src/ZF/CardinalArith.thy
author paulson
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(*  Title:      ZF/CardinalArith.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1994  University of Cambridge
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*)
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header{*Cardinal Arithmetic Without the Axiom of Choice*}
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theory CardinalArith imports Cardinal OrderArith ArithSimp Finite begin
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definition
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  InfCard       :: "i=>o"  where
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    "InfCard(i) == Card(i) & nat \<le> i"
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definition
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  cmult         :: "[i,i]=>i"       (infixl "|*|" 70)  where
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    "i |*| j == |i*j|"
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definition
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  cadd          :: "[i,i]=>i"       (infixl "|+|" 65)  where
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    "i |+| j == |i+j|"
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definition
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  csquare_rel   :: "i=>i"  where
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    "csquare_rel(K) ==
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          rvimage(K*K,
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                  lam <x,y>:K*K. <x \<union> y, x, y>,
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                  rmult(K,Memrel(K), K*K, rmult(K,Memrel(K), K,Memrel(K))))"
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definition
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  jump_cardinal :: "i=>i"  where
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    --{*This def is more complex than Kunen's but it more easily proved to
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        be a cardinal*}
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    "jump_cardinal(K) ==
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         \<Union>X\<in>Pow(K). {z. r \<in> Pow(K*K), well_ord(X,r) & z = ordertype(X,r)}"
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definition
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  csucc         :: "i=>i"  where
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    --{*needed because @{term "jump_cardinal(K)"} might not be the successor
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        of @{term K}*}
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    "csucc(K) == LEAST L. Card(L) & K<L"
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notation (xsymbols)
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  cadd  (infixl "\<oplus>" 65) and
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  cmult  (infixl "\<otimes>" 70)
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notation (HTML)
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  cadd  (infixl "\<oplus>" 65) and
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  cmult  (infixl "\<otimes>" 70)
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lemma Card_Union [simp,intro,TC]:
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  assumes A: "\<And>x. x\<in>A \<Longrightarrow> Card(x)" shows "Card(\<Union>(A))"
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proof (rule CardI)
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  show "Ord(\<Union>A)" using A
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    by (simp add: Card_is_Ord)
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next
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  fix j
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  assume j: "j < \<Union>A"
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  hence "\<exists>c\<in>A. j < c & Card(c)" using A
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    by (auto simp add: lt_def intro: Card_is_Ord)
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  then obtain c where c: "c\<in>A" "j < c" "Card(c)"
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    by blast
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  hence jls: "j \<prec> c"
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    by (simp add: lt_Card_imp_lesspoll)
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  { assume eqp: "j \<approx> \<Union>A"
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    have  "c \<lesssim> \<Union>A" using c
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      by (blast intro: subset_imp_lepoll)
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    also have "... \<approx> j"  by (rule eqpoll_sym [OF eqp])
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    also have "... \<prec> c"  by (rule jls)
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    finally have "c \<prec> c" .
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    hence False
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      by auto
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  } thus "\<not> j \<approx> \<Union>A" by blast
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qed
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lemma Card_UN: "(!!x. x \<in> A ==> Card(K(x))) ==> Card(\<Union>x\<in>A. K(x))"
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  by blast
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lemma Card_OUN [simp,intro,TC]:
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     "(!!x. x \<in> A ==> Card(K(x))) ==> Card(\<Union>x<A. K(x))"
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  by (auto simp add: OUnion_def Card_0)
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lemma in_Card_imp_lesspoll: "[| Card(K); b \<in> K |] ==> b \<prec> K"
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apply (unfold lesspoll_def)
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apply (simp add: Card_iff_initial)
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apply (fast intro!: le_imp_lepoll ltI leI)
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done
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subsection{*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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proof (unfold eqpoll_def, rule exI)
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  show "(\<lambda>z\<in>A+B. case(Inr,Inl,z)) \<in> bij(A+B, B+A)"
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    by (auto intro: lam_bijective [where d = "case(Inr,Inl)"])
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qed
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lemma cadd_commute: "i \<oplus> j = j \<oplus> i"
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apply (unfold cadd_def)
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apply (rule sum_commute_eqpoll [THEN cardinal_cong])
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done
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subsubsection{*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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text{*Unconditional version requires AC*}
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lemma well_ord_cadd_assoc:
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  assumes i: "well_ord(i,ri)" and j: "well_ord(j,rj)" and k: "well_ord(k,rk)"
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  shows "(i \<oplus> j) \<oplus> k = i \<oplus> (j \<oplus> k)"
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proof (unfold cadd_def, rule cardinal_cong)
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  have "|i + j| + k \<approx> (i + j) + k"
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    by (blast intro: sum_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_refl well_ord_radd i j)
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  also have "...  \<approx> i + (j + k)"
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    by (rule sum_assoc_eqpoll)
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  also have "...  \<approx> i + |j + k|"
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    by (blast intro: sum_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_refl well_ord_radd j k eqpoll_sym)
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  finally show "|i + j| + k \<approx> i + |j + k|" .
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qed
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subsubsection{*0 is the identity for addition*}
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lemma sum_0_eqpoll: "0+A \<approx> A"
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apply (unfold eqpoll_def)
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apply (rule exI)
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apply (rule bij_0_sum)
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done
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lemma cadd_0 [simp]: "Card(K) ==> 0 \<oplus> K = K"
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apply (unfold cadd_def)
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apply (simp add: sum_0_eqpoll [THEN cardinal_cong] Card_cardinal_eq)
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done
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subsubsection{*Addition by another cardinal*}
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lemma sum_lepoll_self: "A \<lesssim> A+B"
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proof (unfold lepoll_def, rule exI)
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  show "(\<lambda>x\<in>A. Inl (x)) \<in> inj(A, A + B)"
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    by (simp add: inj_def)
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qed
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(*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
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   156
lemma cadd_le_self:
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   157
  assumes K: "Card(K)" and L: "Ord(L)" shows "K \<le> (K \<oplus> L)"
5e1bcfdcb175 Rewrote some induction proofs to be structured
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   158
proof (unfold cadd_def)
5e1bcfdcb175 Rewrote some induction proofs to be structured
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   159
  have "K \<le> |K|"
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   160
    by (rule Card_cardinal_le [OF K])
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5e1bcfdcb175 Rewrote some induction proofs to be structured
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   161
  moreover have "|K| \<le> |K + L|" using K L
5e1bcfdcb175 Rewrote some induction proofs to be structured
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   162
    by (blast intro: well_ord_lepoll_imp_Card_le sum_lepoll_self
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   163
                     well_ord_radd well_ord_Memrel Card_is_Ord)
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   164
  ultimately show "K \<le> |K + L|"
2b6e55924af3 replacing ":" by "\<in>"
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   165
    by (blast intro: le_trans)
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5e1bcfdcb175 Rewrote some induction proofs to be structured
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   166
qed
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   167
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   168
subsubsection{*Monotonicity of addition*}
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   169
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   170
lemma sum_lepoll_mono:
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e29378f347e4 conversion of Cardinal, CardinalArith
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   171
     "[| A \<lesssim> C;  B \<lesssim> D |] ==> A + B \<lesssim> C + D"
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   172
apply (unfold lepoll_def)
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e29378f347e4 conversion of Cardinal, CardinalArith
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diff changeset
   173
apply (elim exE)
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diff changeset
   174
apply (rule_tac x = "\<lambda>z\<in>A+B. case (%w. Inl(f`w), %y. Inr(fa`y), z)" in exI)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
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parents: 13216
diff changeset
   175
apply (rule_tac d = "case (%w. Inl(converse(f) `w), %y. Inr(converse(fa) ` y))"
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   176
       in lam_injective)
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   177
apply (typecheck add: inj_is_fun, auto)
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diff changeset
   178
done
6104bd4088a2 conversion of CardinalArith to Isar script
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diff changeset
   179
6104bd4088a2 conversion of CardinalArith to Isar script
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   180
lemma cadd_le_mono:
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   181
    "[| K' \<le> K;  L' \<le> L |] ==> (K' \<oplus> L') \<le> (K \<oplus> L)"
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   182
apply (unfold cadd_def)
6104bd4088a2 conversion of CardinalArith to Isar script
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parents: 13161
diff changeset
   183
apply (safe dest!: le_subset_iff [THEN iffD1])
6104bd4088a2 conversion of CardinalArith to Isar script
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parents: 13161
diff changeset
   184
apply (rule well_ord_lepoll_imp_Card_le)
6104bd4088a2 conversion of CardinalArith to Isar script
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parents: 13161
diff changeset
   185
apply (blast intro: well_ord_radd well_ord_Memrel)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   186
apply (blast intro: sum_lepoll_mono subset_imp_lepoll)
6104bd4088a2 conversion of CardinalArith to Isar script
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parents: 13161
diff changeset
   187
done
6104bd4088a2 conversion of CardinalArith to Isar script
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parents: 13161
diff changeset
   188
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   189
subsubsection{*Addition of finite cardinals is "ordinary" addition*}
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   190
6104bd4088a2 conversion of CardinalArith to Isar script
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   191
lemma sum_succ_eqpoll: "succ(A)+B \<approx> succ(A+B)"
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   192
apply (unfold eqpoll_def)
6104bd4088a2 conversion of CardinalArith to Isar script
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parents: 13161
diff changeset
   193
apply (rule exI)
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parents: 45602
diff changeset
   194
apply (rule_tac c = "%z. if z=Inl (A) then A+B else z"
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parents: 13161
diff changeset
   195
            and d = "%z. if z=A+B then Inl (A) else z" in lam_bijective)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
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diff changeset
   196
   apply simp_all
13216
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parents: 13161
diff changeset
   197
apply (blast dest: sym [THEN eq_imp_not_mem] elim: mem_irrefl)+
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
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diff changeset
   198
done
6104bd4088a2 conversion of CardinalArith to Isar script
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diff changeset
   199
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diff changeset
   200
(*Pulling the  succ(...)  outside the |...| requires m, n \<in> nat  *)
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   201
(*Unconditional version requires AC*)
6104bd4088a2 conversion of CardinalArith to Isar script
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diff changeset
   202
lemma cadd_succ_lemma:
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5e1bcfdcb175 Rewrote some induction proofs to be structured
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   203
  assumes "Ord(m)" "Ord(n)" shows "succ(m) \<oplus> n = |succ(m \<oplus> n)|"
5e1bcfdcb175 Rewrote some induction proofs to be structured
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diff changeset
   204
proof (unfold cadd_def)
5e1bcfdcb175 Rewrote some induction proofs to be structured
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diff changeset
   205
  have [intro]: "m + n \<approx> |m + n|" using assms
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
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diff changeset
   206
    by (blast intro: eqpoll_sym well_ord_cardinal_eqpoll well_ord_radd well_ord_Memrel)
13216
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paulson
parents: 13161
diff changeset
   207
46952
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
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diff changeset
   208
  have "|succ(m) + n| = |succ(m + n)|"
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   209
    by (rule sum_succ_eqpoll [THEN cardinal_cong])
2b6e55924af3 replacing ":" by "\<in>"
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parents: 46952
diff changeset
   210
  also have "... = |succ(|m + n|)|"
46952
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   211
    by (blast intro: succ_eqpoll_cong cardinal_cong)
5e1bcfdcb175 Rewrote some induction proofs to be structured
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parents: 46935
diff changeset
   212
  finally show "|succ(m) + n| = |succ(|m + n|)|" .
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   213
qed
5e1bcfdcb175 Rewrote some induction proofs to be structured
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parents: 46935
diff changeset
   214
5e1bcfdcb175 Rewrote some induction proofs to be structured
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parents: 46935
diff changeset
   215
lemma nat_cadd_eq_add:
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2b6e55924af3 replacing ":" by "\<in>"
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diff changeset
   216
  assumes m: "m \<in> nat" and [simp]: "n \<in> nat" shows"m \<oplus> n = m #+ n"
46952
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   217
using m
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
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diff changeset
   218
proof (induct m)
5e1bcfdcb175 Rewrote some induction proofs to be structured
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diff changeset
   219
  case 0 thus ?case by (simp add: nat_into_Card cadd_0)
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
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diff changeset
   220
next
5e1bcfdcb175 Rewrote some induction proofs to be structured
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parents: 46935
diff changeset
   221
  case (succ m) thus ?case by (simp add: cadd_succ_lemma nat_into_Card Card_cardinal_eq)
5e1bcfdcb175 Rewrote some induction proofs to be structured
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diff changeset
   222
qed
13216
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parents: 13161
diff changeset
   223
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
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diff changeset
   224
13356
c9cfe1638bf2 improved presentation markup
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diff changeset
   225
subsection{*Cardinal multiplication*}
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   226
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   227
subsubsection{*Cardinal multiplication is commutative*}
13216
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diff changeset
   228
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
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diff changeset
   229
lemma prod_commute_eqpoll: "A*B \<approx> B*A"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
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diff changeset
   230
apply (unfold eqpoll_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   231
apply (rule exI)
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parents: 45602
diff changeset
   232
apply (rule_tac c = "%<x,y>.<y,x>" and d = "%<x,y>.<y,x>" in lam_bijective,
c656222c4dc1 mathematical symbols instead of ASCII
paulson
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   233
       auto)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
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diff changeset
   234
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   235
46821
ff6b0c1087f2 Using mathematical notation for <-> and cardinal arithmetic
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   236
lemma cmult_commute: "i \<otimes> j = j \<otimes> i"
13216
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diff changeset
   237
apply (unfold cmult_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   238
apply (rule prod_commute_eqpoll [THEN cardinal_cong])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   239
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   240
14883
ca000a495448 Groups, Rings and supporting lemmas
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parents: 14565
diff changeset
   241
subsubsection{*Cardinal multiplication is associative*}
13216
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paulson
parents: 13161
diff changeset
   242
6104bd4088a2 conversion of CardinalArith to Isar script
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diff changeset
   243
lemma prod_assoc_eqpoll: "(A*B)*C \<approx> A*(B*C)"
6104bd4088a2 conversion of CardinalArith to Isar script
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diff changeset
   244
apply (unfold eqpoll_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   245
apply (rule exI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   246
apply (rule prod_assoc_bij)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   247
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   248
46901
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
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   249
text{*Unconditional version requires AC*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
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diff changeset
   250
lemma well_ord_cmult_assoc:
46901
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diff changeset
   251
  assumes i: "well_ord(i,ri)" and j: "well_ord(j,rj)" and k: "well_ord(k,rk)"
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   252
  shows "(i \<otimes> j) \<otimes> k = i \<otimes> (j \<otimes> k)"
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   253
proof (unfold cmult_def, rule cardinal_cong)
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   254
  have "|i * j| * k \<approx> (i * j) * k"
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   255
    by (blast intro: prod_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_refl well_ord_rmult i j)
46901
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   256
  also have "...  \<approx> i * (j * k)"
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   257
    by (rule prod_assoc_eqpoll)
46901
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   258
  also have "...  \<approx> i * |j * k|"
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   259
    by (blast intro: prod_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_refl well_ord_rmult j k eqpoll_sym)
46901
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   260
  finally show "|i * j| * k \<approx> i * |j * k|" .
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   261
qed
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   262
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   263
subsubsection{*Cardinal multiplication distributes over addition*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   264
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   265
lemma sum_prod_distrib_eqpoll: "(A+B)*C \<approx> (A*C)+(B*C)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   266
apply (unfold eqpoll_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   267
apply (rule exI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   268
apply (rule sum_prod_distrib_bij)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   269
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   270
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   271
lemma well_ord_cadd_cmult_distrib:
46901
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   272
  assumes i: "well_ord(i,ri)" and j: "well_ord(j,rj)" and k: "well_ord(k,rk)"
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   273
  shows "(i \<oplus> j) \<otimes> k = (i \<otimes> k) \<oplus> (j \<otimes> k)"
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   274
proof (unfold cadd_def cmult_def, rule cardinal_cong)
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   275
  have "|i + j| * k \<approx> (i + j) * k"
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   276
    by (blast intro: prod_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_refl well_ord_radd i j)
46901
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   277
  also have "...  \<approx> i * k + j * k"
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   278
    by (rule sum_prod_distrib_eqpoll)
46901
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   279
  also have "...  \<approx> |i * k| + |j * k|"
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   280
    by (blast intro: sum_eqpoll_cong well_ord_cardinal_eqpoll well_ord_rmult i j k eqpoll_sym)
46901
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   281
  finally show "|i + j| * k \<approx> |i * k| + |j * k|" .
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   282
qed
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   283
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   284
subsubsection{*Multiplication by 0 yields 0*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   285
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   286
lemma prod_0_eqpoll: "0*A \<approx> 0"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   287
apply (unfold eqpoll_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   288
apply (rule exI)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   289
apply (rule lam_bijective, safe)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   290
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   291
46821
ff6b0c1087f2 Using mathematical notation for <-> and cardinal arithmetic
paulson
parents: 46820
diff changeset
   292
lemma cmult_0 [simp]: "0 \<otimes> i = 0"
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   293
by (simp add: cmult_def prod_0_eqpoll [THEN cardinal_cong])
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   294
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   295
subsubsection{*1 is the identity for multiplication*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   296
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   297
lemma prod_singleton_eqpoll: "{x}*A \<approx> A"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   298
apply (unfold eqpoll_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   299
apply (rule exI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   300
apply (rule singleton_prod_bij [THEN bij_converse_bij])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   301
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   302
46821
ff6b0c1087f2 Using mathematical notation for <-> and cardinal arithmetic
paulson
parents: 46820
diff changeset
   303
lemma cmult_1 [simp]: "Card(K) ==> 1 \<otimes> K = K"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   304
apply (unfold cmult_def succ_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   305
apply (simp add: prod_singleton_eqpoll [THEN cardinal_cong] Card_cardinal_eq)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   306
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   307
13356
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   308
subsection{*Some inequalities for multiplication*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   309
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   310
lemma prod_square_lepoll: "A \<lesssim> A*A"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   311
apply (unfold lepoll_def inj_def)
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   312
apply (rule_tac x = "\<lambda>x\<in>A. <x,x>" in exI, simp)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   313
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   314
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   315
(*Could probably weaken the premise to well_ord(K,r), or remove using AC*)
46821
ff6b0c1087f2 Using mathematical notation for <-> and cardinal arithmetic
paulson
parents: 46820
diff changeset
   316
lemma cmult_square_le: "Card(K) ==> K \<le> K \<otimes> K"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   317
apply (unfold cmult_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   318
apply (rule le_trans)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   319
apply (rule_tac [2] well_ord_lepoll_imp_Card_le)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   320
apply (rule_tac [3] prod_square_lepoll)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   321
apply (simp add: le_refl Card_is_Ord Card_cardinal_eq)
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   322
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
   323
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   324
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   325
subsubsection{*Multiplication by a non-zero cardinal*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   326
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   327
lemma prod_lepoll_self: "b \<in> B ==> A \<lesssim> A*B"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   328
apply (unfold lepoll_def inj_def)
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   329
apply (rule_tac x = "\<lambda>x\<in>A. <x,b>" in exI, simp)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   330
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   331
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   332
(*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
   333
lemma cmult_le_self:
46821
ff6b0c1087f2 Using mathematical notation for <-> and cardinal arithmetic
paulson
parents: 46820
diff changeset
   334
    "[| Card(K);  Ord(L);  0<L |] ==> K \<le> (K \<otimes> L)"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   335
apply (unfold cmult_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   336
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
   337
  apply assumption
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   338
 apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   339
apply (blast intro: prod_lepoll_self ltD)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   340
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   341
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   342
subsubsection{*Monotonicity of multiplication*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   343
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   344
lemma prod_lepoll_mono:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   345
     "[| A \<lesssim> C;  B \<lesssim> D |] ==> A * B  \<lesssim>  C * D"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   346
apply (unfold lepoll_def)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   347
apply (elim exE)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   348
apply (rule_tac x = "lam <w,y>:A*B. <f`w, fa`y>" in exI)
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   349
apply (rule_tac d = "%<w,y>. <converse (f) `w, converse (fa) `y>"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   350
       in lam_injective)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   351
apply (typecheck add: inj_is_fun, auto)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   352
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   353
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   354
lemma cmult_le_mono:
46821
ff6b0c1087f2 Using mathematical notation for <-> and cardinal arithmetic
paulson
parents: 46820
diff changeset
   355
    "[| K' \<le> K;  L' \<le> L |] ==> (K' \<otimes> L') \<le> (K \<otimes> L)"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   356
apply (unfold cmult_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   357
apply (safe dest!: le_subset_iff [THEN iffD1])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   358
apply (rule well_ord_lepoll_imp_Card_le)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   359
 apply (blast intro: well_ord_rmult well_ord_Memrel)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   360
apply (blast intro: prod_lepoll_mono subset_imp_lepoll)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   361
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   362
13356
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   363
subsection{*Multiplication of finite cardinals is "ordinary" multiplication*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   364
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   365
lemma prod_succ_eqpoll: "succ(A)*B \<approx> B + A*B"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   366
apply (unfold eqpoll_def)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   367
apply (rule exI)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   368
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
   369
            and d = "case (%y. <A,y>, %z. z)" in lam_bijective)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   370
apply safe
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   371
apply (simp_all add: succI2 if_type mem_imp_not_eq)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   372
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   373
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   374
(*Unconditional version requires AC*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   375
lemma cmult_succ_lemma:
46821
ff6b0c1087f2 Using mathematical notation for <-> and cardinal arithmetic
paulson
parents: 46820
diff changeset
   376
    "[| Ord(m);  Ord(n) |] ==> succ(m) \<otimes> n = n \<oplus> (m \<otimes> n)"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   377
apply (unfold cmult_def cadd_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   378
apply (rule prod_succ_eqpoll [THEN cardinal_cong, THEN trans])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   379
apply (rule cardinal_cong [symmetric])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   380
apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   381
apply (blast intro: well_ord_rmult well_ord_Memrel)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   382
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   383
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   384
lemma nat_cmult_eq_mult: "[| m \<in> nat;  n \<in> nat |] ==> m \<otimes> n = m#*n"
13244
7b37e218f298 moving some results around
paulson
parents: 13221
diff changeset
   385
apply (induct_tac m)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   386
apply (simp_all add: cmult_succ_lemma nat_cadd_eq_add)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   387
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   388
46821
ff6b0c1087f2 Using mathematical notation for <-> and cardinal arithmetic
paulson
parents: 46820
diff changeset
   389
lemma cmult_2: "Card(n) ==> 2 \<otimes> n = n \<oplus> n"
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   390
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
   391
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   392
lemma sum_lepoll_prod:
46901
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   393
  assumes C: "2 \<lesssim> C" shows "B+B \<lesssim> C*B"
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   394
proof -
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   395
  have "B+B \<lesssim> 2*B"
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   396
    by (simp add: sum_eq_2_times)
46901
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   397
  also have "... \<lesssim> C*B"
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   398
    by (blast intro: prod_lepoll_mono lepoll_refl C)
46901
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   399
  finally show "B+B \<lesssim> C*B" .
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   400
qed
13216
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
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   409
  \<lambda>z\<in>cons(u,A). if z=u then 0 else if z \<in> nat then succ(z) else z
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   410
  and inverse %y. if y \<in> nat then nat_case(u, %z. z, y) else y.  \
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   411
  If f \<in> inj(nat,A) then range(f) behaves like the natural numbers.*)
13216
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)
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   415
apply (rule_tac x =
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   416
          "\<lambda>z\<in>cons (u,A).
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   417
             if z=u then f`0
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   418
             else if z \<in> range (f) then f`succ (converse (f) `z) else z"
13216
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 =
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   421
          "%y. if y \<in> range(f) then nat_case (u, %z. f`z, converse(f) `y)
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   422
                              else y"
13216
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*)
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   436
lemma nat_succ_eqpoll: "nat \<subseteq> A ==> succ(A) \<approx> A"
13216
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:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   452
    "[| InfCard(K);  Card(L) |] ==> InfCard(K \<union> L)"
13216
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:
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   478
    "[| well_ord(A,r);  x \<in> 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)
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   482
apply (erule ordermap_bij [THEN bij_is_inj, THEN restrict_bij,
13221
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
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   489
lemma well_ord_csquare:
46901
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   490
  assumes K: "Ord(K)" shows "well_ord(K*K, csquare_rel(K))"
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   491
proof (unfold csquare_rel_def, rule well_ord_rvimage)
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   492
  show "(\<lambda>\<langle>x,y\<rangle>\<in>K \<times> K. \<langle>x \<union> y, x, y\<rangle>) \<in> inj(K \<times> K, K \<times> K \<times> K)" using K
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   493
    by (force simp add: inj_def intro: lam_type Un_least_lt [THEN ltD] ltI)
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   494
next
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   495
  show "well_ord(K \<times> K \<times> K, rmult(K, Memrel(K), K \<times> K, rmult(K, Memrel(K), K, Memrel(K))))"
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   496
    using K by (blast intro: well_ord_rmult well_ord_Memrel)
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   497
qed
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   498
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   499
subsubsection{*Characterising initial segments of the well-ordering*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   500
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   501
lemma csquareD:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   502
 "[| <<x,y>, <z,z>> \<in> csquare_rel(K);  x<K;  y<K;  z<K |] ==> x \<le> z & y \<le> z"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   503
apply (unfold csquare_rel_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   504
apply (erule rev_mp)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   505
apply (elim ltE)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   506
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
   507
apply (safe elim!: mem_irrefl intro!: Un_upper1_le Un_upper2_le)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   508
apply (simp_all add: lt_def succI2)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   509
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   510
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   511
lemma pred_csquare_subset:
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   512
    "z<K ==> Order.pred(K*K, <z,z>, csquare_rel(K)) \<subseteq> succ(z)*succ(z)"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   513
apply (unfold Order.pred_def)
46901
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   514
apply (safe del: SigmaI dest!: csquareD)
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   515
apply (unfold lt_def, auto)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   516
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   517
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   518
lemma csquare_ltI:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   519
 "[| x<z;  y<z;  z<K |] ==>  <<x,y>, <z,z>> \<in> csquare_rel(K)"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   520
apply (unfold csquare_rel_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   521
apply (subgoal_tac "x<K & y<K")
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   522
 prefer 2 apply (blast intro: lt_trans)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   523
apply (elim ltE)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   524
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
   525
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   526
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   527
(*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
   528
lemma csquare_or_eqI:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   529
 "[| x \<le> z;  y \<le> z;  z<K |] ==> <<x,y>, <z,z>> \<in> csquare_rel(K) | x=z & y=z"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   530
apply (unfold csquare_rel_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   531
apply (subgoal_tac "x<K & y<K")
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   532
 prefer 2 apply (blast intro: lt_trans1)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   533
apply (elim ltE)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   534
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
   535
apply (elim succE)
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   536
apply (simp_all add: subset_Un_iff [THEN iff_sym]
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   537
                     subset_Un_iff2 [THEN iff_sym] OrdmemD)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   538
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   539
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   540
subsubsection{*The cardinality of initial segments*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   541
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   542
lemma ordermap_z_lt:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   543
      "[| Limit(K);  x<K;  y<K;  z=succ(x \<union> y) |] ==>
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   544
          ordermap(K*K, csquare_rel(K)) ` <x,y> <
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   545
          ordermap(K*K, csquare_rel(K)) ` <z,z>"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   546
apply (subgoal_tac "z<K & well_ord (K*K, csquare_rel (K))")
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   547
prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   548
                              Limit_is_Ord [THEN well_ord_csquare], clarify)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   549
apply (rule csquare_ltI [THEN ordermap_mono, THEN ltI])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   550
apply (erule_tac [4] well_ord_is_wf)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   551
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
   552
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   553
46901
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   554
text{*Kunen: "each @{term"\<langle>x,y\<rangle> \<in> K \<times> K"} has no more than @{term"z \<times> z"} predecessors..." (page 29) *}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   555
lemma ordermap_csquare_le:
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   556
  assumes K: "Limit(K)" and x: "x<K" and y: " y<K"
46907
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   557
  defines "z \<equiv> succ(x \<union> y)"
46901
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   558
  shows "|ordermap(K \<times> K, csquare_rel(K)) ` \<langle>x,y\<rangle>| \<le> |succ(z)| \<otimes> |succ(z)|"
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   559
proof (unfold cmult_def, rule well_ord_lepoll_imp_Card_le)
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   560
  show "well_ord(|succ(z)| \<times> |succ(z)|,
46901
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   561
                 rmult(|succ(z)|, Memrel(|succ(z)|), |succ(z)|, Memrel(|succ(z)|)))"
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   562
    by (blast intro: Ord_cardinal well_ord_Memrel well_ord_rmult)
46901
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   563
next
46907
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   564
  have zK: "z<K" using x y K z_def
46901
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   565
    by (blast intro: Un_least_lt Limit_has_succ)
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   566
  hence oz: "Ord(z)" by (elim ltE)
46901
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   567
  have "ordermap(K \<times> K, csquare_rel(K)) ` \<langle>x,y\<rangle> \<lesssim> ordermap(K \<times> K, csquare_rel(K)) ` \<langle>z,z\<rangle>"
46907
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   568
    using z_def
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   569
    by (blast intro: ordermap_z_lt leI le_imp_lepoll K x y)
46901
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   570
  also have "... \<approx>  Order.pred(K \<times> K, \<langle>z,z\<rangle>, csquare_rel(K))"
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   571
    proof (rule ordermap_eqpoll_pred)
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   572
      show "well_ord(K \<times> K, csquare_rel(K))" using K
46901
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   573
        by (rule Limit_is_Ord [THEN well_ord_csquare])
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   574
    next
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   575
      show "\<langle>z, z\<rangle> \<in> K \<times> K" using zK
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   576
        by (blast intro: ltD)
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   577
    qed
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   578
  also have "...  \<lesssim> succ(z) \<times> succ(z)" using zK
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   579
    by (rule pred_csquare_subset [THEN subset_imp_lepoll])
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   580
  also have "... \<approx> |succ(z)| \<times> |succ(z)|" using oz
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   581
    by (blast intro: prod_eqpoll_cong Ord_succ Ord_cardinal_eqpoll eqpoll_sym)
46901
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   582
  finally show "ordermap(K \<times> K, csquare_rel(K)) ` \<langle>x,y\<rangle> \<lesssim> |succ(z)| \<times> |succ(z)|" .
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   583
qed
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   584
46901
1382bba4b7a5 More structured proofs about cardinal arithmetic
paulson
parents: 46841
diff changeset
   585
text{*Kunen: "... so the order type is @{text"\<le>"} K" *}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   586
lemma ordertype_csquare_le:
46907
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   587
  assumes IK: "InfCard(K)" and eq: "\<And>y. y\<in>K \<Longrightarrow> InfCard(y) \<Longrightarrow> y \<otimes> y = y"
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   588
  shows "ordertype(K*K, csquare_rel(K)) \<le> K"
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   589
proof -
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   590
  have  CK: "Card(K)" using IK by (rule InfCard_is_Card)
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   591
  hence OK: "Ord(K)"  by (rule Card_is_Ord)
46907
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   592
  moreover have "Ord(ordertype(K \<times> K, csquare_rel(K)))" using OK
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   593
    by (rule well_ord_csquare [THEN Ord_ordertype])
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   594
  ultimately show ?thesis
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   595
  proof (rule all_lt_imp_le)
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   596
    fix i
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   597
    assume i: "i < ordertype(K \<times> K, csquare_rel(K))"
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   598
    hence Oi: "Ord(i)" by (elim ltE)
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   599
    obtain x y where x: "x \<in> K" and y: "y \<in> K"
46907
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   600
                 and ieq: "i = ordermap(K \<times> K, csquare_rel(K)) ` \<langle>x,y\<rangle>"
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   601
      using i by (auto simp add: ordertype_unfold elim: ltE)
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   602
    hence xy: "Ord(x)" "Ord(y)" "x < K" "y < K" using OK
46907
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   603
      by (blast intro: Ord_in_Ord ltI)+
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   604
    hence ou: "Ord(x \<union> y)"
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   605
      by (simp add: Ord_Un)
46907
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   606
    show "i < K"
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   607
      proof (rule Card_lt_imp_lt [OF _ Oi CK])
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   608
        have "|i| \<le> |succ(succ(x \<union> y))| \<otimes> |succ(succ(x \<union> y))|" using IK xy
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   609
          by (auto simp add: ieq intro: InfCard_is_Limit [THEN ordermap_csquare_le])
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   610
        moreover have "|succ(succ(x \<union> y))| \<otimes> |succ(succ(x \<union> y))| < K"
46907
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   611
          proof (cases rule: Ord_linear2 [OF ou Ord_nat])
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   612
            assume "x \<union> y < nat"
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   613
            hence "|succ(succ(x \<union> y))| \<otimes> |succ(succ(x \<union> y))| \<in> nat"
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   614
              by (simp add: lt_def nat_cmult_eq_mult nat_succI mult_type
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   615
                         nat_into_Card [THEN Card_cardinal_eq]  Ord_nat)
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   616
            also have "... \<subseteq> K" using IK
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   617
              by (simp add: InfCard_def le_imp_subset)
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   618
            finally show "|succ(succ(x \<union> y))| \<otimes> |succ(succ(x \<union> y))| < K"
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   619
              by (simp add: ltI OK)
46907
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   620
          next
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   621
            assume natxy: "nat \<le> x \<union> y"
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   622
            hence seq: "|succ(succ(x \<union> y))| = |x \<union> y|" using xy
46907
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   623
              by (simp add: le_imp_subset nat_succ_eqpoll [THEN cardinal_cong] le_succ_iff)
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   624
            also have "... < K" using xy
46907
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   625
              by (simp add: Un_least_lt Ord_cardinal_le [THEN lt_trans1])
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   626
            finally have "|succ(succ(x \<union> y))| < K" .
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   627
            moreover have "InfCard(|succ(succ(x \<union> y))|)" using xy natxy
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   628
              by (simp add: seq InfCard_def Card_cardinal nat_le_cardinal)
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   629
            ultimately show ?thesis  by (simp add: eq ltD)
46907
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   630
          qed
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   631
        ultimately show "|i| < K" by (blast intro: lt_trans1)
46907
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   632
    qed
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   633
  qed
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   634
qed
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   635
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   636
(*Main result: Kunen's Theorem 10.12*)
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   637
lemma InfCard_csquare_eq:
46907
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   638
  assumes IK: "InfCard(K)" shows "InfCard(K) ==> K \<otimes> K = K"
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   639
proof -
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   640
  have  OK: "Ord(K)" using IK by (simp add: Card_is_Ord InfCard_is_Card)
46935
38ecb2dc3636 structured case and induct rules
paulson
parents: 46907
diff changeset
   641
  show "InfCard(K) ==> K \<otimes> K = K" using OK
38ecb2dc3636 structured case and induct rules
paulson
parents: 46907
diff changeset
   642
  proof (induct rule: trans_induct)
38ecb2dc3636 structured case and induct rules
paulson
parents: 46907
diff changeset
   643
    case (step i)
38ecb2dc3636 structured case and induct rules
paulson
parents: 46907
diff changeset
   644
    show "i \<otimes> i = i"
38ecb2dc3636 structured case and induct rules
paulson
parents: 46907
diff changeset
   645
    proof (rule le_anti_sym)
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   646
      have "|i \<times> i| = |ordertype(i \<times> i, csquare_rel(i))|"
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   647
        by (rule cardinal_cong,
46935
38ecb2dc3636 structured case and induct rules
paulson
parents: 46907
diff changeset
   648
          simp add: step.hyps well_ord_csquare [THEN ordermap_bij, THEN bij_imp_eqpoll])
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   649
      hence "i \<otimes> i \<le> ordertype(i \<times> i, csquare_rel(i))"
46935
38ecb2dc3636 structured case and induct rules
paulson
parents: 46907
diff changeset
   650
        by (simp add: step.hyps cmult_def Ord_cardinal_le well_ord_csquare [THEN Ord_ordertype])
38ecb2dc3636 structured case and induct rules
paulson
parents: 46907
diff changeset
   651
      moreover
38ecb2dc3636 structured case and induct rules
paulson
parents: 46907
diff changeset
   652
      have "ordertype(i \<times> i, csquare_rel(i)) \<le> i" using step
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   653
        by (simp add: ordertype_csquare_le)
46935
38ecb2dc3636 structured case and induct rules
paulson
parents: 46907
diff changeset
   654
      ultimately show "i \<otimes> i \<le> i" by (rule le_trans)
38ecb2dc3636 structured case and induct rules
paulson
parents: 46907
diff changeset
   655
    next
38ecb2dc3636 structured case and induct rules
paulson
parents: 46907
diff changeset
   656
      show "i \<le> i \<otimes> i" using step
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   657
        by (blast intro: cmult_square_le InfCard_is_Card)
46907
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   658
    qed
46935
38ecb2dc3636 structured case and induct rules
paulson
parents: 46907
diff changeset
   659
  qed
46907
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   660
qed
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   661
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   662
(*Corollary for arbitrary well-ordered sets (all sets, assuming AC)*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   663
lemma well_ord_InfCard_square_eq:
46907
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   664
  assumes r: "well_ord(A,r)" and I: "InfCard(|A|)" shows "A \<times> A \<approx> A"
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   665
proof -
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   666
  have "A \<times> A \<approx> |A| \<times> |A|"
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   667
    by (blast intro: prod_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_sym r)
46907
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   668
  also have "... \<approx> A"
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   669
    proof (rule well_ord_cardinal_eqE [OF _ r])
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   670
      show "well_ord(|A| \<times> |A|, rmult(|A|, Memrel(|A|), |A|, Memrel(|A|)))"
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   671
        by (blast intro: Ord_cardinal well_ord_rmult well_ord_Memrel r)
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   672
    next
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   673
      show "||A| \<times> |A|| = |A|" using InfCard_csquare_eq I
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   674
        by (simp add: cmult_def)
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   675
    qed
46907
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   676
  finally show ?thesis .
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   677
qed
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   678
13356
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   679
lemma InfCard_square_eqpoll: "InfCard(K) ==> K \<times> K \<approx> K"
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   680
apply (rule well_ord_InfCard_square_eq)
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   681
 apply (erule InfCard_is_Card [THEN Card_is_Ord, THEN well_ord_Memrel])
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   682
apply (simp add: InfCard_is_Card [THEN Card_cardinal_eq])
13356
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   683
done
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   684
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   685
lemma Inf_Card_is_InfCard: "[| ~Finite(i); Card(i) |] ==> InfCard(i)"
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   686
by (simp add: InfCard_def Card_is_Ord [THEN nat_le_infinite_Ord])
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   687
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   688
subsubsection{*Toward's Kunen's Corollary 10.13 (1)*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   689
46821
ff6b0c1087f2 Using mathematical notation for <-> and cardinal arithmetic
paulson
parents: 46820
diff changeset
   690
lemma InfCard_le_cmult_eq: "[| InfCard(K);  L \<le> K;  0<L |] ==> K \<otimes> L = K"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   691
apply (rule le_anti_sym)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   692
 prefer 2
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   693
 apply (erule ltE, blast intro: cmult_le_self InfCard_is_Card)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   694
apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   695
apply (rule cmult_le_mono [THEN le_trans], assumption+)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   696
apply (simp add: InfCard_csquare_eq)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   697
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   698
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   699
(*Corollary 10.13 (1), for cardinal multiplication*)
46821
ff6b0c1087f2 Using mathematical notation for <-> and cardinal arithmetic
paulson
parents: 46820
diff changeset
   700
lemma InfCard_cmult_eq: "[| InfCard(K);  InfCard(L) |] ==> K \<otimes> L = K \<union> L"
13784
b9f6154427a4 tidying (by script)
paulson
parents: 13615
diff changeset
   701
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
   702
apply (typecheck add: InfCard_is_Card Card_is_Ord)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   703
apply (rule cmult_commute [THEN ssubst])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   704
apply (rule Un_commute [THEN ssubst])
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   705
apply (simp_all add: InfCard_is_Limit [THEN Limit_has_0] InfCard_le_cmult_eq
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   706
                     subset_Un_iff2 [THEN iffD1] le_imp_subset)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   707
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   708
46821
ff6b0c1087f2 Using mathematical notation for <-> and cardinal arithmetic
paulson
parents: 46820
diff changeset
   709
lemma InfCard_cdouble_eq: "InfCard(K) ==> K \<oplus> K = K"
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   710
apply (simp add: cmult_2 [symmetric] InfCard_is_Card cmult_commute)
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   711
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
   712
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   713
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   714
(*Corollary 10.13 (1), for cardinal addition*)
46821
ff6b0c1087f2 Using mathematical notation for <-> and cardinal arithmetic
paulson
parents: 46820
diff changeset
   715
lemma InfCard_le_cadd_eq: "[| InfCard(K);  L \<le> K |] ==> K \<oplus> L = K"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   716
apply (rule le_anti_sym)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   717
 prefer 2
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   718
 apply (erule ltE, blast intro: cadd_le_self InfCard_is_Card)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   719
apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   720
apply (rule cadd_le_mono [THEN le_trans], assumption+)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   721
apply (simp add: InfCard_cdouble_eq)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   722
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   723
46821
ff6b0c1087f2 Using mathematical notation for <-> and cardinal arithmetic
paulson
parents: 46820
diff changeset
   724
lemma InfCard_cadd_eq: "[| InfCard(K);  InfCard(L) |] ==> K \<oplus> L = K \<union> L"
13784
b9f6154427a4 tidying (by script)
paulson
parents: 13615
diff changeset
   725
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
   726
apply (typecheck add: InfCard_is_Card Card_is_Ord)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   727
apply (rule cadd_commute [THEN ssubst])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   728
apply (rule Un_commute [THEN ssubst])
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   729
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
   730
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   731
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   732
(*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
   733
  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
   734
  might be  InfCard(K) ==> |list(K)| = K.
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   735
*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   736
27517
c055e1d49285 Fixed (harmless) typo in closing *}.
ballarin
parents: 24893
diff changeset
   737
subsection{*For Every Cardinal Number There Exists A Greater One*}
13356
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   738
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   739
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
   740
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   741
lemma Ord_jump_cardinal: "Ord(jump_cardinal(K))"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   742
apply (unfold jump_cardinal_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   743
apply (rule Ord_is_Transset [THEN [2] OrdI])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   744
 prefer 2 apply (blast intro!: Ord_ordertype)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   745
apply (unfold Transset_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   746
apply (safe del: subsetI)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   747
apply (simp add: ordertype_pred_unfold, safe)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   748
apply (rule UN_I)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   749
apply (rule_tac [2] ReplaceI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   750
   prefer 4 apply (blast intro: well_ord_subset elim!: predE)+
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   751
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   752
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   753
(*Allows selective unfolding.  Less work than deriving intro/elim rules*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   754
lemma jump_cardinal_iff:
46821
ff6b0c1087f2 Using mathematical notation for <-> and cardinal arithmetic
paulson
parents: 46820
diff changeset
   755
     "i \<in> jump_cardinal(K) \<longleftrightarrow>
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   756
      (\<exists>r X. r \<subseteq> K*K & X \<subseteq> K & well_ord(X,r) & i = ordertype(X,r))"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   757
apply (unfold jump_cardinal_def)
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   758
apply (blast del: subsetI)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   759
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   760
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   761
(*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   762
lemma K_lt_jump_cardinal: "Ord(K) ==> K < jump_cardinal(K)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   763
apply (rule Ord_jump_cardinal [THEN [2] ltI])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   764
apply (rule jump_cardinal_iff [THEN iffD2])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   765
apply (rule_tac x="Memrel(K)" in exI)
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   766
apply (rule_tac x=K in exI)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   767
apply (simp add: ordertype_Memrel well_ord_Memrel)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   768
apply (simp add: Memrel_def subset_iff)
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
(*The proof by contradiction: the bijection f yields a wellordering of X
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   772
  whose ordertype is jump_cardinal(K).  *)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   773
lemma Card_jump_cardinal_lemma:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   774
     "[| well_ord(X,r);  r \<subseteq> K * K;  X \<subseteq> K;
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   775
         f \<in> bij(ordertype(X,r), jump_cardinal(K)) |]
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   776
      ==> jump_cardinal(K) \<in> jump_cardinal(K)"
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   777
apply (subgoal_tac "f O ordermap (X,r) \<in> bij (X, jump_cardinal (K))")
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   778
 prefer 2 apply (blast intro: comp_bij ordermap_bij)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   779
apply (rule jump_cardinal_iff [THEN iffD2])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   780
apply (intro exI conjI)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   781
apply (rule subset_trans [OF rvimage_type Sigma_mono], assumption+)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   782
apply (erule bij_is_inj [THEN well_ord_rvimage])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   783
apply (rule Ord_jump_cardinal [THEN well_ord_Memrel])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   784
apply (simp add: well_ord_Memrel [THEN [2] bij_ordertype_vimage]
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   785
                 ordertype_Memrel Ord_jump_cardinal)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   786
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   787
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   788
(*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
   789
lemma Card_jump_cardinal: "Card(jump_cardinal(K))"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   790
apply (rule Ord_jump_cardinal [THEN CardI])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   791
apply (unfold eqpoll_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   792
apply (safe dest!: ltD jump_cardinal_iff [THEN iffD1])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   793
apply (blast intro: Card_jump_cardinal_lemma [THEN mem_irrefl])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   794
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   795
13356
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   796
subsection{*Basic Properties of Successor Cardinals*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   797
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   798
lemma csucc_basic: "Ord(K) ==> Card(csucc(K)) & K < csucc(K)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   799
apply (unfold csucc_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   800
apply (rule LeastI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   801
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
   802
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   803
45602
2a858377c3d2 eliminated obsolete "standard";
wenzelm
parents: 39159
diff changeset
   804
lemmas Card_csucc = csucc_basic [THEN conjunct1]
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   805
45602
2a858377c3d2 eliminated obsolete "standard";
wenzelm
parents: 39159
diff changeset
   806
lemmas lt_csucc = csucc_basic [THEN conjunct2]
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   807
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   808
lemma Ord_0_lt_csucc: "Ord(K) ==> 0 < csucc(K)"
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   809
by (blast intro: Ord_0_le lt_csucc lt_trans1)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   810
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   811
lemma csucc_le: "[| Card(L);  K<L |] ==> csucc(K) \<le> L"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   812
apply (unfold csucc_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   813
apply (rule Least_le)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   814
apply (blast intro: Card_is_Ord)+
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   815
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   816
46821
ff6b0c1087f2 Using mathematical notation for <-> and cardinal arithmetic
paulson
parents: 46820
diff changeset
   817
lemma lt_csucc_iff: "[| Ord(i); Card(K) |] ==> i < csucc(K) \<longleftrightarrow> |i| \<le> K"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   818
apply (rule iffI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   819
apply (rule_tac [2] Card_lt_imp_lt)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   820
apply (erule_tac [2] lt_trans1)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   821
apply (simp_all add: lt_csucc Card_csucc Card_is_Ord)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   822
apply (rule notI [THEN not_lt_imp_le])
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   823
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
   824
apply (rule Ord_cardinal_le [THEN lt_trans1])
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   825
apply (simp_all add: Ord_cardinal Card_is_Ord)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   826
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   827
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   828
lemma Card_lt_csucc_iff:
46821
ff6b0c1087f2 Using mathematical notation for <-> and cardinal arithmetic
paulson
parents: 46820
diff changeset
   829
     "[| Card(K'); Card(K) |] ==> K' < csucc(K) \<longleftrightarrow> K' \<le> K"
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   830
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
   831
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   832
lemma InfCard_csucc: "InfCard(K) ==> InfCard(csucc(K))"
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   833
by (simp add: InfCard_def Card_csucc Card_is_Ord
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   834
              lt_csucc [THEN leI, THEN [2] le_trans])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   835
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
subsubsection{*Removing elements from a finite set decreases its cardinality*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   838
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   839
lemma Finite_imp_cardinal_cons [simp]:
46952
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   840
  assumes FA: "Finite(A)" and a: "a\<notin>A" shows "|cons(a,A)| = succ(|A|)"
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   841
proof -
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   842
  { fix X
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   843
    have "Finite(X) ==> a \<notin> X \<Longrightarrow> cons(a,X) \<lesssim> X \<Longrightarrow> False"
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   844
      proof (induct X rule: Finite_induct)
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   845
        case 0 thus False  by (simp add: lepoll_0_iff)
46952
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   846
      next
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   847
        case (cons x Y)
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   848
        hence "cons(x, cons(a, Y)) \<lesssim> cons(x, Y)" by (simp add: cons_commute)
46952
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   849
        hence "cons(a, Y) \<lesssim> Y" using cons        by (blast dest: cons_lepoll_consD)
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   850
        thus False using cons by auto
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   851
      qed
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   852
  }
46952
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   853
  hence [simp]: "~ cons(a,A) \<lesssim> A" using a FA by auto
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   854
  have [simp]: "|A| \<approx> A" using Finite_imp_well_ord [OF FA]
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   855
    by (blast intro: well_ord_cardinal_eqpoll)
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   856
  have "(\<mu> i. i \<approx> cons(a, A)) = succ(|A|)"
46952
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   857
    proof (rule Least_equality [OF _ _ notI])
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   858
      show "succ(|A|) \<approx> cons(a, A)"
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   859
        by (simp add: succ_def cons_eqpoll_cong mem_not_refl a)
46952
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   860
    next
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   861
      show "Ord(succ(|A|))" by simp
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   862
    next
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   863
      fix i
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   864
      assume i: "i \<le> |A|" "i \<approx> cons(a, A)"
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   865
      have "cons(a, A) \<approx> i" by (rule eqpoll_sym) (rule i)
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   866
      also have "... \<lesssim> |A|" by (rule le_imp_lepoll) (rule i)
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   867
      also have "... \<approx> A"   by simp
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   868
      finally have "cons(a, A) \<lesssim> A" .
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   869
      thus False by simp
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   870
    qed
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   871
  thus ?thesis by (simp add: cardinal_def)
46952
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   872
qed
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   873
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   874
lemma Finite_imp_succ_cardinal_Diff:
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   875
     "[| Finite(A);  a \<in> A |] ==> succ(|A-{a}|) = |A|"
13784
b9f6154427a4 tidying (by script)
paulson
parents: 13615
diff changeset
   876
apply (rule_tac b = A in cons_Diff [THEN subst], assumption)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   877
apply (simp add: Finite_imp_cardinal_cons Diff_subset [THEN subset_Finite])
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   878
apply (simp add: cons_Diff)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   879
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   880
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   881
lemma Finite_imp_cardinal_Diff: "[| Finite(A);  a \<in> A |] ==> |A-{a}| < |A|"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   882
apply (rule succ_leE)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   883
apply (simp add: Finite_imp_succ_cardinal_Diff)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   884
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   885
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   886
lemma Finite_cardinal_in_nat [simp]: "Finite(A) ==> |A| \<in> nat"
46952
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   887
proof (induct rule: Finite_induct)
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   888
  case 0 thus ?case by (simp add: cardinal_0)
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   889
next
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   890
  case (cons x A) thus ?case by (simp add: Finite_imp_cardinal_cons)
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   891
qed
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   892
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   893
lemma card_Un_Int:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   894
     "[|Finite(A); Finite(B)|] ==> |A| #+ |B| = |A \<union> B| #+ |A \<inter> B|"
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   895
apply (erule Finite_induct, simp)
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   896
apply (simp add: Finite_Int cons_absorb Un_cons Int_cons_left)
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   897
done
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   898
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   899
lemma card_Un_disjoint:
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   900
     "[|Finite(A); Finite(B); A \<inter> B = 0|] ==> |A \<union> B| = |A| #+ |B|"
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   901
by (simp add: Finite_Un card_Un_Int)
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   902
46952
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   903
lemma card_partition:
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   904
  assumes FC: "Finite(C)"
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   905
  shows
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   906
     "Finite (\<Union> C) \<Longrightarrow>
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   907
        (\<forall>c\<in>C. |c| = k) \<Longrightarrow>
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   908
        (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = 0) \<Longrightarrow>
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   909
        k #* |C| = |\<Union> C|"
46952
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   910
using FC
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   911
proof (induct rule: Finite_induct)
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   912
  case 0 thus ?case by simp
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   913
next
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   914
  case (cons x B)
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   915
  hence "x \<inter> \<Union>B = 0" by auto
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   916
  thus ?case using cons
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   917
    by (auto simp add: card_Un_disjoint)
5e1bcfdcb175 Rewrote some induction proofs to be structured
paulson
parents: 46935
diff changeset
   918
qed
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   919
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   920
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   921
subsubsection{*Theorems by Krzysztof Grabczewski, proofs by lcp*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   922
45602
2a858377c3d2 eliminated obsolete "standard";
wenzelm
parents: 39159
diff changeset
   923
lemmas nat_implies_well_ord = nat_into_Ord [THEN well_ord_Memrel]
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   924
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   925
lemma nat_sum_eqpoll_sum:
46907
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   926
  assumes m: "m \<in> nat" and n: "n \<in> nat" shows "m + n \<approx> m #+ n"
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   927
proof -
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   928
  have "m + n \<approx> |m+n|" using m n
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46952
diff changeset
   929
    by (blast intro: nat_implies_well_ord well_ord_radd well_ord_cardinal_eqpoll eqpoll_sym)
46907
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   930
  also have "... = m #+ n" using m n
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   931
    by (simp add: nat_cadd_eq_add [symmetric] cadd_def)
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   932
  finally show ?thesis .
eea3eb057fea Structured proofs concerning the square of an infinite cardinal
paulson
parents: 46901
diff changeset
   933
qed
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   934
46935
38ecb2dc3636 structured case and induct rules
paulson
parents: 46907
diff changeset
   935
lemma Ord_subset_natD [rule_format]: "Ord(i) ==> i \<subseteq> nat \<Longrightarrow> i \<in> nat | i=nat"
38ecb2dc3636 structured case and induct rules
paulson
parents: 46907
diff changeset
   936
proof (induct i rule: trans_induct3)
38ecb2dc3636 structured case and induct rules
paulson
parents: 46907
diff changeset
   937
  case 0 thus ?case by auto
38ecb2dc3636 structured case and induct rules
paulson
parents: 46907
diff changeset
   938
next
38ecb2dc3636 structured case and induct rules
paulson
parents: 46907
diff changeset
   939
  case (succ i) thus ?case by auto
38ecb2dc3636 structured case and induct rules
paulson
parents: 46907
diff changeset
   940
next
38ecb2dc3636 structured case and induct rules
paulson
parents: 46907
diff changeset
   941
  case (limit l) thus ?case
38ecb2dc3636 structured case and induct rules
paulson
parents: 46907
diff changeset
   942
    by (blast dest: nat_le_Limit le_imp_subset)
38ecb2dc3636 structured case and induct rules
paulson
parents: 46907
diff changeset
   943
qed
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   944
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45602
diff changeset
   945
lemma Ord_nat_subset_into_Card: "[| Ord(i); i \<subseteq> nat |] ==> Card(i)"
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   946
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
   947
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   948
end