src/HOL/Basic_BNFs.thy
author blanchet
Thu Mar 06 15:29:18 2014 +0100 (2014-03-06)
changeset 55944 7ab8f003fe41
parent 55943 5c2df04e97d1
child 55945 e96383acecf9
permissions -rw-r--r--
renamed 'prod_rel' to 'rel_prod'
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(*  Title:      HOL/Basic_BNFs.thy
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    Author:     Dmitriy Traytel, TU Muenchen
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    Author:     Andrei Popescu, TU Muenchen
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    Author:     Jasmin Blanchette, TU Muenchen
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    Copyright   2012
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Registration of basic types as bounded natural functors.
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*)
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header {* Registration of Basic Types as Bounded Natural Functors *}
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theory Basic_BNFs
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imports BNF_Def
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begin
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definition setl :: "'a + 'b \<Rightarrow> 'a set" where
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"setl x = (case x of Inl z => {z} | _ => {})"
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definition setr :: "'a + 'b \<Rightarrow> 'b set" where
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"setr x = (case x of Inr z => {z} | _ => {})"
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lemmas sum_set_defs = setl_def[abs_def] setr_def[abs_def]
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definition
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   rel_sum :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> 'c + 'd \<Rightarrow> bool"
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where
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   "rel_sum R1 R2 x y =
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     (case (x, y) of (Inl x, Inl y) \<Rightarrow> R1 x y
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     | (Inr x, Inr y) \<Rightarrow> R2 x y
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     | _ \<Rightarrow> False)"
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lemma rel_sum_simps[simp]:
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  "rel_sum R1 R2 (Inl a1) (Inl b1) = R1 a1 b1"
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  "rel_sum R1 R2 (Inl a1) (Inr b2) = False"
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  "rel_sum R1 R2 (Inr a2) (Inl b1) = False"
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  "rel_sum R1 R2 (Inr a2) (Inr b2) = R2 a2 b2"
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  unfolding rel_sum_def by simp_all
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bnf "'a + 'b"
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  map: map_sum
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  sets: setl setr
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  bd: natLeq
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  wits: Inl Inr
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  rel: rel_sum
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proof -
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  show "map_sum id id = id" by (rule map_sum.id)
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next
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  fix f1 :: "'o \<Rightarrow> 's" and f2 :: "'p \<Rightarrow> 't" and g1 :: "'s \<Rightarrow> 'q" and g2 :: "'t \<Rightarrow> 'r"
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  show "map_sum (g1 o f1) (g2 o f2) = map_sum g1 g2 o map_sum f1 f2"
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    by (rule map_sum.comp[symmetric])
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next
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  fix x and f1 :: "'o \<Rightarrow> 'q" and f2 :: "'p \<Rightarrow> 'r" and g1 g2
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  assume a1: "\<And>z. z \<in> setl x \<Longrightarrow> f1 z = g1 z" and
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         a2: "\<And>z. z \<in> setr x \<Longrightarrow> f2 z = g2 z"
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  thus "map_sum f1 f2 x = map_sum g1 g2 x"
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  proof (cases x)
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    case Inl thus ?thesis using a1 by (clarsimp simp: setl_def)
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  next
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    case Inr thus ?thesis using a2 by (clarsimp simp: setr_def)
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  qed
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next
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  fix f1 :: "'o \<Rightarrow> 'q" and f2 :: "'p \<Rightarrow> 'r"
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  show "setl o map_sum f1 f2 = image f1 o setl"
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    by (rule ext, unfold o_apply) (simp add: setl_def split: sum.split)
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next
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  fix f1 :: "'o \<Rightarrow> 'q" and f2 :: "'p \<Rightarrow> 'r"
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  show "setr o map_sum f1 f2 = image f2 o setr"
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    by (rule ext, unfold o_apply) (simp add: setr_def split: sum.split)
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next
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  show "card_order natLeq" by (rule natLeq_card_order)
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next
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  show "cinfinite natLeq" by (rule natLeq_cinfinite)
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next
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  fix x :: "'o + 'p"
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  show "|setl x| \<le>o natLeq"
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    apply (rule ordLess_imp_ordLeq)
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    apply (rule finite_iff_ordLess_natLeq[THEN iffD1])
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    by (simp add: setl_def split: sum.split)
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next
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  fix x :: "'o + 'p"
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  show "|setr x| \<le>o natLeq"
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    apply (rule ordLess_imp_ordLeq)
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    apply (rule finite_iff_ordLess_natLeq[THEN iffD1])
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    by (simp add: setr_def split: sum.split)
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next
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  fix R1 R2 S1 S2
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  show "rel_sum R1 R2 OO rel_sum S1 S2 \<le> rel_sum (R1 OO S1) (R2 OO S2)"
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    by (auto simp: rel_sum_def split: sum.splits)
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next
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  fix R S
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  show "rel_sum R S =
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        (Grp {x. setl x \<subseteq> Collect (split R) \<and> setr x \<subseteq> Collect (split S)} (map_sum fst fst))\<inverse>\<inverse> OO
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        Grp {x. setl x \<subseteq> Collect (split R) \<and> setr x \<subseteq> Collect (split S)} (map_sum snd snd)"
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  unfolding setl_def setr_def rel_sum_def Grp_def relcompp.simps conversep.simps fun_eq_iff
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  by (fastforce split: sum.splits)
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qed (auto simp: sum_set_defs)
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definition fsts :: "'a \<times> 'b \<Rightarrow> 'a set" where
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"fsts x = {fst x}"
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definition snds :: "'a \<times> 'b \<Rightarrow> 'b set" where
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"snds x = {snd x}"
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lemmas prod_set_defs = fsts_def[abs_def] snds_def[abs_def]
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definition
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  rel_prod :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'b \<times> 'd \<Rightarrow> bool"
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where
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  "rel_prod R1 R2 = (\<lambda>(a, b) (c, d). R1 a c \<and> R2 b d)"
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lemma rel_prod_apply [simp]:
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  "rel_prod R1 R2 (a, b) (c, d) \<longleftrightarrow> R1 a c \<and> R2 b d"
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  by (simp add: rel_prod_def)
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bnf "'a \<times> 'b"
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  map: map_prod
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  sets: fsts snds
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  bd: natLeq
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  rel: rel_prod
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proof (unfold prod_set_defs)
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  show "map_prod id id = id" by (rule map_prod.id)
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next
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  fix f1 f2 g1 g2
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  show "map_prod (g1 o f1) (g2 o f2) = map_prod g1 g2 o map_prod f1 f2"
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    by (rule map_prod.comp[symmetric])
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next
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  fix x f1 f2 g1 g2
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  assume "\<And>z. z \<in> {fst x} \<Longrightarrow> f1 z = g1 z" "\<And>z. z \<in> {snd x} \<Longrightarrow> f2 z = g2 z"
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  thus "map_prod f1 f2 x = map_prod g1 g2 x" by (cases x) simp
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next
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  fix f1 f2
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  show "(\<lambda>x. {fst x}) o map_prod f1 f2 = image f1 o (\<lambda>x. {fst x})"
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    by (rule ext, unfold o_apply) simp
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next
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  fix f1 f2
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  show "(\<lambda>x. {snd x}) o map_prod f1 f2 = image f2 o (\<lambda>x. {snd x})"
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    by (rule ext, unfold o_apply) simp
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next
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  show "card_order natLeq" by (rule natLeq_card_order)
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next
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  show "cinfinite natLeq" by (rule natLeq_cinfinite)
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next
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  fix x
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  show "|{fst x}| \<le>o natLeq"
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    by (rule ordLess_imp_ordLeq) (simp add: finite_iff_ordLess_natLeq[symmetric])
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next
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  fix x
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  show "|{snd x}| \<le>o natLeq"
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    by (rule ordLess_imp_ordLeq) (simp add: finite_iff_ordLess_natLeq[symmetric])
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next
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  fix R1 R2 S1 S2
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  show "rel_prod R1 R2 OO rel_prod S1 S2 \<le> rel_prod (R1 OO S1) (R2 OO S2)" by auto
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next
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  fix R S
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  show "rel_prod R S =
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        (Grp {x. {fst x} \<subseteq> Collect (split R) \<and> {snd x} \<subseteq> Collect (split S)} (map_prod fst fst))\<inverse>\<inverse> OO
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        Grp {x. {fst x} \<subseteq> Collect (split R) \<and> {snd x} \<subseteq> Collect (split S)} (map_prod snd snd)"
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  unfolding prod_set_defs rel_prod_def Grp_def relcompp.simps conversep.simps fun_eq_iff
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  by auto
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qed
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bnf "'a \<Rightarrow> 'b"
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  map: "op \<circ>"
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  sets: range
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  bd: "natLeq +c |UNIV :: 'a set|"
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  rel: "fun_rel op ="
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proof
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  fix f show "id \<circ> f = id f" by simp
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next
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  fix f g show "op \<circ> (g \<circ> f) = op \<circ> g \<circ> op \<circ> f"
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  unfolding comp_def[abs_def] ..
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next
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  fix x f g
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  assume "\<And>z. z \<in> range x \<Longrightarrow> f z = g z"
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  thus "f \<circ> x = g \<circ> x" by auto
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next
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  fix f show "range \<circ> op \<circ> f = op ` f \<circ> range"
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  unfolding image_def comp_def[abs_def] by auto
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next
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  show "card_order (natLeq +c |UNIV| )" (is "_ (_ +c ?U)")
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  apply (rule card_order_csum)
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  apply (rule natLeq_card_order)
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  by (rule card_of_card_order_on)
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(*  *)
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  show "cinfinite (natLeq +c ?U)"
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    apply (rule cinfinite_csum)
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    apply (rule disjI1)
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    by (rule natLeq_cinfinite)
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next
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  fix f :: "'d => 'a"
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  have "|range f| \<le>o | (UNIV::'d set) |" (is "_ \<le>o ?U") by (rule card_of_image)
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  also have "?U \<le>o natLeq +c ?U" by (rule ordLeq_csum2) (rule card_of_Card_order)
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  finally show "|range f| \<le>o natLeq +c ?U" .
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next
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  fix R S
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  show "fun_rel op = R OO fun_rel op = S \<le> fun_rel op = (R OO S)" by (auto simp: fun_rel_def)
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next
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  fix R
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  show "fun_rel op = R =
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        (Grp {x. range x \<subseteq> Collect (split R)} (op \<circ> fst))\<inverse>\<inverse> OO
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         Grp {x. range x \<subseteq> Collect (split R)} (op \<circ> snd)"
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  unfolding fun_rel_def Grp_def fun_eq_iff relcompp.simps conversep.simps subset_iff image_iff
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    comp_apply mem_Collect_eq split_beta bex_UNIV
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  proof (safe, unfold fun_eq_iff[symmetric])
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    fix x xa a b c xb y aa ba
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    assume *: "x = a" "xa = c" "a = ba" "b = aa" "c = (\<lambda>x. snd (b x))" "ba = (\<lambda>x. fst (aa x))" and
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       **: "\<forall>t. (\<exists>x. t = aa x) \<longrightarrow> R (fst t) (snd t)"
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    show "R (x y) (xa y)" unfolding * by (rule mp[OF spec[OF **]]) blast
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  qed force
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qed
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end