src/HOL/BNF/Basic_BNFs.thy
author traytel
Mon Jul 15 15:50:39 2013 +0200 (2013-07-15)
changeset 52660 7f7311d04727
parent 52635 4f84b730c489
child 53026 e1a548c11845
permissions -rw-r--r--
killed unused theorems
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(*  Title:      HOL/BNF/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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lemma wpull_id: "wpull UNIV B1 B2 id id id id"
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unfolding wpull_def by simp
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lemmas natLeq_card_order = natLeq_Card_order[unfolded Field_natLeq]
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lemma ctwo_card_order: "card_order ctwo"
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using Card_order_ctwo by (unfold ctwo_def Field_card_of)
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lemma natLeq_cinfinite: "cinfinite natLeq"
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unfolding cinfinite_def Field_natLeq by (rule nat_infinite)
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lemma wpull_Grp_def: "wpull A B1 B2 f1 f2 p1 p2 \<longleftrightarrow> Grp B1 f1 OO (Grp B2 f2)\<inverse>\<inverse> \<le> (Grp A p1)\<inverse>\<inverse> OO Grp A p2"
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  unfolding wpull_def Grp_def by auto
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bnf ID: "id :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" ["\<lambda>x. {x}"] "\<lambda>_:: 'a. natLeq" ["id :: 'a \<Rightarrow> 'a"]
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  "id :: ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
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apply (auto simp: Grp_def fun_eq_iff relcompp.simps natLeq_card_order natLeq_cinfinite)
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apply (rule ordLess_imp_ordLeq[OF finite_ordLess_infinite[OF _ natLeq_Well_order]])
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apply (auto simp add: Field_card_of Field_natLeq card_of_well_order_on)[3]
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done
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bnf DEADID: "id :: 'a \<Rightarrow> 'a" [] "\<lambda>_:: 'a. natLeq +c |UNIV :: 'a set|" ["SOME x :: 'a. True"]
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  "op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool"
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by (auto simp add: wpull_Grp_def Grp_def
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  card_order_csum natLeq_card_order card_of_card_order_on
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  cinfinite_csum natLeq_cinfinite)
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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 sum_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a + 'c \<Rightarrow> 'b + 'd \<Rightarrow> bool" where
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"sum_rel \<phi> \<psi> x y =
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 (case x of Inl a1 \<Rightarrow> (case y of Inl a2 \<Rightarrow> \<phi> a1 a2 | Inr _ \<Rightarrow> False)
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          | Inr b1 \<Rightarrow> (case y of Inl _ \<Rightarrow> False | Inr b2 \<Rightarrow> \<psi> b1 b2))"
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bnf sum_map [setl, setr] "\<lambda>_::'a + 'b. natLeq" [Inl, Inr] sum_rel
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proof -
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  show "sum_map id id = id" by (rule sum_map.id)
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next
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  fix f1 f2 g1 g2
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  show "sum_map (g1 o f1) (g2 o f2) = sum_map g1 g2 o sum_map f1 f2"
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    by (rule sum_map.comp[symmetric])
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next
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  fix x f1 f2 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 "sum_map f1 f2 x = sum_map 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 f2
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  show "setl o sum_map 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 f2
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  show "setr o sum_map 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
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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
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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 A1 A2 B11 B12 B21 B22 f11 f12 f21 f22 p11 p12 p21 p22
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  assume "wpull A1 B11 B21 f11 f21 p11 p21" "wpull A2 B12 B22 f12 f22 p12 p22"
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  hence
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    pull1: "\<And>b1 b2. \<lbrakk>b1 \<in> B11; b2 \<in> B21; f11 b1 = f21 b2\<rbrakk> \<Longrightarrow> \<exists>a \<in> A1. p11 a = b1 \<and> p21 a = b2"
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    and pull2: "\<And>b1 b2. \<lbrakk>b1 \<in> B12; b2 \<in> B22; f12 b1 = f22 b2\<rbrakk> \<Longrightarrow> \<exists>a \<in> A2. p12 a = b1 \<and> p22 a = b2"
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    unfolding wpull_def by blast+
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  show "wpull {x. setl x \<subseteq> A1 \<and> setr x \<subseteq> A2}
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  {x. setl x \<subseteq> B11 \<and> setr x \<subseteq> B12} {x. setl x \<subseteq> B21 \<and> setr x \<subseteq> B22}
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  (sum_map f11 f12) (sum_map f21 f22) (sum_map p11 p12) (sum_map p21 p22)"
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    (is "wpull ?in ?in1 ?in2 ?mapf1 ?mapf2 ?mapp1 ?mapp2")
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  proof (unfold wpull_def)
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    { fix B1 B2
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      assume *: "B1 \<in> ?in1" "B2 \<in> ?in2" "?mapf1 B1 = ?mapf2 B2"
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      have "\<exists>A \<in> ?in. ?mapp1 A = B1 \<and> ?mapp2 A = B2"
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      proof (cases B1)
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        case (Inl b1)
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        { fix b2 assume "B2 = Inr b2"
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          with Inl *(3) have False by simp
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        } then obtain b2 where Inl': "B2 = Inl b2" by (cases B2) (simp, blast)
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        with Inl * have "b1 \<in> B11" "b2 \<in> B21" "f11 b1 = f21 b2"
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        by (simp add: setl_def)+
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        with pull1 obtain a where "a \<in> A1" "p11 a = b1" "p21 a = b2" by blast+
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        with Inl Inl' have "Inl a \<in> ?in" "?mapp1 (Inl a) = B1 \<and> ?mapp2 (Inl a) = B2"
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        by (simp add: sum_set_defs)+
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        thus ?thesis by blast
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      next
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        case (Inr b1)
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        { fix b2 assume "B2 = Inl b2"
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          with Inr *(3) have False by simp
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        } then obtain b2 where Inr': "B2 = Inr b2" by (cases B2) (simp, blast)
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        with Inr * have "b1 \<in> B12" "b2 \<in> B22" "f12 b1 = f22 b2"
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        by (simp add: sum_set_defs)+
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        with pull2 obtain a where "a \<in> A2" "p12 a = b1" "p22 a = b2" by blast+
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        with Inr Inr' have "Inr a \<in> ?in" "?mapp1 (Inr a) = B1 \<and> ?mapp2 (Inr a) = B2"
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        by (simp add: sum_set_defs)+
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        thus ?thesis by blast
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      qed
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    }
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    thus "\<forall>B1 B2. B1 \<in> ?in1 \<and> B2 \<in> ?in2 \<and> ?mapf1 B1 = ?mapf2 B2 \<longrightarrow>
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      (\<exists>A \<in> ?in. ?mapp1 A = B1 \<and> ?mapp2 A = B2)" by fastforce
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  qed
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next
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  fix R S
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  show "sum_rel R S =
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        (Grp {x. setl x \<subseteq> Collect (split R) \<and> setr x \<subseteq> Collect (split S)} (sum_map fst fst))\<inverse>\<inverse> OO
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        Grp {x. setl x \<subseteq> Collect (split R) \<and> setr x \<subseteq> Collect (split S)} (sum_map snd snd)"
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  unfolding setl_def setr_def sum_rel_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 prod_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'b \<times> 'd \<Rightarrow> bool" where
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"prod_rel \<phi> \<psi> p1 p2 = (case p1 of (a1, b1) \<Rightarrow> case p2 of (a2, b2) \<Rightarrow> \<phi> a1 a2 \<and> \<psi> b1 b2)"
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bnf map_pair [fsts, snds] "\<lambda>_::'a \<times> 'b. natLeq" [Pair] prod_rel
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proof (unfold prod_set_defs)
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  show "map_pair id id = id" by (rule map_pair.id)
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next
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  fix f1 f2 g1 g2
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  show "map_pair (g1 o f1) (g2 o f2) = map_pair g1 g2 o map_pair f1 f2"
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    by (rule map_pair.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_pair f1 f2 x = map_pair 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_pair 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_pair 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 (metis ordLess_imp_ordLeq finite_iff_ordLess_natLeq finite.emptyI finite_insert)
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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 (metis ordLess_imp_ordLeq finite_iff_ordLess_natLeq finite.emptyI finite_insert)
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next
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  fix A1 A2 B11 B12 B21 B22 f11 f12 f21 f22 p11 p12 p21 p22
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  assume "wpull A1 B11 B21 f11 f21 p11 p21" "wpull A2 B12 B22 f12 f22 p12 p22"
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  thus "wpull {x. {fst x} \<subseteq> A1 \<and> {snd x} \<subseteq> A2}
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    {x. {fst x} \<subseteq> B11 \<and> {snd x} \<subseteq> B12} {x. {fst x} \<subseteq> B21 \<and> {snd x} \<subseteq> B22}
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   (map_pair f11 f12) (map_pair f21 f22) (map_pair p11 p12) (map_pair p21 p22)"
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    unfolding wpull_def by simp fast
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next
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  fix R S
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  show "prod_rel R S =
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        (Grp {x. {fst x} \<subseteq> Collect (split R) \<and> {snd x} \<subseteq> Collect (split S)} (map_pair 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_pair snd snd)"
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  unfolding prod_set_defs prod_rel_def Grp_def relcompp.simps conversep.simps fun_eq_iff
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  by auto
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qed simp+
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(* Categorical version of pullback: *)
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lemma wpull_cat:
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assumes p: "wpull A B1 B2 f1 f2 p1 p2"
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and c: "f1 o q1 = f2 o q2"
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and r: "range q1 \<subseteq> B1" "range q2 \<subseteq> B2"
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obtains h where "range h \<subseteq> A \<and> q1 = p1 o h \<and> q2 = p2 o h"
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proof-
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  have *: "\<forall>d. \<exists>a \<in> A. p1 a = q1 d & p2 a = q2 d"
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  proof safe
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    fix d
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    have "f1 (q1 d) = f2 (q2 d)" using c unfolding comp_def[abs_def] by (rule fun_cong)
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    moreover
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    have "q1 d : B1" "q2 d : B2" using r unfolding image_def by auto
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    ultimately show "\<exists>a \<in> A. p1 a = q1 d \<and> p2 a = q2 d"
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      using p unfolding wpull_def by auto
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  qed
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  then obtain h where "!! d. h d \<in> A & p1 (h d) = q1 d & p2 (h d) = q2 d" by metis
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  thus ?thesis using that by fastforce
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qed
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lemma card_of_bounded_range:
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  "|{f :: 'd \<Rightarrow> 'a. range f \<subseteq> B}| \<le>o |Func (UNIV :: 'd set) B|" (is "|?LHS| \<le>o |?RHS|")
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proof -
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  let ?f = "\<lambda>f. %x. if f x \<in> B then f x else undefined"
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  have "inj_on ?f ?LHS" unfolding inj_on_def
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  proof (unfold fun_eq_iff, safe)
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    fix g :: "'d \<Rightarrow> 'a" and f :: "'d \<Rightarrow> 'a" and x
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    assume "range f \<subseteq> B" "range g \<subseteq> B" and eq: "\<forall>x. ?f f x = ?f g x"
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    hence "f x \<in> B" "g x \<in> B" by auto
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    with eq have "Some (f x) = Some (g x)" by metis
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    thus "f x = g x" by simp
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  qed
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  moreover have "?f ` ?LHS \<subseteq> ?RHS" unfolding Func_def by fastforce
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  ultimately show ?thesis using card_of_ordLeq by fast
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qed
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bnf "op \<circ>" [range] "\<lambda>_:: 'a \<Rightarrow> 'b. natLeq +c |UNIV :: 'a set|" ["%c x::'b::type. c::'a::type"]
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  "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 A B1 B2 f1 f2 p1 p2 assume p: "wpull A B1 B2 f1 f2 p1 p2"
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  show "wpull {h. range h \<subseteq> A} {g1. range g1 \<subseteq> B1} {g2. range g2 \<subseteq> B2}
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    (op \<circ> f1) (op \<circ> f2) (op \<circ> p1) (op \<circ> p2)"
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  unfolding wpull_def
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  proof safe
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    fix g1 g2 assume r: "range g1 \<subseteq> B1" "range g2 \<subseteq> B2"
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    and c: "f1 \<circ> g1 = f2 \<circ> g2"
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    show "\<exists>h \<in> {h. range h \<subseteq> A}. p1 \<circ> h = g1 \<and> p2 \<circ> h = g2"
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    using wpull_cat[OF p c r] by simp metis
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  qed
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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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  by auto (force, metis pair_collapse)
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qed auto
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end