(* ========================================================================= *) (* Additional topology theory. *) (* *) (* (c) Copyright, John Harrison 1998-2013 *) (* ========================================================================= *) needs "Multivariate/realanalysis.ml";; (* ------------------------------------------------------------------------- *) (* Injective map into R is also an open map w.r.t. the universe, and this *) (* is actually an implication in both directions for an interval. Compare *) (* the local form in INJECTIVE_INTO_1D_IMP_OPEN_MAP (not a bi-implication). *) (* ------------------------------------------------------------------------- *) let INJECTIVE_EQ_1D_OPEN_MAP_UNIV = prove (`!f:real^1->real^1 s. f continuous_on s /\ is_interval s ==> ((!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) <=> (!t. open t /\ t SUBSET s ==> open(IMAGE f t)))`, REPEAT STRIP_TAC THEN EQ_TAC THEN REPEAT STRIP_TAC THENL [ONCE_REWRITE_TAC[OPEN_SUBOPEN] THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN X_GEN_TAC `x:real^1` THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_CBALL]) THEN DISCH_THEN(MP_TAC o SPEC `x:real^1`) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[BALL_1] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `IMAGE (f:real^1->real^1) (segment (x - lift d,x + lift d))` THEN MP_TAC(ISPECL [`f:real^1->real^1`; `x - lift d`; `x + lift d`] CONTINUOUS_INJECTIVE_IMAGE_OPEN_SEGMENT_1) THEN REWRITE_TAC[SEGMENT_1; DROP_ADD; DROP_SUB; LIFT_DROP] THEN ASM_CASES_TAC `drop x - d <= drop x + d` THENL [ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM SEGMENT_1]; ASM_REAL_ARITH_TAC] THEN ANTS_TAC THENL [ASM_MESON_TAC[SUBSET; CONTINUOUS_ON_SUBSET]; ALL_TAC] THEN REPEAT STRIP_TAC THENL [ASM_REWRITE_TAC[OPEN_SEGMENT_1]; MATCH_MP_TAC FUN_IN_IMAGE THEN REWRITE_TAC[IN_INTERVAL_1] THEN REWRITE_TAC[DROP_ADD; DROP_SUB; LIFT_DROP] THEN ASM_REAL_ARITH_TAC; MATCH_MP_TAC IMAGE_SUBSET THEN ASM_MESON_TAC[INTERVAL_OPEN_SUBSET_CLOSED; SUBSET_TRANS]]; MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THEN MP_TAC(ISPECL [`f:real^1->real^1`; `x:real^1`; `y:real^1`] CONTINUOUS_IVT_LOCAL_EXTREMUM) THEN ASM_REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [ASM_MESON_TAC[CONVEX_CONTAINS_SEGMENT_EQ; IS_INTERVAL_CONVEX_1; CONTINUOUS_ON_SUBSET]; DISCH_THEN(X_CHOOSE_TAC `z:real^1`) THEN FIRST_ASSUM(MP_TAC o SPEC `segment(x:real^1,y)`) THEN REWRITE_TAC[OPEN_SEGMENT_1; NOT_IMP] THEN CONJ_TAC THENL [ASM_MESON_TAC[CONVEX_CONTAINS_SEGMENT; IS_INTERVAL_CONVEX_1; SUBSET_TRANS; SEGMENT_OPEN_SUBSET_CLOSED]; FIRST_X_ASSUM(CONJUNCTS_THEN ASSUME_TAC) THEN REWRITE_TAC[open_def; FORALL_IN_IMAGE] THEN DISCH_THEN(MP_TAC o SPEC `z:real^1`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN FIRST_X_ASSUM DISJ_CASES_TAC THENL [DISCH_THEN(MP_TAC o SPEC `(f:real^1->real^1) z + lift(e / &2)`); DISCH_THEN(MP_TAC o SPEC `(f:real^1->real^1) z - lift(e / &2)`)] THEN ASM_REWRITE_TAC[NORM_ARITH `dist(a + b:real^N,a) = norm b`; NORM_ARITH `dist(a - b:real^N,a) = norm b`; NORM_LIFT; REAL_ARITH `abs(e / &2) < e <=> &0 < e`] THEN REWRITE_TAC[IN_IMAGE] THEN DISCH_THEN(X_CHOOSE_THEN `w:real^1` (STRIP_ASSUME_TAC o GSYM)) THEN FIRST_X_ASSUM(MP_TAC o SPEC `w:real^1`) THEN ASM_SIMP_TAC[REWRITE_RULE[SUBSET] SEGMENT_OPEN_SUBSET_CLOSED] THEN REWRITE_TAC[DROP_ADD; DROP_SUB; LIFT_DROP] THEN ASM_REAL_ARITH_TAC]]]);; (* ------------------------------------------------------------------------- *) (* Map f:S^m->S^n for m < n is nullhomotopic. *) (* ------------------------------------------------------------------------- *) let INESSENTIAL_SPHEREMAP_LOWDIM_GEN = prove (`!f:real^M->real^N s t. convex s /\ bounded s /\ convex t /\ bounded t /\ aff_dim s < aff_dim t /\ f continuous_on relative_frontier s /\ IMAGE f (relative_frontier s) SUBSET (relative_frontier t) ==> ?c. homotopic_with (\z. T) (relative_frontier s,relative_frontier t) f (\x. c)`, let lemma1 = prove (`!f:real^N->real^N s t. subspace s /\ subspace t /\ dim s < dim t /\ s SUBSET t /\ f differentiable_on sphere(vec 0,&1) INTER s ==> ~(IMAGE f (sphere(vec 0,&1) INTER s) = sphere(vec 0,&1) INTER t)`, REPEAT STRIP_TAC THEN ABBREV_TAC `(g:real^N->real^N) = \x. norm(x) % (f:real^N->real^N)(inv(norm x) % x)` THEN SUBGOAL_THEN `(g:real^N->real^N) differentiable_on s DELETE (vec 0)` ASSUME_TAC THENL [EXPAND_TAC "g" THEN MATCH_MP_TAC DIFFERENTIABLE_ON_MUL THEN SIMP_TAC[o_DEF; DIFFERENTIABLE_ON_NORM; IN_DELETE] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC DIFFERENTIABLE_ON_COMPOSE THEN CONJ_TAC THENL [MATCH_MP_TAC DIFFERENTIABLE_ON_MUL THEN REWRITE_TAC[DIFFERENTIABLE_ON_ID] THEN SUBGOAL_THEN `lift o (\x:real^N. inv(norm x)) = (lift o inv o drop) o (\x. lift(norm x))` SUBST1_TAC THENL [REWRITE_TAC[o_DEF; LIFT_DROP]; ALL_TAC] THEN MATCH_MP_TAC DIFFERENTIABLE_ON_COMPOSE THEN SIMP_TAC[DIFFERENTIABLE_ON_NORM; IN_DELETE] THEN MATCH_MP_TAC DIFFERENTIABLE_AT_IMP_DIFFERENTIABLE_ON THEN SIMP_TAC[FORALL_IN_IMAGE; IN_DELETE; GSYM REAL_DIFFERENTIABLE_AT] THEN REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM ETA_AX] THEN MATCH_MP_TAC REAL_DIFFERENTIABLE_INV_ATREAL THEN ASM_REWRITE_TAC[REAL_DIFFERENTIABLE_ID; NORM_EQ_0]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] DIFFERENTIABLE_ON_SUBSET)) THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_SPHERE_0; IN_INTER; SUBSPACE_MUL; NORM_MUL; IN_DELETE] THEN SIMP_TAC[REAL_ABS_INV; REAL_ABS_NORM; REAL_MUL_LINV; NORM_EQ_0]]; ALL_TAC] THEN SUBGOAL_THEN `IMAGE (g:real^N->real^N) (s DELETE vec 0) = t DELETE (vec 0)` ASSUME_TAC THENL [UNDISCH_TAC `IMAGE (f:real^N->real^N) (sphere (vec 0,&1) INTER s) = sphere (vec 0,&1) INTER t` THEN REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_DELETE; IN_INTER; IN_SPHERE_0] THEN EXPAND_TAC "g" THEN REWRITE_TAC[IN_IMAGE; IN_INTER; IN_SPHERE_0] THEN SIMP_TAC[IN_DELETE; VECTOR_MUL_EQ_0; NORM_EQ_0] THEN MATCH_MP_TAC(TAUT `(p ==> r) /\ (p ==> q ==> s) ==> p /\ q ==> r /\ s`) THEN CONJ_TAC THENL [ALL_TAC; DISCH_TAC] THEN DISCH_THEN(fun th -> X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN MP_TAC(SPEC `inv(norm x) % x:real^N` th)) THEN ASM_SIMP_TAC[SUBSPACE_MUL; NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM; REAL_MUL_LINV; NORM_EQ_0; NORM_ARITH `norm x = &1 ==> ~(x:real^N = vec 0)`] THEN DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `norm(x:real^N) % y:real^N` THEN ASM_SIMP_TAC[SUBSPACE_MUL; NORM_MUL; REAL_ABS_NORM; REAL_MUL_RID] THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; NORM_EQ_0] THEN ASM_REWRITE_TAC[VECTOR_MUL_LID; VECTOR_MUL_EQ_0; NORM_EQ_0] THEN ASM_SIMP_TAC[NORM_ARITH `norm x = &1 ==> ~(x:real^N = vec 0)`] THEN UNDISCH_THEN `inv(norm x) % x = (f:real^N->real^N) y` (SUBST1_TAC o SYM) THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; NORM_EQ_0] THEN REWRITE_TAC[VECTOR_MUL_LID]; ALL_TAC] THEN MP_TAC(ISPECL [`t:real^N->bool`; `(:real^N)`] DIM_SUBSPACE_ORTHOGONAL_TO_VECTORS) THEN ASM_REWRITE_TAC[SUBSPACE_UNIV; DIM_UNIV; IN_UNIV; SUBSET_UNIV] THEN ABBREV_TAC `t' = {y:real^N | !x. x IN t ==> orthogonal x y}` THEN DISCH_TAC THEN SUBGOAL_THEN `subspace(t':real^N->bool)` ASSUME_TAC THENL [EXPAND_TAC "t'" THEN REWRITE_TAC[SUBSPACE_ORTHOGONAL_TO_VECTORS]; ALL_TAC] THEN SUBGOAL_THEN `?fst snd. linear fst /\ linear snd /\ (!z. fst(z) IN t /\ snd z IN t' /\ fst z + snd z = z) /\ (!x y:real^N. x IN t /\ y IN t' ==> fst(x + y) = x /\ snd(x + y) = y)` STRIP_ASSUME_TAC THENL [MP_TAC(ISPEC `t:real^N->bool` ORTHOGONAL_SUBSPACE_DECOMP_EXISTS) THEN REWRITE_TAC[SKOLEM_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `fst:real^N->real^N` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `snd:real^N->real^N` THEN DISCH_THEN(MP_TAC o GSYM) THEN ASM_SIMP_TAC[SPAN_OF_SUBSPACE; FORALL_AND_THM] THEN STRIP_TAC THEN MATCH_MP_TAC(TAUT `r /\ (r ==> p /\ q /\ s) ==> p /\ q /\ r /\ s`) THEN CONJ_TAC THENL [EXPAND_TAC "t'" THEN REWRITE_TAC[IN_ELIM_THM] THEN ASM_MESON_TAC[ORTHOGONAL_SYM]; DISCH_TAC] THEN MATCH_MP_TAC(TAUT `r /\ (r ==> p /\ q) ==> p /\ q /\ r`) THEN CONJ_TAC THENL [REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC ORTHOGONAL_SUBSPACE_DECOMP_UNIQUE THEN MAP_EVERY EXISTS_TAC [`t:real^N->bool`; `t':real^N->bool`] THEN ASM_SIMP_TAC[SPAN_OF_SUBSPACE] THEN ASM SET_TAC[]; DISCH_TAC] THEN REWRITE_TAC[linear] THEN MATCH_MP_TAC(TAUT `(p /\ r) /\ (q /\ s) ==> (p /\ q) /\ (r /\ s)`) THEN REWRITE_TAC[AND_FORALL_THM] THEN CONJ_TAC THEN REPEAT GEN_TAC THEN MATCH_MP_TAC ORTHOGONAL_SUBSPACE_DECOMP_UNIQUE THEN MAP_EVERY EXISTS_TAC [`t:real^N->bool`; `t':real^N->bool`] THEN ASM_SIMP_TAC[SPAN_OF_SUBSPACE; SUBSPACE_ADD; SUBSPACE_MUL] THEN (CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN ASM_REWRITE_TAC[GSYM VECTOR_ADD_LDISTRIB] THEN ONCE_REWRITE_TAC[VECTOR_ARITH `(x + y) + (x' + y'):real^N = (x + x') + (y + y')`] THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`\x:real^N. (g:real^N->real^N)(fst x) + snd x`; `{x + y:real^N | x IN (s DELETE vec 0) /\ y IN t'}`] NEGLIGIBLE_DIFFERENTIABLE_IMAGE_NEGLIGIBLE) THEN REWRITE_TAC[LE_REFL; NOT_IMP] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_LOWDIM THEN MP_TAC(ISPECL [`s:real^N->bool`; `t':real^N->bool`] DIM_SUMS_INTER) THEN ASM_REWRITE_TAC[IN_DELETE] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ARITH_RULE `t' + t = n ==> s < t /\ d' <= d /\ i = 0 ==> d + i = s + t' ==> d' < n`)) THEN ASM_REWRITE_TAC[DIM_EQ_0] THEN CONJ_TAC THENL [MATCH_MP_TAC DIM_SUBSET THEN SET_TAC[]; EXPAND_TAC "t'"] THEN REWRITE_TAC[SUBSET; IN_INTER; IN_SING; IN_ELIM_THM] THEN ASM_MESON_TAC[SUBSET; ORTHOGONAL_REFL]; MATCH_MP_TAC DIFFERENTIABLE_ON_ADD THEN ASM_SIMP_TAC[DIFFERENTIABLE_ON_LINEAR] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC DIFFERENTIABLE_ON_COMPOSE THEN ASM_SIMP_TAC[DIFFERENTIABLE_ON_LINEAR] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] DIFFERENTIABLE_ON_SUBSET)) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN RULE_ASSUM_TAC(REWRITE_RULE[SUBSET]) THEN ASM_SIMP_TAC[IN_DELETE]; SUBGOAL_THEN `~negligible {x + y | x IN IMAGE (g:real^N->real^N) (s DELETE vec 0) /\ y IN t'}` MP_TAC THENL [ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `negligible(t':real^N->bool)` MP_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_LOWDIM THEN ASM_ARITH_TAC; REWRITE_TAC[TAUT `p ==> ~q <=> ~(p /\ q)`]] THEN REWRITE_TAC[GSYM NEGLIGIBLE_UNION_EQ] THEN MP_TAC NOT_NEGLIGIBLE_UNIV THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_UNION; IN_UNIV; IN_ELIM_THM; IN_DELETE] THEN X_GEN_TAC `z:real^N` THEN REWRITE_TAC[TAUT `p \/ q <=> ~p ==> q`] THEN DISCH_TAC THEN EXISTS_TAC `(fst:real^N->real^N) z` THEN EXISTS_TAC `(snd:real^N->real^N) z` THEN ASM_SIMP_TAC[] THEN ASM_MESON_TAC[VECTOR_ADD_LID]; REWRITE_TAC[CONTRAPOS_THM] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] NEGLIGIBLE_SUBSET) THEN REWRITE_TAC[SUBSET; FORALL_IN_GSPEC; IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE; IN_DELETE] THEN X_GEN_TAC `x:real^N` THEN REPEAT DISCH_TAC THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `x + y:real^N` THEN RULE_ASSUM_TAC(REWRITE_RULE[SUBSET]) THEN ASM_SIMP_TAC[] THEN ASM SET_TAC[]]]) in let lemma2 = prove (`!f:real^N->real^N s t. subspace s /\ subspace t /\ dim s < dim t /\ s SUBSET t /\ f continuous_on sphere(vec 0,&1) INTER s /\ IMAGE f (sphere(vec 0,&1) INTER s) SUBSET sphere(vec 0,&1) INTER t ==> ?c. homotopic_with (\x. T) (sphere(vec 0,&1) INTER s,sphere(vec 0,&1) INTER t) f (\x. c)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^N->real^N`; `sphere(vec 0:real^N,&1) INTER s`; `&1 / &2`; `t:real^N->bool`;] STONE_WEIERSTRASS_VECTOR_POLYNOMIAL_FUNCTION_SUBSPACE) THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_SIMP_TAC[COMPACT_INTER_CLOSED; COMPACT_SPHERE; CLOSED_SUBSPACE] THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE]] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^N->real^N` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `!x. x IN sphere(vec 0,&1) INTER s ==> ~((g:real^N->real^N) x = vec 0)` ASSUME_TAC THENL [X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_SPHERE_0] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_SPHERE_0]) THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[IN_INTER; IN_SPHERE_0] THEN CONV_TAC NORM_ARITH; ALL_TAC] THEN SUBGOAL_THEN `(g:real^N->real^N) differentiable_on sphere(vec 0,&1) INTER s` ASSUME_TAC THENL [ASM_SIMP_TAC[DIFFERENTIABLE_ON_VECTOR_POLYNOMIAL_FUNCTION]; ALL_TAC] THEN ABBREV_TAC `(h:real^N->real^N) = \x. inv(norm(g x)) % g x` THEN SUBGOAL_THEN `!x. x IN sphere(vec 0,&1) INTER s ==> (h:real^N->real^N) x IN sphere(vec 0,&1) INTER t` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN EXPAND_TAC "h" THEN ASM_SIMP_TAC[SUBSPACE_MUL; IN_INTER; IN_SPHERE_0; NORM_MUL] THEN REWRITE_TAC[REAL_ABS_INV; REAL_ABS_NORM] THEN ASM_SIMP_TAC[REAL_MUL_LINV; NORM_EQ_0; GSYM IN_SPHERE_0]; ALL_TAC] THEN SUBGOAL_THEN `(h:real^N->real^N) differentiable_on sphere(vec 0,&1) INTER s` ASSUME_TAC THENL [EXPAND_TAC "h" THEN MATCH_MP_TAC DIFFERENTIABLE_ON_MUL THEN ASM_SIMP_TAC[DIFFERENTIABLE_ON_VECTOR_POLYNOMIAL_FUNCTION; o_DEF] THEN SUBGOAL_THEN `(\x. lift(inv(norm((g:real^N->real^N) x)))) = (lift o inv o drop) o (\x. lift(norm x)) o (g:real^N->real^N)` SUBST1_TAC THENL [REWRITE_TAC[o_DEF; LIFT_DROP]; ALL_TAC] THEN MATCH_MP_TAC DIFFERENTIABLE_ON_COMPOSE THEN CONJ_TAC THENL [MATCH_MP_TAC DIFFERENTIABLE_ON_COMPOSE THEN ASM_SIMP_TAC[DIFFERENTIABLE_ON_VECTOR_POLYNOMIAL_FUNCTION] THEN MATCH_MP_TAC DIFFERENTIABLE_ON_NORM THEN ASM_REWRITE_TAC[SET_RULE `~(z IN IMAGE f s) <=> !x. x IN s ==> ~(f x = z)`]; MATCH_MP_TAC DIFFERENTIABLE_AT_IMP_DIFFERENTIABLE_ON THEN REWRITE_TAC[GSYM REAL_DIFFERENTIABLE_AT] THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_SPHERE_0] THEN X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `x:real^N = vec 0` THEN ASM_REWRITE_TAC[NORM_0; REAL_OF_NUM_EQ; ARITH_EQ] THEN DISCH_TAC THEN REWRITE_TAC[GSYM REAL_DIFFERENTIABLE_AT; o_THM] THEN GEN_REWRITE_TAC LAND_CONV [GSYM ETA_AX] THEN MATCH_MP_TAC REAL_DIFFERENTIABLE_INV_ATREAL THEN ASM_SIMP_TAC[REAL_DIFFERENTIABLE_ID; NORM_EQ_0; IN_SPHERE_0]]; ALL_TAC] THEN SUBGOAL_THEN `?c. homotopic_with (\z. T) (sphere(vec 0,&1) INTER s,sphere(vec 0,&1) INTER t) (h:real^N->real^N) (\x. c)` MP_TAC THENL [ALL_TAC; MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] HOMOTOPIC_WITH_TRANS) THEN SUBGOAL_THEN `homotopic_with (\z. T) (sphere(vec 0:real^N,&1) INTER s,t DELETE (vec 0:real^N)) f g` MP_TAC THENL [MATCH_MP_TAC HOMOTOPIC_WITH_LINEAR THEN ASM_SIMP_TAC[CONTINUOUS_ON_VECTOR_POLYNOMIAL_FUNCTION] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[SET_RULE `s SUBSET t DELETE v <=> s SUBSET t /\ ~(v IN s)`] THEN CONJ_TAC THENL [REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN MATCH_MP_TAC HULL_MINIMAL THEN ASM_SIMP_TAC[SUBSPACE_IMP_CONVEX] THEN ASM SET_TAC[]; DISCH_THEN(MP_TAC o MATCH_MP SEGMENT_BOUND) THEN SUBGOAL_THEN `(f:real^N->real^N) x IN sphere(vec 0,&1) /\ norm(f x - g x) < &1/ &2` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[IN_SPHERE_0] THEN CONV_TAC NORM_ARITH]; DISCH_THEN(MP_TAC o ISPECL [`\y:real^N. inv(norm y) % y`; `sphere(vec 0:real^N,&1) INTER t`] o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMOTOPIC_COMPOSE_CONTINUOUS_LEFT)) THEN ASM_REWRITE_TAC[o_DEF] THEN ANTS_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_MUL THEN REWRITE_TAC[o_DEF; CONTINUOUS_ON_ID] THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN SIMP_TAC[IN_DELETE; NORM_EQ_0] THEN REWRITE_TAC[REWRITE_RULE[o_DEF] CONTINUOUS_ON_LIFT_NORM]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_DELETE; IN_INTER] THEN ASM_SIMP_TAC[SUBSPACE_MUL; IN_SPHERE_0; NORM_MUL; REAL_ABS_MUL] THEN SIMP_TAC[REAL_ABS_INV; REAL_ABS_NORM; REAL_MUL_LINV; NORM_EQ_0]]; MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_WITH_EQ) THEN RULE_ASSUM_TAC(REWRITE_RULE [SUBSET; IN_INTER; FORALL_IN_IMAGE; IN_SPHERE_0]) THEN ASM_SIMP_TAC[IN_SPHERE_0; IN_INTER; REAL_INV_1; VECTOR_MUL_LID]]]] THEN SUBGOAL_THEN `?c. c IN (sphere(vec 0,&1) INTER t) DIFF (IMAGE (h:real^N->real^N) (sphere(vec 0,&1) INTER s))` MP_TAC THENL [MATCH_MP_TAC(SET_RULE `t SUBSET s /\ ~(t = s) ==> ?a. a IN s DIFF t`) THEN CONJ_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC lemma1] THEN ASM_REWRITE_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM; IN_INTER; IN_DIFF; IN_IMAGE] THEN REWRITE_TAC[SET_RULE `~(?x. P x /\ x IN s /\ x IN t) <=> (!x. x IN s INTER t ==> ~(P x))`] THEN X_GEN_TAC `c:real^N` THEN STRIP_TAC] THEN EXISTS_TAC `--c:real^N` THEN SUBGOAL_THEN `homotopic_with (\z. T) (sphere(vec 0:real^N,&1) INTER s,t DELETE (vec 0:real^N)) h (\x. --c)` MP_TAC THENL [MATCH_MP_TAC HOMOTOPIC_WITH_LINEAR THEN ASM_SIMP_TAC[DIFFERENTIABLE_IMP_CONTINUOUS_ON; CONTINUOUS_ON_CONST] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[SET_RULE `s SUBSET t DELETE v <=> s SUBSET t /\ ~(v IN s)`] THEN CONJ_TAC THENL [REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN MATCH_MP_TAC HULL_MINIMAL THEN ASM_SIMP_TAC[SUBSPACE_IMP_CONVEX; INSERT_SUBSET; SUBSPACE_NEG] THEN ASM SET_TAC[]; DISCH_TAC THEN MP_TAC(ISPECL [`(h:real^N->real^N) x`; `vec 0:real^N`; `--c:real^N`] MIDPOINT_BETWEEN) THEN ASM_REWRITE_TAC[BETWEEN_IN_SEGMENT; DIST_0; NORM_NEG] THEN SUBGOAL_THEN `((h:real^N->real^N) x) IN sphere(vec 0,&1) /\ (c:real^N) IN sphere(vec 0,&1)` MP_TAC THENL [ASM SET_TAC[]; SIMP_TAC[IN_SPHERE_0]] THEN STRIP_TAC THEN REWRITE_TAC[midpoint; VECTOR_ARITH `vec 0:real^N = inv(&2) % (x + --y) <=> x = y`] THEN ASM SET_TAC[]]; DISCH_THEN(MP_TAC o ISPECL [`\y:real^N. inv(norm y) % y`; `sphere(vec 0:real^N,&1) INTER t`] o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMOTOPIC_COMPOSE_CONTINUOUS_LEFT)) THEN ASM_REWRITE_TAC[o_DEF] THEN ANTS_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_MUL THEN REWRITE_TAC[o_DEF; CONTINUOUS_ON_ID] THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN SIMP_TAC[IN_DELETE; NORM_EQ_0] THEN REWRITE_TAC[REWRITE_RULE[o_DEF] CONTINUOUS_ON_LIFT_NORM]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_DELETE; IN_INTER] THEN ASM_SIMP_TAC[SUBSPACE_MUL; IN_SPHERE_0; NORM_MUL; REAL_ABS_MUL] THEN SIMP_TAC[REAL_ABS_INV; REAL_ABS_NORM; REAL_MUL_LINV; NORM_EQ_0]]; MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_WITH_EQ) THEN RULE_ASSUM_TAC(REWRITE_RULE [SUBSET; IN_INTER; FORALL_IN_IMAGE; IN_SPHERE_0]) THEN ASM_SIMP_TAC[IN_SPHERE_0; IN_INTER; REAL_INV_1; VECTOR_MUL_LID; NORM_NEG]]]) in let lemma3 = prove (`!s:real^M->bool u:real^N->bool. bounded s /\ convex s /\ subspace u /\ aff_dim s <= &(dim u) ==> ?t. subspace t /\ t SUBSET u /\ (~(s = {}) ==> aff_dim t = aff_dim s) /\ (relative_frontier s) homeomorphic (sphere(vec 0,&1) INTER t)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THENL [STRIP_TAC THEN EXISTS_TAC `{vec 0:real^N}` THEN ASM_REWRITE_TAC[SUBSPACE_TRIVIAL; RELATIVE_FRONTIER_EMPTY] THEN ASM_SIMP_TAC[HOMEOMORPHIC_EMPTY; SET_RULE `s INTER {a} = {} <=> ~(a IN s)`; IN_SPHERE_0; NORM_0; SING_SUBSET; SUBSPACE_0] THEN CONV_TAC REAL_RAT_REDUCE_CONV; FIRST_X_ASSUM(X_CHOOSE_THEN `a:real^M` MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN GEOM_ORIGIN_TAC `a:real^M` THEN SIMP_TAC[AFF_DIM_DIM_0; HULL_INC; INT_OF_NUM_LE; GSYM DIM_UNIV] THEN REPEAT STRIP_TAC] THEN FIRST_ASSUM(MP_TAC o MATCH_MP CHOOSE_SUBSPACE_OF_SUBSPACE) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real^N->bool` THEN ASM_SIMP_TAC[SPAN_OF_SUBSPACE; AFF_DIM_DIM_SUBSPACE; INT_OF_NUM_EQ] THEN STRIP_TAC THEN TRANS_TAC HOMEOMORPHIC_TRANS `relative_frontier(ball(vec 0:real^N,&1) INTER t)` THEN CONJ_TAC THENL [MATCH_MP_TAC HOMEOMORPHIC_RELATIVE_FRONTIERS_CONVEX_BOUNDED_SETS THEN ASM_SIMP_TAC[CONVEX_INTER; BOUNDED_INTER; BOUNDED_BALL; SUBSPACE_IMP_CONVEX; CONVEX_BALL] THEN ONCE_REWRITE_TAC[INTER_COMM] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP SUBSPACE_0) THEN SUBGOAL_THEN `~(t INTER ball(vec 0:real^N,&1) = {})` ASSUME_TAC THENL [REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `vec 0:real^N` THEN ASM_REWRITE_TAC[IN_INTER; CENTRE_IN_BALL; REAL_LT_01]; ASM_SIMP_TAC[AFF_DIM_CONVEX_INTER_OPEN; OPEN_BALL; SUBSPACE_IMP_CONVEX] THEN ASM_SIMP_TAC[AFF_DIM_DIM_0; HULL_INC]]; MATCH_MP_TAC(MESON[HOMEOMORPHIC_REFL] `s = t ==> s homeomorphic t`) THEN SIMP_TAC[GSYM FRONTIER_BALL; REAL_LT_01] THEN MATCH_MP_TAC RELATIVE_FRONTIER_CONVEX_INTER_AFFINE THEN ASM_SIMP_TAC[CONVEX_BALL; SUBSPACE_IMP_AFFINE; GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `vec 0:real^N` THEN ASM_SIMP_TAC[CENTRE_IN_BALL; INTERIOR_OPEN; OPEN_BALL; SUBSPACE_0; IN_INTER; REAL_LT_01]]) in ONCE_REWRITE_TAC[MESON[] `(!a b c. P a b c) <=> (!b c a. P a b c)`] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[IMP_IMP] THEN ONCE_REWRITE_TAC[RIGHT_IMP_FORALL_THM] THEN REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC] THEN REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THENL [ASM_SIMP_TAC[HOMOTOPIC_WITH; RELATIVE_FRONTIER_EMPTY; PCROSS_EMPTY; NOT_IN_EMPTY; IMAGE_CLAUSES; CONTINUOUS_ON_EMPTY]; ALL_TAC] THEN ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_SIMP_TAC[AFF_DIM_EMPTY; GSYM INT_NOT_LE; AFF_DIM_GE] THEN STRIP_TAC THEN MP_TAC(ISPECL [`t:real^N->bool`; `(:real^N)`] lemma3) THEN ASM_REWRITE_TAC[DIM_UNIV; SUBSPACE_UNIV; AFF_DIM_LE_UNIV] THEN DISCH_THEN(X_CHOOSE_THEN `t':real^N->bool` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMEOMORPHIC_IMP_HOMOTOPY_EQUIVALENT) THEN DISCH_THEN(fun th -> REWRITE_TAC[MATCH_MP HOMOTOPY_EQUIVALENT_HOMOTOPIC_TRIVIALITY_NULL th]) THEN MP_TAC(ISPECL [`s:real^M->bool`; `t':real^N->bool`] lemma3) THEN ASM_SIMP_TAC[GSYM AFF_DIM_DIM_SUBSPACE] THEN ANTS_TAC THENL [ASM_INT_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `s':real^N->bool` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMEOMORPHIC_IMP_HOMOTOPY_EQUIVALENT) THEN DISCH_THEN(fun th -> REWRITE_TAC[MATCH_MP HOMOTOPY_EQUIVALENT_COHOMOTOPIC_TRIVIALITY_NULL th]) THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC lemma2 THEN ASM_SIMP_TAC[GSYM INT_OF_NUM_LT; GSYM AFF_DIM_DIM_SUBSPACE] THEN ASM_INT_ARITH_TAC);; let INESSENTIAL_SPHEREMAP_LOWDIM = prove (`!f:real^M->real^N a r b s. dimindex(:M) < dimindex(:N) /\ f continuous_on sphere(a,r) /\ IMAGE f (sphere(a,r)) SUBSET (sphere(b,s)) ==> ?c. homotopic_with (\z. T) (sphere(a,r),sphere(b,s)) f (\x. c)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s <= &0` THEN ASM_SIMP_TAC[NULLHOMOTOPIC_INTO_CONTRACTIBLE; CONTRACTIBLE_SPHERE] THEN ASM_CASES_TAC `r <= &0` THEN ASM_SIMP_TAC[NULLHOMOTOPIC_FROM_CONTRACTIBLE; CONTRACTIBLE_SPHERE] THEN ASM_SIMP_TAC[GSYM FRONTIER_CBALL; INTERIOR_CBALL; BALL_EQ_EMPTY; CONV_RULE(RAND_CONV SYM_CONV) (SPEC_ALL RELATIVE_FRONTIER_NONEMPTY_INTERIOR)] THEN STRIP_TAC THEN MATCH_MP_TAC INESSENTIAL_SPHEREMAP_LOWDIM_GEN THEN ASM_REWRITE_TAC[CONVEX_CBALL; BOUNDED_CBALL; AFF_DIM_CBALL] THEN ASM_REWRITE_TAC[GSYM REAL_NOT_LE; INT_OF_NUM_LT]);; let HOMEOMORPHIC_SPHERES_EQ,HOMOTOPY_EQUIVALENT_SPHERES_EQ = (CONJ_PAIR o prove) (`(!a:real^M b:real^N r s. sphere(a,r) homeomorphic sphere(b,s) <=> r < &0 /\ s < &0 \/ r = &0 /\ s = &0 \/ &0 < r /\ &0 < s /\ dimindex(:M) = dimindex(:N)) /\ (!a:real^M b:real^N r s. sphere(a,r) homotopy_equivalent sphere(b,s) <=> r < &0 /\ s < &0 \/ r = &0 /\ s = &0 \/ &0 < r /\ &0 < s /\ dimindex(:M) = dimindex(:N))`, let lemma = prove (`!a:real^M r b:real^N s. dimindex(:M) < dimindex(:N) /\ &0 < r /\ &0 < s ==> ~(sphere(a,r) homotopy_equivalent sphere(b,s))`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o ISPEC `sphere(a:real^M,r)` o MATCH_MP HOMOTOPY_EQUIVALENT_HOMOTOPIC_TRIVIALITY) THEN MATCH_MP_TAC(TAUT `~p /\ q ==> (p <=> q) ==> F`) THEN CONJ_TAC THENL [SUBGOAL_THEN `~(sphere(a:real^M,r) = {})` MP_TAC THENL [REWRITE_TAC[SPHERE_EQ_EMPTY] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM]] THEN X_GEN_TAC `c:real^M` THEN DISCH_TAC THEN DISCH_THEN(MP_TAC o SPECL[`\a:real^M. a`; `(\a. c):real^M->real^M`]) THEN SIMP_TAC[CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID; IMAGE_ID; SUBSET_REFL] THEN REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `~(contractible(sphere(a:real^M,r)))` MP_TAC THENL [REWRITE_TAC[CONTRACTIBLE_SPHERE] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[contractible] THEN MESON_TAC[]]; MAP_EVERY X_GEN_TAC [`f:real^M->real^N`; `g:real^M->real^N`] THEN STRIP_TAC THEN MP_TAC(ISPEC `g:real^M->real^N` INESSENTIAL_SPHEREMAP_LOWDIM) THEN MP_TAC(ISPEC `f:real^M->real^N` INESSENTIAL_SPHEREMAP_LOWDIM) THEN ASM_REWRITE_TAC[IMP_IMP; AND_FORALL_THM] THEN DISCH_THEN (MP_TAC o SPECL [`a:real^M`; `r:real`; `b:real^N`; `s:real`]) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; IMP_CONJ; RIGHT_IMP_FORALL_THM] THEN REPEAT GEN_TAC THEN REWRITE_TAC[IMP_IMP] THEN DISCH_THEN (fun th -> CONJUNCTS_THEN (ASSUME_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_SUBSET) th THEN MP_TAC th) THEN MATCH_MP_TAC(MESON[HOMOTOPIC_WITH_TRANS; HOMOTOPIC_WITH_SYM] `homotopic_with p (s,t) c d ==> homotopic_with p (s,t) f c /\ homotopic_with p (s,t) g d ==> homotopic_with p (s,t) f g`) THEN REWRITE_TAC[HOMOTOPIC_CONSTANT_MAPS] THEN DISJ2_TAC THEN MP_TAC(ISPECL [`b:real^N`; `s:real`] PATH_CONNECTED_SPHERE) THEN ANTS_TAC THENL [FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ARITH_RULE `m < n ==> 1 <= m ==> 2 <= n`)) THEN REWRITE_TAC[DIMINDEX_GE_1]; REWRITE_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT] THEN DISCH_THEN MATCH_MP_TAC THEN SUBGOAL_THEN `~(sphere(a:real^M,r) = {})` MP_TAC THENL [REWRITE_TAC[SPHERE_EQ_EMPTY] THEN ASM_REAL_ARITH_TAC; ASM SET_TAC[]]]]) in REWRITE_TAC[AND_FORALL_THM] THEN REPEAT GEN_TAC THEN MATCH_MP_TAC(TAUT `(r ==> p) /\ (q ==> r) /\ (p ==> q) ==> (r <=> q) /\ (p <=> q)`) THEN REWRITE_TAC[HOMEOMORPHIC_IMP_HOMOTOPY_EQUIVALENT] THEN ASM_CASES_TAC `r < &0` THEN ASM_SIMP_TAC[SPHERE_EMPTY; SPHERE_EQ_EMPTY; HOMEOMORPHIC_EMPTY; HOMOTOPY_EQUIVALENT_EMPTY] THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_CASES_TAC `s < &0` THEN ASM_SIMP_TAC[SPHERE_EMPTY; SPHERE_EQ_EMPTY; HOMEOMORPHIC_EMPTY; HOMOTOPY_EQUIVALENT_EMPTY] THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_CASES_TAC `r = &0` THEN ASM_SIMP_TAC[SPHERE_SING; REAL_LT_REFL; HOMEOMORPHIC_SING; HOMOTOPY_EQUIVALENT_SING; CONTRACTIBLE_SPHERE; ONCE_REWRITE_RULE[HOMOTOPY_EQUIVALENT_SYM] HOMOTOPY_EQUIVALENT_SING] THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_CASES_TAC `s = &0` THEN ASM_SIMP_TAC[SPHERE_SING; REAL_LT_REFL; HOMEOMORPHIC_SING; HOMOTOPY_EQUIVALENT_SING; CONTRACTIBLE_SPHERE; ONCE_REWRITE_RULE[HOMOTOPY_EQUIVALENT_SYM] HOMOTOPY_EQUIVALENT_SING] THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `&0 < r /\ &0 < s` STRIP_ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ASM_REWRITE_TAC[]] THEN CONJ_TAC THENL [DISCH_THEN(fun th -> let t = `?a:real^M b:real^N. ~(sphere(a,r) homeomorphic sphere(b,s))` in MP_TAC(DISCH t (GEOM_EQUAL_DIMENSION_RULE th (ASSUME t)))) THEN ASM_SIMP_TAC[HOMEOMORPHIC_SPHERES] THEN MESON_TAC[]; ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[ARITH_RULE `~(m:num = n) <=> m < n \/ n < m`] THEN STRIP_TAC THENL [ALL_TAC; ONCE_REWRITE_TAC[HOMOTOPY_EQUIVALENT_SYM]] THEN ASM_SIMP_TAC[lemma]]);; let SIMPLY_CONNECTED_SPHERE_GEN = prove (`!s. convex s /\ bounded s /\ &3 <= aff_dim s ==> simply_connected(relative_frontier s)`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[SIMPLY_CONNECTED_EQ_CONTRACTIBLE_CIRCLEMAP; PATH_CONNECTED_SPHERE_GEN; INT_ARITH `&3:int <= x ==> ~(x = &1)`] THEN SUBGOAL_THEN `sphere(vec 0:real^2,&1) = relative_frontier(cball(vec 0,&1))` SUBST1_TAC THENL [REWRITE_TAC[RELATIVE_FRONTIER_CBALL; REAL_OF_NUM_EQ; ARITH]; ALL_TAC] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC INESSENTIAL_SPHEREMAP_LOWDIM_GEN THEN ASM_REWRITE_TAC[CONVEX_CBALL; BOUNDED_CBALL; AFF_DIM_CBALL] THEN REWRITE_TAC[DIMINDEX_2; REAL_LT_01] THEN ASM_INT_ARITH_TAC);; let SIMPLY_CONNECTED_SPHERE = prove (`!a:real^N r. 3 <= dimindex(:N) ==> simply_connected(sphere(a,r))`, REPEAT STRIP_TAC THEN REPEAT_TCL DISJ_CASES_THEN ASSUME_TAC (REAL_ARITH `r < &0 \/ r = &0 \/ &0 < r`) THEN ASM_SIMP_TAC[SPHERE_EMPTY; SIMPLY_CONNECTED_EMPTY] THEN ASM_SIMP_TAC[SPHERE_SING; CONVEX_SING; CONVEX_IMP_SIMPLY_CONNECTED] THEN MP_TAC(ISPEC `cball(a:real^N,r)` SIMPLY_CONNECTED_SPHERE_GEN) THEN ASM_SIMP_TAC[AFF_DIM_CBALL; RELATIVE_FRONTIER_CBALL; CONVEX_CBALL; BOUNDED_CBALL; REAL_LT_IMP_NE; INT_OF_NUM_LE]);; let SIMPLY_CONNECTED_PUNCTURED_CONVEX = prove (`!s a:real^N. convex s /\ &3 <= aff_dim s ==> simply_connected(s DELETE a)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `(a:real^N) IN relative_interior s` THENL [ALL_TAC; MATCH_MP_TAC CONTRACTIBLE_IMP_SIMPLY_CONNECTED THEN MATCH_MP_TAC CONTRACTIBLE_CONVEX_TWEAK_BOUNDARY_POINTS THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPEC `s:real^N->bool` RELATIVE_INTERIOR_SUBSET) THEN MP_TAC(ISPEC `s:real^N->bool` CLOSURE_SUBSET) THEN ASM SET_TAC[]] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_RELATIVE_INTERIOR_CBALL]) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC)) THEN MP_TAC(ISPECL [`cball(a:real^N,e) INTER affine hull s`; `s:real^N->bool`; `a:real^N`] HOMOTOPY_EQUIVALENT_RELATIVE_FRONTIER_PUNCTURED_CONVEX) THEN ANTS_TAC THENL [ALL_TAC; MATCH_MP_TAC(MESON[HOMOTOPY_EQUIVALENT_SIMPLE_CONNECTEDNESS] `simply_connected s ==> s homotopy_equivalent t ==> simply_connected t`) THEN MATCH_MP_TAC SIMPLY_CONNECTED_SPHERE_GEN] THEN ASM_SIMP_TAC[CONVEX_INTER; AFFINE_AFFINE_HULL; AFFINE_IMP_CONVEX; CONVEX_CBALL; BOUNDED_INTER; BOUNDED_CBALL] THEN REPEAT CONJ_TAC THENL [W(MP_TAC o PART_MATCH (lhs o rand) RELATIVE_INTERIOR_CONVEX_INTER_AFFINE o rand o snd) THEN REWRITE_TAC[CONVEX_CBALL; AFFINE_AFFINE_HULL; INTERIOR_CBALL] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM; IN_INTER] THEN DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN ASM_SIMP_TAC[CENTRE_IN_BALL] THEN ANTS_TAC THENL [ALL_TAC; DISCH_THEN SUBST1_TAC THEN ASM_SIMP_TAC[CENTRE_IN_BALL; IN_INTER]] THEN ASM_MESON_TAC[SUBSET; HULL_SUBSET; RELATIVE_INTERIOR_SUBSET]; REWRITE_TAC[relative_frontier] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET u ==> c = s ==> c DIFF i SUBSET u`)) THEN REWRITE_TAC[CLOSURE_EQ] THEN MATCH_MP_TAC CLOSED_INTER THEN REWRITE_TAC[CLOSED_AFFINE_HULL; CLOSED_CBALL]; ONCE_REWRITE_TAC[INTER_COMM] THEN W(MP_TAC o PART_MATCH (lhs o rand) AFFINE_HULL_AFFINE_INTER_NONEMPTY_INTERIOR o rand o snd); ONCE_REWRITE_TAC[INTER_COMM] THEN W(MP_TAC o PART_MATCH (lhs o rand) AFF_DIM_CONVEX_INTER_NONEMPTY_INTERIOR o rand o snd)] THEN ASM_SIMP_TAC[AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL; GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM; IN_INTER] THEN DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN ASM_SIMP_TAC[INTERIOR_CBALL; CENTRE_IN_BALL; HULL_INC; HULL_SUBSET; AFF_DIM_AFFINE_HULL]);; let SIMPLY_CONNECTED_PUNCTURED_UNIVERSE = prove (`!a. 3 <= dimindex(:N) ==> simply_connected((:real^N) DELETE a)`, GEN_TAC THEN DISCH_THEN(MP_TAC o SPECL [`a:real^N`; `&1`] o MATCH_MP SIMPLY_CONNECTED_SPHERE) THEN MATCH_MP_TAC EQ_IMP THEN MATCH_MP_TAC HOMOTOPY_EQUIVALENT_SIMPLE_CONNECTEDNESS THEN MP_TAC(ISPECL [`cball(a:real^N,&1)`; `a:real^N`] HOMOTOPY_EQUIVALENT_RELATIVE_FRONTIER_PUNCTURED_AFFINE_HULL) THEN REWRITE_TAC[CONVEX_CBALL; BOUNDED_CBALL; RELATIVE_INTERIOR_CBALL; RELATIVE_FRONTIER_CBALL] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN SIMP_TAC[CENTRE_IN_BALL; AFFINE_HULL_NONEMPTY_INTERIOR; INTERIOR_CBALL; BALL_EQ_EMPTY; REAL_OF_NUM_LE; ARITH; REAL_LT_01]);; let SIMPLY_CONNECTED_CONVEX_DIFF_FINITE = prove (`!s t:real^N->bool. convex s /\ &3 <= aff_dim s /\ FINITE t ==> simply_connected(s DIFF t)`, let lemma = prove (`!P. (?u v. P u /\ P v /\ ~(u = v)) /\ (!c. P c ==> ~(s INTER {x:real^N | x$k = c} = {})) ==> ?u v. u IN s INTER {x | P(x$k)} /\ v IN s INTER {x | P(x$k)} /\ ~(u = v)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(fun th -> MP_TAC(SPEC `u:real` th) THEN MP_TAC(SPEC `v:real` th)) THEN ASM SET_TAC[]) in ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN GEN_TAC THEN WF_INDUCT_TAC `CARD(t:real^N->bool)` THEN X_GEN_TAC `s:real^N->bool` THEN STRIP_TAC THEN ONCE_REWRITE_TAC[SET_RULE `s DIFF t = s DIFF (s INTER t)`] THEN REPEAT_TCL DISJ_CASES_THEN STRIP_ASSUME_TAC (SET_RULE `s INTER t = {} \/ ?a:real^N. s INTER t = {a} \/ ?a b. ~(a = b) /\ a IN s /\ a IN t /\ b IN s /\ b IN t`) THEN ASM_SIMP_TAC[CONVEX_IMP_SIMPLY_CONNECTED; SIMPLY_CONNECTED_PUNCTURED_CONVEX; DIFF_EMPTY; SET_RULE `s DIFF {a} = s DELETE a`] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [CART_EQ]) THEN REWRITE_TAC[NOT_IMP; LEFT_IMP_EXISTS_THM; NOT_FORALL_THM] THEN X_GEN_TAC `k:num` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (REAL_ARITH `~(x = y) ==> x < y \/ y < x`)) THEN POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN ONCE_REWRITE_TAC[REWRITE_RULE[IMP_CONJ_ALT] IMP_IMP] THEN MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`b:real^N`; `a:real^N`] THEN MATCH_MP_TAC(MESON[] `(!a b. R a b ==> R b a) /\ (!a b. P a b ==> R a b) ==> !a b. P a b \/ P b a ==> R a b`) THEN CONJ_TAC THENL [REWRITE_TAC[CONJ_ACI]; REPEAT STRIP_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[RIGHT_IMP_FORALL_THM; IMP_IMP]) THEN SUBGOAL_THEN `!s t. s DIFF t = {x | x IN s /\ x$k < (b:real^N)$k} DIFF {x | x IN t /\ x$k < b$k} UNION {x:real^N | x IN s /\ (a:real^N)$k < x$k} DIFF {x | x IN t /\ a$k < x$k}` (fun th -> ONCE_REWRITE_TAC[th] THEN ASSUME_TAC(GSYM th)) THENL [FIRST_ASSUM(MP_TAC o MATCH_MP (REAL_ARITH `a < b ==> !x. a < x \/ x < b`)) THEN SET_TAC[]; MATCH_MP_TAC SIMPLY_CONNECTED_UNION THEN ASM_REWRITE_TAC[]] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[SET_RULE `{x | x IN s /\ P x} DIFF {x | x IN t /\ P x} = (s DIFF t) INTER {x | P x}`] THEN MATCH_MP_TAC OPEN_IN_OPEN_INTER THEN REWRITE_TAC[OPEN_HALFSPACE_COMPONENT_LT]; REWRITE_TAC[SET_RULE `{x | x IN s /\ P x} DIFF {x | x IN t /\ P x} = (s DIFF t) INTER {x | P x}`] THEN MATCH_MP_TAC OPEN_IN_OPEN_INTER THEN REWRITE_TAC[GSYM real_gt; OPEN_HALFSPACE_COMPONENT_GT]; FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[SET_RULE `{x | x IN s /\ P x} = s INTER {x | P x}`; FINITE_INTER; CONVEX_INTER; CONVEX_HALFSPACE_COMPONENT_LT] THEN CONJ_TAC THENL [MATCH_MP_TAC CARD_PSUBSET THEN ASM SET_TAC[REAL_LT_REFL]; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (INT_ARITH `&3:int <= x ==> y = x ==> &3 <= y`)) THEN MATCH_MP_TAC AFF_DIM_CONVEX_INTER_OPEN THEN ASM_REWRITE_TAC[OPEN_HALFSPACE_COMPONENT_LT] THEN ASM SET_TAC[]; FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[SET_RULE `{x | x IN s /\ P x} = s INTER {x | P x}`; FINITE_INTER; CONVEX_INTER; REWRITE_RULE[real_gt] CONVEX_HALFSPACE_COMPONENT_GT] THEN CONJ_TAC THENL [MATCH_MP_TAC CARD_PSUBSET THEN ASM SET_TAC[REAL_LT_REFL]; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (INT_ARITH `&3:int <= x ==> y = x ==> &3 <= y`)) THEN MATCH_MP_TAC AFF_DIM_CONVEX_INTER_OPEN THEN ASM_REWRITE_TAC[REWRITE_RULE[real_gt] OPEN_HALFSPACE_COMPONENT_GT] THEN ASM SET_TAC[]; ALL_TAC; ALL_TAC] THEN REWRITE_TAC[SET_RULE `{x | x IN s /\ P x} DIFF {x | x IN t /\ P x} = (s DIFF t) INTER {x | P x}`] THEN REWRITE_TAC[SET_RULE `(s INTER u) INTER (s INTER v) = s INTER (u INTER v)`; SET_RULE `(s DIFF t) INTER u = (s INTER u) DIFF t`] THEN REWRITE_TAC[SET_RULE `s INTER u DIFF s INTER t = s INTER u DIFF t`] THENL [MATCH_MP_TAC PATH_CONNECTED_CONVEX_DIFF_COUNTABLE THEN ASM_SIMP_TAC[FINITE_IMP_COUNTABLE; CONVEX_INTER; COLLINEAR_AFF_DIM; CONVEX_HALFSPACE_COMPONENT_LT; REWRITE_RULE[real_gt] CONVEX_HALFSPACE_COMPONENT_GT] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (INT_ARITH `&3:int <= x ==> y = x ==> ~(y <= &1)`)) THEN MATCH_MP_TAC AFF_DIM_CONVEX_INTER_OPEN THEN ASM_SIMP_TAC[OPEN_INTER; OPEN_HALFSPACE_COMPONENT_LT; REWRITE_RULE[real_gt] OPEN_HALFSPACE_COMPONENT_GT] THEN MATCH_MP_TAC(MESON[INFINITE; FINITE_EMPTY] `INFINITE s ==> ~(s = {})`); REWRITE_TAC[SET_RULE `s DIFF t = {} <=> s SUBSET t`] THEN MATCH_MP_TAC(MESON[FINITE_SUBSET; INFINITE] `INFINITE s /\ FINITE t ==> ~(s SUBSET t)`) THEN ASM_REWRITE_TAC[]] THEN (ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 5) [CONNECTED_FINITE_IFF_SING; INFINITE; CONVEX_CONNECTED; CONVEX_INTER; CONVEX_HALFSPACE_COMPONENT_LT; REWRITE_RULE[real_gt] CONVEX_HALFSPACE_COMPONENT_GT] THEN MATCH_MP_TAC(SET_RULE `!u v. u IN s /\ v IN s /\ ~(u = v) ==> ~(s = {} \/ ?z. s = {z})`) THEN REWRITE_TAC[SET_RULE `{x | P x} INTER {x | Q x} = {x | Q x /\ P x}`] THEN MP_TAC lemma THEN DISCH_THEN MATCH_MP_TAC THEN CONJ_TAC THENL [EXISTS_TAC `a$k + &1 / &3 * ((b:real^N)$k - (a:real^N)$k)` THEN EXISTS_TAC `a$k + &2 / &3 * ((b:real^N)$k - (a:real^N)$k)` THEN ASM_REAL_ARITH_TAC; X_GEN_TAC `c:real` THEN STRIP_TAC THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER; IN_ELIM_THM] THEN SUBGOAL_THEN `!x:real^N. x$k = basis k dot x` (fun t -> SIMP_TAC[t]) THENL [ASM_MESON_TAC[DOT_BASIS]; MATCH_MP_TAC CONNECTED_IVT_HYPERPLANE] THEN MAP_EVERY EXISTS_TAC [`a:real^N`; `b:real^N`] THEN ASM_SIMP_TAC[CONVEX_CONNECTED; DOT_BASIS; REAL_LT_IMP_LE]]));; (* ------------------------------------------------------------------------- *) (* Some technical lemmas about extending maps from cell complexes. *) (* ------------------------------------------------------------------------- *) let EXTEND_MAP_CELL_COMPLEX_TO_SPHERE, EXTEND_MAP_CELL_COMPLEX_TO_SPHERE_COFINITE = (CONJ_PAIR o prove) (`(!f:real^M->real^N m s t. FINITE m /\ (!c. c IN m ==> polytope c /\ aff_dim c < aff_dim t) /\ (!c1 c2. c1 IN m /\ c2 IN m ==> c1 INTER c2 face_of c1 /\ c1 INTER c2 face_of c2) /\ s SUBSET UNIONS m /\ closed s /\ convex t /\ bounded t /\ f continuous_on s /\ IMAGE f s SUBSET relative_frontier t ==> ?g. g continuous_on UNIONS m /\ IMAGE g (UNIONS m) SUBSET relative_frontier t /\ !x. x IN s ==> g x = f x) /\ (!f:real^M->real^N m s t. FINITE m /\ (!c. c IN m ==> polytope c /\ aff_dim c <= aff_dim t) /\ (!c1 c2. c1 IN m /\ c2 IN m ==> c1 INTER c2 face_of c1 /\ c1 INTER c2 face_of c2) /\ s SUBSET UNIONS m /\ closed s /\ convex t /\ bounded t /\ f continuous_on s /\ IMAGE f s SUBSET relative_frontier t ==> ?k g. FINITE k /\ DISJOINT k s /\ g continuous_on (UNIONS m DIFF k) /\ IMAGE g (UNIONS m DIFF k) SUBSET relative_frontier t /\ !x. x IN s ==> g x = f x)`, let wemma = prove (`!h:real^M->real^N k t f. (!s. s IN f ==> ?g. g continuous_on s /\ IMAGE g s SUBSET t /\ !x. x IN s INTER k ==> g x = h x) /\ FINITE f /\ (!s. s IN f ==> closed s) /\ (!s t. s IN f /\ t IN f /\ ~(s = t) ==> (s INTER t) SUBSET k) ==> ?g. g continuous_on (UNIONS f) /\ IMAGE g (UNIONS f) SUBSET t /\ !x. x IN (UNIONS f) INTER k ==> g x = h x`, REPLICATE_TAC 3 GEN_TAC THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r ==> s <=> q ==> p /\ r ==> s`] THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN REWRITE_TAC[UNIONS_0; IMAGE_CLAUSES; EMPTY_SUBSET; CONTINUOUS_ON_EMPTY; INTER_EMPTY; NOT_IN_EMPTY] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_INSERT] THEN REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC; SUBSET_REFL] THEN MAP_EVERY X_GEN_TAC [`s:real^M->bool`; `u:(real^M->bool)->bool`] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC (REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC)) THEN ASM_SIMP_TAC[UNIONS_INSERT] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^M->real^N` STRIP_ASSUME_TAC) THEN ASM_CASES_TAC `(s:real^M->bool) UNION UNIONS u = UNIONS u` THENL [ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(X_CHOOSE_THEN `f:real^M->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\x. if x IN s then (f:real^M->real^N) x else g x` THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_ON_CASES THEN ASM_SIMP_TAC[CLOSED_UNIONS] THEN ASM SET_TAC[]) in let lemma = prove (`!h:real^M->real^N k t f. (!s. s IN f ==> ?g. g continuous_on s /\ IMAGE g s SUBSET t /\ !x. x IN s INTER k ==> g x = h x) /\ FINITE f /\ (!s. s IN f ==> closed s) /\ (!s t. s IN f /\ t IN f /\ ~(s SUBSET t) /\ ~(t SUBSET s) ==> (s INTER t) SUBSET k) ==> ?g. g continuous_on (UNIONS f) /\ IMAGE g (UNIONS f) SUBSET t /\ !x. x IN (UNIONS f) INTER k ==> g x = h x`, REPEAT STRIP_TAC THEN FIRST_ASSUM(SUBST1_TAC o SYM o MATCH_MP UNIONS_MAXIMAL_SETS) THEN MATCH_MP_TAC wemma THEN ASM_SIMP_TAC[FINITE_RESTRICT; IN_ELIM_THM] THEN ASM SET_TAC[]) in let zemma = prove (`!f:real^M->real^N m n t. FINITE m /\ (!c. c IN m ==> polytope c) /\ n SUBSET m /\ (!c. c IN m DIFF n ==> aff_dim c < aff_dim t) /\ (!c1 c2. c1 IN m /\ c2 IN m ==> (c1 INTER c2) face_of c1 /\ (c1 INTER c2) face_of c2) /\ convex t /\ bounded t /\ f continuous_on (UNIONS n) /\ IMAGE f (UNIONS n) SUBSET relative_frontier t ==> ?g. g continuous_on (UNIONS m) /\ IMAGE g (UNIONS m) SUBSET relative_frontier t /\ (!x. x IN UNIONS n ==> g x = f x)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `m DIFF n:(real^M->bool)->bool = {}` THENL [SUBGOAL_THEN `(UNIONS m:real^M->bool) SUBSET UNIONS n` ASSUME_TAC THENL [ASM SET_TAC[]; EXISTS_TAC `f:real^M->real^N`] THEN REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ASM SET_TAC[]]; ALL_TAC] THEN SUBGOAL_THEN `!i. &i <= aff_dim t ==> ?g. g continuous_on (UNIONS (n UNION {d | ?c. c IN m /\ d face_of c /\ aff_dim d < &i})) /\ IMAGE g (UNIONS (n UNION {d | ?c. c IN m /\ d face_of c /\ aff_dim d < &i})) SUBSET relative_frontier t /\ (!x. x IN UNIONS n ==> g x = (f:real^M->real^N) x)` MP_TAC THENL [ALL_TAC; MP_TAC(ISPEC `aff_dim(t:real^N->bool)` INT_OF_NUM_EXISTS) THEN MATCH_MP_TAC(TAUT `q /\ (p ==> r) ==> (p <=> q) ==> r`) THEN CONJ_TAC THENL [ASM_MESON_TAC[AFF_DIM_GE; MEMBER_NOT_EMPTY; INT_ARITH `--(&1):int <= s /\ s < t ==> &0 <= t`]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_TAC `i:num`) THEN DISCH_THEN(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[INT_LE_REFL] THEN SUBGOAL_THEN `UNIONS (n UNION {d | ?c. c IN m /\ d face_of c /\ aff_dim d < &i}) = UNIONS m:real^M->bool` (fun th -> REWRITE_TAC[th]) THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [MATCH_MP_TAC UNIONS_MONO THEN REWRITE_TAC[IN_UNION] THEN REWRITE_TAC[TAUT `p \/ q ==> r <=> (p ==> r) /\ (q ==> r)`] THEN REWRITE_TAC[FORALL_AND_THM; FORALL_IN_GSPEC] THEN CONJ_TAC THENL [ASM_MESON_TAC[SUBSET]; GEN_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[FACE_OF_IMP_SUBSET]; MATCH_MP_TAC SUBSET_UNIONS THEN REWRITE_TAC[SUBSET; IN_UNION] THEN X_GEN_TAC `d:real^M->bool` THEN DISCH_TAC THEN ASM_CASES_TAC `(d:real^M->bool) IN n` THEN ASM_SIMP_TAC[IN_ELIM_THM] THEN EXISTS_TAC `d:real^M->bool` THEN ASM_SIMP_TAC[FACE_OF_REFL; POLYTOPE_IMP_CONVEX] THEN ASM SET_TAC[]]] THEN MATCH_MP_TAC num_INDUCTION THEN CONJ_TAC THENL [REWRITE_TAC[INT_ARITH `d < &0 <=> (--(&1) <= d ==> d:int = --(&1))`] THEN REWRITE_TAC[AFF_DIM_GE; AFF_DIM_EQ_MINUS1] THEN SUBGOAL_THEN `{d:real^M->bool| ?c. c IN m /\ d face_of c /\ d = {}} = {{}}` (fun th -> REWRITE_TAC[th]) THENL [GEN_REWRITE_TAC I [EXTENSION] THEN X_GEN_TAC `d:real^M->bool` THEN REWRITE_TAC[IN_SING; IN_ELIM_THM] THEN ASM_CASES_TAC `d:real^M->bool = {}` THEN ASM_REWRITE_TAC[EMPTY_FACE_OF] THEN ASM SET_TAC[]; REWRITE_TAC[UNIONS_UNION; UNIONS_1; UNION_EMPTY] THEN ASM_MESON_TAC[]]; ALL_TAC] THEN X_GEN_TAC `p:num` THEN REWRITE_TAC[GSYM INT_OF_NUM_SUC] THEN REWRITE_TAC[INT_ARITH `p + &1 <= x <=> p:int < x`] THEN DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN ASM_SIMP_TAC[INT_LT_IMP_LE] THEN DISCH_THEN(X_CHOOSE_THEN `h:real^M->real^N` STRIP_ASSUME_TAC) THEN REWRITE_TAC[INT_ARITH `x:int < p + &1 <=> x <= p`] THEN SUBGOAL_THEN `~(t:real^N->bool = {})` ASSUME_TAC THENL [ASM_MESON_TAC[AFF_DIM_EMPTY; INT_ARITH `~(&p:int < --(&1))`]; ALL_TAC] THEN SUBGOAL_THEN `~(relative_frontier t:real^N->bool = {})` ASSUME_TAC THENL [ASM_REWRITE_TAC[RELATIVE_FRONTIER_EQ_EMPTY] THEN DISCH_TAC THEN MP_TAC(ISPEC `t:real^N->bool` AFFINE_BOUNDED_EQ_LOWDIM) THEN ASM_REWRITE_TAC[] THEN ASM_INT_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `!d. d IN n UNION {d | ?c. c IN m /\ d face_of c /\ aff_dim d <= &p} ==> ?g. (g:real^M->real^N) continuous_on d /\ IMAGE g d SUBSET relative_frontier t /\ !x. x IN d INTER UNIONS (n UNION {d | ?c. c IN m /\ d face_of c /\ aff_dim d < &p}) ==> g x = h x` MP_TAC THENL [X_GEN_TAC `d:real^M->bool` THEN ASM_CASES_TAC `(d:real^M->bool) SUBSET UNIONS (n UNION {d | ?c. c IN m /\ d face_of c /\ aff_dim d < &p})` THENL [DISCH_THEN(K ALL_TAC) THEN EXISTS_TAC `h:real^M->real^N` THEN REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ASM SET_TAC[]]; ALL_TAC] THEN ASM_CASES_TAC `?a:real^M. d = {a}` THENL [FIRST_X_ASSUM(X_CHOOSE_THEN `a:real^M` SUBST_ALL_TAC) THEN DISCH_THEN(K ALL_TAC) THEN ASM_SIMP_TAC[CONTINUOUS_ON_SING; SET_RULE `~({a} SUBSET s) ==> ~(x IN {a} INTER s)`] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_INSERT; NOT_IN_EMPTY] THEN MATCH_MP_TAC(MESON[] `(?c. P(\x. c)) ==> (?f. P f)`) THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `~(d:real^M->bool = {})` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o MATCH_MP (SET_RULE `~(s SUBSET UNIONS f) ==> ~(s IN f)`)) THEN REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `~(d IN s UNION t) /\ d IN s UNION u ==> ~(d IN s) /\ d IN u DIFF t`)) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `d IN {d | ?c. c IN m /\ d face_of c /\ aff_dim d <= &p} DIFF {d | ?c. c IN m /\ d face_of c /\ aff_dim d < &p} ==> ?c. c IN m /\ d face_of c /\ (aff_dim d <= &p /\ ~(aff_dim d < &p))`)) THEN REWRITE_TAC[INT_ARITH `d:int <= p /\ ~(d < p) <=> d = p`] THEN DISCH_THEN(X_CHOOSE_THEN `c:real^M->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`h:real^M->real^N`; `relative_frontier d:real^M->bool`; `t:real^N->bool`] NULLHOMOTOPIC_INTO_RELATIVE_FRONTIER_EXTENSION) THEN ASM_REWRITE_TAC[CLOSED_RELATIVE_FRONTIER; RELATIVE_FRONTIER_EQ_EMPTY] THEN SUBGOAL_THEN `relative_frontier d SUBSET UNIONS {e:real^M->bool | e face_of c /\ aff_dim e < &p}` ASSUME_TAC THENL [W(MP_TAC o PART_MATCH (lhs o rand) RELATIVE_FRONTIER_OF_POLYHEDRON o lhand o snd) THEN ANTS_TAC THENL [ASM_MESON_TAC[POLYTOPE_IMP_POLYHEDRON; FACE_OF_POLYTOPE_POLYTOPE]; DISCH_THEN SUBST1_TAC] THEN MATCH_MP_TAC SUBSET_UNIONS THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_GSPEC; IN_ELIM_THM; facet_of] THEN X_GEN_TAC `f:real^M->bool` THEN REPEAT STRIP_TAC THENL [ASM_MESON_TAC[FACE_OF_TRANS]; INT_ARITH_TAC]; ALL_TAC] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; ASM_MESON_TAC[AFFINE_BOUNDED_EQ_TRIVIAL; FACE_OF_POLYTOPE_POLYTOPE; POLYTOPE_IMP_BOUNDED]; ASM SET_TAC[]]; ALL_TAC] THEN MATCH_MP_TAC(TAUT `p /\ (q ==> r) ==> (p <=> q) ==> r`) THEN CONJ_TAC THENL [MATCH_MP_TAC INESSENTIAL_SPHEREMAP_LOWDIM_GEN THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[FACE_OF_POLYTOPE_POLYTOPE; POLYTOPE_IMP_CONVEX]; ASM_MESON_TAC[FACE_OF_POLYTOPE_POLYTOPE; POLYTOPE_IMP_BOUNDED]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; ASM SET_TAC[]]; MATCH_MP_TAC MONO_EXISTS] THEN X_GEN_TAC `g:real^M->real^N` THEN STRIP_TAC THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV]; ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[INTER_UNIONS] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `(!x. x IN s ==> P x) ==> t SUBSET s ==> !x. x IN t ==> P x`)) THEN REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC] THEN X_GEN_TAC `e:real^M->bool` THEN DISCH_TAC THEN MATCH_MP_TAC FACE_OF_SUBSET_RELATIVE_FRONTIER THEN CONJ_TAC THENL [MATCH_MP_TAC(MESON[] `(d INTER e) face_of d /\ (d INTER e) face_of e ==> (d INTER e) face_of d`) THEN MATCH_MP_TAC FACE_OF_INTER_SUBFACE THEN EXISTS_TAC `c:real^M->bool` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_UNION]) THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN ASM_MESON_TAC[FACE_OF_REFL; SUBSET; POLYTOPE_IMP_CONVEX]; REWRITE_TAC[SET_RULE `d INTER e = d <=> d SUBSET e`] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_UNION]) THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(DISJ_CASES_THEN MP_TAC) THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_MESON_TAC[AFF_DIM_SUBSET; INT_NOT_LE]]; ALL_TAC] THEN DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] lemma)) THEN ANTS_TAC THENL [ALL_TAC; MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[] THEN ASM SET_TAC[]] THEN CONJ_TAC THENL [REWRITE_TAC[FINITE_UNION] THEN CONJ_TAC THENL [ASM_MESON_TAC[FINITE_SUBSET]; ALL_TAC] THEN MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `UNIONS {{d:real^M->bool | d face_of c} | c IN m}` THEN CONJ_TAC THENL [REWRITE_TAC[FINITE_UNIONS; FORALL_IN_GSPEC] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN ASM_SIMP_TAC[FINITE_IMAGE] THEN ASM_MESON_TAC[FINITE_POLYTOPE_FACES]; REWRITE_TAC[UNIONS_GSPEC] THEN SET_TAC[]]; ALL_TAC] THEN CONJ_TAC THENL [REWRITE_TAC[IN_UNION; IN_ELIM_THM] THEN ASM_MESON_TAC[FACE_OF_IMP_CLOSED; POLYTOPE_IMP_CLOSED; POLYTOPE_IMP_CONVEX; SUBSET]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`d:real^M->bool`; `e:real^M->bool`] THEN REWRITE_TAC[IN_UNION; IN_ELIM_THM] THEN DISCH_THEN(CONJUNCTS_THEN2 (DISJ_CASES_THEN2 ASSUME_TAC (X_CHOOSE_THEN `c:real^M->bool` STRIP_ASSUME_TAC)) MP_TAC) THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(CONJUNCTS_THEN2 (DISJ_CASES_THEN2 ASSUME_TAC (X_CHOOSE_THEN `k:real^M->bool` STRIP_ASSUME_TAC)) MP_TAC) THENL [ASM SET_TAC[]; STRIP_TAC] THEN REWRITE_TAC[UNIONS_UNION] THEN MATCH_MP_TAC(SET_RULE `s SUBSET u ==> s SUBSET t UNION u`) THEN MATCH_MP_TAC(SET_RULE `x IN s ==> x SUBSET UNIONS s`) THEN REWRITE_TAC[IN_ELIM_THM] THEN EXISTS_TAC `c:real^M->bool` THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `d INTER e face_of (d:real^M->bool) /\ d INTER e face_of e` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[FACE_OF_INTER_SUBFACE]; ALL_TAC] THEN CONJ_TAC THENL [ASM_MESON_TAC[FACE_OF_TRANS]; ALL_TAC] THEN TRANS_TAC INT_LTE_TRANS `aff_dim(d:real^M->bool)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC FACE_OF_AFF_DIM_LT THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[POLYTOPE_IMP_CONVEX; FACE_OF_IMP_CONVEX]; ASM SET_TAC[]]) in let memma = prove (`!h:real^M->real^N k t u f. (!s. s IN f ==> ?a g. ~(a IN u) /\ g continuous_on (s DELETE a) /\ IMAGE g (s DELETE a) SUBSET t /\ !x. x IN s INTER k ==> g x = h x) /\ FINITE f /\ (!s. s IN f ==> closed s) /\ (!s t. s IN f /\ t IN f /\ ~(s = t) ==> (s INTER t) SUBSET k) ==> ?c g. FINITE c /\ DISJOINT c u /\ CARD c <= CARD f /\ g continuous_on (UNIONS f DIFF c) /\ IMAGE g (UNIONS f DIFF c) SUBSET t /\ !x. x IN (UNIONS f DIFF c) INTER k ==> g x = h x`, REPLICATE_TAC 4 GEN_TAC THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r ==> s <=> q ==> p /\ r ==> s`] THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN REWRITE_TAC[UNIONS_0; IMAGE_CLAUSES; EMPTY_SUBSET; CONTINUOUS_ON_EMPTY; INTER_EMPTY; NOT_IN_EMPTY; EMPTY_DIFF] THEN CONJ_TAC THENL [MESON_TAC[DISJOINT_EMPTY; FINITE_EMPTY; CARD_CLAUSES; LE_REFL]; ALL_TAC] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_INSERT] THEN REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC; SUBSET_REFL] THEN MAP_EVERY X_GEN_TAC [`s:real^M->bool`; `u:(real^M->bool)->bool`] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC (REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC)) THEN ASM_SIMP_TAC[UNIONS_INSERT] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`c:real^M->bool`; `g:real^M->real^N`] THEN STRIP_TAC THEN ASM_SIMP_TAC[CARD_CLAUSES] THEN ASM_CASES_TAC `(s:real^M->bool) UNION UNIONS u = UNIONS u` THENL [ASM_SIMP_TAC[] THEN ASM_MESON_TAC[ARITH_RULE `x <= y ==> x <= SUC y`]; ALL_TAC] THEN FIRST_X_ASSUM(X_CHOOSE_THEN `a:real^M` (X_CHOOSE_THEN `f:real^M->real^N` STRIP_ASSUME_TAC)) THEN EXISTS_TAC `(a:real^M) INSERT c` THEN ASM_SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; RIGHT_EXISTS_AND_THM] THEN REPEAT CONJ_TAC THENL [ASM SET_TAC[]; ASM_ARITH_TAC; ALL_TAC] THEN EXISTS_TAC `\x. if x IN s then (f:real^M->real^N) x else g x` THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `(s DIFF ((a:real^M) INSERT c)) UNION (UNIONS u DIFF ((a:real^M) INSERT c))` THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[CLOSED_IN_CLOSED] THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[]; REWRITE_TAC[CLOSED_IN_CLOSED] THEN EXISTS_TAC `UNIONS u:real^M->bool` THEN ASM_SIMP_TAC[CLOSED_UNIONS]; 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 (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)); ALL_TAC] THEN ASM SET_TAC[]) in let temma = prove (`!h:real^M->real^N k t u f. (!s. s IN f ==> ?a g. ~(a IN u) /\ g continuous_on (s DELETE a) /\ IMAGE g (s DELETE a) SUBSET t /\ !x. x IN s INTER k ==> g x = h x) /\ FINITE f /\ (!s. s IN f ==> closed s) /\ (!s t. s IN f /\ t IN f /\ ~(s SUBSET t) /\ ~(t SUBSET s) ==> (s INTER t) SUBSET k) ==> ?c g. FINITE c /\ DISJOINT c u /\ CARD c <= CARD f /\ g continuous_on (UNIONS f DIFF c) /\ IMAGE g (UNIONS f DIFF c) SUBSET t /\ !x. x IN (UNIONS f DIFF c) INTER k ==> g x = h x`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`h:real^M->real^N`; `k:real^M->bool`; `t:real^N->bool`; `u:real^M->bool`; `{t:real^M->bool | t IN f /\ (!u. u IN f ==> ~(t PSUBSET u))}`] memma) THEN ASM_SIMP_TAC[FINITE_RESTRICT; IN_ELIM_THM; UNIONS_MAXIMAL_SETS] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] LE_TRANS)) THEN MATCH_MP_TAC CARD_SUBSET THEN ASM_SIMP_TAC[] THEN SET_TAC[]) in let bemma = prove (`!f:real^M->real^N m n t. FINITE m /\ (!c. c IN m ==> polytope c) /\ n SUBSET m /\ (!c. c IN m DIFF n ==> aff_dim c <= aff_dim t) /\ (!c1 c2. c1 IN m /\ c2 IN m ==> (c1 INTER c2) face_of c1 /\ (c1 INTER c2) face_of c2) /\ convex t /\ bounded t /\ f continuous_on (UNIONS n) /\ IMAGE f (UNIONS n) SUBSET relative_frontier t ==> ?k g. FINITE k /\ DISJOINT k (UNIONS n) /\ CARD k <= CARD m /\ g continuous_on (UNIONS m DIFF k) /\ IMAGE g (UNIONS m DIFF k) SUBSET relative_frontier t /\ (!x. x IN UNIONS n ==> g x = f x)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `n UNION {d:real^M->bool | ?c. c IN m DIFF n /\ d face_of c /\ aff_dim d < aff_dim(t:real^N->bool)}`; `n:(real^M->bool)->bool`; `t:real^N->bool`] zemma) THEN ASM_REWRITE_TAC[SUBSET_UNION; SET_RULE `(n UNION m) DIFF n = m DIFF n`] THEN SIMP_TAC[IN_DIFF; IN_ELIM_THM; LEFT_IMP_EXISTS_THM; LEFT_AND_EXISTS_THM] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [ASM_REWRITE_TAC[FINITE_UNION] THEN CONJ_TAC THENL [ASM_MESON_TAC[FINITE_SUBSET]; ALL_TAC] THEN MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `UNIONS {{d:real^M->bool | d face_of c} | c IN m}` THEN CONJ_TAC THENL [REWRITE_TAC[FINITE_UNIONS; FORALL_IN_GSPEC] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN ASM_SIMP_TAC[FINITE_IMAGE] THEN ASM_MESON_TAC[FINITE_POLYTOPE_FACES]; REWRITE_TAC[UNIONS_GSPEC] THEN SET_TAC[]]; REWRITE_TAC[IN_UNION; IN_ELIM_THM] THEN ASM_MESON_TAC[FACE_OF_POLYTOPE_POLYTOPE; SUBSET]; REWRITE_TAC[IN_UNION; IN_ELIM_THM] THEN ASM_MESON_TAC[FACE_OF_INTER_SUBFACE; SUBSET; FACE_OF_REFL; POLYTOPE_IMP_CONVEX; FACE_OF_IMP_CONVEX]]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `h:real^M->real^N` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `!d. d IN m ==> ?a g. ~(a IN UNIONS n) /\ (g:real^M->real^N) continuous_on (d DELETE a) /\ IMAGE g (d DELETE a) SUBSET relative_frontier t /\ !x. x IN d INTER UNIONS (n UNION {d | ?c. (c IN m /\ ~(c IN n)) /\ d face_of c /\ aff_dim d < aff_dim t}) ==> g x = h x` MP_TAC THENL [X_GEN_TAC `d:real^M->bool` THEN DISCH_TAC THEN ASM_CASES_TAC `(d:real^M->bool) SUBSET UNIONS(n UNION {d | ?c. (c IN m /\ ~(c IN n)) /\ d face_of c /\ aff_dim d < aff_dim(t:real^N->bool)})` THENL [SUBGOAL_THEN `~(UNIONS n = (:real^M))` MP_TAC THENL [MATCH_MP_TAC(MESON[NOT_BOUNDED_UNIV] `bounded s ==> ~(s = UNIV)`) THEN MATCH_MP_TAC BOUNDED_UNIONS THEN ASM_MESON_TAC[POLYTOPE_IMP_BOUNDED; SUBSET; FINITE_SUBSET]; GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [EXTENSION]] THEN REWRITE_TAC[IN_UNIV; NOT_FORALL_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^M` THEN STRIP_TAC THEN EXISTS_TAC `h:real^M->real^N` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SET_RULE `s SUBSET t ==> s DELETE a SUBSET t`]; ASM SET_TAC[]]; ALL_TAC] THEN ASM_CASES_TAC `(d:real^M->bool) IN n` THENL [ASM SET_TAC[]; ALL_TAC] THEN DISJ_CASES_THEN MP_TAC (SPEC `relative_interior(d:real^M->bool) = {}` EXCLUDED_MIDDLE) THENL [ASM_SIMP_TAC[RELATIVE_INTERIOR_EQ_EMPTY; POLYTOPE_IMP_CONVEX] THEN ASM SET_TAC[]; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY]] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^M` THEN STRIP_TAC THEN SUBGOAL_THEN `relative_frontier d SUBSET UNIONS {e:real^M->bool | e face_of d /\ aff_dim e < aff_dim(t:real^N->bool)}` ASSUME_TAC THENL [W(MP_TAC o PART_MATCH (lhs o rand) RELATIVE_FRONTIER_OF_POLYHEDRON o lhand o snd) THEN ANTS_TAC THENL [ASM_MESON_TAC[POLYTOPE_IMP_POLYHEDRON; FACE_OF_POLYTOPE_POLYTOPE]; DISCH_THEN SUBST1_TAC] THEN MATCH_MP_TAC SUBSET_UNIONS THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_GSPEC; IN_ELIM_THM; facet_of] THEN ASM_SIMP_TAC[INT_ARITH `d - &1:int < t <=> d <= t`; IN_DIFF]; ALL_TAC] THEN MP_TAC(ISPECL [`d:real^M->bool`; `a:real^M`] RELATIVE_FRONTIER_RETRACT_OF_PUNCTURED_AFFINE_HULL) THEN ASM_SIMP_TAC[POLYTOPE_IMP_CONVEX; POLYTOPE_IMP_BOUNDED] THEN REWRITE_TAC[retract_of; LEFT_IMP_EXISTS_THM; retraction] THEN X_GEN_TAC `r:real^M->real^M` THEN STRIP_TAC THEN EXISTS_TAC `(h:real^M->real^N) o (r:real^M->real^M)` THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[IN_UNIONS] THEN DISCH_THEN(X_CHOOSE_THEN `e:real^M->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `e INTER d face_of e /\ e INTER d face_of (d:real^M->bool)` MP_TAC THENL [ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] FACE_OF_SUBSET_RELATIVE_FRONTIER) o CONJUNCT2) THEN REWRITE_TAC[NOT_IMP; relative_frontier] THEN MP_TAC(ISPEC `d:real^M->bool` RELATIVE_INTERIOR_SUBSET) THEN ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN SIMP_TAC[HULL_SUBSET; SET_RULE `s SUBSET t ==> s DELETE a SUBSET t DELETE a`]; REWRITE_TAC[IMAGE_o] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE h t SUBSET u ==> s SUBSET t ==> IMAGE h s SUBSET u`)); SIMP_TAC[INTER_UNIONS; o_THM] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `(!x. x IN s ==> r x = x) ==> t SUBSET s ==> !x. x IN t ==> h(r x) = h x`)) THEN REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC] THEN X_GEN_TAC `e:real^M->bool` THEN DISCH_TAC THEN MATCH_MP_TAC FACE_OF_SUBSET_RELATIVE_FRONTIER THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC(MESON[] `(d INTER e) face_of d /\ (d INTER e) face_of e ==> (d INTER e) face_of d`) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_UNION]) THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THENL [ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN MATCH_MP_TAC FACE_OF_INTER_SUBFACE THEN MAP_EVERY EXISTS_TAC [`d:real^M->bool`; `c:real^M->bool`] THEN ASM_SIMP_TAC[FACE_OF_REFL; POLYTOPE_IMP_CONVEX]] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE r (h DELETE a) SUBSET t ==> d SUBSET h /\ t SUBSET u ==> IMAGE r (d DELETE a) SUBSET u`)) THEN REWRITE_TAC[HULL_SUBSET] THEN ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] temma)) THEN ANTS_TAC THENL [ALL_TAC; REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]] THEN ASM_SIMP_TAC[POLYTOPE_IMP_CLOSED] THEN MAP_EVERY X_GEN_TAC [`d:real^M->bool`; `e:real^M->bool`] THEN STRIP_TAC THEN REWRITE_TAC[UNIONS_UNION] THEN ASM_CASES_TAC `(d:real^M->bool) IN n` THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `x IN s ==> x SUBSET t UNION UNIONS s`) THEN REWRITE_TAC[IN_ELIM_THM] THEN EXISTS_TAC `d:real^M->bool` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `d INTER e:real^M->bool = d` THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[] THEN TRANS_TAC INT_LTE_TRANS `aff_dim(d:real^M->bool)` THEN ASM_SIMP_TAC[IN_DIFF] THEN MATCH_MP_TAC FACE_OF_AFF_DIM_LT THEN ASM_MESON_TAC[POLYTOPE_IMP_CONVEX]) in CONJ_TAC THENL [REPEAT STRIP_TAC THEN SUBGOAL_THEN `compact(s:real^M->bool)` ASSUME_TAC THENL [ASM_SIMP_TAC[COMPACT_EQ_BOUNDED_CLOSED] THEN ASM_MESON_TAC[BOUNDED_SUBSET; BOUNDED_UNIONS; POLYTOPE_IMP_BOUNDED]; ALL_TAC] THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`; `relative_frontier t:real^N->bool`] NEIGHBOURHOOD_EXTENSION_INTO_ANR) THEN ASM_SIMP_TAC[LEFT_FORALL_IMP_THM; ENR_IMP_ANR; ENR_RELATIVE_FRONTIER_CONVEX] THEN 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 [`s:real^M->bool`; `(:real^M) DIFF v`] SEPARATE_COMPACT_CLOSED) THEN ASM_SIMP_TAC[GSYM OPEN_CLOSED; IN_DIFF; IN_UNIV] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ONCE_REWRITE_TAC[TAUT `p /\ ~q ==> r <=> p /\ ~r ==> q`] THEN REWRITE_TAC[REAL_NOT_LE; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN MP_TAC(ISPECL [`m:(real^M->bool)->bool`; `aff_dim(t:real^N->bool) - &1`; `d:real`] CELL_COMPLEX_SUBDIVISION_EXISTS) THEN ASM_SIMP_TAC[INT_ARITH `x:int <= t - &1 <=> x < t`] THEN DISCH_THEN(X_CHOOSE_THEN `n:(real^M->bool)->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`g:real^M->real^N`; `n:(real^M->bool)->bool`; `{c:real^M->bool | c IN n /\ c SUBSET v}`; `t:real^N->bool`] zemma) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_SIMP_TAC[SUBSET_RESTRICT; IN_DIFF] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; ASM SET_TAC[]]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `h:real^M->real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN TRANS_TAC EQ_TRANS `(g:real^M->real^N) x` THEN CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `(x:real^M) IN UNIONS n` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[IN_UNIONS] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^M->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN REWRITE_TAC[SUBSET] THEN X_GEN_TAC `y:real^M` THEN DISCH_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN EXISTS_TAC `x:real^M` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `diameter(c:real^M->bool)` THEN ASM_SIMP_TAC[dist] THEN MATCH_MP_TAC DIAMETER_BOUNDED_BOUND THEN ASM_SIMP_TAC[POLYTOPE_IMP_BOUNDED]]; REPEAT STRIP_TAC THEN SUBGOAL_THEN `compact(s:real^M->bool)` ASSUME_TAC THENL [ASM_SIMP_TAC[COMPACT_EQ_BOUNDED_CLOSED] THEN ASM_MESON_TAC[BOUNDED_SUBSET; BOUNDED_UNIONS; POLYTOPE_IMP_BOUNDED]; ALL_TAC] THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`; `relative_frontier t:real^N->bool`] NEIGHBOURHOOD_EXTENSION_INTO_ANR) THEN ASM_SIMP_TAC[LEFT_FORALL_IMP_THM; ENR_IMP_ANR; ENR_RELATIVE_FRONTIER_CONVEX] THEN 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 [`s:real^M->bool`; `(:real^M) DIFF v`] SEPARATE_COMPACT_CLOSED) THEN ASM_SIMP_TAC[GSYM OPEN_CLOSED; IN_DIFF; IN_UNIV] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ONCE_REWRITE_TAC[TAUT `p /\ ~q ==> r <=> p /\ ~r ==> q`] THEN REWRITE_TAC[REAL_NOT_LE; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN MP_TAC(ISPECL [`m:(real^M->bool)->bool`; `aff_dim(t:real^N->bool)`; `d:real`] CELL_COMPLEX_SUBDIVISION_EXISTS) THEN ASM_SIMP_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `n:(real^M->bool)->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`g:real^M->real^N`; `n:(real^M->bool)->bool`; `{c:real^M->bool | c IN n /\ c SUBSET v}`; `t:real^N->bool`] bemma) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_SIMP_TAC[SUBSET_RESTRICT; IN_DIFF] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; ASM SET_TAC[]]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:real^M->bool` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `h:real^M->real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `DISJOINT k u ==> s SUBSET u ==> DISJOINT k s`)) THEN REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC; X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN TRANS_TAC EQ_TRANS `(g:real^M->real^N) x` THEN CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[]] THEN (SUBGOAL_THEN `(x:real^M) IN UNIONS n` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[IN_UNIONS] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^M->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN REWRITE_TAC[SUBSET] THEN X_GEN_TAC `y:real^M` THEN DISCH_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN EXISTS_TAC `x:real^M` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `diameter(c:real^M->bool)` THEN ASM_SIMP_TAC[dist] THEN MATCH_MP_TAC DIAMETER_BOUNDED_BOUND THEN ASM_SIMP_TAC[POLYTOPE_IMP_BOUNDED])]]);; (* ------------------------------------------------------------------------- *) (* Special cases and corollaries involving spheres. *) (* ------------------------------------------------------------------------- *) let EXTEND_MAP_AFFINE_TO_SPHERE_COFINITE_SIMPLE = prove (`!f:real^M->real^N s t u. compact s /\ convex u /\ bounded u /\ aff_dim t <= aff_dim u /\ s SUBSET t /\ f continuous_on s /\ IMAGE f s SUBSET relative_frontier u ==> ?k g. FINITE k /\ k SUBSET t /\ DISJOINT k s /\ g continuous_on (t DIFF k) /\ IMAGE g (t DIFF k) SUBSET relative_frontier u /\ !x. x IN s ==> g x = f x`, let lemma = prove (`!f:A->B->bool P k. INFINITE {x | P x} /\ FINITE k /\ (!x y. P x /\ P y /\ ~(x = y) ==> DISJOINT (f x) (f y)) ==> ?x. P x /\ DISJOINT k (f x)`, REWRITE_TAC[INFINITE] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[SET_RULE `(?x. P x /\ DISJOINT k (f x)) <=> ~(!x. ?y. P x ==> y IN k /\ y IN f x)`] THEN REWRITE_TAC[SKOLEM_THM] THEN DISCH_THEN(X_CHOOSE_TAC `g:A->B`) THEN MP_TAC(ISPECL [`g:A->B`; `{x:A | P x}`] FINITE_IMAGE_INJ_EQ) THEN ASM_REWRITE_TAC[IN_ELIM_THM; NOT_IMP] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] FINITE_SUBSET)) THEN ASM SET_TAC[]) in SUBGOAL_THEN `!f:real^M->real^N s t u. compact s /\ convex u /\ bounded u /\ aff_dim t <= aff_dim u /\ s SUBSET t /\ f continuous_on s /\ IMAGE f s SUBSET relative_frontier u ==> ?k g. FINITE k /\ DISJOINT k s /\ g continuous_on (t DIFF k) /\ IMAGE g (t DIFF k) SUBSET relative_frontier u /\ !x. x IN s ==> g x = f x` MP_TAC THENL [ALL_TAC; REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^M->real^N` THEN DISCH_THEN(X_CHOOSE_THEN `k:real^M->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `k INTER t:real^M->bool` THEN ASM_SIMP_TAC[FINITE_INTER; INTER_SUBSET] THEN REPEAT CONJ_TAC THENL [ASM SET_TAC[]; 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[]] THEN SUBGOAL_THEN `!f:real^M->real^N s t u. compact s /\ s SUBSET t /\ affine t /\ convex u /\ bounded u /\ aff_dim t <= aff_dim u /\ f continuous_on s /\ IMAGE f s SUBSET relative_frontier u ==> ?k g. FINITE k /\ DISJOINT k s /\ g continuous_on (t DIFF k) /\ IMAGE g (t DIFF k) SUBSET relative_frontier u /\ !x. x IN s ==> g x = f x` ASSUME_TAC THENL [ALL_TAC; REPEAT STRIP_TAC THEN SUBGOAL_THEN `?k g. FINITE k /\ DISJOINT k s /\ g continuous_on (affine hull t DIFF k) /\ IMAGE g (affine hull t DIFF k) SUBSET relative_frontier u /\ !x. x IN s ==> g x = (f:real^M->real^N) x` MP_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[AFF_DIM_AFFINE_HULL; AFFINE_AFFINE_HULL] THEN TRANS_TAC SUBSET_TRANS `t:real^M->bool` THEN ASM_REWRITE_TAC[HULL_SUBSET]; REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)); FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS)) THEN MATCH_MP_TAC IMAGE_SUBSET] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s DIFF k SUBSET t DIFF k`) THEN REWRITE_TAC[HULL_SUBSET]]] THEN REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THENL [ASM_CASES_TAC `relative_frontier(u:real^N->bool) = {}` THENL [RULE_ASSUM_TAC(REWRITE_RULE[RELATIVE_FRONTIER_EQ_EMPTY]) THEN UNDISCH_TAC `bounded(u:real^N->bool)` THEN ASM_SIMP_TAC[AFFINE_BOUNDED_EQ_LOWDIM] THEN DISCH_TAC THEN SUBGOAL_THEN `aff_dim(t:real^M->bool) <= &0` MP_TAC THENL [ASM_INT_ARITH_TAC; ALL_TAC] THEN SIMP_TAC[AFF_DIM_GE; INT_ARITH `--(&1):int <= x ==> (x <= &0 <=> x = --(&1) \/ x = &0)`] THEN REWRITE_TAC[AFF_DIM_EQ_MINUS1; AFF_DIM_EQ_0] THEN DISCH_THEN(DISJ_CASES_THEN2 ASSUME_TAC (X_CHOOSE_TAC `a:real^M`)) THEN EXISTS_TAC `{a:real^M}` THEN ASM_REWRITE_TAC[DISJOINT_EMPTY; FINITE_SING; NOT_IN_EMPTY; EMPTY_DIFF; DIFF_EQ_EMPTY; IMAGE_CLAUSES; CONTINUOUS_ON_EMPTY; EMPTY_SUBSET]; EXISTS_TAC `{}:real^M->bool` THEN FIRST_X_ASSUM(X_CHOOSE_TAC `y:real^N` o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN ASM_SIMP_TAC[FINITE_EMPTY; DISJOINT_EMPTY; NOT_IN_EMPTY; DIFF_EMPTY] THEN EXISTS_TAC `(\x. y):real^M->real^N` THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN ASM SET_TAC[]]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o MATCH_MP COMPACT_IMP_BOUNDED) THEN DISCH_THEN(MP_TAC o MATCH_MP BOUNDED_SUBSET_CLOSED_INTERVAL_SYMMETRIC) THEN REWRITE_TAC[INSERT_SUBSET] THEN DISCH_THEN(X_CHOOSE_THEN `b:real^M` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`f:real^M->real^N`; `{interval[--(b + vec 1):real^M,b + vec 1] INTER t}`; `s:real^M->bool`; `u:real^N->bool`] EXTEND_MAP_CELL_COMPLEX_TO_SPHERE_COFINITE) THEN SUBGOAL_THEN `interval[--b,b] SUBSET interval[--(b + vec 1):real^M,b + vec 1]` ASSUME_TAC THENL [REWRITE_TAC[SUBSET_INTERVAL; VECTOR_ADD_COMPONENT; VECTOR_NEG_COMPONENT; VEC_COMPONENT] THEN REAL_ARITH_TAC; ALL_TAC] THEN ASM_REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FINITE_SING] THEN REWRITE_TAC[FORALL_IN_INSERT; NOT_IN_EMPTY; IMP_IMP] THEN REWRITE_TAC[INTER_IDEMPOT; UNIONS_1; FACE_OF_REFL_EQ; SUBSET_INTER] THEN ANTS_TAC THENL [ASM_SIMP_TAC[HULL_SUBSET; COMPACT_IMP_CLOSED] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC POLYTOPE_INTER_POLYHEDRON THEN ASM_SIMP_TAC[POLYTOPE_INTERVAL; AFFINE_IMP_POLYHEDRON]; TRANS_TAC INT_LE_TRANS `aff_dim(t:real^M->bool)` THEN ASM_SIMP_TAC[AFF_DIM_SUBSET; INTER_SUBSET]; ASM_SIMP_TAC[CONVEX_INTER; CONVEX_INTERVAL; AFFINE_IMP_CONVEX]; ASM SET_TAC[]]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`k:real^M->bool`; `g:real^M->real^N`] THEN STRIP_TAC THEN EXISTS_TAC `k:real^M->bool` THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `?d:real. (&1 / &2 <= d /\ d <= &1) /\ DISJOINT k (frontier(interval[--(b + lambda i. d):real^M, (b + lambda i. d)]))` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC lemma THEN ASM_SIMP_TAC[INFINITE; FINITE_REAL_INTERVAL; REAL_NOT_LE] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN MATCH_MP_TAC REAL_WLOG_LT THEN REWRITE_TAC[] THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`x:real`; `y:real`] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[frontier] THEN MATCH_MP_TAC(SET_RULE `c SUBSET i' ==> DISJOINT (c DIFF i) (c' DIFF i')`) THEN REWRITE_TAC[INTERIOR_INTERVAL; CLOSURE_INTERVAL] THEN SIMP_TAC[SUBSET_INTERVAL; VECTOR_NEG_COMPONENT; VECTOR_ADD_COMPONENT; LAMBDA_BETA] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN ABBREV_TAC `c:real^M = b + lambda i. d` THEN SUBGOAL_THEN `interval[--b:real^M,b] SUBSET interval(--c,c) /\ interval[--b:real^M,b] SUBSET interval[--c,c] /\ interval[--c,c] SUBSET interval[--(b + vec 1):real^M,b + vec 1]` STRIP_ASSUME_TAC THENL [REWRITE_TAC[SUBSET_INTERVAL] THEN EXPAND_TAC "c" THEN REPEAT CONJ_TAC THEN SIMP_TAC[VECTOR_NEG_COMPONENT; VECTOR_ADD_COMPONENT; LAMBDA_BETA] THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[VEC_COMPONENT] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN EXISTS_TAC `(g:real^M->real^N) o closest_point (interval[--c,c] INTER t)` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_CLOSEST_POINT THEN ASM_SIMP_TAC[CONVEX_INTER; CLOSED_INTER; CLOSED_INTERVAL; CLOSED_AFFINE; AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL; CONVEX_INTERVAL] THEN ASM SET_TAC[]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET))]; REWRITE_TAC[IMAGE_o] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS)) THEN MATCH_MP_TAC IMAGE_SUBSET; X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN REWRITE_TAC[o_THM] THEN TRANS_TAC EQ_TRANS `(g:real^M->real^N) x` THEN CONJ_TAC THENL [AP_TERM_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CLOSEST_POINT_SELF THEN ASM_SIMP_TAC[IN_INTER; HULL_INC] THEN ASM SET_TAC[]] THEN (REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_DIFF] THEN X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `closest_point s x IN s /\ s SUBSET u ==> closest_point s x IN u`) THEN CONJ_TAC THENL [MATCH_MP_TAC CLOSEST_POINT_IN_SET; ASM SET_TAC[]] THEN ASM_SIMP_TAC[CLOSED_INTER; CLOSED_INTERVAL; CLOSED_AFFINE] THEN ASM SET_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `x IN interval[--c:real^M,c]` THEN ASM_SIMP_TAC[CLOSEST_POINT_SELF; IN_INTER] THEN MATCH_MP_TAC(SET_RULE `closest_point s x IN relative_frontier s /\ DISJOINT k (relative_frontier s) ==> ~(closest_point s x IN k)`) THEN CONJ_TAC THENL [MATCH_MP_TAC CLOSEST_POINT_IN_RELATIVE_FRONTIER THEN ASM_SIMP_TAC[CLOSED_INTER; CLOSED_AFFINE; CLOSED_INTERVAL] THEN CONJ_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[IN_DIFF]] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[SUBSET; RELATIVE_INTERIOR_SUBSET; IN_INTER]] THEN ONCE_REWRITE_TAC[INTER_COMM] THEN W(MP_TAC o PART_MATCH (lhs o rand) AFFINE_HULL_CONVEX_INTER_NONEMPTY_INTERIOR o rand o snd) THEN ASM_SIMP_TAC[HULL_HULL; AFFINE_AFFINE_HULL; AFFINE_IMP_CONVEX] THEN ASM_SIMP_TAC[HULL_P] THEN ANTS_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[INTERIOR_INTERVAL] THEN ASM SET_TAC[]; W(MP_TAC o PART_MATCH (lhs o rand) RELATIVE_FRONTIER_CONVEX_INTER_AFFINE o rand o snd) THEN ANTS_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[CONVEX_INTERVAL; AFFINE_AFFINE_HULL; INTERIOR_INTERVAL] THEN ASM SET_TAC[]]));; let EXTEND_MAP_AFFINE_TO_SPHERE_COFINITE_GEN = prove (`!f:real^M->real^N s t u p. compact s /\ convex u /\ bounded u /\ affine t /\ aff_dim t <= aff_dim u /\ s SUBSET t /\ f continuous_on s /\ IMAGE f s SUBSET relative_frontier u /\ (!c. c IN components(t DIFF s) /\ bounded c ==> ~(c INTER p = {})) ==> ?k g. FINITE k /\ k SUBSET p /\ k SUBSET t /\ DISJOINT k s /\ g continuous_on (t DIFF k) /\ IMAGE g (t DIFF k) SUBSET relative_frontier u /\ !x. x IN s ==> g x = f x`, let lemma0 = prove (`!u t s v. closed_in (subtopology euclidean u) v /\ t SUBSET u /\ s = v INTER t ==> closed_in (subtopology euclidean t) s`, REPEAT GEN_TAC THEN REWRITE_TAC[CLOSED_IN_CLOSED; LEFT_AND_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN SET_TAC[]) in let lemma1 = prove (`!f:A->B->bool P k. INFINITE {x | P x} /\ FINITE k /\ (!x y. P x /\ P y /\ ~(x = y) ==> DISJOINT (f x) (f y)) ==> ?x. P x /\ DISJOINT k (f x)`, REWRITE_TAC[INFINITE] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[SET_RULE `(?x. P x /\ DISJOINT k (f x)) <=> ~(!x. ?y. P x ==> y IN k /\ y IN f x)`] THEN REWRITE_TAC[SKOLEM_THM] THEN DISCH_THEN(X_CHOOSE_TAC `g:A->B`) THEN MP_TAC(ISPECL [`g:A->B`; `{x:A | P x}`] FINITE_IMAGE_INJ_EQ) THEN ASM_REWRITE_TAC[IN_ELIM_THM; NOT_IMP] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] FINITE_SUBSET)) THEN ASM SET_TAC[]) in let lemma2 = prove (`!f:real^M->real^N s t k p u. FINITE k /\ affine u /\ f continuous_on ((u:real^M->bool) DIFF k) /\ IMAGE f ((u:real^M->bool) DIFF k) SUBSET t /\ (!c. c IN components((u:real^M->bool) DIFF s) /\ ~(c INTER k = {}) ==> ~(c INTER p = {})) /\ closed_in (subtopology euclidean u) s /\ DISJOINT k s /\ k SUBSET u ==> ?g. g continuous_on ((u:real^M->bool) DIFF p) /\ IMAGE g ((u:real^M->bool) DIFF p) SUBSET t /\ !x. x IN s ==> g x = f x`, REPEAT GEN_TAC THEN ASM_CASES_TAC `k:real^M->bool = {}` THENL [ASM_REWRITE_TAC[DIFF_EMPTY] THEN REPEAT STRIP_TAC THEN EXISTS_TAC `f:real^M->real^N` THEN REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_DIFF]; ASM SET_TAC[]]; STRIP_TAC] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN SUBGOAL_THEN `~(((u:real^M->bool) DIFF s) INTER k = {})` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV o LAND_CONV o LAND_CONV) [UNIONS_COMPONENTS] THEN REWRITE_TAC[INTER_UNIONS; EMPTY_UNIONS; FORALL_IN_GSPEC] THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `co:real^M->bool` THEN STRIP_TAC THEN SUBGOAL_THEN `locally connected (u:real^M->bool)` ASSUME_TAC THENL [ASM_SIMP_TAC[AFFINE_IMP_CONVEX; CONVEX_IMP_LOCALLY_CONNECTED]; ALL_TAC] THEN SUBGOAL_THEN `!c. c IN components ((u:real^M->bool) DIFF s) /\ ~(c INTER k = {}) ==> ?a g. a IN c /\ a IN p /\ g continuous_on (s UNION (c DELETE a)) /\ IMAGE g (s UNION (c DELETE a)) SUBSET t /\ !x. x IN s ==> g x = (f:real^M->real^N) x` MP_TAC THENL [X_GEN_TAC `c:real^M->bool` THEN STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN FIRST_X_ASSUM(MP_TAC o SPEC `c:real^M->bool`) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^M` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `open_in (subtopology euclidean u) (c:real^M->bool)` MP_TAC THENL [MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `u DIFF s:real^M->bool` THEN ASM_SIMP_TAC[OPEN_IN_DIFF; OPEN_IN_REFL] THEN MATCH_MP_TAC OPEN_IN_COMPONENTS_LOCALLY_CONNECTED THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC LOCALLY_OPEN_SUBSET THEN EXISTS_TAC `u:real^M->bool` THEN ASM_SIMP_TAC[OPEN_IN_DIFF; OPEN_IN_REFL]; DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th)] THEN REWRITE_TAC[OPEN_IN_CONTAINS_CBALL] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `a:real^M`)) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `ball(a:real^M,d) INTER u SUBSET c` ASSUME_TAC THENL [ASM_MESON_TAC[BALL_SUBSET_CBALL; SUBSET_TRANS; SET_RULE `b SUBSET c ==> b INTER u SUBSET c INTER u`]; ALL_TAC] THEN MP_TAC(ISPECL [`ball(a:real^M,d) INTER u`; `c:real^M->bool`; `s UNION c:real^M->bool`; `c INTER k:real^M->bool`] HOMEOMORPHISM_GROUPING_POINTS_EXISTS_GEN) THEN ASM_REWRITE_TAC[INTER_SUBSET; SUBSET_UNION; UNION_SUBSET] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_SUBSET_TRANS THEN EXISTS_TAC `u:real^M->bool` THEN ASM_SIMP_TAC[HULL_MINIMAL; HULL_SUBSET]; MP_TAC(ISPECL [`c:real^M->bool`; `u:real^M->bool`] AFFINE_HULL_OPEN_IN) THEN ASM_SIMP_TAC[HULL_P] THEN ASM SET_TAC[]; REWRITE_TAC[HULL_SUBSET]; ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]; ASM_MESON_TAC[FINITE_SUBSET; INTER_SUBSET]; MATCH_MP_TAC OPEN_IN_SUBSET_TRANS THEN EXISTS_TAC `u:real^M->bool` THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[OPEN_IN_OPEN_INTER; OPEN_BALL; INTER_COMM]; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN EXISTS_TAC `a:real^M` THEN REWRITE_TAC[CENTRE_IN_BALL] THEN ASM SET_TAC[]]; REWRITE_TAC[IN_INTER; LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`h:real^M->real^M`; `k:real^M->real^M`] THEN REWRITE_TAC[homeomorphism] THEN STRIP_TAC THEN MP_TAC(ISPECL [`cball(a:real^M,d) INTER u`; `a:real^M`] RELATIVE_FRONTIER_RETRACT_OF_PUNCTURED_AFFINE_HULL) THEN MP_TAC(ISPECL [`cball(a:real^M,d)`; `u:real^M->bool`] RELATIVE_INTERIOR_CONVEX_INTER_AFFINE) THEN MP_TAC(ISPECL [`cball(a:real^M,d)`; `u:real^M->bool`] RELATIVE_FRONTIER_CONVEX_INTER_AFFINE) THEN MP_TAC(ISPECL [`u:real^M->bool`; `cball(a:real^M,d)`] (ONCE_REWRITE_RULE[INTER_COMM] AFFINE_HULL_AFFINE_INTER_NONEMPTY_INTERIOR)) THEN ASM_SIMP_TAC[CONVEX_CBALL; FRONTIER_CBALL; INTERIOR_CBALL] THEN SUBGOAL_THEN `a IN ball(a:real^M,d) INTER u` ASSUME_TAC THENL [ASM_REWRITE_TAC[CENTRE_IN_BALL; IN_INTER] THEN ASM SET_TAC[]; ALL_TAC] THEN REPLICATE_TAC 3 (ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN SUBST1_TAC]) THEN ASM_SIMP_TAC[CONVEX_INTER; CONVEX_CBALL; AFFINE_IMP_CONVEX] THEN ANTS_TAC THENL [ASM_MESON_TAC[BOUNDED_SUBSET; INTER_SUBSET; BOUNDED_CBALL]; ALL_TAC] THEN ASM_REWRITE_TAC[retract_of; retraction] THEN DISCH_THEN(X_CHOOSE_THEN `r:real^M->real^M` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(f:real^M->real^N) o (k:real^M->real^M) o (\x. if x IN ball(a,d) then r x else x)` THEN REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN REWRITE_TAC[o_THM] THEN COND_CASES_TAC THENL [ASM SET_TAC[]; AP_TERM_TAC THEN ASM SET_TAC[]]] THEN ABBREV_TAC `j = \x:real^M. if x IN ball(a,d) then r x else x` THEN SUBGOAL_THEN `(j:real^M->real^M) continuous_on ((u:real^M->bool) DELETE a)` ASSUME_TAC THENL [EXPAND_TAC "j" THEN SUBGOAL_THEN `u DELETE (a:real^M) = (cball(a,d) DELETE a) INTER u UNION ((u:real^M->bool) DIFF ball(a,d))` (fun th -> SUBST1_TAC th THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL THEN SUBST1_TAC(SYM th)) THENL [MP_TAC(ISPECL [`a:real^M`; `d:real`] BALL_SUBSET_CBALL) THEN ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[IN_DIFF; IN_INTER; IN_DELETE; CONTINUOUS_ON_ID] THEN REPEAT CONJ_TAC THENL [ALL_TAC; ALL_TAC; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; REWRITE_TAC[GSYM BALL_UNION_SPHERE] THEN ASM SET_TAC[]] THEN REWRITE_TAC[CLOSED_IN_CLOSED] THENL [EXISTS_TAC `cball(a:real^M,d)` THEN REWRITE_TAC[CLOSED_CBALL]; EXISTS_TAC `(:real^M) DIFF ball(a,d)` THEN REWRITE_TAC[GSYM OPEN_CLOSED; OPEN_BALL]] THEN MP_TAC(ISPECL [`a:real^M`; `d:real`] BALL_SUBSET_CBALL) THEN MP_TAC(ISPECL [`a:real^M`; `d:real`] CENTRE_IN_BALL) THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `IMAGE (j:real^M->real^M) (s UNION c DELETE a) SUBSET (s UNION c DIFF ball(a,d))` ASSUME_TAC THENL [EXPAND_TAC "j" THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN COND_CASES_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN SUBGOAL_THEN `(r:real^M->real^M) x IN sphere(a,d)` MP_TAC THENL [MP_TAC(ISPECL [`a:real^M`; `d:real`] CENTRE_IN_BALL) THEN ASM SET_TAC[]; REWRITE_TAC[GSYM CBALL_DIFF_BALL] THEN ASM SET_TAC[]]; ALL_TAC] THEN CONJ_TAC THENL [REPEAT(MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THENL [ASM SET_TAC[]; ASM SET_TAC[]; ALL_TAC]; ONCE_REWRITE_TAC[IMAGE_o] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE f u SUBSET t ==> s SUBSET u ==> IMAGE f s SUBSET t`))] THEN REWRITE_TAC[IMAGE_o] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET u ==> IMAGE f u SUBSET t ==> IMAGE f s SUBSET t`)) THEN REWRITE_TAC[SUBSET; IN_UNIV; IN_DIFF; FORALL_IN_IMAGE] THEN ASM SET_TAC[]; ALL_TAC] THEN GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:(real^M->bool)->real^M`; `h:(real^M->bool)->real^M->real^N`] THEN DISCH_TAC THEN MP_TAC(ISPECL [`h:(real^M->bool)->real^M->real^N`; `\c:real^M->bool. s UNION (c DELETE (a c))`; `s UNION UNIONS { c DELETE (a c) | c IN components ((u:real^M->bool) DIFF s) /\ ~(c INTER k = {})}`; `{c | c IN components ((u:real^M->bool) DIFF s) /\ ~(c INTER k = {})}`] PASTING_LEMMA_EXISTS_CLOSED) THEN SUBGOAL_THEN `FINITE {c | c IN components((u:real^M->bool) DIFF s) /\ ~(c INTER k = {})}` ASSUME_TAC THENL [MP_TAC(ISPECL [`\c:real^M->bool. c INTER k`; `{c | c IN components ((u:real^M->bool) DIFF s) /\ ~(c INTER k = {})}`] FINITE_IMAGE_INJ_EQ) THEN REWRITE_TAC[IN_ELIM_THM] THEN ANTS_TAC THENL [MESON_TAC[COMPONENTS_EQ; SET_RULE `s INTER k = t INTER k /\ ~(s INTER k = {}) ==> ~(s INTER t = {})`]; DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[GSYM SIMPLE_IMAGE; IN_ELIM_THM]] THEN MP_TAC(ISPEC `{c INTER k |c| c IN components((u:real^M->bool) DIFF s) /\ ~(c INTER k = {})}` FINITE_UNIONS) THEN MATCH_MP_TAC(TAUT `p ==> (p <=> q /\ r) ==> q`) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] FINITE_SUBSET)) THEN REWRITE_TAC[UNIONS_GSPEC] THEN ASM SET_TAC[]; ALL_TAC] THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [REWRITE_TAC[UNIONS_GSPEC] THEN ASM SET_TAC[]; X_GEN_TAC `c:real^M->bool` THEN DISCH_TAC THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC lemma0 THEN MAP_EVERY EXISTS_TAC [`u:real^M->bool`; `s UNION c:real^M->bool`] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CLOSED_IN_UNION_COMPLEMENT_COMPONENT THEN ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[UNION_SUBSET; UNIONS_SUBSET; FORALL_IN_GSPEC] THEN MESON_TAC[IN_COMPONENTS_SUBSET; SET_RULE `c SUBSET u DIFF s ==> c DELETE a SUBSET u`]; ASM_SIMP_TAC[CLOSED_UNION_COMPLEMENT_COMPONENT; UNIONS_GSPEC] THEN MATCH_MP_TAC(SET_RULE `~(a IN t) /\ c DELETE a SUBSET t ==> s UNION c DELETE a = (s UNION c) INTER (s UNION t)`) THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[IN_ELIM_THM; IN_DELETE] THEN DISCH_THEN(X_CHOOSE_THEN `c':real^M->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`(u:real^M->bool) DIFF s`; `c:real^M->bool`; `c':real^M->bool`] COMPONENTS_EQ) THEN ASM_CASES_TAC `c':real^M->bool = c` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN ASM SET_TAC[]]; MAP_EVERY X_GEN_TAC [`c1:real^M->bool`; `c2:real^M->bool`; `x:real^M`] THEN STRIP_TAC THEN ASM_CASES_TAC `c2:real^M->bool = c1` THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(MP_TAC o MATCH_MP (SET_RULE `x IN u INTER (s UNION c1 DELETE a) INTER (s UNION c2 DELETE b) ==> (c1 INTER c2 = {}) ==> x IN s`)) THEN ANTS_TAC THENL [ASM_MESON_TAC[COMPONENTS_EQ]; ASM_SIMP_TAC[]]]; DISCH_THEN(X_CHOOSE_THEN `g:real^M->real^N` STRIP_ASSUME_TAC)] THEN MP_TAC (ISPECL [`\x. x IN s UNION UNIONS {c | c IN components((u:real^M->bool) DIFF s) /\ c INTER k = {}}`; `f:real^M->real^N`; `g:real^M->real^N`; `s UNION UNIONS {c | c IN components((u:real^M->bool) DIFF s) /\ c INTER k = {}}`; `s UNION UNIONS { c DELETE (a c) | c IN components((u:real^M->bool) DIFF s) /\ ~(c INTER k = {})}`] CONTINUOUS_ON_CASES_LOCAL) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [MATCH_MP_TAC lemma0 THEN EXISTS_TAC `u:real^M->bool` THEN EXISTS_TAC `u DIFF UNIONS {c DELETE a c | c IN components ((u:real^M->bool) DIFF s) /\ ~(c INTER k = {})}` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CLOSED_IN_DIFF THEN REWRITE_TAC[CLOSED_IN_REFL] THEN MATCH_MP_TAC OPEN_IN_UNIONS THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `c:real^M->bool` THEN DISCH_TAC THEN MATCH_MP_TAC OPEN_IN_DELETE THEN MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `u DIFF s:real^M->bool` THEN ASM_SIMP_TAC[OPEN_IN_DIFF; OPEN_IN_REFL] THEN MATCH_MP_TAC OPEN_IN_COMPONENTS_LOCALLY_CONNECTED THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC LOCALLY_OPEN_SUBSET THEN EXISTS_TAC `u:real^M->bool` THEN ASM_SIMP_TAC[OPEN_IN_DIFF; OPEN_IN_REFL]; ASM_REWRITE_TAC[UNION_SUBSET] THEN REWRITE_TAC[UNIONS_SUBSET; IN_ELIM_THM] THEN MESON_TAC[IN_COMPONENTS_SUBSET; SET_RULE `c SUBSET u DIFF s ==> c DELETE a SUBSET u /\ c SUBSET u`]; REWRITE_TAC[SET_RULE `(s UNION t) UNION (s UNION u) = (s UNION t) UNION u`] THEN MATCH_MP_TAC(SET_RULE `s SUBSET u /\ t INTER s = {} ==> s = (u DIFF t) INTER (s UNION t)`) THEN CONJ_TAC THENL [ASM_REWRITE_TAC[UNION_SUBSET] THEN REWRITE_TAC[UNIONS_SUBSET; IN_ELIM_THM] THEN MESON_TAC[IN_COMPONENTS_SUBSET; SET_RULE `c SUBSET u DIFF s ==> c DELETE a SUBSET u /\ c SUBSET u`]; ALL_TAC] THEN REWRITE_TAC[EMPTY_UNION; SET_RULE `c INTER (s UNION t) = (s INTER c) UNION (c INTER t)`] THEN CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `t SUBSET UNIV DIFF s ==> s INTER t = {}`) THEN REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC] THEN GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 (MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET) MP_TAC) THEN ASM SET_TAC[]; REWRITE_TAC[INTER_UNIONS; EMPTY_UNIONS; FORALL_IN_GSPEC] THEN X_GEN_TAC `c:real^M->bool` THEN STRIP_TAC THEN X_GEN_TAC `c':real^M->bool` THEN STRIP_TAC THEN MP_TAC(ISPECL [`(u:real^M->bool) DIFF s`; `c:real^M->bool`; `c':real^M->bool`] COMPONENTS_EQ) THEN ASM_CASES_TAC `c':real^M->bool = c` THENL [ASM_MESON_TAC[]; ASM SET_TAC[]]]]; MATCH_MP_TAC lemma0 THEN EXISTS_TAC `u:real^M->bool` THEN EXISTS_TAC `UNIONS {s UNION c |c| c IN components ((u:real^M->bool) DIFF s) /\ ~(c INTER k = {})}` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CLOSED_IN_UNIONS THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN ASM_SIMP_TAC[FINITE_IMAGE] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CLOSED_IN_UNION_COMPLEMENT_COMPONENT THEN ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[UNION_SUBSET] THEN REWRITE_TAC[UNIONS_SUBSET; IN_ELIM_THM] THEN MESON_TAC[IN_COMPONENTS_SUBSET; SET_RULE `c SUBSET u DIFF s ==> c DELETE a SUBSET u /\ c SUBSET u`]; MATCH_MP_TAC(SET_RULE `t SUBSET u /\ u INTER s SUBSET t ==> t = u INTER (s UNION t)`) THEN CONJ_TAC THENL [REWRITE_TAC[UNIONS_GSPEC] THEN ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `u INTER t SUBSET s ==> u INTER (s UNION t) SUBSET s UNION v`) THEN MATCH_MP_TAC(SET_RULE `((UNIV DIFF s) INTER t) INTER u SUBSET s ==> t INTER u SUBSET s`) THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV o TOP_DEPTH_CONV) [INTER_UNIONS] THEN REWRITE_TAC[SET_RULE `{g x | x IN {f y | P y}} = {g(f y) | P y}`] THEN REWRITE_TAC[SET_RULE `(UNIV DIFF s) INTER (s UNION c) = c DIFF s`] THEN REWRITE_TAC[SET_RULE `t INTER u SUBSET s <=> t INTER ((UNIV DIFF s) INTER u) = {}`] THEN ONCE_REWRITE_TAC[INTER_UNIONS] THEN REWRITE_TAC[EMPTY_UNIONS; FORALL_IN_GSPEC; INTER_UNIONS] THEN X_GEN_TAC `c:real^M->bool` THEN STRIP_TAC THEN X_GEN_TAC `c':real^M->bool` THEN STRIP_TAC THEN MP_TAC(ISPECL [`(u:real^M->bool) DIFF s`; `c:real^M->bool`; `c':real^M->bool`] COMPONENTS_EQ) THEN ASM_CASES_TAC `c':real^M->bool = c` THENL [ASM_MESON_TAC[]; ASM SET_TAC[]]]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[UNION_SUBSET] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[UNIONS_SUBSET; IN_ELIM_THM] THEN GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 (MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET) MP_TAC) THEN ASM SET_TAC[]; REWRITE_TAC[TAUT `p /\ ~p <=> F`] THEN X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_UNION] THEN ASM_CASES_TAC `(x:real^M) IN s` THEN ASM_REWRITE_TAC[] THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[UNIONS_GSPEC; IN_ELIM_THM; IN_DELETE] THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `c:real^M->bool`) (X_CHOOSE_TAC `c':real^M->bool`)) THEN MP_TAC(ISPECL [`(u:real^M->bool) DIFF s`; `c:real^M->bool`; `c':real^M->bool`] COMPONENTS_EQ) THEN ASM_CASES_TAC `c':real^M->bool = c` THENL [ASM_MESON_TAC[]; ASM SET_TAC[]]]; MATCH_MP_TAC(MESON[CONTINUOUS_ON_SUBSET] `t SUBSET s /\ P f ==> f continuous_on s ==> ?g. g continuous_on t /\ P g`) THEN REWRITE_TAC[] THEN CONJ_TAC THENL [REWRITE_TAC[SET_RULE `(s UNION t) UNION (s UNION u) = s UNION (t UNION u)`] THEN MATCH_MP_TAC(SET_RULE `(u DIFF s) DIFF p SUBSET t ==> u DIFF p SUBSET s UNION t`) THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [UNIONS_COMPONENTS] THEN REWRITE_TAC[UNIONS_GSPEC] THEN ASM SET_TAC[]; SIMP_TAC[IN_UNION]] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_DIFF; IN_UNION; IN_UNIV] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN ASM_CASES_TAC `(x:real^M) IN s` THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_REWRITE_TAC[IN_UNIONS; IN_ELIM_THM] THEN COND_CASES_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `x IN ((u:real^M->bool) DIFF s)` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [UNIONS_COMPONENTS] THEN REWRITE_TAC[IN_UNIONS] THEN DISCH_THEN(X_CHOOSE_THEN `c:real^M->bool` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN DISCH_THEN(MP_TAC o SPEC `c:real^M->bool`) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^M`; `c:real^M->bool`]) THEN ASM_REWRITE_TAC[UNIONS_GSPEC] THEN ASM SET_TAC[]]) in let lemma3 = prove (`!f:real^M->real^N s t u p. compact s /\ convex u /\ bounded u /\ affine t /\ aff_dim t <= aff_dim u /\ s SUBSET t /\ f continuous_on s /\ IMAGE f s SUBSET relative_frontier u /\ (!c. c IN components(t DIFF s) ==> ~(c INTER p = {})) ==> ?k g. FINITE k /\ k SUBSET p /\ k SUBSET t /\ DISJOINT k s /\ g continuous_on (t DIFF k) /\ IMAGE g (t DIFF k) SUBSET relative_frontier u /\ !x. x IN s ==> g x = f x`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`; `t:real^M->bool`; `u:real^N->bool`] EXTEND_MAP_AFFINE_TO_SPHERE_COFINITE_SIMPLE) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`k:real^M->bool`; `g:real^M->real^N`] THEN STRIP_TAC THEN SUBGOAL_THEN `!x. ?y. x IN k ==> ?c. c IN components (t DIFF s:real^M->bool) /\ x IN c /\ y IN c /\ y IN p` MP_TAC THENL [X_GEN_TAC `x:real^M` THEN REWRITE_TAC[RIGHT_EXISTS_IMP_THM] THEN DISCH_TAC THEN SUBGOAL_THEN `(x:real^M) IN (t DIFF s)` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [UNIONS_COMPONENTS] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN REWRITE_TAC[IN_UNIONS; RIGHT_EXISTS_AND_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN ASM SET_TAC[]; REWRITE_TAC[SKOLEM_THM] THEN DISCH_THEN(X_CHOOSE_THEN `h:real^M->real^M` (LABEL_TAC "*"))] THEN EXISTS_TAC `IMAGE (h:real^M->real^M) k` THEN MP_TAC(ISPECL [`g:real^M->real^N`; `s:real^M->bool`; `relative_frontier u:real^N->bool`; `k:real^M->bool`; `IMAGE (h:real^M->real^M) k`; `t:real^M->bool`] lemma2) THEN ASM_SIMP_TAC[AFFINE_AFFINE_HULL; FINITE_IMAGE] THEN ANTS_TAC THENL [CONJ_TAC THENL [X_GEN_TAC `c:real^M->bool` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN ONCE_REWRITE_TAC[INTER_COMM] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; EXISTS_IN_IMAGE; IN_INTER] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN REMOVE_THEN "*" (MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `c':real^M->bool` THEN STRIP_TAC THEN MP_TAC(ISPECL [`(t:real^M->bool) DIFF s`; `c:real^M->bool`; `c':real^M->bool`] COMPONENTS_EQ) THEN ASM_CASES_TAC `c':real^M->bool = c` THENL [ALL_TAC; ASM SET_TAC[]] THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; MATCH_MP_TAC CLOSED_IN_SUBSET_TRANS THEN EXISTS_TAC `(:real^M)` THEN ASM_REWRITE_TAC[SUBTOPOLOGY_UNIV; GSYM CLOSED_IN] THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED; SUBSET_UNIV]]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `h:real^M->real^N` THEN STRIP_TAC THEN ASM_SIMP_TAC[] THEN REWRITE_TAC[SET_RULE `DISJOINT s t <=> !x. x IN s ==> ~(x IN t)`] THEN ASM_REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN ASM_MESON_TAC[IN_COMPONENTS_SUBSET; SUBSET; IN_DIFF]]) in REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THENL [ASM_CASES_TAC `relative_frontier(u:real^N->bool) = {}` THENL [RULE_ASSUM_TAC(REWRITE_RULE[RELATIVE_FRONTIER_EQ_EMPTY]) THEN UNDISCH_TAC `bounded(u:real^N->bool)` THEN ASM_SIMP_TAC[AFFINE_BOUNDED_EQ_LOWDIM] THEN DISCH_TAC THEN SUBGOAL_THEN `aff_dim(t:real^M->bool) <= &0` MP_TAC THENL [ASM_INT_ARITH_TAC; ALL_TAC] THEN SIMP_TAC[AFF_DIM_GE; INT_ARITH `--(&1):int <= x ==> (x <= &0 <=> x = --(&1) \/ x = &0)`] THEN REWRITE_TAC[AFF_DIM_EQ_MINUS1; AFF_DIM_EQ_0] THEN DISCH_THEN(DISJ_CASES_THEN2 ASSUME_TAC (X_CHOOSE_TAC `a:real^M`)) THENL [EXISTS_TAC `{}:real^M->bool` THEN ASM_REWRITE_TAC[EMPTY_DIFF; FINITE_EMPTY; CONTINUOUS_ON_EMPTY; IMAGE_CLAUSES; NOT_IN_EMPTY] THEN SET_TAC[]; FIRST_X_ASSUM(MP_TAC o SPEC `{a:real^M}`) THEN ASM_REWRITE_TAC[DIFF_EMPTY; IN_COMPONENTS_SELF] THEN REWRITE_TAC[CONNECTED_SING; NOT_INSERT_EMPTY; BOUNDED_SING] THEN DISCH_TAC THEN EXISTS_TAC `{a:real^M}` THEN ASM_REWRITE_TAC[DIFF_EQ_EMPTY; CONTINUOUS_ON_EMPTY; NOT_IN_EMPTY; FINITE_SING; IMAGE_CLAUSES; EMPTY_SUBSET] THEN ASM SET_TAC[]]; EXISTS_TAC `{}:real^M->bool` THEN FIRST_X_ASSUM(X_CHOOSE_TAC `y:real^N` o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN ASM_SIMP_TAC[FINITE_EMPTY; DISJOINT_EMPTY; NOT_IN_EMPTY; DIFF_EMPTY] THEN EXISTS_TAC `(\x. y):real^M->real^N` THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN ASM SET_TAC[]]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o MATCH_MP COMPACT_IMP_BOUNDED) THEN DISCH_THEN(MP_TAC o MATCH_MP BOUNDED_SUBSET_CLOSED_INTERVAL_SYMMETRIC) THEN REWRITE_TAC[INSERT_SUBSET] THEN DISCH_THEN(X_CHOOSE_THEN `b:real^M` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`; `t:real^M->bool`; `u:real^N->bool`; `p UNION (UNIONS {c | c IN components (t DIFF s) /\ ~bounded c} DIFF interval[--(b + vec 1):real^M,b + vec 1])`] lemma3) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [X_GEN_TAC `c:real^M->bool` THEN STRIP_TAC THEN ASM_CASES_TAC `bounded(c:real^M->bool)` THENL [FIRST_X_ASSUM(MP_TAC o SPEC `c:real^M->bool`) THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `~(c SUBSET interval[--(b + vec 1):real^M,b + vec 1])` MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN ASM_MESON_TAC[BOUNDED_SUBSET; BOUNDED_INTERVAL]; ALL_TAC] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`k:real^M->bool`; `g:real^M->real^N`] THEN STRIP_TAC THEN EXISTS_TAC `k INTER interval[--(b + vec 1):real^M,b + vec 1]` THEN ASM_SIMP_TAC[FINITE_INTER; RIGHT_EXISTS_AND_THM] THEN REPEAT(CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN SUBGOAL_THEN `interval[--b,b] SUBSET interval[--(b + vec 1):real^M,b + vec 1]` ASSUME_TAC THENL [REWRITE_TAC[SUBSET_INTERVAL; VECTOR_ADD_COMPONENT; VECTOR_NEG_COMPONENT; VEC_COMPONENT] THEN REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `?d:real. (&1 / &2 <= d /\ d <= &1) /\ DISJOINT k (frontier(interval[--(b + lambda i. d):real^M, (b + lambda i. d)]))` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC lemma1 THEN ASM_SIMP_TAC[INFINITE; FINITE_REAL_INTERVAL; REAL_NOT_LE] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN MATCH_MP_TAC REAL_WLOG_LT THEN REWRITE_TAC[] THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`x:real`; `y:real`] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[frontier] THEN MATCH_MP_TAC(SET_RULE `c SUBSET i' ==> DISJOINT (c DIFF i) (c' DIFF i')`) THEN REWRITE_TAC[INTERIOR_INTERVAL; CLOSURE_INTERVAL] THEN SIMP_TAC[SUBSET_INTERVAL; VECTOR_NEG_COMPONENT; VECTOR_ADD_COMPONENT; LAMBDA_BETA] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN ABBREV_TAC `c:real^M = b + lambda i. d` THEN SUBGOAL_THEN `interval[--b:real^M,b] SUBSET interval(--c,c) /\ interval[--b:real^M,b] SUBSET interval[--c,c] /\ interval[--c,c] SUBSET interval[--(b + vec 1):real^M,b + vec 1]` STRIP_ASSUME_TAC THENL [REWRITE_TAC[SUBSET_INTERVAL] THEN EXPAND_TAC "c" THEN REPEAT CONJ_TAC THEN SIMP_TAC[VECTOR_NEG_COMPONENT; VECTOR_ADD_COMPONENT; LAMBDA_BETA] THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[VEC_COMPONENT] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN EXISTS_TAC `(g:real^M->real^N) o closest_point (interval[--c,c] INTER t)` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_CLOSEST_POINT THEN ASM_SIMP_TAC[CONVEX_INTER; CLOSED_INTER; CLOSED_INTERVAL; CLOSED_AFFINE; AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL; CONVEX_INTERVAL] THEN ASM SET_TAC[]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET))]; REWRITE_TAC[IMAGE_o] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS)) THEN MATCH_MP_TAC IMAGE_SUBSET; X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN REWRITE_TAC[o_THM] THEN TRANS_TAC EQ_TRANS `(g:real^M->real^N) x` THEN CONJ_TAC THENL [AP_TERM_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CLOSEST_POINT_SELF THEN ASM_SIMP_TAC[IN_INTER; HULL_INC] THEN ASM SET_TAC[]] THEN (REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_DIFF] THEN X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `closest_point s x IN s /\ s SUBSET u ==> closest_point s x IN u`) THEN CONJ_TAC THENL [MATCH_MP_TAC CLOSEST_POINT_IN_SET; ASM SET_TAC[]] THEN ASM_SIMP_TAC[CLOSED_INTER; CLOSED_INTERVAL; CLOSED_AFFINE] THEN ASM SET_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `x IN interval[--c:real^M,c]` THEN ASM_SIMP_TAC[CLOSEST_POINT_SELF; IN_INTER] THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `closest_point s x IN relative_frontier s /\ DISJOINT k (relative_frontier s) ==> ~(closest_point s x IN k)`) THEN CONJ_TAC THENL [MATCH_MP_TAC CLOSEST_POINT_IN_RELATIVE_FRONTIER THEN ASM_SIMP_TAC[CLOSED_INTER; CLOSED_AFFINE; CLOSED_INTERVAL] THEN CONJ_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[IN_DIFF]] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[SUBSET; RELATIVE_INTERIOR_SUBSET; IN_INTER]] THEN ONCE_REWRITE_TAC[INTER_COMM] THEN W(MP_TAC o PART_MATCH (lhs o rand) AFFINE_HULL_CONVEX_INTER_NONEMPTY_INTERIOR o rand o snd) THEN ASM_SIMP_TAC[HULL_HULL; AFFINE_AFFINE_HULL; AFFINE_IMP_CONVEX] THEN ASM_SIMP_TAC[HULL_P] THEN ANTS_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[INTERIOR_INTERVAL] THEN ASM SET_TAC[]; W(MP_TAC o PART_MATCH (lhs o rand) RELATIVE_FRONTIER_CONVEX_INTER_AFFINE o rand o snd) THEN ANTS_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[CONVEX_INTERVAL; AFFINE_AFFINE_HULL; INTERIOR_INTERVAL] THEN ASM SET_TAC[]]));; let EXTEND_MAP_AFFINE_TO_SPHERE_COFINITE = prove (`!f:real^M->real^N s t a r p. compact s /\ affine t /\ aff_dim t <= &(dimindex(:N)) /\ s SUBSET t /\ &0 <= r /\ f continuous_on s /\ IMAGE f s SUBSET sphere(a,r) /\ (!c. c IN components(t DIFF s) /\ bounded c ==> ~(c INTER p = {})) ==> ?k g. FINITE k /\ k SUBSET p /\ k SUBSET t /\ DISJOINT k s /\ g continuous_on (t DIFF k) /\ IMAGE g (t DIFF k) SUBSET sphere(a,r) /\ !x. x IN s ==> g x = f x`, REPEAT GEN_TAC THEN ASM_CASES_TAC `r = &0` THENL [ASM_SIMP_TAC[SPHERE_SING] THEN STRIP_TAC THEN EXISTS_TAC `{}:real^M->bool` THEN EXISTS_TAC `(\x. a):real^M->real^N` THEN REWRITE_TAC[CONTINUOUS_ON_CONST; FINITE_EMPTY] THEN ASM SET_TAC[]; MP_TAC(ISPECL [`a:real^N`; `r:real`] RELATIVE_FRONTIER_CBALL) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN STRIP_TAC THEN MATCH_MP_TAC EXTEND_MAP_AFFINE_TO_SPHERE_COFINITE_GEN THEN ASM_REWRITE_TAC[CONVEX_CBALL; BOUNDED_CBALL; AFF_DIM_CBALL] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC]);; let EXTEND_MAP_UNIV_TO_SPHERE_COFINITE = prove (`!f:real^M->real^N s a r p. dimindex(:M) <= dimindex(:N) /\ &0 <= r /\ compact s /\ f continuous_on s /\ IMAGE f s SUBSET sphere(a,r) /\ (!c. c IN components((:real^M) DIFF s) /\ bounded c ==> ~(c INTER p = {})) ==> ?k g. FINITE k /\ k SUBSET p /\ DISJOINT k s /\ g continuous_on ((:real^M) DIFF k) /\ IMAGE g ((:real^M) DIFF k) SUBSET sphere(a,r) /\ !x. x IN s ==> g x = f x`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`; `(:real^M)`; `a:real^N`; `r:real`; `p:real^M->bool`] EXTEND_MAP_AFFINE_TO_SPHERE_COFINITE) THEN ASM_REWRITE_TAC[AFFINE_UNIV; SUBSET_UNIV; AFF_DIM_UNIV; INT_OF_NUM_LE]);; let EXTEND_MAP_UNIV_TO_SPHERE_NO_BOUNDED_COMPONENT = prove (`!f:real^M->real^N s a r. dimindex(:M) <= dimindex(:N) /\ &0 <= r /\ compact s /\ f continuous_on s /\ IMAGE f s SUBSET sphere(a,r) /\ (!c. c IN components((:real^M) DIFF s) ==> ~bounded c) ==> ?g. g continuous_on (:real^M) /\ IMAGE g (:real^M) SUBSET sphere(a,r) /\ !x. x IN s ==> g x = f x`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`; `a:real^N`; `r:real`; `{}:real^M->bool`] EXTEND_MAP_UNIV_TO_SPHERE_COFINITE) THEN ASM_SIMP_TAC[IMP_CONJ; SUBSET_EMPTY; RIGHT_EXISTS_AND_THM] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN REWRITE_TAC[UNWIND_THM2; FINITE_EMPTY; DISJOINT_EMPTY; DIFF_EMPTY] THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[]);; let EXTEND_MAP_SPHERE_TO_SPHERE_GEN = prove (`!f:real^M->real^N c s t. closed c /\ c SUBSET relative_frontier s /\ convex s /\ bounded s /\ convex t /\ bounded t /\ aff_dim s <= aff_dim t /\ f continuous_on c /\ IMAGE f c SUBSET relative_frontier t ==> ?g. g continuous_on (relative_frontier s) /\ IMAGE g (relative_frontier s) SUBSET relative_frontier t /\ !x. x IN c ==> g x = f x`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?p:real^M->bool. polytope p /\ aff_dim p = aff_dim(s:real^M->bool)` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC CHOOSE_POLYTOPE THEN ASM_REWRITE_TAC[AFF_DIM_GE; AFF_DIM_LE_UNIV]; ALL_TAC] THEN MP_TAC(ISPECL [`s:real^M->bool`; `p:real^M->bool`] HOMEOMORPHIC_RELATIVE_FRONTIERS_CONVEX_BOUNDED_SETS) THEN ASM_SIMP_TAC[POLYTOPE_IMP_CONVEX; POLYTOPE_IMP_BOUNDED; homeomorphic] THEN REWRITE_TAC[HOMEOMORPHISM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`h:real^M->real^M`; `k:real^M->real^M`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`(f:real^M->real^N) o (k:real^M->real^M)`; `{f:real^M->bool | f face_of p /\ ~(f = p)}`; `IMAGE (h:real^M->real^M) c`; `t:real^N->bool`] EXTEND_MAP_CELL_COMPLEX_TO_SPHERE) THEN ASM_SIMP_TAC[FORALL_IN_IMAGE] THEN ASM_SIMP_TAC[GSYM RELATIVE_FRONTIER_OF_POLYHEDRON_ALT; POLYTOPE_IMP_POLYHEDRON] THEN REWRITE_TAC[IN_ELIM_THM; GSYM IMAGE_o; o_THM] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `{f:real^M->bool | f face_of p}` THEN ASM_SIMP_TAC[FINITE_POLYTOPE_FACES] THEN SET_TAC[]; ASM_MESON_TAC[FACE_OF_POLYTOPE_POLYTOPE; FACE_OF_AFF_DIM_LT; POLYTOPE_IMP_CONVEX; INT_LTE_TRANS]; ASM_MESON_TAC[FACE_OF_INTER; FACE_OF_SUBSET; INTER_SUBSET; FACE_OF_INTER; FACE_OF_IMP_SUBSET]; ASM SET_TAC[]; MATCH_MP_TAC COMPACT_IMP_CLOSED THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ALL_TAC] THEN ASM_REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] BOUNDED_SUBSET)) THEN ASM_SIMP_TAC[BOUNDED_RELATIVE_FRONTIER]; 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[]]; DISCH_THEN(X_CHOOSE_THEN `g:real^M->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(g:real^M->real^N) o (h:real^M->real^M)` THEN REWRITE_TAC[IMAGE_o; o_THM] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_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 EXTEND_MAP_SPHERE_TO_SPHERE = prove (`!f:real^M->real^N c a r b s. dimindex(:M) <= dimindex(:N) /\ closed c /\ c SUBSET sphere(a,r) /\ f continuous_on c /\ IMAGE f c SUBSET sphere(b,s) /\ (&0 <= r /\ c = {} ==> &0 <= s) ==> ?g. g continuous_on sphere(a,r) /\ IMAGE g (sphere(a,r)) SUBSET sphere(b,s) /\ !x. x IN c ==> g x = f x`, REPEAT GEN_TAC THEN ASM_CASES_TAC `r < &0` THEN ASM_SIMP_TAC[SPHERE_EMPTY; NOT_IN_EMPTY; CONTINUOUS_ON_EMPTY; IMAGE_CLAUSES; EMPTY_SUBSET] THENL [MESON_TAC[]; ASM_REWRITE_TAC[GSYM REAL_NOT_LT]] THEN ASM_CASES_TAC `sphere(b:real^N,s) = {}` THENL [FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SPHERE_EQ_EMPTY]) THEN ASM SET_TAC[]; FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [SPHERE_EQ_EMPTY])] THEN REWRITE_TAC[REAL_NOT_LE] THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LT]) THEN ASM_CASES_TAC `r = &0` THEN ASM_SIMP_TAC[SPHERE_SING; CONTINUOUS_ON_SING; REAL_LE_REFL] THENL [ASM_CASES_TAC `c:real^M->bool = {}` THENL [DISCH_THEN(K ALL_TAC) THEN MATCH_MP_TAC(MESON[] `(?c. P(\x. c)) ==> ?f. P f`) THEN ASM SET_TAC[]; DISCH_TAC THEN EXISTS_TAC `f:real^M->real^N` THEN ASM SET_TAC[]]; ALL_TAC] THEN ASM_CASES_TAC `s = &0` THENL [ASM_SIMP_TAC[SPHERE_SING] THEN STRIP_TAC THEN EXISTS_TAC `(\x. b):real^M->real^N` THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN ASM SET_TAC[]; ALL_TAC] THEN STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `c:real^M->bool`; `cball(a:real^M,r)`; `cball(b:real^N,s)`] EXTEND_MAP_SPHERE_TO_SPHERE_GEN) THEN ASM_REWRITE_TAC[CONVEX_CBALL; BOUNDED_CBALL; AFF_DIM_CBALL; RELATIVE_FRONTIER_CBALL] THEN DISCH_THEN MATCH_MP_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[INT_OF_NUM_LE]) THEN ASM_REAL_ARITH_TAC);; let EXTEND_MAP_SPHERE_TO_SPHERE_COFINITE_GEN = prove (`!f:real^M->real^N s t u p. convex t /\ bounded t /\ convex u /\ bounded u /\ aff_dim t <= aff_dim u + &1 /\ closed s /\ s SUBSET relative_frontier t /\ f continuous_on s /\ IMAGE f s SUBSET relative_frontier u /\ (!c. c IN components(relative_frontier t DIFF s) ==> ~(c INTER p = {})) ==> ?k g. FINITE k /\ k SUBSET p /\ k SUBSET relative_frontier t /\ DISJOINT k s /\ g continuous_on (relative_frontier t DIFF k) /\ IMAGE g (relative_frontier t DIFF k) SUBSET relative_frontier u /\ !x. x IN s ==> g x = f x`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s = (relative_frontier t:real^M->bool)` THENL [ASM_REWRITE_TAC[] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`{}:real^M->bool`; `f:real^M->real^N`] THEN ASM_REWRITE_TAC[FINITE_EMPTY; DIFF_EMPTY] THEN SET_TAC[]; POP_ASSUM MP_TAC] THEN ASM_CASES_TAC `relative_frontier t:real^M->bool = {}` THENL [ASM SET_TAC[]; REPEAT STRIP_TAC] THEN SUBGOAL_THEN `?c q:real^M. c IN components (relative_frontier t DIFF s) /\ q IN c /\ q IN relative_frontier t /\ ~(q IN s) /\ q IN p` STRIP_ASSUME_TAC THENL [MP_TAC(ISPEC `(relative_frontier t:real^M->bool) DIFF s` UNIONS_COMPONENTS) THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `s = u ==> ~(s = {}) ==> ~(u = {})`)) THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[EMPTY_UNIONS]] THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^M->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o SPEC `c:real^M->bool`) THEN ASM_REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[GSYM IN_DIFF] THEN ASM_MESON_TAC[SUBSET; IN_COMPONENTS_SUBSET]; ALL_TAC] THEN SUBGOAL_THEN `?af. affine af /\ aff_dim(t:real^M->bool) = aff_dim(af:real^M->bool) + &1` STRIP_ASSUME_TAC THENL [MP_TAC(ISPECL [`(:real^M)`; `aff_dim(t:real^M->bool) - &1`] CHOOSE_AFFINE_SUBSET) THEN REWRITE_TAC[SUBSET_UNIV; AFFINE_UNIV] THEN ANTS_TAC THENL [MATCH_MP_TAC(INT_ARITH `&0:int <= t /\ t <= n ==> --a <= t - a /\ t - &1 <= n`) THEN REWRITE_TAC[AFF_DIM_LE_UNIV; AFF_DIM_UNIV; AFF_DIM_POS_LE] THEN ASM_MESON_TAC[RELATIVE_FRONTIER_EMPTY; NOT_IN_EMPTY]; MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[] THEN INT_ARITH_TAC]; ALL_TAC] THEN MP_TAC(ISPECL [`t:real^M->bool`; `af:real^M->bool`; `q:real^M`] HOMEOMORPHIC_PUNCTURED_SPHERE_AFFINE_GEN) THEN ASM_REWRITE_TAC[homeomorphic; homeomorphism; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`h:real^M->real^M`; `k:real^M->real^M`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`(f:real^M->real^N) o (k:real^M->real^M)`; `IMAGE (h:real^M->real^M) s`; `(af:real^M->bool)`; `u:real^N->bool`; `IMAGE (h:real^M->real^M) (p INTER relative_frontier t DELETE q)`] EXTEND_MAP_AFFINE_TO_SPHERE_COFINITE_GEN) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; ASM_MESON_TAC[COMPACT_EQ_BOUNDED_CLOSED; BOUNDED_SUBSET; COMPACT_RELATIVE_FRONTIER_BOUNDED]]; ASM_INT_ARITH_TAC; ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; REWRITE_TAC[IMAGE_o] THEN ASM SET_TAC[]; X_GEN_TAC `l:real^M->bool` THEN STRIP_TAC THEN SUBGOAL_THEN `~(l:real^M->bool = {})` ASSUME_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_NONEMPTY]; ALL_TAC] THEN SUBGOAL_THEN `?x:real^M. x IN l` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `l SUBSET af DIFF IMAGE (h:real^M->real^M) s` ASSUME_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_SUBSET]; ALL_TAC] THEN SUBGOAL_THEN `connected(l:real^M->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]; ALL_TAC] THEN SUBGOAL_THEN `?r. r IN components (relative_frontier t DIFF s) /\ IMAGE (k:real^M->real^M) l SUBSET r` STRIP_ASSUME_TAC THENL [REWRITE_TAC[IN_COMPONENTS; LEFT_AND_EXISTS_THM] THEN EXISTS_TAC `connected_component (relative_frontier t DIFF s) ((k:real^M->real^M) x)` THEN EXISTS_TAC `(k:real^M->real^M) x` THEN REWRITE_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL THEN ASM_SIMP_TAC[FUN_IN_IMAGE] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPEC `r:real^M->bool`) THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC LAND_CONV [GSYM MEMBER_NOT_EMPTY] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; IN_INTER] THEN X_GEN_TAC `z:real^M` THEN STRIP_TAC THEN SUBGOAL_THEN `r SUBSET ((relative_frontier t:real^M->bool) DIFF s)` ASSUME_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_SUBSET]; ALL_TAC] THEN SUBGOAL_THEN `connected(r:real^M->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]; ALL_TAC] THEN ASM_CASES_TAC `(q:real^M) IN r` THENL [ALL_TAC; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `(h:real^M->real^M) z` THEN REWRITE_TAC[IN_INTER] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC(SET_RULE `!s. x IN s /\ s SUBSET t ==> x IN t`) THEN EXISTS_TAC `IMAGE (h:real^M->real^M) r` THEN ASM_SIMP_TAC[FUN_IN_IMAGE] THEN MATCH_MP_TAC COMPONENTS_MAXIMAL THEN EXISTS_TAC `af DIFF IMAGE (h:real^M->real^M) s` THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_DIFF; IN_ELIM_THM] THEN X_GEN_TAC `y:real^M` THEN DISCH_TAC THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[SET_RULE `~(h y IN IMAGE h s) <=> !y'. y' IN s ==> ~(h y = h y')`] THEN X_GEN_TAC `y':real^M` THEN DISCH_TAC THEN DISCH_THEN(MP_TAC o AP_TERM `k:real^M->real^M`) THEN MATCH_MP_TAC(MESON[] `k(h y) = y /\ k(h y') = y' /\ ~(y = y') ==> k(h y) = k(h y') ==> F`) THEN ASM SET_TAC[]; ASM SET_TAC[]]] THEN SUBGOAL_THEN `?n. open_in (subtopology euclidean (relative_frontier t)) n /\ (q:real^M) IN n /\ n INTER IMAGE (k:real^M->real^M) l = {}` STRIP_ASSUME_TAC THENL [EXISTS_TAC `relative_frontier t DIFF IMAGE (k:real^M->real^M) (closure l)` THEN SUBGOAL_THEN `closure l SUBSET (af:real^M->bool)` ASSUME_TAC THENL [MATCH_MP_TAC CLOSURE_MINIMAL THEN ASM_SIMP_TAC[CLOSED_AFFINE] THEN ASM SET_TAC[]; ALL_TAC] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_DIFF THEN REWRITE_TAC[OPEN_IN_REFL] THEN MATCH_MP_TAC CLOSED_SUBSET THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC COMPACT_IMP_CLOSED THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[COMPACT_CLOSURE] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; ASM SET_TAC[]; MP_TAC(ISPEC `l:real^M->bool` CLOSURE_SUBSET) THEN SET_TAC[]]; ALL_TAC] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN SUBGOAL_THEN `?w. connected w /\ w SUBSET r DELETE q /\ (k:real^M->real^M) x IN w /\ ~((n DELETE q) INTER w = {})` STRIP_ASSUME_TAC THENL [ALL_TAC; MATCH_MP_TAC(TAUT `F ==> p`) THEN SUBGOAL_THEN `IMAGE (h:real^M->real^M) w SUBSET l` MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC COMPONENTS_MAXIMAL THEN EXISTS_TAC `af DIFF IMAGE (h:real^M->real^M) s` THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_DIFF; IN_ELIM_THM] THEN X_GEN_TAC `y:real^M` THEN DISCH_TAC THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[SET_RULE `~(h y IN IMAGE h s) <=> !y'. y' IN s ==> ~(h y = h y')`] THEN X_GEN_TAC `y':real^M` THEN DISCH_TAC THEN DISCH_THEN(MP_TAC o AP_TERM `k:real^M->real^M`) THEN MATCH_MP_TAC(MESON[] `k(h y) = y /\ k(h y') = y' /\ ~(y = y') ==> k(h y) = k(h y') ==> F`) THEN ASM SET_TAC[]; ASM SET_TAC[]]] THEN SUBGOAL_THEN `path_connected(r:real^M->bool)` MP_TAC THENL [W(MP_TAC o PART_MATCH (lhand o rand) PATH_CONNECTED_EQ_CONNECTED_LPC o snd) THEN ANTS_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN MATCH_MP_TAC LOCALLY_OPEN_SUBSET THEN EXISTS_TAC `(relative_frontier t:real^M->bool)` THEN ASM_SIMP_TAC[LOCALLY_PATH_CONNECTED_SPHERE_GEN] THEN MATCH_MP_TAC OPEN_IN_TRANS THEN EXISTS_TAC `(relative_frontier t:real^M->bool) DIFF s` THEN CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_COMPONENTS_LOCALLY_CONNECTED THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC LOCALLY_OPEN_SUBSET THEN EXISTS_TAC `(relative_frontier t:real^M->bool)` THEN ASM_SIMP_TAC[LOCALLY_CONNECTED_SPHERE_GEN]; ALL_TAC] THEN MATCH_MP_TAC OPEN_IN_DIFF THEN REWRITE_TAC[OPEN_IN_REFL] THEN MATCH_MP_TAC CLOSED_SUBSET THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN REWRITE_TAC[PATH_CONNECTED_ARCWISE] THEN DISCH_THEN(MP_TAC o SPECL [`(k:real^M->real^M) x`; `q:real^M`]) THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^1->real^M` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC o GEN_REWRITE_RULE I [arc]) THEN DISCH_TAC THEN SUBGOAL_THEN `open_in (subtopology euclidean (interval[vec 0,vec 1])) {x | x IN interval[vec 0,vec 1] /\ (g:real^1->real^M) x IN n}` MP_TAC THENL [MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE_GEN THEN EXISTS_TAC `(relative_frontier t:real^M->bool)` THEN ASM_REWRITE_TAC[GSYM path; GSYM path_image] THEN ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[OPEN_IN_CONTAINS_CBALL] THEN REWRITE_TAC[IN_ELIM_THM; SUBSET_RESTRICT] THEN DISCH_THEN(MP_TAC o SPEC `vec 1:real^1`) THEN REWRITE_TAC[ENDS_IN_UNIT_INTERVAL] THEN ANTS_TAC THENL [ASM_MESON_TAC[pathfinish]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `r:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN ABBREV_TAC `t' = lift(&1 - min (&1 / &2) r)` THEN SUBGOAL_THEN `t' IN interval[vec 0:real^1,vec 1]` ASSUME_TAC THENL [EXPAND_TAC "t'" THEN SIMP_TAC[IN_INTERVAL_1; LIFT_DROP; DROP_VEC] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN GEN_REWRITE_TAC LAND_CONV [SUBSET] THEN DISCH_THEN(MP_TAC o SPEC `t':real^1`) THEN ASM_REWRITE_TAC[IN_INTER; IN_ELIM_THM; IN_CBALL; DIST_REAL; DROP_VEC; GSYM drop] THEN ANTS_TAC THENL [EXPAND_TAC "t'" THEN REWRITE_TAC[LIFT_DROP] THEN ASM_REAL_ARITH_TAC; DISCH_TAC] THEN EXISTS_TAC `IMAGE (g:real^1->real^M) (interval[vec 0,t'])` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN REWRITE_TAC[CONNECTED_INTERVAL] THEN MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `interval[vec 0:real^1,vec 1]` THEN ASM_REWRITE_TAC[GSYM path; SUBSET_INTERVAL_1] THEN ASM_REWRITE_TAC[REAL_LE_REFL; GSYM IN_INTERVAL_1]; REWRITE_TAC[SET_RULE `s SUBSET t DELETE q <=> s SUBSET t /\ !x. x IN s ==> ~(x = q)`] THEN CONJ_TAC THENL [TRANS_TAC SUBSET_TRANS `IMAGE (g:real^1->real^M) (interval[vec 0,vec 1])` THEN CONJ_TAC THENL [MATCH_MP_TAC IMAGE_SUBSET THEN ASM_REWRITE_TAC[REAL_LE_REFL; GSYM IN_INTERVAL_1; SUBSET_INTERVAL_1]; ASM_REWRITE_TAC[GSYM path_image]]; REWRITE_TAC[FORALL_IN_IMAGE] THEN X_GEN_TAC `t'':real^1` THEN DISCH_TAC THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [SYM th]) THEN REWRITE_TAC[pathfinish] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`t'':real^1`; `vec 1:real^1`]) THEN ASM_REWRITE_TAC[GSYM DROP_EQ] THEN UNDISCH_TAC `t'' IN interval[vec 0:real^1,t']` THEN EXPAND_TAC "t'" THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; LIFT_DROP] THEN ASM_REAL_ARITH_TAC]; REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `vec 0:real^1` THEN CONJ_TAC THENL [ASM_MESON_TAC[pathstart]; ALL_TAC] THEN EXPAND_TAC "t'" THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; LIFT_DROP] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN ONCE_REWRITE_TAC[INTER_COMM] THEN REWRITE_TAC[EXISTS_IN_IMAGE; IN_INTER] THEN EXISTS_TAC `t':real^1` THEN CONJ_TAC THENL [EXPAND_TAC "t'" THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; LIFT_DROP] THEN ASM_REAL_ARITH_TAC; ASM_REWRITE_TAC[IN_DELETE] THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [SYM th]) THEN REWRITE_TAC[pathfinish] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`t':real^1`; `vec 1:real^1`]) THEN ASM_REWRITE_TAC[GSYM DROP_EQ] THEN EXPAND_TAC "t'" THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; LIFT_DROP] THEN ASM_REAL_ARITH_TAC]]]; ALL_TAC] THEN ASM_SIMP_TAC[DOT_BASIS; LE_REFL; DIMINDEX_GE_1; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`tk:real^M->bool`; `g:real^M->real^N`] THEN REWRITE_TAC[o_THM] THEN STRIP_TAC THEN EXISTS_TAC `q INSERT IMAGE (k:real^M->real^M) tk` THEN EXISTS_TAC `(g:real^M->real^N) o (h:real^M->real^M)` THEN ASM_SIMP_TAC[FINITE_INSERT; FINITE_IMAGE; o_THM] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `a IN t /\ s SUBSET t DELETE a ==> a INSERT s SUBSET t`) THEN ASM_REWRITE_TAC[] THEN TRANS_TAC SUBSET_TRANS `p INTER (relative_frontier t:real^M->bool) DELETE q` THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `t SUBSET IMAGE h s ==> IMAGE k (IMAGE h s) SUBSET s ==> IMAGE k t SUBSET s`)) THEN REWRITE_TAC[GSYM IMAGE_o] THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> f x = x) ==> IMAGE f s SUBSET s`) THEN REWRITE_TAC[o_THM] THEN ASM SET_TAC[]; ASM SET_TAC[]; ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; REWRITE_TAC[IMAGE_o] THEN ASM SET_TAC[]; ASM SET_TAC[]]);; let EXTEND_MAP_SPHERE_TO_SPHERE_COFINITE = prove (`!f:real^M->real^N s a d b e p. dimindex(:M) <= dimindex(:N) + 1 /\ (&0 < d /\ s = {} ==> &0 <= e) /\ closed s /\ s SUBSET sphere(a,d) /\ f continuous_on s /\ IMAGE f s SUBSET sphere(b,e) /\ (!c. c IN components(sphere(a,d) DIFF s) ==> ~(c INTER p = {})) ==> ?k g. FINITE k /\ k SUBSET p /\ k SUBSET sphere(a,d) /\ DISJOINT k s /\ g continuous_on (sphere(a,d) DIFF k) /\ IMAGE g (sphere(a,d) DIFF k) SUBSET sphere(b,e) /\ !x. x IN s ==> g x = f x`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s = sphere(a:real^M,d)` THENL [ASM_REWRITE_TAC[] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`{}:real^M->bool`; `f:real^M->real^N`] THEN ASM_REWRITE_TAC[FINITE_EMPTY; DIFF_EMPTY] THEN SET_TAC[]; POP_ASSUM MP_TAC] THEN ASM_CASES_TAC `d < &0` THENL [ASM_SIMP_TAC[SPHERE_EMPTY] THEN SET_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `d = &0` THENL [ASM_SIMP_TAC[SPHERE_SING] THEN ASM_CASES_TAC `s:real^M->bool = {}` THENL [ASM_REWRITE_TAC[]; ASM SET_TAC[]] THEN REPEAT STRIP_TAC THEN EXISTS_TAC `{a:real^M}` THEN REWRITE_TAC[FINITE_SING; CONTINUOUS_ON_EMPTY; DIFF_EQ_EMPTY] THEN FIRST_X_ASSUM(MP_TAC o SPEC `{a:real^M}`) THEN REWRITE_TAC[DIFF_EMPTY; IN_COMPONENTS_SELF; CONNECTED_SING] THEN REWRITE_TAC[IMAGE_CLAUSES] THEN SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `&0 < d` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_CASES_TAC `e = &0` THENL [ASM_SIMP_TAC[SPHERE_SING] THEN REPEAT STRIP_TAC THEN EXISTS_TAC `{}:real^M->bool` THEN EXISTS_TAC `(\x. b):real^M->real^N` THEN REWRITE_TAC[CONTINUOUS_ON_CONST; FINITE_EMPTY] THEN ASM SET_TAC[]; REPEAT STRIP_TAC] THEN SUBGOAL_THEN `&0 <= e` ASSUME_TAC THENL [ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_SIMP_TAC[] THEN MP_TAC(SYM(ISPECL [`b:real^N`; `e:real`] SPHERE_EQ_EMPTY)) THEN SIMP_TAC[GSYM REAL_NOT_LT] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `&0 < e` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`; `cball(a:real^M,d)`; `cball(b:real^N,e)`; `p:real^M->bool`] EXTEND_MAP_SPHERE_TO_SPHERE_COFINITE_GEN) THEN ASM_REWRITE_TAC[CONVEX_CBALL; BOUNDED_CBALL] THEN REWRITE_TAC[AFF_DIM_CBALL] THEN MP_TAC(ISPECL [`a:real^M`; `d:real`] RELATIVE_FRONTIER_CBALL) THEN MP_TAC(ISPECL [`b:real^N`; `e:real`] RELATIVE_FRONTIER_CBALL) THEN ASM_REWRITE_TAC[] THEN REPEAT(DISCH_THEN SUBST1_TAC) THEN ASM_REWRITE_TAC[INT_OF_NUM_ADD; INT_OF_NUM_LE]);; (* ------------------------------------------------------------------------- *) (* Borsuk-style characterization of separation. *) (* ------------------------------------------------------------------------- *) let CONTINUOUS_ON_BORSUK_MAP = prove (`!s a:real^N. ~(a IN s) ==> (\x. inv(norm (x - a)) % (x - a)) continuous_on s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN SIMP_TAC[o_DEF] THEN CONJ_TAC THENL [MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV); ALL_TAC] THEN SIMP_TAC[CONTINUOUS_ON_LIFT_NORM_COMPOSE; CONTINUOUS_ON_SUB; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST] THEN REWRITE_TAC[NORM_EQ_0; VECTOR_SUB_EQ] THEN ASM_MESON_TAC[]);; let BORSUK_MAP_INTO_SPHERE = prove (`!s a:real^N. IMAGE (\x. inv(norm (x - a)) % (x - a)) s SUBSET sphere(vec 0,&1) <=> ~(a IN s)`, REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_SPHERE_0] THEN REWRITE_TAC[NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN REWRITE_TAC[REAL_FIELD `inv x * x = &1 <=> ~(x = &0)`] THEN REWRITE_TAC[NORM_EQ_0; VECTOR_SUB_EQ] THEN MESON_TAC[]);; let BORSUK_MAPS_HOMOTOPIC_IN_PATH_COMPONENT = prove (`!s a b. path_component ((:real^N) DIFF s) a b ==> homotopic_with (\x. T) (s,sphere(vec 0,&1)) (\x. inv(norm(x - a)) % (x - a)) (\x. inv(norm(x - b)) % (x - b))`, REPEAT GEN_TAC THEN REWRITE_TAC[path_component; LEFT_IMP_EXISTS_THM] THEN REWRITE_TAC[path; path_image; pathstart; pathfinish; SUBSET; FORALL_IN_IMAGE; IN_UNIV; IN_DIFF] THEN X_GEN_TAC `g:real^1->real^N` THEN STRIP_TAC THEN SIMP_TAC[HOMOTOPIC_WITH] THEN EXISTS_TAC `\z. inv(norm(sndcart z - g(fstcart z))) % (sndcart z - (g:real^1->real^N)(fstcart z))` THEN ASM_SIMP_TAC[FSTCART_PASTECART; SNDCART_PASTECART; IN_SPHERE_0; SUBSET; FORALL_IN_IMAGE; FORALL_IN_PCROSS] THEN 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[FORALL_IN_PCROSS; FSTCART_PASTECART; SNDCART_PASTECART; NORM_EQ_0; VECTOR_SUB_EQ] THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_LIFT_NORM_COMPOSE; ASM_MESON_TAC[]]; ALL_TAC] THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_SNDCART] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_FSTCART] THEN REWRITE_TAC[IMAGE_FSTCART_PCROSS] THEN ASM_MESON_TAC[CONTINUOUS_ON_EMPTY]; REPEAT STRIP_TAC THEN REWRITE_TAC[NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN MATCH_MP_TAC REAL_MUL_LINV THEN ASM_REWRITE_TAC[NORM_EQ_0; VECTOR_SUB_EQ] THEN ASM_MESON_TAC[]]);; let NON_EXTENSIBLE_BORSUK_MAP = prove (`!s c a:real^N. compact s /\ c IN components((:real^N) DIFF s) /\ bounded c /\ a IN c ==> ~(?g. g continuous_on (s UNION c) /\ IMAGE g (s UNION c) SUBSET sphere (vec 0,&1) /\ (!x. x IN s ==> g x = inv(norm(x - a)) % (x - a)))`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN REWRITE_TAC[SUBSET] THEN DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN ASM_REWRITE_TAC[IN_DIFF] THEN STRIP_TAC THEN SUBGOAL_THEN `c = connected_component ((:real^N) DIFF s) a` SUBST_ALL_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS; CONNECTED_COMPONENT_EQ]; ALL_TAC] THEN MP_TAC(ISPECL [`s UNION connected_component ((:real^N) DIFF s) a`; `a:real^N`] BOUNDED_SUBSET_BALL) THEN ASM_SIMP_TAC[BOUNDED_UNION; COMPACT_IMP_BOUNDED] THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o SPEC `a:real^N` o MATCH_MP NO_RETRACTION_CBALL) THEN REWRITE_TAC[retract_of; retraction] THEN EXISTS_TAC `\x. if x IN connected_component ((:real^N) DIFF s) a then a + r % g(x) else a + r % inv(norm(x - a)) % (x - a)` THEN REWRITE_TAC[SPHERE_SUBSET_CBALL] THEN REPEAT CONJ_TAC THENL [SUBGOAL_THEN `cball(a:real^N,r) = (s UNION connected_component ((:real^N) DIFF s) a) UNION (cball(a,r) DIFF connected_component ((:real^N) DIFF s) a)` SUBST1_TAC THENL [MP_TAC(ISPECL [`a:real^N`; `r:real`] BALL_SUBSET_CBALL) THEN ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC CONTINUOUS_ON_CASES THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CLOSED_UNION_COMPLEMENT_COMPONENT THEN ASM_SIMP_TAC[IN_COMPONENTS; COMPACT_IMP_CLOSED; IN_UNIV; IN_DIFF] THEN ASM_MESON_TAC[]; MATCH_MP_TAC CLOSED_DIFF THEN ASM_SIMP_TAC[CLOSED_CBALL; OPEN_CONNECTED_COMPONENT; GSYM closed; COMPACT_IMP_CLOSED]; MATCH_MP_TAC CONTINUOUS_ON_ADD THEN SIMP_TAC[CONTINUOUS_ON_CONST] THEN MATCH_MP_TAC CONTINUOUS_ON_CMUL THEN ASM_SIMP_TAC[CONTINUOUS_ON_CONST]; MATCH_MP_TAC CONTINUOUS_ON_ADD THEN SIMP_TAC[CONTINUOUS_ON_CONST] THEN MATCH_MP_TAC CONTINUOUS_ON_CMUL THEN MATCH_MP_TAC CONTINUOUS_ON_BORSUK_MAP THEN ASM_SIMP_TAC[CENTRE_IN_CBALL; IN_DIFF; REAL_LT_IMP_LE] THEN REWRITE_TAC[IN] THEN REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ] THEN ASM_REWRITE_TAC[IN_DIFF; IN_UNIV]; REPEAT STRIP_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM SET_TAC[]]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[IN_SPHERE; NORM_ARITH `dist(a:real^N,a + x) = norm x`; NORM_MUL] THEN ASM_SIMP_TAC[REAL_ABS_INV; REAL_ABS_NORM; VECTOR_SUB_EQ; REAL_FIELD `&0 < r ==> abs r = r /\ (r * x = r <=> x = &1)`; REAL_FIELD `inv x * x = &1 <=> ~(x = &0)`; NORM_EQ_0] THENL [ONCE_REWRITE_TAC[GSYM IN_SPHERE_0] THEN ASM SET_TAC[]; UNDISCH_TAC `~(x IN connected_component ((:real^N) DIFF s) a)` THEN SIMP_TAC[CONTRAPOS_THM; IN] THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ; IN_DIFF; IN_UNIV]]; SIMP_TAC[IN_SPHERE; ONCE_REWRITE_RULE[NORM_SUB] dist] THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; REAL_LT_IMP_NZ] THEN REWRITE_TAC[VECTOR_ARITH `a + &1 % (x - a):real^N = x`] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s UNION t SUBSET u ==> !x. x IN t /\ ~(x IN u) ==> wev`)) THEN EXISTS_TAC `x:real^N` THEN ASM_REWRITE_TAC[ONCE_REWRITE_RULE[NORM_SUB] dist; IN_BALL; REAL_LT_REFL]]);; let BORSUK_MAP_ESSENTIAL_BOUNDED_COMPONENT = prove (`!s a. compact s /\ ~(a IN s) ==> (bounded(connected_component ((:real^N) DIFF s) a) <=> ~(?c. homotopic_with (\x. T) (s,sphere(vec 0:real^N,&1)) (\x. inv(norm(x - a)) % (x - a)) (\x. c)))`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [ASM_SIMP_TAC[DIFF_EMPTY; CONNECTED_COMPONENT_UNIV; NOT_BOUNDED_UNIV] THEN SIMP_TAC[HOMOTOPIC_WITH; NOT_IN_EMPTY; PCROSS_EMPTY; IMAGE_CLAUSES; CONTINUOUS_ON_EMPTY; EMPTY_SUBSET]; ALL_TAC] THEN EQ_TAC THENL [ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[] THEN REPEAT DISCH_TAC THEN MP_TAC(ISPECL [`\x:real^N. inv(norm(x - a)) % (x - a)`; `s:real^N->bool`; `vec 0:real^N`; `&1`] NULLHOMOTOPIC_INTO_SPHERE_EXTENSION) THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED; NOT_IMP; CONTINUOUS_ON_BORSUK_MAP; BORSUK_MAP_INTO_SPHERE] THEN MP_TAC(ISPECL [`s:real^N->bool`; `connected_component ((:real^N) DIFF s) a`; `a:real^N`] NON_EXTENSIBLE_BORSUK_MAP) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [GEN_REWRITE_TAC RAND_CONV [IN] THEN REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ] THEN ASM_REWRITE_TAC[IN_COMPONENTS; IN_DIFF; IN_UNIV] THEN ASM_MESON_TAC[]; REWRITE_TAC[CONTRAPOS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV]; SET_TAC[]]]; ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[] THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `vec 0:real^N` o MATCH_MP BOUNDED_SUBSET_BALL o MATCH_MP COMPACT_IMP_BOUNDED) THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `?b. b IN connected_component ((:real^N) DIFF s) a /\ ~(b IN ball(vec 0,r))` MP_TAC THENL [REWRITE_TAC[SET_RULE `(?b. b IN s /\ ~(b IN t)) <=> ~(s SUBSET t)`] THEN ASM_MESON_TAC[BOUNDED_SUBSET; BOUNDED_BALL]; DISCH_THEN(X_CHOOSE_THEN `b:real^N` STRIP_ASSUME_TAC)] THEN SUBGOAL_THEN `?c. homotopic_with (\x. T) (ball(vec 0:real^N,r),sphere (vec 0,&1)) (\x. inv (norm (x - b)) % (x - b)) (\x. c)` MP_TAC THENL [MATCH_MP_TAC NULLHOMOTOPIC_FROM_CONTRACTIBLE THEN ASM_SIMP_TAC[CONTINUOUS_ON_BORSUK_MAP; BORSUK_MAP_INTO_SPHERE; CONVEX_IMP_CONTRACTIBLE; CONVEX_BALL]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^N` THEN STRIP_TAC] THEN MATCH_MP_TAC HOMOTOPIC_WITH_TRANS THEN EXISTS_TAC `\x:real^N. inv(norm (x - b)) % (x - b)` THEN CONJ_TAC THENL [MATCH_MP_TAC BORSUK_MAPS_HOMOTOPIC_IN_PATH_COMPONENT THEN ASM_SIMP_TAC[OPEN_PATH_CONNECTED_COMPONENT; GSYM closed; COMPACT_IMP_CLOSED] THEN ASM_MESON_TAC[IN]; ASM_MESON_TAC[HOMOTOPIC_WITH_SUBSET_LEFT]]]);; let HOMOTOPIC_BORSUK_MAPS_IN_BOUNDED_COMPONENT = prove (`!s a b. compact s /\ ~(a IN s) /\ ~(b IN s) /\ bounded (connected_component ((:real^N) DIFF s) a) /\ homotopic_with (\x. T) (s,sphere(vec 0,&1)) (\x. inv(norm(x - a)) % (x - a)) (\x. inv(norm(x - b)) % (x - b)) ==> connected_component ((:real^N) DIFF s) a b`, REPEAT STRIP_TAC THEN GEN_REWRITE_TAC I [GSYM IN] THEN MP_TAC(ISPECL [`s:real^N->bool`; `connected_component ((:real^N) DIFF s) a`; `a:real^N`] NON_EXTENSIBLE_BORSUK_MAP) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [GEN_REWRITE_TAC RAND_CONV [IN] THEN REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ] THEN ASM_REWRITE_TAC[IN_COMPONENTS; IN_DIFF; IN_UNIV] THEN ASM_MESON_TAC[]; ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM]] THEN DISCH_TAC THEN REWRITE_TAC[] THEN MATCH_MP_TAC BORSUK_HOMOTOPY_EXTENSION THEN EXISTS_TAC `\x:real^N. inv(norm(x - b)) % (x - b)` THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED; ANR_SPHERE; CLOSED_SUBSET; SUBSET_UNION] THEN ONCE_REWRITE_TAC[HOMOTOPIC_WITH_SYM] THEN ASM_SIMP_TAC[CONTINUOUS_ON_BORSUK_MAP; IN_UNION; BORSUK_MAP_INTO_SPHERE] THEN REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN MATCH_MP_TAC CLOSED_UNION_COMPLEMENT_COMPONENT THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED; IN_COMPONENTS; IN_DIFF; IN_UNIV] THEN ASM_MESON_TAC[]);; let BORSUK_MAPS_HOMOTOPIC_IN_CONNECTED_COMPONENT_EQ = prove (`!s a b. 2 <= dimindex(:N) /\ compact s /\ ~(a IN s) /\ ~(b IN s) ==> (homotopic_with (\x. T) (s,sphere(vec 0,&1)) (\x. inv(norm(x - a)) % (x - a)) (\x. inv(norm(x - b)) % (x - b)) <=> connected_component ((:real^N) DIFF s) a b)`, REPEAT STRIP_TAC THEN EQ_TAC THENL [DISCH_TAC; ASM_SIMP_TAC[GSYM OPEN_PATH_CONNECTED_COMPONENT; GSYM closed; COMPACT_IMP_CLOSED] THEN REWRITE_TAC[BORSUK_MAPS_HOMOTOPIC_IN_PATH_COMPONENT]] THEN ASM_CASES_TAC `bounded(connected_component ((:real^N) DIFF s) a)` THENL [MATCH_MP_TAC HOMOTOPIC_BORSUK_MAPS_IN_BOUNDED_COMPONENT THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `bounded(connected_component ((:real^N) DIFF s) b)` THENL [ONCE_REWRITE_TAC[CONNECTED_COMPONENT_SYM_EQ] THEN MATCH_MP_TAC HOMOTOPIC_BORSUK_MAPS_IN_BOUNDED_COMPONENT THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[HOMOTOPIC_WITH_SYM] THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`(:real^N) DIFF s`; `a:real^N`; `b:real^N`] COBOUNDED_UNIQUE_UNBOUNDED_COMPONENT) THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_EQ_EQ; IN_DIFF; IN_UNIV; SET_RULE `UNIV DIFF (UNIV DIFF s) = s`] THEN ASM_SIMP_TAC[COMPACT_IMP_BOUNDED]);; let BORSUK_SEPARATION_THEOREM_GEN = prove (`!s:real^N->bool. compact s ==> ((!c. c IN components((:real^N) DIFF s) ==> ~bounded c) <=> (!f. f continuous_on s /\ IMAGE f s SUBSET sphere(vec 0:real^N,&1) ==> ?c. homotopic_with (\x. T) (s,sphere(vec 0,&1)) f (\x. c)))`, REPEAT STRIP_TAC THEN EQ_TAC THENL [ALL_TAC; ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[NOT_FORALL_THM; components; EXISTS_IN_GSPEC; NOT_IMP; IN_UNIV; IN_DIFF] THEN DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\x:real^N. inv(norm(x - a)) % (x - a)` THEN ASM_SIMP_TAC[GSYM BORSUK_MAP_ESSENTIAL_BOUNDED_COMPONENT; CONTINUOUS_ON_BORSUK_MAP; BORSUK_MAP_INTO_SPHERE]] THEN DISCH_TAC THEN X_GEN_TAC `f:real^N->real^N` THEN STRIP_TAC THEN MP_TAC(ISPECL [`f:real^N->real^N`; `s:real^N->bool`; `vec 0:real^N`; `&1:real`] EXTEND_MAP_UNIV_TO_SPHERE_NO_BOUNDED_COMPONENT) THEN ASM_REWRITE_TAC[LE_REFL; REAL_POS] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^N->real^N` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`g:real^N->real^N`; `(:real^N)`; `sphere(vec 0:real^N,&1)`] NULLHOMOTOPIC_FROM_CONTRACTIBLE) THEN ASM_REWRITE_TAC[CONTRACTIBLE_UNIV] THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN DISCH_THEN(MP_TAC o SPEC `s:real^N->bool` o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMOTOPIC_WITH_SUBSET_LEFT)) THEN REWRITE_TAC[SUBSET_UNIV] THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_WITH_EQ) THEN ASM_SIMP_TAC[]);; let BORSUK_SEPARATION_THEOREM = prove (`!s:real^N->bool. 2 <= dimindex(:N) /\ compact s ==> (connected((:real^N) DIFF s) <=> !f. f continuous_on s /\ IMAGE f s SUBSET sphere(vec 0:real^N,&1) ==> ?c. homotopic_with (\x. T) (s,sphere(vec 0,&1)) f (\x. c))`, SIMP_TAC[GSYM BORSUK_SEPARATION_THEOREM_GEN] THEN X_GEN_TAC `s:real^N->bool` THEN STRIP_TAC THEN EQ_TAC THENL [DISCH_TAC THEN MP_TAC(ISPEC `(:real^N) DIFF s` COMPONENTS_EQ_SING) THEN MP_TAC(ISPEC `(:real^N) DIFF s` COBOUNDED_IMP_UNBOUNDED) THEN ASM_CASES_TAC `(:real^N) DIFF s = {}` THEN ASM_SIMP_TAC[COMPACT_IMP_BOUNDED; SET_RULE `UNIV DIFF (UNIV DIFF s) = s`; BOUNDED_EMPTY; FORALL_IN_INSERT; NOT_IN_EMPTY]; REWRITE_TAC[components; FORALL_IN_GSPEC; IN_DIFF; IN_UNIV] THEN DISCH_TAC THEN REWRITE_TAC[CONNECTED_EQ_CONNECTED_COMPONENT_EQ] THEN REWRITE_TAC[IN_DIFF; IN_UNIV] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC COBOUNDED_UNIQUE_UNBOUNDED_COMPONENT THEN ASM_SIMP_TAC[COMPACT_IMP_BOUNDED; SET_RULE `UNIV DIFF (UNIV DIFF s) = s`]]);; let HOMOTOPY_EQUIVALENT_SEPARATION = prove (`!s t. compact s /\ compact t /\ s homotopy_equivalent t ==> (connected((:real^N) DIFF s) <=> connected((:real^N) DIFF t))`, let special = prove (`!s:real^1->bool. bounded s /\ connected((:real^1) DIFF s) ==> s = {}`, REWRITE_TAC[GSYM IS_INTERVAL_CONNECTED_1] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP BOUNDED_SUBSET_OPEN_INTERVAL) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; EXTENSION; NOT_IN_EMPTY] THEN MAP_EVERY X_GEN_TAC [`a:real^1`; `b:real^1`] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IS_INTERVAL_1]) THEN DISCH_THEN(MP_TAC o SPECL [`a:real^1`; `b:real^1`]) THEN REWRITE_TAC[IN_UNIV; IN_DIFF; SUBSET; IN_INTERVAL_1] THEN MESON_TAC[REAL_LT_REFL; REAL_LT_IMP_LE]) in REPEAT STRIP_TAC THEN SUBGOAL_THEN `1 <= dimindex(:N)` MP_TAC THENL [REWRITE_TAC[DIMINDEX_GE_1]; REWRITE_TAC[ARITH_RULE `1 <= n <=> n = 1 \/ 2 <= n`] THEN REWRITE_TAC[GSYM DIMINDEX_1]] THEN STRIP_TAC THENL [ASSUME_TAC(GEOM_EQUAL_DIMENSION_RULE(ASSUME `dimindex(:N) = dimindex(:1)`) special) THEN EQ_TAC THEN DISCH_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPEC `s:real^N->bool`); FIRST_X_ASSUM(MP_TAC o SPEC `t:real^N->bool`)] THEN ASM_SIMP_TAC[COMPACT_IMP_BOUNDED] THEN DISCH_TAC THEN UNDISCH_TAC `(s:real^N->bool) homotopy_equivalent (t:real^N->bool)` THEN ASM_REWRITE_TAC[HOMOTOPY_EQUIVALENT_EMPTY] THEN DISCH_TAC THEN ASM_REWRITE_TAC[CONNECTED_UNIV; DIFF_EMPTY]; REPEAT STRIP_TAC THEN ASM_SIMP_TAC[BORSUK_SEPARATION_THEOREM] THEN MATCH_MP_TAC HOMOTOPY_EQUIVALENT_COHOMOTOPIC_TRIVIALITY_NULL THEN ASM_REWRITE_TAC[]]);; let JORDAN_BROUWER_SEPARATION = prove (`!s a:real^N r. &0 < r /\ s homeomorphic sphere(a,r) ==> ~connected((:real^N) DIFF s)`, REPEAT GEN_TAC THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `sphere(a:real^N,r)`] HOMOTOPY_EQUIVALENT_SEPARATION) THEN ANTS_TAC THENL [ASM_MESON_TAC[HOMEOMORPHIC_COMPACTNESS; COMPACT_SPHERE; HOMEOMORPHIC_IMP_HOMOTOPY_EQUIVALENT]; DISCH_THEN SUBST1_TAC] THEN DISCH_TAC THEN MP_TAC(ISPECL [`(:real^N) DIFF sphere(a,r)`; `ball(a:real^N,r)`] CONNECTED_INTER_FRONTIER) THEN ASM_SIMP_TAC[FRONTIER_BALL; NOT_IMP] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[GSYM CBALL_DIFF_BALL] THEN MATCH_MP_TAC(SET_RULE `~(b = {}) ==> ~((UNIV DIFF (c DIFF b)) INTER b = {})`) THEN ASM_SIMP_TAC[BALL_EQ_EMPTY; REAL_NOT_LE]; MATCH_MP_TAC(SET_RULE `~(s UNION t = UNIV) ==> ~(UNIV DIFF t DIFF s = {})`) THEN REWRITE_TAC[BALL_UNION_SPHERE] THEN MESON_TAC[BOUNDED_CBALL; NOT_BOUNDED_UNIV]; SET_TAC[]]);; let JORDAN_BROUWER_FRONTIER = prove (`!s t a:real^N r. 2 <= dimindex(:N) /\ s homeomorphic sphere(a,r) /\ t IN components((:real^N) DIFF s) ==> frontier t = s`, let lemma = prove (`!s a r. 2 <= dimindex(:N) /\ &0 < r /\ s PSUBSET sphere(a,r) ==> connected((:real^N) DIFF s)`, REWRITE_TAC[PSUBSET_ALT; SUBSET; IN_SPHERE; GSYM REAL_LE_ANTISYM] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `(:real^N) DIFF s = {x:real^N | dist(a,x) <= r /\ ~(x IN s)} UNION {x:real^N | r <= dist(a,x) /\ ~(x IN s)}` SUBST1_TAC THENL [SET_TAC[REAL_LE_TOTAL]; MATCH_MP_TAC CONNECTED_UNION] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_INTERMEDIATE_CLOSURE THEN EXISTS_TAC `ball(a:real^N,r)` THEN ASM_SIMP_TAC[CONNECTED_BALL; CLOSURE_BALL; SUBSET; IN_BALL; IN_CBALL; IN_ELIM_THM] THEN ASM_MESON_TAC[REAL_LT_IMP_LE; REAL_NOT_LE]; MATCH_MP_TAC CONNECTED_INTERMEDIATE_CLOSURE THEN EXISTS_TAC `(:real^N) DIFF cball(a,r)` THEN REWRITE_TAC[CLOSURE_COMPLEMENT; SUBSET; IN_DIFF; IN_UNIV; IN_BALL; IN_CBALL; IN_ELIM_THM; INTERIOR_CBALL] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[REAL_LT_IMP_LE; REAL_NOT_LE]] THEN MATCH_MP_TAC CONNECTED_OPEN_DIFF_CBALL THEN ASM_REWRITE_TAC[SUBSET_UNIV; CONNECTED_UNIV; OPEN_UNIV]; ASM SET_TAC[]]) in MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `c:real^N->bool`; `a:real^N`; `r:real`] THEN ASM_CASES_TAC `r < &0` THENL [ASM_SIMP_TAC[SPHERE_EMPTY; HOMEOMORPHIC_EMPTY; IMP_CONJ; DIFF_EMPTY] THEN SIMP_TAC[snd(EQ_IMP_RULE(SPEC_ALL COMPONENTS_EQ_SING)); UNIV_NOT_EMPTY; CONNECTED_UNIV; IN_SING; FRONTIER_UNIV]; ALL_TAC] THEN ASM_CASES_TAC `r = &0` THENL [ASM_SIMP_TAC[HOMEOMORPHIC_FINITE_STRONG; SPHERE_SING; FINITE_SING] THEN SIMP_TAC[CARD_CLAUSES; FINITE_EMPTY; GSYM HAS_SIZE; NOT_IN_EMPTY] THEN REWRITE_TAC[HAS_SIZE_CLAUSES; UNWIND_THM2; NOT_IN_EMPTY; IMP_CONJ] THEN SIMP_TAC[LEFT_IMP_EXISTS_THM; CONNECTED_PUNCTURED_UNIVERSE; IN_SING; snd(EQ_IMP_RULE(SPEC_ALL COMPONENTS_EQ_SING)); FRONTIER_SING; SET_RULE `UNIV DIFF s = {} <=> s = UNIV`; FRONTIER_COMPLEMENT; MESON[BOUNDED_SING; NOT_BOUNDED_UNIV] `~((:real^N) = {a})`]; ALL_TAC] THEN SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC FRONTIER_MINIMAL_SEPARATING_CLOSED THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMEOMORPHIC_COMPACTNESS) THEN SIMP_TAC[COMPACT_SPHERE; COMPACT_IMP_CLOSED] THEN DISCH_TAC THEN CONJ_TAC THENL [ASM_MESON_TAC[JORDAN_BROUWER_SEPARATION]; ALL_TAC] THEN 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 [`f:real^N->real^N`; `g:real^N->real^N`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`t:real^N->bool`; `IMAGE (f:real^N->real^N) t`] HOMOTOPY_EQUIVALENT_SEPARATION) THEN ANTS_TAC THENL [MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [ASM_MESON_TAC[COMPACT_EQ_BOUNDED_CLOSED; BOUNDED_SUBSET; PSUBSET]; DISCH_TAC THEN SUBGOAL_THEN `t homeomorphic (IMAGE (f:real^N->real^N) t)` MP_TAC THENL [REWRITE_TAC[homeomorphic] THEN MAP_EVERY EXISTS_TAC [`f:real^N->real^N`; `g:real^N->real^N`] THEN ASM_REWRITE_TAC[HOMEOMORPHISM] 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[]; ASM_MESON_TAC[HOMEOMORPHIC_COMPACTNESS; HOMEOMORPHIC_IMP_HOMOTOPY_EQUIVALENT]]]; DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC lemma THEN MAP_EVERY EXISTS_TAC [`a:real^N`; `r:real`] THEN ASM SET_TAC[]]);; let JORDAN_BROUWER_NONSEPARATION = prove (`!s t a:real^N r. 2 <= dimindex(:N) /\ s homeomorphic sphere(a,r) /\ t PSUBSET s ==> connected((:real^N) DIFF t)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `!c. c IN components((:real^N) DIFF s) ==> connected(c UNION (s DIFF t))` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC CONNECTED_INTERMEDIATE_CLOSURE THEN EXISTS_TAC `c:real^N->bool` THEN CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]; ALL_TAC] THEN CONJ_TAC THENL [SET_TAC[]; REWRITE_TAC[UNION_SUBSET; CLOSURE_SUBSET]] THEN SUBGOAL_THEN `s:real^N->bool = frontier c` SUBST1_TAC THENL [ASM_MESON_TAC[JORDAN_BROUWER_FRONTIER]; ALL_TAC] THEN REWRITE_TAC[frontier] THEN SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `~(components((:real^N) DIFF s) = {})` ASSUME_TAC THENL [REWRITE_TAC[COMPONENTS_EQ_EMPTY; SET_RULE `UNIV DIFF s = {} <=> s = UNIV`] THEN ASM_MESON_TAC[NOT_BOUNDED_UNIV; COMPACT_EQ_BOUNDED_CLOSED; HOMEOMORPHIC_COMPACTNESS; COMPACT_SPHERE]; ALL_TAC] THEN SUBGOAL_THEN `(:real^N) DIFF t = UNIONS {c UNION (s DIFF t) | c | c IN components((:real^N) DIFF s)}` SUBST1_TAC THENL [MP_TAC(ISPEC `(:real^N) DIFF s` UNIONS_COMPONENTS) THEN REWRITE_TAC[UNIONS_GSPEC] THEN ASM SET_TAC[]; MATCH_MP_TAC CONNECTED_UNIONS THEN ASM_REWRITE_TAC[FORALL_IN_GSPEC] THEN REWRITE_TAC[INTERS_GSPEC] THEN ASM SET_TAC[]]);; let JORDAN_BROUWER_ACCESSIBILITY = prove (`!s c a:real^N r v x. 2 <= dimindex(:N) /\ s homeomorphic sphere(a,r) /\ c IN components((:real^N) DIFF s) /\ x IN c /\ open_in (subtopology euclidean s) v /\ ~(v = {}) ==> ?g. arc g /\ IMAGE g (interval[vec 0,vec 1] DELETE (vec 1)) SUBSET c /\ pathstart g = x /\ pathfinish g IN v`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMEOMORPHIC_COMPACTNESS) THEN REWRITE_TAC[COMPACT_SPHERE] THEN REWRITE_TAC[closed; COMPACT_EQ_BOUNDED_CLOSED] THEN STRIP_TAC THEN MATCH_MP_TAC DENSE_ACCESSIBLE_FRONTIER_POINTS_CONNECTED THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[JORDAN_BROUWER_FRONTIER; OPEN_COMPONENTS; IN_COMPONENTS_CONNECTED]);; let CONNECTED_COMPLEMENT_CONTRACTIBLE = prove (`!s. 2 <= dimindex(:N) /\ compact s /\ contractible s ==> 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 MP_TAC(ISPECL [`s:real^N->bool`; `{a:real^N}`] HOMOTOPY_EQUIVALENT_SEPARATION) THEN ASM_SIMP_TAC[HOMOTOPY_EQUIVALENT_SING; COMPACT_SING; AR_SING; CONNECTED_COMPLEMENT_ABSOLUTE_RETRACT]);; (* ------------------------------------------------------------------------- *) (* Invariance of domain and corollaries. *) (* ------------------------------------------------------------------------- *) let INVARIANCE_OF_DOMAIN = prove (`!f:real^N->real^N s. f continuous_on s /\ open s /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> open(IMAGE f s)`, let lemma = prove (`!f:real^N->real^N a r. f continuous_on cball(a,r) /\ &0 < r /\ (!x y. x IN cball(a,r) /\ y IN cball(a,r) /\ f x = f y ==> x = y) ==> open(IMAGE f (ball(a,r)))`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `dimindex(:N) = 1` THENL [MP_TAC(ISPECL [`(:real^N)`; `(:real^1)`] ISOMETRIES_SUBSPACES) THEN ASM_SIMP_TAC[SUBSPACE_UNIV; DIM_UNIV; DIMINDEX_1; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`h:real^N->real^1`; `k:real^1->real^N`] THEN REWRITE_TAC[IN_UNIV] THEN STRIP_TAC THEN MP_TAC(ISPECL [`(h:real^N->real^1) o f o (k:real^1->real^N)`; `IMAGE (h:real^N->real^1) (cball(a,r))`] INJECTIVE_EQ_1D_OPEN_MAP_UNIV) THEN MATCH_MP_TAC(TAUT `p /\ q /\ r /\ (s ==> t) ==> (p /\ q ==> (r <=> s)) ==> t`) THEN REPEAT CONJ_TAC THENL [REPEAT(MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC) THEN ASM_SIMP_TAC[LINEAR_CONTINUOUS_ON; GSYM IMAGE_o] THEN ASM_REWRITE_TAC[o_DEF; IMAGE_ID]; REWRITE_TAC[IS_INTERVAL_CONNECTED_1] THEN MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN ASM_SIMP_TAC[LINEAR_CONTINUOUS_ON; CONNECTED_CBALL]; ASM_SIMP_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE; o_DEF] THEN ASM SET_TAC[]; DISCH_THEN(MP_TAC o SPEC `IMAGE (h:real^N->real^1) (ball(a,r))`) THEN ASM_SIMP_TAC[IMAGE_SUBSET; BALL_SUBSET_CBALL; GSYM IMAGE_o] THEN ANTS_TAC THENL [MP_TAC(ISPECL [`a:real^N`; `r:real`] OPEN_BALL); ALL_TAC] THEN MATCH_MP_TAC EQ_IMP THENL [CONV_TAC SYM_CONV; REWRITE_TAC[GSYM o_ASSOC] THEN ONCE_REWRITE_TAC[IMAGE_o] THEN ASM_REWRITE_TAC[o_DEF; ETA_AX]] THEN MATCH_MP_TAC OPEN_BIJECTIVE_LINEAR_IMAGE_EQ THEN ASM_MESON_TAC[]]; FIRST_ASSUM(MP_TAC o MATCH_MP (ARITH_RULE `~(n = 1) ==> 1 <= n ==> 2 <= n`)) THEN REWRITE_TAC[DIMINDEX_GE_1] THEN DISCH_TAC] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`IMAGE (f:real^N->real^N) (sphere(a,r))`; `a:real^N`; `r:real`] JORDAN_BROUWER_SEPARATION) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN MATCH_MP_TAC HOMEOMORPHIC_COMPACT THEN EXISTS_TAC `f:real^N->real^N` THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET; SPHERE_SUBSET_CBALL; COMPACT_SPHERE]; DISCH_TAC] THEN MP_TAC(ISPEC `(:real^N) DIFF IMAGE f (sphere(a:real^N,r))` COBOUNDED_HAS_BOUNDED_COMPONENT) THEN ASM_REWRITE_TAC[SET_RULE `UNIV DIFF (UNIV DIFF s) = s`] THEN ANTS_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET; SPHERE_SUBSET_CBALL; COMPACT_SPHERE; COMPACT_CONTINUOUS_IMAGE; COMPACT_IMP_BOUNDED]; DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` STRIP_ASSUME_TAC)] THEN SUBGOAL_THEN `IMAGE (f:real^N->real^N) (ball(a,r)) = c` SUBST1_TAC THENL [ALL_TAC; FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] OPEN_COMPONENTS)) THEN REWRITE_TAC[GSYM closed] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET; SPHERE_SUBSET_CBALL; COMPACT_SPHERE; COMPACT_CONTINUOUS_IMAGE; COMPACT_IMP_CLOSED]] THEN MATCH_MP_TAC(SET_RULE `~(c = {}) /\ (~(c INTER t = {}) ==> t SUBSET c) /\ c SUBSET t ==> t = c`) THEN REPEAT STRIP_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_NONEMPTY]; FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] COMPONENTS_MAXIMAL)) THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN REWRITE_TAC[CONNECTED_BALL] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; BALL_SUBSET_CBALL]; REWRITE_TAC[GSYM CBALL_DIFF_SPHERE] THEN MP_TAC(ISPECL [`a:real^N`; `r:real`] SPHERE_SUBSET_CBALL) THEN ASM SET_TAC[]]; FIRST_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN FIRST_ASSUM(MP_TAC o SPEC `(:real^N) DIFF IMAGE f (cball(a:real^N,r))` o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] COMPONENTS_MAXIMAL)) THEN SIMP_TAC[SET_RULE `UNIV DIFF t SUBSET UNIV DIFF s <=> s SUBSET t`; IMAGE_SUBSET; SPHERE_SUBSET_CBALL] THEN MATCH_MP_TAC(TAUT `p /\ ~r /\ (~q ==> s) ==> (p /\ q ==> r) ==> s`) THEN REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC(INST_TYPE [`:N`,`:M`] CONNECTED_COMPLEMENT_HOMEOMORPHIC_CONVEX_COMPACT) THEN EXISTS_TAC `cball(a:real^N,r)` THEN ASM_REWRITE_TAC[CONVEX_CBALL; COMPACT_CBALL] THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN MATCH_MP_TAC HOMEOMORPHIC_COMPACT THEN EXISTS_TAC `f:real^N->real^N` THEN ASM_REWRITE_TAC[COMPACT_CBALL]; DISCH_THEN(MP_TAC o MATCH_MP(REWRITE_RULE[IMP_CONJ_ALT] BOUNDED_SUBSET)) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC COBOUNDED_IMP_UNBOUNDED THEN REWRITE_TAC[SET_RULE `UNIV DIFF (UNIV DIFF s) = s`] THEN ASM_MESON_TAC[COMPACT_IMP_BOUNDED; COMPACT_CONTINUOUS_IMAGE; COMPACT_CBALL]; REWRITE_TAC[GSYM CBALL_DIFF_SPHERE] THEN ASM SET_TAC[]]]) in REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[OPEN_SUBOPEN] THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_CBALL]) THEN DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `r:real` THEN STRIP_TAC THEN EXISTS_TAC `IMAGE (f:real^N->real^N) (ball(a,r))` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC lemma THEN ASM_MESON_TAC[SUBSET; CONTINUOUS_ON_SUBSET]; ASM_SIMP_TAC[FUN_IN_IMAGE; CENTRE_IN_BALL]; MATCH_MP_TAC IMAGE_SUBSET THEN ASM_MESON_TAC[BALL_SUBSET_CBALL; SUBSET_TRANS]]);; let INVARIANCE_OF_DOMAIN_SUBSPACES = prove (`!f:real^M->real^N u v s. subspace u /\ subspace v /\ dim v <= dim u /\ f continuous_on s /\ IMAGE f s SUBSET v /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\ open_in (subtopology euclidean u) s ==> open_in (subtopology euclidean v) (IMAGE f s)`, let lemma0 = prove (`!f:real^M->real^M s u. subspace s /\ dim s = dimindex(:N) /\ f continuous_on u /\ IMAGE f u SUBSET s /\ (!x y. x IN u /\ y IN u /\ f x = f y ==> x = y) /\ open_in (subtopology euclidean s) u ==> open_in (subtopology euclidean s) (IMAGE f u)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`(:real^N)`; `s:real^M->bool`] HOMEOMORPHIC_SUBSPACES) THEN ASM_REWRITE_TAC[DIM_UNIV; SUBSPACE_UNIV] THEN REWRITE_TAC[homeomorphic; homeomorphism; LEFT_IMP_EXISTS_THM; IN_UNIV] THEN MAP_EVERY X_GEN_TAC [`h:real^N->real^M`; `k:real^M->real^N`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`(k:real^M->real^N) o f o (h:real^N->real^M)`; `IMAGE (k:real^M->real^N) u`] INVARIANCE_OF_DOMAIN) THEN REWRITE_TAC[GSYM IMAGE_o; o_THM] THEN SUBGOAL_THEN `!t. open t <=> open_in (subtopology euclidean (IMAGE (k:real^M->real^N) s)) t` (fun th -> REWRITE_TAC[th]) THENL [ASM_REWRITE_TAC[SUBTOPOLOGY_UNIV; GSYM OPEN_IN]; ALL_TAC] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [REPEAT(MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN FIRST_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN REWRITE_TAC[IMAGE_o] THEN ASM SET_TAC[]; MATCH_MP_TAC HOMEOMORPHISM_IMP_OPEN_MAP THEN MAP_EVERY EXISTS_TAC [`h:real^N->real^M`; `s:real^M->bool`] THEN ASM_REWRITE_TAC[homeomorphism]; REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN FIRST_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ASM_SIMP_TAC[SUBSET] THEN REPEAT STRIP_TAC THEN AP_TERM_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM SET_TAC[]]; ASM_REWRITE_TAC[] THEN DISCH_TAC THEN SUBGOAL_THEN `IMAGE f u = IMAGE (h:real^N->real^M) (IMAGE ((k o f o h) o (k:real^M->real^N)) u)` SUBST1_TAC THENL [REWRITE_TAC[GSYM IMAGE_o] THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> f x = g x) ==> IMAGE f s = IMAGE g s`) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET)) THEN ASM_SIMP_TAC[SUBSET; o_THM] THEN ASM SET_TAC[]; MATCH_MP_TAC HOMEOMORPHISM_IMP_OPEN_MAP THEN MAP_EVERY EXISTS_TAC [`k:real^M->real^N`; `(:real^N)`] THEN ASM_REWRITE_TAC[homeomorphism]]]) in let lemma1 = prove (`!f:real^N->real^N s u. subspace s /\ f continuous_on u /\ IMAGE f u SUBSET s /\ (!x y. x IN u /\ y IN u /\ f x = f y ==> x = y) /\ open_in (subtopology euclidean s) u ==> open_in (subtopology euclidean s) (IMAGE f u)`, REWRITE_TAC[INJECTIVE_ON_ALT] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ABBREV_TAC `s' = {y:real^N | !x. x IN s ==> orthogonal x y}` THEN SUBGOAL_THEN `subspace(s':real^N->bool)` ASSUME_TAC THENL [EXPAND_TAC "s'" THEN REWRITE_TAC[SUBSPACE_ORTHOGONAL_TO_VECTORS]; FIRST_ASSUM(ASSUME_TAC o MATCH_MP SUBSPACE_IMP_NONEMPTY)] THEN ABBREV_TAC `g:real^(N,N)finite_sum->real^(N,N)finite_sum = \z. pastecart (f(fstcart z)) (sndcart z)` THEN SUBGOAL_THEN `g continuous_on ((u:real^N->bool) PCROSS s') /\ IMAGE g (u PCROSS s') SUBSET (s:real^N->bool) PCROSS (s':real^N->bool) /\ (!w z. w IN u PCROSS s' /\ z IN u PCROSS s' ==> (g w = g z <=> w = z))` STRIP_ASSUME_TAC THENL [EXPAND_TAC "g" THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_PASTECART THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_SNDCART] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; IMAGE_FSTCART_PCROSS] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[CONTINUOUS_ON_EMPTY]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_PCROSS] THEN SIMP_TAC[PASTECART_IN_PCROSS; SNDCART_PASTECART; FSTCART_PASTECART] THEN ASM SET_TAC[]; EXPAND_TAC "g" THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[FORALL_IN_PCROSS; FSTCART_PASTECART; SNDCART_PASTECART] THEN ASM_SIMP_TAC[PASTECART_INJ]]; ALL_TAC] THEN SUBGOAL_THEN `open_in (subtopology euclidean (s PCROSS s')) (IMAGE (g:real^(N,N)finite_sum->real^(N,N)finite_sum) (u PCROSS s'))` MP_TAC THENL [MATCH_MP_TAC lemma0 THEN ASM_SIMP_TAC[SUBSPACE_PCROSS; OPEN_IN_PCROSS_EQ; OPEN_IN_REFL] THEN CONJ_TAC THENL [ASM_SIMP_TAC[DIM_PCROSS]; ASM_MESON_TAC[]] THEN MP_TAC(ISPECL [`s:real^N->bool`; `(:real^N)`] DIM_SUBSPACE_ORTHOGONAL_TO_VECTORS) THEN ASM_REWRITE_TAC[SUBSET_UNIV; SUBSPACE_UNIV; IN_UNIV; DIM_UNIV] THEN ARITH_TAC; SUBGOAL_THEN `IMAGE (g:real^(N,N)finite_sum->real^(N,N)finite_sum) (u PCROSS s') = IMAGE f u PCROSS s'` SUBST1_TAC THENL [EXPAND_TAC "g" THEN REWRITE_TAC[EXTENSION; EXISTS_PASTECART; PASTECART_IN_PCROSS; IN_IMAGE; FORALL_PASTECART; PASTECART_IN_PCROSS; FSTCART_PASTECART; SNDCART_PASTECART; PASTECART_INJ] THEN ASM SET_TAC[]; ASM_SIMP_TAC[OPEN_IN_PCROSS_EQ; IMAGE_EQ_EMPTY] THEN STRIP_TAC THEN ASM_SIMP_TAC[IMAGE_CLAUSES; OPEN_IN_EMPTY]]]) in REWRITE_TAC[INJECTIVE_ON_ALT] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN MP_TAC(ISPECL [`u:real^M->bool`; `dim(v:real^N->bool)`] CHOOSE_SUBSPACE_OF_SUBSPACE) THEN ASM_SIMP_TAC[SPAN_OF_SUBSPACE] THEN DISCH_THEN(X_CHOOSE_THEN `v:real^M->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`v:real^N->bool`; `v:real^M->bool`] HOMEOMORPHIC_SUBSPACES) THEN ASM_REWRITE_TAC[homeomorphic; homeomorphism; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`h:real^N->real^M`; `k:real^M->real^N`] THEN STRIP_TAC THEN SUBGOAL_THEN `IMAGE (f:real^M->real^N) s = IMAGE (k:real^M->real^N) (IMAGE ((h:real^N->real^M) o f) s)` SUBST1_TAC THENL [REWRITE_TAC[GSYM IMAGE_o] THEN MATCH_MP_TAC(SET_RULE `(!x. x IN u ==> f x = g x) ==> IMAGE f u = IMAGE g u`) THEN RULE_ASSUM_TAC(REWRITE_RULE[SUBSET]) THEN ASM_SIMP_TAC[SUBSET; o_THM] THEN ASM SET_TAC[]; MATCH_MP_TAC HOMEOMORPHISM_IMP_OPEN_MAP THEN MAP_EVERY EXISTS_TAC [`h:real^N->real^M`; `v:real^M->bool`] THEN ASM_REWRITE_TAC[homeomorphism] THEN MATCH_MP_TAC OPEN_IN_SUBSET_TRANS THEN EXISTS_TAC `u:real^M->bool` THEN ASM_REWRITE_TAC[IMAGE_o] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[GSYM IMAGE_o] THEN MATCH_MP_TAC lemma1 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_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]]);; let INVARIANCE_OF_DIMENSION_SUBSPACES = prove (`!f:real^M->real^N u v s. subspace u /\ subspace v /\ ~(s = {}) /\ open_in (subtopology euclidean u) s /\ f continuous_on s /\ IMAGE f s SUBSET v /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> dim u <= dim v`, REWRITE_TAC[GSYM NOT_LT] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`u:real^M->bool`; `dim(v:real^N->bool)`] CHOOSE_SUBSPACE_OF_SUBSPACE) THEN ASM_SIMP_TAC[SPAN_OF_SUBSPACE; LE_LT] THEN DISCH_THEN(X_CHOOSE_THEN `t:real^M->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`v:real^N->bool`; `t:real^M->bool`] HOMEOMORPHIC_SUBSPACES) THEN FIRST_X_ASSUM(SUBST_ALL_TAC o SYM) THEN ASM_REWRITE_TAC[homeomorphic; homeomorphism; NOT_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`h:real^N->real^M`; `k:real^M->real^N`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`(h:real^N->real^M) o (f:real^M->real^N)`; `u:real^M->bool`; `u:real^M->bool`; `s:real^M->bool`] INVARIANCE_OF_DOMAIN_SUBSPACES) THEN ASM_REWRITE_TAC[LE_LT; NOT_IMP] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; REWRITE_TAC[IMAGE_o] THEN ASM SET_TAC[]; REWRITE_TAC[o_THM] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `IMAGE ((h:real^N->real^M) o (f:real^M->real^N)) s SUBSET t` ASSUME_TAC THENL [REWRITE_TAC[IMAGE_o] THEN ASM SET_TAC[]; ALL_TAC] THEN ABBREV_TAC `w = IMAGE ((h:real^N->real^M) o (f:real^M->real^N)) s` THEN DISCH_TAC THEN UNDISCH_TAC `dim(t:real^M->bool) < dim(u:real^M->bool)` THEN REWRITE_TAC[NOT_LT] THEN MP_TAC(ISPECL [`w:real^M->bool`; `u:real^M->bool`] DIM_OPEN_IN) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_MESON_TAC[IMAGE_EQ_EMPTY]; DISCH_THEN(SUBST1_TAC o SYM)] THEN ASM_SIMP_TAC[DIM_SUBSET]);; let INVARIANCE_OF_DOMAIN_AFFINE_SETS = prove (`!f:real^M->real^N u v s. affine u /\ affine v /\ aff_dim v <= aff_dim u /\ f continuous_on s /\ IMAGE f s SUBSET v /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\ open_in (subtopology euclidean u) s ==> open_in (subtopology euclidean v) (IMAGE f s)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_REWRITE_TAC[IMAGE_CLAUSES; OPEN_IN_EMPTY; INJECTIVE_ON_ALT] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `?a:real^M b:real^N. a IN s /\ a IN u /\ b IN v` STRIP_ASSUME_TAC THENL [FIRST_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`(\x. --b + x) o (f:real^M->real^N) o (\x. a + x)`; `IMAGE (\x:real^M. --a + x) u`; `IMAGE (\x:real^N. --b + x) v`; `IMAGE (\x:real^M. --a + x) s`] INVARIANCE_OF_DOMAIN_SUBSPACES) THEN REWRITE_TAC[IMAGE_o; INJECTIVE_ON_ALT; OPEN_IN_TRANSLATION_EQ] THEN SIMP_TAC[IMP_CONJ; GSYM INT_OF_NUM_LE; GSYM AFF_DIM_DIM_SUBSPACE] THEN ASM_REWRITE_TAC[AFF_DIM_TRANSLATION_EQ; RIGHT_FORALL_IMP_THM] THEN SIMP_TAC[FORALL_IN_IMAGE; o_THM; GSYM IMAGE_o; IMP_IMP; GSYM CONJ_ASSOC] THEN ANTS_TAC THENL [ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC AFFINE_IMP_SUBSPACE THEN ASM_REWRITE_TAC[AFFINE_TRANSLATION_EQ] THEN REWRITE_TAC[IN_IMAGE; VECTOR_ARITH `vec 0:real^N = --a + x <=> x = a`] THEN ASM_MESON_TAC[]; REPEAT CONJ_TAC THENL [REPEAT(MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID]); REWRITE_TAC[GSYM o_ASSOC] THEN REWRITE_TAC[IMAGE_o] THEN MATCH_MP_TAC IMAGE_SUBSET; REWRITE_TAC[VECTOR_ARITH `a + x:real^N = a + y <=> x = y`]]]; ALL_TAC] THEN ASM_SIMP_TAC[VECTOR_ARITH `a + --a + x:real^N = x`; GSYM IMAGE_o; o_DEF; IMAGE_ID; ETA_AX]);; let INVARIANCE_OF_DIMENSION_AFFINE_SETS = prove (`!f:real^M->real^N u v s. affine u /\ affine v /\ ~(s = {}) /\ open_in (subtopology euclidean u) s /\ f continuous_on s /\ IMAGE f s SUBSET v /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> aff_dim u <= aff_dim v`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_REWRITE_TAC[IMAGE_CLAUSES; OPEN_IN_EMPTY; INJECTIVE_ON_ALT] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `?a:real^M b:real^N. a IN s /\ a IN u /\ b IN v` STRIP_ASSUME_TAC THENL [FIRST_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`(\x. --b + x) o (f:real^M->real^N) o (\x. a + x)`; `IMAGE (\x:real^M. --a + x) u`; `IMAGE (\x:real^N. --b + x) v`; `IMAGE (\x:real^M. --a + x) s`] INVARIANCE_OF_DIMENSION_SUBSPACES) THEN REWRITE_TAC[IMAGE_o; INJECTIVE_ON_ALT; OPEN_IN_TRANSLATION_EQ] THEN SIMP_TAC[IMP_CONJ; GSYM INT_OF_NUM_LE; GSYM AFF_DIM_DIM_SUBSPACE] THEN ASM_REWRITE_TAC[AFF_DIM_TRANSLATION_EQ; RIGHT_FORALL_IMP_THM] THEN SIMP_TAC[FORALL_IN_IMAGE; o_THM; GSYM IMAGE_o; IMP_IMP; GSYM CONJ_ASSOC] THEN DISCH_THEN MATCH_MP_TAC THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC AFFINE_IMP_SUBSPACE THEN ASM_REWRITE_TAC[AFFINE_TRANSLATION_EQ] THEN REWRITE_TAC[IN_IMAGE; VECTOR_ARITH `vec 0:real^N = --a + x <=> x = a`] THEN ASM_MESON_TAC[]; ASM_REWRITE_TAC[IMAGE_EQ_EMPTY] THEN REPEAT CONJ_TAC THENL [REPEAT(MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID]); REWRITE_TAC[GSYM o_ASSOC] THEN REWRITE_TAC[IMAGE_o] THEN MATCH_MP_TAC IMAGE_SUBSET; REWRITE_TAC[VECTOR_ARITH `a + x:real^N = a + y <=> x = y`]]] THEN ASM_SIMP_TAC[VECTOR_ARITH `a + --a + x:real^N = x`; GSYM IMAGE_o; o_DEF; IMAGE_ID; ETA_AX]);; let INVARIANCE_OF_DIMENSION = prove (`!f:real^M->real^N s. f continuous_on s /\ open s /\ ~(s = {}) /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> dimindex(:M) <= dimindex(:N)`, REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM DIM_UNIV] THEN MATCH_MP_TAC INVARIANCE_OF_DIMENSION_SUBSPACES THEN MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `s:real^M->bool`] THEN ASM_REWRITE_TAC[SUBSPACE_UNIV; SUBSET_UNIV; SUBTOPOLOGY_UNIV; GSYM OPEN_IN]);; let CONTINUOUS_INJECTIVE_IMAGE_SUBSPACE_DIM_LE = prove (`!f:real^M->real^N s t. subspace s /\ subspace t /\ f continuous_on s /\ IMAGE f s SUBSET t /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> dim(s) <= dim(t)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC INVARIANCE_OF_DIMENSION_SUBSPACES THEN MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `s:real^M->bool`] THEN ASM_REWRITE_TAC[OPEN_IN_REFL] THEN ASM_SIMP_TAC[SUBSPACE_IMP_NONEMPTY]);; let INVARIANCE_OF_DIMENSION_CONVEX_DOMAIN = prove (`!f:real^M->real^N s t. convex s /\ f continuous_on s /\ IMAGE f s SUBSET affine hull t /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> aff_dim(s) <= aff_dim(t)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_REWRITE_TAC[AFF_DIM_EMPTY; AFF_DIM_GE] THEN MP_TAC(ISPECL [`f:real^M->real^N`; `affine hull s:real^M->bool`; `affine hull t:real^N->bool`; `relative_interior s:real^M->bool`] INVARIANCE_OF_DIMENSION_AFFINE_SETS) THEN ASM_REWRITE_TAC[AFFINE_AFFINE_HULL; AFF_DIM_AFFINE_HULL; OPEN_IN_RELATIVE_INTERIOR] THEN DISCH_THEN MATCH_MP_TAC THEN CONJ_TAC THENL [ASM_MESON_TAC[RELATIVE_INTERIOR_EQ_EMPTY]; ALL_TAC] THEN ASSUME_TAC(ISPEC `s:real^M->bool` RELATIVE_INTERIOR_SUBSET) THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ASM SET_TAC[]]);; let HOMEOMORPHIC_CONVEX_SETS = prove (`!s:real^M->bool t:real^N->bool. convex s /\ convex t /\ s homeomorphic t ==> aff_dim s = aff_dim t`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphic]) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; GSYM INT_LE_ANTISYM; homeomorphism] THEN MAP_EVERY X_GEN_TAC [`f:real^M->real^N`; `g:real^N->real^M`] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC INVARIANCE_OF_DIMENSION_CONVEX_DOMAIN THENL [EXISTS_TAC `f:real^M->real^N`; EXISTS_TAC `g:real^N->real^M`] THEN ASM_REWRITE_TAC[HULL_SUBSET] THEN ASM SET_TAC[]);; let HOMEOMORPHIC_CONVEX_COMPACT_SETS_EQ = prove (`!s:real^M->bool t:real^N->bool. convex s /\ compact s /\ convex t /\ compact t ==> (s homeomorphic t <=> aff_dim s = aff_dim t)`, MESON_TAC[HOMEOMORPHIC_CONVEX_SETS; HOMEOMORPHIC_CONVEX_COMPACT_SETS]);; let INVARIANCE_OF_DOMAIN_GEN = prove (`!f:real^M->real^N s. dimindex(:N) <= dimindex(:M) /\ f continuous_on s /\ open s /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> open(IMAGE f s)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `(:real^M)`; `(:real^N)`; `s:real^M->bool`] INVARIANCE_OF_DOMAIN_SUBSPACES) THEN ASM_REWRITE_TAC[SUBTOPOLOGY_UNIV; GSYM OPEN_IN; SUBSPACE_UNIV; DIM_UNIV; SUBSET_UNIV]);; let INJECTIVE_INTO_1D_IMP_OPEN_MAP_UNIV = prove (`!f:real^N->real^1 s t. f continuous_on s /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\ open t /\ t SUBSET s ==> open (IMAGE f t)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC INVARIANCE_OF_DOMAIN_GEN THEN ASM_REWRITE_TAC[DIMINDEX_1; DIMINDEX_GE_1] THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ASM SET_TAC[]]);; let CONTINUOUS_ON_INVERSE_OPEN = prove (`!f:real^M->real^N g s. dimindex(:N) <= dimindex(:M) /\ f continuous_on s /\ open s /\ (!x. x IN s ==> g(f x) = x) ==> g continuous_on IMAGE f s`, REPEAT STRIP_TAC THEN REWRITE_TAC[CONTINUOUS_OPEN_IN_PREIMAGE_EQ] THEN X_GEN_TAC `t:real^M->bool` THEN DISCH_TAC THEN SUBGOAL_THEN `{x | x IN IMAGE f s /\ g x IN t} = IMAGE (f:real^M->real^N) (s INTER t)` SUBST1_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC OPEN_OPEN_IN_TRANS] THEN REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN CONJ_TAC THEN MATCH_MP_TAC INVARIANCE_OF_DOMAIN_GEN THEN ASM_SIMP_TAC[OPEN_INTER; IN_INTER] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; INTER_SUBSET]);; let CONTINUOUS_ON_INVERSE_INTO_1D = prove (`!f:real^N->real^1 g s t. f continuous_on s /\ (path_connected s \/ compact s \/ open s) /\ IMAGE f s = t /\ (!x. x IN s ==> g(f x) = x) ==> g continuous_on t`, REPEAT STRIP_TAC THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_INVERSE_OPEN_MAP THEN MAP_EVERY EXISTS_TAC [`f:real^N->real^1`; `s:real^N->bool`] THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(SUBST1_TAC o SYM) THEN MATCH_MP_TAC INJECTIVE_INTO_1D_IMP_OPEN_MAP THEN ASM SET_TAC[]; ASM_MESON_TAC[CONTINUOUS_ON_INVERSE]; FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN MATCH_MP_TAC CONTINUOUS_ON_INVERSE_OPEN THEN ASM_REWRITE_TAC[DIMINDEX_1; DIMINDEX_GE_1]]);; let REAL_CONTINUOUS_ON_INVERSE = prove (`!f g s. f real_continuous_on s /\ (is_realinterval s \/ real_compact s \/ real_open s) /\ (!x. x IN s ==> g(f x) = x) ==> g real_continuous_on (IMAGE f s)`, REPEAT GEN_TAC THEN REWRITE_TAC[REAL_CONTINUOUS_ON; real_compact; REAL_OPEN; IS_REALINTERVAL_IS_INTERVAL] THEN DISCH_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_INVERSE_INTO_1D THEN MAP_EVERY EXISTS_TAC [`lift o f o drop`; `IMAGE lift s`] THEN ASM_REWRITE_TAC[GSYM IS_INTERVAL_PATH_CONNECTED_1] THEN ASM_SIMP_TAC[FORALL_IN_IMAGE; o_DEF; LIFT_DROP; GSYM IMAGE_o]);; let REAL_CONTINUOUS_ON_INVERSE_ALT = prove (`!f g s t. f real_continuous_on s /\ (is_realinterval s \/ real_compact s \/ real_open s) /\ IMAGE f s = t /\ (!x. x IN s ==> g(f x) = x) ==> g real_continuous_on t`, MESON_TAC[REAL_CONTINUOUS_ON_INVERSE]);; let INVARIANCE_OF_DOMAIN_HOMEOMORPHISM = prove (`!f:real^M->real^N s. dimindex(:N) <= dimindex(:M) /\ f continuous_on s /\ open s /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> ?g. homeomorphism (s,IMAGE f s) (f,g)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INJECTIVE_ON_LEFT_INVERSE]) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^N->real^M` THEN DISCH_TAC THEN ASM_REWRITE_TAC[homeomorphism] THEN ASM_SIMP_TAC[CONTINUOUS_ON_INVERSE_OPEN] THEN ASM SET_TAC[]);; let INVARIANCE_OF_DOMAIN_HOMEOMORPHIC = prove (`!f:real^M->real^N s. dimindex(:N) <= dimindex(:M) /\ f continuous_on s /\ open s /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> s homeomorphic (IMAGE f s)`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP INVARIANCE_OF_DOMAIN_HOMEOMORPHISM) THEN REWRITE_TAC[homeomorphic] THEN MESON_TAC[]);; let HOMEOMORPHIC_INTERVALS_EQ = prove (`(!a b:real^M c d:real^N. interval[a,b] homeomorphic interval[c,d] <=> aff_dim(interval[a,b]) = aff_dim(interval[c,d])) /\ (!a b:real^M c d:real^N. interval[a,b] homeomorphic interval(c,d) <=> interval[a,b] = {} /\ interval(c,d) = {}) /\ (!a b:real^M c d:real^N. interval(a,b) homeomorphic interval[c,d] <=> interval(a,b) = {} /\ interval[c,d] = {}) /\ (!a b:real^M c d:real^N. interval(a,b) homeomorphic interval(c,d) <=> interval(a,b) = {} /\ interval(c,d) = {} \/ ~(interval(a,b) = {}) /\ ~(interval(c,d) = {}) /\ dimindex(:M) = dimindex(:N))`, SIMP_TAC[HOMEOMORPHIC_CONVEX_COMPACT_SETS_EQ; CONVEX_INTERVAL; COMPACT_INTERVAL] THEN REPEAT STRIP_TAC THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[HOMEOMORPHIC_EMPTY] THENL [FIRST_ASSUM(MP_TAC o MATCH_MP HOMEOMORPHIC_COMPACTNESS) THEN REWRITE_TAC[COMPACT_INTERVAL_EQ] THEN ASM_MESON_TAC[HOMEOMORPHIC_EMPTY]; FIRST_ASSUM(MP_TAC o MATCH_MP HOMEOMORPHIC_COMPACTNESS) THEN REWRITE_TAC[COMPACT_INTERVAL_EQ] THEN ASM_MESON_TAC[HOMEOMORPHIC_EMPTY]; MATCH_MP_TAC(TAUT `(p <=> q) /\ (~p /\ ~q ==> r) ==> p /\ q \/ ~p /\ ~q /\ r`) THEN CONJ_TAC THENL [ASM_MESON_TAC[HOMEOMORPHIC_EMPTY]; STRIP_TAC] THEN REWRITE_TAC[GSYM LE_ANTISYM] THEN CONJ_TAC THEN MATCH_MP_TAC INVARIANCE_OF_DIMENSION THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphic]) THENL [ALL_TAC; GEN_REWRITE_TAC LAND_CONV [SWAP_EXISTS_THM]] THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN REWRITE_TAC[homeomorphism] THEN STRIP_TAC THENL [EXISTS_TAC `interval(a:real^M,b)`; EXISTS_TAC `interval(c:real^N,d)`] THEN ASM_REWRITE_TAC[OPEN_INTERVAL] THEN ASM SET_TAC[]; TRANS_TAC HOMEOMORPHIC_TRANS `IMAGE ((\x. lambda i. x$i):real^M->real^N) (interval(a,b))` THEN CONJ_TAC THENL [MATCH_MP_TAC INVARIANCE_OF_DOMAIN_HOMEOMORPHIC THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[LE_REFL]; MATCH_MP_TAC LINEAR_CONTINUOUS_ON THEN SIMP_TAC[linear; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; LAMBDA_BETA; CART_EQ]; REWRITE_TAC[OPEN_INTERVAL]; SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN ASM_MESON_TAC[]]; ALL_TAC] THEN SUBGOAL_THEN `IMAGE ((\x. lambda i. x$i):real^M->real^N) (interval(a,b)) = interval((lambda i. a$i),(lambda i. b$i))` SUBST1_TAC THENL [MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN SIMP_TAC[IN_INTERVAL; LAMBDA_BETA] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN EXISTS_TAC `(lambda i. (y:real^N)$i):real^M` THEN SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN FIRST_ASSUM(SUBST1_TAC o SYM) THEN SIMP_TAC[CART_EQ; LAMBDA_BETA]; MATCH_MP_TAC HOMEOMORPHIC_OPEN_INTERVALS THEN GEN_REWRITE_TAC I [TAUT `(p <=> q) <=> (~p <=> ~q)`] THEN SIMP_TAC[INTERVAL_NE_EMPTY; LAMBDA_BETA] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INTERVAL_NE_EMPTY])) THEN ASM_MESON_TAC[]]]);; let CONTINUOUS_IMAGE_SUBSET_INTERIOR = prove (`!f:real^M->real^N s. f continuous_on s /\ dimindex(:N) <= dimindex(:M) /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> IMAGE f (interior s) SUBSET interior(IMAGE f s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC INTERIOR_MAXIMAL THEN SIMP_TAC[IMAGE_SUBSET; INTERIOR_SUBSET] THEN ASM_CASES_TAC `interior s:real^M->bool = {}` THENL [ASM_REWRITE_TAC[INTERIOR_EMPTY; OPEN_EMPTY; IMAGE_CLAUSES]; MATCH_MP_TAC INVARIANCE_OF_DOMAIN_GEN] THEN ASM_REWRITE_TAC[OPEN_INTERIOR] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; INTERIOR_SUBSET; SUBSET]);; let HOMEOMORPHIC_INTERIORS_SAME_DIMENSION = prove (`!s:real^M->bool t:real^N->bool. dimindex(:M) = dimindex(:N) /\ s homeomorphic t ==> (interior s) homeomorphic (interior t)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHIC_MINIMAL]) THEN REWRITE_TAC[HOMEOMORPHIC_MINIMAL] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^M->real^N` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^N->real^M` THEN STRIP_TAC THEN ASM_SIMP_TAC[REWRITE_RULE[SUBSET] INTERIOR_SUBSET] THEN REWRITE_TAC[SET_RULE `(!x. x IN s ==> f x IN t) <=> IMAGE f s SUBSET t`] THEN REPEAT CONJ_TAC THENL [SUBGOAL_THEN `t = IMAGE (f:real^M->real^N) s` SUBST1_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_IMAGE_SUBSET_INTERIOR THEN ASM_MESON_TAC[LE_REFL]]; SUBGOAL_THEN `s = IMAGE (g:real^N->real^M) t` SUBST1_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_IMAGE_SUBSET_INTERIOR THEN ASM_MESON_TAC[LE_REFL]]; ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; INTERIOR_SUBSET]; ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; INTERIOR_SUBSET]]);; let HOMEOMORPHIC_INTERIORS = prove (`!s:real^M->bool t:real^N->bool. s homeomorphic t /\ (interior s = {} <=> interior t = {}) ==> (interior s) homeomorphic (interior t)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `interior t:real^N->bool = {}` THEN ASM_SIMP_TAC[HOMEOMORPHIC_EMPTY] THEN STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_INTERIORS_SAME_DIMENSION THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM (STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [HOMEOMORPHIC_MINIMAL]) THEN REWRITE_TAC[GSYM LE_ANTISYM] THEN CONJ_TAC THEN MATCH_MP_TAC INVARIANCE_OF_DIMENSION THENL [MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `interior s:real^M->bool`]; MAP_EVERY EXISTS_TAC [`g:real^N->real^M`; `interior t:real^N->bool`]] THEN ASM_REWRITE_TAC[OPEN_INTERIOR] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; INTERIOR_SUBSET; SUBSET]);; let HOMEOMORPHIC_FRONTIERS_SAME_DIMENSION = prove (`!s:real^M->bool t:real^N->bool. dimindex(:M) = dimindex(:N) /\ s homeomorphic t /\ closed s /\ closed t ==> (frontier s) homeomorphic (frontier t)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHIC_MINIMAL]) THEN REWRITE_TAC[HOMEOMORPHIC_MINIMAL] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^M->real^N` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^N->real^M` THEN ASM_SIMP_TAC[REWRITE_RULE[SUBSET] FRONTIER_SUBSET_CLOSED] THEN STRIP_TAC THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[FRONTIER_SUBSET_CLOSED; CONTINUOUS_ON_SUBSET]] THEN ASM_SIMP_TAC[frontier; CLOSURE_CLOSED] THEN SUBGOAL_THEN `(!x:real^M. x IN interior s ==> f x IN interior t) /\ (!y:real^N. y IN interior t ==> g y IN interior s)` MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[SET_RULE `(!x. x IN s ==> f x IN t) <=> IMAGE f s SUBSET t`] THEN CONJ_TAC THENL [SUBGOAL_THEN `t = IMAGE (f:real^M->real^N) s` SUBST1_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_IMAGE_SUBSET_INTERIOR THEN ASM_MESON_TAC[LE_REFL]]; SUBGOAL_THEN `s = IMAGE (g:real^N->real^M) t` SUBST1_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_IMAGE_SUBSET_INTERIOR THEN ASM_MESON_TAC[LE_REFL]]]);; let HOMEOMORPHIC_FRONTIERS = prove (`!s:real^M->bool t:real^N->bool. s homeomorphic t /\ closed s /\ closed t /\ (interior s = {} <=> interior t = {}) ==> (frontier s) homeomorphic (frontier t)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `interior t:real^N->bool = {}` THENL [ASM_SIMP_TAC[frontier; CLOSURE_CLOSED; DIFF_EMPTY]; STRIP_TAC] THEN MATCH_MP_TAC HOMEOMORPHIC_FRONTIERS_SAME_DIMENSION THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM (STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [HOMEOMORPHIC_MINIMAL]) THEN REWRITE_TAC[GSYM LE_ANTISYM] THEN CONJ_TAC THEN MATCH_MP_TAC INVARIANCE_OF_DIMENSION THENL [MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `interior s:real^M->bool`]; MAP_EVERY EXISTS_TAC [`g:real^N->real^M`; `interior t:real^N->bool`]] THEN ASM_REWRITE_TAC[OPEN_INTERIOR] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; INTERIOR_SUBSET; SUBSET]);; let CONTINUOUS_IMAGE_SUBSET_RELATIVE_INTERIOR = prove (`!f:real^M->real^N s t. f continuous_on s /\ IMAGE f s SUBSET t /\ aff_dim t <= aff_dim s /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> IMAGE f (relative_interior s) SUBSET relative_interior(IMAGE f s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC RELATIVE_INTERIOR_MAXIMAL THEN SIMP_TAC[IMAGE_SUBSET; RELATIVE_INTERIOR_SUBSET] THEN MATCH_MP_TAC INVARIANCE_OF_DOMAIN_AFFINE_SETS THEN EXISTS_TAC `affine hull s:real^M->bool` THEN ASM_REWRITE_TAC[AFFINE_AFFINE_HULL; AFF_DIM_AFFINE_HULL] THEN REWRITE_TAC[OPEN_IN_RELATIVE_INTERIOR] THEN CONJ_TAC THENL [ASM_MESON_TAC[AFF_DIM_SUBSET; INT_LE_TRANS]; ALL_TAC] THEN ASSUME_TAC(ISPEC `s:real^M->bool` RELATIVE_INTERIOR_SUBSET) THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `IMAGE (f:real^M->real^N) s` THEN SIMP_TAC[IMAGE_SUBSET; RELATIVE_INTERIOR_SUBSET; HULL_SUBSET]);; let HOMEOMORPHIC_RELATIVE_INTERIORS_SAME_DIMENSION = prove (`!s:real^M->bool t:real^N->bool. aff_dim s = aff_dim t /\ s homeomorphic t ==> (relative_interior s) homeomorphic (relative_interior t)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHIC_MINIMAL]) THEN REWRITE_TAC[HOMEOMORPHIC_MINIMAL] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^M->real^N` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^N->real^M` THEN STRIP_TAC THEN ASM_SIMP_TAC[REWRITE_RULE[SUBSET] RELATIVE_INTERIOR_SUBSET] THEN REWRITE_TAC[SET_RULE `(!x. x IN s ==> f x IN t) <=> IMAGE f s SUBSET t`] THEN REPEAT CONJ_TAC THENL [SUBGOAL_THEN `t = IMAGE (f:real^M->real^N) s` SUBST1_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_IMAGE_SUBSET_RELATIVE_INTERIOR THEN EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[INT_LE_REFL] THEN ASM SET_TAC[]]; SUBGOAL_THEN `s = IMAGE (g:real^N->real^M) t` SUBST1_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_IMAGE_SUBSET_RELATIVE_INTERIOR THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[INT_LE_REFL] THEN ASM SET_TAC[]]; ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; RELATIVE_INTERIOR_SUBSET]; ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; RELATIVE_INTERIOR_SUBSET]]);; let HOMEOMORPHIC_RELATIVE_INTERIORS = prove (`!s:real^M->bool t:real^N->bool. s homeomorphic t /\ (relative_interior s = {} <=> relative_interior t = {}) ==> (relative_interior s) homeomorphic (relative_interior t)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `relative_interior t:real^N->bool = {}` THEN ASM_SIMP_TAC[HOMEOMORPHIC_EMPTY] THEN STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_RELATIVE_INTERIORS_SAME_DIMENSION THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM (STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [HOMEOMORPHIC_MINIMAL]) THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN REWRITE_TAC[GSYM INT_LE_ANTISYM] THEN CONJ_TAC THEN MATCH_MP_TAC INVARIANCE_OF_DIMENSION_AFFINE_SETS THENL [MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `relative_interior s:real^M->bool`]; MAP_EVERY EXISTS_TAC [`g:real^N->real^M`; `relative_interior t:real^N->bool`]] THEN ASM_REWRITE_TAC[OPEN_IN_RELATIVE_INTERIOR; AFFINE_AFFINE_HULL] THEN (REPEAT CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; RELATIVE_INTERIOR_SUBSET]; ASM_MESON_TAC[RELATIVE_INTERIOR_SUBSET; HULL_SUBSET; SET_RULE `(!x. x IN s ==> f x IN t) /\ s' SUBSET s /\ t SUBSET t' ==> IMAGE f s' SUBSET t'`]; ASM_MESON_TAC[RELATIVE_INTERIOR_SUBSET; SUBSET]]));; let HOMEOMORPHIC_RELATIVE_BOUNDARIES_SAME_DIMENSION = prove (`!s:real^M->bool t:real^N->bool. aff_dim s = aff_dim t /\ s homeomorphic t ==> (s DIFF relative_interior s) homeomorphic (t DIFF relative_interior t)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHIC_MINIMAL]) THEN REWRITE_TAC[HOMEOMORPHIC_MINIMAL] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^M->real^N` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^N->real^M` THEN STRIP_TAC THEN ASM_SIMP_TAC[IN_DIFF] THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[SUBSET_DIFF; CONTINUOUS_ON_SUBSET]] THEN SUBGOAL_THEN `(!x:real^M. x IN relative_interior s ==> f x IN relative_interior t) /\ (!y:real^N. y IN relative_interior t ==> g y IN relative_interior s)` MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[SET_RULE `(!x. x IN s ==> f x IN t) <=> IMAGE f s SUBSET t`] THEN CONJ_TAC THENL [SUBGOAL_THEN `t = IMAGE (f:real^M->real^N) s` SUBST1_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_IMAGE_SUBSET_RELATIVE_INTERIOR THEN EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[INT_LE_REFL] THEN ASM SET_TAC[]]; SUBGOAL_THEN `s = IMAGE (g:real^N->real^M) t` SUBST1_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_IMAGE_SUBSET_RELATIVE_INTERIOR THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[INT_LE_REFL] THEN ASM SET_TAC[]]]);; let HOMEOMORPHIC_RELATIVE_BOUNDARIES = prove (`!s:real^M->bool t:real^N->bool. s homeomorphic t /\ (relative_interior s = {} <=> relative_interior t = {}) ==> (s DIFF relative_interior s) homeomorphic (t DIFF relative_interior t)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `relative_interior t:real^N->bool = {}` THEN ASM_SIMP_TAC[DIFF_EMPTY] THEN STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_RELATIVE_BOUNDARIES_SAME_DIMENSION THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM (STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [HOMEOMORPHIC_MINIMAL]) THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN REWRITE_TAC[GSYM INT_LE_ANTISYM] THEN CONJ_TAC THEN MATCH_MP_TAC INVARIANCE_OF_DIMENSION_AFFINE_SETS THENL [MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `relative_interior s:real^M->bool`]; MAP_EVERY EXISTS_TAC [`g:real^N->real^M`; `relative_interior t:real^N->bool`]] THEN ASM_REWRITE_TAC[OPEN_IN_RELATIVE_INTERIOR; AFFINE_AFFINE_HULL] THEN (REPEAT CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; RELATIVE_INTERIOR_SUBSET]; ASM_MESON_TAC[RELATIVE_INTERIOR_SUBSET; HULL_SUBSET; SET_RULE `(!x. x IN s ==> f x IN t) /\ s' SUBSET s /\ t SUBSET t' ==> IMAGE f s' SUBSET t'`]; ASM_MESON_TAC[RELATIVE_INTERIOR_SUBSET; SUBSET]]));; let UNIFORMLY_CONTINUOUS_HOMEOMORPHISM_UNIV_TRIVIAL = prove (`!f g s:real^N->bool. homeomorphism (s,(:real^N)) (f,g) /\ f uniformly_continuous_on s ==> s = (:real^N)`, REPEAT GEN_TAC THEN REWRITE_TAC[homeomorphism; IN_UNIV] THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY] THENL [SET_TAC[]; STRIP_TAC] THEN MP_TAC(ISPEC `s:real^N->bool` CLOPEN) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN CONJ_TAC THENL [REWRITE_TAC[GSYM COMPLETE_EQ_CLOSED; complete] THEN X_GEN_TAC `x:num->real^N` THEN STRIP_TAC THEN SUBGOAL_THEN `cauchy ((f:real^N->real^N) o x)` MP_TAC THENL [ASM_MESON_TAC[UNIFORMLY_CONTINUOUS_IMP_CAUCHY_CONTINUOUS]; ALL_TAC] THEN REWRITE_TAC[GSYM CONVERGENT_EQ_CAUCHY] THEN DISCH_THEN(X_CHOOSE_TAC `l:real^N`) THEN EXISTS_TAC `(g:real^N->real^N) l` THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC LIM_TRANSFORM_EVENTUALLY THEN EXISTS_TAC `(g:real^N->real^N) o (f:real^N->real^N) o (x:num->real^N)` THEN REWRITE_TAC[o_DEF] THEN CONJ_TAC THENL [MATCH_MP_TAC ALWAYS_EVENTUALLY THEN ASM SET_TAC[]; MATCH_MP_TAC LIM_CONTINUOUS_FUNCTION THEN ASM_SIMP_TAC[GSYM o_DEF] THEN ASM_MESON_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_AT; OPEN_UNIV; IN_UNIV]]; FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [SYM th]) THEN MATCH_MP_TAC INVARIANCE_OF_DOMAIN THEN ASM_REWRITE_TAC[OPEN_UNIV] THEN ASM SET_TAC[]]);; let INVARIANCE_OF_DOMAIN_SPHERE_AFFINE_SET_GEN = prove (`!f:real^M->real^N u s t. f continuous_on s /\ IMAGE f s SUBSET t /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\ bounded u /\ convex u /\ affine t /\ aff_dim t < aff_dim u /\ open_in (subtopology euclidean (relative_frontier u)) s ==> open_in (subtopology euclidean t) (IMAGE f s)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `relative_frontier u:real^M->bool = {}` THEN ASM_SIMP_TAC[OPEN_IN_SUBTOPOLOGY_EMPTY; IMAGE_CLAUSES; OPEN_IN_EMPTY] THEN STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN SUBGOAL_THEN `?b c:real^M. b IN relative_frontier u /\ c IN relative_frontier u /\ ~(b = c)` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC(SET_RULE `~(s = {} \/ ?x. s = {x}) ==> ?a b. a IN s /\ b IN s /\ ~(a = b)`) THEN ASM_MESON_TAC[RELATIVE_FRONTIER_NOT_SING]; ALL_TAC] THEN MP_TAC(ISPECL [`(:real^M)`; `aff_dim(u:real^M->bool) - &1`] CHOOSE_AFFINE_SUBSET) THEN REWRITE_TAC[SUBSET_UNIV; AFFINE_UNIV] THEN ANTS_TAC THENL [MATCH_MP_TAC(INT_ARITH `&0:int <= t /\ t <= n ==> --a <= t - a /\ t - &1 <= n`) THEN REWRITE_TAC[AFF_DIM_LE_UNIV; AFF_DIM_UNIV; AFF_DIM_POS_LE] THEN ASM_MESON_TAC[RELATIVE_FRONTIER_EMPTY; NOT_IN_EMPTY]; DISCH_THEN(X_CHOOSE_THEN `af:real^M->bool` STRIP_ASSUME_TAC)] THEN MP_TAC(ISPECL [`u:real^M->bool`; `af:real^M->bool`] HOMEOMORPHIC_PUNCTURED_SPHERE_AFFINE_GEN) THEN ASM_REWRITE_TAC[INT_ARITH `x - a + a:int = x`] THEN DISCH_THEN(fun th -> MP_TAC(SPEC `c:real^M` th) THEN MP_TAC(SPEC `b:real^M` th)) THEN ASM_REWRITE_TAC[homeomorphic; homeomorphism; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`g:real^M->real^M`; `h:real^M->real^M`] THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`j:real^M->real^M`; `k:real^M->real^M`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`(f:real^M->real^N) o (k:real^M->real^M)`; `(af:real^M->bool)`; `t:real^N->bool`; `IMAGE (j:real^M->real^M) (s DELETE c)`] INVARIANCE_OF_DOMAIN_AFFINE_SETS) THEN MP_TAC(ISPECL [`(f:real^M->real^N) o (h:real^M->real^M)`; `(af:real^M->bool)`; `t:real^N->bool`; `IMAGE (g:real^M->real^M) (s DELETE b)`] INVARIANCE_OF_DOMAIN_AFFINE_SETS) THEN ASM_REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN ASM_REWRITE_TAC[IMP_IMP; INT_ARITH `x:int <= y - &1 <=> x < y`] THEN MATCH_MP_TAC(TAUT `(p1 /\ p2) /\ (q1 /\ q2 ==> r) ==> (p1 ==> q1) /\ (p2 ==> q2) ==> r`) THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; REWRITE_TAC[IMAGE_o] THEN ASM SET_TAC[]; RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; IN_DELETE]) THEN ASM_SIMP_TAC[o_THM; IN_DELETE; IMP_CONJ] THEN ASM_MESON_TAC[]; MATCH_MP_TAC HOMEOMORPHISM_IMP_OPEN_MAP THEN MAP_EVERY EXISTS_TAC [`h:real^M->real^M`; `relative_frontier u DELETE (b:real^M)`] THEN ASM_SIMP_TAC[homeomorphism; DOT_BASIS; DIMINDEX_GE_1; LE_REFL] THEN ASM_REWRITE_TAC[IN_ELIM_THM; OPEN_IN_OPEN] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN MATCH_MP_TAC MONO_EXISTS THEN SET_TAC[]; MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; REWRITE_TAC[IMAGE_o] THEN ASM SET_TAC[]; RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; IN_DELETE]) THEN ASM_SIMP_TAC[o_THM; IN_DELETE; IMP_CONJ] THEN ASM_MESON_TAC[]; MATCH_MP_TAC HOMEOMORPHISM_IMP_OPEN_MAP THEN MAP_EVERY EXISTS_TAC [`k:real^M->real^M`; `relative_frontier u DELETE (c:real^M)`] THEN ASM_SIMP_TAC[homeomorphism; DOT_BASIS; DIMINDEX_GE_1; LE_REFL] THEN ASM_REWRITE_TAC[IN_ELIM_THM; OPEN_IN_OPEN] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN MATCH_MP_TAC MONO_EXISTS THEN SET_TAC[]; DISCH_THEN(MP_TAC o MATCH_MP OPEN_IN_UNION) THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `IMAGE (f:real^M->real^N) ((s DELETE b) UNION (s DELETE c))` THEN CONJ_TAC THENL [REWRITE_TAC[IMAGE_UNION] THEN BINOP_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[IMAGE_o] THEN AP_TERM_TAC THEN ASM SET_TAC[]]);; let INVARIANCE_OF_DOMAIN_SPHERE_AFFINE_SET = prove (`!f:real^M->real^N a r s t. f continuous_on s /\ IMAGE f s SUBSET t /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\ ~(r = &0) /\ affine t /\ aff_dim t < &(dimindex(:M)) /\ open_in (subtopology euclidean (sphere(a,r))) s ==> open_in (subtopology euclidean t) (IMAGE f s)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `sphere(a:real^M,r) = {}` THEN ASM_SIMP_TAC[OPEN_IN_SUBTOPOLOGY_EMPTY; OPEN_IN_EMPTY; IMAGE_CLAUSES] THEN RULE_ASSUM_TAC(REWRITE_RULE[SPHERE_EQ_EMPTY; REAL_NOT_LT]) THEN STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `cball(a:real^M,r)`; `s:real^M->bool`; `t:real^N->bool`] INVARIANCE_OF_DOMAIN_SPHERE_AFFINE_SET_GEN) THEN ASM_REWRITE_TAC[AFF_DIM_CBALL; RELATIVE_FRONTIER_CBALL; BOUNDED_CBALL; CONVEX_CBALL] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC);; let NO_EMBEDDING_SPHERE_LOWDIM = prove (`!f:real^M->real^N a r. &0 < r /\ f continuous_on sphere(a,r) /\ (!x y. x IN sphere(a,r) /\ y IN sphere(a,r) /\ f x = f y ==> x = y) ==> dimindex(:M) <= dimindex(:N)`, REWRITE_TAC[GSYM NOT_LT] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPEC `IMAGE (f:real^M->real^N) (sphere(a:real^M,r))` COMPACT_OPEN) THEN ASM_SIMP_TAC[COMPACT_CONTINUOUS_IMAGE; IMAGE_EQ_EMPTY; COMPACT_SPHERE; SPHERE_EQ_EMPTY; REAL_ARITH `&0 < r ==> ~(r < &0)`] THEN ONCE_REWRITE_TAC[OPEN_IN] THEN ONCE_REWRITE_TAC[GSYM SUBTOPOLOGY_UNIV] THEN MATCH_MP_TAC INVARIANCE_OF_DOMAIN_SPHERE_AFFINE_SET THEN MAP_EVERY EXISTS_TAC [`a:real^M`; `r:real`] THEN ASM_REWRITE_TAC[AFFINE_UNIV; SUBSET_UNIV; AFF_DIM_UNIV; OPEN_IN_REFL; INT_OF_NUM_LT] THEN ASM_REAL_ARITH_TAC);; let EMPTY_INTERIOR_LOWDIM_GEN = prove (`!s:real^N->bool t:real^M->bool. dimindex(:M) < dimindex(:N) /\ s homeomorphic t ==> interior s = {}`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`(:real^M)`; `(:real^N)`] ISOMETRY_SUBSET_SUBSPACE) THEN ASM_SIMP_TAC[SUBSPACE_UNIV; DIM_UNIV; LT_IMP_LE; IN_UNIV; SUBSET_UNIV] THEN DISCH_THEN(X_CHOOSE_THEN `h:real^M->real^N` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC(MESON[HOMEOMORPHIC_EMPTY] `!t. interior(t:real^N->bool) homeomorphic interior(s:real^N->bool) /\ interior t = {} ==> interior s = {}`) THEN EXISTS_TAC `IMAGE (h:real^M->real^N) t` THEN CONJ_TAC THENL [MATCH_MP_TAC HOMEOMORPHIC_INTERIORS_SAME_DIMENSION THEN REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHIC_SYM]) THEN MATCH_MP_TAC EQ_IMP THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_LEFT_EQ THEN ASM_MESON_TAC[PRESERVES_NORM_INJECTIVE]; MATCH_MP_TAC(SET_RULE `!t. s SUBSET t /\ t = {} ==> s = {}`) THEN EXISTS_TAC `interior(IMAGE (h:real^M->real^N) (:real^M))` THEN SIMP_TAC[SUBSET_INTERIOR; SET_RULE `IMAGE f s SUBSET IMAGE f UNIV`] THEN MATCH_MP_TAC EMPTY_INTERIOR_LOWDIM THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] LET_TRANS)) THEN REWRITE_TAC[GSYM DIM_UNIV] THEN MATCH_MP_TAC EQ_IMP_LE THEN MATCH_MP_TAC DIM_INJECTIVE_LINEAR_IMAGE THEN ASM_MESON_TAC[PRESERVES_NORM_INJECTIVE]]);; let EMPTY_INTERIOR_LOWDIM_GEN_LE = prove (`!s:real^N->bool t:real^M->bool. dimindex(:M) <= dimindex(:N) /\ interior t = {} /\ s homeomorphic t ==> interior s = {}`, REWRITE_TAC[LE_LT] THEN REPEAT STRIP_TAC THENL [ASM_MESON_TAC[EMPTY_INTERIOR_LOWDIM_GEN]; ASM_MESON_TAC[HOMEOMORPHIC_INTERIORS_SAME_DIMENSION; HOMEOMORPHIC_EMPTY]]);; let INTERIOR_ARC_IMAGE = prove (`!g:real^1->real^N. 2 <= dimindex(:N) /\ arc g ==> interior(path_image g) = {}`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`path_image(g:real^1->real^N)`; `interval[vec 0:real^1,vec 1]`] EMPTY_INTERIOR_LOWDIM_GEN) THEN DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[DIMINDEX_1] THEN CONJ_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC HOMEOMORPHIC_ARC_IMAGE_INTERVAL THEN ASM_REWRITE_TAC[DROP_VEC; REAL_LT_01]);; let INTERIOR_SIMPLE_PATH_IMAGE = prove (`!g:real^1->real^N. 2 <= dimindex(:N) /\ simple_path g ==> interior(path_image g) = {}`, REPEAT STRIP_TAC THEN FIRST_ASSUM(DISJ_CASES_TAC o MATCH_MP SIMPLE_PATH_CASES) THEN ASM_SIMP_TAC[INTERIOR_ARC_IMAGE] THEN MP_TAC(ISPECL [`path_image(g:real^1->real^N)`; `sphere(vec 0:real^2,&1)`] EMPTY_INTERIOR_LOWDIM_GEN_LE) THEN ASM_REWRITE_TAC[DIMINDEX_2; INTERIOR_SPHERE] THEN DISCH_THEN MATCH_MP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_SIMPLE_PATH_IMAGE_CIRCLE THEN ASM_REWRITE_TAC[REAL_LT_01]);; (* ------------------------------------------------------------------------- *) (* Dimension-based conditions for various homeomorphisms. *) (* ------------------------------------------------------------------------- *) let HOMEOMORPHIC_SUBSPACES_EQ = prove (`!s:real^M->bool t:real^N->bool. subspace s /\ subspace t ==> (s homeomorphic t <=> dim s = dim t)`, REPEAT STRIP_TAC THEN EQ_TAC THENL [ALL_TAC; ASM_MESON_TAC[HOMEOMORPHIC_SUBSPACES]] THEN REWRITE_TAC[homeomorphic; HOMEOMORPHISM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f:real^M->real^N`; `g:real^N->real^M`] THEN STRIP_TAC THEN REWRITE_TAC[GSYM LE_ANTISYM] THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_INJECTIVE_IMAGE_SUBSPACE_DIM_LE THEN ASM_MESON_TAC[]);; let HOMEOMORPHIC_AFFINE_SETS_EQ = prove (`!s:real^M->bool t:real^N->bool. affine s /\ affine t ==> (s homeomorphic t <=> aff_dim s = aff_dim t)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_REWRITE_TAC[AFF_DIM_EMPTY; AFF_DIM_EQ_MINUS1; HOMEOMORPHIC_EMPTY] THEN POP_ASSUM MP_TAC THEN GEN_REWRITE_TAC (funpow 3 RAND_CONV) [EQ_SYM_EQ] THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_EQ_MINUS1; HOMEOMORPHIC_EMPTY] THEN POP_ASSUM MP_TAC THEN REWRITE_TAC [GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM; RIGHT_IMP_FORALL_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^N`] THEN GEOM_ORIGIN_TAC `a:real^M` THEN GEOM_ORIGIN_TAC `b:real^N` THEN SIMP_TAC[AFFINE_EQ_SUBSPACE; HOMEOMORPHIC_SUBSPACES_EQ; AFF_DIM_DIM_0; HULL_INC; INT_OF_NUM_EQ] THEN MESON_TAC[]);; let HOMEOMORPHIC_HYPERPLANES_EQ = prove (`!a:real^M b c:real^N d. ~(a = vec 0) /\ ~(c = vec 0) ==> ({x | a dot x = b} homeomorphic {x | c dot x = d} <=> dimindex(:M) = dimindex(:N))`, SIMP_TAC[HOMEOMORPHIC_AFFINE_SETS_EQ; AFFINE_HYPERPLANE] THEN SIMP_TAC[AFF_DIM_HYPERPLANE; INT_OF_NUM_EQ; INT_ARITH `x - &1:int = y - &1 <=> x = y`]);; let HOMEOMORPHIC_UNIV_UNIV = prove (`(:real^M) homeomorphic (:real^N) <=> dimindex(:M) = dimindex(:N)`, SIMP_TAC[HOMEOMORPHIC_SUBSPACES_EQ; DIM_UNIV; SUBSPACE_UNIV]);; let HOMEOMORPHIC_CBALLS_EQ = prove (`!a:real^M b:real^N r s. cball(a,r) homeomorphic cball(b,s) <=> r < &0 /\ s < &0 \/ r = &0 /\ s = &0 \/ &0 < r /\ &0 < s /\ dimindex(:M) = dimindex(:N)`, let lemma = let d = `dimindex(:M) = dimindex(:N)` and t = `?a:real^M b:real^N. ~(cball(a,r) homeomorphic cball(b,s))` in DISCH d (DISCH t (GEOM_EQUAL_DIMENSION_RULE (ASSUME d) (ASSUME t))) in REPEAT GEN_TAC THEN ASM_CASES_TAC `r < &0` THENL [ASM_SIMP_TAC[CBALL_EMPTY; HOMEOMORPHIC_EMPTY; CBALL_EQ_EMPTY] THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_CASES_TAC `r = &0` THEN ASM_REWRITE_TAC[REAL_LT_REFL] THENL [ASM_SIMP_TAC[CBALL_TRIVIAL; FINITE_SING; HOMEOMORPHIC_FINITE_STRONG] THEN REWRITE_TAC[FINITE_CBALL] THEN ASM_CASES_TAC `s < &0` THEN ASM_SIMP_TAC[CBALL_EMPTY; CARD_CLAUSES; FINITE_EMPTY; NOT_IN_EMPTY; ARITH; REAL_LT_IMP_NE] THEN ASM_CASES_TAC `s = &0` THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN ASM_SIMP_TAC[CBALL_TRIVIAL; CARD_CLAUSES; FINITE_EMPTY; NOT_IN_EMPTY; REAL_LE_REFL; ARITH]; ALL_TAC] THEN SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_CASES_TAC `s <= &0` THEN ASM_SIMP_TAC[HOMEOMORPHIC_FINITE_STRONG; FINITE_CBALL] THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `&0 < s` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_REWRITE_TAC[] THEN EQ_TAC THENL [REWRITE_TAC[homeomorphic; HOMEOMORPHISM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f:real^M->real^N`; `g:real^N->real^M`] THEN STRIP_TAC THEN REWRITE_TAC[GSYM LE_ANTISYM] THEN CONJ_TAC THEN MATCH_MP_TAC INVARIANCE_OF_DIMENSION THENL [MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `ball(a:real^M,r)`] THEN MP_TAC(ISPECL [`a:real^M`; `r:real`] BALL_SUBSET_CBALL); MAP_EVERY EXISTS_TAC [`g:real^N->real^M`; `ball(b:real^N,s)`] THEN MP_TAC(ISPECL [`b:real^N`; `s:real`] BALL_SUBSET_CBALL)] THEN ASM_REWRITE_TAC[BALL_EQ_EMPTY; OPEN_BALL; REAL_NOT_LE] THEN ASM_MESON_TAC[SUBSET; CONTINUOUS_ON_SUBSET]; DISCH_THEN(MP_TAC o MATCH_MP lemma) THEN GEN_REWRITE_TAC LAND_CONV [GSYM CONTRAPOS_THM] THEN REWRITE_TAC[NOT_EXISTS_THM] THEN ASM_SIMP_TAC[HOMEOMORPHIC_CBALLS]]);; let HOMEOMORPHIC_BALLS_EQ = prove (`!a:real^M b:real^N r s. ball(a,r) homeomorphic ball(b,s) <=> r <= &0 /\ s <= &0 \/ &0 < r /\ &0 < s /\ dimindex(:M) = dimindex(:N)`, let lemma = let d = `dimindex(:M) = dimindex(:N)` and t = `?a:real^M b:real^N. ~(ball(a,r) homeomorphic ball(b,s))` in DISCH d (DISCH t (GEOM_EQUAL_DIMENSION_RULE (ASSUME d) (ASSUME t))) in REPEAT GEN_TAC THEN ASM_CASES_TAC `r <= &0` THENL [ASM_SIMP_TAC[BALL_EMPTY; HOMEOMORPHIC_EMPTY; BALL_EQ_EMPTY] THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_CASES_TAC `s <= &0` THENL [ASM_SIMP_TAC[BALL_EMPTY; HOMEOMORPHIC_EMPTY; BALL_EQ_EMPTY] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LE]) THEN ASM_REWRITE_TAC[] THEN EQ_TAC THENL [REWRITE_TAC[homeomorphic; HOMEOMORPHISM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f:real^M->real^N`; `g:real^N->real^M`] THEN STRIP_TAC THEN REWRITE_TAC[GSYM LE_ANTISYM] THEN CONJ_TAC THEN MATCH_MP_TAC INVARIANCE_OF_DIMENSION THENL [MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `ball(a:real^M,r)`]; MAP_EVERY EXISTS_TAC [`g:real^N->real^M`; `ball(b:real^N,s)`]] THEN ASM_REWRITE_TAC[BALL_EQ_EMPTY; OPEN_BALL; REAL_NOT_LE] THEN ASM SET_TAC[]; DISCH_THEN(MP_TAC o MATCH_MP lemma) THEN GEN_REWRITE_TAC LAND_CONV [GSYM CONTRAPOS_THM] THEN REWRITE_TAC[NOT_EXISTS_THM] THEN ASM_SIMP_TAC[HOMEOMORPHIC_BALLS]]);; let SIMPLY_CONNECTED_SPHERE_EQ = prove (`!a:real^N r. simply_connected(sphere(a,r)) <=> 3 <= dimindex(:N) \/ r <= &0`, let hslemma = prove (`!a:real^M r b:real^N s. dimindex(:M) = dimindex(:N) ==> &0 < r /\ &0 < s ==> (sphere(a,r) homeomorphic sphere(b,s))`, REPEAT STRIP_TAC THEN FIRST_ASSUM(fun th -> let t = `?a:real^M b:real^N. ~(sphere(a,r) homeomorphic sphere(b,s))` in MP_TAC(DISCH t (GEOM_EQUAL_DIMENSION_RULE th (ASSUME t)))) THEN ASM_SIMP_TAC[HOMEOMORPHIC_SPHERES] THEN MESON_TAC[]) in REPEAT GEN_TAC THEN ASM_CASES_TAC `r < &0` THEN ASM_SIMP_TAC[SPHERE_EMPTY; REAL_LT_IMP_LE; SIMPLY_CONNECTED_EMPTY] THEN ASM_CASES_TAC `r = &0` THEN ASM_SIMP_TAC[SPHERE_SING; REAL_LE_REFL; CONVEX_IMP_SIMPLY_CONNECTED; CONVEX_SING] THEN ASM_REWRITE_TAC[REAL_LE_LT] THEN EQ_TAC THEN REWRITE_TAC[SIMPLY_CONNECTED_SPHERE] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[ARITH_RULE `~(3 <= n) <=> (1 <= n ==> n = 1 \/ n = 2)`] THEN REWRITE_TAC[DIMINDEX_GE_1] THEN STRIP_TAC THENL [DISCH_THEN(MP_TAC o MATCH_MP SIMPLY_CONNECTED_IMP_CONNECTED) THEN ASM_REWRITE_TAC[CONNECTED_SPHERE_EQ; ARITH] THEN ASM_REAL_ARITH_TAC; RULE_ASSUM_TAC(REWRITE_RULE[GSYM DIMINDEX_2]) THEN FIRST_ASSUM(MP_TAC o ISPECL [`a:real^N`; `r:real`; `vec 0:real^2`; `&1:real`] o MATCH_MP hslemma) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(SUBST1_TAC o MATCH_MP HOMEOMORPHIC_SIMPLY_CONNECTED_EQ) THEN REWRITE_TAC[SIMPLY_CONNECTED_EQ_CONTRACTIBLE_CIRCLEMAP] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `\x:real^2. x`) THEN REWRITE_TAC[CONTINUOUS_ON_ID; IMAGE_ID; SUBSET_REFL] THEN REWRITE_TAC[GSYM contractible; CONTRACTIBLE_SPHERE] THEN CONV_TAC REAL_RAT_REDUCE_CONV]);; let SIMPLY_CONNECTED_PUNCTURED_UNIVERSE_EQ = prove (`!a. simply_connected((:real^N) DELETE a) <=> 3 <= dimindex(:N)`, GEN_TAC THEN TRANS_TAC EQ_TRANS `simply_connected(sphere(a:real^N,&1))` THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[SIMPLY_CONNECTED_SPHERE_EQ]] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN MATCH_MP_TAC HOMOTOPY_EQUIVALENT_SIMPLE_CONNECTEDNESS THEN MP_TAC(ISPECL [`cball(a:real^N,&1)`; `a:real^N`] HOMOTOPY_EQUIVALENT_RELATIVE_FRONTIER_PUNCTURED_AFFINE_HULL) THEN REWRITE_TAC[CONVEX_CBALL; BOUNDED_CBALL; RELATIVE_INTERIOR_CBALL; RELATIVE_FRONTIER_CBALL] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN SIMP_TAC[CENTRE_IN_BALL; AFFINE_HULL_NONEMPTY_INTERIOR; INTERIOR_CBALL; BALL_EQ_EMPTY; REAL_OF_NUM_LE; ARITH; REAL_LT_01]);; let NOT_SIMPLY_CONNECTED_CIRCLE = prove (`!a:real^2 r. &0 < r ==> ~simply_connected(sphere(a,r))`, REWRITE_TAC[SIMPLY_CONNECTED_SPHERE_EQ; DIMINDEX_2; ARITH] THEN REAL_ARITH_TAC);; (* ------------------------------------------------------------------------- *) (* The exponential function as a covering map. *) (* ------------------------------------------------------------------------- *) let COVERING_SPACE_CEXP_PUNCTURED_PLANE = prove (`covering_space((:complex),cexp) ((:complex) DIFF {Cx(&0)})`, SIMP_TAC[covering_space; IN_UNIV; CONTINUOUS_ON_CEXP; IN_DIFF; IN_SING] THEN CONJ_TAC THENL [SET_TAC[CEXP_CLOG; CEXP_NZ]; ALL_TAC] THEN SIMP_TAC[OPEN_IN_OPEN_EQ; OPEN_DIFF; OPEN_UNIV; CLOSED_SING] THEN SIMP_TAC[SUBSET_UNIV; SET_RULE `s SUBSET UNIV DIFF {a} <=> ~(a IN s)`] THEN X_GEN_TAC `z:complex` THEN DISCH_TAC THEN EXISTS_TAC `IMAGE cexp (ball(clog z,&1))` THEN REWRITE_TAC[SET_RULE `~(z IN IMAGE f s) <=> !x. x IN s ==> ~(f x = z)`] THEN REWRITE_TAC[CEXP_NZ] THEN CONJ_TAC THENL [REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `clog z` THEN ASM_SIMP_TAC[CEXP_CLOG; CENTRE_IN_BALL; REAL_LT_01]; ALL_TAC] THEN SUBGOAL_THEN `!x y. x IN cball(clog z,&1) /\ y IN cball(clog z,&1) /\ cexp x = cexp y ==> x = y` ASSUME_TAC THENL [REWRITE_TAC[IN_CBALL] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC COMPLEX_EQ_CEXP THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `norm(x - y:complex)` THEN REWRITE_TAC[GSYM IM_SUB; COMPLEX_NORM_GE_RE_IM] THEN MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `&2` THEN CONJ_TAC THENL [REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC NORM_ARITH; MP_TAC PI_APPROX_32 THEN REAL_ARITH_TAC]; ALL_TAC] THEN CONJ_TAC THENL [MATCH_MP_TAC INVARIANCE_OF_DOMAIN THEN REWRITE_TAC[OPEN_BALL; CONTINUOUS_ON_CEXP] THEN ASM_MESON_TAC[SUBSET; BALL_SUBSET_CBALL]; ALL_TAC] THEN MP_TAC(ISPECL [`cball(clog z,&1)`; `cexp`; `IMAGE cexp (cball(clog z,&1))`] HOMEOMORPHISM_COMPACT) THEN ASM_REWRITE_TAC[COMPACT_CBALL; CONTINUOUS_ON_CEXP] THEN REWRITE_TAC[homeomorphism; LEFT_IMP_EXISTS_THM; FORALL_IN_IMAGE] THEN X_GEN_TAC `l:complex->complex` THEN STRIP_TAC THEN EXISTS_TAC `{ IMAGE (\x. x + Cx (&2 * n * pi) * ii) (ball(clog z,&1)) | integer n}` THEN SIMP_TAC[FORALL_IN_GSPEC; OPEN_BALL; ONCE_REWRITE_RULE[VECTOR_ADD_SYM] OPEN_TRANSLATION] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[UNIONS_GSPEC; IN_IMAGE; CEXP_EQ] THEN SET_TAC[]; REWRITE_TAC[pairwise; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `m:real` THEN DISCH_TAC THEN X_GEN_TAC `n:real` THEN DISCH_TAC THEN ASM_CASES_TAC `m:real = n` THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN REWRITE_TAC[IN_BALL; dist; SET_RULE `DISJOINT (IMAGE f s) (IMAGE g s) <=> !x y. x IN s /\ y IN s ==> ~(f x = g y)`] THEN REPEAT GEN_TAC THEN MATCH_MP_TAC(NORM_ARITH `&2 <= norm(m - n) ==> norm(c - x) < &1 /\ norm(c - y) < &1 ==> ~(x + m = y + n)`) THEN REWRITE_TAC[GSYM COMPLEX_SUB_RDISTRIB; COMPLEX_NORM_MUL] THEN REWRITE_TAC[COMPLEX_NORM_II; GSYM CX_SUB; COMPLEX_NORM_CX] THEN REWRITE_TAC[GSYM REAL_SUB_LDISTRIB; GSYM REAL_SUB_RDISTRIB] THEN REWRITE_TAC[REAL_ABS_MUL; REAL_ABS_NUM; REAL_ABS_PI; REAL_MUL_RID] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `&2 * &1 * pi` THEN CONJ_TAC THENL [MP_TAC PI_APPROX_32 THEN REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC REAL_LE_LMUL THEN SIMP_TAC[REAL_LE_RMUL_EQ; PI_POS; REAL_POS] THEN MATCH_MP_TAC REAL_ABS_INTEGER_LEMMA THEN ASM_SIMP_TAC[REAL_SUB_0; INTEGER_CLOSED]; X_GEN_TAC `n:real` THEN DISCH_TAC THEN EXISTS_TAC `(\x. x + Cx(&2 * n * pi) * ii) o (l:complex->complex)` THEN ASM_REWRITE_TAC[CONTINUOUS_ON_CEXP; o_THM; IMAGE_o; FORALL_IN_IMAGE] THEN RULE_ASSUM_TAC(REWRITE_RULE[INJECTIVE_ON_ALT]) THEN ASM_SIMP_TAC[CEXP_ADD; CEXP_INTEGER_2PI; COMPLEX_MUL_RID; REWRITE_RULE[SUBSET] BALL_SUBSET_CBALL] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `(!x. e(f x) = e x) ==> IMAGE e (IMAGE f s) = IMAGE e s`) THEN ASM_SIMP_TAC[CEXP_ADD; CEXP_INTEGER_2PI; COMPLEX_MUL_RID]; MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> l(e x) = x) ==> IMAGE t (IMAGE l (IMAGE e s)) = IMAGE t s`) THEN ASM_SIMP_TAC[REWRITE_RULE[SUBSET] BALL_SUBSET_CBALL]; MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST] THEN ASM_MESON_TAC[BALL_SUBSET_CBALL; IMAGE_SUBSET; CONTINUOUS_ON_SUBSET]]]);; (* ------------------------------------------------------------------------- *) (* Hence the Borsukian results about mappings into circle. *) (* ------------------------------------------------------------------------- *) let INESSENTIAL_EQ_CONTINUOUS_LOGARITHM = prove (`!f:real^N->complex s. (?a. homotopic_with (\h. T) (s,(:complex) DIFF {Cx(&0)}) f (\t. a)) <=> (?g. g continuous_on s /\ (!x. x IN s ==> f x = cexp(g x)))`, REPEAT GEN_TAC THEN EQ_TAC THENL [DISCH_THEN(CHOOSE_THEN (MP_TAC o CONJ COVERING_SPACE_CEXP_PUNCTURED_PLANE)) THEN DISCH_THEN(MP_TAC o MATCH_MP COVERING_SPACE_LIFT_INESSENTIAL_FUNCTION) THEN REWRITE_TAC[SUBSET_UNIV] THEN MESON_TAC[]; DISCH_THEN(X_CHOOSE_THEN `g:real^N->complex` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `?a. homotopic_with (\h. T) (s,(:complex) DIFF {Cx(&0)}) (cexp o g) (\x:real^N. a)` MP_TAC THENL [MATCH_MP_TAC NULLHOMOTOPIC_THROUGH_CONTRACTIBLE THEN EXISTS_TAC `(:complex)` THEN ASM_REWRITE_TAC[SUBSET_UNIV] THEN ASM_SIMP_TAC[STARLIKE_IMP_CONTRACTIBLE; STARLIKE_UNIV] THEN REWRITE_TAC[CONTINUOUS_ON_CEXP; SUBSET; FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_UNIV; IN_DIFF; IN_SING; CEXP_NZ]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:complex` THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_WITH_EQ) THEN ASM_SIMP_TAC[o_THM]]]);; let INESSENTIAL_IMP_CONTINUOUS_LOGARITHM_CIRCLE = prove (`!f:real^N->complex s. (?a. homotopic_with (\h. T) (s,sphere(vec 0,&1)) f (\t. a)) ==> ?g. g continuous_on s /\ !x. x IN s ==> f x = cexp(g x)`, REPEAT GEN_TAC THEN SIMP_TAC[sphere; GSYM INESSENTIAL_EQ_CONTINUOUS_LOGARITHM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:complex` THEN REWRITE_TAC[homotopic_with] THEN MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] SUBSET_TRANS)) THEN SIMP_TAC[SUBSET; DIST_0; FORALL_IN_GSPEC; IN_UNIV; IN_DIFF; IN_SING] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN SIMP_TAC[COMPLEX_NORM_CX] THEN REAL_ARITH_TAC);; let INESSENTIAL_EQ_CONTINUOUS_LOGARITHM_CIRCLE = prove (`!f:real^N->complex s. (?a. homotopic_with (\h. T) (s,sphere(vec 0,&1)) f (\t. a)) <=> (?g. (Cx o g) continuous_on s /\ !x. x IN s ==> f x = cexp(ii * Cx(g x)))`, REPEAT GEN_TAC THEN EQ_TAC THENL [DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP INESSENTIAL_IMP_CONTINUOUS_LOGARITHM_CIRCLE) THEN DISCH_THEN(X_CHOOSE_THEN `g:real^N->complex` STRIP_ASSUME_TAC) THEN EXISTS_TAC `Im o (g:real^N->complex)` THEN CONJ_TAC THENL [REWRITE_TAC[o_ASSOC] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[CONTINUOUS_ON_CX_IM]; FIRST_X_ASSUM(CHOOSE_THEN (MP_TAC o CONJUNCT1 o MATCH_MP HOMOTOPIC_WITH_IMP_SUBSET)) THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_SPHERE_0; NORM_CEXP] THEN REWRITE_TAC[EULER; o_THM; RE_MUL_II; IM_MUL_II] THEN SIMP_TAC[RE_CX; IM_CX; REAL_NEG_0; REAL_EXP_0]]; DISCH_THEN(X_CHOOSE_THEN `g:real^N->real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `?a. homotopic_with (\h. T) (s,sphere(vec 0,&1)) ((cexp o (\z. ii * z)) o (Cx o g)) (\x:real^N. a)` MP_TAC THENL [MATCH_MP_TAC NULLHOMOTOPIC_THROUGH_CONTRACTIBLE THEN EXISTS_TAC `{z | Im z = &0}` THEN ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[CONTINUOUS_ON_COMPOSE; CONTINUOUS_ON_CEXP; CONJ_ASSOC; CONTINUOUS_ON_COMPLEX_LMUL; CONTINUOUS_ON_ID] THEN CONJ_TAC THENL [REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM; IN_SPHERE_0; o_THM; IM_CX] THEN SIMP_TAC[NORM_CEXP; RE_MUL_II; REAL_EXP_0; REAL_NEG_0]; MATCH_MP_TAC STARLIKE_IMP_CONTRACTIBLE THEN MATCH_MP_TAC CONVEX_IMP_STARLIKE THEN CONJ_TAC THENL [REWRITE_TAC[IM_DEF; CONVEX_STANDARD_HYPERPLANE]; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_ELIM_THM] THEN MESON_TAC[IM_CX]]]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:complex` THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_WITH_EQ) THEN ASM_SIMP_TAC[o_THM]]]);; let HOMOTOPIC_CIRCLEMAPS_DIV,HOMOTOPIC_CIRCLEMAPS_DIV_1 = (CONJ_PAIR o prove) (`(!f g:real^N->real^2 s. homotopic_with (\x. T) (s,sphere(vec 0,&1)) f g <=> f continuous_on s /\ IMAGE f s SUBSET sphere(vec 0,&1) /\ g continuous_on s /\ IMAGE g s SUBSET sphere(vec 0,&1) /\ ?c. homotopic_with (\x. T) (s,sphere(vec 0,&1)) (\x. f x / g x) (\x. c)) /\ (!f g:real^N->real^2 s. homotopic_with (\x. T) (s,sphere(vec 0,&1)) f g <=> f continuous_on s /\ IMAGE f s SUBSET sphere(vec 0,&1) /\ g continuous_on s /\ IMAGE g s SUBSET sphere(vec 0,&1) /\ homotopic_with (\x. T) (s,sphere(vec 0,&1)) (\x. f x / g x) (\x. Cx(&1)))`, let lemma = prove (`!f g h:real^N->real^2 s. homotopic_with (\x. T) (s,sphere(vec 0,&1)) f g ==> h continuous_on s /\ (!x. x IN s ==> h(x) IN sphere(vec 0,&1)) ==> homotopic_with (\x. T) (s,sphere(vec 0,&1)) (\x. f x * h x) (\x. g x * h x)`, REWRITE_TAC[IN_SPHERE_0] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homotopic_with]) THEN ASM_SIMP_TAC[HOMOTOPIC_WITH; LEFT_IMP_EXISTS_THM] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_SPHERE_0; FORALL_IN_PCROSS] THEN X_GEN_TAC `k:real^((1,N)finite_sum)->real^2` THEN STRIP_TAC THEN EXISTS_TAC `\z. (k:real^(1,N)finite_sum->real^2) z * h(sndcart z)` THEN ASM_SIMP_TAC[COMPLEX_NORM_MUL; SNDCART_PASTECART; REAL_MUL_LID] THEN ASM_REWRITE_TAC[SNDCART_PASTECART] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPLEX_MUL THEN ASM_REWRITE_TAC[GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_SNDCART; IMAGE_SNDCART_PCROSS] THEN ASM_REWRITE_TAC[UNIT_INTERVAL_NONEMPTY]) in REWRITE_TAC[AND_FORALL_THM] THEN REPEAT GEN_TAC THEN MATCH_MP_TAC (TAUT `(q <=> r) /\ (p <=> r) ==> (p <=> q) /\ (p <=> r)`) THEN CONJ_TAC THENL [REPEAT(MATCH_MP_TAC(TAUT `(p ==> (q <=> r)) ==> (p /\ q <=> p /\ r)`) THEN DISCH_TAC) THEN EQ_TAC THENL [ALL_TAC; DISCH_TAC THEN EXISTS_TAC `Cx(&1)` THEN ASM_MESON_TAC[]] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `c:complex` 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 ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPECL [`vec 0:real^2`; `&1`] PATH_CONNECTED_SPHERE) THEN REWRITE_TAC[DIMINDEX_2; LE_REFL; PATH_CONNECTED_IFF_PATH_COMPONENT] THEN DISCH_THEN MATCH_MP_TAC THEN CONJ_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[IN_SPHERE_0; COMPLEX_NORM_CX; REAL_ABS_NUM]]; EQ_TAC THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP lemma) THENL [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 DISCH_THEN(MP_TAC o SPEC `\x. inv((g:real^N->complex) x)`); DISCH_THEN(MP_TAC o SPEC `g:real^N->complex`)] THEN RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; FORALL_IN_IMAGE; IN_SPHERE_0]) THEN ASM_SIMP_TAC[IN_SPHERE_0; COMPLEX_NORM_INV; REAL_INV_1] THEN ASM_SIMP_TAC[GSYM COMPLEX_NORM_ZERO; REAL_OF_NUM_EQ; ARITH_EQ; CONTINUOUS_ON_COMPLEX_INV] THEN ASM_REWRITE_TAC[SUBSET; IN_SPHERE_0; FORALL_IN_IMAGE] THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_WITH_EQ) THEN ASM_SIMP_TAC[COMPLEX_DIV_RMUL; COMPLEX_MUL_LID; COMPLEX_MUL_RINV; GSYM complex_div; COMPLEX_DIV_REFL; GSYM COMPLEX_NORM_ZERO; REAL_OF_NUM_EQ; ARITH_EQ]]);; (* ------------------------------------------------------------------------- *) (* In particular, complex logs exist on various "well-behaved" sets. *) (* ------------------------------------------------------------------------- *) let CONTINUOUS_LOGARITHM_ON_CONTRACTIBLE = prove (`!f:real^N->complex s. f continuous_on s /\ contractible s /\ (!x. x IN s ==> ~(f x = Cx(&0))) ==> ?g. g continuous_on s /\ !x. x IN s ==> f x = cexp(g x)`, REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM INESSENTIAL_EQ_CONTINUOUS_LOGARITHM] THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC NULLHOMOTOPIC_FROM_CONTRACTIBLE THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);; let CONTINUOUS_LOGARITHM_ON_SIMPLY_CONNECTED = prove (`!f:real^N->complex s. f continuous_on s /\ simply_connected s /\ locally path_connected s /\ (!x. x IN s ==> ~(f x = Cx(&0))) ==> ?g. g continuous_on s /\ !x. x IN s ==> f x = cexp(g x)`, REPEAT STRIP_TAC THEN MP_TAC (ISPECL [`f:real^N->complex`; `s:real^N->bool`] (MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] COVERING_SPACE_LIFT) COVERING_SPACE_CEXP_PUNCTURED_PLANE)) THEN ASM_REWRITE_TAC[IN_UNIV] THEN ASM SET_TAC[]);; let CONTINUOUS_LOGARITHM_ON_CBALL = prove (`!f:real^N->complex a r. f continuous_on cball(a,r) /\ (!z. z IN cball(a,r) ==> ~(f z = Cx(&0))) ==> ?h. h continuous_on cball(a,r) /\ !z. z IN cball(a,r) ==> f z = cexp(h z)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `cball(a:real^N,r) = {}` THEN ASM_REWRITE_TAC[CONTINUOUS_ON_EMPTY; NOT_IN_EMPTY] THEN MATCH_MP_TAC CONTINUOUS_LOGARITHM_ON_CONTRACTIBLE THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC STARLIKE_IMP_CONTRACTIBLE THEN MATCH_MP_TAC CONVEX_IMP_STARLIKE THEN ASM_REWRITE_TAC[CONVEX_CBALL]);; let CONTINUOUS_LOGARITHM_ON_BALL = prove (`!f:real^N->complex a r. f continuous_on ball(a,r) /\ (!x. x IN ball(a,r) ==> ~(f x = Cx(&0))) ==> ?h. h continuous_on ball(a,r) /\ !x. x IN ball(a,r) ==> f x = cexp(h x)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `ball(a:real^N,r) = {}` THEN ASM_REWRITE_TAC[CONTINUOUS_ON_EMPTY; NOT_IN_EMPTY] THEN MATCH_MP_TAC CONTINUOUS_LOGARITHM_ON_CONTRACTIBLE THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC STARLIKE_IMP_CONTRACTIBLE THEN MATCH_MP_TAC CONVEX_IMP_STARLIKE THEN ASM_REWRITE_TAC[CONVEX_BALL]);; let CONTINUOUS_SQRT_ON_CONTRACTIBLE = prove (`!f:real^N->complex s. f continuous_on s /\ contractible s /\ (!x. x IN s ==> ~(f x = Cx(&0))) ==> ?g. g continuous_on s /\ !x. x IN s ==> f x = (g x) pow 2`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP CONTINUOUS_LOGARITHM_ON_CONTRACTIBLE) THEN DISCH_THEN(X_CHOOSE_THEN `g:real^N->complex` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\z:real^N. cexp(g z / Cx(&2))` THEN ASM_SIMP_TAC[GSYM CEXP_N; COMPLEX_RING `Cx(&2) * z / Cx(&2) = z`] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN REWRITE_TAC[CONTINUOUS_ON_CEXP] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPLEX_DIV THEN ASM_SIMP_TAC[CONTINUOUS_ON_CONST] THEN CONV_TAC COMPLEX_RING);; let CONTINUOUS_SQRT_ON_SIMPLY_CONNECTED = prove (`!f:real^N->complex s. f continuous_on s /\ simply_connected s /\ locally path_connected s /\ (!x. x IN s ==> ~(f x = Cx(&0))) ==> ?g. g continuous_on s /\ !x. x IN s ==> f x = (g x) pow 2`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP CONTINUOUS_LOGARITHM_ON_SIMPLY_CONNECTED) THEN DISCH_THEN(X_CHOOSE_THEN `g:real^N->complex` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\z:real^N. cexp(g z / Cx(&2))` THEN ASM_SIMP_TAC[GSYM CEXP_N; COMPLEX_RING `Cx(&2) * z / Cx(&2) = z`] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN REWRITE_TAC[CONTINUOUS_ON_CEXP] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPLEX_DIV THEN ASM_SIMP_TAC[CONTINUOUS_ON_CONST] THEN CONV_TAC COMPLEX_RING);; (* ------------------------------------------------------------------------- *) (* Analogously, holomorphic logarithms and square roots. *) (* ------------------------------------------------------------------------- *) let CONTRACTIBLE_IMP_HOLOMORPHIC_LOG,SIMPLY_CONNECTED_IMP_HOLOMORPHIC_LOG = (CONJ_PAIR o prove) (`(!s:complex->bool. contractible s ==> !f. f holomorphic_on s /\ (!z. z IN s ==> ~(f z = Cx(&0))) ==> ?g. g holomorphic_on s /\ !z. z IN s ==> f z = cexp(g z)) /\ (!s:complex->bool. simply_connected s /\ locally path_connected s ==> !f. f holomorphic_on s /\ (!z. z IN s ==> ~(f z = Cx(&0))) ==> ?g. g holomorphic_on s /\ !z. z IN s ==> f z = cexp(g z))`, REPEAT STRIP_TAC THENL [MP_TAC(ISPECL [`f:complex->complex`; `s:complex->bool`] CONTINUOUS_LOGARITHM_ON_CONTRACTIBLE); MP_TAC(ISPECL [`f:complex->complex`; `s:complex->bool`] CONTINUOUS_LOGARITHM_ON_SIMPLY_CONNECTED)] THEN ASM_SIMP_TAC[HOLOMORPHIC_ON_IMP_CONTINUOUS_ON] THEN (MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:complex->complex` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN UNDISCH_TAC `f holomorphic_on s` THEN REWRITE_TAC[holomorphic_on] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `z:complex` THEN ASM_CASES_TAC `(z:complex) IN s` THEN ASM_REWRITE_TAC[HAS_COMPLEX_DERIVATIVE_WITHIN] THEN DISCH_THEN(X_CHOOSE_THEN `f':complex` MP_TAC) THEN DISCH_THEN(MP_TAC o ISPECL [`\x. (cexp(g x) - cexp(g z)) / (x - z)`; `&1`] o MATCH_MP (REWRITE_RULE [TAUT `p /\ q /\ r ==> s <=> r ==> p /\ q ==> s`] LIM_TRANSFORM_WITHIN)) THEN ASM_SIMP_TAC[REAL_LT_01] THEN DISCH_THEN(MP_TAC o SPECL [`\x:complex. if g x = g z then cexp(g z) else (cexp(g x) - cexp(g z)) / (g x - g z)`; `cexp(g(z:complex))`] o MATCH_MP (REWRITE_RULE[IMP_CONJ] LIM_COMPLEX_DIV)) THEN REWRITE_TAC[CEXP_NZ] THEN ANTS_TAC THENL [SUBGOAL_THEN `(\x. if g x = g z then cexp(g z) else (cexp(g x) - cexp(g(z:complex))) / (g x - g z)) = (\y. if y = g z then cexp(g z) else (cexp y - cexp(g z)) / (y - g z)) o g` SUBST1_TAC THENL [REWRITE_TAC[o_DEF]; ALL_TAC] THEN MATCH_MP_TAC LIM_COMPOSE_AT THEN EXISTS_TAC `(g:complex->complex) z` THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON]; REWRITE_TAC[EVENTUALLY_TRUE]; ONCE_REWRITE_TAC[LIM_AT_ZERO] THEN SIMP_TAC[COMPLEX_VEC_0; COMPLEX_ADD_SUB; COMPLEX_EQ_ADD_LCANCEL_0] THEN MP_TAC(SPEC `cexp(g(z:complex))` (MATCH_MP LIM_COMPLEX_LMUL LIM_CEXP_MINUS_1)) THEN REWRITE_TAC[COMPLEX_MUL_RID] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_TRANSFORM_EVENTUALLY) THEN SIMP_TAC[EVENTUALLY_AT; GSYM DIST_NZ; CEXP_ADD] THEN EXISTS_TAC `&1` THEN REWRITE_TAC[REAL_LT_01] THEN SIMPLE_COMPLEX_ARITH_TAC]; DISCH_THEN(fun th -> EXISTS_TAC `f' / cexp(g(z:complex))` THEN MP_TAC th) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_TRANSFORM_EVENTUALLY) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN]) THEN DISCH_THEN(MP_TAC o SPEC `z:complex`) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[CONTINUOUS_WITHIN; tendsto] THEN DISCH_THEN(MP_TAC o SPEC `&2 * pi`) THEN REWRITE_TAC[REAL_ARITH `&0 < &2 * x <=> &0 < x`; PI_POS] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN X_GEN_TAC `w:complex` THEN REWRITE_TAC[dist] THEN DISCH_TAC THEN COND_CASES_TAC THENL [ASM_REWRITE_TAC[COMPLEX_SUB_REFL; complex_div; COMPLEX_MUL_LZERO]; ASM_CASES_TAC `w:complex = z` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `~(cexp(g(w:complex)) = cexp(g z))` MP_TAC THENL [UNDISCH_TAC `~((g:complex->complex) w = g z)` THEN REWRITE_TAC[CONTRAPOS_THM] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] COMPLEX_EQ_CEXP) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] REAL_LET_TRANS)) THEN REWRITE_TAC[GSYM IM_SUB; COMPLEX_NORM_GE_RE_IM]; REPEAT(FIRST_X_ASSUM(MP_TAC o check(is_neg o concl))) THEN CONV_TAC COMPLEX_FIELD]]]));; let CONTRACTIBLE_IMP_HOLOMORPHIC_SQRT,SIMPLY_CONNECTED_IMP_HOLOMORPHIC_SQRT = (CONJ_PAIR o prove) (`(!s:complex->bool. contractible s ==> !f. f holomorphic_on s /\ (!z. z IN s ==> ~(f z = Cx(&0))) ==> ?g. g holomorphic_on s /\ !z. z IN s ==> f z = g z pow 2) /\ (!s:complex->bool. simply_connected s /\ locally path_connected s ==> !f. f holomorphic_on s /\ (!z. z IN s ==> ~(f z = Cx(&0))) ==> ?g. g holomorphic_on s /\ !z. z IN s ==> f z = g z pow 2)`, CONJ_TAC THEN GEN_TAC THENL [DISCH_THEN(ASSUME_TAC o MATCH_MP CONTRACTIBLE_IMP_HOLOMORPHIC_LOG); DISCH_THEN(ASSUME_TAC o MATCH_MP SIMPLY_CONNECTED_IMP_HOLOMORPHIC_LOG)] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `f:complex->complex`) THEN ASM_SIMP_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `g:complex->complex` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\z:complex. cexp(g z / Cx(&2))` THEN ASM_SIMP_TAC[GSYM CEXP_N; COMPLEX_RING `Cx(&2) * z / Cx(&2) = z`] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC HOLOMORPHIC_ON_COMPOSE THEN REWRITE_TAC[HOLOMORPHIC_ON_CEXP] THEN MATCH_MP_TAC HOLOMORPHIC_ON_DIV THEN ASM_SIMP_TAC[HOLOMORPHIC_ON_CONST] THEN CONV_TAC COMPLEX_RING);; (* ------------------------------------------------------------------------- *) (* Related theorems about holomorphic inverse cosines. *) (* ------------------------------------------------------------------------- *) let CONTRACTIBLE_IMP_HOLOMORPHIC_ACS = prove (`!f s. f holomorphic_on s /\ contractible s /\ (!z. z IN s ==> ~(f z = Cx(&1)) /\ ~(f z = --Cx(&1))) ==> ?g. g holomorphic_on s /\ !z. z IN s ==> f z = ccos(g z)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `\z:complex. Cx(&1) - f(z) pow 2` o MATCH_MP CONTRACTIBLE_IMP_HOLOMORPHIC_SQRT) THEN ASM_SIMP_TAC[HOLOMORPHIC_ON_SUB; HOLOMORPHIC_ON_CONST; HOLOMORPHIC_ON_POW; COMPLEX_RING `~(Cx(&1) - z pow 2 = Cx(&0)) <=> ~(z = Cx(&1)) /\ ~(z = --Cx(&1))`] THEN REWRITE_TAC[COMPLEX_RING `Cx(&1) - w pow 2 = z pow 2 <=> (w + ii * z) * (w - ii * z) = Cx(&1)`] THEN DISCH_THEN(X_CHOOSE_THEN `g:complex->complex` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o SPEC `\z:complex. f(z) + ii * g(z)` o MATCH_MP CONTRACTIBLE_IMP_HOLOMORPHIC_LOG) THEN ASM_SIMP_TAC[HOLOMORPHIC_ON_ADD; HOLOMORPHIC_ON_MUL; HOLOMORPHIC_ON_CONST; COMPLEX_RING `(a + b) * (a - b) = Cx(&1) ==> ~(a + b = Cx(&0))`] THEN DISCH_THEN(X_CHOOSE_THEN `h:complex->complex` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\z:complex. --ii * h(z)` THEN ASM_SIMP_TAC[HOLOMORPHIC_ON_MUL; HOLOMORPHIC_ON_CONST; ccos] THEN X_GEN_TAC `z:complex` THEN DISCH_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `z:complex`)) THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (COMPLEX_FIELD `a * b = Cx(&1) ==> b = inv a`)) THEN ASM_SIMP_TAC[GSYM CEXP_NEG] THEN FIRST_X_ASSUM(ASSUME_TAC o SYM) THEN DISCH_THEN(ASSUME_TAC o SYM) THEN ASM_REWRITE_TAC[COMPLEX_RING `ii * --ii * z = z`; COMPLEX_RING `--ii * --ii * z = --z`] THEN CONV_TAC COMPLEX_RING);; let CONTRACTIBLE_IMP_HOLOMORPHIC_ACS_BOUNDED = prove (`!f s a. f holomorphic_on s /\ contractible s /\ a IN s /\ (!z. z IN s ==> ~(f z = Cx(&1)) /\ ~(f z = --Cx(&1))) ==> ?g. g holomorphic_on s /\ norm(g a) <= pi + norm(f a) /\ !z. z IN s ==> f z = ccos(g z)`, let lemma = prove (`!w. ?v. ccos(v) = w /\ norm(v) <= pi + norm(w)`, GEN_TAC THEN EXISTS_TAC `cacs w` THEN ABBREV_TAC `v = cacs w` THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [ASM_MESON_TAC[CCOS_CACS]; DISCH_THEN(SUBST1_TAC o SYM)] THEN SIMP_TAC[NORM_LE_SQUARE; PI_POS_LE; NORM_POS_LE; REAL_LE_ADD] THEN MATCH_MP_TAC(REAL_ARITH `&0 <= b * c /\ a <= b pow 2 + c pow 2 ==> a <= (b + c) pow 2`) THEN SIMP_TAC[REAL_LE_MUL; PI_POS_LE; NORM_POS_LE] THEN REWRITE_TAC[COMPLEX_SQNORM; GSYM NORM_POW_2; NORM_CCOS_POW_2] THEN MATCH_MP_TAC REAL_LE_ADD2 THEN REWRITE_TAC[GSYM REAL_LE_SQUARE_ABS] THEN EXPAND_TAC "v" THEN REWRITE_TAC[REAL_ABS_PI; RE_CACS_BOUND] THEN MATCH_MP_TAC(REAL_ARITH `&0 <= c /\ x <= (d / &2) pow 2 ==> x <= c + d pow 2 / &4`) THEN REWRITE_TAC[REAL_LE_POW_2; GSYM REAL_LE_SQUARE_ABS; REAL_LE_ABS_SINH]) in REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:complex->complex`; `s:complex->bool`] CONTRACTIBLE_IMP_HOLOMORPHIC_ACS) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `g:complex->complex` STRIP_ASSUME_TAC) THEN MP_TAC(SPEC `(f:complex->complex) a` lemma) THEN DISCH_THEN(X_CHOOSE_THEN `b:complex` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `ccos b = ccos(g(a:complex))` MP_TAC THENL [ASM_MESON_TAC[]; REWRITE_TAC[CCOS_EQ]] THEN DISCH_THEN(X_CHOOSE_THEN `n:real` (STRIP_ASSUME_TAC o GSYM)) THENL [EXISTS_TAC `\z:complex. g z + Cx(&2 * n * pi)`; EXISTS_TAC `\z:complex. --(g z) + Cx(&2 * n * pi)`] THEN ASM_SIMP_TAC[HOLOMORPHIC_ON_ADD; HOLOMORPHIC_ON_NEG; HOLOMORPHIC_ON_CONST] THEN CONV_TAC(ONCE_DEPTH_CONV SYM_CONV) THEN REWRITE_TAC[CCOS_EQ] THEN ASM_MESON_TAC[]);; (* ------------------------------------------------------------------------- *) (* Extension property for inessential maps. This almost follows from *) (* INESSENTIAL_NEIGHBOURHOOD_EXTENSION except that here we don't need to *) (* assume that t is closed in s. *) (* ------------------------------------------------------------------------- *) let INESSENTIAL_NEIGHBOURHOOD_EXTENSION_LOGARITHM = prove (`!f:real^N->complex s t. f continuous_on s /\ t SUBSET s /\ (?g. g continuous_on t /\ !x. x IN t ==> f x = cexp(g x)) ==> ?u. t SUBSET u /\ open_in (subtopology euclidean s) u /\ (?g. g continuous_on u /\ !x. x IN u ==> f x = cexp(g x))`, REPEAT GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(X_CHOOSE_THEN `h:real^N->complex` (STRIP_ASSUME_TAC o GSYM)) THEN SUBGOAL_THEN `!x. x IN t ==> ?d. &0 < d /\ (!y. y IN s /\ dist(x,y) < d ==> norm(f y / f x - Cx(&1)) < &1 / &7) /\ (!z:real^N. z IN t /\ dist(x,z) < &2 * d ==> norm(h z - h x) < &1 / &5)` MP_TAC THENL [REPEAT STRIP_TAC THEN UNDISCH_TAC `(h:real^N->complex) continuous_on t` THEN GEN_REWRITE_TAC LAND_CONV [CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[continuous_within] THEN DISCH_THEN(MP_TAC o SPEC `&1 / &5`) THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [dist] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `~((f:real^N->complex) x = Cx(&0))` ASSUME_TAC THENL [ASM_MESON_TAC[CEXP_NZ]; ALL_TAC] THEN SUBGOAL_THEN `(\y:real^N. f y / f x) continuous (at x within s)` MP_TAC THENL [REWRITE_TAC[complex_div] THEN MATCH_MP_TAC CONTINUOUS_COMPLEX_MUL THEN REWRITE_TAC[CONTINUOUS_CONST] THEN ASM_MESON_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN; SUBSET]; REWRITE_TAC[continuous_within] THEN DISCH_THEN(MP_TAC o SPEC `&1 / &7`) THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_SIMP_TAC[COMPLEX_DIV_REFL; dist] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC)] THEN EXISTS_TAC `min d (e / &2)` THEN ASM_REWRITE_TAC[REAL_LT_MIN; REAL_HALF] THEN CONJ_TAC THENL [ASM_MESON_TAC[NORM_SUB]; REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ONCE_REWRITE_TAC[NORM_SUB] THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_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 `d:real^N->real` THEN DISCH_THEN(LABEL_TAC "*")] THEN ABBREV_TAC `u = \x. s INTER ball(x:real^N,d x)` THEN ABBREV_TAC `g = \x y. h(x:real^N) + clog(f y / f x)` THEN SUBGOAL_THEN `(!x:real^N. x IN t ==> x IN u x) /\ (!x. x IN t ==> open_in (subtopology euclidean s) (u x))` STRIP_ASSUME_TAC THENL [EXPAND_TAC "u" THEN ASM_SIMP_TAC[IN_INTER; CENTRE_IN_BALL; OPEN_IN_OPEN_INTER; OPEN_BALL] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `!x:real^N y:real^N. x IN t /\ y IN u x ==> cexp(g x y) = f y` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN EXPAND_TAC "g" THEN REWRITE_TAC[CEXP_ADD] THEN ASM_SIMP_TAC[] THEN REMOVE_THEN "*" (MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `y:real^N` o el 1 o CONJUNCTS) THEN MP_TAC(ASSUME `y IN (u:real^N->real^N->bool) x`) THEN EXPAND_TAC "u" THEN REWRITE_TAC[IN_INTER; IN_BALL] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o MATCH_MP (NORM_ARITH `norm(x - y) < &1 / &7 ==> norm(y) = &1 ==> ~(x = vec 0)`)) THEN SIMP_TAC[COMPLEX_NORM_CX; REAL_ABS_NUM; CEXP_CLOG; COMPLEX_VEC_0] THEN SIMP_TAC[COMPLEX_DIV_LMUL; COMPLEX_DIV_EQ_0; DE_MORGAN_THM]; ALL_TAC] THEN MP_TAC(ISPECL [`g:real^N->real^N->complex`; `u:real^N->real^N->bool`; `UNIONS {(u:real^N->real^N->bool) x | x IN t}`; `t:real^N->bool`] PASTING_LEMMA_EXISTS) THEN REWRITE_TAC[SUBSET_REFL] THEN ANTS_TAC THENL [CONJ_TAC THENL [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_SUBSET_TRANS THEN EXISTS_TAC `s:real^N->bool` THEN ASM_SIMP_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC] THEN ASM_MESON_TAC[OPEN_IN_IMP_SUBSET]; EXPAND_TAC "g" THEN REWRITE_TAC[] THEN MATCH_MP_TAC CONTINUOUS_ON_ADD THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THENL [REWRITE_TAC[complex_div] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPLEX_RMUL THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; INTER_SUBSET]; MATCH_MP_TAC CONTINUOUS_ON_CLOG THEN REWRITE_TAC[IMP_CONJ; FORALL_IN_IMAGE; IN_INTER] THEN X_GEN_TAC `y:real^N` THEN REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[COMPLEX_RING `z = (z - Cx(&1)) + Cx(&1)`] THEN REWRITE_TAC[RE_ADD; RE_CX] THEN MATCH_MP_TAC(REAL_ARITH `abs x < &1 ==> &0 < x + &1`) THEN MATCH_MP_TAC(MESON[COMPLEX_NORM_GE_RE_IM; REAL_LET_TRANS] `norm z < &1 ==> abs(Re z) < &1`) THEN MATCH_MP_TAC(REAL_ARITH `x < &1 / &7 ==> x < &1`) THEN REMOVE_THEN "*" (MP_TAC o SPEC `x:real^N`) THEN ASM_SIMP_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `y:real^N` o el 1 o CONJUNCTS) THEN MP_TAC(ASSUME `y IN (u:real^N->real^N->bool) x`) THEN EXPAND_TAC "u" THEN REWRITE_TAC[IN_INTER; IN_BALL] THEN STRIP_TAC THEN ASM_REWRITE_TAC[]]]; MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`; `x:real^N`] THEN REWRITE_TAC[IN_INTER] THEN STRIP_TAC THEN MATCH_MP_TAC COMPLEX_EQ_CEXP THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN EXPAND_TAC "g" THEN REWRITE_TAC[IM_ADD] THEN MATCH_MP_TAC(REAL_ARITH `&5 < a /\ abs(ha - hb) < &1 / &5 /\ abs(fa) < &2 /\ abs(fb) < &2 ==> abs((ha + fa) - (hb + fb)) < a`) THEN CONJ_TAC THENL [MP_TAC PI_APPROX_32 THEN REAL_ARITH_TAC; ALL_TAC] THEN CONJ_TAC THENL [REWRITE_TAC[GSYM IM_SUB] THEN MATCH_MP_TAC(MESON[COMPLEX_NORM_GE_RE_IM; REAL_LET_TRANS] `norm z < a ==> abs(Im z) < a`) THEN MP_TAC(ASSUME `x IN (u:real^N->real^N->bool) b`) THEN MP_TAC(ASSUME `x IN (u:real^N->real^N->bool) a`) THEN EXPAND_TAC "u" THEN REWRITE_TAC[IMP_IMP; IN_INTER; IN_BALL] THEN DISCH_THEN(MP_TAC o MATCH_MP (TAUT `(p /\ q) /\ (p /\ r) ==> q /\ r`)) THEN DISCH_THEN(MP_TAC o MATCH_MP (NORM_ARITH `dist(a,x) < d /\ dist(b,x) < e ==> dist(a,b) < &2 * d \/ dist(a,b) < &2 * e`)) THEN STRIP_TAC THENL [REMOVE_THEN "*" (MP_TAC o SPEC `a:real^N`); REMOVE_THEN "*" (MP_TAC o SPEC `b:real^N`)] THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o last o CONJUNCTS) THEN ASM_MESON_TAC[NORM_SUB; DIST_SYM]; CONJ_TAC THEN TRANS_TAC REAL_LT_TRANS `pi / &2` THEN (CONJ_TAC THENL [ALL_TAC; MP_TAC PI_APPROX_32 THEN REAL_ARITH_TAC]) THEN MATCH_MP_TAC RE_CLOG_POS_LT_IMP THEN ONCE_REWRITE_TAC[COMPLEX_RING `z = (z - Cx(&1)) + Cx(&1)`] THEN REWRITE_TAC[RE_ADD; RE_CX] THEN MATCH_MP_TAC(REAL_ARITH `abs x < &1 ==> &0 < x + &1`) THEN MATCH_MP_TAC(MESON[COMPLEX_NORM_GE_RE_IM; REAL_LET_TRANS] `norm z < &1 ==> abs(Re z) < &1`) THEN MATCH_MP_TAC(REAL_ARITH `x < &1 / &7 ==> x < &1`) THENL [REMOVE_THEN "*" (MP_TAC o SPEC `a:real^N`); REMOVE_THEN "*" (MP_TAC o SPEC `b:real^N`)] THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `x:real^N` o el 1 o CONJUNCTS) THEN DISCH_THEN MATCH_MP_TAC THENL [MP_TAC(ASSUME `x IN (u:real^N->real^N->bool) a`); MP_TAC(ASSUME `x IN (u:real^N->real^N->bool) b`)] THEN EXPAND_TAC "u" THEN REWRITE_TAC[IN_INTER; IN_BALL] THEN STRIP_TAC THEN ASM_REWRITE_TAC[]]]; DISCH_THEN(X_CHOOSE_THEN `h':real^N->complex` STRIP_ASSUME_TAC) THEN EXISTS_TAC `UNIONS {(u:real^N->real^N->bool) x | x IN t}` THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[OPEN_IN_UNIONS; FORALL_IN_GSPEC] THEN EXISTS_TAC `h':real^N->complex` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[FORALL_IN_UNIONS; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`y:real^N`; `x:real^N`]) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN ASM_MESON_TAC[]]);; (* ------------------------------------------------------------------------- *) (* The "borsukian" property of sets. This doesn't seem to have a standard *) (* name. Kuratowski uses "contractible with respect to [S^1]" while *) (* Whyburn uses "property b". It's closely related to unicoherence. *) (* ------------------------------------------------------------------------- *) let borsukian = new_definition `borsukian(s:real^N->bool) <=> !f. f continuous_on s /\ IMAGE f s SUBSET ((:real^2) DIFF {Cx(&0)}) ==> ?a. homotopic_with (\h. T) (s,(:real^2) DIFF {Cx(&0)}) f (\x. a)`;; let BORSUKIAN_RETRACTION_GEN = prove (`!s:real^M->bool t:real^N->bool h k. h continuous_on s /\ IMAGE h s = t /\ k continuous_on t /\ IMAGE k t SUBSET s /\ (!y. y IN t ==> h(k y) = y) /\ borsukian s ==> borsukian t`, REPEAT GEN_TAC THEN REWRITE_TAC[borsukian] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o check (is_forall o concl)) THEN PURE_ONCE_REWRITE_TAC[TAUT `p /\ q <=> p /\ q /\ T`] THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] COHOMOTOPICALLY_TRIVIAL_RETRACTION_NULL_GEN) THEN REWRITE_TAC[] THEN ASM_MESON_TAC[]);; let RETRACT_OF_BORSUKIAN = prove (`!s t:real^N->bool. borsukian t /\ s retract_of t ==> borsukian 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] BORSUKIAN_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 ASM_REWRITE_TAC[] THEN EXISTS_TAC `\x:real^N. x` THEN ASM_REWRITE_TAC[CONTINUOUS_ON_ID; IMAGE_ID]);; let HOMEOMORPHIC_BORSUKIAN = prove (`!s:real^M->bool t:real^N->bool. s homeomorphic t /\ borsukian s ==> borsukian t`, REWRITE_TAC[homeomorphic; homeomorphism] THEN MESON_TAC[BORSUKIAN_RETRACTION_GEN; SUBSET_REFL]);; let HOMEOMORPHIC_BORSUKIAN_EQ = prove (`!s:real^M->bool t:real^N->bool. s homeomorphic t ==> (borsukian s <=> borsukian t)`, REPEAT STRIP_TAC THEN EQ_TAC THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] HOMEOMORPHIC_BORSUKIAN) THEN ASM_MESON_TAC[HOMEOMORPHIC_SYM]);; let BORSUKIAN_TRANSLATION = prove (`!a:real^N s. borsukian (IMAGE (\x. a + x) s) <=> borsukian s`, REPEAT GEN_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_BORSUKIAN_EQ THEN REWRITE_TAC[HOMEOMORPHIC_TRANSLATION_SELF]);; add_translation_invariants [BORSUKIAN_TRANSLATION];; let BORSUKIAN_INJECTIVE_LINEAR_IMAGE = prove (`!f:real^M->real^N s. linear f /\ (!x y. f x = f y ==> x = y) ==> (borsukian(IMAGE f s) <=> borsukian s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_BORSUKIAN_EQ THEN ASM_MESON_TAC[HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_SELF; HOMEOMORPHIC_REFL]);; add_linear_invariants [BORSUKIAN_INJECTIVE_LINEAR_IMAGE];; let HOMOTOPY_EQUIVALENT_BORSUKIANNESS = prove (`!s:real^M->bool t:real^N->bool. s homotopy_equivalent t ==> (borsukian s <=> borsukian t)`, REPEAT STRIP_TAC THEN REWRITE_TAC[borsukian] THEN MATCH_MP_TAC HOMOTOPY_EQUIVALENT_COHOMOTOPIC_TRIVIALITY_NULL THEN ASM_REWRITE_TAC[]);; let BORSUKIAN_ALT = prove (`!s:real^N->bool. borsukian s <=> !f g:real^N->real^2. f continuous_on s /\ IMAGE f s SUBSET ((:real^2) DIFF {Cx(&0)}) /\ g continuous_on s /\ IMAGE g s SUBSET ((:real^2) DIFF {Cx(&0)}) ==> homotopic_with (\h. T) (s,(:real^2) DIFF {Cx (&0)}) f g`, REWRITE_TAC[borsukian; HOMOTOPIC_TRIVIALITY] THEN SIMP_TAC[PATH_CONNECTED_PUNCTURED_UNIVERSE; DIMINDEX_2; LE_REFL]);; let BORSUKIAN_CONTINUOUS_LOGARITHM = prove (`!s:real^N->bool. borsukian s <=> !f. f continuous_on s /\ IMAGE f s SUBSET ((:real^2) DIFF {Cx(&0)}) ==> ?g. g continuous_on s /\ (!x. x IN s ==> f(x) = cexp(g x))`, REWRITE_TAC[borsukian; INESSENTIAL_EQ_CONTINUOUS_LOGARITHM]);; let BORSUKIAN_CONTINUOUS_LOGARITHM_CIRCLE = prove (`!s:real^N->bool. borsukian s <=> !f. f continuous_on s /\ IMAGE f s SUBSET sphere(Cx(&0),&1) ==> ?g. g continuous_on s /\ (!x. x IN s ==> f(x) = cexp(g x))`, GEN_TAC THEN REWRITE_TAC[BORSUKIAN_CONTINUOUS_LOGARITHM] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; COMPLEX_IN_SPHERE_0; SET_RULE `IMAGE f s SUBSET UNIV DIFF {a} <=> !z. z IN s ==> ~(f z = a)`] THEN EQ_TAC THEN DISCH_TAC THEN X_GEN_TAC `f:real^N->complex` THEN STRIP_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `x:real^N` THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[COMPLEX_NORM_0] THEN REAL_ARITH_TAC; FIRST_X_ASSUM(MP_TAC o SPEC `\x:real^N. f(x) / Cx(norm(f x))`) THEN ASM_SIMP_TAC[COMPLEX_NORM_DIV; COMPLEX_NORM_CX; REAL_ABS_NORM; REAL_DIV_REFL; NORM_EQ_0; COMPLEX_NORM_ZERO] THEN ANTS_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPLEX_DIV THEN ASM_REWRITE_TAC[CX_INJ; COMPLEX_NORM_ZERO; CONTINUOUS_ON_CX_LIFT] THEN ASM_SIMP_TAC[CONTINUOUS_ON_LIFT_NORM_COMPOSE]; ASM_SIMP_TAC[CX_INJ; COMPLEX_NORM_ZERO; COMPLEX_FIELD `~(z = Cx(&0)) ==> (w / z = u <=> w = z * u)`] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^N->complex` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\x. clog(Cx(norm(f x:complex))) + (g:real^N->complex)(x)` THEN ASM_SIMP_TAC[CEXP_ADD; CEXP_CLOG; CX_INJ; COMPLEX_NORM_ZERO] THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONTINUOUS_ON_ADD THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_SIMP_TAC[CONTINUOUS_ON_CX_LIFT; CONTINUOUS_ON_LIFT_NORM_COMPOSE] THEN MATCH_MP_TAC CONTINUOUS_ON_CLOG THEN ASM_SIMP_TAC[IMP_CONJ; FORALL_IN_IMAGE; RE_CX; COMPLEX_NORM_NZ]]]);; let BORSUKIAN_CONTINUOUS_LOGARITHM_CIRCLE_CX = prove (`!s:real^N->bool. borsukian s <=> !f. f continuous_on s /\ IMAGE f s SUBSET sphere(Cx(&0),&1) ==> ?g. (Cx o g) continuous_on s /\ (!x. x IN s ==> f x = cexp(ii * Cx(g x)))`, GEN_TAC THEN REWRITE_TAC[BORSUKIAN_CONTINUOUS_LOGARITHM_CIRCLE] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; COMPLEX_IN_SPHERE_0] THEN EQ_TAC THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `f:real^N->complex` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THENL [X_GEN_TAC `g:real^N->complex` THEN STRIP_TAC THEN EXISTS_TAC `Im o (g:real^N->complex)` THEN ASM_SIMP_TAC[CONTINUOUS_ON_CX_IM; CONTINUOUS_ON_COMPOSE; o_ASSOC] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`)) THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `(f:real^N->complex) x = cexp(g x)` THEN ASM_REWRITE_TAC[NORM_CEXP; o_DEF; REAL_EXP_EQ_1] THEN DISCH_TAC THEN AP_TERM_TAC THEN ASM_REWRITE_TAC[COMPLEX_EQ; RE_MUL_II; IM_MUL_II; RE_CX; IM_CX] THEN REWRITE_TAC[REAL_NEG_0]; X_GEN_TAC `g:real^N->real` THEN STRIP_TAC THEN EXISTS_TAC `\x:real^N. ii * Cx(g x)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPLEX_LMUL THEN ASM_REWRITE_TAC[GSYM o_DEF]]);; let BORSUKIAN_CIRCLE = prove (`!s:real^N->bool. borsukian s <=> !f. f continuous_on s /\ IMAGE f s SUBSET sphere(Cx(&0),&1) ==> ?a. homotopic_with (\h. T) (s,sphere(Cx(&0),&1)) f (\x. a)`, REWRITE_TAC[GSYM COMPLEX_VEC_0] THEN REWRITE_TAC[INESSENTIAL_EQ_CONTINUOUS_LOGARITHM_CIRCLE] THEN REWRITE_TAC[BORSUKIAN_CONTINUOUS_LOGARITHM_CIRCLE_CX] THEN REWRITE_TAC[COMPLEX_VEC_0]);; let BORSUKIAN_CIRCLE_ALT = prove (`!s:real^N->bool. borsukian s <=> !f g:real^N->real^2. f continuous_on s /\ IMAGE f s SUBSET sphere(Cx(&0),&1) /\ g continuous_on s /\ IMAGE g s SUBSET sphere(Cx(&0),&1) ==> homotopic_with (\h. T) (s,sphere(Cx(&0),&1)) f g`, REWRITE_TAC[BORSUKIAN_CIRCLE; HOMOTOPIC_TRIVIALITY] THEN SIMP_TAC[PATH_CONNECTED_SPHERE; DIMINDEX_2; LE_REFL]);; let CONTRACTIBLE_IMP_BORSUKIAN = prove (`!s:real^N->bool. contractible s ==> borsukian s`, SIMP_TAC[borsukian; CONTRACTIBLE_IMP_PATH_CONNECTED] THEN MESON_TAC[NULLHOMOTOPIC_FROM_CONTRACTIBLE]);; let CONIC_IMP_BORSUKIAN = prove (`!s:real^N->bool. conic s ==> borsukian s`, MESON_TAC[CONIC_IMP_CONTRACTIBLE; CONTRACTIBLE_IMP_BORSUKIAN]);; let SIMPLY_CONNECTED_IMP_BORSUKIAN = prove (`!s:real^N->bool. simply_connected s /\ locally path_connected s ==> borsukian s`, SIMP_TAC[BORSUKIAN_CONTINUOUS_LOGARITHM] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_LOGARITHM_ON_SIMPLY_CONNECTED THEN ASM SET_TAC[]);; let STARLIKE_IMP_BORSUKIAN = prove (`!s:real^N->bool. starlike s ==> borsukian s`, SIMP_TAC[CONTRACTIBLE_IMP_BORSUKIAN; STARLIKE_IMP_CONTRACTIBLE]);; let BORSUKIAN_EMPTY = prove (`borsukian({}:real^N->bool)`, SIMP_TAC[CONTRACTIBLE_IMP_BORSUKIAN; CONTRACTIBLE_EMPTY]);; let BORSUKIAN_UNIV = prove (`borsukian(:real^N)`, SIMP_TAC[CONTRACTIBLE_IMP_BORSUKIAN; CONTRACTIBLE_UNIV]);; let CONVEX_IMP_BORSUKIAN = prove (`!s:real^N->bool. convex s ==> borsukian s`, MESON_TAC[STARLIKE_IMP_BORSUKIAN; CONVEX_IMP_STARLIKE; BORSUKIAN_EMPTY]);; let BORSUKIAN_1_GEN = prove (`!s:real^N->bool. (dimindex(:N) = 1 \/ ?r:real^1->bool. s homeomorphic r) ==> borsukian s`, GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[BORSUKIAN_CIRCLE] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC COHOMOTOPICALLY_TRIVIAL_1D THEN ASM_REWRITE_TAC[ANR_SPHERE; CONNECTED_SPHERE_EQ] THEN REWRITE_TAC[DIMINDEX_2; LE_REFL]);; let BORSUKIAN_1 = prove (`!s:real^1->bool. borsukian s`, GEN_TAC THEN MATCH_MP_TAC BORSUKIAN_1_GEN THEN REWRITE_TAC[DIMINDEX_1]);; let BORSUKIAN_SPHERE = prove (`!a:real^N r. borsukian(sphere(a,r)) <=> r <= &0 \/ ~(dimindex(:N) = 2)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `r < &0` THEN ASM_SIMP_TAC[SPHERE_EMPTY; BORSUKIAN_EMPTY; REAL_LT_IMP_LE] THEN ASM_CASES_TAC `r = &0` THEN ASM_SIMP_TAC[REAL_LT_REFL; SPHERE_SING; CONVEX_IMP_BORSUKIAN; CONVEX_SING; GSYM REAL_NOT_LT] THEN SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ASM_REWRITE_TAC[]] THEN ASM_CASES_TAC `dimindex(:N) = 1` THENL [ASM_SIMP_TAC[ARITH; BORSUKIAN_1_GEN]; ALL_TAC] THEN ASM_CASES_TAC `dimindex(:N) = 2` THEN ASM_REWRITE_TAC[] THENL [SUBGOAL_THEN `~borsukian(sphere(Cx(&0),&1))` MP_TAC THENL [REWRITE_TAC[BORSUKIAN_CIRCLE] THEN DISCH_THEN(MP_TAC o SPEC `\z:complex. z`) THEN REWRITE_TAC[CONTINUOUS_ON_ID; IMAGE_ID; SUBSET_REFL] THEN ASM_REWRITE_TAC[GSYM contractible; CONTRACTIBLE_SPHERE] THEN REAL_ARITH_TAC; REWRITE_TAC[CONTRAPOS_THM] THEN MATCH_MP_TAC (ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHIC_BORSUKIAN) THEN REWRITE_TAC[HOMEOMORPHIC_SPHERES_EQ] THEN ASM_REWRITE_TAC[DIMINDEX_2; REAL_LT_01]]; MATCH_MP_TAC SIMPLY_CONNECTED_IMP_BORSUKIAN THEN ASM_SIMP_TAC[SIMPLY_CONNECTED_SPHERE] THEN REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] (GSYM dist)] THEN ASM_SIMP_TAC[LOCALLY_PATH_CONNECTED_SPHERE; SIMPLY_CONNECTED_SPHERE_EQ] THEN DISJ1_TAC THEN MATCH_MP_TAC(ARITH_RULE `1 <= n /\ ~(n = 1) /\ ~(n = 2) ==> 3 <= n`) THEN ASM_REWRITE_TAC[DIMINDEX_GE_1]]);; let BORSUKIAN_OPEN_UNION = prove (`!s t:real^N->bool. open_in (subtopology euclidean (s UNION t)) s /\ open_in (subtopology euclidean (s UNION t)) t /\ borsukian s /\ borsukian t /\ connected(s INTER t) ==> borsukian(s UNION t)`, REPEAT GEN_TAC THEN SIMP_TAC[BORSUKIAN_CONTINUOUS_LOGARITHM] THEN STRIP_TAC THEN X_GEN_TAC `f:real^N->complex` THEN STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `f:real^N->complex`)) THEN ANTS_TAC THENL [CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNION]; ASM SET_TAC[]]; DISCH_THEN(X_CHOOSE_THEN `g:real^N->complex` STRIP_ASSUME_TAC)] THEN ANTS_TAC THENL [CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNION]; ASM SET_TAC[]]; DISCH_THEN(X_CHOOSE_THEN `h:real^N->complex` STRIP_ASSUME_TAC)] THEN ASM_CASES_TAC `s INTER t:real^N->bool = {}` THENL [EXISTS_TAC `(\x. if x IN s then g x else h x):real^N->complex` THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL_OPEN THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`(\x. g x - h x):real^N->complex`; `s INTER t:real^N->bool`] CONTINUOUS_DISCRETE_RANGE_CONSTANT) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_SUB THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; INTER_SUBSET]; X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_INTER] THEN STRIP_TAC THEN EXISTS_TAC `&2 * pi` THEN REWRITE_TAC[REAL_ARITH `&0 < &2 * x <=> &0 < x`; PI_POS] THEN X_GEN_TAC `y:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[REAL_NOT_LE] THEN ONCE_REWRITE_TAC[COMPLEX_RING `a - b:complex = c - d <=> a - c = b - d`] THEN DISCH_TAC THEN MATCH_MP_TAC COMPLEX_EQ_CEXP THEN REWRITE_TAC[CEXP_SUB] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] REAL_LET_TRANS)) THEN GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [COMPLEX_RING `(a - b) - (c - d):complex = (a - c) - (b - d)`] THEN REWRITE_TAC[GSYM IM_SUB; COMPLEX_NORM_GE_RE_IM]]; REWRITE_TAC[IN_INTER; COMPLEX_EQ_SUB_RADD] THEN DISCH_THEN(X_CHOOSE_TAC `a:complex`) THEN EXISTS_TAC `(\x. if x IN s then g x else a + h x):real^N->complex` THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL_OPEN THEN ASM_SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_CONST] THEN ASM SET_TAC[]; GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL [ASM_MESON_TAC[]; REWRITE_TAC[CEXP_ADD]] THEN SUBGOAL_THEN `?y:real^N. y IN s /\ y IN t` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `cexp(a + h(y:real^N)) = cexp(h y)` MP_TAC THENL [ASM_MESON_TAC[]; REWRITE_TAC[CEXP_ADD]] THEN SIMP_TAC[COMPLEX_RING `a * z = z <=> a = Cx(&1) \/ z = Cx(&0)`; CEXP_NZ; COMPLEX_MUL_LID] THEN ASM SET_TAC[]]]);; let BORSUKIAN_CLOSED_UNION = prove (`!s t:real^N->bool. closed_in (subtopology euclidean (s UNION t)) s /\ closed_in (subtopology euclidean (s UNION t)) t /\ borsukian s /\ borsukian t /\ connected(s INTER t) ==> borsukian(s UNION t)`, REPEAT GEN_TAC THEN SIMP_TAC[BORSUKIAN_CONTINUOUS_LOGARITHM] THEN STRIP_TAC THEN X_GEN_TAC `f:real^N->complex` THEN STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `f:real^N->complex`)) THEN ANTS_TAC THENL [CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNION]; ASM SET_TAC[]]; DISCH_THEN(X_CHOOSE_THEN `g:real^N->complex` STRIP_ASSUME_TAC)] THEN ANTS_TAC THENL [CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNION]; ASM SET_TAC[]]; DISCH_THEN(X_CHOOSE_THEN `h:real^N->complex` STRIP_ASSUME_TAC)] THEN ASM_CASES_TAC `s INTER t:real^N->bool = {}` THENL [EXISTS_TAC `(\x. if x IN s then g x else h x):real^N->complex` THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`(\x. g x - h x):real^N->complex`; `s INTER t:real^N->bool`] CONTINUOUS_DISCRETE_RANGE_CONSTANT) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_SUB THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; INTER_SUBSET]; X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_INTER] THEN STRIP_TAC THEN EXISTS_TAC `&2 * pi` THEN REWRITE_TAC[REAL_ARITH `&0 < &2 * x <=> &0 < x`; PI_POS] THEN X_GEN_TAC `y:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[REAL_NOT_LE] THEN ONCE_REWRITE_TAC[COMPLEX_RING `a - b:complex = c - d <=> a - c = b - d`] THEN DISCH_TAC THEN MATCH_MP_TAC COMPLEX_EQ_CEXP THEN REWRITE_TAC[CEXP_SUB] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] REAL_LET_TRANS)) THEN GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [COMPLEX_RING `(a - b) - (c - d):complex = (a - c) - (b - d)`] THEN REWRITE_TAC[GSYM IM_SUB; COMPLEX_NORM_GE_RE_IM]]; REWRITE_TAC[IN_INTER; COMPLEX_EQ_SUB_RADD] THEN DISCH_THEN(X_CHOOSE_TAC `a:complex`) THEN EXISTS_TAC `(\x. if x IN s then g x else a + h x):real^N->complex` THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL THEN ASM_SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_CONST] THEN ASM SET_TAC[]; GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL [ASM_MESON_TAC[]; REWRITE_TAC[CEXP_ADD]] THEN SUBGOAL_THEN `?y:real^N. y IN s /\ y IN t` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `cexp(a + h(y:real^N)) = cexp(h y)` MP_TAC THENL [ASM_MESON_TAC[]; REWRITE_TAC[CEXP_ADD]] THEN SIMP_TAC[COMPLEX_RING `a * z = z <=> a = Cx(&1) \/ z = Cx(&0)`; CEXP_NZ; COMPLEX_MUL_LID] THEN ASM SET_TAC[]]]);; let BORSUKIAN_SEPARATION_COMPACT = prove (`!s:real^2->bool. compact s ==> (borsukian s <=> connected((:real^2) DIFF s))`, SIMP_TAC[BORSUKIAN_CIRCLE; BORSUK_SEPARATION_THEOREM; DIMINDEX_2; LE_REFL; COMPLEX_VEC_0]);; let BORSUKIAN_COMPONENTWISE_EQ = prove (`!s:real^N->bool. locally connected s \/ compact s ==> (borsukian s <=> !c. c IN components s ==> borsukian c)`, GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[BORSUKIAN_ALT] THEN MATCH_MP_TAC COHOMOTOPICALLY_TRIVIAL_ON_COMPONENTS THEN ASM_SIMP_TAC[OPEN_IMP_ANR; OPEN_DIFF; OPEN_UNIV; CLOSED_SING]);; let BORSUKIAN_COMPONENTWISE = prove (`!s:real^N->bool. (locally connected s \/ compact s) /\ (!c. c IN components s ==> borsukian c) ==> borsukian s`, MESON_TAC[BORSUKIAN_COMPONENTWISE_EQ]);; let BORSUKIAN_MONOTONE_IMAGE_COMPACT = prove (`!f:real^M->real^N s t. f continuous_on s /\ IMAGE f s = t /\ compact s /\ (!y. y IN t ==> connected {x | x IN s /\ f x = y}) /\ borsukian s ==> borsukian t`, REPEAT STRIP_TAC THEN REWRITE_TAC[BORSUKIAN_CONTINUOUS_LOGARITHM] THEN X_GEN_TAC `g:real^N->complex` THEN STRIP_TAC THEN FIRST_X_ASSUM (MP_TAC o GEN_REWRITE_RULE I [BORSUKIAN_CONTINUOUS_LOGARITHM]) THEN DISCH_THEN(MP_TAC o SPEC `(g:real^N->complex) o (f:real^M->real^N)`) THEN ASM_SIMP_TAC[IMAGE_o; CONTINUOUS_ON_COMPOSE; o_THM] THEN DISCH_THEN(X_CHOOSE_THEN `h:real^M->complex` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `!y. ?x. y IN t ==> x IN s /\ (f:real^M->real^N) x = y` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `f':real^N->real^M` THEN STRIP_TAC THEN EXISTS_TAC `(h:real^M->complex) o (f':real^N->real^M)` THEN REWRITE_TAC[o_THM] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_FROM_CLOSED_GRAPH THEN EXISTS_TAC `IMAGE (h:real^M->complex) s` THEN ASM_SIMP_TAC[COMPACT_CONTINUOUS_IMAGE; IMAGE_o] THEN CONJ_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[o_THM]] THEN SUBGOAL_THEN `{pastecart x ((h:real^M->complex) ((f':real^N->real^M) x)) | x IN t} = {p | ?x. x IN s /\ pastecart x p IN {z | z IN s PCROSS UNIV /\ (sndcart z - pastecart (f(fstcart z)) (h(fstcart z))) IN {vec 0}}}` SUBST1_TAC THENL [ALL_TAC; MATCH_MP_TAC CLOSED_COMPACT_PROJECTION THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE THEN ASM_SIMP_TAC[CLOSED_UNIV; CLOSED_PCROSS; COMPACT_IMP_CLOSED] THEN REWRITE_TAC[CLOSED_SING] THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_SNDCART] THEN MATCH_MP_TAC CONTINUOUS_ON_PASTECART THEN REWRITE_TAC[GSYM o_DEF] THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; IMAGE_FSTCART_PCROSS] THEN ASM_REWRITE_TAC[UNIV_NOT_EMPTY]] THEN REWRITE_TAC[IN_ELIM_THM; PASTECART_IN_PCROSS; FSTCART_PASTECART; SNDCART_PASTECART; IN_UNIV; IN_SING; VECTOR_SUB_EQ] THEN REWRITE_TAC[EXTENSION; FORALL_PASTECART; IN_ELIM_THM] THEN REWRITE_TAC[CONJ_ASSOC; PASTECART_INJ] THEN MAP_EVERY X_GEN_TAC [`y:real^N`; `z:complex`] THEN ONCE_REWRITE_TAC[TAUT `(p /\ q) /\ r <=> q /\ p /\ r`] THEN REWRITE_TAC[UNWIND_THM1] THEN EQ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `w:real^M` STRIP_ASSUME_TAC) THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?a. !x. x IN {x | x IN s /\ (f:real^M->real^N) x = y} ==> h x - h(f' y):complex = a` MP_TAC THENL [ALL_TAC; REWRITE_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `a:complex` THEN DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN DISCH_THEN(MP_TAC o SPEC `(f':real^N->real^M) y`) THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[VECTOR_SUB_REFL]] THEN DISCH_THEN(SUBST_ALL_TAC o SYM) THEN RULE_ASSUM_TAC(REWRITE_RULE[VECTOR_SUB_EQ]) THEN ASM SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_DISCRETE_RANGE_CONSTANT THEN REWRITE_TAC[IN_ELIM_THM] THEN REPEAT CONJ_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_ON_SUB THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN SET_TAC[]; ALL_TAC] THEN X_GEN_TAC `v:real^M` THEN STRIP_TAC THEN EXISTS_TAC `&2 * pi` THEN REWRITE_TAC[REAL_ARITH `&0 < &2 * x <=> &0 < x`; PI_POS] THEN X_GEN_TAC `u:real^M` THEN REWRITE_TAC[COMPLEX_RING `a - x:complex = b - x <=> a = b`] THEN DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[REAL_NOT_LE] THEN REWRITE_TAC[COMPLEX_RING `(a - x) - (b - x):complex = a - b`] THEN DISCH_TAC THEN MATCH_MP_TAC COMPLEX_EQ_CEXP THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[GSYM IM_SUB] THEN ASM_MESON_TAC[REAL_LET_TRANS; COMPLEX_NORM_GE_RE_IM]);; let BORSUKIAN_OPEN_MAP_IMAGE_COMPACT = prove (`!f:real^M->real^N s t. f continuous_on s /\ IMAGE f s = t /\ compact s /\ (!u. open_in (subtopology euclidean s) u ==> open_in (subtopology euclidean t) (IMAGE f u)) /\ borsukian s ==> borsukian t`, REPEAT GEN_TAC THEN REWRITE_TAC[BORSUKIAN_CONTINUOUS_LOGARITHM_CIRCLE_CX] THEN STRIP_TAC THEN X_GEN_TAC `g:real^N->complex` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `(g:real^N->complex) o (f:real^M->real^N)`) THEN ASM_SIMP_TAC[IMAGE_o; CONTINUOUS_ON_COMPOSE; o_THM] THEN DISCH_THEN(X_CHOOSE_THEN `h:real^M->real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `!y. ?x. y IN t ==> x IN s /\ (f:real^M->real^N) x = y /\ (!x'. x' IN s /\ f x' = y ==> h x <= h x')` MP_TAC THENL [REWRITE_TAC[RIGHT_EXISTS_IMP_THM] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPEC `{ h x:real | x IN s /\ (f:real^M->real^N) x = y}` COMPACT_ATTAINS_INF) THEN REWRITE_TAC[FORALL_IN_GSPEC; EXISTS_IN_GSPEC; GSYM CONJ_ASSOC] THEN DISCH_THEN MATCH_MP_TAC THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN REWRITE_TAC[GSYM IMAGE_o] THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN CONJ_TAC THENL [REWRITE_TAC[o_DEF; GSYM CONTINUOUS_ON_CX_LIFT] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN SET_TAC[]; ONCE_REWRITE_TAC[SET_RULE `x = y <=> x IN {y}`] THEN MATCH_MP_TAC PROPER_MAP_FROM_COMPACT THEN ASM_MESON_TAC[CLOSED_IN_SING; SUBSET_REFL]]; REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM]] THEN X_GEN_TAC `k:real^N->real^M` THEN DISCH_TAC THEN EXISTS_TAC `(h:real^M->real) o (k:real^N->real^M)` THEN REWRITE_TAC[o_THM] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN REWRITE_TAC[continuous_on] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(ISPECL [`Cx o (h:real^M->real)`; `s:real^M->bool`] COMPACT_UNIFORMLY_CONTINUOUS) THEN ASM_REWRITE_TAC[uniformly_continuous_on] THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[o_THM; DIST_CX] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`\y. {x | x IN s /\ (f:real^M->real^N) x = y}`; `s:real^M->bool`; `t:real^N->bool`] UPPER_LOWER_HEMICONTINUOUS_EXPLICIT) THEN ASM_SIMP_TAC[GSYM CLOSED_MAP_IFF_UPPER_HEMICONTINUOUS_PREIMAGE; GSYM OPEN_MAP_IFF_LOWER_HEMICONTINUOUS_PREIMAGE; SUBSET_REFL; SUBSET_RESTRICT] THEN ANTS_TAC THENL [ASM_MESON_TAC[CONTINUOUS_IMP_CLOSED_MAP]; ALL_TAC] THEN DISCH_THEN(MP_TAC o SPECL [`y:real^N`; `d:real`]) THEN ASM_REWRITE_TAC[FORALL_IN_GSPEC; EXISTS_IN_GSPEC] THEN ANTS_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC COMPACT_IMP_BOUNDED; ASM SET_TAC[]] THEN MATCH_MP_TAC CLOSED_IN_COMPACT THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[SET_RULE `x IN s /\ f x = y <=> x IN s /\ f x IN {y}`] THEN MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE_GEN THEN EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[CLOSED_IN_SING; SUBSET_REFL]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `q:real` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "*"))] THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `y':real^N` THEN STRIP_TAC THEN REMOVE_THEN "*" (MP_TAC o SPEC `y':real^N`) THEN ANTS_TAC THENL [ASM_MESON_TAC[DIST_SYM]; ALL_TAC] THEN DISCH_THEN(CONJUNCTS_THEN2 (MP_TAC o SPEC `(k:real^N->real^M) y`) (MP_TAC o SPEC `(k:real^N->real^M) y'`)) THEN ASM_SIMP_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `w:real^M` STRIP_ASSUME_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `w':real^M` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o SPEC `y':real^N`) THEN FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN ASM_SIMP_TAC[] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(MP_TAC o SPEC `w:real^M`) THEN ASM_SIMP_TAC[] THEN DISCH_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(MP_TAC o SPEC `w':real^M`) THEN ASM_SIMP_TAC[] THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o SPECL [`w:real^M`; `(k:real^N->real^M) y'`]) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`w':real^M`; `(k:real^N->real^M) y`]) THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC);; (* ------------------------------------------------------------------------- *) (* Unicoherence (closed). *) (* ------------------------------------------------------------------------- *) let unicoherent = new_definition `unicoherent(u:real^N->bool) <=> !s t. connected s /\ connected t /\ s UNION t = u /\ closed_in (subtopology euclidean u) s /\ closed_in (subtopology euclidean u) t ==> connected (s INTER t)`;; let HOMEOMORPHIC_UNICOHERENT = prove (`!s:real^M->bool t:real^N->bool. s homeomorphic t /\ unicoherent s ==> unicoherent t`, 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 [`f:real^M->real^N`; `g:real^N->real^M`] THEN STRIP_TAC THEN REWRITE_TAC[unicoherent] THEN MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `v:real^N->bool`] THEN STRIP_TAC THEN SUBGOAL_THEN `u INTER v = IMAGE (f:real^M->real^N) (IMAGE (g:real^N->real^M) u INTER IMAGE g v)` SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [unicoherent]) THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r /\ s <=> r /\ (p /\ q) /\ s`] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]; CONJ_TAC THEN MATCH_MP_TAC HOMEOMORPHISM_IMP_CLOSED_MAP THEN MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `t:real^N->bool`] THEN ASM_REWRITE_TAC[homeomorphism]]);; let HOMEOMORPHIC_UNICOHERENT_EQ = prove (`!s:real^M->bool t:real^N->bool. s homeomorphic t ==> (unicoherent s <=> unicoherent t)`, REPEAT STRIP_TAC THEN EQ_TAC THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] HOMEOMORPHIC_UNICOHERENT) THEN ASM_MESON_TAC[HOMEOMORPHIC_SYM]);; let UNICOHERENT_TRANSLATION = prove (`!a:real^N s. unicoherent (IMAGE (\x. a + x) s) <=> unicoherent s`, REPEAT GEN_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_UNICOHERENT_EQ THEN REWRITE_TAC[HOMEOMORPHIC_TRANSLATION_SELF]);; add_translation_invariants [UNICOHERENT_TRANSLATION];; let UNICOHERENT_INJECTIVE_LINEAR_IMAGE = prove (`!f:real^M->real^N s. linear f /\ (!x y. f x = f y ==> x = y) ==> (unicoherent(IMAGE f s) <=> unicoherent s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_UNICOHERENT_EQ THEN ASM_MESON_TAC[HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_SELF; HOMEOMORPHIC_REFL]);; add_linear_invariants [UNICOHERENT_INJECTIVE_LINEAR_IMAGE];; let BORSUKIAN_IMP_UNICOHERENT = prove (`!u:real^N->bool. borsukian u ==> unicoherent u`, GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[unicoherent] THEN SUBGOAL_THEN `!f. f continuous_on u /\ IMAGE f u SUBSET sphere(vec 0,&1) ==> ?a. homotopic_with (\h. T) (u,(:complex) DIFF {Cx (&0)}) (f:real^N->complex) (\t. a)` MP_TAC THENL [FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [BORSUKIAN_CIRCLE]) THEN X_GEN_TAC `f:real^N->complex` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `f:real^N->complex`) THEN ASM_REWRITE_TAC[GSYM COMPLEX_VEC_0] THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_WITH_SUBSET_RIGHT) THEN REWRITE_TAC[SET_RULE `s SUBSET UNIV DIFF {a} <=> ~(a IN s)`] THEN REWRITE_TAC[IN_SPHERE; DIST_REFL] THEN REAL_ARITH_TAC; POP_ASSUM(K ALL_TAC)] THEN REWRITE_TAC[sphere; DIST_0; INESSENTIAL_EQ_CONTINUOUS_LOGARITHM] THEN REPEAT STRIP_TAC THEN SIMP_TAC[CONNECTED_CLOSED_IN_EQ; NOT_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`v:real^N->bool`; `w:real^N->bool`] THEN STRIP_TAC THEN SUBGOAL_THEN `closed_in (subtopology euclidean u) (v:real^N->bool) /\ closed_in (subtopology euclidean u) (w:real^N->bool)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[CLOSED_IN_INTER; CLOSED_IN_TRANS]; ALL_TAC] THEN MP_TAC(ISPECL [`v:real^N->bool`; `w:real^N->bool`; `u:real^N->bool`; `vec 0:real^1`; `vec 1:real^1`] URYSOHN_LOCAL) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `q:real^N->real^1` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `?g:real^N->real^2. g continuous_on u /\ IMAGE g u SUBSET {x | norm x = &1} /\ (!x. x IN s ==> g(x) = cexp(Cx pi * ii * Cx(drop(q x)))) /\ (!x. x IN t ==> g(x) = inv(cexp(Cx pi * ii * Cx(drop(q x)))))` (DESTRUCT_TAC "@g. cont circle s t") THENL [EXISTS_TAC `\x. if (x:real^N) IN s then cexp(Cx pi * ii * Cx(drop(q x))) else inv(cexp(Cx pi * ii * Cx(drop(q x))))` THEN SUBGOAL_THEN `!x:real^N. x IN s INTER t ==> cexp(Cx pi * ii * Cx(drop(q x))) = inv(cexp(Cx pi * ii * Cx(drop (q x))))` ASSUME_TAC THENL [SUBST1_TAC(SYM(ASSUME `v UNION w:real^N->bool = s INTER t`)) THEN REWRITE_TAC[IN_UNION] THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[] THEN REWRITE_TAC[DROP_VEC; COMPLEX_MUL_RZERO; CEXP_0; COMPLEX_INV_1] THEN REWRITE_TAC[COMPLEX_MUL_RID; EULER] THEN REWRITE_TAC[RE_MUL_CX; IM_MUL_CX; RE_MUL_II; IM_MUL_II] THEN REWRITE_TAC[RE_II; IM_II; REAL_MUL_RZERO; REAL_MUL_RID] THEN REWRITE_TAC[REAL_EXP_0; COMPLEX_MUL_LID; COS_PI; SIN_PI] THEN REWRITE_TAC[COMPLEX_MUL_RZERO; COMPLEX_ADD_RID] THEN CONV_TAC COMPLEX_RING; ALL_TAC] THEN SIMP_TAC[] THEN REPEAT CONJ_TAC THENL [EXPAND_TAC "u" THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL THEN ASM_REWRITE_TAC[SET_RULE `P /\ ~P \/ x IN t /\ x IN s <=> x IN s INTER t`] THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC CONTINUOUS_ON_COMPLEX_INV THEN REWRITE_TAC[CEXP_NZ]] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN REWRITE_TAC[CONTINUOUS_ON_CEXP] THEN REWRITE_TAC[COMPLEX_MUL_ASSOC] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPLEX_MUL THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN MATCH_MP_TAC CONTINUOUS_ON_CX_DROP THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNION]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN REWRITE_TAC[COMPLEX_NORM_INV; NORM_CEXP] THEN REWRITE_TAC[RE_MUL_CX; RE_MUL_II; IM_CX] THEN REWRITE_TAC[REAL_MUL_RZERO; REAL_NEG_0; REAL_EXP_0; REAL_INV_1]; GEN_TAC THEN DISCH_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]]; FIRST_X_ASSUM(MP_TAC o SPEC `g:real^N->complex`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `h:real^N->complex` STRIP_ASSUME_TAC)] THEN SUBGOAL_THEN `(?n. integer n /\ !x:real^N. x IN s ==> h(x) - Cx pi * ii * Cx (drop (q x)) = Cx(&2 * n * pi) * ii) /\ (?n. integer n /\ !x:real^N. x IN t ==> h(x) + Cx pi * ii * Cx (drop (q x)) = Cx(&2 * n * pi) * ii)` (CONJUNCTS_THEN2 (X_CHOOSE_THEN `m:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) (X_CHOOSE_THEN `n:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC))) THENL [CONJ_TAC THEN MATCH_MP_TAC(MESON[] `(?x. x IN s) /\ (!x. x IN s ==> ?n. P n /\ f x = k n) /\ (?a. !x. x IN s ==> f x = a) ==> (?n. P n /\ !x. x IN s ==> f x = k n)`) THEN (CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN (CONJ_TAC THENL [REWRITE_TAC[COMPLEX_RING `a + b:complex = c <=> a = --b + c`; COMPLEX_RING `a - b:complex = c <=> a = b + c`] THEN REWRITE_TAC[GSYM CEXP_EQ; CEXP_NEG] THEN ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(LABEL_TAC "*") THEN MATCH_MP_TAC CONTINUOUS_DISCRETE_RANGE_CONSTANT THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [(MATCH_MP_TAC CONTINUOUS_ON_ADD ORELSE MATCH_MP_TAC CONTINUOUS_ON_SUB) THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNION]; ALL_TAC] THEN REWRITE_TAC[COMPLEX_MUL_ASSOC] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPLEX_MUL THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN MATCH_MP_TAC CONTINUOUS_ON_CX_DROP THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNION]; X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN EXISTS_TAC `&2 * pi` THEN REWRITE_TAC[REAL_ARITH `&0 < &2 * x <=> &0 < x`; PI_POS] THEN X_GEN_TAC `y:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REMOVE_THEN "*" (fun th -> MP_TAC(SPEC `y:real^N` th) THEN MP_TAC(SPEC `x:real^N` th)) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[COMPLEX_EQ_MUL_RCANCEL; II_NZ; GSYM COMPLEX_SUB_RDISTRIB; COMPLEX_NORM_MUL; CX_INJ; COMPLEX_NORM_II; REAL_MUL_RID] THEN REWRITE_TAC[GSYM CX_SUB; COMPLEX_NORM_CX] THEN REWRITE_TAC[REAL_EQ_MUL_LCANCEL; GSYM REAL_SUB_LDISTRIB] THEN REWRITE_TAC[GSYM REAL_SUB_RDISTRIB; REAL_ABS_MUL] THEN REWRITE_TAC[REAL_EQ_MUL_RCANCEL; PI_NZ; REAL_ABS_PI] THEN REWRITE_TAC[REAL_ABS_NUM; REAL_OF_NUM_EQ; ARITH_EQ] THEN DISCH_TAC THEN REWRITE_TAC[REAL_ARITH `&2 * p <= &2 * a * p <=> &0 <= &2 * p * (a - &1)`] THEN MATCH_MP_TAC REAL_LE_MUL THEN REWRITE_TAC[REAL_POS] THEN MATCH_MP_TAC REAL_LE_MUL THEN REWRITE_TAC[PI_POS_LE; REAL_SUB_LE] THEN MATCH_MP_TAC REAL_ABS_INTEGER_LEMMA THEN ASM_SIMP_TAC[INTEGER_CLOSED; REAL_SUB_0]]); ALL_TAC] THEN GEN_REWRITE_TAC I [TAUT `p ==> q ==> F <=> ~(p /\ q)`] THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `(!x. x IN s ==> P x) /\ (!x. x IN t ==> Q x) ==> ~(v = {}) /\ ~(w = {}) /\ v UNION w SUBSET s INTER t ==> ~(!y z. y IN v /\ z IN w ==> ~(P y /\ Q y /\ P z /\ Q z))`)) THEN ANTS_TAC THENL [ASM SET_TAC[]; ASM_SIMP_TAC[]] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP (COMPLEX_RING `y + p = n /\ y - p = m /\ z + q = n /\ z - q = m ==> q:complex = p`)) THEN REWRITE_TAC[DROP_VEC; COMPLEX_MUL_RZERO; COMPLEX_ENTIRE; CX_INJ] THEN REWRITE_TAC[PI_NZ; II_NZ; REAL_OF_NUM_EQ; ARITH_EQ]);; let CONTRACTIBLE_IMP_UNICOHERENT = prove (`!u:real^N->bool. contractible u ==> unicoherent u`, SIMP_TAC[BORSUKIAN_IMP_UNICOHERENT; CONTRACTIBLE_IMP_BORSUKIAN]);; let CONVEX_IMP_UNICOHERENT = prove (`!u:real^N->bool. convex u ==> unicoherent u`, SIMP_TAC[BORSUKIAN_IMP_UNICOHERENT; CONVEX_IMP_BORSUKIAN]);; let UNICOHERENT_UNIV = prove (`unicoherent(:real^N)`, SIMP_TAC[CONVEX_IMP_UNICOHERENT; CONVEX_UNIV]);; let UNICOHERENT_MONOTONE_IMAGE_COMPACT = prove (`!f:real^M->real^N s t. f continuous_on s /\ IMAGE f s = t /\ compact s /\ (!y. y IN t ==> connected {x | x IN s /\ f x = y}) /\ unicoherent s ==> unicoherent t`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `compact(t:real^N->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[COMPACT_CONTINUOUS_IMAGE]; REWRITE_TAC[unicoherent]] THEN MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `v:real^N->bool`] THEN ASM_SIMP_TAC[CLOSED_IN_CLOSED_EQ; COMPACT_IMP_CLOSED] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [unicoherent]) THEN DISCH_THEN(MP_TAC o SPECL [`{x | x IN s /\ (f:real^M->real^N) x IN u}`; `{x | x IN s /\ (f:real^M->real^N) x IN v}`]) THEN ASM_SIMP_TAC[CLOSED_IN_CLOSED_EQ; COMPACT_IMP_CLOSED; SUBSET_RESTRICT; CONTINUOUS_CLOSED_PREIMAGE; CONJ_ASSOC] THEN REWRITE_TAC[IMP_CONJ_ALT] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`; `t:real^N->bool`] CONNECTED_CLOSED_MONOTONE_PREIMAGE) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_MESON_TAC[CONTINUOUS_IMP_CLOSED_MAP]; ALL_TAC] THEN DISCH_TAC THEN ASM_SIMP_TAC[] THEN DISCH_THEN(MP_TAC o ISPEC `f:real^M->real^N` o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] CONNECTED_CONTINUOUS_IMAGE)) THEN ANTS_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN SET_TAC[]; MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN ASM SET_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Several common variants of unicoherence for R^n. *) (* ------------------------------------------------------------------------- *) let CONNECTED_FRONTIER_SIMPLE = prove (`!s. connected(s) /\ connected((:real^N) DIFF s) ==> connected(frontier s)`, REPEAT STRIP_TAC THEN REWRITE_TAC[FRONTIER_CLOSURES] THEN MATCH_MP_TAC(REWRITE_RULE[unicoherent] UNICOHERENT_UNIV) THEN REWRITE_TAC[SUBTOPOLOGY_UNIV; GSYM CLOSED_IN] THEN ASM_SIMP_TAC[CLOSED_CLOSURE; CONNECTED_CLOSURE] THEN 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[]);; let CONNECTED_FRONTIER_COMPONENT_COMPLEMENT = prove (`!s c:real^N->bool. connected s /\ c IN components((:real^N) DIFF s) ==> connected(frontier c)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CONNECTED_FRONTIER_SIMPLE THEN CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]; MATCH_MP_TAC COMPONENT_COMPLEMENT_CONNECTED THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[SUBSET_UNIV; CONNECTED_UNIV]]);; let CONNECTED_FRONTIER_DISJOINT = prove (`!s t:real^N->bool. connected s /\ connected t /\ DISJOINT s t /\ frontier s SUBSET frontier t ==> connected(frontier s)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s = (:real^N)` THEN ASM_REWRITE_TAC[FRONTIER_UNIV; CONNECTED_EMPTY] THEN SUBGOAL_THEN `?c. c IN components((:real^N) DIFF s) /\ t SUBSET c` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC EXISTS_COMPONENT_SUPERSET THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `frontier s:real^N->bool = frontier c` SUBST1_TAC THENL [ALL_TAC; ASM_MESON_TAC[CONNECTED_FRONTIER_COMPONENT_COMPLEMENT]] THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[frontier; IN_DIFF] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET] o MATCH_MP SUBSET_CLOSURE) THEN ASM_MESON_TAC[SUBSET; frontier; IN_DIFF]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [GSYM FRONTIER_COMPLEMENT]) THEN REWRITE_TAC[frontier] THEN MATCH_MP_TAC(SET_RULE `u SUBSET t ==> x IN s DIFF t ==> ~(x IN u)`) THEN MATCH_MP_TAC SUBSET_INTERIOR THEN ASM_MESON_TAC[IN_COMPONENTS_SUBSET]]; GEN_REWRITE_TAC RAND_CONV [GSYM FRONTIER_COMPLEMENT] THEN ASM_MESON_TAC[FRONTIER_OF_COMPONENTS_SUBSET]]);; let SEPARATION_BY_COMPONENT_CLOSED_POINTWISE = prove (`!s a b. closed s /\ ~connected_component ((:real^N) DIFF s) a b ==> ?c. c IN components s /\ ~connected_component((:real^N) DIFF c) a b`, REPEAT GEN_TAC THEN DISCH_TAC THEN ASM_CASES_TAC `(a:real^N) IN s` THENL [EXISTS_TAC `connected_component s (a:real^N)` THEN ASM_REWRITE_TAC[IN_COMPONENTS] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(MP_TAC o MATCH_MP CONNECTED_COMPONENT_IN) THEN REWRITE_TAC[IN_UNIV; IN_DIFF] THEN REWRITE_TAC[IN] THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ]; ALL_TAC] THEN ASM_CASES_TAC `(b:real^N) IN s` THENL [EXISTS_TAC `connected_component s (b:real^N)` THEN ASM_REWRITE_TAC[IN_COMPONENTS] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(MP_TAC o MATCH_MP CONNECTED_COMPONENT_IN) THEN REWRITE_TAC[IN_UNIV; IN_DIFF] THEN REWRITE_TAC[IN] THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o MATCH_MP CLOSED_IRREDUCIBLE_SEPARATOR) THEN DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `?c:real^N->bool. c IN components s /\ t SUBSET c` MP_TAC THENL [MATCH_MP_TAC EXISTS_COMPONENT_SUPERSET THEN ASM_REWRITE_TAC[]; MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `~(t b) ==> s SUBSET t ==> ~(s b)`)) THEN REWRITE_TAC[ETA_AX] THEN MATCH_MP_TAC CONNECTED_COMPONENT_MONO THEN ASM SET_TAC[]] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`connected_component ((:real^N) DIFF t) a`; `connected_component ((:real^N) DIFF t) b`] CONNECTED_FRONTIER_DISJOINT) THEN REWRITE_TAC[CONNECTED_CONNECTED_COMPONENT; CONNECTED_COMPONENT_DISJOINT] THEN ASM_REWRITE_TAC[IN] THEN ONCE_REWRITE_TAC[CONNECTED_COMPONENT_SYM_EQ] THEN SUBGOAL_THEN `frontier(connected_component ((:real^N) DIFF t) a) = t /\ frontier(connected_component ((:real^N) DIFF t) b) = t` (fun th -> ASM_REWRITE_TAC[th; SUBSET_REFL]) THEN CONJ_TAC THEN MATCH_MP_TAC FRONTIER_MINIMAL_SEPARATING_CLOSED_POINTWISE THENL [EXISTS_TAC `b:real^N`; EXISTS_TAC `a:real^N`] THEN ASM_SIMP_TAC[] THEN ONCE_REWRITE_TAC[CONNECTED_COMPONENT_SYM_EQ] THEN ASM_SIMP_TAC[] THEN ASM SET_TAC[]);; let SEPARATION_BY_COMPONENT_CLOSED = prove (`!s. closed s /\ ~connected((:real^N) DIFF s) ==> ?c. c IN components s /\ ~connected((:real^N) DIFF c)`, REWRITE_TAC[CONNECTED_IFF_CONNECTED_COMPONENT; IN_DIFF; IN_UNIV] THEN MP_TAC SEPARATION_BY_COMPONENT_CLOSED_POINTWISE THEN MATCH_MP_TAC MONO_FORALL THEN MESON_TAC[REWRITE_RULE[SUBSET] IN_COMPONENTS_SUBSET]);; let SEPARATION_BY_UNION_CLOSED_POINTWISE = prove (`!s t a b. closed s /\ closed t /\ DISJOINT s t /\ connected_component ((:real^N) DIFF s) a b /\ connected_component ((:real^N) DIFF t) a b ==> connected_component ((:real^N) DIFF (s UNION t)) a b`, REPEAT GEN_TAC THEN REPLICATE_TAC 3 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(CONJUNCTS_THEN (fun th -> ASSUME_TAC th THEN MP_TAC(MATCH_MP CONNECTED_COMPONENT_IN th))) THEN REWRITE_TAC[IN_DIFF; IN_UNIV] THEN REPEAT STRIP_TAC THEN GEN_REWRITE_TAC I [TAUT `p <=> ~ ~ p`] THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] SEPARATION_BY_COMPONENT_CLOSED_POINTWISE)) THEN ASM_SIMP_TAC[CLOSED_UNION; NOT_EXISTS_THM] THEN X_GEN_TAC `c:real^N->bool` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[] THEN SUBGOAL_THEN `(c:real^N->bool) SUBSET s \/ c SUBSET t` STRIP_ASSUME_TAC THENL [FIRST_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN FIRST_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_CONNECTED) THEN REWRITE_TAC[CONNECTED_CLOSED; NOT_EXISTS_THM] THEN DISCH_THEN(MP_TAC o SPECL [`s:real^N->bool`; `t:real^N->bool`]) THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; UNDISCH_TAC `connected_component ((:real^N) DIFF s) a b`; UNDISCH_TAC `connected_component ((:real^N) DIFF t) a b`] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s b ==> t b`) THEN REWRITE_TAC[ETA_AX] THEN MATCH_MP_TAC CONNECTED_COMPONENT_MONO THEN ASM SET_TAC[]);; let SEPARATION_BY_UNION_CLOSED = prove (`!s t:real^N->bool. closed s /\ closed t /\ DISJOINT s t /\ connected((:real^N) DIFF s) /\ connected((:real^N) DIFF t) ==> connected((:real^N) DIFF (s UNION t))`, SIMP_TAC[CONNECTED_IFF_CONNECTED_COMPONENT; IN_DIFF; IN_UNION; IN_UNIV] THEN MESON_TAC[SEPARATION_BY_UNION_CLOSED_POINTWISE]);; let OPEN_UNICOHERENT_UNIV = prove (`!s t. open s /\ open t /\ connected s /\ connected t /\ s UNION t = (:real^N) ==> connected(s INTER t)`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[SET_RULE `s INTER t = UNIV DIFF ((UNIV DIFF s) UNION (UNIV DIFF t))`] THEN MATCH_MP_TAC SEPARATION_BY_UNION_CLOSED THEN ASM_SIMP_TAC[GSYM OPEN_CLOSED; SET_RULE `UNIV DIFF (UNIV DIFF s) = s`] THEN ASM SET_TAC[]);; let SEPARATION_BY_COMPONENT_OPEN = prove (`!s. open s /\ ~connected((:real^N) DIFF s) ==> ?c. c IN components s /\ ~connected((:real^N) DIFF c)`, let lemma = prove (`!s t u. closed s /\ closed t /\ s INTER t = {} /\ connected u /\ ~(u INTER s = {}) /\ ~(u INTER t = {}) ==> ?c. c IN components((:real^N) DIFF (s UNION t)) /\ ~(c INTER u = {}) /\ ~(frontier c INTER s = {}) /\ ~(frontier c INTER t = {})`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[MESON[] `(?x. P x /\ Q x /\ R x) <=> ~(!x. P x /\ Q x ==> ~R x)`] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONNECTED_CLOSED]) THEN REWRITE_TAC[] THEN MAP_EVERY EXISTS_TAC [`s UNION UNIONS {c | c IN components((:real^N) DIFF (s UNION t)) /\ frontier c SUBSET s}`; `t UNION UNIONS {c | c IN components((:real^N) DIFF (s UNION t)) /\ frontier c SUBSET t}`] THEN REPLICATE_TAC 2 (CONJ_TAC THENL [REWRITE_TAC[GSYM FRONTIER_SUBSET_EQ] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s SUBSET t UNION u`) THEN MATCH_MP_TAC(MATCH_MP (REWRITE_RULE[IMP_CONJ] SUBSET_TRANS) (SPEC_ALL FRONTIER_UNION_SUBSET)) THEN ASM_REWRITE_TAC[UNION_SUBSET; FRONTIER_SUBSET_EQ] THEN MATCH_MP_TAC(MATCH_MP (REWRITE_RULE[IMP_CONJ] SUBSET_TRANS) (SPEC_ALL FRONTIER_UNIONS_SUBSET_CLOSURE)) THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN ASM_REWRITE_TAC[UNIONS_SUBSET] THEN SIMP_TAC[FORALL_IN_GSPEC]; ALL_TAC]) THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `(s UNION t) UNION UNIONS {c | c IN components((:real^N) DIFF (s UNION t)) /\ ~(c INTER u = {})}` THEN CONJ_TAC THENL [MP_TAC(ISPEC `(:real^N) DIFF (s UNION t)` UNIONS_COMPONENTS) THEN SET_TAC[]; MATCH_MP_TAC(SET_RULE `c SUBSET d UNION e ==> (s UNION t) UNION c SUBSET (s UNION d) UNION (t UNION e)`) THEN REWRITE_TAC[GSYM UNIONS_UNION] THEN MATCH_MP_TAC SUBSET_UNIONS THEN ONCE_REWRITE_TAC[SUBSET] THEN REWRITE_TAC[IN_ELIM_THM; IN_UNION] THEN X_GEN_TAC `c:real^N->bool` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `c:real^N->bool`) THEN ASM_REWRITE_TAC[DE_MORGAN_THM] THEN MATCH_MP_TAC(SET_RULE `c SUBSET s UNION t ==> c INTER s = {} \/ c INTER t = {} ==> c SUBSET s \/ c SUBSET t`) THEN FIRST_ASSUM(MP_TAC o MATCH_MP FRONTIER_OF_COMPONENTS_SUBSET) THEN REWRITE_TAC[FRONTIER_COMPLEMENT] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS) THEN ASM_SIMP_TAC[FRONTIER_SUBSET_EQ; CLOSED_UNION]]; MATCH_MP_TAC(SET_RULE `c UNION d SUBSET UNIV DIFF (s UNION t) /\ s INTER t = {} /\ DISJOINT c d ==> (s UNION c) INTER (t UNION d) INTER u = {}`) THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [REWRITE_TAC[GSYM UNIONS_UNION] THEN GEN_REWRITE_TAC RAND_CONV [UNIONS_COMPONENTS] THEN MATCH_MP_TAC SUBSET_UNIONS THEN SET_TAC[]; MATCH_MP_TAC(SET_RULE `(!s. s IN c ==> !t. t IN c' ==> s INTER t = {}) ==> DISJOINT (UNIONS c) (UNIONS c')`) THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN MP_TAC(ISPEC `(:real^N) DIFF (s UNION t)` COMPONENTS_NONOVERLAP) THEN SIMP_TAC[] THEN DISCH_THEN(K ALL_TAC) THEN X_GEN_TAC `c:real^N->bool` THEN STRIP_TAC THEN X_GEN_TAC `c':real^N->bool` THEN ASM_CASES_TAC `c':real^N->bool = c` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `c SUBSET s ==> s INTER t = {} /\ ~(c = {}) ==> ~(c SUBSET t)`)) THEN ASM_REWRITE_TAC[FRONTIER_EQ_EMPTY] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_NONEMPTY) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN ASM SET_TAC[]]; ASM SET_TAC[]; ASM SET_TAC[]]) in GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_SIMP_TAC[CONNECTED_CLOSED_SET; GSYM OPEN_CLOSED; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`t:real^N->bool`; `u:real^N->bool`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`t:real^N->bool`; `u:real^N->bool`; `(:real^N)`] lemma) THEN ASM_REWRITE_TAC[CONNECTED_UNIV; SET_RULE `UNIV DIFF (UNIV DIFF s) = s`] THEN ANTS_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS] THEN X_GEN_TAC `c:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN MP_TAC(ISPEC `c:real^N->bool` CONNECTED_FRONTIER_SIMPLE) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP IN_COMPONENTS_CONNECTED) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[CONNECTED_CLOSED] THEN MAP_EVERY EXISTS_TAC [`t:real^N->bool`; `u:real^N->bool`] THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN FIRST_ASSUM(MP_TAC o MATCH_MP FRONTIER_OF_COMPONENTS_SUBSET) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS) THEN ONCE_REWRITE_TAC[GSYM FRONTIER_COMPLEMENT] THEN ASM_REWRITE_TAC[FRONTIER_SUBSET_EQ; GSYM OPEN_CLOSED]);; let SEPARATION_BY_UNION_OPEN = prove (`!s t:real^N->bool. open s /\ open t /\ DISJOINT s t /\ connected((:real^N) DIFF s) /\ connected((:real^N) DIFF t) ==> connected((:real^N) DIFF (s UNION t))`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[SET_RULE `UNIV DIFF (s UNION t) = (UNIV DIFF s) INTER (UNIV DIFF t)`] THEN MATCH_MP_TAC(REWRITE_RULE[unicoherent] UNICOHERENT_UNIV) THEN REWRITE_TAC[SUBTOPOLOGY_UNIV; GSYM CLOSED_IN] THEN ASM_REWRITE_TAC[GSYM OPEN_CLOSED] THEN ASM SET_TAC[]);; let CONNECTED_INTER_DISJOINT_OPEN_FRONTIERS = prove (`!s t:real^N->bool. open s /\ connected s /\ open t /\ connected t /\ DISJOINT (frontier s) (frontier t) ==> connected(s INTER t)`, let lemma = prove (`~(f = {}) ==> s UNION UNIONS f = UNIONS {s UNION c | c IN f}`, REWRITE_TAC[UNIONS_GSPEC] THEN SET_TAC[]) in REPEAT STRIP_TAC THEN MAP_EVERY ASM_CASES_TAC [`s:real^N->bool = {}`; `t:real^N->bool = {}`] THEN ASM_REWRITE_TAC[INTER_EMPTY; CONNECTED_EMPTY] THEN MAP_EVERY ASM_CASES_TAC [`s = (:real^N)`; `t = (:real^N)`] THEN ASM_REWRITE_TAC[INTER_UNIV; CONNECTED_UNIV] THEN ASM_CASES_TAC `s INTER t:real^N->bool = {}` THEN ASM_REWRITE_TAC[CONNECTED_EMPTY] THEN MP_TAC(ISPECL [`s UNION UNIONS {c | c IN components((:real^N) DIFF closure t) /\ ~(c INTER s = {})}`; `t UNION UNIONS {c | c IN components((:real^N) DIFF closure s) /\ ~(c INTER t = {})}`] OPEN_UNICOHERENT_UNIV) THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [MATCH_MP_TAC OPEN_UNION THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC OPEN_UNIONS THEN REWRITE_TAC[IN_ELIM_THM] THEN MESON_TAC[OPEN_COMPONENTS; closed; CLOSED_CLOSURE]; MATCH_MP_TAC OPEN_UNION THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC OPEN_UNIONS THEN REWRITE_TAC[IN_ELIM_THM] THEN MESON_TAC[OPEN_COMPONENTS; closed; CLOSED_CLOSURE]; MATCH_MP_TAC(MESON[] `(s = {} \/ ~(s = {}) ==> connected(u UNION UNIONS s)) ==> connected(u UNION UNIONS s)`) THEN STRIP_TAC THEN ASM_REWRITE_TAC[UNION_EMPTY; UNIONS_0] THEN ASM_SIMP_TAC[lemma] THEN MATCH_MP_TAC CONNECTED_UNIONS THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN CONJ_TAC THENL [ASM_MESON_TAC[CONNECTED_UNION; IN_COMPONENTS_CONNECTED; UNION_COMM]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ ~(s = {}) ==> ~(t = {})`) THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[SUBSET_INTERS] THEN REWRITE_TAC[FORALL_IN_GSPEC; SUBSET_UNION]; MATCH_MP_TAC(MESON[] `(s = {} \/ ~(s = {}) ==> connected(u UNION UNIONS s)) ==> connected(u UNION UNIONS s)`) THEN STRIP_TAC THEN ASM_REWRITE_TAC[UNION_EMPTY; UNIONS_0] THEN ASM_SIMP_TAC[lemma] THEN MATCH_MP_TAC CONNECTED_UNIONS THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN CONJ_TAC THENL [ASM_MESON_TAC[CONNECTED_UNION; IN_COMPONENTS_CONNECTED; UNION_COMM]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ ~(s = {}) ==> ~(t = {})`) THEN EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[SUBSET_INTERS] THEN REWRITE_TAC[FORALL_IN_GSPEC; SUBSET_UNION]; GEN_REWRITE_TAC I [EXTENSION] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_UNION; UNIONS_GSPEC; IN_UNIV; IN_ELIM_THM] THEN ASM_CASES_TAC `(x:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `(x:real^N) IN t` THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(MP_TAC o SPEC `x:real^N` o MATCH_MP (SET_RULE `DISJOINT s t ==> !x. ~(x IN s) \/ ~(x IN t)`)) THEN ASM_SIMP_TAC[frontier; INTERIOR_OPEN; IN_DIFF] THEN STRIP_TAC THENL [SUBGOAL_THEN `x IN UNIONS(components((:real^N) DIFF closure s))` MP_TAC THENL [ASM_REWRITE_TAC[GSYM UNIONS_COMPONENTS; IN_DIFF; IN_UNIV]; ALL_TAC] THEN REWRITE_TAC[IN_UNIONS] THEN DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` STRIP_ASSUME_TAC) THEN ASM_CASES_TAC `c INTER t:real^N->bool = {}` THENL [ALL_TAC; ASM_MESON_TAC[]] THEN SUBGOAL_THEN `c INTER closure(t:real^N->bool) = {}` ASSUME_TAC THENL [ASM_MESON_TAC[OPEN_INTER_CLOSURE_EQ_EMPTY; OPEN_COMPONENTS; closed; CLOSED_CLOSURE]; ALL_TAC] THEN SUBGOAL_THEN `x IN UNIONS(components((:real^N) DIFF closure t))` MP_TAC THENL [ASM_REWRITE_TAC[GSYM UNIONS_COMPONENTS; IN_DIFF; IN_UNIV] THEN ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[IN_UNIONS] THEN DISCH_THEN(X_CHOOSE_THEN `d:real^N->bool` STRIP_ASSUME_TAC) THEN ASM_CASES_TAC `d INTER s:real^N->bool = {}` THENL [ALL_TAC; ASM_MESON_TAC[]] THEN SUBGOAL_THEN `d INTER closure(s:real^N->bool) = {}` ASSUME_TAC THENL [ASM_MESON_TAC[OPEN_INTER_CLOSURE_EQ_EMPTY; OPEN_COMPONENTS; closed; CLOSED_CLOSURE]; ALL_TAC]; SUBGOAL_THEN `x IN UNIONS(components((:real^N) DIFF closure t))` MP_TAC THENL [ASM_REWRITE_TAC[GSYM UNIONS_COMPONENTS; IN_DIFF; IN_UNIV]; ALL_TAC] THEN REWRITE_TAC[IN_UNIONS] THEN DISCH_THEN(X_CHOOSE_THEN `d:real^N->bool` STRIP_ASSUME_TAC) THEN ASM_CASES_TAC `d INTER s:real^N->bool = {}` THENL [ALL_TAC; ASM_MESON_TAC[]] THEN SUBGOAL_THEN `d INTER closure(s:real^N->bool) = {}` ASSUME_TAC THENL [ASM_MESON_TAC[OPEN_INTER_CLOSURE_EQ_EMPTY; OPEN_COMPONENTS; closed; CLOSED_CLOSURE]; ALL_TAC] THEN SUBGOAL_THEN `x IN UNIONS(components((:real^N) DIFF closure s))` MP_TAC THENL [ASM_REWRITE_TAC[GSYM UNIONS_COMPONENTS; IN_DIFF; IN_UNIV] THEN ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[IN_UNIONS] THEN DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` STRIP_ASSUME_TAC) THEN ASM_CASES_TAC `c INTER t:real^N->bool = {}` THENL [ALL_TAC; ASM_MESON_TAC[]] THEN SUBGOAL_THEN `c INTER closure(t:real^N->bool) = {}` ASSUME_TAC THENL [ASM_MESON_TAC[OPEN_INTER_CLOSURE_EQ_EMPTY; OPEN_COMPONENTS; closed; CLOSED_CLOSURE]; ALL_TAC]] THEN (FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `DISJOINT s t ==> !c d. ~(c = {}) /\ c SUBSET s /\ d SUBSET t /\ c = d ==> p`)) THEN MAP_EVERY EXISTS_TAC [`frontier c:real^N->bool`; `frontier d:real^N->bool`] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[FRONTIER_EQ_EMPTY; DE_MORGAN_THM] THEN ASM_MESON_TAC[IN_COMPONENTS_NONEMPTY; IN_COMPONENTS_SUBSET; SET_RULE `s SUBSET UNIV DIFF t /\ s = UNIV ==> t = {}`; CLOSURE_EQ_EMPTY]; ASM_MESON_TAC[FRONTIER_OF_COMPONENTS_SUBSET;FRONTIER_COMPLEMENT; FRONTIER_CLOSURE_SUBSET; SUBSET_TRANS]; ASM_MESON_TAC[FRONTIER_OF_COMPONENTS_SUBSET;FRONTIER_COMPLEMENT; FRONTIER_CLOSURE_SUBSET; SUBSET_TRANS]; AP_TERM_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THEN MATCH_MP_TAC COMPONENTS_MAXIMAL THENL [EXISTS_TAC `(:real^N) DIFF closure t`; EXISTS_TAC `(:real^N) DIFF closure s`] THEN ASM_REWRITE_TAC[] THEN (CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]; ASM SET_TAC[]])])]; ALL_TAC] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN MATCH_MP_TAC(SET_RULE `s INTER t' = {} /\ t INTER s' = {} /\ s' INTER t' = {} ==> (s UNION s') INTER (t UNION t') = s INTER t`) THEN REWRITE_TAC[INTER_UNIONS; EMPTY_UNIONS; FORALL_IN_GSPEC; UNIONS_SUBSET] THEN REPEAT CONJ_TAC THEN X_GEN_TAC `d:real^N->bool` THENL [MP_TAC(ISPEC `s:real^N->bool` CLOSURE_SUBSET) THEN MP_TAC(ISPECL [`(:real^N) DIFF closure s`; `d:real^N->bool`] IN_COMPONENTS_SUBSET) THEN SET_TAC[]; MP_TAC(ISPEC `t:real^N->bool` CLOSURE_SUBSET) THEN MP_TAC(ISPECL [`(:real^N) DIFF closure t`; `d:real^N->bool`] IN_COMPONENTS_SUBSET) THEN SET_TAC[]; STRIP_TAC THEN X_GEN_TAC `c:real^N->bool` THEN STRIP_TAC] THEN MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `DISJOINT s t ==> !c d. c SUBSET s /\ d SUBSET t /\ ~(c INTER d = {}) ==> F`)) THEN MAP_EVERY EXISTS_TAC [`frontier c:real^N->bool`; `frontier d:real^N->bool`] THEN REPEAT(CONJ_TAC THENL [ASM_MESON_TAC[FRONTIER_OF_COMPONENTS_SUBSET;FRONTIER_COMPLEMENT; FRONTIER_CLOSURE_SUBSET; SUBSET_TRANS]; ALL_TAC]) THEN MATCH_MP_TAC CONNECTED_INTER_FRONTIER THEN CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_FRONTIER_COMPONENT_COMPLEMENT THEN EXISTS_TAC `closure s:real^N->bool` THEN ASM_MESON_TAC[CONNECTED_CLOSURE]; ALL_TAC] THEN ONCE_REWRITE_TAC[SET_RULE `c DIFF d = c INTER (UNIV DIFF d)`] THEN ONCE_REWRITE_TAC[INTER_COMM] THEN CONJ_TAC THEN MATCH_MP_TAC CONNECTED_INTER_FRONTIER THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]; ALL_TAC; MATCH_MP_TAC COMPONENT_COMPLEMENT_CONNECTED THEN EXISTS_TAC `closure t:real^N->bool` THEN ASM_SIMP_TAC[CONNECTED_UNIV; SUBSET_UNIV; CONNECTED_CLOSURE]; ALL_TAC; ALL_TAC] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET)) THEN MP_TAC(ISPEC `s:real^N->bool` CLOSURE_SUBSET) THEN MP_TAC(ISPEC `t:real^N->bool` CLOSURE_SUBSET) THEN ASM SET_TAC[]);; let NONSEPARATION_BY_COMPONENT_EQ = prove (`!s. (open s \/ closed s) ==> ((!c. c IN components s ==> connected((:real^N) DIFF c)) <=> connected((:real^N) DIFF s))`, REPEAT STRIP_TAC THEN EQ_TAC THEN REPEAT STRIP_TAC THENL [ASM_MESON_TAC[SEPARATION_BY_COMPONENT_OPEN]; ALL_TAC; ASM_MESON_TAC[SEPARATION_BY_COMPONENT_CLOSED]; ALL_TAC] THEN MATCH_MP_TAC COMPONENT_COMPLEMENT_CONNECTED THEN EXISTS_TAC `(:real^N) DIFF s` THEN ASM_REWRITE_TAC[CONNECTED_UNIV; SUBSET_UNIV; SET_RULE `UNIV DIFF (UNIV DIFF s) = s`]);; let CONNECTED_COMMON_FRONTIER_DOMAINS = prove (`!s t c:real^N->bool. open s /\ connected s /\ open t /\ connected t /\ ~(s = t) /\ frontier s = c /\ frontier t = c ==> connected c`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`] COMMON_FRONTIER_DOMAINS) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`] CONNECTED_FRONTIER_DISJOINT) THEN ASM_REWRITE_TAC[SUBSET_REFL]);; (* ------------------------------------------------------------------------- *) (* Another interesting equivalent of an inessential mapping into C-{0} *) (* ------------------------------------------------------------------------- *) let INESSENTIAL_EQ_EXTENSIBLE = prove (`!f s. closed s ==> ((?a. homotopic_with (\h. T) (s,(:complex) DIFF {Cx(&0)}) f (\t. a)) <=> (?g. g continuous_on (:real^N) /\ (!x. x IN s ==> g x = f x) /\ (!x. ~(g x = Cx(&0)))))`, REPEAT GEN_TAC THEN DISCH_TAC THEN EQ_TAC THENL [DISCH_THEN(X_CHOOSE_TAC `a:complex`) THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [EXISTS_TAC `\x:real^N. Cx(&1)` THEN ASM_REWRITE_TAC[CONTINUOUS_ON_CONST; NOT_IN_EMPTY] THEN CONV_TAC COMPLEX_RING; ALL_TAC] 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 FIRST_ASSUM(MP_TAC o SPEC `(:real^N)` o MATCH_MP(ONCE_REWRITE_RULE[IMP_CONJ_ALT] (REWRITE_RULE[CONJ_ASSOC] BORSUK_HOMOTOPY_EXTENSION)) o GEN_REWRITE_RULE I [HOMOTOPIC_WITH_SYM]) THEN ASM_REWRITE_TAC[GSYM CLOSED_IN; SUBTOPOLOGY_UNIV] THEN SIMP_TAC[OPEN_IMP_ANR; OPEN_DIFF; OPEN_UNIV; CLOSED_SING] THEN ASM_SIMP_TAC[CLOSED_UNIV; CONTINUOUS_ON_CONST] THEN ANTS_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS] THEN ASM SET_TAC[]; DISCH_THEN(X_CHOOSE_THEN `g:real^N->complex` STRIP_ASSUME_TAC) THEN REWRITE_TAC[INESSENTIAL_EQ_CONTINUOUS_LOGARITHM] THEN MP_TAC(ISPECL [`vec 0:real^N`; `&1`] HOMEOMORPHIC_BALL_UNIV) THEN REWRITE_TAC[REAL_LT_01; homeomorphic; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`h:real^N->real^N`; `k:real^N->real^N`] THEN REWRITE_TAC[homeomorphism; IN_UNIV] THEN STRIP_TAC THEN MP_TAC(ISPECL [`(g:real^N->complex) o (h:real^N->real^N)`; `vec 0:real^N`; `&1`] CONTINUOUS_LOGARITHM_ON_BALL) THEN ASM_SIMP_TAC[CONTINUOUS_ON_COMPOSE; o_THM] THEN DISCH_THEN(X_CHOOSE_THEN `j:real^N->complex` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(j:real^N->complex) o (k:real^N->real^N)` THEN ASM_SIMP_TAC[o_THM] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_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[]]);; (* ------------------------------------------------------------------------- *) (* Another simple case where sphere maps are nullhomotopic. *) (* ------------------------------------------------------------------------- *) let INESSENTIAL_SPHEREMAP_2 = prove (`!f:real^M->real^N a r b s. 2 < dimindex(:M) /\ dimindex(:N) = 2 /\ f continuous_on sphere(a,r) /\ IMAGE f (sphere(a,r)) SUBSET (sphere(b,s)) ==> ?c. homotopic_with (\z. T) (sphere(a,r),sphere(b,s)) f (\x. c)`, let lemma = prove (`!f:real^N->real^2 a r. 2 < dimindex(:N) /\ f continuous_on sphere(a,r) /\ IMAGE f (sphere(a,r)) SUBSET (sphere(vec 0,&1)) ==> ?c. homotopic_with (\z. T) (sphere(a,r),sphere(vec 0,&1)) f (\x. c)`, REPEAT STRIP_TAC THEN REWRITE_TAC[INESSENTIAL_EQ_CONTINUOUS_LOGARITHM_CIRCLE] THEN MP_TAC(ISPECL [`f:real^N->real^2`; `sphere(a:real^N,r)`] CONTINUOUS_LOGARITHM_ON_SIMPLY_CONNECTED) THEN ASM_SIMP_TAC[SIMPLY_CONNECTED_SPHERE_EQ; LOCALLY_PATH_CONNECTED_SPHERE] THEN ANTS_TAC THENL [ASM_REWRITE_TAC[ARITH_RULE `3 <= n <=> 2 < n`] THEN FIRST_X_ASSUM (MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE f s SUBSET t ==> (!x. P x ==> ~(x IN t)) ==> !x. x IN s ==> ~P(f x)`)) THEN SIMP_TAC[COMPLEX_NORM_0; IN_SPHERE_0] THEN REAL_ARITH_TAC; DISCH_THEN(X_CHOOSE_THEN `g:real^N->real^2` STRIP_ASSUME_TAC) THEN EXISTS_TAC `Im o (g:real^N->real^2)` THEN CONJ_TAC THENL [REWRITE_TAC[o_ASSOC] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[CONTINUOUS_ON_CX_IM]; X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN ASM_SIMP_TAC[] THEN AP_TERM_TAC THEN REWRITE_TAC[o_DEF; COMPLEX_EQ; RE_MUL_II; IM_MUL_II; RE_CX; IM_CX] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_SIMP_TAC[IN_SPHERE_0; NORM_CEXP; REAL_EXP_EQ_1] THEN REAL_ARITH_TAC]]) and hslemma = prove (`!a:real^M r b:real^N s. dimindex(:M) = dimindex(:N) /\ &0 < r /\ &0 < s ==> (sphere(a,r) homeomorphic sphere(b,s))`, REPEAT STRIP_TAC THEN FIRST_ASSUM(fun th -> let t = `?a:real^M b:real^N. ~(sphere(a,r) homeomorphic sphere(b,s))` in MP_TAC(DISCH t (GEOM_EQUAL_DIMENSION_RULE th (ASSUME t)))) THEN ASM_SIMP_TAC[HOMEOMORPHIC_SPHERES] THEN MESON_TAC[]) in REPEAT STRIP_TAC THEN ASM_CASES_TAC `s <= &0` THEN ASM_SIMP_TAC[NULLHOMOTOPIC_INTO_CONTRACTIBLE; CONTRACTIBLE_SPHERE] THEN RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LE]) THEN SUBGOAL_THEN `(sphere(b:real^N,s)) homeomorphic (sphere(vec 0:real^2,&1))` MP_TAC THENL [ASM_SIMP_TAC[hslemma; REAL_LT_01; DIMINDEX_2]; REWRITE_TAC[homeomorphic; LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`h:real^N->real^2`; `k:real^2->real^N`] THEN REWRITE_TAC[homeomorphism] THEN STRIP_TAC THEN MP_TAC(ISPECL [`(h:real^N->real^2) o (f:real^M->real^N)`; `a:real^M`; `r:real`] lemma) THEN ASM_REWRITE_TAC[IMAGE_o] THEN ANTS_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE; ASM SET_TAC[]] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; DISCH_THEN(X_CHOOSE_THEN `c:real^2` (fun th -> EXISTS_TAC `(k:real^2->real^N) c` THEN MP_TAC th)) THEN DISCH_THEN(MP_TAC o ISPEC `k:real^2->real^N` o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HOMOTOPIC_COMPOSE_CONTINUOUS_LEFT)) THEN DISCH_THEN(MP_TAC o SPEC `sphere(b:real^N,s)`) THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_WITH_EQ) THEN REWRITE_TAC[o_DEF] THEN ASM SET_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Janiszewski's theorem. *) (* ------------------------------------------------------------------------- *) let JANISZEWSKI = prove (`!s t a b:real^2. compact s /\ closed t /\ connected(s INTER t) /\ connected_component ((:real^2) DIFF s) a b /\ connected_component ((:real^2) DIFF t) a b ==> connected_component ((:real^2) DIFF (s UNION t)) a b`, let lemma = prove (`!s t a b:real^2. compact s /\ compact t /\ connected(s INTER t) /\ connected_component ((:real^2) DIFF s) a b /\ connected_component ((:real^2) DIFF t) a b ==> connected_component ((:real^2) DIFF (s UNION t)) a b`, REPEAT GEN_TAC THEN REPLICATE_TAC 3 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN FIRST_X_ASSUM(CONJUNCTS_THEN(MP_TAC o MATCH_MP CONNECTED_COMPONENT_IN)) THEN REWRITE_TAC[IN_DIFF; IN_UNIV] THEN STRIP_TAC THEN STRIP_TAC THEN ASM_SIMP_TAC[GSYM BORSUK_MAPS_HOMOTOPIC_IN_CONNECTED_COMPONENT_EQ; DIMINDEX_2; LE_REFL; COMPACT_UNION; IN_UNION] THEN ONCE_REWRITE_TAC[HOMOTOPIC_CIRCLEMAPS_DIV] THEN REWRITE_TAC[INESSENTIAL_EQ_CONTINUOUS_LOGARITHM_CIRCLE] THEN ASM_SIMP_TAC[BORSUK_MAP_INTO_SPHERE; CONTINUOUS_ON_BORSUK_MAP; IN_UNION] THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `g:real^2->real` STRIP_ASSUME_TAC) (X_CHOOSE_THEN `h:real^2->real` STRIP_ASSUME_TAC)) THEN SUBGOAL_THEN `closed_in (subtopology euclidean (s UNION t)) s /\ closed_in (subtopology euclidean (s UNION t)) (t:real^2->bool)` STRIP_ASSUME_TAC THENL [REWRITE_TAC[CLOSED_IN_CLOSED] THEN CONJ_TAC THENL [EXISTS_TAC `s:real^2->bool`; EXISTS_TAC `t:real^2->bool`] THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED] THEN ASM SET_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `s INTER t:real^2->bool = {}` THENL [EXISTS_TAC `(\x. if x IN s then g x else h x):real^2->real` THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN REWRITE_TAC[o_DEF; COND_RAND] THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL THEN ASM_REWRITE_TAC[GSYM o_DEF] THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`\x:real^2. lift(g x) - lift(h x)`; `s INTER t:real^2->bool`] CONTINUOUS_DISCRETE_RANGE_CONSTANT) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_SUB THEN REWRITE_TAC[GSYM CONTINUOUS_ON_CX_LIFT] THEN REWRITE_TAC[GSYM o_DEF] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; INTER_SUBSET]; REWRITE_TAC[o_DEF]] THEN X_GEN_TAC `x:real^2` THEN REWRITE_TAC[IN_INTER] THEN STRIP_TAC THEN EXISTS_TAC `&2 * pi` THEN REWRITE_TAC[REAL_ARITH `&0 < &2 * x <=> &0 < x`; PI_POS] THEN X_GEN_TAC `y:real^2` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[REAL_NOT_LE] THEN REWRITE_TAC[GSYM LIFT_SUB; LIFT_EQ; NORM_LIFT] THEN DISCH_TAC THEN ONCE_REWRITE_TAC[REAL_RING `a - b:real = c - d <=> a - c = b - d`] THEN REWRITE_TAC[GSYM CX_INJ] THEN MATCH_MP_TAC(COMPLEX_RING `ii * w = ii * z ==> w = z`) THEN MATCH_MP_TAC COMPLEX_EQ_CEXP THEN CONJ_TAC THENL [REWRITE_TAC[IM_MUL_II; RE_CX] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[CX_SUB; COMPLEX_SUB_LDISTRIB; CEXP_SUB] THEN ASM_MESON_TAC[]]; REWRITE_TAC[EXISTS_LIFT; GSYM LIFT_SUB; LIFT_EQ; IN_INTER] THEN REWRITE_TAC[REAL_EQ_SUB_RADD; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `z:real` THEN DISCH_TAC THEN EXISTS_TAC `(\x. if x IN s then g x else z + h x):real^2->real` THEN CONJ_TAC THENL [REWRITE_TAC[o_DEF; COND_RAND] THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL THEN ASM_SIMP_TAC[TAUT `~(p /\ ~p)`; CX_ADD; GSYM o_DEF] THEN REWRITE_TAC[o_DEF; CX_ADD] THEN MATCH_MP_TAC CONTINUOUS_ON_ADD THEN ASM_REWRITE_TAC[CONTINUOUS_ON_CONST; GSYM o_DEF]; X_GEN_TAC `x:real^2` THEN REWRITE_TAC[] THEN COND_CASES_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[] THEN DISCH_TAC THEN SUBGOAL_THEN `?w:real^2. cexp(ii * Cx(h w)) = cexp (ii * Cx(z + h w))` (CHOOSE_THEN MP_TAC) THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[CX_ADD; COMPLEX_ADD_LDISTRIB; CEXP_ADD] THEN REWRITE_TAC[COMPLEX_FIELD `a = b * a <=> a = Cx(&0) \/ b = Cx(&1)`; CEXP_NZ]]]) in REPEAT STRIP_TAC THEN SUBGOAL_THEN `?c:real^2->bool. compact c /\ connected c /\ a IN c /\ b IN c /\ c INTER t = {}` STRIP_ASSUME_TAC THENL [SUBGOAL_THEN `path_component((:real^2) DIFF t) a b` MP_TAC THENL [ASM_MESON_TAC[OPEN_PATH_CONNECTED_COMPONENT; closed; COMPACT_IMP_CLOSED]; REWRITE_TAC[path_component; SET_RULE `s SUBSET UNIV DIFF t <=> s INTER t = {}`]] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^1->real^2` STRIP_ASSUME_TAC) THEN EXISTS_TAC `path_image(g:real^1->real^2)` THEN ASM_SIMP_TAC[CONNECTED_PATH_IMAGE; COMPACT_PATH_IMAGE] THEN ASM_MESON_TAC[PATHSTART_IN_PATH_IMAGE; PATHFINISH_IN_PATH_IMAGE]; ALL_TAC] THEN MP_TAC(ISPECL [`c UNION s:real^2->bool`; `vec 0:real^2`] BOUNDED_SUBSET_BALL) THEN ASM_SIMP_TAC[BOUNDED_UNION; COMPACT_IMP_BOUNDED; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `r:real` THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^2->bool`; `(t INTER cball(vec 0,r)) UNION sphere(vec 0:real^2,r)`; `a:real^2`; `b:real^2`] lemma) THEN ASM_SIMP_TAC[COMPACT_UNION; CLOSED_INTER_COMPACT; COMPACT_SPHERE; COMPACT_CBALL] THEN ANTS_TAC THENL [CONJ_TAC THENL [UNDISCH_TAC `connected(s INTER t:real^2->bool)` THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC; REWRITE_TAC[connected_component] THEN EXISTS_TAC `c:real^2->bool`] THEN MP_TAC(ISPECL [`vec 0:real^2`; `r:real`] CBALL_DIFF_SPHERE) THEN ASM SET_TAC[]; REWRITE_TAC[connected_component] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^2->bool` THEN SIMP_TAC[SET_RULE `s SUBSET UNIV DIFF t <=> s INTER t = {}`] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`u:real^2->bool`; `cball(vec 0:real^2,r)`] CONNECTED_INTER_FRONTIER) THEN ASM_REWRITE_TAC[FRONTIER_CBALL] THEN MP_TAC(ISPECL [`vec 0:real^2`; `r:real`] BALL_SUBSET_CBALL) THEN ASM SET_TAC[]]);; let JANISZEWSKI_GEN = prove (`!s t a b:real^N. dimindex(:N) <= 2 /\ compact s /\ closed t /\ connected(s INTER t) /\ connected_component ((:real^N) DIFF s) a b /\ connected_component ((:real^N) DIFF t) a b ==> connected_component ((:real^N) DIFF (s UNION t)) a b`, REPEAT GEN_TAC THEN ASM_CASES_TAC `dimindex(:N) = 1` THENL [ASM_SIMP_TAC[CONNECTED_COMPONENT_1_GEN] THEN SET_TAC[]; ASM_SIMP_TAC[ARITH_RULE `1 <= n /\ ~(n = 1) ==> (n <= 2 <=> n = 2)`; DIMINDEX_GE_1] THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN REWRITE_TAC[GSYM DIMINDEX_2] THEN DISCH_THEN(fun th -> MATCH_ACCEPT_TAC(GEOM_EQUAL_DIMENSION_RULE th JANISZEWSKI))]);; let JANISZEWSKI_CONNECTED = prove (`!s t:real^2->bool. compact s /\ closed t /\ connected(s INTER t) /\ connected ((:real^2) DIFF s) /\ connected ((:real^2) DIFF t) ==> connected((:real^2) DIFF (s UNION t))`, REPEAT GEN_TAC THEN REPLICATE_TAC 3 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN REWRITE_TAC[CONNECTED_IFF_CONNECTED_COMPONENT] THEN REWRITE_TAC[IN_DIFF; IN_UNIV; IN_UNION] THEN ASM_MESON_TAC[JANISZEWSKI]);; let JANISZEWSKI_DUAL = prove (`!s t:real^2->bool. compact s /\ compact t /\ connected s /\ connected t /\ connected((:real^2) DIFF (s UNION t)) ==> connected(s INTER t)`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `s UNION t:real^2->bool` BORSUKIAN_IMP_UNICOHERENT) THEN ASM_SIMP_TAC[BORSUKIAN_SEPARATION_COMPACT; COMPACT_UNION; unicoherent] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THEN MATCH_MP_TAC CLOSED_SUBSET THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED] THEN SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* The Jordan Curve theorem. *) (* ------------------------------------------------------------------------- *) let JORDAN_CURVE_THEOREM = prove (`!c:real^1->real^2. simple_path c /\ pathfinish c = pathstart c ==> ?ins out. ~(ins = {}) /\ open ins /\ connected ins /\ ~(out = {}) /\ open out /\ connected out /\ bounded ins /\ ~bounded out /\ ins INTER out = {} /\ ins UNION out = (:real^2) DIFF path_image c /\ frontier ins = path_image c /\ frontier out = path_image c`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `path_image(c:real^1->real^2) homeomorphic sphere(vec 0:real^2,&1)` ASSUME_TAC THENL [ASM_SIMP_TAC[HOMEOMORPHIC_SIMPLE_PATH_IMAGE_CIRCLE; REAL_LT_01]; FIRST_ASSUM(ASSUME_TAC o MATCH_MP SIMPLE_PATH_IMP_PATH) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP COMPACT_PATH_IMAGE) THEN ABBREV_TAC `s:real^2->bool = path_image c`] THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] JORDAN_BROUWER_SEPARATION)) THEN REWRITE_TAC[REAL_LT_01] THEN DISCH_TAC THEN MP_TAC(ISPEC `(:real^2) DIFF s` COBOUNDED_UNBOUNDED_COMPONENTS) THEN MP_TAC(ISPEC `(:real^2) DIFF s` COBOUNDED_HAS_BOUNDED_COMPONENT) THEN ASM_SIMP_TAC[SET_RULE `UNIV DIFF (UNIV DIFF s) = s`; COMPACT_IMP_BOUNDED; DIMINDEX_2; LE_REFL; IMP_IMP] THEN REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `ins:real^2->bool` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `out:real^2->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REPLICATE_TAC 5 (GEN_REWRITE_TAC I [CONJ_ASSOC]) THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_NONEMPTY; IN_COMPONENTS_CONNECTED; OPEN_COMPONENTS; closed; COMPACT_IMP_CLOSED]; STRIP_TAC] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [ASM_MESON_TAC[COMPONENTS_EQ]; DISCH_TAC] THEN MATCH_MP_TAC(TAUT `q /\ (q ==> p) ==> p /\ q`) THEN CONJ_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC JORDAN_BROUWER_FRONTIER THEN REWRITE_TAC[DIMINDEX_2; LE_REFL] THEN ASM_MESON_TAC[]; STRIP_TAC] THEN GEN_REWRITE_TAC RAND_CONV [UNIONS_COMPONENTS] THEN REWRITE_TAC[GSYM UNIONS_2] THEN AP_TERM_TAC THEN MATCH_MP_TAC(SET_RULE `a IN s /\ b IN s /\ (!c. c IN s /\ ~(c = a) /\ ~(c = b) ==> F) ==> {a,b} = s`) THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `mid:real^2->bool` THEN STRIP_TAC THEN SUBGOAL_THEN `frontier mid:real^2->bool = s` ASSUME_TAC THENL [MATCH_MP_TAC JORDAN_BROUWER_FRONTIER THEN REWRITE_TAC[DIMINDEX_2; LE_REFL] THEN ASM_MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `open(mid:real^2->bool) /\ connected mid /\ ~(mid = {})` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[IN_COMPONENTS_NONEMPTY; IN_COMPONENTS_CONNECTED; OPEN_COMPONENTS; closed; COMPACT_IMP_CLOSED]; ALL_TAC] THEN SUBGOAL_THEN `?a b:real^2. a IN s /\ b IN s /\ ~(a = b) /\ ?g. arc g /\ pathstart g = a /\ pathfinish g = b /\ path_image g DIFF {a,b} SUBSET mid` STRIP_ASSUME_TAC THENL [SUBGOAL_THEN `?a b:real^2. a IN s /\ b IN s /\ ~(a = b)` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC(SET_RULE `(!c. s SUBSET {c} ==> F) ==> ?a b. a IN s /\ b IN s /\ ~(a = b)`) THEN ASM_MESON_TAC[INFINITE_SIMPLE_PATH_IMAGE; INFINITE; FINITE_SING; FINITE_SUBSET]; ALL_TAC] THEN MP_TAC(ISPECL [`mid:real^2->bool`; `s INTER ball(a:real^2,dist(a,b))`; `s INTER ball(b:real^2,dist(a,b))`] DENSE_ACCESSIBLE_FRONTIER_POINT_PAIRS) THEN ASM_SIMP_TAC[OPEN_IN_OPEN_INTER; OPEN_BALL] THEN ANTS_TAC THENL [SUBGOAL_THEN `a IN ball(a:real^2,dist(a,b)) /\ b IN ball(b,dist(a,b)) /\ ~(a IN ball(b,dist(a,b))) /\ ~(b IN ball(a,dist(a,b)))` MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN ASM_REWRITE_TAC[IN_BALL; DIST_REFL; GSYM DIST_NZ] THEN REWRITE_TAC[DIST_SYM] THEN REAL_ARITH_TAC; REWRITE_TAC[IN_INTER; LEFT_IMP_EXISTS_THM]] THEN X_GEN_TAC `g:real^1->real^2` THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`pathstart g:real^2`; `pathfinish g:real^2`] THEN ASM_SIMP_TAC[ARC_DISTINCT_ENDS] THEN EXISTS_TAC `g:real^1->real^2` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS)) THEN REWRITE_TAC[OPEN_CLOSED_INTERVAL_1; path_image; pathstart; pathfinish] THEN SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`c:real^1->real^2`; `a:real^2`; `b:real^2`] EXISTS_DOUBLE_ARC) THEN ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u:real^1->real^2`; `d:real^1->real^2`] THEN STRIP_TAC THEN SUBGOAL_THEN `?x:real^2 y:real^2. x IN ins /\ y IN out` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`(path_image u UNION path_image g):real^2->bool`; `(path_image d UNION path_image g):real^2->bool`; `x:real^2`; `y:real^2`] JANISZEWSKI) THEN ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 5) [COMPACT_UNION; COMPACT_IMP_CLOSED; COMPACT_PATH_IMAGE; ARC_IMP_PATH; NOT_IMP] THEN REPEAT CONJ_TAC THENL [SUBGOAL_THEN `(path_image u UNION path_image g) INTER (path_image d UNION path_image g) = path_image(g:real^1->real^2)` (fun th -> ASM_SIMP_TAC[CONNECTED_ARC_IMAGE; th]) THEN MATCH_MP_TAC(SET_RULE `u INTER d SUBSET s ==> (u UNION s) INTER (d UNION s) = s`) THEN ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET] THEN ASM_MESON_TAC[PATHSTART_IN_PATH_IMAGE; PATHFINISH_IN_PATH_IMAGE; ARC_IMP_PATH]; REWRITE_TAC[connected_component] THEN EXISTS_TAC `ins UNION out UNION (s DIFF path_image u):real^2->bool` THEN ASM_REWRITE_TAC[IN_UNION] THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[SET_RULE `s UNION t UNION u = (s UNION u) UNION (t UNION u)`] THEN MATCH_MP_TAC CONNECTED_UNION THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_INTERMEDIATE_CLOSURE THEN EXISTS_TAC `ins:real^2->bool` THEN ASM_SIMP_TAC[UNION_SUBSET; CLOSURE_UNION_FRONTIER; SUBSET_UNION] THEN ASM SET_TAC[]; MATCH_MP_TAC CONNECTED_INTERMEDIATE_CLOSURE THEN EXISTS_TAC `out:real^2->bool` THEN ASM_SIMP_TAC[UNION_SUBSET; CLOSURE_UNION_FRONTIER; SUBSET_UNION] THEN ASM SET_TAC[]; MATCH_MP_TAC(SET_RULE `~(u = {}) ==> ~((s UNION u) INTER (t UNION u) = {})`) THEN SUBGOAL_THEN `~(path_image d SUBSET {a:real^2,b})` MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] FINITE_SUBSET)) THEN REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY] THEN ASM_SIMP_TAC[INFINITE_ARC_IMAGE; GSYM INFINITE]]; SUBGOAL_THEN `ins INTER out = {} /\ ins INTER mid = {} /\ (mid:real^2->bool) INTER out = {}` MP_TAC THENL [ASM_MESON_TAC[COMPONENTS_NONOVERLAP]; ALL_TAC] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET)) THEN ASM SET_TAC[]]; REWRITE_TAC[connected_component] THEN EXISTS_TAC `ins UNION out UNION (s DIFF path_image d):real^2->bool` THEN ASM_REWRITE_TAC[IN_UNION] THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[SET_RULE `s UNION t UNION u = (s UNION u) UNION (t UNION u)`] THEN MATCH_MP_TAC CONNECTED_UNION THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_INTERMEDIATE_CLOSURE THEN EXISTS_TAC `ins:real^2->bool` THEN ASM_SIMP_TAC[UNION_SUBSET; CLOSURE_UNION_FRONTIER; SUBSET_UNION] THEN ASM SET_TAC[]; MATCH_MP_TAC CONNECTED_INTERMEDIATE_CLOSURE THEN EXISTS_TAC `out:real^2->bool` THEN ASM_SIMP_TAC[UNION_SUBSET; CLOSURE_UNION_FRONTIER; SUBSET_UNION] THEN ASM SET_TAC[]; MATCH_MP_TAC(SET_RULE `~(u = {}) ==> ~((s UNION u) INTER (t UNION u) = {})`) THEN SUBGOAL_THEN `~(path_image u SUBSET {a:real^2,b})` MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] FINITE_SUBSET)) THEN REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY] THEN ASM_SIMP_TAC[INFINITE_ARC_IMAGE; GSYM INFINITE]]; SUBGOAL_THEN `ins INTER out = {} /\ ins INTER mid = {} /\ (mid:real^2->bool) INTER out = {}` MP_TAC THENL [ASM_MESON_TAC[COMPONENTS_NONOVERLAP]; ALL_TAC] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET)) THEN ASM SET_TAC[]]; SUBGOAL_THEN `~(connected_component ((:real^2) DIFF s) x y)` MP_TAC THENL [REWRITE_TAC[connected_component] THEN DISCH_THEN(X_CHOOSE_THEN `t:real^2->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`(:real^2) DIFF s`; `t:real^2->bool`] COMPONENTS_MAXIMAL) THEN DISCH_THEN(fun th -> MP_TAC(SPEC `ins:real^2->bool` th) THEN MP_TAC(SPEC `out:real^2->bool` th)) THEN ASM SET_TAC[]; REWRITE_TAC[CONTRAPOS_THM] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s y ==> t y`) THEN REWRITE_TAC[ETA_AX] THEN MATCH_MP_TAC CONNECTED_COMPONENT_MONO THEN ASM SET_TAC[]]]);; let JORDAN_DISCONNECTED = prove (`!c. simple_path c /\ pathfinish c = pathstart c ==> ~connected((:real^2) DIFF path_image c)`, GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[connected] THEN FIRST_ASSUM(MP_TAC o MATCH_MP JORDAN_CURVE_THEOREM) THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);; let JORDAN_INSIDE_OUTSIDE = prove (`!c:real^1->real^2. simple_path c /\ pathfinish c = pathstart c ==> ~(inside(path_image c) = {}) /\ open(inside(path_image c)) /\ connected(inside(path_image c)) /\ ~(outside(path_image c) = {}) /\ open(outside(path_image c)) /\ connected(outside(path_image c)) /\ bounded(inside(path_image c)) /\ ~bounded(outside(path_image c)) /\ inside(path_image c) INTER outside(path_image c) = {} /\ inside(path_image c) UNION outside(path_image c) = (:real^2) DIFF path_image c /\ frontier(inside(path_image c)) = path_image c /\ frontier(outside(path_image c)) = path_image c`, GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP JORDAN_CURVE_THEOREM) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`ins:real^2->bool`; `out:real^2->bool`] THEN STRIP_TAC THEN SUBGOAL_THEN `inside(path_image c) :real^2->bool = ins /\ outside(path_image c):real^2->bool = out ` (fun th -> ASM_REWRITE_TAC[th]) THEN MATCH_MP_TAC INSIDE_OUTSIDE_UNIQUE THEN ASM_SIMP_TAC[JORDAN_DISCONNECTED]);; let JORDAN_COMPONENTS = prove (`!g. simple_path g /\ pathfinish g = pathstart g ==> components((:real^2) DIFF path_image g) = {inside(path_image g),outside(path_image g)}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC COMPONENTS_OPEN_UNIQUE THEN REWRITE_TAC[UNIONS_2; PAIRWISE_INSERT; NOT_IN_EMPTY; FORALL_IN_INSERT; IMP_CONJ; PAIRWISE_EMPTY] THEN MP_TAC(ISPEC `g:real^1->real^2` JORDAN_INSIDE_OUTSIDE) THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPEC `path_image g:complex->bool` INSIDE_INTER_OUTSIDE) THEN REPLICATE_TAC 2 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* Triple-curve or "theta-curve" theorem. Proof that there is no fourth *) (* component taken from Kuratowski's Topology vol 2, para 61, II. *) (* ------------------------------------------------------------------------- *) let THETA_CURVE_INSIDE_CASES = prove (`!c1 c2 c3 a b:real^2. arc c1 /\ pathstart c1 = a /\ pathfinish c1 = b /\ arc c2 /\ pathstart c2 = a /\ pathfinish c2 = b /\ arc c3 /\ pathstart c3 = a /\ pathfinish c3 = b /\ path_image c1 INTER path_image c2 = {a,b} /\ path_image c2 INTER path_image c3 = {a,b} /\ path_image c3 INTER path_image c1 = {a,b} ==> path_image c1 DIFF {a,b} SUBSET inside(path_image c2 UNION path_image c3) \/ path_image c2 DIFF {a,b} SUBSET inside(path_image c3 UNION path_image c1) \/ path_image c3 DIFF {a,b} SUBSET inside(path_image c1 UNION path_image c2)`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `c3 ++ reversepath c1:real^1->real^2` JORDAN_INSIDE_OUTSIDE) THEN MP_TAC(ISPEC `c2 ++ reversepath c3:real^1->real^2` JORDAN_INSIDE_OUTSIDE) THEN MP_TAC(ISPEC `c1 ++ reversepath c2:real^1->real^2` JORDAN_INSIDE_OUTSIDE) THEN ASM_SIMP_TAC[PATHSTART_JOIN; PATHFINISH_JOIN; PATHSTART_REVERSEPATH; PATHFINISH_REVERSEPATH; SIMPLE_PATH_JOIN_LOOP_EQ; ARC_REVERSEPATH_EQ; PATH_IMAGE_REVERSEPATH; SUBSET_REFL; PATH_IMAGE_JOIN; PATH_IMAGE_REVERSEPATH] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[TAUT `p \/ q <=> ~(~p /\ ~q)`] THEN REWRITE_TAC[SET_RULE `s SUBSET t <=> s DIFF t = {}`] THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN (MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] (REWRITE_RULE[CONJ_ASSOC] CONNECTED_INTER_FRONTIER)))) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[TAUT `p /\ ~q ==> ~r <=> p /\ r ==> q`] THEN PURE_ONCE_REWRITE_TAC[TAUT `~p <=> p ==> F`] THEN REPEAT(ANTS_TAC THENL [CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN ASM_MESON_TAC[CONNECTED_SIMPLE_PATH_ENDLESS; ARC_IMP_SIMPLE_PATH]; DISCH_TAC]) THEN SUBGOAL_THEN `inside(path_image c1 UNION path_image c2:real^2->bool) IN components((:real^2) DIFF (path_image c1 UNION path_image c2 UNION path_image c3)) /\ inside(path_image c2 UNION path_image c3:real^2->bool) IN components((:real^2) DIFF (path_image c1 UNION path_image c2 UNION path_image c3)) /\ inside(path_image c3 UNION path_image c1:real^2->bool) IN components((:real^2) DIFF (path_image c1 UNION path_image c2 UNION path_image c3))` STRIP_ASSUME_TAC THENL [REPEAT CONJ_TAC THEN MATCH_MP_TAC CLOPEN_IN_COMPONENTS THEN ASM_REWRITE_TAC[] THEN (CONJ_TAC THENL [ASM_REWRITE_TAC[CLOSED_IN_INTER_CLOSURE; CLOSURE_UNION_FRONTIER]; MATCH_MP_TAC OPEN_SUBSET THEN ASM_REWRITE_TAC[]]) THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`closure(inside(path_image c1 UNION path_image c2)):real^2->bool`; `closure(inside(path_image c2 UNION path_image c3)):real^2->bool`] JANISZEWSKI_CONNECTED) THEN ASM_REWRITE_TAC[COMPACT_CLOSURE; CLOSED_CLOSURE; NOT_IMP] THEN REPEAT CONJ_TAC THENL [MP_TAC(ISPEC `c2:real^1->real^2` CONNECTED_PATH_IMAGE) THEN ASM_SIMP_TAC[ARC_IMP_PATH] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN ASM_REWRITE_TAC[CLOSURE_UNION_FRONTIER] THEN MATCH_MP_TAC(SET_RULE `i INTER (c1 UNION c2 UNION c3) = {} /\ j INTER (c1 UNION c2 UNION c3) = {} /\ i INTER j = {} /\ c1 INTER c3 SUBSET c2 ==> c2 = (i UNION c1 UNION c2) INTER (j UNION c2 UNION c3)`) THEN REPEAT CONJ_TAC THENL [REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET)) THEN SET_TAC[]; REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET)) THEN SET_TAC[]; MP_TAC(ISPEC `(:real^2) DIFF (path_image c1 UNION path_image c2 UNION path_image c3)` COMPONENTS_NONOVERLAP) THEN ASM_SIMP_TAC[] THEN DISCH_THEN(K ALL_TAC) THEN DISCH_THEN(MP_TAC o AP_TERM `frontier:(real^2->bool)->real^2->bool`) THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPEC `c1:real^1->real^2` NONEMPTY_SIMPLE_PATH_ENDLESS) THEN ASM_SIMP_TAC[ARC_IMP_SIMPLE_PATH] THEN ASM SET_TAC[]; ASM SET_TAC[]]; UNDISCH_TAC `connected(outside(path_image c1 UNION path_image c2):real^2->bool)` THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN ASM_REWRITE_TAC[CLOSURE_UNION_FRONTIER] THEN ASM SET_TAC[]; UNDISCH_TAC `connected(outside(path_image c2 UNION path_image c3):real^2->bool)` THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN ASM_REWRITE_TAC[CLOSURE_UNION_FRONTIER] THEN ASM SET_TAC[]; (MP_TAC o ASSUME) `inside(path_image c3 UNION path_image c1:real^2->bool) IN components((:real^2) DIFF (path_image c1 UNION path_image c2 UNION path_image c3))` THEN REWRITE_TAC[IN_COMPONENTS_MAXIMAL] THEN DISCH_THEN(MP_TAC o last o CONJUNCTS) THEN MATCH_MP_TAC(MESON[] `R s /\ ~(s = i) /\ P s /\ Q s ==> (!c. P c /\ Q c /\ R c /\ connected c ==> c = i) ==> ~connected s`) THEN CONJ_TAC THENL [ASM_REWRITE_TAC[CLOSURE_UNION_FRONTIER] THEN ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [DISCH_THEN(MP_TAC o AP_TERM `bounded:(real^2->bool)->bool`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC COBOUNDED_IMP_UNBOUNDED THEN REWRITE_TAC[SET_RULE `UNIV DIFF (UNIV DIFF s) = s`] THEN ASM_REWRITE_TAC[BOUNDED_UNION; BOUNDED_CLOSURE_EQ]; ALL_TAC] THEN ASM_REWRITE_TAC[CLOSURE_UNION_FRONTIER; SET_RULE `(i UNION c1 UNION c2) UNION (j UNION c2 UNION c3) = (i UNION j) UNION (c1 UNION c2 UNION c3)`] THEN MATCH_MP_TAC(SET_RULE `i3 SUBSET UNIV DIFF c /\ ~(i3 = {}) /\ i1 INTER i3 = {} /\ i2 INTER i3 = {} ==> ~(UNIV DIFF ((i1 UNION i2) UNION c) = {}) /\ i3 SUBSET UNIV DIFF ((i1 UNION i2) UNION c)`) THEN ASM_SIMP_TAC[IN_COMPONENTS_SUBSET; IN_COMPONENTS_NONEMPTY] THEN MP_TAC(ISPEC `(:real^2) DIFF (path_image c1 UNION path_image c2 UNION path_image c3)` COMPONENTS_NONOVERLAP) THEN ASM_SIMP_TAC[] THEN DISCH_THEN(K ALL_TAC) THEN CONJ_TAC THEN DISCH_THEN(MP_TAC o AP_TERM `frontier:(real^2->bool)->real^2->bool`) THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPEC `c2:real^1->real^2` NONEMPTY_SIMPLE_PATH_ENDLESS) THEN ASM_SIMP_TAC[ARC_IMP_SIMPLE_PATH] THEN ASM SET_TAC[]]);; let SPLIT_INSIDE_SIMPLE_CLOSED_CURVE = prove (`!c1 c2 c a b:real^2. ~(a = b) /\ simple_path c1 /\ pathstart c1 = a /\ pathfinish c1 = b /\ simple_path c2 /\ pathstart c2 = a /\ pathfinish c2 = b /\ simple_path c /\ pathstart c = a /\ pathfinish c = b /\ path_image c1 INTER path_image c2 = {a,b} /\ path_image c1 INTER path_image c = {a,b} /\ path_image c2 INTER path_image c = {a,b} /\ ~(path_image c INTER inside(path_image c1 UNION path_image c2) = {}) ==> inside(path_image c1 UNION path_image c) INTER inside(path_image c2 UNION path_image c) = {} /\ inside(path_image c1 UNION path_image c) UNION inside(path_image c2 UNION path_image c) UNION (path_image c DIFF {a,b}) = inside(path_image c1 UNION path_image c2)`, REPEAT GEN_TAC THEN STRIP_TAC THEN MAP_EVERY (MP_TAC o C ISPEC JORDAN_INSIDE_OUTSIDE) [`(c1 ++ reversepath c2):real^1->real^2`; `(c1 ++ reversepath c):real^1->real^2`; `(c2 ++ reversepath c):real^1->real^2`] THEN ASM_SIMP_TAC[PATHSTART_JOIN; PATHFINISH_JOIN; PATHSTART_REVERSEPATH; PATHFINISH_REVERSEPATH; SIMPLE_PATH_JOIN_LOOP; SIMPLE_PATH_IMP_ARC; PATH_IMAGE_JOIN; SIMPLE_PATH_IMP_PATH; PATH_IMAGE_REVERSEPATH; SIMPLE_PATH_REVERSEPATH; ARC_REVERSEPATH; SUBSET_REFL] THEN REPLICATE_TAC 3 STRIP_TAC THEN SUBGOAL_THEN `path_image(c:real^1->real^2) INTER outside(path_image c1 UNION path_image c2) = {}` ASSUME_TAC THENL [MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THEN SUBGOAL_THEN `connected(path_image(c:real^1->real^2) DIFF {pathstart c,pathfinish c})` MP_TAC THENL [ASM_SIMP_TAC[CONNECTED_SIMPLE_PATH_ENDLESS]; ALL_TAC] THEN ASM_REWRITE_TAC[connected] THEN MAP_EVERY EXISTS_TAC [`inside(path_image c1 UNION path_image c2):real^2->bool`; `outside(path_image c1 UNION path_image c2):real^2->bool`] THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `outside(path_image c1 UNION path_image c2) SUBSET outside(path_image c1 UNION path_image (c:real^1->real^2)) /\ outside(path_image c1 UNION path_image c2) SUBSET outside(path_image c2 UNION path_image c)` STRIP_ASSUME_TAC THENL [CONJ_TAC THENL [ALL_TAC; GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [UNION_COMM]] THEN MATCH_MP_TAC OUTSIDE_UNION_OUTSIDE_UNION THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[UNION_COMM] THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `path_image(c1:real^1->real^2) INTER inside(path_image c2 UNION path_image c) = {}` ASSUME_TAC THENL [MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THEN SUBGOAL_THEN `frontier(outside(path_image c1 UNION path_image c2)):real^2->bool = frontier(outside(path_image c2 UNION path_image c))` MP_TAC THENL [AP_TERM_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [UNION_COMM] THEN MATCH_MP_TAC OUTSIDE_UNION_OUTSIDE_UNION THEN MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THEN SUBGOAL_THEN `connected(path_image(c1:real^1->real^2) DIFF {pathstart c1,pathfinish c1})` MP_TAC THENL [ASM_SIMP_TAC[CONNECTED_SIMPLE_PATH_ENDLESS]; ALL_TAC] THEN ASM_REWRITE_TAC[connected] THEN MAP_EVERY EXISTS_TAC [`inside(path_image c2 UNION path_image c):real^2->bool`; `outside(path_image c2 UNION path_image c):real^2->bool`] THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; MP_TAC(ISPEC `c:real^1->real^2` NONEMPTY_SIMPLE_PATH_ENDLESS) THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]]; ALL_TAC] THEN SUBGOAL_THEN `path_image(c2:real^1->real^2) INTER inside(path_image c1 UNION path_image c) = {}` ASSUME_TAC THENL [MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THEN SUBGOAL_THEN `frontier(outside(path_image c1 UNION path_image c2)):real^2->bool = frontier(outside(path_image c1 UNION path_image c))` MP_TAC THENL [AP_TERM_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC OUTSIDE_UNION_OUTSIDE_UNION THEN MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THEN SUBGOAL_THEN `connected(path_image(c2:real^1->real^2) DIFF {pathstart c2,pathfinish c2})` MP_TAC THENL [ASM_SIMP_TAC[CONNECTED_SIMPLE_PATH_ENDLESS]; ALL_TAC] THEN ASM_REWRITE_TAC[connected] THEN MAP_EVERY EXISTS_TAC [`inside(path_image c1 UNION path_image c):real^2->bool`; `outside(path_image c1 UNION path_image c):real^2->bool`] THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; MP_TAC(ISPEC `c:real^1->real^2` NONEMPTY_SIMPLE_PATH_ENDLESS) THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]]; ALL_TAC] THEN SUBGOAL_THEN `inside(path_image c1 UNION path_image (c:real^1->real^2)) SUBSET inside(path_image c1 UNION path_image c2) /\ inside(path_image c2 UNION path_image (c:real^1->real^2)) SUBSET inside(path_image c1 UNION path_image c2)` STRIP_ASSUME_TAC THENL [CONJ_TAC THEN REWRITE_TAC[INSIDE_OUTSIDE] THEN REWRITE_TAC[SET_RULE `UNIV DIFF t SUBSET UNIV DIFF s <=> s SUBSET t`] THENL [ALL_TAC; GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [UNION_COMM]] THEN MATCH_MP_TAC(SET_RULE `out1 SUBSET out2 /\ c2 DIFF (c1 UNION c) SUBSET out2 ==> (c1 UNION c2) UNION out1 SUBSET (c1 UNION c) UNION out2`) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[OUTSIDE_INSIDE] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `inside(path_image c1 UNION path_image c :real^2->bool) SUBSET outside(path_image c2 UNION path_image c) /\ inside(path_image c2 UNION path_image c) SUBSET outside(path_image c1 UNION path_image c)` STRIP_ASSUME_TAC THENL [REWRITE_TAC[SUBSET] THEN CONJ_TAC THEN X_GEN_TAC `x:real^2` THEN DISCH_TAC THENL [SUBGOAL_THEN `?z:real^2. z IN path_image c1 /\ z IN outside(path_image c2 UNION path_image c)` (CHOOSE_THEN (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THENL [REWRITE_TAC[OUTSIDE_INSIDE; IN_DIFF; IN_UNION; IN_UNIV] THEN MP_TAC(ISPEC `c1:real^1->real^2` NONEMPTY_SIMPLE_PATH_ENDLESS) THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN REWRITE_TAC[OUTSIDE; IN_ELIM_THM; CONTRAPOS_THM] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN MATCH_MP_TAC CONNECTED_COMPONENT_EQ THEN REWRITE_TAC[IN] THEN MP_TAC(ASSUME `open(outside(path_image c2 UNION path_image c):real^2->bool)`) THEN REWRITE_TAC[OPEN_CONTAINS_BALL] THEN DISCH_THEN(MP_TAC o SPEC `z:real^2`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN MP_TAC(ASSUME `frontier(inside(path_image c1 UNION path_image c):real^2->bool) = path_image c1 UNION path_image c`) THEN GEN_REWRITE_TAC LAND_CONV [EXTENSION] THEN DISCH_THEN(MP_TAC o SPEC `z:real^2`) THEN REWRITE_TAC[frontier] THEN ASM_SIMP_TAC[IN_UNION; IN_DIFF; CLOSURE_APPROACHABLE; IN_ELIM_THM] THEN DISCH_THEN(MP_TAC o SPEC `e:real` o CONJUNCT1) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `w:real^2` THEN STRIP_TAC THEN MATCH_MP_TAC CONNECTED_COMPONENT_TRANS THEN EXISTS_TAC `w:real^2` THEN REWRITE_TAC[connected_component] THEN CONJ_TAC THENL [EXISTS_TAC `outside(path_image c2 UNION path_image c:real^2->bool)` THEN ASM_REWRITE_TAC[SET_RULE `s SUBSET UNIV DIFF t <=> s INTER t = {}`; OUTSIDE_NO_OVERLAP] THEN FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN ASM_REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] IN_BALL]; EXISTS_TAC `inside(path_image c1 UNION path_image c:real^2->bool)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(SET_RULE `inside(c1 UNION c) INTER (c1 UNION c) = {} /\ c2 INTER inside(c1 UNION c) = {} ==> inside(c1 UNION c) SUBSET UNIV DIFF (c2 UNION c)`) THEN ASM_REWRITE_TAC[INSIDE_NO_OVERLAP]]; SUBGOAL_THEN `?z:real^2. z IN path_image c2 /\ z IN outside(path_image c1 UNION path_image c)` (CHOOSE_THEN (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THENL [REWRITE_TAC[OUTSIDE_INSIDE; IN_DIFF; IN_UNION; IN_UNIV] THEN MP_TAC(ISPEC `c2:real^1->real^2` NONEMPTY_SIMPLE_PATH_ENDLESS) THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN REWRITE_TAC[OUTSIDE; IN_ELIM_THM; CONTRAPOS_THM] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN MATCH_MP_TAC CONNECTED_COMPONENT_EQ THEN REWRITE_TAC[IN] THEN MP_TAC(ASSUME `open(outside(path_image c1 UNION path_image c):real^2->bool)`) THEN REWRITE_TAC[OPEN_CONTAINS_BALL] THEN DISCH_THEN(MP_TAC o SPEC `z:real^2`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN MP_TAC(ASSUME `frontier(inside(path_image c2 UNION path_image c):real^2->bool) = path_image c2 UNION path_image c`) THEN GEN_REWRITE_TAC LAND_CONV [EXTENSION] THEN DISCH_THEN(MP_TAC o SPEC `z:real^2`) THEN REWRITE_TAC[frontier] THEN ASM_SIMP_TAC[IN_UNION; IN_DIFF; CLOSURE_APPROACHABLE; IN_ELIM_THM] THEN DISCH_THEN(MP_TAC o SPEC `e:real` o CONJUNCT1) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `w:real^2` THEN STRIP_TAC THEN MATCH_MP_TAC CONNECTED_COMPONENT_TRANS THEN EXISTS_TAC `w:real^2` THEN REWRITE_TAC[connected_component] THEN CONJ_TAC THENL [EXISTS_TAC `outside(path_image c1 UNION path_image c:real^2->bool)` THEN ASM_REWRITE_TAC[SET_RULE `s SUBSET UNIV DIFF t <=> s INTER t = {}`; OUTSIDE_NO_OVERLAP] THEN FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN ASM_REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] IN_BALL]; EXISTS_TAC `inside(path_image c2 UNION path_image c:real^2->bool)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(SET_RULE `inside(c2 UNION c) INTER (c2 UNION c) = {} /\ c1 INTER inside(c2 UNION c) = {} ==> inside(c2 UNION c) SUBSET UNIV DIFF (c1 UNION c)`) THEN ASM_REWRITE_TAC[INSIDE_NO_OVERLAP]]]; ALL_TAC] THEN CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `!u. s SUBSET u /\ t INTER u = {} ==> s INTER t = {}`) THEN EXISTS_TAC `outside(path_image c2 UNION path_image c):real^2->bool` THEN ASM_REWRITE_TAC[INSIDE_INTER_OUTSIDE]; ALL_TAC] THEN SUBGOAL_THEN `outside (path_image c1 UNION path_image c) INTER outside (path_image c2 UNION path_image c):real^2->bool SUBSET outside (path_image c1 UNION path_image c2)` MP_TAC THENL [ALL_TAC; ONCE_REWRITE_TAC[SET_RULE `s INTER t = u <=> (UNIV DIFF s) UNION (UNIV DIFF t) = UNIV DIFF u`] THEN REWRITE_TAC[GSYM UNION_WITH_INSIDE] THEN ASM SET_TAC[]] THEN MATCH_MP_TAC COMPONENTS_MAXIMAL THEN EXISTS_TAC `(:real^2) DIFF (path_image c1 UNION path_image c2)` THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [ASM_REWRITE_TAC[OUTSIDE_IN_COMPONENTS]; DISCH_TAC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MP_TAC(ISPECL [`closure(inside(path_image c1 UNION path_image c)):real^2->bool`; `closure(inside(path_image c2 UNION path_image c)):real^2->bool`] JANISZEWSKI_CONNECTED) THEN ASM_REWRITE_TAC[COMPACT_CLOSURE; CLOSED_CLOSURE] THEN ASM_REWRITE_TAC[CLOSURE_UNION_FRONTIER; SET_RULE `UNIV DIFF (UNIV DIFF s) = s`; ONCE_REWRITE_RULE[UNION_COMM] UNION_WITH_INSIDE] THEN REWRITE_TAC[SET_RULE `UNIV DIFF ((UNIV DIFF s) UNION (UNIV DIFF t)) = s INTER t`] THEN DISCH_THEN MATCH_MP_TAC THEN SUBGOAL_THEN `connected(path_image c:real^2->bool)` MP_TAC THENL [ASM_SIMP_TAC[CONNECTED_SIMPLE_PATH_IMAGE]; ALL_TAC] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN REWRITE_TAC[GSYM UNION_WITH_INSIDE] THEN ASM SET_TAC[]);;