src/HOL/Lifting_Set.thy
author haftmann
Wed Apr 09 10:04:31 2014 +0200 (2014-04-09)
changeset 56482 39ac12b655ab
parent 56218 1c3f1f2431f9
child 56518 beb3b6851665
permissions -rw-r--r--
parametricity transfer rule for INFIMUM, SUPREMUM
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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 rel_set :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool"
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  where "rel_set 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 rel_setI:
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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 "rel_set R A B"
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  using assms unfolding rel_set_def by simp
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lemma rel_setD1: "\<lbrakk> rel_set R A B; x \<in> A \<rbrakk> \<Longrightarrow> \<exists>y \<in> B. R x y"
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  and rel_setD2: "\<lbrakk> rel_set R A B; y \<in> B \<rbrakk> \<Longrightarrow> \<exists>x \<in> A. R x y"
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by(simp_all add: rel_set_def)
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lemma rel_set_conversep [simp]: "rel_set A\<inverse>\<inverse> = (rel_set A)\<inverse>\<inverse>"
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  unfolding rel_set_def by auto
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lemma rel_set_eq [relator_eq]: "rel_set (op =) = (op =)"
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  unfolding rel_set_def fun_eq_iff by auto
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lemma rel_set_mono[relator_mono]:
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  assumes "A \<le> B"
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  shows "rel_set A \<le> rel_set B"
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using assms unfolding rel_set_def by blast
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lemma rel_set_OO[relator_distr]: "rel_set R OO rel_set S = rel_set (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: rel_set_def, fast)
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  apply (simp add: rel_set_def, fast)
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  apply (simp add: rel_set_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 (rel_set T) = (\<lambda>A. Ball A R)"
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using assms unfolding rel_set_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 left_total_rel_set[reflexivity_rule]: 
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  "left_total A \<Longrightarrow> left_total (rel_set A)"
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  unfolding left_total_def rel_set_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_rel_set[reflexivity_rule]: 
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  "left_unique A \<Longrightarrow> left_unique (rel_set A)"
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  unfolding left_unique_def rel_set_def
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  by fast
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lemma right_total_rel_set [transfer_rule]:
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  "right_total A \<Longrightarrow> right_total (rel_set A)"
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using left_total_rel_set[of "A\<inverse>\<inverse>"] by simp
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lemma right_unique_rel_set [transfer_rule]:
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  "right_unique A \<Longrightarrow> right_unique (rel_set A)"
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  unfolding right_unique_def rel_set_def by fast
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lemma bi_total_rel_set [transfer_rule]:
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  "bi_total A \<Longrightarrow> bi_total (rel_set A)"
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by(simp add: bi_total_conv_left_right left_total_rel_set right_total_rel_set)
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lemma bi_unique_rel_set [transfer_rule]:
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  "bi_unique A \<Longrightarrow> bi_unique (rel_set A)"
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  unfolding bi_unique_def rel_set_def by fast
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lemma set_invariant_commute [invariant_commute]:
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  "rel_set (Lifting.invariant P) = Lifting.invariant (\<lambda>A. Ball A P)"
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  unfolding fun_eq_iff rel_set_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 (rel_set R) (image Abs) (image Rep) (rel_set T)"
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  using assms unfolding Quotient_alt_def4
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  apply (simp add: rel_set_OO[symmetric])
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  apply (simp add: rel_set_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]: "(rel_set A) {} {}"
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  unfolding rel_set_def by simp
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lemma insert_transfer [transfer_rule]:
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  "(A ===> rel_set A ===> rel_set A) insert insert"
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  unfolding rel_fun_def rel_set_def by auto
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lemma union_transfer [transfer_rule]:
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  "(rel_set A ===> rel_set A ===> rel_set A) union union"
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  unfolding rel_fun_def rel_set_def by auto
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lemma Union_transfer [transfer_rule]:
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  "(rel_set (rel_set A) ===> rel_set A) Union Union"
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  unfolding rel_fun_def rel_set_def by simp fast
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lemma image_transfer [transfer_rule]:
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  "((A ===> B) ===> rel_set A ===> rel_set B) image image"
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  unfolding rel_fun_def rel_set_def by simp fast
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lemma UNION_transfer [transfer_rule]:
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  "(rel_set A ===> (A ===> rel_set B) ===> rel_set B) UNION UNION"
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  unfolding Union_image_eq [symmetric, abs_def] by transfer_prover
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lemma Ball_transfer [transfer_rule]:
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  "(rel_set A ===> (A ===> op =) ===> op =) Ball Ball"
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  unfolding rel_set_def rel_fun_def by fast
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lemma Bex_transfer [transfer_rule]:
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  "(rel_set A ===> (A ===> op =) ===> op =) Bex Bex"
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  unfolding rel_set_def rel_fun_def by fast
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lemma Pow_transfer [transfer_rule]:
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  "(rel_set A ===> rel_set (rel_set A)) Pow Pow"
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  apply (rule rel_funI, rename_tac X Y, rule rel_setI)
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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: rel_set_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: rel_set_def, fast)
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  done
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lemma rel_set_transfer [transfer_rule]:
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  "((A ===> B ===> op =) ===> rel_set A ===> rel_set B ===> op =) rel_set rel_set"
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  unfolding rel_fun_def rel_set_def by fast
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lemma bind_transfer [transfer_rule]:
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  "(rel_set A ===> (A ===> rel_set B) ===> rel_set B) Set.bind Set.bind"
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  unfolding bind_UNION [abs_def] by transfer_prover
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lemma INF_parametric [transfer_rule]:
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  "(rel_set A ===> (A ===> HOL.eq) ===> HOL.eq) INFIMUM INFIMUM"
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  unfolding INF_def [abs_def] by transfer_prover
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lemma SUP_parametric [transfer_rule]:
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  "(rel_set R ===> (R ===> HOL.eq) ===> HOL.eq) SUPREMUM SUPREMUM"
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  unfolding SUP_def [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 ===> rel_set A ===> op =) (op \<in>) (op \<in>)"
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  using assms unfolding rel_fun_def rel_set_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 =) ===> rel_set A) (\<lambda>P. Collect (\<lambda>x. P x \<and> Domainp A x)) Collect"
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  using assms unfolding right_total_def rel_set_def rel_fun_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 =) ===> rel_set A) Collect Collect"
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  using assms unfolding rel_fun_def rel_set_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 "(rel_set A ===> rel_set A ===> rel_set A) inter inter"
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  using assms unfolding rel_fun_def rel_set_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 "(rel_set A ===> rel_set A ===> rel_set A) (op -) (op -)"
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  using assms unfolding rel_fun_def rel_set_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 "(rel_set A ===> rel_set 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 "(rel_set A) (Collect (Domainp A)) UNIV"
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  using assms unfolding right_total_def rel_set_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 "(rel_set A) UNIV UNIV"
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  using assms unfolding rel_set_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 "(rel_set A ===> rel_set 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 "(rel_set A ===> rel_set 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 "(rel_set (rel_set A) ===> rel_set 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 "(rel_set (rel_set A) ===> rel_set 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=) ===> rel_set A ===> rel_set A) Set.filter Set.filter"
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  unfolding Set.filter_def[abs_def] rel_fun_def rel_set_def by blast
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lemma bi_unique_rel_set_lemma:
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  assumes "bi_unique R" and "rel_set 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: rel_set_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: rel_set_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: rel_set_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> (rel_set A ===> op =) finite finite"
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  by (rule rel_funI, erule (1) bi_unique_rel_set_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> (rel_set A ===> op =) card card"
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  by (rule rel_funI, erule (1) bi_unique_rel_set_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) ===> rel_set B ===> rel_set 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 =) ===> rel_set A ===> op =) setsum setsum"
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proof(rule rel_funI)+
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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: "rel_set 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: rel_setD1[OF ST] rel_setD2[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
Andreas@53927
   300
      assume "t1 \<in> T" "t2 \<in> T" "?f t1 = ?f t2"
blanchet@55938
   301
      from ST `t1 \<in> T` obtain s1 where "A s1 t1" "s1 \<in> S" by(auto dest: rel_setD2)
Andreas@53927
   302
      hence "?f t1 = s1" by(auto dest: bi_uniqueDl[OF assms])
Andreas@53927
   303
      moreover
blanchet@55938
   304
      from ST `t2 \<in> T` obtain s2 where "A s2 t2" "s2 \<in> S" by(auto dest: rel_setD2)
Andreas@53927
   305
      hence "?f t2 = s2" by(auto dest: bi_uniqueDl[OF assms])
Andreas@53927
   306
      ultimately have "s1 = s2" using `?f t1 = ?f t2` by simp
Andreas@53927
   307
      with `A s1 t1` `A s2 t2` show "t1 = t2" by(auto dest: bi_uniqueDr[OF assms])
Andreas@53927
   308
    qed
Andreas@53927
   309
Andreas@53927
   310
    fix t
Andreas@53927
   311
    assume "t \<in> T"
blanchet@55938
   312
    with ST obtain s where "A s t" "s \<in> S" by(auto dest: rel_setD2)
Andreas@53927
   313
    hence "?f t = s" by(auto dest: bi_uniqueDl[OF assms])
blanchet@55945
   314
    moreover from fg `A s t` have "f s = g t" by(rule rel_funD)
Andreas@53927
   315
    ultimately show "g t = f (?f t)" by simp
Andreas@53927
   316
  qed
Andreas@53927
   317
qed
Andreas@53927
   318
kuncar@53012
   319
end
kuncar@53012
   320
kuncar@53012
   321
end