src/ZF/CardinalArith.thy
author wenzelm
Fri Oct 09 20:26:03 2015 +0200 (2015-10-09)
changeset 61378 3e04c9ca001a
parent 60770 240563fbf41d
child 61394 6142b282b164
permissions -rw-r--r--
discontinued specific HTML syntax;
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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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section\<open>Cardinal Arithmetic Without the Axiom of Choice\<close>
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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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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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    --\<open>This def is more complex than Kunen's but it more easily proved to
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        be a cardinal\<close>
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    "jump_cardinal(K) ==
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         \<Union>X\<in>Pow(K). {z. r \<in> 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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    --\<open>needed because @{term "jump_cardinal(K)"} might not be the successor
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        of @{term K}\<close>
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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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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 \<in> A ==> Card(K(x))) ==> Card(\<Union>x\<in>A. K(x))"
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  by blast
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lemma Card_OUN [simp,intro,TC]:
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     "(!!x. x \<in> 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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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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subsection\<open>Cardinal addition\<close>
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text\<open>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.\<close>
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subsubsection\<open>Cardinal addition is commutative\<close>
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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)
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apply (rule sum_commute_eqpoll [THEN cardinal_cong])
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done
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subsubsection\<open>Cardinal addition is associative\<close>
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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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text\<open>Unconditional version requires AC\<close>
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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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subsubsection\<open>0 is the identity for addition\<close>
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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\<open>Addition by another cardinal\<close>
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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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(*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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  assumes K: "Card(K)" and L: "Ord(L)" shows "K \<le> (K \<oplus> L)"
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proof (unfold cadd_def)
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  have "K \<le> |K|"
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    by (rule Card_cardinal_le [OF K])
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  moreover have "|K| \<le> |K + L|" using K L
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    by (blast intro: well_ord_lepoll_imp_Card_le sum_lepoll_self
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                     well_ord_radd well_ord_Memrel Card_is_Ord)
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  ultimately show "K \<le> |K + L|"
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    by (blast intro: le_trans)
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qed
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subsubsection\<open>Monotonicity of addition\<close>
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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
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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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subsubsection\<open>Addition of finite cardinals is "ordinary" addition\<close>
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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 \<in> nat  *)
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(*Unconditional version requires AC*)
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lemma cadd_succ_lemma:
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  assumes "Ord(m)" "Ord(n)" shows "succ(m) \<oplus> n = |succ(m \<oplus> n)|"
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proof (unfold cadd_def)
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  have [intro]: "m + n \<approx> |m + n|" using assms
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    by (blast intro: eqpoll_sym well_ord_cardinal_eqpoll well_ord_radd well_ord_Memrel)
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  have "|succ(m) + n| = |succ(m + n)|"
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    by (rule sum_succ_eqpoll [THEN cardinal_cong])
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  also have "... = |succ(|m + n|)|"
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    by (blast intro: succ_eqpoll_cong cardinal_cong)
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  finally show "|succ(m) + n| = |succ(|m + n|)|" .
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qed
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lemma nat_cadd_eq_add:
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  assumes m: "m \<in> nat" and [simp]: "n \<in> nat" shows"m \<oplus> n = m #+ n"
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using m
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proof (induct m)
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  case 0 thus ?case by (simp add: nat_into_Card cadd_0)
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next
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  case (succ m) thus ?case by (simp add: cadd_succ_lemma nat_into_Card Card_cardinal_eq)
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qed
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subsection\<open>Cardinal multiplication\<close>
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subsubsection\<open>Cardinal multiplication is commutative\<close>
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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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       auto)
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done
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lemma cmult_commute: "i \<otimes> j = j \<otimes> 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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subsubsection\<open>Cardinal multiplication is associative\<close>
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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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text\<open>Unconditional version requires AC\<close>
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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)"
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  shows "(i \<otimes> j) \<otimes> k = i \<otimes> (j \<otimes> k)"
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proof (unfold cmult_def, rule cardinal_cong)
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  have "|i * j| * k \<approx> (i * j) * k"
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    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)"
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    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)
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  finally show "|i * j| * k \<approx> i * |j * k|" .
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qed
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subsubsection\<open>Cardinal multiplication distributes over addition\<close>
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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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  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) \<otimes> k = (i \<otimes> k) \<oplus> (j \<otimes> k)"
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proof (unfold cadd_def cmult_def, rule cardinal_cong)
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  have "|i + j| * k \<approx> (i + j) * k"
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    by (blast intro: prod_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_refl well_ord_radd i j)
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  also have "...  \<approx> i * k + j * k"
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    by (rule sum_prod_distrib_eqpoll)
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  also have "...  \<approx> |i * k| + |j * k|"
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    by (blast intro: sum_eqpoll_cong well_ord_cardinal_eqpoll well_ord_rmult i j k eqpoll_sym)
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  finally show "|i + j| * k \<approx> |i * k| + |j * k|" .
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qed
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subsubsection\<open>Multiplication by 0 yields 0\<close>
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lemma prod_0_eqpoll: "0*A \<approx> 0"
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apply (unfold eqpoll_def)
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apply (rule exI)
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apply (rule lam_bijective, safe)
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done
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lemma cmult_0 [simp]: "0 \<otimes> i = 0"
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by (simp add: cmult_def prod_0_eqpoll [THEN cardinal_cong])
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subsubsection\<open>1 is the identity for multiplication\<close>
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lemma prod_singleton_eqpoll: "{x}*A \<approx> A"
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apply (unfold eqpoll_def)
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apply (rule exI)
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apply (rule singleton_prod_bij [THEN bij_converse_bij])
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done
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lemma cmult_1 [simp]: "Card(K) ==> 1 \<otimes> K = K"
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apply (unfold cmult_def succ_def)
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apply (simp add: prod_singleton_eqpoll [THEN cardinal_cong] Card_cardinal_eq)
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done
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subsection\<open>Some inequalities for multiplication\<close>
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lemma prod_square_lepoll: "A \<lesssim> A*A"
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apply (unfold lepoll_def inj_def)
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apply (rule_tac x = "\<lambda>x\<in>A. <x,x>" in exI, simp)
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done
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(*Could probably weaken the premise to well_ord(K,r), or remove using AC*)
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   312
lemma cmult_square_le: "Card(K) ==> K \<le> K \<otimes> K"
paulson@13216
   313
apply (unfold cmult_def)
paulson@13216
   314
apply (rule le_trans)
paulson@13216
   315
apply (rule_tac [2] well_ord_lepoll_imp_Card_le)
paulson@13216
   316
apply (rule_tac [3] prod_square_lepoll)
paulson@13221
   317
apply (simp add: le_refl Card_is_Ord Card_cardinal_eq)
paulson@13221
   318
apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord)
paulson@13216
   319
done
paulson@13216
   320
wenzelm@60770
   321
subsubsection\<open>Multiplication by a non-zero cardinal\<close>
paulson@13216
   322
paulson@46953
   323
lemma prod_lepoll_self: "b \<in> B ==> A \<lesssim> A*B"
paulson@13216
   324
apply (unfold lepoll_def inj_def)
paulson@46820
   325
apply (rule_tac x = "\<lambda>x\<in>A. <x,b>" in exI, simp)
paulson@13216
   326
done
paulson@13216
   327
paulson@13216
   328
(*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
paulson@13216
   329
lemma cmult_le_self:
paulson@46821
   330
    "[| Card(K);  Ord(L);  0<L |] ==> K \<le> (K \<otimes> L)"
paulson@13216
   331
apply (unfold cmult_def)
paulson@13216
   332
apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le])
paulson@13221
   333
  apply assumption
paulson@13216
   334
 apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord)
paulson@13216
   335
apply (blast intro: prod_lepoll_self ltD)
paulson@13216
   336
done
paulson@13216
   337
wenzelm@60770
   338
subsubsection\<open>Monotonicity of multiplication\<close>
paulson@13216
   339
paulson@13216
   340
lemma prod_lepoll_mono:
paulson@13216
   341
     "[| A \<lesssim> C;  B \<lesssim> D |] ==> A * B  \<lesssim>  C * D"
paulson@13216
   342
apply (unfold lepoll_def)
paulson@13221
   343
apply (elim exE)
paulson@13216
   344
apply (rule_tac x = "lam <w,y>:A*B. <f`w, fa`y>" in exI)
paulson@46820
   345
apply (rule_tac d = "%<w,y>. <converse (f) `w, converse (fa) `y>"
paulson@13216
   346
       in lam_injective)
paulson@13221
   347
apply (typecheck add: inj_is_fun, auto)
paulson@13216
   348
done
paulson@13216
   349
paulson@13216
   350
lemma cmult_le_mono:
paulson@46821
   351
    "[| K' \<le> K;  L' \<le> L |] ==> (K' \<otimes> L') \<le> (K \<otimes> L)"
paulson@13216
   352
apply (unfold cmult_def)
paulson@13216
   353
apply (safe dest!: le_subset_iff [THEN iffD1])
paulson@13216
   354
apply (rule well_ord_lepoll_imp_Card_le)
paulson@13216
   355
 apply (blast intro: well_ord_rmult well_ord_Memrel)
paulson@13216
   356
apply (blast intro: prod_lepoll_mono subset_imp_lepoll)
paulson@13216
   357
done
paulson@13216
   358
wenzelm@60770
   359
subsection\<open>Multiplication of finite cardinals is "ordinary" multiplication\<close>
paulson@13216
   360
paulson@13216
   361
lemma prod_succ_eqpoll: "succ(A)*B \<approx> B + A*B"
paulson@13216
   362
apply (unfold eqpoll_def)
paulson@13221
   363
apply (rule exI)
paulson@13216
   364
apply (rule_tac c = "%<x,y>. if x=A then Inl (y) else Inr (<x,y>)"
paulson@13216
   365
            and d = "case (%y. <A,y>, %z. z)" in lam_bijective)
paulson@13216
   366
apply safe
paulson@13216
   367
apply (simp_all add: succI2 if_type mem_imp_not_eq)
paulson@13216
   368
done
paulson@13216
   369
paulson@13216
   370
(*Unconditional version requires AC*)
paulson@13216
   371
lemma cmult_succ_lemma:
paulson@46821
   372
    "[| Ord(m);  Ord(n) |] ==> succ(m) \<otimes> n = n \<oplus> (m \<otimes> n)"
paulson@13216
   373
apply (unfold cmult_def cadd_def)
paulson@13216
   374
apply (rule prod_succ_eqpoll [THEN cardinal_cong, THEN trans])
paulson@13216
   375
apply (rule cardinal_cong [symmetric])
paulson@13216
   376
apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
paulson@13216
   377
apply (blast intro: well_ord_rmult well_ord_Memrel)
paulson@13216
   378
done
paulson@13216
   379
paulson@46953
   380
lemma nat_cmult_eq_mult: "[| m \<in> nat;  n \<in> nat |] ==> m \<otimes> n = m#*n"
paulson@13244
   381
apply (induct_tac m)
paulson@13221
   382
apply (simp_all add: cmult_succ_lemma nat_cadd_eq_add)
paulson@13216
   383
done
paulson@13216
   384
paulson@46821
   385
lemma cmult_2: "Card(n) ==> 2 \<otimes> n = n \<oplus> n"
paulson@13221
   386
by (simp add: cmult_succ_lemma Card_is_Ord cadd_commute [of _ 0])
paulson@13216
   387
paulson@46953
   388
lemma sum_lepoll_prod:
paulson@46901
   389
  assumes C: "2 \<lesssim> C" shows "B+B \<lesssim> C*B"
paulson@46901
   390
proof -
paulson@46901
   391
  have "B+B \<lesssim> 2*B"
paulson@46953
   392
    by (simp add: sum_eq_2_times)
paulson@46901
   393
  also have "... \<lesssim> C*B"
paulson@46953
   394
    by (blast intro: prod_lepoll_mono lepoll_refl C)
paulson@46901
   395
  finally show "B+B \<lesssim> C*B" .
paulson@46901
   396
qed
paulson@13216
   397
paulson@13216
   398
lemma lepoll_imp_sum_lepoll_prod: "[| A \<lesssim> B; 2 \<lesssim> A |] ==> A+B \<lesssim> A*B"
paulson@13221
   399
by (blast intro: sum_lepoll_mono sum_lepoll_prod lepoll_trans lepoll_refl)
paulson@13216
   400
paulson@13216
   401
wenzelm@60770
   402
subsection\<open>Infinite Cardinals are Limit Ordinals\<close>
paulson@13216
   403
paulson@13216
   404
(*This proof is modelled upon one assuming nat<=A, with injection
paulson@46820
   405
  \<lambda>z\<in>cons(u,A). if z=u then 0 else if z \<in> nat then succ(z) else z
paulson@46953
   406
  and inverse %y. if y \<in> nat then nat_case(u, %z. z, y) else y.  \
paulson@46953
   407
  If f \<in> inj(nat,A) then range(f) behaves like the natural numbers.*)
paulson@13216
   408
lemma nat_cons_lepoll: "nat \<lesssim> A ==> cons(u,A) \<lesssim> A"
paulson@13216
   409
apply (unfold lepoll_def)
paulson@13216
   410
apply (erule exE)
paulson@46820
   411
apply (rule_tac x =
paulson@46820
   412
          "\<lambda>z\<in>cons (u,A).
paulson@46820
   413
             if z=u then f`0
paulson@46953
   414
             else if z \<in> range (f) then f`succ (converse (f) `z) else z"
paulson@13216
   415
       in exI)
paulson@13216
   416
apply (rule_tac d =
paulson@46953
   417
          "%y. if y \<in> range(f) then nat_case (u, %z. f`z, converse(f) `y)
paulson@46820
   418
                              else y"
paulson@13216
   419
       in lam_injective)
paulson@13216
   420
apply (fast intro!: if_type apply_type intro: inj_is_fun inj_converse_fun)
paulson@13216
   421
apply (simp add: inj_is_fun [THEN apply_rangeI]
paulson@13216
   422
                 inj_converse_fun [THEN apply_rangeI]
paulson@13216
   423
                 inj_converse_fun [THEN apply_funtype])
paulson@13216
   424
done
paulson@13216
   425
paulson@13216
   426
lemma nat_cons_eqpoll: "nat \<lesssim> A ==> cons(u,A) \<approx> A"
paulson@13216
   427
apply (erule nat_cons_lepoll [THEN eqpollI])
paulson@13216
   428
apply (rule subset_consI [THEN subset_imp_lepoll])
paulson@13216
   429
done
paulson@13216
   430
paulson@13216
   431
(*Specialized version required below*)
paulson@46820
   432
lemma nat_succ_eqpoll: "nat \<subseteq> A ==> succ(A) \<approx> A"
paulson@13216
   433
apply (unfold succ_def)
paulson@13216
   434
apply (erule subset_imp_lepoll [THEN nat_cons_eqpoll])
paulson@13216
   435
done
paulson@13216
   436
paulson@13216
   437
lemma InfCard_nat: "InfCard(nat)"
paulson@13216
   438
apply (unfold InfCard_def)
paulson@13216
   439
apply (blast intro: Card_nat le_refl Card_is_Ord)
paulson@13216
   440
done
paulson@13216
   441
paulson@13216
   442
lemma InfCard_is_Card: "InfCard(K) ==> Card(K)"
paulson@13216
   443
apply (unfold InfCard_def)
paulson@13216
   444
apply (erule conjunct1)
paulson@13216
   445
done
paulson@13216
   446
paulson@13216
   447
lemma InfCard_Un:
paulson@46820
   448
    "[| InfCard(K);  Card(L) |] ==> InfCard(K \<union> L)"
paulson@13216
   449
apply (unfold InfCard_def)
paulson@13216
   450
apply (simp add: Card_Un Un_upper1_le [THEN [2] le_trans]  Card_is_Ord)
paulson@13216
   451
done
paulson@13216
   452
paulson@13216
   453
(*Kunen's Lemma 10.11*)
paulson@13216
   454
lemma InfCard_is_Limit: "InfCard(K) ==> Limit(K)"
paulson@13216
   455
apply (unfold InfCard_def)
paulson@13216
   456
apply (erule conjE)
paulson@13216
   457
apply (frule Card_is_Ord)
paulson@13216
   458
apply (rule ltI [THEN non_succ_LimitI])
paulson@13216
   459
apply (erule le_imp_subset [THEN subsetD])
paulson@13216
   460
apply (safe dest!: Limit_nat [THEN Limit_le_succD])
paulson@13216
   461
apply (unfold Card_def)
paulson@13216
   462
apply (drule trans)
paulson@13216
   463
apply (erule le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong])
paulson@13216
   464
apply (erule Ord_cardinal_le [THEN lt_trans2, THEN lt_irrefl])
paulson@13221
   465
apply (rule le_eqI, assumption)
paulson@13216
   466
apply (rule Ord_cardinal)
paulson@13216
   467
done
paulson@13216
   468
paulson@13216
   469
paulson@13216
   470
(*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***)
paulson@13216
   471
paulson@13216
   472
(*A general fact about ordermap*)
paulson@13216
   473
lemma ordermap_eqpoll_pred:
paulson@46953
   474
    "[| well_ord(A,r);  x \<in> A |] ==> ordermap(A,r)`x \<approx> Order.pred(A,x,r)"
paulson@13216
   475
apply (unfold eqpoll_def)
paulson@13216
   476
apply (rule exI)
paulson@13221
   477
apply (simp add: ordermap_eq_image well_ord_is_wf)
paulson@46820
   478
apply (erule ordermap_bij [THEN bij_is_inj, THEN restrict_bij,
paulson@13221
   479
                           THEN bij_converse_bij])
paulson@13216
   480
apply (rule pred_subset)
paulson@13216
   481
done
paulson@13216
   482
wenzelm@60770
   483
subsubsection\<open>Establishing the well-ordering\<close>
paulson@13216
   484
paulson@46953
   485
lemma well_ord_csquare:
paulson@46901
   486
  assumes K: "Ord(K)" shows "well_ord(K*K, csquare_rel(K))"
paulson@46901
   487
proof (unfold csquare_rel_def, rule well_ord_rvimage)
paulson@46901
   488
  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
paulson@46901
   489
    by (force simp add: inj_def intro: lam_type Un_least_lt [THEN ltD] ltI)
paulson@46901
   490
next
paulson@46901
   491
  show "well_ord(K \<times> K \<times> K, rmult(K, Memrel(K), K \<times> K, rmult(K, Memrel(K), K, Memrel(K))))"
paulson@46901
   492
    using K by (blast intro: well_ord_rmult well_ord_Memrel)
paulson@46901
   493
qed
paulson@13216
   494
wenzelm@60770
   495
subsubsection\<open>Characterising initial segments of the well-ordering\<close>
paulson@13216
   496
paulson@13216
   497
lemma csquareD:
paulson@46820
   498
 "[| <<x,y>, <z,z>> \<in> csquare_rel(K);  x<K;  y<K;  z<K |] ==> x \<le> z & y \<le> z"
paulson@13216
   499
apply (unfold csquare_rel_def)
paulson@13216
   500
apply (erule rev_mp)
paulson@13216
   501
apply (elim ltE)
paulson@13221
   502
apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
paulson@13216
   503
apply (safe elim!: mem_irrefl intro!: Un_upper1_le Un_upper2_le)
paulson@13221
   504
apply (simp_all add: lt_def succI2)
paulson@13216
   505
done
paulson@13216
   506
paulson@46820
   507
lemma pred_csquare_subset:
paulson@46820
   508
    "z<K ==> Order.pred(K*K, <z,z>, csquare_rel(K)) \<subseteq> succ(z)*succ(z)"
paulson@13216
   509
apply (unfold Order.pred_def)
paulson@46901
   510
apply (safe del: SigmaI dest!: csquareD)
paulson@46820
   511
apply (unfold lt_def, auto)
paulson@13216
   512
done
paulson@13216
   513
paulson@13216
   514
lemma csquare_ltI:
paulson@46820
   515
 "[| x<z;  y<z;  z<K |] ==>  <<x,y>, <z,z>> \<in> csquare_rel(K)"
paulson@13216
   516
apply (unfold csquare_rel_def)
paulson@13216
   517
apply (subgoal_tac "x<K & y<K")
paulson@46820
   518
 prefer 2 apply (blast intro: lt_trans)
paulson@13216
   519
apply (elim ltE)
paulson@13221
   520
apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
paulson@13216
   521
done
paulson@13216
   522
paulson@13216
   523
(*Part of the traditional proof.  UNUSED since it's harder to prove & apply *)
paulson@13216
   524
lemma csquare_or_eqI:
paulson@46820
   525
 "[| x \<le> z;  y \<le> z;  z<K |] ==> <<x,y>, <z,z>> \<in> csquare_rel(K) | x=z & y=z"
paulson@13216
   526
apply (unfold csquare_rel_def)
paulson@13216
   527
apply (subgoal_tac "x<K & y<K")
paulson@46820
   528
 prefer 2 apply (blast intro: lt_trans1)
paulson@13216
   529
apply (elim ltE)
paulson@13221
   530
apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
paulson@13216
   531
apply (elim succE)
paulson@46820
   532
apply (simp_all add: subset_Un_iff [THEN iff_sym]
paulson@13221
   533
                     subset_Un_iff2 [THEN iff_sym] OrdmemD)
paulson@13216
   534
done
paulson@13216
   535
wenzelm@60770
   536
subsubsection\<open>The cardinality of initial segments\<close>
paulson@13216
   537
paulson@13216
   538
lemma ordermap_z_lt:
paulson@46820
   539
      "[| Limit(K);  x<K;  y<K;  z=succ(x \<union> y) |] ==>
paulson@13216
   540
          ordermap(K*K, csquare_rel(K)) ` <x,y> <
paulson@13216
   541
          ordermap(K*K, csquare_rel(K)) ` <z,z>"
paulson@13216
   542
apply (subgoal_tac "z<K & well_ord (K*K, csquare_rel (K))")
paulson@13216
   543
prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ
paulson@46820
   544
                              Limit_is_Ord [THEN well_ord_csquare], clarify)
paulson@13216
   545
apply (rule csquare_ltI [THEN ordermap_mono, THEN ltI])
paulson@13216
   546
apply (erule_tac [4] well_ord_is_wf)
paulson@13216
   547
apply (blast intro!: Un_upper1_le Un_upper2_le Ord_ordermap elim!: ltE)+
paulson@13216
   548
done
paulson@13216
   549
wenzelm@60770
   550
text\<open>Kunen: "each @{term"\<langle>x,y\<rangle> \<in> K \<times> K"} has no more than @{term"z \<times> z"} predecessors..." (page 29)\<close>
paulson@13216
   551
lemma ordermap_csquare_le:
paulson@46953
   552
  assumes K: "Limit(K)" and x: "x<K" and y: " y<K"
paulson@46907
   553
  defines "z \<equiv> succ(x \<union> y)"
paulson@46901
   554
  shows "|ordermap(K \<times> K, csquare_rel(K)) ` \<langle>x,y\<rangle>| \<le> |succ(z)| \<otimes> |succ(z)|"
paulson@46901
   555
proof (unfold cmult_def, rule well_ord_lepoll_imp_Card_le)
paulson@46953
   556
  show "well_ord(|succ(z)| \<times> |succ(z)|,
paulson@46901
   557
                 rmult(|succ(z)|, Memrel(|succ(z)|), |succ(z)|, Memrel(|succ(z)|)))"
paulson@46953
   558
    by (blast intro: Ord_cardinal well_ord_Memrel well_ord_rmult)
paulson@46901
   559
next
paulson@46907
   560
  have zK: "z<K" using x y K z_def
paulson@46901
   561
    by (blast intro: Un_least_lt Limit_has_succ)
paulson@46953
   562
  hence oz: "Ord(z)" by (elim ltE)
paulson@46901
   563
  have "ordermap(K \<times> K, csquare_rel(K)) ` \<langle>x,y\<rangle> \<lesssim> ordermap(K \<times> K, csquare_rel(K)) ` \<langle>z,z\<rangle>"
paulson@46907
   564
    using z_def
paulson@46953
   565
    by (blast intro: ordermap_z_lt leI le_imp_lepoll K x y)
paulson@46901
   566
  also have "... \<approx>  Order.pred(K \<times> K, \<langle>z,z\<rangle>, csquare_rel(K))"
paulson@46901
   567
    proof (rule ordermap_eqpoll_pred)
paulson@46953
   568
      show "well_ord(K \<times> K, csquare_rel(K))" using K
paulson@46901
   569
        by (rule Limit_is_Ord [THEN well_ord_csquare])
paulson@46901
   570
    next
paulson@46901
   571
      show "\<langle>z, z\<rangle> \<in> K \<times> K" using zK
paulson@46901
   572
        by (blast intro: ltD)
paulson@46901
   573
    qed
paulson@46901
   574
  also have "...  \<lesssim> succ(z) \<times> succ(z)" using zK
paulson@46901
   575
    by (rule pred_csquare_subset [THEN subset_imp_lepoll])
paulson@46901
   576
  also have "... \<approx> |succ(z)| \<times> |succ(z)|" using oz
paulson@46953
   577
    by (blast intro: prod_eqpoll_cong Ord_succ Ord_cardinal_eqpoll eqpoll_sym)
paulson@46901
   578
  finally show "ordermap(K \<times> K, csquare_rel(K)) ` \<langle>x,y\<rangle> \<lesssim> |succ(z)| \<times> |succ(z)|" .
paulson@46901
   579
qed
paulson@13216
   580
wenzelm@60770
   581
text\<open>Kunen: "... so the order type is @{text"\<le>"} K"\<close>
paulson@13216
   582
lemma ordertype_csquare_le:
paulson@46907
   583
  assumes IK: "InfCard(K)" and eq: "\<And>y. y\<in>K \<Longrightarrow> InfCard(y) \<Longrightarrow> y \<otimes> y = y"
paulson@46907
   584
  shows "ordertype(K*K, csquare_rel(K)) \<le> K"
paulson@46907
   585
proof -
paulson@46953
   586
  have  CK: "Card(K)" using IK by (rule InfCard_is_Card)
paulson@46953
   587
  hence OK: "Ord(K)"  by (rule Card_is_Ord)
paulson@46907
   588
  moreover have "Ord(ordertype(K \<times> K, csquare_rel(K)))" using OK
paulson@46907
   589
    by (rule well_ord_csquare [THEN Ord_ordertype])
paulson@46907
   590
  ultimately show ?thesis
paulson@46907
   591
  proof (rule all_lt_imp_le)
paulson@46907
   592
    fix i
paulson@46907
   593
    assume i: "i < ordertype(K \<times> K, csquare_rel(K))"
paulson@46907
   594
    hence Oi: "Ord(i)" by (elim ltE)
paulson@46953
   595
    obtain x y where x: "x \<in> K" and y: "y \<in> K"
paulson@46907
   596
                 and ieq: "i = ordermap(K \<times> K, csquare_rel(K)) ` \<langle>x,y\<rangle>"
paulson@46907
   597
      using i by (auto simp add: ordertype_unfold elim: ltE)
paulson@46953
   598
    hence xy: "Ord(x)" "Ord(y)" "x < K" "y < K" using OK
paulson@46907
   599
      by (blast intro: Ord_in_Ord ltI)+
paulson@46907
   600
    hence ou: "Ord(x \<union> y)"
paulson@46953
   601
      by (simp add: Ord_Un)
paulson@46907
   602
    show "i < K"
paulson@46907
   603
      proof (rule Card_lt_imp_lt [OF _ Oi CK])
paulson@46907
   604
        have "|i| \<le> |succ(succ(x \<union> y))| \<otimes> |succ(succ(x \<union> y))|" using IK xy
paulson@46907
   605
          by (auto simp add: ieq intro: InfCard_is_Limit [THEN ordermap_csquare_le])
paulson@46953
   606
        moreover have "|succ(succ(x \<union> y))| \<otimes> |succ(succ(x \<union> y))| < K"
paulson@46907
   607
          proof (cases rule: Ord_linear2 [OF ou Ord_nat])
paulson@46907
   608
            assume "x \<union> y < nat"
paulson@46907
   609
            hence "|succ(succ(x \<union> y))| \<otimes> |succ(succ(x \<union> y))| \<in> nat"
paulson@46907
   610
              by (simp add: lt_def nat_cmult_eq_mult nat_succI mult_type
paulson@46907
   611
                         nat_into_Card [THEN Card_cardinal_eq]  Ord_nat)
paulson@46907
   612
            also have "... \<subseteq> K" using IK
paulson@46907
   613
              by (simp add: InfCard_def le_imp_subset)
paulson@46953
   614
            finally show "|succ(succ(x \<union> y))| \<otimes> |succ(succ(x \<union> y))| < K"
paulson@46953
   615
              by (simp add: ltI OK)
paulson@46907
   616
          next
paulson@46907
   617
            assume natxy: "nat \<le> x \<union> y"
paulson@46953
   618
            hence seq: "|succ(succ(x \<union> y))| = |x \<union> y|" using xy
paulson@46907
   619
              by (simp add: le_imp_subset nat_succ_eqpoll [THEN cardinal_cong] le_succ_iff)
paulson@46953
   620
            also have "... < K" using xy
paulson@46907
   621
              by (simp add: Un_least_lt Ord_cardinal_le [THEN lt_trans1])
paulson@46907
   622
            finally have "|succ(succ(x \<union> y))| < K" .
paulson@46907
   623
            moreover have "InfCard(|succ(succ(x \<union> y))|)" using xy natxy
paulson@46907
   624
              by (simp add: seq InfCard_def Card_cardinal nat_le_cardinal)
paulson@46953
   625
            ultimately show ?thesis  by (simp add: eq ltD)
paulson@46907
   626
          qed
paulson@46953
   627
        ultimately show "|i| < K" by (blast intro: lt_trans1)
paulson@46907
   628
    qed
paulson@46907
   629
  qed
paulson@46907
   630
qed
paulson@13216
   631
paulson@13216
   632
(*Main result: Kunen's Theorem 10.12*)
paulson@46953
   633
lemma InfCard_csquare_eq:
paulson@46907
   634
  assumes IK: "InfCard(K)" shows "InfCard(K) ==> K \<otimes> K = K"
paulson@46907
   635
proof -
paulson@46953
   636
  have  OK: "Ord(K)" using IK by (simp add: Card_is_Ord InfCard_is_Card)
paulson@46935
   637
  show "InfCard(K) ==> K \<otimes> K = K" using OK
paulson@46935
   638
  proof (induct rule: trans_induct)
paulson@46935
   639
    case (step i)
paulson@46935
   640
    show "i \<otimes> i = i"
paulson@46935
   641
    proof (rule le_anti_sym)
paulson@46953
   642
      have "|i \<times> i| = |ordertype(i \<times> i, csquare_rel(i))|"
paulson@46953
   643
        by (rule cardinal_cong,
paulson@46935
   644
          simp add: step.hyps well_ord_csquare [THEN ordermap_bij, THEN bij_imp_eqpoll])
paulson@46953
   645
      hence "i \<otimes> i \<le> ordertype(i \<times> i, csquare_rel(i))"
paulson@46935
   646
        by (simp add: step.hyps cmult_def Ord_cardinal_le well_ord_csquare [THEN Ord_ordertype])
paulson@46935
   647
      moreover
paulson@46935
   648
      have "ordertype(i \<times> i, csquare_rel(i)) \<le> i" using step
paulson@46953
   649
        by (simp add: ordertype_csquare_le)
paulson@46935
   650
      ultimately show "i \<otimes> i \<le> i" by (rule le_trans)
paulson@46935
   651
    next
paulson@46935
   652
      show "i \<le> i \<otimes> i" using step
paulson@46953
   653
        by (blast intro: cmult_square_le InfCard_is_Card)
paulson@46907
   654
    qed
paulson@46935
   655
  qed
paulson@46907
   656
qed
paulson@13216
   657
paulson@13216
   658
(*Corollary for arbitrary well-ordered sets (all sets, assuming AC)*)
paulson@13216
   659
lemma well_ord_InfCard_square_eq:
paulson@46907
   660
  assumes r: "well_ord(A,r)" and I: "InfCard(|A|)" shows "A \<times> A \<approx> A"
paulson@46907
   661
proof -
paulson@46907
   662
  have "A \<times> A \<approx> |A| \<times> |A|"
paulson@46953
   663
    by (blast intro: prod_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_sym r)
paulson@46907
   664
  also have "... \<approx> A"
paulson@46907
   665
    proof (rule well_ord_cardinal_eqE [OF _ r])
paulson@46907
   666
      show "well_ord(|A| \<times> |A|, rmult(|A|, Memrel(|A|), |A|, Memrel(|A|)))"
paulson@46907
   667
        by (blast intro: Ord_cardinal well_ord_rmult well_ord_Memrel r)
paulson@46907
   668
    next
paulson@46907
   669
      show "||A| \<times> |A|| = |A|" using InfCard_csquare_eq I
paulson@46907
   670
        by (simp add: cmult_def)
paulson@46953
   671
    qed
paulson@46907
   672
  finally show ?thesis .
paulson@46907
   673
qed
paulson@13216
   674
paulson@13356
   675
lemma InfCard_square_eqpoll: "InfCard(K) ==> K \<times> K \<approx> K"
paulson@46820
   676
apply (rule well_ord_InfCard_square_eq)
paulson@46820
   677
 apply (erule InfCard_is_Card [THEN Card_is_Ord, THEN well_ord_Memrel])
paulson@46820
   678
apply (simp add: InfCard_is_Card [THEN Card_cardinal_eq])
paulson@13356
   679
done
paulson@13356
   680
paulson@47101
   681
lemma Inf_Card_is_InfCard: "[| Card(i); ~ Finite(i) |] ==> InfCard(i)"
paulson@13356
   682
by (simp add: InfCard_def Card_is_Ord [THEN nat_le_infinite_Ord])
paulson@13356
   683
wenzelm@60770
   684
subsubsection\<open>Toward's Kunen's Corollary 10.13 (1)\<close>
paulson@13216
   685
paulson@46821
   686
lemma InfCard_le_cmult_eq: "[| InfCard(K);  L \<le> K;  0<L |] ==> K \<otimes> L = K"
paulson@13216
   687
apply (rule le_anti_sym)
paulson@13216
   688
 prefer 2
paulson@13216
   689
 apply (erule ltE, blast intro: cmult_le_self InfCard_is_Card)
paulson@13216
   690
apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl])
paulson@13216
   691
apply (rule cmult_le_mono [THEN le_trans], assumption+)
paulson@13216
   692
apply (simp add: InfCard_csquare_eq)
paulson@13216
   693
done
paulson@13216
   694
paulson@13216
   695
(*Corollary 10.13 (1), for cardinal multiplication*)
paulson@46821
   696
lemma InfCard_cmult_eq: "[| InfCard(K);  InfCard(L) |] ==> K \<otimes> L = K \<union> L"
paulson@13784
   697
apply (rule_tac i = K and j = L in Ord_linear_le)
paulson@13216
   698
apply (typecheck add: InfCard_is_Card Card_is_Ord)
paulson@13216
   699
apply (rule cmult_commute [THEN ssubst])
paulson@13216
   700
apply (rule Un_commute [THEN ssubst])
paulson@46820
   701
apply (simp_all add: InfCard_is_Limit [THEN Limit_has_0] InfCard_le_cmult_eq
paulson@13221
   702
                     subset_Un_iff2 [THEN iffD1] le_imp_subset)
paulson@13216
   703
done
paulson@13216
   704
paulson@46821
   705
lemma InfCard_cdouble_eq: "InfCard(K) ==> K \<oplus> K = K"
paulson@13221
   706
apply (simp add: cmult_2 [symmetric] InfCard_is_Card cmult_commute)
paulson@13221
   707
apply (simp add: InfCard_le_cmult_eq InfCard_is_Limit Limit_has_0 Limit_has_succ)
paulson@13216
   708
done
paulson@13216
   709
paulson@13216
   710
(*Corollary 10.13 (1), for cardinal addition*)
paulson@46821
   711
lemma InfCard_le_cadd_eq: "[| InfCard(K);  L \<le> K |] ==> K \<oplus> L = K"
paulson@13216
   712
apply (rule le_anti_sym)
paulson@13216
   713
 prefer 2
paulson@13216
   714
 apply (erule ltE, blast intro: cadd_le_self InfCard_is_Card)
paulson@13216
   715
apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl])
paulson@13216
   716
apply (rule cadd_le_mono [THEN le_trans], assumption+)
paulson@13216
   717
apply (simp add: InfCard_cdouble_eq)
paulson@13216
   718
done
paulson@13216
   719
paulson@46821
   720
lemma InfCard_cadd_eq: "[| InfCard(K);  InfCard(L) |] ==> K \<oplus> L = K \<union> L"
paulson@13784
   721
apply (rule_tac i = K and j = L in Ord_linear_le)
paulson@13216
   722
apply (typecheck add: InfCard_is_Card Card_is_Ord)
paulson@13216
   723
apply (rule cadd_commute [THEN ssubst])
paulson@13216
   724
apply (rule Un_commute [THEN ssubst])
paulson@13221
   725
apply (simp_all add: InfCard_le_cadd_eq subset_Un_iff2 [THEN iffD1] le_imp_subset)
paulson@13216
   726
done
paulson@13216
   727
paulson@13216
   728
(*The other part, Corollary 10.13 (2), refers to the cardinality of the set
paulson@13216
   729
  of all n-tuples of elements of K.  A better version for the Isabelle theory
paulson@13216
   730
  might be  InfCard(K) ==> |list(K)| = K.
paulson@13216
   731
*)
paulson@13216
   732
wenzelm@60770
   733
subsection\<open>For Every Cardinal Number There Exists A Greater One\<close>
paulson@13356
   734
wenzelm@60770
   735
text\<open>This result is Kunen's Theorem 10.16, which would be trivial using AC\<close>
paulson@13216
   736
paulson@13216
   737
lemma Ord_jump_cardinal: "Ord(jump_cardinal(K))"
paulson@13216
   738
apply (unfold jump_cardinal_def)
paulson@13216
   739
apply (rule Ord_is_Transset [THEN [2] OrdI])
paulson@13216
   740
 prefer 2 apply (blast intro!: Ord_ordertype)
paulson@13216
   741
apply (unfold Transset_def)
paulson@13216
   742
apply (safe del: subsetI)
paulson@13221
   743
apply (simp add: ordertype_pred_unfold, safe)
paulson@13216
   744
apply (rule UN_I)
paulson@13216
   745
apply (rule_tac [2] ReplaceI)
paulson@13216
   746
   prefer 4 apply (blast intro: well_ord_subset elim!: predE)+
paulson@13216
   747
done
paulson@13216
   748
paulson@13216
   749
(*Allows selective unfolding.  Less work than deriving intro/elim rules*)
paulson@13216
   750
lemma jump_cardinal_iff:
paulson@46821
   751
     "i \<in> jump_cardinal(K) \<longleftrightarrow>
paulson@46820
   752
      (\<exists>r X. r \<subseteq> K*K & X \<subseteq> K & well_ord(X,r) & i = ordertype(X,r))"
paulson@13216
   753
apply (unfold jump_cardinal_def)
paulson@46820
   754
apply (blast del: subsetI)
paulson@13216
   755
done
paulson@13216
   756
paulson@13216
   757
(*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*)
paulson@13216
   758
lemma K_lt_jump_cardinal: "Ord(K) ==> K < jump_cardinal(K)"
paulson@13216
   759
apply (rule Ord_jump_cardinal [THEN [2] ltI])
paulson@13216
   760
apply (rule jump_cardinal_iff [THEN iffD2])
paulson@13216
   761
apply (rule_tac x="Memrel(K)" in exI)
paulson@46820
   762
apply (rule_tac x=K in exI)
paulson@13216
   763
apply (simp add: ordertype_Memrel well_ord_Memrel)
paulson@13216
   764
apply (simp add: Memrel_def subset_iff)
paulson@13216
   765
done
paulson@13216
   766
paulson@13216
   767
(*The proof by contradiction: the bijection f yields a wellordering of X
paulson@13216
   768
  whose ordertype is jump_cardinal(K).  *)
paulson@13216
   769
lemma Card_jump_cardinal_lemma:
paulson@46820
   770
     "[| well_ord(X,r);  r \<subseteq> K * K;  X \<subseteq> K;
paulson@46820
   771
         f \<in> bij(ordertype(X,r), jump_cardinal(K)) |]
paulson@46820
   772
      ==> jump_cardinal(K) \<in> jump_cardinal(K)"
paulson@46820
   773
apply (subgoal_tac "f O ordermap (X,r) \<in> bij (X, jump_cardinal (K))")
paulson@13216
   774
 prefer 2 apply (blast intro: comp_bij ordermap_bij)
paulson@13216
   775
apply (rule jump_cardinal_iff [THEN iffD2])
paulson@13216
   776
apply (intro exI conjI)
paulson@13221
   777
apply (rule subset_trans [OF rvimage_type Sigma_mono], assumption+)
paulson@13216
   778
apply (erule bij_is_inj [THEN well_ord_rvimage])
paulson@13216
   779
apply (rule Ord_jump_cardinal [THEN well_ord_Memrel])
paulson@13216
   780
apply (simp add: well_ord_Memrel [THEN [2] bij_ordertype_vimage]
paulson@13216
   781
                 ordertype_Memrel Ord_jump_cardinal)
paulson@13216
   782
done
paulson@13216
   783
paulson@13216
   784
(*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*)
paulson@13216
   785
lemma Card_jump_cardinal: "Card(jump_cardinal(K))"
paulson@13216
   786
apply (rule Ord_jump_cardinal [THEN CardI])
paulson@13216
   787
apply (unfold eqpoll_def)
paulson@13216
   788
apply (safe dest!: ltD jump_cardinal_iff [THEN iffD1])
paulson@13216
   789
apply (blast intro: Card_jump_cardinal_lemma [THEN mem_irrefl])
paulson@13216
   790
done
paulson@13216
   791
wenzelm@60770
   792
subsection\<open>Basic Properties of Successor Cardinals\<close>
paulson@13216
   793
paulson@13216
   794
lemma csucc_basic: "Ord(K) ==> Card(csucc(K)) & K < csucc(K)"
paulson@13216
   795
apply (unfold csucc_def)
paulson@13216
   796
apply (rule LeastI)
paulson@13216
   797
apply (blast intro: Card_jump_cardinal K_lt_jump_cardinal Ord_jump_cardinal)+
paulson@13216
   798
done
paulson@13216
   799
wenzelm@45602
   800
lemmas Card_csucc = csucc_basic [THEN conjunct1]
paulson@13216
   801
wenzelm@45602
   802
lemmas lt_csucc = csucc_basic [THEN conjunct2]
paulson@13216
   803
paulson@13216
   804
lemma Ord_0_lt_csucc: "Ord(K) ==> 0 < csucc(K)"
paulson@13221
   805
by (blast intro: Ord_0_le lt_csucc lt_trans1)
paulson@13216
   806
paulson@46820
   807
lemma csucc_le: "[| Card(L);  K<L |] ==> csucc(K) \<le> L"
paulson@13216
   808
apply (unfold csucc_def)
paulson@13216
   809
apply (rule Least_le)
paulson@13216
   810
apply (blast intro: Card_is_Ord)+
paulson@13216
   811
done
paulson@13216
   812
paulson@46821
   813
lemma lt_csucc_iff: "[| Ord(i); Card(K) |] ==> i < csucc(K) \<longleftrightarrow> |i| \<le> K"
paulson@13216
   814
apply (rule iffI)
paulson@13216
   815
apply (rule_tac [2] Card_lt_imp_lt)
paulson@13216
   816
apply (erule_tac [2] lt_trans1)
paulson@13216
   817
apply (simp_all add: lt_csucc Card_csucc Card_is_Ord)
paulson@13216
   818
apply (rule notI [THEN not_lt_imp_le])
paulson@13221
   819
apply (rule Card_cardinal [THEN csucc_le, THEN lt_trans1, THEN lt_irrefl], assumption)
paulson@13216
   820
apply (rule Ord_cardinal_le [THEN lt_trans1])
paulson@46820
   821
apply (simp_all add: Ord_cardinal Card_is_Ord)
paulson@13216
   822
done
paulson@13216
   823
paulson@13216
   824
lemma Card_lt_csucc_iff:
paulson@46821
   825
     "[| Card(K'); Card(K) |] ==> K' < csucc(K) \<longleftrightarrow> K' \<le> K"
paulson@13221
   826
by (simp add: lt_csucc_iff Card_cardinal_eq Card_is_Ord)
paulson@13216
   827
paulson@13216
   828
lemma InfCard_csucc: "InfCard(K) ==> InfCard(csucc(K))"
paulson@46820
   829
by (simp add: InfCard_def Card_csucc Card_is_Ord
paulson@13216
   830
              lt_csucc [THEN leI, THEN [2] le_trans])
paulson@13216
   831
paulson@13216
   832
wenzelm@60770
   833
subsubsection\<open>Removing elements from a finite set decreases its cardinality\<close>
paulson@13216
   834
paulson@14883
   835
lemma Finite_imp_cardinal_cons [simp]:
paulson@46952
   836
  assumes FA: "Finite(A)" and a: "a\<notin>A" shows "|cons(a,A)| = succ(|A|)"
paulson@46952
   837
proof -
paulson@46952
   838
  { fix X
paulson@46952
   839
    have "Finite(X) ==> a \<notin> X \<Longrightarrow> cons(a,X) \<lesssim> X \<Longrightarrow> False"
paulson@46952
   840
      proof (induct X rule: Finite_induct)
paulson@46953
   841
        case 0 thus False  by (simp add: lepoll_0_iff)
paulson@46952
   842
      next
paulson@46953
   843
        case (cons x Y)
paulson@46953
   844
        hence "cons(x, cons(a, Y)) \<lesssim> cons(x, Y)" by (simp add: cons_commute)
paulson@46952
   845
        hence "cons(a, Y) \<lesssim> Y" using cons        by (blast dest: cons_lepoll_consD)
paulson@46952
   846
        thus False using cons by auto
paulson@46952
   847
      qed
paulson@46953
   848
  }
paulson@46952
   849
  hence [simp]: "~ cons(a,A) \<lesssim> A" using a FA by auto
paulson@46952
   850
  have [simp]: "|A| \<approx> A" using Finite_imp_well_ord [OF FA]
paulson@46952
   851
    by (blast intro: well_ord_cardinal_eqpoll)
paulson@46953
   852
  have "(\<mu> i. i \<approx> cons(a, A)) = succ(|A|)"
paulson@46952
   853
    proof (rule Least_equality [OF _ _ notI])
paulson@46953
   854
      show "succ(|A|) \<approx> cons(a, A)"
paulson@46953
   855
        by (simp add: succ_def cons_eqpoll_cong mem_not_refl a)
paulson@46952
   856
    next
paulson@46952
   857
      show "Ord(succ(|A|))" by simp
paulson@46952
   858
    next
paulson@46952
   859
      fix i
paulson@46952
   860
      assume i: "i \<le> |A|" "i \<approx> cons(a, A)"
paulson@46952
   861
      have "cons(a, A) \<approx> i" by (rule eqpoll_sym) (rule i)
paulson@46952
   862
      also have "... \<lesssim> |A|" by (rule le_imp_lepoll) (rule i)
paulson@46952
   863
      also have "... \<approx> A"   by simp
paulson@46952
   864
      finally have "cons(a, A) \<lesssim> A" .
paulson@46952
   865
      thus False by simp
paulson@46952
   866
    qed
paulson@46953
   867
  thus ?thesis by (simp add: cardinal_def)
paulson@46952
   868
qed
paulson@13216
   869
paulson@13221
   870
lemma Finite_imp_succ_cardinal_Diff:
paulson@46953
   871
     "[| Finite(A);  a \<in> A |] ==> succ(|A-{a}|) = |A|"
paulson@13784
   872
apply (rule_tac b = A in cons_Diff [THEN subst], assumption)
paulson@13221
   873
apply (simp add: Finite_imp_cardinal_cons Diff_subset [THEN subset_Finite])
paulson@13221
   874
apply (simp add: cons_Diff)
paulson@13216
   875
done
paulson@13216
   876
paulson@46953
   877
lemma Finite_imp_cardinal_Diff: "[| Finite(A);  a \<in> A |] ==> |A-{a}| < |A|"
paulson@13216
   878
apply (rule succ_leE)
paulson@13221
   879
apply (simp add: Finite_imp_succ_cardinal_Diff)
paulson@13216
   880
done
paulson@13216
   881
paulson@46820
   882
lemma Finite_cardinal_in_nat [simp]: "Finite(A) ==> |A| \<in> nat"
paulson@46952
   883
proof (induct rule: Finite_induct)
paulson@46952
   884
  case 0 thus ?case by (simp add: cardinal_0)
paulson@46952
   885
next
paulson@46952
   886
  case (cons x A) thus ?case by (simp add: Finite_imp_cardinal_cons)
paulson@46952
   887
qed
paulson@13216
   888
paulson@14883
   889
lemma card_Un_Int:
paulson@46820
   890
     "[|Finite(A); Finite(B)|] ==> |A| #+ |B| = |A \<union> B| #+ |A \<inter> B|"
paulson@46820
   891
apply (erule Finite_induct, simp)
paulson@14883
   892
apply (simp add: Finite_Int cons_absorb Un_cons Int_cons_left)
paulson@14883
   893
done
paulson@14883
   894
paulson@46820
   895
lemma card_Un_disjoint:
paulson@46820
   896
     "[|Finite(A); Finite(B); A \<inter> B = 0|] ==> |A \<union> B| = |A| #+ |B|"
paulson@14883
   897
by (simp add: Finite_Un card_Un_Int)
paulson@14883
   898
paulson@46952
   899
lemma card_partition:
paulson@46952
   900
  assumes FC: "Finite(C)"
paulson@46952
   901
  shows
paulson@46952
   902
     "Finite (\<Union> C) \<Longrightarrow>
paulson@46952
   903
        (\<forall>c\<in>C. |c| = k) \<Longrightarrow>
paulson@46952
   904
        (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = 0) \<Longrightarrow>
paulson@14883
   905
        k #* |C| = |\<Union> C|"
paulson@46952
   906
using FC
paulson@46952
   907
proof (induct rule: Finite_induct)
paulson@46952
   908
  case 0 thus ?case by simp
paulson@46952
   909
next
paulson@46952
   910
  case (cons x B)
paulson@46952
   911
  hence "x \<inter> \<Union>B = 0" by auto
paulson@46952
   912
  thus ?case using cons
paulson@46952
   913
    by (auto simp add: card_Un_disjoint)
paulson@46952
   914
qed
paulson@14883
   915
paulson@14883
   916
wenzelm@60770
   917
subsubsection\<open>Theorems by Krzysztof Grabczewski, proofs by lcp\<close>
paulson@13216
   918
wenzelm@45602
   919
lemmas nat_implies_well_ord = nat_into_Ord [THEN well_ord_Memrel]
paulson@13216
   920
paulson@46953
   921
lemma nat_sum_eqpoll_sum:
paulson@46907
   922
  assumes m: "m \<in> nat" and n: "n \<in> nat" shows "m + n \<approx> m #+ n"
paulson@46907
   923
proof -
paulson@46907
   924
  have "m + n \<approx> |m+n|" using m n
paulson@46953
   925
    by (blast intro: nat_implies_well_ord well_ord_radd well_ord_cardinal_eqpoll eqpoll_sym)
paulson@46907
   926
  also have "... = m #+ n" using m n
paulson@46907
   927
    by (simp add: nat_cadd_eq_add [symmetric] cadd_def)
paulson@46907
   928
  finally show ?thesis .
paulson@46907
   929
qed
paulson@13216
   930
paulson@46935
   931
lemma Ord_subset_natD [rule_format]: "Ord(i) ==> i \<subseteq> nat \<Longrightarrow> i \<in> nat | i=nat"
paulson@46935
   932
proof (induct i rule: trans_induct3)
paulson@46935
   933
  case 0 thus ?case by auto
paulson@46935
   934
next
paulson@46935
   935
  case (succ i) thus ?case by auto
paulson@46935
   936
next
paulson@46935
   937
  case (limit l) thus ?case
paulson@46935
   938
    by (blast dest: nat_le_Limit le_imp_subset)
paulson@46935
   939
qed
paulson@13216
   940
paulson@46820
   941
lemma Ord_nat_subset_into_Card: "[| Ord(i); i \<subseteq> nat |] ==> Card(i)"
paulson@13221
   942
by (blast dest: Ord_subset_natD intro: Card_nat nat_into_Card)
paulson@13216
   943
lcp@437
   944
end