src/ZF/CardinalArith.thy
author paulson
Tue Mar 06 16:06:52 2012 +0000 (2012-03-06)
changeset 46821 ff6b0c1087f2
parent 46820 c656222c4dc1
child 46841 49b91b716cbe
permissions -rw-r--r--
Using mathematical notation for <-> and cardinal arithmetic
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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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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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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]: "(\<forall>x\<in>A. Card(x)) ==> Card(\<Union>(A))"
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apply (rule CardI)
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 apply (simp add: Card_is_Ord)
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apply (clarify dest!: ltD)
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apply (drule bspec, assumption)
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apply (frule lt_Card_imp_lesspoll, blast intro: ltI Card_is_Ord)
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apply (drule eqpoll_sym [THEN eqpoll_imp_lepoll])
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apply (drule lesspoll_trans1, assumption)
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apply (subgoal_tac "B \<lesssim> \<Union>A")
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 apply (drule lesspoll_trans1, assumption, blast)
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apply (blast intro: subset_imp_lepoll)
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done
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lemma Card_UN: "(!!x. x:A ==> Card(K(x))) ==> Card(\<Union>x\<in>A. K(x))"
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by (blast intro: Card_Union)
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lemma Card_OUN [simp,intro,TC]:
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     "(!!x. x:A ==> Card(K(x))) ==> Card(\<Union>x<A. K(x))"
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by (simp add: OUnion_def Card_0)
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lemma n_lesspoll_nat: "n \<in> nat ==> n \<prec> nat"
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apply (unfold lesspoll_def)
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apply (rule conjI)
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apply (erule OrdmemD [THEN subset_imp_lepoll], rule Ord_nat)
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apply (rule notI)
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apply (erule eqpollE)
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apply (rule succ_lepoll_natE)
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apply (blast intro: nat_succI [THEN OrdmemD, THEN subset_imp_lepoll]
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                    lepoll_trans, assumption)
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done
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lemma in_Card_imp_lesspoll: "[| Card(K); b \<in> K |] ==> b \<prec> K"
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apply (unfold lesspoll_def)
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apply (simp add: Card_iff_initial)
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apply (fast intro!: le_imp_lepoll ltI leI)
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done
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lemma lesspoll_lemma: "[| ~ A \<prec> B; C \<prec> B |] ==> A - C \<noteq> 0"
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apply (unfold lesspoll_def)
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apply (fast dest!: Diff_eq_0_iff [THEN iffD1, THEN subset_imp_lepoll]
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            intro!: eqpollI elim: notE
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            elim!: eqpollE lepoll_trans)
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done
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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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lemma sum_commute_eqpoll: "A+B \<approx> B+A"
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apply (unfold eqpoll_def)
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apply (rule exI)
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apply (rule_tac c = "case(Inr,Inl)" and d = "case(Inr,Inl)" in lam_bijective)
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apply auto
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done
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lemma cadd_commute: "i \<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{*Cardinal addition is associative*}
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lemma sum_assoc_eqpoll: "(A+B)+C \<approx> A+(B+C)"
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apply (unfold eqpoll_def)
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apply (rule exI)
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apply (rule sum_assoc_bij)
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done
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(*Unconditional version requires AC*)
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lemma well_ord_cadd_assoc:
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    "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |]
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     ==> (i \<oplus> j) \<oplus> k = i \<oplus> (j \<oplus> k)"
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apply (unfold cadd_def)
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apply (rule cardinal_cong)
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apply (rule eqpoll_trans)
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 apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
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 apply (blast intro: well_ord_radd )
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apply (rule sum_assoc_eqpoll [THEN eqpoll_trans])
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apply (rule eqpoll_sym)
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apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
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apply (blast intro: well_ord_radd )
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done
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subsubsection{*0 is the identity for addition*}
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lemma sum_0_eqpoll: "0+A \<approx> A"
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apply (unfold eqpoll_def)
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apply (rule exI)
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apply (rule bij_0_sum)
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done
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lemma cadd_0 [simp]: "Card(K) ==> 0 \<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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apply (unfold lepoll_def inj_def)
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apply (rule_tac x = "\<lambda>x\<in>A. Inl (x) " in exI)
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apply simp
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done
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(*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
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lemma cadd_le_self:
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    "[| Card(K);  Ord(L) |] ==> K \<le> (K \<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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subsubsection{*Monotonicity of addition*}
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lemma sum_lepoll_mono:
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     "[| A \<lesssim> C;  B \<lesssim> D |] ==> A + B \<lesssim> C + D"
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apply (unfold lepoll_def)
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apply (elim exE)
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apply (rule_tac x = "\<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{*Addition of finite cardinals is "ordinary" addition*}
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lemma sum_succ_eqpoll: "succ(A)+B \<approx> succ(A+B)"
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apply (unfold eqpoll_def)
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apply (rule exI)
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apply (rule_tac c = "%z. if z=Inl (A) then A+B else z"
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            and d = "%z. if z=A+B then Inl (A) else z" in lam_bijective)
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   apply simp_all
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apply (blast dest: sym [THEN eq_imp_not_mem] elim: mem_irrefl)+
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done
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(*Pulling the  succ(...)  outside the |...| requires m, n: nat  *)
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(*Unconditional version requires AC*)
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lemma cadd_succ_lemma:
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    "[| Ord(m);  Ord(n) |] ==> succ(m) \<oplus> n = |succ(m \<oplus> n)|"
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apply (unfold cadd_def)
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apply (rule sum_succ_eqpoll [THEN cardinal_cong, THEN trans])
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apply (rule succ_eqpoll_cong [THEN cardinal_cong])
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apply (rule well_ord_cardinal_eqpoll [THEN eqpoll_sym])
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apply (blast intro: well_ord_radd well_ord_Memrel)
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done
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lemma nat_cadd_eq_add: "[| m: nat;  n: nat |] ==> m \<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])
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apply (simp add: cadd_succ_lemma nat_into_Card [THEN Card_cardinal_eq])
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done
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subsection{*Cardinal multiplication*}
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subsubsection{*Cardinal multiplication is commutative*}
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(*Easier to prove the two directions separately*)
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lemma prod_commute_eqpoll: "A*B \<approx> B*A"
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apply (unfold eqpoll_def)
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apply (rule exI)
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apply (rule_tac c = "%<x,y>.<y,x>" and d = "%<x,y>.<y,x>" in lam_bijective,
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       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{*Cardinal multiplication is associative*}
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lemma prod_assoc_eqpoll: "(A*B)*C \<approx> A*(B*C)"
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apply (unfold eqpoll_def)
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apply (rule exI)
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apply (rule prod_assoc_bij)
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done
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(*Unconditional version requires AC*)
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lemma well_ord_cmult_assoc:
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    "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |]
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     ==> (i \<otimes> j) \<otimes> k = i \<otimes> (j \<otimes> k)"
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apply (unfold cmult_def)
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apply (rule cardinal_cong)
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apply (rule eqpoll_trans)
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 apply (rule prod_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
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 apply (blast intro: well_ord_rmult)
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apply (rule prod_assoc_eqpoll [THEN eqpoll_trans])
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apply (rule eqpoll_sym)
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apply (rule prod_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
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apply (blast intro: well_ord_rmult)
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done
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subsubsection{*Cardinal multiplication distributes over addition*}
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lemma sum_prod_distrib_eqpoll: "(A+B)*C \<approx> (A*C)+(B*C)"
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apply (unfold eqpoll_def)
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apply (rule exI)
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apply (rule sum_prod_distrib_bij)
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done
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lemma well_ord_cadd_cmult_distrib:
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    "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |]
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     ==> (i \<oplus> j) \<otimes> k = (i \<otimes> k) \<oplus> (j \<otimes> k)"
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apply (unfold cadd_def cmult_def)
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apply (rule cardinal_cong)
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apply (rule eqpoll_trans)
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 apply (rule prod_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
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apply (blast intro: well_ord_radd)
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apply (rule sum_prod_distrib_eqpoll [THEN eqpoll_trans])
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apply (rule eqpoll_sym)
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apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll
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                                well_ord_cardinal_eqpoll])
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apply (blast intro: well_ord_rmult)+
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done
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subsubsection{*Multiplication by 0 yields 0*}
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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{*1 is the identity for multiplication*}
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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{*Some inequalities for multiplication*}
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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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lemma cmult_square_le: "Card(K) ==> K \<le> K \<otimes> K"
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apply (unfold cmult_def)
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apply (rule le_trans)
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apply (rule_tac [2] well_ord_lepoll_imp_Card_le)
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apply (rule_tac [3] prod_square_lepoll)
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apply (simp add: le_refl Card_is_Ord Card_cardinal_eq)
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apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord)
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done
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paulson@14883
   326
subsubsection{*Multiplication by a non-zero cardinal*}
paulson@13216
   327
paulson@13216
   328
lemma prod_lepoll_self: "b: B ==> A \<lesssim> A*B"
paulson@13216
   329
apply (unfold lepoll_def inj_def)
paulson@46820
   330
apply (rule_tac x = "\<lambda>x\<in>A. <x,b>" in exI, simp)
paulson@13216
   331
done
paulson@13216
   332
paulson@13216
   333
(*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
paulson@13216
   334
lemma cmult_le_self:
paulson@46821
   335
    "[| Card(K);  Ord(L);  0<L |] ==> K \<le> (K \<otimes> L)"
paulson@13216
   336
apply (unfold cmult_def)
paulson@13216
   337
apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le])
paulson@13221
   338
  apply assumption
paulson@13216
   339
 apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord)
paulson@13216
   340
apply (blast intro: prod_lepoll_self ltD)
paulson@13216
   341
done
paulson@13216
   342
paulson@14883
   343
subsubsection{*Monotonicity of multiplication*}
paulson@13216
   344
paulson@13216
   345
lemma prod_lepoll_mono:
paulson@13216
   346
     "[| A \<lesssim> C;  B \<lesssim> D |] ==> A * B  \<lesssim>  C * D"
paulson@13216
   347
apply (unfold lepoll_def)
paulson@13221
   348
apply (elim exE)
paulson@13216
   349
apply (rule_tac x = "lam <w,y>:A*B. <f`w, fa`y>" in exI)
paulson@46820
   350
apply (rule_tac d = "%<w,y>. <converse (f) `w, converse (fa) `y>"
paulson@13216
   351
       in lam_injective)
paulson@13221
   352
apply (typecheck add: inj_is_fun, auto)
paulson@13216
   353
done
paulson@13216
   354
paulson@13216
   355
lemma cmult_le_mono:
paulson@46821
   356
    "[| K' \<le> K;  L' \<le> L |] ==> (K' \<otimes> L') \<le> (K \<otimes> L)"
paulson@13216
   357
apply (unfold cmult_def)
paulson@13216
   358
apply (safe dest!: le_subset_iff [THEN iffD1])
paulson@13216
   359
apply (rule well_ord_lepoll_imp_Card_le)
paulson@13216
   360
 apply (blast intro: well_ord_rmult well_ord_Memrel)
paulson@13216
   361
apply (blast intro: prod_lepoll_mono subset_imp_lepoll)
paulson@13216
   362
done
paulson@13216
   363
paulson@13356
   364
subsection{*Multiplication of finite cardinals is "ordinary" multiplication*}
paulson@13216
   365
paulson@13216
   366
lemma prod_succ_eqpoll: "succ(A)*B \<approx> B + A*B"
paulson@13216
   367
apply (unfold eqpoll_def)
paulson@13221
   368
apply (rule exI)
paulson@13216
   369
apply (rule_tac c = "%<x,y>. if x=A then Inl (y) else Inr (<x,y>)"
paulson@13216
   370
            and d = "case (%y. <A,y>, %z. z)" in lam_bijective)
paulson@13216
   371
apply safe
paulson@13216
   372
apply (simp_all add: succI2 if_type mem_imp_not_eq)
paulson@13216
   373
done
paulson@13216
   374
paulson@13216
   375
(*Unconditional version requires AC*)
paulson@13216
   376
lemma cmult_succ_lemma:
paulson@46821
   377
    "[| Ord(m);  Ord(n) |] ==> succ(m) \<otimes> n = n \<oplus> (m \<otimes> n)"
paulson@13216
   378
apply (unfold cmult_def cadd_def)
paulson@13216
   379
apply (rule prod_succ_eqpoll [THEN cardinal_cong, THEN trans])
paulson@13216
   380
apply (rule cardinal_cong [symmetric])
paulson@13216
   381
apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
paulson@13216
   382
apply (blast intro: well_ord_rmult well_ord_Memrel)
paulson@13216
   383
done
paulson@13216
   384
paulson@46821
   385
lemma nat_cmult_eq_mult: "[| m: nat;  n: nat |] ==> m \<otimes> n = m#*n"
paulson@13244
   386
apply (induct_tac m)
paulson@13221
   387
apply (simp_all add: cmult_succ_lemma nat_cadd_eq_add)
paulson@13216
   388
done
paulson@13216
   389
paulson@46821
   390
lemma cmult_2: "Card(n) ==> 2 \<otimes> n = n \<oplus> n"
paulson@13221
   391
by (simp add: cmult_succ_lemma Card_is_Ord cadd_commute [of _ 0])
paulson@13216
   392
paulson@13216
   393
lemma sum_lepoll_prod: "2 \<lesssim> C ==> B+B \<lesssim> C*B"
paulson@46820
   394
apply (rule lepoll_trans)
paulson@46820
   395
apply (rule sum_eq_2_times [THEN equalityD1, THEN subset_imp_lepoll])
paulson@46820
   396
apply (erule prod_lepoll_mono)
paulson@46820
   397
apply (rule lepoll_refl)
paulson@13216
   398
done
paulson@13216
   399
paulson@13216
   400
lemma lepoll_imp_sum_lepoll_prod: "[| A \<lesssim> B; 2 \<lesssim> A |] ==> A+B \<lesssim> A*B"
paulson@13221
   401
by (blast intro: sum_lepoll_mono sum_lepoll_prod lepoll_trans lepoll_refl)
paulson@13216
   402
paulson@13216
   403
paulson@13356
   404
subsection{*Infinite Cardinals are Limit Ordinals*}
paulson@13216
   405
paulson@13216
   406
(*This proof is modelled upon one assuming nat<=A, with injection
paulson@46820
   407
  \<lambda>z\<in>cons(u,A). if z=u then 0 else if z \<in> nat then succ(z) else z
paulson@13216
   408
  and inverse %y. if y:nat then nat_case(u, %z. z, y) else y.  \
paulson@13216
   409
  If f: inj(nat,A) then range(f) behaves like the natural numbers.*)
paulson@13216
   410
lemma nat_cons_lepoll: "nat \<lesssim> A ==> cons(u,A) \<lesssim> A"
paulson@13216
   411
apply (unfold lepoll_def)
paulson@13216
   412
apply (erule exE)
paulson@46820
   413
apply (rule_tac x =
paulson@46820
   414
          "\<lambda>z\<in>cons (u,A).
paulson@46820
   415
             if z=u then f`0
paulson@46820
   416
             else if z: range (f) then f`succ (converse (f) `z) else z"
paulson@13216
   417
       in exI)
paulson@13216
   418
apply (rule_tac d =
paulson@46820
   419
          "%y. if y: range(f) then nat_case (u, %z. f`z, converse(f) `y)
paulson@46820
   420
                              else y"
paulson@13216
   421
       in lam_injective)
paulson@13216
   422
apply (fast intro!: if_type apply_type intro: inj_is_fun inj_converse_fun)
paulson@13216
   423
apply (simp add: inj_is_fun [THEN apply_rangeI]
paulson@13216
   424
                 inj_converse_fun [THEN apply_rangeI]
paulson@13216
   425
                 inj_converse_fun [THEN apply_funtype])
paulson@13216
   426
done
paulson@13216
   427
paulson@13216
   428
lemma nat_cons_eqpoll: "nat \<lesssim> A ==> cons(u,A) \<approx> A"
paulson@13216
   429
apply (erule nat_cons_lepoll [THEN eqpollI])
paulson@13216
   430
apply (rule subset_consI [THEN subset_imp_lepoll])
paulson@13216
   431
done
paulson@13216
   432
paulson@13216
   433
(*Specialized version required below*)
paulson@46820
   434
lemma nat_succ_eqpoll: "nat \<subseteq> A ==> succ(A) \<approx> A"
paulson@13216
   435
apply (unfold succ_def)
paulson@13216
   436
apply (erule subset_imp_lepoll [THEN nat_cons_eqpoll])
paulson@13216
   437
done
paulson@13216
   438
paulson@13216
   439
lemma InfCard_nat: "InfCard(nat)"
paulson@13216
   440
apply (unfold InfCard_def)
paulson@13216
   441
apply (blast intro: Card_nat le_refl Card_is_Ord)
paulson@13216
   442
done
paulson@13216
   443
paulson@13216
   444
lemma InfCard_is_Card: "InfCard(K) ==> Card(K)"
paulson@13216
   445
apply (unfold InfCard_def)
paulson@13216
   446
apply (erule conjunct1)
paulson@13216
   447
done
paulson@13216
   448
paulson@13216
   449
lemma InfCard_Un:
paulson@46820
   450
    "[| InfCard(K);  Card(L) |] ==> InfCard(K \<union> L)"
paulson@13216
   451
apply (unfold InfCard_def)
paulson@13216
   452
apply (simp add: Card_Un Un_upper1_le [THEN [2] le_trans]  Card_is_Ord)
paulson@13216
   453
done
paulson@13216
   454
paulson@13216
   455
(*Kunen's Lemma 10.11*)
paulson@13216
   456
lemma InfCard_is_Limit: "InfCard(K) ==> Limit(K)"
paulson@13216
   457
apply (unfold InfCard_def)
paulson@13216
   458
apply (erule conjE)
paulson@13216
   459
apply (frule Card_is_Ord)
paulson@13216
   460
apply (rule ltI [THEN non_succ_LimitI])
paulson@13216
   461
apply (erule le_imp_subset [THEN subsetD])
paulson@13216
   462
apply (safe dest!: Limit_nat [THEN Limit_le_succD])
paulson@13216
   463
apply (unfold Card_def)
paulson@13216
   464
apply (drule trans)
paulson@13216
   465
apply (erule le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong])
paulson@13216
   466
apply (erule Ord_cardinal_le [THEN lt_trans2, THEN lt_irrefl])
paulson@13221
   467
apply (rule le_eqI, assumption)
paulson@13216
   468
apply (rule Ord_cardinal)
paulson@13216
   469
done
paulson@13216
   470
paulson@13216
   471
paulson@13216
   472
(*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***)
paulson@13216
   473
paulson@13216
   474
(*A general fact about ordermap*)
paulson@13216
   475
lemma ordermap_eqpoll_pred:
paulson@13269
   476
    "[| well_ord(A,r);  x:A |] ==> ordermap(A,r)`x \<approx> Order.pred(A,x,r)"
paulson@13216
   477
apply (unfold eqpoll_def)
paulson@13216
   478
apply (rule exI)
paulson@13221
   479
apply (simp add: ordermap_eq_image well_ord_is_wf)
paulson@46820
   480
apply (erule ordermap_bij [THEN bij_is_inj, THEN restrict_bij,
paulson@13221
   481
                           THEN bij_converse_bij])
paulson@13216
   482
apply (rule pred_subset)
paulson@13216
   483
done
paulson@13216
   484
paulson@14883
   485
subsubsection{*Establishing the well-ordering*}
paulson@13216
   486
paulson@13216
   487
lemma csquare_lam_inj:
paulson@46820
   488
     "Ord(K) ==> (lam <x,y>:K*K. <x \<union> y, x, y>) \<in> inj(K*K, K*K*K)"
paulson@13216
   489
apply (unfold inj_def)
paulson@13216
   490
apply (force intro: lam_type Un_least_lt [THEN ltD] ltI)
paulson@13216
   491
done
paulson@13216
   492
paulson@13216
   493
lemma well_ord_csquare: "Ord(K) ==> well_ord(K*K, csquare_rel(K))"
paulson@13216
   494
apply (unfold csquare_rel_def)
paulson@13221
   495
apply (rule csquare_lam_inj [THEN well_ord_rvimage], assumption)
paulson@13216
   496
apply (blast intro: well_ord_rmult well_ord_Memrel)
paulson@13216
   497
done
paulson@13216
   498
paulson@14883
   499
subsubsection{*Characterising initial segments of the well-ordering*}
paulson@13216
   500
paulson@13216
   501
lemma csquareD:
paulson@46820
   502
 "[| <<x,y>, <z,z>> \<in> csquare_rel(K);  x<K;  y<K;  z<K |] ==> x \<le> z & y \<le> z"
paulson@13216
   503
apply (unfold csquare_rel_def)
paulson@13216
   504
apply (erule rev_mp)
paulson@13216
   505
apply (elim ltE)
paulson@13221
   506
apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
paulson@13216
   507
apply (safe elim!: mem_irrefl intro!: Un_upper1_le Un_upper2_le)
paulson@13221
   508
apply (simp_all add: lt_def succI2)
paulson@13216
   509
done
paulson@13216
   510
paulson@46820
   511
lemma pred_csquare_subset:
paulson@46820
   512
    "z<K ==> Order.pred(K*K, <z,z>, csquare_rel(K)) \<subseteq> succ(z)*succ(z)"
paulson@13216
   513
apply (unfold Order.pred_def)
paulson@13216
   514
apply (safe del: SigmaI succCI)
paulson@13216
   515
apply (erule csquareD [THEN conjE])
paulson@46820
   516
apply (unfold lt_def, auto)
paulson@13216
   517
done
paulson@13216
   518
paulson@13216
   519
lemma csquare_ltI:
paulson@46820
   520
 "[| x<z;  y<z;  z<K |] ==>  <<x,y>, <z,z>> \<in> csquare_rel(K)"
paulson@13216
   521
apply (unfold csquare_rel_def)
paulson@13216
   522
apply (subgoal_tac "x<K & y<K")
paulson@46820
   523
 prefer 2 apply (blast intro: lt_trans)
paulson@13216
   524
apply (elim ltE)
paulson@13221
   525
apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
paulson@13216
   526
done
paulson@13216
   527
paulson@13216
   528
(*Part of the traditional proof.  UNUSED since it's harder to prove & apply *)
paulson@13216
   529
lemma csquare_or_eqI:
paulson@46820
   530
 "[| x \<le> z;  y \<le> z;  z<K |] ==> <<x,y>, <z,z>> \<in> csquare_rel(K) | x=z & y=z"
paulson@13216
   531
apply (unfold csquare_rel_def)
paulson@13216
   532
apply (subgoal_tac "x<K & y<K")
paulson@46820
   533
 prefer 2 apply (blast intro: lt_trans1)
paulson@13216
   534
apply (elim ltE)
paulson@13221
   535
apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
paulson@13216
   536
apply (elim succE)
paulson@46820
   537
apply (simp_all add: subset_Un_iff [THEN iff_sym]
paulson@13221
   538
                     subset_Un_iff2 [THEN iff_sym] OrdmemD)
paulson@13216
   539
done
paulson@13216
   540
paulson@14883
   541
subsubsection{*The cardinality of initial segments*}
paulson@13216
   542
paulson@13216
   543
lemma ordermap_z_lt:
paulson@46820
   544
      "[| Limit(K);  x<K;  y<K;  z=succ(x \<union> y) |] ==>
paulson@13216
   545
          ordermap(K*K, csquare_rel(K)) ` <x,y> <
paulson@13216
   546
          ordermap(K*K, csquare_rel(K)) ` <z,z>"
paulson@13216
   547
apply (subgoal_tac "z<K & well_ord (K*K, csquare_rel (K))")
paulson@13216
   548
prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ
paulson@46820
   549
                              Limit_is_Ord [THEN well_ord_csquare], clarify)
paulson@13216
   550
apply (rule csquare_ltI [THEN ordermap_mono, THEN ltI])
paulson@13216
   551
apply (erule_tac [4] well_ord_is_wf)
paulson@13216
   552
apply (blast intro!: Un_upper1_le Un_upper2_le Ord_ordermap elim!: ltE)+
paulson@13216
   553
done
paulson@13216
   554
paulson@13216
   555
(*Kunen: "each <x,y>: K*K has no more than z*z predecessors..." (page 29) *)
paulson@13216
   556
lemma ordermap_csquare_le:
paulson@46820
   557
  "[| Limit(K);  x<K;  y<K;  z=succ(x \<union> y) |]
paulson@46821
   558
   ==> | ordermap(K*K, csquare_rel(K)) ` <x,y> | \<le> |succ(z)| \<otimes> |succ(z)|"
paulson@13216
   559
apply (unfold cmult_def)
paulson@13216
   560
apply (rule well_ord_rmult [THEN well_ord_lepoll_imp_Card_le])
paulson@13216
   561
apply (rule Ord_cardinal [THEN well_ord_Memrel])+
paulson@13216
   562
apply (subgoal_tac "z<K")
paulson@13216
   563
 prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ)
paulson@46820
   564
apply (rule ordermap_z_lt [THEN leI, THEN le_imp_lepoll, THEN lepoll_trans],
paulson@13221
   565
       assumption+)
paulson@13216
   566
apply (rule ordermap_eqpoll_pred [THEN eqpoll_imp_lepoll, THEN lepoll_trans])
paulson@13216
   567
apply (erule Limit_is_Ord [THEN well_ord_csquare])
paulson@13216
   568
apply (blast intro: ltD)
paulson@13216
   569
apply (rule pred_csquare_subset [THEN subset_imp_lepoll, THEN lepoll_trans],
paulson@13216
   570
            assumption)
paulson@13216
   571
apply (elim ltE)
paulson@13216
   572
apply (rule prod_eqpoll_cong [THEN eqpoll_sym, THEN eqpoll_imp_lepoll])
paulson@13216
   573
apply (erule Ord_succ [THEN Ord_cardinal_eqpoll])+
paulson@13216
   574
done
paulson@13216
   575
paulson@46820
   576
(*Kunen: "... so the order type is @{text"\<le>"} K" *)
paulson@13216
   577
lemma ordertype_csquare_le:
paulson@46821
   578
     "[| InfCard(K);  \<forall>y\<in>K. InfCard(y) \<longrightarrow> y \<otimes> y = y |]
paulson@46820
   579
      ==> ordertype(K*K, csquare_rel(K)) \<le> K"
paulson@13216
   580
apply (frule InfCard_is_Card [THEN Card_is_Ord])
paulson@13221
   581
apply (rule all_lt_imp_le, assumption)
paulson@13216
   582
apply (erule well_ord_csquare [THEN Ord_ordertype])
paulson@13216
   583
apply (rule Card_lt_imp_lt)
paulson@13216
   584
apply (erule_tac [3] InfCard_is_Card)
paulson@13216
   585
apply (erule_tac [2] ltE)
paulson@13216
   586
apply (simp add: ordertype_unfold)
paulson@13216
   587
apply (safe elim!: ltE)
paulson@13216
   588
apply (subgoal_tac "Ord (xa) & Ord (ya)")
paulson@13221
   589
 prefer 2 apply (blast intro: Ord_in_Ord, clarify)
paulson@46820
   590
(*??WHAT A MESS!*)
paulson@13216
   591
apply (rule InfCard_is_Limit [THEN ordermap_csquare_le, THEN lt_trans1],
paulson@46820
   592
       (assumption | rule refl | erule ltI)+)
paulson@46820
   593
apply (rule_tac i = "xa \<union> ya" and j = nat in Ord_linear2,
paulson@13216
   594
       simp_all add: Ord_Un Ord_nat)
paulson@46820
   595
prefer 2 (*case @{term"nat \<le> (xa \<union> ya)"} *)
paulson@46820
   596
 apply (simp add: le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong]
paulson@13216
   597
                  le_succ_iff InfCard_def Card_cardinal Un_least_lt Ord_Un
paulson@13216
   598
                ltI nat_le_cardinal Ord_cardinal_le [THEN lt_trans1, THEN ltD])
paulson@46820
   599
(*the finite case: @{term"xa \<union> ya < nat"} *)
paulson@13784
   600
apply (rule_tac j = nat in lt_trans2)
paulson@13216
   601
 apply (simp add: lt_def nat_cmult_eq_mult nat_succI mult_type
paulson@13216
   602
                  nat_into_Card [THEN Card_cardinal_eq]  Ord_nat)
paulson@13216
   603
apply (simp add: InfCard_def)
paulson@13216
   604
done
paulson@13216
   605
paulson@13216
   606
(*Main result: Kunen's Theorem 10.12*)
paulson@46821
   607
lemma InfCard_csquare_eq: "InfCard(K) ==> K \<otimes> K = K"
paulson@13216
   608
apply (frule InfCard_is_Card [THEN Card_is_Ord])
paulson@13216
   609
apply (erule rev_mp)
paulson@46820
   610
apply (erule_tac i=K in trans_induct)
paulson@13216
   611
apply (rule impI)
paulson@13216
   612
apply (rule le_anti_sym)
paulson@13216
   613
apply (erule_tac [2] InfCard_is_Card [THEN cmult_square_le])
paulson@13216
   614
apply (rule ordertype_csquare_le [THEN [2] le_trans])
paulson@46820
   615
apply (simp add: cmult_def Ord_cardinal_le
paulson@13221
   616
                 well_ord_csquare [THEN Ord_ordertype]
paulson@46820
   617
                 well_ord_csquare [THEN ordermap_bij, THEN bij_imp_eqpoll,
paulson@13221
   618
                                   THEN cardinal_cong], assumption+)
paulson@13216
   619
done
paulson@13216
   620
paulson@13216
   621
(*Corollary for arbitrary well-ordered sets (all sets, assuming AC)*)
paulson@13216
   622
lemma well_ord_InfCard_square_eq:
paulson@13216
   623
     "[| well_ord(A,r);  InfCard(|A|) |] ==> A*A \<approx> A"
paulson@13216
   624
apply (rule prod_eqpoll_cong [THEN eqpoll_trans])
paulson@13216
   625
apply (erule well_ord_cardinal_eqpoll [THEN eqpoll_sym])+
paulson@13216
   626
apply (rule well_ord_cardinal_eqE)
paulson@13221
   627
apply (blast intro: Ord_cardinal well_ord_rmult well_ord_Memrel, assumption)
paulson@13221
   628
apply (simp add: cmult_def [symmetric] InfCard_csquare_eq)
paulson@13216
   629
done
paulson@13216
   630
paulson@13356
   631
lemma InfCard_square_eqpoll: "InfCard(K) ==> K \<times> K \<approx> K"
paulson@46820
   632
apply (rule well_ord_InfCard_square_eq)
paulson@46820
   633
 apply (erule InfCard_is_Card [THEN Card_is_Ord, THEN well_ord_Memrel])
paulson@46820
   634
apply (simp add: InfCard_is_Card [THEN Card_cardinal_eq])
paulson@13356
   635
done
paulson@13356
   636
paulson@13356
   637
lemma Inf_Card_is_InfCard: "[| ~Finite(i); Card(i) |] ==> InfCard(i)"
paulson@13356
   638
by (simp add: InfCard_def Card_is_Ord [THEN nat_le_infinite_Ord])
paulson@13356
   639
paulson@14883
   640
subsubsection{*Toward's Kunen's Corollary 10.13 (1)*}
paulson@13216
   641
paulson@46821
   642
lemma InfCard_le_cmult_eq: "[| InfCard(K);  L \<le> K;  0<L |] ==> K \<otimes> L = K"
paulson@13216
   643
apply (rule le_anti_sym)
paulson@13216
   644
 prefer 2
paulson@13216
   645
 apply (erule ltE, blast intro: cmult_le_self InfCard_is_Card)
paulson@13216
   646
apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl])
paulson@13216
   647
apply (rule cmult_le_mono [THEN le_trans], assumption+)
paulson@13216
   648
apply (simp add: InfCard_csquare_eq)
paulson@13216
   649
done
paulson@13216
   650
paulson@13216
   651
(*Corollary 10.13 (1), for cardinal multiplication*)
paulson@46821
   652
lemma InfCard_cmult_eq: "[| InfCard(K);  InfCard(L) |] ==> K \<otimes> L = K \<union> L"
paulson@13784
   653
apply (rule_tac i = K and j = L in Ord_linear_le)
paulson@13216
   654
apply (typecheck add: InfCard_is_Card Card_is_Ord)
paulson@13216
   655
apply (rule cmult_commute [THEN ssubst])
paulson@13216
   656
apply (rule Un_commute [THEN ssubst])
paulson@46820
   657
apply (simp_all add: InfCard_is_Limit [THEN Limit_has_0] InfCard_le_cmult_eq
paulson@13221
   658
                     subset_Un_iff2 [THEN iffD1] le_imp_subset)
paulson@13216
   659
done
paulson@13216
   660
paulson@46821
   661
lemma InfCard_cdouble_eq: "InfCard(K) ==> K \<oplus> K = K"
paulson@13221
   662
apply (simp add: cmult_2 [symmetric] InfCard_is_Card cmult_commute)
paulson@13221
   663
apply (simp add: InfCard_le_cmult_eq InfCard_is_Limit Limit_has_0 Limit_has_succ)
paulson@13216
   664
done
paulson@13216
   665
paulson@13216
   666
(*Corollary 10.13 (1), for cardinal addition*)
paulson@46821
   667
lemma InfCard_le_cadd_eq: "[| InfCard(K);  L \<le> K |] ==> K \<oplus> L = K"
paulson@13216
   668
apply (rule le_anti_sym)
paulson@13216
   669
 prefer 2
paulson@13216
   670
 apply (erule ltE, blast intro: cadd_le_self InfCard_is_Card)
paulson@13216
   671
apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl])
paulson@13216
   672
apply (rule cadd_le_mono [THEN le_trans], assumption+)
paulson@13216
   673
apply (simp add: InfCard_cdouble_eq)
paulson@13216
   674
done
paulson@13216
   675
paulson@46821
   676
lemma InfCard_cadd_eq: "[| InfCard(K);  InfCard(L) |] ==> K \<oplus> L = K \<union> L"
paulson@13784
   677
apply (rule_tac i = K and j = L in Ord_linear_le)
paulson@13216
   678
apply (typecheck add: InfCard_is_Card Card_is_Ord)
paulson@13216
   679
apply (rule cadd_commute [THEN ssubst])
paulson@13216
   680
apply (rule Un_commute [THEN ssubst])
paulson@13221
   681
apply (simp_all add: InfCard_le_cadd_eq subset_Un_iff2 [THEN iffD1] le_imp_subset)
paulson@13216
   682
done
paulson@13216
   683
paulson@13216
   684
(*The other part, Corollary 10.13 (2), refers to the cardinality of the set
paulson@13216
   685
  of all n-tuples of elements of K.  A better version for the Isabelle theory
paulson@13216
   686
  might be  InfCard(K) ==> |list(K)| = K.
paulson@13216
   687
*)
paulson@13216
   688
ballarin@27517
   689
subsection{*For Every Cardinal Number There Exists A Greater One*}
paulson@13356
   690
paulson@13356
   691
text{*This result is Kunen's Theorem 10.16, which would be trivial using AC*}
paulson@13216
   692
paulson@13216
   693
lemma Ord_jump_cardinal: "Ord(jump_cardinal(K))"
paulson@13216
   694
apply (unfold jump_cardinal_def)
paulson@13216
   695
apply (rule Ord_is_Transset [THEN [2] OrdI])
paulson@13216
   696
 prefer 2 apply (blast intro!: Ord_ordertype)
paulson@13216
   697
apply (unfold Transset_def)
paulson@13216
   698
apply (safe del: subsetI)
paulson@13221
   699
apply (simp add: ordertype_pred_unfold, safe)
paulson@13216
   700
apply (rule UN_I)
paulson@13216
   701
apply (rule_tac [2] ReplaceI)
paulson@13216
   702
   prefer 4 apply (blast intro: well_ord_subset elim!: predE)+
paulson@13216
   703
done
paulson@13216
   704
paulson@13216
   705
(*Allows selective unfolding.  Less work than deriving intro/elim rules*)
paulson@13216
   706
lemma jump_cardinal_iff:
paulson@46821
   707
     "i \<in> jump_cardinal(K) \<longleftrightarrow>
paulson@46820
   708
      (\<exists>r X. r \<subseteq> K*K & X \<subseteq> K & well_ord(X,r) & i = ordertype(X,r))"
paulson@13216
   709
apply (unfold jump_cardinal_def)
paulson@46820
   710
apply (blast del: subsetI)
paulson@13216
   711
done
paulson@13216
   712
paulson@13216
   713
(*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*)
paulson@13216
   714
lemma K_lt_jump_cardinal: "Ord(K) ==> K < jump_cardinal(K)"
paulson@13216
   715
apply (rule Ord_jump_cardinal [THEN [2] ltI])
paulson@13216
   716
apply (rule jump_cardinal_iff [THEN iffD2])
paulson@13216
   717
apply (rule_tac x="Memrel(K)" in exI)
paulson@46820
   718
apply (rule_tac x=K in exI)
paulson@13216
   719
apply (simp add: ordertype_Memrel well_ord_Memrel)
paulson@13216
   720
apply (simp add: Memrel_def subset_iff)
paulson@13216
   721
done
paulson@13216
   722
paulson@13216
   723
(*The proof by contradiction: the bijection f yields a wellordering of X
paulson@13216
   724
  whose ordertype is jump_cardinal(K).  *)
paulson@13216
   725
lemma Card_jump_cardinal_lemma:
paulson@46820
   726
     "[| well_ord(X,r);  r \<subseteq> K * K;  X \<subseteq> K;
paulson@46820
   727
         f \<in> bij(ordertype(X,r), jump_cardinal(K)) |]
paulson@46820
   728
      ==> jump_cardinal(K) \<in> jump_cardinal(K)"
paulson@46820
   729
apply (subgoal_tac "f O ordermap (X,r) \<in> bij (X, jump_cardinal (K))")
paulson@13216
   730
 prefer 2 apply (blast intro: comp_bij ordermap_bij)
paulson@13216
   731
apply (rule jump_cardinal_iff [THEN iffD2])
paulson@13216
   732
apply (intro exI conjI)
paulson@13221
   733
apply (rule subset_trans [OF rvimage_type Sigma_mono], assumption+)
paulson@13216
   734
apply (erule bij_is_inj [THEN well_ord_rvimage])
paulson@13216
   735
apply (rule Ord_jump_cardinal [THEN well_ord_Memrel])
paulson@13216
   736
apply (simp add: well_ord_Memrel [THEN [2] bij_ordertype_vimage]
paulson@13216
   737
                 ordertype_Memrel Ord_jump_cardinal)
paulson@13216
   738
done
paulson@13216
   739
paulson@13216
   740
(*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*)
paulson@13216
   741
lemma Card_jump_cardinal: "Card(jump_cardinal(K))"
paulson@13216
   742
apply (rule Ord_jump_cardinal [THEN CardI])
paulson@13216
   743
apply (unfold eqpoll_def)
paulson@13216
   744
apply (safe dest!: ltD jump_cardinal_iff [THEN iffD1])
paulson@13216
   745
apply (blast intro: Card_jump_cardinal_lemma [THEN mem_irrefl])
paulson@13216
   746
done
paulson@13216
   747
paulson@13356
   748
subsection{*Basic Properties of Successor Cardinals*}
paulson@13216
   749
paulson@13216
   750
lemma csucc_basic: "Ord(K) ==> Card(csucc(K)) & K < csucc(K)"
paulson@13216
   751
apply (unfold csucc_def)
paulson@13216
   752
apply (rule LeastI)
paulson@13216
   753
apply (blast intro: Card_jump_cardinal K_lt_jump_cardinal Ord_jump_cardinal)+
paulson@13216
   754
done
paulson@13216
   755
wenzelm@45602
   756
lemmas Card_csucc = csucc_basic [THEN conjunct1]
paulson@13216
   757
wenzelm@45602
   758
lemmas lt_csucc = csucc_basic [THEN conjunct2]
paulson@13216
   759
paulson@13216
   760
lemma Ord_0_lt_csucc: "Ord(K) ==> 0 < csucc(K)"
paulson@13221
   761
by (blast intro: Ord_0_le lt_csucc lt_trans1)
paulson@13216
   762
paulson@46820
   763
lemma csucc_le: "[| Card(L);  K<L |] ==> csucc(K) \<le> L"
paulson@13216
   764
apply (unfold csucc_def)
paulson@13216
   765
apply (rule Least_le)
paulson@13216
   766
apply (blast intro: Card_is_Ord)+
paulson@13216
   767
done
paulson@13216
   768
paulson@46821
   769
lemma lt_csucc_iff: "[| Ord(i); Card(K) |] ==> i < csucc(K) \<longleftrightarrow> |i| \<le> K"
paulson@13216
   770
apply (rule iffI)
paulson@13216
   771
apply (rule_tac [2] Card_lt_imp_lt)
paulson@13216
   772
apply (erule_tac [2] lt_trans1)
paulson@13216
   773
apply (simp_all add: lt_csucc Card_csucc Card_is_Ord)
paulson@13216
   774
apply (rule notI [THEN not_lt_imp_le])
paulson@13221
   775
apply (rule Card_cardinal [THEN csucc_le, THEN lt_trans1, THEN lt_irrefl], assumption)
paulson@13216
   776
apply (rule Ord_cardinal_le [THEN lt_trans1])
paulson@46820
   777
apply (simp_all add: Ord_cardinal Card_is_Ord)
paulson@13216
   778
done
paulson@13216
   779
paulson@13216
   780
lemma Card_lt_csucc_iff:
paulson@46821
   781
     "[| Card(K'); Card(K) |] ==> K' < csucc(K) \<longleftrightarrow> K' \<le> K"
paulson@13221
   782
by (simp add: lt_csucc_iff Card_cardinal_eq Card_is_Ord)
paulson@13216
   783
paulson@13216
   784
lemma InfCard_csucc: "InfCard(K) ==> InfCard(csucc(K))"
paulson@46820
   785
by (simp add: InfCard_def Card_csucc Card_is_Ord
paulson@13216
   786
              lt_csucc [THEN leI, THEN [2] le_trans])
paulson@13216
   787
paulson@13216
   788
paulson@14883
   789
subsubsection{*Removing elements from a finite set decreases its cardinality*}
paulson@13216
   790
paulson@46820
   791
lemma Fin_imp_not_cons_lepoll: "A: Fin(U) ==> x\<notin>A \<longrightarrow> ~ cons(x,A) \<lesssim> A"
paulson@13216
   792
apply (erule Fin_induct)
paulson@13221
   793
apply (simp add: lepoll_0_iff)
paulson@13216
   794
apply (subgoal_tac "cons (x,cons (xa,y)) = cons (xa,cons (x,y))")
paulson@13221
   795
apply simp
paulson@13221
   796
apply (blast dest!: cons_lepoll_consD, blast)
paulson@13216
   797
done
paulson@13216
   798
paulson@14883
   799
lemma Finite_imp_cardinal_cons [simp]:
paulson@46820
   800
     "[| Finite(A);  a\<notin>A |] ==> |cons(a,A)| = succ(|A|)"
paulson@13216
   801
apply (unfold cardinal_def)
paulson@13216
   802
apply (rule Least_equality)
paulson@13216
   803
apply (fold cardinal_def)
paulson@13221
   804
apply (simp add: succ_def)
paulson@13216
   805
apply (blast intro: cons_eqpoll_cong well_ord_cardinal_eqpoll
paulson@13216
   806
             elim!: mem_irrefl  dest!: Finite_imp_well_ord)
paulson@13216
   807
apply (blast intro: Card_cardinal Card_is_Ord)
paulson@13216
   808
apply (rule notI)
paulson@13221
   809
apply (rule Finite_into_Fin [THEN Fin_imp_not_cons_lepoll, THEN mp, THEN notE],
paulson@13221
   810
       assumption, assumption)
paulson@13216
   811
apply (erule eqpoll_sym [THEN eqpoll_imp_lepoll, THEN lepoll_trans])
paulson@13216
   812
apply (erule le_imp_lepoll [THEN lepoll_trans])
paulson@13216
   813
apply (blast intro: well_ord_cardinal_eqpoll [THEN eqpoll_imp_lepoll]
paulson@13216
   814
             dest!: Finite_imp_well_ord)
paulson@13216
   815
done
paulson@13216
   816
paulson@13216
   817
paulson@13221
   818
lemma Finite_imp_succ_cardinal_Diff:
paulson@13221
   819
     "[| Finite(A);  a:A |] ==> succ(|A-{a}|) = |A|"
paulson@13784
   820
apply (rule_tac b = A in cons_Diff [THEN subst], assumption)
paulson@13221
   821
apply (simp add: Finite_imp_cardinal_cons Diff_subset [THEN subset_Finite])
paulson@13221
   822
apply (simp add: cons_Diff)
paulson@13216
   823
done
paulson@13216
   824
paulson@13216
   825
lemma Finite_imp_cardinal_Diff: "[| Finite(A);  a:A |] ==> |A-{a}| < |A|"
paulson@13216
   826
apply (rule succ_leE)
paulson@13221
   827
apply (simp add: Finite_imp_succ_cardinal_Diff)
paulson@13216
   828
done
paulson@13216
   829
paulson@46820
   830
lemma Finite_cardinal_in_nat [simp]: "Finite(A) ==> |A| \<in> nat"
paulson@14883
   831
apply (erule Finite_induct)
paulson@14883
   832
apply (auto simp add: cardinal_0 Finite_imp_cardinal_cons)
paulson@14883
   833
done
paulson@13216
   834
paulson@14883
   835
lemma card_Un_Int:
paulson@46820
   836
     "[|Finite(A); Finite(B)|] ==> |A| #+ |B| = |A \<union> B| #+ |A \<inter> B|"
paulson@46820
   837
apply (erule Finite_induct, simp)
paulson@14883
   838
apply (simp add: Finite_Int cons_absorb Un_cons Int_cons_left)
paulson@14883
   839
done
paulson@14883
   840
paulson@46820
   841
lemma card_Un_disjoint:
paulson@46820
   842
     "[|Finite(A); Finite(B); A \<inter> B = 0|] ==> |A \<union> B| = |A| #+ |B|"
paulson@14883
   843
by (simp add: Finite_Un card_Un_Int)
paulson@14883
   844
paulson@14883
   845
lemma card_partition [rule_format]:
paulson@46820
   846
     "Finite(C) ==>
paulson@46820
   847
        Finite (\<Union> C) \<longrightarrow>
paulson@46820
   848
        (\<forall>c\<in>C. |c| = k) \<longrightarrow>
paulson@46820
   849
        (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = 0) \<longrightarrow>
paulson@14883
   850
        k #* |C| = |\<Union> C|"
paulson@14883
   851
apply (erule Finite_induct, auto)
paulson@46820
   852
apply (subgoal_tac " x \<inter> \<Union>B = 0")
paulson@14883
   853
apply (auto simp add: card_Un_disjoint Finite_Union
paulson@14883
   854
       subset_Finite [of _ "\<Union> (cons(x,F))"])
paulson@14883
   855
done
paulson@14883
   856
paulson@14883
   857
paulson@14883
   858
subsubsection{*Theorems by Krzysztof Grabczewski, proofs by lcp*}
paulson@13216
   859
wenzelm@45602
   860
lemmas nat_implies_well_ord = nat_into_Ord [THEN well_ord_Memrel]
paulson@13216
   861
paulson@13216
   862
lemma nat_sum_eqpoll_sum: "[| m:nat; n:nat |] ==> m + n \<approx> m #+ n"
paulson@13216
   863
apply (rule eqpoll_trans)
paulson@13216
   864
apply (rule well_ord_radd [THEN well_ord_cardinal_eqpoll, THEN eqpoll_sym])
paulson@13216
   865
apply (erule nat_implies_well_ord)+
paulson@13221
   866
apply (simp add: nat_cadd_eq_add [symmetric] cadd_def eqpoll_refl)
paulson@13216
   867
done
paulson@13216
   868
paulson@46820
   869
lemma Ord_subset_natD [rule_format]: "Ord(i) ==> i \<subseteq> nat \<longrightarrow> i \<in> nat | i=nat"
paulson@13221
   870
apply (erule trans_induct3, auto)
paulson@13216
   871
apply (blast dest!: nat_le_Limit [THEN le_imp_subset])
paulson@13216
   872
done
paulson@13216
   873
paulson@46820
   874
lemma Ord_nat_subset_into_Card: "[| Ord(i); i \<subseteq> nat |] ==> Card(i)"
paulson@13221
   875
by (blast dest: Ord_subset_natD intro: Card_nat nat_into_Card)
paulson@13216
   876
paulson@13216
   877
lemma Finite_Diff_sing_eq_diff_1: "[| Finite(A); x:A |] ==> |A-{x}| = |A| #- 1"
paulson@13216
   878
apply (rule succ_inject)
paulson@13216
   879
apply (rule_tac b = "|A|" in trans)
paulson@13615
   880
 apply (simp add: Finite_imp_succ_cardinal_Diff)
paulson@13216
   881
apply (subgoal_tac "1 \<lesssim> A")
paulson@13221
   882
 prefer 2 apply (blast intro: not_0_is_lepoll_1)
paulson@13221
   883
apply (frule Finite_imp_well_ord, clarify)
paulson@13216
   884
apply (drule well_ord_lepoll_imp_Card_le)
paulson@13615
   885
 apply (auto simp add: cardinal_1)
paulson@13216
   886
apply (rule trans)
paulson@13615
   887
 apply (rule_tac [2] diff_succ)
paulson@13615
   888
  apply (auto simp add: Finite_cardinal_in_nat)
paulson@13216
   889
done
paulson@13216
   890
paulson@13221
   891
lemma cardinal_lt_imp_Diff_not_0 [rule_format]:
paulson@46820
   892
     "Finite(B) ==> \<forall>A. |B|<|A| \<longrightarrow> A - B \<noteq> 0"
paulson@13221
   893
apply (erule Finite_induct, auto)
paulson@13221
   894
apply (case_tac "Finite (A)")
paulson@13221
   895
 apply (subgoal_tac [2] "Finite (cons (x, B))")
paulson@13221
   896
  apply (drule_tac [2] B = "cons (x, B) " in Diff_Finite)
paulson@13221
   897
   apply (auto simp add: Finite_0 Finite_cons)
paulson@13216
   898
apply (subgoal_tac "|B|<|A|")
paulson@13221
   899
 prefer 2 apply (blast intro: lt_trans Ord_cardinal)
paulson@13216
   900
apply (case_tac "x:A")
paulson@13221
   901
 apply (subgoal_tac [2] "A - cons (x, B) = A - B")
paulson@13221
   902
  apply auto
paulson@46820
   903
apply (subgoal_tac "|A| \<le> |cons (x, B) |")
paulson@13221
   904
 prefer 2
paulson@46820
   905
 apply (blast dest: Finite_cons [THEN Finite_imp_well_ord]
paulson@13216
   906
              intro: well_ord_lepoll_imp_Card_le subset_imp_lepoll)
paulson@13216
   907
apply (auto simp add: Finite_imp_cardinal_cons)
paulson@13216
   908
apply (auto dest!: Finite_cardinal_in_nat simp add: le_iff)
paulson@13216
   909
apply (blast intro: lt_trans)
paulson@13216
   910
done
paulson@13216
   911
lcp@437
   912
end