src/HOL/Lifting_Set.thy
author kuncar
Fri Sep 27 14:43:26 2013 +0200 (2013-09-27)
changeset 53952 b2781a3ce958
parent 53945 4191acef9d0e
child 54257 5c7a3b6b05a9
permissions -rw-r--r--
new parametricity rules and useful lemmas
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(*  Title:      HOL/Lifting_Set.thy
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    Author:     Brian Huffman and Ondrej Kuncar
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*)
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header {* Setup for Lifting/Transfer for the set type *}
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theory Lifting_Set
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imports Lifting
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begin
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subsection {* Relator and predicator properties *}
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definition set_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool"
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  where "set_rel R = (\<lambda>A B. (\<forall>x\<in>A. \<exists>y\<in>B. R x y) \<and> (\<forall>y\<in>B. \<exists>x\<in>A. R x y))"
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lemma set_relI:
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  assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. R x y"
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  assumes "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. R x y"
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  shows "set_rel R A B"
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  using assms unfolding set_rel_def by simp
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lemma set_relD1: "\<lbrakk> set_rel R A B; x \<in> A \<rbrakk> \<Longrightarrow> \<exists>y \<in> B. R x y"
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  and set_relD2: "\<lbrakk> set_rel R A B; y \<in> B \<rbrakk> \<Longrightarrow> \<exists>x \<in> A. R x y"
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by(simp_all add: set_rel_def)
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lemma set_rel_conversep [simp]: "set_rel A\<inverse>\<inverse> = (set_rel A)\<inverse>\<inverse>"
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  unfolding set_rel_def by auto
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lemma set_rel_eq [relator_eq]: "set_rel (op =) = (op =)"
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  unfolding set_rel_def fun_eq_iff by auto
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lemma set_rel_mono[relator_mono]:
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  assumes "A \<le> B"
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  shows "set_rel A \<le> set_rel B"
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using assms unfolding set_rel_def by blast
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lemma set_rel_OO[relator_distr]: "set_rel R OO set_rel S = set_rel (R OO S)"
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  apply (rule sym)
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  apply (intro ext, rename_tac X Z)
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  apply (rule iffI)
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  apply (rule_tac b="{y. (\<exists>x\<in>X. R x y) \<and> (\<exists>z\<in>Z. S y z)}" in relcomppI)
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  apply (simp add: set_rel_def, fast)
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  apply (simp add: set_rel_def, fast)
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  apply (simp add: set_rel_def, fast)
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  done
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lemma Domainp_set[relator_domain]:
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  assumes "Domainp T = R"
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  shows "Domainp (set_rel T) = (\<lambda>A. Ball A R)"
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using assms unfolding set_rel_def Domainp_iff[abs_def]
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apply (intro ext)
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apply (rule iffI) 
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apply blast
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apply (rename_tac A, rule_tac x="{y. \<exists>x\<in>A. T x y}" in exI, fast)
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done
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lemma reflp_set_rel[reflexivity_rule]: "reflp R \<Longrightarrow> reflp (set_rel R)"
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  unfolding reflp_def set_rel_def by fast
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lemma left_total_set_rel[reflexivity_rule]: 
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  "left_total A \<Longrightarrow> left_total (set_rel A)"
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  unfolding left_total_def set_rel_def
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  apply safe
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  apply (rename_tac X, rule_tac x="{y. \<exists>x\<in>X. A x y}" in exI, fast)
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done
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lemma left_unique_set_rel[reflexivity_rule]: 
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  "left_unique A \<Longrightarrow> left_unique (set_rel A)"
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  unfolding left_unique_def set_rel_def
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  by fast
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lemma right_total_set_rel [transfer_rule]:
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  "right_total A \<Longrightarrow> right_total (set_rel A)"
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using left_total_set_rel[of "A\<inverse>\<inverse>"] by simp
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lemma right_unique_set_rel [transfer_rule]:
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  "right_unique A \<Longrightarrow> right_unique (set_rel A)"
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  unfolding right_unique_def set_rel_def by fast
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lemma bi_total_set_rel [transfer_rule]:
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  "bi_total A \<Longrightarrow> bi_total (set_rel A)"
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by(simp add: bi_total_conv_left_right left_total_set_rel right_total_set_rel)
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lemma bi_unique_set_rel [transfer_rule]:
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  "bi_unique A \<Longrightarrow> bi_unique (set_rel A)"
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  unfolding bi_unique_def set_rel_def by fast
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lemma set_invariant_commute [invariant_commute]:
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  "set_rel (Lifting.invariant P) = Lifting.invariant (\<lambda>A. Ball A P)"
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  unfolding fun_eq_iff set_rel_def Lifting.invariant_def Ball_def by fast
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subsection {* Quotient theorem for the Lifting package *}
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lemma Quotient_set[quot_map]:
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  assumes "Quotient R Abs Rep T"
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  shows "Quotient (set_rel R) (image Abs) (image Rep) (set_rel T)"
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  using assms unfolding Quotient_alt_def4
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  apply (simp add: set_rel_OO[symmetric])
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  apply (simp add: set_rel_def, fast)
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  done
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subsection {* Transfer rules for the Transfer package *}
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subsubsection {* Unconditional transfer rules *}
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context
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begin
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interpretation lifting_syntax .
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lemma empty_transfer [transfer_rule]: "(set_rel A) {} {}"
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  unfolding set_rel_def by simp
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lemma insert_transfer [transfer_rule]:
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  "(A ===> set_rel A ===> set_rel A) insert insert"
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  unfolding fun_rel_def set_rel_def by auto
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lemma union_transfer [transfer_rule]:
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  "(set_rel A ===> set_rel A ===> set_rel A) union union"
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  unfolding fun_rel_def set_rel_def by auto
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lemma Union_transfer [transfer_rule]:
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  "(set_rel (set_rel A) ===> set_rel A) Union Union"
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  unfolding fun_rel_def set_rel_def by simp fast
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lemma image_transfer [transfer_rule]:
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  "((A ===> B) ===> set_rel A ===> set_rel B) image image"
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  unfolding fun_rel_def set_rel_def by simp fast
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lemma UNION_transfer [transfer_rule]:
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  "(set_rel A ===> (A ===> set_rel B) ===> set_rel B) UNION UNION"
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  unfolding SUP_def [abs_def] by transfer_prover
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lemma Ball_transfer [transfer_rule]:
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  "(set_rel A ===> (A ===> op =) ===> op =) Ball Ball"
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  unfolding set_rel_def fun_rel_def by fast
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lemma Bex_transfer [transfer_rule]:
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  "(set_rel A ===> (A ===> op =) ===> op =) Bex Bex"
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  unfolding set_rel_def fun_rel_def by fast
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lemma Pow_transfer [transfer_rule]:
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  "(set_rel A ===> set_rel (set_rel A)) Pow Pow"
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  apply (rule fun_relI, rename_tac X Y, rule set_relI)
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  apply (rename_tac X', rule_tac x="{y\<in>Y. \<exists>x\<in>X'. A x y}" in rev_bexI, clarsimp)
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  apply (simp add: set_rel_def, fast)
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  apply (rename_tac Y', rule_tac x="{x\<in>X. \<exists>y\<in>Y'. A x y}" in rev_bexI, clarsimp)
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  apply (simp add: set_rel_def, fast)
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  done
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lemma set_rel_transfer [transfer_rule]:
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  "((A ===> B ===> op =) ===> set_rel A ===> set_rel B ===> op =)
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    set_rel set_rel"
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  unfolding fun_rel_def set_rel_def by fast
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lemma SUPR_parametric [transfer_rule]:
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  "(set_rel R ===> (R ===> op =) ===> op =) SUPR SUPR"
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proof(rule fun_relI)+
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  fix A B f and g :: "'b \<Rightarrow> 'c"
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  assume AB: "set_rel R A B"
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    and fg: "(R ===> op =) f g"
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  show "SUPR A f = SUPR B g"
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    by(rule SUPR_eq)(auto 4 4 dest: set_relD1[OF AB] set_relD2[OF AB] fun_relD[OF fg] intro: rev_bexI)
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qed
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lemma bind_transfer [transfer_rule]:
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  "(set_rel A ===> (A ===> set_rel B) ===> set_rel B) Set.bind Set.bind"
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unfolding bind_UNION[abs_def] by transfer_prover
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subsubsection {* Rules requiring bi-unique, bi-total or right-total relations *}
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lemma member_transfer [transfer_rule]:
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  assumes "bi_unique A"
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  shows "(A ===> set_rel A ===> op =) (op \<in>) (op \<in>)"
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  using assms unfolding fun_rel_def set_rel_def bi_unique_def by fast
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lemma right_total_Collect_transfer[transfer_rule]:
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  assumes "right_total A"
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  shows "((A ===> op =) ===> set_rel A) (\<lambda>P. Collect (\<lambda>x. P x \<and> Domainp A x)) Collect"
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  using assms unfolding right_total_def set_rel_def fun_rel_def Domainp_iff by fast
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lemma Collect_transfer [transfer_rule]:
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  assumes "bi_total A"
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  shows "((A ===> op =) ===> set_rel A) Collect Collect"
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  using assms unfolding fun_rel_def set_rel_def bi_total_def by fast
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lemma inter_transfer [transfer_rule]:
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  assumes "bi_unique A"
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  shows "(set_rel A ===> set_rel A ===> set_rel A) inter inter"
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  using assms unfolding fun_rel_def set_rel_def bi_unique_def by fast
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lemma Diff_transfer [transfer_rule]:
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  assumes "bi_unique A"
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  shows "(set_rel A ===> set_rel A ===> set_rel A) (op -) (op -)"
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  using assms unfolding fun_rel_def set_rel_def bi_unique_def
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  unfolding Ball_def Bex_def Diff_eq
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  by (safe, simp, metis, simp, metis)
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lemma subset_transfer [transfer_rule]:
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  assumes [transfer_rule]: "bi_unique A"
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  shows "(set_rel A ===> set_rel A ===> op =) (op \<subseteq>) (op \<subseteq>)"
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  unfolding subset_eq [abs_def] by transfer_prover
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lemma right_total_UNIV_transfer[transfer_rule]: 
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  assumes "right_total A"
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  shows "(set_rel A) (Collect (Domainp A)) UNIV"
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  using assms unfolding right_total_def set_rel_def Domainp_iff by blast
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lemma UNIV_transfer [transfer_rule]:
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  assumes "bi_total A"
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  shows "(set_rel A) UNIV UNIV"
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  using assms unfolding set_rel_def bi_total_def by simp
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lemma right_total_Compl_transfer [transfer_rule]:
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  assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
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  shows "(set_rel A ===> set_rel A) (\<lambda>S. uminus S \<inter> Collect (Domainp A)) uminus"
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  unfolding Compl_eq [abs_def]
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  by (subst Collect_conj_eq[symmetric]) transfer_prover
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lemma Compl_transfer [transfer_rule]:
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  assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
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  shows "(set_rel A ===> set_rel A) uminus uminus"
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  unfolding Compl_eq [abs_def] by transfer_prover
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lemma right_total_Inter_transfer [transfer_rule]:
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  assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
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  shows "(set_rel (set_rel A) ===> set_rel A) (\<lambda>S. Inter S \<inter> Collect (Domainp A)) Inter"
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  unfolding Inter_eq[abs_def]
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  by (subst Collect_conj_eq[symmetric]) transfer_prover
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lemma Inter_transfer [transfer_rule]:
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  assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
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  shows "(set_rel (set_rel A) ===> set_rel A) Inter Inter"
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  unfolding Inter_eq [abs_def] by transfer_prover
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lemma filter_transfer [transfer_rule]:
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  assumes [transfer_rule]: "bi_unique A"
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  shows "((A ===> op=) ===> set_rel A ===> set_rel A) Set.filter Set.filter"
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  unfolding Set.filter_def[abs_def] fun_rel_def set_rel_def by blast
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lemma bi_unique_set_rel_lemma:
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  assumes "bi_unique R" and "set_rel R X Y"
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  obtains f where "Y = image f X" and "inj_on f X" and "\<forall>x\<in>X. R x (f x)"
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proof
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  let ?f = "\<lambda>x. THE y. R x y"
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  from assms show f: "\<forall>x\<in>X. R x (?f x)"
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    apply (clarsimp simp add: set_rel_def)
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    apply (drule (1) bspec, clarify)
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    apply (rule theI2, assumption)
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    apply (simp add: bi_unique_def)
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    apply assumption
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    done
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  from assms show "Y = image ?f X"
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    apply safe
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    apply (clarsimp simp add: set_rel_def)
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    apply (drule (1) bspec, clarify)
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    apply (rule image_eqI)
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    apply (rule the_equality [symmetric], assumption)
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    apply (simp add: bi_unique_def)
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    apply assumption
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    apply (clarsimp simp add: set_rel_def)
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    apply (frule (1) bspec, clarify)
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    apply (rule theI2, assumption)
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    apply (clarsimp simp add: bi_unique_def)
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    apply (simp add: bi_unique_def, metis)
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    done
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  show "inj_on ?f X"
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    apply (rule inj_onI)
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    apply (drule f [rule_format])
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    apply (drule f [rule_format])
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    apply (simp add: assms(1) [unfolded bi_unique_def])
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    done
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qed
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lemma finite_transfer [transfer_rule]:
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  "bi_unique A \<Longrightarrow> (set_rel A ===> op =) finite finite"
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  by (rule fun_relI, erule (1) bi_unique_set_rel_lemma,
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    auto dest: finite_imageD)
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lemma card_transfer [transfer_rule]:
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  "bi_unique A \<Longrightarrow> (set_rel A ===> op =) card card"
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  by (rule fun_relI, erule (1) bi_unique_set_rel_lemma, simp add: card_image)
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lemma vimage_parametric [transfer_rule]:
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  assumes [transfer_rule]: "bi_total A" "bi_unique B"
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  shows "((A ===> B) ===> set_rel B ===> set_rel A) vimage vimage"
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unfolding vimage_def[abs_def] by transfer_prover
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lemma setsum_parametric [transfer_rule]:
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  assumes "bi_unique A"
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  shows "((A ===> op =) ===> set_rel A ===> op =) setsum setsum"
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proof(rule fun_relI)+
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  fix f :: "'a \<Rightarrow> 'c" and g S T
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  assume fg: "(A ===> op =) f g"
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    and ST: "set_rel A S T"
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  show "setsum f S = setsum g T"
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  proof(rule setsum_reindex_cong)
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    let ?f = "\<lambda>t. THE s. A s t"
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    show "S = ?f ` T"
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      by(blast dest: set_relD1[OF ST] set_relD2[OF ST] bi_uniqueDl[OF assms] 
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           intro: rev_image_eqI the_equality[symmetric] subst[rotated, where P="\<lambda>x. x \<in> S"])
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    show "inj_on ?f T"
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    proof(rule inj_onI)
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      fix t1 t2
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      assume "t1 \<in> T" "t2 \<in> T" "?f t1 = ?f t2"
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      from ST `t1 \<in> T` obtain s1 where "A s1 t1" "s1 \<in> S" by(auto dest: set_relD2)
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      hence "?f t1 = s1" by(auto dest: bi_uniqueDl[OF assms])
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      moreover
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      from ST `t2 \<in> T` obtain s2 where "A s2 t2" "s2 \<in> S" by(auto dest: set_relD2)
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      hence "?f t2 = s2" by(auto dest: bi_uniqueDl[OF assms])
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      ultimately have "s1 = s2" using `?f t1 = ?f t2` by simp
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      with `A s1 t1` `A s2 t2` show "t1 = t2" by(auto dest: bi_uniqueDr[OF assms])
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    qed
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    fix t
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    assume "t \<in> T"
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    with ST obtain s where "A s t" "s \<in> S" by(auto dest: set_relD2)
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    hence "?f t = s" by(auto dest: bi_uniqueDl[OF assms])
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    moreover from fg `A s t` have "f s = g t" by(rule fun_relD)
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    ultimately show "g t = f (?f t)" by simp
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  qed
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qed
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kuncar@53012
   324
end
kuncar@53012
   325
kuncar@53012
   326
end