINUS1 = prove (`!s:real^N->bool. aff_dim s = -- &1 <=> s = {}`, GEN_TAC THEN EQ_TAC THEN SIMP_TAC[AFF_DIM_EMPTY] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN MATCH_MP_TAC(INT_ARITH `&0:int <= n ==> ~(n = -- &1)`) THEN MATCH_MP_TAC INT_LE_TRANS THEN EXISTS_TAC `aff_dim {a:real^N}` THEN ASM_SIMP_TAC[AFF_DIM_SUBSET; SING_SUBSET] THEN REWRITE_TAC[AFF_DIM_SING; INT_LE_REFL]);; let AFF_DIM_POS_LE = prove (`!s:real^N->bool. &0 <= aff_dim s <=> ~(s = {})`, GEN_TAC THEN REWRITE_TAC[GSYM AFF_DIM_EQ_MINUS1] THEN MP_TAC(ISPEC `s:real^N->bool` AFF_DIM_GE) THEN INT_ARITH_TAC);; let AFF_DIM_EQ_0 = prove (`!s:real^N->bool. aff_dim s = &0 <=> ?a. s = {a}`, GEN_TAC THEN EQ_TAC THEN SIMP_TAC[AFF_DIM_SING; LEFT_IMP_EXISTS_THM] THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[AFF_DIM_EMPTY] THEN CONV_TAC INT_REDUCE_CONV THEN DISCH_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 MATCH_MP_TAC(SET_RULE `(!b. ~(b = a) /\ {a,b} SUBSET s ==> F) ==> a IN s ==> s = {a}`) THEN X_GEN_TAC `b:real^N` THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP AFF_DIM_SUBSET) THEN MP_TAC(ISPECL [`a:real^N`; `b:real^N`] AFF_DIM_2) THEN ASM_SIMP_TAC[] THEN INT_ARITH_TAC);; let CONNECTED_IMP_PERFECT_AFF_DIM = prove (`!s x:real^N. connected s /\ ~(aff_dim s = &0) /\ x IN s ==> x limit_point_of s`, REWRITE_TAC[AFF_DIM_EQ_0; CONNECTED_IMP_PERFECT]);; let AFF_DIM_UNIV = prove (`aff_dim(:real^N) = &(dimindex(:N))`, SIMP_TAC[AFF_DIM_DIM_SUBSPACE; SUBSPACE_UNIV; DIM_UNIV]);; let AFF_DIM_EQ_AFFINE_HULL = prove (`!s t:real^N->bool. s SUBSET t /\ aff_dim t <= aff_dim s ==> affine hull s = affine hull t`, MATCH_MP_TAC SET_PROVE_CASES THEN SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_EQ_MINUS1; AFF_DIM_GE; INT_ARITH `a:int <= x ==> (x <= a <=> x = a)`] THEN X_GEN_TAC `a:real^N` THEN GEOM_ORIGIN_TAC `a:real^N` THEN SIMP_TAC[INSERT_SUBSET; IMP_CONJ; AFF_DIM_DIM_0; IN_INSERT; DIM_EQ_SPAN; HULL_INC; AFFINE_HULL_EQ_SPAN; INT_OF_NUM_LE]);; let AFF_DIM_SUMS_INTER = prove (`!s t:real^N->bool. affine s /\ affine t /\ ~(s INTER t = {}) ==> aff_dim {x + y | x IN s /\ y IN t} = (aff_dim s + aff_dim t) - aff_dim(s INTER t)`, REWRITE_TAC[TAUT `a /\ b /\ c ==> d <=> c ==> a /\ b ==> d`] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM] THEN GEN_REWRITE_TAC BINDER_CONV [SWAP_FORALL_THM] THEN GEN_REWRITE_TAC I [SWAP_FORALL_THM] THEN X_GEN_TAC `a:real^N` THEN GEOM_ORIGIN_TAC `a:real^N` THEN REWRITE_TAC[VECTOR_ARITH `(a + x) + (a + y):real^N = &2 % a + (x + y)`] THEN ONCE_REWRITE_TAC[SET_RULE `{a + x + y:real^N | x IN s /\ y IN t} = IMAGE (\x. a + x) {x + y | x IN s /\ y IN t}`] THEN REWRITE_TAC[AFF_DIM_TRANSLATION_EQ; IN_INTER] THEN MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `t:real^N->bool`] THEN STRIP_TAC THEN STRIP_TAC THEN SUBGOAL_THEN `(vec 0:real^N) IN {x + y | x IN s /\ y IN t}` ASSUME_TAC THENL [REWRITE_TAC[IN_ELIM_THM] THEN REPEAT(EXISTS_TAC `vec 0:real^N`) THEN ASM_REWRITE_TAC[VECTOR_ADD_LID]; ALL_TAC] THEN ASM_SIMP_TAC[AFF_DIM_DIM_0; HULL_INC; IN_INTER] THEN REWRITE_TAC[INT_EQ_SUB_LADD; INT_OF_NUM_ADD; INT_OF_NUM_EQ] THEN MATCH_MP_TAC DIM_SUMS_INTER THEN ASM_SIMP_TAC[AFFINE_IMP_SUBSPACE]);; let AFF_DIM_PSUBSET = prove (`!s t. (affine hull s) PSUBSET (affine hull t) ==> aff_dim s < aff_dim t`, ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN SIMP_TAC[PSUBSET; AFF_DIM_SUBSET; INT_LT_LE] THEN MESON_TAC[INT_EQ_IMP_LE; AFF_DIM_EQ_AFFINE_HULL; HULL_HULL]);; let AFF_DIM_EQ_FULL_GEN = prove (`!s t:real^N->bool. s SUBSET t ==> (aff_dim s = aff_dim t <=> affine hull s = affine hull t)`, REPEAT STRIP_TAC THEN EQ_TAC THENL [ALL_TAC; MESON_TAC[AFF_DIM_AFFINE_HULL]] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC AFF_DIM_EQ_AFFINE_HULL THEN ASM_REWRITE_TAC[INT_LE_REFL]);; let AFF_DIM_EQ_FULL = prove (`!s. aff_dim s = &(dimindex(:N)) <=> affine hull s = (:real^N)`, SIMP_TAC[GSYM AFF_DIM_UNIV; SUBSET_UNIV; AFF_DIM_EQ_FULL_GEN] THEN REWRITE_TAC[AFFINE_HULL_UNIV]);; let AFF_DIM_LE_UNIV = prove (`!s:real^N->bool. aff_dim s <= &(dimindex(:N))`, GEN_TAC THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_UNIV] THEN MATCH_MP_TAC AFF_DIM_SUBSET THEN REWRITE_TAC[SUBSET_UNIV]);; let AFFINE_INDEPENDENT_IFF_CARD = prove (`!s:real^N->bool. ~affine_dependent s <=> FINITE s /\ aff_dim s = &(CARD s) - &1`, GEN_TAC THEN EQ_TAC THEN SIMP_TAC[AFF_DIM_AFFINE_INDEPENDENT; AFFINE_INDEPENDENT_IMP_FINITE] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[] THEN DISCH_TAC THEN X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC (ISPEC `s:real^N->bool` AFFINE_BASIS_EXISTS) THEN MATCH_MP_TAC(ARITH_RULE `!b:int. a <= b - &1 /\ b < s ==> ~(a = s - &1)`) THEN EXISTS_TAC `&(CARD(b:real^N->bool)):int` THEN CONJ_TAC THENL [ASM_MESON_TAC[AFF_DIM_LE_CARD; FINITE_SUBSET; AFF_DIM_AFFINE_HULL]; REWRITE_TAC[INT_OF_NUM_LT] THEN MATCH_MP_TAC CARD_PSUBSET THEN ASM_REWRITE_TAC[PSUBSET] THEN ASM_MESON_TAC[]]);; let AFFINE_HULL_CONVEX_INTER_NONEMPTY_INTERIOR = prove (`!s t:real^N->bool. convex s /\ ~(s INTER interior t = {}) ==> affine hull (s INTER t) = affine hull s`, REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; RIGHT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `t:real^N->bool`; `a:real^N`] THEN GEOM_ORIGIN_TAC `a:real^N` THEN REWRITE_TAC[IN_INTER] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN SIMP_TAC[HULL_MONO; INTER_SUBSET] THEN SIMP_TAC[SUBSET_HULL; AFFINE_AFFINE_HULL] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP (SIMP_RULE[SUBSET] INTERIOR_SUBSET)) THEN ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC; IN_INTER] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERIOR_CBALL]) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; SUBSET; IN_CBALL_0] THEN X_GEN_TAC `e:real` THEN STRIP_TAC THEN REWRITE_TAC[EXTENSION; IN_UNIV] THEN X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `x:real^N = vec 0` THEN ASM_SIMP_TAC[SPAN_SUPERSET; IN_INTER] THEN DISCH_TAC THEN ABBREV_TAC `k = min (&1 / &2) (e / norm(x:real^N))` THEN SUBGOAL_THEN `&0 < k /\ k < &1` STRIP_ASSUME_TAC THENL [EXPAND_TAC "k" THEN ASM_SIMP_TAC[REAL_LT_MIN; REAL_LT_DIV; NORM_POS_LT; REAL_MIN_LT] THEN CONV_TAC REAL_RAT_REDUCE_CONV; ALL_TAC] THEN SUBGOAL_THEN `x:real^N = inv k % k % x` SUBST1_TAC THENL [ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; VECTOR_MUL_LID; REAL_LT_IMP_NZ]; ALL_TAC] THEN MATCH_MP_TAC SPAN_MUL THEN MATCH_MP_TAC SPAN_SUPERSET THEN REWRITE_TAC[IN_INTER] THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[VECTOR_ARITH `k % x:real^N = (&1 - k) % vec 0 + k % x`] THEN MATCH_MP_TAC IN_CONVEX_SET THEN ASM_SIMP_TAC[REAL_LT_IMP_LE]; FIRST_X_ASSUM MATCH_MP_TAC THEN EXPAND_TAC "k" THEN ASM_SIMP_TAC[NORM_MUL; GSYM REAL_LE_RDIV_EQ; NORM_POS_LT] THEN ASM_REAL_ARITH_TAC]);; let AFFINE_HULL_CONVEX_INTER_OPEN = prove (`!s t:real^N->bool. convex s /\ open t /\ ~(s INTER t = {}) ==> affine hull (s INTER t) = affine hull s`, ASM_SIMP_TAC[AFFINE_HULL_CONVEX_INTER_NONEMPTY_INTERIOR; INTERIOR_OPEN]);; let AFFINE_HULL_AFFINE_INTER_NONEMPTY_INTERIOR = prove (`!s t:real^N->bool. affine s /\ ~(s INTER interior t = {}) ==> affine hull (s INTER t) = s`, SIMP_TAC[AFFINE_HULL_CONVEX_INTER_NONEMPTY_INTERIOR; AFFINE_IMP_CONVEX; HULL_P]);; let AFFINE_HULL_AFFINE_INTER_OPEN = prove (`!s t:real^N->bool. affine s /\ open t /\ ~(s INTER t = {}) ==> affine hull (s INTER t) = s`, SIMP_TAC[AFFINE_HULL_AFFINE_INTER_NONEMPTY_INTERIOR; INTERIOR_OPEN]);; let CONVEX_AND_AFFINE_INTER_OPEN = prove (`!s t u:real^N->bool. convex s /\ affine t /\ open u /\ s INTER u = t INTER u /\ ~(s INTER u = {}) ==> affine hull s = t`, REPEAT STRIP_TAC THEN MATCH_MP_TAC(MESON[] `!u v. x = u /\ u = v /\ v = y ==> x = y`) THEN MAP_EVERY EXISTS_TAC [`affine hull (s INTER u:real^N->bool)`; `affine hull t:real^N->bool`] THEN REPEAT CONJ_TAC THENL [CONV_TAC SYM_CONV THEN MATCH_MP_TAC AFFINE_HULL_CONVEX_INTER_OPEN THEN ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[] THEN MATCH_MP_TAC AFFINE_HULL_CONVEX_INTER_OPEN THEN ASM_SIMP_TAC[AFFINE_IMP_CONVEX] THEN ASM SET_TAC[]; ASM_REWRITE_TAC[AFFINE_HULL_EQ]]);; let AFFINE_HULL_CONVEX_INTER_OPEN_IN = prove (`!s t:real^N->bool. convex s /\ open_in (subtopology euclidean (affine hull s)) t /\ ~(s INTER t = {}) ==> affine hull (s INTER t) = affine hull s`, REPEAT STRIP_TAC 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_SIMP_TAC[SET_RULE `s SUBSET t ==> s INTER t INTER u = s INTER u`; HULL_SUBSET] THEN MATCH_MP_TAC AFFINE_HULL_CONVEX_INTER_OPEN THEN ASM SET_TAC[]);; let AFFINE_HULL_AFFINE_INTER_OPEN_IN = prove (`!s t:real^N->bool. affine s /\ open_in (subtopology euclidean s) t /\ ~(s INTER t = {}) ==> affine hull (s INTER t) = s`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`affine hull s:real^N->bool`; `t:real^N->bool`] AFFINE_HULL_CONVEX_INTER_OPEN_IN) THEN ASM_SIMP_TAC[HULL_HULL; AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL; HULL_P]);; let AFFINE_HULL_OPEN_IN_CONVEX = prove (`!s t:real^N->bool. convex s /\ open_in (subtopology euclidean s) t /\ ~(t = {}) ==> affine hull t = affine hull s`, REPEAT STRIP_TAC 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[] THEN MATCH_MP_TAC AFFINE_HULL_CONVEX_INTER_OPEN THEN REWRITE_TAC[AFFINE_AFFINE_HULL] THEN ASM SET_TAC[]);; let AFFINE_HULL_OPEN_IN = prove (`!s t:real^N->bool. open_in (subtopology euclidean (affine hull t)) s /\ ~(s = {}) ==> affine hull s = affine hull t`, REPEAT STRIP_TAC THEN GEN_REWRITE_TAC RAND_CONV [GSYM HULL_HULL] THEN MATCH_MP_TAC AFFINE_HULL_OPEN_IN_CONVEX THEN ASM_SIMP_TAC[AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL]);; let AFFINE_HULL_OPEN_IN_AFFINE = prove (`!u s:real^N->bool. affine u /\ open_in (subtopology euclidean u) s /\ ~(s = {}) ==> affine hull s = u`, MESON_TAC[AFFINE_HULL_OPEN_IN; AFFINE_HULL_EQ]);; let AFFINE_HULL_OPEN = prove (`!s. open s /\ ~(s = {}) ==> affine hull s = (:real^N)`, GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN SUBST1_TAC(SET_RULE `s = (:real^N) INTER s`) THEN ASM_SIMP_TAC[AFFINE_HULL_CONVEX_INTER_OPEN; CONVEX_UNIV] THEN REWRITE_TAC[AFFINE_HULL_UNIV]);; let AFFINE_HULL_NONEMPTY_INTERIOR = prove (`!s. ~(interior s = {}) ==> affine hull s = (:real^N)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ s = UNIV ==> t = UNIV`) THEN EXISTS_TAC `affine hull (interior s:real^N->bool)` THEN SIMP_TAC[HULL_MONO; INTERIOR_SUBSET] THEN ASM_SIMP_TAC[AFFINE_HULL_OPEN; OPEN_INTERIOR]);; let AFF_DIM_OPEN = prove (`!s:real^N->bool. open s /\ ~(s = {}) ==> aff_dim s = &(dimindex(:N))`, SIMP_TAC[AFF_DIM_EQ_FULL; AFFINE_HULL_OPEN]);; let AFF_DIM_NONEMPTY_INTERIOR = prove (`!s:real^N->bool. ~(interior s = {}) ==> aff_dim s = &(dimindex(:N))`, SIMP_TAC[AFF_DIM_EQ_FULL; AFFINE_HULL_NONEMPTY_INTERIOR]);; let EMPTY_INTERIOR_AFF_DIM = prove (`!s:real^N->bool. aff_dim s < &(dimindex(:N)) ==> interior s = {}`, MESON_TAC[AFF_DIM_NONEMPTY_INTERIOR; INT_LT_ANTISYM]);; let SPAN_OPEN = prove (`!s. open s /\ ~(s = {}) ==> span s = (:real^N)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ s = UNIV ==> t = UNIV`) THEN EXISTS_TAC `affine hull s:real^N->bool` THEN ASM_SIMP_TAC[AFFINE_HULL_OPEN; AFFINE_HULL_SUBSET_SPAN]);; let DIM_OPEN = prove (`!s:real^N->bool. open s /\ ~(s = {}) ==> dim s = dimindex(:N)`, SIMP_TAC[DIM_EQ_FULL; SPAN_OPEN]);; let AFF_DIM_INSERT = prove (`!a:real^N s. aff_dim (a INSERT s) = if a IN affine hull s then aff_dim s else aff_dim s + &1`, ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN MATCH_MP_TAC SET_PROVE_CASES THEN SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_SING; AFFINE_HULL_EMPTY; NOT_IN_EMPTY] THEN CONV_TAC INT_REDUCE_CONV THEN REWRITE_TAC[RIGHT_IMP_FORALL_THM] THEN MAP_EVERY X_GEN_TAC [`b:real^N`; `s:real^N->bool`; `a:real^N`] THEN GEOM_ORIGIN_TAC `b:real^N` THEN SIMP_TAC[AFFINE_HULL_EQ_SPAN; AFF_DIM_DIM_0; HULL_INC; IN_INSERT] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `s:real^N->bool`] THEN DISCH_THEN(K ALL_TAC) THEN SPEC_TAC(`(vec 0:real^N) INSERT s`,`s:real^N->bool`) THEN SIMP_TAC[DIM_INSERT; INT_OF_NUM_ADD] THEN MESON_TAC[]);; let AFF_DIM_DIM = prove (`!s:real^N->bool. aff_dim s = if vec 0 IN affine hull s then &(dim s) else &(dim s) - &1`, GEN_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[AFF_DIM_DIM_0] THEN MP_TAC(ISPECL [`vec 0:real^N`; `s:real^N->bool`] AFF_DIM_INSERT) THEN ASM_SIMP_TAC[AFF_DIM_DIM_0; IN_INSERT; HULL_INC; DIM_INSERT_0] THEN INT_ARITH_TAC);; let AFF_DIM_CONIC_HULL_DIM = prove (`!s:real^N->bool. aff_dim (conic hull s) = if s = {} then -- &1 else &(dim s)`, GEN_TAC THEN REWRITE_TAC[AFF_DIM_DIM; AFFINE_HULL_CONIC_HULL] THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; SPAN_INSERT_0; HULL_INC; IN_INSERT; SPAN_0; DIM_CONIC_HULL; NOT_IN_EMPTY; DIM_EMPTY; INT_SUB_LZERO]);; let AFFINE_BOUNDED_EQ_TRIVIAL = prove (`!s:real^N->bool. affine s ==> (bounded s <=> s = {} \/ ?a. s = {a})`, GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[BOUNDED_EMPTY] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_THEN `b:real^N` MP_TAC) THEN GEOM_ORIGIN_TAC `b:real^N` THEN SIMP_TAC[AFFINE_EQ_SUBSPACE] THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[SUBSPACE_BOUNDED_EQ_TRIVIAL] THEN FIRST_ASSUM(MP_TAC o MATCH_MP SUBSPACE_0) THEN SET_TAC[]);; let AFFINE_BOUNDED_EQ_LOWDIM = prove (`!s:real^N->bool. affine s ==> (bounded s <=> aff_dim s <= &0)`, SIMP_TAC[AFF_DIM_GE; INT_ARITH `--(&1):int <= x ==> (x <= &0 <=> x = --(&1) \/ x = &0)`] THEN SIMP_TAC[AFF_DIM_EQ_0; AFF_DIM_EQ_MINUS1; AFFINE_BOUNDED_EQ_TRIVIAL]);; let COLLINEAR_AFF_DIM = prove (`!s:real^N->bool. collinear s <=> aff_dim s <= &1`, GEN_TAC THEN EQ_TAC THENL [REWRITE_TAC[COLLINEAR_AFFINE_HULL; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u:real^N`; `v:real^N`] THEN STRIP_TAC THEN MATCH_MP_TAC INT_LE_TRANS THEN EXISTS_TAC `aff_dim{u:real^N,v}` THEN CONJ_TAC THENL [ASM_MESON_TAC[AFF_DIM_SUBSET; AFF_DIM_AFFINE_HULL]; MATCH_MP_TAC INT_LE_TRANS THEN EXISTS_TAC `&(CARD{u:real^N,v}) - &1:int` THEN SIMP_TAC[AFF_DIM_LE_CARD; FINITE_INSERT; FINITE_EMPTY] THEN REWRITE_TAC[INT_ARITH `x - &1:int <= &1 <=> x <= &2`; INT_OF_NUM_LE] THEN SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN ARITH_TAC]; ONCE_REWRITE_TAC[GSYM COLLINEAR_AFFINE_HULL_COLLINEAR; GSYM AFF_DIM_AFFINE_HULL] THEN MP_TAC(ISPEC `s:real^N->bool` AFFINE_BASIS_EXISTS) THEN DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` (STRIP_ASSUME_TAC o GSYM)) THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [AFFINE_INDEPENDENT_IFF_CARD]) THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_REWRITE_TAC[COLLINEAR_AFFINE_HULL_COLLINEAR; AFF_DIM_AFFINE_HULL] THEN REWRITE_TAC[INT_ARITH `x - &1:int <= &1 <=> x <= &2`; INT_OF_NUM_LE] THEN ASM_SIMP_TAC[COLLINEAR_SMALL]]);; let COPLANAR_AFF_DIM = prove (`!s:real^N->bool. coplanar s <=> aff_dim s <= &2`, GEN_TAC THEN EQ_TAC THENL [REWRITE_TAC[coplanar; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`; `c:real^N`] THEN DISCH_TAC THEN TRANS_TAC INT_LE_TRANS `aff_dim(affine hull {a:real^N,b,c})` THEN ASM_SIMP_TAC[AFF_DIM_SUBSET] THEN REWRITE_TAC[AFF_DIM_AFFINE_HULL; AFF_DIM_INSERT] THEN REWRITE_TAC[AFFINE_HULL_EMPTY; NOT_IN_EMPTY; AFF_DIM_EMPTY] THEN INT_ARITH_TAC; DISCH_TAC THEN ONCE_REWRITE_TAC[GSYM COPLANAR_AFFINE_HULL_COPLANAR] THEN MP_TAC(ISPEC `s:real^N->bool` AFF_DIM) THEN DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 (SUBST1_TAC o SYM) MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 (ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) SUBST_ALL_TAC) THEN REWRITE_TAC[COPLANAR_AFFINE_HULL_COPLANAR] THEN MATCH_MP_TAC COPLANAR_SMALL THEN ASM_REWRITE_TAC[GSYM INT_OF_NUM_LE] THEN ASM_INT_ARITH_TAC]);; let HOMEOMORPHIC_AFFINE_SETS = prove (`!s:real^M->bool t:real^N->bool. affine s /\ affine t /\ aff_dim s = aff_dim t ==> s homeomorphic t`, REPEAT GEN_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_EQ_MINUS1; HOMEOMORPHIC_EMPTY] THEN POP_ASSUM MP_TAC THEN GEN_REWRITE_TAC (RAND_CONV o LAND_CONV o ONCE_DEPTH_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; AFF_DIM_DIM_0; HULL_INC; INT_OF_NUM_EQ] THEN MESON_TAC[HOMEOMORPHIC_SUBSPACES]);; let AFF_DIM_OPEN_IN = prove (`!s t:real^N->bool. ~(s = {}) /\ open_in (subtopology euclidean t) s /\ affine t ==> aff_dim s = aff_dim t`, REPEAT GEN_TAC THEN REWRITE_TAC[IMP_CONJ; GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `a:real^N` THEN GEOM_ORIGIN_TAC `a:real^N` THEN REPEAT GEN_TAC THEN STRIP_TAC THEN DISCH_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN SUBGOAL_THEN `(vec 0:real^N) IN t` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[AFF_DIM_DIM_0; HULL_INC; AFFINE_EQ_SUBSPACE] THEN DISCH_TAC THEN AP_TERM_TAC THEN ASM_SIMP_TAC[GSYM LE_ANTISYM; DIM_SUBSET] THEN SUBGOAL_THEN `?e. &0 < e /\ cball(vec 0:real^N,e) INTER t SUBSET s` MP_TAC THENL [FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN FIRST_X_ASSUM(MP_TAC o SPEC `vec 0:real^N` o GEN_REWRITE_RULE I [OPEN_CONTAINS_CBALL]) THEN ASM SET_TAC[]; REWRITE_TAC[SUBSET; IN_INTER; IN_CBALL_0] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC)] THEN FIRST_ASSUM(MP_TAC o MATCH_MP ORTHONORMAL_BASIS_SUBSPACE) THEN DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`s:real^N->bool`; `IMAGE (\x:real^N. e % x) b`] INDEPENDENT_CARD_LE_DIM) THEN RULE_ASSUM_TAC(REWRITE_RULE[HAS_SIZE]) THEN ASM_SIMP_TAC[CARD_IMAGE_INJ; VECTOR_MUL_LCANCEL; REAL_LT_IMP_NZ] THEN ANTS_TAC THENL [REWRITE_TAC[SUBSET]; MESON_TAC[]] THEN CONJ_TAC THENL [REWRITE_TAC[FORALL_IN_IMAGE] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[NORM_MUL] THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; MATCH_MP_TAC SUBSPACE_MUL] THEN ASM SET_TAC[]; MATCH_MP_TAC INDEPENDENT_INJECTIVE_IMAGE THEN ASM_SIMP_TAC[VECTOR_MUL_LCANCEL; REAL_LT_IMP_NZ; LINEAR_SCALING]]);; let DIM_OPEN_IN = prove (`!s t:real^N->bool. ~(s = {}) /\ open_in (subtopology euclidean t) s /\ subspace t ==> dim s = dim t`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ASM_SIMP_TAC[GSYM LE_ANTISYM; DIM_SUBSET] THEN REWRITE_TAC[GSYM INT_OF_NUM_LE] THEN MATCH_MP_TAC INT_LE_TRANS THEN EXISTS_TAC `aff_dim(s:real^N->bool)` THEN REWRITE_TAC[AFF_DIM_LE_DIM] THEN ASM_SIMP_TAC[GSYM AFF_DIM_DIM_SUBSPACE] THEN MATCH_MP_TAC INT_EQ_IMP_LE THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC AFF_DIM_OPEN_IN THEN ASM_SIMP_TAC[SUBSPACE_IMP_AFFINE]);; let AFF_DIM_CONVEX_INTER_NONEMPTY_INTERIOR = prove (`!s t:real^N->bool. convex s /\ ~(s INTER interior t = {}) ==> aff_dim(s INTER t) = aff_dim s`, ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN ASM_SIMP_TAC[AFFINE_HULL_CONVEX_INTER_NONEMPTY_INTERIOR] THEN REWRITE_TAC[AFF_DIM_AFFINE_HULL]);; let AFF_DIM_CONVEX_INTER_OPEN = prove (`!s t:real^N->bool. convex s /\ open t /\ ~(s INTER t = {}) ==> aff_dim(s INTER t) = aff_dim s`, ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN ASM_SIMP_TAC[AFFINE_HULL_CONVEX_INTER_OPEN] THEN REWRITE_TAC[AFF_DIM_AFFINE_HULL]);; let AFF_DIM_NONEMPTY_INTERIOR_OF = prove (`!u s:real^N->bool. s SUBSET u /\ affine u /\ ~((subtopology euclidean u) interior_of s = {}) ==> aff_dim s = aff_dim u`, SIMP_TAC[GSYM INT_LE_ANTISYM; AFF_DIM_SUBSET] THEN REPEAT STRIP_TAC THEN TRANS_TAC INT_LE_TRANS `aff_dim((subtopology euclidean u) interior_of s:real^N->bool)` THEN SIMP_TAC[AFF_DIM_SUBSET; INTERIOR_OF_SUBSET] THEN MATCH_MP_TAC INT_EQ_IMP_LE THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC AFF_DIM_OPEN_IN THEN ASM_REWRITE_TAC[OPEN_IN_INTERIOR_OF]);; let EMPTY_INTERIOR_OF_AFF_DIM = prove (`!u s:real^N->bool. affine u /\ aff_dim s < aff_dim u ==> (subtopology euclidean u) interior_of s = {}`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[INTERIOR_OF_RESTRICT] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN MP_TAC(ISPECL [`u:real^N->bool`; `u INTER s:real^N->bool`] AFF_DIM_NONEMPTY_INTERIOR_OF) THEN ASM_REWRITE_TAC[INTER_SUBSET] THEN MATCH_MP_TAC(TAUT `~q ==> (~p ==> q) ==> p`) THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (INT_ARITH `s:int < u ==> t <= s ==> ~(t = u)`)) THEN SIMP_TAC[AFF_DIM_SUBSET; INTER_SUBSET]);; let AFFINE_HULL_HALFSPACE_LT = prove (`!a b. affine hull {x | a dot x < b} = if a = vec 0 /\ b <= &0 then {} else (:real^N)`, REPEAT GEN_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[AFFINE_HULL_EQ_EMPTY; HALFSPACE_EQ_EMPTY_LT; AFFINE_HULL_OPEN; OPEN_HALFSPACE_LT]);; let AFFINE_HULL_HALFSPACE_LE = prove (`!a b. affine hull {x | a dot x <= b} = if a = vec 0 /\ b < &0 then {} else (:real^N)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `a:real^N = vec 0` THENL [ASM_SIMP_TAC[DOT_LZERO; SET_RULE `{x | p} = if p then UNIV else {}`] THEN COND_CASES_TAC THEN ASM_SIMP_TAC[AFFINE_HULL_EMPTY; AFFINE_HULL_UNIV] THEN COND_CASES_TAC THEN REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC; ASM_SIMP_TAC[GSYM CLOSURE_HALFSPACE_LT; AFFINE_HULL_CLOSURE] THEN ASM_REWRITE_TAC[AFFINE_HULL_HALFSPACE_LT]]);; let AFFINE_HULL_HALFSPACE_GT = prove (`!a b. affine hull {x | a dot x > b} = if a = vec 0 /\ b >= &0 then {} else (:real^N)`, REPEAT GEN_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[AFFINE_HULL_EQ_EMPTY; HALFSPACE_EQ_EMPTY_GT; AFFINE_HULL_OPEN; OPEN_HALFSPACE_GT]);; let AFFINE_HULL_HALFSPACE_GE = prove (`!a b. affine hull {x | a dot x >= b} = if a = vec 0 /\ b > &0 then {} else (:real^N)`, REPEAT GEN_TAC THEN MP_TAC(ISPECL [`--a:real^N`; `--b:real`] AFFINE_HULL_HALFSPACE_LE) THEN SIMP_TAC[real_ge; DOT_LNEG; REAL_LE_NEG2; VECTOR_NEG_EQ_0] THEN REWRITE_TAC[REAL_ARITH `--b < &0 <=> b > &0`]);; let AFF_DIM_HALFSPACE_LT = prove (`!a:real^N b. aff_dim {x | a dot x < b} = if a = vec 0 /\ b <= &0 then --(&1) else &(dimindex(:N))`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN SIMP_TAC[AFFINE_HULL_HALFSPACE_LT] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[AFF_DIM_EMPTY; AFF_DIM_UNIV]);; let AFF_DIM_HALFSPACE_LE = prove (`!a:real^N b. aff_dim {x | a dot x <= b} = if a = vec 0 /\ b < &0 then --(&1) else &(dimindex(:N))`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN SIMP_TAC[AFFINE_HULL_HALFSPACE_LE] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[AFF_DIM_EMPTY; AFF_DIM_UNIV]);; let AFF_DIM_HALFSPACE_GT = prove (`!a:real^N b. aff_dim {x | a dot x > b} = if a = vec 0 /\ b >= &0 then --(&1) else &(dimindex(:N))`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN SIMP_TAC[AFFINE_HULL_HALFSPACE_GT] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[AFF_DIM_EMPTY; AFF_DIM_UNIV]);; let AFF_DIM_HALFSPACE_GE = prove (`!a:real^N b. aff_dim {x | a dot x >= b} = if a = vec 0 /\ b > &0 then --(&1) else &(dimindex(:N))`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN SIMP_TAC[AFFINE_HULL_HALFSPACE_GE] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[AFF_DIM_EMPTY; AFF_DIM_UNIV]);; let CHOOSE_AFFINE_SUBSET = prove (`!s:real^N->bool d. affine s /\ --(&1) <= d /\ d <= aff_dim s ==> ?t. affine t /\ t SUBSET s /\ aff_dim t = d`, REPEAT GEN_TAC THEN ASM_CASES_TAC `d:int = --(&1)` THENL [STRIP_TAC THEN EXISTS_TAC `{}:real^N->bool` THEN ASM_REWRITE_TAC[EMPTY_SUBSET; AFFINE_EMPTY; AFF_DIM_EMPTY]; ASM_SIMP_TAC[INT_ARITH `~(d:int = --(&1)) ==> (--(&1) <= d <=> &0 <= d)`] THEN POP_ASSUM(K ALL_TAC)] THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [ASM_REWRITE_TAC[AFF_DIM_EMPTY] THEN INT_ARITH_TAC; POP_ASSUM MP_TAC] THEN GEN_REWRITE_TAC LAND_CONV [GSYM MEMBER_NOT_EMPTY] THEN DISCH_THEN(X_CHOOSE_THEN `a:real^N` MP_TAC) THEN GEOM_ORIGIN_TAC `a:real^N` THEN SIMP_TAC[IMP_CONJ; AFF_DIM_DIM_SUBSPACE; AFFINE_EQ_SUBSPACE] THEN REPEAT GEN_TAC THEN DISCH_TAC THEN DISCH_TAC THEN REWRITE_TAC[GSYM INT_OF_NUM_EXISTS] THEN DISCH_THEN(X_CHOOSE_THEN `n:num` SUBST1_TAC) THEN REWRITE_TAC[INT_OF_NUM_LE] THEN DISCH_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `n:num`] CHOOSE_SUBSPACE_OF_SUBSPACE) THEN ASM_SIMP_TAC[SPAN_OF_SUBSPACE] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real^N->bool` THEN ASM_SIMP_TAC[AFF_DIM_DIM_SUBSPACE; SUBSPACE_IMP_AFFINE]);; let NONEMPTY_AFFINE_EXISTS = prove (`!n a:real^N. &0 <= n /\ n <= &(dimindex(:N)) ==> ?s. affine s /\ a IN s /\ aff_dim s = n`, REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `a:real^N` THEN X_GEN_TAC `n:int` THEN REWRITE_TAC[IMP_CONJ; GSYM INT_OF_NUM_EXISTS] THEN DISCH_THEN(X_CHOOSE_THEN `m:num` SUBST1_TAC) THEN REWRITE_TAC[INT_OF_NUM_LE] THEN STRIP_TAC THEN SUBGOAL_THEN `?s:real^N->bool. subspace s /\ dim s = m` MP_TAC THENL [ASM_SIMP_TAC[SUBSPACE_EXISTS]; MATCH_MP_TAC MONO_EXISTS] THEN SIMP_TAC[AFF_DIM_DIM_SUBSPACE; SUBSPACE_IMP_AFFINE; SUBSPACE_0]);; let AFFINE_EXISTS = prove (`!n. -- &1 <= n /\ n <= &(dimindex(:N)) ==> ?s:real^N->bool. affine s /\ aff_dim s = n`, REPEAT GEN_TAC THEN ASM_CASES_TAC `&0:int <= n` THENL [ASM_MESON_TAC[NONEMPTY_AFFINE_EXISTS]; ALL_TAC] THEN STRIP_TAC THEN EXISTS_TAC `{}:real^N->bool` THEN REWRITE_TAC[AFFINE_EMPTY; AFF_DIM_EMPTY] THEN ASM_INT_ARITH_TAC);; let AFF_DIM_CONIC_HULL = prove (`!s:real^N->bool. aff_dim(conic hull s) = if s = {} \/ vec 0 IN affine hull s then aff_dim s else aff_dim s + &1`, GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM AFF_DIM_AFFINE_HULL] THEN REWRITE_TAC[AFFINE_HULL_CONIC_HULL] THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[AFFINE_HULL_EMPTY; NOT_IN_EMPTY] THEN REWRITE_TAC[AFF_DIM_AFFINE_HULL; AFF_DIM_INSERT]);; let AFF_DIM_PCROSS = prove (`!s:real^M->bool t:real^N->bool. ~(s = {}) /\ ~(t = {}) ==> aff_dim(s PCROSS t) = aff_dim s + aff_dim t`, REPEAT GEN_TAC THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IMP_CONJ; LEFT_IMP_EXISTS_THM] THEN REWRITE_TAC[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 REPEAT STRIP_TAC THEN ASM_SIMP_TAC[AFF_DIM_DIM_0; HULL_INC] THEN ASM_REWRITE_TAC[INT_OF_NUM_ADD] THEN W(MP_TAC o PART_MATCH (rand o rand) DIM_PCROSS_STRONG o rand o rand o snd) THEN ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN(SUBST1_TAC o SYM)] THEN TRANS_TAC EQ_TRANS `aff_dim(IMAGE (\z. pastecart (a:real^M) (b:real^N) + z) (s PCROSS t))` THEN CONJ_TAC THENL [AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_IMAGE; FORALL_PASTECART; PASTECART_IN_PCROSS; EXISTS_PASTECART; PASTECART_ADD; PASTECART_INJ] THEN MESON_TAC[]; REWRITE_TAC[AFF_DIM_TRANSLATION_EQ] THEN MATCH_MP_TAC AFF_DIM_DIM_0 THEN MATCH_MP_TAC HULL_INC THEN ASM_REWRITE_TAC[GSYM PASTECART_VEC; PASTECART_IN_PCROSS]]);; let AFF_DIM_UNION = prove (`!s t:real^N->bool. affine s /\ affine t /\ ~(s INTER t = {}) ==> aff_dim(s UNION t) = (aff_dim s + aff_dim t) - aff_dim(s INTER t)`, REPEAT GEN_TAC THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `z:real^N` THEN GEOM_ORIGIN_TAC `z:real^N` THEN REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN REWRITE_TAC[IN_INTER] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN ASM_SIMP_TAC[AFFINE_EQ_SUBSPACE; AFF_DIM_DIM_0; HULL_INC; IN_INTER; IN_UNION] THEN REWRITE_TAC[INT_ARITH `a:int = b - c <=> a + c = b`] THEN REWRITE_TAC[INT_OF_NUM_ADD; INT_OF_NUM_EQ; DIM_UNION_INTER]);; let COPLANAR_INTERSECTING_LINES = prove (`!a b c d z:real^N. collinear {a,z,b} /\ collinear {c,z,d} ==> coplanar {z,a,b,c,d}`, REWRITE_TAC[COPLANAR_AFF_DIM] THEN REPEAT STRIP_TAC THEN TRANS_TAC INT_LE_TRANS `aff_dim(affine hull {a:real^N,z,b} UNION affine hull {c,z,d})` THEN SIMP_TAC[AFF_DIM_SUBSET; INSERT_SUBSET; EMPTY_SUBSET; IN_UNION; HULL_INC; IN_INSERT] THEN W(MP_TAC o PART_MATCH (lhand o rand) AFF_DIM_UNION o lhand o snd) THEN REWRITE_TAC[AFFINE_AFFINE_HULL; AFF_DIM_AFFINE_HULL] THEN SUBGOAL_THEN `z IN affine hull {a:real^N,z,b} /\ z IN affine hull {c,z,d}` STRIP_ASSUME_TAC THENL [SIMP_TAC[HULL_INC; IN_INSERT]; ALL_TAC] THEN ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN MATCH_MP_TAC(INT_ARITH `a:int <= &1 /\ b <= &1 /\ &0 <= c ==> (a + b) - c <= &2`) THEN ASM_REWRITE_TAC[GSYM COLLINEAR_AFF_DIM; AFF_DIM_POS_LE] THEN ASM SET_TAC[]);; let ISOMETRIC_HOMEOMORPHISM_AFFINE = prove (`!s:real^M->bool t:real^N->bool. affine s /\ affine t /\ aff_dim s = aff_dim t ==> ?f g. homeomorphism (s,t) (f,g) /\ (!x y. x IN s /\ y IN s ==> dist(f x,f y) = dist(x,y)) /\ (!x y. x IN t /\ y IN t ==> dist(g x,g y) = dist(x,y))`, REPEAT GEN_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THENL [ASM_SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_EQ_MINUS1; NOT_IN_EMPTY] THEN REWRITE_TAC[HOMEOMORPHIC_EMPTY; GSYM homeomorphic]; ALL_TAC] THEN ASM_CASES_TAC `s:real^M->bool = {}` THENL [ASM_MESON_TAC[AFF_DIM_EMPTY; AFF_DIM_EQ_MINUS1]; STRIP_TAC] THEN MP_TAC(ISPEC `t:real^N->bool` AFFINE_TRANSLATION_SUBSPACE) THEN MP_TAC(ISPEC `s:real^M->bool` AFFINE_TRANSLATION_SUBSPACE) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^M`; `s':real^M->bool`] THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`b:real^N`; `t':real^N->bool`] THEN STRIP_TAC THEN UNDISCH_TAC `aff_dim(s:real^M->bool) = aff_dim(t:real^N->bool)` THEN ASM_REWRITE_TAC[AFF_DIM_TRANSLATION_EQ] THEN ASM_SIMP_TAC[AFF_DIM_DIM_SUBSPACE; INT_OF_NUM_EQ] THEN DISCH_TAC THEN MP_TAC(ISPECL [`s':real^M->bool`; `t':real^N->bool`] ISOMETRIES_SUBSPACES) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f:real^M->real^N`; `g:real^N->real^M`] THEN STRIP_TAC THEN EXISTS_TAC `\x. b + (f:real^M->real^N) (--a + x)` THEN EXISTS_TAC `\x. a + (g:real^N->real^M) (--b + x)` THEN REWRITE_TAC[HOMEOMORPHISM; FORALL_IN_IMAGE_2; FORALL_IN_IMAGE] THEN REWRITE_TAC[GSYM IMAGE_o; o_DEF] THEN REWRITE_TAC[VECTOR_ARITH `--a + a + x:real^N = x`] THEN ASM_SIMP_TAC[VECTOR_ARITH `a + x:real^N = a + y <=> x = y`] THEN REWRITE_TAC[NORM_ARITH `dist(a + x:real^N,a + y) = dist(x,y)`] THEN ASM_SIMP_TAC[dist; GSYM LINEAR_SUB; SUBSPACE_SUB] THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THEN (CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]]) THEN MATCH_MP_TAC CONTINUOUS_ON_ADD THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_SIMP_TAC[LINEAR_CONTINUOUS_ON] THEN MATCH_MP_TAC CONTINUOUS_ON_ADD THEN REWRITE_TAC[CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID]);; (* ------------------------------------------------------------------------- *) (* Existence of a rigid transform between congruent sets. *) (* ------------------------------------------------------------------------- *) let RIGID_TRANSFORMATION_BETWEEN_CONGRUENT_SETS = prove (`!x:A->real^N y:A->real^N s. (!i j. i IN s /\ j IN s ==> dist(x i,x j) = dist(y i,y j)) ==> ?a f. orthogonal_transformation f /\ !i. i IN s ==> y i = a + f(x i)`, let lemma = prove (`!x:(real^N)^M y:(real^N)^M. (!i j. 1 <= i /\ i <= dimindex(:M) /\ 1 <= j /\ j <= dimindex(:M) ==> dist(x$i,x$j) = dist(y$i,y$j)) ==> ?a f. orthogonal_transformation f /\ !i. 1 <= i /\ i <= dimindex(:M) ==> y$i = a + f(x$i)`, REPEAT STRIP_TAC THEN ABBREV_TAC `(X:real^M^N) = lambda i j. (x:real^N^M)$j$i - x$1$i` THEN ABBREV_TAC `(Y:real^M^N) = lambda i j. (y:real^N^M)$j$i - y$1$i` THEN SUBGOAL_THEN `transp(X:real^M^N) ** X = transp(Y:real^M^N) ** Y` ASSUME_TAC THENL [REWRITE_TAC[MATRIX_MUL_LTRANSP_DOT_COLUMN] THEN MAP_EVERY EXPAND_TAC ["X"; "Y"] THEN SIMP_TAC[CART_EQ; column; LAMBDA_BETA; dot] THEN REWRITE_TAC[GSYM VECTOR_SUB_COMPONENT; GSYM dot] THEN REWRITE_TAC[DOT_NORM_SUB; VECTOR_ARITH `(x - a) - (y - a):real^N = x - y`] THEN ASM_SIMP_TAC[GSYM dist; DIMINDEX_GE_1; LE_REFL]; ALL_TAC] THEN SUBGOAL_THEN `?M:real^N^N. orthogonal_matrix M /\ (Y:real^M^N) = M ** (X:real^M^N)` (CHOOSE_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THENL [ALL_TAC; GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [CART_EQ] THEN MAP_EVERY EXPAND_TAC ["X"; "Y"] THEN SIMP_TAC[LAMBDA_BETA; matrix_mul] THEN REWRITE_TAC[REAL_ARITH `x - y:real = z <=> x = y + z`] THEN STRIP_TAC THEN EXISTS_TAC `(y:real^N^M)$1 - (M:real^N^N) ** (x:real^N^M)$1` THEN EXISTS_TAC `\x:real^N. (M:real^N^N) ** x` THEN ASM_SIMP_TAC[ORTHOGONAL_TRANSFORMATION_MATRIX; MATRIX_OF_MATRIX_VECTOR_MUL; MATRIX_VECTOR_MUL_LINEAR] THEN SIMP_TAC[CART_EQ; matrix_vector_mul; LAMBDA_BETA; VECTOR_ADD_COMPONENT] THEN ASM_SIMP_TAC[REAL_SUB_LDISTRIB; SUM_SUB_NUMSEG] THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; REAL_ARITH `a + y - b:real = a - z + y <=> z = b`] THEN SIMP_TAC[LAMBDA_BETA]] THEN MP_TAC(ISPEC `transp(X:real^M^N) ** X` SYMMETRIC_MATRIX_DIAGONALIZABLE_EXPLICIT) THEN REWRITE_TAC[symmetric_matrix] THEN REWRITE_TAC[MATRIX_TRANSP_MUL; TRANSP_TRANSP; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`P:real^M^M`; `d:num->real`] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(fun th -> MP_TAC th THEN ASM_REWRITE_TAC[] THEN MP_TAC th) THEN REWRITE_TAC[MATRIX_MUL_ASSOC; GSYM MATRIX_TRANSP_MUL] THEN REWRITE_TAC[GSYM MATRIX_MUL_ASSOC; LEFT_IMP_EXISTS_THM] THEN REWRITE_TAC[IMP_IMP] THEN GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [CART_EQ] THEN SIMP_TAC[MATRIX_MUL_LTRANSP_DOT_COLUMN; LAMBDA_BETA] THEN STRIP_TAC THEN MP_TAC(ISPECL [`\i. column i ((X:real^M^N) ** (P:real^M^M))`; `\i. column i ((Y:real^M^N) ** (P:real^M^M))`; `1..dimindex(:M)`] ORTHOGONAL_TRANSFORMATION_BETWEEN_ORTHOGONAL_SETS) THEN REWRITE_TAC[IN_NUMSEG] THEN ANTS_TAC THENL [ASM_SIMP_TAC[pairwise; IN_NUMSEG; NORM_EQ; orthogonal]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `f:real^N->real^N` (STRIP_ASSUME_TAC o GSYM)) THEN EXISTS_TAC `matrix(f:real^N->real^N)` THEN CONJ_TAC THENL [ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION_MATRIX]; ALL_TAC] THEN SUBGOAL_THEN `!M:real^M^N. M = M ** (P:real^M^M) ** transp P` (fun th -> GEN_REWRITE_TAC BINOP_CONV [th]) THENL [ASM_MESON_TAC[orthogonal_matrix; MATRIX_MUL_RID]; REWRITE_TAC[MATRIX_MUL_ASSOC] THEN AP_THM_TAC THEN AP_TERM_TAC] THEN REWRITE_TAC[GSYM MATRIX_MUL_ASSOC] THEN ASM_SIMP_TAC[MATRIX_EQUAL_COLUMNS] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [orthogonal_transformation]) THEN DISCH_THEN(ASSUME_TAC o GSYM o MATCH_MP MATRIX_WORKS o CONJUNCT1) THEN ASM_REWRITE_TAC[] THEN SIMP_TAC[CART_EQ; matrix_vector_mul; column; LAMBDA_BETA] THEN X_GEN_TAC `j:num` THEN STRIP_TAC THEN GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [matrix_mul] THEN ASM_SIMP_TAC[LAMBDA_BETA]) in REPEAT GEN_TAC THEN ASM_CASES_TAC `s:A->bool = {}` THENL [REPEAT STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`vec 0:real^N`; `\x:real^N. x`] THEN ASM_REWRITE_TAC[NOT_IN_EMPTY; ORTHOGONAL_TRANSFORMATION_ID]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_TAC `m:A`) THEN DISCH_TAC] THEN SUBGOAL_THEN `?r. IMAGE r (1..dimindex(:(N,1)finite_sum)) SUBSET s /\ affine hull (IMAGE (y o r) (1..dimindex(:(N,1)finite_sum))) = affine hull (IMAGE (y:A->real^N) s)` MP_TAC THENL [REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN SIMP_TAC[IMAGE_o; TAUT `p /\ q <=> ~(p ==> ~q)`; HULL_MONO; IMAGE_SUBSET] THEN REWRITE_TAC[NOT_IMP] THEN MP_TAC(ISPEC `IMAGE (y:A->real^N) s` AFFINE_BASIS_EXISTS) THEN DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [AFFINE_INDEPENDENT_IFF_CARD]) THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [FINITE_INDEX_NUMSEG]) THEN DISCH_THEN(X_CHOOSE_THEN `f:num->real^N` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `CARD(b:real^N->bool) <= dimindex(:(N,1)finite_sum)` ASSUME_TAC THENL [REWRITE_TAC[GSYM INT_OF_NUM_LE] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (INT_ARITH `a:int = c - &1 ==> a + &1 <= n ==> c <= n`)) THEN REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1; GSYM INT_OF_NUM_ADD] THEN REWRITE_TAC[INT_LE_RADD; AFF_DIM_LE_UNIV]; ALL_TAC] THEN UNDISCH_TAC `b SUBSET IMAGE (y:A->real^N) s` THEN ONCE_ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_IMAGE] THEN GEN_REWRITE_TAC (LAND_CONV o DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; IN_NUMSEG] THEN DISCH_THEN(X_CHOOSE_THEN `r:num->A` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\i. if i <= CARD(b:real^N->bool) then r i else (m:A)` THEN CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN UNDISCH_THEN `affine hull b:real^N->bool = affine hull IMAGE y (s:A->bool)` (SUBST1_TAC o SYM) THEN REWRITE_TAC[GSYM SUBSET] THEN MATCH_MP_TAC HULL_MONO THEN ONCE_ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_NUMSEG] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN REWRITE_TAC[GSYM IMAGE_o] THEN REWRITE_TAC[IN_IMAGE; IN_NUMSEG] THEN EXISTS_TAC `i:num` THEN ASM_REWRITE_TAC[o_THM] THEN ASM_MESON_TAC[LE_TRANS]; REWRITE_TAC[SUBSET; IN_NUMSEG; FORALL_IN_IMAGE] THEN STRIP_TAC THEN MP_TAC(ISPECL [`(lambda i. x(r i:A)):real^N^(N,1)finite_sum`; `(lambda i. y(r i:A)):real^N^(N,1)finite_sum`] lemma) THEN ASM_SIMP_TAC[LAMBDA_BETA] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^N->real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `k:A` THEN STRIP_TAC THEN SUBGOAL_THEN `!z. z IN affine hull IMAGE (y o (r:num->A)) (1..dimindex(:(N,1)finite_sum)) ==> dist(z,y k) = dist(z,a + (f:real^N->real^N)(x k))` MP_TAC THENL [MATCH_MP_TAC SAME_DISTANCES_TO_AFFINE_HULL THEN REWRITE_TAC[FORALL_IN_IMAGE; o_THM; IN_NUMSEG] THEN X_GEN_TAC `j:num` THEN STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `dist(x(r(j:num)),(x:A->real^N) k)` THEN CONJ_TAC THENL [CONV_TAC SYM_CONV THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[]; REWRITE_TAC[dist] THEN ASM_SIMP_TAC[NORM_ARITH `(a + x) - (a + y):real^N = x - y`] THEN ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION; LINEAR_SUB]]; ASM_SIMP_TAC[NORM_ARITH `a:real^N = b <=> dist(a:real^N,a) = dist(a,b)`] THEN DISCH_THEN MATCH_MP_TAC THEN MATCH_MP_TAC HULL_INC THEN REWRITE_TAC[IN_IMAGE; IN_NUMSEG] THEN ASM_MESON_TAC[]]]);; let RIGID_TRANSFORMATION_BETWEEN_CONGRUENT_SETS_STRONG = prove (`!x:A->real^N y:A->real^N s t. t SUBSET s /\ affine hull (IMAGE y t) = affine hull (IMAGE y s) /\ (!i j. i IN s /\ j IN t ==> dist(x i,x j) = dist(y i,y j)) ==> ?a f. orthogonal_transformation f /\ !i. i IN s ==> y i = a + f(x i)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`x:A->real^N`; `y:A->real^N`; `t:A->bool`] RIGID_TRANSFORMATION_BETWEEN_CONGRUENT_SETS) THEN ANTS_TAC THENL [ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^N->real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `i:A` THEN DISCH_TAC THEN SUBGOAL_THEN `!z. z IN affine hull (IMAGE (y:A->real^N) t) ==> dist(z,y i) = dist(z,a + (f:real^N->real^N)(x i))` MP_TAC THENL [MATCH_MP_TAC SAME_DISTANCES_TO_AFFINE_HULL THEN REWRITE_TAC[FORALL_IN_IMAGE; o_THM; IN_NUMSEG] THEN X_GEN_TAC `j:A` THEN STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `dist(a + f(x(j:A):real^N):real^N,a + f(x i))` THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN REWRITE_TAC[NORM_ARITH `dist(a + x:real^N,a + y) = dist(x,y)`] THEN ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION_ISOMETRY; DIST_SYM]; ASM_SIMP_TAC[NORM_ARITH `a:real^N = b <=> dist(a:real^N,a) = dist(a,b)`] THEN DISCH_THEN MATCH_MP_TAC THEN MATCH_MP_TAC HULL_INC THEN REWRITE_TAC[IN_IMAGE] THEN ASM_MESON_TAC[]]);; let RIGID_TRANSFORMATION_BETWEEN_3 = prove (`!a b c a' b' c':real^N. dist(a,b) = dist(a',b') /\ dist(b,c) = dist(b',c') /\ dist(c,a) = dist(c',a') ==> ?k f. orthogonal_transformation f /\ a' = k + f a /\ b' = k + f b /\ c' = k + f c`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`FST:real^N#real^N->real^N`; `SND:real^N#real^N->real^N`; `{(a:real^N,a':real^N), (b,b'), (c,c')}`] RIGID_TRANSFORMATION_BETWEEN_CONGRUENT_SETS) THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_INSERT] THEN REWRITE_TAC[NOT_IN_EMPTY; IMP_IMP] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_MESON_TAC[DIST_REFL; DIST_SYM]);; let RIGID_TRANSFORMATION_BETWEEN_2 = prove (`!a b a' b':real^N. dist(a,b) = dist(a',b') ==> ?k f. orthogonal_transformation f /\ a' = k + f a /\ b' = k + f b`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`a:real^N`; `b:real^N`; `a:real^N`; `a':real^N`; `b':real^N`; `a':real^N`] RIGID_TRANSFORMATION_BETWEEN_3) THEN ASM_MESON_TAC[DIST_EQ_0; DIST_SYM]);; (* ------------------------------------------------------------------------- *) (* Caratheodory's theorem. *) (* ------------------------------------------------------------------------- *) let CONVEX_HULL_CARATHEODORY_AFF_DIM = prove (`!p. convex hull p = {y:real^N | ?s u. FINITE s /\ s SUBSET p /\ &(CARD s) <= aff_dim p + &1 /\ (!x. x IN s ==> &0 <= u x) /\ sum s u = &1 /\ vsum s (\v. u v % v) = y}`, GEN_TAC THEN REWRITE_TAC[CONVEX_HULL_EXPLICIT] THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `y:real^N` THEN EQ_TAC THENL [ALL_TAC; MESON_TAC[]] THEN MATCH_MP_TAC(TAUT `!q. (p ==> q) /\ (q ==> r) ==> (p ==> r)`) THEN EXISTS_TAC `?n s u. CARD s = n /\ FINITE s /\ s SUBSET p /\ (!x. x IN s ==> &0 <= u x) /\ sum s u = &1 /\ vsum s (\v. u v % v) = (y:real^N)` THEN CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN GEN_REWRITE_TAC LAND_CONV [num_WOP] THEN DISCH_THEN(X_CHOOSE_THEN `n:num` MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC I [GSYM INT_NOT_LT] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `n - 1`) THEN REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [MATCH_MP_TAC(ARITH_RULE `~(n = 0) ==> n - 1 < n`) THEN DISCH_THEN SUBST_ALL_TAC THEN UNDISCH_TAC `aff_dim(p:real^N->bool) + &1 < &0` THEN REWRITE_TAC[INT_ARITH `p + &1:int < &0 <=> ~(-- &1 <= p)`] THEN REWRITE_TAC[AFF_DIM_GE]; ALL_TAC] THEN MP_TAC(ISPEC `s:real^N->bool` AFF_DIM_AFFINE_INDEPENDENT) THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `~(aff_dim(s:real^N->bool) = &n - &1)` ASSUME_TAC THENL [FIRST_ASSUM(MP_TAC o MATCH_MP AFF_DIM_SUBSET) THEN UNDISCH_TAC `aff_dim(p:real^N->bool) + &1 < &n` THEN INT_ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[AFFINE_DEPENDENT_EXPLICIT_FINITE] THEN DISCH_THEN(X_CHOOSE_THEN `w:real^N->real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `?t. (!v:real^N. v IN s ==> u(v) + t * w(v) >= &0) /\ ?a. a IN s /\ u(a) + t * w(a) = &0` STRIP_ASSUME_TAC THENL [ABBREV_TAC `i = IMAGE (\v. u(v) / --w(v)) {v:real^N | v IN s /\ w v < &0}` THEN EXISTS_TAC `inf i` THEN MP_TAC(SPEC `i:real->bool` INF_FINITE) THEN ANTS_TAC THENL [EXPAND_TAC "i" THEN ASM_SIMP_TAC[FINITE_IMAGE; FINITE_RESTRICT; IMAGE_EQ_EMPTY] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_ELIM_THM] THEN MP_TAC(ISPECL [`w:real^N->real`; `s:real^N->bool`] SUM_ZERO_EXISTS) THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]; ALL_TAC] THEN ABBREV_TAC `t = inf i` THEN EXPAND_TAC "i" THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_IMAGE; IN_ELIM_THM] THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) MP_TAC) THEN SIMP_TAC[REAL_LE_RDIV_EQ; REAL_ARITH `x < &0 ==> &0 < --x`; real_ge] THEN REWRITE_TAC[REAL_ARITH `t * --w <= u <=> &0 <= u + t * w`] THEN STRIP_TAC THEN CONJ_TAC THENL [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN DISJ_CASES_TAC(REAL_ARITH `(w:real^N->real) x < &0 \/ &0 <= w x`) THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC REAL_LE_ADD THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_DIV THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC(REAL_ARITH `w < &0 ==> &0 <= --w`) THEN ASM_REWRITE_TAC[]; EXISTS_TAC `a:real^N` THEN ASM_REWRITE_TAC[] THEN UNDISCH_TAC `w(a:real^N) < &0` THEN CONV_TAC REAL_FIELD]; ALL_TAC] THEN MAP_EVERY EXISTS_TAC [`s DELETE (a:real^N)`; `(\v. u(v) + t * w(v)):real^N->real`] THEN ASM_SIMP_TAC[SUM_DELETE; VSUM_DELETE; CARD_DELETE; FINITE_DELETE] THEN ASM_SIMP_TAC[SUM_ADD; VECTOR_ADD_RDISTRIB; VSUM_ADD] THEN ASM_SIMP_TAC[GSYM VECTOR_MUL_ASSOC; SUM_LMUL; VSUM_LMUL] THEN REPEAT CONJ_TAC THENL [ASM SET_TAC[]; ASM SET_TAC[real_ge]; REAL_ARITH_TAC; VECTOR_ARITH_TAC]);; let CARATHEODORY_AFF_DIM = prove (`!p. convex hull p = {x:real^N | ?s. FINITE s /\ s SUBSET p /\ &(CARD s) <= aff_dim p + &1 /\ x IN convex hull s}`, REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN REPEAT GEN_TAC THEN EQ_TAC THENL [GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [CONVEX_HULL_CARATHEODORY_AFF_DIM] THEN REWRITE_TAC[IN_ELIM_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN ASM_MESON_TAC[HULL_SUBSET; CONVEX_EXPLICIT; CONVEX_CONVEX_HULL]; MESON_TAC[SUBSET; HULL_MONO]]);; let CONVEX_HULL_CARATHEODORY = prove (`!p. convex hull p = {y:real^N | ?s u. FINITE s /\ s SUBSET p /\ CARD(s) <= dimindex(:N) + 1 /\ (!x. x IN s ==> &0 <= u x) /\ sum s u = &1 /\ vsum s (\v. u v % v) = y}`, GEN_TAC THEN REWRITE_TAC[EXTENSION] THEN X_GEN_TAC `y:real^N` THEN EQ_TAC THENL [REWRITE_TAC[CONVEX_HULL_CARATHEODORY_AFF_DIM; IN_ELIM_THM] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN STRIP_TAC THEN ASM_REWRITE_TAC[GSYM INT_OF_NUM_LE; GSYM INT_OF_NUM_ADD] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (INT_ARITH `a:int <= x + &1 ==> x <= y ==> a <= y + &1`)) THEN REWRITE_TAC[AFF_DIM_LE_UNIV]; REWRITE_TAC[CONVEX_HULL_EXPLICIT; IN_ELIM_THM] THEN MESON_TAC[]]);; let CARATHEODORY = prove (`!p. convex hull p = {x:real^N | ?s. FINITE s /\ s SUBSET p /\ CARD(s) <= dimindex(:N) + 1 /\ x IN convex hull s}`, REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN REPEAT GEN_TAC THEN EQ_TAC THENL [GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [CONVEX_HULL_CARATHEODORY] THEN REWRITE_TAC[IN_ELIM_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN ASM_MESON_TAC[HULL_SUBSET; CONVEX_EXPLICIT; CONVEX_CONVEX_HULL]; MESON_TAC[SUBSET; HULL_MONO]]);; (* ------------------------------------------------------------------------- *) (* Some results on decomposing convex hulls, e.g. simplicial subdivision. *) (* ------------------------------------------------------------------------- *) let AFFINE_HULL_INTER,CONVEX_HULL_INTER = (CONJ_PAIR o prove) (`(!s t:real^N->bool. ~(affine_dependent(s UNION t)) ==> affine hull s INTER affine hull t = affine hull (s INTER t)) /\ (!s t:real^N->bool. ~(affine_dependent (s UNION t)) ==> convex hull s INTER convex hull t = convex hull (s INTER t))`, CONJ_TAC THEN (REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN REWRITE_TAC[FINITE_UNION] THEN STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[SUBSET_INTER] THEN SIMP_TAC[HULL_MONO; INTER_SUBSET] THEN REWRITE_TAC[SUBSET; AFFINE_HULL_FINITE; CONVEX_HULL_FINITE; IN_ELIM_THM; IN_INTER] THEN X_GEN_TAC `y:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) (X_CHOOSE_THEN `v:real^N->real` STRIP_ASSUME_TAC)) THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [AFFINE_DEPENDENT_EXPLICIT]) THEN REWRITE_TAC[NOT_EXISTS_THM] THEN DISCH_THEN(MP_TAC o SPEC `(s UNION t):real^N->bool`) THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN DISCH_THEN(MP_TAC o SPEC `\x:real^N. (if x IN s then u x else &0) - (if x IN t then v x else &0)`) THEN ASM_SIMP_TAC[SUM_SUB; FINITE_UNION; VSUM_SUB; VECTOR_SUB_RDISTRIB] THEN REWRITE_TAC[MESON[] `(if p then a else b) % x = (if p then a % x else b % x)`] THEN ASM_SIMP_TAC[SUM_CASES; VSUM_CASES; VECTOR_MUL_LZERO; FINITE_UNION] THEN ASM_REWRITE_TAC[SUM_0; VSUM_0; SET_RULE `{x | x IN (s UNION t) /\ x IN s} = s`; SET_RULE `{x | x IN (s UNION t) /\ x IN t} = t`] THEN MATCH_MP_TAC(TAUT `a /\ c /\ (~b ==> d) ==> ~(a /\ b /\ c) ==> d`) THEN REPEAT CONJ_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC; ALL_TAC] THEN DISCH_TAC THEN EXISTS_TAC `u:real^N->real` THEN ASM_SIMP_TAC[] THEN CONJ_TAC THEN MATCH_MP_TAC EQ_TRANS THENL [EXISTS_TAC `sum s (u:real^N->real)`; EXISTS_TAC `vsum s (\x. (u:real^N->real) x % x)`] THEN (CONJ_TAC THENL [ALL_TAC; FIRST_X_ASSUM ACCEPT_TAC]) THEN CONV_TAC SYM_CONV THENL [MATCH_MP_TAC SUM_EQ_SUPERSET; MATCH_MP_TAC VSUM_EQ_SUPERSET] THEN ASM_SIMP_TAC[FINITE_INTER; INTER_SUBSET; IN_INTER] THEN X_GEN_TAC `x:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[VECTOR_MUL_EQ_0] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[REAL_SUB_RZERO] THEN ASM SET_TAC[]));; let AFFINE_HULL_INTERS = prove (`!s:(real^N->bool)->bool. ~(affine_dependent(UNIONS s)) ==> affine hull (INTERS s) = INTERS {affine hull t | t IN s}`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(fun th -> MP_TAC th THEN MP_TAC(MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE th)) THEN SPEC_TAC(`s:(real^N->bool)->bool`,`s:(real^N->bool)->bool`) THEN REWRITE_TAC[FINITE_UNIONS; IMP_CONJ] THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN REWRITE_TAC[UNIONS_0; INTERS_0; UNIONS_INSERT; INTERS_INSERT; SET_RULE `{f x | x IN {}} = {}`; AFFINE_HULL_UNIV] THEN MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `f:(real^N->bool)->bool`] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN REWRITE_TAC[FORALL_IN_INSERT] THEN DISCH_THEN(fun th -> STRIP_TAC THEN STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [UNDISCH_TAC `~affine_dependent((s UNION UNIONS f):real^N->bool)` THEN REWRITE_TAC[CONTRAPOS_THM] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] AFFINE_DEPENDENT_MONO) THEN SET_TAC[]; DISCH_TAC] THEN ASM_CASES_TAC `f:(real^N->bool)->bool = {}` THENL [ASM_REWRITE_TAC[INTERS_0; INTER_UNIV; IN_SING] THEN REWRITE_TAC[SET_RULE `{f x | x = a} = {f a}`; INTERS_1]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (rhs o rand) AFFINE_HULL_INTER o lhand o snd) THEN ANTS_TAC THENL [UNDISCH_TAC `~affine_dependent((s UNION UNIONS f):real^N->bool)` THEN REWRITE_TAC[CONTRAPOS_THM] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] AFFINE_DEPENDENT_MONO) THEN UNDISCH_TAC `~(f:(real^N->bool)->bool = {})` THEN SET_TAC[]; DISCH_THEN(SUBST1_TAC o SYM)] THEN REWRITE_TAC[SET_RULE `{f x | x IN (a INSERT s)} = (f a) INSERT {f x | x IN s}`] THEN ASM_REWRITE_TAC[INTERS_INSERT]);; let CONVEX_HULL_INTERS = prove (`!s:(real^N->bool)->bool. ~(affine_dependent(UNIONS s)) ==> convex hull (INTERS s) = INTERS {convex hull t | t IN s}`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(fun th -> MP_TAC th THEN MP_TAC(MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE th)) THEN SPEC_TAC(`s:(real^N->bool)->bool`,`s:(real^N->bool)->bool`) THEN REWRITE_TAC[FINITE_UNIONS; IMP_CONJ] THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN REWRITE_TAC[UNIONS_0; INTERS_0; UNIONS_INSERT; INTERS_INSERT; SET_RULE `{f x | x IN {}} = {}`; CONVEX_HULL_UNIV] THEN MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `f:(real^N->bool)->bool`] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN REWRITE_TAC[FORALL_IN_INSERT] THEN DISCH_THEN(fun th -> STRIP_TAC THEN STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [UNDISCH_TAC `~affine_dependent((s UNION UNIONS f):real^N->bool)` THEN REWRITE_TAC[CONTRAPOS_THM] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] AFFINE_DEPENDENT_MONO) THEN SET_TAC[]; DISCH_TAC] THEN ASM_CASES_TAC `f:(real^N->bool)->bool = {}` THENL [ASM_REWRITE_TAC[INTERS_0; INTER_UNIV; IN_SING] THEN REWRITE_TAC[SET_RULE `{f x | x = a} = {f a}`; INTERS_1]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (rhs o rand) CONVEX_HULL_INTER o lhand o snd) THEN ANTS_TAC THENL [UNDISCH_TAC `~affine_dependent((s UNION UNIONS f):real^N->bool)` THEN REWRITE_TAC[CONTRAPOS_THM] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] AFFINE_DEPENDENT_MONO) THEN UNDISCH_TAC `~(f:(real^N->bool)->bool = {})` THEN SET_TAC[]; DISCH_THEN(SUBST1_TAC o SYM)] THEN REWRITE_TAC[SET_RULE `{f x | x IN (a INSERT s)} = (f a) INSERT {f x | x IN s}`] THEN ASM_REWRITE_TAC[INTERS_INSERT]);; let IN_CONVEX_HULL_EXCHANGE = prove (`!s a x:real^N. a IN convex hull s /\ x IN convex hull s ==> ?b. b IN s /\ x IN convex hull (a INSERT (s DELETE b))`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `(a:real^N) IN s` THENL [EXISTS_TAC `a:real^N` THEN ASM_SIMP_TAC[INSERT_DELETE]; ALL_TAC] THEN ASM_CASES_TAC `FINITE(s:real^N->bool) /\ CARD s <= dimindex(:N) + 1` THENL [ALL_TAC; UNDISCH_TAC `(x:real^N) IN convex hull s` THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [CARATHEODORY] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; IN_ELIM_THM] THEN X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN ASM_CASES_TAC `t:real^N->bool = s` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?b:real^N. b IN s /\ ~(b IN t)` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `b:real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN UNDISCH_TAC `(x:real^N) IN convex hull t` THEN SPEC_TAC(`x:real^N`,`x:real^N`) THEN REWRITE_TAC[GSYM SUBSET] THEN MATCH_MP_TAC HULL_MONO THEN ASM SET_TAC[]] THEN MP_TAC(ASSUME `(a:real^N) IN convex hull s`) THEN MP_TAC(ASSUME `(x:real^N) IN convex hull s`) THEN REWRITE_TAC[CONVEX_HULL_FINITE; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `v:real^N->real` THEN STRIP_TAC THEN X_GEN_TAC `u:real^N->real` THEN STRIP_TAC THEN ASM_CASES_TAC `?b. b IN s /\ (v:real^N->real) b = &0` THENL [FIRST_X_ASSUM(fun th -> MP_TAC th THEN MATCH_MP_TAC MONO_EXISTS) THEN X_GEN_TAC `b:real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `\x:real^N. if x = a then &0 else v x` THEN ASM_SIMP_TAC[FORALL_IN_INSERT; REAL_LE_REFL] THEN ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; FINITE_DELETE] THEN ASM_REWRITE_TAC[IN_DELETE] THEN ASM_SIMP_TAC[SUM_DELETE; VSUM_DELETE; COND_ID] THEN ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN ASM_SIMP_TAC[SUM_CASES; VSUM_CASES; REAL_LE_REFL; COND_ID] THEN REWRITE_TAC[VECTOR_MUL_LZERO; SUM_0; VSUM_0] THEN ASM_SIMP_TAC[SET_RULE `~(a IN s) ==> {x | x IN s /\ ~(x = a)} = s`] THEN CONJ_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN REWRITE_TAC[TAUT `~(a /\ b) <=> a ==> ~b`] THEN DISCH_TAC THEN MP_TAC(ISPEC `IMAGE (\b. (u:real^N->real) b / v b) s` SUP_FINITE) THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [ASM_MESON_TAC[CONVEX_HULL_EMPTY; NOT_IN_EMPTY]; ALL_TAC] THEN ASM_SIMP_TAC[FINITE_IMAGE; IMAGE_EQ_EMPTY; FORALL_IN_IMAGE] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN REWRITE_TAC[IN_IMAGE] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `b:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC) THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `!b. b IN s ==> &0 < (v:real^N->real) b` ASSUME_TAC THENL [ASM_SIMP_TAC[REAL_LT_LE]; ALL_TAC] THEN SUBGOAL_THEN `&0 < (u:real^N->real) b /\ &0 < v b` STRIP_ASSUME_TAC THENL [ASM_SIMP_TAC[REAL_LT_LE] THEN UNDISCH_TAC `!x. x IN s ==> (u:real^N->real) x / v x <= u b / v b` THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ] THEN REWRITE_TAC[real_div; REAL_MUL_LZERO] THEN ASM_SIMP_TAC[REAL_ARITH `&0 <= x ==> (x <= &0 <=> x = &0)`] THEN DISCH_TAC THEN UNDISCH_TAC `sum s (u:real^N->real) = &1` THEN MATCH_MP_TAC(REAL_ARITH `x = &0 ==> x = &1 ==> F`) THEN ASM_SIMP_TAC[SUM_EQ_0]; ALL_TAC] THEN EXISTS_TAC `(\x. if x = a then v b / u b else v x - (v b / u b) * u x): real^N->real` THEN ASM_SIMP_TAC[FORALL_IN_INSERT; REAL_LE_DIV; REAL_LT_IMP_LE] THEN ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; FINITE_DELETE] THEN ASM_SIMP_TAC[SUM_DELETE; VSUM_DELETE; IN_DELETE] THEN ASM_SIMP_TAC[SUM_CASES; VSUM_CASES; FINITE_DELETE] THEN ASM_SIMP_TAC[SET_RULE `~(a IN s) ==> {x | x IN s /\ ~(x = a)} = s`; SET_RULE `~(a IN s) ==> {x | x IN s /\ x = a} = {}`] THEN REWRITE_TAC[VSUM_CLAUSES; SUM_CLAUSES] THEN ASM_CASES_TAC `b:real^N = a` THENL [ASM_MESON_TAC[]; ASM_REWRITE_TAC[]] THEN ASM_SIMP_TAC[VECTOR_SUB_RDISTRIB; VSUM_SUB; SUM_SUB] THEN REWRITE_TAC[GSYM VECTOR_MUL_ASSOC; VECTOR_ADD_LID; REAL_ADD_LID] THEN ASM_SIMP_TAC[SUM_LMUL; VSUM_LMUL] THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[REAL_DIV_RMUL; REAL_LT_IMP_NZ] THEN REPEAT CONJ_TAC THENL [ALL_TAC; REAL_ARITH_TAC; VECTOR_ARITH_TAC] THEN X_GEN_TAC `c:real^N` THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ASM_CASES_TAC `(u:real^N->real) c = &0` THENL [ASM_SIMP_TAC[REAL_MUL_RZERO; REAL_SUB_RZERO]; ALL_TAC] THEN REWRITE_TAC[REAL_SUB_LE] THEN ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ; REAL_LT_LE] THEN ONCE_REWRITE_TAC[GSYM REAL_INV_DIV] THEN MATCH_MP_TAC REAL_LE_INV2 THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_LE]);; let IN_CONVEX_HULL_EXCHANGE_UNIQUE = prove (`!s t t' a x:real^N. ~(affine_dependent s) /\ a IN convex hull s /\ t SUBSET s /\ t' SUBSET s /\ x IN convex hull (a INSERT t) /\ x IN convex hull (a INSERT t') ==> x IN convex hull (a INSERT (t INTER t'))`, REPEAT GEN_TAC THEN ASM_CASES_TAC `(a:real^N) IN s` THENL [REPEAT STRIP_TAC THEN REWRITE_TAC[SET_RULE `a INSERT (s INTER t) = (a INSERT s) INTER (a INSERT t)`] THEN W(MP_TAC o PART_MATCH (rand o rand) CONVEX_HULL_INTER o rand o snd) THEN ANTS_TAC THENL [UNDISCH_TAC `~(affine_dependent(s:real^N->bool))` THEN REWRITE_TAC[CONTRAPOS_THM] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] AFFINE_DEPENDENT_MONO); DISCH_THEN(SUBST1_TAC o SYM)] THEN ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN REWRITE_TAC[CONVEX_HULL_FINITE; IN_ELIM_THM] THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `b:real^N->real` STRIP_ASSUME_TAC) MP_TAC) THEN REPLICATE_TAC 2 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN SUBGOAL_THEN `~((a:real^N) IN t) /\ ~(a IN t')` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `FINITE(t:real^N->bool) /\ FINITE(t':real^N->bool)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[FINITE_SUBSET]; ALL_TAC] THEN ASM_SIMP_TAC[AFFINE_HULL_FINITE_STEP_GEN; REAL_LE_ADD; REAL_ARITH `&0 <= a / &2 <=> &0 <= a`] THEN REWRITE_TAC[IMP_CONJ; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u':real`; `u:real^N->real`] THEN REPEAT DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`v':real`; `v:real^N->real`] THEN REPEAT DISCH_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [AFFINE_DEPENDENT_EXPLICIT]) THEN REWRITE_TAC[NOT_EXISTS_THM] THEN DISCH_THEN(MP_TAC o SPEC `s:real^N->bool`) THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN DISCH_THEN(MP_TAC o SPEC `\x:real^N. (if x IN t then u x else &0) - (if x IN t' then v x else &0) + (u' - v') * b x`) THEN ASM_SIMP_TAC[SUM_ADD; VSUM_ADD; SUM_LMUL; VSUM_LMUL; VECTOR_ADD_RDISTRIB] THEN ASM_SIMP_TAC[SUM_SUB; VSUM_SUB; VECTOR_SUB_RDISTRIB] THEN REWRITE_TAC[MESON[] `(if p then a else b) % x = (if p then a % x else b % x)`] THEN ASM_SIMP_TAC[SUM_CASES; VSUM_CASES; VECTOR_MUL_LZERO; SUM_0; VSUM_0] THEN ASM_SIMP_TAC[SET_RULE `t SUBSET s ==> {x | x IN s /\ x IN t} = t`] THEN ASM_SIMP_TAC[SUM_ADD; SUM_LMUL; VSUM_ADD; VSUM_LMUL; VECTOR_ADD_RDISTRIB; GSYM VECTOR_MUL_ASSOC] THEN MATCH_MP_TAC(TAUT `a /\ c /\ (~b ==> d) ==> ~(a /\ b /\ c) ==> d`) THEN REPEAT CONJ_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC; ALL_TAC] THEN DISCH_TAC THEN SUBGOAL_THEN `(!x. x IN s ==> (if x IN t then u x else &0) <= (if x IN t' then v x else &0)) \/ (!x:real^N. x IN s ==> (if x IN t' then v x else &0) <= (if x IN t then u x else &0))` (DISJ_CASES_THEN(LABEL_TAC "*")) THENL [MP_TAC(REAL_ARITH `&0 <= (u' - v') \/ &0 <= (v' - u')`) THEN MATCH_MP_TAC MONO_OR THEN CONJ_TAC THEN DISCH_TAC THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN DISCH_THEN(MP_TAC o SPEC `y:real^N`) THEN ASM_REWRITE_TAC[] THENL [MATCH_MP_TAC(REAL_ARITH `&0 <= c ==> a - b + c = &0 ==> a <= b`); MATCH_MP_TAC(REAL_ARITH `&0 <= --c ==> a - b + c = &0 ==> b <= a`)] THEN ASM_SIMP_TAC[REAL_LE_MUL; GSYM REAL_MUL_LNEG; REAL_NEG_SUB]; EXISTS_TAC `(\x. if x = a then u' else u x):real^N->real`; EXISTS_TAC `(\x. if x = a then v' else v x):real^N->real`] THEN ASM_SIMP_TAC[FORALL_IN_INSERT] THEN (CONJ_TAC THENL [ASM_MESON_TAC[IN_INTER]; ALL_TAC]) THEN ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; FINITE_INTER] THEN ASM_REWRITE_TAC[IN_INTER] THEN REWRITE_TAC[REAL_ARITH `u' + u = &1 <=> u = &1 - u'`; VECTOR_ARITH `u' + u:real^N = y <=> u = y - u'`] THEN (CONJ_TAC THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [GSYM th]) THEN CONV_TAC SYM_CONV THENL [MATCH_MP_TAC SUM_EQ_SUPERSET; MATCH_MP_TAC VSUM_EQ_SUPERSET]) THEN ASM_SIMP_TAC[FINITE_INTER; INTER_SUBSET; IN_INTER] THEN (CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN X_GEN_TAC `y:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[VECTOR_MUL_EQ_0] THEN DISCH_TAC THEN REMOVE_THEN "*" (MP_TAC o SPEC `y:real^N`) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM REAL_LE_ANTISYM] THEN ASM SET_TAC[]);; let CONVEX_HULL_EXCHANGE_UNION = prove (`!s a:real^N. a IN convex hull s ==> convex hull s = UNIONS {convex hull (a INSERT (s DELETE b)) |b| b IN s}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN REWRITE_TAC[UNIONS_IMAGE] THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN ASM_MESON_TAC[IN_CONVEX_HULL_EXCHANGE]; REWRITE_TAC[SUBSET; FORALL_IN_UNIONS; FORALL_IN_GSPEC; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[GSYM SUBSET] THEN ASM_SIMP_TAC[SUBSET_HULL; CONVEX_CONVEX_HULL] THEN ASM_REWRITE_TAC[INSERT_SUBSET] THEN MESON_TAC[HULL_INC; SUBSET; IN_DELETE]]);; let CONVEX_HULL_EXCHANGE_INTER = prove (`!s a:real^N t t'. ~affine_dependent s /\ a IN convex hull s /\ t SUBSET s /\ t' SUBSET s ==> (convex hull (a INSERT t)) INTER (convex hull (a INSERT t')) = convex hull (a INSERT (t INTER t'))`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [REWRITE_TAC[SUBSET; IN_INTER] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC IN_CONVEX_HULL_EXCHANGE_UNIQUE THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[]; REWRITE_TAC[SUBSET_INTER] THEN CONJ_TAC THEN MATCH_MP_TAC HULL_MONO THEN SET_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Representing affine hull as hyperplane or finite intersection of them. *) (* ------------------------------------------------------------------------- *) let AFF_DIM_EQ_INTER_HYPERPLANE = prove (`!s t:real^N->bool. affine s /\ affine t /\ t SUBSET s /\ aff_dim t + &1 = aff_dim s ==> ?a b. ~(a = vec 0) /\ {x | a dot x = b} INTER s = t`, REPEAT GEN_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THENL [ASM_REWRITE_TAC[AFF_DIM_EMPTY; INT_ARITH `--a + a:int = b <=> b = &0`] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o last o CONJUNCTS)) THEN REWRITE_TAC[AFF_DIM_EQ_0; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `a:real^N` THEN DISCH_THEN SUBST1_TAC THEN MAP_EVERY EXISTS_TAC [`basis 1:real^N`; `basis 1 dot (a:real^N) + &1`] THEN SIMP_TAC[BASIS_NONZERO; LE_REFL; DIMINDEX_GE_1; REAL_ARITH `x:real = x + &1 <=> F`; IN_ELIM_THM; SET_RULE `s INTER {a} = {} <=> ~(a IN s)`]; 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 `z:real^N` THEN GEN_GEOM_ORIGIN_TAC `z:real^N` ["a"] THEN REPEAT GEN_TAC THEN DISCH_TAC THEN ASM_CASES_TAC `(vec 0:real^N) IN s` THENL [ALL_TAC; ASM SET_TAC[]] THEN ASM_SIMP_TAC[AFFINE_EQ_SUBSPACE; AFF_DIM_DIM; HULL_INC] THEN REWRITE_TAC[INT_OF_NUM_ADD; INT_OF_NUM_EQ] THEN DISCH_THEN(MP_TAC o MATCH_MP LOWDIM_EQ_INTER_HYPERPLANE) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN ASM_REWRITE_TAC[DOT_RADD] THEN EXISTS_TAC `(a:real^N) dot z` THEN REWRITE_TAC[REAL_EQ_ADD_LCANCEL_0]]);; let AFF_DIM_EQ_HYPERPLANE = prove (`!s. aff_dim s = &(dimindex(:N)) - &1 <=> ?a b. ~(a = vec 0) /\ affine hull s = {x:real^N | a dot x = b}`, GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [ASM_REWRITE_TAC[AFF_DIM_EMPTY; INT_ARITH `--a:int = b - a <=> b = &0`] THEN SIMP_TAC[INT_OF_NUM_EQ; LE_1; DIMINDEX_GE_1; AFFINE_HULL_EMPTY] THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY; NOT_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real`] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(MP_TAC o SPEC `(b / (a dot a)) % a:real^N`) THEN ASM_SIMP_TAC[DOT_RMUL; REAL_DIV_RMUL; DOT_EQ_0]; 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 `c:real^N` THEN GEN_GEOM_ORIGIN_TAC `c:real^N` ["a"] THEN SIMP_TAC[AFF_DIM_DIM_0; HULL_INC] THEN SIMP_TAC[INT_OF_NUM_SUB; DIMINDEX_GE_1; INT_OF_NUM_EQ] THEN SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC; DIM_EQ_HYPERPLANE] THEN REPEAT STRIP_TAC THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `a:real^N` THEN REWRITE_TAC[] THEN ASM_CASES_TAC `a:real^N = vec 0` THEN ASM_REWRITE_TAC[DOT_RADD; REAL_ARITH `a + b:real = c <=> b = c - a`] THEN EQ_TAC THEN STRIP_TAC THENL [EXISTS_TAC `(a:real^N) dot c` THEN ASM_REWRITE_TAC[REAL_SUB_REFL]; ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o AP_TERM `\s. (vec 0:real^N) IN s`) THEN ASM_SIMP_TAC[SPAN_SUPERSET; IN_ELIM_THM; DOT_RZERO]]]);; let AFF_DIM_HYPERPLANE = prove (`!a b. ~(a = vec 0) ==> aff_dim {x:real^N | a dot x = b} = &(dimindex(:N)) - &1`, REPEAT STRIP_TAC THEN REWRITE_TAC[AFF_DIM_EQ_HYPERPLANE] THEN MAP_EVERY EXISTS_TAC [`a:real^N`; `b:real`] THEN ASM_REWRITE_TAC[AFFINE_HULL_EQ; AFFINE_HYPERPLANE]);; let BOUNDED_HYPERPLANE_EQ_TRIVIAL = prove (`!a b. bounded {x:real^N | a dot x = b} <=> if a = vec 0 then ~(b = &0) else dimindex(:N) = 1`, REPEAT GEN_TAC THEN ASM_CASES_TAC `a:real^N = vec 0` THEN ASM_REWRITE_TAC[DOT_LZERO] THENL [ASM_CASES_TAC `b = &0` THEN ASM_REWRITE_TAC[EMPTY_GSPEC; BOUNDED_EMPTY] THEN REWRITE_TAC[NOT_BOUNDED_UNIV; SET_RULE `{x | T} = UNIV`]; ASM_SIMP_TAC[AFFINE_BOUNDED_EQ_LOWDIM; AFF_DIM_HYPERPLANE; AFFINE_HYPERPLANE] THEN REWRITE_TAC[INT_ARITH `a - &1:int <= &0 <=> a <= &1`; INT_OF_NUM_LE] THEN MATCH_MP_TAC(ARITH_RULE `1 <= n ==> (n <= 1 <=> n = 1)`) THEN REWRITE_TAC[DIMINDEX_GE_1]]);; let AFFINE_HULL_FINITE_INTERSECTION_HYPERPLANES = prove (`!s:real^N->bool. ?f. FINITE f /\ &(CARD f) + aff_dim s = &(dimindex(:N)) /\ affine hull s = INTERS f /\ (!h. h IN f ==> ?a b. ~(a = vec 0) /\ h = {x | a dot x = b})`, GEN_TAC THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN MP_TAC(ISPEC `s:real^N->bool` AFFINE_BASIS_EXISTS) THEN DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN MP_TAC(ISPECL [`b:real^N->bool`; `(:real^N)`] EXTEND_TO_AFFINE_BASIS) THEN ASM_REWRITE_TAC[SUBSET_UNIV; AFFINE_HULL_UNIV] THEN DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `FINITE(c:real^N->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[AFFINE_INDEPENDENT_IMP_FINITE]; ALL_TAC] THEN REWRITE_TAC[GSYM AFF_DIM_UNIV] THEN FIRST_ASSUM(SUBST1_TAC o SYM) THEN REWRITE_TAC[AFF_DIM_AFFINE_HULL] THEN ASM_SIMP_TAC[AFF_DIM_AFFINE_INDEPENDENT; CARD_DIFF] THEN REWRITE_TAC[INT_ARITH `f + b - &1:int = c - &1 <=> f = c - b`] THEN ASM_SIMP_TAC[INT_OF_NUM_SUB; CARD_SUBSET; GSYM CARD_DIFF; INT_OF_NUM_EQ] THEN ASM_CASES_TAC `c:real^N->bool = b` THENL [EXISTS_TAC `{}:(real^N->bool)->bool` THEN ASM_REWRITE_TAC[CARD_CLAUSES; FINITE_RULES; NOT_IN_EMPTY; INTERS_0; DIFF_EQ_EMPTY] THEN ASM_MESON_TAC[]; ALL_TAC] THEN EXISTS_TAC `{affine hull (c DELETE a) |a| (a:real^N) IN (c DIFF b)}` THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN REWRITE_TAC[SIMPLE_IMAGE] THEN ASM_SIMP_TAC[FINITE_IMAGE; FINITE_DIFF] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CARD_IMAGE_INJ THEN ASM_SIMP_TAC[FINITE_DIFF] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN REWRITE_TAC[IN_DIFF] THEN STRIP_TAC THEN ASM_CASES_TAC `x:real^N = y` THEN ASM_REWRITE_TAC[] THEN UNDISCH_TAC `~affine_dependent(c:real^N->bool)` THEN REWRITE_TAC[affine_dependent] THEN EXISTS_TAC `x:real^N` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC HULL_INC THEN ASM SET_TAC[]; ONCE_REWRITE_TAC[GSYM o_DEF] THEN REWRITE_TAC[IMAGE_o] THEN ONCE_REWRITE_TAC[GSYM SIMPLE_IMAGE] THEN W(MP_TAC o PART_MATCH (rhs o rand) AFFINE_HULL_INTERS o rand o snd) THEN ANTS_TAC THENL [MATCH_MP_TAC AFFINE_INDEPENDENT_SUBSET THEN EXISTS_TAC `c:real^N->bool` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET; FORALL_IN_UNIONS; FORALL_IN_IMAGE; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN SET_TAC[]; DISCH_THEN(SUBST1_TAC o SYM) THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[IN_INTERS; FORALL_IN_IMAGE] THEN ASM SET_TAC[]]; REWRITE_TAC[GSYM AFF_DIM_EQ_HYPERPLANE] THEN ASM_SIMP_TAC[IN_DIFF; AFFINE_INDEPENDENT_DELETE; AFF_DIM_AFFINE_INDEPENDENT; CARD_DELETE] THEN REWRITE_TAC[GSYM AFF_DIM_UNIV] THEN FIRST_ASSUM(SUBST1_TAC o SYM) THEN REWRITE_TAC[AFF_DIM_AFFINE_HULL] THEN ASM_SIMP_TAC[AFF_DIM_AFFINE_INDEPENDENT; CARD_DIFF] THEN REPEAT STRIP_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC(GSYM INT_OF_NUM_SUB) THEN MATCH_MP_TAC(ARITH_RULE `~(c = 0) ==> 1 <= c`) THEN ASM_SIMP_TAC[CARD_EQ_0] THEN ASM SET_TAC[]]);; let AFFINE_HYPERPLANE_SUMS_EQ_UNIV = prove (`!a b s. affine s /\ ~(s INTER {v | a dot v = b} = {}) /\ ~(s DIFF {v | a dot v = b} = {}) ==> {x + y | x IN s /\ y IN {v | a dot v = b}} = (:real^N)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `a:real^N = vec 0` THENL [ASM_REWRITE_TAC[DOT_LZERO] THEN SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[TAUT `a /\ b /\ c ==> d <=> b ==> a /\ c ==> d`] THEN GEN_REWRITE_TAC LAND_CONV [GSYM MEMBER_NOT_EMPTY] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; IN_ELIM_THM] THEN X_GEN_TAC `c:real^N` THEN ONCE_REWRITE_TAC[SET_RULE `{x + y:real^N | x IN s /\ P y} = {z | ?x y. x IN s /\ P y /\ z = x + y}`] THEN GEOM_ORIGIN_TAC `c:real^N` THEN REPEAT GEN_TAC THEN REWRITE_TAC[DOT_RADD; REAL_ARITH `b dot c + a = d <=> a = d - b dot c`] THEN REWRITE_TAC[IN_INTER; IN_ELIM_THM; DOT_RZERO] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (SUBST1_TAC o SYM)) THEN ASM_SIMP_TAC[AFFINE_EQ_SUBSPACE; HULL_INC] THEN STRIP_TAC THEN REWRITE_TAC[VECTOR_ARITH `c + z:real^N = (c + x) + (c + y) <=> z = c + x + y`] THEN REWRITE_TAC[SET_RULE `{z | ?x y. x IN s /\ P y /\ z = c + x + y} = IMAGE (\x. c + x) {x + y:real^N | x IN s /\ y IN {v | P v}}`] THEN MATCH_MP_TAC(SET_RULE `!f. (!x. g(f x) = x) /\ s = UNIV ==> IMAGE g s = UNIV`) THEN EXISTS_TAC `\x:real^N. x - c` THEN REWRITE_TAC[VECTOR_ARITH `c + x - c:real^N = x`] THEN MATCH_MP_TAC(MESON[SPAN_EQ_SELF] `subspace s /\ span s = t ==> s = t`) THEN CONJ_TAC THENL [ASM_SIMP_TAC[SUBSPACE_SUMS; SUBSPACE_HYPERPLANE]; ALL_TAC] THEN REWRITE_TAC[GSYM DIM_EQ_FULL] THEN REWRITE_TAC[GSYM LE_ANTISYM; DIM_SUBSET_UNIV] THEN MATCH_MP_TAC(ARITH_RULE `m - 1 < n ==> m <= n`) THEN MATCH_MP_TAC LET_TRANS THEN EXISTS_TAC `dim {x:real^N | a dot x = &0}` THEN CONJ_TAC THENL [ASM_SIMP_TAC[DIM_HYPERPLANE; LE_REFL]; ALL_TAC] THEN MATCH_MP_TAC DIM_PSUBSET THEN ASM_SIMP_TAC[snd(EQ_IMP_RULE(SPEC_ALL SPAN_EQ_SELF)); SUBSPACE_SUMS; SUBSPACE_HYPERPLANE] THEN REWRITE_TAC[PSUBSET; SUBSET; FORALL_IN_GSPEC] THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN CONJ_TAC THENL [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MAP_EVERY EXISTS_TAC [`vec 0:real^N`; `x:real^N`] THEN ASM_SIMP_TAC[SUBSPACE_0; VECTOR_ADD_LID]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN REWRITE_TAC[NOT_FORALL_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:real^N` THEN SIMP_TAC[IN_DIFF; IN_ELIM_THM] THEN DISCH_TAC THEN MAP_EVERY EXISTS_TAC [`x:real^N`; `vec 0:real^N`] THEN ASM_REWRITE_TAC[DOT_RZERO; VECTOR_ADD_RID]]);; let AFF_DIM_AFFINE_INTER_HYPERPLANE = prove (`!a b s:real^N->bool. affine s ==> aff_dim(s INTER {x | a dot x = b}) = if s INTER {v | a dot v = b} = {} then -- &1 else if s SUBSET {v | a dot v = b} then aff_dim s else aff_dim s - &1`, REPEAT GEN_TAC THEN ASM_CASES_TAC `a:real^N = vec 0` THENL [ASM_REWRITE_TAC[DOT_LZERO] THEN ASM_CASES_TAC `b = &0` THEN ASM_REWRITE_TAC[EMPTY_GSPEC; INTER_EMPTY; AFF_DIM_EMPTY] THEN SIMP_TAC[SET_RULE `{x | T} = UNIV`; IN_UNIV; INTER_UNIV; SUBSET_UNIV] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[AFF_DIM_EMPTY]; STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[AFF_DIM_EMPTY] THEN COND_CASES_TAC THENL [AP_TERM_TAC THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`s:real^N->bool`; `{x:real^N | a dot x = b}`] AFF_DIM_SUMS_INTER) THEN ASM_SIMP_TAC[AFFINE_HYPERPLANE; AFF_DIM_HYPERPLANE] THEN ASM_SIMP_TAC[AFFINE_HYPERPLANE_SUMS_EQ_UNIV; AFF_DIM_UNIV; SET_RULE `~(s SUBSET t) ==> ~(s DIFF t = {})`] THEN SPEC_TAC(`aff_dim (s INTER {x:real^N | a dot x = b})`,`i:int`) THEN INT_ARITH_TAC]);; let AFF_DIM_LT_FULL = prove (`!s. aff_dim s < &(dimindex(:N)) <=> ~(affine hull s = (:real^N))`, GEN_TAC THEN REWRITE_TAC[GSYM AFF_DIM_EQ_FULL] THEN MP_TAC(ISPEC `s:real^N->bool` AFF_DIM_LE_UNIV) THEN ARITH_TAC);; let AFF_LOWDIM_SUBSET_HYPERPLANE = prove (`!s:real^N->bool. aff_dim s < &(dimindex(:N)) ==> ?a b. ~(a = vec 0) /\ s SUBSET {x | a dot x = b}`, MATCH_MP_TAC SET_PROVE_CASES THEN CONJ_TAC THENL [DISCH_TAC THEN EXISTS_TAC `basis 1:real^N` THEN SIMP_TAC[EMPTY_SUBSET; BASIS_NONZERO; LE_REFL; DIMINDEX_GE_1]; MAP_EVERY X_GEN_TAC [`c:real^N`; `s:real^N->bool`] THEN CONV_TAC(ONCE_DEPTH_CONV(GEN_ALPHA_CONV `a:real^N`)) THEN GEN_GEOM_ORIGIN_TAC `c:real^N` ["a"] THEN SIMP_TAC[AFF_DIM_DIM_0; HULL_INC; IN_INSERT; INT_OF_NUM_LT] THEN REPEAT GEN_TAC THEN DISCH_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP LOWDIM_SUBSET_HYPERPLANE) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^N` THEN STRIP_TAC THEN EXISTS_TAC `(u:real^N) dot c` THEN ASM_REWRITE_TAC[DOT_RADD; REAL_EQ_ADD_LCANCEL_0] THEN ASM_MESON_TAC[SPAN_INC; SUBSET_TRANS]]);; let COLLINEAR_HYPERPLANE_2 = prove (`!a:real^N b. dimindex(:N) <= 2 /\ ~(a = vec 0) ==> collinear {x | a dot x = b}`, SIMP_TAC[COLLINEAR_AFF_DIM; AFF_DIM_HYPERPLANE; GSYM INT_OF_NUM_LE] THEN INT_ARITH_TAC);; let COLLINEAR_STANDARD_HYPERPLANE_2 = prove (`!k b. dimindex(:N) <= 2 ==> collinear {x:real^N | x$k = b}`, REPEAT GEN_TAC THEN SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i` CHOOSE_TAC THENL [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN MP_TAC(ISPECL [`basis i:real^N`; `b:real`] COLLINEAR_HYPERPLANE_2) THEN ASM_SIMP_TAC[DOT_BASIS; BASIS_NONZERO]);; (* ------------------------------------------------------------------------- *) (* Existence of rotation into general position w.r.t the axes. *) (* ------------------------------------------------------------------------- *) let ROTATION_TO_GENERAL_POSITION_EXISTS_GEN = prove (`!n s:real^N->bool. n <= dimindex(:N) /\ COUNTABLE s /\ s SUBSET span(IMAGE basis (1..n)) ==> ?f. orthogonal_transformation f /\ det(matrix f) = &1 /\ (!x. (!i. 1 <= i /\ i <= n ==> x$i = &0) ==> f x = x) /\ IMAGE f (span(IMAGE basis (1..n))) = span(IMAGE basis (1..n)) /\ pairwise (\x y. !i. 1 <= i /\ i <= n ==> ~(f x$i = f y$i)) s`, let lemma0 = prove (`!s:real^N->bool k. affine s /\ &2 <= aff_dim s /\ COUNTABLE k /\ (!a. a IN k ==> ~(s SUBSET {x | orthogonal a x})) ==> UNIONS {{x | x IN s /\ orthogonal a x} | a IN k} UNION UNIONS {{x | x IN s /\ x IN span {a}} | a IN k} PSUBSET s`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[AFF_DIM_EMPTY] THEN CONV_TAC INT_REDUCE_CONV THEN STRIP_TAC THEN REWRITE_TAC[GSYM UNIONS_UNION] THEN MATCH_MP_TAC(SET_RULE `t SUBSET s /\ ~(s SUBSET t) ==> t PSUBSET s`) THEN REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC; FORALL_IN_UNION; SUBSET_RESTRICT] THEN DISCH_THEN(MP_TAC o ISPEC `subtopology euclidean (s:real^N->bool)` o MATCH_MP INTERIOR_OF_MONO) THEN MATCH_MP_TAC(SET_RULE `~(s = {}) /\ t = {} ==> s SUBSET t ==> F`) THEN CONJ_TAC THENL [REWRITE_TAC[INTERIOR_OF_EQ_EMPTY_COMPLEMENT] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN ASM_REWRITE_TAC[DIFF_EQ_EMPTY; CLOSURE_OF_EMPTY]; ALL_TAC] THEN MATCH_MP_TAC NOWHERE_DENSE_COUNTABLE_UNIONS_CLOSED_IN THEN ASM_SIMP_TAC[SIMPLE_IMAGE; COUNTABLE_IMAGE; FORALL_IN_IMAGE; COUNTABLE_UNION; FORALL_IN_UNION] THEN ASM_SIMP_TAC[CLOSED_IMP_LOCALLY_COMPACT; CLOSED_AFFINE] THEN REWRITE_TAC[orthogonal; SET_RULE `{x | x IN s /\ P x} = s INTER {x | P x}`] THEN REWRITE_TAC[SET_RULE `{x | x IN s} = s`] THEN SIMP_TAC[CLOSED_IN_CLOSED_INTER; CLOSED_HYPERPLANE; CLOSED_SPAN] THEN REWRITE_TAC[AND_FORALL_THM] THEN X_GEN_TAC `a:real^N` THEN ASM_CASES_TAC `(a:real^N) IN k` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THEN MATCH_MP_TAC EMPTY_INTERIOR_OF_AFF_DIM THEN ASM_REWRITE_TAC[] THENL [ASM_SIMP_TAC[AFF_DIM_AFFINE_INTER_HYPERPLANE] THEN ASM_SIMP_TAC[GSYM orthogonal] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[INT_ARITH `x - &1:int < x`] THEN REWRITE_TAC[INT_LT_LE; AFF_DIM_GE] THEN ASM_MESON_TAC[AFF_DIM_EQ_MINUS1]; TRANS_TAC INT_LET_TRANS `aff_dim(span{a:real^N})` THEN SIMP_TAC[AFF_DIM_SUBSET; INTER_SUBSET] THEN SIMP_TAC[AFF_DIM_DIM_SUBSPACE; SUBSPACE_SPAN] THEN REWRITE_TAC[DIM_SPAN; DIM_SING] THEN COND_CASES_TAC THEN ASM_INT_ARITH_TAC]) in let lemma1 = prove (`!n s:real^N->bool. n <= dimindex(:N) /\ COUNTABLE s /\ s SUBSET span(IMAGE basis (1..n)) DELETE vec 0 ==> ?f. orthogonal_transformation f /\ (!x. (!i. 1 <= i /\ i <= n ==> x$i = &0) ==> f x = x) /\ IMAGE f (span(IMAGE basis (1..n))) = span(IMAGE basis (1..n)) /\ !x i. x IN s /\ 1 <= i /\ i <= n ==> ~(f x$i = &0)`, MATCH_MP_TAC num_INDUCTION THEN CONJ_TAC THENL [CONV_TAC(ONCE_DEPTH_CONV NUMSEG_CONV) THEN REWRITE_TAC[IMAGE_CLAUSES; SPAN_EMPTY] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[ARITH_RULE `~(1 <= i /\ i <= 0)`] THEN EXISTS_TAC `\x:real^N. x` THEN REWRITE_TAC[ORTHOGONAL_TRANSFORMATION_ID]; X_GEN_TAC `n:num` THEN REWRITE_TAC[SUBSET_DELETE] THEN STRIP_TAC] THEN X_GEN_TAC `s:real^N->bool` THEN STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [EXISTS_TAC `\x:real^N. x` THEN ASM_REWRITE_TAC[IMAGE_ID; NOT_IN_EMPTY; ORTHOGONAL_TRANSFORMATION_ID]; ALL_TAC] THEN ASM_CASES_TAC `n = 0` THENL [EXISTS_TAC `\x:real^N. x` THEN ASM_REWRITE_TAC[ORTHOGONAL_TRANSFORMATION_ID; IMAGE_ID] THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN CONV_TAC NUM_REDUCE_CONV THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM; FORALL_1] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N` o REWRITE_RULE[SUBSET]) THEN ASM_REWRITE_TAC[IN_SPAN_IMAGE_BASIS] THEN SUBGOAL_THEN `~(x:real^N = vec 0)` MP_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[CART_EQ]] THEN REWRITE_TAC[VEC_COMPONENT; IN_NUMSEG; VEC_COMPONENT] THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN CONV_TAC NUM_REDUCE_CONV THEN DISCH_THEN(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[LE_ANTISYM] THEN ASM_MESON_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`span(IMAGE basis (1..SUC n)):real^N->bool`; `s:real^N->bool`] lemma0) THEN SIMP_TAC[SUBSPACE_IMP_AFFINE; SUBSPACE_SPAN; NONEMPTY_SPAN] THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [SIMP_TAC[AFF_DIM_DIM_SUBSPACE; SUBSPACE_SPAN; DIM_SPAN] THEN REWRITE_TAC[DIM_BASIS_IMAGE; INT_OF_NUM_LE] THEN REWRITE_TAC[CARD_NUMSEG_1; INTER_NUMSEG; ARITH_RULE `MAX 1 1 = 1`] THEN CONJ_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `s SUBSET u /\ (!a. a IN s ==> ~(a IN f a)) ==> !a. a IN s ==> ~(u SUBSET f a)`) THEN ASM_REWRITE_TAC[IN_ELIM_THM; ORTHOGONAL_REFL] THEN ASM SET_TAC[]; DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `t PSUBSET s ==> ?a. a IN s /\ ~(a IN t)`)) THEN REWRITE_TAC[UNIONS_GSPEC; IN_ELIM_THM; IN_UNION; DE_MORGAN_THM] THEN DISCH_THEN(X_CHOOSE_THEN `a:real^N` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN ASM_REWRITE_TAC[NOT_EXISTS_THM; TAUT `~(p /\ q) <=> p ==> ~q`]] THEN REWRITE_TAC[ONCE_REWRITE_RULE[ORTHOGONAL_SYM] orthogonal] THEN ASM_CASES_TAC `a:real^N = vec 0` THENL [ASM_MESON_TAC[DOT_LZERO; MEMBER_NOT_EMPTY]; STRIP_TAC] THEN MP_TAC(ISPECL [`span(IMAGE basis (1..SUC n)):real^N->bool`; `inv(norm a) % a:real^N`; `basis(SUC n):real^N`] ORTHOGONAL_TRANSFORMATION_EXISTS_GEN) THEN REWRITE_TAC[SUBSPACE_SPAN] THEN ANTS_TAC THENL [ASM_SIMP_TAC[SPAN_MUL; NORM_BASIS; ARITH_RULE `1 <= SUC n`] THEN REWRITE_TAC[NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN ASM_SIMP_TAC[REAL_MUL_LINV; NORM_EQ_0] THEN MATCH_MP_TAC SPAN_SUPERSET THEN MATCH_MP_TAC FUN_IN_IMAGE THEN REWRITE_TAC[IN_NUMSEG] THEN ARITH_TAC; DISCH_THEN(X_CHOOSE_THEN `h:real^N->real^N` STRIP_ASSUME_TAC)] THEN FIRST_X_ASSUM(MP_TAC o SPEC `IMAGE (\x. x - x$(SUC n) % basis(SUC n)) (IMAGE (h:real^N->real^N) s)`) THEN ASM_SIMP_TAC[COUNTABLE_IMAGE] THEN ANTS_TAC THENL [CONJ_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN CONJ_TAC THENL [REWRITE_TAC[VECTOR_SUB_EQ; SET_RULE `~(a IN IMAGE f (IMAGE g s)) <=> !x. x IN s ==> ~(f(g x) = a)`] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN SPEC_TAC(`(h:real^N->real^N) x$(SUC n)`,`b:real`) THEN GEN_TAC THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (funpow 3 RAND_CONV) [SYM th]) THEN ASM_SIMP_TAC[GSYM LINEAR_CMUL; ORTHOGONAL_TRANSFORMATION_LINEAR] THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN DISCH_TAC THEN SUBGOAL_THEN `(x:real^N) IN span {a} /\ ~(a IN span {x})` MP_TAC THENL [ASM_SIMP_TAC[] THEN REWRITE_TAC[SPAN_SING; IN_ELIM_THM; IN_UNIV] THEN ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION_INJECTIVE]; MP_TAC(MESON[INSERT_AC] `collinear{vec 0:real^N,a,x} <=> collinear{vec 0,x,a}`) THEN REWRITE_TAC[COLLINEAR_SPAN] THEN ASM_MESON_TAC[]]; TRANS_TAC SUBSET_TRANS `IMAGE (\x:real^N. x - x$(SUC n) % basis(SUC n)) (span (IMAGE basis (1..SUC n)))` THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_SPAN_IMAGE_BASIS] THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; IN_NUMSEG] THEN X_GEN_TAC `x:real^N` THEN SIMP_TAC[VECTOR_MUL_COMPONENT; BASIS_COMPONENT] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `i:num` THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_SUB_REFL; REAL_MUL_RID] THEN REWRITE_TAC[REAL_SUB_RZERO; REAL_MUL_RZERO] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC]; DISCH_THEN(X_CHOOSE_THEN `f:real^N->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(f:real^N->real^N) o (h:real^N->real^N)` THEN ASM_SIMP_TAC[IMAGE_o; ORTHOGONAL_TRANSFORMATION_COMPOSE] THEN ASM_REWRITE_TAC[o_THM] THEN REPEAT CONJ_TAC THENL [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN TRANS_TAC EQ_TRANS `(f:real^N->real^N) x` THEN CONJ_TAC THENL [AP_TERM_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[ORTHOGONAL_BASIS; ARITH_RULE `1 <= SUC n`; LE_REFL] THEN REWRITE_TAC[ORTHOGONAL_MUL] THEN DISJ2_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_SPAN_IMAGE_BASIS]) THEN REWRITE_TAC[IN_NUMSEG] THEN STRIP_TAC THEN REWRITE_TAC[orthogonal; dot] THEN MATCH_MP_TAC SUM_EQ_0 THEN X_GEN_TAC `i:num` THEN REWRITE_TAC[IN_NUMSEG] THEN STRIP_TAC THEN ASM_MESON_TAC[REAL_ENTIRE]; ASM_MESON_TAC[ARITH_RULE `i <= n ==> i <= SUC n`]]; REWRITE_TAC[NUMSEG_CLAUSES; ARITH_RULE `1 <= SUC n`] THEN ONCE_REWRITE_TAC[SET_RULE `a INSERT s = s UNION {a}`] THEN REWRITE_TAC[SPAN_UNION; IMAGE_UNION] THEN REWRITE_TAC[SET_RULE `IMAGE f {p x y | P x y} = {f(p x y) | P x y}`] THEN ASM_SIMP_TAC[LINEAR_ADD; ORTHOGONAL_TRANSFORMATION_LINEAR] THEN MATCH_MP_TAC(SET_RULE `IMAGE f s = u /\ (!x. x IN t ==> f x = x) ==> {(f:real^N->real^N) x + f y | x IN s /\ y IN t} = {x + y | x IN u /\ y IN t}`) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[IMAGE_CLAUSES; SPAN_SING; FORALL_IN_GSPEC; IN_UNIV] THEN ASM_SIMP_TAC[LINEAR_CMUL; ORTHOGONAL_TRANSFORMATION_LINEAR] THEN GEN_TAC THEN AP_TERM_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN SUBGOAL_THEN `!i. i <= n ==> i <= dimindex(:N)` MP_TAC THENL [ASM_ARITH_TAC; SIMP_TAC[BASIS_COMPONENT]] THEN SIMP_TAC[ARITH_RULE `i <= n ==> ~(i = SUC n)`]; MAP_EVERY X_GEN_TAC [`x:real^N`; `i:num`] THEN REWRITE_TAC[LE] THEN STRIP_TAC THENL [ALL_TAC; RULE_ASSUM_TAC(REWRITE_RULE [IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE]) THEN RULE_ASSUM_TAC(REWRITE_RULE [IMP_IMP; RIGHT_IMP_FORALL_THM; GSYM CONJ_ASSOC]) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `i:num`]) THEN ASM_REWRITE_TAC[CONTRAPOS_THM; VECTOR_SUB_COMPONENT] THEN ASM_SIMP_TAC[LINEAR_SUB; ORTHOGONAL_TRANSFORMATION_LINEAR; VECTOR_SUB_COMPONENT] THEN DISCH_THEN(K ALL_TAC) THEN MATCH_MP_TAC(REAL_ARITH `y$i = &0 /\ (f:real^N->real^N)(y)$i = y$i ==> &0 - f(y)$i = &0`) THEN CONJ_TAC THENL [SUBGOAL_THEN `i <= dimindex(:N)` MP_TAC THENL [ASM_ARITH_TAC; ALL_TAC]; AP_THM_TAC THEN AP_TERM_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN SUBGOAL_THEN `!j. j <= n ==> j <= dimindex(:N)` MP_TAC THENL [ASM_ARITH_TAC; ALL_TAC]] THEN ASM_SIMP_TAC[VECTOR_MUL_COMPONENT; BASIS_COMPONENT; REAL_MUL_RZERO; ARITH_RULE `i <= n ==> ~(i = SUC n)`]] THEN SUBGOAL_THEN `(f:real^N->real^N)(h(x:real^N)) = f(h x - h x$(SUC n) % basis(SUC n)) + h x$(SUC n) % basis(SUC n)` SUBST1_TAC THENL [W(MP_TAC o PART_MATCH (lhand o rand) LINEAR_SUB o lhand o rand o snd) THEN ASM_SIMP_TAC[ORTHOGONAL_TRANSFORMATION_LINEAR] THEN DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC(VECTOR_ARITH `y:real^N = z ==> x = x - y + z`) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN SUBGOAL_THEN `!j. j <= n ==> j <= dimindex(:N)` MP_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[VECTOR_MUL_COMPONENT; BASIS_COMPONENT; REAL_MUL_RZERO; ARITH_RULE `i <= n ==> ~(i = SUC n)`]; ASM_REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN ASM_SIMP_TAC[BASIS_COMPONENT; ARITH_RULE `1 <= SUC n`] THEN MATCH_MP_TAC(REAL_ARITH `x = &0 /\ ~(y = &0) ==> ~(x + y * &1 = &0)`) THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP(SET_RULE `IMAGE f b = b ==> x IN b /\ (!y. y IN b ==> y$SUC n = &0) ==> (f:real^N->real^N) x $SUC n = &0`)) THEN REWRITE_TAC[IN_SPAN_IMAGE_BASIS] THEN ASM_SIMP_TAC[IN_NUMSEG; ARITH_RULE `1 <= SUC n /\ ~(SUC n <= n)`] THEN SUBGOAL_THEN `(h:real^N->real^N) x IN span(IMAGE basis (1..SUC n))` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[IN_SPAN_IMAGE_BASIS] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `j:num` THEN ASM_CASES_TAC `1 <= j` THEN ASM_REWRITE_TAC[VECTOR_SUB_COMPONENT] THEN ASM_CASES_TAC `j <= dimindex(:N)` THEN ASM_REWRITE_TAC[VECTOR_MUL_COMPONENT; IN_NUMSEG] THEN ASM_SIMP_TAC[BASIS_COMPONENT] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_MUL_RID; REAL_SUB_REFL] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[REAL_MUL_RZERO; REAL_SUB_RZERO] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; SUBGOAL_THEN `~(h(inv(norm a) % a) dot (h:real^N->real^N) x = &0)` MP_TAC THENL [ALL_TAC; ASM_SIMP_TAC[DOT_BASIS; ARITH_RULE `1 <= SUC n`]] THEN RULE_ASSUM_TAC(REWRITE_RULE[orthogonal_transformation]) THEN ASM_REWRITE_TAC[DOT_LMUL; REAL_ENTIRE; REAL_INV_EQ_0; NORM_EQ_0] THEN ASM_MESON_TAC[orthogonal]]]]]) in REPEAT STRIP_TAC THEN ASM_CASES_TAC `n = 0` THENL [EXISTS_TAC `\x:real^N. x` THEN ASM_REWRITE_TAC[ORTHOGONAL_TRANSFORMATION_ID; MATRIX_ID; DET_I] THEN CONV_TAC(ONCE_DEPTH_CONV NUMSEG_CONV) THEN REWRITE_TAC[IMAGE_CLAUSES; SPAN_EMPTY] THEN REWRITE_TAC[pairwise] THEN ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC(MESON[] `(!f. P f /\ R f ==> ?f'. P f' /\ Q f' /\ R f') /\ (?f. P f /\ R f) ==> ?f. P f /\ Q f /\ R f`) THEN CONJ_TAC THENL [X_GEN_TAC `f:real^N->real^N` THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP ORTHOGONAL_MATRIX_MATRIX) THEN DISCH_THEN(STRIP_ASSUME_TAC o MATCH_MP DET_ORTHOGONAL_MATRIX) THENL [ASM_MESON_TAC[]; ALL_TAC] THEN EXISTS_TAC `reflect_along (basis 1:real^N) o (f:real^N->real^N)` THEN ASM_SIMP_TAC[ORTHOGONAL_TRANSFORMATION_COMPOSE; ORTHOGONAL_TRANSFORMATION_REFLECT_ALONG] THEN ASM_SIMP_TAC[MATRIX_COMPOSE; ORTHOGONAL_TRANSFORMATION_LINEAR; LINEAR_REFLECT_ALONG; DET_MUL] THEN SUBGOAL_THEN `!i. i <= n ==> i <= dimindex(:N)` MP_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[DET_MATRIX_REFLECT_ALONG; BASIS_NONZERO; o_THM; IMAGE_o; CART_EQ; DIMINDEX_GE_1; LE_REFL; REFLECT_ALONG_BASIS_COMPONENT] THEN DISCH_TAC THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REPEAT CONJ_TAC THENL [GEN_TAC THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[LE_1; LE_REFL] THEN CONV_TAC REAL_RAT_REDUCE_CONV; MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN CONJ_TAC THENL [MESON_TAC[REFLECT_ALONG_INVOLUTION]; ALL_TAC] THEN REWRITE_TAC[IN_SPAN_IMAGE_BASIS] THEN SIMP_TAC[REFLECT_ALONG_BASIS_COMPONENT; DIMINDEX_GE_1; LE_REFL] THEN ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN REWRITE_TAC[REAL_NEG_EQ_0] THEN GEN_TAC THEN AP_TERM_TAC THEN ABS_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] PAIRWISE_IMP)) THEN REWRITE_TAC[] THEN REPEAT GEN_TAC THEN REPEAT DISCH_TAC THEN GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_EQ_NEG2]]; MP_TAC(ISPECL [`n:num`; `{x - y:real^N | x IN s /\ y IN s} DELETE (vec 0)`] lemma1) THEN ASM_REWRITE_TAC[IN_DELETE] THEN ASM_SIMP_TAC[COUNTABLE_PRODUCT_DEPENDENT; COUNTABLE_DELETE] THEN ANTS_TAC THENL [MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s DELETE a SUBSET t DELETE a`) THEN REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC SPAN_SUB THEN ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_GSPEC] THEN REWRITE_TAC[VECTOR_SUB_EQ; pairwise] THEN GEN_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(MP_TAC o MATCH_MP LINEAR_SUB o MATCH_MP ORTHOGONAL_TRANSFORMATION_LINEAR) THEN DISCH_THEN(fun th -> REWRITE_TAC[th; VECTOR_SUB_COMPONENT]) THEN REWRITE_TAC[REAL_SUB_0] THEN MESON_TAC[]]]);; let ROTATION_TO_GENERAL_POSITION_EXISTS = prove (`!s:real^N->bool. COUNTABLE s ==> ?f. orthogonal_transformation f /\ det(matrix f) = &1 /\ pairwise (\x y. !i. 1 <= i /\ i <= dimindex(:N) ==> ~(f x$i = f y$i)) s`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`dimindex(:N)`; `s:real^N->bool`] ROTATION_TO_GENERAL_POSITION_EXISTS_GEN) THEN ASM_REWRITE_TAC[LE_REFL] THEN REWRITE_TAC[GSYM SIMPLE_IMAGE; IN_NUMSEG; SPAN_STDBASIS; SUBSET_UNIV] THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[]);; (* ------------------------------------------------------------------------- *) (* Openness and compactness are preserved by convex hull operation. *) (* ------------------------------------------------------------------------- *) let OPEN_CONVEX_HULL = prove (`!s:real^N->bool. open s ==> open(convex hull s)`, REPEAT GEN_TAC THEN REWRITE_TAC[CONVEX_HULL_EXPLICIT; OPEN_CONTAINS_CBALL] THEN REWRITE_TAC[IN_ELIM_THM; SUBSET; LEFT_IMP_EXISTS_THM] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `t:real^N->bool`; `u:real^N->real`] THEN STRIP_TAC THEN SUBGOAL_THEN `?b. !x:real^N. x IN t ==> &0 < b(x) /\ cball(x,b(x)) SUBSET s` STRIP_ASSUME_TAC THENL [REWRITE_TAC[GSYM SKOLEM_THM] THEN ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN ABBREV_TAC `i = IMAGE (b:real^N->real) t` THEN EXISTS_TAC `inf i` THEN MP_TAC(SPEC `i:real->bool` INF_FINITE) THEN EXPAND_TAC "i" THEN ASM_REWRITE_TAC[FORALL_IN_IMAGE; IN_IMAGE] THEN ANTS_TAC THENL [EXPAND_TAC "i" THEN CONJ_TAC THENL [ASM_SIMP_TAC[FINITE_IMAGE]; ALL_TAC] THEN REWRITE_TAC[IMAGE_EQ_EMPTY] THEN ASM_MESON_TAC[SUM_CLAUSES; REAL_ARITH `~(&1 = &0)`]; ALL_TAC] THEN STRIP_TAC THEN CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_CBALL; dist] THEN DISCH_TAC THEN EXISTS_TAC `IMAGE (\v:real^N. v + (y - a)) t` THEN EXISTS_TAC `\v. (u:real^N->real)(v - (y - a))` THEN ASM_SIMP_TAC[FINITE_IMAGE; FORALL_IN_IMAGE; SUM_IMAGE; VSUM_IMAGE; VECTOR_ARITH `v + a:real^N = w + a <=> v = w`] THEN ASM_REWRITE_TAC[o_DEF; VECTOR_ARITH `(v + a) - a:real^N = v`] THEN ASM_REWRITE_TAC[VECTOR_ADD_LDISTRIB; ETA_AX] THEN ASM_SIMP_TAC[VSUM_ADD; VSUM_RMUL] THEN CONJ_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN X_GEN_TAC `z:real^N` THEN STRIP_TAC THEN SUBGOAL_THEN `z + (y - a) IN cball(z:real^N,b z)` (fun th -> ASM_MESON_TAC[th; SUBSET]) THEN REWRITE_TAC[IN_CBALL; dist; NORM_ARITH `norm(z - (z + a - y)) = norm(y - a)`] THEN ASM_MESON_TAC[REAL_LE_TRANS]);; let COMPACT_CONVEX_COMBINATIONS = prove (`!s t. compact s /\ compact t ==> compact { (&1 - u) % x + u % y :real^N | &0 <= u /\ u <= &1 /\ x IN s /\ y IN t}`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `{ (&1 - u) % x + u % y :real^N | &0 <= u /\ u <= &1 /\ x IN s /\ y IN t} = IMAGE (\z. (&1 - drop(fstcart z)) % fstcart(sndcart z) + drop(fstcart z) % sndcart(sndcart z)) { pastecart u w | u IN interval[vec 0,vec 1] /\ w IN { pastecart x y | x IN s /\ y IN t} }` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_IMAGE] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM; LEFT_AND_EXISTS_THM] THEN CONV_TAC(ONCE_DEPTH_CONV UNWIND_CONV) THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN REWRITE_TAC[IN_INTERVAL_1; GSYM EXISTS_DROP; DROP_VEC] THEN MESON_TAC[]; ALL_TAC] THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_SIMP_TAC[COMPACT_PCROSS; GSYM PCROSS; COMPACT_INTERVAL] THEN MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN X_GEN_TAC `z:real^(1,(N,N)finite_sum)finite_sum` THEN DISCH_THEN(K ALL_TAC) THEN REWRITE_TAC[PCROSS] THEN MATCH_MP_TAC CONTINUOUS_ADD THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_MUL THEN REWRITE_TAC[o_DEF; LIFT_SUB; LIFT_DROP] THEN CONJ_TAC THEN TRY(MATCH_MP_TAC CONTINUOUS_SUB) THEN REWRITE_TAC[CONTINUOUS_CONST] THEN MATCH_MP_TAC LINEAR_CONTINUOUS_AT THEN REWRITE_TAC[LINEAR_FSTCART; LINEAR_SNDCART; ETA_AX] THEN GEN_REWRITE_TAC RAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC LINEAR_COMPOSE THEN REWRITE_TAC[LINEAR_FSTCART; LINEAR_SNDCART]);; let COMPACT_CONVEX_HULL = prove (`!s:real^N->bool. compact s ==> compact(convex hull s)`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[CARATHEODORY] THEN SPEC_TAC(`dimindex(:N) + 1`,`n:num`) THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [ASM_REWRITE_TAC[SUBSET_EMPTY] THEN CONV_TAC(ONCE_DEPTH_CONV UNWIND_CONV) THEN REWRITE_TAC[CONVEX_HULL_EMPTY; NOT_IN_EMPTY] THEN REWRITE_TAC[SET_RULE `{x | F} = {}`; COMPACT_EMPTY]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_TAC `w:real^N`) THEN INDUCT_TAC THENL [SUBGOAL_THEN `{x:real^N | ?t. FINITE t /\ t SUBSET s /\ CARD t <= 0 /\ x IN convex hull t} = {}` (fun th -> REWRITE_TAC[th; COMPACT_EMPTY]) THEN REWRITE_TAC[EXTENSION; NOT_IN_EMPTY; LE; IN_ELIM_THM] THEN MESON_TAC[CARD_EQ_0; CONVEX_HULL_EMPTY; NOT_IN_EMPTY]; ALL_TAC] THEN ASM_CASES_TAC `n = 0` THENL [ASM_REWRITE_TAC[ARITH_RULE `s <= SUC 0 <=> s = 0 \/ s = 1`] THEN UNDISCH_TAC `compact(s:real^N->bool)` THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN REWRITE_TAC[TAUT `a /\ b /\ (c \/ d) /\ e <=> (a /\ c) /\ (b /\ e) \/ (a /\ d) /\ (b /\ e)`] THEN REWRITE_TAC[GSYM HAS_SIZE; num_CONV `1`; HAS_SIZE_CLAUSES] THEN REWRITE_TAC[EXISTS_OR_THM; LEFT_AND_EXISTS_THM; RIGHT_AND_EXISTS_THM] THEN CONV_TAC(TOP_DEPTH_CONV UNWIND_CONV) THEN REWRITE_TAC[NOT_IN_EMPTY; CONVEX_HULL_EMPTY] THEN REWRITE_TAC[CONVEX_HULL_SING] THEN SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `{x:real^N | ?t. FINITE t /\ t SUBSET s /\ CARD t <= SUC n /\ x IN convex hull t} = { (&1 - u) % x + u % y :real^N | &0 <= u /\ u <= &1 /\ x IN s /\ y IN {x | ?t. FINITE t /\ t SUBSET s /\ CARD t <= n /\ x IN convex hull t}}` (fun th -> ASM_SIMP_TAC[th; COMPACT_CONVEX_COMBINATIONS]) THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN EQ_TAC THENL [ALL_TAC; REWRITE_TAC[LEFT_IMP_EXISTS_THM; RIGHT_AND_EXISTS_THM; LEFT_AND_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u:real^N`; `c:real`; `v:real^N`; `t:real^N->bool`] THEN STRIP_TAC THEN EXISTS_TAC `(u:real^N) INSERT t` THEN ASM_REWRITE_TAC[FINITE_INSERT; INSERT_SUBSET] THEN ASM_SIMP_TAC[CARD_CLAUSES] THEN CONJ_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC IN_CONVEX_SET THEN ASM_REWRITE_TAC[CONVEX_CONVEX_HULL] THEN CONJ_TAC THEN ASM_MESON_TAC[HULL_SUBSET; SUBSET; IN_INSERT; HULL_MONO]] THEN DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` STRIP_ASSUME_TAC) THEN ASM_CASES_TAC `CARD(t:real^N->bool) <= n` THENL [MAP_EVERY EXISTS_TAC [`w:real^N`; `&1`; `x:real^N`] THEN ASM_REWRITE_TAC[REAL_POS; REAL_LE_REFL] THEN CONJ_TAC THENL [ASM_MESON_TAC[]; VECTOR_ARITH_TAC]; ALL_TAC] THEN SUBGOAL_THEN `(t:real^N->bool) HAS_SIZE (SUC n)` MP_TAC THENL [ASM_REWRITE_TAC[HAS_SIZE] THEN ASM_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[HAS_SIZE_CLAUSES] THEN DISCH_THEN(X_CHOOSE_THEN `a:real^N` (X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC)) THEN FIRST_X_ASSUM SUBST_ALL_TAC THEN UNDISCH_TAC `(x:real^N) IN convex hull (a INSERT u)` THEN RULE_ASSUM_TAC(REWRITE_RULE[FINITE_INSERT]) THEN ASM_CASES_TAC `(u:real^N->bool) = {}` THENL [ASM_REWRITE_TAC[CONVEX_HULL_SING; IN_SING] THEN DISCH_THEN SUBST_ALL_TAC THEN MAP_EVERY EXISTS_TAC [`a:real^N`; `&1`; `a:real^N`] THEN ASM_REWRITE_TAC[REAL_POS; REAL_LE_REFL] THEN CONJ_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN EXISTS_TAC `{a:real^N}` THEN SIMP_TAC[FINITE_RULES] THEN REWRITE_TAC[CONVEX_HULL_SING; IN_SING] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SIMP_TAC[CARD_CLAUSES; FINITE_RULES; NOT_IN_EMPTY] THEN UNDISCH_TAC `~(n = 0)` THEN ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[CONVEX_HULL_INSERT; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`c:real`; `d:real`; `z:real^N`] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`a:real^N`; `d:real`; `z:real^N`] THEN FIRST_X_ASSUM(SUBST_ALL_TAC o MATCH_MP (REAL_ARITH `c + d = &1 ==> c = (&1 - d)`)) THEN ASM_REWRITE_TAC[REAL_ARITH `d <= &1 <=> &0 <= &1 - d`] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN UNDISCH_TAC `CARD ((a:real^N) INSERT u) <= SUC n` THEN ASM_SIMP_TAC[CARD_CLAUSES; LE_SUC]);; let FINITE_IMP_COMPACT_CONVEX_HULL = prove (`!s:real^N->bool. FINITE s ==> compact(convex hull s)`, SIMP_TAC[FINITE_IMP_COMPACT; COMPACT_CONVEX_HULL]);; let CONVEX_HULL_INTERIOR_SUBSET = prove (`!s:real^N->bool. convex hull (interior s) SUBSET interior (convex hull s)`, GEN_TAC THEN MATCH_MP_TAC INTERIOR_MAXIMAL THEN SIMP_TAC[OPEN_CONVEX_HULL; OPEN_INTERIOR; HULL_MONO; INTERIOR_SUBSET]);; (* ------------------------------------------------------------------------- *) (* Extremal points of a simplex are some vertices. *) (* ------------------------------------------------------------------------- *) let SIMPLEX_FURTHEST_LT = prove (`!a:real^N s. FINITE s ==> !x. x IN (convex hull s) /\ ~(x IN s) ==> ?y. y IN (convex hull s) /\ norm(x - a) < norm(y - a)`, GEN_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN REWRITE_TAC[CONVEX_HULL_EMPTY; NOT_IN_EMPTY] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `s:real^N->bool`] THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [ASM_REWRITE_TAC[CONVEX_HULL_SING; IN_SING] THEN MESON_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[CONVEX_HULL_INSERT] THEN STRIP_TAC THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_ELIM_THM; LEFT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u:real`; `v:real`; `b:real^N`] THEN ASM_CASES_TAC `y:real^N IN (convex hull s)` THENL [REWRITE_TAC[IN_INSERT; DE_MORGAN_THM] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^N` THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`&0`; `&1`; `c:real^N`] THEN ASM_REWRITE_TAC[REAL_ADD_LID; REAL_POS] THEN VECTOR_ARITH_TAC; ALL_TAC] THEN ASM_CASES_TAC `u = &0` THENL [ASM_SIMP_TAC[REAL_ADD_LID; VECTOR_MUL_LZERO; VECTOR_ADD_LID] THEN ASM_MESON_TAC[VECTOR_MUL_LID]; ALL_TAC] THEN ASM_CASES_TAC `v = &0` THENL [ASM_SIMP_TAC[REAL_ADD_RID; VECTOR_MUL_LZERO; VECTOR_ADD_RID] THEN ASM_CASES_TAC `u = &1` THEN ASM_REWRITE_TAC[VECTOR_MUL_LID] THEN ASM_CASES_TAC `y = a:real^N` THEN ASM_REWRITE_TAC[IN_INSERT] THEN ASM_MESON_TAC[]; ALL_TAC] THEN REWRITE_TAC[IN_INSERT; DE_MORGAN_THM] THEN STRIP_TAC THEN MP_TAC(SPECL [`u:real`; `v:real`] REAL_DOWN2) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[REAL_LT_LE]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `w:real` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`a:real^N`; `y:real^N`; `w % (x - b):real^N`] DIST_INCREASES_ONLINE) THEN ANTS_TAC THENL [ASM_SIMP_TAC[VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ] THEN REWRITE_TAC[VECTOR_ARITH `(x - y = vec 0) <=> (x = y)`] THEN DISCH_THEN SUBST_ALL_TAC THEN UNDISCH_TAC `~(y:real^N IN convex hull s)` THEN ASM_REWRITE_TAC[GSYM VECTOR_ADD_RDISTRIB; VECTOR_MUL_LID]; ALL_TAC] THEN ASM_REWRITE_TAC[dist; real_gt] THEN REWRITE_TAC[VECTOR_ARITH `((u % x + v % b) + w % (x - b) = (u + w) % x + (v - w) % b) /\ ((u % x + v % b) - w % (x - b) = (u - w) % x + (v + w) % b)`] THEN STRIP_TAC THENL [MAP_EVERY EXISTS_TAC [`(u + w) % x + (v - w) % b:real^N`; `u + w`; `v - w`; `b:real^N`]; MAP_EVERY EXISTS_TAC [`(u - w) % x + (v + w) % b:real^N`; `u - w`; `v + w`; `b:real^N`]] THEN ONCE_REWRITE_TAC[NORM_SUB] THEN ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[REAL_LE_ADD; REAL_LT_IMP_LE; REAL_SUB_LE] THEN UNDISCH_TAC `u + v = &1` THEN REAL_ARITH_TAC);; let SIMPLEX_FURTHEST_LE = prove (`!a:real^N s. FINITE s /\ ~(s = {}) ==> ?y. y IN s /\ !x. x IN (convex hull s) ==> norm(x - a) <= norm(y - a)`, REPEAT GEN_TAC THEN DISCH_TAC THEN MP_TAC(ISPEC `convex hull (s:real^N->bool)` DISTANCE_ATTAINS_SUP) THEN DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN ANTS_TAC THENL [ASM_SIMP_TAC[FINITE_IMP_COMPACT_CONVEX_HULL] THEN ASM_MESON_TAC[SUBSET_EMPTY; HULL_SUBSET]; ALL_TAC] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN REWRITE_TAC[dist] THEN ASM_MESON_TAC[SIMPLEX_FURTHEST_LT; REAL_NOT_LE]);; let SIMPLEX_FURTHEST_LE_EXISTS = prove (`!a:real^N s. FINITE s ==> !x. x IN (convex hull s) ==> ?y. y IN s /\ norm(x - a) <= norm(y - a)`, MESON_TAC[NOT_IN_EMPTY; CONVEX_HULL_EMPTY; SIMPLEX_FURTHEST_LE]);; let SIMPLEX_EXTREMAL_LE = prove (`!s:real^N->bool. FINITE s /\ ~(s = {}) ==> ?u v. u IN s /\ v IN s /\ !x y. x IN convex hull s /\ y IN convex hull s ==> norm(x - y) <= norm(u - v)`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `convex hull (s:real^N->bool)` COMPACT_SUP_MAXDISTANCE) THEN ANTS_TAC THENL [ASM_SIMP_TAC[FINITE_IMP_COMPACT_CONVEX_HULL] THEN ASM_MESON_TAC[SUBSET_EMPTY; HULL_SUBSET]; ALL_TAC] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN SIMP_TAC[] THEN ASM_MESON_TAC[SIMPLEX_FURTHEST_LT; REAL_NOT_LE; NORM_SUB]);; let SIMPLEX_EXTREMAL_LE_EXISTS = prove (`!s:real^N->bool x y. FINITE s /\ x IN convex hull s /\ y IN convex hull s ==> ?u v. u IN s /\ v IN s /\ norm(x - y) <= norm(u - v)`, MESON_TAC[NOT_IN_EMPTY; CONVEX_HULL_EMPTY; SIMPLEX_EXTREMAL_LE]);; (* ------------------------------------------------------------------------- *) (* Closest point of a convex set is unique, with a continuous projection. *) (* ------------------------------------------------------------------------- *) let CLOSER_POINTS_LEMMA = prove (`!y:real^N z. y dot z > &0 ==> ?u. &0 < u /\ !v. &0 < v /\ v <= u ==> norm(v % z - y) < norm y`, REWRITE_TAC[NORM_LT; DOT_LSUB; DOT_RSUB; DOT_LMUL; DOT_RMUL; REAL_SUB_LDISTRIB; real_gt] THEN REPEAT GEN_TAC THEN REWRITE_TAC[REAL_ARITH `(a - b) - (c - d) < d <=> a < b + c`] THEN STRIP_TAC THEN SUBST1_TAC(VECTOR_ARITH `(z:real^N) dot y = y dot z`) THEN SIMP_TAC[GSYM REAL_ADD_LDISTRIB; REAL_LT_LMUL_EQ] THEN EXISTS_TAC `(y dot (z:real^N)) / (z dot z)` THEN SUBGOAL_THEN `&0 < z dot (z:real^N)` ASSUME_TAC THENL [ASM_MESON_TAC[DOT_POS_LT; DOT_RZERO; REAL_LT_REFL]; ALL_TAC] THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_LE_RDIV_EQ] THEN ASM_SIMP_TAC[REAL_ARITH `&0 < y /\ x <= y ==> x < y + y`; REAL_LT_MUL]);; let CLOSER_POINT_LEMMA = prove (`!x y z. (y - x) dot (z - x) > &0 ==> ?u. &0 < u /\ u <= &1 /\ dist(x + u % (z - x),y) < dist(x,y)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP CLOSER_POINTS_LEMMA) THEN ONCE_REWRITE_TAC[DIST_SYM] THEN REWRITE_TAC[dist; NORM_LT] THEN REWRITE_TAC[VECTOR_ARITH `(y - (x + z)) dot (y - (x + z)) = (z - (y - x)) dot (z - (y - x))`] THEN DISCH_THEN(X_CHOOSE_THEN `u:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `min u (&1)` THEN ASM_SIMP_TAC[REAL_LT_MIN; REAL_MIN_LE; REAL_LT_01; REAL_LE_REFL]);; let ANY_CLOSEST_POINT_DOT = prove (`!s a x y:real^N. convex s /\ closed s /\ x IN s /\ y IN s /\ (!z. z IN s ==> dist(a,x) <= dist(a,z)) ==> (a - x) dot (y - x) <= &0`, REPEAT STRIP_TAC THEN REWRITE_TAC[REAL_ARITH `x <= &0 <=> ~(x > &0)`] THEN DISCH_THEN(MP_TAC o MATCH_MP CLOSER_POINT_LEMMA) THEN DISCH_THEN(X_CHOOSE_THEN `u:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[REAL_NOT_LT] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[VECTOR_ARITH `x + u % (y - x) = (&1 - u) % x + u % y`] THEN MATCH_MP_TAC IN_CONVEX_SET THEN ASM_SIMP_TAC[REAL_LT_IMP_LE]);; let ANY_CLOSEST_POINT_UNIQUE = prove (`!s a x y:real^N. convex s /\ closed s /\ x IN s /\ y IN s /\ (!z. z IN s ==> dist(a,x) <= dist(a,z)) /\ (!z. z IN s ==> dist(a,y) <= dist(a,z)) ==> x = y`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN REWRITE_TAC[GSYM NORM_LE_0; NORM_LE_SQUARE] THEN SUBGOAL_THEN `(a - x:real^N) dot (y - x) <= &0 /\ (a - y) dot (x - y) <= &0` MP_TAC THENL [ASM_MESON_TAC[ANY_CLOSEST_POINT_DOT]; ALL_TAC] THEN REWRITE_TAC[NORM_LT; DOT_LSUB; DOT_RSUB] THEN REAL_ARITH_TAC);; let CLOSEST_POINT_UNIQUE = prove (`!s a x:real^N. convex s /\ closed s /\ x IN s /\ (!z. z IN s ==> dist(a,x) <= dist(a,z)) ==> x = closest_point s a`, REPEAT STRIP_TAC THEN MATCH_MP_TAC ANY_CLOSEST_POINT_UNIQUE THEN MAP_EVERY EXISTS_TAC [`s:real^N->bool`; `a:real^N`] THEN ASM_MESON_TAC[CLOSEST_POINT_EXISTS; MEMBER_NOT_EMPTY]);; let CLOSEST_POINT_DOT = prove (`!s a x:real^N. convex s /\ closed s /\ x IN s ==> (a - closest_point s a) dot (x - closest_point s a) <= &0`, REPEAT STRIP_TAC THEN MATCH_MP_TAC ANY_CLOSEST_POINT_DOT THEN EXISTS_TAC `s:real^N->bool` THEN ASM_MESON_TAC[CLOSEST_POINT_EXISTS; MEMBER_NOT_EMPTY]);; let CLOSEST_POINT_LT = prove (`!s a x. convex s /\ closed s /\ x IN s /\ ~(x = closest_point s a) ==> dist(a,closest_point s a) < dist(a,x)`, REPEAT GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN REWRITE_TAC[GSYM REAL_NOT_LE; CONTRAPOS_THM] THEN DISCH_TAC THEN MATCH_MP_TAC CLOSEST_POINT_UNIQUE THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[CLOSEST_POINT_LE; REAL_LE_TRANS]);; let CLOSEST_POINT_LIPSCHITZ = prove (`!s x y:real^N. convex s /\ closed s /\ ~(s = {}) ==> dist(closest_point s x,closest_point s y) <= dist(x,y)`, REPEAT STRIP_TAC THEN REWRITE_TAC[dist; NORM_LE] THEN SUBGOAL_THEN `(x - closest_point s x :real^N) dot (closest_point s y - closest_point s x) <= &0 /\ (y - closest_point s y) dot (closest_point s x - closest_point s y) <= &0` MP_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC ANY_CLOSEST_POINT_DOT THEN EXISTS_TAC `s:real^N->bool` THEN ASM_MESON_TAC[CLOSEST_POINT_EXISTS]; MP_TAC(ISPEC `(x - closest_point s x :real^N) - (y - closest_point s y)` DOT_POS_LE) THEN REWRITE_TAC[NORM_LT; DOT_LSUB; DOT_RSUB; DOT_SYM] THEN REAL_ARITH_TAC]);; let CONTINUOUS_AT_CLOSEST_POINT = prove (`!s x. convex s /\ closed s /\ ~(s = {}) ==> (closest_point s) continuous (at x)`, REPEAT STRIP_TAC THEN REWRITE_TAC[continuous_at] THEN ASM_MESON_TAC[CLOSEST_POINT_LIPSCHITZ; REAL_LET_TRANS]);; let CONTINUOUS_ON_CLOSEST_POINT = prove (`!s t. convex s /\ closed s /\ ~(s = {}) ==> (closest_point s) continuous_on t`, MESON_TAC[CONTINUOUS_AT_IMP_CONTINUOUS_ON; CONTINUOUS_AT_CLOSEST_POINT]);; let CLOSEST_POINT_TRANSLATION = prove (`!s a:real^N. convex s /\ closed s /\ ~(s = {}) ==> closest_point (IMAGE (\x. a + x) s) (a + x) = a + closest_point s x`, INTRO_TAC "!s a; cvx cld nempty" THEN MATCH_MP_TAC (GSYM CLOSEST_POINT_UNIQUE) THEN ASM_SIMP_TAC[CONVEX_TRANSLATION; CLOSED_TRANSLATION; IN_IMAGE] THEN CONJ_TAC THENL [EXISTS_TAC `closest_point s (x:real^N)` THEN ASM_SIMP_TAC[CLOSEST_POINT_IN_SET]; ALL_TAC] THEN INTRO_TAC "!z; @y. zdef yhp" THEN REMOVE_THEN "zdef" SUBST1_TAC THEN SUBGOAL_THEN `dist(x:real^N,closest_point s x) <= dist(x,y)` MP_TAC THENL [ASM_SIMP_TAC[CLOSEST_POINT_LE]; ALL_TAC] THEN NORM_ARITH_TAC);; (* ------------------------------------------------------------------------- *) (* Relating closest points and orthogonality. *) (* ------------------------------------------------------------------------- *) let ANY_CLOSEST_POINT_AFFINE_ORTHOGONAL = prove (`!s a b:real^N. affine s /\ b IN s /\ (!x. x IN s ==> dist(a,b) <= dist(a,x)) ==> (!x. x IN s ==> orthogonal (x - b) (a - b))`, REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `b:real^N` THEN REWRITE_TAC[DIST_0; VECTOR_SUB_RZERO; orthogonal; dist; NORM_LE] THEN REWRITE_TAC[DOT_LSUB] THEN REWRITE_TAC[DOT_RSUB] THEN REWRITE_TAC[DOT_SYM; REAL_ARITH `a <= a - y - (y - x) <=> &2 * y <= x`] THEN REPEAT STRIP_TAC THEN ASM_CASES_TAC `x:real^N = vec 0` THEN ASM_REWRITE_TAC[DOT_RZERO] THEN FIRST_X_ASSUM(fun th -> MP_TAC(SPEC `vec 0 + --((a dot x) / (x dot x)) % (x - vec 0:real^N)` th) THEN MP_TAC(SPEC `vec 0 + (a dot x) / (x dot x) % (x - vec 0:real^N)` th)) THEN ASM_SIMP_TAC[IN_AFFINE_ADD_MUL_DIFF] THEN REWRITE_TAC[VECTOR_SUB_RZERO; VECTOR_ADD_LID; DOT_RMUL] THEN REWRITE_TAC[DOT_LMUL; IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `&2 * x * a <= b * c * z /\ &2 * --x * a <= --b * --c * z ==> &2 * abs(x * a) <= b * c * z`)) THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN ASM_SIMP_TAC[REAL_NOT_LE; REAL_DIV_RMUL; DOT_EQ_0] THEN MATCH_MP_TAC(REAL_ARITH `~(x = &0) ==> x < &2 * abs x`) THEN RULE_ASSUM_TAC(REWRITE_RULE[GSYM DOT_EQ_0]) THEN REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_FIELD);; let ORTHOGONAL_ANY_CLOSEST_POINT = prove (`!s a b:real^N. b IN s /\ (!x. x IN s ==> orthogonal (x - b) (a - b)) ==> (!x. x IN s ==> dist(a,b) <= dist(a,x))`, REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `b:real^N` THEN REWRITE_TAC[dist; NORM_LE; orthogonal; VECTOR_SUB_RZERO] THEN SIMP_TAC[DOT_LSUB; DOT_RSUB; DOT_SYM] THEN REWRITE_TAC[DOT_POS_LE; REAL_ARITH `a <= a - &0 - (&0 - x) <=> &0 <= x`]);; let CLOSEST_POINT_AFFINE_ORTHOGONAL = prove (`!s a:real^N x. affine s /\ ~(s = {}) /\ x IN s ==> orthogonal (x - closest_point s a) (a - closest_point s a)`, GEN_TAC THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN DISCH_TAC THEN DISCH_TAC THEN GEN_TAC THEN MATCH_MP_TAC ANY_CLOSEST_POINT_AFFINE_ORTHOGONAL THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CLOSEST_POINT_EXISTS THEN ASM_SIMP_TAC[CLOSED_AFFINE]);; let CLOSEST_POINT_AFFINE_ORTHOGONAL_EQ = prove (`!s a b:real^N. affine s /\ b IN s ==> (closest_point s a = b <=> !x. x IN s ==> orthogonal (x - b) (a - b))`, REPEAT STRIP_TAC THEN EQ_TAC THENL [ASM_MESON_TAC[CLOSEST_POINT_AFFINE_ORTHOGONAL; MEMBER_NOT_EMPTY]; DISCH_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC CLOSEST_POINT_UNIQUE THEN ASM_SIMP_TAC[CLOSED_AFFINE; AFFINE_IMP_CONVEX] THEN MATCH_MP_TAC ORTHOGONAL_ANY_CLOSEST_POINT THEN ASM_REWRITE_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Using "closest_point" to give orthogonal projection onto a subspace *) (* ------------------------------------------------------------------------- *) let CLOSEST_POINT_SUBSPACE_ORTHOGONAL_EQ = prove (`!s a b:real^N. subspace s ==> (closest_point s a = b <=> b IN s /\ (!x. x IN s ==> orthogonal (a - b) x))`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `(b:real^N) IN s` THENL [ASM_SIMP_TAC[CLOSEST_POINT_AFFINE_ORTHOGONAL_EQ; SUBSPACE_IMP_AFFINE]; ASM_MESON_TAC[CLOSEST_POINT_IN_SET; CLOSED_SUBSPACE; SUBSPACE_IMP_NONEMPTY]] THEN GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [ORTHOGONAL_SYM] THEN EQ_TAC THEN DISCH_TAC THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPEC `b + x:real^N`); FIRST_X_ASSUM(MP_TAC o SPEC `x - b:real^N`)] THEN ASM_SIMP_TAC[SUBSPACE_SUB; SUBSPACE_ADD; VECTOR_ADD_SUB]);; let CLOSEST_POINT_SUBSPACE_ORTHOGONAL = prove (`!s a b:real^N. subspace s /\ b IN s ==> orthogonal (a - closest_point s a) b`, MESON_TAC[CLOSEST_POINT_SUBSPACE_ORTHOGONAL_EQ]);; let LINEAR_CLOSEST_POINT = prove (`!s:real^N->bool. subspace s ==> linear(closest_point s)`, REPEAT STRIP_TAC THEN REWRITE_TAC[linear] THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[CLOSEST_POINT_SUBSPACE_ORTHOGONAL_EQ] THEN ASM_SIMP_TAC[SUBSPACE_ADD; SUBSPACE_MUL; CLOSEST_POINT_IN_SET; CLOSED_SUBSPACE; SUBSPACE_IMP_NONEMPTY] THENL [REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[VECTOR_ARITH `(x + y) - (a + b):real^N = (x - a) + (y - b)`] THEN MATCH_MP_TAC(el 8 (CONJUNCTS ORTHOGONAL_CLAUSES)); REWRITE_TAC[GSYM VECTOR_SUB_LDISTRIB; ORTHOGONAL_MUL]] THEN ASM_MESON_TAC[CLOSEST_POINT_SUBSPACE_ORTHOGONAL_EQ]);; let SELF_ADJOINT_CLOSEST_POINT = prove (`!s:real^N->bool. subspace s ==> adjoint(closest_point s) = closest_point s`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `closest_point(s:real^N->bool)` ORTHOGONAL_PROJECTION_EQ_SELF_ADJOINT_IDEMPOTENT) THEN ASM_SIMP_TAC[ORTHOGONAL_PROJECTION_ALT; LINEAR_CLOSEST_POINT; ETA_AX] THEN MATCH_MP_TAC(TAUT `p ==> (p <=> q /\ r) ==> q`) THEN REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM VECTOR_NEG_SUB] THEN REWRITE_TAC[ORTHOGONAL_LNEG] THEN MATCH_MP_TAC CLOSEST_POINT_SUBSPACE_ORTHOGONAL THEN ASM_SIMP_TAC[CLOSEST_POINT_IN_SET; CLOSED_SUBSPACE; SUBSPACE_IMP_NONEMPTY]);; let CLOSEST_POINT_IDEMPOTENT = prove (`!s:real^N->bool. closed s ==> closest_point s o closest_point s = closest_point s`, REPEAT STRIP_TAC THEN REWRITE_TAC[FUN_EQ_THM; o_THM] THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [ASM_REWRITE_TAC[closest_point; NOT_IN_EMPTY]; ASM_SIMP_TAC[CLOSEST_POINT_SELF; CLOSEST_POINT_IN_SET]]);; let MATRIX_INV_PROJECTION_IMAGE,MATRIX_INV_PROJECTION_IMAGE_ALT = (CONJ_PAIR o prove) (`(!A:real^M^N. A ** matrix_inv A = matrix(closest_point (IMAGE (\x. A ** x) UNIV))) /\ (!A:real^M^N x. (A ** matrix_inv A) ** x = closest_point (IMAGE (\x. A ** x) UNIV) x)`, REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN SIMP_TAC[MATRIX_EQ; MATRIX_WORKS; GSYM MATRIX_VECTOR_MUL_ASSOC; LINEAR_CLOSEST_POINT; MATRIX_VECTOR_MUL_LINEAR; SUBSPACE_LINEAR_IMAGE; SUBSPACE_UNIV; CLOSEST_POINT_SUBSPACE_ORTHOGONAL_EQ] THEN REPEAT GEN_TAC THEN (CONJ_TAC THENL [SET_TAC[]; REWRITE_TAC[FORALL_IN_IMAGE]]) THEN REWRITE_TAC[MOORE_PENROSE_PSEUDOINVERSE]);; (* ------------------------------------------------------------------------- *) (* Stronger separating hyperplane results for affine sets / affine hulls. *) (* ------------------------------------------------------------------------- *) let SEPARATING_HYPERPLANE_AFFINE_AFFINE = prove (`!s t:real^N->bool. affine s /\ affine t /\ ~(s = {}) /\ ~(t = {}) /\ DISJOINT s t ==> ?a b c. ~(a = vec 0) /\ b < c /\ (!x. x IN s ==> a dot x = b) /\ (!x. x IN t ==> a dot x = c)`, SUBGOAL_THEN `!s t:real^N->bool. affine s /\ affine t /\ ~(s = {}) /\ ~(t = {}) /\ DISJOINT s t ==> ?a b c. ~(a = vec 0) /\ ~(b = c) /\ (!x. x IN s ==> a dot x = b) /\ (!x. x IN t ==> a dot x = c)` MP_TAC THENL [ALL_TAC; REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real`;` c:real`] THEN STRIP_TAC THEN FIRST_X_ASSUM(DISJ_CASES_TAC o MATCH_MP (REAL_ARITH `~(x = y) ==> x < y \/ y < x`)) THENL [MAP_EVERY EXISTS_TAC [`a:real^N`; `b:real`; `c:real`]; MAP_EVERY EXISTS_TAC [`--a:real^N`; `--b:real`; `--c:real`]] THEN ASM_REWRITE_TAC[REAL_LT_NEG2; DOT_LNEG; REAL_EQ_NEG2] THEN ASM_REWRITE_TAC[VECTOR_NEG_EQ_0]] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `?u v. u IN s /\ v IN t /\ !x y:real^N. x IN s /\ y IN t ==> dist(u,v) <= dist(x,y)` STRIP_ASSUME_TAC THENL [MP_TAC(ISPECL [`{x - y:real^N | x IN s /\ y IN t}`; `vec 0:real^N`] CLOSEST_POINT_EXISTS) THEN ASM_SIMP_TAC[CLOSED_AFFINE; AFFINE_DIFFERENCES; DIST_0] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[FORALL_IN_GSPEC; dist] THEN REWRITE_TAC[IN_ELIM_THM] THEN MESON_TAC[]; ALL_TAC] THEN ABBREV_TAC `d:real^N = u - v` THEN EXISTS_TAC `d:real^N` THEN REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [EXPAND_TAC "d" THEN REWRITE_TAC[VECTOR_SUB_EQ] THEN ASM SET_TAC[]; DISCH_TAC] THEN MAP_EVERY EXISTS_TAC [`(d:real^N) dot u`; `(d:real^N) dot v`] THEN ONCE_REWRITE_TAC[GSYM REAL_SUB_0] THEN REWRITE_TAC[GSYM DOT_RSUB] THEN ASM_REWRITE_TAC[DOT_EQ_0] THEN CONJ_TAC THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THENL [MP_TAC(ISPECL [`s:real^N->bool`; `v:real^N`; `u:real^N`] ANY_CLOSEST_POINT_AFFINE_ORTHOGONAL); MP_TAC(ISPECL [`t:real^N->bool`; `u:real^N`; `v:real^N`] ANY_CLOSEST_POINT_AFFINE_ORTHOGONAL)] THEN (ANTS_TAC THENL [ASM_MESON_TAC[DIST_SYM]; ALL_TAC]) THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[ORTHOGONAL_SYM] THEN ASM_REWRITE_TAC[GSYM orthogonal] THEN GEN_REWRITE_TAC LAND_CONV [GSYM ORTHOGONAL_LNEG] THEN ASM_REWRITE_TAC[VECTOR_NEG_SUB]);; let SEPARATING_HYPERPLANE_AFFINE_HULLS = prove (`!s t:real^N->bool. ~(s = {}) /\ ~(t = {}) /\ DISJOINT (affine hull s) (affine hull t) ==> ?a b c. ~(a = vec 0) /\ b < c /\ (!x. x IN s ==> a dot x = b) /\ (!x. x IN t ==> a dot x = c)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`affine hull s:real^N->bool`; `affine hull t:real^N->bool`] SEPARATING_HYPERPLANE_AFFINE_AFFINE) THEN ASM_REWRITE_TAC[AFFINE_HULL_EQ_EMPTY; AFFINE_AFFINE_HULL] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN MESON_TAC[HULL_INC]);; (* ------------------------------------------------------------------------- *) (* Various point-to-set separating/supporting hyperplane theorems. *) (* ------------------------------------------------------------------------- *) let SUPPORTING_HYPERPLANE_COMPACT_POINT_SUP = prove (`!a c s:real^N->bool. compact s /\ ~(s = {}) ==> ?b y. y IN s /\ a dot (y - c) = b /\ (!x. x IN s ==> a dot (x - c) <= b)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`\x:real^N. a dot (x - c)`; `s:real^N->bool`] CONTINUOUS_ATTAINS_SUP) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ALL_TAC; MESON_TAC[]] THEN SUBGOAL_THEN `(\x:real^N. a dot (x - c)) = (\x. a dot x) o (\x. x - c)` SUBST1_TAC THENL [REWRITE_TAC[o_DEF]; ALL_TAC] THEN REWRITE_TAC[o_ASSOC] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_LIFT_DOT; CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID]);; let SUPPORTING_HYPERPLANE_COMPACT_POINT_INF = prove (`!a c s:real^N->bool. compact s /\ ~(s = {}) ==> ?b y. y IN s /\ a dot (y - c) = b /\ (!x. x IN s ==> a dot (x - c) >= b)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`--a:real^N`; `c:real^N`; `s:real^N->bool`] SUPPORTING_HYPERPLANE_COMPACT_POINT_SUP) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `b:real` (fun th -> EXISTS_TAC `--b:real` THEN MP_TAC th)) THEN REWRITE_TAC[DOT_LNEG; REAL_ARITH `x >= -- b <=> --x <= b`] THEN REWRITE_TAC[REAL_NEG_EQ]);; let SUPPORTING_HYPERPLANE_CLOSED_POINT = prove (`!s z:real^N. convex s /\ closed s /\ ~(s = {}) /\ ~(z IN s) ==> ?a b y. a dot z < b /\ y IN s /\ (a dot y = b) /\ (!x. x IN s ==> a dot x >= b)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `z:real^N`] DISTANCE_ATTAINS_INF) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `y - z:real^N` THEN EXISTS_TAC `(y - z:real^N) dot y` THEN EXISTS_TAC `y:real^N` THEN ONCE_REWRITE_TAC[GSYM REAL_SUB_LT] THEN ASM_REWRITE_TAC[GSYM DOT_RSUB; DOT_POS_LT; VECTOR_SUB_EQ] THEN CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN SUBGOAL_THEN `!u. &0 <= u /\ u <= &1 ==> dist(z:real^N,y) <= dist(z,(&1 - u) % y + u % x)` MP_TAC THENL [ASM_MESON_TAC[CONVEX_ALT]; ALL_TAC] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN REWRITE_TAC[real_ge; REAL_NOT_LE; NOT_FORALL_THM; NOT_IMP] THEN GEN_REWRITE_TAC LAND_CONV [REAL_ARITH `x < y <=> y - x > &0`] THEN REWRITE_TAC[VECTOR_ARITH `(a - b) dot x - (a - b) dot y = (b - a) dot (y - x)`] THEN DISCH_THEN(MP_TAC o MATCH_MP CLOSER_POINT_LEMMA) THEN REWRITE_TAC[VECTOR_ARITH `y + u % (x - y) = (&1 - u) % y + u % x`] THEN MESON_TAC[REAL_LT_IMP_LE]);; let SEPARATING_HYPERPLANE_CLOSED_POINT_INSET = prove (`!s z:real^N. convex s /\ closed s /\ ~(s = {}) /\ ~(z IN s) ==> ?a b. a IN s /\ (a - z) dot z < b /\ (!x. x IN s ==> (a - z) dot x > b)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `z:real^N`] DISTANCE_ATTAINS_INF) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `y:real^N` THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `(y - z:real^N) dot z + norm(y - z) pow 2 / &2` THEN SUBGOAL_THEN `&0 < norm(y - z:real^N)` ASSUME_TAC THENL [ASM_MESON_TAC[NORM_POS_LT; VECTOR_SUB_EQ]; ALL_TAC] THEN ASM_SIMP_TAC[REAL_LT_ADDR; REAL_LT_DIV; REAL_POW_LT; REAL_OF_NUM_LT; ARITH] THEN REWRITE_TAC[NORM_POW_2; REAL_ARITH `a > b + c <=> c < a - b`] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN SIMP_TAC[REAL_LT_LDIV_EQ; REAL_OF_NUM_LT; ARITH] THEN ONCE_REWRITE_TAC[GSYM REAL_SUB_LT] THEN REWRITE_TAC[VECTOR_ARITH `((y - z) dot x - (y - z) dot z) * &2 - (y - z) dot (y - z) = &2 * ((y - z) dot (x - y)) + (y - z) dot (y - z)`] THEN MATCH_MP_TAC(REAL_ARITH `~(--x > &0) /\ &0 < y ==> &0 < &2 * x + y`) THEN ASM_SIMP_TAC[GSYM NORM_POW_2; REAL_POW_LT] THEN REWRITE_TAC[GSYM DOT_LNEG; VECTOR_NEG_SUB] THEN DISCH_THEN(MP_TAC o MATCH_MP CLOSER_POINT_LEMMA) THEN REWRITE_TAC[NOT_EXISTS_THM] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN REWRITE_TAC[REAL_NOT_LT] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[VECTOR_ARITH `y + u % (x - y) = (&1 - u) % y + u % x`] THEN ASM_MESON_TAC[CONVEX_ALT; REAL_LT_IMP_LE]);; let SEPARATING_HYPERPLANE_CLOSED_0_INSET = prove (`!s:real^N->bool. convex s /\ closed s /\ ~(s = {}) /\ ~(vec 0 IN s) ==> ?a b. a IN s /\ ~(a = vec 0) /\ &0 < b /\ (!x. x IN s ==> a dot x > b)`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP SEPARATING_HYPERPLANE_CLOSED_POINT_INSET) THEN REWRITE_TAC[DOT_RZERO; real_gt] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN SIMP_TAC[VECTOR_SUB_RZERO] THEN ASM_MESON_TAC[]);; let SEPARATING_HYPERPLANE_CLOSED_POINT = prove (`!s z:real^N. convex s /\ closed s /\ ~(z IN s) ==> ?a b. a dot z < b /\ (!x. x IN s ==> a dot x > b)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [MAP_EVERY EXISTS_TAC [`--z:real^N`; `&1`] THEN SIMP_TAC[DOT_LNEG; REAL_ARITH `&0 <= x ==> --x < &1`; DOT_POS_LE] THEN ASM_MESON_TAC[NOT_IN_EMPTY]; ALL_TAC] THEN ASM_MESON_TAC[SEPARATING_HYPERPLANE_CLOSED_POINT_INSET]);; let SEPARATING_HYPERPLANE_CLOSED_0 = prove (`!s:real^N->bool. convex s /\ closed s /\ ~(vec 0 IN s) ==> ?a b. ~(a = vec 0) /\ &0 < b /\ (!x. x IN s ==> a dot x > b)`, REPEAT GEN_TAC THEN DISCH_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [EXISTS_TAC `basis 1:real^N` THEN EXISTS_TAC `&1` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY; REAL_LT_01; GSYM NORM_POS_LT] THEN ASM_SIMP_TAC[NORM_BASIS; DIMINDEX_GE_1; LE_REFL; REAL_LT_01]; FIRST_X_ASSUM(MP_TAC o MATCH_MP SEPARATING_HYPERPLANE_CLOSED_POINT) THEN REWRITE_TAC[DOT_RZERO; real_gt] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN ASM_MESON_TAC[MEMBER_NOT_EMPTY; DOT_LZERO; REAL_LT_ANTISYM]]);; (* ------------------------------------------------------------------------- *) (* Now set-to-set for closed/compact sets. *) (* ------------------------------------------------------------------------- *) let SEPARATING_HYPERPLANE_CLOSED_COMPACT = prove (`!s t. convex s /\ closed s /\ convex t /\ compact t /\ ~(t = {}) /\ DISJOINT s t ==> ?a:real^N b. (!x. x IN s ==> a dot x < b) /\ (!x. x IN t ==> a dot x > b)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN FIRST_ASSUM(MP_TAC o MATCH_MP COMPACT_IMP_BOUNDED) THEN REWRITE_TAC[BOUNDED_POS] THEN DISCH_THEN(X_CHOOSE_THEN `b:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `?z:real^N. norm(z) = b + &1` CHOOSE_TAC THENL [ASM_SIMP_TAC[VECTOR_CHOOSE_SIZE; REAL_ARITH `&0 < b ==> &0 <= b + &1`]; ALL_TAC] THEN MP_TAC(SPECL [`t:real^N->bool`; `z:real^N`] SEPARATING_HYPERPLANE_CLOSED_POINT) THEN ANTS_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED] THEN ASM_MESON_TAC[REAL_ARITH `~(b + &1 <= b)`]; ALL_TAC] THEN MP_TAC(ISPECL [`{x - y:real^N | x IN s /\ y IN t}`; `vec 0 :real^N`] SEPARATING_HYPERPLANE_CLOSED_POINT) THEN ASM_SIMP_TAC[CLOSED_COMPACT_DIFFERENCES; CONVEX_DIFFERENCES] THEN ANTS_TAC THENL [REWRITE_TAC[IN_ELIM_THM] THEN ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN REWRITE_TAC[VECTOR_SUB_EQ] THEN ASM_MESON_TAC[DISJOINT; NOT_IN_EMPTY; IN_INTER; EXTENSION]; ALL_TAC] THEN SIMP_TAC[DOT_RZERO; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real`] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN GEN_REWRITE_TAC LAND_CONV [SWAP_FORALL_THM] THEN GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [SWAP_FORALL_THM] THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[LEFT_FORALL_IMP_THM; EXISTS_REFL; DOT_RSUB] THEN REWRITE_TAC[real_gt; REAL_LT_SUB_LADD] THEN DISCH_TAC THEN EXISTS_TAC `--a:real^N` THEN MP_TAC(SPEC `IMAGE (\x:real^N. a dot x) t` SUP) THEN ABBREV_TAC `k = sup (IMAGE (\x:real^N. a dot x) t)` THEN ASM_REWRITE_TAC[FORALL_IN_IMAGE; IMAGE_EQ_EMPTY] THEN ANTS_TAC THENL [ASM_MESON_TAC[REAL_ARITH `b + x < y ==> x <= y - b`; MEMBER_NOT_EMPTY]; ALL_TAC] THEN STRIP_TAC THEN EXISTS_TAC `--(k + b / &2)` THEN REWRITE_TAC[DOT_LNEG; REAL_LT_NEG2] THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH; REAL_ARITH `&0 < b /\ x <= k ==> x < k + b`] THEN FIRST_X_ASSUM(MP_TAC o SPEC `k - b / &2`) THEN ASM_SIMP_TAC[REAL_ARITH `k <= k - b2 <=> ~(&0 < b2)`; REAL_LT_DIV; REAL_OF_NUM_LT; ARITH; NOT_FORALL_THM; LEFT_IMP_EXISTS_THM; NOT_IMP] THEN X_GEN_TAC `y:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN MATCH_MP_TAC(REAL_ARITH `!b. (b2 + b2 = b) /\ b + ay < ax ==> ~(ay <= k - b2) ==> k + b2 < ax`) THEN ASM_MESON_TAC[REAL_HALF]);; let SEPARATING_HYPERPLANE_COMPACT_CLOSED = prove (`!s t. convex s /\ compact s /\ ~(s = {}) /\ convex t /\ closed t /\ DISJOINT s t ==> ?a:real^N b. (!x. x IN s ==> a dot x < b) /\ (!x. x IN t ==> a dot x > b)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`t:real^N->bool`; `s:real^N->bool`] SEPARATING_HYPERPLANE_CLOSED_COMPACT) THEN ANTS_TAC THENL [ASM_MESON_TAC[DISJOINT_SYM]; ALL_TAC] THEN REWRITE_TAC[real_gt] THEN DISCH_THEN(X_CHOOSE_THEN `a:real^N` (X_CHOOSE_THEN `b:real` STRIP_ASSUME_TAC)) THEN MAP_EVERY EXISTS_TAC [`--a:real^N`; `--b:real`] THEN ASM_REWRITE_TAC[REAL_LT_NEG2; DOT_LNEG]);; let SEPARATING_HYPERPLANE_COMPACT_CLOSED_NONZERO = prove (`!s t:real^N->bool. convex s /\ compact s /\ ~(s = {}) /\ convex t /\ closed t /\ DISJOINT s t ==> ?a b. ~(a = vec 0) /\ (!x. x IN s ==> a dot x < b) /\ (!x. x IN t ==> a dot x > b)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THENL [ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN STRIP_TAC THEN EXISTS_TAC `basis 1:real^N` THEN SUBGOAL_THEN `bounded(IMAGE (\x:real^N. lift(basis 1 dot x)) s)` MP_TAC THENL [MATCH_MP_TAC COMPACT_IMP_BOUNDED THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_SIMP_TAC[REWRITE_RULE[o_DEF] CONTINUOUS_ON_LIFT_DOT]; REWRITE_TAC[BOUNDED_POS_LT; FORALL_IN_IMAGE; NORM_LIFT] THEN SIMP_TAC[BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL] THEN MESON_TAC[REAL_ARITH `abs x < b ==> x < b`]]; STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`] SEPARATING_HYPERPLANE_COMPACT_CLOSED) THEN ASM_REWRITE_TAC[] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN ASM_CASES_TAC `a:real^N = vec 0` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[DOT_LZERO; real_gt] THEN ASM_MESON_TAC[REAL_LT_ANTISYM; MEMBER_NOT_EMPTY]]);; let SEPARATING_HYPERPLANE_COMPACT_COMPACT = prove (`!s t:real^N->bool. convex s /\ compact s /\ convex t /\ compact t /\ DISJOINT s t ==> ?a b. ~(a = vec 0) /\ (!x. x IN s ==> a dot x < b) /\ (!x. x IN t ==> a dot x > b)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN STRIP_TAC THEN EXISTS_TAC `--basis 1:real^N` THEN SUBGOAL_THEN `bounded(IMAGE (\x:real^N. lift(basis 1 dot x)) t)` MP_TAC THENL [MATCH_MP_TAC COMPACT_IMP_BOUNDED THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_SIMP_TAC[REWRITE_RULE[o_DEF] CONTINUOUS_ON_LIFT_DOT]; REWRITE_TAC[BOUNDED_POS_LT; FORALL_IN_IMAGE; NORM_LIFT] THEN SIMP_TAC[VECTOR_NEG_EQ_0; BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL] THEN DISCH_THEN(X_CHOOSE_THEN `b:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `--b:real` THEN REWRITE_TAC[DOT_LNEG] THEN REWRITE_TAC[REAL_ARITH `--x > --y <=> x < y`] THEN ASM_MESON_TAC[REAL_ARITH `abs x < b ==> x < b`]]; STRIP_TAC THEN MATCH_MP_TAC SEPARATING_HYPERPLANE_COMPACT_CLOSED_NONZERO THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED]]);; (* ------------------------------------------------------------------------- *) (* General case without assuming closure and getting non-strict separation. *) (* ------------------------------------------------------------------------- *) let SEPARATING_HYPERPLANE_SET_0_INSPAN = prove (`!s:real^N->bool. convex s /\ ~(s = {}) /\ ~(vec 0 IN s) ==> ?a b. a IN span s /\ ~(a = vec 0) /\ !x. x IN s ==> &0 <= a dot x`, REPEAT STRIP_TAC THEN ABBREV_TAC `k = \c:real^N. {x | &0 <= c dot x}` THEN SUBGOAL_THEN `~((span s INTER frontier(cball(vec 0:real^N,&1))) INTER (INTERS (IMAGE k (s:real^N->bool))) = {})` MP_TAC THENL [ALL_TAC; SIMP_TAC[EXTENSION; NOT_IN_EMPTY; IN_INTER; IN_INTERS; NOT_FORALL_THM; FORALL_IN_IMAGE; FRONTIER_CBALL; REAL_LT_01] THEN EXPAND_TAC "k" THEN REWRITE_TAC[IN_SPHERE_0; IN_ELIM_THM; NORM_NEG] THEN MESON_TAC[NORM_EQ_0; REAL_ARITH `~(&1 = &0)`; DOT_SYM]] THEN MATCH_MP_TAC COMPACT_IMP_FIP THEN SIMP_TAC[COMPACT_CBALL; COMPACT_FRONTIER; FORALL_IN_IMAGE; CLOSED_INTER_COMPACT; CLOSED_SPAN] THEN CONJ_TAC THENL [EXPAND_TAC "k" THEN REWRITE_TAC[GSYM real_ge; CLOSED_HALFSPACE_GE]; ALL_TAC] THEN REWRITE_TAC[FINITE_SUBSET_IMAGE] THEN GEN_TAC THEN DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` MP_TAC) THEN ASM_CASES_TAC `c:real^N->bool = {}` THENL [ASM_SIMP_TAC[INTERS_0; INTER_UNIV; IMAGE_CLAUSES] THEN DISCH_THEN(K ALL_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 SUBGOAL_THEN `~(a:real^N = vec 0)` ASSUME_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `inv(norm a) % a:real^N` THEN ASM_SIMP_TAC[IN_INTER; FRONTIER_CBALL; SPAN_CLAUSES; IN_SPHERE_0] THEN REWRITE_TAC[DIST_0; NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN ASM_SIMP_TAC[REAL_MUL_LINV; NORM_EQ_0]; ALL_TAC] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPEC `convex hull (c:real^N->bool)` SEPARATING_HYPERPLANE_CLOSED_0_INSET) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[CONVEX_HULL_EQ_EMPTY] THEN ASM_MESON_TAC[CONVEX_CONVEX_HULL; SUBSET; SUBSET_HULL; HULL_SUBSET; FINITE_IMP_COMPACT_CONVEX_HULL; COMPACT_IMP_CLOSED]; ALL_TAC] THEN REWRITE_TAC[DOT_RZERO; real_gt] THEN DISCH_THEN(X_CHOOSE_THEN `a:real^N` (X_CHOOSE_THEN `b:real` STRIP_ASSUME_TAC)) THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER; IN_INTERS; FORALL_IN_IMAGE] THEN EXPAND_TAC "k" THEN SIMP_TAC[IN_ELIM_THM; FRONTIER_CBALL; REAL_LT_01] THEN REWRITE_TAC[dist; VECTOR_SUB_LZERO; NORM_NEG] THEN EXISTS_TAC `inv(norm(a)) % a:real^N` THEN REWRITE_TAC[DOT_RMUL] THEN SUBGOAL_THEN `(a:real^N) IN s` ASSUME_TAC THENL [ASM_MESON_TAC[SUBSET; HULL_MINIMAL]; ASM_SIMP_TAC[SPAN_CLAUSES]] THEN REWRITE_TAC[IN_SPHERE_0; VECTOR_SUB_LZERO; NORM_NEG; NORM_MUL] THEN REWRITE_TAC[REAL_ABS_INV; REAL_ABS_NORM] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[GSYM real_div] THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_EQ_LDIV_EQ; NORM_POS_LT] THEN REWRITE_TAC[REAL_MUL_LZERO; REAL_MUL_LID] THEN ASM_MESON_TAC[REAL_LT_IMP_LE; REAL_LE_TRANS; HULL_SUBSET; SUBSET; DOT_SYM]);; let SEPARATING_HYPERPLANE_SET_POINT_INAFF = prove (`!s z:real^N. convex s /\ ~(s = {}) /\ ~(z IN s) ==> ?a b. (z + a) IN affine hull (z INSERT s) /\ ~(a = vec 0) /\ a dot z <= b /\ (!x. x IN s ==> a dot x >= b)`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `IMAGE (\x:real^N. --z + x) s` SEPARATING_HYPERPLANE_SET_0_INSPAN) THEN ASM_SIMP_TAC[FORALL_IN_IMAGE; CONVEX_TRANSLATION; IMAGE_EQ_EMPTY] THEN REWRITE_TAC[IN_IMAGE; VECTOR_ARITH `vec 0:real^N = --z + x <=> x = z`] THEN ASM_SIMP_TAC[UNWIND_THM2; AFFINE_HULL_INSERT_SPAN; IN_ELIM_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN REWRITE_TAC[GSYM SIMPLE_IMAGE; VECTOR_ARITH `--x + y:real^N = y - x`] THEN STRIP_TAC THEN ASM_REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN EXISTS_TAC `(a:real^N) dot z` THEN REWRITE_TAC[REAL_LE_REFL] THEN ASM_REWRITE_TAC[REAL_ARITH `x >= y <=> &0 <= x - y`; GSYM DOT_RSUB]);; let SEPARATING_HYPERPLANE_SET_0 = prove (`!s:real^N->bool. convex s /\ ~(vec 0 IN s) ==> ?a. ~(a = vec 0) /\ !x. x IN s ==> &0 <= a dot x`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN MESON_TAC[BASIS_NONZERO; LE_REFL; DIMINDEX_GE_1]; ASM_MESON_TAC[SEPARATING_HYPERPLANE_SET_0_INSPAN]]);; let SEPARATING_HYPERPLANE_SETS = prove (`!s t. convex s /\ convex t /\ ~(s = {}) /\ ~(t = {}) /\ DISJOINT s t ==> ?a:real^N b. ~(a = vec 0) /\ (!x. x IN s ==> a dot x <= b) /\ (!x. x IN t ==> a dot x >= b)`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `{y - x:real^N | y IN t /\ x IN s}` SEPARATING_HYPERPLANE_SET_0) THEN ASM_SIMP_TAC[CONVEX_DIFFERENCES] THEN ANTS_TAC THENL [REWRITE_TAC[IN_ELIM_THM] THEN ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN REWRITE_TAC[VECTOR_SUB_EQ] THEN ASM_MESON_TAC[DISJOINT; NOT_IN_EMPTY; IN_INTER; EXTENSION]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN SIMP_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN GEN_REWRITE_TAC LAND_CONV [SWAP_FORALL_THM] THEN GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [SWAP_FORALL_THM] THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[LEFT_FORALL_IMP_THM; EXISTS_REFL; DOT_RSUB; REAL_SUB_LE] THEN DISCH_TAC THEN MP_TAC(SPEC `IMAGE (\x:real^N. a dot x) s` SUP) THEN ABBREV_TAC `k = sup (IMAGE (\x:real^N. a dot x) s)` THEN ASM_REWRITE_TAC[FORALL_IN_IMAGE; IMAGE_EQ_EMPTY; real_ge] THEN ANTS_TAC THENL [ASM_MESON_TAC[MEMBER_NOT_EMPTY]; ASM_MESON_TAC[]]);; (* ------------------------------------------------------------------------- *) (* More convexity generalities. *) (* ------------------------------------------------------------------------- *) let UNBOUNDED_COMPLEMENT_COMPONENT_CONVEX = prove (`!s c:real^N->bool. convex s /\ c IN components((:real^N) DIFF s) ==> ~bounded c`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_SIMP_TAC[DIFF_EMPTY; COMPONENTS_UNIV; IN_SING; NOT_BOUNDED_UNIV] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_NONEMPTY) THEN GEN_REWRITE_TAC LAND_CONV [GSYM MEMBER_NOT_EMPTY] THEN DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN ASM_CASES_TAC `(a:real^N) IN s` THENL [ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`s:real^N->bool`; `{a:real^N}`] SEPARATING_HYPERPLANE_SETS) THEN ASM_REWRITE_TAC[CONVEX_SING; NOT_IMP] THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`v:real^N`; `d:real`] THEN STRIP_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] BOUNDED_SUBSET)) THEN DISCH_THEN(MP_TAC o SPEC `a INSERT {x:real^N | ~(v dot x <= d)}`) THEN REWRITE_TAC[BOUNDED_INSERT] THEN ASM_REWRITE_TAC[BOUNDED_HALFSPACE_GT; GSYM real_gt; REAL_NOT_LE] THEN MATCH_MP_TAC COMPONENTS_MAXIMAL THEN EXISTS_TAC `(:real^N) DIFF s` THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [SIMP_TAC[CONNECTED_INSERT; CONVEX_HALFSPACE_GT; CONVEX_CONNECTED] THEN DISJ2_TAC THEN ASM_SIMP_TAC[CLOSURE_HALFSPACE_GT] THEN ASM SET_TAC[]; ASM_REWRITE_TAC[INSERT_SUBSET; IN_DIFF; IN_UNIV] THEN ASM_REWRITE_TAC[real_gt; IN_ELIM_THM; REAL_NOT_LT; SET_RULE `s SUBSET UNIV DIFF t <=> !x. x IN t ==> ~(x IN s)`]; ASM SET_TAC[]]);; let UNBOUNDED_COMPLEMENT_CONVEX = prove (`!c. convex c /\ ~(c = (:real^N)) ==> ~bounded((:real^N) DIFF c)`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `(:real^N) DIFF c` COMPONENTS_EQ_EMPTY) THEN ASM_REWRITE_TAC[SET_RULE `UNIV DIFF s = {} <=> s = UNIV`] THEN REWRITE_TAC[EXTENSION; NOT_IN_EMPTY] THEN X_GEN_TAC `k:real^N->bool` THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] UNBOUNDED_COMPLEMENT_COMPONENT_CONVEX)) THEN ASM_MESON_TAC[BOUNDED_SUBSET; IN_COMPONENTS_SUBSET]);; let COMPONENTS_CONVEX_COMPLEMENT_CONTAINS_HALFSPACE = prove (`!s c. convex s /\ c IN components((:real^N) DIFF s) ==> ?a b. ~(a = vec 0) /\ {x | a dot x <= b} SUBSET c`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [ASM_SIMP_TAC[DIFF_EMPTY; COMPONENTS_UNIV; IN_SING; SUBSET_UNIV] THEN MESON_TAC[BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL]; STRIP_TAC] THEN FIRST_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_NONEMPTY) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; GSYM MEMBER_NOT_EMPTY] THEN X_GEN_TAC `z:real^N` THEN DISCH_TAC THEN FIRST_ASSUM (MP_TAC o REWRITE_RULE[SUBSET] o MATCH_MP IN_COMPONENTS_SUBSET) THEN DISCH_THEN(MP_TAC o SPEC `z:real^N`) THEN ASM_REWRITE_TAC[IN_DIFF; IN_UNIV] THEN DISCH_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `z:real^N`] SEPARATING_HYPERPLANE_SET_POINT_INAFF) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN DISCH_THEN(X_CHOOSE_THEN `b:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `b - &1` THEN ASM_REWRITE_TAC[] THEN TRANS_TAC SUBSET_TRANS `z INSERT {x:real^N | a dot x < b}` THEN CONJ_TAC THENL [SIMP_TAC[SUBSET; IN_ELIM_THM; IN_INSERT] THEN REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC COMPONENTS_MAXIMAL THEN EXISTS_TAC `(:real^N) DIFF s` THEN ASM_SIMP_TAC[CONVEX_HALFSPACE_LT; CONVEX_CONNECTED; CONNECTED_INSERT; CLOSURE_HALFSPACE_LT; IN_ELIM_THM] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN ASM_REWRITE_TAC[INSERT_SUBSET; IN_UNIV; IN_DIFF] THEN ONCE_REWRITE_TAC[SET_RULE `{x | P x} SUBSET UNIV DIFF s <=> s SUBSET {x | ~P x}`] THEN ASM_REWRITE_TAC[SUBSET; IN_ELIM_THM; REAL_ARITH `~(a < b) <=> a >= b`]);; let CARD_COMPONENTS_COMPLEMENT_CONVEX,FINITE_COMPONENTS_COMPLEMENT_CONVEX = (CONJ_PAIR o prove) (`(!s. convex s ==> CARD(components((:real^N) DIFF s)) <= 2) /\ (!s. convex s ==> FINITE(components((:real^N) DIFF s)))`, REWRITE_TAC[AND_FORALL_THM] THEN X_GEN_TAC `s:real^N->bool` THEN ASM_CASES_TAC `convex(s:real^N->bool)` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[ARITH_RULE `x <= 2 <=> ~(3 <= x)`] THEN REWRITE_TAC[TAUT `~p /\ q <=> ~(q ==> p)`] THEN DISCH_THEN(MP_TAC o MATCH_MP CHOOSE_SUBSET_STRONG) THEN REWRITE_TAC[HAS_SIZE_CONV `s HAS_SIZE 3`] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM; NOT_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`t:(real^N->bool)->bool`; `c1:real^N->bool`; `c2:real^N->bool`; `c3:real^N->bool`] THEN ASM_CASES_TAC `t = {c1:real^N->bool,c2,c3}` THEN ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET] THEN STRIP_TAC THEN MP_TAC(ISPEC `(:real^N) DIFF s` COMPONENTS_EQ) THEN DISCH_THEN(fun th -> MP_TAC(SPECL [`c1:real^N->bool`; `c2:real^N->bool`] th) THEN MP_TAC(SPECL [`c2:real^N->bool`; `c3:real^N->bool`] th) THEN MP_TAC(SPECL [`c3:real^N->bool`; `c1:real^N->bool`] th)) THEN ASM_REWRITE_TAC[GSYM DISJOINT] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPEC `s:real^N->bool` COMPONENTS_CONVEX_COMPLEMENT_CONTAINS_HALFSPACE) THEN DISCH_THEN(fun th -> MP_TAC(SPEC `c3:real^N->bool` th) THEN MP_TAC(SPEC `c2:real^N->bool` th) THEN MP_TAC(SPEC `c1:real^N->bool` th)) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; NOT_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a1:real^N`; `b1:real`] THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`a2:real^N`; `b2:real`] THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`a3:real^N`; `b3:real`] THEN STRIP_TAC THEN SUBGOAL_THEN `?c. &0 < c /\ (a3:real^N = c % a1 \/ a2 = c % a1 \/ a3:real^N = c % a2)` MP_TAC THENL [MP_TAC(ISPECL [`a1:real^N`; `a3:real^N`; `b1:real`; `b3:real`] (el 6 (CONJUNCTS DISJOINT_HALFSPACES_IMP_COLLINEAR))) THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`a1:real^N`; `a2:real^N`; `b1:real`; `b2:real`] (el 6 (CONJUNCTS DISJOINT_HALFSPACES_IMP_COLLINEAR))) THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_REWRITE_TAC[COLLINEAR_LEMMA_ALT; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `c2:real` THEN ASM_CASES_TAC `c2 = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN DISCH_THEN(ASSUME_TAC o SYM) THEN X_GEN_TAC `c3:real` THEN ASM_CASES_TAC `c3 = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN DISCH_THEN(ASSUME_TAC o SYM) THEN ASM_CASES_TAC `&0 < c2` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `&0 < c3` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN EXISTS_TAC `c3 / c2:real` THEN CONJ_TAC THENL [REWRITE_TAC[REAL_ARITH `&0 < x / y <=> &0 < --x * --inv y`] THEN MATCH_MP_TAC REAL_LT_MUL THEN REWRITE_TAC[GSYM REAL_INV_NEG; REAL_LT_INV_EQ] THEN ASM_REAL_ARITH_TAC; REPEAT DISJ2_TAC THEN MAP_EVERY EXPAND_TAC ["a2"; "a3"] THEN REWRITE_TAC[VECTOR_MUL_ASSOC; VECTOR_MUL_RCANCEL] THEN DISJ1_TAC THEN UNDISCH_TAC `~(c2 = &0)` THEN CONV_TAC REAL_FIELD]; ALL_TAC] THEN STRIP_TAC THENL [SUBGOAL_THEN `DISJOINT {x:real^N | a1 dot x <= b1} {x | a3 dot x <= b3}` MP_TAC THENL [ASM SET_TAC[]; ASM_REWRITE_TAC[]]; SUBGOAL_THEN `DISJOINT {x:real^N | a1 dot x <= b1} {x | a2 dot x <= b2}` MP_TAC THENL [ASM SET_TAC[]; ASM_REWRITE_TAC[]]; SUBGOAL_THEN `DISJOINT {x:real^N | a2 dot x <= b2} {x | a3 dot x <= b3}` MP_TAC THENL [ASM SET_TAC[]; ASM_REWRITE_TAC[]]] THEN REWRITE_TAC[DOT_LMUL] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ] THEN MATCH_MP_TAC(SET_RULE `{x | f x <= min a b} SUBSET {x | f x <= a} /\ {x | f x <= min a b} SUBSET {x | f x <= b} /\ ~({x | f x <= min a b} = {}) ==> ~DISJOINT {x | f x <= a} {x | f x <= b}`) THEN ASM_REWRITE_TAC[HALFSPACE_EQ_EMPTY_LE; SUBSET; IN_ELIM_THM] THEN REAL_ARITH_TAC);; let CONVEX_CLOSURE = prove (`!s:real^N->bool. convex s ==> convex(closure s)`, REWRITE_TAC[convex; CLOSURE_SEQUENTIAL] THEN GEN_TAC THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `u:real`; `v:real`] THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `a:num->real^N`) MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `b:num->real^N`) MP_TAC) THEN STRIP_TAC THEN EXISTS_TAC `\n:num. u % a(n) + v % b(n) :real^N` THEN ASM_SIMP_TAC[LIM_ADD; LIM_CMUL]);; let CONVEX_INTERIOR = prove (`!s:real^N->bool. convex s ==> convex(interior s)`, REWRITE_TAC[CONVEX_ALT; IN_INTERIOR; SUBSET; IN_BALL; dist] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `d:real`) MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `e:real`) STRIP_ASSUME_TAC) THEN EXISTS_TAC `min d e` THEN ASM_REWRITE_TAC[REAL_LT_MIN] THEN X_GEN_TAC `z:real^N` THEN STRIP_TAC THEN SUBST1_TAC(VECTOR_ARITH `z:real^N = (&1 - u) % (z - u % (y - x)) + u % (z + (&1 - u) % (y - x))`) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[VECTOR_ARITH `x - (z - u % (y - x)) = ((&1 - u) % x + u % y) - z:real^N`; VECTOR_ARITH `y - (z + (&1 - u) % (y - x)) = ((&1 - u) % x + u % y) - z:real^N`]);; let CONVEX_HULL_CLOSURE_SUBSET = prove (`!s:real^N->bool. convex hull (closure s) SUBSET closure(convex hull s)`, GEN_TAC THEN MATCH_MP_TAC HULL_MINIMAL THEN SIMP_TAC[SUBSET_CLOSURE; HULL_SUBSET; CONVEX_CLOSURE; CONVEX_CONVEX_HULL]);; let CONVEX_HULL_CLOSURE = prove (`!s. bounded s ==> convex hull (closure s) = closure(convex hull s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN SIMP_TAC[CONVEX_HULL_CLOSURE_SUBSET] THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN SIMP_TAC[HULL_MONO; CLOSURE_SUBSET] THEN MATCH_MP_TAC COMPACT_IMP_CLOSED THEN MATCH_MP_TAC COMPACT_CONVEX_HULL THEN ASM_REWRITE_TAC[COMPACT_CLOSURE]);; let SUPPORTING_HYPERPLANE_POINT = prove (`!s z:real^N. convex s /\ ~(s = {}) /\ ~(z IN s) ==> ?a b y. ~(a = vec 0) /\ a dot z <= b /\ y IN closure s /\ a dot y = b /\ !x. x IN closure s ==> a dot x >= b`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `(z:real^N) IN closure s` THENL [MP_TAC(ISPECL [`{z:real^N}`; `s:real^N->bool`] SEPARATING_HYPERPLANE_SETS) THEN ASM_REWRITE_TAC[CONVEX_SING] THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real`] THEN REWRITE_TAC[FORALL_IN_INSERT; NOT_IN_EMPTY] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`a:real^N`; `a dot (z:real^N)`; `z:real^N`] THEN ASM_REWRITE_TAC[REAL_LE_REFL] THEN ONCE_REWRITE_TAC[SET_RULE `a dot x >= b <=> x IN {x | a dot x >= b}`] THEN MATCH_MP_TAC FORALL_IN_CLOSURE THEN ASM_REWRITE_TAC[IN_ELIM_THM; CONTINUOUS_ON_ID; CLOSED_HALFSPACE_GE] THEN ASM_MESON_TAC[real_ge; REAL_LE_TRANS]; MP_TAC(ISPECL [`closure s:real^N->bool`; `z:real^N`] SUPPORTING_HYPERPLANE_CLOSED_POINT)THEN ASM_SIMP_TAC[CLOSED_CLOSURE; CONVEX_CLOSURE; CLOSURE_EQ_EMPTY] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN SIMP_TAC[REAL_LT_IMP_LE] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN SIMP_TAC[DOT_LZERO] THEN MESON_TAC[REAL_LT_REFL]]);; let CONVEX_ON_SETDIST = prove (`!s t:real^N->bool. convex t ==> (\x. setdist ({x},t)) convex_on s`, SUBGOAL_THEN `!s t:real^N->bool. convex t /\ closed t ==> (\x. setdist ({x},t)) convex_on s` MP_TAC THENL [ALL_TAC; REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`s:real^N->bool`; `closure t:real^N->bool`]) THEN ASM_SIMP_TAC[CLOSED_CLOSURE; SETDIST_CLOSURE; CONVEX_CLOSURE]] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[convex_on] THEN ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_SIMP_TAC[SETDIST_EMPTY; REAL_MUL_RZERO; REAL_ADD_RID; REAL_LE_REFL] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `u:real`; `v:real`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`{x:real^N}`; `t:real^N->bool`] SETDIST_COMPACT_CLOSED) THEN MP_TAC(ISPECL [`{y:real^N}`; `t:real^N->bool`] SETDIST_COMPACT_CLOSED) THEN ASM_REWRITE_TAC[NOT_INSERT_EMPTY; COMPACT_SING; UNWIND_THM2; SETDIST_CLOSURE; CLOSURE_EQ_EMPTY; RIGHT_EXISTS_AND_THM; IN_SING] THEN DISCH_THEN(X_CHOOSE_THEN `y':real^N` (STRIP_ASSUME_TAC o GSYM)) THEN DISCH_THEN(X_CHOOSE_THEN `x':real^N` (STRIP_ASSUME_TAC o GSYM)) THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `dist(u % x + v % y:real^N,u % x' + v % y')` THEN CONJ_TAC THENL [MATCH_MP_TAC SETDIST_LE_DIST THEN REWRITE_TAC[IN_SING] THEN ASM_MESON_TAC[convex]; REWRITE_TAC[dist] THEN MATCH_MP_TAC(NORM_ARITH `norm(a - a':real^N) + norm(b - b') <= r ==> norm((a + b) - (a' + b')) <= r`) THEN ASM_REWRITE_TAC[GSYM VECTOR_SUB_LDISTRIB; NORM_MUL; dist] THEN ASM_REWRITE_TAC[real_abs; REAL_LE_REFL]]);; (* ------------------------------------------------------------------------- *) (* Moving and scaling convex hulls. *) (* ------------------------------------------------------------------------- *) let CONVEX_HULL_TRANSLATION = prove (`!a:real^N s. convex hull (IMAGE (\x. a + x) s) = IMAGE (\x. a + x) (convex hull s)`, REPEAT GEN_TAC THEN MATCH_MP_TAC HULL_IMAGE THEN REWRITE_TAC[CONVEX_TRANSLATION_EQ; CONVEX_CONVEX_HULL] THEN REWRITE_TAC[VECTOR_ARITH `a + x:real^N = y <=> x = y - a`; EXISTS_REFL] THEN VECTOR_ARITH_TAC);; add_translation_invariants [CONVEX_HULL_TRANSLATION];; let CONVEX_HULL_SCALING = prove (`!s:real^N->bool c. convex hull (IMAGE (\x. c % x) s) = IMAGE (\x. c % x) (convex hull s)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `c = &0` THENL [ASM_SIMP_TAC[IMAGE_CONST; VECTOR_MUL_LZERO; CONVEX_HULL_EQ_EMPTY] THEN COND_CASES_TAC THEN REWRITE_TAC[CONVEX_HULL_EMPTY; CONVEX_HULL_SING]; ALL_TAC] THEN MATCH_MP_TAC HULL_IMAGE THEN ASM_SIMP_TAC[CONVEX_SCALING_EQ; CONVEX_CONVEX_HULL] THEN REWRITE_TAC[VECTOR_ARITH `c % x = c % y <=> c % (x - y) = vec 0`] THEN ASM_SIMP_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ] THEN X_GEN_TAC `x:real^N` THEN EXISTS_TAC `inv c % x:real^N` THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; VECTOR_MUL_LID]);; let CONVEX_HULL_AFFINITY = prove (`!s a:real^N c. convex hull (IMAGE (\x. c % x + a) s) = IMAGE (\x. c % x + a) (convex hull s)`, REPEAT GEN_TAC THEN REWRITE_TAC[AFFINITY_SCALING_TRANSLATION] THEN ASM_SIMP_TAC[IMAGE_o; CONVEX_HULL_TRANSLATION; CONVEX_HULL_SCALING]);; (* ------------------------------------------------------------------------- *) (* Convex set as intersection of halfspaces. *) (* ------------------------------------------------------------------------- *) let CONVEX_HALFSPACE_INTERSECTION = prove (`!s. closed(s:real^N->bool) /\ convex s ==> s = INTERS {h | s SUBSET h /\ ?a b. h = {x | a dot x <= b}}`, REPEAT STRIP_TAC THEN GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[IN_INTERS] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_ELIM_THM] THEN REWRITE_TAC[MESON[] `(!t. (P t /\ ?a b. t = x a b) ==> Q t) <=> (!a b. P(x a b) ==> Q(x a b))`] THEN EQ_TAC THENL [SET_TAC[]; ALL_TAC] THEN STRIP_TAC THEN MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `x:real^N`] SEPARATING_HYPERPLANE_CLOSED_POINT) THEN ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`--a:real^N`; `--b:real`]) THEN ASM_SIMP_TAC[SUBSET; IN_ELIM_THM; DOT_LNEG; NOT_IMP] THEN ASM_SIMP_TAC[REAL_LE_NEG2; REAL_LT_NEG2; REAL_NOT_LE; REAL_ARITH `a > b ==> b <= a`]);; (* ------------------------------------------------------------------------- *) (* Polar dual of a set. *) (* ------------------------------------------------------------------------- *) let polar_dual = new_definition `polar_dual s = {x:real^N | !u. u IN s ==> u dot x >= -- &1}`;; let POLAR_DUAL = prove (`!s:real^N->bool. polar_dual s = INTERS {{x | u dot x >= -- &1} | u IN s}`, REWRITE_TAC[polar_dual; INTERS_GSPEC] THEN SET_TAC[]);; let CLOSED_POLAR_DUAL = prove (`!s:real^N->bool. closed(polar_dual s)`, GEN_TAC THEN REWRITE_TAC[POLAR_DUAL] THEN MATCH_MP_TAC CLOSED_INTERS THEN REWRITE_TAC[FORALL_IN_GSPEC; CLOSED_HALFSPACE_GE]);; let CONVEX_POLAR_DUAL = prove (`!s:real^N->bool. convex(polar_dual s)`, GEN_TAC THEN REWRITE_TAC[POLAR_DUAL] THEN MATCH_MP_TAC CONVEX_INTERS THEN REWRITE_TAC[FORALL_IN_GSPEC; CONVEX_HALFSPACE_GE]);; let POLAR_DUAL_0 = prove (`!s:real^N->bool. vec 0 IN polar_dual s`, REWRITE_TAC[polar_dual; IN_ELIM_THM; DOT_RZERO] THEN CONV_TAC REAL_RAT_REDUCE_CONV);; let POLAR_DUAL_EMPTY = prove (`polar_dual {} = (:real^N)`, REWRITE_TAC[polar_dual; NOT_IN_EMPTY; UNIV_GSPEC]);; let POLAR_DUAL_SING = prove (`polar_dual {vec 0} = (:real^N)`, REWRITE_TAC[polar_dual; FORALL_IN_INSERT; NOT_IN_EMPTY; DOT_LZERO] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN SET_TAC[]);; let POLAR_DUAL_UNIV = prove (`polar_dual (:real^N) = {vec 0}`, REWRITE_TAC[polar_dual; IN_UNIV; EXTENSION; IN_ELIM_THM; IN_SING] THEN X_GEN_TAC `x:real^N` THEN EQ_TAC THEN SIMP_TAC[DOT_RZERO] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN GEN_REWRITE_TAC I [GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN DISCH_THEN(MP_TAC o SPEC `--(&2 / (x dot x)) % x:real^N`) THEN REWRITE_TAC[DOT_LMUL; REAL_ARITH `~(--x * y >= --a) <=> a < x * y`] THEN ASM_SIMP_TAC[REAL_DIV_RMUL; DOT_EQ_0] THEN REAL_ARITH_TAC);; let POLAR_DUAL_ANTIMONO = prove (`!s t:real^N->bool. s SUBSET t ==> polar_dual t SUBSET polar_dual s`, REWRITE_TAC[polar_dual] THEN SET_TAC[]);; let POLAR_DUAL_UNION = prove (`!s t:real^N->bool. polar_dual(s UNION t) = polar_dual s INTER polar_dual t`, REWRITE_TAC[polar_dual] THEN SET_TAC[]);; let POLAR_DUAL_SCALING = prove (`!a s:real^N->bool. ~(a = &0) ==> polar_dual {a % x | x IN s} = {inv a % x | x IN polar_dual s}`, REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN REWRITE_TAC[SIMPLE_IMAGE] THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN CONJ_TAC THENL [X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN EXISTS_TAC `a % y:real^N` THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; VECTOR_MUL_LID; REAL_MUL_LINV]; REWRITE_TAC[polar_dual; FORALL_IN_IMAGE; IN_ELIM_THM; DOT_RMUL] THEN ASM_SIMP_TAC[DOT_LMUL; REAL_FIELD `~(a = &0) ==> inv a * a * b = b`]]);; let POLAR_DUAL_UNIT_CBALL = prove (`polar_dual(cball(vec 0:real^N,&1)) = cball(vec 0,&1)`, REWRITE_TAC[polar_dual; IN_CBALL_0; EXTENSION; IN_ELIM_THM] THEN REWRITE_TAC[NORM_LE_SQUARE] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN GEN_REWRITE_TAC I [MESON[VECTOR_NEG_NEG] `(!x:real^N. P x) <=> (!x. P(--x))`] THEN REWRITE_TAC[DOT_RNEG; DOT_LNEG; REAL_NEG_NEG] THEN REWRITE_TAC[REAL_ARITH `--x >= -- &1 <=> x <= &1`] THEN X_GEN_TAC `x:real^N` THEN EQ_TAC THENL [ASM_CASES_TAC `x:real^N = vec 0` THEN ASM_REWRITE_TAC[DOT_LZERO; REAL_POS] THEN DISCH_THEN(MP_TAC o SPEC `inv(norm x) % x:real^N`) THEN REWRITE_TAC[DOT_LMUL; DOT_RMUL; GSYM REAL_POW_2; REAL_MUL_ASSOC] THEN REWRITE_TAC[REAL_POW_INV; NORM_POW_2] THEN ASM_SIMP_TAC[REAL_MUL_LINV; DOT_EQ_0; REAL_LE_REFL] THEN ASM_SIMP_TAC[GSYM NORM_POW_2; NORM_EQ_0; REAL_FIELD `~(x = &0) ==> inv x * x pow 2 = x`] THEN REWRITE_TAC[ABS_SQUARE_LE_1; REAL_ABS_NORM]; REWRITE_TAC[GSYM NORM_POW_2; ABS_SQUARE_LE_1; REAL_ABS_NORM] THEN REWRITE_TAC[RIGHT_IMP_FORALL_THM] THEN GEN_TAC THEN MATCH_MP_TAC(REAL_ARITH `y <= u * x /\ (u <= &1 /\ x <= &1 ==> u * x <= &1 * &1) ==> x <= &1 ==> u <= &1 ==> y <= &1`) THEN REWRITE_TAC[NORM_CAUCHY_SCHWARZ] THEN SIMP_TAC[REAL_LE_MUL2; NORM_POS_LE]]);; let POLAR_DUAL_CBALL = prove (`!r. &0 < r ==> polar_dual(cball(vec 0:real^N,r)) = cball(vec 0,inv r)`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[MESON[CBALL_SCALING; REAL_MUL_RID; VECTOR_MUL_RZERO] `&0 < r ==> cball(vec 0,r) = IMAGE (\x. r % x) (cball(vec 0,&1))`] THEN ASM_SIMP_TAC[GSYM SIMPLE_IMAGE; POLAR_DUAL_SCALING; REAL_LT_IMP_NZ] THEN REWRITE_TAC[POLAR_DUAL_UNIT_CBALL] THEN ASM_SIMP_TAC[SIMPLE_IMAGE; GSYM CBALL_SCALING; REAL_LT_INV_EQ] THEN REWRITE_TAC[REAL_MUL_RID; VECTOR_MUL_RZERO]);; let POLAR_DUAL_POLAR_DUAL_GEN = prove (`!s:real^N->bool. polar_dual(polar_dual s) = closure(convex hull (vec 0 INSERT s))`, GEN_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[TAUT `p ==> q <=> ~(p /\ ~q)`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`closure (convex hull (vec 0 INSERT s)):real^N->bool`; `x:real^N`] SEPARATING_HYPERPLANE_CLOSED_POINT) THEN ASM_SIMP_TAC[CLOSED_CLOSURE; CONVEX_CONVEX_HULL; CONVEX_CLOSURE] THEN REWRITE_TAC[NOT_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real`] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `vec 0:real^N`) THEN SIMP_TAC[IN_INSERT; HULL_INC; CLOSURE_INC] THEN REWRITE_TAC[real_gt; DOT_RZERO] THEN STRIP_TAC THEN UNDISCH_TAC `(x:real^N) IN polar_dual(polar_dual s)` THEN REWRITE_TAC[polar_dual; IN_ELIM_THM] THEN DISCH_THEN(MP_TAC o SPEC `inv(-- b) % a:real^N`) THEN REWRITE_TAC[DOT_LMUL; DOT_RMUL] THEN REWRITE_TAC[REAL_ARITH `inv x * y:real = y / x`] THEN ASM_SIMP_TAC[real_ge; REAL_LE_RDIV_EQ; REAL_ARITH `&0 < --b <=> b < &0`; REAL_ARITH `--x * --y:real = x * y`; REAL_MUL_LID] THEN ASM_REWRITE_TAC[GSYM REAL_NOT_LT] THEN X_GEN_TAC `u:real^N` THEN DISCH_TAC THEN MATCH_MP_TAC(REAL_ARITH `a > b ==> ~(a < b)`) THEN ONCE_REWRITE_TAC[DOT_SYM] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[IN_INSERT; CLOSURE_INC; HULL_INC]; MATCH_MP_TAC CLOSURE_MINIMAL THEN REWRITE_TAC[CLOSED_POLAR_DUAL] THEN MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[CONVEX_POLAR_DUAL] THEN REWRITE_TAC[INSERT_SUBSET; POLAR_DUAL_0] THEN REWRITE_TAC[SUBSET; polar_dual; IN_ELIM_THM] THEN MESON_TAC[DOT_SYM]]);; let POLAR_DUAL_POLAR_DUAL_EQ = prove (`!s:real^N->bool. polar_dual(polar_dual s) = s <=> closed s /\ convex s /\ vec 0 IN s`, GEN_TAC THEN REWRITE_TAC[POLAR_DUAL_POLAR_DUAL_GEN] THEN EQ_TAC THENL [DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (SET_RULE `closure t = s ==> (!x. x IN s ==> x IN t) /\ t SUBSET closure t ==> t = s`)) THEN SIMP_TAC[IN_INSERT; HULL_INC; CLOSURE_SUBSET] THEN DISCH_TAC THEN CONJ_TAC THENL [ASM_MESON_TAC[CLOSURE_EQ]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o MATCH_MP (SET_RULE `convex hull t = s ==> (!x. x IN s ==> x IN t) /\ t SUBSET convex hull t ==> t = s`)) THEN SIMP_TAC[HULL_SUBSET; IN_INSERT] THEN REWRITE_TAC[GSYM CONVEX_HULL_EQ] THEN REPEAT STRIP_TAC THENL [ASM_MESON_TAC[]; ASM SET_TAC[]]; SIMP_TAC[SET_RULE `a IN s ==> a INSERT s = s`; HULL_P] THEN SIMP_TAC[CLOSURE_CLOSED]]);; let POLAR_DUAL_POLAR_DUAL = prove (`!s:real^N->bool. closed s /\ convex s /\ vec 0 IN s ==> polar_dual(polar_dual s) = s`, REWRITE_TAC[POLAR_DUAL_POLAR_DUAL_EQ]);; (* ------------------------------------------------------------------------- *) (* Radon's theorem (from Lars Schewe). *) (* ------------------------------------------------------------------------- *) let RADON_EX_LEMMA = prove (`!(c:real^N->bool). FINITE c /\ affine_dependent c ==> (?u. sum c u = &0 /\ (?v. v IN c /\ ~(u v = &0)) /\ vsum c (\v. u v % v) = (vec 0):real^N)`, REWRITE_TAC[AFFINE_DEPENDENT_EXPLICIT] THEN REPEAT STRIP_TAC THEN EXISTS_TAC `\v:real^N. if v IN s then u v else &0` THEN ASM_SIMP_TAC[GSYM SUM_RESTRICT_SET] THEN ASM_SIMP_TAC[COND_RAND;COND_RATOR; VECTOR_MUL_LZERO;GSYM VSUM_RESTRICT_SET] THEN ASM_SIMP_TAC[SET_RULE `s SUBSET c ==> {x | x IN c /\ x IN s} = s`] THEN EXISTS_TAC `v:real^N` THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);; let RADON_S_LEMMA = prove (`!(s:A->bool) f. FINITE s /\ sum s f = &0 ==> sum {x | x IN s /\ &0 < f x} f = -- sum {x | x IN s /\ f x < &0} f`, REWRITE_TAC[REAL_ARITH `a = --b <=> a + b = &0`] THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[FINITE_RESTRICT;GSYM SUM_UNION; REWRITE_RULE [REAL_ARITH `&0 < f x ==> ~(f x < &0)`] (SET_RULE `(!x:A. &0 < f x ==> ~(f x < &0)) ==> DISJOINT {x | x IN s /\ &0 < f x} {x | x IN s /\ f x < &0}`)] THEN MATCH_MP_TAC (REAL_ARITH `!a b.a = &0 /\ a + b = &0 ==> b = &0`) THEN EXISTS_TAC `sum {x:A | x IN s /\ f x = &0} f` THEN CONJ_TAC THENL [ASM_SIMP_TAC[SUM_RESTRICT_SET] THEN REWRITE_TAC[COND_ID;SUM_0]; ALL_TAC] THEN SUBGOAL_THEN `DISJOINT {x:A | x IN s /\ f x = &0} ({x | x IN s /\ &0 < f x} UNION {x | x IN s /\ f x < &0})` ASSUME_TAC THENL [REWRITE_TAC[DISJOINT;UNION;INTER;IN_ELIM_THM;EXTENSION;NOT_IN_EMPTY] THEN REAL_ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[FINITE_UNION;FINITE_RESTRICT;GSYM SUM_UNION] THEN FIRST_X_ASSUM (SUBST1_TAC o GSYM) THEN MATCH_MP_TAC (MESON[] `a = b ==> sum a f = sum b f`) THEN REWRITE_TAC[EXTENSION;IN_ELIM_THM;UNION] THEN MESON_TAC[REAL_LT_TOTAL]);; let RADON_V_LEMMA = prove (`!(s:A->bool) f g. FINITE s /\ vsum s f = vec 0 /\ (!x. g x = &0 ==> f x = vec 0) ==> (vsum {x | x IN s /\ &0 < g x} f) :real^N = -- vsum {x | x IN s /\ g x < &0} f`, REWRITE_TAC[VECTOR_ARITH `a:real^N = --b <=> a + b = vec 0`] THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[FINITE_RESTRICT;GSYM VSUM_UNION; REWRITE_RULE [REAL_ARITH `&0 < f x ==> ~(f x < &0)`] (SET_RULE `(!x:A. &0 < f x ==> ~(f x < &0)) ==> DISJOINT {x | x IN s /\ &0 < f x} {x | x IN s /\ f x < &0}`)] THEN MATCH_MP_TAC (VECTOR_ARITH `!a b. (a:real^N) = vec 0 /\ a + b = vec 0 ==> b = vec 0`) THEN EXISTS_TAC `(vsum {x:A | x IN s /\ g x = &0} f):real^N` THEN CONJ_TAC THENL [ASM_SIMP_TAC[VSUM_RESTRICT_SET;COND_ID;VSUM_0];ALL_TAC] THEN SUBGOAL_THEN `DISJOINT {x:A | x IN s /\ g x = &0} ({x | x IN s /\ &0 < g x} UNION {x | x IN s /\ g x < &0})` ASSUME_TAC THENL [REWRITE_TAC[DISJOINT;UNION;INTER;IN_ELIM_THM;EXTENSION;NOT_IN_EMPTY] THEN REAL_ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[FINITE_UNION;FINITE_RESTRICT;GSYM VSUM_UNION] THEN FIRST_X_ASSUM (SUBST1_TAC o GSYM) THEN MATCH_MP_TAC (MESON[] `a = b ==> vsum a f = vsum b f`) THEN REWRITE_TAC[EXTENSION;IN_ELIM_THM;UNION] THEN MESON_TAC[REAL_LT_TOTAL]);; let RADON_PARTITION = prove (`!(c:real^N->bool). FINITE c /\ affine_dependent c ==> ?(m:real^N->bool) (p:real^N->bool). (DISJOINT m p) /\ (m UNION p = c) /\ ~(DISJOINT (convex hull m) (convex hull p))`, REPEAT STRIP_TAC THEN MP_TAC (ISPEC `c:real^N->bool` RADON_EX_LEMMA) THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`{v:real^N | v IN c /\ u v <= &0}`; `{v:real^N | v IN c /\ u v > &0}`] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[DISJOINT;INTER; IN_ELIM_THM;REAL_ARITH `x <= &0 <=> ~(x > &0)`] THEN SET_TAC[]; REWRITE_TAC[UNION;IN_ELIM_THM;REAL_ARITH `x <= &0 <=> ~(x > &0)`] THEN SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `~(sum {x:real^N | x IN c /\ u x > &0} u = &0)` ASSUME_TAC THENL [MATCH_MP_TAC (REAL_ARITH `a > &0 ==> ~(a = &0)`) THEN REWRITE_TAC[REAL_ARITH `a > &0 <=> &0 < a`] THEN MATCH_MP_TAC (REWRITE_RULE[SUM_0] (ISPEC `\x. &0` SUM_LT_ALL)) THEN ASM_SIMP_TAC[FINITE_RESTRICT;IN_ELIM_THM;EXTENSION;NOT_IN_EMPTY] THEN REWRITE_TAC[MESON[]`~(!x. ~(P x /\ Q x)) = ?x. P x /\ Q x`] THEN ASM_CASES_TAC `&0 < u (v:real^N)` THENL [ASM SET_TAC[];ALL_TAC] THEN POP_ASSUM MP_TAC THEN POP_ASSUM (K ALL_TAC) THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[IMP_IMP;REAL_ARITH `~(a = &0) /\ ~(&0 < a) <=> a < &0`] THEN DISCH_TAC THEN REWRITE_TAC[MESON[REAL_NOT_LT] `(?x:real^N. P x /\ &0 < u x) <=> (!x. P x ==> u x <= &0) ==> F`] THEN DISCH_TAC THEN MP_TAC (ISPECL [`u:real^N->real`;`\x:real^N. &0`;`c:real^N->bool`] SUM_LT) THEN ASM_REWRITE_TAC[SUM_0;REAL_ARITH `~(&0 < &0)`] THEN ASM_MESON_TAC[];ALL_TAC] THEN REWRITE_TAC[SET_RULE `~DISJOINT a b <=> ?y. y IN a /\ y IN b`] THEN EXISTS_TAC `&1 / (sum {x:real^N | x IN c /\ u x > &0} u) % vsum {x:real^N | x IN c /\ u x > &0} (\x. u x % x)` THEN REWRITE_TAC[CONVEX_HULL_EXPLICIT;IN_ELIM_THM] THEN CONJ_TAC THENL [MAP_EVERY EXISTS_TAC [`{v:real^N | v IN c /\ u v < &0}`; `\y:real^N. &1 / (sum {x:real^N | x IN c /\ u x > &0} u) * (--(u y))`] THEN ASM_SIMP_TAC[FINITE_RESTRICT;SUBSET;IN_ELIM_THM] THEN REPEAT CONJ_TAC THENL [REAL_ARITH_TAC; REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_MUL THEN CONJ_TAC THENL [ALL_TAC; ASM_REWRITE_TAC[REAL_NEG_GE0;REAL_LE_LT]] THEN MATCH_MP_TAC REAL_LE_DIV THEN REWRITE_TAC[REAL_LE_01] THEN MATCH_MP_TAC SUM_POS_LE THEN ASM_SIMP_TAC[FINITE_RESTRICT;IN_ELIM_THM] THEN REAL_ARITH_TAC; ASM_SIMP_TAC[FINITE_RESTRICT;SUM_LMUL] THEN MATCH_MP_TAC (REAL_FIELD `!a. ~(a = &0) /\ a * b = a * c ==> b = c`) THEN EXISTS_TAC `sum {x:real^N | x IN c /\ u x > &0} u` THEN REWRITE_TAC[SUM_LMUL] THEN ASM_SIMP_TAC[REAL_FIELD `~(a = &0) ==> a * &1 / a * b = b`] THEN REWRITE_TAC[SUM_NEG;REAL_MUL_RID] THEN REWRITE_TAC[REAL_ARITH `a > &0 <=> &0 < a`] THEN MATCH_MP_TAC (GSYM RADON_S_LEMMA) THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN REWRITE_TAC[GSYM VECTOR_MUL_ASSOC;VSUM_LMUL;VECTOR_MUL_LCANCEL] THEN REWRITE_TAC[VECTOR_MUL_LNEG;VSUM_NEG] THEN DISJ2_TAC THEN MATCH_MP_TAC (REWRITE_RULE[REAL_ARITH `&0 < a <=> a > &0`] (GSYM RADON_V_LEMMA)) THEN ASM_REWRITE_TAC[] THEN MESON_TAC[VECTOR_MUL_LZERO];ALL_TAC] THEN MAP_EVERY EXISTS_TAC [`{v:real^N | v IN c /\ u v > &0}`; `\y:real^N. &1 / (sum {x:real^N | x IN c /\ u x > &0} u) * (u y)`] THEN ASM_SIMP_TAC[FINITE_RESTRICT;SUBSET;IN_ELIM_THM] THEN REPEAT CONJ_TAC THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_MUL THEN CONJ_TAC THENL [ALL_TAC; ASM_SIMP_TAC[REAL_ARITH `a > &0 ==> &0 <= a`]] THEN MATCH_MP_TAC REAL_LE_DIV THEN REWRITE_TAC[REAL_LE_01] THEN MATCH_MP_TAC SUM_POS_LE THEN ASM_SIMP_TAC[FINITE_RESTRICT;IN_ELIM_THM] THEN REAL_ARITH_TAC; ASM_SIMP_TAC[FINITE_RESTRICT;SUM_LMUL] THEN MATCH_MP_TAC (REAL_FIELD `!a. ~(a = &0) /\ a * b = a * c ==> b = c`) THEN EXISTS_TAC `sum {x:real^N | x IN c /\ u x > &0} u` THEN REWRITE_TAC[SUM_LMUL] THEN ASM_SIMP_TAC[REAL_FIELD `~(a = &0) ==> a * &1 / a * b = b`] THEN REWRITE_TAC[SUM_NEG;REAL_MUL_RID] THEN REWRITE_TAC[REAL_ARITH `a > &0 <=> &0 < a`] THEN MATCH_MP_TAC (GSYM RADON_S_LEMMA) THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN REWRITE_TAC[GSYM VECTOR_MUL_ASSOC;VSUM_LMUL;VECTOR_MUL_LCANCEL] THEN REWRITE_TAC[VECTOR_MUL_LNEG;VSUM_NEG] THEN DISJ2_TAC THEN MATCH_MP_TAC (REWRITE_RULE[REAL_ARITH `&0 < a <=> a > &0`] (GSYM RADON_V_LEMMA)) THEN ASM_REWRITE_TAC[] THEN MESON_TAC[VECTOR_MUL_LZERO]);; let RADON = prove (`!(c:real^N->bool). affine_dependent c ==> ?(m:real^N->bool) (p:real^N->bool). m SUBSET c /\ p SUBSET c /\ DISJOINT m p /\ ~(DISJOINT (convex hull m) (convex hull p))`, REPEAT STRIP_TAC THEN MP_TAC (ISPEC `c:real^N->bool` AFFINE_DEPENDENT_EXPLICIT) THEN ASM_SIMP_TAC[] THEN REPEAT STRIP_TAC THEN MP_TAC (ISPEC `s:real^N->bool` RADON_PARTITION) THEN ANTS_TAC THENL [ASM_SIMP_TAC[AFFINE_DEPENDENT_EXPLICIT] THEN MAP_EVERY EXISTS_TAC [`s:real^N->bool`;`u:real^N->real`] THEN ASM SET_TAC[];ALL_TAC] THEN DISCH_THEN STRIP_ASSUME_TAC THEN MAP_EVERY EXISTS_TAC [`m:real^N->bool`;`p:real^N->bool`] THEN ASM SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* Helly's theorem. *) (* ------------------------------------------------------------------------- *) let HELLY_INDUCT = prove (`!n f. f HAS_SIZE n /\ n >= dimindex(:N) + 1 /\ (!s:real^N->bool. s IN f ==> convex s) /\ (!t. t SUBSET f /\ CARD(t) = dimindex(:N) + 1 ==> ~(INTERS t = {})) ==> ~(INTERS f = {})`, INDUCT_TAC THEN REWRITE_TAC[ARITH_RULE `~(0 >= n + 1)`] THEN GEN_TAC THEN POP_ASSUM(LABEL_TAC "*") THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HAS_SIZE_SUC]) THEN STRIP_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[HAS_SIZE]) THEN FIRST_X_ASSUM(DISJ_CASES_TAC o MATCH_MP (ARITH_RULE `SUC n >= m + 1 ==> m = n \/ n >= m + 1`)) THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[HAS_SIZE]) THEN ASM_SIMP_TAC[CARD_CLAUSES; SUBSET_REFL] THEN ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `?X. !s:real^N->bool. s IN f ==> X(s) IN INTERS (f DELETE s)` STRIP_ASSUME_TAC THENL [REWRITE_TAC[GSYM SKOLEM_THM; MEMBER_NOT_EMPTY; RIGHT_EXISTS_IMP_THM] THEN GEN_TAC THEN STRIP_TAC THEN REMOVE_THEN "*" MATCH_MP_TAC THEN ASM_SIMP_TAC[FINITE_DELETE; CARD_DELETE] THEN ASM SET_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `?s t:real^N->bool. s IN f /\ t IN f /\ ~(s = t) /\ X s:real^N = X t` THENL [FIRST_X_ASSUM(CHOOSE_THEN STRIP_ASSUME_TAC) THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `(X:(real^N->bool)->real^N) t` THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC ONCE_DEPTH_CONV [MATCH_MP (SET_RULE`~(s = t) ==> INTERS f = INTERS(f DELETE s) INTER INTERS(f DELETE t)`) th]) THEN REWRITE_TAC[IN_INTER] THEN ASM_MESON_TAC[]; ALL_TAC] THEN MP_TAC(ISPEC `IMAGE (X:(real^N->bool)->real^N) f` RADON_PARTITION) THEN ANTS_TAC THENL [ASM_SIMP_TAC[FINITE_IMAGE] THEN MATCH_MP_TAC AFFINE_DEPENDENT_BIGGERSET THEN ASM_SIMP_TAC[FINITE_IMAGE] THEN MATCH_MP_TAC(ARITH_RULE `!f n. n >= d + 1 /\ f = SUC n /\ c = f ==> c >= d + 2`) THEN MAP_EVERY EXISTS_TAC [`CARD(f:(real^N->bool)->bool)`; `n:num`] THEN REPEAT(CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC]) THEN MATCH_MP_TAC CARD_IMAGE_INJ THEN ASM_MESON_TAC[]; ALL_TAC] THEN ONCE_REWRITE_TAC[SET_RULE `P /\ m UNION p = s /\ Q <=> m SUBSET s /\ p SUBSET s /\ m UNION p = s /\ P /\ Q`] THEN REWRITE_TAC[SUBSET_IMAGE; DISJOINT] THEN REWRITE_TAC[MESON[] `(?m p. (?u. P u /\ m = t u) /\ (?u. P u /\ p = t u) /\ Q m p) ==> r <=> (!u v. P u /\ P v /\ Q (t u) (t v) ==> r)`] THEN MAP_EVERY X_GEN_TAC [`g:(real^N->bool)->bool`; `h:(real^N->bool)->bool`] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN SUBGOAL_THEN `(f:(real^N->bool)->bool) = h UNION g` SUBST1_TAC THENL [MATCH_MP_TAC SUBSET_ANTISYM THEN ASM_REWRITE_TAC[UNION_SUBSET] THEN REWRITE_TAC[SUBSET; IN_UNION] THEN X_GEN_TAC `s:real^N->bool` THEN DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN DISCH_THEN(MP_TAC o ISPEC `X:(real^N->bool)->real^N` o MATCH_MP FUN_IN_IMAGE) THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM th]) THEN ONCE_REWRITE_TAC[DISJ_SYM] THEN REWRITE_TAC[IN_UNION; IN_IMAGE] THEN MATCH_MP_TAC MONO_OR THEN ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `g SUBSET INTERS g' /\ h SUBSET INTERS h' ==> ~(g INTER h = {}) ==> ~(INTERS(g' UNION h') = {})`) THEN FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP (SET_RULE `IMAGE X s INTER IMAGE X t = {} ==> s INTER t = {}`)) THEN CONJ_TAC THEN MATCH_MP_TAC HULL_MINIMAL THEN (CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[SUBSET; CONVEX_INTERS]]) THEN REWRITE_TAC[SUBSET; IN_INTERS; FORALL_IN_IMAGE] THEN ASM SET_TAC[]);; let HELLY = prove (`!f:(real^N->bool)->bool. FINITE f /\ CARD(f) >= dimindex(:N) + 1 /\ (!s. s IN f ==> convex s) /\ (!t. t SUBSET f /\ CARD(t) = dimindex(:N) + 1 ==> ~(INTERS t = {})) ==> ~(INTERS f = {})`, GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC HELLY_INDUCT THEN ASM_REWRITE_TAC[HAS_SIZE] THEN ASM_MESON_TAC[]);; let HELLY_ALT = prove (`!f:(real^N->bool)->bool. FINITE f /\ (!s. s IN f ==> convex s) /\ (!t. t SUBSET f /\ CARD(t) <= dimindex(:N) + 1 ==> ~(INTERS t = {})) ==> ~(INTERS f = {})`, GEN_TAC THEN STRIP_TAC THEN ASM_CASES_TAC `CARD(f:(real^N->bool)->bool) < dimindex(:N) + 1` THEN ASM_SIMP_TAC[SUBSET_REFL; LT_IMP_LE] THEN MATCH_MP_TAC HELLY THEN ASM_SIMP_TAC[GE; GSYM NOT_LT] THEN ASM_MESON_TAC[LE_REFL]);; let HELLY_CLOSED_ALT = prove (`!f:(real^N->bool)->bool. (!s. s IN f ==> convex s /\ closed s) /\ (?s. s IN f /\ bounded s) /\ (!t. t SUBSET f /\ FINITE t /\ CARD(t) <= dimindex(:N) + 1 ==> ~(INTERS t = {})) ==> ~(INTERS f = {})`, GEN_TAC THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN MATCH_MP_TAC CLOSED_FIP THEN ASM_SIMP_TAC[] THEN X_GEN_TAC `g:(real^N->bool)->bool` THEN STRIP_TAC THEN MATCH_MP_TAC HELLY_ALT THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; ASM_MESON_TAC[SUBSET_TRANS; FINITE_SUBSET]]);; let HELLY_COMPACT_ALT = prove (`!f:(real^N->bool)->bool. (!s. s IN f ==> convex s /\ compact s) /\ (!t. t SUBSET f /\ FINITE t /\ CARD(t) <= dimindex(:N) + 1 ==> ~(INTERS t = {})) ==> ~(INTERS f = {})`, GEN_TAC THEN STRIP_TAC THEN ASM_CASES_TAC `f:(real^N->bool)->bool = {}` THEN ASM_REWRITE_TAC[INTERS_0; UNIV_NOT_EMPTY] THEN MATCH_MP_TAC HELLY_CLOSED_ALT THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED] THEN ASM_MESON_TAC[MEMBER_NOT_EMPTY; COMPACT_IMP_BOUNDED]);; let HELLY_CLOSED = prove (`!f:(real^N->bool)->bool. (FINITE f ==> CARD f >= dimindex (:N) + 1) /\ (!s. s IN f ==> convex s /\ closed s) /\ (?s. s IN f /\ bounded s) /\ (!t. t SUBSET f /\ FINITE t /\ CARD(t) = dimindex(:N) + 1 ==> ~(INTERS t = {})) ==> ~(INTERS f = {})`, GEN_TAC THEN REWRITE_TAC[GE] THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN MATCH_MP_TAC HELLY_CLOSED_ALT THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `g:(real^N->bool)->bool` THEN STRIP_TAC THEN MP_TAC(ISPECL [`dimindex(:N) + 1`; `g:(real^N->bool)->bool`; `f:(real^N->bool)->bool`] CHOOSE_SUBSET_BETWEEN) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `h:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ ~(s = {}) ==> ~(t = {})`) THEN EXISTS_TAC `INTERS h: real^N->bool` THEN CONJ_TAC THENL [ASM SET_TAC[]; FIRST_X_ASSUM MATCH_MP_TAC] THEN ASM_MESON_TAC[HAS_SIZE]);; let HELLY_COMPACT = prove (`!f:(real^N->bool)->bool. (FINITE f ==> CARD f >= dimindex (:N) + 1) /\ (!s. s IN f ==> convex s /\ compact s) /\ (!t. t SUBSET f /\ FINITE t /\ CARD(t) = dimindex(:N) + 1 ==> ~(INTERS t = {})) ==> ~(INTERS f = {})`, GEN_TAC THEN STRIP_TAC THEN ASM_CASES_TAC `f:(real^N->bool)->bool = {}` THEN ASM_REWRITE_TAC[INTERS_0; UNIV_NOT_EMPTY] THEN MATCH_MP_TAC HELLY_CLOSED THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED] THEN ASM_MESON_TAC[MEMBER_NOT_EMPTY; COMPACT_IMP_BOUNDED]);; (* ------------------------------------------------------------------------- *) (* Kirchberger's theorem *) (* ------------------------------------------------------------------------- *) let KIRCHBERGER = prove (`!s t:real^N->bool. compact s /\ compact t /\ (!s' t'. s' SUBSET s /\ t' SUBSET t /\ FINITE s' /\ FINITE t' /\ CARD(s') + CARD(t') <= dimindex(:N) + 2 ==> ?a b. (!x. x IN s' ==> a dot x < b) /\ (!x. x IN t' ==> a dot x > b)) ==> ?a b. ~(a = vec 0) /\ (!x. x IN s ==> a dot x < b) /\ (!x. x IN t ==> a dot x > b)`, let lemma = prove (`(!x. x IN convex hull s ==> a dot x < b) /\ (!x. x IN convex hull t ==> a dot x > b) <=> (!x. x IN s ==> a dot x < b) /\ (!x. x IN t ==> a dot x > b)`, REWRITE_TAC[SET_RULE `(!x. x IN s ==> P x) <=> s SUBSET {x | P x}`] THEN SIMP_TAC[SUBSET_HULL; CONVEX_HALFSPACE_LT; CONVEX_HALFSPACE_GT]) and KIRCH_LEMMA = prove (`!s t:real^N->bool. FINITE s /\ FINITE t /\ (!s' t'. s' SUBSET s /\ t' SUBSET t /\ CARD(s') + CARD(t') <= dimindex(:N) + 2 ==> ?a b. (!x. x IN s' ==> a dot x < b) /\ (!x. x IN t' ==> a dot x > b)) ==> ?a b. (!x. x IN s ==> a dot x < b) /\ (!x. x IN t ==> a dot x > b)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`IMAGE (\r. {z:real^(N,1)finite_sum | fstcart z dot r < drop(sndcart z)}) s UNION IMAGE (\r. {z:real^(N,1)finite_sum | fstcart z dot r > drop(sndcart z)}) t`] HELLY_ALT) THEN REWRITE_TAC[FORALL_SUBSET_UNION; IN_UNION; IMP_CONJ] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM; FORALL_SUBSET_IMAGE] THEN ASM_SIMP_TAC[FINITE_UNION; FINITE_IMAGE; INTERS_UNION] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; INTERS_IMAGE; IN_INTER; EXISTS_PASTECART; IN_ELIM_PASTECART_THM; FSTCART_PASTECART; SNDCART_PASTECART] THEN REWRITE_TAC[TAUT `p \/ q ==> r <=> (p ==> r) /\ (q ==> r)`] THEN REWRITE_TAC[FORALL_AND_THM; FORALL_IN_IMAGE; RIGHT_IMP_FORALL_THM] THEN REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC; GSYM EXISTS_DROP] THEN DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [REWRITE_TAC[REAL_ARITH `a > b <=> --a < --b`; GSYM DOT_RNEG] THEN REWRITE_TAC[convex; IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_GSPEC] THEN SIMP_TAC[PASTECART_ADD; GSYM PASTECART_CMUL; IN_ELIM_PASTECART_THM] THEN SIMP_TAC[DOT_LADD; DOT_LMUL; DROP_ADD; DROP_CMUL; GSYM FORALL_DROP] THEN REWRITE_TAC[REAL_ARITH `--(a * x + b * y):real = a * --x + b * --y`] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REAL_ARITH `u + v = &1 ==> &0 <= u /\ &0 <= v ==> u = &0 /\ v = &1 \/ u = &1 /\ v = &0 \/ &0 < u /\ &0 < v`)) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ASM_REWRITE_TAC[REAL_MUL_LZERO; REAL_MUL_LID; REAL_ADD_LID; REAL_ADD_RID] THEN MATCH_MP_TAC REAL_LT_ADD2 THEN ASM_SIMP_TAC[REAL_LT_LMUL_EQ]; REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1; ARITH_RULE `(n + 1) + 1 = n + 2`] THEN MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `v:real^N->bool`] THEN DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN SUBGOAL_THEN `FINITE(u:real^N->bool) /\ FINITE(v:real^N->bool)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[FINITE_SUBSET]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhs o rand) CARD_UNION o lhand o lhand o snd) THEN ASM_SIMP_TAC[FINITE_IMAGE] THEN ANTS_TAC THENL [REWRITE_TAC[SET_RULE `IMAGE f s INTER IMAGE g t = {} <=> !x y. x IN s /\ y IN t ==> ~(f x = g y)`] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN REWRITE_TAC[EXTENSION; FORALL_PASTECART; IN_ELIM_PASTECART_THM] THEN DISCH_THEN(MP_TAC o SPEC `vec 0:real^N`) THEN REWRITE_TAC[GSYM FORALL_DROP; DOT_LZERO] THEN DISCH_THEN(MP_TAC o SPEC `&1`) THEN REAL_ARITH_TAC; DISCH_THEN SUBST1_TAC] THEN DISCH_THEN(fun th -> FIRST_X_ASSUM MATCH_MP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(ARITH_RULE `a = a' /\ b = b' ==> a + b <= n + 2 ==> a' + b' <= n + 2`) THEN CONJ_TAC THEN MATCH_MP_TAC CARD_IMAGE_INJ THEN ASM_REWRITE_TAC[EXTENSION; FORALL_PASTECART; IN_ELIM_PASTECART_THM] THEN SIMP_TAC[GSYM FORALL_DROP; real_gt; VECTOR_EQ_LDOT; MESON[REAL_LT_TOTAL; REAL_LT_REFL] `((!y:real. a < y <=> b < y) <=> a = b) /\ ((!y:real. y < a <=> y < b) <=> a = b)`]]) in REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM lemma] THEN MATCH_MP_TAC SEPARATING_HYPERPLANE_COMPACT_COMPACT THEN ASM_SIMP_TAC[CONVEX_CONVEX_HULL; COMPACT_CONVEX_HULL; CONVEX_HULL_EQ_EMPTY] THEN SUBGOAL_THEN `!s' t'. (s':real^N->bool) SUBSET s /\ t' SUBSET t /\ FINITE s' /\ CARD(s') <= dimindex(:N) + 1 /\ FINITE t' /\ CARD(t') <= dimindex(:N) + 1 ==> DISJOINT (convex hull s') (convex hull t')` MP_TAC THENL [REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s':real^N->bool`; `t':real^N->bool`] KIRCH_LEMMA) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[SUBSET; FINITE_SUBSET]; ONCE_REWRITE_TAC[GSYM lemma] THEN SET_TAC[REAL_LT_ANTISYM; real_gt]]; POP_ASSUM_LIST(K ALL_TAC) THEN STRIP_TAC THEN REWRITE_TAC[SET_RULE `DISJOINT s t <=> !x. x IN s /\ x IN t ==> F`] THEN X_GEN_TAC `x:real^N` THEN ONCE_REWRITE_TAC[CARATHEODORY] THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `s':real^N->bool` STRIP_ASSUME_TAC) (X_CHOOSE_THEN `t':real^N->bool` STRIP_ASSUME_TAC)) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`s':real^N->bool`; `t':real^N->bool`]) THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Convex hull is "preserved" by a linear function. *) (* ------------------------------------------------------------------------- *) let CONVEX_HULL_LINEAR_IMAGE = prove (`!f s. linear f ==> convex hull (IMAGE f s) = IMAGE f (convex hull s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN CONJ_TAC THEN MATCH_MP_TAC HULL_INDUCT THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN SIMP_TAC[FUN_IN_IMAGE; HULL_INC] THEN REWRITE_TAC[convex; IN_ELIM_THM] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THENL [FIRST_ASSUM(fun th -> REWRITE_TAC[GSYM(MATCH_MP LINEAR_CMUL th)]) THEN FIRST_ASSUM(fun th -> REWRITE_TAC[GSYM(MATCH_MP LINEAR_ADD th)]) THEN REWRITE_TAC[IN_IMAGE] THEN MESON_TAC[REWRITE_RULE[convex] CONVEX_CONVEX_HULL]; ASM_SIMP_TAC[LINEAR_ADD; LINEAR_CMUL] THEN MESON_TAC[REWRITE_RULE[convex] CONVEX_CONVEX_HULL]]);; add_linear_invariants [CONVEX_HULL_LINEAR_IMAGE];; let IN_CONVEX_HULL_LINEAR_IMAGE = prove (`!f:real^M->real^N s x. linear f /\ x IN convex hull s ==> (f x) IN convex hull (IMAGE f s)`, SIMP_TAC[CONVEX_HULL_LINEAR_IMAGE] THEN SET_TAC[]);; let CONIC_CONVEX_HULL = prove (`!s:real^N->bool. conic s ==> conic(convex hull s)`, SIMP_TAC[CONIC_IMAGE_MULTIPLE_EQ; GSYM CONVEX_HULL_LINEAR_IMAGE; LINEAR_SCALING; HULL_MONO]);; let CONIC_HULL_CONVEX_HULL = prove (`!s:real^N->bool. conic hull (convex hull s) = convex hull (conic hull s)`, GEN_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THEN MATCH_MP_TAC HULL_MINIMAL THEN SIMP_TAC[CONVEX_CONIC_HULL; HULL_SUBSET; CONVEX_CONVEX_HULL; HULL_MONO; CONIC_CONVEX_HULL; CONIC_CONIC_HULL]);; (* ------------------------------------------------------------------------- *) (* Convexity of general and special intervals. *) (* ------------------------------------------------------------------------- *) let IS_INTERVAL_CONVEX = prove (`!s:real^N->bool. is_interval s ==> convex s`, REWRITE_TAC[is_interval; convex] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM MATCH_MP_TAC THEN MAP_EVERY EXISTS_TAC [`x:real^N`; `y:real^N`] THEN ASM_SIMP_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN GEN_TAC THEN STRIP_TAC THEN DISJ_CASES_TAC(SPECL [`(x:real^N)$i`; `(y:real^N)$i`] REAL_LE_TOTAL) THENL [DISJ1_TAC; DISJ2_TAC] THEN MATCH_MP_TAC(REAL_ARITH `&1 * a <= b /\ b <= &1 * c ==> a <= b /\ b <= c`) THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN ASM_SIMP_TAC[GSYM VECTOR_MUL_COMPONENT; VECTOR_ADD_RDISTRIB; VECTOR_ADD_COMPONENT] THEN ASM_SIMP_TAC[VECTOR_MUL_COMPONENT; REAL_LE_LMUL; REAL_LE_LADD; REAL_LE_RADD]);; let IS_INTERVAL_CONNECTED = prove (`!s:real^N->bool. is_interval s ==> connected s`, MESON_TAC[IS_INTERVAL_CONVEX; CONVEX_CONNECTED]);; let IS_INTERVAL_CONNECTED_1 = prove (`!s:real^1->bool. is_interval s <=> connected s`, GEN_TAC THEN EQ_TAC THEN REWRITE_TAC[IS_INTERVAL_CONNECTED] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[IS_INTERVAL_1; connected; NOT_FORALL_THM; LEFT_IMP_EXISTS_THM; NOT_IMP; FORALL_LIFT; LIFT_DROP] THEN MAP_EVERY X_GEN_TAC [`a:real`; `b:real`; `x:real`] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`{z:real^1 | basis 1 dot z < x}`; `{z:real^1 | basis 1 dot z > x}`] THEN REWRITE_TAC[OPEN_HALFSPACE_LT; OPEN_HALFSPACE_GT] THEN SIMP_TAC[SUBSET; EXTENSION; IN_UNION; IN_INTER; GSYM drop; NOT_FORALL_THM; real_gt; NOT_IN_EMPTY; IN_ELIM_THM; DOT_BASIS; DIMINDEX_1; ARITH] THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[REAL_LT_TOTAL; LIFT_DROP]; REAL_ARITH_TAC; EXISTS_TAC `lift a`; EXISTS_TAC `lift b`] THEN ASM_REWRITE_TAC[REAL_LT_LE; LIFT_DROP] THEN ASM_MESON_TAC[]);; let CONVEX_INTERVAL = prove (`!a b:real^N. convex(interval [a,b]) /\ convex(interval (a,b))`, SIMP_TAC[IS_INTERVAL_CONVEX; IS_INTERVAL_INTERVAL]);; let CONNECTED_INTERVAL = prove (`(!a b:real^N. connected(interval[a,b])) /\ (!a b:real^N. connected(interval(a,b)))`, SIMP_TAC[CONVEX_CONNECTED; CONVEX_INTERVAL]);; let CONVEX_CONNECTED_COLLINEAR = prove (`!s:real^N->bool. collinear s ==> (convex s <=> connected s)`, REPEAT STRIP_TAC THEN EQ_TAC THEN REWRITE_TAC[CONVEX_CONNECTED] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [COLLINEAR_AFFINE_HULL]) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u:real^N`; `v:real^N`] THEN GEOM_ORIGIN_TAC `u:real^N` THEN SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC; IN_INSERT; SPAN_INSERT_0] THEN GEOM_BASIS_MULTIPLE_TAC 1 `v:real^N` THEN GEN_TAC THEN DISCH_TAC THEN GEN_TAC THEN REWRITE_TAC[SPAN_SPECIAL_SCALE] THEN COND_CASES_TAC THENL [REWRITE_TAC[SPAN_EMPTY; SET_RULE `s SUBSET {a} <=> s = {} \/ s = {a}`] THEN STRIP_TAC THEN ASM_REWRITE_TAC[CONVEX_EMPTY; CONVEX_SING]; DISCH_TAC THEN REWRITE_TAC[CONVEX_CONTAINS_SEGMENT; connected; NOT_EXISTS_THM] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN STRIP_TAC THEN ASM_CASES_TAC `a:real^N = b` THEN ASM_REWRITE_TAC[SEGMENT_REFL; SING_SUBSET] THEN REWRITE_TAC[SUBSET; IN_SEGMENT; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `u:real`] THEN MAP_EVERY ASM_CASES_TAC [`u = &0`; `u = &1`] THEN ASM_SIMP_TAC[VECTOR_MUL_LZERO; VECTOR_MUL_LID; REAL_SUB_REFL; REAL_SUB_RZERO; VECTOR_ADD_LID; VECTOR_ADD_RID] THEN ASM_REWRITE_TAC[REAL_LE_LT] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`{y:real^N | basis 1 dot y < basis 1 dot (x:real^N)}`; `{y:real^N | basis 1 dot y > basis 1 dot (x:real^N)}`]) THEN REWRITE_TAC[OPEN_HALFSPACE_LT; OPEN_HALFSPACE_GT] THEN MATCH_MP_TAC(TAUT `q /\ r /\ (~p ==> s) ==> ~(p /\ q /\ r) ==> s`) THEN CONJ_TAC THENL [REWRITE_TAC[EXTENSION; IN_INTER; IN_ELIM_THM; NOT_IN_EMPTY] THEN REWRITE_TAC[CONJ_ASSOC; REAL_ARITH `~(x:real < a /\ x > a)`]; ALL_TAC] THEN REWRITE_TAC[real_gt] THEN ONCE_REWRITE_TAC[GSYM REAL_SUB_LT] THEN REWRITE_TAC[GSYM DOT_RSUB; SET_RULE `~(s SUBSET {x | P x} UNION {x | Q x}) <=> ?x. x IN s /\ ~(P x \/ Q x)`] THEN SUBGOAL_THEN `!p q:real^N. p IN span {basis 1} /\ q IN span {basis 1} /\ basis 1 dot p = basis 1 dot q ==> p = q` ASSUME_TAC THENL [SIMP_TAC[SPAN_SING; IMP_CONJ; LEFT_IMP_EXISTS_THM; IN_ELIM_THM] THEN SIMP_TAC[DOT_RMUL; BASIS_NONZERO; DOT_BASIS_BASIS; DIMINDEX_GE_1; LE_REFL; REAL_MUL_RID]; ALL_TAC] THEN SUBGOAL_THEN `(x:real^N) IN span {basis 1}` ASSUME_TAC THENL [ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SPAN_ADD THEN CONJ_TAC THEN MATCH_MP_TAC SPAN_MUL THEN ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC(SET_RULE `(a:real^N) IN s \/ b IN s ==> ~(s = {})`) THEN ASM_REWRITE_TAC[IN_INTER; IN_ELIM_THM; DOT_RADD; DOT_RMUL; VECTOR_ARITH `((&1 - u) % a + u % b) - b:real^N = (u - &1) % (b - a)`; VECTOR_ARITH `((&1 - u) % a + u % b) - a:real^N = u % (b - a)`; VECTOR_ARITH `b - ((&1 - u) % a + u % b):real^N = (u - &1) % (a - b)`; VECTOR_ARITH `a - ((&1 - u) % a + u % b):real^N = u % (a - b)`] THEN MATCH_MP_TAC(REAL_ARITH `(&0 < x ==> &0 < u * x) /\ (&0 < --x ==> &0 < (&1 - u) * --x) /\ ~(x = &0) ==> &0 < u * x \/ &0 < (u - &1) * x`) THEN ASM_SIMP_TAC[REAL_LT_MUL; REAL_SUB_LT] THEN REWRITE_TAC[DOT_RSUB; REAL_SUB_0]; REWRITE_TAC[DOT_RSUB; REAL_ARITH `~(&0 < x - y \/ &0 < y - x) <=> y = x`]] THEN ASM SET_TAC[]]);; let CONVEX_EQ_CONVEX_LINE_INTERSECTION = prove (`!s:real^N->bool. convex s <=> !a b. convex(s INTER affine hull {a,b})`, GEN_TAC THEN EQ_TAC THEN SIMP_TAC[CONVEX_INTER; AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL] THEN REWRITE_TAC[CONVEX_CONTAINS_SEGMENT] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a:real^N`; `b:real^N`; `a:real^N`; `b:real^N`]) THEN ASM_SIMP_TAC[IN_INTER; HULL_INC; IN_INSERT] THEN SET_TAC[]);; let CONVEX_EQ_CONNECTED_LINE_INTERSECTION = prove (`!s:real^N->bool. convex s <=> !a b. connected(s INTER affine hull {a,b})`, GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [CONVEX_EQ_CONVEX_LINE_INTERSECTION] THEN REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN MATCH_MP_TAC CONVEX_CONNECTED_COLLINEAR THEN MATCH_MP_TAC COLLINEAR_SUBSET THEN EXISTS_TAC `affine hull {a:real^N,b}` THEN REWRITE_TAC[COLLINEAR_AFFINE_HULL_COLLINEAR; COLLINEAR_2] THEN SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* On real^1, is_interval, convex and connected are all equivalent. *) (* ------------------------------------------------------------------------- *) let IS_INTERVAL_CONVEX_1 = prove (`!s:real^1->bool. is_interval s <=> convex s`, MESON_TAC[IS_INTERVAL_CONVEX; CONVEX_CONNECTED; IS_INTERVAL_CONNECTED_1]);; let CONVEX_CONNECTED_1 = prove (`!s:real^1->bool. convex s <=> connected s`, REWRITE_TAC[GSYM IS_INTERVAL_CONVEX_1; GSYM IS_INTERVAL_CONNECTED_1]);; let CONNECTED_CONVEX_1 = prove (`!s:real^1->bool. connected s <=> convex s`, REWRITE_TAC[GSYM IS_INTERVAL_CONVEX_1; GSYM IS_INTERVAL_CONNECTED_1]);; let CONNECTED_COMPACT_INTERVAL_1 = prove (`!s:real^1->bool. connected s /\ compact s <=> ?a b. s = interval[a,b]`, REWRITE_TAC[GSYM IS_INTERVAL_CONNECTED_1; IS_INTERVAL_COMPACT]);; let CONVEX_CONNECTED_1_GEN = prove (`!s:real^N->bool. dimindex(:N) = 1 ==> (convex s <=> connected s)`, REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[GSYM DIMINDEX_1] THEN DISCH_THEN(ACCEPT_TAC o C GEOM_EQUAL_DIMENSION_RULE CONVEX_CONNECTED_1));; let CONNECTED_CONVEX_1_GEN = prove (`!s:real^N->bool. dimindex(:N) = 1 ==> (connected s <=> convex s)`, SIMP_TAC[CONVEX_CONNECTED_1_GEN]);; let COMPACT_CONVEX_COLLINEAR_SEGMENT_ALT = prove (`!s:real^N->bool. ~(s = {}) /\ compact s /\ connected s /\ collinear s ==> ?a b. s = segment[a,b]`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [COLLINEAR_AFFINE_HULL]) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`z:real^N`; `w:real^N`] THEN REPEAT(POP_ASSUM MP_TAC) THEN GEOM_ORIGIN_TAC `z:real^N` THEN REPEAT GEN_TAC THEN GEOM_BASIS_MULTIPLE_TAC 1 `w:real^N` THEN REPEAT STRIP_TAC THEN MP_TAC(ISPEC `IMAGE (\x:real^N. lift(x$1)) s` CONNECTED_COMPACT_INTERVAL_1) THEN ASM_SIMP_TAC[CONNECTED_CONTINUOUS_IMAGE; COMPACT_CONTINUOUS_IMAGE; LINEAR_LIFT_COMPONENT; LINEAR_CONTINUOUS_ON] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^1`; `b:real^1`] THEN ASM_CASES_TAC `interval[a:real^1,b] = {}` THEN ASM_REWRITE_TAC[IMAGE_EQ_EMPTY] THEN STRIP_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[INTERVAL_NE_EMPTY_1]) THEN MAP_EVERY EXISTS_TAC [`drop a % basis 1:real^N`; `drop b % basis 1:real^N`] THEN ASM_SIMP_TAC[SEGMENT_SCALAR_MULTIPLE; REAL_ARITH `a <= b ==> (a <= x /\ x <= b \/ b <= x /\ x <= a <=> a <= x /\ x <= b)`] THEN ONCE_REWRITE_TAC[MESON[LIFT_DROP] `a <= x /\ x <= b <=> a <= drop(lift x) /\ drop(lift x) <= b`] THEN REWRITE_TAC[GSYM IN_INTERVAL_1] THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN MATCH_MP_TAC(SET_RULE `(!x. lift(drop x) = x) /\ (!x. drop(lift x) = x) /\ (!x. x IN s ==> f(drop(g x)) = x) ==> s = {f y | lift y IN IMAGE g s}`) THEN REWRITE_TAC[LIFT_DROP] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET t ==> (!x. x IN t ==> P x) ==> (!x. x IN s ==> P x)`)) THEN MATCH_MP_TAC HULL_INDUCT THEN SIMP_TAC[FORALL_IN_INSERT; NOT_IN_EMPTY; VECTOR_MUL_COMPONENT; VEC_COMPONENT; DIMINDEX_GE_1; LE_REFL; VECTOR_MUL_LZERO; BASIS_COMPONENT; REAL_MUL_RID; affine; IN_ELIM_THM; VECTOR_ADD_COMPONENT; VECTOR_ADD_RDISTRIB; GSYM VECTOR_MUL_ASSOC]);; let COMPACT_CONVEX_COLLINEAR_SEGMENT = prove (`!s:real^N->bool. ~(s = {}) /\ compact s /\ convex s /\ collinear s ==> ?a b. s = segment[a,b]`, MESON_TAC[COMPACT_CONVEX_COLLINEAR_SEGMENT_ALT; CONVEX_CONNECTED_COLLINEAR]);; let IN_CONVEX_HULL_SEGMENT_1,IN_CONVEX_HULL_INTERVAL_1 = (CONJ_PAIR o prove) (`(!s:real^1->bool x. x IN convex hull s <=> ?a b. a IN s /\ b IN s /\ x IN segment[a,b]) /\ (!s:real^1->bool x. x IN convex hull s <=> ?a b. a IN s /\ b IN s /\ x IN interval[a,b])`, REWRITE_TAC[AND_FORALL_THM] THEN REPEAT GEN_TAC THEN MATCH_MP_TAC(TAUT `(r ==> q) /\ (q ==> p) /\ (p ==> r) ==> (p <=> q) /\ (p <=> r)`) THEN REPEAT CONJ_TAC THENL [MESON_TAC[SUBSET; INTERVAL_SUBSET_SEGMENT_1]; MESON_TAC[REWRITE_RULE[SUBSET] CONVEX_CONTAINS_SEGMENT; HULL_INC; CONVEX_CONVEX_HULL]; DISCH_TAC THEN REWRITE_TAC[IN_INTERVAL_1] THEN ONCE_REWRITE_TAC[TAUT `a /\ b /\ p /\ q <=> (a /\ p) /\ (b /\ q)`] THEN REWRITE_TAC[RIGHT_EXISTS_AND_THM; LEFT_EXISTS_AND_THM] THEN ONCE_REWRITE_TAC[SET_RULE `(?x. x IN s /\ P x) <=> ~(s SUBSET {x | ~P x})`] THEN CONJ_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP (MESON[HULL_MONO] `s SUBSET t ==> convex hull s SUBSET convex hull t`)) THEN REWRITE_TAC[SUBSET; REAL_NOT_LE] THEN DISCH_THEN(MP_TAC o SPEC `x:real^1`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(MESON[] `convex hull s = s /\ ~(x IN s) ==> ~(x IN convex hull s)`) THEN REWRITE_TAC[IN_ELIM_THM; REAL_LT_REFL] THEN MATCH_MP_TAC HULL_P THEN REWRITE_TAC[GSYM IS_INTERVAL_CONVEX_1; IS_INTERVAL_1; IN_ELIM_THM] THEN REAL_ARITH_TAC]);; (* ------------------------------------------------------------------------- *) (* Jung's theorem. *) (* Proof taken from http://cstheory.wordpress.com/2010/08/07/jungs-theorem/ *) (* ------------------------------------------------------------------------- *) let JUNG = prove (`!s:real^N->bool r. bounded s /\ sqrt(&(dimindex(:N)) / &(2 * dimindex(:N) + 2)) * diameter s <= r ==> ?a. s SUBSET cball(a,r)`, let lemma = prove (`&0 < x /\ x <= y ==> (x - &1) / x <= (y - &1) / y`, SIMP_TAC[REAL_LE_LDIV_EQ] THEN REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[REAL_ARITH `x / y * z:real = (x * z) / y`] THEN SUBGOAL_THEN `&0 < y` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ASM_SIMP_TAC[REAL_LE_RDIV_EQ]] THEN ASM_REAL_ARITH_TAC) in REPEAT STRIP_TAC THEN SUBGOAL_THEN `&0 <= r` ASSUME_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS)) THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_SIMP_TAC[DIAMETER_POS_LE] THEN SIMP_TAC[SQRT_POS_LE; REAL_LE_DIV; REAL_POS]; ALL_TAC] THEN MP_TAC(ISPEC `IMAGE (\x:real^N. cball(x,r)) s` HELLY_COMPACT_ALT) THEN REWRITE_TAC[FORALL_IN_IMAGE; COMPACT_CBALL; CONVEX_CBALL] THEN REWRITE_TAC[TAUT `p /\ q /\ r ==> s <=> q /\ p ==> r ==> s`] THEN REWRITE_TAC[FORALL_FINITE_SUBSET_IMAGE] THEN REWRITE_TAC[INTERS_IMAGE; GSYM MEMBER_NOT_EMPTY] THEN REWRITE_TAC[SUBSET; IN_CBALL; IN_ELIM_THM] THEN ANTS_TAC THENL [ALL_TAC; MESON_TAC[DIST_SYM]] THEN X_GEN_TAC `t:real^N->bool` THEN REWRITE_TAC[GSYM SUBSET] THEN STRIP_TAC THEN ASM_SIMP_TAC[CARD_IMAGE_INJ; EQ_BALLS; GSYM REAL_NOT_LE] THEN UNDISCH_TAC `FINITE(t:real^N->bool)` THEN SUBGOAL_THEN `bounded(t:real^N->bool)` MP_TAC THENL [ASM_MESON_TAC[BOUNDED_SUBSET]; ALL_TAC] THEN UNDISCH_TAC `&0 <= r` THEN SUBGOAL_THEN `sqrt(&(dimindex(:N)) / &(2 * dimindex(:N) + 2)) * diameter(t:real^N->bool) <= r` MP_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS)) THEN MATCH_MP_TAC REAL_LE_LMUL THEN ASM_SIMP_TAC[DIAMETER_SUBSET; SQRT_POS_LE; REAL_POS; REAL_LE_DIV]; POP_ASSUM_LIST(K ALL_TAC) THEN SPEC_TAC(`t:real^N->bool`,`s:real^N->bool`) THEN REPEAT STRIP_TAC] THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN MP_TAC(ISPEC `{d | &0 <= d /\ ?a:real^N. s SUBSET cball(a,d)}` INF) THEN ABBREV_TAC `d = inf {d | &0 <= d /\ ?a:real^N. s SUBSET cball(a,d)}` THEN REWRITE_TAC[IN_ELIM_THM] THEN ANTS_TAC THENL [REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_ELIM_THM] THEN ASM_MESON_TAC[BOUNDED_SUBSET_CBALL; REAL_LT_IMP_LE]; DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "P") (LABEL_TAC "M"))] THEN SUBGOAL_THEN `&0 <= d` ASSUME_TAC THENL [ASM_MESON_TAC[REAL_LE_REFL]; ALL_TAC] THEN SUBGOAL_THEN `?a:real^N. s SUBSET cball(a,d)` MP_TAC THENL [SUBGOAL_THEN `!n. ?a:real^N. s SUBSET cball(a,d + inv(&n + &1))` MP_TAC THENL [X_GEN_TAC `n:num` THEN REMOVE_THEN "M" (MP_TAC o SPEC `d + inv(&n + &1)`) THEN REWRITE_TAC[REAL_ARITH `d + i <= d <=> ~(&0 < i)`] THEN REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; REAL_NOT_LE] THEN MESON_TAC[SUBSET_CBALL; REAL_LT_IMP_LE; SUBSET_TRANS]; ALL_TAC] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; SKOLEM_THM] THEN X_GEN_TAC `aa:num->real^N` THEN DISCH_TAC THEN SUBGOAL_THEN `?t. compact t /\ !n. (aa:num->real^N) n IN t` MP_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPEC `vec 0:real^N` o MATCH_MP BOUNDED_SUBSET_CBALL) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; SUBSET; IN_CBALL_0] THEN X_GEN_TAC `B:real` THEN STRIP_TAC THEN EXISTS_TAC `cball(vec 0:real^N,B + d + &1)` THEN REWRITE_TAC[COMPACT_CBALL; IN_CBALL_0] THEN X_GEN_TAC `n:num` THEN RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; IN_CBALL]) THEN MATCH_MP_TAC(NORM_ARITH `(?x:real^N. norm(x) <= B /\ dist(a,x) <= d) ==> norm(a) <= B + d`) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `d + inv(&n + &1)` THEN ASM_SIMP_TAC[REAL_LE_LADD] THEN MATCH_MP_TAC REAL_INV_LE_1 THEN REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[compact; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `t:real^N->bool` THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN DISCH_THEN(MP_TAC o SPEC `aa:num->real^N`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN DISCH_THEN(X_CHOOSE_THEN `r:num->num` STRIP_ASSUME_TAC) THEN REWRITE_TAC[SUBSET; IN_CBALL] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[GSYM REAL_NOT_LT] THEN DISCH_TAC THEN MP_TAC(SPEC `(dist(a:real^N,x) - d) / &2` REAL_ARCH_INV) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIM_SEQUENTIALLY]) THEN DISCH_THEN(MP_TAC o SPEC `(dist(a:real^N,x) - d) / &2`) THEN ASM_SIMP_TAC[REAL_SUB_LT; REAL_HALF; o_THM] THEN DISCH_THEN(X_CHOOSE_THEN `N1:num` STRIP_ASSUME_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `N2:num` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE BINDER_CONV [SUBSET]) THEN DISCH_THEN(MP_TAC o SPECL [`(r:num->num)(N1 + N2)`; `x:real^N`]) THEN ASM_REWRITE_TAC[IN_CBALL; REAL_NOT_LE] THEN FIRST_X_ASSUM(MP_TAC o SPEC `N1 + N2:num`) THEN ASM_REWRITE_TAC[LE_ADD] THEN SUBGOAL_THEN `inv(&(r (N1 + N2:num)) + &1) < (dist(a:real^N,x) - d) / &2` MP_TAC THENL [ALL_TAC; NORM_ARITH_TAC] THEN MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `inv(&N2)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_INV2 THEN CONJ_TAC THENL [ASM_MESON_TAC[REAL_LT_INV_EQ]; ALL_TAC] THEN REWRITE_TAC[REAL_OF_NUM_LE; REAL_OF_NUM_ADD] THEN MATCH_MP_TAC(ARITH_RULE `N1 + N2 <= r(N1 + N2) ==> N2 <= r(N1 + N2) + 1`) THEN ASM_MESON_TAC[MONOTONE_BIGGER]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN ONCE_REWRITE_TAC[DIST_SYM] THEN REWRITE_TAC[GSYM IN_CBALL; GSYM SUBSET] THEN DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS) THEN MATCH_MP_TAC SUBSET_CBALL THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `a * s <= r ==> d <= a * s ==> d <= r`)) THEN UNDISCH_THEN `&0 <= r` (K ALL_TAC) THEN REMOVE_THEN "M" (K ALL_TAC) THEN FIRST_X_ASSUM(K ALL_TAC o SYM) THEN REMOVE_THEN "P" MP_TAC THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN ABBREV_TAC `n = CARD(s:real^N->bool)` THEN SUBGOAL_THEN `(s:real^N->bool) HAS_SIZE n` MP_TAC THENL [ASM_REWRITE_TAC[HAS_SIZE]; ALL_TAC] THEN UNDISCH_THEN `CARD(s:real^N->bool) = n` (K ALL_TAC) THEN POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN SPEC_TAC(`d:real`,`r:real`) THEN GEN_TAC THEN GEOM_ORIGIN_TAC `a:real^N` THEN SIMP_TAC[HAS_SIZE] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(SUBST_ALL_TAC o SYM) THEN ABBREV_TAC `t = {x:real^N | x IN s /\ norm(x) = r}` THEN SUBGOAL_THEN `FINITE(t:real^N->bool)` ASSUME_TAC THENL [EXPAND_TAC "t" THEN ASM_SIMP_TAC[FINITE_RESTRICT]; ALL_TAC] THEN SUBGOAL_THEN `(vec 0:real^N) IN convex hull t` MP_TAC THENL [MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THEN MP_TAC(ISPEC `convex hull t:real^N->bool` SEPARATING_HYPERPLANE_CLOSED_0) THEN ASM_SIMP_TAC[CONVEX_CONVEX_HULL; NOT_IMP; COMPACT_CONVEX_HULL; FINITE_IMP_COMPACT; COMPACT_IMP_CLOSED] THEN REWRITE_TAC[NOT_EXISTS_THM; TAUT `~(p /\ q) <=> p ==> ~q`] THEN X_GEN_TAC `v:real^N` THEN ABBREV_TAC `k = CARD(s:real^N->bool)` THEN SUBGOAL_THEN `(s:real^N->bool) HAS_SIZE k` MP_TAC THENL [ASM_REWRITE_TAC[HAS_SIZE]; ALL_TAC] THEN UNDISCH_THEN `CARD(s:real^N->bool) = k` (K ALL_TAC) THEN POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN GEOM_BASIS_MULTIPLE_TAC 1 `v:real^N` THEN X_GEN_TAC `m:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_ARITH `&0 <= x <=> x = &0 \/ &0 < x`] THEN STRIP_TAC THEN ASM_REWRITE_TAC[VECTOR_MUL_EQ_0] THEN ASM_SIMP_TAC[BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL; REAL_LT_IMP_NZ] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[HAS_SIZE] THEN DISCH_THEN(SUBST_ALL_TAC o SYM) THEN X_GEN_TAC `b:real` THEN DISCH_TAC THEN ASM_SIMP_TAC[DOT_LMUL; DOT_BASIS; DIMINDEX_GE_1; LE_REFL] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[real_gt; GSYM REAL_LT_LDIV_EQ] THEN SUBGOAL_THEN `&0 < b / m` MP_TAC THENL [ASM_SIMP_TAC[REAL_LT_DIV]; UNDISCH_THEN `&0 < b` (K ALL_TAC) THEN SPEC_TAC(`b / m:real`,`b:real`)] THEN X_GEN_TAC `b:real` THEN DISCH_TAC THEN DISCH_TAC THEN SUBGOAL_THEN `!x:real^N e. &0 < e /\ e < b /\ x IN t ==> norm(x - e % basis 1) < r` ASSUME_TAC THENL [MAP_EVERY X_GEN_TAC [`x:real^N`; `e:real`] THEN STRIP_TAC THEN SUBGOAL_THEN `r = norm(x:real^N)` SUBST1_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[NORM_LT; dot]] THEN SIMP_TAC[SUM_CLAUSES_LEFT; DIMINDEX_GE_1] THEN SIMP_TAC[VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT; BASIS_COMPONENT; DIMINDEX_GE_1; LE_REFL; ARITH_RULE `2 <= n ==> 1 <= n /\ ~(n = 1)`; ARITH] THEN REWRITE_TAC[REAL_MUL_RZERO; REAL_SUB_RZERO; REAL_LT_RADD] THEN REWRITE_TAC[GSYM REAL_POW_2; GSYM REAL_LT_SQUARE_ABS] THEN MATCH_MP_TAC(REAL_ARITH `!b. &0 < e /\ e < b /\ b < x ==> abs(x - e * &1) < abs x`) THEN EXISTS_TAC `b:real` THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[HULL_INC]; ALL_TAC] THEN SUBGOAL_THEN `?d. &0 < d /\ !x:real^N a. x IN (s DIFF t) /\ norm(a) < d ==> norm(x - a) < r` STRIP_ASSUME_TAC THENL [ASM_CASES_TAC `s DIFF t:real^N->bool = {}` THENL [ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN MESON_TAC[REAL_LT_01]; ALL_TAC] THEN EXISTS_TAC `inf (IMAGE (\x:real^N. r - norm x) (s DIFF t))` THEN SUBGOAL_THEN `FINITE(s DIFF t:real^N->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[FINITE_DIFF]; ALL_TAC] THEN ASM_SIMP_TAC[REAL_LT_INF_FINITE; FINITE_IMAGE; IMAGE_EQ_EMPTY] THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN SIMP_TAC [NORM_ARITH `norm a < r - norm x ==> norm(x - a:real^N) < r`] THEN EXPAND_TAC "t" THEN REWRITE_TAC[IN_DIFF; IN_ELIM_THM; REAL_SUB_LT] THEN RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; IN_CBALL_0]) THEN ASM_MESON_TAC[REAL_LT_LE]; ALL_TAC] THEN SUBGOAL_THEN `?a. !x. x IN s ==> norm(x - a:real^N) < r` STRIP_ASSUME_TAC THENL [EXISTS_TAC `min (b / &2) (d / &2) % basis 1:real^N` THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN ASM_CASES_TAC `(x:real^N) IN t` THENL [MATCH_MP_TAC(ASSUME `!x:real^N e. &0 < e /\ e < b /\ x IN t ==> norm (x - e % basis 1) < r`) THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC; MATCH_MP_TAC(ASSUME `!x:real^N a. x IN s DIFF t /\ norm a < d ==> norm (x - a) < r`) THEN ASM_SIMP_TAC[IN_DIFF; NORM_MUL; LE_REFL; NORM_BASIS; DIMINDEX_GE_1] THEN ASM_REAL_ARITH_TAC]; SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL [ASM_MESON_TAC[MEMBER_NOT_EMPTY; NORM_ARITH `norm(x:real^N) < r ==> &0 < r`]; ALL_TAC] THEN UNDISCH_THEN `!x a:real^N. &0 <= x /\ s SUBSET cball (a,x) ==> r <= x` (MP_TAC o SPECL [`max (&0) (r - inf (IMAGE (\x:real^N. r - norm(x - a)) s))`; `a:real^N`]) THEN ASM_SIMP_TAC[REAL_ARITH `&0 < r ==> (r <= max (&0) a <=> r <= a)`] THEN REWRITE_TAC[SUBSET; IN_CBALL; REAL_ARITH `a <= max a b`] THEN REWRITE_TAC[NOT_IMP; REAL_ARITH `~(r <= r - x) <=> &0 < x`] THEN ASM_SIMP_TAC[REAL_LT_INF_FINITE; FINITE_IMAGE; IMAGE_EQ_EMPTY] THEN ASM_REWRITE_TAC[FORALL_IN_IMAGE; REAL_SUB_LT] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MATCH_MP_TAC(REAL_ARITH `d <= b ==> d <= max a b`) THEN ONCE_REWRITE_TAC[REAL_ARITH `a <= b - c <=> c <= b - a`] THEN ASM_SIMP_TAC[REAL_INF_LE_FINITE; FINITE_IMAGE; IMAGE_EQ_EMPTY] THEN REWRITE_TAC[EXISTS_IN_IMAGE; ONCE_REWRITE_RULE[NORM_SUB] dist] THEN ASM_MESON_TAC[REAL_LE_REFL]]; ALL_TAC] THEN ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_REWRITE_TAC[CONVEX_HULL_EMPTY; NOT_IN_EMPTY] THEN REWRITE_TAC[CONVEX_HULL_FINITE; IN_ELIM_THM] THEN DISCH_THEN(X_CHOOSE_THEN `l:real^N->real` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sqrt((&(dimindex (:N)) / &(2 * dimindex (:N) + 2)) * diameter(s:real^N->bool) pow 2)` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_LE_RSQRT; ASM_SIMP_TAC[SQRT_MUL; DIAMETER_POS_LE; REAL_POW_LE; REAL_LE_DIV; REAL_POS; POW_2_SQRT; REAL_LE_REFL]] THEN SUBGOAL_THEN `sum t (\y:real^N. &2 * r pow 2) <= sum t (\y. (&1 - l y) * diameter(s:real^N->bool) pow 2)` MP_TAC THENL [MATCH_MP_TAC SUM_LE THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sum (t DELETE x) (\x:real^N. l(x)) * diameter(s:real^N->bool) pow 2` THEN CONJ_TAC THENL [ALL_TAC; ASM_SIMP_TAC[SUM_DELETE; ETA_AX; REAL_LE_REFL]] THEN REWRITE_TAC[GSYM SUM_RMUL] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sum (t DELETE x) (\y:real^N. l y * norm(y - x) pow 2)` THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC SUM_LE THEN ASM_REWRITE_TAC[FINITE_DELETE; IN_DELETE] THEN X_GEN_TAC `y:real^N` THEN STRIP_TAC THEN MATCH_MP_TAC REAL_LE_LMUL THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC REAL_POW_LE2 THEN REWRITE_TAC[NORM_POS_LE] THEN MATCH_MP_TAC DIAMETER_BOUNDED_BOUND THEN ASM SET_TAC[]] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sum t (\y:real^N. l y * norm (y - x) pow 2)` THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC REAL_EQ_IMP_LE THEN MATCH_MP_TAC SUM_EQ_SUPERSET THEN ASM_REWRITE_TAC[FINITE_DELETE] THEN CONJ_TAC THENL [SET_TAC[]; REWRITE_TAC[IN_DELETE]] THEN SIMP_TAC[TAUT `p /\ ~(p /\ ~q) <=> p /\ q`] THEN REWRITE_TAC[VECTOR_SUB_REFL; NORM_0] THEN REAL_ARITH_TAC] THEN REWRITE_TAC[NORM_POW_2; VECTOR_ARITH `(y - x:real^N) dot (y - x) = (x dot x + y dot y) - &2 * x dot y`] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sum t (\y:real^N. l y * (&2 * r pow 2 - &2 * (x dot y)))` THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC REAL_EQ_IMP_LE THEN MATCH_MP_TAC SUM_EQ THEN UNDISCH_TAC `(x:real^N) IN t` THEN EXPAND_TAC "t" THEN REWRITE_TAC[IN_DELETE; IN_ELIM_THM] THEN SIMP_TAC[NORM_EQ_SQUARE; NORM_POW_2] THEN REAL_ARITH_TAC] THEN REWRITE_TAC[REAL_ARITH `x * (&2 * y - &2 * z) = &2 * (x * y - x * z)`] THEN REWRITE_TAC[SUM_LMUL] THEN MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[REAL_POS] THEN ASM_SIMP_TAC[SUM_SUB; FINITE_DELETE; SUM_RMUL] THEN REWRITE_TAC[GSYM DOT_RMUL] THEN ASM_SIMP_TAC[GSYM DOT_RSUM; DOT_RZERO] THEN REAL_ARITH_TAC; ASM_SIMP_TAC[SUM_CONST; SUM_RMUL; SUM_SUB] THEN REWRITE_TAC[REAL_OF_NUM_MUL; MULT_CLAUSES] THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [REAL_MUL_SYM] THEN SUBGOAL_THEN `&0 < &(CARD(t:real^N->bool) * 2)` ASSUME_TAC THENL [REWRITE_TAC[REAL_OF_NUM_LT; ARITH_RULE `0 < n * 2 <=> ~(n = 0)`] THEN ASM_SIMP_TAC[CARD_EQ_0]; ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN REWRITE_TAC[REAL_ARITH `(a * b) / c:real = a / c * b`] THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[REAL_LE_POW_2] THEN REWRITE_TAC[ARITH_RULE `2 * n + 2 = (n + 1) * 2`; GSYM REAL_OF_NUM_MUL; real_div; REAL_INV_MUL; REAL_MUL_ASSOC] THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[GSYM real_div] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN SUBGOAL_THEN `&(dimindex(:N)) = &(dimindex(:N) + 1) - &1` SUBST1_TAC THENL [REWRITE_TAC[GSYM REAL_OF_NUM_ADD] THEN REAL_ARITH_TAC; MATCH_MP_TAC lemma THEN ASM_SIMP_TAC[REAL_OF_NUM_LE; REAL_OF_NUM_LT; CARD_EQ_0; ARITH_RULE `0 < n <=> ~(n = 0)`] THEN MATCH_MP_TAC LE_TRANS THEN EXISTS_TAC `CARD(s:real^N->bool)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CARD_SUBSET THEN ASM SET_TAC[]]]]);; (* ------------------------------------------------------------------------- *) (* Kirszbraun's theorem (proof from Federer's "Geometric Measure Theory") *) (* ------------------------------------------------------------------------- *) let KIRSZBRAUN = prove (`!f:real^M->real^N s B. &0 <= B /\ (!x y. x IN s /\ y IN s ==> norm(f x - f y) <= B * norm(x - y)) ==> ?g. (!x y. norm(g x - g y) <= B * norm(x - y)) /\ (!x. x IN s ==> g x = f x)`, let lemma1 = prove (`!p Y c. compact p /\ ~(p = {}) /\ p SUBSET (:real^N) PCROSS {r | &0 < drop r} /\ (\t. {y | !a r. pastecart a r IN p ==> norm(y - a) <= drop r * t}) = Y /\ inf {t | &0 <= t /\ ~(Y t = {})} = c ==> ?b. b IN Y c /\ b IN convex hull {a | ?r. pastecart a r IN p /\ norm(b - a) = drop r * c}`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `!t t'. t <= t' ==> (Y:real->real^N->bool) t SUBSET Y t'` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN EXPAND_TAC "Y" THEN REWRITE_TAC[IN_ELIM_THM] THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN GEN_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `p SUBSET s ==> (!a r. pastecart a r IN s /\ P a r ==> Q a r) ==> (!a r. pastecart a r IN p ==> P a r) ==> (!a r. pastecart a r IN p ==> Q a r)`)) THEN REPEAT GEN_TAC THEN REWRITE_TAC[PASTECART_IN_PCROSS] THEN REWRITE_TAC[IN_UNIV; IN_ELIM_THM; LIFT_DROP] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[GSYM REAL_LE_LDIV_EQ] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `~({t | &0 <= t /\ ~(Y t:real^N->bool = {})} = {})` ASSUME_TAC THENL [SUBGOAL_THEN `bounded (IMAGE (\z. lift(norm(fstcart z:real^N) / drop(sndcart z))) p)` MP_TAC THENL [MATCH_MP_TAC COMPACT_IMP_BOUNDED THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[real_div; LIFT_CMUL] THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN SIMP_TAC[o_DEF; CONTINUOUS_ON_LIFT_NORM_COMPOSE; LINEAR_CONTINUOUS_ON; LINEAR_FSTCART] THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN SIMP_TAC[LIFT_DROP; ETA_AX; LINEAR_CONTINUOUS_ON; LINEAR_SNDCART] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET t ==> (!x. x IN t ==> P x) ==> (!x. x IN s ==> P x)`)) THEN REWRITE_TAC[FORALL_IN_PCROSS; IN_ELIM_THM; SNDCART_PASTECART] THEN REAL_ARITH_TAC; REWRITE_TAC[BOUNDED_POS; LEFT_IMP_EXISTS_THM; FORALL_IN_IMAGE] THEN X_GEN_TAC `B:real` THEN REWRITE_TAC[FORALL_PASTECART; FSTCART_PASTECART; SNDCART_PASTECART] THEN STRIP_TAC THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXPAND_TAC "Y" THEN REWRITE_TAC[IN_ELIM_THM] THEN EXISTS_TAC `B:real` THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN EXISTS_TAC `vec 0:real^N` THEN REWRITE_TAC[NORM_ARITH `norm(vec 0 - x:real^N) = norm x`] THEN REWRITE_TAC[FORALL_LIFT; LIFT_DROP] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `r:real`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a:real^N`; `lift r`]) THEN FIRST_X_ASSUM(MP_TAC o SPEC `pastecart (a:real^N) (lift r)` o REWRITE_RULE[SUBSET]) THEN ASM_REWRITE_TAC[PASTECART_IN_PCROSS; IN_ELIM_THM; NORM_LIFT] THEN SIMP_TAC[REAL_ABS_DIV; REAL_ABS_NORM; LIFT_DROP] THEN SIMP_TAC[REAL_ARITH `&0 < r ==> abs r = r`; REAL_LE_LDIV_EQ] THEN REAL_ARITH_TAC]; ALL_TAC] THEN MP_TAC(ISPEC `IMAGE (Y:real->real^N->bool) {t | c < t}` COMPACT_CHAIN) THEN REWRITE_TAC[FORALL_IN_IMAGE_2] THEN SIMP_TAC[FORALL_IN_IMAGE; IN_ELIM_THM] THEN ANTS_TAC THENL [CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[REAL_LE_TOTAL]] THEN X_GEN_TAC `t:real` THEN REPEAT STRIP_TAC THENL [SUBGOAL_THEN `Y t = INTERS(IMAGE (\z.cball(fstcart z:real^N,drop(sndcart z) * t)) p)` SUBST1_TAC THENL [EXPAND_TAC "Y" THEN REWRITE_TAC[INTERS_IMAGE] THEN GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[IN_ELIM_THM; FORALL_PASTECART; IN_CBALL] THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; dist] THEN MESON_TAC[NORM_SUB]; MATCH_MP_TAC COMPACT_INTERS THEN ASM_REWRITE_TAC[IMAGE_EQ_EMPTY; FORALL_IN_IMAGE; COMPACT_CBALL]]; UNDISCH_TAC `c:real < t` THEN REWRITE_TAC[REAL_NOT_LT] THEN EXPAND_TAC "c" THEN MATCH_MP_TAC REAL_LE_INF THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN ASM_MESON_TAC[REAL_LE_TOTAL; SUBSET_EMPTY]]; ALL_TAC] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; INTERS_IMAGE; IN_ELIM_THM] THEN REWRITE_TAC[MEMBER_NOT_EMPTY] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `b:real^N` THEN DISCH_TAC THEN SUBGOAL_THEN `b IN (Y:real->real^N->bool) c` ASSUME_TAC THENL [SUBGOAL_THEN `(Y:real->real^N->bool) c = INTERS (IMAGE Y {t | c < t})` SUBST1_TAC THENL [ALL_TAC; REWRITE_TAC[INTERS_IMAGE] THEN ASM SET_TAC[]] THEN REWRITE_TAC[INTERS_IMAGE] THEN MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN CONJ_TAC THENL [ASM_MESON_TAC[SUBSET; REAL_LT_IMP_LE]; ALL_TAC] THEN X_GEN_TAC `y:real^N` THEN EXPAND_TAC "Y" THEN REWRITE_TAC[IN_ELIM_THM] THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[MESON[] `(!x. P x ==> !a r. Q a r ==> R a r x) <=> (!a r. Q a r ==> !x. P x ==> R a r x)`] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `a:real^N` THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `r:real^1` THEN DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN FIRST_X_ASSUM(MP_TAC o SPEC `pastecart (a:real^N) (r:real^1)` o GEN_REWRITE_RULE I [SUBSET]) THEN ASM_REWRITE_TAC[PASTECART_IN_PCROSS; IN_UNIV; IN_ELIM_THM] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN SIMP_TAC[GSYM REAL_LE_LDIV_EQ] THEN DISCH_TAC THEN GEN_REWRITE_TAC I [GSYM CONTRAPOS_THM] THEN REWRITE_TAC[REAL_NOT_LE; NOT_IMP; NOT_FORALL_THM] THEN MESON_TAC[REAL_ARITH `a < b ==> a < (a + b) / &2 /\ (a + b) / &2 < b`]; ASM_REWRITE_TAC[]] THEN SUBGOAL_THEN `!t. t < c ==> (Y:real->real^N->bool) t = {}` ASSUME_TAC THENL [GEN_TAC THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN REWRITE_TAC[REAL_NOT_LT] THEN ASM_CASES_TAC `&0 <= t` THENL [EXPAND_TAC "c" THEN MATCH_MP_TAC INF_LE_ELEMENT THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN MESON_TAC[]; UNDISCH_TAC `~((Y:real->real^N->bool) t = {})` THEN REWRITE_TAC[GSYM REAL_NOT_LT; CONTRAPOS_THM] THEN DISCH_TAC THEN EXPAND_TAC "Y" THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY] THEN X_GEN_TAC `x:real^N` THEN SUBGOAL_THEN `?a:real^N r:real^1. pastecart a r IN p` MP_TAC THENL [ASM_REWRITE_TAC[GSYM EXISTS_PASTECART; MEMBER_NOT_EMPTY]; REWRITE_TAC[NOT_FORALL_THM; NOT_IMP]] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `r:real^1` THEN STRIP_TAC THEN ASM_REWRITE_TAC[REAL_NOT_LE] THEN MATCH_MP_TAC(REAL_ARITH `&0 < --x /\ &0 <= y ==> x < y`) THEN REWRITE_TAC[NORM_POS_LE; GSYM REAL_MUL_RNEG] THEN MATCH_MP_TAC REAL_LT_MUL THEN FIRST_X_ASSUM(MP_TAC o SPEC `pastecart (a:real^N) (r:real^1)` o GEN_REWRITE_RULE I [SUBSET]) THEN ASM_REWRITE_TAC[PASTECART_IN_PCROSS; IN_UNIV; IN_ELIM_THM] THEN ASM_REAL_ARITH_TAC]; ALL_TAC] THEN SUBGOAL_THEN `&0 <= c` ASSUME_TAC THENL [EXPAND_TAC "c" THEN MATCH_MP_TAC REAL_LE_INF THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(Y:real->real^N->bool) c = {b}` ASSUME_TAC THENL [MATCH_MP_TAC(SET_RULE `b IN s /\ (!y z. y IN s /\ z IN s ==> y = z) ==> s = {b}`) THEN ASM_REWRITE_TAC[] THEN MAP_EVERY X_GEN_TAC [`y:real^N`; `z:real^N`] THEN EXPAND_TAC "Y" THEN REWRITE_TAC[IN_ELIM_THM] THEN ASM_CASES_TAC `c = &0` THENL [ASM_REWRITE_TAC[REAL_MUL_RZERO; NORM_ARITH `norm(x - y:real^N) <= &0 <=> x = y`] THEN SUBGOAL_THEN `?a:real^N r:real^1. pastecart a r IN p` MP_TAC THENL [ASM_REWRITE_TAC[GSYM EXISTS_PASTECART; MEMBER_NOT_EMPTY]; SET_TAC[]]; ALL_TAC] THEN SUBGOAL_THEN `&0 < c` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN STRIP_TAC THEN SUBGOAL_THEN `bounded(IMAGE sndcart (p:real^(N,1)finite_sum->bool))` MP_TAC THENL [MATCH_MP_TAC COMPACT_IMP_BOUNDED THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_SNDCART]; REWRITE_TAC[BOUNDED_POS; FORALL_IN_IMAGE; LEFT_IMP_EXISTS_THM]] THEN X_GEN_TAC `B:real` THEN REWRITE_TAC[FORALL_PASTECART; SNDCART_PASTECART] THEN STRIP_TAC THEN MP_TAC(ASSUME `!t. t < c ==> (Y:real->real^N->bool) t = {}`) THEN REWRITE_TAC[NOT_FORALL_THM; GSYM MEMBER_NOT_EMPTY; NOT_IMP] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN REWRITE_TAC[NOT_FORALL_THM; GSYM MEMBER_NOT_EMPTY; NOT_IMP] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN EXISTS_TAC `midpoint(y:real^N,z)` THEN EXPAND_TAC "Y" THEN EXISTS_TAC `sqrt(c pow 2 - dist(y:real^N,z) pow 2 / B pow 2 / &4)` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_LT_LSQRT THEN ASM_REWRITE_TAC[REAL_ARITH `x - a < x <=> &0 < a`] THEN REPEAT(MATCH_MP_TAC REAL_LT_DIV THEN CONJ_TAC) THEN ASM_SIMP_TAC[REAL_POW_LT; GSYM DIST_NZ] THEN REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[IN_ELIM_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `r:real^1`] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `pastecart (a:real^N) (r:real^1)` o REWRITE_RULE[SUBSET]) THEN ASM_REWRITE_TAC[PASTECART_IN_PCROSS; IN_ELIM_THM; NORM_LIFT] THEN SIMP_TAC[REAL_ABS_DIV; REAL_ABS_NORM; LIFT_DROP; IN_UNIV] THEN DISCH_TAC THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[GSYM REAL_LE_LDIV_EQ] THEN MATCH_MP_TAC REAL_LE_RSQRT THEN REWRITE_TAC[REAL_POW_DIV] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_POW_LT] THEN TRANS_TAC REAL_LE_TRANS `(norm(y - a) pow 2 + norm(z - a) pow 2) / &2 - norm(y - z:real^N) pow 2 / &4` THEN CONJ_TAC THENL [REWRITE_TAC[NORM_POW_2; midpoint] THEN REWRITE_TAC[DOT_LSUB; DOT_RSUB; DOT_LADD; DOT_RADD; DOT_LMUL; DOT_RMUL] THEN REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC(REAL_ARITH `x <= a * r /\ y <= a * r /\ r * w <= z ==> (x + y) / &2 - z <= (a - w) * r`) THEN ASM_REWRITE_TAC[GSYM REAL_POW_MUL; GSYM REAL_LE_SQUARE_ABS] THEN REWRITE_TAC[REAL_ABS_MUL; REAL_ABS_NORM] THEN ASM_SIMP_TAC[real_abs; REAL_LT_IMP_LE] THEN REPEAT(CONJ_TAC THENL [ASM_MESON_TAC[REAL_MUL_SYM]; ALL_TAC]) THEN REWRITE_TAC[dist; REAL_ARITH `r * d / b / &4 <= d / &4 <=> d * (r / b) <= d * &1`] THEN MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[REAL_LE_POW_2] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_POW_LT; REAL_MUL_LID] THEN REWRITE_TAC[GSYM REAL_LE_SQUARE_ABS] THEN REWRITE_TAC[GSYM NORM_1] THEN ASM_SIMP_TAC[real_abs; REAL_LT_IMP_LE] THEN ASM_MESON_TAC[]; ALL_TAC] THEN GEN_REWRITE_TAC I [SET_RULE `x IN s <=> ~(DISJOINT {x} s)`] THEN DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] (REWRITE_RULE[CONJ_ASSOC] SEPARATING_HYPERPLANE_COMPACT_CLOSED_NONZERO))) THEN REWRITE_TAC[CONVEX_SING; NOT_INSERT_EMPTY; COMPACT_SING] THEN REWRITE_TAC[CONVEX_CONVEX_HULL; FORALL_IN_INSERT; NOT_IN_EMPTY] THEN REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [MATCH_MP_TAC COMPACT_IMP_CLOSED THEN MATCH_MP_TAC COMPACT_CONVEX_HULL THEN SUBGOAL_THEN `{a | ?r. pastecart a r IN p /\ norm(b - a) = drop r * c} = IMAGE fstcart {z | z IN p /\ lift(norm(b - fstcart z:real^N) - drop(sndcart z) * c) IN {vec 0}}` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; FORALL_PASTECART; IN_ELIM_THM; IN_IMAGE; EXISTS_PASTECART; FSTCART_PASTECART; SNDCART_PASTECART] THEN REWRITE_TAC[IN_SING; GSYM DROP_EQ; DROP_VEC; LIFT_DROP] THEN REWRITE_TAC[REAL_SUB_0] THEN MESON_TAC[]; MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_FSTCART] THEN MATCH_MP_TAC PROPER_MAP_FROM_COMPACT THEN EXISTS_TAC `(:real^1)` THEN ASM_REWRITE_TAC[SUBTOPOLOGY_UNIV; GSYM CLOSED_IN; CLOSED_SING] THEN REWRITE_TAC[SUBSET_UNIV; LIFT_SUB; LIFT_CMUL] THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN SIMP_TAC[CONTINUOUS_ON_MUL; CONTINUOUS_ON_CONST; o_DEF; LIFT_DROP; LINEAR_CONTINUOUS_ON; LINEAR_SNDCART; ETA_AX] THEN MATCH_MP_TAC CONTINUOUS_ON_LIFT_NORM_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; LINEAR_CONTINUOUS_ON; LINEAR_FSTCART]]; ALL_TAC] THEN REWRITE_TAC[NOT_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u:real^N`; `k:real`] THEN DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; real_gt; REAL_NOT_LT] THEN MP_TAC(ISPEC `{ p INTER {z | abs(norm(b - fstcart z:real^N) - drop(sndcart z) * c) <= e} INTER {z | u dot (fstcart z) - k <= e} | &0 < e}` COMPACT_CHAIN) THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM; IMP_CONJ; FORALL_IN_GSPEC] THEN REWRITE_TAC[IMP_IMP; RIGHT_IMP_FORALL_THM] THEN ANTS_TAC THENL [ALL_TAC; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; EXISTS_PASTECART] THEN REWRITE_TAC[INTERS_GSPEC; IN_ELIM_THM] THEN REWRITE_TAC[IN_INTER; IN_ELIM_PASTECART_THM] THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [TAUT `p ==> q /\ r <=> (p ==> q) /\ (p ==> r)`] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; FORALL_AND_THM] THEN X_GEN_TAC `r:real^1` THEN REWRITE_TAC[MESON[REAL_LT_IMP_LE; REAL_LE_TRANS; REAL_ARITH `~(x <= &0) ==> &0 < x / &2 /\ ~(x <= x / &2)`] `(!e. &0 < e ==> x <= e) <=> x <= &0`] THEN REWRITE_TAC[REAL_ARITH `abs(x - y) <= &0 <=> x = y`] THEN REWRITE_TAC[REAL_ARITH `x - y <= &0 <=> x <= y`] THEN DISCH_THEN(CONJUNCTS_THEN2 (MP_TAC o SPEC `&1`) STRIP_ASSUME_TAC) THEN ASM_REWRITE_TAC[REAL_LT_01] THEN DISCH_TAC THEN MATCH_MP_TAC HULL_INC THEN REWRITE_TAC[IN_ELIM_THM] THEN EXISTS_TAC `r:real^1` THEN ASM_REWRITE_TAC[]] THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC REAL_WLOG_LE THEN CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN REPEAT STRIP_TAC THEN DISJ1_TAC THEN REWRITE_TAC[SUBSET; IN_INTER; IN_ELIM_THM] THEN ASM_MESON_TAC[REAL_LT_IMP_LE; REAL_LE_TRANS]] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN CONJ_TAC THENL [MATCH_MP_TAC COMPACT_INTER_CLOSED THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CLOSED_INTER THEN CONJ_TAC THENL [REWRITE_TAC[GSYM NORM_LIFT; GSYM IN_CBALL_0] THEN MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE_UNIV THEN REWRITE_TAC[CLOSED_CBALL; LIFT_SUB; LIFT_CMUL] THEN GEN_TAC THEN MATCH_MP_TAC CONTINUOUS_SUB THEN SIMP_TAC[CONTINUOUS_MUL; CONTINUOUS_CONST; o_DEF; LIFT_DROP; LINEAR_CONTINUOUS_AT; LINEAR_SNDCART; ETA_AX] THEN MATCH_MP_TAC CONTINUOUS_LIFT_NORM_COMPOSE THEN SIMP_TAC[CONTINUOUS_SUB; CONTINUOUS_CONST; LINEAR_CONTINUOUS_AT; LINEAR_FSTCART]; ONCE_REWRITE_TAC[MESON[LIFT_DROP] `a - k = drop(lift(a - k))`] THEN ONCE_REWRITE_TAC[SET_RULE `drop x <= a <=> x IN {x | drop x <= a}`] THEN MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE_UNIV THEN REWRITE_TAC[drop; LIFT_SUB; CLOSED_HALFSPACE_COMPONENT_LE] THEN GEN_TAC THEN MATCH_MP_TAC CONTINUOUS_SUB THEN REWRITE_TAC[CONTINUOUS_CONST] THEN MATCH_MP_TAC CONTINUOUS_LIFT_DOT2 THEN SIMP_TAC[CONTINUOUS_CONST; LINEAR_CONTINUOUS_AT; LINEAR_FSTCART]]; ALL_TAC] THEN SUBGOAL_THEN `?ee. &0 < ee /\ ee < e / norm(u:real^N) /\ ee < e / norm(u) pow 2` STRIP_ASSUME_TAC THENL [REWRITE_TAC[GSYM REAL_LT_MIN] THEN MATCH_MP_TAC(MESON[REAL_ARITH `&0 < y ==> &0 < y / &2 /\ y / &2 < y`] `&0 < y ==> ?x. &0 < x /\ x < y`) THEN ASM_SIMP_TAC[REAL_LT_MIN; REAL_POW_LT; REAL_LT_DIV; NORM_POS_LT]; ALL_TAC] THEN SUBGOAL_THEN `~((b + ee % u:real^N) IN Y(c:real))` MP_TAC THENL [ASM_REWRITE_TAC[IN_SING; VECTOR_ARITH `b + e:real^N = b <=> e = vec 0`; VECTOR_MUL_EQ_0] THEN ASM_REAL_ARITH_TAC; EXPAND_TAC "Y" THEN REWRITE_TAC[IN_ELIM_THM]] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER; IN_ELIM_THM] THEN REWRITE_TAC[EXISTS_PASTECART; FSTCART_PASTECART; SNDCART_PASTECART] THEN REWRITE_TAC[NOT_FORALL_THM; IMP_CONJ] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `r:real^1` THEN REWRITE_TAC[NOT_IMP] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(MP_TAC o MATCH_MP (SET_RULE `s = {a} ==> a IN s`)) THEN EXPAND_TAC "Y" THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(MP_TAC o SPECL [`a:real^N`; `r:real^1`]) THEN ASM_REWRITE_TAC[IMP_IMP] THEN DISCH_TAC THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `d < k ==> &0 < e /\ a <= b /\ u - d <= e /\ b <= a + e ==> abs(a - b) <= e /\ u - k <= e`)) THEN ASM_REWRITE_TAC[GSYM DOT_RSUB] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `b <= rc /\ ~(b' <= rc) ==> b' - b <= e /\ u <= v ==> u <= v /\ rc <= b + e`)) THEN REWRITE_TAC[VECTOR_ARITH `(b + e % u) - a:real^N = (b - a) + e % u`] THEN MATCH_MP_TAC(NORM_ARITH `norm(y) <= e /\ u <= v ==> norm(x + y:real^N) - norm(x) <= e /\ u <= v`) THEN REWRITE_TAC[NORM_MUL] THEN CONJ_TAC THENL [ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ; NORM_POS_LT] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REAL_ARITH `n <= rc /\ ~(m <= rc) ==> n < m`)) THEN REWRITE_TAC[NORM_LT; VECTOR_ARITH `(b + e % u) - a:real^N = (b - a) + e % u`] THEN ONCE_REWRITE_TAC[VECTOR_ARITH `(a + b:real^N) dot (a + b) = a dot a + b dot b + &2 * b dot a`] THEN REWRITE_TAC[REAL_ARITH `a < a + x + y <=> --y < x`] THEN REWRITE_TAC[GSYM REAL_MUL_RNEG; GSYM DOT_RNEG; VECTOR_NEG_SUB] THEN REWRITE_TAC[DOT_LMUL] THEN REWRITE_TAC[DOT_RMUL] THEN REWRITE_TAC[REAL_ARITH `&2 * x < e * e * p <=> x < e * e * p / &2`] THEN ASM_SIMP_TAC[REAL_LT_LMUL_EQ] THEN MATCH_MP_TAC(REAL_ARITH `b <= c ==> a < b ==> a <= c`) THEN MATCH_MP_TAC(REAL_ARITH `a * b < c /\ &0 < c ==> a * b / &2 <= c`) THEN ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; GSYM NORM_POW_2; NORM_POS_LT; REAL_POW_LT]) in let lemma2 = prove (`!f:real^M->real^N s. (!x y. x IN s /\ y IN s ==> norm(f x - f y) <= norm(x - y)) ==> ?g. (!x y. norm(g x - g y) <= norm(x - y)) /\ (!x. x IN s ==> g x = f x)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?g:real^M->real^N t. s SUBSET t /\ (!x. x IN s ==> g x = f x) /\ (!x y. x IN t /\ y IN t ==> norm(g x - g y) <= norm(x - y)) /\ !h u. t SUBSET u /\ (!x. x IN t ==> h x = g x) /\ (!x y. x IN u /\ y IN u ==> norm(h x - h y) <= norm(x - y)) ==> u = t` MP_TAC THENL [MP_TAC(ISPEC `\r. (!x y x' y'. r(x,y) /\ r(x',y') ==> norm(y' - y) <= norm(x' - x)) /\ (!x. x IN s ==> r(x,(f:real^M->real^N) x))` ZL_SUBSETS_UNIONS_NONEMPTY) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [CONJ_TAC THENL [EXISTS_TAC `\(x,y). x IN s /\ (f:real^M->real^N) x = y` THEN SIMP_TAC[] THEN ASM_MESON_TAC[]; REWRITE_TAC[UNIONS; IN_ELIM_THM; SUBSET; FORALL_PAIR_THM] THEN SET_TAC[]]; DISCH_THEN(X_CHOOSE_THEN `r:real^M#real^N->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `!x:real^M y z:real^N. r(x,y) /\ r(x,z) ==> y = z` ASSUME_TAC THENL [ASM_MESON_TAC[NORM_ARITH `norm(x - y:real^N) <= norm(z - z:real^M) ==> x = y`]; ALL_TAC] THEN EXISTS_TAC `\x:real^M. @y:real^N. r(x,y)` THEN EXISTS_TAC `IMAGE FST (r:real^M#real^N->bool)` THEN REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[SUBSET; IN_IMAGE; EXISTS_PAIR_THM] THEN ASM SET_TAC[]; ASM_MESON_TAC[]; REWRITE_TAC[FORALL_IN_IMAGE_2; FORALL_PAIR_THM] THEN ASM SET_TAC[]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_PAIR_THM] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `{(x,(h:real^M->real^N) x) | x IN u}`) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [ALL_TAC; DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[EXTENSION] THEN REWRITE_TAC[EXISTS_PAIR_THM; IN_ELIM_PAIR_THM; IN_IMAGE] THEN REWRITE_TAC[IN_ELIM_THM; PAIR_EQ] THEN ASM SET_TAC[]] THEN REWRITE_TAC[IN_ELIM_THM; PAIR_EQ; SUBSET] THEN REWRITE_TAC[FORALL_PAIR_THM] THEN ASM SET_TAC[]]]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^M->real^N` THEN DISCH_THEN(X_CHOOSE_THEN `t:real^M->bool` STRIP_ASSUME_TAC)] THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `t = (:real^M)` THENL [ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `~(t = UNIV) ==> (!y. ~(y IN t) ==> F) ==> P`)) THEN X_GEN_TAC `y:real^M` THEN DISCH_TAC THEN SUBGOAL_THEN `?z. !x. x IN t ==> norm(z - (g:real^M->real^N) x) <= norm(y - x)` STRIP_ASSUME_TAC THENL [ALL_TAC; FIRST_X_ASSUM(MP_TAC o SPECL [`\x. if x = y then z else (g:real^M->real^N) x`; `(y:real^M) INSERT t`]) THEN ASM_REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; NOT_IMP] THEN REWRITE_TAC[FORALL_IN_INSERT; GSYM CONJ_ASSOC] THEN REWRITE_TAC[NORM_0; VECTOR_SUB_REFL; REAL_LE_REFL] THEN REPLICATE_TAC 3 (CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REPEAT(GEN_TAC THEN STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN TRY CONJ_TAC) THEN ASM_REWRITE_TAC[VECTOR_SUB_REFL; NORM_0; NORM_POS_LE; REAL_LE_REFL] THEN ASM_MESON_TAC[NORM_SUB]] THEN MP_TAC(ISPEC `IMAGE (\x. cball((g:real^M->real^N) x,norm(x - y))) t` COMPACT_FIP) THEN REWRITE_TAC[FORALL_IN_IMAGE; COMPACT_CBALL] THEN ANTS_TAC THENL [REWRITE_TAC[FORALL_FINITE_SUBSET_IMAGE]; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; INTERS_IMAGE] THEN REWRITE_TAC[IN_ELIM_THM; IN_CBALL; dist] THEN MESON_TAC[NORM_SUB]] THEN X_GEN_TAC `c:real^M->bool` THEN STRIP_TAC THEN ASM_CASES_TAC `c:real^M->bool = {}` THEN ASM_REWRITE_TAC[IMAGE_CLAUSES; INTERS_0; UNIV_NOT_EMPTY] THEN MP_TAC(SPEC `IMAGE (\x. pastecart ((g:real^M->real^N) x) (lift(norm(x - y)))) c` lemma1) THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN ANTS_TAC THENL [ASM_SIMP_TAC[FINITE_IMP_COMPACT; FINITE_IMAGE]; ALL_TAC] THEN ASM_REWRITE_TAC[IMAGE_EQ_EMPTY] THEN ANTS_TAC THENL [REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_PASTECART; PASTECART_IN_PCROSS; IN_UNIV; IN_ELIM_THM; LIFT_DROP; NORM_POS_LT; VECTOR_SUB_EQ] THEN ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(MP_TAC o MATCH_MP (MESON[] `(!y. a = y ==> p y) ==> (?y. a = y /\ p y)`)) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `Y:real->real^N->bool` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(MP_TAC o MATCH_MP (MESON[] `(!y. a = y ==> p y) ==> (?y. a = y /\ p y)`)) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `q:real` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; INTERS_IMAGE; IN_INTER] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `b:real^N` THEN STRIP_TAC THEN SUBGOAL_THEN `q <= &1` ASSUME_TAC THENL [ALL_TAC; UNDISCH_TAC `b IN (Y:real->real^N->bool) q` THEN EXPAND_TAC "Y" THEN REWRITE_TAC[IN_ELIM_THM; IN_IMAGE; LEFT_IMP_EXISTS_THM] THEN ONCE_REWRITE_TAC[MESON[] `(!a r x. P a r x) <=> (!x a r. P a r x)`] THEN REWRITE_TAC[PASTECART_INJ; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[FORALL_UNWIND_THM2; IN_CBALL; LIFT_DROP] THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] dist] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN GEN_REWRITE_TAC RAND_CONV [GSYM REAL_MUL_RID] THEN MATCH_MP_TAC REAL_LE_LMUL THEN ASM_REWRITE_TAC[NORM_POS_LE]] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN]) THEN REWRITE_TAC[IN_ELIM_THM; IN_IMAGE; PASTECART_INJ] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN REWRITE_TAC[GSYM CONJ_ASSOC; LEFT_AND_EXISTS_THM] THEN ONCE_REWRITE_TAC[MESON[] `(?r x. P x /\ r = f x /\ Q r x) <=> (?x. P x /\ Q (f x) x)`] THEN REWRITE_TAC[SET_RULE `{a | ?x. a = g x /\ x IN c /\ P a x} = IMAGE g {x | x IN c /\ P (g x) x}`] THEN REWRITE_TAC[LIFT_DROP] THEN REWRITE_TAC[CONVEX_HULL_IMAGE_LT; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`d:real^M->bool`; `l:real^M->real`] THEN STRIP_TAC THEN SUBGOAL_THEN `!a:real^M. a IN d ==> &0 <= l a` ASSUME_TAC THENL [ASM_MESON_TAC[REAL_LT_IMP_LE]; ALL_TAC] THEN MATCH_MP_TAC(REAL_ARITH `abs x <= abs(&1) ==> x <= &1`) THEN REWRITE_TAC[REAL_LE_SQUARE_ABS] THEN REWRITE_TAC[REAL_POW_ONE] THEN SUBGOAL_THEN `&2 * q pow 2 * norm(vsum d (\x:real^M. l x % (x - y))) pow 2 + (q pow 2 - &1) * sum d (\x. sum d (\z. l x * l z * norm(x - z) pow 2)) <= &0` MP_TAC THENL [ALL_TAC; ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[REAL_NOT_LE] THEN DISCH_TAC THEN MATCH_MP_TAC(REAL_ARITH `&0 <= x /\ &0 <= y /\ ~(x = &0 /\ y = &0) ==> &0 < x + y`) THEN REPEAT CONJ_TAC THENL [REPEAT(MATCH_MP_TAC REAL_LE_MUL THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]) THEN REWRITE_TAC[REAL_LE_POW_2]; MATCH_MP_TAC REAL_LE_MUL THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; MATCH_MP_TAC SUM_POS_LE] THEN REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC SUM_POS_LE THEN REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN REPEAT(MATCH_MP_TAC REAL_LE_MUL THEN CONJ_TAC) THEN REWRITE_TAC[REAL_LE_POW_2] THEN ASM SET_TAC[]; REWRITE_TAC[REAL_ENTIRE] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN MATCH_MP_TAC(TAUT `~p /\ ~r /\ (s ==> ~q) ==> ~((p \/ q) /\ (r \/ s))`) THEN REPEAT(CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]) THEN DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] (REWRITE_RULE[CONJ_ASSOC] SUM_POS_EQ_0))) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC SUM_POS_LE THEN REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN REPEAT(MATCH_MP_TAC REAL_LE_MUL THEN CONJ_TAC) THEN REWRITE_TAC[REAL_LE_POW_2] THEN ASM SET_TAC[]; ALL_TAC] THEN DISJ_CASES_THEN MP_TAC (SET_RULE `d:real^M->bool = {} \/ ?a. a IN d`) THENL [ASM_MESON_TAC[SUM_CLAUSES; REAL_ARITH `~(&1 = &0)`]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_TAC `x:real^M`) THEN DISCH_THEN(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] (REWRITE_RULE[CONJ_ASSOC] SUM_POS_EQ_0))) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [REPEAT STRIP_TAC THEN REPEAT(MATCH_MP_TAC REAL_LE_MUL THEN CONJ_TAC) THEN REWRITE_TAC[REAL_LE_POW_2] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `!x:real^M. x IN d ==> ~(l x = &0)` MP_TAC THENL [ASM_MESON_TAC[REAL_LT_IMP_NZ]; SIMP_TAC[REAL_ENTIRE]] THEN UNDISCH_TAC `(x:real^M) IN d` THEN SIMP_TAC[] THEN REWRITE_TAC[REAL_POW_EQ_0; NORM_EQ_0] THEN CONV_TAC NUM_REDUCE_CONV THEN REWRITE_TAC[VECTOR_SUB_EQ] THEN REPLICATE_TAC 3 DISCH_TAC THEN UNDISCH_TAC `sum (d:real^M->bool) l = &1` THEN SUBGOAL_THEN `d = {x:real^M}` SUBST1_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[VSUM_SING; SUM_SING]] THEN REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ; DE_MORGAN_THM] THEN DISCH_THEN SUBST1_TAC THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM SET_TAC[]]] THEN REWRITE_TAC[NORM_POW_2] THEN UNDISCH_TAC `FINITE(d:real^M->bool)` THEN SIMP_TAC[DOT_LSUM] THEN SIMP_TAC[DOT_RSUM] THEN SIMP_TAC[GSYM SUM_LMUL] THEN SIMP_TAC[GSYM SUM_ADD] THEN REWRITE_TAC[GSYM NORM_POW_2] THEN REWRITE_TAC[DOT_LMUL] THEN REWRITE_TAC[DOT_RMUL] THEN REWRITE_TAC[REAL_ARITH `&2 * a * x * y * z + b * x * y * w = x * y * (&2 * a * z + b * w)`] THEN REWRITE_TAC[REAL_ARITH `&2 * q pow 2 * x = &2 * q * q * x`] THEN ONCE_REWRITE_TAC[GSYM DOT_RMUL] THEN ONCE_REWRITE_TAC[GSYM DOT_LMUL] THEN REWRITE_TAC[DOT_NORM_SUB; REAL_ARITH `&2 * x / &2 = x`] THEN REWRITE_TAC[VECTOR_ARITH `q % (x - y) - q % (x' - y):real^N = q % (x - x')`] THEN REWRITE_TAC[NORM_MUL; REAL_POW_MUL; REAL_POW2_ABS] THEN REWRITE_TAC[REAL_ARITH `(a - q * y) + (q - &1) * y = a - y`] THEN DISCH_TAC THEN TRANS_TAC REAL_LE_TRANS `sum d (\x. sum d (\y. (l x * l y) * ((norm(g x - b) pow 2 + norm(g y - b) pow 2) - norm((g:real^M->real^N) x - g y) pow 2)))` THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_LE THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN MATCH_MP_TAC SUM_LE THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `z:real^M` THEN DISCH_TAC THEN REWRITE_TAC[GSYM REAL_MUL_ASSOC] THEN REPEAT(MATCH_MP_TAC REAL_LE_LMUL THEN CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC]) THEN MATCH_MP_TAC(REAL_ARITH `c' <= c /\ a' = a /\ b' = b ==> (a + b) - c <= (a' + b') - c'`) THEN REWRITE_TAC[GSYM REAL_LE_SQUARE_ABS; REAL_ABS_NORM] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[GSYM REAL_POW_MUL; GSYM REAL_LE_SQUARE_ABS] THEN CONJ_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN GEN_REWRITE_TAC LAND_CONV [NORM_SUB] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `!x y. (g:real^M->real^N) x - g y = (g x - b) - (g y - b)` MP_TAC THENL [CONV_TAC VECTOR_ARITH; ALL_TAC] THEN DISCH_THEN(fun th -> REWRITE_TAC[th]) THEN ONCE_REWRITE_TAC[REAL_ARITH `(x * y) * (a - b) = &2 * (x * y * (a - b) / &2)`] THEN REWRITE_TAC[GSYM DOT_NORM_SUB] THEN ONCE_REWRITE_TAC[GSYM DOT_RMUL] THEN ONCE_REWRITE_TAC[GSYM DOT_LMUL] THEN REWRITE_TAC[SUM_LMUL] THEN MATCH_MP_TAC(REAL_ARITH `x = &0 ==> &2 * x <= &0`) THEN UNDISCH_TAC `FINITE(d:real^M->bool)` THEN SIMP_TAC[GSYM DOT_RSUM] THEN SIMP_TAC[GSYM DOT_LSUM] THEN REWRITE_TAC[DOT_EQ_0; VECTOR_SUB_LDISTRIB] THEN SIMP_TAC[VSUM_SUB; VSUM_RMUL] THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN CONV_TAC VECTOR_ARITH) in REPEAT GEN_TAC THEN ASM_CASES_TAC `B = &0` THENL [ASM_REWRITE_TAC[REAL_LE_REFL; REAL_MUL_LZERO; NORM_ARITH `norm(x - y:real^N) <= &0 <=> x = y`] THEN REWRITE_TAC[MESON[] `(!x y. x IN s /\ y IN s ==> f x = f y) <=> (?a. !x. x IN s ==> f x = a)`] THEN DISCH_THEN(X_CHOOSE_TAC `b:real^N`) THEN EXISTS_TAC `(\a. b):real^M->real^N` THEN ASM_SIMP_TAC[REAL_LE_MUL; NORM_POS_LE; VECTOR_SUB_REFL; NORM_0]; STRIP_TAC] THEN SUBGOAL_THEN `&0 < B` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN MP_TAC(ISPECL [`\x. inv(B) % (f:real^M->real^N) x`; `s:real^M->bool`] lemma2) THEN REWRITE_TAC[GSYM VECTOR_SUB_LDISTRIB; NORM_MUL] THEN ASM_SIMP_TAC[real_abs; REAL_LE_INV_EQ; REAL_LT_IMP_LE] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ] THEN ANTS_TAC THENL [ASM_MESON_TAC[REAL_MUL_SYM]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^M->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\x. B % (g:real^M->real^N) x` THEN REWRITE_TAC[GSYM VECTOR_SUB_LDISTRIB; NORM_MUL] THEN ASM_SIMP_TAC[real_abs; REAL_LE_INV_EQ; REAL_LT_IMP_LE] THEN ASM_SIMP_TAC[REAL_LE_LMUL_EQ; VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[REAL_MUL_RINV; REAL_LT_IMP_NZ; VECTOR_MUL_LID]);; let LIPSCHITZ_EXTENSION_EXISTS = prove (`!f:real^M->real^N s. (?B. !x y. x IN s /\ y IN s ==> norm(f x - f y) <= B * norm(x - y)) ==> ?g. (?B. !x y. norm(g x - g y) <= B * norm(x - y)) /\ (!x. x IN s ==> g x = f x)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`; `abs B + &1`] KIRSZBRAUN) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [ALL_TAC; MESON_TAC[]] THEN ASM_MESON_TAC[REAL_LE_RMUL; REAL_LE_TRANS; NORM_POS_LE; REAL_ARITH `&0 <= abs B + &1 /\ B <= abs B + &1`]);; (* ------------------------------------------------------------------------- *) (* The Dugundji extension theorem, and Tietze variants as corollaries. *) (* ------------------------------------------------------------------------- *) let DUGUNDJI = prove (`!f:real^M->real^N c u s. convex c /\ ~(c = {}) /\ closed_in (subtopology euclidean u) s /\ f continuous_on s /\ IMAGE f s SUBSET c ==> ?g. g continuous_on u /\ IMAGE g u SUBSET c /\ !x. x IN s ==> g x = f x`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_TAC `y:real^N`) THEN EXISTS_TAC `(\x. y):real^M->real^N` THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`u DIFF s:real^M->bool`; `{ ball(x:real^M,setdist({x},s) / &2) |x| x IN u DIFF s}`] PARACOMPACT) THEN REWRITE_TAC[FORALL_IN_GSPEC; EXISTS_IN_GSPEC; OPEN_BALL] THEN ANTS_TAC THENL [REWRITE_TAC[SUBSET; IN_DIFF; IN_ELIM_THM; UNIONS_GSPEC] THEN X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN EXISTS_TAC `x:real^M` THEN ASM_REWRITE_TAC[CENTRE_IN_BALL] THEN MATCH_MP_TAC(REAL_ARITH `&0 <= x /\ ~(x = &0) ==> &0 < x / &2`) THEN ASM_MESON_TAC[SETDIST_POS_LE; SETDIST_EQ_0_CLOSED_IN]; DISCH_THEN(X_CHOOSE_THEN `c:(real^M->bool)->bool` STRIP_ASSUME_TAC)] THEN SUBGOAL_THEN `!t. t IN c ==> ?v a:real^M. v IN u /\ ~(v IN s) /\ a IN s /\ t SUBSET ball(v,setdist({v},s) / &2) /\ dist(v,a) <= &2 * setdist({v},s)` MP_TAC THENL [X_GEN_TAC `t:real^M->bool` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `t:real^M->bool`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `v:real^M` THEN REWRITE_TAC[IN_DIFF] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPECL[`{v:real^M}`; `s:real^M->bool`; `&2 * setdist({v:real^M},s)`] REAL_SETDIST_LT_EXISTS) THEN ASM_SIMP_TAC[NOT_INSERT_EMPTY; SETDIST_POS_LE; REAL_ARITH `&0 <= x ==> (x < &2 * x <=> ~(x = &0))`] THEN ASM_MESON_TAC[REAL_LT_IMP_LE; IN_SING; SETDIST_EQ_0_CLOSED_IN]; 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 [`vv:(real^M->bool)->real^M`; `aa:(real^M->bool)->real^M`] THEN STRIP_TAC] THEN SUBGOAL_THEN `!t v:real^M. t IN c /\ v IN t ==> setdist({vv t},s) <= &2 * setdist({v},s)` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `t:real^M->bool`) THEN ASM_REWRITE_TAC[SUBSET] THEN DISCH_THEN(MP_TAC o el 3 o CONJUNCTS) THEN DISCH_THEN(MP_TAC o SPEC `v:real^M`) THEN ASM_REWRITE_TAC[IN_BALL] THEN MP_TAC(ISPECL [`s:real^M->bool`; `(vv:(real^M->bool)->real^M) t`; `v:real^M`] SETDIST_SING_TRIANGLE) THEN REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `!t v a:real^M. t IN c /\ v IN t /\ a IN s ==> dist(a,aa t) <= &6 * dist(a,v)` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`t:real^M->bool`; `v:real^M`]) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o SPEC `t:real^M->bool`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o funpow 3 CONJUNCT2) THEN REWRITE_TAC[IMP_CONJ; SUBSET; IN_BALL] THEN DISCH_THEN(MP_TAC o SPEC `v:real^M`) THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPECL [`{v:real^M}`; `s:real^M->bool`; `v:real^M`; `a:real^M`] SETDIST_LE_DIST) THEN ASM_REWRITE_TAC[IN_SING] THEN CONV_TAC NORM_ARITH; ALL_TAC] THEN MP_TAC(ISPECL [`c:(real^M->bool)->bool`; `u DIFF s:real^M->bool`] SUBORDINATE_PARTITION_OF_UNITY) THEN ASM_SIMP_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `h:(real^M->bool)->real^M->real` THEN STRIP_TAC THEN EXISTS_TAC `\x. if x IN s then (f:real^M->real^N) x else vsum c (\t:real^M->bool. h t x % f(aa t))` THEN SIMP_TAC[] THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN COND_CASES_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC CONVEX_VSUM_STRONG THEN ASM SET_TAC[]] THEN REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN X_GEN_TAC `a:real^M` THEN DISCH_TAC THEN ASM_CASES_TAC `(a:real^M) IN s` THENL [ALL_TAC; MATCH_MP_TAC CONTINUOUS_TRANSFORM_WITHIN_OPEN_IN THEN MAP_EVERY EXISTS_TAC [`\x:real^M. vsum c (\t:real^M->bool. h t x % (f:real^M->real^N) (aa t))`; `u DIFF s:real^M->bool`] THEN ASM_SIMP_TAC[OPEN_IN_DIFF; OPEN_IN_REFL; IN_DIFF] THEN FIRST_X_ASSUM(MP_TAC o SPEC `a:real^M`) THEN ASM_REWRITE_TAC[IN_DIFF] THEN DISCH_THEN(X_CHOOSE_THEN `n:real^M->bool` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC CONTINUOUS_TRANSFORM_WITHIN_OPEN_IN THEN MAP_EVERY EXISTS_TAC [`\x. vsum {u | u IN c /\ ~(!x:real^M. x IN n ==> h u x = &0)} (\t:real^M->bool. h t x % (f:real^M->real^N) (aa t))`; `(u DIFF s) INTER n:real^M->bool`] THEN ASM_SIMP_TAC[OPEN_IN_DIFF; OPEN_IN_REFL; OPEN_IN_INTER_OPEN; IN_INTER; IN_DIFF] THEN CONJ_TAC THENL [REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC VSUM_SUPERSET THEN REWRITE_TAC[VECTOR_MUL_EQ_0] THEN ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_VSUM THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_VMUL THEN FIRST_X_ASSUM(MP_TAC o SPEC `t:real^M->bool`) THEN ASM_REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN DISCH_THEN(MP_TAC o SPEC `a:real^M` o CONJUNCT1) THEN ASM_REWRITE_TAC[IN_DIFF; ETA_AX] THEN REWRITE_TAC[CONTINUOUS_WITHIN] THEN MATCH_MP_TAC EQ_IMP THEN MATCH_MP_TAC LIM_TRANSFORM_WITHIN_SET THEN SUBGOAL_THEN `open_in (subtopology euclidean u) (u DIFF s:real^M->bool)` MP_TAC THENL [ASM_SIMP_TAC[OPEN_IN_DIFF; OPEN_IN_REFL]; ALL_TAC] THEN REWRITE_TAC[EVENTUALLY_AT; OPEN_IN_CONTAINS_BALL] THEN DISCH_THEN(MP_TAC o SPEC `a:real^M` o CONJUNCT2) THEN ASM_REWRITE_TAC[IN_DIFF] THEN MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[SUBSET; IN_BALL; IN_INTER; IN_DIFF] THEN MESON_TAC[DIST_SYM]]] THEN ASM_REWRITE_TAC[CONTINUOUS_WITHIN_OPEN] THEN X_GEN_TAC `w:real^N->bool` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_BALL]) THEN DISCH_THEN(MP_TAC o SPEC `(f:real^M->real^N) a`) 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 [CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN]) THEN DISCH_THEN(MP_TAC o SPEC `a:real^M`) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[continuous_within] 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 EXISTS_TAC `ball(a:real^M,d / &6)` THEN ASM_REWRITE_TAC[CENTRE_IN_BALL; OPEN_BALL] THEN ASM_REWRITE_TAC[REAL_ARITH `&0 < e / &6 <=> &0 < e`] THEN X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_BALL] THEN STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN COND_CASES_TAC THENL [REWRITE_TAC[IN_BALL] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC CONVEX_VSUM_STRONG THEN ASM_SIMP_TAC[CONVEX_BALL; IN_DIFF] THEN X_GEN_TAC `t:real^M->bool` THEN DISCH_TAC THEN ASM_CASES_TAC `(x:real^M) IN t` THENL [DISJ2_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[IN_BALL] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (NORM_ARITH `dist(a:real^M,v) < d / &6 ==> dist(a,a') <= &6 * dist(a,v) ==> dist(a',a) < d`)) THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);; let TIETZE = prove (`!f:real^M->real^N u s B. &0 <= B /\ closed_in (subtopology euclidean u) s /\ f continuous_on s /\ (!x. x IN s ==> norm(f x) <= B) ==> ?g. g continuous_on u /\ (!x. x IN s ==> g x = f x) /\ (!x. x IN u ==> norm(g x) <= B)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `cball(vec 0:real^N,B)`; `u:real^M->bool`; `s:real^M->bool`] DUGUNDJI) THEN ASM_REWRITE_TAC[CONVEX_CBALL; CBALL_EQ_EMPTY; REAL_NOT_LT] THEN ASM_REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_CBALL_0] THEN MESON_TAC[]);; let TIETZE_CLOSED_INTERVAL = prove (`!f:real^M->real^N u s a b. ~(interval[a,b] = {}) /\ closed_in (subtopology euclidean u) s /\ f continuous_on s /\ (!x. x IN s ==> f x IN interval[a,b]) ==> ?g. g continuous_on u /\ (!x. x IN s ==> g x = f x) /\ (!x. x IN u ==> g(x) IN interval[a,b])`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `interval[a:real^N,b]`; `u:real^M->bool`; `s:real^M->bool`] DUGUNDJI) THEN ASM_REWRITE_TAC[CONVEX_INTERVAL; SUBSET; FORALL_IN_IMAGE] THEN MESON_TAC[]);; let TIETZE_CLOSED_INTERVAL_1 = prove (`!f:real^N->real^1 u s a b. drop a <= drop b /\ closed_in (subtopology euclidean u) s /\ f continuous_on s /\ (!x. x IN s ==> f x IN interval[a,b]) ==> ?g. g continuous_on u /\ (!x. x IN s ==> g x = f x) /\ (!x. x IN u ==> g(x) IN interval[a,b])`, REPEAT STRIP_TAC THEN MATCH_MP_TAC TIETZE_CLOSED_INTERVAL THEN ASM_REWRITE_TAC[INTERVAL_NE_EMPTY_1]);; let TIETZE_OPEN_INTERVAL = prove (`!f:real^M->real^N u s a b. ~(interval(a,b) = {}) /\ closed_in (subtopology euclidean u) s /\ f continuous_on s /\ (!x. x IN s ==> f x IN interval(a,b)) ==> ?g. g continuous_on u /\ (!x. x IN s ==> g x = f x) /\ (!x. x IN u ==> g(x) IN interval(a,b))`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `interval(a:real^N,b)`; `u:real^M->bool`; `s:real^M->bool`] DUGUNDJI) THEN ASM_REWRITE_TAC[CONVEX_INTERVAL; SUBSET; FORALL_IN_IMAGE] THEN MESON_TAC[]);; let TIETZE_OPEN_INTERVAL_1 = prove (`!f:real^N->real^1 u s a b. drop a < drop b /\ closed_in (subtopology euclidean u) s /\ f continuous_on s /\ (!x. x IN s ==> f x IN interval(a,b)) ==> ?g. g continuous_on u /\ (!x. x IN s ==> g x = f x) /\ (!x. x IN u ==> g(x) IN interval(a,b))`, REPEAT STRIP_TAC THEN MATCH_MP_TAC TIETZE_OPEN_INTERVAL THEN ASM_REWRITE_TAC[INTERVAL_NE_EMPTY_1]);; let TIETZE_UNBOUNDED = prove (`!f:real^M->real^N u s. closed_in (subtopology euclidean u) s /\ f continuous_on s ==> ?g. g continuous_on u /\ (!x. x IN s ==> g x = f x)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `(:real^N)`; `u:real^M->bool`; `s:real^M->bool`] DUGUNDJI) THEN ASM_REWRITE_TAC[CONVEX_UNIV; UNIV_NOT_EMPTY; SUBSET_UNIV]);; (* ------------------------------------------------------------------------- *) (* Convex cones and corresponding hulls. *) (* ------------------------------------------------------------------------- *) let convex_cone = new_definition `convex_cone s <=> ~(s = {}) /\ convex s /\ conic s`;; let CONVEX_CONE = prove (`!s:real^N->bool. convex_cone s <=> vec 0 IN s /\ (!x y. x IN s /\ y IN s ==> (x + y) IN s) /\ (!x c. x IN s /\ &0 <= c ==> (c % x) IN s)`, GEN_TAC THEN REWRITE_TAC[convex_cone; GSYM conic] THEN ASM_CASES_TAC `conic(s:real^N->bool)` THEN ASM_SIMP_TAC[CONIC_CONTAINS_0] THEN AP_TERM_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[conic]) THEN REWRITE_TAC[convex] THEN EQ_TAC THEN ASM_SIMP_TAC[REAL_SUB_LE] 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 [`&2 % (x:real^N)`; `&2 % (y:real^N)`; `&1 / &2`; `&1 / &2`]) THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_SIMP_TAC[VECTOR_MUL_LID; REAL_POS]);; let CONVEX_CONE_ADD = prove (`!s x y. convex_cone s /\ x IN s /\ y IN s ==> (x + y) IN s`, MESON_TAC[CONVEX_CONE]);; let CONVEX_CONE_MUL = prove (`!s c x. convex_cone s /\ &0 <= c /\ x IN s ==> (c % x) IN s`, MESON_TAC[CONVEX_CONE]);; let CONVEX_CONE_NONEMPTY = prove (`!s. convex_cone s ==> ~(s = {})`, MESON_TAC[CONVEX_CONE; MEMBER_NOT_EMPTY]);; let CONVEX_CONE_LINEAR_IMAGE = prove (`!f:real^M->real^N s. convex_cone s /\ linear f ==> convex_cone(IMAGE f s)`, SIMP_TAC[convex_cone; CONVEX_LINEAR_IMAGE; IMAGE_EQ_EMPTY; CONIC_LINEAR_IMAGE]);; let CONVEX_CONE_LINEAR_IMAGE_EQ = prove (`!f:real^M->real^N s. linear f /\ (!x y. f x = f y ==> x = y) ==> (convex_cone(IMAGE f s) <=> convex_cone s)`, REWRITE_TAC[convex_cone] THEN MESON_TAC[IMAGE_EQ_EMPTY; CONVEX_LINEAR_IMAGE_EQ; CONIC_LINEAR_IMAGE_EQ]);; add_linear_invariants [CONVEX_CONE_LINEAR_IMAGE_EQ];; let CONVEX_CONE_HALFSPACE_GE = prove (`!a. convex_cone {x | a dot x >= &0}`, SIMP_TAC[CONVEX_CONE; real_ge; IN_ELIM_THM; DOT_RZERO; DOT_RADD; DOT_RMUL; REAL_LE_ADD; REAL_LE_MUL; REAL_LE_REFL]);; let CONVEX_CONE_HALFSPACE_LE = prove (`!a. convex_cone {x | a dot x <= &0}`, REWRITE_TAC[REAL_ARITH `x <= &0 <=> &0 <= --x`; GSYM DOT_LNEG] THEN REWRITE_TAC[GSYM real_ge; CONVEX_CONE_HALFSPACE_GE]);; let CONVEX_CONE_CONTAINS_0 = prove (`!s:real^N->bool. convex_cone s ==> vec 0 IN s`, SIMP_TAC[CONVEX_CONE]);; let CONVEX_CONE_INTERS = prove (`!f. (!s:real^N->bool. s IN f ==> convex_cone s) ==> convex_cone(INTERS f)`, SIMP_TAC[convex_cone; CONIC_INTERS; CONVEX_INTERS] THEN REWRITE_TAC[GSYM convex_cone] THEN GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `vec 0:real^N` THEN ASM_SIMP_TAC[IN_INTERS; CONVEX_CONE_CONTAINS_0]);; let CONVEX_CONE_CONVEX_CONE_HULL = prove (`!s. convex_cone(convex_cone hull s)`, SIMP_TAC[P_HULL; CONVEX_CONE_INTERS]);; let CONVEX_CONVEX_CONE_HULL = prove (`!s. convex(convex_cone hull s)`, MESON_TAC[CONVEX_CONE_CONVEX_CONE_HULL; convex_cone]);; let CONIC_CONVEX_CONE_HULL = prove (`!s. conic(convex_cone hull s)`, MESON_TAC[CONVEX_CONE_CONVEX_CONE_HULL; convex_cone]);; let CONVEX_CONE_HULL_NONEMPTY = prove (`!s. ~(convex_cone hull s = {})`, MESON_TAC[CONVEX_CONE_CONVEX_CONE_HULL; convex_cone]);; let CONVEX_CONE_HULL_CONTAINS_0 = prove (`!s. vec 0 IN convex_cone hull s`, MESON_TAC[CONVEX_CONE_CONVEX_CONE_HULL; CONVEX_CONE]);; let CONVEX_CONE_HULL_ADD = prove (`!s x y:real^N. x IN convex_cone hull s /\ y IN convex_cone hull s ==> x + y IN convex_cone hull s`, MESON_TAC[CONVEX_CONE; CONVEX_CONE_CONVEX_CONE_HULL]);; let CONVEX_CONE_HULL_MUL = prove (`!s c x:real^N. &0 <= c /\ x IN convex_cone hull s ==> (c % x) IN convex_cone hull s`, MESON_TAC[CONVEX_CONE; CONVEX_CONE_CONVEX_CONE_HULL]);; let CONVEX_CONE_SUMS = prove (`!s t. convex_cone s /\ convex_cone t ==> convex_cone {x + y:real^N | x IN s /\ y IN t}`, SIMP_TAC[convex_cone; CONIC_SUMS; CONVEX_SUMS] THEN SET_TAC[]);; let CONVEX_CONE_PCROSS = prove (`!s:real^M->bool t:real^N->bool. convex_cone s /\ convex_cone t ==> convex_cone(s PCROSS t)`, SIMP_TAC[convex_cone; CONVEX_PCROSS; CONIC_PCROSS; PCROSS_EQ_EMPTY]);; let CONVEX_CONE_PCROSS_EQ = prove (`!s:real^M->bool t:real^N->bool. convex_cone(s PCROSS t) <=> convex_cone s /\ convex_cone t`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THENL [ASM_REWRITE_TAC[PCROSS_EMPTY; convex_cone]; ALL_TAC] THEN ASM_CASES_TAC `t:real^N->bool = {}` THENL [ASM_REWRITE_TAC[PCROSS_EMPTY; convex_cone]; ALL_TAC] THEN EQ_TAC THEN REWRITE_TAC[CONVEX_CONE_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)`] CONVEX_CONE_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)`] CONVEX_CONE_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 CONVEX_CONE_HULL_UNION = prove (`!s t. convex_cone hull(s UNION t) = {x + y:real^N | x IN convex_cone hull s /\ y IN convex_cone hull t}`, REPEAT GEN_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [MATCH_MP_TAC HULL_MINIMAL THEN SIMP_TAC[CONVEX_CONE_SUMS; CONVEX_CONE_CONVEX_CONE_HULL] THEN REWRITE_TAC[SUBSET; IN_UNION; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THENL [MAP_EVERY EXISTS_TAC [`x:real^N`; `vec 0:real^N`] THEN ASM_SIMP_TAC[HULL_INC; CONVEX_CONE_HULL_CONTAINS_0; VECTOR_ADD_RID]; MAP_EVERY EXISTS_TAC [`vec 0:real^N`; `x:real^N`] THEN ASM_SIMP_TAC[HULL_INC; CONVEX_CONE_HULL_CONTAINS_0; VECTOR_ADD_LID]]; REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CONVEX_CONE_HULL_ADD THEN ASM_MESON_TAC[HULL_MONO; SUBSET_UNION; SUBSET]]);; let CONVEX_CONE_SING = prove (`convex_cone {vec 0}`, SIMP_TAC[CONVEX_CONE; IN_SING; VECTOR_ADD_LID; VECTOR_MUL_RZERO]);; let CONVEX_HULL_SUBSET_CONVEX_CONE_HULL = prove (`!s. convex hull s SUBSET convex_cone hull s`, GEN_TAC THEN MATCH_MP_TAC HULL_ANTIMONO THEN SIMP_TAC[convex_cone; SUBSET; IN]);; let CONIC_HULL_SUBSET_CONVEX_CONE_HULL = prove (`!s. conic hull s SUBSET convex_cone hull s`, GEN_TAC THEN MATCH_MP_TAC HULL_ANTIMONO THEN SIMP_TAC[convex_cone; SUBSET; IN]);; let CONVEX_CONE_HULL_SEPARATE_NONEMPTY = prove (`!s:real^N->bool. ~(s = {}) ==> convex_cone hull s = conic hull (convex hull s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THEN MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[CONIC_CONVEX_CONE_HULL; CONVEX_HULL_SUBSET_CONVEX_CONE_HULL] THEN ASM_SIMP_TAC[CONVEX_CONIC_HULL; CONVEX_CONVEX_HULL; CONIC_CONIC_HULL; convex_cone; CONIC_HULL_EQ_EMPTY; CONVEX_HULL_EQ_EMPTY] THEN ASM_MESON_TAC[HULL_SUBSET; SUBSET_REFL; SUBSET_TRANS]);; let CONVEX_CONE_HULL_EMPTY = prove (`convex_cone hull {} = {vec 0}`, MATCH_MP_TAC HULL_UNIQUE THEN REWRITE_TAC[CONVEX_CONE_CONTAINS_0; EMPTY_SUBSET; CONVEX_CONE_SING; SET_RULE `{a} SUBSET s <=> a IN s`; CONVEX_CONE_CONTAINS_0]);; let CONVEX_CONE_HULL_SEPARATE = prove (`!s:real^N->bool. convex_cone hull s = vec 0 INSERT conic hull (convex hull s)`, GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_SIMP_TAC[CONVEX_CONE_HULL_EMPTY; CONVEX_HULL_EMPTY; CONIC_HULL_EMPTY] THEN ASM_SIMP_TAC[CONVEX_CONE_HULL_SEPARATE_NONEMPTY] THEN MATCH_MP_TAC(SET_RULE `a IN s ==> s = a INSERT s`) THEN ASM_SIMP_TAC[CONIC_CONTAINS_0; CONIC_CONIC_HULL] THEN ASM_REWRITE_TAC[CONIC_HULL_EQ_EMPTY; CONVEX_HULL_EQ_EMPTY]);; let CONVEX_CONE_HULL_CONVEX_HULL_NONEMPTY = prove (`!s:real^N->bool. ~(s = {}) ==> convex_cone hull s = {c % x | &0 <= c /\ x IN convex hull s}`, SIMP_TAC[CONVEX_CONE_HULL_SEPARATE_NONEMPTY; CONIC_HULL_EXPLICIT]);; let CONVEX_CONE_HULL_CONVEX_HULL = prove (`!s:real^N->bool. convex_cone hull s = vec 0 INSERT {c % x | &0 <= c /\ x IN convex hull s}`, REWRITE_TAC[CONVEX_CONE_HULL_SEPARATE; CONIC_HULL_EXPLICIT]);; let CONVEX_CONE_HULL_LINEAR_IMAGE = prove (`!f:real^M->real^N s. linear f ==> convex_cone hull (IMAGE f s) = IMAGE f (convex_cone hull s)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^M-> bool = {}` THEN ASM_SIMP_TAC[CONVEX_CONE_HULL_SEPARATE_NONEMPTY; IMAGE_EQ_EMPTY; CONVEX_HULL_LINEAR_IMAGE; CONIC_HULL_LINEAR_IMAGE] THEN REWRITE_TAC[IMAGE_CLAUSES; CONVEX_CONE_HULL_EMPTY] THEN MATCH_MP_TAC(SET_RULE `f x = y ==> {y} = {f x}`) THEN ASM_MESON_TAC[LINEAR_0]);; add_linear_invariants [CONVEX_CONE_HULL_LINEAR_IMAGE];; let SUBSPACE_IMP_CONVEX_CONE = prove (`!s. subspace s ==> convex_cone s`, SIMP_TAC[subspace; CONVEX_CONE]);; let CONVEX_CONE_SPAN = prove (`!s. convex_cone(span s)`, SIMP_TAC[convex_cone; CONVEX_SPAN; CONIC_SPAN; GSYM MEMBER_NOT_EMPTY] THEN MESON_TAC[SPAN_0]);; let CONVEX_CONE_NEGATIONS = prove (`!s. convex_cone s ==> convex_cone (IMAGE (--) s)`, SIMP_TAC[convex_cone; IMAGE_EQ_EMPTY; CONIC_NEGATIONS; CONVEX_NEGATIONS]);; let SUBSPACE_CONVEX_CONE_SYMMETRIC = prove (`!s:real^N->bool. subspace s <=> convex_cone s /\ (!x. x IN s ==> --x IN s)`, GEN_TAC THEN REWRITE_TAC[subspace; CONVEX_CONE] THEN EQ_TAC THEN STRIP_TAC THEN ASM_SIMP_TAC[] THENL [ASM_MESON_TAC[VECTOR_ARITH `--x:real^N = -- &1 % x`]; MAP_EVERY X_GEN_TAC [`c:real`; `x:real^N`] THEN DISCH_TAC THEN DISJ_CASES_TAC(SPEC `c:real` REAL_LE_NEGTOTAL) THEN ASM_SIMP_TAC[] THEN ASM_MESON_TAC[VECTOR_ARITH `c % x:real^N = --(--c % x)`]]);; let SPAN_CONVEX_CONE_ALLSIGNS = prove (`!s:real^N->bool. span s = convex_cone hull (s UNION IMAGE (--) s)`, GEN_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [MATCH_MP_TAC SPAN_SUBSET_SUBSPACE THEN CONJ_TAC THENL [MESON_TAC[HULL_SUBSET; SUBSET_UNION; SUBSET_TRANS]; ALL_TAC] THEN REWRITE_TAC[SUBSPACE_CONVEX_CONE_SYMMETRIC; CONVEX_CONE_CONVEX_CONE_HULL] THEN MATCH_MP_TAC HULL_INDUCT THEN CONJ_TAC THENL [X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_UNION; IN_IMAGE] THEN DISCH_TAC THEN MATCH_MP_TAC HULL_INC THEN REWRITE_TAC[IN_UNION; IN_IMAGE] THEN ASM_MESON_TAC[VECTOR_NEG_NEG]; SUBGOAL_THEN `!s. {x:real^N | (--x) IN s} = IMAGE (--) s` (fun th -> SIMP_TAC[th; CONVEX_CONE_NEGATIONS; CONVEX_CONE_CONVEX_CONE_HULL]) THEN GEN_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN REWRITE_TAC[IN_ELIM_THM] THEN MESON_TAC[VECTOR_NEG_NEG]]; MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[CONVEX_CONE_SPAN] THEN REWRITE_TAC[UNION_SUBSET; SPAN_INC] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN MESON_TAC[SPAN_SUPERSET; SPAN_NEG]]);; (* ------------------------------------------------------------------------- *) (* Epigraphs of convex functions. *) (* ------------------------------------------------------------------------- *) let epigraph = new_definition `epigraph s (f:real^N->real) = {xy:real^((N,1)finite_sum) | fstcart xy IN s /\ f(fstcart xy) <= drop(sndcart xy)}`;; let IN_EPIGRAPH = prove (`!x y. (pastecart x (lift y)) IN epigraph s f <=> x IN s /\ f(x) <= y`, REWRITE_TAC[epigraph; IN_ELIM_THM; FSTCART_PASTECART; SNDCART_PASTECART; LIFT_DROP]);; let CONVEX_EPIGRAPH = prove (`!f s. f convex_on s /\ convex s <=> convex(epigraph s f)`, REWRITE_TAC[convex; convex_on; IN_ELIM_THM; SNDCART_ADD; SNDCART_CMUL; epigraph; FSTCART_ADD; FSTCART_CMUL; FORALL_PASTECART; FSTCART_PASTECART; SNDCART_PASTECART] THEN REWRITE_TAC[GSYM FORALL_DROP; DROP_ADD; DROP_CMUL] THEN MESON_TAC[REAL_LE_REFL; REAL_LE_ADD2; REAL_LE_LMUL; REAL_LE_TRANS]);; let CONVEX_EPIGRAPH_CONVEX = prove (`!f s. convex s ==> (f convex_on s <=> convex(epigraph s f))`, REWRITE_TAC[GSYM CONVEX_EPIGRAPH] THEN CONV_TAC TAUT);; let CONVEX_ON_EPIGRAPH_SLICE_LE = prove (`!f:real^N->real s a. f convex_on s /\ convex s ==> convex {x | x IN s /\ f(x) <= a}`, SIMP_TAC[convex_on; convex; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(fun th -> W(MP_TAC o PART_MATCH (lhand o rand) th o lhand o snd)) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN MATCH_MP_TAC REAL_CONVEX_BOUND_LE THEN ASM_REWRITE_TAC[]);; let CONVEX_ON_EPIGRAPH_SLICE_LT = prove (`!f:real^N->real s a. f convex_on s /\ convex s ==> convex {x | x IN s /\ f(x) < a}`, SIMP_TAC[convex_on; convex; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(fun th -> W(MP_TAC o PART_MATCH (lhand o rand) th o lhand o snd)) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LET_TRANS) THEN MATCH_MP_TAC REAL_CONVEX_BOUND_LT THEN ASM_REWRITE_TAC[]);; let CONVEX_ON_SUP = prove (`!t:A->bool s:real^N->bool. convex s /\ (!i. i IN t ==> f i convex_on s) /\ (!x. x IN s ==> ?B. !i. i IN t ==> f i x <= B) ==> (\x. sup {f i x | i IN t}) convex_on s`, REPEAT GEN_TAC THEN ASM_CASES_TAC `t:A->bool = {}` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY; CONVEX_ON_CONST; SET_RULE `{f i x | i | F} = {}`] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN ASM_SIMP_TAC[CONVEX_EPIGRAPH_CONVEX] THEN DISCH_TAC THEN SUBGOAL_THEN `convex(INTERS {epigraph (s:real^N->bool) (f i) | (i:A) IN t})` MP_TAC THENL [ASM_SIMP_TAC[CONVEX_INTERS; FORALL_IN_GSPEC]; ALL_TAC] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; FORALL_PASTECART; INTERS_GSPEC; IN_ELIM_THM] THEN REWRITE_TAC[epigraph; IN_ELIM_THM; FSTCART_PASTECART; SNDCART_PASTECART] THEN REWRITE_TAC[GSYM FORALL_DROP] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real`] THEN ASM_CASES_TAC `(x:real^N) IN s` THEN ASM_REWRITE_TAC[] THENL [CONV_TAC SYM_CONV; ASM_MESON_TAC[MEMBER_NOT_EMPTY]] THEN W(MP_TAC o PART_MATCH (lhs o rand) REAL_SUP_LE_EQ o lhand o snd) THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN DISCH_THEN MATCH_MP_TAC THEN ASM SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* Use this to derive general bound property of convex function. *) (* ------------------------------------------------------------------------- *) let FORALL_OF_PASTECART = prove (`(!p. P (fstcart o p) (sndcart o p)) <=> (!x:A->B^M y:A->B^N. P x y)`, EQ_TAC THENL [ALL_TAC; MESON_TAC[]] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `\a:A. pastecart (x a :B^M) (y a :B^N)`) THEN REWRITE_TAC[o_DEF; FSTCART_PASTECART; SNDCART_PASTECART; ETA_AX]);; let FORALL_OF_DROP = prove (`(!v. P (drop o v)) <=> (!x:A->real. P x)`, EQ_TAC THENL [ALL_TAC; MESON_TAC[]] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `\a:A. lift(x a)`) THEN REWRITE_TAC[o_DEF; LIFT_DROP; ETA_AX]);; let CONVEX_ON_JENSEN = prove (`!f:real^N->real s. convex s ==> (f convex_on s <=> !k u x. (!i:num. 1 <= i /\ i <= k ==> &0 <= u(i) /\ x(i) IN s) /\ (sum (1..k) u = &1) ==> f(vsum (1..k) (\i. u(i) % x(i))) <= sum (1..k) (\i. u(i) * f(x(i))))`, let lemma = prove (`(!x. P x ==> (Q x = R x)) ==> (!x. P x) ==> ((!x. Q x) <=> (!x. R x))`, MESON_TAC[]) in REPEAT STRIP_TAC THEN FIRST_ASSUM (fun th -> REWRITE_TAC[MATCH_MP CONVEX_EPIGRAPH_CONVEX th]) THEN REWRITE_TAC[CONVEX_INDEXED; epigraph] THEN SIMP_TAC[IN_ELIM_THM; SNDCART_ADD; SNDCART_CMUL; FINITE_NUMSEG; FSTCART_ADD; FSTCART_CMUL; FORALL_PASTECART; DROP_CMUL; FSTCART_PASTECART; SNDCART_PASTECART; FSTCART_VSUM; SNDCART_VSUM; DROP_VSUM; o_DEF] THEN REWRITE_TAC[GSYM(ISPEC `fstcart` o_THM); GSYM(ISPEC `sndcart` o_THM)] THEN REWRITE_TAC[GSYM(ISPEC `drop` o_THM)] THEN REWRITE_TAC[FORALL_OF_PASTECART; FORALL_OF_DROP] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONVEX_INDEXED]) THEN REPEAT(MATCH_MP_TAC lemma THEN GEN_TAC) THEN SIMP_TAC[] THEN REWRITE_TAC[TAUT `a ==> b /\ c <=> (a ==> b) /\ (a ==> c)`] THEN REWRITE_TAC[FORALL_AND_THM] THEN DISCH_THEN(K ALL_TAC) THEN EQ_TAC THEN SIMP_TAC[REAL_LE_REFL] THEN DISCH_THEN(fun th -> REPEAT STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REAL_ARITH `a <= b ==> x <= a ==> x <= b`) THEN ASM_SIMP_TAC[SUM_LE_NUMSEG; REAL_LE_LMUL]);; let CONVEX_ON_IMP_JENSEN = prove (`!f:real^N->real s k:A->bool u x. f convex_on s /\ convex s /\ FINITE k /\ (!i. i IN k ==> &0 <= u i /\ x i IN s) /\ sum k u = &1 ==> f(vsum k (\i. u i % x i)) <= sum k (\i. u i * f(x i))`, REPEAT GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [FINITE_INDEX_NUMSEG]) THEN ABBREV_TAC `n = CARD(k:A->bool)` THEN REWRITE_TAC[INJECTIVE_ON_ALT] THEN DISCH_THEN(X_CHOOSE_THEN `g:num->A` (CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC)) THEN ASM_SIMP_TAC[VSUM_IMAGE; SUM_IMAGE; FINITE_NUMSEG; IMP_CONJ; o_DEF] THEN DISCH_TAC THEN MP_TAC(ISPECL [`f:real^N->real`; `s:real^N->bool`] CONVEX_ON_JENSEN) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[GSYM IN_NUMSEG] THEN ASM SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* Another intermediate value theorem formulation. *) (* ------------------------------------------------------------------------- *) let IVT_INCREASING_COMPONENT_ON_1 = prove (`!f:real^1->real^N a b y k. drop a <= drop b /\ 1 <= k /\ k <= dimindex(:N) /\ f continuous_on interval[a,b] /\ f(a)$k <= y /\ y <= f(b)$k ==> ?x. x IN interval[a,b] /\ f(x)$k = y`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`IMAGE (f:real^1->real^N) (interval[a,b])`] CONNECTED_IVT_COMPONENT) THEN REWRITE_TAC[EXISTS_IN_IMAGE] THEN DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[RIGHT_EXISTS_AND_THM; EXISTS_IN_IMAGE] THEN ASM_SIMP_TAC[CONNECTED_CONTINUOUS_IMAGE; CONVEX_CONNECTED; CONVEX_INTERVAL] THEN EXISTS_TAC `a:real^1` THEN ASM_REWRITE_TAC[IN_INTERVAL_1; REAL_LE_REFL] THEN EXISTS_TAC `b:real^1` THEN ASM_REWRITE_TAC[IN_INTERVAL_1; REAL_LE_REFL]);; let IVT_INCREASING_COMPONENT_1 = prove (`!f:real^1->real^N a b y k. drop a <= drop b /\ 1 <= k /\ k <= dimindex(:N) /\ (!x. x IN interval[a,b] ==> f continuous at x) /\ f(a)$k <= y /\ y <= f(b)$k ==> ?x. x IN interval[a,b] /\ f(x)$k = y`, REPEAT STRIP_TAC THEN MATCH_MP_TAC IVT_INCREASING_COMPONENT_ON_1 THEN ASM_SIMP_TAC[CONTINUOUS_AT_IMP_CONTINUOUS_ON]);; let IVT_DECREASING_COMPONENT_ON_1 = prove (`!f:real^1->real^N a b y k. drop a <= drop b /\ 1 <= k /\ k <= dimindex(:N) /\ f continuous_on interval[a,b] /\ f(b)$k <= y /\ y <= f(a)$k ==> ?x. x IN interval[a,b] /\ f(x)$k = y`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_EQ_NEG2] THEN ASM_SIMP_TAC[GSYM VECTOR_NEG_COMPONENT] THEN MATCH_MP_TAC IVT_INCREASING_COMPONENT_ON_1 THEN ASM_SIMP_TAC[VECTOR_NEG_COMPONENT; CONTINUOUS_ON_NEG; REAL_LE_NEG2]);; let IVT_DECREASING_COMPONENT_1 = prove (`!f:real^1->real^N a b y k. drop a <= drop b /\ 1 <= k /\ k <= dimindex(:N) /\ (!x. x IN interval[a,b] ==> f continuous at x) /\ f(b)$k <= y /\ y <= f(a)$k ==> ?x. x IN interval[a,b] /\ f(x)$k = y`, REPEAT STRIP_TAC THEN MATCH_MP_TAC IVT_DECREASING_COMPONENT_ON_1 THEN ASM_SIMP_TAC[CONTINUOUS_AT_IMP_CONTINUOUS_ON]);; (* ------------------------------------------------------------------------- *) (* A bound within a convex hull, and so an interval. *) (* ------------------------------------------------------------------------- *) let CONVEX_ON_CONVEX_HULL_BOUND = prove (`!f s b. f convex_on (convex hull s) /\ (!x:real^N. x IN s ==> f(x) <= b) ==> !x. x IN convex hull s ==> f(x) <= b`, REPEAT GEN_TAC THEN SIMP_TAC[CONVEX_ON_JENSEN; CONVEX_CONVEX_HULL] THEN STRIP_TAC THEN GEN_TAC THEN REWRITE_TAC[CONVEX_HULL_INDEXED] THEN REWRITE_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`k:num`; `u:num->real`; `v:num->real^N`] THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sum(1..k) (\i. u i * f(v i :real^N))` THEN CONJ_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[SUBSET; HULL_SUBSET]; ALL_TAC] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sum(1..k) (\i. u i * b)` THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_LE_NUMSEG THEN ASM_SIMP_TAC[REAL_LE_LMUL]; ALL_TAC] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[SUM_LMUL] THEN ASM_MESON_TAC[REAL_LE_REFL; REAL_MUL_RID]);; let CONVEX_ON_CONVEX_HULL_BOUND_EQ = prove (`!f s:real^N->bool b. f convex_on convex hull s ==> ((!x. x IN convex hull s ==> f x <= b) <=> (!x. x IN s ==> f x <= b))`, MESON_TAC[CONVEX_ON_CONVEX_HULL_BOUND; HULL_INC]);; let DIST_CONVEX_HULL_BOUND_EQ = prove (`!s a:real^N d. (!x. x IN convex hull s ==> dist(a,x) <= d) <=> (!x. x IN s ==> dist(a,x) <= d)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CONVEX_ON_CONVEX_HULL_BOUND_EQ THEN REWRITE_TAC[CONVEX_DISTANCE]);; let DIST_CONVEX_HULL_BOUND_2 = prove (`!s:real^N->bool d. (!x y. x IN convex hull s /\ y IN convex hull s ==> dist(x,y) <= d) <=> (!x y. x IN s /\ y IN s ==> dist(x,y) <= d)`, MESON_TAC[DIST_CONVEX_HULL_BOUND_EQ; DIST_SYM]);; let DIAMETER_CONVEX_HULL = prove (`!s:real^N->bool. diameter(convex hull s) = diameter s`, GEN_TAC THEN REWRITE_TAC[diameter; CONVEX_HULL_EQ_EMPTY] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUP_EQ THEN REWRITE_TAC[FORALL_IN_GSPEC; GSYM dist; DIST_CONVEX_HULL_BOUND_2]);; let DIAMETER_SIMPLEX = prove (`!s:real^N->bool. ~(s = {}) ==> diameter(convex hull s) = sup { dist(x,y) | x IN s /\ y IN s}`, REWRITE_TAC[DIAMETER_CONVEX_HULL] THEN SIMP_TAC[diameter; dist]);; let UNIT_INTERVAL_CONVEX_HULL = prove (`interval [vec 0,vec 1:real^N] = convex hull {x:real^N | !i. 1 <= i /\ i <= dimindex(:N) ==> ((x$i = &0) \/ (x$i = &1))}`, let lemma = prove (`FINITE {i | 1 <= i /\ i <= n /\ P(i)} /\ CARD {i | 1 <= i /\ i <= n /\ P(i)} <= n`, CONJ_TAC THENL [MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `1..n`; GEN_REWRITE_TAC RAND_CONV [ARITH_RULE `x = (x + 1) - 1`] THEN REWRITE_TAC[GSYM CARD_NUMSEG] THEN MATCH_MP_TAC CARD_SUBSET] THEN SIMP_TAC[FINITE_NUMSEG; IN_NUMSEG; SUBSET; IN_ELIM_THM]) in MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[CONVEX_INTERVAL; SUBSET; IN_INTERVAL; IN_ELIM_THM] THEN SIMP_TAC[VEC_COMPONENT] THEN MESON_TAC[REAL_LE_REFL; REAL_POS]] THEN SUBGOAL_THEN `!n x:real^N. x IN interval[vec 0,vec 1] /\ n <= dimindex(:N) /\ CARD {i | 1 <= i /\ i <= dimindex(:N) /\ ~(x$i = &0)} <= n ==> x IN convex hull {x:real^N | !i. 1 <= i /\ i <= dimindex(:N) ==> ((x$i = &0) \/ (x$i = &1))}` MP_TAC THENL [ALL_TAC; REWRITE_TAC[SUBSET] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN EXISTS_TAC `dimindex(:N)` THEN ASM_REWRITE_TAC[LE_REFL; lemma]] THEN INDUCT_TAC THEN X_GEN_TAC `x:real^N` THENL [SIMP_TAC[LE; lemma; CARD_EQ_0] THEN GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY; BETA_THM] THEN REWRITE_TAC[TAUT `~(a /\ b /\ c) <=> a /\ b ==> ~c`] THEN STRIP_TAC THEN SUBGOAL_THEN `x = vec 0:real^N` SUBST1_TAC THENL [ASM_SIMP_TAC[CART_EQ; VEC_COMPONENT]; ALL_TAC] THEN MATCH_MP_TAC(REWRITE_RULE[SUBSET] HULL_SUBSET) THEN SIMP_TAC[IN_ELIM_THM; VEC_COMPONENT]; ALL_TAC] THEN ASM_CASES_TAC `{i | 1 <= i /\ i <= dimindex(:N) /\ ~((x:real^N)$i = &0)} = {}` THENL [DISCH_THEN(K ALL_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [EXTENSION]) THEN GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY; BETA_THM] THEN REWRITE_TAC[TAUT `~(a /\ b /\ c) <=> a /\ b ==> ~c`] THEN STRIP_TAC THEN SUBGOAL_THEN `x = vec 0:real^N` SUBST1_TAC THENL [ASM_SIMP_TAC[CART_EQ; VEC_COMPONENT]; ALL_TAC] THEN MATCH_MP_TAC(REWRITE_RULE[SUBSET] HULL_SUBSET) THEN SIMP_TAC[IN_ELIM_THM; VEC_COMPONENT]; ALL_TAC] THEN MP_TAC(ISPEC `IMAGE (\i. x$i) {i | 1 <= i /\ i <= dimindex(:N) /\ ~((x:real^N)$i = &0)}` INF_FINITE) THEN ABBREV_TAC `xi = inf (IMAGE (\i. x$i) {i | 1 <= i /\ i <= dimindex(:N) /\ ~((x:real^N)$i = &0)})` THEN ASM_SIMP_TAC[FINITE_IMAGE; IMAGE_EQ_EMPTY; lemma] THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [IN_IMAGE; IN_ELIM_THM] THEN REWRITE_TAC[] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN FIRST_X_ASSUM(X_CHOOSE_THEN `i:num` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM SUBST_ALL_TAC THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `&0 <= (x:real^N)$i /\ x$i <= &1` STRIP_ASSUME_TAC THENL [UNDISCH_TAC `x:real^N IN interval [vec 0,vec 1]` THEN ASM_SIMP_TAC[IN_INTERVAL; VEC_COMPONENT]; ALL_TAC] THEN FIRST_X_ASSUM(DISJ_CASES_TAC o MATCH_MP (REAL_ARITH `x <= &1 ==> (x = &1) \/ x < &1`)) THENL [SUBGOAL_THEN `x = lambda i. if (x:real^N)$i = &0 then &0 else &1` SUBST1_TAC THENL [UNDISCH_TAC `x:real^N IN interval [vec 0,vec 1]` THEN ASM_SIMP_TAC[CART_EQ; IN_INTERVAL; VEC_COMPONENT; LAMBDA_BETA] THEN ASM_MESON_TAC[REAL_LE_ANTISYM]; ALL_TAC] THEN MATCH_MP_TAC(REWRITE_RULE[SUBSET] HULL_SUBSET) THEN SIMP_TAC[IN_ELIM_THM; LAMBDA_BETA] THEN MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `x:real^N = x$i % (lambda j. if x$j = &0 then &0 else &1) + (&1 - x$i) % (lambda j. if x$j = &0 then &0 else (x$j - x$i) / (&1 - x$i))` SUBST1_TAC THENL [SIMP_TAC[CART_EQ; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; LAMBDA_BETA; VEC_COMPONENT] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_MUL_RZERO; REAL_ADD_LID] THEN ASM_SIMP_TAC[REAL_DIV_LMUL; ARITH_RULE `x < &1 ==> ~(&1 - x = &0)`] THEN REPEAT STRIP_TAC THEN REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC(REWRITE_RULE[convex] CONVEX_CONVEX_HULL) THEN ASM_SIMP_TAC[REAL_ARITH `x < &1 ==> &0 <= &1 - x`; REAL_ARITH `x + &1 - x = &1`] THEN CONJ_TAC THENL [MATCH_MP_TAC(REWRITE_RULE[SUBSET] HULL_SUBSET) THEN SIMP_TAC[LAMBDA_BETA; IN_ELIM_THM] THEN MESON_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[ARITH_RULE `SUC k <= n ==> k <= n`] THEN CONJ_TAC THENL [SIMP_TAC[IN_INTERVAL; LAMBDA_BETA; VEC_COMPONENT] THEN GEN_TAC THEN STRIP_TAC THEN COND_CASES_TAC THEN REWRITE_TAC[REAL_LE_REFL; REAL_POS] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ; REAL_ARITH `x < &1 ==> &0 < &1 - x`] THEN ASM_REWRITE_TAC[REAL_MUL_LZERO; REAL_SUB_LE; REAL_MUL_LID] THEN ASM_SIMP_TAC[REAL_ARITH `a - b <= &1 - b <=> a <= &1`] THEN UNDISCH_TAC `x:real^N IN interval [vec 0,vec 1]` THEN ASM_SIMP_TAC[CART_EQ; IN_INTERVAL; VEC_COMPONENT; LAMBDA_BETA]; ALL_TAC] THEN MATCH_MP_TAC LE_TRANS THEN EXISTS_TAC `CARD({i | 1 <= i /\ i <= dimindex(:N) /\ ~((x:real^N)$i = &0)} DELETE i)` THEN CONJ_TAC THENL [MATCH_MP_TAC CARD_SUBSET THEN REWRITE_TAC[lemma; FINITE_DELETE] THEN REWRITE_TAC[SUBSET; IN_DELETE; IN_ELIM_THM] THEN GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN ASM_SIMP_TAC[LAMBDA_BETA] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[CONTRAPOS_THM] THEN SIMP_TAC[real_div; REAL_SUB_REFL; REAL_MUL_LZERO]; SIMP_TAC[lemma; CARD_DELETE] THEN COND_CASES_TAC THEN ASM_SIMP_TAC[ARITH_RULE `x <= SUC n ==> x - 1 <= n`] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_ELIM_THM]) THEN ASM_MESON_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Representation of any interval as a finite convex hull. *) (* ------------------------------------------------------------------------- *) let CLOSED_INTERVAL_AS_CONVEX_HULL = prove (`!a b:real^N. ?s. FINITE s /\ interval[a,b] = convex hull s`, REPEAT GEN_TAC THEN ASM_CASES_TAC `interval[a:real^N,b] = {}` THENL [ASM_MESON_TAC[CONVEX_HULL_EMPTY; FINITE_EMPTY]; ALL_TAC] THEN ASM_SIMP_TAC[CLOSED_INTERVAL_IMAGE_UNIT_INTERVAL] THEN SUBGOAL_THEN `?s:real^N->bool. FINITE s /\ interval[vec 0,vec 1] = convex hull s` STRIP_ASSUME_TAC THENL [EXISTS_TAC `{x:real^N | !i. 1 <= i /\ i <= dimindex(:N) ==> ((x$i = &0) \/ (x$i = &1))}` THEN REWRITE_TAC[UNIT_INTERVAL_CONVEX_HULL] THEN MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `IMAGE (\s. (lambda i. if i IN s then &1 else &0):real^N) {t | t SUBSET (1..dimindex(:N))}` THEN ASM_SIMP_TAC[FINITE_POWERSET; FINITE_IMAGE; FINITE_NUMSEG] THEN REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_IMAGE] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN EXISTS_TAC `{i | 1 <= i /\ i <= dimindex(:N) /\ ((x:real^N)$i = &1)}` THEN SIMP_TAC[CART_EQ; IN_ELIM_THM; IN_NUMSEG; LAMBDA_BETA] THEN ASM_MESON_TAC[]; EXISTS_TAC `IMAGE (\x:real^N. a + x) (IMAGE (\x. (lambda i. ((b:real^N)$i - a$i) * x$i)) (s:real^N->bool))` THEN ASM_SIMP_TAC[FINITE_IMAGE; CONVEX_HULL_TRANSLATION] THEN AP_TERM_TAC THEN MATCH_MP_TAC(GSYM CONVEX_HULL_LINEAR_IMAGE) THEN SIMP_TAC[linear; CART_EQ; LAMBDA_BETA; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN REPEAT STRIP_TAC THEN REAL_ARITH_TAC]);; (* ------------------------------------------------------------------------- *) (* Characterizations of convex functions in terms of secants. *) (* ------------------------------------------------------------------------- *) let [CONVEX_ON_LEFT_SECANT_MUL; CONVEX_ON_RIGHT_SECANT_MUL; CONVEX_ON_MID_SECANT_MUL] = (CONJUNCTS o prove) (`(!f s:real^N->bool. f convex_on s <=> !a b x. a IN s /\ b IN s /\ x IN segment[a,b] ==> (f x - f a) * norm(b - a) <= (f b - f a) * norm(x - a)) /\ (!f s:real^N->bool. f convex_on s <=> !a b x. a IN s /\ b IN s /\ x IN segment[a,b] ==> (f b - f a) * norm(b - x) <= (f b - f x) * norm(b - a)) /\ (!f s:real^N->bool. f convex_on s <=> !a b x. a IN s /\ b IN s /\ x IN segment[a,b] ==> (f x - f a) * norm (b - x) <= (f b - f x) * norm(x - a))`, REPEAT CONJ_TAC THEN REPEAT GEN_TAC THEN REWRITE_TAC[convex_on] THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `a:real^N` THEN REWRITE_TAC[] THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `b:real^N` THEN REWRITE_TAC[] THEN ASM_CASES_TAC `(a:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `(b:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[IN_SEGMENT; LEFT_IMP_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `u:real` THEN REWRITE_TAC[] THEN REWRITE_TAC[TAUT `a /\ x = y <=> x = y /\ a`; TAUT `a /\ x = y /\ b <=> x = y /\ a /\ b`] THEN REWRITE_TAC[REAL_ARITH `v + u = &1 <=> v = &1 - u`] THEN REWRITE_TAC[FORALL_UNWIND_THM2; IMP_CONJ] THEN REWRITE_TAC[REAL_SUB_LE] THEN ASM_CASES_TAC `&0 <= u` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `u <= &1` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[VECTOR_ARITH `((&1 - u) % a + u % b) - a:real^N = u % (b - a)`; VECTOR_ARITH `b - ((&1 - u) % a + u % b):real^N = (&1 - u) % (b - a)`] THEN REWRITE_TAC[NORM_MUL; REAL_MUL_ASSOC] THEN (ASM_CASES_TAC `b:real^N = a` THENL [ASM_REWRITE_TAC[VECTOR_SUB_REFL; REAL_SUB_REFL; VECTOR_ARITH `(&1 - u) % a + u % a:real^N = a`] THEN REAL_ARITH_TAC; ASM_SIMP_TAC[REAL_LE_RMUL_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN ASM_SIMP_TAC[REAL_ARITH `&0 <= u /\ u <= &1 ==> abs u = u /\ abs(&1 - u) = &1 - u`] THEN REAL_ARITH_TAC]));; let [CONVEX_ON_LEFT_SECANT; CONVEX_ON_RIGHT_SECANT; CONVEX_ON_MID_SECANT] = (CONJUNCTS o prove) (`(!f s:real^N->bool. f convex_on s <=> !a b x. a IN s /\ b IN s /\ x IN segment(a,b) ==> (f x - f a) / norm(x - a) <= (f b - f a) / norm(b - a)) /\ (!f s:real^N->bool. f convex_on s <=> !a b x. a IN s /\ b IN s /\ x IN segment(a,b) ==> (f b - f a) / norm(b - a) <= (f b - f x) / norm(b - x)) /\ (!f s:real^N->bool. f convex_on s <=> !a b x. a IN s /\ b IN s /\ x IN segment(a,b) ==> (f x - f a) / norm (x - a) <= (f b - f x) / norm(b - x))`, REPEAT CONJ_TAC THEN REPEAT GEN_TAC THENL [REWRITE_TAC[CONVEX_ON_LEFT_SECANT_MUL]; REWRITE_TAC[CONVEX_ON_RIGHT_SECANT_MUL]; REWRITE_TAC[CONVEX_ON_MID_SECANT_MUL]] THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `a:real^N` THEN REWRITE_TAC[] THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `b:real^N` THEN REWRITE_TAC[] THEN ASM_CASES_TAC `(a:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `(b:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `a:real^N = b` THEN ASM_REWRITE_TAC[SEGMENT_REFL; NOT_IN_EMPTY; REAL_SUB_REFL; VECTOR_SUB_REFL; NORM_0; REAL_MUL_LZERO; REAL_MUL_RZERO; REAL_LE_REFL] THEN REWRITE_TAC[IN_SING; FORALL_UNWIND_THM2; REAL_LE_REFL] THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[] THEN REWRITE_TAC[open_segment; IN_DIFF; IN_INSERT; NOT_IN_EMPTY] THEN MAP_EVERY ASM_CASES_TAC [`x:real^N = a`; `x:real^N = b`] THEN ASM_REWRITE_TAC[REAL_LE_REFL; REAL_SUB_REFL; VECTOR_SUB_REFL; NORM_0; REAL_MUL_LZERO; REAL_MUL_RZERO] THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; GSYM REAL_LE_LDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN AP_TERM_TAC THEN REAL_ARITH_TAC);; let CONVEX_ON_SECANTS_1_IMP = prove (`!f s a b c d. f convex_on s /\ a IN s /\ b IN s /\ c IN s /\ d IN s /\ drop a < drop b /\ drop b <= drop c /\ drop c < drop d ==> (f b - f a) / (drop b - drop a) <= (f d - f c) / (drop d - drop c)`, REWRITE_TAC[CONVEX_ON_MID_SECANT] THEN REPEAT STRIP_TAC THEN ASM_CASES_TAC `c:real^1 = b` THENL [FIRST_X_ASSUM SUBST_ALL_TAC THEN SUBGOAL_THEN `drop a <= drop d` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a:real^1`; `d:real^1`; `b:real^1`]) THEN ASM_SIMP_TAC[SEGMENT_1; NORM_1; DROP_SUB; REAL_LT_IMP_LE; real_abs; REAL_SUB_LT; IN_INTERVAL_1]; SUBGOAL_THEN `drop b < drop c` ASSUME_TAC THENL [ASM_REWRITE_TAC[REAL_LT_LE; DROP_EQ]; ALL_TAC]] THEN SUBGOAL_THEN `drop a <= drop c /\ drop b <= drop d` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN TRANS_TAC REAL_LE_TRANS `(f c - f b) / (drop c - drop b)` THEN CONJ_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPECL [`a:real^1`; `c:real^1`; `b:real^1`]); FIRST_X_ASSUM(MP_TAC o SPECL [`b:real^1`; `d:real^1`; `c:real^1`])] THEN ASM_SIMP_TAC[SEGMENT_1; NORM_1; DROP_SUB; REAL_LT_IMP_LE; real_abs; REAL_SUB_LT; IN_INTERVAL_1]);; let CONVEX_ON_SECANTS_1 = prove (`!f s. is_interval s ==> (f convex_on s <=> !a b c d. a IN s /\ b IN s /\ c IN s /\ d IN s /\ drop a < drop b /\ drop b <= drop c /\ drop c < drop d ==> (f b - f a) / (drop b - drop a) <= (f d - f c) / (drop d - drop c))`, REPEAT STRIP_TAC THEN EQ_TAC THENL [MESON_TAC[CONVEX_ON_SECANTS_1_IMP]; DISCH_TAC] THEN REWRITE_TAC[CONVEX_ON_MID_SECANT] THEN MAP_EVERY X_GEN_TAC [`a:real^1`; `b:real^1`; `x:real^1`] THEN DISJ_CASES_THEN MP_TAC (REAL_ARITH `drop a = drop b \/ drop a < drop b \/ drop b < drop a`) THENL [ASM_MESON_TAC[DROP_EQ; SEGMENT_REFL; NOT_IN_EMPTY]; ALL_TAC] THEN STRIP_TAC THEN ASM_SIMP_TAC[SEGMENT_1; IN_INTERVAL_1; REAL_LT_IMP_LE; REAL_ARITH `a < b ==> ~(b <= a)`] THEN STRIP_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPECL [`a:real^1`; `x:real^1`; `x:real^1`; `b:real^1`]) THEN ASM_SIMP_TAC[NORM_1; DROP_SUB; REAL_LT_IMP_LE; real_abs; REAL_SUB_LT; REAL_LE_REFL] THEN ASM_MESON_TAC[IS_INTERVAL_1; REAL_LT_IMP_LE]; FIRST_X_ASSUM(MP_TAC o SPECL [`b:real^1`; `x:real^1`; `x:real^1`; `a:real^1`]) THEN ASM_SIMP_TAC[NORM_1; DROP_SUB; REAL_LT_IMP_LE; real_abs; REAL_SUB_LT; REAL_LE_REFL; REAL_SUB_LE; REAL_ARITH `a < b ==> ~(b <= a)`] THEN REWRITE_TAC[REAL_NEG_SUB] THEN ANTS_TAC THENL [ASM_MESON_TAC[IS_INTERVAL_1; REAL_LT_IMP_LE]; REAL_ARITH_TAC]]);; (* ------------------------------------------------------------------------- *) (* Starlike sets and more stuff about line segments. *) (* ------------------------------------------------------------------------- *) let starlike = new_definition `starlike s <=> ?a. a IN s /\ !x. x IN s ==> segment[a,x] SUBSET s`;; let CONVEX_IMP_STARLIKE = prove (`!s. convex s /\ ~(s = {}) ==> starlike s`, REWRITE_TAC[CONVEX_CONTAINS_SEGMENT; starlike; GSYM MEMBER_NOT_EMPTY] THEN MESON_TAC[]);; let CONIC_IMP_STARLIKE = prove (`!s:real^N->bool. conic s /\ ~(s = {}) ==> starlike s`, REPEAT STRIP_TAC THEN REWRITE_TAC[starlike] THEN EXISTS_TAC `vec 0:real^N` THEN SUBGOAL_THEN `(vec 0:real^N) IN s` ASSUME_TAC THENL [ASM_MESON_TAC[CONIC_CONTAINS_0]; ASM_REWRITE_TAC[]] THEN REWRITE_TAC[SUBSET; segment; FORALL_IN_GSPEC] THEN RULE_ASSUM_TAC(REWRITE_RULE[conic]) THEN ASM_SIMP_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID]);; let SEGMENT_CONVEX_HULL = prove (`!a b. segment[a,b] = convex hull {a,b}`, REPEAT GEN_TAC THEN SIMP_TAC[CONVEX_HULL_INSERT; CONVEX_HULL_SING; NOT_INSERT_EMPTY] THEN REWRITE_TAC[IN_SING; RIGHT_EXISTS_AND_THM; UNWIND_THM2] THEN REWRITE_TAC[segment; EXTENSION; IN_ELIM_THM] THEN REWRITE_TAC[REAL_ARITH `u + v = &1 <=> u = &1 - v`] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN ONCE_REWRITE_TAC[TAUT `a /\ b /\ c /\ d <=> c /\ a /\ b /\ d`] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN REWRITE_TAC[UNWIND_THM2] THEN REWRITE_TAC[REAL_LE_SUB_LADD; REAL_ADD_LID] THEN MESON_TAC[]);; let CONTINUOUS_INCREASING_IMAGE_INTERVAL_1 = prove (`!f:real^1->real^1 a b. ~(interval[a,b] = {}) /\ f continuous_on interval[a,b] /\ (!x y. x IN interval[a,b] /\ y IN interval[a,b] /\ drop x <= drop y ==> drop(f x) <= drop(f y)) ==> IMAGE f (interval[a,b]) = interval[f a,f b]`, REWRITE_TAC[INTERVAL_NE_EMPTY_1; IN_INTERVAL_1] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN CONJ_TAC THENL [ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTERVAL_1] THEN ASM_MESON_TAC[REAL_LE_REFL; REAL_LE_TRANS]; ALL_TAC] THEN TRANS_TAC SUBSET_TRANS `segment[(f:real^1->real^1) a,f b]` THEN CONJ_TAC THENL [ASM_SIMP_TAC[SEGMENT_1; REAL_LE_REFL; SUBSET_REFL]; ALL_TAC] THEN REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN MATCH_MP_TAC HULL_MINIMAL THEN ASM_SIMP_TAC[INSERT_SUBSET; EMPTY_SUBSET; FUN_IN_IMAGE; ENDS_IN_INTERVAL; INTERVAL_NE_EMPTY_1; CONVEX_CONNECTED_1] THEN ASM_SIMP_TAC[CONNECTED_CONTINUOUS_IMAGE; CONNECTED_INTERVAL]);; let CONTINUOUS_DECREASING_IMAGE_INTERVAL_1 = prove (`!f:real^1->real^1 a b. ~(interval[a,b] = {}) /\ f continuous_on interval[a,b] /\ (!x y. x IN interval[a,b] /\ y IN interval[a,b] /\ drop x <= drop y ==> drop(f y) <= drop(f x)) ==> IMAGE f (interval[a,b]) = interval[f b,f a]`, REWRITE_TAC[INTERVAL_NE_EMPTY_1; IN_INTERVAL_1] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN CONJ_TAC THENL [ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTERVAL_1] THEN ASM_MESON_TAC[REAL_LE_REFL; REAL_LE_TRANS]; ALL_TAC] THEN TRANS_TAC SUBSET_TRANS `segment[(f:real^1->real^1) b,f a]` THEN CONJ_TAC THENL [ASM_SIMP_TAC[SEGMENT_1; REAL_LE_REFL; SUBSET_REFL]; ALL_TAC] THEN REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN MATCH_MP_TAC HULL_MINIMAL THEN ASM_SIMP_TAC[INSERT_SUBSET; EMPTY_SUBSET; FUN_IN_IMAGE; ENDS_IN_INTERVAL; INTERVAL_NE_EMPTY_1; CONVEX_CONNECTED_1] THEN ASM_SIMP_TAC[CONNECTED_CONTINUOUS_IMAGE; CONNECTED_INTERVAL]);; let SEGMENT_FURTHEST_LE = prove (`!a b x y:real^N. x IN segment[a,b] ==> norm(y - x) <= norm(y - a) \/ norm(y - x) <= norm(y - b)`, REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`y:real^N`; `{a:real^N,b}`] SIMPLEX_FURTHEST_LE) THEN ASM_REWRITE_TAC[FINITE_INSERT; FINITE_RULES; NOT_INSERT_EMPTY] THEN REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ASM_MESON_TAC[NORM_SUB]);; let SEGMENT_BOUND = prove (`!a b x:real^N. x IN segment[a,b] ==> norm(x - a) <= norm(b - a) /\ norm(x - b) <= norm(b - a)`, REPEAT STRIP_TAC THEN MP_TAC(SPECL [`a:real^N`; `b:real^N`; `x:real^N`] SEGMENT_FURTHEST_LE) THENL [DISCH_THEN(MP_TAC o SPEC `a:real^N`); DISCH_THEN(MP_TAC o SPEC `b:real^N`)] THEN REWRITE_TAC[VECTOR_SUB_REFL; NORM_0] THEN ASM_MESON_TAC[NORM_POS_LE; REAL_LE_TRANS; NORM_SUB]);; let BETWEEN_IN_CONVEX_HULL = prove (`!x a b:real^N. between x (a,b) <=> x IN convex hull {a,b}`, REWRITE_TAC[BETWEEN_IN_SEGMENT; SEGMENT_CONVEX_HULL]);; let STARLIKE_LINEAR_IMAGE = prove (`!f s. starlike s /\ linear f ==> starlike(IMAGE f s)`, REWRITE_TAC[starlike; FORALL_IN_IMAGE; EXISTS_IN_IMAGE] THEN SIMP_TAC[CLOSED_SEGMENT_LINEAR_IMAGE] THEN SET_TAC[]);; let STARLIKE_LINEAR_IMAGE_EQ = prove (`!f s. linear f /\ (!x y. f x = f y ==> x = y) ==> (starlike (IMAGE f s) <=> starlike s)`, MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE STARLIKE_LINEAR_IMAGE));; add_linear_invariants [STARLIKE_LINEAR_IMAGE_EQ];; let STARLIKE_TRANSLATION_EQ = prove (`!a s. starlike (IMAGE (\x. a + x) s) <=> starlike s`, REWRITE_TAC[starlike] THEN GEOM_TRANSLATE_TAC[]);; add_translation_invariants [STARLIKE_TRANSLATION_EQ];; let BETWEEN_LINEAR_IMAGE_EQ = prove (`!f x y z. linear f /\ (!x y. f x = f y ==> x = y) ==> (between (f x) (f y,f z) <=> between x (y,z))`, SIMP_TAC[BETWEEN_IN_SEGMENT; CLOSED_SEGMENT_LINEAR_IMAGE] THEN SET_TAC[]);; add_linear_invariants [BETWEEN_LINEAR_IMAGE_EQ];; let STARLIKE_CLOSURE = prove (`!s:real^N->bool. starlike s ==> starlike(closure s)`, GEN_TAC THEN REWRITE_TAC[starlike; SUBSET; segment; FORALL_IN_GSPEC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN STRIP_TAC THEN ASM_SIMP_TAC[REWRITE_RULE[SUBSET] CLOSURE_SUBSET] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[SUBSET; CLOSURE_APPROACHABLE] THEN DISCH_TAC THEN X_GEN_TAC `u:real` 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[] THEN DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(&1 - u) % a + u % y:real^N` THEN ASM_SIMP_TAC[dist; NORM_MUL; VECTOR_ARITH `(v % a + u % y) - (v % a + u % z):real^N = u % (y - z)`] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] REAL_LET_TRANS)) THEN REWRITE_TAC[dist; REAL_ARITH `u * n <= n <=> &0 <= n * (&1 - u)`] THEN MATCH_MP_TAC REAL_LE_MUL THEN REWRITE_TAC[NORM_POS_LE] THEN ASM_REAL_ARITH_TAC);; let STARLIKE_UNIV = prove (`starlike(:real^N)`, MESON_TAC[CONVEX_IMP_STARLIKE; CONVEX_UNIV; BOUNDED_EMPTY; NOT_BOUNDED_UNIV]);; let STARLIKE_PCROSS = prove (`!s:real^M->bool t:real^N->bool. starlike s /\ starlike t ==> starlike(s PCROSS t)`, SIMP_TAC[starlike; EXISTS_IN_PCROSS; SUBSET; IN_SEGMENT] THEN REPEAT GEN_TAC THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN REWRITE_TAC[FORALL_IN_PCROSS; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[FORALL_UNWIND_THM2; IMP_IMP] THEN REWRITE_TAC[GSYM PASTECART_CMUL; PASTECART_ADD] THEN REWRITE_TAC[PASTECART_IN_PCROSS] THEN MESON_TAC[]);; let STARLIKE_PCROSS_EQ = prove (`!s:real^M->bool t:real^N->bool. starlike(s PCROSS t) <=> starlike s /\ starlike t`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THENL [ASM_REWRITE_TAC[PCROSS_EMPTY] THEN MESON_TAC[starlike; NOT_IN_EMPTY]; ALL_TAC] THEN ASM_CASES_TAC `t:real^N->bool = {}` THENL [ASM_REWRITE_TAC[PCROSS_EMPTY] THEN MESON_TAC[starlike; NOT_IN_EMPTY]; ALL_TAC] THEN EQ_TAC THEN REWRITE_TAC[STARLIKE_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)`] STARLIKE_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)`] STARLIKE_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 BETWEEN_DIST_LT = prove (`!r a b c:real^N. dist(c,a) < r /\ dist(c,b) < r /\ between x (a,b) ==> dist(c,x) < r`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `convex hull {a,b} SUBSET ball(c:real^N,r)` MP_TAC THENL [MATCH_MP_TAC HULL_MINIMAL THEN ASM_REWRITE_TAC[CONVEX_BALL; INSERT_SUBSET; EMPTY_SUBSET; IN_BALL]; ASM_SIMP_TAC[SUBSET; GSYM BETWEEN_IN_CONVEX_HULL; IN_BALL]]);; let BETWEEN_DIST_LE = prove (`!r a b c:real^N. dist(c,a) <= r /\ dist(c,b) <= r /\ between x (a,b) ==> dist(c,x) <= r`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `convex hull {a,b} SUBSET cball(c:real^N,r)` MP_TAC THENL [MATCH_MP_TAC HULL_MINIMAL THEN ASM_REWRITE_TAC[CONVEX_CBALL; INSERT_SUBSET; EMPTY_SUBSET; IN_CBALL]; ASM_SIMP_TAC[SUBSET; GSYM BETWEEN_IN_CONVEX_HULL; IN_CBALL]]);; let BETWEEN_NORM_LT = prove (`!r a b x:real^N. norm a < r /\ norm b < r /\ between x (a,b) ==> norm x < r`, REWRITE_TAC[GSYM(CONJUNCT2(SPEC_ALL DIST_0)); BETWEEN_DIST_LT]);; let BETWEEN_NORM_LE = prove (`!r a b x:real^N. norm a <= r /\ norm b <= r /\ between x (a,b) ==> norm x <= r`, REWRITE_TAC[GSYM(CONJUNCT2(SPEC_ALL DIST_0)); BETWEEN_DIST_LE]);; let UNION_SEGMENT = prove (`!a b c:real^N. b IN segment[a,c] ==> segment[a,b] UNION segment[b,c] = segment[a,c]`, REPEAT GEN_TAC THEN ASM_CASES_TAC `c:real^N = a` THENL [ASM_SIMP_TAC[SEGMENT_REFL; IN_SING; UNION_IDEMPOT]; ONCE_REWRITE_TAC[UNION_COMM] THEN REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN DISCH_THEN(SUBST1_TAC o MATCH_MP CONVEX_HULL_EXCHANGE_UNION) THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN REWRITE_TAC[IMAGE_CLAUSES; UNIONS_2] THEN BINOP_TAC THEN AP_TERM_TAC THEN ASM SET_TAC[]]);; let CONVEX_STARCENTRES = prove (`!s:real^N->bool. convex {a | a IN s /\ !x. x IN s ==> segment[a,x] SUBSET s}`, GEN_TAC THEN REWRITE_TAC[CONVEX_CONTAINS_SEGMENT; IN_ELIM_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN REWRITE_TAC[SUBSET; RIGHT_IMP_FORALL_THM; IMP_IMP; GSYM CONJ_ASSOC] THEN X_GEN_TAC `c:real^N` THEN DISCH_THEN(DESTRUCT_TAC "a aseg b bseg c") THEN REWRITE_TAC[IN_ELIM_THM] THEN CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`y:real^N`; `z:real^N`] THEN STRIP_TAC THEN SUBGOAL_THEN `(z:real^N) IN convex hull {a,b,y}` MP_TAC THENL [ONCE_REWRITE_TAC[SET_RULE `{a,b,y} = {y,b,a}`]; ALL_TAC] THEN ONCE_REWRITE_TAC[CONVEX_HULL_INSERT_SEGMENTS] THEN REWRITE_TAC[NOT_INSERT_EMPTY; GSYM SEGMENT_CONVEX_HULL] THENL [ONCE_REWRITE_TAC[SEGMENT_SYM]; ALL_TAC] THEN ASM SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* It might occasionally be handy to use midpoint convexity only. *) (* ------------------------------------------------------------------------- *) let MIDPOINT_CONVEX_SET = prove (`!s:real^N->bool. open s \/ closed s ==> (convex s <=> !a b. a IN s /\ b IN s ==> midpoint(a,b) IN s)`, GEN_TAC THEN DISCH_TAC THEN EQ_TAC THEN SIMP_TAC[MIDPOINT_IN_CONVEX] THEN DISCH_TAC THEN REWRITE_TAC[CONVEX_CONTAINS_SEGMENT] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN STRIP_TAC THEN ASM_CASES_TAC `b:real^N = a` THENL [ASM_REWRITE_TAC[SEGMENT_REFL; SING_SUBSET]; ALL_TAC] THEN REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM DISJ_CASES_TAC THENL [SUBGOAL_THEN `?e. &0 < e /\ ball(a:real^N,e) SUBSET s /\ ball(b,e) SUBSET s` STRIP_ASSUME_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_BALL]) THEN DISCH_THEN(fun th -> MP_TAC(SPEC `b:real^N` th) THEN MP_TAC(SPEC `a:real^N` th)) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN X_GEN_TAC `e:real` THEN STRIP_TAC THEN EXISTS_TAC `min d e:real` THEN ASM_REWRITE_TAC[BALL_MIN_INTER; REAL_LT_MIN] THEN ASM SET_TAC[]; ALL_TAC]; FIRST_ASSUM(fun th -> ONCE_REWRITE_TAC[GSYM(MATCH_MP CLOSED_APPROACHABLE th)]) THEN X_GEN_TAC `e:real` THEN DISCH_TAC] THEN MP_TAC(ISPECL [`inv(&2)`; `e / dist(a:real^N,b)`] REAL_ARCH_POW_INV) THEN ASM_SIMP_TAC[REAL_LT_DIV; DIST_POS_LT; REAL_POW_INV] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN DISCH_THEN(X_CHOOSE_TAC `N:num`) THENL [SUBGOAL_THEN `!n. ?c d:real^N. ball(c,e) SUBSET s /\ ball(d,e) SUBSET s /\ x IN segment[c,d] /\ dist(c,d) <= dist(a:real^N,b) / &2 pow n` (MP_TAC o SPEC `N:num`) THENL [INDUCT_TAC THEN REWRITE_TAC[real_pow; REAL_DIV_1] THENL [ASM_MESON_TAC[REAL_LE_REFL]; ALL_TAC] THEN FIRST_X_ASSUM(X_CHOOSE_THEN `c:real^N` MP_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `d:real^N` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`c:real^N`; `midpoint(c:real^N,d)`; `d:real^N`] UNION_SEGMENT) THEN REWRITE_TAC[MIDPOINT_IN_SEGMENT] THEN DISCH_THEN(SUBST_ALL_TAC o SYM) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_UNION]) THEN DISCH_THEN(DISJ_CASES_THEN MP_TAC) THEN MATCH_MP_TAC(MESON[] `(x IN segment[a,b] ==> P a b) ==> x IN segment[a,b] ==> ?c d. P c d`) THEN DISCH_TAC THEN ASM_SIMP_TAC[DIST_MIDPOINT; real_div; REAL_INV_MUL] THEN (CONJ_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC]) THEN REWRITE_TAC[SUBSET; IN_BALL] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`c + (y - midpoint(c,d)):real^N`; `d + (y - midpoint(c,d)):real^N`]) THEN REWRITE_TAC[midpoint; VECTOR_ARITH `inv(&2) % ((c + y - inv (&2) % (c + d)) + d + y - inv (&2) % (c + d)) = (y:real^N)`] THEN DISCH_THEN MATCH_MP_TAC THEN (CONJ_TAC THENL [UNDISCH_TAC `ball(c:real^N,e) SUBSET s`; UNDISCH_TAC `ball(d:real^N,e) SUBSET s`] THEN REWRITE_TAC[SUBSET; IN_BALL] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[GSYM midpoint; NORM_ARITH `dist(c:real^N,c + x - y) = dist(y,x)`]); REWRITE_TAC[LEFT_IMP_EXISTS_THM; SUBSET; IN_BALL] THEN MAP_EVERY X_GEN_TAC [`c:real^N`; `d:real^N`] THEN DISCH_THEN(CONJUNCTS_THEN2 MATCH_MP_TAC STRIP_ASSUME_TAC)]; SUBGOAL_THEN `!n. ?c d:real^N. c IN s /\ d IN s /\ x IN segment[c,d] /\ dist(c,d) <= dist(a:real^N,b) / &2 pow n` (MP_TAC o SPEC `N:num`) THENL [INDUCT_TAC THEN REWRITE_TAC[real_pow; REAL_DIV_1] THENL [ASM_MESON_TAC[REAL_LE_REFL]; ALL_TAC] THEN FIRST_X_ASSUM(X_CHOOSE_THEN `c:real^N` MP_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `d:real^N` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`c:real^N`; `midpoint(c:real^N,d)`; `d:real^N`] UNION_SEGMENT) THEN REWRITE_TAC[MIDPOINT_IN_SEGMENT] THEN DISCH_THEN(SUBST_ALL_TAC o SYM) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_UNION]) THEN DISCH_THEN(DISJ_CASES_THEN MP_TAC) THEN MATCH_MP_TAC(MESON[] `(x IN segment[a,b] ==> P a b) ==> x IN segment[a,b] ==> ?c d. P c d`) THEN DISCH_TAC THEN ASM_SIMP_TAC[DIST_MIDPOINT; real_div; REAL_INV_MUL] THEN ASM_REAL_ARITH_TAC; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^N` THEN DISCH_THEN(X_CHOOSE_THEN `d:real^N` STRIP_ASSUME_TAC) THEN ASM_REWRITE_TAC[]]] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `dist(c,d) <= m ==> dist(c,x) <= dist(c,d) /\ m < e ==> dist(c:real^N,x) < e`)) THEN (CONJ_TAC THENL [ASM_MESON_TAC[DIST_IN_CLOSED_SEGMENT; DIST_SYM]; REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] real_div] THEN ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; DIST_POS_LT]]));; (* ------------------------------------------------------------------------- *) (* Eliminate scalings when 0 is not in the affine hull. *) (* ------------------------------------------------------------------------- *) let COLLINEAR_DESCALE = prove (`!a b c x y z:real^N. ~(a = &0) /\ ~(c = &0) /\ collinear {a % x,b % y,c % z} /\ ~(vec 0 IN affine hull {x,y,z}) ==> collinear {x,y,z}`, REPEAT GEN_TAC THEN REWRITE_TAC[AFFINE_HULL_0_3_EXPLICIT; COLLINEAR_3_EXPLICIT] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; IMP_CONJ; VECTOR_MUL_ASSOC] THEN DISCH_TAC THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`a':real`; `b':real`; `c':real`] THEN REPEAT STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`a' * a:real`; `b' * b:real`; `c' * c:real`] THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN REWRITE_TAC[NOT_EXISTS_THM] THEN DISCH_THEN(MP_TAC o SPECL [`a' * a:real`; `b' * b:real`; `c' * c:real`]) THEN ASM_SIMP_TAC[] THEN FIRST_X_ASSUM(K ALL_TAC o check ((=) `vec 0:real^N` o rand o concl)) THEN REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_RING);; let CLOSED_SEGMENT_DESCALE = prove (`!a b c x y z:real^N. &0 < a /\ &0 <= b /\ &0 < c /\ (b % y) IN segment[a % x,c % z] /\ ~(vec 0 IN affine hull {x,y,z}) ==> y IN segment[x,z]`, REPEAT GEN_TAC THEN REWRITE_TAC[AFFINE_HULL_0_3_EXPLICIT] THEN REWRITE_TAC[CONVEX_HULL_2; SEGMENT_CONVEX_HULL; IN_ELIM_THM] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; IMP_CONJ; VECTOR_MUL_ASSOC] THEN REPLICATE_TAC 3 DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`u:real`; `v:real`] THEN REWRITE_TAC[VECTOR_ARITH `b % y:real^N = a + c <=> a + (--b) % y + c = vec 0`] THEN REPEAT STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`u * a / b:real`; `v * c / b:real`] THEN ASM_SIMP_TAC[REAL_LE_MUL; REAL_LE_DIV; REAL_LT_IMP_LE] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN REWRITE_TAC[NOT_EXISTS_THM] THEN DISCH_THEN(MP_TAC o SPECL [`u * a:real`; `--b:real`; `v * c:real`]) THEN ASM_SIMP_TAC[] THEN ASM_CASES_TAC `b = &0` THENL [ASM_SIMP_TAC[REAL_NEG_0; REAL_ADD_LID; REAL_LE_MUL; REAL_LT_IMP_LE; REAL_ARITH `&0 <= x /\ &0 <= y ==> (x + y = &0 <=> x = &0 /\ y = &0)`; REAL_ENTIRE; REAL_LT_IMP_NZ] THEN ASM_REAL_ARITH_TAC; DISCH_THEN(fun th -> CONJ_TAC THEN MP_TAC th) THENL [UNDISCH_TAC `~(b = &0)` THEN CONV_TAC REAL_FIELD; DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o AP_TERM `(%) (inv b):real^N->real^N`) THEN REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[REAL_MUL_RNEG; REAL_MUL_LINV] THEN REWRITE_TAC[real_div; REAL_MUL_AC] THEN CONV_TAC VECTOR_ARITH]]);; let OPEN_SEGMENT_DESCALE = prove (`!a b c x y z:real^N. &0 < a /\ &0 <= b /\ &0 < c /\ (b % y) IN segment(a % x,c % z) /\ ~(vec 0 IN affine hull {x,y,z}) /\ ~(x = y /\ z = y) ==> y IN segment(x,z)`, REWRITE_TAC[CONJ_ASSOC] THEN MATCH_MP_TAC(MESON[] `(!a b c x y z. P a b c x y z ==> P c b a z y x) /\ (!a b c x y z. P a b c x y z /\ x = y ==> z = y) /\ (!a b c x y z. P a b c x y z /\ ~(x = y) /\ ~(z = y) ==> Q x y z) ==> !a b c x y z. P a b c x y z /\ ~(x = y /\ z = y) ==> Q x y z`) THEN REWRITE_TAC[GSYM CONJ_ASSOC] THEN REPEAT CONJ_TAC THENL [MESON_TAC[SEGMENT_SYM; INSERT_AC]; REPEAT GEN_TAC THEN ASM_CASES_TAC `y:real^N = x` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `b % x:real^N = c % z` THEN ASM_REWRITE_TAC[ENDS_NOT_IN_SEGMENT] THEN ASM_CASES_TAC `z:real^N = x` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SET_RULE `{x,x,z} = {x,z}`; AFFINE_HULL_0_2_EXPLICIT] THEN REWRITE_TAC[NOT_EXISTS_THM; IN_SEGMENT] THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN FIRST_X_ASSUM(X_CHOOSE_THEN `u:real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`(&1 - u) * a - b`; `u * c:real`]) THEN ASM_REWRITE_TAC[VECTOR_ARITH `(a - b) % x + z:real^N = vec 0 <=> b % x = a % x + z`] THEN CONJ_TAC THENL [CONV_TAC VECTOR_ARITH; ALL_TAC] THEN UNDISCH_TAC `~(b % x:real^N = c % z)` THEN ASM_REWRITE_TAC[CONTRAPOS_THM; VECTOR_ARITH `y + u % c % z = c % z <=> y = (&1 - u) % c % z`] THEN REWRITE_TAC[REAL_ARITH `a - b + c = &0 <=> b = a + c`] THEN DISCH_TAC THEN UNDISCH_TAC `b % x:real^N = (&1 - u) % a % x + u % c % z` THEN ASM_REWRITE_TAC[VECTOR_ADD_RDISTRIB; GSYM VECTOR_MUL_ASSOC] THEN REWRITE_TAC[VECTOR_ARITH `a + x:real^N = a + y <=> x = y`] THEN REWRITE_TAC[VECTOR_MUL_LCANCEL] THEN ASM_SIMP_TAC[REAL_LT_IMP_NZ]; ONCE_REWRITE_TAC[segment] THEN REWRITE_TAC[IN_DIFF; IN_INSERT; NOT_IN_EMPTY] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[CLOSED_SEGMENT_DESCALE]]);; (* ------------------------------------------------------------------------- *) (* Shrinking towards the interior of a convex set. *) (* ------------------------------------------------------------------------- *) let IN_INTERIOR_CONVEX_SHRINK = prove (`!s e x c:real^N. convex s /\ c IN interior s /\ x IN s /\ &0 < e /\ e <= &1 ==> x - e % (x - c) IN interior s`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERIOR]) THEN REWRITE_TAC[IN_INTERIOR; SUBSET; IN_BALL; dist] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `e * d:real` THEN ASM_SIMP_TAC[REAL_LT_MUL] THEN X_GEN_TAC `y':real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `(&1 / e) % y' - ((&1 - e) / e) % x:real^N`) THEN ANTS_TAC THENL [UNDISCH_TAC `norm (x - e % (x - c) - y':real^N) < e * d` THEN SUBGOAL_THEN `x - e % (x - c) - y':real^N = e % (c - (&1 / e % y' - (&1 - e) / e % x))` SUBST1_TAC THENL [ASM_SIMP_TAC[VECTOR_SUB_LDISTRIB; VECTOR_MUL_ASSOC; REAL_DIV_LMUL; REAL_LT_IMP_NZ] THEN VECTOR_ARITH_TAC; ASM_SIMP_TAC[NORM_MUL; REAL_LT_LMUL_EQ; real_abs; REAL_LT_IMP_LE]]; DISCH_TAC THEN SUBGOAL_THEN `y' = (&1 - (&1 - e)) % (&1 / e % y' - (&1 - e) / e % x) + (&1 - e) % x:real^N` SUBST1_TAC THENL [ASM_SIMP_TAC[REAL_ARITH `&1 - (&1 - e) = e`; VECTOR_SUB_LDISTRIB; VECTOR_MUL_ASSOC; REAL_DIV_LMUL; REAL_LT_IMP_NZ] THEN VECTOR_ARITH_TAC; FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_ALT]) THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC]]);; let IN_INTERIOR_CLOSURE_CONVEX_SHRINK = prove (`!s e x c:real^N. convex s /\ c IN interior s /\ x IN closure s /\ &0 < e /\ e <= &1 ==> x - e % (x - c) IN interior s`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERIOR]) THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `?y:real^N. y IN s /\ norm(y - x) * (&1 - e) < e * d` STRIP_ASSUME_TAC THENL [ASM_CASES_TAC `(x:real^N) IN s` THENL [EXISTS_TAC `x:real^N` THEN ASM_SIMP_TAC[REAL_LT_MUL; VECTOR_SUB_REFL; NORM_0; REAL_MUL_LZERO]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [closure]) THEN ASM_REWRITE_TAC[IN_UNION; IN_ELIM_THM; LIMPT_APPROACHABLE; dist] THEN FIRST_ASSUM(DISJ_CASES_TAC o MATCH_MP (REAL_ARITH `e <= &1 ==> e = &1 \/ e < &1`)) THEN ASM_SIMP_TAC[REAL_SUB_REFL; GSYM REAL_LT_RDIV_EQ; REAL_SUB_LT] THENL [DISCH_THEN(MP_TAC o SPEC `&1`) THEN REWRITE_TAC[REAL_MUL_RZERO; REAL_LT_01]; DISCH_THEN(MP_TAC o SPEC `(e * d) / (&1 - e)`)] THEN ASM_SIMP_TAC[REAL_LT_RDIV_EQ; REAL_SUB_LT; REAL_MUL_LZERO; REAL_LT_MUL; REAL_MUL_LID] THEN MATCH_MP_TAC MONO_EXISTS THEN MESON_TAC[]; ALL_TAC] THEN ABBREV_TAC `z:real^N = c + ((&1 - e) / e) % (x - y)` THEN SUBGOAL_THEN `x - e % (x - c):real^N = y - e % (y - z)` SUBST1_TAC THENL [EXPAND_TAC "z" THEN REWRITE_TAC[VECTOR_SUB_LDISTRIB; VECTOR_ADD_LDISTRIB] THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_DIV_LMUL; REAL_LT_IMP_NZ] THEN VECTOR_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC IN_INTERIOR_CONVEX_SHRINK THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET] o MATCH_MP SUBSET_INTERIOR) THEN SIMP_TAC[INTERIOR_OPEN; OPEN_BALL] THEN REWRITE_TAC[IN_BALL; dist] THEN EXPAND_TAC "z" THEN REWRITE_TAC[NORM_ARITH `norm(c - (c + x)) = norm(x)`] THEN REWRITE_TAC[NORM_MUL; REAL_ABS_DIV] THEN ASM_SIMP_TAC[real_abs; REAL_LT_IMP_LE; REAL_SUB_LE] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[real_div; REAL_MUL_ASSOC] THEN ASM_SIMP_TAC[GSYM real_div; REAL_LT_LDIV_EQ] THEN ASM_MESON_TAC[REAL_MUL_SYM; NORM_SUB]);; let IN_INTERIOR_CLOSURE_CONVEX_SEGMENT = prove (`!s a b:real^N. convex s /\ a IN interior s /\ b IN closure s ==> segment(a,b) SUBSET interior s`, REWRITE_TAC[SUBSET; IN_SEGMENT] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[VECTOR_ARITH `(&1 - u) % a + u % b:real^N = b - (&1 - u) % (b - a)`] THEN MATCH_MP_TAC IN_INTERIOR_CLOSURE_CONVEX_SHRINK THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC);; (* ------------------------------------------------------------------------- *) (* Relative interior of a set. *) (* ------------------------------------------------------------------------- *) let relative_interior = new_definition `relative_interior s = {x | ?t. open_in (subtopology euclidean (affine hull s)) t /\ x IN t /\ t SUBSET s}`;; let relative_frontier = new_definition `relative_frontier s = closure s DIFF relative_interior s`;; let RELATIVE_INTERIOR_INTERIOR_OF = prove (`!s:real^N->bool. relative_interior s = subtopology euclidean (affine hull s) interior_of s`, REWRITE_TAC[interior_of; relative_interior]);; let RELATIVE_FRONTIER_FRONTIER_OF = prove (`!s:real^N->bool. relative_frontier s = subtopology euclidean (affine hull s) frontier_of s`, GEN_TAC THEN REWRITE_TAC[relative_frontier] THEN REWRITE_TAC[frontier_of; RELATIVE_INTERIOR_INTERIOR_OF] THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[CLOSURE_OF_SUBTOPOLOGY; EUCLIDEAN_CLOSURE_OF] THEN SIMP_TAC[HULL_SUBSET; SET_RULE `s SUBSET t ==> t INTER s = s`; CLOSURE_SUBSET_AFFINE_HULL]);; let RELATIVE_INTERIOR = prove (`!s. relative_interior s = {x | x IN s /\ ?t. open t /\ x IN t /\ t INTER (affine hull s) SUBSET s}`, REWRITE_TAC[EXTENSION; relative_interior; IN_ELIM_THM] THEN REPEAT GEN_TAC THEN REWRITE_TAC[OPEN_IN_OPEN; LEFT_AND_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN ONCE_REWRITE_TAC[TAUT `(a /\ b) /\ c /\ d <=> b /\ a /\ c /\ d`] THEN REWRITE_TAC[UNWIND_THM2; SUBSET; IN_INTER; RIGHT_AND_EXISTS_THM] THEN AP_TERM_TAC THEN ABS_TAC THEN MESON_TAC[HULL_INC]);; let RELATIVE_INTERIOR_EQ = prove (`!s. relative_interior s = s <=> open_in(subtopology euclidean (affine hull s)) s`, GEN_TAC THEN REWRITE_TAC[EXTENSION; relative_interior; IN_ELIM_THM] THEN GEN_REWRITE_TAC RAND_CONV [OPEN_IN_SUBOPEN] THEN MESON_TAC[SUBSET]);; let RELATIVE_INTERIOR_OPEN_IN = prove (`!s. open_in(subtopology euclidean (affine hull s)) s ==> relative_interior s = s`, REWRITE_TAC[RELATIVE_INTERIOR_EQ]);; let RELATIVE_INTERIOR_EMPTY = prove (`relative_interior {} = {}`, SIMP_TAC[RELATIVE_INTERIOR_OPEN_IN; OPEN_IN_EMPTY]);; let RELATIVE_FRONTIER_EMPTY = prove (`relative_frontier {} = {}`, REWRITE_TAC[relative_frontier; CLOSURE_EMPTY; EMPTY_DIFF]);; let RELATIVE_INTERIOR_AFFINE = prove (`!s:real^N->bool. affine s ==> relative_interior s = s`, SIMP_TAC[RELATIVE_INTERIOR_EQ; OPEN_IN_SUBTOPOLOGY_REFL; HULL_P] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN; SUBSET_UNIV]);; let RELATIVE_INTERIOR_UNIV = prove (`!s. relative_interior(affine hull s) = affine hull s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC RELATIVE_INTERIOR_OPEN_IN THEN REWRITE_TAC[HULL_HULL; OPEN_IN_SUBTOPOLOGY_REFL] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN; SUBSET_UNIV]);; let OPEN_IN_RELATIVE_INTERIOR = prove (`!s. open_in (subtopology euclidean (affine hull s)) (relative_interior s)`, GEN_TAC THEN REWRITE_TAC[relative_interior] THEN GEN_REWRITE_TAC I [OPEN_IN_SUBOPEN] THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN MESON_TAC[]);; let RELATIVE_INTERIOR_SUBSET = prove (`!s. (relative_interior s) SUBSET s`, REWRITE_TAC[SUBSET; relative_interior; IN_ELIM_THM] THEN MESON_TAC[]);; let RELATIVE_FRONTIER_SUBSET = prove (`!s:real^N->bool. closed s ==> relative_frontier s SUBSET s`, REWRITE_TAC[GSYM CLOSURE_SUBSET_EQ; relative_frontier] THEN SET_TAC[]);; let RELATIVE_FRONTIER_SUBSET_EQ = prove (`!s:real^N->bool. relative_frontier s SUBSET s <=> closed s`, GEN_TAC THEN REWRITE_TAC[GSYM CLOSURE_SUBSET_EQ; relative_frontier] THEN MP_TAC(ISPEC `s:real^N->bool` RELATIVE_INTERIOR_SUBSET) THEN SET_TAC[]);; let BOUNDED_RELATIVE_INTERIOR = prove (`!s:real^N->bool. bounded s ==> bounded(relative_interior s)`, MESON_TAC[BOUNDED_SUBSET; RELATIVE_INTERIOR_SUBSET]);; let OPEN_IN_SET_RELATIVE_INTERIOR = prove (`!s:real^N->bool. open_in (subtopology euclidean s) (relative_interior s)`, GEN_TAC THEN MATCH_MP_TAC OPEN_IN_SUBSET_TRANS THEN EXISTS_TAC `affine hull s:real^N->bool` THEN REWRITE_TAC[OPEN_IN_RELATIVE_INTERIOR; RELATIVE_INTERIOR_SUBSET; HULL_SUBSET]);; let SUBSET_RELATIVE_INTERIOR = prove (`!s t. s SUBSET t /\ affine hull s = affine hull t ==> (relative_interior s) SUBSET (relative_interior t)`, REWRITE_TAC[relative_interior; SUBSET; IN_ELIM_THM] THEN MESON_TAC[]);; let RELATIVE_INTERIOR_CLOSURE_SUBSET = prove (`!s. relative_interior s SUBSET relative_interior(closure s)`, SIMP_TAC[SUBSET_RELATIVE_INTERIOR; CLOSURE_SUBSET; AFFINE_HULL_CLOSURE]);; let RELATIVE_INTERIOR_MAXIMAL = prove (`!s t. t SUBSET s /\ open_in(subtopology euclidean (affine hull s)) t ==> t SUBSET (relative_interior s)`, REWRITE_TAC[relative_interior; SUBSET; IN_ELIM_THM] THEN MESON_TAC[]);; let RELATIVE_INTERIOR_UNIQUE = prove (`!s t. t SUBSET s /\ open_in(subtopology euclidean (affine hull s)) t /\ (!t'. t' SUBSET s /\ open_in(subtopology euclidean (affine hull s)) t' ==> t' SUBSET t) ==> (relative_interior s = t)`, MESON_TAC[SUBSET_ANTISYM; RELATIVE_INTERIOR_MAXIMAL; RELATIVE_INTERIOR_SUBSET; OPEN_IN_RELATIVE_INTERIOR]);; let IN_RELATIVE_INTERIOR = prove (`!x:real^N s. x IN relative_interior s <=> x IN s /\ ?e. &0 < e /\ (ball(x,e) INTER (affine hull s)) SUBSET s`, REPEAT GEN_TAC THEN REWRITE_TAC[relative_interior; IN_ELIM_THM] THEN REWRITE_TAC[OPEN_IN_OPEN; LEFT_AND_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN ONCE_REWRITE_TAC[TAUT `(a /\ b) /\ c /\ d <=> b /\ a /\ c /\ d`] THEN REWRITE_TAC[UNWIND_THM2; SUBSET; IN_INTER] THEN EQ_TAC THENL [ASM_MESON_TAC[SUBSET; OPEN_CONTAINS_BALL]; STRIP_TAC THEN EXISTS_TAC `ball(x:real^N,e)` THEN ASM_SIMP_TAC[OPEN_BALL; CENTRE_IN_BALL; HULL_INC]]);; let IN_RELATIVE_INTERIOR_CBALL = prove (`!x:real^N s. x IN relative_interior s <=> x IN s /\ ?e. &0 < e /\ (cball(x,e) INTER affine hull s) SUBSET s`, REPEAT GEN_TAC THEN REWRITE_TAC[IN_RELATIVE_INTERIOR] THEN AP_TERM_TAC THEN EQ_TAC THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THENL [EXISTS_TAC `e / &2` THEN ASM_REWRITE_TAC[REAL_HALF] THEN MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `ball(x:real^N,e) INTER affine hull s` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET; IN_INTER; IN_BALL; IN_CBALL] THEN ASM_SIMP_TAC[REAL_ARITH `&0 < e /\ x <= e / &2 ==> x < e`]; EXISTS_TAC `e:real` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `cball(x:real^N,e) INTER affine hull s` THEN ASM_REWRITE_TAC[] THEN SIMP_TAC[SUBSET; IN_INTER; IN_BALL; IN_CBALL; REAL_LT_IMP_LE]]);; let RELATIVE_INTERIOR_CONVEX_INTER_OPEN = prove (`!s t:real^N->bool. convex s /\ open t /\ ~(s INTER t = {}) ==> relative_interior(s INTER t) = relative_interior s INTER t`, REPEAT STRIP_TAC THEN MATCH_MP_TAC RELATIVE_INTERIOR_UNIQUE THEN ASM_SIMP_TAC[AFFINE_HULL_CONVEX_INTER_OPEN; SUBSET_INTER; INTER_SUBSET] THEN REPEAT CONJ_TAC THENL [MP_TAC(ISPEC `s:real^N->bool` RELATIVE_INTERIOR_SUBSET) THEN SET_TAC[]; MP_TAC(ISPEC `s:real^N->bool` OPEN_IN_RELATIVE_INTERIOR) THEN REWRITE_TAC[OPEN_IN_OPEN] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `t INTER u:real^N->bool` THEN ASM_SIMP_TAC[OPEN_INTER] THEN ASM SET_TAC[]; MESON_TAC[RELATIVE_INTERIOR_MAXIMAL]]);; let CONIC_HULL_EQ_SPAN,CONIC_HULL_EQ_AFFINE_HULL = (CONJ_PAIR o prove) (`(!s:real^N->bool. vec 0 IN relative_interior s ==> conic hull s = span s) /\ (!s:real^N->bool. vec 0 IN relative_interior s ==> conic hull s = affine hull s)`, SIMP_TAC[AND_FORALL_THM; TAUT `(p ==> q) /\ (p ==> r) <=> p ==> q /\ r`] THEN GEN_TAC THEN DISCH_TAC THEN FIRST_X_ASSUM(CONJUNCTS_THEN ASSUME_TAC o GEN_REWRITE_RULE I [IN_RELATIVE_INTERIOR_CBALL]) THEN MATCH_MP_TAC(SET_RULE `a = s /\ c SUBSET s /\ a SUBSET c ==> c = s /\ c = a`) THEN ASM_SIMP_TAC[CONIC_HULL_SUBSET_SPAN; AFFINE_HULL_EQ_SPAN_EQ; HULL_INC] THEN REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN ASM_CASES_TAC `x:real^N = vec 0` THEN ASM_SIMP_TAC[HULL_INC] THEN FIRST_X_ASSUM(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPEC `e / norm x % x:real^N` o REWRITE_RULE[SUBSET]) THEN ASM_SIMP_TAC[IN_INTER; IN_CBALL_0; NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM; REAL_DIV_RMUL; NORM_EQ_0] THEN ASM_SIMP_TAC[REAL_ARITH `&0 < e ==> abs e <= e`] THEN ANTS_TAC THENL [ASM_MESON_TAC[SPAN_MUL; AFFINE_HULL_EQ_SPAN; HULL_INC]; DISCH_TAC] THEN REWRITE_TAC[CONIC_HULL_EXPLICIT; IN_ELIM_THM] THEN MAP_EVERY EXISTS_TAC [`norm(x:real^N) / e`; `e / norm x % x:real^N`] THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_LE_DIV; NORM_POS_LE; REAL_LT_IMP_LE] THEN ASM_SIMP_TAC[VECTOR_MUL_LID; NORM_EQ_0; REAL_FIELD `~(x = &0) /\ &0 < e ==> x / e * e / x = &1`]);; let CONIC_HULL_EQ_SPAN_EQ = prove (`!s:real^N->bool. vec 0 IN relative_interior(conic hull s) <=> conic hull s = span s`, GEN_TAC THEN EQ_TAC THEN DISCH_TAC THENL [MP_TAC(ISPEC `conic hull s:real^N->bool` CONIC_HULL_EQ_SPAN) THEN ASM_REWRITE_TAC[SPAN_CONIC_HULL; HULL_HULL]; ASM_SIMP_TAC[RELATIVE_INTERIOR_AFFINE; AFFINE_SPAN; SPAN_0]]);; let OPEN_IN_SUBSET_RELATIVE_INTERIOR = prove (`!s t. open_in(subtopology euclidean (affine hull t)) s ==> (s SUBSET relative_interior t <=> s SUBSET t)`, MESON_TAC[RELATIVE_INTERIOR_MAXIMAL; RELATIVE_INTERIOR_SUBSET; SUBSET_TRANS]);; let RELATIVE_INTERIOR_TRANSLATION = prove (`!a:real^N s. relative_interior (IMAGE (\x. a + x) s) = IMAGE (\x. a + x) (relative_interior s)`, REWRITE_TAC[relative_interior; OPEN_IN_OPEN] THEN GEOM_TRANSLATE_TAC[]);; add_translation_invariants [RELATIVE_INTERIOR_TRANSLATION];; let RELATIVE_FRONTIER_TRANSLATION = prove (`!a:real^N s. relative_frontier (IMAGE (\x. a + x) s) = IMAGE (\x. a + x) (relative_frontier s)`, REWRITE_TAC[relative_frontier] THEN GEOM_TRANSLATE_TAC[]);; add_translation_invariants [RELATIVE_FRONTIER_TRANSLATION];; let RELATIVE_INTERIOR_INJECTIVE_LINEAR_IMAGE = prove (`!f:real^M->real^N s. linear f /\ (!x y. f x = f y ==> x = y) ==> relative_interior(IMAGE f s) = IMAGE f (relative_interior s)`, REPEAT GEN_TAC THEN DISCH_TAC THEN ASM_SIMP_TAC[relative_interior; AFFINE_HULL_LINEAR_IMAGE] THEN ONCE_REWRITE_TAC[TAUT `a /\ b /\ c <=> c /\ a /\ b`] THEN REWRITE_TAC[EXISTS_SUBSET_IMAGE] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_INJECTIVE_LINEAR_IMAGE) THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);; add_linear_invariants [RELATIVE_INTERIOR_INJECTIVE_LINEAR_IMAGE];; let RELATIVE_FRONTIER_INJECTIVE_LINEAR_IMAGE = prove (`!f:real^M->real^N s. linear f /\ (!x y. f x = f y ==> x = y) ==> relative_frontier(IMAGE f s) = IMAGE f (relative_frontier s)`, REWRITE_TAC[relative_frontier] THEN GEOM_TRANSFORM_TAC[]);; add_linear_invariants [RELATIVE_FRONTIER_INJECTIVE_LINEAR_IMAGE];; let RELATIVE_INTERIOR_RELATIVE_INTERIOR = prove (`!s:real^N->bool. relative_interior(relative_interior s) = relative_interior s`, GEN_TAC THEN REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; RELATIVE_INTERIOR_SUBSET] THEN REWRITE_TAC[SUBSET] THEN SIMP_TAC[IN_RELATIVE_INTERIOR] THEN X_GEN_TAC `x:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `e:real` THEN ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(SET_RULE `a' SUBSET a /\ (b INTER a SUBSET s ==> b INTER a SUBSET i) ==> b INTER a SUBSET s ==> b INTER a' SUBSET i`) THEN SIMP_TAC[HULL_MONO; RELATIVE_INTERIOR_SUBSET] THEN REWRITE_TAC[SUBSET; IN_INTER; IN_BALL] THEN DISCH_TAC THEN X_GEN_TAC `y:real^N` THEN STRIP_TAC THEN ASM_SIMP_TAC[IN_RELATIVE_INTERIOR] THEN EXISTS_TAC `e - dist(x:real^N,y)` THEN ASM_REWRITE_TAC[REAL_SUB_LT] THEN REWRITE_TAC[IN_BALL; IN_INTER; SUBSET] THEN X_GEN_TAC `z:real^N` THEN STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM REAL_NOT_LE])) THEN CONV_TAC NORM_ARITH);; let RELATIVE_INTERIOR_EQ_EMPTY = prove (`!s:real^N->bool. convex s ==> (relative_interior s = {} <=> s = {})`, SUBGOAL_THEN `!s:real^N->bool. vec 0 IN s /\ convex s ==> ~(relative_interior s = {})` ASSUME_TAC THENL [ALL_TAC; GEN_TAC THEN DISCH_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[RELATIVE_INTERIOR_EMPTY] 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_X_ASSUM(MP_TAC o SPEC `IMAGE (\x:real^N. --a + x) s`) THEN REWRITE_TAC[CONVEX_TRANSLATION_EQ; RELATIVE_INTERIOR_TRANSLATION] THEN ASM_REWRITE_TAC[IMAGE_EQ_EMPTY; IN_IMAGE] THEN DISCH_THEN MATCH_MP_TAC THEN EXISTS_TAC `a:real^N` THEN ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC] THEN GEN_TAC THEN STRIP_TAC THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_RELATIVE_INTERIOR] THEN ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC] THEN X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC (ISPEC `s:real^N->bool` BASIS_EXISTS) THEN SUBGOAL_THEN `span(s:real^N->bool) = span b` SUBST_ALL_TAC THENL [ASM_SIMP_TAC[SPAN_EQ] THEN ASM_MESON_TAC[SPAN_INC; SUBSET_TRANS]; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[HAS_SIZE]) THEN ABBREV_TAC `n = dim(s:real^N->bool)` THEN SUBGOAL_THEN `!c. (!v. v IN b ==> &0 <= c(v)) /\ sum b c <= &1 ==> vsum b (\v:real^N. c(v) % v) IN s` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN SUBGOAL_THEN `vsum (vec 0 INSERT b :real^N->bool) (\v. (if v = vec 0 then &1 - sum b c else c v) % v) IN s` MP_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_EXPLICIT]) THEN ASM_SIMP_TAC[INSERT_SUBSET; FINITE_INSERT; SUM_CLAUSES; INDEPENDENT_NONZERO; IN_INSERT] THEN CONJ_TAC THENL [ASM_MESON_TAC[REAL_SUB_LE]; ALL_TAC] THEN REWRITE_TAC[REAL_ARITH `&1 - x + y = &1 <=> x = y`] THEN MATCH_MP_TAC SUM_EQ THEN ASM_MESON_TAC[INDEPENDENT_NONZERO]; MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN ASM_SIMP_TAC[VSUM_CLAUSES; INDEPENDENT_NONZERO] THEN REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN MATCH_MP_TAC VSUM_EQ THEN ASM_MESON_TAC[INDEPENDENT_NONZERO]]; ALL_TAC] THEN ABBREV_TAC `a:real^N = vsum b (\v. inv(&2 * &n + &1) % v)` THEN EXISTS_TAC `a:real^N` THEN CONJ_TAC THENL [EXPAND_TAC "a" THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[SUM_CONST; REAL_LE_INV_EQ; REAL_ARITH `&0 < &2 * &n + &1`; GSYM real_div; REAL_LT_IMP_LE; REAL_LE_LDIV_EQ] THEN REAL_ARITH_TAC; ALL_TAC] THEN MP_TAC(ISPECL [`b:real^N->bool`; `inv(&2 * &n + &1)`] BASIS_COORDINATES_CONTINUOUS) THEN ASM_REWRITE_TAC[REAL_LT_INV_EQ] THEN ANTS_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN ASM_SIMP_TAC[SUBSET; IN_INTER; IMP_CONJ_ALT] THEN ASM_SIMP_TAC[SPAN_FINITE; LEFT_IMP_EXISTS_THM; IN_ELIM_THM] THEN GEN_TAC THEN X_GEN_TAC `u:real^N->real` THEN DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[IN_BALL; dist] THEN EXPAND_TAC "a" THEN ASM_SIMP_TAC[GSYM VSUM_SUB] THEN DISCH_THEN(fun th -> FIRST_X_ASSUM MATCH_MP_TAC THEN MP_TAC th) THEN REWRITE_TAC[GSYM VECTOR_SUB_RDISTRIB] THEN DISCH_THEN(fun th -> FIRST_X_ASSUM(MP_TAC o C MATCH_MP th)) THEN REWRITE_TAC[REAL_ARITH `abs(x - y) < x <=> &0 < y /\ abs(y) < &2 * x`] THEN SIMP_TAC[REAL_LT_IMP_LE] THEN DISCH_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `&(CARD(b:real^N->bool)) * &2 * inv(&2 * &n + &1)` THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_BOUND THEN ASM_SIMP_TAC[REAL_ARITH `abs x < a ==> x <= a`]; ASM_REWRITE_TAC[REAL_MUL_ASSOC] THEN REWRITE_TAC[GSYM real_div] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_ARITH `&0 < &2 * &n + &1`] THEN REAL_ARITH_TAC]);; let AFF_DIM_NONEMPTY_INTERIOR_OF_EQ = prove (`!u s:real^N->bool. convex s /\ affine u /\ s SUBSET u ==> (aff_dim s = aff_dim u <=> s = {} /\ u = {} \/ ~((subtopology euclidean u) interior_of s = {}))`, REPEAT GEN_TAC THEN ASM_CASES_TAC `u:real^N->bool = {}` THEN ASM_SIMP_TAC[SUBSET_EMPTY] THEN REPEAT STRIP_TAC THEN EQ_TAC THENL [ALL_TAC; ASM_MESON_TAC[AFF_DIM_NONEMPTY_INTERIOR_OF]] THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [ASM_MESON_TAC[AFF_DIM_EQ_MINUS1]; ALL_TAC] THEN ASM_SIMP_TAC[AFF_DIM_EQ_FULL_GEN; HULL_P] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[GSYM RELATIVE_INTERIOR_INTERIOR_OF] THEN ASM_SIMP_TAC[RELATIVE_INTERIOR_EQ_EMPTY]);; let RELATIVE_INTERIOR_INTERIOR = prove (`!s. affine hull s = (:real^N) ==> relative_interior s = interior s`, SIMP_TAC[relative_interior; interior; SUBTOPOLOGY_UNIV; OPEN_IN]);; let RELATIVE_INTERIOR_OPEN = prove (`!s:real^N->bool. open s ==> relative_interior s = s`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[RELATIVE_INTERIOR_EMPTY] THEN ASM_SIMP_TAC[RELATIVE_INTERIOR_INTERIOR; AFFINE_HULL_OPEN; INTERIOR_EQ]);; let RELATIVE_INTERIOR_NONEMPTY_INTERIOR = prove (`!s. ~(interior s = {}) ==> relative_interior s = interior s`, MESON_TAC[RELATIVE_INTERIOR_INTERIOR; AFFINE_HULL_NONEMPTY_INTERIOR]);; let RELATIVE_FRONTIER_NONEMPTY_INTERIOR = prove (`!s. ~(interior s = {}) ==> relative_frontier s = frontier s`, SIMP_TAC[relative_frontier; frontier; RELATIVE_INTERIOR_NONEMPTY_INTERIOR]);; let RELATIVE_FRONTIER_FRONTIER = prove (`!s. affine hull s = (:real^N) ==> relative_frontier s = frontier s`, SIMP_TAC[relative_frontier; frontier; RELATIVE_INTERIOR_INTERIOR]);; let RELATIVE_FRONTIER_OPEN = prove (`!s:real^N->bool. open s ==> relative_frontier s = frontier s`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[FRONTIER_EMPTY; RELATIVE_FRONTIER_EMPTY] THEN MATCH_MP_TAC RELATIVE_FRONTIER_NONEMPTY_INTERIOR THEN ASM_SIMP_TAC[INTERIOR_OPEN]);; let AFFINE_HULL_CONVEX_HULL = prove (`!s. affine hull (convex hull s) = affine hull s`, GEN_TAC THEN MATCH_MP_TAC HULL_UNIQUE THEN REWRITE_TAC[AFFINE_AFFINE_HULL; CONVEX_HULL_SUBSET_AFFINE_HULL] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC HULL_MINIMAL THEN ASM_MESON_TAC[SUBSET_TRANS; HULL_SUBSET]);; let INTERIOR_SIMPLEX_NONEMPTY = prove (`!s:real^N->bool. independent s /\ s HAS_SIZE (dimindex(:N)) ==> ?a. a IN interior(convex hull (vec 0 INSERT s))`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `convex hull (vec 0 INSERT s):real^N->bool` RELATIVE_INTERIOR_EQ_EMPTY) THEN ASM_SIMP_TAC[AFFINE_HULL_CONVEX_HULL] THEN REWRITE_TAC[CONVEX_HULL_EQ_EMPTY; CONVEX_CONVEX_HULL; NOT_INSERT_EMPTY] THEN REWRITE_TAC[MEMBER_NOT_EMPTY] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC RELATIVE_INTERIOR_INTERIOR THEN SIMP_TAC[AFFINE_HULL_EQ_SPAN; IN_INSERT; HULL_INC] THEN MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ s = UNIV ==> t = UNIV`) THEN EXISTS_TAC `span s:real^N->bool` THEN CONJ_TAC THENL [MATCH_MP_TAC SPAN_MONO THEN MATCH_MP_TAC(SET_RULE `(a INSERT s) SUBSET P hull (a INSERT s) ==> s SUBSET P hull (a INSERT s)`) THEN REWRITE_TAC[HULL_SUBSET]; MATCH_MP_TAC(SET_RULE `UNIV SUBSET s ==> s = UNIV`) THEN MATCH_MP_TAC CARD_GE_DIM_INDEPENDENT THEN ASM_REWRITE_TAC[DIM_UNIV; SUBSET_UNIV] THEN ASM_MESON_TAC[LE_REFL;HAS_SIZE]]);; let INTERIOR_SUBSET_RELATIVE_INTERIOR = prove (`!s. interior s SUBSET relative_interior s`, REWRITE_TAC[SUBSET; IN_INTERIOR; IN_RELATIVE_INTERIOR; IN_INTER] THEN MESON_TAC[CENTRE_IN_BALL]);; let RELATIVE_FRONTIER_SUBSET_FRONTIER = prove (`!s:real^N->bool. relative_frontier s SUBSET frontier s`, GEN_TAC THEN REWRITE_TAC[relative_frontier; frontier] THEN MP_TAC(ISPEC `s:real^N->bool` INTERIOR_SUBSET_RELATIVE_INTERIOR) THEN SET_TAC[]);; let CONVEX_RELATIVE_INTERIOR = prove (`!s:real^N->bool. convex s ==> convex(relative_interior s)`, REWRITE_TAC[CONVEX_ALT; IN_RELATIVE_INTERIOR; IN_INTER; SUBSET; IN_BALL; dist] THEN GEN_TAC THEN DISCH_TAC THEN REPEAT GEN_TAC THEN REWRITE_TAC[TAUT `(a /\ b) /\ (c /\ d) /\ e ==> f <=> a /\ c /\ e ==> b /\ d ==> f`] THEN STRIP_TAC THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC(MESON[] `(!d e. P d /\ Q e ==> R(min d e)) ==> (?e. P e) /\ (?e. Q e) ==> (?e. R e)`) THEN REPEAT GEN_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[REAL_LT_MIN] THEN X_GEN_TAC `z:real^N` THEN STRIP_TAC THEN SUBST1_TAC(VECTOR_ARITH `z:real^N = (&1 - u) % (z - u % (y - x)) + u % (z + (&1 - u) % (y - x))`) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(CONJUNCTS_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN REWRITE_TAC[GSYM IMP_CONJ_ALT] THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THEN DISCH_THEN MATCH_MP_TAC THEN (CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `norm x < e ==> norm x = y ==> y < e`)) THEN AP_TERM_TAC THEN VECTOR_ARITH_TAC; REWRITE_TAC[VECTOR_ARITH `a - b % c:real^N = a + --b % c`] THEN MATCH_MP_TAC IN_AFFINE_ADD_MUL_DIFF THEN ASM_SIMP_TAC[AFFINE_AFFINE_HULL; HULL_INC]]));; let IN_RELATIVE_INTERIOR_CONVEX_SHRINK = prove (`!s e x c:real^N. convex s /\ c IN relative_interior s /\ x IN s /\ &0 < e /\ e <= &1 ==> x - e % (x - c) IN relative_interior s`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_RELATIVE_INTERIOR]) THEN REWRITE_TAC[IN_RELATIVE_INTERIOR; SUBSET; IN_INTER; IN_BALL; dist] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN CONJ_TAC THENL [REWRITE_TAC[VECTOR_ARITH `x - e % (x - c):real^N = (&1 - e) % x + e % c`] THEN FIRST_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [convex]) THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN EXISTS_TAC `e * d:real` THEN ASM_SIMP_TAC[REAL_LT_MUL] THEN X_GEN_TAC `y':real^N` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `(&1 / e) % y' - ((&1 - e) / e) % x:real^N`) THEN ANTS_TAC THENL [CONJ_TAC THENL [UNDISCH_TAC `norm (x - e % (x - c) - y':real^N) < e * d` THEN SUBGOAL_THEN `x - e % (x - c) - y':real^N = e % (c - (&1 / e % y' - (&1 - e) / e % x))` SUBST1_TAC THENL [ASM_SIMP_TAC[VECTOR_SUB_LDISTRIB; VECTOR_MUL_ASSOC; REAL_DIV_LMUL; REAL_LT_IMP_NZ] THEN VECTOR_ARITH_TAC; ASM_SIMP_TAC[NORM_MUL; REAL_LT_LMUL_EQ; real_abs; REAL_LT_IMP_LE]]; REWRITE_TAC[real_div; REAL_SUB_RDISTRIB] THEN ASM_SIMP_TAC[REAL_MUL_RINV; REAL_LT_IMP_NZ] THEN REWRITE_TAC[VECTOR_ARITH `a % y - (b - c) % x:real^N = (c - b) % x + a % y`] THEN MATCH_MP_TAC(REWRITE_RULE[AFFINE_ALT] AFFINE_AFFINE_HULL) THEN ASM_SIMP_TAC[HULL_INC]]; DISCH_TAC THEN SUBGOAL_THEN `y' = (&1 - (&1 - e)) % (&1 / e % y' - (&1 - e) / e % x) + (&1 - e) % x:real^N` SUBST1_TAC THENL [ASM_SIMP_TAC[REAL_ARITH `&1 - (&1 - e) = e`; VECTOR_SUB_LDISTRIB; VECTOR_MUL_ASSOC; REAL_DIV_LMUL; REAL_LT_IMP_NZ] THEN VECTOR_ARITH_TAC; FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_ALT]) THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC]]);; let IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SHRINK = prove (`!s e x c:real^N. convex s /\ c IN relative_interior s /\ x IN closure s /\ &0 < e /\ e <= &1 ==> x - e % (x - c) IN relative_interior s`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_RELATIVE_INTERIOR]) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `?y:real^N. y IN s /\ norm(y - x) * (&1 - e) < e * d` STRIP_ASSUME_TAC THENL [ASM_CASES_TAC `(x:real^N) IN s` THENL [EXISTS_TAC `x:real^N` THEN ASM_SIMP_TAC[REAL_LT_MUL; VECTOR_SUB_REFL; NORM_0; REAL_MUL_LZERO]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [closure]) THEN ASM_REWRITE_TAC[IN_UNION; IN_ELIM_THM; LIMPT_APPROACHABLE; dist] THEN FIRST_ASSUM(DISJ_CASES_TAC o MATCH_MP (REAL_ARITH `e <= &1 ==> e = &1 \/ e < &1`)) THEN ASM_SIMP_TAC[REAL_SUB_REFL; GSYM REAL_LT_RDIV_EQ; REAL_SUB_LT] THENL [DISCH_THEN(MP_TAC o SPEC `&1`) THEN REWRITE_TAC[REAL_MUL_RZERO; REAL_LT_01]; DISCH_THEN(MP_TAC o SPEC `(e * d) / (&1 - e)`)] THEN ASM_SIMP_TAC[REAL_LT_RDIV_EQ; REAL_SUB_LT; REAL_MUL_LZERO; REAL_LT_MUL; REAL_MUL_LID] THEN MATCH_MP_TAC MONO_EXISTS THEN MESON_TAC[]; ALL_TAC] THEN ABBREV_TAC `z:real^N = c + ((&1 - e) / e) % (x - y)` THEN SUBGOAL_THEN `x - e % (x - c):real^N = y - e % (y - z)` SUBST1_TAC THENL [EXPAND_TAC "z" THEN REWRITE_TAC[VECTOR_SUB_LDISTRIB; VECTOR_ADD_LDISTRIB] THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_DIV_LMUL; REAL_LT_IMP_NZ] THEN VECTOR_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC IN_RELATIVE_INTERIOR_CONVEX_SHRINK THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `dist(c:real^N,z) < d` ASSUME_TAC THENL [EXPAND_TAC "z" THEN REWRITE_TAC[NORM_ARITH `dist(c:real^N,c + x) = norm x`] THEN REWRITE_TAC[NORM_MUL; REAL_ABS_DIV] THEN ONCE_REWRITE_TAC[NORM_SUB] THEN REWRITE_TAC[REAL_ARITH `a / b * c:real = (c * a) / b`] THEN ASM_SIMP_TAC[real_abs; REAL_SUB_LE; REAL_LT_IMP_LE; REAL_LT_LDIV_EQ] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `(z:real^N) IN affine hull s` ASSUME_TAC THENL [EXPAND_TAC "z" THEN MATCH_MP_TAC IN_AFFINE_ADD_MUL_DIFF THEN ASM_SIMP_TAC[AFFINE_AFFINE_HULL; HULL_INC] THEN MATCH_MP_TAC(SET_RULE `!t. x IN t /\ t = s ==> x IN s`) THEN EXISTS_TAC `closure(affine hull s):real^N->bool` THEN SIMP_TAC[CLOSURE_EQ; CLOSED_AFFINE_HULL] THEN ASM_MESON_TAC[SUBSET_CLOSURE; HULL_INC; SUBSET]; ALL_TAC] THEN ASM_REWRITE_TAC[IN_RELATIVE_INTERIOR] THEN CONJ_TAC THENL [ASM_MESON_TAC[IN_BALL; IN_INTER; SUBSET]; ALL_TAC] THEN EXISTS_TAC `d - dist(c:real^N,z)` THEN ASM_REWRITE_TAC[REAL_SUB_LT] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS)) THEN REWRITE_TAC[SUBSET; IN_INTER] THEN GEN_TAC THEN MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN UNDISCH_TAC `dist(c:real^N,z) < d` THEN REWRITE_TAC[IN_BALL] THEN NORM_ARITH_TAC);; let IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT = prove (`!s a b:real^N. convex s /\ a IN relative_interior s /\ b IN closure s ==> segment(a,b) SUBSET relative_interior s`, REWRITE_TAC[SUBSET; IN_SEGMENT] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[VECTOR_ARITH `(&1 - u) % a + u % b:real^N = b - (&1 - u) % (b - a)`] THEN MATCH_MP_TAC IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SHRINK THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC);; let INTER_RELATIVE_FRONTIER_CONIC_HULL = prove (`!s t:real^N->bool. convex s /\ vec 0 IN relative_interior s /\ t SUBSET relative_frontier s ==> t = relative_frontier s INTER conic hull t`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[GSYM SUBSET_ANTISYM_EQ; SUBSET_INTER; HULL_SUBSET] THEN REWRITE_TAC[SUBSET; IN_INTER; CONIC_HULL_EXPLICIT; IMP_CONJ_ALT] THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN MAP_EVERY X_GEN_TAC [`c:real`; `x:real^N`] THEN ASM_CASES_TAC `c = &0` THENL [ASM_REWRITE_TAC[relative_frontier; IN_DIFF; VECTOR_MUL_LZERO]; ASM_REWRITE_TAC[REAL_LE_LT]] THEN REPEAT_TCL DISJ_CASES_THEN ASSUME_TAC (ISPECL [`c:real`; `&1`] REAL_LT_TOTAL) THEN ASM_SIMP_TAC[VECTOR_MUL_LID] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC(TAUT `F ==> p`) THEN MP_TAC (ISPECL [`s:real^N->bool`; `vec 0:real^N`] IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT) THEN ASM_REWRITE_TAC[] THENL [DISCH_THEN(MP_TAC o SPEC `x:real^N`); DISCH_THEN(MP_TAC o SPEC `c % x:real^N`)] THEN (ANTS_TAC THENL [ASM_MESON_TAC[relative_frontier; IN_DIFF; SUBSET]; ALL_TAC]) THEN REWRITE_TAC[SUBSET; IN_SEGMENT; VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN MATCH_MP_TAC(MESON[] `P /\ (?u. &0 < u /\ u < &1 /\ ~Q(u % y)) ==> ~(!x. P /\ (?u. &0 < u /\ u < &1 /\ x = u % y) ==> Q x)`) THEN (CONJ_TAC THENL [ASM_MESON_TAC[relative_frontier; IN_DIFF; SUBSET]; ALL_TAC]) THENL [EXISTS_TAC `c:real`; EXISTS_TAC `inv c:real`] THEN ASM_SIMP_TAC[REAL_LT_INV_EQ; REAL_INV_LT_1; VECTOR_MUL_ASSOC; VECTOR_MUL_LID; REAL_MUL_LINV] THEN ASM_MESON_TAC[relative_frontier; IN_DIFF; SUBSET]);; let INTER_CONVEX_HULL_INSERT_RELATIVE_EXTERIOR = prove (`!c t s z:real^N. convex c /\ t SUBSET c /\ z IN relative_interior c /\ DISJOINT s (relative_interior c) ==> s INTER (convex hull (z INSERT t)) = s INTER (convex hull t)`, REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN SIMP_TAC[HULL_MONO; SET_RULE `s SUBSET a INSERT s`; SET_RULE `t SUBSET u ==> s INTER t SUBSET s INTER u`] THEN REWRITE_TAC[CONVEX_HULL_INSERT_SEGMENTS; SUBSET] THEN X_GEN_TAC `x:real^N` THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[IN_INTER; UNIONS_GSPEC; IN_ELIM_THM] THEN ASM_CASES_TAC `(x:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`c:real^N->bool`; `z:real^N`; `y:real^N`] IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[SUBSET] CLOSURE_SUBSET) THEN ASM_MESON_TAC[HULL_MINIMAL; SUBSET]; REWRITE_TAC[SUBSET] THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ONCE_REWRITE_TAC[segment] THEN ASM_REWRITE_TAC[IN_DIFF; IN_INSERT; NOT_IN_EMPTY] THEN ASM SET_TAC[]]);; let CONVEX_OPEN_SEGMENT_CASES = prove (`!s a b:real^N. convex s /\ a IN closure s /\ b IN closure s ==> segment(a,b) SUBSET relative_frontier s \/ segment(a,b) SUBSET relative_interior s`, REPEAT STRIP_TAC THEN REWRITE_TAC[relative_frontier] THEN MATCH_MP_TAC(SET_RULE `s SUBSET c /\ (!a. a IN i /\ a IN s ==> s SUBSET i) ==> s SUBSET c DIFF i \/ s SUBSET i`) THEN CONJ_TAC THENL [ASM_MESON_TAC[CONVEX_CONTAINS_OPEN_SEGMENT; CONVEX_CLOSURE]; X_GEN_TAC `c:real^N` THEN ONCE_REWRITE_TAC[segment]] THEN REWRITE_TAC[IN_DIFF; IN_INSERT; DE_MORGAN_THM; NOT_IN_EMPTY] THEN STRIP_TAC THEN FIRST_ASSUM(SUBST1_TAC o SYM o MATCH_MP UNION_SEGMENT) THEN MP_TAC(ISPECL [`s:real^N->bool`; `c:real^N`] IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT) THEN DISCH_THEN(fun th -> MP_TAC(SPEC `b:real^N` th) THEN MP_TAC(SPEC `a:real^N` th)) THEN ASM_REWRITE_TAC[SEGMENT_SYM; CONJUNCT2 segment] THEN ASM SET_TAC[]);; let CONVEX_OPEN_SEGMENT_CASES_ALT = prove (`!s a b:real^N. convex s /\ a IN closure s /\ b IN closure s ==> segment (a,b) SUBSET frontier s \/ segment (a,b) SUBSET interior s`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `interior s:real^N->bool = {}` THENL [DISJ1_TAC THEN ASM_REWRITE_TAC[frontier; DIFF_EMPTY] THEN ASM_MESON_TAC[CONVEX_CONTAINS_OPEN_SEGMENT; CONVEX_CLOSURE]; MP_TAC(ISPECL [`s:real^N->bool`; `a:real^N`; `b:real^N`] CONVEX_OPEN_SEGMENT_CASES) THEN ASM_SIMP_TAC[RELATIVE_INTERIOR_NONEMPTY_INTERIOR; RELATIVE_FRONTIER_NONEMPTY_INTERIOR]]);; let SEGMENT_SUBSET_RELATIVE_FRONTIER_CONVEX = prove (`!s a b c:real^N. convex s /\ c IN segment(a,b) /\ {a,b,c} SUBSET relative_frontier s ==> segment[a,b] SUBSET relative_frontier s`, REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[SEGMENT_CLOSED_OPEN; UNION_SUBSET] THEN ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET] THEN MP_TAC(ISPECL [`s:real^N->bool`; `a:real^N`; `b:real^N`] CONVEX_OPEN_SEGMENT_CASES) THEN RULE_ASSUM_TAC(REWRITE_RULE[relative_frontier; IN_DIFF]) THEN ASM SET_TAC[]);; let RELATIVE_INTERIOR_SING = prove (`!a. relative_interior {a} = {a}`, GEN_TAC THEN MATCH_MP_TAC(SET_RULE `s SUBSET {a} /\ ~(s = {}) ==> s = {a}`) THEN SIMP_TAC[RELATIVE_INTERIOR_SUBSET; RELATIVE_INTERIOR_EQ_EMPTY; CONVEX_SING] THEN SET_TAC[]);; let RELATIVE_FRONTIER_SING = prove (`!a:real^N. relative_frontier {a} = {}`, REWRITE_TAC[relative_frontier; RELATIVE_INTERIOR_SING; CLOSURE_SING] THEN SET_TAC[]);; let RELATIVE_INTERIOR_CBALL = prove (`!a r. relative_interior(cball(a,r)) = if r = &0 then {a} else ball(a,r)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `r < &0` THEN ASM_SIMP_TAC[REAL_LT_IMP_NE; CBALL_EMPTY; BALL_EMPTY; RELATIVE_INTERIOR_EMPTY; REAL_LT_IMP_LE] THEN COND_CASES_TAC THEN ASM_SIMP_TAC[CBALL_SING; RELATIVE_INTERIOR_SING] THEN REWRITE_TAC[GSYM INTERIOR_CBALL] THEN MATCH_MP_TAC RELATIVE_INTERIOR_NONEMPTY_INTERIOR THEN ASM_REWRITE_TAC[INTERIOR_CBALL; BALL_EQ_EMPTY] THEN ASM_REAL_ARITH_TAC);; let RELATIVE_INTERIOR_BALL = prove (`!a r. relative_interior(ball(a,r)) = ball(a,r)`, SIMP_TAC[RELATIVE_INTERIOR_OPEN; OPEN_BALL]);; let RELATIVE_FRONTIER_CBALL = prove (`!a:real^N r. relative_frontier(cball(a,r)) = if r = &0 then {} else sphere(a,r)`, REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[CBALL_SING; RELATIVE_FRONTIER_SING] THEN ASM_CASES_TAC `r < &0` THEN ASM_SIMP_TAC[CBALL_EMPTY; SPHERE_EMPTY; RELATIVE_FRONTIER_EMPTY] THEN SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[RELATIVE_FRONTIER_NONEMPTY_INTERIOR; INTERIOR_CBALL; BALL_EQ_EMPTY; GSYM REAL_NOT_LT; FRONTIER_CBALL]);; let RELATIVE_FRONTIER_BALL = prove (`!a:real^N r. relative_frontier(ball(a,r)) = if r = &0 then {} else sphere(a,r)`, REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[BALL_EMPTY; REAL_LE_REFL; RELATIVE_FRONTIER_EMPTY] THEN ASM_CASES_TAC `r < &0` THEN ASM_SIMP_TAC[BALL_EMPTY; REAL_LT_IMP_LE; SPHERE_EMPTY; RELATIVE_FRONTIER_EMPTY] THEN SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[RELATIVE_FRONTIER_NONEMPTY_INTERIOR; INTERIOR_OPEN; OPEN_BALL; BALL_EQ_EMPTY; GSYM REAL_NOT_LT; FRONTIER_BALL]);; let DIFFERENT_NORM_3_COLLINEAR_POINTS = prove (`!a b x:real^N. ~(x IN segment(a,b) /\ norm(a) = norm(b) /\ norm(x) = norm(b))`, REPEAT GEN_TAC THEN ASM_CASES_TAC `a:real^N = b` THEN ASM_SIMP_TAC[SEGMENT_REFL; NOT_IN_EMPTY; OPEN_SEGMENT_ALT] THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN (CONJUNCTS_THEN2 (X_CHOOSE_THEN `u:real` STRIP_ASSUME_TAC) MP_TAC) THEN ASM_REWRITE_TAC[NORM_EQ] THEN REWRITE_TAC[VECTOR_ARITH `(x + y:real^N) dot (x + y) = x dot x + &2 * x dot y + y dot y`] THEN REWRITE_TAC[DOT_LMUL; DOT_RMUL] THEN DISCH_THEN(CONJUNCTS_THEN2 (ASSUME_TAC o SYM) MP_TAC) THEN UNDISCH_TAC `~(a:real^N = b)` THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM VECTOR_SUB_EQ] THEN REWRITE_TAC[GSYM DOT_EQ_0; VECTOR_ARITH `(a - b:real^N) dot (a - b) = a dot a + b dot b - &2 * a dot b`] THEN ASM_REWRITE_TAC[REAL_RING `a + a - &2 * ab = &0 <=> ab = a`] THEN SIMP_TAC[REAL_RING `(&1 - u) * (&1 - u) * a + &2 * (&1 - u) * u * x + u * u * a = a <=> x = a \/ u = &0 \/ u = &1`] THEN ASM_REAL_ARITH_TAC);; let OPEN_SEGMENT_SUBSET_BALL = prove (`!a r u v:real^N. u IN cball(a,r) /\ v IN cball(a,r) ==> segment(u,v) SUBSET ball(a,r)`, REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `a:real^N` THEN REPEAT GEN_TAC THEN ASM_CASES_TAC `u:real^N = v` THEN ASM_REWRITE_TAC[SEGMENT_REFL; EMPTY_SUBSET] THEN ASM_CASES_TAC `r < &0` THEN ASM_SIMP_TAC[CBALL_EMPTY; NOT_IN_EMPTY] THEN ASM_CASES_TAC `r = &0` THEN ASM_SIMP_TAC[CBALL_SING; IN_SING] THENL [ASM_MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; STRIP_TAC] THEN ASM_CASES_TAC `u IN ball(vec 0:real^N,r)` THENL [MP_TAC(ISPECL [`ball(vec 0:real^N,r)`; `u:real^N`; `v:real^N`] IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT) THEN ASM_SIMP_TAC[CONVEX_BALL; RELATIVE_INTERIOR_BALL; CLOSURE_BALL]; ALL_TAC] THEN ASM_CASES_TAC `v IN ball(vec 0:real^N,r)` THENL [MP_TAC(ISPECL [`ball(vec 0:real^N,r)`; `v:real^N`; `u:real^N`] IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT) THEN ASM_SIMP_TAC[CONVEX_BALL; RELATIVE_INTERIOR_BALL; CLOSURE_BALL] THEN REWRITE_TAC[SEGMENT_SYM]; ALL_TAC] THEN MP_TAC(ISPECL [`ball(vec 0:real^N,r)`; `u:real^N`; `v:real^N`] CONVEX_OPEN_SEGMENT_CASES) THEN ASM_SIMP_TAC[CLOSURE_BALL; CONVEX_BALL] THEN DISCH_THEN(DISJ_CASES_THEN MP_TAC) THEN ASM_REWRITE_TAC[RELATIVE_INTERIOR_BALL; RELATIVE_FRONTIER_BALL] THEN REWRITE_TAC[SUBSET] THEN DISCH_THEN(MP_TAC o SPEC `midpoint(u,v):real^N`) THEN ASM_REWRITE_TAC[MIDPOINT_IN_SEGMENT] THEN DISCH_TAC THEN MP_TAC(ISPECL [`u:real^N`; `v:real^N`; `midpoint(u,v):real^N`] DIFFERENT_NORM_3_COLLINEAR_POINTS) THEN ASM_REWRITE_TAC[MIDPOINT_IN_SEGMENT] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_SPHERE_0; IN_BALL_0; IN_CBALL_0]) THEN ASM_REAL_ARITH_TAC);; let STARLIKE_CONVEX_TWEAK_BOUNDARY_POINTS = prove (`!s t:real^N->bool. convex s /\ ~(s = {}) /\ relative_interior s SUBSET t /\ t SUBSET closure s ==> starlike t`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `~(relative_interior s:real^N->bool = {})` MP_TAC THENL [ASM_SIMP_TAC[RELATIVE_INTERIOR_EQ_EMPTY]; REWRITE_TAC[starlike]] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN REPEAT STRIP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `a IN s /\ b IN s /\ segment[a,b] DIFF {a,b} SUBSET s ==> segment[a:real^N,b] SUBSET s`) THEN ASM_REWRITE_TAC[GSYM open_segment] THEN ASM_MESON_TAC[IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT; SUBSET]);; let RELATIVE_INTERIOR_PROLONG = prove (`!s x y:real^N. x IN relative_interior s /\ y IN s ==> ?t. &1 < t /\ (y + t % (x - y)) IN s`, REPEAT GEN_TAC THEN REWRITE_TAC[IN_RELATIVE_INTERIOR_CBALL; IN_ELIM_THM] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC)) THEN ASM_CASES_TAC `y:real^N = x` THENL [ASM_REWRITE_TAC[VECTOR_ARITH `y + t % (x - x):real^N = y`] THEN EXISTS_TAC `&2` THEN CONV_TAC REAL_RAT_REDUCE_CONV; EXISTS_TAC `&1 + e / norm(x - y:real^N)` THEN ASM_SIMP_TAC[REAL_LT_ADDR; REAL_LT_DIV; NORM_POS_LT; VECTOR_SUB_EQ] THEN REWRITE_TAC[VECTOR_ARITH `y + (&1 + e) % (x - y):real^N = x + e % (x - y)`] THEN FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET]) THEN ASM_SIMP_TAC[AFFINE_AFFINE_HULL; IN_INTER; IN_AFFINE_ADD_MUL_DIFF; HULL_INC; IN_CBALL] THEN REWRITE_TAC[NORM_ARITH `dist(x:real^N,x + y) = norm y`] THEN REWRITE_TAC[NORM_MUL; 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 RELATIVE_INTERIOR_CONVEX_PROLONG = prove (`!s. convex s ==> relative_interior s = {x:real^N | x IN s /\ !y. y IN s ==> ?t. &1 < t /\ (y + t % (x - y)) IN s}`, REPEAT STRIP_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN EQ_TAC THENL [SIMP_TAC[RELATIVE_INTERIOR_PROLONG] THEN MESON_TAC[SUBSET; RELATIVE_INTERIOR_SUBSET]; STRIP_TAC THEN SUBGOAL_THEN `?y:real^N. y IN relative_interior s` STRIP_ASSUME_TAC THENL [ASM_SIMP_TAC[MEMBER_NOT_EMPTY; RELATIVE_INTERIOR_EQ_EMPTY] THEN ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN ANTS_TAC THENL [ASM_MESON_TAC[RELATIVE_INTERIOR_SUBSET; SUBSET]; ALL_TAC] THEN ASM_CASES_TAC `y:real^N = x` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `t:real` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`s:real^N->bool`; `y:real^N`; `y + t % (x - y):real^N`] IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT) THEN ANTS_TAC THENL [ASM_MESON_TAC[SUBSET; CLOSURE_SUBSET]; ALL_TAC] THEN REWRITE_TAC[SUBSET] THEN DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[IN_SEGMENT; IN_ELIM_THM] THEN ASM_REWRITE_TAC[VECTOR_ARITH `y:real^N = y + x <=> x = vec 0`; VECTOR_ARITH `(&1 - u) % y + u % (y + t % (x - y)):real^N = y + t % u % (x - y)`] THEN ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ] THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN EXISTS_TAC `inv t:real` THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; REAL_LT_INV_EQ; REAL_INV_LT_1; REAL_LT_IMP_NZ; REAL_ARITH `&1 < x ==> &0 < x`] THEN VECTOR_ARITH_TAC]);; let RELATIVE_INTERIOR_EQ_CLOSURE = prove (`!s:real^N->bool. relative_interior s = closure s <=> affine s`, GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[RELATIVE_INTERIOR_EMPTY; CLOSURE_EMPTY; AFFINE_EMPTY] THEN EQ_TAC THEN SIMP_TAC[RELATIVE_INTERIOR_AFFINE; CLOSURE_CLOSED; CLOSED_AFFINE] THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `relative_interior s = closure s ==> relative_interior s SUBSET s /\ s SUBSET closure s ==> relative_interior s = s /\ closure s = s`)) THEN REWRITE_TAC[RELATIVE_INTERIOR_SUBSET; CLOSURE_SUBSET] THEN REWRITE_TAC[RELATIVE_INTERIOR_EQ; CLOSURE_EQ; GSYM AFFINE_HULL_EQ] THEN STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `~(s = {}) ==> s = {} \/ s = a ==> a = s`)) THEN MP_TAC(ISPEC `affine hull s:real^N->bool` CONNECTED_CLOPEN) THEN SIMP_TAC[AFFINE_IMP_CONVEX; CONVEX_CONNECTED; AFFINE_AFFINE_HULL] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CLOSED_SUBSET THEN ASM_REWRITE_TAC[HULL_SUBSET]);; let RAY_TO_RELATIVE_FRONTIER = prove (`!s a l:real^N. bounded s /\ a IN relative_interior s /\ (a + l) IN affine hull s /\ ~(l = vec 0) ==> ?d. &0 < d /\ (a + d % l) IN relative_frontier s /\ !e. &0 <= e /\ e < d ==> (a + e % l) IN relative_interior s`, REPEAT STRIP_TAC THEN REWRITE_TAC[relative_frontier] THEN MP_TAC(ISPEC `{d | &0 < d /\ ~((a + d % l:real^N) IN relative_interior(s))}` INF) THEN ABBREV_TAC `d = inf {d | &0 < d /\ ~((a + d % l:real^N) IN relative_interior(s))}` THEN SUBGOAL_THEN `?e. &0 < e /\ !d. &0 <= d /\ d < e ==> (a + d % l:real^N) IN relative_interior s` (X_CHOOSE_THEN `k:real` (LABEL_TAC "0")) THENL [MP_TAC(ISPEC `s:real^N->bool` OPEN_IN_RELATIVE_INTERIOR) THEN REWRITE_TAC[open_in; GSYM MEMBER_NOT_EMPTY; IN_ELIM_THM] THEN DISCH_THEN(MP_TAC o SPEC `a:real^N` o CONJUNCT2) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `e / norm(l:real^N)` THEN ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT] THEN X_GEN_TAC `x:real` THEN STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN CONJ_TAC THENL [MATCH_MP_TAC IN_AFFINE_ADD_MUL THEN ASM_REWRITE_TAC[AFFINE_AFFINE_HULL] THEN ASM_MESON_TAC[SUBSET; HULL_SUBSET; RELATIVE_INTERIOR_SUBSET]; REWRITE_TAC[NORM_ARITH `dist(a + x:real^N,a) = norm x`] THEN ASM_SIMP_TAC[NORM_MUL; GSYM REAL_LT_RDIV_EQ; NORM_POS_LT] THEN ASM_REAL_ARITH_TAC]; ALL_TAC] THEN ANTS_TAC THENL [REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_ELIM_THM] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[REAL_NOT_LT; REAL_LT_IMP_LE]] THEN FIRST_X_ASSUM(MP_TAC o SPEC `a:real^N` o MATCH_MP BOUNDED_SUBSET_BALL) THEN REWRITE_TAC[SUBSET; IN_BALL] THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `B / norm(l:real^N)` THEN ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT] THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[SUBSET] RELATIVE_INTERIOR_SUBSET)) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE BINDER_CONV [GSYM CONTRAPOS_THM]) THEN REWRITE_TAC[REAL_NOT_LT] THEN DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[NORM_ARITH `dist(a:real^N,a + x) = norm x`] THEN ASM_SIMP_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM; REAL_DIV_RMUL; NORM_EQ_0] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "1") (LABEL_TAC "2")) THEN EXISTS_TAC `d:real` THEN MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_LTE_TRANS THEN EXISTS_TAC `k:real` THEN ASM_MESON_TAC[REAL_NOT_LT; REAL_LT_IMP_LE]; DISCH_TAC] THEN MATCH_MP_TAC(TAUT `b /\ (b ==> a) ==> a /\ b`) THEN CONJ_TAC THENL [REWRITE_TAC[REAL_LE_LT] THEN ASM_MESON_TAC[VECTOR_ARITH `a + &0 % l:real^N = a`; REAL_NOT_LT; REAL_LT_IMP_LE]; DISCH_TAC] THEN REWRITE_TAC[IN_DIFF] THEN CONJ_TAC THENL [REWRITE_TAC[CLOSURE_APPROACHABLE] THEN X_GEN_TAC `x:real` THEN DISCH_TAC THEN EXISTS_TAC `a + (d - min d (x / &2 / norm(l:real^N))) % l` THEN CONJ_TAC THENL [MATCH_MP_TAC(REWRITE_RULE[SUBSET] RELATIVE_INTERIOR_SUBSET) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC(REAL_ARITH `&0 < x /\ &0 < d ==> d - min d x < d`) THEN ASM_SIMP_TAC[REAL_HALF; REAL_LT_DIV; NORM_POS_LT]; REWRITE_TAC[NORM_ARITH `dist(a + x:real^N,a + y) = norm(x - y)`] THEN REWRITE_TAC[GSYM VECTOR_SUB_RDISTRIB; NORM_MUL] THEN ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; NORM_POS_LT] THEN MATCH_MP_TAC(REAL_ARITH `&0 < x /\ x < y /\ &0 < d ==> abs((d - min d x) - d) < y`) THEN REWRITE_TAC[REAL_ARITH `x / &2 / y < x / y <=> &0 < x / y`] THEN ASM_SIMP_TAC[REAL_HALF; REAL_LT_DIV; NORM_POS_LT]]; DISCH_TAC THEN MP_TAC(ISPEC `s:real^N->bool` OPEN_IN_RELATIVE_INTERIOR) THEN REWRITE_TAC[open_in; GSYM MEMBER_NOT_EMPTY; IN_ELIM_THM] THEN DISCH_THEN(MP_TAC o SPEC `a + d % l:real^N` o CONJUNCT2) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` (CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "3"))) THEN REMOVE_THEN "2" (MP_TAC o SPEC `d + e / norm(l:real^N)`) THEN ASM_SIMP_TAC[NOT_IMP; REAL_ARITH `~(d + l <= d) <=> &0 < l`; REAL_LT_DIV; NORM_POS_LT] THEN X_GEN_TAC `x:real` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[REAL_NOT_LE] THEN DISCH_TAC THEN ASM_CASES_TAC `x < d` THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN REMOVE_THEN "3" MATCH_MP_TAC THEN CONJ_TAC THENL [MATCH_MP_TAC IN_AFFINE_ADD_MUL THEN ASM_REWRITE_TAC[AFFINE_AFFINE_HULL] THEN ASM_MESON_TAC[SUBSET; HULL_SUBSET; RELATIVE_INTERIOR_SUBSET]; REWRITE_TAC[NORM_ARITH `dist(a + x:real^N,a + y) = norm(x - y)`] THEN REWRITE_TAC[GSYM VECTOR_SUB_RDISTRIB; NORM_MUL] THEN ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; NORM_POS_LT] THEN ASM_REAL_ARITH_TAC]]]);; let RAY_TO_FRONTIER = prove (`!s a l:real^N. bounded s /\ a IN interior s /\ ~(l = vec 0) ==> ?d. &0 < d /\ (a + d % l) IN frontier s /\ !e. &0 <= e /\ e < d ==> (a + e % l) IN interior s`, REPEAT STRIP_TAC THEN REWRITE_TAC[frontier] THEN SUBGOAL_THEN `interior s:real^N->bool = relative_interior s` SUBST1_TAC THENL [ALL_TAC; REWRITE_TAC[GSYM relative_frontier] THEN MATCH_MP_TAC RAY_TO_RELATIVE_FRONTIER THEN ASM_REWRITE_TAC[]] THEN ASM_MESON_TAC[NOT_IN_EMPTY; RELATIVE_INTERIOR_NONEMPTY_INTERIOR; IN_UNIV; AFFINE_HULL_NONEMPTY_INTERIOR]);; let SEGMENT_TO_RELATIVE_FRONTIER = prove (`!s x y:real^N. convex s /\ bounded s /\ x IN relative_interior s /\ y IN s /\ ~(x = y /\ s = {x}) ==> ?z. z IN relative_frontier s /\ y IN segment[x,z] /\ segment(x,z) SUBSET relative_interior s`, SUBGOAL_THEN `!s x y:real^N. convex s /\ bounded s /\ x IN relative_interior s /\ y IN s /\ ~(x = y) ==> ?z. z IN relative_frontier s /\ y IN segment[x,z] /\ segment(x,z) SUBSET relative_interior s` ASSUME_TAC THENL [ALL_TAC; REPEAT GEN_TAC THEN ASM_CASES_TAC `x:real^N = y` THEN ASM_SIMP_TAC[] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE `~(s = {a}) ==> a IN s ==> ?b. ~(b = a) /\ b IN s`)) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `w:real^N` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`s:real^N->bool`; `y:real^N`; `w:real^N`]) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[ENDS_IN_SEGMENT]] THEN REPEAT STRIP_TAC THEN MP_TAC(SPECL [`s:real^N->bool`; `x:real^N`; `y - x:real^N`] RAY_TO_RELATIVE_FRONTIER) THEN ASM_REWRITE_TAC[VECTOR_ARITH `x + (y - x):real^N = y`; VECTOR_SUB_EQ] THEN ASM_SIMP_TAC[HULL_INC] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `x + d % (y - x):real^N` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[relative_frontier; IN_DIFF]] THEN REWRITE_TAC[IN_SEGMENT] THEN EXISTS_TAC `inv(d:real)` THEN ASM_SIMP_TAC[VECTOR_ADD_LDISTRIB; REAL_LE_INV_EQ; REAL_LT_IMP_LE; VECTOR_MUL_ASSOC; REAL_MUL_LINV; REAL_LT_IMP_NZ] THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_INV_LE_1; CONV_TAC VECTOR_ARITH] THEN REWRITE_TAC[GSYM REAL_NOT_LT] THEN DISCH_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `x:real^N`; `y:real^N`] IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT) THEN ASM_SIMP_TAC[NOT_IMP; REWRITE_RULE[SUBSET] CLOSURE_SUBSET] THEN REWRITE_TAC[SUBSET; IN_SEGMENT] THEN DISCH_THEN(MP_TAC o SPEC `x + d % (y - x):real^N`) THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_DIFF; relative_frontier]) THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `d:real` THEN ASM_REWRITE_TAC[] THEN CONV_TAC VECTOR_ARITH);; let SEGMENT_TO_RELATIVE_FRONTIER_SIMPLE = prove (`!s x:real^N. bounded s /\ x IN s /\ ~(s = {x}) ==> ?a b. a IN relative_frontier s /\ b IN relative_frontier s /\ x IN segment[a,b]`, REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `x:real^N` THEN REPEAT STRIP_TAC THEN ASM_CASES_TAC `(vec 0:real^N) IN relative_frontier s` THENL [ASM_MESON_TAC[SEGMENT_REFL; IN_SING]; ALL_TAC] THEN UNDISCH_TAC `~((vec 0:real^N) IN relative_frontier s)` THEN ASM_SIMP_TAC[relative_frontier; IN_DIFF; CLOSURE_INC] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE `~(s = {a}) ==> a IN s ==> ?b. b IN s /\ ~(b = a)`)) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `z:real^N` THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `vec 0:real^N`; `z:real^N`] RAY_TO_RELATIVE_FRONTIER) THEN ASM_SIMP_TAC[VECTOR_ADD_LID; HULL_INC] THEN DISCH_THEN(X_CHOOSE_THEN `u:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `u % z:real^N` THEN ASM_REWRITE_TAC[GSYM relative_frontier; GSYM IN_DIFF] THEN MP_TAC(ISPECL [`s:real^N->bool`; `vec 0:real^N`; `--z:real^N`] RAY_TO_RELATIVE_FRONTIER) THEN ASM_SIMP_TAC[VECTOR_ADD_LID; HULL_INC; VECTOR_NEG_EQ_0] THEN ANTS_TAC THENL [SUBST1_TAC(VECTOR_ARITH `--z:real^N = vec 0 - &1 % (z - vec 0)`) THEN ASM_SIMP_TAC[IN_AFFINE_SUB_MUL_DIFF; AFFINE_AFFINE_HULL; HULL_INC]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `v:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `v % (--z):real^N` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[IN_SEGMENT] THEN EXISTS_TAC `u:real / (u + v)` THEN REWRITE_TAC[VECTOR_MUL_ASSOC; VECTOR_MUL_RNEG] THEN REWRITE_TAC[GSYM VECTOR_MUL_LNEG; GSYM VECTOR_ADD_RDISTRIB] THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_LE_LDIV_EQ; REAL_LT_ADD] THEN CONV_TAC(RAND_CONV(RAND_CONV SYM_CONV)) THEN ASM_REWRITE_TAC[VECTOR_MUL_EQ_0] THEN MAP_EVERY UNDISCH_TAC [`&0 < u`; `&0 < v`] THEN CONV_TAC REAL_FIELD);; let SEGMENT_TO_FRONTIER_SIMPLE = prove (`!s x:real^N. bounded s /\ x IN s ==> ?a b. a IN frontier s /\ b IN frontier s /\ x IN segment[a,b]`, REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `x:real^N` THEN REPEAT GEN_TAC THEN ASM_CASES_TAC `s = {vec 0:real^N}` THENL [ASM_REWRITE_TAC[FRONTIER_SING; IN_SING] THEN MESON_TAC[SEGMENT_REFL; IN_SING]; REPEAT STRIP_TAC] THEN ASM_CASES_TAC `(vec 0:real^N) IN frontier s` THENL [ASM_MESON_TAC[SEGMENT_REFL; IN_SING]; ALL_TAC] THEN UNDISCH_TAC `~((vec 0:real^N) IN frontier s)` THEN ASM_SIMP_TAC[frontier; IN_DIFF; CLOSURE_INC] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE `~(s = {a}) ==> a IN s ==> ?b. b IN s /\ ~(b = a)`)) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `z:real^N` THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `vec 0:real^N`; `z:real^N`] RAY_TO_FRONTIER) THEN ASM_SIMP_TAC[VECTOR_ADD_LID; HULL_INC] THEN DISCH_THEN(X_CHOOSE_THEN `u:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `u % z:real^N` THEN ASM_REWRITE_TAC[GSYM frontier; GSYM IN_DIFF] THEN MP_TAC(ISPECL [`s:real^N->bool`; `vec 0:real^N`; `--z:real^N`] RAY_TO_FRONTIER) THEN ASM_SIMP_TAC[VECTOR_ADD_LID; HULL_INC; VECTOR_NEG_EQ_0] THEN DISCH_THEN(X_CHOOSE_THEN `v:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `v % (--z):real^N` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[IN_SEGMENT] THEN EXISTS_TAC `u:real / (u + v)` THEN REWRITE_TAC[VECTOR_MUL_ASSOC; VECTOR_MUL_RNEG] THEN REWRITE_TAC[GSYM VECTOR_MUL_LNEG; GSYM VECTOR_ADD_RDISTRIB] THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_LE_LDIV_EQ; REAL_LT_ADD] THEN CONV_TAC(RAND_CONV(RAND_CONV SYM_CONV)) THEN ASM_REWRITE_TAC[VECTOR_MUL_EQ_0] THEN MAP_EVERY UNDISCH_TAC [`&0 < u`; `&0 < v`] THEN CONV_TAC REAL_FIELD);; let SUBSET_CONVEX_HULL_RELATIVE_FRONTIER = prove (`!s:real^N->bool. bounded s /\ ~(?a. s = {a}) ==> s SUBSET convex hull (relative_frontier s)`, REPEAT STRIP_TAC THEN REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `x:real^N`] SEGMENT_TO_RELATIVE_FRONTIER_SIMPLE) THEN ANTS_TAC THENL [ASM SET_TAC[]; STRIP_TAC] THEN SUBGOAL_THEN `segment[a:real^N,b] SUBSET convex hull (relative_frontier s)` (fun th -> MP_TAC th THEN ASM SET_TAC[]) THEN REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN MATCH_MP_TAC HULL_MINIMAL THEN ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; CONVEX_CONVEX_HULL] THEN ASM_SIMP_TAC[HULL_INC]);; let SUBSET_CONVEX_HULL_FRONTIER = prove (`!s:real^N->bool. bounded s ==> s SUBSET convex hull (frontier s)`, REPEAT STRIP_TAC THEN REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `x:real^N`] SEGMENT_TO_FRONTIER_SIMPLE) THEN ANTS_TAC THENL [ASM SET_TAC[]; STRIP_TAC] THEN SUBGOAL_THEN `segment[a:real^N,b] SUBSET convex hull (frontier s)` (fun th -> MP_TAC th THEN ASM SET_TAC[]) THEN REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN MATCH_MP_TAC HULL_MINIMAL THEN ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; CONVEX_CONVEX_HULL] THEN ASM_SIMP_TAC[HULL_INC]);; let AFFINE_HULL_RELATIVE_FRONTIER_BOUNDED = prove (`!s:real^N->bool. bounded s /\ ~(?a. s = {a}) ==> affine hull (relative_frontier s) = affine hull s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [GEN_REWRITE_TAC RAND_CONV [GSYM AFFINE_HULL_CLOSURE] THEN MATCH_MP_TAC HULL_MONO THEN REWRITE_TAC[relative_frontier] THEN SET_TAC[]; MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[AFFINE_AFFINE_HULL] THEN TRANS_TAC SUBSET_TRANS `convex hull (relative_frontier s):real^N->bool` THEN REWRITE_TAC[CONVEX_HULL_SUBSET_AFFINE_HULL] THEN MATCH_MP_TAC SUBSET_CONVEX_HULL_RELATIVE_FRONTIER THEN ASM_REWRITE_TAC[]]);; let KREIN_MILMAN_RELATIVE_FRONTIER = prove (`!s:real^N->bool. convex s /\ compact s /\ ~(?a. s = {a}) ==> s = convex hull (relative_frontier s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN ASM_SIMP_TAC[SUBSET_CONVEX_HULL_RELATIVE_FRONTIER; COMPACT_IMP_BOUNDED] THEN ASM_SIMP_TAC[SUBSET_HULL] THEN ASM_SIMP_TAC[relative_frontier; CLOSURE_CLOSED; COMPACT_IMP_CLOSED] THEN SET_TAC[]);; let KREIN_MILMAN_RELATIVE_BOUNDARY = prove (`!s:real^N->bool. convex s /\ compact s /\ ~(?a. s = {a}) ==> s = convex hull (s DIFF relative_interior s)`, GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP KREIN_MILMAN_RELATIVE_FRONTIER) THEN ASM_SIMP_TAC[relative_frontier; CLOSURE_CLOSED; COMPACT_IMP_CLOSED]);; let KREIN_MILMAN_FRONTIER = prove (`!s:real^N->bool. convex s /\ compact s ==> s = convex hull (frontier s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN ASM_SIMP_TAC[SUBSET_CONVEX_HULL_FRONTIER; COMPACT_IMP_BOUNDED] THEN ASM_SIMP_TAC[SUBSET_HULL] THEN ASM_SIMP_TAC[frontier; CLOSURE_CLOSED; COMPACT_IMP_CLOSED] THEN SET_TAC[]);; let RELATIVE_FRONTIER_NOT_SING = prove (`!s a:real^N. bounded s ==> ~(relative_frontier s = {a})`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[RELATIVE_FRONTIER_EMPTY; NOT_INSERT_EMPTY] 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 ASM_CASES_TAC `s = {z:real^N}` THEN ASM_REWRITE_TAC[RELATIVE_FRONTIER_SING; NOT_INSERT_EMPTY] THEN SUBGOAL_THEN `?w:real^N. w IN s /\ ~(w = z)` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; REPEAT STRIP_TAC] THEN SUBGOAL_THEN `~((w:real^N) IN relative_frontier s /\ z IN relative_frontier s)` MP_TAC THENL [ASM SET_TAC[]; DISCH_TAC] THEN MAP_EVERY UNDISCH_TAC [`relative_frontier s = {a:real^N}`; `bounded(s:real^N->bool)`; `~(w:real^N = z)`; `(z:real^N) IN s`; `(w:real^N) IN s`; `~((w:real^N) IN relative_frontier s /\ z IN relative_frontier s)`] THEN POP_ASSUM_LIST(K ALL_TAC) THEN REWRITE_TAC[DE_MORGAN_THM] THEN MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`z:real^N`; `w:real^N`] THEN MATCH_MP_TAC(MESON[] `(!w z. Q w z <=> Q z w) /\ (!w z. P z ==> Q w z) ==> !w z. P w \/ P z ==> Q w z`) THEN CONJ_TAC THENL [MESON_TAC[]; REPEAT GEN_TAC] THEN DISCH_THEN(fun th -> REPEAT STRIP_TAC THEN MP_TAC th) THEN REWRITE_TAC[relative_frontier; IN_DIFF] THEN CONJ_TAC THENL [ASM_MESON_TAC[SUBSET; CLOSURE_SUBSET]; DISCH_TAC] THEN MP_TAC(GEN `d:real` (ISPECL [`s:real^N->bool`; `z:real^N`; `d % (w - z):real^N`] RAY_TO_RELATIVE_FRONTIER)) THEN ASM_SIMP_TAC[VECTOR_SUB_EQ; IN_AFFINE_ADD_MUL_DIFF; AFFINE_AFFINE_HULL; HULL_INC; VECTOR_MUL_EQ_0] THEN DISCH_THEN(fun th -> MP_TAC(SPEC `&1` th) THEN MP_TAC(SPEC `--(&1)` th)) THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[IN_SING] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` (STRIP_ASSUME_TAC o GSYM)) THEN ASM_REWRITE_TAC[VECTOR_MUL_RCANCEL; VECTOR_MUL_ASSOC; VECTOR_SUB_EQ; VECTOR_ARITH `a + x:real^N = a + y <=> x = y`] THEN ASM_REAL_ARITH_TAC);; let RELATIVE_INTERIOR_PCROSS = prove (`!s:real^M->bool t:real^N->bool. relative_interior(s PCROSS t) = relative_interior s PCROSS relative_interior t`, REPEAT STRIP_TAC THEN MAP_EVERY ASM_CASES_TAC [`s:real^M->bool = {}`; `t:real^N->bool = {}`] THEN ASM_REWRITE_TAC[PCROSS_EMPTY; RELATIVE_INTERIOR_EMPTY] THEN REWRITE_TAC[relative_interior; AFFINE_HULL_PCROSS] THEN REWRITE_TAC[EXTENSION; FORALL_PASTECART; IN_ELIM_THM; PASTECART_IN_PCROSS] THEN MAP_EVERY X_GEN_TAC [`x:real^M`; `y:real^N`] THEN EQ_TAC THENL [ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> r /\ q /\ p`] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^(M,N)finite_sum->bool` (CONJUNCTS_THEN ASSUME_TAC)) THEN FIRST_ASSUM(MP_TAC o MATCH_MP PASTECART_IN_INTERIOR_SUBTOPOLOGY) THEN REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN W(MP_TAC o PART_MATCH (funpow 3 rand) SUBSET_PCROSS o snd) THEN ASM SET_TAC[]; DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `v:real^M->bool` STRIP_ASSUME_TAC) (X_CHOOSE_THEN `w:real^N->bool` STRIP_ASSUME_TAC)) THEN EXISTS_TAC `(v:real^M->bool) PCROSS (w:real^N->bool)` THEN ASM_SIMP_TAC[PASTECART_IN_PCROSS; SUBSET_PCROSS; OPEN_IN_PCROSS]]);; let RELATIVE_FRONTIER_EQ_EMPTY = prove (`!s:real^N->bool. relative_frontier s = {} <=> affine s`, GEN_TAC THEN REWRITE_TAC[relative_frontier] THEN REWRITE_TAC[GSYM RELATIVE_INTERIOR_EQ_CLOSURE] THEN MP_TAC(ISPEC `s:real^N->bool` RELATIVE_INTERIOR_SUBSET) THEN MP_TAC(ISPEC `s:real^N->bool` CLOSURE_SUBSET) THEN SET_TAC[]);; let DIAMETER_BOUNDED_BOUND_LT = prove (`!s x y:real^N. bounded s /\ x IN relative_interior s /\ y IN closure s /\ ~(diameter s = &0) ==> norm(x - y) < diameter s`, let lemma = prove (`!s x y:real^N. bounded s /\ x IN relative_interior s /\ y IN s /\ ~(diameter s = &0) ==> norm(x - y) < diameter s`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM (MP_TAC o GEN_REWRITE_RULE I [IN_RELATIVE_INTERIOR_CBALL]) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC)) THEN ASM_SIMP_TAC[REAL_LT_LE; DIAMETER_BOUNDED_BOUND] THEN ASM_CASES_TAC `y:real^N = x` THEN ASM_SIMP_TAC[VECTOR_SUB_REFL; NORM_0] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN DISCH_THEN(MP_TAC o SPEC `x + e / norm(x - y) % (x - y):real^N`) THEN REWRITE_TAC[NOT_IMP; IN_INTER] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[IN_CBALL; NORM_ARITH `dist(x:real^M,x + y) = norm y`] THEN ASM_SIMP_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM; REAL_DIV_RMUL; NORM_EQ_0; VECTOR_SUB_EQ] THEN ASM_REAL_ARITH_TAC; MATCH_MP_TAC IN_AFFINE_ADD_MUL_DIFF THEN ASM_SIMP_TAC[HULL_INC; AFFINE_AFFINE_HULL]; DISCH_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `x + e / norm(x - y) % (x - y):real^N`; `y:real^N`] DIAMETER_BOUNDED_BOUND) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN REWRITE_TAC[VECTOR_ARITH `(x + e % (x - y)) - y:real^N = (&1 + e) % (x - y)`] THEN SIMP_TAC[NORM_MUL; REAL_ARITH `~(a * n <= n) <=> &0 < n * (a - &1)`] THEN MATCH_MP_TAC REAL_LT_MUL THEN ASM_REWRITE_TAC[NORM_POS_LT; VECTOR_SUB_EQ] THEN MATCH_MP_TAC(REAL_ARITH `&0 < e ==> &0 < abs(&1 + e) - &1`) THEN MATCH_MP_TAC REAL_LT_DIV THEN ASM_REWRITE_TAC[NORM_POS_LT; VECTOR_SUB_EQ]]) in REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`closure s:real^N->bool`; `x:real^N`; `y:real^N`] lemma) THEN ASM_SIMP_TAC[DIAMETER_CLOSURE; BOUNDED_CLOSURE] THEN DISCH_THEN MATCH_MP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `x IN s ==> s SUBSET t ==> x IN t`)) THEN MATCH_MP_TAC SUBSET_RELATIVE_INTERIOR THEN REWRITE_TAC[CLOSURE_SUBSET; AFFINE_HULL_CLOSURE]);; let DIAMETER_ATTAINED_RELATIVE_FRONTIER = prove (`!s:real^N->bool. bounded s /\ ~(diameter s = &0) ==> ?x y. x IN relative_frontier s /\ y IN relative_frontier s /\ norm(x - y) = diameter s`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[DIAMETER_EMPTY; relative_frontier] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPEC `closure s:real^N->bool` DIAMETER_COMPACT_ATTAINED) THEN ASM_SIMP_TAC[COMPACT_CLOSURE; CLOSURE_EQ_EMPTY; DIAMETER_CLOSURE] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN STRIP_TAC THEN ASM_REWRITE_TAC[IN_DIFF] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPEC `s:real^N->bool` DIAMETER_BOUNDED_BOUND_LT) THENL [DISCH_THEN(MP_TAC o SPECL [`x:real^N`; `y:real^N`]); DISCH_THEN(MP_TAC o SPECL [`y:real^N`; `x:real^N`])] THEN ASM_MESON_TAC[REAL_LT_REFL; NORM_SUB]);; let DIAMETER_RELATIVE_FRONTIER = prove (`!s:real^N->bool. bounded s /\ ~(?a. s = {a}) ==> diameter(relative_frontier s) = diameter s`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[RELATIVE_FRONTIER_EMPTY] THEN REWRITE_TAC[relative_frontier] THEN GEN_REWRITE_TAC RAND_CONV [GSYM DIAMETER_CLOSURE] THEN ASM_SIMP_TAC[GSYM REAL_LE_ANTISYM] THEN ASM_SIMP_TAC[SUBSET_DIFF; DIAMETER_SUBSET; BOUNDED_CLOSURE] THEN ASM_SIMP_TAC[DIAMETER_CLOSURE] THEN MP_TAC(ISPEC `s:real^N->bool` DIAMETER_ATTAINED_RELATIVE_FRONTIER) THEN ASM_SIMP_TAC[DIAMETER_EQ_0; relative_frontier] THEN STRIP_TAC THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN MATCH_MP_TAC DIAMETER_BOUNDED_BOUND THEN ASM_SIMP_TAC[BOUNDED_CLOSURE; BOUNDED_DIFF]);; let DIAMETER_ATTAINED_FRONTIER = prove (`!s:real^N->bool. bounded s /\ ~(diameter s = &0) ==> ?x y. x IN frontier s /\ y IN frontier s /\ norm(x - y) = diameter s`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP DIAMETER_ATTAINED_RELATIVE_FRONTIER) THEN REWRITE_TAC[frontier; relative_frontier; IN_DIFF] THEN MESON_TAC[REWRITE_RULE[SUBSET] INTERIOR_SUBSET_RELATIVE_INTERIOR]);; let DIAMETER_FRONTIER = prove (`!s:real^N->bool. bounded s ==> diameter(frontier s) = diameter s`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `?a:real^N. s = {a}` THENL [ASM_MESON_TAC[FRONTIER_SING]; ALL_TAC] THEN MATCH_MP_TAC(REAL_ARITH `!r. r <= f /\ f <= s /\ r = s ==> f = s`) THEN EXISTS_TAC `diameter(closure s DIFF relative_interior s:real^N->bool)` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC DIAMETER_SUBSET THEN ASM_SIMP_TAC[BOUNDED_FRONTIER] THEN REWRITE_TAC[frontier] THEN MP_TAC(ISPEC `s:real^N->bool` INTERIOR_SUBSET_RELATIVE_INTERIOR) THEN SET_TAC[]; GEN_REWRITE_TAC RAND_CONV [GSYM DIAMETER_CLOSURE] THEN MATCH_MP_TAC DIAMETER_SUBSET THEN ASM_SIMP_TAC[BOUNDED_CLOSURE; frontier; SUBSET_DIFF]; ASM_SIMP_TAC[DIAMETER_RELATIVE_FRONTIER; GSYM relative_frontier]]);; let CLOSEST_POINT_IN_RELATIVE_INTERIOR = prove (`!s x:real^N. closed s /\ ~(s = {}) /\ x IN affine hull s ==> ((closest_point s x) IN relative_interior s <=> x IN relative_interior s)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `(x:real^N) IN s` THEN ASM_SIMP_TAC[CLOSEST_POINT_SELF] THEN MATCH_MP_TAC(TAUT `~q /\ ~p ==> (p <=> q)`) THEN CONJ_TAC THENL [ASM_MESON_TAC[RELATIVE_INTERIOR_SUBSET; SUBSET]; STRIP_TAC] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_RELATIVE_INTERIOR_CBALL]) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `~(closest_point s (x:real^N) = x)` ASSUME_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`s:real^N->bool`; `x:real^N`; `closest_point s x - (min (&1) (e / norm(closest_point s x - x))) % (closest_point s x - x):real^N`] CLOSEST_POINT_LE) THEN ASM_REWRITE_TAC[dist; NOT_IMP; VECTOR_ARITH `x - (y - e % (y - x)):real^N = (&1 - e) % (x - y)`] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN REWRITE_TAC[IN_CBALL; IN_INTER] THEN CONJ_TAC THENL [REWRITE_TAC[NORM_ARITH `dist(a:real^N,a - x) = norm x`] THEN REWRITE_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM] THEN ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN MATCH_MP_TAC(REAL_ARITH `&0 <= a ==> abs(min (&1) a) <= a`) THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; REAL_LE_DIV; NORM_POS_LE]; MATCH_MP_TAC IN_AFFINE_SUB_MUL_DIFF THEN ASM_SIMP_TAC[AFFINE_AFFINE_HULL; HULL_INC]]; REWRITE_TAC[NORM_MUL; REAL_ARITH `~(n <= a * n) <=> &0 < (&1 - a) * n`] THEN MATCH_MP_TAC REAL_LT_MUL THEN ASM_SIMP_TAC[NORM_POS_LT; VECTOR_SUB_EQ] THEN MATCH_MP_TAC(REAL_ARITH `&0 < e /\ e <= &1 ==> &0 < &1 - abs(&1 - e)`) THEN REWRITE_TAC[REAL_MIN_LE; REAL_LT_MIN; REAL_LT_01; REAL_LE_REFL] THEN ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT; VECTOR_SUB_EQ]]);; let CLOSEST_POINT_IN_RELATIVE_FRONTIER = prove (`!s x:real^N. closed s /\ ~(s = {}) /\ x IN affine hull s DIFF relative_interior s ==> closest_point s x IN relative_frontier s`, SIMP_TAC[relative_frontier; IN_DIFF; CLOSEST_POINT_IN_RELATIVE_INTERIOR] THEN MESON_TAC[CLOSURE_SUBSET; CLOSEST_POINT_IN_SET; SUBSET]);; let IN_RELATIVE_INTERIOR_IN_OPEN_SEGMENT = prove (`!s x a:real^N. convex s /\ x IN relative_interior s /\ a IN affine hull s /\ ~(x = a) ==> ?b. b IN s /\ x IN segment(a,b)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_RELATIVE_INTERIOR_CBALL]) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC)) THEN EXISTS_TAC `x + d / norm(x - a) % (x - a:real^N)` THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET]) THEN REWRITE_TAC[IN_INTER; IN_CBALL] THEN ASM_SIMP_TAC[IN_AFFINE_ADD_MUL_DIFF; HULL_INC; AFFINE_AFFINE_HULL] THEN REWRITE_TAC[NORM_ARITH `dist(x:real^N,x + y) = norm y`] THEN REWRITE_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM] THEN ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_EQ_0; VECTOR_SUB_EQ] THEN ASM_REAL_ARITH_TAC; ONCE_REWRITE_TAC[segment] THEN ASM_REWRITE_TAC[IN_DIFF; IN_INSERT; NOT_IN_EMPTY] THEN REWRITE_TAC[VECTOR_ARITH `a:real^N = a + x <=> x = vec 0`] THEN ASM_SIMP_TAC[VECTOR_MUL_EQ_0; REAL_DIV_EQ_0; NORM_EQ_0; VECTOR_SUB_EQ; REAL_LT_IMP_NZ] THEN ASM_REWRITE_TAC[IN_SEGMENT; VECTOR_MUL_EQ_0; VECTOR_SUB_EQ; VECTOR_ARITH `x:real^N = (&1 - u) % a + u % (x + v % (x - a)) <=> (&1 - u * (&1 + v)) % (x - a) = vec 0`] THEN ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT; VECTOR_SUB_EQ; REAL_FIELD `&0 < d ==> (&1 - u * (&1 + d) = &0 <=> u = inv(&1 + d))`] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> r /\ p /\ q`] THEN ASM_SIMP_TAC[UNWIND_THM2; REAL_LE_INV_EQ; REAL_INV_LE_1; REAL_ARITH `&0 < d ==> &1 <= &1 + d /\ &0 <= &1 + d`; REAL_LT_DIV; NORM_POS_LT; VECTOR_SUB_EQ]]);; let IN_RELATIVE_INTERIOR_IN_OPEN_SEGMENT_STRONG = prove (`!s x a:real^N. convex s /\ x IN relative_interior s /\ a IN affine hull s /\ ~(x = a) ==> ?b. b IN relative_interior s /\ x IN segment(a,b)`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(X_CHOOSE_THEN `b:real^N` STRIP_ASSUME_TAC o MATCH_MP IN_RELATIVE_INTERIOR_IN_OPEN_SEGMENT) THEN EXISTS_TAC `midpoint(x:real^N,b)` THEN CONJ_TAC THENL [MP_TAC(ISPECL [`s:real^N->bool`; `x:real^N`; `b:real^N`] IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT) THEN ASM_SIMP_TAC[REWRITE_RULE[SUBSET] CLOSURE_SUBSET] THEN DISCH_THEN(MATCH_MP_TAC o REWRITE_RULE[SUBSET]) THEN ASM_REWRITE_TAC[MIDPOINT_IN_SEGMENT] THEN ASM_MESON_TAC[ENDS_NOT_IN_SEGMENT]; ONCE_REWRITE_TAC[segment] THEN ASM_REWRITE_TAC[IN_DIFF; IN_INSERT; NOT_IN_EMPTY] THEN CONJ_TAC THENL [REWRITE_TAC[GSYM BETWEEN_IN_SEGMENT] THEN MATCH_MP_TAC BETWEEN_TRANS_2 THEN EXISTS_TAC `b:real^N` THEN REWRITE_TAC[BETWEEN_MIDPOINT] THEN ASM_MESON_TAC[BETWEEN_IN_SEGMENT; segment; IN_DIFF; SEGMENT_SYM]; CONV_TAC(RAND_CONV SYM_CONV) THEN REWRITE_TAC[MIDPOINT_EQ_ENDPOINT] THEN ASM_MESON_TAC[ENDS_NOT_IN_SEGMENT]]]);; let IN_RELATIVE_INTERIOR_IN_OPEN_SEGMENT_EQ = prove (`!s x:real^N. convex s ==> (x IN relative_interior s <=> ~(s = {}) /\ !a. a IN s /\ ~(a = x) ==> ?b. b IN s /\ x IN segment(a,b))`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[RELATIVE_INTERIOR_EMPTY; NOT_IN_EMPTY] THEN EQ_TAC THEN STRIP_TAC THENL [ASM_MESON_TAC[HULL_INC; IN_RELATIVE_INTERIOR_IN_OPEN_SEGMENT]; ALL_TAC] THEN MP_TAC(ISPEC `s:real^N->bool` RELATIVE_INTERIOR_EQ_EMPTY) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `z:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `z:real^N`) THEN ASM_CASES_TAC `x:real^N = z` THEN ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[REWRITE_RULE[SUBSET] RELATIVE_INTERIOR_SUBSET] THEN DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`s:real^N->bool`; `z:real^N`; `y:real^N`] IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT) THEN ASM_SIMP_TAC[REWRITE_RULE[SUBSET] CLOSURE_SUBSET] THEN ASM SET_TAC[]);; let INTER_RELATIVE_INTERIOR_SUBSET = prove (`!s t:real^N->bool. convex s /\ convex t ==> relative_interior s INTER relative_interior t SUBSET relative_interior(s INTER t)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s INTER t:real^N->bool = {}` THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `i1 INTER i2 = {} ==> k1 SUBSET i1 /\ k2 SUBSET i2 ==> k1 INTER k2 SUBSET u`)) THEN REWRITE_TAC[RELATIVE_INTERIOR_SUBSET]; REWRITE_TAC[SUBSET; IN_INTER] THEN X_GEN_TAC `x:real^N`] THEN ASM_SIMP_TAC[IN_RELATIVE_INTERIOR_IN_OPEN_SEGMENT_EQ; CONVEX_INTER] THEN ONCE_REWRITE_TAC[RIGHT_AND_FORALL_THM] THEN REWRITE_TAC[AND_FORALL_THM; IN_INTER] 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(CONJUNCTS_THEN2 (CONJUNCTS_THEN2 ASSUME_TAC (X_CHOOSE_THEN `u:real^N` STRIP_ASSUME_TAC)) (CONJUNCTS_THEN2 ASSUME_TAC (X_CHOOSE_THEN `v:real^N` STRIP_ASSUME_TAC))) THEN MP_TAC(ISPECL [`y:real^N`; `u:real^N`; `v:real^N`; `x:real^N`] BETWEEN_RESTRICTED_CASES) THEN ANTS_TAC THENL [ASM_MESON_TAC[BETWEEN_IMP_COLLINEAR; INSERT_AC; BETWEEN_IN_SEGMENT; REWRITE_RULE[SUBSET] SEGMENT_OPEN_SUBSET_CLOSED]; REWRITE_TAC[BETWEEN_IN_SEGMENT] THEN STRIP_TAC THENL [EXISTS_TAC `u:real^N` THEN MP_TAC(ISPEC `t:real^N->bool` CONVEX_CONTAINS_SEGMENT) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPECL [`y:real^N`; `v:real^N`]) THEN ASM_REWRITE_TAC[SUBSET] THEN ASM SET_TAC[]; EXISTS_TAC `v:real^N` THEN MP_TAC(ISPEC `s:real^N->bool` CONVEX_CONTAINS_SEGMENT) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPECL [`y:real^N`; `u:real^N`]) THEN ASM_REWRITE_TAC[SUBSET] THEN ASM SET_TAC[]]]);; let RELATIVE_INTERIOR_INTER = prove (`!s t:real^N->bool. convex s /\ convex t /\ ~(relative_interior s INTER relative_interior t = {}) ==> relative_interior(s INTER t) = relative_interior s INTER relative_interior t`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[GSYM SUBSET_ANTISYM_EQ; INTER_RELATIVE_INTERIOR_SUBSET] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN REWRITE_TAC[IN_INTER; SUBSET; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `v:real^N` THEN STRIP_TAC THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MP_TAC(ISPECL [`s INTER t:real^N->bool`; `x:real^N`; `v:real^N`] IN_RELATIVE_INTERIOR_IN_OPEN_SEGMENT) THEN ASM_CASES_TAC `x:real^N = v` THEN ASM_SIMP_TAC[CONVEX_INTER; IN_INTER] THEN ANTS_TAC THENL [MATCH_MP_TAC HULL_INC THEN MP_TAC(ISPEC `s:real^N->bool` RELATIVE_INTERIOR_SUBSET) THEN MP_TAC(ISPEC `t:real^N->bool` RELATIVE_INTERIOR_SUBSET) THEN ASM SET_TAC[]; DISCH_THEN(X_CHOOSE_THEN `z:real^N` STRIP_ASSUME_TAC) THEN CONJ_TAC THENL [MP_TAC(ISPECL [`s:real^N->bool`; `v:real^N`; `z:real^N`] IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT); MP_TAC(ISPECL [`t:real^N->bool`; `v:real^N`; `z:real^N`] IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT)] THEN ASM_SIMP_TAC[REWRITE_RULE[SUBSET] CLOSURE_SUBSET] THEN ASM SET_TAC[]]);; let SUBSET_RELATIVE_INTERIOR_INTERSECTING_CONVEX = prove (`!s t:real^N->bool. convex s /\ convex t /\ s SUBSET t /\ ~(s INTER relative_interior t = {}) ==> relative_interior s SUBSET relative_interior t`, REPEAT STRIP_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 `a:real^N` THEN STRIP_TAC THEN REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `x:real^N = a` THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `x:real^N`; `a:real^N`] IN_RELATIVE_INTERIOR_IN_OPEN_SEGMENT) THEN ASM_SIMP_TAC[HULL_INC] THEN DISCH_THEN(X_CHOOSE_THEN `b:real^N` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> a IN s ==> a IN t`) THEN MATCH_MP_TAC IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT THEN ASM_MESON_TAC[SUBSET; CLOSURE_SUBSET]);; let CONVEX_HULL_SPHERE = prove (`!a:real^N r. convex hull (sphere(a,r)) = cball(a,r)`, REPEAT GEN_TAC THEN REWRITE_TAC[GSYM FRONTIER_CBALL] THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC KREIN_MILMAN_FRONTIER THEN REWRITE_TAC[CONVEX_CBALL; COMPACT_CBALL]);; let SPHERE_SUBSET_CONVEX = prove (`!s a:real^N r. convex s ==> (sphere(a,r) SUBSET s <=> cball(a,r) SUBSET s)`, REWRITE_TAC[GSYM CONVEX_HULL_SPHERE] THEN CONV_TAC(ONCE_DEPTH_CONV SYM_CONV) THEN REWRITE_TAC[SUBSET_HULL]);; let DIAMETER_SPHERE = prove (`!a:real^N r. diameter(sphere(a,r)) = if r < &0 then &0 else &2 * r`, REWRITE_TAC[GSYM FRONTIER_CBALL] THEN ASM_SIMP_TAC[DIAMETER_FRONTIER; BOUNDED_CBALL; DIAMETER_CBALL]);; (* ------------------------------------------------------------------------- *) (* Small move from (relative frontier of) convex set. *) (* ------------------------------------------------------------------------- *) let CONVEX_NEARBY_IN_SCALING = prove (`!s:real^N->bool r. convex s /\ vec 0 IN relative_interior s /\ &1 < r ==> ?d. &0 < d /\ !x y. x IN s /\ y IN affine hull s /\ dist(x,y) <= d ==> y IN IMAGE (\x. r % x) s`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_RELATIVE_INTERIOR_CBALL]) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC] THEN DISCH_THEN(X_CHOOSE_THEN `a:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `a * (r - &1)` THEN ASM_SIMP_TAC[REAL_LT_MUL; REAL_SUB_LT] THEN REPEAT STRIP_TAC THEN SUBST1_TAC(REAL_ARITH `r = &1 + (r - &1)`) THEN ASM_SIMP_TAC[GSYM CONVEX_SUMS_MULTIPLES; VECTOR_MUL_LID; REAL_POS; REAL_SUB_LE; REAL_LT_IMP_LE] THEN REWRITE_TAC[IN_ELIM_THM] THEN MAP_EVERY EXISTS_TAC [`x:real^N`; `inv(r - &1) % (y - x):real^N`] THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; REAL_LT_IMP_NZ; REAL_SUB_LT] THEN CONJ_TAC THENL [ALL_TAC; CONV_TAC VECTOR_ARITH] THEN FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN REWRITE_TAC[IN_CBALL_0; IN_INTER; NORM_MUL; REAL_ABS_INV] THEN ASM_SIMP_TAC[SPAN_MUL; SPAN_SUB; SPAN_SUPERSET] THEN ASM_SIMP_TAC[real_abs; REAL_SUB_LE; REAL_LT_IMP_LE] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ONCE_REWRITE_TAC[NORM_SUB] THEN ASM_SIMP_TAC[GSYM real_div; REAL_LE_LDIV_EQ; REAL_SUB_LT] THEN ASM_REWRITE_TAC[GSYM dist]);; let CONVEX_NEARBY_IN_SCALING_RELATIVE_INTERIOR = prove (`!s:real^N->bool r. convex s /\ vec 0 IN relative_interior s /\ &1 < r ==> ?d. &0 < d /\ !x y. x IN s /\ y IN affine hull s /\ dist(x,y) <= d ==> y IN IMAGE (\x. r % x) (relative_interior s)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `(r + &1) / &2`] CONVEX_NEARBY_IN_SCALING) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] 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^N`; `y:real^N`] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `y:real^N`]) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> x IN s ==> x IN t`) THEN SUBGOAL_THEN `(\x:real^N. (r + &1) / &2 % x) = (\x. r % x) o (\x. (r + &1) / (&2 * r) % x)` SUBST1_TAC THENL [REWRITE_TAC[o_DEF; VECTOR_MUL_ASSOC] THEN ABS_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN UNDISCH_TAC `&1 < r` THEN CONV_TAC REAL_FIELD; REWRITE_TAC[IMAGE_o] THEN MATCH_MP_TAC IMAGE_SUBSET] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `z:real^N` THEN STRIP_TAC THEN ONCE_REWRITE_TAC[VECTOR_ARITH `a % x:real^N = x - (&1 - a) % (x - vec 0)`] THEN MATCH_MP_TAC IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SHRINK THEN ASM_SIMP_TAC[CLOSURE_INC] THEN MATCH_MP_TAC(REAL_ARITH `&0 < x /\ x < &1 ==> &0 < &1 - x /\ &1 - x <= &1`) THEN ASM_SIMP_TAC[REAL_LT_LDIV_EQ; REAL_LT_RDIV_EQ; REAL_ARITH `&1 < x ==> &0 < &2 * x`] THEN ASM_REAL_ARITH_TAC);; let CONVEX_NEARBY_NOT_IN_SCALING = prove (`!s:real^N->bool r. convex s /\ vec 0 IN relative_interior s /\ &0 < r /\ r < &1 ==> ?d. &0 < d /\ !x y. x IN relative_frontier s /\ dist(x,y) <= d ==> ~(y IN IMAGE (\x. r % x) s)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `inv r:real`] CONVEX_NEARBY_IN_SCALING_RELATIVE_INTERIOR) THEN ASM_SIMP_TAC[REAL_INV_1_LT] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `d * r:real` THEN ASM_SIMP_TAC[REAL_LT_MUL] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN REWRITE_TAC[relative_frontier; IN_DIFF] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`inv r % y:real^N`; `inv r % x:real^N`]) THEN ASM_REWRITE_TAC[DIST_MUL; IN_IMAGE; NOT_IMP; REAL_ABS_INV] THEN ASM_SIMP_TAC[VECTOR_MUL_LCANCEL; REAL_LT_IMP_NZ; REAL_LT_INV_EQ] THEN ASM_SIMP_TAC[real_abs; REAL_LT_IMP_LE; UNWIND_THM1; CONTRAPOS_THM] THEN DISCH_THEN(X_CHOOSE_THEN `z:real^N` (CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC)) THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; REAL_LT_IMP_NZ] THEN ASM_REWRITE_TAC[VECTOR_MUL_LID] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_SIMP_TAC[GSYM real_div; REAL_LE_LDIV_EQ] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP (REWRITE_RULE[SUBSET] RELATIVE_INTERIOR_SUBSET)) THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[SUBSET] CLOSURE_SUBSET_AFFINE_HULL)) THEN ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC; SPAN_MUL]);; (* ------------------------------------------------------------------------- *) (* Basic closure properties for "is_interval". *) (* ------------------------------------------------------------------------- *) let IS_INTERVAL_RELATIVE_INTERIOR = prove (`!s:real^N->bool. is_interval s ==> is_interval(relative_interior s)`, REWRITE_TAC[is_interval; IN_RELATIVE_INTERIOR_CBALL] THEN GEN_TAC THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`; `x:real^N`] THEN DISCH_THEN(CONJUNCTS_THEN2 (CONJUNCTS_THEN2 ASSUME_TAC (X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC)) MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 (CONJUNCTS_THEN2 ASSUME_TAC (X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC)) ASSUME_TAC) THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [ASM_METIS_TAC[]; DISCH_TAC] THEN EXISTS_TAC `min d e:real` THEN ASM_REWRITE_TAC[REAL_LT_MIN] THEN REWRITE_TAC[SUBSET; IN_CBALL; IN_INTER] THEN X_GEN_TAC `y:real^N` THEN STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN MAP_EVERY EXISTS_TAC [`a + (y - x):real^N`; `b + (y - x):real^N`] THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_SUB_COMPONENT] THEN REWRITE_TAC[REAL_ARITH `a + y - x <= y <=> a <= x`] THEN ASM_REWRITE_TAC[REAL_ARITH `y <= b + y - x <=> x <= b`] THEN CONJ_TAC THENL [UNDISCH_TAC `cball(a:real^N,d) INTER affine hull s SUBSET s`; UNDISCH_TAC `cball(b:real^N,e) INTER affine hull s SUBSET s`] THEN REWRITE_TAC[SUBSET] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_CBALL; IN_INTER] THEN (CONJ_TAC THENL [UNDISCH_TAC `dist(x:real^N,y) <= min d e` THEN CONV_TAC NORM_ARITH; ONCE_REWRITE_TAC[VECTOR_ARITH `a + b:real^N = a + &1 % b`] THEN MATCH_MP_TAC IN_AFFINE_ADD_MUL_DIFF THEN ASM_SIMP_TAC[AFFINE_AFFINE_HULL; HULL_INC]]));; let IS_INTERVAL_INTERIOR = prove (`!s:real^N->bool. is_interval s ==> is_interval(interior s)`, GEN_TAC THEN ASM_CASES_TAC `interior s:real^N->bool = {}` THEN ASM_REWRITE_TAC[IS_INTERVAL_EMPTY] THEN ASM_SIMP_TAC[GSYM RELATIVE_INTERIOR_NONEMPTY_INTERIOR] THEN REWRITE_TAC[IS_INTERVAL_RELATIVE_INTERIOR]);; let IS_INTERVAL_CLOSURE = prove (`!s:real^N->bool. is_interval s ==> is_interval(closure s)`, let lemma = prove (`!a b u v. (u <= x /\ x <= v \/ v <= x /\ x <= u) /\ abs(a - u) < e /\ abs(b - v) < e ==> ?y. (a <= y /\ y <= b \/ b <= y /\ y <= a) /\ abs(y - x) < e`, REPEAT GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC(MESON[] `!a b c. P a \/ P b \/ P c ==> ?x. P x`) THEN MAP_EVERY EXISTS_TAC [`x:real`; `a:real`; `b:real`] THEN ASM_REAL_ARITH_TAC) in REWRITE_TAC[is_interval; CLOSURE_APPROACHABLE] THEN GEN_TAC THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`; `x:real^N`] THEN REWRITE_TAC[CONJ_ASSOC; AND_FORALL_THM] THEN STRIP_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `e / &(dimindex(:N))`) THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; DIMINDEX_GE_1; LE_1] THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `u:real^N` STRIP_ASSUME_TAC) (X_CHOOSE_THEN `v:real^N` STRIP_ASSUME_TAC)) THEN SUBGOAL_THEN `!i. 1 <= i /\ i <= dimindex(:N) ==> ?y. ((u:real^N)$i <= y /\ y <= (v:real^N)$i \/ v$i <= y /\ y <= u$i) /\ abs(y - (x:real^N)$i) < e / &(dimindex(:N))` MP_TAC THENL [X_GEN_TAC `i:num` THEN STRIP_TAC THEN MATCH_MP_TAC lemma THEN MAP_EVERY EXISTS_TAC [`(a:real^N)$i`; `(b:real^N)$i`] THEN ASM_SIMP_TAC[GSYM VECTOR_SUB_COMPONENT] THEN CONJ_TAC THEN W(MP_TAC o PART_MATCH lhand COMPONENT_LE_NORM o lhand o snd) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LET_TRANS) THEN ASM_REWRITE_TAC[GSYM dist]; REWRITE_TAC[LAMBDA_SKOLEM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`u:real^N`; `v:real^N`; `y:real^N`]) THEN ASM_SIMP_TAC[] THEN DISCH_TAC THEN REWRITE_TAC[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_LET_TRANS) THEN MATCH_MP_TAC SUM_BOUND_LT_GEN THEN ASM_SIMP_TAC[FINITE_NUMSEG; IN_NUMSEG; CARD_NUMSEG_1; VECTOR_SUB_COMPONENT; NUMSEG_EMPTY; NOT_LT; DIMINDEX_GE_1]]);; (* ------------------------------------------------------------------------- *) (* Shrinking space to a ball while preserving convexity. *) (* ------------------------------------------------------------------------- *) let CONVEX_PREIMAGE_CONCAVE_SCALING = prove (`!f s t:real^N->bool. convex s /\ convex t /\ vec 0 IN s /\ (\x. --f x) convex_on t /\ (!x. x IN t ==> &0 < f x) ==> convex {x | x IN t /\ (inv(f x) % x) IN s}`, REPEAT GEN_TAC THEN STRIP_TAC THEN REWRITE_TAC[CONVEX_CONTAINS_SEGMENT; IN_ELIM_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN STRIP_TAC THEN REWRITE_TAC[SET_RULE `s SUBSET {x | x IN t /\ Q x} <=> s SUBSET t /\ s SUBSET {x | Q x}`] THEN CONJ_TAC THENL [ASM_MESON_TAC[CONVEX_CONTAINS_SEGMENT]; ALL_TAC] THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MATCH_MP_TAC(SET_RULE `!s. x IN s /\ s SUBSET t ==> x IN t`) THEN EXISTS_TAC `convex hull {vec 0:real^N,inv(f a) % a,inv(f b) % b}` THEN CONJ_TAC THENL [REWRITE_TAC[CONVEX_HULL_3; IN_ELIM_THM; VECTOR_MUL_RZERO] THEN REWRITE_TAC[TAUT `(p /\ q /\ r /\ s) /\ t <=> s /\ q /\ r /\ p /\ t`] THEN ONCE_REWRITE_TAC[MESON[] `(?a b c. P a b c) <=> (?b c a. P a b c)`] THEN REWRITE_TAC[REAL_ARITH `u + v = &1 <=> u = &1 - v`; UNWIND_THM2] THEN REWRITE_TAC[VECTOR_ADD_LID; REAL_SUB_LE] THEN SUBGOAL_THEN `(x:real^N) IN t` ASSUME_TAC THENL [ASM_MESON_TAC[CONVEX_CONTAINS_SEGMENT; SUBSET]; ALL_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 o GSYM)) THEN EXISTS_TAC `(&1 - u) * (f:real^N->real) a / f x` THEN EXISTS_TAC `u * (f:real^N->real) b / f x` THEN ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 5) [REAL_LE_MUL; REAL_LE_DIV; REAL_SUB_LE; REAL_LT_IMP_LE] THEN CONJ_TAC THENL [ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_ARITH `u * a / c + v * b / c:real = (u * a + v * b) / c`] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a:real^N`; `b:real^N`; `&1 - u`; `u:real`] o GEN_REWRITE_RULE I [convex_on]) THEN ASM_REWRITE_TAC[ENDS_IN_SEGMENT; REAL_SUB_LE] THEN REAL_ARITH_TAC; ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_FIELD `&0 < a ==> (u * a / b) * inv a = inv b * u`] THEN ASM_REWRITE_TAC[GSYM VECTOR_MUL_ASSOC; GSYM VECTOR_ADD_LDISTRIB]]; MATCH_MP_TAC HULL_MINIMAL THEN ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET]]);; let CONVEXITY_PRESERVING_SHRINK_0 = prove (`?f g. homeomorphism ((:real^N),ball(vec 0:real^N,&1)) (f,g) /\ (!s. conic hull (IMAGE f s) = conic hull s) /\ (!s. vec 0 IN s ==> vec 0 IN IMAGE f s) /\ (!s. convex s /\ vec 0 IN s ==> convex(IMAGE f s)) /\ (!s. vec 0 IN relative_interior s ==> vec 0 IN relative_interior(IMAGE f s))`, ABBREV_TAC `f = \x:real^N. inv(&1 + norm x) % x` THEN ABBREV_TAC `g = \x:real^N. inv(&1 - norm x) % x` THEN MAP_EVERY EXISTS_TAC [`f:real^N->real^N`; `g:real^N->real^N`] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [REWRITE_TAC[HOMEOMORPHISM; SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM] THEN MAP_EVERY EXPAND_TAC ["f"; "g"] THEN REWRITE_TAC[NORM_MUL] THEN REWRITE_TAC[IN_BALL_0; REAL_ABS_INV; IN_UNIV] THEN SIMP_TAC[real_abs; NORM_MUL; REAL_LE_INV_EQ; REAL_SUB_LE; REAL_LT_IMP_LE; NORM_ARITH `&0 <= &1 + norm(x:real^N)`] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_MUL THEN REWRITE_TAC[CONTINUOUS_ON_ID; o_DEF] THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN REWRITE_TAC[LIFT_ADD; NORM_ARITH `~(&1 + norm(x:real^N) = &0)`] THEN SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_CONST; REWRITE_RULE[o_DEF] CONTINUOUS_ON_LIFT_NORM]; ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[VECTOR_MUL_ASSOC; GSYM real_div] THEN SIMP_TAC[REAL_LT_LDIV_EQ; NORM_ARITH `&0 < &1 + norm(x:real^N)`] THEN REAL_ARITH_TAC; MATCH_MP_TAC CONTINUOUS_ON_MUL THEN REWRITE_TAC[CONTINUOUS_ON_ID; o_DEF] THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN SIMP_TAC[LIFT_SUB; IN_ELIM_THM; REAL_SUB_0; REAL_LT_IMP_NE] THEN SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; REWRITE_RULE[o_DEF] CONTINUOUS_ON_LIFT_NORM] THEN SIMP_TAC[IN_BALL_0; REAL_LT_IMP_NE]; X_GEN_TAC `x:real^N` THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN GEN_REWRITE_TAC RAND_CONV [GSYM VECTOR_MUL_LID] THEN AP_THM_TAC THEN AP_TERM_TAC THEN MP_TAC(ISPEC `x:real^N` NORM_POS_LE) THEN CONV_TAC REAL_FIELD; GEN_TAC THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN MATCH_MP_TAC(MESON[VECTOR_MUL_LID] `(P ==> a = &1) ==> P ==> a % y = y`) THEN CONV_TAC REAL_FIELD]; DISCH_TAC] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [X_GEN_TAC `s:real^N->bool` THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THEN MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[CONIC_CONIC_HULL] THENL [EXPAND_TAC "f" THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN SIMP_TAC[HULL_INC; REAL_LE_INV_EQ; REWRITE_RULE[conic] CONIC_CONIC_HULL; NORM_ARITH `&0 <= &1 + norm(x:real^N)`]; EXPAND_TAC "f" THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[CONIC_HULL_EXPLICIT; IN_ELIM_THM; IN_IMAGE] THEN EXISTS_TAC `&1 + norm(x:real^N)` THEN EXISTS_TAC `inv(&1 + norm x) % x:real^N` THEN SIMP_TAC[VECTOR_MUL_ASSOC; VECTOR_MUL_LID; REAL_MUL_RINV; REAL_LT_IMP_NZ; REAL_LT_IMP_LE; NORM_ARITH `&0 < &1 + norm(x:real^N)`] THEN ASM_MESON_TAC[]]; DISCH_TAC] THEN REWRITE_TAC[CONJ_ASSOC] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [REWRITE_TAC[AND_FORALL_THM] THEN X_GEN_TAC `s:real^N->bool` THEN SUBGOAL_THEN `IMAGE f s = {x | x IN ball(vec 0,&1) /\ (g:real^N->real^N) x IN s}` SUBST1_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphism]) THEN SET_TAC[]; EXPAND_TAC "g" THEN REWRITE_TAC[IN_ELIM_THM; VECTOR_MUL_RZERO] THEN ASM_REWRITE_TAC[CENTRE_IN_BALL; REAL_LT_01] THEN STRIP_TAC] THEN MATCH_MP_TAC CONVEX_PREIMAGE_CONCAVE_SCALING THEN ASM_REWRITE_TAC[CONVEX_BALL] THEN EXPAND_TAC "g" THEN SIMP_TAC[IN_BALL_0; REAL_LT_INV_EQ; REAL_SUB_LT] THEN SIMP_TAC[convex_on; REAL_LE_LADD; REAL_ARITH `--(&1 - z) <= u * --(&1 - x) + v * --(&1 - y) <=> (u + v) + z <= &1 + (u * x + v * y)`] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC NORM_TRIANGLE_LE THEN ASM_REWRITE_TAC[NORM_MUL; real_abs; REAL_LE_REFL]; STRIP_TAC] THEN X_GEN_TAC `s:real^N->bool` THEN REWRITE_TAC[IN_RELATIVE_INTERIOR] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC] THEN ONCE_REWRITE_TAC[GSYM SPAN_CONIC_HULL] THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SPAN_CONIC_HULL] THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `open_in (subtopology euclidean (ball(vec 0,&1))) (IMAGE (f:real^N->real^N) (ball(vec 0,r)))` MP_TAC THENL [MATCH_MP_TAC HOMEOMORPHISM_IMP_OPEN_MAP THEN MAP_EVERY EXISTS_TAC [`g:real^N->real^N`; `(:real^N)`] THEN ASM_REWRITE_TAC[SUBTOPOLOGY_UNIV; GSYM OPEN_IN; OPEN_BALL]; SIMP_TAC[OPEN_IN_OPEN_EQ; OPEN_BALL]] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN REWRITE_TAC[OPEN_CONTAINS_BALL] THEN DISCH_THEN(MP_TAC o SPEC `vec 0:real^N`) THEN ASM_SIMP_TAC[CENTRE_IN_BALL] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `bd SUBSET IMAGE f br ==> br INTER ss SUBSET s /\ IMAGE f br INTER ss SUBSET IMAGE f (br INTER ss) ==> bd INTER ss SUBSET IMAGE f s`)) THEN ASM_REWRITE_TAC[] THEN TRANS_TAC SUBSET_TRANS `IMAGE (f:real^N->real^N) (ball(vec 0,r)) INTER IMAGE f (span s)` THEN CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `(!x. f x IN ss ==> x IN ss) ==> IMAGE f br INTER ss SUBSET IMAGE f br INTER IMAGE f ss`); FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphism]) THEN ASM SET_TAC[]] THEN X_GEN_TAC `x:real^N` THEN EXPAND_TAC "f" THEN REWRITE_TAC[] THEN DISCH_TAC THEN SUBGOAL_THEN `x:real^N = (&1 + norm x) % inv(&1 + norm x) % x` SUBST1_TAC THENL [ALL_TAC; ASM_SIMP_TAC[SPAN_MUL]] THEN SIMP_TAC[VECTOR_MUL_ASSOC; VECTOR_MUL_LID; REAL_MUL_RINV; NORM_ARITH `~(&1 + norm(x:real^N) = &0)`]);; (* ------------------------------------------------------------------------- *) (* Some convexity-related properties of Hausdorff distance *) (* ------------------------------------------------------------------------- *) let HAUSDIST_CONVEX_HULLS = prove (`!s t:real^N->bool. bounded s /\ bounded t ==> hausdist(convex hull s,convex hull t) <= hausdist(s,t)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [ASM_MESON_TAC[HAUSDIST_EMPTY; CONVEX_HULL_EMPTY; REAL_LE_REFL]; ALL_TAC] THEN ASM_CASES_TAC `t:real^N->bool = {}` THENL [ASM_MESON_TAC[HAUSDIST_EMPTY; CONVEX_HULL_EMPTY; REAL_LE_REFL]; ALL_TAC] THEN MATCH_MP_TAC REAL_HAUSDIST_LE THEN ASM_REWRITE_TAC[CONVEX_HULL_EQ_EMPTY] THEN CONJ_TAC THEN MATCH_MP_TAC CONVEX_ON_CONVEX_HULL_BOUND THEN CONJ_TAC THEN SIMP_TAC[CONVEX_ON_SETDIST; CONVEX_CONVEX_HULL] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_HAUSDIST THEN ASM_REWRITE_TAC[LEFT_EXISTS_AND_THM; RIGHT_EXISTS_AND_THM; CONJ_ASSOC] THEN (CONJ_TAC THENL [CONJ_TAC; ASM_MESON_TAC[SETDIST_SUBSET_RIGHT; HULL_SUBSET]]) THEN MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`] BOUNDED_DIFFS) THEN ASM_REWRITE_TAC[bounded; FORALL_IN_GSPEC; GSYM dist] THEN MATCH_MP_TAC MONO_EXISTS THEN ASM_MESON_TAC[SETDIST_LE_DIST; dist; DIST_SYM; REAL_LE_TRANS; MEMBER_NOT_EMPTY; IN_SING]);; let HAUSDIST_SUMS = prove (`!s t:real^N->bool u. bounded s /\ bounded t /\ convex s /\ convex t /\ bounded u /\ ~(s = {}) /\ ~(t = {}) /\ ~(u = {}) ==> hausdist({x + e | x IN s /\ e IN u}, {y + e | y IN t /\ e IN u}) = hausdist(s,t)`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[MESON[HAUSDIST_CLOSURE] `hausdist(s:real^N->bool,t) = hausdist(closure s,closure t)`] THEN SIMP_TAC[CLOSURE_SUMS] THEN SIMP_TAC[CLOSURE_CLOSED; CLOSED_CBALL; GSYM COMPACT_CLOSURE] THEN ONCE_REWRITE_TAC[GSYM CLOSURE_EQ_EMPTY] THEN ASM_CASES_TAC `convex(closure s:real^N->bool) /\ convex(closure t:real^N->bool)` THENL [POP_ASSUM MP_TAC; ASM_MESON_TAC[CONVEX_CLOSURE]] THEN ASM_CASES_TAC `convex(s:real^N->bool)` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `convex(t:real^N->bool)` THEN ASM_REWRITE_TAC[] THEN POP_ASSUM_LIST(K ALL_TAC) THEN SPEC_TAC(`closure u:real^N->bool`,`u:real^N->bool`) THEN SPEC_TAC(`closure t:real^N->bool`,`t:real^N->bool`) THEN SPEC_TAC(`closure s:real^N->bool`,`s:real^N->bool`) THEN REWRITE_TAC[GSYM REAL_LE_ANTISYM] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_HAUSDIST_LE_SUMS THEN MAP_EVERY ABBREV_TAC [`a = hausdist(s:real^N->bool,t)`; `b = hausdist({x + e:real^N | x IN s /\ e IN u}, {y + e | y IN t /\ e IN u})`] THEN ASM_REWRITE_TAC[CBALL_EQ_EMPTY; REAL_NOT_LT; SET_RULE `{f x y | x IN s /\ y IN t} = {} <=> s = {} \/ t = {}`] THENL [REWRITE_TAC[SUMS_ASSOC] THEN GEN_REWRITE_TAC (BINOP_CONV o RAND_CONV o RAND_CONV o ONCE_DEPTH_CONV) [SUMS_SYM] THEN REWRITE_TAC[GSYM SUMS_ASSOC] THEN CONJ_TAC THEN MATCH_MP_TAC(SET_RULE `s SUBSET s' ==> {f x y | x IN s /\ y IN t} SUBSET {f x y | x IN s' /\ y IN t}`) THEN EXPAND_TAC "a" THENL [ALL_TAC; ONCE_REWRITE_TAC[HAUSDIST_SYM]] THEN MATCH_MP_TAC HAUSDIST_COMPACT_SUMS THEN ASM_SIMP_TAC[COMPACT_IMP_BOUNDED]; CONJ_TAC THEN MATCH_MP_TAC SUBSET_SUMS_RCANCEL THEN EXISTS_TAC `u:real^N->bool` THEN ASM_SIMP_TAC[CLOSED_COMPACT_SUMS; COMPACT_CBALL; COMPACT_IMP_CLOSED; CONVEX_CBALL; COMPACT_IMP_BOUNDED; CONVEX_SUMS; REAL_NOT_LT] THEN REWRITE_TAC[SUMS_ASSOC] THEN GEN_REWRITE_TAC (RAND_CONV o RAND_CONV o ONCE_DEPTH_CONV) [SUMS_SYM] THEN REWRITE_TAC[GSYM SUMS_ASSOC] THEN EXPAND_TAC "b" THENL [ALL_TAC; ONCE_REWRITE_TAC[HAUSDIST_SYM]] THEN MATCH_MP_TAC HAUSDIST_COMPACT_SUMS THEN ASM_SIMP_TAC[BOUNDED_SUMS; COMPACT_SUMS; COMPACT_CBALL; COMPACT_IMP_BOUNDED; CBALL_EQ_EMPTY; REAL_NOT_LT; SET_RULE `{f x y | x IN s /\ y IN t} = {} <=> s = {} \/ t = {}`]]);; let HAUSDIST_COMPLEMENTS_CONVEX_EXPLICIT = prove (`!s t d x:real^N. convex s /\ bounded s /\ ~(s = {}) /\ bounded t /\ ~(x IN s) /\ hausdist(s,t) < d ==> ?y. ~(y IN t) /\ dist(x,y) < d`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `x:real^N`] SUPPORTING_HYPERPLANE_POINT) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `bb:real`; `y:real^N`] THEN STRIP_TAC THEN FIRST_X_ASSUM(SUBST_ALL_TAC o SYM) THEN ASM_CASES_TAC `x IN closure((:real^N) DIFF t)` THENL [ONCE_REWRITE_TAC[DIST_SYM] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CLOSURE_APPROACHABLE]) THEN REWRITE_TAC[IN_UNIV; IN_DIFF] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_MESON_TAC[REAL_LET_TRANS; HAUSDIST_POS_LE]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE (RAND_CONV o RAND_CONV) [CLOSURE_COMPLEMENT]) THEN REWRITE_TAC[IN_DIFF; IN_UNIV] THEN DISCH_TAC] THEN MP_TAC(ISPECL [`t:real^N->bool`; `x:real^N`; `--a:real^N`] RAY_TO_FRONTIER) THEN ASM_REWRITE_TAC[VECTOR_NEG_EQ_0] THEN DISCH_THEN(X_CHOOSE_THEN `l:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `?w:real^N. w IN s /\ dist(x + l % --a,w) < d` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC REAL_LT_HAUSDIST_POINT_EXISTS THEN EXISTS_TAC `closure t:real^N->bool` THEN ASM_SIMP_TAC[HAUSDIST_CLOSURE; BOUNDED_CLOSURE] THEN ASM_MESON_TAC[frontier; IN_DIFF; HAUSDIST_SYM]; ALL_TAC] THEN SUBGOAL_THEN `(x + l % --a) IN frontier((:real^N) DIFF t)` MP_TAC THENL [ASM_REWRITE_TAC[FRONTIER_COMPLEMENT]; REWRITE_TAC[frontier]] THEN REWRITE_TAC[IN_DIFF; CLOSURE_APPROACHABLE] THEN DISCH_THEN(MP_TAC o SPEC `d - dist(x:real^N,x + l % --a)` o CONJUNCT1) THEN ASM_REWRITE_TAC[IN_UNIV; REAL_SUB_LT] THEN ANTS_TAC THENL [ALL_TAC; MESON_TAC[NORM_ARITH `dist(y:real^N,z) < d - dist(x,z) ==> dist(x,y) < d`]] THEN SUBGOAL_THEN `ball(x + l % --a:real^N,dist(x,x + l % --a)) INTER closure s = {}` MP_TAC THENL [FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `(!y. y IN t ==> P y) ==> (!x. x IN u ==> ~P x) ==> u INTER t = {}`)) THEN X_GEN_TAC `v:real^N` THEN REWRITE_TAC[IN_BALL; REAL_NOT_LE] THEN STRIP_TAC THEN MP_TAC(ISPECL [`a:real^N`; `v - (x + l % --a):real^N`] NORM_CAUCHY_SCHWARZ_ABS) THEN REWRITE_TAC[DOT_RSUB; DOT_RADD; DOT_RNEG; DOT_RMUL] THEN MATCH_MP_TAC(REAL_ARITH `d < l * a /\ x <= y ==> abs(v - (x + l * --a)) <= d ==> ~(v >= y)`) THEN ASM_REWRITE_TAC[] THEN TRANS_TAC REAL_LTE_TRANS `norm(a:real^N) * dist(x:real^N,x + l % --a)` THEN ASM_SIMP_TAC[REAL_LT_LMUL_EQ; NORM_POS_LT; NORM_ARITH `norm(a - b:real^N) = dist(b,a)`] THEN REWRITE_TAC[NORM_ARITH `dist(x:real^N,x + y) = norm y`] THEN REWRITE_TAC[DOT_RMUL; NORM_MUL; NORM_NEG; DOT_RNEG] THEN ASM_SIMP_TAC[GSYM NORM_POW_2; REAL_POW_2; REAL_MUL_AC; real_abs; REAL_LT_IMP_LE; REAL_LE_REFL]; DISCH_THEN(MP_TAC o SPEC `w:real^N` o MATCH_MP (SET_RULE `s INTER t = {} ==> !w. w IN t ==> ~(w IN s)`)) THEN ASM_SIMP_TAC[REWRITE_RULE[SUBSET] CLOSURE_SUBSET; IN_BALL] THEN ASM_REAL_ARITH_TAC]);; let HAUSDIST_COMPLEMENTS_CONVEX_LE = prove (`!s t:real^N->bool. convex s /\ bounded s /\ convex t /\ bounded t /\ ~(s = {}) /\ ~(t = {}) ==> hausdist((:real^N) DIFF s,(:real^N) DIFF t) <= hausdist(s,t)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `(:real^N) DIFF s = {}` THEN ASM_REWRITE_TAC[HAUSDIST_EMPTY; HAUSDIST_POS_LE] THEN ASM_CASES_TAC `(:real^N) DIFF t = {}` THEN ASM_REWRITE_TAC[HAUSDIST_EMPTY; HAUSDIST_POS_LE] THEN MATCH_MP_TAC REAL_HAUSDIST_LE THEN ASM_REWRITE_TAC[IN_DIFF; IN_UNIV] THEN CONJ_TAC THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MATCH_MP_TAC(MESON[REAL_NOT_LT; REAL_LT_REFL] `(!z:real. y < z ==> x < z) ==> x <= y`) THEN X_GEN_TAC `d:real` THEN DISCH_TAC THENL [MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`] HAUSDIST_COMPLEMENTS_CONVEX_EXPLICIT); MP_TAC(ISPECL [`t:real^N->bool`; `s:real^N->bool`] HAUSDIST_COMPLEMENTS_CONVEX_EXPLICIT)] THEN DISCH_THEN(MP_TAC o SPECL [`d:real`; `x:real^N`]) THEN (ANTS_TAC THENL [ASM_MESON_TAC[HAUSDIST_SYM]; ALL_TAC]) THEN DISCH_THEN(X_CHOOSE_THEN `y:real^N` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] REAL_LET_TRANS) THEN MATCH_MP_TAC SETDIST_LE_DIST THEN ASM SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* The Blaschke selection principle and related results. *) (* ------------------------------------------------------------------------- *) let CONVEX_HAUSDIST_LIMIT = prove (`!s:(num->real^N->bool) t. eventually (\n. bounded(s n) /\ convex(s n) /\ ~(s n = {})) sequentially /\ compact t /\ ((\n. lift(hausdist(s n,t))) --> vec 0) sequentially ==> convex t`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_REWRITE_TAC[CONVEX_EMPTY] THEN SUBGOAL_THEN `hausdist(convex hull t:real^N->bool,t) = &0` MP_TAC THENL [ALL_TAC; ASM_SIMP_TAC[HAUSDIST_EQ_0; COMPACT_IMP_CLOSED; CONVEX_HULL_EQ_EMPTY; CLOSURE_CLOSED; COMPACT_IMP_BOUNDED; COMPACT_CONVEX_HULL] THEN REWRITE_TAC[CONVEX_HULL_EQ]] THEN REWRITE_TAC[GSYM LIFT_EQ; LIFT_NUM] THEN MATCH_MP_TAC(MESON[LIM_CONST; LIM_UNIQUE; TRIVIAL_LIMIT_SEQUENTIALLY] `((\x. a) --> b) sequentially ==> a = b`) THEN MATCH_MP_TAC LIM_NULL_COMPARISON THEN EXISTS_TAC `\n. hausdist(convex hull t,convex hull ((s:num->real^N->bool) n)) + hausdist(s n,t)` THEN REWRITE_TAC[NORM_LIFT; REAL_ABS_HAUSDIST] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] EVENTUALLY_MONO)) THEN SIMP_TAC[HULL_P] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC HAUSDIST_TRIANGLE THEN ASM_SIMP_TAC[COMPACT_CONVEX_HULL; COMPACT_IMP_BOUNDED]; REWRITE_TAC[LIFT_ADD] THEN MATCH_MP_TAC LIM_NULL_ADD THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] LIM_NULL_COMPARISON)) THEN REWRITE_TAC[NORM_LIFT; REAL_ABS_HAUSDIST] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] EVENTUALLY_MONO)) THEN REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN GEN_REWRITE_TAC RAND_CONV [HAUSDIST_SYM] THEN MATCH_MP_TAC HAUSDIST_CONVEX_HULLS THEN ASM_SIMP_TAC[COMPACT_IMP_BOUNDED]]);; let COMPLETE_HAUSDIST_CONVEX = prove (`!f:num->(real^N->bool) c. closed c /\ (!n. bounded(f n) /\ convex(f n) /\ ~(f n = {}) /\ f n SUBSET c) /\ (!e. &0 < e ==> ?N. !m n. m >= N /\ n >= N ==> hausdist(f m,f n) < e) ==> ?s. compact s /\ convex s /\ ~(s = {}) /\ s SUBSET c /\ ((\n. lift(hausdist(f n,s))) --> vec 0) sequentially`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:num->(real^N->bool)`; `c:real^N->bool`] COMPLETE_HAUSDIST) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONVEX_HAUSDIST_LIMIT THEN EXISTS_TAC `f:num->real^N->bool` THEN ASM_REWRITE_TAC[EVENTUALLY_SEQUENTIALLY]);; let COMPLETE_HAUSDIST_CONVEX_UNIV = prove (`!f:num->(real^N->bool). (!n. bounded(f n) /\ convex(f n) /\ ~(f n = {})) /\ (!e. &0 < e ==> ?N. !m n. m >= N /\ n >= N ==> hausdist(f m,f n) < e) ==> ?s. compact s /\ convex s /\ ~(s = {}) /\ ((\n. lift(hausdist(f n,s))) --> vec 0) sequentially`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:num->(real^N->bool)`; `(:real^N)`] COMPLETE_HAUSDIST_CONVEX) THEN ASM_REWRITE_TAC[SUBSET_UNIV; CLOSED_UNIV]);; let BLASCHKE = prove (`!f:num->(real^N->bool) c. compact c /\ (!n. convex(f n) /\ ~(f n = {}) /\ f n SUBSET c) ==> ?r s. (!m n. m < n ==> r m < r n) /\ compact s /\ convex s /\ ~(s = {}) /\ s SUBSET c /\ ((\n. lift(hausdist(f(r n),s))) --> vec 0) sequentially`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:num->(real^N->bool)`; `c:real^N->bool`] COMPACT_HAUSDIST) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `r:num->num` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `s:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONVEX_HAUSDIST_LIMIT THEN EXISTS_TAC `(f:num->real^N->bool) o (r:num->num)` THEN ASM_REWRITE_TAC[o_THM; EVENTUALLY_SEQUENTIALLY] THEN ASM_MESON_TAC[BOUNDED_SUBSET; COMPACT_IMP_BOUNDED]);; let BLASCHKE_UNIV = prove (`!f:num->(real^N->bool) c. bounded c /\ (!n. convex(f n) /\ ~(f n = {}) /\ f n SUBSET c) ==> ?r s. (!m n. m < n ==> r m < r n) /\ compact s /\ convex s /\ ~(s = {}) /\ ((\n. lift(hausdist(f(r n),s))) --> vec 0) sequentially`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:num->(real^N->bool)`; `closure c:real^N->bool`] BLASCHKE) THEN ASM_REWRITE_TAC[COMPACT_CLOSURE] THEN ANTS_TAC THENL [ALL_TAC; MESON_TAC[]] THEN ASM_MESON_TAC[SUBSET; CLOSURE_SUBSET]);; (* ------------------------------------------------------------------------- *) (* Interior, relative interior and closure interrelations. *) (* ------------------------------------------------------------------------- *) let CONVEX_CLOSURE_INTERIOR = prove (`!s:real^N->bool. convex s /\ ~(interior s = {}) ==> closure(interior s) = closure s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN SIMP_TAC[SUBSET_CLOSURE; INTERIOR_SUBSET] 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[SUBSET] THEN X_GEN_TAC `b:real^N` THEN DISCH_TAC THEN ASM_CASES_TAC `b:real^N = a` THENL [ASM_MESON_TAC[CLOSURE_SUBSET; SUBSET]; ALL_TAC] THEN REWRITE_TAC[closure; IN_UNION; IN_ELIM_THM] THEN DISJ2_TAC THEN REWRITE_TAC[LIMPT_APPROACHABLE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN EXISTS_TAC `b - min (e / &2 / norm(b - a)) (&1) % (b - a):real^N` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC IN_INTERIOR_CLOSURE_CONVEX_SHRINK THEN ASM_REWRITE_TAC[REAL_MIN_LE; REAL_LT_MIN; REAL_LE_REFL; REAL_LT_01]; REWRITE_TAC[VECTOR_ARITH `b - x:real^N = b <=> x = vec 0`] THEN ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ] THEN MATCH_MP_TAC(REAL_ARITH `&0 < x ==> ~(min x (&1) = &0)`); REWRITE_TAC[NORM_ARITH `dist(b - x:real^N,b) = norm x`] THEN REWRITE_TAC[NORM_MUL] THEN MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `e / &2 / norm(b - a:real^N) * norm(b - a)` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE] THEN MATCH_MP_TAC(REAL_ARITH `&0 < x ==> abs(min x (&1)) <= x`); ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_POS_LT; REAL_LT_IMP_NZ; VECTOR_SUB_EQ] THEN ASM_REAL_ARITH_TAC]] THEN ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT; REAL_OF_NUM_LT; VECTOR_SUB_EQ; ARITH]);; let EMPTY_INTERIOR_SUBSET_HYPERPLANE = prove (`!s. convex s /\ interior s = {} ==> ?a:real^N b. ~(a = vec 0) /\ s SUBSET {x | a dot x = b}`, let lemma = prove (`!s. convex s /\ (vec 0) IN s /\ interior s = {} ==> ?a:real^N b. ~(a = vec 0) /\ s SUBSET {x | a dot x = b}`, GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN SUBGOAL_THEN `~(relative_interior(s:real^N->bool) = {})` MP_TAC THENL [ASM_MESON_TAC[RELATIVE_INTERIOR_EQ_EMPTY; MEMBER_NOT_EMPTY]; ALL_TAC] THEN ASM_REWRITE_TAC[CONTRAPOS_THM] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC RELATIVE_INTERIOR_INTERIOR THEN ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC] THEN ONCE_REWRITE_TAC[GSYM SPAN_UNIV] THEN MATCH_MP_TAC DIM_EQ_SPAN THEN REWRITE_TAC[SUBSET_UNIV; DIM_UNIV; GSYM NOT_LT] THEN DISCH_THEN(MP_TAC o MATCH_MP LOWDIM_SUBSET_HYPERPLANE) THEN DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN EXISTS_TAC `&0` THEN ASM_MESON_TAC[SUBSET_TRANS; SPAN_INC]) in GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [ASM_MESON_TAC[EMPTY_SUBSET; BASIS_NONZERO; LE_REFL; DIMINDEX_GE_1]; ALL_TAC] THEN 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 MP_TAC(ISPEC `IMAGE (\x:real^N. --a + x) s` lemma) THEN ASM_REWRITE_TAC[CONVEX_TRANSLATION_EQ; INTERIOR_TRANSLATION; IMAGE_EQ_EMPTY; IN_IMAGE; UNWIND_THM2; VECTOR_ARITH `vec 0:real^N = --a + x <=> x = a`] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^N` THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM; DOT_RADD] THEN MESON_TAC[REAL_ARITH `a + x:real = b <=> x = b - a`]);; let CONVEX_INTERIOR_CLOSURE = prove (`!s:real^N->bool. convex s ==> interior(closure s) = interior s`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `interior(s:real^N->bool) = {}` THENL [MP_TAC(ISPEC `s:real^N->bool` EMPTY_INTERIOR_SUBSET_HYPERPLANE) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real`] THEN STRIP_TAC THEN MATCH_MP_TAC(SET_RULE `!t. s SUBSET t /\ t = {} ==> s = {}`) THEN EXISTS_TAC `interior {x:real^N | a dot x = b}` THEN CONJ_TAC THENL [ALL_TAC; ASM_SIMP_TAC[INTERIOR_HYPERPLANE]] THEN MATCH_MP_TAC SUBSET_INTERIOR THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN ASM_REWRITE_TAC[CLOSED_HYPERPLANE]; ALL_TAC] THEN FIRST_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 MATCH_MP_TAC SUBSET_ANTISYM THEN SIMP_TAC[SUBSET_INTERIOR; CLOSURE_SUBSET] THEN REWRITE_TAC[SUBSET] THEN X_GEN_TAC `b:real^N` THEN DISCH_TAC THEN MP_TAC(ASSUME `(b:real^N) IN interior(closure s)`) THEN GEN_REWRITE_TAC LAND_CONV [IN_INTERIOR_CBALL] THEN REWRITE_TAC[SUBSET; IN_CBALL; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `e:real` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_CASES_TAC `b:real^N = a` THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `b + e / norm(b - a) % (b - a):real^N`) THEN ASM_SIMP_TAC[NORM_ARITH `dist(b:real^N,b + e) = norm e`; NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM; REAL_DIV_RMUL; NORM_EQ_0; VECTOR_SUB_EQ; REAL_ARITH `&0 < e ==> abs e <= e`] THEN DISCH_TAC THEN SUBGOAL_THEN `b = (b + e / norm(b - a) % (b - a)) - e / norm(b - a) / (&1 + e / norm(b - a)) % ((b + e / norm(b - a) % (b - a)) - a):real^N` SUBST1_TAC THENL [REWRITE_TAC[VECTOR_ARITH `b = (b + e % (b - a)) - d % ((b + e % (b - a)) - a) <=> (e - d * (&1 + e)) % (b - a) = vec 0`] THEN ASM_REWRITE_TAC[VECTOR_SUB_EQ; VECTOR_MUL_EQ_0]; MATCH_MP_TAC IN_INTERIOR_CLOSURE_CONVEX_SHRINK] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LT_DIV; NORM_POS_LT; VECTOR_SUB_EQ; REAL_ARITH `&0 < x ==> &0 < &1 + x`; REAL_ARITH `&0 < x ==> ~(&1 + x = &0)`; REAL_MUL_LID; REAL_ADD_RDISTRIB; REAL_DIV_RMUL; REAL_LT_IMP_NZ; REAL_LE_ADDL; NORM_POS_LE; REAL_SUB_REFL]);; let FRONTIER_CLOSURE_CONVEX = prove (`!s:real^N->bool. convex s ==> frontier(closure s) = frontier s`, SIMP_TAC[frontier; CLOSURE_CLOSURE; CONVEX_INTERIOR_CLOSURE]);; let CONVEX_CLOSURE_RELATIVE_INTERIOR = prove (`!s:real^N->bool. convex s ==> closure(relative_interior s) = closure s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN SIMP_TAC[SUBSET_CLOSURE; RELATIVE_INTERIOR_SUBSET] THEN ASM_CASES_TAC `relative_interior(s:real^N->bool) = {}` THENL [ASM_MESON_TAC[RELATIVE_INTERIOR_EQ_EMPTY; SUBSET_REFL]; ALL_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[SUBSET] THEN X_GEN_TAC `b:real^N` THEN DISCH_TAC THEN ASM_CASES_TAC `b:real^N = a` THENL [ASM_MESON_TAC[CLOSURE_SUBSET; SUBSET]; ALL_TAC] THEN REWRITE_TAC[closure; IN_UNION; IN_ELIM_THM] THEN DISJ2_TAC THEN REWRITE_TAC[LIMPT_APPROACHABLE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN EXISTS_TAC `b - min (e / &2 / norm(b - a)) (&1) % (b - a):real^N` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SHRINK THEN ASM_REWRITE_TAC[REAL_MIN_LE; REAL_LT_MIN; REAL_LE_REFL; REAL_LT_01]; REWRITE_TAC[VECTOR_ARITH `b - x:real^N = b <=> x = vec 0`] THEN ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ] THEN MATCH_MP_TAC(REAL_ARITH `&0 < x ==> ~(min x (&1) = &0)`); REWRITE_TAC[NORM_ARITH `dist(b - x:real^N,b) = norm x`] THEN REWRITE_TAC[NORM_MUL] THEN MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `e / &2 / norm(b - a:real^N) * norm(b - a)` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE] THEN MATCH_MP_TAC(REAL_ARITH `&0 < x ==> abs(min x (&1)) <= x`); ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_POS_LT; REAL_LT_IMP_NZ; VECTOR_SUB_EQ] THEN ASM_REAL_ARITH_TAC]] THEN ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT; REAL_OF_NUM_LT; VECTOR_SUB_EQ; ARITH]);; let OPEN_IN_CONVEX_MEETS_RELATIVE_INTERIOR = prove (`!u s:real^N->bool. convex u /\ open_in (subtopology euclidean u) s /\ ~(s = {}) ==> ~(s INTER relative_interior u = {})`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(X_CHOOSE_THEN `v:real^N->bool` STRIP_ASSUME_TAC o GSYM o REWRITE_RULE[OPEN_IN_OPEN]) THEN MP_TAC(ISPECL [`v:real^N->bool`; `relative_interior u:real^N->bool`] OPEN_INTER_CLOSURE_EQ_EMPTY) THEN ASM_SIMP_TAC[CONVEX_CLOSURE_RELATIVE_INTERIOR] THEN MP_TAC(ISPEC `u:real^N->bool` RELATIVE_INTERIOR_SUBSET) THEN MP_TAC(ISPEC `u:real^N->bool` CLOSURE_SUBSET) THEN ASM SET_TAC[]);; let OPEN_SUBSET_CLOSURE_CONVEX = prove (`!u s:real^N->bool. open u /\ convex s ==> (u SUBSET closure s <=> u SUBSET interior s)`, REPEAT STRIP_TAC THEN EQ_TAC THENL [ALL_TAC; MESON_TAC[INTERIOR_SUBSET; CLOSURE_SUBSET; SUBSET]] THEN DISCH_THEN(MP_TAC o MATCH_MP SUBSET_INTERIOR) THEN ASM_SIMP_TAC[CONVEX_INTERIOR_CLOSURE; INTERIOR_OPEN]);; let SETDIST_RELATIVE_INTERIOR = prove (`(!s t. convex s ==> setdist(relative_interior s,t) = setdist(s,t)) /\ (!s t. convex t ==> setdist(s,relative_interior t) = setdist(s,t))`, MESON_TAC[CONVEX_CLOSURE_RELATIVE_INTERIOR; SETDIST_CLOSURE]);; let HAUSDIST_RELATIVE_INTERIOR = prove (`(!s t. convex s ==> hausdist(relative_interior s,t) = hausdist(s,t)) /\ (!s t. convex t ==> hausdist(s,relative_interior t) = hausdist(s,t))`, MESON_TAC[CONVEX_CLOSURE_RELATIVE_INTERIOR; HAUSDIST_CLOSURE]);; let AFFINE_HULL_RELATIVE_INTERIOR = prove (`!s. convex s ==> affine hull (relative_interior s) = affine hull s`, MESON_TAC[CONVEX_CLOSURE_RELATIVE_INTERIOR; AFFINE_HULL_CLOSURE]);; let AFF_DIM_RELATIVE_INTERIOR = prove (`!s:real^N->bool. convex s ==> aff_dim(relative_interior s) = aff_dim s`, ASM_MESON_TAC[AFF_DIM_AFFINE_HULL; AFFINE_HULL_RELATIVE_INTERIOR]);; let CONVEX_RELATIVE_INTERIOR_CLOSURE = prove (`!s:real^N->bool. convex s ==> relative_interior(closure s) = relative_interior s`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[CLOSURE_EMPTY; RELATIVE_INTERIOR_EMPTY] THEN SUBGOAL_THEN `?a:real^N. a IN relative_interior s` STRIP_ASSUME_TAC THENL [ASM_SIMP_TAC[MEMBER_NOT_EMPTY; RELATIVE_INTERIOR_EQ_EMPTY]; ALL_TAC] THEN MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[SUBSET] THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[IN_RELATIVE_INTERIOR; AFFINE_HULL_CLOSURE; SUBSET] THEN MESON_TAC[CLOSURE_SUBSET; SUBSET]] THEN X_GEN_TAC `b:real^N` THEN DISCH_TAC THEN MP_TAC(ASSUME `(b:real^N) IN relative_interior(closure s)`) THEN GEN_REWRITE_TAC LAND_CONV [IN_RELATIVE_INTERIOR_CBALL] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[SUBSET; IN_CBALL; IN_INTER; LEFT_IMP_EXISTS_THM; AFFINE_HULL_CLOSURE] THEN X_GEN_TAC `e:real` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_CASES_TAC `b:real^N = a` THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `b + e / norm(b - a) % (b - a):real^N`) THEN ASM_SIMP_TAC[NORM_ARITH `dist(b:real^N,b + e) = norm e`; NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM; REAL_DIV_RMUL; NORM_EQ_0; VECTOR_SUB_EQ; REAL_ARITH `&0 < e ==> abs e <= e`] THEN ANTS_TAC THENL [MATCH_MP_TAC IN_AFFINE_ADD_MUL_DIFF THEN ASM_MESON_TAC[SUBSET; AFFINE_AFFINE_HULL; RELATIVE_INTERIOR_SUBSET; CLOSURE_SUBSET_AFFINE_HULL; HULL_INC]; ALL_TAC] THEN DISCH_TAC THEN SUBGOAL_THEN `b = (b + e / norm(b - a) % (b - a)) - e / norm(b - a) / (&1 + e / norm(b - a)) % ((b + e / norm(b - a) % (b - a)) - a):real^N` SUBST1_TAC THENL [REWRITE_TAC[VECTOR_ARITH `b = (b + e % (b - a)) - d % ((b + e % (b - a)) - a) <=> (e - d * (&1 + e)) % (b - a) = vec 0`] THEN ASM_REWRITE_TAC[VECTOR_SUB_EQ; VECTOR_MUL_EQ_0]; MATCH_MP_TAC IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SHRINK] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LT_DIV; NORM_POS_LT; VECTOR_SUB_EQ; REAL_ARITH `&0 < x ==> &0 < &1 + x`; REAL_ARITH `&0 < x ==> ~(&1 + x = &0)`; REAL_MUL_LID; REAL_ADD_RDISTRIB; REAL_DIV_RMUL; REAL_LT_IMP_NZ; REAL_LE_ADDL; NORM_POS_LE; REAL_SUB_REFL]);; let RELATIVE_FRONTIER_CLOSURE = prove (`!s. convex s ==> relative_frontier(closure s) = relative_frontier s`, SIMP_TAC[relative_frontier; CLOSURE_CLOSURE; CONVEX_RELATIVE_INTERIOR_CLOSURE]);; let RELATIVE_FRONTIER_RELATIVE_INTERIOR = prove (`!s:real^N->bool. convex s ==> relative_frontier(relative_interior s) = relative_frontier s`, ASM_MESON_TAC[RELATIVE_FRONTIER_CLOSURE; CONVEX_CLOSURE_RELATIVE_INTERIOR; CONVEX_RELATIVE_INTERIOR]);; let CONNECTED_INTER_RELATIVE_FRONTIER = prove (`!s t:real^N->bool. connected s /\ s SUBSET affine hull t /\ ~(s INTER t = {}) /\ ~(s DIFF t = {}) ==> ~(s INTER relative_frontier t = {})`, REWRITE_TAC[relative_frontier] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONNECTED_OPEN_IN]) THEN REWRITE_TAC[] THEN MAP_EVERY EXISTS_TAC [`s INTER relative_interior t:real^N->bool`; `s DIFF closure t:real^N->bool`] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_SUBTOPOLOGY_INTER_SUBSET THEN EXISTS_TAC `affine hull t:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC OPEN_IN_INTER THEN REWRITE_TAC[OPEN_IN_RELATIVE_INTERIOR; OPEN_IN_SUBTOPOLOGY_REFL] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN; SUBSET_UNIV]; ONCE_REWRITE_TAC[SET_RULE `s DIFF t = s INTER (UNIV DIFF t)`] THEN MATCH_MP_TAC OPEN_IN_OPEN_INTER THEN REWRITE_TAC[GSYM closed; CLOSED_CLOSURE]; ASM SET_TAC[]; MATCH_MP_TAC(SET_RULE `i SUBSET t /\ t SUBSET c ==> (s INTER i) INTER (s DIFF c) = {}`) THEN REWRITE_TAC[RELATIVE_INTERIOR_SUBSET; CLOSURE_SUBSET]; MP_TAC(ISPEC `t:real^N->bool` CLOSURE_SUBSET) THEN ASM SET_TAC[]; MP_TAC(ISPEC `t:real^N->bool` RELATIVE_INTERIOR_SUBSET) THEN ASM SET_TAC[]]);; let CLOSED_RELATIVE_FRONTIER = prove (`!s:real^N->bool. closed(relative_frontier s)`, REPEAT GEN_TAC THEN REWRITE_TAC[relative_frontier] THEN MATCH_MP_TAC CLOSED_IN_CLOSED_TRANS THEN EXISTS_TAC `affine hull s:real^N->bool` THEN REWRITE_TAC[CLOSED_AFFINE_HULL] THEN MATCH_MP_TAC CLOSED_IN_DIFF THEN REWRITE_TAC[OPEN_IN_RELATIVE_INTERIOR] THEN MATCH_MP_TAC CLOSED_SUBSET THEN REWRITE_TAC[CLOSED_CLOSURE] THEN MATCH_MP_TAC(SET_RULE `s SUBSET closure t /\ closure t = t ==> s SUBSET t`) THEN SIMP_TAC[SUBSET_CLOSURE; HULL_SUBSET; CLOSURE_EQ; CLOSED_AFFINE_HULL]);; let CLOSED_RELATIVE_BOUNDARY = prove (`!s. closed s ==> closed(s DIFF relative_interior s)`, MESON_TAC[CLOSED_RELATIVE_FRONTIER; relative_frontier; CLOSURE_CLOSED]);; let COMPACT_RELATIVE_BOUNDARY = prove (`!s. compact s ==> compact(s DIFF relative_interior s)`, SIMP_TAC[COMPACT_EQ_BOUNDED_CLOSED; CLOSED_RELATIVE_BOUNDARY; BOUNDED_DIFF]);; let BOUNDED_RELATIVE_FRONTIER = prove (`!s:real^N->bool. bounded s ==> bounded(relative_frontier s)`, REWRITE_TAC[relative_frontier] THEN MESON_TAC[BOUNDED_CLOSURE; BOUNDED_SUBSET; SUBSET_DIFF]);; let COMPACT_RELATIVE_FRONTIER_BOUNDED = prove (`!s:real^N->bool. bounded s ==> compact(relative_frontier s)`, SIMP_TAC[COMPACT_EQ_BOUNDED_CLOSED; CLOSED_RELATIVE_FRONTIER; BOUNDED_RELATIVE_FRONTIER]);; let COMPACT_RELATIVE_FRONTIER = prove (`!s:real^N->bool. compact s ==> compact(relative_frontier s)`, SIMP_TAC[COMPACT_RELATIVE_FRONTIER_BOUNDED; COMPACT_IMP_BOUNDED]);; let CONVEX_SAME_RELATIVE_INTERIOR_CLOSURE = prove (`!s t. convex s /\ convex t ==> (relative_interior s = relative_interior t <=> closure s = closure t)`, MESON_TAC[CONVEX_CLOSURE_RELATIVE_INTERIOR; CONVEX_RELATIVE_INTERIOR_CLOSURE]);; let CONVEX_SAME_RELATIVE_INTERIOR_CLOSURE_STRADDLE = prove (`!s t. convex s /\ convex t ==> (relative_interior s = relative_interior t <=> relative_interior s SUBSET t /\ t SUBSET closure s)`, MESON_TAC[CONVEX_CLOSURE_RELATIVE_INTERIOR; CONVEX_RELATIVE_INTERIOR_CLOSURE; SUBSET_CLOSURE; SUBSET_ANTISYM; RELATIVE_INTERIOR_SUBSET; CLOSURE_SUBSET; CLOSURE_CLOSURE]);; let RELATIVE_INTERIOR_LINEAR_IMAGE_CONVEX = prove (`!f:real^M->real^N s. linear f /\ convex s ==> relative_interior(IMAGE f s) = IMAGE f (relative_interior s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [SUBGOAL_THEN `relative_interior (IMAGE f (relative_interior s)) = relative_interior (IMAGE (f:real^M->real^N) s)` (fun th -> REWRITE_TAC[SYM th; RELATIVE_INTERIOR_SUBSET]) THEN ASM_SIMP_TAC[CONVEX_SAME_RELATIVE_INTERIOR_CLOSURE_STRADDLE; CONVEX_RELATIVE_INTERIOR; CONVEX_LINEAR_IMAGE] THEN CONJ_TAC THENL [MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `IMAGE (f:real^M->real^N) (relative_interior s)` THEN SIMP_TAC[RELATIVE_INTERIOR_SUBSET; IMAGE_SUBSET]; MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `IMAGE (f:real^M->real^N) (closure(relative_interior s))` THEN ASM_SIMP_TAC[CLOSURE_LINEAR_IMAGE_SUBSET] THEN ASM_SIMP_TAC[CONVEX_CLOSURE_RELATIVE_INTERIOR] THEN MATCH_MP_TAC IMAGE_SUBSET THEN REWRITE_TAC[CLOSURE_SUBSET]]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `z:real^M` THEN DISCH_TAC THEN ASM_SIMP_TAC[RELATIVE_INTERIOR_CONVEX_PROLONG; CONVEX_LINEAR_IMAGE] THEN REWRITE_TAC[IN_ELIM_THM; FORALL_IN_IMAGE] THEN CONJ_TAC THENL [MATCH_MP_TAC FUN_IN_IMAGE THEN ASM_MESON_TAC[SUBSET; RELATIVE_INTERIOR_SUBSET]; ALL_TAC] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN MP_TAC(ISPECL [`s:real^M->bool`; `z:real^M`; `x:real^M`] RELATIVE_INTERIOR_PROLONG) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real` THEN MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o ISPEC `f:real^M->real^N` o MATCH_MP FUN_IN_IMAGE) THEN ASM_MESON_TAC[LINEAR_ADD; LINEAR_SUB; LINEAR_CMUL]]);; let RELATIVE_INTERIOR_LINEAR_PREIMAGE_CONVEX = prove (`!f:real^M->real^N s. linear f /\ convex s /\ ~({x | f(x) IN relative_interior s} = {}) ==> relative_interior {x | f(x) IN s} = {x | f(x) IN relative_interior s}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `IMAGE f s SUBSET t ==> s SUBSET {x | f x IN t}`) THEN ASM_SIMP_TAC[GSYM RELATIVE_INTERIOR_LINEAR_IMAGE_CONVEX; CONVEX_LINEAR_PREIMAGE; CONVEX_RELATIVE_INTERIOR] THEN MATCH_MP_TAC SUBSET_RELATIVE_INTERIOR_INTERSECTING_CONVEX THEN ASM_SIMP_TAC[CONVEX_LINEAR_IMAGE; CONVEX_LINEAR_PREIMAGE] THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER; EXISTS_IN_IMAGE] THEN REWRITE_TAC[IN_ELIM_THM] THEN MP_TAC(ISPEC `s:real^N->bool` RELATIVE_INTERIOR_SUBSET) THEN ASM SET_TAC[]; MATCH_MP_TAC RELATIVE_INTERIOR_MAXIMAL THEN CONJ_TAC THENL [MP_TAC(ISPEC `s:real^N->bool` RELATIVE_INTERIOR_SUBSET) THEN SET_TAC[]; MATCH_MP_TAC OPEN_IN_SUBSET_TRANS THEN EXISTS_TAC `{x | (f:real^M->real^N) x IN affine hull s}` THEN REPEAT CONJ_TAC THENL [SUBGOAL_THEN `{x | f x IN relative_interior s} = {x | x IN {x | (f:real^M->real^N) x IN affine hull s} /\ f x IN relative_interior s}` SUBST1_TAC THENL [REWRITE_TAC[IN_ELIM_THM] THEN MATCH_MP_TAC(SET_RULE `relative_interior s SUBSET s /\ s SUBSET affine hull s ==> {x | f x IN relative_interior s} = {x | f x IN affine hull s /\ f x IN relative_interior s}`) THEN REWRITE_TAC[RELATIVE_INTERIOR_SUBSET; HULL_SUBSET]; MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE_GEN THEN EXISTS_TAC `affine hull s:real^N->bool` THEN ASM_SIMP_TAC[OPEN_IN_RELATIVE_INTERIOR; LINEAR_CONTINUOUS_ON] THEN SET_TAC[]]; REWRITE_TAC[SUBSET] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC HULL_INC THEN MP_TAC(ISPEC `s:real^N->bool` RELATIVE_INTERIOR_SUBSET) THEN ASM SET_TAC[]; MATCH_MP_TAC HULL_MINIMAL THEN ASM_SIMP_TAC[AFFINE_LINEAR_PREIMAGE; AFFINE_AFFINE_HULL] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> {x | f x IN s} SUBSET {x | f x IN t}`) THEN REWRITE_TAC[HULL_SUBSET]]]]);; let RELATIVE_INTERIOR_SUMS = prove (`!s t:real^N->bool. convex s /\ convex t ==> relative_interior {x + y | x IN s /\ y IN t} = {x + y | x IN relative_interior s /\ y IN relative_interior t}`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `!s t. {x + y:real^N | x IN s /\ y IN t} = IMAGE (\z. fstcart z + sndcart z) (s PCROSS t)` (fun th -> REWRITE_TAC[th]) THENL [REWRITE_TAC[EXTENSION; IN_IMAGE; IN_ELIM_THM; EXISTS_PASTECART] THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; PASTECART_IN_PCROSS] THEN MESON_TAC[]; ASM_SIMP_TAC[RELATIVE_INTERIOR_LINEAR_IMAGE_CONVEX; CONVEX_PCROSS; LINEAR_COMPOSE_ADD; LINEAR_FSTCART; LINEAR_SNDCART] THEN REWRITE_TAC[RELATIVE_INTERIOR_PCROSS]]);; let CLOSURE_INTERS_CONVEX = prove (`!f:(real^N->bool)->bool. (!s. s IN f ==> convex s) /\ ~(INTERS(IMAGE relative_interior f) = {}) ==> closure(INTERS f) = INTERS(IMAGE closure f)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[CLOSURE_INTERS_SUBSET] THEN REWRITE_TAC[SUBSET; IN_INTERS; FORALL_IN_IMAGE] THEN X_GEN_TAC `b:real^N` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN REWRITE_TAC[INTERS_IMAGE; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN REWRITE_TAC[CLOSURE_APPROACHABLE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN ASM_CASES_TAC `b:real^N = a` THENL [EXISTS_TAC `a:real^N` THEN ASM_REWRITE_TAC[DIST_REFL; IN_INTERS] THEN ASM_MESON_TAC[SUBSET; RELATIVE_INTERIOR_SUBSET]; ALL_TAC] THEN EXISTS_TAC `b - min (&1 / &2) (e / &2 / norm(b - a)) % (b - a):real^N` THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[NORM_ARITH `dist(b - a:real^N,b) = norm a`; NORM_MUL] THEN ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN MATCH_MP_TAC(REAL_ARITH `&0 < a /\ &0 < x /\ x < y ==> abs(min a x) < y`) THEN ASM_SIMP_TAC[REAL_LT_DIV2_EQ; REAL_HALF; REAL_LT_DIV; NORM_POS_LT; VECTOR_SUB_EQ] THEN ASM_REAL_ARITH_TAC] THEN REWRITE_TAC[IN_INTERS] THEN X_GEN_TAC `s:real^N->bool` THEN DISCH_TAC THEN MATCH_MP_TAC (MESON[RELATIVE_INTERIOR_SUBSET; SUBSET] `!x. x IN relative_interior s ==> x IN s`) THEN MATCH_MP_TAC IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SHRINK THEN ASM_SIMP_TAC[REAL_LT_MIN; REAL_HALF; REAL_LT_DIV; NORM_POS_LT; VECTOR_SUB_EQ] THEN REAL_ARITH_TAC);; let CLOSURE_INTERS_CONVEX_OPEN = prove (`!f:(real^N->bool)->bool. (!s. s IN f ==> convex s /\ open s) ==> closure(INTERS f) = if INTERS f = {} then {} else INTERS(IMAGE closure f)`, REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[CLOSURE_EMPTY] THEN MATCH_MP_TAC CLOSURE_INTERS_CONVEX THEN ASM_SIMP_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `~(s = {}) ==> s = t ==> ~(t = {})`)) THEN AP_TERM_TAC THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> f x = x) ==> s = IMAGE f s`) THEN ASM_SIMP_TAC[RELATIVE_INTERIOR_OPEN; INTERIOR_EQ]);; let CLOSURE_INTER_CONVEX = prove (`!s t:real^N->bool. convex s /\ convex t /\ ~(relative_interior s INTER relative_interior t = {}) ==> closure(s INTER t) = closure(s) INTER closure(t)`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `{s:real^N->bool,t}` CLOSURE_INTERS_CONVEX) THEN ASM_SIMP_TAC[IMAGE_CLAUSES; INTERS_2] THEN ASM_REWRITE_TAC[FORALL_IN_INSERT; NOT_IN_EMPTY]);; let CLOSURE_INTER_CONVEX_OPEN = prove (`!s t. convex s /\ open s /\ convex t /\ open t ==> closure(s INTER t) = if s INTER t = {} then {} else closure(s) INTER closure(t)`, REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[CLOSURE_EMPTY] THEN MATCH_MP_TAC CLOSURE_INTER_CONVEX THEN ASM_SIMP_TAC[RELATIVE_INTERIOR_OPEN]);; let CLOSURE_CONVEX_INTER_SUPERSET = prove (`!s t:real^N->bool. convex s /\ ~(interior s = {}) /\ interior s SUBSET closure t ==> closure(s INTER t) = closure s`, REPEAT GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN SIMP_TAC[SUBSET_CLOSURE; INTER_SUBSET; SUBSET_INTER] THEN MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `closure(interior s):real^N->bool` THEN CONJ_TAC THENL [ASM_SIMP_TAC[CONVEX_CLOSURE_INTERIOR; SUBSET_REFL]; ASM_SIMP_TAC[GSYM CLOSURE_OPEN_INTER_SUPERSET; OPEN_INTERIOR] THEN MATCH_MP_TAC SUBSET_CLOSURE THEN MP_TAC(ISPEC `s:real^N->bool` INTERIOR_SUBSET) THEN SET_TAC[]]);; let CLOSURE_DYADIC_RATIONALS_IN_CONVEX_SET = prove (`!s:real^N->bool. convex s /\ ~(interior s = {}) ==> closure(s INTER { inv(&2 pow n) % x | n,x | !i. 1 <= i /\ i <= dimindex(:N) ==> integer(x$i) }) = closure s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CLOSURE_CONVEX_INTER_SUPERSET THEN ASM_REWRITE_TAC[CLOSURE_DYADIC_RATIONALS; SUBSET_UNIV]);; let CLOSURE_RATIONALS_IN_CONVEX_SET = prove (`!s:real^N->bool. convex s /\ ~(interior s = {}) ==> closure(s INTER { x | !i. 1 <= i /\ i <= dimindex(:N) ==> rational(x$i) }) = closure s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CLOSURE_CONVEX_INTER_SUPERSET THEN ASM_REWRITE_TAC[CLOSURE_RATIONAL_COORDINATES; SUBSET_UNIV]);; let RELATIVE_INTERIOR_CONVEX_INTER_AFFINE = prove (`!s t:real^N->bool. convex s /\ affine t /\ ~(interior s INTER t = {}) ==> relative_interior(s INTER t) = interior s INTER t`, REPEAT GEN_TAC THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; RIGHT_AND_EXISTS_THM] THEN DISCH_THEN(X_CHOOSE_THEN `a:real^N` MP_TAC) THEN GEOM_ORIGIN_TAC `a:real^N` THEN REWRITE_TAC[IN_INTER] THEN REPEAT GEN_TAC THEN ASM_CASES_TAC `(vec 0:real^N) IN t` THEN ASM_SIMP_TAC[AFFINE_EQ_SUBSPACE] THEN STRIP_TAC THEN GEN_REWRITE_TAC I [EXTENSION] THEN X_GEN_TAC `x:real^N` THEN MP_TAC(ISPECL [`t:real^N->bool`; `s:real^N->bool`] (ONCE_REWRITE_RULE[INTER_COMM] AFFINE_HULL_AFFINE_INTER_NONEMPTY_INTERIOR)) THEN ASM_SIMP_TAC[SUBSPACE_IMP_AFFINE; IN_RELATIVE_INTERIOR_CBALL] THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[IN_INTER; IN_INTERIOR_CBALL]] THEN DISCH_THEN SUBST1_TAC THEN ASM_CASES_TAC `(x:real^N) IN t` THEN ASM_REWRITE_TAC[] THEN SIMP_TAC[SUBSET; IN_INTER] THEN ASM_CASES_TAC `(x:real^N) IN s` THENL [ASM_REWRITE_TAC[]; ASM_MESON_TAC[CENTRE_IN_CBALL; REAL_LT_IMP_LE]] THEN EQ_TAC THENL [REWRITE_TAC[IN_CBALL]; MESON_TAC[]] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN ASM_CASES_TAC `x:real^N = vec 0` THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERIOR_CBALL]) THEN ASM_REWRITE_TAC[SUBSET; IN_CBALL]; ALL_TAC] THEN MP_TAC(ISPECL [`s:real^N->bool`; `vec 0:real^N`; `(&1 + e / norm x) % x:real^N`] IN_INTERIOR_CLOSURE_CONVEX_SEGMENT) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [MATCH_MP_TAC(REWRITE_RULE[SUBSET] CLOSURE_SUBSET) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[SUBSPACE_MUL] THEN REWRITE_TAC[VECTOR_ADD_RDISTRIB; VECTOR_MUL_LID; NORM_ARITH `dist(a:real^N,a + x) = norm x`] THEN ASM_SIMP_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM; REAL_DIV_RMUL; NORM_EQ_0] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[SUBSET; IN_INTERIOR_CBALL; IN_CBALL] THEN DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[IN_SEGMENT] THEN CONV_TAC(ONCE_DEPTH_CONV SYM_CONV) THEN ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN EXISTS_TAC `inv(&1 + e / norm(x:real^N))` THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_LT_DIV; NORM_POS_LT; VECTOR_MUL_LID; REAL_LT_INV_EQ; REAL_MUL_LINV; REAL_INV_LT_1; REAL_ARITH `&0 < x ==> &1 < &1 + x /\ &0 < &1 + x /\ ~(&1 + x = &0)`]]);; let CONNECTED_WITH_RELATIVE_INTERIOR_OPEN_IN_CONVEX = prove (`!c s:real^N->bool. convex c /\ connected s /\ open_in (subtopology euclidean c) s ==> connected(relative_interior c INTER s)`, REPEAT STRIP_TAC THEN REWRITE_TAC[CONNECTED; NOT_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 [CONNECTED_CLOSED]) THEN REWRITE_TAC[] THEN MAP_EVERY EXISTS_TAC [`closure(relative_interior c INTER u):real^N->bool`; `closure(relative_interior c INTER v):real^N->bool`] THEN REWRITE_TAC[CLOSED_CLOSURE] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[GSYM CLOSURE_UNION] THEN TRANS_TAC SUBSET_TRANS `closure(relative_interior c INTER s):real^N->bool` THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[INTER_COMM] THEN MP_TAC(ISPECL [`s:real^N->bool`; `relative_interior c:real^N->bool`; `c:real^N->bool`] CLOSURE_OPEN_IN_INTER_CLOSURE) THEN REWRITE_TAC[RELATIVE_INTERIOR_SUBSET] THEN ASM_SIMP_TAC[CONVEX_CLOSURE_RELATIVE_INTERIOR] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN TRANS_TAC SUBSET_TRANS `closure s:real^N->bool` THEN REWRITE_TAC[CLOSURE_SUBSET] THEN MATCH_MP_TAC SUBSET_CLOSURE THEN FIRST_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN MP_TAC(ISPEC `c:real^N->bool` CLOSURE_SUBSET) THEN SET_TAC[]; MATCH_MP_TAC SUBSET_CLOSURE THEN ASM SET_TAC[]]; ALL_TAC; MP_TAC(ISPEC`relative_interior c INTER u:real^N->bool` CLOSURE_SUBSET) THEN ASM SET_TAC[]; MP_TAC(ISPEC`relative_interior c INTER v:real^N->bool` CLOSURE_SUBSET) THEN ASM SET_TAC[]] THEN REWRITE_TAC[EXTENSION; IN_INTER; NOT_IN_EMPTY] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC 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 `x:real^N`)) THEN ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN X_GEN_TAC `r:real` THEN STRIP_TAC THEN SUBGOAL_THEN `connected(ball(x:real^N,r) INTER relative_interior c)` MP_TAC THENL [ASM_SIMP_TAC[CONVEX_CONNECTED; CONVEX_INTER; CONVEX_BALL; CONVEX_RELATIVE_INTERIOR]; REWRITE_TAC[connected]] THEN MAP_EVERY EXISTS_TAC [`u:real^N->bool`; `v:real^N->bool`] THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [MP_TAC(ISPEC `c:real^N->bool` RELATIVE_INTERIOR_SUBSET) THEN ASM SET_TAC[]; CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]] THEN ONCE_REWRITE_TAC[SET_RULE `u INTER b INTER i = b INTER u INTER i`] THEN MP_TAC(ISPEC `ball(x:real^N,r)` OPEN_INTER_CLOSURE_EQ_EMPTY) THEN REWRITE_TAC[OPEN_BALL] THEN DISCH_THEN(fun th -> ONCE_REWRITE_TAC[GSYM th]) THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN CONJ_TAC THEN EXISTS_TAC `x:real^N` THEN ASM_REWRITE_TAC[CENTRE_IN_BALL] THEN ONCE_REWRITE_TAC[INTER_COMM] THEN ASM_REWRITE_TAC[]);; let RELATIVE_INTERIOR_CBALL_INTER_AFFINE = prove (`!s a:real^N r. affine s /\ a IN s /\ ~(r = &0) ==> relative_interior(cball(a,r) INTER s) = ball(a,r) INTER s`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `r < &0` THENL [ASM_SIMP_TAC[CBALL_EMPTY; BALL_EMPTY; REAL_LT_IMP_LE; INTER_EMPTY] THEN REWRITE_TAC[RELATIVE_INTERIOR_EMPTY]; W(MP_TAC o PART_MATCH (lhand o rand) RELATIVE_INTERIOR_CONVEX_INTER_AFFINE o lhand o snd) THEN REWRITE_TAC[INTERIOR_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]);; (* ------------------------------------------------------------------------- *) (* Lemmas about extending nondecreasing functions. *) (* ------------------------------------------------------------------------- *) let NONDECREASING_EXTENDS_TO_CONVEX_HULL = prove (`!f s. (!x y. x IN s /\ y IN s /\ drop x <= drop y ==> drop(f x) <= drop(f y)) ==> ?g. (!x y. x IN convex hull s /\ y IN convex hull s /\ drop x <= drop y ==> drop(g x) <= drop(g y)) /\ (!x. x IN s ==> g x = f x)`, REPEAT STRIP_TAC THEN EXISTS_TAC `\x. lift(sup {drop(f u) | u IN s /\ drop u <= drop x})` THEN REWRITE_TAC[GSYM DROP_EQ; LIFT_DROP] THEN CONJ_TAC THENL [MAP_EVERY X_GEN_TAC [`x:real^1`; `y:real^1`] THEN STRIP_TAC THEN MATCH_MP_TAC REAL_SUP_LE_SUBSET THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN REWRITE_TAC[FORALL_IN_IMAGE; IMAGE_EQ_EMPTY] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_ELIM_THM] THEN REPEAT CONJ_TAC THENL [UNDISCH_TAC `(x:real^1) IN convex hull s` THEN REWRITE_TAC[IN_CONVEX_HULL_INTERVAL_1; IN_INTERVAL_1] THEN MESON_TAC[]; MATCH_MP_TAC IMAGE_SUBSET THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN ASM_MESON_TAC[REAL_LE_TRANS]; UNDISCH_TAC `(y:real^1) IN convex hull s` THEN REWRITE_TAC[IN_CONVEX_HULL_INTERVAL_1; IN_INTERVAL_1] THEN ASM_MESON_TAC[REAL_LE_TRANS]]; X_GEN_TAC `x:real^1` THEN DISCH_TAC THEN REWRITE_TAC[GSYM DROP_EQ; LIFT_DROP] THEN MATCH_MP_TAC SUP_UNIQUE THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN ASM_MESON_TAC[REAL_LE_TRANS; REAL_LE_REFL]]);; let NONDECREASING_EXTENDS_FROM_DENSE = prove (`!f s. closure s = (:real^1) /\ closure(IMAGE f s) = (:real^1) /\ (!x y. x IN s /\ y IN s /\ drop x <= drop y ==> drop(f x) <= drop(f y)) ==> ?g. (!x y. drop x <= drop y ==> drop(g x) <= drop(g y)) /\ (!x. x IN s ==> g x = f x) /\ g continuous_on (:real^1) /\ IMAGE g (:real^1) = (:real^1)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^1->real^1`; `s:real^1->bool`] NONDECREASING_EXTENDS_TO_CONVEX_HULL) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^1->real^1` THEN MP_TAC(ISPEC `s:real^1->bool` CONVEX_HULL_CLOSURE_SUBSET) THEN ASM_REWRITE_TAC[CONVEX_HULL_UNIV] THEN DISCH_THEN(MP_TAC o MATCH_MP SUBSET_INTERIOR) THEN SIMP_TAC[CONVEX_INTERIOR_CLOSURE; CONVEX_CONVEX_HULL] THEN SIMP_TAC[INTERIOR_UNIV; OPEN_UNIV; INTERIOR_MAXIMAL_EQ] THEN DISCH_THEN(ASSUME_TAC o MATCH_MP (SET_RULE `UNIV SUBSET s ==> s = UNIV`)) THEN ASM_REWRITE_TAC[IN_UNIV] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN X_GEN_TAC `x:real^1` THEN REWRITE_TAC[IN_UNIV; continuous_at] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN UNDISCH_TAC `closure(IMAGE (f:real^1->real^1) s) = UNIV` THEN REWRITE_TAC[EXTENSION; IN_UNIV; EXISTS_IN_IMAGE; CLOSURE_APPROACHABLE] THEN ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN DISCH_THEN(fun th -> MP_TAC(SPEC `g(x:real^1) + lift(e / &2)` th) THEN MP_TAC(SPEC `g(x:real^1) - lift(e / &2)` th)) THEN SIMP_TAC[DIST_1; DROP_ADD; DROP_SUB; LIFT_DROP; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `a:real^1` THEN STRIP_TAC THEN X_GEN_TAC `b:real^1` THEN STRIP_TAC THEN SUBGOAL_THEN `(f:real^1->real^1) a = g a /\ f b = g b` (CONJUNCTS_THEN SUBST_ALL_TAC) THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `drop a < drop x /\ drop x < drop b` STRIP_ASSUME_TAC THENL [REWRITE_TAC[GSYM REAL_NOT_LE] THEN CONJ_TAC THEN DISCH_THEN(ANTE_RES_THEN MP_TAC) THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN EXISTS_TAC `min (drop x - drop a) (drop b - drop x)` THEN ASM_REWRITE_TAC[REAL_LT_MIN; REAL_SUB_LT] THEN X_GEN_TAC `y:real^1` THEN STRIP_TAC THEN SUBGOAL_THEN `drop((g:real^1->real^1) a) <= drop(g y) /\ drop(g y) <= drop(g b)` MP_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REAL_ARITH_TAC; DISCH_THEN(MP_TAC o MATCH_MP(REWRITE_RULE[IMP_CONJ] CONNECTED_CONTINUOUS_IMAGE)) THEN REWRITE_TAC[GSYM CONVEX_CONNECTED_1; CONVEX_UNIV] THEN DISCH_TAC THEN MATCH_MP_TAC(SET_RULE `interior(closure s) SUBSET s /\ UNIV SUBSET interior(closure s) ==> s = UNIV`) THEN SIMP_TAC[INTERIOR_MAXIMAL_EQ; OPEN_UNIV] THEN ASM_SIMP_TAC[CONVEX_INTERIOR_CLOSURE; INTERIOR_SUBSET] THEN TRANS_TAC SUBSET_TRANS `closure(IMAGE (f:real^1->real^1) s)` THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC SUBSET_CLOSURE] THEN ASM SET_TAC[]]);; let INCREASING_EXTENDS_FROM_DENSE = prove (`!f s. closure s = (:real^1) /\ closure(IMAGE f s) = (:real^1) /\ (!x y. x IN s /\ y IN s /\ drop x < drop y ==> drop(f x) < drop(f y)) ==> ?g. (!x y. drop(g x) < drop(g y) <=> drop x < drop y) /\ (!x. x IN s ==> g x = f x) /\ g continuous_on (:real^1) /\ IMAGE g (:real^1) = (:real^1)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^1->real^1`; `s:real^1->bool`] NONDECREASING_EXTENDS_FROM_DENSE) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_MESON_TAC[REAL_LE_LT; DROP_EQ]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^1->real^1` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MAP_EVERY X_GEN_TAC [`x:real^1`; `y:real^1`] THEN EQ_TAC THEN ASM_REWRITE_TAC[GSYM REAL_NOT_LE; CONTRAPOS_THM] THEN REWRITE_TAC[GSYM REAL_NOT_LT; CONTRAPOS_THM] THEN DISCH_TAC THEN ASM_SIMP_TAC[REAL_LT_LE; REAL_LT_IMP_LE] THEN UNDISCH_TAC `closure s = (:real^1)` THEN REWRITE_TAC[EXTENSION; IN_UNIV; CLOSURE_APPROACHABLE] THEN ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN DISCH_THEN(MP_TAC o SPEC `dist(x:real^1,y) / &4`) THEN DISCH_THEN(fun th -> MP_TAC(SPEC `y - inv(&4) % (y - x):real^1` th) THEN MP_TAC(SPEC `x + inv(&4) % (y - x):real^1` th)) THEN ASM_SIMP_TAC[GSYM DIST_NZ; GSYM DROP_EQ; REAL_LT_IMP_NE; REAL_ARITH `&0 < x / &4 <=> &0 < x`] THEN SIMP_TAC[DIST_1; DROP_SUB; DROP_ADD; LIFT_DROP; DROP_CMUL] THEN DISCH_THEN(X_CHOOSE_THEN `x':real^1` STRIP_ASSUME_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `y':real^1` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `drop(g x) <= drop(g(x':real^1)) /\ drop(g x') < drop(g y') /\ drop(g y') <= drop(g y)` MP_TAC THENL [ALL_TAC; REAL_ARITH_TAC] THEN REPEAT CONJ_TAC THEN TRY(FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REAL_ARITH_TAC) THEN ASM_SIMP_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC);; (* ------------------------------------------------------------------------- *) (* More basics about segments. *) (* ------------------------------------------------------------------------- *) let BOUNDED_SEGMENT = prove (`(!a b:real^N. bounded(segment[a,b])) /\ (!a b:real^N. bounded(segment(a,b)))`, REWRITE_TAC[AND_FORALL_THM] THEN REPEAT GEN_TAC THEN MATCH_MP_TAC(MESON[BOUNDED_SUBSET] `bounded s /\ t SUBSET s ==> bounded s /\ bounded t`) THEN REWRITE_TAC[SEGMENT_OPEN_SUBSET_CLOSED] THEN MATCH_MP_TAC COMPACT_IMP_BOUNDED THEN REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN MATCH_MP_TAC COMPACT_CONVEX_HULL THEN SIMP_TAC[COMPACT_INSERT; COMPACT_EMPTY]);; let SEGMENT_IMAGE_INTERVAL = prove (`(!a b. segment[a,b] = IMAGE (\u. (&1 - drop u) % a + drop u % b) (interval[vec 0,vec 1])) /\ (!a b. ~(a = b) ==> segment(a,b) = IMAGE (\u. (&1 - drop u) % a + drop u % b) (interval(vec 0,vec 1)))`, REPEAT STRIP_TAC THEN REWRITE_TAC[EXTENSION; IN_IMAGE; IN_INTERVAL_1; IN_SEGMENT] THEN ASM_REWRITE_TAC[GSYM EXISTS_DROP; DROP_VEC] THEN MESON_TAC[]);; let CLOSURE_SEGMENT = prove (`(!a b:real^N. closure(segment[a,b]) = segment[a,b]) /\ (!a b:real^N. closure(segment(a,b)) = if a = b then {} else segment[a,b])`, REPEAT STRIP_TAC THENL [ASM_MESON_TAC[CLOSURE_EQ; COMPACT_IMP_CLOSED; SEGMENT_CONVEX_HULL; COMPACT_CONVEX_HULL; COMPACT_INSERT; COMPACT_EMPTY]; ALL_TAC] THEN REPEAT GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[SEGMENT_REFL; CLOSURE_EMPTY] THEN ASM_SIMP_TAC[SEGMENT_IMAGE_INTERVAL] THEN ASM_SIMP_TAC[CONV_RULE(RAND_CONV SYM_CONV) (SPEC_ALL CLOSURE_OPEN_INTERVAL); INTERVAL_EQ_EMPTY_1; DROP_VEC; REAL_ARITH `~(&1 <= &0)`] THEN SUBGOAL_THEN `(\u. (&1 - drop u) % a + drop u % (b:real^N)) = (\x. a + x) o (\u. drop u % (b - a))` SUBST1_TAC THENL [REWRITE_TAC[FUN_EQ_THM; o_THM] THEN VECTOR_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[IMAGE_o; CLOSURE_TRANSLATION] THEN AP_TERM_TAC THEN MATCH_MP_TAC CLOSURE_INJECTIVE_LINEAR_IMAGE THEN ASM_REWRITE_TAC[VECTOR_MUL_RCANCEL; VECTOR_SUB_EQ; DROP_EQ] THEN REWRITE_TAC[linear; DROP_ADD; DROP_CMUL] THEN VECTOR_ARITH_TAC);; let CLOSED_SEGMENT = prove (`(!a b:real^N. closed(segment[a,b])) /\ (!a b:real^N. closed(segment(a,b)) <=> a = b)`, REWRITE_TAC[GSYM CLOSURE_EQ; CLOSURE_SEGMENT] THEN REPEAT GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[SEGMENT_REFL] THEN MESON_TAC[ENDS_NOT_IN_SEGMENT; ENDS_IN_SEGMENT]);; let COMPACT_SEGMENT = prove (`(!a b:real^N. compact(segment[a,b])) /\ (!a b:real^N. compact(segment(a,b)) <=> a = b)`, REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED; CLOSED_SEGMENT; BOUNDED_SEGMENT]);; let AFFINE_HULL_SEGMENT = prove (`(!a b:real^N. affine hull (segment [a,b]) = affine hull {a,b}) /\ (!a b:real^N. affine hull (segment(a,b)) = if a = b then {} else affine hull {a,b})`, REWRITE_TAC[SEGMENT_CONVEX_HULL; AFFINE_HULL_CONVEX_HULL] THEN REPEAT GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM AFFINE_HULL_CLOSURE] THEN REWRITE_TAC[CLOSURE_SEGMENT] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[AFFINE_HULL_EMPTY] THEN REWRITE_TAC[SEGMENT_CONVEX_HULL; AFFINE_HULL_CONVEX_HULL]);; let SEGMENT_AS_BALL = prove (`(!a b. segment[a:real^N,b] = affine hull {a,b} INTER cball(inv(&2) % (a + b),norm(b - a) / &2)) /\ (!a b. segment(a:real^N,b) = affine hull {a,b} INTER ball(inv(&2) % (a + b),norm(b - a) / &2))`, REPEAT STRIP_TAC THEN (ASM_CASES_TAC `b:real^N = a` THEN ASM_REWRITE_TAC[SEGMENT_REFL; VECTOR_SUB_REFL; NORM_0] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[BALL_TRIVIAL; CBALL_TRIVIAL] THENL [REWRITE_TAC[INTER_EMPTY; INSERT_AC] THEN REWRITE_TAC[VECTOR_ARITH `&1 / &2 % (a + a) = a`] THEN REWRITE_TAC[SET_RULE `a = b INTER a <=> a SUBSET b`; HULL_SUBSET]; ASM_REWRITE_TAC[EXTENSION; IN_SEGMENT; IN_INTER; AFFINE_HULL_2] THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_ELIM_THM] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN REWRITE_TAC[REAL_ARITH `u + v:real = &1 <=> u = &1 - v`] THEN REWRITE_TAC[UNWIND_THM2] THEN REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `u:real` THEN REWRITE_TAC[] THEN ASM_CASES_TAC `y:real^N = (&1 - u) % a + u % b` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[IN_BALL; IN_CBALL; dist; VECTOR_ARITH `&1 / &2 % (a + b) - ((&1 - u) % a + u % b):real^N = (&1 / &2 - u) % (b - a)`] THEN ASM_SIMP_TAC[NORM_MUL; REAL_LT_MUL_EQ; REAL_LE_MUL_EQ; NORM_POS_LT; VECTOR_SUB_EQ; REAL_ARITH `a * n < n / &2 <=> &0 < n * (inv(&2) - a)`; REAL_ARITH `a * n <= n / &2 <=> &0 <= n * (inv(&2) - a)`] THEN REAL_ARITH_TAC]));; let CONVEX_SEGMENT = prove (`(!a b. convex(segment[a,b])) /\ (!a b. convex(segment(a,b)))`, REWRITE_TAC[SEGMENT_AS_BALL] THEN SIMP_TAC[CONVEX_INTER; CONVEX_BALL; CONVEX_CBALL; AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL]);; let RELATIVE_INTERIOR_SEGMENT = prove (`(!a b:real^N. relative_interior(segment[a,b]) = if a = b then {a} else segment(a,b)) /\ (!a b:real^N. relative_interior(segment(a,b)) = segment(a,b))`, MATCH_MP_TAC(TAUT `b /\ (b ==> a) ==> a /\ b`) THEN CONJ_TAC THENL [REPEAT GEN_TAC THEN ASM_CASES_TAC `a:real^N = b` THEN ASM_REWRITE_TAC[SEGMENT_REFL; RELATIVE_INTERIOR_EMPTY] THEN REWRITE_TAC[RELATIVE_INTERIOR_EQ; OPEN_IN_OPEN] THEN ASM_REWRITE_TAC[AFFINE_HULL_SEGMENT] THEN EXISTS_TAC `ball(inv(&2) % (a + b):real^N,norm(b - a) / &2)` THEN REWRITE_TAC[OPEN_BALL; SEGMENT_AS_BALL]; REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[SEGMENT_REFL; RELATIVE_INTERIOR_SING] THEN MP_TAC(ISPECL [`a:real^N`; `b:real^N`] (CONJUNCT2 CLOSURE_SEGMENT)) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [GSYM th]) THEN MATCH_MP_TAC CONVEX_RELATIVE_INTERIOR_CLOSURE THEN REWRITE_TAC[CONVEX_SEGMENT]]);; let OPEN_IN_SEGMENT = prove (`!s a b:real^N. segment(a,b) SUBSET s /\ s SUBSET affine hull (segment(a,b)) ==> open_in (subtopology euclidean s) (segment(a,b))`, REPEAT STRIP_TAC THEN MATCH_MP_TAC OPEN_IN_SUBSET_TRANS THEN EXISTS_TAC `affine hull (segment(a:real^N,b))` THEN ASM_MESON_TAC[OPEN_IN_RELATIVE_INTERIOR; RELATIVE_INTERIOR_SEGMENT]);; let AFF_DIM_SEGMENT = prove (`(!a b:real^N. aff_dim(segment[a,b]) = if a = b then &0 else &1) /\ (!a b:real^N. aff_dim(segment(a,b)) = if a = b then -- &1 else &1)`, REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[SEGMENT_REFL; AFF_DIM_EMPTY; AFF_DIM_SING] THEN MP_TAC(ISPECL [`a:real^N`; `b:real^N`] (CONJUNCT1 RELATIVE_INTERIOR_SEGMENT)) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN SIMP_TAC[AFF_DIM_RELATIVE_INTERIOR; CONVEX_SEGMENT] THEN REWRITE_TAC[AFF_DIM_CONVEX_HULL; SEGMENT_CONVEX_HULL] THEN ASM_REWRITE_TAC[AFF_DIM_2]);; let CONVEX_SEMIOPEN_SEGMENT = prove (`(!a b:real^N. convex(segment[a,b] DELETE a)) /\ (!a b:real^N. convex(segment[a,b] DELETE b))`, MATCH_MP_TAC(TAUT `(a ==> b) /\ a ==> a /\ b`) THEN CONJ_TAC THENL [MESON_TAC[SEGMENT_SYM]; ALL_TAC] THEN REPEAT GEN_TAC THEN ASM_CASES_TAC `b:real^N = a` THEN ASM_SIMP_TAC[SEGMENT_REFL; SET_RULE `{a} DELETE a = {}`; CONVEX_EMPTY] THEN REWRITE_TAC[CONVEX_ALT; IN_DELETE] THEN SIMP_TAC[REWRITE_RULE[CONVEX_ALT] CONVEX_SEGMENT] THEN REWRITE_TAC[IN_SEGMENT] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC; GSYM VECTOR_ADD_ASSOC] THEN ASM_REWRITE_TAC[VECTOR_ARITH `x % a + y % b + z % a + w % b:real^N = a <=> (&1 - x - z) % a = (w + y) % b`] THEN ASM_REWRITE_TAC[VECTOR_MUL_LCANCEL; REAL_ARITH `&1 - (&1 - u) * (&1 - v) - u * (&1 - w) = u * w + (&1 - u) * v`] THEN ASM_SIMP_TAC[REAL_LE_MUL; REAL_SUB_LE; REAL_ARITH `&0 <= x /\ &0 <= y ==> (x + y = &0 <=> x = &0 /\ y = &0)`] THEN REWRITE_TAC[REAL_ENTIRE; REAL_ARITH `&1 - x = &0 <=> x = &1`] THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `(u = &0 \/ w = &0) /\ (u = &1 \/ v = &0) ==> u = &0 /\ v = &0 \/ u = &1 /\ w = &0 \/ v = &0 /\ w = &0`)) THEN DISCH_THEN(REPEAT_TCL DISJ_CASES_THEN (CONJUNCTS_THEN SUBST_ALL_TAC)) THEN ASM_MESON_TAC[VECTOR_ARITH `(&1 - &0) % a + &0 % b:real^N = a`]);; let CONNECTED_SEMIOPEN_SEGMENT = prove (`(!a b:real^N. connected(segment[a,b] DELETE a)) /\ (!a b:real^N. connected(segment[a,b] DELETE b))`, SIMP_TAC[CONVEX_CONNECTED; CONVEX_SEMIOPEN_SEGMENT]);; let SEGMENT_EQ_EMPTY = prove (`(!a b:real^N. ~(segment[a,b] = {})) /\ (!a b:real^N. segment(a,b) = {} <=> a = b)`, REWRITE_TAC[SEGMENT_CONVEX_HULL; CONVEX_HULL_EQ_EMPTY; NOT_INSERT_EMPTY] THEN REPEAT GEN_TAC THEN ASM_CASES_TAC `a:real^N = b` THEN ASM_REWRITE_TAC[SEGMENT_REFL] THEN ASM_MESON_TAC[NOT_IN_EMPTY; MIDPOINT_IN_SEGMENT]);; let FINITE_SEGMENT = prove (`(!a b:real^N. FINITE(segment[a,b]) <=> a = b) /\ (!a b:real^N. FINITE(segment(a,b)) <=> a = b)`, REWRITE_TAC[open_segment; SET_RULE `s DIFF {a,b} = s DELETE a DELETE b`] THEN REWRITE_TAC[FINITE_DELETE] THEN REPEAT GEN_TAC THEN ASM_CASES_TAC `a:real^N = b` THEN ASM_REWRITE_TAC[SEGMENT_REFL; FINITE_SING] THEN REWRITE_TAC[SEGMENT_IMAGE_INTERVAL] THEN W(MP_TAC o PART_MATCH (lhs o rand) FINITE_IMAGE_INJ_EQ o rand o snd) THEN ANTS_TAC THENL [REWRITE_TAC[VECTOR_ARITH `(&1 - u) % a + u % b:real^N = (&1 - v) % a + v % b <=> (u - v) % (b - a) = vec 0`] THEN ASM_SIMP_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ; REAL_SUB_0; DROP_EQ]; DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[FINITE_INTERVAL_1] THEN REWRITE_TAC[DROP_VEC] THEN REAL_ARITH_TAC]);; let SEGMENT_EQ_SING = prove (`(!a b c:real^N. segment[a,b] = {c} <=> a = c /\ b = c) /\ (!a b c:real^N. ~(segment(a,b) = {c}))`, REWRITE_TAC[SEGMENT_CONVEX_HULL; CONVEX_HULL_EQ_SING] THEN CONJ_TAC THENL [SET_TAC[]; REPEAT GEN_TAC] THEN ASM_CASES_TAC `a:real^N = b` THEN ASM_REWRITE_TAC[SEGMENT_REFL; NOT_INSERT_EMPTY] THEN DISCH_TAC THEN MP_TAC(ISPECL [`a:real^N`; `b:real^N`] (CONJUNCT2 FINITE_SEGMENT)) THEN ASM_REWRITE_TAC[FINITE_SING]);; let SEGMENT_SUBSET_RELATIVE_FRONTIER_CONVEX_GEN = prove (`!s a b c:real^N. convex s /\ collinear{a,b,c} /\ ~(a = b) /\ ~(a = c) /\ ~(b = c) /\ {a,b,c} SUBSET relative_frontier s ==> convex hull {a,b,c} SUBSET relative_frontier s`, REWRITE_TAC[COLLINEAR_BETWEEN_CASES] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPEC `s:real^N->bool` SEGMENT_SUBSET_RELATIVE_FRONTIER_CONVEX) THENL [DISCH_THEN(MP_TAC o SPECL [`b:real^N`; `c:real^N`; `a:real^N`]); DISCH_THEN(MP_TAC o SPECL [`c:real^N`; `a:real^N`; `b:real^N`]); DISCH_THEN(MP_TAC o SPECL [`a:real^N`; `b:real^N`; `c:real^N`])] THEN REWRITE_TAC[open_segment; IN_DIFF; IN_INSERT; NOT_IN_EMPTY] THEN REWRITE_TAC[GSYM BETWEEN_IN_SEGMENT] THEN (ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] SUBSET_TRANS) THEN MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[CONVEX_SEGMENT] THEN REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; ENDS_IN_SEGMENT] THEN ASM_REWRITE_TAC[GSYM BETWEEN_IN_SEGMENT]);; let SUBSET_SEGMENT_OPEN_CLOSED = prove (`!a b c d:real^N. segment(a,b) SUBSET segment(c,d) <=> a = b \/ segment[a,b] SUBSET segment[c,d]`, REPEAT GEN_TAC THEN EQ_TAC THENL [ASM_CASES_TAC `a:real^N = b` THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o MATCH_MP SUBSET_CLOSURE) THEN ASM_REWRITE_TAC[CLOSURE_SEGMENT] THEN COND_CASES_TAC THEN REWRITE_TAC[SUBSET_EMPTY; SEGMENT_EQ_EMPTY]; ALL_TAC] THEN DISCH_THEN(DISJ_CASES_THEN2 SUBST1_TAC MP_TAC) THEN REWRITE_TAC[SEGMENT_REFL; EMPTY_SUBSET] THEN ABBREV_TAC `m:real^N = d - c` THEN POP_ASSUM MP_TAC THEN GEOM_NORMALIZE_TAC `m:real^N` THEN SIMP_TAC[VECTOR_SUB_EQ; SEGMENT_REFL; SEGMENT_EQ_SING; SEGMENT_EQ_EMPTY; SET_RULE `s SUBSET {a} <=> s = {a} \/ s = {}`; SUBSET_REFL] THEN X_GEN_TAC `m:real^N` THEN DISCH_TAC THEN REPEAT GEN_TAC THEN DISCH_THEN(SUBST_ALL_TAC o SYM) THEN POP_ASSUM MP_TAC THEN GEOM_ORIGIN_TAC `c:real^N` THEN GEOM_BASIS_MULTIPLE_TAC 1 `d:real^N` THEN X_GEN_TAC `d:real` THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN SIMP_TAC[VECTOR_SUB_RZERO; NORM_MUL; NORM_BASIS; DIMINDEX_GE_1; LE_REFL] THEN ASM_REWRITE_TAC[real_abs; REAL_MUL_RID] THEN DISCH_THEN SUBST_ALL_TAC THEN POP_ASSUM(K ALL_TAC) THEN DISCH_TAC THEN SUBGOAL_THEN `collinear{vec 0:real^N,&1 % basis 1,x} /\ collinear{vec 0:real^N,&1 % basis 1,y}` MP_TAC THENL [ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {a,c,b}`] THEN CONJ_TAC THEN MATCH_MP_TAC BETWEEN_IMP_COLLINEAR THEN REWRITE_TAC[BETWEEN_IN_SEGMENT] THEN ASM_MESON_TAC[SUBSET; ENDS_IN_SEGMENT]; ALL_TAC] THEN SIMP_TAC[COLLINEAR_LEMMA_ALT; BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL; VECTOR_ARITH `&1 % x:real^N = vec 0 <=> x = vec 0`] THEN REWRITE_TAC[IMP_CONJ; VECTOR_MUL_ASSOC; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `a:real` THEN REWRITE_TAC[REAL_MUL_RID] THEN DISCH_THEN SUBST_ALL_TAC THEN X_GEN_TAC `b:real` THEN DISCH_THEN SUBST_ALL_TAC THEN POP_ASSUM MP_TAC THEN SUBST1_TAC(VECTOR_ARITH `vec 0:real^N = &0 % basis 1`) THEN ASM_SIMP_TAC[SEGMENT_SCALAR_MULTIPLE; VECTOR_MUL_RCANCEL; BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL; SET_RULE `(!x y. x % v = y % v <=> x = y) ==> ({x % v | P x} SUBSET {x % v | Q x} <=> {x | P x} SUBSET {x | Q x})`] THEN REWRITE_TAC[REAL_ARITH `a <= x /\ x <= b \/ b <= x /\ x <= a <=> min a b <= x /\ x <= max a b`; REAL_ARITH `a < x /\ x < b \/ b < x /\ x < a <=> min a b < x /\ x < max a b`] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN DISCH_TAC THEN X_GEN_TAC `x:real` THEN FIRST_X_ASSUM(fun th -> MAP_EVERY (MP_TAC o C SPEC th) [`min (a:real) b`; `max (a:real) b`]) THEN REAL_ARITH_TAC);; let SUBSET_SEGMENT = prove (`(!a b c d:real^N. segment[a,b] SUBSET segment[c,d] <=> a IN segment[c,d] /\ b IN segment[c,d]) /\ (!a b c d:real^N. segment[a,b] SUBSET segment(c,d) <=> a IN segment(c,d) /\ b IN segment(c,d)) /\ (!a b c d:real^N. segment(a,b) SUBSET segment[c,d] <=> a = b \/ a IN segment[c,d] /\ b IN segment[c,d]) /\ (!a b c d:real^N. segment(a,b) SUBSET segment(c,d) <=> a = b \/ a IN segment[c,d] /\ b IN segment[c,d])`, MATCH_MP_TAC(TAUT `(a /\ b) /\ (a /\ b ==> c) ==> a /\ b /\ c`) THEN CONJ_TAC THENL [REPEAT STRIP_TAC THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [SEGMENT_CONVEX_HULL] THEN SIMP_TAC[SUBSET_HULL; CONVEX_SEGMENT] THEN SET_TAC[]; STRIP_TAC THEN ASM_REWRITE_TAC[SUBSET_SEGMENT_OPEN_CLOSED] THEN REPEAT GEN_TAC THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `closure(segment(a:real^N,b)) SUBSET segment[c,d]` THEN CONJ_TAC THENL [SIMP_TAC[CLOSURE_MINIMAL_EQ; CLOSED_SEGMENT]; ALL_TAC] THEN REWRITE_TAC[CLOSURE_SEGMENT] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[EMPTY_SUBSET]]);; let INTERIOR_SEGMENT = prove (`(!a b:real^N. interior(segment[a,b]) = if 2 <= dimindex(:N) then {} else segment(a,b)) /\ (!a b:real^N. interior(segment(a,b)) = if 2 <= dimindex(:N) then {} else segment(a,b))`, REWRITE_TAC[AND_FORALL_THM] THEN REPEAT GEN_TAC THEN ASM_CASES_TAC `2 <= dimindex(:N)` THEN ASM_REWRITE_TAC[] THENL [MATCH_MP_TAC(SET_RULE `t SUBSET s /\ s = {} ==> s = {} /\ t = {}`) THEN SIMP_TAC[SEGMENT_OPEN_SUBSET_CLOSED; SUBSET_INTERIOR] THEN REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN MATCH_MP_TAC EMPTY_INTERIOR_CONVEX_HULL THEN REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY] THEN FIRST_ASSUM (MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] LE_TRANS)) THEN SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN ARITH_TAC; ASM_CASES_TAC `a:real^N = b` THEN ASM_SIMP_TAC[SEGMENT_REFL; INTERIOR_EMPTY; EMPTY_INTERIOR_FINITE; FINITE_SING] THEN SUBGOAL_THEN `affine hull (segment[a,b]) = (:real^N) /\ affine hull (segment(a,b)) = (:real^N)` (fun th -> ASM_SIMP_TAC[th; GSYM RELATIVE_INTERIOR_INTERIOR; RELATIVE_INTERIOR_SEGMENT]) THEN ASM_REWRITE_TAC[AFFINE_HULL_SEGMENT] THEN MATCH_MP_TAC AFFINE_INDEPENDENT_SPAN_GT THEN REWRITE_TAC[AFFINE_INDEPENDENT_2] THEN ASM_SIMP_TAC[CARD_CLAUSES; FINITE_RULES; IN_INSERT; NOT_IN_EMPTY] THEN ASM_ARITH_TAC]);; let FRONTIER_SEGMENT = prove (`(!a b:real^N. frontier(segment[a,b]) = if 2 <= dimindex(:N) then segment[a,b] else {a,b}) /\ (!a b:real^N. frontier(segment(a,b)) = if a = b then {} else if 2 <= dimindex(:N) then segment[a,b] else {a,b})`, REPEAT GEN_TAC THEN REWRITE_TAC[frontier; INTERIOR_SEGMENT; CLOSURE_SEGMENT] THEN ASM_CASES_TAC `2 <= dimindex(:N)` THEN ASM_REWRITE_TAC[DIFF_EMPTY] THEN SIMP_TAC[SEGMENT_REFL] THEN REWRITE_TAC[SEGMENT_CLOSED_OPEN] THEN REPEAT STRIP_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[SEGMENT_REFL; DIFF_EMPTY]) THEN MP_TAC(ISPECL [`a:real^N`; `b:real^N`] ENDS_NOT_IN_SEGMENT) THEN SET_TAC[]);; let SEGMENT_EQ = prove (`(!a b c d:real^N. segment[a,b] = segment[c,d] <=> {a,b} = {c,d}) /\ (!a b c d:real^N. ~(segment[a,b] = segment(c,d))) /\ (!a b c d:real^N. ~(segment(a,b) = segment[c,d])) /\ (!a b c d:real^N. segment(a,b) = segment(c,d) <=> a = b /\ c = d \/ {a,b} = {c,d})`, MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL [REPEAT GEN_TAC THEN EQ_TAC THENL [DISCH_THEN(fun th -> MP_TAC th THEN MP_TAC(AP_TERM `\s:real^N->bool. s DIFF relative_interior s` th)) THEN REWRITE_TAC[RELATIVE_INTERIOR_SEGMENT] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[SEGMENT_REFL]) THEN SIMP_TAC[ENDS_IN_SEGMENT; open_segment; SET_RULE `a IN s /\ b IN s ==> s DIFF (s DIFF {a,b}) = {a,b}`] THEN ASM SET_TAC[SEGMENT_EQ_SING]; SIMP_TAC[SEGMENT_CONVEX_HULL]]; DISCH_TAC] THEN MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL [REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o AP_TERM `closed:(real^N->bool)->bool`) THEN REWRITE_TAC[CONJUNCT1 CLOSED_SEGMENT] THEN REWRITE_TAC[GSYM CLOSURE_EQ; CLOSURE_SEGMENT] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL [ASM SET_TAC[SEGMENT_EQ_EMPTY]; REWRITE_TAC[open_segment; ENDS_IN_SEGMENT; SET_RULE `s = s DIFF {a,b} <=> ~(a IN s) /\ ~(b IN s)`]]; DISCH_TAC THEN CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN REPEAT GEN_TAC THEN ASM_CASES_TAC `c:real^N = d` THEN ASM_REWRITE_TAC[SEGMENT_EQ_EMPTY; SEGMENT_REFL] THENL [ASM SET_TAC[]; ALL_TAC] THEN CONV_TAC(BINOP_CONV SYM_CONV)THEN ASM_CASES_TAC `a:real^N = b` THEN ASM_REWRITE_TAC[SEGMENT_EQ_EMPTY; SEGMENT_REFL] THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; SUBSET_SEGMENT_OPEN_CLOSED] THEN ASM_REWRITE_TAC[SUBSET_ANTISYM_EQ]]);; let COLLINEAR_SEGMENT = prove (`(!a b:real^N. collinear(segment[a,b])) /\ (!a b:real^N. collinear(segment(a,b)))`, REWRITE_TAC[AND_FORALL_THM] THEN REPEAT GEN_TAC THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [REWRITE_TAC[COLLINEAR_AFFINE_HULL] THEN MAP_EVERY EXISTS_TAC [`a:real^N`; `b:real^N`] THEN REWRITE_TAC[SEGMENT_CONVEX_HULL; CONVEX_HULL_SUBSET_AFFINE_HULL]; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] COLLINEAR_SUBSET) THEN REWRITE_TAC[SEGMENT_OPEN_SUBSET_CLOSED]]);; let INTER_SEGMENT = prove (`!a b c:real^N. b IN segment[a,c] \/ ~collinear{a,b,c} ==> segment[a,b] INTER segment[b,c] = {b}`, REPEAT GEN_TAC THEN ASM_CASES_TAC `c:real^N = a` THENL [ASM_SIMP_TAC[SEGMENT_REFL; IN_SING; INTER_IDEMPOT; INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN DISCH_THEN(DISJ_CASES_THEN MP_TAC) THENL [REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN DISCH_TAC THEN MP_TAC(ISPECL [`{a:real^N,c}`; `b:real^N`; `{a:real^N}`; `{c:real^N}`] CONVEX_HULL_EXCHANGE_INTER) THEN ASM_REWRITE_TAC[AFFINE_INDEPENDENT_2] THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[INSERT_AC]] THEN DISCH_THEN SUBST1_TAC THEN ASM_SIMP_TAC[SET_RULE `~(a = c) ==> {a} INTER {c} = {}`] THEN REWRITE_TAC[CONVEX_HULL_SING]; ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `~(s INTER t = {b}) ==> b IN s /\ b IN t ==> ?a. ~(a = b) /\ a IN s /\ b IN s /\ a IN t /\ b IN t`)) THEN ANTS_TAC THENL [REWRITE_TAC[ENDS_IN_SEGMENT]; ALL_TAC] THEN REWRITE_TAC[GSYM BETWEEN_IN_SEGMENT; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `d:real^N` THEN STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP BETWEEN_IMP_COLLINEAR)) THEN MATCH_MP_TAC COLLINEAR_3_TRANS THEN EXISTS_TAC `d:real^N` THEN REPEAT(POP_ASSUM MP_TAC) THEN SIMP_TAC[INSERT_AC]]);; let CONVEX_LINE_INTERSECTION_UNIQUE_CLOSED = prove (`!s a b:real^N. convex s /\ closed s /\ a IN relative_frontier s /\ b IN relative_frontier s /\ ~(segment(a,b) INTER relative_interior s = {}) ==> s INTER (affine hull {a,b}) = segment[a,b]`, REPEAT GEN_TAC THEN ASM_CASES_TAC `a:real^N = b` THEN ASM_REWRITE_TAC[SEGMENT_REFL; EMPTY_SUBSET; INTER_EMPTY] THEN STRIP_TAC THEN SUBGOAL_THEN `(a:real^N) IN s /\ (b:real^N) IN s` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[RELATIVE_FRONTIER_SUBSET; SUBSET]; ALL_TAC] THEN REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; SUBSET_INTER] THEN ASM_SIMP_TAC[SEGMENT_SUBSET_CONVEX] THEN REWRITE_TAC[SEGMENT_CONVEX_HULL; CONVEX_HULL_SUBSET_AFFINE_HULL] THEN REWRITE_TAC[SUBSET; GSYM SEGMENT_CONVEX_HULL; IN_INTER] THEN ASM_SIMP_TAC[GSYM COLLINEAR_3_IN_AFFINE_HULL] THEN X_GEN_TAC `c:real^N` THEN STRIP_TAC THEN MAP_EVERY ASM_CASES_TAC [`c:real^N = a`; `c:real^N = b`] THEN ASM_REWRITE_TAC[ENDS_IN_SEGMENT] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [COLLINEAR_BETWEEN_CASES]) THEN REWRITE_TAC[BETWEEN_IN_SEGMENT; SEGMENT_CLOSED_OPEN] THEN ASM_REWRITE_TAC[IN_UNION; IN_INSERT; NOT_IN_EMPTY] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; IN_INTER] THEN X_GEN_TAC `z:real^N` THEN STRIP_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPECL [`s:real^N->bool`; `z:real^N`; `c:real^N`] IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT) THEN ASM_SIMP_TAC[CLOSURE_INC] THEN MATCH_MP_TAC(SET_RULE `(?a. ~(a IN t) /\ a IN s) ==> s SUBSET t ==> P`) THENL [EXISTS_TAC `a:real^N`; EXISTS_TAC `b:real^N`] THEN RULE_ASSUM_TAC(REWRITE_RULE[relative_frontier; IN_DIFF]) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[open_segment; IN_DIFF; IN_INSERT; NOT_IN_EMPTY] THEN (CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[ENDS_NOT_IN_SEGMENT]]) THEN RULE_ASSUM_TAC(REWRITE_RULE [open_segment; IN_DIFF; IN_INSERT; NOT_IN_EMPTY; GSYM BETWEEN_IN_SEGMENT]) THEN REWRITE_TAC[GSYM BETWEEN_IN_SEGMENT] THEN ASM_MESON_TAC[BETWEEN_TRANS_2; BETWEEN_SYM]);; let CONVEX_LINE_INTERSECTION_UNIQUE_OPEN_IN = prove (`!s a b:real^N. convex s /\ open_in (subtopology euclidean (affine hull s)) s /\ a IN relative_frontier s /\ b IN relative_frontier s /\ ~(segment(a,b) INTER s = {}) ==> s INTER (affine hull {a,b}) = segment(a,b)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [REWRITE_TAC[open_segment; SET_RULE `s SUBSET t DIFF {a,b} <=> (~(a IN s) /\ ~(b IN s)) /\ s SUBSET t`] THEN CONJ_TAC THENL [REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [relative_frontier])) THEN ASM_SIMP_TAC[RELATIVE_INTERIOR_OPEN_IN] THEN SET_TAC[]; TRANS_TAC SUBSET_TRANS `closure s INTER affine hull {a:real^N,b}` THEN CONJ_TAC THENL [MP_TAC(ISPEC `s:real^N->bool` CLOSURE_SUBSET) THEN SET_TAC[]; MATCH_MP_TAC(SET_RULE `s = t ==> s SUBSET t`)] THEN MATCH_MP_TAC CONVEX_LINE_INTERSECTION_UNIQUE_CLOSED THEN ASM_SIMP_TAC[CLOSED_CLOSURE; CONVEX_CLOSURE; RELATIVE_FRONTIER_CLOSURE; CONVEX_RELATIVE_INTERIOR_CLOSURE; RELATIVE_INTERIOR_OPEN_IN]]; REWRITE_TAC[SUBSET_INTER; SEGMENT_SUBSET_LINE] THEN MP_TAC(ISPECL [`s:real^N->bool`; `a:real^N`; `b:real^N`] CONVEX_OPEN_SEGMENT_CASES) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [relative_frontier])) THEN ASM_SIMP_TAC[RELATIVE_INTERIOR_OPEN_IN; IN_DIFF; relative_frontier] THEN ASM SET_TAC[]]);; let CONVEX_LINE_INTERSECTION_UNIQUE_OPEN = prove (`!s a b:real^N. convex s /\ open s /\ a IN relative_frontier s /\ b IN relative_frontier s /\ ~(segment(a,b) INTER s = {}) ==> s INTER (affine hull {a,b}) = segment(a,b)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CONVEX_LINE_INTERSECTION_UNIQUE_OPEN_IN THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC OPEN_SUBSET THEN ASM_REWRITE_TAC[HULL_SUBSET]);; let CONVEX_LINE_INTERSECTIONS = prove (`!s a b:real^N. convex s /\ a IN relative_frontier s /\ b IN relative_frontier s /\ ~(segment(a,b) INTER relative_interior s = {}) ==> ~(a = b) /\ closure s INTER affine hull {a,b} = segment[a,b] /\ relative_interior s INTER affine hull {a,b} = segment(a,b) /\ relative_frontier s INTER affine hull {a,b} = {a,b}`, REPEAT GEN_TAC THEN ASM_CASES_TAC `a:real^N = b` THEN ASM_REWRITE_TAC[SEGMENT_REFL; INTER_EMPTY] THEN STRIP_TAC THEN MATCH_MP_TAC(TAUT `(p /\ q ==> r) /\ p /\ q ==> p /\ q /\ r`) THEN REPEAT CONJ_TAC THENL [STRIP_TAC THEN ASM_REWRITE_TAC[relative_frontier; SET_RULE `(s DIFF t) INTER u = (s INTER u) DIFF (t INTER u)`] THEN REWRITE_TAC[open_segment] THEN MATCH_MP_TAC(SET_RULE `t SUBSET s ==> s DIFF (s DIFF t) = t`) THEN REWRITE_TAC[INSERT_SUBSET; ENDS_IN_SEGMENT; EMPTY_SUBSET]; MATCH_MP_TAC CONVEX_LINE_INTERSECTION_UNIQUE_CLOSED THEN ASM_SIMP_TAC[RELATIVE_FRONTIER_CLOSURE; CONVEX_RELATIVE_INTERIOR_CLOSURE; CLOSED_CLOSURE; CONVEX_CLOSURE]; MATCH_MP_TAC CONVEX_LINE_INTERSECTION_UNIQUE_OPEN_IN THEN ASM_SIMP_TAC[RELATIVE_FRONTIER_RELATIVE_INTERIOR; OPEN_IN_RELATIVE_INTERIOR; CONVEX_RELATIVE_INTERIOR; AFFINE_HULL_RELATIVE_INTERIOR]]);; let CONVEX_LINE_INTERSECTIONS_ALT = prove (`!s a b:real^N. convex s /\ a IN relative_frontier s /\ b IN relative_frontier s /\ ~(segment(a,b) SUBSET relative_frontier s) ==> ~(a = b) /\ closure s INTER affine hull {a,b} = segment[a,b] /\ relative_interior s INTER affine hull {a,b} = segment(a,b) /\ relative_frontier s INTER affine hull {a,b} = {a,b}`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC CONVEX_LINE_INTERSECTIONS THEN ASM_REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[relative_frontier; IN_DIFF]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `~(s SUBSET c DIFF d) ==> s SUBSET c ==> ~(s INTER d = {})`)) THEN TRANS_TAC SUBSET_TRANS `segment[a:real^N,b]` THEN REWRITE_TAC[SEGMENT_OPEN_SUBSET_CLOSED] THEN ASM_SIMP_TAC[CONVEX_CONTAINS_SEGMENT_IMP; CONVEX_CLOSURE]);; (* ------------------------------------------------------------------------- *) (* Theorems about strips between bounds on a component. *) (* ------------------------------------------------------------------------- *) let CLOSED_STRIP_COMPONENT_LE = prove (`!a k. closed {x:real^N | abs(x$k) <= a}`, REWRITE_TAC[REAL_ARITH `abs(x) <= a <=> x <= a /\ x >= --a`] THEN REWRITE_TAC[SET_RULE `{x | P x /\ Q x} = {x | P x} INTER {x | Q x}`] THEN SIMP_TAC[CLOSED_HALFSPACE_COMPONENT_LE; CLOSED_HALFSPACE_COMPONENT_GE; CLOSED_INTER]);; let OPEN_STRIP_COMPONENT_LT = prove (`!a k. open {x:real^N | abs(x$k) < a}`, REWRITE_TAC[REAL_ARITH `abs(x) < a <=> x < a /\ x > --a`] THEN REWRITE_TAC[SET_RULE `{x | P x /\ Q x} = {x | P x} INTER {x | Q x}`] THEN SIMP_TAC[OPEN_HALFSPACE_COMPONENT_LT; OPEN_HALFSPACE_COMPONENT_GT; OPEN_INTER]);; let INTERIOR_STRIP_COMPONENT_LE = prove (`!a k. interior {x:real^N | abs(x$k) <= a} = {x | abs(x$k) < a}`, REWRITE_TAC[REAL_ARITH `abs(x) <= a <=> x <= a /\ x >= --a`; REAL_ARITH `abs(x) < a <=> x < a /\ x > --a`] THEN REWRITE_TAC[SET_RULE `{x | P x /\ Q x} = {x | P x} INTER {x | Q x}`] THEN REWRITE_TAC[INTERIOR_INTER; INTERIOR_HALFSPACE_COMPONENT_LE; INTERIOR_HALFSPACE_COMPONENT_GE]);; let CLOSURE_STRIP_COMPONENT_LT = prove (`!a k. closure {x:real^N | abs(x$k) < a} = if a = &0 then {} else {x | abs(x$k) <= a}`, REPEAT GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_ARITH `~(abs x < &0)`; EMPTY_GSPEC; CLOSURE_EMPTY] THEN ASM_CASES_TAC `a < &0` THEN ASM_SIMP_TAC[REAL_ARITH `a < &0 ==> ~(abs x < a) /\ ~(abs x <= a)`; EMPTY_GSPEC; CLOSURE_EMPTY] THEN REWRITE_TAC[GSYM INTERIOR_STRIP_COMPONENT_LE] THEN GEN_REWRITE_TAC RAND_CONV [GSYM(MATCH_MP CLOSURE_CLOSED (SPEC_ALL CLOSED_STRIP_COMPONENT_LE))] THEN MATCH_MP_TAC CONVEX_CLOSURE_INTERIOR THEN REWRITE_TAC[CONVEX_STRIP_COMPONENT_LE; GSYM MEMBER_NOT_EMPTY] THEN REWRITE_TAC[INTERIOR_STRIP_COMPONENT_LE; IN_ELIM_THM] THEN EXISTS_TAC `vec 0:real^N` THEN REWRITE_TAC[VEC_COMPONENT] THEN ASM_REAL_ARITH_TAC);; let FRONTIER_STRIP_COMPONENT_LE = prove (`!a k. frontier {x:real^N | abs(x$k) <= a} = {x | abs(x$k) = a}`, SIMP_TAC[frontier; CLOSED_STRIP_COMPONENT_LE; CLOSURE_CLOSED; INTERIOR_STRIP_COMPONENT_LE] THEN REWRITE_TAC[IN_DIFF; EXTENSION; IN_ELIM_THM] THEN REAL_ARITH_TAC);; let FRONTIER_STRIP_COMPONENT_LT = prove (`!a k. frontier {x:real^N | abs(x$k) < a} = if a = &0 then {} else {x | abs(x$k) = a}`, SIMP_TAC[frontier; OPEN_STRIP_COMPONENT_LT; INTERIOR_OPEN; CLOSURE_STRIP_COMPONENT_LT] THEN REPEAT GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[EMPTY_DIFF] THEN REWRITE_TAC[IN_DIFF; EXTENSION; IN_ELIM_THM] THEN REAL_ARITH_TAC);; (* ------------------------------------------------------------------------- *) (* Lower-dimensional affine subsets are nowhere dense. *) (* ------------------------------------------------------------------------- *) let DENSE_COMPLEMENT_SUBSPACE = prove (`!s t:real^N->bool. dim t < dim s /\ subspace s ==> closure(s DIFF t) = s`, SUBGOAL_THEN `!s t:real^N->bool. dim t < dim s /\ subspace s /\ t SUBSET s ==> closure(s DIFF t) = s` MP_TAC THENL [ALL_TAC; MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `s:real^N->bool` THEN DISCH_TAC THEN X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `s INTER t:real^N->bool`) THEN ASM_REWRITE_TAC[SET_RULE `s DIFF (s INTER t) = s DIFF t`] THEN DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[INTER_SUBSET] THEN TRANS_TAC LET_TRANS `dim(t:real^N->bool)` THEN ASM_SIMP_TAC[DIM_SUBSET; INTER_SUBSET]] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`t:real^N->bool`; `s:real^N->bool`] ORTHOGONAL_TO_SUBSPACE_EXISTS_GEN) THEN ASM_SIMP_TAC[PSUBSET; SPAN_MONO] THEN ANTS_TAC THENL [ASM_MESON_TAC[LT_REFL; DIM_SPAN]; ASM_SIMP_TAC[SPAN_OF_SUBSPACE]] THEN DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [MATCH_MP_TAC CLOSURE_MINIMAL THEN ASM_SIMP_TAC[CLOSED_SUBSPACE] THEN SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[SUBSET; IN_DIFF; CLOSURE_APPROACHABLE] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN ASM_CASES_TAC `(x:real^N) IN t` THENL [ALL_TAC; EXISTS_TAC `x:real^N` THEN ASM_REWRITE_TAC[DIST_REFL]] THEN EXISTS_TAC `x + e / &2 / norm(a) % a:real^N` THEN REPEAT CONJ_TAC THENL [ASM_SIMP_TAC[SUBSPACE_ADD; SUBSPACE_MUL]; DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `(x + e / &2 / norm(a) % a) + -- &1 % x:real^N`) THEN ASM_SIMP_TAC[NOT_IMP; SPAN_ADD; SPAN_MUL; SPAN_SUPERSET] THEN REWRITE_TAC[VECTOR_ARITH `(x + a) + -- &1 % x:real^N = a`] THEN ASM_REWRITE_TAC[ORTHOGONAL_MUL; ORTHOGONAL_REFL; REAL_DIV_EQ_0; NORM_EQ_0] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[NORM_ARITH `dist(x + a:real^N,x) = norm a`] THEN REWRITE_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM; REAL_ABS_NUM] THEN ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_EQ_0] THEN ASM_REAL_ARITH_TAC]);; let DENSE_COMPLEMENT_AFFINE = prove (`!s t:real^N->bool. aff_dim t < aff_dim s /\ affine s ==> closure(s DIFF t) = s`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s INTER t:real^N->bool = {}` THENL [REPEAT STRIP_TAC THEN TRANS_TAC EQ_TRANS `closure s:real^N->bool` THEN CONJ_TAC THENL [AP_TERM_TAC THEN ASM SET_TAC[]; ASM_SIMP_TAC[CLOSURE_CLOSED; CLOSED_AFFINE]]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; IN_INTER] THEN X_GEN_TAC `z:real^N` THEN GEOM_ORIGIN_TAC `z:real^N` THEN SIMP_TAC[AFFINE_EQ_SUBSPACE; AFF_DIM_DIM_0; HULL_INC; INT_OF_NUM_LT] THEN MESON_TAC[DENSE_COMPLEMENT_SUBSPACE]]);; let DENSE_COMPLEMENT_OPEN_IN_AFFINE_HULL = prove (`!s t:real^N->bool. aff_dim t < aff_dim s /\ open_in (subtopology euclidean (affine hull s)) s ==> closure(s DIFF t) = closure s`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `affine hull s DIFF t:real^N->bool`; `affine hull s:real^N->bool`] CLOSURE_OPEN_IN_INTER_CLOSURE) THEN ASM_REWRITE_TAC[OPEN_IN_RELATIVE_INTERIOR; SUBSET_DIFF] THEN ASM_SIMP_TAC[DENSE_COMPLEMENT_AFFINE; AFFINE_AFFINE_HULL; AFF_DIM_AFFINE_HULL] THEN CONV_TAC(RAND_CONV SYM_CONV) THEN MATCH_MP_TAC EQ_IMP THEN BINOP_TAC THEN AP_TERM_TAC THEN MP_TAC(ISPECL [`affine:(real^N->bool)->bool`; `s:real^N->bool`] HULL_SUBSET) THEN SET_TAC[]);; let DENSE_COMPLEMENT_CONVEX = prove (`!s t:real^N->bool. aff_dim t < aff_dim s /\ convex s ==> closure(s DIFF t) = closure s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN ASM_SIMP_TAC[SUBSET_CLOSURE; SUBSET_DIFF] THEN MP_TAC(ISPECL [`relative_interior s:real^N->bool`; `t:real^N->bool`] DENSE_COMPLEMENT_OPEN_IN_AFFINE_HULL) THEN ASM_SIMP_TAC[OPEN_IN_RELATIVE_INTERIOR; AFF_DIM_RELATIVE_INTERIOR; AFFINE_HULL_RELATIVE_INTERIOR; CONVEX_CLOSURE_RELATIVE_INTERIOR] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN MATCH_MP_TAC SUBSET_CLOSURE THEN MP_TAC(ISPEC `s:real^N->bool` RELATIVE_INTERIOR_SUBSET) THEN SET_TAC[]);; let DENSE_COMPLEMENT_CONVEX_CLOSED = prove (`!s t:real^N->bool. aff_dim t < aff_dim s /\ convex s /\ closed s ==> closure(s DIFF t) = s`, ASM_SIMP_TAC[DENSE_COMPLEMENT_CONVEX; CLOSURE_CLOSED]);; (* ------------------------------------------------------------------------- *) (* Homeomorphism of all convex compact sets with same affine dimension, and *) (* in particular all those with nonempty interior. *) (* ------------------------------------------------------------------------- *) let COMPACT_FRONTIER_LINE_LEMMA = prove (`!s x. compact s /\ (vec 0 IN s) /\ ~(x = vec 0 :real^N) ==> ?u. &0 <= u /\ (u % x) IN frontier s /\ !v. u < v ==> ~((v % x) IN s)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP COMPACT_IMP_BOUNDED) THEN REWRITE_TAC[BOUNDED_POS] THEN DISCH_THEN(X_CHOOSE_THEN `b:real` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`{y:real^N | ?u. &0 <= u /\ u <= b / norm(x) /\ (y = u % x)} INTER s`; `vec 0:real^N`] DISTANCE_ATTAINS_SUP) THEN ANTS_TAC THENL [CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `vec 0:real^N` THEN ASM_REWRITE_TAC[IN_INTER; IN_ELIM_THM] THEN EXISTS_TAC `&0` THEN ASM_SIMP_TAC[VECTOR_MUL_LZERO; REAL_LE_REFL; REAL_LT_IMP_LE; REAL_LT_DIV; NORM_POS_LT]] THEN MATCH_MP_TAC COMPACT_INTER THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `{y:real^N | ?u. &0 <= u /\ u <= b / norm(x) /\ (y = u % x)} = IMAGE (\u. drop u % x) (interval [vec 0,lambda i. b / norm(x:real^N)])` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_IMAGE; IN_INTERVAL] THEN SIMP_TAC[LAMBDA_BETA] THEN SIMP_TAC[DIMINDEX_1; ARITH_RULE `1 <= i /\ i <= 1 <=> (i = 1)`] THEN REWRITE_TAC[GSYM drop; LEFT_FORALL_IMP_THM; EXISTS_REFL; DROP_VEC] THEN REWRITE_TAC[EXISTS_LIFT; LIFT_DROP] THEN MESON_TAC[]; ALL_TAC] THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN REWRITE_TAC[COMPACT_INTERVAL] THEN MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_VMUL THEN REWRITE_TAC[o_DEF; LIFT_DROP; CONTINUOUS_AT_ID]; ALL_TAC] THEN REWRITE_TAC[IN_INTER; IN_ELIM_THM; LEFT_AND_EXISTS_THM] THEN ONCE_REWRITE_TAC[TAUT `(a /\ b /\ c) /\ d <=> c /\ a /\ b /\ d`] THEN SIMP_TAC[LEFT_IMP_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN GEN_REWRITE_TAC (BINDER_CONV o ONCE_DEPTH_CONV) [SWAP_FORALL_THM] THEN SIMP_TAC[IMP_CONJ] THEN REWRITE_TAC[LEFT_FORALL_IMP_THM; EXISTS_REFL] THEN REWRITE_TAC[IMP_IMP] THEN REWRITE_TAC[LEFT_FORALL_IMP_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real` THEN REWRITE_TAC[dist; VECTOR_SUB_LZERO; NORM_NEG; NORM_MUL] THEN ASM_SIMP_TAC[REAL_LE_RMUL_EQ; NORM_POS_LT] THEN DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN ASM_SIMP_TAC[real_abs] THEN REPEAT STRIP_TAC THENL [REWRITE_TAC[FRONTIER_STRADDLE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN CONJ_TAC THENL [EXISTS_TAC `u % x :real^N` THEN ASM_REWRITE_TAC[DIST_REFL]; ALL_TAC] THEN EXISTS_TAC `(u + (e / &2) / norm(x)) % x :real^N` THEN REWRITE_TAC[dist; VECTOR_ARITH `u % x - (u + a) % x = --(a % x)`] THEN ASM_SIMP_TAC[NORM_NEG; NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM; NORM_EQ_0; REAL_DIV_RMUL; REAL_ABS_NUM; REAL_LT_LDIV_EQ; REAL_OF_NUM_LT; ARITH; REAL_ARITH `abs e < e * &2 <=> &0 < e`] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `u + (e / &2) / norm(x:real^N)`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REAL_ARITH `&0 < e /\ &0 <= u /\ u + e <= b ==> ~(&0 <= u + e /\ u + e <= b ==> u + e <= u)`) THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH; NORM_POS_LT] THEN FIRST_X_ASSUM(MP_TAC o SPEC `(u + (e / &2) / norm(x:real^N)) % x`) THEN ASM_SIMP_TAC[NORM_MUL; GSYM REAL_LE_RDIV_EQ; NORM_POS_LT] THEN REAL_ARITH_TAC; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPEC `v:real`) THEN ASM_REWRITE_TAC[GSYM REAL_NOT_LT] THEN ASM_REWRITE_TAC[REAL_NOT_LT] THEN CONJ_TAC THENL [ASM_MESON_TAC[REAL_LET_TRANS; REAL_LT_IMP_LE]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPEC `v % x:real^N`) THEN ASM_SIMP_TAC[NORM_MUL; GSYM REAL_LE_RDIV_EQ; NORM_POS_LT] THEN REAL_ARITH_TAC);; let STARLIKE_COMPACT_PROJECTIVE = prove (`!s:real^N->bool a. compact s /\ a IN relative_interior s /\ (!x. x IN s ==> segment(a,x) SUBSET relative_interior s) ==> s DIFF relative_interior s homeomorphic sphere(a,&1) INTER affine hull s /\ s homeomorphic cball(a,&1) INTER affine hull s /\ relative_interior s homeomorphic (ball(a,&1)) INTER affine hull s`, REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `a:real^N` THEN REWRITE_TAC[SUBSET; IMP_IMP; RIGHT_IMP_FORALL_THM] THEN GEN_TAC THEN STRIP_TAC THEN SUBGOAL_THEN `!x:real^N u. x IN s /\ &0 <= u /\ u < &1 ==> (u % x) IN relative_interior s` ASSUME_TAC THENL [REWRITE_TAC[REAL_ARITH `&0 <= u <=> u = &0 \/ &0 < u`] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN ASM_CASES_TAC `x:real^N = vec 0` THEN ASM_REWRITE_TAC[VECTOR_MUL_RZERO] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_SEGMENT] THEN REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN ASM_MESON_TAC[]; FIRST_X_ASSUM(K ALL_TAC o SPECL [`x:real^N`; `x:real^N`])] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP (REWRITE_RULE[SUBSET] RELATIVE_INTERIOR_SUBSET)) THEN ABBREV_TAC `proj = \x:real^N. inv(norm(x)) % x` THEN SUBGOAL_THEN `!x:real^N y. (proj(x) = proj(y):real^N) /\ (norm x = norm y) <=> (x = y)` ASSUME_TAC THENL [REPEAT GEN_TAC THEN EQ_TAC THENL [ALL_TAC; MESON_TAC[]] THEN ASM_CASES_TAC `y:real^N = vec 0` THEN ASM_SIMP_TAC[NORM_EQ_0; NORM_0] THEN ASM_CASES_TAC `x:real^N = vec 0` THENL [ASM_MESON_TAC[NORM_EQ_0]; ALL_TAC] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN EXPAND_TAC "proj" THEN REWRITE_TAC[] THEN ASM_REWRITE_TAC[VECTOR_ARITH `a % x = a % y <=> a % (x - y):real^N = vec 0`] THEN ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; REAL_INV_EQ_0; NORM_EQ_0; VECTOR_SUB_EQ]; ALL_TAC] THEN SUBGOAL_THEN `(!x. x IN affine hull s ==> proj x IN affine hull s) /\ (!x. ~(x = vec 0) ==> norm(proj x) = &1) /\ (!x:real^N. proj x = vec 0 <=> x = vec 0)` STRIP_ASSUME_TAC THENL [EXPAND_TAC "proj" THEN REWRITE_TAC[NORM_MUL; VECTOR_MUL_EQ_0] THEN REWRITE_TAC[REAL_INV_EQ_0; NORM_EQ_0; REAL_ABS_INV; REAL_ABS_NORM] THEN SIMP_TAC[REAL_MUL_LINV; NORM_EQ_0] THEN REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM VECTOR_ADD_LID] THEN MATCH_MP_TAC IN_AFFINE_ADD_MUL THEN ASM_SIMP_TAC[AFFINE_AFFINE_HULL; VECTOR_ADD_LID; HULL_INC]; ALL_TAC] THEN SUBGOAL_THEN `(proj:real^N->real^N) continuous_on (UNIV DELETE vec 0)` ASSUME_TAC THENL [MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN REWRITE_TAC[IN_DELETE; IN_UNIV] THEN EXPAND_TAC "proj" THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_MUL THEN ASM_SIMP_TAC[CONTINUOUS_AT_ID] THEN REWRITE_TAC[GSYM(ISPEC `lift` o_DEF); GSYM(ISPEC `inv:real->real` o_DEF)] THEN MATCH_MP_TAC CONTINUOUS_AT_INV THEN ASM_REWRITE_TAC[NORM_EQ_0; VECTOR_SUB_EQ; CONTINUOUS_AT_LIFT_NORM]; ALL_TAC] THEN SUBGOAL_THEN `!a x. &0 < a ==> (proj:real^N->real^N)(a % x) = proj x` ASSUME_TAC THENL [REPEAT GEN_TAC THEN EXPAND_TAC "proj" THEN REWRITE_TAC[NORM_MUL; REAL_INV_MUL; VECTOR_MUL_ASSOC] THEN SIMP_TAC[REAL_FIELD `&0 < a ==> (inv(a) * x) * a = x`; real_abs; REAL_LT_IMP_LE]; ALL_TAC] THEN ABBREV_TAC `usph = {x:real^N | x IN affine hull s /\ norm x = &1}` THEN SUBGOAL_THEN ` sphere(vec 0:real^N,&1) INTER affine hull s = usph` SUBST1_TAC THENL [EXPAND_TAC "usph" THEN REWRITE_TAC[EXTENSION; IN_INTER; IN_SPHERE_0] THEN SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `!x. x IN affine hull s /\ ~(x = vec 0) ==> (proj:real^N->real^N) x IN usph` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?surf. homeomorphism (s DIFF relative_interior s,usph) (proj:real^N->real^N,surf)` MP_TAC THENL [MATCH_MP_TAC HOMEOMORPHISM_COMPACT THEN ASM_SIMP_TAC[COMPACT_RELATIVE_BOUNDARY] THEN REPEAT CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_DIFF] THEN EXPAND_TAC "usph" THEN REWRITE_TAC[IN_ELIM_THM] THEN ASM_MESON_TAC[HULL_INC]; MAP_EVERY EXPAND_TAC ["proj"; "usph"] THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `x:real^N = vec 0` THEN ASM_REWRITE_TAC[NORM_0; REAL_OF_NUM_EQ; ARITH_EQ] THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `vec 0:real^N`; `x:real^N`] RAY_TO_RELATIVE_FRONTIER) THEN REWRITE_TAC[relative_frontier] THEN ASM_SIMP_TAC[COMPACT_IMP_BOUNDED; CLOSURE_CLOSED; COMPACT_IMP_CLOSED; VECTOR_ADD_LID] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN EXPAND_TAC "proj" THEN REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `d % x:real^N` THEN ASM_REWRITE_TAC[NORM_MUL] THEN ASM_SIMP_TAC[REAL_MUL_RID; real_abs; REAL_LT_IMP_LE] THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; REAL_LT_IMP_NZ; VECTOR_MUL_LID]]; MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN REWRITE_TAC[IN_DIFF] THEN STRIP_TAC THEN ASM_CASES_TAC `x:real^N = vec 0` THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `y:real^N = vec 0` THENL [ASM SET_TAC[]; ALL_TAC] THEN UNDISCH_TAC `(proj:real^N->real^N) x = proj y` THEN EXPAND_TAC "proj" THEN REPEAT_TCL DISJ_CASES_THEN ASSUME_TAC (REAL_ARITH `norm(x:real^N) = norm(y:real^N) \/ norm x < norm y \/ norm y < norm x`) THENL [ASM_REWRITE_TAC[VECTOR_MUL_LCANCEL; REAL_INV_EQ_0; NORM_EQ_0]; DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`y:real^N`; `norm(x:real^N) / norm(y:real^N)`]); DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `norm(y:real^N) / norm(x:real^N)`])] THEN ASM_SIMP_TAC[REAL_LE_DIV; NORM_POS_LE; REAL_LT_LDIV_EQ; NORM_POS_LT; REAL_MUL_LID] THEN ASM_REWRITE_TAC[real_div; GSYM VECTOR_MUL_ASSOC] THENL [FIRST_X_ASSUM(SUBST1_TAC o SYM); ALL_TAC] THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; NORM_EQ_0] THEN ASM_REWRITE_TAC[VECTOR_MUL_LID]]; DISCH_THEN(X_CHOOSE_TAC `surf:real^N->real^N`)] THEN CONJ_TAC THENL [ASM_MESON_TAC[homeomorphic]; ALL_TAC] THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN REWRITE_TAC[homeomorphic] THEN MATCH_MP_TAC(MESON[] `(?x. P x /\ Q x) ==> (?x. P x) /\ (?x. Q x)`) THEN EXISTS_TAC `\x:real^N. norm(x) % (surf:real^N->real^N)(proj(x))` THEN MATCH_MP_TAC(TAUT `(p ==> q) /\ p ==> p /\ q`) THEN CONJ_TAC THENL [MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^N->real^N` THEN ONCE_REWRITE_TAC[TAUT `p ==> q <=> p ==> p ==> q`] THEN GEN_REWRITE_TAC LAND_CONV [homeomorphism] THEN REWRITE_TAC[] THEN STRIP_TAC THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] HOMEOMORPHISM_OF_SUBSETS) THEN SIMP_TAC[RELATIVE_INTERIOR_SUBSET; BALL_SUBSET_CBALL; SET_RULE `b SUBSET c ==> b INTER s SUBSET c INTER s`] THEN ONCE_REWRITE_TAC[GSYM CBALL_DIFF_SPHERE] THEN REWRITE_TAC[SET_RULE `(s DIFF t) INTER u = (s INTER u) DIFF {x | x IN u /\ x IN t}`] THEN ASM_REWRITE_TAC[IN_SPHERE_0] THEN MATCH_MP_TAC(SET_RULE `(!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\ u SUBSET s /\ IMAGE f s DIFF IMAGE f u = v ==> IMAGE f (s DIFF u) = v`) THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [EXPAND_TAC "usph" THEN SIMP_TAC[SUBSET; IN_ELIM_THM; IN_INTER; IN_CBALL_0; REAL_LE_REFL]; MATCH_MP_TAC(SET_RULE `t SUBSET s /\ u = s DIFF t ==> s DIFF u = t`) THEN REWRITE_TAC[RELATIVE_INTERIOR_SUBSET] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphism]) THEN DISCH_THEN(SUBST1_TAC o SYM o el 4 o CONJUNCTS) THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> f x = g x) ==> IMAGE f s = IMAGE g s`) THEN MAP_EVERY EXPAND_TAC ["usph"; "proj"] THEN SIMP_TAC[IN_ELIM_THM; REAL_INV_1; REAL_MUL_LID; VECTOR_MUL_LID]]; ALL_TAC] THEN MATCH_MP_TAC HOMEOMORPHISM_COMPACT THEN SIMP_TAC[COMPACT_INTER_CLOSED; CLOSED_AFFINE_HULL; COMPACT_CBALL] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHISM]) THEN STRIP_TAC THEN UNDISCH_THEN `(proj:real^N->real^N) continuous_on s DIFF relative_interior s` (K ALL_TAC) THEN REWRITE_TAC[] THEN CONJ_TAC THENL [REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN; IN_INTER] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN ASM_CASES_TAC `x = vec 0:real^N` THENL [ASM_REWRITE_TAC[CONTINUOUS_WITHIN; VECTOR_MUL_LZERO; NORM_0] THEN MATCH_MP_TAC LIM_NULL_VMUL_BOUNDED THEN FIRST_ASSUM(MP_TAC o MATCH_MP COMPACT_IMP_BOUNDED) THEN REWRITE_TAC[BOUNDED_POS] THEN MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THENL [REWRITE_TAC[LIM_WITHIN; o_THM; DIST_0; NORM_LIFT; REAL_ABS_NORM] THEN MESON_TAC[]; REWRITE_TAC[EVENTUALLY_WITHIN] THEN EXISTS_TAC `&1` THEN REWRITE_TAC[REAL_LT_01; IN_INTER; DIST_0; NORM_POS_LT] THEN ASM SET_TAC[]]; MATCH_MP_TAC CONTINUOUS_WITHIN_SUBSET THEN EXISTS_TAC `affine hull s:real^N->bool` THEN REWRITE_TAC[INTER_SUBSET] THEN MATCH_MP_TAC CONTINUOUS_MUL THEN SIMP_TAC[CONTINUOUS_LIFT_NORM_COMPOSE; CONTINUOUS_WITHIN_ID; o_DEF] THEN SUBGOAL_THEN `((surf:real^N->real^N) o (proj:real^N->real^N)) continuous_on (affine hull s DELETE vec 0)` MP_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN SIMP_TAC[SUBSET; IN_DELETE; IN_UNIV; FORALL_IN_IMAGE] THEN EXPAND_TAC "usph" THEN ASM_SIMP_TAC[IN_ELIM_THM]; SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_SIMP_TAC[IN_DELETE] THEN REWRITE_TAC[CONTINUOUS_WITHIN; o_DEF] THEN MATCH_MP_TAC EQ_IMP THEN MATCH_MP_TAC LIM_TRANSFORM_WITHIN_SET THEN REWRITE_TAC[EVENTUALLY_AT] THEN EXISTS_TAC `norm(x:real^N)` THEN ASM_REWRITE_TAC[IN_DELETE; IN_INTER; IN_CBALL; NORM_POS_LT] THEN X_GEN_TAC `y:real^N` THEN ASM_CASES_TAC `(y:real^N) IN affine hull s` THEN ASM_REWRITE_TAC[] THEN CONV_TAC NORM_ARITH]]; ALL_TAC] THEN CONJ_TAC THENL [ALL_TAC; MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN ASM_CASES_TAC `y:real^N = vec 0` THENL [ASM_SIMP_TAC[VECTOR_MUL_LZERO; VECTOR_MUL_EQ_0; NORM_0; NORM_EQ_0] THEN ASM SET_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `x:real^N = vec 0` THENL [CONV_TAC(ONCE_DEPTH_CONV SYM_CONV) THEN ASM_SIMP_TAC[VECTOR_MUL_LZERO; VECTOR_MUL_EQ_0; NORM_0; NORM_EQ_0] THEN ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[IN_INTER; IN_CBALL_0] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC)) THEN DISCH_THEN(fun th -> MP_TAC th THEN MP_TAC(AP_TERM `proj:real^N->real^N` th)) THEN ASM_SIMP_TAC[NORM_POS_LT; VECTOR_MUL_RCANCEL] THEN ASM SET_TAC[]] THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTER; IN_CBALL_0] THEN X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `x:real^N = vec 0` THEN ASM_REWRITE_TAC[NORM_0; VECTOR_MUL_LZERO; IN_INTER] THEN REWRITE_TAC[IN_CBALL_0; REAL_LE_LT] THEN STRIP_TAC THENL [MATCH_MP_TAC(REWRITE_RULE[SUBSET] RELATIVE_INTERIOR_SUBSET) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[NORM_POS_LE] THEN ASM SET_TAC[]; ASM_REWRITE_TAC[VECTOR_MUL_LID] THEN ASM SET_TAC[]]; ALL_TAC] THEN REWRITE_TAC[SUBSET; IN_IMAGE; IN_CBALL_0; IN_INTER] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN ASM_CASES_TAC `x:real^N = vec 0` THENL [EXISTS_TAC `vec 0:real^N` THEN ASM_SIMP_TAC[NORM_0; VECTOR_MUL_LZERO; HULL_INC; REAL_POS]; ALL_TAC] THEN SUBGOAL_THEN `!x. x IN usph ==> ~((surf:real^N->real^N) x = vec 0)` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN EXISTS_TAC `inv(norm(surf(proj x:real^N):real^N)) % x:real^N` THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC LAND_CONV [GSYM th]) THEN REWRITE_TAC[GSYM CONJ_ASSOC] THEN ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 5) [NORM_POS_LT; REAL_LT_INV_EQ; HULL_INC; REAL_LT_MUL; NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC(REAL_FIELD `~(y = &0) ==> x = (inv y * x) * y`) THEN ASM_SIMP_TAC[NORM_EQ_0; HULL_INC]; ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 5) [GSYM real_div; REAL_LE_LDIV_EQ; NORM_POS_LT; HULL_INC; REAL_MUL_LID] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `norm(surf(proj x:real^N):real^N) / norm(x:real^N)`]) THEN ASM_SIMP_TAC[REAL_LE_DIV; NORM_POS_LE; REAL_LT_LDIV_EQ; NORM_POS_LT] THEN GEN_REWRITE_TAC LAND_CONV [GSYM CONTRAPOS_THM] THEN REWRITE_TAC[REAL_NOT_LT; REAL_MUL_LID] THEN DISCH_THEN MATCH_MP_TAC THEN SUBGOAL_THEN `norm(surf(proj x)) / norm x % x:real^N = surf(proj x:real^N)` SUBST1_TAC THENL [FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC I [GSYM th]) THEN ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 5) [NORM_POS_LT; REAL_LT_INV_EQ; HULL_INC; REAL_LT_MUL; NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM; REAL_ABS_DIV; REAL_LT_DIV; REAL_DIV_RMUL; NORM_EQ_0]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE f s SUBSET t DIFF u ==> x IN s ==> ~(f x IN u)`)) THEN ASM_SIMP_TAC[HULL_INC]]; GEN_REWRITE_TAC LAND_CONV [GSYM VECTOR_ADD_LID] THEN MATCH_MP_TAC IN_AFFINE_ADD_MUL THEN ASM_SIMP_TAC[AFFINE_AFFINE_HULL; VECTOR_ADD_LID; HULL_INC]]);; let [HOMEOMORPHIC_CONVEX_COMPACT_SETS; HOMEOMORPHIC_RELATIVE_INTERIORS_CONVEX_COMPACT_SETS; HOMEOMORPHIC_RELATIVE_FRONTIERS_CONVEX_BOUNDED_SETS] = (CONJUNCTS o prove) (`(!s:real^M->bool t:real^N->bool. convex s /\ compact s /\ convex t /\ compact t /\ aff_dim s = aff_dim t ==> s homeomorphic t) /\ (!s:real^M->bool t:real^N->bool. convex s /\ compact s /\ convex t /\ compact t /\ aff_dim s = aff_dim t ==> relative_interior s homeomorphic relative_interior t) /\ (!s:real^M->bool t:real^N->bool. convex s /\ bounded s /\ convex t /\ bounded t /\ aff_dim s = aff_dim t ==> relative_frontier s homeomorphic relative_frontier t)`, let lemma = prove (`!s:real^M->bool t:real^N->bool. convex s /\ compact s /\ convex t /\ compact t /\ aff_dim s = aff_dim t ==> (s DIFF relative_interior s) homeomorphic (t DIFF relative_interior t) /\ s homeomorphic t /\ relative_interior s homeomorphic relative_interior t`, REPEAT GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN ASM_CASES_TAC `relative_interior t:real^N->bool = {}` THENL [UNDISCH_TAC `relative_interior t:real^N->bool = {}` THEN ASM_SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_EQ_MINUS1; RELATIVE_INTERIOR_EMPTY; EMPTY_DIFF; HOMEOMORPHIC_EMPTY; RELATIVE_INTERIOR_EQ_EMPTY]; FIRST_X_ASSUM(X_CHOOSE_THEN `b:real^N` MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY])] THEN CONV_TAC(ONCE_DEPTH_CONV SYM_CONV) THEN ASM_CASES_TAC `relative_interior s:real^M->bool = {}` THENL [UNDISCH_TAC `relative_interior s:real^M->bool = {}` THEN ASM_SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_EQ_MINUS1; RELATIVE_INTERIOR_EMPTY; EMPTY_DIFF; HOMEOMORPHIC_EMPTY; RELATIVE_INTERIOR_EQ_EMPTY]; FIRST_X_ASSUM(X_CHOOSE_THEN `a:real^M` MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY])] THEN REPEAT(POP_ASSUM MP_TAC) THEN GEOM_ORIGIN_TAC `b:real^N` THEN REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `a:real^M` THEN REPEAT GEN_TAC THEN REPEAT DISCH_TAC THEN MP_TAC(ISPECL [`s:real^M->bool`; `vec 0:real^M`] STARLIKE_COMPACT_PROJECTIVE) THEN MP_TAC(ISPECL [`t:real^N->bool`; `vec 0:real^N`] STARLIKE_COMPACT_PROJECTIVE) THEN ASM_SIMP_TAC[IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT; REWRITE_RULE[SUBSET] CLOSURE_SUBSET] THEN MATCH_MP_TAC(TAUT `(p ==> q ==> r) /\ (p' ==> q' ==> r') /\ (p'' ==> q'' ==> r'') ==> p /\ p' /\ p'' ==> q /\ q' /\ q'' ==> r /\ r' /\ r''`) THEN REPEAT CONJ_TAC THEN DISCH_THEN(fun th -> MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] HOMEOMORPHIC_TRANS) THEN MP_TAC(ONCE_REWRITE_RULE[HOMEOMORPHIC_SYM] th)) THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHIC_TRANS) THEN REPEAT(FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP (REWRITE_RULE[SUBSET] RELATIVE_INTERIOR_SUBSET))) THEN FIRST_X_ASSUM(MP_TAC o SYM) THEN ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC; AFF_DIM_DIM_0] THEN REWRITE_TAC[INT_OF_NUM_EQ] THEN DISCH_TAC THEN MP_TAC(ISPECL [`span s:real^M->bool`; `span t:real^N->bool`] ISOMETRIES_SUBSPACES) THEN ASM_REWRITE_TAC[SUBSPACE_SPAN; DIM_SPAN; homeomorphic; HOMEOMORPHISM] 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 SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTER; IN_BALL_0; IN_CBALL_0; IN_SPHERE_0] THEN SIMP_TAC[LINEAR_CONTINUOUS_ON] THEN ASM SET_TAC[]) in SIMP_TAC[lemma; relative_frontier] THEN REPEAT STRIP_TAC THEN MP_TAC(SPECL [`closure s:real^M->bool`; `closure t:real^N->bool`] lemma) THEN ASM_SIMP_TAC[CONVEX_CLOSURE; COMPACT_CLOSURE; AFF_DIM_CLOSURE] THEN ASM_SIMP_TAC[CONVEX_RELATIVE_INTERIOR_CLOSURE]);; let HOMEOMORPHIC_CONVEX_COMPACT = prove (`!s:real^N->bool t:real^N->bool. convex s /\ compact s /\ ~(interior s = {}) /\ convex t /\ compact t /\ ~(interior t = {}) ==> s homeomorphic t`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_CONVEX_COMPACT_SETS THEN ASM_SIMP_TAC[AFF_DIM_NONEMPTY_INTERIOR]);; let HOMEOMORPHIC_CONVEX_COMPACT_CBALL = prove (`!s:real^N->bool b:real^N e. convex s /\ compact s /\ ~(interior s = {}) /\ &0 < e ==> s homeomorphic cball(b,e)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_CONVEX_COMPACT THEN ASM_REWRITE_TAC[COMPACT_CBALL; INTERIOR_CBALL; CONVEX_CBALL] THEN ASM_REWRITE_TAC[BALL_EQ_EMPTY; REAL_NOT_LE]);; let HOMEOMORPHIC_CLOSED_INTERVALS = prove (`!a b:real^N c d:real^N. ~(interval(a,b) = {}) /\ ~(interval(c,d) = {}) ==> interval[a,b] homeomorphic interval[c,d]`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_CONVEX_COMPACT THEN REWRITE_TAC[CONVEX_INTERVAL; COMPACT_INTERVAL] THEN ASM_REWRITE_TAC[INTERIOR_CLOSED_INTERVAL]);; (* ------------------------------------------------------------------------- *) (* Hence homeomorphism of convex open sets of same affine dimension. *) (* ------------------------------------------------------------------------- *) let HOMEOMORPHIC_RELATIVELY_OPEN_CONVEX_SETS = prove (`!s:real^M->bool t:real^N->bool. convex s /\ open_in (subtopology euclidean (affine hull s)) s /\ convex t /\ open_in (subtopology euclidean (affine hull t)) t /\ aff_dim s = aff_dim t ==> s homeomorphic t`, let lemma = prove (`!s:real^N->bool. convex s /\ open_in (subtopology euclidean (affine hull s)) s ==> ?t:real^N->bool. convex t /\ bounded t /\ aff_dim t = aff_dim s /\ open_in (subtopology euclidean (affine hull t)) t /\ s homeomorphic t`, GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [STRIP_TAC THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[BOUNDED_EMPTY; HOMEOMORPHIC_REFL]; POP_ASSUM MP_TAC THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `z:real^N` THEN GEOM_ORIGIN_TAC `z:real^N` THEN REPEAT STRIP_TAC] THEN X_CHOOSE_THEN `f:real^N->real^N` MP_TAC CONVEXITY_PRESERVING_SHRINK_0 THEN DISCH_THEN(X_CHOOSE_THEN `g:real^N->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `IMAGE (f:real^N->real^N) s` THEN ASM_SIMP_TAC[] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `ball(vec 0:real^N,&1)` THEN REWRITE_TAC[BOUNDED_BALL] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphism]) THEN SET_TAC[]; ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC] THEN ONCE_REWRITE_TAC[GSYM SPAN_CONIC_HULL] THEN ASM_REWRITE_TAC[]; FIRST_X_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 REWRITE_TAC[OPEN_IN_OPEN] THEN EXISTS_TAC `IMAGE (f:real^N->real^N) u` THEN CONJ_TAC THENL [FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_IMP_OPEN_MAP)) THEN ASM_SIMP_TAC[OPEN_IN_OPEN_EQ; OPEN_UNIV; OPEN_BALL; SUBSET_UNIV]; MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [REWRITE_TAC[SUBSET_INTER; HULL_SUBSET] THEN ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GSYM) THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC] THEN ONCE_REWRITE_TAC[GSYM SPAN_CONIC_HULL] THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SPAN_CONIC_HULL] THEN MATCH_MP_TAC(SET_RULE `(!x y. f x = f y ==> x = y) /\ (!x. f(x) IN v ==> x IN v) ==> v INTER u = s ==> v INTER IMAGE f u SUBSET IMAGE f s`) THEN CONJ_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphism]) THEN SET_TAC[]; ALL_TAC] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN SUBGOAL_THEN `conic hull (IMAGE (f:real^N->real^N) {x}) = conic hull {x}` MP_TAC THENL [ASM_REWRITE_TAC[]; ALL_TAC] THEN REWRITE_TAC[CONIC_HULL_EXPLICIT; IMAGE_CLAUSES] THEN REWRITE_TAC[SET_RULE `{f a x | P a /\ x IN {b}} = {f a b | P a}`] THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `{c % f x | P c} = {c % x | P c} ==> P(&1) ==> ?c. &1 % x = c % f x`)) THEN REWRITE_TAC[REAL_POS; LEFT_IMP_EXISTS_THM; VECTOR_MUL_LID] THEN ASM_MESON_TAC[SPAN_MUL]]; FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [homeomorphism]) THEN REWRITE_TAC[homeomorphism; homeomorphic] THEN MAP_EVERY EXISTS_TAC [`f:real^N->real^N`; `g:real^N->real^N`] THEN REPEAT CONJ_TAC THEN TRY(FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET))) THEN ASM SET_TAC[]]) in REPEAT STRIP_TAC THEN MP_TAC(ISPEC `s:real^M->bool` lemma) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `s':real^M->bool` THEN STRIP_TAC THEN TRANS_TAC HOMEOMORPHIC_TRANS `s':real^M->bool` THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPEC `t:real^N->bool` lemma) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `t':real^N->bool` THEN STRIP_TAC THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN TRANS_TAC HOMEOMORPHIC_TRANS `t':real^N->bool` THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPECL [`closure t':real^N->bool`; `closure s':real^M->bool`] HOMEOMORPHIC_RELATIVE_INTERIORS_CONVEX_COMPACT_SETS) THEN ASM_SIMP_TAC[CONVEX_CLOSURE; COMPACT_CLOSURE; AFF_DIM_CLOSURE] THEN ASM_SIMP_TAC[CONVEX_RELATIVE_INTERIOR_CLOSURE; RELATIVE_INTERIOR_OPEN_IN]);; let HOMEOMORPHIC_CONVEX_OPEN_SETS = prove (`!s:real^N->bool t:real^N->bool. convex s /\ open s /\ convex t /\ open t /\ (s = {} <=> t = {}) ==> s homeomorphic t`, REPEAT GEN_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_SIMP_TAC[HOMEOMORPHIC_REFL] THEN STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_RELATIVELY_OPEN_CONVEX_SETS THEN ASM_SIMP_TAC[OPEN_SUBSET; HULL_SUBSET; AFF_DIM_OPEN]);; (* ------------------------------------------------------------------------- *) (* More refined Lipschitz homeomorphisms between relative frontiers. *) (* ------------------------------------------------------------------------- *) let LIPSCHITZ_CONVEX_SPHERICAL_PROJECTION_EXPLICIT = prove (`!r s x y:real^N. convex s /\ &0 < r /\ vec 0 IN s /\ ball(vec 0,r) INTER affine hull s SUBSET relative_interior s /\ x IN relative_frontier s /\ y IN relative_frontier s ==> dist(inv(norm x) % x,inv(norm y) % y) <= inv r * dist(x,y)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`norm(x:real^N)`; `norm(y:real^N)`; `inv(norm x) % x:real^N`; `inv(norm y) % y:real^N`] DIST_DESCALE) THEN SUBGOAL_THEN `~(x:real^N = vec 0) /\ ~(y:real^N = vec 0)` STRIP_ASSUME_TAC THENL [CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s INTER t SUBSET u ==> z IN s /\ z IN t /\ ~(x IN u) ==> ~(x = z)`)) THEN ASM_SIMP_TAC[HULL_INC; CENTRE_IN_BALL] THEN ASM_MESON_TAC[relative_frontier; IN_DIFF]; ALL_TAC] THEN ASM_SIMP_TAC[NORM_POS_LE; NORM_MUL; VECTOR_MUL_ASSOC; REAL_MUL_LINV; REAL_MUL_RINV; NORM_EQ_0; REAL_ABS_INV; REAL_ABS_NORM] THEN REWRITE_TAC[VECTOR_MUL_LID; real_ge] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ; REAL_LT_MIN; NORM_POS_LT] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN REWRITE_TAC[real_div] THEN MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[DIST_POS_LE] THEN MATCH_MP_TAC REAL_LE_INV2 THEN ASM_REWRITE_TAC[REAL_LE_MIN] THEN CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s INTER t SUBSET u ==> (r <= norm(z:real^N) <=> ~(z IN s)) /\ z IN t /\ ~(z IN u) ==> r <= norm z`)) THEN RULE_ASSUM_TAC(REWRITE_RULE[relative_frontier; IN_DIFF]) THEN ASM_REWRITE_TAC[IN_BALL_0; REAL_NOT_LT] THEN ASM_MESON_TAC[AFFINE_HULL_CLOSURE; HULL_INC]);; let LIPSCHITZ_CONVEX_SPHERICAL_PROJECTION = prove (`!s:real^N->bool. convex s /\ vec 0 IN relative_interior s ==> ?B. !x y. x IN relative_frontier s /\ y IN relative_frontier s ==> dist(inv(norm x) % x,inv(norm y) % y) <= B * dist(x,y)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_RELATIVE_INTERIOR]) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `inv r:real` THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC LIPSCHITZ_CONVEX_SPHERICAL_PROJECTION_EXPLICIT THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN W(MP_TAC o PART_MATCH (lhand o rand) OPEN_IN_SUBSET_RELATIVE_INTERIOR o snd) THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[INTER_COMM] THEN SIMP_TAC[OPEN_IN_OPEN_INTER; OPEN_BALL]);; let INVERSE_LIPSCHITZ_CONVEX_SPHERICAL_PROJECTION_EXPLICIT = prove (`!r R s x y:real^N. convex s /\ &0 < r /\ vec 0 IN s /\ ball(vec 0,r) INTER affine hull s SUBSET relative_interior s /\ s SUBSET cball(vec 0,R) /\ x IN relative_frontier s /\ y IN relative_frontier s ==> dist(inv(norm x) % x,inv(norm y) % y) >= r / R pow 2 * dist(x,y)`, let lemma0 = prove (`!x y:real^N. orthogonal x y ==> norm(x) <= norm(x + y)`, REPEAT GEN_TAC THEN DISCH_THEN(ASSUME_TAC o MATCH_MP NORM_ADD_PYTHAGOREAN) THEN REWRITE_TAC[NORM_LE_SQUARE] THEN ASM_REWRITE_TAC[NORM_POS_LE; GSYM NORM_POW_2] THEN REWRITE_TAC[REAL_LE_ADDR; REAL_LE_POW_2]) in let lemma1 = prove (`!a b x y:real^N. &0 <= a /\ &0 <= b /\ x dot y <= &0 ==> dist(a % x,b % y) >= min a b * dist(x,y)`, REPEAT STRIP_TAC THEN REWRITE_TAC[dist; NORM_GE_SQUARE] THEN DISJ2_TAC THEN REWRITE_TAC[REAL_POW_MUL; NORM_POW_2] THEN REWRITE_TAC[VECTOR_ARITH `(x - y:real^N) dot (x - y) = x dot x + y dot y + &2 * --(x dot y)`] THEN REWRITE_TAC[DOT_RMUL] THEN REWRITE_TAC[DOT_LMUL; real_ge] THEN REWRITE_TAC[GSYM REAL_MUL_RNEG; REAL_ADD_LDISTRIB; REAL_MUL_ASSOC] THEN REPEAT(MATCH_MP_TAC REAL_LE_ADD2 THEN CONJ_TAC) THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[DOT_POS_LE] THEN ASM_REWRITE_TAC[REAL_NEG_GE0] THEN REWRITE_TAC[REAL_ARITH `a * &2 <= (&2 * x) * y <=> a <= x * y`] THEN REWRITE_TAC[REAL_POW_2] THEN MATCH_MP_TAC REAL_LE_MUL2 THEN ASM_REAL_ARITH_TAC) in let lemma2 = prove (`!a b w x y:real^N. &0 <= a /\ &0 <= b /\ between w (x,y) /\ orthogonal w (x - y) ==> dist(a % x,b % y) >= min a b * dist(x,y)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`a:real`; `b:real`; `x - w:real^N`; `y - w:real^N`] lemma1) THEN ASM_REWRITE_TAC[real_ge] THEN ANTS_TAC THENL [FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [BETWEEN_IN_SEGMENT]) THEN SIMP_TAC[IN_SEGMENT; LEFT_IMP_EXISTS_THM] THEN REWRITE_TAC[VECTOR_ARITH `x - ((&1 - u) % x + u % y):real^N = u % (x - y) /\ y - ((&1 - u) % x + u % y):real^N = (u - &1) % (x - y)`] THEN REWRITE_TAC[DOT_RMUL] THEN REWRITE_TAC[DOT_LMUL] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[REAL_ARITH `(u - &1) * x <= &0 <=> &0 <= (&1 - u) * x`] THEN REPEAT(MATCH_MP_TAC REAL_LE_MUL THEN CONJ_TAC) THEN REWRITE_TAC[DOT_POS_LE] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[NORM_ARITH `dist(x - w:real^N,y - w) = dist(x,y)`] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN REWRITE_TAC[dist] THEN SUBST1_TAC(VECTOR_ARITH `a % x - b % y:real^N = (a % (x - w) - b % (y - w)) + (a - b) % w`) THEN MATCH_MP_TAC lemma0 THEN REWRITE_TAC[ORTHOGONAL_MUL] THEN DISJ2_TAC THEN MATCH_MP_TAC(last(CONJUNCTS ORTHOGONAL_CLAUSES)) THEN CONJ_TAC THEN REWRITE_TAC[ORTHOGONAL_MUL] THEN DISJ2_TAC THEN ONCE_REWRITE_TAC[ORTHOGONAL_SYM] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [BETWEEN_IN_SEGMENT]) THEN SIMP_TAC[IN_SEGMENT; LEFT_IMP_EXISTS_THM] THEN REWRITE_TAC[VECTOR_ARITH `x - ((&1 - u) % x + u % y):real^N = u % (x - y) /\ y - ((&1 - u) % x + u % y):real^N = (u - &1) % (x - y)`] THEN GEN_TAC THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN ASM_REWRITE_TAC[ORTHOGONAL_MUL]]) in let mainlemma_2d = prove (`collinear {z:real^2,x,x'} /\ collinear {w,x,y} /\ orthogonal (z - w) (x - y) /\ orthogonal (y - x') (z - x') /\ ~(x' = z) /\ ~(y = w) ==> dist(z,w) * dist(x,y) = dist(y,x') * dist(z,x)`, REPEAT GEN_TAC THEN REWRITE_TAC[PAIRWISE; ALL] THEN GEOM_ORIGIN_TAC `x:real^2` THEN REWRITE_TAC[GSYM DIST_EQ_0] THEN REPEAT GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC(REAL_ARITH `&0 <= x /\ &0 <= y /\ abs x = abs y ==> x = y`) THEN SIMP_TAC[DIST_POS_LE; REAL_LE_MUL; REAL_EQ_SQUARE_ABS; REAL_POW_MUL] THEN POP_ASSUM MP_TAC THEN ONCE_REWRITE_TAC[REAL_RING `x = &0 <=> x pow 2 = &0`] THEN REWRITE_TAC[COLLINEAR_3_2D; dist; orthogonal; NORM_POW_2] THEN REWRITE_TAC[DOT_2; VECTOR_SUB_COMPONENT; VEC_COMPONENT] THEN CONV_TAC REAL_RING) in let mainlemma = prove (`collinear {z:real^N,x,x'} /\ collinear {w,x,y} /\ orthogonal (z - w) (x - y) /\ orthogonal (y - x') (z - x') /\ ~(x' = z) /\ ~(y = w) ==> dist(z,w) * dist(x,y) = dist(y,x') * dist(z,x)`, ASM_CASES_TAC `dimindex(:N) <= dimindex(:2)` THENL [MP_TAC(DISCH_ALL(GEOM_DROP_DIMENSION_RULE (ASSUME `dimindex(:N) <= dimindex(:2)`) (GEN_ALL mainlemma_2d))) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN MATCH_ACCEPT_TAC; RULE_ASSUM_TAC(REWRITE_RULE[NOT_LE])] THEN GEOM_ORIGIN_TAC `z:real^N` THEN REPEAT GEN_TAC THEN DISCH_TAC THEN SUBGOAL_THEN `?f:real^2->real^N. linear f /\ span {vec 0:real^N,x,y,w,x'} SUBSET IMAGE f (:real^2) /\ (!x. norm(f x) = norm x)` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC ISOMETRY_UNIV_SUPERSET_SUBSPACE THEN ASM_SIMP_TAC[LT_IMP_LE; SUBSPACE_SPAN; DIM_SPAN; DIMINDEX_2] THEN SIMP_TAC[GSYM INT_OF_NUM_LE; GSYM AFF_DIM_DIM_0; HULL_INC; IN_INSERT] THEN TRANS_TAC INT_LE_TRANS `aff_dim(affine hull {vec 0:real^N,x,x'} UNION affine hull {w,x,y})` THEN CONJ_TAC THENL [MATCH_MP_TAC AFF_DIM_SUBSET THEN SIMP_TAC[INSERT_SUBSET; EMPTY_SUBSET; IN_UNION; HULL_INC; IN_INSERT]; W(MP_TAC o PART_MATCH (lhand o rand) AFF_DIM_UNION o lhand o snd) THEN REWRITE_TAC[AFF_DIM_AFFINE_HULL; AFFINE_AFFINE_HULL] THEN MATCH_MP_TAC(TAUT `p /\ (p /\ q ==> r) ==> (p ==> q) ==> r`) THEN CONJ_TAC THENL [REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN MESON_TAC[HULL_INC; IN_INSERT]; DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC SUBST1_TAC)] THEN MATCH_MP_TAC(INT_ARITH `x:int <= &1 /\ y <= &1 /\ &0 <= z ==> (x + y) - z <= &2`) THEN ASM_REWRITE_TAC[GSYM COLLINEAR_AFF_DIM; AFF_DIM_POS_LE]]; FIRST_X_ASSUM(MP_TAC o check (is_conj o concl))] THEN FIRST_ASSUM(MP_TAC o MATCH_MP (SET_RULE `span s SUBSET t ==> s SUBSET span s ==> s SUBSET t`)) THEN REWRITE_TAC[SPAN_INC] THEN REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET] THEN REWRITE_TAC[IN_IMAGE; IN_UNIV] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `!x y. (f:real^2->real^N) x = f y ==> x = y` ASSUME_TAC THENL [ASM_MESON_TAC[PRESERVES_NORM_INJECTIVE]; ALL_TAC] THEN MP_TAC(end_itlist CONJ (mapfilter (ISPEC `f:real^2->real^N`) (!invariant_under_linear))) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN SUBST1_TAC(SET_RULE `{} = IMAGE (f:real^2->real^N) {}`) THEN ASM_REWRITE_TAC[] THEN MATCH_ACCEPT_TAC mainlemma_2d) in REPEAT GEN_TAC THEN ASM_CASES_TAC `R < &0` THENL [ASM_SIMP_TAC[CBALL_EMPTY] THEN SET_TAC[]; RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LT])] THEN STRIP_TAC THEN REWRITE_TAC[real_ge] THEN SUBGOAL_THEN `~(x:real^N = vec 0) /\ ~(y:real^N = vec 0)` STRIP_ASSUME_TAC THENL [CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s INTER t SUBSET u ==> z IN s /\ z IN t /\ ~(x IN u) ==> ~(x = z)`)) THEN ASM_SIMP_TAC[HULL_INC; CENTRE_IN_BALL] THEN ASM_MESON_TAC[relative_frontier; IN_DIFF]; ALL_TAC] THEN SUBGOAL_THEN `r <= norm(x:real^N) /\ r <= norm(y:real^N)` STRIP_ASSUME_TAC THENL [CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s INTER t SUBSET u ==> (r <= norm(z:real^N) <=> ~(z IN s)) /\ z IN t /\ ~(z IN u) ==> r <= norm z`)) THEN RULE_ASSUM_TAC(REWRITE_RULE[relative_frontier; IN_DIFF]) THEN ASM_REWRITE_TAC[IN_BALL_0; REAL_NOT_LT] THEN ASM_MESON_TAC[AFFINE_HULL_CLOSURE; HULL_INC]; ALL_TAC] THEN SUBGOAL_THEN `norm(x:real^N) <= R /\ norm(y:real^N) <= R` STRIP_ASSUME_TAC THENL [CONJ_TAC THEN UNDISCH_TAC `s SUBSET cball(vec 0:real^N,R)` THEN DISCH_THEN(MP_TAC o MATCH_MP SUBSET_CLOSURE) THEN REWRITE_TAC[CLOSURE_CBALL; SUBSET; IN_CBALL_0] THEN ASM_MESON_TAC[relative_frontier; IN_DIFF]; ALL_TAC] THEN SUBGOAL_THEN `r <= R /\ &0 < R` STRIP_ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_CASES_TAC `(x:real^N) dot y <= &0` THENL [W(MP_TAC o PART_MATCH (lhand o rand) lemma1 o rand o snd) THEN ASM_REWRITE_TAC[REAL_LE_INV_EQ; real_ge; NORM_POS_LE] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] REAL_LE_TRANS) THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[DIST_POS_LE] THEN TRANS_TAC REAL_LE_TRANS `inv R:real` THEN CONJ_TAC THENL [REWRITE_TAC[real_div; REAL_INV_MUL; REAL_POW_2] THEN GEN_REWRITE_TAC RAND_CONV [GSYM REAL_MUL_LID] THEN REWRITE_TAC[REAL_MUL_ASSOC] THEN MATCH_MP_TAC REAL_LE_RMUL THEN ASM_SIMP_TAC[REAL_LE_INV_EQ; GSYM real_div; REAL_LE_LDIV_EQ] THEN ASM_REWRITE_TAC[REAL_MUL_LID]; REWRITE_TAC[REAL_LE_MIN] THEN CONJ_TAC THEN MATCH_MP_TAC REAL_LE_INV2 THEN ASM_SIMP_TAC[NORM_POS_LT]]; RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LE])] THEN ASM_CASES_TAC `x:real^N = y` THEN ASM_REWRITE_TAC[DIST_REFL; REAL_MUL_RZERO; REAL_LE_REFL] THEN MP_TAC(GEN `v:real^N` (ISPECL [`affine hull {x:real^N,y}`; `vec 0:real^N`; `v:real^N`] CLOSEST_POINT_AFFINE_ORTHOGONAL)) THEN MP_TAC(ISPECL [`affine hull {x:real^N,y}`; `vec 0:real^N`] CLOSEST_POINT_EXISTS) THEN ABBREV_TAC `w = closest_point (affine hull {x, y}) (vec 0:real^N)` THEN FIRST_X_ASSUM(K ALL_TAC o SYM) THEN REWRITE_TAC[CLOSED_AFFINE_HULL; AFFINE_HULL_EQ_EMPTY; NOT_INSERT_EMPTY] THEN SIMP_TAC[AFFINE_AFFINE_HULL; HULL_INC; IN_INSERT] THEN REWRITE_TAC[DIST_0; VECTOR_SUB_LZERO; ORTHOGONAL_RNEG] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `orthogonal (x - y:real^N) w` MP_TAC THENL [SUBST1_TAC(VECTOR_ARITH `x - y:real^N = (x - w) - (y - w)`) THEN MATCH_MP_TAC(last(CONJUNCTS ORTHOGONAL_CLAUSES)) THEN ASM_SIMP_TAC[HULL_INC; IN_INSERT]; UNDISCH_THEN `!v:real^N. v IN affine hull {x, y} ==> orthogonal (v - w) w` (K ALL_TAC) THEN DISCH_TAC] THEN MP_TAC(fst(EQ_IMP_RULE(ISPECL [`w:real^N`; `x:real^N`; `y:real^N`] COLLINEAR_BETWEEN_CASES))) THEN ANTS_TAC THENL [ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {b,c,a}`] THEN MATCH_MP_TAC AFFINE_HULL_3_IMP_COLLINEAR THEN ASM_REWRITE_TAC[]; DISCH_THEN(MP_TAC o MATCH_MP (TAUT `p \/ q \/ r ==> (q /\ ~p \/ r /\ ~p) \/ p`))] THEN DISCH_THEN(DISJ_CASES_THEN MP_TAC) THENL [GEN_REWRITE_TAC (LAND_CONV o RAND_CONV o ONCE_DEPTH_CONV) [BETWEEN_SYM] THEN DISCH_THEN(fun th -> POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN MP_TAC th) THEN MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`y:real^N`; `x:real^N`] THEN MATCH_MP_TAC(MESON[] `(!x y. R x y <=> R y x) /\ (!x y. P x y ==> R x y) ==> (!x y. P x y \/ P y x ==> R x y)`) THEN CONJ_TAC THENL [REWRITE_TAC[INSERT_AC; DIST_SYM; EQ_SYM_EQ; DOT_SYM; MESON[ORTHOGONAL_LNEG; VECTOR_NEG_SUB] `orthogonal (x - y:real^N) w <=> orthogonal (y - x) w`] THEN REWRITE_TAC[CONJ_ACI]; REPEAT GEN_TAC THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [BETWEEN_SYM] THEN ASM_CASES_TAC `w:real^N = x` THEN ASM_REWRITE_TAC[BETWEEN_REFL] THEN ASM_CASES_TAC `w:real^N = y` THEN ASM_REWRITE_TAC[BETWEEN_REFL] THEN REPEAT STRIP_TAC] THEN TRANS_TAC REAL_LE_TRANS `r / R pow 2 * dist(x:real^N,y)` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[DIST_POS_LE] THEN REAL_ARITH_TAC; ALL_TAC] THEN TRANS_TAC REAL_LE_TRANS `abs(inv(norm y)) * dist(norm y / norm x % x:real^N,y)` THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[dist; GSYM NORM_MUL; VECTOR_SUB_LDISTRIB] THEN ASM_SIMP_TAC[REAL_LE_REFL; VECTOR_MUL_ASSOC; NORM_EQ_0; REAL_FIELD `~(x = &0) /\ ~(y = &0) ==> inv y * y / x = inv x`]] THEN REWRITE_TAC[real_div; REAL_INV_POW] THEN REWRITE_TAC[REAL_ARITH `(r * inv(R) pow 2) * d:real = inv(R) * r / R * d`] THEN MATCH_MP_TAC REAL_LE_MUL2 THEN ASM_REWRITE_TAC[REAL_LE_INV_EQ] THEN ASM_SIMP_TAC[REAL_LE_MUL; REAL_LE_DIV; REAL_LT_IMP_LE; DIST_POS_LE] THEN CONJ_TAC THENL [REWRITE_TAC[REAL_ABS_INV; REAL_ABS_NORM] THEN MATCH_MP_TAC REAL_LE_INV2 THEN ASM_REWRITE_TAC[NORM_POS_LT] THEN UNDISCH_TAC `s SUBSET cball(vec 0:real^N,R)` THEN DISCH_THEN(MP_TAC o MATCH_MP SUBSET_CLOSURE) THEN REWRITE_TAC[CLOSURE_CBALL; SUBSET; IN_CBALL_0] THEN ASM_MESON_TAC[relative_frontier; IN_DIFF]; ALL_TAC] THEN ABBREV_TAC `x' = closest_point (affine hull {vec 0,x}) (y:real^N)` THEN MP_TAC(GEN `v:real^N` (ISPECL [`affine hull {vec 0:real^N,x}`; `y:real^N`; `v:real^N`] CLOSEST_POINT_AFFINE_ORTHOGONAL)) THEN MP_TAC(ISPECL [`affine hull {vec 0:real^N,x}`; `y:real^N`] CLOSEST_POINT_EXISTS) THEN SIMP_TAC[CLOSED_AFFINE_HULL; AFFINE_HULL_EQ_EMPTY; NOT_INSERT_EMPTY] THEN ASM_SIMP_TAC[AFFINE_AFFINE_HULL; HULL_INC; IN_INSERT] THEN REPEAT STRIP_TAC THEN TRANS_TAC REAL_LE_TRANS `dist(y:real^N,x')` THEN CONJ_TAC THENL [ALL_TAC; GEN_REWRITE_TAC RAND_CONV [DIST_SYM] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[AFFINE_HULL_2; IN_ELIM_THM; VECTOR_MUL_RZERO] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN EXISTS_TAC `norm(y:real^N) * inv(norm(x:real^N))` THEN REWRITE_TAC[VECTOR_ADD_LID; REAL_ARITH `u + v = &1 <=> u = &1 - v`] THEN REWRITE_TAC[EXISTS_REFL]] THEN ASM_CASES_TAC `R = &0` THENL [ASM_REWRITE_TAC[real_div; REAL_INV_0; REAL_MUL_RZERO] THEN REWRITE_TAC[REAL_MUL_LZERO; DIST_POS_LE]; SUBGOAL_THEN `&0 < R` ASSUME_TAC THENL [ASM_REWRITE_TAC[REAL_LT_LE]; ALL_TAC]] THEN ONCE_REWRITE_TAC[REAL_ARITH `r / R * x:real = (r * x) / R`] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ] THEN TRANS_TAC REAL_LE_TRANS `norm(w:real^N) * dist(x:real^N,y)` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[DIST_POS_LE] THEN REWRITE_TAC[GSYM IN_BALL_0; GSYM REAL_NOT_LT] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `b INTER a SUBSET r ==> w IN a /\ ~(w IN r) ==> ~(w IN b)`)) THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `w IN s ==> s SUBSET t ==> w IN t`)) THEN MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[AFFINE_AFFINE_HULL] THEN ONCE_REWRITE_TAC[GSYM AFFINE_HULL_CLOSURE] THEN REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET] THEN RULE_ASSUM_TAC(REWRITE_RULE[relative_frontier; IN_DIFF]) THEN ASM_SIMP_TAC[HULL_INC]; DISCH_TAC THEN UNDISCH_TAC `(x:real^N) IN relative_frontier s` THEN REWRITE_TAC[relative_frontier; IN_DIFF; DE_MORGAN_THM] THEN DISJ2_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `w:real^N`; `y:real^N`] IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT) THEN ANTS_TAC THENL [ASM_MESON_TAC[relative_frontier; IN_DIFF]; ALL_TAC] THEN REWRITE_TAC[SUBSET] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[open_segment; IN_DIFF; IN_INSERT; NOT_IN_EMPTY] THEN ASM_REWRITE_TAC[GSYM BETWEEN_IN_SEGMENT]]; ALL_TAC] THEN TRANS_TAC REAL_LE_TRANS `dist(y:real^N,x') * norm(x:real^N)` THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[DIST_POS_LE] THEN UNDISCH_TAC `s SUBSET cball(vec 0:real^N,R)` THEN DISCH_THEN(MP_TAC o MATCH_MP SUBSET_CLOSURE) THEN REWRITE_TAC[CLOSURE_CBALL; SUBSET; IN_CBALL_0] THEN ASM_MESON_TAC[relative_frontier; IN_DIFF]] THEN REWRITE_TAC[NORM_ARITH `norm(w:real^N) = dist(vec 0,w)`] THEN MATCH_MP_TAC REAL_EQ_IMP_LE THEN MATCH_MP_TAC mainlemma THEN ASM_REWRITE_TAC[VECTOR_SUB_LZERO; ORTHOGONAL_LNEG; ORTHOGONAL_RNEG] THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[AFFINE_HULL_3_IMP_COLLINEAR; INSERT_AC]; ASM_MESON_TAC[AFFINE_HULL_3_IMP_COLLINEAR; INSERT_AC]; ASM_MESON_TAC[ORTHOGONAL_SYM]; ONCE_REWRITE_TAC[ORTHOGONAL_SYM] THEN GEN_REWRITE_TAC LAND_CONV [GSYM VECTOR_SUB_RZERO] THEN ONCE_REWRITE_TAC[GSYM VECTOR_NEG_SUB] THEN REWRITE_TAC[ORTHOGONAL_LNEG] THEN REWRITE_TAC[VECTOR_NEG_SUB] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN SIMP_TAC[HULL_INC; IN_INSERT]; DISCH_THEN SUBST_ALL_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[VECTOR_SUB_RZERO; orthogonal]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `&0 < x dot y ==> x dot y = &0 ==> F`)) THEN ASM_SIMP_TAC[HULL_INC; IN_INSERT]]; DISCH_TAC THEN MP_TAC(ISPECL [`inv(norm(x:real^N))`; `inv(norm(y:real^N))`; `w:real^N`; `x:real^N`; `y:real^N`] lemma2) THEN ASM_REWRITE_TAC[REAL_LE_INV_EQ; NORM_POS_LE; real_ge] THEN ANTS_TAC THENL [ASM_MESON_TAC[ORTHOGONAL_SYM]; ALL_TAC] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] REAL_LE_TRANS) THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[DIST_POS_LE] THEN TRANS_TAC REAL_LE_TRANS `inv R:real` THEN CONJ_TAC THENL [REWRITE_TAC[real_div; REAL_INV_MUL; REAL_POW_2] THEN GEN_REWRITE_TAC RAND_CONV [GSYM REAL_MUL_LID] THEN REWRITE_TAC[REAL_MUL_ASSOC] THEN MATCH_MP_TAC REAL_LE_RMUL THEN ASM_SIMP_TAC[REAL_LE_INV_EQ; GSYM real_div; REAL_LE_LDIV_EQ] THEN ASM_REWRITE_TAC[REAL_MUL_LID]; REWRITE_TAC[REAL_LE_MIN] THEN CONJ_TAC THEN MATCH_MP_TAC REAL_LE_INV2 THEN ASM_SIMP_TAC[NORM_POS_LT]]]);; let INVERSE_LIPSCHITZ_CONVEX_SPHERICAL_PROJECTION = prove (`!s:real^N->bool. convex s /\ bounded s /\ vec 0 IN relative_interior s ==> ?B. &0 < B /\ !x y. x IN relative_frontier s /\ y IN relative_frontier s ==> dist(inv(norm x) % x,inv(norm y) % y) >= B * dist(x,y)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [BOUNDED_POS]) THEN REWRITE_TAC[GSYM IN_CBALL_0; GSYM SUBSET] THEN DISCH_THEN(X_CHOOSE_THEN `R:real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_RELATIVE_INTERIOR]) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(r:real) / R pow 2` THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_POW_LT] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC INVERSE_LIPSCHITZ_CONVEX_SPHERICAL_PROJECTION_EXPLICIT THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN W(MP_TAC o PART_MATCH (lhand o rand) OPEN_IN_SUBSET_RELATIVE_INTERIOR o snd) THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[INTER_COMM] THEN SIMP_TAC[OPEN_IN_OPEN_INTER; OPEN_BALL]);; let BILIPSCHITZ_HOMEOMORPHISM_SPHERICAL_PROJECTION = prove (`!s:real^N->bool. convex s /\ bounded s /\ vec 0 IN relative_interior s ==> ?g. homeomorphism (relative_frontier s,sphere(vec 0,&1) INTER affine hull s) ((\x. inv(norm x) % x),g) /\ (?B. !x y. x IN relative_frontier s /\ y IN relative_frontier s ==> norm(inv(norm x) % x - inv(norm y) % y) <= B * norm(x - y)) /\ (?B. !x y. x IN sphere(vec 0,&1) INTER affine hull s /\ y IN sphere(vec 0,&1) INTER affine hull s ==> norm(g x - g y) <= B * norm(x - y))`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `s:real^N->bool` LIPSCHITZ_CONVEX_SPHERICAL_PROJECTION) THEN ASM_REWRITE_TAC[dist; LIPSCHITZ_ON_POS] THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN MP_TAC(ISPEC `s:real^N->bool` INVERSE_LIPSCHITZ_CONVEX_SPHERICAL_PROJECTION) THEN ASM_REWRITE_TAC[real_ge; dist] THEN DISCH_THEN(X_CHOOSE_THEN `b:real` STRIP_ASSUME_TAC) THEN ONCE_REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN EXISTS_TAC `B:real` THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPECL [`relative_frontier s:real^N->bool`; `\x:real^N. inv(norm x) % x`; `sphere(vec 0:real^N,&1) INTER affine hull s`] HOMEOMORPHISM_COMPACT) THEN ASM_SIMP_TAC[COMPACT_RELATIVE_FRONTIER_BOUNDED] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_MUL THEN REWRITE_TAC[o_DEF; CONTINUOUS_ON_ID] THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN SIMP_TAC[CONTINUOUS_ON_LIFT_NORM_COMPOSE; CONTINUOUS_ON_ID] THEN REWRITE_TAC[NORM_EQ_0; relative_frontier; IN_DIFF] THEN ASM_MESON_TAC[]; REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; SUBSET] THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_INTER; IN_SPHERE_0] THEN CONJ_TAC THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THENL [CONJ_TAC THENL [REWRITE_TAC[NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN MATCH_MP_TAC REAL_MUL_LINV THEN REWRITE_TAC[NORM_EQ_0] THEN ASM_MESON_TAC[relative_frontier; IN_DIFF]; GEN_REWRITE_TAC LAND_CONV [GSYM VECTOR_ADD_LID] THEN MATCH_MP_TAC IN_AFFINE_ADD_MUL THEN REWRITE_TAC[AFFINE_AFFINE_HULL; VECTOR_ADD_LID] THEN ASM_MESON_TAC[AFFINE_HULL_CLOSURE; relative_frontier; IN_DIFF; IN_RELATIVE_INTERIOR; HULL_INC]]; REWRITE_TAC[IN_IMAGE] THEN MP_TAC(ISPECL [`s:real^N->bool`; `vec 0:real^N`; `x:real^N`] RAY_TO_RELATIVE_FRONTIER) THEN ASM_REWRITE_TAC[VECTOR_ADD_LID] THEN ANTS_TAC THENL [ASM_MESON_TAC[NORM_ARITH `~(norm(vec 0:real^N) = &1)`]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `a:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `a % x:real^N` THEN ASM_REWRITE_TAC[NORM_MUL] THEN ASM_SIMP_TAC[real_abs; REAL_LT_IMP_LE; REAL_MUL_RID; VECTOR_MUL_ASSOC; REAL_MUL_LINV; REAL_LT_IMP_NZ; VECTOR_MUL_LID]]; MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `y:real^N`])) THEN ASM_REWRITE_TAC[VECTOR_SUB_REFL; NORM_0] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ] THEN REWRITE_TAC[real_div; REAL_MUL_LZERO] THEN SIMP_TAC[NORM_ARITH `norm(x - y:real^N) <= &0 <=> x = y`]]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^N->real^N` THEN STRIP_TAC THEN EXISTS_TAC `inv b:real` THEN ASM_REWRITE_TAC[REAL_LT_INV_EQ] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphism]) THEN REWRITE_TAC[] THEN STRIP_TAC THEN SUBST1_TAC(SYM(ASSUME `IMAGE (\x:real^N. inv (norm x) % x) (relative_frontier s) = sphere (vec 0,&1) INTER affine hull s`)) THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN ASM_SIMP_TAC[] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[GSYM real_div; REAL_LE_RDIV_EQ] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[]]);; let BILIPSCHITZ_HOMEOMORPHISM_RELATIVE_FRONTIERS = prove (`!s:real^M->bool t:real^N->bool. convex s /\ bounded s /\ convex t /\ bounded t /\ aff_dim s = aff_dim t ==> ?f g. homeomorphism (relative_frontier s,relative_frontier t) (f,g) /\ (?B. !x y. x IN relative_frontier s /\ y IN relative_frontier s ==> norm(f x - f y) <= B * norm(x - y)) /\ (?B. !x y. x IN relative_frontier t /\ y IN relative_frontier t ==> norm(g x - g y) <= B * norm(x - y))`, let lemma1 = prove (`!s:real^N->bool t:real^N->bool. convex s /\ bounded s /\ convex t /\ bounded t /\ vec 0 IN relative_interior s /\ vec 0 IN relative_interior t /\ affine hull s = affine hull t ==> ?f g. homeomorphism (relative_frontier s,relative_frontier t) (f,g) /\ (?B. !x y. x IN relative_frontier s /\ y IN relative_frontier s ==> norm(f x - f y) <= B * norm(x - y)) /\ (?B. !x y. x IN relative_frontier t /\ y IN relative_frontier t ==> norm(g x - g y) <= B * norm(x - y))`, REPEAT STRIP_TAC THEN MAP_EVERY (MP_TAC o C SPEC BILIPSCHITZ_HOMEOMORPHISM_SPHERICAL_PROJECTION) [`t:real^N->bool`; `s:real^N->bool`] THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; IMP_CONJ] THEN REWRITE_TAC[IMP_IMP] THEN ABBREV_TAC `n (x:real^N) = inv(norm x) % x` THEN FIRST_X_ASSUM(K ALL_TAC o SPEC `vec 0:real^N`) THEN REWRITE_TAC[ETA_AX] THEN MAP_EVERY (fun t -> X_GEN_TAC t THEN STRIP_TAC) [`f:real^N->real^N`; `B:real`; `C:real`; `f':real^N->real^N`; `B':real`; `C':real`] THEN MAP_EVERY EXISTS_TAC [`(f':real^N->real^N) o (n:real^N->real^N)`; `(f:real^N->real^N) o (n:real^N->real^N)`] THEN CONJ_TAC THENL [MATCH_MP_TAC HOMEOMORPHISM_COMPOSE THEN EXISTS_TAC `sphere(vec 0:real^N,&1) INTER affine hull t` THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[HOMEOMORPHISM_SYM] THEN ASM_REWRITE_TAC[]; CONJ_TAC THEN REWRITE_TAC[o_DEF] THEN MATCH_MP_TAC LIPSCHITZ_ON_COMPOSE THEN EXISTS_TAC `sphere(vec 0:real^N,&1) INTER affine hull t` THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN ASM_MESON_TAC[]]) in let lemma2 = prove (`!s:real^M->bool t:real^N->bool. convex s /\ bounded s /\ convex t /\ bounded t /\ vec 0 IN relative_interior s /\ vec 0 IN relative_interior t /\ dim s = dim t ==> ?f g. homeomorphism (relative_frontier s,relative_frontier t) (f,g) /\ (?B. !x y. x IN relative_frontier s /\ y IN relative_frontier s ==> norm(f x - f y) <= B * norm(x - y)) /\ (?B. !x y. x IN relative_frontier t /\ y IN relative_frontier t ==> norm(g x - g y) <= B * norm(x - y))`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`span s:real^M->bool`; `span t:real^N->bool`] ISOMETRIES_SUBSPACES) THEN ASM_REWRITE_TAC[DIM_SPAN; SUBSPACE_SPAN; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`h:real^M->real^N`; `k:real^N->real^M`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`IMAGE (h:real^M->real^N) s`; `t:real^N->bool`] lemma1) THEN ASM_SIMP_TAC[CONVEX_LINEAR_IMAGE; BOUNDED_LINEAR_IMAGE; RELATIVE_INTERIOR_LINEAR_IMAGE_CONVEX; AFFINE_HULL_LINEAR_IMAGE] THEN ANTS_TAC THENL [CONJ_TAC THENL [ASM_MESON_TAC[LINEAR_0; FUN_IN_IMAGE]; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_RELATIVE_INTERIOR]) THEN ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`f:real^N->real^N`; `g:real^N->real^N`] THEN SUBGOAL_THEN `relative_frontier (IMAGE h s) = IMAGE (h:real^M->real^N) (relative_frontier s)` SUBST1_TAC THENL [REWRITE_TAC[relative_frontier] THEN ASM_SIMP_TAC[RELATIVE_INTERIOR_LINEAR_IMAGE_CONVEX] THEN ASM_SIMP_TAC[CLOSURE_BOUNDED_LINEAR_IMAGE] THEN MP_TAC(ISPEC `s:real^M->bool` RELATIVE_INTERIOR_SUBSET) THEN MP_TAC(ISPEC `s:real^M->bool` CLOSURE_INC) THEN MP_TAC(ISPEC `s:real^M->bool` CLOSURE_SUBSET_SPAN) THEN ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[FORALL_IN_IMAGE_2] THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN EXISTS_TAC `(f:real^N->real^N) o (h:real^M->real^N)` THEN EXISTS_TAC `(k:real^N->real^M) o (g:real^N->real^N)` THEN ASM_REWRITE_TAC[o_THM] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC HOMEOMORPHISM_COMPOSE THEN EXISTS_TAC `IMAGE (h:real^M->real^N) (relative_frontier s)` THEN ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[HOMEOMORPHISM; LINEAR_CONTINUOUS_ON] THEN REWRITE_TAC[FORALL_IN_IMAGE; GSYM IMAGE_o; o_DEF; SUBSET] THEN MP_TAC(ISPEC `s:real^M->bool` CLOSURE_SUBSET_SPAN) THEN REWRITE_TAC[relative_frontier] THEN ASM SET_TAC[]; SUBGOAL_THEN `!x y. x IN relative_frontier s /\ y IN relative_frontier s ==> norm((h:real^M->real^N) x - h y) = norm(x - y)` (fun th -> ASM_MESON_TAC[LINEAR_SUB; th]) THEN ASM_SIMP_TAC[GSYM LINEAR_SUB] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN MATCH_MP_TAC SPAN_SUB THEN RULE_ASSUM_TAC(REWRITE_RULE[relative_frontier; IN_DIFF]) THEN ASM_MESON_TAC[REWRITE_RULE[SUBSET] CLOSURE_SUBSET_SPAN]; SUBGOAL_THEN `!x y. x IN relative_frontier t /\ y IN relative_frontier t ==> ((g:real^N->real^N) x - g y) IN span t` (fun th -> ASM_MESON_TAC[LINEAR_SUB; th]) THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC SPAN_SUB THEN CONJ_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHISM]) THEN DISCH_THEN(MP_TAC o el 3 o CONJUNCTS) THEN MATCH_MP_TAC(SET_RULE `x IN s /\ t SUBSET u ==> IMAGE f s SUBSET t ==> f x IN u`) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [SYM th]) THEN MATCH_MP_TAC IMAGE_SUBSET THEN REWRITE_TAC[relative_frontier] THEN MATCH_MP_TAC(SET_RULE `s SUBSET u ==> s DIFF t SUBSET u`) THEN REWRITE_TAC[CLOSURE_SUBSET_SPAN]]) in REPEAT GEN_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THENL [ASM_SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_EQ_MINUS1; RELATIVE_FRONTIER_EMPTY; NOT_IN_EMPTY; HOMEOMORPHIC_EMPTY; GSYM homeomorphic]; ALL_TAC] THEN ASM_CASES_TAC `s:real^M->bool = {}` THENL [ASM_MESON_TAC[AFF_DIM_EMPTY; AFF_DIM_EQ_MINUS1]; STRIP_TAC] THEN SUBGOAL_THEN `~(relative_interior(s:real^M->bool) = {}) /\ ~(relative_interior(t:real^N->bool) = {})` MP_TAC THENL [ASM_MESON_TAC[RELATIVE_INTERIOR_EQ_EMPTY]; ALL_TAC] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `a:real^M`) (X_CHOOSE_TAC `b:real^N`)) THEN MP_TAC(ISPECL [`IMAGE (\x:real^M. --a + x) s`; `IMAGE (\x:real^N. --b + x) t`] lemma2) THEN ASM_REWRITE_TAC[CONVEX_TRANSLATION_EQ; BOUNDED_TRANSLATION_EQ; RELATIVE_INTERIOR_TRANSLATION; AFFINE_HULL_TRANSLATION] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [IN_IMAGE] THEN REWRITE_TAC[VECTOR_ARITH `vec 0:real^N = --a + x <=> x = a`] THEN ASM_REWRITE_TAC[UNWIND_THM2] THEN ANTS_TAC THENL [MATCH_MP_TAC(MESON[INT_OF_NUM_EQ] `aff_dim s = aff_dim t /\ aff_dim s = &(dim s) /\ aff_dim t = &(dim t) ==> dim(s:real^M->bool) = dim(t:real^N->bool)`) THEN CONJ_TAC THENL [ASM_REWRITE_TAC[AFF_DIM_TRANSLATION_EQ]; ALL_TAC] THEN CONJ_TAC THEN MATCH_MP_TAC AFF_DIM_DIM_0 THEN MATCH_MP_TAC HULL_INC THEN REWRITE_TAC[IN_IMAGE; VECTOR_ARITH `vec 0:real^N = --a + x <=> x = a`] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_RELATIVE_INTERIOR]) THEN ASM_REWRITE_TAC[UNWIND_THM2]; ALL_TAC] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; RELATIVE_FRONTIER_TRANSLATION] THEN REWRITE_TAC[FORALL_IN_IMAGE_2] THEN REWRITE_TAC[VECTOR_ARITH `(a + x) - (a + y):real^N = x - y`] THEN MAP_EVERY X_GEN_TAC [`f:real^M->real^N`; `g:real^N->real^M`] THEN STRIP_TAC THEN EXISTS_TAC `\x. b + (f:real^M->real^N) (--a + x)` THEN EXISTS_TAC `\x. a + (g:real^N->real^M) (--b + x)` THEN REWRITE_TAC[VECTOR_ARITH `(a + x) - (a + y):real^N = x - y`] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphism]) THEN REWRITE_TAC[HOMEOMORPHISM; FORALL_IN_IMAGE; GSYM IMAGE_o] THEN SIMP_TAC[VECTOR_ARITH `--a + a + x:real^N = x`; VECTOR_ARITH `a + --a + x:real^N = x`] THEN REWRITE_TAC[o_DEF] THEN STRIP_TAC THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_ADD THEN ASM_SIMP_TAC[CONTINUOUS_ON_CONST] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID]; GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM o_DEF] THEN ASM_REWRITE_TAC[IMAGE_o] THEN REWRITE_TAC[GSYM IMAGE_o; o_DEF] THEN REWRITE_TAC[VECTOR_ARITH `a + --a + x:real^N = x`; IMAGE_ID; SUBSET_REFL]; MATCH_MP_TAC CONTINUOUS_ON_ADD THEN ASM_SIMP_TAC[CONTINUOUS_ON_CONST] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID]; GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM o_DEF] THEN ASM_REWRITE_TAC[IMAGE_o] THEN REWRITE_TAC[GSYM IMAGE_o; o_DEF] THEN REWRITE_TAC[VECTOR_ARITH `a + --a + x:real^N = x`; IMAGE_ID; SUBSET_REFL]]);; (* ------------------------------------------------------------------------- *) (* More about affine dimension of special sets. *) (* ------------------------------------------------------------------------- *) let AFF_DIM_NONEMPTY_INTERIOR_EQ = prove (`!s:real^N->bool. convex s ==> (aff_dim s = &(dimindex (:N)) <=> ~(interior s = {}))`, REPEAT STRIP_TAC THEN EQ_TAC THEN ASM_SIMP_TAC[AFF_DIM_NONEMPTY_INTERIOR] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPEC `s:real^N->bool` EMPTY_INTERIOR_SUBSET_HYPERPLANE) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP AFF_DIM_SUBSET) THEN ASM_SIMP_TAC[AFF_DIM_HYPERPLANE] THEN INT_ARITH_TAC);; let AFF_DIM_BALL = prove (`!a:real^N r. aff_dim(ball(a,r)) = if &0 < r then &(dimindex(:N)) else --(&1)`, REPEAT GEN_TAC THEN COND_CASES_TAC THENL [MATCH_MP_TAC AFF_DIM_OPEN THEN ASM_REWRITE_TAC[OPEN_BALL; BALL_EQ_EMPTY; REAL_NOT_LE]; RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LT; GSYM BALL_EQ_EMPTY]) THEN ASM_REWRITE_TAC[AFF_DIM_EMPTY]]);; let AFF_DIM_CBALL = prove (`!a:real^N r. aff_dim(cball(a,r)) = if &0 < r then &(dimindex(:N)) else if r = &0 then &0 else --(&1)`, REPEAT GEN_TAC THEN REPEAT COND_CASES_TAC THENL [MATCH_MP_TAC AFF_DIM_NONEMPTY_INTERIOR THEN ASM_REWRITE_TAC[INTERIOR_CBALL; BALL_EQ_EMPTY] THEN ASM_REAL_ARITH_TAC; ASM_SIMP_TAC[CBALL_SING; AFF_DIM_SING]; MATCH_MP_TAC(MESON[AFF_DIM_EMPTY] `s = {} ==> aff_dim s = --(&1)`) THEN REWRITE_TAC[CBALL_EQ_EMPTY] THEN ASM_REAL_ARITH_TAC]);; let AFF_DIM_INTERVAL = prove (`(!a b:real^N. aff_dim(interval[a,b]) = if interval[a,b] = {} then --(&1) else &(CARD {i | 1 <= i /\ i <= dimindex(:N) /\ a$i < b$i})) /\ (!a b:real^N. aff_dim(interval(a,b)) = if interval(a,b) = {} then --(&1) else &(dimindex(:N)))`, REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_OPEN; OPEN_INTERVAL] THEN POP_ASSUM MP_TAC THEN GEOM_ORIGIN_TAC `a:real^N` THEN REPEAT STRIP_TAC THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VEC_COMPONENT; REAL_LT_LADD] THEN ASM_SIMP_TAC[AFF_DIM_DIM_0; HULL_INC; ENDS_IN_INTERVAL] THEN AP_TERM_TAC THEN ONCE_REWRITE_TAC[GSYM DIM_SPAN] THEN MATCH_MP_TAC DIM_UNIQUE THEN EXISTS_TAC `{basis i:real^N | 1 <= i /\ i <= dimindex(:N) /\ &0 < (b:real^N)$i}` THEN RULE_ASSUM_TAC(REWRITE_RULE[INTERVAL_NE_EMPTY; VEC_COMPONENT]) THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN SUBGOAL_THEN `basis i:real^N = inv(b$i) % (b:real^N)$i % basis i` SUBST1_TAC THENL [ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; REAL_LT_IMP_NZ] THEN REWRITE_TAC[VECTOR_MUL_LID]; MATCH_MP_TAC SPAN_MUL THEN MATCH_MP_TAC SPAN_SUPERSET THEN SIMP_TAC[IN_INTERVAL; VECTOR_MUL_COMPONENT; BASIS_COMPONENT] THEN X_GEN_TAC `j:num` THEN REWRITE_TAC[VEC_COMPONENT] THEN COND_CASES_TAC THEN ASM_SIMP_TAC[REAL_MUL_RZERO; REAL_MUL_RID; REAL_LE_REFL]]; MATCH_MP_TAC SPAN_SUBSET_SUBSPACE THEN REWRITE_TAC[SUBSPACE_SPAN; SUBSET; IN_INTERVAL; VEC_COMPONENT] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM BASIS_EXPANSION] THEN MATCH_MP_TAC SPAN_VSUM THEN REWRITE_TAC[FINITE_NUMSEG] THEN X_GEN_TAC `i:num` THEN REWRITE_TAC[IN_NUMSEG] THEN STRIP_TAC THEN ASM_CASES_TAC `&0 < (b:real^N)$i` THENL [MATCH_MP_TAC SPAN_MUL THEN MATCH_MP_TAC SPAN_SUPERSET THEN ASM SET_TAC[]; SUBGOAL_THEN `(x:real^N)$i = &0` (fun th -> REWRITE_TAC[th; VECTOR_MUL_LZERO; SPAN_0]) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `i:num`)) THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC]; MATCH_MP_TAC PAIRWISE_ORTHOGONAL_INDEPENDENT THEN REWRITE_TAC[SET_RULE `~(a IN {f x | P x}) <=> !x. P x ==> ~(f x = a)`] THEN SIMP_TAC[BASIS_NONZERO; pairwise; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN SIMP_TAC[FORALL_IN_GSPEC; BASIS_INJ_EQ; ORTHOGONAL_BASIS_BASIS]; GEN_REWRITE_TAC LAND_CONV [SIMPLE_IMAGE_GEN] THEN MATCH_MP_TAC HAS_SIZE_IMAGE_INJ THEN SIMP_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_GSPEC; BASIS_INJ_EQ; HAS_SIZE] THEN SIMP_TAC[CONJ_ASSOC; GSYM IN_NUMSEG; FINITE_RESTRICT; FINITE_NUMSEG]]);; (* ------------------------------------------------------------------------- *) (* A complete graph of |R|-many vertices can be embedded in R^3 with the *) (* edges as straight-line segments that intersect only at common endpoints. *) (* Basically, you just scatter the points onto the twisted cubic. *) (* ------------------------------------------------------------------------- *) let GRAPH_EMBEDS_IN_R3 = prove (`!s:A->bool. s <=_c (:real) ==> ?v:A->real^3. (!a b. a IN s /\ b IN s ==> (v a = v b <=> a = b)) /\ (!a b c d. ~({v a,v b} = {v c,v d}) ==> segment[v a,v b] INTER segment[v c,v d] SUBSET {v a,v b} INTER {v c,v d})`, SUBGOAL_THEN `?v:real->real^3. (!a b. v a = v b <=> a = b) /\ (!a b c d. ~({v a,v b} = {v c,v d}) ==> segment[v a,v b] INTER segment[v c,v d] SUBSET {v a,v b} INTER {v c,v d})` STRIP_ASSUME_TAC THENL [ALL_TAC; X_GEN_TAC `s:A->bool` THEN REWRITE_TAC[le_c; IN_UNIV; INJECTIVE_ON_ALT] THEN DISCH_THEN(X_CHOOSE_THEN `f:A->real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(v:real->real^3) o (f:A->real)` THEN ASM_REWRITE_TAC[o_THM]] THEN ABBREV_TAC `v:real->real^3 = \x. vector[x; x pow 2; x pow 3]` THEN EXISTS_TAC `v:real->real^3` THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [REWRITE_TAC[GSYM INJECTIVE_ALT] THEN EXPAND_TAC "v" THEN SIMP_TAC[CART_EQ; FORALL_3; DIMINDEX_3; VECTOR_3]; DISCH_TAC] THEN SUBGOAL_THEN `!a b c d. PAIRWISE (\x y. ~(x = y)) [a;b;c;d] ==> aff_dim (IMAGE (v:real->real^3) {a,b,c,d}) = &3` (LABEL_TAC "COPLANAR") THENL [REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`(v:real->real^3) a`; `IMAGE (v:real->real^3) {a,b,c,d}`] AFF_DIM_DIM_AFFINE_DIFFS) THEN SIMP_TAC[FUN_IN_IMAGE; IN_INSERT] THEN DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[SIMPLE_IMAGE; IMAGE_CLAUSES; INT_OF_NUM_EQ] THEN REWRITE_TAC[VECTOR_SUB_REFL; DIM_INSERT_0] THEN W(MP_TAC o PART_MATCH (lhand o rand o rand) INDEPENDENT_EQ_DIM_EQ_CARD o lhand o snd) THEN REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY] THEN SUBGOAL_THEN `~(det(vector[(v:real->real^3) b - (v:real->real^3) a; (v:real->real^3) c - (v:real->real^3) a; (v:real->real^3) d - (v:real->real^3) a]:real^3^3) = &0)` MP_TAC THENL [EXPAND_TAC "v" THEN REWRITE_TAC[DET_3; VECTOR_3; VECTOR_SUB_COMPONENT] THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[PAIRWISE; ALL] THEN CONV_TAC REAL_RING; DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[GSYM CONTRAPOS_THM] DET_DEPENDENT_ROWS))] THEN REWRITE_TAC[rows] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN REWRITE_TAC[GSYM numseg; DIMINDEX_3; NUMSEG_CONV `1..3`] THEN SIMP_TAC[row; IMAGE_CLAUSES; DIMINDEX_3; LAMBDA_ETA; VECTOR_3] THEN DISCH_TAC THEN ASM_REWRITE_TAC[independent] THEN DISCH_THEN SUBST1_TAC THEN SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN ASM_REWRITE_TAC[VECTOR_ARITH `x - z:real^3 = y - z <=> x = y`] THEN RULE_ASSUM_TAC(REWRITE_RULE[PAIRWISE; ALL]) THEN ASM_REWRITE_TAC[] THEN CONV_TAC NUM_REDUCE_CONV; ALL_TAC] THEN SUBGOAL_THEN `!a b c. collinear(IMAGE (v:real->real^3) {a,b,c}) ==> ~PAIRWISE (\x y. ~(x = y)) [a;b;c]` (LABEL_TAC "COLLINEAR") THENL [MAP_EVERY X_GEN_TAC [`a:real`; `b:real`; `c:real`] THEN GEN_REWRITE_TAC I [GSYM CONTRAPOS_THM] THEN REWRITE_TAC[] THEN REPEAT DISCH_TAC THEN REMOVE_THEN "COPLANAR" (MP_TAC o SPECL [`abs a + abs b + abs c + &1:real`; `a:real`; `b:real`; `c:real`]) THEN REWRITE_TAC[NOT_IMP] THEN ONCE_REWRITE_TAC[PAIRWISE] THEN ASM_REWRITE_TAC[ALL] THEN CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN ONCE_REWRITE_TAC[IMAGE_CLAUSES] THEN REWRITE_TAC[AFF_DIM_INSERT] THEN RULE_ASSUM_TAC(REWRITE_RULE[COLLINEAR_AFF_DIM]) THEN ASM_INT_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC(METIS[] `(!a b c d. P a b c d ==> P c d a b /\ P b a c d /\ P a b d c) /\ (!a b c d. ~(a = b) /\ ~(a = c) /\ ~(a = d) /\ ~(b = c) /\ ~(b = d) /\ ~(c = d) ==> P a b c d) /\ (!a b c. P a b c c) /\ (!a b c. ~(a = b) ==> P a b b c) ==> !a b c d. P a b c d`) THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[SET_RULE `{a,b} = {b,a}`; SEGMENT_SYM; INTER_ACI] THEN MESON_TAC[]; MAP_EVERY X_GEN_TAC [`a:real`; `b:real`; `c:real`; `d:real`] THEN STRIP_TAC THEN DISCH_TAC THEN REWRITE_TAC[SUBSET; IN_INTER] THEN X_GEN_TAC `z:real^3` THEN STRIP_TAC THEN REMOVE_THEN "COPLANAR" (MP_TAC o SPECL [`a:real`; `b:real`; `c:real`; `d:real`]) THEN ASM_REWRITE_TAC[PAIRWISE; ALL] THEN MATCH_MP_TAC(TAUT `~p ==> p ==> q`) THEN MATCH_MP_TAC(INT_ARITH `!y:int. x <= y /\ y <= &2 ==> ~(x = &3)`) THEN EXISTS_TAC `aff_dim(z INSERT IMAGE (v:real->real^3) {a,b,c,d})` THEN SIMP_TAC[AFF_DIM_SUBSET; SET_RULE `s SUBSET a INSERT s`] THEN REWRITE_TAC[IMAGE_CLAUSES; GSYM COPLANAR_AFF_DIM] THEN MATCH_MP_TAC COPLANAR_INTERSECTING_LINES THEN ASM_SIMP_TAC[BETWEEN_IMP_COLLINEAR; BETWEEN_IN_SEGMENT]; REWRITE_TAC[SET_RULE `{a,a} = {a}`; SEGMENT_REFL] THEN MAP_EVERY X_GEN_TAC [`a:real`; `b:real`; `c:real`] THEN DISCH_THEN(K ALL_TAC) THEN REWRITE_TAC[SET_RULE `s INTER {a} SUBSET t INTER {a} <=> a IN s ==> a IN t`] THEN ASM_CASES_TAC `b:real = a` THEN ASM_SIMP_TAC[SEGMENT_REFL; IN_INSERT; NOT_IN_EMPTY] THEN STRIP_TAC THEN REMOVE_THEN "COLLINEAR" (MP_TAC o SPECL [`a:real`; `c:real`; `b:real`]) THEN ASM_SIMP_TAC[IMAGE_CLAUSES; BETWEEN_IMP_COLLINEAR; BETWEEN_IN_SEGMENT] THEN REWRITE_TAC[PAIRWISE; ALL] THEN ASM_MESON_TAC[]; MAP_EVERY X_GEN_TAC [`a:real`; `b:real`; `c:real`] THEN DISCH_TAC THEN ASM_CASES_TAC `collinear {(v:real->real^3) a,v b,v c}` THENL [ALL_TAC; ASM_SIMP_TAC[INTER_SEGMENT] THEN SET_TAC[]] THEN ASM_CASES_TAC `c:real = a` THENL [ASM SET_TAC[]; DISCH_THEN(K ALL_TAC)] THEN REMOVE_THEN "COLLINEAR" (MP_TAC o SPECL [`a:real`; `b:real`; `c:real`]) THEN ASM_REWRITE_TAC[IMAGE_CLAUSES; PAIRWISE; ALL] THEN DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[SEGMENT_REFL] THEN SET_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Deducing convexity from midpoint convexity in common cases. *) (* ------------------------------------------------------------------------- *) let MIDPOINT_CONVEX_DYADIC_RATIONALS = prove (`!f:real^N->real s. (!x y. x IN s /\ y IN s ==> midpoint(x,y) IN s /\ f(midpoint(x,y)) <= (f(x) + f(y)) / &2) ==> !n m p x y. x IN s /\ y IN s /\ m + p = 2 EXP n ==> (&m / &2 pow n % x + &p / &2 pow n % y) IN s /\ f(&m / &2 pow n % x + &p / &2 pow n % y) <= &m / &2 pow n * f x + &p / &2 pow n * f y`, REPEAT GEN_TAC THEN DISCH_TAC THEN INDUCT_TAC THENL [REWRITE_TAC[ARITH_RULE `m + p = 2 EXP 0 <=> m = 0 /\ p = 1 \/ m = 1 /\ p = 0`] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[VECTOR_MUL_LID; VECTOR_MUL_LZERO; VECTOR_ADD_LID; VECTOR_ADD_RID] THEN REAL_ARITH_TAC; MATCH_MP_TAC WLOG_LE THEN CONJ_TAC THENL [REWRITE_TAC[VECTOR_ADD_SYM; REAL_ADD_SYM; ADD_SYM] THEN MESON_TAC[]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`m:num`; `p:num`] THEN DISCH_TAC THEN REPEAT GEN_TAC THEN REWRITE_TAC[EXP; real_pow] THEN STRIP_TAC THEN REWRITE_TAC[real_div; REAL_INV_MUL] THEN ONCE_REWRITE_TAC[REAL_ARITH `x * inv(&2) * y = inv(&2) * x * y`] THEN ONCE_REWRITE_TAC[GSYM REAL_MUL_ASSOC; GSYM VECTOR_MUL_ASSOC] THEN REWRITE_TAC[GSYM REAL_ADD_LDISTRIB; GSYM VECTOR_ADD_LDISTRIB] THEN SUBGOAL_THEN `2 EXP n <= p` ASSUME_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `&p * inv(&2 pow n) = &(p - 2 EXP n) * inv(&2 pow n) + &1` SUBST1_TAC THENL [ASM_SIMP_TAC[GSYM REAL_OF_NUM_SUB; GSYM REAL_OF_NUM_POW] THEN ASM_SIMP_TAC[REAL_SUB_RDISTRIB; REAL_MUL_RINV; REAL_LT_IMP_NZ; REAL_LT_POW2] THEN REAL_ARITH_TAC; REWRITE_TAC[VECTOR_ADD_RDISTRIB; REAL_ADD_RDISTRIB] THEN REWRITE_TAC[VECTOR_MUL_LID; REAL_MUL_LID] THEN REWRITE_TAC[VECTOR_ADD_ASSOC; REAL_ADD_ASSOC] THEN REWRITE_TAC[GSYM midpoint; GSYM real_div] THEN FIRST_X_ASSUM(fun th -> W(MP_TAC o PART_MATCH (lhand o rand) th o lhand o snd)) THEN FIRST_X_ASSUM(fun th -> W(MP_TAC o PART_MATCH (lhand o rand) th o funpow 3 lhand o snd)) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_ARITH_TAC; SIMP_TAC[] THEN REAL_ARITH_TAC]]]);; let CONTINUOUS_MIDPOINT_CONVEX = prove (`!f:real^N->real s. (lift o f) continuous_on s /\ convex s /\ (!x y. x IN s /\ y IN s ==> f(midpoint(x,y)) <= (f(x) + f(y)) / &2) ==> f convex_on s`, REWRITE_TAC[midpoint] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[convex_on] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[GSYM IMP_CONJ_ALT] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN REWRITE_TAC[REAL_ARITH `u + v = &1 <=> v = &1 - u`; IMP_CONJ] THEN REWRITE_TAC[FORALL_UNWIND_THM2; REAL_SUB_LE] THEN REWRITE_TAC[FORALL_DROP; GSYM DROP_VEC; IMP_IMP; GSYM IN_INTERVAL_1] THEN MP_TAC(ISPEC `interval[vec 0:real^1,vec 1]` CLOSURE_DYADIC_RATIONALS_IN_CONVEX_SET) THEN SIMP_TAC[CONVEX_INTERVAL; INTERIOR_CLOSED_INTERVAL; CLOSURE_CLOSED; CLOSED_INTERVAL; UNIT_INTERVAL_NONEMPTY] THEN REWRITE_TAC[DIMINDEX_1; FORALL_1; GSYM drop] THEN DISCH_THEN(fun th -> SUBST1_TAC(SYM th) THEN ASSUME_TAC th) THEN ONCE_REWRITE_TAC[REAL_ARITH `a <= b <=> a - b <= &0`] THEN MATCH_MP_TAC CONTINUOUS_LE_ON_CLOSURE THEN REWRITE_TAC[IN_INTER; IMP_CONJ_ALT; FORALL_IN_GSPEC] THEN FIRST_X_ASSUM SUBST1_TAC THEN REWRITE_TAC[IN_INTERVAL_1; DROP_CMUL; GSYM FORALL_DROP; DROP_VEC] THEN CONJ_TAC THENL [REWRITE_TAC[o_DEF; LIFT_SUB; LIFT_ADD; LIFT_CMUL] THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN CONJ_TAC THENL [REPLICATE_TAC 2 (ONCE_REWRITE_TAC[GSYM o_DEF]) THEN REWRITE_TAC[o_ASSOC] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THENL [ALL_TAC; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTERVAL_1; GSYM FORALL_DROP; DROP_VEC] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [convex]) THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC]; ALL_TAC] THEN MATCH_MP_TAC CONTINUOUS_ON_ADD THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN SIMP_TAC[o_DEF; LIFT_DROP; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST; LIFT_SUB; CONTINUOUS_ON_SUB]; MAP_EVERY X_GEN_TAC [`n:num`; `i:real`] THEN ASM_SIMP_TAC[REAL_LE_MUL_EQ; REAL_LT_INV_EQ; REAL_LT_POW2] THEN ASM_CASES_TAC `&0 <= i` THEN ASM_SIMP_TAC[INTEGER_POS] THEN DISCH_THEN(X_CHOOSE_THEN `m:num` SUBST_ALL_TAC) THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LT_POW2; REAL_MUL_LID] THEN GEN_REWRITE_TAC (LAND_CONV o DEPTH_CONV) [REAL_OF_NUM_POW; REAL_OF_NUM_LE] THEN DISCH_TAC THEN MP_TAC(ISPECL [`f:real^N->real`; `s:real^N->bool`] MIDPOINT_CONVEX_DYADIC_RATIONALS) THEN ANTS_TAC THENL [ASM_SIMP_TAC[midpoint] THEN REWRITE_TAC[VECTOR_ADD_LDISTRIB] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [convex]) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC; DISCH_THEN(MP_TAC o SPECL [`n:num`; `m:num`; `2 EXP n - m`; `x:real^N`; `y:real^N`]) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_ARITH_TAC; DISCH_THEN(MP_TAC o CONJUNCT2)] THEN ASM_SIMP_TAC[GSYM REAL_OF_NUM_SUB; GSYM REAL_OF_NUM_POW] THEN ASM_SIMP_TAC[REAL_LT_POW2; REAL_FIELD `&0 < y ==> (y - x) / y = &1 - x / y`] THEN REAL_ARITH_TAC]]);; (* ------------------------------------------------------------------------- *) (* Slightly shaper separating/supporting hyperplane results. *) (* ------------------------------------------------------------------------- *) let SEPARATING_HYPERPLANE_RELATIVE_INTERIORS = prove (`!s t. convex s /\ convex t /\ ~(s = {} /\ t = (:real^N) \/ s = (:real^N) /\ t = {}) /\ DISJOINT (relative_interior s) (relative_interior t) ==> ?a b. ~(a = vec 0) /\ (!x. x IN s ==> a dot x <= b) /\ (!x. x IN t ==> a dot x >= b)`, REPEAT GEN_TAC THEN MAP_EVERY ASM_CASES_TAC [`s:real^N->bool = {}`; `t:real^N->bool = {}`] THEN ASM_REWRITE_TAC[NOT_IN_EMPTY; UNIV_NOT_EMPTY; CONVEX_EMPTY; RELATIVE_INTERIOR_EMPTY] THEN STRIP_TAC THENL [EXISTS_TAC `basis 1:real^N` THEN SIMP_TAC[BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL]; FIRST_X_ASSUM(X_CHOOSE_TAC `x:real^N` o MATCH_MP (SET_RULE `~(s = UNIV) ==> ?a. ~(a IN s)`)) THEN MP_TAC(ISPECL [`t:real^N->bool`; `x:real^N`] SEPARATING_HYPERPLANE_SET_POINT_INAFF) THEN ASM_MESON_TAC[]; FIRST_X_ASSUM(X_CHOOSE_TAC `x:real^N` o MATCH_MP (SET_RULE `~(s = UNIV) ==> ?a. ~(a IN s)`)) THEN MP_TAC(ISPECL [`s:real^N->bool`; `x:real^N`] SEPARATING_HYPERPLANE_SET_POINT_INAFF) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; real_ge] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real`] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`--a:real^N`; `--b:real`] THEN ASM_REWRITE_TAC[VECTOR_NEG_EQ_0; DOT_LNEG; REAL_LE_NEG2]; MP_TAC(ISPECL [`relative_interior s:real^N->bool`; `relative_interior t:real^N->bool`] SEPARATING_HYPERPLANE_SETS) THEN ASM_SIMP_TAC[RELATIVE_INTERIOR_EQ_EMPTY; CONVEX_RELATIVE_INTERIOR] THEN SIMP_TAC[real_ge] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `b:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THEN MATCH_MP_TAC (MESON[CONVEX_CLOSURE_RELATIVE_INTERIOR; CLOSURE_SUBSET; SUBSET] `convex s /\ (!x. x IN closure(relative_interior s) ==> P x) ==> !x. x IN s ==> P x`) THEN ASM_REWRITE_TAC[] THENL [MATCH_MP_TAC CONTINUOUS_LE_ON_CLOSURE; MATCH_MP_TAC CONTINUOUS_GE_ON_CLOSURE] THEN ASM_REWRITE_TAC[CONTINUOUS_ON_LIFT_DOT]]);; let SUPPORTING_HYPERPLANE_RELATIVE_BOUNDARY = prove (`!s x:real^N. convex s /\ x IN s /\ ~(x IN relative_interior s) ==> ?a. ~(a = vec 0) /\ (!y. y IN s ==> a dot x <= a dot y) /\ (!y. y IN relative_interior s ==> a dot x < a dot y)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`relative_interior s:real^N->bool`; `x:real^N`] SEPARATING_HYPERPLANE_SET_POINT_INAFF) THEN ASM_SIMP_TAC[CONVEX_SING; CONVEX_RELATIVE_INTERIOR; RELATIVE_INTERIOR_EQ_EMPTY; real_ge] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN REWRITE_TAC[FORALL_IN_INSERT; NOT_IN_EMPTY] THEN DISCH_THEN(X_CHOOSE_THEN `b:real` STRIP_ASSUME_TAC) THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL [X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN MP_TAC(ISPECL [`lift o (\x:real^N. a dot x)`; `relative_interior s:real^N->bool`; `y:real^N`; `(a:real^N) dot x`; `1`] CONTINUOUS_ON_CLOSURE_COMPONENT_GE) THEN REWRITE_TAC[CONTINUOUS_ON_LIFT_DOT; GSYM drop; o_THM; LIFT_DROP] THEN ASM_SIMP_TAC[CONVEX_CLOSURE_RELATIVE_INTERIOR] THEN ASM_MESON_TAC[CLOSURE_SUBSET; REAL_LE_TRANS; SUBSET]; DISCH_TAC] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN REWRITE_TAC[REAL_LT_LE] THEN CONJ_TAC THENL [ASM_MESON_TAC[REAL_LE_TRANS]; ALL_TAC] THEN DISCH_TAC THEN UNDISCH_TAC `(y:real^N) IN relative_interior s` THEN REWRITE_TAC[IN_RELATIVE_INTERIOR_CBALL] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; SUBSET; IN_INTER; IN_CBALL] THEN X_GEN_TAC `e:real` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(MP_TAC o SPEC `y + --(e / norm(a)) % ((x + a) - x):real^N`) THEN REWRITE_TAC[NOT_IMP] THEN REPEAT CONJ_TAC THENL [SIMP_TAC[NORM_ARITH `dist(y:real^N,y + e) = norm e`; VECTOR_ADD_SUB] THEN REWRITE_TAC[NORM_MUL; REAL_ABS_NEG; REAL_ABS_DIV; REAL_ABS_NORM] THEN ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_EQ_0] THEN ASM_REAL_ARITH_TAC; MATCH_MP_TAC IN_AFFINE_ADD_MUL_DIFF THEN ASM_SIMP_TAC[AFFINE_AFFINE_HULL; HULL_INC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `x IN s ==> s SUBSET t ==> x IN t`)) THEN MATCH_MP_TAC HULL_MONO THEN ASM_REWRITE_TAC[INSERT_SUBSET; RELATIVE_INTERIOR_SUBSET]; REWRITE_TAC[VECTOR_ADD_SUB] THEN DISCH_TAC THEN UNDISCH_TAC `!y:real^N. y IN s ==> a dot x <= a dot y` THEN DISCH_THEN(MP_TAC o SPEC `y + --(e / norm(a)) % a:real^N`) THEN ASM_REWRITE_TAC[DOT_RMUL; DOT_RNEG; DOT_RADD] THEN MATCH_MP_TAC(REAL_ARITH `&0 < x * y ==> ~(a <= a + --x * y)`) THEN ASM_SIMP_TAC[REAL_LT_MUL; REAL_LT_DIV; NORM_POS_LT; DOT_POS_LT]]);; let SUPPORTING_HYPERPLANE_RELATIVE_FRONTIER = prove (`!s x:real^N. convex s /\ x IN relative_frontier s ==> ?a. ~(a = vec 0) /\ (!y. y IN closure s ==> a dot x <= a dot y) /\ (!y. y IN relative_interior s ==> a dot x < a dot y)`, REWRITE_TAC[relative_frontier; IN_DIFF] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`closure s:real^N->bool`; `x:real^N`] SUPPORTING_HYPERPLANE_RELATIVE_BOUNDARY) THEN ASM_SIMP_TAC[CONVEX_CLOSURE; CONVEX_RELATIVE_INTERIOR_CLOSURE]);; let SUPPORTING_HYPERPLANE_FRONTIER = prove (`!s x:real^N. convex s /\ x IN frontier s ==> ?a. ~(a = vec 0) /\ !y. y IN closure s ==> a dot x <= a dot y`, REPEAT GEN_TAC THEN ASM_CASES_TAC `interior s:real^N->bool = {}` THENL [STRIP_TAC THEN MP_TAC(ISPEC `s:real^N->bool` EMPTY_INTERIOR_SUBSET_HYPERPLANE) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP SUBSET_CLOSURE) THEN ASM_REWRITE_TAC[SUBSET; CLOSURE_HYPERPLANE; IN_ELIM_THM] THEN RULE_ASSUM_TAC(REWRITE_RULE[frontier; IN_DIFF]) THEN ASM_SIMP_TAC[REAL_LE_REFL]; ASM_SIMP_TAC[GSYM RELATIVE_FRONTIER_NONEMPTY_INTERIOR] THEN DISCH_THEN(MP_TAC o MATCH_MP SUPPORTING_HYPERPLANE_RELATIVE_FRONTIER) THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Containment of rays in unbounded convex sets. *) (* ------------------------------------------------------------------------- *) let UNBOUNDED_CONVEX_CLOSED_CONTAINS_RAY = prove (`!s a:real^N. convex s /\ ~bounded s /\ closed s /\ a IN s ==> ?l. ~(l = vec 0) /\ !t. &0 <= t ==> (a + t % l) IN s`, GEN_GEOM_ORIGIN_TAC `a:real^N` ["l"] THEN REWRITE_TAC[VECTOR_ADD_LID] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [BOUNDED_POS]) THEN REWRITE_TAC[NOT_EXISTS_THM; TAUT `~(p /\ q) <=> p ==> ~q`] THEN DISCH_THEN(MP_TAC o GEN `n:num` o SPEC `&n + &1:real`) THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; REAL_ARITH `&0 < &n + &1`] THEN REWRITE_TAC[REAL_NOT_LE; SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `x:num->real^N` THEN REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC THEN SUBGOAL_THEN `!n. ~((x:num->real^N) n = vec 0)` ASSUME_TAC THENL [ASM_MESON_TAC[NORM_ARITH `~(&n + &1 < norm(vec 0:real^N))`]; ALL_TAC] THEN MP_TAC(ISPEC `sphere(vec 0:real^N,&1)` compact) THEN REWRITE_TAC[COMPACT_SPHERE] THEN DISCH_THEN(MP_TAC o SPEC `\n. inv(norm(x n)) % (x:num->real^N) n`) THEN ASM_SIMP_TAC[IN_SPHERE_0; NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM; REAL_MUL_LINV; NORM_EQ_0; o_DEF] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `l:real^N` THEN DISCH_THEN(X_CHOOSE_THEN `r:num->num` STRIP_ASSUME_TAC) THEN CONJ_TAC THENL [ASM_MESON_TAC[NORM_ARITH `~(norm(vec 0:real^N) = &1)`]; ALL_TAC] THEN X_GEN_TAC `t:real` THEN DISCH_TAC THEN MATCH_MP_TAC CLOSED_CONTAINS_SEQUENTIAL_LIMIT THEN SUBGOAL_THEN `?N:num. !n. N <= n ==> t / norm(x n:real^N) <= &1` STRIP_ASSUME_TAC THENL [ASM_SIMP_TAC[REAL_LE_LDIV_EQ; NORM_POS_LT] THEN MP_TAC(SPEC `t:real` REAL_ARCH_SIMPLE) THEN MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[GSYM REAL_OF_NUM_LE; REAL_MUL_LID] THEN ASM_MESON_TAC[REAL_ARITH `t <= m /\ m <= n /\ n + &1 < x ==> t <= x`]; EXISTS_TAC `\n:num. t / norm((x:num->real^N)(r(N + n))) % x(r(N + n))` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [X_GEN_TAC `n:num` THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONVEX_ALT]) THEN DISCH_THEN(MP_TAC o SPEC `vec 0:real^N`) THEN ASM_REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_SIMP_TAC[REAL_LE_DIV; NORM_POS_LE] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `N + n:num` o MATCH_MP MONOTONE_BIGGER) THEN ARITH_TAC; REWRITE_TAC[real_div; GSYM VECTOR_MUL_ASSOC] THEN MATCH_MP_TAC LIM_CMUL THEN ONCE_REWRITE_TAC[ADD_SYM] THEN FIRST_ASSUM(MP_TAC o SPEC `N:num` o MATCH_MP SEQ_OFFSET) THEN ASM_REWRITE_TAC[]]]);; let CONVEX_CLOSED_CONTAINS_SAME_RAY = prove (`!s a b l:real^N. convex s /\ closed s /\ b IN s /\ (!t. &0 <= t ==> (a + t % l) IN s) ==> !t. &0 <= t ==> (b + t % l) IN s`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `&0`) THEN REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_RID] THEN DISCH_TAC THEN MATCH_MP_TAC(ISPEC `sequentially` LIM_IN_CLOSED_SET) THEN EXISTS_TAC `\n. (&1 - t / (&n + &1)) % b + t / (&n + &1) % (a + (&n + &1) % l):real^N` THEN ASM_REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN CONJ_TAC THENL [REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN MP_TAC(SPEC `t:real` REAL_ARCH_SIMPLE) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN DISCH_TAC THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_ALT]) THEN ASM_SIMP_TAC[REAL_LE_DIV; REAL_ARITH `&0 <= &n + &1`] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_ARITH `&0 < &n + &1`] THEN RULE_ASSUM_TAC(REWRITE_RULE[GSYM REAL_OF_NUM_LE]) THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[VECTOR_ARITH `(&1 - u) % b + u % c:real^N = b + u % (c - b)`] THEN MATCH_MP_TAC LIM_ADD THEN REWRITE_TAC[LIM_CONST] THEN REWRITE_TAC[VECTOR_ADD_LDISTRIB; VECTOR_SUB_LDISTRIB] THEN SIMP_TAC[VECTOR_MUL_ASSOC; REAL_FIELD `t / (&n + &1) * (&n + &1) = t`] THEN SIMP_TAC[VECTOR_ARITH `(v % a + b) - v % c:real^N = b + v % (a - c)`] THEN GEN_REWRITE_TAC LAND_CONV [GSYM VECTOR_ADD_RID] THEN MATCH_MP_TAC LIM_ADD THEN REWRITE_TAC[LIM_CONST] THEN REWRITE_TAC[real_div; VECTOR_ARITH `(x * y) % a:real^N = y % x % a`] THEN MATCH_MP_TAC LIM_NULL_VMUL_BOUNDED THEN EXISTS_TAC `norm(t % (a - b):real^N)` THEN REWRITE_TAC[REAL_LE_REFL; EVENTUALLY_TRUE; o_DEF] THEN MP_TAC(MATCH_MP SEQ_OFFSET SEQ_HARMONIC) THEN SIMP_TAC[REAL_OF_NUM_ADD]]);; let UNBOUNDED_CONVEX_CLOSED_CONTAINS_RAYS = prove (`!s:real^N->bool. convex s /\ ~bounded s /\ closed s ==> ?l. ~(l = vec 0) /\ !a t. a IN s /\ &0 <= t ==> (a + t % l) IN s`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[BOUNDED_EMPTY] THEN RULE_ASSUM_TAC(REWRITE_RULE[GSYM MEMBER_NOT_EMPTY]) THEN ASM_MESON_TAC[UNBOUNDED_CONVEX_CLOSED_CONTAINS_RAY; CONVEX_CLOSED_CONTAINS_SAME_RAY]);; let RELATIVE_INTERIOR_UNBOUNDED_CONVEX_CONTAINS_RAY = prove (`!s a:real^N. convex s /\ ~bounded s /\ a IN relative_interior s ==> ?l. ~(l = vec 0) /\ !t. &0 <= t ==> (a + t % l) IN relative_interior s`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`closure s:real^N->bool`; `a:real^N`] UNBOUNDED_CONVEX_CLOSED_CONTAINS_RAY) THEN ASM_SIMP_TAC[CONVEX_CLOSURE; CLOSED_CLOSURE] THEN ANTS_TAC THENL [ASM_MESON_TAC[BOUNDED_SUBSET; SUBSET; CLOSURE_SUBSET; RELATIVE_INTERIOR_SUBSET]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `l:real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[VECTOR_ARITH `a + t % l:real^N = (a + (&2 * t) % l) - inv(&2) % ((a + (&2 * t) % l) - a)`] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SHRINK THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REAL_ARITH_TAC]);; let RELATIVE_INTERIOR_CONVEX_CONTAINS_SAME_RAY = prove (`!s a b l:real^N. convex s /\ b IN relative_interior s /\ (!t. &0 <= t ==> (a + t % l) IN relative_interior s) ==> !t. &0 <= t ==> (b + t % l) IN relative_interior s`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`closure s:real^N->bool`; `a:real^N`; `b:real^N`; `l:real^N`] CONVEX_CLOSED_CONTAINS_SAME_RAY) THEN ASM_SIMP_TAC[CONVEX_CLOSURE; CLOSED_CLOSURE] THEN ANTS_TAC THENL [ASM_MESON_TAC[BOUNDED_SUBSET; SUBSET; CLOSURE_SUBSET; RELATIVE_INTERIOR_SUBSET]; DISCH_TAC THEN ONCE_REWRITE_TAC[VECTOR_ARITH `a + t % l:real^N = (a + (&2 * t) % l) - inv(&2) % ((a + (&2 * t) % l) - a)`] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SHRINK THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REAL_ARITH_TAC]);; let RELATIVE_INTERIOR_UNBOUNDED_CONVEX_CONTAINS_RAYS = prove (`!s:real^N->bool. convex s /\ ~bounded s ==> ?l. ~(l = vec 0) /\ !a t. a IN relative_interior s /\ &0 <= t ==> (a + t % l) IN relative_interior s`, REPEAT GEN_TAC THEN ASM_CASES_TAC `relative_interior s:real^N->bool = {}` THENL [ASM_MESON_TAC[RELATIVE_INTERIOR_EQ_EMPTY; BOUNDED_EMPTY]; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[GSYM MEMBER_NOT_EMPTY]) THEN ASM_MESON_TAC[RELATIVE_INTERIOR_UNBOUNDED_CONVEX_CONTAINS_RAY; RELATIVE_INTERIOR_CONVEX_CONTAINS_SAME_RAY]);; (* ------------------------------------------------------------------------- *) (* Explicit formulas for interior and relative interior of convex hull. *) (* ------------------------------------------------------------------------- *) let EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL = prove (`!s. FINITE s ==> {y:real^N | ?u. (!x. x IN s ==> &0 < u x /\ u x < &1) /\ sum s u = &1 /\ vsum s (\x. u x % x) = y} SUBSET relative_interior(convex hull s)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[SUM_CLAUSES; REAL_OF_NUM_EQ; ARITH_EQ] THEN REWRITE_TAC[EMPTY_GSPEC; EMPTY_SUBSET] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC RELATIVE_INTERIOR_MAXIMAL THEN REWRITE_TAC[AFFINE_HULL_CONVEX_HULL] THEN CONJ_TAC THENL [REWRITE_TAC[CONVEX_HULL_FINITE; SUBSET; IN_ELIM_THM] THEN GEN_TAC THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[REAL_LT_IMP_LE]; ALL_TAC] THEN REWRITE_TAC[open_in; IN_ELIM_THM] THEN CONJ_TAC THENL [REWRITE_TAC[AFFINE_HULL_FINITE; SUBSET; IN_ELIM_THM] THEN GEN_TAC THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[REAL_LT_IMP_LE]; ALL_TAC] THEN X_GEN_TAC `y:real^N` THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) THEN ABBREV_TAC `e = inf (IMAGE (\x:real^N. min (&1 - u x) (u x)) s)` THEN SUBGOAL_THEN `&0 < e` ASSUME_TAC THENL [EXPAND_TAC "e" THEN ASM_SIMP_TAC[REAL_LT_INF_FINITE; FINITE_IMAGE; IMAGE_EQ_EMPTY] THEN ASM_SIMP_TAC[REAL_LT_MIN; REAL_SUB_LT; FORALL_IN_IMAGE]; ALL_TAC] THEN MP_TAC(ISPEC `IMAGE (\z:real^N. z - y) (affine hull s)` BASIS_EXISTS) THEN REWRITE_TAC[SUBSET_IMAGE] THEN DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` (CONJUNCTS_THEN2 (X_CHOOSE_THEN `c:real^N->bool` (STRIP_ASSUME_TAC o GSYM)) MP_TAC)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; HAS_SIZE] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN ASM_SIMP_TAC[SPAN_FINITE; IN_ELIM_THM] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM] THEN DISCH_THEN(X_CHOOSE_TAC `compo:real^N->real^N->real`) THEN FIRST_ASSUM(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC o MATCH_MP BASIS_COORDINATES_LIPSCHITZ) THEN SUBGOAL_THEN `!i. i IN b ==> ?u. sum s u = &0 /\ vsum s (\x:real^N. u x % x) = i` MP_TAC THENL [EXPAND_TAC "b" THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN SUBGOAL_THEN `(x:real^N) IN affine hull s` MP_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[AFFINE_HULL_FINITE; IN_ELIM_THM]] THEN DISCH_THEN(X_CHOOSE_THEN `v:real^N->real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(\x. v x - u x):real^N->real` THEN ASM_SIMP_TAC[SUM_SUB; VSUM_SUB; VECTOR_SUB_RDISTRIB] THEN REWRITE_TAC[REAL_SUB_REFL; VECTOR_SUB_RZERO]; GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM; SKOLEM_THM; FORALL_AND_THM; TAUT `(a ==> b /\ c) <=> (a ==> b) /\ (a ==> c)`] THEN DISCH_THEN(X_CHOOSE_THEN `w:real^N->real^N->real` STRIP_ASSUME_TAC)] THEN EXISTS_TAC `e / B / (&1 + sum (b:real^N->bool) (\i. abs(sup(IMAGE (abs o w i) (s:real^N->bool)))))` THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_ARITH `&0 <= x ==> &0 < &1 + x`; SUM_POS_LE; REAL_ABS_POS] THEN X_GEN_TAC `z:real^N` THEN STRIP_TAC THEN EXISTS_TAC `\x:real^N. u x + sum (b:real^N->bool) (\i. compo (z:real^N) i * w i x)` THEN REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [ALL_TAC; ASM_SIMP_TAC[SUM_ADD; REAL_ARITH `&1 + x = &1 <=> x = &0`] THEN W(MP_TAC o PART_MATCH (lhs o rand) SUM_SWAP o lhand o snd) THEN ASM_REWRITE_TAC[FINITE_NUMSEG] THEN DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC SUM_EQ_0 THEN ASM_SIMP_TAC[SUM_LMUL; ETA_AX; REAL_MUL_RZERO; SUM_0]; ASM_SIMP_TAC[VSUM_ADD; VECTOR_ADD_RDISTRIB] THEN ONCE_REWRITE_TAC[VECTOR_ARITH `y + w:real^N = z <=> w = z - y`] THEN ASM_SIMP_TAC[GSYM VSUM_LMUL; GSYM VSUM_RMUL; GSYM VECTOR_MUL_ASSOC] THEN W(MP_TAC o PART_MATCH (lhs o rand) VSUM_SWAP o lhand o snd) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN ASM_SIMP_TAC[VSUM_LMUL] THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `vsum b (\v:real^N. compo (z:real^N) v % v)` THEN CONJ_TAC THENL [ALL_TAC; ASM_SIMP_TAC[]] THEN MATCH_MP_TAC VSUM_EQ THEN ASM_SIMP_TAC[]] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[] THEN MATCH_MP_TAC(REAL_ARITH `abs(x) < min u (&1 - u) ==> &0 < u + x /\ u + x < &1`) THEN MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `B * norm(z - y:real^N) * sum (b:real^N->bool) (\i. abs(sup(IMAGE (abs o w i) (s:real^N->bool))))` THEN CONJ_TAC THENL [REWRITE_TAC[GSYM SUM_LMUL] THEN MATCH_MP_TAC SUM_ABS_LE THEN ASM_REWRITE_TAC[REAL_ABS_MUL; REAL_MUL_ASSOC] THEN X_GEN_TAC `i:real^N` THEN STRIP_TAC THEN MATCH_MP_TAC REAL_LE_MUL2 THEN REWRITE_TAC[REAL_ABS_POS] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPECL [`(compo:real^N->real^N->real) z`; `i:real^N`]) THEN ASM_SIMP_TAC[]; MATCH_MP_TAC(REAL_ARITH `x <= a ==> x <= abs a`) THEN ASM_SIMP_TAC[REAL_LE_SUP_FINITE; FINITE_IMAGE; IMAGE_EQ_EMPTY] THEN REWRITE_TAC[EXISTS_IN_IMAGE; o_THM] THEN ASM_MESON_TAC[REAL_LE_REFL]]; ALL_TAC] THEN GEN_REWRITE_TAC LAND_CONV [REAL_MUL_SYM] THEN ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ] THEN MATCH_MP_TAC(REAL_ARITH `&0 <= x /\ x * (&1 + e) < d ==> x * e < d`) THEN REWRITE_TAC[NORM_POS_LE] THEN ASM_SIMP_TAC[NORM_POS_LE; GSYM REAL_LT_RDIV_EQ; REAL_ARITH `&0 <= x ==> &0 < &1 + x`; SUM_POS_LE; REAL_ABS_POS] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (NORM_ARITH `dist(z:real^N,y) < k ==> k <= d ==> norm(z - y) < d`)) THEN ASM_SIMP_TAC[REAL_LE_DIV2_EQ; REAL_ARITH `&0 <= x ==> &0 < &1 + x`; SUM_POS_LE; REAL_ABS_POS] THEN EXPAND_TAC "e" THEN ASM_SIMP_TAC[REAL_INF_LE_FINITE; FINITE_IMAGE; IMAGE_EQ_EMPTY] THEN REWRITE_TAC[EXISTS_IN_IMAGE] THEN EXISTS_TAC `x:real^N` THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);; let EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL_MINIMAL = prove (`!s. FINITE s ==> {y:real^N | ?u. (!x. x IN s ==> &0 < u x) /\ sum s u = &1 /\ vsum s (\x. u x % x) = y} SUBSET relative_interior(convex hull s)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[SUM_CLAUSES; REAL_OF_NUM_EQ; ARITH_EQ] THEN REWRITE_TAC[EMPTY_GSPEC; EMPTY_SUBSET] 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 ASM_CASES_TAC `s = {a:real^N}` THENL [ASM_REWRITE_TAC[SUM_SING; VSUM_SING; FORALL_IN_INSERT; NOT_IN_EMPTY] THEN REWRITE_TAC[RELATIVE_INTERIOR_SING; CONVEX_HULL_SING] THEN REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_SING] THEN MESON_TAC[VECTOR_MUL_LID]; FIRST_ASSUM(MP_TAC o MATCH_MP EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] SUBSET_TRANS) THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `w:real^N` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^N->real` THEN STRIP_TAC THEN ASM_SIMP_TAC[] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN SUBGOAL_THEN `?y:real^N. y IN s /\ ~(y = x)` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `sum {x,y} u <= sum s (u:real^N->real)` MP_TAC THENL [MATCH_MP_TAC SUM_SUBSET_SIMPLE THEN ASM_SIMP_TAC[AFFINE_INDEPENDENT_IMP_FINITE; REAL_LT_IMP_LE; IN_DIFF] THEN ASM SET_TAC[]; ASM_SIMP_TAC[SUM_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN MATCH_MP_TAC(REAL_ARITH `&0 < y ==> x + y + &0 <= &1 ==> x < &1`) THEN ASM_SIMP_TAC[]]]);; let RELATIVE_INTERIOR_CONVEX_HULL_EXPLICIT = prove (`!s. ~(affine_dependent s) ==> relative_interior(convex hull s) = {y:real^N | ?u. (!x. x IN s ==> &0 < u x) /\ sum s u = &1 /\ vsum s (\x. u x % x) = y}`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN MATCH_MP_TAC SUBSET_ANTISYM THEN ASM_SIMP_TAC[EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL_MINIMAL] THEN ASM_CASES_TAC `?a:real^N. s = {a}` THENL [FIRST_X_ASSUM(CHOOSE_THEN SUBST1_TAC) THEN ASM_REWRITE_TAC[SUM_SING; VSUM_SING; CONVEX_HULL_SING; RELATIVE_INTERIOR_SING] THEN REWRITE_TAC[IN_ELIM_THM; SUBSET; IN_SING] THEN REPEAT STRIP_TAC THEN EXISTS_TAC `\x:real^N. &1` THEN ASM_REWRITE_TAC[VECTOR_MUL_LID; REAL_LT_01]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `relative_interior s SUBSET s /\ (!x. x IN s /\ ~(x IN t) ==> ~(x IN relative_interior s)) ==> relative_interior s SUBSET t`) THEN REWRITE_TAC[RELATIVE_INTERIOR_SUBSET] THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_RELATIVE_INTERIOR] THEN REWRITE_TAC[AFFINE_HULL_CONVEX_HULL; IN_ELIM_THM; NOT_EXISTS_THM] THEN REWRITE_TAC[CONVEX_HULL_FINITE; IN_ELIM_THM] THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) (MP_TAC o SPEC `u:real^N->real`)) THEN ASM_REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; IN_RELATIVE_INTERIOR; DE_MORGAN_THM; SUBSET; IN_ELIM_THM; IN_BALL; IN_INTER] THEN DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN DISJ2_TAC THEN DISCH_THEN(X_CHOOSE_THEN `e:real` (CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "*"))) THEN SUBGOAL_THEN `(u:real^N->real) a = &0` ASSUME_TAC THENL [ASM_SIMP_TAC[REAL_ARITH `&0 <= x /\ ~(&0 < x) ==> x = &0`]; ALL_TAC] THEN SUBGOAL_THEN `?b:real^N. b IN s /\ ~(b = a)` STRIP_ASSUME_TAC THENL [ASM SET_TAC[];ALL_TAC] THEN SUBGOAL_THEN `?d. &0 < d /\ norm(d % (a - b):real^N) < e` STRIP_ASSUME_TAC THENL [EXISTS_TAC `e / &2 / norm(a - b:real^N)` THEN ASM_SIMP_TAC[NORM_MUL; REAL_LT_DIV; REAL_OF_NUM_LT; ARITH; NORM_POS_LT; REAL_ABS_DIV; REAL_ABS_NORM; REAL_ABS_NUM; REAL_DIV_RMUL; REAL_LT_IMP_NZ; VECTOR_SUB_EQ] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN REMOVE_THEN "*" (MP_TAC o SPEC `y - d % (a - b):real^N`) THEN ASM_REWRITE_TAC[NORM_ARITH `dist(a:real^N,a - b) = norm b`] THEN REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [MATCH_MP_TAC IN_AFFINE_SUB_MUL_DIFF THEN ASM_SIMP_TAC[HULL_INC; AFFINE_AFFINE_HULL] THEN REWRITE_TAC[AFFINE_HULL_FINITE; IN_ELIM_THM] THEN EXISTS_TAC `u:real^N->real` THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `v:real^N->real` STRIP_ASSUME_TAC) THEN UNDISCH_TAC `~(affine_dependent(s:real^N->bool))` THEN ASM_SIMP_TAC[AFFINE_DEPENDENT_EXPLICIT_FINITE] THEN EXISTS_TAC `\x:real^N. (v x - u x) - (if x = a then --d else if x = b then d else &0)` THEN REWRITE_TAC[VECTOR_SUB_RDISTRIB; MESON[] `(if p then a else b) % x = (if p then a % x else b % x)`] THEN ASM_SIMP_TAC[SUM_SUB; VSUM_SUB] THEN ASM_SIMP_TAC[VSUM_CASES; SUM_CASES; FINITE_RESTRICT; IN_ELIM_THM] THEN ASM_SIMP_TAC[SET_RULE `a IN s ==> {x | x IN s /\ x = a} = {a}`; SET_RULE `b IN s /\ ~(b = a) ==> {x | (x IN s /\ ~(x = a)) /\ x = b} = {b}`] THEN ASM_SIMP_TAC[VECTOR_MUL_LZERO; SUM_0; VSUM_0; SUM_SING; VSUM_SING] THEN CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN CONJ_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN EXISTS_TAC `a:real^N` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o SPEC `a:real^N`) THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC);; let EXPLICIT_SUBSET_INTERIOR_CONVEX_HULL = prove (`!s. FINITE s /\ affine hull s = (:real^N) ==> {y | ?u. (!x. x IN s ==> &0 < u x /\ u x < &1) /\ sum s u = &1 /\ vsum s (\x. u x % x) = y} SUBSET interior(convex hull s)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL) THEN ASM_SIMP_TAC[RELATIVE_INTERIOR_INTERIOR; AFFINE_HULL_CONVEX_HULL]);; let EXPLICIT_SUBSET_INTERIOR_CONVEX_HULL_MINIMAL = prove (`!s. FINITE s /\ affine hull s = (:real^N) ==> {y | ?u. (!x. x IN s ==> &0 < u x) /\ sum s u = &1 /\ vsum s (\x. u x % x) = y} SUBSET interior(convex hull s)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL_MINIMAL) THEN ASM_SIMP_TAC[RELATIVE_INTERIOR_INTERIOR; AFFINE_HULL_CONVEX_HULL]);; let INTERIOR_CONVEX_HULL_EXPLICIT_MINIMAL = prove (`!s:real^N->bool. ~(affine_dependent s) ==> interior(convex hull s) = if CARD(s) <= dimindex(:N) then {} else {y | ?u. (!x. x IN s ==> &0 < u x) /\ sum s u = &1 /\ vsum s (\x. u x % x) = y}`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN COND_CASES_TAC THEN ASM_SIMP_TAC[EMPTY_INTERIOR_CONVEX_HULL] THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `relative_interior(convex hull s):real^N->bool` THEN CONJ_TAC THENL [CONV_TAC SYM_CONV THEN MATCH_MP_TAC RELATIVE_INTERIOR_INTERIOR THEN REWRITE_TAC[AFFINE_HULL_CONVEX_HULL] THEN MATCH_MP_TAC AFFINE_INDEPENDENT_SPAN_GT THEN ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC; ASM_SIMP_TAC[RELATIVE_INTERIOR_CONVEX_HULL_EXPLICIT]]);; let INTERIOR_CONVEX_HULL_EXPLICIT = prove (`!s:real^N->bool. ~(affine_dependent s) ==> interior(convex hull s) = if CARD(s) <= dimindex(:N) then {} else {y | ?u. (!x. x IN s ==> &0 < u x /\ u x < &1) /\ sum s u = &1 /\ vsum s (\x. u x % x) = y}`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[INTERIOR_CONVEX_HULL_EXPLICIT_MINIMAL] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `v:real^N` THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `u:real^N->real` THEN EQ_TAC THEN STRIP_TAC THEN ASM_SIMP_TAC[] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MP_TAC(ISPEC `s:real^N->bool` CHOOSE_SUBSET) THEN ASM_SIMP_TAC[AFFINE_INDEPENDENT_IMP_FINITE] THEN DISCH_THEN(MP_TAC o SPEC `2`) THEN ANTS_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ARITH_RULE `~(c <= n) ==> 1 <= n ==> 2 <= c`)) THEN REWRITE_TAC[DIMINDEX_GE_1]; ALL_TAC] THEN CONV_TAC(ONCE_DEPTH_CONV HAS_SIZE_CONV) THEN REWRITE_TAC[SUBSET] THEN DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(X_CHOOSE_THEN `a:real^N` (X_CHOOSE_THEN `b:real^N` STRIP_ASSUME_TAC)) THEN SUBGOAL_THEN `?y:real^N. y IN s /\ ~(y = x)` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `sum {x,y} u <= sum s (u:real^N->real)` MP_TAC THENL [MATCH_MP_TAC SUM_SUBSET_SIMPLE THEN ASM_SIMP_TAC[AFFINE_INDEPENDENT_IMP_FINITE; REAL_LT_IMP_LE; IN_DIFF] THEN ASM SET_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[SUM_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN MATCH_MP_TAC(REAL_ARITH `&0 < y ==> x + y + &0 <= &1 ==> x < &1`) THEN ASM_SIMP_TAC[]);; let DISJOINT_RELATIVE_INTERIOR_CONVEX_HULL = prove (`!s:real^N->bool. ~affine_dependent s /\ ~(?a. s = {a}) ==> relative_interior(convex hull s) INTER s = {}`, REWRITE_TAC[SET_RULE `s INTER t = {} <=> !x. x IN t ==> ~(x IN s)`] THEN GEN_TAC THEN REWRITE_TAC[RIGHT_IMP_FORALL_THM] THEN X_GEN_TAC `z:real^N` THEN GEOM_ORIGIN_TAC `z:real^N` THEN REPEAT GEN_TAC THEN STRIP_TAC THEN STRIP_TAC THEN ASM_SIMP_TAC[RELATIVE_INTERIOR_CONVEX_HULL_EXPLICIT; IN_ELIM_THM] THEN FIRST_X_ASSUM(fun th -> STRIP_ASSUME_TAC(GEN_REWRITE_RULE I [AFFINE_INDEPENDENT_IFF_CARD] th) THEN MP_TAC th) THEN ASM_SIMP_TAC[AFFINE_DEPENDENT_EXPLICIT_FINITE; CONTRAPOS_THM] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\x. (if x = vec 0 then -- &1 else &0) + (u:real^N->real) x` THEN ASM_SIMP_TAC[VECTOR_ADD_RDISTRIB; VSUM_ADD; SUM_ADD] THEN REWRITE_TAC[COND_RAND; COND_RATOR; VECTOR_MUL_LZERO] THEN ASM_SIMP_TAC[SUM_DELTA; VSUM_DELTA; VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN REWRITE_TAC[REAL_ADD_LINV; REAL_ADD_LID] THEN MATCH_MP_TAC(MESON[] `(?x. ~(x = a) /\ q x) ==> ?x. if x = a then p x else q x`) THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE `~(?a. s = {a}) ==> vec 0 IN s ==> ?b. ~(b = vec 0) /\ b IN s`)) THEN ASM_MESON_TAC[REAL_LT_IMP_NZ]);; let INTERIOR_CONVEX_HULL_3_MINIMAL = prove (`!a b c:real^2. ~collinear{a,b,c} ==> interior(convex hull {a,b,c}) = {v | ?x y z. &0 < x /\ &0 < y /\ &0 < z /\ x + y + z = &1 /\ x % a + y % b + z % c = v}`, REWRITE_TAC[COLLINEAR_3_EQ_AFFINE_DEPENDENT; DE_MORGAN_THM] THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[INTERIOR_CONVEX_HULL_EXPLICIT_MINIMAL] THEN ASM_SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN CONV_TAC(LAND_CONV(RATOR_CONV(LAND_CONV(ONCE_DEPTH_CONV(REWRITE_CONV [IN_INSERT; NOT_IN_EMPTY]))))) THEN ASM_REWRITE_TAC[DIMINDEX_2; ARITH] THEN SIMP_TAC[FINITE_INSERT; FINITE_UNION; FINITE_EMPTY; RIGHT_EXISTS_AND_THM; AFFINE_HULL_FINITE_STEP_GEN; REAL_LT_ADD; REAL_HALF] THEN REWRITE_TAC[REAL_ARITH `&1 - a - b - c = &0 <=> a + b + c = &1`; VECTOR_ARITH `y - a - b - c:real^N = vec 0 <=> a + b + c = y`]);; let INTERIOR_CONVEX_HULL_3 = prove (`!a b c:real^2. ~collinear{a,b,c} ==> interior(convex hull {a,b,c}) = {v | ?x y z. &0 < x /\ x < &1 /\ &0 < y /\ y < &1 /\ &0 < z /\ z < &1 /\ x + y + z = &1 /\ x % a + y % b + z % c = v}`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[INTERIOR_CONVEX_HULL_3_MINIMAL] THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN GEN_TAC THEN REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC);; (* ------------------------------------------------------------------------- *) (* Similar results for closure and (relative or absolute) frontier. *) (* ------------------------------------------------------------------------- *) let CLOSURE_CONVEX_HULL = prove (`!s. compact s ==> closure(convex hull s) = convex hull s`, SIMP_TAC[CLOSURE_CLOSED; COMPACT_IMP_CLOSED; COMPACT_CONVEX_HULL]);; let RELATIVE_FRONTIER_CONVEX_HULL_EXPLICIT = prove (`!s:real^N->bool. ~(affine_dependent s) ==> relative_frontier(convex hull s) = {y | ?u. (!x. x IN s ==> &0 <= u x) /\ (?x. x IN s /\ u x = &0) /\ sum s u = &1 /\ vsum s (\x. u x % x) = y}`, REPEAT STRIP_TAC THEN REWRITE_TAC[relative_frontier; UNIONS_GSPEC] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN ASM_SIMP_TAC[CLOSURE_CONVEX_HULL; FINITE_IMP_COMPACT] THEN ASM_SIMP_TAC[CONVEX_HULL_FINITE; RELATIVE_INTERIOR_CONVEX_HULL_EXPLICIT] THEN GEN_REWRITE_TAC I [EXTENSION] THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_DIFF; IN_ELIM_THM] THEN EQ_TAC THENL [DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN DISCH_THEN(MP_TAC o SPEC `u:real^N->real`) THEN ASM_REWRITE_TAC[NOT_FORALL_THM; NOT_IMP] THEN DISCH_THEN(CHOOSE_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN ASM_SIMP_TAC[REAL_ARITH `&0 <= x ==> (~(&0 < x) <=> x = &0)`] THEN DISCH_TAC THEN EXISTS_TAC `u:real^N->real` THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]; DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` (REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC)) THEN CONJ_TAC THENL [EXISTS_TAC `u:real^N->real` THEN ASM_SIMP_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `v:real^N->real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [AFFINE_DEPENDENT_EXPLICIT]) THEN REWRITE_TAC[] THEN MAP_EVERY EXISTS_TAC [`s:real^N->bool`; `(\x. u x - v x):real^N->real`] THEN ASM_SIMP_TAC[SUBSET_REFL; VECTOR_SUB_RDISTRIB; SUM_SUB; VSUM_SUB] THEN REWRITE_TAC[REAL_SUB_0; VECTOR_SUB_EQ] THEN ASM_MESON_TAC[REAL_LT_REFL]]);; let FRONTIER_CONVEX_HULL_EXPLICIT = prove (`!s:real^N->bool. ~(affine_dependent s) ==> frontier(convex hull s) = {y | ?u. (!x. x IN s ==> &0 <= u x) /\ (dimindex(:N) < CARD s ==> ?x. x IN s /\ u x = &0) /\ sum s u = &1 /\ vsum s (\x. u x % x) = y}`, REPEAT STRIP_TAC THEN REWRITE_TAC[frontier] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN DISJ_CASES_TAC (ARITH_RULE `CARD(s:real^N->bool) <= dimindex(:N) \/ dimindex(:N) < CARD(s:real^N->bool)`) THENL [ASM_SIMP_TAC[GSYM NOT_LE; INTERIOR_CONVEX_HULL_EXPLICIT] THEN ASM_SIMP_TAC[CLOSURE_CONVEX_HULL; FINITE_IMP_COMPACT; DIFF_EMPTY] THEN REWRITE_TAC[CONVEX_HULL_FINITE]; ASM_SIMP_TAC[GSYM RELATIVE_FRONTIER_CONVEX_HULL_EXPLICIT] THEN REWRITE_TAC[relative_frontier] THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC RELATIVE_INTERIOR_INTERIOR THEN MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ s = UNIV ==> t = UNIV`) THEN EXISTS_TAC `affine hull s:real^N->bool` THEN ASM_SIMP_TAC[AFFINE_INDEPENDENT_SPAN_GT; HULL_MONO; HULL_SUBSET]]);; let RELATIVE_FRONTIER_OF_CONVEX_HULL = prove (`!s:real^N->bool. ~(affine_dependent s) ==> relative_frontier(convex hull s) = UNIONS { convex hull (s DELETE a) |a| a IN s }`, REPEAT STRIP_TAC THEN REWRITE_TAC[UNIONS_GSPEC] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN ASM_SIMP_TAC[RELATIVE_FRONTIER_CONVEX_HULL_EXPLICIT] THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM; CONVEX_HULL_FINITE] THEN X_GEN_TAC `y:real^N` THEN EQ_TAC THENL [DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `u:real^N->real` THEN ASM_SIMP_TAC[IN_DELETE; SUM_DELETE; VSUM_DELETE; REAL_SUB_RZERO] THEN VECTOR_ARITH_TAC; REWRITE_TAC[IN_DELETE] THEN DISCH_THEN(X_CHOOSE_THEN `a:real^N` (CONJUNCTS_THEN2 ASSUME_TAC (X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC))) THEN EXISTS_TAC `(\x. if x = a then &0 else u x):real^N->real` THEN ASM_SIMP_TAC[COND_RAND; COND_RATOR; REAL_LE_REFL; COND_ID] THEN CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[SUM_CASES; VSUM_CASES; VECTOR_MUL_LZERO] THEN ASM_SIMP_TAC[GSYM DELETE; SUM_0; VSUM_0; REAL_ADD_LID; VECTOR_ADD_LID]]);; let FRONTIER_CONVEX_HULL_CASES = prove (`!s:real^N->bool. ~(affine_dependent s) ==> frontier(convex hull s) = if CARD(s) <= dimindex(:N) then convex hull s else UNIONS { convex hull (s DELETE a) |a| a IN s }`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN ASM_SIMP_TAC[frontier; CLOSURE_CONVEX_HULL; FINITE_IMP_COMPACT] THEN COND_CASES_TAC THENL [ASM_SIMP_TAC[INTERIOR_CONVEX_HULL_EXPLICIT; DIFF_EMPTY]; ALL_TAC] THEN ASM_SIMP_TAC[GSYM RELATIVE_FRONTIER_OF_CONVEX_HULL] THEN ASM_SIMP_TAC[relative_frontier; frontier; CLOSURE_CONVEX_HULL; FINITE_IMP_COMPACT] THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN RULE_ASSUM_TAC(REWRITE_RULE[NOT_LE]) THEN MATCH_MP_TAC RELATIVE_INTERIOR_INTERIOR THEN MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ s = UNIV ==> t = UNIV`) THEN EXISTS_TAC `affine hull s:real^N->bool` THEN ASM_SIMP_TAC[AFFINE_INDEPENDENT_SPAN_GT; HULL_MONO; HULL_SUBSET]);; let IN_FRONTIER_CONVEX_HULL = prove (`!s x:real^N. FINITE s /\ CARD s <= dimindex(:N) + 1 /\ x IN s ==> x IN frontier(convex hull s)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `affine_dependent(s:real^N->bool)` THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [affine_dependent]) THEN DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN ASM_SIMP_TAC[frontier; CLOSURE_CONVEX_HULL; FINITE_IMP_COMPACT] THEN ASM_SIMP_TAC[HULL_INC; IN_DIFF] THEN MATCH_MP_TAC(SET_RULE `!t. s SUBSET t /\ t = {} ==> ~(x IN s)`) THEN EXISTS_TAC `interior(affine hull s):real^N->bool` THEN SIMP_TAC[SUBSET_INTERIOR; CONVEX_HULL_SUBSET_AFFINE_HULL] THEN SUBGOAL_THEN `s = (a:real^N) INSERT (s DELETE a)` SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[HULL_REDUNDANT] THEN MATCH_MP_TAC EMPTY_INTERIOR_AFFINE_HULL THEN ASM_SIMP_TAC[FINITE_DELETE; CARD_DELETE] THEN ASM_ARITH_TAC; ASM_SIMP_TAC[FRONTIER_CONVEX_HULL_CASES] THEN COND_CASES_TAC THEN ASM_SIMP_TAC[HULL_INC] THEN REWRITE_TAC[UNIONS_GSPEC; IN_ELIM_THM] THEN SUBGOAL_THEN `?y:real^N. y IN s /\ ~(y = x)` MP_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `x IN s ==> ~(s = {x}) ==> ?y. y IN s /\ ~(y = x)`)) THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_LE]) THEN ASM_SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN REWRITE_TAC[NOT_LT; NOT_IN_EMPTY; ARITH_SUC; DIMINDEX_GE_1]; MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC HULL_INC THEN ASM SET_TAC[]]]);; let NOT_IN_INTERIOR_CONVEX_HULL = prove (`!s x:real^N. FINITE s /\ CARD s <= dimindex(:N) + 1 /\ x IN s ==> ~(x IN interior(convex hull s))`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP IN_FRONTIER_CONVEX_HULL) THEN SIMP_TAC[frontier; IN_DIFF]);; let INTERIOR_CONVEX_HULL_EQ_EMPTY = prove (`!s:real^N->bool. s HAS_SIZE (dimindex(:N) + 1) ==> (interior(convex hull s) = {} <=> affine_dependent s)`, REPEAT GEN_TAC THEN REWRITE_TAC[HAS_SIZE] THEN STRIP_TAC THEN ASM_CASES_TAC `affine_dependent(s:real^N->bool)` THENL [ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [affine_dependent]) THEN DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN ASM_SIMP_TAC[frontier; CLOSURE_CONVEX_HULL; FINITE_IMP_COMPACT] THEN ASM_SIMP_TAC[HULL_INC; IN_DIFF] THEN MATCH_MP_TAC(SET_RULE `!t. s SUBSET t /\ t = {} ==> s = {}`) THEN EXISTS_TAC `interior(affine hull s):real^N->bool` THEN SIMP_TAC[SUBSET_INTERIOR; CONVEX_HULL_SUBSET_AFFINE_HULL] THEN SUBGOAL_THEN `s = (a:real^N) INSERT (s DELETE a)` SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[HULL_REDUNDANT] THEN MATCH_MP_TAC EMPTY_INTERIOR_AFFINE_HULL THEN ASM_SIMP_TAC[FINITE_DELETE; CARD_DELETE] THEN ASM_ARITH_TAC; ASM_SIMP_TAC[INTERIOR_CONVEX_HULL_EXPLICIT_MINIMAL] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; ARITH_RULE `~(n + 1 <= n)`] THEN EXISTS_TAC `vsum s (\x:real^N. inv(&(dimindex(:N)) + &1) % x)` THEN REWRITE_TAC[IN_ELIM_THM] THEN EXISTS_TAC `\x:real^N. inv(&(dimindex(:N)) + &1)` THEN ASM_SIMP_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN ASM_SIMP_TAC[SUM_CONST; GSYM REAL_OF_NUM_ADD] THEN CONV_TAC REAL_FIELD]);; (* ------------------------------------------------------------------------- *) (* Similar things in special case (could use above as lemmas here instead). *) (* ------------------------------------------------------------------------- *) let SIMPLEX_EXPLICIT = prove (`!s:real^N->bool. FINITE s /\ ~(vec 0 IN s) ==> convex hull (vec 0 INSERT s) = { y | ?u. (!x. x IN s ==> &0 <= u x) /\ sum s u <= &1 /\ vsum s (\x. u x % x) = y}`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[CONVEX_HULL_FINITE; FINITE_INSERT] THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `y:real^N` THEN ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; IN_INSERT] THEN REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN EQ_TAC THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) THENL [EXISTS_TAC `u:real^N->real` THEN ASM_SIMP_TAC[REAL_LE_REFL] THEN FIRST_X_ASSUM(MP_TAC o SPEC `vec 0:real^N`) THEN REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC; EXISTS_TAC `\x:real^N. if x = vec 0 then &1 - sum (s:real^N->bool) u else u(x)` THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `x:real^N = vec 0` THEN ASM_REWRITE_TAC[REAL_SUB_LE]; MATCH_MP_TAC(REAL_ARITH `s = t ==> &1 - s + t = &1`) THEN MATCH_MP_TAC SUM_EQ THEN ASM_MESON_TAC[]; FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN MATCH_MP_TAC VSUM_EQ THEN ASM_MESON_TAC[]]]);; let STD_SIMPLEX = prove (`convex hull (vec 0 INSERT { basis i | 1 <= i /\ i <= dimindex(:N)}) = {x:real^N | (!i. 1 <= i /\ i <= dimindex(:N) ==> &0 <= x$i) /\ sum (1..dimindex(:N)) (\i. x$i) <= &1 }`, W(MP_TAC o PART_MATCH (lhs o rand) SIMPLEX_EXPLICIT o lhs o snd) THEN ANTS_TAC THENL [REWRITE_TAC[SIMPLE_IMAGE; GSYM IN_NUMSEG] THEN SIMP_TAC[FINITE_IMAGE; FINITE_NUMSEG; IN_IMAGE] THEN REWRITE_TAC[IN_NUMSEG] THEN MESON_TAC[BASIS_NONZERO]; ALL_TAC] THEN DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[EXTENSION] THEN ONCE_REWRITE_TAC[IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[SIMPLE_IMAGE; GSYM IN_NUMSEG] THEN SUBGOAL_THEN `!u. sum (IMAGE (basis:num->real^N) (1..dimindex(:N))) u = sum (1..dimindex(:N)) (u o basis)` (fun th -> REWRITE_TAC[th]) THENL [GEN_TAC THEN MATCH_MP_TAC SUM_IMAGE THEN REWRITE_TAC[IN_NUMSEG] THEN REWRITE_TAC[GSYM CONJ_ASSOC; BASIS_INJ]; ALL_TAC] THEN SUBGOAL_THEN `!u. vsum (IMAGE (basis:num->real^N) (1..dimindex(:N))) u = vsum (1..dimindex(:N)) ((u:real^N->real^N) o basis)` (fun th -> REWRITE_TAC[th]) THENL [GEN_TAC THEN MATCH_MP_TAC VSUM_IMAGE THEN REWRITE_TAC[IN_NUMSEG] THEN REWRITE_TAC[GSYM CONJ_ASSOC; BASIS_INJ; FINITE_NUMSEG]; ALL_TAC] THEN REWRITE_TAC[o_DEF; BASIS_EXPANSION_UNIQUE; FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_NUMSEG] THEN EQ_TAC THENL [DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) THEN CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `x <= &1 ==> x = y ==> y <= &1`)) THEN MATCH_MP_TAC SUM_EQ THEN ASM_SIMP_TAC[IN_NUMSEG]; STRIP_TAC THEN EXISTS_TAC `\y:real^N. y dot x` THEN ASM_SIMP_TAC[DOT_BASIS]]);; let INTERIOR_STD_SIMPLEX = prove (`interior (convex hull (vec 0 INSERT { basis i | 1 <= i /\ i <= dimindex(:N)})) = {x:real^N | (!i. 1 <= i /\ i <= dimindex(:N) ==> &0 < x$i) /\ sum (1..dimindex(:N)) (\i. x$i) < &1 }`, REWRITE_TAC[EXTENSION; IN_INTERIOR; IN_ELIM_THM; STD_SIMPLEX] THEN REWRITE_TAC[SUBSET; IN_BALL; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN EQ_TAC THENL [DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o SPEC `x:real^N`) THEN REWRITE_TAC[DIST_REFL] THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ASM_SIMP_TAC[REAL_LT_LE] THEN CONJ_TAC THENL [X_GEN_TAC `k:num` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x - (e / &2) % basis k:real^N`) THEN REWRITE_TAC[NORM_ARITH `dist(x,x - e) = norm(e)`; NORM_MUL] THEN ASM_SIMP_TAC[NORM_BASIS; REAL_ARITH `&0 < e ==> abs(e / &2) * &1 < e`; VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT] THEN DISCH_THEN(MP_TAC o SPEC `k:num` o CONJUNCT1) THEN ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[BASIS_COMPONENT] THEN UNDISCH_TAC `&0 < e` THEN REAL_ARITH_TAC; FIRST_X_ASSUM(MP_TAC o SPEC `x + (e / &2) % basis 1:real^N`) THEN REWRITE_TAC[NORM_ARITH `dist(x,x + e) = norm(e)`; NORM_MUL] THEN ASM_SIMP_TAC[NORM_BASIS; LE_REFL; DIMINDEX_GE_1] THEN ASM_SIMP_TAC[REAL_ARITH `&0 < e ==> abs(e / &2) * &1 < e`] THEN DISCH_THEN(MP_TAC o CONJUNCT2) THEN MATCH_MP_TAC(REAL_ARITH `x < y ==> y <= &1 ==> ~(x = &1)`) THEN MATCH_MP_TAC SUM_LT THEN REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN ONCE_REWRITE_TAC[TAUT `(a /\ b) /\ c <=> ~(a /\ b ==> ~c)`] THEN SIMP_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; BASIS_COMPONENT] THEN CONJ_TAC THENL [GEN_TAC THEN COND_CASES_TAC; EXISTS_TAC `1` THEN REWRITE_TAC[LE_REFL; DIMINDEX_GE_1]] THEN ASM_REAL_ARITH_TAC]; STRIP_TAC THEN EXISTS_TAC `min (inf(IMAGE (\i. (x:real^N)$i) (1..dimindex(:N)))) ((&1 - sum (1..dimindex(:N)) (\i. x$i)) / &(dimindex(:N)))` THEN ASM_SIMP_TAC[REAL_LT_MIN] THEN SIMP_TAC[REAL_LT_INF_FINITE; FINITE_IMAGE; FINITE_NUMSEG; IMAGE_EQ_EMPTY; NUMSEG_EMPTY; GSYM NOT_LE; DIMINDEX_GE_1] THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN ASM_SIMP_TAC[REAL_LT_RDIV_EQ; REAL_OF_NUM_LT; ARITH_RULE `0 < x <=> 1 <= x`; DIMINDEX_GE_1] THEN ASM_REWRITE_TAC[IN_NUMSEG; REAL_MUL_LZERO; REAL_SUB_LT] THEN REPEAT(POP_ASSUM(K ALL_TAC)) THEN X_GEN_TAC `y:real^N` THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `k:num` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REAL_ARITH `abs(xk - yk) <= d ==> d < xk ==> &0 <= yk`); GEN_REWRITE_TAC (LAND_CONV o LAND_CONV o RAND_CONV o RAND_CONV) [GSYM CARD_NUMSEG_1] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN SIMP_TAC[GSYM SUM_CONST; FINITE_NUMSEG] THEN MATCH_MP_TAC(REAL_ARITH `s2 <= s0 + s1 ==> s0 < &1 - s1 ==> s2 <= &1`) THEN REWRITE_TAC[GSYM SUM_ADD_NUMSEG] THEN MATCH_MP_TAC SUM_LE_NUMSEG THEN REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC(REAL_ARITH `abs(y - x) <= z ==> x <= z + y`)] THEN ASM_SIMP_TAC[GSYM VECTOR_SUB_COMPONENT; dist; COMPONENT_LE_NORM]]);; (* ------------------------------------------------------------------------- *) (* Barycentres. *) (* ------------------------------------------------------------------------- *) let barycentre = new_definition `barycentre s = if FINITE s then vsum s (\x. inv(&(CARD s)) % x) else vec 0`;; let BARYCENTRE_0 = prove (`barycentre {} = vec 0`, REWRITE_TAC[barycentre; FINITE_EMPTY; VSUM_CLAUSES]);; let BARYCENTRE_1 = prove (`!a:real^N. barycentre {a} = a`, REWRITE_TAC[barycentre; VSUM_SING; FINITE_SING; NOT_INSERT_EMPTY] THEN REWRITE_TAC[CARD_SING; REAL_INV_1; VECTOR_MUL_LID]);; let BARYCENTRE_2 = prove (`!a b:real^N. barycentre {a,b} = midpoint(a,b)`, SIMP_TAC[barycentre; FINITE_INSERT; FINITE_EMPTY; NOT_INSERT_EMPTY; VSUM_CLAUSES; NOT_IN_EMPTY; IN_SING; midpoint] THEN REPEAT GEN_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[CARD_CLAUSES; FINITE_RULES; IN_SING; NOT_IN_EMPTY] THEN CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC VECTOR_ARITH);; let BARYCENTRE_IN_RELATIVE_INTERIOR = prove (`!s:real^N->bool. FINITE s /\ ~(s = {}) ==> (barycentre s) IN relative_interior(convex hull s)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET] o MATCH_MP EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL_MINIMAL) THEN REWRITE_TAC[IN_ELIM_THM] THEN EXISTS_TAC `\x:real^N. inv(&(CARD(s:real^N->bool)))` THEN ASM_SIMP_TAC[SUM_CONST; barycentre; REAL_LT_INV_EQ] THEN REWRITE_TAC[REAL_OF_NUM_LT; ARITH_RULE `0 < n <=> ~(n = 0)`] THEN ASM_SIMP_TAC[CARD_EQ_0; REAL_OF_NUM_EQ; REAL_MUL_RINV]);; let BARYCENTRE_IN_CONVEX_HULL = prove (`!s:real^N->bool. FINITE s /\ ~(s = {}) ==> (barycentre s) IN (convex hull s)`, MESON_TAC[BARYCENTRE_IN_RELATIVE_INTERIOR; SUBSET; RELATIVE_INTERIOR_SUBSET]);; let BARYCENTRE_IN_AFFINE_HULL = prove (`!s:real^N->bool. FINITE s /\ ~(s = {}) ==> (barycentre s) IN (affine hull s)`, MESON_TAC[BARYCENTRE_IN_CONVEX_HULL; SUBSET; CONVEX_HULL_SUBSET_AFFINE_HULL]);; let BARYCENTRE_TRANSLATION = prove (`!a:real^N s. barycentre (IMAGE (\x. a + x) s) = (if FINITE s /\ ~(s = {}) then a else vec 0) + barycentre s`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[BARYCENTRE_0; IMAGE_CLAUSES; VECTOR_ADD_RID] THEN REWRITE_TAC[barycentre] THEN SIMP_TAC[FINITE_IMAGE_INJ_EQ; CARD_IMAGE_INJ; VSUM_IMAGE; VECTOR_ARITH `a + x:real^N = a + y <=> x = y`] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[VECTOR_ADD_RID] THEN ASM_SIMP_TAC[o_DEF; VECTOR_ADD_LDISTRIB; VSUM_ADD; VSUM_CONST] THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; CARD_EQ_0; REAL_OF_NUM_EQ; VECTOR_MUL_LID]);; add_translation_invariants [BARYCENTRE_TRANSLATION];; let BARYCENTRE_LINEAR_IMAGE = prove (`!f:real^M->real^N s. linear f /\ (!x y. f x = f y ==> x = y) ==> barycentre (IMAGE f s) = f(barycentre s)`, REWRITE_TAC[INJECTIVE_ALT] THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[barycentre; IMAGE_EQ_EMPTY; FINITE_IMAGE_INJ_EQ; CARD_IMAGE_INJ; VSUM_IMAGE] THEN COND_CASES_TAC THENL [ALL_TAC; ASM_MESON_TAC[LINEAR_0]] THEN ASM_SIMP_TAC[LINEAR_VSUM; o_DEF] THEN MATCH_MP_TAC VSUM_EQ THEN ASM_MESON_TAC[LINEAR_CMUL]);; add_linear_invariants [BARYCENTRE_LINEAR_IMAGE];; let BARYCENTRE_NOT_IN_SET = prove (`!s:real^N->bool. ~affine_dependent s /\ ~(?a. s = {a}) ==> ~(barycentre s IN s)`, GEN_TAC THEN DISCH_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN FIRST_ASSUM(MP_TAC o MATCH_MP DISJOINT_RELATIVE_INTERIOR_CONVEX_HULL) THEN MATCH_MP_TAC(SET_RULE `b IN i ==> i INTER s = {} ==> ~(b IN s)`) THEN MATCH_MP_TAC BARYCENTRE_IN_RELATIVE_INTERIOR THEN ASM_MESON_TAC[AFFINE_INDEPENDENT_IFF_CARD]);; (* ------------------------------------------------------------------------- *) (* Construction of regular polyhedra with given parameters. *) (* ------------------------------------------------------------------------- *) let REGULAR_POLYTOPE_DIST_BARYCENTRE = prove (`!s:real^N->bool n r. s HAS_SIZE n /\ (!x y. x IN s /\ y IN s /\ ~(x = y) ==> dist(x,y) = r) ==> !x. x IN s ==> dist(barycentre s,x) = sqrt((&n - &1) / (&2 * &n)) * r`, REPEAT GEN_TAC THEN ASM_CASES_TAC `&0 <= r` THENL [POP_ASSUM MP_TAC; ASM_SIMP_TAC[NORM_ARITH `~(&0 <= r) ==> ~(dist(x,y) = r)`] THEN ASM_CASES_TAC `(?a:real^N. s = {a})` THENL [ALL_TAC; ASM SET_TAC[]] THEN FIRST_X_ASSUM(CHOOSE_THEN SUBST1_TAC) THEN REWRITE_TAC[HAS_SIZE; CARD_SING] THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN ASM_REWRITE_TAC[] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_SIMP_TAC[BARYCENTRE_1; IN_SING; SQRT_0; DIST_REFL; REAL_MUL_LZERO]] THEN ABBREV_TAC `z:real^N = barycentre s` THEN POP_ASSUM MP_TAC THEN GEOM_ORIGIN_TAC `z:real^N` THEN REPEAT GEN_TAC THEN ASM_CASES_TAC `FINITE(s:real^N->bool)` THEN ASM_REWRITE_TAC[HAS_SIZE] THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN REWRITE_TAC[VECTOR_ARITH `a + x:real^N = a + y <=> x = y`] THEN ASM_SIMP_TAC[barycentre] THEN DISCH_THEN(fun th -> DISCH_TAC THEN STRIP_TAC THEN MP_TAC th) THEN ASM_CASES_TAC `n = 0` THENL [ASM_MESON_TAC[CARD_EQ_0]; ALL_TAC] THEN ASM_SIMP_TAC[VSUM_LMUL; VECTOR_MUL_EQ_0; REAL_INV_EQ_0; REAL_OF_NUM_EQ] THEN STRIP_TAC THEN SUBGOAL_THEN `!x. x IN s ==> sum s (\y:real^N. dist(x,y) pow 2) = (&n - &1) * r pow 2` MP_TAC THENL [REPEAT STRIP_TAC THEN TRANS_TAC EQ_TRANS `sum s (\y:real^N. if y = x then &0 else r pow 2)` THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_EQ THEN REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[DIST_REFL] THEN CONV_TAC REAL_RAT_REDUCE_CONV; ASM_SIMP_TAC[SUM_CASES; GSYM DELETE; SUM_0; SUM_CONST; FINITE_DELETE; REAL_ADD_LID; CARD_DELETE; GSYM REAL_OF_NUM_SUB; LE_1]]; REWRITE_TAC[DIST_0] THEN REWRITE_TAC[dist; NORM_POW_2] THEN REWRITE_TAC[VECTOR_ARITH `(x - y:real^N) dot (x - y) = x dot x - &2 * x dot y + y dot y`] THEN ASM_SIMP_TAC[SUM_SUB; SUM_ADD; SUM_CONST; SUM_LMUL] THEN ASM_SIMP_TAC[GSYM DOT_RSUM; DOT_RZERO; REAL_ARITH `x - &2 * &0 + y = x + y`] THEN DISCH_THEN(LABEL_TAC "*")] THEN ASM_REWRITE_TAC[NORM_EQ_SQUARE] THEN SUBGOAL_THEN `sum s (\x. &n * (x dot x) + sum s (\y. y dot y)) = sum s (\x:real^N. (&n - &1) * r pow 2)` MP_TAC THENL [ASM_SIMP_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[SUM_ADD; SUM_LMUL; SUM_CONST; REAL_OF_NUM_EQ; REAL_RING `n * s + n * s = (n - &1) * n * r <=> n = &0 \/ s = (n - &1) * r / &2`] THEN DISCH_TAC THEN REMOVE_THEN "*" MP_TAC THEN ASM_SIMP_TAC[REAL_OF_NUM_EQ; REAL_FIELD `~(n = &0) ==> (n * x + a * r / &2 = a * r <=> x = (a / (&2 * n)) * r)`] THEN DISCH_THEN(K ALL_TAC) THEN REWRITE_TAC[REAL_POW_MUL] THEN REWRITE_TAC[REAL_RING `x * r:real = y * r <=> y = x \/ r = &0`] THEN REWRITE_TAC[SQRT_POW2] THEN REPEAT STRIP_TAC THENL [MATCH_MP_TAC REAL_LE_MUL THEN REWRITE_TAC[SQRT_LE_0]; ALL_TAC] THEN ASM_SIMP_TAC[REAL_LE_DIV; REAL_LE_MUL; REAL_POS; REAL_SUB_LE; REAL_OF_NUM_LE; LE_1]);; let REGULAR_POLYTOPE_EXISTS = prove (`!r s:real^N->bool n. &n <= aff_dim s + &1 /\ &0 < r ==> ?k. k HAS_SIZE n /\ ~affine_dependent k /\ k SUBSET affine hull s /\ (!x y. x IN k /\ y IN k /\ ~(x = y) ==> dist(x,y) = r)`, GEN_TAC THEN GEN_TAC THEN INDUCT_TAC THEN REWRITE_TAC[HAS_SIZE_0; UNWIND_THM2; AFFINE_INDEPENDENT_EMPTY] THEN REWRITE_TAC[EMPTY_SUBSET; NOT_IN_EMPTY; GSYM INT_OF_NUM_SUC] THEN DISCH_THEN(fun th -> POP_ASSUM MP_TAC THEN STRIP_ASSUME_TAC th) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_INT_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `k:real^N->bool` STRIP_ASSUME_TAC) THEN ASM_CASES_TAC `n = 0` THENL [UNDISCH_TAC `&n + &1 <= aff_dim(s:real^N->bool) + &1` THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[AFF_DIM_EMPTY] THENL [INT_ARITH_TAC; ALL_TAC] THEN DISCH_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 EXISTS_TAC `{a:real^N}` THEN ASM_SIMP_TAC[ARITH; AFFINE_INDEPENDENT_1; SING_SUBSET; HULL_INC] THEN REWRITE_TAC[HAS_SIZE_CONV `s HAS_SIZE 1`] THEN SET_TAC[]; ALL_TAC] THEN POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN ABBREV_TAC `z:real^N = barycentre k` THEN POP_ASSUM MP_TAC THEN GEOM_ORIGIN_TAC `z:real^N` THEN REPEAT GEN_TAC THEN ASM_CASES_TAC `FINITE(k:real^N->bool)` THENL [ASM_REWRITE_TAC[]; ASM_MESON_TAC[HAS_SIZE]] THEN ASM_CASES_TAC `k:real^N->bool = {}` THENL [ASM_REWRITE_TAC[HAS_SIZE; CARD_CLAUSES] THEN MESON_TAC[]; ALL_TAC] THEN ASM_REWRITE_TAC[VECTOR_ARITH `a + x:real^N = a + v <=> x = v`] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`k:real^N->bool`; `n:num`; `r:real`] REGULAR_POLYTOPE_DIST_BARYCENTRE) THEN ASM_REWRITE_TAC[DIST_0] THEN DISCH_TAC THEN MP_TAC(ISPECL [`k:real^N->bool`; `s:real^N->bool`] ORTHOGONAL_TO_SUBSPACE_EXISTS_GEN) THEN MP_TAC(ISPEC `k:real^N->bool` BARYCENTRE_IN_AFFINE_HULL) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN SUBGOAL_THEN `(vec 0:real^N) IN affine hull s` ASSUME_TAC THENL [ASM_MESON_TAC[SUBSET; HULL_HULL; HULL_MONO]; ALL_TAC] THEN ASM_SIMP_TAC[GSYM AFFINE_HULL_EQ_SPAN] THEN ANTS_TAC THENL [REWRITE_TAC[PSUBSET] THEN CONJ_TAC THENL [ASM_MESON_TAC[HULL_HULL; HULL_MONO]; ALL_TAC] THEN DISCH_THEN(MP_TAC o AP_TERM `aff_dim:(real^N->bool)->int`) THEN REWRITE_TAC[AFF_DIM_AFFINE_HULL] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (INT_ARITH `n + &1:int <= s + &1 ==> k <= n - &1 ==> ~(k = s)`)) THEN ASM_MESON_TAC[AFF_DIM_LE_CARD; HAS_SIZE]; DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC)] THEN ABBREV_TAC `b = (sqrt((&n + &1) / (&2 * &n)) * r) % inv(norm a) % a:real^N` THEN EXISTS_TAC `(b:real^N) INSERT k` THEN SUBGOAL_THEN `norm(b:real^N) = sqrt((&n + &1) / (&2 * &n)) * r` ASSUME_TAC THENL [EXPAND_TAC "b" THEN REWRITE_TAC[NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN ASM_SIMP_TAC[REAL_MUL_LINV; NORM_EQ_0; REAL_MUL_RID] THEN REWRITE_TAC[REAL_ABS_REFL] THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; SQRT_LE_0] THEN SIMP_TAC[REAL_LE_DIV; REAL_LE_MUL; REAL_LE_ADD; REAL_POS]; ALL_TAC] THEN SUBGOAL_THEN `~(b:real^N = vec 0)` ASSUME_TAC THENL [ASM_SIMP_TAC[GSYM NORM_EQ_0; REAL_ENTIRE; REAL_LT_IMP_NZ] THEN REWRITE_TAC[SQRT_EQ_0; REAL_DIV_EQ_0] THEN REWRITE_TAC[REAL_ENTIRE; REAL_OF_NUM_ADD; REAL_OF_NUM_EQ] THEN ASM_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `!y:real^N. y IN affine hull k ==> orthogonal b y` ASSUME_TAC THENL [EXPAND_TAC "b" THEN REWRITE_TAC[ORTHOGONAL_MUL] THEN ASM_SIMP_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(b:real^N) IN affine hull s` ASSUME_TAC THENL [SUBST1_TAC(VECTOR_ARITH `b:real^N = vec 0 + b`) THEN EXPAND_TAC "b" THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN MATCH_MP_TAC IN_AFFINE_ADD_MUL THEN ASM_REWRITE_TAC[AFFINE_AFFINE_HULL; VECTOR_ADD_LID]; ALL_TAC] THEN SUBGOAL_THEN `~((b:real^N) IN affine hull k)` ASSUME_TAC THENL [ASM_MESON_TAC[ORTHOGONAL_REFL]; ALL_TAC] THEN REPEAT CONJ_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[HAS_SIZE]) THEN ASM_SIMP_TAC[HAS_SIZE; FINITE_INSERT; CARD_CLAUSES] THEN ASM_MESON_TAC[HULL_INC]; MATCH_MP_TAC AFFINE_INDEPENDENT_INSERT THEN ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[INSERT_SUBSET]; REWRITE_TAC[GSYM pairwise; PAIRWISE_INSERT] THEN ASM_REWRITE_TAC[pairwise; NORM_ARITH `dist(b:real^N,y) = r /\ dist(y,b) = r <=> dist(b,y) = r`] THEN X_GEN_TAC `c:real^N` THEN STRIP_TAC THEN ASM_SIMP_TAC[NORM_EQ_SQUARE; dist; REAL_LT_IMP_LE; GSYM NORM_POW_2] THEN MP_TAC(ISPECL [`b:real^N`; `--c:real^N`] NORM_ADD_PYTHAGOREAN) THEN ASM_SIMP_TAC[ORTHOGONAL_CLAUSES; HULL_INC] THEN ASM_SIMP_TAC[GSYM VECTOR_SUB; NORM_NEG] THEN DISCH_THEN SUBST1_TAC THEN ASM_SIMP_TAC[SQRT_POW_2; REAL_LE_DIV; REAL_LE_MUL; REAL_POS; REAL_LE_ADD; REAL_LT_IMP_LE; REAL_POW_MUL; REAL_SUB_LE; REAL_OF_NUM_LE; LE_1] THEN UNDISCH_TAC `~(n = 0)` THEN REWRITE_TAC[GSYM REAL_OF_NUM_EQ] THEN CONV_TAC REAL_FIELD]);; let REGULAR_POLYTOPE_WITH_BARYCENTRE_EXISTS_ALT = prove (`!r s:real^N->bool a n. &n <= aff_dim s + &1 /\ &0 < r /\ a IN affine hull s /\ ~(n = 0) ==> ?k. k HAS_SIZE n /\ ~affine_dependent k /\ k SUBSET affine hull s /\ barycentre k = a /\ (!x y. x IN k /\ y IN k /\ ~(x = y) ==> dist(x,y) = r) /\ (!x. x IN k ==> dist(a,x) = sqrt((&n - &1) / (&2 * &n)) * r)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`r:real`; `s:real^N->bool`; `n:num`] REGULAR_POLYTOPE_EXISTS) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `k:real^N->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `IMAGE (\x:real^N. (a - barycentre k) + x) k` THEN REWRITE_TAC[CONJ_ASSOC] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[REGULAR_POLYTOPE_DIST_BARYCENTRE]] THEN ASM_REWRITE_TAC[AFFINE_DEPENDENT_TRANSLATION_EQ] THEN ASM_SIMP_TAC[HAS_SIZE_IMAGE_INJ_EQ; VECTOR_ARITH `a + x:real^N = a + y <=> x = y`] THEN RULE_ASSUM_TAC(REWRITE_RULE[HAS_SIZE]) THEN ASM_CASES_TAC `k:real^N->bool = {}` THENL [ASM_MESON_TAC[CARD_EQ_0]; ALL_TAC] THEN ASM_REWRITE_TAC[BARYCENTRE_TRANSLATION; VECTOR_ARITH `(a - b) + b:real^N = a`] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[VECTOR_ARITH `a + x:real^N = a + y <=> x = y`] THEN ASM_SIMP_TAC[NORM_ARITH `dist(a + x:real^N,a + y) = dist(x,y)`] THEN REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[VECTOR_ARITH `x - y:real^N = &1 % (x - y)`] THEN MATCH_MP_TAC IN_AFFINE_MUL_DIFF_ADD THEN ASM_REWRITE_TAC[AFFINE_AFFINE_HULL] THEN ASM_MESON_TAC[BARYCENTRE_IN_AFFINE_HULL; SUBSET; HULL_MONO; HULL_HULL]);; let REGULAR_POLYTOPE_WITH_BARYCENTRE_EXISTS = prove (`!r s:real^N->bool a n. &n <= aff_dim s + &1 /\ &0 < r /\ a IN affine hull s /\ 1 < n ==> ?k. k HAS_SIZE n /\ ~affine_dependent k /\ k SUBSET affine hull s /\ barycentre k = a /\ (!x. x IN k ==> dist(a,x) = r) /\ (!x y. x IN k /\ y IN k /\ ~(x = y) ==> dist(x,y) = sqrt((&2 * &n) / (&n - &1)) * r)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`sqrt((&2 * &n) / (&n - &1)) * r`; `s:real^N->bool`; `a:real^N`; `n:num`] REGULAR_POLYTOPE_WITH_BARYCENTRE_EXISTS_ALT) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [CONJ_TAC THENL [ALL_TAC; ASM_ARITH_TAC] THEN MATCH_MP_TAC REAL_LT_MUL THEN ASM_REWRITE_TAC[SQRT_LT_0] THEN MATCH_MP_TAC REAL_LT_DIV THEN RULE_ASSUM_TAC(REWRITE_RULE[GSYM REAL_OF_NUM_LT]) THEN ASM_REAL_ARITH_TAC; MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN STRIP_TAC THEN ASM_SIMP_TAC[] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC(REAL_RING `a * b = &1 ==> a * b * y = y`) THEN REWRITE_TAC[GSYM SQRT_MUL] THEN MATCH_MP_TAC(MESON[SQRT_1] `x = &1 ==> sqrt x = &1`) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM REAL_OF_NUM_LT]) THEN CONV_TAC REAL_FIELD]);; (* ------------------------------------------------------------------------- *) (* Continuity of convex functions and related results. *) (* ------------------------------------------------------------------------- *) let CONVEX_IMP_LOCALLY_BOUNDED = prove (`!f s a:real^N. f convex_on s /\ a IN relative_interior s ==> ?e B. &0 < e /\ &0 < B /\ cball(a,e) INTER affine hull s SUBSET s /\ !x. x IN cball(a,e) INTER affine hull s ==> abs(f x) <= B`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_RELATIVE_INTERIOR_CBALL]) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC)) THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `aff_dim(s:real^N->bool) = &0` THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [AFF_DIM_EQ_0]) THEN DISCH_THEN(X_CHOOSE_THEN `b:real^N` SUBST_ALL_TAC) THEN ASM_CASES_TAC `b:real^N = a` THEN ASM_REWRITE_TAC[IN_SING] THEN REWRITE_TAC[AFFINE_HULL_SING; INTER_SUBSET; IN_INTER; IN_SING] THEN MESON_TAC[REAL_ARITH `abs x <= abs x + &1 /\ &0 < abs x + &1`]; ALL_TAC] THEN SUBGOAL_THEN `&1 <= aff_dim(s:real^N->bool)` ASSUME_TAC THENL [MATCH_MP_TAC(INT_ARITH `-- &1:int <= x /\ ~(x = &0) /\ ~(x = -- &1) ==> &1 <= x`) THEN ASM_REWRITE_TAC[AFF_DIM_GE; AFF_DIM_EQ_MINUS1]; ALL_TAC] THEN SUBGOAL_THEN `?n. aff_dim(s:real^N->bool) = &n - &1` STRIP_ASSUME_TAC THENL [REWRITE_TAC[INT_EQ_SUB_LADD; INT_OF_NUM_EXISTS] THEN REWRITE_TAC[GSYM INT_LE_SUB_RADD; AFF_DIM_GE; INT_SUB_LZERO]; ALL_TAC] THEN SUBGOAL_THEN `1 < n /\ ~(n = 0)` STRIP_ASSUME_TAC THENL [REWRITE_TAC[GSYM INT_OF_NUM_EQ; GSYM INT_OF_NUM_LT] THEN ASM_INT_ARITH_TAC; ALL_TAC] THEN MP_TAC(ISPECL [`e:real`; `s:real^N->bool`; `a:real^N`; `n:num`] REGULAR_POLYTOPE_WITH_BARYCENTRE_EXISTS) THEN ASM_SIMP_TAC[HULL_INC; INT_SUB_ADD; INT_LE_REFL] THEN DISCH_THEN(X_CHOOSE_THEN `k:real^N->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`f:real^N->real`; `k:real^N->bool`] UPPER_BOUND_FINITE_SET_REAL) THEN ANTS_TAC THENL [ASM_MESON_TAC[HAS_SIZE]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `b:real` MP_TAC) THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] CONVEX_ON_CONVEX_HULL_BOUND)) THEN SUBGOAL_THEN `(f:real^N->real) convex_on convex hull k` ASSUME_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONVEX_ON_SUBSET)) THEN 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 [MATCH_MP_TAC HULL_MINIMAL THEN ASM_SIMP_TAC[CONVEX_CBALL; SUBSET; IN_CBALL; REAL_LE_REFL]; ASM_MESON_TAC[HULL_MONO; HULL_HULL; SUBSET; CONVEX_HULL_SUBSET_AFFINE_HULL]]; ASM_REWRITE_TAC[] THEN DISCH_TAC] THEN MP_TAC(SPEC `k:real^N->bool` BARYCENTRE_IN_RELATIVE_INTERIOR) THEN ANTS_TAC THENL [ASM_MESON_TAC[HAS_SIZE; CARD_CLAUSES]; ASM_SIMP_TAC[]] THEN REWRITE_TAC[IN_RELATIVE_INTERIOR_CBALL; AFFINE_HULL_CONVEX_HULL] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN SUBGOAL_THEN `affine hull k:real^N->bool = affine hull s` ASSUME_TAC THENL [GEN_REWRITE_TAC RAND_CONV [GSYM HULL_HULL] THEN MATCH_MP_TAC AFF_DIM_EQ_AFFINE_HULL THEN ASM_REWRITE_TAC[AFF_DIM_AFFINE_HULL] THEN MP_TAC(ISPECL [`k:real^N->bool`; `k:real^N->bool`] AFF_DIM_UNIQUE) THEN RULE_ASSUM_TAC(REWRITE_RULE[HAS_SIZE]) THEN ASM_SIMP_TAC[INT_LE_REFL]; ASM_REWRITE_TAC[]] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `&1 + abs(b + &2 * abs(f(a:real^N)))` THEN REPEAT CONJ_TAC THENL [REAL_ARITH_TAC; SUBGOAL_THEN `convex hull k SUBSET cball(a:real^N,e)` MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC HULL_MINIMAL THEN ASM_SIMP_TAC[CONVEX_CBALL; SUBSET; IN_CBALL; REAL_LE_REFL]; X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_INTER] THEN STRIP_TAC THEN SUBGOAL_THEN `f convex_on cball(a:real^N,d) INTER affine hull s` MP_TAC THENL [ASM_MESON_TAC[CONVEX_ON_SUBSET]; SIMP_TAC[convex_on]] THEN DISCH_THEN(MP_TAC o SPECL [`x:real^N`; `&2 % a - x:real^N`; `&1 / &2`; `&1 / &2`]) THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[IN_INTER] THEN REWRITE_TAC[VECTOR_ARITH `&1 / &2 % x + &1 / &2 % (&2 % a - x):real^N = a`] THEN UNDISCH_TAC `!x:real^N. x IN convex hull k ==> f x <= b` THEN DISCH_THEN(fun th -> MP_TAC(SPEC `&2 % a - x:real^N` th) THEN MP_TAC(SPEC `x:real^N` th) THEN MP_TAC(SPEC `a:real^N` th)) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ANTS_TAC THENL [ASM SET_TAC[]; DISCH_TAC] THEN SUBGOAL_THEN `&2 % a - x IN cball(a:real^N,d) /\ &2 % a - x IN affine hull s` STRIP_ASSUME_TAC THENL [REWRITE_TAC[IN_CBALL; NORM_ARITH `dist(a:real^N,&2 % a - x) = dist(a,x)`] THEN ASM_REWRITE_TAC[GSYM IN_CBALL] THEN REWRITE_TAC[VECTOR_ARITH `&2 % a - x:real^N = &1 % (a - x) + a`] THEN MATCH_MP_TAC IN_AFFINE_MUL_DIFF_ADD THEN ASM_REWRITE_TAC[AFFINE_AFFINE_HULL] THEN ASM_MESON_TAC[SUBSET; CONVEX_HULL_SUBSET_AFFINE_HULL]; ANTS_TAC THENL [ASM SET_TAC[]; DISCH_TAC] THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC]]);; let CONVEX_IMP_LOCALLY_LIPSCHITZ = prove (`!f s a:real^N. f convex_on s /\ a IN relative_interior s ==> ?e B. &0 < e /\ &0 < B /\ cball(a,e) INTER affine hull s SUBSET s /\ !x y. x IN cball(a,e) INTER affine hull s /\ y IN cball(a,e) INTER affine hull s ==> abs(f x - f y) <= B * norm(x - y)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^N->real`; `s:real^N->bool`; `a:real^N`] CONVEX_IMP_LOCALLY_BOUNDED) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`e:real`; `B:real`] THEN STRIP_TAC THEN EXISTS_TAC `e / &2` THEN ASM_REWRITE_TAC[REAL_HALF] THEN EXISTS_TAC `&4 * B / e` THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_MUL; REAL_ARITH `&0 < &4`] THEN MP_TAC(ISPECL [`a:real^N`; `e / &2`; `e:real`] SUBSET_CBALL) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; DISCH_TAC] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(MESON[] `!i. (!x y. i x y \/ i y x) /\ (!x y. P x y ==> P y x) /\ (!x y. i x y ==> P x y) ==> !x y. P x y`) THEN EXISTS_TAC `\x y. (f:real^N->real) x <= f y` THEN REWRITE_TAC[REAL_LE_TOTAL] THEN CONJ_TAC THENL [REWRITE_TAC[NORM_SUB; REAL_ABS_SUB] THEN MESON_TAC[]; ALL_TAC] THEN SIMP_TAC[REAL_ARITH `x <= y ==> abs(x - y) = y - x`] THEN REPEAT STRIP_TAC THEN ASM_CASES_TAC `y:real^N = x` THEN ASM_REWRITE_TAC[REAL_SUB_REFL; VECTOR_SUB_REFL; NORM_0; REAL_MUL_RZERO; REAL_LE_REFL] THEN SUBGOAL_THEN `?z:real^N. dist(a,z) = e /\ z IN affine hull s /\ y IN segment[x,z]` STRIP_ASSUME_TAC THENL [MP_TAC(ISPECL [`cball(a:real^N,e)`; `x:real^N`; `y - x:real^N`] RAY_TO_FRONTIER) THEN REWRITE_TAC[INTERIOR_CBALL; FRONTIER_CBALL] THEN ASM_REWRITE_TAC[IN_BALL; VECTOR_SUB_EQ; BOUNDED_CBALL] THEN ANTS_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[IN_INTER; IN_CBALL]) THEN ASM_REAL_ARITH_TAC; DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC)] THEN EXISTS_TAC `x + d % (y - x):real^N` THEN ASM_REWRITE_TAC[GSYM IN_SPHERE] THEN CONJ_TAC THENL [ASM_MESON_TAC[IN_INTER; AFFINE_AFFINE_HULL; IN_AFFINE_ADD_MUL_DIFF]; REWRITE_TAC[IN_SEGMENT]] THEN ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ; VECTOR_ARITH `y:real^N = (&1 - u) % x + u % (x + d % (y - x)) <=> (u * d - &1) % (y - x) = vec 0`] THEN EXISTS_TAC `inv(d):real` THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; REAL_LT_IMP_NZ; REAL_MUL_LINV; REAL_SUB_REFL; REAL_LE_INV_EQ] THEN MATCH_MP_TAC REAL_INV_LE_1 THEN REWRITE_TAC[GSYM REAL_NOT_LT] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_SPHERE]) THEN REWRITE_TAC[dist; VECTOR_ARITH `a - (x + d % (y - x)) = --((&1 - d) % (x - a) + d % (y - a))`] THEN MATCH_MP_TAC(NORM_ARITH `norm(x:real^N) + norm(y) < e ==> ~(norm(--(x + y)) = e)`) THEN REWRITE_TAC[NORM_MUL] THEN TRANS_TAC REAL_LET_TRANS `abs(&1 - d) * e / &2 + abs d * e / &2` THEN CONJ_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN MATCH_MP_TAC REAL_LE_ADD2 THEN CONJ_TAC THEN MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[REAL_ABS_POS] THEN REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] (GSYM dist)] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTER; IN_CBALL]) THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONVEX_ON_LEFT_SECANT_MUL]) THEN DISCH_THEN(MP_TAC o SPECL [`x:real^N`; `z:real^N`; `y:real^N`]) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET]) THEN ASM_MESON_TAC[SUBSET; IN_CBALL; REAL_LE_REFL; IN_INTER]; ALL_TAC] THEN SUBGOAL_THEN `e / &2 <= norm(z - x:real^N)` MP_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (NORM_ARITH `dist(a:real^N,z) = e ==> dist(a,x) <= e / &2 ==> e / &2 <= norm(z - x)`)) THEN ASM_MESON_TAC[IN_INTER; IN_CBALL]; ALL_TAC] THEN ASM_CASES_TAC `z:real^N = x` THENL [ASM_REWRITE_TAC[VECTOR_SUB_REFL; NORM_0] THEN ASM_REAL_ARITH_TAC; DISCH_TAC] THEN ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN REWRITE_TAC[REAL_ARITH `(a * b) / c:real = a / c * b`] THEN GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [NORM_SUB] THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN TRANS_TAC REAL_LE_TRANS `(&4 * B / e) * e / &2` THEN CONJ_TAC THENL [ASM_SIMP_TAC[REAL_FIELD `&0 < e ==> (&4 * B / e) * e / &2 = &2 * B`] THEN MATCH_MP_TAC(REAL_ARITH `abs(x) <= B /\ abs(y) <= B ==> x - y <= &2 * B`) THEN CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[SUBSET; IN_CBALL; REAL_LE_REFL; IN_INTER]; MATCH_MP_TAC REAL_LE_LMUL THEN ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[REAL_LE_DIV; REAL_LE_MUL; REAL_POS; REAL_LT_IMP_LE]]);; let CONVEX_ON_CONTINUOUS_ON_RELATIVE_INTERIOR = prove (`!f s:real^N->bool. f convex_on s ==> lift o f continuous_on relative_interior s`, REPEAT STRIP_TAC THEN REWRITE_TAC[continuous_on] THEN X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(ISPECL [`f:real^N->real`; `s:real^N->bool`; `a:real^N`] CONVEX_IMP_LOCALLY_LIPSCHITZ) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`d:real`; `B:real`] THEN STRIP_TAC THEN EXISTS_TAC `min d (e / B)` THEN ASM_SIMP_TAC[REAL_LT_MIN; REAL_LT_DIV; o_DEF; DIST_LIFT] THEN X_GEN_TAC `b:real^N` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a:real^N`; `b:real^N`]) THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_RELATIVE_INTERIOR]) THEN ASM_SIMP_TAC[IN_INTER; HULL_INC; IN_CBALL; DIST_REFL] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC(REAL_ARITH `d < e ==> abs(a - b) <= d ==> abs(b - a) < e`) THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ] THEN ASM_REWRITE_TAC[NORM_ARITH `norm(a - b:real^N) = dist(b,a)`]);; let CONVEX_ON_CONTINUOUS = prove (`!f s:real^N->bool. open s /\ f convex_on s ==> lift o f continuous_on s`, MESON_TAC[RELATIVE_INTERIOR_OPEN; CONVEX_ON_CONTINUOUS_ON_RELATIVE_INTERIOR]);; let CONVEX_IMP_LIPSCHITZ = prove (`!f:real^N->real s t. f convex_on t /\ compact s /\ s SUBSET relative_interior t ==> ?B. &0 < B /\ !x y. x IN s /\ y IN s ==> abs(f x - f y) <= B * norm(x - y)`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[MESON[] `(?x. P x) <=> ~(!x. ~P x)`] THEN DISCH_TAC THEN SUBGOAL_THEN `?x y a b:real^N. (!n. x n IN s) /\ (!n. y n IN s) /\ a IN s /\ b IN s /\ (x --> a) sequentially /\ (y --> b) sequentially /\ (!n. abs(f(x n) - f(y n)) > &n * norm(x n - y n))` STRIP_ASSUME_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN `n:num` o SPEC `&n + &1:real`) THEN REWRITE_TAC[REAL_ARITH `&0 < &n + &1`; NOT_FORALL_THM; NOT_IMP] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM; GSYM CONJ_ASSOC] THEN MAP_EVERY X_GEN_TAC [`x:num->real^N`; `y:num->real^N`] THEN REWRITE_TAC[FORALL_AND_THM; REAL_NOT_LE] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `x:num->real^N` o GEN_REWRITE_RULE I [compact]) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `r:num->num`] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `(y:num->real^N) o (r:num->num)` o GEN_REWRITE_RULE I [compact]) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; o_THM] THEN MAP_EVERY X_GEN_TAC [`b:real^N`; `q:num->num`] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`(x:num->real^N) o (r:num->num) o (q:num->num)`; `(y:num->real^N) o (r:num->num) o (q:num->num)`; `a:real^N`; `b:real^N`] THEN ASM_SIMP_TAC[o_ASSOC; o_THM; LIM_SUBSEQUENCE] THEN X_GEN_TAC `n:num` THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM REAL_NOT_LE] o SPEC `(r:num->num) ((q:num->num) n)`) THEN MATCH_MP_TAC(REAL_ARITH `b <= a ==> ~(x <= a) ==> x > b`) THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE] THEN MATCH_MP_TAC(REAL_ARITH `a <= b ==> a <= b + &1`) THEN REWRITE_TAC[REAL_OF_NUM_LE] THEN SPEC_TAC(`n:num`,`n:num`) THEN MATCH_MP_TAC MONOTONE_BIGGER THEN ASM_MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `b:real^N = a` SUBST_ALL_TAC THENL [ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THEN EXISTS_TAC `\n. (y:num->real^N) n - x n` THEN ASM_SIMP_TAC[LIM_SUB; TRIVIAL_LIMIT_SEQUENTIALLY] THEN MATCH_MP_TAC LIM_NULL_COMPARISON THEN EXISTS_TAC `\n. abs(f(y n) - f(x n:real^N)) / &n` THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN CONJ_TAC THENL [EXISTS_TAC `1` THEN SIMP_TAC[REAL_LE_RDIV_EQ; REAL_OF_NUM_LT; LE_1] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ONCE_REWRITE_TAC[NORM_SUB; REAL_ABS_SUB] THEN ASM_SIMP_TAC[REAL_ARITH `b:real > a ==> a <= b`]; REWRITE_TAC[real_div; LIFT_CMUL] THEN SUBST1_TAC(VECTOR_ARITH `vec 0:real^1 = abs(f(b:real^N) - f a) % vec 0`) THEN MATCH_MP_TAC LIM_MUL THEN REWRITE_TAC[SEQ_HARMONIC; o_DEF] THEN REWRITE_TAC[GSYM NORM_LIFT; LIFT_SUB] THEN MATCH_MP_TAC LIM_NORM THEN MATCH_MP_TAC LIM_SUB THEN CONJ_TAC THEN MP_TAC(ISPECL [`lift o (f:real^N->real)`; `relative_interior t:real^N->bool`] CONTINUOUS_WITHIN_SEQUENTIALLY) THEN REWRITE_TAC[o_DEF] THENL [DISCH_THEN(MP_TAC o SPEC `b:real^N`); DISCH_THEN(MP_TAC o SPEC `a:real^N`)] THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o fst o EQ_IMP_RULE) THEN (ANTS_TAC THENL [ALL_TAC; DISCH_THEN MATCH_MP_TAC THEN ASM_MESON_TAC[SUBSET]]) THEN FIRST_ASSUM(MP_TAC o MATCH_MP CONVEX_ON_CONTINUOUS_ON_RELATIVE_INTERIOR) THEN REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN; o_DEF] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_MESON_TAC[SUBSET]]; ALL_TAC] THEN MP_TAC(SPECL [`f:real^N->real`; `t:real^N->bool`; `a:real^N`] CONVEX_IMP_LOCALLY_LIPSCHITZ) THEN ANTS_TAC THENL [ASM SET_TAC[]; SIMP_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`d:real`; `B:real`] THEN STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIM_SEQUENTIALLY])) THEN REWRITE_TAC[TAUT `a ==> ~b <=> ~(a /\ b)`; AND_FORALL_THM] THEN DISCH_THEN(MP_TAC o SPEC `d:real`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `N1:num`) (X_CHOOSE_TAC `N2:num`)) THEN MP_TAC(SPEC `max B (max (&N1) (&N2))` REAL_ARCH_SIMPLE) THEN REWRITE_TAC[REAL_MAX_LE; REAL_OF_NUM_LE] THEN DISCH_THEN(X_CHOOSE_THEN `n:num` STRIP_ASSUME_TAC) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `n:num`)) THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`(x:num->real^N) n`; `(y:num->real^N) n`]) THEN ASM_REWRITE_TAC[IN_CBALL; IN_INTER] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; NOT_IMP] THEN CONJ_TAC THENL [ASM_MESON_TAC[SUBSET; HULL_INC; RELATIVE_INTERIOR_SUBSET]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `x > a ==> b <= a ==> ~(x <= b)`)) THEN ASM_SIMP_TAC[REAL_LE_RMUL; NORM_POS_LE]]);; let CONVEX_BOUNDS_LEMMA = prove (`!f x:real^N e. f convex_on cball(x,e) /\ (!y. y IN cball(x,e) ==> f(y) <= b) ==> !y. y IN cball(x,e) ==> abs(f(y)) <= b + &2 * abs(f(x))`, REPEAT GEN_TAC THEN ASM_CASES_TAC `&0 <= e` THENL [ALL_TAC; REWRITE_TAC[IN_CBALL] THEN ASM_MESON_TAC[DIST_POS_LE; REAL_LE_TRANS]] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [convex_on]) THEN DISCH_THEN(MP_TAC o SPECL [`y:real^N`; `&2 % x - y:real^N`; `&1 / &2`; `&1 / &2`]) THEN REWRITE_TAC[GSYM VECTOR_ADD_LDISTRIB; GSYM REAL_ADD_LDISTRIB] THEN REWRITE_TAC[VECTOR_ARITH `y + x - y = x:real^N`] THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ABBREV_TAC `z = &2 % x - y:real^N` THEN SUBGOAL_THEN `z:real^N IN cball(x,e)` ASSUME_TAC THENL [UNDISCH_TAC `y:real^N IN cball(x,e)` THEN EXPAND_TAC "z" THEN REWRITE_TAC[dist; IN_CBALL] THEN REWRITE_TAC[VECTOR_ARITH `x - (&2 % x - y) = y - x`] THEN REWRITE_TAC[NORM_SUB]; ALL_TAC] THEN ASM_REWRITE_TAC[VECTOR_MUL_LID] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[real_div; REAL_MUL_LID] THEN REWRITE_TAC[GSYM real_div] THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_OF_NUM_LT; ARITH] THEN FIRST_X_ASSUM(fun th -> MAP_EVERY (MP_TAC o C SPEC th) [`y:real^N`; `z:real^N`]) THEN ASM_REWRITE_TAC[CENTRE_IN_CBALL] THEN REAL_ARITH_TAC);; let CONVEX_IMP_BOUNDED_ON_INTERVAL = prove (`!f:real^1->real a b. f convex_on interval[a,b] ==> ?B. &0 < B /\ !x. x IN interval[a,b] ==> abs(f x) <= B`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `interval[a:real^1,b] = {}` THENL [ASM_MESON_TAC[NOT_IN_EMPTY; REAL_LT_01]; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[INTERVAL_NE_EMPTY_1]) THEN SUBGOAL_THEN `?B. !x:real^1. x IN interval[a,b] ==> f(x) <= B` STRIP_ASSUME_TAC THENL [EXISTS_TAC `max (f(a:real^1)) (f b)` THEN MP_TAC(ISPECL [`f:real^1->real`; `{a:real^1,b}`; `max (f(a:real^1)) (f b)`] CONVEX_ON_CONVEX_HULL_BOUND) THEN ASM_REWRITE_TAC[GSYM SEGMENT_CONVEX_HULL; SEGMENT_1] THEN DISCH_THEN MATCH_MP_TAC THEN SIMP_TAC[FORALL_IN_INSERT; NOT_IN_EMPTY] THEN REAL_ARITH_TAC; EXISTS_TAC `(&1 + abs B) + &2 * abs(f(midpoint(a,b):real^1))` THEN CONJ_TAC THENL [REAL_ARITH_TAC; ASM_REWRITE_TAC[INTERVAL_1]] THEN MATCH_MP_TAC CONVEX_BOUNDS_LEMMA THEN MAP_EVERY UNDISCH_TAC [`f convex_on interval[a:real^1,b]`; `!x:real^1. x IN interval[a,b] ==> f x <= B`] THEN ASM_SIMP_TAC[INTERVAL_1; REAL_ARITH `x <= b ==> x <= &1 + abs b`]]);; (* ------------------------------------------------------------------------- *) (* A convex function on R^1 is "piecewise monotone" in this precise sense. *) (* ------------------------------------------------------------------------- *) let CONVEX_IMP_PIECEWISE_MONOTONE = prove (`!f s. f convex_on s /\ is_interval s ==> (!x y. x IN interior s /\ y IN interior s /\ drop x <= drop y ==> f x <= f y) \/ (!x y. x IN interior s /\ y IN interior s /\ drop x <= drop y ==> f y <= f x) \/ ?a. a IN interior s /\ (!x y. x IN s /\ y IN s /\ drop x <= drop y /\ drop y <= drop a ==> f y <= f x) /\ (!x y. x IN s /\ y IN s /\ drop a <= drop x /\ drop x <= drop y ==> f x <= f y)`, REPEAT STRIP_TAC THEN REWRITE_TAC[FORALL_LIFT; LIFT_DROP; DISJ_ASSOC] THEN REWRITE_TAC[REAL_NON_MONOTONE] THEN REWRITE_TAC[TAUT `~p \/ q <=> p ==> q`; LEFT_IMP_EXISTS_THM] THEN REWRITE_TAC[FORALL_DROP; LIFT_DROP] THEN MAP_EVERY X_GEN_TAC [`u:real^1`; `b:real^1`; `v:real^1`] THEN STRIP_TAC THENL [MATCH_MP_TAC(TAUT `F ==> p`) THEN MP_TAC(ISPECL [`f:real^1->real`; `s:real^1->bool`; `u:real^1`; `v:real^1`; `b:real^1`] CONVEX_LOWER_SEGMENT) THEN ASM_SIMP_TAC[NOT_IMP; REWRITE_RULE[SUBSET] INTERIOR_SUBSET] THEN REWRITE_TAC[SEGMENT_1] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL_1] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `interval[u:real^1,v] SUBSET interior s` ASSUME_TAC THENL [ASM_MESON_TAC[INTERVAL_SUBSET_IS_INTERVAL; IS_INTERVAL_CONVEX_1; CONVEX_INTERIOR]; ALL_TAC] THEN MP_TAC(ISPECL [`f:real^1->real`; `interval[u:real^1,v]`] CONTINUOUS_ATTAINS_INF) THEN REWRITE_TAC[COMPACT_INTERVAL] THEN ANTS_TAC THENL [REWRITE_TAC[INTERVAL_NE_EMPTY_1] THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `interior s:real^1->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONVEX_ON_CONTINUOUS THEN ASM_MESON_TAC[CONVEX_ON_SUBSET; INTERIOR_SUBSET; OPEN_INTERIOR]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^1`] THEN STRIP_TAC THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [ASM SET_TAC[]; DISCH_TAC] THEN SUBGOAL_THEN `!x. x IN s ==> (f:real^1->real) a <= f x` ASSUME_TAC THENL [MATCH_MP_TAC CONVEX_LOCAL_GLOBAL_MINIMUM THEN EXISTS_TAC `interval(u:real^1,v)` THEN ASM_REWRITE_TAC[OPEN_INTERVAL] THEN CONJ_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_SIMP_TAC[IN_INTERVAL_1; REAL_LT_LE; DROP_EQ] THEN ASM_MESON_TAC[REAL_LE_LT; REAL_NOT_LE]; ASM_MESON_TAC[INTERVAL_OPEN_SUBSET_CLOSED; SUBSET; INTERIOR_SUBSET]]; ALL_TAC] THEN SUBGOAL_THEN `!x y. x IN s /\ y IN s /\ x IN segment[a,y] ==> (f:real^1->real) x <= f y` MP_TAC THENL [REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^1->real`; `s:real^1->bool`; `a:real^1`; `y:real^1`; `x:real^1`] CONVEX_LOWER_SEGMENT) THEN ASM_SIMP_TAC[real_max] THEN ASM_MESON_TAC[INTERIOR_SUBSET; SUBSET]; REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[SEGMENT_1] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL_1] THEN ASM_REAL_ARITH_TAC]);;