(* ========================================================================= *) (* Basic setup of singular homology. *) (* ========================================================================= *) needs "Library/frag.ml";; (* Free Abelian groups *) needs "Library/grouptheory.ml";; (* Basic group theory machinery *) needs "Multivariate/metric.ml";; (* General topology *) (* ------------------------------------------------------------------------- *) (* Standard simplices, all of which are topological subspaces of R^num. *) (* ------------------------------------------------------------------------- *) let standard_simplex = new_definition `standard_simplex p = { x:num->real | (!i. &0 <= x i /\ x i <= &1) /\ (!i. p < i ==> x i = &0) /\ sum (0..p) x = &1}`;; let TOPSPACE_STANDARD_SIMPLEX = prove (`!p. topspace(subtopology (product_topology (:num) (\i. euclideanreal)) (standard_simplex p)) = standard_simplex p`, REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; TOPSPACE_PRODUCT_TOPOLOGY] THEN REWRITE_TAC[o_DEF; TOPSPACE_EUCLIDEANREAL; CARTESIAN_PRODUCT_UNIV] THEN REWRITE_TAC[INTER_UNIV]);; let BASIS_IN_STANDARD_SIMPLEX = prove (`!p i. (\j. if j = i then &1 else &0) IN standard_simplex p <=> i <= p`, REPEAT STRIP_TAC THEN REWRITE_TAC[standard_simplex; IN_ELIM_THM] THEN REWRITE_TAC[SUM_DELTA; IN_NUMSEG; LE_0] THEN ASM_MESON_TAC[REAL_POS; REAL_LE_REFL; NOT_LE; REAL_ARITH `~(&1 = &0)`]);; let NONEMPTY_STANDARD_SIMPLEX = prove (`!p. ~(standard_simplex p = {})`, GEN_TAC THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `(\i. if i = 0 then &1 else &0):num->real` THEN REWRITE_TAC[BASIS_IN_STANDARD_SIMPLEX; LE_0]);; let STANDARD_SIMPLEX_0 = prove (`standard_simplex 0 = {(\j. if j = 0 then &1 else &0)}`, REWRITE_TAC[EXTENSION; standard_simplex; IN_ELIM_THM; IN_SING] THEN GEN_TAC THEN GEN_REWRITE_TAC RAND_CONV [FUN_EQ_THM] THEN REWRITE_TAC[SUM_SING_NUMSEG] THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[LE_1] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_ARITH_TAC);; let STANDARD_SIMPLEX_MONO = prove (`!p q. p <= q ==> standard_simplex p SUBSET standard_simplex q`, REPEAT STRIP_TAC THEN REWRITE_TAC[SUBSET; standard_simplex; IN_ELIM_THM] THEN X_GEN_TAC `x:num->real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[LET_TRANS]; ALL_TAC] THEN FIRST_ASSUM(SUBST1_TAC o SYM) THEN MATCH_MP_TAC SUM_SUPERSET THEN ASM_SIMP_TAC[IN_NUMSEG; LE_0; NOT_LE; SUBSET] THEN ASM_ARITH_TAC);; let CLOSED_IN_STANDARD_SIMPLEX = prove (`!p. closed_in (product_topology (:num) (\i. euclideanreal)) (standard_simplex p)`, GEN_TAC THEN ABBREV_TAC `top = product_topology (:num) (\i. euclideanreal)` THEN SUBGOAL_THEN `standard_simplex p = INTERS { {x | x IN topspace top /\ x i IN real_interval[&0,&1]} | i IN (:num)} INTER INTERS { {x | x IN topspace top /\ x i IN {&0}} | p < i} INTER {x | x IN topspace top /\ sum(0..p) x IN {&1}}` SUBST1_TAC THENL [GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[standard_simplex; IN_ELIM_THM; INTERS_GSPEC; IN_INTER] THEN EXPAND_TAC "top" THEN REWRITE_TAC[TOPSPACE_PRODUCT_TOPOLOGY] THEN REWRITE_TAC[CARTESIAN_PRODUCT_UNIV; o_DEF; TOPSPACE_EUCLIDEANREAL] THEN REWRITE_TAC[IN_SING; IN_UNIV; IN_ELIM_THM; IN_REAL_INTERVAL]; REPEAT(MATCH_MP_TAC CLOSED_IN_INTER THEN CONJ_TAC) THEN TRY(MATCH_MP_TAC CLOSED_IN_INTERS) THEN REWRITE_TAC[SET_RULE `~({f x | P x} = {}) <=> ?x. P x`] THEN REWRITE_TAC[FORALL_IN_GSPEC; IN_UNIV] THEN TRY(CONJ_TAC THENL [MESON_TAC[LT]; ALL_TAC]) THEN TRY(X_GEN_TAC `i:num`) THEN TRY DISCH_TAC THEN MATCH_MP_TAC CLOSED_IN_CONTINUOUS_MAP_PREIMAGE THEN EXISTS_TAC `euclideanreal` THEN REWRITE_TAC[GSYM REAL_CLOSED_IN] THEN REWRITE_TAC[REAL_CLOSED_SING; REAL_CLOSED_REAL_INTERVAL] THEN EXPAND_TAC "top" THEN SIMP_TAC[CONTINUOUS_MAP_PRODUCT_PROJECTION; IN_UNIV] THEN MATCH_MP_TAC CONTINUOUS_MAP_SUM THEN SIMP_TAC[CONTINUOUS_MAP_PRODUCT_PROJECTION; IN_UNIV] THEN REWRITE_TAC[FINITE_NUMSEG]]);; let COMPACT_IN_STANDARD_SIMPLEX = prove (`!p. compact_in (product_topology (:num) (\i. euclideanreal)) (standard_simplex p)`, GEN_TAC THEN MATCH_MP_TAC CLOSED_COMPACT_IN THEN EXISTS_TAC `cartesian_product (:num) (\i. real_interval[&0,&1])` THEN REWRITE_TAC[COMPACT_IN_CARTESIAN_PRODUCT] THEN REWRITE_TAC[CLOSED_IN_STANDARD_SIMPLEX] THEN REWRITE_TAC[COMPACT_IN_EUCLIDEANREAL_INTERVAL] THEN REWRITE_TAC[SUBSET; standard_simplex; cartesian_product] THEN REWRITE_TAC[IN_UNIV; EXTENSIONAL_UNIV; IN_ELIM_THM] THEN SIMP_TAC[IN_REAL_INTERVAL]);; let CONVEX_STANDARD_SIMPLEX = prove (`!p x y u. x IN standard_simplex p /\ y IN standard_simplex p /\ &0 <= u /\ u <= &1 ==> (\i. (&1 - u) * x i + u * y i) IN standard_simplex p`, REPEAT GEN_TAC THEN REWRITE_TAC[standard_simplex; IN_ELIM_THM] THEN STRIP_TAC THEN ASM_SIMP_TAC[SUM_ADD_NUMSEG; SUM_LMUL] THEN CONJ_TAC THENL [ALL_TAC; REAL_ARITH_TAC] THEN ASM_SIMP_TAC[REAL_LE_ADD; REAL_LE_MUL; REAL_SUB_LE] THEN GEN_TAC THEN MATCH_MP_TAC REAL_CONVEX_BOUND_LE THEN ASM_SIMP_TAC[REAL_SUB_LE] THEN REAL_ARITH_TAC);; let PATH_CONNECTED_IN_STANDARD_SIMPLEX = prove (`!p. path_connected_in (product_topology (:num) (\i. euclideanreal)) (standard_simplex p)`, GEN_TAC THEN REWRITE_TAC[path_connected_in; path_connected_space; path_in] THEN REWRITE_TAC[TOPSPACE_STANDARD_SIMPLEX; TOPSPACE_PRODUCT_TOPOLOGY; CONTINUOUS_MAP_IN_SUBTOPOLOGY; o_DEF; TOPSPACE_EUCLIDEANREAL; TOPSPACE_EUCLIDEANREAL_SUBTOPOLOGY; CARTESIAN_PRODUCT_UNIV; SUBSET; FORALL_IN_IMAGE; IN_UNIV] THEN MAP_EVERY X_GEN_TAC [`x:num->real`; `y:num->real`] THEN STRIP_TAC THEN EXISTS_TAC `(\u i. (&1 - u) * x i + u * y i):real->num->real` THEN ASM_SIMP_TAC[IN_REAL_INTERVAL; CONVEX_STANDARD_SIMPLEX] THEN REWRITE_TAC[CONTINUOUS_MAP_COMPONENTWISE; FUN_EQ_THM] THEN REWRITE_TAC[SET_RULE `s SUBSET P <=> !x. x IN s ==> P x`] THEN REWRITE_TAC[EXTENSIONAL_UNIV; IN_UNIV] THEN CONJ_TAC THENL [GEN_TAC; REAL_ARITH_TAC] THEN MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_ADD THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_MUL THEN CONJ_TAC THEN TRY(MATCH_MP_TAC CONTINUOUS_MAP_REAL_SUB) THEN REWRITE_TAC[CONTINUOUS_MAP_REAL_CONST; CONTINUOUS_MAP_ID]);; let CONNECTED_IN_STANDARD_SIMPLEX = prove (`!p. connected_in (product_topology (:num) (\i. euclideanreal)) (standard_simplex p)`, GEN_TAC THEN MATCH_MP_TAC PATH_CONNECTED_IN_IMP_CONNECTED_IN THEN REWRITE_TAC[PATH_CONNECTED_IN_STANDARD_SIMPLEX]);; (* ------------------------------------------------------------------------- *) (* Face map. *) (* ------------------------------------------------------------------------- *) let simplicial_face = new_definition `simplicial_face k (x:num->real) = \i. if i < k then x i else if i = k then &0 else x(i - 1)`;; let SIMPLICIAL_FACE_IN_STANDARD_SIMPLEX = prove (`!p k (x:num->real). 1 <= p /\ k <= p /\ x IN standard_simplex (p - 1) ==> (simplicial_face k x) IN standard_simplex p`, REWRITE_TAC[standard_simplex; simplicial_face; IN_ELIM_THM] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN REPEAT CONJ_TAC THENL [GEN_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_POS]); GEN_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN DISCH_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; ASM_SIMP_TAC[SUM_CASES; FINITE_NUMSEG; IN_NUMSEG; ARITH_RULE `(0 <= i /\ i <= p) /\ ~(i < k) <=> k <= i /\ i <= p`] THEN REWRITE_TAC[GSYM numseg] THEN ASM_SIMP_TAC[SUM_CLAUSES_LEFT; REAL_ADD_LID] THEN SUBGOAL_THEN `p = (p - 1) + 1` SUBST1_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[SPEC `1` SUM_OFFSET; ADD_SUB] THEN SIMP_TAC[ARITH_RULE `k <= i ==> ~(i + 1 = k)`] THEN REWRITE_TAC[GSYM IN_NUMSEG] THEN W(MP_TAC o PART_MATCH (rand o rand) SUM_UNION o lhand o snd) THEN ASM_SIMP_TAC[FINITE_NUMSEG; FINITE_RESTRICT] THEN ANTS_TAC THENL [SIMP_TAC[DISJOINT; EXTENSION; IN_INTER; NOT_IN_EMPTY; IN_ELIM_THM] THEN REWRITE_TAC[IN_NUMSEG] THEN ARITH_TAC; DISCH_THEN(SUBST1_TAC o SYM)] THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [SYM th]) THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION] THEN REWRITE_TAC[IN_UNION; IN_ELIM_THM; IN_NUMSEG] THEN ASM_ARITH_TAC]);; (* ------------------------------------------------------------------------- *) (* Singular simplices, forcing canonicity outside the intended domain. *) (* ------------------------------------------------------------------------- *) let singular_simplex = new_definition `singular_simplex (p,top) (f:(num->real)->A) <=> continuous_map(subtopology (product_topology (:num) (\i. euclideanreal)) (standard_simplex p), top) f /\ EXTENSIONAL (standard_simplex p) f`;; let SINGULAR_SIMPLEX_EMPTY = prove (`!p (top:A topology) f. topspace top = {} ==> ~(singular_simplex (p,top) f)`, REPEAT GEN_TAC THEN GEN_REWRITE_TAC I [GSYM CONTRAPOS_THM] THEN REWRITE_TAC[singular_simplex; CONTINUOUS_MAP] THEN MATCH_MP_TAC(SET_RULE `~(s = {}) ==> (IMAGE f s SUBSET t /\ P) /\ Q ==> ~(t = {})`) THEN REWRITE_TAC[TOPSPACE_STANDARD_SIMPLEX; NONEMPTY_STANDARD_SIMPLEX]);; let SINGULAR_SIMPLEX_MONO = prove (`!p (top:A topology) s t f. t SUBSET s /\ singular_simplex (p,subtopology top t) f ==> singular_simplex (p,subtopology top s) f`, REWRITE_TAC[singular_simplex; CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN MESON_TAC[SUBSET_TRANS]);; let SINGULAR_SIMPLEX_SUBTOPOLOGY = prove (`!p (top:A topology) s f. singular_simplex (p,subtopology top s) f <=> singular_simplex (p,top) f /\ IMAGE f (standard_simplex p) SUBSET s`, REWRITE_TAC[singular_simplex; CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; TOPSPACE_PRODUCT_TOPOLOGY] THEN REWRITE_TAC[o_DEF; TOPSPACE_EUCLIDEANREAL; CARTESIAN_PRODUCT_UNIV] THEN REWRITE_TAC[INTER_UNIV; CONJ_ACI]);; (* ------------------------------------------------------------------------- *) (* Singular face. *) (* ------------------------------------------------------------------------- *) let singular_face = new_definition `singular_face p k f = RESTRICTION (standard_simplex (p - 1)) (f o simplicial_face k)`;; let SINGULAR_SIMPLEX_SINGULAR_FACE = prove (`!p (top:A topology) k f. singular_simplex (p,top) f /\ 1 <= p /\ k <= p ==> singular_simplex (p - 1,top) (singular_face p k f)`, REPEAT GEN_TAC THEN REWRITE_TAC[singular_simplex] THEN STRIP_TAC THEN REWRITE_TAC[singular_face; REWRITE_RULE[IN] RESTRICTION_IN_EXTENSIONAL] THEN SIMP_TAC[RESTRICTION_CONTINUOUS_MAP; TOPSPACE_STANDARD_SIMPLEX; SUBSET_REFL] THEN MATCH_MP_TAC CONTINUOUS_MAP_COMPOSE THEN EXISTS_TAC `subtopology (product_topology (:num) (\i. euclideanreal)) (standard_simplex p)` THEN ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY; TOPSPACE_STANDARD_SIMPLEX; SUBSET; FORALL_IN_IMAGE; SIMPLICIAL_FACE_IN_STANDARD_SIMPLEX] THEN REWRITE_TAC[CONTINUOUS_MAP_COMPONENTWISE; IN_UNIV] THEN REWRITE_TAC[SUBSET; EXTENSIONAL_UNIV; IN] THEN X_GEN_TAC `i:num` THEN REWRITE_TAC[simplicial_face] THEN MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN MAP_EVERY ASM_CASES_TAC [`i:num < k`; `i:num = k`] THEN ASM_SIMP_TAC[CONTINUOUS_MAP_PRODUCT_PROJECTION; IN_UNIV] THEN REWRITE_TAC[CONTINUOUS_MAP_CONST; TOPSPACE_EUCLIDEANREAL; IN_UNIV]);; (* ------------------------------------------------------------------------- *) (* Singular chains. *) (* ------------------------------------------------------------------------- *) let singular_chain = new_definition `singular_chain (p,top) (c:((num->real)->A)frag) <=> frag_support c SUBSET singular_simplex(p,top)`;; let SINGULAR_CHAIN_EMPTY = prove (`!p (top:A topology) c. topspace top = {} ==> (singular_chain(p,top) c <=> c = frag_0)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `c:((num->real)->A)frag = frag_0` THEN ASM_REWRITE_TAC[singular_chain; FRAG_SUPPORT_0; EMPTY_SUBSET] THEN ASM_SIMP_TAC[FRAG_SUPPORT_EQ_EMPTY; SINGULAR_SIMPLEX_EMPTY; SET_RULE `(!x. ~t x) ==> (s SUBSET t <=> s = {})`]);; let SINGULAR_CHAIN_MONO = prove (`!p (top:A topology) s t c. t SUBSET s /\ singular_chain (p,subtopology top t) c ==> singular_chain (p,subtopology top s) c`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[singular_chain] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS) THEN REWRITE_TAC[SUBSET; IN] THEN ASM_MESON_TAC[SINGULAR_SIMPLEX_MONO]);; let SINGULAR_CHAIN_SUBTOPOLOGY = prove (`!p (top:A topology) s c. singular_chain (p,subtopology top s) c <=> singular_chain (p,top) c /\ !f. f IN frag_support c ==> IMAGE f (standard_simplex p) SUBSET s`, SIMP_TAC[singular_chain; SET_RULE `s SUBSET P <=> !x. x IN s ==> P x`] THEN REWRITE_TAC[SINGULAR_SIMPLEX_SUBTOPOLOGY] THEN SET_TAC[]);; let SINGULAR_CHAIN_0 = prove (`!p (top:A topology). singular_chain (p,top) frag_0`, REWRITE_TAC[singular_chain; FRAG_SUPPORT_0; EMPTY_SUBSET]);; let SINGULAR_CHAIN_OF = prove (`!p (top:A topology) c. singular_chain (p,top) (frag_of c) <=> singular_simplex (p,top) c`, REWRITE_TAC[singular_chain; FRAG_SUPPORT_OF] THEN SET_TAC[]);; let SINGULAR_CHAIN_CMUL = prove (`!p (top:A topology) a c. singular_chain (p,top) c ==> singular_chain (p,top) (frag_cmul a c)`, REWRITE_TAC[singular_chain] THEN MESON_TAC[FRAG_SUPPORT_CMUL; SUBSET_TRANS]);; let SINGULAR_CHAIN_NEG = prove (`!p (top:A topology) c. singular_chain (p,top) (frag_neg c) <=> singular_chain (p,top) c`, REWRITE_TAC[singular_chain; FRAG_SUPPORT_NEG]);; let SINGULAR_CHAIN_ADD = prove (`!p (top:A topology) c1 c2. singular_chain (p,top) c1 /\ singular_chain (p,top) c2 ==> singular_chain (p,top) (frag_add c1 c2)`, REWRITE_TAC[singular_chain] THEN MESON_TAC[FRAG_SUPPORT_ADD; SUBSET_TRANS; UNION_SUBSET]);; let SINGULAR_CHAIN_SUB = prove (`!p (top:A topology) c1 c2. singular_chain (p,top) c1 /\ singular_chain (p,top) c2 ==> singular_chain (p,top) (frag_sub c1 c2)`, REWRITE_TAC[singular_chain] THEN MESON_TAC[FRAG_SUPPORT_SUB; SUBSET_TRANS; UNION_SUBSET]);; let SINGULAR_CHAIN_SUM = prove (`!p (top:A topology) f k. (!c. c IN k ==> singular_chain (p,top) (f c)) ==> singular_chain (p,top) (iterate frag_add k f)`, REWRITE_TAC[singular_chain] THEN REPEAT STRIP_TAC THEN W(MP_TAC o PART_MATCH lhand FRAG_SUPPORT_SUM o lhand o snd) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS) THEN ASM_REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC]);; let SINGULAR_CHAIN_EXTEND = prove (`!p top (f:((num->real)->A)->((num->real)->B)frag) x. (!c. c IN frag_support x ==> singular_chain (p,top) (f c)) ==> singular_chain (p,top) (frag_extend f x)`, REPEAT STRIP_TAC THEN REWRITE_TAC[frag_extend] THEN MATCH_MP_TAC SINGULAR_CHAIN_SUM THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[SINGULAR_CHAIN_CMUL]);; (* ------------------------------------------------------------------------- *) (* Boundary homomorphism for singular chains. *) (* ------------------------------------------------------------------------- *) let chain_boundary = new_definition `chain_boundary p (c:((num->real)->A)frag) = if p = 0 then frag_0 else frag_extend (\f. iterate frag_add (0..p) (\k. frag_cmul (--(&1) pow k) (frag_of(singular_face p k f)))) c`;; let SINGULAR_CHAIN_BOUNDARY = prove (`!p (top:A topology) c. singular_chain (p,top) c ==> singular_chain (p - 1,top) (chain_boundary p c)`, REPEAT STRIP_TAC THEN REWRITE_TAC[chain_boundary] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[SINGULAR_CHAIN_0] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP (ARITH_RULE `~(p = 0) ==> 1 <= p`)) THEN MATCH_MP_TAC SINGULAR_CHAIN_EXTEND THEN REPEAT STRIP_TAC THEN REWRITE_TAC[] THEN MATCH_MP_TAC SINGULAR_CHAIN_SUM THEN REWRITE_TAC[IN_NUMSEG] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC SINGULAR_CHAIN_CMUL THEN REWRITE_TAC[SINGULAR_CHAIN_OF] THEN MATCH_MP_TAC SINGULAR_SIMPLEX_SINGULAR_FACE THEN RULE_ASSUM_TAC(REWRITE_RULE[singular_chain]) THEN ASM SET_TAC[]);; let SINGULAR_CHAIN_BOUNDARY_ALT = prove (`!p (top:A topology) c. singular_chain (p + 1,top) c ==> singular_chain (p,top) (chain_boundary (p + 1) c)`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP SINGULAR_CHAIN_BOUNDARY) THEN REWRITE_TAC[ADD_SUB]);; let CHAIN_BOUNDARY_0 = prove (`!p. chain_boundary p frag_0 :((num->real)->A)frag = frag_0`, GEN_TAC THEN REWRITE_TAC[chain_boundary] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[FRAG_EXTEND_0]);; let CHAIN_BOUNDARY_CMUL = prove (`!p a c:((num->real)->A)frag. chain_boundary p (frag_cmul a c) = frag_cmul a (chain_boundary p c)`, REPEAT GEN_TAC THEN REWRITE_TAC[chain_boundary] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[FRAG_EXTEND_CMUL] THEN CONV_TAC FRAG_MODULE);; let CHAIN_BOUNDARY_NEG = prove (`!p c:((num->real)->A)frag. chain_boundary p (frag_neg c) = frag_neg (chain_boundary p c)`, REWRITE_TAC[FRAG_MODULE `frag_neg x = frag_cmul (-- &1) x`] THEN REWRITE_TAC[CHAIN_BOUNDARY_CMUL]);; let CHAIN_BOUNDARY_ADD = prove (`!p c1 c2:((num->real)->A)frag. chain_boundary p (frag_add c1 c2) = frag_add (chain_boundary p c1) (chain_boundary p c2)`, REPEAT GEN_TAC THEN REWRITE_TAC[chain_boundary] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[FRAG_EXTEND_ADD] THEN CONV_TAC FRAG_MODULE);; let CHAIN_BOUNDARY_SUB = prove (`!p c1 c2:((num->real)->A)frag. chain_boundary p (frag_sub c1 c2) = frag_sub (chain_boundary p c1) (chain_boundary p c2)`, REWRITE_TAC[FRAG_MODULE `frag_sub x y = frag_add x (frag_neg y)`] THEN REWRITE_TAC[CHAIN_BOUNDARY_ADD; SINGULAR_CHAIN_NEG; CHAIN_BOUNDARY_NEG]);; let CHAIN_BOUNDARY_SUM = prove (`!p g:K->((num->real)->A)frag k. FINITE k ==> chain_boundary p (iterate frag_add k g) = iterate frag_add k (chain_boundary p o g)`, GEN_TAC THEN GEN_TAC THEN CONV_TAC(ONCE_DEPTH_CONV SYM_CONV) THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN SIMP_TAC[MATCH_MP ITERATE_CLAUSES MONOIDAL_FRAG_ADD] THEN REWRITE_TAC[NEUTRAL_FRAG_ADD; CHAIN_BOUNDARY_0] THEN REWRITE_TAC[o_THM; CHAIN_BOUNDARY_ADD]);; let CHAIN_BOUNDARY_OF = prove (`!p f:(num->real)->A. chain_boundary p (frag_of f) = if p = 0 then frag_0 else iterate frag_add (0..p) (\k. frag_cmul (-- &1 pow k) (frag_of(singular_face p k f)))`, REWRITE_TAC[chain_boundary; FRAG_EXTEND_OF]);; (* ------------------------------------------------------------------------- *) (* Factoring out chains in a subtopology for relative homology. *) (* ------------------------------------------------------------------------- *) let mod_subset = new_definition `mod_subset (p,top) c1 c2 <=> singular_chain (p,top) (frag_sub c1 c2)`;; let MOD_SUBSET_EMPTY = prove (`!p (top:A topology) c1 c2. (c1 == c2) (mod_subset (p,subtopology top {})) <=> c1 = c2`, SIMP_TAC[cong; mod_subset; TOPSPACE_SUBTOPOLOGY; INTER_EMPTY; SINGULAR_CHAIN_EMPTY] THEN REPEAT STRIP_TAC THEN CONV_TAC FRAG_MODULE);; let MOD_SUBSET_REFL = prove (`!p (top:A topology) c. (c == c) (mod_subset(p,top))`, REWRITE_TAC[cong; mod_subset; FRAG_MODULE `frag_sub x x = frag_0`] THEN REWRITE_TAC[SINGULAR_CHAIN_0]);; let MOD_SUBSET_CMUL = prove (`!p (top:A topology) a c1 c2. (c1 == c2) (mod_subset(p,top)) ==> (frag_cmul a c1 == frag_cmul a c2) (mod_subset(p,top))`, REPEAT GEN_TAC THEN REWRITE_TAC[cong; mod_subset] THEN DISCH_THEN(MP_TAC o SPEC `a:int` o MATCH_MP SINGULAR_CHAIN_CMUL) THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN CONV_TAC FRAG_MODULE);; let MOD_SUBSET_ADD = prove (`!p (top:A topology) c1 c2 d1 d2. (c1 == c2) (mod_subset(p,top)) /\ (d1 == d2) (mod_subset(p,top)) ==> (frag_add c1 d1 == frag_add c2 d2) (mod_subset(p,top))`, REPEAT GEN_TAC THEN REWRITE_TAC[cong; mod_subset] THEN DISCH_THEN(MP_TAC o MATCH_MP SINGULAR_CHAIN_ADD) THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN CONV_TAC FRAG_MODULE);; (* ------------------------------------------------------------------------- *) (* Relative cycles Z_p(top,s) where top is a topology and s a subset. *) (* ------------------------------------------------------------------------- *) let singular_relcycle = new_definition `singular_relcycle(p,top,s) c <=> singular_chain (p,top) c /\ (chain_boundary p c == frag_0) (mod_subset(p-1,subtopology top s))`;; let SINGULAR_RELCYCLE_RESTRICT = prove (`!p (top:A topology) s. singular_relcycle(p,top,s) = singular_relcycle(p,top,topspace top INTER s)`, REWRITE_TAC[FUN_EQ_THM; singular_relcycle; GSYM SUBTOPOLOGY_RESTRICT]);; let SINGULAR_RELCYCLE = prove (`!p (top:A topology) s c. singular_relcycle(p,top,s) c <=> singular_chain (p,top) c /\ singular_chain (p-1,subtopology top s) (chain_boundary p c)`, REWRITE_TAC[singular_relcycle; cong; mod_subset] THEN REWRITE_TAC[FRAG_MODULE `frag_sub x frag_0 = x`]);; let SINGULAR_RELCYCLE_0 = prove (`!p (top:A topology) s. singular_relcycle (p,top,s) frag_0`, REWRITE_TAC[singular_relcycle; SINGULAR_CHAIN_0; CHAIN_BOUNDARY_0] THEN REWRITE_TAC[MOD_SUBSET_REFL]);; let SINGULAR_RELCYCLE_CMUL = prove (`!p (top:A topology) s a c. singular_relcycle (p,top,s) c ==> singular_relcycle (p,top,s) (frag_cmul a c)`, SIMP_TAC[singular_relcycle; SINGULAR_CHAIN_CMUL] THEN REWRITE_TAC[CHAIN_BOUNDARY_CMUL] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `a:int` o MATCH_MP MOD_SUBSET_CMUL) THEN REWRITE_TAC[FRAG_MODULE `frag_cmul x frag_0 = frag_0`]);; let SINGULAR_RELCYCLE_NEG = prove (`!p (top:A topology) s c. singular_relcycle (p,top,s) (frag_neg c) <=> singular_relcycle (p,top,s) c`, MESON_TAC[SINGULAR_RELCYCLE_CMUL; FRAG_MODULE `frag_neg(frag_neg x) = x`; FRAG_MODULE `frag_neg x = frag_cmul (-- &1) x`]);; let SINGULAR_RELCYCLE_ADD = prove (`!p (top:A topology) s c1 c2. singular_relcycle (p,top,s) c1 /\ singular_relcycle (p,top,s) c2 ==> singular_relcycle (p,top,s) (frag_add c1 c2)`, REPEAT GEN_TAC THEN SIMP_TAC[singular_relcycle; SINGULAR_CHAIN_ADD] THEN DISCH_THEN(CONJUNCTS_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN REWRITE_TAC[GSYM IMP_CONJ_ALT] THEN DISCH_THEN(MP_TAC o MATCH_MP MOD_SUBSET_ADD) THEN REWRITE_TAC[CHAIN_BOUNDARY_ADD; FRAG_MODULE `frag_add x frag_0 = x`]);; let SINGULAR_RELCYCLE_SUM = prove (`!p top s (f:K->((num->real)->A)frag) k. FINITE k /\ (!c. c IN k ==> singular_relcycle (p,top,s) (f c)) ==> singular_relcycle (p,top,s) (iterate frag_add k f)`, REPLICATE_TAC 4 GEN_TAC THEN REWRITE_TAC[IMP_CONJ] THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN SIMP_TAC[MATCH_MP ITERATE_CLAUSES MONOIDAL_FRAG_ADD] THEN REWRITE_TAC[NEUTRAL_FRAG_ADD; SINGULAR_RELCYCLE_0] THEN SIMP_TAC[FORALL_IN_INSERT; SINGULAR_RELCYCLE_ADD]);; let SINGULAR_RELCYCLE_SUB = prove (`!p (top:A topology) s c1 c2. singular_relcycle (p,top,s) c1 /\ singular_relcycle (p,top,s) c2 ==> singular_relcycle (p,top,s) (frag_sub c1 c2)`, REWRITE_TAC[FRAG_MODULE `frag_sub x y = frag_add x (frag_neg y)`] THEN SIMP_TAC[SINGULAR_RELCYCLE_ADD; SINGULAR_RELCYCLE_NEG]);; let SINGULAR_CYCLE = prove (`!p (top:A topology) c. singular_relcycle(p,top,{}) c <=> singular_chain(p,top) c /\ chain_boundary p c = frag_0`, REWRITE_TAC[singular_relcycle; MOD_SUBSET_EMPTY]);; let SINGULAR_CYCLE_MONO = prove (`!p (top:A topology) s t c. t SUBSET s /\ singular_relcycle (p,subtopology top t,{}) c ==> singular_relcycle (p,subtopology top s,{}) c`, REWRITE_TAC[SINGULAR_CYCLE] THEN MESON_TAC[SINGULAR_CHAIN_MONO]);; (* ------------------------------------------------------------------------- *) (* Relative boundaries B_p(top,s), where top is a topology and s a subset. *) (* ------------------------------------------------------------------------- *) let singular_relboundary = new_definition `singular_relboundary(p,top,s) c <=> ?d. singular_chain (p + 1,top) d /\ (chain_boundary (p + 1) d == c) (mod_subset (p,subtopology top s))`;; let SINGULAR_RELBOUNDARY_RESTRICT = prove (`!p (top:A topology) s. singular_relboundary(p,top,s) = singular_relboundary(p,top,topspace top INTER s)`, REWRITE_TAC[FUN_EQ_THM; singular_relboundary; GSYM SUBTOPOLOGY_RESTRICT]);; let SINGULAR_RELBOUNDARY_ALT = prove (`!p (top:A topology) s c. singular_relboundary(p,top,s) c <=> ?d e. singular_chain (p + 1,top) d /\ singular_chain(p,subtopology top s) e /\ chain_boundary (p + 1) d = frag_add c e`, REWRITE_TAC[singular_relboundary; cong; mod_subset] THEN REWRITE_TAC[FRAG_MODULE `d = frag_add c e <=> e = frag_sub d c`] THEN MESON_TAC[]);; let SINGULAR_RELBOUNDARY = prove (`!p (top:A topology) s c. singular_relboundary(p,top,s) c <=> ?d e. singular_chain (p + 1,top) d /\ singular_chain(p,subtopology top s) e /\ frag_add (chain_boundary (p + 1) d) e = c`, REWRITE_TAC[SINGULAR_RELBOUNDARY_ALT; FRAG_MODULE `d = frag_add c e <=> frag_add d (frag_neg e) = c`] THEN MESON_TAC[SINGULAR_CHAIN_NEG; FRAG_MODULE `frag_neg(frag_neg x) = x`]);; let SINGULAR_BOUNDARY = prove (`!p (top:A topology) c. singular_relboundary (p,top,{}) c <=> ?d. singular_chain(p+1,top) d /\ chain_boundary (p+1) d = c`, REWRITE_TAC[singular_relboundary; MOD_SUBSET_EMPTY]);; let SINGULAR_BOUNDARY_IMP_CHAIN = prove (`!p (top:A topology) c. singular_relboundary (p,top,{}) c ==> singular_chain(p,top) c`, REWRITE_TAC[singular_relboundary; MOD_SUBSET_EMPTY] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN FIRST_ASSUM(MP_TAC o MATCH_MP SINGULAR_CHAIN_BOUNDARY) THEN REWRITE_TAC[ADD_SUB]);; let SINGULAR_BOUNDARY_MONO = prove (`!p (top:A topology) s t c. t SUBSET s /\ singular_relboundary (p,subtopology top t,{}) c ==> singular_relboundary (p,subtopology top s,{}) c`, REWRITE_TAC[SINGULAR_BOUNDARY] THEN MESON_TAC[SINGULAR_CHAIN_MONO]);; let SINGULAR_RELBOUNDARY_IMP_CHAIN = prove (`!p (top:A topology) s c. singular_relboundary (p,top,s) c ==> singular_chain(p,top) c`, REPEAT GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [SINGULAR_RELBOUNDARY] THEN STRIP_TAC THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN MATCH_MP_TAC SINGULAR_CHAIN_ADD THEN ASM_SIMP_TAC[SINGULAR_CHAIN_BOUNDARY_ALT] THEN ONCE_REWRITE_TAC[GSYM SUBTOPOLOGY_TOPSPACE] THEN MATCH_MP_TAC SINGULAR_CHAIN_MONO THEN EXISTS_TAC `topspace top INTER s:A->bool` THEN REWRITE_TAC[INTER_SUBSET; GSYM SUBTOPOLOGY_SUBTOPOLOGY] THEN ASM_REWRITE_TAC[SUBTOPOLOGY_TOPSPACE]);; let SINGULAR_CHAIN_IMP_RELBOUNDARY = prove (`!p (top:A topology) s c. singular_chain (p,subtopology top s) c ==> singular_relboundary (p,top,s) c`, REWRITE_TAC[singular_relboundary] THEN REPEAT STRIP_TAC THEN EXISTS_TAC `frag_0:((num->real)->A)frag` THEN REWRITE_TAC[SINGULAR_CHAIN_0; cong; mod_subset] THEN MATCH_MP_TAC SINGULAR_CHAIN_SUB THEN ASM_REWRITE_TAC[SINGULAR_CHAIN_0; CHAIN_BOUNDARY_0]);; let SINGULAR_RELBOUNDARY_0 = prove (`!p (top:A topology) s. singular_relboundary (p,top,s) frag_0`, REPEAT GEN_TAC THEN REWRITE_TAC[singular_relboundary] THEN EXISTS_TAC `frag_0:((num->real)->A)frag` THEN REWRITE_TAC[SINGULAR_CHAIN_0; CHAIN_BOUNDARY_0; MOD_SUBSET_REFL]);; let SINGULAR_RELBOUNDARY_CMUL = prove (`!p (top:A topology) s a c. singular_relboundary (p,top,s) c ==> singular_relboundary (p,top,s) (frag_cmul a c)`, REWRITE_TAC[singular_relboundary] THEN MESON_TAC[SINGULAR_CHAIN_CMUL; CHAIN_BOUNDARY_CMUL; MOD_SUBSET_CMUL]);; let SINGULAR_RELBOUNDARY_NEG = prove (`!p (top:A topology) s c. singular_relboundary (p,top,s) (frag_neg c) <=> singular_relboundary (p,top,s) c`, MESON_TAC[SINGULAR_RELBOUNDARY_CMUL; FRAG_MODULE `frag_neg(frag_neg x) = x`; FRAG_MODULE `frag_neg x = frag_cmul (-- &1) x`]);; let SINGULAR_RELBOUNDARY_ADD = prove (`!p (top:A topology) s c1 c2. singular_relboundary (p,top,s) c1 /\ singular_relboundary (p,top,s) c2 ==> singular_relboundary (p,top,s) (frag_add c1 c2)`, REWRITE_TAC[singular_relboundary] THEN MESON_TAC[SINGULAR_CHAIN_ADD; CHAIN_BOUNDARY_ADD; MOD_SUBSET_ADD]);; let SINGULAR_RELBOUNDARY_SUB = prove (`!p (top:A topology) s c1 c2. singular_relboundary (p,top,s) c1 /\ singular_relboundary (p,top,s) c2 ==> singular_relboundary (p,top,s) (frag_sub c1 c2)`, REWRITE_TAC[FRAG_MODULE `frag_sub x y = frag_add x (frag_neg y)`] THEN SIMP_TAC[SINGULAR_RELBOUNDARY_ADD; SINGULAR_RELBOUNDARY_NEG]);; (* ------------------------------------------------------------------------- *) (* The (relative) homology relation. *) (* ------------------------------------------------------------------------- *) let homologous_rel = new_definition `homologous_rel (p,top,s) (c1:((num->real)->A)frag) c2 <=> singular_relboundary (p,top,s) (frag_sub c1 c2)`;; let HOMOLOGOUS_REL_RESTRICT = prove (`!p (top:A topology) s. homologous_rel (p,top,s) = homologous_rel (p,top,topspace top INTER s)`, REWRITE_TAC[FUN_EQ_THM; homologous_rel; GSYM SINGULAR_RELBOUNDARY_RESTRICT]);; let HOMOLOGOUS_REL_REFL = prove (`!p (top:A topology) s c. homologous_rel (p,top,s) c c`, REWRITE_TAC[homologous_rel; FRAG_MODULE `frag_sub x x = frag_0`] THEN REWRITE_TAC[SINGULAR_RELBOUNDARY_0]);; let HOMOLOGOUS_REL_SYM = prove (`!p (top:A topology) s c1 c2. homologous_rel (p,top,s) c1 c2 <=> homologous_rel (p,top,s) c2 c1`, REPEAT GEN_TAC THEN REWRITE_TAC[homologous_rel] THEN GEN_REWRITE_TAC RAND_CONV [GSYM SINGULAR_RELBOUNDARY_NEG] THEN REWRITE_TAC[FRAG_MODULE `frag_neg(frag_sub x y) = frag_sub y x`]);; let HOMOLOGOUS_REL_TRANS = prove (`!p (top:A topology) s c1 c2 c3. homologous_rel (p,top,s) c1 c2 /\ homologous_rel (p,top,s) c2 c3 ==> homologous_rel (p,top,s) c1 c3`, REPEAT GEN_TAC THEN REWRITE_TAC[homologous_rel] THEN DISCH_THEN(MP_TAC o MATCH_MP SINGULAR_RELBOUNDARY_ADD) THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN CONV_TAC FRAG_MODULE);; let HOMOLOGOUS_REL_EQ = prove (`!p (top:A topology) s c1 c2. homologous_rel (p,top,s) c1 = homologous_rel (p,top,s) c2 <=> homologous_rel (p,top,s) c1 c2`, REWRITE_TAC[FUN_EQ_THM] THEN MESON_TAC[HOMOLOGOUS_REL_REFL; HOMOLOGOUS_REL_SYM; HOMOLOGOUS_REL_TRANS]);; let HOMOLOGOUS_REL_SINGULAR_CHAIN = prove (`!p (top:A topology) s c1 c2. homologous_rel (p,top,s) c1 c2 ==> (singular_chain (p,top) c1 <=> singular_chain (p,top) c2)`, REPLICATE_TAC 3 GEN_TAC THEN MATCH_MP_TAC(MESON[] `(!x y. R x y ==> R y x) /\ (!x y. R x y ==> P y ==> P x) ==> (!x y. R x y ==> (P x <=> P y))`) THEN CONJ_TAC THENL [MESON_TAC[HOMOLOGOUS_REL_SYM]; REPEAT GEN_TAC] THEN REWRITE_TAC[homologous_rel] THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP SINGULAR_RELBOUNDARY_IMP_CHAIN) THEN REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP SINGULAR_CHAIN_ADD) THEN REWRITE_TAC[FRAG_MODULE `frag_add (frag_sub c1 c2) c2 = c1`]);; let HOMOLOGOUS_REL_ADD = prove (`!p (top:A topology) s c1 c1' c2 c2'. homologous_rel (p,top,s) c1 c1' /\ homologous_rel (p,top,s) c2 c2' ==> homologous_rel (p,top,s) (frag_add c1 c2) (frag_add c1' c2')`, REPEAT GEN_TAC THEN REWRITE_TAC[homologous_rel] THEN DISCH_THEN(MP_TAC o MATCH_MP SINGULAR_RELBOUNDARY_ADD) THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN CONV_TAC FRAG_MODULE);; let HOMOLOGOUS_REL_SUB = prove (`!p (top:A topology) s c1 c1' c2 c2'. homologous_rel (p,top,s) c1 c1' /\ homologous_rel (p,top,s) c2 c2' ==> homologous_rel (p,top,s) (frag_sub c1 c2) (frag_sub c1' c2')`, REPEAT GEN_TAC THEN REWRITE_TAC[homologous_rel] THEN DISCH_THEN(MP_TAC o MATCH_MP SINGULAR_RELBOUNDARY_SUB) THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN CONV_TAC FRAG_MODULE);; let HOMOLOGOUS_REL_SUM = prove (`!p top s f (g:K->((num->real)->A)frag) k. (!i. i IN k ==> homologous_rel(p,top,s) (f i) (g i)) /\ FINITE {i | i IN k /\ ~(f i = frag_0)} /\ FINITE {i | i IN k /\ ~(g i = frag_0)} ==> homologous_rel (p,top,s) (iterate frag_add k f) (iterate frag_add k g)`, REWRITE_TAC[GSYM FINITE_UNION] THEN REPEAT STRIP_TAC THEN ABBREV_TAC `l = {i | i IN k /\ ~((f:K->((num->real)->A)frag) i = frag_0)} UNION {i | i IN k /\ ~((g:K->((num->real)->A)frag) i = frag_0)}` THEN SUBGOAL_THEN `iterate frag_add k (f:K->((num->real)->A)frag) = iterate frag_add l f /\ iterate frag_add k (g:K->((num->real)->A)frag) = iterate frag_add l g` (CONJUNCTS_THEN SUBST1_TAC) THENL [ONCE_REWRITE_TAC[GSYM ITERATE_SUPPORT] THEN EXPAND_TAC "l" THEN REWRITE_TAC[support; NEUTRAL_FRAG_ADD] THEN CONJ_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `!i. i IN l ==> homologous_rel (p,top,s) ((f:K->((num->real)->A)frag) i) (g i)` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN UNDISCH_TAC `FINITE(l:K->bool)` THEN POP_ASSUM_LIST(K ALL_TAC) THEN SPEC_TAC(`l:K->bool`,`l:K->bool`) THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN SIMP_TAC[FORALL_IN_INSERT; MATCH_MP ITERATE_CLAUSES MONOIDAL_FRAG_ADD] THEN SIMP_TAC[HOMOLOGOUS_REL_REFL; HOMOLOGOUS_REL_ADD]);; let CHAIN_HOMOTOPIC_IMP_HOMOLOGOUS_REL = prove (`!p (top:A topology) s (top':B topology) t h h' f g. (!c. singular_chain (p,top) c ==> singular_chain(p + 1,top') (h c)) /\ (!c. singular_chain (p - 1,subtopology top s) c ==> singular_chain(p,subtopology top' t) (h' c)) /\ (!c. singular_chain (p,top) c ==> frag_add (chain_boundary (p + 1) (h c)) (h'(chain_boundary p c)) = frag_sub (f c) (g c)) ==> !c. singular_relcycle (p,top,s) c ==> homologous_rel (p,top',t) (f c) (g c)`, REPEAT GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN X_GEN_TAC `c:((num->real)->A)frag` THEN REWRITE_TAC[singular_relcycle; cong; mod_subset] THEN REWRITE_TAC[FRAG_MODULE `frag_sub x frag_0 = x`] THEN STRIP_TAC THEN REWRITE_TAC[homologous_rel; singular_relboundary] THEN EXISTS_TAC `(h:((num->real)->A)frag->((num->real)->B)frag) c` THEN ASM_SIMP_TAC[cong; mod_subset; SINGULAR_CHAIN_NEG; FRAG_MODULE `frag_sub a (frag_add a b) = frag_neg b`]);; (* ------------------------------------------------------------------------- *) (* Show that all boundaries are cycles, the key "chain complex" property. *) (* ------------------------------------------------------------------------- *) let CHAIN_BOUNDARY_BOUNDARY = prove (`!p (top:A topology) c. singular_chain (p,top) c ==> chain_boundary (p - 1) (chain_boundary p c) = frag_0`, GEN_TAC THEN GEN_TAC THEN ASM_CASES_TAC `p - 1 = 0` THENL [ASM_REWRITE_TAC[chain_boundary]; ALL_TAC] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP (ARITH_RULE `~(p - 1 = 0) ==> 2 <= p`)) THEN REWRITE_TAC[singular_chain] THEN MATCH_MP_TAC FRAG_INDUCTION THEN SIMP_TAC[CHAIN_BOUNDARY_0; CHAIN_BOUNDARY_SUB] THEN REWRITE_TAC[FRAG_MODULE `frag_sub x frag_0 = x`] THEN X_GEN_TAC `g:(num->real)->A` THEN REWRITE_TAC[IN] THEN DISCH_TAC THEN ASM_SIMP_TAC[CHAIN_BOUNDARY_OF; ARITH_RULE `2 <= p ==> ~(p = 0)`] THEN SIMP_TAC[CHAIN_BOUNDARY_SUM; FINITE_NUMSEG; o_DEF] THEN REWRITE_TAC[CHAIN_BOUNDARY_CMUL] THEN ASM_REWRITE_TAC[chain_boundary; FRAG_EXTEND_OF] THEN SIMP_TAC[FRAG_CMUL_SUM; FINITE_NUMSEG; o_DEF; FRAG_MODULE `frag_cmul a (frag_cmul b c) = frag_cmul (a * b) c`] THEN REWRITE_TAC[GSYM INT_POW_ADD] THEN SIMP_TAC[MATCH_MP ITERATE_ITERATE_PRODUCT MONOIDAL_FRAG_ADD; FINITE_NUMSEG] THEN ONCE_REWRITE_TAC[SET_RULE `{x,y | P x y} = {x,y | P x y /\ (y:num) < x} UNION {x,y | P x y /\ ~(y < x)}`] THEN W(MP_TAC o PART_MATCH (lhand o rand) (MATCH_MP ITERATE_UNION MONOIDAL_FRAG_ADD) o lhand o snd) THEN SIMP_TAC[PAIR_EQ; SET_RULE `(!a b a' b'. f a b = f a' b' <=> a = a' /\ b = b') ==> DISJOINT {f x y | P x y /\ Q x y} {f x y | P x y /\ ~Q x y}`] THEN ONCE_REWRITE_TAC[SET_RULE `{i,j | (i IN s /\ j IN t) /\ P i j} = {i,j | i IN s /\ j IN {k | k IN t /\ P i k}}`] THEN SIMP_TAC[FINITE_PRODUCT_DEPENDENT; FINITE_RESTRICT; FINITE_NUMSEG] THEN DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC(FRAG_MODULE `frag_cmul (-- &1) c1 = c2 ==> frag_add c1 c2 = frag_0`) THEN REWRITE_TAC[FRAG_CMUL_SUM; o_DEF] THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [LAMBDA_PAIR_THM] THEN REWRITE_TAC[GSYM(CONJUNCT2 INT_POW); FRAG_MODULE `frag_cmul a (frag_cmul b c) = frag_cmul (a * b) c`] THEN REWRITE_TAC[IN_NUMSEG; IN_ELIM_THM; LE_0; NOT_LT; GSYM CONJ_ASSOC] THEN REWRITE_TAC[ARITH_RULE `i <= p /\ j <= p - 1 /\ j < i <=> j < i /\ i <= p`] THEN REWRITE_TAC[ARITH_RULE `i <= p /\ j <= p - 1 /\ i <= j <=> i <= j /\ j <= p - 1`] THEN SUBGOAL_THEN `{i,j | j < i /\ i <= p} = IMAGE (\(j,i). i+1,j) {i,j | i <= j /\ j <= p - 1}` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; IN_IMAGE; EXISTS_PAIR_THM] THEN REWRITE_TAC[FORALL_PAIR_THM; IN_ELIM_PAIR_THM] THEN MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN REWRITE_TAC[PAIR_EQ] THEN EQ_TAC THENL [STRIP_TAC; ASM_ARITH_TAC] THEN MAP_EVERY EXISTS_TAC [`j:num`; `i - 1`] THEN ASM_ARITH_TAC; ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhand o rand) (MATCH_MP ITERATE_IMAGE MONOIDAL_FRAG_ADD) o lhand o snd) THEN ANTS_TAC THENL [REWRITE_TAC[FORALL_PAIR_THM; PAIR_EQ] THEN ARITH_TAC; DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[o_DEF]] THEN MATCH_MP_TAC(MATCH_MP ITERATE_EQ MONOIDAL_FRAG_ADD) THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN STRIP_TAC THEN REWRITE_TAC[ARITH_RULE `SUC((j + 1) + i) = SUC(SUC(i + j))`] THEN REWRITE_TAC[INT_POW; INT_MUL_LNEG; INT_MUL_LID; INT_NEG_NEG] THEN AP_TERM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM; singular_face] THEN X_GEN_TAC `t:(num->real)` THEN ASM_CASES_TAC `(t:(num->real)) IN standard_simplex (p - 1 - 1)` THEN ASM_REWRITE_TAC[RESTRICTION; o_THM] THEN SUBGOAL_THEN `simplicial_face i (t:num->real) IN standard_simplex (p - 1) /\ simplicial_face j (t:num->real) IN standard_simplex (p - 1)` STRIP_ASSUME_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC SIMPLICIAL_FACE_IN_STANDARD_SIMPLEX THEN ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC; ASM_REWRITE_TAC[] THEN AP_TERM_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[standard_simplex; IN_ELIM_THM]) THEN REPEAT(FIRST_X_ASSUM(CONJUNCTS_THEN STRIP_ASSUME_TAC)) THEN REWRITE_TAC[simplicial_face; FUN_EQ_THM] THEN X_GEN_TAC `k:num` THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN ASM_ARITH_TAC);; let CHAIN_BOUNDARY_BOUNDARY_ALT = prove (`!p (top:A topology) c. singular_chain (p + 1,top) c ==> chain_boundary p (chain_boundary (p + 1) c) = frag_0`, REPEAT GEN_TAC THEN REWRITE_TAC[ARITH_RULE `p + 2 = (p + 1) + 1`] THEN DISCH_THEN(MP_TAC o MATCH_MP CHAIN_BOUNDARY_BOUNDARY) THEN REWRITE_TAC[ADD_SUB]);; let SINGULAR_RELBOUNDARY_IMP_RELCYCLE = prove (`!p s (top:A topology) c. singular_relboundary(p,top,s) c ==> singular_relcycle(p,top,s) c`, REPEAT GEN_TAC THEN REWRITE_TAC[SINGULAR_RELBOUNDARY; SINGULAR_RELCYCLE] THEN STRIP_TAC THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN REWRITE_TAC[CHAIN_BOUNDARY_ADD] THEN CONJ_TAC THEN MATCH_MP_TAC SINGULAR_CHAIN_ADD THEN ASM_SIMP_TAC[SINGULAR_CHAIN_BOUNDARY; SINGULAR_CHAIN_BOUNDARY_ALT] THEN FIRST_ASSUM(SUBST1_TAC o MATCH_MP CHAIN_BOUNDARY_BOUNDARY_ALT) THEN REWRITE_TAC[SINGULAR_CHAIN_0] THEN ONCE_REWRITE_TAC[GSYM SUBTOPOLOGY_TOPSPACE] THEN MATCH_MP_TAC SINGULAR_CHAIN_MONO THEN EXISTS_TAC `topspace top INTER s:A->bool` THEN REWRITE_TAC[INTER_SUBSET; GSYM SUBTOPOLOGY_SUBTOPOLOGY] THEN ASM_REWRITE_TAC[SUBTOPOLOGY_TOPSPACE]);; let HOMOLOGOUS_REL_SINGULAR_RELCYCLE = prove (`!p (top:A topology) s c1 c2. homologous_rel (p,top,s) c1 c2 ==> (singular_relcycle (p,top,s) c1 <=> singular_relcycle (p,top,s) c2)`, REPLICATE_TAC 3 GEN_TAC THEN MATCH_MP_TAC(MESON[] `(!x y. R x y ==> R y x) /\ (!x y. R x y ==> P y ==> P x) ==> (!x y. R x y ==> (P x <=> P y))`) THEN CONJ_TAC THENL [MESON_TAC[HOMOLOGOUS_REL_SYM]; REPEAT GEN_TAC] THEN DISCH_TAC THEN REWRITE_TAC[singular_relcycle] THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [ASM_MESON_TAC[HOMOLOGOUS_REL_SINGULAR_CHAIN]; ALL_TAC] THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[homologous_rel] THEN DISCH_THEN(MP_TAC o MATCH_MP SINGULAR_RELBOUNDARY_IMP_RELCYCLE) THEN REWRITE_TAC[singular_relcycle; CHAIN_BOUNDARY_SUB] THEN DISCH_THEN(MP_TAC o CONJUNCT2) THEN REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP MOD_SUBSET_ADD) THEN REWRITE_TAC[FRAG_MODULE `frag_add (frag_sub c1 c2) c2 = c1`] THEN REWRITE_TAC[FRAG_MODULE `frag_add frag_0 x = x`]);; (* ------------------------------------------------------------------------- *) (* Operations induced by a continuous map g between topological spaces. *) (* ------------------------------------------------------------------------- *) let simplex_map = new_definition `simplex_map p (g:A->B) (c:(num->real)->A) = RESTRICTION (standard_simplex p) (g o c)`;; let SINGULAR_SIMPLEX_SIMPLEX_MAP = prove (`!p top top' f g:A->B. singular_simplex (p,top) f /\ continuous_map (top,top') g ==> singular_simplex (p,top') (simplex_map p g f)`, REWRITE_TAC[singular_simplex; simplex_map] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[REWRITE_RULE[IN] RESTRICTION_IN_EXTENSIONAL] THEN SIMP_TAC[RESTRICTION_CONTINUOUS_MAP; TOPSPACE_SUBTOPOLOGY; INTER_SUBSET] THEN MATCH_MP_TAC CONTINUOUS_MAP_COMPOSE THEN ASM_MESON_TAC[]);; let SIMPLEX_MAP_EQ = prove (`!p (f:A->B) g top c. singular_simplex (p,top) c /\ (!x. x IN topspace top ==> f x = g x) ==> simplex_map p f c = simplex_map p g c`, REWRITE_TAC[simplex_map; singular_simplex; FUN_EQ_THM] THEN SIMP_TAC[RESTRICTION; o_DEF; EXTENSIONAL; IN_ELIM_THM] THEN REWRITE_TAC[continuous_map] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; TOPSPACE_PRODUCT_TOPOLOGY] THEN REWRITE_TAC[o_DEF; TOPSPACE_EUCLIDEANREAL; CARTESIAN_PRODUCT_UNIV] THEN SET_TAC[]);; let SIMPLEX_MAP_ID_GEN = prove (`!p f top (c:(num->real)->A). singular_simplex (p,top) c /\ (!x. x IN topspace top ==> f x = x) ==> simplex_map p f c = c`, REWRITE_TAC[simplex_map; singular_simplex; FUN_EQ_THM] THEN SIMP_TAC[RESTRICTION; o_DEF; EXTENSIONAL; IN_ELIM_THM] THEN REWRITE_TAC[continuous_map] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; TOPSPACE_PRODUCT_TOPOLOGY] THEN REWRITE_TAC[o_DEF; TOPSPACE_EUCLIDEANREAL; CARTESIAN_PRODUCT_UNIV] THEN SET_TAC[]);; let SIMPLEX_MAP_I = prove (`!p. simplex_map p (I:A->A) = RESTRICTION (standard_simplex p:(num->real)->bool)`, REWRITE_TAC[simplex_map; FUN_EQ_THM; I_O_ID]);; let SIMPLEX_MAP_COMPOSE = prove (`!p (g:A->B) (h:B->C). simplex_map p (h o g) :((num->real)->A)->((num->real)->C) = simplex_map p h o simplex_map p g`, REWRITE_TAC[simplex_map; FUN_EQ_THM; o_THM; RESTRICTION_COMPOSE_RIGHT] THEN REWRITE_TAC[o_ASSOC]);; let SINGULAR_FACE_SIMPLEX_MAP = prove (`!p k f:A->B c. 1 <= p /\ k <= p ==> singular_face p k (simplex_map p f c) = simplex_map (p - 1) f (c o simplicial_face k)`, REWRITE_TAC[FUN_EQ_THM; simplex_map; singular_face] THEN SIMP_TAC[SIMPLICIAL_FACE_IN_STANDARD_SIMPLEX; RESTRICTION; o_THM]);; let chain_map = new_definition `chain_map p (g:A->B) (c:((num->real)->A)frag) = frag_extend (frag_of o simplex_map p g) c`;; let SINGULAR_CHAIN_CHAIN_MAP = prove (`!p top top' (g:A->B) c. singular_chain (p,top) c /\ continuous_map (top,top') g ==> singular_chain (p,top') (chain_map p g c)`, REPEAT STRIP_TAC THEN REWRITE_TAC[chain_map] THEN MATCH_MP_TAC SINGULAR_CHAIN_EXTEND THEN REWRITE_TAC[SINGULAR_CHAIN_OF; o_THM] THEN RULE_ASSUM_TAC(REWRITE_RULE[singular_chain]) THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC SINGULAR_SIMPLEX_SIMPLEX_MAP THEN ASM SET_TAC[]);; let CHAIN_MAP_0 = prove (`!p (g:A->B). chain_map p g frag_0 = frag_0`, REWRITE_TAC[chain_map; FRAG_EXTEND_0]);; let CHAIN_MAP_OF = prove (`!p (g:A->B) f. chain_map p g (frag_of f) = frag_of (simplex_map p g f)`, REWRITE_TAC[chain_map; FRAG_EXTEND_OF; o_THM]);; let CHAIN_MAP_CMUL = prove (`!p (g:A->B) a c. chain_map p g (frag_cmul a c) = frag_cmul a (chain_map p g c)`, REWRITE_TAC[chain_map; FRAG_EXTEND_CMUL]);; let CHAIN_MAP_NEG = prove (`!p (g:A->B) c. chain_map p g (frag_neg c) = frag_neg(chain_map p g c)`, REWRITE_TAC[chain_map; FRAG_EXTEND_NEG]);; let CHAIN_MAP_ADD = prove (`!p (g:A->B) c1 c2. chain_map p g (frag_add c1 c2) = frag_add (chain_map p g c1) (chain_map p g c2)`, REWRITE_TAC[chain_map; FRAG_EXTEND_ADD]);; let CHAIN_MAP_SUB = prove (`!p (g:A->B) c1 c2. chain_map p g (frag_sub c1 c2) = frag_sub (chain_map p g c1) (chain_map p g c2)`, REWRITE_TAC[chain_map; FRAG_EXTEND_SUB]);; let CHAIN_MAP_SUM = prove (`!p (g:A->B) (f:K->((num->real)->A)frag) k. FINITE k ==> chain_map p g (iterate frag_add k f) = iterate frag_add k (chain_map p g o f)`, SIMP_TAC[chain_map; FRAG_EXTEND_SUM; o_DEF]);; let CHAIN_MAP_EQ = prove (`!p (f:A->B) g top c. singular_chain (p,top) c /\ (!x. x IN topspace top ==> f x = g x) ==> chain_map p f c = chain_map p g c`, REPLICATE_TAC 4 GEN_TAC THEN REWRITE_TAC[IMP_CONJ] THEN REWRITE_TAC[singular_chain] THEN MATCH_MP_TAC FRAG_INDUCTION THEN SIMP_TAC[CHAIN_MAP_0; CHAIN_MAP_SUB; CHAIN_MAP_OF] THEN MESON_TAC[SIMPLEX_MAP_EQ; IN]);; let CHAIN_MAP_ID_GEN = prove (`!p f top (c:((num->real)->A)frag). singular_chain (p,top) c /\ (!x. x IN topspace top ==> f x = x) ==> chain_map p f c = c`, REPLICATE_TAC 3 GEN_TAC THEN REWRITE_TAC[IMP_CONJ] THEN REWRITE_TAC[singular_chain] THEN MATCH_MP_TAC FRAG_INDUCTION THEN SIMP_TAC[CHAIN_MAP_0; CHAIN_MAP_SUB; CHAIN_MAP_OF] THEN MESON_TAC[SIMPLEX_MAP_ID_GEN; IN]);; let CHAIN_MAP_ID = prove (`!p top (c:((num->real)->B)frag). singular_chain (p,top) c ==> chain_map p (\x. x) c = c`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CHAIN_MAP_ID_GEN THEN REWRITE_TAC[] THEN ASM_MESON_TAC[]);; let CHAIN_MAP_I = prove (`!p. chain_map p I = frag_extend (frag_of o RESTRICTION (standard_simplex p))`, REWRITE_TAC[FUN_EQ_THM; chain_map; SIMPLEX_MAP_I; o_DEF]);; let CHAIN_MAP_COMPOSE = prove (`!p (g:A->B) (h:B->C). chain_map p (h o g):((num->real)->A)frag->((num->real)->C)frag = chain_map p h o chain_map p g`, REPEAT GEN_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN MATCH_MP_TAC(MESON[SUBSET_UNIV] `(!c. frag_support c SUBSET (:B) ==> P c) ==> (!c. P c)`) THEN MATCH_MP_TAC FRAG_INDUCTION THEN SIMP_TAC[CHAIN_MAP_SUB; CHAIN_MAP_0; CHAIN_MAP_OF; o_THM] THEN REWRITE_TAC[SIMPLEX_MAP_COMPOSE; o_THM]);; let SINGULAR_SIMPLEX_CHAIN_MAP_I = prove (`!p top f:(num->real)->A. singular_simplex (p,top) f ==> chain_map p f (frag_of (RESTRICTION (standard_simplex p) I)) = frag_of f`, REPEAT GEN_TAC THEN REWRITE_TAC[singular_simplex; EXTENSIONAL; IN_ELIM_THM] THEN DISCH_THEN(ASSUME_TAC o CONJUNCT2) THEN REWRITE_TAC[CHAIN_MAP_OF; simplex_map] THEN REWRITE_TAC[RESTRICTION_COMPOSE_RIGHT; I_O_ID] THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM; RESTRICTION] THEN ASM_MESON_TAC[]);; let CHAIN_BOUNDARY_CHAIN_MAP = prove (`!p top g:A->B c:((num->real)->A)frag. singular_chain (p,top) c ==> chain_boundary p (chain_map p g c) = chain_map (p - 1) g (chain_boundary p c)`, REPLICATE_TAC 3 GEN_TAC THEN REWRITE_TAC[singular_chain] THEN MATCH_MP_TAC FRAG_INDUCTION THEN REWRITE_TAC[CHAIN_MAP_0; CHAIN_BOUNDARY_0] THEN SIMP_TAC[CHAIN_BOUNDARY_SUB; CHAIN_MAP_SUB] THEN X_GEN_TAC `f:(num->real)->A` THEN REWRITE_TAC[IN] THEN DISCH_TAC THEN REWRITE_TAC[CHAIN_MAP_OF; CHAIN_BOUNDARY_OF] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[CHAIN_MAP_0] THEN SIMP_TAC[CHAIN_MAP_SUM; FINITE_NUMSEG; o_DEF; CHAIN_MAP_CMUL] THEN REWRITE_TAC[CHAIN_MAP_OF] THEN MATCH_MP_TAC(MATCH_MP ITERATE_EQ MONOIDAL_FRAG_ADD) THEN REWRITE_TAC[IN_NUMSEG; o_THM] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[singular_face; simplex_map; FUN_EQ_THM] THEN REWRITE_TAC[RESTRICTION_COMPOSE_RIGHT] THEN ASM_SIMP_TAC[RESTRICTION; o_THM; SIMPLICIAL_FACE_IN_STANDARD_SIMPLEX; LE_1]);; let SINGULAR_RELCYCLE_CHAIN_MAP = prove (`!p top s top' t g:A->B c:((num->real)->A)frag. singular_relcycle (p,top,s) c /\ continuous_map (top,top') g /\ IMAGE g s SUBSET t ==> singular_relcycle (p,top',t) (chain_map p g c)`, REWRITE_TAC[SINGULAR_RELCYCLE] THEN REPEAT STRIP_TAC THENL [ASM_MESON_TAC[SINGULAR_CHAIN_CHAIN_MAP]; ALL_TAC] THEN MP_TAC(ISPECL [`p:num`; `top:A topology`; `g:A->B`] CHAIN_BOUNDARY_CHAIN_MAP) THEN ASM_SIMP_TAC[] THEN DISCH_THEN(K ALL_TAC) THEN MATCH_MP_TAC SINGULAR_CHAIN_CHAIN_MAP THEN EXISTS_TAC `subtopology top (s:A->bool)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONTINUOUS_MAP_INTO_SUBTOPOLOGY THEN ASM_SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; TOPSPACE_SUBTOPOLOGY] THEN ASM SET_TAC[]);; let SINGULAR_RELBOUNDARY_CHAIN_MAP = prove (`!p top s top' t g:A->B c:((num->real)->A)frag. singular_relboundary (p,top,s) c /\ continuous_map (top,top') g /\ IMAGE g s SUBSET t ==> singular_relboundary (p,top',t) (chain_map p g c)`, REPEAT GEN_TAC THEN REWRITE_TAC[SINGULAR_RELBOUNDARY] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN INTRO_TAC "!d e" THEN STRIP_TAC THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN MAP_EVERY EXISTS_TAC [`chain_map (p + 1) (g:A->B) d`; `chain_map p (g:A->B) e`] THEN REWRITE_TAC[CHAIN_MAP_ADD; CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[CHAIN_BOUNDARY_CHAIN_MAP; ADD_SUB]] THEN CONJ_TAC THEN MATCH_MP_TAC SINGULAR_CHAIN_CHAIN_MAP THENL [ASM_MESON_TAC[]; EXISTS_TAC `subtopology top (s:A->bool)`] THEN ASM_SIMP_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY; TOPSPACE_SUBTOPOLOGY; CONTINUOUS_MAP_FROM_SUBTOPOLOGY] THEN ASM SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* Homology of one-point spaces degenerates except for p = 0. *) (* ------------------------------------------------------------------------- *) let SINGULAR_SIMPLEX_SING = prove (`!p top a f:(num->real)->A. topspace top = {a} ==> (singular_simplex(p,top) f <=> f = RESTRICTION (standard_simplex p) (\x. a))`, REPEAT STRIP_TAC THEN REWRITE_TAC[singular_simplex] THEN EQ_TAC THEN STRIP_TAC THENL [FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] (REWRITE_RULE[IN] EXTENSIONAL_EQ))) THEN REWRITE_TAC[REWRITE_RULE[IN] RESTRICTION_IN_EXTENSIONAL] THEN FIRST_ASSUM(MP_TAC o MATCH_MP CONTINUOUS_MAP_IMAGE_SUBSET_TOPSPACE) THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; TOPSPACE_PRODUCT_TOPOLOGY] THEN REWRITE_TAC[CARTESIAN_PRODUCT_UNIV; o_DEF; TOPSPACE_EUCLIDEANREAL] THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_SING; INTER_UNIV] THEN SIMP_TAC[IN; RESTRICTION]; ASM_REWRITE_TAC[REWRITE_RULE[IN] RESTRICTION_IN_EXTENSIONAL] THEN ASM_SIMP_TAC[RESTRICTION_CONTINUOUS_MAP; TOPSPACE_SUBTOPOLOGY; INTER_SUBSET; CONTINUOUS_MAP_CONST; IN_SING]]);; let SINGULAR_CHAIN_SING = prove (`!p top a c:((num->real)->A)frag. topspace top = {a} ==> (singular_chain(p,top) c <=> ?b. c = frag_cmul b (frag_of(RESTRICTION (standard_simplex p) (\x. a))))`, REPEAT STRIP_TAC THEN REWRITE_TAC[singular_chain] THEN REWRITE_TAC[SET_RULE `s SUBSET t <=> !x. x IN s ==> t x`] THEN ASM_SIMP_TAC[SINGULAR_SIMPLEX_SING; SET_RULE `(!x. x IN s ==> x = a) <=> s SUBSET {a}`] THEN EQ_TAC THENL [ALL_TAC; MESON_TAC[FRAG_SUPPORT_CMUL; FRAG_SUPPORT_OF; SUBSET_TRANS]] THEN REWRITE_TAC[FRAG_SUPPORT_EQ_EMPTY; SET_RULE `s SUBSET {a} <=> s = {} \/ s = {a}`] THEN STRIP_TAC THENL [ASM_MESON_TAC[FRAG_MODULE `frag_cmul (&0) x = frag_0`]; ALL_TAC] THEN GEN_REWRITE_TAC (BINDER_CONV o LAND_CONV) [FRAG_EXPANSION] THEN ASM_REWRITE_TAC[frag_extend; MATCH_MP ITERATE_SING MONOIDAL_FRAG_ADD] THEN REWRITE_TAC[o_THM] THEN MESON_TAC[]);; let CHAIN_BOUNDARY_OF_SING = prove (`!p top a c:((num->real)->A)frag. topspace top = {a} /\ singular_chain (p,top) c ==> chain_boundary p c = if p = 0 \/ ODD p then frag_0 else frag_extend (\f. frag_of(RESTRICTION (standard_simplex (p - 1)) (\x. a))) c`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `p = 0` THEN ASM_REWRITE_TAC[chain_boundary] THEN TRANS_TAC EQ_TRANS `frag_extend (\f. if ODD p then frag_0 else frag_of(RESTRICTION (standard_simplex (p - 1)) ((\x. a):(num->real)->A))) (c:((num->real)->A)frag)` THEN CONJ_TAC THENL [MATCH_MP_TAC FRAG_EXTEND_EQ; ASM_CASES_TAC `ODD p` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC FRAG_EXTEND_EQ_0 THEN REWRITE_TAC[]] THEN RULE_ASSUM_TAC(REWRITE_RULE[singular_chain]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET t ==> (!x. t x ==> P x) ==> (!x. x IN s ==> P x)`)) THEN REWRITE_TAC[] THEN X_GEN_TAC `f:(num->real)->A` THEN STRIP_TAC THEN TRANS_TAC EQ_TRANS `iterate frag_add (0..p) (\k. frag_cmul (-- &1 pow k) (frag_of(RESTRICTION (standard_simplex (p - 1)) ((\x. a):(num->real)->A))))` THEN CONJ_TAC THENL [MATCH_MP_TAC(MATCH_MP ITERATE_EQ MONOIDAL_FRAG_ADD) THEN X_GEN_TAC `k:num` THEN REWRITE_TAC[IN_NUMSEG] THEN STRIP_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN ASM_SIMP_TAC[GSYM SINGULAR_SIMPLEX_SING] THEN MATCH_MP_TAC SINGULAR_SIMPLEX_SINGULAR_FACE THEN ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC; ABBREV_TAC `c:((num->real)->A)frag = frag_of(RESTRICTION (standard_simplex (p - 1)) (\x. a))` THEN SPEC_TAC(`p:num`,`p:num`) THEN POP_ASSUM_LIST(K ALL_TAC) THEN MATCH_MP_TAC num_INDUCTION THEN SIMP_TAC[MATCH_MP ITERATE_CLAUSES_NUMSEG MONOIDAL_FRAG_ADD; LE_0] THEN SIMP_TAC[ODD] THEN CONJ_TAC THENL [CONV_TAC FRAG_MODULE; ALL_TAC] THEN X_GEN_TAC `p:num` THEN ASM_CASES_TAC `ODD p` THEN ASM_SIMP_TAC[INT_POW_NEG; GSYM NOT_ODD; ODD; INT_POW_ONE] THEN DISCH_THEN(K ALL_TAC) THEN CONV_TAC FRAG_MODULE]);; let SINGULAR_CYCLE_SING = prove (`!p top a c:((num->real)->A)frag. topspace top = {a} ==> (singular_relcycle (p,top,{}) c <=> singular_chain (p,top) c /\ (p = 0 \/ ODD p \/ c = frag_0))`, REPEAT STRIP_TAC THEN REWRITE_TAC[SINGULAR_CYCLE] THEN MATCH_MP_TAC(TAUT `(p ==> (q <=> r)) ==> (p /\ q <=> p /\ r)`) THEN DISCH_TAC THEN REWRITE_TAC[DISJ_ASSOC] THEN MP_TAC(ISPECL [`p:num`; `top:A topology`; `a:A`] CHAIN_BOUNDARY_OF_SING) THEN ASM_SIMP_TAC[] THEN DISCH_THEN(K ALL_TAC) THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPECL [`p:num`; `top:A topology`; `a:A`; `c:((num->real)->A)frag`] SINGULAR_CHAIN_SING) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(CHOOSE_THEN SUBST1_TAC) THEN REWRITE_TAC[FRAG_EXTEND_CMUL; FRAG_EXTEND_OF; FRAG_OF_NONZERO; FRAG_MODULE `frag_cmul b c = frag_0 <=> b = &0 \/ c = frag_0`]);; let SINGULAR_BOUNDARY_SING = prove (`!p top a c:((num->real)->A)frag. topspace top = {a} ==> (singular_relboundary (p,top,{}) c <=> singular_chain (p,top) c /\ (ODD p \/ c = frag_0))`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `singular_chain(p,top) (c:((num->real)->A)frag)` THENL [ASM_REWRITE_TAC[]; ASM_MESON_TAC[SINGULAR_BOUNDARY_IMP_CHAIN]] THEN REWRITE_TAC[SINGULAR_BOUNDARY; TAUT `p /\ q <=> ~(p ==> ~q)`] THEN MP_TAC(ISPECL [`p + 1`; `top:A topology`; `a:A`] CHAIN_BOUNDARY_OF_SING) THEN ASM_SIMP_TAC[] THEN DISCH_THEN(K ALL_TAC) THEN REWRITE_TAC[ADD_EQ_0; ODD_ADD; NOT_IMP] THEN CONV_TAC NUM_REDUCE_CONV THEN MP_TAC(ISPECL [`p + 1`; `top:A topology`; `a:A`] SINGULAR_CHAIN_SING) THEN ASM_SIMP_TAC[LEFT_AND_EXISTS_THM] THEN DISCH_THEN(K ALL_TAC) THEN GEN_REWRITE_TAC LAND_CONV [SWAP_EXISTS_THM] THEN REWRITE_TAC[UNWIND_THM2; FRAG_EXTEND_CMUL; FRAG_EXTEND_OF; ADD_SUB] THEN ASM_CASES_TAC `ODD p` THEN ASM_REWRITE_TAC[] THEN CONV_TAC(RAND_CONV(ONCE_DEPTH_CONV SYM_CONV)) THEN ASM_MESON_TAC[GSYM SINGULAR_CHAIN_SING]);; let SINGULAR_BOUNDARY_EQ_CYCLE_SING = prove (`!p top a c:((num->real)->A)frag. topspace top = {a} /\ 1 <= p ==> (singular_relboundary (p,top,{}) c <=> singular_relcycle (p,top,{}) c)`, METIS_TAC[LE_1; SINGULAR_CYCLE_SING; SINGULAR_BOUNDARY_SING]);; (* ------------------------------------------------------------------------- *) (* Simplicial chains, effectively those resulting from linear maps. *) (* We still allow the map to be singular, so the name is questionable. *) (* These are intended as building-blocks for singular subdivision, rather *) (* than as a basis for simplicial homology. *) (* ------------------------------------------------------------------------- *) let oriented_simplex = new_definition `oriented_simplex p l = RESTRICTION (standard_simplex p) (\x i:num. sum(0..p) (\j. l j i * x j))`;; let simplicial_simplex = new_definition `simplicial_simplex (p,s) f <=> singular_simplex (p,subtopology (product_topology (:num) (\i. euclideanreal)) s) f /\ ?l. f = oriented_simplex p l`;; let SIMPLICIAL_SIMPLEX = prove (`!p s f. simplicial_simplex (p,s) f <=> IMAGE f (standard_simplex p) SUBSET s /\ ?l. f = oriented_simplex p l`, REPEAT GEN_TAC THEN REWRITE_TAC[simplicial_simplex; SINGULAR_SIMPLEX_SUBTOPOLOGY] THEN MATCH_MP_TAC(TAUT `(r ==> p) ==> ((p /\ q) /\ r <=> q /\ r)`) THEN SIMP_TAC[LEFT_IMP_EXISTS_THM; singular_simplex; oriented_simplex] THEN SIMP_TAC[RESTRICTION_CONTINUOUS_MAP; TOPSPACE_SUBTOPOLOGY; INTER_SUBSET] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[REWRITE_RULE[IN] RESTRICTION_IN_EXTENSIONAL] THEN REWRITE_TAC[CONTINUOUS_MAP_COMPONENTWISE; IN_UNIV] THEN REWRITE_TAC[SUBSET; IN; EXTENSIONAL_UNIV] THEN X_GEN_TAC `k:num` THEN MATCH_MP_TAC CONTINUOUS_MAP_SUM THEN REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN X_GEN_TAC `n:num` THEN STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_LMUL THEN MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN SIMP_TAC[CONTINUOUS_MAP_PRODUCT_PROJECTION; IN_UNIV]);; let SIMPLICIAL_SIMPLEX_EMPTY = prove (`!p f. ~(simplicial_simplex (p,{}) f)`, SIMP_TAC[simplicial_simplex; TOPSPACE_SUBTOPOLOGY; INTER_EMPTY; SINGULAR_SIMPLEX_EMPTY]);; let simplicial_chain = new_definition `simplicial_chain (p,s) c <=> frag_support c SUBSET simplicial_simplex (p,s)`;; let SIMPLICIAL_CHAIN_0 = prove (`!p s. simplicial_chain (p,s) frag_0`, REWRITE_TAC[simplicial_chain; FRAG_SUPPORT_0; EMPTY_SUBSET]);; let SIMPLICIAL_CHAIN_OF = prove (`!p s c. simplicial_chain (p,s) (frag_of c) <=> simplicial_simplex (p,s) c`, REWRITE_TAC[simplicial_chain; FRAG_SUPPORT_OF] THEN SET_TAC[]);; let SIMPLICIAL_CHAIN_CMUL = prove (`!p s a c. simplicial_chain (p,s) c ==> simplicial_chain (p,s) (frag_cmul a c)`, REWRITE_TAC[simplicial_chain] THEN MESON_TAC[FRAG_SUPPORT_CMUL; SUBSET_TRANS]);; let SIMPLICIAL_CHAIN_SUB = prove (`!p s c1 c2. simplicial_chain (p,s) c1 /\ simplicial_chain (p,s) c2 ==> simplicial_chain (p,s) (frag_sub c1 c2)`, REWRITE_TAC[simplicial_chain] THEN MESON_TAC[FRAG_SUPPORT_SUB; SUBSET_TRANS; UNION_SUBSET]);; let SIMPLICIAL_CHAIN_SUM = prove (`!p s f k. (!c. c IN k ==> simplicial_chain (p,s) (f c)) ==> simplicial_chain (p,s) (iterate frag_add k f)`, REWRITE_TAC[simplicial_chain] THEN REPEAT STRIP_TAC THEN W(MP_TAC o PART_MATCH lhand FRAG_SUPPORT_SUM o lhand o snd) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS) THEN ASM_REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC]);; let SIMPLICIAL_SIMPLEX_ORIENTED_SIMPLEX = prove (`!p s l. simplicial_simplex (p,s) (oriented_simplex p l) <=> IMAGE (\x i. sum (0..p) (\j. l j i * x j)) (standard_simplex p) SUBSET s`, REPEAT GEN_TAC THEN REWRITE_TAC[SIMPLICIAL_SIMPLEX] THEN MATCH_MP_TAC(TAUT `q /\ (p <=> p') ==> (p /\ q <=> p')`) THEN CONJ_TAC THENL [MESON_TAC[]; REWRITE_TAC[oriented_simplex]] THEN SIMP_TAC[SUBSET; FORALL_IN_IMAGE; RESTRICTION]);; let SIMPLICIAL_IMP_SINGULAR_SIMPLEX = prove (`!p s f. simplicial_simplex (p,s) f ==> singular_simplex (p,subtopology (product_topology (:num) (\i. euclideanreal)) s) f`, SIMP_TAC[simplicial_simplex]);; let SIMPLICIAL_IMP_SINGULAR_CHAIN = prove (`!p s c. simplicial_chain (p,s) c ==> singular_chain (p,subtopology (product_topology (:num) (\i. euclideanreal)) s) c`, REWRITE_TAC[simplicial_chain; singular_chain] THEN SIMP_TAC[SUBSET; IN; SIMPLICIAL_IMP_SINGULAR_SIMPLEX]);; let ORIENTED_SIMPLEX_EQ = prove (`!p l l'. oriented_simplex p l = oriented_simplex p l' <=> !i. i <= p ==> l i = l' i`, REPEAT GEN_TAC THEN REWRITE_TAC[oriented_simplex] THEN EQ_TAC THENL [DISCH_TAC THEN X_GEN_TAC `i:num` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o C AP_THM `(\j:num. if j = i then &1 else &0)`) THEN ASM_REWRITE_TAC[BASIS_IN_STANDARD_SIMPLEX; RESTRICTION] THEN SIMP_TAC[COND_RAND; REAL_MUL_RZERO; SUM_DELTA] THEN ASM_REWRITE_TAC[IN_NUMSEG; LE_0; REAL_MUL_RID; ETA_AX]; DISCH_TAC THEN AP_TERM_TAC THEN REPEAT ABS_TAC THEN MATCH_MP_TAC SUM_EQ THEN ASM_SIMP_TAC[IN_NUMSEG; LE_0]]);; let SINGULAR_FACE_ORIENTED_SIMPLEX = prove (`!p k l. 1 <= p /\ k <= p ==> singular_face p k (oriented_simplex p l) = oriented_simplex (p - 1) (\j. if j < k then l j else l (j + 1))`, REPEAT STRIP_TAC THEN REWRITE_TAC[singular_face; oriented_simplex] THEN REWRITE_TAC[FUN_EQ_THM; RESTRICTION] THEN MAP_EVERY X_GEN_TAC [`x:num->real`; `j:num`] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[RESTRICTION; o_DEF; SIMPLICIAL_FACE_IN_STANDARD_SIMPLEX] THEN REWRITE_TAC[simplicial_face] THEN REPLICATE_TAC 2 (REWRITE_TAC[COND_RATOR] THEN ONCE_REWRITE_TAC[COND_RAND]) THEN SIMP_TAC[SUM_CASES; FINITE_NUMSEG; FINITE_RESTRICT] THEN REWRITE_TAC[REAL_MUL_RZERO; REAL_ADD_LID; SUM_0] THEN SIMP_TAC[SUM_CASES; FINITE_NUMSEG; FINITE_RESTRICT] THEN REWRITE_TAC[REAL_MUL_RZERO; REAL_ADD_LID; SUM_0] THEN REWRITE_TAC[IN_NUMSEG; LE_0; IN_ELIM_THM] THEN SUBGOAL_THEN `!i. i <= p - 1 /\ i < k <=> i <= p /\ i < k` (fun th -> REWRITE_TAC[th]) THENL [ASM_ARITH_TAC; ALL_TAC] THEN AP_TERM_TAC THEN SUBGOAL_THEN `!i. (i <= p /\ ~(i < k)) /\ ~(i = k) <=> k + 1 <= i /\ i <= (p - 1) + 1` (fun th -> REWRITE_TAC[th]) THENL [ASM_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[GSYM numseg; NOT_LT; SUM_OFFSET; ADD_SUB] THEN BINOP_TAC THEN REWRITE_TAC[] THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_NUMSEG] THEN ASM_ARITH_TAC);; let SIMPLICIAL_SIMPLEX_SINGULAR_FACE = prove (`!p s k f. simplicial_simplex (p,s) f /\ 1 <= p /\ k <= p ==> simplicial_simplex (p - 1,s) (singular_face p k f)`, SIMP_TAC[simplicial_simplex; SINGULAR_SIMPLEX_SINGULAR_FACE] THEN REPEAT GEN_TAC THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN FIRST_X_ASSUM(X_CHOOSE_THEN `m:num->num->real` SUBST_ALL_TAC) THEN ASM_REWRITE_TAC[singular_face; oriented_simplex] THEN ASM_SIMP_TAC[RESTRICTION_COMPOSE_LEFT; SUBSET; FORALL_IN_IMAGE; SIMPLICIAL_FACE_IN_STANDARD_SIMPLEX] THEN EXISTS_TAC `\i. if i < k then (m:num->num->real) i else m (i + 1)` THEN REWRITE_TAC[simplicial_face; o_DEF] THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN MAP_EVERY X_GEN_TAC [`x:num->real`; `i:num`] THEN REPLICATE_TAC 2 (ONCE_REWRITE_TAC[COND_RAND] THEN GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [COND_RATOR]) THEN SIMP_TAC[SUM_CASES; FINITE_NUMSEG] THEN ASM_SIMP_TAC[IN_NUMSEG; LE_0; ARITH_RULE `k <= p ==> (j <= p - 1 /\ j < k <=> j <= p /\ j < k)`] THEN AP_TERM_TAC THEN REWRITE_TAC[REAL_MUL_RZERO; NOT_LT] THEN REWRITE_TAC[ARITH_RULE `j:num <= p /\ k <= j <=> k <= j /\ j <= p`] THEN SIMP_TAC[SUM_CASES; FINITE_NUMSEG; GSYM numseg] THEN REWRITE_TAC[SUM_0; REAL_ADD_LID; IN_NUMSEG] THEN ASM_SIMP_TAC[ARITH_RULE `1 <= p ==> ((k <= j /\ j <= p) /\ ~(j = k) <=> k + 1 <= j /\ j <= (p - 1) + 1)`] THEN REWRITE_TAC[GSYM numseg; SUM_OFFSET; ADD_SUB]);; let SIMPLICIAL_CHAIN_BOUNDARY = prove (`!p s c. simplicial_chain (p,s) c ==> simplicial_chain (p - 1,s) (chain_boundary p c)`, GEN_TAC THEN GEN_TAC THEN REWRITE_TAC[simplicial_chain] THEN MATCH_MP_TAC FRAG_INDUCTION THEN REWRITE_TAC[CHAIN_BOUNDARY_0; FRAG_SUPPORT_0; EMPTY_SUBSET] THEN REWRITE_TAC[CHAIN_BOUNDARY_SUB; GSYM UNION_SUBSET] THEN CONJ_TAC THENL [REWRITE_TAC[IN]; MESON_TAC[SUBSET; FRAG_SUPPORT_SUB]] THEN X_GEN_TAC `f:(num->real)->(num->real)` THEN DISCH_TAC THEN REWRITE_TAC[GSYM simplicial_chain] THEN REWRITE_TAC[chain_boundary; FRAG_EXTEND_OF] THEN COND_CASES_TAC THEN REWRITE_TAC[SIMPLICIAL_CHAIN_0] THEN MATCH_MP_TAC SIMPLICIAL_CHAIN_SUM THEN X_GEN_TAC `i:num` THEN REWRITE_TAC[IN_NUMSEG; LE_0] THEN DISCH_TAC THEN MATCH_MP_TAC SIMPLICIAL_CHAIN_CMUL THEN REWRITE_TAC[SIMPLICIAL_CHAIN_OF; simplicial_simplex] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [simplicial_simplex]) THEN ASM_SIMP_TAC[SINGULAR_SIMPLEX_SINGULAR_FACE; LE_1] THEN STRIP_TAC THEN ASM_SIMP_TAC[SINGULAR_FACE_ORIENTED_SIMPLEX; LE_1] THEN MESON_TAC[]);; (* ------------------------------------------------------------------------- *) (* The cone construction on simplicial simplices. *) (* ------------------------------------------------------------------------- *) let simplex_cone = let exth = prove (`?cone. !p v l. cone p v (oriented_simplex p l) = oriented_simplex (p + 1) (\i. if i = 0 then v else l(i - 1))`, REWRITE_TAC[GSYM SKOLEM_THM] THEN REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM FUN_EQ_THM] THEN GEN_REWRITE_TAC (BINDER_CONV o LAND_CONV) [GSYM o_DEF] THEN CONV_TAC(ONCE_DEPTH_CONV SYM_CONV) THEN GEN_REWRITE_TAC I [GSYM FUNCTION_FACTORS_LEFT] THEN REWRITE_TAC[ORIENTED_SIMPLEX_EQ] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC) in new_specification ["simplex_cone"] exth;; let SIMPLICIAL_SIMPLEX_SIMPLEX_CONE = prove (`!p s t v f. simplicial_simplex (p,s) f /\ (!x u. &0 <= u /\ u <= &1 /\ x IN s ==> (\i. (&1 - u) * v i + u * x i) IN t) ==> simplicial_simplex (p + 1,t) (simplex_cone p v f)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `s:(num->real)->bool = {}` THEN ASM_REWRITE_TAC[SIMPLICIAL_SIMPLEX_EMPTY] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN SIMP_TAC[SIMPLICIAL_SIMPLEX; IMP_CONJ_ALT; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `l:num->num->real` THEN DISCH_THEN SUBST1_TAC THEN SIMP_TAC[SUBSET; FORALL_IN_IMAGE; simplex_cone] THEN DISCH_THEN(fun th -> CONJ_TAC THENL [MP_TAC th; MESON_TAC[]]) THEN SIMP_TAC[oriented_simplex; RESTRICTION; standard_simplex; IN_ELIM_THM] THEN DISCH_TAC THEN X_GEN_TAC `x:num->real` THEN STRIP_TAC THEN SIMP_TAC[SUM_CLAUSES_LEFT; LE_0] THEN REWRITE_TAC[SUM_OFFSET] THEN REWRITE_TAC[ADD_SUB; ARITH_RULE `~(i + 1 = 0)`] THEN ASM_CASES_TAC `(x:num->real) 0 = &1` THENL [FIRST_X_ASSUM(X_CHOOSE_TAC `a:num->real` o REWRITE_RULE[GSYM MEMBER_NOT_EMPTY]) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a:num->real`; `&0`]) THEN ASM_REWRITE_TAC[REAL_POS] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[REAL_SUB_RZERO; REAL_MUL_LID; REAL_MUL_RID] THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN REWRITE_TAC[REAL_MUL_LZERO] THEN MATCH_MP_TAC SUM_EQ_0_NUMSEG THEN X_GEN_TAC `k:num` THEN STRIP_TAC THEN REWRITE_TAC[REAL_ENTIRE] THEN DISJ2_TAC THEN UNDISCH_TAC `sum (0..p + 1) x = &1` THEN SIMP_TAC[SUM_CLAUSES_LEFT; LE_0] THEN REWRITE_TAC[SUM_OFFSET] THEN ASM_REWRITE_TAC[REAL_ARITH `&1 + x = &1 <=> x = &0`] THEN DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] SUM_POS_EQ_0_NUMSEG)) THEN ASM_REWRITE_TAC[LE_0] THEN ASM_MESON_TAC[]; ASM_SIMP_TAC[REAL_FIELD `~(x = &1) ==> v * x + s = (&1 - (&1 - x)) * v + (&1 - x) * inv(&1 - x) * s`] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[REAL_SUB_LE; REAL_ARITH `&1 - x <= &1 <=> &0 <= x`] THEN REWRITE_TAC[GSYM SUM_LMUL] THEN ONCE_REWRITE_TAC[REAL_ARITH `inv x * y * z:real = y * z / x`] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ; REAL_LT_LE; REAL_SUB_LT; ARITH_RULE `p < i ==> p + 1 < i + 1`] THEN ASM_REWRITE_TAC[REAL_MUL_LZERO; real_div; REAL_MUL_LID] THEN UNDISCH_TAC `sum (0..p + 1) x = &1` THEN MP_TAC(ARITH_RULE `0 <= p + 1`) THEN SIMP_TAC[SUM_CLAUSES_LEFT] THEN DISCH_THEN(K ALL_TAC) THEN REWRITE_TAC[SUM_OFFSET] THEN REWRITE_TAC[REAL_ARITH `x + s = &1 <=> s = &1 - x`] THEN DISCH_TAC THEN ASM_SIMP_TAC[SUM_RMUL; REAL_MUL_RINV; REAL_SUB_0] THEN X_GEN_TAC `n:num` THEN ASM_CASES_TAC `p + 1 < n + 1` THEN ASM_SIMP_TAC[REAL_SUB_LE] THEN TRANS_TAC REAL_LE_TRANS `sum {n} (\i. (x:num->real)(i + 1))` THEN CONJ_TAC THENL [REWRITE_TAC[SUM_SING; REAL_LE_REFL]; ALL_TAC] THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN MATCH_MP_TAC SUM_SUBSET_SIMPLE THEN ASM_REWRITE_TAC[FINITE_NUMSEG; SING_SUBSET; IN_NUMSEG] THEN ASM_ARITH_TAC]);; let simplicial_cone = new_definition `simplicial_cone p v = frag_extend (frag_of o simplex_cone p v)`;; let SIMPLICIAL_CHAIN_SIMPLICIAL_CONE = prove (`!p s t v c. simplicial_chain (p,s) c /\ (!x u. &0 <= u /\ u <= &1 /\ x IN s ==> (\i. (&1 - u) * v i + u * x i) IN t) ==> simplicial_chain (p + 1,t) (simplicial_cone p v c)`, REWRITE_TAC[IMP_CONJ_ALT; RIGHT_FORALL_IMP_THM] THEN REPEAT GEN_TAC THEN REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC] THEN DISCH_TAC THEN REWRITE_TAC[simplicial_chain] THEN REWRITE_TAC[simplicial_cone] THEN MATCH_MP_TAC FRAG_INDUCTION THEN REWRITE_TAC[FRAG_EXTEND_0; FRAG_EXTEND_SUB; FRAG_EXTEND_OF] THEN REWRITE_TAC[FRAG_SUPPORT_0; EMPTY_SUBSET; FRAG_SUPPORT_OF; o_THM] THEN REWRITE_TAC[GSYM UNION_SUBSET] THEN CONJ_TAC THENL [REWRITE_TAC[SING_SUBSET] THEN REWRITE_TAC[IN] THEN ASM_MESON_TAC[SIMPLICIAL_SIMPLEX_SIMPLEX_CONE]; MESON_TAC[SUBSET; FRAG_SUPPORT_SUB]]);; let CHAIN_BOUNDARY_SIMPLICIAL_CONE_OF = prove (`!p s v f. simplicial_simplex (p,s) f ==> chain_boundary (p + 1) (simplicial_cone p v (frag_of f)) = frag_sub (frag_of f) (if p = 0 then frag_of(RESTRICTION (standard_simplex p) (\u. v)) else simplicial_cone (p - 1) v (chain_boundary p (frag_of f)))`, REPLICATE_TAC 3 GEN_TAC THEN REWRITE_TAC[simplicial_simplex; IMP_CONJ_ALT; LEFT_IMP_EXISTS_THM] THEN GEN_REWRITE_TAC I [SWAP_FORALL_THM] THEN REWRITE_TAC[FORALL_UNWIND_THM2] THEN X_GEN_TAC `l:num->num->real` THEN DISCH_THEN(K ALL_TAC) THEN REWRITE_TAC[simplicial_cone; FRAG_EXTEND_OF; o_THM; CHAIN_BOUNDARY_OF] THEN REWRITE_TAC[ADD_EQ_0] THEN CONV_TAC NUM_REDUCE_CONV THEN ASM_CASES_TAC `p = 0` THEN ASM_REWRITE_TAC[] THENL [SIMP_TAC[GSYM ADD1; MATCH_MP ITERATE_CLAUSES_NUMSEG MONOIDAL_FRAG_ADD] THEN CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC INT_REDUCE_CONV THEN SIMP_TAC[simplex_cone; SINGULAR_FACE_ORIENTED_SIMPLEX; ARITH] THEN REWRITE_TAC[CONJUNCT1 LT; ADD_EQ_0] THEN CONV_TAC NUM_REDUCE_CONV THEN SIMP_TAC[ARITH_RULE `j < 1 <=> j = 0`; ADD_SUB; ETA_AX; FRAG_MODULE `frag_add (frag_cmul (&1) x) (frag_cmul (-- &1) y) = frag_sub x y`] THEN AP_TERM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[oriented_simplex] THEN SIMP_TAC[SUM_SING_NUMSEG; RESTRICTION; FUN_EQ_THM; standard_simplex; IN_ELIM_THM; REAL_MUL_RID; ETA_AX]; GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM(REWRITE_RULE[LE_0] (SPEC `0` NUMSEG_LREC))] THEN SIMP_TAC[MATCH_MP ITERATE_CLAUSES MONOIDAL_FRAG_ADD; FINITE_NUMSEG] THEN REWRITE_TAC[ARITH_RULE `~(0 + 1 <= 0)`; IN_NUMSEG; FRAG_MODULE `frag_add (frag_cmul (-- &1 pow 0) x) y = frag_sub x (frag_cmul (-- &1) y)`] THEN BINOP_TAC THENL [AP_TERM_TAC THEN SIMP_TAC[simplex_cone; SINGULAR_FACE_ORIENTED_SIMPLEX; LE_0; ADD_SUB; ARITH_RULE `1 <= p + 1 /\ ~(j + 1 = 0)`] THEN REWRITE_TAC[ORIENTED_SIMPLEX_EQ; CONJUNCT1 LT]; REWRITE_TAC[FRAG_CMUL_SUM; NUMSEG_OFFSET_IMAGE] THEN W(MP_TAC o PART_MATCH (lhand o rand) (MATCH_MP ITERATE_IMAGE MONOIDAL_FRAG_ADD) o lhand o snd) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [ARITH_TAC; DISCH_THEN SUBST1_TAC] THEN SIMP_TAC[FRAG_EXTEND_SUM; FINITE_NUMSEG] THEN MATCH_MP_TAC (MATCH_MP ITERATE_EQ MONOIDAL_FRAG_ADD) THEN X_GEN_TAC `k:num` THEN REWRITE_TAC[IN_NUMSEG; o_THM] THEN DISCH_TAC THEN REWRITE_TAC[FRAG_EXTEND_CMUL; FRAG_MODULE `frag_cmul a (frag_cmul b c) = frag_cmul (a * b) c`] THEN REWRITE_TAC[INT_POW_ADD; INT_MUL_LNEG; INT_MUL_RNEG; INT_POW_1] THEN REWRITE_TAC[INT_NEG_NEG; INT_MUL_RID; INT_MUL_LID] THEN AP_TERM_TAC THEN REWRITE_TAC[o_THM; FRAG_EXTEND_OF] THEN AP_TERM_TAC THEN ASM_SIMP_TAC[simplex_cone; SINGULAR_FACE_ORIENTED_SIMPLEX; LE_1; LE_ADD_RCANCEL; ARITH_RULE `1 <= k + 1`; ADD_SUB; SUB_ADD; ORIENTED_SIMPLEX_EQ] THEN REPEAT STRIP_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN ASM_ARITH_TAC]]);; let CHAIN_BOUNDARY_SIMPLICIAL_CONE = prove (`!p s v c. simplicial_chain (p,s) c ==> chain_boundary (p + 1) (simplicial_cone p v c) = frag_sub c (if p = 0 then frag_extend (\f. frag_of(RESTRICTION (standard_simplex p) (\u. v))) c else simplicial_cone (p - 1) v (chain_boundary p c))`, REPLICATE_TAC 3 GEN_TAC THEN REWRITE_TAC[simplicial_chain] THEN MATCH_MP_TAC FRAG_INDUCTION THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[CHAIN_BOUNDARY_0; simplicial_cone; FRAG_EXTEND_0] THEN REWRITE_TAC[COND_ID] THEN CONV_TAC FRAG_MODULE; REWRITE_TAC[IN; FRAG_EXTEND_OF; o_THM] THEN GEN_TAC THEN DISCH_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) CHAIN_BOUNDARY_SIMPLICIAL_CONE_OF o lhand o snd) THEN ASM_REWRITE_TAC[]; REPEAT GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[CHAIN_BOUNDARY_SUB; simplicial_cone; FRAG_EXTEND_SUB] THEN REWRITE_TAC[GSYM simplicial_cone] THEN REWRITE_TAC[COND_ID] THEN CONV_TAC FRAG_MODULE]);; let SIMPLEX_MAP_ORIENTED_SIMPLEX = prove (`!p q r g l s. simplicial_simplex(p,standard_simplex q) (oriented_simplex p l) /\ simplicial_simplex(r,s) g /\ q <= r ==> simplex_map p g (oriented_simplex p l) = oriented_simplex p (g o l)`, REPEAT STRIP_TAC THEN REWRITE_TAC[oriented_simplex; simplex_map; RESTRICTION_COMPOSE_RIGHT] THEN REWRITE_TAC[o_DEF] THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `x:num->real` THEN REWRITE_TAC[RESTRICTION] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN MP_TAC(ASSUME `simplicial_simplex (r,s) g`) THEN REWRITE_TAC[simplicial_simplex] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `m:num->num->real` SUBST_ALL_TAC) THEN REWRITE_TAC[oriented_simplex] THEN GEN_REWRITE_TAC LAND_CONV [RESTRICTION] THEN UNDISCH_TAC `simplicial_simplex (p,standard_simplex q) (oriented_simplex p l)` THEN REWRITE_TAC[SIMPLICIAL_SIMPLEX_ORIENTED_SIMPLEX] THEN FIRST_ASSUM(MP_TAC o REWRITE_RULE[SUBSET] o MATCH_MP STANDARD_SIMPLEX_MONO) THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE] THEN DISCH_TAC THEN DISCH_THEN(MP_TAC o GEN `i:num` o SPEC `(\j. if j = i then &1 else &0):num->real`) THEN REWRITE_TAC[BASIS_IN_STANDARD_SIMPLEX; COND_RAND] THEN SIMP_TAC[REAL_MUL_RZERO; SUM_DELTA; IN_NUMSEG; LE_0; REAL_MUL_RID] THEN ASM_SIMP_TAC[ETA_AX; RESTRICTION; GSYM SUM_LMUL; GSYM SUM_RMUL] THEN DISCH_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `k:num` THEN REWRITE_TAC[] THEN GEN_REWRITE_TAC LAND_CONV [SUM_SWAP_NUMSEG] THEN REWRITE_TAC[GSYM REAL_MUL_ASSOC]);; let CHAIN_MAP_SIMPLICIAL_CONE = prove (`!s p q r v c g. simplicial_simplex (r,s) g /\ simplicial_chain (p,standard_simplex q) c /\ v IN standard_simplex q /\ q <= r ==> chain_map (p + 1) g (simplicial_cone p v c) = simplicial_cone p (g v) (chain_map p g c)`, REWRITE_TAC[chain_map; simplicial_cone; FRAG_EXTEND_COMPOSE] THEN REWRITE_TAC[simplicial_chain] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC FRAG_EXTEND_EQ THEN REPEAT STRIP_TAC THEN REWRITE_TAC[o_THM] THEN AP_TERM_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `f:(num->real)->(num->real)` o REWRITE_RULE[SUBSET]) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[IN] THEN DISCH_TAC THEN MP_TAC(ASSUME `simplicial_simplex (p,standard_simplex q) f`) THEN REWRITE_TAC[simplicial_simplex] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `m:num->num->real` SUBST_ALL_TAC) THEN REWRITE_TAC[simplex_cone] THEN MP_TAC(ISPECL [`p + 1`; `q:num`; `r:num`; `g:(num->real)->(num->real)`; `(\i. if i = 0 then v else m (i - 1)):num->num->real`; `s:(num->real)->bool`] SIMPLEX_MAP_ORIENTED_SIMPLEX) THEN REWRITE_TAC[LE_REFL] THEN ANTS_TAC THENL [ASM_REWRITE_TAC[GSYM ADD1; GSYM simplex_cone] THEN REWRITE_TAC[ADD1] THEN MATCH_MP_TAC SIMPLICIAL_SIMPLEX_SIMPLEX_CONE THEN EXISTS_TAC `standard_simplex q` THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CONVEX_STANDARD_SIMPLEX THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `x IN s ==> s SUBSET t ==> x IN t`)) THEN MATCH_MP_TAC STANDARD_SIMPLEX_MONO THEN ARITH_TAC; DISCH_THEN SUBST1_TAC] THEN MP_TAC(ISPECL [`p:num`; `q:num`; `r:num`; `g:(num->real)->(num->real)`; `m:num->num->real`; `s:(num->real)->bool`] SIMPLEX_MAP_ORIENTED_SIMPLEX) THEN ASM_REWRITE_TAC[GSYM ADD1; ARITH_RULE `p <= SUC p`] THEN DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[simplex_cone] THEN REWRITE_TAC[o_DEF; COND_RAND; ADD1]);; (* ------------------------------------------------------------------------- *) (* Barycentric subdivision of a linear ("simplicial") simplex's image. *) (* ------------------------------------------------------------------------- *) let simplicial_vertex = new_definition `simplicial_vertex i (f:(num->real)->(num->real)) = f(\j. if j = i then &1 else &0)`;; let SIMPLICIAL_VERTEX_ORIENTED_SIMPLEX = prove (`!i p l. simplicial_vertex i (oriented_simplex p l) = if i <= p then l i else ARB`, REPEAT GEN_TAC THEN REWRITE_TAC[simplicial_vertex; oriented_simplex] THEN REWRITE_TAC[RESTRICTION; BASIS_IN_STANDARD_SIMPLEX] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN SIMP_TAC[COND_RAND; REAL_MUL_RZERO; SUM_DELTA] THEN ASM_REWRITE_TAC[IN_NUMSEG; LE_0; REAL_MUL_RID; ETA_AX]);; let simplicial_subdivision = new_recursive_definition num_RECURSION `simplicial_subdivision 0 = I /\ simplicial_subdivision (SUC p) = frag_extend (\f. simplicial_cone p (\i. sum(0..SUC p) (\j. simplicial_vertex j f i) / (&p + &2)) (simplicial_subdivision p (chain_boundary (SUC p) (frag_of f))))`;; let SIMPLICIAL_SUBDIVISION_0 = prove (`!p. simplicial_subdivision p frag_0 = frag_0`, INDUCT_TAC THEN REWRITE_TAC[simplicial_subdivision; FRAG_EXTEND_0; I_THM]);; let SIMPLICIAL_SUBDIVISION_SUB = prove (`!p c1 c2. simplicial_subdivision p (frag_sub c1 c2) = frag_sub (simplicial_subdivision p c1) (simplicial_subdivision p c2)`, INDUCT_TAC THEN REWRITE_TAC[simplicial_subdivision; FRAG_EXTEND_SUB; I_THM]);; let SIMPLICIAL_SUBDIVISION_OF = prove (`!p f. simplicial_subdivision p (frag_of f) = if p = 0 then frag_of f else simplicial_cone (p - 1) (\i. sum(0..p) (\j. simplicial_vertex j f i) / (&p + &1)) (simplicial_subdivision (p - 1) (chain_boundary p (frag_of f)))`, INDUCT_TAC THEN REWRITE_TAC[simplicial_subdivision; I_THM] THEN REWRITE_TAC[NOT_SUC; SUC_SUB1; FRAG_EXTEND_OF] THEN REWRITE_TAC[REAL_OF_NUM_ADD; ARITH_RULE `SUC p + 1 = p + 2`]);; let SIMPLICIAL_CHAIN_SIMPLICIAL_SUBDIVISION = prove (`!p s c. simplicial_chain (p,s) c ==> simplicial_chain (p,s) (simplicial_subdivision p c)`, MATCH_MP_TAC num_INDUCTION THEN CONJ_TAC THENL [SIMP_TAC[simplicial_subdivision; I_THM]; ALL_TAC] THEN X_GEN_TAC `p:num` THEN DISCH_TAC THEN X_GEN_TAC `s:(num->real)->bool` THEN REWRITE_TAC[simplicial_chain] THEN MATCH_MP_TAC FRAG_INDUCTION THEN REWRITE_TAC[SIMPLICIAL_SUBDIVISION_0; SIMPLICIAL_SUBDIVISION_SUB] THEN REWRITE_TAC[FRAG_SUPPORT_0; EMPTY_SUBSET; GSYM UNION_SUBSET] THEN CONJ_TAC THENL [ALL_TAC; MESON_TAC[SUBSET; FRAG_SUPPORT_SUB]] THEN REWRITE_TAC[simplicial_subdivision; FRAG_EXTEND_OF] THEN REWRITE_TAC[IN; GSYM simplicial_chain; ADD1] THEN X_GEN_TAC `f:(num->real)->(num->real)` THEN DISCH_TAC THEN MATCH_MP_TAC SIMPLICIAL_CHAIN_SIMPLICIAL_CONE THEN EXISTS_TAC `IMAGE (f:(num->real)->(num->real)) (standard_simplex(p + 1))` THEN REWRITE_TAC[] THEN CONJ_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [ARITH_RULE `p = (p + 1) - 1`] THEN MATCH_MP_TAC SIMPLICIAL_CHAIN_BOUNDARY THEN REWRITE_TAC[SIMPLICIAL_CHAIN_OF] THEN POP_ASSUM MP_TAC THEN SIMP_TAC[SIMPLICIAL_SIMPLEX; SUBSET_REFL]; GEN_REWRITE_TAC I [SWAP_FORALL_THM] THEN X_GEN_TAC `u:real` THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REPEAT DISCH_TAC THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN X_GEN_TAC `x:num->real` THEN DISCH_TAC THEN FIRST_X_ASSUM(CONJUNCTS_THEN MP_TAC o REWRITE_RULE[SIMPLICIAL_SIMPLEX]) THEN DISCH_THEN(X_CHOOSE_THEN `l:num->num->real` SUBST1_TAC) THEN SIMP_TAC[SIMPLICIAL_VERTEX_ORIENTED_SIMPLEX] THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE; oriented_simplex; RESTRICTION] THEN REWRITE_TAC[GSYM SUM_LMUL; real_div; GSYM SUM_RMUL; GSYM SUM_ADD_NUMSEG; REAL_ARITH `v * l * i + u * l * x:real = l * (v * i + u * x)`] THEN DISCH_TAC THEN ONCE_REWRITE_TAC[SET_RULE `0..n = {x | x IN 0..n /\ x IN 0..n}`] THEN REWRITE_TAC[SUM_RESTRICT_SET; MESON[REAL_MUL_RZERO] `(if p then x * y else &0) = x * (if p then y else &0)`] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN UNDISCH_TAC `x IN standard_simplex (p + 1)` THEN REWRITE_TAC[standard_simplex; IN_ELIM_THM] THEN SIMP_TAC[SUM_ADD_NUMSEG; SUM_LMUL; IN_NUMSEG; GSYM NOT_LT] THEN STRIP_TAC THEN REWRITE_TAC[CONJUNCT1 LT; SUM_CONST_NUMSEG] THEN REWRITE_TAC[GSYM REAL_OF_NUM_ADD; SUB_0] THEN CONJ_TAC THENL [X_GEN_TAC `k:num`; CONV_TAC REAL_FIELD] THEN REWRITE_TAC[NOT_LT] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_SIMP_TAC[REAL_LE_ADD; REAL_LE_MUL; REAL_SUB_LE; REAL_LE_INV_EQ; REAL_POS] THEN MATCH_MP_TAC REAL_CONVEX_BOUND_LE THEN ASM_REWRITE_TAC[REAL_SUB_LE; REAL_ARITH `&1 - u + u = &1`] THEN MATCH_MP_TAC REAL_INV_LE_1 THEN REAL_ARITH_TAC]);; let CHAIN_BOUNDARY_SIMPLICIAL_SUBDIVISION = prove (`!p s c. simplicial_chain (p,s) c ==> chain_boundary p (simplicial_subdivision p c) = simplicial_subdivision (p - 1) (chain_boundary p c)`, GEN_REWRITE_TAC I [SWAP_FORALL_THM] THEN GEN_TAC THEN MATCH_MP_TAC num_INDUCTION THEN CONJ_TAC THENL [REWRITE_TAC[chain_boundary; SIMPLICIAL_SUBDIVISION_0]; ALL_TAC] THEN X_GEN_TAC `p:num` THEN DISCH_TAC THEN REWRITE_TAC[simplicial_chain] THEN MATCH_MP_TAC FRAG_INDUCTION THEN REWRITE_TAC[SIMPLICIAL_SUBDIVISION_0; SIMPLICIAL_SUBDIVISION_SUB; CHAIN_BOUNDARY_0; CHAIN_BOUNDARY_SUB] THEN CONJ_TAC THENL [ALL_TAC; CONV_TAC FRAG_MODULE] THEN X_GEN_TAC `f:(num->real)->(num->real)` THEN REWRITE_TAC[IN] THEN DISCH_TAC THEN REWRITE_TAC[simplicial_subdivision; FRAG_EXTEND_OF] THEN REWRITE_TAC[ADD1; ADD_SUB] THEN W(MP_TAC o PART_MATCH (lhand o rand) CHAIN_BOUNDARY_SIMPLICIAL_CONE o lhand o snd) THEN ANTS_TAC THENL [MATCH_MP_TAC SIMPLICIAL_CHAIN_SIMPLICIAL_SUBDIVISION THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [ARITH_RULE `p = (p + 1) - 1`] THEN MATCH_MP_TAC SIMPLICIAL_CHAIN_BOUNDARY THEN ASM_REWRITE_TAC[GSYM ADD1; SIMPLICIAL_CHAIN_OF]; DISCH_THEN SUBST1_TAC] THEN ASM_CASES_TAC `p = 0` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(FRAG_MODULE `y = frag_0 ==> frag_sub x y = x`) THENL [REWRITE_TAC[CHAIN_BOUNDARY_OF] THEN CONV_TAC NUM_REDUCE_CONV THEN REWRITE_TAC[simplicial_subdivision; I_THM] THEN SIMP_TAC[FRAG_EXTEND_SUM; FINITE_NUMSEG; o_DEF] THEN REWRITE_TAC[FRAG_EXTEND_CMUL; FRAG_EXTEND_OF] THEN CONV_TAC (LAND_CONV (LAND_CONV (RAND_CONV num_CONV))) THEN SIMP_TAC[MATCH_MP ITERATE_CLAUSES_NUMSEG MONOIDAL_FRAG_ADD] THEN REWRITE_TAC[LE_0; INT_POW] THEN CONV_TAC FRAG_MODULE; ALL_TAC] THEN FIRST_ASSUM(fun th -> W(MP_TAC o PART_MATCH (lhand o rand) th o rand o lhand o snd)) THEN ANTS_TAC THENL [GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [ARITH_RULE `p = (p + 1) - 1`] THEN MATCH_MP_TAC SIMPLICIAL_CHAIN_BOUNDARY THEN ASM_REWRITE_TAC[GSYM ADD1; SIMPLICIAL_CHAIN_OF]; DISCH_THEN SUBST1_TAC] THEN REWRITE_TAC[simplicial_cone; o_DEF] THEN MATCH_MP_TAC (MESON[FRAG_EXTEND_0] `c = frag_0 ==> frag_extend f c = frag_0`) THEN MATCH_MP_TAC(MESON[SIMPLICIAL_SUBDIVISION_0] `c = frag_0 ==> simplicial_subdivision q c = frag_0`) THEN MATCH_MP_TAC CHAIN_BOUNDARY_BOUNDARY_ALT THEN MATCH_MP_TAC(MESON[SIMPLICIAL_IMP_SINGULAR_CHAIN] `simplicial_chain (p,s) c ==> ?s. singular_chain(p,s) c`) THEN ASM_REWRITE_TAC[GSYM ADD1; SIMPLICIAL_CHAIN_OF]);; let SIMPLICIAL_SUBDIVISION_SHRINKS = prove (`!s k p d c. simplicial_chain(p,s) c /\ (!f x y. f IN frag_support c /\ x IN standard_simplex p /\ y IN standard_simplex p ==> abs(f x k - f y k) <= d) ==> (!f x y. f IN frag_support(simplicial_subdivision p c) /\ x IN standard_simplex p /\ y IN standard_simplex p ==> abs(f x k - f y k) <= &p / (&p + &1) * d)`, GEN_TAC THEN GEN_TAC THEN MATCH_MP_TAC num_INDUCTION THEN REWRITE_TAC[simplicial_subdivision] THEN CONJ_TAC THENL [REPEAT GEN_TAC THEN DISCH_THEN(K ALL_TAC) THEN REPEAT GEN_TAC THEN SIMP_TAC[STANDARD_SIMPLEX_0; IN_SING; REAL_SUB_REFL] THEN REAL_ARITH_TAC; X_GEN_TAC `p:num` THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`d:real`; `c:((num->real)->(num->real))frag`] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN DISCH_TAC] THEN W(MP_TAC o PART_MATCH lhand FRAG_SUPPORT_FRAG_EXTEND o rand o lhand o snd o dest_forall o snd) THEN MATCH_MP_TAC(SET_RULE `t SUBSET P ==> s SUBSET t ==> (!x. x IN s ==> P x)`) THEN REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [simplicial_chain]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `(!x. x IN s ==> P x) ==> (!x. t x /\ P x ==> Q x) ==> s SUBSET t ==> !x. x IN s ==> Q x`)) THEN X_GEN_TAC `f:(num->real)->(num->real)` THEN STRIP_TAC THEN REWRITE_TAC[simplicial_cone] THEN W(MP_TAC o PART_MATCH lhand FRAG_SUPPORT_FRAG_EXTEND o lhand o snd) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS) THEN REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC] THEN X_GEN_TAC `g:(num->real)->(num->real)` THEN DISCH_TAC THEN REWRITE_TAC[o_THM; FRAG_SUPPORT_OF; SING_SUBSET] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`d:real`; `chain_boundary (SUC p) (frag_of(f:(num->real)->(num->real)))`]) THEN ANTS_TAC THENL [CONJ_TAC THENL [GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM SUC_SUB1] THEN MATCH_MP_TAC SIMPLICIAL_CHAIN_BOUNDARY THEN ASM_REWRITE_TAC[SIMPLICIAL_CHAIN_OF]; SUBGOAL_THEN `simplicial_chain (SUC p,IMAGE f (standard_simplex(SUC p))) (frag_of f)` MP_TAC THENL [REWRITE_TAC[SIMPLICIAL_CHAIN_OF] THEN ASM_MESON_TAC[SIMPLICIAL_SIMPLEX; SUBSET_REFL]; DISCH_THEN(MP_TAC o MATCH_MP SIMPLICIAL_CHAIN_BOUNDARY) THEN REWRITE_TAC[SUC_SUB1; simplicial_chain; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN MATCH_MP_TAC(SET_RULE `(!x. t x ==> P x) ==> s SUBSET t ==> !x. x IN s ==> P x`) THEN REWRITE_TAC[SIMPLICIAL_SIMPLEX] THEN ASM SET_TAC[]]]; DISCH_THEN(MP_TAC o SPEC `g:(num->real)->(num->real)`) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC] THEN SUBGOAL_THEN `&0 <= d` ASSUME_TAC THENL [ASM_MESON_TAC[NONEMPTY_STANDARD_SIMPLEX; MEMBER_NOT_EMPTY; REAL_ARITH `abs x <= d ==> &0 <= d`]; ALL_TAC] THEN SUBGOAL_THEN `simplicial_simplex (p,IMAGE f (standard_simplex(SUC p))) g` MP_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `g IN s ==> s SUBSET P ==> P g`)) THEN REWRITE_TAC[GSYM simplicial_chain; ETA_AX] THEN MATCH_MP_TAC SIMPLICIAL_CHAIN_SIMPLICIAL_SUBDIVISION THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM SUC_SUB1] THEN MATCH_MP_TAC SIMPLICIAL_CHAIN_BOUNDARY THEN REWRITE_TAC[SIMPLICIAL_CHAIN_OF] THEN ASM_MESON_TAC[SIMPLICIAL_SIMPLEX; SUBSET_REFL]; DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN MP_TAC(ASSUME `simplicial_simplex (SUC p,s) f`)] THEN REWRITE_TAC[SIMPLICIAL_SIMPLEX; IMP_CONJ; LEFT_IMP_EXISTS_THM] THEN DISCH_TAC THEN X_GEN_TAC `l:num->num->real` THEN DISCH_THEN(ASSUME_TAC o SYM) THEN DISCH_TAC THEN X_GEN_TAC `m:num->num->real` THEN DISCH_THEN(ASSUME_TAC o SYM) THEN ONCE_REWRITE_TAC[IN] THEN EXPAND_TAC "g" THEN MAP_EVERY X_GEN_TAC [`x:num->real`; `y:num->real`] THEN REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(fun th -> STRIP_ASSUME_TAC th THEN MP_TAC th) THEN REWRITE_TAC[standard_simplex; IN_ELIM_THM] THEN STRIP_TAC THEN EXPAND_TAC "f" THEN REWRITE_TAC[SIMPLICIAL_VERTEX_ORIENTED_SIMPLEX] THEN SIMP_TAC[] THEN REWRITE_TAC[simplex_cone] THEN ASM_REWRITE_TAC[oriented_simplex; RESTRICTION; GSYM ADD1] THEN MATCH_MP_TAC REAL_CONVEX_SUM_BOUND_LE THEN ASM_REWRITE_TAC[ADD1; IN_NUMSEG; LE_0] THEN X_GEN_TAC `j:num` THEN DISCH_TAC THEN ONCE_REWRITE_TAC[REAL_ABS_SUB] THEN MATCH_MP_TAC REAL_CONVEX_SUM_BOUND_LE THEN ASM_REWRITE_TAC[GSYM ADD1; IN_NUMSEG; LE_0] THEN UNDISCH_TAC `j:num <= p + 1` THEN SPEC_TAC(`j:num`,`j:num`) THEN REWRITE_TAC[RIGHT_IMP_FORALL_THM; IMP_IMP] THEN MATCH_MP_TAC WLOG_LT THEN REWRITE_TAC[REAL_SUB_REFL] THEN REPEAT CONJ_TAC THENL [REPEAT STRIP_TAC THEN REWRITE_TAC[REAL_ABS_NUM] THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_SIMP_TAC[REAL_LE_DIV; REAL_LE_ADD; REAL_POS]; REWRITE_TAC[ADD1; REAL_ABS_SUB; CONJ_ACI]; MAP_EVERY X_GEN_TAC [`m:num`; `n:num`]] THEN DISCH_TAC THEN REWRITE_TAC[ADD1] THEN STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP (ARITH_RULE `m < n ==> ~(n = 0)`)) THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `!i. i <= p ==> (m:num->num->real) i IN IMAGE g (standard_simplex p)` (LABEL_TAC "*") THENL [REPEAT STRIP_TAC THEN EXPAND_TAC "g" THEN REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `(\j. if j = i then &1 else &0):num->real` THEN SIMP_TAC[BASIS_IN_STANDARD_SIMPLEX; oriented_simplex; RESTRICTION] THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[COND_RAND] THEN SIMP_TAC[REAL_MUL_RZERO; SUM_DELTA] THEN ASM_REWRITE_TAC[IN_NUMSEG; LE_0; ETA_AX; REAL_MUL_RID]; ALL_TAC] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL [ALL_TAC; REMOVE_THEN "*" (fun th -> MP_TAC(SPEC `n - 1` th) THEN MP_TAC(SPEC `m - 1` th)) THEN ASM_SIMP_TAC[ARITH_RULE `~(n = 0) /\ n <= p + 1 ==> n - 1 <= p`] THEN REWRITE_TAC[IN_IMAGE; LEFT_IMP_EXISTS_THM] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN TRANS_TAC REAL_LE_TRANS `&p / (&p + &1) * d` THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC REAL_LE_RMUL THEN ASM_REWRITE_TAC[GSYM REAL_OF_NUM_ADD] THEN SIMP_TAC[REAL_LE_RDIV_EQ; REAL_ARITH `&0 < (&p + &1) + &1`] THEN REWRITE_TAC[REAL_ARITH `a / b * c:real = (a * c) / b`] THEN SIMP_TAC[REAL_LE_LDIV_EQ; REAL_ARITH `&0 < &p + &1`] THEN REAL_ARITH_TAC] THEN SUBGOAL_THEN `(m:num->num->real) (n - 1) IN IMAGE f (standard_simplex (SUC p))` MP_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; REWRITE_TAC[IN_IMAGE; LEFT_IMP_EXISTS_THM]] THEN X_GEN_TAC `z:num->real` THEN EXPAND_TAC "f" THEN SIMP_TAC[oriented_simplex; RESTRICTION] THEN DISCH_THEN(MP_TAC o CONJUNCT2) THEN REWRITE_TAC[standard_simplex; IN_ELIM_THM; ADD1] THEN STRIP_TAC THEN MATCH_MP_TAC REAL_CONVEX_SUM_BOUND_LE THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `i:num` THEN REWRITE_TAC[IN_NUMSEG; LE_0] THEN DISCH_TAC THEN SUBGOAL_THEN `(l:num->num->real) i k = sum(0..p+1) (\j. l i k) / (&p + &2)` SUBST1_TAC THENL [REWRITE_TAC[SUM_CONST_NUMSEG; SUB_0; GSYM REAL_OF_NUM_ADD] THEN CONV_TAC REAL_FIELD; REWRITE_TAC[real_div; GSYM SUM_RMUL; GSYM SUM_SUB_NUMSEG]] THEN W(MP_TAC o PART_MATCH lhand SUM_ABS_NUMSEG o lhand o snd) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN TRANS_TAC REAL_LE_TRANS `sum((0..p+1) DELETE i) (\j. abs(l i (k:num) - l j k) / (&p + &2))` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_EQ_IMP_LE THEN REWRITE_TAC[real_div; GSYM REAL_SUB_RDISTRIB] THEN REWRITE_TAC[GSYM real_div; REAL_ABS_DIV] THEN REWRITE_TAC[REAL_ARITH `abs(&p + &2) = &p + &2`] THEN MATCH_MP_TAC SUM_SUPERSET THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `x IN s /\ ~(x IN s DELETE a) ==> x = a`)) THEN SIMP_TAC[REAL_SUB_REFL; real_div; REAL_MUL_LZERO; REAL_ABS_NUM]; MATCH_MP_TAC SUM_BOUND_GEN THEN SIMP_TAC[CARD_DELETE; FINITE_NUMSEG] THEN ASM_REWRITE_TAC[FINITE_DELETE; FINITE_NUMSEG; IN_NUMSEG; LE_0] THEN REWRITE_TAC[CARD_NUMSEG; GSYM REAL_OF_NUM_ADD; ARITH_RULE `(x + 1) - 0 - 1 = x`] THEN REWRITE_TAC[SET_RULE `s DELETE a = {} <=> s SUBSET {a}`] THEN REWRITE_TAC[GSYM NUMSEG_SING; SUBSET_NUMSEG] THEN CONJ_TAC THENL [ARITH_TAC; ALL_TAC] THEN X_GEN_TAC `q:num` THEN REWRITE_TAC[IN_DELETE; IN_NUMSEG; LE_0] THEN STRIP_TAC THEN SIMP_TAC[REAL_LE_LDIV_EQ; REAL_ARITH `&0 < &p + &2`] THEN REWRITE_TAC[REAL_FIELD `(((&p + &1) * inv((&p + &1) + &1)) * d) / (&p + &1) * (&p + &2) = d`] THEN SUBGOAL_THEN `!r. r <= p + 1 ==> (l:num->num->real) r IN IMAGE f (standard_simplex(SUC p))` MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN X_GEN_TAC `r:num` THEN DISCH_TAC THEN EXPAND_TAC "f" THEN REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `(\j. if j = r then &1 else &0):num->real` THEN REWRITE_TAC[RESTRICTION; BASIS_IN_STANDARD_SIMPLEX; oriented_simplex] THEN ASM_REWRITE_TAC[ADD1] THEN REWRITE_TAC[COND_RAND] THEN SIMP_TAC[REAL_MUL_RZERO; SUM_DELTA] THEN ASM_REWRITE_TAC[IN_NUMSEG; LE_0; ETA_AX; REAL_MUL_RID]]);; (* ------------------------------------------------------------------------- *) (* Now singular subdivision. *) (* ------------------------------------------------------------------------- *) let singular_subdivision = new_definition `singular_subdivision p = frag_extend (\f. chain_map p f (simplicial_subdivision p (frag_of(RESTRICTION (standard_simplex p) I))))`;; let SINGULAR_SUBDIVISION_0 = prove (`!p. singular_subdivision p frag_0 = frag_0`, REWRITE_TAC[singular_subdivision; FRAG_EXTEND_0]);; let SINGULAR_SUBDIVISION_SUB = prove (`!p c1 c2. singular_subdivision p (frag_sub c1 c2) = frag_sub (singular_subdivision p c1) (singular_subdivision p c2)`, REWRITE_TAC[singular_subdivision; FRAG_EXTEND_SUB]);; let SINGULAR_SUBDIVISION_ADD = prove (`!p c1 c2. singular_subdivision p (frag_add c1 c2) = frag_add (singular_subdivision p c1) (singular_subdivision p c2)`, REWRITE_TAC[singular_subdivision; FRAG_EXTEND_ADD]);; let SIMPLICIAL_SIMPLEX_I = prove (`!p s. simplicial_simplex (p,s) (RESTRICTION (standard_simplex p) I) <=> standard_simplex p SUBSET s`, REPEAT GEN_TAC THEN REWRITE_TAC[simplicial_simplex; singular_simplex] THEN SIMP_TAC[RESTRICTION_CONTINUOUS_MAP; TOPSPACE_SUBTOPOLOGY; INTER_UNIV; CONTINUOUS_MAP_IN_SUBTOPOLOGY; TOPSPACE_PRODUCT_TOPOLOGY; o_DEF; CARTESIAN_PRODUCT_UNIV; SUBSET_REFL; TOPSPACE_EUCLIDEANREAL] THEN SIMP_TAC[I_DEF; IMAGE_ID; CONTINUOUS_MAP_ID; CONTINUOUS_MAP_FROM_SUBTOPOLOGY; REWRITE_RULE[IN] RESTRICTION_IN_EXTENSIONAL] THEN SIMP_TAC[SUBSET; FORALL_IN_IMAGE; RESTRICTION] THEN MATCH_MP_TAC(TAUT `q ==> (p /\ q <=> p)`) THEN EXISTS_TAC `(\i j. if i = j then &1 else &0):num->num->real` THEN REWRITE_TAC[oriented_simplex; RESTRICTION_EXTENSION] THEN X_GEN_TAC `x:num->real` THEN REWRITE_TAC[standard_simplex; IN_ELIM_THM] THEN STRIP_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN REWRITE_TAC[COND_RATOR; COND_RAND; REAL_MUL_LZERO; REAL_MUL_LID] THEN SIMP_TAC[SUM_DELTA; IN_NUMSEG; LE_0] THEN ASM_MESON_TAC[NOT_LT]);; let SINGULAR_CHAIN_SINGULAR_SUBDIVISION = prove (`!p s c:((num->real)->A)frag. singular_chain (p,s) c ==> singular_chain (p,s) (singular_subdivision p c)`, REPEAT STRIP_TAC THEN REWRITE_TAC[singular_subdivision] THEN MATCH_MP_TAC SINGULAR_CHAIN_EXTEND THEN X_GEN_TAC `f:(num->real)->A` THEN REWRITE_TAC[IN] THEN DISCH_TAC THEN MATCH_MP_TAC SINGULAR_CHAIN_CHAIN_MAP THEN EXISTS_TAC `subtopology (product_topology (:num) (\i. euclideanreal)) (standard_simplex p)` THEN CONJ_TAC THENL [MATCH_MP_TAC SIMPLICIAL_IMP_SINGULAR_CHAIN THEN MATCH_MP_TAC SIMPLICIAL_CHAIN_SIMPLICIAL_SUBDIVISION THEN REWRITE_TAC[SIMPLICIAL_CHAIN_OF; SIMPLICIAL_SIMPLEX_I; SUBSET_REFL]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [singular_chain]) THEN ASM_SIMP_TAC[SUBSET; IN; singular_simplex]]);; let NATURALITY_SINGULAR_SUBDIVISION = prove (`!p s (g:A->B) c. singular_chain (p,s) c ==> singular_subdivision p (chain_map p g c) = chain_map p g (singular_subdivision p c)`, REPLICATE_TAC 3 GEN_TAC THEN REWRITE_TAC[singular_chain] THEN MATCH_MP_TAC FRAG_INDUCTION THEN REWRITE_TAC[singular_subdivision; FRAG_EXTEND_0; CHAIN_MAP_0] THEN REWRITE_TAC[FRAG_EXTEND_SUB; CHAIN_MAP_SUB] THEN REWRITE_TAC[GSYM singular_subdivision] THEN SIMP_TAC[] THEN X_GEN_TAC `f:(num->real)->A` THEN REWRITE_TAC[IN] THEN DISCH_TAC THEN REWRITE_TAC[singular_subdivision; CHAIN_MAP_OF; FRAG_EXTEND_OF] THEN GEN_REWRITE_TAC RAND_CONV [GSYM o_THM] THEN MP_TAC(ISPECL [`p:num`; `standard_simplex p:(num->real)->bool`; `frag_of (RESTRICTION (standard_simplex p:(num->real)->bool) I)`] SIMPLICIAL_CHAIN_SIMPLICIAL_SUBDIVISION) THEN REWRITE_TAC[SIMPLICIAL_CHAIN_OF; SIMPLICIAL_SIMPLEX_I; SUBSET_REFL] THEN SPEC_TAC(`simplicial_subdivision p (frag_of (RESTRICTION (standard_simplex p:(num->real)->bool) I))`, `d:((num->real)->(num->real))frag`) THEN REWRITE_TAC[simplicial_chain; o_DEF] THEN MATCH_MP_TAC FRAG_INDUCTION THEN SIMP_TAC[CHAIN_MAP_0; CHAIN_MAP_SUB; CHAIN_MAP_OF] THEN REWRITE_TAC[IN; SIMPLICIAL_SIMPLEX] THEN REWRITE_TAC[simplex_map; RESTRICTION_COMPOSE_RIGHT] THEN SIMP_TAC[RESTRICTION_COMPOSE_LEFT; SUBSET_REFL] THEN REWRITE_TAC[o_ASSOC]);; let SIMPLICIAL_CHAIN_CHAIN_MAP = prove (`!p q s f c. simplicial_simplex (q,s) f /\ simplicial_chain (p,standard_simplex q) c ==> simplicial_chain (p,s) (chain_map p f c)`, REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REPEAT GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[simplicial_chain] THEN MATCH_MP_TAC FRAG_INDUCTION THEN REWRITE_TAC[GSYM simplicial_chain] THEN REWRITE_TAC[CHAIN_MAP_0; CHAIN_MAP_SUB; CHAIN_MAP_OF] THEN SIMP_TAC[SIMPLICIAL_CHAIN_0; SIMPLICIAL_CHAIN_SUB; SIMPLICIAL_CHAIN_OF] THEN X_GEN_TAC `g:(num->real)->(num->real)` THEN REWRITE_TAC[IN] THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[simplicial_simplex] THEN REWRITE_TAC[IMP_CONJ; LEFT_IMP_EXISTS_THM] THEN DISCH_TAC THEN X_GEN_TAC `l:num->num->real` THEN DISCH_TAC THEN DISCH_TAC THEN X_GEN_TAC `m:num->num->real` THEN DISCH_TAC THEN CONJ_TAC THENL [MATCH_MP_TAC SINGULAR_SIMPLEX_SIMPLEX_MAP THEN EXISTS_TAC `subtopology (product_topology (:num) (\i. euclideanreal)) (standard_simplex q)` THEN ASM_REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[singular_simplex]) THEN ASM_MESON_TAC[]; ASM_REWRITE_TAC[oriented_simplex; simplex_map] THEN REWRITE_TAC[RESTRICTION_COMPOSE_RIGHT] THEN EXISTS_TAC `\(j:num) (k:num). sum (0..q) (\i. l i k * m j i)` THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `x:num->real` THEN REWRITE_TAC[RESTRICTION] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[o_DEF; RESTRICTION] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o CONJUNCT1 o REWRITE_RULE[singular_simplex])) THEN REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN REWRITE_TAC[TOPSPACE_PRODUCT_TOPOLOGY; TOPSPACE_SUBTOPOLOGY; o_DEF; TOPSPACE_EUCLIDEANREAL; CARTESIAN_PRODUCT_UNIV] THEN ASM_REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; INTER_UNIV] THEN ASM_SIMP_TAC[oriented_simplex; RESTRICTION] THEN REPEAT(DISCH_THEN(K ALL_TAC)) THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `k:num` THEN REWRITE_TAC[GSYM SUM_LMUL] THEN GEN_REWRITE_TAC LAND_CONV [SUM_SWAP_NUMSEG] THEN REWRITE_TAC[REAL_MUL_ASSOC; SUM_RMUL]]);; let SINGULAR_SUBDIVISION_SIMPLICIAL_SIMPLEX = prove (`!p s c. simplicial_chain (p,s) c ==> singular_subdivision p c = simplicial_subdivision p c`, MATCH_MP_TAC num_INDUCTION THEN CONJ_TAC THENL [REWRITE_TAC[singular_subdivision; simplicial_subdivision; I_THM] THEN X_GEN_TAC `s:(num->real)->bool` THEN REWRITE_TAC[simplicial_chain] THEN MATCH_MP_TAC FRAG_INDUCTION THEN SIMP_TAC[FRAG_EXTEND_0; FRAG_EXTEND_SUB; FRAG_EXTEND_OF] THEN REWRITE_TAC[IN] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC SINGULAR_SIMPLEX_CHAIN_MAP_I THEN FIRST_ASSUM(MP_TAC o MATCH_MP SIMPLICIAL_IMP_SINGULAR_SIMPLEX) THEN MESON_TAC[]; ALL_TAC] THEN X_GEN_TAC `p:num` THEN DISCH_TAC THEN X_GEN_TAC `s:(num->real)->bool` THEN REWRITE_TAC[simplicial_chain] THEN MATCH_MP_TAC FRAG_INDUCTION THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[simplicial_subdivision; singular_subdivision; FRAG_EXTEND_0]; X_GEN_TAC `f:(num->real)->(num->real)` THEN REWRITE_TAC[IN] THEN DISCH_TAC; REWRITE_TAC[simplicial_subdivision; singular_subdivision] THEN SIMP_TAC[FRAG_EXTEND_SUB]] THEN REWRITE_TAC[singular_subdivision; FRAG_EXTEND_OF] THEN REWRITE_TAC[SIMPLICIAL_SUBDIVISION_OF; NOT_SUC; SUC_SUB1] THEN MP_TAC(ISPECL [`s:(num->real)->bool`; `p:num`; `p + 1`; `p + 1`] CHAIN_MAP_SIMPLICIAL_CONE) THEN REWRITE_TAC[ADD1] THEN DISCH_THEN(fun th -> W(MP_TAC o PART_MATCH (lhand o rand) th o lhand o snd)) THEN ASM_REWRITE_TAC[GSYM ADD1; LE_REFL] THEN ANTS_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC SIMPLICIAL_CHAIN_SIMPLICIAL_SUBDIVISION THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM SUC_SUB1] THEN MATCH_MP_TAC SIMPLICIAL_CHAIN_BOUNDARY THEN REWRITE_TAC[SIMPLICIAL_CHAIN_OF; SIMPLICIAL_SIMPLEX_I; SUBSET_REFL]; REWRITE_TAC[simplicial_vertex; RESTRICTION; I_THM] THEN SIMP_TAC[BASIS_IN_STANDARD_SIMPLEX] THEN ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN REWRITE_TAC[SUM_DELTA; IN_NUMSEG; LE_0] THEN SIMP_TAC[standard_simplex; IN_ELIM_THM; GSYM NOT_LE] THEN REWRITE_TAC[real_div; REAL_MUL_LZERO; SUM_CONST_NUMSEG] THEN REWRITE_TAC[SUB_0; GSYM REAL_OF_NUM_ADD; GSYM REAL_OF_NUM_SUC] THEN CONJ_TAC THENL [ALL_TAC; CONV_TAC REAL_FIELD] THEN GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_MUL_LZERO; REAL_POS; REAL_MUL_LID] THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_LE_INV_EQ; REAL_POS] THEN MATCH_MP_TAC REAL_INV_LE_1 THEN REWRITE_TAC[REAL_OF_NUM_LE] THEN ARITH_TAC]; DISCH_THEN SUBST1_TAC] THEN BINOP_TAC THENL [REWRITE_TAC[RESTRICTION; simplicial_vertex; BASIS_IN_STANDARD_SIMPLEX] THEN SIMP_TAC[I_THM] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [EQ_SYM_EQ] THEN SIMP_TAC[SUM_DELTA] THEN FIRST_ASSUM(MP_TAC o CONJUNCT2 o REWRITE_RULE[simplicial_simplex]) THEN DISCH_THEN(X_CHOOSE_THEN `m:num->num->real` SUBST1_TAC) THEN REWRITE_TAC[oriented_simplex; RESTRICTION; BASIS_IN_STANDARD_SIMPLEX] THEN REWRITE_TAC[MESON[REAL_MUL_RZERO; REAL_MUL_RID] `a * (if p then &1 else &0) = if p then a else &0`] THEN SIMP_TAC[SUM_DELTA; IN_NUMSEG; LE_0] THEN SIMP_TAC[standard_simplex; IN_ELIM_THM; GSYM NOT_LE] THEN REWRITE_TAC[real_div; REAL_MUL_LZERO; SUM_CONST_NUMSEG] THEN REWRITE_TAC[SUB_0; GSYM REAL_OF_NUM_ADD; GSYM REAL_OF_NUM_SUC] THEN COND_CASES_TAC THENL [REWRITE_TAC[GSYM SUM_RMUL] THEN ABS_TAC THEN MATCH_MP_TAC SUM_EQ_NUMSEG THEN SIMP_TAC[REAL_MUL_LID]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (TAUT `~p ==> p ==> q`)) THEN CONJ_TAC THENL [ALL_TAC; CONV_TAC REAL_FIELD] THEN GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_MUL_LZERO; REAL_POS; REAL_MUL_LID] THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_LE_INV_EQ; REAL_POS] THEN MATCH_MP_TAC REAL_INV_LE_1 THEN REWRITE_TAC[REAL_OF_NUM_LE] THEN ARITH_TAC]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o MATCH_MP SIMPLICIAL_IMP_SINGULAR_SIMPLEX) THEN DISCH_THEN(MP_TAC o MATCH_MP SINGULAR_SIMPLEX_CHAIN_MAP_I) THEN DISCH_THEN(SUBST1_TAC o SYM) THEN MP_TAC(ISPECL [`SUC p`; `standard_simplex (SUC p)`] SIMPLICIAL_SIMPLEX_I) THEN REWRITE_TAC[SUBSET_REFL] THEN DISCH_THEN(MP_TAC o MATCH_MP SIMPLICIAL_IMP_SINGULAR_SIMPLEX) THEN REWRITE_TAC[GSYM SINGULAR_CHAIN_OF] THEN DISCH_THEN(fun th -> REWRITE_TAC[MATCH_MP CHAIN_BOUNDARY_CHAIN_MAP th]) THEN REWRITE_TAC[SUC_SUB1] THEN FIRST_ASSUM(MP_TAC o SPECL [`standard_simplex (SUC p)`; `chain_boundary (SUC p) (frag_of (RESTRICTION (standard_simplex (SUC p)) I))`]) THEN ANTS_TAC THENL [GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM SUC_SUB1] THEN MATCH_MP_TAC SIMPLICIAL_CHAIN_BOUNDARY THEN REWRITE_TAC[SIMPLICIAL_CHAIN_OF; SIMPLICIAL_SIMPLEX_I; SUBSET_REFL]; DISCH_THEN(SUBST1_TAC o SYM)] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`s:(num->real)->bool`; `chain_map p (f:(num->real)->(num->real)) (chain_boundary (SUC p) (frag_of (RESTRICTION (standard_simplex (SUC p)) I)))`]) THEN ANTS_TAC THENL [MATCH_MP_TAC SIMPLICIAL_CHAIN_CHAIN_MAP THEN EXISTS_TAC `SUC p` THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM SUC_SUB1] THEN MATCH_MP_TAC SIMPLICIAL_CHAIN_BOUNDARY THEN REWRITE_TAC[SIMPLICIAL_CHAIN_OF; SIMPLICIAL_SIMPLEX_I; SUBSET_REFL]; DISCH_THEN(SUBST1_TAC o SYM)] THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC NATURALITY_SINGULAR_SUBDIVISION THEN EXISTS_TAC `subtopology (product_topology (:num) (\i. euclideanreal)) (standard_simplex (SUC p))` THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM SUC_SUB1] THEN MATCH_MP_TAC SINGULAR_CHAIN_BOUNDARY THEN REWRITE_TAC[SINGULAR_CHAIN_OF] THEN MATCH_MP_TAC SIMPLICIAL_IMP_SINGULAR_SIMPLEX THEN REWRITE_TAC[SIMPLICIAL_SIMPLEX_I; SUBSET_REFL]);; let NATURALITY_SIMPLICIAL_SUBDIVISION = prove (`!p q s g c. simplicial_chain(p,standard_simplex q) c /\ simplicial_simplex (q,s) g ==> simplicial_subdivision p (chain_map p g c) = chain_map p g (simplicial_subdivision p c)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(SUBST1_TAC o SYM o MATCH_MP SINGULAR_SUBDIVISION_SIMPLICIAL_SIMPLEX) THEN FIRST_ASSUM(fun th -> REWRITE_TAC[GSYM (MATCH_MP NATURALITY_SINGULAR_SUBDIVISION (MATCH_MP SIMPLICIAL_IMP_SINGULAR_CHAIN th))]) THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC SINGULAR_SUBDIVISION_SIMPLICIAL_SIMPLEX THEN EXISTS_TAC `s:(num->real)->bool` THEN MATCH_MP_TAC SIMPLICIAL_CHAIN_CHAIN_MAP THEN EXISTS_TAC `q:num` THEN ASM_REWRITE_TAC[]);; let CHAIN_BOUNDARY_SINGULAR_SUBDIVISION = prove (`!p s c:((num->real)->A)frag. singular_chain (p,s) c ==> chain_boundary p (singular_subdivision p c) = singular_subdivision (p - 1) (chain_boundary p c)`, REPLICATE_TAC 2 GEN_TAC THEN REWRITE_TAC[singular_chain] THEN MATCH_MP_TAC FRAG_INDUCTION THEN REWRITE_TAC[singular_subdivision; FRAG_EXTEND_0; CHAIN_BOUNDARY_0] THEN REWRITE_TAC[FRAG_EXTEND_SUB; CHAIN_BOUNDARY_SUB; FRAG_EXTEND_OF] THEN REWRITE_TAC[GSYM singular_subdivision] THEN SIMP_TAC[] THEN X_GEN_TAC `f:(num->real)->A` THEN REWRITE_TAC[IN] THEN DISCH_TAC THEN MP_TAC(ISPECL [`p:num`; `subtopology (product_topology (:num) (\i. euclideanreal)) (standard_simplex p)`; `f:(num->real)->A`] CHAIN_BOUNDARY_CHAIN_MAP) THEN SIMP_TAC[SIMPLICIAL_CHAIN_SIMPLICIAL_SUBDIVISION; SIMPLICIAL_IMP_SINGULAR_CHAIN; SIMPLICIAL_CHAIN_OF; SIMPLICIAL_SIMPLEX_I; SUBSET_REFL] THEN DISCH_THEN(K ALL_TAC) THEN MP_TAC(ISPECL [`p:num`; `standard_simplex p:(num->real)->bool`] CHAIN_BOUNDARY_SIMPLICIAL_SUBDIVISION) THEN DISCH_THEN(fun th -> W(MP_TAC o PART_MATCH (lhand o rand) th o rand o lhand o snd)) THEN REWRITE_TAC[SIMPLICIAL_CHAIN_OF; SIMPLICIAL_SIMPLEX_I; SUBSET_REFL] THEN DISCH_THEN SUBST1_TAC THEN MP_TAC(ISPECL [`p - 1`; `standard_simplex p`; `chain_boundary p (frag_of (RESTRICTION (standard_simplex p) I))`] SINGULAR_SUBDIVISION_SIMPLICIAL_SIMPLEX) THEN ANTS_TAC THENL [MATCH_MP_TAC SIMPLICIAL_CHAIN_BOUNDARY THEN REWRITE_TAC[SIMPLICIAL_SIMPLEX_I; SIMPLICIAL_CHAIN_OF; SUBSET_REFL]; DISCH_THEN(SUBST1_TAC o SYM)] THEN MP_TAC(ISPECL [`p - 1`; `subtopology (product_topology (:num) (\i. euclideanreal)) (standard_simplex p)`; `f:(num->real)->A`; `chain_boundary p (frag_of (RESTRICTION (standard_simplex p) I))`] NATURALITY_SINGULAR_SUBDIVISION) THEN ANTS_TAC THENL [MATCH_MP_TAC SINGULAR_CHAIN_BOUNDARY THEN REWRITE_TAC[SINGULAR_CHAIN_OF] THEN MATCH_MP_TAC SIMPLICIAL_IMP_SINGULAR_SIMPLEX THEN REWRITE_TAC[SIMPLICIAL_SIMPLEX_I; SUBSET_REFL]; DISCH_THEN(SUBST1_TAC o SYM)] THEN AP_TERM_TAC THEN MP_TAC(ISPECL [`p:num`; `subtopology (product_topology (:num) (\i. euclideanreal)) (standard_simplex p)`; `f:(num->real)->A`; `frag_of (RESTRICTION (standard_simplex p) I)`] CHAIN_BOUNDARY_CHAIN_MAP) THEN REWRITE_TAC[SINGULAR_CHAIN_OF] THEN ANTS_TAC THENL [MATCH_MP_TAC SIMPLICIAL_IMP_SINGULAR_SIMPLEX THEN REWRITE_TAC[SIMPLICIAL_SIMPLEX_I; SUBSET_REFL]; DISCH_THEN(SUBST1_TAC o SYM)] THEN AP_TERM_TAC THEN MATCH_MP_TAC SINGULAR_SIMPLEX_CHAIN_MAP_I THEN ASM_MESON_TAC[]);; let SINGULAR_SUBDIVISION_ZERO = prove (`!s c. singular_chain(0,s) c ==> singular_subdivision 0 c = c`, GEN_TAC THEN REWRITE_TAC[singular_chain] THEN MATCH_MP_TAC FRAG_INDUCTION THEN SIMP_TAC[singular_subdivision; FRAG_EXTEND_0; FRAG_EXTEND_SUB] THEN REWRITE_TAC[simplicial_subdivision; FRAG_EXTEND_OF] THEN REWRITE_TAC[I_THM; CHAIN_MAP_OF; simplex_map] THEN REWRITE_TAC[RESTRICTION_COMPOSE_RIGHT; I_O_ID] THEN REWRITE_TAC[IN; singular_simplex] THEN REWRITE_TAC[EXTENSIONAL; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM; RESTRICTION] THEN ASM_MESON_TAC[]);; let CHAIN_HOMOTOPIC_SIMPLICIAL_SUBDIVISION = prove (`?h. (!p. h p frag_0 = frag_0) /\ (!p c1 c2. h p (frag_sub c1 c2) = frag_sub (h p c1) (h p c2)) /\ (!p q r g c. simplicial_chain (p,standard_simplex q) c /\ simplicial_simplex (q,standard_simplex r) g ==> chain_map (p + 1) g (h p c) = h p (chain_map p g c)) /\ (!p q c. simplicial_chain (p,standard_simplex q) c ==> simplicial_chain (p + 1,standard_simplex q) (h p c)) /\ (!p q c. simplicial_chain (p,standard_simplex q) c ==> frag_add (chain_boundary (p + 1) (h p c)) (h (p - 1) (chain_boundary p c)) = frag_sub (simplicial_subdivision p c) c)`, REPEAT STRIP_TAC THEN (STRIP_ASSUME_TAC o prove_recursive_functions_exist num_RECURSION) `h 0 = (\x. frag_0) /\ !p. h(SUC p) = frag_extend (\f. simplicial_cone (SUC p) (\i. sum (0..SUC p) (\j. simplicial_vertex j f i) / (&(SUC p) + &1)) (frag_sub (frag_sub (simplicial_subdivision (SUC p) (frag_of f)) (frag_of f)) (h p (chain_boundary (SUC p) (frag_of f)))))` THEN EXISTS_TAC `h:num->((num->real)->num->real)frag->((num->real)->num->real)frag` THEN GEN_REWRITE_TAC I [CONJ_ASSOC] THEN CONJ_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC num_INDUCTION THEN ASM_REWRITE_TAC[FRAG_EXTEND_0; FRAG_EXTEND_SUB] THEN CONV_TAC FRAG_MODULE; ALL_TAC] THEN REWRITE_TAC[AND_FORALL_THM] THEN GEN_REWRITE_TAC I [SWAP_FORALL_THM] THEN X_GEN_TAC `s:num` THEN ONCE_REWRITE_TAC[MESON[] `(!r g c. P r c g) <=> (!c r g. P r c g)`] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; AND_FORALL_THM; TAUT `(p ==> q) /\ (p ==> r) <=> p ==> q /\ r`] THEN MATCH_MP_TAC num_INDUCTION THEN CONV_TAC NUM_REDUCE_CONV THEN CONJ_TAC THENL [ASM_REWRITE_TAC[CONJUNCT1 simplicial_subdivision; I_THM; CHAIN_MAP_0] THEN REWRITE_TAC[SIMPLICIAL_CHAIN_0; CHAIN_BOUNDARY_0] THEN REPEAT STRIP_TAC THEN CONV_TAC FRAG_MODULE; X_GEN_TAC `p:num`] THEN GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [RIGHT_IMP_FORALL_THM; TAUT `p ==> q /\ r <=> (p ==> q) /\ (p ==> r)`; FORALL_AND_THM; IMP_IMP] THEN STRIP_TAC THEN SUBGOAL_THEN `!p. (h:num->((num->real)->num->real)frag->((num->real)->num->real)frag) p frag_0 = frag_0` ASSUME_TAC THENL [MATCH_MP_TAC num_INDUCTION THEN ASM_REWRITE_TAC[FRAG_EXTEND_0]; ALL_TAC] THEN SUBGOAL_THEN `!p c1 c2. (h:num->((num->real)->num->real)frag->((num->real)->num->real)frag) p (frag_sub c1 c2) = frag_sub (h p c1) (h p c2)` ASSUME_TAC THENL [MATCH_MP_TAC num_INDUCTION THEN ASM_REWRITE_TAC[FRAG_EXTEND_SUB] THEN CONV_TAC FRAG_MODULE; ALL_TAC] THEN ABBREV_TAC `q = SUC p` THEN REWRITE_TAC[simplicial_chain] THEN MATCH_MP_TAC FRAG_INDUCTION THEN REWRITE_TAC[GSYM simplicial_chain] THEN ASM_REWRITE_TAC[CHAIN_BOUNDARY_0; SIMPLICIAL_SUBDIVISION_0; CHAIN_MAP_0] THEN ASM_SIMP_TAC[CHAIN_BOUNDARY_SUB; SIMPLICIAL_SUBDIVISION_SUB; CHAIN_MAP_SUB] THEN REWRITE_TAC[SIMPLICIAL_CHAIN_0] THEN CONJ_TAC THENL [CONV_TAC FRAG_MODULE; ALL_TAC] THEN CONJ_TAC THENL [X_GEN_TAC `f:(num->real)->(num->real)` THEN REWRITE_TAC[IN] THEN DISCH_TAC THEN UNDISCH_THEN `SUC p = q` (SUBST_ALL_TAC o SYM) THEN ASM_REWRITE_TAC[SUC_SUB1; FRAG_EXTEND_OF]; REWRITE_TAC[FRAG_MODULE `frag_add x y = z <=> x = frag_sub z y`] THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[SIMPLICIAL_CHAIN_SUB] THENL [ASM_MESON_TAC[]; CONV_TAC FRAG_MODULE]] THEN SUBGOAL_THEN `(\i. sum (0..SUC p) (\j. simplicial_vertex j f i) / (&(SUC p) + &1)) IN standard_simplex s` ASSUME_TAC THENL [MP_TAC(ASSUME `simplicial_simplex (SUC p,standard_simplex s) f`) THEN REWRITE_TAC[simplicial_simplex] THEN DISCH_THEN(X_CHOOSE_THEN `m:num->num->real` SUBST_ALL_TAC o CONJUNCT2) THEN ASM_SIMP_TAC[SIMPLICIAL_VERTEX_ORIENTED_SIMPLEX] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SIMPLICIAL_SIMPLEX_ORIENTED_SIMPLEX]) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN DISCH_THEN(MP_TAC o GEN `i:num` o SPEC `(\j. if j = i then &1 else &0):num->real`) THEN REWRITE_TAC[BASIS_IN_STANDARD_SIMPLEX; COND_RAND] THEN SIMP_TAC[REAL_MUL_RZERO; SUM_DELTA; IN_NUMSEG; LE_0; REAL_MUL_RID] THEN ASM_SIMP_TAC[ETA_AX; RESTRICTION; GSYM SUM_LMUL; GSYM SUM_RMUL] THEN SPEC_TAC(`SUC p`,`n:num`) THEN MATCH_MP_TAC num_INDUCTION THEN REWRITE_TAC[SUM_CLAUSES_NUMSEG; LE_0; REAL_ADD_LID; REAL_DIV_1] THEN ASM_SIMP_TAC[ETA_AX; LE_0; GSYM REAL_OF_NUM_SUC; REAL_FIELD `(a + b) / ((&n + &1) + &1) = (&1 - inv(&n + &2)) * ((a / (&n + &1))) + inv(&n + &2) * b`] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CONVEX_STANDARD_SIMPLEX THEN ASM_SIMP_TAC[ETA_AX; LE_REFL; REAL_LE_INV_EQ; REAL_INV_LE_1; REAL_ARITH `&1 <= &n + &2 /\ &0 <= &n + &2`] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[ARITH_RULE `i <= n ==> i <= SUC n`]; ALL_TAC] THEN REPEAT CONJ_TAC THENL [MAP_EVERY X_GEN_TAC [`r:num`; `g:(num->real)->(num->real)`] THEN DISCH_TAC THEN MP_TAC(ISPECL [`standard_simplex r`; `SUC p`; `s:num`; `s:num`] CHAIN_MAP_SIMPLICIAL_CONE) THEN DISCH_THEN(fun th -> W(MP_TAC o PART_MATCH (lhand o rand) th o lhand o snd)) THEN ASM_REWRITE_TAC[LE_REFL] THEN ANTS_TAC THENL [REPEAT(MATCH_MP_TAC SIMPLICIAL_CHAIN_SUB THEN CONJ_TAC) THEN ASM_SIMP_TAC[SIMPLICIAL_CHAIN_SIMPLICIAL_SUBDIVISION; SIMPLICIAL_CHAIN_OF] THEN REWRITE_TAC[ADD1] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [ARITH_RULE `p = (p + 1) - 1`] THEN MATCH_MP_TAC SIMPLICIAL_CHAIN_BOUNDARY THEN ASM_REWRITE_TAC[GSYM ADD1; SIMPLICIAL_CHAIN_OF]; DISCH_THEN SUBST1_TAC] THEN REWRITE_TAC[CHAIN_MAP_OF; FRAG_EXTEND_OF] THEN BINOP_TAC THENL [MP_TAC(ASSUME `simplicial_simplex (s,standard_simplex r) g`) THEN REWRITE_TAC[simplicial_simplex] THEN DISCH_THEN(X_CHOOSE_THEN `m:num->num->real` SUBST_ALL_TAC o CONJUNCT2) THEN ASM_REWRITE_TAC[oriented_simplex; RESTRICTION] THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `k:num` THEN REWRITE_TAC[real_div; SUM_RMUL; REAL_MUL_ASSOC] THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[GSYM oriented_simplex; GSYM SUM_LMUL] THEN GEN_REWRITE_TAC LAND_CONV [SUM_SWAP_NUMSEG] THEN MATCH_MP_TAC SUM_EQ_NUMSEG THEN X_GEN_TAC `m:num` THEN STRIP_TAC THEN REWRITE_TAC[] THEN MP_TAC(ASSUME `simplicial_simplex (SUC p,standard_simplex s) f`) THEN REWRITE_TAC[simplicial_simplex] THEN DISCH_THEN(X_CHOOSE_THEN `l:num->num->real` SUBST_ALL_TAC o CONJUNCT2) THEN MP_TAC(ISPECL [`SUC p`; `s:num`; `s:num`; `oriented_simplex s m`; `l:num->num->real`; `standard_simplex r`] SIMPLEX_MAP_ORIENTED_SIMPLEX) THEN ASM_REWRITE_TAC[LE_REFL] THEN DISCH_THEN SUBST1_TAC THEN ASM_REWRITE_TAC[SIMPLICIAL_VERTEX_ORIENTED_SIMPLEX] THEN MP_TAC(ASSUME `simplicial_simplex (SUC p,standard_simplex s) (oriented_simplex (SUC p) l)`) THEN REWRITE_TAC[SIMPLICIAL_SIMPLEX_ORIENTED_SIMPLEX; SUBSET] THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN DISCH_THEN(MP_TAC o SPEC `(\i. if i = m then &1 else &0):num->real`) THEN ASM_REWRITE_TAC[BASIS_IN_STANDARD_SIMPLEX; COND_RAND] THEN SIMP_TAC[REAL_MUL_RZERO; SUM_DELTA; REAL_MUL_RID] THEN ASM_REWRITE_TAC[IN_NUMSEG; oriented_simplex; o_THM; RESTRICTION] THEN SIMP_TAC[ETA_AX]; REWRITE_TAC[CHAIN_MAP_SUB] THEN REPEAT BINOP_TAC THEN REWRITE_TAC[CHAIN_MAP_OF] THENL [REWRITE_TAC[GSYM CHAIN_MAP_OF] THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC NATURALITY_SIMPLICIAL_SUBDIVISION THEN REWRITE_TAC[SIMPLICIAL_CHAIN_OF] THEN ASM_MESON_TAC[]; REWRITE_TAC[ADD1] THEN FIRST_X_ASSUM(MP_TAC o SPEC `r:num` o GEN_REWRITE_RULE I [SWAP_FORALL_THM]) THEN DISCH_THEN(fun th -> W(MP_TAC o PART_MATCH (lhand o rand) th o lhand o snd)) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [ARITH_RULE `p = (p + 1) - 1`] THEN MATCH_MP_TAC SIMPLICIAL_CHAIN_BOUNDARY THEN ASM_REWRITE_TAC[GSYM ADD1; SIMPLICIAL_CHAIN_OF]; DISCH_THEN SUBST1_TAC] THEN AP_TERM_TAC THEN REWRITE_TAC[GSYM CHAIN_MAP_OF] THEN CONV_TAC SYM_CONV THEN GEN_REWRITE_TAC (RAND_CONV o RATOR_CONV o LAND_CONV) [ARITH_RULE `p = (p + 1) - 1`] THEN MATCH_MP_TAC CHAIN_BOUNDARY_CHAIN_MAP THEN REWRITE_TAC[GSYM ADD1; SINGULAR_CHAIN_OF] THEN ASM_MESON_TAC[SIMPLICIAL_IMP_SINGULAR_SIMPLEX]]]; MATCH_MP_TAC SIMPLICIAL_CHAIN_SIMPLICIAL_CONE THEN EXISTS_TAC `standard_simplex s` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [REPEAT(MATCH_MP_TAC SIMPLICIAL_CHAIN_SUB THEN CONJ_TAC) THEN ASM_SIMP_TAC[SIMPLICIAL_CHAIN_SIMPLICIAL_SUBDIVISION; SIMPLICIAL_CHAIN_OF] THEN REWRITE_TAC[ADD1] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [ARITH_RULE `p = (p + 1) - 1`] THEN MATCH_MP_TAC SIMPLICIAL_CHAIN_BOUNDARY THEN ASM_REWRITE_TAC[GSYM ADD1; SIMPLICIAL_CHAIN_OF]; REPEAT STRIP_TAC THEN MATCH_MP_TAC CONVEX_STANDARD_SIMPLEX THEN ASM_REWRITE_TAC[]]; MP_TAC(ISPECL [`SUC p`; `standard_simplex s`] CHAIN_BOUNDARY_SIMPLICIAL_CONE) THEN DISCH_THEN(fun th -> W(MP_TAC o PART_MATCH (lhand o rand) th o lhand o lhand o snd)) THEN ASM_REWRITE_TAC[NOT_SUC] THEN ANTS_TAC THENL [REPEAT(MATCH_MP_TAC SIMPLICIAL_CHAIN_SUB THEN CONJ_TAC) THEN ASM_SIMP_TAC[SIMPLICIAL_CHAIN_SIMPLICIAL_SUBDIVISION; SIMPLICIAL_CHAIN_OF] THEN REWRITE_TAC[ADD1] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [ARITH_RULE `p = (p + 1) - 1`] THEN MATCH_MP_TAC SIMPLICIAL_CHAIN_BOUNDARY THEN ASM_REWRITE_TAC[GSYM ADD1; SIMPLICIAL_CHAIN_OF]; DISCH_THEN SUBST1_TAC] THEN MATCH_MP_TAC(FRAG_MODULE `z = frag_0 ==> frag_add (frag_sub (frag_sub (frag_sub ff f) t) z) t = frag_sub ff f`) THEN REWRITE_TAC[simplicial_cone] THEN MATCH_MP_TAC(MESON[FRAG_EXTEND_0] `z = frag_0 ==> frag_extend f z = frag_0`) THEN REWRITE_TAC[CHAIN_BOUNDARY_SUB] THEN FIRST_ASSUM(MP_TAC o SPEC `chain_boundary (SUC p) (frag_of(f:(num->real)->(num->real)))`) THEN ANTS_TAC THENL [GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM SUC_SUB1] THEN MATCH_MP_TAC SIMPLICIAL_CHAIN_BOUNDARY THEN ASM_REWRITE_TAC[SIMPLICIAL_CHAIN_OF]; REWRITE_TAC[ADD1]] THEN DISCH_THEN(SUBST1_TAC o MATCH_MP (FRAG_MODULE `frag_add w z = x ==> w = frag_sub x z`)) THEN REWRITE_TAC[CHAIN_BOUNDARY_SUB] THEN MP_TAC(ISPECL [`p + 1`; `subtopology (product_topology (:num) (\i. euclideanreal)) (standard_simplex s)`; `frag_of(f:(num->real)->(num->real))`] CHAIN_BOUNDARY_BOUNDARY) THEN ASM_SIMP_TAC[SIMPLICIAL_IMP_SINGULAR_CHAIN; SIMPLICIAL_CHAIN_OF; GSYM ADD1; SUC_SUB1] THEN DISCH_THEN SUBST1_TAC THEN MP_TAC(ISPECL [`SUC p`; `standard_simplex s`; `frag_of(f:(num->real)->(num->real))`] CHAIN_BOUNDARY_SIMPLICIAL_SUBDIVISION) THEN ASM_REWRITE_TAC[SIMPLICIAL_CHAIN_OF; SUC_SUB1] THEN CONV_TAC FRAG_MODULE]);; let CHAIN_HOMOTOPIC_SINGULAR_SUBDIVISION = prove (`?h:num->((num->real)->A)frag->((num->real)->A)frag. (!p. h p frag_0 = frag_0) /\ (!p c1 c2. h p (frag_sub c1 c2) = frag_sub (h p c1) (h p c2)) /\ (!p top c. singular_chain (p,top) c ==> singular_chain (p + 1,top) (h p c)) /\ (!p top c. singular_chain (p,top) c ==> frag_add (chain_boundary (p + 1) (h p c)) (h (p - 1) (chain_boundary p c)) = frag_sub (singular_subdivision p c) c)`, X_CHOOSE_THEN `h:num->((num->real)->(num->real))frag->((num->real)->(num->real))frag` STRIP_ASSUME_TAC CHAIN_HOMOTOPIC_SIMPLICIAL_SUBDIVISION THEN ABBREV_TAC `k:num->((num->real)->A)frag->((num->real)->A)frag = \p. frag_extend (\f. chain_map (p + 1) (f:(num->real)->A) (h p (frag_of(RESTRICTION (standard_simplex p) (I:(num->real)->(num->real))))))` THEN EXISTS_TAC `k:num->((num->real)->A)frag->((num->real)->A)frag` THEN SUBGOAL_THEN `!p. (k:num->((num->real)->A)frag->((num->real)->A)frag) p frag_0 = frag_0` ASSUME_TAC THENL [EXPAND_TAC "k" THEN REWRITE_TAC[FRAG_EXTEND_0]; ALL_TAC] THEN SUBGOAL_THEN `!p c1 c2. (k:num->((num->real)->A)frag->((num->real)->A)frag) p (frag_sub c1 c2) = frag_sub (k p c1) (k p c2)` ASSUME_TAC THENL [EXPAND_TAC "k" THEN REWRITE_TAC[FRAG_EXTEND_SUB]; ASM_REWRITE_TAC[]] THEN REWRITE_TAC[AND_FORALL_THM; TAUT `(p ==> q) /\ (p ==> r) <=> p ==> q /\ r`] THEN MAP_EVERY X_GEN_TAC [`p:num`; `s:A topology`] THEN REWRITE_TAC[singular_chain] THEN MATCH_MP_TAC FRAG_INDUCTION THEN ASM_REWRITE_TAC[GSYM singular_chain] THEN ASM_SIMP_TAC[CHAIN_BOUNDARY_0; SINGULAR_CHAIN_0; CHAIN_BOUNDARY_SUB; singular_subdivision; FRAG_EXTEND_0; FRAG_EXTEND_SUB] THEN REWRITE_TAC[GSYM singular_subdivision] THEN SIMP_TAC[SINGULAR_CHAIN_SUB] THEN CONJ_TAC THENL [CONV_TAC FRAG_MODULE; ALL_TAC] THEN CONJ_TAC THENL [X_GEN_TAC `f:(num->real)->A` THEN REWRITE_TAC[IN] THEN DISCH_TAC; REWRITE_TAC[FRAG_MODULE `frag_add x y = z <=> x = frag_sub z y`] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONV_TAC FRAG_MODULE] THEN EXPAND_TAC "k" THEN REWRITE_TAC[FRAG_EXTEND_OF; ADD_SUB] THEN CONJ_TAC THENL [MATCH_MP_TAC SINGULAR_CHAIN_CHAIN_MAP THEN EXISTS_TAC `subtopology (product_topology (:num) (\i. euclideanreal)) (standard_simplex p)` THEN RULE_ASSUM_TAC(REWRITE_RULE[singular_simplex]) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SIMPLICIAL_IMP_SINGULAR_CHAIN THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[SIMPLICIAL_CHAIN_OF; SIMPLICIAL_SIMPLEX_I; SUBSET_REFL]; ALL_TAC] THEN MP_TAC(ISPECL [`p + 1`; `subtopology (product_topology (:num) (\i. euclideanreal)) (standard_simplex p)`; `f:(num->real)->A`] CHAIN_BOUNDARY_CHAIN_MAP) THEN DISCH_THEN(fun th -> W(MP_TAC o PART_MATCH (lhand o rand) th o lhand o lhand o snd)) THEN ANTS_TAC THENL [MATCH_MP_TAC SIMPLICIAL_IMP_SINGULAR_CHAIN THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[SIMPLICIAL_CHAIN_OF; SIMPLICIAL_SIMPLEX_I; SUBSET_REFL]; DISCH_THEN SUBST1_TAC] THEN REWRITE_TAC[ADD_SUB] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE (funpow 3 BINDER_CONV o RAND_CONV) [FRAG_MODULE `frag_add x y = z <=> x = frag_sub z y`]) THEN DISCH_THEN(MP_TAC o SPECL [`p:num`; `p:num`; `frag_of(RESTRICTION (standard_simplex p) I)`]) THEN REWRITE_TAC[SIMPLICIAL_CHAIN_OF; SIMPLICIAL_SIMPLEX_I; SUBSET_REFL] THEN DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[FRAG_MODULE `frag_add x y = z <=> x = frag_sub z y`] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[CHAIN_MAP_SUB; CHAIN_MAP_OF] THEN MATCH_MP_TAC(FRAG_MODULE `b = y /\ frag_sub a c = frag_sub x z ==> frag_sub (frag_sub a b) c = frag_sub (frag_sub x y) z`) THEN CONJ_TAC THENL [REWRITE_TAC[GSYM CHAIN_MAP_OF] THEN MATCH_MP_TAC SINGULAR_SIMPLEX_CHAIN_MAP_I THEN ASM_MESON_TAC[]; REWRITE_TAC[GSYM CHAIN_MAP_SUB; singular_subdivision]] THEN REWRITE_TAC[FRAG_EXTEND_OF; CHAIN_MAP_SUB] THEN AP_TERM_TAC THEN REWRITE_TAC[chain_boundary] THEN ASM_REWRITE_TAC[CHAIN_MAP_0; FRAG_EXTEND_0] THEN ASM_CASES_TAC `p = 0` THEN ASM_REWRITE_TAC[FRAG_EXTEND_0; CHAIN_MAP_0] THEN ASM_SIMP_TAC[SUB_ADD; LE_1; FRAG_EXTEND_OF] THEN SIMP_TAC[FRAG_EXTEND_SUM; FINITE_NUMSEG] THEN REWRITE_TAC[o_DEF; FRAG_EXTEND_CMUL; FRAG_EXTEND_OF] THEN TRANS_TAC EQ_TRANS `chain_map p (f:(num->real)->A) (iterate frag_add (0..p) (\k. frag_cmul (-- &1 pow k) (chain_map p (singular_face p k I) ((h:num->((num->real)->(num->real))frag ->((num->real)->(num->real))frag) (p - 1) (frag_of (RESTRICTION (standard_simplex (p - 1)) I))))))` THEN CONJ_TAC THENL [ASM_REWRITE_TAC[] THEN AP_TERM_TAC THEN SUBGOAL_THEN `!k z. FINITE k ==> (h:num->((num->real)->(num->real))frag ->((num->real)->(num->real))frag) (p - 1) (iterate frag_add (k:num->bool) z) = iterate frag_add k (h (p - 1) o z)` (fun th -> SIMP_TAC[th; FINITE_NUMSEG]) THENL [GEN_REWRITE_TAC I [SWAP_FORALL_THM] THEN GEN_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN SIMP_TAC[MATCH_MP ITERATE_CLAUSES MONOIDAL_FRAG_ADD] THEN ASM_REWRITE_TAC[NEUTRAL_FRAG_ADD; o_DEF] THEN ASM_SIMP_TAC[FRAG_MODULE `frag_add x y = frag_sub x (frag_sub frag_0 y)`]; ALL_TAC] THEN MATCH_MP_TAC(MATCH_MP ITERATE_EQ MONOIDAL_FRAG_ADD) THEN X_GEN_TAC `k:num` THEN REWRITE_TAC[IN_NUMSEG; o_DEF] THEN STRIP_TAC THEN SUBGOAL_THEN `!k c. (h:num->((num->real)->(num->real))frag ->((num->real)->(num->real))frag) (p - 1) (frag_cmul (-- &1 pow k) c) = frag_cmul (-- &1 pow k) (h (p - 1) c)` (fun th -> REWRITE_TAC[th]) THENL [REPEAT GEN_TAC THEN REWRITE_TAC[INT_POW_NEG; INT_POW_ONE] THEN COND_CASES_TAC THEN REWRITE_TAC[FRAG_MODULE `frag_cmul (&1) c = c`] THEN ASM_REWRITE_TAC[FRAG_MODULE `frag_cmul (-- &1) c = frag_sub frag_0 c`]; AP_TERM_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`p - 1`; `p - 1`; `p:num`; `singular_face p k (RESTRICTION (standard_simplex p) I)`; `frag_of(RESTRICTION (standard_simplex (p - 1)) (I:(num->real)->(num->real)))`]) THEN ASM_REWRITE_TAC[SIMPLICIAL_CHAIN_OF; SIMPLICIAL_SIMPLEX_I] THEN REWRITE_TAC[SUBSET_REFL] THEN ANTS_TAC THENL [MATCH_MP_TAC SIMPLICIAL_SIMPLEX_SINGULAR_FACE THEN ASM_SIMP_TAC[LE_1; SIMPLICIAL_SIMPLEX_I; SUBSET_REFL]; MATCH_MP_TAC(MESON[] `x = y' /\ y = x' ==> x = y ==> x' = y'`) THEN REWRITE_TAC[CHAIN_MAP_OF] THEN CONJ_TAC THENL [ASM_SIMP_TAC[LE_1; SUB_ADD]; ASM_SIMP_TAC[SUB_ADD; LE_1] THEN AP_TERM_TAC THEN REWRITE_TAC[CHAIN_MAP_OF] THEN AP_TERM_TAC THEN REWRITE_TAC[singular_face; simplex_map; RESTRICTION; FUN_EQ_THM; o_THM; I_THM] THEN ASM_SIMP_TAC[LE_1; SIMPLICIAL_FACE_IN_STANDARD_SIMPLEX]]]; SIMP_TAC[CHAIN_MAP_SUM; FINITE_NUMSEG] THEN MATCH_MP_TAC(MATCH_MP ITERATE_EQ MONOIDAL_FRAG_ADD) THEN X_GEN_TAC `k:num` THEN REWRITE_TAC[IN_NUMSEG] THEN STRIP_TAC THEN REWRITE_TAC[o_DEF; CHAIN_MAP_CMUL] THEN AP_TERM_TAC THEN REWRITE_TAC[REWRITE_RULE[FUN_EQ_THM; o_THM] (GSYM CHAIN_MAP_COMPOSE)] THEN ASM_REWRITE_TAC[singular_face; I_O_ID]] THEN (SUBGOAL_THEN `simplicial_chain((p-1)+1,standard_simplex(p - 1)) ((h:num->((num->real)->(num->real))frag ->((num->real)->(num->real))frag) (p - 1) (frag_of (RESTRICTION (standard_simplex (p - 1)) I)))` MP_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[SIMPLICIAL_CHAIN_OF; SIMPLICIAL_SIMPLEX_I; SUBSET_REFL]; ASM_SIMP_TAC[SUB_ADD; LE_1]] THEN SPEC_TAC(`(h:num->((num->real)->(num->real))frag ->((num->real)->(num->real))frag) (p - 1) (frag_of (RESTRICTION (standard_simplex (p - 1)) I))`, `c:((num->real)->(num->real))frag`) THEN REWRITE_TAC[simplicial_chain] THEN MATCH_MP_TAC FRAG_INDUCTION THEN SIMP_TAC[CHAIN_MAP_0; CHAIN_MAP_SUB] THEN X_GEN_TAC `f:(num->real)->(num->real)` THEN REWRITE_TAC[IN] THEN REWRITE_TAC[CHAIN_MAP_OF; singular_simplex; simplicial_simplex] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; CONTINUOUS_MAP_IN_SUBTOPOLOGY; TOPSPACE_PRODUCT_TOPOLOGY; TOPSPACE_SUBTOPOLOGY; INTER_UNIV; TOPSPACE_EUCLIDEANREAL; CARTESIAN_PRODUCT_UNIV; o_DEF] THEN STRIP_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM; simplex_map] THEN ASM_SIMP_TAC[singular_face; I_THM; o_THM; RESTRICTION; SIMPLICIAL_FACE_IN_STANDARD_SIMPLEX; LE_1]));; let HOMOLOGOUS_REL_SINGULAR_SUBDIVISION = prove (`!p s t c:((num->real)->A)frag. singular_relcycle (p,s,t) c ==> homologous_rel (p,s,t) (singular_subdivision p c) c`, REPLICATE_TAC 3 GEN_TAC THEN ASM_CASES_TAC `p = 0` THENL [ASM_MESON_TAC[singular_relcycle; SINGULAR_SUBDIVISION_ZERO; HOMOLOGOUS_REL_REFL]; ALL_TAC] THEN MATCH_MP_TAC CHAIN_HOMOTOPIC_IMP_HOMOLOGOUS_REL THEN X_CHOOSE_THEN `k:num->((num->real)->A)frag->((num->real)->A)frag` STRIP_ASSUME_TAC CHAIN_HOMOTOPIC_SINGULAR_SUBDIVISION THEN EXISTS_TAC `(k:num->((num->real)->A)frag->((num->real)->A)frag) p` THEN EXISTS_TAC `(k:num->((num->real)->A)frag->((num->real)->A)frag) (p - 1)` THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[SUB_ADD; LE_1]);; (* ------------------------------------------------------------------------- *) (* Excision argument that we keep doing singular subdivision *) (* ------------------------------------------------------------------------- *) let ITERATED_SINGULAR_SUBDIVISION = prove (`!p s n c. singular_chain(p,s) c ==> ITER n (singular_subdivision p) c = frag_extend (\f:(num->real)->A. chain_map p f (ITER n (simplicial_subdivision p) (frag_of(RESTRICTION (standard_simplex p) I)))) c`, GEN_TAC THEN GEN_TAC THEN CONV_TAC(ONCE_DEPTH_CONV SYM_CONV) THEN MATCH_MP_TAC num_INDUCTION THEN REWRITE_TAC[ITER] THEN CONJ_TAC THENL [ALL_TAC; X_GEN_TAC `n:num` THEN DISCH_TAC] THEN REWRITE_TAC[singular_chain] THEN MATCH_MP_TAC FRAG_INDUCTION THEN REWRITE_TAC[GSYM singular_chain] THENL [SIMP_TAC[FRAG_EXTEND_0; FRAG_EXTEND_SUB; FRAG_EXTEND_OF] THEN REWRITE_TAC[IN; SINGULAR_SIMPLEX_CHAIN_MAP_I]; ALL_TAC] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[singular_subdivision; FRAG_EXTEND_0] THEN MATCH_MP_TAC(MESON[FRAG_EXTEND_0] `c = frag_0 ==> frag_0 = frag_extend f c`) THEN SPEC_TAC(`n:num`,`n:num`) THEN INDUCT_TAC THEN ASM_REWRITE_TAC[ITER; FRAG_EXTEND_0]; ALL_TAC; REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[FRAG_EXTEND_SUB] THEN SPEC_TAC(`n:num`,`n:num`) THEN MATCH_MP_TAC num_INDUCTION THEN SIMP_TAC[ITER; singular_subdivision; GSYM FRAG_EXTEND_SUB]] THEN X_GEN_TAC `f:(num->real)->A` THEN GEN_REWRITE_TAC LAND_CONV [IN] THEN DISCH_TAC THEN REWRITE_TAC[FRAG_EXTEND_OF] THEN FIRST_X_ASSUM(MP_TAC o SPEC `frag_of(f:(num->real)->A)`) THEN ASM_REWRITE_TAC[SINGULAR_CHAIN_OF; FRAG_EXTEND_OF] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN MP_TAC(ISPECL [`p:num`; `subtopology (product_topology (:num) (\i. euclideanreal)) (standard_simplex p)`; `f:(num->real)->A`] NATURALITY_SINGULAR_SUBDIVISION) THEN DISCH_THEN(fun th -> W(MP_TAC o PART_MATCH (lhand o rand) th o rand o snd)) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [MATCH_MP_TAC SIMPLICIAL_IMP_SINGULAR_CHAIN THEN SPEC_TAC(`n:num`,`n:num`) THEN INDUCT_TAC THEN REWRITE_TAC[ITER] THEN ASM_SIMP_TAC[SIMPLICIAL_CHAIN_OF; SIMPLICIAL_SIMPLEX_I; SUBSET_REFL] THEN MATCH_MP_TAC SIMPLICIAL_CHAIN_SIMPLICIAL_SUBDIVISION THEN ASM_REWRITE_TAC[]; DISCH_THEN SUBST1_TAC] THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC SINGULAR_SUBDIVISION_SIMPLICIAL_SIMPLEX THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `standard_simplex p` THEN SPEC_TAC(`n:num`,`n:num`) THEN INDUCT_TAC THEN SIMP_TAC[ITER; SIMPLICIAL_CHAIN_OF; SIMPLICIAL_SIMPLEX_I; SUBSET_REFL] THEN MATCH_MP_TAC SIMPLICIAL_CHAIN_SIMPLICIAL_SUBDIVISION THEN ASM_REWRITE_TAC[]);; let CHAIN_HOMOTOPIC_ITERATED_SINGULAR_SUBDIVISION = prove (`!n. ?h:num->((num->real)->A)frag->((num->real)->A)frag. (!p. h p frag_0 = frag_0) /\ (!p c1 c2. h p (frag_sub c1 c2) = frag_sub (h p c1) (h p c2)) /\ (!p top c. singular_chain (p,top) c ==> singular_chain (p + 1,top) (h p c)) /\ (!p top c. singular_chain (p,top) c ==> frag_add (chain_boundary (p + 1) (h p c)) (h (p - 1) (chain_boundary p c)) = frag_sub (ITER n (singular_subdivision p) c) c)`, INDUCT_TAC THEN REWRITE_TAC[ITER] THENL [EXISTS_TAC `(\p x. frag_0):num->((num->real)->A)frag->((num->real)->A)frag` THEN REWRITE_TAC[SINGULAR_CHAIN_0; CHAIN_BOUNDARY_0] THEN REPEAT STRIP_TAC THEN CONV_TAC FRAG_MODULE; ALL_TAC] THEN FIRST_X_ASSUM(X_CHOOSE_THEN `k:num->((num->real)->A)frag->((num->real)->A)frag` STRIP_ASSUME_TAC) THEN X_CHOOSE_THEN `h:num->((num->real)->A)frag->((num->real)->A)frag` STRIP_ASSUME_TAC CHAIN_HOMOTOPIC_SINGULAR_SUBDIVISION THEN EXISTS_TAC `(\p c. frag_add (singular_subdivision (p + 1) (k p c)) (h p c)) :num->((num->real)->A)frag->((num->real)->A)frag` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [REWRITE_TAC[SINGULAR_SUBDIVISION_0] THEN REPEAT STRIP_TAC THEN CONV_TAC FRAG_MODULE; ALL_TAC] THEN CONJ_TAC THENL [REWRITE_TAC[SINGULAR_SUBDIVISION_SUB] THEN REPEAT STRIP_TAC THEN CONV_TAC FRAG_MODULE; ALL_TAC] THEN REWRITE_TAC[AND_FORALL_THM] THEN MAP_EVERY X_GEN_TAC [`p:num`; `top:A topology`; `c:((num->real)->A)frag`] THEN ASM_SIMP_TAC[SINGULAR_CHAIN_ADD; SINGULAR_CHAIN_SINGULAR_SUBDIVISION] THEN DISCH_TAC THEN ONCE_REWRITE_TAC[FRAG_MODULE `frag_sub (singular_subdivision p (ITER n (singular_subdivision p) c)) c = frag_add (frag_sub (singular_subdivision p (ITER n (singular_subdivision p) c)) (singular_subdivision p c)) (frag_sub (singular_subdivision p c) c)`] THEN FIRST_X_ASSUM(fun th -> W(MP_TAC o PART_MATCH (rand o rand) th o rand o rand o snd)) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[GSYM SINGULAR_SUBDIVISION_SUB] THEN FIRST_X_ASSUM(fun th -> W(MP_TAC o PART_MATCH (rand o rand) th o rand o lhand o rand o snd)) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN SUBGOAL_THEN `singular_subdivision (p - 1 + 1) (k (p - 1) (chain_boundary p c)) = singular_subdivision p ((k:num->((num->real)->A)frag->((num->real)->A)frag) (p - 1) (chain_boundary p c))` SUBST1_TAC THENL [ASM_CASES_TAC `p = 0` THEN ASM_SIMP_TAC[SUB_ADD; LE_1] THEN ASM_REWRITE_TAC[chain_boundary; SINGULAR_SUBDIVISION_0]; ALL_TAC] THEN REWRITE_TAC[CHAIN_BOUNDARY_ADD; SINGULAR_SUBDIVISION_ADD] THEN MP_TAC(ISPECL [`p + 1`; `top:A topology`] CHAIN_BOUNDARY_SINGULAR_SUBDIVISION) THEN ASM_SIMP_TAC[] THEN DISCH_THEN(K ALL_TAC) THEN ASM_REWRITE_TAC[ADD_SUB] THEN REPEAT STRIP_TAC THEN CONV_TAC FRAG_MODULE);; let SUFFICIENT_ITERATED_SINGULAR_SUBDIVISION_EXISTS = prove (`!p top u c:((num->real)->A)frag. (!v. v IN u ==> open_in top v) /\ topspace top SUBSET UNIONS u /\ singular_chain (p,top) c ==> ?n. !m f. n <= m /\ f IN frag_support (ITER m (singular_subdivision p) c) ==> ?v. v IN u /\ IMAGE f (standard_simplex p) SUBSET v`, let llemma = prove (`!p c. standard_simplex p SUBSET UNIONS c /\ (!u. u IN c ==> open_in (product_topology (:num) (\i. euclideanreal)) u) ==> ?d. &0 < d /\ !k. k SUBSET standard_simplex p /\ (!x y i. x IN k /\ y IN k /\ i <= p ==> abs (x i - y i) <= d) ==> ?u. u IN c /\ k SUBSET u`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `!x. x IN standard_simplex p ==> ?e u. &0 < e /\ u IN c /\ x IN u /\ !y. (!i. i <= p ==> abs(y i - x i) <= &2 * e) /\ (!i. p < i ==> y i = &0) ==> y IN u` MP_TAC THENL [REPEAT STRIP_TAC THEN GEN_REWRITE_TAC I [SWAP_EXISTS_THM] THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:num->real` o REWRITE_RULE[SUBSET]) THEN ASM_REWRITE_TAC[IN_UNIONS] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:(num->real)->bool` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `u:(num->real)->bool`) THEN ASM_REWRITE_TAC[OPEN_IN_PRODUCT_TOPOLOGY_ALT] THEN DISCH_THEN(MP_TAC o SPEC `x:num->real`) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN REWRITE_TAC[TOPSPACE_EUCLIDEANREAL; cartesian_product; IN_UNIV] THEN REWRITE_TAC[EXTENSIONAL_UNIV; IN_ELIM_THM] THEN X_GEN_TAC `v:num->(real->bool)` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE BINDER_CONV [GSYM REAL_OPEN_IN]) THEN REWRITE_TAC[real_open] THEN DISCH_THEN(MP_TAC o GEN `i:num` o SPECL[`i:num`; `(x:num->real) i`]) THEN ASM_REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM; FORALL_AND_THM] THEN X_GEN_TAC `d:num->real` THEN STRIP_TAC THEN EXISTS_TAC `inf ((&1) INSERT IMAGE (d:num->real) {i | ~(v i = (:real))}) / &3` THEN REWRITE_TAC[REAL_ARITH `&0 < x / &3 <=> &0 < x`] THEN REWRITE_TAC[REAL_ARITH `x <= &2 * y / &3 <=> &3 / &2 * x <= y`] THEN ASM_SIMP_TAC[REAL_LT_INF_FINITE; REAL_LE_INF_FINITE; NOT_INSERT_EMPTY; FINITE_IMAGE; FINITE_INSERT] THEN ASM_SIMP_TAC[FORALL_IN_IMAGE; FORALL_IN_INSERT; REAL_LT_01] THEN X_GEN_TAC `y:num->real` THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN MP_TAC(ASSUME `x IN standard_simplex p`) THEN REWRITE_TAC[IN_ELIM_THM; standard_simplex] THEN STRIP_TAC THEN X_GEN_TAC `i:num` THEN ASM_CASES_TAC `p:num < i` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[NOT_LT]) THEN ASM_CASES_TAC `(v:num->real->bool) i = UNIV` THEN ASM_REWRITE_TAC[IN_UNIV] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN MATCH_MP_TAC(REAL_ARITH `&3 / &2 * x <= y /\ &0 < y ==> x < y`) THEN ASM_SIMP_TAC[]; GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM]] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`e:(num->real)->real`; `v:(num->real)->(num->real)->bool`] THEN DISCH_TAC THEN MP_TAC(SPEC `p:num` COMPACT_IN_STANDARD_SIMPLEX) THEN REWRITE_TAC[compact_in] THEN DISCH_THEN(MP_TAC o SPEC `{cartesian_product (:num) (\i. if i <= p then real_interval(x i - e x,x i + e x) else (:real)) | x | x IN standard_simplex p}` o CONJUNCT2) THEN REWRITE_TAC[SIMPLE_IMAGE; FORALL_IN_IMAGE; EXISTS_FINITE_SUBSET_IMAGE] THEN ANTS_TAC THENL [REWRITE_TAC[SUBSET; UNIONS_IMAGE; IN_ELIM_THM] THEN CONJ_TAC THEN X_GEN_TAC `x:num->real` THEN DISCH_TAC THENL [REWRITE_TAC[OPEN_IN_CARTESIAN_PRODUCT_GEN] THEN DISJ2_TAC THEN REWRITE_TAC[TOPSPACE_EUCLIDEANREAL; IN_UNIV] THEN CONJ_TAC THENL [MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `{i:num | i <= p}` THEN REWRITE_TAC[FINITE_NUMSEG_LE; SUBSET; IN_ELIM_THM] THEN MESON_TAC[]; GEN_TAC THEN REWRITE_TAC[GSYM REAL_OPEN_IN] THEN MESON_TAC[REAL_OPEN_REAL_INTERVAL; REAL_OPEN_UNIV]]; EXISTS_TAC `x:num->real` THEN ASM_REWRITE_TAC[cartesian_product; IN_ELIM_THM; EXTENSIONAL_UNIV] THEN GEN_TAC THEN COND_CASES_TAC THEN REWRITE_TAC[IN_UNIV] THEN ASM_SIMP_TAC[IN_REAL_INTERVAL; REAL_ARITH `x - e < x /\ x < x + e <=> &0 < e`]]; DISCH_THEN(X_CHOOSE_THEN `s:(num->real)->bool` MP_TAC)] THEN ASM_CASES_TAC `s:(num->real)->bool = {}` THEN ASM_REWRITE_TAC[IMAGE_CLAUSES; UNIONS_0; SUBSET_EMPTY] THEN REWRITE_TAC[NONEMPTY_STANDARD_SIMPLEX] THEN STRIP_TAC THEN EXISTS_TAC `inf(IMAGE (e:(num->real)->real) s)` THEN ASM_SIMP_TAC[REAL_LT_INF_FINITE; REAL_LE_INF_FINITE; FINITE_IMAGE; IMAGE_EQ_EMPTY] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN X_GEN_TAC `k:(num->real)->bool` THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN STRIP_TAC THEN ASM_CASES_TAC `k:(num->real)->bool = {}` THENL [ASM_REWRITE_TAC[EMPTY_SUBSET] THEN ASM_MESON_TAC[MEMBER_NOT_EMPTY; NONEMPTY_STANDARD_SIMPLEX]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY])] THEN DISCH_THEN(X_CHOOSE_THEN `x:num->real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `x IN standard_simplex p` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [UNIONS_IMAGE]) THEN GEN_REWRITE_TAC LAND_CONV [SUBSET] THEN DISCH_THEN(MP_TAC o SPEC `x:num->real`) THEN ASM_REWRITE_TAC[IN_ELIM_THM; cartesian_product; EXTENSIONAL_UNIV] THEN DISCH_THEN(X_CHOOSE_THEN `a:num->real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN REWRITE_TAC[IN_UNIV] THEN ONCE_REWRITE_TAC[COND_RAND] THEN REWRITE_TAC[COND_EXPAND; FORALL_AND_THM; IN_UNIV] THEN REWRITE_TAC[TAUT `~p \/ q <=> p ==> q`; IN_REAL_INTERVAL] THEN REWRITE_TAC[REAL_ARITH `a - e < x /\ x < a + e <=> abs(x - a) < e`] THEN DISCH_TAC THEN EXISTS_TAC `(v:(num->real)->(num->real)->bool) a` THEN SUBGOAL_THEN `a IN standard_simplex p` (fun th -> ASSUME_TAC th THEN FIRST_X_ASSUM(MP_TAC o C MATCH_MP th)) THENL [ASM SET_TAC[]; STRIP_TAC THEN ASM_REWRITE_TAC[]] THEN REWRITE_TAC[SUBSET] THEN X_GEN_TAC `y:num->real` THEN DISCH_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN SUBGOAL_THEN `y IN standard_simplex p` (fun th -> ASSUME_TAC th THEN MP_TAC th) THENL [ASM SET_TAC[]; SIMP_TAC[standard_simplex; IN_ELIM_THM]] THEN DISCH_THEN(K ALL_TAC) THEN X_GEN_TAC `i:num` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:num->real`; `y:num->real`; `i:num`]) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `a:num->real`) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC) in REPEAT GEN_TAC THEN ASM_CASES_TAC `c:((num->real)->A)frag = frag_0` THENL [STRIP_TAC THEN EXISTS_TAC `0` THEN SUBGOAL_THEN `!k. ITER k (singular_subdivision p) frag_0:((num->real)->A)frag = frag_0` (fun th -> ASM_REWRITE_TAC[th; FRAG_SUPPORT_0; NOT_IN_EMPTY]) THEN INDUCT_TAC THEN ASM_REWRITE_TAC[ITER; SINGULAR_SUBDIVISION_0]; ALL_TAC] THEN ASM_CASES_TAC `topspace top:A->bool = {}` THENL [ASM_MESON_TAC[SINGULAR_CHAIN_EMPTY]; ALL_TAC] THEN ASM_CASES_TAC `u:(A->bool)->bool = {}` THEN ASM_REWRITE_TAC[UNIONS_0; SUBSET_EMPTY] THEN STRIP_TAC THEN SUBGOAL_THEN `?d. &0 < d /\ !f k. f IN frag_support c /\ k SUBSET standard_simplex p /\ (!x y i. x IN k /\ y IN k /\ i <= p ==> abs(x i - y i) <= d) ==> ?v. v IN u /\ IMAGE (f:(num->real)->A) k SUBSET v` STRIP_ASSUME_TAC THENL [SUBGOAL_THEN `!f:(num->real)->A. f IN frag_support c ==> ?e. &0 < e /\ !k. k SUBSET standard_simplex p /\ (!x y i. x IN k /\ y IN k /\ i <= p ==> abs(x i - y i) <= e) ==> ?v. v IN u /\ IMAGE f k SUBSET v` MP_TAC THENL [REPEAT STRIP_TAC THEN SUBGOAL_THEN `!v. v IN u ==> ?t. open_in (product_topology (:num) (\i. euclideanreal)) t /\ {x | x IN standard_simplex p /\ (f:(num->real)->A) x IN v} = t INTER standard_simplex p` MP_TAC THENL [REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM OPEN_IN_SUBTOPOLOGY] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [singular_chain]) THEN REWRITE_TAC[SUBSET] THEN DISCH_THEN(MP_TAC o SPEC `f:(num->real)->A`) THEN ASM_REWRITE_TAC[singular_simplex; IN; continuous_map] THEN DISCH_THEN(MP_TAC o SPEC `v:A->bool` o CONJUNCT2 o CONJUNCT1) THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; TOPSPACE_PRODUCT_TOPOLOGY] THEN REWRITE_TAC[o_DEF; TOPSPACE_EUCLIDEANREAL; CARTESIAN_PRODUCT_UNIV] THEN ASM_SIMP_TAC[INTER_UNIV]; GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [RIGHT_IMP_EXISTS_THM]] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `g:(A->bool)->((num->real)->bool)` THEN DISCH_TAC THEN MP_TAC(SPECL [`p:num`; `IMAGE (g:(A->bool)->((num->real)->bool)) u`] llemma) THEN REWRITE_TAC[FORALL_IN_IMAGE; EXISTS_IN_IMAGE] THEN ASM_SIMP_TAC[UNIONS_IMAGE] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [singular_chain]) THEN REWRITE_TAC[SUBSET] THEN DISCH_THEN(MP_TAC o SPEC `f:(num->real)->A`) THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC LAND_CONV [IN] THEN REWRITE_TAC[singular_simplex] THEN DISCH_THEN(MP_TAC o MATCH_MP CONTINUOUS_MAP_IMAGE_SUBSET_TOPSPACE o CONJUNCT1) THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; TOPSPACE_PRODUCT_TOPOLOGY] THEN REWRITE_TAC[o_DEF; TOPSPACE_EUCLIDEANREAL; CARTESIAN_PRODUCT_UNIV] THEN REWRITE_TAC[INTER_UNIV] THEN DISCH_TAC THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `k:(num->real)->bool` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN ASM SET_TAC[]; GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `e:((num->real)->A)->real` THEN DISCH_TAC THEN EXISTS_TAC `inf {e f | (f:(num->real)->A) IN frag_support c}` THEN REWRITE_TAC[SIMPLE_IMAGE] THEN ASM_SIMP_TAC[REAL_LT_INF_FINITE; FINITE_FRAG_SUPPORT; FRAG_SUPPORT_EQ_EMPTY; FINITE_IMAGE; IMAGE_EQ_EMPTY; REAL_LE_INF_FINITE] THEN ASM_SIMP_TAC[FORALL_IN_IMAGE]]; ALL_TAC] THEN MP_TAC(SPECL [`&p / (&p + &1)`; `d:real`] REAL_ARCH_POW_INV) THEN SIMP_TAC[REAL_LT_LDIV_EQ; REAL_LT_RDIV_EQ; REAL_POS; REAL_ARITH `&0 <= x ==> &0 < x + &1`; REAL_ABS_POS] THEN ASM_REWRITE_TAC[REAL_MUL_LZERO] THEN ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN X_GEN_TAC `m:num` THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN DISCH_TAC THEN ABBREV_TAC `c':((num->real)->A)frag = ITER m (singular_subdivision p) c` THEN MP_TAC(ISPECL [`p:num`; `top:A topology`; `m:num`; `c:((num->real)->A)frag`] ITERATED_SINGULAR_SUBDIVISION) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN W(MP_TAC o PART_MATCH lhand FRAG_SUPPORT_FRAG_EXTEND o rand o lhand o snd o dest_forall o snd) THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> P x) ==> top SUBSET s ==> (!x. x IN top ==> P x)`) THEN REWRITE_TAC[FORALL_IN_UNIONS] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_GSPEC] THEN X_GEN_TAC `f:(num->real)->A` THEN DISCH_TAC THEN REWRITE_TAC[chain_map] THEN W(MP_TAC o PART_MATCH lhand FRAG_SUPPORT_FRAG_EXTEND o rand o lhand o snd o dest_forall o snd) THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> P x) ==> top SUBSET s ==> (!x. x IN top ==> P x)`) THEN REWRITE_TAC[FORALL_IN_UNIONS] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_GSPEC] THEN REWRITE_TAC[FRAG_SUPPORT_OF; o_THM; FORALL_IN_INSERT; NOT_IN_EMPTY] THEN SIMP_TAC[simplex_map; IMAGE_RESTRICTION; SUBSET_REFL] THEN X_GEN_TAC `d:(num->real)->(num->real)` THEN DISCH_THEN(LABEL_TAC "*") THEN REWRITE_TAC[IMAGE_o] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[MESON[] `P /\ (!x y i. Q x y i) <=> !i. P /\ (!x y. Q x y i)`] THEN X_GEN_TAC `i:num` THEN SUBGOAL_THEN `!n. simplicial_chain (p,standard_simplex p) (ITER n (simplicial_subdivision p) (frag_of (RESTRICTION (standard_simplex p) I))) /\ !f x y. f IN frag_support(ITER n (simplicial_subdivision p) (frag_of (RESTRICTION (standard_simplex p) I))) /\ x IN standard_simplex p /\ y IN standard_simplex p ==> abs (f x i - f y i) <= (&p / (&p + &1)) pow n` MP_TAC THENL [MATCH_MP_TAC num_INDUCTION THEN REWRITE_TAC[ITER] THEN CONJ_TAC THENL [REWRITE_TAC[SIMPLICIAL_CHAIN_OF; SIMPLICIAL_SIMPLEX_I; SUBSET_REFL] THEN SIMP_TAC[FRAG_SUPPORT_OF; IN_SING; RESTRICTION; real_pow; I_THM] THEN REWRITE_TAC[standard_simplex; IN_ELIM_THM] THEN MESON_TAC[REAL_ARITH `&0 <= x /\ &0 <= y /\ x <= &1 /\ y <= &1 ==> abs(x - y) <= &1`]; GEN_TAC THEN STRIP_TAC THEN ASM_SIMP_TAC[SIMPLICIAL_CHAIN_SIMPLICIAL_SUBDIVISION] THEN REWRITE_TAC[real_pow] THEN MATCH_MP_TAC SIMPLICIAL_SUBDIVISION_SHRINKS THEN ASM_MESON_TAC[]]; DISCH_THEN(MP_TAC o SPEC `m:num`)] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN REWRITE_TAC[simplicial_chain] THEN GEN_REWRITE_TAC LAND_CONV [SUBSET] THEN DISCH_THEN(MP_TAC o SPEC `d:(num->real)->(num->real)`) THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC LAND_CONV [IN] THEN SIMP_TAC[SIMPLICIAL_SIMPLEX] THEN DISCH_THEN(K ALL_TAC) THEN ASM_CASES_TAC `i:num <= p` THEN ASM_REWRITE_TAC[FORALL_IN_IMAGE_2] THEN REPEAT STRIP_TAC THEN TRANS_TAC REAL_LE_TRANS `(&p / (&p + &1)) pow n` THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN TRANS_TAC REAL_LE_TRANS `(&p / (&p + &1)) pow m` THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC REAL_POW_MONO_INV THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ; REAL_ARITH `&0 < &p + &1`] THEN REAL_ARITH_TAC);; let SMALL_HOMOLOGOUS_REL_RELCYCLE_EXISTS = prove (`!p top s u c:((num->real)->A)frag. (!v. v IN u ==> open_in top v) /\ topspace top SUBSET UNIONS u /\ singular_relcycle (p,top,s) c ==> ?c'. singular_relcycle (p,top,s) c' /\ homologous_rel (p,top,s) c c' /\ !f. f IN frag_support c' ==> ?v. v IN u /\ IMAGE f (standard_simplex p) SUBSET v`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o CONJUNCT1 o REWRITE_RULE[singular_relcycle]) THEN MP_TAC(ISPECL [`p:num`; `top:A topology`; `u:(A->bool)->bool`; `c:((num->real)->A)frag`] SUFFICIENT_ITERATED_SINGULAR_SUBDIVISION_EXISTS) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `n:num` (MP_TAC o SPEC `n:num`)) THEN ABBREV_TAC `c':((num->real)->A)frag = ITER n (singular_subdivision p) c` THEN REWRITE_TAC[LE_REFL] THEN DISCH_TAC THEN EXISTS_TAC `c':((num->real)->A)frag` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(TAUT `(q ==> p) /\ q ==> p /\ q`) THEN CONJ_TAC THENL [ASM_MESON_TAC[HOMOLOGOUS_REL_SINGULAR_RELCYCLE]; ALL_TAC] THEN EXPAND_TAC "c'" THEN SPEC_TAC(`n:num`,`m:num`) THEN INDUCT_TAC THEN ASM_REWRITE_TAC[ITER; HOMOLOGOUS_REL_REFL] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMOLOGOUS_REL_TRANS)) THEN ONCE_REWRITE_TAC[HOMOLOGOUS_REL_SYM] THEN MATCH_MP_TAC HOMOLOGOUS_REL_SINGULAR_SUBDIVISION THEN ASM_MESON_TAC[HOMOLOGOUS_REL_SINGULAR_RELCYCLE]);; let EXCISED_CHAIN_EXISTS = prove (`!p top s t u c:((num->real)->A)frag. top closure_of u SUBSET top interior_of t /\ t SUBSET s /\ singular_chain(p,subtopology top s) c ==> ?n d e. singular_chain(p,subtopology top (s DIFF u)) d /\ singular_chain(p,subtopology top t) e /\ ITER n (singular_subdivision p) c = frag_add d e`, SUBGOAL_THEN `!p top s t u c:((num->real)->A)frag. top closure_of u SUBSET top interior_of t /\ u SUBSET topspace top /\ t SUBSET s /\ s SUBSET topspace top /\ singular_chain(p,subtopology top s) c ==> ?n d e. singular_chain(p,subtopology top (s DIFF u)) d /\ singular_chain(p,subtopology top t) e /\ ITER n (singular_subdivision p) c = frag_add d e` MP_TAC THENL [REPEAT STRIP_TAC; REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`p:num`; `top:A topology`; `topspace top INTER s:A->bool`; `topspace top INTER t:A->bool`; `topspace top INTER u:A->bool`; `c:((num->real)->A)frag`]) THEN REWRITE_TAC[SET_RULE `s INTER t DIFF s INTER u = s INTER (t DIFF u)`] THEN REWRITE_TAC[GSYM SUBTOPOLOGY_RESTRICT; INTER_SUBSET] THEN ASM_REWRITE_TAC[GSYM INTERIOR_OF_RESTRICT; GSYM CLOSURE_OF_RESTRICT] THEN DISCH_THEN MATCH_MP_TAC THEN ASM SET_TAC[]] THEN MP_TAC(ISPECL [`p:num`; `subtopology top (s:A->bool)`; `{s INTER top interior_of t:A->bool,s DIFF top closure_of u}`; `c:((num->real)->A)frag`] SUFFICIENT_ITERATED_SINGULAR_SUBDIVISION_EXISTS) THEN ASM_REWRITE_TAC[FORALL_IN_INSERT; NOT_IN_EMPTY] THEN ANTS_TAC THENL [SIMP_TAC[OPEN_IN_SUBTOPOLOGY_INTER_OPEN; OPEN_IN_INTERIOR_OF] THEN ASM_SIMP_TAC[OPEN_IN_SUBTOPOLOGY_DIFF_CLOSED; CLOSED_IN_CLOSURE_OF] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; UNIONS_2] THEN ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:num`] THEN DISCH_THEN(MP_TAC o SPEC `n:num`) THEN REWRITE_TAC[LE_REFL] THEN ABBREV_TAC `c':((num->real)->A)frag = ITER n (singular_subdivision p) c` THEN REWRITE_TAC[EXISTS_IN_INSERT; NOT_IN_EMPTY] THEN DISCH_TAC THEN ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN REWRITE_TAC[singular_chain] THEN MATCH_MP_TAC FRAG_SPLIT THEN REWRITE_TAC[SUBSET; IN_UNION] THEN X_GEN_TAC `f:(num->real)->A` THEN DISCH_TAC THEN GEN_REWRITE_TAC I [DISJ_SYM] THEN FIRST_X_ASSUM(MP_TAC o SPEC `f:(num->real)->A`) THEN SUBGOAL_THEN `singular_chain(p,subtopology top s:A topology) c'` (MP_TAC o REWRITE_RULE[singular_chain; SUBSET]) THENL [EXPAND_TAC "c'" THEN SPEC_TAC(`n:num`,`m:num`) THEN INDUCT_TAC THEN ASM_SIMP_TAC[ITER; SINGULAR_CHAIN_0] THEN ASM_SIMP_TAC[SINGULAR_CHAIN_SINGULAR_SUBDIVISION]; DISCH_THEN(MP_TAC o SPEC `f:(num->real)->A`) THEN ASM_SIMP_TAC[IN; SINGULAR_SIMPLEX_SUBTOPOLOGY; SUBSET_INTER] THEN STRIP_TAC THEN MATCH_MP_TAC MONO_OR THEN CONJ_TAC THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS) THEN REWRITE_TAC[INTERIOR_OF_SUBSET] THEN MP_TAC(ISPECL [`top:A topology`; `u:A->bool`] CLOSURE_OF_SUBSET) THEN ASM SET_TAC[]]);; let EXCISED_RELCYCLE_EXISTS = prove (`!p top s t u c:((num->real)->A)frag. top closure_of u SUBSET top interior_of t /\ t SUBSET s /\ singular_relcycle (p,subtopology top s,t) c ==> ?c'. singular_relcycle (p,subtopology top (s DIFF u),t DIFF u) c' /\ homologous_rel (p,subtopology top s,t) c c'`, REWRITE_TAC[singular_relcycle; cong; mod_subset] THEN REWRITE_TAC[IMP_CONJ; FRAG_MODULE `frag_sub x frag_0 = x`] THEN SIMP_TAC[SUBTOPOLOGY_SUBTOPOLOGY; SET_RULE `t SUBSET s ==> s INTER t = t /\ (s DIFF u) INTER (t DIFF u) = t DIFF u`] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`p:num`; `top:A topology`; `s:A->bool`; `t:A->bool`; `u:A->bool`; `c:((num->real)->A)frag`] EXCISED_CHAIN_EXISTS) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`n:num`; `d:((num->real)->A)frag`; `e:((num->real)->A)frag`] THEN STRIP_TAC THEN EXISTS_TAC `d:((num->real)->A)frag` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [SUBGOAL_THEN `singular_chain(p - 1,subtopology top (t:A->bool)) (chain_boundary p d) /\ singular_chain(p - 1,subtopology top (s DIFF u)) (chain_boundary p d)` MP_TAC THENL [ASM_SIMP_TAC[SINGULAR_CHAIN_BOUNDARY] THEN FIRST_X_ASSUM(SUBST1_TAC o MATCH_MP (FRAG_MODULE `c' = frag_add d e ==> d = frag_sub c' e`)) THEN REWRITE_TAC[CHAIN_BOUNDARY_SUB] THEN MATCH_MP_TAC SINGULAR_CHAIN_SUB THEN ASM_SIMP_TAC[SINGULAR_CHAIN_BOUNDARY] THEN SPEC_TAC(`n:num`,`m:num`) THEN INDUCT_TAC THEN ASM_REWRITE_TAC[ITER] THEN MP_TAC(ISPECL [`p:num`; `subtopology top s:A topology`; `ITER m (singular_subdivision p) c:((num->real)->A)frag`] CHAIN_BOUNDARY_SINGULAR_SUBDIVISION) THEN ANTS_TAC THENL [SPEC_TAC(`m:num`,`r:num`) THEN INDUCT_TAC THEN ASM_SIMP_TAC[SINGULAR_CHAIN_0; ITER] THEN ASM_SIMP_TAC[SINGULAR_CHAIN_SINGULAR_SUBDIVISION]; DISCH_THEN SUBST1_TAC] THEN ASM_SIMP_TAC[SINGULAR_CHAIN_SINGULAR_SUBDIVISION]; REWRITE_TAC[SINGULAR_CHAIN_SUBTOPOLOGY] THEN ASM SET_TAC[]]; FIRST_X_ASSUM(SUBST1_TAC o MATCH_MP (FRAG_MODULE `c' = frag_add d e ==> d = frag_sub c' e`)) THEN GEN_REWRITE_TAC LAND_CONV [FRAG_MODULE `x = frag_sub x frag_0`] THEN MATCH_MP_TAC HOMOLOGOUS_REL_SUB THEN CONJ_TAC THENL [SPEC_TAC(`n:num`,`m:num`) THEN INDUCT_TAC THEN ASM_REWRITE_TAC[ITER; HOMOLOGOUS_REL_REFL] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMOLOGOUS_REL_TRANS)) THEN ONCE_REWRITE_TAC[HOMOLOGOUS_REL_SYM] THEN MATCH_MP_TAC HOMOLOGOUS_REL_SINGULAR_SUBDIVISION THEN FIRST_ASSUM(SUBST1_TAC o SYM o MATCH_MP HOMOLOGOUS_REL_SINGULAR_RELCYCLE) THEN REWRITE_TAC[singular_relcycle; cong; mod_subset] THEN ASM_REWRITE_TAC[FRAG_MODULE `frag_sub x frag_0 = x`]; REWRITE_TAC[homologous_rel] THEN MATCH_MP_TAC SINGULAR_RELBOUNDARY_SUB THEN REWRITE_TAC[SINGULAR_RELBOUNDARY_0] THEN MATCH_MP_TAC SINGULAR_CHAIN_IMP_RELBOUNDARY] THEN ASM_SIMP_TAC[SUBTOPOLOGY_SUBTOPOLOGY; SET_RULE `t SUBSET s ==> s INTER t = t`]]);; (* ------------------------------------------------------------------------- *) (* Homotopy invariance. *) (* ------------------------------------------------------------------------- *) let HOMOTOPIC_IMP_HOMOLOGOUS_REL_CHAIN_MAPS = prove (`!p f g:A->B s t u v c:((num->real)->A)frag. homotopic_with (\h. IMAGE h t SUBSET v) (s,u) f g /\ singular_relcycle (p,s,t) c ==> homologous_rel (p,u,v) (chain_map p f c) (chain_map p g c)`, REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REPEAT GEN_TAC THEN DISCH_TAC THEN SUBGOAL_THEN `?prism:num->(((num->real)->A)frag)->(((num->real)->B)frag). (!q. prism q frag_0 = frag_0) /\ (!q c. singular_chain (q,s) c ==> singular_chain (q + 1,u) (prism q c)) /\ (!q c. singular_chain (q,subtopology s t) c ==> singular_chain (q + 1,subtopology u v) (prism q c)) /\ (!q c. singular_chain (q,s) c ==> chain_boundary (q + 1) (prism q c) = frag_sub (frag_sub (chain_map q g c) (chain_map q f c)) (prism (q - 1) (chain_boundary q c)))` STRIP_ASSUME_TAC THENL [ALL_TAC; X_GEN_TAC `c:((num->real)->A)frag` THEN REWRITE_TAC[singular_relcycle; homologous_rel; singular_relboundary; cong; mod_subset; FRAG_MODULE `frag_sub x frag_0 = x`] THEN STRIP_TAC THEN EXISTS_TAC `frag_neg((prism:num->(((num->real)->A)frag)->((num->real)->B)frag) p c)` THEN ASM_SIMP_TAC[SINGULAR_CHAIN_NEG; CHAIN_BOUNDARY_NEG] THEN REWRITE_TAC[FRAG_MODULE `frag_sub (frag_neg (frag_sub (frag_sub g f) d)) (frag_sub f g) = d`] THEN ASM_CASES_TAC `p = 0` THENL [ASM_REWRITE_TAC[chain_boundary; SINGULAR_CHAIN_0]; ALL_TAC] THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [MATCH_MP (ARITH_RULE `~(p = 0) ==> p = (p - 1) + 1`) th]) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[]] THEN FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_CONTINUOUS_MAPS) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homotopic_with]) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; RIGHT_IMP_FORALL_THM; IMP_IMP] THEN DISCH_THEN(X_CHOOSE_THEN `h:real#A->B` STRIP_ASSUME_TAC) THEN MAP_EVERY ABBREV_TAC [`vv:num->num->real = \j i. if i = j + 1 then &1 else &0`; `ww:num->num->real = \j i. if i = 0 \/ i = j + 1 then &1 else &0`; `simp = \q i. oriented_simplex (q + 1) (\j. if j <= i then vv j else ww(j - 1))`; `prism:num->((num->real)->A)->((num->real)->B)frag = \q c. iterate frag_add (0..q) (\i. frag_cmul (int_pow (-- &1) i) (frag_of (simplex_map (q + 1) (\z. h(z 0,c(z o SUC))) (simp q i:(num->real)->(num->real)))))`] THEN EXISTS_TAC `\q. frag_extend((prism:num->((num->real)->A)->((num->real)->B)frag) q)` THEN REWRITE_TAC[FRAG_EXTEND_0] THEN SUBGOAL_THEN `!q i. i <= q ==> simplicial_simplex (q + 1,{x | x 0 IN real_interval[&0,&1] /\ (x o SUC) IN standard_simplex q}) ((simp:num->num->(num->real)->(num->real)) q i)` ASSUME_TAC THENL [MAP_EVERY X_GEN_TAC [`q:num`; `i:num`] THEN DISCH_TAC THEN EXPAND_TAC "simp" THEN REWRITE_TAC[SIMPLICIAL_SIMPLEX_ORIENTED_SIMPLEX] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM; o_DEF] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN X_GEN_TAC `x:num->real` THEN DISCH_TAC THEN MAP_EVERY EXPAND_TAC ["vv"; "ww"] THEN REWRITE_TAC[NOT_SUC; ARITH_RULE `~(0 = j + 1)`] THEN REWRITE_TAC[standard_simplex; IN_ELIM_THM; IN_REAL_INTERVAL] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [standard_simplex]) THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN CONJ_TAC THENL [REWRITE_TAC[REAL_ARITH `&0 <= x /\ x <= &1 <=> abs(x - &0) <= &1 /\ abs(x - &1) <= &1`] THEN CONJ_TAC THEN MATCH_MP_TAC REAL_CONVEX_SUM_BOUND_LE THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC; ALL_TAC] THEN CONJ_TAC THENL [X_GEN_TAC `j:num` THEN REWRITE_TAC[REAL_ARITH `&0 <= x /\ x <= &1 <=> abs(x - &0) <= &1 /\ abs(x - &1) <= &1`] THEN CONJ_TAC THEN MATCH_MP_TAC REAL_CONVEX_SUM_BOUND_LE THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC; ALL_TAC] THEN CONJ_TAC THENL [X_GEN_TAC `j:num` THEN DISCH_TAC THEN MATCH_MP_TAC SUM_EQ_0_NUMSEG THEN X_GEN_TAC `k:num` THEN STRIP_TAC THEN REWRITE_TAC[] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_MUL_LZERO]) THEN REWRITE_TAC[REAL_MUL_LID] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN SIMP_TAC[SUM_CASES; FINITE_NUMSEG; FINITE_RESTRICT] THEN REWRITE_TAC[REAL_MUL_LZERO; SUM_0; REAL_MUL_LID; ETA_AX] THEN REWRITE_TAC[IN_ELIM_THM; REAL_ADD_RID] THEN REWRITE_TAC[IN_NUMSEG; LE_0; GSYM CONJ_ASSOC] THEN REWRITE_TAC[ARITH_RULE `(j <= q + 1 /\ j <= i /\ SUC k = j + 1 <=> j = k /\ k <= q + 1 /\ k <= i) /\ (j <= q + 1 /\ ~(j <= i) /\ SUC k = j - 1 + 1 <=> j = k + 1 /\ k + 1 <= q + 1 /\ ~(k + 1 <= i))`] THEN REWRITE_TAC[SET_RULE `{x | x = a /\ P} = if P then {a} else {}`] THEN ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN REWRITE_TAC[SUM_ADD_NUMSEG; SUM_SING; SUM_CLAUSES] THEN REWRITE_TAC[GSYM SUM_RESTRICT_SET] THEN REWRITE_TAC[IN_NUMSEG; LE_0] THEN ASM_SIMP_TAC[ARITH_RULE `i <= q ==> (k <= q /\ k <= q + 1 /\ k <= i <=> 0 <= k /\ k <= i)`] THEN ASM_SIMP_TAC[ARITH_RULE `k <= q /\ k + 1 <= q + 1 /\ ~(k + 1 <= i) <=> i <= k /\ k <= q`] THEN REWRITE_TAC[GSYM numseg] THEN REWRITE_TAC[GSYM(SPEC `1` SUM_OFFSET)] THEN MP_TAC(ISPECL [`x:num->real`; `0`; `i:num`; `(q + 1) - i`] SUM_ADD_SPLIT) THEN ASM_SIMP_TAC[ARITH_RULE `i <= q ==> i + (q + 1) - i = q + 1`; LE_0]]; ALL_TAC] THEN CONJ_TAC THENL [X_GEN_TAC `q:num` THEN REWRITE_TAC[singular_chain] THEN MATCH_MP_TAC FRAG_INDUCTION THEN REWRITE_TAC[GSYM singular_chain] THEN REWRITE_TAC[FRAG_EXTEND_0; SINGULAR_CHAIN_0; FRAG_EXTEND_OF] THEN SIMP_TAC[FRAG_EXTEND_SUB; SINGULAR_CHAIN_SUB] THEN X_GEN_TAC `m:(num->real)->A` THEN GEN_REWRITE_TAC LAND_CONV [IN] THEN DISCH_TAC THEN EXPAND_TAC "prism" THEN MATCH_MP_TAC SINGULAR_CHAIN_SUM THEN X_GEN_TAC `i:num` THEN REWRITE_TAC[IN_NUMSEG; LE_0] THEN DISCH_TAC THEN MATCH_MP_TAC SINGULAR_CHAIN_CMUL THEN REWRITE_TAC[SINGULAR_CHAIN_OF] THEN MATCH_MP_TAC SINGULAR_SIMPLEX_SIMPLEX_MAP THEN EXISTS_TAC `subtopology (product_topology (:num) (\i. euclideanreal)) {x | x 0 IN real_interval[&0,&1] /\ (x o SUC) IN standard_simplex q}` THEN ASM_SIMP_TAC[SIMPLICIAL_IMP_SINGULAR_SIMPLEX] THEN GEN_REWRITE_TAC RAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_MAP_COMPOSE THEN EXISTS_TAC `prod_topology (subtopology euclideanreal (real_interval [&0,&1])) (s:(A)topology)` THEN ASM_REWRITE_TAC[CONTINUOUS_MAP_PAIRWISE; o_DEF] THEN CONJ_TAC THENL [REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY] THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN MATCH_MP_TAC CONTINUOUS_MAP_PRODUCT_PROJECTION THEN REWRITE_TAC[IN_UNIV]; GEN_REWRITE_TAC RAND_CONV [GSYM o_DEF] THEN FIRST_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [singular_simplex]) THEN MATCH_MP_TAC CONTINUOUS_MAP_COMPOSE THEN EXISTS_TAC `subtopology (product_topology (:num) (\i. euclideanreal)) (standard_simplex q)` THEN ASM_REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY] THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN REWRITE_TAC[CONTINUOUS_MAP_COMPONENTWISE; IN_UNIV] THEN REWRITE_TAC[SUBSET; IN; EXTENSIONAL_UNIV] THEN GEN_TAC THEN MATCH_MP_TAC CONTINUOUS_MAP_PRODUCT_PROJECTION THEN REWRITE_TAC[IN_UNIV]]; ALL_TAC] THEN CONJ_TAC THENL [X_GEN_TAC `q:num` THEN REWRITE_TAC[singular_chain] THEN MATCH_MP_TAC FRAG_INDUCTION THEN REWRITE_TAC[GSYM singular_chain] THEN REWRITE_TAC[FRAG_EXTEND_0; SINGULAR_CHAIN_0; FRAG_EXTEND_OF] THEN SIMP_TAC[FRAG_EXTEND_SUB; SINGULAR_CHAIN_SUB] THEN X_GEN_TAC `m:(num->real)->A` THEN GEN_REWRITE_TAC LAND_CONV [IN] THEN DISCH_TAC THEN EXPAND_TAC "prism" THEN MATCH_MP_TAC SINGULAR_CHAIN_SUM THEN X_GEN_TAC `i:num` THEN REWRITE_TAC[IN_NUMSEG; LE_0] THEN DISCH_TAC THEN MATCH_MP_TAC SINGULAR_CHAIN_CMUL THEN REWRITE_TAC[SINGULAR_CHAIN_OF] THEN MATCH_MP_TAC SINGULAR_SIMPLEX_SIMPLEX_MAP THEN EXISTS_TAC `subtopology (product_topology (:num) (\i. euclideanreal)) {x | x 0 IN real_interval[&0,&1] /\ (x o SUC) IN standard_simplex q}` THEN ASM_SIMP_TAC[SIMPLICIAL_IMP_SINGULAR_SIMPLEX] THEN GEN_REWRITE_TAC RAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_MAP_COMPOSE THEN EXISTS_TAC `prod_topology (subtopology euclideanreal (real_interval [&0,&1])) (subtopology s t:(A)topology)` THEN ASM_REWRITE_TAC[CONTINUOUS_MAP_PAIRWISE; o_DEF] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY] THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN MATCH_MP_TAC CONTINUOUS_MAP_PRODUCT_PROJECTION THEN REWRITE_TAC[IN_UNIV]; GEN_REWRITE_TAC RAND_CONV [GSYM o_DEF] THEN FIRST_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [singular_simplex]) THEN MATCH_MP_TAC CONTINUOUS_MAP_COMPOSE THEN EXISTS_TAC `subtopology (product_topology (:num) (\i. euclideanreal)) (standard_simplex q)` THEN ASM_REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY] THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN REWRITE_TAC[CONTINUOUS_MAP_COMPONENTWISE; IN_UNIV] THEN REWRITE_TAC[SUBSET; IN; EXTENSIONAL_UNIV] THEN GEN_TAC THEN MATCH_MP_TAC CONTINUOUS_MAP_PRODUCT_PROJECTION THEN REWRITE_TAC[IN_UNIV]; REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[SUBTOPOLOGY_RESTRICT] THEN REWRITE_TAC[GSYM SUBTOPOLOGY_CROSS] THEN MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY_MONO THEN EXISTS_TAC `real_interval [&0,&1] CROSS (topspace s:A->bool)` THEN ASM_REWRITE_TAC[SUBTOPOLOGY_CROSS; SUBTOPOLOGY_TOPSPACE] THEN REWRITE_TAC[SUBSET_CROSS; INTER_SUBSET]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_PAIR_THM; IN_CROSS; TOPSPACE_SUBTOPOLOGY; TOPSPACE_PROD_TOPOLOGY; IN_INTER] THEN ASM_SIMP_TAC[]]]; ALL_TAC] THEN X_GEN_TAC `q:num` THEN REWRITE_TAC[singular_chain] THEN MATCH_MP_TAC FRAG_INDUCTION THEN REWRITE_TAC[GSYM singular_chain] THEN REWRITE_TAC[FRAG_EXTEND_0; CHAIN_BOUNDARY_0; CHAIN_MAP_0] THEN REWRITE_TAC[FRAG_MODULE `frag_sub x x = frag_0`] THEN REWRITE_TAC[FRAG_EXTEND_SUB; CHAIN_BOUNDARY_SUB; CHAIN_MAP_SUB] THEN CONJ_TAC THENL [ALL_TAC; REPEAT GEN_TAC THEN CONV_TAC FRAG_MODULE] THEN X_GEN_TAC `a:(num->real)->A` THEN GEN_REWRITE_TAC LAND_CONV [IN] THEN DISCH_TAC THEN REWRITE_TAC[CHAIN_MAP_OF; FRAG_EXTEND_OF] THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o RAND_CONV o funpow 2 RATOR_CONV) [SYM th]) THEN SIMP_TAC[CHAIN_BOUNDARY_SUM; FINITE_NUMSEG; o_DEF; CHAIN_BOUNDARY_CMUL] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [chain_boundary] THEN REWRITE_TAC[FRAG_EXTEND_OF; ARITH_RULE `~(q + 1 = 0)`] THEN REWRITE_TAC[FRAG_CMUL_SUM] THEN SIMP_TAC[MATCH_MP ITERATE_ITERATE_PRODUCT MONOIDAL_FRAG_ADD; FINITE_NUMSEG] THEN ONCE_REWRITE_TAC[SET_RULE `{i,j | i IN 0..q /\ j IN 0..q + 1} = {i,j | i IN 0..q /\ j IN {j | j IN 0..q + 1 /\ j <= i}} UNION {i,j | i IN 0..q /\ j IN {j | j IN 0..q + 1 /\ ~(j <= i)}}`] THEN W(MP_TAC o PART_MATCH (lhand o rand) (MATCH_MP ITERATE_UNION MONOIDAL_FRAG_ADD) o lhand o snd) THEN ANTS_TAC THENL [SIMP_TAC[FINITE_PRODUCT_DEPENDENT; FINITE_NUMSEG; FINITE_RESTRICT] THEN REWRITE_TAC[DISJOINT; EXTENSION; IN_INTER; NOT_IN_EMPTY] THEN REWRITE_TAC[FORALL_PAIR_THM; IN_ELIM_PAIR_THM] THEN SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN SUBGOAL_THEN `!i. {j | j IN 0..q + 1 /\ ~(j <= i)} = i+1..q+1` (fun th -> REWRITE_TAC[th]) THENL [REWRITE_TAC[EXTENSION; IN_NUMSEG; IN_ELIM_THM] THEN ARITH_TAC; ALL_TAC] THEN SIMP_TAC[GSYM(MATCH_MP ITERATE_ITERATE_PRODUCT MONOIDAL_FRAG_ADD); FINITE_NUMSEG; FINITE_RESTRICT] THEN REWRITE_TAC[NUMSEG_OFFSET_IMAGE] THEN SIMP_TAC[MATCH_MP ITERATE_IMAGE MONOIDAL_FRAG_ADD; EQ_ADD_RCANCEL] THEN REWRITE_TAC[o_DEF] THEN SIMP_TAC[MATCH_MP ITERATE_ITERATE_PRODUCT MONOIDAL_FRAG_ADD; FINITE_NUMSEG; FINITE_RESTRICT] THEN ONCE_REWRITE_TAC[SET_RULE `{i,j | i IN s /\ j IN t i} = IMAGE (\i. (i,i)) {i | i IN s /\ i IN t i} UNION {i,j | i IN s /\ j IN (t i DELETE i)}`] THEN MAP_EVERY (fun conv -> W(MP_TAC o PART_MATCH (lhand o rand) (MATCH_MP ITERATE_UNION MONOIDAL_FRAG_ADD) o conv o lhand o snd) THEN ANTS_TAC THENL [SIMP_TAC[FINITE_IMAGE; FINITE_PRODUCT_DEPENDENT; FINITE_NUMSEG; FINITE_DELETE; FINITE_RESTRICT] THEN REWRITE_TAC[SET_RULE `DISJOINT s t <=> !x. x IN s ==> ~(x IN t)`] THEN REWRITE_TAC[FORALL_IN_IMAGE; FORALL_IN_GSPEC; IN_ELIM_PAIR_THM] THEN SET_TAC[]; DISCH_THEN SUBST1_TAC]) [lhand;rand] THEN MATCH_MP_TAC(FRAG_MODULE `frag_add x1 y1 = w /\ frag_add x2 y2 = frag_neg z ==> frag_add (frag_add x1 x2) (frag_add y1 y2) = frag_sub w z`) THEN CONJ_TAC THENL [SIMP_TAC[MATCH_MP ITERATE_IMAGE MONOIDAL_FRAG_ADD; PAIR_EQ] THEN REWRITE_TAC[o_DEF; IN_ELIM_THM; IN_NUMSEG] THEN REWRITE_TAC[LE_REFL; LE_0; ARITH_RULE `i <= q /\ i <= q + 1 <=> i <= q`] THEN ONCE_REWRITE_TAC[ARITH_RULE `i <= k <=> 0 <= i /\ i <= k`] THEN REWRITE_TAC[GSYM numseg] THEN REWRITE_TAC[INT_POW_ADD; INT_POW_1; FRAG_MODULE `frag_cmul a (frag_cmul b c) = frag_cmul (a * b) c`] THEN REWRITE_TAC[INT_MUL_ASSOC; GSYM INT_POW_ADD] THEN SIMP_TAC[INT_POW_NEG; EVEN_ADD; INT_POW_ONE] THEN CONV_TAC INT_REDUCE_CONV THEN REWRITE_TAC[FRAG_MODULE `frag_cmul (&1) x = x`] THEN SUBGOAL_THEN `0..q = 0 INSERT IMAGE (\i. i + 1) {i | i < q}` (fun th -> GEN_REWRITE_TAC (LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV) [th]) THENL [SIMP_TAC[GSYM NUMSEG_LREC; LE_0] THEN AP_TERM_TAC THEN ASM_CASES_TAC `q = 0` THENL [ASM_REWRITE_TAC[CONJUNCT1 LT; EMPTY_GSPEC; IMAGE_CLAUSES] THEN REWRITE_TAC[NUMSEG_EMPTY] THEN CONV_TAC NUM_REDUCE_CONV; ASM_SIMP_TAC[ARITH_RULE `~(q = 0) ==> (i < q <=> 0 <= i /\ i <= q - 1)`] THEN REWRITE_TAC[GSYM numseg; GSYM NUMSEG_OFFSET_IMAGE] THEN AP_TERM_TAC THEN ASM_ARITH_TAC]; ALL_TAC] THEN SUBGOAL_THEN `0..q = q INSERT {i | i < q}` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; IN_INSERT; IN_NUMSEG; IN_ELIM_THM] THEN ARITH_TAC; ALL_TAC] THEN SIMP_TAC[MATCH_MP ITERATE_CLAUSES MONOIDAL_FRAG_ADD; FINITE_IMAGE; FINITE_NUMSEG_LT] THEN REWRITE_TAC[IN_ELIM_THM; LT_REFL; IN_IMAGE; ARITH_RULE `~(0 = i + 1)`] THEN SIMP_TAC[MATCH_MP ITERATE_IMAGE MONOIDAL_FRAG_ADD; EQ_ADD_RCANCEL] THEN MATCH_MP_TAC(FRAG_MODULE `(x1 = w /\ frag_neg y1 = z) /\ frag_cmul (-- &1) x2 = y2 ==> frag_add (frag_add x1 x2) (frag_add y1 y2) = frag_sub w z`) THEN CONJ_TAC THENL [SIMP_TAC[SINGULAR_FACE_SIMPLEX_MAP; LE_REFL; LE_0; ADD_SUB; FRAG_MODULE `frag_neg (frag_cmul (-- &1) x) = x`; ARITH_RULE `1 <= p + 1`] THEN REWRITE_TAC[simplex_map] THEN CONJ_TAC THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `x:num->real` THEN REWRITE_TAC[RESTRICTION] THEN ASM_CASES_TAC `x IN standard_simplex q` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[o_DEF] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (MESON[] `(!x. h(a,x) = c x) ==> w = a /\ z = y ==> h(w,z) = c y`)) THEN (CONJ_TAC THENL [ALL_TAC; AP_TERM_TAC]) THEN EXPAND_TAC "simp" THEN REWRITE_TAC[oriented_simplex] THEN ASM_SIMP_TAC[RESTRICTION; SIMPLICIAL_FACE_IN_STANDARD_SIMPLEX; ADD_SUB; LE_REFL; LE_0; ARITH_RULE `1 <= p + 1`] THEN REWRITE_TAC[simplicial_face; CONJUNCT1 LE; CONJUNCT1 LT] THEN REWRITE_TAC[ARITH_RULE `j < q + 1 <=> j <= q`] THEN REWRITE_TAC[MESON[] `(if p then f1 else f2) a * (if p then y1 else y2):real = if p then f1 a * y1 else f2 a * y2`] THEN MAP_EVERY EXPAND_TAC ["vv"; "ww"] THEN REWRITE_TAC[NOT_SUC; ARITH_RULE `~(0 = j + 1)`; REAL_MUL_LZERO; REAL_MUL_RZERO] THEN SIMP_TAC[SUM_CASES; FINITE_NUMSEG; SUM_0; REAL_ADD_LID] THEN REWRITE_TAC[MESON[REAL_MUL_LZERO; REAL_MUL_RZERO; REAL_MUL_LID] `(if p then &1 else &0) * (if q then &0 else x):real = if p /\ ~q then x else &0`] THEN REWRITE_TAC[COND_RAND; COND_RATOR] THEN SIMP_TAC[SUM_CASES; FINITE_NUMSEG; FINITE_RESTRICT] THEN REWRITE_TAC[REAL_MUL_LZERO; REAL_MUL_RZERO; SUM_0] THEN REWRITE_TAC[REAL_ADD_LID; REAL_ADD_RID] THEN REWRITE_TAC[IN_ELIM_THM; IN_NUMSEG; LE_0] THEN REWRITE_TAC[EMPTY_GSPEC; CONJUNCT1 SUM_CLAUSES; ARITH_RULE `~((j <= q + 1 /\ ~(j <= q)) /\ ~(j = q + 1))`] THEN REWRITE_TAC[ARITH_RULE `(j <= q + 1 /\ ~(j = 0)) /\ SUC k = j - 1 + 1 <=> j = k + 1 /\ k <= q`] THEN REWRITE_TAC[EMPTY_GSPEC; CONJUNCT1 SUM_CLAUSES; REAL_ADD_RID; ARITH_RULE `~((j <= q + 1 /\ ~(j <= q)) /\ k = j - 1 + 1 /\ ~(j = q + 1))`] THEN REWRITE_TAC[ARITH_RULE `(j <= q + 1 /\ j <= q) /\ SUC k = j + 1 <=> j = k /\ k <= q`] THEN REWRITE_TAC[SET_RULE `{x | x = a /\ P} = if P then {a} else {}`] THEN ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN REWRITE_TAC[CONJUNCT1 SUM_CLAUSES; SUM_SING] THEN REWRITE_TAC[ADD_SUB; REAL_MUL_LID; ARITH_RULE `j <= q + 1 /\ ~(j = 0) <=> 0 + 1 <= j /\ j <= q + 1`] THEN REWRITE_TAC[GSYM numseg; SUM_OFFSET; ADD_SUB; ETA_AX] THEN UNDISCH_TAC `x IN standard_simplex q` THEN SIMP_TAC[standard_simplex; IN_ELIM_THM] THEN STRIP_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN REWRITE_TAC[GSYM NOT_LT] THEN ASM_MESON_TAC[]; REWRITE_TAC[GSYM FRAG_CMUL_SUM] THEN AP_TERM_TAC THEN MATCH_MP_TAC(MATCH_MP ITERATE_EQ MONOIDAL_FRAG_ADD) THEN X_GEN_TAC `i:num` THEN REWRITE_TAC[o_THM; IN_ELIM_THM] THEN DISCH_TAC THEN AP_TERM_TAC THEN ASM_SIMP_TAC[SINGULAR_FACE_SIMPLEX_MAP; ARITH_RULE `1 <= q + 1`; ARITH_RULE `i < q ==> i + 1 <= q + 1`] THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `x:num->real` THEN REWRITE_TAC[simplex_map; RESTRICTION; ADD_SUB] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC BINOP_CONV [o_THM] THEN AP_TERM_TAC THEN EXPAND_TAC "simp" THEN REWRITE_TAC[o_THM; oriented_simplex] THEN ASM_SIMP_TAC[RESTRICTION; SIMPLICIAL_FACE_IN_STANDARD_SIMPLEX; ARITH_RULE `1 <= p + 1`; ADD_SUB; ARITH_RULE `i < q ==> i + 1 <= q + 1`] THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `k:num` THEN REWRITE_TAC[] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN MATCH_MP_TAC SUM_EQ_NUMSEG THEN X_GEN_TAC `j:num` THEN STRIP_TAC THEN REWRITE_TAC[] THEN ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN ASM_CASES_TAC `j:num <= i` THEN ASM_SIMP_TAC[ARITH_RULE `j <= i ==> j <= i + 1`] THEN ASM_SIMP_TAC[ARITH_RULE `~(j <= i) ==> (j <= i + 1 <=> j = i + 1)`] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN MAP_EVERY EXPAND_TAC ["vv"; "ww"] THEN REWRITE_TAC[ADD_SUB; simplicial_face; LT_REFL; REAL_MUL_RZERO]]; ALL_TAC] THEN REWRITE_TAC[chain_boundary] THEN ASM_CASES_TAC `q = 0` THEN ASM_REWRITE_TAC[] THENL [REWRITE_TAC[IN_ELIM_THM; IN_NUMSEG; FRAG_EXTEND_0; IN_DELETE; LE_0] THEN REWRITE_TAC[ARITH_RULE `~(x <= 0 /\ (j <= 0 + 1 /\ j <= x) /\ ~(j = x))`; ARITH_RULE `~(x <= 0 /\ (x <= j /\ j <= 0) /\ ~(j = x))`] THEN REWRITE_TAC[SET_RULE `{i,j | F} = {}`] THEN REWRITE_TAC[MATCH_MP ITERATE_CLAUSES MONOIDAL_FRAG_ADD] THEN REWRITE_TAC[NEUTRAL_FRAG_ADD] THEN CONV_TAC FRAG_MODULE; ALL_TAC] THEN REWRITE_TAC[FRAG_EXTEND_OF] THEN SIMP_TAC[FRAG_EXTEND_SUM; FINITE_NUMSEG] THEN REWRITE_TAC[o_DEF; FRAG_EXTEND_CMUL; FRAG_EXTEND_OF] THEN EXPAND_TAC "prism" THEN REWRITE_TAC[FRAG_CMUL_SUM; FRAG_MODULE `frag_neg c = frag_cmul (-- &1) c`] THEN REWRITE_TAC[FRAG_MODULE `frag_cmul a (frag_cmul b c) = frag_cmul (a * b) c`; GSYM INT_POW_ADD; GSYM(CONJUNCT2 INT_POW)] THEN W(MP_TAC o PART_MATCH (lhand o rand) (MATCH_MP ITERATE_SWAP MONOIDAL_FRAG_ADD) o rand o snd) THEN REWRITE_TAC[FINITE_NUMSEG] THEN DISCH_THEN SUBST1_TAC THEN ASM_SIMP_TAC[SUB_ADD; LE_1] THEN SIMP_TAC[MATCH_MP ITERATE_ITERATE_PRODUCT MONOIDAL_FRAG_ADD; FINITE_NUMSEG] THEN SUBGOAL_THEN `{i,j | i IN 0..q-1 /\ j IN 0..q} = {i,j | i IN 0..q-1 /\ j IN {j | j IN 0..q /\ j <= i}} UNION {i,j | i IN 0..q /\ j IN {j | j IN 0..q /\ i < j}}` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; FORALL_PAIR_THM; IN_UNION; IN_ELIM_PAIR_THM] THEN REWRITE_TAC[IN_ELIM_THM; IN_NUMSEG] THEN ARITH_TAC; ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhand o rand) (MATCH_MP ITERATE_UNION MONOIDAL_FRAG_ADD) o rand o snd) THEN ANTS_TAC THENL [SIMP_TAC[FINITE_PRODUCT_DEPENDENT; FINITE_NUMSEG; FINITE_RESTRICT] THEN REWRITE_TAC[SET_RULE `DISJOINT s t <=> !x. x IN s ==> ~(x IN t)`] THEN REWRITE_TAC[FORALL_IN_IMAGE; FORALL_IN_GSPEC; IN_ELIM_PAIR_THM] THEN REWRITE_TAC[IN_ELIM_THM; IN_NUMSEG] THEN ARITH_TAC; DISCH_THEN SUBST1_TAC] THEN BINOP_TAC THENL [MATCH_MP_TAC(MATCH_MP ITERATE_EQ_GENERAL_INVERSES MONOIDAL_FRAG_ADD) THEN EXISTS_TAC `\(a:num,b:num). (a - 1,b)` THEN EXISTS_TAC `\(a:num,b:num). (a + 1,b)` THEN REWRITE_TAC[IN_DELETE; IMP_CONJ; FORALL_IN_GSPEC] THEN REWRITE_TAC[IN_ELIM_PAIR_THM; IN_NUMSEG; PAIR_EQ; LE_0] THEN REWRITE_TAC[IN_ELIM_THM] THEN CONJ_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN REPLICATE_TAC 3 STRIP_TAC THEN REPEAT(CONJ_TAC THENL [ASM_ARITH_TAC; ALL_TAC]) THEN SUBGOAL_THEN `SUC j + i - 1 = i + j` SUBST1_TAC THENL [ASM_ARITH_TAC; AP_TERM_TAC THEN AP_TERM_TAC] THEN ASM_SIMP_TAC[SINGULAR_FACE_SIMPLEX_MAP; ARITH_RULE `1 <= q + 1`] THEN REWRITE_TAC[ADD_SUB] THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN REWRITE_TAC[simplex_map; RESTRICTION] THEN X_GEN_TAC `x:num->real` THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[singular_face; o_THM; RESTRICTION] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`q - 1`; `i - 1`]) THEN ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[SIMPLICIAL_SIMPLEX; SUBSET; FORALL_IN_IMAGE] THEN ASM_SIMP_TAC[SUB_ADD; LE_1; IN_ELIM_THM] THEN DISCH_THEN(K ALL_TAC) THEN AP_TERM_TAC THEN REWRITE_TAC[PAIR_EQ] THEN CONJ_TAC THENL [ALL_TAC; AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `k:num` THEN REWRITE_TAC[o_THM]] THEN EXPAND_TAC "simp" THEN REWRITE_TAC[oriented_simplex; RESTRICTION] THEN ASM_SIMP_TAC[SUB_ADD; LE_1] THEN ASM_SIMP_TAC[SIMPLICIAL_FACE_IN_STANDARD_SIMPLEX; ADD_SUB; ARITH_RULE `1 <= j + 1`] THEN MAP_EVERY EXPAND_TAC ["vv"; "ww"] THEN REWRITE_TAC[simplicial_face; o_DEF] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN REWRITE_TAC[ARITH_RULE `~(0 = k + 1)`; NOT_SUC] THENL [ONCE_REWRITE_TAC[MESON[] `(if p then x else y) * z:real = if p then x * z else y * z`] THEN SIMP_TAC[REAL_MUL_LZERO; FINITE_NUMSEG; SUM_CASES; SUM_0] THEN REWRITE_TAC[REAL_MUL_LID; REAL_ADD_LID] THEN SIMP_TAC[SUM_CASES; FINITE_NUMSEG; FINITE_RESTRICT] THEN REWRITE_TAC[SUM_0; REAL_ADD_LID] THEN REWRITE_TAC[IN_ELIM_THM; IN_NUMSEG; LE_1; NOT_LE; LE_0; NOT_LT] THEN GEN_REWRITE_TAC (RAND_CONV o RAND_CONV o RAND_CONV) [GSYM o_DEF] THEN W(MP_TAC o PART_MATCH (rand o rand) SUM_IMAGE o rand o rand o snd) THEN REWRITE_TAC[IN_ELIM_THM] THEN ANTS_TAC THENL [ARITH_TAC; DISCH_THEN(SUBST1_TAC o SYM)] THEN REWRITE_TAC[IMAGE; IN_ELIM_THM; ARITH_RULE `(((x <= q + 1 /\ i < x) /\ j <= x) /\ ~(x = j)) /\ y = x - 1 <=> x = y + 1 /\ y <= q /\ i <= y /\ j <= y`] THEN REWRITE_TAC[UNWIND_THM2; GSYM CONJ_ASSOC] THEN W(MP_TAC o PART_MATCH (rand o rand) SUM_UNION o rand o snd) THEN ANTS_TAC THENL [REWRITE_TAC[CONJ_ASSOC; DISJOINT; EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY; IN_INTER; IN_ELIM_THM] THEN CONJ_TAC THENL [ALL_TAC; ARITH_TAC] THEN CONJ_TAC THEN MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `0..q+1` THEN REWRITE_TAC[FINITE_NUMSEG; SUBSET; IN_NUMSEG; IN_ELIM_THM] THEN ARITH_TAC; DISCH_THEN(SUBST1_TAC o SYM)] THEN REWRITE_TAC[ETA_AX] THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_UNION; IN_ELIM_THM] THEN ASM_ARITH_TAC; ASM_CASES_TAC `k:num = j` THEN ASM_REWRITE_TAC[LT_REFL] THENL [CONV_TAC SYM_CONV THEN MATCH_MP_TAC SUM_EQ_0_NUMSEG THEN X_GEN_TAC `m:num` THEN STRIP_TAC THEN REWRITE_TAC[] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[MESON[] `(if p then if q then a else b else if r then a else b) = (if p /\ q \/ ~p /\ r then a else b)`] THEN ONCE_REWRITE_TAC[REAL_ARITH `(if p then a else b) * x:real = if p then a * x else b * x`] THEN REWRITE_TAC[REAL_MUL_LZERO; GSYM SUM_RESTRICT_SET; REAL_MUL_LID] THEN SIMP_TAC[SUM_CASES; FINITE_NUMSEG; FINITE_RESTRICT] THEN REWRITE_TAC[SUM_0; REAL_ADD_LID] THEN GEN_REWRITE_TAC (RAND_CONV o RAND_CONV o RAND_CONV) [GSYM o_DEF] THEN W(MP_TAC o PART_MATCH (rand o rand) SUM_IMAGE o rand o rand o snd) THEN REWRITE_TAC[IN_ELIM_THM] THEN ANTS_TAC THENL [ARITH_TAC; DISCH_THEN(SUBST1_TAC o SYM)] THEN REWRITE_TAC[IN_NUMSEG; ADD1; EQ_ADD_RCANCEL; IN_ELIM_THM] THEN REWRITE_TAC[GSYM CONJ_ASSOC; LE_0; NOT_LE; NOT_LT] THEN REWRITE_TAC[IMAGE; IN_ELIM_THM; ARITH_RULE `j:num <= k /\ ~(k = j) <=> j < k`] THEN REWRITE_TAC[GSYM CONJ_ASSOC; ARITH_RULE `j < k /\ m = k - 1 <=> j <= m /\ k = m + 1`] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r /\ s <=> s /\ p /\ q /\ r`] THEN REWRITE_TAC[UNWIND_THM2; ETA_AX] THEN W(MP_TAC o PART_MATCH (rand o rand) SUM_UNION o rand o snd) THEN ANTS_TAC THENL [REWRITE_TAC[CONJ_ASSOC; DISJOINT; EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY; IN_INTER; IN_ELIM_THM] THEN CONJ_TAC THENL [ALL_TAC; ARITH_TAC] THEN CONJ_TAC THEN MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `0..q+1` THEN REWRITE_TAC[FINITE_NUMSEG; SUBSET; IN_NUMSEG; IN_ELIM_THM] THEN ARITH_TAC; DISCH_THEN(SUBST1_TAC o SYM)] THEN ASM_CASES_TAC `k:num < j` THEN ASM_REWRITE_TAC[] THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_UNION; IN_ELIM_THM; ADD_SUB] THEN (SUBGOAL_THEN `1 <= i` MP_TAC THENL [ASM_ARITH_TAC; ALL_TAC]) THEN SIMP_TAC[ARITH_RULE `1 <= i ==> (j <= i - 1 <=> j < i)`; ARITH_RULE `1 <= i ==> (i - 1 <= k <=> i <= k + 1)`] THENL [ASM_ARITH_TAC; DISCH_TAC] THEN SUBGOAL_THEN `1 <= k` MP_TAC THENL [ASM_ARITH_TAC; SIMP_TAC[SUB_ADD]] THEN SIMP_TAC[ARITH_RULE `1 <= k ==> (k - 1 = j <=> k = j + 1)`] THEN ASM_ARITH_TAC]; REWRITE_TAC[IN_ELIM_THM; IN_NUMSEG; IN_DELETE; LE_0] THEN REWRITE_TAC[ARITH_RULE `i <= q /\ (i <= j /\ j <= q) /\ ~(j = i) <=> i:num <= q /\ j <= q /\ i < j`] THEN MATCH_MP_TAC(MATCH_MP ITERATE_EQ MONOIDAL_FRAG_ADD) THEN REWRITE_TAC[FORALL_PAIR_THM; IN_ELIM_PAIR_THM] THEN MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN STRIP_TAC THEN REWRITE_TAC[ARITH_RULE `SUC j + i = i + j + 1`] THEN AP_TERM_TAC THEN AP_TERM_TAC THEN ASM_SIMP_TAC[SINGULAR_FACE_SIMPLEX_MAP; ARITH_RULE `1 <= q + 1`; ARITH_RULE `j <= q ==> j + 1 <= q + 1`] THEN REWRITE_TAC[ADD_SUB] THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN REWRITE_TAC[simplex_map; RESTRICTION] THEN X_GEN_TAC `x:num->real` THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[singular_face; o_THM; RESTRICTION] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`q - 1`; `i:num`]) THEN ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[SIMPLICIAL_SIMPLEX; SUBSET; FORALL_IN_IMAGE] THEN ASM_SIMP_TAC[SUB_ADD; LE_1; IN_ELIM_THM] THEN DISCH_THEN(K ALL_TAC) THEN AP_TERM_TAC THEN REWRITE_TAC[PAIR_EQ] THEN CONJ_TAC THENL [ALL_TAC; AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `k:num` THEN REWRITE_TAC[o_THM]] THEN EXPAND_TAC "simp" THEN REWRITE_TAC[oriented_simplex; RESTRICTION] THEN ASM_SIMP_TAC[SUB_ADD; LE_1] THEN ASM_SIMP_TAC[SIMPLICIAL_FACE_IN_STANDARD_SIMPLEX; ADD_SUB; LE_ADD_RCANCEL; ARITH_RULE `1 <= j + 1`] THEN MAP_EVERY EXPAND_TAC ["vv"; "ww"] THEN REWRITE_TAC[simplicial_face; o_DEF] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN REWRITE_TAC[ARITH_RULE `~(0 = k + 1)`; NOT_SUC] THENL [ONCE_REWRITE_TAC[MESON[] `(if p then x else y) * z:real = if p then x * z else y * z`] THEN SIMP_TAC[REAL_MUL_LZERO; FINITE_NUMSEG; SUM_CASES; SUM_0] THEN REWRITE_TAC[REAL_MUL_LID; REAL_ADD_LID] THEN SIMP_TAC[SUM_CASES; FINITE_NUMSEG; FINITE_RESTRICT] THEN REWRITE_TAC[SUM_0; REAL_ADD_LID] THEN REWRITE_TAC[IN_ELIM_THM; IN_NUMSEG; LE_1; NOT_LE; LE_0; NOT_LT] THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV o RAND_CONV) [GSYM o_DEF] THEN W(MP_TAC o PART_MATCH (rand o rand) SUM_IMAGE o rand o lhand o snd) THEN REWRITE_TAC[IN_ELIM_THM] THEN ANTS_TAC THENL [ARITH_TAC; DISCH_THEN(SUBST1_TAC o SYM)] THEN REWRITE_TAC[IMAGE; IN_ELIM_THM; ARITH_RULE `(((x <= q + 1 /\ i < x) /\ j + 1 <= x) /\ ~(x = j + 1)) /\ y = x - 1 <=> x = y + 1 /\ y <= q /\ i <= y /\ j < y`] THEN REWRITE_TAC[UNWIND_THM2; GSYM CONJ_ASSOC] THEN W(MP_TAC o PART_MATCH (rand o rand) SUM_UNION o lhand o snd) THEN ANTS_TAC THENL [REWRITE_TAC[CONJ_ASSOC; DISJOINT; EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY; IN_INTER; IN_ELIM_THM] THEN CONJ_TAC THENL [ALL_TAC; ARITH_TAC] THEN CONJ_TAC THEN MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `0..q+1` THEN REWRITE_TAC[FINITE_NUMSEG; SUBSET; IN_NUMSEG; IN_ELIM_THM] THEN ARITH_TAC; DISCH_THEN(SUBST1_TAC o SYM)] THEN REWRITE_TAC[ETA_AX] THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_UNION; IN_ELIM_THM] THEN ASM_ARITH_TAC; ASM_CASES_TAC `k:num = j` THEN ASM_REWRITE_TAC[LT_REFL] THENL [MATCH_MP_TAC SUM_EQ_0_NUMSEG THEN X_GEN_TAC `m:num` THEN STRIP_TAC THEN REWRITE_TAC[] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[MESON[] `(if p then if q then a else b else if r then a else b) = (if p /\ q \/ ~p /\ r then a else b)`] THEN ONCE_REWRITE_TAC[REAL_ARITH `(if p then a else b) * x:real = if p then a * x else b * x`] THEN REWRITE_TAC[REAL_MUL_LZERO; GSYM SUM_RESTRICT_SET; REAL_MUL_LID] THEN SIMP_TAC[SUM_CASES; FINITE_NUMSEG; FINITE_RESTRICT] THEN REWRITE_TAC[SUM_0; REAL_ADD_LID] THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV o RAND_CONV) [GSYM o_DEF] THEN W(MP_TAC o PART_MATCH (rand o rand) SUM_IMAGE o rand o lhand o snd) THEN REWRITE_TAC[IN_ELIM_THM] THEN ANTS_TAC THENL [ARITH_TAC; DISCH_THEN(SUBST1_TAC o SYM)] THEN REWRITE_TAC[IN_NUMSEG; ADD1; EQ_ADD_RCANCEL; IN_ELIM_THM] THEN REWRITE_TAC[GSYM CONJ_ASSOC; LE_0; NOT_LE; NOT_LT] THEN REWRITE_TAC[IMAGE; IN_ELIM_THM; ARITH_RULE `j:num <= k /\ ~(k = j) <=> j < k`] THEN REWRITE_TAC[GSYM CONJ_ASSOC; ARITH_RULE `j + 1 < k /\ m = k - 1 <=> j < m /\ k = m + 1`] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r /\ s <=> s /\ p /\ q /\ r`] THEN REWRITE_TAC[UNWIND_THM2; ETA_AX] THEN W(MP_TAC o PART_MATCH (rand o rand) SUM_UNION o lhand o snd) THEN ANTS_TAC THENL [REWRITE_TAC[CONJ_ASSOC; DISJOINT; EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY; IN_INTER; IN_ELIM_THM] THEN CONJ_TAC THENL [ALL_TAC; ARITH_TAC] THEN CONJ_TAC THEN MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `0..q+1` THEN REWRITE_TAC[FINITE_NUMSEG; SUBSET; IN_NUMSEG; IN_ELIM_THM] THEN ARITH_TAC; DISCH_THEN(SUBST1_TAC o SYM)] THEN ASM_CASES_TAC `k:num < j` THEN ASM_REWRITE_TAC[] THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_UNION; IN_ELIM_THM; ADD_SUB] THEN REWRITE_TAC[ARITH_RULE `i < j /\ k = j - 1 <=> i < j /\ j = k + 1`] THENL [ASM_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `1 <= k` MP_TAC THENL [ASM_ARITH_TAC; SIMP_TAC[SUB_ADD]] THEN SIMP_TAC[ARITH_RULE `1 <= k ==> (k - 1 = j <=> k = j + 1)`] THEN ASM_ARITH_TAC]]);; (* ------------------------------------------------------------------------- *) (* Now actually connect to group theory and set up homology groups. Note *) (* that we define homomogy groups for all *integers* p, since this seems to *) (* avoid some special-case reasoning, though they are trivial for p < 0. *) (* ------------------------------------------------------------------------- *) let chain_group = new_definition `chain_group (p,top:A topology) = free_abelian_group (singular_simplex(p,top))`;; let CHAIN_GROUP = prove (`(!p top:A topology. group_carrier(chain_group(p,top)) = singular_chain(p,top)) /\ (!p top:A topology. group_id(chain_group(p,top)) = frag_0) /\ (!p top:A topology. group_inv(chain_group(p,top)) = frag_neg) /\ (!p top:A topology. group_mul(chain_group(p,top)) = frag_add)`, REWRITE_TAC[chain_group; FREE_ABELIAN_GROUP] THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN REWRITE_TAC[IN; singular_chain]);; let ABELIAN_CHAIN_GROUP = prove (`!p top:A topology. abelian_group(chain_group (p,top))`, REWRITE_TAC[chain_group; ABELIAN_FREE_ABELIAN_GROUP]);; let SUBGROUP_SINGULAR_RELCYCLE = prove (`!p top s:A->bool. singular_relcycle(p,top,s) subgroup_of chain_group(p,top)`, REPEAT GEN_TAC THEN REWRITE_TAC[subgroup_of; CHAIN_GROUP] THEN REWRITE_TAC[SUBSET; IN; SINGULAR_RELCYCLE_0] THEN REWRITE_TAC[SINGULAR_RELCYCLE_ADD; SINGULAR_RELCYCLE_NEG] THEN SIMP_TAC[singular_relcycle]);; let relcycle_group = new_definition `relcycle_group(p,top:A topology,s) = subgroup_generated (chain_group(p,top)) (singular_relcycle(p,top,s))`;; let RELCYCLE_GROUP = prove (`(!p top s:A->bool. group_carrier(relcycle_group(p,top,s)) = singular_relcycle(p,top,s)) /\ (!p top s:A->bool. group_id(relcycle_group(p,top,s)) = frag_0) /\ (!p top s:A->bool. group_inv(relcycle_group(p,top,s)) = frag_neg) /\ (!p top s:A->bool. group_mul(relcycle_group(p,top,s)) = frag_add)`, SIMP_TAC[relcycle_group; CARRIER_SUBGROUP_GENERATED_SUBGROUP; SUBGROUP_SINGULAR_RELCYCLE] THEN REWRITE_TAC[SUBGROUP_GENERATED; CHAIN_GROUP]);; let ABELIAN_RELCYCLE_GROUP = prove (`!p top s:A->bool. abelian_group(relcycle_group(p,top,s))`, SIMP_TAC[relcycle_group; ABELIAN_SUBGROUP_GENERATED; ABELIAN_CHAIN_GROUP]);; let RELCYCLE_GROUP_RESTRICT = prove (`!p top s:A->bool. relcycle_group(p,top,s) = relcycle_group(p,top,topspace top INTER s)`, REWRITE_TAC[relcycle_group; GSYM SINGULAR_RELCYCLE_RESTRICT]);; let relative_homology_group = new_definition `relative_homology_group(p,top:A topology,s) = if p < &0 then singleton_group ARB else quotient_group (relcycle_group(num_of_int p,top,s)) (singular_relboundary(num_of_int p,top,s))`;; let homology_group = new_definition `homology_group(p,top:A topology) = relative_homology_group(p,top,{})`;; let RELATIVE_HOMOLOGY_GROUP_RESTRICT = prove (`!p top s:A->bool. relative_homology_group(p,top,s) = relative_homology_group(p,top,topspace top INTER s)`, REPEAT GEN_TAC THEN REWRITE_TAC[relative_homology_group] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[relcycle_group] THEN REWRITE_TAC[GSYM SINGULAR_RELBOUNDARY_RESTRICT; GSYM SINGULAR_RELCYCLE_RESTRICT]);; let NONTRIVIAL_RELATIVE_HOMOLOGY_GROUP = prove (`!p top s:A->bool. relative_homology_group(&p,top,s) = quotient_group (relcycle_group(p,top,s)) (singular_relboundary(p,top,s))`, REWRITE_TAC[relative_homology_group; INT_ARITH `~(&p:int < &0)`] THEN REWRITE_TAC[NUM_OF_INT_OF_NUM]);; let TRIVIAL_RELATIVE_HOMOLOGY_GROUP = prove (`!p top s:A->bool. p < &0 ==> trivial_group(relative_homology_group(p,top,s))`, SIMP_TAC[relative_homology_group; TRIVIAL_GROUP_SINGLETON_GROUP]);; let SUBGROUP_SINGULAR_RELBOUNDARY = prove (`!p top s:A->bool. singular_relboundary(p,top,s) subgroup_of chain_group(p,top)`, REPEAT GEN_TAC THEN REWRITE_TAC[subgroup_of; CHAIN_GROUP] THEN REWRITE_TAC[SUBSET; IN; SINGULAR_RELBOUNDARY_0] THEN REWRITE_TAC[SINGULAR_RELBOUNDARY_ADD; SINGULAR_RELBOUNDARY_NEG] THEN REWRITE_TAC[SINGULAR_RELBOUNDARY_IMP_CHAIN]);; let SUBGROUP_SINGULAR_RELBOUNDARY_RELCYCLE = prove (`!p top s:A->bool. singular_relboundary(p,top,s) subgroup_of relcycle_group(p,top,s)`, REPEAT GEN_TAC THEN REWRITE_TAC[relcycle_group] THEN MATCH_MP_TAC SUBGROUP_OF_SUBGROUP_GENERATED THEN REWRITE_TAC[SUBGROUP_SINGULAR_RELBOUNDARY; SUBSET; IN] THEN REWRITE_TAC[SINGULAR_RELBOUNDARY_IMP_RELCYCLE]);; let NORMAL_SUBGROUP_SINGULAR_RELBOUNDARY_RELCYCLE = prove (`!p top s:A->bool. singular_relboundary(p,top,s) normal_subgroup_of relcycle_group(p,top,s)`, SIMP_TAC[ABELIAN_GROUP_NORMAL_SUBGROUP; ABELIAN_RELCYCLE_GROUP; SUBGROUP_SINGULAR_RELBOUNDARY_RELCYCLE]);; let RIGHT_COSET_SINGULAR_RELBOUNDARY = prove (`!p top s:A->bool. right_coset (relcycle_group(p,top,s)) (singular_relboundary (p,top,s)) = homologous_rel (p,top,s)`, REPEAT GEN_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN MAP_EVERY X_GEN_TAC [`c1:((num->real)->A)frag`; `c2:((num->real)->A)frag`] THEN GEN_REWRITE_TAC LAND_CONV [GSYM IN] THEN REWRITE_TAC[right_coset; homologous_rel; group_setmul] THEN REWRITE_TAC[IN_IMAGE; RELCYCLE_GROUP; SET_RULE `{f x y | x IN s /\ y IN {a}} = IMAGE (\x. f x a) s`] THEN REWRITE_TAC[UNWIND_THM2; FRAG_MODULE `c2 = frag_add x c1 <=> x = frag_neg(frag_sub c1 c2)`] THEN REWRITE_TAC[IN; SINGULAR_RELBOUNDARY_NEG]);; let RELATIVE_HOMOLOGY_GROUP = prove (`(!p top s:A->bool. group_carrier(relative_homology_group(&p,top,s)) = {homologous_rel (p,top,s) c | c | singular_relcycle (p,top,s) c}) /\ (!p top s:A->bool. group_id(relative_homology_group(&p,top,s)) = singular_relboundary (p,top,s)) /\ (!p top s:A->bool. group_inv(relative_homology_group(&p,top,s)) = \r. {frag_neg c | c IN r}) /\ (!p top s:A->bool. group_mul(relative_homology_group(&p,top,s)) = \r1 r2. {frag_add c1 c2 | c1 IN r1 /\ c2 IN r2})`, REWRITE_TAC[AND_FORALL_THM] THEN REPEAT GEN_TAC THEN REWRITE_TAC[NONTRIVIAL_RELATIVE_HOMOLOGY_GROUP] THEN SIMP_TAC[QUOTIENT_GROUP; RIGHT_COSET_SINGULAR_RELBOUNDARY; RELCYCLE_GROUP; NORMAL_SUBGROUP_SINGULAR_RELBOUNDARY_RELCYCLE] THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[group_setinv; group_setmul; RELCYCLE_GROUP; FUN_EQ_THM]);; let HOMOLOGOUS_REL_EQ_RELBOUNDARY = prove (`!p top (s:A->bool) c. homologous_rel(p,top,s) c = singular_relboundary(p,top,s) <=> singular_relboundary(p,top,s) c`, REPEAT GEN_TAC THEN REWRITE_TAC[FUN_EQ_THM; homologous_rel] THEN MESON_TAC[SINGULAR_RELBOUNDARY_SUB; SINGULAR_RELBOUNDARY_0; FRAG_MODULE `frag_sub c c:((num->real)->A)frag = frag_0`; FRAG_MODULE `d:((num->real)->A)frag = frag_sub c (frag_sub c d)`]);; (* ------------------------------------------------------------------------- *) (* Lift the boundary and induced maps to homology groups. We totalize both *) (* quite aggressively to the appropriate group identity in all "undefined" *) (* situations, which makes several of the properties cleaner and simpler. *) (* ------------------------------------------------------------------------- *) let GROUP_HOMOMORPHISM_CHAIN_BOUNDARY = prove (`!p top s:A->bool. group_homomorphism (relcycle_group(p,top,s), relcycle_group(p-1,subtopology top s,{})) (chain_boundary p)`, REPEAT GEN_TAC THEN REWRITE_TAC[group_homomorphism; RELCYCLE_GROUP; CHAIN_BOUNDARY_0; SUBSET] THEN REWRITE_TAC[CHAIN_BOUNDARY_NEG; CHAIN_BOUNDARY_ADD; FORALL_IN_IMAGE] THEN REWRITE_TAC[IN; singular_relcycle] THEN X_GEN_TAC `c:((num->real)->A)frag` THEN SIMP_TAC[cong; mod_subset; FRAG_MODULE `frag_sub x frag_0 = x`] THEN DISCH_THEN(SUBST1_TAC o MATCH_MP CHAIN_BOUNDARY_BOUNDARY o CONJUNCT1) THEN REWRITE_TAC[SINGULAR_CHAIN_0]);; let [HOM_BOUNDARY_DEFAULT; GROUP_HOMOMORPHISM_HOM_BOUNDARY; HOM_BOUNDARY_CHAIN_BOUNDARY; HOM_BOUNDARY_RESTRICT; HOM_BOUNDARY; HOM_BOUNDARY_TRIVIAL] = let oth = prove (`?d. !p (top:A topology) s. group_homomorphism (relative_homology_group (&p,top,s), homology_group (&(p - 1),subtopology top s)) (d p (top,s)) /\ !c. singular_relcycle (p,top,s) c ==> d p (top,s) (homologous_rel (p,top,s) c) = homologous_rel (p - 1,subtopology top s,{}) (chain_boundary p c)`, REWRITE_TAC[homology_group] THEN REWRITE_TAC[GSYM SKOLEM_THM] THEN REWRITE_TAC[EXISTS_CURRY] THEN REWRITE_TAC[GSYM SKOLEM_THM] THEN REPEAT GEN_TAC THEN MP_TAC(ISPECL [`relcycle_group(p,top:A topology,s)`; `relative_homology_group(&(p-1),subtopology top s:A topology,{})`; `singular_relboundary(p,top:A topology,s)`; `right_coset (relcycle_group(p-1,subtopology top s:A topology,{})) (singular_relboundary(p - 1,subtopology top s,{})) o (chain_boundary p:((num->real)->A)frag->((num->real)->A)frag)`] QUOTIENT_GROUP_UNIVERSAL_EXPLICIT) THEN REWRITE_TAC[GSYM NONTRIVIAL_RELATIVE_HOMOLOGY_GROUP] THEN GEN_REWRITE_TAC LAND_CONV [IMP_CONJ] THEN ANTS_TAC THENL [MATCH_MP_TAC GROUP_HOMOMORPHISM_COMPOSE THEN EXISTS_TAC `relcycle_group (p-1,subtopology top s:A topology,{})` THEN REWRITE_TAC[NONTRIVIAL_RELATIVE_HOMOLOGY_GROUP] THEN ASM_SIMP_TAC[GROUP_HOMOMORPHISM_RIGHT_COSET; NORMAL_SUBGROUP_SINGULAR_RELBOUNDARY_RELCYCLE] THEN REWRITE_TAC[GROUP_HOMOMORPHISM_CHAIN_BOUNDARY]; REWRITE_TAC[NORMAL_SUBGROUP_SINGULAR_RELBOUNDARY_RELCYCLE]] THEN REWRITE_TAC[RIGHT_COSET_SINGULAR_RELBOUNDARY] THEN REWRITE_TAC[RELCYCLE_GROUP] THEN ANTS_TAC THENL [ALL_TAC; REWRITE_TAC[IN; o_THM]] THEN REWRITE_TAC[HOMOLOGOUS_REL_EQ; o_DEF] THEN REWRITE_TAC[homologous_rel; GSYM CHAIN_BOUNDARY_SUB] THEN MAP_EVERY X_GEN_TAC [`c1:((num->real)->A)frag`; `c2:((num->real)->A)frag`] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 (K ALL_TAC) MP_TAC)) THEN SPEC_TAC(`frag_sub c1 c2:((num->real)->A)frag`, `c:((num->real)->A)frag`) THEN GEN_TAC THEN ASM_CASES_TAC `p = 0` THENL [ASM_REWRITE_TAC[chain_boundary; SINGULAR_RELBOUNDARY_0]; ALL_TAC] THEN REWRITE_TAC[singular_relboundary; SINGULAR_BOUNDARY] THEN ASM_SIMP_TAC[SUB_ADD; LE_1; cong; mod_subset] THEN DISCH_THEN(X_CHOOSE_THEN `d:((num->real)->A)frag` STRIP_ASSUME_TAC) THEN EXISTS_TAC `frag_neg(frag_sub (chain_boundary (p + 1) d) c): ((num->real)->A)frag` THEN ASM_SIMP_TAC[SINGULAR_CHAIN_NEG] THEN REWRITE_TAC[CHAIN_BOUNDARY_SUB; CHAIN_BOUNDARY_NEG] THEN REWRITE_TAC[FRAG_MODULE `frag_neg(frag_sub x y) = y <=> x = frag_0`] THEN ASM_MESON_TAC[CHAIN_BOUNDARY_BOUNDARY_ALT]) in let eth = prove (`?d. (!p (top:A topology) s. group_homomorphism (relative_homology_group (p,top,s), homology_group (p - &1,subtopology top s)) (d p (top,s))) /\ (!p (top:A topology) s c. singular_relcycle (p,top,s) c /\ 1 <= p ==> d (&p) (top,s) (homologous_rel (p,top,s) c) = homologous_rel (p - 1,subtopology top s,{}) (chain_boundary p c)) /\ (!p. p <= &0 ==> d p = \q r. ARB)`, MATCH_MP_TAC(MESON[] `(?f. P(\p. if p <= &0 then \q r. ARB else f(num_of_int p))) ==> ?f. P f`) THEN MP_TAC oth THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN DISCH_TAC THEN ASM_SIMP_TAC[INT_OF_NUM_LE; ARITH_RULE `1 <= p ==> ~(p <= 0)`; NUM_OF_INT_OF_NUM] THEN MAP_EVERY X_GEN_TAC [`p:int`; `top:A topology`; `s:A->bool`] THEN COND_CASES_TAC THENL [ASM_SIMP_TAC[homology_group; relative_homology_group; INT_ARITH `p:int <= &0 ==> p - &1 < &0`] THEN REWRITE_TAC[group_homomorphism; SINGLETON_GROUP] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_SING]; FIRST_X_ASSUM(MP_TAC o CONJUNCT1 o SPECL [`num_of_int p`; `top:A topology`; `s:A->bool`]) THEN ASM_SIMP_TAC[INT_OF_NUM_OF_INT; INT_ARITH `~(p:int <= &0) ==> &0 <= p`; GSYM INT_OF_NUM_SUB; GSYM INT_OF_NUM_LE; INT_ARITH `~(p:int <= &0) ==> &1 <= p`]]) in let fth = prove (`?d. (!p (top:A topology) s c. ~(c IN group_carrier(relative_homology_group (p,top,s))) ==> d p (top,s) c = group_id(homology_group (p - &1,subtopology top s))) /\ (!p (top:A topology) s. group_homomorphism (relative_homology_group (p,top,s), homology_group (p - &1,subtopology top s)) (d p (top,s))) /\ (!p (top:A topology) s c. singular_relcycle (p,top,s) c /\ 1 <= p ==> d (&p) (top,s) (homologous_rel (p,top,s) c) = homologous_rel (p - 1,subtopology top s,{}) (chain_boundary p c)) /\ (!p (top:A topology) s. d p (top,s) = d p (top,topspace top INTER s)) /\ (!p (top:A topology) s c. d p (top,s) c IN group_carrier(homology_group (p - &1,subtopology top s))) /\ (!p. p <= &0 ==> d p = \q r. ARB)`, REWRITE_TAC[CONJ_ASSOC] THEN MATCH_MP_TAC(MESON[] `(!x. Q x ==> R x) /\ (?x. P x /\ Q x) ==> ?x. (P x /\ Q x) /\ R x`) THEN CONJ_TAC THENL [GEN_TAC THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `p:int` THEN ASM_CASES_TAC `p:int <= &0` THEN ASM_REWRITE_TAC[homology_group; relative_homology_group] THEN ASM_REWRITE_TAC[INT_ARITH `p - &1:int < &0 <=> p <= &0`] THEN REWRITE_TAC[FUN_EQ_THM; SINGLETON_GROUP; IN_SING] THEN REWRITE_TAC[FORALL_PAIR_THM]; REWRITE_TAC[GSYM CONJ_ASSOC]] THEN MP_TAC eth THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN INTRO_TAC "!d; hom rel triv" THEN EXISTS_TAC `\p (top:A topology,s) c. if c IN group_carrier(relative_homology_group (p,top,s)) then d p (top,topspace top INTER s) c else group_id(homology_group (p - &1,subtopology top s))` THEN REWRITE_TAC[homology_group; GSYM RELATIVE_HOMOLOGY_GROUP_RESTRICT] THEN REWRITE_TAC[SET_RULE `u INTER u INTER s = u INTER s`] THEN REWRITE_TAC[GSYM SUBTOPOLOGY_RESTRICT] THEN REPEAT CONJ_TAC THENL [REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]; MAP_EVERY X_GEN_TAC [`p:int`; `top:A topology`; `s:A->bool`] THEN REMOVE_THEN "hom" (MP_TAC o SPECL [`p:int`; `top:A topology`; `topspace top INTER s:A->bool`]) THEN REWRITE_TAC[homology_group; GSYM RELATIVE_HOMOLOGY_GROUP_RESTRICT] THEN REWRITE_TAC[GSYM SUBTOPOLOGY_RESTRICT] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] GROUP_HOMOMORPHISM_EQ) THEN SIMP_TAC[]; REPEAT GEN_TAC THEN DISCH_THEN(fun th -> STRIP_ASSUME_TAC th THEN MP_TAC th) THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [SINGULAR_RELCYCLE_RESTRICT] THEN DISCH_THEN(ANTE_RES_THEN MP_TAC) THEN REWRITE_TAC[GSYM HOMOLOGOUS_REL_RESTRICT; GSYM SUBTOPOLOGY_RESTRICT] THEN DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[RELATIVE_HOMOLOGY_GROUP] THEN COND_CASES_TAC THEN REWRITE_TAC[] THEN ASM SET_TAC[]; MAP_EVERY X_GEN_TAC [`p:int`; `top:A topology`; `s:A->bool`] THEN REMOVE_THEN "hom" (MP_TAC o SPECL [`p:int`; `top:A topology`; `topspace top INTER s:A->bool`]) THEN REWRITE_TAC[group_homomorphism; homology_group] THEN REWRITE_TAC[GSYM RELATIVE_HOMOLOGY_GROUP_RESTRICT] THEN REWRITE_TAC[GSYM SUBTOPOLOGY_RESTRICT] THEN MATCH_MP_TAC(SET_RULE `(!x. (x IN s ==> f x IN t) ==> P x) ==> IMAGE f s SUBSET t /\ Q ==> !x. P x`) THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[GROUP_ID]]) in let dth = new_specification ["hom_boundary"] fth in CONJUNCTS dth;; let GROUP_HOMOMORPHISM_CHAIN_MAP = prove (`!p top s top' t (f:A->B). continuous_map (top,top') f /\ IMAGE f s SUBSET t ==> group_homomorphism (relcycle_group (p,top,s),relcycle_group (p,top',t)) (chain_map p f)`, REPEAT STRIP_TAC THEN REWRITE_TAC[group_homomorphism; RELCYCLE_GROUP] THEN REWRITE_TAC[CHAIN_MAP_0; CHAIN_MAP_NEG; CHAIN_MAP_ADD] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN] THEN ASM_MESON_TAC[SINGULAR_RELCYCLE_CHAIN_MAP]);; let [HOM_INDUCED_DEFAULT; GROUP_HOMOMORPHISM_HOM_INDUCED; HOM_INDUCED_CHAIN_MAP_GEN; HOM_INDUCED_RESTRICT; HOM_INDUCED; HOM_INDUCED_TRIVIAL] = let oth = prove (`?hom_relmap. (!p top s top' t (f:A->B). continuous_map(top,top') f /\ IMAGE f (topspace top INTER s) SUBSET t ==> group_homomorphism(relative_homology_group(&p,top,s), relative_homology_group(&p,top',t)) (hom_relmap p (top,s) (top',t) f)) /\ (!p top s top' t (f:A->B) c. continuous_map(top,top') f /\ IMAGE f (topspace top INTER s) SUBSET t /\ singular_relcycle (p,top,s) c ==> hom_relmap p (top,s) (top',t) f (homologous_rel (p,top,s) c) = homologous_rel (p,top',t) (chain_map p f c))`, REWRITE_TAC[AND_FORALL_THM] THEN REWRITE_TAC[EXISTS_CURRY; GSYM SKOLEM_THM] THEN REPEAT GEN_TAC THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[IMP_IMP; TAUT `(p ==> q) /\ (p ==> r) <=> p ==> q /\ r`] THEN REWRITE_TAC[RIGHT_EXISTS_IMP_THM] THEN STRIP_TAC THEN MP_TAC(ISPECL [`relcycle_group(p,top:A topology,s)`; `relative_homology_group(&p,top':B topology,t)`; `singular_relboundary(p,top:A topology,s)`; `right_coset (relcycle_group(p,top',t)) (singular_relboundary(p,top',t)) o chain_map p (f:A->B)`] QUOTIENT_GROUP_UNIVERSAL_EXPLICIT) THEN REWRITE_TAC[GSYM NONTRIVIAL_RELATIVE_HOMOLOGY_GROUP] THEN GEN_REWRITE_TAC LAND_CONV [IMP_CONJ] THEN ANTS_TAC THENL [MATCH_MP_TAC GROUP_HOMOMORPHISM_COMPOSE THEN EXISTS_TAC `relcycle_group (p,top':B topology,t)` THEN REWRITE_TAC[NONTRIVIAL_RELATIVE_HOMOLOGY_GROUP] THEN ASM_SIMP_TAC[GROUP_HOMOMORPHISM_RIGHT_COSET; NORMAL_SUBGROUP_SINGULAR_RELBOUNDARY_RELCYCLE] THEN ONCE_REWRITE_TAC[RELCYCLE_GROUP_RESTRICT] THEN MATCH_MP_TAC GROUP_HOMOMORPHISM_CHAIN_MAP THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM (MP_TAC o MATCH_MP CONTINUOUS_MAP_IMAGE_SUBSET_TOPSPACE) THEN ASM SET_TAC[]; REWRITE_TAC[NORMAL_SUBGROUP_SINGULAR_RELBOUNDARY_RELCYCLE]] THEN REWRITE_TAC[RIGHT_COSET_SINGULAR_RELBOUNDARY] THEN REWRITE_TAC[RELCYCLE_GROUP] THEN ANTS_TAC THENL [ALL_TAC; REWRITE_TAC[IN; o_THM]] THEN REWRITE_TAC[HOMOLOGOUS_REL_EQ; o_DEF] THEN REWRITE_TAC[homologous_rel; GSYM CHAIN_MAP_SUB] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC SINGULAR_RELBOUNDARY_CHAIN_MAP THEN MAP_EVERY EXISTS_TAC [`top:A topology`; `topspace top INTER s:A->bool`] THEN ASM_REWRITE_TAC[GSYM SINGULAR_RELBOUNDARY_RESTRICT]) in let eth = prove (`?hom_relmap. (!p top s top' t (f:A->B). continuous_map(top,top') f /\ IMAGE f (topspace top INTER s) SUBSET t ==> group_homomorphism(relative_homology_group(p,top,s), relative_homology_group(p,top',t)) (hom_relmap p (top,s) (top',t) f)) /\ (!p top s top' t (f:A->B) c. continuous_map(top,top') f /\ IMAGE f (topspace top INTER s) SUBSET t /\ singular_relcycle (p,top,s) c ==> hom_relmap (&p) (top,s) (top',t) f (homologous_rel (p,top,s) c) = homologous_rel (p,top',t) (chain_map p f c)) /\ (!p. p < &0 ==> hom_relmap p = \q r s t. ARB)`, MATCH_MP_TAC(MESON[] `(?f. P(\p. if p < &0 then \q r s t. ARB else f(num_of_int p))) ==> ?f. P f`) THEN MP_TAC oth THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN DISCH_TAC THEN ASM_SIMP_TAC[INT_ARITH `~(&p:int < &0)`; NUM_OF_INT_OF_NUM] THEN MAP_EVERY X_GEN_TAC [`p:int`; `top:A topology`; `s:A->bool`; `top':B topology`; `t:B->bool`; `f:A->B`] THEN STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL [ASM_REWRITE_TAC[relative_homology_group; group_homomorphism] THEN REWRITE_TAC[SINGLETON_GROUP] THEN SET_TAC[]; FIRST_X_ASSUM(MP_TAC o SPEC `num_of_int p` o CONJUNCT1) THEN ASM_SIMP_TAC[INT_OF_NUM_OF_INT; INT_ARITH `~(p:int < &0) ==> &0 <= p`]]) in let fth = prove (`?hom_relmap. (!p top s top' t (f:A->B) c. ~(continuous_map (top,top') f /\ IMAGE f (topspace top INTER s) SUBSET t /\ c IN group_carrier(relative_homology_group (p,top,s))) ==> hom_relmap p (top,s) (top',t) f c = group_id(relative_homology_group (p,top',t))) /\ (!p top s top' t (f:A->B). group_homomorphism(relative_homology_group(p,top,s), relative_homology_group(p,top',t)) (hom_relmap p (top,s) (top',t) f)) /\ (!p top s top' t (f:A->B) c. continuous_map(top,top') f /\ IMAGE f (topspace top INTER s) SUBSET t /\ singular_relcycle (p,top,s) c ==> hom_relmap (&p) (top,s) (top',t) f (homologous_rel (p,top,s) c) = homologous_rel (p,top',t) (chain_map p f c)) /\ (!p top s top' t. hom_relmap p (top,s) (top',t) = hom_relmap p (top,topspace top INTER s) (top',topspace top' INTER t)) /\ (!p top s top' f t c. hom_relmap p (top,s) (top',t) f c IN group_carrier(relative_homology_group(p,top',t))) /\ (!p. p < &0 ==> hom_relmap p = \q r s t. ARB)`, REWRITE_TAC[CONJ_ASSOC] THEN MATCH_MP_TAC(MESON[] `(!x. Q x ==> R x) /\ (?x. P x /\ Q x) ==> ?x. (P x /\ Q x) /\ R x`) THEN CONJ_TAC THENL [GEN_TAC THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `p:int` THEN ASM_CASES_TAC `p:int < &0` THEN ASM_REWRITE_TAC[homology_group; relative_homology_group] THEN REWRITE_TAC[FUN_EQ_THM; SINGLETON_GROUP; IN_SING] THEN SIMP_TAC[FORALL_PAIR_THM]; REWRITE_TAC[GSYM CONJ_ASSOC]] THEN MP_TAC eth THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN INTRO_TAC "!hom_relmap; hom rel triv" THEN EXISTS_TAC `\p (top:A topology,s) (top':B topology,t) (f:A->B) c. if continuous_map (top,top') f /\ IMAGE f (topspace top INTER s) SUBSET t /\ c IN group_carrier(relative_homology_group (p,top,s)) then hom_relmap p (top,topspace top INTER s) (top',topspace top' INTER t) f c else group_id(relative_homology_group (p,top',t))` THEN REWRITE_TAC[homology_group; GSYM RELATIVE_HOMOLOGY_GROUP_RESTRICT] THEN REWRITE_TAC[SET_RULE `u INTER u INTER s = u INTER s`] THEN REWRITE_TAC[GSYM SUBTOPOLOGY_RESTRICT] THEN REPEAT CONJ_TAC THENL [REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]; INTRO_TAC "!p top s top' t f" THEN ASM_CASES_TAC ` continuous_map (top,top') (f:A->B)` THEN ASM_REWRITE_TAC[GROUP_HOMOMORPHISM_TRIVIAL] THEN ASM_CASES_TAC `IMAGE (f:A->B) (topspace top INTER s) SUBSET t` THEN ASM_REWRITE_TAC[GROUP_HOMOMORPHISM_TRIVIAL] THEN REMOVE_THEN "hom" (MP_TAC o SPECL [`p:int`; `top:A topology`; `topspace top INTER s:A->bool`; `top':B topology`; `topspace top' INTER t:B->bool`; `f:A->B`]) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[continuous_map]) THEN ASM SET_TAC[]; REWRITE_TAC[GSYM RELATIVE_HOMOLOGY_GROUP_RESTRICT] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] GROUP_HOMOMORPHISM_EQ) THEN SIMP_TAC[]]; REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REPEAT GEN_TAC THEN REPEAT DISCH_TAC THEN ASM_REWRITE_TAC[] THEN REMOVE_THEN "rel" (MP_TAC o SPECL [`p:num`; `top:A topology`; `topspace top INTER s:A->bool`; `top':B topology`; `topspace top' INTER t:B->bool`; `f:A->B`]) THEN ASM_REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN ANTS_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[continuous_map]) THEN ASM SET_TAC[]; REWRITE_TAC[GSYM SINGULAR_RELCYCLE_RESTRICT]] THEN SIMP_TAC[GSYM HOMOLOGOUS_REL_RESTRICT] THEN DISCH_THEN(K ALL_TAC) THEN GEN_TAC THEN REWRITE_TAC[RELATIVE_HOMOLOGY_GROUP] THEN COND_CASES_TAC THEN REWRITE_TAC[] THEN ASM SET_TAC[]; REPEAT GEN_TAC THEN REPEAT ABS_TAC THEN AP_THM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[continuous_map] THEN SET_TAC[]; INTRO_TAC "!p top s top' t f c" THEN COND_CASES_TAC THEN REWRITE_TAC[GROUP_ID] THEN REMOVE_THEN "hom" (MP_TAC o SPECL [`p:int`; `top:A topology`; `topspace top INTER s:A->bool`; `top':B topology`; `topspace top' INTER t:B->bool`; `f:A->B`]) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[continuous_map]) THEN ASM SET_TAC[]; REWRITE_TAC[group_homomorphism] THEN REWRITE_TAC[GSYM RELATIVE_HOMOLOGY_GROUP_RESTRICT] THEN ASM SET_TAC[]]]) in let dth = new_specification ["hom_induced"] fth in CONJUNCTS dth;; let HOM_INDUCED_CHAIN_MAP = prove (`!p top s top' t (f:A->B) c. continuous_map (top,top') f /\ IMAGE f s SUBSET t /\ singular_relcycle (p,top,s) c ==> hom_induced (&p) (top,s) (top',t) f (homologous_rel (p,top,s) c) = homologous_rel (p,top',t) (chain_map p f c)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOM_INDUCED_CHAIN_MAP_GEN THEN ASM SET_TAC[]);; let HOM_INDUCED_EQ = prove (`!p top s top' t (f:A->B) g. (!x. x IN topspace top ==> f x = g x) ==> hom_induced p (top,s) (top',t) f = hom_induced p (top,s) (top',t) g`, MATCH_MP_TAC(MESON[INT_OF_NUM_OF_INT; INT_NOT_LT] `(!x. x < &0 ==> P x) /\ (!p. P(&p)) ==> !x:int. P x`) THEN CONJ_TAC THENL [SIMP_TAC[HOM_INDUCED_TRIVIAL]; ALL_TAC] THEN REPEAT STRIP_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `c:((num->real)->A)frag->bool` THEN SUBGOAL_THEN `continuous_map (top,top') (f:A->B) /\ IMAGE f (topspace top INTER s) SUBSET t /\ c IN group_carrier (relative_homology_group (&p,top,s)) <=> continuous_map (top,top') (g:A->B) /\ IMAGE g (topspace top INTER s) SUBSET t /\ c IN group_carrier (relative_homology_group (&p,top,s))` MP_TAC THENL [BINOP_TAC THENL [ASM_MESON_TAC[CONTINUOUS_MAP_EQ]; ASM SET_TAC[]]; ALL_TAC] THEN REWRITE_TAC[TAUT `(p <=> q) <=> ~p /\ ~q \/ p /\ q`] THEN STRIP_TAC THENL [ASM_MESON_TAC[HOM_INDUCED_DEFAULT]; ALL_TAC] THEN UNDISCH_TAC `c IN group_carrier (relative_homology_group (&p,top,s:A->bool))` THEN SPEC_TAC(`c:((num->real)->A)frag->bool`,`c:((num->real)->A)frag->bool`) THEN REWRITE_TAC[RELATIVE_HOMOLOGY_GROUP; FORALL_IN_GSPEC] THEN ONCE_REWRITE_TAC[HOMOLOGOUS_REL_RESTRICT; SINGULAR_RELCYCLE_RESTRICT] THEN ONCE_REWRITE_TAC[HOM_INDUCED_RESTRICT] THEN REPEAT STRIP_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) HOM_INDUCED_CHAIN_MAP o lhand o snd) THEN ASM_REWRITE_TAC[SUBSET_INTER] THEN ANTS_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[continuous_map]) THEN ASM SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN W(MP_TAC o PART_MATCH (lhand o rand) HOM_INDUCED_CHAIN_MAP o rand o snd) THEN ASM_REWRITE_TAC[SUBSET_INTER] THEN ANTS_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[continuous_map]) THEN ASM SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN AP_TERM_TAC THEN MATCH_MP_TAC CHAIN_MAP_EQ THEN ASM_MESON_TAC[singular_relcycle; SINGULAR_RELCYCLE_RESTRICT]);; (* ------------------------------------------------------------------------- *) (* First prove we get functors into abelian groups with the boundary map *) (* being a natural transformation between them, and prove Eilenberg-Steenrod *) (* axioms (we also prove additivity a bit later on if one counts that). *) (* *) (* 1. Exact sequence from the inclusions and boundary map *) (* H_{p+1}(X) --(j')--> H_{p+1}(X,A) --(d')--> H_p(A) --(i')--> H_p(X) *) (* *) (* 2. Dimension axiom: H_p(X) is trivial for one-point X and p =/= 0 *) (* *) (* 3. Homotopy invariance of the induced map *) (* *) (* 4. Excision: inclusion (X - U,A - U) --(i')--> (X,A) induces an *) (* isomorphism when cl(U) SUBSET int(A) *) (* ------------------------------------------------------------------------- *) let ABELIAN_RELATIVE_HOMOLOGY_GROUP = prove (`!p top s:A->bool. abelian_group(relative_homology_group(p,top,s))`, REPEAT GEN_TAC THEN REWRITE_TAC[relative_homology_group] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[ABELIAN_SINGLETON_GROUP] THEN SIMP_TAC[ABELIAN_QUOTIENT_GROUP; SUBGROUP_SINGULAR_RELBOUNDARY_RELCYCLE; ABELIAN_RELCYCLE_GROUP]);; let HOM_INDUCED_ID_GEN = prove (`!p top (f:A->A) s c. continuous_map(top,top) f /\ (!x. x IN topspace top ==> f x = x) /\ c IN group_carrier (relative_homology_group (p,top,s)) ==> hom_induced p (top,s) (top,s) f c = c`, MATCH_MP_TAC(MESON[INT_OF_NUM_OF_INT; INT_NOT_LT] `(!x. x < &0 ==> P x) /\ (!p. P(&p)) ==> !x:int. P x`) THEN CONJ_TAC THENL [SIMP_TAC[relative_homology_group; SINGLETON_GROUP; HOM_INDUCED_TRIVIAL; IN_SING]; ALL_TAC] THEN REWRITE_TAC[NONTRIVIAL_RELATIVE_HOMOLOGY_GROUP] THEN SIMP_TAC[QUOTIENT_GROUP; NORMAL_SUBGROUP_SINGULAR_RELBOUNDARY_RELCYCLE] THEN REWRITE_TAC[FORALL_IN_GSPEC; IMP_CONJ; RIGHT_FORALL_IMP_THM; RIGHT_COSET_SINGULAR_RELBOUNDARY] THEN REWRITE_TAC[RELCYCLE_GROUP; IN] THEN REPEAT STRIP_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) HOM_INDUCED_CHAIN_MAP_GEN o lhand o snd) THEN ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN AP_TERM_TAC THEN MATCH_MP_TAC CHAIN_MAP_ID_GEN THEN ASM_MESON_TAC[singular_relcycle; IN]);; let HOM_INDUCED_ID = prove (`!p top (s:A->bool) c. c IN group_carrier (relative_homology_group (p,top,s)) ==> hom_induced p (top,s) (top,s) (\x. x) c = c`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOM_INDUCED_ID_GEN THEN ASM_SIMP_TAC[CONTINUOUS_MAP_ID]);; let HOM_INDUCED_COMPOSE = prove (`!p top s top' t top'' u (f:A->B) (g:B->C). continuous_map(top,top') f /\ IMAGE f s SUBSET t /\ continuous_map(top',top'') g /\ IMAGE g t SUBSET u ==> hom_induced p (top,s) (top'',u) (g o f) = hom_induced p (top',t) (top'',u) g o hom_induced p (top,s) (top',t) f`, MATCH_MP_TAC(MESON[INT_OF_NUM_OF_INT; INT_NOT_LT] `(!x. x < &0 ==> P x) /\ (!p. P(&p)) ==> !x:int. P x`) THEN CONJ_TAC THENL [SIMP_TAC[HOM_INDUCED_TRIVIAL; o_DEF]; ALL_TAC] THEN REPEAT STRIP_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN MATCH_MP_TAC(MESON[] `!P. (!x. ~P x ==> Q x) /\ (!x. P x ==> Q x) ==> !x. Q x`) THEN EXISTS_TAC `\c. c IN group_carrier(relative_homology_group(&p,top:A topology,s))` THEN CONJ_TAC THENL [SIMP_TAC[HOM_INDUCED_DEFAULT; o_DEF; REWRITE_RULE[group_homomorphism] GROUP_HOMOMORPHISM_HOM_INDUCED]; REWRITE_TAC[RELATIVE_HOMOLOGY_GROUP; FORALL_IN_GSPEC; o_THM]] THEN X_GEN_TAC `c:((num->real)->A)frag` THEN DISCH_TAC THEN ASM_SIMP_TAC[HOM_INDUCED_CHAIN_MAP] THEN W(MP_TAC o PART_MATCH (lhand o rand) HOM_INDUCED_CHAIN_MAP o rand o snd) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_MESON_TAC[SINGULAR_RELCYCLE_CHAIN_MAP]; DISCH_THEN SUBST1_TAC] THEN W(MP_TAC o PART_MATCH (lhand o rand) HOM_INDUCED_CHAIN_MAP o lhand o snd) THEN ASM_REWRITE_TAC[IMAGE_o] THEN ANTS_TAC THENL [CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_MAP_COMPOSE]; ASM SET_TAC[]]; DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[CHAIN_MAP_COMPOSE; o_THM]]);; let NATURALITY_HOM_INDUCED = prove (`!p top s top' t (f:A->B). continuous_map(top,top') f /\ IMAGE f s SUBSET t ==> hom_boundary p (top',t) o hom_induced p (top,s) (top',t) f = hom_induced (p - &1) (subtopology top s,{}) (subtopology top' t,{}) f o hom_boundary p (top,s)`, X_GEN_TAC `q:int` THEN ASM_CASES_TAC `q:int <= &0` THENL [ASM_SIMP_TAC[o_DEF; HOM_BOUNDARY_TRIVIAL; HOM_INDUCED_TRIVIAL; INT_ARITH `q:int <= &0 ==> q - &1 < &0`]; FIRST_ASSUM(MP_TAC o MATCH_MP (INT_ARITH `~(q:int <= &0) ==> &0 <= q /\ &1 <= q`)) THEN SPEC_TAC(`q:int`,`q:int`) THEN REWRITE_TAC[IMP_CONJ; GSYM INT_FORALL_POS; INT_OF_NUM_LE; LE_0] THEN X_GEN_TAC `p:num` THEN DISCH_TAC] THEN REPEAT STRIP_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN MATCH_MP_TAC(MESON[] `!P. (!x. ~P x ==> Q x) /\ (!x. P x ==> Q x) ==> !x. Q x`) THEN EXISTS_TAC `\c. c IN group_carrier(relative_homology_group(&p,top:A topology,s))` THEN CONJ_TAC THENL [SIMP_TAC[HOM_INDUCED_DEFAULT; HOM_BOUNDARY_DEFAULT; o_DEF; homology_group; REWRITE_RULE[group_homomorphism] (CONJ GROUP_HOMOMORPHISM_HOM_INDUCED GROUP_HOMOMORPHISM_HOM_BOUNDARY)]; REWRITE_TAC[RELATIVE_HOMOLOGY_GROUP; FORALL_IN_GSPEC; o_THM]] THEN X_GEN_TAC `c:((num->real)->A)frag` THEN DISCH_TAC THEN ASM_SIMP_TAC[HOM_BOUNDARY_CHAIN_BOUNDARY; HOM_INDUCED_CHAIN_MAP] THEN W(MP_TAC o PART_MATCH (lhand o rand) HOM_BOUNDARY_CHAIN_BOUNDARY o lhand o snd) THEN ANTS_TAC THENL [ASM_MESON_TAC[SINGULAR_RELCYCLE_CHAIN_MAP]; ALL_TAC] THEN DISCH_THEN SUBST1_TAC THEN ASM_SIMP_TAC[INT_OF_NUM_SUB] THEN FIRST_ASSUM(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC o REWRITE_RULE[singular_relcycle]) THEN SIMP_TAC[cong; mod_subset; FRAG_MODULE `frag_sub x frag_0 = x`] THEN DISCH_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) HOM_INDUCED_CHAIN_MAP o rand o snd) THEN ASM_REWRITE_TAC[IMAGE_CLAUSES; SUBSET_REFL] THEN ANTS_TAC THENL [ASM_REWRITE_TAC[SINGULAR_CYCLE] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[CHAIN_BOUNDARY_BOUNDARY]] THEN ASM_SIMP_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY; TOPSPACE_SUBTOPOLOGY; CONTINUOUS_MAP_FROM_SUBTOPOLOGY] THEN ASM SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN AP_TERM_TAC THEN MATCH_MP_TAC CHAIN_BOUNDARY_CHAIN_MAP THEN ASM_MESON_TAC[]);; let HOMOLOGY_EXACTNESS_AXIOM_1 = prove (`!p top s:A->bool. group_exactness(homology_group(p,top), relative_homology_group(p,top,s), homology_group(p - &1,subtopology top s)) (hom_induced p (top,{}) (top,s) (\x. x), hom_boundary p (top,s))`, REWRITE_TAC[group_exactness; GROUP_HOMOMORPHISM_HOM_BOUNDARY] THEN SIMP_TAC[homology_group; GROUP_HOMOMORPHISM_HOM_INDUCED; CONTINUOUS_MAP_ID; IMAGE_CLAUSES; EMPTY_SUBSET] THEN REWRITE_TAC[group_image; group_kernel] THEN MATCH_MP_TAC(MESON[INT_OF_NUM_OF_INT; INT_NOT_LT] `(!x. x < &0 ==> P x) /\ (!p. P(&p)) ==> !x:int. P x`) THEN CONJ_TAC THENL [SIMP_TAC[relative_homology_group; SINGLETON_GROUP; HOM_INDUCED_TRIVIAL; HOM_BOUNDARY_TRIVIAL; INT_ARITH `q:int < &0 ==> q <= &0 /\ q - &1 < &0`] THEN SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC num_INDUCTION THEN CONJ_TAC THENL [SIMP_TAC[HOM_BOUNDARY_TRIVIAL; INT_LE_REFL] THEN SIMP_TAC[relative_homology_group; INT_ARITH `&0 - &1:int < &0`] THEN REWRITE_TAC[GSYM relative_homology_group; SINGLETON_GROUP] THEN REPEAT GEN_TAC THEN REWRITE_TAC[SET_RULE `s = {x | x IN t} <=> s SUBSET t /\ t SUBSET s`] THEN MP_TAC(ISPECL [`&0:int`; `top:A topology`; `{}:A->bool`; `top:A topology`; `s:A->bool`; `\x:A. x`] GROUP_HOMOMORPHISM_HOM_INDUCED) THEN REWRITE_TAC[CONTINUOUS_MAP_ID; IMAGE_CLAUSES; EMPTY_SUBSET] THEN SIMP_TAC[group_homomorphism; homology_group] THEN DISCH_THEN(K ALL_TAC) THEN REWRITE_TAC[SUBSET; RELATIVE_HOMOLOGY_GROUP; IN_IMAGE] THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `c:((num->real)->A)frag` THEN REWRITE_TAC[singular_relcycle] THEN STRIP_TAC THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[EXISTS_IN_GSPEC] THEN EXISTS_TAC `c:((num->real)->A)frag` THEN ASM_REWRITE_TAC[singular_relcycle; chain_boundary; MOD_SUBSET_REFL] THEN ASM_SIMP_TAC[HOM_INDUCED_CHAIN_MAP; CONTINUOUS_MAP_ID; IMAGE_CLAUSES; EMPTY_SUBSET; singular_relcycle; chain_boundary; MOD_SUBSET_REFL] THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC CHAIN_MAP_ID THEN ASM_MESON_TAC[]; ALL_TAC] THEN X_GEN_TAC `p:num` THEN DISCH_THEN(K ALL_TAC) THEN REPEAT GEN_TAC THEN REWRITE_TAC[GSYM INT_OF_NUM_SUC; INT_ARITH `(p + &1) - &1:int = p`] THEN REWRITE_TAC[INT_OF_NUM_ADD; RELATIVE_HOMOLOGY_GROUP] THEN REWRITE_TAC[SET_RULE `{x | x IN {f y | P y} /\ Q x} = {f y |y| P y /\ Q(f y)}`] THEN REWRITE_TAC[SET_RULE `IMAGE g {f y | P y} = { g(f y) | P y}`] THEN TRANS_TAC EQ_TRANS `{ homologous_rel (p+1,top:A topology,s) c | c | singular_relcycle (p+1,top,{}) c}` THEN CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `(!x. P x ==> f x = g x) ==> {f x | P x} = {g x | P x}`) THEN X_GEN_TAC `c:((num->real)->A)frag` THEN DISCH_TAC THEN ASM_SIMP_TAC[HOM_INDUCED_CHAIN_MAP; CONTINUOUS_MAP_ID; IMAGE_CLAUSES; CONTINUOUS_MAP_FROM_SUBTOPOLOGY; EMPTY_SUBSET] THEN AP_TERM_TAC THEN MATCH_MP_TAC CHAIN_MAP_ID THEN ASM_MESON_TAC[singular_relcycle]; ALL_TAC] THEN TRANS_TAC EQ_TRANS `{ homologous_rel (p+1,top:A topology,s) c | c | singular_relcycle (p+1,top,s) c /\ singular_relboundary (p,subtopology top s,{}) (chain_boundary (p+1) c)}` THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC(SET_RULE `(!x. P x ==> (Q x <=> R x)) ==> {f x | P x /\ Q x} = {f x | P x /\ R x}`) THEN SIMP_TAC[HOM_BOUNDARY_CHAIN_BOUNDARY; ARITH_RULE `1 <= p + 1`] THEN X_GEN_TAC `c:((num->real)->A)frag` THEN DISCH_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN GEN_REWRITE_TAC RAND_CONV [GSYM FUN_EQ_THM] THEN REWRITE_TAC[ETA_AX; ADD_SUB; HOMOLOGOUS_REL_EQ_RELBOUNDARY] THEN REWRITE_TAC[SINGULAR_BOUNDARY]] THEN MATCH_MP_TAC(SET_RULE `(!x. P x ==> ?y. Q y /\ f x = f y) /\ (!x. Q x ==> ?y. P y /\ f x = f y) ==> {f x | P x} = {f x | Q x}`) THEN REWRITE_TAC[HOMOLOGOUS_REL_EQ] THEN CONJ_TAC THEN X_GEN_TAC `c:((num->real)->A)frag` THEN STRIP_TAC THENL [EXISTS_TAC `c:((num->real)->A)frag` THEN REWRITE_TAC[HOMOLOGOUS_REL_REFL] THEN RULE_ASSUM_TAC(REWRITE_RULE[SINGULAR_CYCLE]) THEN ASM_REWRITE_TAC[SINGULAR_RELBOUNDARY_0; singular_relcycle] THEN REWRITE_TAC[cong; mod_subset; FRAG_MODULE `frag_sub x x = frag_0`] THEN REWRITE_TAC[SINGULAR_CHAIN_0]; FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SINGULAR_BOUNDARY]) THEN DISCH_THEN(X_CHOOSE_THEN `d:((num->real)->A)frag` STRIP_ASSUME_TAC) THEN EXISTS_TAC `frag_sub c d:((num->real)->A)frag` THEN ASM_REWRITE_TAC[SINGULAR_CYCLE] THEN ASM_REWRITE_TAC[CHAIN_BOUNDARY_SUB; FRAG_MODULE `frag_sub x x = frag_0`] THEN CONJ_TAC THENL [MATCH_MP_TAC SINGULAR_CHAIN_SUB THEN CONJ_TAC THENL [ASM_MESON_TAC[singular_relcycle]; ALL_TAC] THEN ASM_MESON_TAC[SINGULAR_CHAIN_SUBTOPOLOGY]; REWRITE_TAC[homologous_rel; singular_relboundary] THEN EXISTS_TAC `frag_0:((num->real)->A)frag` THEN REWRITE_TAC[SINGULAR_CHAIN_0; CHAIN_BOUNDARY_0] THEN ASM_REWRITE_TAC[cong; mod_subset; SINGULAR_CHAIN_NEG; FRAG_MODULE `frag_sub frag_0 (frag_sub c (frag_sub c d)) = frag_neg d`]]]);; let HOMOLOGY_EXACTNESS_AXIOM_2 = prove (`!p top s:A->bool. group_exactness(relative_homology_group(p,top,s), homology_group(p - &1,subtopology top s), homology_group(p - &1,top)) (hom_boundary p (top,s), hom_induced (p - &1) (subtopology top s,{}) (top,{}) (\x. x))`, REWRITE_TAC[group_exactness; GROUP_HOMOMORPHISM_HOM_BOUNDARY] THEN SIMP_TAC[homology_group; GROUP_HOMOMORPHISM_HOM_INDUCED; CONTINUOUS_MAP_FROM_SUBTOPOLOGY; GROUP_HOMOMORPHISM_HOM_BOUNDARY; CONTINUOUS_MAP_ID; IMAGE_CLAUSES; EMPTY_SUBSET] THEN REWRITE_TAC[group_image; group_kernel] THEN X_GEN_TAC `q:int` THEN ASM_CASES_TAC `q:int <= &0` THENL [ASM_SIMP_TAC[HOM_BOUNDARY_TRIVIAL; HOM_INDUCED_TRIVIAL; INT_ARITH `q:int <= &0 ==> q - &1 < &0`; relative_homology_group] THEN REWRITE_TAC[GSYM relative_homology_group; SINGLETON_GROUP] THEN REPEAT GEN_TAC THEN MATCH_MP_TAC(SET_RULE `(?x. x IN s) ==> IMAGE (\x. a) s = {x | x IN {a}}`) THEN MESON_TAC[GROUP_ID]; FIRST_ASSUM(MP_TAC o MATCH_MP (INT_ARITH `~(q:int <= &0) ==> &0 <= q /\ &1 <= q`)) THEN SPEC_TAC(`q:int`,`q:int`) THEN REWRITE_TAC[IMP_CONJ; GSYM INT_FORALL_POS; INT_OF_NUM_LE; LE_0] THEN MATCH_MP_TAC num_INDUCTION THEN CONV_TAC NUM_REDUCE_CONV] THEN X_GEN_TAC `p:num` THEN REPEAT(DISCH_THEN(K ALL_TAC)) THEN REWRITE_TAC[GSYM INT_OF_NUM_SUC; INT_ARITH `(p + &1) - &1:int = p`] THEN POP_ASSUM_LIST(K ALL_TAC) THEN REWRITE_TAC[INT_OF_NUM_ADD] THEN REPEAT GEN_TAC THEN REWRITE_TAC[homology_group; RELATIVE_HOMOLOGY_GROUP] THEN REWRITE_TAC[SET_RULE `{x | x IN {f y | P y} /\ Q x} = {f y |y| P y /\ Q(f y)}`] THEN REWRITE_TAC[SET_RULE `IMAGE g {f y | P y} = { g(f y) | P y}`] THEN TRANS_TAC EQ_TRANS `{ homologous_rel (p,subtopology top s,{}) (chain_boundary (p + 1) c) | c | singular_relcycle (p+1,top:A topology,s) c}` THEN CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `(!x. P x ==> f x = g x) ==> {f x | P x} = {g x | P x}`) THEN X_GEN_TAC `c:((num->real)->A)frag` THEN DISCH_TAC THEN ASM_SIMP_TAC[HOM_BOUNDARY_CHAIN_BOUNDARY; ADD_SUB; ARITH_RULE `1 <= p + 1`] THEN REWRITE_TAC[FUN_EQ_THM]; ALL_TAC] THEN TRANS_TAC EQ_TRANS `{ homologous_rel (p,subtopology top s:A topology,{}) c | c | singular_relcycle (p,subtopology top s,{}) c /\ singular_relboundary (p,top,{}) c}` THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC(SET_RULE `(!x. P x ==> (Q x <=> R x)) ==> {f x | P x /\ Q x} = {f x | P x /\ R x}`) THEN SIMP_TAC[HOM_INDUCED_CHAIN_MAP; CONTINUOUS_MAP_ID; IMAGE_CLAUSES; CONTINUOUS_MAP_FROM_SUBTOPOLOGY; EMPTY_SUBSET] THEN REWRITE_TAC[HOMOLOGOUS_REL_EQ_RELBOUNDARY] THEN REPEAT STRIP_TAC THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC CHAIN_MAP_ID THEN ASM_MESON_TAC[singular_relcycle]] THEN REWRITE_TAC[SINGULAR_BOUNDARY] THEN REWRITE_TAC[SET_RULE `{f x | P x /\ ?y. Q y /\ g y = x} = {f(g z) |z| Q z /\ P(g z)}`] THEN REWRITE_TAC[SINGULAR_CYCLE] THEN REWRITE_TAC[singular_relcycle; GSYM CONJ_ASSOC] THEN MATCH_MP_TAC(SET_RULE `(!x. P x ==> (Q x <=> R x)) ==> {f x | P x /\ Q x} = {f x | P x /\ R x}`) THEN X_GEN_TAC `c:((num->real)->A)frag` THEN DISCH_TAC THEN FIRST_ASSUM(SUBST1_TAC o MATCH_MP CHAIN_BOUNDARY_BOUNDARY_ALT) THEN REWRITE_TAC[ADD_SUB; cong; mod_subset; FRAG_MODULE `frag_sub x frag_0 = x`]);; let HOMOLOGY_EXACTNESS_AXIOM_3 = prove (`!p top s:A->bool. group_exactness(homology_group(p,subtopology top s), homology_group(p,top), relative_homology_group(p,top,s)) (hom_induced p (subtopology top s,{}) (top,{}) (\x. x), hom_induced p (top,{}) (top,s) (\x. x))`, MATCH_MP_TAC(MESON[INT_OF_NUM_OF_INT; INT_NOT_LT] `(!x. x < &0 ==> P x) /\ (!p. P(&p)) ==> !x:int. P x`) THEN CONJ_TAC THENL [SIMP_TAC[group_exactness; homology_group; relative_homology_group; HOM_INDUCED_TRIVIAL; group_image; group_kernel; group_homomorphism; SINGLETON_GROUP] THEN SET_TAC[]; ALL_TAC] THEN REPEAT GEN_TAC THEN REWRITE_TAC[homology_group; group_exactness] THEN REWRITE_TAC[GROUP_HOMOMORPHISM_HOM_INDUCED] THEN REWRITE_TAC[group_image; group_kernel; RELATIVE_HOMOLOGY_GROUP] THEN REWRITE_TAC[SET_RULE `{x | x IN {f y | P y} /\ Q x} = {f y |y| P y /\ Q(f y)}`] THEN REWRITE_TAC[SET_RULE `IMAGE g {f y | P y} = { g(f y) | P y}`] THEN TRANS_TAC EQ_TRANS `{ homologous_rel (p,top:A topology,{}) c | c | singular_relcycle (p,subtopology top s,{}) c}` THEN CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `(!x. P x ==> f x = g x) ==> {f x | P x} = {g x | P x}`) THEN X_GEN_TAC `c:((num->real)->A)frag` THEN DISCH_TAC THEN ASM_SIMP_TAC[HOM_INDUCED_CHAIN_MAP; CONTINUOUS_MAP_ID; IMAGE_CLAUSES; CONTINUOUS_MAP_FROM_SUBTOPOLOGY; EMPTY_SUBSET] THEN AP_TERM_TAC THEN MATCH_MP_TAC CHAIN_MAP_ID THEN ASM_MESON_TAC[singular_relcycle]; ALL_TAC] THEN TRANS_TAC EQ_TRANS `{ homologous_rel (p,top:A topology,{}) c | c | singular_relcycle (p,top,{}) c /\ singular_relboundary (p,top,s) c}` THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC(SET_RULE `(!x. P x ==> (Q x <=> R x)) ==> {f x | P x /\ Q x} = {f x | P x /\ R x}`) THEN REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM HOMOLOGOUS_REL_EQ_RELBOUNDARY] THEN AP_THM_TAC THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN ASM_SIMP_TAC[HOM_INDUCED_CHAIN_MAP; CONTINUOUS_MAP_ID; IMAGE_CLAUSES; EMPTY_SUBSET] THEN AP_TERM_TAC THEN MATCH_MP_TAC CHAIN_MAP_ID THEN ASM_MESON_TAC[singular_relcycle]] THEN MATCH_MP_TAC(SET_RULE `(!x. P x ==> ?y. Q y /\ f x = f y) /\ (!x. Q x ==> ?y. P y /\ f x = f y) ==> {f x | P x} = {f x | Q x}`) THEN REWRITE_TAC[HOMOLOGOUS_REL_EQ] THEN CONJ_TAC THEN X_GEN_TAC `c:((num->real)->A)frag` THEN STRIP_TAC THENL [EXISTS_TAC `c:((num->real)->A)frag` THEN REWRITE_TAC[HOMOLOGOUS_REL_REFL] THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[GSYM SUBTOPOLOGY_TOPSPACE] THEN MATCH_MP_TAC SINGULAR_CYCLE_MONO THEN EXISTS_TAC `topspace top INTER s:A->bool` THEN REWRITE_TAC[INTER_SUBSET; GSYM SUBTOPOLOGY_RESTRICT] THEN ASM_REWRITE_TAC[GSYM SINGULAR_CYCLE]; MATCH_MP_TAC SINGULAR_CHAIN_IMP_RELBOUNDARY THEN ASM_MESON_TAC[singular_relcycle]]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SINGULAR_RELBOUNDARY_ALT]) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`e:((num->real)->A)frag`; `d:((num->real)->A)frag`] THEN STRIP_TAC THEN EXISTS_TAC `frag_neg d:((num->real)->A)frag` THEN REWRITE_TAC[homologous_rel; SINGULAR_RELCYCLE_NEG; FRAG_MODULE `frag_sub c (frag_neg d) = frag_add c d`] THEN CONJ_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[SINGULAR_CYCLE]) THEN ASM_REWRITE_TAC[SINGULAR_CYCLE] THEN FIRST_ASSUM(SUBST1_TAC o MATCH_MP (FRAG_MODULE `e = frag_add c d ==> d = frag_sub e c`)) THEN ASM_REWRITE_TAC[CHAIN_BOUNDARY_SUB] THEN REWRITE_TAC[FRAG_MODULE `frag_sub x frag_0 = x`] THEN ASM_MESON_TAC[CHAIN_BOUNDARY_BOUNDARY_ALT]; FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN REWRITE_TAC[SINGULAR_BOUNDARY] THEN ASM_MESON_TAC[]]]);; let HOMOLOGY_DIMENSION_AXIOM = prove (`!p top (a:A). topspace top = {a} /\ ~(p = &0) ==> trivial_group(homology_group(p,top))`, MATCH_MP_TAC(MESON[INT_OF_NUM_OF_INT; INT_NOT_LT] `(!x. x < &0 ==> P x) /\ (!p. P(&p)) ==> !x:int. P x`) THEN SIMP_TAC[homology_group; INT_OF_NUM_EQ; TRIVIAL_RELATIVE_HOMOLOGY_GROUP] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[trivial_group] THEN REWRITE_TAC[SET_RULE `s = {a} <=> a IN s /\ !b. b IN s ==> b = a`] THEN REWRITE_TAC[GROUP_ID] THEN REWRITE_TAC[RELATIVE_HOMOLOGY_GROUP] THEN REWRITE_TAC[FORALL_IN_GSPEC; HOMOLOGOUS_REL_EQ_RELBOUNDARY] THEN ASM_MESON_TAC[SINGULAR_BOUNDARY_EQ_CYCLE_SING; LE_1]);; let HOMOLOGY_HOMOTOPY_AXIOM = prove (`!p top s top' t (f:A->B) g. homotopic_with (\h. IMAGE h s SUBSET t) (top,top') f g ==> hom_induced p (top,s) (top',t) f = hom_induced p (top,s) (top',t) g`, MATCH_MP_TAC(MESON[INT_OF_NUM_OF_INT; INT_NOT_LT] `(!x. x < &0 ==> P x) /\ (!p. P(&p)) ==> !x:int. P x`) THEN SIMP_TAC[HOM_INDUCED_TRIVIAL] THEN REPEAT STRIP_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN MATCH_MP_TAC(MESON[] `!P. (!x. ~P x ==> Q x) /\ (!x. P x ==> Q x) ==> !x. Q x`) THEN EXISTS_TAC `\c. c IN group_carrier(relative_homology_group(&p,top:A topology,s))` THEN SIMP_TAC[HOM_INDUCED_DEFAULT] THEN REWRITE_TAC[RELATIVE_HOMOLOGY_GROUP; FORALL_IN_GSPEC] THEN REWRITE_TAC[RELCYCLE_GROUP; IN] THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_PROPERTY) THEN REWRITE_TAC[] THEN STRIP_TAC THEN FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_CONTINUOUS_MAPS) THEN ASM_SIMP_TAC[HOM_INDUCED_CHAIN_MAP] THEN REWRITE_TAC[HOMOLOGOUS_REL_EQ] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMOTOPIC_IMP_HOMOLOGOUS_REL_CHAIN_MAPS THEN ASM_MESON_TAC[]);; let HOMOLOGY_EXCISION_AXIOM = prove (`!p top s t (u:A->bool). top closure_of u SUBSET top interior_of t /\ t SUBSET s ==> group_isomorphism (relative_homology_group(p,subtopology top (s DIFF u),t DIFF u), relative_homology_group(p,subtopology top s,t)) (hom_induced p (subtopology top (s DIFF u),t DIFF u) (subtopology top s,t) (\x. x))`, MATCH_MP_TAC(MESON[INT_OF_NUM_OF_INT; INT_NOT_LT] `(!x. x < &0 ==> P x) /\ (!p. P(&p)) ==> !x:int. P x`) THEN CONJ_TAC THENL [SIMP_TAC[relative_homology_group; HOM_INDUCED_TRIVIAL] THEN REWRITE_TAC[GROUP_ISOMORPHISM; group_homomorphism; SINGLETON_GROUP] THEN SET_TAC[]; ALL_TAC] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[GROUP_ISOMORPHISM] THEN REWRITE_TAC[group_homomorphism; SET_RULE `(IMAGE f s SUBSET t /\ P) /\ IMAGE f s = t /\ Q <=> (IMAGE f s SUBSET t /\ P) /\ t SUBSET IMAGE f s /\ Q`] THEN REWRITE_TAC[GSYM group_homomorphism] THEN SUBGOAL_THEN `continuous_map (subtopology top (s DIFF u),subtopology top s) (\x. x) /\ IMAGE (\x:A. x) (t DIFF u) SUBSET t` STRIP_ASSUME_TAC THENL [CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN ONCE_REWRITE_TAC[SET_RULE `s DIFF u = s INTER (s DIFF u)`] THEN SIMP_TAC[GSYM SUBTOPOLOGY_SUBTOPOLOGY; CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_ID]; ALL_TAC] THEN ASM_SIMP_TAC[GROUP_HOMOMORPHISM_HOM_INDUCED] THEN CONJ_TAC THENL [GEN_REWRITE_TAC I [SUBSET] THEN REWRITE_TAC[NONTRIVIAL_RELATIVE_HOMOLOGY_GROUP] THEN SIMP_TAC[QUOTIENT_GROUP; NORMAL_SUBGROUP_SINGULAR_RELBOUNDARY_RELCYCLE] THEN REWRITE_TAC[FORALL_IN_GSPEC; RIGHT_COSET_SINGULAR_RELBOUNDARY] THEN X_GEN_TAC `c:((num->real)->A)frag` THEN GEN_REWRITE_TAC LAND_CONV [IN] THEN REWRITE_TAC[RELCYCLE_GROUP] THEN DISCH_TAC THEN REWRITE_TAC[IN_IMAGE] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[EXISTS_IN_GSPEC] THEN MP_TAC(ISPECL [`p:num`; `top:A topology`; `s:A->bool`; `t:A->bool`; `u:A->bool`; `c:((num->real)->A)frag`] EXCISED_RELCYCLE_EXISTS) THEN ASM_REWRITE_TAC[IN] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:((num->real)->A)frag` THEN STRIP_TAC THEN ASM_SIMP_TAC[HOM_INDUCED_CHAIN_MAP] THEN FIRST_X_ASSUM(SUBST1_TAC o GEN_REWRITE_RULE I [GSYM HOMOLOGOUS_REL_EQ]) THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC CHAIN_MAP_ID THEN ASM_MESON_TAC[singular_relcycle]; REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[NONTRIVIAL_RELATIVE_HOMOLOGY_GROUP] THEN SIMP_TAC[QUOTIENT_GROUP; NORMAL_SUBGROUP_SINGULAR_RELBOUNDARY_RELCYCLE] THEN REWRITE_TAC[FORALL_IN_GSPEC; RIGHT_COSET_SINGULAR_RELBOUNDARY] THEN REWRITE_TAC[RELCYCLE_GROUP; IN] THEN X_GEN_TAC `c:((num->real)->A)frag` THEN DISCH_TAC THEN X_GEN_TAC `d:((num->real)->A)frag` THEN DISCH_TAC THEN ASM_SIMP_TAC[HOM_INDUCED_CHAIN_MAP] THEN SUBGOAL_THEN `singular_chain(p,subtopology top (s DIFF u:A->bool)) c /\ singular_chain(p,subtopology top (s DIFF u)) d` MP_TAC THENL [ASM_MESON_TAC[singular_relcycle]; ALL_TAC] THEN DISCH_THEN(CONJUNCTS_THEN(SUBST1_TAC o MATCH_MP CHAIN_MAP_ID)) THEN REWRITE_TAC[HOMOLOGOUS_REL_EQ] THEN REWRITE_TAC[homologous_rel] THEN MAP_EVERY UNDISCH_TAC [`singular_relcycle (p,subtopology top (s DIFF u:A->bool),t DIFF u) d`; `singular_relcycle (p,subtopology top (s DIFF u:A->bool),t DIFF u) c`] THEN GEN_REWRITE_TAC I [IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP SINGULAR_RELCYCLE_SUB) THEN SPEC_TAC(`frag_sub c d:((num->real)->A)frag`, `c:((num->real)->A)frag`) THEN X_GEN_TAC `c:((num->real)->A)frag` THEN REWRITE_TAC[singular_relcycle; cong; mod_subset] THEN REWRITE_TAC[FRAG_MODULE `frag_sub x frag_0 = x`] THEN ASM_SIMP_TAC[SUBTOPOLOGY_SUBTOPOLOGY; SET_RULE `t SUBSET s ==> (s DIFF u) INTER (t DIFF u) = t DIFF u`] THEN STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [SINGULAR_RELBOUNDARY_ALT] THEN ASM_SIMP_TAC[LEFT_IMP_EXISTS_THM; SUBTOPOLOGY_SUBTOPOLOGY; SET_RULE `t SUBSET s ==> s INTER t = t`] THEN MAP_EVERY X_GEN_TAC [`d:((num->real)->A)frag`; `e:((num->real)->A)frag`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`p + 1`; `top:A topology`; `s:A->bool`; `t:A->bool`; `u:A->bool`; `d:((num->real)->A)frag`] EXCISED_CHAIN_EXISTS) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`n:num`; `f:((num->real)->A)frag`; `g:((num->real)->A)frag`] THEN STRIP_TAC THEN X_CHOOSE_THEN `h:num->((num->real)->A)frag->((num->real)->A)frag` MP_TAC (SPEC `n:num` CHAIN_HOMOTOPIC_ITERATED_SINGULAR_SUBDIVISION) THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(MP_TAC o SPECL [`p + 1`; `subtopology top (s:A->bool)`; `d:((num->real)->A)frag`]) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o AP_TERM `chain_boundary (p + 1):((num->real)->A)frag->((num->real)->A)frag`) THEN REWRITE_TAC[CHAIN_BOUNDARY_ADD; CHAIN_BOUNDARY_SUB] THEN MP_TAC(ISPECL [`p + 1`; `subtopology top s:A topology`] CHAIN_BOUNDARY_BOUNDARY_ALT) THEN ASM_SIMP_TAC[FRAG_MODULE `frag_add frag_0 c = c`; ADD_SUB] THEN DISCH_THEN(K ALL_TAC) THEN SUBST1_TAC(FRAG_MODULE `frag_add c e:((num->real)->A)frag = frag_sub c (frag_sub frag_0 e)`) THEN ASM_REWRITE_TAC[CHAIN_BOUNDARY_SUB; CHAIN_BOUNDARY_0] THEN REWRITE_TAC[FRAG_MODULE `frag_sub x (frag_sub frag_0 y) = frag_add x y`] THEN REWRITE_TAC[FRAG_MODULE `frag_add c' e' = frag_sub (frag_add f g) (frag_add c e) <=> c = frag_add (frag_sub f c') (frag_sub (frag_sub g e) e')`] THEN REWRITE_TAC[GSYM CHAIN_BOUNDARY_SUB] THEN ONCE_REWRITE_TAC[FRAG_MODULE `c = frag_add c1 (frag_sub c2 c3) <=> c1 = frag_add c (frag_sub c3 c2)`] THEN REWRITE_TAC[SINGULAR_RELBOUNDARY_ALT] THEN MAP_EVERY ABBREV_TAC [`c1 = frag_sub f ((h:num->((num->real)->A)frag->((num->real)->A)frag) p c)`; `c2:((num->real)->A)frag = frag_sub (chain_boundary (p + 1) (h p e)) (frag_sub (chain_boundary (p + 1) g) e)`] THEN DISCH_TAC THEN REWRITE_TAC[SINGULAR_RELBOUNDARY_ALT] THEN MAP_EVERY EXISTS_TAC [`c1:((num->real)->A)frag`; `c2:((num->real)->A)frag`] THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [ASM_MESON_TAC[SINGULAR_CHAIN_SUB]; DISCH_TAC] THEN SUBGOAL_THEN `singular_chain (p,subtopology top (s DIFF u:A->bool)) c2 /\ singular_chain (p,subtopology top t) c2` MP_TAC THENL [CONJ_TAC THENL [FIRST_X_ASSUM(SUBST1_TAC o MATCH_MP (FRAG_MODULE `c1 = frag_add c c2 ==> c2 = frag_sub c1 c`)) THEN MATCH_MP_TAC SINGULAR_CHAIN_SUB THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[SINGULAR_CHAIN_BOUNDARY; ADD_SUB]; EXPAND_TAC "c2" THEN REPEAT(MATCH_MP_TAC SINGULAR_CHAIN_SUB THEN CONJ_TAC) THEN ASM_MESON_TAC[SINGULAR_CHAIN_BOUNDARY; ADD_SUB]]; REWRITE_TAC[SINGULAR_CHAIN_SUBTOPOLOGY] THEN ASM SET_TAC[]]]);; (* ------------------------------------------------------------------------- *) (* Additivity axiom; not in the original Eilenberg-Steenrod list but usually *) (* included nowadays, following Milnor's "On Axiomatic Homology Theory". *) (* ------------------------------------------------------------------------- *) let GROUP_ISOMORPHISM_CHAIN_GROUP_SUM = prove (`!p top u:(A->bool)->bool. pairwise DISJOINT u /\ UNIONS u = topspace top /\ (!c t. compact_in top c /\ path_connected_in top c /\ t IN u /\ ~(DISJOINT c t) ==> c SUBSET t) ==> group_isomorphism (sum_group u (\s. chain_group(p,subtopology top s)), chain_group(p,top)) (iterate frag_add u)`, REPEAT STRIP_TAC THEN REWRITE_TAC[chain_group] THEN W(MP_TAC o PART_MATCH (lhand o lhand o rand) GROUP_ISOMORPHISM_FREE_ABELIAN_GROUP_SUM o lhand o lhand o snd) THEN ANTS_TAC THENL [REWRITE_TAC[pairwise] THEN MAP_EVERY X_GEN_TAC [`s:A->bool`; `t:A->bool`] THEN STRIP_TAC THEN REWRITE_TAC[SET_RULE `DISJOINT s t <=> !x. s x /\ t x ==> F`] THEN X_GEN_TAC `f:(num->real)->A` THEN REWRITE_TAC[SINGULAR_SIMPLEX_SUBTOPOLOGY] THEN DISCH_THEN(CONJUNCTS_THEN(MP_TAC o CONJUNCT2)) THEN MATCH_MP_TAC(SET_RULE `DISJOINT s t /\ ~(p = {}) ==> IMAGE f p SUBSET t ==> IMAGE f p SUBSET s ==> F`) THEN REWRITE_TAC[NONEMPTY_STANDARD_SIMPLEX] THEN ASM_MESON_TAC[pairwise]; MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN REPLICATE_TAC 3 AP_TERM_TAC THEN REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; UNIONS_SUBSET] THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN REWRITE_TAC[UNIONS_GSPEC; IN_ELIM_THM; SUBSET; SET_RULE `c IN singular_simplex (p,top) <=> singular_simplex (p,top) c`] THEN SIMP_TAC[SINGULAR_SIMPLEX_SUBTOPOLOGY] THEN X_GEN_TAC `f:(num->real)->A` THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV o RAND_CONV) [GSYM SUBTOPOLOGY_TOPSPACE] THEN REWRITE_TAC[SINGULAR_SIMPLEX_SUBTOPOLOGY] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o MATCH_MP (SET_RULE `IMAGE f p SUBSET s ==> ~(p = {}) ==> ?x. x IN s /\ x IN IMAGE f p`))) THEN REWRITE_TAC[NONEMPTY_STANDARD_SIMPLEX] THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN REWRITE_TAC[EXISTS_IN_UNIONS] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:A->bool` THEN REWRITE_TAC[RIGHT_EXISTS_AND_THM; SET_RULE `(?x. x IN t /\ x IN s) <=> ~(DISJOINT s t)`] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [MATCH_MP_TAC IMAGE_COMPACT_IN; MATCH_MP_TAC PATH_CONNECTED_IN_CONTINUOUS_MAP_IMAGE] THEN EXISTS_TAC `subtopology (product_topology (:num) (\i. euclideanreal)) (standard_simplex p)` THEN RULE_ASSUM_TAC(REWRITE_RULE[singular_simplex]) THEN ASM_REWRITE_TAC[COMPACT_IN_SUBTOPOLOGY; PATH_CONNECTED_IN_SUBTOPOLOGY; COMPACT_IN_STANDARD_SIMPLEX; PATH_CONNECTED_IN_STANDARD_SIMPLEX; SUBSET_REFL]]);; let GROUP_ISOMORPHISM_CYCLE_GROUP_SUM = prove (`!p top u:(A->bool)->bool. pairwise DISJOINT u /\ UNIONS u = topspace top /\ (!c t. compact_in top c /\ path_connected_in top c /\ t IN u /\ ~(DISJOINT c t) ==> c SUBSET t) ==> group_isomorphism (sum_group u (\t. relcycle_group(p,subtopology top t,{})), relcycle_group(p,top,{})) (iterate frag_add u)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `p = 0` THENL [SUBGOAL_THEN `!top:A topology. relcycle_group(p,top,{}) = chain_group(p,top)` (fun th -> REWRITE_TAC[th]) THENL [REWRITE_TAC[GROUPS_EQ; RELCYCLE_GROUP; CHAIN_GROUP] THEN REWRITE_TAC[FUN_EQ_THM; SINGULAR_CYCLE] THEN ASM_REWRITE_TAC[chain_boundary]; MATCH_MP_TAC GROUP_ISOMORPHISM_CHAIN_GROUP_SUM THEN ASM_REWRITE_TAC[]]; ALL_TAC] THEN REWRITE_TAC[relcycle_group] THEN SIMP_TAC[SUM_GROUP_SUBGROUP_GENERATED; SUBGROUP_SINGULAR_RELCYCLE] THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [SUBGROUP_GENERATED_RESTRICT] THEN MATCH_MP_TAC GROUP_ISOMORPHISM_BETWEEN_SUBGROUPS THEN REWRITE_TAC[INTER_SUBSET] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [MATCH_MP_TAC GROUP_ISOMORPHISM_CHAIN_GROUP_SUM THEN ASM_REWRITE_TAC[]; REWRITE_TAC[group_isomorphism; group_isomorphisms; group_homomorphism; LEFT_IMP_EXISTS_THM]] THEN GEN_TAC THEN MATCH_MP_TAC(SET_RULE `s SUBSET u /\ t SUBSET v /\ (!x. x IN u ==> (f x IN t <=> x IN s)) ==> (IMAGE f u SUBSET v /\ P) /\ (IMAGE f' v SUBSET u /\ Q) /\ (!x. x IN u ==> f'(f x) = x) /\ (!y. y IN v ==> f(f' y) = y) ==> IMAGE f s = t`) THEN SIMP_TAC[INTER_SUBSET; IN_INTER] THEN CONJ_TAC THENL [REWRITE_TAC[CHAIN_GROUP; SUBSET; IN] THEN SIMP_TAC[SINGULAR_CYCLE]; X_GEN_TAC `z:(A->bool)->((num->real)->A)frag`] THEN REWRITE_TAC[SUM_GROUP_CLAUSES; cartesian_product; IN_ELIM_THM; CHAIN_GROUP] THEN REWRITE_TAC[SET_RULE `z IN singular_relcycle q <=> singular_relcycle q z`; SET_RULE `z IN singular_chain q <=> singular_chain q z`] THEN REWRITE_TAC[SINGULAR_CYCLE] THEN STRIP_TAC THEN ASM_SIMP_TAC[] THEN ONCE_REWRITE_TAC[GSYM ITERATE_SUPPORT] THEN REWRITE_TAC[support; NEUTRAL_FRAG_ADD] THEN MATCH_MP_TAC(TAUT `p /\ (q <=> r) ==> (p /\ q <=> r)`) THEN CONJ_TAC THENL [MATCH_MP_TAC SINGULAR_CHAIN_SUM THEN RULE_ASSUM_TAC(REWRITE_RULE[SINGULAR_CHAIN_SUBTOPOLOGY]) THEN ASM_SIMP_TAC[IN_ELIM_THM]; ASM_SIMP_TAC[CHAIN_BOUNDARY_SUM]] THEN MP_TAC(ISPECL [`p - 1`; `top:A topology`; `u:(A->bool)->bool`] GROUP_ISOMORPHISM_CHAIN_GROUP_SUM) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o MATCH_MP GROUP_ISOMORPHISM_IMP_MONOMORPHISM) THEN REWRITE_TAC[GROUP_MONOMORPHISM_ALT_EQ] THEN DISCH_THEN(MP_TAC o SPEC `RESTRICTION u (chain_boundary p o (z:(A->bool)->((num->real)->A)frag))` o CONJUNCT2) THEN REWRITE_TAC[SUM_GROUP_CLAUSES; CHAIN_GROUP; RESTRICTION_EXTENSION] THEN REWRITE_TAC[IN_ELIM_THM; RESTRICTION_IN_CARTESIAN_PRODUCT; o_THM] THEN ANTS_TAC THENL [CONJ_TAC THENL [REPEAT STRIP_TAC THEN REWRITE_TAC[IN] THEN ASM_SIMP_TAC[SINGULAR_CHAIN_BOUNDARY]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] FINITE_SUBSET)) THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `t:A->bool` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[CONTRAPOS_THM; RESTRICTION; o_THM] THEN SIMP_TAC[CHAIN_BOUNDARY_0]]; DISCH_THEN(SUBST1_TAC o SYM) THEN AP_THM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC(MESON[] `iterate f u (RESTRICTION u x) = iterate f u x /\ iterate f u x = y ==> y = iterate f u (RESTRICTION u x)`) THEN CONJ_TAC THENL [MATCH_MP_TAC(MATCH_MP ITERATE_EQ MONOIDAL_FRAG_ADD) THEN SIMP_TAC[RESTRICTION]; MATCH_MP_TAC(MATCH_MP ITERATE_SUPERSET MONOIDAL_FRAG_ADD) THEN SIMP_TAC[SUBSET_RESTRICT; IN_ELIM_THM; IMP_CONJ] THEN SIMP_TAC[o_THM; NEUTRAL_FRAG_ADD; CHAIN_BOUNDARY_0]]]);; let HOMOLOGY_ADDITIVITY_AXIOM_GEN = prove (`!p top u:(A->bool)->bool. pairwise DISJOINT u /\ UNIONS u = topspace top /\ (!c t. compact_in top c /\ path_connected_in top c /\ t IN u /\ ~(DISJOINT c t) ==> c SUBSET t) ==> group_isomorphism (sum_group u (\s. homology_group(p,subtopology top s)), homology_group(p,top)) (\x. iterate (group_add (homology_group(p,top))) u (\v. hom_induced p (subtopology top v,{}) (top,{}) (\z. z) (x v)))`, MATCH_MP_TAC(MESON[INT_OF_NUM_OF_INT; INT_NOT_LT] `(!x. x < &0 ==> P x) /\ (!p. P(&p)) ==> !x:int. P x`) THEN CONJ_TAC THENL [SIMP_TAC[homology_group; relative_homology_group] THEN ASM_SIMP_TAC[HOM_INDUCED_TRIVIAL] THEN REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM ITERATE_SUPPORT] THEN REWRITE_TAC[support; NEUTRAL_GROUP_ADD; SINGLETON_GROUP; EMPTY_GSPEC] THEN SIMP_TAC[ITERATE_CLAUSES; MONOIDAL_GROUP_ADD; ABELIAN_SINGLETON_GROUP] THEN REWRITE_TAC[NEUTRAL_GROUP_ADD; GROUP_ISOMORPHISM_TRIVIAL] THEN REWRITE_TAC[TRIVIAL_GROUP_SINGLETON_GROUP] THEN REWRITE_TAC[sum_group] THEN MATCH_MP_TAC TRIVIAL_GROUP_SUBGROUP_GENERATED THEN REWRITE_TAC[TRIVIAL_PRODUCT_GROUP; TRIVIAL_GROUP_SINGLETON_GROUP]; REPEAT STRIP_TAC] THEN REWRITE_TAC[GSYM GROUP_MONOMORPHISM_EPIMORPHISM; GROUP_EPIMORPHISM_ALT; GROUP_MONOMORPHISM_ALT] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q /\ r) ==> (p /\ q) /\ (p /\ r)`) THEN CONJ_TAC THENL [MATCH_MP_TAC GROUP_HOMOMORPHISM_EQ THEN EXISTS_TAC `\x. group_sum (homology_group (&p,top:A topology)) u (\v. hom_induced (&p) (subtopology top v,{}) (top,{}) (\z. z) (x v))` THEN CONJ_TAC THENL [MATCH_MP_TAC ABELIAN_GROUP_HOMOMORPHISM_GROUP_SUM THEN REWRITE_TAC[homology_group; ETA_AX; ABELIAN_RELATIVE_HOMOLOGY_GROUP] THEN REWRITE_TAC[GROUP_HOMOMORPHISM_HOM_INDUCED]; REPEAT GEN_TAC THEN REWRITE_TAC[] THEN DISCH_TAC THEN MATCH_MP_TAC(GSYM ABELIAN_GROUP_SUM_ITERATE) THEN ASM_REWRITE_TAC[homology_group; ETA_AX; ABELIAN_RELATIVE_HOMOLOGY_GROUP; HOM_INDUCED]]; DISCH_TAC] THEN REWRITE_TAC[group_image] THEN SUBGOAL_THEN `group_carrier(sum_group u (\s:A->bool. homology_group (&p,subtopology top s))) = IMAGE (\x. RESTRICTION u (\s. homologous_rel (p,subtopology top s,{}) (x s))) (group_carrier (sum_group u (\s. relcycle_group(p,subtopology top s,{}))))` SUBST1_TAC THENL [REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; SUBSET; FORALL_IN_IMAGE] THEN REWRITE_TAC[RELATIVE_HOMOLOGY_GROUP; SUM_GROUP_CLAUSES; homology_group] THEN REWRITE_TAC[cartesian_product; IN_ELIM_THM] THEN REWRITE_TAC[REWRITE_RULE[IN] RESTRICTION_IN_EXTENSIONAL] THEN CONJ_TAC THENL [ALL_TAC; ONCE_REWRITE_TAC[TAUT `p /\ ~q <=> ~(p ==> q)`] THEN SIMP_TAC[RESTRICTION; HOMOLOGOUS_REL_EQ_RELBOUNDARY] THEN REWRITE_TAC[RELCYCLE_GROUP] THEN GEN_TAC THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [MESON_TAC[IN]; ALL_TAC] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] FINITE_SUBSET) THEN REWRITE_TAC[SUBSET; IN_ELIM_THM; NOT_IMP] THEN MESON_TAC[SINGULAR_RELBOUNDARY_0]] THEN X_GEN_TAC `z:(A->bool)->((num->real)->A)frag->bool` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE BINDER_CONV [RIGHT_IMP_EXISTS_THM]) THEN REWRITE_TAC[SKOLEM_THM; IN_IMAGE; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `c:(A->bool)->((num->real)->A)frag` THEN STRIP_TAC THEN EXISTS_TAC `RESTRICTION u (\s. if singular_relboundary (p,subtopology top s,{}) (c s) then frag_0 else (c:(A->bool)->((num->real)->A)frag) s)` THEN REWRITE_TAC[IN_ELIM_THM; REWRITE_RULE[IN] RESTRICTION_IN_EXTENSIONAL] THEN GEN_REWRITE_TAC LAND_CONV [FUN_EQ_THM] THEN ONCE_REWRITE_TAC[TAUT `p /\ ~q <=> ~(p ==> q)`] THEN SIMP_TAC[RESTRICTION] THEN REPEAT CONJ_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[EXTENSIONAL; IN_ELIM_THM]) THEN GEN_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[HOMOLOGOUS_REL_EQ] THEN ASM_REWRITE_TAC[homologous_rel; FRAG_MODULE `frag_sub c frag_0 = c`]; REPEAT STRIP_TAC THEN REWRITE_TAC[RELCYCLE_GROUP; IN] THEN COND_CASES_TAC THEN ASM_SIMP_TAC[SINGULAR_RELCYCLE_0]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] FINITE_SUBSET)) THEN REWRITE_TAC[SUBSET; NOT_IMP; IN_ELIM_THM] THEN GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_SIMP_TAC[CONTRAPOS_THM] THEN SIMP_TAC[HOMOLOGOUS_REL_EQ_RELBOUNDARY] THEN REWRITE_TAC[RELCYCLE_GROUP]]; ALL_TAC] THEN REWRITE_TAC[IMP_CONJ; FORALL_IN_IMAGE; EXISTS_IN_IMAGE] THEN REWRITE_TAC[GSYM IMAGE_o; o_DEF] THEN REWRITE_TAC[SET_RULE `t SUBSET IMAGE f s <=> !y. y IN t ==> ~(!x. x IN s ==> ~(f x = y))`] THEN SUBGOAL_THEN `!x. x IN group_carrier (sum_group u (\s. relcycle_group (p,subtopology top s,{}))) ==> iterate (group_add (homology_group (&p,top))) u (\v. hom_induced (&p) (subtopology top v,{}) (top,{}) (\z:A. z) (RESTRICTION u (\s. homologous_rel (p,subtopology top s,{}) (x s)) v)) = homologous_rel(p,top,{}) (iterate frag_add u x)` (fun th -> SIMP_TAC[th]) THENL [REWRITE_TAC[SUM_GROUP_CLAUSES; cartesian_product; IN_ELIM_THM] THEN REWRITE_TAC[RELCYCLE_GROUP; SET_RULE `x IN singular_relcycle q <=> singular_relcycle q x`] THEN X_GEN_TAC `z:(A->bool)->((num->real)->A)frag` THEN STRIP_TAC THEN TRANS_TAC EQ_TRANS `iterate (group_add (homology_group (&p,top:A topology))) u (\s:A->bool. homologous_rel (p,top,{}) (z s))` THEN CONJ_TAC THENL [MATCH_MP_TAC(REWRITE_RULE[RIGHT_IMP_FORALL_THM; IMP_IMP] ITERATE_EQ) THEN REWRITE_TAC[MONOIDAL_GROUP_ADD; ABELIAN_RELATIVE_HOMOLOGY_GROUP; homology_group] THEN SIMP_TAC[RESTRICTION] THEN X_GEN_TAC `s:A->bool` THEN DISCH_TAC THEN ASM_SIMP_TAC[HOM_INDUCED_CHAIN_MAP; CONTINUOUS_MAP_ID; CONTINUOUS_MAP_FROM_SUBTOPOLOGY; IMAGE_CLAUSES; EMPTY_SUBSET] THEN AP_TERM_TAC THEN MATCH_MP_TAC CHAIN_MAP_ID THEN ASM_MESON_TAC[SINGULAR_CYCLE]; ALL_TAC] THEN TRANS_TAC EQ_TRANS `iterate (group_add (homology_group (&p,top:A topology))) {s:A->bool | s IN u /\ ~(z s = frag_0)} (\s. homologous_rel (p,top,{}) (z s))` THEN CONJ_TAC THENL [MATCH_MP_TAC(REWRITE_RULE[RIGHT_IMP_FORALL_THM; IMP_IMP] ITERATE_SUPERSET) THEN REWRITE_TAC[MONOIDAL_GROUP_ADD; ABELIAN_RELATIVE_HOMOLOGY_GROUP; homology_group; SUBSET_RESTRICT; IN_ELIM_THM] THEN SIMP_TAC[IMP_CONJ; RELATIVE_HOMOLOGY_GROUP; NEUTRAL_GROUP_ADD] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[HOMOLOGOUS_REL_EQ_RELBOUNDARY] THEN REWRITE_TAC[SINGULAR_RELBOUNDARY_0]; ALL_TAC] THEN GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [GSYM ITERATE_SUPPORT] THEN REWRITE_TAC[support; NEUTRAL_FRAG_ADD] THEN SUBGOAL_THEN `!s:A->bool. s IN {i | i IN u /\ ~(z i = frag_0)} ==> singular_relcycle (p,subtopology top s,{}) (z s)` MP_TAC THENL [ASM_SIMP_TAC[IN_ELIM_THM]; ALL_TAC] THEN UNDISCH_TAC `FINITE {s:A->bool | s IN u /\ ~(z s:((num->real)->A)frag = frag_0)}` THEN SPEC_TAC(`{s:A->bool | s IN u /\ ~(z s:((num->real)->A)frag = frag_0)}`, `v:(A->bool)->bool`) THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN REWRITE_TAC[NOT_IN_EMPTY; FORALL_IN_INSERT] THEN SIMP_TAC[ITERATE_CLAUSES; MONOIDAL_FRAG_ADD; MONOIDAL_GROUP_ADD; homology_group; ABELIAN_RELATIVE_HOMOLOGY_GROUP] THEN REWRITE_TAC[NEUTRAL_GROUP_ADD; NEUTRAL_FRAG_ADD] THEN REWRITE_TAC[RELATIVE_HOMOLOGY_GROUP] THEN CONV_TAC(LAND_CONV SYM_CONV) THEN REWRITE_TAC[HOMOLOGOUS_REL_EQ_RELBOUNDARY] THEN REWRITE_TAC[SINGULAR_RELBOUNDARY_0] THEN MAP_EVERY X_GEN_TAC [`s:A->bool`; `v:(A->bool)->bool`] THEN REPEAT STRIP_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) GROUP_ADD_EQ_MUL o lhand o snd) THEN ANTS_TAC THENL [REWRITE_TAC[RELATIVE_HOMOLOGY_GROUP] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN REWRITE_TAC[ETA_AX] THEN CONJ_TAC THEN MATCH_MP_TAC FUN_IN_IMAGE THEN REWRITE_TAC[IN_ELIM_THM] THEN ASM_MESON_TAC[SINGULAR_CYCLE; SINGULAR_CHAIN_SUBTOPOLOGY; SINGULAR_RELCYCLE_SUM]; DISCH_THEN SUBST1_TAC] THEN REWRITE_TAC[relative_homology_group] THEN REWRITE_TAC[NUM_OF_INT_OF_NUM; INT_ARITH `~(&p:int < &0)`] THEN REWRITE_TAC[GSYM RIGHT_COSET_SINGULAR_RELBOUNDARY] THEN W(MP_TAC o PART_MATCH (lhand o rand) QUOTIENT_GROUP_MUL o lhand o snd) THEN REWRITE_TAC[NORMAL_SUBGROUP_SINGULAR_RELBOUNDARY_RELCYCLE] THEN ANTS_TAC THENL [REWRITE_TAC[RELCYCLE_GROUP] THEN REWRITE_TAC[IN] THEN ASM_MESON_TAC[SINGULAR_CYCLE; SINGULAR_CHAIN_SUBTOPOLOGY; SINGULAR_RELCYCLE_SUM]; DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[RELCYCLE_GROUP]]; ALL_TAC] THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP] THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[homology_group; RELATIVE_HOMOLOGY_GROUP; FORALL_IN_GSPEC] THEN MP_TAC(ISPECL [`p:num`; `top:A topology`; `u:(A->bool)->bool`] GROUP_ISOMORPHISM_CYCLE_GROUP_SUM) THEN ASM_REWRITE_TAC[GSYM GROUP_MONOMORPHISM_EPIMORPHISM] THEN REWRITE_TAC[group_epimorphism] THEN DISCH_THEN(MP_TAC o MATCH_MP(SET_RULE `IMAGE f s = t ==> !y. y IN t ==> ?x. x IN s /\ f x = y`) o last o CONJUNCTS) THEN REWRITE_TAC[RELCYCLE_GROUP] THEN MESON_TAC[IN]] THEN REWRITE_TAC[SUM_GROUP_CLAUSES; RESTRICTION_EXTENSION; IN_ELIM_THM] THEN REWRITE_TAC[homology_group; RELATIVE_HOMOLOGY_GROUP] THEN REWRITE_TAC[HOMOLOGOUS_REL_EQ_RELBOUNDARY; RELCYCLE_GROUP] THEN SIMP_TAC[cartesian_product; IN_ELIM_THM; SINGULAR_BOUNDARY; SET_RULE `x IN singular_relcycle p <=> singular_relcycle p x`] THEN X_GEN_TAC `z:(A->bool)->((num->real)->A)frag` THEN STRIP_TAC THEN DISCH_THEN(X_CHOOSE_THEN `d:((num->real)->A)frag` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`p + 1`; `top:A topology`; `u:(A->bool)->bool`] GROUP_ISOMORPHISM_CHAIN_GROUP_SUM) THEN ASM_REWRITE_TAC[GSYM GROUP_MONOMORPHISM_EPIMORPHISM] THEN STRIP_TAC THEN FIRST_X_ASSUM(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC o GEN_REWRITE_RULE I [group_epimorphism]) THEN GEN_REWRITE_TAC LAND_CONV [EXTENSION] THEN DISCH_THEN(MP_TAC o SPEC `d:((num->real)->A)frag`) THEN REWRITE_TAC[SET_RULE `(y IN IMAGE f s <=> x IN t) <=> ((?x. x IN s /\ f x = y) <=> t x)`] THEN ASM_REWRITE_TAC[CHAIN_GROUP; SUM_GROUP_CLAUSES; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `w:(A->bool)->((num->real)->A)frag` THEN REWRITE_TAC[cartesian_product; IN_ELIM_THM; SET_RULE `x IN singular_chain p <=> singular_chain p x`] THEN DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC (SUBST_ALL_TAC o SYM)) THEN MP_TAC(ISPECL [`p:num`; `top:A topology`; `u:(A->bool)->bool`] GROUP_ISOMORPHISM_CHAIN_GROUP_SUM) THEN ASM_REWRITE_TAC[GSYM GROUP_MONOMORPHISM_EPIMORPHISM] THEN REWRITE_TAC[group_monomorphism] THEN DISCH_THEN(MP_TAC o SPECL [`RESTRICTION u (\s. chain_boundary (p + 1) ((w:(A->bool)->((num->real)->A)frag) s))`; `z:(A->bool)->((num->real)->A)frag`] o CONJUNCT2 o CONJUNCT1) THEN REWRITE_TAC[SUM_GROUP_CLAUSES; IN_ELIM_THM; RESTRICTION_IN_CARTESIAN_PRODUCT] THEN ASM_REWRITE_TAC[CHAIN_GROUP; cartesian_product; IN_ELIM_THM; SET_RULE `x IN singular_chain p <=> singular_chain p x`] THEN ASM_SIMP_TAC[SINGULAR_CHAIN_BOUNDARY_ALT] THEN ANTS_TAC THENL [ALL_TAC; GEN_REWRITE_TAC LAND_CONV [FUN_EQ_THM] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `s:A->bool` THEN ASM_CASES_TAC `(s:A->bool) IN u` THEN ASM_REWRITE_TAC[RESTRICTION] THEN ASM_MESON_TAC[]] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[SET_RULE `p /\ ~q <=> ~(p ==> q)`] THEN SIMP_TAC[RESTRICTION] THEN MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `{s:A->bool | s IN u /\ ~(w s:((num->real)->A)frag = frag_0)}` THEN ASM_REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN MESON_TAC[CHAIN_BOUNDARY_0]; ASM_MESON_TAC[SINGULAR_CYCLE]; ALL_TAC] THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [SYM th]) THEN GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM ITERATE_SUPPORT] THEN REWRITE_TAC[support; NEUTRAL_FRAG_ADD] THEN ASM_SIMP_TAC[CHAIN_BOUNDARY_SUM] THEN TRANS_TAC EQ_TRANS `iterate frag_add u (chain_boundary (p + 1) o (w:(A->bool)->((num->real)->A)frag))` THEN CONJ_TAC THENL [MATCH_MP_TAC(REWRITE_RULE[RIGHT_IMP_FORALL_THM; IMP_IMP] ITERATE_EQ) THEN SIMP_TAC[MONOIDAL_FRAG_ADD; RESTRICTION; o_THM]; MATCH_MP_TAC(REWRITE_RULE[RIGHT_IMP_FORALL_THM; IMP_IMP] ITERATE_SUPERSET) THEN REWRITE_TAC[SUBSET_RESTRICT; MONOIDAL_FRAG_ADD] THEN SIMP_TAC[NEUTRAL_FRAG_ADD; IMP_CONJ; IN_ELIM_THM; o_THM] THEN REWRITE_TAC[CHAIN_BOUNDARY_0]]);; let HOMOLOGY_ADDITIVITY_AXIOM = prove (`!p top u:(A->bool)->bool. (!v. v IN u ==> open_in top v) /\ pairwise DISJOINT u /\ UNIONS u = topspace top ==> group_isomorphism (sum_group u (\s. homology_group(p,subtopology top s)), homology_group(p,top)) (\x. iterate (group_add (homology_group(p,top))) u (\v. hom_induced p (subtopology top v,{}) (top,{}) (\z. z) (x v)))`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMOLOGY_ADDITIVITY_AXIOM_GEN THEN ASM_REWRITE_TAC[] THEN MAP_EVERY X_GEN_TAC [`c:A->bool`; `t:A->bool`] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP PATH_CONNECTED_IN_IMP_CONNECTED_IN) THEN REWRITE_TAC[CONNECTED_IN] THEN MATCH_MP_TAC(TAUT `(p /\ ~r ==> q) ==> p /\ ~q ==> r`) THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`t:A->bool`; `UNIONS (u DIFF {t:A->bool})`] THEN ASM_SIMP_TAC[OPEN_IN_UNIONS; IN_DIFF] THEN ASM_SIMP_TAC[GSYM DIFF_UNIONS_PAIRWISE_DISJOINT; SING_SUBSET] THEN ASM SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* Special properties of singular homology, in particular the fact that the *) (* zeroth homology group is isomorphic to the free abelian group generated *) (* by the path components, and so the "coefficient group" is the integers. *) (* ------------------------------------------------------------------------- *) let GROUP_ISOMORPHISM_INTEGER_ZEROTH_HOMOLOGY_GROUP = prove (`!(top:A topology) f. path_connected_space top /\ singular_simplex(0,top) f ==> group_isomorphism (integer_group,homology_group(&0,top)) (group_zpow (homology_group(&0,top)) (homologous_rel(0,top,{}) (frag_of f)))`, let lemma = prove (`!(top:A topology) f f'. path_connected_space top /\ singular_simplex (0,top) f /\ singular_simplex (0,top) f' ==> homologous_rel (0,top,{}) (frag_of f) (frag_of f')`, REPEAT GEN_TAC THEN REWRITE_TAC[singular_simplex; STANDARD_SIMPLEX_0] THEN ABBREV_TAC `p:num->real = \j. if j = 0 then &1 else &0` THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPECL [`(f:(num->real)->A) p`; `(f':(num->real)->A) p`] o GEN_REWRITE_RULE I [path_connected_space]) THEN ANTS_TAC THENL [REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [continuous_map])) THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; TOPSPACE_PRODUCT_TOPOLOGY] THEN REWRITE_TAC[o_DEF; TOPSPACE_EUCLIDEANREAL; CARTESIAN_PRODUCT_UNIV] THEN SET_TAC[]; REWRITE_TAC[path_in; LEFT_IMP_EXISTS_THM]] THEN X_GEN_TAC `g:real->A` THEN STRIP_TAC THEN REWRITE_TAC[homologous_rel; SINGULAR_BOUNDARY] THEN EXISTS_TAC `frag_of(RESTRICTION (standard_simplex 1) ((g:real->A) o (\x:num->real. x 0)))` THEN REWRITE_TAC[SINGULAR_CHAIN_OF; CHAIN_BOUNDARY_OF] THEN CONV_TAC NUM_REDUCE_CONV THEN REWRITE_TAC[NUMSEG_CONV `0..1`] THEN SIMP_TAC[ITERATE_CLAUSES; MONOIDAL_FRAG_ADD; FINITE_INSERT; FINITE_EMPTY; NOT_IN_EMPTY; IN_INSERT; NEUTRAL_FRAG_ADD] THEN CONV_TAC NUM_REDUCE_CONV THEN CONJ_TAC THENL [REWRITE_TAC[singular_simplex] THEN REWRITE_TAC[REWRITE_RULE[IN] RESTRICTION_IN_EXTENSIONAL] THEN SIMP_TAC[RESTRICTION_CONTINUOUS_MAP; TOPSPACE_SUBTOPOLOGY; INTER_SUBSET] THEN MATCH_MP_TAC CONTINUOUS_MAP_COMPOSE THEN EXISTS_TAC `subtopology euclideanreal (real_interval [&0,&1])` THEN ASM_REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN SIMP_TAC[CONTINUOUS_MAP_PRODUCT_PROJECTION; IN_UNIV; CONTINUOUS_MAP_FROM_SUBTOPOLOGY; TOPSPACE_SUBTOPOLOGY] THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> f x IN t) ==> IMAGE f (u INTER s) SUBSET t`) THEN SIMP_TAC[standard_simplex; IN_REAL_INTERVAL; IN_ELIM_THM]; CONV_TAC INT_REDUCE_CONV THEN MATCH_MP_TAC(FRAG_MODULE `x = x' /\ y = y' ==> frag_add (frag_cmul (&1) x) (frag_add (frag_cmul (-- &1) y) frag_0) = frag_sub x' y'`) THEN CONJ_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[singular_face] THEN ASM_REWRITE_TAC[SUB_REFL; STANDARD_SIMPLEX_0] THEN RULE_ASSUM_TAC(REWRITE_RULE[EXTENSIONAL; IN_ELIM_THM; IN_SING]) THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `x:num->real` THEN ASM_CASES_TAC `x:num->real = p` THEN ASM_SIMP_TAC[RESTRICTION; IN_SING; o_THM] THEN REWRITE_TAC[simplicial_face; CONJUNCT1 LT] THEN CONV_TAC NUM_REDUCE_CONV THEN ASM_REWRITE_TAC[standard_simplex; IN_ELIM_THM] THEN UNDISCH_THEN `x:num->real = p` SUBST1_TAC THEN REWRITE_TAC[num_CONV `1`; SUM_CLAUSES_NUMSEG] THEN EXPAND_TAC "p" THEN REWRITE_TAC[] THEN CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(MESON[] `p ==> (if p then x else y) = x`) THEN REPEAT STRIP_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_ARITH_TAC]) in REPEAT STRIP_TAC THEN SUBGOAL_THEN `singular_relcycle(0,top:A topology,{}) (frag_of f)` ASSUME_TAC THENL [ASM_REWRITE_TAC[SINGULAR_CYCLE; SINGULAR_CHAIN_OF] THEN REWRITE_TAC[chain_boundary]; ALL_TAC] THEN ABBREV_TAC `q = homologous_rel(0,top:A topology,{}) (frag_of f)` THEN SUBGOAL_THEN `q IN group_carrier (homology_group (&0,top:A topology))` ASSUME_TAC THENL [REWRITE_TAC[homology_group; RELATIVE_HOMOLOGY_GROUP] THEN EXPAND_TAC "q" THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN REWRITE_TAC[ETA_AX] THEN MATCH_MP_TAC FUN_IN_IMAGE THEN ASM_REWRITE_TAC[IN_ELIM_THM]; ALL_TAC] THEN REWRITE_TAC[GSYM GROUP_MONOMORPHISM_EPIMORPHISM; GROUP_EPIMORPHISM_ALT; GROUP_MONOMORPHISM_ALT] THEN ASM_SIMP_TAC[GROUP_HOMOMORPHISM_GROUP_ZPOW] THEN SUBGOAL_THEN `group_zpow (homology_group (&0,top)) q = \n. homologous_rel(0,top:A topology,{}) (frag_cmul n (frag_of f))` SUBST1_TAC THENL [GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `n:int` THEN EXPAND_TAC "q" THEN SIMP_TAC[GSYM RIGHT_COSET_SINGULAR_RELBOUNDARY] THEN SIMP_TAC[homology_group; relative_homology_group; INT_LT_REFL; NUM_OF_INT_OF_NUM] THEN ASM_SIMP_TAC[QUOTIENT_GROUP_ZPOW; CONJUNCT1 RELCYCLE_GROUP; IN; NORMAL_SUBGROUP_SINGULAR_RELBOUNDARY_RELCYCLE] THEN AP_TERM_TAC THEN REWRITE_TAC[GROUP_ZPOW_SUBGROUP_GENERATED; relcycle_group] THEN REWRITE_TAC[FREE_ABELIAN_GROUP_ZPOW; chain_group]; ALL_TAC] THEN REWRITE_TAC[INTEGER_GROUP; IN_UNIV; homology_group] THEN REWRITE_TAC[group_image; SUBSET; RELATIVE_HOMOLOGY_GROUP] THEN REWRITE_TAC[FORALL_IN_GSPEC; INTEGER_GROUP; IN_UNIV] THEN REWRITE_TAC[HOMOLOGOUS_REL_EQ_RELBOUNDARY] THEN CONJ_TAC THENL [X_GEN_TAC `n:int` THEN REWRITE_TAC[SINGULAR_BOUNDARY; ADD_CLAUSES] THEN SUBGOAL_THEN `!d. singular_chain (1,top:A topology) d ==> frag_extend (\x. frag_of(f:(num->real)->A)) (chain_boundary 1 d) = frag_0` MP_TAC THENL [REWRITE_TAC[singular_chain] THEN MATCH_MP_TAC FRAG_INDUCTION THEN REWRITE_TAC[GSYM singular_chain] THEN REWRITE_TAC[FRAG_EXTEND_0; CHAIN_BOUNDARY_0] THEN SIMP_TAC[FRAG_EXTEND_SUB; CHAIN_BOUNDARY_SUB] THEN REWRITE_TAC[FRAG_MODULE `frag_sub x frag_0 = x`; IN] THEN X_GEN_TAC `g:(num->real)->A` THEN DISCH_TAC THEN REWRITE_TAC[CHAIN_BOUNDARY_OF] THEN CONV_TAC NUM_REDUCE_CONV THEN SIMP_TAC[MATCH_MP ITERATE_CLAUSES_NUMSEG MONOIDAL_FRAG_ADD; num_CONV `1`; LE_0; FRAG_EXTEND_ADD; FRAG_EXTEND_CMUL; FRAG_EXTEND_OF] THEN CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC INT_REDUCE_CONV THEN CONV_TAC FRAG_MODULE; REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[FRAG_EXTEND_OF; FRAG_EXTEND_CMUL; FRAG_MODULE `frag_cmul n t = frag_0 <=> n = &0 \/ t = frag_0`] THEN REWRITE_TAC[FRAG_OF_NONZERO]]; REWRITE_TAC[SINGULAR_CYCLE; IN_IMAGE; IN_UNIV; HOMOLOGOUS_REL_EQ] THEN MATCH_MP_TAC(MESON[] `(!x. P x ==> R x) ==> (!x. P x /\ Q x ==> R x)`) THEN REWRITE_TAC[singular_chain] THEN MATCH_MP_TAC FRAG_INDUCTION THEN REPEAT CONJ_TAC THENL [EXISTS_TAC `&0:int` THEN REWRITE_TAC[HOMOLOGOUS_REL_REFL; FRAG_MODULE `frag_cmul (&0) x = frag_0`]; ALL_TAC; REWRITE_TAC[LEFT_AND_EXISTS_THM; RIGHT_AND_EXISTS_THM] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP HOMOLOGOUS_REL_SUB) THEN REWRITE_TAC[FRAG_MODULE `frag_sub (frag_cmul a c) (frag_cmul b c) = frag_cmul (a - b) c`] THEN MESON_TAC[]] THEN REWRITE_TAC[IN] THEN X_GEN_TAC `f':(num->real)->A` THEN DISCH_TAC THEN EXISTS_TAC `&1:int` THEN REWRITE_TAC[FRAG_MODULE `frag_cmul (&1) x = x`] THEN ASM_MESON_TAC[lemma]]);; let ISOMORPHIC_GROUP_INTEGER_ZEROTH_HOMOLOGY_GROUP = prove (`!top:A topology. path_connected_space top /\ ~(topspace top = {}) ==> homology_group(&0,top) isomorphic_group integer_group`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?f. singular_simplex(0,top:A topology) f` STRIP_ASSUME_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_TAC `a:A`) THEN EXISTS_TAC `RESTRICTION (standard_simplex 0) (\x. (a:A))` THEN SIMP_TAC[singular_simplex; RESTRICTION_CONTINUOUS_MAP; TOPSPACE_SUBTOPOLOGY; INTER_SUBSET; CONTINUOUS_MAP_CONST; REWRITE_RULE[IN] RESTRICTION_IN_EXTENSIONAL] THEN ASM_REWRITE_TAC[]; ONCE_REWRITE_TAC[ISOMORPHIC_GROUP_SYM] THEN REWRITE_TAC[isomorphic_group] THEN ASM_MESON_TAC[GROUP_ISOMORPHISM_INTEGER_ZEROTH_HOMOLOGY_GROUP]]);; let HOMOLOGY_COEFFICIENTS = prove (`!top (a:A). topspace top = {a} ==> homology_group(&0,top) isomorphic_group integer_group`, REPEAT STRIP_TAC THEN MATCH_MP_TAC ISOMORPHIC_GROUP_INTEGER_ZEROTH_HOMOLOGY_GROUP THEN ASM_REWRITE_TAC[GSYM PATH_CONNECTED_IN_TOPSPACE; NOT_INSERT_EMPTY] THEN ASM_REWRITE_TAC[PATH_CONNECTED_IN_SING; IN_SING]);; let ZEROTH_HOMOLOGY_GROUP = prove (`!top:A topology. homology_group(&0,top) isomorphic_group free_abelian_group (path_components_of top)`, GEN_TAC THEN TRANS_TAC ISOMORPHIC_GROUP_TRANS `sum_group (path_components_of top) (\s:A->bool. homology_group(&0,subtopology top s))` THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[ISOMORPHIC_GROUP_SYM] THEN MP_TAC(ISPECL [`&0:int`; `top:A topology`; `path_components_of(top:A topology)`] HOMOLOGY_ADDITIVITY_AXIOM_GEN) THEN REWRITE_TAC[isomorphic_group] THEN ANTS_TAC THENL [ALL_TAC; MESON_TAC[]] THEN REWRITE_TAC[PAIRWISE_DISJOINT_PATH_COMPONENTS_OF] THEN REWRITE_TAC[UNIONS_PATH_COMPONENTS_OF] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC PATH_COMPONENTS_OF_MAXIMAL THEN EXISTS_TAC `top:A topology` THEN ASM SET_TAC[]; TRANS_TAC ISOMORPHIC_GROUP_TRANS `sum_group (path_components_of (top:A topology)) (\i. integer_group)` THEN REWRITE_TAC[ISOMORPHIC_SUM_INTEGER_GROUP] THEN MATCH_MP_TAC ISOMORPHIC_GROUP_SUM_GROUP THEN REWRITE_TAC[] THEN X_GEN_TAC `s:A->bool` THEN DISCH_TAC THEN MATCH_MP_TAC ISOMORPHIC_GROUP_INTEGER_ZEROTH_HOMOLOGY_GROUP THEN ASM_SIMP_TAC[TOPSPACE_SUBTOPOLOGY_SUBSET; PATH_COMPONENTS_OF_SUBSET; REWRITE_RULE[path_connected_in] PATH_CONNECTED_IN_PATH_COMPONENTS_OF] THEN ASM_MESON_TAC[NONEMPTY_PATH_COMPONENTS_OF]]);; let ISOMORPHIC_HOMOLOGY_IMP_PATH_COMPONENTS = prove (`!(top:A topology) (top':B topology). homology_group(&0,top) isomorphic_group homology_group(&0,top') ==> path_components_of top =_c path_components_of top'`, REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM ISOMORPHIC_FREE_ABELIAN_GROUPS] THEN TRANS_TAC ISOMORPHIC_GROUP_TRANS `homology_group(&0,(top':B topology))` THEN REWRITE_TAC[ZEROTH_HOMOLOGY_GROUP] THEN GEN_REWRITE_TAC I [ISOMORPHIC_GROUP_SYM] THEN TRANS_TAC ISOMORPHIC_GROUP_TRANS `homology_group(&0,(top:A topology))` THEN REWRITE_TAC[ZEROTH_HOMOLOGY_GROUP] THEN GEN_REWRITE_TAC I [ISOMORPHIC_GROUP_SYM] THEN ASM_REWRITE_TAC[]);; let ISOMORPHIC_HOMOLOGY_IMP_PATH_CONNECTEDNESS = prove (`!(top:A topology) (top':B topology). homology_group(&0,top) isomorphic_group homology_group(&0,top') ==> (path_connected_space top <=> path_connected_space top')`, REPEAT STRIP_TAC THEN REWRITE_TAC[PATH_CONNECTED_SPACE_IFF_COMPONENTS_SUBSET_SING] THEN REWRITE_TAC[GSYM CARD_LE_SING] THEN MATCH_MP_TAC CARD_LE_CONG THEN REWRITE_TAC[CARD_EQ_REFL] THEN MATCH_MP_TAC ISOMORPHIC_HOMOLOGY_IMP_PATH_COMPONENTS THEN ASM_REWRITE_TAC[]);; (* ------------------------------------------------------------------------- *) (* More basic properties of homology groups, deduced from the E-S axioms. *) (* ------------------------------------------------------------------------- *) let TRIVIAL_HOMOLOGY_GROUP = prove (`!p top:A topology. p < &0 ==> trivial_group(homology_group(p,top))`, REWRITE_TAC[TRIVIAL_RELATIVE_HOMOLOGY_GROUP; homology_group]);; let ABELIAN_HOMOLOGY_GROUP = prove (`!p top:A topology. abelian_group(homology_group(p,top))`, REWRITE_TAC[homology_group; ABELIAN_RELATIVE_HOMOLOGY_GROUP]);; let GROUP_HOMOMORPHISM_HOM_INDUCED_EMPTY = prove (`!p top top' f:A->B. group_homomorphism (homology_group (p,top),homology_group (p,top')) (hom_induced p (top,{}) (top',{}) f)`, REWRITE_TAC[GROUP_HOMOMORPHISM_HOM_INDUCED; homology_group]);; let HOM_INDUCED_COMPOSE_EMPTY = prove (`!p top top' top'' (f:A->B) (g:B->C). continuous_map(top,top') f /\ continuous_map(top',top'') g ==> hom_induced p (top,{}) (top'',{}) (g o f) = hom_induced p (top',{}) (top'',{}) g o hom_induced p (top,{}) (top',{}) f`, REWRITE_TAC[homology_group] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC HOM_INDUCED_COMPOSE THEN ASM_REWRITE_TAC[IMAGE_CLAUSES; EMPTY_SUBSET]);; let HOMOLOGY_HOMOTOPY_EMPTY = prove (`!p (top:A topology) (top':B topology) f g. homotopic_with (\h. T) (top,top') f g ==> hom_induced p (top,{}) (top',{}) f = hom_induced p (top,{}) (top',{}) g`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMOLOGY_HOMOTOPY_AXIOM THEN ASM_REWRITE_TAC[GSYM homology_group; IMAGE_CLAUSES; EMPTY_SUBSET]);; let HOMOTOPY_EQUIVALENCE_RELATIVE_HOMOLOGY_GROUP_ISOMORPHISMS = prove (`!p top top' s t (f:A->B) g. continuous_map (top,top') f /\ IMAGE f s SUBSET t /\ continuous_map (top',top) g /\ IMAGE g t SUBSET s /\ homotopic_with (\h. IMAGE h s SUBSET s) (top,top) (g o f) I /\ homotopic_with (\k. IMAGE k t SUBSET t) (top',top') (f o g) I ==> group_isomorphisms (relative_homology_group (p,top,s), relative_homology_group (p,top',t)) (hom_induced p (top,s) (top',t) f, hom_induced p (top',t) (top,s) g)`, REPEAT GEN_TAC THEN REPLICATE_TAC 4 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_TAC THEN REWRITE_TAC[group_isomorphisms] THEN REWRITE_TAC[GROUP_HOMOMORPHISM_HOM_INDUCED] THEN FIRST_ASSUM(CONJUNCTS_THEN (MP_TAC o SPEC `p:int` o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMOLOGY_HOMOTOPY_AXIOM))) THEN REWRITE_TAC[GSYM IMP_CONJ_ALT; homology_group] THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THEN GEN_REWRITE_TAC LAND_CONV [FUN_EQ_THM] THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC EQ_IMP THEN BINOP_TAC THEN ASM_SIMP_TAC[I_DEF; HOM_INDUCED_ID] THEN GEN_REWRITE_TAC RAND_CONV [GSYM o_THM] THEN AP_THM_TAC THEN MATCH_MP_TAC HOM_INDUCED_COMPOSE THEN ASM_REWRITE_TAC[]);; let HOMOTOPY_EQUIVALENCE_RELATIVE_HOMOLOGY_GROUP_ISOMORPHISM = prove (`!p top top' s t (f:A->B) g. continuous_map (top,top') f /\ IMAGE f s SUBSET t /\ continuous_map (top',top) g /\ IMAGE g t SUBSET s /\ homotopic_with (\h. IMAGE h s SUBSET s) (top,top) (g o f) I /\ homotopic_with (\k. IMAGE k t SUBSET t) (top',top') (f o g) I ==> group_isomorphism (relative_homology_group (p,top,s), relative_homology_group (p,top',t)) (hom_induced p (top,s) (top',t) f)`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP HOMOTOPY_EQUIVALENCE_RELATIVE_HOMOLOGY_GROUP_ISOMORPHISMS) THEN REWRITE_TAC[group_isomorphism] THEN MESON_TAC[]);; let HOMOTOPY_EQUIVALENCE_HOMOLOGY_GROUP_ISOMORPHISM = prove (`!p top top' (f:A->B) g. continuous_map (top,top') f /\ continuous_map (top',top) g /\ homotopic_with (\h. T) (top,top) (g o f) I /\ homotopic_with (\k. T) (top',top') (f o g) I ==> group_isomorphism (homology_group (p,top),homology_group (p,top')) (hom_induced p (top,{}) (top',{}) f)`, REPEAT STRIP_TAC THEN REWRITE_TAC[homology_group] THEN MATCH_MP_TAC HOMOTOPY_EQUIVALENCE_RELATIVE_HOMOLOGY_GROUP_ISOMORPHISM THEN EXISTS_TAC `g:B->A` THEN ASM_REWRITE_TAC[IMAGE_CLAUSES; EMPTY_SUBSET]);; let HOMOTOPY_EQUIVALENT_SPACE_IMP_ISOMORPHIC_RELATIVE_HOMOLOGY_GROUPS = prove (`!p top top' s t (f:A->B) g. continuous_map (top,top') f /\ IMAGE f s SUBSET t /\ continuous_map (top',top) g /\ IMAGE g t SUBSET s /\ homotopic_with (\h. IMAGE h s SUBSET s) (top,top) (g o f) I /\ homotopic_with (\k. IMAGE k t SUBSET t) (top',top') (f o g) I ==> relative_homology_group (p,top,s) isomorphic_group relative_homology_group (p,top',t)`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP HOMOTOPY_EQUIVALENCE_RELATIVE_HOMOLOGY_GROUP_ISOMORPHISMS) THEN REWRITE_TAC[isomorphic_group; group_isomorphism] THEN MESON_TAC[]);; let HOMOTOPY_EQUIVALENT_SPACE_IMP_ISOMORPHIC_HOMOLOGY_GROUPS = prove (`!p (top:A topology) (top':B topology). top homotopy_equivalent_space top' ==> homology_group(p,top) isomorphic_group homology_group(p,top')`, REWRITE_TAC[homotopy_equivalent_space] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[homology_group] THEN MATCH_MP_TAC HOMOTOPY_EQUIVALENT_SPACE_IMP_ISOMORPHIC_RELATIVE_HOMOLOGY_GROUPS THEN REWRITE_TAC[IMAGE_CLAUSES; EMPTY_SUBSET] THEN ASM_MESON_TAC[]);; let HOMEOMORPHIC_SPACE_IMP_ISOMORPHIC_HOMOLOGY_GROUPS = prove (`!p (top:A topology) (top':B topology). top homeomorphic_space top' ==> homology_group(p,top) isomorphic_group homology_group(p,top')`, SIMP_TAC[HOMOTOPY_EQUIVALENT_SPACE_IMP_ISOMORPHIC_HOMOLOGY_GROUPS; HOMEOMORPHIC_IMP_HOMOTOPY_EQUIVALENT_SPACE]);; let TRIVIAL_RELATIVE_HOMOLOGY_GROUP_GEN = prove (`!p top s f:A->A. continuous_map (top,subtopology top s) f /\ homotopic_with (\h. T) (subtopology top s,subtopology top s) f I /\ homotopic_with (\k. T) (top,top) f I ==> trivial_group(relative_homology_group(p,top,s))`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`hom_induced p (subtopology top s:A topology,{}) (top,{}) (\x. x)`; `hom_induced p (top:A topology,{}) (top,s) (\x. x)`; `hom_boundary p (top:A topology,s)`; `hom_induced (p - &1) (subtopology top s:A topology,{}) (top,{}) (\x. x)`; `homology_group (p,subtopology top (s:A->bool))`; `homology_group (p,top:A topology)`; `relative_homology_group (p,top,s:A->bool)`; `homology_group (p - &1,subtopology top (s:A->bool))`; `homology_group (p - &1,top:A topology)`] GROUP_EXACTNESS_IMP_TRIVIALITY) THEN REWRITE_TAC[HOMOLOGY_EXACTNESS_AXIOM_1; HOMOLOGY_EXACTNESS_AXIOM_2; HOMOLOGY_EXACTNESS_AXIOM_3] THEN DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[homology_group] THEN CONJ_TAC THEN MATCH_MP_TAC HOMOTOPY_EQUIVALENCE_RELATIVE_HOMOLOGY_GROUP_ISOMORPHISM THEN SIMP_TAC[IMAGE_CLAUSES; EMPTY_SUBSET] THEN SIMP_TAC[CONTINUOUS_MAP_ID; CONTINUOUS_MAP_FROM_SUBTOPOLOGY] THEN REWRITE_TAC[o_DEF; ETA_AX] THEN ASM_MESON_TAC[]);; let TRIVIAL_RELATIVE_HOMOLOGY_GROUP_TOPSPACE = prove (`!p top:A topology. trivial_group(relative_homology_group(p,top,topspace top))`, REPEAT GEN_TAC THEN MATCH_MP_TAC TRIVIAL_RELATIVE_HOMOLOGY_GROUP_GEN THEN EXISTS_TAC `\x:A. x` THEN REWRITE_TAC[HOMOTOPIC_WITH_REFL; I_DEF; SUBTOPOLOGY_TOPSPACE] THEN REWRITE_TAC[CONTINUOUS_MAP_ID]);; let TRIVIAL_RELATIVE_HOMOLOGY_GROUP_EMPTY = prove (`!p top s:A->bool. topspace top = {} ==> trivial_group(relative_homology_group(p,top,s))`, ONCE_REWRITE_TAC[RELATIVE_HOMOLOGY_GROUP_RESTRICT] THEN SIMP_TAC[INTER_EMPTY] THEN MESON_TAC[TRIVIAL_RELATIVE_HOMOLOGY_GROUP_TOPSPACE]);; let TRIVIAL_HOMOLOGY_GROUP_EMPTY = prove (`!p top:A topology. topspace top = {} ==> trivial_group(homology_group(p,top))`, REWRITE_TAC[homology_group; TRIVIAL_RELATIVE_HOMOLOGY_GROUP_EMPTY]);; let TRIVIAL_HOMOLOGY_GROUP_CONTRACTIBLE_SPACE = prove (`!p top:A topology. contractible_space top /\ ~(p = &0) ==> trivial_group(homology_group(p,top))`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN REWRITE_TAC[CONTRACTIBLE_EQ_HOMOTOPY_EQUIVALENT_SINGLETON_SUBTOPOLOGY] THEN DISCH_THEN(DISJ_CASES_THEN MP_TAC) THEN REWRITE_TAC[TRIVIAL_HOMOLOGY_GROUP_EMPTY] THEN DISCH_THEN(X_CHOOSE_THEN `a:A` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(SUBST1_TAC o MATCH_MP ISOMORPHIC_GROUP_TRIVIALITY o SPEC`p:int` o MATCH_MP HOMOTOPY_EQUIVALENT_SPACE_IMP_ISOMORPHIC_HOMOLOGY_GROUPS) THEN MATCH_MP_TAC HOMOLOGY_DIMENSION_AXIOM THEN EXISTS_TAC `a:A` THEN ASM_REWRITE_TAC[TOPSPACE_SUBTOPOLOGY] THEN ASM SET_TAC[]);; let HOMEOMORPHIC_MAPS_RELATIVE_HOMOLOGY_GROUP_ISOMORPHISMS = prove (`!p top top' s t (f:A->B) g. homeomorphic_maps (top,top') (f,g) /\ IMAGE f s SUBSET t /\ IMAGE g t SUBSET s ==> group_isomorphisms (relative_homology_group (p,top,s), relative_homology_group (p,top',t)) (hom_induced p (top,s) (top',t) f, hom_induced p (top',t) (top,s) g)`, ONCE_REWRITE_TAC[HOM_INDUCED_RESTRICT; RELATIVE_HOMOLOGY_GROUP_RESTRICT] THEN REWRITE_TAC[homeomorphic_maps] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMOTOPY_EQUIVALENCE_RELATIVE_HOMOLOGY_GROUP_ISOMORPHISMS THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[continuous_map]) THEN ASM SET_TAC[]; CONJ_TAC THEN MATCH_MP_TAC HOMOTOPIC_WITH_EQUAL THEN ASM_SIMP_TAC[o_THM; SUBSET; FORALL_IN_IMAGE; I_THM; IN_INTER] THEN ASM_MESON_TAC[CONTINUOUS_MAP_COMPOSE]]);; let HOMEOMORPHIC_MAP_RELATIVE_HOMOLOGY_GROUP_ISOMORPHISM = prove (`!p top top' s t (f:A->B). homeomorphic_map (top,top') f /\ s SUBSET topspace top /\ IMAGE f s = t ==> group_isomorphism (relative_homology_group (p,top,s), relative_homology_group (p,top',t)) (hom_induced p (top,s) (top',t) f)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHIC_MAP_MAPS]) THEN DISCH_THEN(X_CHOOSE_TAC `g:B->A`) THEN REWRITE_TAC[group_isomorphism] THEN EXISTS_TAC `hom_induced p (top',t) (top,s) (g:B->A)` THEN MATCH_MP_TAC HOMEOMORPHIC_MAPS_RELATIVE_HOMOLOGY_GROUP_ISOMORPHISMS THEN ASM_REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[homeomorphic_maps; continuous_map]) THEN ASM SET_TAC[]);; let GROUP_MONOMORPHISM_HOM_INDUCED_SECTION_MAP = prove (`!p top top' f:A->B. section_map (top,top') f ==> group_monomorphism (homology_group (p,top), homology_group (p,top')) (hom_induced p (top,{}) (top',{}) f)`, REPEAT GEN_TAC THEN REWRITE_TAC[section_map; retraction_maps; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `g:B->A` THEN STRIP_TAC THEN MATCH_MP_TAC GROUP_MONOMORPHISM_LEFT_INVERTIBLE THEN EXISTS_TAC `hom_induced p (top',{}) (top,{}) (g:B->A)` THEN ASM_SIMP_TAC[GROUP_HOMOMORPHISM_HOM_INDUCED_EMPTY] THEN MP_TAC(GSYM(ISPECL [`p:int`; `top:A topology`; `top':B topology`; `top:A topology`; `f:A->B`; `g:B->A`] HOM_INDUCED_COMPOSE_EMPTY)) THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC LAND_CONV [FUN_EQ_THM] THEN SIMP_TAC[o_THM] THEN DISCH_THEN(K ALL_TAC) THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC HOM_INDUCED_ID_GEN THEN ASM_REWRITE_TAC[o_THM; GSYM homology_group] THEN ASM_MESON_TAC[CONTINUOUS_MAP_COMPOSE]);; let GROUP_EPIMORPHISM_HOM_INDUCED_RETRACTION_MAP = prove (`!p top top' f:A->B. retraction_map (top,top') f ==> group_epimorphism (homology_group (p,top), homology_group (p,top')) (hom_induced p (top,{}) (top',{}) f)`, REPEAT GEN_TAC THEN REWRITE_TAC[retraction_map; retraction_maps; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `g:B->A` THEN STRIP_TAC THEN MATCH_MP_TAC GROUP_EPIMORPHISM_RIGHT_INVERTIBLE THEN EXISTS_TAC `hom_induced p (top',{}) (top,{}) (g:B->A)` THEN ASM_SIMP_TAC[GROUP_HOMOMORPHISM_HOM_INDUCED_EMPTY] THEN MP_TAC(GSYM(ISPECL [`p:int`; `top':B topology`; `top:A topology`; `top':B topology`; `g:B->A`; `f:A->B`] HOM_INDUCED_COMPOSE_EMPTY)) THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC LAND_CONV [FUN_EQ_THM] THEN SIMP_TAC[o_THM] THEN DISCH_THEN(K ALL_TAC) THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC HOM_INDUCED_ID_GEN THEN ASM_REWRITE_TAC[o_THM; GSYM homology_group] THEN ASM_MESON_TAC[CONTINUOUS_MAP_COMPOSE]);; let HOMEOMORPHIC_MAP_HOMOLOGY_GROUP_ISOMORPHISM = prove (`!p top top' f:A->B. homeomorphic_map (top,top') f ==> group_isomorphism (homology_group (p,top), homology_group (p,top')) (hom_induced p (top,{}) (top',{}) f)`, REWRITE_TAC[GSYM SECTION_AND_RETRACTION_EQ_HOMEOMORPHIC_MAP] THEN REWRITE_TAC[GSYM GROUP_MONOMORPHISM_EPIMORPHISM] THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[GROUP_MONOMORPHISM_HOM_INDUCED_SECTION_MAP; GROUP_EPIMORPHISM_HOM_INDUCED_RETRACTION_MAP]);; let GROUP_MONOMORPHISM_HOM_INDUCED_INCLUSION = prove (`!p top s:A->bool. s = {} \/ s retract_of_space top ==> group_monomorphism (homology_group(p,subtopology top s),homology_group(p,top)) (hom_induced p (subtopology top s,{}) (top,{}) (\x. x))`, REWRITE_TAC[RETRACT_OF_SPACE_SECTION_MAP] THEN REPEAT STRIP_TAC THENL [W(MP_TAC o PART_MATCH (lhand o rand) GROUP_MONOMORPHISM_FROM_TRIVIAL_GROUP o snd) THEN ASM_SIMP_TAC[TRIVIAL_HOMOLOGY_GROUP_EMPTY; TOPSPACE_SUBTOPOLOGY; INTER_EMPTY] THEN ASM_SIMP_TAC[homology_group; GROUP_HOMOMORPHISM_HOM_INDUCED; CONTINUOUS_MAP_ID; CONTINUOUS_MAP_FROM_SUBTOPOLOGY; IMAGE_CLAUSES; EMPTY_SUBSET]; MATCH_MP_TAC GROUP_MONOMORPHISM_HOM_INDUCED_SECTION_MAP THEN ASM_REWRITE_TAC[]]);; let TRIVIAL_HOMOMORPHISM_HOM_BOUNDARY_INCLUSION = prove (`!p top s:A->bool. s = {} \/ s retract_of_space top ==> trivial_homomorphism (relative_homology_group(p,top,s), homology_group (p - &1,subtopology top s)) (hom_boundary p (top,s))`, REPEAT GEN_TAC THEN MP_TAC(CONJ (SPEC `p:int` HOMOLOGY_EXACTNESS_AXIOM_1) (SPEC `p:int` HOMOLOGY_EXACTNESS_AXIOM_2)) THEN REWRITE_TAC[AND_FORALL_THM] THEN DISCH_THEN(MP_TAC o SPECL [`top:A topology`; `s:A->bool`]) THEN DISCH_THEN(SUBST1_TAC o SYM o MATCH_MP GROUP_EXACTNESS_MONOMORPHISM_EQ_TRIVIALITY) THEN REWRITE_TAC[GROUP_MONOMORPHISM_HOM_INDUCED_INCLUSION]);; let GROUP_EPIMORPHISM_HOM_INDUCED_RELATIVIZATION = prove (`!p top s:A->bool. s = {} \/ s retract_of_space top ==> group_epimorphism (homology_group(p,top),relative_homology_group(p,top,s)) (hom_induced p (top,{}) (top,s) (\x. x))`, REPEAT GEN_TAC THEN MP_TAC(CONJ (SPEC `p:int` HOMOLOGY_EXACTNESS_AXIOM_1) (SPEC `p:int` HOMOLOGY_EXACTNESS_AXIOM_2)) THEN REWRITE_TAC[AND_FORALL_THM] THEN DISCH_THEN(MP_TAC o SPECL [`top:A topology`; `s:A->bool`]) THEN DISCH_THEN(SUBST1_TAC o MATCH_MP GROUP_EXACTNESS_EPIMORPHISM_EQ_TRIVIALITY) THEN REWRITE_TAC[TRIVIAL_HOMOMORPHISM_HOM_BOUNDARY_INCLUSION]);; let SHORT_EXACT_SEQUENCE_HOM_INDUCED_INCLUSION = prove (`!p top s:A->bool. s = {} \/ s retract_of_space top ==> short_exact_sequence (homology_group(p,subtopology top s), homology_group(p,top), relative_homology_group(p,top,s)) (hom_induced p (subtopology top s,{}) (top,{}) (\x. x), hom_induced p (top,{}) (top,s) (\x. x))`, REWRITE_TAC[short_exact_sequence; HOMOLOGY_EXACTNESS_AXIOM_3] THEN SIMP_TAC[GROUP_MONOMORPHISM_HOM_INDUCED_INCLUSION] THEN REWRITE_TAC[GROUP_EPIMORPHISM_HOM_INDUCED_RELATIVIZATION]);; let GROUP_ISOMORPHISMS_HOMOLOGY_GROUP_PROD_RETRACT = prove (`!p top s:A->bool. s = {} \/ s retract_of_space top ==> ?h k. h subgroup_of homology_group(p,top) /\ k subgroup_of homology_group(p,top) /\ group_isomorphism (prod_group (subgroup_generated (homology_group(p,top)) h) (subgroup_generated (homology_group(p,top)) k), (homology_group(p,top))) (\(x,y). group_mul(homology_group(p,top)) x y) /\ group_isomorphism (homology_group(p,subtopology top s), subgroup_generated (homology_group(p,top)) h) (hom_induced p (subtopology top s,{}) (top,{}) (\x. x)) /\ group_isomorphism (subgroup_generated (homology_group(p,top)) k, relative_homology_group(p,top,s)) (hom_induced p (top,{}) (top,s) (\x. x))`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `p:int` o MATCH_MP SHORT_EXACT_SEQUENCE_HOM_INDUCED_INCLUSION) THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] SPLITTING_LEMMA_LEFT)) THEN REWRITE_TAC[LEFT_FORALL_IMP_THM; IMP_IMP] THEN ANTS_TAC THENL [FIRST_X_ASSUM(DISJ_CASES_THEN2 SUBST1_TAC ASSUME_TAC) THENL [EXISTS_TAC `(\x. group_id(homology_group(p,subtopology top {}))) :(((num->real)->A)frag->bool)->((num->real)->A)frag->bool` THEN REWRITE_TAC[GROUP_HOMOMORPHISM_TRIVIAL] THEN MATCH_MP_TAC(SET_RULE `s = {a} ==> !x. x IN s ==> a = x`) THEN REWRITE_TAC[GSYM trivial_group] THEN MATCH_MP_TAC TRIVIAL_HOMOLOGY_GROUP_EMPTY THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; INTER_EMPTY]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [retract_of_space]) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (X_CHOOSE_THEN `r:A->A` STRIP_ASSUME_TAC)) THEN EXISTS_TAC `hom_induced p (top,{}) (subtopology top s,{}) (r:A->A)` THEN ASM_SIMP_TAC[GROUP_HOMOMORPHISM_HOM_INDUCED_EMPTY] THEN REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_THM] THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o rator o lhand o snd) THEN REWRITE_TAC[IMAGE_CLAUSES; EMPTY_SUBSET] THEN ASM_SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_ID] THEN ASM_REWRITE_TAC[GSYM homology_group] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[o_DEF; ETA_AX] THEN MATCH_MP_TAC HOM_INDUCED_ID_GEN THEN ASM_REWRITE_TAC[GSYM homology_group] THEN ASM_SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; TOPSPACE_SUBTOPOLOGY] THEN ASM SET_TAC[]]; REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN REWRITE_TAC[normal_subgroup_of] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[GROUP_ISOMORPHISM_GROUP_MUL; ABELIAN_HOMOLOGY_GROUP]]);; let ISOMORPHIC_GROUP_HOMOLOGY_GROUP_PROD_RETRACT = prove (`!p top s:A->bool. s = {} \/ s retract_of_space top ==> homology_group(p,top) isomorphic_group prod_group (homology_group(p,subtopology top s)) (relative_homology_group(p,top,s))`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `p:int` o MATCH_MP GROUP_ISOMORPHISMS_HOMOLOGY_GROUP_PROD_RETRACT) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`h:(((num->real)->A)frag->bool)->bool`; `k:(((num->real)->A)frag->bool)->bool`] THEN STRIP_TAC THEN TRANS_TAC ISOMORPHIC_GROUP_TRANS `prod_group (subgroup_generated (homology_group(p,top:A topology)) h) (subgroup_generated (homology_group(p,top)) k)` THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[ISOMORPHIC_GROUP_SYM] THEN REWRITE_TAC[isomorphic_group] THEN ASM_MESON_TAC[]; MATCH_MP_TAC ISOMORPHIC_GROUP_PROD_GROUPS THEN GEN_REWRITE_TAC LAND_CONV [ISOMORPHIC_GROUP_SYM] THEN REWRITE_TAC[isomorphic_group] THEN ASM_MESON_TAC[]]);; let HOMOLOGY_ADDITIVITY_EXPLICIT = prove (`!p top s t:A->bool. open_in top s /\ open_in top t /\ DISJOINT s t /\ s UNION t = topspace top ==> group_isomorphism (prod_group (homology_group (p,subtopology top s)) (homology_group (p,subtopology top t)), homology_group (p,top)) (\(a,b). group_mul (homology_group (p,top)) (hom_induced p (subtopology top s,{}) (top,{}) (\x. x) a) (hom_induced p (subtopology top t,{}) (top,{}) (\x. x) b))`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `closed_in top (s:A->bool) /\ closed_in top t` STRIP_ASSUME_TAC THENL [ASM_SIMP_TAC[closed_in; OPEN_IN_SUBSET] THEN CONJ_TAC THENL [UNDISCH_TAC `open_in top (t:A->bool)`; UNDISCH_TAC `open_in top (s:A->bool)`] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`hom_induced p (top,{}) (top,t) (\x:A. x)`; `hom_induced p (top,{}) (top,s) (\x:A. x)`; `hom_induced p (subtopology top s,{}) (top,t) (\x:A. x)`; `hom_induced p (subtopology top s,{}) (top,{}) (\x:A. x)`; `hom_induced p (subtopology top t,{}) (top,{}) (\x:A. x)`; `hom_induced p (subtopology top t,{}) (top,s) (\x:A. x)`; `homology_group(p,subtopology top s:A topology)`; `homology_group(p,subtopology top t:A topology)`; `relative_homology_group(p,top,t:A->bool)`; `relative_homology_group(p,top,s:A->bool)`; `homology_group(p,top:A topology)`] EXACT_SEQUENCE_SUM_LEMMA) THEN ANTS_TAC THENL [ALL_TAC; SIMP_TAC[]] THEN REWRITE_TAC[ABELIAN_HOMOLOGY_GROUP; HOMOLOGY_EXACTNESS_AXIOM_3] THEN GEN_REWRITE_TAC I [CONJ_ASSOC] THEN CONJ_TAC THENL [CONJ_TAC THENL [MP_TAC(ISPECL [`p:int`; `top:A topology`; `s UNION t:A->bool`; `t:A->bool`; `t:A->bool`] HOMOLOGY_EXCISION_AXIOM); MP_TAC(ISPECL [`p:int`; `top:A topology`; `s UNION t:A->bool`; `s:A->bool`; `s:A->bool`] HOMOLOGY_EXCISION_AXIOM)] THEN REWRITE_TAC[DIFF_EQ_EMPTY; SUBSET_UNION] THEN ASM_SIMP_TAC[SET_RULE `DISJOINT s t ==> (s UNION t) DIFF t = s /\ (s UNION t) DIFF s = t`] THEN ASM_REWRITE_TAC[SUBTOPOLOGY_TOPSPACE; GSYM homology_group] THEN ASM_SIMP_TAC[CLOSURE_OF_CLOSED_IN; INTERIOR_OF_OPEN_IN; SUBSET_REFL]; REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_THM] THEN ASM_SIMP_TAC[GSYM HOM_INDUCED_COMPOSE; IMAGE_CLAUSES; EMPTY_SUBSET; CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_ID] THEN REWRITE_TAC[o_DEF]]);; (* ------------------------------------------------------------------------- *) (* Generalize exact homology sequence to triples (derived from E-S axioms) *) (* ------------------------------------------------------------------------- *) let hom_relboundary = new_definition `hom_relboundary p (top:A topology,s,t) = hom_induced (p - &1) (subtopology top s,{}) (subtopology top s,t) (\x. x) o hom_boundary p (top,s)`;; let GROUP_HOMOMORPHISM_HOM_RELBOUNDARY = prove (`!p top s t:A->bool. group_homomorphism (relative_homology_group (p,top,s), relative_homology_group (p - &1,subtopology top s,t)) (hom_relboundary p (top,s,t))`, REPEAT GEN_TAC THEN REWRITE_TAC[hom_relboundary] THEN MATCH_MP_TAC GROUP_HOMOMORPHISM_COMPOSE THEN EXISTS_TAC `homology_group(p - &1,subtopology top (s:A->bool))` THEN REWRITE_TAC[GROUP_HOMOMORPHISM_HOM_BOUNDARY; GROUP_HOMOMORPHISM_HOM_INDUCED; homology_group]);; let HOM_RELBOUNDARY = prove (`!p top s t c. hom_relboundary p (top,s,t) c IN group_carrier (relative_homology_group (p - &1,subtopology top s,t))`, REWRITE_TAC[hom_relboundary; o_THM; HOM_INDUCED]);; let HOM_RELBOUNDARY_EMPTY = prove (`!p top s:A->bool. hom_relboundary p (top,s,{}) = hom_boundary p (top,s)`, SIMP_TAC[hom_relboundary; o_DEF; HOM_INDUCED_ID; HOM_BOUNDARY; GSYM homology_group; ETA_AX]);; let NATURALITY_HOM_INDUCED_RELBOUNDARY = prove (`!p top s t top' u v (f:A->B). continuous_map(top,top') f /\ IMAGE f s SUBSET u /\ IMAGE f t SUBSET v ==> hom_relboundary p (top',u,v) o hom_induced p (top,s) (top',u) f = hom_induced (p - &1) (subtopology top s,t) (subtopology top' u,v) f o hom_relboundary p (top,s,t)`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[hom_relboundary; GSYM o_ASSOC; NATURALITY_HOM_INDUCED] THEN REWRITE_TAC[o_ASSOC] THEN AP_THM_TAC THEN AP_TERM_TAC THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o lhand o snd) THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o rand o rand o snd) THEN ASM_SIMP_TAC[IMAGE_CLAUSES; IMAGE_ID; EMPTY_SUBSET; INTER_SUBSET; CONTINUOUS_MAP_ID; CONTINUOUS_MAP_IN_SUBTOPOLOGY; CONTINUOUS_MAP_FROM_SUBTOPOLOGY; TOPSPACE_SUBTOPOLOGY; SET_RULE `IMAGE f s SUBSET t ==> IMAGE f (u INTER s) SUBSET t`] THEN REPEAT(DISCH_THEN(SUBST1_TAC o SYM)) THEN REWRITE_TAC[o_DEF]);; let HOMOLOGY_EXACTNESS_TRIPLE_1 = prove (`!p top s t:A->bool. t SUBSET s ==> group_exactness(relative_homology_group(p,top,t), relative_homology_group(p,top,s), relative_homology_group(p - &1,subtopology top s,t)) (hom_induced p (top,t) (top,s) (\x. x), hom_relboundary p (top,s,t))`, REPEAT STRIP_TAC THEN REWRITE_TAC[group_exactness; GROUP_HOMOMORPHISM_HOM_RELBOUNDARY; GROUP_HOMOMORPHISM_HOM_INDUCED] THEN REWRITE_TAC[group_image; group_kernel] THEN MATCH_MP_TAC SUBSET_ANTISYM THEN GEN_REWRITE_TAC BINOP_CONV [SUBSET] THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_IMAGE; IN_ELIM_THM] THEN CONJ_TAC THENL [X_GEN_TAC `b:((num->real)->A)frag->bool` THEN DISCH_TAC THEN REWRITE_TAC[HOM_INDUCED; hom_relboundary] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_THM] THEN REWRITE_TAC[GSYM o_ASSOC] THEN ASM_SIMP_TAC[NATURALITY_HOM_INDUCED; IMAGE_ID; CONTINUOUS_MAP_ID] THEN REWRITE_TAC[o_THM] THEN MP_TAC(ISPECL [`p - &1:int`; `subtopology top (s:A->bool)`; `t:A->bool`] HOMOLOGY_EXACTNESS_AXIOM_3) THEN ASM_SIMP_TAC[SUBTOPOLOGY_SUBTOPOLOGY; SET_RULE `t SUBSET s ==> s INTER t = t`] THEN REWRITE_TAC[group_exactness; group_kernel; group_image] THEN DISCH_THEN(MATCH_MP_TAC o MATCH_MP (SET_RULE `P /\ Q /\ IMAGE f s = {y | y IN t /\ g y = z} ==> !x. x IN s ==> g(f x) = z`)) THEN REWRITE_TAC[HOM_BOUNDARY]; ALL_TAC] THEN X_GEN_TAC `x:((num->real)->A)frag->bool` THEN REWRITE_TAC[hom_relboundary; o_THM] THEN STRIP_TAC THEN MP_TAC(ISPECL [`p - &1:int`; `subtopology top (s:A->bool)`; `t:A->bool`] HOMOLOGY_EXACTNESS_AXIOM_3) THEN REWRITE_TAC[group_exactness; group_kernel; group_image] THEN DISCH_THEN(MP_TAC o SPEC `hom_boundary p (top,s:A->bool) x` o MATCH_MP (SET_RULE `P /\ Q /\ IMAGE f s = {y | y IN t /\ g y = z} ==> !x. x IN t /\ g x = z ==> ?y. y IN s /\ f y = x`)) THEN ASM_REWRITE_TAC[HOM_BOUNDARY] THEN ASM_SIMP_TAC[SUBTOPOLOGY_SUBTOPOLOGY; SET_RULE `t SUBSET s ==> s INTER t = t`] THEN DISCH_THEN(X_CHOOSE_THEN `y:((num->real)->A)frag->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`p:int`; `top:A topology`; `t:A->bool`] HOMOLOGY_EXACTNESS_AXIOM_2) THEN REWRITE_TAC[group_exactness; group_kernel; group_image] THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `P /\ Q /\ IMAGE f s = {y | y IN t /\ g y = z} ==> !x. x IN t /\ g x = z ==> ?y. y IN s /\ f y = x`)) THEN DISCH_THEN(MP_TAC o SPEC `y:((num->real)->A)frag->bool`) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [MP_TAC(ISPECL [`p:int`; `top:A topology`; `s:A->bool`] HOMOLOGY_EXACTNESS_AXIOM_2) THEN REWRITE_TAC[group_exactness; group_kernel; group_image] THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `P /\ Q /\ IMAGE f s = {y | y IN t /\ g y = z} ==> !x. x IN s ==> g(f x) = z`)) THEN DISCH_THEN(MP_TAC o SPEC `x:((num->real)->A)frag->bool`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EQ_TRANS) THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [SYM th]) THEN GEN_REWRITE_TAC RAND_CONV [GSYM o_THM] THEN AP_THM_TAC THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o rand o snd) THEN SIMP_TAC[CONTINUOUS_MAP_ID; CONTINUOUS_MAP_IN_SUBTOPOLOGY; CONTINUOUS_MAP_FROM_SUBTOPOLOGY; IMAGE_ID; EMPTY_SUBSET] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY] THEN ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN(SUBST1_TAC o SYM)] THEN REWRITE_TAC[o_DEF]; DISCH_THEN(X_CHOOSE_THEN `z:((num->real)->A)frag->bool` STRIP_ASSUME_TAC)] THEN MP_TAC(ISPECL [`p:int`; `top:A topology`; `s:A->bool`] HOMOLOGY_EXACTNESS_AXIOM_1) THEN REWRITE_TAC[group_exactness; group_kernel; group_image] THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `P /\ Q /\ IMAGE f s = {y | y IN t /\ g y = z} ==> !x. x IN t /\ g x = z ==> ?y. y IN s /\ f y = x`)) THEN DISCH_THEN(MP_TAC o SPEC `group_div (relative_homology_group (p,top,s)) x (hom_induced p (top,t) (top,s) (\x:A. x) z)`) THEN ASM_SIMP_TAC[GROUP_DIV; HOM_INDUCED; HOM_BOUNDARY; MATCH_MP GROUP_HOMOMORPHISM_DIV (SPEC_ALL GROUP_HOMOMORPHISM_HOM_BOUNDARY); GROUP_RULE `group_div G x y = group_id G <=> y = x`] THEN ANTS_TAC THENL [GEN_REWRITE_TAC LAND_CONV [GSYM o_THM] THEN ASM_SIMP_TAC[NATURALITY_HOM_INDUCED; IMAGE_ID; CONTINUOUS_MAP_ID] THEN ASM_REWRITE_TAC[o_THM]; DISCH_THEN(X_CHOOSE_THEN `w:((num->real)->A)frag->bool` STRIP_ASSUME_TAC)] THEN EXISTS_TAC `group_mul (relative_homology_group (p,top,t)) z (hom_induced p (top,{}) (top,t) (\x:A. x) w)` THEN ASM_SIMP_TAC[GROUP_MUL; HOM_INDUCED; MATCH_MP GROUP_HOMOMORPHISM_MUL (SPEC_ALL GROUP_HOMOMORPHISM_HOM_INDUCED)] THEN W(MP_TAC o PART_MATCH (lhand o rand) (REWRITE_RULE[abelian_group] ABELIAN_RELATIVE_HOMOLOGY_GROUP) o rand o snd) THEN REWRITE_TAC[HOM_INDUCED] THEN DISCH_THEN SUBST1_TAC THEN ASM_SIMP_TAC[GROUP_MUL; HOM_INDUCED; GROUP_RULE `x = group_mul G z y <=> group_div G x y = z`] THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC LAND_CONV [SYM th]) THEN GEN_REWRITE_TAC RAND_CONV [GSYM o_THM] THEN AP_THM_TAC THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o rand o snd) THEN SIMP_TAC[CONTINUOUS_MAP_ID; CONTINUOUS_MAP_IN_SUBTOPOLOGY; CONTINUOUS_MAP_FROM_SUBTOPOLOGY; IMAGE_ID; EMPTY_SUBSET] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY] THEN ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN(SUBST1_TAC o SYM)] THEN REWRITE_TAC[o_DEF]);; let HOMOLOGY_EXACTNESS_TRIPLE_2 = prove (`!p top s t:A->bool. t SUBSET s ==> group_exactness(relative_homology_group(p,top,s), relative_homology_group(p - &1,subtopology top s,t), relative_homology_group(p - &1,top,t)) (hom_relboundary p (top,s,t), hom_induced (p - &1) (subtopology top s,t) (top,t) (\x. x))`, REPEAT STRIP_TAC THEN REWRITE_TAC[group_exactness; GROUP_HOMOMORPHISM_HOM_RELBOUNDARY; GROUP_HOMOMORPHISM_HOM_INDUCED] THEN REWRITE_TAC[group_image; group_kernel] THEN MATCH_MP_TAC SUBSET_ANTISYM THEN GEN_REWRITE_TAC BINOP_CONV [SUBSET] THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_IMAGE; IN_ELIM_THM] THEN CONJ_TAC THENL [REWRITE_TAC[HOM_RELBOUNDARY] THEN X_GEN_TAC `c:((num->real)->A)frag->bool` THEN DISCH_TAC THEN REWRITE_TAC[hom_relboundary] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_THM] THEN TRANS_TAC EQ_TRANS `(hom_induced (p - &1) (top,{}) (top,t) (\x:A. x) o hom_induced (p - &1) (subtopology top s,{}) (top,{}) (\x. x) o hom_boundary p (top,s)) c` THEN CONJ_TAC THENL [AP_THM_TAC THEN REWRITE_TAC[o_ASSOC] THEN AP_THM_TAC THEN AP_TERM_TAC THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o rand o snd) THEN SIMP_TAC[IMAGE_ID; EMPTY_SUBSET; CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_ID] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o lhand o snd) THEN REWRITE_TAC[CONTINUOUS_MAP_ID; IMAGE_ID; EMPTY_SUBSET] THEN SIMP_TAC[SUBSET_REFL; CONTINUOUS_MAP_ID; CONTINUOUS_MAP_FROM_SUBTOPOLOGY]; ONCE_REWRITE_TAC[o_THM] THEN MATCH_MP_TAC(MESON[group_homomorphism] `!G. group_homomorphism(G,H) f /\ x = group_id G ==> f x = group_id H`) THEN EXISTS_TAC `homology_group(p - &1,top:A topology)` THEN REWRITE_TAC[homology_group; GROUP_HOMOMORPHISM_HOM_INDUCED] THEN MP_TAC(ISPECL[`p:int`; `top:A topology`; `s:A->bool`] HOMOLOGY_EXACTNESS_AXIOM_2) THEN REWRITE_TAC[group_exactness; group_kernel; group_image; o_THM] THEN REWRITE_TAC[GSYM homology_group] THEN MATCH_MP_TAC(SET_RULE `x IN s ==> P /\ Q /\ IMAGE f s = {y | y IN t /\ g y = z} ==> g(f x) = z`) THEN ASM SET_TAC[]]; ALL_TAC] THEN X_GEN_TAC `x:((num->real)->A)frag->bool` THEN STRIP_TAC THEN MP_TAC(ISPECL[`p - &1:int`; `subtopology top s:A topology`; `t:A->bool`] HOMOLOGY_EXACTNESS_AXIOM_1) THEN REWRITE_TAC[group_exactness; group_kernel; group_image] THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `P /\ Q /\ IMAGE f s = {y | y IN t /\ g y = z} ==> !x. x IN t /\ g x = z ==> ?y. y IN s /\ f y = x`)) THEN DISCH_THEN(MP_TAC o SPEC `x:((num->real)->A)frag->bool`) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [MP_TAC(ISPECL [`p - &1:int`; `subtopology top s:A topology`; `t:A->bool`; `top:A topology`; `t:A->bool`; `\x:A. x`] NATURALITY_HOM_INDUCED) THEN REWRITE_TAC[IMAGE_ID; SUBSET_REFL] THEN SIMP_TAC[CONTINUOUS_MAP_ID; CONTINUOUS_MAP_FROM_SUBTOPOLOGY] THEN ASM_SIMP_TAC[SUBTOPOLOGY_SUBTOPOLOGY; SET_RULE `t SUBSET s ==> s INTER t = t`] THEN DISCH_THEN(MP_TAC o C AP_THM `x:((num->real)->A)frag->bool`) THEN ASM_REWRITE_TAC[o_THM; REWRITE_RULE[group_homomorphism] GROUP_HOMOMORPHISM_HOM_BOUNDARY] THEN DISCH_THEN SUBST1_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC HOM_INDUCED_ID THEN W(MP_TAC o PART_MATCH lhand HOM_BOUNDARY o lhand o snd) THEN ASM_SIMP_TAC[homology_group; SUBTOPOLOGY_SUBTOPOLOGY; SET_RULE `t SUBSET s ==> s INTER t = t`]; DISCH_THEN(X_CHOOSE_THEN `y:((num->real)->A)frag->bool` STRIP_ASSUME_TAC)] THEN MP_TAC(ISPECL [`p - &1:int`; `top:A topology`; `t:A->bool`] HOMOLOGY_EXACTNESS_AXIOM_3) THEN REWRITE_TAC[group_exactness; group_kernel; group_image] THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `P /\ Q /\ IMAGE f s = {y | y IN t /\ g y = z} ==> !x. x IN t /\ g x = z ==> ?y. y IN s /\ f y = x`)) THEN DISCH_THEN(MP_TAC o SPEC `hom_induced (p - &1) (subtopology top s,{}) (top,{}) (\x:A. x) y`) THEN REWRITE_TAC[HOM_INDUCED; homology_group] THEN GEN_REWRITE_TAC (funpow 3 LAND_CONV) [GSYM o_THM] THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o rator o lhand o lhand o lhand o snd) THEN REWRITE_TAC[IMAGE_ID; EMPTY_SUBSET] THEN SIMP_TAC[CONTINUOUS_MAP_ID; CONTINUOUS_MAP_FROM_SUBTOPOLOGY] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN ASM_REWRITE_TAC[o_DEF] THEN MP_TAC(ISPECL [`p - &1:int`; `subtopology top s:A topology`; `{}:A->bool`; `subtopology top s:A topology`; `t:A->bool`; `top:A topology`; `t:A->bool`; `\x:A. x`; `\x:A. x`] HOM_INDUCED_COMPOSE) THEN REWRITE_TAC[IMAGE_ID; EMPTY_SUBSET; SUBSET_REFL] THEN SIMP_TAC[CONTINUOUS_MAP_ID; CONTINUOUS_MAP_FROM_SUBTOPOLOGY] THEN REWRITE_TAC[o_DEF] THEN DISCH_THEN SUBST1_TAC THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `z:((num->real)->A)frag->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`p:int`; `top:A topology`; `s:A->bool`] HOMOLOGY_EXACTNESS_AXIOM_2) THEN REWRITE_TAC[group_exactness; group_kernel; group_image] THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `P /\ Q /\ IMAGE f s = {y | y IN t /\ g y = z} ==> !x. x IN t /\ g x = z ==> ?y. y IN s /\ f y = x`)) THEN DISCH_THEN(MP_TAC o SPEC `group_div (homology_group(p - &1,subtopology top s)) y (hom_induced (p - &1) (subtopology top t,{}) (subtopology top s,{}) (\x:A. x) z)`) THEN RULE_ASSUM_TAC(REWRITE_RULE[homology_group]) THEN ASM_SIMP_TAC[GROUP_DIV; HOM_INDUCED; homology_group; MATCH_MP GROUP_HOMOMORPHISM_DIV (SPEC_ALL GROUP_HOMOMORPHISM_HOM_INDUCED); GROUP_RULE `group_div G x y = group_id G <=> y = x`] THEN GEN_REWRITE_TAC (funpow 3 LAND_CONV) [GSYM o_THM] THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o rator o lhand o lhand o lhand o snd) THEN SIMP_TAC[IMAGE_ID; EMPTY_SUBSET; CONTINUOUS_MAP_ID; CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY] THEN ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN(SUBST1_TAC o SYM)] THEN ASM_REWRITE_TAC[o_DEF] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `w:((num->real)->A)frag->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[hom_relboundary; o_THM] THEN ASM_SIMP_TAC[HOM_INDUCED; MATCH_MP GROUP_HOMOMORPHISM_DIV (SPEC_ALL GROUP_HOMOMORPHISM_HOM_INDUCED)] THEN ASM_SIMP_TAC[HOM_INDUCED; GROUP_RULE `x = group_div G x y <=> y = group_id G`] THEN MP_TAC(ISPECL[`p - &1:int`; `subtopology top s:A topology`; `t:A->bool`] HOMOLOGY_EXACTNESS_AXIOM_3) THEN ASM_SIMP_TAC[SUBTOPOLOGY_SUBTOPOLOGY; SET_RULE `t SUBSET s ==> s INTER t = t`] THEN REWRITE_TAC[group_exactness; group_kernel; group_image; o_THM] THEN REWRITE_TAC[GSYM homology_group] THEN MATCH_MP_TAC(SET_RULE `x IN s ==> P /\ Q /\ IMAGE f s = {y | y IN t /\ g y = z} ==> g(f x) = z`) THEN ASM_REWRITE_TAC[homology_group]);; let HOMOLOGY_EXACTNESS_TRIPLE_3 = prove (`!p top s t:A->bool. t SUBSET s ==> group_exactness(relative_homology_group(p,subtopology top s,t), relative_homology_group(p,top,t), relative_homology_group(p,top,s)) (hom_induced p (subtopology top s,t) (top,t) (\x. x), hom_induced p (top,t) (top,s) (\x. x))`, REPEAT STRIP_TAC THEN REWRITE_TAC[group_exactness; GROUP_HOMOMORPHISM_HOM_RELBOUNDARY; GROUP_HOMOMORPHISM_HOM_INDUCED] THEN REWRITE_TAC[group_image; group_kernel] THEN MATCH_MP_TAC SUBSET_ANTISYM THEN GEN_REWRITE_TAC BINOP_CONV [SUBSET] THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_IMAGE; IN_ELIM_THM] THEN REWRITE_TAC[HOM_INDUCED; HOM_RELBOUNDARY] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [X_GEN_TAC `c:((num->real)->A)frag->bool` THEN DISCH_TAC THEN TRANS_TAC EQ_TRANS `hom_induced p (subtopology top s,s) (top,s) (\x:A. x) (hom_induced p (subtopology top s,t) (subtopology top s,s) (\x. x) c)` THEN CONJ_TAC THENL [GEN_REWRITE_TAC BINOP_CONV [GSYM o_THM] THEN AP_THM_TAC THEN ASM_SIMP_TAC[GSYM HOM_INDUCED_COMPOSE; CONTINUOUS_MAP_FROM_SUBTOPOLOGY; IMAGE_ID; SUBSET_REFL; CONTINUOUS_MAP_ID]; MATCH_MP_TAC(MESON[group_homomorphism] `!G. group_homomorphism(G,H) f /\ x = group_id G ==> f x = group_id H`) THEN EXISTS_TAC `relative_homology_group (p,subtopology top (s:A->bool),s)` THEN REWRITE_TAC[GROUP_HOMOMORPHISM_HOM_INDUCED] THEN MP_TAC(ISPECL [`p:int`; `subtopology top (s:A->bool)`] TRIVIAL_RELATIVE_HOMOLOGY_GROUP_TOPSPACE) THEN REWRITE_TAC[trivial_group] THEN MATCH_MP_TAC(SET_RULE `x IN c /\ a = b ==> c = {b} ==> x = a`) THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY] THEN ONCE_REWRITE_TAC[SET_RULE `u INTER s = (u INTER s) INTER s`] THEN ONCE_REWRITE_TAC[GSYM TOPSPACE_SUBTOPOLOGY] THEN REWRITE_TAC[GSYM RELATIVE_HOMOLOGY_GROUP_RESTRICT; HOM_INDUCED]]; DISCH_TAC] THEN X_GEN_TAC `x:((num->real)->A)frag->bool` THEN STRIP_TAC THEN ABBREV_TAC `b = hom_boundary p (top:A topology,t) x` THEN SUBGOAL_THEN `b IN group_carrier(homology_group(p - &1,subtopology top (t:A->bool)))` ASSUME_TAC THENL [EXPAND_TAC "b" THEN REWRITE_TAC[HOM_BOUNDARY]; ALL_TAC] THEN MP_TAC(ISPECL [`p:int`; `top:A topology`; `t:A->bool`; `top:A topology`; `s:A->bool`; `\x:A. x`] NATURALITY_HOM_INDUCED) THEN ASM_SIMP_TAC[CONTINUOUS_MAP_ID; IMAGE_ID] THEN DISCH_THEN(MP_TAC o C AP_THM `x:((num->real)->A)frag->bool`) THEN ASM_REWRITE_TAC[o_THM; REWRITE_RULE[group_homomorphism] GROUP_HOMOMORPHISM_HOM_BOUNDARY] THEN DISCH_THEN(ASSUME_TAC o SYM) THEN MP_TAC(ISPECL [`p:int`; `subtopology top s:A topology`; `t:A->bool`] HOMOLOGY_EXACTNESS_AXIOM_2) THEN REWRITE_TAC[group_exactness; group_image; group_kernel] THEN DISCH_THEN(MP_TAC o GEN_REWRITE_RULE I [EXTENSION] o last o CONJUNCTS) THEN DISCH_THEN(MP_TAC o SPEC `b:((num->real)->A)frag->bool`) THEN REWRITE_TAC[IN_IMAGE; IN_ELIM_THM; SUBTOPOLOGY_SUBTOPOLOGY] THEN ASM_SIMP_TAC[SET_RULE `t SUBSET s ==> s INTER t = t`] THEN DISCH_THEN(X_CHOOSE_THEN `u:((num->real)->A)frag->bool` (STRIP_ASSUME_TAC o GSYM)) THEN ABBREV_TAC `y = group_div (relative_homology_group(p,top,t)) x (hom_induced p (subtopology top s,t) (top,t) (\x:A. x) u)` THEN SUBGOAL_THEN `y IN group_carrier(relative_homology_group(p,top:A topology,t))` ASSUME_TAC THENL [EXPAND_TAC "y" THEN MATCH_MP_TAC GROUP_DIV THEN ASM_REWRITE_TAC[HOM_INDUCED]; ALL_TAC] THEN MP_TAC(ISPECL [`p:int`; `top:A topology`; `t:A->bool`] HOMOLOGY_EXACTNESS_AXIOM_1) THEN REWRITE_TAC[group_exactness; group_image; group_kernel] THEN DISCH_THEN(MP_TAC o GEN_REWRITE_RULE I [EXTENSION] o last o CONJUNCTS) THEN DISCH_THEN(MP_TAC o SPEC `y:((num->real)->A)frag->bool`) THEN DISCH_THEN(MP_TAC o snd o EQ_IMP_RULE) THEN ASM_REWRITE_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM; IN_IMAGE] THEN ANTS_TAC THENL [EXPAND_TAC "y" THEN ASM_SIMP_TAC[HOM_INDUCED; MATCH_MP GROUP_HOMOMORPHISM_DIV (SPEC_ALL GROUP_HOMOMORPHISM_HOM_BOUNDARY)] THEN ASM_SIMP_TAC[GROUP_DIV_EQ_ID; HOM_INDUCED; HOM_BOUNDARY] THEN CONV_TAC SYM_CONV THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_THM] THEN SIMP_TAC[NATURALITY_HOM_INDUCED; IMAGE_ID; CONTINUOUS_MAP_ID; CONTINUOUS_MAP_FROM_SUBTOPOLOGY; SUBSET_REFL] THEN ASM_REWRITE_TAC[o_THM; SUBTOPOLOGY_SUBTOPOLOGY] THEN ASM_SIMP_TAC[SET_RULE `t SUBSET s ==> s INTER t = t`] THEN ASM_SIMP_TAC[HOM_INDUCED_ID; GSYM homology_group]; DISCH_THEN(X_CHOOSE_THEN `z:((num->real)->A)frag->bool` (STRIP_ASSUME_TAC o GSYM))] THEN MP_TAC(ISPECL [`p:int`; `top:A topology`; `s:A->bool`] HOMOLOGY_EXACTNESS_AXIOM_3) THEN REWRITE_TAC[group_exactness; group_image; group_kernel] THEN DISCH_THEN(MP_TAC o GEN_REWRITE_RULE I [EXTENSION] o last o CONJUNCTS) THEN DISCH_THEN(MP_TAC o SPEC `z:((num->real)->A)frag->bool`) THEN DISCH_THEN(MP_TAC o snd o EQ_IMP_RULE) THEN ASM_REWRITE_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM; IN_IMAGE] THEN ANTS_TAC THENL [TRANS_TAC EQ_TRANS `(hom_induced p (top,t) (top,s) (\x:A. x) o hom_induced p (top,{}) (top,t) (\x. x)) z` THEN CONJ_TAC THENL [ASM_SIMP_TAC[GSYM HOM_INDUCED_COMPOSE; CONTINUOUS_MAP_ID; IMAGE_ID; EMPTY_SUBSET] THEN REWRITE_TAC[o_DEF]; ASM_REWRITE_TAC[o_THM]] THEN SUBST1_TAC(SYM(ASSUME `group_div (relative_homology_group (p,top,t)) x (hom_induced p (subtopology top s,t) (top,t) (\x:A. x) u) = y`)) THEN ASM_SIMP_TAC[HOM_INDUCED; MATCH_MP GROUP_HOMOMORPHISM_DIV (SPEC_ALL GROUP_HOMOMORPHISM_HOM_INDUCED)] THEN REWRITE_TAC[GROUP_RULE `group_div G (group_id G) (group_id G) = group_id G`]; DISCH_THEN(X_CHOOSE_THEN `w:((num->real)->A)frag->bool` (STRIP_ASSUME_TAC o GSYM)) THEN EXISTS_TAC `group_mul (relative_homology_group (p,subtopology top s,t)) (hom_induced p (subtopology top s,{}) (subtopology top s,t) (\x:A. x) w) u` THEN ASM_SIMP_TAC[GROUP_MUL; HOM_INDUCED; MATCH_MP GROUP_HOMOMORPHISM_MUL (SPEC_ALL GROUP_HOMOMORPHISM_HOM_INDUCED)] THEN ASM_SIMP_TAC[GROUP_RULE `x = group_mul G z u <=> z = group_div G x u`; HOM_INDUCED] THEN EXPAND_TAC "y" THEN EXPAND_TAC "z" THEN GEN_REWRITE_TAC BINOP_CONV [GSYM o_THM] THEN SIMP_TAC[GSYM HOM_INDUCED_COMPOSE; IMAGE_ID; SUBSET_REFL; EMPTY_SUBSET; CONTINUOUS_MAP_ID; CONTINUOUS_MAP_FROM_SUBTOPOLOGY]]);; (* ------------------------------------------------------------------------- *) (* Reduced homology. *) (* ------------------------------------------------------------------------- *) let reduced_homology_group = new_definition `reduced_homology_group(p,(top:A topology)) = subgroup_generated (homology_group(p,top)) (group_kernel (homology_group(p,top), homology_group(p,discrete_topology {one})) (hom_induced p (top,{}) (discrete_topology {one},{}) (\x. one)))`;; let GROUP_CARRIER_REDUCED_HOMOLOGY_GROUP = prove (`!p top:A topology. group_carrier (reduced_homology_group(p,top)) = group_kernel (homology_group(p,top), homology_group(p,discrete_topology {one})) (hom_induced p (top,{}) (discrete_topology {one},{}) (\x. one))`, SIMP_TAC[reduced_homology_group; CARRIER_SUBGROUP_GENERATED_SUBGROUP; SUBGROUP_GROUP_KERNEL; GROUP_HOMOMORPHISM_HOM_INDUCED_EMPTY; CONTINUOUS_MAP_CONST; TOPSPACE_DISCRETE_TOPOLOGY; IN_SING]);; let GROUP_CARRIER_REDUCED_HOMOLOGY_GROUP_SUBSET = prove (`!p top:A topology. group_carrier (reduced_homology_group(p,top)) SUBSET group_carrier (homology_group(p,top))`, REWRITE_TAC[GROUP_CARRIER_SUBGROUP_GENERATED_SUBSET; reduced_homology_group]);; let UN_REDUCED_HOMOLOGY_GROUP = prove (`!p top:A topology. ~(p = &0) ==> reduced_homology_group(p,top) = homology_group(p,top)`, REPEAT STRIP_TAC THEN REWRITE_TAC[reduced_homology_group] THEN MATCH_MP_TAC(MESON[SUBGROUP_GENERATED_GROUP_CARRIER] `s = group_carrier G ==> subgroup_generated G s = G`) THEN MATCH_MP_TAC GROUP_KERNEL_TO_TRIVIAL_GROUP THEN SIMP_TAC[GROUP_HOMOMORPHISM_HOM_INDUCED_EMPTY; CONTINUOUS_MAP_CONST; TOPSPACE_DISCRETE_TOPOLOGY; IN_SING] THEN MATCH_MP_TAC HOMOLOGY_DIMENSION_AXIOM THEN EXISTS_TAC `one` THEN ASM_REWRITE_TAC[TOPSPACE_DISCRETE_TOPOLOGY]);; let TRIVIAL_REDUCED_HOMOLOGY_GROUP = prove (`!p top:A topology. p < &0 ==> trivial_group(reduced_homology_group(p,top))`, SIMP_TAC[UN_REDUCED_HOMOLOGY_GROUP; INT_LT_IMP_NE] THEN REWRITE_TAC[TRIVIAL_HOMOLOGY_GROUP]);; let GROUP_HOMOMORPHISM_HOM_INDUCED_REDUCED = prove (`!p top top' f:A->B. group_homomorphism (reduced_homology_group (p,top),reduced_homology_group (p,top')) (hom_induced p (top,{}) (top',{}) f)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `continuous_map(top,top') (f:A->B)` THENL [ALL_TAC; SUBGOAL_THEN `hom_induced p (top,{}) (top',{}) (f:A->B) = \c. group_id(reduced_homology_group (p,top'))` (fun th -> REWRITE_TAC[th; GROUP_HOMOMORPHISM_TRIVIAL]) THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN ASM_SIMP_TAC[HOM_INDUCED_DEFAULT] THEN REWRITE_TAC[reduced_homology_group; homology_group] THEN REWRITE_TAC[SUBGROUP_GENERATED]] THEN REWRITE_TAC[reduced_homology_group] THEN ASM_SIMP_TAC[GROUP_HOMOMORPHISM_INTO_SUBGROUP_EQ; GROUP_HOMOMORPHISM_HOM_INDUCED_EMPTY; GROUP_HOMOMORPHISM_FROM_SUBGROUP_GENERATED; CONTINUOUS_MAP_CONST; TOPSPACE_DISCRETE_TOPOLOGY; IN_SING; CARRIER_SUBGROUP_GENERATED_SUBGROUP; SUBGROUP_GROUP_KERNEL] THEN REWRITE_TAC[group_kernel] THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> k(f x) = h x) /\ IMAGE f s SUBSET t ==> IMAGE f {x | x IN s /\ h x = z} SUBSET {y | y IN t /\ k y = z}`) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; HOM_INDUCED; homology_group] THEN REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_THM] THEN ASM_SIMP_TAC[GSYM HOM_INDUCED_COMPOSE_EMPTY; CONTINUOUS_MAP_CONST; TOPSPACE_DISCRETE_TOPOLOGY; IN_SING] THEN REWRITE_TAC[o_DEF]);; let HOM_INDUCED_REDUCED = prove (`!p top top' f c. c IN group_carrier(reduced_homology_group(p,top)) ==> hom_induced p (top,{}) (top',{}) f c IN group_carrier(reduced_homology_group (p,top'))`, REWRITE_TAC[REWRITE_RULE[group_homomorphism; SUBSET; FORALL_IN_IMAGE] GROUP_HOMOMORPHISM_HOM_INDUCED_REDUCED]);; let GROUP_HOMOMORPHISM_HOM_BOUNDARY_REDUCED = prove (`!p top s:A->bool. group_homomorphism (relative_homology_group (p,top,s), reduced_homology_group (p - &1,subtopology top s)) (hom_boundary p (top,s))`, REPEAT GEN_TAC THEN REWRITE_TAC[reduced_homology_group] THEN MATCH_MP_TAC GROUP_HOMOMORPHISM_INTO_SUBGROUP THEN REWRITE_TAC[GROUP_HOMOMORPHISM_HOM_BOUNDARY] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; group_kernel; IN_ELIM_THM] THEN X_GEN_TAC `c:((num->real)->A)frag->bool` THEN DISCH_TAC THEN REWRITE_TAC[HOM_BOUNDARY] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_THM] THEN MP_TAC(ISPECL [`p:int`; `top:A topology`; `s:A->bool`; `discrete_topology {one}`; `{one}`; `\x:A. one`] NATURALITY_HOM_INDUCED) THEN ASM_REWRITE_TAC[CONTINUOUS_MAP_CONST; TOPSPACE_DISCRETE_TOPOLOGY] THEN REWRITE_TAC[SUBTOPOLOGY_DISCRETE_TOPOLOGY; INTER_IDEMPOT] THEN ANTS_TAC THENL [SET_TAC[]; DISCH_THEN(SUBST1_TAC o SYM)] THEN MP_TAC(ISPECL [`relative_homology_group(p,discrete_topology {one},{one})`; `homology_group (p - &1,discrete_topology {one})`; `hom_boundary p (discrete_topology {one},{one})`] GROUP_IMAGE_FROM_TRIVIAL_GROUP) THEN MP_TAC(ISPECL [`p:int`; `discrete_topology {one}`; `{one}`] GROUP_HOMOMORPHISM_HOM_BOUNDARY) THEN REWRITE_TAC[SUBTOPOLOGY_DISCRETE_TOPOLOGY; INTER_IDEMPOT] THEN DISCH_THEN(fun th -> REWRITE_TAC[th]) THEN ANTS_TAC THENL [MESON_TAC[TRIVIAL_RELATIVE_HOMOLOGY_GROUP_TOPSPACE; TOPSPACE_DISCRETE_TOPOLOGY]; REWRITE_TAC[group_image; o_THM] THEN MATCH_MP_TAC(SET_RULE `x IN s ==> IMAGE f s = {z} ==> f x = z`) THEN REWRITE_TAC[HOM_INDUCED]]);; let HOMOTOPY_EQUIVALENCE_REDUCED_HOMOLOGY_GROUP_ISOMORPHISMS = prove (`!p top top' (f:A->B) g. continuous_map (top,top') f /\ continuous_map (top',top) g /\ homotopic_with (\h. T) (top,top) (g o f) I /\ homotopic_with (\k. T) (top',top') (f o g) I ==> group_isomorphisms (reduced_homology_group (p,top), reduced_homology_group (p,top')) (hom_induced p (top,{}) (top',{}) f, hom_induced p (top',{}) (top,{}) g)`, REPEAT GEN_TAC THEN REPLICATE_TAC 2 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_TAC THEN REWRITE_TAC[group_isomorphisms] THEN REWRITE_TAC[GROUP_HOMOMORPHISM_HOM_INDUCED_REDUCED] THEN FIRST_ASSUM(CONJUNCTS_THEN (MP_TAC o SPEC `p:int` o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMOLOGY_HOMOTOPY_EMPTY))) THEN REWRITE_TAC[GSYM IMP_CONJ_ALT] THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THEN GEN_REWRITE_TAC LAND_CONV [FUN_EQ_THM] THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC EQ_IMP THEN BINOP_TAC THEN ASM_SIMP_TAC[I_DEF; HOM_INDUCED_ID; REWRITE_RULE[SUBSET; homology_group] GROUP_CARRIER_REDUCED_HOMOLOGY_GROUP_SUBSET] THEN GEN_REWRITE_TAC RAND_CONV [GSYM o_THM] THEN AP_THM_TAC THEN MATCH_MP_TAC HOM_INDUCED_COMPOSE THEN ASM_REWRITE_TAC[IMAGE_CLAUSES; EMPTY_SUBSET]);; let HOMOTOPY_EQUIVALENCE_REDUCED_HOMOLOGY_GROUP_ISOMORPHISM = prove (`!p top top' (f:A->B) g. continuous_map (top,top') f /\ continuous_map (top',top) g /\ homotopic_with (\h. T) (top,top) (g o f) I /\ homotopic_with (\k. T) (top',top') (f o g) I ==> group_isomorphism (reduced_homology_group (p,top), reduced_homology_group (p,top')) (hom_induced p (top,{}) (top',{}) f)`, REPEAT STRIP_TAC THEN REWRITE_TAC[group_isomorphism] THEN EXISTS_TAC `hom_induced p (top',{}) (top,{}) (g:B->A)` THEN ASM_SIMP_TAC[HOMOTOPY_EQUIVALENCE_REDUCED_HOMOLOGY_GROUP_ISOMORPHISMS]);; let HOMOTOPY_EQUIVALENT_SPACE_IMP_ISOMORPHIC_REDUCED_HOMOLOGY_GROUPS = prove (`!p (top:A topology) (top':B topology). top homotopy_equivalent_space top' ==> reduced_homology_group(p,top) isomorphic_group reduced_homology_group(p,top')`, REPEAT GEN_TAC THEN REWRITE_TAC[homotopy_equivalent_space; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f:A->B`; `g:B->A`] THEN STRIP_TAC THEN REWRITE_TAC[isomorphic_group] THEN EXISTS_TAC `hom_induced p (top,{}) (top',{}) (f:A->B)` THEN MATCH_MP_TAC HOMOTOPY_EQUIVALENCE_REDUCED_HOMOLOGY_GROUP_ISOMORPHISM THEN ASM_MESON_TAC[]);; let HOMEOMORPHIC_SPACE_IMP_ISOMORPHIC_REDUCED_HOMOLOGY_GROUPS = prove (`!p (top:A topology) (top':B topology). top homeomorphic_space top' ==> reduced_homology_group(p,top) isomorphic_group reduced_homology_group(p,top')`, SIMP_TAC[HOMOTOPY_EQUIVALENT_SPACE_IMP_ISOMORPHIC_REDUCED_HOMOLOGY_GROUPS; HOMEOMORPHIC_IMP_HOMOTOPY_EQUIVALENT_SPACE]);; let TRIVIAL_REDUCED_HOMOLOGY_GROUP_EMPTY = prove (`!p top:A topology. topspace top = {} ==> trivial_group(reduced_homology_group(p,top))`, SIMP_TAC[reduced_homology_group; TRIVIAL_GROUP_SUBGROUP_GENERATED; TRIVIAL_HOMOLOGY_GROUP_EMPTY]);; let HOMOLOGY_DIMENSION_REDUCED = prove (`!p top a:A. topspace top = {a} ==> trivial_group (reduced_homology_group (p,top))`, REPEAT STRIP_TAC THEN REWRITE_TAC[reduced_homology_group] THEN MATCH_MP_TAC TRIVIAL_GROUP_SUBGROUP_GENERATED_TRIVIAL THEN MATCH_MP_TAC(MESON[GROUP_ISOMORPHISM_GROUP_KERNEL_GROUP_IMAGE; SUBSET_REFL] `group_isomorphism (G,G') f ==> group_kernel(G,G') f SUBSET {group_id G}`) THEN MATCH_MP_TAC HOMEOMORPHIC_MAP_HOMOLOGY_GROUP_ISOMORPHISM THEN REWRITE_TAC[HOMEOMORPHIC_MAP_MAPS] THEN EXISTS_TAC `(\x. a):1->A` THEN ASM_REWRITE_TAC[homeomorphic_maps; CONTINUOUS_MAP_CONST; TOPSPACE_DISCRETE_TOPOLOGY] THEN SIMP_TAC[IN_SING]);; let TRIVIAL_REDUCED_HOMOLOGY_GROUP_CONTRACTIBLE_SPACE = prove (`!p top:A topology. contractible_space top ==> trivial_group (reduced_homology_group (p,top))`, REPEAT GEN_TAC THEN REWRITE_TAC[CONTRACTIBLE_EQ_HOMOTOPY_EQUIVALENT_SINGLETON_SUBTOPOLOGY] THEN DISCH_THEN(DISJ_CASES_THEN MP_TAC) THEN REWRITE_TAC[TRIVIAL_REDUCED_HOMOLOGY_GROUP_EMPTY] THEN DISCH_THEN(X_CHOOSE_THEN `a:A` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(SUBST1_TAC o MATCH_MP ISOMORPHIC_GROUP_TRIVIALITY o SPEC`p:int` o MATCH_MP HOMOTOPY_EQUIVALENT_SPACE_IMP_ISOMORPHIC_REDUCED_HOMOLOGY_GROUPS) THEN MATCH_MP_TAC HOMOLOGY_DIMENSION_REDUCED THEN EXISTS_TAC `a:A` THEN ASM_REWRITE_TAC[TOPSPACE_SUBTOPOLOGY] THEN ASM SET_TAC[]);; let GROUP_IMAGE_REDUCED_HOMOLOGY_GROUP = prove (`!p top s:A->bool. ~(topspace top INTER s = {}) ==> group_image (reduced_homology_group (p,top), relative_homology_group (p,top,s)) (hom_induced p (top,{}) (top,s) (\x. x)) = group_image (homology_group (p,top), relative_homology_group (p,top,s)) (hom_induced p (top,{}) (top,s) (\x. x))`, let lemma = prove(`(\y. a) o f = \x. a`,REWRITE_TAC[o_DEF]) in REPEAT STRIP_TAC THEN SIMP_TAC[reduced_homology_group; GROUP_IMAGE_FROM_SUBGROUP_GENERATED; SUBGROUP_GROUP_KERNEL; GROUP_HOMOMORPHISM_HOM_INDUCED_EMPTY; CONTINUOUS_MAP_CONST; TOPSPACE_DISCRETE_TOPOLOGY; IN_SING] THEN REWRITE_TAC[SET_RULE `s INTER t = s <=> s SUBSET t`] THEN REWRITE_TAC[group_image; SUBSET; FORALL_IN_IMAGE] THEN REWRITE_TAC[group_kernel; IN_ELIM_THM; IN_IMAGE] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN REWRITE_TAC[IN_INTER; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `a:A` THEN DISCH_TAC THEN X_GEN_TAC `y:((num->real)->A)frag->bool` THEN DISCH_TAC THEN EXISTS_TAC `group_div (homology_group (p,top)) y (hom_induced p (discrete_topology {one},{}) (top,{}) (\x. a) (hom_induced p (top,{}) (discrete_topology {one},{}) (\x:A. one) y))` THEN MP_TAC(ISPECL [`p:int`;`discrete_topology {one}`; `top:A topology`; `(\x. a):1->A`] GROUP_HOMOMORPHISM_HOM_INDUCED_EMPTY) THEN MP_TAC(ISPECL [`p:int`; `top:A topology`; `discrete_topology {one}`; `\x:A. one`] GROUP_HOMOMORPHISM_HOM_INDUCED_EMPTY) THEN MP_TAC(ISPECL [`p:int`; `top:A topology`; `{}:A->bool`; `top:A topology`; `s:A->bool`; `\x:A. x`] GROUP_HOMOMORPHISM_HOM_INDUCED) THEN MP_TAC(ISPECL [`p:int`;`discrete_topology {one}`; `{one}`; `top:A topology`; `s:A->bool`; `(\x. a):1->A`] GROUP_HOMOMORPHISM_HOM_INDUCED) THEN MP_TAC(ISPECL [`p:int`; `top:A topology`; `s:A->bool`; `discrete_topology {one}`; `{one}`; `\x:A. one`] GROUP_HOMOMORPHISM_HOM_INDUCED) THEN MP_TAC(ISPECL [`p:int`;`discrete_topology {one}`; `{}:1->bool`; `discrete_topology {one}`; `{one}`; `\x:1. x`] GROUP_HOMOMORPHISM_HOM_INDUCED) THEN SIMP_TAC[CONTINUOUS_MAP_ID; CONTINUOUS_MAP_CONST] THEN ASM_REWRITE_TAC[IMAGE_CLAUSES; EMPTY_SUBSET] THEN ASM_REWRITE_TAC[TOPSPACE_DISCRETE_TOPOLOGY; IN_SING; SING_SUBSET] THEN REWRITE_TAC[SET_RULE `IMAGE (\x. a) s SUBSET {a}`] THEN REWRITE_TAC[group_homomorphism; GSYM homology_group] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN REPEAT STRIP_TAC THENL [ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 4) [group_div; GROUP_INV] THEN MATCH_MP_TAC(MESON[GROUP_MUL_RID] `x IN group_carrier G /\ y = group_id G ==> x = group_mul G x y`) THEN ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 4) [GROUP_INV_EQ_ID] THEN TRANS_TAC EQ_TRANS `hom_induced p (discrete_topology {one},{one}) (top,s) (\x. (a:A)) (hom_induced p (discrete_topology {one},{}) (discrete_topology {one},{one}) (\x:1. x) (hom_induced p (top,{}) (discrete_topology {one},{}) (\x:A. one) y))` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [GEN_REWRITE_TAC LAND_CONV [GSYM o_THM] THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o rator o lhand o snd) THEN REWRITE_TAC[IMAGE_CLAUSES; EMPTY_SUBSET; CONTINUOUS_MAP_CONST] THEN REWRITE_TAC[CONTINUOUS_MAP_ID] THEN ASM_SIMP_TAC[GSYM homology_group] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN GEN_REWRITE_TAC RAND_CONV [GSYM o_THM] THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o rator o rand o snd) THEN REWRITE_TAC[IMAGE_CLAUSES; EMPTY_SUBSET; CONTINUOUS_MAP_CONST] THEN ASM_REWRITE_TAC[CONTINUOUS_MAP_ID; SING_SUBSET] THEN ASM_SIMP_TAC[GSYM homology_group] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[o_DEF]; FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (MESON[] `f w = z ==> x = w ==> f x = z`)) THEN MP_TAC(ISPECL [`p:int`; `discrete_topology {one}`] TRIVIAL_RELATIVE_HOMOLOGY_GROUP_TOPSPACE) THEN REWRITE_TAC[trivial_group; TOPSPACE_DISCRETE_TOPOLOGY] THEN ASM SET_TAC[]]; MATCH_MP_TAC GROUP_DIV THEN ASM_SIMP_TAC[]; ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 4) [group_div; GROUP_INV] THEN ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 4) [GSYM group_div; GROUP_DIV_EQ_ID] THEN REPLICATE_TAC 2 (GEN_REWRITE_TAC RAND_CONV [GSYM o_THM]) THEN ASM_SIMP_TAC[GSYM HOM_INDUCED_COMPOSE; CONTINUOUS_MAP_CONST; TOPSPACE_DISCRETE_TOPOLOGY; IN_SING; IMAGE_CLAUSES; EMPTY_SUBSET; lemma]]);; let HOMOLOGY_EXACTNESS_REDUCED_1 = prove (`!p top s:A->bool. ~(topspace top INTER s = {}) ==> group_exactness(reduced_homology_group(p,top), relative_homology_group(p,top,s), reduced_homology_group(p - &1,subtopology top s)) (hom_induced p (top,{}) (top,s) (\x. x), hom_boundary p (top,s))`, REWRITE_TAC[group_exactness; GROUP_HOMOMORPHISM_HOM_BOUNDARY_REDUCED] THEN REPEAT STRIP_TAC THENL [REWRITE_TAC[reduced_homology_group] THEN MATCH_MP_TAC GROUP_HOMOMORPHISM_FROM_SUBGROUP_GENERATED THEN REWRITE_TAC[homology_group; GROUP_HOMOMORPHISM_HOM_INDUCED]; ALL_TAC] THEN REWRITE_TAC[reduced_homology_group; GROUP_KERNEL_TO_SUBGROUP_GENERATED; GROUP_IMAGE_TO_SUBGROUP_GENERATED] THEN REWRITE_TAC[GSYM reduced_homology_group] THEN MP_TAC(ISPECL [`p:int`; `top:A topology`; `s:A->bool`] HOMOLOGY_EXACTNESS_AXIOM_1) THEN REWRITE_TAC[group_exactness] THEN DISCH_THEN(SUBST1_TAC o SYM o last o CONJUNCTS) THEN MATCH_MP_TAC GROUP_IMAGE_REDUCED_HOMOLOGY_GROUP THEN ASM_REWRITE_TAC[]);; let HOMOLOGY_EXACTNESS_REDUCED_2 = prove (`!p top s:A->bool. group_exactness(relative_homology_group(p,top,s), reduced_homology_group(p - &1,subtopology top s), reduced_homology_group(p - &1,top)) (hom_boundary p (top,s), hom_induced (p - &1) (subtopology top s,{}) (top,{}) (\x. x))`, REWRITE_TAC[group_exactness; GROUP_HOMOMORPHISM_HOM_BOUNDARY_REDUCED] THEN REPEAT GEN_TAC THEN REWRITE_TAC[GROUP_HOMOMORPHISM_HOM_INDUCED_REDUCED] THEN REWRITE_TAC[reduced_homology_group; GROUP_KERNEL_TO_SUBGROUP_GENERATED; GROUP_IMAGE_TO_SUBGROUP_GENERATED] THEN REWRITE_TAC[GSYM reduced_homology_group] THEN MP_TAC(ISPECL [`p:int`; `top:A topology`; `s:A->bool`] HOMOLOGY_EXACTNESS_AXIOM_2) THEN REWRITE_TAC[group_exactness] THEN DISCH_THEN(MP_TAC o SYM o last o CONJUNCTS) THEN SIMP_TAC[reduced_homology_group; GROUP_KERNEL_FROM_SUBGROUP_GENERATED; SUBGROUP_GROUP_KERNEL; GROUP_HOMOMORPHISM_HOM_INDUCED_EMPTY; CONTINUOUS_MAP_CONST; TOPSPACE_DISCRETE_TOPOLOGY; IN_SING] THEN DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[SET_RULE `s = s INTER t <=> s SUBSET t`] THEN MP_TAC(ISPECL [`p:int`; `top:A topology`; `s:A->bool`] GROUP_HOMOMORPHISM_HOM_BOUNDARY_REDUCED) THEN REWRITE_TAC[group_homomorphism; group_image] THEN DISCH_THEN(MP_TAC o CONJUNCT1) THEN REWRITE_TAC[GROUP_CARRIER_REDUCED_HOMOLOGY_GROUP]);; let HOMOLOGY_EXACTNESS_REDUCED_3 = prove (`!p top s:A->bool. group_exactness(reduced_homology_group(p,subtopology top s), reduced_homology_group(p,top), relative_homology_group(p,top,s)) (hom_induced p (subtopology top s,{}) (top,{}) (\x. x), hom_induced p (top,{}) (top,s) (\x. x))`, REWRITE_TAC[group_exactness; GROUP_HOMOMORPHISM_HOM_INDUCED_REDUCED] THEN REPEAT STRIP_TAC THENL [REWRITE_TAC[reduced_homology_group] THEN MATCH_MP_TAC GROUP_HOMOMORPHISM_FROM_SUBGROUP_GENERATED THEN REWRITE_TAC[homology_group; GROUP_HOMOMORPHISM_HOM_INDUCED]; ALL_TAC] THEN REWRITE_TAC[reduced_homology_group; GROUP_KERNEL_TO_SUBGROUP_GENERATED; GROUP_IMAGE_TO_SUBGROUP_GENERATED] THEN REWRITE_TAC[GSYM reduced_homology_group] THEN MP_TAC(ISPECL [`p:int`; `top:A topology`; `s:A->bool`] HOMOLOGY_EXACTNESS_AXIOM_3) THEN REWRITE_TAC[group_exactness] THEN DISCH_THEN(MP_TAC o last o CONJUNCTS) THEN MATCH_MP_TAC(SET_RULE `!u. s' SUBSET u /\ s' SUBSET s /\ u INTER t = t' /\ u INTER s SUBSET s' ==> s = t ==> s' = t'`) THEN EXISTS_TAC `group_carrier(reduced_homology_group (p,top:A topology))` THEN CONJ_TAC THENL [MP_TAC(ISPECL [`p:int`; `subtopology top s:A topology`; `top:A topology`; `\x:A. x`] GROUP_HOMOMORPHISM_HOM_INDUCED_REDUCED) THEN SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_ID] THEN SIMP_TAC[group_image; group_homomorphism]; ALL_TAC] THEN CONJ_TAC THENL [REWRITE_TAC[reduced_homology_group; group_image] THEN MATCH_MP_TAC IMAGE_SUBSET THEN REWRITE_TAC[GROUP_CARRIER_SUBGROUP_GENERATED_SUBSET]; ALL_TAC] THEN REWRITE_TAC[GROUP_CARRIER_REDUCED_HOMOLOGY_GROUP] THEN SIMP_TAC[reduced_homology_group; GROUP_KERNEL_FROM_SUBGROUP_GENERATED; SUBGROUP_GROUP_KERNEL; GROUP_HOMOMORPHISM_HOM_INDUCED_EMPTY; CONTINUOUS_MAP_CONST; TOPSPACE_DISCRETE_TOPOLOGY; IN_SING; GROUP_IMAGE_FROM_SUBGROUP_GENERATED] THEN CONJ_TAC THENL [MATCH_ACCEPT_TAC INTER_COMM; ALL_TAC] THEN REWRITE_TAC[group_image] THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s /\ f x IN k ==> x IN c) ==> k INTER IMAGE f s SUBSET IMAGE f s INTER IMAGE f c`) THEN X_GEN_TAC `x:((num->real)->A)frag->bool` THEN REWRITE_TAC[group_kernel; IN_ELIM_THM] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM o_THM] THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o rator o lhand o lhand o snd) THEN REWRITE_TAC[IMAGE_CLAUSES; EMPTY_SUBSET; CONTINUOUS_MAP_CONST] THEN ASM_REWRITE_TAC[TOPSPACE_DISCRETE_TOPOLOGY; IN_SING] THEN ASM_REWRITE_TAC[GSYM homology_group] THEN SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_ID; o_DEF]);; (* ------------------------------------------------------------------------- *) (* More homology properties of deformations, retracts, contractible spaces. *) (* ------------------------------------------------------------------------- *) let GROUP_ISOMORPHISM_RELATIVE_HOMOLOGY_OF_CONTRACTIBLE = prove (`!p top s:A->bool. contractible_space top /\ ~(topspace top INTER s = {}) ==> group_isomorphism (relative_homology_group(p,top,s), reduced_homology_group(p - &1,subtopology top s)) (hom_boundary p (top,s))`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`hom_induced p (top,{}) (top,s) (\x:A. x)`; `hom_boundary p (top:A topology,s)`; `hom_induced (p - &1) (subtopology top s,{}) (top,{}) (\x:A. x)`; `reduced_homology_group (p,top:A topology)`; `relative_homology_group (p,top:A topology,s)`; `reduced_homology_group (p - &1,subtopology top s:A topology)`; `reduced_homology_group (p - &1,top:A topology)`] VERY_SHORT_EXACT_SEQUENCE) THEN ASM_SIMP_TAC[TRIVIAL_REDUCED_HOMOLOGY_GROUP_CONTRACTIBLE_SPACE] THEN DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[HOMOLOGY_EXACTNESS_REDUCED_2] THEN ASM_SIMP_TAC[HOMOLOGY_EXACTNESS_REDUCED_1]);; let ISOMORPHIC_GROUP_RELATIVE_HOMOLOGY_OF_CONTRACTIBLE = prove (`!p top s:A->bool. contractible_space top /\ ~(topspace top INTER s = {}) ==> relative_homology_group(p,top,s) isomorphic_group reduced_homology_group(p - &1,subtopology top s)`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o SPEC `p:int` o MATCH_MP GROUP_ISOMORPHISM_RELATIVE_HOMOLOGY_OF_CONTRACTIBLE) THEN REWRITE_TAC[isomorphic_group] THEN MESON_TAC[]);; let ISOMORPHIC_GROUP_REDUCED_HOMOLOGY_OF_CONTRACTIBLE = prove (`!p top s:A->bool. contractible_space top /\ ~(topspace top INTER s = {}) ==> reduced_homology_group(p,subtopology top s) isomorphic_group relative_homology_group(p + &1,top,s)`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[ISOMORPHIC_GROUP_SYM] THEN DISCH_THEN(MP_TAC o SPEC `p + &1:int` o MATCH_MP ISOMORPHIC_GROUP_RELATIVE_HOMOLOGY_OF_CONTRACTIBLE) THEN REWRITE_TAC[INT_ARITH `(x + &1) - &1:int = x`]);; let GROUP_ISOMORPHISM_REDUCED_HOMOLOGY_BY_CONTRACTIBLE = prove (`!p top s:A->bool. contractible_space(subtopology top s) /\ ~(topspace top INTER s = {}) ==> group_isomorphism (reduced_homology_group(p,top), relative_homology_group(p,top,s)) (hom_induced p (top,{}) (top,s) (\x. x))`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`hom_induced p (subtopology top s,{}) (top,{}) (\x:A. x)`; `hom_induced p (top,{}) (top,s) (\x:A. x)`; `hom_boundary p (top:A topology,s)`; `reduced_homology_group (p,subtopology top s:A topology)`; `reduced_homology_group (p,top:A topology)`; `relative_homology_group (p,top:A topology,s)`; `reduced_homology_group (p - &1,subtopology top s:A topology)`] VERY_SHORT_EXACT_SEQUENCE) THEN ASM_SIMP_TAC[TRIVIAL_REDUCED_HOMOLOGY_GROUP_CONTRACTIBLE_SPACE] THEN DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[HOMOLOGY_EXACTNESS_REDUCED_3] THEN ASM_SIMP_TAC[HOMOLOGY_EXACTNESS_REDUCED_1]);; let ISOMORPHIC_GROUP_REDUCED_HOMOLOGY_BY_CONTRACTIBLE = prove (`!p top s:A->bool. contractible_space(subtopology top s) /\ ~(topspace top INTER s = {}) ==> reduced_homology_group(p,top) isomorphic_group relative_homology_group(p,top,s)`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o SPEC `p:int` o MATCH_MP GROUP_ISOMORPHISM_REDUCED_HOMOLOGY_BY_CONTRACTIBLE) THEN REWRITE_TAC[isomorphic_group] THEN MESON_TAC[]);; let ISOMORPHIC_GROUP_RELATIVE_HOMOLOGY_BY_CONTRACTIBLE = prove (`!p top s:A->bool. contractible_space(subtopology top s) /\ ~(topspace top INTER s = {}) ==> relative_homology_group(p,top,s) isomorphic_group reduced_homology_group(p,top)`, ONCE_REWRITE_TAC[ISOMORPHIC_GROUP_SYM] THEN REWRITE_TAC[ISOMORPHIC_GROUP_REDUCED_HOMOLOGY_BY_CONTRACTIBLE]);; let ISOMORPHIC_GROUP_REDUCED_HOMOLOGY_BY_SING = prove (`!p top a:A. a IN topspace top ==> reduced_homology_group(p,top) isomorphic_group relative_homology_group(p,top,{a})`, REPEAT STRIP_TAC THEN MATCH_MP_TAC ISOMORPHIC_GROUP_REDUCED_HOMOLOGY_BY_CONTRACTIBLE THEN REWRITE_TAC[CONTRACTIBLE_SPACE_SUBTOPOLOGY_SING] THEN ASM SET_TAC[]);; let ISOMORPHIC_GROUP_RELATIVE_HOMOLOGY_BY_SING = prove (`!p top a:A. a IN topspace top ==> relative_homology_group(p,top,{a}) isomorphic_group reduced_homology_group(p,top)`, ONCE_REWRITE_TAC[ISOMORPHIC_GROUP_SYM] THEN REWRITE_TAC[ISOMORPHIC_GROUP_REDUCED_HOMOLOGY_BY_SING]);; let REDUCED_HOMOLOGY_GROUP_PAIR = prove (`!p top a b:A. t1_space top /\ a IN topspace top /\ b IN topspace top /\ ~(a = b) ==> reduced_homology_group(p,subtopology top {a,b}) isomorphic_group homology_group(p,subtopology top {a})`, REPEAT STRIP_TAC THEN TRANS_TAC ISOMORPHIC_GROUP_TRANS `relative_homology_group(p,subtopology top {a,b},{b:A})` THEN CONJ_TAC THENL [MATCH_MP_TAC ISOMORPHIC_GROUP_REDUCED_HOMOLOGY_BY_SING THEN ASM_REWRITE_TAC[IN_INTER; TOPSPACE_SUBTOPOLOGY; IN_INSERT]; ONCE_REWRITE_TAC[ISOMORPHIC_GROUP_SYM] THEN REWRITE_TAC[isomorphic_group; homology_group] THEN MP_TAC(ISPECL [`p:int`; `subtopology top {a:A,b}`; `{a:A,b}`; `{b:A}`; `{b:A}`] HOMOLOGY_EXCISION_AXIOM) THEN REWRITE_TAC[SUBTOPOLOGY_SUBTOPOLOGY; DIFF_EQ_EMPTY; INTER_IDEMPOT] THEN ASM_SIMP_TAC[SET_RULE `~(a = b) ==> {a, b} INTER ({a, b} DIFF {b}) = {a}`] THEN ANTS_TAC THENL [ALL_TAC; MESON_TAC[]] THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN ASM_SIMP_TAC[SUBTOPOLOGY_EQ_DISCRETE_TOPOLOGY_FINITE; TOPSPACE_SUBTOPOLOGY; INTER_SUBSET; INSERT_SUBSET; EMPTY_SUBSET; FINITE_INSERT; FINITE_EMPTY] THEN SIMP_TAC[DISCRETE_TOPOLOGY_CLOSURE_OF; DISCRETE_TOPOLOGY_INTERIOR_OF] THEN SET_TAC[]]);; let DEFORMATION_RETRACTION_RELATIVE_HOMOLOGY_GROUP_ISOMORPHISMS = prove (`!p top top' u v (r:A->B) s. retraction_maps(top,top') (r,s) /\ IMAGE r u SUBSET v /\ IMAGE s v SUBSET u /\ homotopic_with (\h. IMAGE h u SUBSET u) (top,top) (s o r) I ==> group_isomorphisms (relative_homology_group (p,top,u), relative_homology_group (p,top',v)) (hom_induced p (top,u) (top',v) r, hom_induced p (top',v) (top,u) s)`, REWRITE_TAC[retraction_maps] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMOTOPY_EQUIVALENCE_RELATIVE_HOMOLOGY_GROUP_ISOMORPHISMS THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC HOMOTOPIC_WITH_EQUAL THEN ASM_REWRITE_TAC[o_THM; I_DEF; IMAGE_ID; SUBSET_REFL; IMAGE_o] THEN CONJ_TAC THENL [ASM SET_TAC[]; ASM_MESON_TAC[CONTINUOUS_MAP_COMPOSE]]);; let DEFORMATION_RETRACT_RELATIVE_HOMOLOGY_GROUP_ISOMORPHISMS = prove (`!p top top' u v (r:A->A). retraction_maps(top,top') (r,I) /\ v SUBSET u /\ IMAGE r u SUBSET v /\ homotopic_with (\h. IMAGE h u SUBSET u) (top,top) r I ==> group_isomorphisms (relative_homology_group (p,top,u), relative_homology_group (p,top',v)) (hom_induced p (top,u) (top',v) r, hom_induced p (top',v) (top,u) (\x. x))`, REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM I_DEF] THEN MATCH_MP_TAC DEFORMATION_RETRACTION_RELATIVE_HOMOLOGY_GROUP_ISOMORPHISMS THEN ASM_REWRITE_TAC[I_O_ID; IMAGE_I]);; let DEFORMATION_RETRACT_RELATIVE_HOMOLOGY_GROUP_ISOMORPHISM = prove (`!p top top' u v (r:A->A). retraction_maps(top,top') (r,I) /\ v SUBSET u /\ IMAGE r u SUBSET v /\ homotopic_with (\h. IMAGE h u SUBSET u) (top,top) r I ==> group_isomorphism (relative_homology_group (p,top,u), relative_homology_group (p,top',v)) (hom_induced p (top,u) (top',v) r)`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o SPEC `p:int` o MATCH_MP DEFORMATION_RETRACT_RELATIVE_HOMOLOGY_GROUP_ISOMORPHISMS) THEN REWRITE_TAC[GROUP_ISOMORPHISMS_IMP_ISOMORPHISM]);; let DEFORMATION_RETRACT_RELATIVE_HOMOLOGY_GROUP_ISOMORPHISM_ID = prove (`!p top top' u v (r:A->A). retraction_maps(top,top') (r,I) /\ v SUBSET u /\ IMAGE r u SUBSET v /\ homotopic_with (\h. IMAGE h u SUBSET u) (top,top) r I ==> group_isomorphism (relative_homology_group (p,top',v), relative_homology_group (p,top,u)) (hom_induced p (top',v) (top,u) (\x. x))`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o SPEC `p:int` o MATCH_MP DEFORMATION_RETRACT_RELATIVE_HOMOLOGY_GROUP_ISOMORPHISMS) THEN REWRITE_TAC[GROUP_ISOMORPHISMS_IMP_ISOMORPHISM_ALT]);; let DEFORMATION_RETRACTION_IMP_ISOMORPHIC_RELATIVE_HOMOLOGY_GROUPS = prove (`!p top top' u v (r:A->B) s. retraction_maps(top,top') (r,s) /\ IMAGE r u SUBSET v /\ IMAGE s v SUBSET u /\ homotopic_with (\h. IMAGE h u SUBSET u) (top,top) (s o r) I ==> relative_homology_group (p,top,u) isomorphic_group relative_homology_group (p,top',v)`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o SPEC `p:int` o MATCH_MP DEFORMATION_RETRACTION_RELATIVE_HOMOLOGY_GROUP_ISOMORPHISMS) THEN REWRITE_TAC[group_isomorphism; isomorphic_group] THEN MESON_TAC[]);; let DEFORMATION_RETRACTION_IMP_ISOMORPHIC_HOMOLOGY_GROUPS = prove (`!p top top' (r:A->B) s. retraction_maps(top,top') (r,s) /\ homotopic_with (\h. T) (top,top) (s o r) I ==> homology_group (p,top) isomorphic_group homology_group (p,top')`, REPEAT STRIP_TAC THEN REWRITE_TAC[homology_group] THEN MATCH_MP_TAC DEFORMATION_RETRACTION_IMP_ISOMORPHIC_RELATIVE_HOMOLOGY_GROUPS THEN REWRITE_TAC[IMAGE_CLAUSES; EMPTY_SUBSET] THEN ASM_MESON_TAC[]);; let DEFORMATION_RETRACT_IMP_ISOMORPHIC_RELATIVE_HOMOLOGY_GROUPS = prove (`!p top top' u v (r:A->A). retraction_maps(top,top') (r,I) /\ v SUBSET u /\ IMAGE r u SUBSET v /\ homotopic_with (\h. IMAGE h u SUBSET u) (top,top) r I ==> relative_homology_group (p,top,u) isomorphic_group relative_homology_group (p,top',v)`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o SPEC `p:int` o MATCH_MP DEFORMATION_RETRACT_RELATIVE_HOMOLOGY_GROUP_ISOMORPHISMS) THEN REWRITE_TAC[group_isomorphism; isomorphic_group] THEN MESON_TAC[]);; let DEFORMATION_RETRACT_IMP_ISOMORPHIC_HOMOLOGY_GROUPS = prove (`!p top top' (r:A->A). retraction_maps(top,top') (r,I) /\ homotopic_with (\h. T) (top,top) r I ==> homology_group (p,top) isomorphic_group homology_group (p,top')`, REPEAT STRIP_TAC THEN REWRITE_TAC[homology_group] THEN MATCH_MP_TAC DEFORMATION_RETRACT_IMP_ISOMORPHIC_RELATIVE_HOMOLOGY_GROUPS THEN REWRITE_TAC[IMAGE_CLAUSES; EMPTY_SUBSET] THEN ASM_MESON_TAC[]);; let ISOMORPHIC_GROUP_HOMOLOGY_BY_SING = prove (`(!top a:A. a IN topspace top ==> homology_group (&0,top) isomorphic_group prod_group integer_group (relative_homology_group(&0,top,{a}))) /\ (!p top a:A. a IN topspace top /\ ~(p = &0) ==> homology_group (p,top) isomorphic_group relative_homology_group(p,top,{a}))`, REPEAT STRIP_TAC THENL [MP_TAC(ISPECL [`&0:int`; `top:A topology`; `{a:A}`] ISOMORPHIC_GROUP_HOMOLOGY_GROUP_PROD_RETRACT) THEN ASM_REWRITE_TAC[RETRACT_OF_SPACE_SING] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] ISOMORPHIC_GROUP_TRANS) THEN MATCH_MP_TAC ISOMORPHIC_GROUP_PROD_GROUPS THEN REWRITE_TAC[ISOMORPHIC_GROUP_REFL] THEN MATCH_MP_TAC HOMOLOGY_COEFFICIENTS THEN EXISTS_TAC `a:A` THEN ASM_SIMP_TAC[TOPSPACE_SUBTOPOLOGY_SUBSET; SING_SUBSET]; ASM_SIMP_TAC[GSYM UN_REDUCED_HOMOLOGY_GROUP] THEN ONCE_REWRITE_TAC[ISOMORPHIC_GROUP_SYM] THEN ASM_SIMP_TAC[ISOMORPHIC_GROUP_RELATIVE_HOMOLOGY_BY_SING]]);; let GROUP_EPIMORPHISM_HOM_INDUCED_INCLUSION = prove (`!p top s:A->bool. (?f. homotopic_with (\x. T) (top,top) I f /\ IMAGE f (topspace top) SUBSET s) ==> group_epimorphism (homology_group(p,subtopology top s),homology_group(p,top)) (hom_induced p (subtopology top s,{}) (top,{}) (\x. x))`, REPEAT STRIP_TAC THEN FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_CONTINUOUS_MAPS) THEN MATCH_MP_TAC GROUP_EPIMORPHISM_RIGHT_INVERTIBLE THEN EXISTS_TAC `hom_induced p (top,{}) (subtopology top s,{}) (f:A->A)` THEN REWRITE_TAC[GROUP_HOMOMORPHISM_HOM_INDUCED; homology_group] THEN GEN_TAC THEN DISCH_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_THM] THEN ASM_SIMP_TAC[GSYM HOM_INDUCED_COMPOSE; IMAGE_CLAUSES; EMPTY_SUBSET; CONTINUOUS_MAP_IN_SUBTOPOLOGY; CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_ID] THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [GSYM(MATCH_MP HOM_INDUCED_ID th)]) THEN AP_THM_TAC THEN REWRITE_TAC[o_DEF; ETA_AX; GSYM I_DEF] THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC HOMOLOGY_HOMOTOPY_EMPTY THEN ASM_REWRITE_TAC[]);; let TRIVIAL_HOMOMORPHISM_HOM_INDUCED_RELATIVIZATION = prove (`!p top s:A->bool. (?f. homotopic_with (\x. T) (top,top) I f /\ IMAGE f (topspace top) SUBSET s) ==> trivial_homomorphism (homology_group(p,top),relative_homology_group(p,top,s)) (hom_induced p (top,{}) (top,s) (\x. x))`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o SPEC `p:int` o MATCH_MP GROUP_EPIMORPHISM_HOM_INDUCED_INCLUSION) THEN MP_TAC(CONJ (ISPECL [`p:int`; `top:A topology`; `s:A->bool`] HOMOLOGY_EXACTNESS_AXIOM_3) (ISPECL [`p:int`; `top:A topology`; `s:A->bool`] HOMOLOGY_EXACTNESS_AXIOM_1)) THEN DISCH_THEN(SUBST1_TAC o MATCH_MP GROUP_EXACTNESS_EPIMORPHISM_EQ_TRIVIALITY) THEN REWRITE_TAC[]);; let GROUP_MONOMORPHISM_HOM_BOUNDARY_INCLUSION = prove (`!p top s:A->bool. (?f. homotopic_with (\x. T) (top,top) I f /\ IMAGE f (topspace top) SUBSET s) ==> group_monomorphism (relative_homology_group(p,top,s), homology_group (p - &1,subtopology top s)) (hom_boundary p (top,s))`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o SPEC `p:int` o MATCH_MP TRIVIAL_HOMOMORPHISM_HOM_INDUCED_RELATIVIZATION) THEN MP_TAC(CONJ (ISPECL [`p:int`; `top:A topology`; `s:A->bool`] HOMOLOGY_EXACTNESS_AXIOM_3) (ISPECL [`p:int`; `top:A topology`; `s:A->bool`] HOMOLOGY_EXACTNESS_AXIOM_1)) THEN REWRITE_TAC[GSYM homology_group] THEN DISCH_THEN(SUBST1_TAC o MATCH_MP GROUP_EXACTNESS_MONOMORPHISM_EQ_TRIVIALITY) THEN REWRITE_TAC[]);; let SHORT_EXACT_SEQUENCE_HOM_INDUCED_RELATIVIZATION = prove (`!p top s:A->bool. (?f. homotopic_with (\x. T) (top,top) I f /\ IMAGE f (topspace top) SUBSET s) ==> short_exact_sequence (relative_homology_group(p,top,s), homology_group(p - &1,subtopology top s), homology_group(p - &1,top)) (hom_boundary p (top,s), hom_induced (p - &1) (subtopology top s,{}) (top,{}) (\x. x))`, REWRITE_TAC[short_exact_sequence; HOMOLOGY_EXACTNESS_AXIOM_2] THEN SIMP_TAC[GROUP_EPIMORPHISM_HOM_INDUCED_INCLUSION] THEN SIMP_TAC[GROUP_MONOMORPHISM_HOM_BOUNDARY_INCLUSION]);; let GROUP_ISOMORPHISMS_HOMOLOGY_GROUP_PROD_DEFORMATION = prove (`!p top s:A->bool. (?f. homotopic_with (\x. T) (top,top) I f /\ IMAGE f (topspace top) SUBSET s) ==> ?h k. h subgroup_of homology_group (p,subtopology top s) /\ k subgroup_of homology_group (p,subtopology top s) /\ group_isomorphism (prod_group (subgroup_generated (homology_group(p,subtopology top s)) h) (subgroup_generated (homology_group(p,subtopology top s)) k), homology_group(p,subtopology top s)) (\(x,y). group_mul(homology_group(p,subtopology top s)) x y) /\ group_isomorphism (relative_homology_group (p + &1,top,s), subgroup_generated (homology_group (p,subtopology top s)) h) (hom_boundary (p + &1) (top,s)) /\ group_isomorphism (subgroup_generated (homology_group (p,subtopology top s)) k, homology_group (p,top)) (hom_induced p (subtopology top s,{}) (top,{}) (\x. x))`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `p + &1:int` o MATCH_MP SHORT_EXACT_SEQUENCE_HOM_INDUCED_RELATIVIZATION) THEN REWRITE_TAC[INT_ARITH `(p + &1) - &1:int = p`] THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] SPLITTING_LEMMA_RIGHT)) THEN FIRST_X_ASSUM(X_CHOOSE_THEN `f:A->A` STRIP_ASSUME_TAC) THEN DISCH_THEN(MP_TAC o SPEC `hom_induced p (top,{}) (subtopology top s,{}) (f:A->A)`) THEN REWRITE_TAC[GROUP_HOMOMORPHISM_HOM_INDUCED; homology_group] THEN ANTS_TAC THENL [FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_CONTINUOUS_MAPS) THEN GEN_TAC THEN DISCH_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_THM] THEN ASM_SIMP_TAC[GSYM HOM_INDUCED_COMPOSE; IMAGE_CLAUSES; EMPTY_SUBSET; CONTINUOUS_MAP_IN_SUBTOPOLOGY; CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_ID] THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [GSYM(MATCH_MP HOM_INDUCED_ID th)]) THEN AP_THM_TAC THEN REWRITE_TAC[o_DEF; ETA_AX; GSYM I_DEF] THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC HOMOLOGY_HOMOTOPY_EMPTY THEN ASM_REWRITE_TAC[]; REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN REWRITE_TAC[normal_subgroup_of] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[GROUP_ISOMORPHISM_GROUP_MUL; ABELIAN_RELATIVE_HOMOLOGY_GROUP]]);; let ISOMORPHIC_GROUP_HOMOLOGY_GROUP_PROD_DEFORMATION = prove (`!p top s:A->bool. (?f. homotopic_with (\x. T) (top,top) I f /\ IMAGE f (topspace top) SUBSET s) ==> homology_group(p,subtopology top s) isomorphic_group prod_group (homology_group(p,top)) (relative_homology_group(p + &1,top,s))`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[ISOMORPHIC_GROUP_PROD_GROUP_SWAP_RIGHT] THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `p:int` o MATCH_MP GROUP_ISOMORPHISMS_HOMOLOGY_GROUP_PROD_DEFORMATION) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`h:(((num->real)->A)frag->bool)->bool`; `k:(((num->real)->A)frag->bool)->bool`] THEN STRIP_TAC THEN TRANS_TAC ISOMORPHIC_GROUP_TRANS `prod_group (subgroup_generated (homology_group(p,subtopology top s:A topology)) h) (subgroup_generated (homology_group(p,subtopology top s)) k)` THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[ISOMORPHIC_GROUP_SYM] THEN REWRITE_TAC[isomorphic_group] THEN ASM_MESON_TAC[]; MATCH_MP_TAC ISOMORPHIC_GROUP_PROD_GROUPS THEN GEN_REWRITE_TAC LAND_CONV [ISOMORPHIC_GROUP_SYM] THEN REWRITE_TAC[isomorphic_group] THEN ASM_MESON_TAC[]]);; let ISOMORPHIC_GROUP_HOMOLOGY_CONTRACTIBLE_SPACE_SUBTOPOLOGY = prove (`(!top s:A->bool. contractible_space top /\ s SUBSET topspace top /\ ~(s = {}) ==> homology_group (&0,subtopology top s) isomorphic_group prod_group integer_group (relative_homology_group(&1,top,s))) /\ (!p top s:A->bool. ~(p = &0) /\ contractible_space top /\ s SUBSET topspace top /\ ~(s = {}) ==> homology_group (p,subtopology top s) isomorphic_group relative_homology_group(p + &1,top,s))`, SIMP_TAC[GSYM UN_REDUCED_HOMOLOGY_GROUP; ISOMORPHIC_GROUP_REDUCED_HOMOLOGY_OF_CONTRACTIBLE; SET_RULE `s SUBSET u ==> u INTER s = s`] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`&0:int`; `top:A topology`; `s:A->bool`] ISOMORPHIC_GROUP_HOMOLOGY_GROUP_PROD_DEFORMATION) THEN REWRITE_TAC[INT_ADD_LID] THEN ANTS_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_TAC `a:A`) THEN EXISTS_TAC `(\x. a):A->A` THEN ASM_REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; I_DEF] THEN ASM_MESON_TAC[CONTRACTIBLE_SPACE_ALT; SUBSET]; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] ISOMORPHIC_GROUP_TRANS) THEN MATCH_MP_TAC ISOMORPHIC_GROUP_PROD_GROUPS THEN REWRITE_TAC[ISOMORPHIC_GROUP_REFL] THEN MATCH_MP_TAC ISOMORPHIC_GROUP_INTEGER_ZEROTH_HOMOLOGY_GROUP THEN ASM_SIMP_TAC[CONTRACTIBLE_IMP_PATH_CONNECTED_SPACE] THEN ASM SET_TAC[]]);; let TRIVIAL_RELATIVE_HOMOLOGY_GROUP_CONTRACTIBLE_SPACES = prove (`!p top s:A->bool. contractible_space top /\ contractible_space(subtopology top s) /\ ~(topspace top INTER s = {}) ==> trivial_group(relative_homology_group(p,top,s))`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`p:int`; `top:A topology`; `s:A->bool`] ISOMORPHIC_GROUP_REDUCED_HOMOLOGY_BY_CONTRACTIBLE) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(SUBST1_TAC o SYM o MATCH_MP ISOMORPHIC_GROUP_TRIVIALITY) THEN ASM_SIMP_TAC[TRIVIAL_REDUCED_HOMOLOGY_GROUP_CONTRACTIBLE_SPACE]);; let TRIVIAL_RELATIVE_HOMOLOGY_GROUP_ALT = prove (`!p top s f:A->A. continuous_map (top,subtopology top s) f /\ homotopic_with (\k. IMAGE k s SUBSET s) (top,top) f I ==> trivial_group (relative_homology_group (p,top,s))`, REPEAT STRIP_TAC THEN MATCH_MP_TAC TRIVIAL_RELATIVE_HOMOLOGY_GROUP_GEN THEN EXISTS_TAC `f:A->A` THEN ASM_REWRITE_TAC[homotopic_with] THEN CONJ_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homotopic_with]) THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[] THEN REWRITE_TAC[CONJUNCT2 PROD_TOPOLOGY_SUBTOPOLOGY] THEN SIMP_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY; CONTINUOUS_MAP_FROM_SUBTOPOLOGY] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; TOPSPACE_SUBTOPOLOGY; IN_INTER; IN_CROSS; FORALL_PAIR_THM; TOPSPACE_PROD_TOPOLOGY] THEN SIMP_TAC[]);; let GROUP_ISOMORPHISM_HOM_INDUCED_RELATIVIZATION_CONTRACTIBLE = prove (`!p top s t:A->bool. contractible_space(subtopology top s) /\ contractible_space(subtopology top t) /\ t SUBSET s /\ ~(topspace top INTER t = {}) ==> group_isomorphism (relative_homology_group(p,top,t), relative_homology_group(p,top,s)) (hom_induced p (top,t) (top,s) (\x. x))`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`hom_induced p (subtopology top s,t) (top,t) (\x:A. x)`; `hom_induced p (top,t) (top,s) (\x:A. x)`; `hom_relboundary p (top,s:A->bool,t)`; `relative_homology_group(p,subtopology top (s:A->bool),t)`; `relative_homology_group(p,top:A topology,t)`; `relative_homology_group(p,top:A topology,s)`; `relative_homology_group(p - &1,subtopology top (s:A->bool),t)`] VERY_SHORT_EXACT_SEQUENCE) THEN ASM_SIMP_TAC[HOMOLOGY_EXACTNESS_TRIPLE_1; HOMOLOGY_EXACTNESS_TRIPLE_3] THEN DISCH_THEN MATCH_MP_TAC THEN CONJ_TAC THEN MATCH_MP_TAC TRIVIAL_RELATIVE_HOMOLOGY_GROUP_CONTRACTIBLE_SPACES THEN ASM_SIMP_TAC[TOPSPACE_SUBTOPOLOGY; SUBTOPOLOGY_SUBTOPOLOGY; SET_RULE `t SUBSET s ==> s INTER t = t`; SET_RULE `t SUBSET s ==> (u INTER s) INTER t = u INTER t`]);; let ISOMORPHIC_RELATIVE_HOMOLOGY_GROUPS_RELATIVIZATION_CONTRACTIBLE = prove (`!p top s t:A->bool. contractible_space(subtopology top s) /\ contractible_space(subtopology top t) /\ t SUBSET s /\ ~(topspace top INTER t = {}) ==> relative_homology_group(p,top,t) isomorphic_group relative_homology_group(p,top,s)`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o SPEC `p:int` o MATCH_MP GROUP_ISOMORPHISM_HOM_INDUCED_RELATIVIZATION_CONTRACTIBLE) THEN REWRITE_TAC[GROUP_ISOMORPHISM_IMP_ISOMORPHIC]);; let GROUP_ISOMORPHISM_HOM_INDUCED_INCLUSION_CONTRACTIBLE = prove (`!p top s t:A->bool. contractible_space top /\ contractible_space(subtopology top s) /\ t SUBSET s /\ ~(topspace top INTER s = {}) ==> group_isomorphism (relative_homology_group(p,subtopology top s,t), relative_homology_group(p,top,t)) (hom_induced p (subtopology top s,t) (top,t) (\x. x))`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`hom_relboundary (p + &1) (top:A topology,s,t)`; `hom_induced p (subtopology top s,t) (top,t) (\x:A. x)`; `hom_induced p (top,t) (top,s) (\x:A. x)`; `relative_homology_group(p + &1,top,s:A->bool)`; `relative_homology_group(p,subtopology top (s:A->bool),t)`; `relative_homology_group(p,top:A topology,t)`; `relative_homology_group(p,top,s:A->bool)`] VERY_SHORT_EXACT_SEQUENCE) THEN ASM_SIMP_TAC[HOMOLOGY_EXACTNESS_TRIPLE_3] THEN DISCH_THEN MATCH_MP_TAC THEN CONJ_TAC THENL [ABBREV_TAC `q:int = p + &1` THEN SUBGOAL_THEN `p:int = q - &1` SUBST1_TAC THENL [ASM_INT_ARITH_TAC; ASM_SIMP_TAC[HOMOLOGY_EXACTNESS_TRIPLE_2]]; CONJ_TAC THEN MATCH_MP_TAC TRIVIAL_RELATIVE_HOMOLOGY_GROUP_CONTRACTIBLE_SPACES THEN ASM_REWRITE_TAC[]]);; let ISOMORPHIC_RELATIVE_HOMOLOGY_GROUPS_INCLUSION_CONTRACTIBLE = prove (`!p top s t:A->bool. contractible_space top /\ contractible_space(subtopology top s) /\ t SUBSET s /\ ~(topspace top INTER s = {}) ==> relative_homology_group(p,subtopology top s,t) isomorphic_group relative_homology_group(p,top,t)`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o SPEC `p:int` o MATCH_MP GROUP_ISOMORPHISM_HOM_INDUCED_INCLUSION_CONTRACTIBLE) THEN REWRITE_TAC[GROUP_ISOMORPHISM_IMP_ISOMORPHIC]);; let GROUP_ISOMORPHISM_HOM_RELBOUNDARY_CONTRACTIBLE = prove (`!p top s t:A->bool. contractible_space top /\ contractible_space(subtopology top t) /\ t SUBSET s /\ ~(topspace top INTER t = {}) ==> group_isomorphism (relative_homology_group(p,top,s), relative_homology_group(p - &1,subtopology top s,t)) (hom_relboundary p (top,s,t))`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`hom_induced p (top,t) (top,s) (\x:A. x)`; `hom_relboundary p (top:A topology,s,t)`; `hom_induced (p - &1) (subtopology top s,t) (top,t) (\x:A. x)`; `relative_homology_group(p,top,t:A->bool)`; `relative_homology_group(p,top,s:A->bool)`; `relative_homology_group(p - &1,subtopology top s,t:A->bool)`; `relative_homology_group(p - &1,top,t:A->bool)`] VERY_SHORT_EXACT_SEQUENCE) THEN ASM_SIMP_TAC[HOMOLOGY_EXACTNESS_TRIPLE_1; HOMOLOGY_EXACTNESS_TRIPLE_2] THEN DISCH_THEN MATCH_MP_TAC THEN CONJ_TAC THEN MATCH_MP_TAC TRIVIAL_RELATIVE_HOMOLOGY_GROUP_CONTRACTIBLE_SPACES THEN ASM_REWRITE_TAC[]);; let ISOMORPHIC_RELATIVE_HOMOLOGY_GROUPS_RELBOUNDARY_CONTRACTIBLE = prove (`!p top s t:A->bool. contractible_space top /\ contractible_space(subtopology top t) /\ t SUBSET s /\ ~(topspace top INTER t = {}) ==> relative_homology_group(p,top,s) isomorphic_group relative_homology_group(p - &1,subtopology top s,t)`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o SPEC `p:int` o MATCH_MP GROUP_ISOMORPHISM_HOM_RELBOUNDARY_CONTRACTIBLE) THEN REWRITE_TAC[GROUP_ISOMORPHISM_IMP_ISOMORPHIC]);; let ISOMORPHIC_RELATIVE_CONTRACTIBLE_SPACE_IMP_HOMOLOGY_GROUPS = prove (`!(top:A topology) (top':B topology) s t. contractible_space top /\ contractible_space top' /\ s SUBSET topspace top /\ t SUBSET topspace top' /\ (s = {} <=> t = {}) /\ (!p. relative_homology_group(p,top,s) isomorphic_group relative_homology_group(p,top',t)) ==> !p. homology_group(p,subtopology top s) isomorphic_group homology_group(p,subtopology top' t)`, let tac t = TRANS_TAC ISOMORPHIC_GROUP_TRANS t THEN ASM_SIMP_TAC[ISOMORPHIC_GROUP_HOMOLOGY_CONTRACTIBLE_SPACE_SUBTOPOLOGY] THEN GEN_REWRITE_TAC I [ISOMORPHIC_GROUP_SYM] in REPEAT GEN_TAC THEN ASM_CASES_TAC `t:B->bool = {}` THEN ASM_SIMP_TAC[TRIVIAL_HOMOLOGY_GROUP_EMPTY; TOPSPACE_SUBTOPOLOGY; INTER_EMPTY; ISOMORPHIC_TO_TRIVIAL_GROUP] THEN REPEAT STRIP_TAC THEN ASM_CASES_TAC `p:int = &0` THENL [tac `prod_group integer_group (relative_homology_group (&1,top:A topology,s))` THEN tac `prod_group integer_group (relative_homology_group (&1,top':B topology,t))` THEN ASM_SIMP_TAC[ISOMORPHIC_GROUP_PROD_GROUPS; ISOMORPHIC_GROUP_REFL]; tac `relative_homology_group (p + &1,top:A topology,s)` THEN tac `relative_homology_group (p + &1,top':B topology,t)` THEN ASM_SIMP_TAC[ISOMORPHIC_GROUP_PROD_GROUPS; ISOMORPHIC_GROUP_REFL]]);; (* ------------------------------------------------------------------------- *) (* Homology groups of spheres. *) (* ------------------------------------------------------------------------- *) let GROUP_ISOMORPHISM_REDUCED_HOMOLOGY_GROUP_LOWER_HEMISPHERE = prove (`!p n k. k IN 1..n+1 ==> group_isomorphism (reduced_homology_group (p,nsphere n), relative_homology_group (p,nsphere n,{x | x k <= &0})) (hom_induced p (nsphere n,{}) (nsphere n,{x | x k <= &0}) (\x. x))`, REPEAT STRIP_TAC THEN MATCH_MP_TAC GROUP_ISOMORPHISM_REDUCED_HOMOLOGY_BY_CONTRACTIBLE THEN ASM_SIMP_TAC[CONTRACTIBLE_SPACE_LOWER_HEMISPHERE; GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `(\i. if i = k then -- &1 else &0):num->real` THEN REWRITE_TAC[NSPHERE; TOPSPACE_SUBTOPOLOGY; IN_INTER; IN_ELIM_THM] THEN REWRITE_TAC[TOPSPACE_PRODUCT_TOPOLOGY; o_DEF; TOPSPACE_EUCLIDEANREAL] THEN REWRITE_TAC[CARTESIAN_PRODUCT_UNIV; IN_UNIV] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN REWRITE_TAC[COND_RAND; COND_RATOR] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[SUM_DELTA]);; let GROUP_ISOMORPHISM_RELATIVE_HOMOLOGY_GROUP_UPPER_HEMISPHERE = prove (`!p n k. group_isomorphism (relative_homology_group (p,subtopology (nsphere n) {x | x k >= &0},{x | x k = &0}), relative_homology_group (p,nsphere n,{x | x k <= &0})) (hom_induced p (subtopology (nsphere n) {x | x k >= &0},{x | x k = &0}) (nsphere n,{x | x k <= &0}) (\x. x))`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`p:int`; `nsphere n`; `(:num->real)`; `{x:num->real | x k <= &0}`; `{x:num->real | x k < -- &1 / &2}`] HOMOLOGY_EXCISION_AXIOM) THEN REWRITE_TAC[SET_RULE `UNIV DIFF {x | P x} = {x | ~P x}`] THEN REWRITE_TAC[SET_RULE `{x | P x} DIFF {x | Q x} = {x | ~Q x /\ P x}`] THEN REWRITE_TAC[REAL_NOT_LT; SUBSET_UNIV; SUBTOPOLOGY_UNIV] THEN ANTS_TAC THENL [TRANS_TAC SUBSET_TRANS `{x | x IN topspace(nsphere n) /\ x k IN {a | a <= -- &1 / &2}}` THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[CLOSURE_OF_RESTRICT] THEN MATCH_MP_TAC CLOSURE_OF_MINIMAL THEN CONJ_TAC THENL [SIMP_TAC[SUBSET; IN_ELIM_THM; IN_INTER] THEN REAL_ARITH_TAC; MATCH_MP_TAC CLOSED_IN_CONTINUOUS_MAP_PREIMAGE THEN EXISTS_TAC `euclideanreal` THEN REWRITE_TAC[REAL_CLOSED_HALFSPACE_LE; GSYM REAL_CLOSED_IN] THEN REWRITE_TAC[NSPHERE] THEN MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN SIMP_TAC[CONTINUOUS_MAP_PRODUCT_PROJECTION; IN_UNIV]]; ALL_TAC] THEN TRANS_TAC SUBSET_TRANS `{x | x IN topspace(nsphere n) /\ x k IN {a | a < &0}}` THEN CONJ_TAC THENL [SIMP_TAC[SUBSET; IN_ELIM_THM; IN_INTER] THEN REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC INTERIOR_OF_MAXIMAL THEN CONJ_TAC THENL [SIMP_TAC[SUBSET; IN_ELIM_THM; IN_INTER] THEN REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC OPEN_IN_CONTINUOUS_MAP_PREIMAGE THEN EXISTS_TAC `euclideanreal` THEN REWRITE_TAC[REAL_OPEN_HALFSPACE_LT; GSYM REAL_OPEN_IN] THEN REWRITE_TAC[NSPHERE] THEN MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN SIMP_TAC[CONTINUOUS_MAP_PRODUCT_PROJECTION; IN_UNIV]; ALL_TAC] THEN MP_TAC(ISPECL [`p:int`; `subtopology (nsphere n) {x | -- &1 / &2 <= x k}`; `subtopology (nsphere n) {x | &0 <= x k}`; `topspace(subtopology (nsphere n) {x | -- &1 / &2 <= x k}) INTER {x:num->real | -- &1 / &2 <= x k /\ x k <= &0}`; `topspace(subtopology (nsphere n) {x | &0 <= x k}) INTER {x:num->real | x k = &0}`] DEFORMATION_RETRACT_RELATIVE_HOMOLOGY_GROUP_ISOMORPHISM_ID) THEN REWRITE_TAC[GSYM HOM_INDUCED_RESTRICT; GSYM RELATIVE_HOMOLOGY_GROUP_RESTRICT] THEN REWRITE_TAC[LEFT_FORALL_IMP_THM] THEN ANTS_TAC THENL [ALL_TAC; REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP GROUP_ISOMORPHISM_COMPOSE) THEN REWRITE_TAC[real_ge] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o lhand o snd) THEN ANTS_TAC THENL [ALL_TAC; SIMP_TAC[o_DEF]] THEN SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_IN_SUBTOPOLOGY; CONTINUOUS_MAP_ID; IMAGE_ID] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; SUBSET; IN_INTER; IN_ELIM_THM] THEN REAL_ARITH_TAC] THEN ABBREV_TAC `h = \(t,x). let y = max (x k) (--t) in \i. if i:num = k then y else sqrt(&1 - y pow 2) / sqrt(&1 - x k pow 2) * x i` THEN SUBGOAL_THEN `!t x. &0 <= t /\ t <= &1 /\ x IN topspace(nsphere n) /\ --t <= x k ==> h(t,x) = x` ASSUME_TAC THENL [REWRITE_TAC[NSPHERE; TOPSPACE_SUBTOPOLOGY; IN_INTER; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN EXPAND_TAC "h" THEN REWRITE_TAC[] THEN ASM_SIMP_TAC[REAL_ARITH `t <= x ==> max x t = x`] THEN CONV_TAC(ONCE_DEPTH_CONV let_CONV) THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `i:num` THEN REWRITE_TAC[] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `(x:num->real) k pow 2 = &1` THEN ASM_SIMP_TAC[REAL_DIV_REFL; REAL_MUL_LID; SQRT_EQ_0; REAL_SUB_0] THEN ASM_CASES_TAC `k IN 1..n+1` THENL [ALL_TAC; ASM_MESON_TAC[REAL_ARITH `~(&0 pow 2 = &1)`]] THEN MATCH_MP_TAC(REAL_RING `x = &0 ==> a * x = x`) THEN ASM_CASES_TAC `i IN 1..n+1` THEN ASM_SIMP_TAC[] THEN REWRITE_TAC[MESON[REAL_LT_POW_2] `x = &0 <=> ~(&0 < x pow 2)`] THEN DISCH_TAC THEN UNDISCH_TAC `sum (1..n + 1) (\i. x i pow 2) = &1` THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `a = &1 ==> a < b ==> b = &1 ==> F`)) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `&0 < x ==> x + y <= s ==> y < s`)) THEN TRANS_TAC REAL_LE_TRANS `sum {i:num,k} (\j. x j pow 2)` THEN CONJ_TAC THENL [SIMP_TAC[SUM_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; REAL_ADD_RID; REAL_LE_REFL]; MATCH_MP_TAC SUM_SUBSET_SIMPLE THEN ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; FINITE_NUMSEG] THEN REWRITE_TAC[REAL_LE_POW_2]]; ALL_TAC] THEN SUBGOAL_THEN `continuous_map (prod_topology (subtopology euclideanreal (real_interval [&0,&1])) (subtopology (nsphere n) {x | -- &1 / &2 <= x k}), nsphere n) h` ASSUME_TAC THENL [GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [NSPHERE] THEN REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN CONJ_TAC THENL [REWRITE_TAC[CONTINUOUS_MAP_COMPONENTWISE_UNIV]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; TOPSPACE_SUBTOPOLOGY] THEN REWRITE_TAC[FORALL_PAIR_THM; TOPSPACE_PROD_TOPOLOGY; IN_CROSS] THEN MAP_EVERY X_GEN_TAC [`t:real`; `x:num->real`] THEN REWRITE_TAC[NSPHERE; TOPSPACE_EUCLIDEANREAL_SUBTOPOLOGY] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; IN_INTER; IN_ELIM_THM] THEN REWRITE_TAC[IN_REAL_INTERVAL] THEN STRIP_TAC THEN ASM_CASES_TAC `--t <= (x:num->real) k` THEN ASM_SIMP_TAC[NSPHERE; TOPSPACE_SUBTOPOLOGY; IN_INTER; IN_ELIM_THM] THEN ASM_CASES_TAC `~(&0 <= (x:num->real) k)` THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN SUBGOAL_THEN `k IN 1..n+1` ASSUME_TAC THENL [ASM_MESON_TAC[REAL_LE_REFL]; ALL_TAC] THEN EXPAND_TAC "h" THEN REWRITE_TAC[] THEN ABBREV_TAC `y = max ((x:num->real) k) (--t)` THEN CONV_TAC(ONCE_DEPTH_CONV let_CONV) THEN ASM_SIMP_TAC[REAL_MUL_RZERO] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN REWRITE_TAC[COND_RAND; COND_RATOR; REAL_POW_MUL] THEN SIMP_TAC[SUM_CASES; FINITE_NUMSEG; SUM_LMUL; FINITE_RESTRICT] THEN ASM_SIMP_TAC[SET_RULE `k IN s ==> {x | x IN s /\ x = k} = {k}`] THEN REWRITE_TAC[SUM_SING; REAL_POW_DIV] THEN SUBGOAL_THEN `(x:num->real) k pow 2 <= &1 pow 2 /\ y pow 2 <= &1 pow 2` MP_TAC THENL [REWRITE_TAC[GSYM REAL_LE_SQUARE_ABS] THEN ASM_REAL_ARITH_TAC; CONV_TAC REAL_RAT_REDUCE_CONV THEN STRIP_TAC] THEN SUBGOAL_THEN `abs((x:num->real) k) <= &1 /\ abs y <= &1` STRIP_ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[SQRT_POW_2; REAL_SUB_LE; GSYM DELETE] THEN ASM_SIMP_TAC[SUM_DELETE; FINITE_NUMSEG] THEN MATCH_MP_TAC(REAL_FIELD `~(x = &1) ==> y + (&1 - y) / (&1 - x) * (&1 - x) = &1`) THEN MATCH_MP_TAC(REAL_ARITH `x < &1 pow 2 ==> ~(x = &1)`) THEN REWRITE_TAC[GSYM REAL_LT_SQUARE_ABS] THEN ASM_REAL_ARITH_TAC] THEN X_GEN_TAC `i:num` THEN MATCH_MP_TAC CONTINUOUS_MAP_EQ THEN EXISTS_TAC `\(t,x). if &0 <= x k then x i else (h:real#(num->real)->(num->real)) (t,x) i` THEN CONJ_TAC THENL [REWRITE_TAC[FORALL_PAIR_THM; TOPSPACE_PROD_TOPOLOGY; IN_CROSS] THEN REWRITE_TAC[TOPSPACE_EUCLIDEANREAL_SUBTOPOLOGY; IN_REAL_INTERVAL] THEN MAP_EVERY X_GEN_TAC [`t:real`; `x:num->real`] THEN REWRITE_TAC[NSPHERE; TOPSPACE_SUBTOPOLOGY; IN_INTER; IN_ELIM_THM] THEN STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN CONV_TAC SYM_CONV THEN AP_THM_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[NSPHERE; TOPSPACE_SUBTOPOLOGY; IN_INTER; IN_ELIM_THM] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN ONCE_REWRITE_TAC[LAMBDA_PAIR] THEN MATCH_MP_TAC CONTINUOUS_MAP_CASES_LE THEN REWRITE_TAC[CONTINUOUS_MAP_REAL_CONST] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[CONTINUOUS_MAP_OF_SND] THEN REWRITE_TAC[NSPHERE] THEN DISJ2_TAC THEN REPEAT(MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY) THEN MATCH_MP_TAC CONTINUOUS_MAP_PRODUCT_PROJECTION THEN REWRITE_TAC[IN_UNIV]; MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN REWRITE_TAC[CONTINUOUS_MAP_OF_SND] THEN REWRITE_TAC[NSPHERE] THEN DISJ2_TAC THEN REPEAT(MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY) THEN MATCH_MP_TAC CONTINUOUS_MAP_PRODUCT_PROJECTION THEN REWRITE_TAC[IN_UNIV]; ALL_TAC; REWRITE_TAC[FORALL_PAIR_THM; TOPSPACE_PROD_TOPOLOGY; IN_CROSS] THEN REWRITE_TAC[TOPSPACE_EUCLIDEANREAL_SUBTOPOLOGY; IN_REAL_INTERVAL] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; IN_INTER; IN_ELIM_THM] THEN REPEAT GEN_TAC THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN AP_THM_TAC THEN CONV_TAC SYM_CONV THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC] THEN GEN_REWRITE_TAC RAND_CONV [LAMBDA_PAIR_THM] THEN EXPAND_TAC "h" THEN REWRITE_TAC[] THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN REWRITE_TAC[] THEN GEN_REWRITE_TAC RAND_CONV [LAMBDA_PAIR] THEN ASM_CASES_TAC `i:num = k` THEN ASM_REWRITE_TAC[] THENL [MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_MAX THEN REWRITE_TAC[CONTINUOUS_MAP_OF_FST; CONTINUOUS_MAP_OF_SND] THEN CONJ_TAC THEN DISJ2_TAC THEN GEN_REWRITE_TAC RAND_CONV [GSYM ETA_AX] THEN SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_REAL_NEG; CONTINUOUS_MAP_ID] THEN REWRITE_TAC[NSPHERE] THEN REPEAT(MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY) THEN MATCH_MP_TAC CONTINUOUS_MAP_PRODUCT_PROJECTION THEN REWRITE_TAC[IN_UNIV]; ALL_TAC] THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_MUL THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN REWRITE_TAC[CONTINUOUS_MAP_OF_SND] THEN REWRITE_TAC[NSPHERE] THEN DISJ2_TAC THEN REPEAT(MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY) THEN MATCH_MP_TAC CONTINUOUS_MAP_PRODUCT_PROJECTION THEN REWRITE_TAC[IN_UNIV]] THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_DIV THEN REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; IN_INTER; IN_ELIM_THM; SQRT_EQ_0; FORALL_PAIR_THM; IN_CROSS; TOPSPACE_PROD_TOPOLOGY] THEN REWRITE_TAC[REAL_RING `&1 - x pow 2 = &0 <=> x = &1 \/ x = -- &1`] THEN REAL_ARITH_TAC] THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN MATCH_MP_TAC CONTINUOUS_MAP_SQRT THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_SUB THEN REWRITE_TAC[CONTINUOUS_MAP_REAL_CONST] THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_POW THENL [MATCH_MP_TAC CONTINUOUS_MAP_REAL_MAX; ALL_TAC] THEN REWRITE_TAC[CONTINUOUS_MAP_REAL_NEG_EQ; CONTINUOUS_MAP_OF_SND; CONTINUOUS_MAP_OF_FST] THEN REPEAT CONJ_TAC THEN DISJ2_TAC THEN GEN_REWRITE_TAC RAND_CONV [GSYM ETA_AX] THEN SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_REAL_NEG; CONTINUOUS_MAP_ID] THEN REWRITE_TAC[NSPHERE] THEN REPEAT(MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY) THEN MATCH_MP_TAC CONTINUOUS_MAP_PRODUCT_PROJECTION THEN REWRITE_TAC[IN_UNIV] THEN AP_THM_TAC THEN CONV_TAC SYM_CONV THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN EXISTS_TAC `(h:real#(num->real)->(num->real)) o (\x. &0,x)` THEN REWRITE_TAC[retraction_maps] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] CONTINUOUS_MAP_COMPOSE)) THEN REWRITE_TAC[CONTINUOUS_MAP_PAIRWISE; o_DEF; CONTINUOUS_MAP_ID] THEN REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN REWRITE_TAC[CONTINUOUS_MAP_REAL_CONST; SUBSET; FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_REAL_INTERVAL] THEN REAL_ARITH_TAC; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; TOPSPACE_SUBTOPOLOGY] THEN EXPAND_TAC "h" THEN REWRITE_TAC[IN_ELIM_THM; IN_INTER; o_DEF] THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN REWRITE_TAC[] THEN REAL_ARITH_TAC]; REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY; I_DEF] THEN SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_ID] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; TOPSPACE_SUBTOPOLOGY] THEN REWRITE_TAC[IN_INTER; IN_ELIM_THM] THEN REAL_ARITH_TAC; REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; IN_INTER; IN_ELIM_THM; o_DEF; I_DEF] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[REAL_POS; REAL_NEG_0]; SIMP_TAC[TOPSPACE_SUBTOPOLOGY; IN_INTER; IN_ELIM_THM; SUBSET] THEN REAL_ARITH_TAC; REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; SET_RULE `(s INTER {x | P x}) INTER {x | Q x} = s INTER {x | P x /\ Q x}`] THEN REWRITE_TAC[REAL_ARITH `&0 <= x /\ x = &0 <=> x = &0`] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; SUBSET_INTER] THEN CONJ_TAC THENL [REWRITE_TAC[IMAGE_o] THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP CONTINUOUS_MAP_IMAGE_SUBSET_TOPSPACE) THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> IMAGE h t SUBSET u ==> IMAGE h s SUBSET u`) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; TOPSPACE_PROD_TOPOLOGY] THEN SIMP_TAC[IN_CROSS; IN_INTER; IN_ELIM_THM; TOPSPACE_SUBTOPOLOGY; TOPSPACE_EUCLIDEANREAL; IN_UNIV; IN_REAL_INTERVAL] THEN CONV_TAC REAL_RAT_REDUCE_CONV; EXPAND_TAC "h" THEN REWRITE_TAC[SUBSET; o_DEF; FORALL_IN_IMAGE] THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN REWRITE_TAC[IN_INTER; IN_ELIM_THM] THEN REAL_ARITH_TAC]; W(MP_TAC o PART_MATCH (lhand o rand) HOMOTOPIC_WITH o snd) THEN ANTS_TAC THENL [REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; SUBSET; FORALL_IN_IMAGE] THEN SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN EXISTS_TAC `h:real#(num->real)->(num->real)`] THEN REWRITE_TAC[o_THM; I_THM; TOPSPACE_SUBTOPOLOGY; IN_INTER] THEN REPEAT CONJ_TAC THENL [ASM_REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN EXPAND_TAC "h" THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_PAIR_THM] THEN REWRITE_TAC[TOPSPACE_PROD_TOPOLOGY; IN_CROSS; TOPSPACE_SUBTOPOLOGY; IN_INTER; IN_ELIM_THM] THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN REWRITE_TAC[] THEN REAL_ARITH_TAC; REWRITE_TAC[IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC; X_GEN_TAC `t:real` THEN REWRITE_TAC[IN_REAL_INTERVAL] THEN STRIP_TAC THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM o_DEF] THEN REWRITE_TAC[SET_RULE `(s INTER {x | P x}) INTER {x | P x /\ Q x} = s INTER {x | P x /\ Q x}`] THEN FIRST_ASSUM(MP_TAC o MATCH_MP CONTINUOUS_MAP_IMAGE_SUBSET_TOPSPACE) THEN REWRITE_TAC[IMAGE_o; TOPSPACE_PROD_TOPOLOGY; TOPSPACE_SUBTOPOLOGY] THEN REWRITE_TAC[TOPSPACE_EUCLIDEANREAL; INTER_UNIV] THEN MATCH_MP_TAC(SET_RULE `i SUBSET p /\ (!x. x IN i ==> h x IN v) ==> IMAGE h p SUBSET u ==> IMAGE h i SUBSET u INTER v`) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_CROSS; IN_ELIM_THM; IN_INTER] THEN ASM_SIMP_TAC[IN_REAL_INTERVAL] THEN EXPAND_TAC "h" THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC]);; let GROUP_ISOMORPHISM_UPPER_HEMISPHERE_REDUCED_HOMOLOGY_GROUP = prove (`!p n. group_isomorphism (relative_homology_group (p + &1,subtopology (nsphere (n + 1)) {x | x(n+2) >= &0}, {x | x(n+2) = &0}), reduced_homology_group(p,nsphere n)) (hom_boundary (p + &1) (subtopology (nsphere (n + 1)) {x | x(n+2) >= &0}, {x | x(n+2) = &0}))`, REPEAT GEN_TAC THEN SUBGOAL_THEN `nsphere n = subtopology (subtopology (nsphere (n + 1)) {x | x(n+2) >= &0}) {x | x(n+2) = &0}` SUBST1_TAC THENL [GEN_REWRITE_TAC LAND_CONV [GSYM SUBTOPOLOGY_NSPHERE_EQUATOR] THEN REWRITE_TAC[SUBTOPOLOGY_SUBTOPOLOGY] THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_INTER] THEN REAL_ARITH_TAC; ALL_TAC] THEN ABBREV_TAC `q:int = p + &1` THEN SUBGOAL_THEN `p:int = q - &1` SUBST1_TAC THENL [EXPAND_TAC "q" THEN INT_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC GROUP_ISOMORPHISM_RELATIVE_HOMOLOGY_OF_CONTRACTIBLE THEN CONJ_TAC THENL [MATCH_MP_TAC CONTRACTIBLE_SPACE_UPPER_HEMISPHERE THEN REWRITE_TAC[IN_NUMSEG] THEN ARITH_TAC; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER; IN_ELIM_THM; real_ge; IN_UNIV; TOPSPACE_SUBTOPOLOGY; TOPSPACE_PRODUCT_TOPOLOGY; NSPHERE; o_DEF; TOPSPACE_EUCLIDEANREAL; CARTESIAN_PRODUCT_UNIV] THEN EXISTS_TAC `(\i. if i = n + 1 then &1 else &0):num->real` THEN ASM_REWRITE_TAC[ARITH_RULE `(n + 1) + 1 = n + 2`] THEN REWRITE_TAC[ARITH_RULE `~(n + 2 = n + 1)`] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN CONJ_TAC THENL [REWRITE_TAC[COND_RAND; COND_RATOR] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[SUM_DELTA; IN_NUMSEG] THEN COND_CASES_TAC THEN REWRITE_TAC[] THEN ASM_ARITH_TAC; GEN_TAC THEN REWRITE_TAC[IN_NUMSEG] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC]]);; let GROUP_ISOMORPHISM_REDUCED_HOMOLOGY_GROUP_UPPER_HEMISPHERE = prove (`!p n k. k IN 1..n+1 ==> group_isomorphism (reduced_homology_group (p,nsphere n), relative_homology_group (p,nsphere n,{x | x k >= &0})) (hom_induced p (nsphere n,{}) (nsphere n,{x | x k >= &0}) (\x. x))`, REPEAT STRIP_TAC THEN MATCH_MP_TAC GROUP_ISOMORPHISM_REDUCED_HOMOLOGY_BY_CONTRACTIBLE THEN ASM_SIMP_TAC[CONTRACTIBLE_SPACE_UPPER_HEMISPHERE; GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `(\i. if i = k then &1 else &0):num->real` THEN REWRITE_TAC[NSPHERE; TOPSPACE_SUBTOPOLOGY; IN_INTER; IN_ELIM_THM] THEN REWRITE_TAC[TOPSPACE_PRODUCT_TOPOLOGY; o_DEF; TOPSPACE_EUCLIDEANREAL] THEN REWRITE_TAC[CARTESIAN_PRODUCT_UNIV; IN_UNIV] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN REWRITE_TAC[COND_RAND; COND_RATOR] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[SUM_DELTA]);; let GROUP_ISOMORPHISM_RELATIVE_HOMOLOGY_GROUP_LOWER_HEMISPHERE = prove (`!p n k. group_isomorphism (relative_homology_group (p,subtopology (nsphere n) {x | x k <= &0},{x | x k = &0}), relative_homology_group (p,nsphere n,{x | x k >= &0})) (hom_induced p (subtopology (nsphere n) {x | x k <= &0},{x | x k = &0}) (nsphere n,{x | x k >= &0}) (\x. x))`, REPEAT GEN_TAC THEN MP_TAC(SPEC_ALL GROUP_ISOMORPHISM_RELATIVE_HOMOLOGY_GROUP_UPPER_HEMISPHERE) THEN MATCH_MP_TAC(MESON[GROUP_ISOMORPHISM_COMPOSE] `!f h. group_isomorphism(G',G) f /\ group_isomorphism(H,H') h /\ h o g o f = k ==> group_isomorphism(G,H) g ==> group_isomorphism(G',H') k`) THEN ABBREV_TAC `r = \(x:num->real) i. if i = k then --x i else x i` THEN MP_TAC(SPECL [`n:num`; `k:num`] CONTINUOUS_MAP_NSPHERE_REFLECTION) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN SUBGOAL_THEN `!x:num->real. r(r x) = x` ASSUME_TAC THENL [EXPAND_TAC "r" THEN SIMP_TAC[FUN_EQ_THM; REAL_NEG_NEG] THEN MESON_TAC[]; ALL_TAC] THEN MAP_EVERY EXISTS_TAC [`hom_induced p (subtopology (nsphere n) {x | x k <= &0},{x | x k = &0}) (subtopology (nsphere n) {x | x k >= &0},{x | x k = &0}) r`; `hom_induced p (nsphere n,{x | x k <= &0}) (nsphere n,{x | x k >= &0}) r`] THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [CONJ_TAC THEN ONCE_REWRITE_TAC[HOM_INDUCED_RESTRICT; RELATIVE_HOMOLOGY_GROUP_RESTRICT] THEN MATCH_MP_TAC HOMEOMORPHIC_MAP_RELATIVE_HOMOLOGY_GROUP_ISOMORPHISM THEN (CONJ_TAC THENL [REWRITE_TAC[HOMEOMORPHIC_MAP_MAPS] THEN EXISTS_TAC `r:(num->real)->(num->real)` THEN ASM_REWRITE_TAC[homeomorphic_maps]; ALL_TAC]); W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o rand o lhand o snd) THEN SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_ID] THEN ANTS_TAC THENL [ALL_TAC; DISCH_THEN(SUBST1_TAC o SYM) THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o lhand o snd) THEN ANTS_TAC THENL [ALL_TAC; DISCH_THEN(SUBST1_TAC o SYM) THEN ASM_SIMP_TAC[o_DEF]]]] THEN REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY; o_DEF; ETA_AX] THEN ASM_SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; SET_RULE `(!x. r(r x) = x) ==> (IMAGE r s = t <=> IMAGE r s SUBSET t /\ IMAGE r t SUBSET s)`] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTER; IN_ELIM_THM; TOPSPACE_SUBTOPOLOGY] THEN FIRST_X_ASSUM(ASSUME_TAC o CONJUNCT1 o REWRITE_RULE[continuous_map]) THEN ASM_SIMP_TAC[] THEN EXPAND_TAC "r" THEN REWRITE_TAC[] THEN REAL_ARITH_TAC);; let GROUP_ISOMORPHISM_LOWER_HEMISPHERE_REDUCED_HOMOLOGY_GROUP = prove (`!p n. group_isomorphism (relative_homology_group (p + &1,subtopology (nsphere (n + 1)) {x | x(n+2) <= &0}, {x | x(n+2) = &0}), reduced_homology_group(p,nsphere n)) (hom_boundary (p + &1) (subtopology (nsphere (n + 1)) {x | x(n+2) <= &0}, {x | x(n+2) = &0}))`, REPEAT GEN_TAC THEN SUBGOAL_THEN `nsphere n = subtopology (subtopology (nsphere (n + 1)) {x | x(n+2) <= &0}) {x | x(n+2) = &0}` SUBST1_TAC THENL [REWRITE_TAC[NSPHERE; SUBTOPOLOGY_SUBTOPOLOGY] THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [EXTENSION] THEN X_GEN_TAC `x:num->real` THEN REWRITE_TAC[IN_INTER; IN_ELIM_THM; GSYM CONJ_ASSOC] THEN REWRITE_TAC[ARITH_RULE `(n + 1) + 1 = SUC(n + 1)`; SUM_CLAUSES_NUMSEG] THEN REWRITE_TAC[ARITH_RULE `1 <= SUC n`; NUMSEG_CLAUSES] THEN REWRITE_TAC[ARITH_RULE `SUC(n + 1) = n + 2`; IN_INSERT] THEN REWRITE_TAC[REAL_ARITH `x >= &0 /\ x = &0 <=> x = &0`; IN_NUMSEG] THEN ASM_CASES_TAC `(x:num->real)(n + 2) = &0` THENL [ALL_TAC; ASM_MESON_TAC[ARITH_RULE `~(n + 2 <= n + 1)`]] THEN ASM_REWRITE_TAC[] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[REAL_ADD_RID] THEN ASM_MESON_TAC[]; ALL_TAC] THEN ABBREV_TAC `q:int = p + &1` THEN SUBGOAL_THEN `p:int = q - &1` SUBST1_TAC THENL [EXPAND_TAC "q" THEN INT_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC GROUP_ISOMORPHISM_RELATIVE_HOMOLOGY_OF_CONTRACTIBLE THEN CONJ_TAC THENL [MATCH_MP_TAC CONTRACTIBLE_SPACE_LOWER_HEMISPHERE THEN REWRITE_TAC[IN_NUMSEG] THEN ARITH_TAC; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER; IN_ELIM_THM; real_ge; IN_UNIV; TOPSPACE_SUBTOPOLOGY; TOPSPACE_PRODUCT_TOPOLOGY; NSPHERE; o_DEF; TOPSPACE_EUCLIDEANREAL; CARTESIAN_PRODUCT_UNIV] THEN EXISTS_TAC `(\i. if i = n + 1 then &1 else &0):num->real` THEN ASM_REWRITE_TAC[ARITH_RULE `(n + 1) + 1 = n + 2`] THEN REWRITE_TAC[ARITH_RULE `~(n + 2 = n + 1)`] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN CONJ_TAC THENL [REWRITE_TAC[COND_RAND; COND_RATOR] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[SUM_DELTA; IN_NUMSEG] THEN COND_CASES_TAC THEN REWRITE_TAC[] THEN ASM_ARITH_TAC; GEN_TAC THEN REWRITE_TAC[IN_NUMSEG] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC]]);; let REDUCED_HOMOLOGY_GROUP_NSPHERE_STEP = prove (`!p n. ?f. group_isomorphism(reduced_homology_group (p,nsphere n), reduced_homology_group(p + &1,nsphere (n + 1))) f /\ !c. c IN group_carrier(reduced_homology_group(p,nsphere n)) ==> hom_induced (p + &1) (nsphere(n + 1),{}) (nsphere(n + 1),{}) (\x i. if i = 1 then --x i else x i) (f c) = f (hom_induced p (nsphere n,{}) (nsphere n,{}) (\x i. if i = 1 then --x i else x i) c)`, let isomorphism_sym = prove (`!G1 G2 (r:A->A) (r':A->A). (?f. group_isomorphism(G1,G2) f /\ !x. x IN group_carrier G1 ==> r'(f x) = f(r x)) ==> (!x. x IN group_carrier G1 ==> r x IN group_carrier G1) ==> (?f. group_isomorphism(G2,G1) f /\ !x. x IN group_carrier G2 ==> r(f x) = f(r' x))`, REPEAT GEN_TAC THEN REWRITE_TAC[TAUT `p ==> q ==> r <=> q ==> p ==> r`] THEN DISCH_TAC THEN REWRITE_TAC[group_isomorphism; LEFT_AND_EXISTS_THM] THEN GEN_REWRITE_TAC RAND_CONV [SWAP_EXISTS_THM] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN REWRITE_TAC[group_isomorphisms; group_homomorphism] THEN ASM SET_TAC[]) and isomorphism_trans = prove (`!G1 G2 G3 (r:A->A) (r':A->A) (r'':A->A). (?f. group_isomorphism(G1,G2) f /\ !x. x IN group_carrier G1 ==> r'(f x) = f(r x)) /\ (?f. group_isomorphism(G2,G3) f /\ !x. x IN group_carrier G2 ==> r''(f x) = f(r' x)) ==> (?f. group_isomorphism(G1,G3) f /\ !x. x IN group_carrier G1 ==> r''(f x) = f(r x))`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `f:A->A` STRIP_ASSUME_TAC) (X_CHOOSE_THEN `g:A->A` STRIP_ASSUME_TAC)) THEN EXISTS_TAC `(g:A->A) o (f:A->A)` THEN CONJ_TAC THENL [ASM_MESON_TAC[GROUP_ISOMORPHISM_COMPOSE]; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[GROUP_ISOMORPHISM]) THEN REWRITE_TAC[o_DEF] THEN ASM SET_TAC[]) in REPEAT GEN_TAC THEN ABBREV_TAC `r = \(x:num->real) i. if i = 1 then --x i else x i` THEN SUBGOAL_THEN `!n. continuous_map (nsphere n,nsphere n) r` ASSUME_TAC THENL [EXPAND_TAC "r" THEN REWRITE_TAC[CONTINUOUS_MAP_NSPHERE_REFLECTION]; ALL_TAC] THEN MATCH_MP_TAC isomorphism_trans THEN MAP_EVERY EXISTS_TAC [`relative_homology_group(p + &1, subtopology (nsphere (n + 1)) {x | x (n + 2) >= &0}, {x | x (n + 2) = &0})`; `hom_induced (p + &1) (subtopology (nsphere (n + 1)) {x | x (n + 2) >= &0}, {x | x (n + 2) = &0}) (subtopology (nsphere (n + 1)) {x | x (n + 2) >= &0}, {x | x (n + 2) = &0}) r`] THEN CONJ_TAC THENL [MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_IMP] isomorphism_sym) THEN REWRITE_TAC[HOM_INDUCED] THEN EXISTS_TAC `hom_boundary (p + &1) (subtopology (nsphere (n + 1)) {x | x (n + 2) >= &0}, {x | x (n + 2) = &0})` THEN SIMP_TAC[GROUP_ISOMORPHISM_UPPER_HEMISPHERE_REDUCED_HOMOLOGY_GROUP] THEN REPEAT STRIP_TAC THEN GEN_REWRITE_TAC BINOP_CONV [GSYM o_THM] THEN AP_THM_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) NATURALITY_HOM_INDUCED o rand o snd) THEN REWRITE_TAC[INT_ARITH `(p + &1) - &1:int = p`] THEN ANTS_TAC THENL [ASM_SIMP_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY; CONTINUOUS_MAP_FROM_SUBTOPOLOGY] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTER; IN_ELIM_THM; TOPSPACE_SUBTOPOLOGY] THEN EXPAND_TAC "r" THEN SIMP_TAC[ARITH_RULE `~(n + 2 = 1)`]; DISCH_THEN SUBST1_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[SUBTOPOLOGY_SUBTOPOLOGY] THEN REWRITE_TAC[SET_RULE `{x | P x} INTER {x | Q x} = {x | P x /\ Q x}`] THEN REWRITE_TAC[REAL_ARITH `x:real >= &0 /\ x = &0 <=> x = &0`] THEN REWRITE_TAC[SUBTOPOLOGY_NSPHERE_EQUATOR]]; ALL_TAC] THEN MATCH_MP_TAC isomorphism_trans THEN MAP_EVERY EXISTS_TAC [`relative_homology_group(p + &1,nsphere (n + 1),{x | x(n + 2) <= &0})`; `hom_induced (p + &1) (nsphere (n + 1),{x | x (n + 2) <= &0}) (nsphere (n + 1),{x | x (n + 2) <= &0}) r`] THEN CONJ_TAC THENL [EXISTS_TAC `hom_induced (p + &1) (subtopology (nsphere(n + 1)) {x | x(n+2) >= &0},{x | x(n+2) = &0}) (nsphere(n+1),{x | x(n+2) <= &0}) (\x. x)` THEN SIMP_TAC[GROUP_ISOMORPHISM_RELATIVE_HOMOLOGY_GROUP_UPPER_HEMISPHERE] THEN REPEAT STRIP_TAC THEN GEN_REWRITE_TAC BINOP_CONV [GSYM o_THM] THEN AP_THM_TAC THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o rand o snd) THEN ASM_SIMP_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY; CONTINUOUS_MAP_ID; CONTINUOUS_MAP_FROM_SUBTOPOLOGY] THEN ANTS_TAC THENL [EXPAND_TAC "r" THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_ELIM_THM; TOPSPACE_SUBTOPOLOGY; IN_INTER] THEN SIMP_TAC[ARITH_RULE `~(n + 2 = 1)`; REAL_LE_REFL]; DISCH_THEN(SUBST1_TAC o SYM)] THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o lhand o snd) THEN ANTS_TAC THENL [ALL_TAC; SIMP_TAC[o_DEF]] THEN ASM_SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_ID] THEN EXPAND_TAC "r" THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN SIMP_TAC[IN_ELIM_THM; ARITH_RULE `~(n + 2 = 1)`; REAL_LE_REFL]; MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_IMP] isomorphism_sym) THEN SIMP_TAC[HOM_INDUCED_REDUCED] THEN EXISTS_TAC `hom_induced (p + &1) (nsphere(n + 1),{}) (nsphere (n + 1),{x | x(n + 2) <= &0}) (\x. x)` THEN CONJ_TAC THENL [MATCH_MP_TAC GROUP_ISOMORPHISM_REDUCED_HOMOLOGY_GROUP_LOWER_HEMISPHERE THEN REWRITE_TAC[IN_NUMSEG] THEN ARITH_TAC; REPEAT STRIP_TAC THEN GEN_REWRITE_TAC BINOP_CONV [GSYM o_THM] THEN AP_THM_TAC THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o rand o snd) THEN ASM_REWRITE_TAC[IMAGE_CLAUSES; CONTINUOUS_MAP_ID; EMPTY_SUBSET] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o lhand o snd) THEN ASM_REWRITE_TAC[IMAGE_CLAUSES; CONTINUOUS_MAP_ID; EMPTY_SUBSET] THEN ANTS_TAC THENL [ALL_TAC; SIMP_TAC[o_DEF]] THEN EXPAND_TAC "r" THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM] THEN REWRITE_TAC[ARITH_RULE `~(n + 2 = 1)`]]]);; let REDUCED_HOMOLOGY_GROUP_NSPHERE = prove (`(!n. reduced_homology_group(&n,nsphere n) isomorphic_group integer_group) /\ (!n p. ~(p = &n) ==> trivial_group(reduced_homology_group(p,nsphere n)))`, ONCE_REWRITE_TAC[MESON[] `(!n. P (&n:int) n) /\ (!n p. ~(p = &n) ==> Q p n) <=> !n p. (p = &n ==> P p n) /\ (~(p = &n) ==> Q p n)`] THEN MATCH_MP_TAC num_INDUCTION THEN CONJ_TAC THENL [X_GEN_TAC `p:int` THEN SUBGOAL_THEN `subtopology (product_topology (:num) (\i. euclideanreal)) {(\i. if i = 1 then &1 else &0), (\i. if i = 1 then -- &1 else &0)} = nsphere 0` ASSUME_TAC THENL [REWRITE_TAC[NSPHERE; ADD_CLAUSES; NUMSEG_SING; SUM_SING; IN_SING] THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_INSERT; NOT_IN_EMPTY] THEN X_GEN_TAC `x:num->real` THEN REWRITE_TAC[FUN_EQ_THM; IN_ELIM_THM] THEN REWRITE_TAC[REAL_RING `x pow 2 = &1 <=> x = &1 \/ x = -- &1`] THEN MESON_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`p:int`; `product_topology (:num) (\i. euclideanreal)`; `(\i. if i = 1 then &1 else &0):num->real`; `(\i. if i = 1 then -- &1 else &0):num->real`] REDUCED_HOMOLOGY_GROUP_PAIR) THEN REWRITE_TAC[T1_SPACE_PRODUCT_TOPOLOGY; TOPSPACE_PRODUCT_TOPOLOGY] THEN REWRITE_TAC[T1_SPACE_EUCLIDEANREAL; o_DEF; TOPSPACE_EUCLIDEANREAL] THEN ASM_REWRITE_TAC[CARTESIAN_PRODUCT_UNIV; IN_UNIV] THEN ANTS_TAC THENL [REWRITE_TAC[FUN_EQ_THM] THEN DISCH_THEN(MP_TAC o SPEC `1`) THEN REWRITE_TAC[] THEN CONV_TAC REAL_RAT_REDUCE_CONV; ALL_TAC] THEN ABBREV_TAC `a:num->real = \i. if i = 1 then &1 else &0` THEN ASM_CASES_TAC `p:int = &0` THEN ASM_REWRITE_TAC[] THENL [MP_TAC(ISPECL [`subtopology (product_topology (:num) (\i. euclideanreal)) {a}`; `a:num->real`] HOMOLOGY_COEFFICIENTS); MP_TAC(ISPECL [`p:int`; `subtopology (product_topology (:num) (\i. euclideanreal)) {a}`; `a:num->real`] HOMOLOGY_DIMENSION_AXIOM)] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; TOPSPACE_PRODUCT_TOPOLOGY; o_DEF] THEN REWRITE_TAC[TOPSPACE_EUCLIDEANREAL; CARTESIAN_PRODUCT_UNIV] THEN ASM_REWRITE_TAC[INTER_UNIV; GSYM IMP_CONJ_ALT] THEN REWRITE_TAC[ISOMORPHIC_GROUP_TRANS; IMP_CONJ] THEN DISCH_THEN(SUBST1_TAC o MATCH_MP ISOMORPHIC_GROUP_TRIVIALITY) THEN REWRITE_TAC[]; ALL_TAC] THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN X_GEN_TAC `p:int` THEN REWRITE_TAC[GSYM INT_OF_NUM_SUC] THEN FIRST_X_ASSUM(MP_TAC o SPEC `p - &1:int`) THEN SUBST1_TAC(INT_ARITH `p:int = (p - &1) + &1`) THEN REWRITE_TAC[INT_EQ_ADD_RCANCEL; INT_ARITH `(x + &1) - &1:int = x`] THEN SPEC_TAC(`p - &1:int`,`p:int`) THEN X_GEN_TAC `p:int` THEN REWRITE_TAC[ADD1] THEN SUBGOAL_THEN `reduced_homology_group (p + &1,nsphere (n + 1)) isomorphic_group reduced_homology_group (p,nsphere n)` MP_TAC THENL [ONCE_REWRITE_TAC[ISOMORPHIC_GROUP_SYM] THEN REWRITE_TAC[isomorphic_group] THEN MESON_TAC[REDUCED_HOMOLOGY_GROUP_NSPHERE_STEP]; ASM_CASES_TAC `p:int = &n` THEN ASM_REWRITE_TAC[IMP_IMP; ISOMORPHIC_GROUP_TRANS] THEN REWRITE_TAC[IMP_CONJ] THEN DISCH_THEN(SUBST1_TAC o MATCH_MP ISOMORPHIC_GROUP_TRIVIALITY) THEN REWRITE_TAC[]]);; let CYCLIC_REDUCED_HOMOLOGY_GROUP_NSPHERE = prove (`!p n. cyclic_group(reduced_homology_group(p,nsphere n))`, ASM_MESON_TAC[REDUCED_HOMOLOGY_GROUP_NSPHERE; TRIVIAL_IMP_CYCLIC_GROUP; ISOMORPHIC_GROUP_CYCLICITY; CYCLIC_INTEGER_GROUP]);; let TRIVIAL_REDUCED_HOMOLOGY_GROUP_NSPHERE = prove (`!p n. trivial_group(reduced_homology_group (p,nsphere n)) <=> ~(p = &n)`, MESON_TAC[NON_TRIVIAL_INTEGER_GROUP; REDUCED_HOMOLOGY_GROUP_NSPHERE; ISOMORPHIC_GROUP_TRIVIALITY]);; let NON_CONTRACTIBLE_SPACE_NSPHERE = prove (`!n. ~(contractible_space(nsphere n))`, REWRITE_TAC[CONTRACTIBLE_EQ_HOMOTOPY_EQUIVALENT_SINGLETON_SUBTOPOLOGY] THEN GEN_TAC THEN REWRITE_TAC[NONEMPTY_NSPHERE] THEN DISCH_THEN(X_CHOOSE_THEN `a:num->real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP ISOMORPHIC_GROUP_TRIVIALITY o SPEC `&n:int` o MATCH_MP HOMOTOPY_EQUIVALENT_SPACE_IMP_ISOMORPHIC_REDUCED_HOMOLOGY_GROUPS) THEN REWRITE_TAC[TRIVIAL_REDUCED_HOMOLOGY_GROUP_NSPHERE] THEN MATCH_MP_TAC HOMOLOGY_DIMENSION_REDUCED THEN EXISTS_TAC `a:num->real` THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY] THEN ASM SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* Brouwer degree of a map f:S^n->S^n. *) (* ------------------------------------------------------------------------- *) let brouwer_degree2 = new_definition `brouwer_degree2 p f = @d. !x. x IN group_carrier(reduced_homology_group(&p,nsphere p)) ==> hom_induced (&p) (nsphere p,{}) (nsphere p,{}) f x = group_zpow (reduced_homology_group (&p,nsphere p)) x d`;; let BROUWER_DEGREE2_EQ = prove (`!p f g. (!x. x IN topspace(nsphere p) ==> f x = g x) ==> brouwer_degree2 p f = brouwer_degree2 p g`, REPEAT GEN_TAC THEN DISCH_THEN(ASSUME_TAC o MATCH_MP HOM_INDUCED_EQ) THEN ASM_REWRITE_TAC[brouwer_degree2]);; let BROUWER_DEGREE2 = prove (`!p f x. x IN group_carrier(reduced_homology_group(&p,nsphere p)) ==> hom_induced (&p) (nsphere p,{}) (nsphere p,{}) f x = group_zpow (reduced_homology_group (&p,nsphere p)) x (brouwer_degree2 p f)`, GEN_TAC THEN GEN_TAC THEN REWRITE_TAC[brouwer_degree2] THEN CONV_TAC SELECT_CONV THEN ASM_CASES_TAC `continuous_map(nsphere p,nsphere p) f` THENL [ALL_TAC; EXISTS_TAC `&0:int` THEN ASM_SIMP_TAC[HOM_INDUCED_DEFAULT; GROUP_ZPOW_POW; group_pow] THEN SIMP_TAC[reduced_homology_group; SUBGROUP_GENERATED; homology_group]] THEN MP_TAC(ISPECL [`&p:int`; `p:num`] CYCLIC_REDUCED_HOMOLOGY_GROUP_NSPHERE) THEN REWRITE_TAC[cyclic_group; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `a:((num->real)->num->real)frag->bool` THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP CARRIER_SUBGROUP_GENERATED_BY_SING) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(ASSUME_TAC o SYM) THEN FIRST_ASSUM(MP_TAC o SPEC `hom_induced (&p) (nsphere p,{}) (nsphere p,{}) f a` o MATCH_MP(SET_RULE `s = t ==> !x. x IN t ==> x IN s`)) THEN ASM_SIMP_TAC[HOM_INDUCED_REDUCED; IN_ELIM_THM; IN_UNIV] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:int` THEN STRIP_TAC THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (BINDER_CONV o LAND_CONV o RAND_CONV) [GSYM th]) THEN REWRITE_TAC[FORALL_IN_GSPEC; IN_UNIV] THEN X_GEN_TAC `n:int` THEN ASM_SIMP_TAC[GSYM GROUP_ZPOW_MUL] THEN ONCE_REWRITE_TAC[INT_MUL_SYM] THEN ASM_SIMP_TAC[GROUP_ZPOW_MUL] THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (RAND_CONV o LAND_CONV) [SYM th]) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_IMP; RIGHT_IMP_FORALL_THM] GROUP_HOMOMORPHISM_ZPOW) THEN ASM_REWRITE_TAC[GROUP_HOMOMORPHISM_HOM_INDUCED_REDUCED]);; let BROUWER_DEGREE2_IFF = prove (`!p f x d. continuous_map (nsphere p,nsphere p) f /\ x IN group_carrier(reduced_homology_group(&p,nsphere p)) ==> (hom_induced (&p) (nsphere p,{}) (nsphere p,{}) f x = group_zpow (reduced_homology_group (&p,nsphere p)) x d <=> x = group_id(reduced_homology_group(&p,nsphere p)) \/ brouwer_degree2 p f = d)`, REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REPEAT GEN_TAC THEN DISCH_TAC THEN MP_TAC(ISPECL [`&p:int`; `p:num`] CYCLIC_REDUCED_HOMOLOGY_GROUP_NSPHERE) THEN REWRITE_TAC[cyclic_group; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `a:((num->real)->num->real)frag->bool` THEN STRIP_TAC THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (BINDER_CONV o LAND_CONV o funpow 2 RAND_CONV) [GSYM th]) THEN ASM_SIMP_TAC[CARRIER_SUBGROUP_GENERATED_BY_SING; FORALL_IN_GSPEC; BROUWER_DEGREE2; GROUP_ZPOW] THEN ASM_SIMP_TAC[IN_UNIV; GSYM GROUP_ZPOW_MUL; GROUP_ZPOW_EQ_ALT] THEN REPEAT GEN_TAC THEN ASM_SIMP_TAC[GROUP_ZPOW_EQ_ID] THEN MATCH_MP_TAC(INTEGER_RULE `d:int = &0 ==> (d divides n * a - n * b <=> d divides n \/ b = a)`) THEN ASM_SIMP_TAC[INT_OF_NUM_EQ; GSYM INFINITE_CYCLIC_SUBGROUP_ORDER] THEN MP_TAC INFINITE_INTEGER_GROUP THEN MATCH_MP_TAC EQ_IMP THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC ISOMORPHIC_GROUP_INFINITENESS THEN REWRITE_TAC[REDUCED_HOMOLOGY_GROUP_NSPHERE]);; let BROUWER_DEGREE2_UNIQUE = prove (`!p f d. continuous_map (nsphere p,nsphere p) f /\ (!x. x IN group_carrier(reduced_homology_group(&p,nsphere p)) ==> hom_induced (&p) (nsphere p,{}) (nsphere p,{}) f x = group_zpow (reduced_homology_group (&p,nsphere p)) x d) ==> brouwer_degree2 p f = d`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`&p:int`; `p:num`] CYCLIC_REDUCED_HOMOLOGY_GROUP_NSPHERE) THEN REWRITE_TAC[cyclic_group; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `a:((num->real)->num->real)frag->bool` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `a:((num->real)->num->real)frag->bool`) THEN ASM_SIMP_TAC[BROUWER_DEGREE2; GROUP_ZPOW_EQ_ALT] THEN MATCH_MP_TAC(INTEGER_RULE `d:int = &0 ==> d divides b - a ==> a = b`) THEN ASM_SIMP_TAC[INT_OF_NUM_EQ; GSYM INFINITE_CYCLIC_SUBGROUP_ORDER] THEN MP_TAC INFINITE_INTEGER_GROUP THEN MATCH_MP_TAC EQ_IMP THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC ISOMORPHIC_GROUP_INFINITENESS THEN REWRITE_TAC[REDUCED_HOMOLOGY_GROUP_NSPHERE]);; let BROUWER_DEGREE2_UNIQUE_GENERATOR = prove (`!p f d a. continuous_map (nsphere p,nsphere p) f /\ subgroup_generated (reduced_homology_group (&p,nsphere p)) {a} = reduced_homology_group (&p,nsphere p) /\ hom_induced (&p) (nsphere p,{}) (nsphere p,{}) f a = group_zpow (reduced_homology_group (&p,nsphere p)) a d ==> brouwer_degree2 p f = d`, REPEAT GEN_TAC THEN REPLICATE_TAC 2 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN ASM_CASES_TAC `a IN group_carrier(reduced_homology_group (&p,nsphere p))` THENL [ASM_SIMP_TAC[BROUWER_DEGREE2; GROUP_ZPOW_EQ_ALT] THEN MATCH_MP_TAC(INTEGER_RULE `d:int = &0 ==> d divides b - a ==> a = b`) THEN ASM_SIMP_TAC[INT_OF_NUM_EQ; GSYM INFINITE_CYCLIC_SUBGROUP_ORDER] THEN MP_TAC INFINITE_INTEGER_GROUP THEN MATCH_MP_TAC EQ_IMP THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC ISOMORPHIC_GROUP_INFINITENESS THEN REWRITE_TAC[REDUCED_HOMOLOGY_GROUP_NSPHERE]; MP_TAC(SPECL [`&p:int`; `p:num`] TRIVIAL_REDUCED_HOMOLOGY_GROUP_NSPHERE) THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN REWRITE_TAC[] THEN REWRITE_TAC[TRIVIAL_GROUP_SUBGROUP_GENERATED_EQ] THEN ASM SET_TAC[]]);; let BROUWER_DEGREE2_HOMOTOPIC = prove (`!p f g. homotopic_with (\x. T) (nsphere p,nsphere p) f g ==> brouwer_degree2 p f = brouwer_degree2 p g`, REPEAT STRIP_TAC THEN FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_CONTINUOUS_MAPS) THEN FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMOLOGY_HOMOTOPY_EMPTY)) THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC BROUWER_DEGREE2_UNIQUE THEN ASM_SIMP_TAC[REWRITE_RULE[SUBSET] GROUP_CARRIER_REDUCED_HOMOLOGY_GROUP_SUBSET] THEN ASM_SIMP_TAC[BROUWER_DEGREE2]);; let BROUWER_DEGREE2_ID = prove (`!p. brouwer_degree2 p (\x. x) = &1`, GEN_TAC THEN MATCH_MP_TAC BROUWER_DEGREE2_UNIQUE THEN REWRITE_TAC[CONTINUOUS_MAP_ID] THEN SIMP_TAC[INT_MUL_LID;HOM_INDUCED_ID; GROUP_ZPOW_1; REWRITE_RULE[SUBSET; homology_group] GROUP_CARRIER_REDUCED_HOMOLOGY_GROUP_SUBSET]);; let BROUWER_DEGREE2_COMPOSE = prove (`!p f g. continuous_map (nsphere p,nsphere p) f /\ continuous_map (nsphere p,nsphere p) g ==> brouwer_degree2 p (g o f) = brouwer_degree2 p g * brouwer_degree2 p f`, REPEAT STRIP_TAC THEN MATCH_MP_TAC BROUWER_DEGREE2_UNIQUE THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_MAP_COMPOSE]; ALL_TAC] THEN ONCE_REWRITE_TAC[INT_MUL_SYM] THEN SIMP_TAC[GROUP_ZPOW_MUL; GSYM BROUWER_DEGREE2; HOM_INDUCED_REDUCED] THEN REPEAT STRIP_TAC THEN GEN_REWRITE_TAC RAND_CONV [GSYM o_THM] THEN AP_THM_TAC THEN MATCH_MP_TAC HOM_INDUCED_COMPOSE THEN ASM_REWRITE_TAC[IMAGE_CLAUSES; EMPTY_SUBSET]);; let BROUWER_DEGREE2_HOMOTOPY_EQUIVALENCE = prove (`!p f g. continuous_map (nsphere p,nsphere p) f /\ continuous_map (nsphere p,nsphere p) g /\ homotopic_with (\x. T) (nsphere p,nsphere p) (f o g) I ==> abs(brouwer_degree2 p f) = &1 /\ abs(brouwer_degree2 p g) = &1 /\ brouwer_degree2 p g = brouwer_degree2 p f`, REPEAT GEN_TAC THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP BROUWER_DEGREE2_HOMOTOPIC) THEN ASM_SIMP_TAC[BROUWER_DEGREE2_COMPOSE; I_DEF; BROUWER_DEGREE2_ID] THEN REWRITE_TAC[INT_MUL_EQ_1] THEN INT_ARITH_TAC);; let BROUWER_DEGREE2_HOMEOMORPHIC_MAPS = prove (`!p f g. homeomorphic_maps (nsphere p,nsphere p) (f,g) ==> abs(brouwer_degree2 p f) = &1 /\ abs(brouwer_degree2 p g) = &1 /\ brouwer_degree2 p g = brouwer_degree2 p f`, REPEAT GEN_TAC THEN REWRITE_TAC[homeomorphic_maps] THEN STRIP_TAC THEN MATCH_MP_TAC BROUWER_DEGREE2_HOMOTOPY_EQUIVALENCE THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC HOMOTOPIC_WITH_EQUAL THEN ASM_REWRITE_TAC[o_THM; I_THM] THEN ASM_MESON_TAC[CONTINUOUS_MAP_COMPOSE]);; let BROUWER_DEGREE2_RETRACTION_MAP = prove (`!p f. retraction_map(nsphere p,nsphere p) f ==> abs(brouwer_degree2 p f) = &1`, REPEAT GEN_TAC THEN REWRITE_TAC[retraction_map; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `g:(num->real)->(num->real)` THEN REWRITE_TAC[retraction_maps] THEN STRIP_TAC THEN MP_TAC(ISPECL [`p:num`; `f:(num->real)->(num->real)`; `g:(num->real)->(num->real)`] BROUWER_DEGREE2_HOMOTOPY_EQUIVALENCE) THEN ANTS_TAC THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC HOMOTOPIC_WITH_EQUAL THEN ASM_REWRITE_TAC[o_THM; I_THM] THEN ASM_MESON_TAC[CONTINUOUS_MAP_COMPOSE]);; let BROUWER_DEGREE2_SECTION_MAP = prove (`!p f. section_map (nsphere p,nsphere p) f ==> abs(brouwer_degree2 p f) = &1`, REPEAT GEN_TAC THEN REWRITE_TAC[section_map; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `g:(num->real)->(num->real)` THEN REWRITE_TAC[retraction_maps] THEN STRIP_TAC THEN MP_TAC(ISPECL [`p:num`; `g:(num->real)->(num->real)`; `f:(num->real)->(num->real)`] BROUWER_DEGREE2_HOMOTOPY_EQUIVALENCE) THEN ANTS_TAC THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC HOMOTOPIC_WITH_EQUAL THEN ASM_REWRITE_TAC[o_THM; I_THM] THEN ASM_MESON_TAC[CONTINUOUS_MAP_COMPOSE]);; let BROUWER_DEGREE2_HOMEOMORPHIC_MAP = prove (`!p f. homeomorphic_map (nsphere p,nsphere p) f ==> abs(brouwer_degree2 p f) = &1`, SIMP_TAC[GSYM SECTION_AND_RETRACTION_EQ_HOMEOMORPHIC_MAP; BROUWER_DEGREE2_SECTION_MAP]);; let BROUWER_DEGREE2_NULLHOMOTOPIC = prove (`!p f a. homotopic_with (\x. T) (nsphere p,nsphere p) f (\x. a) ==> brouwer_degree2 p f = &0`, REPEAT STRIP_TAC THEN FIRST_ASSUM(SUBST1_TAC o MATCH_MP BROUWER_DEGREE2_HOMOTOPIC) THEN FIRST_X_ASSUM(STRIP_ASSUME_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_CONTINUOUS_MAPS) THEN MATCH_MP_TAC BROUWER_DEGREE2_UNIQUE THEN ASM_REWRITE_TAC[GROUP_ZPOW_0] THEN X_GEN_TAC `c:((num->real)->num->real)frag->bool` THEN DISCH_TAC THEN SUBGOAL_THEN `hom_induced (&p) (nsphere p,{}) (nsphere p,{}) (\x. a) = hom_induced (&p) (subtopology (nsphere p) {a},{}) (nsphere p,{}) (\x. x) o hom_induced (&p) (nsphere p,{}) (subtopology (nsphere p) {a},{}) (\x. a)` SUBST1_TAC THENL [W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o rand o snd) THEN ASM_REWRITE_TAC[IMAGE_CLAUSES; EMPTY_SUBSET; SET_RULE `IMAGE (\x. a) s SUBSET {a}`; CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_ID] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[o_DEF]; REWRITE_TAC[o_THM] THEN MATCH_MP_TAC (MESON[REWRITE_RULE[group_homomorphism] GROUP_HOMOMORPHISM_HOM_INDUCED_REDUCED] `a = group_id(reduced_homology_group(p,top)) ==> hom_induced p (top,{}) (top',{}) f a = group_id(reduced_homology_group(p,top'))`) THEN MP_TAC(ISPECL [`&p:int`; `subtopology (nsphere p) {a}`] TRIVIAL_REDUCED_HOMOLOGY_GROUP_CONTRACTIBLE_SPACE) THEN SIMP_TAC[CONTRACTIBLE_SPACE_SUBTOPOLOGY_SING; TRIVIAL_GROUP_SUBSET] THEN REWRITE_TAC[SUBSET; IN_SING] THEN DISCH_THEN MATCH_MP_TAC THEN MATCH_MP_TAC HOM_INDUCED_REDUCED THEN ASM_REWRITE_TAC[]]);; let BROUWER_DEGREE2_CONST = prove (`!p a. brouwer_degree2 p (\x. a) = &0`, REPEAT GEN_TAC THEN ASM_CASES_TAC `continuous_map(nsphere p,nsphere p) (\x. a)` THENL [MATCH_MP_TAC BROUWER_DEGREE2_NULLHOMOTOPIC THEN EXISTS_TAC `a:num->real` THEN ASM_REWRITE_TAC[HOMOTOPIC_WITH_REFL]; ALL_TAC] THEN ASM_SIMP_TAC[brouwer_degree2; HOM_INDUCED_DEFAULT] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [EQ_SYM_EQ] THEN SUBGOAL_THEN `group_id(relative_homology_group(&p,nsphere p,{})) = group_id(reduced_homology_group(&p,nsphere p))` SUBST1_TAC THENL [REWRITE_TAC[reduced_homology_group; GSYM homology_group] THEN REWRITE_TAC[CONJUNCT2 SUBGROUP_GENERATED]; SIMP_TAC[GROUP_ZPOW_EQ_ID]] THEN MATCH_MP_TAC SELECT_UNIQUE THEN X_GEN_TAC `d:int` THEN EQ_TAC THEN SIMP_TAC[INTEGER_RULE `!x:int. x divides &0`] THEN MP_TAC(ISPECL [`&p:int`; `p:num`] CYCLIC_REDUCED_HOMOLOGY_GROUP_NSPHERE) THEN REWRITE_TAC[cyclic_group; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `c:((num->real)->num->real)frag->bool` THEN STRIP_TAC THEN DISCH_THEN(MP_TAC o SPEC `c:((num->real)->num->real)frag->bool`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(INTEGER_RULE `c = &0 ==> c divides d ==> d = &0`) THEN ASM_SIMP_TAC[INT_OF_NUM_EQ; GSYM INFINITE_CYCLIC_SUBGROUP_ORDER] THEN MP_TAC INFINITE_INTEGER_GROUP THEN MATCH_MP_TAC EQ_IMP THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC ISOMORPHIC_GROUP_INFINITENESS THEN REWRITE_TAC[REDUCED_HOMOLOGY_GROUP_NSPHERE]);; let BROUWER_DEGREE2_NONSURJECTIVE = prove (`!p f. continuous_map(nsphere p,nsphere p) f /\ ~(IMAGE f (topspace(nsphere p)) = topspace(nsphere p)) ==> brouwer_degree2 p f = &0`, REPEAT STRIP_TAC THEN MATCH_MP_TAC BROUWER_DEGREE2_NULLHOMOTOPIC THEN MATCH_MP_TAC NULLHOMOTOPIC_NONSURJECTIVE_SPHERE_MAP THEN ASM_REWRITE_TAC[]);; let BROUWER_DEGREE2_REFLECTION = prove (`!p. brouwer_degree2 p (\x i. if i = 1 then --x i else x i) = -- &1`, ABBREV_TAC `r = \(x:num->real) i. if i = 1 then --x i else x i` THEN MP_TAC(GEN `p:num` (SPECL [`p:num`; `1`] CONTINUOUS_MAP_NSPHERE_REFLECTION)) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN MATCH_MP_TAC num_INDUCTION THEN CONJ_TAC THENL [ALL_TAC; X_GEN_TAC `p:num` THEN DISCH_TAC THEN MATCH_MP_TAC BROUWER_DEGREE2_UNIQUE THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPECL [`&p:int`; `p:num`] REDUCED_HOMOLOGY_GROUP_NSPHERE_STEP) THEN ASM_REWRITE_TAC[ADD1; GSYM INT_OF_NUM_ADD] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SYM o el 1 o CONJUNCTS o GEN_REWRITE_RULE I [GROUP_ISOMORPHISM]) THEN REWRITE_TAC[ADD1; GSYM INT_OF_NUM_ADD] THEN DISCH_THEN SUBST1_TAC THEN ASM_SIMP_TAC[FORALL_IN_IMAGE; BROUWER_DEGREE2] THEN FIRST_ASSUM(MP_TAC o MATCH_MP GROUP_HOMOMORPHISM_ZPOW o MATCH_MP GROUP_ISOMORPHISM_IMP_HOMOMORPHISM) THEN SIMP_TAC[]] THEN SUBGOAL_THEN `!c. c IN group_carrier(reduced_homology_group(&0,nsphere 0)) ==> hom_induced (&0) (nsphere 0,{}) (nsphere 0,{}) r c = group_inv (homology_group(&0,nsphere 0)) c` ASSUME_TAC THENL [ALL_TAC; MATCH_MP_TAC BROUWER_DEGREE2_UNIQUE THEN ASM_SIMP_TAC[] THEN SIMP_TAC[GROUP_ZPOW_POW; GROUP_POW_1] THEN REWRITE_TAC[reduced_homology_group; CONJUNCT2 SUBGROUP_GENERATED]] THEN X_GEN_TAC `c:((num->real)->num->real)frag->bool` THEN REWRITE_TAC[GROUP_CARRIER_REDUCED_HOMOLOGY_GROUP] THEN REWRITE_TAC[group_kernel; IN_ELIM_THM] THEN STRIP_TAC THEN MAP_EVERY ABBREV_TAC [`p:num->real = \i. if i = 1 then &1 else &0`; `n:num->real = \i. if i = 1 then -- &1 else &0`] THEN SUBGOAL_THEN `topspace(nsphere 0) = {p,n}` ASSUME_TAC THENL [REWRITE_TAC[NSPHERE] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; TOPSPACE_PRODUCT_TOPOLOGY; o_DEF] THEN SIMP_TAC[TOPSPACE_EUCLIDEANREAL; CARTESIAN_PRODUCT_UNIV; INTER_UNIV] THEN REWRITE_TAC[ADD_CLAUSES; NUMSEG_SING; SUM_SING; IN_SING] THEN MAP_EVERY EXPAND_TAC ["p"; "n"] THEN REWRITE_TAC[SET_RULE `s = {a,b} <=> !x. x IN s <=> x = a \/ x = b`] THEN REWRITE_TAC[REAL_RING `(x:real) pow 2 = &1 <=> x = &1 \/ x = -- &1`] THEN REWRITE_TAC[IN_ELIM_THM] THEN REWRITE_TAC[FUN_EQ_THM] THEN MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `~(p:num->real = n)` ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["p"; "n"] THEN REWRITE_TAC[FUN_EQ_THM] THEN DISCH_THEN(MP_TAC o SPEC `1`) THEN REWRITE_TAC[] THEN CONV_TAC REAL_RAT_REDUCE_CONV; ALL_TAC] THEN SUBGOAL_THEN `r(p:num->real) = n /\ r n = p` STRIP_ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["r"; "n"; "p"] THEN SIMP_TAC[REAL_NEG_NEG]; ALL_TAC] THEN SUBGOAL_THEN `!x:num->real. r(r x) = x` ASSUME_TAC THENL [EXPAND_TAC "r" THEN SIMP_TAC[FUN_EQ_THM; REAL_NEG_NEG] THEN MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?d. d IN group_carrier(homology_group(&0,subtopology (nsphere 0) {p})) /\ group_div (homology_group (&0,nsphere 0)) (hom_induced (&0) (subtopology (nsphere 0) {p},{}) (nsphere 0,{}) (\x. x) d) (hom_induced (&0) (subtopology (nsphere 0) {p},{}) (nsphere 0,{}) r d) = c` STRIP_ASSUME_TAC THENL [MP_TAC(ISPECL [`&0:int`; `nsphere 0`; `{p:num->real}`; `{n:num->real}`] HOMOLOGY_ADDITIVITY_EXPLICIT) THEN ANTS_TAC THENL [ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MP_TAC(fst(EQ_IMP_RULE(ISPECL [`nsphere 0`; `{p:num->real,n}`] DISCRETE_TOPOLOGY_UNIQUE))) THEN ANTS_TAC THENL [ALL_TAC; SIMP_TAC[IN_INSERT]] THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC FINITE_T1_SPACE_IMP_DISCRETE_TOPOLOGY THEN ASM_REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY] THEN REWRITE_TAC[NSPHERE] THEN MATCH_MP_TAC T1_SPACE_SUBTOPOLOGY THEN REWRITE_TAC[T1_SPACE_PRODUCT_TOPOLOGY; T1_SPACE_EUCLIDEANREAL]; REWRITE_TAC[GROUP_ISOMORPHISM] THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `P /\ IMAGE f s = t /\ Q ==> !y. y IN t ==> ?x. x IN s /\ f x = y`))] THEN DISCH_THEN(MP_TAC o SPEC `c:((num->real)->num->real)frag->bool`) THEN ASM_REWRITE_TAC[PROD_GROUP; EXISTS_PAIR_THM; IN_CROSS] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:((num->real)->num->real)frag->bool` THEN DISCH_THEN(X_CHOOSE_THEN `d':((num->real)->num->real)frag->bool` STRIP_ASSUME_TAC) THEN ASM_REWRITE_TAC[group_div] THEN EXPAND_TAC "c" THEN AP_TERM_TAC THEN SUBGOAL_THEN `group_epimorphism(homology_group(&0,subtopology (nsphere 0) {p}), homology_group(&0,subtopology (nsphere 0) {n})) (hom_induced (&0) (subtopology (nsphere 0) {p},{}) (subtopology (nsphere 0) {n},{}) r)` MP_TAC THENL [MATCH_MP_TAC GROUP_EPIMORPHISM_HOM_INDUCED_RETRACTION_MAP THEN REWRITE_TAC[retraction_map] THEN EXISTS_TAC `r:(num->real)->num->real` THEN REWRITE_TAC[retraction_maps; CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN ASM_SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY] THEN ASM SET_TAC[]; REWRITE_TAC[group_epimorphism] THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `P /\ IMAGE f s = t ==> !y. y IN t ==> ?x. x IN s /\ f x = y`))] THEN DISCH_THEN(MP_TAC o SPEC `d':((num->real)->num->real)frag->bool`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `e:((num->real)->num->real)frag->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `group_monomorphism(homology_group(&0,subtopology (nsphere 0) {p}), homology_group(&0,discrete_topology {one})) (hom_induced (&0) (subtopology (nsphere 0) {p},{}) (discrete_topology {one},{}) (\x. one))` MP_TAC THENL [MATCH_MP_TAC GROUP_MONOMORPHISM_HOM_INDUCED_SECTION_MAP THEN REWRITE_TAC[section_map; retraction_maps] THEN EXISTS_TAC `(\w. p):1->num->real` THEN SIMP_TAC[TOPSPACE_SUBTOPOLOGY; IN_INTER; IN_SING] THEN REWRITE_TAC[CONTINUOUS_MAP_CONST; TOPSPACE_SUBTOPOLOGY] THEN REWRITE_TAC[TOPSPACE_DISCRETE_TOPOLOGY; IN_SING] THEN ASM SET_TAC[]; REWRITE_TAC[GROUP_MONOMORPHISM_ALT]] THEN DISCH_THEN(MP_TAC o SPEC `group_mul (homology_group (&0,subtopology (nsphere 0) {p})) d e` o CONJUNCT2) THEN ASM_SIMP_TAC[GROUP_MUL; GSYM GROUP_LINV_EQ] THEN ANTS_TAC THENL [FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [SYM th]) THEN W(MP_TAC o PART_MATCH rand GROUP_HOMOMORPHISM_HOM_INDUCED_EMPTY o rator o lhand o snd) THEN SIMP_TAC[CONTINUOUS_MAP_CONST; TOPSPACE_DISCRETE_TOPOLOGY; IN_SING] THEN ASM_SIMP_TAC[group_homomorphism] THEN DISCH_THEN(K ALL_TAC) THEN EXPAND_TAC "c" THEN W(MP_TAC o PART_MATCH rand GROUP_HOMOMORPHISM_HOM_INDUCED_EMPTY o rator o rand o snd) THEN SIMP_TAC[CONTINUOUS_MAP_CONST; TOPSPACE_DISCRETE_TOPOLOGY; IN_SING] THEN REWRITE_TAC[group_homomorphism] THEN DISCH_THEN(fun th -> W(MP_TAC o PART_MATCH (lhand o rand) (last(CONJUNCTS th)) o rand o snd)) THEN ANTS_TAC THENL [CONJ_TAC THEN W(MP_TAC o PART_MATCH rand GROUP_HOMOMORPHISM_HOM_INDUCED_EMPTY o rator o lhand o snd) THEN SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_ID] THEN REWRITE_TAC[GROUP_HOMOMORPHISM; SUBSET; FORALL_IN_IMAGE] THEN DISCH_THEN(MATCH_MP_TAC o CONJUNCT1) THEN ASM_REWRITE_TAC[]; DISCH_THEN SUBST1_TAC] THEN BINOP_TAC THEN GEN_REWRITE_TAC RAND_CONV [GSYM o_THM] THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE_EMPTY o rator o rand o snd) THEN ASM_SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_ID; IN_SING; CONTINUOUS_MAP_CONST; TOPSPACE_DISCRETE_TOPOLOGY] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[o_DEF] THEN EXPAND_TAC "d'" THEN GEN_REWRITE_TAC RAND_CONV [GSYM o_THM] THEN AP_THM_TAC THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE_EMPTY o rand o snd) THEN ASM_SIMP_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY; CONTINUOUS_MAP_CONST; CONTINUOUS_MAP_FROM_SUBTOPOLOGY] THEN SIMP_TAC[TOPSPACE_DISCRETE_TOPOLOGY; IN_SING; SUBSET; FORALL_IN_IMAGE; TOPSPACE_SUBTOPOLOGY; IN_INTER] THEN ASM_REWRITE_TAC[]; DISCH_THEN(SUBST_ALL_TAC o SYM)] THEN REWRITE_TAC[o_DEF] THEN EXPAND_TAC "d'" THEN GEN_REWRITE_TAC RAND_CONV [GSYM o_THM] THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE_EMPTY o rator o rand o snd) THEN ASM_SIMP_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY; CONTINUOUS_MAP_ID; GROUP_INV; CONTINUOUS_MAP_FROM_SUBTOPOLOGY] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY] THEN ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN(SUBST1_TAC o SYM)] THEN W(MP_TAC o PART_MATCH rand GROUP_HOMOMORPHISM_HOM_INDUCED_EMPTY o rator o rand o snd) THEN ASM_SIMP_TAC[ETA_AX; o_DEF; CONTINUOUS_MAP_FROM_SUBTOPOLOGY] THEN ASM_SIMP_TAC[group_homomorphism]; EXPAND_TAC "c" THEN W(MP_TAC o PART_MATCH (lhand o rand) (GROUP_RULE `group_inv G (group_div G x y) = group_div G y x`) o rand o snd) THEN ANTS_TAC THENL [CONJ_TAC THEN W(MP_TAC o PART_MATCH rand GROUP_HOMOMORPHISM_HOM_INDUCED_EMPTY o rator o lhand o snd) THEN ASM_SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_ID] THEN ASM_SIMP_TAC[group_homomorphism; SUBSET; FORALL_IN_IMAGE]; DISCH_THEN SUBST1_TAC] THEN W(MP_TAC o PART_MATCH rand GROUP_HOMOMORPHISM_HOM_INDUCED_EMPTY o rator o lhand o snd) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o MATCH_MP GROUP_HOMOMORPHISM_DIV) THEN EXPAND_TAC "c" THEN DISCH_THEN(fun th -> W(MP_TAC o PART_MATCH (lhand o rand) th o lhand o snd)) THEN REWRITE_TAC[HOM_INDUCED; homology_group] THEN DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[GSYM homology_group] THEN BINOP_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_THM] THEN AP_THM_TAC THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE_EMPTY o lhand o snd) THEN ASM_SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_ID] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN ASM_REWRITE_TAC[o_DEF; ETA_AX]]);; (* ------------------------------------------------------------------------- *) (* Degree invariance mod 2 for map between pairs (S^n,S^{n-1}) *) (* ------------------------------------------------------------------------- *) let BORSUK_ODD_MAPPING_DEGREE_STEP = prove (`!f n. continuous_map (nsphere n,nsphere n) f /\ (!x. x IN topspace(nsphere n) ==> (f o (\x i. --x i)) x = ((\x i. --x i) o f) x) /\ IMAGE f (topspace(nsphere(n - 1))) SUBSET topspace(nsphere(n - 1)) ==> (brouwer_degree2 n f == brouwer_degree2 (n - 1) f) (mod &2)`, REPEAT GEN_TAC THEN DISJ_CASES_TAC(ARITH_RULE `n = 0 \/ 1 <= n`) THENL [ASM_REWRITE_TAC[] THEN CONV_TAC NUM_REDUCE_CONV THEN REWRITE_TAC[INTEGER_RULE `(x:int == x) (mod b)`]; ALL_TAC] THEN ABBREV_TAC `neg = \(x:num->real) i. --x i` THEN STRIP_TAC THEN MAP_EVERY ABBREV_TAC [`upper = \n. {x:num->real | x(n+1) >= &0}`; `lower = \n. {x:num->real | x(n+1) <= &0}`; `equator = \n. {x:num->real | x(n+1) = &0}`; `usphere = \n. subtopology (nsphere n) (upper n)`; `lsphere = \n. subtopology (nsphere n) (lower n)`] THEN SUBGOAL_THEN `subtopology (nsphere n) (equator n) = nsphere(n - 1) /\ subtopology (lsphere n) (equator n) = nsphere(n - 1) /\ subtopology (usphere n) (equator n) = nsphere(n - 1)` STRIP_ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["lsphere"; "usphere"; "equator"; "lower"; "upper"] THEN REWRITE_TAC[SUBTOPOLOGY_SUBTOPOLOGY; SET_RULE `{x | P x} INTER {x | Q x} = {x | P x /\ Q x}`] THEN REWRITE_TAC[REAL_ARITH `x <= &0 /\ x = &0 <=> x = &0`] THEN REWRITE_TAC[REAL_ARITH `x >= &0 /\ x = &0 <=> x = &0`] THEN GEN_REWRITE_TAC RAND_CONV [GSYM SUBTOPOLOGY_NSPHERE_EQUATOR] THEN ASM_SIMP_TAC[SUB_ADD; ARITH_RULE `1 <= n ==> n - 1 + 2 = n + 1`]; ALL_TAC] THEN SUBGOAL_THEN `continuous_map(nsphere(n - 1),nsphere(n - 1)) f` ASSUME_TAC THENL [UNDISCH_TAC `IMAGE f (topspace(nsphere(n-1))) SUBSET topspace(nsphere(n - 1))` THEN ONCE_REWRITE_TAC[GSYM SUBTOPOLOGY_NSPHERE_EQUATOR] THEN ASM_SIMP_TAC[SUB_ADD; CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN ASM_SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY] THEN SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `!m x. neg x IN topspace(nsphere m) <=> x IN topspace(nsphere m)` ASSUME_TAC THENL [REWRITE_TAC[nsphere; TOPSPACE_SUBTOPOLOGY] THEN EXPAND_TAC "neg" THEN REWRITE_TAC[IN_INTER; IN_ELIM_THM; TOPSPACE_EUCLIDEAN_SPACE] THEN REWRITE_TAC[REAL_NEG_EQ_0; REAL_ARITH `(--x:real) pow 2 = x pow 2`]; ALL_TAC] THEN SUBGOAL_THEN `!m. continuous_map(nsphere m,nsphere m) neg` ASSUME_TAC THENL [GEN_TAC THEN REWRITE_TAC[NSPHERE; CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN EXPAND_TAC "neg" THEN REWRITE_TAC[CONTINUOUS_MAP_COMPONENTWISE_UNIV] THEN GEN_TAC THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_NEG THEN SIMP_TAC[CONTINUOUS_MAP_PRODUCT_PROJECTION; IN_UNIV]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; TOPSPACE_SUBTOPOLOGY] THEN EXPAND_TAC "neg" THEN REWRITE_TAC[IN_INTER; IN_ELIM_THM] THEN SIMP_TAC[REAL_NEG_EQ_0; REAL_ARITH `(--x:real) pow 2 = x pow 2`]]; ALL_TAC] THEN MP_TAC(ISPECL [`&n - &1:int`; `n - 1`] CYCLIC_REDUCED_HOMOLOGY_GROUP_NSPHERE) THEN REWRITE_TAC[cyclic_group; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `z:((num->real)->num->real)frag->bool` THEN STRIP_TAC THEN SUBGOAL_THEN `?zp. zp IN group_carrier(relative_homology_group(&n,lsphere n,equator n)) /\ hom_boundary (&n) (lsphere n,equator n:(num->real)->bool) zp = z /\ subgroup_generated (relative_homology_group(&n,lsphere n,equator n)) {zp} = relative_homology_group(&n,lsphere n,equator n)` STRIP_ASSUME_TAC THENL [MP_TAC(ISPECL [`&n - &1:int`; `n - 1`] GROUP_ISOMORPHISM_LOWER_HEMISPHERE_REDUCED_HOMOLOGY_GROUP) THEN ASM_SIMP_TAC[SUB_ADD; INT_SUB_ADD; ARITH_RULE `1 <= n ==> n - 1 + 2 = n + 1`] THEN MAP_EVERY EXPAND_TAC ["equator"; "lsphere"; "lower"] THEN REWRITE_TAC[group_isomorphism] THEN MATCH_MP_TAC(MESON[] `(!g. P g ==> ?x. Q(g x)) ==> (?g. P g) ==> ?y. Q y`) THEN REWRITE_TAC[group_isomorphisms] THEN REPEAT STRIP_TAC THEN EXISTS_TAC `z:((num->real)->num->real)frag->bool` THEN ASM_SIMP_TAC[] THEN CONJ_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[group_homomorphism]) THEN ASM SET_TAC[]; ONCE_REWRITE_TAC[SET_RULE `{g z} = IMAGE g {z}`] THEN REWRITE_TAC[GROUPS_EQ; CONJUNCT2 SUBGROUP_GENERATED] THEN FIRST_ASSUM(fun th -> W(MP_TAC o PART_MATCH (lhand o rand) (MATCH_MP (REWRITE_RULE[IMP_CONJ] SUBGROUP_GENERATED_BY_HOMOMORPHIC_IMAGE) th) o lhand o snd)) THEN ASM_REWRITE_TAC[SING_SUBSET] THEN DISCH_THEN SUBST1_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `(!x. x IN s ==> g'(g x) = x) ==> (!y. y IN t ==> g(g' y) = y) /\ IMAGE g s SUBSET t /\ IMAGE g' t SUBSET s ==> IMAGE g s = t`)) THEN RULE_ASSUM_TAC(REWRITE_RULE[group_homomorphism]) THEN ASM SET_TAC[]]; ALL_TAC] THEN SUBGOAL_THEN `?zn. zn IN group_carrier(relative_homology_group(&n,usphere n,equator n)) /\ hom_boundary (&n) (usphere n,equator n:(num->real)->bool) zn = z /\ subgroup_generated (relative_homology_group(&n,usphere n,equator n)) {zn} = relative_homology_group(&n,usphere n,equator n)` STRIP_ASSUME_TAC THENL [MP_TAC(ISPECL [`&n - &1:int`; `n - 1`] GROUP_ISOMORPHISM_UPPER_HEMISPHERE_REDUCED_HOMOLOGY_GROUP) THEN ASM_SIMP_TAC[SUB_ADD; INT_SUB_ADD; ARITH_RULE `1 <= n ==> n - 1 + 2 = n + 1`] THEN MAP_EVERY EXPAND_TAC ["equator"; "usphere"; "upper"] THEN REWRITE_TAC[group_isomorphism] THEN MATCH_MP_TAC(MESON[] `(!g. P g ==> ?x. Q(g x)) ==> (?g. P g) ==> ?y. Q y`) THEN REWRITE_TAC[group_isomorphisms] THEN REPEAT STRIP_TAC THEN EXISTS_TAC `z:((num->real)->num->real)frag->bool` THEN ASM_SIMP_TAC[] THEN CONJ_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[group_homomorphism]) THEN ASM SET_TAC[]; ONCE_REWRITE_TAC[SET_RULE `{g z} = IMAGE g {z}`] THEN REWRITE_TAC[GROUPS_EQ; CONJUNCT2 SUBGROUP_GENERATED] THEN FIRST_ASSUM(fun th -> W(MP_TAC o PART_MATCH (lhand o rand) (MATCH_MP (REWRITE_RULE[IMP_CONJ] SUBGROUP_GENERATED_BY_HOMOMORPHIC_IMAGE) th) o lhand o snd)) THEN ASM_REWRITE_TAC[SING_SUBSET] THEN DISCH_THEN SUBST1_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `(!x. x IN s ==> g'(g x) = x) ==> (!y. y IN t ==> g(g' y) = y) /\ IMAGE g s SUBSET t /\ IMAGE g' t SUBSET s ==> IMAGE g s = t`)) THEN RULE_ASSUM_TAC(REWRITE_RULE[group_homomorphism]) THEN ASM SET_TAC[]]; ALL_TAC] THEN MAP_EVERY ABBREV_TAC [`wp = hom_induced (&n) (lsphere n,equator n) (nsphere n,upper n) (\x. x) zp`; `wn = hom_induced (&n) (usphere n,equator n) (nsphere n,lower n) (\x. x) zn`] THEN SUBGOAL_THEN `wp IN group_carrier(relative_homology_group(&n,nsphere n,upper n)) /\ wn IN group_carrier(relative_homology_group(&n,nsphere n,lower n))` STRIP_ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["wp"; "wn"] THEN REWRITE_TAC[HOM_INDUCED]; ALL_TAC] THEN SUBGOAL_THEN `?vp. vp IN group_carrier(reduced_homology_group(&n,nsphere n)) /\ hom_induced (&n) (nsphere n,{}) (nsphere n,upper n) (\x. x) vp = wp` STRIP_ASSUME_TAC THENL [MP_TAC(ISPECL [`&n:int`; `n:num`; `n + 1`] GROUP_ISOMORPHISM_REDUCED_HOMOLOGY_GROUP_UPPER_HEMISPHERE) THEN REWRITE_TAC[IN_NUMSEG] THEN ANTS_TAC THENL [ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[GROUP_ISOMORPHISM] THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `P /\ IMAGE f s = t /\ Q ==> !y. y IN t ==> ?x. x IN s /\ f x = y`)) THEN MAP_EVERY EXPAND_TAC ["usphere"; "upper"] THEN REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `x IN s ==> s = t ==> x IN t`)) THEN EXPAND_TAC "upper" THEN REWRITE_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?vn. vn IN group_carrier(reduced_homology_group(&n,nsphere n)) /\ hom_induced (&n) (nsphere n,{}) (nsphere n,lower n) (\x. x) vn = wn` STRIP_ASSUME_TAC THENL [MP_TAC(ISPECL [`&n:int`; `n:num`; `n + 1`] GROUP_ISOMORPHISM_REDUCED_HOMOLOGY_GROUP_LOWER_HEMISPHERE) THEN REWRITE_TAC[IN_NUMSEG] THEN ANTS_TAC THENL [ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[GROUP_ISOMORPHISM] THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `P /\ IMAGE f s = t /\ Q ==> !y. y IN t ==> ?x. x IN s /\ f x = y`)) THEN MAP_EVERY EXPAND_TAC ["usphere"; "lower"] THEN REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `x IN s ==> s = t ==> x IN t`)) THEN EXPAND_TAC "lower" THEN REWRITE_TAC[]; ALL_TAC] THEN MAP_EVERY ABBREV_TAC [`up = hom_induced (&n) (lsphere n,equator n) (nsphere n,equator n) (\x. x) zp`; `un = hom_induced (&n) (usphere n,equator n) (nsphere n,equator n) (\x. x) zn`] THEN SUBGOAL_THEN `up IN group_carrier(relative_homology_group(&n,nsphere n,equator n)) /\ un IN group_carrier(relative_homology_group(&n,nsphere n,equator n))` STRIP_ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["up"; "un"] THEN REWRITE_TAC[HOM_INDUCED]; ALL_TAC] THEN MP_TAC(ISPECL [`hom_induced (&n) (nsphere n,equator n) (nsphere n,upper n) (\x. x)`; `hom_induced (&n) (nsphere n,equator n) (nsphere n,lower n) (\x. x)`; `hom_induced (&n) (lsphere n,equator n) (nsphere n,upper n) (\x. x)`; `hom_induced (&n) (lsphere n,equator n) (nsphere n,equator n) (\x. x)`; `hom_induced (&n) (usphere n,equator n) (nsphere n,equator n) (\x. x)`; `hom_induced (&n) (usphere n,equator n) (nsphere n,lower n) (\x. x)`; `relative_homology_group(&n,lsphere n:(num->real)topology,equator n)`; `relative_homology_group(&n,usphere n:(num->real)topology,equator n)`; `relative_homology_group(&n,nsphere n,upper n)`; `relative_homology_group(&n,nsphere n,lower n)`; `relative_homology_group(&n,nsphere n,equator n)`] EXACT_SEQUENCE_SUM_LEMMA) THEN REWRITE_TAC[ABELIAN_RELATIVE_HOMOLOGY_GROUP] THEN ANTS_TAC THENL [MAP_EVERY EXPAND_TAC ["usphere"; "lsphere"; "equator"; "lower"; "upper"] THEN REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [MATCH_ACCEPT_TAC GROUP_ISOMORPHISM_RELATIVE_HOMOLOGY_GROUP_LOWER_HEMISPHERE; MATCH_ACCEPT_TAC GROUP_ISOMORPHISM_RELATIVE_HOMOLOGY_GROUP_UPPER_HEMISPHERE; MATCH_MP_TAC HOMOLOGY_EXACTNESS_TRIPLE_3 THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN REAL_ARITH_TAC; MATCH_MP_TAC HOMOLOGY_EXACTNESS_TRIPLE_3 THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN REAL_ARITH_TAC; REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_THM] THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o rator o lhand o snd) THEN REWRITE_TAC[IMAGE_ID; SUBSET_REFL] THEN ANTS_TAC THENL [SIMP_TAC[CONTINUOUS_MAP_ID; CONTINUOUS_MAP_FROM_SUBTOPOLOGY] THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN REAL_ARITH_TAC; DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[o_DEF]]; REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_THM] THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o rator o lhand o snd) THEN REWRITE_TAC[IMAGE_ID; SUBSET_REFL] THEN ANTS_TAC THENL [SIMP_TAC[CONTINUOUS_MAP_ID; CONTINUOUS_MAP_FROM_SUBTOPOLOGY] THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN REAL_ARITH_TAC; DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[o_DEF]]]; DISCH_THEN(LABEL_TAC "PRI" o CONJUNCT1)] THEN SUBGOAL_THEN `?a b. hom_induced (&n) (nsphere n,equator n) (nsphere n,equator n) f up = group_mul (relative_homology_group(&n,nsphere n,equator n)) (group_zpow (relative_homology_group(&n,nsphere n,equator n)) up a) (group_zpow (relative_homology_group(&n,nsphere n,equator n)) un b)` STRIP_ASSUME_TAC THENL [REMOVE_THEN "PRI" MP_TAC THEN REWRITE_TAC[GROUP_ISOMORPHISM] THEN DISCH_THEN(MP_TAC o SPEC `hom_induced (&n) (nsphere n,equator n) (nsphere n,equator n) f up` o MATCH_MP (SET_RULE `P /\ IMAGE f s = t /\ Q ==> !y. y IN t ==> ?x. x IN s /\ f x = y`)) THEN REWRITE_TAC[PROD_GROUP; EXISTS_PAIR_THM; IN_CROSS; HOM_INDUCED] THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o funpow 2 BINDER_CONV o LAND_CONV o LAND_CONV o RAND_CONV o RAND_CONV) [SYM th]) THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o funpow 2 BINDER_CONV o LAND_CONV o RAND_CONV o RAND_CONV o RAND_CONV) [SYM th]) THEN ASM_SIMP_TAC[CARRIER_SUBGROUP_GENERATED_BY_SING] THEN REWRITE_TAC[GSYM CONJ_ASSOC; RIGHT_EXISTS_AND_THM; EXISTS_IN_GSPEC] THEN REWRITE_TAC[IN_UNIV] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:int` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `b:int` THEN DISCH_THEN(SUBST1_TAC o SYM) THEN MAP_EVERY EXPAND_TAC ["up"; "un"] THEN REWRITE_TAC[] THEN BINOP_TAC THEN MATCH_MP_TAC(REWRITE_RULE[RIGHT_IMP_FORALL_THM; IMP_IMP] GROUP_HOMOMORPHISM_ZPOW) THEN ASM_REWRITE_TAC[GROUP_HOMOMORPHISM_HOM_INDUCED]; ALL_TAC] THEN SUBGOAL_THEN `brouwer_degree2 (n - 1) f = a + b /\ brouwer_degree2 n f = a - b` (CONJUNCTS_THEN SUBST1_TAC) THENL [ALL_TAC; CONV_TAC INTEGER_RULE] THEN SUBGOAL_THEN `hom_boundary (&n) (nsphere n,equator n) up = z /\ hom_boundary (&n) (nsphere n,equator n) un = z` STRIP_ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["up"; "un"] THEN CONJ_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_THM] THEN W(MP_TAC o PART_MATCH (lhand o rand) NATURALITY_HOM_INDUCED o rator o lhand o snd) THEN (REWRITE_TAC[IMAGE_ID; SUBSET_REFL] THEN ANTS_TAC THENL [MAP_EVERY EXPAND_TAC ["lsphere"; "usphere"] THEN SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_ID]; DISCH_THEN SUBST1_TAC]) THEN ASM_REWRITE_TAC[o_THM] THEN MATCH_MP_TAC HOM_INDUCED_ID THEN REWRITE_TAC[GSYM homology_group] THEN ASM_SIMP_TAC[REWRITE_RULE[SUBSET] GROUP_CARRIER_REDUCED_HOMOLOGY_GROUP_SUBSET]; ALL_TAC] THEN CONJ_TAC THENL [MATCH_MP_TAC BROUWER_DEGREE2_UNIQUE_GENERATOR THEN EXISTS_TAC `z:((num->real)->num->real)frag->bool` THEN ASM_SIMP_TAC[GSYM INT_OF_NUM_SUB] THEN SUBST1_TAC(SYM(ASSUME `hom_boundary (&n) (nsphere n,equator n) up = z`)) THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_THM] THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPECL [`&n:int`; `nsphere n`; `topspace(nsphere n) INTER equator n`; `nsphere n`; `topspace(nsphere n) INTER equator n`; `f:(num->real)->(num->real)`] NATURALITY_HOM_INDUCED) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[GSYM TOPSPACE_SUBTOPOLOGY]; ALL_TAC] THEN REWRITE_TAC[GSYM SUBTOPOLOGY_RESTRICT] THEN ONCE_REWRITE_TAC[HOM_INDUCED_RESTRICT] THEN REWRITE_TAC[SET_RULE `s INTER s INTER t = s INTER t`] THEN REWRITE_TAC[GSYM HOM_INDUCED_RESTRICT; GSYM HOM_BOUNDARY_RESTRICT] THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN ASM_REWRITE_TAC[o_THM] THEN ASM_SIMP_TAC[MATCH_MP GROUP_HOMOMORPHISM_MUL (SPEC_ALL GROUP_HOMOMORPHISM_HOM_BOUNDARY); GROUP_ZPOW; GROUP_ZPOW_ADD; MATCH_MP GROUP_HOMOMORPHISM_ZPOW (SPEC_ALL GROUP_HOMOMORPHISM_HOM_BOUNDARY)] THEN REWRITE_TAC[reduced_homology_group] THEN REWRITE_TAC[CONJUNCT2 SUBGROUP_GENERATED; GROUP_ZPOW_SUBGROUP_GENERATED]; ALL_TAC] THEN ABBREV_TAC `u = group_div (relative_homology_group (&n,nsphere n,equator n)) up un` THEN SUBGOAL_THEN `u IN group_carrier (relative_homology_group (&n,nsphere n,equator n))` ASSUME_TAC THENL [ASM_MESON_TAC[GROUP_DIV]; ALL_TAC] THEN MP_TAC(ISPECL [`relative_homology_group (&n,nsphere n,equator n)`; `u:((num->real)->num->real)frag->bool`; `brouwer_degree2 n f`; `a - b:int`] GROUP_ZPOW_EQ_ALT) THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `group_element_order (relative_homology_group (&n,nsphere n,equator n)) u = 0` SUBST1_TAC THENL [ASM_SIMP_TAC[GROUP_ELEMENT_ORDER_EQ_0] THEN X_GEN_TAC `d:num` THEN DISCH_TAC THEN DISCH_THEN(MP_TAC o AP_TERM `hom_induced (&n) (nsphere n,equator n) (nsphere n,upper n) (\x. x)`) THEN ASM_SIMP_TAC[MATCH_MP GROUP_HOMOMORPHISM_POW (SPEC_ALL GROUP_HOMOMORPHISM_HOM_INDUCED); REWRITE_RULE[group_homomorphism] GROUP_HOMOMORPHISM_HOM_INDUCED] THEN SUBGOAL_THEN `hom_induced (&n) (nsphere n,equator n) (nsphere n,upper n) (\x. x) u = wp` SUBST1_TAC THENL [EXPAND_TAC "u" THEN ASM_SIMP_TAC[MATCH_MP GROUP_HOMOMORPHISM_DIV (SPEC_ALL GROUP_HOMOMORPHISM_HOM_INDUCED)] THEN ASM_SIMP_TAC[HOM_INDUCED; GROUP_RULE `group_div G a b = c <=> group_mul G c b = group_mul G a (group_id G)`] THEN BINOP_TAC THENL [SUBST1_TAC(SYM(ASSUME `hom_induced (&n) (lsphere n,equator n) (nsphere n,upper n) (\x. x) zp = wp`)) THEN SUBST1_TAC(SYM(ASSUME `hom_induced (&n) (lsphere n,equator n) (nsphere n,equator n) (\x. x) zp = up`)) THEN GEN_REWRITE_TAC RAND_CONV [GSYM o_THM] THEN AP_THM_TAC THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o rand o snd) THEN REWRITE_TAC[o_DEF] THEN DISCH_THEN MATCH_MP_TAC THEN EXPAND_TAC "lsphere" THEN REWRITE_TAC[IMAGE_ID; SUBSET_REFL] THEN SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_ID] THEN MAP_EVERY EXPAND_TAC ["equator"; "upper"] THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN REAL_ARITH_TAC; MP_TAC(ISPECL [`&n:int`; `nsphere n`; `(upper:num->(num->real)->bool) n`; `(equator:num->(num->real)->bool) n`] HOMOLOGY_EXACTNESS_TRIPLE_3) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [MAP_EVERY EXPAND_TAC ["equator"; "upper"] THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `subtopology (nsphere n) (upper n) = usphere n` SUBST1_TAC THENL [EXPAND_TAC "usphere" THEN REWRITE_TAC[]; ALL_TAC] THEN REWRITE_TAC[group_exactness; group_image; group_kernel] THEN MATCH_MP_TAC(SET_RULE `a IN IMAGE f s ==> P /\ Q /\ IMAGE f s = {x | x IN t /\ g x = z} ==> g a = z`) THEN REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `zn:((num->real)->num->real)frag->bool` THEN ASM_REWRITE_TAC[]]; UNDISCH_TAC `~(d = 0)` THEN SPEC_TAC(`d:num`,`d:num`) THEN ASM_SIMP_TAC[GSYM GROUP_ELEMENT_ORDER_EQ_0] THEN ASM_SIMP_TAC[GSYM INFINITE_CYCLIC_SUBGROUP_ORDER] THEN SUBGOAL_THEN `subgroup_generated (relative_homology_group(&n,nsphere n,upper n)) {wp} = relative_homology_group(&n,nsphere n,upper n)` SUBST1_TAC THENL [REWRITE_TAC[GROUPS_EQ; CONJUNCT2 SUBGROUP_GENERATED] THEN MP_TAC(ISPECL [`relative_homology_group(&n,lsphere n,equator n:(num->real)->bool)`; `relative_homology_group(&n,nsphere n,upper n)`; `hom_induced (&n) (lsphere n,equator n) (nsphere n,upper n) (\x. x)`; `{zp:((num->real)->num->real)frag->bool}`] SUBGROUP_GENERATED_BY_HOMOMORPHIC_IMAGE) THEN ASM_REWRITE_TAC[IMAGE_CLAUSES; SING_SUBSET] THEN REWRITE_TAC[GROUP_HOMOMORPHISM_HOM_INDUCED] THEN DISCH_THEN SUBST1_TAC THEN MP_TAC(ISPECL [`&n:int`; `n:num`; `n + 1`] GROUP_ISOMORPHISM_RELATIVE_HOMOLOGY_GROUP_LOWER_HEMISPHERE) THEN MAP_EVERY EXPAND_TAC ["lsphere"; "equator"; "lower"; "upper"] THEN SIMP_TAC[GROUP_ISOMORPHISM]; MP_TAC INFINITE_INTEGER_GROUP THEN MATCH_MP_TAC EQ_IMP THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC ISOMORPHIC_GROUP_INFINITENESS THEN TRANS_TAC ISOMORPHIC_GROUP_TRANS `reduced_homology_group(&n,nsphere n)` THEN REWRITE_TAC[REDUCED_HOMOLOGY_GROUP_NSPHERE] THEN ONCE_REWRITE_TAC[ISOMORPHIC_GROUP_SYM] THEN MP_TAC(ISPECL [`&n:int`; `n:num`; `n + 1`] GROUP_ISOMORPHISM_REDUCED_HOMOLOGY_GROUP_UPPER_HEMISPHERE) THEN REWRITE_TAC[IN_NUMSEG] THEN ANTS_TAC THENL [ARITH_TAC; ALL_TAC] THEN DISCH_THEN(MP_TAC o MATCH_MP GROUP_ISOMORPHISM_IMP_ISOMORPHIC) THEN EXPAND_TAC "upper" THEN REWRITE_TAC[]]]; ALL_TAC] THEN REWRITE_TAC[INTEGER_RULE `&0 divides a - b <=> b = a`] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN TRANS_TAC EQ_TRANS `hom_induced (&n) (nsphere n,equator n) (nsphere n,equator n) f u` THEN CONJ_TAC THENL [MP_TAC(ISPECL [`&n:int`; `nsphere n`; `(equator:num->(num->real)->bool) n`] HOMOLOGY_EXACTNESS_REDUCED_1) THEN ASM_REWRITE_TAC[GSYM TOPSPACE_SUBTOPOLOGY] THEN ANTS_TAC THENL [REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; NSPHERE; TOPSPACE_SUBTOPOLOGY] THEN EXISTS_TAC `\i. if i = 1 then &1:real else &0` THEN REWRITE_TAC[IN_INTER; TOPSPACE_PRODUCT_TOPOLOGY; IN_ELIM_THM; o_DEF] THEN REWRITE_TAC[TOPSPACE_EUCLIDEANREAL; CARTESIAN_PRODUCT_UNIV; IN_UNIV] THEN ASM_SIMP_TAC[SUB_ADD; LE_REFL; COND_RAND; COND_RATOR; IN_NUMSEG] THEN REWRITE_TAC[COND_ID] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_SIMP_TAC[SUM_DELTA; IN_NUMSEG; LE_REFL]; REWRITE_TAC[group_exactness; group_image; group_kernel] THEN DISCH_THEN(MP_TAC o ISPEC `u:((num->real)->num->real)frag->bool` o MATCH_MP (SET_RULE `P /\ Q /\ IMAGE f s = t ==> !y. y IN t ==> ?x. x IN s /\ f x = y`)) THEN ASM_REWRITE_TAC[IN_ELIM_THM]] THEN ANTS_TAC THENL [EXPAND_TAC "u" THEN ASM_SIMP_TAC[MATCH_MP GROUP_HOMOMORPHISM_DIV (SPEC_ALL GROUP_HOMOMORPHISM_HOM_BOUNDARY)] THEN MP_TAC(MATCH_MP GROUP_DIV_REFL (ASSUME `z IN group_carrier(reduced_homology_group(&n - &1,nsphere(n-1)))`)) THEN REWRITE_TAC[reduced_homology_group; homology_group; GROUP_DIV_SUBGROUP_GENERATED]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN X_GEN_TAC `v:((num->real)->num->real)frag->bool` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (SUBST1_TAC o SYM)) THEN W(MP_TAC o PART_MATCH rand GROUP_HOMOMORPHISM_HOM_INDUCED o rator o lhand o lhand o snd) THEN DISCH_THEN(MP_TAC o MATCH_MP GROUP_HOMOMORPHISM_ZPOW) THEN DISCH_THEN(fun th -> W(MP_TAC o PART_MATCH (rand o rand) th o lhand o snd)) THEN ANTS_TAC THENL [MATCH_MP_TAC(REWRITE_RULE[SUBSET; homology_group] GROUP_CARRIER_REDUCED_HOMOLOGY_GROUP_SUBSET) THEN ASM_REWRITE_TAC[]; DISCH_THEN(SUBST1_TAC o SYM)] THEN SUBGOAL_THEN `group_zpow (relative_homology_group(&n,nsphere n,{})) = group_zpow (reduced_homology_group(&n,nsphere n))` SUBST1_TAC THENL [REWRITE_TAC[reduced_homology_group; homology_group] THEN REWRITE_TAC[GROUP_ZPOW_SUBGROUP_GENERATED]; ASM_SIMP_TAC[GSYM BROUWER_DEGREE2]] THEN GEN_REWRITE_TAC BINOP_CONV [GSYM o_THM] THEN AP_THM_TAC THEN ONCE_REWRITE_TAC[HOM_INDUCED_RESTRICT] THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o rand o snd) THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o lhand o rand o snd) THEN MATCH_MP_TAC(TAUT `(q /\ s ==> t) /\ (p /\ r) ==> (p ==> q) ==> (r ==> s) ==> t`) THEN CONJ_TAC THENL [DISCH_THEN(CONJUNCTS_THEN(SUBST1_TAC o SYM)) THEN REWRITE_TAC[o_DEF; ETA_AX]; ALL_TAC] THEN REWRITE_TAC[INTER_EMPTY; IMAGE_CLAUSES; EMPTY_SUBSET] THEN ASM_REWRITE_TAC[GSYM TOPSPACE_SUBTOPOLOGY] THEN REWRITE_TAC[CONTINUOUS_MAP_ID]; ALL_TAC] THEN EXPAND_TAC "u" THEN ASM_SIMP_TAC[MATCH_MP GROUP_HOMOMORPHISM_DIV (SPEC_ALL GROUP_HOMOMORPHISM_HOM_INDUCED)] THEN SUBGOAL_THEN `hom_induced (&n) (nsphere n,equator n) (nsphere n,equator n) f un = group_mul (relative_homology_group (&n,nsphere n,equator n)) (group_zpow (relative_homology_group (&n,nsphere n,equator n)) un a) (group_zpow (relative_homology_group (&n,nsphere n,equator n)) up b)` SUBST1_TAC THENL [ALL_TAC; ASM_SIMP_TAC[GROUP_ZPOW_SUB] THEN EXPAND_TAC "u" THEN ASM_SIMP_TAC[ABELIAN_GROUP_DIV_ZPOW; ABELIAN_RELATIVE_HOMOLOGY_GROUP] THEN ASM_SIMP_TAC[group_div; GROUP_INV_MUL; GROUP_ZPOW; GROUP_INV; GROUP_INV_INV] THEN ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 5) [REWRITE_RULE[ABELIAN_GROUP_MUL_AC] ABELIAN_RELATIVE_HOMOLOGY_GROUP; GROUP_INV; GROUP_ZPOW; GROUP_MUL]] THEN SUBGOAL_THEN `hom_induced (&n) (nsphere n,equator n) (nsphere n,equator n) neg up = group_zpow (relative_homology_group(&n,nsphere n,equator n)) un (brouwer_degree2 (n - 1) neg)` ASSUME_TAC THENL [EXPAND_TAC "un" THEN ASM_SIMP_TAC[MATCH_MP(GSYM GROUP_HOMOMORPHISM_ZPOW) (SPEC_ALL GROUP_HOMOMORPHISM_HOM_INDUCED)] THEN EXPAND_TAC "up" THEN TRANS_TAC EQ_TRANS `hom_induced (&n) (usphere n,equator n) (nsphere n,equator n) (\x. x) (hom_induced (&n) (lsphere n,equator n) (usphere n,equator n) neg zp)` THEN CONJ_TAC THENL [GEN_REWRITE_TAC BINOP_CONV [GSYM o_THM] THEN AP_THM_TAC THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o rand o snd) THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o lhand o rand o snd) THEN MATCH_MP_TAC(TAUT `(p /\ r) /\ (q /\ s ==> t) ==> (p ==> q) ==> (r ==> s) ==> t`) THEN CONJ_TAC THENL [MAP_EVERY EXPAND_TAC ["lsphere"; "usphere"; "lower"; "upper"] THEN SIMP_TAC[CONTINUOUS_MAP_ID; CONTINUOUS_MAP_FROM_SUBTOPOLOGY] THEN ASM_SIMP_TAC[IMAGE_ID; SUBSET_REFL; CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN ASM_SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; SUBSET] THEN EXPAND_TAC "neg" THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN EXPAND_TAC "equator" THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY] THEN REWRITE_TAC[IN_INTER; IN_ELIM_THM] THEN REAL_ARITH_TAC; DISCH_THEN(CONJUNCTS_THEN(SUBST1_TAC o SYM)) THEN REWRITE_TAC[o_DEF; ETA_AX]]; AP_TERM_TAC] THEN MP_TAC(ISPECL [`&n - &1:int`; `n - 1`] GROUP_ISOMORPHISM_UPPER_HEMISPHERE_REDUCED_HOMOLOGY_GROUP) THEN ASM_SIMP_TAC[SUB_ADD; INT_SUB_ADD; ARITH_RULE `1 <= n ==> n - 1 + 2 = n + 1`] THEN SUBGOAL_THEN `subtopology (nsphere n) {x | x (n + 1) >= &0} = usphere n /\ {x | x (n + 1) = &0} = equator n` (fun th -> ASM_REWRITE_TAC[th]) THENL [MAP_EVERY EXPAND_TAC ["equator"; "usphere"; "upper"] THEN REWRITE_TAC[]; REWRITE_TAC[GROUP_ISOMORPHISM]] THEN DISCH_THEN(MATCH_MP_TAC o last o CONJUNCTS) THEN ASM_SIMP_TAC[HOM_INDUCED; GROUP_ZPOW; MATCH_MP GROUP_HOMOMORPHISM_ZPOW (SPEC_ALL GROUP_HOMOMORPHISM_HOM_BOUNDARY)] THEN MP_TAC(ISPECL [`n - 1`; `neg:(num->real)->(num->real)`; `z:((num->real)->num->real)frag->bool`] BROUWER_DEGREE2) THEN ASM_SIMP_TAC[GSYM INT_OF_NUM_SUB] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [reduced_homology_group] THEN REWRITE_TAC[GROUP_ZPOW_SUBGROUP_GENERATED] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN SUBST1_TAC(SYM(ASSUME `hom_boundary (&n) (lsphere n,equator n:(num->real)->bool) zp = z`)) THEN GEN_REWRITE_TAC BINOP_CONV [GSYM o_THM] THEN AP_THM_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) NATURALITY_HOM_INDUCED o lhand o snd) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN MAP_EVERY EXPAND_TAC ["lsphere"; "usphere"; "lower"; "upper"] THEN SIMP_TAC[CONTINUOUS_MAP_ID; CONTINUOUS_MAP_FROM_SUBTOPOLOGY] THEN ASM_SIMP_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY; SUBSET; FORALL_IN_IMAGE] THEN ASM_SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; TOPSPACE_SUBTOPOLOGY] THEN MAP_EVERY EXPAND_TAC ["neg"; "equator"] THEN REWRITE_TAC[IN_INTER; IN_ELIM_THM] THEN REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `brouwer_degree2 (n - 1) neg pow 2 = &1` ASSUME_TAC THENL [REWRITE_TAC[INT_POW_2; INT_MUL_EQ_1] THEN REWRITE_TAC[INT_ARITH `x:int = &1 \/ x = -- &1 <=> abs x = &1`] THEN MATCH_MP_TAC BROUWER_DEGREE2_HOMEOMORPHIC_MAP THEN REWRITE_TAC[HOMEOMORPHIC_MAP_MAPS] THEN EXISTS_TAC `neg:(num->real)->(num->real)` THEN ASM_REWRITE_TAC[homeomorphic_maps] THEN EXPAND_TAC "neg" THEN REWRITE_TAC[REAL_NEG_NEG; ETA_AX]; ALL_TAC] THEN SUBGOAL_THEN `hom_induced (&n) (nsphere n,equator n) (nsphere n,equator n) neg un = group_zpow (relative_homology_group(&n,nsphere n,equator n)) up (brouwer_degree2 (n - 1) neg)` ASSUME_TAC THENL [MATCH_MP_TAC(MESON[] `f(f y) = y /\ f y = x ==> f x = y`) THEN CONJ_TAC THENL [GEN_REWRITE_TAC LAND_CONV [GSYM o_THM] THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o rator o lhand o snd) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [MAP_EVERY EXPAND_TAC ["neg"; "equator"] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM] THEN REAL_ARITH_TAC; DISCH_THEN(SUBST1_TAC o SYM)] THEN MATCH_MP_TAC HOM_INDUCED_ID_GEN THEN ASM_SIMP_TAC[GROUP_ZPOW] THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_MAP_COMPOSE]; ALL_TAC] THEN EXPAND_TAC "neg" THEN REWRITE_TAC[o_DEF; REAL_NEG_NEG; ETA_AX]; ALL_TAC] THEN ASM_SIMP_TAC[MATCH_MP GROUP_HOMOMORPHISM_ZPOW (SPEC_ALL GROUP_HOMOMORPHISM_HOM_INDUCED)] THEN ASM_SIMP_TAC[GSYM GROUP_ZPOW_MUL; GSYM INT_POW_2; GROUP_ZPOW_1]; ALL_TAC] THEN TRANS_TAC EQ_TRANS `hom_induced (&n) (nsphere n,equator n) (nsphere n,equator n) f (group_zpow (relative_homology_group (&n,nsphere n,equator n)) (hom_induced (&n) (nsphere n,equator n) (nsphere n,equator n) neg up) (brouwer_degree2 (n - 1) neg))` THEN CONJ_TAC THENL [AP_TERM_TAC THEN ASM_SIMP_TAC[GSYM GROUP_ZPOW_MUL] THEN ASM_SIMP_TAC[GSYM INT_POW_2; GROUP_ZPOW_1]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhand o rand) (MATCH_MP GROUP_HOMOMORPHISM_ZPOW (SPEC_ALL GROUP_HOMOMORPHISM_HOM_INDUCED)) o lhand o snd) THEN ANTS_TAC THENL [REWRITE_TAC[HOM_INDUCED]; DISCH_THEN SUBST1_TAC] THEN TRANS_TAC EQ_TRANS `group_zpow (relative_homology_group (&n,nsphere n,equator n)) (hom_induced (&n) (nsphere n,equator n) (nsphere n,equator n) neg (hom_induced (&n) (nsphere n,equator n) (nsphere n,equator n) f up)) (brouwer_degree2 (n - 1) neg)` THEN CONJ_TAC THENL [AP_THM_TAC THEN AP_TERM_TAC THEN GEN_REWRITE_TAC BINOP_CONV [GSYM o_THM] THEN AP_THM_TAC THEN ONCE_REWRITE_TAC[HOM_INDUCED_RESTRICT] THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o rand o snd) THEN W(MP_TAC o PART_MATCH (rand o rand) HOM_INDUCED_COMPOSE o lhand o rand o snd) THEN MATCH_MP_TAC(TAUT `(p /\ r) /\ (q /\ s ==> t) ==> (p ==> q) ==> (r ==> s) ==> t`) THEN CONJ_TAC THENL [ASM_REWRITE_TAC[GSYM TOPSPACE_SUBTOPOLOGY] THEN ASM_MESON_TAC[CONTINUOUS_MAP_IMAGE_SUBSET_TOPSPACE]; DISCH_THEN(CONJUNCTS_THEN(SUBST1_TAC o SYM)) THEN MATCH_MP_TAC HOM_INDUCED_EQ THEN ASM_SIMP_TAC[]]; ALL_TAC] THEN ASM_SIMP_TAC[MATCH_MP GROUP_HOMOMORPHISM_ZPOW (SPEC_ALL GROUP_HOMOMORPHISM_HOM_INDUCED); MATCH_MP GROUP_HOMOMORPHISM_MUL (SPEC_ALL GROUP_HOMOMORPHISM_HOM_INDUCED); GROUP_ZPOW] THEN ASM_SIMP_TAC[ABELIAN_GROUP_MUL_ZPOW; GROUP_ZPOW; GSYM GROUP_ZPOW_MUL; GROUP_MUL; ABELIAN_RELATIVE_HOMOLOGY_GROUP] THEN REWRITE_TAC[INT_ARITH `d * a * d:int = a * d pow 2`] THEN ASM_SIMP_TAC[INT_MUL_RID; GROUP_ZPOW_1]);; (* ------------------------------------------------------------------------- *) (* General Jordan-Brouwer separation theorem and invariance of dimension. *) (* ------------------------------------------------------------------------- *) let RELATIVE_HOMOLOGY_GROUP_EUCLIDEAN_COMPLEMENT_STEP = prove (`!p n k s. closed_in (euclidean_space n) s ==> relative_homology_group (p,euclidean_space n,topspace(euclidean_space n) DIFF s) isomorphic_group relative_homology_group (p + &k,euclidean_space (n+k), topspace(euclidean_space (n+k)) DIFF s)`, SUBGOAL_THEN `!p n s. closed_in (euclidean_space (n + 1)) s /\ (!x y. (!i. ~(i = n + 1) ==> x i = y i) /\ x IN s ==> y IN s) ==> relative_homology_group (p,euclidean_space n,topspace(euclidean_space n) DIFF s) isomorphic_group relative_homology_group (p + &1,euclidean_space (n+1), topspace(euclidean_space (n+1)) DIFF {x | x IN s /\ x(n + 1) = &0})` ASSUME_TAC THENL [ALL_TAC; GEN_TAC THEN GEN_TAC THEN GEN_REWRITE_TAC I [SWAP_FORALL_THM] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC num_INDUCTION THEN REWRITE_TAC[ADD_CLAUSES; INT_ADD_RID; ISOMORPHIC_GROUP_REFL] THEN X_GEN_TAC `m:num` THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] ISOMORPHIC_GROUP_TRANS) THEN SUBGOAL_THEN `closed_in (euclidean_space (m + n)) s` MP_TAC THEN UNDISCH_TAC `closed_in (euclidean_space n) s` THENL [REWRITE_TAC[euclidean_space; CLOSED_IN_SUBTOPOLOGY] THEN DISCH_THEN(X_CHOOSE_THEN `c:(num->real)->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `c INTER topspace(euclidean_space n)` THEN ASM_SIMP_TAC[CLOSED_IN_INTER; CLOSED_IN_EUCLIDEAN_SPACE] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SPACE] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> c INTER s = (c INTER s) INTER t`) THEN REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_NUMSEG] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; DISCH_THEN(K ALL_TAC) THEN REWRITE_TAC[GSYM INT_OF_NUM_SUC; INT_ADD_ASSOC; ADD_CLAUSES] THEN SPEC_TAC(`p + &m:int`,`p:int`) THEN REWRITE_TAC[ADD1] THEN SUBST1_TAC(ARITH_RULE `m + n:num = n + m`) THEN SPEC_TAC(`n + m:num`,`n:num`)] THEN REPEAT STRIP_TAC THEN ABBREV_TAC `s' = {x | x IN topspace(euclidean_space(n + 1)) /\ (\i. if i <= n then x i else &0) IN s}` THEN FIRST_X_ASSUM(MP_TAC o SPECL [`p:int`; `n:num`; `s':(num->real)->bool`]) THEN ANTS_TAC THENL [EXPAND_TAC "s'" THEN CONJ_TAC THENL [MATCH_MP_TAC CLOSED_IN_CONTINUOUS_MAP_PREIMAGE THEN EXISTS_TAC `euclidean_space n` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[euclidean_space] THEN REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN REWRITE_TAC[CONTINUOUS_MAP_COMPONENTWISE_UNIV; SUBSET] THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_ELIM_THM] THEN CONJ_TAC THENL [X_GEN_TAC `k:num` THEN MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN ASM_CASES_TAC `k:num <= n` THEN ASM_REWRITE_TAC[CONTINUOUS_MAP_REAL_CONST] THEN SIMP_TAC[CONTINUOUS_MAP_PRODUCT_PROJECTION; IN_UNIV]; REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; IN_INTER; IN_ELIM_THM] THEN REWRITE_TAC[IN_NUMSEG] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC]; REWRITE_TAC[IN_ELIM_THM; TOPSPACE_EUCLIDEAN_SPACE; IN_NUMSEG] THEN REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [ASM_MESON_TAC[LE_REFL; ARITH_RULE `1 <= n + 1`]; MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN ABS_TAC THEN COND_CASES_TAC THEN REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC]]; MATCH_MP_TAC EQ_IMP THEN BINOP_TAC THEN REPLICATE_TAC 3 AP_TERM_TAC THEN EXPAND_TAC "s'" THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SPACE] THEN MATCH_MP_TAC(SET_RULE `(!x. x IN u ==> (x IN s <=> x IN t)) ==> u DIFF s = u DIFF t`) THEN SIMP_TAC[IN_ELIM_THM; IN_NUMSEG; ARITH_RULE `~(1 <= i /\ i <= n + 1) ==> ~(1 <= i /\ i <= n)`] THENL [REPEAT STRIP_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN ASM_MESON_TAC[]; X_GEN_TAC `x:num->real` THEN DISCH_TAC THEN EQ_TAC THENL [DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `i:num` THEN ASM_CASES_TAC `i:num <= n` THEN ASM_REWRITE_TAC[] THEN CONV_TAC SYM_CONV THEN ASM_CASES_TAC `i = n + 1` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_SUBSET) THEN REWRITE_TAC[SUBSET] THEN DISCH_THEN(MP_TAC o SPEC `x:num->real`) THEN ASM_REWRITE_TAC[TOPSPACE_EUCLIDEAN_SPACE; IN_ELIM_THM] THEN REWRITE_TAC[IN_NUMSEG] THEN STRIP_TAC THEN ASM_SIMP_TAC[ARITH_RULE `~(n + 1 <= n)`] THEN UNDISCH_TAC `(x:num->real) IN s` THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN ASM_MESON_TAC[]]]]] THEN REPEAT STRIP_TAC THEN MAP_EVERY ABBREV_TAC [`lo = {x | x IN topspace(euclidean_space (n + 1)) /\ x(n+1) < (if x IN s then &0 else &1)}`; `hi = {x | x IN topspace(euclidean_space (n + 1)) /\ x(n+1) > (if x IN s then &0 else -- &1)}`] THEN SUBGOAL_THEN `lo INTER hi = {x | x IN topspace(euclidean_space (n + 1)) DIFF s /\ x(n+1) IN real_interval(-- &1,&1)}` ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["lo"; "hi"] THEN REWRITE_TAC[EXTENSION; IN_INTER; IN_ELIM_THM; IN_DIFF] THEN X_GEN_TAC `x:num->real` THEN ASM_CASES_TAC `x IN topspace(euclidean_space (n + 1))` THEN ASM_REWRITE_TAC[IN_REAL_INTERVAL] THEN ASM_CASES_TAC `(x:num->real) IN s` THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `lo UNION hi = topspace(euclidean_space (n + 1)) DIFF {x | x IN s /\ x(n+1) = &0}` ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["lo"; "hi"] THEN REWRITE_TAC[EXTENSION; IN_UNION; IN_ELIM_THM; IN_DIFF] THEN X_GEN_TAC `x:num->real` THEN ASM_CASES_TAC `x IN topspace(euclidean_space (n + 1))` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `(x:num->real) IN s` THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC; ALL_TAC] THEN ABBREV_TAC `ret = \c (x:num->real) i. if i = n + 1 then c else x i` THEN SUBGOAL_THEN `!t:real. continuous_map (product_topology (:num) (\i. euclideanreal), product_topology (:num) (\i. euclideanreal)) (ret t)` ASSUME_TAC THENL [GEN_TAC THEN EXPAND_TAC "ret" THEN REWRITE_TAC[CONTINUOUS_MAP_COMPONENTWISE_UNIV] THEN X_GEN_TAC `k:num` THEN ASM_CASES_TAC `k = n + 1` THEN ASM_REWRITE_TAC[CONTINUOUS_MAP_REAL_CONST] THEN SIMP_TAC[CONTINUOUS_MAP_PRODUCT_PROJECTION; IN_UNIV]; ALL_TAC] THEN ABBREV_TAC `squashable = \t s. !x t'. x IN s /\ (x(n+1):real <= t' /\ t' <= t \/ t <= t' /\ t' <= x(n+1)) ==> ret t' x IN s` THEN SUBGOAL_THEN `!t:real. squashable t (topspace(euclidean_space(n + 1))):bool` ASSUME_TAC THENL [GEN_TAC THEN EXPAND_TAC "squashable" THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SPACE] THEN EXPAND_TAC "ret" THEN SIMP_TAC[IN_NUMSEG; IN_ELIM_THM] THEN MESON_TAC[LE_REFL; ARITH_RULE `1 <= n + 1`]; ALL_TAC] THEN SUBGOAL_THEN `!u t. squashable t u ==> retraction_maps (subtopology (euclidean_space (n + 1)) u, subtopology (euclidean_space (n + 1)) {x | x IN u /\ x(n+1) = t}) (ret t,I)` ASSUME_TAC THENL [EXPAND_TAC "squashable" THEN REPEAT STRIP_TAC THEN REWRITE_TAC[retraction_maps] THEN ASM_SIMP_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY; I_DEF; CONTINUOUS_MAP_ID; CONTINUOUS_MAP_FROM_SUBTOPOLOGY; euclidean_space] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; TOPSPACE_SUBTOPOLOGY; IN_ELIM_THM; IN_INTER] THEN FIRST_X_ASSUM(MP_TAC o GEN `x:num->real` o SPECL [`x:num->real`; `t:real`]) THEN SIMP_TAC[REAL_LE_REFL; REAL_LE_TOTAL] THEN DISCH_TAC THEN EXPAND_TAC "ret" THEN SIMP_TAC[IN_NUMSEG] THEN REPEAT STRIP_TAC THEN TRY(GEN_REWRITE_TAC I [FUN_EQ_THM]) THEN REPEAT GEN_TAC THEN REWRITE_TAC[] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `!u v t. squashable t u /\ squashable (t:real) v ==> homotopic_with (\k. IMAGE k v SUBSET v) (subtopology (euclidean_space (n + 1)) u, subtopology (euclidean_space (n + 1)) u) (ret t) I` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN REWRITE_TAC[homotopic_with] THEN EXISTS_TAC `(\(z,x). ret ((&1 - z) * t + z * x(n+1)) x) :real#(num->real)->num->real` THEN REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[REAL_MUL_LID; REAL_MUL_LZERO; REAL_ADD_RID; REAL_ADD_LID] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY; euclidean_space] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; TOPSPACE_SUBTOPOLOGY] THEN CONJ_TAC THENL [REWRITE_TAC[SUBTOPOLOGY_SUBTOPOLOGY] THEN REWRITE_TAC[GSYM SUBTOPOLOGY_CROSS] THEN MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN EXPAND_TAC "ret" THEN REWRITE_TAC[CONTINUOUS_MAP_COMPONENTWISE_UNIV] THEN REWRITE_TAC[LAMBDA_PAIR_THM] THEN X_GEN_TAC `k:num` THEN ASM_CASES_TAC `k = n + 1` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[LAMBDA_PAIR] THENL [MATCH_MP_TAC CONTINUOUS_MAP_REAL_ADD THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_MUL THEN REWRITE_TAC[CONTINUOUS_MAP_FST; CONTINUOUS_MAP_REAL_CONST] THEN TRY(MATCH_MP_TAC CONTINUOUS_MAP_REAL_SUB) THEN REWRITE_TAC[CONTINUOUS_MAP_FST; CONTINUOUS_MAP_REAL_CONST]; ALL_TAC] THEN REWRITE_TAC[CONTINUOUS_MAP_OF_SND] THEN SIMP_TAC[CONTINUOUS_MAP_PRODUCT_PROJECTION; IN_UNIV]; REWRITE_TAC[FORALL_PAIR_THM; TOPSPACE_PROD_TOPOLOGY; TOPSPACE_SUBTOPOLOGY; IN_CROSS; IN_INTER] THEN MAP_EVERY X_GEN_TAC [`z:real`; `x:num->real`] THEN REWRITE_TAC[IN_ELIM_THM; IN_REAL_INTERVAL; IN_NUMSEG] THEN EXPAND_TAC "ret" THEN SIMP_TAC[] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN REWRITE_TAC[] THEN ASM_ARITH_TAC]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_PAIR_THM] THEN MAP_EVERY X_GEN_TAC [`z:real`; `x:num->real`] THEN REWRITE_TAC[TOPSPACE_PROD_TOPOLOGY; IN_CROSS] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; IN_REAL_INTERVAL; IN_INTER] THEN STRIP_TAC THEN UNDISCH_TAC `(squashable:real->((num->real)->bool)->bool) t u` THEN EXPAND_TAC "squashable" THEN REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[REAL_ARITH `b <= (&1 - t) * a + t * b /\ (&1 - t) * a + t * b <= a \/ a <= (&1 - t) * a + t * b /\ (&1 - t) * a + t * b <= b <=> abs(t * (a - b)) + abs((&1 - t) * (a - b)) = &1 * abs(a - b)`] THEN REWRITE_TAC[REAL_ABS_MUL; GSYM REAL_ADD_RDISTRIB] THEN AP_THM_TAC THEN AP_TERM_TAC THEN ASM_REAL_ARITH_TAC; EXPAND_TAC "ret" THEN REWRITE_TAC[I_DEF; FUN_EQ_THM] THEN MESON_TAC[]; X_GEN_TAC `z:real` THEN REWRITE_TAC[IN_REAL_INTERVAL] THEN STRIP_TAC THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `x:num->real` THEN DISCH_TAC THEN UNDISCH_TAC `(squashable:real->((num->real)->bool)->bool) t v` THEN EXPAND_TAC "squashable" THEN REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[REAL_ARITH `b <= (&1 - t) * a + t * b /\ (&1 - t) * a + t * b <= a \/ a <= (&1 - t) * a + t * b /\ (&1 - t) * a + t * b <= b <=> abs(t * (a - b)) + abs((&1 - t) * (a - b)) = &1 * abs(a - b)`] THEN REWRITE_TAC[REAL_ABS_MUL; GSYM REAL_ADD_RDISTRIB] THEN AP_THM_TAC THEN AP_TERM_TAC THEN ASM_REAL_ARITH_TAC]; ALL_TAC] THEN SUBGOAL_THEN `contractible_space(subtopology (euclidean_space(n+1)) hi)` ASSUME_TAC THENL [SUBGOAL_THEN `contractible_space(subtopology (euclidean_space(n+1)) {x | x(n+1) = &1})` MP_TAC THENL [REWRITE_TAC[contractible_space] THEN EXISTS_TAC `(\i. if i = n+1 then &1 else &0):num->real` THEN ONCE_REWRITE_TAC[HOMOTOPIC_WITH_SYM] THEN SIMP_TAC[HOMOTOPIC_WITH] THEN EXISTS_TAC `(\(z,x) i. if i = n + 1 then &1 else z * x i) :real#(num->real)->num->real` THEN REWRITE_TAC[euclidean_space] THEN REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY; SUBSET; FORALL_IN_IMAGE] THEN REWRITE_TAC[TOPSPACE_PROD_TOPOLOGY; IN_CROSS; FORALL_PAIR_THM] THEN SIMP_TAC[TOPSPACE_SUBTOPOLOGY; IN_ELIM_THM; IN_INTER] THEN REWRITE_TAC[REAL_MUL_LID; REAL_MUL_LZERO] THEN REWRITE_TAC[FUN_EQ_THM; IN_NUMSEG; REAL_MUL_RZERO; GSYM CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[LE_REFL; ARITH_RULE `1 <= n + 1`]] THEN REWRITE_TAC[CONTINUOUS_MAP_COMPONENTWISE_UNIV] THEN X_GEN_TAC `k:num` THEN REWRITE_TAC[LAMBDA_PAIR_THM] THEN ASM_CASES_TAC `k = n + 1` THEN ASM_REWRITE_TAC[CONTINUOUS_MAP_REAL_CONST; LAMBDA_PAIR] THEN REWRITE_TAC[SUBTOPOLOGY_SUBTOPOLOGY] THEN REWRITE_TAC[GSYM SUBTOPOLOGY_CROSS] THEN MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_MUL THEN REWRITE_TAC[CONTINUOUS_MAP_FST; CONTINUOUS_MAP_OF_SND] THEN SIMP_TAC[CONTINUOUS_MAP_PRODUCT_PROJECTION; IN_UNIV]; MATCH_MP_TAC EQ_IMP THEN MATCH_MP_TAC HOMOTOPY_EQUIVALENT_SPACE_CONTRACTIBILITY THEN ONCE_REWRITE_TAC[HOMOTOPY_EQUIVALENT_SPACE_SYM] THEN MATCH_MP_TAC DEFORMATION_RETRACT_IMP_HOMOTOPY_EQUIVALENT_SPACE THEN EXISTS_TAC `(ret:real->(num->real)->num->real) (&1)` THEN FIRST_X_ASSUM(MP_TAC o SPECL [`hi:(num->real)->bool`; `(:num->real)`; `&1`]) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`hi:(num->real)->bool`; `&1`]) THEN SUBGOAL_THEN `squashable (&1) (hi:(num->real)->bool):bool` ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["hi"; "squashable"; "ret"] THEN REWRITE_TAC[IN_ELIM_THM; TOPSPACE_EUCLIDEAN_SPACE; IN_NUMSEG] THEN MAP_EVERY X_GEN_TAC [`x:num->real`; `z:real`] THEN SIMP_TAC[ARITH_RULE `~(1 <= i /\ i <= n + 1) ==> ~(i = n + 1)`] THEN SUBGOAL_THEN `(\i. if i = n + 1 then z else x i) IN s <=> (x:num->real) IN s` SUBST1_TAC THENL [EQ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[IMP_CONJ]) THEN SIMP_TAC[ARITH_RULE `i <= n ==> ~(i = n + 1)`]; ASM_CASES_TAC `(x:num->real) IN s` THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC]; ASM_REWRITE_TAC[] THEN EXPAND_TAC "squashable" THEN REWRITE_TAC[IN_UNIV; SUBSET_UNIV; GSYM IMP_CONJ_ALT] THEN MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN REPLICATE_TAC 2 AP_TERM_TAC THEN REWRITE_TAC[euclidean_space; SUBTOPOLOGY_SUBTOPOLOGY] THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_INTER; IN_ELIM_THM] THEN EXPAND_TAC "hi" THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SPACE] THEN X_GEN_TAC `x:num->real` THEN REWRITE_TAC[IN_ELIM_THM; IN_NUMSEG] THEN ASM_CASES_TAC `(x:num->real) (n + 1) = &1` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(TAUT `q ==> (p /\ p /\ q <=> p)`) THEN COND_CASES_TAC THEN CONV_TAC REAL_RAT_REDUCE_CONV]]; ALL_TAC] THEN TRANS_TAC ISOMORPHIC_GROUP_TRANS `relative_homology_group(p,euclidean_space (n+1),lo INTER hi)` THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[ISOMORPHIC_GROUP_SYM] THEN MATCH_MP_TAC DEFORMATION_RETRACT_IMP_ISOMORPHIC_RELATIVE_HOMOLOGY_GROUPS THEN EXISTS_TAC `(ret:real->(num->real)->num->real) (&0)` THEN REPEAT CONJ_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPECL [`(:num->real)`; `&0`]) THEN EXPAND_TAC "squashable" THEN REWRITE_TAC[IN_UNIV] THEN REWRITE_TAC[SUBTOPOLOGY_UNIV] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[euclidean_space; SUBTOPOLOGY_SUBTOPOLOGY] THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_INTER] THEN REWRITE_TAC[IN_NUMSEG] THEN X_GEN_TAC `n:num->real` THEN EQ_TAC THEN SIMP_TAC[ARITH_RULE `~(n + 1 <= n)`] THEN STRIP_TAC THEN X_GEN_TAC `i:num` THEN ASM_CASES_TAC `i = n + 1` THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET; IN_ELIM_THM; euclidean_space; IN_REAL_INTERVAL; IN_DIFF; TOPSPACE_SUBTOPOLOGY; IN_INTER; IN_NUMSEG] THEN REWRITE_TAC[ARITH_RULE `~(1 <= i /\ i <= n + 1) <=> ~(1 <= i /\ i <= n) /\ ~(i = n + 1)`] THEN X_GEN_TAC `x:num->real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[ARITH_RULE `~(n + 1 <= n)`] THEN CONV_TAC REAL_RAT_REDUCE_CONV; ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM; IN_DIFF] THEN X_GEN_TAC `x:num->real` THEN REWRITE_TAC[IN_REAL_INTERVAL] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[CONTRAPOS_THM] THEN CONJ_TAC THENL [DISCH_TAC THEN FIRST_ASSUM(MP_TAC o SPECL [`(:num->real)`; `&0`]) THEN EXPAND_TAC "squashable" THEN REWRITE_TAC[IN_UNIV] THEN REWRITE_TAC[continuous_map; retraction_maps; SUBTOPOLOGY_UNIV] THEN DISCH_THEN(MP_TAC o SPEC `x:num->real` o hd o CONJUNCTS) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[euclidean_space] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; IN_INTER; IN_ELIM_THM] THEN REWRITE_TAC[IN_NUMSEG; ARITH_RULE `~(1 <= i /\ i <= n + 1) <=> ~(1 <= i /\ i <= n) /\ ~(i = n + 1)`] THEN MESON_TAC[]; FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[IMP_CONJ]) THEN X_GEN_TAC `i:num` THEN EXPAND_TAC "ret" THEN MESON_TAC[ARITH_RULE `~(n + 1 <= n)`]]; ONCE_REWRITE_TAC[GSYM SUBTOPOLOGY_TOPSPACE] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `squashable (&0) (topspace (euclidean_space (n + 1))):bool` MP_TAC THENL [ASM_REWRITE_TAC[]; EXPAND_TAC "squashable"] THEN SIMP_TAC[IN_DIFF; IN_ELIM_THM; IN_REAL_INTERVAL] THEN DISCH_THEN(K ALL_TAC) THEN REPEAT GEN_TAC THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN CONJ_TAC THENL [UNDISCH_TAC `~((x:num->real) IN s)` THEN REWRITE_TAC[CONTRAPOS_THM] THEN FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[IMP_CONJ]) THEN X_GEN_TAC `i:num` THEN EXPAND_TAC "ret" THEN MESON_TAC[ARITH_RULE `~(n + 1 <= n)`]; EXPAND_TAC "ret" THEN REWRITE_TAC[] THEN ASM_ARITH_TAC]]; ALL_TAC] THEN TRANS_TAC ISOMORPHIC_GROUP_TRANS `relative_homology_group (p,subtopology (euclidean_space (n+1)) hi,lo INTER hi)` THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[ISOMORPHIC_GROUP_SYM] THEN MATCH_MP_TAC ISOMORPHIC_RELATIVE_HOMOLOGY_GROUPS_INCLUSION_CONTRACTIBLE THEN ASM_REWRITE_TAC[CONTRACTIBLE_EUCLIDEAN_SPACE] THEN EXPAND_TAC "hi" THEN REWRITE_TAC[SUBSET; GSYM MEMBER_NOT_EMPTY; IN_INTER; IN_ELIM_THM] THEN SIMP_TAC[IN_DIFF; IN_REAL_INTERVAL; real_gt] THEN EXISTS_TAC `(\i. if i = n + 1 then &1 else &0):num->real` THEN REWRITE_TAC[TAUT `p /\ p /\ q <=> p /\ q`] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SPACE; IN_ELIM_THM; IN_NUMSEG] THEN SIMP_TAC[ARITH_RULE `~(1 <= i /\ i <= n + 1) ==> ~(i = n + 1)`] THEN COND_CASES_TAC THEN CONV_TAC REAL_RAT_REDUCE_CONV; ALL_TAC] THEN TRANS_TAC ISOMORPHIC_GROUP_TRANS `relative_homology_group (p,subtopology (euclidean_space (n+1)) (lo UNION hi),lo)` THEN CONJ_TAC THENL [MP_TAC(ISPECL [`p:int`; `subtopology(euclidean_space (n + 1)) (lo UNION hi)`; `lo UNION hi:(num->real)->bool`; `lo:(num->real)->bool`; `topspace(subtopology(euclidean_space (n + 1)) (lo UNION hi)) DIFF hi`] HOMOLOGY_EXCISION_AXIOM) THEN REWRITE_TAC[SUBSET_UNION; CLOSURE_OF_COMPLEMENT] THEN SUBGOAL_THEN `lo SUBSET topspace(euclidean_space(n+1)) /\ hi SUBSET topspace(euclidean_space(n+1))` MP_TAC THENL [MAP_EVERY EXPAND_TAC ["hi"; "lo"] THEN SET_TAC[]; REWRITE_TAC[TOPSPACE_SUBTOPOLOGY]] THEN SIMP_TAC[SET_RULE `l SUBSET u /\ h SUBSET u ==> (l UNION h) DIFF ((u INTER (l UNION h)) DIFF h) = h /\ l DIFF ((u INTER (l UNION h)) DIFF h) = l INTER h`] THEN REWRITE_TAC[SUBTOPOLOGY_SUBTOPOLOGY; SET_RULE `s INTER s = s`] THEN REWRITE_TAC[SET_RULE `(l UNION h) INTER h = h`] THEN STRIP_TAC THEN ANTS_TAC THENL [ALL_TAC; REWRITE_TAC[GROUP_ISOMORPHISM_IMP_ISOMORPHIC]] THEN MATCH_MP_TAC(SET_RULE `h' = h /\ l' = l ==> (u INTER (l UNION h)) DIFF h' SUBSET l'`) THEN SUBGOAL_THEN `open_in (euclidean_space(n + 1)) lo /\ open_in (euclidean_space(n + 1)) hi` MP_TAC THENL [ALL_TAC; SIMP_TAC[INTERIOR_OF_SUBTOPOLOGY_OPEN; OPEN_IN_UNION] THEN SIMP_TAC[INTERIOR_OF_OPEN_IN] THEN SET_TAC[]] THEN MAP_EVERY EXPAND_TAC ["lo"; "hi"] THEN ONCE_REWRITE_TAC[OPEN_IN_SUBOPEN] THEN REWRITE_TAC[AND_FORALL_THM; IN_ELIM_THM] THEN X_GEN_TAC `x:num->real` THEN ASM_CASES_TAC `(x:num->real) IN s` THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THENL [EXISTS_TAC `{x | x IN topspace(euclidean_space(n+1)) /\ x(n + 1) < &0}` THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[SET_RULE `x n < &0 <=> x n IN {y | y < &0}`] THEN MATCH_MP_TAC OPEN_IN_CONTINUOUS_MAP_PREIMAGE THEN EXISTS_TAC `euclideanreal` THEN REWRITE_TAC[REAL_OPEN_HALFSPACE_LT; GSYM REAL_OPEN_IN] THEN REWRITE_TAC[euclidean_space] THEN MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN SIMP_TAC[CONTINUOUS_MAP_PRODUCT_PROJECTION; IN_UNIV]; EXPAND_TAC "lo" THEN SIMP_TAC[SUBSET; IN_ELIM_THM] THEN REAL_ARITH_TAC]; EXISTS_TAC `{x | x IN topspace(euclidean_space(n+1)) /\ x(n + 1) > &0}` THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[SET_RULE `x n > &0 <=> x n IN {y | y > &0}`] THEN MATCH_MP_TAC OPEN_IN_CONTINUOUS_MAP_PREIMAGE THEN EXISTS_TAC `euclideanreal` THEN REWRITE_TAC[REAL_OPEN_HALFSPACE_GT; GSYM REAL_OPEN_IN] THEN REWRITE_TAC[euclidean_space] THEN MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN SIMP_TAC[CONTINUOUS_MAP_PRODUCT_PROJECTION; IN_UNIV]; EXPAND_TAC "hi" THEN SIMP_TAC[SUBSET; IN_ELIM_THM] THEN REAL_ARITH_TAC]; EXISTS_TAC `{x | x IN topspace(euclidean_space(n+1)) /\ x(n + 1) < &1} INTER (topspace(euclidean_space(n+1)) DIFF s)` THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_INTER THEN ASM_SIMP_TAC[OPEN_IN_DIFF; OPEN_IN_TOPSPACE] THEN ONCE_REWRITE_TAC[SET_RULE `x n < &1 <=> x n IN {y | y < &1}`] THEN MATCH_MP_TAC OPEN_IN_CONTINUOUS_MAP_PREIMAGE THEN EXISTS_TAC `euclideanreal` THEN REWRITE_TAC[REAL_OPEN_HALFSPACE_LT; GSYM REAL_OPEN_IN] THEN REWRITE_TAC[euclidean_space] THEN MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN SIMP_TAC[CONTINUOUS_MAP_PRODUCT_PROJECTION; IN_UNIV]; EXPAND_TAC "lo" THEN SIMP_TAC[SUBSET; IN_ELIM_THM; IN_DIFF; IN_INTER] THEN ASM_REWRITE_TAC[]]; EXISTS_TAC `{x | x IN topspace(euclidean_space(n+1)) /\ x(n + 1) > -- &1} INTER (topspace(euclidean_space(n+1)) DIFF s)` THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN CONJ_TAC THENL [MATCH_MP_TAC OPEN_IN_INTER THEN ASM_SIMP_TAC[OPEN_IN_DIFF; OPEN_IN_TOPSPACE] THEN ONCE_REWRITE_TAC[SET_RULE `x n > -- &1 <=> x n IN {y | y > -- &1}`] THEN MATCH_MP_TAC OPEN_IN_CONTINUOUS_MAP_PREIMAGE THEN EXISTS_TAC `euclideanreal` THEN REWRITE_TAC[REAL_OPEN_HALFSPACE_GT; GSYM REAL_OPEN_IN] THEN REWRITE_TAC[euclidean_space] THEN MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN SIMP_TAC[CONTINUOUS_MAP_PRODUCT_PROJECTION; IN_UNIV]; EXPAND_TAC "hi" THEN SIMP_TAC[SUBSET; IN_ELIM_THM; IN_DIFF; IN_INTER] THEN ASM_REWRITE_TAC[]]]; ALL_TAC] THEN ASM_REWRITE_TAC[] THEN ABBREV_TAC `q:int = p + &1` THEN SUBGOAL_THEN `p:int = q - &1` SUBST1_TAC THENL [EXPAND_TAC "q" THEN INT_ARITH_TAC; ALL_TAC] THEN ONCE_REWRITE_TAC[ISOMORPHIC_GROUP_SYM] THEN MATCH_MP_TAC ISOMORPHIC_RELATIVE_HOMOLOGY_GROUPS_RELBOUNDARY_CONTRACTIBLE THEN REWRITE_TAC[CONTRACTIBLE_EUCLIDEAN_SPACE] THEN CONJ_TAC THENL [UNDISCH_TAC `contractible_space(subtopology (euclidean_space(n+1)) hi)` THEN MATCH_MP_TAC EQ_IMP THEN MATCH_MP_TAC HOMEOMORPHIC_SPACE_CONTRACTIBILITY THEN REWRITE_TAC[homeomorphic_space] THEN REPEAT(EXISTS_TAC `\(x:num->real) i. if i = n + 1 then --(x i) else x i`) THEN REWRITE_TAC[homeomorphic_maps; FUN_EQ_THM] THEN SIMP_TAC[REAL_NEG_NEG] THEN GEN_REWRITE_TAC I [CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; MESON_TAC[]] THEN MAP_EVERY EXPAND_TAC ["hi"; "lo"] THEN REWRITE_TAC[euclidean_space; CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN REWRITE_TAC[GSYM euclidean_space; TOPSPACE_EUCLIDEAN_SPACE; TOPSPACE_SUBTOPOLOGY] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTER; IN_ELIM_THM] THEN REWRITE_TAC[MESON[REAL_NEG_EQ_0] `(if p then --x:real else y) = &0 <=> (if p then x else y) = &0`] THEN REWRITE_TAC[MESON[] `(if i = n then x n else x i) = x i`] THEN SUBGOAL_THEN `!(x:num->real) a. (\i. if i = n + 1 then a else x i) IN s <=> x IN s` (fun th -> REWRITE_TAC[th]) THENL [REPEAT GEN_TAC THEN EQ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[IMP_CONJ]) THEN MESON_TAC[ARITH_RULE `~(n + 1 <= n)`]; ALL_TAC] THEN REWRITE_TAC[REAL_ARITH `--x > (if p then &0 else -- &1) <=> x < (if p then &0 else &1)`] THEN REWRITE_TAC[REAL_ARITH `--x < (if p then &0 else &1) <=> x > (if p then &0 else -- &1)`] THEN SIMP_TAC[] THEN REWRITE_TAC[CONTINUOUS_MAP_COMPONENTWISE_UNIV] THEN REWRITE_TAC[TOPSPACE_PRODUCT_TOPOLOGY; o_DEF; TOPSPACE_EUCLIDEANREAL] THEN REWRITE_TAC[CARTESIAN_PRODUCT_UNIV; IN_UNIV] THEN REWRITE_TAC[euclidean_space] THEN CONJ_TAC THEN X_GEN_TAC `k:num` THEN REPEAT(MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY) THEN ASM_CASES_TAC `k = n + 1` THEN ASM_SIMP_TAC[CONTINUOUS_MAP_PRODUCT_PROJECTION; IN_UNIV; CONTINUOUS_MAP_REAL_NEG_EQ]; MAP_EVERY EXPAND_TAC ["lo"; "hi"] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; SUBSET; IN_ELIM_THM; IN_INTER; IN_DIFF] THEN SIMP_TAC[TAUT `p /\ p /\ q <=> p /\ q`] THEN CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN EXISTS_TAC `(\i. if i = n + 1 then -- &1 else &0)` THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SPACE; IN_ELIM_THM; IN_NUMSEG] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_ARITH_TAC]);; let ISOMORPHIC_RELATIVE_HOMOLOGY_GROUPS_EUCLIDEAN_COMPLEMENTS = prove (`!p n s t. closed_in (euclidean_space n) s /\ closed_in (euclidean_space n) t /\ (subtopology (euclidean_space n) s) homeomorphic_space (subtopology (euclidean_space n) t) ==> relative_homology_group (p,euclidean_space n,topspace(euclidean_space n) DIFF s) isomorphic_group relative_homology_group (p,euclidean_space n,topspace(euclidean_space n) DIFF t)`, let lemma1 = prove (`!f s m n. closed_in (euclidean_space m) s /\ continuous_map(subtopology (euclidean_space m) s,euclidean_space n) f ==> ?g. continuous_map(euclidean_space m,euclidean_space n) g /\ !x. x IN s ==> g x = f x`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `!i:num. ?g. continuous_map(euclidean_space m,euclideanreal) g /\ !x. x IN s ==> g x = f x i` MP_TAC THENL [GEN_TAC THEN MP_TAC(ISPEC `euclidean_space m` NORMAL_SPACE_EQ_TIETZE) THEN SIMP_TAC[METRIZABLE_IMP_NORMAL_SPACE; METRIZABLE_EUCLIDEAN_SPACE] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN UNDISCH_TAC `continuous_map(subtopology (euclidean_space m) s, euclidean_space n) f` THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV o RAND_CONV) [euclidean_space] THEN SIMP_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY; CONTINUOUS_MAP_COMPONENTWISE] THEN SIMP_TAC[IN_UNIV]; REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM; FORALL_AND_THM] THEN X_GEN_TAC `g:num->(num->real)->real` THEN STRIP_TAC THEN EXISTS_TAC `\x i. if i IN 1..n then (g:num->(num->real)->real) i x else &0` THEN UNDISCH_TAC `continuous_map(subtopology (euclidean_space m) s, euclidean_space n) f` THEN SUBST1_TAC(SPEC `n:num` euclidean_space) THEN REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM] THEN ASM_REWRITE_TAC[CONTINUOUS_MAP_COMPONENTWISE] THEN REWRITE_TAC[SET_RULE `IMAGE f s SUBSET P <=> !x. x IN s ==> P(f x)`] THEN REWRITE_TAC[EXTENSIONAL_UNIV; IN_UNIV] THEN STRIP_TAC THEN SIMP_TAC[] THEN CONJ_TAC THENL [X_GEN_TAC `i:num` THEN ASM_CASES_TAC `i IN 1..n` THEN ASM_REWRITE_TAC[ETA_AX; CONTINUOUS_MAP_REAL_CONST]; GEN_TAC THEN DISCH_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `i:num` THEN ASM_CASES_TAC `i IN 1..n` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_SUBSET) THEN RULE_ASSUM_TAC(REWRITE_RULE[TOPSPACE_SUBTOPOLOGY]) THEN ASM SET_TAC[]]]) in let lemma2 = prove (`!f s t m n. closed_in (euclidean_space m) s /\ closed_in (euclidean_space n) t /\ homeomorphic_map (subtopology (euclidean_space m) s,subtopology (euclidean_space n) t) f ==> ?g. homeomorphic_map(prod_topology (euclidean_space m) (euclidean_space n), prod_topology (euclidean_space n) (euclidean_space m)) g /\ !a. a IN s ==> g(a,(\i. &0)) = (f a,(\i. &0))`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHIC_MAP_MAPS]) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; homeomorphic_maps] THEN ASM_SIMP_TAC[CLOSED_IN_SUBSET; TOPSPACE_SUBTOPOLOGY; SET_RULE `s SUBSET t ==> t INTER s = s`] THEN X_GEN_TAC `g:(num->real)->(num->real)` THEN STRIP_TAC THEN MP_TAC(SPECL [`f:(num->real)->(num->real)`; `s:(num->real)->bool`; `m:num`; `n:num`] lemma1) THEN ANTS_TAC THENL [ASM_MESON_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `f':(num->real)->(num->real)` STRIP_ASSUME_TAC) THEN MP_TAC(SPECL [`g:(num->real)->(num->real)`; `t:(num->real)->bool`; `n:num`; `m:num`] lemma1) THEN ANTS_TAC THENL [ASM_MESON_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `g':(num->real)->(num->real)` STRIP_ASSUME_TAC) THEN MAP_EVERY ABBREV_TAC [`p:(num->real)#(num->real)->(num->real)#(num->real) = \(x,y). (x,(\i. y i + f' x i))`; `p':(num->real)#(num->real)->(num->real)#(num->real) = \(x,y). (x,(\i. y i - f' x i))`; `q:(num->real)#(num->real)->(num->real)#(num->real) = \(x,y). (x,(\i. y i + g' x i))`; `q':(num->real)#(num->real)->(num->real)#(num->real) = \(x,y). (x,(\i. y i - g' x i))`] THEN SUBGOAL_THEN `homeomorphic_maps (prod_topology (euclidean_space m) (euclidean_space n), prod_topology (euclidean_space m) (euclidean_space n)) (p,p') /\ homeomorphic_maps (prod_topology (euclidean_space n) (euclidean_space m), prod_topology (euclidean_space n) (euclidean_space m)) (q,q')` STRIP_ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["p"; "p'"; "q"; "q'"] THEN REWRITE_TAC[homeomorphic_maps; FORALL_PAIR_THM] THEN REWRITE_TAC[REAL_ARITH `(x + y) - y:real = x`] THEN REWRITE_TAC[REAL_ARITH `(x - y) + y:real = x`; ETA_AX] THEN REWRITE_TAC[CONTINUOUS_MAP_PAIRWISE; o_DEF] THEN ONCE_REWRITE_TAC[LAMBDA_PAIR_THM] THEN REWRITE_TAC[] THEN REWRITE_TAC[LAMBDA_PAIR] THEN REPEAT CONJ_TAC THEN TRY(MATCH_MP_TAC CONTINUOUS_MAP_EUCLIDEAN_SPACE_ADD THEN CONJ_TAC) THEN TRY(MATCH_MP_TAC CONTINUOUS_MAP_EUCLIDEAN_SPACE_SUB THEN CONJ_TAC) THEN REWRITE_TAC[ETA_AX; CONTINUOUS_MAP_FST; CONTINUOUS_MAP_SND] THEN ASM_REWRITE_TAC[CONTINUOUS_MAP_OF_FST]; ALL_TAC] THEN REWRITE_TAC[HOMEOMORPHIC_MAP_MAPS; LEFT_AND_EXISTS_THM] THEN MAP_EVERY EXISTS_TAC [`(q':(num->real)#(num->real)->(num->real)#(num->real)) o (\(x,y). y,x) o (p:(num->real)#(num->real)->(num->real)#(num->real))`; `(p':(num->real)#(num->real)->(num->real)#(num->real)) o (\(x,y). y,x) o (q:(num->real)#(num->real)->(num->real)#(num->real))`] THEN CONJ_TAC THENL [RULE_ASSUM_TAC(ONCE_REWRITE_RULE[HOMEOMORPHIC_MAPS_SYM]) THEN GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [o_ASSOC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] HOMEOMORPHIC_MAPS_COMPOSE)) THEN RULE_ASSUM_TAC(ONCE_REWRITE_RULE[HOMEOMORPHIC_MAPS_SYM]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HOMEOMORPHIC_MAPS_COMPOSE)) THEN REWRITE_TAC[homeomorphic_maps; FORALL_PAIR_THM] THEN REWRITE_TAC[CONTINUOUS_MAP_PAIRWISE; o_DEF] THEN ONCE_REWRITE_TAC[LAMBDA_PAIR_THM] THEN REWRITE_TAC[] THEN REWRITE_TAC[LAMBDA_PAIR] THEN REWRITE_TAC[ETA_AX] THEN REWRITE_TAC[CONTINUOUS_MAP_FST; CONTINUOUS_MAP_SND]; MAP_EVERY EXPAND_TAC ["q'"; "p"] THEN REWRITE_TAC[o_THM; REAL_ADD_LID] THEN ASM_SIMP_TAC[ETA_AX; PAIR_EQ] THEN REPEAT STRIP_TAC THEN ABS_TAC THEN REWRITE_TAC[REAL_SUB_0] THEN AP_THM_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONTINUOUS_MAP_IN_SUBTOPOLOGY])) THEN ASM_SIMP_TAC[CLOSED_IN_SUBSET; TOPSPACE_SUBTOPOLOGY; SET_RULE `s SUBSET t ==> t INTER s = s`] THEN ASM SET_TAC[]]) in REPEAT STRIP_TAC THEN MP_TAC(SPECL [`p:int`; `n:num`; `n:num`; `s:(num->real)->bool`] RELATIVE_HOMOLOGY_GROUP_EUCLIDEAN_COMPLEMENT_STEP) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] ISOMORPHIC_GROUP_TRANS) THEN ONCE_REWRITE_TAC[ISOMORPHIC_GROUP_SYM] THEN MP_TAC(SPECL [`p:int`; `n:num`; `n:num`; `t:(num->real)->bool`] RELATIVE_HOMOLOGY_GROUP_EUCLIDEAN_COMPLEMENT_STEP) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] ISOMORPHIC_GROUP_TRANS) THEN ONCE_REWRITE_TAC[ISOMORPHIC_GROUP_SYM] THEN REWRITE_TAC[GSYM MULT_2] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHIC_SPACE]) THEN DISCH_THEN(X_CHOOSE_THEN `f:(num->real)->(num->real)` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`f:(num->real)->(num->real)`; `s:(num->real)->bool`; `t:(num->real)->bool`; `n:num`; `n:num`] lemma2) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `g:(num->real)#(num->real)->(num->real)#(num->real)` THEN STRIP_TAC THEN MAP_EVERY ABBREV_TAC [`h:(num->real)->(num->real)#(num->real) = \x. (\i. if 1 <= i /\ i <= n then x i else &0), (\j. if 1 <= j /\ j <= n then x(n + j) else &0)`; `k:(num->real)#(num->real)->num->real = \(x,y) i. if 1 <= i /\ i <= 2 * n then if i <= n then x i else y(i - n) else &0`] THEN SUBGOAL_THEN `homeomorphic_maps (euclidean_space(2 * n), prod_topology (euclidean_space n) (euclidean_space n)) (h,k)` ASSUME_TAC THENL [REWRITE_TAC[homeomorphic_maps; TOPSPACE_EUCLIDEAN_SPACE; TOPSPACE_PROD_TOPOLOGY; FORALL_PAIR_THM; IN_CROSS] THEN MAP_EVERY EXPAND_TAC ["h"; "k"] THEN REWRITE_TAC[CONTINUOUS_MAP_PAIRWISE; o_DEF; CONTINUOUS_MAP_COMPONENTWISE_EUCLIDEAN_SPACE] THEN REWRITE_TAC[PAIR_EQ; FUN_EQ_THM] THEN SIMP_TAC[euclidean_space; CONTINUOUS_MAP_IN_SUBTOPOLOGY; CONTINUOUS_MAP_PRODUCT_PROJECTION; IN_UNIV; CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_COMPONENTWISE_UNIV] THEN REWRITE_TAC[GSYM CONJ_ASSOC] THEN CONJ_TAC THENL [X_GEN_TAC `k:num` THEN REWRITE_TAC[LAMBDA_PAIR] THEN ASM_CASES_TAC `1 <= k /\ k <= 2 * n` THEN ASM_REWRITE_TAC[CONTINUOUS_MAP_REAL_CONST] THEN ASM_CASES_TAC `k:num <= n` THEN ASM_REWRITE_TAC[CONTINUOUS_MAP_OF_FST; CONTINUOUS_MAP_OF_SND] THEN SIMP_TAC[CONTINUOUS_MAP_PRODUCT_PROJECTION; IN_UNIV; CONTINUOUS_MAP_FROM_SUBTOPOLOGY]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; TOPSPACE_PROD_TOPOLOGY] THEN REWRITE_TAC[GSYM euclidean_space; TOPSPACE_EUCLIDEAN_SPACE] THEN REWRITE_TAC[FORALL_PAIR_THM; IN_CROSS; IN_ELIM_THM] THEN REWRITE_TAC[IN_NUMSEG] THEN REPEAT STRIP_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN TRY ASM_ARITH_TAC THEN REWRITE_TAC[REAL_ARITH `&0 = x <=> x = &0`] THEN TRY(AP_TERM_TAC THEN ASM_ARITH_TAC) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC]; ALL_TAC] THEN ABBREV_TAC `g' = (k:(num->real)#(num->real)->num->real) o g o (h:(num->real)->(num->real)#(num->real))` THEN REWRITE_TAC[isomorphic_group] THEN EXISTS_TAC `hom_induced (p + &n) (euclidean_space(2 * n),topspace(euclidean_space(2 * n)) DIFF s) (euclidean_space(2 * n),topspace(euclidean_space(2 * n)) DIFF t) g'` THEN MATCH_MP_TAC HOMEOMORPHIC_MAP_RELATIVE_HOMOLOGY_GROUP_ISOMORPHISM THEN REWRITE_TAC[SUBSET_DIFF] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [EXPAND_TAC "g'" THEN MATCH_MP_TAC HOMEOMORPHIC_MAP_COMPOSE THEN ASM_MESON_TAC[HOMEOMORPHIC_MAPS_MAP; HOMEOMORPHIC_MAP_COMPOSE]; DISCH_TAC THEN MATCH_MP_TAC(SET_RULE `s SUBSET u /\ t SUBSET u /\ IMAGE g u = u /\ (!x y. x IN u /\ y IN u /\ g x = g y ==> x = y) /\ IMAGE g s = t ==> IMAGE g (u DIFF s) = u DIFF t`) THEN GEN_REWRITE_TAC I [CONJ_ASSOC] THEN CONJ_TAC THENL [CONJ_TAC THENL [UNDISCH_TAC `closed_in (euclidean_space n) s`; UNDISCH_TAC `closed_in (euclidean_space n) t`] THEN DISCH_THEN(MP_TAC o MATCH_MP CLOSED_IN_SUBSET) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS) THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SPACE; IN_NUMSEG] THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN MESON_TAC[ARITH_RULE `i <= n ==> i <= 2 * n`]; ALL_TAC] THEN GEN_REWRITE_TAC I [CONJ_ASSOC] THEN CONJ_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[HOMEOMORPHIC_EQ_EVERYTHING_MAP]) THEN ASM SET_TAC[]; SUBGOAL_THEN `t = IMAGE (f:(num->real)->num->real) s` SUBST1_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[HOMEOMORPHIC_EQ_EVERYTHING_MAP; TOPSPACE_SUBTOPOLOGY]) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_SUBSET)) THEN ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> f x = g x) ==> IMAGE f s = IMAGE g s`) THEN EXPAND_TAC "g'" THEN REWRITE_TAC[o_THM] THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> k(h(f x)) = f x) /\ (!x. x IN s ==> g(h x) = h(f x)) ==> !x. x IN s ==> k(g(h x)) = f x`) THEN CONJ_TAC THENL [X_GEN_TAC `x:num->real` THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [homeomorphic_maps]) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN SUBGOAL_THEN `f (x:num->real) IN topspace(euclidean_space n)` MP_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[HOMEOMORPHIC_EQ_EVERYTHING_MAP; TOPSPACE_SUBTOPOLOGY]) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_SUBSET)) THEN ASM SET_TAC[]; REWRITE_TAC[TOPSPACE_EUCLIDEAN_SPACE; IN_ELIM_THM; IN_NUMSEG] THEN MESON_TAC[ARITH_RULE `i <= n ==> i <= 2 * n`]]; X_GEN_TAC `x:num->real` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:num->real`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC EQ_IMP THEN BINOP_TAC THENL [AP_TERM_TAC THEN EXPAND_TAC "h" THEN REWRITE_TAC[PAIR_EQ; FUN_EQ_THM] THEN SUBGOAL_THEN `(x:num->real) IN topspace(euclidean_space n)` MP_TAC THENL [ASM_MESON_TAC[CLOSED_IN_SUBSET; SUBSET]; ALL_TAC]; EXPAND_TAC "h" THEN REWRITE_TAC[PAIR_EQ; FUN_EQ_THM] THEN SUBGOAL_THEN `f (x:num->real) IN topspace(euclidean_space n)` MP_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[HOMEOMORPHIC_EQ_EVERYTHING_MAP; TOPSPACE_SUBTOPOLOGY]) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_SUBSET)) THEN ASM SET_TAC[]; ALL_TAC]] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SPACE; IN_ELIM_THM] THEN REWRITE_TAC[IN_NUMSEG] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[REAL_ARITH `&0 = x <=> x = &0`] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC]]]);; let INVARIANCE_OF_DIMENSION_CLOSED_IN_EUCLIDEAN_SPACE = prove (`!n s. closed_in (euclidean_space n) s ==> (subtopology (euclidean_space n) s homeomorphic_space euclidean_space n <=> s = topspace(euclidean_space n))`, let lemma = prove (`!n s t:(num->real)->bool. s SUBSET t /\ ~(s = {}) /\ t SUBSET topspace(euclidean_space n) /\ (!a b u. a IN s /\ b IN t /\ &0 < u /\ u < &1 ==> (\i. (&1 - u) * a i + u * b i) IN s) ==> path_connected_in (euclidean_space n) t`, REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[path_connected_in] THEN SUBGOAL_THEN `t SUBSET topspace(euclidean_space n)` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[PATH_CONNECTED_SPACE_IFF_PATH_COMPONENT] THEN ASM_SIMP_TAC[TOPSPACE_SUBTOPOLOGY_SUBSET] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_THEN `a:num->real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `!x. x IN t ==> path_component_of (subtopology (euclidean_space n) t) a x` MP_TAC THENL [ALL_TAC; ASM_MESON_TAC[PATH_COMPONENT_OF_SYM; PATH_COMPONENT_OF_TRANS]] THEN X_GEN_TAC `b:num->real` THEN DISCH_TAC THEN REWRITE_TAC[path_component_of] THEN EXISTS_TAC `(\t i. (&1 - t) * a i + t * b i):real->num->real` THEN REWRITE_TAC[REAL_SUB_REFL; REAL_SUB_RZERO; REAL_MUL_LZERO] THEN REWRITE_TAC[REAL_MUL_LID; REAL_ADD_LID; REAL_ADD_RID; ETA_AX] THEN REWRITE_TAC[euclidean_space; SUBTOPOLOGY_SUBTOPOLOGY] THEN REWRITE_TAC[PATH_IN_SUBTOPOLOGY] THEN ASM_SIMP_TAC[GSYM TOPSPACE_SUBTOPOLOGY] THEN CONJ_TAC THENL [REWRITE_TAC[path_in; CONTINUOUS_MAP_COMPONENTWISE_UNIV] THEN GEN_TAC THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_ADD THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_RMUL THEN MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN REWRITE_TAC[CONTINUOUS_MAP_ID] THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_SUB THEN REWRITE_TAC[CONTINUOUS_MAP_ID; CONTINUOUS_MAP_REAL_CONST]; ASM_SIMP_TAC[IN_REAL_INTERVAL; GSYM TOPSPACE_EUCLIDEAN_SPACE; SET_RULE `t SUBSET u ==> u INTER t = t`] THEN REWRITE_TAC[REAL_ARITH `&0 <= x /\ x <= &1 <=> x = &0 \/ x = &1 \/ &0 < x /\ x < &1`] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[REAL_SUB_REFL; REAL_SUB_RZERO; REAL_MUL_LZERO] THEN REWRITE_TAC[REAL_MUL_LID; REAL_ADD_LID; REAL_ADD_RID; ETA_AX] THEN ASM SET_TAC[]]) in REPEAT STRIP_TAC THEN EQ_TAC THEN SIMP_TAC[SUBTOPOLOGY_TOPSPACE; HOMEOMORPHIC_SPACE_REFL] THEN DISCH_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_SUBSET) THEN MATCH_MP_TAC(SET_RULE `s SUBSET t /\ (!x. ~(~(x IN s) /\ x IN t)) ==> s = t`) THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `a:num->real` THEN STRIP_TAC THEN SUBGOAL_THEN `closed_in (euclidean_space (n + 1)) (topspace(euclidean_space n))` ASSUME_TAC THENL [REWRITE_TAC[euclidean_space] THEN MATCH_MP_TAC CLOSED_IN_SUBSET_TOPSPACE THEN REWRITE_TAC[GSYM euclidean_space; CLOSED_IN_EUCLIDEAN_SPACE] THEN REWRITE_TAC[GSYM TOPSPACE_EUCLIDEAN_SPACE; SUBSET_EUCLIDEAN_SPACE] THEN ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `subtopology (euclidean_space(n + 1)) (topspace(euclidean_space n)) = euclidean_space n` ASSUME_TAC THENL [REWRITE_TAC[euclidean_space; SUBTOPOLOGY_SUBTOPOLOGY] THEN AP_TERM_TAC THEN REWRITE_TAC[GSYM euclidean_space] THEN REWRITE_TAC[GSYM TOPSPACE_EUCLIDEAN_SPACE] THEN REWRITE_TAC[SET_RULE `t INTER s = s <=> s SUBSET t`] THEN REWRITE_TAC[SUBSET_EUCLIDEAN_SPACE] THEN ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `closed_in (euclidean_space(n + 1)) s` ASSUME_TAC THENL [ASM_MESON_TAC[CLOSED_IN_TRANS_FULL]; ALL_TAC] THEN MP_TAC(ISPECL [`subtopology (euclidean_space (n + 1)) (topspace(euclidean_space(n + 1)) DIFF s)`; `subtopology (euclidean_space (n + 1)) (topspace(euclidean_space(n + 1)) DIFF topspace(euclidean_space n))`] ISOMORPHIC_HOMOLOGY_IMP_PATH_CONNECTEDNESS) THEN MATCH_MP_TAC(TAUT `p /\ ~r /\ q ==> (p ==> (q <=> r)) ==> F`) THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC(REWRITE_RULE[RIGHT_IMP_FORALL_THM] ISOMORPHIC_RELATIVE_CONTRACTIBLE_SPACE_IMP_HOMOLOGY_GROUPS) THEN REWRITE_TAC[CONTRACTIBLE_EUCLIDEAN_SPACE; SUBSET_DIFF] THEN CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `t SUBSET s /\ s SUBSET u /\ ~(u SUBSET s) ==> (u DIFF t = {} <=> u DIFF s = {})`) THEN ASM_REWRITE_TAC[SUBSET_EUCLIDEAN_SPACE] THEN ARITH_TAC; GEN_TAC THEN MATCH_MP_TAC ISOMORPHIC_RELATIVE_HOMOLOGY_GROUPS_EUCLIDEAN_COMPLEMENTS THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `subtopology (euclidean_space (n + 1)) s = subtopology (euclidean_space n) s` (fun th -> ASM_REWRITE_TAC[th]) THEN REWRITE_TAC[euclidean_space; SUBTOPOLOGY_SUBTOPOLOGY] THEN AP_TERM_TAC THEN REWRITE_TAC[GSYM TOPSPACE_EUCLIDEAN_SPACE] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t /\ t SUBSET u ==> u INTER s = t INTER s`) THEN ASM_REWRITE_TAC[SUBSET_EUCLIDEAN_SPACE] THEN ARITH_TAC]; REWRITE_TAC[GSYM PATH_CONNECTED_IN_TOPSPACE] THEN DISCH_THEN(MP_TAC o ISPECL [`\x:num->real. x(n + 1)`; `euclideanreal`] o MATCH_MP(REWRITE_RULE[IMP_CONJ_ALT] PATH_CONNECTED_IN_CONTINUOUS_MAP_IMAGE)) THEN SIMP_TAC[euclidean_space; CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_PRODUCT_PROJECTION; IN_UNIV] THEN REWRITE_TAC[GSYM euclidean_space; TOPSPACE_EUCLIDEAN_SPACE] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; TOPSPACE_EUCLIDEAN_SPACE] THEN REWRITE_TAC[PATH_CONNECTED_IN_EUCLIDEANREAL; is_realinterval] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_INTER; IN_DIFF; IN_ELIM_THM; IN_IMAGE] THEN REWRITE_TAC[TAUT `p /\ p /\ ~q <=> p /\ ~q`] THEN DISCH_THEN(MP_TAC o SPEC `(\i. if i = n + 1 then -- &1 else &0):num->real`) THEN REWRITE_TAC[NOT_IMP; IN_NUMSEG] THEN REPEAT CONJ_TAC THENL [GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ARITH_TAC; DISCH_THEN(MP_TAC o SPEC `n + 1`) THEN REWRITE_TAC[] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ARITH_TAC; DISCH_THEN(MP_TAC o SPEC `(\i. if i = n + 1 then &1 else &0):num->real`)] THEN REWRITE_TAC[NOT_IMP] THEN REPEAT CONJ_TAC THENL [GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ARITH_TAC; DISCH_THEN(MP_TAC o SPEC `n + 1`) THEN REWRITE_TAC[] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ARITH_TAC; DISCH_THEN(MP_TAC o SPEC `&0:real`)] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[ARITH_RULE `i <= n + 1 <=> i = n + 1 \/ i <= n`] THEN MESON_TAC[]; REWRITE_TAC[GSYM PATH_CONNECTED_IN_TOPSPACE] THEN REWRITE_TAC[PATH_CONNECTED_IN_SUBTOPOLOGY; TOPSPACE_SUBTOPOLOGY] THEN REWRITE_TAC[SUBSET_REFL; SET_RULE `u INTER (u DIFF s) = u DIFF s`] THEN SUBGOAL_THEN `!t. t = t INTER {x:num->real | x(n+1) <= &0} UNION t INTER {x:num->real | x(n+1) >= &0}` (fun th -> GEN_REWRITE_TAC RAND_CONV [th]) THENL [GEN_TAC THEN MATCH_MP_TAC(SET_RULE `(!x. P x \/ Q x) ==> t = t INTER {x | P x} UNION t INTER {x | Q x}`) THEN REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC PATH_CONNECTED_IN_UNION THEN REWRITE_TAC[CONJ_ASSOC; SET_RULE `(s INTER {x | P x}) INTER (s INTER {x | Q x}) = s INTER {x | P x /\ Q x}`] THEN SUBGOAL_THEN `topspace(euclidean_space(n + 1)) INTER {x | x(n+1) = &0} = topspace(euclidean_space n)` ASSUME_TAC THENL [REWRITE_TAC[TOPSPACE_EUCLIDEAN_SPACE; IN_NUMSEG] THEN REWRITE_TAC[EXTENSION; IN_INTER; IN_ELIM_THM; ARITH_RULE `1 <= i /\ i <= n + 1 <=> 1 <= i /\ i <= n \/ i = n + 1`] THEN MESON_TAC[ARITH_RULE `~(n + 1 <= n)`]; ALL_TAC] THEN REWRITE_TAC[REAL_ARITH `x <= &0 /\ x >= &0 <=> x = &0`] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN CONJ_TAC THEN MATCH_MP_TAC lemma THENL [EXISTS_TAC `topspace(euclidean_space(n + 1)) INTER {x | x(n+1) < &0}`; EXISTS_TAC `topspace(euclidean_space(n + 1)) INTER {x | x(n+1) > &0}`] THEN REWRITE_TAC[SUBSET; IN_INTER; TOPSPACE_EUCLIDEAN_SPACE; IN_ELIM_THM; IN_DIFF; real_gt; real_ge; GSYM MEMBER_NOT_EMPTY] THEN SIMP_TAC[REAL_LT_IMP_LE; REAL_MUL_RZERO; REAL_ADD_LID] THEN (CONJ_TAC THENL [MP_TAC(REAL_LT_REFL) THEN ASM SET_TAC[]; ALL_TAC]) THEN CONJ_TAC THENL [EXISTS_TAC `(\i. if i = n + 1 then -- &1 else &0):num->real` THEN REWRITE_TAC[IN_NUMSEG] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ARITH_TAC; REWRITE_TAC[GSYM REAL_NEG_GE0; GSYM REAL_NEG_GT0] THEN REWRITE_TAC[GSYM REAL_MUL_RNEG; REAL_NEG_ADD] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC(REAL_ARITH `&0 < a /\ &0 <= b ==> &0 < a + b`) THEN ASM_SIMP_TAC[REAL_LE_MUL_EQ; REAL_LT_MUL_EQ; REAL_SUB_LT]; EXISTS_TAC `(\i. if i = n + 1 then &1 else &0):num->real` THEN REWRITE_TAC[IN_NUMSEG] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ARITH_TAC; REPEAT STRIP_TAC THEN MATCH_MP_TAC(REAL_ARITH `&0 < a /\ &0 <= b ==> &0 < a + b`) THEN ASM_SIMP_TAC[REAL_LE_MUL_EQ; REAL_LT_MUL_EQ; REAL_SUB_LT]]]);; let ISOMORPHIC_HOMOLOGY_GROUPS_EUCLIDEAN_COMPLEMENTS = prove (`!p n s t. closed_in (euclidean_space n) s /\ closed_in (euclidean_space n) t /\ (subtopology (euclidean_space n) s) homeomorphic_space (subtopology (euclidean_space n) t) ==> homology_group(p,subtopology (euclidean_space n) (topspace(euclidean_space n) DIFF s)) isomorphic_group homology_group(p,subtopology (euclidean_space n) (topspace(euclidean_space n) DIFF t))`, REPEAT STRIP_TAC THEN MATCH_MP_TAC(REWRITE_RULE[RIGHT_IMP_FORALL_THM] ISOMORPHIC_RELATIVE_CONTRACTIBLE_SPACE_IMP_HOMOLOGY_GROUPS) THEN ASM_SIMP_TAC[ISOMORPHIC_RELATIVE_HOMOLOGY_GROUPS_EUCLIDEAN_COMPLEMENTS] THEN REWRITE_TAC[CONTRACTIBLE_EUCLIDEAN_SPACE; SUBSET_DIFF] THEN ASM_SIMP_TAC[CLOSED_IN_SUBSET; SET_RULE `s SUBSET u ==> (u DIFF s = {} <=> s = u)`] THEN ASM_SIMP_TAC[GSYM INVARIANCE_OF_DIMENSION_CLOSED_IN_EUCLIDEAN_SPACE] THEN EQ_TAC THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] HOMEOMORPHIC_SPACE_TRANS) THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SPACE_SYM] THEN ASM_REWRITE_TAC[]);; let CARD_EQ_PATH_COMPONENTS_EUCLIDEAN_COMPLEMENTS = prove (`!n s t. closed_in (euclidean_space n) s /\ closed_in (euclidean_space n) t /\ (subtopology (euclidean_space n) s) homeomorphic_space (subtopology (euclidean_space n) t) ==> path_components_of (subtopology (euclidean_space n) (topspace(euclidean_space n) DIFF s)) =_c path_components_of (subtopology (euclidean_space n) (topspace(euclidean_space n) DIFF t))`, REPEAT STRIP_TAC THEN MATCH_MP_TAC ISOMORPHIC_HOMOLOGY_IMP_PATH_COMPONENTS THEN ASM_SIMP_TAC[ISOMORPHIC_HOMOLOGY_GROUPS_EUCLIDEAN_COMPLEMENTS]);; let PATH_CONNECTED_IN_EUCLIDEAN_COMPLEMENTS = prove (`!n s t. closed_in (euclidean_space n) s /\ closed_in (euclidean_space n) t /\ (subtopology (euclidean_space n) s) homeomorphic_space (subtopology (euclidean_space n) t) ==> (path_connected_in (euclidean_space n) (topspace(euclidean_space n) DIFF s) <=> path_connected_in (euclidean_space n) (topspace(euclidean_space n) DIFF t))`, REPEAT STRIP_TAC THEN REWRITE_TAC[path_connected_in; SUBSET_DIFF] THEN MATCH_MP_TAC ISOMORPHIC_HOMOLOGY_IMP_PATH_CONNECTEDNESS THEN ASM_SIMP_TAC[ISOMORPHIC_HOMOLOGY_GROUPS_EUCLIDEAN_COMPLEMENTS]);; let CARD_EQ_CONNECTED_COMPONENTS_EUCLIDEAN_COMPLEMENTS = prove (`!n s t. closed_in (euclidean_space n) s /\ closed_in (euclidean_space n) t /\ (subtopology (euclidean_space n) s) homeomorphic_space (subtopology (euclidean_space n) t) ==> connected_components_of (subtopology (euclidean_space n) (topspace(euclidean_space n) DIFF s)) =_c connected_components_of (subtopology (euclidean_space n) (topspace(euclidean_space n) DIFF t))`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP CARD_EQ_PATH_COMPONENTS_EUCLIDEAN_COMPLEMENTS) THEN MATCH_MP_TAC EQ_IMP THEN BINOP_TAC THEN MATCH_MP_TAC PATH_COMPONENTS_EQ_CONNECTED_COMPONENTS_OF THEN MATCH_MP_TAC LOCALLY_PATH_CONNECTED_SPACE_OPEN_SUBSET THEN REWRITE_TAC[LOCALLY_PATH_CONNECTED_EUCLIDEAN_SPACE] THEN ASM_SIMP_TAC[OPEN_IN_DIFF; OPEN_IN_TOPSPACE]);; let CONNECTED_IN_EUCLIDEAN_COMPLEMENTS = prove (`!n s t. closed_in (euclidean_space n) s /\ closed_in (euclidean_space n) t /\ (subtopology (euclidean_space n) s) homeomorphic_space (subtopology (euclidean_space n) t) ==> (connected_in (euclidean_space n) (topspace(euclidean_space n) DIFF s) <=> connected_in (euclidean_space n) (topspace(euclidean_space n) DIFF t))`, REPEAT STRIP_TAC THEN REWRITE_TAC[connected_in; SUBSET_DIFF] THEN REWRITE_TAC[CONNECTED_SPACE_IFF_COMPONENTS_SUBSET_SING] THEN REWRITE_TAC[GSYM CARD_LE_SING] THEN MATCH_MP_TAC CARD_LE_CONG THEN REWRITE_TAC[CARD_EQ_REFL] THEN MATCH_MP_TAC CARD_EQ_CONNECTED_COMPONENTS_EUCLIDEAN_COMPLEMENTS THEN ASM_REWRITE_TAC[]);; let INVARIANCE_OF_DIMENSION_EUCLIDEAN_SPACE = prove (`!m n. euclidean_space m homeomorphic_space euclidean_space n <=> m = n`, MATCH_MP_TAC WLOG_LT THEN REWRITE_TAC[HOMEOMORPHIC_SPACE_REFL] THEN CONJ_TAC THENL [REWRITE_TAC[HOMEOMORPHIC_SPACE_SYM; EQ_SYM_EQ]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN DISCH_TAC THEN ASM_SIMP_TAC[LT_IMP_NE] THEN SUBGOAL_THEN `euclidean_space m = subtopology (euclidean_space n) (topspace(euclidean_space m))` SUBST1_TAC THENL [REWRITE_TAC[euclidean_space; TOPSPACE_EUCLIDEAN_SPACE] THEN REWRITE_TAC[SUBTOPOLOGY_SUBTOPOLOGY] THEN AP_TERM_TAC THEN REWRITE_TAC[SET_RULE `s = t INTER s <=> s SUBSET t`] THEN REWRITE_TAC[GSYM TOPSPACE_EUCLIDEAN_SPACE; SUBSET_EUCLIDEAN_SPACE] THEN ASM_SIMP_TAC[LT_IMP_LE]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhand o rand) INVARIANCE_OF_DIMENSION_CLOSED_IN_EUCLIDEAN_SPACE o rand o snd) THEN ANTS_TAC THENL [REWRITE_TAC[euclidean_space] THEN MATCH_MP_TAC CLOSED_IN_SUBSET_TOPSPACE THEN REWRITE_TAC[GSYM euclidean_space; CLOSED_IN_EUCLIDEAN_SPACE] THEN REWRITE_TAC[GSYM TOPSPACE_EUCLIDEAN_SPACE; SUBSET_EUCLIDEAN_SPACE] THEN ASM_SIMP_TAC[LT_IMP_LE]; DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; SUBSET_EUCLIDEAN_SPACE] THEN ASM_ARITH_TAC]);; let INVARIANCE_OF_DOMAIN_EUCLIDEAN_SPACE = prove (`!n u f. open_in (euclidean_space n) u /\ continuous_map(subtopology (euclidean_space n) u,euclidean_space n) f /\ (!x y. x IN u /\ y IN u /\ f x = f y ==> x = y) ==> open_in (euclidean_space n) (IMAGE f u)`, let ISOMORPHIC_HOMOLOGY_GROUPS_EUCLIDEAN_COMPLEMENTS = prove (`!p n s t. closed_in (euclidean_space n) s /\ closed_in (euclidean_space n) t /\ ~(s = topspace(euclidean_space n)) /\ ~(t = topspace(euclidean_space n)) /\ (subtopology (euclidean_space n) s) homeomorphic_space (subtopology (euclidean_space n) t) ==> homology_group(p,subtopology (euclidean_space n) (topspace(euclidean_space n) DIFF s)) isomorphic_group homology_group(p,subtopology (euclidean_space n) (topspace(euclidean_space n) DIFF t))`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `p:int = &0` THEN ASM_REWRITE_TAC[] THENL [MP_TAC(ISPECL [`euclidean_space n`; `topspace(euclidean_space n) DIFF s`] (CONJUNCT1 ISOMORPHIC_GROUP_HOMOLOGY_CONTRACTIBLE_SPACE_SUBTOPOLOGY)) THEN REWRITE_TAC[CONTRACTIBLE_EUCLIDEAN_SPACE; SUBSET_DIFF] THEN ASM_SIMP_TAC[CLOSED_IN_SUBSET; SET_RULE `s SUBSET t /\ ~(s = t) ==> ~(t DIFF s = {})`] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] ISOMORPHIC_GROUP_TRANS) THEN ONCE_REWRITE_TAC[ISOMORPHIC_GROUP_SYM] THEN MP_TAC(ISPECL [`euclidean_space n`; `topspace(euclidean_space n) DIFF t`] (CONJUNCT1 ISOMORPHIC_GROUP_HOMOLOGY_CONTRACTIBLE_SPACE_SUBTOPOLOGY)) THEN REWRITE_TAC[CONTRACTIBLE_EUCLIDEAN_SPACE; SUBSET_DIFF] THEN ASM_SIMP_TAC[CLOSED_IN_SUBSET; SET_RULE `s SUBSET t /\ ~(s = t) ==> ~(t DIFF s = {})`] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] ISOMORPHIC_GROUP_TRANS) THEN MATCH_MP_TAC ISOMORPHIC_GROUP_PROD_GROUPS THEN REWRITE_TAC[ISOMORPHIC_GROUP_REFL] THEN ONCE_REWRITE_TAC[ISOMORPHIC_GROUP_SYM] THEN MATCH_MP_TAC ISOMORPHIC_RELATIVE_HOMOLOGY_GROUPS_EUCLIDEAN_COMPLEMENTS THEN ASM_REWRITE_TAC[]; MP_TAC(ISPECL [`p:int`; `euclidean_space n`; `topspace(euclidean_space n) DIFF s`] (CONJUNCT2 ISOMORPHIC_GROUP_HOMOLOGY_CONTRACTIBLE_SPACE_SUBTOPOLOGY)) THEN REWRITE_TAC[CONTRACTIBLE_EUCLIDEAN_SPACE; SUBSET_DIFF] THEN ASM_SIMP_TAC[CLOSED_IN_SUBSET; SET_RULE `s SUBSET t /\ ~(s = t) ==> ~(t DIFF s = {})`] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] ISOMORPHIC_GROUP_TRANS) THEN ONCE_REWRITE_TAC[ISOMORPHIC_GROUP_SYM] THEN MP_TAC(ISPECL [`p:int`; `euclidean_space n`; `topspace(euclidean_space n) DIFF t`] (CONJUNCT2 ISOMORPHIC_GROUP_HOMOLOGY_CONTRACTIBLE_SPACE_SUBTOPOLOGY)) THEN REWRITE_TAC[CONTRACTIBLE_EUCLIDEAN_SPACE; SUBSET_DIFF] THEN ASM_SIMP_TAC[CLOSED_IN_SUBSET; SET_RULE `s SUBSET t /\ ~(s = t) ==> ~(t DIFF s = {})`] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] ISOMORPHIC_GROUP_TRANS) THEN ONCE_REWRITE_TAC[ISOMORPHIC_GROUP_SYM] THEN MATCH_MP_TAC ISOMORPHIC_RELATIVE_HOMOLOGY_GROUPS_EUCLIDEAN_COMPLEMENTS THEN ASM_REWRITE_TAC[]]) in let CARD_EQ_PATH_COMPONENTS_EUCLIDEAN_COMPLEMENTS = prove (`!n s t. closed_in (euclidean_space n) s /\ closed_in (euclidean_space n) t /\ ~(s = topspace(euclidean_space n)) /\ ~(t = topspace(euclidean_space n)) /\ (subtopology (euclidean_space n) s) homeomorphic_space (subtopology (euclidean_space n) t) ==> path_components_of (subtopology (euclidean_space n) (topspace(euclidean_space n) DIFF s)) =_c path_components_of (subtopology (euclidean_space n) (topspace(euclidean_space n) DIFF t))`, REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM ISOMORPHIC_FREE_ABELIAN_GROUPS] THEN MP_TAC(ISPEC `subtopology (euclidean_space n) (topspace (euclidean_space n) DIFF s)` ZEROTH_HOMOLOGY_GROUP) THEN GEN_REWRITE_TAC LAND_CONV [ISOMORPHIC_GROUP_SYM] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] ISOMORPHIC_GROUP_TRANS) THEN GEN_REWRITE_TAC I [ISOMORPHIC_GROUP_SYM] THEN MP_TAC(ISPEC `subtopology (euclidean_space n) (topspace (euclidean_space n) DIFF t)` ZEROTH_HOMOLOGY_GROUP) THEN GEN_REWRITE_TAC LAND_CONV [ISOMORPHIC_GROUP_SYM] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] ISOMORPHIC_GROUP_TRANS) THEN GEN_REWRITE_TAC I [ISOMORPHIC_GROUP_SYM] THEN MATCH_MP_TAC ISOMORPHIC_HOMOLOGY_GROUPS_EUCLIDEAN_COMPLEMENTS THEN ASM_REWRITE_TAC[]) in let PATH_CONNECTED_IN_EUCLIDEAN_COMPLEMENTS = prove (`!n s t. closed_in (euclidean_space n) s /\ closed_in (euclidean_space n) t /\ ~(s = topspace(euclidean_space n)) /\ ~(t = topspace(euclidean_space n)) /\ (subtopology (euclidean_space n) s) homeomorphic_space (subtopology (euclidean_space n) t) ==> (path_connected_in (euclidean_space n) (topspace(euclidean_space n) DIFF s) <=> path_connected_in (euclidean_space n) (topspace(euclidean_space n) DIFF t))`, REPEAT STRIP_TAC THEN REWRITE_TAC[path_connected_in; SUBSET_DIFF] THEN REWRITE_TAC[PATH_CONNECTED_SPACE_IFF_COMPONENTS_SUBSET_SING] THEN REWRITE_TAC[GSYM CARD_LE_SING] THEN MATCH_MP_TAC CARD_LE_CONG THEN REWRITE_TAC[CARD_EQ_REFL] THEN MATCH_MP_TAC CARD_EQ_PATH_COMPONENTS_EUCLIDEAN_COMPLEMENTS THEN ASM_REWRITE_TAC[]) in let BIGLEMMA = prove (`!n h s. ~(n = 0) /\ compact_in (euclidean_space n) s /\ continuous_map (subtopology (euclidean_space n) s,euclidean_space n) h /\ (!x y. x IN s /\ y IN s ==> (h x = h y <=> x = y)) ==> (path_connected_in (euclidean_space n) (topspace(euclidean_space n) DIFF IMAGE h s) <=> path_connected_in (euclidean_space n) (topspace(euclidean_space n) DIFF s))`, REPEAT STRIP_TAC THEN MATCH_MP_TAC PATH_CONNECTED_IN_EUCLIDEAN_COMPLEMENTS THEN REWRITE_TAC[CONJ_ASSOC] THEN MATCH_MP_TAC(TAUT `q /\ (q ==> p) ==> p /\ q`) THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[HOMEOMORPHIC_SPACE_SYM] THEN REWRITE_TAC[HOMEOMORPHIC_SPACE] THEN EXISTS_TAC `h:(num->real)->(num->real)` THEN MATCH_MP_TAC CONTINUOUS_IMP_HOMEOMORPHIC_MAP THEN ASM_REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN ASM_SIMP_TAC[COMPACT_IN_SUBSET_TOPSPACE; TOPSPACE_SUBTOPOLOGY_SUBSET] THEN ASM_SIMP_TAC[COMPACT_SPACE_SUBTOPOLOGY; SUBSET_REFL] THEN SIMP_TAC[HAUSDORFF_EUCLIDEAN_SPACE; HAUSDORFF_SPACE_SUBTOPOLOGY] THEN RULE_ASSUM_TAC(REWRITE_RULE[continuous_map; TOPSPACE_SUBTOPOLOGY]) THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP COMPACT_IN_SUBSET_TOPSPACE) THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY] THEN ASM SET_TAC[]; DISCH_THEN(MP_TAC o MATCH_MP HOMEOMORPHIC_COMPACT_SPACE)] THEN ASM_SIMP_TAC[COMPACT_SPACE_SUBTOPOLOGY] THEN W(MP_TAC o PART_MATCH (rand o rand) COMPACT_IN_SUBSPACE o lhand o snd) THEN SUBGOAL_THEN `IMAGE (h:(num->real)->(num->real)) s SUBSET topspace(euclidean_space n)` ASSUME_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[continuous_map; TOPSPACE_SUBTOPOLOGY]) THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP COMPACT_IN_SUBSET_TOPSPACE) THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY] THEN ASM SET_TAC[]; ASM_REWRITE_TAC[] THEN DISCH_THEN(SUBST1_TAC o SYM)] THEN DISCH_TAC THEN ASM_SIMP_TAC[COMPACT_IN_IMP_CLOSED_IN; HAUSDORFF_EUCLIDEAN_SPACE] THEN CONJ_TAC THEN DISCH_THEN(MP_TAC o AP_TERM `compact_in (euclidean_space n)`) THEN ASM_REWRITE_TAC[GSYM compact_space; COMPACT_EUCLIDEAN_SPACE]) in let lemma = prove (`!u c d:A->bool. (?t. t IN u /\ c SUBSET t) /\ (?t. t IN u /\ d SUBSET t) /\ UNIONS u = c UNION d /\ (!t. t IN u ==> ~(t = {})) /\ pairwise DISJOINT u /\ ~(?t. u SUBSET {t}) ==> c IN u`, REPEAT GEN_TAC THEN REWRITE_TAC[IMP_CONJ; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `c':A->bool` THEN REPEAT DISCH_TAC THEN X_GEN_TAC `d':A->bool` THEN REPEAT DISCH_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[pairwise]) THEN ASM_CASES_TAC `c':A->bool = d'` THENL [FIRST_X_ASSUM(MP_TAC o SPEC `d':A->bool` o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN MATCH_MP_TAC(TAUT `p ==> ~p ==> q`) THEN ASM_REWRITE_TAC[] THEN ASM_REWRITE_TAC[SING_SUBSET; GSYM SUBSET_ANTISYM_EQ] THEN REWRITE_TAC[SUBSET; IN_SING] THEN ASM SET_TAC[]; SUBGOAL_THEN `c:A->bool = c'` (fun th -> ASM_REWRITE_TAC[th]) THEN ASM SET_TAC[]]) in GEN_TAC THEN ASM_CASES_TAC `n = 0` THENL [ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `euclidean_space 0 = discrete_topology {\i. &0}` SUBST1_TAC THENL [REWRITE_TAC[SUBTOPOLOGY_EQ_DISCRETE_TOPOLOGY_SING] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SPACE] THEN REWRITE_TAC[NUMSEG_CLAUSES; ARITH; NOT_IN_EMPTY] THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_SING] THEN REWRITE_TAC[FUN_EQ_THM]; REWRITE_TAC[OPEN_IN_DISCRETE_TOPOLOGY; continuous_map] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; TOPSPACE_DISCRETE_TOPOLOGY] THEN SET_TAC[]]; ALL_TAC] THEN ABBREV_TAC `enorm = \x. sqrt(sum(1..n) (\i. x i pow 2))` THEN ABBREV_TAC `zero:num->real = \i. &0` THEN SUBGOAL_THEN `zero IN topspace(euclidean_space n)` ASSUME_TAC THENL [EXPAND_TAC "zero" THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SPACE; IN_ELIM_THM]; ALL_TAC] THEN SUBGOAL_THEN `(enorm:(num->real)->real) zero = &0` ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["enorm"; "zero"] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[SUM_0; SQRT_0]; ALL_TAC] THEN SUBGOAL_THEN `continuous_map(euclidean_space n,euclideanreal) enorm` ASSUME_TAC THENL [EXPAND_TAC "enorm" THEN MATCH_MP_TAC CONTINUOUS_MAP_SQRT THEN MATCH_MP_TAC CONTINUOUS_MAP_SUM THEN REWRITE_TAC[IN_NUMSEG; FINITE_NUMSEG; euclidean_space] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_POW THEN SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; IN_UNIV; CONTINUOUS_MAP_PRODUCT_PROJECTION]; ALL_TAC] THEN SUBGOAL_THEN `!x. &0 <= (enorm:(num->real)->real) x` ASSUME_TAC THENL [GEN_TAC THEN EXPAND_TAC "enorm" THEN MATCH_MP_TAC SQRT_POS_LE THEN MATCH_MP_TAC SUM_POS_LE THEN REWRITE_TAC[REAL_LE_POW_2]; ALL_TAC] THEN SUBGOAL_THEN `!(x:num->real) i. 1 <= i /\ i <= n ==> abs(x i) <= enorm x` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN TRANS_TAC REAL_LE_TRANS `sqrt(sum {i:num} (\i. x i pow 2))` THEN CONJ_TAC THENL [REWRITE_TAC[SUM_SING; POW_2_SQRT_ABS; REAL_LE_REFL]; ALL_TAC] THEN EXPAND_TAC "enorm" THEN REWRITE_TAC[] THEN MATCH_MP_TAC SQRT_MONO_LE THEN MATCH_MP_TAC SUM_SUBSET_SIMPLE THEN ASM_REWRITE_TAC[REAL_LE_POW_2; SING_SUBSET; FINITE_NUMSEG; IN_NUMSEG]; ALL_TAC] THEN MAP_EVERY ABBREV_TAC [`B = \r:real. {x | x IN topspace(euclidean_space n) /\ enorm x < r}`; `C = \r:real. {x | x IN topspace(euclidean_space n) /\ enorm x <= r}`; `S = \r:real. {x | x IN topspace(euclidean_space n) /\ enorm x = r}`] THEN SUBGOAL_THEN `(!r. (B:real->(num->real)->bool) r SUBSET C r) /\ (!r. S r SUBSET C r) /\ (!r. DISJOINT (S r) (B r)) /\ (!r. B r UNION S r = C r)` STRIP_ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["C"; "B"; "S"] THEN REWRITE_TAC[SUBSET; EXTENSION; DISJOINT; IN_INTER; IN_ELIM_THM; IN_UNION; NOT_IN_EMPTY] THEN REPEAT CONJ_TAC THEN MAP_EVERY X_GEN_TAC [`r:real`; `x:num->real`] THEN ASM_CASES_TAC `x IN topspace(euclidean_space n)` THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `!r f. &0 < r /\ continuous_map(subtopology (euclidean_space n) (C r), euclidean_space n) f /\ (!x y. x IN C r /\ y IN C r /\ f x = f y ==> x = y) ==> open_in (euclidean_space n) (IMAGE f (B r))` ASSUME_TAC THENL [ALL_TAC; MAP_EVERY X_GEN_TAC [`u:(num->real)->bool`; `f:(num->real)->(num->real)`] THEN STRIP_TAC THEN GEN_REWRITE_TAC I [OPEN_IN_SUBOPEN]THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN X_GEN_TAC `x:num->real` THEN DISCH_TAC THEN SUBGOAL_THEN `x IN topspace(euclidean_space n)` ASSUME_TAC THENL [FIRST_X_ASSUM(MP_TAC o MATCH_MP OPEN_IN_SUBSET) THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?r. &0 < r /\ !y. y IN topspace(euclidean_space n) /\ (!i. 1 <= i /\ i <= n ==> abs(y i - x i) < r) ==> y IN u` STRIP_ASSUME_TAC THENL [UNDISCH_TAC `open_in (euclidean_space n) u` THEN REWRITE_TAC[OPEN_IN_SUBTOPOLOGY; euclidean_space] THEN REWRITE_TAC[GSYM euclidean_space] THEN REWRITE_TAC[GSYM TOPSPACE_EUCLIDEAN_SPACE] THEN DISCH_THEN(X_CHOOSE_THEN `v:(num->real)->bool` (CONJUNCTS_THEN2 MP_TAC (ASSUME_TAC o SYM))) THEN REWRITE_TAC[OPEN_IN_PRODUCT_TOPOLOGY_ALT] THEN DISCH_THEN(MP_TAC o SPEC `x:num->real`) THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[TOPSPACE_EUCLIDEANREAL; IN_UNIV; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `t:num->real->bool` THEN REWRITE_TAC[GSYM REAL_OPEN_IN; real_open] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[cartesian_product; SUBSET; EXTENSIONAL_UNIV] THEN REWRITE_TAC[IN_UNIV; IN_ELIM_THM] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC)) THEN GEN_REWRITE_TAC LAND_CONV [SWAP_FORALL_THM] THEN DISCH_THEN(MP_TAC o GEN `i:num` o SPECL [`(x:num->real) i`; `i:num`]) THEN ASM_REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM; FORALL_AND_THM] THEN X_GEN_TAC `B:num->real` THEN STRIP_TAC THEN EXISTS_TAC `inf((&1) INSERT IMAGE B {i:num | ~(t i = (:real))})` THEN ASM_SIMP_TAC[REAL_LT_INF_FINITE; NOT_INSERT_EMPTY; FINITE_INSERT; FINITE_IMAGE] THEN ASM_REWRITE_TAC[FORALL_IN_INSERT; FORALL_IN_IMAGE] THEN REWRITE_TAC[REAL_LT_01; IN_ELIM_THM] THEN X_GEN_TAC `y:num->real` THEN STRIP_TAC THEN EXPAND_TAC "u" THEN REWRITE_TAC[IN_INTER] THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN X_GEN_TAC `i:num` THEN ASM_CASES_TAC `t(i:num) = (:real)` THEN ASM_REWRITE_TAC[IN_UNIV] THEN ASM_CASES_TAC `1 <= i /\ i <= n` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN MAP_EVERY UNDISCH_TAC [`x IN topspace(euclidean_space n)`; `y IN topspace(euclidean_space n)`] THEN SIMP_TAC[TOPSPACE_EUCLIDEAN_SPACE; IN_ELIM_THM; IMP_IMP; IN_NUMSEG] THEN DISCH_THEN(CONJUNCTS_THEN (MP_TAC o SPEC `i:num`)) THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `!y:num->real. y IN C(r / &2) ==> (\i. x i + y i) IN u` ASSUME_TAC THENL [X_GEN_TAC `y:num->real` THEN EXPAND_TAC "C" THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN CONJ_TAC THENL [MAP_EVERY UNDISCH_TAC [`x IN topspace(euclidean_space n)`; `y IN topspace(euclidean_space n)`] THEN SIMP_TAC[TOPSPACE_EUCLIDEAN_SPACE; IN_ELIM_THM; REAL_ADD_LID]; REWRITE_TAC[REAL_ADD_SUB] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `y <= r / &2 ==> &0 < r /\ x <= y ==> x < r`)) THEN ASM_MESON_TAC[]]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`r / &2`; `(f:(num->real)->(num->real)) o (\y i. x i + y i)`]) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[o_THM; REAL_HALF] THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_MAP_COMPOSE THEN EXISTS_TAC `subtopology (euclidean_space n) u` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; TOPSPACE_SUBTOPOLOGY] THEN ASM_SIMP_TAC[IN_INTER] THEN MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY THEN MATCH_MP_TAC CONTINUOUS_MAP_EUCLIDEAN_SPACE_ADD THEN REWRITE_TAC[CONTINUOUS_MAP_ID; ETA_AX; CONTINUOUS_MAP_CONST] THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP OPEN_IN_SUBSET) THEN ASM SET_TAC[]; MAP_EVERY X_GEN_TAC [`w:num->real`; `z:num->real`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`(\i. x i + w i):num->real`; `(\i. x i + z i):num->real`]) THEN ASM_SIMP_TAC[] THEN REWRITE_TAC[FUN_EQ_THM; REAL_EQ_ADD_LCANCEL]]; REWRITE_TAC[IMAGE_o] THEN MATCH_MP_TAC(MESON[] `P s ==> open_in top s ==> ?u. open_in top u /\ P u`) THEN CONJ_TAC THENL [MATCH_MP_TAC FUN_IN_IMAGE; ASM SET_TAC[]] THEN REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `zero:num->real` THEN EXPAND_TAC "B" THEN REWRITE_TAC[IN_ELIM_THM] THEN ASM_REWRITE_TAC[REAL_HALF] THEN EXPAND_TAC "zero" THEN REWRITE_TAC[REAL_ADD_RID; ETA_AX]]] THEN FIRST_ASSUM(DISJ_CASES_TAC o MATCH_MP (ARITH_RULE `~(n = 0) ==> n = 1 \/ 2 <= n`)) THENL [FIRST_X_ASSUM SUBST_ALL_TAC THEN REPEAT STRIP_TAC THEN MAP_EVERY ABBREV_TAC [`e:real->num->real = \x i. if i = 1 then x else &0`; `e':(num->real)->real = \x. x 1`] THEN SUBGOAL_THEN `homeomorphic_maps(euclideanreal,euclidean_space 1) (e,e')` ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["e"; "e'"] THEN REWRITE_TAC[homeomorphic_maps] THEN REWRITE_TAC[euclidean_space; CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN REWRITE_TAC[GSYM euclidean_space; TOPSPACE_EUCLIDEAN_SPACE] THEN REWRITE_TAC[NUMSEG_SING; IN_SING; SUBSET; FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_ELIM_THM] THEN SIMP_TAC[FUN_EQ_THM] THEN REWRITE_TAC[CONTINUOUS_MAP_COMPONENTWISE_UNIV] THEN SIMP_TAC[euclidean_space; CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_PRODUCT_PROJECTION; IN_UNIV] THEN REWRITE_TAC[FUN_EQ_THM] THEN CONJ_TAC THENL [ALL_TAC; MESON_TAC[]] THEN X_GEN_TAC `k:num` THEN ASM_CASES_TAC `k = 1` THEN ASM_REWRITE_TAC[CONTINUOUS_MAP_REAL_CONST; CONTINUOUS_MAP_ID]; ALL_TAC] THEN MP_TAC(ISPECL [`(e':(num->real)->real) o f o (e:real->num->real)`; `real_interval(--r,r)`] INJECTIVE_EQ_REAL_OPEN_MAP_EUCLIDEANREAL) THEN REWRITE_TAC[IS_REALINTERVAL_INTERVAL] THEN ANTS_TAC THENL [FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [homeomorphic_maps]) THEN MATCH_MP_TAC CONTINUOUS_MAP_COMPOSE THEN EXISTS_TAC `euclidean_space 1` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONTINUOUS_MAP_COMPOSE THEN EXISTS_TAC `subtopology (euclidean_space 1) (C(r:real))` THEN ASM_REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN ASM_SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; SUBSET] THEN REWRITE_TAC[FORALL_IN_IMAGE; TOPSPACE_SUBTOPOLOGY; IN_INTER] THEN X_GEN_TAC `x:real` THEN REWRITE_TAC[IN_REAL_INTERVAL] THEN STRIP_TAC THEN EXPAND_TAC "C" THEN REWRITE_TAC[IN_ELIM_THM] THEN CONJ_TAC THENL [ASM_MESON_TAC[continuous_map]; ALL_TAC] THEN EXPAND_TAC "enorm" THEN REWRITE_TAC[NUMSEG_SING; SUM_SING] THEN EXPAND_TAC "e" THEN REWRITE_TAC[POW_2_SQRT_ABS] THEN ASM_REAL_ARITH_TAC; DISCH_THEN(MP_TAC o fst o EQ_IMP_RULE)] THEN ANTS_TAC THENL [MAP_EVERY X_GEN_TAC [`x:real`; `y:real`] THEN REWRITE_TAC[IN_REAL_INTERVAL; REAL_BOUNDS_LT; o_THM] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphic_maps]) THEN REWRITE_TAC[TOPSPACE_EUCLIDEANREAL; IN_UNIV] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`(e:real->num->real) x`; `(e:real->num->real) y`]) THEN ANTS_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN REWRITE_TAC[CONJ_ASSOC] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [MAP_EVERY EXPAND_TAC ["C"; "enorm"] THEN REWRITE_TAC[IN_ELIM_THM] THEN REWRITE_TAC[NUMSEG_SING; SUM_SING; POW_2_SQRT_ABS] THEN RULE_ASSUM_TAC(REWRITE_RULE[continuous_map]) THEN ASM_SIMP_TAC[TOPSPACE_EUCLIDEANREAL; IN_UNIV] THEN EXPAND_TAC "e" THEN REWRITE_TAC[] THEN ASM_SIMP_TAC[REAL_LT_IMP_LE]; DISCH_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (MESON[] `e' a = e' b ==> e(e' a) = a /\ e(e' b) = b ==> a = b`)) THEN CONJ_TAC THEN FIRST_ASSUM MATCH_MP_TAC THEN REPEAT(FIRST_X_ASSUM(ASSUME_TAC o CONJUNCT1 o GEN_REWRITE_RULE I [continuous_map])) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; IN_INTER] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[TOPSPACE_EUCLIDEANREAL; IN_UNIV]]; DISCH_THEN(MP_TAC o SPEC `real_interval(--r,r)`) THEN REWRITE_TAC[REAL_OPEN_REAL_INTERVAL; SUBSET_REFL]] THEN REWRITE_TAC[IMAGE_o] THEN SUBGOAL_THEN `IMAGE (e:real->num->real) (real_interval(--r,r)) = B r` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; IN_IMAGE] THEN MAP_EVERY EXPAND_TAC ["B"; "e"; "enorm"] THEN REWRITE_TAC[IN_REAL_INTERVAL; REAL_BOUNDS_LT] THEN REWRITE_TAC[NUMSEG_SING; SUM_SING; POW_2_SQRT_ABS] THEN X_GEN_TAC `x:num->real` THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SPACE; IN_ELIM_THM] THEN REWRITE_TAC[NUMSEG_SING; IN_SING; FUN_EQ_THM] THEN MESON_TAC[]; ALL_TAC] THEN REWRITE_TAC[REAL_OPEN_IN] THEN MATCH_MP_TAC EQ_IMP THEN MATCH_MP_TAC HOMEOMORPHIC_MAP_OPENNESS THEN CONJ_TAC THENL [ASM_MESON_TAC[HOMEOMORPHIC_MAPS_MAP]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o CONJUNCT1 o REWRITE_RULE[continuous_map]) THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; SUBSET; FORALL_IN_IMAGE; IN_INTER] THEN DISCH_TAC THEN X_GEN_TAC `x:num->real` THEN STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN UNDISCH_TAC `x IN (B:real->(num->real)->bool) r` THEN EXPAND_TAC "B" THEN SIMP_TAC[IN_ELIM_THM]; ALL_TAC] THEN SUBGOAL_THEN `(!r:real. closed_in (euclidean_space n) (C r)) /\ (!r:real. closed_in (euclidean_space n) (S r))` STRIP_ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["C"; "S"] THEN REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[SET_RULE `{x | P x /\ R (f x) z} = {x | P x /\ f x IN {y | R y z}}`] THEN MATCH_MP_TAC CLOSED_IN_CONTINUOUS_MAP_PREIMAGE THEN EXISTS_TAC `euclideanreal` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SING_GSPEC; GSYM REAL_CLOSED_IN; REAL_CLOSED_SING] THEN REWRITE_TAC[REAL_CLOSED_HALFSPACE_LE]; ALL_TAC] THEN SUBGOAL_THEN `(!r:real. compact_in (euclidean_space n) (C r)) /\ (!r:real. compact_in (euclidean_space n) (S r))` STRIP_ASSUME_TAC THENL [REPEAT STRIP_TAC THEN REWRITE_TAC[euclidean_space; COMPACT_IN_SUBTOPOLOGY] THEN REWRITE_TAC[GSYM TOPSPACE_EUCLIDEAN_SPACE] THEN ASM_SIMP_TAC[OPEN_IN_SUBSET; CLOSED_IN_SUBSET] THEN MATCH_MP_TAC CLOSED_COMPACT_IN THEN EXISTS_TAC `cartesian_product (:num) (\i. real_interval[--(abs r),abs r])` THEN REWRITE_TAC[COMPACT_IN_CARTESIAN_PRODUCT] THEN REWRITE_TAC[COMPACT_IN_EUCLIDEANREAL_INTERVAL] THEN (CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[CLOSED_IN_TRANS_FULL; CLOSED_IN_EUCLIDEAN_SPACE; TOPSPACE_EUCLIDEAN_SPACE; euclidean_space]]) THEN MAP_EVERY EXPAND_TAC ["C"; "S"] THEN REWRITE_TAC[SUBSET; cartesian_product; IN_ELIM_THM; EXTENSIONAL_UNIV; TOPSPACE_EUCLIDEAN_SPACE; IN_NUMSEG] THEN X_GEN_TAC `x:num->real` THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `i:num` THEN REWRITE_TAC[IN_UNIV; IN_REAL_INTERVAL; REAL_BOUNDS_LE] THEN ASM_CASES_TAC `1 <= i /\ i <= n` THEN ASM_SIMP_TAC[REAL_ABS_POS; REAL_ABS_NUM] THEN TRANS_TAC REAL_LE_TRANS `(enorm:(num->real)->real) x` THEN ASM_SIMP_TAC[] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`r:real`; `h:(num->real)->(num->real)`] THEN REWRITE_TAC[INJECTIVE_ON_ALT] THEN STRIP_TAC THEN SUBGOAL_THEN `IMAGE (h:(num->real)->(num->real)) (B(r:real)) IN path_components_of (subtopology (euclidean_space n) (topspace(euclidean_space n) DIFF IMAGE h (S r)))` MP_TAC THENL [ALL_TAC; SPEC_TAC(`IMAGE (h:(num->real)->(num->real)) (B(r:real))`, `c:(num->real)->bool`) THEN MP_TAC(SPEC `n:num` LOCALLY_PATH_CONNECTED_EUCLIDEAN_SPACE) THEN REWRITE_TAC[LOCALLY_PATH_CONNECTED_SPACE_OPEN_PATH_COMPONENTS] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN DISCH_THEN MATCH_MP_TAC THEN MATCH_MP_TAC OPEN_IN_DIFF THEN REWRITE_TAC[OPEN_IN_TOPSPACE] THEN MATCH_MP_TAC COMPACT_IN_IMP_CLOSED_IN THEN REWRITE_TAC[HAUSDORFF_EUCLIDEAN_SPACE] THEN MATCH_MP_TAC IMAGE_COMPACT_IN THEN EXISTS_TAC `subtopology (euclidean_space n) (C(r:real))` THEN ASM_REWRITE_TAC[] THEN ASM_REWRITE_TAC[COMPACT_IN_SUBTOPOLOGY]] THEN MATCH_MP_TAC lemma THEN REWRITE_TAC[NONEMPTY_PATH_COMPONENTS_OF] THEN REWRITE_TAC[PAIRWISE_DISJOINT_PATH_COMPONENTS_OF] THEN REWRITE_TAC[GSYM PATH_CONNECTED_SPACE_IFF_COMPONENTS_SUBSET_SING] THEN REWRITE_TAC[UNIONS_PATH_COMPONENTS_OF] THEN EXISTS_TAC `topspace(euclidean_space n) DIFF IMAGE (h:(num->real)->(num->real)) (C(r:real))` THEN SUBGOAL_THEN `~(topspace(subtopology (euclidean_space n) (topspace (euclidean_space n) DIFF IMAGE h ((S:real->(num->real)->bool) r))) = {})` ASSUME_TAC THENL [REWRITE_TAC[TOPSPACE_SUBTOPOLOGY] THEN MATCH_MP_TAC(SET_RULE `s SUBSET u /\ ~(s = u) ==> ~(u INTER (u DIFF s) = {})`) THEN CONJ_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE [continuous_map; closed_in; TOPSPACE_SUBTOPOLOGY]) THEN ASM SET_TAC[]; DISCH_THEN(MP_TAC o AP_TERM `compact_in (euclidean_space n)`)] THEN ASM_REWRITE_TAC[GSYM compact_space; COMPACT_EUCLIDEAN_SPACE] THEN MATCH_MP_TAC IMAGE_COMPACT_IN THEN EXISTS_TAC `subtopology (euclidean_space n) (C(r:real))` THEN ASM_REWRITE_TAC[] THEN ASM_REWRITE_TAC[COMPACT_IN_SUBTOPOLOGY]; ALL_TAC] THEN SUBGOAL_THEN `!P. is_realinterval {x | P x} ==> path_connected_in (euclidean_space n) {x | x IN topspace(euclidean_space n) /\ P(enorm x)}` ASSUME_TAC THENL [SUBGOAL_THEN `!x. x IN topspace(euclidean_space n) ==> (enorm x = &0 <=> x = zero)` ASSUME_TAC THENL [X_GEN_TAC `x:num->real` THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SPACE; IN_ELIM_THM; IN_NUMSEG] THEN DISCH_TAC THEN EQ_TAC THEN ASM_SIMP_TAC[] THEN REWRITE_TAC[FUN_EQ_THM; RIGHT_IMP_FORALL_THM] THEN X_GEN_TAC `i:num` THEN EXPAND_TAC "zero" THEN REWRITE_TAC[] THEN ASM_MESON_TAC[REAL_ARITH `abs(x) <= y ==> y = &0 ==> x = &0`]; ALL_TAC] THEN ABBREV_TAC `mul = \a (x:num->real) i. a * x i` THEN SUBGOAL_THEN `!a x:num->real. enorm(mul a x) = abs a * enorm x` ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["enorm"; "mul"] THEN REWRITE_TAC[REAL_POW_MUL; SUM_LMUL; SQRT_MUL] THEN REWRITE_TAC[POW_2_SQRT_ABS]; ALL_TAC] THEN SUBGOAL_THEN `!(a:real) x. x IN topspace(euclidean_space n) ==> mul a x IN topspace(euclidean_space n)` ASSUME_TAC THENL [EXPAND_TAC "mul" THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SPACE] THEN SIMP_TAC[IN_ELIM_THM; REAL_MUL_RZERO]; ALL_TAC] THEN SUBGOAL_THEN `!r:real. &0 <= r ==> path_connected_in (euclidean_space n) (S r)` (LABEL_TAC "*") THENL [ALL_TAC; X_GEN_TAC `P:real->bool` THEN ONCE_REWRITE_TAC[SET_RULE `{x | x IN s /\ P(f x)} = {x | x IN s /\ f x IN {y | P y}}`] THEN SPEC_TAC(`{x:real | P x}`,`t:real->bool`) THEN REPEAT STRIP_TAC THEN REWRITE_TAC[path_connected_in; SUBSET_RESTRICT] THEN MATCH_MP_TAC(MESON[PATH_CONNECTED_SPACE_IFF_PATH_COMPONENT] `!f:A->real. (!m n x y. f x = m /\ f y = n /\ x IN topspace top /\ y IN topspace top ==> path_component_of top x y) ==> path_connected_space top`) THEN EXISTS_TAC `enorm:(num->real)->real` THEN MATCH_MP_TAC REAL_WLOG_LT THEN REPEAT CONJ_TAC THENL [MAP_EVERY X_GEN_TAC [`d:real`; `x:num->real`; `y:num->real`] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; IN_INTER; IN_ELIM_THM] THEN STRIP_TAC THEN REMOVE_THEN "*" (MP_TAC o SPEC `d:real`) THEN ANTS_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN REWRITE_TAC[path_connected_in] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[PATH_CONNECTED_SPACE_IFF_PATH_COMPONENT] THEN DISCH_THEN(MP_TAC o SPECL [`x:num->real`; `y:num->real`]) THEN EXPAND_TAC "S" THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; IN_INTER; IN_ELIM_THM] THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] PATH_COMPONENT_OF_MONO) THEN ASM SET_TAC[]; MESON_TAC[PATH_COMPONENT_OF_SYM]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`r1:real`; `r2:real`] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`x1:num->real`; `x2:num->real`] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; IN_INTER; IN_ELIM_THM] THEN STRIP_TAC THEN SUBGOAL_THEN `&0 <= r1 /\ &0 <= r2` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `&0 < r2` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC PATH_COMPONENT_OF_TRANS THEN EXISTS_TAC `(mul:real->(num->real)->(num->real)) (r1 / r2) x2` THEN CONJ_TAC THENL [REMOVE_THEN "*" (MP_TAC o SPEC `r1:real`) THEN ASM_REWRITE_TAC[path_connected_in] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[PATH_CONNECTED_SPACE_IFF_PATH_COMPONENT] THEN DISCH_THEN(MP_TAC o SPECL [`x1:num->real`; `(mul:real->(num->real)->num->real) (r1 / r2) x2`]) THEN EXPAND_TAC "S" THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; IN_INTER; IN_ELIM_THM] THEN ASM_SIMP_TAC[REAL_LE_DIV; REAL_ARITH `&0 <= x ==> abs x = x`] THEN ASM_SIMP_TAC[REAL_DIV_RMUL; REAL_LT_IMP_NZ] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] PATH_COMPONENT_OF_MONO) THEN ASM SET_TAC[]; ONCE_REWRITE_TAC[PATH_COMPONENT_OF_SYM] THEN REWRITE_TAC[path_component_of]] THEN EXISTS_TAC `\t. (mul:real->(num->real)->(num->real)) (((&1 - t) * r2 + t * r1) / r2) x2` THEN REWRITE_TAC[REAL_SUB_REFL; REAL_MUL_LZERO; REAL_SUB_RZERO] THEN REWRITE_TAC[REAL_MUL_LID; REAL_ADD_LID; REAL_ADD_RID] THEN ASM_SIMP_TAC[REAL_DIV_REFL; REAL_LT_IMP_NZ] THEN CONJ_TAC THENL [ALL_TAC; EXPAND_TAC "mul" THEN REWRITE_TAC[REAL_MUL_LID; ETA_AX]] THEN REWRITE_TAC[SET_RULE `{x | x IN s /\ P x} = s INTER {x | P x}`] THEN REWRITE_TAC[GSYM SUBTOPOLOGY_RESTRICT] THEN REWRITE_TAC[PATH_IN_SUBTOPOLOGY] THEN CONJ_TAC THENL [REWRITE_TAC[path_in; euclidean_space] THEN REWRITE_TAC[GSYM TOPSPACE_EUCLIDEAN_SPACE] THEN REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE] THEN REWRITE_TAC[CONTINUOUS_MAP_COMPONENTWISE_UNIV] THEN X_GEN_TAC `i:num` THEN EXPAND_TAC "mul" THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_MUL THEN REWRITE_TAC[CONTINUOUS_MAP_REAL_CONST] THEN REWRITE_TAC[real_div] THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_RMUL THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_ADD THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_RMUL THEN SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_ID] THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_SUB THEN SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_ID] THEN REWRITE_TAC[CONTINUOUS_MAP_REAL_CONST]; X_GEN_TAC `d:real` THEN REWRITE_TAC[IN_REAL_INTERVAL] THEN STRIP_TAC THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN SUBGOAL_THEN `abs(((&1 - d) * r2 + d * r1) / r2) * r2 = (&1 - d) * r2 + d * r1` SUBST1_TAC THENL [ASM_SIMP_TAC[REAL_ABS_DIV; REAL_ARITH `&0 < x ==> abs x = x`] THEN ASM_SIMP_TAC[REAL_DIV_RMUL; REAL_LT_IMP_NZ; REAL_ABS_REFL] THEN MATCH_MP_TAC REAL_LE_ADD THEN CONJ_TAC THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [is_realinterval]) THEN MAP_EVERY EXISTS_TAC [`r1:real`; `r2:real`] THEN REPLICATE_TAC 2 (CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC]) THEN ONCE_REWRITE_TAC[REAL_ADD_SYM] THEN MATCH_MP_TAC REAL_CONVEX_BOUNDS_LE THEN ASM_REAL_ARITH_TAC]] THEN X_GEN_TAC `d:real` THEN DISCH_TAC THEN ASM_CASES_TAC `d:real = &0` THENL [SUBGOAL_THEN `(S:real->(num->real)->bool) d = {zero}` SUBST1_TAC THENL [EXPAND_TAC "S" THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN ASM SET_TAC[]; REWRITE_TAC[GSYM PATH_CONNECTED_IN_TOPSPACE] THEN ASM_SIMP_TAC[TOPSPACE_SUBTOPOLOGY_SUBSET; SING_SUBSET; PATH_CONNECTED_IN_SING; IN_SING]]; ALL_TAC] THEN SUBGOAL_THEN `&0 < d` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[path_connected_in] THEN CONJ_TAC THENL [EXPAND_TAC "S" THEN ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[PATH_CONNECTED_SPACE_IFF_PATH_COMPONENT] THEN ABBREV_TAC `neg:(num->real)->(num->real) = mul (-- &1:real)` THEN SUBGOAL_THEN `!x. (neg:(num->real)->(num->real)) (neg x) = x` ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["neg"; "mul"] THEN REWRITE_TAC[FUN_EQ_THM] THEN REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC(METIS[PATH_COMPONENT_OF_TRANS; PATH_COMPONENT_OF_SYM; PATH_COMPONENT_OF_REFL] `!n. (!x. n(n x) = x) /\ (?a b. a IN topspace top /\ b IN topspace top /\ ~(a = b) /\ ~(n a = b)) /\ (!x y. x IN topspace top /\ y IN topspace top /\ ~(x = n y) ==> path_component_of top x y) ==> !x y. x IN topspace top /\ y IN topspace top ==> path_component_of top x y`) THEN EXISTS_TAC `neg:(num->real)->(num->real)` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [EXPAND_TAC "S" THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; IN_INTER; IN_ELIM_THM] THEN MAP_EVERY EXISTS_TAC [`(\i. if i = 1 then d else &0):num->real`; `(\i. if i = 2 then d else &0):num->real`] THEN REWRITE_TAC[TOPSPACE_EUCLIDEAN_SPACE; IN_ELIM_THM; IN_NUMSEG] THEN EXPAND_TAC "enorm" THEN REWRITE_TAC[] THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[COND_RATOR; COND_RAND] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[SUM_DELTA; IN_NUMSEG; LE_REFL] THEN ASM_SIMP_TAC[ARITH_RULE `2 <= n ==> 1 <= n`] THEN REWRITE_TAC[POW_2_SQRT_ABS] THEN CONV_TAC NUM_REDUCE_CONV THEN ASM_REWRITE_TAC[COND_ID; REAL_ABS_REFL; POW_2_SQRT_ABS] THEN REWRITE_TAC[FUN_EQ_THM] THEN MAP_EVERY EXPAND_TAC ["neg"; "mul"] THEN REWRITE_TAC[] THEN CONJ_TAC THEN DISCH_THEN(MP_TAC o SPEC `1`) THEN CONV_TAC NUM_REDUCE_CONV THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`x:num->real`; `y:num->real`] THEN EXPAND_TAC "S" THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; IN_INTER; IN_ELIM_THM] THEN STRIP_TAC THEN REWRITE_TAC[path_component_of; path_in] THEN EXISTS_TAC `(\x. (mul:real->(num->real)->(num->real)) (d / enorm x) x) o (\t i. (&1 - t) * x i + t * y i)` THEN REWRITE_TAC[o_THM] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[REAL_MUL_LZERO; REAL_MUL_LID] THEN REWRITE_TAC[REAL_ADD_LID; REAL_ADD_RID; ETA_AX] THEN ASM_SIMP_TAC[REAL_DIV_REFL] THEN CONJ_TAC THENL [ALL_TAC; EXPAND_TAC "mul" THEN REWRITE_TAC[FUN_EQ_THM; REAL_MUL_LID]] THEN MATCH_MP_TAC CONTINUOUS_MAP_COMPOSE THEN EXISTS_TAC `subtopology (euclidean_space n) (UNIV DIFF {zero})` THEN CONJ_TAC THENL [REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN CONJ_TAC THENL [REWRITE_TAC[euclidean_space; CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN CONJ_TAC THENL [REWRITE_TAC[CONTINUOUS_MAP_COMPONENTWISE_UNIV] THEN X_GEN_TAC `i:num` THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_ADD THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_RMUL THEN SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_ID] THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_SUB THEN SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; CONTINUOUS_MAP_ID] THEN REWRITE_TAC[CONTINUOUS_MAP_REAL_CONST]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_NUMSEG; IN_ELIM_THM] THEN RULE_ASSUM_TAC(REWRITE_RULE [TOPSPACE_EUCLIDEAN_SPACE; IN_ELIM_THM; IN_NUMSEG]) THEN ASM_SIMP_TAC[REAL_MUL_RZERO; REAL_ADD_RID]]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `q:real` THEN REWRITE_TAC[TOPSPACE_EUCLIDEANREAL_SUBTOPOLOGY; IN_REAL_INTERVAL] THEN STRIP_TAC THEN REWRITE_TAC[IN_DIFF; IN_UNIV; IN_SING] THEN EXPAND_TAC "zero" THEN REWRITE_TAC[FUN_EQ_THM] THEN DISCH_TAC THEN SUBGOAL_THEN `(mul:real->(num->real)->(num->real)) (&1 - q) x = mul (--q) y` MP_TAC THENL [EXPAND_TAC "mul" THEN REWRITE_TAC[FUN_EQ_THM] THEN ASM_REWRITE_TAC[REAL_ARITH `a:real = --b * c <=> a + b * c = &0`]; DISCH_THEN(fun th -> MP_TAC th THEN MP_TAC th)] THEN DISCH_THEN(MP_TAC o AP_TERM `enorm:(num->real)->real`) THEN ASM_REWRITE_TAC[REAL_EQ_MUL_RCANCEL] THEN ASM_SIMP_TAC[REAL_ARITH `&0 <= q /\ q <= &1 ==> (abs(&1 - q) = abs(--q) <=> q = &1 / &2)`] THEN DISCH_THEN SUBST1_TAC THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN UNDISCH_TAC `~(x = (neg:(num->real)->(num->real)) y)` THEN MAP_EVERY EXPAND_TAC ["neg"; "mul"] THEN REWRITE_TAC[CONTRAPOS_THM] THEN REWRITE_TAC[FUN_EQ_THM] THEN MATCH_MP_TAC MONO_FORALL THEN REAL_ARITH_TAC]; REWRITE_TAC[SET_RULE `{x | x IN s /\ P x} = s INTER {x | P x}`] THEN REWRITE_TAC[GSYM SUBTOPOLOGY_RESTRICT] THEN REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN CONJ_TAC THENL [REWRITE_TAC[euclidean_space; CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN REWRITE_TAC[GSYM euclidean_space] THEN REWRITE_TAC[GSYM TOPSPACE_EUCLIDEAN_SPACE] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; TOPSPACE_SUBTOPOLOGY] THEN ASM_SIMP_TAC[IN_INTER] THEN EXPAND_TAC "mul" THEN REWRITE_TAC[CONTINUOUS_MAP_COMPONENTWISE_UNIV] THEN X_GEN_TAC `i:num` THEN MATCH_MP_TAC CONTINUOUS_MAP_REAL_MUL THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_MAP_REAL_DIV THEN REWRITE_TAC[CONTINUOUS_MAP_REAL_CONST] THEN ASM_SIMP_TAC[CONTINUOUS_MAP_FROM_SUBTOPOLOGY; IN_INTER; TOPSPACE_SUBTOPOLOGY; IN_DIFF; IN_SING]; REWRITE_TAC[euclidean_space] THEN REPEAT(MATCH_MP_TAC CONTINUOUS_MAP_FROM_SUBTOPOLOGY) THEN ASM_SIMP_TAC[CONTINUOUS_MAP_PRODUCT_PROJECTION; IN_UNIV]]; ASM_REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM] THEN ASM_SIMP_TAC[real_abs; REAL_LE_DIV] THEN REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; IN_INTER; IN_DIFF; IN_SING] THEN ASM_SIMP_TAC[REAL_DIV_RMUL]]]; ALL_TAC] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC EXISTS_PATH_COMPONENT_OF_SUPERSET THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[PATH_CONNECTED_IN_SUBTOPOLOGY] THEN CONJ_TAC THENL [ALL_TAC; RULE_ASSUM_TAC(REWRITE_RULE [continuous_map; closed_in; TOPSPACE_SUBTOPOLOGY]) THEN ASM SET_TAC[]] THEN MATCH_MP_TAC PATH_CONNECTED_IN_CONTINUOUS_MAP_IMAGE THEN EXISTS_TAC `subtopology (euclidean_space n) (C(r:real))` THEN ASM_REWRITE_TAC[PATH_CONNECTED_IN_SUBTOPOLOGY] THEN EXPAND_TAC "B" THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[is_realinterval; IN_ELIM_THM] THEN REAL_ARITH_TAC; MATCH_MP_TAC EXISTS_PATH_COMPONENT_OF_SUPERSET THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[PATH_CONNECTED_IN_SUBTOPOLOGY] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN W(MP_TAC o PART_MATCH (lhand o rand) BIGLEMMA o snd) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN EXPAND_TAC "C" THEN REWRITE_TAC[SET_RULE `u DIFF {x | x IN u /\ P x} = {x | x IN u /\ ~P x}`] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[is_realinterval; IN_ELIM_THM] THEN REAL_ARITH_TAC; REWRITE_TAC[TOPSPACE_SUBTOPOLOGY] THEN RULE_ASSUM_TAC(REWRITE_RULE [continuous_map; closed_in; TOPSPACE_SUBTOPOLOGY]) THEN ASM SET_TAC[]; REWRITE_TAC[GSYM PATH_CONNECTED_IN_TOPSPACE] THEN REWRITE_TAC[PATH_CONNECTED_IN_SUBTOPOLOGY; TOPSPACE_SUBTOPOLOGY] THEN REWRITE_TAC[INTER_SUBSET] THEN REWRITE_TAC[SET_RULE `u INTER (u DIFF s) = u DIFF s`] THEN W(MP_TAC o PART_MATCH (lhand o rand) BIGLEMMA o rand o snd) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN SUBGOAL_THEN `subtopology (euclidean_space n) (S(r:real)) = subtopology (subtopology (euclidean_space n) (C r)) (S r)` (fun th -> ASM_SIMP_TAC[th; CONTINUOUS_MAP_FROM_SUBTOPOLOGY]) THEN REWRITE_TAC[SUBTOPOLOGY_SUBTOPOLOGY] THEN AP_TERM_TAC THEN ASM SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN DISCH_THEN(MP_TAC o SPEC `euclideanreal` o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] (ISPEC `enorm:(num->real)->real` PATH_CONNECTED_IN_CONTINUOUS_MAP_IMAGE))) THEN ASM_REWRITE_TAC[PATH_CONNECTED_IN_EUCLIDEANREAL] THEN REWRITE_TAC[is_realinterval; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_DIFF] THEN EXPAND_TAC "S" THEN DISCH_THEN(MP_TAC o SPEC `zero:num->real`) THEN ASM_REWRITE_TAC[IN_ELIM_THM; NOT_IMP] THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN DISCH_THEN (MP_TAC o SPEC `(\i. if i = 1 then r + &1 else &0):num->real`) THEN REWRITE_TAC[TAUT `p /\ ~(p /\ q) <=> p /\ ~q`; NOT_IMP] THEN SUBGOAL_THEN `enorm((\i. if i = 1 then r + &1 else &0):num->real) = r + &1` SUBST1_TAC THENL [EXPAND_TAC "enorm" THEN REWRITE_TAC[] THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[COND_RATOR; COND_RAND] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[SUM_DELTA; IN_NUMSEG; LE_REFL] THEN ASM_SIMP_TAC[ARITH_RULE `2 <= n ==> 1 <= n`] THEN REWRITE_TAC[POW_2_SQRT_ABS] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[REAL_ARITH `~(r + &1 = r)`]] THEN CONJ_TAC THENL [REWRITE_TAC[TOPSPACE_EUCLIDEAN_SPACE] THEN REWRITE_TAC[IN_ELIM_THM; IN_NUMSEG] THEN X_GEN_TAC `i:num` THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC; DISCH_THEN(MP_TAC o SPEC `r:real`) THEN REWRITE_TAC[NOT_IMP] THEN REPEAT(CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]) THEN SET_TAC[]]]);; let INVARIANCE_OF_DOMAIN_EUCLIDEAN_SPACE_EMBEDDING_MAP = prove (`!n u f. open_in (euclidean_space n) u /\ continuous_map(subtopology (euclidean_space n) u,euclidean_space n) f /\ (!x y. x IN u /\ y IN u /\ f x = f y ==> x = y) ==> embedding_map(subtopology (euclidean_space n) u,euclidean_space n) f`, REWRITE_TAC[INJECTIVE_ON_ALT] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC INJECTIVE_OPEN_IMP_EMBEDDING_MAP THEN ASM_SIMP_TAC[TOPSPACE_SUBTOPOLOGY; IN_INTER] THEN REWRITE_TAC[open_map; OPEN_IN_SUBTOPOLOGY_ALT; FORALL_IN_GSPEC] THEN X_GEN_TAC `v:(num->real)->bool` THEN DISCH_TAC THEN MATCH_MP_TAC INVARIANCE_OF_DOMAIN_EUCLIDEAN_SPACE THEN REWRITE_TAC[INJECTIVE_ON_ALT] THEN ASM_SIMP_TAC[IN_INTER; OPEN_IN_INTER] THEN ASM_MESON_TAC[SUBTOPOLOGY_SUBTOPOLOGY; CONTINUOUS_MAP_FROM_SUBTOPOLOGY]);; let INVARIANCE_OF_DOMAIN_EUCLIDEAN_SPACE_GEN = prove (`!m n u f. n <= m /\ open_in (euclidean_space m) u /\ continuous_map(subtopology (euclidean_space m) u,euclidean_space n) f /\ (!x y. x IN u /\ y IN u /\ f x = f y ==> x = y) ==> open_in (euclidean_space n) (IMAGE f u)`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN SUBGOAL_THEN `euclidean_space n = subtopology (euclidean_space m) (topspace(euclidean_space n))` SUBST1_TAC THENL [REWRITE_TAC[euclidean_space; SUBTOPOLOGY_SUBTOPOLOGY] THEN AP_TERM_TAC THEN REWRITE_TAC[GSYM euclidean_space] THEN REWRITE_TAC[GSYM TOPSPACE_EUCLIDEAN_SPACE] THEN REWRITE_TAC[SET_RULE `n = m INTER n <=> n SUBSET m`] THEN ASM_REWRITE_TAC[SUBSET_EUCLIDEAN_SPACE]; REWRITE_TAC[CONTINUOUS_MAP_IN_SUBTOPOLOGY] THEN STRIP_TAC THEN MATCH_MP_TAC OPEN_IN_SUBSET_TOPSPACE THEN CONJ_TAC THENL [MATCH_MP_TAC INVARIANCE_OF_DOMAIN_EUCLIDEAN_SPACE THEN ASM_REWRITE_TAC[]; ASM_MESON_TAC[TOPSPACE_SUBTOPOLOGY_SUBSET; OPEN_IN_SUBSET]]]);; let INVARIANCE_OF_DOMAIN_EUCLIDEAN_SPACE_EMBEDDING_MAP_GEN = prove (`!m n u f. n <= m /\ open_in (euclidean_space m) u /\ continuous_map(subtopology (euclidean_space m) u,euclidean_space n) f /\ (!x y. x IN u /\ y IN u /\ f x = f y ==> x = y) ==> embedding_map(subtopology (euclidean_space m) u,euclidean_space n) f`, REWRITE_TAC[INJECTIVE_ON_ALT] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC INJECTIVE_OPEN_IMP_EMBEDDING_MAP THEN ASM_SIMP_TAC[TOPSPACE_SUBTOPOLOGY; IN_INTER] THEN REWRITE_TAC[open_map; OPEN_IN_SUBTOPOLOGY_ALT; FORALL_IN_GSPEC] THEN X_GEN_TAC `v:(num->real)->bool` THEN DISCH_TAC THEN MATCH_MP_TAC INVARIANCE_OF_DOMAIN_EUCLIDEAN_SPACE_GEN THEN EXISTS_TAC `m:num` THEN REWRITE_TAC[INJECTIVE_ON_ALT] THEN ASM_SIMP_TAC[IN_INTER; OPEN_IN_INTER] THEN ASM_MESON_TAC[SUBTOPOLOGY_SUBTOPOLOGY; CONTINUOUS_MAP_FROM_SUBTOPOLOGY]);;