src/ZF/CardinalArith.thy
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Sun, 07 Oct 2007 21:19:31 +0200
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(*  Title:      ZF/CardinalArith.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1994  University of Cambridge
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*)
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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 Un y, x, y>, 
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		  rmult(K,Memrel(K), K*K, rmult(K,Memrel(K), K,Memrel(K))))"
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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 output)
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  cadd  (infixl "\<oplus>" 65) and
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  cmult  (infixl "\<otimes>" 70)
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notation (HTML output)
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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]: "(ALL x:A. Card(x)) ==> Card(Union(A))"
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apply (rule CardI) 
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 apply (simp add: Card_is_Ord) 
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apply (clarify dest!: ltD)
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apply (drule bspec, assumption) 
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apply (frule lt_Card_imp_lesspoll, blast intro: ltI Card_is_Ord) 
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apply (drule eqpoll_sym [THEN eqpoll_imp_lepoll])
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apply (drule lesspoll_trans1, assumption) 
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apply (subgoal_tac "B \<lesssim> \<Union>A")
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 apply (drule lesspoll_trans1, assumption, blast) 
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apply (blast intro: subset_imp_lepoll) 
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done
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lemma Card_UN: "(!!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 |+| j = j |+| i"
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apply (unfold cadd_def)
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apply (rule sum_commute_eqpoll [THEN cardinal_cong])
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done
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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 |+| j) |+| k = i |+| (j |+| k)"
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apply (unfold cadd_def)
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apply (rule cardinal_cong)
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apply (rule eqpoll_trans)
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 apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
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 apply (blast intro: well_ord_radd ) 
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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 |+| 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 = "lam x: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 |+| 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 = "lam z:A+B. case (%w. Inl(f`w), %y. Inr(fa`y), z)" in exI)
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apply (rule_tac d = "case (%w. Inl(converse(f) `w), %y. Inr(converse(fa) ` y))"
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       in lam_injective)
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apply (typecheck add: inj_is_fun, auto)
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done
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lemma cadd_le_mono:
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    "[| K' le K;  L' le L |] ==> (K' |+| L') le (K |+| L)"
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apply (unfold cadd_def)
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apply (safe dest!: le_subset_iff [THEN iffD1])
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apply (rule well_ord_lepoll_imp_Card_le)
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apply (blast intro: well_ord_radd well_ord_Memrel)
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apply (blast intro: sum_lepoll_mono subset_imp_lepoll)
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done
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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) |+| n = |succ(m |+| n)|"
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apply (unfold cadd_def)
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apply (rule sum_succ_eqpoll [THEN cardinal_cong, THEN trans])
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apply (rule succ_eqpoll_cong [THEN cardinal_cong])
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apply (rule well_ord_cardinal_eqpoll [THEN eqpoll_sym])
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apply (blast intro: well_ord_radd well_ord_Memrel)
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done
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lemma nat_cadd_eq_add: "[| m: nat;  n: nat |] ==> m |+| n = m#+n"
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apply (induct_tac m)
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apply (simp add: nat_into_Card [THEN cadd_0])
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apply (simp add: cadd_succ_lemma nat_into_Card [THEN Card_cardinal_eq])
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done
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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 |*| j = j |*| i"
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apply (unfold cmult_def)
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apply (rule prod_commute_eqpoll [THEN cardinal_cong])
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done
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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 |*| j) |*| k = i |*| (j |*| k)"
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apply (unfold cmult_def)
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apply (rule cardinal_cong)
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apply (rule eqpoll_trans) 
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 apply (rule prod_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
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 apply (blast intro: well_ord_rmult)
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apply (rule prod_assoc_eqpoll [THEN eqpoll_trans])
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apply (rule eqpoll_sym) 
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apply (rule prod_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
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apply (blast intro: well_ord_rmult)
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done
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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 |+| j) |*| k = (i |*| k) |+| (j |*| k)"
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apply (unfold cadd_def cmult_def)
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apply (rule cardinal_cong)
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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 |*| 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 |*| 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 = "lam x: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 |*| K"
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   320
apply (unfold cmult_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   321
apply (rule le_trans)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   322
apply (rule_tac [2] well_ord_lepoll_imp_Card_le)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   323
apply (rule_tac [3] prod_square_lepoll)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   324
apply (simp add: le_refl Card_is_Ord Card_cardinal_eq)
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   325
apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   326
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   327
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   328
subsubsection{*Multiplication by a non-zero cardinal*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   329
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   330
lemma prod_lepoll_self: "b: B ==> A \<lesssim> A*B"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   331
apply (unfold lepoll_def inj_def)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   332
apply (rule_tac x = "lam x:A. <x,b>" in exI, simp)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   333
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   334
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   335
(*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   336
lemma cmult_le_self:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   337
    "[| Card(K);  Ord(L);  0<L |] ==> K le (K |*| L)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   338
apply (unfold cmult_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   339
apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le])
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   340
  apply assumption
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   341
 apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   342
apply (blast intro: prod_lepoll_self ltD)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   343
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   344
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   345
subsubsection{*Monotonicity of multiplication*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   346
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   347
lemma prod_lepoll_mono:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   348
     "[| A \<lesssim> C;  B \<lesssim> D |] ==> A * B  \<lesssim>  C * D"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   349
apply (unfold lepoll_def)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   350
apply (elim exE)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   351
apply (rule_tac x = "lam <w,y>:A*B. <f`w, fa`y>" in exI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   352
apply (rule_tac d = "%<w,y>. <converse (f) `w, converse (fa) `y>" 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   353
       in lam_injective)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   354
apply (typecheck add: inj_is_fun, auto)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   355
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   356
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   357
lemma cmult_le_mono:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   358
    "[| K' le K;  L' le L |] ==> (K' |*| L') le (K |*| L)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   359
apply (unfold cmult_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   360
apply (safe dest!: le_subset_iff [THEN iffD1])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   361
apply (rule well_ord_lepoll_imp_Card_le)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   362
 apply (blast intro: well_ord_rmult well_ord_Memrel)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   363
apply (blast intro: prod_lepoll_mono subset_imp_lepoll)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   364
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   365
13356
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   366
subsection{*Multiplication of finite cardinals is "ordinary" multiplication*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   367
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   368
lemma prod_succ_eqpoll: "succ(A)*B \<approx> B + A*B"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   369
apply (unfold eqpoll_def)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   370
apply (rule exI)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   371
apply (rule_tac c = "%<x,y>. if x=A then Inl (y) else Inr (<x,y>)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   372
            and d = "case (%y. <A,y>, %z. z)" in lam_bijective)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   373
apply safe
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   374
apply (simp_all add: succI2 if_type mem_imp_not_eq)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   375
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   376
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   377
(*Unconditional version requires AC*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   378
lemma cmult_succ_lemma:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   379
    "[| Ord(m);  Ord(n) |] ==> succ(m) |*| n = n |+| (m |*| n)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   380
apply (unfold cmult_def cadd_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   381
apply (rule prod_succ_eqpoll [THEN cardinal_cong, THEN trans])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   382
apply (rule cardinal_cong [symmetric])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   383
apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   384
apply (blast intro: well_ord_rmult well_ord_Memrel)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   385
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   386
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   387
lemma nat_cmult_eq_mult: "[| m: nat;  n: nat |] ==> m |*| n = m#*n"
13244
7b37e218f298 moving some results around
paulson
parents: 13221
diff changeset
   388
apply (induct_tac m)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   389
apply (simp_all add: cmult_succ_lemma nat_cadd_eq_add)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   390
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   391
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   392
lemma cmult_2: "Card(n) ==> 2 |*| n = n |+| n"
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   393
by (simp add: cmult_succ_lemma Card_is_Ord cadd_commute [of _ 0])
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   394
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   395
lemma sum_lepoll_prod: "2 \<lesssim> C ==> B+B \<lesssim> C*B"
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   396
apply (rule lepoll_trans) 
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   397
apply (rule sum_eq_2_times [THEN equalityD1, THEN subset_imp_lepoll]) 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   398
apply (erule prod_lepoll_mono) 
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   399
apply (rule lepoll_refl) 
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   400
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   401
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   402
lemma lepoll_imp_sum_lepoll_prod: "[| A \<lesssim> B; 2 \<lesssim> A |] ==> A+B \<lesssim> A*B"
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   403
by (blast intro: sum_lepoll_mono sum_lepoll_prod lepoll_trans lepoll_refl)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   404
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   405
13356
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   406
subsection{*Infinite Cardinals are Limit Ordinals*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   407
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   408
(*This proof is modelled upon one assuming nat<=A, with injection
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   409
  lam z:cons(u,A). if z=u then 0 else if z : nat then succ(z) else z
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   410
  and inverse %y. if y:nat then nat_case(u, %z. z, y) else y.  \
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   411
  If f: inj(nat,A) then range(f) behaves like the natural numbers.*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   412
lemma nat_cons_lepoll: "nat \<lesssim> A ==> cons(u,A) \<lesssim> A"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   413
apply (unfold lepoll_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   414
apply (erule exE)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   415
apply (rule_tac x = 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   416
          "lam z:cons (u,A).
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   417
             if z=u then f`0 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   418
             else if z: range (f) then f`succ (converse (f) `z) else z" 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   419
       in exI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   420
apply (rule_tac d =
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   421
          "%y. if y: range(f) then nat_case (u, %z. f`z, converse(f) `y) 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   422
                              else y" 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   423
       in lam_injective)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   424
apply (fast intro!: if_type apply_type intro: inj_is_fun inj_converse_fun)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   425
apply (simp add: inj_is_fun [THEN apply_rangeI]
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   426
                 inj_converse_fun [THEN apply_rangeI]
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   427
                 inj_converse_fun [THEN apply_funtype])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   428
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   429
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   430
lemma nat_cons_eqpoll: "nat \<lesssim> A ==> cons(u,A) \<approx> A"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   431
apply (erule nat_cons_lepoll [THEN eqpollI])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   432
apply (rule subset_consI [THEN subset_imp_lepoll])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   433
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   434
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   435
(*Specialized version required below*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   436
lemma nat_succ_eqpoll: "nat <= A ==> succ(A) \<approx> A"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   437
apply (unfold succ_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   438
apply (erule subset_imp_lepoll [THEN nat_cons_eqpoll])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   439
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   440
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   441
lemma InfCard_nat: "InfCard(nat)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   442
apply (unfold InfCard_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   443
apply (blast intro: Card_nat le_refl Card_is_Ord)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   444
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   445
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   446
lemma InfCard_is_Card: "InfCard(K) ==> Card(K)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   447
apply (unfold InfCard_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   448
apply (erule conjunct1)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   449
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   450
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   451
lemma InfCard_Un:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   452
    "[| InfCard(K);  Card(L) |] ==> InfCard(K Un L)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   453
apply (unfold InfCard_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   454
apply (simp add: Card_Un Un_upper1_le [THEN [2] le_trans]  Card_is_Ord)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   455
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   456
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   457
(*Kunen's Lemma 10.11*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   458
lemma InfCard_is_Limit: "InfCard(K) ==> Limit(K)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   459
apply (unfold InfCard_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   460
apply (erule conjE)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   461
apply (frule Card_is_Ord)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   462
apply (rule ltI [THEN non_succ_LimitI])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   463
apply (erule le_imp_subset [THEN subsetD])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   464
apply (safe dest!: Limit_nat [THEN Limit_le_succD])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   465
apply (unfold Card_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   466
apply (drule trans)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   467
apply (erule le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   468
apply (erule Ord_cardinal_le [THEN lt_trans2, THEN lt_irrefl])
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   469
apply (rule le_eqI, assumption)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   470
apply (rule Ord_cardinal)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   471
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   472
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   473
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   474
(*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   475
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   476
(*A general fact about ordermap*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   477
lemma ordermap_eqpoll_pred:
13269
3ba9be497c33 Tidying and introduction of various new theorems
paulson
parents: 13244
diff changeset
   478
    "[| well_ord(A,r);  x:A |] ==> ordermap(A,r)`x \<approx> Order.pred(A,x,r)"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   479
apply (unfold eqpoll_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   480
apply (rule exI)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   481
apply (simp add: ordermap_eq_image well_ord_is_wf)
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   482
apply (erule ordermap_bij [THEN bij_is_inj, THEN restrict_bij, 
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   483
                           THEN bij_converse_bij])
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   484
apply (rule pred_subset)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   485
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   486
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   487
subsubsection{*Establishing the well-ordering*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   488
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   489
lemma csquare_lam_inj:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   490
     "Ord(K) ==> (lam <x,y>:K*K. <x Un y, x, y>) : inj(K*K, K*K*K)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   491
apply (unfold inj_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   492
apply (force intro: lam_type Un_least_lt [THEN ltD] ltI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   493
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   494
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   495
lemma well_ord_csquare: "Ord(K) ==> well_ord(K*K, csquare_rel(K))"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   496
apply (unfold csquare_rel_def)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   497
apply (rule csquare_lam_inj [THEN well_ord_rvimage], assumption)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   498
apply (blast intro: well_ord_rmult well_ord_Memrel)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   499
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   500
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   501
subsubsection{*Characterising initial segments of the well-ordering*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   502
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   503
lemma csquareD:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   504
 "[| <<x,y>, <z,z>> : csquare_rel(K);  x<K;  y<K;  z<K |] ==> x le z & y le z"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   505
apply (unfold csquare_rel_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   506
apply (erule rev_mp)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   507
apply (elim ltE)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   508
apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   509
apply (safe elim!: mem_irrefl intro!: Un_upper1_le Un_upper2_le)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   510
apply (simp_all add: lt_def succI2)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   511
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   512
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   513
lemma pred_csquare_subset: 
13269
3ba9be497c33 Tidying and introduction of various new theorems
paulson
parents: 13244
diff changeset
   514
    "z<K ==> Order.pred(K*K, <z,z>, csquare_rel(K)) <= succ(z)*succ(z)"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   515
apply (unfold Order.pred_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   516
apply (safe del: SigmaI succCI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   517
apply (erule csquareD [THEN conjE])
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   518
apply (unfold lt_def, auto) 
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   519
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   520
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   521
lemma csquare_ltI:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   522
 "[| x<z;  y<z;  z<K |] ==>  <<x,y>, <z,z>> : csquare_rel(K)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   523
apply (unfold csquare_rel_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   524
apply (subgoal_tac "x<K & y<K")
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   525
 prefer 2 apply (blast intro: lt_trans) 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   526
apply (elim ltE)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   527
apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   528
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   529
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   530
(*Part of the traditional proof.  UNUSED since it's harder to prove & apply *)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   531
lemma csquare_or_eqI:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   532
 "[| x le z;  y le z;  z<K |] ==> <<x,y>, <z,z>> : csquare_rel(K) | x=z & y=z"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   533
apply (unfold csquare_rel_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   534
apply (subgoal_tac "x<K & y<K")
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   535
 prefer 2 apply (blast intro: lt_trans1) 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   536
apply (elim ltE)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   537
apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   538
apply (elim succE)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   539
apply (simp_all add: subset_Un_iff [THEN iff_sym] 
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   540
                     subset_Un_iff2 [THEN iff_sym] OrdmemD)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   541
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   542
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   543
subsubsection{*The cardinality of initial segments*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   544
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   545
lemma ordermap_z_lt:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   546
      "[| Limit(K);  x<K;  y<K;  z=succ(x Un y) |] ==>
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   547
          ordermap(K*K, csquare_rel(K)) ` <x,y> <
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   548
          ordermap(K*K, csquare_rel(K)) ` <z,z>"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   549
apply (subgoal_tac "z<K & well_ord (K*K, csquare_rel (K))")
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   550
prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   551
                              Limit_is_Ord [THEN well_ord_csquare], clarify) 
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   552
apply (rule csquare_ltI [THEN ordermap_mono, THEN ltI])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   553
apply (erule_tac [4] well_ord_is_wf)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   554
apply (blast intro!: Un_upper1_le Un_upper2_le Ord_ordermap elim!: ltE)+
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   555
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   556
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   557
(*Kunen: "each <x,y>: K*K has no more than z*z predecessors..." (page 29) *)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   558
lemma ordermap_csquare_le:
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   559
  "[| Limit(K);  x<K;  y<K;  z=succ(x Un y) |]
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   560
   ==> | ordermap(K*K, csquare_rel(K)) ` <x,y> | le  |succ(z)| |*| |succ(z)|"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   561
apply (unfold cmult_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   562
apply (rule well_ord_rmult [THEN well_ord_lepoll_imp_Card_le])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   563
apply (rule Ord_cardinal [THEN well_ord_Memrel])+
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   564
apply (subgoal_tac "z<K")
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   565
 prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   566
apply (rule ordermap_z_lt [THEN leI, THEN le_imp_lepoll, THEN lepoll_trans], 
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   567
       assumption+)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   568
apply (rule ordermap_eqpoll_pred [THEN eqpoll_imp_lepoll, THEN lepoll_trans])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   569
apply (erule Limit_is_Ord [THEN well_ord_csquare])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   570
apply (blast intro: ltD)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   571
apply (rule pred_csquare_subset [THEN subset_imp_lepoll, THEN lepoll_trans],
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   572
            assumption)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   573
apply (elim ltE)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   574
apply (rule prod_eqpoll_cong [THEN eqpoll_sym, THEN eqpoll_imp_lepoll])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   575
apply (erule Ord_succ [THEN Ord_cardinal_eqpoll])+
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   576
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   577
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   578
(*Kunen: "... so the order type <= K" *)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   579
lemma ordertype_csquare_le:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   580
     "[| InfCard(K);  ALL y:K. InfCard(y) --> y |*| y = y |] 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   581
      ==> ordertype(K*K, csquare_rel(K)) le K"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   582
apply (frule InfCard_is_Card [THEN Card_is_Ord])
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   583
apply (rule all_lt_imp_le, assumption)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   584
apply (erule well_ord_csquare [THEN Ord_ordertype])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   585
apply (rule Card_lt_imp_lt)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   586
apply (erule_tac [3] InfCard_is_Card)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   587
apply (erule_tac [2] ltE)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   588
apply (simp add: ordertype_unfold)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   589
apply (safe elim!: ltE)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   590
apply (subgoal_tac "Ord (xa) & Ord (ya)")
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   591
 prefer 2 apply (blast intro: Ord_in_Ord, clarify)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   592
(*??WHAT A MESS!*)  
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   593
apply (rule InfCard_is_Limit [THEN ordermap_csquare_le, THEN lt_trans1],
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   594
       (assumption | rule refl | erule ltI)+) 
13784
b9f6154427a4 tidying (by script)
paulson
parents: 13615
diff changeset
   595
apply (rule_tac i = "xa Un ya" and j = nat in Ord_linear2,
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   596
       simp_all add: Ord_Un Ord_nat)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   597
prefer 2 (*case nat le (xa Un ya) *)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   598
 apply (simp add: le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong] 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   599
                  le_succ_iff InfCard_def Card_cardinal Un_least_lt Ord_Un
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   600
                ltI nat_le_cardinal Ord_cardinal_le [THEN lt_trans1, THEN ltD])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   601
(*the finite case: xa Un ya < nat *)
13784
b9f6154427a4 tidying (by script)
paulson
parents: 13615
diff changeset
   602
apply (rule_tac j = nat in lt_trans2)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   603
 apply (simp add: lt_def nat_cmult_eq_mult nat_succI mult_type
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   604
                  nat_into_Card [THEN Card_cardinal_eq]  Ord_nat)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   605
apply (simp add: InfCard_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   606
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   607
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   608
(*Main result: Kunen's Theorem 10.12*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   609
lemma InfCard_csquare_eq: "InfCard(K) ==> K |*| K = K"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   610
apply (frule InfCard_is_Card [THEN Card_is_Ord])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   611
apply (erule rev_mp)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   612
apply (erule_tac i=K in trans_induct) 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   613
apply (rule impI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   614
apply (rule le_anti_sym)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   615
apply (erule_tac [2] InfCard_is_Card [THEN cmult_square_le])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   616
apply (rule ordertype_csquare_le [THEN [2] le_trans])
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   617
apply (simp add: cmult_def Ord_cardinal_le   
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   618
                 well_ord_csquare [THEN Ord_ordertype]
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   619
                 well_ord_csquare [THEN ordermap_bij, THEN bij_imp_eqpoll, 
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   620
                                   THEN cardinal_cong], assumption+)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   621
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   622
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   623
(*Corollary for arbitrary well-ordered sets (all sets, assuming AC)*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   624
lemma well_ord_InfCard_square_eq:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   625
     "[| well_ord(A,r);  InfCard(|A|) |] ==> A*A \<approx> A"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   626
apply (rule prod_eqpoll_cong [THEN eqpoll_trans])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   627
apply (erule well_ord_cardinal_eqpoll [THEN eqpoll_sym])+
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   628
apply (rule well_ord_cardinal_eqE)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   629
apply (blast intro: Ord_cardinal well_ord_rmult well_ord_Memrel, assumption)
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   630
apply (simp add: cmult_def [symmetric] InfCard_csquare_eq)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   631
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   632
13356
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   633
lemma InfCard_square_eqpoll: "InfCard(K) ==> K \<times> K \<approx> K"
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   634
apply (rule well_ord_InfCard_square_eq)  
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   635
 apply (erule InfCard_is_Card [THEN Card_is_Ord, THEN well_ord_Memrel]) 
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   636
apply (simp add: InfCard_is_Card [THEN Card_cardinal_eq]) 
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   637
done
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   638
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   639
lemma Inf_Card_is_InfCard: "[| ~Finite(i); Card(i) |] ==> InfCard(i)"
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   640
by (simp add: InfCard_def Card_is_Ord [THEN nat_le_infinite_Ord])
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   641
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   642
subsubsection{*Toward's Kunen's Corollary 10.13 (1)*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   643
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   644
lemma InfCard_le_cmult_eq: "[| InfCard(K);  L le K;  0<L |] ==> K |*| L = K"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   645
apply (rule le_anti_sym)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   646
 prefer 2
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   647
 apply (erule ltE, blast intro: cmult_le_self InfCard_is_Card)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   648
apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   649
apply (rule cmult_le_mono [THEN le_trans], assumption+)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   650
apply (simp add: InfCard_csquare_eq)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   651
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   652
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   653
(*Corollary 10.13 (1), for cardinal multiplication*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   654
lemma InfCard_cmult_eq: "[| InfCard(K);  InfCard(L) |] ==> K |*| L = K Un L"
13784
b9f6154427a4 tidying (by script)
paulson
parents: 13615
diff changeset
   655
apply (rule_tac i = K and j = L in Ord_linear_le)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   656
apply (typecheck add: InfCard_is_Card Card_is_Ord)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   657
apply (rule cmult_commute [THEN ssubst])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   658
apply (rule Un_commute [THEN ssubst])
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   659
apply (simp_all add: InfCard_is_Limit [THEN Limit_has_0] InfCard_le_cmult_eq 
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   660
                     subset_Un_iff2 [THEN iffD1] le_imp_subset)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   661
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   662
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   663
lemma InfCard_cdouble_eq: "InfCard(K) ==> K |+| K = K"
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   664
apply (simp add: cmult_2 [symmetric] InfCard_is_Card cmult_commute)
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   665
apply (simp add: InfCard_le_cmult_eq InfCard_is_Limit Limit_has_0 Limit_has_succ)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   666
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   667
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   668
(*Corollary 10.13 (1), for cardinal addition*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   669
lemma InfCard_le_cadd_eq: "[| InfCard(K);  L le K |] ==> K |+| L = K"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   670
apply (rule le_anti_sym)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   671
 prefer 2
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   672
 apply (erule ltE, blast intro: cadd_le_self InfCard_is_Card)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   673
apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   674
apply (rule cadd_le_mono [THEN le_trans], assumption+)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   675
apply (simp add: InfCard_cdouble_eq)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   676
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   677
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   678
lemma InfCard_cadd_eq: "[| InfCard(K);  InfCard(L) |] ==> K |+| L = K Un L"
13784
b9f6154427a4 tidying (by script)
paulson
parents: 13615
diff changeset
   679
apply (rule_tac i = K and j = L in Ord_linear_le)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   680
apply (typecheck add: InfCard_is_Card Card_is_Ord)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   681
apply (rule cadd_commute [THEN ssubst])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   682
apply (rule Un_commute [THEN ssubst])
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   683
apply (simp_all add: InfCard_le_cadd_eq subset_Un_iff2 [THEN iffD1] le_imp_subset)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   684
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   685
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   686
(*The other part, Corollary 10.13 (2), refers to the cardinality of the set
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   687
  of all n-tuples of elements of K.  A better version for the Isabelle theory
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   688
  might be  InfCard(K) ==> |list(K)| = K.
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   689
*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   690
13356
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   691
subsection{*For Every Cardinal Number There Exists A Greater One}
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   692
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   693
text{*This result is Kunen's Theorem 10.16, which would be trivial using AC*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   694
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   695
lemma Ord_jump_cardinal: "Ord(jump_cardinal(K))"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   696
apply (unfold jump_cardinal_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   697
apply (rule Ord_is_Transset [THEN [2] OrdI])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   698
 prefer 2 apply (blast intro!: Ord_ordertype)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   699
apply (unfold Transset_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   700
apply (safe del: subsetI)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   701
apply (simp add: ordertype_pred_unfold, safe)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   702
apply (rule UN_I)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   703
apply (rule_tac [2] ReplaceI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   704
   prefer 4 apply (blast intro: well_ord_subset elim!: predE)+
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   705
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   706
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   707
(*Allows selective unfolding.  Less work than deriving intro/elim rules*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   708
lemma jump_cardinal_iff:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   709
     "i : jump_cardinal(K) <->
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   710
      (EX r X. r <= K*K & X <= K & well_ord(X,r) & i = ordertype(X,r))"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   711
apply (unfold jump_cardinal_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   712
apply (blast del: subsetI) 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   713
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   714
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   715
(*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   716
lemma K_lt_jump_cardinal: "Ord(K) ==> K < jump_cardinal(K)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   717
apply (rule Ord_jump_cardinal [THEN [2] ltI])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   718
apply (rule jump_cardinal_iff [THEN iffD2])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   719
apply (rule_tac x="Memrel(K)" in exI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   720
apply (rule_tac x=K in exI)  
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   721
apply (simp add: ordertype_Memrel well_ord_Memrel)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   722
apply (simp add: Memrel_def subset_iff)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   723
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   724
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   725
(*The proof by contradiction: the bijection f yields a wellordering of X
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   726
  whose ordertype is jump_cardinal(K).  *)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   727
lemma Card_jump_cardinal_lemma:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   728
     "[| well_ord(X,r);  r <= K * K;  X <= K;
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   729
         f : bij(ordertype(X,r), jump_cardinal(K)) |]
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   730
      ==> jump_cardinal(K) : jump_cardinal(K)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   731
apply (subgoal_tac "f O ordermap (X,r) : bij (X, jump_cardinal (K))")
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   732
 prefer 2 apply (blast intro: comp_bij ordermap_bij)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   733
apply (rule jump_cardinal_iff [THEN iffD2])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   734
apply (intro exI conjI)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   735
apply (rule subset_trans [OF rvimage_type Sigma_mono], assumption+)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   736
apply (erule bij_is_inj [THEN well_ord_rvimage])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   737
apply (rule Ord_jump_cardinal [THEN well_ord_Memrel])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   738
apply (simp add: well_ord_Memrel [THEN [2] bij_ordertype_vimage]
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   739
                 ordertype_Memrel Ord_jump_cardinal)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   740
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   741
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   742
(*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   743
lemma Card_jump_cardinal: "Card(jump_cardinal(K))"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   744
apply (rule Ord_jump_cardinal [THEN CardI])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   745
apply (unfold eqpoll_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   746
apply (safe dest!: ltD jump_cardinal_iff [THEN iffD1])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   747
apply (blast intro: Card_jump_cardinal_lemma [THEN mem_irrefl])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   748
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   749
13356
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   750
subsection{*Basic Properties of Successor Cardinals*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   751
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   752
lemma csucc_basic: "Ord(K) ==> Card(csucc(K)) & K < csucc(K)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   753
apply (unfold csucc_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   754
apply (rule LeastI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   755
apply (blast intro: Card_jump_cardinal K_lt_jump_cardinal Ord_jump_cardinal)+
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   756
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   757
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   758
lemmas Card_csucc = csucc_basic [THEN conjunct1, standard]
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   759
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   760
lemmas lt_csucc = csucc_basic [THEN conjunct2, standard]
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   761
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   762
lemma Ord_0_lt_csucc: "Ord(K) ==> 0 < csucc(K)"
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   763
by (blast intro: Ord_0_le lt_csucc lt_trans1)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   764
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   765
lemma csucc_le: "[| Card(L);  K<L |] ==> csucc(K) le L"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   766
apply (unfold csucc_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   767
apply (rule Least_le)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   768
apply (blast intro: Card_is_Ord)+
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   769
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   770
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   771
lemma lt_csucc_iff: "[| Ord(i); Card(K) |] ==> i < csucc(K) <-> |i| le K"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   772
apply (rule iffI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   773
apply (rule_tac [2] Card_lt_imp_lt)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   774
apply (erule_tac [2] lt_trans1)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   775
apply (simp_all add: lt_csucc Card_csucc Card_is_Ord)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   776
apply (rule notI [THEN not_lt_imp_le])
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   777
apply (rule Card_cardinal [THEN csucc_le, THEN lt_trans1, THEN lt_irrefl], assumption)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   778
apply (rule Ord_cardinal_le [THEN lt_trans1])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   779
apply (simp_all add: Ord_cardinal Card_is_Ord) 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   780
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   781
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   782
lemma Card_lt_csucc_iff:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   783
     "[| Card(K'); Card(K) |] ==> K' < csucc(K) <-> K' le K"
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   784
by (simp add: lt_csucc_iff Card_cardinal_eq Card_is_Ord)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   785
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   786
lemma InfCard_csucc: "InfCard(K) ==> InfCard(csucc(K))"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   787
by (simp add: InfCard_def Card_csucc Card_is_Ord 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   788
              lt_csucc [THEN leI, THEN [2] le_trans])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   789
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   790
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   791
subsubsection{*Removing elements from a finite set decreases its cardinality*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   792
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   793
lemma Fin_imp_not_cons_lepoll: "A: Fin(U) ==> x~:A --> ~ cons(x,A) \<lesssim> A"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   794
apply (erule Fin_induct)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   795
apply (simp add: lepoll_0_iff)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   796
apply (subgoal_tac "cons (x,cons (xa,y)) = cons (xa,cons (x,y))")
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   797
apply simp
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   798
apply (blast dest!: cons_lepoll_consD, blast)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   799
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   800
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   801
lemma Finite_imp_cardinal_cons [simp]:
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   802
     "[| Finite(A);  a~:A |] ==> |cons(a,A)| = succ(|A|)"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   803
apply (unfold cardinal_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   804
apply (rule Least_equality)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   805
apply (fold cardinal_def)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   806
apply (simp add: succ_def)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   807
apply (blast intro: cons_eqpoll_cong well_ord_cardinal_eqpoll
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   808
             elim!: mem_irrefl  dest!: Finite_imp_well_ord)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   809
apply (blast intro: Card_cardinal Card_is_Ord)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   810
apply (rule notI)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   811
apply (rule Finite_into_Fin [THEN Fin_imp_not_cons_lepoll, THEN mp, THEN notE],
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   812
       assumption, assumption)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   813
apply (erule eqpoll_sym [THEN eqpoll_imp_lepoll, THEN lepoll_trans])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   814
apply (erule le_imp_lepoll [THEN lepoll_trans])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   815
apply (blast intro: well_ord_cardinal_eqpoll [THEN eqpoll_imp_lepoll]
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   816
             dest!: Finite_imp_well_ord)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   817
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   818
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   819
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   820
lemma Finite_imp_succ_cardinal_Diff:
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   821
     "[| Finite(A);  a:A |] ==> succ(|A-{a}|) = |A|"
13784
b9f6154427a4 tidying (by script)
paulson
parents: 13615
diff changeset
   822
apply (rule_tac b = A in cons_Diff [THEN subst], assumption)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   823
apply (simp add: Finite_imp_cardinal_cons Diff_subset [THEN subset_Finite])
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   824
apply (simp add: cons_Diff)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   825
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   826
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   827
lemma Finite_imp_cardinal_Diff: "[| Finite(A);  a:A |] ==> |A-{a}| < |A|"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   828
apply (rule succ_leE)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   829
apply (simp add: Finite_imp_succ_cardinal_Diff)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   830
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   831
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   832
lemma Finite_cardinal_in_nat [simp]: "Finite(A) ==> |A| : nat"
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   833
apply (erule Finite_induct)
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   834
apply (auto simp add: cardinal_0 Finite_imp_cardinal_cons)
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   835
done
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   836
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   837
lemma card_Un_Int:
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   838
     "[|Finite(A); Finite(B)|] ==> |A| #+ |B| = |A Un B| #+ |A Int B|"
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   839
apply (erule Finite_induct, simp) 
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   840
apply (simp add: Finite_Int cons_absorb Un_cons Int_cons_left)
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   841
done
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   842
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   843
lemma card_Un_disjoint: 
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   844
     "[|Finite(A); Finite(B); A Int B = 0|] ==> |A Un B| = |A| #+ |B|" 
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   845
by (simp add: Finite_Un card_Un_Int)
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   846
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   847
lemma card_partition [rule_format]:
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   848
     "Finite(C) ==>  
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   849
        Finite (\<Union> C) -->  
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   850
        (\<forall>c\<in>C. |c| = k) -->   
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   851
        (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = 0) -->  
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   852
        k #* |C| = |\<Union> C|"
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   853
apply (erule Finite_induct, auto)
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   854
apply (subgoal_tac " x \<inter> \<Union>B = 0")  
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   855
apply (auto simp add: card_Un_disjoint Finite_Union
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   856
       subset_Finite [of _ "\<Union> (cons(x,F))"])
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   857
done
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   858
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   859
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   860
subsubsection{*Theorems by Krzysztof Grabczewski, proofs by lcp*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   861
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   862
lemmas nat_implies_well_ord = nat_into_Ord [THEN well_ord_Memrel, standard]
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   863
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   864
lemma nat_sum_eqpoll_sum: "[| m:nat; n:nat |] ==> m + n \<approx> m #+ n"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   865
apply (rule eqpoll_trans)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   866
apply (rule well_ord_radd [THEN well_ord_cardinal_eqpoll, THEN eqpoll_sym])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   867
apply (erule nat_implies_well_ord)+
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   868
apply (simp add: nat_cadd_eq_add [symmetric] cadd_def eqpoll_refl)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   869
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   870
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   871
lemma Ord_subset_natD [rule_format]: "Ord(i) ==> i <= nat --> i : nat | i=nat"
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   872
apply (erule trans_induct3, auto)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   873
apply (blast dest!: nat_le_Limit [THEN le_imp_subset])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   874
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   875
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   876
lemma Ord_nat_subset_into_Card: "[| Ord(i); i <= nat |] ==> Card(i)"
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   877
by (blast dest: Ord_subset_natD intro: Card_nat nat_into_Card)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   878
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   879
lemma Finite_Diff_sing_eq_diff_1: "[| Finite(A); x:A |] ==> |A-{x}| = |A| #- 1"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   880
apply (rule succ_inject)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   881
apply (rule_tac b = "|A|" in trans)
13615
449a70d88b38 Numerous cosmetic changes, prompted by the new simplifier
paulson
parents: 13356
diff changeset
   882
 apply (simp add: Finite_imp_succ_cardinal_Diff)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   883
apply (subgoal_tac "1 \<lesssim> A")
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   884
 prefer 2 apply (blast intro: not_0_is_lepoll_1)
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   885
apply (frule Finite_imp_well_ord, clarify)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   886
apply (drule well_ord_lepoll_imp_Card_le)
13615
449a70d88b38 Numerous cosmetic changes, prompted by the new simplifier
paulson
parents: 13356
diff changeset
   887
 apply (auto simp add: cardinal_1)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   888
apply (rule trans)
13615
449a70d88b38 Numerous cosmetic changes, prompted by the new simplifier
paulson
parents: 13356
diff changeset
   889
 apply (rule_tac [2] diff_succ)
449a70d88b38 Numerous cosmetic changes, prompted by the new simplifier
paulson
parents: 13356
diff changeset
   890
  apply (auto simp add: Finite_cardinal_in_nat)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   891
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   892
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   893
lemma cardinal_lt_imp_Diff_not_0 [rule_format]:
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   894
     "Finite(B) ==> ALL A. |B|<|A| --> A - B ~= 0"
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   895
apply (erule Finite_induct, auto)
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   896
apply (case_tac "Finite (A)")
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   897
 apply (subgoal_tac [2] "Finite (cons (x, B))")
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   898
  apply (drule_tac [2] B = "cons (x, B) " in Diff_Finite)
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   899
   apply (auto simp add: Finite_0 Finite_cons)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   900
apply (subgoal_tac "|B|<|A|")
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   901
 prefer 2 apply (blast intro: lt_trans Ord_cardinal)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   902
apply (case_tac "x:A")
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   903
 apply (subgoal_tac [2] "A - cons (x, B) = A - B")
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   904
  apply auto
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   905
apply (subgoal_tac "|A| le |cons (x, B) |")
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   906
 prefer 2
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   907
 apply (blast dest: Finite_cons [THEN Finite_imp_well_ord] 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   908
              intro: well_ord_lepoll_imp_Card_le subset_imp_lepoll)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   909
apply (auto simp add: Finite_imp_cardinal_cons)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   910
apply (auto dest!: Finite_cardinal_in_nat simp add: le_iff)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   911
apply (blast intro: lt_trans)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   912
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   913
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   914
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   915
ML{*
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   916
val InfCard_def = thm "InfCard_def"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   917
val cmult_def = thm "cmult_def"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   918
val cadd_def = thm "cadd_def"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   919
val jump_cardinal_def = thm "jump_cardinal_def"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   920
val csucc_def = thm "csucc_def"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   921
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   922
val sum_commute_eqpoll = thm "sum_commute_eqpoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   923
val cadd_commute = thm "cadd_commute";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   924
val sum_assoc_eqpoll = thm "sum_assoc_eqpoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   925
val well_ord_cadd_assoc = thm "well_ord_cadd_assoc";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   926
val sum_0_eqpoll = thm "sum_0_eqpoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   927
val cadd_0 = thm "cadd_0";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   928
val sum_lepoll_self = thm "sum_lepoll_self";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   929
val cadd_le_self = thm "cadd_le_self";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   930
val sum_lepoll_mono = thm "sum_lepoll_mono";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   931
val cadd_le_mono = thm "cadd_le_mono";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   932
val eq_imp_not_mem = thm "eq_imp_not_mem";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   933
val sum_succ_eqpoll = thm "sum_succ_eqpoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   934
val nat_cadd_eq_add = thm "nat_cadd_eq_add";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   935
val prod_commute_eqpoll = thm "prod_commute_eqpoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   936
val cmult_commute = thm "cmult_commute";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   937
val prod_assoc_eqpoll = thm "prod_assoc_eqpoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   938
val well_ord_cmult_assoc = thm "well_ord_cmult_assoc";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   939
val sum_prod_distrib_eqpoll = thm "sum_prod_distrib_eqpoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   940
val well_ord_cadd_cmult_distrib = thm "well_ord_cadd_cmult_distrib";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   941
val prod_0_eqpoll = thm "prod_0_eqpoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   942
val cmult_0 = thm "cmult_0";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   943
val prod_singleton_eqpoll = thm "prod_singleton_eqpoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   944
val cmult_1 = thm "cmult_1";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   945
val prod_lepoll_self = thm "prod_lepoll_self";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   946
val cmult_le_self = thm "cmult_le_self";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   947
val prod_lepoll_mono = thm "prod_lepoll_mono";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   948
val cmult_le_mono = thm "cmult_le_mono";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   949
val prod_succ_eqpoll = thm "prod_succ_eqpoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   950
val nat_cmult_eq_mult = thm "nat_cmult_eq_mult";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   951
val cmult_2 = thm "cmult_2";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   952
val sum_lepoll_prod = thm "sum_lepoll_prod";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   953
val lepoll_imp_sum_lepoll_prod = thm "lepoll_imp_sum_lepoll_prod";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   954
val nat_cons_lepoll = thm "nat_cons_lepoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   955
val nat_cons_eqpoll = thm "nat_cons_eqpoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   956
val nat_succ_eqpoll = thm "nat_succ_eqpoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   957
val InfCard_nat = thm "InfCard_nat";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   958
val InfCard_is_Card = thm "InfCard_is_Card";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   959
val InfCard_Un = thm "InfCard_Un";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   960
val InfCard_is_Limit = thm "InfCard_is_Limit";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   961
val ordermap_eqpoll_pred = thm "ordermap_eqpoll_pred";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   962
val ordermap_z_lt = thm "ordermap_z_lt";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   963
val InfCard_le_cmult_eq = thm "InfCard_le_cmult_eq";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   964
val InfCard_cmult_eq = thm "InfCard_cmult_eq";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   965
val InfCard_cdouble_eq = thm "InfCard_cdouble_eq";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   966
val InfCard_le_cadd_eq = thm "InfCard_le_cadd_eq";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   967
val InfCard_cadd_eq = thm "InfCard_cadd_eq";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   968
val Ord_jump_cardinal = thm "Ord_jump_cardinal";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   969
val jump_cardinal_iff = thm "jump_cardinal_iff";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   970
val K_lt_jump_cardinal = thm "K_lt_jump_cardinal";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   971
val Card_jump_cardinal = thm "Card_jump_cardinal";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   972
val csucc_basic = thm "csucc_basic";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   973
val Card_csucc = thm "Card_csucc";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   974
val lt_csucc = thm "lt_csucc";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   975
val Ord_0_lt_csucc = thm "Ord_0_lt_csucc";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   976
val csucc_le = thm "csucc_le";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   977
val lt_csucc_iff = thm "lt_csucc_iff";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   978
val Card_lt_csucc_iff = thm "Card_lt_csucc_iff";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   979
val InfCard_csucc = thm "InfCard_csucc";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   980
val Finite_into_Fin = thm "Finite_into_Fin";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   981
val Fin_into_Finite = thm "Fin_into_Finite";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   982
val Finite_Fin_iff = thm "Finite_Fin_iff";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   983
val Finite_Un = thm "Finite_Un";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   984
val Finite_Union = thm "Finite_Union";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   985
val Finite_induct = thm "Finite_induct";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   986
val Fin_imp_not_cons_lepoll = thm "Fin_imp_not_cons_lepoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   987
val Finite_imp_cardinal_cons = thm "Finite_imp_cardinal_cons";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   988
val Finite_imp_succ_cardinal_Diff = thm "Finite_imp_succ_cardinal_Diff";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   989
val Finite_imp_cardinal_Diff = thm "Finite_imp_cardinal_Diff";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   990
val nat_implies_well_ord = thm "nat_implies_well_ord";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   991
val nat_sum_eqpoll_sum = thm "nat_sum_eqpoll_sum";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   992
val Diff_sing_Finite = thm "Diff_sing_Finite";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   993
val Diff_Finite = thm "Diff_Finite";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   994
val Ord_subset_natD = thm "Ord_subset_natD";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   995
val Ord_nat_subset_into_Card = thm "Ord_nat_subset_into_Card";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   996
val Finite_cardinal_in_nat = thm "Finite_cardinal_in_nat";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   997
val Finite_Diff_sing_eq_diff_1 = thm "Finite_Diff_sing_eq_diff_1";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   998
val cardinal_lt_imp_Diff_not_0 = thm "cardinal_lt_imp_Diff_not_0";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   999
*}
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
  1000
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
  1001
end