(* ========================================================================= *) (* Transfer of homological definition of Brouwer degree to our Multivariate *) (* context, used to get some key results about homotopy of linear mappings *) (* and so all the usual things like Brouwer's fixed-point theorem. *) (* *) (* (c) Copyright, John Harrison 2017-2018 *) (* ========================================================================= *) needs "Multivariate/homology.ml";; needs "Multivariate/polytope.ml";; (* ------------------------------------------------------------------------- *) (* Transfer of Brouwer degree from product topology setting. *) (* ------------------------------------------------------------------------- *) let brouwer_degree1 = new_definition `brouwer_degree1 n (f:real^N->real^N) = if 1 <= n /\ n <= dimindex(:N) then brouwer_degree2 (n - 1) ((\x i. if 1 <= i /\ i <= n then x$i else &0) o f o (\x. lambda i. if 1 <= i /\ i <= n then x i else &0)) else &1`;; let brouwer_degree = new_definition `brouwer_degree (f:real^N->real^N) = brouwer_degree1 (dimindex(:N)) f`;; let BROUWER_DEGREE1_EQ = prove (`!n f g:real^N->real^N. (!x. x IN sphere(vec 0,&1) INTER span(IMAGE basis (1..n)) ==> f x = g x) ==> brouwer_degree1 n f = brouwer_degree1 n g`, REPEAT STRIP_TAC THEN REWRITE_TAC[brouwer_degree1] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN MP_TAC(SPEC `n:num` HOMEOMORPHIC_MAPS_NSPHERE_EUCLIDEAN_SPHERE) THEN MAP_EVERY ABBREV_TAC [`h:(num->real)->real^N = \x. lambda i. if 1 <= i /\ i <= n then x i else &0`; `h':real^N->num->real = \x i. if 1 <= i /\ i <= n then x$i else &0`] THEN ASM_REWRITE_TAC[homeomorphic_maps] THEN STRIP_TAC THEN MATCH_MP_TAC BROUWER_DEGREE2_EQ THEN REPEAT STRIP_TAC THEN REWRITE_TAC[o_THM] THEN AP_TERM_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE [continuous_map; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY]) THEN ASM SET_TAC[]);; let BROUWER_DEGREE1_ID = prove (`!n. brouwer_degree1 n (\x:real^N. x) = &1`, GEN_TAC THEN REWRITE_TAC[brouwer_degree1] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN MP_TAC(SPEC `n:num` HOMEOMORPHIC_MAPS_NSPHERE_EUCLIDEAN_SPHERE) THEN MAP_EVERY ABBREV_TAC [`h:(num->real)->real^N = \x. lambda i. if 1 <= i /\ i <= n then x i else &0`; `h':real^N->num->real = \x i. if 1 <= i /\ i <= n then x$i else &0`] THEN ASM_REWRITE_TAC[homeomorphic_maps] THEN STRIP_TAC THEN SUBST1_TAC(SYM(SPEC `n - 1` BROUWER_DEGREE2_ID)) THEN MATCH_MP_TAC BROUWER_DEGREE2_EQ THEN ASM_SIMP_TAC[o_THM]);; let BROUWER_DEGREE1_COMPOSE = prove (`!n f g:real^N->real^N. f continuous_on (sphere(vec 0,&1) INTER span(IMAGE basis (1..n))) /\ g continuous_on (sphere(vec 0,&1) INTER span(IMAGE basis (1..n))) /\ IMAGE f (sphere(vec 0,&1) INTER span(IMAGE basis (1..n))) SUBSET (sphere(vec 0,&1) INTER span(IMAGE basis (1..n))) /\ IMAGE g (sphere(vec 0,&1) INTER span(IMAGE basis (1..n))) SUBSET (sphere(vec 0,&1) INTER span(IMAGE basis (1..n))) ==> brouwer_degree1 n (g o f) = brouwer_degree1 n g * brouwer_degree1 n f`, REPEAT STRIP_TAC THEN REWRITE_TAC[brouwer_degree1] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[INT_MUL_LID] THEN MP_TAC(SPEC `n:num` HOMEOMORPHIC_MAPS_NSPHERE_EUCLIDEAN_SPHERE) THEN MAP_EVERY ABBREV_TAC [`h:(num->real)->real^N = \x. lambda i. if 1 <= i /\ i <= n then x i else &0`; `h':real^N->num->real = \x i. if 1 <= i /\ i <= n then x$i else &0`] THEN ASM_REWRITE_TAC[homeomorphic_maps] THEN STRIP_TAC THEN W(MP_TAC o PART_MATCH (rand o rand) BROUWER_DEGREE2_COMPOSE o rand o snd) THEN ANTS_TAC THENL [ASM_MESON_TAC[CONTINUOUS_MAP_COMPOSE; CONTINUOUS_MAP_EUCLIDEAN2]; DISCH_THEN(SUBST1_TAC o SYM)] THEN MATCH_MP_TAC BROUWER_DEGREE2_EQ THEN REPEAT STRIP_TAC THEN REWRITE_TAC[o_THM] THEN AP_TERM_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE [continuous_map; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY]) THEN ASM SET_TAC[]);; let BROUWER_DEGREE1_HOMOTOPIC = prove (`!n f g:real^N->real^N. homotopic_with (\x. T) (subtopology euclidean (sphere(vec 0,&1) INTER span(IMAGE basis (1..n))), subtopology euclidean (sphere(vec 0,&1) INTER span(IMAGE basis (1..n)))) f g ==> brouwer_degree1 n f = brouwer_degree1 n g`, REPEAT STRIP_TAC THEN REWRITE_TAC[brouwer_degree1] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN MP_TAC(SPEC `n:num` HOMEOMORPHIC_MAPS_NSPHERE_EUCLIDEAN_SPHERE) THEN MAP_EVERY ABBREV_TAC [`h:(num->real)->real^N = \x. lambda i. if 1 <= i /\ i <= n then x i else &0`; `h':real^N->num->real = \x i. if 1 <= i /\ i <= n then x$i else &0`] THEN ASM_REWRITE_TAC[homeomorphic_maps] THEN STRIP_TAC THEN MATCH_MP_TAC BROUWER_DEGREE2_HOMOTOPIC THEN ASM_MESON_TAC[HOMOTOPIC_COMPOSE_CONTINUOUS_MAP_LEFT; HOMOTOPIC_COMPOSE_CONTINUOUS_MAP_RIGHT]);; let BROUWER_DEGREE1_CONST = prove (`!n a:real^N. 1 <= n /\ n <= dimindex(:N) ==> brouwer_degree1 n (\x. a) = &0`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[brouwer_degree1; o_DEF;BROUWER_DEGREE2_CONST]);; let BROUWER_DEGREE1_REFLECT_ALONG = prove (`!n a:real^N. 1 <= n /\ n <= dimindex(:N) /\ a IN span(IMAGE basis (1..n)) DELETE vec 0 ==> brouwer_degree1 n (reflect_along a) = -- &1`, REWRITE_TAC[IN_DELETE; IN_SPAN_IMAGE_BASIS] THEN REPEAT STRIP_TAC THEN TRANS_TAC EQ_TRANS `brouwer_degree1 n (reflect_along (basis 1:real^N))` THEN CONJ_TAC THENL [MATCH_MP_TAC BROUWER_DEGREE1_HOMOTOPIC THEN MATCH_MP_TAC HOMOTOPIC_WITH_REFLECTIONS_ALONG THEN ASM_SIMP_TAC[BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTER] THEN ASM_REWRITE_TAC[IN_SPHERE_0; NORM_REFLECT_ALONG; IN_SPAN_IMAGE_BASIS] THEN SIMP_TAC[IN_SEGMENT; LEFT_IMP_EXISTS_THM] THEN ASM_SIMP_TAC[reflect_along; VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT; VECTOR_ADD_COMPONENT] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN GEN_TAC THEN STRIP_TAC THEN SIMP_TAC[IN_NUMSEG; IMP_CONJ] THEN X_GEN_TAC `j:num` THEN ASM_CASES_TAC `j = 1` THENL [ASM_ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[BASIS_COMPONENT] THEN REAL_ARITH_TAC; SUBST1_TAC(SYM(SPEC `n - 1` BROUWER_DEGREE2_REFLECTION)) THEN ASM_REWRITE_TAC[brouwer_degree1] THEN MATCH_MP_TAC BROUWER_DEGREE2_EQ THEN REWRITE_TAC[NSPHERE; TOPSPACE_SUBTOPOLOGY; IN_INTER; IN_ELIM_THM] THEN X_GEN_TAC `x:num->real` THEN ASM_SIMP_TAC[SUB_ADD; IN_NUMSEG] THEN STRIP_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `i:num` THEN REWRITE_TAC[o_THM] THEN ASM_CASES_TAC `1 <= i /\ i <= n` THEN ASM_REWRITE_TAC[] THENL [ALL_TAC; ASM_MESON_TAC[INT_NEG_0; LE_REFL]] THEN SUBGOAL_THEN `i <= dimindex(:N)` MP_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[REFLECT_ALONG_BASIS_COMPONENT; DIMINDEX_GE_1; LE_REFL; LAMBDA_BETA]]);; let BROUWER_DEGREE1_NONSURJECTIVE = prove (`!n (f:real^N->real^N). 1 <= n /\ n <= dimindex(:N) /\ f continuous_on (sphere(vec 0,&1) INTER span(IMAGE basis (1..n))) /\ IMAGE f (sphere(vec 0,&1) INTER span(IMAGE basis (1..n))) PSUBSET (sphere(vec 0,&1) INTER span(IMAGE basis (1..n))) ==> brouwer_degree1 n f = &0`, REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[brouwer_degree1] THEN MATCH_MP_TAC BROUWER_DEGREE2_NONSURJECTIVE THEN MP_TAC(SPEC `n:num` HOMEOMORPHIC_MAPS_NSPHERE_EUCLIDEAN_SPHERE) THEN MAP_EVERY ABBREV_TAC [`h:(num->real)->real^N = \x. lambda i. if 1 <= i /\ i <= n then x i else &0`; `h':real^N->num->real = \x i. if 1 <= i /\ i <= n then x$i else &0`] THEN ASM_REWRITE_TAC[homeomorphic_maps] THEN STRIP_TAC THEN CONJ_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[PSUBSET]) THEN ASM_MESON_TAC[CONTINUOUS_MAP_COMPOSE; CONTINUOUS_MAP_EUCLIDEAN2]; ALL_TAC] THEN FIRST_X_ASSUM(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC o GEN_REWRITE_RULE I [PSUBSET_ALT]) THEN DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC(SET_RULE `!a. a IN t /\ ~(a IN s) ==> ~(s = t)`) THEN EXISTS_TAC `(h':real^N->num->real) a` THEN RULE_ASSUM_TAC(REWRITE_RULE [continuous_map; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY]) THEN ASM_SIMP_TAC[IMAGE_o] THEN ASM SET_TAC[]);; let BROUWER_DEGREE_EQ = prove (`!f g:real^N->real^N. (!x. x IN sphere(vec 0,&1) ==> f x = g x) ==> brouwer_degree f = brouwer_degree g`, REPEAT STRIP_TAC THEN REWRITE_TAC[brouwer_degree] THEN MATCH_MP_TAC BROUWER_DEGREE1_EQ THEN REWRITE_TAC[GSYM SIMPLE_IMAGE; IN_NUMSEG; SPAN_STDBASIS] THEN ASM_REWRITE_TAC[IN_INTER; IN_UNIV]);; let BROUWER_DEGREE_ID = prove (`brouwer_degree (\x:real^N. x) = &1`, REWRITE_TAC[brouwer_degree; BROUWER_DEGREE1_ID]);; let BROUWER_DEGREE_COMPOSE = prove (`!f g:real^N->real^N. f continuous_on sphere(vec 0,&1) /\ g continuous_on sphere(vec 0,&1) /\ IMAGE f (sphere(vec 0,&1)) SUBSET sphere(vec 0,&1) /\ IMAGE g (sphere(vec 0,&1)) SUBSET sphere(vec 0,&1) ==> brouwer_degree (g o f) = brouwer_degree g * brouwer_degree f`, REPEAT STRIP_TAC THEN REWRITE_TAC[brouwer_degree] THEN MATCH_MP_TAC BROUWER_DEGREE1_COMPOSE THEN REWRITE_TAC[GSYM SIMPLE_IMAGE; IN_NUMSEG; SPAN_STDBASIS; INTER_UNIV] THEN ASM_REWRITE_TAC[SIMPLE_IMAGE]);; let BROUWER_DEGREE_HOMOTOPIC = prove (`!f g:real^N->real^N. homotopic_with (\x. T) (subtopology euclidean (sphere(vec 0,&1)), subtopology euclidean (sphere(vec 0,&1))) f g ==> brouwer_degree f = brouwer_degree g`, REPEAT STRIP_TAC THEN REWRITE_TAC[brouwer_degree] THEN MATCH_MP_TAC BROUWER_DEGREE1_HOMOTOPIC THEN REWRITE_TAC[GSYM SIMPLE_IMAGE; IN_NUMSEG; SPAN_STDBASIS; INTER_UNIV] THEN ASM_REWRITE_TAC[SIMPLE_IMAGE]);; let BROUWER_DEGREE_CONST = prove (`!a:real^N. brouwer_degree (\x. a) = &0`, REPEAT STRIP_TAC THEN REWRITE_TAC[brouwer_degree] THEN MATCH_MP_TAC BROUWER_DEGREE1_CONST THEN REWRITE_TAC[DIMINDEX_GE_1; LE_REFL]);; let BROUWER_DEGREE_REFLECT_ALONG = prove (`!a:real^N. ~(a = vec 0) ==> brouwer_degree (reflect_along a) = -- &1`, REPEAT STRIP_TAC THEN REWRITE_TAC[brouwer_degree] THEN MATCH_MP_TAC BROUWER_DEGREE1_REFLECT_ALONG THEN ASM_REWRITE_TAC[DIMINDEX_GE_1; LE_REFL] THEN REWRITE_TAC[GSYM SIMPLE_IMAGE; IN_NUMSEG; SPAN_STDBASIS] THEN ASM_REWRITE_TAC[IN_UNIV; IN_DELETE]);; let BROUWER_DEGREE_NONSURJECTIVE = prove (`!(f:real^N->real^N). f continuous_on sphere(vec 0,&1) /\ IMAGE f (sphere(vec 0,&1)) PSUBSET sphere(vec 0,&1) ==> brouwer_degree f = &0`, REPEAT STRIP_TAC THEN REWRITE_TAC[brouwer_degree] THEN MATCH_MP_TAC BROUWER_DEGREE1_NONSURJECTIVE THEN REWRITE_TAC[GSYM SIMPLE_IMAGE; IN_NUMSEG; SPAN_STDBASIS; INTER_UNIV] THEN ASM_REWRITE_TAC[DIMINDEX_GE_1; LE_REFL; SIMPLE_IMAGE]);; let BROUWER_DEGREE_ORTHOGONAL_TRANSFORMATION = prove (`!(f:real^N->real^N). orthogonal_transformation f ==> real_of_int(brouwer_degree f) = det(matrix f)`, REPEAT STRIP_TAC THEN MP_TAC (ISPEC `f:real^N->real^N` HOMOTOPIC_WITH_ORTHOGONAL_TRANSFORMATIONS) THEN FIRST_ASSUM(DISJ_CASES_TAC o MATCH_MP DET_ORTHOGONAL_MATRIX o MATCH_MP ORTHOGONAL_MATRIX_MATRIX) THENL [DISCH_THEN(MP_TAC o SPEC `I:real^N->real^N`); DISCH_THEN(MP_TAC o SPEC `reflect_along(basis 1):real^N->real^N`)] THEN ASM_SIMP_TAC[MATRIX_I; DET_I; ORTHOGONAL_TRANSFORMATION_I; DET_MATRIX_REFLECT_ALONG; ORTHOGONAL_TRANSFORMATION_REFLECT_ALONG; BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL] THEN DISCH_THEN(MP_TAC o SPEC `\f:real^N->real^N. T` o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMOTOPIC_WITH_MONO)) THEN REWRITE_TAC[] THEN DISCH_THEN(SUBST1_TAC o MATCH_MP BROUWER_DEGREE_HOMOTOPIC) THEN SIMP_TAC[BROUWER_DEGREE_ID; I_DEF; BROUWER_DEGREE_REFLECT_ALONG; BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL] THEN REWRITE_TAC[int_neg_th; int_of_num_th]);; (* ------------------------------------------------------------------------- *) (* Hence the key theorem about homotopy of linear maps. *) (* ------------------------------------------------------------------------- *) let HOMOTOPIC_ORTHOGONAL_TRANSFORMATIONS = prove (`!f g:real^N->real^N. orthogonal_transformation f /\ orthogonal_transformation g ==> (homotopic_with (\x. T) (subtopology euclidean (sphere (vec 0,&1)), subtopology euclidean (sphere (vec 0,&1))) f g <=> det(matrix f) = det(matrix g))`, REPEAT STRIP_TAC THEN EQ_TAC THENL [ASM_SIMP_TAC[GSYM BROUWER_DEGREE_ORTHOGONAL_TRANSFORMATION] THEN REWRITE_TAC[GSYM int_eq; BROUWER_DEGREE_HOMOTOPIC]; DISCH_TAC THEN MATCH_MP_TAC HOMOTOPIC_WITH_MONO THEN EXISTS_TAC `orthogonal_transformation:(real^N->real^N)->bool` THEN ASM_REWRITE_TAC[HOMOTOPIC_WITH_ORTHOGONAL_TRANSFORMATIONS]]);; let HOMOTOPIC_ORTHOGONAL_TRANSFORMATIONS_ALT = prove (`!f g:real^N->real^N. orthogonal_transformation f /\ orthogonal_transformation g ==> (homotopic_with (\x. T) (subtopology euclidean ((:real^N) DELETE vec 0), subtopology euclidean ((:real^N) DELETE vec 0)) f g <=> det(matrix f) = det(matrix g))`, REPEAT STRIP_TAC THEN EQ_TAC THENL [ASM_SIMP_TAC[GSYM HOMOTOPIC_ORTHOGONAL_TRANSFORMATIONS] THEN DISCH_TAC THEN MATCH_MP_TAC HOMOTOPIC_WITH_EQ THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN MAP_EVERY EXISTS_TAC [`(\x. inv(norm x) % x) o (f:real^N->real^N)`; `(\x. inv(norm x) % x) o (g:real^N->real^N)`] THEN RULE_ASSUM_TAC(REWRITE_RULE[ORTHOGONAL_TRANSFORMATION]) THEN ASM_SIMP_TAC[IN_SPHERE_0; o_THM; REAL_INV_1; VECTOR_MUL_LID] THEN MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE THEN MAP_EVERY EXISTS_TAC [`\f:real^N->real^N. T`; `\f:real^N->real^N. T`; `(:real^N) DELETE vec 0`] THEN REWRITE_TAC[HOMOTOPIC_WITH_REFL; CONTINUOUS_MAP_EUCLIDEAN2] THEN SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_UNIV; IN_DELETE; IN_SPHERE_0] THEN REWRITE_TAC[NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN SIMP_TAC[REAL_MUL_LINV; NORM_EQ_0] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMOTOPIC_WITH_RESTRICT)) THEN REWRITE_TAC[SET_RULE `s SUBSET UNIV DELETE z <=> ~(z IN s)`] THEN REWRITE_TAC[CONTRAPOS_THM; IN_SPHERE_0; NORM_0] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN MATCH_MP_TAC(SET_RULE `~(z IN s) ==> !f. a IN IMAGE f s ==> a IN IMAGE f (UNIV DELETE z)`) THEN REWRITE_TAC[CONTRAPOS_THM; IN_SPHERE_0; NORM_0] THEN CONV_TAC REAL_RAT_REDUCE_CONV; MATCH_MP_TAC CONTINUOUS_ON_MUL THEN REWRITE_TAC[CONTINUOUS_ON_ID] THEN REWRITE_TAC[o_DEF] THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN SIMP_TAC[IN_DELETE; NORM_EQ_0] THEN REWRITE_TAC[REWRITE_RULE[o_DEF] CONTINUOUS_ON_LIFT_NORM]]; DISCH_TAC THEN MATCH_MP_TAC HOMOTOPIC_WITH_MONO THEN EXISTS_TAC `orthogonal_transformation:(real^N->real^N)->bool` THEN ASM_REWRITE_TAC[HOMOTOPIC_WITH_ORTHOGONAL_TRANSFORMATIONS_ALT]]);; let HOMOTOPIC_ORTHOGONAL_TRANSFORMATIONS_IMP = prove (`!f g:real^N->real^N. orthogonal_transformation f /\ orthogonal_transformation g /\ homotopic_with (\x. T) (subtopology euclidean (sphere (vec 0,&1)), subtopology euclidean (sphere (vec 0,&1))) f g ==> det(matrix f) = det(matrix g)`, SIMP_TAC[GSYM HOMOTOPIC_ORTHOGONAL_TRANSFORMATIONS]);; let HOMOTOPIC_LINEAR_MAPS_IMP = prove (`!f g:real^N->real^N. linear f /\ linear g /\ homotopic_with (\x. T) (subtopology euclidean ((:real^N) DELETE vec 0), subtopology euclidean ((:real^N) DELETE vec 0)) f g ==> real_sgn(det(matrix f)) = real_sgn(det(matrix g))`, let lemma = prove (`!f:real^N->real^N. linear f /\ ~(det(matrix f) = &0) ==> ?P. positive_definite P /\ orthogonal_transformation ((\x. P ** x) o f)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM INVERTIBLE_DET_NZ]) THEN REWRITE_TAC[LEFT_POLAR_DECOMPOSITION_INVERTIBLE; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`U:real^N^N`; `P:real^N^N`] THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN EXISTS_TAC `matrix_inv P:real^N^N` THEN ASM_REWRITE_TAC[POSITIVE_DEFINITE_INV; o_DEF] THEN FIRST_ASSUM(fun th -> REWRITE_TAC[GSYM(MATCH_MP MATRIX_WORKS th)]) THEN ASM_SIMP_TAC[MATRIX_MUL_ASSOC; MATRIX_VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[MATRIX_INV; POSITIVE_DEFINITE_IMP_INVERTIBLE] THEN ASM_SIMP_TAC[MATRIX_MUL_LID; MATRIX_WORKS] THEN ASM_REWRITE_TAC[GSYM ORTHOGONAL_MATRIX_TRANSFORMATION]) in REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_SUBSET) THEN REWRITE_TAC[SET_RULE `IMAGE f (UNIV DELETE w) SUBSET UNIV DELETE z <=> ~(?x. ~(x = w) /\ f x = z)`] THEN ASM_SIMP_TAC[GSYM MATRIX_WORKS; HOMOGENEOUS_LINEAR_EQUATIONS_DET] THEN STRIP_TAC THEN MP_TAC(ISPEC `g:real^N->real^N` lemma) THEN MP_TAC(ISPEC `f:real^N->real^N` lemma) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `P:real^N^N` THEN STRIP_TAC THEN X_GEN_TAC `Q:real^N^N` THEN STRIP_TAC THEN MP_TAC(ISPECL [`(\x. (P:real^N^N) ** x) o (f:real^N->real^N)`; `(\x. (Q:real^N^N) ** x) o (g:real^N->real^N)`] HOMOTOPIC_ORTHOGONAL_TRANSFORMATIONS_ALT) THEN ASM_SIMP_TAC[MATRIX_COMPOSE; MATRIX_VECTOR_MUL_LINEAR] THEN REWRITE_TAC[DET_MUL; MATRIX_OF_MATRIX_VECTOR_MUL] THEN DISCH_THEN(MP_TAC o fst o EQ_IMP_RULE) THEN ANTS_TAC THENL [MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE THEN MAP_EVERY EXISTS_TAC [`\f:real^N->real^N. T`; `\f:real^N->real^N. linear f /\ positive_definite(matrix f)`; `(:real^N) DELETE vec 0`] THEN ASM_REWRITE_TAC[HOMOTOPIC_WITH_LINEAR_POSITIVE_DEFINITE_MAPS] THEN ASM_REWRITE_TAC[MATRIX_OF_MATRIX_VECTOR_MUL; MATRIX_VECTOR_MUL_LINEAR]; DISCH_THEN(MP_TAC o AP_TERM `real_sgn`) THEN REWRITE_TAC[REAL_SGN_MUL] THEN MATCH_MP_TAC(REAL_RING `x = &1 /\ y = &1 ==> x * a = y * b ==> a = b`) THEN ASM_SIMP_TAC[REAL_SGN_EQ; real_gt; DET_POSITIVE_DEFINITE]]);; let HOMOTOPIC_LINEAR_MAPS_ALT = prove (`!f g:real^N->real^N. linear f /\ linear g /\ homotopic_with (\x. T) (subtopology euclidean ((:real^N) DELETE vec 0), subtopology euclidean ((:real^N) DELETE vec 0)) f g ==> &0 < det(matrix f) * det(matrix g)`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_SGN_INEQS] THEN MP_TAC(ISPECL [`f:real^N->real^N`; `g:real^N->real^N`] HOMOTOPIC_LINEAR_MAPS_IMP) THEN ASM_SIMP_TAC[REAL_SGN_MUL; GSYM REAL_POW_2; REAL_LT_POW_2] THEN DISCH_TAC THEN REWRITE_TAC[REAL_SGN_INEQS] THEN FIRST_ASSUM(MP_TAC o CONJUNCT2 o MATCH_MP HOMOTOPIC_WITH_IMP_SUBSET) THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `IMAGE f (UNIV DELETE a) SUBSET UNIV DELETE a ==> !x. f x = a ==> x = a`)) THEN ASM_SIMP_TAC[GSYM LINEAR_INJECTIVE_0; MATRIX_INVERTIBLE; GSYM INVERTIBLE_DET_NZ] THEN ASM_MESON_TAC[LINEAR_INJECTIVE_LEFT_INVERSE; LINEAR_INVERSE_LEFT]);; (* ------------------------------------------------------------------------- *) (* Hairy ball theorem and relatives. *) (* ------------------------------------------------------------------------- *) let FIXPOINT_HOMOTOPIC_IDENTITY_SPHERE = prove (`!f:real^N->real^N. ODD(dimindex(:N)) /\ homotopic_with (\x. T) (subtopology euclidean (sphere(vec 0,&1)), subtopology euclidean (sphere(vec 0,&1))) (\x. x) f ==> ?x. x IN sphere(vec 0,&1) /\ f x = x`, REPEAT STRIP_TAC THEN MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THEN MP_TAC(ISPECL [`f:real^N->real^N`; `\x:real^N. --x`; `sphere(vec 0:real^N,&1)`; `&1`] HOMOTOPIC_NON_ANTIPODAL_SPHEREMAPS) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_CONTINUOUS) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_SUBSET) THEN ASM_REWRITE_TAC[NOT_IMP] THEN REPEAT CONJ_TAC THENL [SIMP_TAC[CONTINUOUS_ON_NEG; CONTINUOUS_ON_ID]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_SPHERE_0; NORM_NEG]; ASM_MESON_TAC[VECTOR_NEG_NEG]; DISCH_THEN(MP_TAC o SPEC `\x:real^N. x` o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_WITH_TRANS)) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] (REWRITE_RULE[CONJ_ASSOC] HOMOTOPIC_ORTHOGONAL_TRANSFORMATIONS_IMP))) THEN SIMP_TAC[ORTHOGONAL_TRANSFORMATION_NEG; ORTHOGONAL_TRANSFORMATION_ID; MATRIX_NEG; LINEAR_ID; DET_NEG; MATRIX_ID; DET_I] THEN ASM_REWRITE_TAC[REAL_POW_NEG; REAL_POW_ONE; GSYM NOT_ODD] THEN CONV_TAC REAL_RAT_REDUCE_CONV]);; let FIXPOINT_OR_NEG_MAPPING_SPHERE = prove (`!f:real^N->real^N. ODD(dimindex(:N)) /\ f continuous_on sphere(vec 0,&1) /\ IMAGE f (sphere(vec 0,&1)) SUBSET sphere(vec 0,&1) ==> ?x. x IN sphere(vec 0,&1) /\ (f x = --x \/ f x = x)`, REPEAT STRIP_TAC THEN REWRITE_TAC[LEFT_OR_DISTRIB; EXISTS_OR_THM] THEN MATCH_MP_TAC(TAUT `(~p ==> q) ==> p \/ q`) THEN DISCH_TAC THEN MATCH_MP_TAC FIXPOINT_HOMOTOPIC_IDENTITY_SPHERE THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC HOMOTOPIC_NON_ANTIPODAL_SPHEREMAPS THEN ASM_REWRITE_TAC[IMAGE_ID; SUBSET_REFL; CONTINUOUS_ON_ID] THEN ASM_MESON_TAC[VECTOR_NEG_NEG]);; let HAIRY_BALL_THEOREM_ALT,HAIRY_BALL_THEOREM = (CONJ_PAIR o prove) (`(!r. (?f. f continuous_on sphere(vec 0:real^N,r) /\ (!x. x IN sphere(vec 0,r) ==> ~(f x = vec 0) /\ orthogonal x (f x))) <=> r <= &0 \/ EVEN(dimindex(:N))) /\ (!r. (?f. f continuous_on sphere(vec 0:real^N,r) /\ IMAGE f (sphere(vec 0,r)) SUBSET sphere(vec 0,r) /\ (!x. x IN sphere(vec 0,r) ==> ~(f x = vec 0) /\ orthogonal x (f x))) <=> r < &0 \/ &0 < r /\ EVEN(dimindex(:N)))`, REWRITE_TAC[AND_FORALL_THM] THEN X_GEN_TAC `r:real` THEN ASM_CASES_TAC `r < &0` THEN ASM_SIMP_TAC[SPHERE_EMPTY; NOT_IN_EMPTY; IMAGE_CLAUSES; EMPTY_SUBSET; CONTINUOUS_ON_EMPTY; REAL_LT_IMP_LE] THEN ASM_CASES_TAC `r = &0` THEN ASM_REWRITE_TAC[REAL_LE_REFL; REAL_LT_REFL] THENL [SIMP_TAC[SPHERE_SING; FORALL_IN_INSERT; NOT_IN_EMPTY; SUBSET; FORALL_IN_IMAGE] THEN CONJ_TAC THENL [ALL_TAC; MESON_TAC[IN_SING]] THEN EXISTS_TAC `(\x. basis 1):real^N->real^N` THEN SIMP_TAC[CONTINUOUS_ON_CONST; ORTHOGONAL_0; BASIS_NONZERO; LE_REFL; DIMINDEX_GE_1]; ALL_TAC] THEN SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ASM_SIMP_TAC[GSYM REAL_NOT_LT]] THEN MATCH_MP_TAC(TAUT `(q ==> p) /\ (p ==> r) /\ (r ==> q) ==> (p <=> r) /\ (q <=> r)`) THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[]; REWRITE_TAC[GSYM NOT_ODD] THEN REPEAT STRIP_TAC THEN MP_TAC(SPEC `\x. inv(norm(f(r % x))) % (f:real^N->real^N) (r % x)` FIXPOINT_OR_NEG_MAPPING_SPHERE) THEN ASM_REWRITE_TAC[NOT_IMP] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_MUL THEN CONJ_TAC THENL [REWRITE_TAC[o_DEF] THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_LIFT_NORM_COMPOSE; X_GEN_TAC `x:real^N` THEN FIRST_X_ASSUM(MP_TAC o SPEC `r % x:real^N`) THEN ASM_SIMP_TAC[NORM_MUL; real_abs; REAL_LT_IMP_LE; NORM_EQ_0; IN_SPHERE_0; REAL_MUL_RID]]; ALL_TAC] THEN ONCE_REWRITE_TAC[GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_SIMP_TAC[GSYM SPHERE_SCALING; CONTINUOUS_ON_CMUL; CONTINUOUS_ON_ID; VECTOR_MUL_RZERO; REAL_MUL_RID]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_SPHERE_0] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN REWRITE_TAC[NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN MATCH_MP_TAC REAL_MUL_LINV THEN ASM_SIMP_TAC[NORM_MUL; real_abs; REAL_LT_IMP_LE; NORM_EQ_0; IN_SPHERE_0; REAL_MUL_RID]; REWRITE_TAC[IN_SPHERE_0; VECTOR_ARITH `a:real^N = --x <=> --a = x`] THEN DISCH_THEN(X_CHOOSE_THEN `x:real^N` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPEC `r % x:real^N`) THEN ASM_SIMP_TAC[NORM_MUL; real_abs; REAL_LT_IMP_LE; NORM_EQ_0; IN_SPHERE_0; REAL_MUL_RID] THEN ASM_SIMP_TAC[ORTHOGONAL_MUL; REAL_LT_IMP_NZ] THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (RAND_CONV o RAND_CONV o LAND_CONV) [SYM th]) THEN REWRITE_TAC[ORTHOGONAL_MUL; ORTHOGONAL_LNEG; ORTHOGONAL_REFL; REAL_INV_EQ_0; NORM_EQ_0] THEN CONV_TAC TAUT]; REWRITE_TAC[EVEN_EXISTS] THEN DISCH_THEN(X_CHOOSE_TAC `n:num`) THEN EXISTS_TAC `(\x. lambda i. if EVEN(i) then --(x$(i-1)) else x$(i+1)): real^N->real^N` THEN CONJ_TAC THENL [MATCH_MP_TAC LINEAR_CONTINUOUS_ON THEN SIMP_TAC[linear; CART_EQ; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; LAMBDA_BETA; REAL_NEG_ADD; GSYM REAL_MUL_RNEG] THEN MESON_TAC[]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_SPHERE_0; GSYM DOT_EQ_0] THEN SIMP_TAC[orthogonal; dot; LAMBDA_BETA; NORM_EQ_SQUARE]] THEN SUBGOAL_THEN `1..dimindex(:N) = 2*0+1..(2 * (n - 1) + 1) + 1` SUBST1_TAC THENL [BINOP_TAC THEN REWRITE_TAC[ADD_CLAUSES; MULT_CLAUSES] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ARITH_RULE `m = 2 * n ==> 1 <= m ==> m = (2 * (n - 1) + 1) + 1`)) THEN REWRITE_TAC[DIMINDEX_GE_1]; REWRITE_TAC[SUM_OFFSET; SUM_PAIR]] THEN REWRITE_TAC[EVEN_ADD; EVEN_MULT; ARITH; ADD_SUB] THEN REWRITE_TAC[REAL_ARITH `a + --x * --y:real = x * y + a`] THEN ASM_SIMP_TAC[REAL_POW_EQ_0; REAL_LT_IMP_NZ] THEN REWRITE_TAC[REAL_ARITH `x + y * --z = x - z * y`; REAL_SUB_REFL; SUM_0]]);; let CONTINUOUS_FUNCTION_HAS_EIGENVALUES_ODD_DIM = prove (`!f:real^N->real^N. ODD(dimindex(:N)) /\ f continuous_on sphere(vec 0:real^N,&1) ==> ?v c. v IN sphere(vec 0,&1) /\ f v = c % v`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `!v. norm v = &1 ==> ~((f:real^N->real^N) v = vec 0)` THENL [ALL_TAC; ASM_MESON_TAC[VECTOR_MUL_LZERO; IN_SPHERE_0]] THEN MP_TAC(ISPEC `\x. inv(norm(f x)) % (f:real^N->real^N) x` FIXPOINT_OR_NEG_MAPPING_SPHERE) THEN ASM_REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_SPHERE_0] THEN ANTS_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_MUL THEN ASM_REWRITE_TAC[o_DEF] THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN ASM_SIMP_TAC[CONTINUOUS_ON_LIFT_NORM_COMPOSE; IN_SPHERE_0]; REWRITE_TAC[NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM]]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `v:real^N` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o AP_TERM `(%) (norm((f:real^N->real^N) v)):real^N->real^N`)] THEN ASM_SIMP_TAC[VECTOR_MUL_LID; VECTOR_MUL_ASSOC; REAL_MUL_RINV; NORM_EQ_0; REAL_MUL_LINV] THEN ASM_MESON_TAC[VECTOR_MUL_RNEG; VECTOR_MUL_LNEG]);; let EULER_ROTATION_THEOREM_GEN = prove (`!A:real^N^N. ODD(dimindex(:N)) /\ rotation_matrix A ==> ?v. norm v = &1 /\ A ** v = v`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [rotation_matrix]) THEN ASM_CASES_TAC `!v:real^N. v IN sphere (vec 0,&1) ==> ~(A ** v = v)` THENL [ALL_TAC; ASM_MESON_TAC[IN_SPHERE_0]] THEN MP_TAC(ISPECL [`\x:real^N. (A:real^N^N) ** x`; `\x:real^N. --x`; `sphere(vec 0:real^N,&1)`; `&1`] HOMOTOPIC_NON_ANTIPODAL_SPHEREMAPS) THEN ASM_REWRITE_TAC[VECTOR_NEG_NEG] THEN ANTS_TAC THENL [SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_COMPOSE_NEG; LINEAR_ID; MATRIX_VECTOR_MUL_LINEAR] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_SPHERE_0; NORM_NEG] THEN MATCH_MP_TAC(MESON[ORTHOGONAL_TRANSFORMATION] `orthogonal_transformation(f:real^N->real^N) ==> !x. norm x = a ==> norm(f x) = a`) THEN ASM_REWRITE_TAC[GSYM ORTHOGONAL_MATRIX_TRANSFORMATION]; DISCH_THEN(MP_TAC o MATCH_MP(ONCE_REWRITE_RULE[IMP_CONJ_ALT] (REWRITE_RULE[CONJ_ASSOC] HOMOTOPIC_ORTHOGONAL_TRANSFORMATIONS_IMP))) THEN ASM_SIMP_TAC[ORTHOGONAL_TRANSFORMATION_NEG; ORTHOGONAL_TRANSFORMATION_ID; GSYM ORTHOGONAL_MATRIX_TRANSFORMATION] THEN SIMP_TAC[MATRIX_NEG; LINEAR_ID; MATRIX_OF_MATRIX_VECTOR_MUL] THEN ASM_REWRITE_TAC[MATRIX_ID; DET_NEG; DET_I; REAL_POW_NEG; GSYM NOT_ODD] THEN REWRITE_TAC[REAL_POW_ONE] THEN CONV_TAC REAL_RAT_REDUCE_CONV]);; (* ------------------------------------------------------------------------- *) (* Retractions. *) (* ------------------------------------------------------------------------- *) parse_as_infix("retract_of",(12,"right"));; let retraction = new_definition `retraction (s,t) (r:real^N->real^N) <=> t SUBSET s /\ r continuous_on s /\ (IMAGE r s SUBSET t) /\ (!x. x IN t ==> (r x = x))`;; let retract_of = new_definition `t retract_of s <=> ?r. retraction (s,t) r`;; let RETRACTION_MAPS_EUCLIDEAN = prove (`!r s t:real^N->bool. retraction_maps (subtopology euclidean s,subtopology euclidean t) (r,I) <=> retraction (s,t) r`, REWRITE_TAC[retraction_maps; retraction; I_DEF] THEN REWRITE_TAC[CONTINUOUS_MAP_EUCLIDEAN; CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN REWRITE_TAC[CONTINUOUS_ON_ID; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; IMAGE_ID] THEN REWRITE_TAC[CONJ_ACI]);; let RETRACT_OF_SPACE_EUCLIDEAN = prove (`!s t:real^N->bool. t retract_of_space (subtopology euclidean s) <=> t retract_of s`, REWRITE_TAC[retract_of; retract_of_space; retraction] THEN REWRITE_TAC[CONTINUOUS_MAP_EUCLIDEAN2; SUBTOPOLOGY_SUBTOPOLOGY] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN SET_TAC[]);; let RETRACTION = prove (`!s t r. retraction (s,t) r <=> t SUBSET s /\ r continuous_on s /\ IMAGE r s = t /\ (!x. x IN t ==> r x = x)`, REWRITE_TAC[retraction] THEN SET_TAC[]);; let RETRACT_OF_IMP_EXTENSIBLE = prove (`!f:real^M->real^N u s t. s retract_of t /\ f continuous_on s /\ IMAGE f s SUBSET u ==> ?g. g continuous_on t /\ IMAGE g t SUBSET u /\ (!x. x IN s ==> g x = f x)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retract_of]) THEN REWRITE_TAC[RETRACTION; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `r:real^M->real^M` THEN STRIP_TAC THEN EXISTS_TAC `(f:real^M->real^N) o (r:real^M->real^M)` THEN REWRITE_TAC[IMAGE_o; o_THM] THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE; ASM SET_TAC[]] THEN ASM_MESON_TAC[]);; let RETRACTION_IDEMPOTENT = prove (`!r s t. retraction (s,t) r ==> !x. x IN s ==> (r(r(x)) = r(x))`, REWRITE_TAC[retraction; SUBSET; FORALL_IN_IMAGE] THEN MESON_TAC[]);; let IDEMPOTENT_IMP_RETRACTION = prove (`!f:real^N->real^N s. f continuous_on s /\ IMAGE f s SUBSET s /\ (!x. x IN s ==> f(f x) = f x) ==> retraction (s,IMAGE f s) f`, REWRITE_TAC[retraction] THEN SET_TAC[]);; let RETRACTION_SUBSET = prove (`!r s s' t. retraction (s,t) r /\ t SUBSET s' /\ s' SUBSET s ==> retraction (s',t) r`, SIMP_TAC[retraction] THEN MESON_TAC[IMAGE_SUBSET; SUBSET_TRANS; CONTINUOUS_ON_SUBSET]);; let RETRACT_OF_SUBSET = prove (`!s s' t. t retract_of s /\ t SUBSET s' /\ s' SUBSET s ==> t retract_of s'`, REPEAT GEN_TAC THEN REWRITE_TAC[retract_of; LEFT_AND_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN MESON_TAC[RETRACTION_SUBSET]);; let RETRACT_OF_TRANSLATION = prove (`!a t s:real^N->bool. t retract_of s ==> (IMAGE (\x. a + x) t) retract_of (IMAGE (\x. a + x) s)`, REPEAT GEN_TAC THEN REWRITE_TAC[retract_of; retraction] THEN DISCH_THEN(X_CHOOSE_THEN `r:real^N->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(\x:real^N. a + x) o r o (\x:real^N. --a + x)` THEN ASM_SIMP_TAC[IMAGE_SUBSET; FORALL_IN_IMAGE] THEN REPEAT CONJ_TAC THENL [REPEAT(MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID]) THEN ASM_REWRITE_TAC[GSYM IMAGE_o; o_DEF; VECTOR_ARITH `--a + a + x:real^N = x`; IMAGE_ID]; REWRITE_TAC[IMAGE_o] THEN MATCH_MP_TAC IMAGE_SUBSET THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV o ONCE_DEPTH_CONV) [GSYM IMAGE_o] THEN ASM_REWRITE_TAC[o_DEF; VECTOR_ARITH `--a + a + x:real^N = x`; IMAGE_ID]; ASM_SIMP_TAC[o_DEF; VECTOR_ARITH `--a + a + x:real^N = x`]]);; let RETRACT_OF_TRANSLATION_EQ = prove (`!a t s:real^N->bool. (IMAGE (\x. a + x) t) retract_of (IMAGE (\x. a + x) s) <=> t retract_of s`, REPEAT GEN_TAC THEN EQ_TAC THEN REWRITE_TAC[RETRACT_OF_TRANSLATION] THEN DISCH_THEN(MP_TAC o SPEC `--a:real^N` o MATCH_MP RETRACT_OF_TRANSLATION) THEN REWRITE_TAC[GSYM IMAGE_o; o_DEF; IMAGE_ID; VECTOR_ARITH `--a + a + x:real^N = x`]);; add_translation_invariants [RETRACT_OF_TRANSLATION_EQ];; let RETRACT_OF_INJECTIVE_LINEAR_IMAGE = prove (`!f:real^M->real^N s t. linear f /\ (!x y. f x = f y ==> x = y) /\ t retract_of s ==> (IMAGE f t) retract_of (IMAGE f s)`, REPEAT GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN REWRITE_TAC[retract_of; retraction] THEN DISCH_THEN(X_CHOOSE_THEN `r:real^M->real^M` STRIP_ASSUME_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 EXISTS_TAC `(f:real^M->real^N) o r o (g:real^N->real^M)` THEN UNDISCH_THEN `!x y. (f:real^M->real^N) x = f y ==> x = y` (K ALL_TAC) THEN ASM_SIMP_TAC[IMAGE_SUBSET; FORALL_IN_IMAGE] THEN REPEAT CONJ_TAC THENL [REPEAT(MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_SIMP_TAC[LINEAR_CONTINUOUS_ON]) THEN ASM_REWRITE_TAC[GSYM IMAGE_o; o_DEF; IMAGE_ID]; REWRITE_TAC[IMAGE_o] THEN MATCH_MP_TAC IMAGE_SUBSET THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV o ONCE_DEPTH_CONV) [GSYM IMAGE_o] THEN ASM_REWRITE_TAC[o_DEF; IMAGE_ID]; ASM_SIMP_TAC[o_DEF]]);; let RETRACT_OF_LINEAR_IMAGE_EQ = prove (`!f:real^M->real^N s t. linear f /\ (!x y. f x = f y ==> x = y) /\ (!y. ?x. f x = y) ==> ((IMAGE f t) retract_of (IMAGE f s) <=> t retract_of s)`, REPEAT GEN_TAC THEN DISCH_TAC THEN EQ_TAC THENL [DISCH_TAC; ASM_MESON_TAC[RETRACT_OF_INJECTIVE_LINEAR_IMAGE]] THEN FIRST_ASSUM(X_CHOOSE_THEN `h:real^N->real^M` STRIP_ASSUME_TAC o MATCH_MP LINEAR_BIJECTIVE_LEFT_RIGHT_INVERSE) THEN SUBGOAL_THEN `!s. s = IMAGE (h:real^N->real^M) (IMAGE (f:real^M->real^N) s)` (fun th -> ONCE_REWRITE_TAC[th]) THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC RETRACT_OF_INJECTIVE_LINEAR_IMAGE THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]);; add_linear_invariants [RETRACT_OF_LINEAR_IMAGE_EQ];; let RETRACTION_REFL = prove (`!s. retraction (s,s) (\x. x)`, REWRITE_TAC[retraction; IMAGE_ID; SUBSET_REFL; CONTINUOUS_ON_ID]);; let RETRACT_OF_REFL = prove (`!s. s retract_of s`, REWRITE_TAC[retract_of] THEN MESON_TAC[RETRACTION_REFL]);; let RETRACTION_CLOSEST_POINT = prove (`!s t:real^N->bool. convex t /\ closed t /\ ~(t = {}) /\ t SUBSET s ==> retraction (s,t) (closest_point t)`, REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[retraction] THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE; CLOSEST_POINT_SELF; CLOSEST_POINT_IN_SET; CONTINUOUS_ON_CLOSEST_POINT]);; let RETRACT_OF_IMP_SUBSET = prove (`!s t. s retract_of t ==> s SUBSET t`, SIMP_TAC[retract_of; retraction] THEN MESON_TAC[]);; let RETRACT_OF_EMPTY = prove (`(!s:real^N->bool. {} retract_of s <=> s = {}) /\ (!s:real^N->bool. s retract_of {} <=> s = {})`, REWRITE_TAC[retract_of; retraction; SUBSET_EMPTY; IMAGE_CLAUSES] THEN CONJ_TAC THEN X_GEN_TAC `s:real^N->bool` THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY; IMAGE_EQ_EMPTY; CONTINUOUS_ON_EMPTY; SUBSET_REFL]);; let RETRACT_OF_SING = prove (`!s x:real^N. {x} retract_of s <=> x IN s`, REPEAT GEN_TAC THEN REWRITE_TAC[retract_of; RETRACTION] THEN EQ_TAC THENL [SET_TAC[]; ALL_TAC] THEN DISCH_TAC THEN EXISTS_TAC `(\y. x):real^N->real^N` THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN ASM SET_TAC[]);; let RETRACT_OF_OPEN_UNION = prove (`!s t:real^N->bool. open_in (subtopology euclidean (s UNION t)) s /\ open_in (subtopology euclidean (s UNION t)) t /\ DISJOINT s t /\ (s = {} ==> t = {}) ==> s retract_of (s UNION t)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_SIMP_TAC[RETRACT_OF_EMPTY; UNION_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 STRIP_TAC THEN REWRITE_TAC[retract_of; retraction] THEN EXISTS_TAC `\x:real^N. if x IN s then x else a` THEN SIMP_TAC[SUBSET_UNION] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_ON_UNION_LOCAL_OPEN THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_EQ THENL [EXISTS_TAC `\x:real^N. x`; EXISTS_TAC `(\x. a):real^N->real^N`] THEN REWRITE_TAC[CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN ASM SET_TAC[]);; let RETRACT_OF_SEPARATED_UNION = prove (`!s t:real^N->bool. s INTER closure t = {} /\ t INTER closure s = {} /\ (s = {} ==> t = {}) ==> s retract_of (s UNION t)`, REWRITE_TAC[CONJ_ASSOC; SEPARATION_OPEN_IN_UNION] THEN MESON_TAC[RETRACT_OF_OPEN_UNION]);; let RETRACT_OF_CLOSED_UNION = prove (`!s t:real^N->bool. closed_in (subtopology euclidean (s UNION t)) s /\ closed_in (subtopology euclidean (s UNION t)) t /\ DISJOINT s t /\ (s = {} ==> t = {}) ==> s retract_of (s UNION t)`, ONCE_REWRITE_TAC[TAUT `p /\ q /\ r /\ s <=> (r /\ p /\ q) /\ s`] THEN REWRITE_TAC[GSYM SEPARATION_CLOSED_IN_UNION] THEN MESON_TAC[RETRACT_OF_SEPARATED_UNION]);; let RETRACTION_o = prove (`!f g s t u:real^N->bool. retraction (s,t) f /\ retraction (t,u) g ==> retraction (s,u) (g o f)`, REPEAT GEN_TAC THEN REWRITE_TAC[retraction] THEN REPEAT STRIP_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; REWRITE_TAC[IMAGE_o] THEN ASM SET_TAC[]; REWRITE_TAC[o_THM] THEN ASM SET_TAC[]]);; let RETRACT_OF_TRANS = prove (`!s t u:real^N->bool. s retract_of t /\ t retract_of u ==> s retract_of u`, REWRITE_TAC[retract_of] THEN MESON_TAC[RETRACTION_o]);; let CLOSED_IN_RETRACT = prove (`!s t:real^N->bool. s retract_of t ==> closed_in (subtopology euclidean t) s`, REPEAT GEN_TAC THEN REWRITE_TAC[retract_of; retraction] THEN DISCH_THEN(X_CHOOSE_THEN `r:real^N->real^N` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `s = {x:real^N | x IN t /\ lift(norm(r x - x)) = vec 0}` SUBST1_TAC THENL [REWRITE_TAC[GSYM DROP_EQ; DROP_VEC; LIFT_DROP; NORM_EQ_0] THEN REWRITE_TAC[VECTOR_SUB_EQ] THEN ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE_CONSTANT THEN MATCH_MP_TAC CONTINUOUS_ON_LIFT_NORM_COMPOSE THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN ASM_SIMP_TAC[CONTINUOUS_ON_ID]]);; let RETRACT_OF_CONTRACTIBLE = prove (`!s t:real^N->bool. contractible t /\ s retract_of t ==> contractible s`, REPEAT GEN_TAC THEN REWRITE_TAC[contractible; retract_of] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC (X_CHOOSE_TAC `r:real^N->real^N`)) THEN SIMP_TAC[HOMOTOPIC_WITH_EUCLIDEAN_ALT; PCROSS; LEFT_IMP_EXISTS_THM] THEN FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [retraction]) THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `h:real^(1,N)finite_sum->real^N`] THEN RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; FORALL_IN_IMAGE]) THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`(r:real^N->real^N) a`; `(r:real^N->real^N) o (h:real^(1,N)finite_sum->real^N)`] THEN ASM_SIMP_TAC[o_THM; IMAGE_o; SUBSET] THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP(REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; ASM SET_TAC[]]);; let RETRACT_OF_COMPACT = prove (`!s t:real^N->bool. compact t /\ s retract_of t ==> compact s`, REWRITE_TAC[retract_of; RETRACTION] THEN MESON_TAC[COMPACT_CONTINUOUS_IMAGE]);; let RETRACT_OF_CLOSED = prove (`!s t. closed t /\ s retract_of t ==> closed s`, MESON_TAC[CLOSED_IN_CLOSED_EQ; CLOSED_IN_RETRACT]);; let RETRACT_OF_CONNECTED = prove (`!s t:real^N->bool. connected t /\ s retract_of t ==> connected s`, REWRITE_TAC[retract_of; RETRACTION] THEN MESON_TAC[CONNECTED_CONTINUOUS_IMAGE]);; let RETRACT_OF_PATH_CONNECTED = prove (`!s t:real^N->bool. path_connected t /\ s retract_of t ==> path_connected s`, REWRITE_TAC[retract_of; RETRACTION] THEN MESON_TAC[PATH_CONNECTED_CONTINUOUS_IMAGE]);; let RETRACT_OF_SIMPLY_CONNECTED = prove (`!s t:real^N->bool. simply_connected t /\ s retract_of t ==> simply_connected s`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] (REWRITE_RULE[CONJ_ASSOC] SIMPLY_CONNECTED_RETRACTION_GEN)) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retract_of]) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `r:real^N->real^N` THEN REWRITE_TAC[RETRACTION] THEN STRIP_TAC THEN EXISTS_TAC `\x:real^N. x` THEN ASM_REWRITE_TAC[IMAGE_ID; CONTINUOUS_ON_ID]);; let RETRACT_OF_HOMOTOPICALLY_TRIVIAL = prove (`!s t:real^N->bool u:real^M->bool. t retract_of s /\ (!f g. f continuous_on u /\ IMAGE f u SUBSET s /\ g continuous_on u /\ IMAGE g u SUBSET s ==> homotopic_with (\x. T) (subtopology euclidean u,subtopology euclidean s) f g) ==> (!f g. f continuous_on u /\ IMAGE f u SUBSET t /\ g continuous_on u /\ IMAGE g u SUBSET t ==> homotopic_with (\x. T) (subtopology euclidean u,subtopology euclidean t) f g)`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r /\ s <=> p /\ q /\ T /\ r /\ s /\ T`] THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] HOMOTOPICALLY_TRIVIAL_RETRACTION_GEN) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retract_of]) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `r:real^N->real^N` THEN REWRITE_TAC[RETRACTION] THEN STRIP_TAC THEN EXISTS_TAC `\x:real^N. x` THEN ASM_REWRITE_TAC[CONTINUOUS_ON_ID; IMAGE_ID]);; let RETRACT_OF_HOMOTOPICALLY_TRIVIAL_NULL = prove (`!s t:real^N->bool u:real^M->bool. t retract_of s /\ (!f. f continuous_on u /\ IMAGE f u SUBSET s ==> ?c. homotopic_with (\x. T) (subtopology euclidean u,subtopology euclidean s) f (\x. c)) ==> (!f. f continuous_on u /\ IMAGE f u SUBSET t ==> ?c. homotopic_with (\x. T) (subtopology euclidean u,subtopology euclidean t) f (\x. c))`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ONCE_REWRITE_TAC[TAUT `p /\ q <=> p /\ q /\ T`] THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] HOMOTOPICALLY_TRIVIAL_RETRACTION_NULL_GEN) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retract_of]) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `r:real^N->real^N` THEN REWRITE_TAC[RETRACTION] THEN STRIP_TAC THEN EXISTS_TAC `\x:real^N. x` THEN ASM_REWRITE_TAC[CONTINUOUS_ON_ID; IMAGE_ID]);; let RETRACT_OF_COHOMOTOPICALLY_TRIVIAL = prove (`!s t:real^N->bool u:real^M->bool. t retract_of s /\ (!f g. f continuous_on s /\ IMAGE f s SUBSET u /\ g continuous_on s /\ IMAGE g s SUBSET u ==> homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean u) f g) ==> (!f g. f continuous_on t /\ IMAGE f t SUBSET u /\ g continuous_on t /\ IMAGE g t SUBSET u ==> homotopic_with (\x. T) (subtopology euclidean t,subtopology euclidean u) f g)`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r /\ s <=> p /\ q /\ T /\ r /\ s /\ T`] THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] COHOMOTOPICALLY_TRIVIAL_RETRACTION_GEN) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retract_of]) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `r:real^N->real^N` THEN REWRITE_TAC[RETRACTION] THEN STRIP_TAC THEN EXISTS_TAC `\x:real^N. x` THEN ASM_REWRITE_TAC[CONTINUOUS_ON_ID; IMAGE_ID]);; let RETRACT_OF_COHOMOTOPICALLY_TRIVIAL_NULL = prove (`!s t:real^N->bool u:real^M->bool. t retract_of s /\ (!f. f continuous_on s /\ IMAGE f s SUBSET u ==> ?c. homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean u) f (\x. c)) ==> (!f. f continuous_on t /\ IMAGE f t SUBSET u ==> ?c. homotopic_with (\x. T) (subtopology euclidean t,subtopology euclidean u) f (\x. c))`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ONCE_REWRITE_TAC[TAUT `p /\ q <=> p /\ q /\ T`] THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] COHOMOTOPICALLY_TRIVIAL_RETRACTION_NULL_GEN) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retract_of]) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `r:real^N->real^N` THEN REWRITE_TAC[RETRACTION] THEN STRIP_TAC THEN EXISTS_TAC `\x:real^N. x` THEN ASM_REWRITE_TAC[CONTINUOUS_ON_ID; IMAGE_ID]);; let RETRACTION_IMP_QUOTIENT_MAP_EXPLICIT = prove (`!r s t:real^N->bool. retraction (s,t) r ==> !u. u SUBSET t ==> (open_in (subtopology euclidean s) {x | x IN s /\ r x IN u} <=> open_in (subtopology euclidean t) u)`, REPEAT GEN_TAC THEN REWRITE_TAC[RETRACTION] THEN STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_RIGHT_INVERSE_IMP_QUOTIENT_MAP THEN EXISTS_TAC `\x:real^N. x` THEN ASM_REWRITE_TAC[CONTINUOUS_ON_ID; SUBSET_REFL; IMAGE_ID]);; let RETRACT_OF_LOCALLY_CONNECTED = prove (`!s t:real^N->bool. s retract_of t /\ locally connected t ==> locally connected s`, REPEAT GEN_TAC THEN REWRITE_TAC[retract_of] THEN DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN FIRST_ASSUM(SUBST1_TAC o SYM o el 2 o CONJUNCTS o GEN_REWRITE_RULE I [RETRACTION]) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LOCALLY_CONNECTED_QUOTIENT_IMAGE) THEN MATCH_MP_TAC RETRACTION_IMP_QUOTIENT_MAP_EXPLICIT THEN ASM_MESON_TAC[RETRACTION]);; let RETRACT_OF_LOCALLY_PATH_CONNECTED = prove (`!s t:real^N->bool. s retract_of t /\ locally path_connected t ==> locally path_connected s`, REPEAT GEN_TAC THEN REWRITE_TAC[retract_of] THEN DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN FIRST_ASSUM(SUBST1_TAC o SYM o el 2 o CONJUNCTS o GEN_REWRITE_RULE I [RETRACTION]) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LOCALLY_PATH_CONNECTED_QUOTIENT_IMAGE) THEN MATCH_MP_TAC RETRACTION_IMP_QUOTIENT_MAP_EXPLICIT THEN ASM_MESON_TAC[RETRACTION]);; let RETRACT_OF_LOCALLY_COMPACT = prove (`!s t:real^N->bool. locally compact s /\ t retract_of s ==> locally compact t`, MESON_TAC[CLOSED_IN_RETRACT; LOCALLY_COMPACT_CLOSED_IN]);; let RETRACT_OF_PCROSS = prove (`!s:real^M->bool s' t:real^N->bool t'. s retract_of s' /\ t retract_of t' ==> (s PCROSS t) retract_of (s' PCROSS t')`, REPEAT GEN_TAC THEN REWRITE_TAC[PCROSS] THEN REWRITE_TAC[retract_of; retraction; SUBSET; FORALL_IN_IMAGE] THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `f:real^M->real^M` STRIP_ASSUME_TAC) (X_CHOOSE_THEN `g:real^N->real^N` STRIP_ASSUME_TAC)) THEN EXISTS_TAC `\z. pastecart ((f:real^M->real^M) (fstcart z)) ((g:real^N->real^N) (sndcart z))` THEN REWRITE_TAC[FORALL_PASTECART; IN_ELIM_PASTECART_THM] THEN ASM_SIMP_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN MATCH_MP_TAC CONTINUOUS_ON_PASTECART THEN CONJ_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; LINEAR_SNDCART] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN SIMP_TAC[FSTCART_PASTECART; SNDCART_PASTECART]);; let RETRACT_OF_PCROSS_EQ = prove (`!s s':real^M->bool t t':real^N->bool. s PCROSS t retract_of s' PCROSS t' <=> (s = {} \/ t = {}) /\ (s' = {} \/ t' = {}) \/ s retract_of s' /\ t retract_of t'`, REPEAT GEN_TAC THEN MAP_EVERY ASM_CASES_TAC [`s:real^M->bool = {}`; `s':real^M->bool = {}`; `t:real^N->bool = {}`; `t':real^N->bool = {}`] THEN ASM_REWRITE_TAC[PCROSS_EMPTY; RETRACT_OF_EMPTY; PCROSS_EQ_EMPTY] THEN EQ_TAC THEN REWRITE_TAC[RETRACT_OF_PCROSS] THEN REWRITE_TAC[retract_of; retraction; SUBSET; FORALL_IN_PCROSS; FORALL_IN_IMAGE; PASTECART_IN_PCROSS] THEN DISCH_THEN(X_CHOOSE_THEN `r:real^(M,N)finite_sum->real^(M,N)finite_sum` STRIP_ASSUME_TAC) THEN CONJ_TAC THENL [SUBGOAL_THEN `?b:real^N. b IN t` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN EXISTS_TAC `\x. fstcart((r:real^(M,N)finite_sum->real^(M,N)finite_sum) (pastecart x b))` THEN ASM_SIMP_TAC[FSTCART_PASTECART] THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[]; GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[LINEAR_FSTCART; LINEAR_CONTINUOUS_ON] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_PASTECART; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE; PASTECART_IN_PCROSS] THEN ASM_MESON_TAC[MEMBER_NOT_EMPTY]; ASM_MESON_TAC[PASTECART_FST_SND; PASTECART_IN_PCROSS; MEMBER_NOT_EMPTY]]; SUBGOAL_THEN `?a:real^M. a IN s` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN EXISTS_TAC `\x. sndcart((r:real^(M,N)finite_sum->real^(M,N)finite_sum) (pastecart a x))` THEN ASM_SIMP_TAC[SNDCART_PASTECART] THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[]; GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[LINEAR_SNDCART; LINEAR_CONTINUOUS_ON] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_PASTECART; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE; PASTECART_IN_PCROSS] THEN ASM_MESON_TAC[MEMBER_NOT_EMPTY]; ASM_MESON_TAC[PASTECART_FST_SND; PASTECART_IN_PCROSS; MEMBER_NOT_EMPTY]]]);; let HOMOTOPIC_INTO_RETRACT = prove (`!f:real^M->real^N g s t u. IMAGE f s SUBSET t /\ IMAGE g s SUBSET t /\ t retract_of u /\ homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean u) f g ==> homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean t) f g`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMOTOPIC_WITH_EUCLIDEAN]) THEN SIMP_TAC[HOMOTOPIC_WITH_EUCLIDEAN_ALT; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `h:real^(1,M)finite_sum->real^N` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retract_of]) THEN REWRITE_TAC[retraction; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `r:real^N->real^N` THEN STRIP_TAC THEN EXISTS_TAC `(r:real^N->real^N) o (h:real^(1,M)finite_sum->real^N)` THEN ASM_SIMP_TAC[o_THM; IMAGE_o] THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE; ASM SET_TAC[]] THEN CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* Brouwer fixed-point theorem and related results. *) (* ------------------------------------------------------------------------- *) let CONTRACTIBLE_SPHERE = prove (`!a:real^N r. contractible(sphere(a,r)) <=> r <= &0`, GEOM_ORIGIN_TAC `a:real^N` THEN REPEAT GEN_TAC THEN ASM_CASES_TAC `r < &0` THEN ASM_SIMP_TAC[SPHERE_EMPTY; CONTRACTIBLE_EMPTY; REAL_LT_IMP_LE] THEN RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LT]) THEN FIRST_X_ASSUM(X_CHOOSE_THEN `b:real^N` (SUBST1_TAC o SYM) o MATCH_MP VECTOR_CHOOSE_SIZE) THEN REWRITE_TAC[NORM_ARITH `norm(b:real^N) <= &0 <=> b = vec 0`] THEN GEOM_NORMALIZE_TAC `b:real^N` THEN SIMP_TAC[NORM_0; SPHERE_SING; CONTRACTIBLE_SING] THEN X_GEN_TAC `b:real^N` THEN ASM_CASES_TAC `b:real^N = vec 0` THEN ASM_REWRITE_TAC[NORM_0; REAL_OF_NUM_EQ; ARITH_EQ] THEN DISCH_THEN(K ALL_TAC) THEN POP_ASSUM_LIST(K ALL_TAC) THEN DISCH_THEN(MP_TAC o ISPEC `I:real^N->real^N` o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] (REWRITE_RULE[CONJ_ASSOC] HOMOTOPIC_INTO_CONTRACTIBLE))) THEN DISCH_THEN(MP_TAC o SPECL [`reflect_along (basis 1:real^N)`; `sphere(vec 0:real^N,&1)`]) THEN REWRITE_TAC[CONTINUOUS_ON_ID; IMAGE_ID; I_DEF; NOT_IMP] THEN SIMP_TAC[SUBSET_REFL; LINEAR_CONTINUOUS_ON; LINEAR_REFLECT_ALONG; ORTHOGONAL_TRANSFORMATION_SPHERE; ORTHOGONAL_TRANSFORMATION_REFLECT_ALONG] THEN DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] (REWRITE_RULE[CONJ_ASSOC] HOMOTOPIC_ORTHOGONAL_TRANSFORMATIONS_IMP))) THEN REWRITE_TAC[ORTHOGONAL_TRANSFORMATION_REFLECT_ALONG; ORTHOGONAL_TRANSFORMATION_ID] THEN REWRITE_TAC[DET_MATRIX_REFLECT_ALONG; MATRIX_ID; DET_I] THEN SIMP_TAC[BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL] THEN CONV_TAC REAL_RAT_REDUCE_CONV);; let NO_RETRACTION_CBALL = prove (`!a:real^N e. &0 < e ==> ~(sphere(a,e) retract_of cball(a,e))`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] RETRACT_OF_CONTRACTIBLE)) THEN SIMP_TAC[CONVEX_IMP_CONTRACTIBLE; CONVEX_CBALL; CONTRACTIBLE_SPHERE] THEN ASM_REWRITE_TAC[REAL_NOT_LE]);; let BROUWER_BALL = prove (`!f:real^N->real^N a e. &0 < e /\ f continuous_on cball(a,e) /\ IMAGE f (cball(a,e)) SUBSET cball(a,e) ==> ?x. x IN cball(a,e) /\ f x = x`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[MESON[] `(?x. P x /\ Q x) <=> ~(!x. P x ==> ~Q x)`] THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o ISPEC `a:real^N` o MATCH_MP NO_RETRACTION_CBALL) THEN REWRITE_TAC[retract_of; retraction; SPHERE_SUBSET_CBALL] THEN ABBREV_TAC `s = \x:real^N. &4 * ((a - x:real^N) dot (f x - x)) pow 2 + &4 * (e pow 2 - norm(a - x) pow 2) * norm(f x - x) pow 2` THEN SUBGOAL_THEN `!x:real^N. x IN cball(a,e) ==> &0 <= s x` ASSUME_TAC THENL [X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_CBALL; dist] THEN DISCH_TAC THEN EXPAND_TAC "s" THEN REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_ADD THEN CONJ_TAC THEN MATCH_MP_TAC REAL_LE_MUL THEN REWRITE_TAC[REAL_POS; REAL_LE_POW_2] THEN MATCH_MP_TAC REAL_LE_MUL THEN REWRITE_TAC[REAL_LE_POW_2; REAL_SUB_LE] THEN MATCH_MP_TAC REAL_POW_LE2 THEN ASM_REWRITE_TAC[NORM_POS_LE]; ALL_TAC] THEN EXISTS_TAC `\x:real^N. x + (&2 * ((a - x) dot (f x - x)) - sqrt(s x)) / (&2 * ((f x - x) dot (f x - x))) % (f x - x)` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_ADD THEN REWRITE_TAC[CONTINUOUS_ON_ID] THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN ASM_SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_ID; o_DEF] THEN REWRITE_TAC[real_div; LIFT_CMUL] THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN CONJ_TAC THENL [REWRITE_TAC[o_DEF; LIFT_CMUL; LIFT_SUB] THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN ASM_SIMP_TAC[CONTINUOUS_ON_CMUL; CONTINUOUS_ON_SUB; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST; CONTINUOUS_ON_LIFT_DOT2] THEN MATCH_MP_TAC CONTINUOUS_ON_LIFT_SQRT_COMPOSE THEN ASM_REWRITE_TAC[o_DEF] THEN EXPAND_TAC "s" THEN REWRITE_TAC[LIFT_ADD; LIFT_CMUL; LIFT_SUB; NORM_POW_2; REAL_POW_2] THEN REPEAT((MATCH_MP_TAC CONTINUOUS_ON_ADD ORELSE MATCH_MP_TAC CONTINUOUS_ON_SUB ORELSE MATCH_MP_TAC CONTINUOUS_ON_MUL) THEN CONJ_TAC THEN REWRITE_TAC[o_DEF; LIFT_SUB]); MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN ASM_SIMP_TAC[REAL_ENTIRE; DOT_EQ_0; VECTOR_SUB_EQ] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[LIFT_CMUL]] THEN ASM_SIMP_TAC[CONTINUOUS_ON_CMUL; CONTINUOUS_ON_SUB; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST; CONTINUOUS_ON_LIFT_DOT2]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_SPHERE; IN_CBALL; dist; NORM_EQ_SQUARE] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN REWRITE_TAC[VECTOR_ARITH `a - (x + y):real^N = (a - x) - y`] THEN ONCE_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_LMUL] THEN REWRITE_TAC[DOT_RMUL] THEN REWRITE_TAC[REAL_RING `(a + u * u * b) - &2 * u * c = d <=> b * u pow 2 - (&2 * c) * u + (a - d) = &0`] THEN SUBGOAL_THEN `sqrt(s(x:real^N)) pow 2 = s x` MP_TAC THENL [ASM_SIMP_TAC[SQRT_POW_2; IN_CBALL; dist]; ALL_TAC] THEN MATCH_MP_TAC(REAL_FIELD `~(a = &0) /\ e = b pow 2 - &4 * a * c /\ x = (b - s) / (&2 * a) ==> s pow 2 = e ==> a * x pow 2 - b * x + c = &0`) THEN ASM_SIMP_TAC[DOT_EQ_0; VECTOR_SUB_EQ; IN_CBALL; dist] THEN EXPAND_TAC "s" THEN REWRITE_TAC[NORM_POW_2] THEN REAL_ARITH_TAC; X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_SPHERE; dist] THEN DISCH_TAC THEN EXPAND_TAC "s" THEN ASM_REWRITE_TAC[REAL_SUB_REFL] THEN REWRITE_TAC[REAL_MUL_LZERO; REAL_MUL_RZERO; REAL_ADD_RID] THEN REWRITE_TAC[VECTOR_ARITH `x + a:real^N = x <=> a = vec 0`] THEN REWRITE_TAC[VECTOR_MUL_EQ_0; REAL_DIV_EQ_0] THEN REPEAT DISJ1_TAC THEN REWRITE_TAC[REAL_ARITH `&2 * a - s = &0 <=> s = &2 * a`] THEN MATCH_MP_TAC SQRT_UNIQUE THEN CONJ_TAC THENL [ALL_TAC; REAL_ARITH_TAC] THEN REWRITE_TAC[REAL_ARITH `&0 <= &2 * x <=> &0 <= x`] THEN REWRITE_TAC[DOT_NORM_SUB; REAL_ARITH `&0 <= x / &2 <=> &0 <= x`] THEN REWRITE_TAC[VECTOR_ARITH `a - x - (y - x):real^N = a - y`] THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REAL_ARITH `&0 <= b /\ x <= a ==> &0 <= (a + b) - x`) THEN REWRITE_TAC[REAL_LE_POW_2] THEN MATCH_MP_TAC REAL_POW_LE2 THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN REWRITE_TAC[IN_CBALL; FORALL_IN_IMAGE; NORM_POS_LE] THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[dist] THEN CONV_TAC NORM_ARITH]);; let BROUWER = prove (`!f:real^N->real^N s. compact s /\ convex s /\ ~(s = {}) /\ f continuous_on s /\ IMAGE f s SUBSET s ==> ?x. x IN s /\ f x = x`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?e. &0 < e /\ s SUBSET cball(vec 0:real^N,e)` STRIP_ASSUME_TAC THENL [REWRITE_TAC[SUBSET; IN_CBALL; NORM_ARITH `dist(vec 0,x) = norm(x)`] THEN ASM_MESON_TAC[BOUNDED_POS; COMPACT_IMP_BOUNDED]; ALL_TAC] THEN SUBGOAL_THEN `?x:real^N. x IN cball(vec 0,e) /\ (f o closest_point s) x = x` MP_TAC THENL [MATCH_MP_TAC BROUWER_BALL THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [REWRITE_TAC[ETA_AX] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_SIMP_TAC[CONTINUOUS_ON_CLOSEST_POINT; COMPACT_IMP_CLOSED] THEN MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[SUBSET; FORALL_IN_IMAGE]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `x:real^N` THEN REPEAT STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET])) THEN REWRITE_TAC[o_THM; IN_IMAGE] THEN EXISTS_TAC `closest_point s x:real^N` THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED; CLOSEST_POINT_IN_SET]] THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED; CLOSEST_POINT_IN_SET]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[o_THM] THEN STRIP_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; FORALL_IN_IMAGE]) THEN ASM_MESON_TAC[CLOSEST_POINT_SELF; CLOSEST_POINT_IN_SET; COMPACT_IMP_CLOSED]]);; let BROUWER_WEAK = prove (`!f:real^N->real^N s. compact s /\ convex s /\ ~(interior s = {}) /\ f continuous_on s /\ IMAGE f s SUBSET s ==> ?x. x IN s /\ f x = x`, REPEAT STRIP_TAC THEN MATCH_MP_TAC BROUWER THEN ASM_MESON_TAC[INTERIOR_EMPTY]);; let BROUWER_CUBE = prove (`!f:real^N->real^N. f continuous_on (interval [vec 0,vec 1]) /\ IMAGE f (interval [vec 0,vec 1]) SUBSET (interval [vec 0,vec 1]) ==> ?x. x IN interval[vec 0,vec 1] /\ f x = x`, REPEAT STRIP_TAC THEN MATCH_MP_TAC BROUWER THEN ASM_REWRITE_TAC[CONVEX_INTERVAL; COMPACT_INTERVAL; UNIT_INTERVAL_NONEMPTY]);; (* ------------------------------------------------------------------------- *) (* Now we can finally deduce what the topological dimension of R^n is. *) (* Proof following Hurewicz & Wallman's "dimension theory". *) (* ------------------------------------------------------------------------- *) let DIMENSION_EQ_AFF_DIM = prove (`!s:real^N->bool. convex s ==> dimension s = aff_dim s`, REPEAT STRIP_TAC THEN SIMP_TAC[GSYM INT_LE_ANTISYM; DIMENSION_LE_AFF_DIM] THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[DIMENSION_EMPTY; AFF_DIM_EMPTY; INT_LE_REFL] THEN ASM_CASES_TAC `aff_dim(s:real^N->bool) = &0` THENL [ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [AFF_DIM_EQ_0]) THEN SIMP_TAC[LEFT_IMP_EXISTS_THM; DIMENSION_SING; INT_LE_REFL]; ALL_TAC] THEN SUBGOAL_THEN `&0 <= aff_dim(s:real^N->bool) /\ &1 <= aff_dim s` MP_TAC THENL [MP_TAC(ISPEC `s:real^N->bool` AFF_DIM_GE) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [GSYM AFF_DIM_EQ_MINUS1]) THEN ASM_INT_ARITH_TAC; POP_ASSUM_LIST(MP_TAC o end_itlist CONJ)] THEN ABBREV_TAC `nn = aff_dim(s:real^N->bool)` THEN POP_ASSUM MP_TAC THEN ONCE_REWRITE_TAC[TAUT `d ==> p ==> q /\ r ==> s <=> q ==> d /\ p /\ r ==> s`] THEN SPEC_TAC(`nn:int`,`nn:int`) THEN REWRITE_TAC[GSYM INT_FORALL_POS; INT_OF_NUM_EQ; INT_OF_NUM_LE] THEN REPEAT STRIP_TAC THEN TRANS_TAC INT_LE_TRANS `dimension(relative_interior s:real^N->bool)` THEN ASM_SIMP_TAC[DIMENSION_SUBSET; RELATIVE_INTERIOR_SUBSET] THEN MP_TAC(ISPEC `s:real^N->bool` OPEN_IN_RELATIVE_INTERIOR) THEN FIRST_ASSUM(MP_TAC o MATCH_MP AFF_DIM_RELATIVE_INTERIOR) THEN FIRST_ASSUM(SUBST1_TAC o SYM o MATCH_MP AFFINE_HULL_RELATIVE_INTERIOR) THEN FIRST_ASSUM(MP_TAC o MATCH_MP RELATIVE_INTERIOR_EQ_EMPTY) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP CONVEX_RELATIVE_INTERIOR) THEN UNDISCH_TAC `1 <= n` THEN POP_ASSUM_LIST(K ALL_TAC) THEN SPEC_TAC(`relative_interior s:real^N->bool`,`t:real^N->bool`) THEN X_GEN_TAC `u:real^N->bool` THEN REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`u:real^N->bool`; `span(IMAGE basis (1..n)):real^N->bool`] HOMEOMORPHIC_RELATIVELY_OPEN_CONVEX_SETS) THEN SIMP_TAC[AFFINE_HULL_EQ_SPAN; AFF_DIM_DIM_0; HULL_INC; SPAN_0] THEN ASM_REWRITE_TAC[SPAN_SPAN; OPEN_IN_REFL; CONVEX_SPAN] THEN MP_TAC(ISPEC `u:real^N->bool` AFF_DIM_LE_UNIV) THEN ASM_REWRITE_TAC[DIM_SPAN; INT_OF_NUM_EQ; INT_OF_NUM_LE] THEN DISCH_TAC THEN SUBGOAL_THEN `dim(IMAGE basis (1..n):real^N->bool) = n` ASSUME_TAC THENL [REWRITE_TAC[DIM_BASIS_IMAGE] THEN GEN_REWRITE_TAC RAND_CONV [GSYM CARD_NUMSEG_1] THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_INTER; IN_NUMSEG] THEN ASM_ARITH_TAC; ASM_REWRITE_TAC[] THEN DISCH_THEN(SUBST1_TAC o MATCH_MP HOMEOMORPHIC_DIMENSION)] THEN ABBREV_TAC `box = {x:real^N | (!i. 1 <= i /\ i <= dimindex(:N) ==> &0 <= x$i /\ x$i <= &1) /\ (!i. n < i /\ i <= dimindex(:N) ==> x$i = &0)}` THEN TRANS_TAC INT_LE_TRANS `dimension(box:real^N->bool)` THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC DIMENSION_SUBSET THEN EXPAND_TAC "box" THEN REWRITE_TAC[SUBSET; IN_SPAN_IMAGE_BASIS; IN_NUMSEG; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC] THEN FIRST_X_ASSUM(MP_TAC o check (free_in `box:real^N->bool` o concl)) THEN MAP_EVERY UNDISCH_TAC [`n <= dimindex(:N)`; `1 <= n`] THEN POP_ASSUM_LIST(K ALL_TAC) THEN REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[INT_ARITH `n:int <= d <=> ~(d <= n - &1)`] THEN DISCH_TAC THEN SUBGOAL_THEN `~(box:real^N->bool = {}) /\ convex(box:real^N->bool) /\ compact box` STRIP_ASSUME_TAC THENL [CONJ_TAC THENL [EXPAND_TAC "box" THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_ELIM_THM] THEN EXISTS_TAC `vec 0:real^N` THEN REWRITE_TAC[VEC_COMPONENT; REAL_POS]; ALL_TAC] THEN SUBGOAL_THEN `box = interval[vec 0:real^N,vec 1] INTER span(IMAGE basis (1..n))` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; IN_INTER; IN_SPAN_IMAGE_BASIS; IN_INTERVAL] THEN EXPAND_TAC "box" THEN REWRITE_TAC[IN_ELIM_THM; VEC_COMPONENT] THEN GEN_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[IN_NUMSEG] THEN EQ_TAC THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; SIMP_TAC[CONVEX_INTER; CONVEX_INTERVAL; CONVEX_SPAN] THEN SIMP_TAC[COMPACT_INTER_CLOSED; COMPACT_INTERVAL; CLOSED_SPAN]]; ALL_TAC] THEN MAP_EVERY ABBREV_TAC [`l = \i. box INTER {x:real^N | x$i = &0}`; `r = \i. box INTER {x:real^N | x$i = &1}`] THEN SUBGOAL_THEN `(!i:num. 1 <= i /\ i <= n ==> ~(l i:real^N->bool = {})) /\ (!i:num. 1 <= i /\ i <= n ==> ~(r i:real^N->bool = {}))` STRIP_ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["l"; "r"; "box"] THEN REWRITE_TAC[IN_INTER; IN_ELIM_THM; GSYM MEMBER_NOT_EMPTY] THEN CONJ_TAC THEN GEN_TAC THEN STRIP_TAC THENL [EXISTS_TAC `vec 0:real^N`; EXISTS_TAC `(lambda j. if j = i then &1 else &0):real^N`] THEN SIMP_TAC[VEC_COMPONENT; REAL_POS; LAMBDA_BETA] THEN REWRITE_TAC[GSYM CONJ_ASSOC] THEN CONJ_TAC THENL [MESON_TAC[REAL_POS; REAL_LE_REFL]; ALL_TAC] THEN REPEAT STRIP_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) LAMBDA_BETA o lhand o snd) THEN (ANTS_TAC THENL [ASM_ARITH_TAC; DISCH_THEN SUBST_ALL_TAC]) THEN REWRITE_TAC[] THEN COND_CASES_TAC THEN REWRITE_TAC[] THEN ASM_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `?b:num->real^N->bool. (!i. closed_in (subtopology euclidean box) (b i)) /\ (!i. 1 <= i /\ i <= n ==> dimension(box INTER INTERS (IMAGE b (1..i))) <= &n - &i - &1 /\ ?u v. open_in (subtopology euclidean box) u /\ open_in (subtopology euclidean box) v /\ DISJOINT u v /\ u UNION v = box DIFF b i /\ l i SUBSET u /\ r i SUBSET v)` MP_TAC THENL [SIMP_TAC[GSYM NUMSEG_RREC] THEN REWRITE_TAC[IMAGE_CLAUSES; INTERS_INSERT] THEN REWRITE_TAC[AND_FORALL_THM] THEN MATCH_MP_TAC(MATCH_MP WF_REC_EXISTS WF_num) THEN CONJ_TAC THENL [SIMP_TAC[numseg; ARITH_RULE `1 <= i ==> (x <= i - 1 <=> x < i)`] THEN REPEAT STRIP_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN ASM SET_TAC[]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`b:num->real^N->bool`; `i:num`] THEN DISCH_TAC THEN ASM_CASES_TAC `1 <= i /\ i <= n` THEN ASM_REWRITE_TAC[] THENL [ALL_TAC; ASM_MESON_TAC[CLOSED_IN_REFL]] THEN ONCE_REWRITE_TAC[SET_RULE `b INTER s INTER t = s INTER b INTER t`] THEN MATCH_MP_TAC DIMENSION_SEPARATION_THEOREM THEN ASM_REWRITE_TAC[INT_SUB_LE; INT_OF_NUM_LE; INTER_SUBSET] THEN CONJ_TAC THENL [ASM_CASES_TAC `i = 1` THENL [ASM_REWRITE_TAC[] THEN CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC(ONCE_DEPTH_CONV NUMSEG_CONV) THEN ASM_REWRITE_TAC[IMAGE_CLAUSES; INTERS_0; INTER_UNIV]; FIRST_X_ASSUM(MP_TAC o SPEC `i - 1`) THEN ANTS_TAC THENL [ASM_ARITH_TAC; DISCH_THEN(MP_TAC o CONJUNCT2)] THEN ANTS_TAC THENL [ASM_ARITH_TAC; DISCH_THEN(MP_TAC o CONJUNCT1)] THEN ASM_SIMP_TAC[GSYM INT_OF_NUM_SUB; INT_ARITH `n - (i - w) - w:int = n - i`] THEN ASM_SIMP_TAC[GSYM NUMSEG_RREC; ARITH_RULE `1 <= i /\ ~(i = 1) ==> 1 <= i - 1`] THEN REWRITE_TAC[IMAGE_CLAUSES; INTERS_INSERT; INTER_ACI]]; MAP_EVERY EXPAND_TAC ["l"; "r"] THEN SIMP_TAC[CLOSED_IN_CLOSED_INTER; CLOSED_STANDARD_HYPERPLANE] THEN MATCH_MP_TAC(SET_RULE `(!x. P x /\ Q x ==> F) ==> DISJOINT (b INTER {x | P x}) (b INTER {x | Q x})`) THEN REAL_ARITH_TAC]; REWRITE_TAC[RIGHT_AND_EXISTS_THM; SKOLEM_THM; RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[NOT_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`b:num->real^N->bool`; `u:num->real^N->bool`; `v:num->real^N->bool`] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[TAUT `p ==> q /\ r <=> (p ==> q) /\ (p ==> r)`] THEN REWRITE_TAC[FORALL_AND_THM] THEN DISCH_THEN(CONJUNCTS_THEN2 (MP_TAC o SPEC `n:num`) STRIP_ASSUME_TAC) THEN ASM_REWRITE_TAC[LE_REFL; INT_ARITH `n - n - w:int = --w`] THEN REWRITE_TAC[DIMENSION_LE_MINUS1] THEN MATCH_MP_TAC(SET_RULE `t SUBSET s /\ ~(t = {}) ==> ~(s INTER t = {})`) THEN CONJ_TAC THENL [MATCH_MP_TAC INTERS_SUBSET THEN ASM_REWRITE_TAC[IMAGE_EQ_EMPTY; NUMSEG_EMPTY; NOT_LT] THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_NUMSEG] THEN ASM_MESON_TAC[CLOSED_IN_IMP_SUBSET]; DISCH_TAC] THEN SUBGOAL_THEN `!i. 1 <= i /\ i <= n ==> ~(b i:real^N->bool = {})` ASSUME_TAC THENL [X_GEN_TAC `i:num` THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP CONVEX_CONNECTED) THEN REWRITE_TAC[CONNECTED_OPEN_IN] THEN MAP_EVERY EXISTS_TAC [`(u:num->real^N->bool) i`; `(v:num->real^N->bool) i`] THEN ASM_SIMP_TAC[GSYM DISJOINT; DIFF_EMPTY; SUBSET_REFL] THEN ASM SET_TAC[]; ALL_TAC] THEN ABBREV_TAC `(f:real^N->real^N) = \x. x + lambda i. if n < i then &0 else if x IN v i then --setdist({x},b i) else setdist({x},b i)` THEN MP_TAC(ISPECL [`f:real^N->real^N`; `box:real^N->bool`] BROUWER) THEN ASM_REWRITE_TAC[NOT_IMP] THEN REPEAT CONJ_TAC THENL [ONCE_REWRITE_TAC[CONTINUOUS_ON_COMPONENTWISE_LIFT] THEN EXPAND_TAC "f" THEN SIMP_TAC[VECTOR_ADD_COMPONENT; LAMBDA_BETA] THEN X_GEN_TAC `m:num` THEN STRIP_TAC THEN REWRITE_TAC[LIFT_ADD] THEN REWRITE_TAC[GSYM NOT_LE] THEN ASM_CASES_TAC `m:num <= n` THEN ASM_SIMP_TAC[LIFT_NUM; VECTOR_ADD_RID; CONTINUOUS_ON_LIFT_COMPONENT] THEN MATCH_MP_TAC CONTINUOUS_ON_ADD THEN ASM_SIMP_TAC[CONTINUOUS_ON_LIFT_COMPONENT] THEN REWRITE_TAC[COND_RAND] THEN SUBGOAL_THEN `box = (box DIFF (u:num->real^N->bool) m) UNION (box DIFF v m)` (fun th -> SUBST1_TAC th THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL THEN SUBST1_TAC(SYM th)) THENL [ASM SET_TAC[]; ASM_SIMP_TAC[CLOSED_IN_DIFF; CLOSED_IN_REFL]] THEN SIMP_TAC[LIFT_NEG; CONTINUOUS_ON_NEG; CONTINUOUS_ON_LIFT_SETDIST] THEN X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `x IN (b:num->real^N->bool) m` THENL [ASM_SIMP_TAC[SETDIST_SING_IN_SET]; ASM SET_TAC[]] THEN REWRITE_TAC[LIFT_NUM; VECTOR_NEG_0; VECTOR_MUL_RZERO]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `x:real^N` THEN DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN EXPAND_TAC "box" THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN EXPAND_TAC "f" THEN SUBGOAL_THEN `!i. n < i ==> 1 <= i` MP_TAC THENL [ASM_ARITH_TAC; SIMP_TAC[LAMBDA_BETA; VECTOR_ADD_COMPONENT]] THEN DISCH_THEN(K ALL_TAC) THEN ASM_REWRITE_TAC[REAL_ADD_RID] THEN X_GEN_TAC `m:num` THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_ADD_RID] THEN RULE_ASSUM_TAC(REWRITE_RULE[NOT_LT]) THEN STRIP_TAC THEN ASM_CASES_TAC `x IN (b:num->real^N->bool) m` THEN ASM_SIMP_TAC[SETDIST_SING_IN_SET] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_SIMP_TAC[COND_ID; REAL_ADD_RID] THEN SUBGOAL_THEN `x IN (u:num->real^N->bool) m /\ ~(x IN v m) \/ x IN v m /\ ~(x IN u m)` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THENL [ALL_TAC; ONCE_REWRITE_TAC[CONJ_SYM]] THEN (MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [ASM_SIMP_TAC[SETDIST_POS_LE; REAL_LE_ADD; REAL_ARITH `x <= &1 /\ &0 <= y ==> x + --y <= &1`]; DISCH_TAC]) THEN ONCE_REWRITE_TAC[GSYM REAL_NOT_LT] THEN DISCH_TAC THENL [ABBREV_TAC `y:real^N = lambda i. if i = m then &1 else (x:real^N)$i` THEN SUBGOAL_THEN `y IN (r:num->real^N->bool) m` ASSUME_TAC THENL [UNDISCH_TAC `(x:real^N) IN box` THEN MAP_EVERY EXPAND_TAC ["y"; "r"; "box"] THEN REWRITE_TAC[IN_ELIM_THM; IN_INTER] THEN SUBGOAL_THEN `!i. n < i ==> 1 <= i` MP_TAC THENL [ASM_ARITH_TAC; ASM_SIMP_TAC[LAMBDA_BETA]] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[REAL_POS; REAL_LE_REFL] THEN ASM_ARITH_TAC; ALL_TAC]; ABBREV_TAC `y:real^N = lambda i. if i = m then &0 else (x:real^N)$i` THEN SUBGOAL_THEN `y IN (l:num->real^N->bool) m` ASSUME_TAC THENL [UNDISCH_TAC `(x:real^N) IN box` THEN MAP_EVERY EXPAND_TAC ["y"; "l"; "box"] THEN REWRITE_TAC[IN_ELIM_THM; IN_INTER] THEN SUBGOAL_THEN `!i. n < i ==> 1 <= i` MP_TAC THENL [ASM_ARITH_TAC; ASM_SIMP_TAC[LAMBDA_BETA]] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[REAL_POS; REAL_LE_REFL] THEN ASM_ARITH_TAC; ALL_TAC]] THEN (SUBGOAL_THEN `segment[x:real^N,y] SUBSET box` ASSUME_TAC THENL [MATCH_MP_TAC SEGMENT_SUBSET_CONVEX THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`x:real^N`; `y:real^N`] (CONJUNCT1 CONNECTED_SEGMENT)) THEN REWRITE_TAC[CONNECTED_OPEN_IN] THEN MAP_EVERY EXISTS_TAC [`segment[x,y] INTER (u:num->real^N->bool) m`; `segment[x,y] INTER (v:num->real^N->bool) m`] THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC OPEN_IN_SUBTOPOLOGY_INTER_SUBSET THEN EXISTS_TAC `box:real^N->bool` THEN ASM_SIMP_TAC[OPEN_IN_INTER; OPEN_IN_REFL]; ASM_SIMP_TAC[GSYM UNION_OVER_INTER]] THEN CONJ_TAC THENL [ASM_REWRITE_TAC[SUBSET_INTER; SUBSET_REFL; SET_RULE `s SUBSET t DIFF u <=> s SUBSET t /\ !x y. x IN s ==> ~(x IN u)`]; MP_TAC(ISPECL [`x:real^N`; `y:real^N`] ENDS_IN_SEGMENT) THEN ASM SET_TAC[]] THEN X_GEN_TAC `z:real^N` THEN DISCH_TAC THEN ASM_CASES_TAC `z:real^N = x` THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `z:real^N = y` THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `z IN segment(x:real^N,y)` MP_TAC THENL [REWRITE_TAC[open_segment] THEN ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(MP_TAC o CONJUNCT1 o MATCH_MP DIST_IN_OPEN_SEGMENT) THEN GEN_REWRITE_TAC I [GSYM CONTRAPOS_THM] THEN REWRITE_TAC[REAL_NOT_LT] THEN DISCH_TAC THEN TRANS_TAC REAL_LE_TRANS `setdist(b(m:num),{x:real^N})` THEN ASM_SIMP_TAC[SETDIST_LE_DIST; IN_SING] THEN EXPAND_TAC "y" THEN ONCE_REWRITE_TAC[SETDIST_SYM]) THENL [TRANS_TAC REAL_LE_TRANS `&1 - (x:real^N)$m`; TRANS_TAC REAL_LE_TRANS `(x:real^N)$m`] THEN (CONJ_TAC THENL [EXPAND_TAC "y"; ASM_REAL_ARITH_TAC] THEN MATCH_MP_TAC REAL_EQ_IMP_LE THEN REWRITE_TAC[dist] THEN REWRITE_TAC[NORM_EQ_SQUARE] THEN SIMP_TAC[dot; LAMBDA_BETA; VECTOR_SUB_COMPONENT] THEN REWRITE_TAC[COND_RAND] THEN SIMP_TAC[REAL_SUB_REFL; SUM_DELTA; REAL_MUL_LZERO] THEN ASM_REWRITE_TAC[IN_NUMSEG] THEN CONJ_TAC THENL [ALL_TAC; REAL_ARITH_TAC] THEN SUBGOAL_THEN `(x:real^N) IN box` MP_TAC THENL [ASM SET_TAC[]; EXPAND_TAC "box"] THEN REWRITE_TAC[IN_ELIM_THM; REAL_SUB_LE] THEN ASM_MESON_TAC[]); DISCH_THEN(X_CHOOSE_THEN `a:real^N` MP_TAC) THEN EXPAND_TAC "f" THEN SIMP_TAC[CART_EQ; VECTOR_ADD_COMPONENT; LAMBDA_BETA] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE LAND_CONV [INTERS_IMAGE]) THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY] THEN DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN REWRITE_TAC[IN_NUMSEG; CONTRAPOS_THM] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `m:num` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN ASM_REWRITE_TAC[GSYM NOT_LE; REAL_ARITH `a + (if p then --x else x) = a <=> x = &0`] THEN ASM_MESON_TAC[SETDIST_EQ_0_CLOSED_IN]]);; let AFF_DIM_DIMENSION = prove (`!s:real^N->bool. aff_dim s = dimension(affine hull s)`, SIMP_TAC[DIMENSION_EQ_AFF_DIM; AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL] THEN REWRITE_TAC[AFF_DIM_AFFINE_HULL]);; let AFF_DIM_DIMENSION_ALT = prove (`!s:real^N->bool. aff_dim s = dimension(convex hull s)`, SIMP_TAC[DIMENSION_EQ_AFF_DIM; CONVEX_CONVEX_HULL] THEN REWRITE_TAC[AFF_DIM_CONVEX_HULL]);; let DIMENSION_SUBSPACE = prove (`!s:real^N->bool. subspace s ==> dimension s = &(dim s)`, SIMP_TAC[DIMENSION_EQ_AFF_DIM; SUBSPACE_IMP_CONVEX; AFF_DIM_DIM_SUBSPACE]);; let DIM_DIMENSION = prove (`!s:real^N->bool. &(dim s) = dimension(span s)`, SIMP_TAC[DIMENSION_SUBSPACE; DIM_SPAN; SUBSPACE_SPAN]);; let DIMENSION_OPEN_IN_CONVEX = prove (`!u s:real^N->bool. convex u /\ open_in (subtopology euclidean u) s ==> dimension s = if s = {} then -- &1 else aff_dim u`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[GSYM DIMENSION_EQ_AFF_DIM] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[DIMENSION_EMPTY] THEN REWRITE_TAC[GSYM INT_LE_ANTISYM] THEN CONJ_TAC THENL [ASM_MESON_TAC[DIMENSION_SUBSET; OPEN_IN_IMP_SUBSET]; ALL_TAC] THEN MP_TAC(ISPECL [`u:real^N->bool`; `s:real^N->bool`] OPEN_IN_CONVEX_MEETS_RELATIVE_INTERIOR) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_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 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_CONTAINS_BALL]) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `a:real^N`)) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN TRANS_TAC INT_LE_TRANS `dimension(affine hull u INTER ball(a:real^N,min d e))` THEN CONJ_TAC THENL [ASM_SIMP_TAC[DIMENSION_EQ_AFF_DIM; CONVEX_INTER; AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL; CONVEX_BALL] THEN MATCH_MP_TAC INT_EQ_IMP_LE THEN CONV_TAC SYM_CONV THEN GEN_REWRITE_TAC RAND_CONV [GSYM AFF_DIM_AFFINE_HULL] THEN MATCH_MP_TAC AFF_DIM_CONVEX_INTER_OPEN THEN ASM_SIMP_TAC[AFFINE_AFFINE_HULL; AFFINE_IMP_CONVEX; OPEN_BALL] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN EXISTS_TAC `a:real^N` THEN ASM_SIMP_TAC[CENTRE_IN_BALL; REAL_LT_MIN; HULL_INC]; MATCH_MP_TAC DIMENSION_SUBSET THEN REWRITE_TAC[BALL_MIN_INTER] THEN ASM SET_TAC[]]);; let DIMENSION_OPEN = prove (`!s:real^N->bool. open s ==> dimension s = if s = {} then -- &1 else &(dimindex(:N))`, REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM AFF_DIM_UNIV] THEN MATCH_MP_TAC DIMENSION_OPEN_IN_CONVEX THEN ASM_REWRITE_TAC[CONVEX_UNIV; SUBTOPOLOGY_UNIV; GSYM OPEN_IN]);; let DIMENSION_UNIV = prove (`dimension(:real^N) = &(dimindex(:N))`, SIMP_TAC[DIMENSION_OPEN; OPEN_UNIV; UNIV_NOT_EMPTY]);; let DIMENSION_NONEMPTY_INTERIOR = prove (`!s:real^N->bool. ~(interior s = {}) ==> dimension s = &(dimindex(:N))`, REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM INT_LE_ANTISYM] THEN SIMP_TAC[GSYM DIMENSION_UNIV; DIMENSION_SUBSET; SUBSET_UNIV] THEN TRANS_TAC INT_LE_TRANS `dimension(interior(s:real^N->bool))` THEN SIMP_TAC[DIMENSION_SUBSET; INTERIOR_SUBSET] THEN ASM_SIMP_TAC[INT_LE_REFL; DIMENSION_OPEN; OPEN_INTERIOR; DIMENSION_UNIV]);; let DIMENSION_ATMOST_RATIONAL_COORDINATES = prove (`!n. n <= dimindex(:N) ==> dimension {x:real^N | CARD {i | i IN 1..dimindex(:N) /\ rational(x$i)} <= n} = &n`, REWRITE_TAC[GSYM INT_LE_ANTISYM; FORALL_AND_THM; TAUT `p ==> q /\ r <=> (p ==> q) /\ (p ==> r)`] THEN CONJ_TAC THENL [MATCH_MP_TAC num_INDUCTION THEN REWRITE_TAC[LE; LE_0] THEN CONJ_TAC THENL [MP_TAC(SPEC `0` DIMENSION_LE_RATIONAL_COORDINATES) THEN ASM_SIMP_TAC[HAS_SIZE; FINITE_RESTRICT; FINITE_NUMSEG; CONJUNCT1 LE]; X_GEN_TAC `n:num` THEN DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN ANTS_TAC THENL [ASM_ARITH_TAC; DISCH_TAC] THEN REWRITE_TAC[LE; SET_RULE `{x | Q x \/ R x} = {x | Q x} UNION {x | R x}`] THEN W(MP_TAC o PART_MATCH lhand DIMENSION_UNION_LE_BASIC o lhand o snd) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] INT_LE_TRANS) THEN REWRITE_TAC[GSYM INT_OF_NUM_SUC] THEN MATCH_MP_TAC(INT_ARITH `x:int <= &0 /\ y <= n ==> x + y + &1 <= n + &1`) THEN ASM_REWRITE_TAC[] THEN MP_TAC(SPEC `SUC n` DIMENSION_LE_RATIONAL_COORDINATES) THEN ASM_SIMP_TAC[HAS_SIZE; FINITE_RESTRICT; FINITE_NUMSEG]]; X_GEN_TAC `n:num` THEN DISCH_TAC THEN SUBGOAL_THEN `n = dimindex(:N) - (dimindex(:N) - n)` SUBST1_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN MP_TAC(ARITH_RULE `dimindex(:N) - n <= dimindex(:N)`) THEN POP_ASSUM(K ALL_TAC) THEN SPEC_TAC(`dimindex(:N) - n`,`n:num`) THEN SIMP_TAC[GSYM INT_OF_NUM_SUB] THEN MATCH_MP_TAC num_INDUCTION THEN REWRITE_TAC[LE; LE_0] THEN CONJ_TAC THENL [REWRITE_TAC[INT_SUB_RZERO; SUB_0] THEN GEN_REWRITE_TAC LAND_CONV [GSYM DIMENSION_UNIV] THEN MATCH_MP_TAC INT_EQ_IMP_LE THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_UNIV; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN GEN_REWRITE_TAC RAND_CONV [GSYM CARD_NUMSEG_1] THEN MATCH_MP_TAC CARD_SUBSET THEN SIMP_TAC[FINITE_RESTRICT; SUBSET_RESTRICT; FINITE_NUMSEG]; X_GEN_TAC `n:num` THEN DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[GSYM INT_NOT_LT; CONTRAPOS_THM] THEN DISCH_TAC THEN ASM_SIMP_TAC[ARITH_RULE `SUC n <= N ==> (a <= N - n <=> a = N - n \/ a <= N - SUC n)`] THEN REWRITE_TAC[LE; SET_RULE `{x | Q x \/ R x} = {x | Q x} UNION {x | R x}`] THEN W(MP_TAC o PART_MATCH lhand DIMENSION_UNION_LE_BASIC o lhand o snd) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] INT_LET_TRANS) THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (INT_ARITH `d2:int < n2 ==> d1 < N - n2 ==> d1 + d2 + &1 < N`)) THEN REWRITE_TAC[GSYM INT_OF_NUM_SUC; INT_ARITH `x:int < N - n - (N - (n + &1)) <=> x <= &0`] THEN MP_TAC(SPEC `dimindex(:N) - n` DIMENSION_LE_RATIONAL_COORDINATES) THEN SIMP_TAC[HAS_SIZE; FINITE_RESTRICT; FINITE_NUMSEG; LE]]]);; let DIMENSION_COMPLEMENT_RATIONAL_COORDINATES = prove (`dimension((:real^N) DIFF { x | !i. 1 <= i /\ i <= dimindex(:N) ==> rational(x$i)}) = &(dimindex(:N)) - &1`, MP_TAC(SPEC `dimindex(:N) - 1` DIMENSION_ATMOST_RATIONAL_COORDINATES) THEN REWRITE_TAC[ARITH_RULE `n - 1 <= n`] THEN SIMP_TAC[GSYM INT_OF_NUM_SUB; DIMINDEX_GE_1] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [EXTENSION] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_UNIV; IN_DIFF; IN_ELIM_THM] THEN REWRITE_TAC[GSYM IN_NUMSEG; SET_RULE `(!x. P x ==> Q x) <=> {x | P x /\ Q x} = {x | P x}`] THEN SIMP_TAC[GSYM SUBSET_CARD_EQ; FINITE_RESTRICT; FINITE_NUMSEG; CARD_NUMSEG_1; SUBSET_RESTRICT; SET_RULE `{x | x IN s} = s`] THEN MATCH_MP_TAC(ARITH_RULE `1 <= N /\ n <= N ==> (~(n = N) <=> n <= N - 1)`) THEN REWRITE_TAC[DIMINDEX_GE_1] THEN GEN_REWRITE_TAC RAND_CONV [GSYM CARD_NUMSEG_1] THEN MATCH_MP_TAC CARD_SUBSET THEN REWRITE_TAC[SUBSET_RESTRICT; FINITE_NUMSEG]);; let DIMENSION_EQ_FULL_GEN = prove (`!s:real^N->bool. dimension s = aff_dim s <=> s = {} \/ ~(relative_interior s = {})`, let lemma1 = prove (`closure(span(IMAGE basis (1..n)) INTER {x:real^N | !i. 1 <= i /\ i <= dimindex(:N) ==> rational(x$i)}) = span(IMAGE basis (1..n))`, MATCH_MP_TAC SUBSET_ANTISYM THEN SIMP_TAC[CLOSURE_MINIMAL; CLOSED_SPAN; INTER_SUBSET] THEN REWRITE_TAC[SUBSET; IN_SPAN_IMAGE_BASIS] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MP_TAC(SET_RULE `x IN (:real^N)`) THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM CLOSURE_RATIONAL_COORDINATES] THEN REWRITE_TAC[CLOSURE_APPROACHABLE] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `e:real` THEN ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[IN_INTER; IN_ELIM_THM; IN_SPAN_IMAGE_BASIS] THEN DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(lambda i. if i IN 1..n then (y:real^N)$i else &0):real^N` THEN SIMP_TAC[LAMBDA_BETA] THEN CONJ_TAC THENL [ASM_MESON_TAC[RATIONAL_NUM]; ALL_TAC] THEN TRANS_TAC REAL_LET_TRANS `dist(y:real^N,x)` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[dist] THEN MATCH_MP_TAC NORM_LE_COMPONENTWISE THEN X_GEN_TAC `i:num` THEN SIMP_TAC[LAMBDA_BETA; VECTOR_SUB_COMPONENT] THEN STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[] THEN REAL_ARITH_TAC) and lemma2 = prove (`!n. n <= dimindex(:N) ==> dimension(span(IMAGE basis (1..n)) DIFF {x:real^N | !i. i IN 1..dimindex(:N) ==> rational(x$i)}) < &n`, REPEAT STRIP_TAC THEN TRANS_TAC INT_LET_TRANS `dimension(UNIONS {{x:real^N | {i | i IN 1..dimindex (:N) /\ rational (x$i)} HAS_SIZE m} | m IN (dimindex(:N)-n)..dimindex(:N)-1})` THEN CONJ_TAC THENL [MATCH_MP_TAC DIMENSION_SUBSET THEN REWRITE_TAC[UNIONS_GSPEC; SUBSET] THEN SIMP_TAC[HAS_SIZE; FINITE_NUMSEG; FINITE_RESTRICT] THEN REWRITE_TAC[IN_SPAN_IMAGE_BASIS; IN_DIFF; IN_ELIM_THM; IN_NUMSEG] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[UNWIND_THM1] THEN CONJ_TAC THENL [TRANS_TAC LE_TRANS `CARD(n+1..dimindex(:N))` THEN CONJ_TAC THENL [REWRITE_TAC[CARD_NUMSEG] THEN ASM_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC CARD_SUBSET THEN SIMP_TAC[GSYM IN_NUMSEG; FINITE_RESTRICT; FINITE_NUMSEG] THEN REWRITE_TAC[SUBSET; IN_NUMSEG; IN_ELIM_THM] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `i:num`) THEN ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN SIMP_TAC[RATIONAL_NUM] THEN ASM_ARITH_TAC; MATCH_MP_TAC(ARITH_RULE `c < n ==> c <= n - 1`) THEN GEN_REWRITE_TAC RAND_CONV [GSYM CARD_NUMSEG_1] THEN MATCH_MP_TAC CARD_PSUBSET THEN REWRITE_TAC[FINITE_NUMSEG] THEN REWRITE_TAC[numseg] THEN ASM SET_TAC[]]; W(MP_TAC o PART_MATCH (lhand o rand) DIMENSION_LE_UNIONS_ZERODIMENSIONAL o lhand o snd) THEN SIMP_TAC[SIMPLE_IMAGE; FINITE_IMAGE; FINITE_NUMSEG] THEN REWRITE_TAC[FORALL_IN_IMAGE; DIMENSION_LE_RATIONAL_COORDINATES] THEN MATCH_MP_TAC(INT_ARITH `c:int <= n ==> d <= c - &1 ==> d < n`) THEN REWRITE_TAC[INT_OF_NUM_LE] THEN W(MP_TAC o PART_MATCH (lhand o rand) CARD_IMAGE_LE o lhand o snd) THEN REWRITE_TAC[FINITE_NUMSEG] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] LE_TRANS) THEN REWRITE_TAC[CARD_NUMSEG] THEN MATCH_MP_TAC(ARITH_RULE `1 <= N /\ n <= N ==> (N - 1 + 1) - (N - n) <= n`) THEN ASM_REWRITE_TAC[DIMINDEX_GE_1]]) in GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[DIMENSION_EMPTY; AFF_DIM_EMPTY] THEN EQ_TAC THEN DISCH_TAC THENL [DISCH_TAC; MP_TAC(ISPECL [`affine hull s:real^N->bool`; `relative_interior s:real^N->bool`] DIMENSION_OPEN_IN_CONVEX) THEN ASM_REWRITE_TAC[AFF_DIM_AFFINE_HULL; OPEN_IN_RELATIVE_INTERIOR] THEN SIMP_TAC[AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL] THEN MATCH_MP_TAC(INT_ARITH `i:int <= s /\ s <= u ==> i = u ==> s = u`) THEN REWRITE_TAC[DIMENSION_LE_AFF_DIM] THEN SIMP_TAC[DIMENSION_SUBSET; RELATIVE_INTERIOR_SUBSET]] THEN MP_TAC(ISPEC `affine hull s DIFF s:real^N->bool` SEPARABLE) THEN DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `closure c:real^N->bool = affine hull s` ASSUME_TAC THENL [MATCH_MP_TAC SUBSET_ANTISYM THEN SIMP_TAC[CLOSURE_MINIMAL_EQ; CLOSED_AFFINE_HULL] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE LAND_CONV [RELATIVE_INTERIOR_INTERIOR_OF]) THEN REWRITE_TAC[INTERIOR_OF_CLOSURE_OF; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN MATCH_MP_TAC(SET_RULE `t SUBSET u ==> s DIFF t = {} ==> s SUBSET u`) THEN REWRITE_TAC[CLOSURE_OF_SUBTOPOLOGY; EUCLIDEAN_CLOSURE_OF] THEN MATCH_MP_TAC(SET_RULE `t SUBSET u ==> s INTER t SUBSET u`) THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN REWRITE_TAC[CLOSED_CLOSURE] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `affine hull c:real^N->bool = affine hull s` ASSUME_TAC THENL [ASM_MESON_TAC[HULL_HULL; AFFINE_HULL_CLOSURE]; ALL_TAC] THEN SUBGOAL_THEN `aff_dim c <= dimension(affine hull c DIFF c:real^N->bool)` MP_TAC THENL [TRANS_TAC INT_LE_TRANS `dimension(s:real^N->bool)` THEN CONJ_TAC THENL [ASM_MESON_TAC[INT_LE_REFL; AFF_DIM_AFFINE_HULL]; MATCH_MP_TAC DIMENSION_SUBSET THEN SUBGOAL_THEN `(s:real^N->bool) SUBSET affine hull s` MP_TAC THENL [REWRITE_TAC[HULL_SUBSET]; ASM SET_TAC[]]]; REWRITE_TAC[INT_NOT_LE] THEN SUBGOAL_THEN `closure c:real^N->bool = affine hull c` MP_TAC THENL [ASM MESON_TAC[]; UNDISCH_TAC `COUNTABLE(c:real^N->bool)`] THEN SUBGOAL_THEN `~(c:real^N->bool = {})` MP_TAC THENL [ASM_MESON_TAC[AFFINE_HULL_EQ_EMPTY]; POP_ASSUM_LIST(K ALL_TAC)]] THEN SPEC_TAC(`c:real^N->bool`,`c:real^N->bool`) THEN X_GEN_TAC `s:real^N->bool` THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM AFF_DIM_POS_LE]) THEN REWRITE_TAC[GSYM INT_OF_NUM_EXISTS] THEN DISCH_THEN(X_CHOOSE_TAC `n:num`) THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPECL [`s:real^N->bool`; `span(IMAGE basis (1..n)) INTER {x:real^N | !i. i IN 1..dimindex(:N) ==> rational(x$i)}`] HOMEOMORPHISM_MOVING_DENSE_COUNTABLE_SUBSETS_EXISTS) THEN ASM_SIMP_TAC[COUNTABLE_INTER; COUNTABLE_RATIONAL_COORDINATES; IN_NUMSEG] THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN ONCE_REWRITE_TAC[GSYM AFFINE_HULL_CLOSURE] THEN REWRITE_TAC[lemma1] THEN SIMP_TAC[HULL_P; SUBSPACE_SPAN; SUBSPACE_IMP_AFFINE] THEN SIMP_TAC[AFF_DIM_DIM_SUBSPACE; SUBSPACE_SPAN] THEN SIMP_TAC[DIM_SPAN; DIM_BASIS_IMAGE] THEN MP_TAC(ISPEC `s:real^N->bool` AFF_DIM_LE_UNIV) THEN ASM_REWRITE_TAC[INT_OF_NUM_LE] THEN DISCH_TAC THEN ASM_SIMP_TAC[SUBSET_NUMSEG; LE_REFL; CARD_NUMSEG_1; LEFT_IMP_EXISTS_THM; HULL_HULL; SET_RULE `t SUBSET s ==> s INTER t = t`] THEN MAP_EVERY X_GEN_TAC [`f:real^N->real^N`; `g:real^N->real^N`] THEN STRIP_TAC THEN MP_TAC(ISPEC `n:num` lemma2) THEN ASM_REWRITE_TAC[HULL_HULL] THEN MATCH_MP_TAC(INT_ARITH `d':int = d ==> d < n ==> d' < n`) THEN MATCH_MP_TAC HOMEOMORPHIC_DIMENSION THEN REWRITE_TAC[homeomorphic] THEN MAP_EVERY EXISTS_TAC [`f:real^N->real^N`; `g:real^N->real^N`] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN REPEAT(CONJ_TAC THENL [SET_TAC[]; ALL_TAC]) THEN W(MP_TAC o PART_MATCH (lhand o rand) IMAGE_DIFF_INJ_ALT o lhand o snd) THEN REWRITE_TAC[HULL_SUBSET] THEN ANTS_TAC THENL [ALL_TAC; DISCH_THEN SUBST1_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM_REWRITE_TAC[IN_NUMSEG] THEN ASM SET_TAC[]);; let DIMENSION_LT_FULL_GEN = prove (`!s:real^N->bool. dimension s < aff_dim s <=> ~(s = {}) /\ relative_interior s = {}`, REWRITE_TAC[INT_ARITH `s:int < a <=> s <= a /\ ~(s = a)`] THEN REWRITE_TAC[DIMENSION_EQ_FULL_GEN; DIMENSION_LE_AFF_DIM] THEN CONV_TAC TAUT);; let DIMENSION_EQ_FULL_ALT = prove (`!u s:real^N->bool. convex u /\ s SUBSET u ==> (dimension 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[AFF_DIM_EMPTY; SUBSET_EMPTY; DIMENSION_EMPTY] THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[DIMENSION_EMPTY; INTERIOR_OF_EMPTY] THENL [ASM_MESON_TAC[AFF_DIM_EQ_MINUS1]; STRIP_TAC] THEN EQ_TAC THEN DISCH_TAC THENL [FIRST_ASSUM(MP_TAC o SPEC `aff_dim(s:real^N->bool)` o MATCH_MP (INT_ARITH `s = u ==> !a:int. s <= a /\ a <= u ==> a = u /\ s = a`)) THEN REWRITE_TAC[DIMENSION_EQ_FULL_GEN; DIMENSION_LE_AFF_DIM] THEN ASM_SIMP_TAC[AFF_DIM_SUBSET] THEN ASM_SIMP_TAC[AFF_DIM_EQ_FULL_GEN] THEN REWRITE_TAC[RELATIVE_INTERIOR_INTERIOR_OF] THEN SIMP_TAC[IMP_CONJ] THEN DISCH_THEN(K ALL_TAC) THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> ~(s = {}) ==> ~(t = {})`) THEN MATCH_MP_TAC INTERIOR_OF_SUBTOPOLOGY_MONO THEN ASM_REWRITE_TAC[HULL_SUBSET]; ASM_SIMP_TAC[GSYM DIMENSION_EQ_AFF_DIM; GSYM INT_LE_ANTISYM; DIMENSION_SUBSET] THEN TRANS_TAC INT_LE_TRANS `dimension(subtopology euclidean u interior_of s:real^N->bool)` THEN SIMP_TAC[DIMENSION_SUBSET; INTERIOR_OF_SUBSET] THEN MP_TAC(ISPECL [`u:real^N->bool`; `subtopology euclidean u interior_of s:real^N->bool`] DIMENSION_OPEN_IN_CONVEX) THEN ASM_SIMP_TAC[OPEN_IN_INTERIOR_OF; DIMENSION_LE_AFF_DIM]]);; let DIMENSION_LT_FULL_ALT = prove (`!u s:real^N->bool. convex u /\ s SUBSET u ==> (dimension s < aff_dim u <=> ~(u = {}) /\ subtopology euclidean u interior_of s = {})`, REPEAT STRIP_TAC THEN REWRITE_TAC[INT_LT_LE] THEN FIRST_ASSUM(MP_TAC o MATCH_MP DIMENSION_SUBSET) THEN ASM_SIMP_TAC[DIMENSION_EQ_AFF_DIM; DIMENSION_EQ_FULL_ALT] THEN ASM_CASES_TAC `u:real^N->bool = {}` THEN ASM_SIMP_TAC[AFF_DIM_EMPTY; DIMENSION_LE_MINUS1]);; let DIMENSION_EQ_FULL = prove (`!s:real^N->bool. dimension s = &(dimindex(:N)) <=> ~(interior s = {})`, GEN_TAC THEN EQ_TAC THEN REWRITE_TAC[DIMENSION_NONEMPTY_INTERIOR] THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[DIMENSION_EMPTY; INT_ARITH `~(-- &1:int = &n)`] THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `aff_dim(s:real^N->bool)` o MATCH_MP (INT_ARITH `!a. d:int = n ==> d <= a /\ a <= n ==> a = n`)) THEN REWRITE_TAC[DIMENSION_LE_AFF_DIM; AFF_DIM_LE_UNIV] THEN DISCH_TAC THEN MP_TAC(ISPEC `s:real^N->bool` DIMENSION_EQ_FULL_GEN) 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_REWRITE_TAC[GSYM AFF_DIM_EQ_FULL]);; let DIMENSION_LT_FULL = prove (`!s:real^N->bool. dimension s < &(dimindex(:N)) <=> interior s = {}`, REWRITE_TAC[INT_LT_LE; DIMENSION_LE_DIMINDEX; DIMENSION_EQ_FULL]);; let DIMENSION_RELATIVE_FRONTIER_BOUNDED_OPEN = prove (`!u s:real^N->bool. affine u /\ open_in (subtopology euclidean u) s /\ bounded s ==> dimension(relative_frontier s) = if s = {} then -- &1 else aff_dim u - &1`, REPEAT GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[RELATIVE_FRONTIER_EMPTY; DIMENSION_EMPTY] THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `u:real^N->bool`] AFF_DIM_OPEN_IN) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_CASES_TAC `aff_dim(u:real^N->bool) <= &0` THENL [FIRST_X_ASSUM(MP_TAC o MATCH_MP (INT_ARITH `s:int <= &0 ==> -- &1 <= s /\ ~(s = -- &1) ==> s = &0`)) THEN ASM_REWRITE_TAC[AFF_DIM_GE] THEN FIRST_ASSUM(SUBST1_TAC o SYM) THEN REWRITE_TAC[AFF_DIM_EQ_MINUS1] THEN ANTS_TAC THENL [ASM_REWRITE_TAC[]; REWRITE_TAC[AFF_DIM_EQ_0]] THEN SIMP_TAC[AFF_DIM_EQ_0; LEFT_IMP_EXISTS_THM; RELATIVE_FRONTIER_SING] THEN REWRITE_TAC[DIMENSION_EMPTY; AFF_DIM_SING] THEN CONV_TAC INT_REDUCE_CONV; ALL_TAC] THEN MATCH_MP_TAC(INT_ARITH `d:int < n /\ ~(d <= n - &2) ==> d = n - &1`) THEN CONJ_TAC THENL [TRANS_TAC INT_LTE_TRANS `aff_dim(relative_frontier s:real^N->bool)` THEN CONJ_TAC THENL [REWRITE_TAC[DIMENSION_LT_FULL_GEN] THEN CONJ_TAC THENL [ASM_MESON_TAC[RELATIVE_FRONTIER_EQ_EMPTY; AFFINE_BOUNDED_EQ_LOWDIM]; ALL_TAC] THEN REWRITE_TAC[RELATIVE_INTERIOR_INTERIOR_OF] THEN ASM_SIMP_TAC[AFFINE_HULL_RELATIVE_FRONTIER_BOUNDED; GSYM AFF_DIM_EQ_0; INT_ARITH `~(u:int <= &0) ==> ~(u = &0)`] THEN REWRITE_TAC[RELATIVE_FRONTIER_FRONTIER_OF] THEN MATCH_MP_TAC INTERIOR_OF_FRONTIER_OF_EMPTY THEN DISJ1_TAC THEN ASM_MESON_TAC[AFFINE_HULL_OPEN_IN_CONVEX; AFFINE_IMP_CONVEX; HULL_P]; MATCH_MP_TAC AFF_DIM_SUBSET THEN REWRITE_TAC[relative_frontier] THEN MATCH_MP_TAC(SET_RULE `s SUBSET u ==> s DIFF t SUBSET u`) THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN ASM_MESON_TAC[OPEN_IN_IMP_SUBSET; CLOSED_AFFINE]]; DISCH_TAC] THEN SUBGOAL_THEN `relative_frontier s:real^N->bool = subtopology euclidean u frontier_of s` SUBST_ALL_TAC THENL [REWRITE_TAC[RELATIVE_FRONTIER_FRONTIER_OF] THEN ASM_MESON_TAC[AFFINE_HULL_OPEN_IN_AFFINE]; ALL_TAC] THEN MP_TAC(ISPECL [`u:real^N->bool`; `s:real^N->bool`] DIMENSION_OPEN_IN_CONVEX) THEN ASM_SIMP_TAC[AFFINE_IMP_CONVEX] THEN MATCH_MP_TAC(INT_ARITH `x:int <= n - &1 ==> ~(x = n)`) THEN MP_TAC(ISPECL [`u:real^N->bool`; `s:real^N->bool`; `aff_dim(u:real^N->bool) - &1`] DIMENSION_LE_EQ_GENERAL) THEN ANTS_TAC THENL [ASM_MESON_TAC[OPEN_IN_IMP_SUBSET]; DISCH_THEN SUBST1_TAC] THEN CONJ_TAC THENL [ASM_INT_ARITH_TAC; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`v:real^N->bool`; `a:real^N`] THEN POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN GEOM_ORIGIN_TAC `a:real^N` THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `(vec 0:real^N) IN u` ASSUME_TAC THENL [ASM_MESON_TAC[REWRITE_RULE[SUBSET] OPEN_IN_IMP_SUBSET]; ALL_TAC] THEN UNDISCH_TAC `affine(u:real^N->bool)` THEN ASM_SIMP_TAC[AFFINE_EQ_SUBSPACE] THEN DISCH_TAC THEN SUBGOAL_THEN `?c. &0 < c /\ IMAGE (\x:real^N. c % x) (subtopology euclidean u closure_of s) SUBSET s /\ IMAGE (\x:real^N. c % x) (subtopology euclidean u closure_of s) SUBSET v` STRIP_ASSUME_TAC THENL [FIRST_ASSUM(MP_TAC o MATCH_MP BOUNDED_CLOSURE) THEN DISCH_THEN(MP_TAC o MATCH_MP BOUNDED_SUBSET_CBALL) THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC o SPEC `vec 0:real^N`) THEN MP_TAC(ISPECL [`s INTER v:real^N->bool`; `u:real^N->bool`] OPEN_IN_CONTAINS_CBALL) THEN ASM_SIMP_TAC[OPEN_IN_INTER; IN_INTER] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `vec 0:real^N`)) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; GSYM SUBSET_INTER] THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN EXISTS_TAC `d / r:real` THEN ASM_SIMP_TAC[REAL_LT_DIV] THEN TRANS_TAC SUBSET_TRANS `cball(vec 0:real^N,d) INTER u` THEN ASM_REWRITE_TAC[] THEN TRANS_TAC SUBSET_TRANS `IMAGE (\x. d / r % x) (cball(vec 0:real^N,r)) INTER u` THEN CONJ_TAC THENL [REWRITE_TAC[SUBSET_INTER] THEN CONJ_TAC THENL [MATCH_MP_TAC IMAGE_SUBSET THEN TRANS_TAC SUBSET_TRANS `closure s:real^N->bool` THEN ASM_REWRITE_TAC[CLOSURE_OF_SUBTOPOLOGY; EUCLIDEAN_CLOSURE_OF] THEN MATCH_MP_TAC(SET_RULE `u SUBSET s ==> t INTER u SUBSET s`) THEN SIMP_TAC[SUBSET_CLOSURE; INTER_SUBSET]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSPACE_MUL THEN ASM_MESON_TAC[REWRITE_RULE[SUBSET] CLOSURE_OF_SUBSET_SUBTOPOLOGY]]; ASM_SIMP_TAC[GSYM CBALL_SCALING; REAL_LT_DIV] THEN ASM_SIMP_TAC[REAL_DIV_RMUL; REAL_LT_IMP_NZ; VECTOR_MUL_RZERO; SUBSET_REFL]]; ALL_TAC] THEN EXISTS_TAC `IMAGE (\x:real^N. c % x) s` THEN CONJ_TAC THENL [REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `vec 0:real^N` THEN ASM_REWRITE_TAC[VECTOR_MUL_RZERO]; ALL_TAC] THEN CONJ_TAC THENL [TRANS_TAC SUBSET_TRANS `IMAGE (\x:real^N. c % x) (subtopology euclidean u closure_of s)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC IMAGE_SUBSET THEN MATCH_MP_TAC CLOSURE_OF_SUBSET THEN ASM_MESON_TAC[OPEN_IN_SUBSET]; ALL_TAC] THEN SUBGOAL_THEN `u = IMAGE (\x:real^N. c % x) u` (fun th -> SUBST1_TAC th THEN ASSUME_TAC(SYM th)) THENL [ASM_MESON_TAC[CONIC_IMAGE_MULTIPLE; SUBSPACE_IMP_CONIC]; ALL_TAC] THEN CONJ_TAC THENL [W(MP_TAC o PART_MATCH (lhand o rand) OPEN_IN_INJECTIVE_LINEAR_IMAGE o snd) THEN ASM_REWRITE_TAC[LINEAR_SCALING] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_SIMP_TAC[VECTOR_MUL_LCANCEL; REAL_LT_IMP_NZ]; W(MP_TAC o PART_MATCH (lhand o rand) FRONTIER_OF_INJECTIVE_LINEAR_IMAGE o rand o rand o lhand o snd) THEN ASM_SIMP_TAC[LINEAR_SCALING; VECTOR_MUL_LCANCEL; REAL_LT_IMP_NZ] THEN DISCH_THEN SUBST1_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (INT_ARITH `d:int <= n - &2 ==> d' = d ==> d' <= n - &1 - &1`)) THEN ASM_SIMP_TAC[frontier_of; SET_RULE `IMAGE f c SUBSET s ==> s INTER (IMAGE f (c DIFF i)) = IMAGE f (c DIFF i)`] THEN REWRITE_TAC[GSYM frontier_of] THEN MATCH_MP_TAC DIMENSION_LINEAR_IMAGE THEN ASM_SIMP_TAC[LINEAR_SCALING; VECTOR_MUL_LCANCEL; REAL_LT_IMP_NZ]]);; let DIMENSION_FRONTIER_BOUNDED_OPEN = prove (`!u:real^N->bool. open u /\ bounded u ==> dimension(frontier u) = if u = {} then -- &1 else &(dimindex(:N)) - &1`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`(:real^N)`; `u:real^N->bool`] DIMENSION_RELATIVE_FRONTIER_BOUNDED_OPEN) THEN ASM_SIMP_TAC[RELATIVE_FRONTIER_OPEN] THEN ASM_REWRITE_TAC[SUBTOPOLOGY_UNIV; GSYM OPEN_IN; AFFINE_UNIV; AFF_DIM_UNIV]);; let DIMENSION_RELATIVE_FRONTIER_NONDENSE_OPEN = prove (`!u s:real^N->bool. affine u /\ open_in (subtopology euclidean u) s /\ ~(s = {}) /\ ~(subtopology euclidean u closure_of s = u) ==> dimension(relative_frontier s) = aff_dim u - &1`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `bounded(s:real^N->bool)` THEN ASM_SIMP_TAC[DIMENSION_RELATIVE_FRONTIER_BOUNDED_OPEN] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN SUBGOAL_THEN `?z:real^N. z IN u /\ ~(z IN subtopology euclidean u closure_of s)` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC(SET_RULE `s SUBSET u /\ ~(s = u) ==> ?x. x IN u /\ ~(x IN s)`) THEN ASM_REWRITE_TAC[CLOSURE_OF_SUBSET_SUBTOPOLOGY]; ALL_TAC] THEN POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN GEOM_ORIGIN_TAC `z:real^N` THEN REPEAT STRIP_TAC THEN UNDISCH_TAC `affine(u:real^N->bool)` THEN ASM_SIMP_TAC[AFFINE_EQ_SUBSPACE] THEN DISCH_TAC THEN MP_TAC(ISPECL [`u DIFF subtopology euclidean u closure_of s:real^N->bool`; `u:real^N->bool`] OPEN_IN_CONTAINS_CBALL) THEN SIMP_TAC[OPEN_IN_DIFF; OPEN_IN_REFL; CLOSED_IN_CLOSURE_OF] THEN DISCH_THEN(MP_TAC o SPEC `vec 0:real^N` o CONJUNCT2) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; IN_DIFF] THEN X_GEN_TAC `r:real` THEN STRIP_TAC THEN ABBREV_TAC `i = \x:real^N. r pow 2 / norm x pow 2 % x` THEN MP_TAC(ISPECL [`i:real^N->real^N`; `u DELETE (vec 0:real^N)`] INVOLUTION_IMP_HOMEOMORPHISM) THEN ANTS_TAC THENL [EXPAND_TAC "i" THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_MUL THEN REWRITE_TAC[real_div; o_DEF; LIFT_CMUL; CONTINUOUS_ON_ID] THEN MATCH_MP_TAC CONTINUOUS_ON_CMUL THEN REWRITE_TAC[REAL_INV_POW] THEN MATCH_MP_TAC CONTINUOUS_ON_LIFT_POW THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN REWRITE_TAC[IN_DELETE; IN_UNIV; NORM_EQ_0] THEN SIMP_TAC[CONTINUOUS_ON_LIFT_NORM_COMPOSE; CONTINUOUS_ON_ID]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_DELETE; IN_UNIV] THEN ASM_SIMP_TAC[VECTOR_MUL_EQ_0; REAL_DIV_EQ_0; NORM_EQ_0; REAL_POW_EQ_0; REAL_LT_IMP_NZ; ARITH_EQ; SUBSPACE_MUL] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[GSYM NORM_EQ_0] THEN DISCH_TAC THEN GEN_REWRITE_TAC RAND_CONV [GSYM VECTOR_MUL_LID] THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_POW; REAL_ABS_NORM] THEN FIRST_ASSUM(MP_TAC o CONJUNCT2) THEN UNDISCH_TAC `&0 < r` THEN SIMP_TAC[real_abs; REAL_LT_IMP_LE] THEN CONV_TAC REAL_FIELD]; DISCH_TAC] THEN MP_TAC(ISPECL [`u:real^N->bool`; `IMAGE (i:real^N->real^N) s`] DIMENSION_RELATIVE_FRONTIER_BOUNDED_OPEN) THEN ASM_REWRITE_TAC[IMAGE_EQ_EMPTY] THEN ASM_SIMP_TAC[AFFINE_EQ_SUBSPACE] THEN SUBGOAL_THEN `s SUBSET u DELETE (vec 0:real^N)` ASSUME_TAC THENL [ASM_REWRITE_TAC[SUBSET_DELETE] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `~(x IN s) ==> t SUBSET s ==> ~(x IN t)`)) THEN MATCH_MP_TAC CLOSURE_OF_SUBSET THEN ASM_REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY]; ALL_TAC] THEN ASM_CASES_TAC `(vec 0:real^N) IN s` THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `open_in (subtopology euclidean u) (IMAGE (i:real^N->real^N) s)` ASSUME_TAC THENL [TRANS_TAC OPEN_IN_TRANS `u DELETE (vec 0:real^N)` THEN SIMP_TAC[OPEN_IN_DELETE; OPEN_IN_REFL] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_IMP_OPEN_MAP)) THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] OPEN_IN_SUBSET_TRANS)) THEN ASM SET_TAC[]; ASM_REWRITE_TAC[]] THEN ANTS_TAC THENL [MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `cball(vec 0:real^N,r) INTER u` THEN SIMP_TAC[BOUNDED_CBALL; BOUNDED_INTER; SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[IN_INTER] THEN CONJ_TAC THENL [ALL_TAC; RULE_ASSUM_TAC(REWRITE_RULE[HOMEOMORPHISM]) THEN ASM SET_TAC[]] THEN SUBGOAL_THEN `x IN (:real^N) DIFF cball(vec 0,r)` MP_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `b INTER u SUBSET u DIFF c ==> c SUBSET u /\ x IN c ==> x IN UNIV DIFF b`)) THEN REWRITE_TAC[CLOSURE_OF_SUBSET_SUBTOPOLOGY] 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 CLOSURE_OF_SUBSET THEN ASM_REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY]; EXPAND_TAC "i" THEN REWRITE_TAC[IN_UNIV; IN_DIFF; IN_CBALL_0]] THEN REWRITE_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_POW; REAL_ABS_NORM] THEN ASM_SIMP_TAC[REAL_NOT_LE; real_abs; REAL_LT_IMP_LE] THEN ASM_CASES_TAC `x:real^N = vec 0` THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[NORM_EQ_0; REAL_FIELD `~(x = &0) ==> r pow 2 / x pow 2 * x = (r * r) / x`] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_LMUL_EQ; NORM_POS_LT] THEN REWRITE_TAC[REAL_LT_IMP_LE]; DISCH_THEN(SUBST1_TAC o SYM)] THEN REWRITE_TAC[RELATIVE_FRONTIER_FRONTIER_OF] THEN SUBGOAL_THEN `affine hull s:real^N->bool = u` ASSUME_TAC THENL [ASM_MESON_TAC[AFFINE_HULL_OPEN_IN_AFFINE; SUBSPACE_IMP_AFFINE; HULL_P]; ALL_TAC] THEN SUBGOAL_THEN `affine hull (IMAGE (i:real^N->real^N) s) = u` ASSUME_TAC THENL [MATCH_MP_TAC AFFINE_HULL_OPEN_IN_AFFINE THEN ASM_SIMP_TAC[IMAGE_EQ_EMPTY; SUBSPACE_IMP_AFFINE]; ASM_REWRITE_TAC[]] THEN FIRST_ASSUM(MP_TAC o SPECL [`subtopology euclidean u frontier_of s:real^N->bool`; `IMAGE (i:real^N->real^N) (subtopology euclidean u frontier_of s)`] o MATCH_MP(ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [REWRITE_TAC[frontier_of] THEN SUBGOAL_THEN `subtopology euclidean u closure_of s SUBSET (u:real^N->bool)` MP_TAC THENL [REWRITE_TAC[CLOSURE_OF_SUBSET_SUBTOPOLOGY]; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[HOMEOMORPHISM]) THEN ASM SET_TAC[]; DISCH_THEN(SUBST1_TAC o MATCH_MP HOMEOMORPHIC_DIMENSION o MATCH_MP HOMEOMORPHISM_IMP_HOMEOMORPHIC)] THEN MATCH_MP_TAC(MESON[DIMENSION_INSERT] `(?a:real^N. ~(s = {}) /\ ~(t = {}) /\ (a INSERT s = a INSERT t)) ==> dimension s = dimension t`) THEN EXISTS_TAC `vec 0:real^N` THEN REWRITE_TAC[IMAGE_EQ_EMPTY] THEN SUBGOAL_THEN `connected(u:real^N->bool)` MP_TAC THENL [ASM_SIMP_TAC[SUBSPACE_IMP_CONVEX; CONVEX_CONNECTED]; ALL_TAC] THEN GEN_REWRITE_TAC LAND_CONV [CONNECTED_CLOPEN] THEN REWRITE_TAC[ONCE_REWRITE_RULE[CONJ_SYM] CLOPEN_IN_EQ_FRONTIER_OF] THEN GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [GSYM CONTRAPOS_THM] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN REWRITE_TAC[TAUT `~p ==> ~(q /\ r) <=> ~p /\ r ==> ~q`] THEN REWRITE_TAC[DE_MORGAN_THM] THEN DISCH_TAC THEN GEN_REWRITE_TAC I [CONJ_ASSOC] THEN CONJ_TAC THENL [CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IMAGE_EQ_EMPTY] THEN RULE_ASSUM_TAC(REWRITE_RULE[HOMEOMORPHISM]) THEN ASM SET_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[frontier_of; INTERIOR_OF_OPEN_IN] THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> i(i x) = x) /\ t SUBSET s /\ a INSERT (IMAGE i s) = a INSERT u ==> a INSERT IMAGE i (s DIFF t) = a INSERT (u DIFF IMAGE i t)`) THEN REPEAT CONJ_TAC THENL [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN EXPAND_TAC "i" THEN REWRITE_TAC[] THEN ASM_CASES_TAC `x:real^N = vec 0` THEN ASM_REWRITE_TAC[VECTOR_MUL_RZERO] THEN GEN_REWRITE_TAC RAND_CONV [GSYM VECTOR_MUL_LID] THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_POW; REAL_ABS_NORM] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [GSYM NORM_EQ_0]) THEN UNDISCH_TAC `&0 < r` THEN SIMP_TAC[real_abs; REAL_LT_IMP_LE] THEN CONV_TAC REAL_FIELD; MATCH_MP_TAC CLOSURE_OF_SUBSET THEN ASM_MESON_TAC[OPEN_IN_SUBSET]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o SPEC `s:real^N->bool` o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_CLOSURE_OF)) THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[SET_RULE `s DELETE a = s INTER (s DELETE a)`] THEN REWRITE_TAC[GSYM SUBTOPOLOGY_SUBTOPOLOGY] THEN SIMP_TAC[CLOSURE_OF_SUBTOPOLOGY_OPEN; OPEN_IN_DELETE; OPEN_IN_REFL] THEN SIMP_TAC[SET_RULE `c SUBSET u ==> (u DELETE z) INTER c = c DELETE z`; CLOSURE_OF_SUBSET_SUBTOPOLOGY] THEN MATCH_MP_TAC(SET_RULE `z INSERT w = z INSERT y ==> w = x DELETE z ==> z INSERT y = z INSERT x`) THEN MATCH_MP_TAC(SET_RULE `i z = z ==> z INSERT IMAGE i (s DELETE z) = z INSERT IMAGE i s`) THEN EXPAND_TAC "i" THEN REWRITE_TAC[VECTOR_MUL_RZERO]);; let DIMENSION_FRONTIER_NONDENSE_OPEN = prove (`!u:real^N->bool. open u /\ ~(u = {}) /\ ~(closure u = (:real^N)) ==> dimension(frontier u) = &(dimindex(:N)) - &1`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`(:real^N)`; `u:real^N->bool`] DIMENSION_RELATIVE_FRONTIER_NONDENSE_OPEN) THEN ASM_SIMP_TAC[RELATIVE_FRONTIER_OPEN] THEN ASM_REWRITE_TAC[SUBTOPOLOGY_UNIV; GSYM OPEN_IN; AFFINE_UNIV; AFF_DIM_UNIV; EUCLIDEAN_CLOSURE_OF]);; let DIMENSION_RELATIVE_FRONTIER_CONVEX = prove (`!s:real^N->bool. convex s /\ bounded s /\ ~(s = {}) ==> dimension(relative_frontier s) = aff_dim s - &1`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`affine hull s:real^N->bool`; `relative_interior s:real^N->bool`] DIMENSION_RELATIVE_FRONTIER_BOUNDED_OPEN) THEN REWRITE_TAC[AFFINE_AFFINE_HULL; OPEN_IN_RELATIVE_INTERIOR] THEN ASM_SIMP_TAC[BOUNDED_RELATIVE_INTERIOR; RELATIVE_FRONTIER_RELATIVE_INTERIOR; AFF_DIM_AFFINE_HULL; RELATIVE_INTERIOR_EQ_EMPTY]);; let DIMENSION_SPHERE_INTER_AFFINE = prove (`!a:real^N r t. &0 < r /\ affine t /\ a IN t ==> dimension(sphere(a,r) INTER t) = aff_dim t - &1`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[GSYM FRONTIER_CBALL] THEN W(MP_TAC o PART_MATCH (rand o rand) RELATIVE_FRONTIER_CONVEX_INTER_AFFINE o rand o lhand o snd) THEN ANTS_TAC THENL [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]; DISCH_THEN(SUBST1_TAC o SYM)] THEN W(MP_TAC o PART_MATCH (lhand o rand) DIMENSION_RELATIVE_FRONTIER_CONVEX o lhand o snd) THEN ANTS_TAC THENL [SIMP_TAC[BOUNDED_INTER; BOUNDED_CBALL] THEN ASM_SIMP_TAC[CONVEX_INTER; CONVEX_CBALL; AFFINE_IMP_CONVEX] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN ASM_MESON_TAC[CENTRE_IN_CBALL; REAL_LT_IMP_LE]; DISCH_THEN SUBST1_TAC THEN AP_THM_TAC THEN AP_TERM_TAC] THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN AP_TERM_TAC THEN ASM_SIMP_TAC[HULL_P] THEN ONCE_REWRITE_TAC[INTER_COMM] THEN MATCH_MP_TAC AFFINE_HULL_AFFINE_INTER_NONEMPTY_INTERIOR THEN ASM_REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; INTERIOR_CBALL; IN_INTER] THEN ASM_MESON_TAC[CENTRE_IN_BALL]);; let DIMENSION_SPHERE = prove (`!a:real^N r. dimension(sphere(a,r)) = if &0 < r then &(dimindex(:N)) - &1 else if r = &0 then &0 else -- &1`, REPEAT GEN_TAC THEN ASM_CASES_TAC `r:real = &0` THEN ASM_SIMP_TAC[REAL_LT_REFL; SPHERE_SING; DIMENSION_SING] THEN COND_CASES_TAC THEN ASM_SIMP_TAC[DIMENSION_EQ_MINUS1; SPHERE_EQ_EMPTY] THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [SET_RULE `s = s INTER UNIV`] THEN ASM_SIMP_TAC[DIMENSION_SPHERE_INTER_AFFINE; AFFINE_UNIV; IN_UNIV] THEN REWRITE_TAC[AFF_DIM_UNIV]);; (* ------------------------------------------------------------------------- *) (* Nonseparation: a "simple" set of dimension n can't be separated by sets *) (* of dimension <= n - 2. *) (* ------------------------------------------------------------------------- *) let CONNECTED_OPEN_IN_CONVEX_DIFF_LOWDIM = prove (`!c s t:real^N->bool. convex c /\ open_in (subtopology euclidean c) s /\ connected s /\ dimension t <= aff_dim c - &2 ==> connected(s DIFF t)`, let lemma1 = prove (`!u s:real^N->bool. affine u /\ dimension s <= aff_dim u - &2 ==> connected(u DIFF s)`, MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `d:real^N->bool`] THEN STRIP_TAC THEN SUBGOAL_THEN `subtopology euclidean u interior_of d:real^N->bool = {}` ASSUME_TAC THENL [ONCE_REWRITE_TAC[INTERIOR_OF_RESTRICT] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN MP_TAC(ISPECL [`u:real^N->bool`; `u INTER d:real^N->bool`] DIMENSION_LT_FULL_ALT) THEN ASM_SIMP_TAC[AFFINE_IMP_CONVEX; INTER_SUBSET] THEN MATCH_MP_TAC(TAUT `p ==> (p <=> q /\ r) ==> r`) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (INT_ARITH `d:int <= u - &2 ==> d' <= d ==> d' < u`)) THEN SIMP_TAC[DIMENSION_SUBSET; INTER_SUBSET]; ALL_TAC] THEN REWRITE_TAC[CONNECTED_SEPARATION; NOT_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u1:real^N->bool`; `u2:real^N->bool`] THEN STRIP_TAC THEN FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP (SET_RULE `u1 UNION u2 = u DIFF d ==> u INTER u1 = u1 /\ u INTER u2 = u2`)) THEN SUBGOAL_THEN `(u:real^N->bool) SUBSET closure u1 UNION closure u2` ASSUME_TAC THENL [FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INTERIOR_OF_EQ_EMPTY_COMPLEMENT]) THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN SUBST1_TAC(SYM(ASSUME `u1 UNION u2:real^N->bool = u DIFF d`)) THEN REWRITE_TAC[CLOSURE_OF_UNION] THEN ASM_REWRITE_TAC[CLOSURE_OF_SUBTOPOLOGY; EUCLIDEAN_CLOSURE_OF] THEN SET_TAC[]; ALL_TAC] THEN ABBREV_TAC `v:real^N->bool = u DIFF closure u1` THEN MP_TAC(ISPECL [`u:real^N->bool`; `v:real^N->bool`] DIMENSION_RELATIVE_FRONTIER_NONDENSE_OPEN) THEN SUBGOAL_THEN `~(v:real^N->bool = {})` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `open_in (subtopology euclidean u) (v:real^N->bool)` ASSUME_TAC THENL [EXPAND_TAC "v" THEN SIMP_TAC[OPEN_IN_DIFF_CLOSED; CLOSED_CLOSURE]; ALL_TAC] THEN MP_TAC(ISPECL [`u:real^N->bool`; `v:real^N->bool`] AFFINE_HULL_OPEN_IN_AFFINE) THEN ASM_REWRITE_TAC[RELATIVE_FRONTIER_FRONTIER_OF] THEN DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `!t. s SUBSET t /\ t SUBSET u /\ ~(t = u) ==> ~(s = u)`) THEN EXISTS_TAC `subtopology euclidean u closure_of u2:real^N->bool` THEN REWRITE_TAC[CLOSURE_OF_SUBSET_SUBTOPOLOGY] THEN CONJ_TAC THENL [MATCH_MP_TAC CLOSURE_OF_MINIMAL THEN REWRITE_TAC[CLOSED_IN_CLOSURE_OF]; ALL_TAC] THEN ASM_REWRITE_TAC[CLOSURE_OF_SUBTOPOLOGY; EUCLIDEAN_CLOSURE_OF] THEN ASM SET_TAC[]; MATCH_MP_TAC(INT_ARITH `!x. d:int <= x /\ x < n ==> ~(d = n)`) THEN EXISTS_TAC `dimension(d:real^N->bool)` THEN CONJ_TAC THENL [MATCH_MP_TAC DIMENSION_SUBSET; ASM_INT_ARITH_TAC] THEN ASM_SIMP_TAC[frontier_of; INTERIOR_OF_OPEN_IN] THEN MP_TAC(ISPECL [`subtopology euclidean (u:real^N->bool)`; `u DIFF subtopology euclidean u closure_of u1:real^N->bool`; `subtopology euclidean u closure_of u2:real^N->bool`] CLOSURE_OF_MONO) THEN REWRITE_TAC[CLOSURE_OF_CLOSURE_OF] THEN ASM_REWRITE_TAC[CLOSURE_OF_SUBTOPOLOGY; EUCLIDEAN_CLOSURE_OF] THEN ASM_REWRITE_TAC[SET_RULE `u DIFF u INTER s = u DIFF s`] THEN ASM SET_TAC[]]) in let lemma2 = prove (`!u s t:real^N->bool. affine u /\ open_in (subtopology euclidean u) s /\ connected s /\ dimension t <= aff_dim u - &2 ==> connected(s DIFF t)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN MATCH_MP_TAC CONNECTED_CONNECTED_DIFF THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [MP_TAC(ISPECL [`s:real^N->bool`; `u DIFF t:real^N->bool`; `u:real^N->bool`] CLOSURE_OPEN_IN_INTER_CLOSURE) THEN ASM_REWRITE_TAC[SUBSET_DIFF] THEN MP_TAC(ISPECL [`u:real^N->bool`; `u DIFF closure(u DIFF t):real^N->bool`] DIMENSION_OPEN_IN_CONVEX) THEN ASM_SIMP_TAC[AFFINE_IMP_CONVEX; OPEN_IN_DIFF_CLOSED; CLOSED_CLOSURE] THEN COND_CASES_TAC THENL [DISCH_THEN(K ALL_TAC) THEN ASM_SIMP_TAC[SET_RULE `s SUBSET u /\ u DIFF closure(u DIFF t) = {} ==> s INTER closure (u DIFF t) = s`] THEN ASM_SIMP_TAC[SET_RULE `s SUBSET u ==> s INTER (u DIFF t) = s DIFF t`] THEN MESON_TAC[CLOSURE_SUBSET]; MATCH_MP_TAC(TAUT `~p ==> p ==> q`) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (INT_ARITH `d:int <= u - &2 ==> d' <= d ==> ~(d' = u)`)) THEN MATCH_MP_TAC DIMENSION_SUBSET THEN MP_TAC(ISPEC `u DIFF t:real^N->bool` CLOSURE_SUBSET) THEN SET_TAC[]]; X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_CONTAINS_BALL]) THEN DISCH_THEN(MP_TAC o SPEC `x:real^N` o CONJUNCT2) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `ball(x:real^N,r) INTER u` THEN ASM_REWRITE_TAC[CENTRE_IN_BALL; IN_INTER] THEN CONJ_TAC THENL[ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [ASM_MESON_TAC[INTER_COMM; OPEN_BALL; OPEN_IN_OPEN_INTER; OPEN_IN_SUBSET_TRANS]; ALL_TAC] THEN MP_TAC(ISPECL [`ball(x:real^N,r) INTER u`; `u:real^N->bool`] HOMEOMORPHIC_RELATIVELY_OPEN_CONVEX_SETS) THEN ASM_SIMP_TAC[CONVEX_BALL; CONVEX_INTER; AFFINE_IMP_CONVEX] THEN ASM_SIMP_TAC[HULL_P; OPEN_IN_REFL] THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN ONCE_REWRITE_TAC[INTER_COMM] THEN SUBGOAL_THEN `affine hull (u INTER ball(x:real^N,r)) = affine hull u` SUBST1_TAC THENL [MATCH_MP_TAC AFFINE_HULL_CONVEX_INTER_OPEN THEN ASM_SIMP_TAC[AFFINE_IMP_CONVEX; OPEN_BALL; GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `x:real^N` THEN ASM_REWRITE_TAC[IN_INTER; CENTRE_IN_BALL] THEN ASM SET_TAC[]; ASM_SIMP_TAC[HULL_P; OPEN_IN_OPEN_INTER; OPEN_BALL]] THEN REWRITE_TAC[homeomorphic; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f:real^N->real^N`; `g:real^N->real^N`] THEN DISCH_TAC THEN MP_TAC(ISPECL [`u:real^N->bool`; `IMAGE (f:real^N->real^N) (ball(x,r) INTER u INTER t)`] lemma1) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [TRANS_TAC INT_LE_TRANS `dimension(t:real^N->bool)` THEN ASM_REWRITE_TAC[] THEN TRANS_TAC INT_LE_TRANS `dimension(ball(x:real^N,r) INTER u INTER t)` THEN ASM_SIMP_TAC[DIMENSION_SUBSET; INTER_SUBSET; GSYM INTER_ASSOC] THEN MATCH_MP_TAC INT_EQ_IMP_LE THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC HOMEOMORPHIC_DIMENSION; MATCH_MP_TAC EQ_IMP THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC HOMEOMORPHIC_CONNECTEDNESS] THEN REWRITE_TAC[homeomorphic] THEN MAP_EVERY EXISTS_TAC [`f:real^N->real^N`; `g:real^N->real^N`] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN REWRITE_TAC[INTER_SUBSET; SUBSET_DIFF; SUBSET_UNIV] THEN RULE_ASSUM_TAC(REWRITE_RULE[HOMEOMORPHISM]) THEN ASM SET_TAC[]]) in REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[CONNECTED_EMPTY; EMPTY_DIFF] THEN MATCH_MP_TAC CONNECTED_INTERMEDIATE_CLOSURE THEN EXISTS_TAC `(relative_interior c INTER s) DIFF t:real^N->bool` THEN SUBGOAL_THEN `open_in (subtopology euclidean (affine hull c)) (relative_interior c INTER s:real^N->bool)` ASSUME_TAC THENL [TRANS_TAC OPEN_IN_TRANS `relative_interior c:real^N->bool` THEN ASM_REWRITE_TAC[OPEN_IN_RELATIVE_INTERIOR] THEN MATCH_MP_TAC OPEN_IN_SUBTOPOLOGY_INTER_SUBSET THEN EXISTS_TAC `c:real^N->bool` THEN ASM_SIMP_TAC[OPEN_IN_INTER; OPEN_IN_REFL; RELATIVE_INTERIOR_SUBSET]; ALL_TAC] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC lemma2 THEN EXISTS_TAC `affine hull s:real^N->bool` THEN REWRITE_TAC[OPEN_IN_RELATIVE_INTERIOR] THEN ASM_SIMP_TAC[AFF_DIM_AFFINE_HULL; AFFINE_AFFINE_HULL] THEN MP_TAC(ISPECL [`c:real^N->bool`; `s:real^N->bool`] AFFINE_HULL_OPEN_IN_CONVEX) THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN DISCH_THEN SUBST1_TAC THEN ASM_REWRITE_TAC[AFF_DIM_AFFINE_HULL] THEN MATCH_MP_TAC CONNECTED_WITH_RELATIVE_INTERIOR_OPEN_IN_CONVEX THEN ASM_REWRITE_TAC[]; MP_TAC(ISPEC `c:real^N->bool` RELATIVE_INTERIOR_SUBSET) THEN SET_TAC[]; MP_TAC(ISPECL [`relative_interior c INTER s:real^N->bool`; `affine hull c DIFF t:real^N->bool`; `affine hull c:real^N->bool`] CLOSURE_OPEN_IN_INTER_CLOSURE) THEN ASM_REWRITE_TAC[SUBSET_DIFF] THEN MP_TAC(ISPECL [`affine hull c:real^N->bool`; `affine hull c DIFF closure(affine hull c DIFF t):real^N->bool`] DIMENSION_OPEN_IN_CONVEX) THEN ASM_SIMP_TAC[AFFINE_IMP_CONVEX; OPEN_IN_DIFF_CLOSED; CLOSED_CLOSURE; AFFINE_AFFINE_HULL] THEN COND_CASES_TAC THENL [DISCH_THEN(K ALL_TAC) THEN ASM_SIMP_TAC[RELATIVE_INTERIOR_SUBSET; HULL_SUBSET; SET_RULE `relative_interior c SUBSET c /\ c SUBSET u /\ u DIFF closure(u DIFF t) = {} ==> (relative_interior c INTER s) INTER closure (u DIFF t) = relative_interior c INTER s /\ (relative_interior c INTER s) INTER (u DIFF t) = (relative_interior c INTER s) DIFF t`] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN 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 `s:real^N->bool` THEN REWRITE_TAC[SUBSET_DIFF] THEN TRANS_TAC SUBSET_TRANS `closure s:real^N->bool` THEN REWRITE_TAC[CLOSURE_SUBSET] THEN MATCH_MP_TAC SUBSET_CLOSURE THEN MATCH_MP_TAC(SET_RULE `c SUBSET closure c /\ s SUBSET c ==> s SUBSET s INTER closure c`) THEN ASM_MESON_TAC[OPEN_IN_IMP_SUBSET; CLOSURE_SUBSET]; MATCH_MP_TAC(TAUT `~p ==> p ==> q`) THEN REWRITE_TAC[AFF_DIM_AFFINE_HULL] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (INT_ARITH `d:int <= u - &2 ==> d' <= d ==> ~(d' = u)`)) THEN MATCH_MP_TAC DIMENSION_SUBSET THEN MP_TAC(ISPEC `affine hull c DIFF t:real^N->bool` CLOSURE_SUBSET) THEN SET_TAC[]]]);; let CONNECTED_CONVEX_DIFF_LOWDIM = prove (`!s t:real^N->bool. convex s /\ dimension t <= aff_dim s - &2 ==> connected(s DIFF t)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CONNECTED_OPEN_IN_CONVEX_DIFF_LOWDIM THEN EXISTS_TAC `s:real^N->bool` THEN ASM_SIMP_TAC[CONVEX_CONNECTED; OPEN_IN_REFL]);; let CONNECTED_OPEN_IN_DIFF_LOWDIM = prove (`!s t:real^N->bool. open_in (subtopology euclidean (affine hull s)) s /\ connected s /\ dimension t <= aff_dim s - &2 ==> connected(s DIFF t)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CONNECTED_OPEN_IN_CONVEX_DIFF_LOWDIM THEN EXISTS_TAC `affine hull s:real^N->bool` THEN ASM_SIMP_TAC[AFFINE_IMP_CONVEX; AFF_DIM_AFFINE_HULL; AFFINE_AFFINE_HULL]);; let CONNECTED_OPEN_DIFF_LOWDIM = prove (`!s t:real^N->bool. open s /\ connected s /\ dimension t <= &(dimindex(:N)) - &2 ==> connected(s DIFF t)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[EMPTY_DIFF; CONNECTED_EMPTY] THEN MATCH_MP_TAC CONNECTED_OPEN_IN_DIFF_LOWDIM THEN ASM_SIMP_TAC[OPEN_SUBSET; HULL_SUBSET; AFF_DIM_OPEN]);; let CONNECTED_FULL_CONVEX_DIFF_LOWDIM = prove (`!s:real^N->bool t. convex s /\ ~(interior s = {}) /\ dimension t <= &(dimindex(:N)) - &2 ==> connected(s DIFF t)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CONNECTED_CONVEX_DIFF_LOWDIM THEN ASM_SIMP_TAC[AFF_DIM_NONEMPTY_INTERIOR]);; let CONNECTED_UNIV_DIFF_LOWDIM = prove (`!s:real^N->bool. dimension s <= &(dimindex(:N)) - &2 ==> connected((:real^N) DIFF s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CONNECTED_FULL_CONVEX_DIFF_LOWDIM THEN ASM_REWRITE_TAC[CONVEX_UNIV; INTERIOR_UNIV; UNIV_NOT_EMPTY]);; let CONNECTED_FULL_REGULAR_DIFF_LOWDIM = prove (`!s:real^N->bool t. s SUBSET closure(interior s) /\ connected(interior s) /\ dimension t <= &(dimindex(:N)) - &2 ==> connected(s DIFF t)`, let lemma = prove (`!s t:real^N->bool. open s /\ interior t = {} ==> s SUBSET closure(s DIFF t)`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[SET_RULE `s DIFF t = s INTER (UNIV DIFF t)`] THEN ONCE_REWRITE_TAC[SET_RULE `s SUBSET t <=> s INTER t = s`] THEN W(MP_TAC o PART_MATCH (rand o rand) OPEN_INTER_CLOSURE_EQ o lhand o snd) THEN ASM_REWRITE_TAC[CLOSURE_COMPLEMENT] THEN SET_TAC[]) in REPEAT STRIP_TAC THEN MATCH_MP_TAC CONNECTED_INTERMEDIATE_CLOSURE THEN EXISTS_TAC `interior s DIFF t:real^N->bool` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_OPEN_DIFF_LOWDIM THEN ASM_REWRITE_TAC[OPEN_INTERIOR]; MP_TAC(ISPEC `s:real^N->bool` INTERIOR_SUBSET) THEN SET_TAC[]; TRANS_TAC SUBSET_TRANS `closure(interior s):real^N->bool` THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN REWRITE_TAC[CLOSED_CLOSURE] THEN MATCH_MP_TAC lemma THEN REWRITE_TAC[OPEN_INTERIOR] THEN MP_TAC(SPEC `t:real^N->bool` DIMENSION_NONEMPTY_INTERIOR) THEN ASM_CASES_TAC `interior t:real^N->bool = {}` THEN ASM_REWRITE_TAC[] THEN ASM_INT_ARITH_TAC]);; (* ------------------------------------------------------------------------- *) (* Absolute retracts (AR), absolute neighbourhood retracts (ANR) and also *) (* Euclidean neighbourhood retracts (ENR). We define AR and ANR by *) (* specializing the standard definitions for a set in R^n to embedding in *) (* spaces inside R^{n+1}. This turns out to be sufficient (since any set in *) (* R^n can be embedded as a closed subset of a convex subset of R^{n+1}) to *) (* derive the usual definitions, but we need to split them into two *) (* implications because of the lack of type quantifiers. Then ENR turns out *) (* to be equivalent to ANR plus local compactness. *) (* ------------------------------------------------------------------------- *) let AR = new_definition `AR(s:real^N->bool) <=> !u s':real^(N,1)finite_sum->bool. s homeomorphic s' /\ closed_in (subtopology euclidean u) s' ==> s' retract_of u`;; let ANR = new_definition `ANR(s:real^N->bool) <=> !u s':real^(N,1)finite_sum->bool. s homeomorphic s' /\ closed_in (subtopology euclidean u) s' ==> ?t. open_in (subtopology euclidean u) t /\ s' retract_of t`;; let ENR = new_definition `ENR s <=> ?u. open u /\ s retract_of u`;; (* ------------------------------------------------------------------------- *) (* First, show that we do indeed get the "usual" properties of ARs and ANRs. *) (* ------------------------------------------------------------------------- *) let AR_IMP_ABSOLUTE_EXTENSOR = prove (`!f:real^M->real^N u t s. AR s /\ f continuous_on t /\ IMAGE f t SUBSET s /\ closed_in (subtopology euclidean u) t ==> ?g. g continuous_on u /\ IMAGE g u SUBSET s /\ !x. x IN t ==> g x = f x`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?c s':real^(N,1)finite_sum->bool. convex c /\ ~(c = {}) /\ closed_in (subtopology euclidean c) s' /\ (s:real^N->bool) homeomorphic s'` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC HOMEOMORPHIC_CLOSED_IN_CONVEX THEN REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1; GSYM INT_OF_NUM_ADD] THEN REWRITE_TAC[INT_ARITH `x:int < y + &1 <=> x <= y`; AFF_DIM_LE_UNIV]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [AR]) THEN DISCH_THEN(MP_TAC o SPECL [`c:real^(N,1)finite_sum->bool`; `s':real^(N,1)finite_sum->bool`]) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphic]) THEN REWRITE_TAC[homeomorphism; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`g:real^N->real^(N,1)finite_sum`; `h:real^(N,1)finite_sum->real^N`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`(g:real^N->real^(N,1)finite_sum) o (f:real^M->real^N)`; `c:real^(N,1)finite_sum->bool`; `u:real^M->bool`; `t:real^M->bool`] DUGUNDJI) THEN ASM_REWRITE_TAC[IMAGE_o; o_THM] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP RETRACT_OF_IMP_SUBSET) THEN ANTS_TAC THENL [CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `f':real^M->real^(N,1)finite_sum` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retract_of]) THEN REWRITE_TAC[retraction; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `r:real^(N,1)finite_sum->real^(N,1)finite_sum` THEN STRIP_TAC THEN EXISTS_TAC `(h:real^(N,1)finite_sum->real^N) o r o (f':real^M->real^(N,1)finite_sum)` THEN ASM_REWRITE_TAC[IMAGE_o; o_THM] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[o_ASSOC] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]);; let AR_IMP_ABSOLUTE_RETRACT = prove (`!s:real^N->bool u s':real^M->bool. AR s /\ s homeomorphic s' /\ closed_in (subtopology euclidean u) s' ==> s' retract_of u`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphic]) THEN REWRITE_TAC[homeomorphism; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`g:real^N->real^M`; `h:real^M->real^N`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`h:real^M->real^N`; `u:real^M->bool`; `s':real^M->bool`; `s:real^N->bool`] AR_IMP_ABSOLUTE_EXTENSOR) THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN DISCH_THEN(X_CHOOSE_THEN `h':real^M->real^N` STRIP_ASSUME_TAC) THEN REWRITE_TAC[retract_of; retraction] THEN EXISTS_TAC `(g:real^N->real^M) o (h':real^M->real^N)` THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN ASM_SIMP_TAC[o_THM; IMAGE_o] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]);; let AR_IMP_ABSOLUTE_RETRACT_UNIV = prove (`!s:real^N->bool s':real^M->bool. AR s /\ s homeomorphic s' /\ closed s' ==> s' retract_of (:real^M)`, MESON_TAC[AR_IMP_ABSOLUTE_RETRACT; TOPSPACE_EUCLIDEAN; SUBTOPOLOGY_UNIV; OPEN_IN; CLOSED_IN]);; let ABSOLUTE_EXTENSOR_IMP_AR = prove (`!s:real^N->bool. (!f:real^(N,1)finite_sum->real^N u t. f continuous_on t /\ IMAGE f t SUBSET s /\ closed_in (subtopology euclidean u) t ==> ?g. g continuous_on u /\ IMAGE g u SUBSET s /\ !x. x IN t ==> g x = f x) ==> AR s`, REPEAT STRIP_TAC THEN REWRITE_TAC[AR] THEN MAP_EVERY X_GEN_TAC [`u:real^(N,1)finite_sum->bool`; `t:real^(N,1)finite_sum->bool`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphic]) THEN REWRITE_TAC[homeomorphism; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`g:real^N->real^(N,1)finite_sum`; `h:real^(N,1)finite_sum->real^N`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o ISPECL [`h:real^(N,1)finite_sum->real^N`; `u:real^(N,1)finite_sum->bool`; `t:real^(N,1)finite_sum->bool`]) THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN DISCH_THEN(X_CHOOSE_THEN `h':real^(N,1)finite_sum->real^N` STRIP_ASSUME_TAC) THEN REWRITE_TAC[retract_of; retraction] THEN EXISTS_TAC `(g:real^N->real^(N,1)finite_sum) o (h':real^(N,1)finite_sum->real^N)` THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN ASM_SIMP_TAC[o_THM; IMAGE_o] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]);; let AR_EQ_ABSOLUTE_EXTENSOR = prove (`!s:real^N->bool. AR s <=> (!f:real^(N,1)finite_sum->real^N u t. f continuous_on t /\ IMAGE f t SUBSET s /\ closed_in (subtopology euclidean u) t ==> ?g. g continuous_on u /\ IMAGE g u SUBSET s /\ !x. x IN t ==> g x = f x)`, GEN_TAC THEN EQ_TAC THEN SIMP_TAC[AR_IMP_ABSOLUTE_EXTENSOR; ABSOLUTE_EXTENSOR_IMP_AR]);; let AR_IMP_RETRACT = prove (`!s u:real^N->bool. AR s /\ closed_in (subtopology euclidean u) s ==> s retract_of u`, MESON_TAC[AR_IMP_ABSOLUTE_RETRACT; HOMEOMORPHIC_REFL]);; let HOMEOMORPHIC_ARNESS = prove (`!s:real^M->bool t:real^N->bool. s homeomorphic t ==> (AR s <=> AR t)`, let lemma = prove (`!s:real^M->bool t:real^N->bool. s homeomorphic t /\ AR t ==> AR s`, REPEAT STRIP_TAC THEN REWRITE_TAC[AR] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP(ONCE_REWRITE_RULE[IMP_CONJ] AR_IMP_ABSOLUTE_RETRACT)) THEN ASM_REWRITE_TAC[] THEN TRANS_TAC HOMEOMORPHIC_TRANS `s:real^M->bool` THEN ASM_MESON_TAC[HOMEOMORPHIC_SYM]) in REPEAT STRIP_TAC THEN EQ_TAC THEN POP_ASSUM MP_TAC THENL [ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM]; ALL_TAC] THEN ASM_MESON_TAC[lemma]);; let AR_TRANSLATION = prove (`!a:real^N s. AR(IMAGE (\x. a + x) s) <=> AR s`, REPEAT GEN_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_ARNESS THEN REWRITE_TAC[HOMEOMORPHIC_TRANSLATION_SELF]);; add_translation_invariants [AR_TRANSLATION];; let AR_LINEAR_IMAGE_EQ = prove (`!f:real^M->real^N s. linear f /\ (!x y. f x = f y ==> x = y) ==> (AR(IMAGE f s) <=> AR s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_ARNESS THEN ASM_MESON_TAC[HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_SELF]);; add_linear_invariants [AR_LINEAR_IMAGE_EQ];; let HOMEOMORPHISM_ARNESS = prove (`!f:real^M->real^N g s t k. homeomorphism (s,t) (f,g) /\ k SUBSET s ==> (AR(IMAGE f k) <=> AR k)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_ARNESS THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN REWRITE_TAC[homeomorphic] THEN MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `g:real^N->real^M`] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]);; let ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR = prove (`!f:real^M->real^N u t s. ANR s /\ f continuous_on t /\ IMAGE f t SUBSET s /\ closed_in (subtopology euclidean u) t ==> ?v g. t SUBSET v /\ open_in (subtopology euclidean u) v /\ g continuous_on v /\ IMAGE g v SUBSET s /\ !x. x IN t ==> g x = f x`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?c s':real^(N,1)finite_sum->bool. convex c /\ ~(c = {}) /\ closed_in (subtopology euclidean c) s' /\ (s:real^N->bool) homeomorphic s'` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC HOMEOMORPHIC_CLOSED_IN_CONVEX THEN REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1; GSYM INT_OF_NUM_ADD] THEN REWRITE_TAC[INT_ARITH `x:int < y + &1 <=> x <= y`; AFF_DIM_LE_UNIV]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [ANR]) THEN DISCH_THEN(MP_TAC o SPECL [`c:real^(N,1)finite_sum->bool`; `s':real^(N,1)finite_sum->bool`]) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `d:real^(N,1)finite_sum->bool` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphic]) THEN REWRITE_TAC[homeomorphism; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`g:real^N->real^(N,1)finite_sum`; `h:real^(N,1)finite_sum->real^N`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`(g:real^N->real^(N,1)finite_sum) o (f:real^M->real^N)`; `c:real^(N,1)finite_sum->bool`; `u:real^M->bool`; `t:real^M->bool`] DUGUNDJI) THEN ASM_REWRITE_TAC[IMAGE_o; o_THM] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP RETRACT_OF_IMP_SUBSET) THEN ANTS_TAC THENL [CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `f':real^M->real^(N,1)finite_sum` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retract_of]) THEN REWRITE_TAC[retraction; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `r:real^(N,1)finite_sum->real^(N,1)finite_sum` THEN STRIP_TAC THEN EXISTS_TAC `{x | x IN u /\ (f':real^M->real^(N,1)finite_sum) x IN d}` THEN EXISTS_TAC `(h:real^(N,1)finite_sum->real^N) o r o (f':real^M->real^(N,1)finite_sum)` THEN ASM_REWRITE_TAC[IMAGE_o; o_THM] THEN REPEAT CONJ_TAC THENL [REPEAT(FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET)) THEN ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE_GEN THEN ASM_MESON_TAC[]; REPEAT(MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC) THEN REWRITE_TAC[IMAGE_o] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; ASM SET_TAC[]; ASM SET_TAC[]]);; let ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_RETRACT = prove (`!s:real^N->bool u s':real^M->bool. ANR s /\ s homeomorphic s' /\ closed_in (subtopology euclidean u) s' ==> ?v. open_in (subtopology euclidean u) v /\ s' retract_of v`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphic]) THEN REWRITE_TAC[homeomorphism; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`g:real^N->real^M`; `h:real^M->real^N`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`h:real^M->real^N`; `u:real^M->bool`; `s':real^M->bool`; `s:real^N->bool`] ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR) THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `v:real^M->bool` THEN DISCH_THEN(X_CHOOSE_THEN `h':real^M->real^N` STRIP_ASSUME_TAC) THEN ASM_REWRITE_TAC[retract_of; retraction] THEN EXISTS_TAC `(g:real^N->real^M) o (h':real^M->real^N)` THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN ASM_SIMP_TAC[o_THM; IMAGE_o] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]);; let ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_RETRACT_UNIV = prove (`!s:real^N->bool s':real^M->bool. ANR s /\ s homeomorphic s' /\ closed s' ==> ?v. open v /\ s' retract_of v`, MESON_TAC[ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_RETRACT; TOPSPACE_EUCLIDEAN; SUBTOPOLOGY_UNIV; OPEN_IN; CLOSED_IN]);; let ABSOLUTE_NEIGHBOURHOOD_EXTENSOR_IMP_ANR = prove (`!s:real^N->bool. (!f:real^(N,1)finite_sum->real^N u t. f continuous_on t /\ IMAGE f t SUBSET s /\ closed_in (subtopology euclidean u) t ==> ?v g. t SUBSET v /\ open_in (subtopology euclidean u) v /\ g continuous_on v /\ IMAGE g v SUBSET s /\ !x. x IN t ==> g x = f x) ==> ANR s`, REPEAT STRIP_TAC THEN REWRITE_TAC[ANR] THEN MAP_EVERY X_GEN_TAC [`u:real^(N,1)finite_sum->bool`; `t:real^(N,1)finite_sum->bool`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphic]) THEN REWRITE_TAC[homeomorphism; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`g:real^N->real^(N,1)finite_sum`; `h:real^(N,1)finite_sum->real^N`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o ISPECL [`h:real^(N,1)finite_sum->real^N`; `u:real^(N,1)finite_sum->bool`; `t:real^(N,1)finite_sum->bool`]) THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `v:real^(N,1)finite_sum->bool` THEN DISCH_THEN(X_CHOOSE_THEN `h':real^(N,1)finite_sum->real^N` STRIP_ASSUME_TAC) THEN ASM_REWRITE_TAC[retract_of; retraction] THEN EXISTS_TAC `(g:real^N->real^(N,1)finite_sum) o (h':real^(N,1)finite_sum->real^N)` THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN ASM_SIMP_TAC[o_THM; IMAGE_o] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]);; let ANR_EQ_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR = prove (`!s:real^N->bool. ANR s <=> (!f:real^(N,1)finite_sum->real^N u t. f continuous_on t /\ IMAGE f t SUBSET s /\ closed_in (subtopology euclidean u) t ==> ?v g. t SUBSET v /\ open_in (subtopology euclidean u) v /\ g continuous_on v /\ IMAGE g v SUBSET s /\ !x. x IN t ==> g x = f x)`, GEN_TAC THEN EQ_TAC THEN SIMP_TAC[ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR; ABSOLUTE_NEIGHBOURHOOD_EXTENSOR_IMP_ANR]);; let ANR_IMP_ABSOLUTE_CLOSED_NEIGHBOURHOOD_RETRACT = prove (`!s:real^N->bool u s':real^M->bool. ANR s /\ s homeomorphic s' /\ closed_in (subtopology euclidean u) s' ==> ?v w. open_in (subtopology euclidean u) v /\ closed_in (subtopology euclidean u) w /\ s' SUBSET v /\ v SUBSET w /\ s' retract_of w`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?z. open_in (subtopology euclidean u) z /\ (s':real^M->bool) retract_of z` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_RETRACT THEN ASM_MESON_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`s':real^M->bool`; `u DIFF z:real^M->bool`; `u:real^M->bool`] SEPARATION_NORMAL_LOCAL) THEN ASM_SIMP_TAC[CLOSED_IN_INTER; CLOSED_IN_REFL; CLOSED_IN_DIFF] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP RETRACT_OF_IMP_SUBSET) THEN ANTS_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS] THEN X_GEN_TAC `v:real^M->bool` THEN DISCH_THEN(X_CHOOSE_THEN `w:real^M->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `u DIFF w:real^M->bool` THEN ASM_SIMP_TAC[CLOSED_IN_DIFF; CLOSED_IN_REFL] THEN REPEAT(FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET)) THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] RETRACT_OF_SUBSET)) THEN ASM SET_TAC[]);; let ANR_IMP_ABSOLUTE_CLOSED_NEIGHBOURHOOD_EXTENSOR = prove (`!f:real^M->real^N u t s. ANR s /\ f continuous_on t /\ IMAGE f t SUBSET s /\ closed_in (subtopology euclidean u) t ==> ?v w g. open_in (subtopology euclidean u) v /\ closed_in (subtopology euclidean u) w /\ t SUBSET v /\ v SUBSET w /\ g continuous_on w /\ IMAGE g w SUBSET s /\ !x. x IN t ==> g x = f x`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?v g. t SUBSET v /\ open_in (subtopology euclidean u) v /\ g continuous_on v /\ IMAGE g v SUBSET s /\ !x. x IN t ==> g x = (f:real^M->real^N) x` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR THEN ASM_MESON_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`t:real^M->bool`; `u DIFF v:real^M->bool`; `u:real^M->bool`] SEPARATION_NORMAL_LOCAL) THEN ASM_SIMP_TAC[CLOSED_IN_INTER; CLOSED_IN_REFL; CLOSED_IN_DIFF] THEN ANTS_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS] THEN X_GEN_TAC `w:real^M->bool` THEN DISCH_THEN(X_CHOOSE_THEN `z:real^M->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `u DIFF z:real^M->bool` THEN ASM_SIMP_TAC[CLOSED_IN_DIFF; CLOSED_IN_REFL] THEN EXISTS_TAC `g:real^M->real^N` THEN REPEAT(FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET)) THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]);; let ANR_IMP_NEIGHBOURHOOD_RETRACT = prove (`!s:real^N->bool u. ANR s /\ closed_in (subtopology euclidean u) s ==> ?v. open_in (subtopology euclidean u) v /\ s retract_of v`, MESON_TAC[ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_RETRACT; HOMEOMORPHIC_REFL]);; let ANR_IMP_CLOSED_NEIGHBOURHOOD_RETRACT = prove (`!s:real^N->bool u. ANR s /\ closed_in (subtopology euclidean u) s ==> ?v w. open_in (subtopology euclidean u) v /\ closed_in (subtopology euclidean u) w /\ s SUBSET v /\ v SUBSET w /\ s retract_of w`, MESON_TAC[ANR_IMP_ABSOLUTE_CLOSED_NEIGHBOURHOOD_RETRACT; HOMEOMORPHIC_REFL]);; let HOMEOMORPHIC_ANRNESS = prove (`!s:real^M->bool t:real^N->bool. s homeomorphic t ==> (ANR s <=> ANR t)`, let lemma = prove (`!s:real^M->bool t:real^N->bool. s homeomorphic t /\ ANR t ==> ANR s`, REPEAT STRIP_TAC THEN REWRITE_TAC[ANR] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP(ONCE_REWRITE_RULE[IMP_CONJ] ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_RETRACT)) THEN ASM_REWRITE_TAC[] THEN TRANS_TAC HOMEOMORPHIC_TRANS `s:real^M->bool` THEN ASM_MESON_TAC[HOMEOMORPHIC_SYM]) in REPEAT STRIP_TAC THEN EQ_TAC THEN POP_ASSUM MP_TAC THENL [ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM]; ALL_TAC] THEN ASM_MESON_TAC[lemma]);; let ANR_TRANSLATION = prove (`!a:real^N s. ANR(IMAGE (\x. a + x) s) <=> ANR s`, REPEAT GEN_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_ANRNESS THEN REWRITE_TAC[HOMEOMORPHIC_TRANSLATION_SELF]);; add_translation_invariants [ANR_TRANSLATION];; let ANR_LINEAR_IMAGE_EQ = prove (`!f:real^M->real^N s. linear f /\ (!x y. f x = f y ==> x = y) ==> (ANR(IMAGE f s) <=> ANR s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_ANRNESS THEN ASM_MESON_TAC[HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_SELF]);; add_linear_invariants [ANR_LINEAR_IMAGE_EQ];; let HOMEOMORPHISM_ANRNESS = prove (`!f:real^M->real^N g s t k. homeomorphism (s,t) (f,g) /\ k SUBSET s ==> (ANR(IMAGE f k) <=> ANR k)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_ANRNESS THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN REWRITE_TAC[homeomorphic] THEN MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `g:real^N->real^M`] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]);; let HOMOTOPIC_ON_NEIGHBOURHOOD_INTO_ANR = prove (`!f g:real^M->real^N s t v. ANR v /\ f continuous_on s /\ IMAGE f s SUBSET v /\ g continuous_on s /\ IMAGE g s SUBSET v /\ t SUBSET s /\ (!x. x IN t ==> f x = g x) ==> ?u. open_in (subtopology euclidean s) u /\ t SUBSET u /\ homotopic_with (\h. !x. x IN t ==> h x = f x) (subtopology euclidean u,subtopology euclidean v) f g`, REPEAT STRIP_TAC THEN ABBREV_TAC `c = {x | x IN s /\ (f:real^M->real^N) x = g x}` THEN SUBGOAL_THEN `closed_in (subtopology euclidean s) (c:real^M->bool)` ASSUME_TAC THENL [EXPAND_TAC "c" THEN ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE_CONSTANT THEN ASM_SIMP_TAC[CONTINUOUS_ON_SUB]; ALL_TAC] THEN ABBREV_TAC `fg:real^(1,M)finite_sum->real^N = \x. if fstcart x = vec 1 then g(sndcart x) else f(sndcart x)` THEN MP_TAC(ISPECL [`fg:real^(1,M)finite_sum->real^N`; `(interval[vec 0,vec 1] PCROSS s):real^(1,M)finite_sum->bool`; `interval[vec 0,vec 1] PCROSS c UNION {vec 0:real^1,vec 1} PCROSS (s:real^M->bool)`; `v:real^N->bool`] ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR) THEN ASM_SIMP_TAC[CLOSED_IN_PCROSS_EQ; CLOSED_IN_REFL; CLOSED_IN_INSERT; CLOSED_IN_EMPTY; ENDS_IN_UNIT_INTERVAL; CLOSED_IN_UNION] THEN ANTS_TAC THENL [CONJ_TAC THENL [ONCE_REWRITE_TAC[SET_RULE `{a,b} = {a} UNION {b}`] THEN REWRITE_TAC[PCROSS_UNION] THEN REWRITE_TAC[GSYM UNION_ASSOC] THEN ONCE_REWRITE_TAC[UNION_COMM] THEN EXPAND_TAC "fg" THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC CLOSED_IN_SUBSET_TRANS THEN EXISTS_TAC `interval[vec 0:real^1,vec 1] PCROSS (s:real^M->bool)` THEN ASM_SIMP_TAC[CLOSED_IN_PCROSS_EQ; CLOSED_IN_REFL; CLOSED_IN_INSERT; CLOSED_IN_EMPTY; ENDS_IN_UNIT_INTERVAL; CLOSED_IN_UNION] THEN REWRITE_TAC[SUBSET_UNION] THEN REWRITE_TAC[SUBSET_PCROSS; UNION_SUBSET; UNIT_INTERVAL_NONEMPTY; INSERT_SUBSET; EMPTY_SUBSET; ENDS_IN_UNIT_INTERVAL; NOT_INSERT_EMPTY; SUBSET_REFL] THEN ASM_MESON_TAC[CLOSED_IN_IMP_SUBSET]; ONCE_REWRITE_TAC[CONJ_ASSOC]] THEN CONJ_TAC THENL [CONJ_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[LINEAR_SNDCART; LINEAR_CONTINUOUS_ON] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_UNION] THEN SIMP_TAC[FORALL_PASTECART; SNDCART_PASTECART; PASTECART_IN_PCROSS] THEN ASM SET_TAC[]; REWRITE_TAC[FORALL_PASTECART; PASTECART_IN_PCROSS] THEN REWRITE_TAC[PASTECART_IN_PCROSS; IN_UNION] THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN X_GEN_TAC `x:real^1` THEN ASM_CASES_TAC `x:real^1 = vec 1` THEN ASM_REWRITE_TAC[VEC_EQ; IN_INSERT; NOT_IN_EMPTY] THEN CONV_TAC NUM_REDUCE_CONV THEN ASM SET_TAC[]]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_UNION] THEN SIMP_TAC[FORALL_PASTECART; SNDCART_PASTECART; PASTECART_IN_PCROSS] THEN EXPAND_TAC "fg" THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN ASM SET_TAC[]]; REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u2:real^(1,M)finite_sum->bool`; `h:real^(1,M)finite_sum->real^N`] THEN REWRITE_TAC[UNION_SUBSET; FORALL_IN_UNION] THEN REWRITE_TAC[FORALL_PASTECART; SNDCART_PASTECART; PASTECART_IN_PCROSS] THEN STRIP_TAC THEN MP_TAC(ISPECL [`interval[vec 0:real^1,vec 1]`; `c:real^M->bool`; `s:real^M->bool`; `u2:real^(1,M)finite_sum->bool`] TUBE_LEMMA_GEN) THEN ASM_REWRITE_TAC[COMPACT_INTERVAL; UNIT_INTERVAL_NONEMPTY] THEN ANTS_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS] THEN X_GEN_TAC `u:real^M->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhand o rand) HOMOTOPIC_WITH_EUCLIDEAN_ALT o snd) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN EXISTS_TAC `h:real^(1,M)finite_sum->real^N` THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)); FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE h u2 SUBSET v ==> u SUBSET u2 ==> IMAGE h u SUBSET v`))] THEN ASM_REWRITE_TAC[]; ONCE_REWRITE_TAC[CONJ_ASSOC]] THEN SUBGOAL_THEN `!x:real^M. x IN u ==> x IN s` MP_TAC THENL [ASM_MESON_TAC[SUBSET; OPEN_IN_IMP_SUBSET]; ALL_TAC] THEN SUBGOAL_THEN `!x:real^M. x IN t ==> x IN c` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[IN_INSERT] THEN REPEAT DISCH_TAC THEN EXPAND_TAC "fg" THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; VEC_EQ; ARITH_EQ] THEN ASM SET_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Analogous properties of ENRs. *) (* ------------------------------------------------------------------------- *) let ENR_IMP_ABSOLUTE_NEIGHBOURHOOD_RETRACT = prove (`!s:real^M->bool s':real^N->bool u. ENR s /\ s homeomorphic s' /\ s' SUBSET u ==> ?t'. open_in (subtopology euclidean u) t' /\ s' retract_of t'`, REWRITE_TAC[ENR; LEFT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`X:real^M->bool`; `Y:real^N->bool`; `K:real^N->bool`; `U:real^M->bool`] THEN STRIP_TAC THEN SUBGOAL_THEN `locally compact (Y:real^N->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[RETRACT_OF_LOCALLY_COMPACT; OPEN_IMP_LOCALLY_COMPACT; HOMEOMORPHIC_LOCAL_COMPACTNESS]; ALL_TAC] THEN SUBGOAL_THEN `?W:real^N->bool. open_in (subtopology euclidean K) W /\ closed_in (subtopology euclidean W) Y` STRIP_ASSUME_TAC THENL [FIRST_ASSUM(X_CHOOSE_THEN `W:real^N->bool` STRIP_ASSUME_TAC o MATCH_MP LOCALLY_COMPACT_CLOSED_IN_OPEN) THEN EXISTS_TAC `K INTER W:real^N->bool` THEN ASM_SIMP_TAC[OPEN_IN_OPEN_INTER; CLOSED_IN_CLOSED] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CLOSED_IN_CLOSED]) THEN ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphic]) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f:real^M->real^N`; `g:real^N->real^M`] THEN REWRITE_TAC[homeomorphism] THEN STRIP_TAC THEN MP_TAC(ISPECL [`g:real^N->real^M`; `W:real^N->bool`; `Y:real^N->bool`] TIETZE_UNBOUNDED) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `h:real^N->real^M` STRIP_ASSUME_TAC) THEN EXISTS_TAC `{x | x IN W /\ (h:real^N->real^M) x IN U}` THEN CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `W:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE_GEN THEN EXISTS_TAC `(:real^M)` THEN ASM_REWRITE_TAC[SUBTOPOLOGY_UNIV; GSYM OPEN_IN; SUBSET_UNIV]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retract_of]) THEN REWRITE_TAC[retraction; retract_of; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `r:real^M->real^M` THEN STRIP_TAC THEN EXISTS_TAC `(f:real^M->real^N) o r o (h:real^N->real^M)` THEN SUBGOAL_THEN `(W:real^N->bool) SUBSET K /\ Y SUBSET W` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[OPEN_IN_IMP_SUBSET; CLOSED_IN_IMP_SUBSET]; ALL_TAC] THEN REWRITE_TAC[IMAGE_o; o_THM] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REPEAT(MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC) THEN REWRITE_TAC[IMAGE_o] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]);; let ENR_IMP_ABSOLUTE_NEIGHBOURHOOD_RETRACT_UNIV = prove (`!s:real^M->bool s':real^N->bool. ENR s /\ s homeomorphic s' ==> ?t'. open t' /\ s' retract_of t'`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[OPEN_IN] THEN ONCE_REWRITE_TAC[GSYM SUBTOPOLOGY_UNIV] THEN MATCH_MP_TAC ENR_IMP_ABSOLUTE_NEIGHBOURHOOD_RETRACT THEN ASM_MESON_TAC[SUBSET_UNIV]);; let HOMEOMORPHIC_ENRNESS = prove (`!s:real^M->bool t:real^N->bool. s homeomorphic t ==> (ENR s <=> ENR t)`, REPEAT STRIP_TAC THEN EQ_TAC THEN DISCH_TAC THEN REWRITE_TAC[ENR] THENL [MP_TAC(ISPECL [`s:real^M->bool`; `t:real^N->bool`] ENR_IMP_ABSOLUTE_NEIGHBOURHOOD_RETRACT_UNIV); MP_TAC(ISPECL [`t:real^N->bool`; `s:real^M->bool`] ENR_IMP_ABSOLUTE_NEIGHBOURHOOD_RETRACT_UNIV)] THEN ASM_REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_MESON_TAC[HOMEOMORPHIC_SYM]);; let ENR_TRANSLATION = prove (`!a:real^N s. ENR(IMAGE (\x. a + x) s) <=> ENR s`, REPEAT GEN_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_ENRNESS THEN REWRITE_TAC[HOMEOMORPHIC_TRANSLATION_SELF]);; add_translation_invariants [ENR_TRANSLATION];; let ENR_LINEAR_IMAGE_EQ = prove (`!f:real^M->real^N s. linear f /\ (!x y. f x = f y ==> x = y) ==> (ENR(IMAGE f s) <=> ENR s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_ENRNESS THEN ASM_MESON_TAC[HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_SELF]);; add_linear_invariants [ENR_LINEAR_IMAGE_EQ];; let HOMEOMORPHISM_ENRNESS = prove (`!f:real^M->real^N g s t k. homeomorphism (s,t) (f,g) /\ k SUBSET s ==> (ENR(IMAGE f k) <=> ENR k)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_ENRNESS THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN REWRITE_TAC[homeomorphic] THEN MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `g:real^N->real^M`] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* Some relations among the concepts. We also relate AR to being a retract *) (* of UNIV, which is often a more convenient proxy in the closed case. *) (* ------------------------------------------------------------------------- *) let AR_IMP_ANR = prove (`!s:real^N->bool. AR s ==> ANR s`, REWRITE_TAC[AR; ANR] THEN MESON_TAC[OPEN_IN_REFL; CLOSED_IN_IMP_SUBSET]);; let ENR_IMP_ANR = prove (`!s:real^N->bool. ENR s ==> ANR s`, REWRITE_TAC[ANR] THEN MESON_TAC[ENR_IMP_ABSOLUTE_NEIGHBOURHOOD_RETRACT; CLOSED_IN_IMP_SUBSET]);; let ENR_ANR = prove (`!s:real^N->bool. ENR s <=> ANR s /\ locally compact s`, REPEAT(STRIP_TAC ORELSE EQ_TAC) THEN ASM_SIMP_TAC[ENR_IMP_ANR] THENL [ASM_MESON_TAC[ENR; RETRACT_OF_LOCALLY_COMPACT; OPEN_IMP_LOCALLY_COMPACT]; SUBGOAL_THEN `?t. closed t /\ (s:real^N->bool) homeomorphic (t:real^(N,1)finite_sum->bool)` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC LOCALLY_COMPACT_HOMEOMORPHIC_CLOSED THEN ASM_REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1] THEN ARITH_TAC; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [ANR]) THEN DISCH_THEN(MP_TAC o SPECL [`(:real^(N,1)finite_sum)`; `t:real^(N,1)finite_sum->bool`]) THEN ASM_REWRITE_TAC[SUBTOPOLOGY_UNIV; GSYM CLOSED_IN; GSYM OPEN_IN] THEN REWRITE_TAC[GSYM ENR] THEN ASM_MESON_TAC[HOMEOMORPHIC_ENRNESS]]]);; let AR_ANR = prove (`!s:real^N->bool. AR s <=> ANR s /\ contractible s /\ ~(s = {})`, GEN_TAC THEN EQ_TAC THEN STRIP_TAC THEN ASM_SIMP_TAC[AR_IMP_ANR] THENL [CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[AR; HOMEOMORPHIC_EMPTY; RETRACT_OF_EMPTY; FORALL_UNWIND_THM2; CLOSED_IN_EMPTY; UNIV_NOT_EMPTY]] THEN SUBGOAL_THEN `?c s':real^(N,1)finite_sum->bool. convex c /\ ~(c = {}) /\ closed_in (subtopology euclidean c) s' /\ (s:real^N->bool) homeomorphic s'` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC HOMEOMORPHIC_CLOSED_IN_CONVEX THEN REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1; GSYM INT_OF_NUM_ADD] THEN REWRITE_TAC[INT_ARITH `x:int < y + &1 <=> x <= y`; AFF_DIM_LE_UNIV]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [AR]) THEN DISCH_THEN(MP_TAC o SPECL [`c:real^(N,1)finite_sum->bool`; `s':real^(N,1)finite_sum->bool`]) THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[HOMEOMORPHIC_SYM; HOMEOMORPHIC_CONTRACTIBLE; RETRACT_OF_CONTRACTIBLE; CONVEX_IMP_CONTRACTIBLE]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [contractible]) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; HOMOTOPIC_WITH_EUCLIDEAN] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `h:real^(1,N)finite_sum->real^N`] THEN STRIP_TAC THEN REWRITE_TAC[AR_EQ_ABSOLUTE_EXTENSOR] THEN MAP_EVERY X_GEN_TAC [`f:real^(N,1)finite_sum->real^N`; `w:real^(N,1)finite_sum->bool`; `t:real^(N,1)finite_sum->bool`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o ISPECL [`f:real^(N,1)finite_sum->real^N`; `w:real^(N,1)finite_sum->bool`; `t:real^(N,1)finite_sum->bool`] o REWRITE_RULE[ANR_EQ_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR]) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u:real^(N,1)finite_sum->bool`; `g:real^(N,1)finite_sum->real^N`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`t:real^(N,1)finite_sum->bool`; `w DIFF u:real^(N,1)finite_sum->bool`; `w:real^(N,1)finite_sum->bool`] SEPARATION_NORMAL_LOCAL) THEN ASM_SIMP_TAC[CLOSED_IN_DIFF; CLOSED_IN_REFL] THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`v:real^(N,1)finite_sum->bool`; `v':real^(N,1)finite_sum->bool`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`t:real^(N,1)finite_sum->bool`; `w DIFF v:real^(N,1)finite_sum->bool`; `w:real^(N,1)finite_sum->bool`; `vec 0:real^1`; `vec 1:real^1`] URYSOHN_LOCAL) THEN ASM_SIMP_TAC[SEGMENT_1; CLOSED_IN_DIFF; CLOSED_IN_REFL] THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN REWRITE_TAC[DROP_VEC; REAL_POS] THEN X_GEN_TAC `e:real^(N,1)finite_sum->real^1` THEN STRIP_TAC THEN EXISTS_TAC `\x. if (x:real^(N,1)finite_sum) IN w DIFF v then a else (h:real^(1,N)finite_sum->real^N) (pastecart (e x) (g x))` THEN REWRITE_TAC[] THEN CONJ_TAC THENL [SUBGOAL_THEN `w:real^(N,1)finite_sum->bool = (w DIFF v) UNION (w DIFF v')` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(fun th -> GEN_REWRITE_TAC RAND_CONV [th] THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL THEN REWRITE_TAC[GSYM th]) THEN ASM_SIMP_TAC[CLOSED_IN_DIFF; CLOSED_IN_REFL; CONTINUOUS_ON_CONST] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_PASTECART THEN CONJ_TAC; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; PASTECART_IN_PCROSS] THEN ASM SET_TAC[]; ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE] THEN RULE_ASSUM_TAC (REWRITE_RULE[SUBSET; FORALL_IN_IMAGE; FORALL_IN_PCROSS]) THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[IN_DIFF] THEN COND_CASES_TAC THEN ASM SET_TAC[]]);; let ANR_RETRACT_OF_ANR = prove (`!s t:real^N->bool. ANR t /\ s retract_of t ==> ANR s`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN REWRITE_TAC[ANR_EQ_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR] THEN REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retract_of]) THEN REWRITE_TAC[retraction; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `r:real^N->real^N` THEN STRIP_TAC THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN DISCH_THEN(X_CHOOSE_THEN `g:real^(N,1)finite_sum->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(r:real^N->real^N) o (g:real^(N,1)finite_sum->real^N)` THEN ASM_SIMP_TAC[IMAGE_o; o_THM] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]);; let AR_RETRACT_OF_AR = prove (`!s t:real^N->bool. AR t /\ s retract_of t ==> AR s`, REWRITE_TAC[AR_ANR] THEN MESON_TAC[ANR_RETRACT_OF_ANR; RETRACT_OF_CONTRACTIBLE; RETRACT_OF_EMPTY]);; let ENR_RETRACT_OF_ENR = prove (`!s t:real^N->bool. ENR t /\ s retract_of t ==> ENR s`, REWRITE_TAC[ENR] THEN MESON_TAC[RETRACT_OF_TRANS]);; let RETRACT_OF_UNIV = prove (`!s:real^N->bool. s retract_of (:real^N) <=> AR s /\ closed s`, GEN_TAC THEN EQ_TAC THEN REPEAT STRIP_TAC THENL [MATCH_MP_TAC AR_RETRACT_OF_AR THEN EXISTS_TAC `(:real^N)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC ABSOLUTE_EXTENSOR_IMP_AR THEN MESON_TAC[DUGUNDJI; CONVEX_UNIV; UNIV_NOT_EMPTY]; MATCH_MP_TAC RETRACT_OF_CLOSED THEN ASM_MESON_TAC[CLOSED_UNIV]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] AR_IMP_ABSOLUTE_RETRACT)) THEN ASM_REWRITE_TAC[SUBTOPOLOGY_UNIV; GSYM CLOSED_IN; HOMEOMORPHIC_REFL]]);; let COMPACT_AR = prove (`!s. compact s /\ AR s <=> compact s /\ s retract_of (:real^N)`, REWRITE_TAC[RETRACT_OF_UNIV] THEN MESON_TAC[COMPACT_IMP_CLOSED]);; (* ------------------------------------------------------------------------- *) (* More properties of ARs, ANRs and ENRs. *) (* ------------------------------------------------------------------------- *) let NOT_AR_EMPTY = prove (`~(AR({}:real^N->bool))`, REWRITE_TAC[AR_ANR]);; let AR_IMP_NONEMPTY = prove (`!s:real^N->bool. AR s ==> ~(s = {})`, MESON_TAC[NOT_AR_EMPTY]);; let ENR_EMPTY = prove (`ENR {}`, REWRITE_TAC[ENR; RETRACT_OF_EMPTY] THEN MESON_TAC[OPEN_EMPTY]);; let ANR_EMPTY = prove (`ANR {}`, SIMP_TAC[ENR_EMPTY; ENR_IMP_ANR]);; let CONVEX_IMP_AR = prove (`!s:real^N->bool. convex s /\ ~(s = {}) ==> AR s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC ABSOLUTE_EXTENSOR_IMP_AR THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC DUGUNDJI THEN ASM_REWRITE_TAC[]);; let CONVEX_IMP_ANR = prove (`!s:real^N->bool. convex s ==> ANR s`, MESON_TAC[ANR_EMPTY; CONVEX_IMP_AR; AR_IMP_ANR]);; let IS_INTERVAL_IMP_ENR = prove (`!s:real^N->bool. is_interval s ==> ENR s`, SIMP_TAC[ENR_ANR; IS_INTERVAL_IMP_LOCALLY_COMPACT] THEN SIMP_TAC[CONVEX_IMP_ANR; IS_INTERVAL_CONVEX]);; let ENR_CONVEX_CLOSED = prove (`!s:real^N->bool. closed s /\ convex s ==> ENR s`, MESON_TAC[CONVEX_IMP_ANR; ENR_ANR; CLOSED_IMP_LOCALLY_COMPACT]);; let AR_UNIV = prove (`AR(:real^N)`, MESON_TAC[CONVEX_IMP_AR; CONVEX_UNIV; UNIV_NOT_EMPTY]);; let ANR_UNIV = prove (`ANR(:real^N)`, MESON_TAC[CONVEX_IMP_ANR; CONVEX_UNIV]);; let ENR_UNIV = prove (`ENR(:real^N)`, MESON_TAC[ENR_CONVEX_CLOSED; CONVEX_UNIV; CLOSED_UNIV]);; let AR_SING = prove (`!a:real^N. AR {a}`, SIMP_TAC[CONVEX_IMP_AR; CONVEX_SING; NOT_INSERT_EMPTY]);; let ANR_SING = prove (`!a:real^N. ANR {a}`, SIMP_TAC[AR_IMP_ANR; AR_SING]);; let ENR_SING = prove (`!a:real^N. ENR {a}`, SIMP_TAC[ENR_ANR; ANR_SING; CLOSED_IMP_LOCALLY_COMPACT; CLOSED_SING]);; let ANR_OPEN_IN = prove (`!s t:real^N->bool. open_in (subtopology euclidean t) s /\ ANR t ==> ANR s`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[ANR_EQ_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR] THEN REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^(N,1)finite_sum->real^N` THEN DISCH_THEN(X_CHOOSE_THEN `w:real^(N,1)finite_sum->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `{x | x IN w /\ (g:real^(N,1)finite_sum->real^N) x IN s}` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `w:real^(N,1)finite_sum->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE_GEN THEN ASM_MESON_TAC[]; CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]]);; let ENR_OPEN_IN = prove (`!s t:real^N->bool. open_in (subtopology euclidean t) s /\ ENR t ==> ENR s`, REWRITE_TAC[ENR_ANR] THEN MESON_TAC[ANR_OPEN_IN; LOCALLY_OPEN_SUBSET]);; let ANR_NEIGHBORHOOD_RETRACT = prove (`!s t u:real^N->bool. s retract_of t /\ open_in (subtopology euclidean u) t /\ ANR u ==> ANR s`, MESON_TAC[ANR_OPEN_IN; ANR_RETRACT_OF_ANR]);; let ENR_NEIGHBORHOOD_RETRACT = prove (`!s t u:real^N->bool. s retract_of t /\ open_in (subtopology euclidean u) t /\ ENR u ==> ENR s`, MESON_TAC[ENR_OPEN_IN; ENR_RETRACT_OF_ENR]);; let ANR_RELATIVE_INTERIOR = prove (`!s. ANR(s) ==> ANR(relative_interior s)`, MESON_TAC[OPEN_IN_SET_RELATIVE_INTERIOR; ANR_OPEN_IN]);; let ANR_DELETE = prove (`!s a:real^N. ANR(s) ==> ANR(s DELETE a)`, MESON_TAC[ANR_OPEN_IN; OPEN_IN_DELETE; OPEN_IN_REFL]);; let ENR_RELATIVE_INTERIOR = prove (`!s. ENR(s) ==> ENR(relative_interior s)`, MESON_TAC[OPEN_IN_SET_RELATIVE_INTERIOR; ENR_OPEN_IN]);; let ENR_DELETE = prove (`!s a:real^N. ENR(s) ==> ENR(s DELETE a)`, MESON_TAC[ENR_OPEN_IN; OPEN_IN_DELETE; OPEN_IN_REFL]);; let OPEN_IMP_ENR = prove (`!s:real^N->bool. open s ==> ENR s`, REWRITE_TAC[OPEN_IN] THEN ONCE_REWRITE_TAC[GSYM SUBTOPOLOGY_UNIV] THEN MESON_TAC[ENR_UNIV; ENR_OPEN_IN]);; let OPEN_IMP_ANR = prove (`!s:real^N->bool. open s ==> ANR s`, SIMP_TAC[OPEN_IMP_ENR; ENR_IMP_ANR]);; let ANR_BALL = prove (`!a:real^N r. ANR(ball(a,r))`, MESON_TAC[CONVEX_IMP_ANR; CONVEX_BALL]);; let ENR_BALL = prove (`!a:real^N r. ENR(ball(a,r))`, SIMP_TAC[ENR_ANR; ANR_BALL; OPEN_IMP_LOCALLY_COMPACT; OPEN_BALL]);; let AR_BALL = prove (`!a:real^N r. AR(ball(a,r)) <=> &0 < r`, SIMP_TAC[AR_ANR; BALL_EQ_EMPTY; ANR_BALL; CONVEX_BALL; CONVEX_IMP_CONTRACTIBLE; REAL_NOT_LE]);; let ANR_CBALL = prove (`!a:real^N r. ANR(cball(a,r))`, MESON_TAC[CONVEX_IMP_ANR; CONVEX_CBALL]);; let ENR_CBALL = prove (`!a:real^N r. ENR(cball(a,r))`, SIMP_TAC[ENR_ANR; ANR_CBALL; CLOSED_IMP_LOCALLY_COMPACT; CLOSED_CBALL]);; let AR_CBALL = prove (`!a:real^N r. AR(cball(a,r)) <=> &0 <= r`, SIMP_TAC[AR_ANR; CBALL_EQ_EMPTY; ANR_CBALL; CONVEX_CBALL; CONVEX_IMP_CONTRACTIBLE; REAL_NOT_LT]);; let ANR_INTERVAL = prove (`(!a b:real^N. ANR(interval[a,b])) /\ (!a b:real^N. ANR(interval(a,b)))`, SIMP_TAC[CONVEX_IMP_ANR; CONVEX_INTERVAL; CLOSED_INTERVAL; OPEN_IMP_ANR; OPEN_INTERVAL]);; let ENR_INTERVAL = prove (`(!a b:real^N. ENR(interval[a,b])) /\ (!a b:real^N. ENR(interval(a,b)))`, SIMP_TAC[ENR_CONVEX_CLOSED; CONVEX_INTERVAL; CLOSED_INTERVAL; OPEN_IMP_ENR; OPEN_INTERVAL]);; let AR_INTERVAL = prove (`(!a b:real^N. AR(interval[a,b]) <=> ~(interval[a,b] = {})) /\ (!a b:real^N. AR(interval(a,b)) <=> ~(interval(a,b) = {}))`, SIMP_TAC[AR_ANR; ANR_INTERVAL; CONVEX_IMP_CONTRACTIBLE; CONVEX_INTERVAL]);; let ANR_INTERIOR = prove (`!s. ANR(interior s)`, SIMP_TAC[OPEN_INTERIOR; OPEN_IMP_ANR]);; let ENR_INTERIOR = prove (`!s. ENR(interior s)`, SIMP_TAC[OPEN_INTERIOR; OPEN_IMP_ENR]);; let AR_IMP_CONTRACTIBLE = prove (`!s:real^N->bool. AR s ==> contractible s`, SIMP_TAC[AR_ANR]);; let AR_IMP_PATH_CONNECTED = prove (`!s:real^N->bool. AR s ==> path_connected s`, MESON_TAC[AR_IMP_CONTRACTIBLE; CONTRACTIBLE_IMP_PATH_CONNECTED]);; let AR_IMP_CONNECTED = prove (`!s:real^N->bool. AR s ==> connected s`, MESON_TAC[AR_IMP_CONTRACTIBLE; CONTRACTIBLE_IMP_CONNECTED]);; let ENR_IMP_LOCALLY_COMPACT = prove (`!s:real^N->bool. ENR s ==> locally compact s`, SIMP_TAC[ENR_ANR]);; let ANR_IMP_LOCALLY_PATH_CONNECTED = prove (`!s:real^N->bool. ANR s ==> locally path_connected s`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?c s':real^(N,1)finite_sum->bool. convex c /\ ~(c = {}) /\ closed_in (subtopology euclidean c) s' /\ (s:real^N->bool) homeomorphic s'` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC HOMEOMORPHIC_CLOSED_IN_CONVEX THEN REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1; GSYM INT_OF_NUM_ADD] THEN REWRITE_TAC[INT_ARITH `x:int < y + &1 <=> x <= y`; AFF_DIM_LE_UNIV]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [ANR]) THEN DISCH_THEN(MP_TAC o SPECL [`c:real^(N,1)finite_sum->bool`; `s':real^(N,1)finite_sum->bool`]) THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[HOMEOMORPHIC_SYM; HOMEOMORPHIC_LOCAL_PATH_CONNECTEDNESS; RETRACT_OF_LOCALLY_PATH_CONNECTED; CONVEX_IMP_LOCALLY_PATH_CONNECTED; LOCALLY_OPEN_SUBSET]);; let ANR_IMP_LOCALLY_CONNECTED = prove (`!s:real^N->bool. ANR s ==> locally connected s`, SIMP_TAC[ANR_IMP_LOCALLY_PATH_CONNECTED; LOCALLY_PATH_CONNECTED_IMP_LOCALLY_CONNECTED]);; let AR_IMP_LOCALLY_PATH_CONNECTED = prove (`!s:real^N->bool. AR s ==> locally path_connected s`, SIMP_TAC[AR_IMP_ANR; ANR_IMP_LOCALLY_PATH_CONNECTED]);; let AR_IMP_LOCALLY_CONNECTED = prove (`!s:real^N->bool. AR s ==> locally connected s`, SIMP_TAC[AR_IMP_LOCALLY_PATH_CONNECTED; LOCALLY_PATH_CONNECTED_IMP_LOCALLY_CONNECTED]);; let ENR_IMP_LOCALLY_PATH_CONNECTED = prove (`!s:real^N->bool. ENR s ==> locally path_connected s`, SIMP_TAC[ANR_IMP_LOCALLY_PATH_CONNECTED; ENR_IMP_ANR]);; let ENR_IMP_LOCALLY_CONNECTED = prove (`!s:real^N->bool. ENR s ==> locally connected s`, SIMP_TAC[ANR_IMP_LOCALLY_CONNECTED; ENR_IMP_ANR]);; let COUNTABLE_ANR_COMPONENTS = prove (`!s:real^N->bool. ANR s ==> COUNTABLE(components s)`, SIMP_TAC[ANR_IMP_LOCALLY_CONNECTED; COUNTABLE_COMPONENTS]);; let COUNTABLE_ANR_CONNECTED_COMPONENTS = prove (`!s:real^N->bool t. ANR s ==> COUNTABLE {connected_component s x | x IN t}`, SIMP_TAC[ANR_IMP_LOCALLY_CONNECTED; COUNTABLE_CONNECTED_COMPONENTS]);; let COUNTABLE_ANR_PATH_COMPONENTS = prove (`!s:real^N->bool t. ANR s ==> COUNTABLE {path_component s x | x IN t}`, SIMP_TAC[ANR_IMP_LOCALLY_PATH_CONNECTED; COUNTABLE_PATH_COMPONENTS]);; let FINITE_ANR_COMPONENTS = prove (`!s:real^N->bool. ANR s /\ compact s ==> FINITE(components s)`, SIMP_TAC[FINITE_COMPONENTS; ANR_IMP_LOCALLY_CONNECTED]);; let FINITE_ENR_COMPONENTS = prove (`!s:real^N->bool. ENR s /\ compact s ==> FINITE(components s)`, SIMP_TAC[FINITE_COMPONENTS; ENR_IMP_LOCALLY_CONNECTED]);; let ANR_PCROSS = prove (`!s:real^M->bool t:real^N->bool. ANR s /\ ANR t ==> ANR(s PCROSS t)`, REPEAT STRIP_TAC THEN SIMP_TAC[ANR_EQ_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR] THEN MAP_EVERY X_GEN_TAC [`f:real^((M,N)finite_sum,1)finite_sum->real^(M,N)finite_sum`; `u:real^((M,N)finite_sum,1)finite_sum->bool`; `c:real^((M,N)finite_sum,1)finite_sum->bool`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`fstcart o (f:real^((M,N)finite_sum,1)finite_sum->real^(M,N)finite_sum)`; `u:real^((M,N)finite_sum,1)finite_sum->bool`; `c:real^((M,N)finite_sum,1)finite_sum->bool`; `s:real^M->bool`] ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR) THEN MP_TAC(ISPECL [`sndcart o (f:real^((M,N)finite_sum,1)finite_sum->real^(M,N)finite_sum)`; `u:real^((M,N)finite_sum,1)finite_sum->bool`; `c:real^((M,N)finite_sum,1)finite_sum->bool`; `t:real^N->bool`] ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR) THEN ASM_SIMP_TAC[CONTINUOUS_ON_COMPOSE; LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; LINEAR_SNDCART; IMAGE_o] THEN RULE_ASSUM_TAC (REWRITE_RULE[SUBSET; FORALL_IN_IMAGE; PCROSS; IN_ELIM_THM]) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN ANTS_TAC THENL [ASM_MESON_TAC[SNDCART_PASTECART]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`w2:real^((M,N)finite_sum,1)finite_sum->bool`; `h:real^((M,N)finite_sum,1)finite_sum->real^N`] THEN STRIP_TAC THEN ANTS_TAC THENL [ASM_MESON_TAC[FSTCART_PASTECART]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`w1:real^((M,N)finite_sum,1)finite_sum->bool`; `g:real^((M,N)finite_sum,1)finite_sum->real^M`] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`w1 INTER w2:real^((M,N)finite_sum,1)finite_sum->bool`; `\x:real^((M,N)finite_sum,1)finite_sum. pastecart (g x:real^M) (h x:real^N)`] THEN ASM_SIMP_TAC[OPEN_IN_INTER; IN_INTER; o_DEF; PASTECART_IN_PCROSS; PASTECART_FST_SND] THEN MATCH_MP_TAC CONTINUOUS_ON_PASTECART THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; INTER_SUBSET]);; let ANR_PCROSS_EQ = prove (`!s:real^M->bool t:real^N->bool. ANR(s PCROSS t) <=> s = {} \/ t = {} \/ ANR s /\ ANR t`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_REWRITE_TAC[PCROSS_EMPTY; ANR_EMPTY] THEN ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_REWRITE_TAC[PCROSS_EMPTY; ANR_EMPTY] THEN EQ_TAC THEN REWRITE_TAC[ANR_PCROSS] THEN REPEAT STRIP_TAC THENL [UNDISCH_TAC `~(t:real^N->bool = {})` THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `b:real^N` THEN DISCH_TAC THEN SUBGOAL_THEN `ANR ((s:real^M->bool) PCROSS {b:real^N})` MP_TAC THENL [ALL_TAC; MESON_TAC[HOMEOMORPHIC_PCROSS_SING; HOMEOMORPHIC_ANRNESS]]; UNDISCH_TAC `~(s:real^M->bool = {})` THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `a:real^M` THEN DISCH_TAC THEN SUBGOAL_THEN `ANR ({a:real^M} PCROSS (t:real^N->bool))` MP_TAC THENL [ALL_TAC; MESON_TAC[HOMEOMORPHIC_PCROSS_SING; HOMEOMORPHIC_ANRNESS]]] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] ANR_RETRACT_OF_ANR)) THEN REWRITE_TAC[retract_of; retraction] THENL [EXISTS_TAC`\x:real^(M,N)finite_sum. pastecart (fstcart x) (b:real^N)`; EXISTS_TAC`\x:real^(M,N)finite_sum. pastecart (a:real^M) (sndcart x)`] THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_PCROSS; FORALL_IN_IMAGE; IN_SING; FSTCART_PASTECART; SNDCART_PASTECART; PASTECART_IN_PCROSS; CONTINUOUS_ON_PASTECART; LINEAR_FSTCART; LINEAR_SNDCART; LINEAR_CONTINUOUS_ON; CONTINUOUS_ON_CONST]);; let AR_PCROSS = prove (`!s:real^M->bool t:real^N->bool. AR s /\ AR t ==> AR(s PCROSS t)`, SIMP_TAC[AR_ANR; ANR_PCROSS; CONTRACTIBLE_PCROSS; PCROSS_EQ_EMPTY]);; let ENR_PCROSS = prove (`!s:real^M->bool t:real^N->bool. ENR s /\ ENR t ==> ENR(s PCROSS t)`, SIMP_TAC[ENR_ANR; ANR_PCROSS; LOCALLY_COMPACT_PCROSS]);; let ENR_PCROSS_EQ = prove (`!s:real^M->bool t:real^N->bool. ENR(s PCROSS t) <=> s = {} \/ t = {} \/ ENR s /\ ENR t`, REWRITE_TAC[ENR_ANR; ANR_PCROSS_EQ; LOCALLY_COMPACT_PCROSS_EQ] THEN CONV_TAC TAUT);; let AR_PCROSS_EQ = prove (`!s:real^M->bool t:real^N->bool. AR(s PCROSS t) <=> AR s /\ AR t /\ ~(s = {}) /\ ~(t = {})`, SIMP_TAC[AR_ANR; ANR_PCROSS_EQ; CONTRACTIBLE_PCROSS_EQ; PCROSS_EQ_EMPTY] THEN CONV_TAC TAUT);; let AR_CLOSED_UNION_LOCAL = prove (`!s t:real^N->bool. closed_in (subtopology euclidean (s UNION t)) s /\ closed_in (subtopology euclidean (s UNION t)) t /\ AR(s) /\ AR(t) /\ AR(s INTER t) ==> AR(s UNION t)`, let lemma = prove (`!s t u:real^N->bool. closed_in (subtopology euclidean u) s /\ closed_in (subtopology euclidean u) t /\ AR s /\ AR t /\ AR(s INTER t) ==> (s UNION t) retract_of u`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s INTER t:real^N->bool = {}` THENL [ASM_MESON_TAC[NOT_AR_EMPTY]; ALL_TAC] THEN SUBGOAL_THEN `(s:real^N->bool) SUBSET u /\ t SUBSET u` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[CLOSED_IN_IMP_SUBSET]; ALL_TAC] THEN MAP_EVERY ABBREV_TAC [`s' = {x:real^N | x IN u /\ setdist({x},s) <= setdist({x},t)}`; `t' = {x:real^N | x IN u /\ setdist({x},t) <= setdist({x},s)}`; `w = {x:real^N | x IN u /\ setdist({x},s) = setdist({x},t)}`] THEN SUBGOAL_THEN `closed_in (subtopology euclidean u) (s':real^N->bool) /\ closed_in (subtopology euclidean u) (t':real^N->bool)` STRIP_ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["s'"; "t'"] THEN ONCE_REWRITE_TAC[GSYM REAL_SUB_LE] THEN ONCE_REWRITE_TAC[GSYM LIFT_DROP] THEN REWRITE_TAC[SET_RULE `a <= drop(lift x) <=> lift x IN {x | a <= drop x}`] THEN REWRITE_TAC[LIFT_DROP; LIFT_SUB] THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE THEN SIMP_TAC[CLOSED_SING; CONTINUOUS_ON_SUB; CONTINUOUS_ON_LIFT_SETDIST; drop; CLOSED_HALFSPACE_COMPONENT_LE; REWRITE_RULE[real_ge] CLOSED_HALFSPACE_COMPONENT_GE]; ALL_TAC] THEN SUBGOAL_THEN `(s:real^N->bool) SUBSET s' /\ (t:real^N->bool) SUBSET t'` STRIP_ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["s'"; "t'"] THEN SIMP_TAC[SUBSET; IN_ELIM_THM; SETDIST_SING_IN_SET; SETDIST_POS_LE] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(s INTER t:real^N->bool) retract_of w` MP_TAC THENL [MATCH_MP_TAC AR_IMP_ABSOLUTE_RETRACT THEN EXISTS_TAC `s INTER t:real^N->bool` THEN ASM_REWRITE_TAC[HOMEOMORPHIC_REFL] THEN MATCH_MP_TAC CLOSED_IN_SUBSET_TRANS THEN EXISTS_TAC `u:real^N->bool` THEN ASM_SIMP_TAC[CLOSED_IN_INTER] THEN CONJ_TAC THENL [EXPAND_TAC "w"; ASM SET_TAC[]] THEN SIMP_TAC[SUBSET; IN_INTER; IN_ELIM_THM; SETDIST_SING_IN_SET] THEN ASM SET_TAC[]; GEN_REWRITE_TAC LAND_CONV [retract_of] THEN REWRITE_TAC[retraction; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `r0:real^N->real^N` THEN STRIP_TAC] THEN SUBGOAL_THEN `!x:real^N. x IN w ==> (x IN s <=> x IN t)` ASSUME_TAC THENL [EXPAND_TAC "w" THEN REWRITE_TAC[IN_ELIM_THM] THEN GEN_TAC THEN DISCH_THEN(fun th -> EQ_TAC THEN DISCH_TAC THEN MP_TAC th) THEN ASM_SIMP_TAC[SETDIST_SING_IN_SET] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[REAL_ARITH `&0 = setdist p <=> setdist p = &0`] THEN MATCH_MP_TAC(SET_RULE `~(s = {}) /\ (p <=> s = {} \/ x IN s) ==> p ==> x IN s`) THEN (CONJ_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC SETDIST_EQ_0_CLOSED_IN]) THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `s' INTER t':real^N->bool = w` ASSUME_TAC THENL [ASM SET_TAC[REAL_LE_ANTISYM]; ALL_TAC] THEN SUBGOAL_THEN `closed_in (subtopology euclidean u) (w:real^N->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[CLOSED_IN_INTER]; ALL_TAC] THEN ABBREV_TAC `r = \x:real^N. if x IN w then r0 x else x` THEN SUBGOAL_THEN `IMAGE (r:real^N->real^N) (w UNION s) SUBSET s /\ IMAGE (r:real^N->real^N) (w UNION t) SUBSET t` STRIP_ASSUME_TAC THENL [EXPAND_TAC "r" THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(r:real^N->real^N) continuous_on (w UNION s UNION t)` ASSUME_TAC THENL [EXPAND_TAC "r" THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL THEN ASM_REWRITE_TAC[CONTINUOUS_ON_ID] THEN REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN CONJ_TAC THEN MATCH_MP_TAC CLOSED_IN_SUBSET_TRANS THEN EXISTS_TAC `u:real^N->bool` THEN ASM_SIMP_TAC[CLOSED_IN_UNION] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?g:real^N->real^N. g continuous_on u /\ IMAGE g u SUBSET s /\ !x. x IN w UNION s ==> g x = r x` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC AR_IMP_ABSOLUTE_EXTENSOR THEN ASM_SIMP_TAC[CLOSED_IN_UNION] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET; IN_UNION]; ALL_TAC] THEN SUBGOAL_THEN `?h:real^N->real^N. h continuous_on u /\ IMAGE h u SUBSET t /\ !x. x IN w UNION t ==> h x = r x` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC AR_IMP_ABSOLUTE_EXTENSOR THEN ASM_SIMP_TAC[CLOSED_IN_UNION] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET; IN_UNION]; ALL_TAC] THEN REWRITE_TAC[retract_of; retraction] THEN EXISTS_TAC `\x. if x IN s' then (g:real^N->real^N) x else h x` THEN REPEAT CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_UNION] THEN ASM SET_TAC[]; X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_UNION] THEN STRIP_TAC THEN ASM_SIMP_TAC[IN_UNION; COND_ID] THENL [COND_CASES_TAC THENL [EXPAND_TAC "r"; ASM SET_TAC[]]; COND_CASES_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN TRANS_TAC EQ_TRANS `(r:real^N->real^N) x` THEN CONJ_TAC THENL [ASM SET_TAC[]; EXPAND_TAC "r"]] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM SET_TAC[]] THEN SUBGOAL_THEN `u:real^N->bool = s' UNION t'` (fun th -> ONCE_REWRITE_TAC[th] THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL THEN REWRITE_TAC[GSYM th]) THENL [ASM SET_TAC[REAL_LE_TOTAL]; ASM_SIMP_TAC[]] 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[]) THEN REWRITE_TAC[TAUT `p /\ ~p \/ q /\ p <=> p /\ q`] THEN ASM_SIMP_TAC[GSYM IN_INTER; IN_UNION]) in REPEAT STRIP_TAC THEN REWRITE_TAC[AR] THEN MAP_EVERY X_GEN_TAC [`u:real^(N,1)finite_sum->bool`; `c:real^(N,1)finite_sum->bool`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphic]) THEN REWRITE_TAC[homeomorphism; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f:real^N->real^(N,1)finite_sum`; `g:real^(N,1)finite_sum->real^N`] THEN STRIP_TAC THEN SUBGOAL_THEN `closed_in (subtopology euclidean u) {x | x IN c /\ (g:real^(N,1)finite_sum->real^N) x IN s} /\ closed_in (subtopology euclidean u) {x | x IN c /\ (g:real^(N,1)finite_sum->real^N) x IN t}` STRIP_ASSUME_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC CLOSED_IN_TRANS THEN EXISTS_TAC `c:real^(N,1)finite_sum->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE_GEN THEN EXISTS_TAC `s UNION t:real^N->bool` THEN ASM_REWRITE_TAC[SUBSET_REFL]; ALL_TAC] THEN SUBGOAL_THEN `{x | x IN c /\ (g:real^(N,1)finite_sum->real^N) x IN s} UNION {x | x IN c /\ (g:real^(N,1)finite_sum->real^N) x IN t} = c` (fun th -> SUBST1_TAC(SYM th)) THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC lemma THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [UNDISCH_TAC `AR(s:real^N->bool)`; UNDISCH_TAC `AR(t:real^N->bool)`; UNDISCH_TAC `AR(s INTER t:real^N->bool)`] THEN MATCH_MP_TAC EQ_IMP THEN MATCH_MP_TAC HOMEOMORPHIC_ARNESS THEN REWRITE_TAC[homeomorphic; homeomorphism] THEN MAP_EVERY EXISTS_TAC [`f:real^N->real^(N,1)finite_sum`; `g:real^(N,1)finite_sum->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[]);; (* ------------------------------------------------------------------------- *) (* General ANR union lemma (Kuratowski). *) (* ------------------------------------------------------------------------- *) let ANR_UNION_EXTENSION_LEMMA = prove (`!f:real^M->real^N s t u s1 s2 u1 u2. f continuous_on t /\ IMAGE f t SUBSET u /\ ANR u1 /\ ANR u2 /\ ANR(u1 INTER u2) /\ u1 UNION u2 = u /\ closed_in (subtopology euclidean s) t /\ closed_in (subtopology euclidean s) s1 /\ closed_in (subtopology euclidean s) s2 /\ s1 UNION s2 = s /\ IMAGE f (t INTER s1) SUBSET u1 /\ IMAGE f (t INTER s2) SUBSET u2 ==> ?v g. t SUBSET v /\ open_in (subtopology euclidean s) v /\ g continuous_on v /\ IMAGE g v SUBSET u /\ !x. x IN t ==> g x = f x`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?v v' h. t INTER s1 INTER s2 SUBSET v /\ v SUBSET v' /\ open_in (subtopology euclidean (s1 INTER s2)) v /\ closed_in (subtopology euclidean (s1 INTER s2)) v' /\ h continuous_on v' /\ IMAGE h v' SUBSET u1 INTER u2 /\ !x. x IN v' INTER t ==> (h:real^M->real^N) x = f x` STRIP_ASSUME_TAC THENL [MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`; `t INTER s1 INTER s2:real^M->bool`; `u1 INTER u2:real^N->bool`] ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR) THEN ASM_SIMP_TAC[CLOSED_IN_INTER] THEN ANTS_TAC THENL [CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`v:real^M->bool`; `g:real^M->real^N`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`t INTER s1 INTER s2:real^M->bool`; `s DIFF v:real^M->bool`; `s:real^M->bool`] SEPARATION_NORMAL_LOCAL) THEN ASM_SIMP_TAC[CLOSED_IN_INTER; CLOSED_IN_DIFF; CLOSED_IN_REFL] THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`w:real^M->bool`; `w':real^M->bool`] THEN STRIP_TAC THEN EXISTS_TAC `(s1 INTER s2) INTER w:real^M->bool` THEN EXISTS_TAC `(s1 INTER s2) DIFF w':real^M->bool` THEN EXISTS_TAC `g:real^M->real^N` THEN ASM_SIMP_TAC[CLOSED_IN_DIFF; CLOSED_IN_REFL] THEN REPEAT CONJ_TAC THENL [ASM SET_TAC[]; ASM SET_TAC[]; MATCH_MP_TAC OPEN_IN_SUBTOPOLOGY_INTER_SUBSET THEN EXISTS_TAC `s:real^M->bool` THEN ASM_SIMP_TAC[OPEN_IN_INTER; OPEN_IN_REFL] THEN ASM SET_TAC[]; ONCE_REWRITE_TAC[SET_RULE `s DIFF t = s DIFF (s INTER t)`] THEN MATCH_MP_TAC CLOSED_IN_DIFF THEN REWRITE_TAC[CLOSED_IN_REFL] THEN MATCH_MP_TAC OPEN_IN_SUBTOPOLOGY_INTER_SUBSET THEN EXISTS_TAC `s:real^M->bool` THEN ASM_SIMP_TAC[OPEN_IN_INTER; OPEN_IN_REFL] THEN ASM SET_TAC[]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; ASM SET_TAC[]; ASM SET_TAC[]]; ALL_TAC] THEN ABBREV_TAC `k:real^M->bool = (s1 INTER s2) DIFF v` THEN SUBGOAL_THEN `closed_in (subtopology euclidean (s1 INTER s2)) (k:real^M->bool)` ASSUME_TAC THENL [EXPAND_TAC "k" THEN ASM_SIMP_TAC[CLOSED_IN_DIFF; CLOSED_IN_REFL]; ALL_TAC] THEN SUBGOAL_THEN `closed_in (subtopology euclidean s) (k:real^M->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[CLOSED_IN_TRANS; CLOSED_IN_INTER]; ALL_TAC] THEN SUBGOAL_THEN `k INTER t:real^M->bool = {}` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`subtopology euclidean ((t INTER s2) UNION v':real^M->bool)`; `euclidean:(real^N)topology`; `\i. if i = 0 then (f:real^M->real^N) else h`; `\i. if i = 0 then t INTER s2:real^M->bool else v'`; `{0,1}`] PASTING_LEMMA_EXISTS_CLOSED) THEN MP_TAC(ISPECL [`subtopology euclidean ((t INTER s1) UNION v':real^M->bool)`; `euclidean:(real^N)topology`; `\i. if i = 0 then (f:real^M->real^N) else h`; `\i. if i = 0 then t INTER s1:real^M->bool else v'`; `{0,1}`] PASTING_LEMMA_EXISTS_CLOSED) THEN REWRITE_TAC[CONTINUOUS_MAP_EUCLIDEAN; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; SUBTOPOLOGY_SUBTOPOLOGY] THEN ONCE_REWRITE_TAC[TAUT `closed_in a b /\ c <=> ~(closed_in a b ==> ~c)`] THEN SIMP_TAC[ISPEC `euclidean` CLOSED_IN_IMP_SUBSET; SET_RULE `s SUBSET u ==> u INTER s = s`] THEN REWRITE_TAC[NOT_IMP] THEN REWRITE_TAC[SUBSET_UNIV] THEN MAP_EVERY (fun x -> REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY] THEN ANTS_TAC THENL [REWRITE_TAC[SIMPLE_IMAGE; IMAGE_CLAUSES; UNIONS_2] THEN ASM_REWRITE_TAC[ARITH_EQ; SUBSET_REFL; FORALL_IN_INSERT; NOT_IN_EMPTY] THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[TAUT `(p /\ q) /\ r <=> q /\ p /\ r`] THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; INTER_SUBSET]; ALL_TAC] THEN ASM_REWRITE_TAC[SET_RULE `(s UNION t) INTER t = t`] THEN CONJ_TAC THEN MATCH_MP_TAC(MESON[] `u INTER s = s /\ closed_in (subtopology top u) (u INTER s) ==> closed_in (subtopology top u) s`) THEN (CONJ_TAC THENL [SET_TAC[]; ALL_TAC]) THEN MATCH_MP_TAC CLOSED_IN_SUBTOPOLOGY_INTER_SUBSET THEN EXISTS_TAC `s:real^M->bool` THEN ASM_SIMP_TAC[CLOSED_IN_INTER; CLOSED_IN_REFL] THEN TRY(CONJ_TAC THENL [ASM_MESON_TAC[CLOSED_IN_TRANS; CLOSED_IN_INTER; CLOSED_IN_REFL]; ALL_TAC]) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET)) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET)) THEN ASM SET_TAC[]; MATCH_MP_TAC WLOG_LT THEN REWRITE_TAC[] THEN CONJ_TAC THENL [MESON_TAC[INTER_COMM]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN REWRITE_TAC[CONJ_ASSOC] THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN ONCE_REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[IMP_IMP; IN_INSERT; NOT_IN_EMPTY] THEN REWRITE_TAC[ARITH_RULE `m < n /\ (m = 0 \/ m = 1) /\ (n = 0 \/ n = 1) <=> m = 0 /\ n = 1`] THEN STRIP_TAC THEN ASM_REWRITE_TAC[ARITH_EQ] THEN ASM SET_TAC[]]; GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [SWAP_FORALL_THM] THEN GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [FORALL_IN_INSERT; NOT_IN_EMPTY; ARITH_EQ] THEN REWRITE_TAC[SET_RULE `(s UNION t) INTER s = s /\ (s UNION t) INTER t = t`] THEN DISCH_THEN(X_CHOOSE_THEN x STRIP_ASSUME_TAC)]) [`f1:real^M->real^N`; `f2:real^M->real^N`] THEN MP_TAC(ISPECL [`f1:real^M->real^N`; `s:real^M->bool`; `t INTER s1 UNION v':real^M->bool`; `u1:real^N->bool`] ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC CLOSED_IN_UNION THEN ASM_MESON_TAC[CLOSED_IN_TRANS; CLOSED_IN_INTER]; REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`v1:real^M->bool`; `g1:real^M->real^N`] THEN STRIP_TAC] THEN MP_TAC(ISPECL [`f2:real^M->real^N`; `s:real^M->bool`; `t INTER s2 UNION v':real^M->bool`; `u2:real^N->bool`] ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC CLOSED_IN_UNION THEN ASM_MESON_TAC[CLOSED_IN_TRANS; CLOSED_IN_INTER]; REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`v2:real^M->bool`; `g2:real^M->real^N`] THEN STRIP_TAC] THEN MAP_EVERY ABBREV_TAC [`w1:real^M->bool = s1 DIFF v1`; `w2:real^M->bool = s2 DIFF v2`] THEN SUBGOAL_THEN `closed_in (subtopology euclidean s) (w1:real^M->bool) /\ closed_in (subtopology euclidean s) (w2:real^M->bool)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[CLOSED_IN_DIFF]; ALL_TAC] THEN SUBGOAL_THEN `t INTER w1 = {} /\ v' INTER w1:real^M->bool = {} /\ t INTER w2 = {} /\ v' INTER w2 = {}` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ABBREV_TAC `n:real^M->bool = s DIFF (k UNION w1 UNION w2)` THEN EXISTS_TAC `n:real^M->bool` THEN REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET)) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET)) THEN ASM SET_TAC[]; DISCH_TAC] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [EXPAND_TAC "n" THEN MATCH_MP_TAC OPEN_IN_DIFF THEN ASM_SIMP_TAC[OPEN_IN_REFL; CLOSED_IN_UNION]; DISCH_TAC] THEN MP_TAC(ISPECL [`subtopology euclidean (n:real^M->bool)`; `euclidean:(real^N)topology`; `\i. if i = 0 then (g1:real^M->real^N) else g2`; `\i. if i = 0 then s1 INTER n:real^M->bool else s2 INTER n`; `{0,1}`] PASTING_LEMMA_EXISTS_CLOSED) THEN REWRITE_TAC[CONTINUOUS_MAP_EUCLIDEAN; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; SUBTOPOLOGY_SUBTOPOLOGY] THEN ONCE_REWRITE_TAC[TAUT `closed_in a b /\ c <=> ~(closed_in a b ==> ~c)`] THEN SIMP_TAC[ISPEC `euclidean` CLOSED_IN_IMP_SUBSET; SET_RULE `s SUBSET u ==> u INTER s = s`] THEN REWRITE_TAC[NOT_IMP] THEN REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY; SUBSET_UNIV] THEN REWRITE_TAC[FORALL_IN_INSERT; NOT_IN_EMPTY; IMP_CONJ] THEN REWRITE_TAC[ARITH_EQ; IMP_IMP; FORALL_AND_THM] THEN ANTS_TAC THENL [REWRITE_TAC[SIMPLE_IMAGE; IMAGE_CLAUSES; UNIONS_2; GSYM CONJ_ASSOC] THEN REWRITE_TAC[ARITH_EQ] THEN REPEAT CONJ_TAC THENL [ASM SET_TAC[]; ONCE_REWRITE_TAC[INTER_COMM] THEN MATCH_MP_TAC CLOSED_IN_SUBTOPOLOGY_INTER_SUBSET THEN EXISTS_TAC `s:real^M->bool` THEN ASM_SIMP_TAC[CLOSED_IN_INTER; CLOSED_IN_REFL] THEN ASM SET_TAC[]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; ONCE_REWRITE_TAC[INTER_COMM] THEN MATCH_MP_TAC CLOSED_IN_SUBTOPOLOGY_INTER_SUBSET THEN EXISTS_TAC `s:real^M->bool` THEN ASM_SIMP_TAC[CLOSED_IN_INTER; CLOSED_IN_REFL] THEN ASM SET_TAC[]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; MATCH_MP_TAC WLOG_LT THEN REWRITE_TAC[] THEN CONJ_TAC THENL [MESON_TAC[INTER_COMM]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN REWRITE_TAC[CONJ_ASSOC] THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN ONCE_REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[IMP_IMP; IN_INSERT; NOT_IN_EMPTY] THEN REWRITE_TAC[ARITH_RULE `m < n /\ (m = 0 \/ m = 1) /\ (n = 0 \/ n = 1) <=> m = 0 /\ n = 1`] THEN STRIP_TAC THEN ASM_REWRITE_TAC[ARITH_EQ] THEN X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_INTER] THEN STRIP_TAC THEN SUBGOAL_THEN `(x:real^M) IN v` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET)) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET)) THEN ASM SET_TAC[]]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f':real^M->real^N` THEN REWRITE_TAC[SET_RULE `n INTER s INTER n = n INTER s`] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [REWRITE_TAC[SUBSET; FORALL_IN_IMAGE]; ALL_TAC] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN (SUBGOAL_THEN `(x:real^M) IN s1 \/ x IN s2` MP_TAC THENL [ASM SET_TAC[]; STRIP_TAC THEN ASM_SIMP_TAC[IN_INTER] THEN ASM SET_TAC[]])]);; (* ------------------------------------------------------------------------- *) (* Application to closed union. *) (* ------------------------------------------------------------------------- *) let ANR_CLOSED_UNION_LOCAL = prove (`!s t:real^N->bool u. closed_in (subtopology euclidean u) s /\ closed_in (subtopology euclidean u) t /\ ANR(s) /\ ANR(t) /\ ANR(s INTER t) ==> ANR(s UNION t)`, MAP_EVERY X_GEN_TAC [`y1:real^N->bool`; `y2:real^N->bool`; `yn:real^N->bool`] THEN STRIP_TAC THEN SUBGOAL_THEN `closed_in (subtopology euclidean (y1 UNION y2)) (y1:real^N->bool) /\ closed_in (subtopology euclidean (y1 UNION y2)) (y2:real^N->bool)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[CLOSED_IN_SUBSET_TRANS; SUBSET_UNION; UNION_SUBSET; CLOSED_IN_IMP_SUBSET]; REPEAT(FIRST_X_ASSUM(K ALL_TAC o check (free_in `yn:real^N->bool` o concl)))] THEN MATCH_MP_TAC ABSOLUTE_NEIGHBOURHOOD_EXTENSOR_IMP_ANR THEN MAP_EVERY X_GEN_TAC [`f:real^(N,1)finite_sum->real^N`; `s:real^(N,1)finite_sum->bool`; `t:real^(N,1)finite_sum->bool`] THEN STRIP_TAC THEN ASM_CASES_TAC `IMAGE (f:real^(N,1)finite_sum->real^N) t SUBSET y1` THENL [MP_TAC(ISPECL [`f:real^(N,1)finite_sum->real^N`; `s:real^(N,1)finite_sum->bool`; `t:real^(N,1)finite_sum->bool`; `y1:real^N->bool`] ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR) THEN ASM_REWRITE_TAC[] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS) THEN ASM SET_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `IMAGE (f:real^(N,1)finite_sum->real^N) t SUBSET y2` THENL [MP_TAC(ISPECL [`f:real^(N,1)finite_sum->real^N`; `s:real^(N,1)finite_sum->bool`; `t:real^(N,1)finite_sum->bool`; `y2:real^N->bool`] ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR) THEN ASM_REWRITE_TAC[] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS) THEN ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC ANR_UNION_EXTENSION_LEMMA THEN MAP_EVERY ABBREV_TAC [`b1 = {x | x IN s /\ setdist({x}, {x | x IN t /\ (f:real^(N,1)finite_sum->real^N) x IN y1}) <= setdist({x},{x | x IN t /\ f x IN y2})}`; `b2 = {x | x IN s /\ setdist({x}, {x | x IN t /\ (f:real^(N,1)finite_sum->real^N) x IN y2}) <= setdist({x},{x | x IN t /\ f x IN y1})}`] THEN MAP_EVERY EXISTS_TAC [`b1:real^(N,1)finite_sum->bool`; `b2:real^(N,1)finite_sum->bool`; `y1:real^N->bool`; `y2:real^N->bool`] THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [CONJ_TAC THENL [EXPAND_TAC "b1"; EXPAND_TAC "b2"] THEN ONCE_REWRITE_TAC[MESON[LIFT_DROP; REAL_SUB_LE] `x <= y <=> &0 <= drop(lift(y - x))`] THEN ONCE_REWRITE_TAC[SET_RULE `&0 <= drop x <=> x IN {y | &0 <= drop y}`] THEN MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE THEN REWRITE_TAC[drop; GSYM real_ge; CLOSED_HALFSPACE_COMPONENT_GE] THEN SIMP_TAC[LIFT_SUB; CONTINUOUS_ON_SUB; CONTINUOUS_ON_LIFT_SETDIST; CONTINUOUS_ON_CONST]; ALL_TAC] THEN CONJ_TAC THENL [MAP_EVERY EXPAND_TAC ["b1"; "b2"] THEN MP_TAC REAL_LE_TOTAL THEN SET_TAC[]; ALL_TAC] THEN CONJ_TAC THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTER] THEN MAP_EVERY EXPAND_TAC ["b1"; "b2"] THEN X_GEN_TAC `x:real^(N,1)finite_sum` THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THENL [SUBGOAL_THEN `(f:real^(N,1)finite_sum->real^N) x IN y2` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC]; SUBGOAL_THEN `(f:real^(N,1)finite_sum->real^N) x IN y1` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC]] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM REAL_NOT_LT]) THEN ASM_SIMP_TAC[SETDIST_SING_IN_SET; IN_ELIM_THM] THEN REWRITE_TAC[REAL_ARITH `&0 < x <=> &0 <= x /\ ~(x = &0)`] THEN REWRITE_TAC[SETDIST_POS_LE] THEN MP_TAC(ISPEC `s:real^(N,1)finite_sum->bool` SETDIST_EQ_0_CLOSED_IN) THEN DISCH_THEN(fun th -> W(MP_TAC o PART_MATCH (lhand o rand) th o rand o snd)) THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN (ANTS_TAC THENL [ALL_TAC; ASM SET_TAC[]]) THEN MATCH_MP_TAC CLOSED_IN_TRANS THEN EXISTS_TAC `t:real^(N,1)finite_sum->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE_GEN THEN EXISTS_TAC `y1 UNION y2:real^N->bool` THEN ASM_REWRITE_TAC[]);; let ENR_CLOSED_UNION_LOCAL = prove (`!s t u:real^N->bool. closed_in (subtopology euclidean u) s /\ closed_in (subtopology euclidean u) t /\ ENR(s) /\ ENR(t) /\ ENR(s INTER t) ==> ENR(s UNION t)`, REWRITE_TAC[ENR_ANR] THEN MESON_TAC[ANR_CLOSED_UNION_LOCAL; LOCALLY_COMPACT_CLOSED_UNION]);; let AR_CLOSED_UNION = prove (`!s t:real^N->bool. closed s /\ closed t /\ AR(s) /\ AR(t) /\ AR(s INTER t) ==> AR(s UNION t)`, MESON_TAC[AR_CLOSED_UNION_LOCAL; CLOSED_SUBSET; SUBSET_UNION]);; let ANR_CLOSED_UNION = prove (`!s t:real^N->bool. closed s /\ closed t /\ ANR(s) /\ ANR(t) /\ ANR(s INTER t) ==> ANR(s UNION t)`, MESON_TAC[ANR_CLOSED_UNION_LOCAL; CLOSED_SUBSET; SUBSET_UNION]);; let ENR_CLOSED_UNION = prove (`!s t:real^N->bool. closed s /\ closed t /\ ENR(s) /\ ENR(t) /\ ENR(s INTER t) ==> ENR(s UNION t)`, MESON_TAC[ENR_CLOSED_UNION_LOCAL; CLOSED_SUBSET; SUBSET_UNION]);; let ABSOLUTE_RETRACT_UNION = prove (`!s t. s retract_of (:real^N) /\ t retract_of (:real^N) /\ (s INTER t) retract_of (:real^N) ==> (s UNION t) retract_of (:real^N)`, SIMP_TAC[RETRACT_OF_UNIV; AR_CLOSED_UNION; CLOSED_UNION]);; let RETRACT_FROM_UNION_AND_INTER = prove (`!s t:real^N->bool. closed_in (subtopology euclidean (s UNION t)) s /\ closed_in (subtopology euclidean (s UNION t)) t /\ (s UNION t) retract_of u /\ (s INTER t) retract_of t ==> s retract_of u`, REPEAT STRIP_TAC THEN UNDISCH_TAC `(s UNION t) retract_of (u:real^N->bool)` THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] RETRACT_OF_TRANS) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retract_of]) THEN REWRITE_TAC[retraction; retract_of] THEN DISCH_THEN(X_CHOOSE_THEN `r:real^N->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\x:real^N. if x IN s then x else r x` THEN SIMP_TAC[] THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL THEN ASM_REWRITE_TAC[CONTINUOUS_ON_ID] THEN ASM SET_TAC[]);; let AR_FROM_UNION_AND_INTER_LOCAL = prove (`!s t:real^N->bool. closed_in (subtopology euclidean (s UNION t)) s /\ closed_in (subtopology euclidean (s UNION t)) t /\ AR(s UNION t) /\ AR(s INTER t) ==> AR(s) /\ AR(t)`, SUBGOAL_THEN `!s t:real^N->bool. closed_in (subtopology euclidean (s UNION t)) s /\ closed_in (subtopology euclidean (s UNION t)) t /\ AR(s UNION t) /\ AR(s INTER t) ==> AR(s)` MP_TAC THENL [ALL_TAC; MESON_TAC[UNION_COMM; INTER_COMM]] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC AR_RETRACT_OF_AR THEN EXISTS_TAC `s UNION t:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC RETRACT_FROM_UNION_AND_INTER THEN EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[RETRACT_OF_REFL] THEN MATCH_MP_TAC RETRACT_OF_SUBSET THEN EXISTS_TAC `s UNION t:real^N->bool` THEN REWRITE_TAC[INTER_SUBSET; SUBSET_UNION] THEN MATCH_MP_TAC AR_IMP_RETRACT THEN ASM_SIMP_TAC[CLOSED_IN_INTER]);; let AR_FROM_UNION_AND_INTER = prove (`!s t:real^N->bool. closed s /\ closed t /\ AR(s UNION t) /\ AR(s INTER t) ==> AR(s) /\ AR(t)`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC AR_FROM_UNION_AND_INTER_LOCAL THEN ASM_MESON_TAC[CLOSED_SUBSET; SUBSET_UNION]);; let ANR_FROM_UNION_AND_INTER_LOCAL = prove (`!s t:real^N->bool. closed_in (subtopology euclidean (s UNION t)) s /\ closed_in (subtopology euclidean (s UNION t)) t /\ ANR(s UNION t) /\ ANR(s INTER t) ==> ANR(s) /\ ANR(t)`, SUBGOAL_THEN `!s t:real^N->bool. closed_in (subtopology euclidean (s UNION t)) s /\ closed_in (subtopology euclidean (s UNION t)) t /\ ANR(s UNION t) /\ ANR(s INTER t) ==> ANR(s)` MP_TAC THENL [ALL_TAC; MESON_TAC[UNION_COMM; INTER_COMM]] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC ANR_NEIGHBORHOOD_RETRACT THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN EXISTS_TAC `s UNION t:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPECL [`s INTER t:real^N->bool`; `s UNION t:real^N->bool`] ANR_IMP_NEIGHBOURHOOD_RETRACT) THEN ASM_SIMP_TAC[CLOSED_IN_INTER] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP RETRACT_OF_IMP_SUBSET) THEN EXISTS_TAC `s UNION u:real^N->bool` THEN CONJ_TAC THENL [ALL_TAC; SUBGOAL_THEN `s UNION u:real^N->bool = ((s UNION t) DIFF t) UNION u` SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[OPEN_IN_UNION; OPEN_IN_DIFF; OPEN_IN_REFL]] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retract_of]) THEN REWRITE_TAC[retract_of; retraction; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `r:real^N->real^N` THEN STRIP_TAC THEN EXISTS_TAC `\x:real^N. if x IN s then x else r x` THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN SUBGOAL_THEN `s UNION u:real^N->bool = s UNION (u INTER t)` SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL THEN ASM_REWRITE_TAC[CONTINUOUS_ON_ID; CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; INTER_SUBSET]] THEN CONJ_TAC THENL [UNDISCH_TAC `closed_in(subtopology euclidean (s UNION t)) (s:real^N->bool)`; UNDISCH_TAC `closed_in(subtopology euclidean (s UNION t)) (t:real^N->bool)`] THEN REWRITE_TAC[CLOSED_IN_CLOSED] THEN MATCH_MP_TAC MONO_EXISTS THEN ASM SET_TAC[]);; let ANR_FROM_UNION_AND_INTER = prove (`!s t:real^N->bool. closed s /\ closed t /\ ANR(s UNION t) /\ ANR(s INTER t) ==> ANR(s) /\ ANR(t)`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC ANR_FROM_UNION_AND_INTER_LOCAL THEN ASM_MESON_TAC[CLOSED_SUBSET; SUBSET_UNION]);; let ANR_FINITE_UNIONS_CONVEX_CLOSED = prove (`!t:(real^N->bool)->bool. FINITE t /\ (!c. c IN t ==> closed c /\ convex c) ==> ANR(UNIONS t)`, GEN_TAC THEN WF_INDUCT_TAC `CARD(t:(real^N->bool)->bool)` THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[TAUT `p ==> q /\ r ==> s <=> q ==> p ==> r ==> s`] THEN SPEC_TAC(`t:(real^N->bool)->bool`,`t:(real^N->bool)->bool`) THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN REWRITE_TAC[UNIONS_0; UNIONS_INSERT; FORALL_IN_INSERT] THEN REWRITE_TAC[ANR_EMPTY] THEN MAP_EVERY X_GEN_TAC [`c:real^N->bool`; `t:(real^N->bool)->bool`] THEN DISCH_THEN(CONJUNCTS_THEN2 (K ALL_TAC) STRIP_ASSUME_TAC) THEN REWRITE_TAC[IMP_IMP] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC ANR_CLOSED_UNION THEN ASM_SIMP_TAC[CLOSED_UNIONS] THEN ASM_SIMP_TAC[CONVEX_IMP_ANR] THEN REWRITE_TAC[INTER_UNIONS] THEN CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[CARD_CLAUSES] THEN REWRITE_TAC[FORALL_IN_GSPEC; LT_SUC_LE; LE_REFL] THEN ASM_SIMP_TAC[SIMPLE_IMAGE; FINITE_IMAGE; CLOSED_INTER; CONVEX_INTER] THEN ASM_SIMP_TAC[CARD_IMAGE_LE]);; let FINITE_IMP_ANR = prove (`!s:real^N->bool. FINITE s ==> ANR s`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `s = UNIONS {{a:real^N} | a IN s}` SUBST1_TAC THENL [REWRITE_TAC[UNIONS_GSPEC] THEN SET_TAC[]; MATCH_MP_TAC ANR_FINITE_UNIONS_CONVEX_CLOSED THEN ASM_SIMP_TAC[FORALL_IN_IMAGE; SIMPLE_IMAGE; FINITE_IMAGE] THEN REWRITE_TAC[CLOSED_SING; CONVEX_SING]]);; let ANR_INSERT = prove (`!s a:real^N. closed s /\ ANR s ==> ANR(a INSERT s)`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[SET_RULE `a INSERT s = {a} UNION s`] THEN MATCH_MP_TAC ANR_CLOSED_UNION THEN ASM_MESON_TAC[CLOSED_SING; ANR_SING; ANR_EMPTY; SET_RULE `{a} INTER s = {a} \/ {a} INTER s = {}`]);; let ANR_TRIANGULATION = prove (`!tr. triangulation tr ==> ANR(UNIONS tr)`, REWRITE_TAC[triangulation] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC ANR_FINITE_UNIONS_CONVEX_CLOSED THEN ASM_MESON_TAC[SIMPLEX_IMP_CLOSED; SIMPLEX_IMP_CONVEX]);; let ANR_SIMPLICIAL_COMPLEX = prove (`!c. simplicial_complex c ==> ANR(UNIONS c)`, MESON_TAC[ANR_TRIANGULATION; SIMPLICIAL_COMPLEX_IMP_TRIANGULATION]);; let ANR_PATH_COMPONENT_ANR = prove (`!s x:real^N. ANR(s) ==> ANR(path_component s x)`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] ANR_OPEN_IN)) THEN MATCH_MP_TAC OPEN_IN_PATH_COMPONENT_LOCALLY_PATH_CONNECTED THEN ASM_SIMP_TAC[ANR_IMP_LOCALLY_PATH_CONNECTED]);; let ANR_CONNECTED_COMPONENT_ANR = prove (`!s x:real^N. ANR(s) ==> ANR(connected_component s x)`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] ANR_OPEN_IN)) THEN MATCH_MP_TAC OPEN_IN_CONNECTED_COMPONENT_LOCALLY_CONNECTED THEN ASM_SIMP_TAC[ANR_IMP_LOCALLY_CONNECTED]);; let ANR_COMPONENT_ANR = prove (`!s:real^N->bool. ANR s /\ c IN components s ==> ANR c`, REWRITE_TAC[IN_COMPONENTS] THEN MESON_TAC[ANR_CONNECTED_COMPONENT_ANR]);; (* ------------------------------------------------------------------------- *) (* Application to open union. *) (* ------------------------------------------------------------------------- *) let ANR_OPEN_UNION = prove (`!s t u:real^N->bool. open_in (subtopology euclidean u) s /\ open_in (subtopology euclidean u) t /\ ANR(s) /\ ANR(t) ==> ANR(s UNION t)`, MAP_EVERY X_GEN_TAC [`u1:real^N->bool`; `u2:real^N->bool`; `un:real^N->bool`] THEN STRIP_TAC THEN SUBGOAL_THEN `open_in (subtopology euclidean (u1 UNION u2)) (u1:real^N->bool) /\ open_in (subtopology euclidean (u1 UNION u2)) (u2:real^N->bool)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[OPEN_IN_SUBSET_TRANS; SUBSET_UNION; UNION_SUBSET; OPEN_IN_IMP_SUBSET]; REPEAT(FIRST_X_ASSUM(K ALL_TAC o check (free_in `un:real^N->bool` o concl)))] THEN MATCH_MP_TAC ABSOLUTE_NEIGHBOURHOOD_EXTENSOR_IMP_ANR THEN MAP_EVERY X_GEN_TAC [`f:real^(N,1)finite_sum->real^N`; `s:real^(N,1)finite_sum->bool`; `t:real^(N,1)finite_sum->bool`] THEN STRIP_TAC THEN MATCH_MP_TAC ANR_UNION_EXTENSION_LEMMA THEN MAP_EVERY ABBREV_TAC [`t1 = {x | x IN t /\ ~((f:real^(N,1)finite_sum->real^N)(x) IN u1)}`; `t2 = {x | x IN t /\ ~((f:real^(N,1)finite_sum->real^N)(x) IN u2)}`] THEN MP_TAC(ISPECL [`t1:real^(N,1)finite_sum->bool`; `t2:real^(N,1)finite_sum->bool`; `s:real^(N,1)finite_sum->bool`; `vec 1:real^1`; `vec 0:real^1`] URYSOHN_LOCAL) THEN ANTS_TAC THENL [REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN CONJ_TAC THEN MATCH_MP_TAC CLOSED_IN_TRANS THEN EXISTS_TAC `t:real^(N,1)finite_sum->bool` THEN ASM_REWRITE_TAC[] THENL [EXPAND_TAC "t1"; EXPAND_TAC "t2"] THEN FIRST_ASSUM(fun th -> ONCE_REWRITE_TAC [MATCH_MP (SET_RULE `IMAGE f s SUBSET t ==> {x | x IN s /\ ~(f x IN u)} = {x | x IN s /\ f x IN t DIFF u}`) th]) THEN MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE_GEN THEN EXISTS_TAC `u1 UNION u2:real^N->bool` THEN ASM_SIMP_TAC[CLOSED_IN_DIFF; CLOSED_IN_REFL]; DISCH_THEN(X_CHOOSE_THEN `l:real^(N,1)finite_sum->real^1` STRIP_ASSUME_TAC)] THEN MAP_EVERY EXISTS_TAC [`{ x:real^(N,1)finite_sum | x IN s /\ l x IN {y | drop y <= &1 / &2}}`; `{ x:real^(N,1)finite_sum | x IN s /\ l x IN {y | drop y >= &1 / &2}}`; `u1:real^N->bool`; `u2:real^N->bool`] THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC ANR_OPEN_IN THEN EXISTS_TAC `u1:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC OPEN_IN_SUBTOPOLOGY_INTER_SUBSET THEN EXISTS_TAC `u1 UNION u2:real^N->bool` THEN ASM_SIMP_TAC[OPEN_IN_INTER; OPEN_IN_REFL] THEN SET_TAC[]; MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE THEN ASM_REWRITE_TAC[drop; CLOSED_HALFSPACE_COMPONENT_LE]; MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE THEN ASM_REWRITE_TAC[drop; CLOSED_HALFSPACE_COMPONENT_GE]; MP_TAC(REAL_ARITH `!x. x <= &1 / &2 \/ x >= &1 / &2`) THEN SET_TAC[]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTER; IN_ELIM_THM] THEN X_GEN_TAC `x:real^(N,1)finite_sum` THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN ASM_CASES_TAC `(x:real^(N,1)finite_sum) IN t1` THENL [ASM_SIMP_TAC[DROP_VEC] THEN CONV_TAC REAL_RAT_REDUCE_CONV; ASM SET_TAC[]]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTER; IN_ELIM_THM] THEN X_GEN_TAC `x:real^(N,1)finite_sum` THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN ASM_CASES_TAC `(x:real^(N,1)finite_sum) IN t2` THENL [ASM_SIMP_TAC[DROP_VEC] THEN CONV_TAC REAL_RAT_REDUCE_CONV; ASM SET_TAC[]]]);; let ENR_OPEN_UNION = prove (`!s t u:real^N->bool. open_in (subtopology euclidean u) s /\ open_in (subtopology euclidean u) t /\ ENR(s) /\ ENR(t) ==> ENR(s UNION t)`, REWRITE_TAC[ENR_ANR] THEN ASM_MESON_TAC[ANR_OPEN_UNION; LOCALLY_COMPACT_OPEN_UNION]);; let ANR_OPEN_UNIONS = prove (`!f:(real^N->bool)->bool u. (!s. s IN f ==> ANR s) /\ (!s. s IN f ==> open_in (subtopology euclidean u) s) ==> ANR(UNIONS f)`, let lemma1 = prove (`!f:(real^N->bool)->bool. pairwise DISJOINT f /\ (!u. u IN f ==> ANR u) /\ (!u. u IN f ==> open_in (subtopology euclidean (UNIONS f)) u) ==> ANR(UNIONS f)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC ABSOLUTE_NEIGHBOURHOOD_EXTENSOR_IMP_ANR THEN MAP_EVERY X_GEN_TAC [`g:real^(N,1)finite_sum->real^N`; `s:real^(N,1)finite_sum->bool`; `t:real^(N,1)finite_sum->bool`] THEN STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN ABBREV_TAC `a = \u. {x | x IN t /\ (g:real^(N,1)finite_sum->real^N) x IN u}` THEN ASM_CASES_TAC `?u. u IN f /\ (a:(real^N->bool)->real^(N,1)finite_sum->bool) u = t` THENL [FIRST_X_ASSUM(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `ANR(u:real^N->bool)` MP_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN DISCH_THEN(MP_TAC o ISPEC `g:real^(N,1)finite_sum->real^N` o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR)) THEN DISCH_THEN(MP_TAC o SPECL [`s:real^(N,1)finite_sum->bool`; `t:real^(N,1)finite_sum->bool`]) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [UNDISCH_TAC `(a:(real^N->bool)->real^(N,1)finite_sum->bool) u = t` THEN EXPAND_TAC "a" THEN REWRITE_TAC[IN_ELIM_THM] THEN ASM SET_TAC[]; REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN ASM SET_TAC[]]; ALL_TAC] THEN SUBGOAL_THEN `!u. u IN f ==> closed_in (subtopology euclidean s) ((a:(real^N->bool)->real^(N,1)finite_sum->bool) u)` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN EXPAND_TAC "a" THEN MATCH_MP_TAC CLOSED_IN_TRANS THEN EXISTS_TAC `t:real^(N,1)finite_sum->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE_GEN THEN EXISTS_TAC `UNIONS f:real^N->bool` THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `u:real^N->bool = UNIONS f DIFF UNIONS(f DELETE u)` SUBST1_TAC THENL [ASM_SIMP_TAC[DIFF_UNIONS_PAIRWISE_DISJOINT; DELETE_SUBSET] THEN ASM SET_TAC[]; MATCH_MP_TAC CLOSED_IN_DIFF THEN REWRITE_TAC[CLOSED_IN_REFL] THEN MATCH_MP_TAC OPEN_IN_UNIONS THEN ASM_SIMP_TAC[IN_DELETE]]; ALL_TAC] THEN SUBGOAL_THEN `pairwise (\i j. DISJOINT ((a:(real^N->bool)->real^(N,1)finite_sum->bool) i) (a j)) f` ASSUME_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [pairwise]) THEN EXPAND_TAC "a" THEN REWRITE_TAC[pairwise] THEN SET_TAC[]; ALL_TAC] THEN ABBREV_TAC `v = \u. if a u = {} then {} else { x:real^(N,1)finite_sum | x IN s /\ setdist({x},a(u:real^N->bool)) < setdist({x},t DIFF a u)}` THEN SUBGOAL_THEN `!u. u IN f ==> open_in (subtopology euclidean s) ((v:(real^N->bool)->real^(N,1)finite_sum->bool) u)` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN EXPAND_TAC "v" THEN COND_CASES_TAC THEN REWRITE_TAC[OPEN_IN_EMPTY] THEN ONCE_REWRITE_TAC[MESON[LIFT_DROP; REAL_SUB_LT] `x < y <=> &0 < drop(lift(y - x))`] THEN ONCE_REWRITE_TAC[SET_RULE `&0 < drop x <=> x IN {y | &0 < drop y}`] THEN MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE THEN REWRITE_TAC[drop; GSYM real_gt; OPEN_HALFSPACE_COMPONENT_GT] THEN SIMP_TAC[LIFT_SUB; CONTINUOUS_ON_SUB; CONTINUOUS_ON_LIFT_SETDIST]; ALL_TAC] THEN SUBGOAL_THEN `!u. u IN f ==> (a:(real^N->bool)->real^(N,1)finite_sum->bool) u SUBSET v u` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN EXPAND_TAC "v" THEN REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^(N,1)finite_sum` THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN SIMP_TAC[IN_ELIM_THM; SETDIST_SING_IN_SET; SUBSET] THEN REPEAT STRIP_TAC THENL [ASM_MESON_TAC[CLOSED_IN_IMP_SUBSET; SUBSET]; ALL_TAC] THEN REWRITE_TAC[REAL_ARITH `&0 < x <=> &0 <= x /\ ~(x = &0)`] THEN REWRITE_TAC[SETDIST_POS_LE] THEN MP_TAC(ISPEC `t:real^(N,1)finite_sum->bool` SETDIST_EQ_0_CLOSED_IN) THEN DISCH_THEN(fun th -> W(MP_TAC o PART_MATCH (lhand o rand) th o rand o snd)) THEN ASM_REWRITE_TAC[IN_DIFF] THEN ANTS_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC CLOSED_IN_DIFF THEN REWRITE_TAC[CLOSED_IN_REFL] THEN EXPAND_TAC "a" THEN MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE_GEN THEN EXISTS_TAC `UNIONS f:real^N->bool` THEN ASM_SIMP_TAC[]; UNDISCH_TAC `x IN (a:(real^N->bool)->real^(N,1)finite_sum->bool) u` THEN EXPAND_TAC "a" THEN SIMP_TAC[IN_ELIM_THM]]; DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC(SET_RULE `s SUBSET t /\ ~(s = t) ==> ~(t DIFF s = {})`) THEN CONJ_TAC THENL [EXPAND_TAC "a" THEN SET_TAC[]; ASM_MESON_TAC[]]]; ALL_TAC] THEN SUBGOAL_THEN `pairwise (\i j. DISJOINT ((v:(real^N->bool)->real^(N,1)finite_sum->bool) i) (v j)) f` ASSUME_TAC THENL [EXPAND_TAC "v" THEN REWRITE_TAC[pairwise] THEN MAP_EVERY X_GEN_TAC [`u1:real^N->bool`; `u2:real^N->bool`] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[DISJOINT_EMPTY]) THEN STRIP_TAC THEN REWRITE_TAC[DISJOINT; EXTENSION] THEN X_GEN_TAC `x:real^(N,1)finite_sum` THEN REWRITE_TAC[IN_ELIM_THM; NOT_IN_EMPTY; IN_INTER] THEN ASM_CASES_TAC `(x:real^(N,1)finite_sum) IN s` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REAL_ARITH `b <= c /\ d <= a ==> ~(a < b /\ c < d)`) THEN CONJ_TAC THEN MATCH_MP_TAC SETDIST_SUBSET_RIGHT THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t /\ DISJOINT s u ==> s SUBSET t DIFF u`) THEN (CONJ_TAC THENL [EXPAND_TAC "a" THEN SET_TAC[]; ALL_TAC]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[pairwise]) THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `!u. u IN (f:(real^N->bool)->bool) ==> ?v h. a u SUBSET v /\ open_in (subtopology euclidean s) v /\ (h:real^(N,1)finite_sum->real^N) continuous_on v /\ IMAGE h v SUBSET u /\ (!x. x IN a u ==> h x = g x)` MP_TAC THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR THEN ASM_SIMP_TAC[] THEN EXPAND_TAC "a" THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN SET_TAC[]; ASM SET_TAC[]]; GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`w:(real^N->bool)->real^(N,1)finite_sum->bool`; `h:(real^N->bool)->real^(N,1)finite_sum->real^N`] THEN REWRITE_TAC[TAUT `p ==> q /\ r <=> (p ==> q) /\ (p ==> r)`] THEN REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC THEN MP_TAC(ISPECL [`subtopology euclidean (UNIONS(IMAGE (\u. v u INTER (w:(real^N->bool)->real^(N,1)finite_sum->bool) u) f))`; `euclidean:(real^N)topology`; `h:(real^N->bool)->real^(N,1)finite_sum->real^N`; `\u. v u INTER (w:(real^N->bool)->real^(N,1)finite_sum->bool) u`; `f:(real^N->bool)->bool`] PASTING_LEMMA_EXISTS) THEN REWRITE_TAC[CONTINUOUS_MAP_EUCLIDEAN; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; SUBTOPOLOGY_SUBTOPOLOGY] THEN ONCE_REWRITE_TAC[TAUT `open_in a b /\ c <=> ~(open_in a b ==> ~c)`] THEN SIMP_TAC[ISPEC `euclidean` OPEN_IN_IMP_SUBSET; SET_RULE `s SUBSET u ==> u INTER s = s`] THEN REWRITE_TAC[NOT_IMP] THEN REWRITE_TAC[SIMPLE_IMAGE; SUBSET_REFL; SUBSET_UNIV] THEN ANTS_TAC THENL [CONJ_TAC THEN X_GEN_TAC `u:real^N->bool` THENL [DISCH_TAC THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; INTER_SUBSET]] THEN MATCH_MP_TAC OPEN_IN_SUBSET_TRANS THEN EXISTS_TAC `s:real^(N,1)finite_sum->bool` THEN REPEAT CONJ_TAC THENL [ASM_SIMP_TAC[OPEN_IN_INTER]; REWRITE_TAC[UNIONS_IMAGE] THEN ASM SET_TAC[]; REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_IMAGE] THEN ASM_MESON_TAC[OPEN_IN_IMP_SUBSET; SUBSET_TRANS; INTER_SUBSET]]; X_GEN_TAC `u':real^N->bool` THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [pairwise]) THEN DISCH_THEN(MP_TAC o SPECL [`u:real^N->bool`; `u':real^N->bool`]) THEN ASM_CASES_TAC `u:real^N->bool = u'` THEN ASM_REWRITE_TAC[] THEN SET_TAC[]]; GEN_REWRITE_TAC RAND_CONV [SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `h:real^(N,1)finite_sum->real^N` THEN STRIP_TAC THEN EXISTS_TAC `UNIONS(IMAGE (\u. v u INTER (w:(real^N->bool)->real^(N,1)finite_sum->bool) u) f)` THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE g t SUBSET u ==> {x | x IN t /\ g x IN u} SUBSET x ==> t SUBSET x`)) THEN REWRITE_TAC[UNIONS_IMAGE; SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `x:real^(N,1)finite_sum` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[IN_UNIONS; IN_INTER] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^N->bool` THEN STRIP_TAC THEN SUBGOAL_THEN `x IN (a:(real^N->bool)->real^(N,1)finite_sum->bool) u` MP_TAC THENL [EXPAND_TAC "a" THEN REWRITE_TAC[IN_ELIM_THM] THEN ASM SET_TAC[]; ASM SET_TAC[]]; MATCH_MP_TAC OPEN_IN_UNIONS THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN ASM_SIMP_TAC[OPEN_IN_INTER]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_UNIONS; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN X_GEN_TAC `u:real^N->bool` THEN DISCH_TAC THEN X_GEN_TAC `x:real^(N,1)finite_sum` THEN REWRITE_TAC[IN_INTER] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^(N,1)finite_sum`; `u:real^N->bool`]) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[IN_INTER; UNIONS_IMAGE; IN_ELIM_THM] THEN ASM_MESON_TAC[]; ASM SET_TAC[]]; X_GEN_TAC `x:real^(N,1)finite_sum` THEN DISCH_TAC THEN SUBGOAL_THEN `?u. u IN f /\ x IN (a:(real^N->bool)->real^(N,1)finite_sum->bool) u` STRIP_ASSUME_TAC THENL [EXPAND_TAC "a" THEN REWRITE_TAC[IN_ELIM_THM] THEN ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^(N,1)finite_sum`; `u:real^N->bool`]) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[IN_INTER; UNIONS_IMAGE; IN_ELIM_THM] THEN ASM_MESON_TAC[SUBSET]; ASM SET_TAC[]]]]) in let lemma2 = prove (`!f:(real^N->bool)->bool. FINITE f /\ (!u. u IN f ==> ANR u) /\ (!u. u IN f ==> open_in (subtopology euclidean (UNIONS f)) u) ==> ANR(UNIONS f)`, ONCE_REWRITE_TAC[IMP_CONJ] THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN REWRITE_TAC[UNIONS_0; ANR_EMPTY; FORALL_IN_INSERT; UNIONS_INSERT] THEN MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `f:(real^N->bool)->bool`] THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN MATCH_MP_TAC ANR_OPEN_UNION THEN EXISTS_TAC `u UNION UNIONS f:real^N->bool` THEN ASM_SIMP_TAC[OPEN_IN_UNIONS; FORALL_IN_INSERT] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN X_GEN_TAC `v:real^N->bool` THEN DISCH_TAC THEN MATCH_MP_TAC OPEN_IN_SUBSET_TRANS THEN EXISTS_TAC `u UNION UNIONS f:real^N->bool` THEN ASM_SIMP_TAC[] THEN ASM SET_TAC[]) in let lemma3 = prove (`!v:num->real^N->bool. (!n. v(n) SUBSET v(SUC n)) /\ (!n. open_in (subtopology euclidean (UNIONS(IMAGE v (:num)))) (v n)) /\ (!n. ANR(v n)) ==> ANR(UNIONS(IMAGE v (:num)))`, REPEAT STRIP_TAC THEN ABBREV_TAC `s:real^N->bool = UNIONS(IMAGE v (:num))` THEN ASM_CASES_TAC `?n:num. s:real^N->bool = v n` THENL [ASM_MESON_TAC[]; RULE_ASSUM_TAC(REWRITE_RULE[NOT_EXISTS_THM])] THEN ABBREV_TAC `w = \n:num. {x:real^N | x IN s /\ inv(&2 pow n) < setdist({x},s DIFF v n)}` THEN SUBGOAL_THEN `!n. open_in (subtopology euclidean s) ((w:num->real^N->bool) n)` ASSUME_TAC THENL [GEN_TAC THEN EXPAND_TAC "w" THEN ONCE_REWRITE_TAC[MESON[LIFT_DROP; REAL_SUB_LT] `x < y <=> &0 < drop(lift(y - x))`] THEN ONCE_REWRITE_TAC[SET_RULE `&0 < drop x <=> x IN {y | &0 < drop y}`] THEN MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE THEN REWRITE_TAC[drop; GSYM real_gt; OPEN_HALFSPACE_COMPONENT_GT] THEN SIMP_TAC[LIFT_SUB; CONTINUOUS_ON_SUB; CONTINUOUS_ON_LIFT_SETDIST; CONTINUOUS_ON_CONST]; ALL_TAC] THEN SUBGOAL_THEN `!n. (w:num->real^N->bool) n SUBSET v n` ASSUME_TAC THENL [GEN_TAC THEN EXPAND_TAC "w" THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN ONCE_REWRITE_TAC[TAUT `p /\ q ==> r <=> p /\ ~r ==> ~q`] THEN SIMP_TAC[SETDIST_SING_IN_SET; IN_DIFF; REAL_NOT_LT; REAL_LE_INV_EQ] THEN SIMP_TAC[REAL_LT_IMP_LE; REAL_LT_POW2]; ALL_TAC] THEN SUBGOAL_THEN `!n. ANR((w:num->real^N->bool) n)` ASSUME_TAC THENL [GEN_TAC THEN MATCH_MP_TAC ANR_OPEN_IN THEN EXISTS_TAC `(v:num->real^N->bool) n` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC OPEN_IN_SUBSET_TRANS THEN EXISTS_TAC `s:real^N->bool` THEN ASM_MESON_TAC[OPEN_IN_IMP_SUBSET]; ALL_TAC] THEN SUBGOAL_THEN `!n. s INTER closure(w n) SUBSET (w:num->real^N->bool)(SUC n)` ASSUME_TAC THENL [GEN_TAC THEN EXPAND_TAC "w" THEN TRANS_TAC SUBSET_TRANS `{x:real^N | x IN s /\ inv(&2 pow n) <= setdist({x},s DIFF v n)}` THEN CONJ_TAC THENL [MATCH_MP_TAC CLOSURE_MINIMAL_LOCAL THEN SIMP_TAC[SUBSET; IN_ELIM_THM; REAL_LT_IMP_LE] THEN ONCE_REWRITE_TAC[MESON[LIFT_DROP; REAL_SUB_LE] `x <= y <=> &0 <= drop(lift(y - x))`] THEN ONCE_REWRITE_TAC[SET_RULE `&0 <= drop x <=> x IN {y | &0 <= drop y}`] THEN MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE THEN REWRITE_TAC[drop; GSYM real_ge; CLOSED_HALFSPACE_COMPONENT_GE] THEN SIMP_TAC[LIFT_SUB; CONTINUOUS_ON_SUB; CONTINUOUS_ON_LIFT_SETDIST; CONTINUOUS_ON_CONST]; REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REAL_ARITH `b < a /\ x <= y ==> a <= x ==> b < y`) THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_LT_INV2 THEN REWRITE_TAC[REAL_LT_POW2] THEN MATCH_MP_TAC REAL_POW_MONO_LT THEN REWRITE_TAC[REAL_OF_NUM_LT] THEN ARITH_TAC; MATCH_MP_TAC SETDIST_SUBSET_RIGHT THEN ASM SET_TAC[]]]; ALL_TAC] THEN SUBGOAL_THEN `s:real^N->bool = UNIONS(IMAGE w (:num))` SUBST1_TAC THENL [REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; UNIONS_SUBSET] THEN REWRITE_TAC[FORALL_IN_IMAGE; UNIONS_IMAGE; IN_UNIV] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[OPEN_IN_IMP_SUBSET]] THEN EXPAND_TAC "w" THEN REWRITE_TAC[IN_ELIM_THM; SUBSET] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN SUBGOAL_THEN `?n:num. (x:real^N) IN v n` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `&0 < setdist ({x:real^N},s DIFF v(n:num))` MP_TAC THENL [REWRITE_TAC[REAL_ARITH `&0 < x <=> &0 <= x /\ ~(x = &0)`] THEN REWRITE_TAC[SETDIST_POS_LE] THEN MP_TAC(ISPEC `s:real^N->bool` SETDIST_EQ_0_CLOSED_IN) THEN DISCH_THEN(fun th -> W(MP_TAC o PART_MATCH (lhand o rand) th o rand o snd)) THEN ASM_SIMP_TAC[CLOSED_IN_DIFF; CLOSED_IN_REFL] THEN DISCH_THEN SUBST1_TAC THEN ASM SET_TAC[]; DISCH_THEN(MP_TAC o SPEC `inv(&2)` o MATCH_MP (REWRITE_RULE[IMP_CONJ] REAL_ARCH_POW_INV)) THEN ANTS_TAC THENL [REAL_ARITH_TAC; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN X_GEN_TAC `m:num` THEN REWRITE_TAC[REAL_POW_INV] THEN DISCH_TAC THEN EXISTS_TAC `m + n:num` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `a < b ==> x <= a /\ b <= y ==> x < y`)) THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_LT_POW2] THEN MATCH_MP_TAC REAL_POW_MONO THEN REWRITE_TAC[REAL_OF_NUM_LE] THEN ARITH_TAC; MATCH_MP_TAC SETDIST_SUBSET_RIGHT THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> u DIFF t SUBSET u DIFF s`) THEN MATCH_MP_TAC(MESON[LE_ADD; ADD_SYM] `(!m n:num. m <= n ==> v m SUBSET v n) ==> v b SUBSET v(a + b)`) THEN MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN ASM SET_TAC[]]]; ALL_TAC] THEN (STRIP_ASSUME_TAC o prove_general_recursive_function_exists) `?r:num->real^N->bool. r 0 = w 0 /\ r 1 = w 1 /\ (!n. r(n + 2) = w(n + 2) DIFF (s INTER closure(w n)))` THEN SUBGOAL_THEN `!n. open_in (subtopology euclidean (w n)) ((r:num->real^N->bool) n)` ASSUME_TAC THENL [MATCH_MP_TAC num_INDUCTION THEN ASM_REWRITE_TAC[OPEN_IN_REFL] THEN MATCH_MP_TAC num_INDUCTION THEN ASM_REWRITE_TAC[ARITH; OPEN_IN_REFL] THEN X_GEN_TAC `n:num` THEN REPLICATE_TAC 2 (DISCH_THEN(K ALL_TAC)) THEN ASM_REWRITE_TAC[ARITH_RULE `SUC(SUC n) = n + 2`] THEN ONCE_REWRITE_TAC[SET_RULE `s DIFF t = s DIFF (s INTER t)`] THEN MATCH_MP_TAC OPEN_IN_DIFF THEN REWRITE_TAC[OPEN_IN_REFL] THEN MATCH_MP_TAC(MESON[CLOSED_IN_CLOSED_INTER] `closed u /\ s INTER t INTER u = s INTER u ==> closed_in (subtopology euclidean s) (s INTER t INTER u)`) THEN REWRITE_TAC[CLOSED_CLOSURE] THEN MATCH_MP_TAC (SET_RULE `s SUBSET t ==> s INTER t INTER u = s INTER u`) THEN ASM_MESON_TAC[OPEN_IN_IMP_SUBSET]; ALL_TAC] THEN SUBGOAL_THEN `!n. open_in (subtopology euclidean s) ((r:num->real^N->bool) n)` ASSUME_TAC THENL [ASM_MESON_TAC[OPEN_IN_TRANS]; ALL_TAC] THEN SUBGOAL_THEN `!n. ANR((r:num->real^N->bool) n)` ASSUME_TAC THENL [ASM_MESON_TAC[ANR_OPEN_IN]; ALL_TAC] THEN SUBGOAL_THEN `UNIONS (IMAGE w (:num)):real^N->bool = UNIONS(IMAGE r (:num))` SUBST1_TAC THENL [REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN CONJ_TAC THENL [REWRITE_TAC[UNIONS_IMAGE; IN_UNIV; SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN GEN_REWRITE_TAC LAND_CONV [num_WOP] THEN MATCH_MP_TAC MONO_EXISTS THEN MATCH_MP_TAC num_INDUCTION THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC num_INDUCTION THEN ASM_SIMP_TAC[ARITH] THEN X_GEN_TAC `n:num` THEN REPLICATE_TAC 2 (DISCH_THEN(K ALL_TAC)) THEN ASM_REWRITE_TAC[ARITH_RULE `SUC(SUC n) = n + 2`] THEN SIMP_TAC[IN_DIFF; IN_INTER] THEN DISCH_THEN(MP_TAC o SPEC `SUC n` o CONJUNCT2) THEN ANTS_TAC THENL [ARITH_TAC; ASM SET_TAC[]]; MATCH_MP_TAC UNIONS_MONO_IMAGE THEN REWRITE_TAC[IN_UNIV] THEN MATCH_MP_TAC num_INDUCTION THEN ASM_SIMP_TAC[SUBSET_REFL] THEN MATCH_MP_TAC num_INDUCTION THEN ASM_SIMP_TAC[ARITH; SUBSET_REFL] THEN ASM_REWRITE_TAC[ARITH_RULE `SUC(SUC n) = n + 2`] THEN SET_TAC[]]; ALL_TAC] THEN EXPAND_TAC "s" THEN SUBGOAL_THEN `(:num) = IMAGE (\n. 2 * n) (:num) UNION IMAGE (\n. 2 * n + 1) (:num)` (fun th -> ONCE_REWRITE_TAC[th] THEN ASSUME_TAC(SYM th)) THENL [REWRITE_TAC[EXTENSION; IN_UNIV; IN_IMAGE; IN_UNION] THEN REWRITE_TAC[GSYM EVEN_EXISTS; GSYM ADD1; GSYM ODD_EXISTS] THEN REWRITE_TAC[EVEN_OR_ODD]; REWRITE_TAC[IMAGE_UNION; GSYM IMAGE_o; o_DEF; UNIONS_UNION]] THEN MATCH_MP_TAC ANR_OPEN_UNION THEN EXISTS_TAC `UNIONS (IMAGE (\x. r (2 * x)) (:num)) UNION UNIONS (IMAGE (\x. r (2 * x + 1)) (:num)):real^N->bool` THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC OPEN_IN_SUBSET_TRANS THEN EXISTS_TAC `s:real^N->bool` THEN REWRITE_TAC[SUBSET_UNION] THEN ASM_SIMP_TAC[OPEN_IN_UNIONS; FORALL_IN_IMAGE; UNION_SUBSET] THEN REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_IMAGE] THEN ASM_MESON_TAC[OPEN_IN_IMP_SUBSET]; ALL_TAC] THEN CONJ_TAC THEN MATCH_MP_TAC lemma1 THEN ASM_REWRITE_TAC[FORALL_IN_IMAGE; IN_UNIV] THEN (CONJ_TAC THENL [ALL_TAC; X_GEN_TAC `n:num` THEN MATCH_MP_TAC OPEN_IN_SUBSET_TRANS THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[UNIONS_SUBSET] THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN CONJ_TAC THENL [ASM SET_TAC[]; ASM_MESON_TAC[OPEN_IN_IMP_SUBSET]]]) THEN REWRITE_TAC[pairwise; IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_UNIV] THEN MATCH_MP_TAC WLOG_LT THEN REWRITE_TAC[] THEN (CONJ_TAC THENL [MESON_TAC[DISJOINT_SYM]; ALL_TAC]) THEN X_GEN_TAC `m:num` THEN MATCH_MP_TAC num_INDUCTION THEN ASM_REWRITE_TAC[CONJUNCT1 LT; ARITH_RULE `2 * SUC n = 2 * n + 2`; ARITH_RULE `(2 * n + 2) + 1 = (2 * n + 1) + 2`] THEN X_GEN_TAC `n:num` THEN DISCH_THEN(K ALL_TAC) THEN REWRITE_TAC[LT_SUC_LE] THEN DISCH_TAC THEN DISCH_THEN(K ALL_TAC) THENL [FIRST_X_ASSUM(MP_TAC o MATCH_MP (ARITH_RULE `m <= n ==> 2 * m <= 2 * n`)) THEN SPEC_TAC(`2 * n`,`n:num`) THEN SPEC_TAC(`2 * m`,`m:num`); FIRST_X_ASSUM(MP_TAC o MATCH_MP (ARITH_RULE `m <= n ==> 2 * m + 1 <= 2 * n + 1`)) THEN SPEC_TAC(`2 * n + 1`,`n:num`) THEN SPEC_TAC(`2 * m + 1`,`m:num`)] THEN (REPEAT STRIP_TAC THEN MATCH_MP_TAC(SET_RULE `r m SUBSET s /\ r m SUBSET w m /\ w m SUBSET w n /\ w n SUBSET closure(w n) ==> DISJOINT (r m) (w(n + 2) DIFF s INTER closure(w n))`) THEN REWRITE_TAC[CLOSURE_SUBSET] THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[OPEN_IN_IMP_SUBSET]; SPEC_TAC(`m:num`,`p:num`) THEN MATCH_MP_TAC num_INDUCTION THEN ASM_SIMP_TAC[SUBSET_REFL] THEN MATCH_MP_TAC num_INDUCTION THEN ASM_SIMP_TAC[ARITH; SUBSET_REFL] THEN ASM_REWRITE_TAC[ARITH_RULE `SUC(SUC n) = n + 2`] THEN SET_TAC[]; UNDISCH_TAC `m:num <= n` THEN MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`n:num`;` m:num`] THEN MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN REWRITE_TAC[SUBSET_REFL; SUBSET_TRANS] THEN X_GEN_TAC `p:num` THEN TRANS_TAC SUBSET_TRANS `s INTER closure((w:num->real^N->bool) p)` THEN ASM_REWRITE_TAC[SUBSET_INTER; CLOSURE_SUBSET] THEN ASM_MESON_TAC[OPEN_IN_IMP_SUBSET]])) in let lemma4 = prove (`!v:num->real^N->bool. (!n. open_in (subtopology euclidean (UNIONS(IMAGE v (:num)))) (v n)) /\ (!n. ANR(v n)) ==> ANR(UNIONS(IMAGE v (:num)))`, GEN_TAC THEN ABBREV_TAC `u:num->real^N->bool = \n. UNIONS (IMAGE v (0..n))` THEN SUBGOAL_THEN `UNIONS(IMAGE v (:num)):real^N->bool = UNIONS(IMAGE u (:num))` (fun th -> ONCE_REWRITE_TAC[th] THEN RULE_ASSUM_TAC(REWRITE_RULE[th])) THENL [EXPAND_TAC "u" THEN REWRITE_TAC[EXTENSION; UNIONS_IMAGE; IN_UNIV; IN_ELIM_THM] THEN REWRITE_TAC[IN_NUMSEG; LE_0] THEN MESON_TAC[LE_REFL]; REPEAT STRIP_TAC THEN MATCH_MP_TAC lemma3 THEN EXPAND_TAC "u" THEN REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [GEN_TAC THEN MATCH_MP_TAC SUBSET_UNIONS THEN MATCH_MP_TAC IMAGE_SUBSET THEN REWRITE_TAC[SUBSET_NUMSEG] THEN ARITH_TAC; GEN_TAC THEN MATCH_MP_TAC OPEN_IN_UNIONS THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN ASM_REWRITE_TAC[]; GEN_TAC THEN MATCH_MP_TAC lemma2 THEN ASM_SIMP_TAC[FORALL_IN_IMAGE; FINITE_IMAGE; FINITE_NUMSEG] THEN X_GEN_TAC `k:num` THEN REWRITE_TAC[IN_NUMSEG; LE_0] THEN STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(ASSUME_TAC o SPEC `k:num`)) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] OPEN_IN_SUBSET_TRANS)) THEN CONJ_TAC THENL [REWRITE_TAC[UNIONS_IMAGE; IN_NUMSEG; LE_0] THEN ASM SET_TAC[]; REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_IMAGE; IN_NUMSEG; LE_0] THEN X_GEN_TAC `m:num` THEN DISCH_TAC THEN REWRITE_TAC[UNIONS_IMAGE] THEN EXPAND_TAC "u" THEN REWRITE_TAC[UNIONS_IMAGE; IN_UNIV; IN_NUMSEG; LE_0] THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN ASM_MESON_TAC[]]]]) in REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:(real^N->bool)->bool`; `u:real^N->bool`] LINDELOF_OPEN_IN) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `g:(real^N->bool)->bool` THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(SUBST_ALL_TAC o SYM) THEN ASM_CASES_TAC `g:(real^N->bool)->bool = {}` THEN ASM_REWRITE_TAC[UNIONS_0; ANR_EMPTY] THEN MP_TAC(ISPEC `g:(real^N->bool)->bool` COUNTABLE_AS_IMAGE) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `h:num->real^N->bool` THEN DISCH_THEN SUBST_ALL_TAC THEN MATCH_MP_TAC lemma4 THEN CONJ_TAC THENL [GEN_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC OPEN_IN_SUBSET_TRANS THEN EXISTS_TAC `u:real^N->bool` THEN REWRITE_TAC[UNIONS_IMAGE] THEN REPEAT(CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN REWRITE_TAC[SUBSET; IN_UNIV; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_THEN(X_CHOOSE_TAC `n:num`) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `(h:num->real^N->bool) n`)) THEN REPEAT(ANTS_TAC THENL [ASM SET_TAC[]; DISCH_TAC]) THEN ASM_MESON_TAC[OPEN_IN_IMP_SUBSET; SUBSET]);; let ENR_OPEN_UNIONS = prove (`!f:(real^N->bool)->bool u. (!s. s IN f ==> ENR s) /\ (!s. s IN f ==> open_in (subtopology euclidean u) s) ==> ENR(UNIONS f)`, REWRITE_TAC[ENR_ANR] THEN MESON_TAC[ANR_OPEN_UNIONS; LOCALLY_COMPACT_OPEN_UNIONS]);; let LOCALLY_ANR_ALT = prove (`!s:real^N->bool. locally ANR s <=> !v x. open_in (subtopology euclidean s) v /\ x IN v ==> ?u. open_in (subtopology euclidean s) u /\ ANR u /\ x IN u /\ u SUBSET v`, GEN_TAC THEN REWRITE_TAC[locally] THEN EQ_TAC THENL [ALL_TAC; MESON_TAC[SUBSET_REFL]] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `v:real^N->bool` THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `x:real^N` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^N->bool` THEN DISCH_THEN(X_CHOOSE_THEN `w:real^N->bool` STRIP_ASSUME_TAC) THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC ANR_OPEN_IN THEN EXISTS_TAC `w:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC OPEN_IN_SUBSET_TRANS THEN EXISTS_TAC `s:real^N->bool` THEN ASM_MESON_TAC[OPEN_IN_IMP_SUBSET; SUBSET_TRANS]);; let LOCALLY_ANR = prove (`!s:real^N->bool. locally ANR s <=> !x. x IN s ==> ?v. x IN v /\ open_in (subtopology euclidean s) v /\ ANR v`, GEN_TAC THEN REWRITE_TAC[LOCALLY_ANR_ALT] THEN GEN_REWRITE_TAC LAND_CONV [SWAP_FORALL_THM] THEN EQ_TAC THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `x:real^N` THENL [DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC(SPEC `s:real^N->bool` th)) THEN ASM_REWRITE_TAC[OPEN_IN_REFL] THEN MESON_TAC[]; DISCH_THEN(fun th -> X_GEN_TAC `v:real^N->bool` THEN STRIP_TAC THEN MP_TAC th) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN X_GEN_TAC `w:real^N->bool` THEN STRIP_TAC THEN EXISTS_TAC `v INTER w:real^N->bool` THEN ASM_SIMP_TAC[OPEN_IN_INTER; IN_INTER; INTER_SUBSET] THEN MATCH_MP_TAC ANR_OPEN_IN THEN EXISTS_TAC `w:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC OPEN_IN_SUBSET_TRANS THEN EXISTS_TAC `s:real^N->bool` THEN ASM_SIMP_TAC[INTER_SUBSET; OPEN_IN_INTER] THEN ASM_MESON_TAC[OPEN_IN_IMP_SUBSET]]);; let ANR_LOCALLY = prove (`!s:real^N->bool. locally ANR s <=> ANR s`, REPEAT STRIP_TAC THEN REWRITE_TAC[LOCALLY_ANR] THEN EQ_TAC THENL [ALL_TAC; MESON_TAC[OPEN_IN_REFL]] THEN REWRITE_TAC[RIGHT_IMP_EXISTS_THM; SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `f:real^N->real^N->bool` THEN DISCH_TAC THEN SUBGOAL_THEN `UNIONS (IMAGE (f:real^N->real^N->bool) s) = s` ASSUME_TAC THENL [MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_IMAGE] THEN ASM_MESON_TAC[OPEN_IN_IMP_SUBSET]; EXPAND_TAC "s" THEN MATCH_MP_TAC ANR_OPEN_UNIONS THEN ASM_MESON_TAC[FORALL_IN_IMAGE]]);; let LOCALLY_ENR_ALT = prove (`!s:real^N->bool. locally ENR s <=> !v x. open_in (subtopology euclidean s) v /\ x IN v ==> ?u. open_in (subtopology euclidean s) u /\ ENR u /\ x IN u /\ u SUBSET v`, GEN_TAC THEN REWRITE_TAC[locally] THEN EQ_TAC THENL [ALL_TAC; MESON_TAC[SUBSET_REFL]] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `v:real^N->bool` THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `x:real^N` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^N->bool` THEN DISCH_THEN(X_CHOOSE_THEN `w:real^N->bool` STRIP_ASSUME_TAC) THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC ENR_OPEN_IN THEN EXISTS_TAC `w:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC OPEN_IN_SUBSET_TRANS THEN EXISTS_TAC `s:real^N->bool` THEN ASM_MESON_TAC[OPEN_IN_IMP_SUBSET; SUBSET_TRANS]);; let LOCALLY_ENR = prove (`!s:real^N->bool. locally ENR s <=> !x. x IN s ==> ?v. x IN v /\ open_in (subtopology euclidean s) v /\ ENR v`, GEN_TAC THEN REWRITE_TAC[LOCALLY_ENR_ALT] THEN GEN_REWRITE_TAC LAND_CONV [SWAP_FORALL_THM] THEN EQ_TAC THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `x:real^N` THENL [DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC(SPEC `s:real^N->bool` th)) THEN ASM_REWRITE_TAC[OPEN_IN_REFL] THEN MESON_TAC[]; DISCH_THEN(fun th -> X_GEN_TAC `v:real^N->bool` THEN STRIP_TAC THEN MP_TAC th) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN X_GEN_TAC `w:real^N->bool` THEN STRIP_TAC THEN EXISTS_TAC `v INTER w:real^N->bool` THEN ASM_SIMP_TAC[OPEN_IN_INTER; IN_INTER; INTER_SUBSET] THEN MATCH_MP_TAC ENR_OPEN_IN THEN EXISTS_TAC `w:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC OPEN_IN_SUBSET_TRANS THEN EXISTS_TAC `s:real^N->bool` THEN ASM_SIMP_TAC[INTER_SUBSET; OPEN_IN_INTER] THEN ASM_MESON_TAC[OPEN_IN_IMP_SUBSET]]);; let ENR_LOCALLY = prove (`!s:real^N->bool. locally ENR s <=> ENR s`, REPEAT STRIP_TAC THEN REWRITE_TAC[LOCALLY_ENR] THEN EQ_TAC THENL [ALL_TAC; MESON_TAC[OPEN_IN_REFL]] THEN REWRITE_TAC[RIGHT_IMP_EXISTS_THM; SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `f:real^N->real^N->bool` THEN DISCH_TAC THEN SUBGOAL_THEN `UNIONS (IMAGE (f:real^N->real^N->bool) s) = s` ASSUME_TAC THENL [MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_IMAGE] THEN ASM_MESON_TAC[OPEN_IN_IMP_SUBSET]; EXPAND_TAC "s" THEN MATCH_MP_TAC ENR_OPEN_UNIONS THEN ASM_MESON_TAC[FORALL_IN_IMAGE]]);; let ANR_COVERING_SPACE_EQ = prove (`!p:real^M->real^N s c. covering_space (c,p) s ==> (ANR s <=> ANR c)`, REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN ONCE_REWRITE_TAC[GSYM ANR_LOCALLY] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] COVERING_SPACE_LOCALLY_HOMEOMORPHIC_EQ)) THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_ANRNESS THEN REWRITE_TAC[homeomorphic] THEN ASM_MESON_TAC[]);; let ANR_COVERING_SPACE = prove (`!p:real^M->real^N s c. covering_space (c,p) s /\ ANR c ==> ANR s`, MESON_TAC[ANR_COVERING_SPACE_EQ]);; let ENR_COVERING_SPACE_EQ = prove (`!p:real^M->real^N s c. covering_space (c,p) s ==> (ENR s <=> ENR c)`, REWRITE_TAC[ENR_ANR] THEN MESON_TAC[ANR_COVERING_SPACE_EQ; COVERING_SPACE_LOCALLY_COMPACT_EQ]);; let ENR_COVERING_SPACE = prove (`!p:real^M->real^N s c. covering_space (c,p) s /\ ENR c ==> ENR s`, MESON_TAC[ENR_COVERING_SPACE_EQ]);; (* ------------------------------------------------------------------------- *) (* Original ANR material, now for ENRs. Eventually more of this will be *) (* updated and generalized for AR and ANR as well. *) (* ------------------------------------------------------------------------- *) let ENR_BOUNDED = prove (`!s:real^N->bool. bounded s ==> (ENR s <=> ?u. open u /\ bounded u /\ s retract_of u)`, REPEAT STRIP_TAC THEN REWRITE_TAC[ENR] THEN EQ_TAC THENL [ALL_TAC; MESON_TAC[]] THEN FIRST_ASSUM(MP_TAC o SPEC `vec 0:real^N` o MATCH_MP BOUNDED_SUBSET_BALL) THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `ball(vec 0:real^N,r) INTER u` THEN ASM_SIMP_TAC[BOUNDED_INTER; OPEN_INTER; OPEN_BALL; BOUNDED_BALL] THEN MATCH_MP_TAC RETRACT_OF_SUBSET THEN EXISTS_TAC `u:real^N->bool` THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP RETRACT_OF_IMP_SUBSET) THEN ASM SET_TAC[]);; let ABSOLUTE_RETRACT_IMP_AR_GEN = prove (`!s:real^M->bool s':real^N->bool t u. s retract_of t /\ convex t /\ ~(t = {}) /\ s homeomorphic s' /\ closed_in (subtopology euclidean u) s' ==> s' retract_of u`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^M->bool`; `t:real^M->bool`] AR_RETRACT_OF_AR) THEN ASM_SIMP_TAC[CONVEX_IMP_AR] THEN ASM_MESON_TAC[AR_IMP_ABSOLUTE_RETRACT]);; let ABSOLUTE_RETRACT_IMP_AR = prove (`!s s'. s retract_of (:real^M) /\ s homeomorphic s' /\ closed s' ==> s' retract_of (:real^N)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC ABSOLUTE_RETRACT_IMP_AR_GEN THEN MAP_EVERY EXISTS_TAC [`s:real^M->bool`; `(:real^M)`] THEN ASM_REWRITE_TAC[SUBTOPOLOGY_UNIV; GSYM CLOSED_IN] THEN REWRITE_TAC[CONVEX_UNIV; CLOSED_UNIV; UNIV_NOT_EMPTY]);; let HOMEOMORPHIC_COMPACT_ARNESS = prove (`!s s'. s homeomorphic s' ==> (compact s /\ s retract_of (:real^M) <=> compact s' /\ s' retract_of (:real^N))`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `compact(s:real^M->bool) /\ compact(s':real^N->bool)` THENL [ALL_TAC; ASM_MESON_TAC[HOMEOMORPHIC_COMPACTNESS]] THEN ASM_REWRITE_TAC[] THEN EQ_TAC THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] ABSOLUTE_RETRACT_IMP_AR) THEN ASM_MESON_TAC[HOMEOMORPHIC_SYM; COMPACT_IMP_CLOSED]);; let EXTENSION_INTO_AR_LOCAL = prove (`!f:real^M->real^N c s t. f continuous_on c /\ IMAGE f c SUBSET t /\ t retract_of (:real^N) /\ closed_in (subtopology euclidean s) c ==> ?g. g continuous_on s /\ IMAGE g (:real^M) SUBSET t /\ !x. x IN c ==> g x = f x`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`; `c:real^M->bool`] TIETZE_UNBOUNDED) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `g:real^M->real^N` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retract_of]) THEN REWRITE_TAC[retraction] THEN DISCH_THEN(X_CHOOSE_THEN `r:real^N->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(r:real^N->real^N) o (g:real^M->real^N)` THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; REWRITE_TAC[IMAGE_o] THEN ASM SET_TAC[]; REWRITE_TAC[o_THM] THEN ASM SET_TAC[]]);; let EXTENSION_INTO_AR = prove (`!f:real^M->real^N s t. f continuous_on s /\ IMAGE f s SUBSET t /\ t retract_of (:real^N) /\ closed s ==> ?g. g continuous_on (:real^M) /\ IMAGE g (:real^M) SUBSET t /\ !x. x IN s ==> g x = f x`, REPEAT GEN_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`; `(:real^M)`; `t:real^N->bool`] EXTENSION_INTO_AR_LOCAL) THEN REWRITE_TAC[GSYM OPEN_IN; GSYM CLOSED_IN; SUBTOPOLOGY_UNIV]);; let NEIGHBOURHOOD_EXTENSION_INTO_ANR = prove (`!f:real^M->real^N s t. f continuous_on s /\ IMAGE f s SUBSET t /\ ANR t /\ closed s ==> ?v g. s SUBSET v /\ open v /\ g continuous_on v /\ IMAGE g v SUBSET t /\ !x. x IN s ==> g x = f x`, REPEAT GEN_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `(:real^M)`; `s:real^M->bool`; `t:real^N->bool`] ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR) THEN REWRITE_TAC[GSYM OPEN_IN; GSYM CLOSED_IN; SUBTOPOLOGY_UNIV] THEN CONV_TAC TAUT);; let EXTENSION_FROM_COMPONENT = prove (`!f:real^M->real^N s c u. (locally connected s \/ compact s /\ ANR u) /\ c IN components s /\ f continuous_on c /\ IMAGE f c SUBSET u ==> ?g. g continuous_on s /\ IMAGE g s SUBSET u /\ !x. x IN c ==> g x = f x`, REPEAT GEN_TAC THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN SUBGOAL_THEN `?t g. open_in (subtopology euclidean s) t /\ closed_in (subtopology euclidean s) t /\ c SUBSET t /\ (g:real^M->real^N) continuous_on t /\ IMAGE g t SUBSET u /\ !x. x IN c ==> g x = f x` STRIP_ASSUME_TAC THENL [FIRST_X_ASSUM(DISJ_CASES_THEN STRIP_ASSUME_TAC) THENL [MAP_EVERY EXISTS_TAC [`c:real^M->bool`; `f:real^M->real^N`] THEN ASM_SIMP_TAC[SUBSET_REFL; CLOSED_IN_COMPONENT; OPEN_IN_COMPONENTS_LOCALLY_CONNECTED]; MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`; `c:real^M->bool`; `u:real^N->bool`] ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR) THEN ASM_SIMP_TAC[CLOSED_IN_COMPONENT; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`w:real^M->bool`; `g:real^M->real^N`] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN DISCH_THEN(X_CHOOSE_THEN `v:real^M->bool` (STRIP_ASSUME_TAC o GSYM)) THEN MP_TAC(ISPECL [`s:real^M->bool`; `c:real^M->bool`; `v:real^M->bool`] SURA_BURA_CLOPEN_SUBSET) THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED; CLOSED_IMP_LOCALLY_COMPACT] THEN ANTS_TAC THENL [CONJ_TAC THENL [ASM_MESON_TAC[COMPACT_COMPONENTS]; ASM SET_TAC[]]; MATCH_MP_TAC MONO_EXISTS] THEN X_GEN_TAC `k:real^M->bool` THEN STRIP_TAC THEN EXISTS_TAC `g:real^M->real^N` THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CLOSED_SUBSET THEN ASM_MESON_TAC[COMPACT_IMP_CLOSED; OPEN_IN_IMP_SUBSET]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ASM SET_TAC[]; FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ASM SET_TAC[]]]; MP_TAC(ISPECL [`g:real^M->real^N`; `s:real^M->bool`; `t:real^M->bool`; `u:real^N->bool`] EXTENSION_FROM_CLOPEN) THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_NONEMPTY) THEN ANTS_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS] THEN ASM SET_TAC[]]);; let ABSOLUTE_RETRACT_FROM_UNION_AND_INTER = prove (`!s t. (s UNION t) retract_of (:real^N) /\ (s INTER t) retract_of (:real^N) /\ closed s /\ closed t ==> s retract_of (:real^N)`, MESON_TAC[RETRACT_OF_UNIV; AR_FROM_UNION_AND_INTER]);; let COUNTABLE_ENR_COMPONENTS = prove (`!s:real^N->bool. ENR s ==> COUNTABLE(components s)`, SIMP_TAC[ENR_IMP_ANR; COUNTABLE_ANR_COMPONENTS]);; let COUNTABLE_ENR_CONNECTED_COMPONENTS = prove (`!s:real^N->bool t. ENR s ==> COUNTABLE {connected_component s x | x | x IN t}`, SIMP_TAC[ENR_IMP_ANR; COUNTABLE_ANR_CONNECTED_COMPONENTS]);; let COUNTABLE_ENR_PATH_COMPONENTS = prove (`!s:real^N->bool. ENR s ==> COUNTABLE {path_component s x | x | x IN s}`, SIMP_TAC[ENR_IMP_ANR; COUNTABLE_ANR_PATH_COMPONENTS]);; let ENR_FROM_UNION_AND_INTER_GEN = prove (`!s t:real^N->bool. closed_in (subtopology euclidean (s UNION t)) s /\ closed_in (subtopology euclidean (s UNION t)) t /\ ENR(s UNION t) /\ ENR(s INTER t) ==> ENR s`, REWRITE_TAC[ENR_ANR] THEN MESON_TAC[LOCALLY_COMPACT_CLOSED_IN; ANR_FROM_UNION_AND_INTER_LOCAL]);; let ENR_FROM_UNION_AND_INTER = prove (`!s t:real^N->bool. closed s /\ closed t /\ ENR(s UNION t) /\ ENR(s INTER t) ==> ENR s`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC ENR_FROM_UNION_AND_INTER_GEN THEN ASM_MESON_TAC[CLOSED_SUBSET; SUBSET_UNION]);; let ENR_CLOSURE_FROM_FRONTIER = prove (`!s:real^N->bool. ENR(frontier s) ==> ENR(closure s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC ENR_FROM_UNION_AND_INTER THEN EXISTS_TAC `closure((:real^N) DIFF s)` THEN ASM_REWRITE_TAC[CLOSED_CLOSURE; GSYM FRONTIER_CLOSURES] THEN SUBGOAL_THEN `closure s UNION closure ((:real^N) DIFF s) = (:real^N)` (fun th -> REWRITE_TAC[th; ENR_UNIV]) THEN MATCH_MP_TAC(SET_RULE `s SUBSET closure s /\ (:real^N) DIFF s SUBSET closure((:real^N) DIFF s) ==> closure s UNION closure ((:real^N) DIFF s) = (:real^N)`) THEN REWRITE_TAC[CLOSURE_SUBSET]);; let ANR_CLOSURE_FROM_FRONTIER = prove (`!s:real^N->bool. ANR(frontier s) ==> ANR(closure s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC ENR_IMP_ANR THEN MATCH_MP_TAC ENR_CLOSURE_FROM_FRONTIER THEN ASM_SIMP_TAC[ENR_ANR; FRONTIER_CLOSED; CLOSED_IMP_LOCALLY_COMPACT]);; let ENR_FINITE_UNIONS_CONVEX_CLOSED = prove (`!t:(real^N->bool)->bool. FINITE t /\ (!c. c IN t ==> closed c /\ convex c) ==> ENR(UNIONS t)`, SIMP_TAC[ENR_ANR; ANR_FINITE_UNIONS_CONVEX_CLOSED] THEN SIMP_TAC[CLOSED_IMP_LOCALLY_COMPACT; CLOSED_UNIONS]);; let FINITE_IMP_ENR = prove (`!s:real^N->bool. FINITE s ==> ENR s`, SIMP_TAC[FINITE_IMP_ANR; FINITE_IMP_CLOSED; ENR_ANR; CLOSED_IMP_LOCALLY_COMPACT]);; let ENR_INSERT = prove (`!s a:real^N. closed s /\ ENR s ==> ENR(a INSERT s)`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[SET_RULE `a INSERT s = {a} UNION s`] THEN MATCH_MP_TAC ENR_CLOSED_UNION THEN ASM_MESON_TAC[CLOSED_SING; ENR_SING; ENR_EMPTY; SET_RULE `{a} INTER s = {a} \/ {a} INTER s = {}`]);; let ENR_TRIANGULATION = prove (`!tr. triangulation tr ==> ENR(UNIONS tr)`, REWRITE_TAC[triangulation] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC ENR_FINITE_UNIONS_CONVEX_CLOSED THEN ASM_MESON_TAC[SIMPLEX_IMP_CLOSED; SIMPLEX_IMP_CONVEX]);; let ENR_SIMPLICIAL_COMPLEX = prove (`!c. simplicial_complex c ==> ENR(UNIONS c)`, MESON_TAC[ENR_TRIANGULATION; SIMPLICIAL_COMPLEX_IMP_TRIANGULATION]);; let ENR_PATH_COMPONENT_ENR = prove (`!s x:real^N. ENR(s) ==> ENR(path_component s x)`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] ENR_OPEN_IN)) THEN MATCH_MP_TAC OPEN_IN_PATH_COMPONENT_LOCALLY_PATH_CONNECTED THEN MATCH_MP_TAC RETRACT_OF_LOCALLY_PATH_CONNECTED THEN ASM_MESON_TAC[ENR; OPEN_IMP_LOCALLY_PATH_CONNECTED]);; let ENR_CONNECTED_COMPONENT_ENR = prove (`!s x:real^N. ENR(s) ==> ENR(connected_component s x)`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] ENR_OPEN_IN)) THEN MATCH_MP_TAC OPEN_IN_CONNECTED_COMPONENT_LOCALLY_CONNECTED THEN MATCH_MP_TAC RETRACT_OF_LOCALLY_CONNECTED THEN ASM_MESON_TAC[ENR; OPEN_IMP_LOCALLY_CONNECTED]);; let ENR_COMPONENT_ENR = prove (`!s:real^N->bool. ENR s /\ c IN components s ==> ENR c`, REWRITE_TAC[IN_COMPONENTS] THEN MESON_TAC[ENR_CONNECTED_COMPONENT_ENR]);; let ENR_INTER_CLOSED_OPEN = prove (`!s:real^N->bool. ENR s ==> ?t u. closed t /\ open u /\ s = t INTER u`, GEN_TAC THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN REWRITE_TAC[ENR] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^N->bool` THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_RETRACT) THEN REWRITE_TAC[CLOSED_IN_CLOSED] THEN ASM_MESON_TAC[INTER_COMM]);; let ENR_IMP_FSGIMA = prove (`!s:real^N->bool. ENR s ==> fsigma s`, REPEAT STRIP_TAC THEN FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP ENR_INTER_CLOSED_OPEN) THEN ASM_SIMP_TAC[CLOSED_IMP_FSIGMA; OPEN_IMP_FSIGMA; FSIGMA_INTER]);; let ENR_IMP_GDELTA = prove (`!s:real^N->bool. ENR s ==> gdelta s`, REPEAT STRIP_TAC THEN FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP ENR_INTER_CLOSED_OPEN) THEN ASM_SIMP_TAC[CLOSED_IMP_GDELTA; OPEN_IMP_GDELTA; GDELTA_INTER]);; let IS_INTERVAL_IMP_FSIGMA = prove (`!s:real^N->bool. is_interval s ==> fsigma s`, SIMP_TAC[IS_INTERVAL_IMP_ENR; ENR_IMP_FSGIMA]);; let IS_INTERVAL_IMP_GDELTA = prove (`!s:real^N->bool. is_interval s ==> gdelta s`, SIMP_TAC[IS_INTERVAL_IMP_ENR; ENR_IMP_GDELTA]);; let IS_INTERVAL_IMP_BAIRE1_INDICATOR = prove (`!s. is_interval s ==> baire 1 (:real^N) (indicator s)`, SIMP_TAC[BAIRE1_INDICATOR; IS_INTERVAL_IMP_FSIGMA; IS_INTERVAL_IMP_GDELTA]);; let ANR_COMPONENTWISE = prove (`!s:real^N->bool. ANR s <=> COUNTABLE(components s) /\ !c. c IN components s ==> open_in (subtopology euclidean s) c /\ ANR c`, GEN_TAC THEN MATCH_MP_TAC(TAUT `(r ==> p) /\ (p ==> q) /\ (p ==> r) ==> (p <=> q /\ r)`) THEN REWRITE_TAC[COUNTABLE_ANR_COMPONENTS] THEN CONJ_TAC THENL [DISCH_TAC THEN GEN_REWRITE_TAC RAND_CONV [UNIONS_COMPONENTS] THEN MATCH_MP_TAC ANR_OPEN_UNIONS THEN ASM_MESON_TAC[GSYM UNIONS_COMPONENTS]; ASM_MESON_TAC[OPEN_IN_COMPONENTS_LOCALLY_CONNECTED; ANR_IMP_LOCALLY_CONNECTED; ANR_OPEN_IN]]);; let ENR_COMPONENTWISE = prove (`!s:real^N->bool. ENR s <=> COUNTABLE(components s) /\ !c. c IN components s ==> open_in (subtopology euclidean s) c /\ ENR c`, GEN_TAC THEN MATCH_MP_TAC(TAUT `(r ==> p) /\ (p ==> q) /\ (p ==> r) ==> (p <=> q /\ r)`) THEN REWRITE_TAC[COUNTABLE_ENR_COMPONENTS] THEN CONJ_TAC THENL [DISCH_TAC THEN GEN_REWRITE_TAC RAND_CONV [UNIONS_COMPONENTS] THEN MATCH_MP_TAC ENR_OPEN_UNIONS THEN ASM_MESON_TAC[GSYM UNIONS_COMPONENTS]; ASM_MESON_TAC[OPEN_IN_COMPONENTS_LOCALLY_CONNECTED; ENR_IMP_LOCALLY_CONNECTED; ENR_OPEN_IN]]);; let ABSOLUTE_RETRACT_HOMEOMORPHIC_CONVEX_COMPACT = prove (`!s:real^N->bool t u:real^M->bool. s homeomorphic u /\ ~(s = {}) /\ s SUBSET t /\ convex u /\ compact u ==> s retract_of t`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`u:real^M->bool`; `t:real^N->bool`; `s:real^N->bool`] AR_IMP_ABSOLUTE_RETRACT) THEN DISCH_THEN MATCH_MP_TAC THEN ASM_MESON_TAC[CONVEX_IMP_AR; HOMEOMORPHIC_EMPTY; HOMEOMORPHIC_SYM; CLOSED_SUBSET; COMPACT_IMP_CLOSED; HOMEOMORPHIC_COMPACTNESS]);; let ABSOLUTE_RETRACT_PATH_IMAGE_ARC = prove (`!g s:real^N->bool. arc g /\ path_image g SUBSET s ==> (path_image g) retract_of s`, REPEAT STRIP_TAC THEN MP_TAC (ISPECL [`path_image g:real^N->bool`; `s:real^N->bool`; `interval[vec 0:real^1,vec 1:real^1]`] ABSOLUTE_RETRACT_HOMEOMORPHIC_CONVEX_COMPACT) THEN DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[PATH_IMAGE_NONEMPTY] THEN REWRITE_TAC[COMPACT_INTERVAL; CONVEX_INTERVAL] THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN MATCH_MP_TAC HOMEOMORPHIC_COMPACT THEN EXISTS_TAC `g:real^1->real^N` THEN RULE_ASSUM_TAC(REWRITE_RULE[arc; path; path_image]) THEN ASM_REWRITE_TAC[COMPACT_INTERVAL; path_image]);; let AR_ARC_IMAGE = prove (`!g:real^1->real^N. arc g ==> AR(path_image g)`, MESON_TAC[RETRACT_OF_UNIV; SUBSET_UNIV; ABSOLUTE_RETRACT_PATH_IMAGE_ARC]);; let RELATIVE_FRONTIER_DEFORMATION_RETRACT_OF_PUNCTURED_CONVEX = prove (`!s t a:real^N. convex s /\ convex t /\ bounded s /\ a IN relative_interior s /\ relative_frontier s SUBSET t /\ t SUBSET affine hull s ==> ?r. homotopic_with (\x. T) (subtopology euclidean (t DELETE a), subtopology euclidean (t DELETE a)) (\x. x) r /\ retraction (t DELETE a,relative_frontier s) r /\ (!x. ?c. &0 < c /\ r(x) - a = c % (x - a))`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `a:real^N`] RAY_TO_RELATIVE_FRONTIER) THEN ASM_SIMP_TAC[relative_frontier; VECTOR_ADD_LID] THEN DISCH_THEN(MP_TAC o MATCH_MP (MESON[REAL_LT_01] `(!x. P x ==> ?d. &0 < d /\ R d x) ==> !x. ?d. &0 < d /\ (P x ==> R d x)`)) THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM; FORALL_AND_THM] THEN REWRITE_TAC[TAUT `p ==> q /\ r <=> (p ==> q) /\ (p ==> r)`] THEN REWRITE_TAC[FORALL_AND_THM; retraction] THEN X_GEN_TAC `dd:real^N->real` THEN STRIP_TAC THEN EXISTS_TAC `\x:real^N. a + dd(x - a) % (x - a)` THEN SUBGOAL_THEN `((\x:real^N. a + dd x % x) o (\x. x - a)) continuous_on t DELETE a` MP_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `affine hull s DELETE (a:real^N)` THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST] THEN MATCH_MP_TAC CONTINUOUS_ON_ADD THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN SIMP_TAC[VECTOR_ARITH `x - a:real^N = y - a <=> x = y`; VECTOR_SUB_REFL; SET_RULE `(!x y. f x = f y <=> x = y) ==> IMAGE f (s DELETE a) = IMAGE f s DELETE f a`] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPACT_SURFACE_PROJECTION THEN EXISTS_TAC `relative_frontier (IMAGE (\x:real^N. x - a) s)` THEN ASM_SIMP_TAC[COMPACT_RELATIVE_FRONTIER_BOUNDED; VECTOR_ARITH `x - a:real^N = --a + x`; RELATIVE_FRONTIER_TRANSLATION; COMPACT_TRANSLATION_EQ] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `s SUBSET t /\ ~(a IN IMAGE f s) ==> IMAGE f s SUBSET IMAGE f t DELETE a`) THEN REWRITE_TAC[IN_IMAGE; UNWIND_THM2; VECTOR_ARITH `vec 0:real^N = --a + x <=> x = a`] THEN ASM_REWRITE_TAC[relative_frontier; IN_DIFF] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s DIFF u SUBSET t`) THEN REWRITE_TAC[CLOSURE_SUBSET_AFFINE_HULL]; MATCH_MP_TAC SUBSPACE_IMP_CONIC THEN MATCH_MP_TAC AFFINE_IMP_SUBSPACE THEN SIMP_TAC[AFFINE_TRANSLATION; AFFINE_AFFINE_HULL; IN_IMAGE] THEN REWRITE_TAC[UNWIND_THM2; VECTOR_ARITH `vec 0:real^N = --a + x <=> x = a`] THEN ASM_MESON_TAC[SUBSET; HULL_SUBSET; RELATIVE_INTERIOR_SUBSET]; ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN REWRITE_TAC[IN_DELETE; IMP_CONJ; FORALL_IN_IMAGE] THEN REWRITE_TAC[VECTOR_ARITH `--a + x:real^N = vec 0 <=> x = a`] THEN MAP_EVERY X_GEN_TAC [`k:real`; `x:real^N`] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[IN_IMAGE; UNWIND_THM2; relative_frontier; VECTOR_ARITH `y:real^N = --a + x <=> x = a + y`] THEN EQ_TAC THENL [STRIP_TAC; DISCH_THEN(SUBST1_TAC o SYM) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[VECTOR_ARITH `a + --a + x:real^N = x`; VECTOR_ARITH `--a + x:real^N = vec 0 <=> x = a`]] THEN MATCH_MP_TAC(REAL_ARITH `~(a < b) /\ ~(b < a) ==> a = b`) THEN CONJ_TAC THEN DISCH_TAC THENL [ALL_TAC; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `x IN c DIFF i ==> x IN i ==> F`)) THEN RULE_ASSUM_TAC(REWRITE_RULE[IMP_IMP; RIGHT_IMP_FORALL_THM]) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; VECTOR_ARITH `a + --a + x:real^N = x`; VECTOR_ARITH `--a + x:real^N = vec 0 <=> x = a`]] THEN MP_TAC(ISPECL [`s:real^N->bool`; `a:real^N`; `a + k % (--a + x):real^N`] IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT) THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_DIFF]) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET; IN_SEGMENT; NOT_FORALL_THM] THEN EXISTS_TAC `a + dd(--a + x) % (--a + x):real^N` THEN ASM_REWRITE_TAC[VECTOR_ARITH `a:real^N = a + k % (--a + x) <=> k % (x - a) = vec 0`] THEN ASM_SIMP_TAC[VECTOR_SUB_EQ; VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ] THEN REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [EXISTS_TAC `(dd:real^N->real) (--a + x) / k` THEN ASM_SIMP_TAC[REAL_LT_LDIV_EQ; REAL_MUL_LID] THEN REWRITE_TAC[VECTOR_ARITH `a + b:real^N = (&1 - u) % a + u % c <=> b = u % (c - a)`] THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; VECTOR_ADD_SUB; REAL_DIV_RMUL; REAL_LT_IMP_NZ] THEN MATCH_MP_TAC REAL_LT_DIV THEN ASM_REWRITE_TAC[]; MATCH_MP_TAC(SET_RULE `a IN closure s /\ ~(a IN relative_interior s) ==> ~(a IN relative_interior s)`)] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[VECTOR_ARITH `a + --a + x:real^N = x`; VECTOR_ARITH `--a + x:real^N = vec 0 <=> x = a`]]; REWRITE_TAC[o_DEF] THEN STRIP_TAC] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC HOMOTOPIC_WITH_LINEAR THEN ASM_REWRITE_TAC[CONTINUOUS_ON_ID] THEN REWRITE_TAC[segment; SUBSET; FORALL_IN_GSPEC; IN_DELETE] THEN REPEAT(GEN_TAC THEN STRIP_TAC) THEN CONJ_TAC THENL [FIRST_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [convex]) THEN ASM_REWRITE_TAC[REAL_ARITH `&1 - u + u = &1`; REAL_SUB_LE] THEN FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN REWRITE_TAC[relative_frontier] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[VECTOR_ARITH `a + x - a:real^N = x`; VECTOR_SUB_EQ] THEN ASM_MESON_TAC[HULL_SUBSET; RELATIVE_INTERIOR_SUBSET; SUBSET]; ASM_SIMP_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ; VECTOR_ARITH `(&1 - u) % x + u % (a + d % (x - a)):real^N = a <=> (&1 - u + u * d) % (x - a) = vec 0`] THEN MATCH_MP_TAC(REAL_ARITH `&0 <= x /\ &0 <= u /\ u <= &1 /\ ~(x = &0 /\ u = &1) ==> ~(&1 - u + x = &0)`) THEN ASM_SIMP_TAC[REAL_ENTIRE; REAL_ARITH `(u = &0 \/ d = &0) /\ u = &1 <=> d = &0 /\ u = &1`] THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_LE_MUL THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LT_IMP_LE; MATCH_MP_TAC(REAL_ARITH `&0 < x ==> ~(x = &0 /\ u = &1)`)] THEN ASM_REWRITE_TAC[]]; RULE_ASSUM_TAC(REWRITE_RULE[relative_frontier]) THEN ASM SET_TAC[]; ASM_REWRITE_TAC[]; MATCH_MP_TAC(SET_RULE `!s t. s SUBSET t /\ IMAGE f (t DELETE a) SUBSET u ==> IMAGE f (s DELETE a) SUBSET u`) THEN EXISTS_TAC `affine hull s:real^N->bool` THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_DELETE] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[VECTOR_SUB_EQ; VECTOR_ARITH `a + x - a:real^N = x`]; X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_DIFF] THEN STRIP_TAC THEN ASM_CASES_TAC `x:real^N = a` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `dd(x - a:real^N) = &1` (fun th -> REWRITE_TAC[th] THEN CONV_TAC VECTOR_ARITH) THEN MATCH_MP_TAC(REAL_ARITH `~(d < &1) /\ ~(&1 < d) ==> d = &1`) THEN CONJ_TAC THEN DISCH_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `a:real^N`] IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT) THENL [DISCH_THEN(MP_TAC o SPEC `x:real^N`); DISCH_THEN(MP_TAC o SPEC `a + dd(x - a) % (x - a):real^N`)] THEN ASM_REWRITE_TAC[SUBSET; NOT_IMP; IN_SEGMENT; NOT_FORALL_THM] THENL [EXISTS_TAC `a + dd(x - a) % (x - a):real^N` THEN ASM_REWRITE_TAC[VECTOR_SUB_EQ; VECTOR_MUL_EQ_0; REAL_SUB_0; VECTOR_ARITH `a + d % (x - a):real^N = (&1 - u) % a + u % x <=> (u - d) % (x - a) = vec 0`] THEN CONJ_TAC THENL [EXISTS_TAC `(dd:real^N->real)(x - a)` THEN ASM_REWRITE_TAC[]; MATCH_MP_TAC(SET_RULE `x IN closure s DIFF relative_interior s ==> ~(x IN relative_interior s)`)] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[VECTOR_SUB_EQ; VECTOR_ARITH `a + x - a:real^N = x`] THEN ASM_MESON_TAC[CLOSURE_SUBSET_AFFINE_HULL; SUBSET]; CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `x IN closure s DIFF relative_interior s ==> x IN closure s`) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[VECTOR_SUB_EQ; VECTOR_ARITH `a + x - a:real^N = x`] THEN ASM_MESON_TAC[CLOSURE_SUBSET_AFFINE_HULL; SUBSET]; EXISTS_TAC `x:real^N` THEN ASM_SIMP_TAC[VECTOR_SUB_EQ; VECTOR_MUL_EQ_0; VECTOR_ARITH `a = a + d <=> d:real^N = vec 0`; VECTOR_ARITH `x:real^N = (&1 - u) % a + u % (a + d % (x - a)) <=> (u * d - &1) % (x - a) = vec 0`] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; DISCH_TAC] THEN EXISTS_TAC `inv((dd:real^N->real)(x - a))` THEN ASM_SIMP_TAC[REAL_MUL_LINV; REAL_SUB_REFL; REAL_LT_INV_EQ] THEN ASM_SIMP_TAC[REAL_INV_LT_1] THEN ASM_REAL_ARITH_TAC]]; REWRITE_TAC[VECTOR_ADD_SUB] THEN EXISTS_TAC `\x. (dd:real^N->real)(x - a)` THEN ASM_REWRITE_TAC[]]);; let RELATIVE_FRONTIER_RETRACT_OF_PUNCTURED_AFFINE_HULL = prove (`!s a:real^N. convex s /\ bounded s /\ a IN relative_interior s ==> relative_frontier s retract_of (affine hull s DELETE a)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `affine hull s:real^N->bool`; `a:real^N`] RELATIVE_FRONTIER_DEFORMATION_RETRACT_OF_PUNCTURED_CONVEX) THEN ASM_SIMP_TAC[AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL; SUBSET_REFL] THEN REWRITE_TAC[retract_of] THEN ANTS_TAC THENL [ALL_TAC; MESON_TAC[]] THEN REWRITE_TAC[relative_frontier] THEN MATCH_MP_TAC(SET_RULE `s SUBSET u ==> s DIFF t SUBSET u`) THEN REWRITE_TAC[CLOSURE_SUBSET_AFFINE_HULL]);; let RELATIVE_BOUNDARY_RETRACT_OF_PUNCTURED_AFFINE_HULL = prove (`!s a:real^N. convex s /\ compact s /\ a IN relative_interior s ==> (s DIFF relative_interior s) retract_of (affine hull s DELETE a)`, MP_TAC RELATIVE_FRONTIER_RETRACT_OF_PUNCTURED_AFFINE_HULL THEN REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_SIMP_TAC[relative_frontier; COMPACT_IMP_BOUNDED; COMPACT_IMP_CLOSED; CLOSURE_CLOSED]);; let PATH_CONNECTED_SPHERE_GEN = prove (`!s:real^N->bool. convex s /\ bounded s /\ ~(aff_dim s = &1) ==> path_connected(relative_frontier s)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `relative_interior s:real^N->bool = {}` THENL [ASM_MESON_TAC[RELATIVE_INTERIOR_EQ_EMPTY; PATH_CONNECTED_EMPTY; RELATIVE_FRONTIER_EMPTY]; 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 MATCH_MP_TAC RETRACT_OF_PATH_CONNECTED THEN EXISTS_TAC `affine hull s DELETE (a:real^N)` THEN ASM_SIMP_TAC[PATH_CONNECTED_PUNCTURED_CONVEX; AFFINE_AFFINE_HULL; AFFINE_IMP_CONVEX; AFF_DIM_AFFINE_HULL; RELATIVE_FRONTIER_RETRACT_OF_PUNCTURED_AFFINE_HULL]]);; let CONNECTED_SPHERE_GEN = prove (`!s:real^N->bool. convex s /\ bounded s /\ ~(aff_dim s = &1) ==> connected(relative_frontier s)`, SIMP_TAC[PATH_CONNECTED_SPHERE_GEN; PATH_CONNECTED_IMP_CONNECTED]);; let ENR_RELATIVE_FRONTIER_CONVEX = prove (`!s:real^N->bool. bounded s /\ convex s ==> ENR(relative_frontier s)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[ENR; RELATIVE_FRONTIER_EMPTY] THENL [ASM_MESON_TAC[RETRACT_OF_REFL; OPEN_EMPTY]; ALL_TAC] THEN SUBGOAL_THEN `~(relative_interior s:real^N->bool = {})` MP_TAC THENL [ASM_SIMP_TAC[RELATIVE_INTERIOR_EQ_EMPTY]; ALL_TAC] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN EXISTS_TAC `{x | x IN (:real^N) /\ closest_point (affine hull s) x IN ((:real^N) DELETE a)}` THEN CONJ_TAC THENL [REWRITE_TAC[OPEN_IN] THEN ONCE_REWRITE_TAC[GSYM SUBTOPOLOGY_UNIV] THEN MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE_GEN THEN EXISTS_TAC `(:real^N)` THEN SIMP_TAC[OPEN_IN_DELETE; OPEN_IN_REFL; SUBSET_UNIV; ETA_AX]; MATCH_MP_TAC RETRACT_OF_TRANS THEN EXISTS_TAC `(affine hull s) DELETE (a:real^N)` THEN CONJ_TAC THENL [MATCH_MP_TAC RELATIVE_FRONTIER_RETRACT_OF_PUNCTURED_AFFINE_HULL THEN ASM_REWRITE_TAC[]; REWRITE_TAC[retract_of; retraction] THEN EXISTS_TAC `closest_point (affine hull s:real^N->bool)` THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_DELETE] THEN ASM_SIMP_TAC[IN_ELIM_THM; IN_UNIV; CLOSEST_POINT_SELF; CLOSEST_POINT_IN_SET; AFFINE_HULL_EQ_EMPTY; CLOSED_AFFINE_HULL]]] THEN MATCH_MP_TAC CONTINUOUS_ON_CLOSEST_POINT THEN ASM_SIMP_TAC[AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL; CLOSED_AFFINE_HULL; AFFINE_HULL_EQ_EMPTY]);; let ANR_RELATIVE_FRONTIER_CONVEX = prove (`!s:real^N->bool. bounded s /\ convex s ==> ANR(relative_frontier s)`, SIMP_TAC[ENR_IMP_ANR; ENR_RELATIVE_FRONTIER_CONVEX]);; let FRONTIER_RETRACT_OF_PUNCTURED_UNIVERSE = prove (`!s a. convex s /\ bounded s /\ a IN interior s ==> (frontier s) retract_of ((:real^N) DELETE a)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP (SET_RULE `a IN s ==> ~(s = {})`)) THEN MP_TAC(ISPECL [`s:real^N->bool`; `a:real^N`] RELATIVE_FRONTIER_RETRACT_OF_PUNCTURED_AFFINE_HULL) THEN ASM_SIMP_TAC[RELATIVE_FRONTIER_NONEMPTY_INTERIOR; RELATIVE_INTERIOR_NONEMPTY_INTERIOR; AFFINE_HULL_NONEMPTY_INTERIOR]);; let SPHERE_RETRACT_OF_PUNCTURED_UNIVERSE_GEN = prove (`!a r b:real^N. b IN ball(a,r) ==> sphere(a,r) retract_of ((:real^N) DELETE b)`, REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM FRONTIER_CBALL] THEN MATCH_MP_TAC FRONTIER_RETRACT_OF_PUNCTURED_UNIVERSE THEN ASM_REWRITE_TAC[CONVEX_CBALL; BOUNDED_CBALL; INTERIOR_CBALL]);; let SPHERE_RETRACT_OF_PUNCTURED_UNIVERSE = prove (`!a r. &0 < r ==> sphere(a,r) retract_of ((:real^N) DELETE a)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SPHERE_RETRACT_OF_PUNCTURED_UNIVERSE_GEN THEN ASM_REWRITE_TAC[CENTRE_IN_BALL]);; let ENR_SPHERE = prove (`!a:real^N r. ENR(sphere(a,r))`, REPEAT GEN_TAC THEN ASM_CASES_TAC `&0 < r` THENL [REWRITE_TAC[ENR] THEN EXISTS_TAC `(:real^N) DELETE a` THEN ASM_SIMP_TAC[SPHERE_RETRACT_OF_PUNCTURED_UNIVERSE; OPEN_DELETE; OPEN_UNIV]; ASM_MESON_TAC[FINITE_IMP_ENR; REAL_NOT_LE; FINITE_SPHERE]]);; let ANR_SPHERE = prove (`!a:real^N r. ANR(sphere(a,r))`, SIMP_TAC[ENR_SPHERE; ENR_IMP_ANR]);; let LOCALLY_PATH_CONNECTED_SPHERE_GEN = prove (`!s:real^N->bool. bounded s /\ convex s ==> locally path_connected (relative_frontier s)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `relative_interior(s:real^N->bool) = {}` THENL [UNDISCH_TAC `relative_interior(s:real^N->bool) = {}` THEN ASM_SIMP_TAC[RELATIVE_INTERIOR_EQ_EMPTY] THEN REWRITE_TAC[LOCALLY_EMPTY; RELATIVE_FRONTIER_EMPTY]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN MATCH_MP_TAC RETRACT_OF_LOCALLY_PATH_CONNECTED THEN EXISTS_TAC `(affine hull s) DELETE (a:real^N)` THEN ASM_SIMP_TAC[RELATIVE_FRONTIER_RETRACT_OF_PUNCTURED_AFFINE_HULL] THEN MATCH_MP_TAC LOCALLY_OPEN_SUBSET THEN EXISTS_TAC `affine hull s:real^N->bool` THEN SIMP_TAC[OPEN_IN_DELETE; OPEN_IN_REFL] THEN SIMP_TAC[CONVEX_IMP_LOCALLY_PATH_CONNECTED; AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL]]);; let LOCALLY_CONNECTED_SPHERE_GEN = prove (`!s:real^N->bool. bounded s /\ convex s ==> locally connected (relative_frontier s)`, SIMP_TAC[LOCALLY_PATH_CONNECTED_SPHERE_GEN; LOCALLY_PATH_CONNECTED_IMP_LOCALLY_CONNECTED]);; let ABSOLUTE_RETRACTION_CONVEX_CLOSED_RELATIVE = prove (`!s:real^N->bool t. convex s /\ closed s /\ ~(s = {}) /\ s SUBSET t ==> ?r. retraction (t,s) r /\ !x. x IN (affine hull s) DIFF (relative_interior s) ==> r(x) IN relative_frontier s`, REPEAT STRIP_TAC THEN REWRITE_TAC[retraction] THEN EXISTS_TAC `closest_point(s:real^N->bool)` THEN ASM_SIMP_TAC[CONTINUOUS_ON_CLOSEST_POINT; CLOSEST_POINT_SELF] THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE; CLOSEST_POINT_IN_SET] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CLOSEST_POINT_IN_RELATIVE_FRONTIER THEN ASM_MESON_TAC[SUBSET; RELATIVE_INTERIOR_SUBSET]);; let ABSOLUTE_RETRACTION_CONVEX_CLOSED = prove (`!s:real^N->bool t. convex s /\ closed s /\ ~(s = {}) /\ s SUBSET t ==> ?r. retraction (t,s) r /\ (!x. ~(x IN s) ==> r(x) IN frontier s)`, REPEAT STRIP_TAC THEN REWRITE_TAC[retraction] THEN EXISTS_TAC `closest_point(s:real^N->bool)` THEN ASM_SIMP_TAC[CONTINUOUS_ON_CLOSEST_POINT; CLOSEST_POINT_SELF] THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE; CLOSEST_POINT_IN_SET] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CLOSEST_POINT_IN_FRONTIER THEN ASM_MESON_TAC[SUBSET; INTERIOR_SUBSET]);; let ABSOLUTE_RETRACT_CONVEX_CLOSED = prove (`!s:real^N->bool t. convex s /\ closed s /\ ~(s = {}) /\ s SUBSET t ==> s retract_of t`, REWRITE_TAC[retract_of] THEN MESON_TAC[ABSOLUTE_RETRACTION_CONVEX_CLOSED]);; let ABSOLUTE_RETRACT_CONVEX = prove (`!s u:real^N->bool. convex s /\ ~(s = {}) /\ closed_in (subtopology euclidean u) s ==> s retract_of u`, REPEAT STRIP_TAC THEN REWRITE_TAC[retract_of; retraction] THEN MP_TAC(ISPECL [`\x:real^N. x`; `s:real^N->bool`; `u:real^N->bool`; `s:real^N->bool`] DUGUNDJI) THEN ASM_MESON_TAC[CONTINUOUS_ON_ID; IMAGE_ID; SUBSET_REFL; CLOSED_IN_IMP_SUBSET]);; let ENR_PATH_IMAGE_SIMPLE_PATH = prove (`!g:real^1->real^N. simple_path g ==> ENR(path_image g)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `pathfinish g:real^N = pathstart g` THENL [MP_TAC(ISPECL [`g:real^1->real^N`; `vec 0:real^2`; `&1`] HOMEOMORPHIC_SIMPLE_PATH_IMAGE_CIRCLE) THEN ASM_REWRITE_TAC[REAL_LT_01] THEN DISCH_THEN(SUBST1_TAC o MATCH_MP HOMEOMORPHIC_ENRNESS) THEN REWRITE_TAC[ENR_SPHERE]; REWRITE_TAC[ENR] THEN EXISTS_TAC `(:real^N)` THEN REWRITE_TAC[OPEN_UNIV] THEN MATCH_MP_TAC ABSOLUTE_RETRACT_PATH_IMAGE_ARC THEN ASM_REWRITE_TAC[ARC_SIMPLE_PATH; SUBSET_UNIV]]);; let ANR_PATH_IMAGE_SIMPLE_PATH = prove (`!g:real^1->real^N. simple_path g ==> ANR(path_image g)`, SIMP_TAC[ENR_PATH_IMAGE_SIMPLE_PATH; ENR_IMP_ANR]);; (* ------------------------------------------------------------------------- *) (* Borsuk homotopy extension thorem. It's only this late so we can use the *) (* concept of retraction, saying that the domain sets or range set are ANRs. *) (* ------------------------------------------------------------------------- *) let BORSUK_HOMOTOPY_EXTENSION_HOMOTOPIC = prove (`!f:real^M->real^N g s t u. closed_in (subtopology euclidean t) s /\ (ANR s /\ ANR t \/ ANR u) /\ f continuous_on t /\ IMAGE f t SUBSET u /\ homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean u) f g ==> ?g'. homotopic_with (\x. T) (subtopology euclidean t,subtopology euclidean u) f g' /\ g' continuous_on t /\ IMAGE g' t SUBSET u /\ !x. x IN s ==> g'(x) = g(x)`, REPEAT GEN_TAC THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMOTOPIC_WITH_EUCLIDEAN]) THEN REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `h:real^(1,M)finite_sum->real^N` STRIP_ASSUME_TAC) THEN MAP_EVERY ABBREV_TAC [`h' = \z. if sndcart z IN s then (h:real^(1,M)finite_sum->real^N) z else f(sndcart z)`; `B:real^(1,M)finite_sum->bool = {vec 0} PCROSS t UNION interval[vec 0,vec 1] PCROSS s`] THEN SUBGOAL_THEN `closed_in (subtopology euclidean (interval[vec 0:real^1,vec 1] PCROSS t)) ({vec 0} PCROSS (t:real^M->bool)) /\ closed_in (subtopology euclidean (interval[vec 0:real^1,vec 1] PCROSS t)) (interval[vec 0,vec 1] PCROSS s)` STRIP_ASSUME_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC CLOSED_IN_PCROSS THEN ASM_REWRITE_TAC[CLOSED_IN_SING; CLOSED_IN_REFL; ENDS_IN_UNIT_INTERVAL]; ALL_TAC] THEN SUBGOAL_THEN `(h':real^(1,M)finite_sum->real^N) continuous_on B` ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["h'"; "B"] THEN ONCE_REWRITE_TAC[UNION_COMM] THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] CLOSED_IN_SUBSET_TRANS)) THEN REWRITE_TAC[SUBSET_UNION; UNION_SUBSET; SUBSET_PCROSS] THEN ASM_REWRITE_TAC[SING_SUBSET; SUBSET_REFL; ENDS_IN_UNIT_INTERVAL]; ASM_SIMP_TAC[FORALL_PASTECART; PASTECART_IN_PCROSS; IN_SING; SNDCART_PASTECART; TAUT `(p /\ q) /\ ~q <=> F`] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_SIMP_TAC[LINEAR_SNDCART; LINEAR_CONTINUOUS_ON; IMAGE_SNDCART_PCROSS; NOT_INSERT_EMPTY]]; ALL_TAC] THEN SUBGOAL_THEN `IMAGE (h':real^(1,M)finite_sum->real^N) B SUBSET u` ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["h'"; "B"] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_PASTECART; SNDCART_PASTECART; PASTECART_IN_PCROSS; IN_UNION; IN_SING] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[COND_ID] THENL [ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o SIMP_RULE[SUBSET; FORALL_IN_IMAGE]) THEN ASM_REWRITE_TAC[PASTECART_IN_PCROSS; ENDS_IN_UNIT_INTERVAL]; ALL_TAC] THEN SUBGOAL_THEN `?V k:real^(1,M)finite_sum->real^N. B SUBSET V /\ open_in (subtopology euclidean (interval [vec 0,vec 1] PCROSS t)) V /\ k continuous_on V /\ IMAGE k V SUBSET u /\ (!x. x IN B ==> k x = h' x)` STRIP_ASSUME_TAC THENL [FIRST_X_ASSUM(DISJ_CASES_THEN STRIP_ASSUME_TAC) THENL [SUBGOAL_THEN `ANR(B:real^(1,M)finite_sum->bool)` MP_TAC THENL [EXPAND_TAC "B" THEN MATCH_MP_TAC ANR_CLOSED_UNION_LOCAL THEN EXISTS_TAC `{vec 0:real^1} PCROSS (t:real^M->bool) UNION interval[vec 0,vec 1] PCROSS s` THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] CLOSED_IN_SUBSET_TRANS)) THEN REWRITE_TAC[SUBSET_UNION; UNION_SUBSET; SUBSET_PCROSS] THEN ASM_REWRITE_TAC[SING_SUBSET; SUBSET_REFL; ENDS_IN_UNIT_INTERVAL]; ASM_SIMP_TAC[INTER_PCROSS; SET_RULE `s SUBSET t ==> t INTER s = s`; ENDS_IN_UNIT_INTERVAL; SET_RULE `a IN s ==> {a} INTER s = {a}`] THEN REPEAT CONJ_TAC THEN MATCH_MP_TAC ANR_PCROSS THEN ASM_REWRITE_TAC[ANR_INTERVAL; ANR_SING]]; DISCH_THEN(MP_TAC o SPEC `interval[vec 0:real^1,vec 1] PCROSS (t:real^M->bool)` o MATCH_MP(ONCE_REWRITE_RULE[IMP_CONJ] ANR_IMP_NEIGHBOURHOOD_RETRACT)) THEN ANTS_TAC THENL [EXPAND_TAC "B" THEN MATCH_MP_TAC CLOSED_IN_UNION THEN CONJ_TAC THEN MATCH_MP_TAC CLOSED_IN_PCROSS THEN ASM_REWRITE_TAC[CLOSED_IN_REFL; CLOSED_IN_SING; ENDS_IN_UNIT_INTERVAL]; MATCH_MP_TAC MONO_EXISTS] THEN X_GEN_TAC `V:real^(1,M)finite_sum->bool` THEN STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP RETRACT_OF_IMP_SUBSET) THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retract_of]) THEN REWRITE_TAC[retraction; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `r:real^(1,M)finite_sum->real^(1,M)finite_sum` THEN STRIP_TAC THEN EXISTS_TAC `(h':real^(1,M)finite_sum->real^N) o (r:real^(1,M)finite_sum->real^(1,M)finite_sum)` THEN ASM_REWRITE_TAC[IMAGE_o; o_THM] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]]; MATCH_MP_TAC ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR THEN ASM_SIMP_TAC[] THEN EXPAND_TAC "B" THEN ASM_SIMP_TAC[CLOSED_IN_UNION]]; ABBREV_TAC `s' = {x | ?u. u IN interval[vec 0,vec 1] /\ pastecart (u:real^1) (x:real^M) IN interval [vec 0,vec 1] PCROSS t DIFF V}` THEN SUBGOAL_THEN `closed_in (subtopology euclidean t) (s':real^M->bool)` ASSUME_TAC THENL [EXPAND_TAC "s'" THEN MATCH_MP_TAC CLOSED_IN_COMPACT_PROJECTION THEN REWRITE_TAC[COMPACT_INTERVAL] THEN MATCH_MP_TAC CLOSED_IN_DIFF THEN ASM_REWRITE_TAC[CLOSED_IN_REFL]; ALL_TAC] THEN MP_TAC(ISPECL [`s:real^M->bool`; `s':real^M->bool`; `t:real^M->bool`; `vec 1:real^1`; `vec 0:real^1`] URYSOHN_LOCAL) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [EXPAND_TAC "s'" THEN REWRITE_TAC[EXTENSION; IN_INTER; IN_ELIM_THM] THEN REWRITE_TAC[NOT_IN_EMPTY; IN_DIFF; PASTECART_IN_PCROSS] THEN X_GEN_TAC `x:real^M` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `p:real^1` MP_TAC) THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC)) THEN REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN EXPAND_TAC "B" THEN REWRITE_TAC[IN_UNION; PASTECART_IN_PCROSS] THEN ASM SET_TAC[]; ALL_TAC] THEN ONCE_REWRITE_TAC[SEGMENT_SYM] THEN REWRITE_TAC[SEGMENT_1; DROP_VEC; REAL_POS] THEN DISCH_THEN(X_CHOOSE_THEN `a:real^M->real^1` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(\x. (k:real^(1,M)finite_sum->real^N) (pastecart (a x) x))` THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE] THEN REPEAT CONJ_TAC THENL [SIMP_TAC[HOMOTOPIC_WITH_EUCLIDEAN_ALT] THEN EXISTS_TAC `(k:real^(1,M)finite_sum->real^N) o (\z. pastecart (drop(fstcart z) % a(sndcart z)) (sndcart z))` THEN REWRITE_TAC[o_THM; FSTCART_PASTECART; SNDCART_PASTECART] THEN REWRITE_TAC[DROP_VEC; VECTOR_MUL_LZERO; VECTOR_MUL_LID] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_PASTECART THEN SIMP_TAC[LINEAR_SNDCART; LINEAR_CONTINUOUS_ON] THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN SIMP_TAC[o_DEF; LIFT_DROP; LINEAR_FSTCART; LINEAR_CONTINUOUS_ON; ETA_AX] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[LINEAR_SNDCART; LINEAR_CONTINUOUS_ON] THEN ASM_SIMP_TAC[IMAGE_SNDCART_PCROSS; UNIT_INTERVAL_NONEMPTY]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET))]; REWRITE_TAC[IMAGE_o] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE k t SUBSET u ==> s SUBSET t ==> IMAGE k s SUBSET u`)); X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN SUBGOAL_THEN `pastecart (vec 0:real^1) (x:real^M) IN B` MP_TAC THENL [EXPAND_TAC "B" THEN ASM_REWRITE_TAC[IN_UNION; PASTECART_IN_PCROSS; IN_SING]; DISCH_TAC THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `(h':real^(1,M)finite_sum->real^N) (pastecart (vec 0) x)` THEN CONJ_TAC THENL [ASM_MESON_TAC[]; EXPAND_TAC "h'"] THEN ASM_REWRITE_TAC[SNDCART_PASTECART; COND_ID]]] THEN (REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_PCROSS] THEN MAP_EVERY X_GEN_TAC [`p:real^1`; `x:real^M`] THEN STRIP_TAC THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN ASM_CASES_TAC `(x:real^M) IN s'` THENL [ASM_SIMP_TAC[VECTOR_MUL_RZERO] THEN FIRST_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET]) THEN EXPAND_TAC "B" THEN REWRITE_TAC[IN_UNION; PASTECART_IN_PCROSS] THEN ASM_REWRITE_TAC[IN_SING]; UNDISCH_TAC `~((x:real^M) IN s')` THEN EXPAND_TAC "s'" THEN REWRITE_TAC[IN_ELIM_THM; NOT_EXISTS_THM] THEN DISCH_THEN(MP_TAC o SPEC `drop p % (a:real^M->real^1) x`) THEN REWRITE_TAC[PASTECART_IN_PCROSS; IN_DIFF] THEN ASM_REWRITE_TAC[CONJ_ASSOC] THEN MATCH_MP_TAC(TAUT `p ==> ~(p /\ ~q) ==> q`) THEN REWRITE_TAC[IN_INTERVAL_1; DROP_CMUL; DROP_VEC] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN ASM_SIMP_TAC[REAL_LE_MUL; REAL_LE_LMUL; REAL_ARITH `p * a <= p * &1 /\ p <= &1 ==> p * a <= &1`]]); GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_SIMP_TAC[CONTINUOUS_ON_PASTECART; CONTINUOUS_ON_ID] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC; X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET; FORALL_IN_IMAGE]); X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `(h':real^(1,M)finite_sum->real^N) (pastecart (vec 1) x)` THEN CONJ_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC; EXPAND_TAC "h'"] THEN ASM_REWRITE_TAC[SNDCART_PASTECART] THEN EXPAND_TAC "B" THEN REWRITE_TAC[IN_UNION; PASTECART_IN_PCROSS] THEN ASM_REWRITE_TAC[ENDS_IN_UNIT_INTERVAL]] THEN (ASM_CASES_TAC `(x:real^M) IN s'` THEN ASM_SIMP_TAC[] THENL [FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN EXPAND_TAC "B" THEN REWRITE_TAC[IN_UNION; PASTECART_IN_PCROSS] THEN ASM SET_TAC[]; UNDISCH_TAC `~((x:real^M) IN s')` THEN EXPAND_TAC "s'" THEN REWRITE_TAC[IN_ELIM_THM; NOT_EXISTS_THM] THEN DISCH_THEN(MP_TAC o SPEC `(a:real^M->real^1) x`) THEN ASM_SIMP_TAC[PASTECART_IN_PCROSS; IN_DIFF] THEN ASM SET_TAC[]])]);; let BORSUK_HOMOTOPY_EXTENSION = prove (`!f:real^M->real^N g s t u. closed_in (subtopology euclidean t) s /\ (ANR s /\ ANR t \/ ANR u) /\ f continuous_on t /\ IMAGE f t SUBSET u /\ homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean u) f g ==> ?g'. g' continuous_on t /\ IMAGE g' t SUBSET u /\ !x. x IN s ==> g'(x) = g(x)`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP BORSUK_HOMOTOPY_EXTENSION_HOMOTOPIC) THEN MESON_TAC[]);; let NULLHOMOTOPIC_INTO_ANR_EXTENSION = prove (`!f:real^M->real^N s t. closed s /\ f continuous_on s /\ ~(s = {}) /\ IMAGE f s SUBSET t /\ ANR t ==> ((?c. homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean t) f (\x. c)) <=> (?g. g continuous_on (:real^M) /\ IMAGE g (:real^M) SUBSET t /\ !x. x IN s ==> g x = f x))`, REPEAT STRIP_TAC THEN EQ_TAC THEN STRIP_TAC THENL [MATCH_MP_TAC BORSUK_HOMOTOPY_EXTENSION THEN ASM_REWRITE_TAC[SUBTOPOLOGY_UNIV; GSYM CLOSED_IN] THEN ONCE_REWRITE_TAC[HOMOTOPIC_WITH_SYM] THEN EXISTS_TAC `(\x. c):real^M->real^N` THEN ASM_REWRITE_TAC[CLOSED_UNIV; CONTINUOUS_ON_CONST] THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_SUBSET) THEN ASM SET_TAC[]; MP_TAC(ISPECL [`g:real^M->real^N`; `(:real^M)`; `t:real^N->bool`] NULLHOMOTOPIC_FROM_CONTRACTIBLE) THEN ASM_REWRITE_TAC[CONTRACTIBLE_UNIV] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^N` THEN DISCH_TAC THEN MATCH_MP_TAC HOMOTOPIC_WITH_EQ THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN MAP_EVERY EXISTS_TAC [`g:real^M->real^N`; `(\x. c):real^M->real^N`] THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC HOMOTOPIC_WITH_SUBSET_LEFT THEN EXISTS_TAC `(:real^M)` THEN ASM_REWRITE_TAC[SUBSET_UNIV]]);; let NULLHOMOTOPIC_INTO_RELATIVE_FRONTIER_EXTENSION = prove (`!f:real^M->real^N s t. closed s /\ f continuous_on s /\ ~(s = {}) /\ IMAGE f s SUBSET relative_frontier t /\ convex t /\ bounded t ==> ((?c. homotopic_with (\x. T) (subtopology euclidean s, subtopology euclidean (relative_frontier t)) f (\x. c)) <=> (?g. g continuous_on (:real^M) /\ IMAGE g (:real^M) SUBSET relative_frontier t /\ !x. x IN s ==> g x = f x))`, REPEAT STRIP_TAC THEN MATCH_MP_TAC NULLHOMOTOPIC_INTO_ANR_EXTENSION THEN MP_TAC(ISPEC `t:real^N->bool` ANR_RELATIVE_FRONTIER_CONVEX) THEN ASM_REWRITE_TAC[]);; let NULLHOMOTOPIC_INTO_SPHERE_EXTENSION = prove (`!f:real^M->real^N s a r. closed s /\ f continuous_on s /\ ~(s = {}) /\ IMAGE f s SUBSET sphere(a,r) ==> ((?c. homotopic_with (\x. T) (subtopology euclidean s, subtopology euclidean (sphere(a,r))) f (\x. c)) <=> (?g. g continuous_on (:real^M) /\ IMAGE g (:real^M) SUBSET sphere(a,r) /\ !x. x IN s ==> g x = f x))`, REPEAT GEN_TAC THEN MP_TAC(ISPECL [`a:real^N`; `r:real`] RELATIVE_FRONTIER_CBALL) THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL [ASM_SIMP_TAC[SPHERE_SING] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC(TAUT `p /\ q ==> (p <=> q)`) THEN CONJ_TAC THENL [EXISTS_TAC `a:real^N` THEN SIMP_TAC[HOMOTOPIC_WITH_EUCLIDEAN_ALT; PCROSS] THEN EXISTS_TAC `\y:real^(1,M)finite_sum. (a:real^N)`; EXISTS_TAC `(\x. a):real^M->real^N`] THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN ASM SET_TAC[]; DISCH_THEN(SUBST1_TAC o SYM) THEN STRIP_TAC THEN MATCH_MP_TAC NULLHOMOTOPIC_INTO_RELATIVE_FRONTIER_EXTENSION THEN ASM_REWRITE_TAC[CONVEX_CBALL; BOUNDED_CBALL]]);; let ABSOLUTE_RETRACT_CONTRACTIBLE_ANR = prove (`!s u:real^N->bool. closed_in (subtopology euclidean u) s /\ contractible s /\ ~(s = {}) /\ ANR s ==> s retract_of u`, REPEAT STRIP_TAC THEN MATCH_MP_TAC AR_IMP_RETRACT THEN ASM_SIMP_TAC[AR_ANR]);; (* ------------------------------------------------------------------------- *) (* More homotopy extension results and relations to components. *) (* ------------------------------------------------------------------------- *) let HOMOTOPIC_ON_COMPONENTS = prove (`!s t f g:real^M->real^N. locally connected s /\ (!c. c IN components s ==> homotopic_with (\x. T) (subtopology euclidean c,subtopology euclidean t) f g) ==> homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean t) f g`, REPEAT STRIP_TAC THEN GEN_REWRITE_TAC (RATOR_CONV o LAND_CONV o LAND_CONV o RAND_CONV) [UNIONS_COMPONENTS] THEN MATCH_MP_TAC HOMOTOPIC_ON_CLOPEN_UNIONS THEN X_GEN_TAC `c:real^M->bool` THEN DISCH_TAC THEN ASM_SIMP_TAC[GSYM UNIONS_COMPONENTS] THEN ASM_MESON_TAC[CLOSED_IN_COMPONENT; OPEN_IN_COMPONENTS_LOCALLY_CONNECTED]);; let INESSENTIAL_ON_COMPONENTS = prove (`!f:real^M->real^N s t. locally connected s /\ path_connected t /\ (!c. c IN components s ==> ?a. homotopic_with (\x. T) (subtopology euclidean c,subtopology euclidean t) f (\x. a)) ==> ?a. homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean t) f (\x. a)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `components(s:real^M->bool) = {}` THENL [RULE_ASSUM_TAC(REWRITE_RULE[COMPONENTS_EQ_EMPTY]) THEN ASM_SIMP_TAC[HOMOTOPIC_ON_EMPTY; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY]; ALL_TAC] THEN SUBGOAL_THEN `?a:real^N. a IN t` MP_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_TAC `c:real^M->bool`) THEN FIRST_X_ASSUM(MP_TAC o SPEC `c:real^M->bool`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_SUBSET) THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_NONEMPTY) THEN SET_TAC[]; MATCH_MP_TAC MONO_EXISTS] THEN X_GEN_TAC `a:real^N` THEN STRIP_TAC THEN MATCH_MP_TAC HOMOTOPIC_ON_COMPONENTS THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `c:real^M->bool` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `c:real^M->bool`) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `b:real^N` THEN DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_SUBSET) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_WITH_TRANS) THEN REWRITE_TAC[HOMOTOPIC_CONSTANT_MAPS] THEN REWRITE_TAC[PATH_COMPONENT_OF_EUCLIDEAN] THEN DISJ2_TAC THEN FIRST_X_ASSUM (MATCH_MP_TAC o REWRITE_RULE[PATH_CONNECTED_IFF_PATH_COMPONENT]) THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_NONEMPTY) THEN ASM SET_TAC[]);; let HOMOTOPIC_NEIGHBOURHOOD_EXTENSION = prove (`!f g:real^M->real^N s t u. f continuous_on s /\ IMAGE f s SUBSET u /\ g continuous_on s /\ IMAGE g s SUBSET u /\ closed_in (subtopology euclidean s) t /\ ANR u /\ homotopic_with (\x. T) (subtopology euclidean t,subtopology euclidean u) f g ==> ?v. t SUBSET v /\ open_in (subtopology euclidean s) v /\ homotopic_with (\x. T) (subtopology euclidean v,subtopology euclidean u) f g`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMOTOPIC_WITH_EUCLIDEAN]) THEN DISCH_THEN(X_CHOOSE_THEN `h:real^(1,M)finite_sum->real^N` STRIP_ASSUME_TAC) THEN ABBREV_TAC `h' = \z. if fstcart z IN {vec 0} then f(sndcart z) else if fstcart z IN {vec 1} then g(sndcart z) else (h:real^(1,M)finite_sum->real^N) z` THEN MP_TAC(ISPECL [`h':real^(1,M)finite_sum->real^N`; `interval[vec 0:real^1,vec 1] PCROSS (s:real^M->bool)`; `{vec 0:real^1,vec 1} PCROSS (s:real^M->bool) UNION interval[vec 0,vec 1] PCROSS t`; `u:real^N->bool`] ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR) THEN ASM_SIMP_TAC[ENR_IMP_ANR] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [REWRITE_TAC[SET_RULE `{a,b} = {a} UNION {b}`] THEN REWRITE_TAC[PCROSS_UNION; UNION_ASSOC] THEN EXPAND_TAC "h'" THEN REPLICATE_TAC 2 (MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL THEN REPLICATE_TAC 2 (CONJ_TAC THENL [MATCH_MP_TAC CLOSED_IN_SUBSET_TRANS THEN EXISTS_TAC `interval[vec 0:real^1,vec 1] PCROSS (s:real^M->bool)` THEN REWRITE_TAC[SET_RULE `t UNION u SUBSET s UNION t UNION u`] THEN REWRITE_TAC[SUBSET_UNION; UNION_SUBSET; SUBSET_PCROSS] THEN REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; ENDS_IN_UNIT_INTERVAL] THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN TRY(MATCH_MP_TAC CLOSED_IN_UNION THEN CONJ_TAC) THEN MATCH_MP_TAC CLOSED_IN_PCROSS THEN ASM_REWRITE_TAC[CLOSED_IN_REFL] THEN MATCH_MP_TAC CLOSED_SUBSET THEN REWRITE_TAC[SING_SUBSET; ENDS_IN_UNIT_INTERVAL; CLOSED_SING]; ALL_TAC]) THEN REPEAT CONJ_TAC THENL [GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[LINEAR_SNDCART; LINEAR_CONTINUOUS_ON] THEN ASM_REWRITE_TAC[IMAGE_SNDCART_PCROSS; NOT_INSERT_EMPTY]; ASM_REWRITE_TAC[]; REWRITE_TAC[FORALL_PASTECART; IN_UNION; PASTECART_IN_PCROSS] THEN REWRITE_TAC[FSTCART_PASTECART; IN_SING; SNDCART_PASTECART] THEN MAP_EVERY X_GEN_TAC [`x:real^1`; `y:real^M`] THEN ASM_CASES_TAC `x:real^1 = vec 0` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[VEC_EQ; ARITH_EQ; ENDS_IN_UNIT_INTERVAL] THEN ASM_CASES_TAC `x:real^1 = vec 1` THEN ASM_REWRITE_TAC[]]); REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_PASTECART] THEN REWRITE_TAC[IN_UNION; PASTECART_IN_PCROSS; IN_SING; NOT_IN_EMPTY] THEN MAP_EVERY X_GEN_TAC [`x:real^1`; `y:real^M`] THEN REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN EXPAND_TAC "h'" THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; IN_SING] THEN REPEAT(COND_CASES_TAC THENL [ASM SET_TAC[]; ASM_REWRITE_TAC[]]) THEN STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE f s SUBSET u ==> b IN s ==> f b IN u`)) THEN ASM_REWRITE_TAC[PASTECART_IN_PCROSS]; MATCH_MP_TAC CLOSED_IN_UNION THEN CONJ_TAC THEN MATCH_MP_TAC CLOSED_IN_PCROSS THEN ASM_REWRITE_TAC[CLOSED_IN_REFL] THEN MATCH_MP_TAC CLOSED_SUBSET THEN REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; ENDS_IN_UNIT_INTERVAL] THEN SIMP_TAC[CLOSED_INSERT; CLOSED_EMPTY]]; REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`w:real^(1,M)finite_sum->bool`; `k:real^(1,M)finite_sum->real^N`] THEN STRIP_TAC] THEN MP_TAC(ISPECL [`interval[vec 0:real^1,vec 1]`; `t:real^M->bool`; `s:real^M->bool`; `w:real^(1,M)finite_sum->bool`] TUBE_LEMMA_GEN) THEN ASM_REWRITE_TAC[COMPACT_INTERVAL; UNIT_INTERVAL_NONEMPTY] THEN ANTS_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS] THEN X_GEN_TAC `t':real^M->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN SIMP_TAC[HOMOTOPIC_WITH_EUCLIDEAN_ALT] THEN EXISTS_TAC `k:real^(1,M)finite_sum->real^N` THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ALL_TAC] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN CONJ_TAC THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN FIRST_X_ASSUM(fun th -> W(MP_TAC o PART_MATCH (lhs o snd o dest_imp) th o lhs o snd)) THEN REWRITE_TAC[IN_UNION; PASTECART_IN_PCROSS; IN_INSERT] THEN (ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN SUBST1_TAC]) THEN EXPAND_TAC "h'" THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; IN_SING] THEN REWRITE_TAC[VEC_EQ; ARITH_EQ]);; let HOMOTOPIC_ON_COMPONENTS_EQ = prove (`!s t f g:real^M->real^N. (locally connected s \/ compact s /\ ANR t) ==> (homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean t) f g <=> f continuous_on s /\ IMAGE f s SUBSET t /\ g continuous_on s /\ IMAGE g s SUBSET t /\ !c. c IN components s ==> homotopic_with (\x. T) (subtopology euclidean c,subtopology euclidean t) f g)`, REPEAT GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[CONJ_ASSOC] THEN MATCH_MP_TAC(TAUT `(q ==> r) /\ (r ==> (q <=> s)) ==> (q <=> r /\ s)`) THEN CONJ_TAC THENL [MESON_TAC[HOMOTOPIC_WITH_IMP_CONTINUOUS; HOMOTOPIC_WITH_IMP_SUBSET]; ALL_TAC] THEN STRIP_TAC THEN EQ_TAC THENL [MESON_TAC[HOMOTOPIC_WITH_SUBSET_LEFT; IN_COMPONENTS_SUBSET]; ALL_TAC] THEN DISCH_TAC THEN SUBGOAL_THEN `!c. c IN components s ==> ?u. c SUBSET u /\ closed_in (subtopology euclidean s) u /\ open_in (subtopology euclidean s) u /\ homotopic_with (\x. T) (subtopology euclidean u,subtopology euclidean t) (f:real^M->real^N) g` MP_TAC THENL [X_GEN_TAC `c:real^M->bool` THEN DISCH_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN FIRST_X_ASSUM DISJ_CASES_TAC THENL [EXISTS_TAC `c:real^M->bool` THEN ASM_SIMP_TAC[CLOSED_IN_COMPONENT; SUBSET_REFL; OPEN_IN_COMPONENTS_LOCALLY_CONNECTED]; FIRST_X_ASSUM(MP_TAC o SPEC `c:real^M->bool`) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `g:real^M->real^N`; `s:real^M->bool`; `c:real^M->bool`; `t:real^N->bool`] HOMOTOPIC_NEIGHBOURHOOD_EXTENSION) THEN ASM_SIMP_TAC[CLOSED_IN_COMPONENT] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^M->bool` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN DISCH_THEN(X_CHOOSE_THEN `v:real^M->bool` (STRIP_ASSUME_TAC o GSYM)) THEN MP_TAC(ISPECL [`s:real^M->bool`; `c:real^M->bool`; `v:real^M->bool`] SURA_BURA_CLOPEN_SUBSET) THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED; CLOSED_IMP_LOCALLY_COMPACT] THEN ANTS_TAC THENL [CONJ_TAC THENL [ASM_MESON_TAC[COMPACT_COMPONENTS]; ASM SET_TAC[]]; MATCH_MP_TAC MONO_EXISTS] THEN X_GEN_TAC `k:real^M->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN CONJ_TAC THENL [MATCH_MP_TAC CLOSED_SUBSET THEN ASM_MESON_TAC[COMPACT_IMP_CLOSED; OPEN_IN_IMP_SUBSET]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMOTOPIC_WITH_SUBSET_LEFT)) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ASM SET_TAC[]]]; GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `k:(real^M->bool)->(real^M->bool)` THEN DISCH_TAC THEN SUBGOAL_THEN `s = UNIONS (IMAGE k (components(s:real^M->bool)))` (fun th -> SUBST1_TAC th THEN ASSUME_TAC(SYM th)) THENL [MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [GEN_REWRITE_TAC LAND_CONV [UNIONS_COMPONENTS] THEN MATCH_MP_TAC UNIONS_MONO THEN REWRITE_TAC[EXISTS_IN_IMAGE] THEN ASM_MESON_TAC[]; REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_IMAGE] THEN ASM_MESON_TAC[CLOSED_IN_IMP_SUBSET]]; MATCH_MP_TAC HOMOTOPIC_ON_CLOPEN_UNIONS THEN ASM_SIMP_TAC[FORALL_IN_IMAGE]]]);; let INESSENTIAL_ON_COMPONENTS_EQ = prove (`!s t f:real^M->real^N. (locally connected s \/ compact s /\ ANR t) /\ path_connected t ==> ((?a. homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean t) f (\x. a)) <=> f continuous_on s /\ IMAGE f s SUBSET t /\ !c. c IN components s ==> ?a. homotopic_with (\x. T) (subtopology euclidean c,subtopology euclidean t) f (\x. a))`, REPEAT GEN_TAC THEN REWRITE_TAC[CONJ_ASSOC] THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN MATCH_MP_TAC(TAUT `(q ==> r) /\ (r ==> (q <=> s)) ==> (q <=> r /\ s)`) THEN CONJ_TAC THENL [MESON_TAC[HOMOTOPIC_WITH_IMP_CONTINUOUS; HOMOTOPIC_WITH_IMP_SUBSET]; STRIP_TAC] THEN FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP HOMOTOPIC_ON_COMPONENTS_EQ th]) THEN ASM_REWRITE_TAC[CONTINUOUS_ON_CONST] THEN EQ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_SIMP_TAC[COMPONENTS_EMPTY; IMAGE_CLAUSES; NOT_IN_EMPTY; EMPTY_SUBSET] THEN DISCH_TAC THEN SUBGOAL_THEN `?c:real^M->bool. c IN components s` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[MEMBER_NOT_EMPTY; COMPONENTS_EQ_EMPTY]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o SPEC `c:real^M->bool`) THEN ANTS_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS] THEN X_GEN_TAC `a:real^N` THEN DISCH_THEN(ASSUME_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_SUBSET) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_NONEMPTY) THEN CONJ_TAC THENL [ASM SET_TAC[]; X_GEN_TAC `d:real^M->bool`] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `d:real^M->bool`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `b:real^N` MP_TAC) THEN DISCH_THEN(fun th -> ASSUME_TAC(MATCH_MP HOMOTOPIC_WITH_IMP_SUBSET th) THEN MP_TAC th) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_WITH_TRANS) THEN REWRITE_TAC[HOMOTOPIC_CONSTANT_MAPS] THEN REWRITE_TAC[PATH_COMPONENT_OF_EUCLIDEAN] THEN DISJ2_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[PATH_CONNECTED_IFF_PATH_COMPONENT]) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_NONEMPTY)) THEN ASM SET_TAC[]);; let COHOMOTOPICALLY_TRIVIAL_ON_COMPONENTS = prove (`!s:real^M->bool t:real^N->bool. (locally connected s \/ compact s /\ ANR t) ==> ((!f g. f continuous_on s /\ IMAGE f s SUBSET t /\ g continuous_on s /\ IMAGE g s SUBSET t ==> homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean t) f g) <=> (!c. c IN components s ==> (!f g. f continuous_on c /\ IMAGE f c SUBSET t /\ g continuous_on c /\ IMAGE g c SUBSET t ==> homotopic_with (\x. T) (subtopology euclidean c, subtopology euclidean t) f g)))`, REPEAT GEN_TAC THEN DISCH_TAC THEN EQ_TAC THEN REPEAT STRIP_TAC THENL [MP_TAC(ISPECL [`g:real^M->real^N`; `s:real^M->bool`; `c:real^M->bool`; `t:real^N->bool`] EXTENSION_FROM_COMPONENT) THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`; `c:real^M->bool`; `t:real^N->bool`] EXTENSION_FROM_COMPONENT) THEN ANTS_TAC THENL [ASM_MESON_TAC[ENR_IMP_ANR]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `f':real^M->real^N` STRIP_ASSUME_TAC) THEN ANTS_TAC THENL [ASM_MESON_TAC[ENR_IMP_ANR]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `g':real^M->real^N` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`f':real^M->real^N`; `g':real^M->real^N`]) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `c:real^M->bool` o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMOTOPIC_WITH_SUBSET_LEFT)) THEN ASM_SIMP_TAC[IN_COMPONENTS_SUBSET] THEN MATCH_MP_TAC (ONCE_REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_WITH_EQ) THEN ASM_SIMP_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY]; FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP HOMOTOPIC_ON_COMPONENTS_EQ th]) THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `c:real^M->bool` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `c:real^M->bool`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_SUBSET) 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[]]);; let COHOMOTOPICALLY_TRIVIAL_ON_COMPONENTS_NULL = prove (`!s:real^M->bool t:real^N->bool. (locally connected s \/ compact s /\ ANR t) /\ path_connected t ==> ((!f. f continuous_on s /\ IMAGE f s SUBSET t ==> ?a. homotopic_with (\x. T) (subtopology euclidean s, subtopology euclidean t) f (\x. a)) <=> (!c. c IN components s ==> (!f. f continuous_on c /\ IMAGE f c SUBSET t ==> ?a. homotopic_with (\x. T) (subtopology euclidean c, subtopology euclidean t) f (\x. a))))`, REPEAT GEN_TAC THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o MATCH_MP COHOMOTOPICALLY_TRIVIAL_ON_COMPONENTS) THEN ASM_SIMP_TAC[HOMOTOPIC_TRIVIALITY]);; let COHOMOTOPICALLY_TRIVIAL_1D = prove (`!f:real^M->real^N s t. f continuous_on s /\ IMAGE f s SUBSET t /\ ANR t /\ connected t /\ (dimindex(:M) = 1 \/ ?r:real^1->bool. s homeomorphic r) ==> ?a. homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean t) f (\x. a)`, REPEAT GEN_TAC THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN SUBGOAL_THEN `path_connected(t:real^N->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[ANR_IMP_LOCALLY_PATH_CONNECTED; PATH_CONNECTED_EQ_CONNECTED_LPC]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (MESON[] `p \/ q ==> (p ==> q) ==> q`)) THEN ANTS_TAC THENL [REWRITE_TAC[GSYM DIMINDEX_1; GSYM DIM_UNIV] THEN DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] (REWRITE_RULE[CONJ_ASSOC] HOMEOMORPHIC_SUBSPACES))) THEN REWRITE_TAC[SUBSPACE_UNIV; homeomorphic] THEN GEN_REWRITE_TAC RAND_CONV [SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^M->real^1` THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^1->real^M` THEN DISCH_TAC THEN EXISTS_TAC `IMAGE (f:real^M->real^1) s` THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]; DISCH_THEN(X_CHOOSE_TAC `r:real^1->bool`)] THEN SUBGOAL_THEN `!c. c IN components s ==> ?u. closed_in (subtopology euclidean s) u /\ open_in (subtopology euclidean s) u /\ c SUBSET u /\ ?a. homotopic_with (\x. T) (subtopology euclidean u,subtopology euclidean t) (f:real^M->real^N) (\x. a)` MP_TAC THENL [REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN MP_TAC(ISPECL [`f:real^M->real^N`; `c:real^M->bool`; `t:real^N->bool`] NULLHOMOTOPIC_FROM_CONTRACTIBLE) THEN ANTS_TAC THENL [CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ALL_TAC] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphic]) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`g:real^M->real^1`; `h:real^1->real^M`] THEN STRIP_TAC THEN SUBGOAL_THEN `contractible(IMAGE (g:real^M->real^1) c)` MP_TAC THENL [SIMP_TAC[GSYM IS_INTERVAL_CONTRACTIBLE_1; IS_INTERVAL_CONNECTED_1] THEN MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_CONNECTED) THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; homeomorphism]; MATCH_MP_TAC EQ_IMP THEN MATCH_MP_TAC HOMEOMORPHIC_CONTRACTIBLE_EQ THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN REWRITE_TAC[homeomorphic] THEN MAP_EVERY EXISTS_TAC [`g:real^M->real^1`; `h:real^1->real^M`] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHISM_OF_SUBSETS)) THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism]) THEN ASM SET_TAC[]]; DISCH_THEN(X_CHOOSE_TAC `a:real^N`)] THEN MP_TAC(ISPECL [`f:real^M->real^N`; `(\x. a):real^M->real^N`; `s:real^M->bool`; `c:real^M->bool`; `t:real^N->bool`] HOMOTOPIC_NEIGHBOURHOOD_EXTENSION) THEN ASM_REWRITE_TAC[CONTINUOUS_ON_CONST] THEN ASM_SIMP_TAC[CLOSED_IN_COMPONENT] THEN ANTS_TAC THENL [FIRST_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_SUBSET) THEN FIRST_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_NONEMPTY) THEN ASM SET_TAC[]; DISCH_THEN(X_CHOOSE_THEN `u:real^M->bool` STRIP_ASSUME_TAC)] THEN MP_TAC(ISPECL [`s:real^M->bool`; `c:real^M->bool`; `u:real^M->bool`] COMPONENT_INTERMEDIATE_CLOPEN) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `v:real^M->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `a:real^N` THEN ASM_MESON_TAC[HOMOTOPIC_WITH_SUBSET_LEFT]; GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `u:(real^M->bool)->real^M->bool` THEN STRIP_TAC THEN SUBGOAL_THEN `s = UNIONS (IMAGE (u:(real^M->bool)->real^M->bool) (components s))` (fun th -> SUBST1_TAC th THEN ASSUME_TAC (SYM th)) THENL [REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN CONJ_TAC THENL [GEN_REWRITE_TAC LAND_CONV [UNIONS_COMPONENTS] THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM IMAGE_ID] THEN MATCH_MP_TAC UNIONS_MONO_IMAGE THEN ASM_SIMP_TAC[]; REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_IMAGE] THEN ASM_MESON_TAC[CLOSED_IN_IMP_SUBSET]]; MATCH_MP_TAC INESSENTIAL_ON_CLOPEN_UNIONS THEN ASM_SIMP_TAC[FORALL_IN_IMAGE]]]);; (* ------------------------------------------------------------------------- *) (* A few simple lemmas about deformation retracts. *) (* ------------------------------------------------------------------------- *) let DEFORMATION_RETRACTION_COMPOSE = prove (`!s t u r1 r2:real^N->real^N. homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean s) (\x. x) r1 /\ retraction (s,t) r1 /\ homotopic_with (\x. T) (subtopology euclidean t,subtopology euclidean t) (\x. x) r2 /\ retraction (t,u) r2 ==> homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean s) (\x. x) (r2 o r1) /\ retraction (s,u) (r2 o r1)`, REPEAT STRIP_TAC THENL [ALL_TAC; ASM_MESON_TAC[RETRACTION_o]] THEN MATCH_MP_TAC HOMOTOPIC_WITH_TRANS THEN EXISTS_TAC `(\x. x) o (r1:real^N->real^N)` THEN CONJ_TAC THENL [ASM_REWRITE_TAC[o_DEF; ETA_AX]; ALL_TAC] THEN MATCH_MP_TAC HOMOTOPIC_COMPOSE_CONTINUOUS_RIGHT THEN EXISTS_TAC `t:real^N->bool` THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMOTOPIC_WITH_RESTRICT)); ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[retraction]) THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);; let DEFORMATION_RETRACT_TRANS = prove (`!s t u:real^N->bool. (?r. homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean s) (\x. x) r /\ retraction (s,t) r) /\ (?r. homotopic_with (\x. T) (subtopology euclidean t,subtopology euclidean t) (\x. x) r /\ retraction (t,u) r) ==> ?r. homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean s) (\x. x) r /\ retraction (s,u) r`, MESON_TAC[DEFORMATION_RETRACTION_COMPOSE]);; let DEFORMATION_RETRACT_IMP_HOMOTOPY_EQUIVALENT = prove (`!s t:real^N->bool. (?r. homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean s) (\x. x) r /\ retraction(s,t) r) ==> s homotopy_equivalent t`, REWRITE_TAC[GSYM I_DEF; GSYM RETRACTION_MAPS_EUCLIDEAN] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM HOMOTOPY_EQUIVALENT_SPACE_EUCLIDEAN] THEN MATCH_MP_TAC DEFORMATION_RETRACTION_IMP_HOMOTOPY_EQUIVALENT_SPACE THEN ASM_MESON_TAC[I_O_ID; HOMOTOPIC_WITH_SYM]);; let DEFORMATION_RETRACT = prove (`!s t:real^N->bool. (?r. homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean s) (\x. x) r /\ retraction(s,t) r) <=> t retract_of s /\ ?f. homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean s) (\x. x) f /\ IMAGE f s SUBSET t`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`subtopology euclidean (s:real^N->bool)`; `t:real^N->bool`] DEFORMATION_RETRACT_OF_SPACE) THEN REWRITE_TAC[RETRACT_OF_SPACE_EUCLIDEAN; SUBTOPOLOGY_SUBTOPOLOGY; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; RETRACTION_MAPS_EUCLIDEAN] THEN REWRITE_TAC[I_DEF] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN ASM_CASES_TAC `(t:real^N->bool) SUBSET s` THEN ASM_SIMP_TAC[retraction; SET_RULE `t SUBSET s ==> s INTER t = t`]);; let ANR_STRONG_DEFORMATION_RETRACTION = prove (`!s t:real^N->bool. ANR s /\ (?r. homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean s) (\x. x) r /\ retraction(s,t) r) ==> ?r. homotopic_with (\h. !x. x IN t ==> h x = x) (subtopology euclidean s,subtopology euclidean s) (\x. x) r /\ retraction(s,t) r`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMOTOPIC_WITH_EUCLIDEAN]) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `f:real^(1,N)finite_sum->real^N` THEN STRIP_TAC THEN ABBREV_TAC `g:real^(1,(1,N)finite_sum)finite_sum->real^N = \z. if fstcart(sndcart z) = vec 0 then (sndcart(sndcart z)) else if fstcart(sndcart z) = vec 1 then f(pastecart (vec 1 - fstcart z) (f(pastecart (vec 1) (sndcart(sndcart z))))) else f(pastecart (lift(drop(fstcart(sndcart z)) * (&1 - drop (fstcart z)))) (sndcart(sndcart z)))` THEN MP_TAC(ISPECL [`f:real^(1,N)finite_sum->real^N`; `\x. (g:real^(1,(1,N)finite_sum)finite_sum->real^N) (pastecart (vec 1) x)`; `{vec 0:real^1,vec 1} PCROSS (s:real^N->bool) UNION interval[vec 0:real^1,vec 1] PCROSS (t:real^N->bool)`; `interval[vec 0:real^1,vec 1] PCROSS (s:real^N->bool)`; `s:real^N->bool`] BORSUK_HOMOTOPY_EXTENSION) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC CLOSED_IN_UNION THEN CONJ_TAC THEN MATCH_MP_TAC CLOSED_IN_PCROSS THEN REWRITE_TAC[CLOSED_IN_REFL] THENL [ALL_TAC; ASM_MESON_TAC[CLOSED_IN_RETRACT; retract_of]] THEN ONCE_REWRITE_TAC[SET_RULE `{a,b} = {a} UNION {b}`] THEN MATCH_MP_TAC CLOSED_IN_UNION THEN REWRITE_TAC[CLOSED_IN_SING; ENDS_IN_UNIT_INTERVAL]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhand o rand) HOMOTOPIC_WITH_EUCLIDEAN_ALT o snd) THEN ANTS_TAC THENL [FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retraction]) THEN SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN EXISTS_TAC `g:real^(1,(1,N)finite_sum)finite_sum->real^N` THEN EXPAND_TAC "g" THEN REWRITE_TAC[FORALL_PASTECART; FSTCART_PASTECART; SNDCART_PASTECART] THEN ASM_REWRITE_TAC[DROP_VEC; REAL_SUB_RZERO; REAL_MUL_RID; LIFT_DROP; VECTOR_SUB_RZERO; PASTECART_FST_SND; CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[IN_UNION; PASTECART_IN_PCROSS; IN_INSERT; NOT_IN_EMPTY] THEN REPEAT GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retraction]) THEN SET_TAC[]] THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN EXPAND_TAC "g" THEN REWRITE_TAC[FORALL_PASTECART; FSTCART_PASTECART; SNDCART_PASTECART] THEN ASM_REWRITE_TAC[DROP_VEC; REAL_SUB_RZERO; REAL_MUL_RID; LIFT_DROP; VECTOR_SUB_RZERO; PASTECART_FST_SND; CONJ_ASSOC; PASTECART_IN_PCROSS; IN_UNION; IN_INSERT; NOT_IN_EMPTY] THEN MAP_EVERY X_GEN_TAC [`u:real^1`; `v:real^1`; `y:real^N`] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL [FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retraction]) THEN SET_TAC[]; ALL_TAC] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE f s SUBSET t ==> x IN s ==> f x IN t`)) THEN ASM_REWRITE_TAC[PASTECART_IN_PCROSS] THEN (CONJ_TAC THENL [ALL_TAC; FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retraction]) THEN ASM SET_TAC[]]) THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; DROP_VEC]) THEN ASM_REWRITE_TAC[IN_INTERVAL_1; LIFT_DROP; DROP_VEC; DROP_SUB; REAL_SUB_LE; REAL_ARITH `&1 - x <= &1 <=> &0 <= x`] THEN ASM_SIMP_TAC[REAL_LE_MUL; REAL_SUB_LE] THEN GEN_REWRITE_TAC RAND_CONV [GSYM REAL_MUL_LID] THEN MATCH_MP_TAC REAL_LE_MUL2 THEN ASM_REAL_ARITH_TAC] THEN EXPAND_TAC "g" THEN REWRITE_TAC[MESON[] `(if p then x else if q then y else r) = (if p \/ q then if p then x else y else r)`] THEN REWRITE_TAC[PCROSS_UNION] THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[CLOSED_IN_CLOSED] THEN EXISTS_TAC `(:real^1) PCROSS {vec 0:real^1,vec 1} PCROSS (:real^N)` THEN SIMP_TAC[CLOSED_PCROSS_EQ; CLOSED_UNIV; CLOSED_INSERT; CLOSED_EMPTY] THEN REWRITE_TAC[FORALL_PASTECART; PASTECART_IN_PCROSS; IN_UNION; IN_INTER; EXTENSION; IN_UNIV; IN_INSERT; NOT_IN_EMPTY] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retraction]) THEN DISCH_THEN(MP_TAC o CONJUNCT1 o REWRITE_RULE[SUBSET]) THEN MESON_TAC[ENDS_IN_UNIT_INTERVAL]; SUBGOAL_THEN `closed_in (subtopology euclidean s) (t:real^N->bool)` MP_TAC THENL [ASM_MESON_TAC[CLOSED_IN_RETRACT; retract_of]; ALL_TAC] THEN REWRITE_TAC[CLOSED_IN_CLOSED] THEN DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(:real^1) PCROSS (:real^1) PCROSS (c:real^N->bool)` THEN ASM_REWRITE_TAC[CLOSED_PCROSS_EQ; CLOSED_UNIV] THEN REWRITE_TAC[FORALL_PASTECART; PASTECART_IN_PCROSS; IN_UNION; IN_INTER; EXTENSION; IN_UNIV; IN_INSERT; NOT_IN_EMPTY] THEN MESON_TAC[ENDS_IN_UNIT_INTERVAL]; ONCE_REWRITE_TAC[SET_RULE `{a,b} = {a} UNION {b}`] THEN REWRITE_TAC[PCROSS_UNION] THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[CLOSED_IN_CLOSED] THEN EXISTS_TAC `(:real^1) PCROSS {vec 0:real^1} PCROSS (:real^N)` THEN ASM_REWRITE_TAC[CLOSED_PCROSS_EQ; CLOSED_UNIV] THEN REWRITE_TAC[FORALL_PASTECART; PASTECART_IN_PCROSS; IN_UNION; IN_INTER; EXTENSION; IN_UNIV; IN_INSERT; NOT_IN_EMPTY; CLOSED_SING] THEN MESON_TAC[ENDS_IN_UNIT_INTERVAL]; REWRITE_TAC[CLOSED_IN_CLOSED] THEN EXISTS_TAC `(:real^1) PCROSS {vec 1:real^1} PCROSS (:real^N)` THEN ASM_REWRITE_TAC[CLOSED_PCROSS_EQ; CLOSED_UNIV] THEN REWRITE_TAC[FORALL_PASTECART; PASTECART_IN_PCROSS; IN_UNION; IN_INTER; EXTENSION; IN_UNIV; IN_INSERT; NOT_IN_EMPTY; CLOSED_SING] THEN MESON_TAC[ENDS_IN_UNIT_INTERVAL]; SIMP_TAC[CONTINUOUS_ON_SNDCART; LINEAR_CONTINUOUS_ON; LINEAR_SNDCART]; ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_PASTECART THEN SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; LINEAR_CONTINUOUS_ON; LINEAR_FSTCART] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_SNDCART; LINEAR_CONTINUOUS_ON; LINEAR_SNDCART] THEN FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [retraction]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; PASTECART_IN_PCROSS; FORALL_PASTECART; SNDCART_PASTECART] THEN SIMP_TAC[]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; PASTECART_IN_PCROSS; FORALL_PASTECART; SNDCART_PASTECART; FSTCART_PASTECART] THEN SIMP_TAC[IN_INTERVAL_1; DROP_SUB; DROP_VEC] THEN SIMP_TAC[REAL_ARITH `&1 - x <= &1 <=> &0 <= x`; REAL_SUB_LE] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retraction]) THEN SET_TAC[]]; REWRITE_TAC[FORALL_PASTECART; PASTECART_IN_PCROSS; FSTCART_PASTECART; SNDCART_PASTECART; IN_SING] THEN MAP_EVERY X_GEN_TAC [`u:real^1`; `v:real^1`; `y:real^N`] THEN ASM_CASES_TAC `v:real^1 = vec 0` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[VEC_EQ; ARITH_EQ]]; GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_PASTECART THEN SIMP_TAC[CONTINUOUS_ON_SNDCART; LINEAR_CONTINUOUS_ON; LINEAR_SNDCART] THEN REWRITE_TAC[LIFT_CMUL] THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN SIMP_TAC[o_DEF; LIFT_DROP; CONTINUOUS_ON_FSTCART; LINEAR_CONTINUOUS_ON; LINEAR_SNDCART] THEN SIMP_TAC[LIFT_SUB; LIFT_DROP; CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; LINEAR_CONTINUOUS_ON; LINEAR_FSTCART]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; PASTECART_IN_PCROSS; FORALL_PASTECART; SNDCART_PASTECART; FSTCART_PASTECART] THEN REWRITE_TAC[IN_INTERVAL_1; DROP_SUB; DROP_VEC; LIFT_DROP] THEN SIMP_TAC[REAL_LE_MUL; REAL_SUB_LE] THEN REPEAT STRIP_TAC THENL [GEN_REWRITE_TAC RAND_CONV [GSYM REAL_MUL_LID] THEN MATCH_MP_TAC REAL_LE_MUL2 THEN ASM_REAL_ARITH_TAC; FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retraction]) THEN ASM SET_TAC[]]]; REWRITE_TAC[FORALL_PASTECART; PASTECART_IN_PCROSS; FSTCART_PASTECART; SNDCART_PASTECART; IN_SING] THEN MAP_EVERY X_GEN_TAC [`u:real^1`; `v:real^1`; `y:real^N`] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[IN_INSERT; LIFT_DROP; REAL_MUL_LZERO; DROP_VEC; LIFT_NUM] THEN ASM_CASES_TAC `v:real^1 = vec 1` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN ASM_REWRITE_TAC[DROP_VEC; REAL_MUL_LID; LIFT_SUB; LIFT_NUM; LIFT_DROP] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retraction]) THEN ASM SET_TAC[]]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `h:real^(1,N)finite_sum->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(h:real^(1,N)finite_sum->real^N) o pastecart (vec 1)` THEN CONJ_TAC THENL [W(MP_TAC o PART_MATCH (lhand o rand) HOMOTOPIC_WITH_EUCLIDEAN_ALT o snd) THEN ANTS_TAC THENL [FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retraction]) THEN SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN EXISTS_TAC `h:real^(1,N)finite_sum->real^N` THEN ASM_SIMP_TAC[IN_UNION; PASTECART_IN_PCROSS; IN_INSERT; o_THM] THEN EXPAND_TAC "g" THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN ASM_REWRITE_TAC[DROP_VEC; VECTOR_SUB_REFL; REAL_SUB_REFL; REAL_MUL_RZERO; LIFT_NUM] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retraction]) THEN SET_TAC[]; REWRITE_TAC[retraction; o_THM] THEN REPEAT CONJ_TAC THENL [FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retraction]) THEN SET_TAC[]; MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THENL [GEN_REWRITE_TAC LAND_CONV [GSYM ETA_AX] THEN SIMP_TAC[CONTINUOUS_ON_PASTECART; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; PASTECART_IN_PCROSS] THEN SIMP_TAC[ENDS_IN_UNIT_INTERVAL]]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; o_THM]; ALL_TAC] THEN ASM_SIMP_TAC[IN_UNION; IN_INSERT; PASTECART_IN_PCROSS; ENDS_IN_UNIT_INTERVAL] THEN EXPAND_TAC "g" THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; VEC_EQ; ARITH_EQ] THEN ASM_REWRITE_TAC[VECTOR_SUB_REFL] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retraction]) THEN SET_TAC[]]);; let DEFORMATION_RETRACT_OF_CONTRACTIBLE = prove (`!s t:real^N->bool. contractible s /\ t retract_of s ==> ?r. homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean s) (\x. x) r /\ retraction(s,t) r`, REPEAT GEN_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_SIMP_TAC[RETRACT_OF_EMPTY; HOMOTOPIC_ON_EMPTY; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THENL [MESON_TAC[RETRACTION_REFL]; ALL_TAC] THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[DEFORMATION_RETRACT] THEN SUBGOAL_THEN `?a:real^N. a IN t` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[MEMBER_NOT_EMPTY; AR_ANR]; ALL_TAC] THEN EXISTS_TAC `(\x. a):real^N->real^N` THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [contractible]) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `b:real^N` THEN DISCH_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_SUBSET) THEN SUBGOAL_THEN `(a:real^N) IN s` ASSUME_TAC THENL [ASM_MESON_TAC[RETRACT_OF_IMP_SUBSET; SUBSET]; ALL_TAC] THEN SUBGOAL_THEN `(b:real^N) IN s` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMOTOPIC_WITH_TRANS)) THEN REWRITE_TAC[HOMOTOPIC_CONSTANT_MAPS; PATH_COMPONENT_OF_EUCLIDEAN] THEN ASM_MESON_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT; CONTRACTIBLE_IMP_PATH_CONNECTED]);; let AR_DEFORMATION_RETRACT_OF_CONTRACTIBLE = prove (`!s t:real^N->bool. contractible s /\ AR t /\ closed_in (subtopology euclidean s) t ==> ?r. homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean s) (\x. x) r /\ retraction(s,t) r`, MESON_TAC[DEFORMATION_RETRACT_OF_CONTRACTIBLE; AR_IMP_RETRACT]);; let DEFORMATION_RETRACT_OF_CONTRACTIBLE_SING = prove (`!s a:real^N. contractible s /\ a IN s ==> ?r. homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean s) (\x. x) r /\ retraction(s,{a}) r`, REPEAT STRIP_TAC THEN MATCH_MP_TAC AR_DEFORMATION_RETRACT_OF_CONTRACTIBLE THEN ASM_REWRITE_TAC[CLOSED_IN_SING; AR_SING]);; let STRONG_DEFORMATION_RETRACT_OF_AR = prove (`!s t:real^N->bool. AR s /\ t retract_of s ==> ?r. homotopic_with (\h. !x. x IN t ==> h x = x) (subtopology euclidean s,subtopology euclidean s) (\x. x) r /\ retraction(s,t) r`, REPEAT STRIP_TAC THEN MATCH_MP_TAC ANR_STRONG_DEFORMATION_RETRACTION THEN ASM_SIMP_TAC[AR_IMP_ANR] THEN MATCH_MP_TAC DEFORMATION_RETRACT_OF_CONTRACTIBLE THEN ASM_SIMP_TAC[AR_IMP_CONTRACTIBLE]);; let AR_STRONG_DEFORMATION_RETRACT_OF_AR = prove (`!s t:real^N->bool. AR s /\ AR t /\ closed_in (subtopology euclidean s) t ==> ?r. homotopic_with (\h. !x. x IN t ==> h x = x) (subtopology euclidean s,subtopology euclidean s) (\x. x) r /\ retraction(s,t) r`, REPEAT STRIP_TAC THEN MATCH_MP_TAC ANR_STRONG_DEFORMATION_RETRACTION THEN ASM_SIMP_TAC[AR_IMP_ANR] THEN MATCH_MP_TAC AR_DEFORMATION_RETRACT_OF_CONTRACTIBLE THEN ASM_SIMP_TAC[AR_IMP_CONTRACTIBLE]);; let SING_STRONG_DEFORMATION_RETRACT_OF_AR = prove (`!s a:real^N. AR s /\ a IN s ==> ?r. homotopic_with (\h. h a = a) (subtopology euclidean s,subtopology euclidean s) (\x. x) r /\ retraction(s,{a}) r`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `{a:real^N}`] AR_STRONG_DEFORMATION_RETRACT_OF_AR) THEN ASM_REWRITE_TAC[AR_SING; CLOSED_IN_SING] THEN REWRITE_TAC[FORALL_IN_INSERT; NOT_IN_EMPTY]);; let HOMOTOPY_EQUIVALENT_RELATIVE_FRONTIER_PUNCTURED_CONVEX = prove (`!s t a:real^N. convex s /\ bounded s /\ a IN relative_interior s /\ convex t /\ relative_frontier s SUBSET t /\ t SUBSET affine hull s ==> (relative_frontier s) homotopy_equivalent (t DELETE a)`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[HOMOTOPY_EQUIVALENT_SYM] THEN MATCH_MP_TAC DEFORMATION_RETRACT_IMP_HOMOTOPY_EQUIVALENT THEN ASM_MESON_TAC[RELATIVE_FRONTIER_DEFORMATION_RETRACT_OF_PUNCTURED_CONVEX]);; let HOMOTOPY_EQUIVALENT_RELATIVE_FRONTIER_PUNCTURED_AFFINE_HULL = prove (`!s a:real^N. convex s /\ bounded s /\ a IN relative_interior s ==> (relative_frontier s) homotopy_equivalent (affine hull s DELETE a)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMOTOPY_EQUIVALENT_RELATIVE_FRONTIER_PUNCTURED_CONVEX THEN ASM_SIMP_TAC[AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL; SUBSET_REFL] THEN REWRITE_TAC[relative_frontier] THEN MATCH_MP_TAC(SET_RULE `s SUBSET u ==> s DIFF t SUBSET u`) THEN REWRITE_TAC[CLOSURE_SUBSET_AFFINE_HULL]);; let HOMOTOPY_EQUIVALENT_PUNCTURED_UNIV_SPHERE = prove (`!c a:real^N r. &0 < r ==> ((:real^N) DELETE c) homotopy_equivalent sphere(a,r)`, REPEAT GEN_TAC THEN GEN_GEOM_ORIGIN_TAC `c:real^N` ["a"] THEN REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[HOMOTOPY_EQUIVALENT_SYM] THEN TRANS_TAC HOMOTOPY_EQUIVALENT_TRANS `sphere(vec 0:real^N,r)` THEN ASM_SIMP_TAC[HOMEOMORPHIC_SPHERES; HOMEOMORPHIC_IMP_HOMOTOPY_EQUIVALENT] THEN MP_TAC(ISPECL [`cball(vec 0:real^N,r)`; `vec 0:real^N`] HOMOTOPY_EQUIVALENT_RELATIVE_FRONTIER_PUNCTURED_AFFINE_HULL) THEN REWRITE_TAC[CONVEX_CBALL; BOUNDED_CBALL; RELATIVE_FRONTIER_CBALL; RELATIVE_INTERIOR_CBALL] THEN ASM_SIMP_TAC[CENTRE_IN_BALL; REAL_LT_IMP_NZ; AFFINE_HULL_NONEMPTY_INTERIOR; INTERIOR_CBALL; BALL_EQ_EMPTY; REAL_NOT_LE]);; (* ------------------------------------------------------------------------- *) (* Preservation of fixpoints under (more general notion of) retraction. *) (* ------------------------------------------------------------------------- *) let INVERTIBLE_FIXPOINT_PROPERTY = prove (`!s:real^M->bool t:real^N->bool i r. i continuous_on t /\ IMAGE i t SUBSET s /\ r continuous_on s /\ IMAGE r s SUBSET t /\ (!y. y IN t ==> (r(i(y)) = y)) ==> (!f. f continuous_on s /\ IMAGE f s SUBSET s ==> ?x. x IN s /\ (f x = x)) ==> !g. g continuous_on t /\ IMAGE g t SUBSET t ==> ?y. y IN t /\ (g y = y)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `(i:real^N->real^M) o (g:real^N->real^N) o (r:real^M->real^N)`) THEN ANTS_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; CONTINUOUS_ON_COMPOSE; IMAGE_SUBSET; SUBSET_TRANS; IMAGE_o]; RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; FORALL_IN_IMAGE]) THEN REWRITE_TAC[o_THM] THEN ASM_MESON_TAC[]]);; let HOMEOMORPHIC_FIXPOINT_PROPERTY = prove (`!s t. s homeomorphic t ==> ((!f. f continuous_on s /\ IMAGE f s SUBSET s ==> ?x. x IN s /\ (f x = x)) <=> (!g. g continuous_on t /\ IMAGE g t SUBSET t ==> ?y. y IN t /\ (g y = y)))`, REWRITE_TAC[homeomorphic; homeomorphism] THEN REPEAT STRIP_TAC THEN EQ_TAC THEN MATCH_MP_TAC INVERTIBLE_FIXPOINT_PROPERTY THEN ASM_MESON_TAC[SUBSET_REFL]);; let RETRACT_FIXPOINT_PROPERTY = prove (`!s t:real^N->bool. t retract_of s /\ (!f. f continuous_on s /\ IMAGE f s SUBSET s ==> ?x. x IN s /\ (f x = x)) ==> !g. g continuous_on t /\ IMAGE g t SUBSET t ==> ?y. y IN t /\ (g y = y)`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN MATCH_MP_TAC INVERTIBLE_FIXPOINT_PROPERTY THEN EXISTS_TAC `\x:real^N. x` THEN REWRITE_TAC[CONTINUOUS_ON_ID] THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[retract_of] THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN REWRITE_TAC[retraction] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE]);; let FRONTIER_SUBSET_RETRACTION = prove (`!s:real^N->bool t r. bounded s /\ frontier s SUBSET t /\ r continuous_on (closure s) /\ IMAGE r s SUBSET t /\ (!x. x IN t ==> r x = x) ==> s SUBSET t`, ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[SET_RULE `~(s SUBSET t) <=> ?x. x IN s /\ ~(x IN t)`] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM] THEN REPLICATE_TAC 3 GEN_TAC THEN X_GEN_TAC `a:real^N` THEN REPEAT STRIP_TAC THEN ABBREV_TAC `q = \z:real^N. if z IN closure s then r(z) else z` THEN SUBGOAL_THEN `(q:real^N->real^N) continuous_on closure(s) UNION closure((:real^N) DIFF s)` MP_TAC THENL [EXPAND_TAC "q" THEN MATCH_MP_TAC CONTINUOUS_ON_CASES THEN ASM_REWRITE_TAC[CLOSED_CLOSURE; CONTINUOUS_ON_ID] THEN REWRITE_TAC[TAUT `p /\ ~p <=> F`] THEN X_GEN_TAC `z:real^N` THEN REWRITE_TAC[CLOSURE_COMPLEMENT; IN_DIFF; IN_UNIV] THEN STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; frontier; IN_DIFF]) THEN ASM_MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `closure(s) UNION closure((:real^N) DIFF s) = (:real^N)` SUBST1_TAC THENL [MATCH_MP_TAC(SET_RULE `s SUBSET closure s /\ t SUBSET closure t /\ s UNION t = UNIV ==> closure s UNION closure t = UNIV`) THEN REWRITE_TAC[CLOSURE_SUBSET] THEN SET_TAC[]; DISCH_TAC] THEN FIRST_ASSUM(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC o SPEC `a:real^N` o MATCH_MP BOUNDED_SUBSET_BALL o MATCH_MP BOUNDED_CLOSURE) THEN SUBGOAL_THEN `!x. ~((q:real^N->real^N) x = a)` ASSUME_TAC THENL [GEN_TAC THEN EXPAND_TAC "q" THEN COND_CASES_TAC THENL [ASM_CASES_TAC `(x:real^N) IN s` THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(x:real^N) IN t` (fun th -> ASM_MESON_TAC[th]) THEN UNDISCH_TAC `frontier(s:real^N->bool) SUBSET t` THEN REWRITE_TAC[SUBSET; frontier; IN_DIFF] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_MESON_TAC[SUBSET; INTERIOR_SUBSET]; ASM_MESON_TAC[SUBSET; INTERIOR_SUBSET; CLOSURE_SUBSET]]; ALL_TAC] THEN MP_TAC(ISPECL [`a:real^N`; `B:real`] NO_RETRACTION_CBALL) THEN ASM_REWRITE_TAC[retract_of; GSYM FRONTIER_CBALL] THEN EXISTS_TAC `(\y. a + B / norm(y - a) % (y - a)) o (q:real^N->real^N)` THEN REWRITE_TAC[retraction; FRONTIER_SUBSET_EQ; CLOSED_CBALL] THEN REWRITE_TAC[FRONTIER_CBALL; SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN REWRITE_TAC[IN_SPHERE; DIST_0] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV]; ALL_TAC] THEN MATCH_MP_TAC CONTINUOUS_ON_ADD THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST] THEN REWRITE_TAC[o_DEF; real_div; LIFT_CMUL] THEN MATCH_MP_TAC CONTINUOUS_ON_CMUL THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN ASM_REWRITE_TAC[FORALL_IN_IMAGE; NORM_EQ_0; VECTOR_SUB_EQ] THEN SUBGOAL_THEN `(\x:real^N. lift(norm(x - a))) = (lift o norm) o (\x. x - a)` SUBST1_TAC THENL [REWRITE_TAC[FUN_EQ_THM; o_THM]; ALL_TAC] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST] THEN REWRITE_TAC[CONTINUOUS_ON_LIFT_NORM]; REWRITE_TAC[o_THM; NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM; NORM_ARITH `dist(a,a + b) = norm b`] THEN ASM_SIMP_TAC[REAL_DIV_RMUL; VECTOR_SUB_EQ; NORM_EQ_0] THEN ASM_REAL_ARITH_TAC; X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[o_THM] THEN EXPAND_TAC "q" THEN REWRITE_TAC[] THEN COND_CASES_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; IN_BALL]) THEN ASM_MESON_TAC[REAL_LT_REFL]; REWRITE_TAC[NORM_ARITH `norm(x - a) = dist(a,x)`] THEN ASM_SIMP_TAC[REAL_DIV_REFL; REAL_LT_IMP_NZ; VECTOR_MUL_LID] THEN VECTOR_ARITH_TAC]]);; let NO_RETRACTION_FRONTIER_BOUNDED = prove (`!s:real^N->bool. bounded s /\ ~(interior s = {}) ==> ~((frontier s) retract_of s)`, GEN_TAC THEN STRIP_TAC THEN REWRITE_TAC[retract_of; retraction] THEN REWRITE_TAC[FRONTIER_SUBSET_EQ] THEN DISCH_THEN(X_CHOOSE_THEN `r:real^N->real^N` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`s:real^N->bool`; `frontier s:real^N->bool`; `r:real^N->real^N`] FRONTIER_SUBSET_RETRACTION) THEN ASM_SIMP_TAC[CLOSURE_CLOSED; SUBSET_REFL] THEN REWRITE_TAC[frontier] THEN MP_TAC(ISPEC `s:real^N->bool` INTERIOR_SUBSET) THEN ASM SET_TAC[]);; let COMPACT_SUBSET_FRONTIER_RETRACTION = prove (`!f:real^N->real^N s. compact s /\ f continuous_on s /\ (!x. x IN frontier s ==> f x = x) ==> s SUBSET IMAGE f s`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s UNION (IMAGE f s):real^N->bool`; `vec 0:real^N`] BOUNDED_SUBSET_BALL) THEN ASM_SIMP_TAC[BOUNDED_UNION; COMPACT_IMP_BOUNDED; COMPACT_CONTINUOUS_IMAGE; UNION_SUBSET] THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN ABBREV_TAC `g = \x:real^N. if x IN s then f(x) else x` THEN SUBGOAL_THEN `(g:real^N->real^N) continuous_on (:real^N)` ASSUME_TAC THENL [SUBGOAL_THEN `(:real^N) = s UNION closure((:real^N) DIFF s)` SUBST1_TAC THENL [MATCH_MP_TAC(SET_RULE `UNIV DIFF s SUBSET t ==> UNIV = s UNION t`) THEN REWRITE_TAC[CLOSURE_SUBSET]; ALL_TAC] THEN EXPAND_TAC "g" THEN MATCH_MP_TAC CONTINUOUS_ON_CASES THEN ASM_SIMP_TAC[CLOSED_CLOSURE; CONTINUOUS_ON_ID; COMPACT_IMP_CLOSED] THEN REWRITE_TAC[CLOSURE_COMPLEMENT; IN_DIFF; IN_UNIV] THEN REWRITE_TAC[TAUT `p /\ ~p <=> F`] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[frontier; IN_DIFF] THEN ASM_SIMP_TAC[CLOSURE_CLOSED; COMPACT_IMP_CLOSED]; ALL_TAC] THEN REWRITE_TAC[SUBSET] THEN X_GEN_TAC `p:real^N` THEN DISCH_TAC THEN SUBGOAL_THEN `?h:real^N->real^N. retraction (UNIV DELETE p,sphere(vec 0,r)) h` STRIP_ASSUME_TAC THENL [REWRITE_TAC[GSYM retract_of] THEN MATCH_MP_TAC SPHERE_RETRACT_OF_PUNCTURED_UNIVERSE_GEN THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`vec 0:real^N`; `r:real`] NO_RETRACTION_CBALL) THEN ASM_REWRITE_TAC[retract_of; NOT_EXISTS_THM] THEN DISCH_THEN(MP_TAC o SPEC `(h:real^N->real^N) o (g:real^N->real^N)`) THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN REWRITE_TAC[] THEN REWRITE_TAC[retraction] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retraction]) THEN SIMP_TAC[SUBSET; IN_SPHERE; IN_CBALL; REAL_EQ_IMP_LE] THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_DELETE; IN_UNIV; o_THM] THEN STRIP_TAC THEN SUBGOAL_THEN `!x. x IN cball (vec 0,r) ==> ~((g:real^N->real^N) x = p)` ASSUME_TAC THENL [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN EXPAND_TAC "g" THEN COND_CASES_TAC THEN ASM SET_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[] THEN REPEAT STRIP_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM_REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_UNIV; IN_DELETE]; SUBGOAL_THEN `(g:real^N->real^N) x = x` (fun th -> ASM_SIMP_TAC[th]) THEN EXPAND_TAC "g" THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[IN_BALL; REAL_LT_REFL; SUBSET]]);; let NOT_ABSOLUTE_RETRACT_COBOUNDED = prove (`!s. bounded s /\ ((:real^N) DIFF s) retract_of (:real^N) ==> s = {}`, GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> F) ==> s = {}`) THEN X_GEN_TAC `a:real^N` THEN POP_ASSUM MP_TAC THEN GEOM_ORIGIN_TAC `a:real^N` THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `vec 0:real^N` o MATCH_MP BOUNDED_SUBSET_BALL) THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o SPEC `vec 0:real^N` o MATCH_MP NO_RETRACTION_CBALL) THEN REWRITE_TAC[] THEN MATCH_MP_TAC RETRACT_OF_SUBSET THEN EXISTS_TAC `(:real^N)` THEN SIMP_TAC[SUBSET_UNIV; SPHERE_SUBSET_CBALL] THEN MATCH_MP_TAC RETRACT_OF_TRANS THEN EXISTS_TAC `(:real^N) DIFF s` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC RETRACT_OF_SUBSET THEN EXISTS_TAC `(:real^N) DELETE (vec 0)` THEN ASM_SIMP_TAC[SPHERE_RETRACT_OF_PUNCTURED_UNIVERSE] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN REWRITE_TAC[SUBSET; IN_BALL; IN_SPHERE; IN_DIFF; IN_UNIV] THEN MESON_TAC[REAL_LT_REFL]);; (* ------------------------------------------------------------------------- *) (* Bohl-type fixed point theorems. *) (* ------------------------------------------------------------------------- *) let BOHL = prove (`!f s a:real^N. f continuous_on s /\ convex s /\ compact s /\ a IN interior s ==> (?x. x IN s /\ f x = x) \/ (?x. x IN frontier s /\ x IN segment(a,f x))`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY; INTERIOR_EMPTY] THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `affine hull s:real^N->bool`; `a:real^N`] RELATIVE_FRONTIER_DEFORMATION_RETRACT_OF_PUNCTURED_CONVEX) THEN ASM_SIMP_TAC[AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL; COMPACT_IMP_BOUNDED] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP (SET_RULE `a IN s ==> ~(s = {})`)) THEN ASM_SIMP_TAC[RELATIVE_INTERIOR_NONEMPTY_INTERIOR; RELATIVE_FRONTIER_NONEMPTY_INTERIOR] THEN SIMP_TAC[SUBSET_REFL; frontier; CLOSURE_SUBSET_AFFINE_HULL; SET_RULE `s SUBSET u ==> s DIFF t SUBSET u`] THEN DISCH_THEN(X_CHOOSE_THEN `r:real^N->real^N` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retraction]) THEN ASM_SIMP_TAC[AFFINE_HULL_NONEMPTY_INTERIOR; GSYM frontier] THEN STRIP_TAC THEN MP_TAC(ISPECL [`(\x. if x IN s then x else r x) o (f:real^N->real^N)`; `s:real^N->bool`] BROUWER) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `IMAGE (f:real^N->real^N) s = s INTER IMAGE f s UNION ((:real^N) DIFF interior s) INTER IMAGE f s` SUBST1_TAC THENL [MP_TAC(ISPEC `s:real^N->bool` INTERIOR_SUBSET) THEN SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC CONTINUOUS_ON_CASES THEN ASM_SIMP_TAC[CLOSED_INTER_COMPACT; COMPACT_CONTINUOUS_IMAGE; COMPACT_IMP_CLOSED; GSYM OPEN_CLOSED; OPEN_INTERIOR] THEN REWRITE_TAC[CONTINUOUS_ON_ID] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[frontier] THEN MP_TAC(ISPEC `s:real^N->bool` CLOSURE_SUBSET) THEN ASM SET_TAC[]]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[o_DEF] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE r t SUBSET u ==> u SUBSET s /\ y IN t ==> r y IN s`)) THEN ASM_SIMP_TAC[frontier; CLOSURE_CLOSED; COMPACT_IMP_CLOSED] THEN MP_TAC(ISPEC `s:real^N->bool` INTERIOR_SUBSET) THEN ASM SET_TAC[]]; REWRITE_TAC[OR_EXISTS_THM; o_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:real^N` THEN COND_CASES_TAC THEN ASM_SIMP_TAC[] THEN ASM_CASES_TAC `f(x:real^N) = x` THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN SUBGOAL_THEN `~((f:real^N->real^N) x = a)` ASSUME_TAC THENL [MP_TAC(ISPEC `s:real^N->bool` INTERIOR_SUBSET) THEN ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [ASM SET_TAC[]; ASM_REWRITE_TAC[IN_SEGMENT]] THEN FIRST_X_ASSUM(X_CHOOSE_THEN `c:real` MP_TAC o SPEC `(f:real^N->real^N) x`) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN EXISTS_TAC `c:real` THEN ASM_REWRITE_TAC[VECTOR_ARITH `x:real^N = (&1 - c) % a + c % y <=> x - a = c % (y - a)`] THEN REWRITE_TAC[GSYM REAL_NOT_LE] THEN DISCH_TAC THEN UNDISCH_TAC `~((f:real^N->real^N) x IN s)` THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [convex]) THEN DISCH_THEN(MP_TAC o SPECL [`a:real^N`; `x:real^N`; `&1 - inv c`; `inv(c):real`]) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP(REWRITE_RULE[SUBSET] INTERIOR_SUBSET)) THEN ASM_SIMP_TAC[REAL_LE_INV_EQ; REAL_SUB_LE; REAL_LT_IMP_LE; REAL_INV_LE_1; REAL_ARITH `(&1 - u) + u = &1`] THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [MATCH_MP (VECTOR_ARITH `x - a:real^N = y ==> x = a + y`) th]) THEN REWRITE_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[REAL_MUL_LINV; REAL_ARITH `&1 <= c ==> ~(c = &0)`] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN CONV_TAC VECTOR_ARITH]);; let BOHL_ALT = prove (`!f s a. f continuous_on s /\ convex s /\ compact s /\ a IN interior s /\ IMAGE f s SUBSET (:real^N) DELETE a ==> ?x. x IN frontier s /\ a IN segment(x,f x)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`\x:real^N. x + (a - f(x))`; `s:real^N->bool`; `a:real^N`] BOHL) THEN ASM_SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN REWRITE_TAC[VECTOR_ARITH `x + a - y:real^N = x <=> y = a`] THEN DISCH_THEN(DISJ_CASES_THEN MP_TAC) THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[] THEN SIMP_TAC[IN_SEGMENT; VECTOR_ARITH `a:real^N = x + a - y <=> y = x`] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN REPEAT(MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[]) THEN CONV_TAC VECTOR_ARITH);; let BOHL_SIMPLE = prove (`!f:real^N->real^N s a. compact s /\ a IN s /\ f continuous_on s /\ IMAGE f s SUBSET (:real^N) DELETE a ==> ?x. x IN frontier s /\ ~(f x = x)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^N->real^N`; `s:real^N->bool`] COMPACT_SUBSET_FRONTIER_RETRACTION) THEN ASM SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* Some more theorems about connectivity of retract complements. *) (* ------------------------------------------------------------------------- *) let BOUNDED_COMPONENT_RETRACT_COMPLEMENT_MEETS = prove (`!s t c. closed s /\ s retract_of t /\ c IN components((:real^N) DIFF s) /\ bounded c ==> ~(c SUBSET t)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP RETRACT_OF_IMP_SUBSET) THEN SUBGOAL_THEN `frontier(c:real^N->bool) SUBSET s` ASSUME_TAC THENL [TRANS_TAC SUBSET_TRANS `frontier((:real^N) DIFF s)` THEN ASM_SIMP_TAC[FRONTIER_OF_COMPONENTS_SUBSET] THEN REWRITE_TAC[FRONTIER_COMPLEMENT] THEN ASM_SIMP_TAC[frontier; CLOSURE_CLOSED] THEN SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `closure(c:real^N->bool) SUBSET t` ASSUME_TAC THENL [REWRITE_TAC[CLOSURE_UNION_FRONTIER] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(c:real^N->bool) SUBSET s` ASSUME_TAC THENL [MATCH_MP_TAC FRONTIER_SUBSET_RETRACTION THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retract_of]) THEN REWRITE_TAC[retraction] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `r:real^N->real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ASM SET_TAC[]]; FIRST_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_NONEMPTY) THEN ASM SET_TAC[]]);; let COMPONENT_RETRACT_COMPLEMENT_MEETS = prove (`!s t c. closed s /\ s retract_of t /\ bounded t /\ c IN components((:real^N) DIFF s) ==> ~(c SUBSET t)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP RETRACT_OF_IMP_SUBSET) THEN ASM_CASES_TAC `bounded(c:real^N->bool)` THENL [ASM_MESON_TAC[BOUNDED_COMPONENT_RETRACT_COMPLEMENT_MEETS]; ASM_MESON_TAC[BOUNDED_SUBSET]]);; let FINITE_COMPLEMENT_ENR_COMPONENTS = prove (`!s. compact s /\ ENR s ==> FINITE(components((:real^N) DIFF s))`, GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [ASM_SIMP_TAC[DIFF_EMPTY] THEN MESON_TAC[COMPONENTS_EQ_SING; CONNECTED_UNIV; UNIV_NOT_EMPTY; FINITE_SING]; ALL_TAC] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_SIMP_TAC[ENR_BOUNDED; COMPACT_IMP_BOUNDED] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `!c. c IN components((:real^N) DIFF s) ==> ~(c SUBSET u)` ASSUME_TAC THENL [GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC COMPONENT_RETRACT_COMPLEMENT_MEETS THEN ASM_MESON_TAC[COMPACT_IMP_CLOSED]; ALL_TAC] THEN MP_TAC(ISPECL [`u:real^N->bool`; `vec 0:real^N`] BOUNDED_SUBSET_CBALL) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP RETRACT_OF_IMP_SUBSET) THEN MP_TAC(ISPECL [`cball(vec 0:real^N,r) DIFF u`; `(:real^N) DIFF s`] FINITE_COMPONENTS_MEETING_COMPACT_SUBSET) THEN ASM_SIMP_TAC[COMPACT_DIFF; COMPACT_CBALL; OPEN_IMP_LOCALLY_CONNECTED; GSYM closed; COMPACT_IMP_CLOSED] THEN ANTS_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC] THEN MATCH_MP_TAC(SET_RULE `(!c. c IN s ==> P c) ==> {c | c IN s /\ P c} = s`) THEN X_GEN_TAC `c:real^N->bool` THEN DISCH_TAC THEN SUBGOAL_THEN `~(c INTER frontier(u:real^N->bool) = {})` MP_TAC THENL [MATCH_MP_TAC CONNECTED_INTER_FRONTIER THEN CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]; ALL_TAC] THEN ASM_SIMP_TAC[SET_RULE `s DIFF t = {} <=> s SUBSET t`] THEN ONCE_REWRITE_TAC[INTER_COMM] THEN W(MP_TAC o PART_MATCH (rand o rand) OPEN_INTER_CLOSURE_EQ_EMPTY o rand o snd) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[CLOSURE_UNION_FRONTIER] THEN MATCH_MP_TAC(SET_RULE `~(t = {}) /\ t SUBSET u ==> ~(u INTER (s UNION t) = {})`) THEN ASM_REWRITE_TAC[FRONTIER_EQ_EMPTY; DE_MORGAN_THM; GSYM CONJ_ASSOC] THEN CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_NONEMPTY]; ALL_TAC] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN TRANS_TAC SUBSET_TRANS `frontier((:real^N) DIFF s)` THEN ASM_SIMP_TAC[FRONTIER_OF_COMPONENTS_SUBSET] THEN REWRITE_TAC[FRONTIER_COMPLEMENT] THEN ASM_SIMP_TAC[frontier; CLOSURE_CLOSED; COMPACT_IMP_CLOSED] THEN ASM SET_TAC[]; MATCH_MP_TAC(SET_RULE `s SUBSET t ==> ~(c INTER s = {}) ==> ~(c INTER t = {})`) THEN ASM_SIMP_TAC[frontier; INTERIOR_OPEN] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s DIFF u SUBSET t DIFF u`) THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN ASM_REWRITE_TAC[CLOSED_CBALL]]);; let FINITE_COMPLEMENT_ANR_COMPONENTS = prove (`!s. compact s /\ ANR s ==> FINITE(components((:real^N) DIFF s))`, MESON_TAC[FINITE_COMPLEMENT_ENR_COMPONENTS; ENR_ANR; COMPACT_IMP_CLOSED; CLOSED_IMP_LOCALLY_COMPACT]);; let CARD_LE_RETRACT_COMPLEMENT_COMPONENTS = prove (`!s t. compact s /\ s retract_of t /\ bounded t ==> components((:real^N) DIFF s) <=_c components((:real^N) DIFF t)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP RETRACT_OF_IMP_SUBSET) THEN MATCH_MP_TAC(ISPEC `SUBSET` CARD_LE_RELATIONAL_FULL) THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; MAP_EVERY X_GEN_TAC [`d:real^N->bool`; `c:real^N->bool`; `c':real^N->bool`] THEN STRIP_TAC THEN MP_TAC(ISPEC `(:real^N) DIFF s` COMPONENTS_EQ) THEN ASM_SIMP_TAC[] THEN ASM_CASES_TAC `d:real^N->bool = {}` THENL [ALL_TAC; ASM SET_TAC[]] THEN ASM_MESON_TAC[IN_COMPONENTS_NONEMPTY]] THEN X_GEN_TAC `u:real^N->bool` THEN STRIP_TAC THEN SUBGOAL_THEN `~((u:real^N->bool) SUBSET t)` MP_TAC THENL [MATCH_MP_TAC COMPONENT_RETRACT_COMPLEMENT_MEETS THEN ASM_MESON_TAC[COMPACT_EQ_BOUNDED_CLOSED]; ALL_TAC] THEN REWRITE_TAC[SET_RULE `~(s SUBSET t) <=> ?p. p IN s /\ ~(p IN t)`] THEN REWRITE_TAC[components; EXISTS_IN_GSPEC; IN_UNIV; IN_DIFF] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `p:real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `u = connected_component ((:real^N) DIFF s) p` SUBST_ALL_TAC THENL [MP_TAC(ISPECL [`(:real^N) DIFF s`; `u:real^N->bool`] COMPONENTS_EQ) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[components; FORALL_IN_GSPEC; IN_DIFF; IN_UNIV] THEN DISCH_THEN(MP_TAC o SPEC `p:real^N`) THEN ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `p:real^N` THEN ASM_REWRITE_TAC[IN_INTER] THEN REWRITE_TAC[IN] THEN REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ] THEN ASM SET_TAC[]; MATCH_MP_TAC CONNECTED_COMPONENT_MONO THEN ASM SET_TAC[]]);; let CONNECTED_RETRACT_COMPLEMENT = prove (`!s t. compact s /\ s retract_of t /\ bounded t /\ connected((:real^N) DIFF t) ==> connected((:real^N) DIFF s)`, REPEAT GEN_TAC THEN REWRITE_TAC[CONNECTED_EQ_COMPONENTS_SUBSET_SING_EXISTS] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(X_CHOOSE_TAC `u:real^N->bool`) THEN SUBGOAL_THEN `FINITE(components((:real^N) DIFF t))` ASSUME_TAC THENL [ASM_MESON_TAC[FINITE_SUBSET; FINITE_SING]; ALL_TAC] THEN MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`] CARD_LE_RETRACT_COMPLEMENT_COMPONENTS) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN SUBGOAL_THEN `FINITE(components((:real^N) DIFF s)) /\ CARD(components((:real^N) DIFF s)) <= CARD(components((:real^N) DIFF t))` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[CARD_LE_CARD_IMP; CARD_LE_FINITE]; ALL_TAC] THEN REWRITE_TAC[SET_RULE `s SUBSET {a} <=> s = {} \/ s = {a}`] THEN REWRITE_TAC[EXISTS_OR_THM] THEN REWRITE_TAC[GSYM HAS_SIZE_0; GSYM(HAS_SIZE_CONV `s HAS_SIZE 1`)] THEN ASM_REWRITE_TAC[HAS_SIZE; ARITH_RULE `n = 0 \/ n = 1 <=> n <= 1`] THEN TRANS_TAC LE_TRANS `CARD{u:real^N->bool}` THEN CONJ_TAC THENL [TRANS_TAC LE_TRANS `CARD(components((:real^N) DIFF t))` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CARD_SUBSET THEN ASM_REWRITE_TAC[FINITE_SING]; SIMP_TAC[CARD_CLAUSES; FINITE_EMPTY; NOT_IN_EMPTY] THEN ARITH_TAC]);; (* ------------------------------------------------------------------------- *) (* We also get fixpoint properties for suitable ANRs. *) (* ------------------------------------------------------------------------- *) let BROUWER_INESSENTIAL_ANR = prove (`!f:real^N->real^N s. compact s /\ ~(s = {}) /\ ANR s /\ f continuous_on s /\ IMAGE f s SUBSET s /\ (?a. homotopic_with (\x. T) (subtopology euclidean s,subtopology euclidean s) f (\x. a)) ==> ?x. x IN s /\ f x = x`, ONCE_REWRITE_TAC[HOMOTOPIC_WITH_SYM] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(X_CHOOSE_TAC `r:real` o SPEC `vec 0:real^N` o MATCH_MP BOUNDED_SUBSET_CBALL o MATCH_MP COMPACT_IMP_BOUNDED) THEN MP_TAC(ISPECL [`(\x. a):real^N->real^N`; `f:real^N->real^N`; `s:real^N->bool`; `cball(vec 0:real^N,r)`; `s:real^N->bool`] BORSUK_HOMOTOPY_EXTENSION) THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED; CLOSED_SUBSET; CONTINUOUS_ON_CONST; CLOSED_CBALL] THEN FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_SUBSET) THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^N->real^N` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`g:real^N->real^N`; `cball(vec 0:real^N,r)`] BROUWER) THEN ASM_SIMP_TAC[COMPACT_CBALL; CONVEX_CBALL; CBALL_EQ_EMPTY] THEN ASM_SIMP_TAC[REAL_ARITH `&0 < r ==> ~(r < &0)`] THEN ASM SET_TAC[]);; let BROUWER_CONTRACTIBLE_ANR = prove (`!f:real^N->real^N s. compact s /\ contractible s /\ ~(s = {}) /\ ANR s /\ f continuous_on s /\ IMAGE f s SUBSET s ==> ?x. x IN s /\ f x = x`, REPEAT STRIP_TAC THEN MATCH_MP_TAC BROUWER_INESSENTIAL_ANR THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC NULLHOMOTOPIC_FROM_CONTRACTIBLE THEN ASM_REWRITE_TAC[]);; let FIXED_POINT_INESSENTIAL_SPHERE_MAP = prove (`!f a:real^N r c. &0 < r /\ homotopic_with (\x. T) (subtopology euclidean (sphere(a,r)), subtopology euclidean (sphere(a,r))) f (\x. c) ==> ?x. x IN sphere(a,r) /\ f x = x`, REPEAT STRIP_TAC THEN MATCH_MP_TAC BROUWER_INESSENTIAL_ANR THEN REWRITE_TAC[ANR_SPHERE] THEN ASM_SIMP_TAC[SPHERE_EQ_EMPTY; COMPACT_SPHERE; OPEN_DELETE; OPEN_UNIV] THEN FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_CONTINUOUS) THEN FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_SUBSET) THEN ASM_SIMP_TAC[REAL_NOT_LT; REAL_LT_IMP_LE] THEN ASM_MESON_TAC[]);; let BROUWER_AR = prove (`!f s:real^N->bool. compact s /\ AR s /\ f continuous_on s /\ IMAGE f s SUBSET s ==> ?x. x IN s /\ f x = x`, REWRITE_TAC[AR_ANR] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC BROUWER_CONTRACTIBLE_ANR THEN ASM_REWRITE_TAC[]);; let BROUWER_ABSOLUTE_RETRACT = prove (`!f s. compact s /\ s retract_of (:real^N) /\ f continuous_on s /\ IMAGE f s SUBSET s ==> ?x. x IN s /\ f x = x`, REWRITE_TAC[RETRACT_OF_UNIV; AR_ANR] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC BROUWER_CONTRACTIBLE_ANR THEN ASM_REWRITE_TAC[]);; (* ------------------------------------------------------------------------- *) (* This interesting lemma is no longer used for Schauder but we keep it. *) (* ------------------------------------------------------------------------- *) let SCHAUDER_PROJECTION = prove (`!s:real^N->bool e. compact s /\ &0 < e ==> ?t f. FINITE t /\ t SUBSET s /\ f continuous_on s /\ IMAGE f s SUBSET (convex hull t) /\ (!x. x IN s ==> norm(f x - x) < e)`, REPEAT STRIP_TAC THEN FIRST_ASSUM (MP_TAC o SPEC `e:real` o MATCH_MP COMPACT_IMP_TOTALLY_BOUNDED) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ABBREV_TAC `g = \p x:real^N. max (&0) (e - norm(x - p))` THEN SUBGOAL_THEN `!x. x IN s ==> &0 < sum t (\p. (g:real^N->real^N->real) p x)` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC SUM_POS_LT THEN ASM_REWRITE_TAC[] THEN EXPAND_TAC "g" THEN REWRITE_TAC[REAL_ARITH `&0 <= max (&0) b`] THEN REWRITE_TAC[REAL_ARITH `&0 < max (&0) b <=> &0 < b`; REAL_SUB_LT] THEN UNDISCH_TAC `s SUBSET UNIONS (IMAGE (\x:real^N. ball(x,e)) t)` THEN REWRITE_TAC[SUBSET; UNIONS_IMAGE; IN_BALL; IN_ELIM_THM] THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[dist; NORM_SUB]; ALL_TAC] THEN EXISTS_TAC `(\x. inv(sum t (\p. g p x)) % vsum t (\p. g p x % p)):real^N->real^N` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_MUL THEN REWRITE_TAC[o_DEF] THEN CONJ_TAC THENL [MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN ASM_SIMP_TAC[REAL_LT_IMP_NZ; LIFT_SUM; o_DEF]; ALL_TAC] THEN MATCH_MP_TAC CONTINUOUS_ON_VSUM THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THENL [ALL_TAC; MATCH_MP_TAC CONTINUOUS_ON_MUL] THEN REWRITE_TAC[o_DEF; CONTINUOUS_ON_CONST] THEN EXPAND_TAC "g" THEN (SUBGOAL_THEN `(\x. lift (max (&0) (e - norm (x - y:real^N)))) = (\x. (lambda i. max (lift(&0)$i) (lift(e - norm (x - y))$i)))` SUBST1_TAC THENL [SIMP_TAC[CART_EQ; LAMBDA_BETA; FUN_EQ_THM] THEN REWRITE_TAC[DIMINDEX_1; FORALL_1; GSYM drop; LIFT_DROP]; MATCH_MP_TAC CONTINUOUS_ON_MAX] THEN REWRITE_TAC[CONTINUOUS_ON_CONST; LIFT_SUB] THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN REWRITE_TAC[ONCE_REWRITE_RULE[NORM_SUB] (GSYM dist)] THEN REWRITE_TAC[REWRITE_RULE[o_DEF] CONTINUOUS_ON_LIFT_DIST]); REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; GSYM VSUM_LMUL; VECTOR_MUL_ASSOC] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MATCH_MP_TAC CONVEX_VSUM THEN ASM_SIMP_TAC[HULL_INC; CONVEX_CONVEX_HULL; SUM_LMUL] THEN ASM_SIMP_TAC[REAL_LT_IMP_NZ; REAL_MUL_LINV] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_SIMP_TAC[REAL_LE_INV_EQ; REAL_LT_IMP_LE] THEN EXPAND_TAC "g" THEN REAL_ARITH_TAC; X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[] THEN ONCE_REWRITE_TAC[NORM_SUB] THEN REWRITE_TAC[REWRITE_RULE[dist] (GSYM IN_BALL)] THEN REWRITE_TAC[GSYM VSUM_LMUL; VECTOR_MUL_ASSOC] THEN MATCH_MP_TAC CONVEX_VSUM_STRONG THEN ASM_REWRITE_TAC[CONVEX_BALL; SUM_LMUL; REAL_ENTIRE] THEN ASM_SIMP_TAC[REAL_LT_IMP_NZ; REAL_MUL_LINV; REAL_LT_INV_EQ; REAL_LE_MUL_EQ] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN EXPAND_TAC "g" THEN REWRITE_TAC[IN_BALL; dist; NORM_SUB] THEN REAL_ARITH_TAC]);; (* ------------------------------------------------------------------------- *) (* Some other related fixed-point theorems. *) (* ------------------------------------------------------------------------- *) let BROUWER_FACTOR_THROUGH_AR = prove (`!f:real^M->real^N g:real^N->real^M s t. f continuous_on s /\ IMAGE f s SUBSET t /\ g continuous_on t /\ IMAGE g t SUBSET s /\ compact s /\ AR t ==> ?x. x IN s /\ g(f x) = x`, REPEAT STRIP_TAC THEN FIRST_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [COMPACT_EQ_BOUNDED_CLOSED]) THEN FIRST_ASSUM(MP_TAC o SPEC `a:real^M` o MATCH_MP BOUNDED_SUBSET_CBALL) THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`f:real^M->real^N`; `(:real^M)`; `s:real^M->bool`; `t:real^N->bool`] AR_IMP_ABSOLUTE_EXTENSOR) THEN ASM_REWRITE_TAC[SUBTOPOLOGY_UNIV; GSYM CLOSED_IN] THEN DISCH_THEN(X_CHOOSE_THEN `h:real^M->real^N` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`(g:real^N->real^M) o (h:real^M->real^N)`; `a:real^M`; `r:real`] BROUWER_BALL) THEN ASM_REWRITE_TAC[o_THM; IMAGE_o] THEN ANTS_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV; IMAGE_SUBSET]);; let BROUWER_ABSOLUTE_RETRACT_GEN = prove (`!f s:real^N->bool. s retract_of (:real^N) /\ f continuous_on s /\ IMAGE f s SUBSET s /\ bounded(IMAGE f s) ==> ?x. x IN s /\ f x = x`, REWRITE_TAC[RETRACT_OF_UNIV] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`\x:real^N. x`; `f:real^N->real^N`; `closure(IMAGE (f:real^N->real^N) s)`; `s:real^N->bool`] BROUWER_FACTOR_THROUGH_AR) THEN ASM_REWRITE_TAC[CONTINUOUS_ON_ID; COMPACT_CLOSURE; IMAGE_ID] THEN REWRITE_TAC[CLOSURE_SUBSET] THEN MATCH_MP_TAC(TAUT `(p /\ q ==> r) /\ p ==> (p ==> q) ==> r`) THEN CONJ_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC CLOSURE_MINIMAL] THEN ASM_MESON_TAC[RETRACT_OF_CLOSED; CLOSED_UNIV]);; let SCHAUDER_GEN = prove (`!f s t:real^N->bool. AR s /\ f continuous_on s /\ IMAGE f s SUBSET t /\ t SUBSET s /\ compact t ==> ?x. x IN t /\ f x = x`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`\x:real^N. x`; `f:real^N->real^N`; `t:real^N->bool`; `s:real^N->bool`] BROUWER_FACTOR_THROUGH_AR) THEN ASM_REWRITE_TAC[CONTINUOUS_ON_ID; IMAGE_ID]);; let SCHAUDER = prove (`!f s t:real^N->bool. convex s /\ ~(s = {}) /\ t SUBSET s /\ compact t /\ f continuous_on s /\ IMAGE f s SUBSET t ==> ?x. x IN s /\ f x = x`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^N->real^N`; `s:real^N->bool`; `t:real^N->bool`] SCHAUDER_GEN) THEN ASM_SIMP_TAC[CONVEX_IMP_AR] THEN ASM SET_TAC[]);; let SCHAUDER_UNIV = prove (`!f:real^N->real^N. f continuous_on (:real^N) /\ bounded (IMAGE f (:real^N)) ==> ?x. f x = x`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^N->real^N`; `(:real^N)`; `closure(IMAGE (f:real^N->real^N) (:real^N))`] SCHAUDER) THEN ASM_REWRITE_TAC[UNIV_NOT_EMPTY; CONVEX_UNIV; COMPACT_CLOSURE; IN_UNIV] THEN REWRITE_TAC[SUBSET_UNIV; CLOSURE_SUBSET]);; let ROTHE = prove (`!f s:real^N->bool. closed s /\ convex s /\ ~(s = {}) /\ f continuous_on s /\ bounded(IMAGE f s) /\ IMAGE f (frontier s) SUBSET s ==> ?x. x IN s /\ f x = x`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `(:real^N)`] ABSOLUTE_RETRACTION_CONVEX_CLOSED) THEN ASM_REWRITE_TAC[retraction; SUBSET_UNIV] THEN DISCH_THEN(X_CHOOSE_THEN `r:real^N->real^N` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`(r:real^N->real^N) o (f:real^N->real^N)`; `s:real^N->bool`; `IMAGE (r:real^N->real^N) (closure(IMAGE (f:real^N->real^N) s))`] SCHAUDER) THEN ANTS_TAC THENL [ASM_SIMP_TAC[CLOSURE_SUBSET; IMAGE_SUBSET; IMAGE_o] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[COMPACT_CLOSURE]; MATCH_MP_TAC CONTINUOUS_ON_COMPOSE] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[o_THM] THEN STRIP_TAC THEN ASM SET_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Perron-Frobenius theorem. *) (* ------------------------------------------------------------------------- *) let PERRON_FROBENIUS = prove (`!A:real^N^N. (!i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) ==> &0 <= A$i$j) ==> ?v c. norm v = &1 /\ &0 <= c /\ A ** v = c % v`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `?v. ~(v = vec 0) /\ (A:real^N^N) ** v = vec 0` THENL [FIRST_X_ASSUM(X_CHOOSE_THEN `v:real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `inv(norm v) % v:real^N` THEN EXISTS_TAC `&0` THEN ASM_SIMP_TAC[REAL_LE_REFL; NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM; REAL_MUL_LINV; NORM_EQ_0; MATRIX_VECTOR_MUL_RMUL] THEN REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_MUL_RZERO]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN REWRITE_TAC[TAUT `~(~p /\ q) <=> q ==> p`] THEN DISCH_TAC] THEN MP_TAC(ISPECL [`\x:real^N. inv(vec 1 dot (A ** x)) % ((A:real^N^N) ** x)`; `{x:real^N | vec 1 dot x = &1} INTER {x:real^N | !i. 1 <= i /\ i <= dimindex(:N) ==> &0 <= x$i}`] BROUWER) THEN SIMP_TAC[CONVEX_INTER; CONVEX_POSITIVE_ORTHANT; CONVEX_HYPERPLANE] THEN SUBGOAL_THEN `!x. (!i. 1 <= i /\ i <= dimindex(:N) ==> &0 <= x$i) ==> !i. 1 <= i /\ i <= dimindex(:N) ==> &0 <= ((A:real^N^N) ** x)$i` ASSUME_TAC THENL [GEN_TAC THEN STRIP_TAC THEN SIMP_TAC[matrix_vector_mul; LAMBDA_BETA] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC SUM_POS_LE_NUMSEG THEN ASM_MESON_TAC[REAL_LE_MUL]; ALL_TAC] THEN SUBGOAL_THEN `!x. (!i. 1 <= i /\ i <= dimindex(:N) ==> &0 <= x$i) /\ vec 1 dot x = &1 ==> &0 < vec 1 dot ((A:real^N^N) ** x)` ASSUME_TAC THENL [X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `x:real^N = vec 0` THEN ASM_REWRITE_TAC[DOT_RZERO] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[REAL_ARITH `&0 < x <=> &0 <= x /\ ~(x = &0)`] THEN DISCH_TAC THEN REWRITE_TAC[dot; VEC_COMPONENT; REAL_MUL_LID] THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_POS_LE_NUMSEG THEN ASM_MESON_TAC[]; DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] SUM_POS_EQ_0_NUMSEG)) THEN RULE_ASSUM_TAC(REWRITE_RULE[CART_EQ; VEC_COMPONENT]) THEN ASM_MESON_TAC[]]; ALL_TAC] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [SIMP_TAC[COMPACT_EQ_BOUNDED_CLOSED; CLOSED_INTER; CLOSED_HYPERPLANE; CLOSED_POSITIVE_ORTHANT] THEN MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `interval[vec 0:real^N,vec 1]` THEN SIMP_TAC[BOUNDED_INTERVAL; SUBSET; IN_INTER; IN_ELIM_THM; IN_INTERVAL; dot; VEC_COMPONENT; REAL_MUL_LID] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN TRANS_TAC REAL_LE_TRANS `sum {i} (\i. (x:real^N)$i)` THEN CONJ_TAC THENL [REWRITE_TAC[SUM_SING; REAL_LE_REFL]; ALL_TAC] THEN MATCH_MP_TAC SUM_SUBSET_SIMPLE THEN REWRITE_TAC[FINITE_SING; FINITE_NUMSEG] THEN ASM_SIMP_TAC[SING_SUBSET; IN_SING; IN_DIFF; IN_NUMSEG]; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `basis 1:real^N` THEN SIMP_TAC[IN_INTER; IN_ELIM_THM; BASIS_COMPONENT] THEN CONJ_TAC THENL [ALL_TAC; MESON_TAC[REAL_POS]] THEN SIMP_TAC[DOT_BASIS; DIMINDEX_GE_1; LE_REFL; VEC_COMPONENT]; MATCH_MP_TAC CONTINUOUS_ON_MUL THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; MATRIX_VECTOR_MUL_LINEAR; o_DEF] THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN SIMP_TAC[CONTINUOUS_ON_LIFT_DOT2; MATRIX_VECTOR_MUL_LINEAR; CONTINUOUS_ON_CONST; LINEAR_CONTINUOUS_ON] THEN REWRITE_TAC[IN_INTER; IN_ELIM_THM] THEN ASM_MESON_TAC[REAL_LT_REFL]; SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTER; IN_ELIM_THM] THEN REWRITE_TAC[DOT_RMUL] THEN REPEAT STRIP_TAC THENL [MATCH_MP_TAC REAL_MUL_LINV THEN MATCH_MP_TAC REAL_LT_IMP_NZ THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[]; REWRITE_TAC[VECTOR_MUL_COMPONENT] THEN MATCH_MP_TAC REAL_LE_MUL THEN REWRITE_TAC[REAL_LE_INV_EQ] THEN ASM_MESON_TAC[REAL_LT_IMP_LE]]]; REWRITE_TAC[IN_INTER; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `x:real^N = vec 0` THEN ASM_REWRITE_TAC[DOT_RZERO] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN STRIP_TAC THEN EXISTS_TAC `inv(norm x) % x:real^N` THEN EXISTS_TAC `vec 1 dot ((A:real^N^N) ** x)` THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN MATCH_MP_TAC REAL_MUL_LINV THEN ASM_REWRITE_TAC[NORM_EQ_0]; ASM_MESON_TAC[REAL_LT_IMP_LE]; REWRITE_TAC[MATRIX_VECTOR_MUL_RMUL; VECTOR_MUL_ASSOC] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[GSYM VECTOR_MUL_ASSOC] THEN AP_TERM_TAC THEN FIRST_X_ASSUM(SUBST1_TAC o SYM o AP_TERM `(%) (vec 1 dot ((A:real^N^N) ** x)):real^N->real^N`) THEN REWRITE_TAC[VECTOR_MUL_ASSOC; VECTOR_MUL_EQ_0; VECTOR_ARITH `v:real^N = c % v <=> (c - &1) % v = vec 0`] THEN DISJ1_TAC THEN REWRITE_TAC[REAL_SUB_0] THEN MATCH_MP_TAC REAL_MUL_RINV THEN MATCH_MP_TAC REAL_LT_IMP_NZ THEN ASM_MESON_TAC[]]]);; (* ------------------------------------------------------------------------- *) (* Bijections between intervals. *) (* ------------------------------------------------------------------------- *) let interval_bij = new_definition `interval_bij (a:real^N,b:real^N) (u:real^N,v:real^N) (x:real^N) = (lambda i. u$i + (x$i - a$i) / (b$i - a$i) * (v$i - u$i)):real^N`;; let INTERVAL_BIJ_AFFINE = prove (`interval_bij (a,b) (u,v) = \x. (lambda i. (v$i - u$i) / (b$i - a$i) * x$i) + (lambda i. u$i - (v$i - u$i) / (b$i - a$i) * a$i)`, SIMP_TAC[FUN_EQ_THM; CART_EQ; VECTOR_ADD_COMPONENT; LAMBDA_BETA; interval_bij] THEN REAL_ARITH_TAC);; let CONTINUOUS_INTERVAL_BIJ = prove (`!a b u v x. (interval_bij (a:real^N,b:real^N) (u:real^N,v:real^N)) continuous at x`, REPEAT GEN_TAC THEN REWRITE_TAC[INTERVAL_BIJ_AFFINE] THEN MATCH_MP_TAC CONTINUOUS_ADD THEN REWRITE_TAC[CONTINUOUS_CONST] THEN MATCH_MP_TAC LINEAR_CONTINUOUS_AT THEN SIMP_TAC[linear; CART_EQ; LAMBDA_BETA; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN REAL_ARITH_TAC);; let CONTINUOUS_ON_INTERVAL_BIJ = prove (`!a b u v s. interval_bij (a,b) (u,v) continuous_on s`, REPEAT GEN_TAC THEN MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN REWRITE_TAC[CONTINUOUS_INTERVAL_BIJ]);; let IN_INTERVAL_INTERVAL_BIJ = prove (`!a b u v x:real^N. x IN interval[a,b] /\ ~(interval[u,v] = {}) ==> (interval_bij (a,b) (u,v) x) IN interval[u,v]`, SIMP_TAC[IN_INTERVAL; interval_bij; LAMBDA_BETA; INTERVAL_NE_EMPTY] THEN REWRITE_TAC[REAL_ARITH `u <= u + x <=> &0 <= x`; REAL_ARITH `u + x <= v <=> x <= &1 * (v - u)`] THEN REPEAT STRIP_TAC THENL [MATCH_MP_TAC REAL_LE_MUL THEN CONJ_TAC THEN TRY(MATCH_MP_TAC REAL_LE_DIV) THEN ASM_SIMP_TAC[REAL_SUB_LE] THEN ASM_MESON_TAC[REAL_LE_TRANS]; MATCH_MP_TAC REAL_LE_RMUL THEN ASM_SIMP_TAC[REAL_SUB_LE] THEN SUBGOAL_THEN `(a:real^N)$i <= (b:real^N)$i` MP_TAC THENL [ASM_MESON_TAC[REAL_LE_TRANS]; ALL_TAC] THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN STRIP_TAC THENL [ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_SUB_LT] THEN ASM_SIMP_TAC[REAL_ARITH `a <= x /\ x <= b ==> x - a <= &1 * (b - a)`]; ASM_REWRITE_TAC[real_div; REAL_SUB_REFL; REAL_INV_0] THEN REAL_ARITH_TAC]]);; let INTERVAL_BIJ_BIJ = prove (`!a b u v x:real^N. (!i. 1 <= i /\ i <= dimindex(:N) ==> a$i < b$i /\ u$i < v$i) ==> interval_bij (a,b) (u,v) (interval_bij (u,v) (a,b) x) = x`, SIMP_TAC[interval_bij; CART_EQ; LAMBDA_BETA; REAL_ADD_SUB] THEN REPEAT GEN_TAC THEN REWRITE_TAC[AND_FORALL_THM] THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN CONV_TAC REAL_FIELD);; (* ------------------------------------------------------------------------- *) (* Fashoda meet theorem. *) (* ------------------------------------------------------------------------- *) let INFNORM_2 = prove (`infnorm (x:real^2) = max (abs(x$1)) (abs(x$2))`, REWRITE_TAC[infnorm; INFNORM_SET_IMAGE; NUMSEG_CONV `1..2`; DIMINDEX_2] THEN REWRITE_TAC[IMAGE_CLAUSES; GSYM REAL_MAX_SUP]);; let INFNORM_EQ_1_2 = prove (`infnorm (x:real^2) = &1 <=> abs(x$1) <= &1 /\ abs(x$2) <= &1 /\ (x$1 = -- &1 \/ x$1 = &1 \/ x$2 = -- &1 \/ x$2 = &1)`, REWRITE_TAC[INFNORM_2] THEN REAL_ARITH_TAC);; let INFNORM_EQ_1_IMP = prove (`infnorm (x:real^2) = &1 ==> abs(x$1) <= &1 /\ abs(x$2) <= &1`, SIMP_TAC[INFNORM_EQ_1_2]);; let FASHODA_UNIT = prove (`!f:real^1->real^2 g:real^1->real^2. IMAGE f (interval[--vec 1,vec 1]) SUBSET interval[--vec 1,vec 1] /\ IMAGE g (interval[--vec 1,vec 1]) SUBSET interval[--vec 1,vec 1] /\ f continuous_on interval[--vec 1,vec 1] /\ g continuous_on interval[--vec 1,vec 1] /\ f(--vec 1)$1 = -- &1 /\ f(vec 1)$1 = &1 /\ g(--vec 1)$2 = -- &1 /\ g(vec 1)$2 = &1 ==> ?s t. s IN interval[--vec 1,vec 1] /\ t IN interval[--vec 1,vec 1] /\ f(s) = g(t)`, REPEAT STRIP_TAC THEN GEN_REWRITE_TAC I [TAUT `p <=> ~ ~p`] THEN DISCH_THEN(MP_TAC o REWRITE_RULE[NOT_EXISTS_THM]) THEN REWRITE_TAC[TAUT `~(a /\ b /\ c) <=> a /\ b ==> ~c`] THEN DISCH_TAC THEN ABBREV_TAC `sqprojection = \z:real^2. inv(infnorm z) % z` THEN ABBREV_TAC `(negatex:real^2->real^2) = \x. vector[--(x$1); x$2]` THEN SUBGOAL_THEN `!z:real^2. infnorm(negatex z:real^2) = infnorm z` ASSUME_TAC THENL [EXPAND_TAC "negatex" THEN SIMP_TAC[VECTOR_2; INFNORM_2] THEN REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `!z. ~(z = vec 0) ==> infnorm((sqprojection:real^2->real^2) z) = &1` ASSUME_TAC THENL [EXPAND_TAC "sqprojection" THEN REWRITE_TAC[INFNORM_MUL; REAL_ABS_INFNORM; REAL_ABS_INV] THEN SIMP_TAC[REAL_MUL_LINV; INFNORM_EQ_0]; ALL_TAC] THEN MP_TAC(ISPECL [`(\w. (negatex:real^2->real^2) (sqprojection(f(lift(w$1)) - g(lift(w$2)):real^2))) :real^2->real^2`; `interval[--vec 1,vec 1]:real^2->bool`] BROUWER_WEAK) THEN REWRITE_TAC[NOT_IMP; COMPACT_INTERVAL; CONVEX_INTERVAL] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[INTERIOR_CLOSED_INTERVAL; INTERVAL_NE_EMPTY] THEN SIMP_TAC[VEC_COMPONENT; VECTOR_NEG_COMPONENT] THEN REAL_ARITH_TAC; MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_COMPOSE) THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC LINEAR_CONTINUOUS_ON THEN EXPAND_TAC "negatex" THEN SIMP_TAC[linear; VECTOR_2; CART_EQ; FORALL_2; DIMINDEX_2; VECTOR_MUL_COMPONENT; VECTOR_NEG_COMPONENT; VECTOR_ADD_COMPONENT; ARITH] THEN REAL_ARITH_TAC] THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_COMPOSE) THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_SUB THEN CONJ_TAC THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_COMPOSE) THEN SIMP_TAC[CONTINUOUS_ON_LIFT_COMPONENT; DIMINDEX_2; ARITH] THEN MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `interval[--vec 1:real^1,vec 1]`; MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN EXPAND_TAC "sqprojection" THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN X_GEN_TAC `x:real^2` THEN STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_MUL THEN REWRITE_TAC[CONTINUOUS_AT_ID] THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_AT_INV THEN REWRITE_TAC[CONTINUOUS_AT_LIFT_INFNORM; INFNORM_EQ_0; VECTOR_SUB_EQ] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL])] THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTERVAL_1] THEN SIMP_TAC[IN_INTERVAL; DIMINDEX_2; FORALL_2; VEC_COMPONENT; ARITH; VECTOR_NEG_COMPONENT; DROP_NEG; DROP_VEC; LIFT_DROP]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `x:real^2` THEN STRIP_TAC THEN SIMP_TAC[IN_INTERVAL; DIMINDEX_2; FORALL_2; REAL_BOUNDS_LE; VECTOR_NEG_COMPONENT; VEC_COMPONENT; ARITH] THEN MATCH_MP_TAC INFNORM_EQ_1_IMP THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[VECTOR_SUB_EQ] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL]) THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTERVAL_1] THEN SIMP_TAC[IN_INTERVAL; DIMINDEX_2; FORALL_2; VEC_COMPONENT; ARITH; VECTOR_NEG_COMPONENT; DROP_NEG; DROP_VEC; LIFT_DROP]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `x:real^2` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `infnorm(x:real^2) = &1` MP_TAC THENL [FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [SYM th]) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[VECTOR_SUB_EQ] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[IN_INTERVAL_1] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL]) THEN SIMP_TAC[IN_INTERVAL; DIMINDEX_2; FORALL_2; VEC_COMPONENT; ARITH; VECTOR_NEG_COMPONENT; DROP_NEG; DROP_VEC; LIFT_DROP]; ALL_TAC] THEN SUBGOAL_THEN `(!x i. 1 <= i /\ i <= 2 /\ ~(x = vec 0) ==> (&0 < ((sqprojection:real^2->real^2) x)$i <=> &0 < x$i)) /\ (!x i. 1 <= i /\ i <= 2 /\ ~(x = vec 0) ==> ((sqprojection x)$i < &0 <=> x$i < &0))` STRIP_ASSUME_TAC THENL [EXPAND_TAC "sqprojection" THEN SIMP_TAC[VECTOR_MUL_COMPONENT; DIMINDEX_2; ARITH] THEN REWRITE_TAC[GSYM(ONCE_REWRITE_RULE[REAL_MUL_SYM] real_div)] THEN SIMP_TAC[REAL_LT_LDIV_EQ; REAL_LT_RDIV_EQ; INFNORM_POS_LT] THEN REWRITE_TAC[REAL_MUL_LZERO]; ALL_TAC] THEN REWRITE_TAC[INFNORM_EQ_1_2; CONJ_ASSOC] THEN DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC (REPEAT_TCL DISJ_CASES_THEN (fun th -> ASSUME_TAC th THEN MP_TAC th))) THEN MAP_EVERY EXPAND_TAC ["x"; "negatex"] THEN REWRITE_TAC[VECTOR_2] THENL [DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `--x = -- &1 ==> &0 < x`)); DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `--x = &1 ==> x < &0`)); DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `x = -- &1 ==> x < &0`)); DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `x = &1 ==> &0 < x`))] THEN W(fun (_,w) -> FIRST_X_ASSUM(fun th -> MP_TAC(PART_MATCH (lhs o rand) th (lhand w)))) THEN (ANTS_TAC THENL [REWRITE_TAC[VECTOR_SUB_EQ; ARITH] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL]) THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTERVAL_1] THEN SIMP_TAC[IN_INTERVAL; DIMINDEX_2; FORALL_2; VEC_COMPONENT; ARITH; VECTOR_NEG_COMPONENT; DROP_NEG; DROP_VEC; LIFT_DROP] THEN REAL_ARITH_TAC; DISCH_THEN SUBST1_TAC]) THEN ASM_SIMP_TAC[VECTOR_SUB_COMPONENT; DIMINDEX_2; ARITH; LIFT_NEG; LIFT_NUM] THENL [MATCH_MP_TAC(REAL_ARITH `abs(x$1) <= &1 /\ abs(x$2) <= &1 ==> ~(&0 < -- &1 - x$1)`); MATCH_MP_TAC(REAL_ARITH `abs(x$1) <= &1 /\ abs(x$2) <= &1 ==> ~(&1 - x$1 < &0)`); MATCH_MP_TAC(REAL_ARITH `abs(x$1) <= &1 /\ abs(x$2) <= &1 ==> ~(x$2 - -- &1 < &0)`); MATCH_MP_TAC(REAL_ARITH `abs(x$1) <= &1 /\ abs(x$2) <= &1 ==> ~(&0 < x$2 - &1)`)] THEN (SUBGOAL_THEN `!z:real^2. abs(z$1) <= &1 /\ abs(z$2) <= &1 <=> z IN interval[--vec 1,vec 1]` (fun th -> REWRITE_TAC[th]) THENL [SIMP_TAC[IN_INTERVAL; DIMINDEX_2; FORALL_2; VEC_COMPONENT; ARITH; VECTOR_NEG_COMPONENT; DROP_NEG; DROP_VEC; LIFT_DROP] THEN REAL_ARITH_TAC; ALL_TAC]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE f s SUBSET t ==> x IN s ==> f x IN t`)) THEN REWRITE_TAC[IN_INTERVAL_1; DROP_NEG; DROP_VEC; LIFT_DROP] THEN ASM_REWRITE_TAC[REAL_BOUNDS_LE]);; let FASHODA_UNIT_PATH = prove (`!f:real^1->real^2 g:real^1->real^2. path f /\ path g /\ path_image f SUBSET interval[--vec 1,vec 1] /\ path_image g SUBSET interval[--vec 1,vec 1] /\ (pathstart f)$1 = -- &1 /\ (pathfinish f)$1 = &1 /\ (pathstart g)$2 = -- &1 /\ (pathfinish g)$2 = &1 ==> ?z. z IN path_image f /\ z IN path_image g`, SIMP_TAC[path; path_image; pathstart; pathfinish] THEN REPEAT STRIP_TAC THEN ABBREV_TAC `iscale = \z:real^1. inv(&2) % (z + vec 1)` THEN MP_TAC(ISPECL [`(f:real^1->real^2) o (iscale:real^1->real^1)`; `(g:real^1->real^2) o (iscale:real^1->real^1)`] FASHODA_UNIT) THEN SUBGOAL_THEN `IMAGE (iscale:real^1->real^1) (interval[--vec 1,vec 1]) SUBSET interval[vec 0,vec 1]` ASSUME_TAC THENL [REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN EXPAND_TAC "iscale" THEN REWRITE_TAC[IN_INTERVAL_1; DROP_NEG; DROP_VEC; DROP_CMUL; DROP_ADD] THEN REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `(iscale:real^1->real^1) continuous_on interval [--vec 1,vec 1]` ASSUME_TAC THENL [EXPAND_TAC "iscale" THEN SIMP_TAC[CONTINUOUS_ON_CMUL; CONTINUOUS_ON_ID; CONTINUOUS_ON_ADD; CONTINUOUS_ON_CONST]; ALL_TAC] THEN ASM_REWRITE_TAC[IMAGE_o] THEN ANTS_TAC THENL [REPLICATE_TAC 2 (CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN REPLICATE_TAC 2 (CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ALL_TAC]) THEN EXPAND_TAC "iscale" THEN REWRITE_TAC[o_THM] THEN ASM_REWRITE_TAC[VECTOR_ARITH `inv(&2) % (--x + x) = vec 0`; VECTOR_ARITH `inv(&2) % (x + x) = x`]; REWRITE_TAC[o_THM; LEFT_IMP_EXISTS_THM; IN_IMAGE] THEN ASM SET_TAC[]]);; let FASHODA = prove (`!f g a b:real^2. path f /\ path g /\ path_image f SUBSET interval[a,b] /\ path_image g SUBSET interval[a,b] /\ (pathstart f)$1 = a$1 /\ (pathfinish f)$1 = b$1 /\ (pathstart g)$2 = a$2 /\ (pathfinish g)$2 = b$2 ==> ?z. z IN path_image f /\ z IN path_image g`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `~(interval[a:real^2,b] = {})` MP_TAC THENL [FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET t ==> ~(s = {}) ==> ~(t = {})`)) THEN REWRITE_TAC[PATH_IMAGE_NONEMPTY]; ALL_TAC] THEN REWRITE_TAC[INTERVAL_NE_EMPTY; DIMINDEX_2; FORALL_2] THEN STRIP_TAC THEN MP_TAC(ASSUME `(a:real^2)$1 <= (b:real^2)$1`) THEN REWRITE_TAC[REAL_ARITH `a <= b <=> b = a \/ a < b`] THEN STRIP_TAC THENL [SUBGOAL_THEN `?z:real^2. z IN path_image g /\ z$2 = (pathstart f:real^2)$2` MP_TAC THENL [MATCH_MP_TAC CONNECTED_IVT_COMPONENT THEN MAP_EVERY EXISTS_TAC [`pathstart(g:real^1->real^2)`; `pathfinish(g:real^1->real^2)`] THEN ASM_SIMP_TAC[CONNECTED_PATH_IMAGE; PATHSTART_IN_PATH_IMAGE; REAL_LE_REFL; PATHFINISH_IN_PATH_IMAGE; DIMINDEX_2; ARITH] THEN UNDISCH_TAC `path_image f SUBSET interval[a:real^2,b]` THEN REWRITE_TAC[SUBSET; path_image; IN_INTERVAL_1; FORALL_IN_IMAGE] THEN DISCH_THEN(MP_TAC o SPEC `vec 0:real^1`) THEN SIMP_TAC[pathstart] THEN SIMP_TAC[DROP_VEC; REAL_POS; IN_INTERVAL; FORALL_2; DIMINDEX_2]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `z:real^2` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN SIMP_TAC[path_image; IN_IMAGE] THEN EXISTS_TAC `vec 0:real^1` THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; REAL_POS] THEN ASM_REWRITE_TAC[CART_EQ; FORALL_2; DIMINDEX_2; pathstart] THEN SUBGOAL_THEN `(z:real^2) IN interval[a,b] /\ f(vec 0:real^1) IN interval[a,b]` MP_TAC THENL [ASM_MESON_TAC[SUBSET; path_image; IN_IMAGE; PATHSTART_IN_PATH_IMAGE; pathstart]; ASM_REWRITE_TAC[IN_INTERVAL; FORALL_2; DIMINDEX_2] THEN REAL_ARITH_TAC]; ALL_TAC] THEN MP_TAC(ASSUME `(a:real^2)$2 <= (b:real^2)$2`) THEN REWRITE_TAC[REAL_ARITH `a <= b <=> b = a \/ a < b`] THEN STRIP_TAC THENL [SUBGOAL_THEN `?z:real^2. z IN path_image f /\ z$1 = (pathstart g:real^2)$1` MP_TAC THENL [MATCH_MP_TAC CONNECTED_IVT_COMPONENT THEN MAP_EVERY EXISTS_TAC [`pathstart(f:real^1->real^2)`; `pathfinish(f:real^1->real^2)`] THEN ASM_SIMP_TAC[CONNECTED_PATH_IMAGE; PATHSTART_IN_PATH_IMAGE; REAL_LE_REFL; PATHFINISH_IN_PATH_IMAGE; DIMINDEX_2; ARITH] THEN UNDISCH_TAC `path_image g SUBSET interval[a:real^2,b]` THEN REWRITE_TAC[SUBSET; path_image; IN_INTERVAL_1; FORALL_IN_IMAGE] THEN DISCH_THEN(MP_TAC o SPEC `vec 0:real^1`) THEN SIMP_TAC[pathstart] THEN SIMP_TAC[DROP_VEC; REAL_POS; IN_INTERVAL; FORALL_2; DIMINDEX_2]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `z:real^2` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN SIMP_TAC[path_image; IN_IMAGE] THEN EXISTS_TAC `vec 0:real^1` THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; REAL_POS] THEN ASM_REWRITE_TAC[CART_EQ; FORALL_2; DIMINDEX_2; pathstart] THEN SUBGOAL_THEN `(z:real^2) IN interval[a,b] /\ g(vec 0:real^1) IN interval[a,b]` MP_TAC THENL [ASM_MESON_TAC[SUBSET; path_image; IN_IMAGE; PATHSTART_IN_PATH_IMAGE; pathstart]; ASM_REWRITE_TAC[IN_INTERVAL; FORALL_2; DIMINDEX_2] THEN REAL_ARITH_TAC]; ALL_TAC] THEN MP_TAC(ISPECL [`interval_bij (a,b) (--vec 1,vec 1) o (f:real^1->real^2)`; `interval_bij (a,b) (--vec 1,vec 1) o (g:real^1->real^2)`] FASHODA_UNIT_PATH) THEN RULE_ASSUM_TAC(REWRITE_RULE[path; path_image; pathstart; pathfinish]) THEN ASM_REWRITE_TAC[path; path_image; pathstart; pathfinish; o_THM] THEN ANTS_TAC THENL [ASM_SIMP_TAC[CONTINUOUS_ON_COMPOSE; CONTINUOUS_ON_INTERVAL_BIJ] THEN REWRITE_TAC[IMAGE_o] THEN REPLICATE_TAC 2 (CONJ_TAC THENL [REWRITE_TAC[SUBSET] THEN ONCE_REWRITE_TAC[FORALL_IN_IMAGE] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC IN_INTERVAL_INTERVAL_BIJ THEN SIMP_TAC[INTERVAL_NE_EMPTY; VECTOR_NEG_COMPONENT; VEC_COMPONENT] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM SET_TAC[]; ALL_TAC]) THEN ASM_SIMP_TAC[interval_bij; LAMBDA_BETA; DIMINDEX_2; ARITH] THEN ASM_SIMP_TAC[REAL_DIV_REFL; REAL_LT_IMP_NZ; REAL_SUB_LT] THEN REWRITE_TAC[real_div; REAL_SUB_REFL; REAL_MUL_LZERO] THEN SIMP_TAC[VECTOR_NEG_COMPONENT; VEC_COMPONENT; DIMINDEX_2; ARITH] THEN CONV_TAC REAL_RAT_REDUCE_CONV; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `z:real^2` (fun th -> EXISTS_TAC `interval_bij (--vec 1,vec 1) (a,b) (z:real^2)` THEN MP_TAC th)) THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THEN REWRITE_TAC[IMAGE_o] THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> g(f(x)) = x) ==> x IN IMAGE f s ==> g x IN s`) THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC INTERVAL_BIJ_BIJ THEN ASM_SIMP_TAC[FORALL_2; DIMINDEX_2; VECTOR_NEG_COMPONENT; VEC_COMPONENT; ARITH] THEN CONV_TAC REAL_RAT_REDUCE_CONV);; (* ------------------------------------------------------------------------- *) (* Some slightly ad hoc lemmas I use below *) (* ------------------------------------------------------------------------- *) let SEGMENT_VERTICAL = prove (`!a:real^2 b:real^2 x:real^2. a$1 = b$1 ==> (x IN segment[a,b] <=> x$1 = a$1 /\ x$1 = b$1 /\ (a$2 <= x$2 /\ x$2 <= b$2 \/ b$2 <= x$2 /\ x$2 <= a$2))`, GEOM_ORIGIN_TAC `a:real^2` THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VEC_COMPONENT; REAL_LE_LADD; REAL_EQ_ADD_LCANCEL] THEN REPEAT GEN_TAC THEN DISCH_THEN(ASSUME_TAC o SYM) THEN SUBST1_TAC(SYM(ISPEC `b:real^2` BASIS_EXPANSION)) THEN ASM_REWRITE_TAC[DIMINDEX_2; VSUM_2; VECTOR_MUL_LZERO; VECTOR_ADD_LID] THEN SUBST1_TAC(VECTOR_ARITH `vec 0:real^2 = &0 % basis 2`) THEN REWRITE_TAC[SEGMENT_SCALAR_MULTIPLE; IN_ELIM_THM; CART_EQ] THEN REWRITE_TAC[DIMINDEX_2; FORALL_2; VECTOR_MUL_COMPONENT] THEN SIMP_TAC[BASIS_COMPONENT; DIMINDEX_2; ARITH; REAL_MUL_RZERO; REAL_MUL_RID] THEN MESON_TAC[]);; let SEGMENT_HORIZONTAL = prove (`!a:real^2 b:real^2 x:real^2. a$2 = b$2 ==> (x IN segment[a,b] <=> x$2 = a$2 /\ x$2 = b$2 /\ (a$1 <= x$1 /\ x$1 <= b$1 \/ b$1 <= x$1 /\ x$1 <= a$1))`, GEOM_ORIGIN_TAC `a:real^2` THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VEC_COMPONENT; REAL_LE_LADD; REAL_EQ_ADD_LCANCEL] THEN REPEAT GEN_TAC THEN DISCH_THEN(ASSUME_TAC o SYM) THEN SUBST1_TAC(SYM(ISPEC `b:real^2` BASIS_EXPANSION)) THEN ASM_REWRITE_TAC[DIMINDEX_2; VSUM_2; VECTOR_MUL_LZERO; VECTOR_ADD_RID] THEN SUBST1_TAC(VECTOR_ARITH `vec 0:real^2 = &0 % basis 1`) THEN REWRITE_TAC[SEGMENT_SCALAR_MULTIPLE; IN_ELIM_THM; CART_EQ] THEN REWRITE_TAC[DIMINDEX_2; FORALL_2; VECTOR_MUL_COMPONENT] THEN SIMP_TAC[BASIS_COMPONENT; DIMINDEX_2; ARITH; REAL_MUL_RZERO; REAL_MUL_RID] THEN MESON_TAC[]);; (* ------------------------------------------------------------------------- *) (* Useful Fashoda corollary pointed out to me by Tom Hales. *) (* ------------------------------------------------------------------------- *) let FASHODA_INTERLACE = prove (`!f g a b:real^2. path f /\ path g /\ path_image f SUBSET interval[a,b] /\ path_image g SUBSET interval[a,b] /\ (pathstart f)$2 = a$2 /\ (pathfinish f)$2 = a$2 /\ (pathstart g)$2 = a$2 /\ (pathfinish g)$2 = a$2 /\ (pathstart f)$1 < (pathstart g)$1 /\ (pathstart g)$1 < (pathfinish f)$1 /\ (pathfinish f)$1 < (pathfinish g)$1 ==> ?z. z IN path_image f /\ z IN path_image g`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `~(interval[a:real^2,b] = {})` MP_TAC THENL [FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET t ==> ~(s = {}) ==> ~(t = {})`)) THEN REWRITE_TAC[PATH_IMAGE_NONEMPTY]; ALL_TAC] THEN SUBGOAL_THEN `pathstart (f:real^1->real^2) IN interval[a,b] /\ pathfinish f IN interval[a,b] /\ pathstart g IN interval[a,b] /\ pathfinish g IN interval[a,b]` MP_TAC THENL [ASM_MESON_TAC[SUBSET; PATHSTART_IN_PATH_IMAGE; PATHFINISH_IN_PATH_IMAGE]; ALL_TAC] THEN REWRITE_TAC[INTERVAL_NE_EMPTY; IN_INTERVAL; FORALL_2; DIMINDEX_2] THEN REPEAT STRIP_TAC THEN MP_TAC(SPECL [`linepath(vector[a$1 - &2;a$2 - &2],vector[(pathstart f)$1;a$2 - &2]) ++ linepath(vector[(pathstart f)$1;(a:real^2)$2 - &2],pathstart f) ++ (f:real^1->real^2) ++ linepath(pathfinish f,vector[(pathfinish f)$1;a$2 - &2]) ++ linepath(vector[(pathfinish f)$1;a$2 - &2], vector[(b:real^2)$1 + &2;a$2 - &2])`; `linepath(vector[(pathstart g)$1; (pathstart g)$2 - &3],pathstart g) ++ (g:real^1->real^2) ++ linepath(pathfinish g,vector[(pathfinish g)$1;(a:real^2)$2 - &1]) ++ linepath(vector[(pathfinish g)$1;a$2 - &1],vector[b$1 + &1;a$2 - &1]) ++ linepath(vector[b$1 + &1;a$2 - &1],vector[(b:real^2)$1 + &1;b$2 + &3])`; `vector[(a:real^2)$1 - &2; a$2 - &3]:real^2`; `vector[(b:real^2)$1 + &2; b$2 + &3]:real^2`] FASHODA) THEN ASM_SIMP_TAC[PATH_JOIN; PATHSTART_JOIN; PATHFINISH_JOIN; PATH_IMAGE_JOIN; PATHSTART_LINEPATH; PATHFINISH_LINEPATH; PATH_LINEPATH] THEN REWRITE_TAC[VECTOR_2] THEN ANTS_TAC THENL [CONJ_TAC THEN REPEAT(MATCH_MP_TAC (SET_RULE `s SUBSET u /\ t SUBSET u ==> (s UNION t) SUBSET u`) THEN CONJ_TAC) THEN TRY(REWRITE_TAC[PATH_IMAGE_LINEPATH] THEN MATCH_MP_TAC(REWRITE_RULE[CONVEX_CONTAINS_SEGMENT] (CONJUNCT1 (SPEC_ALL CONVEX_INTERVAL))) THEN ASM_REWRITE_TAC[IN_INTERVAL; FORALL_2; DIMINDEX_2; VECTOR_2] THEN ASM_REAL_ARITH_TAC) THEN MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `interval[a:real^2,b:real^2]` THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN REWRITE_TAC[SUBSET_INTERVAL; FORALL_2; DIMINDEX_2; VECTOR_2] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `z:real^2` THEN REWRITE_TAC[PATH_IMAGE_LINEPATH] THEN SUBGOAL_THEN `!f s:real^2->bool. path_image f UNION s = path_image f UNION (s DIFF {pathstart f,pathfinish f})` (fun th -> ONCE_REWRITE_TAC[th] THEN REWRITE_TAC[GSYM UNION_ASSOC] THEN ONCE_REWRITE_TAC[SET_RULE `(s UNION t) UNION u = u UNION t UNION s`] THEN ONCE_REWRITE_TAC[th]) THENL [REWRITE_TAC[EXTENSION; IN_UNION; IN_DIFF; IN_INSERT; NOT_IN_EMPTY] THEN ASM_MESON_TAC[PATHSTART_IN_PATH_IMAGE; PATHFINISH_IN_PATH_IMAGE]; ALL_TAC] THEN REWRITE_TAC[IN_UNION; IN_DIFF; GSYM DISJ_ASSOC; LEFT_OR_DISTRIB; RIGHT_OR_DISTRIB; GSYM CONJ_ASSOC; SET_RULE `~(z IN {x,y}) <=> ~(z = x) /\ ~(z = y)`] THEN DISCH_THEN(REPEAT_TCL DISJ_CASES_THEN MP_TAC) THEN ASM_SIMP_TAC[SEGMENT_VERTICAL; SEGMENT_HORIZONTAL; VECTOR_2] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN UNDISCH_TAC `path_image (f:real^1->real^2) SUBSET interval [a,b]` THEN REWRITE_TAC[SUBSET] THEN DISCH_THEN(MP_TAC o SPEC `z:real^2`) THEN UNDISCH_TAC `path_image (g:real^1->real^2) SUBSET interval [a,b]` THEN REWRITE_TAC[SUBSET] THEN DISCH_THEN(MP_TAC o SPEC `z:real^2`) THEN ASM_REWRITE_TAC[IN_INTERVAL; FORALL_2; DIMINDEX_2] THEN REPEAT(DISCH_THEN(fun th -> if is_imp(concl th) then ALL_TAC else ASSUME_TAC th)) THEN REPEAT(POP_ASSUM MP_TAC) THEN TRY REAL_ARITH_TAC THEN REWRITE_TAC[CART_EQ; FORALL_2; DIMINDEX_2] THEN REAL_ARITH_TAC);; (* ------------------------------------------------------------------------- *) (* Complement in dimension N >= 2 of set homeomorphic to any interval in *) (* any dimension is (path-)connected. This naively generalizes the argument *) (* in Ryuji Maehara's paper "The Jordan curve theorem via the Brouwer *) (* fixed point theorem", American Mathematical Monthly 1984. *) (* ------------------------------------------------------------------------- *) let UNBOUNDED_COMPONENTS_COMPLEMENT_ABSOLUTE_RETRACT = prove (`!s c. compact s /\ AR s /\ c IN components((:real^N) DIFF s) ==> ~bounded c`, REWRITE_TAC[CONJ_ASSOC; COMPACT_AR] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; components; FORALL_IN_GSPEC] THEN GEN_TAC THEN DISCH_TAC THEN DISCH_TAC THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_DIFF; IN_UNIV] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `open((:real^N) DIFF s)` ASSUME_TAC THENL [ASM_SIMP_TAC[GSYM closed; COMPACT_IMP_CLOSED]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retract_of]) THEN REWRITE_TAC[retraction; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `r:real^N->real^N` THEN STRIP_TAC THEN MP_TAC(ISPECL [`connected_component ((:real^N) DIFF s) y`; `s:real^N->bool`; `r:real^N->real^N`] FRONTIER_SUBSET_RETRACTION) THEN ASM_SIMP_TAC[NOT_IMP; INTERIOR_OPEN; OPEN_CONNECTED_COMPONENT] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[frontier] THEN ASM_SIMP_TAC[INTERIOR_OPEN; OPEN_CONNECTED_COMPONENT] THEN REWRITE_TAC[SUBSET; IN_DIFF] THEN X_GEN_TAC `z:real^N` THEN ASM_CASES_TAC `(z:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[IN_CLOSURE_CONNECTED_COMPONENT; IN_UNIV; IN_DIFF] THEN CONV_TAC TAUT; ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV]; ASM SET_TAC[]; MATCH_MP_TAC(SET_RULE `~(c = {}) /\ c SUBSET (:real^N) DIFF s ==> ~(c SUBSET s)`) THEN REWRITE_TAC[CONNECTED_COMPONENT_SUBSET; CONNECTED_COMPONENT_EQ_EMPTY] THEN ASM_REWRITE_TAC[IN_UNIV; IN_DIFF]]);; let CONNECTED_COMPLEMENT_ABSOLUTE_RETRACT = prove (`!s. 2 <= dimindex(:N) /\ compact s /\ AR s ==> connected((:real^N) DIFF s)`, REWRITE_TAC[COMPACT_AR] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[CONNECTED_EQ_CONNECTED_COMPONENT_EQ] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC COBOUNDED_UNIQUE_UNBOUNDED_COMPONENT THEN ASM_SIMP_TAC[COMPL_COMPL; COMPACT_IMP_BOUNDED] THEN CONJ_TAC THEN MATCH_MP_TAC UNBOUNDED_COMPONENTS_COMPLEMENT_ABSOLUTE_RETRACT THEN EXISTS_TAC `s:real^N->bool` THEN REWRITE_TAC[CONJ_ASSOC; COMPACT_AR] THEN ASM_REWRITE_TAC[IN_COMPONENTS] THEN ASM_MESON_TAC[]);; let PATH_CONNECTED_COMPLEMENT_ABSOLUTE_RETRACT = prove (`!s:real^N->bool. 2 <= dimindex(:N) /\ compact s /\ AR s ==> path_connected((:real^N) DIFF s)`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM (MP_TAC o MATCH_MP CONNECTED_COMPLEMENT_ABSOLUTE_RETRACT) THEN MATCH_MP_TAC EQ_IMP THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC PATH_CONNECTED_EQ_CONNECTED THEN REWRITE_TAC[GSYM closed] THEN ASM_MESON_TAC[HOMEOMORPHIC_COMPACTNESS; COMPACT_INTERVAL; COMPACT_IMP_CLOSED]);; let CONNECTED_COMPLEMENT_HOMEOMORPHIC_CONVEX_COMPACT = prove (`!s:real^N->bool t:real^M->bool. 2 <= dimindex(:N) /\ s homeomorphic t /\ convex t /\ compact t ==> connected((:real^N) DIFF s)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[DIFF_EMPTY; CONNECTED_UNIV] THEN MATCH_MP_TAC CONNECTED_COMPLEMENT_ABSOLUTE_RETRACT THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[HOMEOMORPHIC_COMPACTNESS]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMEOMORPHIC_ARNESS) THEN ASM_MESON_TAC[CONVEX_IMP_AR; HOMEOMORPHIC_EMPTY]);; let PATH_CONNECTED_COMPLEMENT_HOMEOMORPHIC_CONVEX_COMPACT = prove (`!s:real^N->bool t:real^M->bool. 2 <= dimindex(:N) /\ s homeomorphic t /\ convex t /\ compact t ==> path_connected((:real^N) DIFF s)`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM (MP_TAC o MATCH_MP CONNECTED_COMPLEMENT_HOMEOMORPHIC_CONVEX_COMPACT) THEN MATCH_MP_TAC EQ_IMP THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC PATH_CONNECTED_EQ_CONNECTED THEN REWRITE_TAC[GSYM closed] THEN ASM_MESON_TAC[HOMEOMORPHIC_COMPACTNESS; COMPACT_INTERVAL; COMPACT_IMP_CLOSED]);; (* ------------------------------------------------------------------------- *) (* In particular, apply all these to the special case of an arc. *) (* ------------------------------------------------------------------------- *) let RETRACTION_ARC = prove (`!p. arc p ==> ?f. f continuous_on (:real^N) /\ IMAGE f (:real^N) SUBSET path_image p /\ (!x. x IN path_image p ==> f x = x)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `(:real^N)` o MATCH_MP (REWRITE_RULE[IMP_CONJ] ABSOLUTE_RETRACT_PATH_IMAGE_ARC)) THEN REWRITE_TAC[SUBSET_UNIV; retract_of; retraction]);; let PATH_CONNECTED_ARC_COMPLEMENT = prove (`!p. 2 <= dimindex(:N) /\ arc p ==> path_connected((:real^N) DIFF path_image p)`, REWRITE_TAC[arc; path] THEN REPEAT STRIP_TAC THEN SIMP_TAC[path_image] THEN MP_TAC(ISPECL [`path_image p:real^N->bool`; `interval[vec 0:real^1,vec 1]`] PATH_CONNECTED_COMPLEMENT_HOMEOMORPHIC_CONVEX_COMPACT) THEN ASM_REWRITE_TAC[CONVEX_INTERVAL; COMPACT_INTERVAL; path_image] THEN DISCH_THEN MATCH_MP_TAC THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN MATCH_MP_TAC HOMEOMORPHIC_COMPACT THEN EXISTS_TAC `p:real^1->real^N` THEN ASM_REWRITE_TAC[COMPACT_INTERVAL]);; let CONNECTED_ARC_COMPLEMENT = prove (`!p. 2 <= dimindex(:N) /\ arc p ==> connected((:real^N) DIFF path_image p)`, SIMP_TAC[PATH_CONNECTED_ARC_COMPLEMENT; PATH_CONNECTED_IMP_CONNECTED]);; let INSIDE_ARC_EMPTY = prove (`!p:real^1->real^N. arc p ==> inside(path_image p) = {}`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `dimindex(:N) = 1` THENL [MATCH_MP_TAC INSIDE_CONVEX THEN ASM_SIMP_TAC[CONVEX_CONNECTED_1_GEN; CONNECTED_PATH_IMAGE; ARC_IMP_PATH]; MATCH_MP_TAC INSIDE_BOUNDED_COMPLEMENT_CONNECTED_EMPTY THEN ASM_SIMP_TAC[BOUNDED_PATH_IMAGE; ARC_IMP_PATH] THEN MATCH_MP_TAC CONNECTED_ARC_COMPLEMENT THEN ASM_REWRITE_TAC[ARITH_RULE `2 <= n <=> 1 <= n /\ ~(n = 1)`] THEN REWRITE_TAC[DIMINDEX_GE_1]]);; let INSIDE_SIMPLE_CURVE_IMP_CLOSED = prove (`!g x:real^N. simple_path g /\ x IN inside(path_image g) ==> pathfinish g = pathstart g`, MESON_TAC[ARC_SIMPLE_PATH; INSIDE_ARC_EMPTY; NOT_IN_EMPTY]);; (* ------------------------------------------------------------------------- *) (* Some nice theorems giving accessibility for ANR complement components *) (* (from Hu's "Theory of Retracts", apparently originally from Borsuk). *) (* ------------------------------------------------------------------------- *) let FINITE_ANR_COMPLEMENT_COMPONENTS_CONCENTRIC = prove (`!s p:real^N a b. compact s /\ ANR s /\ a < b ==> FINITE {c | c IN components(cball(p,b) DIFF s) /\ ~(closure c INTER cball(p,a) = {})}`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `(:real^N)`] ANR_IMP_NEIGHBOURHOOD_RETRACT) THEN REWRITE_TAC[SUBTOPOLOGY_UNIV; GSYM OPEN_IN; GSYM CLOSED_IN] THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retract_of]) THEN REWRITE_TAC[retraction; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `r:real^N->real^N` THEN STRIP_TAC THEN SUBGOAL_THEN `?d. &0 < d /\ {x + e:real^N | x IN s /\ e IN cball(vec 0,d)} SUBSET u /\ !w. w IN {x + e:real^N | x IN s /\ e IN cball(vec 0,d)} ==> dist(w,r w) <= (b - a) / &4` STRIP_ASSUME_TAC THENL [SUBGOAL_THEN `?d. &0 < d /\ {x + e:real^N | x IN s /\ e IN cball(vec 0,d)} SUBSET u` STRIP_ASSUME_TAC THENL [ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[SET_RULE `{f x y | x IN {} /\ P y} SUBSET u`] THENL [MESON_TAC[REAL_LT_01]; ALL_TAC] THEN ASM_CASES_TAC `u = (:real^N)` THEN ASM_REWRITE_TAC[SUBSET_UNIV] THENL [MESON_TAC[REAL_LT_01]; ALL_TAC] THEN EXISTS_TAC `setdist(s,(:real^N) DIFF u) / &2` THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [REWRITE_TAC[REAL_HALF; SETDIST_POS_LT] THEN ASM_SIMP_TAC[SETDIST_EQ_0_COMPACT_CLOSED; GSYM OPEN_CLOSED] THEN ASM SET_TAC[]; REWRITE_TAC[REAL_HALF; SUBSET; FORALL_IN_GSPEC] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `e:real^N`] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[IN_CBALL_0] THEN DISCH_TAC THEN MATCH_MP_TAC(REAL_ARITH `&0 < s /\ s <= e ==> ~(e <= s / &2)`) THEN ASM_REWRITE_TAC[] THEN SUBST1_TAC(NORM_ARITH `norm(e:real^N) = dist(x,x + e)`) THEN MATCH_MP_TAC SETDIST_LE_DIST THEN ASM SET_TAC[]]; SUBGOAL_THEN `(r:real^N->real^N) uniformly_continuous_on {x + e | x IN s /\ e IN cball(vec 0,d)}` MP_TAC THENL [MATCH_MP_TAC COMPACT_UNIFORMLY_CONTINUOUS THEN ASM_SIMP_TAC[COMPACT_SUMS; COMPACT_CBALL] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; REWRITE_TAC[uniformly_continuous_on]] THEN DISCH_THEN(MP_TAC o SPEC `(b - a) / &8`) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `min d (min (e / &2) ((b - a) / &8))` THEN ASM_REWRITE_TAC[REAL_LT_MIN; REAL_HALF] THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN CONJ_TAC THENL [REWRITE_TAC[CBALL_MIN_INTER] THEN ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[FORALL_IN_GSPEC; IN_CBALL_0; REAL_LE_MIN] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN MATCH_MP_TAC(NORM_ARITH `dist(r x,r(x + y)) < e / &8 /\ norm y <= e / &8 /\ r x = x ==> dist(x + y:real^N,r(x + y)) <= e / &4`) THEN REPEAT CONJ_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[NORM_ARITH `&0 < e /\ norm y <= e / &2 ==> dist(x:real^N,x + y) < e`] THEN REWRITE_TAC[IN_ELIM_THM; IN_CBALL_0] THEN CONJ_TAC THEN EXISTS_TAC `x:real^N` THENL [EXISTS_TAC `y:real^N`; EXISTS_TAC `vec 0:real^N`] THEN ASM_SIMP_TAC[NORM_0; VECTOR_ADD_RID; REAL_LT_IMP_LE]; FIRST_ASSUM ACCEPT_TAC; ASM_SIMP_TAC[]]]; ABBREV_TAC `sd = {x + e:real^N | x IN s /\ e IN cball(vec 0,d)}`] THEN SUBGOAL_THEN `(s:real^N->bool) SUBSET interior sd` ASSUME_TAC THENL [TRANS_TAC SUBSET_TRANS `{x + e:real^N | x IN s /\ e IN ball(vec 0,d)}` THEN CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `vec 0 IN t /\ (!x:real^N. f x (vec 0) = x) ==> s SUBSET {f x y | x IN s /\ y IN t}`) THEN ASM_REWRITE_TAC[CENTRE_IN_BALL; VECTOR_ADD_RID]; SIMP_TAC[INTERIOR_MAXIMAL_EQ; OPEN_SUMS; OPEN_BALL] THEN EXPAND_TAC "sd" THEN REWRITE_TAC[GSYM BALL_UNION_SPHERE] THEN SET_TAC[]]; ALL_TAC] THEN SUBGOAL_THEN `(s:real^N->bool) SUBSET sd` ASSUME_TAC THENL [ASM_MESON_TAC[SUBSET_TRANS; INTERIOR_SUBSET]; ALL_TAC] THEN SUBGOAL_THEN `compact(sd:real^N->bool)` ASSUME_TAC THENL [EXPAND_TAC "sd" THEN ASM_SIMP_TAC[COMPACT_SUMS; COMPACT_CBALL]; ALL_TAC] THEN SUBGOAL_THEN `FINITE {c | c IN components(cball(p:real^N,b) DIFF s) /\ ~(c INTER (cball(p,b) DIFF interior sd) = {})}` MP_TAC THENL [MATCH_MP_TAC FINITE_COMPONENTS_MEETING_COMPACT_SUBSET THEN REPEAT CONJ_TAC THENL [SIMP_TAC[COMPACT_DIFF; COMPACT_CBALL; OPEN_INTERIOR]; MATCH_MP_TAC LOCALLY_OPEN_SUBSET THEN EXISTS_TAC `cball(p:real^N,b)` THEN SIMP_TAC[CONVEX_IMP_LOCALLY_CONNECTED; CONVEX_CBALL] THEN ASM_SIMP_TAC[OPEN_IN_DIFF_CLOSED; COMPACT_IMP_CLOSED]; ASM SET_TAC[]]; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] FINITE_SUBSET) THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `c:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN ASM_CASES_TAC `(c:real^N->bool) SUBSET interior sd` THENL [DISCH_THEN(K ALL_TAC); ASM SET_TAC[]]] THEN SUBGOAL_THEN `closure c SUBSET (sd:real^N->bool)` ASSUME_TAC THENL [MATCH_MP_TAC CLOSURE_MINIMAL THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED] THEN ASM_MESON_TAC[INTERIOR_SUBSET; SUBSET_TRANS]; ALL_TAC] THEN SUBGOAL_THEN `frontier c SUBSET (sd:real^N->bool)` ASSUME_TAC THENL [REWRITE_TAC[frontier] THEN ASM SET_TAC[]; ALL_TAC] THEN ABBREV_TAC `h = cball(p:real^N,a + &3 / &4 * (b - a))` THEN SUBGOAL_THEN `(h:real^N->bool) INTER frontier c SUBSET s` ASSUME_TAC THENL [FIRST_ASSUM(MP_TAC o MATCH_MP FRONTIER_OF_COMPONENTS_SUBSET) THEN MATCH_MP_TAC(SET_RULE `h INTER g SUBSET s ==> f SUBSET g ==> h INTER f SUBSET s`) THEN ONCE_REWRITE_TAC[SET_RULE `s DIFF t = s INTER (UNIV DIFF t)`] THEN W(MP_TAC o PART_MATCH lhand FRONTIER_INTER_SUBSET o rand o lhand o snd) THEN MATCH_MP_TAC(SET_RULE `h INTER g SUBSET s ==> f SUBSET g ==> h INTER f SUBSET s`) THEN REWRITE_TAC[FRONTIER_CBALL; UNION_OVER_INTER; UNION_SUBSET] THEN REWRITE_TAC[FRONTIER_COMPLEMENT] THEN ASM_SIMP_TAC[frontier; CLOSURE_CLOSED; COMPACT_IMP_CLOSED] THEN CONJ_TAC THENL [EXPAND_TAC "h"; SET_TAC[]] THEN REWRITE_TAC[SUBSET; IN_CBALL; IN_SPHERE; IN_INTER] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `?g. g continuous_on (h UNION frontier c) /\ (!x. x IN h ==> (g:real^N->real^N) x = vec 0) /\ (!x. x IN frontier c ==> g x = r x - x)` STRIP_ASSUME_TAC THENL [EXISTS_TAC `\x:real^N. if x IN frontier c then r x - x else vec 0` THEN SIMP_TAC[] THEN REWRITE_TAC[COND_RAND; COND_RATOR; VECTOR_SUB_EQ] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN GEN_REWRITE_TAC RAND_CONV [UNION_COMM] THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL THEN EXPAND_TAC "h" THEN SIMP_TAC[CLOSED_SUBSET_EQ; CLOSED_CBALL; FRONTIER_CLOSED] THEN ASM_REWRITE_TAC[VECTOR_SUB_EQ; SUBSET_UNION; CONTINUOUS_ON_CONST] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN REWRITE_TAC[CONTINUOUS_ON_ID] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; ALL_TAC] THEN ABBREV_TAC `D = cball(vec 0:real^N,(b - a) / &4)` THEN SUBGOAL_THEN `IMAGE (g:real^N->real^N) (h UNION frontier c) SUBSET D` ASSUME_TAC THENL [REWRITE_TAC[IMAGE_UNION; UNION_SUBSET] THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE] THEN EXPAND_TAC "D" THEN REWRITE_TAC[CENTRE_IN_CBALL] THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; REWRITE_TAC[IN_CBALL_0]] THEN REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] (GSYM dist)] THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`g:real^N->real^N`; `cball(p:real^N,b)`; `h UNION frontier c:real^N->bool`; `D:real^N->bool`] AR_IMP_ABSOLUTE_EXTENSOR) THEN ASM_REWRITE_TAC[] THEN EXPAND_TAC "D" THEN REWRITE_TAC[AR_CBALL] THEN REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC] THEN REPEAT CONJ_TAC THENL [ASM_REAL_ARITH_TAC; MATCH_MP_TAC CLOSED_SUBSET THEN EXPAND_TAC "h" THEN SIMP_TAC[CLOSED_UNION; FRONTIER_CLOSED; CLOSED_CBALL; UNION_SUBSET] THEN REWRITE_TAC[SUBSET_BALLS; DIST_REFL] THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[frontier] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s DIFF u SUBSET t`) THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN REWRITE_TAC[CLOSED_CBALL] THEN ASM SET_TAC[]; DISCH_THEN(X_CHOOSE_THEN `g':real^N->real^N` STRIP_ASSUME_TAC)] THEN ABBREV_TAC `f:real^N->real^N = \x. r x - g' x` THEN SUBGOAL_THEN `!x:real^N. x IN frontier c ==> f x = x` (LABEL_TAC "1") THENL [EXPAND_TAC "f" THEN REWRITE_TAC[] THEN ASM_SIMP_TAC[IN_UNION] THEN REPEAT STRIP_TAC THEN CONV_TAC VECTOR_ARITH; ALL_TAC] THEN SUBGOAL_THEN `!x. x IN closure c INTER h ==> (f:real^N->real^N) x = r x` (LABEL_TAC "2") THENL [EXPAND_TAC "f" THEN REWRITE_TAC[] THEN ASM_SIMP_TAC[IN_UNION; IN_INTER] THEN REPEAT STRIP_TAC THEN CONV_TAC VECTOR_ARITH; ALL_TAC] THEN SUBGOAL_THEN `!x:real^N. x IN closure c ==> dist(x,f x) <= (b - a) / &2` (LABEL_TAC "3") THENL [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN EXPAND_TAC "f" THEN REWRITE_TAC[] THEN MATCH_MP_TAC(NORM_ARITH `dist(x:real^N,r x) <= e / &4 /\ norm(g x) <= e / &4 ==> dist(x,r x - g x) <= e / &2`) THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; FORALL_IN_IMAGE; IN_CBALL_0]) THEN FIRST_X_ASSUM 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 CLOSURE_MINIMAL THEN REWRITE_TAC[CLOSED_CBALL] THEN ASM SET_TAC[]; ALL_TAC] THEN UNDISCH_TAC `~(closure c INTER cball(p:real^N,a) = {})` THEN PURE_REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN DISCH_THEN(X_CHOOSE_THEN `l:real^N` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CLOSURE_APPROACHABLE]) THEN DISCH_THEN(MP_TAC o SPEC `(b - a) / &5`) THEN REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `q:real^N` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `setdist({q},(:real^N) DIFF h) > (b - a) / &2` ASSUME_TAC THENL [MP_TAC(ISPECL [`(:real^N) DIFF h`; `q:real^N`; `l:real^N`] SETDIST_SING_TRIANGLE) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `d < (b - a) / &5 ==> &3 / &4 * (b - a) <= l ==> abs(q - l) <= d ==> q > (b - a) / &2`)) THEN MATCH_MP_TAC REAL_LE_SETDIST THEN REWRITE_TAC[NOT_INSERT_EMPTY] THEN REWRITE_TAC[SET_RULE `s DIFF t = {} <=> s SUBSET t`; IN_SING] THEN EXPAND_TAC "h" THEN CONJ_TAC THENL [MESON_TAC[BOUNDED_SUBSET; NOT_BOUNDED_UNIV; BOUNDED_CBALL]; REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_UNWIND_THM2] THEN UNDISCH_TAC `l IN cball(p:real^N,a)` THEN REWRITE_TAC[IN_DIFF; IN_UNIV; IN_CBALL] THEN CONV_TAC NORM_ARITH]; ALL_TAC] THEN SUBGOAL_THEN `~(q IN IMAGE (f:real^N->real^N) (closure c))` (LABEL_TAC "4") THENL [REWRITE_TAC[IN_IMAGE; NOT_EXISTS_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN ASM_CASES_TAC `(x:real^N) IN h` THENL [ASM SET_TAC[]; ALL_TAC] THEN REMOVE_THEN "3" (MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `d > e ==> d <= x ==> ~(x <= e)`)) THEN ONCE_REWRITE_TAC[DIST_SYM] THEN MATCH_MP_TAC SETDIST_LE_DIST THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`f:real^N->real^N`; `closure c:real^N->bool`] COMPACT_SUBSET_FRONTIER_RETRACTION) THEN REWRITE_TAC[NOT_IMP] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[COMPACT_CLOSURE] THEN MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `cball(p:real^N,b) DIFF s` THEN ASM_REWRITE_TAC[] THEN SIMP_TAC[BOUNDED_DIFF; BOUNDED_CBALL]; EXPAND_TAC "f" THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THENL [ASM SET_TAC[]; MATCH_MP_TAC CLOSURE_MINIMAL] THEN REWRITE_TAC[CLOSED_CBALL] THEN ASM SET_TAC[]; MP_TAC(ISPEC `c:real^N->bool` FRONTIER_CLOSURE_SUBSET) THEN ASM SET_TAC[]; REWRITE_TAC[SUBSET] THEN DISCH_THEN(MP_TAC o SPEC `q:real^N`) THEN ASM_SIMP_TAC[CLOSURE_INC]]);; let ACCESSIBLE_FRONTIER_ANR_INTER_COMPLEMENT_COMPONENT = prove (`!s c p:real^N b. compact s /\ ANR s /\ c IN components(b DIFF s) /\ p IN frontier c /\ p IN interior b ==> ?g. arc g /\ pathfinish g = p /\ !t. t IN interval[vec 0,vec 1] DELETE (vec 1) ==> g(t) IN c`, let lemma = prove (`!s p:real^N a b c. compact s /\ ANR s /\ &0 < a /\ cball(p,a) SUBSET b /\ c IN components(b DIFF s) /\ p IN frontier c ==> ?d. d IN components(cball(p,a) INTER c) /\ p IN frontier d`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `p:real^N`; `a / &2`; `a:real`] FINITE_ANR_COMPLEMENT_COMPONENTS_CONCENTRIC) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(MP_TAC o SPEC `{d | d IN components(cball(p,a) INTER c) /\ ~(closure d INTER cball(p:real^N,a / &2) = {})}` o MATCH_MP(REWRITE_RULE[IMP_CONJ] FINITE_SUBSET)) THEN ANTS_TAC THENL [REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `d:real^N->bool` THEN MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN DISCH_TAC THEN MP_TAC(ISPECL [`b DIFF s:real^N->bool`; `cball(p:real^N,a)`; `c:real^N->bool`; `d:real^N->bool`] COMPONENTS_INTER_COMPONENTS) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN AP_TERM_TAC THEN ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(MP_TAC o MATCH_MP CLOSURE_UNIONS) THEN DISCH_THEN(MP_TAC o SPEC `p:real^N` o MATCH_MP (SET_RULE `s = t ==> !x. x IN s ==> x IN t`)) THEN ANTS_TAC THENL [SUBGOAL_THEN `p IN closure (UNIONS {d | d IN components (cball(p:real^N,a) INTER c) /\ ~(closure d INTER cball (p,a / &2) = {})} UNION UNIONS {d | d IN components (cball(p,a) INTER c) /\ closure d INTER cball (p,a / &2) = {}})` MP_TAC THENL [REWRITE_TAC[GSYM UNIONS_UNION; GSYM UNIONS_COMPONENTS; SET_RULE `{x | x IN s /\ ~P x} UNION {x | x IN s /\ P x} = s`] THEN MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ x IN s ==> x IN t`) THEN EXISTS_TAC `closure(ball(p:real^N,a) INTER c)` THEN SIMP_TAC[SUBSET_CLOSURE; BALL_SUBSET_CBALL; SET_RULE `s SUBSET t ==> s INTER c SUBSET t INTER c`] THEN W(MP_TAC o PART_MATCH (rand o rand) OPEN_INTER_CLOSURE_SUBSET o rand o snd) THEN REWRITE_TAC[OPEN_BALL] THEN MATCH_MP_TAC(SET_RULE `x IN s ==> s SUBSET t ==> x IN t`) THEN RULE_ASSUM_TAC(REWRITE_RULE[frontier; IN_DIFF]) THEN ASM_REWRITE_TAC[IN_INTER; CENTRE_IN_BALL]; REWRITE_TAC[CLOSURE_UNION; IN_UNION] THEN MATCH_MP_TAC(TAUT `~p ==> q \/ p ==> q`) THEN MATCH_MP_TAC(SET_RULE `!t. ~(x IN t) /\ s SUBSET t ==> ~(x IN s)`) THEN EXISTS_TAC `(:real^N) DIFF ball(p,a / &2)` THEN ASM_REWRITE_TAC[IN_DIFF; IN_UNIV; CENTRE_IN_BALL; REAL_HALF] THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN REWRITE_TAC[GSYM OPEN_CLOSED; OPEN_BALL] THEN REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC] THEN X_GEN_TAC `d:real^N->bool` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE `d INTER cball(x:real^N,r) = {} ==> ball(x,r) SUBSET cball(x,r) ==> ball(x,r) INTER d = {}`)) THEN SIMP_TAC[BALL_SUBSET_CBALL; OPEN_INTER_CLOSURE_EQ_EMPTY; OPEN_BALL] THEN SET_TAC[]]; REWRITE_TAC[IN_UNIONS; EXISTS_IN_GSPEC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[frontier; IN_DIFF] THEN UNDISCH_TAC `(p:real^N) IN frontier c` THEN REWRITE_TAC[frontier] THEN MATCH_MP_TAC(SET_RULE `d SUBSET c ==> p IN s DIFF c ==> ~(p IN d)`) THEN MATCH_MP_TAC SUBSET_INTERIOR THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET)) THEN SET_TAC[]]) in REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERIOR_CBALL]) THEN DISCH_THEN(X_CHOOSE_THEN `a:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `?u. (!n. u n IN components(cball(p:real^N,min a (inv(&2 pow n))) INTER c) /\ p IN frontier(u n)) /\ (!n. u(SUC n) SUBSET u n)` MP_TAC THENL [MATCH_MP_TAC DEPENDENT_CHOICE THEN CONJ_TAC THENL [CONV_TAC REAL_RAT_REDUCE_CONV THEN MP_TAC(ISPECL [`s:real^N->bool`; `p:real^N`; `min a (&1)`; `b:real^N->bool`; `c:real^N->bool`] lemma) THEN ASM_REWRITE_TAC[REAL_LT_MIN; REAL_LT_01] THEN DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[CBALL_MIN_INTER] THEN ASM SET_TAC[]; MAP_EVERY X_GEN_TAC [`n:num`; `d:real^N->bool`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `p:real^N`; `min a (inv(&2 pow (SUC n)))`; `cball(p:real^N,min a (inv(&2 pow n)))`; `d:real^N->bool`] lemma) THEN ASM_REWRITE_TAC[REAL_LT_INV_EQ; REAL_LT_POW2; REAL_LT_MIN] THEN SIMP_TAC[REAL_LT_INV2; REAL_LT_INV_EQ; REAL_LT_POW2; REAL_POW_MONO_LT; REAL_ARITH `&1 < &2`; ARITH_RULE `n < SUC n`; SUBSET_BALLS; DIST_REFL; REAL_ADD_LID; REAL_ARITH `x < y ==> min a x <= min a y`] THEN ANTS_TAC THENL [MP_TAC(ISPECL [`b DIFF s:real^N->bool`; `cball(p:real^N,min a (inv(&2 pow n)))`; `c:real^N->bool`; `d:real^N->bool`] COMPONENTS_INTER_COMPONENTS) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[CBALL_MIN_INTER] THEN ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `e:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[GSYM CONJ_ASSOC] THEN CONJ_TAC THENL [MP_TAC(ISPECL [`cball(p:real^N,min a (inv(&2 pow n))) INTER c`; `cball(p:real^N,min a (inv(&2 pow SUC n)))`; `d:real^N->bool`; `e:real^N->bool`] COMPONENTS_INTER_COMPONENTS) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[CBALL_MIN_INTER] THEN MATCH_MP_TAC(SET_RULE `n SUBSET s ==> (b INTER n) INTER (b INTER s) INTER c = (b INTER n) INTER c`) THEN MATCH_MP_TAC SUBSET_CBALL THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_LT_POW2] THEN MATCH_MP_TAC REAL_POW_MONO THEN REWRITE_TAC[REAL_OF_NUM_LE] THEN ARITH_TAC; REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET)) THEN ASM SET_TAC[]]]]; REWRITE_TAC[FORALL_AND_THM; LEFT_IMP_EXISTS_THM]] THEN X_GEN_TAC `u:num->real^N->bool` THEN STRIP_TAC THEN SUBGOAL_THEN `!n. (u:num->real^N->bool) n SUBSET c` ASSUME_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_SUBSET; SUBSET_TRANS; SUBSET_INTER]; ALL_TAC] THEN SUBGOAL_THEN `!n. u n IN components(cball(p:real^N,min a (inv(&2 pow n))) DIFF s)` ASSUME_TAC THENL [X_GEN_TAC `n:num` THEN MP_TAC(ISPECL [`b DIFF s:real^N->bool`; `cball(p:real^N,min a (inv(&2 pow n)))`; `c:real^N->bool`; `(u:num->real^N->bool) n`] COMPONENTS_INTER_COMPONENTS) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[CBALL_MIN_INTER] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `!n. ~((u:num->real^N->bool) n = {})` MP_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_NONEMPTY]; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; SKOLEM_THM]] THEN DISCH_THEN(X_CHOOSE_THEN `q:num->real^N` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `!n. ?f. (f:real^1->real^N) continuous_on interval[lift(inv(&2 pow (SUC n))),lift(inv(&2 pow n))] /\ IMAGE f (interval[lift(inv(&2 pow (SUC n))),lift(inv(&2 pow n))]) SUBSET u n /\ f(lift(inv(&2 pow n))) = q n /\ f(lift(inv(&2 pow (SUC n)))) = q(SUC n)` MP_TAC THENL [X_GEN_TAC `n:num` THEN SUBGOAL_THEN `path_component (u n) (q n:real^N) (q(SUC n))` MP_TAC THENL [W(MP_TAC o PART_MATCH (lhand o rand) PATH_COMPONENT_EQ_CONNECTED_COMPONENT o rator o snd) THEN ANTS_TAC THENL [MATCH_MP_TAC LOCALLY_OPEN_SUBSET THEN EXISTS_TAC `cball(p:real^N,min a (inv(&2 pow n))) DIFF s` THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC OPEN_IN_COMPONENTS_LOCALLY_CONNECTED THEN ASM_REWRITE_TAC[]] THEN MATCH_MP_TAC LOCALLY_OPEN_SUBSET THEN EXISTS_TAC `cball(p:real^N,min a (inv(&2 pow n)))` THEN ASM_SIMP_TAC[OPEN_IN_DIFF_CLOSED; COMPACT_IMP_CLOSED] THEN SIMP_TAC[CONVEX_IMP_LOCALLY_PATH_CONNECTED; CONVEX_CBALL; CONVEX_IMP_LOCALLY_CONNECTED]; DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[connected_component] THEN EXISTS_TAC `(u:num->real^N->bool) n` THEN CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]; ASM SET_TAC[]]]; ONCE_REWRITE_TAC[PATH_COMPONENT_SYM_EQ] THEN REWRITE_TAC[path_component; path; path_image; pathstart; pathfinish] THEN DISCH_THEN(X_CHOOSE_THEN `f:real^1->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(f:real^1->real^N) o (\x. &2 pow (SUC n) % (x - lift(inv(&2 pow (SUC n)))))` THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_CMUL; CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)); REWRITE_TAC[IMAGE_o] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE f i SUBSET u ==> s SUBSET i ==> IMAGE f s SUBSET u`))] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTERVAL_1; FORALL_LIFT] THEN REWRITE_TAC[LIFT_DROP; DROP_VEC; DROP_CMUL; DROP_SUB] THEN SIMP_TAC[REAL_LT_POW2; REAL_SUB_LDISTRIB; REAL_MUL_RINV; REAL_LT_IMP_NZ] THEN REWRITE_TAC[REAL_SUB_LE; REAL_ARITH `x - &1 <= &1 <=> x <= &2`] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN SIMP_TAC[REAL_LT_POW2; GSYM REAL_LE_LDIV_EQ; GSYM REAL_LE_RDIV_EQ] THEN REWRITE_TAC[real_pow; REAL_INV_MUL; real_div] THEN REAL_ARITH_TAC; REWRITE_TAC[o_THM; GSYM LIFT_SUB; GSYM LIFT_CMUL] THEN ASM_REWRITE_TAC[REAL_SUB_REFL; REAL_MUL_RZERO; LIFT_NUM] THEN REWRITE_TAC[real_pow; REAL_INV_MUL] THEN ASM_SIMP_TAC[REAL_LT_POW2; LIFT_NUM; REAL_FIELD `&0 < x ==> (&2 * x) * (inv x - inv(&2) * inv x) = &1`]]]; REWRITE_TAC[SKOLEM_THM; FORALL_AND_THM] THEN DISCH_THEN(X_CHOOSE_THEN `f:num->real^1->real^N` STRIP_ASSUME_TAC)] THEN MP_TAC(ISPECL [`subtopology euclidean (interval[vec 0:real^1,vec 1] DELETE (vec 0))`; `subtopology euclidean (c:real^N->bool)`; `f:num->real^1->real^N`; `\n. interval[lift(inv(&2 pow (SUC n))),lift(inv(&2 pow n))]`; `(:num)`] PASTING_LEMMA_EXISTS_LOCALLY_FINITE) THEN REWRITE_TAC[CONTINUOUS_MAP_EUCLIDEAN2; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; SUBTOPOLOGY_SUBTOPOLOGY] THEN ONCE_REWRITE_TAC[TAUT `closed_in a b /\ c <=> ~(closed_in a b ==> ~c)`] THEN SIMP_TAC[ISPEC `euclidean` CLOSED_IN_IMP_SUBSET; SET_RULE `s SUBSET u ==> u INTER s = s`] THEN REWRITE_TAC[NOT_IMP] THEN REWRITE_TAC[IN_UNIV] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [X_GEN_TAC `x:real^1` THEN REWRITE_TAC[IN_INTERVAL_1; IN_DELETE] THEN REWRITE_TAC[GSYM DROP_EQ; DROP_VEC; LIFT_DROP] THEN STRIP_TAC THEN MP_TAC(ISPECL [`inv(&2)`; `drop x / &9`] REAL_ARCH_POW_INV) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[REAL_POW_INV] THEN DISCH_THEN(X_CHOOSE_TAC `n:num`) THEN EXISTS_TAC `(interval[vec 0,vec 1] DELETE vec 0) INTER ball(x:real^1,inv(&2 pow n))` THEN SIMP_TAC[OPEN_IN_OPEN_INTER; OPEN_BALL; IN_INTER; CENTRE_IN_BALL] THEN REWRITE_TAC[IN_DELETE; IN_INTERVAL_1; REAL_LT_INV_EQ] THEN ASM_REWRITE_TAC[GSYM DROP_EQ; DROP_VEC; REAL_LT_POW2] THEN MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `{i | ~(interval[lift(inv(&2 pow SUC i)),lift(inv(&2 pow i))] INTER ball(x:real^1,inv(&2 pow n)) = {})}` THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN REWRITE_TAC[BALL_1; DISJOINT_INTERVAL_1] THEN REWRITE_TAC[DE_MORGAN_THM; DROP_ADD; DROP_SUB; LIFT_DROP] THEN REWRITE_TAC[REAL_NOT_LT; REAL_NOT_LE] THEN MATCH_MP_TAC(MESON[FINITE_SUBSET; FINITE_INSERT; FINITE_EMPTY] `(?a b. s SUBSET {a,b}) ==> FINITE s`) THEN MATCH_MP_TAC(SET_RULE `~(?a b c. a IN s /\ b IN s /\ c IN s /\ ~(a = b) /\ ~(a = c) /\ ~(b = c)) ==> ?a b. s SUBSET {a,b}`) THEN MATCH_MP_TAC(MESON[] `(!a b c. a IN s /\ b IN s /\ c IN s /\ ~(a = b) /\ ~(a = c) /\ ~(b = c) ==> ?x y. x IN s /\ y IN s /\ x + 2 <= y) /\ (!x y. x IN s /\ y IN s /\ x + 2 <= y ==> F) ==> ~(?a b c. a IN s /\ b IN s /\ c IN s /\ ~(a = b) /\ ~(a = c) /\ ~(b = c))`) THEN CONJ_TAC THENL [MAP_EVERY X_GEN_TAC [`a:num`; `b:num`; `c:num`] THEN STRIP_TAC THEN SUBGOAL_THEN `?x y. x IN {a,b,c} /\ y IN {a,b,c} /\ x + 2 <= y` MP_TAC THENL [SIMP_TAC[RIGHT_EXISTS_AND_THM; EXISTS_IN_INSERT; NOT_IN_EMPTY] THEN ASM_ARITH_TAC; REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN ASM SET_TAC[]]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`m:num`; `r:num`] THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN SUBGOAL_THEN `&2 * drop x - &2 / &2 pow n < drop x + inv(&2 pow n)` MP_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN TRANS_TAC REAL_LET_TRANS `inv(&2 pow (SUC m))` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `x - inv i < a ==> a <= inv(&2) * b ==> &2 * x - &2 / i <= b`)) THEN REWRITE_TAC[GSYM REAL_INV_MUL; GSYM(CONJUNCT2 real_pow)] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_LT_POW2] THEN MATCH_MP_TAC REAL_POW_MONO THEN REWRITE_TAC[REAL_OF_NUM_LE] THEN ASM_ARITH_TAC; REWRITE_TAC[SUBSET; UNIONS_GSPEC; IN_ELIM_THM; IN_DELETE] THEN X_GEN_TAC `y:real^1` THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN REWRITE_TAC[GSYM DROP_EQ; DROP_VEC; DIST_1; REAL_SUB_RZERO] THEN DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[real_abs] THEN STRIP_TAC THEN MP_TAC(fst(EQ_IMP_RULE(ISPEC`\n. drop y <= inv(&2 pow n)` num_MAX))) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [CONJ_TAC THENL [EXISTS_TAC `0` THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[]; MP_TAC(ISPECL [`inv(&2)`; `drop y`] REAL_ARCH_POW_INV) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `m:num` THEN REWRITE_TAC[REAL_POW_INV] THEN DISCH_TAC THEN X_GEN_TAC `m':num` THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[NOT_LE; REAL_NOT_LE] THEN DISCH_TAC THEN TRANS_TAC REAL_LT_TRANS `inv(&2 pow m)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LT_INV2 THEN REWRITE_TAC[REAL_LT_POW2] THEN MATCH_MP_TAC REAL_POW_MONO_LT THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[]]; MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[LIFT_DROP] THEN X_GEN_TAC `m:num` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(MP_TAC o SPEC `m + 1`) THEN ASM_REWRITE_TAC[ADD1; ARITH_RULE `~(m + 1 <= m)`] THEN REAL_ARITH_TAC]; X_GEN_TAC `n:num` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CLOSED_SUBSET THEN REWRITE_TAC[CLOSED_INTERVAL] THEN REWRITE_TAC[SET_RULE `s SUBSET t DELETE a <=> ~(a IN s) /\ s SUBSET t`] THEN REWRITE_TAC[SUBSET_INTERVAL_1; IN_INTERVAL_1; DROP_VEC; LIFT_DROP] THEN SIMP_TAC[DE_MORGAN_THM; REAL_NOT_LE; REAL_LT_INV_EQ; REAL_LE_INV_EQ; REAL_LT_POW2; REAL_LT_IMP_LE] THEN DISJ2_TAC THEN SIMP_TAC[REAL_INV_LE_1; REAL_LE_POW2] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_LT_POW2] THEN MATCH_MP_TAC REAL_POW_MONO THEN REWRITE_TAC[REAL_OF_NUM_LE] THEN ARITH_TAC; MATCH_MP_TAC WLOG_LT THEN REWRITE_TAC[] THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN STRIP_TAC THEN MATCH_MP_TAC(SET_RULE `(~(t = {}) ==> !x. x IN s /\ x IN t ==> P x) ==> !x. x IN s INTER t ==> P x`) THEN REWRITE_TAC[DISJOINT_INTERVAL_1; DE_MORGAN_THM; LIFT_DROP] THEN ASM_CASES_TAC `SUC m < n` THEN ASM_SIMP_TAC[REAL_LT_INV2; REAL_LT_POW2; REAL_POW_MONO_LT; REAL_ARITH `&1 < &2`] THEN DISCH_THEN(K ALL_TAC) THEN SUBGOAL_THEN `n = SUC m` SUBST_ALL_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[IN_INTER; IN_DELETE; IN_INTERVAL_1; LIFT_DROP; DROP_VEC] THEN X_GEN_TAC `x:real^1` THEN STRIP_TAC THEN SUBGOAL_THEN `drop x = inv(&2 pow (SUC m))` MP_TAC THENL [ASM_REAL_ARITH_TAC; REWRITE_TAC[GSYM LIFT_EQ; LIFT_DROP]] THEN DISCH_THEN SUBST1_TAC THEN ASM_REWRITE_TAC[]]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `h:real^1->real^N` STRIP_ASSUME_TAC) THEN ABBREV_TAC `g = \x. if x = vec 0 then p else (h:real^1->real^N) x` THEN SUBGOAL_THEN `path g /\ pathstart g = (p:real^N) /\ (!t. t IN interval[vec 0,vec 1] DELETE vec 0 ==> g t IN c)` STRIP_ASSUME_TAC THENL [EXPAND_TAC "g" THEN REWRITE_TAC[pathstart; IN_DELETE] THEN SIMP_TAC[] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN SIMP_TAC[path; CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN; CONTINUOUS_WITHIN] THEN X_GEN_TAC `x:real^1` THEN DISCH_TAC THEN ASM_CASES_TAC `x:real^1 = vec 0` THEN ASM_SIMP_TAC[] THENL [REWRITE_TAC[LIM_WITHIN_LE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(ISPECL [`inv(&2)`; `e:real`] REAL_ARCH_POW_INV) THEN ASM_REWRITE_TAC[REAL_POW_INV] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN DISCH_THEN(X_CHOOSE_THEN `n:num` STRIP_ASSUME_TAC) THEN EXISTS_TAC `inv(&2 pow n)` THEN REWRITE_TAC[REAL_LT_POW2; REAL_LT_INV_EQ; GSYM DIST_NZ] THEN EXPAND_TAC "g" THEN SIMP_TAC[] THEN X_GEN_TAC `y:real^1` THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN REWRITE_TAC[GSYM DROP_EQ; DROP_VEC; DIST_1; REAL_SUB_RZERO] THEN DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[real_abs] THEN STRIP_TAC THEN MP_TAC(fst(EQ_IMP_RULE(ISPEC`\n. drop y <= inv(&2 pow n)` num_MAX))) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`inv(&2)`; `drop y`] REAL_ARCH_POW_INV) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `m:num` THEN REWRITE_TAC[REAL_POW_INV] THEN DISCH_TAC THEN X_GEN_TAC `m':num` THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[NOT_LE; REAL_NOT_LE] THEN DISCH_TAC THEN TRANS_TAC REAL_LT_TRANS `inv(&2 pow m)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LT_INV2 THEN REWRITE_TAC[REAL_LT_POW2] THEN MATCH_MP_TAC REAL_POW_MONO_LT THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[]; DISCH_THEN(X_CHOOSE_THEN `m:num` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(fun th -> MP_TAC(SPEC `n:num` th) THEN MP_TAC(SPEC `m + 1` th)) THEN ASM_REWRITE_TAC[REAL_NOT_LE; ARITH_RULE `~(m + 1 <= m)`] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `h y = (f:num->real^1->real^N) m y` SUBST1_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; IN_DELETE; IN_INTER] THEN ASM_REWRITE_TAC[GSYM DROP_EQ; LIFT_DROP; ADD1; DROP_VEC] THEN ASM_SIMP_TAC[REAL_LT_IMP_LE]; TRANS_TAC REAL_LET_TRANS `inv(&2 pow n)` THEN ASM_REWRITE_TAC[] THEN TRANS_TAC REAL_LE_TRANS `inv(&2 pow m)` THEN ASM_SIMP_TAC[REAL_LE_INV2; REAL_LT_INV_EQ; REAL_LT_POW2; REAL_POW_MONO; REAL_ARITH `&1 <= &2`] THEN TRANS_TAC REAL_LE_TRANS `min a (inv(&2 pow m))` THEN CONJ_TAC THENL [ALL_TAC; REAL_ARITH_TAC] THEN REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] (GSYM IN_CBALL)] THEN MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ x IN s ==> x IN t`) THEN EXISTS_TAC `cball(p:real^N,min a (inv(&2 pow m))) DIFF s` THEN REWRITE_TAC[SUBSET_DIFF] THEN MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ x IN s ==> x IN t`) THEN EXISTS_TAC `(u:num->real^N->bool) m` THEN ASM_SIMP_TAC[IN_COMPONENTS_SUBSET] THEN RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; FORALL_IN_IMAGE]) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; LIFT_DROP] THEN ASM_SIMP_TAC[ADD1; REAL_LT_IMP_LE]]]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN]) THEN DISCH_THEN(MP_TAC o SPEC `x:real^1`) THEN ASM_REWRITE_TAC[IN_DELETE; CONTINUOUS_WITHIN] THEN REWRITE_TAC[tendsto] THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[EVENTUALLY_WITHIN_IMP] THEN MP_TAC(ISPECL [`(:real^1) DELETE vec 0`; `x:real^1`] EVENTUALLY_IN_OPEN) THEN ASM_SIMP_TAC[IN_DELETE; IN_UNIV; OPEN_DELETE; OPEN_UNIV] THEN REWRITE_TAC[IMP_IMP; GSYM EVENTUALLY_AND] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN EXPAND_TAC "g" THEN SIMP_TAC[IMP_CONJ]]; MP_TAC(ISPECL [`reversepath g:real^1->real^N`; `pathfinish g:real^N`; `p:real^N`] PATH_CONTAINS_ARC) THEN REWRITE_TAC[PATHSTART_REVERSEPATH; PATHFINISH_REVERSEPATH] THEN ASM_REWRITE_TAC[PATH_REVERSEPATH; PATH_IMAGE_REVERSEPATH] THEN ANTS_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPEC `vec 1:real^1`) THEN REWRITE_TAC[IN_DELETE; ENDS_IN_UNIT_INTERVAL; VEC_EQ; ARITH_EQ] THEN REWRITE_TAC[pathfinish] THEN FIRST_ASSUM(MP_TAC o MATCH_MP FRONTIER_OF_COMPONENTS_SUBSET) THEN ONCE_REWRITE_TAC[SET_RULE `s DIFF t = s INTER (UNIV DIFF t)`] THEN DISCH_THEN(MP_TAC o SPEC `p:real^N` o REWRITE_RULE[SUBSET]) THEN ASM_REWRITE_TAC[FRONTIER_INTER; IN_INTER] THEN REWRITE_TAC[IN_UNION; FRONTIER_CBALL; FRONTIER_COMPLEMENT] THEN ASM_SIMP_TAC[IN_SPHERE; DIST_REFL; REAL_LT_IMP_NZ] THEN ASM_SIMP_TAC[frontier; IN_DIFF; CLOSURE_CLOSED; COMPACT_IMP_CLOSED] THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^1->real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [arc]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `(!x y. x IN i /\ y IN i /\ f x = f y ==> x = y) ==> z IN i /\ IMAGE f i DELETE f z SUBSET c ==> (!x. x IN i DELETE z ==> f x IN c)`)) THEN REWRITE_TAC[ENDS_IN_UNIT_INTERVAL; GSYM path_image] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `a SUBSET g ==> g DELETE z SUBSET u ==> a DELETE z SUBSET u`)) THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM pathfinish] THEN ASM_REWRITE_TAC[path_image] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `(!x. x IN i DELETE z ==> g x IN c) ==> g z = p ==> IMAGE g i DELETE p SUBSET c`)) THEN ASM_MESON_TAC[pathstart]]]);; let ACCESSIBLE_FRONTIER_ANR_COMPLEMENT_COMPONENT = prove (`!s c x y. compact s /\ ANR s /\ c IN components((:real^N) DIFF s) /\ x IN c /\ y IN frontier c ==> ?g. arc g /\ pathstart g = x /\ pathfinish g = y /\ !t. t IN interval[vec 0,vec 1] DELETE (vec 1) ==> g(t) IN c`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] OPEN_COMPONENTS)) THEN ASM_SIMP_TAC[GSYM closed; COMPACT_IMP_CLOSED] THEN DISCH_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `c:real^N->bool`; `y:real^N`; `(:real^N)`] ACCESSIBLE_FRONTIER_ANR_INTER_COMPLEMENT_COMPONENT) THEN ASM_REWRITE_TAC[INTERIOR_UNIV; IN_UNIV] THEN DISCH_THEN(X_CHOOSE_THEN `g2:real^1->real^N` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `path_component c (x:real^N) (pathstart g2)` MP_TAC THENL [ASM_SIMP_TAC[OPEN_PATH_CONNECTED_COMPONENT] THEN REWRITE_TAC[connected_component] THEN EXISTS_TAC `c:real^N->bool` THEN CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]; ALL_TAC] THEN ASM_REWRITE_TAC[SUBSET_REFL; pathstart] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[ENDS_IN_UNIT_INTERVAL; IN_DELETE; VEC_EQ; ARITH_EQ]; REWRITE_TAC[path_component] THEN DISCH_THEN(X_CHOOSE_THEN `g1:real^1->real^N` STRIP_ASSUME_TAC)] THEN ABBREV_TAC `g:real^1->real^N = g1 ++ g2` THEN SUBGOAL_THEN `pathstart g:real^N = x /\ pathfinish g = y` STRIP_ASSUME_TAC THENL [EXPAND_TAC "g" THEN REWRITE_TAC[PATHSTART_JOIN; PATHFINISH_JOIN] THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `path(g:real^1->real^N)` ASSUME_TAC THENL [EXPAND_TAC "g" THEN ASM_SIMP_TAC[PATH_JOIN; ARC_IMP_PATH]; ALL_TAC] THEN SUBGOAL_THEN `!t. t IN interval[vec 0,vec 1] DELETE vec 1 ==> (g:real^1->real^N) t IN c` ASSUME_TAC THENL [X_GEN_TAC `t:real^1` THEN REWRITE_TAC[IN_INTERVAL_1; IN_DELETE; DROP_VEC; GSYM DROP_EQ] THEN STRIP_TAC THEN EXPAND_TAC "g" THEN REWRITE_TAC[joinpaths] THEN COND_CASES_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; FORALL_IN_IMAGE]) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[path_image] THEN MATCH_MP_TAC FUN_IN_IMAGE THEN REWRITE_TAC[IN_INTERVAL_1; DROP_CMUL; DROP_VEC] THEN ASM_REAL_ARITH_TAC; FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[IN_INTERVAL_1; IN_DELETE; DROP_SUB; DROP_CMUL; GSYM DROP_EQ; DROP_VEC] THEN ASM_REAL_ARITH_TAC]; MP_TAC(ISPECL [`g:real^1->real^N`; `x:real^N`; `y:real^N`] PATH_CONTAINS_ARC) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM FRONTIER_DISJOINT_EQ]) THEN ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `h:real^1->real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT(FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [arc])) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `(!x y. x IN i /\ y IN i /\ f x = f y ==> x = y) ==> z IN i /\ IMAGE f i DELETE f z SUBSET c ==> (!x. x IN i DELETE z ==> f x IN c)`)) THEN REWRITE_TAC[ENDS_IN_UNIT_INTERVAL; GSYM path_image] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `a SUBSET g ==> g DELETE z SUBSET u ==> a DELETE z SUBSET u`)) THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM pathfinish] THEN ASM_REWRITE_TAC[path_image] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `(!x. x IN i DELETE z ==> g x IN c) ==> g z = p ==> IMAGE g i DELETE p SUBSET c`)) THEN ASM_MESON_TAC[pathfinish]]]);; (* ------------------------------------------------------------------------- *) (* Some simple consequences for complement connectivity. *) (* ------------------------------------------------------------------------- *) let LPC_INTERMEDIATE_CLOSURE_ANR_COMPLEMENT_COMPONENT = prove (`!s c t. compact s /\ ANR s /\ c IN components ((:real^N) DIFF s) /\ c SUBSET t /\ t SUBSET closure c ==> locally path_connected t`, let lemma = prove (`!s c u p. compact s /\ ANR s /\ c IN components((:real^N) DIFF s) /\ p IN frontier c /\ open u /\ p IN u ==> ?v. open v /\ p IN v /\ v SUBSET u /\ !y. y IN c INTER v ==> path_component ((p INSERT c) INTER u) p y`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_CBALL]) THEN DISCH_THEN(MP_TAC o SPEC `p:real^N`) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `b:real` THEN STRIP_TAC THEN SUBGOAL_THEN `open(c:real^N->bool)` ASSUME_TAC THENL [REPEAT(FIRST_X_ASSUM((MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] OPEN_COMPONENTS)))) THEN ASM_SIMP_TAC[GSYM closed; COMPACT_IMP_CLOSED]; ALL_TAC] THEN SUBGOAL_THEN `?a. &0 < a /\ a < b /\ !d x. d IN components(cball(p,b) DIFF s) /\ x IN d /\ x IN ball(p:real^N,a) ==> p IN closure d` STRIP_ASSUME_TAC THENL [EXISTS_TAC `inf ((b / &2) INSERT IMAGE (\c. setdist({p:real^N},c)) {c | c IN components (cball (p,b) DIFF s) /\ ~(closure c INTER cball (p,b / &2) = {}) /\ ~(p IN closure c)})` THEN MP_TAC(ISPECL [`s:real^N->bool`; `p:real^N`; `b / &2`; `b:real`] FINITE_ANR_COMPLEMENT_COMPONENTS_CONCENTRIC) THEN ASM_REWRITE_TAC[REAL_ARITH `e / &2 < e <=> &0 < e`] THEN DISCH_TAC THEN REWRITE_TAC[IN_BALL] THEN ONCE_REWRITE_TAC[SET_RULE `{x | P x /\ Q x /\ R x} = {x | x IN {y | P y /\ Q y} /\ R x}`] THEN ASM_SIMP_TAC[REAL_LT_INF_FINITE; NOT_INSERT_EMPTY; FINITE_INSERT; FINITE_IMAGE; FINITE_RESTRICT; REAL_INF_LT_FINITE] THEN REWRITE_TAC[EXISTS_IN_INSERT; FORALL_IN_INSERT] THEN ASM_REWRITE_TAC[REAL_HALF; REAL_ARITH `e / &2 < e <=> &0 < e`] THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN ASM_REWRITE_TAC[IMP_CONJ; FORALL_IN_GSPEC] THEN SIMP_TAC[SETDIST_POS_LT; SETDIST_EQ_0_SING] THEN CONJ_TAC THENL [MESON_TAC[IN_COMPONENTS_NONEMPTY]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`d:real^N->bool`; `x:real^N`] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `d:real^N->bool`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(TAUT `p /\ ~q ==> (p ==> ~r ==> q) ==> r`) THEN ASM_SIMP_TAC[REAL_NOT_LT; SETDIST_LE_DIST; IN_SING] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER; IN_CBALL] THEN EXISTS_TAC `x:real^N` THEN ASM_SIMP_TAC[CLOSURE_INC; REAL_LT_IMP_LE]; ALL_TAC] THEN EXISTS_TAC `ball(p:real^N,a)` THEN ASM_REWRITE_TAC[OPEN_BALL; CENTRE_IN_BALL] THEN CONJ_TAC THENL [TRANS_TAC SUBSET_TRANS `cball(p:real^N,b)` THEN ASM_REWRITE_TAC[SUBSET_BALLS; DIST_REFL] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_INTER] THEN STRIP_TAC THEN ABBREV_TAC `d = connected_component (cball(p:real^N,b) DIFF s) x` THEN SUBGOAL_THEN `d IN components(cball(p:real^N,b) DIFF s)` ASSUME_TAC THENL [REWRITE_TAC[components; IN_ELIM_THM; IN_DIFF] THEN EXISTS_TAC `x:real^N` THEN ASM_REWRITE_TAC[IN_CBALL] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_BALL]) THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(x:real^N) IN d` ASSUME_TAC THENL [EXPAND_TAC "d" THEN REWRITE_TAC[IN] THEN REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ] THEN ASM_REWRITE_TAC[IN_CBALL; IN_DIFF] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_BALL]) THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET)) THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(d:real^N->bool) SUBSET c` ASSUME_TAC THENL [MATCH_MP_TAC COMPONENTS_MAXIMAL THEN EXISTS_TAC `(:real^N) DIFF s` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]; ALL_TAC] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET)) THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`s:real^N->bool`; `d:real^N->bool`; `p:real^N`; `cball(p:real^N,b)`] ACCESSIBLE_FRONTIER_ANR_INTER_COMPLEMENT_COMPONENT) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_REWRITE_TAC[INTERIOR_CBALL; CENTRE_IN_BALL] THEN REWRITE_TAC[frontier; IN_DIFF] THEN CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[SUBSET] INTERIOR_SUBSET)) THEN UNDISCH_TAC `(p:real^N) IN frontier c` THEN ASM_SIMP_TAC[frontier; INTERIOR_OPEN] THEN ASM SET_TAC[]; DISCH_THEN(X_CHOOSE_THEN `g:real^1->real^N` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC PATH_COMPONENT_TRANS THEN EXISTS_TAC `pathstart g:real^N` THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[PATH_COMPONENT_SYM_EQ] THEN REWRITE_TAC[path_component] THEN EXISTS_TAC `g:real^1->real^N` THEN ASM_SIMP_TAC[ARC_IMP_PATH] THEN REWRITE_TAC[path_image; SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `r:real^1` THEN DISCH_TAC THEN ASM_CASES_TAC `r:real^1 = vec 1` THENL [RULE_ASSUM_TAC(REWRITE_RULE[pathfinish]) THEN ASM_REWRITE_TAC[IN_INTER; IN_INSERT]; FIRST_X_ASSUM(MP_TAC o SPEC `r:real^1`) THEN ASM_REWRITE_TAC[IN_DELETE] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET)) THEN ASM SET_TAC[]]; MATCH_MP_TAC PATH_COMPONENT_OF_SUBSET THEN EXISTS_TAC `c INTER u:real^N->bool` THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[OPEN_PATH_CONNECTED_COMPONENT; OPEN_INTER] THEN REWRITE_TAC[connected_component] THEN EXISTS_TAC `d:real^N->bool` THEN ASM_REWRITE_TAC[SUBSET_INTER] THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]; REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET)) THEN ASM SET_TAC[]; REWRITE_TAC[pathstart] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_DELETE; ENDS_IN_UNIT_INTERVAL; VEC_EQ] THEN CONV_TAC NUM_REDUCE_CONV]]]) in REPEAT STRIP_TAC THEN SUBGOAL_THEN `open(c:real^N->bool)` ASSUME_TAC THENL [REPEAT(FIRST_X_ASSUM((MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] OPEN_COMPONENTS)))) THEN ASM_SIMP_TAC[GSYM closed; COMPACT_IMP_CLOSED]; ALL_TAC] THEN REWRITE_TAC[LOCALLY_PATH_CONNECTED_IM_KLEINEN] THEN MAP_EVERY X_GEN_TAC [`uu:real^N->bool`; `p:real^N`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM SUBST_ALL_TAC THEN FIRST_X_ASSUM(CONJUNCTS_THEN ASSUME_TAC o REWRITE_RULE[IN_INTER]) THEN SUBGOAL_THEN `(p:real^N) IN closure c` MP_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[CLOSURE_UNION_FRONTIER; IN_UNION]] THEN STRIP_TAC THENL [MP_TAC(ISPEC `c INTER u:real^N->bool` OPEN_IMP_LOCALLY_PATH_CONNECTED) THEN ASM_SIMP_TAC[OPEN_INTER; LOCALLY_PATH_CONNECTED_IM_KLEINEN] THEN DISCH_THEN(MP_TAC o SPECL [`c INTER u:real^N->bool`; `p:real^N`]) THEN ASM_REWRITE_TAC[OPEN_IN_REFL; IN_INTER] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `v:real^N->bool` THEN ASM_SIMP_TAC[OPEN_IN_OPEN_EQ; OPEN_INTER] THEN STRIP_TAC THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC OPEN_SUBSET THEN ASM SET_TAC[]; REWRITE_TAC[GSYM path_component] THEN MP_TAC(ISPECL [`s:real^N->bool`; `c:real^N->bool`; `u:real^N->bool`; `p:real^N`] lemma) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `v:real^N->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `t INTER v:real^N->bool` THEN ASM_SIMP_TAC[OPEN_IN_OPEN_INTER] THEN REPEAT(CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN X_GEN_TAC `q:real^N` THEN REWRITE_TAC[IN_INTER] THEN STRIP_TAC THEN SUBGOAL_THEN `(q:real^N) IN closure c` MP_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[CLOSURE_UNION_FRONTIER; IN_UNION]] THEN STRIP_TAC THENL [MATCH_MP_TAC PATH_COMPONENT_OF_SUBSET THEN EXISTS_TAC `(p:real^N) INSERT c INTER u` THEN ASM SET_TAC[]; MP_TAC(ISPECL [`s:real^N->bool`; `c:real^N->bool`; `v:real^N->bool`; `q:real^N`] lemma) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `w:real^N->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`c:real^N->bool`; `w:real^N->bool`] FRONTIER_OPEN_STRADDLE_INTER) THEN ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN(MP_TAC o CONJUNCT1)] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM; IN_INTER] THEN X_GEN_TAC `r:real^N` THEN STRIP_TAC THEN MATCH_MP_TAC PATH_COMPONENT_TRANS THEN EXISTS_TAC `r:real^N` THEN CONJ_TAC THEN MATCH_MP_TAC PATH_COMPONENT_OF_SUBSET THENL [EXISTS_TAC `(p:real^N) INSERT c INTER u` THEN ASM SET_TAC[]; EXISTS_TAC `(q:real^N) INSERT c INTER v` THEN ONCE_REWRITE_TAC[PATH_COMPONENT_SYM_EQ] THEN ASM SET_TAC[]]]]);; let LPC_INTERMEDIATE_CLOSURE_ANR_COMPLEMENT = prove (`!s t. compact s /\ ANR s /\ (:real^N) DIFF s SUBSET t /\ DISJOINT t (interior s) ==> locally path_connected t`, let lemma = prove (`!s u p:real^N. compact s /\ ANR s /\ p IN frontier s /\ open u /\ p IN u ==> ?v. open v /\ p IN v /\ v SUBSET u /\ !y. y IN v DIFF s ==> path_component (p INSERT (u DIFF s)) p y`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_CBALL]) THEN DISCH_THEN(MP_TAC o SPEC `p:real^N`) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `b:real` THEN STRIP_TAC THEN SUBGOAL_THEN `?a. &0 < a /\ a < b /\ !d x. d IN components(cball(p,b) DIFF s) /\ x IN d /\ x IN ball(p:real^N,a) ==> p IN closure d` STRIP_ASSUME_TAC THENL [EXISTS_TAC `inf ((b / &2) INSERT IMAGE (\c. setdist({p:real^N},c)) {c | c IN components (cball (p,b) DIFF s) /\ ~(closure c INTER cball (p,b / &2) = {}) /\ ~(p IN closure c)})` THEN MP_TAC(ISPECL [`s:real^N->bool`; `p:real^N`; `b / &2`; `b:real`] FINITE_ANR_COMPLEMENT_COMPONENTS_CONCENTRIC) THEN ASM_REWRITE_TAC[REAL_ARITH `e / &2 < e <=> &0 < e`] THEN DISCH_TAC THEN REWRITE_TAC[IN_BALL] THEN ONCE_REWRITE_TAC[SET_RULE `{x | P x /\ Q x /\ R x} = {x | x IN {y | P y /\ Q y} /\ R x}`] THEN ASM_SIMP_TAC[REAL_LT_INF_FINITE; NOT_INSERT_EMPTY; FINITE_INSERT; FINITE_IMAGE; FINITE_RESTRICT; REAL_INF_LT_FINITE] THEN REWRITE_TAC[EXISTS_IN_INSERT; FORALL_IN_INSERT] THEN ASM_REWRITE_TAC[REAL_HALF; REAL_ARITH `e / &2 < e <=> &0 < e`] THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN ASM_REWRITE_TAC[IMP_CONJ; FORALL_IN_GSPEC] THEN SIMP_TAC[SETDIST_POS_LT; SETDIST_EQ_0_SING] THEN CONJ_TAC THENL [MESON_TAC[IN_COMPONENTS_NONEMPTY]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`d:real^N->bool`; `x:real^N`] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `d:real^N->bool`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(TAUT `p /\ ~q ==> (p ==> ~r ==> q) ==> r`) THEN ASM_SIMP_TAC[REAL_NOT_LT; SETDIST_LE_DIST; IN_SING] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER; IN_CBALL] THEN EXISTS_TAC `x:real^N` THEN ASM_SIMP_TAC[CLOSURE_INC; REAL_LT_IMP_LE]; ALL_TAC] THEN EXISTS_TAC `ball(p:real^N,a)` THEN ASM_REWRITE_TAC[OPEN_BALL; CENTRE_IN_BALL] THEN CONJ_TAC THENL [TRANS_TAC SUBSET_TRANS `cball(p:real^N,b)` THEN ASM_REWRITE_TAC[SUBSET_BALLS; DIST_REFL] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_DIFF] THEN STRIP_TAC THEN SUBGOAL_THEN `x IN UNIONS(components(cball(p:real^N,b) DIFF s))` MP_TAC THENL [ASM_REWRITE_TAC[GSYM UNIONS_COMPONENTS; IN_DIFF] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `x IN b ==> b SUBSET c ==> x IN c`)) THEN REWRITE_TAC[SUBSET_BALLS; DIST_REFL] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[IN_UNIONS; LEFT_IMP_EXISTS_THM]] THEN X_GEN_TAC `c:real^N->bool` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`c:real^N->bool`; `x:real^N`]) THEN ASM_REWRITE_TAC[CLOSURE_UNION_FRONTIER; IN_UNION] THEN STRIP_TAC THENL [UNDISCH_TAC `(p:real^N) IN frontier s` THEN ASM_SIMP_TAC[frontier; CLOSURE_CLOSED; COMPACT_IMP_CLOSED] THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`s:real^N->bool`; `c:real^N->bool`; `p:real^N`; `cball(p:real^N,b)`] ACCESSIBLE_FRONTIER_ANR_INTER_COMPLEMENT_COMPONENT) THEN ASM_REWRITE_TAC[INTERIOR_CBALL; CENTRE_IN_BALL] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^1->real^N` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC PATH_COMPONENT_OF_SUBSET THEN EXISTS_TAC `(p:real^N) INSERT c` THEN CONJ_TAC THENL [FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC PATH_COMPONENT_TRANS THEN EXISTS_TAC `pathstart g:real^N` THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[PATH_COMPONENT_SYM_EQ] THEN REWRITE_TAC[path_component] THEN EXISTS_TAC `g:real^1->real^N` THEN ASM_SIMP_TAC[ARC_IMP_PATH] THEN REWRITE_TAC[path_image; SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `r:real^1` THEN DISCH_TAC THEN ASM_CASES_TAC `r:real^1 = vec 1` THENL [RULE_ASSUM_TAC(REWRITE_RULE[pathfinish]) THEN ASM_REWRITE_TAC[IN_INTER; IN_INSERT]; FIRST_X_ASSUM(MP_TAC o SPEC `r:real^1`) THEN ASM_REWRITE_TAC[IN_DELETE] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET)) THEN ASM SET_TAC[]]; MATCH_MP_TAC PATH_COMPONENT_OF_SUBSET THEN EXISTS_TAC `c:real^N->bool` THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhand o rand) PATH_COMPONENT_EQ_CONNECTED_COMPONENT o rator o snd) THEN ANTS_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] LOCALLY_PATH_CONNECTED_COMPONENTS)) THEN MATCH_MP_TAC LOCALLY_OPEN_SUBSET THEN EXISTS_TAC `cball(p:real^N,b)` THEN SIMP_TAC[CONVEX_IMP_LOCALLY_PATH_CONNECTED; CONVEX_CBALL] THEN ASM_SIMP_TAC[OPEN_IN_DIFF_CLOSED; COMPACT_IMP_CLOSED]; DISCH_THEN SUBST1_TAC] THEN REWRITE_TAC[connected_component] THEN EXISTS_TAC `c:real^N->bool` THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]; ALL_TAC] THEN REWRITE_TAC[pathstart] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_DELETE; ENDS_IN_UNIT_INTERVAL; VEC_EQ] THEN CONV_TAC NUM_REDUCE_CONV]) in REPEAT STRIP_TAC THEN REWRITE_TAC[LOCALLY_PATH_CONNECTED_IM_KLEINEN] THEN MAP_EVERY X_GEN_TAC [`uu:real^N->bool`; `p:real^N`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM SUBST_ALL_TAC THEN FIRST_X_ASSUM(CONJUNCTS_THEN ASSUME_TAC o REWRITE_RULE[IN_INTER]) THEN ASM_CASES_TAC `(p:real^N) IN s` THENL [ALL_TAC; MP_TAC(ISPEC `u DIFF s:real^N->bool` OPEN_IMP_LOCALLY_PATH_CONNECTED) THEN ASM_SIMP_TAC[OPEN_DIFF; COMPACT_IMP_CLOSED] THEN REWRITE_TAC[LOCALLY_PATH_CONNECTED_IM_KLEINEN] THEN DISCH_THEN(MP_TAC o SPECL [`u DIFF s:real^N->bool`; `p:real^N`]) THEN ASM_REWRITE_TAC[OPEN_IN_REFL; IN_DIFF] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `v:real^N->bool` THEN ASM_SIMP_TAC[OPEN_IN_OPEN_EQ; OPEN_DIFF; COMPACT_IMP_CLOSED] THEN STRIP_TAC THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC OPEN_SUBSET THEN ASM SET_TAC[]] THEN REWRITE_TAC[GSYM path_component] THEN MP_TAC(ISPECL [`s:real^N->bool`; `u:real^N->bool`; `p:real^N`] lemma) THEN ASM_SIMP_TAC[frontier; CLOSURE_CLOSED; COMPACT_IMP_CLOSED] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `v:real^N->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `t INTER v:real^N->bool` THEN ASM_SIMP_TAC[OPEN_IN_OPEN_INTER] THEN REPEAT(CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN X_GEN_TAC `q:real^N` THEN REWRITE_TAC[IN_INTER] THEN STRIP_TAC THEN ASM_CASES_TAC `(q:real^N) IN s` THENL [MP_TAC(ISPECL [`s:real^N->bool`; `v:real^N->bool`; `q:real^N`] lemma) THEN ASM_SIMP_TAC[frontier; CLOSURE_CLOSED; COMPACT_IMP_CLOSED] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `w:real^N->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`s:real^N->bool`; `w:real^N->bool`] FRONTIER_OPEN_STRADDLE_INTER) THEN ASM_SIMP_TAC[frontier; CLOSURE_CLOSED; COMPACT_IMP_CLOSED] THEN ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN(MP_TAC o CONJUNCT2)] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM; IN_DIFF] THEN X_GEN_TAC `r:real^N` THEN STRIP_TAC THEN MATCH_MP_TAC PATH_COMPONENT_TRANS THEN EXISTS_TAC `r:real^N` THEN CONJ_TAC THEN MATCH_MP_TAC PATH_COMPONENT_OF_SUBSET THENL [EXISTS_TAC `(p:real^N) INSERT (u DIFF s)` THEN ASM SET_TAC[]; EXISTS_TAC `(q:real^N) INSERT (v DIFF s)` THEN ONCE_REWRITE_TAC[PATH_COMPONENT_SYM_EQ] THEN ASM SET_TAC[]]; MATCH_MP_TAC PATH_COMPONENT_OF_SUBSET THEN EXISTS_TAC `(p:real^N) INSERT (u DIFF s)` THEN ASM SET_TAC[]]);; let LPC_SUPERSET_COMPLEMENT_SIMPLE_PATH_IMAGE = prove (`!g s:real^N->bool. 2 <= dimindex(:N) /\ simple_path g /\ (:real^N) DIFF path_image g SUBSET s ==> locally path_connected s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC LPC_INTERMEDIATE_CLOSURE_ANR_COMPLEMENT THEN EXISTS_TAC `path_image g:real^N->bool` THEN ASM_SIMP_TAC[COMPACT_PATH_IMAGE; SIMPLE_PATH_IMP_PATH] THEN ASM_SIMP_TAC[ANR_PATH_IMAGE_SIMPLE_PATH; INTERIOR_SIMPLE_PATH_IMAGE] THEN SET_TAC[]);; let LPC_OPEN_SIMPLE_PATH_COMPLEMENT = prove (`!g. simple_path g ==> locally path_connected ((:real^N) DIFF (path_image g DIFF {pathstart g,pathfinish g}))`, REPEAT STRIP_TAC THEN MATCH_MP_TAC LPC_INTERMEDIATE_CLOSURE_ANR_COMPLEMENT THEN EXISTS_TAC `path_image g:real^N->bool` THEN ASM_SIMP_TAC[COMPACT_PATH_IMAGE; SIMPLE_PATH_IMP_PATH] THEN ASM_SIMP_TAC[ANR_PATH_IMAGE_SIMPLE_PATH] THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `i SUBSET p /\ DISJOINT {a,b} i ==> DISJOINT (UNIV DIFF (p DIFF {a,b})) i`) THEN ASM_SIMP_TAC[INTERIOR_SUBSET; ENDPOINTS_NOT_IN_INTERIOR_SIMPLE_PATH_IMAGE]);; let PATH_CONNECTED_INTERMEDIATE_CLOSURE_ANR_COMPLEMENT_COMPONENT = prove (`!s c t. compact s /\ ANR s /\ c IN components((:real^N) DIFF s) /\ c SUBSET t /\ t SUBSET closure c ==> path_connected t`, REPEAT STRIP_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) PATH_CONNECTED_EQ_CONNECTED_LPC o snd) THEN ANTS_TAC THENL [MATCH_MP_TAC LPC_INTERMEDIATE_CLOSURE_ANR_COMPLEMENT_COMPONENT THEN ASM_MESON_TAC[]; DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC CONNECTED_INTERMEDIATE_CLOSURE THEN EXISTS_TAC `c:real^N->bool` THEN ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]]);; let PATH_CONNECTED_SUPERSET_COMPLEMENT_ARC_IMAGE = prove (`!g s:real^N->bool. 2 <= dimindex(:N) /\ arc g /\ (:real^N) DIFF path_image g SUBSET s ==> path_connected s`, REPEAT STRIP_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) PATH_CONNECTED_EQ_CONNECTED_LPC o snd) THEN ANTS_TAC THENL [MATCH_MP_TAC LPC_SUPERSET_COMPLEMENT_SIMPLE_PATH_IMAGE THEN ASM_MESON_TAC[ARC_IMP_SIMPLE_PATH]; DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC CONNECTED_INTERMEDIATE_CLOSURE THEN EXISTS_TAC `(:real^N) DIFF path_image g` THEN ASM_SIMP_TAC[CONNECTED_ARC_COMPLEMENT; CLOSURE_COMPLEMENT] THEN ASM_SIMP_TAC[INTERIOR_ARC_IMAGE] THEN SET_TAC[]]);; let PATH_CONNECTED_OPEN_ARC_COMPLEMENT = prove (`!g. 2 <= dimindex(:N) /\ arc g ==> path_connected ((:real^N) DIFF (path_image g DIFF {pathstart g,pathfinish g}))`, REPEAT STRIP_TAC THEN MATCH_MP_TAC PATH_CONNECTED_SUPERSET_COMPLEMENT_ARC_IMAGE THEN EXISTS_TAC `g:real^1->real^N` THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);;