author | lcp |
Mon, 22 Aug 1994 11:11:17 +0200 | |
changeset 571 | 0b03ce5b62f7 |
parent 523 | 972119df615e |
child 760 | f0200e91b272 |
permissions | -rw-r--r-- |
(* Title: ZF/CardinalArith.ML ID: $Id$ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1994 University of Cambridge Cardinal arithmetic -- WITHOUT the Axiom of Choice *) open CardinalArith; (*** Elementary properties ***) (*Use AC to discharge first premise*) goal CardinalArith.thy "!!A B. [| well_ord(B,r); A lepoll B |] ==> |A| le |B|"; by (res_inst_tac [("i","|A|"),("j","|B|")] Ord_linear_le 1); by (REPEAT_FIRST (ares_tac [Ord_cardinal, le_eqI])); by (rtac (eqpollI RS cardinal_cong) 1 THEN assume_tac 1); by (rtac lepoll_trans 1); by (rtac (well_ord_cardinal_eqpoll RS eqpoll_sym RS eqpoll_imp_lepoll) 1); by (assume_tac 1); by (etac (le_imp_subset RS subset_imp_lepoll RS lepoll_trans) 1); by (rtac eqpoll_imp_lepoll 1); by (rewtac lepoll_def); by (etac exE 1); by (rtac well_ord_cardinal_eqpoll 1); by (etac well_ord_rvimage 1); by (assume_tac 1); val well_ord_lepoll_imp_le = result(); (*Each element of Fin(A) is equivalent to a natural number*) goal CardinalArith.thy "!!X A. X: Fin(A) ==> EX n:nat. X eqpoll n"; by (eresolve_tac [Fin_induct] 1); by (fast_tac (ZF_cs addIs [eqpoll_refl, nat_0I]) 1); by (fast_tac (ZF_cs addSIs [cons_eqpoll_cong, rewrite_rule [succ_def] nat_succI] addSEs [mem_irrefl]) 1); val Fin_eqpoll = result(); (*** Cardinal addition ***) (** Cardinal addition is commutative **) goalw CardinalArith.thy [eqpoll_def] "A+B eqpoll B+A"; by (rtac exI 1); by (res_inst_tac [("c", "case(Inr, Inl)"), ("d", "case(Inr, Inl)")] lam_bijective 1); by (safe_tac (ZF_cs addSEs [sumE])); by (ALLGOALS (asm_simp_tac case_ss)); val sum_commute_eqpoll = result(); goalw CardinalArith.thy [cadd_def] "i |+| j = j |+| i"; by (rtac (sum_commute_eqpoll RS cardinal_cong) 1); val cadd_commute = result(); (** Cardinal addition is associative **) goalw CardinalArith.thy [eqpoll_def] "(A+B)+C eqpoll A+(B+C)"; by (rtac exI 1); by (res_inst_tac [("c", "case(case(Inl, %y.Inr(Inl(y))), %y. Inr(Inr(y)))"), ("d", "case(%x.Inl(Inl(x)), case(%x.Inl(Inr(x)), Inr))")] lam_bijective 1); by (ALLGOALS (asm_simp_tac (case_ss setloop etac sumE))); val sum_assoc_eqpoll = result(); (*Unconditional version requires AC*) goalw CardinalArith.thy [cadd_def] "!!i j k. [| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==> \ \ (i |+| j) |+| k = i |+| (j |+| k)"; by (rtac cardinal_cong 1); br ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS sum_eqpoll_cong RS eqpoll_trans) 1; by (rtac (sum_assoc_eqpoll RS eqpoll_trans) 2); br ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong RS eqpoll_sym) 2; by (REPEAT (ares_tac [well_ord_radd] 1)); val well_ord_cadd_assoc = result(); (** 0 is the identity for addition **) goalw CardinalArith.thy [eqpoll_def] "0+A eqpoll A"; by (rtac exI 1); by (res_inst_tac [("c", "case(%x.x, %y.y)"), ("d", "Inr")] lam_bijective 1); by (ALLGOALS (asm_simp_tac (case_ss setloop eresolve_tac [sumE,emptyE]))); val sum_0_eqpoll = result(); goalw CardinalArith.thy [cadd_def] "!!K. Card(K) ==> 0 |+| K = K"; by (asm_simp_tac (ZF_ss addsimps [sum_0_eqpoll RS cardinal_cong, Card_cardinal_eq]) 1); val cadd_0 = result(); (** Addition of finite cardinals is "ordinary" addition **) goalw CardinalArith.thy [eqpoll_def] "succ(A)+B eqpoll succ(A+B)"; by (rtac exI 1); by (res_inst_tac [("c", "%z.if(z=Inl(A),A+B,z)"), ("d", "%z.if(z=A+B,Inl(A),z)")] lam_bijective 1); by (ALLGOALS (asm_simp_tac (case_ss addsimps [succI2, mem_imp_not_eq] setloop eresolve_tac [sumE,succE]))); val sum_succ_eqpoll = result(); (*Pulling the succ(...) outside the |...| requires m, n: nat *) (*Unconditional version requires AC*) goalw CardinalArith.thy [cadd_def] "!!m n. [| Ord(m); Ord(n) |] ==> succ(m) |+| n = |succ(m |+| n)|"; by (rtac (sum_succ_eqpoll RS cardinal_cong RS trans) 1); by (rtac (succ_eqpoll_cong RS cardinal_cong) 1); by (rtac (well_ord_cardinal_eqpoll RS eqpoll_sym) 1); by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel] 1)); val cadd_succ_lemma = result(); val [mnat,nnat] = goal CardinalArith.thy "[| m: nat; n: nat |] ==> m |+| n = m#+n"; by (cut_facts_tac [nnat] 1); by (nat_ind_tac "m" [mnat] 1); by (asm_simp_tac (arith_ss addsimps [nat_into_Card RS cadd_0]) 1); by (asm_simp_tac (arith_ss addsimps [nat_into_Ord, cadd_succ_lemma, nat_into_Card RS Card_cardinal_eq]) 1); val nat_cadd_eq_add = result(); (*** Cardinal multiplication ***) (** Cardinal multiplication is commutative **) (*Easier to prove the two directions separately*) goalw CardinalArith.thy [eqpoll_def] "A*B eqpoll B*A"; by (rtac exI 1); by (res_inst_tac [("c", "split(%x y.<y,x>)"), ("d", "split(%x y.<y,x>)")] lam_bijective 1); by (safe_tac ZF_cs); by (ALLGOALS (asm_simp_tac ZF_ss)); val prod_commute_eqpoll = result(); goalw CardinalArith.thy [cmult_def] "i |*| j = j |*| i"; by (rtac (prod_commute_eqpoll RS cardinal_cong) 1); val cmult_commute = result(); (** Cardinal multiplication is associative **) goalw CardinalArith.thy [eqpoll_def] "(A*B)*C eqpoll A*(B*C)"; by (rtac exI 1); by (res_inst_tac [("c", "split(%w z. split(%x y. <x,<y,z>>, w))"), ("d", "split(%x. split(%y z. <<x,y>, z>))")] lam_bijective 1); by (safe_tac ZF_cs); by (ALLGOALS (asm_simp_tac ZF_ss)); val prod_assoc_eqpoll = result(); (*Unconditional version requires AC*) goalw CardinalArith.thy [cmult_def] "!!i j k. [| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==> \ \ (i |*| j) |*| k = i |*| (j |*| k)"; by (rtac cardinal_cong 1); br ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS prod_eqpoll_cong RS eqpoll_trans) 1; by (rtac (prod_assoc_eqpoll RS eqpoll_trans) 2); br ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS prod_eqpoll_cong RS eqpoll_sym) 2; by (REPEAT (ares_tac [well_ord_rmult] 1)); val well_ord_cmult_assoc = result(); (** Cardinal multiplication distributes over addition **) goalw CardinalArith.thy [eqpoll_def] "(A+B)*C eqpoll (A*C)+(B*C)"; by (rtac exI 1); by (res_inst_tac [("c", "split(%x z. case(%y.Inl(<y,z>), %y.Inr(<y,z>), x))"), ("d", "case(split(%x y.<Inl(x),y>), split(%x y.<Inr(x),y>))")] lam_bijective 1); by (safe_tac (ZF_cs addSEs [sumE])); by (ALLGOALS (asm_simp_tac case_ss)); val sum_prod_distrib_eqpoll = result(); goalw CardinalArith.thy [lepoll_def, inj_def] "A lepoll A*A"; by (res_inst_tac [("x", "lam x:A. <x,x>")] exI 1); by (simp_tac (ZF_ss addsimps [lam_type]) 1); val prod_square_lepoll = result(); goalw CardinalArith.thy [cmult_def] "!!K. Card(K) ==> K le K |*| K"; by (rtac le_trans 1); by (rtac well_ord_lepoll_imp_le 2); by (rtac prod_square_lepoll 3); by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel, Card_is_Ord] 2)); by (asm_simp_tac (ZF_ss addsimps [le_refl, Card_is_Ord, Card_cardinal_eq]) 1); val cmult_square_le = result(); (** Multiplication by 0 yields 0 **) goalw CardinalArith.thy [eqpoll_def] "0*A eqpoll 0"; by (rtac exI 1); by (rtac lam_bijective 1); by (safe_tac ZF_cs); val prod_0_eqpoll = result(); goalw CardinalArith.thy [cmult_def] "0 |*| i = 0"; by (asm_simp_tac (ZF_ss addsimps [prod_0_eqpoll RS cardinal_cong, Card_0 RS Card_cardinal_eq]) 1); val cmult_0 = result(); (** 1 is the identity for multiplication **) goalw CardinalArith.thy [eqpoll_def] "{x}*A eqpoll A"; by (rtac exI 1); by (res_inst_tac [("c", "snd"), ("d", "%z.<x,z>")] lam_bijective 1); by (safe_tac ZF_cs); by (ALLGOALS (asm_simp_tac ZF_ss)); val prod_singleton_eqpoll = result(); goalw CardinalArith.thy [cmult_def, succ_def] "!!K. Card(K) ==> 1 |*| K = K"; by (asm_simp_tac (ZF_ss addsimps [prod_singleton_eqpoll RS cardinal_cong, Card_cardinal_eq]) 1); val cmult_1 = result(); (** Multiplication of finite cardinals is "ordinary" multiplication **) goalw CardinalArith.thy [eqpoll_def] "succ(A)*B eqpoll B + A*B"; by (rtac exI 1); by (res_inst_tac [("c", "split(%x y. if(x=A, Inl(y), Inr(<x,y>)))"), ("d", "case(%y. <A,y>, %z.z)")] lam_bijective 1); by (safe_tac (ZF_cs addSEs [sumE])); by (ALLGOALS (asm_simp_tac (case_ss addsimps [succI2, if_type, mem_imp_not_eq]))); val prod_succ_eqpoll = result(); (*Unconditional version requires AC*) goalw CardinalArith.thy [cmult_def, cadd_def] "!!m n. [| Ord(m); Ord(n) |] ==> succ(m) |*| n = n |+| (m |*| n)"; by (rtac (prod_succ_eqpoll RS cardinal_cong RS trans) 1); by (rtac (cardinal_cong RS sym) 1); by (rtac ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong) 1); by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1)); val cmult_succ_lemma = result(); val [mnat,nnat] = goal CardinalArith.thy "[| m: nat; n: nat |] ==> m |*| n = m#*n"; by (cut_facts_tac [nnat] 1); by (nat_ind_tac "m" [mnat] 1); by (asm_simp_tac (arith_ss addsimps [cmult_0]) 1); by (asm_simp_tac (arith_ss addsimps [nat_into_Ord, cmult_succ_lemma, nat_cadd_eq_add]) 1); val nat_cmult_eq_mult = result(); (*** Infinite Cardinals are Limit Ordinals ***) (*This proof is modelled upon one assuming nat<=A, with injection lam z:cons(u,A). if(z=u, 0, if(z : nat, succ(z), z)) and inverse %y. if(y:nat, nat_case(u,%z.z,y), y). If f: inj(nat,A) then range(f) behaves like the natural numbers.*) goalw CardinalArith.thy [lepoll_def] "!!i. nat lepoll A ==> cons(u,A) lepoll A"; by (etac exE 1); by (res_inst_tac [("x", "lam z:cons(u,A). if(z=u, f`0, \ \ if(z: range(f), f`succ(converse(f)`z), z))")] exI 1); by (res_inst_tac [("d", "%y. if(y: range(f), \ \ nat_case(u, %z.f`z, converse(f)`y), y)")] lam_injective 1); by (fast_tac (ZF_cs addSIs [if_type, nat_0I, nat_succI, apply_type] addIs [inj_is_fun, inj_converse_fun]) 1); by (asm_simp_tac (ZF_ss addsimps [inj_is_fun RS apply_rangeI, inj_converse_fun RS apply_rangeI, inj_converse_fun RS apply_funtype, left_inverse, right_inverse, nat_0I, nat_succI, nat_case_0, nat_case_succ] setloop split_tac [expand_if]) 1); val nat_cons_lepoll = result(); goal CardinalArith.thy "!!i. nat lepoll A ==> cons(u,A) eqpoll A"; by (etac (nat_cons_lepoll RS eqpollI) 1); by (rtac (subset_consI RS subset_imp_lepoll) 1); val nat_cons_eqpoll = result(); (*Specialized version required below*) goalw CardinalArith.thy [succ_def] "!!i. nat <= A ==> succ(A) eqpoll A"; by (eresolve_tac [subset_imp_lepoll RS nat_cons_eqpoll] 1); val nat_succ_eqpoll = result(); goalw CardinalArith.thy [InfCard_def] "InfCard(nat)"; by (fast_tac (ZF_cs addIs [Card_nat, le_refl, Card_is_Ord]) 1); val InfCard_nat = result(); goalw CardinalArith.thy [InfCard_def] "!!K. InfCard(K) ==> Card(K)"; by (etac conjunct1 1); val InfCard_is_Card = result(); goalw CardinalArith.thy [InfCard_def] "!!K L. [| InfCard(K); Card(L) |] ==> InfCard(K Un L)"; by (asm_simp_tac (ZF_ss addsimps [Card_Un, Un_upper1_le RSN (2,le_trans), Card_is_Ord]) 1); val InfCard_Un = result(); (*Kunen's Lemma 10.11*) goalw CardinalArith.thy [InfCard_def] "!!K. InfCard(K) ==> Limit(K)"; by (etac conjE 1); by (rtac (ltI RS non_succ_LimitI) 1); by (etac ([asm_rl, nat_0I] MRS (le_imp_subset RS subsetD)) 1); by (etac Card_is_Ord 1); by (safe_tac (ZF_cs addSDs [Limit_nat RS Limit_le_succD])); by (forward_tac [Card_is_Ord RS Ord_succD] 1); by (rewtac Card_def); by (res_inst_tac [("i", "succ(y)")] lt_irrefl 1); by (dtac (le_imp_subset RS nat_succ_eqpoll RS cardinal_cong) 1); (*Tricky combination of substitutions; backtracking needed*) by (etac ssubst 1 THEN etac ssubst 1 THEN rtac Ord_cardinal_le 1); by (assume_tac 1); val InfCard_is_Limit = result(); (*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***) (*A general fact about ordermap*) goalw Cardinal.thy [eqpoll_def] "!!A. [| well_ord(A,r); x:A |] ==> ordermap(A,r)`x eqpoll pred(A,x,r)"; by (rtac exI 1); by (asm_simp_tac (ZF_ss addsimps [ordermap_eq_image, well_ord_is_wf]) 1); by (etac (ordermap_bij RS bij_is_inj RS restrict_bij RS bij_converse_bij) 1); by (rtac pred_subset 1); val ordermap_eqpoll_pred = result(); (** Establishing the well-ordering **) goalw CardinalArith.thy [inj_def] "!!K. Ord(K) ==> \ \ (lam z:K*K. split(%x y. <x Un y, <x, y>>, z)) : inj(K*K, K*K*K)"; by (safe_tac ZF_cs); by (fast_tac (ZF_cs addIs [lam_type, Un_least_lt RS ltD, ltI] addSEs [split_type]) 1); by (asm_full_simp_tac ZF_ss 1); val csquare_lam_inj = result(); goalw CardinalArith.thy [csquare_rel_def] "!!K. Ord(K) ==> well_ord(K*K, csquare_rel(K))"; by (rtac (csquare_lam_inj RS well_ord_rvimage) 1); by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1)); val well_ord_csquare = result(); (** Characterising initial segments of the well-ordering **) goalw CardinalArith.thy [csquare_rel_def] "!!K. [| x<K; y<K; z<K |] ==> \ \ <<x,y>, <z,z>> : csquare_rel(K) --> x le z & y le z"; by (REPEAT (etac ltE 1)); by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff, Un_absorb, Un_least_mem_iff, ltD]) 1); by (safe_tac (ZF_cs addSEs [mem_irrefl] addSIs [Un_upper1_le, Un_upper2_le])); by (ALLGOALS (asm_simp_tac (ZF_ss addsimps [lt_def, succI2, Ord_succ]))); val csquareD_lemma = result(); val csquareD = csquareD_lemma RS mp |> standard; goalw CardinalArith.thy [pred_def] "!!K. z<K ==> pred(K*K, <z,z>, csquare_rel(K)) <= succ(z)*succ(z)"; by (safe_tac (lemmas_cs addSEs [SigmaE])); (*avoids using succCI*) by (rtac (csquareD RS conjE) 1); by (rewtac lt_def); by (assume_tac 4); by (ALLGOALS (fast_tac ZF_cs)); val pred_csquare_subset = result(); goalw CardinalArith.thy [csquare_rel_def] "!!K. [| x<z; y<z; z<K |] ==> \ \ <<x,y>, <z,z>> : csquare_rel(K)"; by (subgoals_tac ["x<K", "y<K"] 1); by (REPEAT (eresolve_tac [asm_rl, lt_trans] 2)); by (REPEAT (etac ltE 1)); by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff, Un_absorb, Un_least_mem_iff, ltD]) 1); val csquare_ltI = result(); (*Part of the traditional proof. UNUSED since it's harder to prove & apply *) goalw CardinalArith.thy [csquare_rel_def] "!!K. [| x le z; y le z; z<K |] ==> \ \ <<x,y>, <z,z>> : csquare_rel(K) | x=z & y=z"; by (subgoals_tac ["x<K", "y<K"] 1); by (REPEAT (eresolve_tac [asm_rl, lt_trans1] 2)); by (REPEAT (etac ltE 1)); by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff, Un_absorb, Un_least_mem_iff, ltD]) 1); by (REPEAT_FIRST (etac succE)); by (ALLGOALS (asm_simp_tac (ZF_ss addsimps [subset_Un_iff RS iff_sym, subset_Un_iff2 RS iff_sym, OrdmemD]))); val csquare_or_eqI = result(); (** The cardinality of initial segments **) goal CardinalArith.thy "!!K. [| InfCard(K); x<K; y<K; z=succ(x Un y) |] ==> \ \ ordermap(K*K, csquare_rel(K)) ` <x,y> lepoll \ \ ordermap(K*K, csquare_rel(K)) ` <z,z>"; by (subgoals_tac ["z<K", "well_ord(K*K, csquare_rel(K))"] 1); by (etac (InfCard_is_Card RS Card_is_Ord RS well_ord_csquare) 2); by (fast_tac (ZF_cs addSIs [Un_least_lt, InfCard_is_Limit, Limit_has_succ]) 2); by (rtac (OrdmemD RS subset_imp_lepoll) 1); by (res_inst_tac [("z1","z")] (csquare_ltI RS ordermap_mono) 1); by (etac well_ord_is_wf 4); by (ALLGOALS (fast_tac (ZF_cs addSIs [Un_upper1_le, Un_upper2_le, Ord_ordermap] addSEs [ltE]))); val ordermap_z_lepoll = result(); (*Kunen: "each <x,y>: K*K has no more than z*z predecessors..." (page 29) *) goalw CardinalArith.thy [cmult_def] "!!K. [| InfCard(K); x<K; y<K; z=succ(x Un y) |] ==> \ \ | ordermap(K*K, csquare_rel(K)) ` <x,y> | le |succ(z)| |*| |succ(z)|"; by (rtac (well_ord_rmult RS well_ord_lepoll_imp_le) 1); by (REPEAT (ares_tac [Ord_cardinal, well_ord_Memrel] 1)); by (subgoals_tac ["z<K"] 1); by (fast_tac (ZF_cs addSIs [Un_least_lt, InfCard_is_Limit, Limit_has_succ]) 2); by (rtac (ordermap_z_lepoll RS lepoll_trans) 1); by (REPEAT_SOME assume_tac); by (rtac (ordermap_eqpoll_pred RS eqpoll_imp_lepoll RS lepoll_trans) 1); by (etac (InfCard_is_Card RS Card_is_Ord RS well_ord_csquare) 1); by (fast_tac (ZF_cs addIs [ltD]) 1); by (rtac (pred_csquare_subset RS subset_imp_lepoll RS lepoll_trans) 1 THEN assume_tac 1); by (REPEAT_FIRST (etac ltE)); by (rtac (prod_eqpoll_cong RS eqpoll_sym RS eqpoll_imp_lepoll) 1); by (REPEAT_FIRST (etac (Ord_succ RS Ord_cardinal_eqpoll))); val ordermap_csquare_le = result(); (*Kunen: "... so the order type <= K" *) goal CardinalArith.thy "!!K. [| InfCard(K); ALL y:K. InfCard(y) --> y |*| y = y |] ==> \ \ ordertype(K*K, csquare_rel(K)) le K"; by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1); by (rtac all_lt_imp_le 1); by (assume_tac 1); by (etac (well_ord_csquare RS Ord_ordertype) 1); by (rtac Card_lt_imp_lt 1); by (etac InfCard_is_Card 3); by (etac ltE 2 THEN assume_tac 2); by (asm_full_simp_tac (ZF_ss addsimps [ordertype_unfold]) 1); by (safe_tac (ZF_cs addSEs [ltE])); by (subgoals_tac ["Ord(xb)", "Ord(y)"] 1); by (REPEAT (eresolve_tac [asm_rl, Ord_in_Ord] 2)); by (rtac (ordermap_csquare_le RS lt_trans1) 1 THEN REPEAT (ares_tac [refl] 1 ORELSE etac ltI 1)); by (res_inst_tac [("i","xb Un y"), ("j","nat")] Ord_linear2 1 THEN REPEAT (ares_tac [Ord_Un, Ord_nat] 1)); (*the finite case: xb Un y < nat *) by (res_inst_tac [("j", "nat")] lt_trans2 1); by (asm_full_simp_tac (FOL_ss addsimps [InfCard_def]) 2); by (asm_full_simp_tac (ZF_ss addsimps [lt_def, nat_cmult_eq_mult, nat_succI, mult_type, nat_into_Card RS Card_cardinal_eq, Ord_nat]) 1); (*case nat le (xb Un y), equivalently InfCard(xb Un y) *) by (asm_full_simp_tac (ZF_ss addsimps [le_imp_subset RS nat_succ_eqpoll RS cardinal_cong, le_succ_iff, InfCard_def, Card_cardinal, Un_least_lt, Ord_Un, ltI, nat_le_cardinal, Ord_cardinal_le RS lt_trans1 RS ltD]) 1); val ordertype_csquare_le = result(); (*This lemma can easily be generalized to premise well_ord(A*A,r) *) goalw CardinalArith.thy [cmult_def] "!!K. Ord(K) ==> K |*| K = |ordertype(K*K, csquare_rel(K))|"; by (rtac cardinal_cong 1); by (rewtac eqpoll_def); by (rtac exI 1); by (etac (well_ord_csquare RS ordermap_bij) 1); val csquare_eq_ordertype = result(); (*Main result: Kunen's Theorem 10.12*) goal CardinalArith.thy "!!K. InfCard(K) ==> K |*| K = K"; by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1); by (etac rev_mp 1); by (trans_ind_tac "K" [] 1); by (rtac impI 1); by (rtac le_anti_sym 1); by (etac (InfCard_is_Card RS cmult_square_le) 2); by (rtac (ordertype_csquare_le RSN (2, le_trans)) 1); by (assume_tac 2); by (assume_tac 2); by (asm_simp_tac (ZF_ss addsimps [csquare_eq_ordertype, Ord_cardinal_le, well_ord_csquare RS Ord_ordertype]) 1); val InfCard_csquare_eq = result(); goal CardinalArith.thy "!!A. [| well_ord(A,r); InfCard(|A|) |] ==> A*A eqpoll A"; by (resolve_tac [prod_eqpoll_cong RS eqpoll_trans] 1); by (REPEAT (etac (well_ord_cardinal_eqpoll RS eqpoll_sym) 1)); by (resolve_tac [well_ord_cardinal_eqE] 1); by (REPEAT (ares_tac [Ord_cardinal, well_ord_rmult, well_ord_Memrel] 1)); by (asm_simp_tac (ZF_ss addsimps [symmetric cmult_def, InfCard_csquare_eq]) 1); val well_ord_InfCard_square_eq = result(); (*** For every cardinal number there exists a greater one [Kunen's Theorem 10.16, which would be trivial using AC] ***) goalw CardinalArith.thy [jump_cardinal_def] "Ord(jump_cardinal(K))"; by (rtac (Ord_is_Transset RSN (2,OrdI)) 1); by (safe_tac (ZF_cs addSIs [Ord_ordertype])); bw Transset_def; by (safe_tac ZF_cs); by (rtac (ordertype_subset RS exE) 1 THEN REPEAT (assume_tac 1)); by (resolve_tac [UN_I] 1); by (resolve_tac [ReplaceI] 2); by (ALLGOALS (fast_tac (ZF_cs addSEs [well_ord_subset]))); val Ord_jump_cardinal = result(); (*Allows selective unfolding. Less work than deriving intro/elim rules*) goalw CardinalArith.thy [jump_cardinal_def] "i : jump_cardinal(K) <-> \ \ (EX r X. r <= K*K & X <= K & well_ord(X,r) & i = ordertype(X,r))"; by (fast_tac subset_cs 1); (*It's vital to avoid reasoning about <=*) val jump_cardinal_iff = result(); (*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*) goal CardinalArith.thy "!!K. Ord(K) ==> K < jump_cardinal(K)"; by (resolve_tac [Ord_jump_cardinal RSN (2,ltI)] 1); by (resolve_tac [jump_cardinal_iff RS iffD2] 1); by (REPEAT_FIRST (ares_tac [exI, conjI, well_ord_Memrel])); by (resolve_tac [subset_refl] 2); by (asm_simp_tac (ZF_ss addsimps [Memrel_def, subset_iff]) 1); by (asm_simp_tac (ZF_ss addsimps [ordertype_Memrel]) 1); val K_lt_jump_cardinal = result(); (*The proof by contradiction: the bijection f yields a wellordering of X whose ordertype is jump_cardinal(K). *) goal CardinalArith.thy "!!K. [| well_ord(X,r); r <= K * K; X <= K; \ \ f : bij(ordertype(X,r), jump_cardinal(K)) \ \ |] ==> jump_cardinal(K) : jump_cardinal(K)"; by (subgoal_tac "f O ordermap(X,r): bij(X, jump_cardinal(K))" 1); by (REPEAT (ares_tac [comp_bij, ordermap_bij] 2)); by (resolve_tac [jump_cardinal_iff RS iffD2] 1); by (REPEAT_FIRST (resolve_tac [exI, conjI])); by (rtac ([rvimage_type, Sigma_mono] MRS subset_trans) 1); by (REPEAT (assume_tac 1)); by (etac (bij_is_inj RS well_ord_rvimage) 1); by (rtac (Ord_jump_cardinal RS well_ord_Memrel) 1); by (asm_simp_tac (ZF_ss addsimps [well_ord_Memrel RSN (2, bij_ordertype_vimage), ordertype_Memrel, Ord_jump_cardinal]) 1); val Card_jump_cardinal_lemma = result(); (*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*) goal CardinalArith.thy "Card(jump_cardinal(K))"; by (rtac (Ord_jump_cardinal RS CardI) 1); by (rewrite_goals_tac [eqpoll_def]); by (safe_tac (ZF_cs addSDs [ltD, jump_cardinal_iff RS iffD1])); by (REPEAT (ares_tac [Card_jump_cardinal_lemma RS mem_irrefl] 1)); val Card_jump_cardinal = result(); (*** Basic properties of successor cardinals ***) goalw CardinalArith.thy [csucc_def] "!!K. Ord(K) ==> Card(csucc(K)) & K < csucc(K)"; by (rtac LeastI 1); by (REPEAT (ares_tac [conjI, Card_jump_cardinal, K_lt_jump_cardinal, Ord_jump_cardinal] 1)); val csucc_basic = result(); val Card_csucc = csucc_basic RS conjunct1 |> standard; val lt_csucc = csucc_basic RS conjunct2 |> standard; goal CardinalArith.thy "!!K. Ord(K) ==> 0 < csucc(K)"; by (resolve_tac [[Ord_0_le, lt_csucc] MRS lt_trans1] 1); by (REPEAT (assume_tac 1)); val Ord_0_lt_csucc = result(); goalw CardinalArith.thy [csucc_def] "!!K L. [| Card(L); K<L |] ==> csucc(K) le L"; by (rtac Least_le 1); by (REPEAT (ares_tac [conjI, Card_is_Ord] 1)); val csucc_le = result(); goal CardinalArith.thy "!!K. [| Ord(i); Card(K) |] ==> i < csucc(K) <-> |i| le K"; by (resolve_tac [iffI] 1); by (resolve_tac [Card_lt_imp_lt] 2); by (eresolve_tac [lt_trans1] 2); by (REPEAT (ares_tac [lt_csucc, Card_csucc, Card_is_Ord] 2)); by (resolve_tac [notI RS not_lt_imp_le] 1); by (resolve_tac [Card_cardinal RS csucc_le RS lt_trans1 RS lt_irrefl] 1); by (assume_tac 1); by (resolve_tac [Ord_cardinal_le RS lt_trans1] 1); by (REPEAT (ares_tac [Ord_cardinal] 1 ORELSE eresolve_tac [ltE, Card_is_Ord] 1)); val lt_csucc_iff = result(); goal CardinalArith.thy "!!K' K. [| Card(K'); Card(K) |] ==> K' < csucc(K) <-> K' le K"; by (asm_simp_tac (ZF_ss addsimps [lt_csucc_iff, Card_cardinal_eq, Card_is_Ord]) 1); val Card_lt_csucc_iff = result(); goalw CardinalArith.thy [InfCard_def] "!!K. InfCard(K) ==> InfCard(csucc(K))"; by (asm_simp_tac (ZF_ss addsimps [Card_csucc, Card_is_Ord, lt_csucc RS leI RSN (2,le_trans)]) 1); val InfCard_csucc = result(); val Limit_csucc = InfCard_csucc RS InfCard_is_Limit |> standard;