(*  Title:      HOL/Lifting_Set.thy

     Author:     Brian Huffman and Ondrej Kuncar

 *)

 header {* Setup for Lifting/Transfer for the set type *}

 theory Lifting_Set

 imports Lifting

 begin

 subsection {* Relator and predicator properties *}

 definition rel_set :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool"

   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))"

 lemma rel_setI:

   assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. R x y"

   assumes "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. R x y"

   shows "rel_set R A B"

   using assms unfolding rel_set_def by simp

 lemma rel_setD1: "\<lbrakk> rel_set R A B; x \<in> A \<rbrakk> \<Longrightarrow> \<exists>y \<in> B. R x y"

   and rel_setD2: "\<lbrakk> rel_set R A B; y \<in> B \<rbrakk> \<Longrightarrow> \<exists>x \<in> A. R x y"

 by(simp_all add: rel_set_def)

 lemma rel_set_conversep [simp]: "rel_set A\<inverse>\<inverse> = (rel_set A)\<inverse>\<inverse>"

   unfolding rel_set_def by auto

 lemma rel_set_eq [relator_eq]: "rel_set (op =) = (op =)"

   unfolding rel_set_def fun_eq_iff by auto

 lemma rel_set_mono[relator_mono]:

   assumes "A \<le> B"

   shows "rel_set A \<le> rel_set B"

 using assms unfolding rel_set_def by blast

 lemma rel_set_OO[relator_distr]: "rel_set R OO rel_set S = rel_set (R OO S)"

   apply (rule sym)

   apply (intro ext, rename_tac X Z)

   apply (rule iffI)

   apply (rule_tac b="{y. (\<exists>x\<in>X. R x y) \<and> (\<exists>z\<in>Z. S y z)}" in relcomppI)

   apply (simp add: rel_set_def, fast)

   apply (simp add: rel_set_def, fast)

   apply (simp add: rel_set_def, fast)

   done

 lemma Domainp_set[relator_domain]:

   "Domainp (rel_set T) = (\<lambda>A. Ball A (Domainp T))"

 unfolding rel_set_def Domainp_iff[abs_def]

 apply (intro ext)

 apply (rule iffI) 

 apply blast

 apply (rename_tac A, rule_tac x="{y. \<exists>x\<in>A. T x y}" in exI, fast)

 done

 lemma left_total_rel_set[transfer_rule]: 

   "left_total A \<Longrightarrow> left_total (rel_set A)"

   unfolding left_total_def rel_set_def

   apply safe

   apply (rename_tac X, rule_tac x="{y. \<exists>x\<in>X. A x y}" in exI, fast)

 done

 lemma left_unique_rel_set[transfer_rule]: 

   "left_unique A \<Longrightarrow> left_unique (rel_set A)"

   unfolding left_unique_def rel_set_def

   by fast

 lemma right_total_rel_set [transfer_rule]:

   "right_total A \<Longrightarrow> right_total (rel_set A)"

 using left_total_rel_set[of "A\<inverse>\<inverse>"] by simp

 lemma right_unique_rel_set [transfer_rule]:

   "right_unique A \<Longrightarrow> right_unique (rel_set A)"

   unfolding right_unique_def rel_set_def by fast

 lemma bi_total_rel_set [transfer_rule]:

   "bi_total A \<Longrightarrow> bi_total (rel_set A)"

 by(simp add: bi_total_alt_def left_total_rel_set right_total_rel_set)

 lemma bi_unique_rel_set [transfer_rule]:

   "bi_unique A \<Longrightarrow> bi_unique (rel_set A)"

   unfolding bi_unique_def rel_set_def by fast

 lemma set_relator_eq_onp [relator_eq_onp]:

   "rel_set (eq_onp P) = eq_onp (\<lambda>A. Ball A P)"

   unfolding fun_eq_iff rel_set_def eq_onp_def Ball_def by fast

 subsection {* Quotient theorem for the Lifting package *}

 lemma Quotient_set[quot_map]:

   assumes "Quotient R Abs Rep T"

   shows "Quotient (rel_set R) (image Abs) (image Rep) (rel_set T)"

   using assms unfolding Quotient_alt_def4

   apply (simp add: rel_set_OO[symmetric])

   apply (simp add: rel_set_def, fast)

   done

 subsection {* Transfer rules for the Transfer package *}

 subsubsection {* Unconditional transfer rules *}

 context

 begin

 interpretation lifting_syntax .

 lemma empty_transfer [transfer_rule]: "(rel_set A) {} {}"

   unfolding rel_set_def by simp

 lemma insert_transfer [transfer_rule]:

   "(A ===> rel_set A ===> rel_set A) insert insert"

   unfolding rel_fun_def rel_set_def by auto

 lemma union_transfer [transfer_rule]:

   "(rel_set A ===> rel_set A ===> rel_set A) union union"

   unfolding rel_fun_def rel_set_def by auto

 lemma Union_transfer [transfer_rule]:

   "(rel_set (rel_set A) ===> rel_set A) Union Union"

   unfolding rel_fun_def rel_set_def by simp fast

 lemma image_transfer [transfer_rule]:

   "((A ===> B) ===> rel_set A ===> rel_set B) image image"

   unfolding rel_fun_def rel_set_def by simp fast

 lemma UNION_transfer [transfer_rule]:

   "(rel_set A ===> (A ===> rel_set B) ===> rel_set B) UNION UNION"

   unfolding Union_image_eq [symmetric, abs_def] by transfer_prover

 lemma Ball_transfer [transfer_rule]:

   "(rel_set A ===> (A ===> op =) ===> op =) Ball Ball"

   unfolding rel_set_def rel_fun_def by fast

 lemma Bex_transfer [transfer_rule]:

   "(rel_set A ===> (A ===> op =) ===> op =) Bex Bex"

   unfolding rel_set_def rel_fun_def by fast

 lemma Pow_transfer [transfer_rule]:

   "(rel_set A ===> rel_set (rel_set A)) Pow Pow"

   apply (rule rel_funI, rename_tac X Y, rule rel_setI)

   apply (rename_tac X', rule_tac x="{y\<in>Y. \<exists>x\<in>X'. A x y}" in rev_bexI, clarsimp)

   apply (simp add: rel_set_def, fast)

   apply (rename_tac Y', rule_tac x="{x\<in>X. \<exists>y\<in>Y'. A x y}" in rev_bexI, clarsimp)

   apply (simp add: rel_set_def, fast)

   done

 lemma rel_set_transfer [transfer_rule]:

   "((A ===> B ===> op =) ===> rel_set A ===> rel_set B ===> op =) rel_set rel_set"

   unfolding rel_fun_def rel_set_def by fast

 lemma bind_transfer [transfer_rule]:

   "(rel_set A ===> (A ===> rel_set B) ===> rel_set B) Set.bind Set.bind"

   unfolding bind_UNION [abs_def] by transfer_prover

 lemma INF_parametric [transfer_rule]:

   "(rel_set A ===> (A ===> HOL.eq) ===> HOL.eq) INFIMUM INFIMUM"

   unfolding INF_def [abs_def] by transfer_prover

 lemma SUP_parametric [transfer_rule]:

   "(rel_set R ===> (R ===> HOL.eq) ===> HOL.eq) SUPREMUM SUPREMUM"

   unfolding SUP_def [abs_def] by transfer_prover

 subsubsection {* Rules requiring bi-unique, bi-total or right-total relations *}

 lemma member_transfer [transfer_rule]:

   assumes "bi_unique A"

   shows "(A ===> rel_set A ===> op =) (op \<in>) (op \<in>)"

   using assms unfolding rel_fun_def rel_set_def bi_unique_def by fast

 lemma right_total_Collect_transfer[transfer_rule]:

   assumes "right_total A"

   shows "((A ===> op =) ===> rel_set A) (\<lambda>P. Collect (\<lambda>x. P x \<and> Domainp A x)) Collect"

   using assms unfolding right_total_def rel_set_def rel_fun_def Domainp_iff by fast

 lemma Collect_transfer [transfer_rule]:

   assumes "bi_total A"

   shows "((A ===> op =) ===> rel_set A) Collect Collect"

   using assms unfolding rel_fun_def rel_set_def bi_total_def by fast

 lemma inter_transfer [transfer_rule]:

   assumes "bi_unique A"

   shows "(rel_set A ===> rel_set A ===> rel_set A) inter inter"

   using assms unfolding rel_fun_def rel_set_def bi_unique_def by fast

 lemma Diff_transfer [transfer_rule]:

   assumes "bi_unique A"

   shows "(rel_set A ===> rel_set A ===> rel_set A) (op -) (op -)"

   using assms unfolding rel_fun_def rel_set_def bi_unique_def

   unfolding Ball_def Bex_def Diff_eq

   by (safe, simp, metis, simp, metis)

 lemma subset_transfer [transfer_rule]:

   assumes [transfer_rule]: "bi_unique A"

   shows "(rel_set A ===> rel_set A ===> op =) (op \<subseteq>) (op \<subseteq>)"

   unfolding subset_eq [abs_def] by transfer_prover

 lemma right_total_UNIV_transfer[transfer_rule]: 

   assumes "right_total A"

   shows "(rel_set A) (Collect (Domainp A)) UNIV"

   using assms unfolding right_total_def rel_set_def Domainp_iff by blast

 lemma UNIV_transfer [transfer_rule]:

   assumes "bi_total A"

   shows "(rel_set A) UNIV UNIV"

   using assms unfolding rel_set_def bi_total_def by simp

 lemma right_total_Compl_transfer [transfer_rule]:

   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"

   shows "(rel_set A ===> rel_set A) (\<lambda>S. uminus S \<inter> Collect (Domainp A)) uminus"

   unfolding Compl_eq [abs_def]

   by (subst Collect_conj_eq[symmetric]) transfer_prover

 lemma Compl_transfer [transfer_rule]:

   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"

   shows "(rel_set A ===> rel_set A) uminus uminus"

   unfolding Compl_eq [abs_def] by transfer_prover

 lemma right_total_Inter_transfer [transfer_rule]:

   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"

   shows "(rel_set (rel_set A) ===> rel_set A) (\<lambda>S. Inter S \<inter> Collect (Domainp A)) Inter"

   unfolding Inter_eq[abs_def]

   by (subst Collect_conj_eq[symmetric]) transfer_prover

 lemma Inter_transfer [transfer_rule]:

   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"

   shows "(rel_set (rel_set A) ===> rel_set A) Inter Inter"

   unfolding Inter_eq [abs_def] by transfer_prover

 lemma filter_transfer [transfer_rule]:

   assumes [transfer_rule]: "bi_unique A"

   shows "((A ===> op=) ===> rel_set A ===> rel_set A) Set.filter Set.filter"

   unfolding Set.filter_def[abs_def] rel_fun_def rel_set_def by blast

 lemma bi_unique_rel_set_lemma:

   assumes "bi_unique R" and "rel_set R X Y"

   obtains f where "Y = image f X" and "inj_on f X" and "\<forall>x\<in>X. R x (f x)"

 proof

   let ?f = "\<lambda>x. THE y. R x y"

   from assms show f: "\<forall>x\<in>X. R x (?f x)"

     apply (clarsimp simp add: rel_set_def)

     apply (drule (1) bspec, clarify)

     apply (rule theI2, assumption)

     apply (simp add: bi_unique_def)

     apply assumption

     done

   from assms show "Y = image ?f X"

     apply safe

     apply (clarsimp simp add: rel_set_def)

     apply (drule (1) bspec, clarify)

     apply (rule image_eqI)

     apply (rule the_equality [symmetric], assumption)

     apply (simp add: bi_unique_def)

     apply assumption

     apply (clarsimp simp add: rel_set_def)

     apply (frule (1) bspec, clarify)

     apply (rule theI2, assumption)

     apply (clarsimp simp add: bi_unique_def)

     apply (simp add: bi_unique_def, metis)

     done

   show "inj_on ?f X"

     apply (rule inj_onI)

     apply (drule f [rule_format])

     apply (drule f [rule_format])

     apply (simp add: assms(1) [unfolded bi_unique_def])

     done

 qed

 lemma finite_transfer [transfer_rule]:

   "bi_unique A \<Longrightarrow> (rel_set A ===> op =) finite finite"

   by (rule rel_funI, erule (1) bi_unique_rel_set_lemma,

     auto dest: finite_imageD)

 lemma card_transfer [transfer_rule]:

   "bi_unique A \<Longrightarrow> (rel_set A ===> op =) card card"

   by (rule rel_funI, erule (1) bi_unique_rel_set_lemma, simp add: card_image)

 lemma vimage_parametric [transfer_rule]:

   assumes [transfer_rule]: "bi_total A" "bi_unique B"

   shows "((A ===> B) ===> rel_set B ===> rel_set A) vimage vimage"

 unfolding vimage_def[abs_def] by transfer_prover

 lemma setsum_parametric [transfer_rule]:

   assumes "bi_unique A"

   shows "((A ===> op =) ===> rel_set A ===> op =) setsum setsum"

 proof(rule rel_funI)+

   fix f :: "'a \<Rightarrow> 'c" and g S T

   assume fg: "(A ===> op =) f g"

     and ST: "rel_set A S T"

   show "setsum f S = setsum g T"

   proof(rule setsum_reindex_cong)

     let ?f = "\<lambda>t. THE s. A s t"

     show "S = ?f ` T"

       by(blast dest: rel_setD1[OF ST] rel_setD2[OF ST] bi_uniqueDl[OF assms] 

            intro: rev_image_eqI the_equality[symmetric] subst[rotated, where P="\<lambda>x. x \<in> S"])

     show "inj_on ?f T"

     proof(rule inj_onI)

       fix t1 t2

       assume "t1 \<in> T" "t2 \<in> T" "?f t1 = ?f t2"

       from ST `t1 \<in> T` obtain s1 where "A s1 t1" "s1 \<in> S" by(auto dest: rel_setD2)

       hence "?f t1 = s1" by(auto dest: bi_uniqueDl[OF assms])

       moreover

       from ST `t2 \<in> T` obtain s2 where "A s2 t2" "s2 \<in> S" by(auto dest: rel_setD2)

       hence "?f t2 = s2" by(auto dest: bi_uniqueDl[OF assms])

       ultimately have "s1 = s2" using `?f t1 = ?f t2` by simp

       with `A s1 t1` `A s2 t2` show "t1 = t2" by(auto dest: bi_uniqueDr[OF assms])

     qed

     fix t

     assume "t \<in> T"

     with ST obtain s where "A s t" "s \<in> S" by(auto dest: rel_setD2)

     hence "?f t = s" by(auto dest: bi_uniqueDl[OF assms])

     moreover from fg `A s t` have "f s = g t" by(rule rel_funD)

     ultimately show "g t = f (?f t)" by simp

   qed

 qed

 end

 end