author | paulson |
Sun, 16 Jun 2002 11:58:54 +0200 | |
changeset 13216 | 6104bd4088a2 |
parent 13161 | a40db0418145 |
child 13221 | e29378f347e4 |
permissions | -rw-r--r-- |
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(* Title: ZF/CardinalArith.thy |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1994 University of Cambridge |
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Cardinal arithmetic -- WITHOUT the Axiom of Choice |
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Note: Could omit proving the algebraic laws for cardinal addition and |
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multiplication. On finite cardinals these operations coincide with |
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addition and multiplication of natural numbers; on infinite cardinals they |
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coincide with union (maximum). Either way we get most laws for free. |
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*) |
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theory CardinalArith = Cardinal + OrderArith + ArithSimp + Finite: |
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constdefs |
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InfCard :: "i=>o" |
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"InfCard(i) == Card(i) & nat le i" |
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cmult :: "[i,i]=>i" (infixl "|*|" 70) |
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"i |*| j == |i*j|" |
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cadd :: "[i,i]=>i" (infixl "|+|" 65) |
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"i |+| j == |i+j|" |
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csquare_rel :: "i=>i" |
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"csquare_rel(K) == |
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rvimage(K*K, |
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lam <x,y>:K*K. <x Un y, x, y>, |
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rmult(K,Memrel(K), K*K, rmult(K,Memrel(K), K,Memrel(K))))" |
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(*This def is more complex than Kunen's but it more easily proved to |
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be a cardinal*) |
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jump_cardinal :: "i=>i" |
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"jump_cardinal(K) == |
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UN X:Pow(K). {z. r: Pow(K*K), well_ord(X,r) & z = ordertype(X,r)}" |
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(*needed because jump_cardinal(K) might not be the successor of K*) |
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csucc :: "i=>i" |
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"csucc(K) == LEAST L. Card(L) & K<L" |
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syntax (xsymbols) |
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"op |+|" :: "[i,i] => i" (infixl "\<oplus>" 65) |
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"op |*|" :: "[i,i] => i" (infixl "\<otimes>" 70) |
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(*** The following really belong early in the development ***) |
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lemma relation_converse_converse [simp]: |
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"relation(r) ==> converse(converse(r)) = r" |
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by (simp add: relation_def, blast) |
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lemma relation_restrict [simp]: "relation(restrict(r,A))" |
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by (simp add: restrict_def relation_def, blast) |
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(*** The following really belong in Order ***) |
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lemma subset_ord_iso_Memrel: |
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"[| f: ord_iso(A,Memrel(B),C,r); A<=B |] ==> f: ord_iso(A,Memrel(A),C,r)" |
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apply (frule ord_iso_is_bij [THEN bij_is_fun, THEN fun_is_rel]) |
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apply (frule ord_iso_trans [OF id_ord_iso_Memrel], assumption) |
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apply (simp add: right_comp_id) |
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done |
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lemma restrict_ord_iso: |
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"[| f \<in> ord_iso(i, Memrel(i), Order.pred(A,a,r), r); a \<in> A; j < i; |
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trans[A](r) |] |
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==> restrict(f,j) \<in> ord_iso(j, Memrel(j), Order.pred(A,f`j,r), r)" |
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apply (frule ltD) |
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apply (frule ord_iso_is_bij [THEN bij_is_fun, THEN apply_type], assumption) |
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apply (frule ord_iso_restrict_pred, assumption) |
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apply (simp add: pred_iff trans_pred_pred_eq lt_pred_Memrel) |
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apply (blast intro!: subset_ord_iso_Memrel le_imp_subset [OF leI]) |
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done |
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lemma restrict_ord_iso2: |
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"[| f \<in> ord_iso(Order.pred(A,a,r), r, i, Memrel(i)); a \<in> A; |
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j < i; trans[A](r) |] |
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==> converse(restrict(converse(f), j)) |
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\<in> ord_iso(Order.pred(A, converse(f)`j, r), r, j, Memrel(j))" |
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by (blast intro: restrict_ord_iso ord_iso_sym ltI) |
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(*** The following really belong in OrderType ***) |
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lemma oadd_eq_0_iff: "[| Ord(i); Ord(j) |] ==> (i ++ j) = 0 <-> i=0 & j=0" |
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apply (erule trans_induct3 [of j]) |
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apply (simp_all add: oadd_Limit) |
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apply (simp add: Union_empty_iff Limit_def lt_def, blast) |
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done |
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lemma oadd_eq_lt_iff: "[| Ord(i); Ord(j) |] ==> 0 < (i ++ j) <-> 0<i | 0<j" |
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by (simp add: Ord_0_lt_iff [symmetric] oadd_eq_0_iff) |
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lemma oadd_lt_self: "[| Ord(i); 0<j |] ==> i < i++j" |
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apply (rule lt_trans2) |
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apply (erule le_refl) |
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apply (simp only: lt_Ord2 oadd_1 [of i, symmetric]) |
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apply (blast intro: succ_leI oadd_le_mono) |
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done |
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lemma oadd_LimitI: "[| Ord(i); Limit(j) |] ==> Limit(i ++ j)" |
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apply (simp add: oadd_Limit) |
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apply (frule Limit_has_1 [THEN ltD]) |
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apply (rule increasing_LimitI) |
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apply (rule Ord_0_lt) |
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apply (blast intro: Ord_in_Ord [OF Limit_is_Ord]) |
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apply (force simp add: Union_empty_iff oadd_eq_0_iff |
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Limit_is_Ord [of j, THEN Ord_in_Ord], auto) |
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apply (rule_tac x="succ(x)" in bexI) |
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apply (simp add: ltI Limit_is_Ord [of j, THEN Ord_in_Ord]) |
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apply (simp add: Limit_def lt_def) |
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done |
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(*** The following really belong in Cardinal ***) |
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lemma lesspoll_not_refl: "~ (i lesspoll i)" |
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by (simp add: lesspoll_def) |
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lemma lesspoll_irrefl [elim!]: "i lesspoll i ==> P" |
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by (simp add: lesspoll_def) |
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lemma Card_Union [simp,intro,TC]: "(ALL x:A. Card(x)) ==> Card(Union(A))" |
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apply (rule CardI) |
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apply (simp add: Card_is_Ord) |
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apply (clarify dest!: ltD) |
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apply (drule bspec, assumption) |
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apply (frule lt_Card_imp_lesspoll, blast intro: ltI Card_is_Ord) |
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apply (drule eqpoll_sym [THEN eqpoll_imp_lepoll]) |
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apply (drule lesspoll_trans1, assumption) |
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apply (subgoal_tac "B \<lesssim> \<Union>A") |
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apply (drule lesspoll_trans1, assumption, blast) |
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apply (blast intro: subset_imp_lepoll) |
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done |
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lemma Card_UN: |
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"(!!x. x:A ==> Card(K(x))) ==> Card(UN x:A. K(x))" |
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by (blast intro: Card_Union) |
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lemma Card_OUN [simp,intro,TC]: |
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"(!!x. x:A ==> Card(K(x))) ==> Card(UN x<A. K(x))" |
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by (simp add: OUnion_def Card_0) |
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lemma n_lesspoll_nat: "n \<in> nat ==> n \<prec> nat" |
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apply (unfold lesspoll_def) |
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apply (rule conjI) |
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apply (erule OrdmemD [THEN subset_imp_lepoll], rule Ord_nat) |
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apply (rule notI) |
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apply (erule eqpollE) |
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apply (rule succ_lepoll_natE) |
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apply (blast intro: nat_succI [THEN OrdmemD, THEN subset_imp_lepoll] |
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lepoll_trans, assumption) |
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done |
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lemma in_Card_imp_lesspoll: "[| Card(K); b \<in> K |] ==> b \<prec> K" |
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apply (unfold lesspoll_def) |
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apply (simp add: Card_iff_initial) |
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apply (fast intro!: le_imp_lepoll ltI leI) |
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done |
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lemma succ_lepoll_imp_not_empty: "succ(x) \<lesssim> y ==> y \<noteq> 0" |
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by (fast dest!: lepoll_0_is_0) |
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lemma eqpoll_succ_imp_not_empty: "x \<approx> succ(n) ==> x \<noteq> 0" |
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by (fast elim!: eqpoll_sym [THEN eqpoll_0_is_0, THEN succ_neq_0]) |
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lemma Finite_Fin_lemma [rule_format]: |
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"n \<in> nat ==> \<forall>A. (A\<approx>n & A \<subseteq> X) --> A \<in> Fin(X)" |
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apply (induct_tac "n") |
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apply (rule allI) |
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apply (fast intro!: Fin.emptyI dest!: eqpoll_imp_lepoll [THEN lepoll_0_is_0]) |
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apply (rule allI) |
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apply (rule impI) |
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apply (erule conjE) |
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apply (rule eqpoll_succ_imp_not_empty [THEN not_emptyE], assumption) |
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apply (frule Diff_sing_eqpoll, assumption) |
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apply (erule allE) |
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apply (erule impE, fast) |
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apply (drule subsetD, assumption) |
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apply (drule Fin.consI, assumption) |
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apply (simp add: cons_Diff) |
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done |
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lemma Finite_Fin: "[| Finite(A); A \<subseteq> X |] ==> A \<in> Fin(X)" |
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by (unfold Finite_def, blast intro: Finite_Fin_lemma) |
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lemma lesspoll_lemma: |
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"[| ~ A \<prec> B; C \<prec> B |] ==> A - C \<noteq> 0" |
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apply (unfold lesspoll_def) |
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apply (fast dest!: Diff_eq_0_iff [THEN iffD1, THEN subset_imp_lepoll] |
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intro!: eqpollI elim: notE |
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elim!: eqpollE lepoll_trans) |
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done |
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lemma eqpoll_imp_Finite_iff: "A \<approx> B ==> Finite(A) <-> Finite(B)" |
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apply (unfold Finite_def) |
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apply (blast intro: eqpoll_trans eqpoll_sym) |
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done |
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(*** Cardinal addition ***) |
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(** Cardinal addition is commutative **) |
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lemma sum_commute_eqpoll: "A+B \<approx> B+A" |
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apply (unfold eqpoll_def) |
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apply (rule exI) |
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apply (rule_tac c = "case(Inr,Inl)" and d = "case(Inr,Inl)" in lam_bijective) |
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apply auto |
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done |
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lemma cadd_commute: "i |+| j = j |+| i" |
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apply (unfold cadd_def) |
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apply (rule sum_commute_eqpoll [THEN cardinal_cong]) |
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done |
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(** Cardinal addition is associative **) |
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lemma sum_assoc_eqpoll: "(A+B)+C \<approx> A+(B+C)" |
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apply (unfold eqpoll_def) |
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apply (rule exI) |
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apply (rule sum_assoc_bij) |
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done |
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(*Unconditional version requires AC*) |
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lemma well_ord_cadd_assoc: |
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"[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] |
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==> (i |+| j) |+| k = i |+| (j |+| k)" |
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apply (unfold cadd_def) |
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apply (rule cardinal_cong) |
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apply (rule eqpoll_trans) |
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apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl]) |
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apply (blast intro: well_ord_radd elim:) |
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apply (rule sum_assoc_eqpoll [THEN eqpoll_trans]) |
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apply (rule eqpoll_sym) |
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apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll]) |
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apply (blast intro: well_ord_radd elim:) |
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done |
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(** 0 is the identity for addition **) |
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lemma sum_0_eqpoll: "0+A \<approx> A" |
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apply (unfold eqpoll_def) |
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apply (rule exI) |
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apply (rule bij_0_sum) |
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done |
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lemma cadd_0 [simp]: "Card(K) ==> 0 |+| K = K" |
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apply (unfold cadd_def) |
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apply (simp add: sum_0_eqpoll [THEN cardinal_cong] Card_cardinal_eq) |
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done |
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(** Addition by another cardinal **) |
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lemma sum_lepoll_self: "A \<lesssim> A+B" |
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apply (unfold lepoll_def inj_def) |
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apply (rule_tac x = "lam x:A. Inl (x) " in exI) |
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apply (simp (no_asm_simp)) |
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done |
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(*Could probably weaken the premises to well_ord(K,r), or removing using AC*) |
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lemma cadd_le_self: |
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"[| Card(K); Ord(L) |] ==> K le (K |+| L)" |
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apply (unfold cadd_def) |
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apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le]) |
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apply assumption; |
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apply (rule_tac [2] sum_lepoll_self) |
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apply (blast intro: well_ord_radd well_ord_Memrel Card_is_Ord) |
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done |
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(** Monotonicity of addition **) |
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lemma sum_lepoll_mono: |
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"[| A \<lesssim> C; B \<lesssim> D |] ==> A + B \<lesssim> C + D" |
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apply (unfold lepoll_def) |
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apply (elim exE); |
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apply (rule_tac x = "lam z:A+B. case (%w. Inl(f`w), %y. Inr(fa`y), z)" in exI) |
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apply (rule_tac d = "case (%w. Inl(converse(f) `w), %y. Inr(converse(fa) `y))" |
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in lam_injective) |
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apply (typecheck add: inj_is_fun) |
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apply auto |
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done |
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lemma cadd_le_mono: |
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"[| K' le K; L' le L |] ==> (K' |+| L') le (K |+| L)" |
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apply (unfold cadd_def) |
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apply (safe dest!: le_subset_iff [THEN iffD1]) |
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apply (rule well_ord_lepoll_imp_Card_le) |
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apply (blast intro: well_ord_radd well_ord_Memrel) |
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apply (blast intro: sum_lepoll_mono subset_imp_lepoll) |
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done |
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(** Addition of finite cardinals is "ordinary" addition **) |
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(*????????????????upair.ML*) |
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lemma eq_imp_not_mem: "a=A ==> a ~: A" |
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apply (blast intro: elim: mem_irrefl); |
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done |
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lemma sum_succ_eqpoll: "succ(A)+B \<approx> succ(A+B)" |
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apply (unfold eqpoll_def) |
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apply (rule exI) |
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apply (rule_tac c = "%z. if z=Inl (A) then A+B else z" |
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and d = "%z. if z=A+B then Inl (A) else z" in lam_bijective) |
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apply (simp_all) |
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apply (blast dest: sym [THEN eq_imp_not_mem] elim: mem_irrefl)+ |
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done |
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(*Pulling the succ(...) outside the |...| requires m, n: nat *) |
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(*Unconditional version requires AC*) |
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lemma cadd_succ_lemma: |
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"[| Ord(m); Ord(n) |] ==> succ(m) |+| n = |succ(m |+| n)|" |
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apply (unfold cadd_def) |
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apply (rule sum_succ_eqpoll [THEN cardinal_cong, THEN trans]) |
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apply (rule succ_eqpoll_cong [THEN cardinal_cong]) |
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apply (rule well_ord_cardinal_eqpoll [THEN eqpoll_sym]) |
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apply (blast intro: well_ord_radd well_ord_Memrel) |
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done |
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lemma nat_cadd_eq_add: "[| m: nat; n: nat |] ==> m |+| n = m#+n" |
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apply (induct_tac "m") |
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apply (simp add: nat_into_Card [THEN cadd_0]) |
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apply (simp add: cadd_succ_lemma nat_into_Card [THEN Card_cardinal_eq]) |
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done |
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(*** Cardinal multiplication ***) |
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(** Cardinal multiplication is commutative **) |
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(*Easier to prove the two directions separately*) |
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lemma prod_commute_eqpoll: "A*B \<approx> B*A" |
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apply (unfold eqpoll_def) |
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apply (rule exI) |
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apply (rule_tac c = "%<x,y>.<y,x>" and d = "%<x,y>.<y,x>" in lam_bijective) |
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apply (auto ); |
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done |
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lemma cmult_commute: "i |*| j = j |*| i" |
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apply (unfold cmult_def) |
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apply (rule prod_commute_eqpoll [THEN cardinal_cong]) |
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done |
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(** Cardinal multiplication is associative **) |
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lemma prod_assoc_eqpoll: "(A*B)*C \<approx> A*(B*C)" |
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apply (unfold eqpoll_def) |
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apply (rule exI) |
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apply (rule prod_assoc_bij) |
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done |
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(*Unconditional version requires AC*) |
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lemma well_ord_cmult_assoc: |
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"[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] |
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==> (i |*| j) |*| k = i |*| (j |*| k)" |
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apply (unfold cmult_def) |
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apply (rule cardinal_cong) |
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apply (rule eqpoll_trans); |
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apply (rule prod_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl]) |
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apply (blast intro: well_ord_rmult) |
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apply (rule prod_assoc_eqpoll [THEN eqpoll_trans]) |
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apply (rule eqpoll_sym); |
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apply (rule prod_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll]) |
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apply (blast intro: well_ord_rmult) |
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done |
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(** Cardinal multiplication distributes over addition **) |
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lemma sum_prod_distrib_eqpoll: "(A+B)*C \<approx> (A*C)+(B*C)" |
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apply (unfold eqpoll_def) |
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apply (rule exI) |
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apply (rule sum_prod_distrib_bij) |
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done |
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lemma well_ord_cadd_cmult_distrib: |
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"[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] |
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==> (i |+| j) |*| k = (i |*| k) |+| (j |*| k)" |
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apply (unfold cadd_def cmult_def) |
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apply (rule cardinal_cong) |
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381 |
apply (rule eqpoll_trans); |
|
382 |
apply (rule prod_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl]) |
|
383 |
apply (blast intro: well_ord_radd) |
|
384 |
apply (rule sum_prod_distrib_eqpoll [THEN eqpoll_trans]) |
|
385 |
apply (rule eqpoll_sym); |
|
386 |
apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll |
|
387 |
well_ord_cardinal_eqpoll]) |
|
388 |
apply (blast intro: well_ord_rmult)+ |
|
389 |
done |
|
390 |
||
391 |
(** Multiplication by 0 yields 0 **) |
|
392 |
||
393 |
lemma prod_0_eqpoll: "0*A \<approx> 0" |
|
394 |
apply (unfold eqpoll_def) |
|
395 |
apply (rule exI) |
|
396 |
apply (rule lam_bijective) |
|
397 |
apply safe |
|
398 |
done |
|
399 |
||
400 |
lemma cmult_0 [simp]: "0 |*| i = 0" |
|
401 |
apply (simp add: cmult_def prod_0_eqpoll [THEN cardinal_cong]) |
|
402 |
done |
|
403 |
||
404 |
(** 1 is the identity for multiplication **) |
|
405 |
||
406 |
lemma prod_singleton_eqpoll: "{x}*A \<approx> A" |
|
407 |
apply (unfold eqpoll_def) |
|
408 |
apply (rule exI) |
|
409 |
apply (rule singleton_prod_bij [THEN bij_converse_bij]) |
|
410 |
done |
|
411 |
||
412 |
lemma cmult_1 [simp]: "Card(K) ==> 1 |*| K = K" |
|
413 |
apply (unfold cmult_def succ_def) |
|
414 |
apply (simp add: prod_singleton_eqpoll [THEN cardinal_cong] Card_cardinal_eq) |
|
415 |
done |
|
416 |
||
417 |
(*** Some inequalities for multiplication ***) |
|
418 |
||
419 |
lemma prod_square_lepoll: "A \<lesssim> A*A" |
|
420 |
apply (unfold lepoll_def inj_def) |
|
421 |
apply (rule_tac x = "lam x:A. <x,x>" in exI) |
|
422 |
apply (simp (no_asm)) |
|
423 |
done |
|
424 |
||
425 |
(*Could probably weaken the premise to well_ord(K,r), or remove using AC*) |
|
426 |
lemma cmult_square_le: "Card(K) ==> K le K |*| K" |
|
427 |
apply (unfold cmult_def) |
|
428 |
apply (rule le_trans) |
|
429 |
apply (rule_tac [2] well_ord_lepoll_imp_Card_le) |
|
430 |
apply (rule_tac [3] prod_square_lepoll) |
|
431 |
apply (simp (no_asm_simp) add: le_refl Card_is_Ord Card_cardinal_eq) |
|
432 |
apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord); |
|
433 |
done |
|
434 |
||
435 |
(** Multiplication by a non-zero cardinal **) |
|
436 |
||
437 |
lemma prod_lepoll_self: "b: B ==> A \<lesssim> A*B" |
|
438 |
apply (unfold lepoll_def inj_def) |
|
439 |
apply (rule_tac x = "lam x:A. <x,b>" in exI) |
|
440 |
apply (simp (no_asm_simp)) |
|
441 |
done |
|
442 |
||
443 |
(*Could probably weaken the premises to well_ord(K,r), or removing using AC*) |
|
444 |
lemma cmult_le_self: |
|
445 |
"[| Card(K); Ord(L); 0<L |] ==> K le (K |*| L)" |
|
446 |
apply (unfold cmult_def) |
|
447 |
apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le]) |
|
448 |
apply assumption; |
|
449 |
apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord) |
|
450 |
apply (blast intro: prod_lepoll_self ltD) |
|
451 |
done |
|
452 |
||
453 |
(** Monotonicity of multiplication **) |
|
454 |
||
455 |
lemma prod_lepoll_mono: |
|
456 |
"[| A \<lesssim> C; B \<lesssim> D |] ==> A * B \<lesssim> C * D" |
|
457 |
apply (unfold lepoll_def) |
|
458 |
apply (elim exE); |
|
459 |
apply (rule_tac x = "lam <w,y>:A*B. <f`w, fa`y>" in exI) |
|
460 |
apply (rule_tac d = "%<w,y>. <converse (f) `w, converse (fa) `y>" |
|
461 |
in lam_injective) |
|
462 |
apply (typecheck add: inj_is_fun) |
|
463 |
apply auto |
|
464 |
done |
|
465 |
||
466 |
lemma cmult_le_mono: |
|
467 |
"[| K' le K; L' le L |] ==> (K' |*| L') le (K |*| L)" |
|
468 |
apply (unfold cmult_def) |
|
469 |
apply (safe dest!: le_subset_iff [THEN iffD1]) |
|
470 |
apply (rule well_ord_lepoll_imp_Card_le) |
|
471 |
apply (blast intro: well_ord_rmult well_ord_Memrel) |
|
472 |
apply (blast intro: prod_lepoll_mono subset_imp_lepoll) |
|
473 |
done |
|
474 |
||
475 |
(*** Multiplication of finite cardinals is "ordinary" multiplication ***) |
|
476 |
||
477 |
lemma prod_succ_eqpoll: "succ(A)*B \<approx> B + A*B" |
|
478 |
apply (unfold eqpoll_def) |
|
479 |
apply (rule exI); |
|
480 |
apply (rule_tac c = "%<x,y>. if x=A then Inl (y) else Inr (<x,y>)" |
|
481 |
and d = "case (%y. <A,y>, %z. z)" in lam_bijective) |
|
482 |
apply safe |
|
483 |
apply (simp_all add: succI2 if_type mem_imp_not_eq) |
|
484 |
done |
|
485 |
||
486 |
(*Unconditional version requires AC*) |
|
487 |
lemma cmult_succ_lemma: |
|
488 |
"[| Ord(m); Ord(n) |] ==> succ(m) |*| n = n |+| (m |*| n)" |
|
489 |
apply (unfold cmult_def cadd_def) |
|
490 |
apply (rule prod_succ_eqpoll [THEN cardinal_cong, THEN trans]) |
|
491 |
apply (rule cardinal_cong [symmetric]) |
|
492 |
apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll]) |
|
493 |
apply (blast intro: well_ord_rmult well_ord_Memrel) |
|
494 |
done |
|
495 |
||
496 |
lemma nat_cmult_eq_mult: "[| m: nat; n: nat |] ==> m |*| n = m#*n" |
|
497 |
apply (induct_tac "m") |
|
498 |
apply (simp (no_asm_simp)) |
|
499 |
apply (simp (no_asm_simp) add: cmult_succ_lemma nat_cadd_eq_add) |
|
500 |
done |
|
501 |
||
502 |
lemma cmult_2: "Card(n) ==> 2 |*| n = n |+| n" |
|
503 |
apply (simp add: cmult_succ_lemma Card_is_Ord cadd_commute [of _ 0]) |
|
504 |
done |
|
505 |
||
506 |
lemma sum_lepoll_prod: "2 \<lesssim> C ==> B+B \<lesssim> C*B" |
|
507 |
apply (rule lepoll_trans); |
|
508 |
apply (rule sum_eq_2_times [THEN equalityD1, THEN subset_imp_lepoll]) |
|
509 |
apply (erule prod_lepoll_mono) |
|
510 |
apply (rule lepoll_refl); |
|
511 |
done |
|
512 |
||
513 |
lemma lepoll_imp_sum_lepoll_prod: "[| A \<lesssim> B; 2 \<lesssim> A |] ==> A+B \<lesssim> A*B" |
|
514 |
apply (blast intro: sum_lepoll_mono sum_lepoll_prod lepoll_trans lepoll_refl) |
|
515 |
done |
|
516 |
||
517 |
||
518 |
(*** Infinite Cardinals are Limit Ordinals ***) |
|
519 |
||
520 |
(*This proof is modelled upon one assuming nat<=A, with injection |
|
521 |
lam z:cons(u,A). if z=u then 0 else if z : nat then succ(z) else z |
|
522 |
and inverse %y. if y:nat then nat_case(u, %z. z, y) else y. \ |
|
523 |
If f: inj(nat,A) then range(f) behaves like the natural numbers.*) |
|
524 |
lemma nat_cons_lepoll: "nat \<lesssim> A ==> cons(u,A) \<lesssim> A" |
|
525 |
apply (unfold lepoll_def) |
|
526 |
apply (erule exE) |
|
527 |
apply (rule_tac x = |
|
528 |
"lam z:cons (u,A). |
|
529 |
if z=u then f`0 |
|
530 |
else if z: range (f) then f`succ (converse (f) `z) else z" |
|
531 |
in exI) |
|
532 |
apply (rule_tac d = |
|
533 |
"%y. if y: range(f) then nat_case (u, %z. f`z, converse(f) `y) |
|
534 |
else y" |
|
535 |
in lam_injective) |
|
536 |
apply (fast intro!: if_type apply_type intro: inj_is_fun inj_converse_fun) |
|
537 |
apply (simp add: inj_is_fun [THEN apply_rangeI] |
|
538 |
inj_converse_fun [THEN apply_rangeI] |
|
539 |
inj_converse_fun [THEN apply_funtype]) |
|
540 |
done |
|
541 |
||
542 |
lemma nat_cons_eqpoll: "nat \<lesssim> A ==> cons(u,A) \<approx> A" |
|
543 |
apply (erule nat_cons_lepoll [THEN eqpollI]) |
|
544 |
apply (rule subset_consI [THEN subset_imp_lepoll]) |
|
545 |
done |
|
546 |
||
547 |
(*Specialized version required below*) |
|
548 |
lemma nat_succ_eqpoll: "nat <= A ==> succ(A) \<approx> A" |
|
549 |
apply (unfold succ_def) |
|
550 |
apply (erule subset_imp_lepoll [THEN nat_cons_eqpoll]) |
|
551 |
done |
|
552 |
||
553 |
lemma InfCard_nat: "InfCard(nat)" |
|
554 |
apply (unfold InfCard_def) |
|
555 |
apply (blast intro: Card_nat le_refl Card_is_Ord) |
|
556 |
done |
|
557 |
||
558 |
lemma InfCard_is_Card: "InfCard(K) ==> Card(K)" |
|
559 |
apply (unfold InfCard_def) |
|
560 |
apply (erule conjunct1) |
|
561 |
done |
|
562 |
||
563 |
lemma InfCard_Un: |
|
564 |
"[| InfCard(K); Card(L) |] ==> InfCard(K Un L)" |
|
565 |
apply (unfold InfCard_def) |
|
566 |
apply (simp add: Card_Un Un_upper1_le [THEN [2] le_trans] Card_is_Ord) |
|
567 |
done |
|
568 |
||
569 |
(*Kunen's Lemma 10.11*) |
|
570 |
lemma InfCard_is_Limit: "InfCard(K) ==> Limit(K)" |
|
571 |
apply (unfold InfCard_def) |
|
572 |
apply (erule conjE) |
|
573 |
apply (frule Card_is_Ord) |
|
574 |
apply (rule ltI [THEN non_succ_LimitI]) |
|
575 |
apply (erule le_imp_subset [THEN subsetD]) |
|
576 |
apply (safe dest!: Limit_nat [THEN Limit_le_succD]) |
|
577 |
apply (unfold Card_def) |
|
578 |
apply (drule trans) |
|
579 |
apply (erule le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong]) |
|
580 |
apply (erule Ord_cardinal_le [THEN lt_trans2, THEN lt_irrefl]) |
|
581 |
apply (rule le_eqI) |
|
582 |
apply assumption; |
|
583 |
apply (rule Ord_cardinal) |
|
584 |
done |
|
585 |
||
586 |
||
587 |
(*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***) |
|
588 |
||
589 |
(*A general fact about ordermap*) |
|
590 |
lemma ordermap_eqpoll_pred: |
|
591 |
"[| well_ord(A,r); x:A |] ==> ordermap(A,r)`x \<approx> pred(A,x,r)" |
|
592 |
apply (unfold eqpoll_def) |
|
593 |
apply (rule exI) |
|
594 |
apply (simp (no_asm_simp) add: ordermap_eq_image well_ord_is_wf) |
|
595 |
apply (erule ordermap_bij [THEN bij_is_inj, THEN restrict_bij, THEN bij_converse_bij]) |
|
596 |
apply (rule pred_subset) |
|
597 |
done |
|
598 |
||
599 |
(** Establishing the well-ordering **) |
|
600 |
||
601 |
lemma csquare_lam_inj: |
|
602 |
"Ord(K) ==> (lam <x,y>:K*K. <x Un y, x, y>) : inj(K*K, K*K*K)" |
|
603 |
apply (unfold inj_def) |
|
604 |
apply (force intro: lam_type Un_least_lt [THEN ltD] ltI) |
|
605 |
done |
|
606 |
||
607 |
lemma well_ord_csquare: "Ord(K) ==> well_ord(K*K, csquare_rel(K))" |
|
608 |
apply (unfold csquare_rel_def) |
|
609 |
apply (rule csquare_lam_inj [THEN well_ord_rvimage]) |
|
610 |
apply assumption; |
|
611 |
apply (blast intro: well_ord_rmult well_ord_Memrel) |
|
612 |
done |
|
613 |
||
614 |
(** Characterising initial segments of the well-ordering **) |
|
615 |
||
616 |
lemma csquareD: |
|
617 |
"[| <<x,y>, <z,z>> : csquare_rel(K); x<K; y<K; z<K |] ==> x le z & y le z" |
|
618 |
apply (unfold csquare_rel_def) |
|
619 |
apply (erule rev_mp) |
|
620 |
apply (elim ltE) |
|
621 |
apply (simp (no_asm_simp) add: rvimage_iff Un_absorb Un_least_mem_iff ltD) |
|
622 |
apply (safe elim!: mem_irrefl intro!: Un_upper1_le Un_upper2_le) |
|
623 |
apply (simp_all (no_asm_simp) add: lt_def succI2) |
|
624 |
done |
|
625 |
||
626 |
lemma pred_csquare_subset: |
|
627 |
"z<K ==> pred(K*K, <z,z>, csquare_rel(K)) <= succ(z)*succ(z)" |
|
628 |
apply (unfold Order.pred_def) |
|
629 |
apply (safe del: SigmaI succCI) |
|
630 |
apply (erule csquareD [THEN conjE]) |
|
631 |
apply (unfold lt_def) |
|
632 |
apply (auto ); |
|
633 |
done |
|
634 |
||
635 |
lemma csquare_ltI: |
|
636 |
"[| x<z; y<z; z<K |] ==> <<x,y>, <z,z>> : csquare_rel(K)" |
|
637 |
apply (unfold csquare_rel_def) |
|
638 |
apply (subgoal_tac "x<K & y<K") |
|
639 |
prefer 2 apply (blast intro: lt_trans) |
|
640 |
apply (elim ltE) |
|
641 |
apply (simp (no_asm_simp) add: rvimage_iff Un_absorb Un_least_mem_iff ltD) |
|
642 |
done |
|
643 |
||
644 |
(*Part of the traditional proof. UNUSED since it's harder to prove & apply *) |
|
645 |
lemma csquare_or_eqI: |
|
646 |
"[| x le z; y le z; z<K |] ==> <<x,y>, <z,z>> : csquare_rel(K) | x=z & y=z" |
|
647 |
apply (unfold csquare_rel_def) |
|
648 |
apply (subgoal_tac "x<K & y<K") |
|
649 |
prefer 2 apply (blast intro: lt_trans1) |
|
650 |
apply (elim ltE) |
|
651 |
apply (simp (no_asm_simp) add: rvimage_iff Un_absorb Un_least_mem_iff ltD) |
|
652 |
apply (elim succE) |
|
653 |
apply (simp_all (no_asm_simp) add: subset_Un_iff [THEN iff_sym] subset_Un_iff2 [THEN iff_sym] OrdmemD) |
|
654 |
done |
|
655 |
||
656 |
(** The cardinality of initial segments **) |
|
657 |
||
658 |
lemma ordermap_z_lt: |
|
659 |
"[| Limit(K); x<K; y<K; z=succ(x Un y) |] ==> |
|
660 |
ordermap(K*K, csquare_rel(K)) ` <x,y> < |
|
661 |
ordermap(K*K, csquare_rel(K)) ` <z,z>" |
|
662 |
apply (subgoal_tac "z<K & well_ord (K*K, csquare_rel (K))") |
|
663 |
prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ |
|
664 |
Limit_is_Ord [THEN well_ord_csquare]) |
|
665 |
apply (clarify ); |
|
666 |
apply (rule csquare_ltI [THEN ordermap_mono, THEN ltI]) |
|
667 |
apply (erule_tac [4] well_ord_is_wf) |
|
668 |
apply (blast intro!: Un_upper1_le Un_upper2_le Ord_ordermap elim!: ltE)+ |
|
669 |
done |
|
670 |
||
671 |
(*Kunen: "each <x,y>: K*K has no more than z*z predecessors..." (page 29) *) |
|
672 |
lemma ordermap_csquare_le: |
|
673 |
"[| Limit(K); x<K; y<K; z=succ(x Un y) |] ==> |
|
674 |
| ordermap(K*K, csquare_rel(K)) ` <x,y> | le |succ(z)| |*| |succ(z)|" |
|
675 |
apply (unfold cmult_def) |
|
676 |
apply (rule well_ord_rmult [THEN well_ord_lepoll_imp_Card_le]) |
|
677 |
apply (rule Ord_cardinal [THEN well_ord_Memrel])+ |
|
678 |
apply (subgoal_tac "z<K") |
|
679 |
prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ) |
|
680 |
apply (rule ordermap_z_lt [THEN leI, THEN le_imp_lepoll, THEN lepoll_trans]) |
|
681 |
apply assumption + |
|
682 |
apply (rule ordermap_eqpoll_pred [THEN eqpoll_imp_lepoll, THEN lepoll_trans]) |
|
683 |
apply (erule Limit_is_Ord [THEN well_ord_csquare]) |
|
684 |
apply (blast intro: ltD) |
|
685 |
apply (rule pred_csquare_subset [THEN subset_imp_lepoll, THEN lepoll_trans], |
|
686 |
assumption) |
|
687 |
apply (elim ltE) |
|
688 |
apply (rule prod_eqpoll_cong [THEN eqpoll_sym, THEN eqpoll_imp_lepoll]) |
|
689 |
apply (erule Ord_succ [THEN Ord_cardinal_eqpoll])+ |
|
690 |
done |
|
691 |
||
692 |
(*Kunen: "... so the order type <= K" *) |
|
693 |
lemma ordertype_csquare_le: |
|
694 |
"[| InfCard(K); ALL y:K. InfCard(y) --> y |*| y = y |] |
|
695 |
==> ordertype(K*K, csquare_rel(K)) le K" |
|
696 |
apply (frule InfCard_is_Card [THEN Card_is_Ord]) |
|
697 |
apply (rule all_lt_imp_le) |
|
698 |
apply assumption |
|
699 |
apply (erule well_ord_csquare [THEN Ord_ordertype]) |
|
700 |
apply (rule Card_lt_imp_lt) |
|
701 |
apply (erule_tac [3] InfCard_is_Card) |
|
702 |
apply (erule_tac [2] ltE) |
|
703 |
apply (simp add: ordertype_unfold) |
|
704 |
apply (safe elim!: ltE) |
|
705 |
apply (subgoal_tac "Ord (xa) & Ord (ya)") |
|
706 |
prefer 2 apply (blast intro: Ord_in_Ord) |
|
707 |
apply (clarify ); |
|
708 |
(*??WHAT A MESS!*) |
|
709 |
apply (rule InfCard_is_Limit [THEN ordermap_csquare_le, THEN lt_trans1], |
|
710 |
(assumption | rule refl | erule ltI)+) |
|
711 |
apply (rule_tac i = "xa Un ya" and j = "nat" in Ord_linear2, |
|
712 |
simp_all add: Ord_Un Ord_nat) |
|
713 |
prefer 2 (*case nat le (xa Un ya) *) |
|
714 |
apply (simp add: le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong] |
|
715 |
le_succ_iff InfCard_def Card_cardinal Un_least_lt Ord_Un |
|
716 |
ltI nat_le_cardinal Ord_cardinal_le [THEN lt_trans1, THEN ltD]) |
|
717 |
(*the finite case: xa Un ya < nat *) |
|
718 |
apply (rule_tac j = "nat" in lt_trans2) |
|
719 |
apply (simp add: lt_def nat_cmult_eq_mult nat_succI mult_type |
|
720 |
nat_into_Card [THEN Card_cardinal_eq] Ord_nat) |
|
721 |
apply (simp add: InfCard_def) |
|
722 |
done |
|
723 |
||
724 |
(*Main result: Kunen's Theorem 10.12*) |
|
725 |
lemma InfCard_csquare_eq: "InfCard(K) ==> K |*| K = K" |
|
726 |
apply (frule InfCard_is_Card [THEN Card_is_Ord]) |
|
727 |
apply (erule rev_mp) |
|
728 |
apply (erule_tac i=K in trans_induct) |
|
729 |
apply (rule impI) |
|
730 |
apply (rule le_anti_sym) |
|
731 |
apply (erule_tac [2] InfCard_is_Card [THEN cmult_square_le]) |
|
732 |
apply (rule ordertype_csquare_le [THEN [2] le_trans]) |
|
733 |
prefer 2 apply (assumption) |
|
734 |
prefer 2 apply (assumption) |
|
735 |
apply (simp (no_asm_simp) add: cmult_def Ord_cardinal_le well_ord_csquare [THEN ordermap_bij, THEN bij_imp_eqpoll, THEN cardinal_cong] well_ord_csquare [THEN Ord_ordertype]) |
|
736 |
done |
|
737 |
||
738 |
(*Corollary for arbitrary well-ordered sets (all sets, assuming AC)*) |
|
739 |
lemma well_ord_InfCard_square_eq: |
|
740 |
"[| well_ord(A,r); InfCard(|A|) |] ==> A*A \<approx> A" |
|
741 |
apply (rule prod_eqpoll_cong [THEN eqpoll_trans]) |
|
742 |
apply (erule well_ord_cardinal_eqpoll [THEN eqpoll_sym])+ |
|
743 |
apply (rule well_ord_cardinal_eqE) |
|
744 |
apply (blast intro: Ord_cardinal well_ord_rmult well_ord_Memrel) |
|
745 |
apply assumption; |
|
746 |
apply (simp (no_asm_simp) add: cmult_def [symmetric] InfCard_csquare_eq) |
|
747 |
done |
|
748 |
||
749 |
(** Toward's Kunen's Corollary 10.13 (1) **) |
|
750 |
||
751 |
lemma InfCard_le_cmult_eq: "[| InfCard(K); L le K; 0<L |] ==> K |*| L = K" |
|
752 |
apply (rule le_anti_sym) |
|
753 |
prefer 2 |
|
754 |
apply (erule ltE, blast intro: cmult_le_self InfCard_is_Card) |
|
755 |
apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl]) |
|
756 |
apply (rule cmult_le_mono [THEN le_trans], assumption+) |
|
757 |
apply (simp add: InfCard_csquare_eq) |
|
758 |
done |
|
759 |
||
760 |
(*Corollary 10.13 (1), for cardinal multiplication*) |
|
761 |
lemma InfCard_cmult_eq: "[| InfCard(K); InfCard(L) |] ==> K |*| L = K Un L" |
|
762 |
apply (rule_tac i = "K" and j = "L" in Ord_linear_le) |
|
763 |
apply (typecheck add: InfCard_is_Card Card_is_Ord) |
|
764 |
apply (rule cmult_commute [THEN ssubst]) |
|
765 |
apply (rule Un_commute [THEN ssubst]) |
|
766 |
apply (simp_all (no_asm_simp) add: InfCard_is_Limit [THEN Limit_has_0] InfCard_le_cmult_eq subset_Un_iff2 [THEN iffD1] le_imp_subset) |
|
767 |
done |
|
768 |
||
769 |
lemma InfCard_cdouble_eq: "InfCard(K) ==> K |+| K = K" |
|
770 |
apply (simp (no_asm_simp) add: cmult_2 [symmetric] InfCard_is_Card cmult_commute) |
|
771 |
apply (simp (no_asm_simp) add: InfCard_le_cmult_eq InfCard_is_Limit Limit_has_0 Limit_has_succ) |
|
772 |
done |
|
773 |
||
774 |
(*Corollary 10.13 (1), for cardinal addition*) |
|
775 |
lemma InfCard_le_cadd_eq: "[| InfCard(K); L le K |] ==> K |+| L = K" |
|
776 |
apply (rule le_anti_sym) |
|
777 |
prefer 2 |
|
778 |
apply (erule ltE, blast intro: cadd_le_self InfCard_is_Card) |
|
779 |
apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl]) |
|
780 |
apply (rule cadd_le_mono [THEN le_trans], assumption+) |
|
781 |
apply (simp add: InfCard_cdouble_eq) |
|
782 |
done |
|
783 |
||
784 |
lemma InfCard_cadd_eq: "[| InfCard(K); InfCard(L) |] ==> K |+| L = K Un L" |
|
785 |
apply (rule_tac i = "K" and j = "L" in Ord_linear_le) |
|
786 |
apply (typecheck add: InfCard_is_Card Card_is_Ord) |
|
787 |
apply (rule cadd_commute [THEN ssubst]) |
|
788 |
apply (rule Un_commute [THEN ssubst]) |
|
789 |
apply (simp_all (no_asm_simp) add: InfCard_le_cadd_eq subset_Un_iff2 [THEN iffD1] le_imp_subset) |
|
790 |
done |
|
791 |
||
792 |
(*The other part, Corollary 10.13 (2), refers to the cardinality of the set |
|
793 |
of all n-tuples of elements of K. A better version for the Isabelle theory |
|
794 |
might be InfCard(K) ==> |list(K)| = K. |
|
795 |
*) |
|
796 |
||
797 |
(*** For every cardinal number there exists a greater one |
|
798 |
[Kunen's Theorem 10.16, which would be trivial using AC] ***) |
|
799 |
||
800 |
lemma Ord_jump_cardinal: "Ord(jump_cardinal(K))" |
|
801 |
apply (unfold jump_cardinal_def) |
|
802 |
apply (rule Ord_is_Transset [THEN [2] OrdI]) |
|
803 |
prefer 2 apply (blast intro!: Ord_ordertype) |
|
804 |
apply (unfold Transset_def) |
|
805 |
apply (safe del: subsetI) |
|
806 |
apply (simp add: ordertype_pred_unfold) |
|
807 |
apply safe |
|
808 |
apply (rule UN_I) |
|
809 |
apply (rule_tac [2] ReplaceI) |
|
810 |
prefer 4 apply (blast intro: well_ord_subset elim!: predE)+ |
|
811 |
done |
|
812 |
||
813 |
(*Allows selective unfolding. Less work than deriving intro/elim rules*) |
|
814 |
lemma jump_cardinal_iff: |
|
815 |
"i : jump_cardinal(K) <-> |
|
816 |
(EX r X. r <= K*K & X <= K & well_ord(X,r) & i = ordertype(X,r))" |
|
817 |
apply (unfold jump_cardinal_def) |
|
818 |
apply (blast del: subsetI) |
|
819 |
done |
|
820 |
||
821 |
(*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*) |
|
822 |
lemma K_lt_jump_cardinal: "Ord(K) ==> K < jump_cardinal(K)" |
|
823 |
apply (rule Ord_jump_cardinal [THEN [2] ltI]) |
|
824 |
apply (rule jump_cardinal_iff [THEN iffD2]) |
|
825 |
apply (rule_tac x="Memrel(K)" in exI) |
|
826 |
apply (rule_tac x=K in exI) |
|
827 |
apply (simp add: ordertype_Memrel well_ord_Memrel) |
|
828 |
apply (simp add: Memrel_def subset_iff) |
|
829 |
done |
|
830 |
||
831 |
(*The proof by contradiction: the bijection f yields a wellordering of X |
|
832 |
whose ordertype is jump_cardinal(K). *) |
|
833 |
lemma Card_jump_cardinal_lemma: |
|
834 |
"[| well_ord(X,r); r <= K * K; X <= K; |
|
835 |
f : bij(ordertype(X,r), jump_cardinal(K)) |] |
|
836 |
==> jump_cardinal(K) : jump_cardinal(K)" |
|
837 |
apply (subgoal_tac "f O ordermap (X,r) : bij (X, jump_cardinal (K))") |
|
838 |
prefer 2 apply (blast intro: comp_bij ordermap_bij) |
|
839 |
apply (rule jump_cardinal_iff [THEN iffD2]) |
|
840 |
apply (intro exI conjI) |
|
841 |
apply (rule subset_trans [OF rvimage_type Sigma_mono]) |
|
842 |
apply assumption+ |
|
843 |
apply (erule bij_is_inj [THEN well_ord_rvimage]) |
|
844 |
apply (rule Ord_jump_cardinal [THEN well_ord_Memrel]) |
|
845 |
apply (simp add: well_ord_Memrel [THEN [2] bij_ordertype_vimage] |
|
846 |
ordertype_Memrel Ord_jump_cardinal) |
|
847 |
done |
|
848 |
||
849 |
(*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*) |
|
850 |
lemma Card_jump_cardinal: "Card(jump_cardinal(K))" |
|
851 |
apply (rule Ord_jump_cardinal [THEN CardI]) |
|
852 |
apply (unfold eqpoll_def) |
|
853 |
apply (safe dest!: ltD jump_cardinal_iff [THEN iffD1]) |
|
854 |
apply (blast intro: Card_jump_cardinal_lemma [THEN mem_irrefl]) |
|
855 |
done |
|
856 |
||
857 |
(*** Basic properties of successor cardinals ***) |
|
858 |
||
859 |
lemma csucc_basic: "Ord(K) ==> Card(csucc(K)) & K < csucc(K)" |
|
860 |
apply (unfold csucc_def) |
|
861 |
apply (rule LeastI) |
|
862 |
apply (blast intro: Card_jump_cardinal K_lt_jump_cardinal Ord_jump_cardinal)+ |
|
863 |
done |
|
864 |
||
865 |
lemmas Card_csucc = csucc_basic [THEN conjunct1, standard] |
|
866 |
||
867 |
lemmas lt_csucc = csucc_basic [THEN conjunct2, standard] |
|
868 |
||
869 |
lemma Ord_0_lt_csucc: "Ord(K) ==> 0 < csucc(K)" |
|
870 |
apply (blast intro: Ord_0_le lt_csucc lt_trans1) |
|
871 |
done |
|
872 |
||
873 |
lemma csucc_le: "[| Card(L); K<L |] ==> csucc(K) le L" |
|
874 |
apply (unfold csucc_def) |
|
875 |
apply (rule Least_le) |
|
876 |
apply (blast intro: Card_is_Ord)+ |
|
877 |
done |
|
878 |
||
879 |
lemma lt_csucc_iff: "[| Ord(i); Card(K) |] ==> i < csucc(K) <-> |i| le K" |
|
880 |
apply (rule iffI) |
|
881 |
apply (rule_tac [2] Card_lt_imp_lt) |
|
882 |
apply (erule_tac [2] lt_trans1) |
|
883 |
apply (simp_all add: lt_csucc Card_csucc Card_is_Ord) |
|
884 |
apply (rule notI [THEN not_lt_imp_le]) |
|
885 |
apply (rule Card_cardinal [THEN csucc_le, THEN lt_trans1, THEN lt_irrefl]) |
|
886 |
apply assumption |
|
887 |
apply (rule Ord_cardinal_le [THEN lt_trans1]) |
|
888 |
apply (simp_all add: Ord_cardinal Card_is_Ord) |
|
889 |
done |
|
890 |
||
891 |
lemma Card_lt_csucc_iff: |
|
892 |
"[| Card(K'); Card(K) |] ==> K' < csucc(K) <-> K' le K" |
|
893 |
by (simp (no_asm_simp) add: lt_csucc_iff Card_cardinal_eq Card_is_Ord) |
|
894 |
||
895 |
lemma InfCard_csucc: "InfCard(K) ==> InfCard(csucc(K))" |
|
896 |
by (simp add: InfCard_def Card_csucc Card_is_Ord |
|
897 |
lt_csucc [THEN leI, THEN [2] le_trans]) |
|
898 |
||
899 |
||
900 |
(*** Finite sets ***) |
|
901 |
||
902 |
lemma Fin_lemma [rule_format]: "n: nat ==> ALL A. A \<approx> n --> A : Fin(A)" |
|
903 |
apply (induct_tac "n") |
|
904 |
apply (simp (no_asm) add: eqpoll_0_iff) |
|
905 |
apply clarify |
|
906 |
apply (subgoal_tac "EX u. u:A") |
|
907 |
apply (erule exE) |
|
908 |
apply (rule Diff_sing_eqpoll [THEN revcut_rl]) |
|
909 |
prefer 2 apply (assumption) |
|
910 |
apply assumption |
|
911 |
apply (rule_tac b = "A" in cons_Diff [THEN subst]) |
|
912 |
apply assumption |
|
913 |
apply (rule Fin.consI) |
|
914 |
apply blast |
|
915 |
apply (blast intro: subset_consI [THEN Fin_mono, THEN subsetD]) |
|
916 |
(*Now for the lemma assumed above*) |
|
917 |
apply (unfold eqpoll_def) |
|
918 |
apply (blast intro: bij_converse_bij [THEN bij_is_fun, THEN apply_type]) |
|
919 |
done |
|
920 |
||
921 |
lemma Finite_into_Fin: "Finite(A) ==> A : Fin(A)" |
|
922 |
apply (unfold Finite_def) |
|
923 |
apply (blast intro: Fin_lemma) |
|
924 |
done |
|
925 |
||
926 |
lemma Fin_into_Finite: "A : Fin(U) ==> Finite(A)" |
|
927 |
apply (fast intro!: Finite_0 Finite_cons elim: Fin_induct) |
|
928 |
done |
|
929 |
||
930 |
lemma Finite_Fin_iff: "Finite(A) <-> A : Fin(A)" |
|
931 |
apply (blast intro: Finite_into_Fin Fin_into_Finite) |
|
932 |
done |
|
933 |
||
934 |
lemma Finite_Un: "[| Finite(A); Finite(B) |] ==> Finite(A Un B)" |
|
935 |
by (blast intro!: Fin_into_Finite Fin_UnI |
|
936 |
dest!: Finite_into_Fin |
|
937 |
intro: Un_upper1 [THEN Fin_mono, THEN subsetD] |
|
938 |
Un_upper2 [THEN Fin_mono, THEN subsetD]) |
|
939 |
||
940 |
lemma Finite_Union: "[| ALL y:X. Finite(y); Finite(X) |] ==> Finite(Union(X))" |
|
941 |
apply (simp add: Finite_Fin_iff) |
|
942 |
apply (rule Fin_UnionI) |
|
943 |
apply (erule Fin_induct) |
|
944 |
apply (simp (no_asm)) |
|
945 |
apply (blast intro: Fin.consI Fin_mono [THEN [2] rev_subsetD]) |
|
946 |
done |
|
947 |
||
948 |
(* Induction principle for Finite(A), by Sidi Ehmety *) |
|
949 |
lemma Finite_induct: |
|
950 |
"[| Finite(A); P(0); |
|
951 |
!! x B. [| Finite(B); x ~: B; P(B) |] ==> P(cons(x, B)) |] |
|
952 |
==> P(A)" |
|
953 |
apply (erule Finite_into_Fin [THEN Fin_induct]) |
|
954 |
apply (blast intro: Fin_into_Finite)+ |
|
955 |
done |
|
956 |
||
957 |
||
958 |
(** Removing elements from a finite set decreases its cardinality **) |
|
959 |
||
960 |
lemma Fin_imp_not_cons_lepoll: "A: Fin(U) ==> x~:A --> ~ cons(x,A) \<lesssim> A" |
|
961 |
apply (erule Fin_induct) |
|
962 |
apply (simp (no_asm) add: lepoll_0_iff) |
|
963 |
apply (subgoal_tac "cons (x,cons (xa,y)) = cons (xa,cons (x,y))") |
|
964 |
apply (simp (no_asm_simp)) |
|
965 |
apply (blast dest!: cons_lepoll_consD) |
|
966 |
apply blast |
|
967 |
done |
|
968 |
||
969 |
lemma Finite_imp_cardinal_cons: "[| Finite(A); a~:A |] ==> |cons(a,A)| = succ(|A|)" |
|
970 |
apply (unfold cardinal_def) |
|
971 |
apply (rule Least_equality) |
|
972 |
apply (fold cardinal_def) |
|
973 |
apply (simp (no_asm) add: succ_def) |
|
974 |
apply (blast intro: cons_eqpoll_cong well_ord_cardinal_eqpoll |
|
975 |
elim!: mem_irrefl dest!: Finite_imp_well_ord) |
|
976 |
apply (blast intro: Card_cardinal Card_is_Ord) |
|
977 |
apply (rule notI) |
|
978 |
apply (rule Finite_into_Fin [THEN Fin_imp_not_cons_lepoll, THEN mp, THEN notE]) |
|
979 |
apply assumption |
|
980 |
apply assumption |
|
981 |
apply (erule eqpoll_sym [THEN eqpoll_imp_lepoll, THEN lepoll_trans]) |
|
982 |
apply (erule le_imp_lepoll [THEN lepoll_trans]) |
|
983 |
apply (blast intro: well_ord_cardinal_eqpoll [THEN eqpoll_imp_lepoll] |
|
984 |
dest!: Finite_imp_well_ord) |
|
985 |
done |
|
986 |
||
987 |
||
988 |
lemma Finite_imp_succ_cardinal_Diff: "[| Finite(A); a:A |] ==> succ(|A-{a}|) = |A|" |
|
989 |
apply (rule_tac b = "A" in cons_Diff [THEN subst]) |
|
990 |
apply assumption |
|
991 |
apply (simp (no_asm_simp) add: Finite_imp_cardinal_cons Diff_subset [THEN subset_Finite]) |
|
992 |
apply (simp (no_asm_simp) add: cons_Diff) |
|
993 |
done |
|
994 |
||
995 |
lemma Finite_imp_cardinal_Diff: "[| Finite(A); a:A |] ==> |A-{a}| < |A|" |
|
996 |
apply (rule succ_leE) |
|
997 |
apply (simp (no_asm_simp) add: Finite_imp_succ_cardinal_Diff) |
|
998 |
done |
|
999 |
||
1000 |
||
1001 |
(** Theorems by Krzysztof Grabczewski, proofs by lcp **) |
|
1002 |
||
1003 |
lemmas nat_implies_well_ord = nat_into_Ord [THEN well_ord_Memrel, standard] |
|
1004 |
||
1005 |
lemma nat_sum_eqpoll_sum: "[| m:nat; n:nat |] ==> m + n \<approx> m #+ n" |
|
1006 |
apply (rule eqpoll_trans) |
|
1007 |
apply (rule well_ord_radd [THEN well_ord_cardinal_eqpoll, THEN eqpoll_sym]) |
|
1008 |
apply (erule nat_implies_well_ord)+ |
|
1009 |
apply (simp (no_asm_simp) add: nat_cadd_eq_add [symmetric] cadd_def eqpoll_refl) |
|
1010 |
done |
|
1011 |
||
1012 |
||
1013 |
(*** Theorems by Sidi Ehmety ***) |
|
1014 |
||
1015 |
(*The contrapositive says ~Finite(A) ==> ~Finite(A-{a}) *) |
|
1016 |
lemma Diff_sing_Finite: "Finite(A - {a}) ==> Finite(A)" |
|
1017 |
apply (unfold Finite_def) |
|
1018 |
apply (case_tac "a:A") |
|
1019 |
apply (subgoal_tac [2] "A-{a}=A") |
|
1020 |
apply auto |
|
1021 |
apply (rule_tac x = "succ (n) " in bexI) |
|
1022 |
apply (subgoal_tac "cons (a, A - {a}) = A & cons (n, n) = succ (n) ") |
|
1023 |
apply (drule_tac a = "a" and b = "n" in cons_eqpoll_cong) |
|
1024 |
apply (auto dest: mem_irrefl) |
|
1025 |
done |
|
1026 |
||
1027 |
(*And the contrapositive of this says |
|
1028 |
[| ~Finite(A); Finite(B) |] ==> ~Finite(A-B) *) |
|
1029 |
lemma Diff_Finite [rule_format (no_asm)]: "Finite(B) ==> Finite(A-B) --> Finite(A)" |
|
1030 |
apply (erule Finite_induct) |
|
1031 |
apply auto |
|
1032 |
apply (case_tac "x:A") |
|
1033 |
apply (subgoal_tac [2] "A-cons (x, B) = A - B") |
|
1034 |
apply (subgoal_tac "A - cons (x, B) = (A - B) - {x}") |
|
1035 |
apply (rotate_tac -1) |
|
1036 |
apply simp |
|
1037 |
apply (drule Diff_sing_Finite) |
|
1038 |
apply auto |
|
1039 |
done |
|
1040 |
||
1041 |
lemma Ord_subset_natD [rule_format (no_asm)]: "Ord(i) ==> i <= nat --> i : nat | i=nat" |
|
1042 |
apply (erule trans_induct3) |
|
1043 |
apply auto |
|
1044 |
apply (blast dest!: nat_le_Limit [THEN le_imp_subset]) |
|
1045 |
done |
|
1046 |
||
1047 |
lemma Ord_nat_subset_into_Card: "[| Ord(i); i <= nat |] ==> Card(i)" |
|
1048 |
apply (blast dest: Ord_subset_natD intro: Card_nat nat_into_Card) |
|
1049 |
done |
|
1050 |
||
1051 |
lemma Finite_cardinal_in_nat [simp]: "Finite(A) ==> |A| : nat" |
|
1052 |
apply (erule Finite_induct) |
|
1053 |
apply (auto simp add: cardinal_0 Finite_imp_cardinal_cons) |
|
1054 |
done |
|
1055 |
||
1056 |
lemma Finite_Diff_sing_eq_diff_1: "[| Finite(A); x:A |] ==> |A-{x}| = |A| #- 1" |
|
1057 |
apply (rule succ_inject) |
|
1058 |
apply (rule_tac b = "|A|" in trans) |
|
1059 |
apply (simp (no_asm_simp) add: Finite_imp_succ_cardinal_Diff) |
|
1060 |
apply (subgoal_tac "1 \<lesssim> A") |
|
1061 |
prefer 2 apply (blast intro: not_0_is_lepoll_1) |
|
1062 |
apply (frule Finite_imp_well_ord) |
|
1063 |
apply clarify |
|
1064 |
apply (rotate_tac -1) |
|
1065 |
apply (drule well_ord_lepoll_imp_Card_le) |
|
1066 |
apply (auto simp add: cardinal_1) |
|
1067 |
apply (rule trans) |
|
1068 |
apply (rule_tac [2] diff_succ) |
|
1069 |
apply (auto simp add: Finite_cardinal_in_nat) |
|
1070 |
done |
|
1071 |
||
1072 |
lemma cardinal_lt_imp_Diff_not_0 [rule_format (no_asm)]: "Finite(B) ==> ALL A. |B|<|A| --> A - B ~= 0" |
|
1073 |
apply (erule Finite_induct) |
|
1074 |
apply auto |
|
1075 |
apply (simp_all add: Finite_imp_cardinal_cons) |
|
1076 |
apply (case_tac "Finite (A) ") |
|
1077 |
apply (subgoal_tac [2] "Finite (cons (x, B))") |
|
1078 |
apply (drule_tac [2] B = "cons (x, B) " in Diff_Finite) |
|
1079 |
apply (auto simp add: Finite_0 Finite_cons) |
|
1080 |
apply (subgoal_tac "|B|<|A|") |
|
1081 |
prefer 2 apply (blast intro: lt_trans Ord_cardinal) |
|
1082 |
apply (case_tac "x:A") |
|
1083 |
apply (subgoal_tac [2] "A - cons (x, B) = A - B") |
|
1084 |
apply auto |
|
1085 |
apply (subgoal_tac "|A| le |cons (x, B) |") |
|
1086 |
prefer 2 |
|
1087 |
apply (blast dest: Finite_cons [THEN Finite_imp_well_ord] |
|
1088 |
intro: well_ord_lepoll_imp_Card_le subset_imp_lepoll) |
|
1089 |
apply (auto simp add: Finite_imp_cardinal_cons) |
|
1090 |
apply (auto dest!: Finite_cardinal_in_nat simp add: le_iff) |
|
1091 |
apply (blast intro: lt_trans) |
|
1092 |
done |
|
1093 |
||
1094 |
||
1095 |
ML{* |
|
1096 |
val InfCard_def = thm "InfCard_def" |
|
1097 |
val cmult_def = thm "cmult_def" |
|
1098 |
val cadd_def = thm "cadd_def" |
|
1099 |
val jump_cardinal_def = thm "jump_cardinal_def" |
|
1100 |
val csucc_def = thm "csucc_def" |
|
1101 |
||
1102 |
val sum_commute_eqpoll = thm "sum_commute_eqpoll"; |
|
1103 |
val cadd_commute = thm "cadd_commute"; |
|
1104 |
val sum_assoc_eqpoll = thm "sum_assoc_eqpoll"; |
|
1105 |
val well_ord_cadd_assoc = thm "well_ord_cadd_assoc"; |
|
1106 |
val sum_0_eqpoll = thm "sum_0_eqpoll"; |
|
1107 |
val cadd_0 = thm "cadd_0"; |
|
1108 |
val sum_lepoll_self = thm "sum_lepoll_self"; |
|
1109 |
val cadd_le_self = thm "cadd_le_self"; |
|
1110 |
val sum_lepoll_mono = thm "sum_lepoll_mono"; |
|
1111 |
val cadd_le_mono = thm "cadd_le_mono"; |
|
1112 |
val eq_imp_not_mem = thm "eq_imp_not_mem"; |
|
1113 |
val sum_succ_eqpoll = thm "sum_succ_eqpoll"; |
|
1114 |
val nat_cadd_eq_add = thm "nat_cadd_eq_add"; |
|
1115 |
val prod_commute_eqpoll = thm "prod_commute_eqpoll"; |
|
1116 |
val cmult_commute = thm "cmult_commute"; |
|
1117 |
val prod_assoc_eqpoll = thm "prod_assoc_eqpoll"; |
|
1118 |
val well_ord_cmult_assoc = thm "well_ord_cmult_assoc"; |
|
1119 |
val sum_prod_distrib_eqpoll = thm "sum_prod_distrib_eqpoll"; |
|
1120 |
val well_ord_cadd_cmult_distrib = thm "well_ord_cadd_cmult_distrib"; |
|
1121 |
val prod_0_eqpoll = thm "prod_0_eqpoll"; |
|
1122 |
val cmult_0 = thm "cmult_0"; |
|
1123 |
val prod_singleton_eqpoll = thm "prod_singleton_eqpoll"; |
|
1124 |
val cmult_1 = thm "cmult_1"; |
|
1125 |
val prod_lepoll_self = thm "prod_lepoll_self"; |
|
1126 |
val cmult_le_self = thm "cmult_le_self"; |
|
1127 |
val prod_lepoll_mono = thm "prod_lepoll_mono"; |
|
1128 |
val cmult_le_mono = thm "cmult_le_mono"; |
|
1129 |
val prod_succ_eqpoll = thm "prod_succ_eqpoll"; |
|
1130 |
val nat_cmult_eq_mult = thm "nat_cmult_eq_mult"; |
|
1131 |
val cmult_2 = thm "cmult_2"; |
|
1132 |
val sum_lepoll_prod = thm "sum_lepoll_prod"; |
|
1133 |
val lepoll_imp_sum_lepoll_prod = thm "lepoll_imp_sum_lepoll_prod"; |
|
1134 |
val nat_cons_lepoll = thm "nat_cons_lepoll"; |
|
1135 |
val nat_cons_eqpoll = thm "nat_cons_eqpoll"; |
|
1136 |
val nat_succ_eqpoll = thm "nat_succ_eqpoll"; |
|
1137 |
val InfCard_nat = thm "InfCard_nat"; |
|
1138 |
val InfCard_is_Card = thm "InfCard_is_Card"; |
|
1139 |
val InfCard_Un = thm "InfCard_Un"; |
|
1140 |
val InfCard_is_Limit = thm "InfCard_is_Limit"; |
|
1141 |
val ordermap_eqpoll_pred = thm "ordermap_eqpoll_pred"; |
|
1142 |
val ordermap_z_lt = thm "ordermap_z_lt"; |
|
1143 |
val InfCard_le_cmult_eq = thm "InfCard_le_cmult_eq"; |
|
1144 |
val InfCard_cmult_eq = thm "InfCard_cmult_eq"; |
|
1145 |
val InfCard_cdouble_eq = thm "InfCard_cdouble_eq"; |
|
1146 |
val InfCard_le_cadd_eq = thm "InfCard_le_cadd_eq"; |
|
1147 |
val InfCard_cadd_eq = thm "InfCard_cadd_eq"; |
|
1148 |
val Ord_jump_cardinal = thm "Ord_jump_cardinal"; |
|
1149 |
val jump_cardinal_iff = thm "jump_cardinal_iff"; |
|
1150 |
val K_lt_jump_cardinal = thm "K_lt_jump_cardinal"; |
|
1151 |
val Card_jump_cardinal = thm "Card_jump_cardinal"; |
|
1152 |
val csucc_basic = thm "csucc_basic"; |
|
1153 |
val Card_csucc = thm "Card_csucc"; |
|
1154 |
val lt_csucc = thm "lt_csucc"; |
|
1155 |
val Ord_0_lt_csucc = thm "Ord_0_lt_csucc"; |
|
1156 |
val csucc_le = thm "csucc_le"; |
|
1157 |
val lt_csucc_iff = thm "lt_csucc_iff"; |
|
1158 |
val Card_lt_csucc_iff = thm "Card_lt_csucc_iff"; |
|
1159 |
val InfCard_csucc = thm "InfCard_csucc"; |
|
1160 |
val Finite_into_Fin = thm "Finite_into_Fin"; |
|
1161 |
val Fin_into_Finite = thm "Fin_into_Finite"; |
|
1162 |
val Finite_Fin_iff = thm "Finite_Fin_iff"; |
|
1163 |
val Finite_Un = thm "Finite_Un"; |
|
1164 |
val Finite_Union = thm "Finite_Union"; |
|
1165 |
val Finite_induct = thm "Finite_induct"; |
|
1166 |
val Fin_imp_not_cons_lepoll = thm "Fin_imp_not_cons_lepoll"; |
|
1167 |
val Finite_imp_cardinal_cons = thm "Finite_imp_cardinal_cons"; |
|
1168 |
val Finite_imp_succ_cardinal_Diff = thm "Finite_imp_succ_cardinal_Diff"; |
|
1169 |
val Finite_imp_cardinal_Diff = thm "Finite_imp_cardinal_Diff"; |
|
1170 |
val nat_implies_well_ord = thm "nat_implies_well_ord"; |
|
1171 |
val nat_sum_eqpoll_sum = thm "nat_sum_eqpoll_sum"; |
|
1172 |
val Diff_sing_Finite = thm "Diff_sing_Finite"; |
|
1173 |
val Diff_Finite = thm "Diff_Finite"; |
|
1174 |
val Ord_subset_natD = thm "Ord_subset_natD"; |
|
1175 |
val Ord_nat_subset_into_Card = thm "Ord_nat_subset_into_Card"; |
|
1176 |
val Finite_cardinal_in_nat = thm "Finite_cardinal_in_nat"; |
|
1177 |
val Finite_Diff_sing_eq_diff_1 = thm "Finite_Diff_sing_eq_diff_1"; |
|
1178 |
val cardinal_lt_imp_Diff_not_0 = thm "cardinal_lt_imp_Diff_not_0"; |
|
1179 |
*} |
|
1180 |
||
437 | 1181 |
end |