author | paulson |
Tue, 13 Mar 2012 12:04:07 +0000 | |
changeset 46901 | 1382bba4b7a5 |
parent 46841 | 49b91b716cbe |
child 46907 | eea3eb057fea |
permissions | -rw-r--r-- |
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(* Title: ZF/CardinalArith.thy |
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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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*) |
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|
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header{*Cardinal Arithmetic Without the Axiom of Choice*} |
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theory CardinalArith imports Cardinal OrderArith ArithSimp Finite begin |
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definition |
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InfCard :: "i=>o" where |
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"InfCard(i) == Card(i) & nat \<le> i" |
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definition |
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cmult :: "[i,i]=>i" (infixl "|*|" 70) where |
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"i |*| j == |i*j|" |
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|
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definition |
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cadd :: "[i,i]=>i" (infixl "|+|" 65) where |
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"i |+| j == |i+j|" |
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definition |
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csquare_rel :: "i=>i" where |
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"csquare_rel(K) == |
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rvimage(K*K, |
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lam <x,y>:K*K. <x \<union> y, x, y>, |
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rmult(K,Memrel(K), K*K, rmult(K,Memrel(K), K,Memrel(K))))" |
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definition |
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jump_cardinal :: "i=>i" where |
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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(K) == |
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\<Union>X\<in>Pow(K). {z. r: Pow(K*K), well_ord(X,r) & z = ordertype(X,r)}" |
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definition |
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csucc :: "i=>i" where |
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--{*needed because @{term "jump_cardinal(K)"} might not be the successor |
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of @{term K}*} |
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"csucc(K) == LEAST L. Card(L) & K<L" |
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notation (xsymbols) |
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cadd (infixl "\<oplus>" 65) and |
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cmult (infixl "\<otimes>" 70) |
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notation (HTML) |
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cadd (infixl "\<oplus>" 65) and |
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cmult (infixl "\<otimes>" 70) |
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lemma Card_Union [simp,intro,TC]: |
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assumes A: "\<And>x. x\<in>A \<Longrightarrow> Card(x)" shows "Card(\<Union>(A))" |
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proof (rule CardI) |
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show "Ord(\<Union>A)" using A |
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by (simp add: Card_is_Ord) |
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next |
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fix j |
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assume j: "j < \<Union>A" |
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hence "\<exists>c\<in>A. j < c & Card(c)" using A |
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by (auto simp add: lt_def intro: Card_is_Ord) |
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then obtain c where c: "c\<in>A" "j < c" "Card(c)" |
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by blast |
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hence jls: "j \<prec> c" |
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by (simp add: lt_Card_imp_lesspoll) |
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{ assume eqp: "j \<approx> \<Union>A" |
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have "c \<lesssim> \<Union>A" using c |
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by (blast intro: subset_imp_lepoll) |
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also have "... \<approx> j" by (rule eqpoll_sym [OF eqp]) |
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also have "... \<prec> c" by (rule jls) |
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finally have "c \<prec> c" . |
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hence False |
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by auto |
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} thus "\<not> j \<approx> \<Union>A" by blast |
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qed |
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lemma Card_UN: "(!!x. x:A ==> Card(K(x))) ==> Card(\<Union>x\<in>A. K(x))" |
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by blast |
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|
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lemma Card_OUN [simp,intro,TC]: |
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"(!!x. x:A ==> Card(K(x))) ==> Card(\<Union>x<A. K(x))" |
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by (auto simp add: OUnion_def Card_0) |
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|
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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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||
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subsection{*Cardinal addition*} |
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text{*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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subsubsection{*Cardinal addition is commutative*} |
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|
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lemma sum_commute_eqpoll: "A+B \<approx> B+A" |
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proof (unfold eqpoll_def, rule exI) |
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show "(\<lambda>z\<in>A+B. case(Inr,Inl,z)) \<in> bij(A+B, B+A)" |
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by (auto intro: lam_bijective [where d = "case(Inr,Inl)"]) |
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qed |
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lemma cadd_commute: "i \<oplus> j = j \<oplus> i" |
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apply (unfold cadd_def) |
107 |
apply (rule sum_commute_eqpoll [THEN cardinal_cong]) |
|
108 |
done |
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||
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subsubsection{*Cardinal addition is associative*} |
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|
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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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||
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text{*Unconditional version requires AC*} |
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lemma well_ord_cadd_assoc: |
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assumes i: "well_ord(i,ri)" and j: "well_ord(j,rj)" and k: "well_ord(k,rk)" |
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shows "(i \<oplus> j) \<oplus> k = i \<oplus> (j \<oplus> k)" |
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proof (unfold cadd_def, rule cardinal_cong) |
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have "|i + j| + k \<approx> (i + j) + k" |
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by (blast intro: sum_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_refl well_ord_radd i j) |
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also have "... \<approx> i + (j + k)" |
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by (rule sum_assoc_eqpoll) |
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also have "... \<approx> i + |j + k|" |
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by (blast intro: sum_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_refl well_ord_radd j k eqpoll_sym) |
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finally show "|i + j| + k \<approx> i + |j + k|" . |
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qed |
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||
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subsubsection{*0 is the identity for addition*} |
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|
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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 \<oplus> 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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subsubsection{*Addition by another cardinal*} |
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lemma sum_lepoll_self: "A \<lesssim> A+B" |
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proof (unfold lepoll_def, rule exI) |
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show "(\<lambda>x\<in>A. Inl (x)) \<in> inj(A, A + B)" |
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by (simp add: inj_def) |
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qed |
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|
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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 \<oplus> 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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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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||
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subsubsection{*Monotonicity of addition*} |
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|
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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 = "\<lambda>z\<in>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, auto) |
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done |
176 |
||
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lemma cadd_le_mono: |
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"[| K' \<le> K; L' \<le> L |] ==> (K' \<oplus> L') \<le> (K \<oplus> 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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||
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subsubsection{*Addition of finite cardinals is "ordinary" addition*} |
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|
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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)+ |
195 |
done |
|
196 |
||
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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) \<oplus> n = |succ(m \<oplus> n)|" |
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apply (unfold cadd_def) |
202 |
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 \<oplus> n = m#+n" |
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apply (induct_tac m) |
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apply (simp add: nat_into_Card [THEN cadd_0]) |
211 |
apply (simp add: cadd_succ_lemma nat_into_Card [THEN Card_cardinal_eq]) |
|
212 |
done |
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213 |
||
214 |
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subsection{*Cardinal multiplication*} |
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|
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subsubsection{*Cardinal multiplication is commutative*} |
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|
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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, |
223 |
auto) |
|
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done |
225 |
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lemma cmult_commute: "i \<otimes> j = j \<otimes> i" |
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apply (unfold cmult_def) |
228 |
apply (rule prod_commute_eqpoll [THEN cardinal_cong]) |
|
229 |
done |
|
230 |
||
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subsubsection{*Cardinal multiplication is associative*} |
13216 | 232 |
|
233 |
lemma prod_assoc_eqpoll: "(A*B)*C \<approx> A*(B*C)" |
|
234 |
apply (unfold eqpoll_def) |
|
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apply (rule exI) |
|
236 |
apply (rule prod_assoc_bij) |
|
237 |
done |
|
238 |
||
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text{*Unconditional version requires AC*} |
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lemma well_ord_cmult_assoc: |
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assumes i: "well_ord(i,ri)" and j: "well_ord(j,rj)" and k: "well_ord(k,rk)" |
242 |
shows "(i \<otimes> j) \<otimes> k = i \<otimes> (j \<otimes> k)" |
|
243 |
proof (unfold cmult_def, rule cardinal_cong) |
|
244 |
have "|i * j| * k \<approx> (i * j) * k" |
|
245 |
by (blast intro: prod_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_refl well_ord_rmult i j) |
|
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also have "... \<approx> i * (j * k)" |
|
247 |
by (rule prod_assoc_eqpoll) |
|
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also have "... \<approx> i * |j * k|" |
|
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by (blast intro: prod_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_refl well_ord_rmult j k eqpoll_sym) |
|
250 |
finally show "|i * j| * k \<approx> i * |j * k|" . |
|
251 |
qed |
|
13216 | 252 |
|
14883 | 253 |
subsubsection{*Cardinal multiplication distributes over addition*} |
13216 | 254 |
|
255 |
lemma sum_prod_distrib_eqpoll: "(A+B)*C \<approx> (A*C)+(B*C)" |
|
256 |
apply (unfold eqpoll_def) |
|
257 |
apply (rule exI) |
|
258 |
apply (rule sum_prod_distrib_bij) |
|
259 |
done |
|
260 |
||
261 |
lemma well_ord_cadd_cmult_distrib: |
|
46901 | 262 |
assumes i: "well_ord(i,ri)" and j: "well_ord(j,rj)" and k: "well_ord(k,rk)" |
263 |
shows "(i \<oplus> j) \<otimes> k = (i \<otimes> k) \<oplus> (j \<otimes> k)" |
|
264 |
proof (unfold cadd_def cmult_def, rule cardinal_cong) |
|
265 |
have "|i + j| * k \<approx> (i + j) * k" |
|
266 |
by (blast intro: prod_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_refl well_ord_radd i j) |
|
267 |
also have "... \<approx> i * k + j * k" |
|
268 |
by (rule sum_prod_distrib_eqpoll) |
|
269 |
also have "... \<approx> |i * k| + |j * k|" |
|
270 |
by (blast intro: sum_eqpoll_cong well_ord_cardinal_eqpoll well_ord_rmult i j k eqpoll_sym) |
|
271 |
finally show "|i + j| * k \<approx> |i * k| + |j * k|" . |
|
272 |
qed |
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13216 | 273 |
|
14883 | 274 |
subsubsection{*Multiplication by 0 yields 0*} |
13216 | 275 |
|
276 |
lemma prod_0_eqpoll: "0*A \<approx> 0" |
|
277 |
apply (unfold eqpoll_def) |
|
278 |
apply (rule exI) |
|
13221 | 279 |
apply (rule lam_bijective, safe) |
13216 | 280 |
done |
281 |
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lemma cmult_0 [simp]: "0 \<otimes> i = 0" |
13221 | 283 |
by (simp add: cmult_def prod_0_eqpoll [THEN cardinal_cong]) |
13216 | 284 |
|
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subsubsection{*1 is the identity for multiplication*} |
13216 | 286 |
|
287 |
lemma prod_singleton_eqpoll: "{x}*A \<approx> A" |
|
288 |
apply (unfold eqpoll_def) |
|
289 |
apply (rule exI) |
|
290 |
apply (rule singleton_prod_bij [THEN bij_converse_bij]) |
|
291 |
done |
|
292 |
||
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|
293 |
lemma cmult_1 [simp]: "Card(K) ==> 1 \<otimes> K = K" |
13216 | 294 |
apply (unfold cmult_def succ_def) |
295 |
apply (simp add: prod_singleton_eqpoll [THEN cardinal_cong] Card_cardinal_eq) |
|
296 |
done |
|
297 |
||
13356 | 298 |
subsection{*Some inequalities for multiplication*} |
13216 | 299 |
|
300 |
lemma prod_square_lepoll: "A \<lesssim> A*A" |
|
301 |
apply (unfold lepoll_def inj_def) |
|
46820 | 302 |
apply (rule_tac x = "\<lambda>x\<in>A. <x,x>" in exI, simp) |
13216 | 303 |
done |
304 |
||
305 |
(*Could probably weaken the premise to well_ord(K,r), or remove using AC*) |
|
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|
306 |
lemma cmult_square_le: "Card(K) ==> K \<le> K \<otimes> K" |
13216 | 307 |
apply (unfold cmult_def) |
308 |
apply (rule le_trans) |
|
309 |
apply (rule_tac [2] well_ord_lepoll_imp_Card_le) |
|
310 |
apply (rule_tac [3] prod_square_lepoll) |
|
13221 | 311 |
apply (simp add: le_refl Card_is_Ord Card_cardinal_eq) |
312 |
apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord) |
|
13216 | 313 |
done |
314 |
||
14883 | 315 |
subsubsection{*Multiplication by a non-zero cardinal*} |
13216 | 316 |
|
317 |
lemma prod_lepoll_self: "b: B ==> A \<lesssim> A*B" |
|
318 |
apply (unfold lepoll_def inj_def) |
|
46820 | 319 |
apply (rule_tac x = "\<lambda>x\<in>A. <x,b>" in exI, simp) |
13216 | 320 |
done |
321 |
||
322 |
(*Could probably weaken the premises to well_ord(K,r), or removing using AC*) |
|
323 |
lemma cmult_le_self: |
|
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|
324 |
"[| Card(K); Ord(L); 0<L |] ==> K \<le> (K \<otimes> L)" |
13216 | 325 |
apply (unfold cmult_def) |
326 |
apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le]) |
|
13221 | 327 |
apply assumption |
13216 | 328 |
apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord) |
329 |
apply (blast intro: prod_lepoll_self ltD) |
|
330 |
done |
|
331 |
||
14883 | 332 |
subsubsection{*Monotonicity of multiplication*} |
13216 | 333 |
|
334 |
lemma prod_lepoll_mono: |
|
335 |
"[| A \<lesssim> C; B \<lesssim> D |] ==> A * B \<lesssim> C * D" |
|
336 |
apply (unfold lepoll_def) |
|
13221 | 337 |
apply (elim exE) |
13216 | 338 |
apply (rule_tac x = "lam <w,y>:A*B. <f`w, fa`y>" in exI) |
46820 | 339 |
apply (rule_tac d = "%<w,y>. <converse (f) `w, converse (fa) `y>" |
13216 | 340 |
in lam_injective) |
13221 | 341 |
apply (typecheck add: inj_is_fun, auto) |
13216 | 342 |
done |
343 |
||
344 |
lemma cmult_le_mono: |
|
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|
345 |
"[| K' \<le> K; L' \<le> L |] ==> (K' \<otimes> L') \<le> (K \<otimes> L)" |
13216 | 346 |
apply (unfold cmult_def) |
347 |
apply (safe dest!: le_subset_iff [THEN iffD1]) |
|
348 |
apply (rule well_ord_lepoll_imp_Card_le) |
|
349 |
apply (blast intro: well_ord_rmult well_ord_Memrel) |
|
350 |
apply (blast intro: prod_lepoll_mono subset_imp_lepoll) |
|
351 |
done |
|
352 |
||
13356 | 353 |
subsection{*Multiplication of finite cardinals is "ordinary" multiplication*} |
13216 | 354 |
|
355 |
lemma prod_succ_eqpoll: "succ(A)*B \<approx> B + A*B" |
|
356 |
apply (unfold eqpoll_def) |
|
13221 | 357 |
apply (rule exI) |
13216 | 358 |
apply (rule_tac c = "%<x,y>. if x=A then Inl (y) else Inr (<x,y>)" |
359 |
and d = "case (%y. <A,y>, %z. z)" in lam_bijective) |
|
360 |
apply safe |
|
361 |
apply (simp_all add: succI2 if_type mem_imp_not_eq) |
|
362 |
done |
|
363 |
||
364 |
(*Unconditional version requires AC*) |
|
365 |
lemma cmult_succ_lemma: |
|
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|
366 |
"[| Ord(m); Ord(n) |] ==> succ(m) \<otimes> n = n \<oplus> (m \<otimes> n)" |
13216 | 367 |
apply (unfold cmult_def cadd_def) |
368 |
apply (rule prod_succ_eqpoll [THEN cardinal_cong, THEN trans]) |
|
369 |
apply (rule cardinal_cong [symmetric]) |
|
370 |
apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll]) |
|
371 |
apply (blast intro: well_ord_rmult well_ord_Memrel) |
|
372 |
done |
|
373 |
||
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|
374 |
lemma nat_cmult_eq_mult: "[| m: nat; n: nat |] ==> m \<otimes> n = m#*n" |
13244 | 375 |
apply (induct_tac m) |
13221 | 376 |
apply (simp_all add: cmult_succ_lemma nat_cadd_eq_add) |
13216 | 377 |
done |
378 |
||
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changeset
|
379 |
lemma cmult_2: "Card(n) ==> 2 \<otimes> n = n \<oplus> n" |
13221 | 380 |
by (simp add: cmult_succ_lemma Card_is_Ord cadd_commute [of _ 0]) |
13216 | 381 |
|
46901 | 382 |
lemma sum_lepoll_prod: |
383 |
assumes C: "2 \<lesssim> C" shows "B+B \<lesssim> C*B" |
|
384 |
proof - |
|
385 |
have "B+B \<lesssim> 2*B" |
|
386 |
by (simp add: sum_eq_2_times) |
|
387 |
also have "... \<lesssim> C*B" |
|
388 |
by (blast intro: prod_lepoll_mono lepoll_refl C) |
|
389 |
finally show "B+B \<lesssim> C*B" . |
|
390 |
qed |
|
13216 | 391 |
|
392 |
lemma lepoll_imp_sum_lepoll_prod: "[| A \<lesssim> B; 2 \<lesssim> A |] ==> A+B \<lesssim> A*B" |
|
13221 | 393 |
by (blast intro: sum_lepoll_mono sum_lepoll_prod lepoll_trans lepoll_refl) |
13216 | 394 |
|
395 |
||
13356 | 396 |
subsection{*Infinite Cardinals are Limit Ordinals*} |
13216 | 397 |
|
398 |
(*This proof is modelled upon one assuming nat<=A, with injection |
|
46820 | 399 |
\<lambda>z\<in>cons(u,A). if z=u then 0 else if z \<in> nat then succ(z) else z |
13216 | 400 |
and inverse %y. if y:nat then nat_case(u, %z. z, y) else y. \ |
401 |
If f: inj(nat,A) then range(f) behaves like the natural numbers.*) |
|
402 |
lemma nat_cons_lepoll: "nat \<lesssim> A ==> cons(u,A) \<lesssim> A" |
|
403 |
apply (unfold lepoll_def) |
|
404 |
apply (erule exE) |
|
46820 | 405 |
apply (rule_tac x = |
406 |
"\<lambda>z\<in>cons (u,A). |
|
407 |
if z=u then f`0 |
|
408 |
else if z: range (f) then f`succ (converse (f) `z) else z" |
|
13216 | 409 |
in exI) |
410 |
apply (rule_tac d = |
|
46820 | 411 |
"%y. if y: range(f) then nat_case (u, %z. f`z, converse(f) `y) |
412 |
else y" |
|
13216 | 413 |
in lam_injective) |
414 |
apply (fast intro!: if_type apply_type intro: inj_is_fun inj_converse_fun) |
|
415 |
apply (simp add: inj_is_fun [THEN apply_rangeI] |
|
416 |
inj_converse_fun [THEN apply_rangeI] |
|
417 |
inj_converse_fun [THEN apply_funtype]) |
|
418 |
done |
|
419 |
||
420 |
lemma nat_cons_eqpoll: "nat \<lesssim> A ==> cons(u,A) \<approx> A" |
|
421 |
apply (erule nat_cons_lepoll [THEN eqpollI]) |
|
422 |
apply (rule subset_consI [THEN subset_imp_lepoll]) |
|
423 |
done |
|
424 |
||
425 |
(*Specialized version required below*) |
|
46820 | 426 |
lemma nat_succ_eqpoll: "nat \<subseteq> A ==> succ(A) \<approx> A" |
13216 | 427 |
apply (unfold succ_def) |
428 |
apply (erule subset_imp_lepoll [THEN nat_cons_eqpoll]) |
|
429 |
done |
|
430 |
||
431 |
lemma InfCard_nat: "InfCard(nat)" |
|
432 |
apply (unfold InfCard_def) |
|
433 |
apply (blast intro: Card_nat le_refl Card_is_Ord) |
|
434 |
done |
|
435 |
||
436 |
lemma InfCard_is_Card: "InfCard(K) ==> Card(K)" |
|
437 |
apply (unfold InfCard_def) |
|
438 |
apply (erule conjunct1) |
|
439 |
done |
|
440 |
||
441 |
lemma InfCard_Un: |
|
46820 | 442 |
"[| InfCard(K); Card(L) |] ==> InfCard(K \<union> L)" |
13216 | 443 |
apply (unfold InfCard_def) |
444 |
apply (simp add: Card_Un Un_upper1_le [THEN [2] le_trans] Card_is_Ord) |
|
445 |
done |
|
446 |
||
447 |
(*Kunen's Lemma 10.11*) |
|
448 |
lemma InfCard_is_Limit: "InfCard(K) ==> Limit(K)" |
|
449 |
apply (unfold InfCard_def) |
|
450 |
apply (erule conjE) |
|
451 |
apply (frule Card_is_Ord) |
|
452 |
apply (rule ltI [THEN non_succ_LimitI]) |
|
453 |
apply (erule le_imp_subset [THEN subsetD]) |
|
454 |
apply (safe dest!: Limit_nat [THEN Limit_le_succD]) |
|
455 |
apply (unfold Card_def) |
|
456 |
apply (drule trans) |
|
457 |
apply (erule le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong]) |
|
458 |
apply (erule Ord_cardinal_le [THEN lt_trans2, THEN lt_irrefl]) |
|
13221 | 459 |
apply (rule le_eqI, assumption) |
13216 | 460 |
apply (rule Ord_cardinal) |
461 |
done |
|
462 |
||
463 |
||
464 |
(*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***) |
|
465 |
||
466 |
(*A general fact about ordermap*) |
|
467 |
lemma ordermap_eqpoll_pred: |
|
13269 | 468 |
"[| well_ord(A,r); x:A |] ==> ordermap(A,r)`x \<approx> Order.pred(A,x,r)" |
13216 | 469 |
apply (unfold eqpoll_def) |
470 |
apply (rule exI) |
|
13221 | 471 |
apply (simp add: ordermap_eq_image well_ord_is_wf) |
46820 | 472 |
apply (erule ordermap_bij [THEN bij_is_inj, THEN restrict_bij, |
13221 | 473 |
THEN bij_converse_bij]) |
13216 | 474 |
apply (rule pred_subset) |
475 |
done |
|
476 |
||
14883 | 477 |
subsubsection{*Establishing the well-ordering*} |
13216 | 478 |
|
46901 | 479 |
lemma well_ord_csquare: |
480 |
assumes K: "Ord(K)" shows "well_ord(K*K, csquare_rel(K))" |
|
481 |
proof (unfold csquare_rel_def, rule well_ord_rvimage) |
|
482 |
show "(\<lambda>\<langle>x,y\<rangle>\<in>K \<times> K. \<langle>x \<union> y, x, y\<rangle>) \<in> inj(K \<times> K, K \<times> K \<times> K)" using K |
|
483 |
by (force simp add: inj_def intro: lam_type Un_least_lt [THEN ltD] ltI) |
|
484 |
next |
|
485 |
show "well_ord(K \<times> K \<times> K, rmult(K, Memrel(K), K \<times> K, rmult(K, Memrel(K), K, Memrel(K))))" |
|
486 |
using K by (blast intro: well_ord_rmult well_ord_Memrel) |
|
487 |
qed |
|
13216 | 488 |
|
14883 | 489 |
subsubsection{*Characterising initial segments of the well-ordering*} |
13216 | 490 |
|
491 |
lemma csquareD: |
|
46820 | 492 |
"[| <<x,y>, <z,z>> \<in> csquare_rel(K); x<K; y<K; z<K |] ==> x \<le> z & y \<le> z" |
13216 | 493 |
apply (unfold csquare_rel_def) |
494 |
apply (erule rev_mp) |
|
495 |
apply (elim ltE) |
|
13221 | 496 |
apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD) |
13216 | 497 |
apply (safe elim!: mem_irrefl intro!: Un_upper1_le Un_upper2_le) |
13221 | 498 |
apply (simp_all add: lt_def succI2) |
13216 | 499 |
done |
500 |
||
46820 | 501 |
lemma pred_csquare_subset: |
502 |
"z<K ==> Order.pred(K*K, <z,z>, csquare_rel(K)) \<subseteq> succ(z)*succ(z)" |
|
13216 | 503 |
apply (unfold Order.pred_def) |
46901 | 504 |
apply (safe del: SigmaI dest!: csquareD) |
46820 | 505 |
apply (unfold lt_def, auto) |
13216 | 506 |
done |
507 |
||
508 |
lemma csquare_ltI: |
|
46820 | 509 |
"[| x<z; y<z; z<K |] ==> <<x,y>, <z,z>> \<in> csquare_rel(K)" |
13216 | 510 |
apply (unfold csquare_rel_def) |
511 |
apply (subgoal_tac "x<K & y<K") |
|
46820 | 512 |
prefer 2 apply (blast intro: lt_trans) |
13216 | 513 |
apply (elim ltE) |
13221 | 514 |
apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD) |
13216 | 515 |
done |
516 |
||
517 |
(*Part of the traditional proof. UNUSED since it's harder to prove & apply *) |
|
518 |
lemma csquare_or_eqI: |
|
46820 | 519 |
"[| x \<le> z; y \<le> z; z<K |] ==> <<x,y>, <z,z>> \<in> csquare_rel(K) | x=z & y=z" |
13216 | 520 |
apply (unfold csquare_rel_def) |
521 |
apply (subgoal_tac "x<K & y<K") |
|
46820 | 522 |
prefer 2 apply (blast intro: lt_trans1) |
13216 | 523 |
apply (elim ltE) |
13221 | 524 |
apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD) |
13216 | 525 |
apply (elim succE) |
46820 | 526 |
apply (simp_all add: subset_Un_iff [THEN iff_sym] |
13221 | 527 |
subset_Un_iff2 [THEN iff_sym] OrdmemD) |
13216 | 528 |
done |
529 |
||
14883 | 530 |
subsubsection{*The cardinality of initial segments*} |
13216 | 531 |
|
532 |
lemma ordermap_z_lt: |
|
46820 | 533 |
"[| Limit(K); x<K; y<K; z=succ(x \<union> y) |] ==> |
13216 | 534 |
ordermap(K*K, csquare_rel(K)) ` <x,y> < |
535 |
ordermap(K*K, csquare_rel(K)) ` <z,z>" |
|
536 |
apply (subgoal_tac "z<K & well_ord (K*K, csquare_rel (K))") |
|
537 |
prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ |
|
46820 | 538 |
Limit_is_Ord [THEN well_ord_csquare], clarify) |
13216 | 539 |
apply (rule csquare_ltI [THEN ordermap_mono, THEN ltI]) |
540 |
apply (erule_tac [4] well_ord_is_wf) |
|
541 |
apply (blast intro!: Un_upper1_le Un_upper2_le Ord_ordermap elim!: ltE)+ |
|
542 |
done |
|
543 |
||
46901 | 544 |
text{*Kunen: "each @{term"\<langle>x,y\<rangle> \<in> K \<times> K"} has no more than @{term"z \<times> z"} predecessors..." (page 29) *} |
13216 | 545 |
lemma ordermap_csquare_le: |
46901 | 546 |
assumes K: "Limit(K)" and x: "x<K" and y: " y<K" and z: "z=succ(x \<union> y)" |
547 |
shows "|ordermap(K \<times> K, csquare_rel(K)) ` \<langle>x,y\<rangle>| \<le> |succ(z)| \<otimes> |succ(z)|" |
|
548 |
proof (unfold cmult_def, rule well_ord_lepoll_imp_Card_le) |
|
549 |
show "well_ord(|succ(z)| \<times> |succ(z)|, |
|
550 |
rmult(|succ(z)|, Memrel(|succ(z)|), |succ(z)|, Memrel(|succ(z)|)))" |
|
551 |
by (blast intro: Ord_cardinal well_ord_Memrel well_ord_rmult) |
|
552 |
next |
|
553 |
have zK: "z<K" using x y z K |
|
554 |
by (blast intro: Un_least_lt Limit_has_succ) |
|
555 |
hence oz: "Ord(z)" by (elim ltE) |
|
556 |
have "ordermap(K \<times> K, csquare_rel(K)) ` \<langle>x,y\<rangle> \<lesssim> ordermap(K \<times> K, csquare_rel(K)) ` \<langle>z,z\<rangle>" |
|
557 |
by (blast intro: ordermap_z_lt leI le_imp_lepoll K x y z) |
|
558 |
also have "... \<approx> Order.pred(K \<times> K, \<langle>z,z\<rangle>, csquare_rel(K))" |
|
559 |
proof (rule ordermap_eqpoll_pred) |
|
560 |
show "well_ord(K \<times> K, csquare_rel(K))" using K |
|
561 |
by (rule Limit_is_Ord [THEN well_ord_csquare]) |
|
562 |
next |
|
563 |
show "\<langle>z, z\<rangle> \<in> K \<times> K" using zK |
|
564 |
by (blast intro: ltD) |
|
565 |
qed |
|
566 |
also have "... \<lesssim> succ(z) \<times> succ(z)" using zK |
|
567 |
by (rule pred_csquare_subset [THEN subset_imp_lepoll]) |
|
568 |
also have "... \<approx> |succ(z)| \<times> |succ(z)|" using oz |
|
569 |
by (blast intro: prod_eqpoll_cong Ord_succ Ord_cardinal_eqpoll eqpoll_sym) |
|
570 |
finally show "ordermap(K \<times> K, csquare_rel(K)) ` \<langle>x,y\<rangle> \<lesssim> |succ(z)| \<times> |succ(z)|" . |
|
571 |
qed |
|
13216 | 572 |
|
46901 | 573 |
text{*Kunen: "... so the order type is @{text"\<le>"} K" *} |
13216 | 574 |
lemma ordertype_csquare_le: |
46821
ff6b0c1087f2
Using mathematical notation for <-> and cardinal arithmetic
paulson
parents:
46820
diff
changeset
|
575 |
"[| InfCard(K); \<forall>y\<in>K. InfCard(y) \<longrightarrow> y \<otimes> y = y |] |
46820 | 576 |
==> ordertype(K*K, csquare_rel(K)) \<le> K" |
13216 | 577 |
apply (frule InfCard_is_Card [THEN Card_is_Ord]) |
13221 | 578 |
apply (rule all_lt_imp_le, assumption) |
13216 | 579 |
apply (erule well_ord_csquare [THEN Ord_ordertype]) |
580 |
apply (rule Card_lt_imp_lt) |
|
581 |
apply (erule_tac [3] InfCard_is_Card) |
|
582 |
apply (erule_tac [2] ltE) |
|
583 |
apply (simp add: ordertype_unfold) |
|
584 |
apply (safe elim!: ltE) |
|
585 |
apply (subgoal_tac "Ord (xa) & Ord (ya)") |
|
13221 | 586 |
prefer 2 apply (blast intro: Ord_in_Ord, clarify) |
46820 | 587 |
(*??WHAT A MESS!*) |
13216 | 588 |
apply (rule InfCard_is_Limit [THEN ordermap_csquare_le, THEN lt_trans1], |
46820 | 589 |
(assumption | rule refl | erule ltI)+) |
590 |
apply (rule_tac i = "xa \<union> ya" and j = nat in Ord_linear2, |
|
13216 | 591 |
simp_all add: Ord_Un Ord_nat) |
46820 | 592 |
prefer 2 (*case @{term"nat \<le> (xa \<union> ya)"} *) |
593 |
apply (simp add: le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong] |
|
13216 | 594 |
le_succ_iff InfCard_def Card_cardinal Un_least_lt Ord_Un |
595 |
ltI nat_le_cardinal Ord_cardinal_le [THEN lt_trans1, THEN ltD]) |
|
46820 | 596 |
(*the finite case: @{term"xa \<union> ya < nat"} *) |
13784 | 597 |
apply (rule_tac j = nat in lt_trans2) |
13216 | 598 |
apply (simp add: lt_def nat_cmult_eq_mult nat_succI mult_type |
599 |
nat_into_Card [THEN Card_cardinal_eq] Ord_nat) |
|
600 |
apply (simp add: InfCard_def) |
|
601 |
done |
|
602 |
||
603 |
(*Main result: Kunen's Theorem 10.12*) |
|
46821
ff6b0c1087f2
Using mathematical notation for <-> and cardinal arithmetic
paulson
parents:
46820
diff
changeset
|
604 |
lemma InfCard_csquare_eq: "InfCard(K) ==> K \<otimes> K = K" |
13216 | 605 |
apply (frule InfCard_is_Card [THEN Card_is_Ord]) |
606 |
apply (erule rev_mp) |
|
46820 | 607 |
apply (erule_tac i=K in trans_induct) |
13216 | 608 |
apply (rule impI) |
609 |
apply (rule le_anti_sym) |
|
610 |
apply (erule_tac [2] InfCard_is_Card [THEN cmult_square_le]) |
|
611 |
apply (rule ordertype_csquare_le [THEN [2] le_trans]) |
|
46820 | 612 |
apply (simp add: cmult_def Ord_cardinal_le |
13221 | 613 |
well_ord_csquare [THEN Ord_ordertype] |
46820 | 614 |
well_ord_csquare [THEN ordermap_bij, THEN bij_imp_eqpoll, |
13221 | 615 |
THEN cardinal_cong], assumption+) |
13216 | 616 |
done |
617 |
||
618 |
(*Corollary for arbitrary well-ordered sets (all sets, assuming AC)*) |
|
619 |
lemma well_ord_InfCard_square_eq: |
|
620 |
"[| well_ord(A,r); InfCard(|A|) |] ==> A*A \<approx> A" |
|
621 |
apply (rule prod_eqpoll_cong [THEN eqpoll_trans]) |
|
622 |
apply (erule well_ord_cardinal_eqpoll [THEN eqpoll_sym])+ |
|
623 |
apply (rule well_ord_cardinal_eqE) |
|
13221 | 624 |
apply (blast intro: Ord_cardinal well_ord_rmult well_ord_Memrel, assumption) |
625 |
apply (simp add: cmult_def [symmetric] InfCard_csquare_eq) |
|
13216 | 626 |
done |
627 |
||
13356 | 628 |
lemma InfCard_square_eqpoll: "InfCard(K) ==> K \<times> K \<approx> K" |
46820 | 629 |
apply (rule well_ord_InfCard_square_eq) |
630 |
apply (erule InfCard_is_Card [THEN Card_is_Ord, THEN well_ord_Memrel]) |
|
631 |
apply (simp add: InfCard_is_Card [THEN Card_cardinal_eq]) |
|
13356 | 632 |
done |
633 |
||
634 |
lemma Inf_Card_is_InfCard: "[| ~Finite(i); Card(i) |] ==> InfCard(i)" |
|
635 |
by (simp add: InfCard_def Card_is_Ord [THEN nat_le_infinite_Ord]) |
|
636 |
||
14883 | 637 |
subsubsection{*Toward's Kunen's Corollary 10.13 (1)*} |
13216 | 638 |
|
46821
ff6b0c1087f2
Using mathematical notation for <-> and cardinal arithmetic
paulson
parents:
46820
diff
changeset
|
639 |
lemma InfCard_le_cmult_eq: "[| InfCard(K); L \<le> K; 0<L |] ==> K \<otimes> L = K" |
13216 | 640 |
apply (rule le_anti_sym) |
641 |
prefer 2 |
|
642 |
apply (erule ltE, blast intro: cmult_le_self InfCard_is_Card) |
|
643 |
apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl]) |
|
644 |
apply (rule cmult_le_mono [THEN le_trans], assumption+) |
|
645 |
apply (simp add: InfCard_csquare_eq) |
|
646 |
done |
|
647 |
||
648 |
(*Corollary 10.13 (1), for cardinal multiplication*) |
|
46821
ff6b0c1087f2
Using mathematical notation for <-> and cardinal arithmetic
paulson
parents:
46820
diff
changeset
|
649 |
lemma InfCard_cmult_eq: "[| InfCard(K); InfCard(L) |] ==> K \<otimes> L = K \<union> L" |
13784 | 650 |
apply (rule_tac i = K and j = L in Ord_linear_le) |
13216 | 651 |
apply (typecheck add: InfCard_is_Card Card_is_Ord) |
652 |
apply (rule cmult_commute [THEN ssubst]) |
|
653 |
apply (rule Un_commute [THEN ssubst]) |
|
46820 | 654 |
apply (simp_all add: InfCard_is_Limit [THEN Limit_has_0] InfCard_le_cmult_eq |
13221 | 655 |
subset_Un_iff2 [THEN iffD1] le_imp_subset) |
13216 | 656 |
done |
657 |
||
46821
ff6b0c1087f2
Using mathematical notation for <-> and cardinal arithmetic
paulson
parents:
46820
diff
changeset
|
658 |
lemma InfCard_cdouble_eq: "InfCard(K) ==> K \<oplus> K = K" |
13221 | 659 |
apply (simp add: cmult_2 [symmetric] InfCard_is_Card cmult_commute) |
660 |
apply (simp add: InfCard_le_cmult_eq InfCard_is_Limit Limit_has_0 Limit_has_succ) |
|
13216 | 661 |
done |
662 |
||
663 |
(*Corollary 10.13 (1), for cardinal addition*) |
|
46821
ff6b0c1087f2
Using mathematical notation for <-> and cardinal arithmetic
paulson
parents:
46820
diff
changeset
|
664 |
lemma InfCard_le_cadd_eq: "[| InfCard(K); L \<le> K |] ==> K \<oplus> L = K" |
13216 | 665 |
apply (rule le_anti_sym) |
666 |
prefer 2 |
|
667 |
apply (erule ltE, blast intro: cadd_le_self InfCard_is_Card) |
|
668 |
apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl]) |
|
669 |
apply (rule cadd_le_mono [THEN le_trans], assumption+) |
|
670 |
apply (simp add: InfCard_cdouble_eq) |
|
671 |
done |
|
672 |
||
46821
ff6b0c1087f2
Using mathematical notation for <-> and cardinal arithmetic
paulson
parents:
46820
diff
changeset
|
673 |
lemma InfCard_cadd_eq: "[| InfCard(K); InfCard(L) |] ==> K \<oplus> L = K \<union> L" |
13784 | 674 |
apply (rule_tac i = K and j = L in Ord_linear_le) |
13216 | 675 |
apply (typecheck add: InfCard_is_Card Card_is_Ord) |
676 |
apply (rule cadd_commute [THEN ssubst]) |
|
677 |
apply (rule Un_commute [THEN ssubst]) |
|
13221 | 678 |
apply (simp_all add: InfCard_le_cadd_eq subset_Un_iff2 [THEN iffD1] le_imp_subset) |
13216 | 679 |
done |
680 |
||
681 |
(*The other part, Corollary 10.13 (2), refers to the cardinality of the set |
|
682 |
of all n-tuples of elements of K. A better version for the Isabelle theory |
|
683 |
might be InfCard(K) ==> |list(K)| = K. |
|
684 |
*) |
|
685 |
||
27517 | 686 |
subsection{*For Every Cardinal Number There Exists A Greater One*} |
13356 | 687 |
|
688 |
text{*This result is Kunen's Theorem 10.16, which would be trivial using AC*} |
|
13216 | 689 |
|
690 |
lemma Ord_jump_cardinal: "Ord(jump_cardinal(K))" |
|
691 |
apply (unfold jump_cardinal_def) |
|
692 |
apply (rule Ord_is_Transset [THEN [2] OrdI]) |
|
693 |
prefer 2 apply (blast intro!: Ord_ordertype) |
|
694 |
apply (unfold Transset_def) |
|
695 |
apply (safe del: subsetI) |
|
13221 | 696 |
apply (simp add: ordertype_pred_unfold, safe) |
13216 | 697 |
apply (rule UN_I) |
698 |
apply (rule_tac [2] ReplaceI) |
|
699 |
prefer 4 apply (blast intro: well_ord_subset elim!: predE)+ |
|
700 |
done |
|
701 |
||
702 |
(*Allows selective unfolding. Less work than deriving intro/elim rules*) |
|
703 |
lemma jump_cardinal_iff: |
|
46821
ff6b0c1087f2
Using mathematical notation for <-> and cardinal arithmetic
paulson
parents:
46820
diff
changeset
|
704 |
"i \<in> jump_cardinal(K) \<longleftrightarrow> |
46820 | 705 |
(\<exists>r X. r \<subseteq> K*K & X \<subseteq> K & well_ord(X,r) & i = ordertype(X,r))" |
13216 | 706 |
apply (unfold jump_cardinal_def) |
46820 | 707 |
apply (blast del: subsetI) |
13216 | 708 |
done |
709 |
||
710 |
(*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*) |
|
711 |
lemma K_lt_jump_cardinal: "Ord(K) ==> K < jump_cardinal(K)" |
|
712 |
apply (rule Ord_jump_cardinal [THEN [2] ltI]) |
|
713 |
apply (rule jump_cardinal_iff [THEN iffD2]) |
|
714 |
apply (rule_tac x="Memrel(K)" in exI) |
|
46820 | 715 |
apply (rule_tac x=K in exI) |
13216 | 716 |
apply (simp add: ordertype_Memrel well_ord_Memrel) |
717 |
apply (simp add: Memrel_def subset_iff) |
|
718 |
done |
|
719 |
||
720 |
(*The proof by contradiction: the bijection f yields a wellordering of X |
|
721 |
whose ordertype is jump_cardinal(K). *) |
|
722 |
lemma Card_jump_cardinal_lemma: |
|
46820 | 723 |
"[| well_ord(X,r); r \<subseteq> K * K; X \<subseteq> K; |
724 |
f \<in> bij(ordertype(X,r), jump_cardinal(K)) |] |
|
725 |
==> jump_cardinal(K) \<in> jump_cardinal(K)" |
|
726 |
apply (subgoal_tac "f O ordermap (X,r) \<in> bij (X, jump_cardinal (K))") |
|
13216 | 727 |
prefer 2 apply (blast intro: comp_bij ordermap_bij) |
728 |
apply (rule jump_cardinal_iff [THEN iffD2]) |
|
729 |
apply (intro exI conjI) |
|
13221 | 730 |
apply (rule subset_trans [OF rvimage_type Sigma_mono], assumption+) |
13216 | 731 |
apply (erule bij_is_inj [THEN well_ord_rvimage]) |
732 |
apply (rule Ord_jump_cardinal [THEN well_ord_Memrel]) |
|
733 |
apply (simp add: well_ord_Memrel [THEN [2] bij_ordertype_vimage] |
|
734 |
ordertype_Memrel Ord_jump_cardinal) |
|
735 |
done |
|
736 |
||
737 |
(*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*) |
|
738 |
lemma Card_jump_cardinal: "Card(jump_cardinal(K))" |
|
739 |
apply (rule Ord_jump_cardinal [THEN CardI]) |
|
740 |
apply (unfold eqpoll_def) |
|
741 |
apply (safe dest!: ltD jump_cardinal_iff [THEN iffD1]) |
|
742 |
apply (blast intro: Card_jump_cardinal_lemma [THEN mem_irrefl]) |
|
743 |
done |
|
744 |
||
13356 | 745 |
subsection{*Basic Properties of Successor Cardinals*} |
13216 | 746 |
|
747 |
lemma csucc_basic: "Ord(K) ==> Card(csucc(K)) & K < csucc(K)" |
|
748 |
apply (unfold csucc_def) |
|
749 |
apply (rule LeastI) |
|
750 |
apply (blast intro: Card_jump_cardinal K_lt_jump_cardinal Ord_jump_cardinal)+ |
|
751 |
done |
|
752 |
||
45602 | 753 |
lemmas Card_csucc = csucc_basic [THEN conjunct1] |
13216 | 754 |
|
45602 | 755 |
lemmas lt_csucc = csucc_basic [THEN conjunct2] |
13216 | 756 |
|
757 |
lemma Ord_0_lt_csucc: "Ord(K) ==> 0 < csucc(K)" |
|
13221 | 758 |
by (blast intro: Ord_0_le lt_csucc lt_trans1) |
13216 | 759 |
|
46820 | 760 |
lemma csucc_le: "[| Card(L); K<L |] ==> csucc(K) \<le> L" |
13216 | 761 |
apply (unfold csucc_def) |
762 |
apply (rule Least_le) |
|
763 |
apply (blast intro: Card_is_Ord)+ |
|
764 |
done |
|
765 |
||
46821
ff6b0c1087f2
Using mathematical notation for <-> and cardinal arithmetic
paulson
parents:
46820
diff
changeset
|
766 |
lemma lt_csucc_iff: "[| Ord(i); Card(K) |] ==> i < csucc(K) \<longleftrightarrow> |i| \<le> K" |
13216 | 767 |
apply (rule iffI) |
768 |
apply (rule_tac [2] Card_lt_imp_lt) |
|
769 |
apply (erule_tac [2] lt_trans1) |
|
770 |
apply (simp_all add: lt_csucc Card_csucc Card_is_Ord) |
|
771 |
apply (rule notI [THEN not_lt_imp_le]) |
|
13221 | 772 |
apply (rule Card_cardinal [THEN csucc_le, THEN lt_trans1, THEN lt_irrefl], assumption) |
13216 | 773 |
apply (rule Ord_cardinal_le [THEN lt_trans1]) |
46820 | 774 |
apply (simp_all add: Ord_cardinal Card_is_Ord) |
13216 | 775 |
done |
776 |
||
777 |
lemma Card_lt_csucc_iff: |
|
46821
ff6b0c1087f2
Using mathematical notation for <-> and cardinal arithmetic
paulson
parents:
46820
diff
changeset
|
778 |
"[| Card(K'); Card(K) |] ==> K' < csucc(K) \<longleftrightarrow> K' \<le> K" |
13221 | 779 |
by (simp add: lt_csucc_iff Card_cardinal_eq Card_is_Ord) |
13216 | 780 |
|
781 |
lemma InfCard_csucc: "InfCard(K) ==> InfCard(csucc(K))" |
|
46820 | 782 |
by (simp add: InfCard_def Card_csucc Card_is_Ord |
13216 | 783 |
lt_csucc [THEN leI, THEN [2] le_trans]) |
784 |
||
785 |
||
14883 | 786 |
subsubsection{*Removing elements from a finite set decreases its cardinality*} |
13216 | 787 |
|
46820 | 788 |
lemma Fin_imp_not_cons_lepoll: "A: Fin(U) ==> x\<notin>A \<longrightarrow> ~ cons(x,A) \<lesssim> A" |
13216 | 789 |
apply (erule Fin_induct) |
13221 | 790 |
apply (simp add: lepoll_0_iff) |
13216 | 791 |
apply (subgoal_tac "cons (x,cons (xa,y)) = cons (xa,cons (x,y))") |
13221 | 792 |
apply simp |
793 |
apply (blast dest!: cons_lepoll_consD, blast) |
|
13216 | 794 |
done |
795 |
||
14883 | 796 |
lemma Finite_imp_cardinal_cons [simp]: |
46820 | 797 |
"[| Finite(A); a\<notin>A |] ==> |cons(a,A)| = succ(|A|)" |
13216 | 798 |
apply (unfold cardinal_def) |
799 |
apply (rule Least_equality) |
|
800 |
apply (fold cardinal_def) |
|
13221 | 801 |
apply (simp add: succ_def) |
13216 | 802 |
apply (blast intro: cons_eqpoll_cong well_ord_cardinal_eqpoll |
803 |
elim!: mem_irrefl dest!: Finite_imp_well_ord) |
|
804 |
apply (blast intro: Card_cardinal Card_is_Ord) |
|
805 |
apply (rule notI) |
|
13221 | 806 |
apply (rule Finite_into_Fin [THEN Fin_imp_not_cons_lepoll, THEN mp, THEN notE], |
807 |
assumption, assumption) |
|
13216 | 808 |
apply (erule eqpoll_sym [THEN eqpoll_imp_lepoll, THEN lepoll_trans]) |
809 |
apply (erule le_imp_lepoll [THEN lepoll_trans]) |
|
810 |
apply (blast intro: well_ord_cardinal_eqpoll [THEN eqpoll_imp_lepoll] |
|
811 |
dest!: Finite_imp_well_ord) |
|
812 |
done |
|
813 |
||
814 |
||
13221 | 815 |
lemma Finite_imp_succ_cardinal_Diff: |
816 |
"[| Finite(A); a:A |] ==> succ(|A-{a}|) = |A|" |
|
13784 | 817 |
apply (rule_tac b = A in cons_Diff [THEN subst], assumption) |
13221 | 818 |
apply (simp add: Finite_imp_cardinal_cons Diff_subset [THEN subset_Finite]) |
819 |
apply (simp add: cons_Diff) |
|
13216 | 820 |
done |
821 |
||
822 |
lemma Finite_imp_cardinal_Diff: "[| Finite(A); a:A |] ==> |A-{a}| < |A|" |
|
823 |
apply (rule succ_leE) |
|
13221 | 824 |
apply (simp add: Finite_imp_succ_cardinal_Diff) |
13216 | 825 |
done |
826 |
||
46820 | 827 |
lemma Finite_cardinal_in_nat [simp]: "Finite(A) ==> |A| \<in> nat" |
14883 | 828 |
apply (erule Finite_induct) |
829 |
apply (auto simp add: cardinal_0 Finite_imp_cardinal_cons) |
|
830 |
done |
|
13216 | 831 |
|
14883 | 832 |
lemma card_Un_Int: |
46820 | 833 |
"[|Finite(A); Finite(B)|] ==> |A| #+ |B| = |A \<union> B| #+ |A \<inter> B|" |
834 |
apply (erule Finite_induct, simp) |
|
14883 | 835 |
apply (simp add: Finite_Int cons_absorb Un_cons Int_cons_left) |
836 |
done |
|
837 |
||
46820 | 838 |
lemma card_Un_disjoint: |
839 |
"[|Finite(A); Finite(B); A \<inter> B = 0|] ==> |A \<union> B| = |A| #+ |B|" |
|
14883 | 840 |
by (simp add: Finite_Un card_Un_Int) |
841 |
||
842 |
lemma card_partition [rule_format]: |
|
46820 | 843 |
"Finite(C) ==> |
844 |
Finite (\<Union> C) \<longrightarrow> |
|
845 |
(\<forall>c\<in>C. |c| = k) \<longrightarrow> |
|
846 |
(\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = 0) \<longrightarrow> |
|
14883 | 847 |
k #* |C| = |\<Union> C|" |
848 |
apply (erule Finite_induct, auto) |
|
46820 | 849 |
apply (subgoal_tac " x \<inter> \<Union>B = 0") |
14883 | 850 |
apply (auto simp add: card_Un_disjoint Finite_Union |
851 |
subset_Finite [of _ "\<Union> (cons(x,F))"]) |
|
852 |
done |
|
853 |
||
854 |
||
855 |
subsubsection{*Theorems by Krzysztof Grabczewski, proofs by lcp*} |
|
13216 | 856 |
|
45602 | 857 |
lemmas nat_implies_well_ord = nat_into_Ord [THEN well_ord_Memrel] |
13216 | 858 |
|
859 |
lemma nat_sum_eqpoll_sum: "[| m:nat; n:nat |] ==> m + n \<approx> m #+ n" |
|
860 |
apply (rule eqpoll_trans) |
|
861 |
apply (rule well_ord_radd [THEN well_ord_cardinal_eqpoll, THEN eqpoll_sym]) |
|
862 |
apply (erule nat_implies_well_ord)+ |
|
13221 | 863 |
apply (simp add: nat_cadd_eq_add [symmetric] cadd_def eqpoll_refl) |
13216 | 864 |
done |
865 |
||
46820 | 866 |
lemma Ord_subset_natD [rule_format]: "Ord(i) ==> i \<subseteq> nat \<longrightarrow> i \<in> nat | i=nat" |
13221 | 867 |
apply (erule trans_induct3, auto) |
13216 | 868 |
apply (blast dest!: nat_le_Limit [THEN le_imp_subset]) |
869 |
done |
|
870 |
||
46820 | 871 |
lemma Ord_nat_subset_into_Card: "[| Ord(i); i \<subseteq> nat |] ==> Card(i)" |
13221 | 872 |
by (blast dest: Ord_subset_natD intro: Card_nat nat_into_Card) |
13216 | 873 |
|
874 |
lemma Finite_Diff_sing_eq_diff_1: "[| Finite(A); x:A |] ==> |A-{x}| = |A| #- 1" |
|
875 |
apply (rule succ_inject) |
|
876 |
apply (rule_tac b = "|A|" in trans) |
|
13615
449a70d88b38
Numerous cosmetic changes, prompted by the new simplifier
paulson
parents:
13356
diff
changeset
|
877 |
apply (simp add: Finite_imp_succ_cardinal_Diff) |
13216 | 878 |
apply (subgoal_tac "1 \<lesssim> A") |
13221 | 879 |
prefer 2 apply (blast intro: not_0_is_lepoll_1) |
880 |
apply (frule Finite_imp_well_ord, clarify) |
|
13216 | 881 |
apply (drule well_ord_lepoll_imp_Card_le) |
13615
449a70d88b38
Numerous cosmetic changes, prompted by the new simplifier
paulson
parents:
13356
diff
changeset
|
882 |
apply (auto simp add: cardinal_1) |
13216 | 883 |
apply (rule trans) |
13615
449a70d88b38
Numerous cosmetic changes, prompted by the new simplifier
paulson
parents:
13356
diff
changeset
|
884 |
apply (rule_tac [2] diff_succ) |
449a70d88b38
Numerous cosmetic changes, prompted by the new simplifier
paulson
parents:
13356
diff
changeset
|
885 |
apply (auto simp add: Finite_cardinal_in_nat) |
13216 | 886 |
done |
887 |
||
13221 | 888 |
lemma cardinal_lt_imp_Diff_not_0 [rule_format]: |
46820 | 889 |
"Finite(B) ==> \<forall>A. |B|<|A| \<longrightarrow> A - B \<noteq> 0" |
13221 | 890 |
apply (erule Finite_induct, auto) |
891 |
apply (case_tac "Finite (A)") |
|
892 |
apply (subgoal_tac [2] "Finite (cons (x, B))") |
|
893 |
apply (drule_tac [2] B = "cons (x, B) " in Diff_Finite) |
|
894 |
apply (auto simp add: Finite_0 Finite_cons) |
|
13216 | 895 |
apply (subgoal_tac "|B|<|A|") |
13221 | 896 |
prefer 2 apply (blast intro: lt_trans Ord_cardinal) |
13216 | 897 |
apply (case_tac "x:A") |
13221 | 898 |
apply (subgoal_tac [2] "A - cons (x, B) = A - B") |
899 |
apply auto |
|
46820 | 900 |
apply (subgoal_tac "|A| \<le> |cons (x, B) |") |
13221 | 901 |
prefer 2 |
46820 | 902 |
apply (blast dest: Finite_cons [THEN Finite_imp_well_ord] |
13216 | 903 |
intro: well_ord_lepoll_imp_Card_le subset_imp_lepoll) |
904 |
apply (auto simp add: Finite_imp_cardinal_cons) |
|
905 |
apply (auto dest!: Finite_cardinal_in_nat simp add: le_iff) |
|
906 |
apply (blast intro: lt_trans) |
|
907 |
done |
|
908 |
||
437 | 909 |
end |