(*  Title:      Quotient_List.thy+
−
    Author:     Cezary Kaliszyk and Christian Urban+
−
*)+
−
theory Quotient_List+
−
imports Quotient Quotient_Syntax List+
−
begin+
−
+
−
section {* Quotient infrastructure for the list type. *}+
−
+
−
fun+
−
  list_rel+
−
where+
−
  "list_rel R [] [] = True"+
−
| "list_rel R (x#xs) [] = False"+
−
| "list_rel R [] (x#xs) = False"+
−
| "list_rel R (x#xs) (y#ys) = (R x y \<and> list_rel R xs ys)"+
−
+
−
declare [[map list = (map, list_rel)]]+
−
+
−
lemma split_list_all:+
−
  shows "(\<forall>x. P x) \<longleftrightarrow> P [] \<and> (\<forall>x xs. P (x#xs))"+
−
  apply(auto)+
−
  apply(case_tac x)+
−
  apply(simp_all)+
−
  done+
−
+
−
lemma map_id[id_simps]:+
−
  shows "map id = id"+
−
  apply(simp add: expand_fun_eq)+
−
  apply(rule allI)+
−
  apply(induct_tac x)+
−
  apply(simp_all)+
−
  done+
−
+
−
+
−
lemma list_rel_reflp:+
−
  shows "equivp R \<Longrightarrow> list_rel R xs xs"+
−
  apply(induct xs)+
−
  apply(simp_all add: equivp_reflp)+
−
  done+
−
+
−
lemma list_rel_symp:+
−
  assumes a: "equivp R"+
−
  shows "list_rel R xs ys \<Longrightarrow> list_rel R ys xs"+
−
  apply(induct xs ys rule: list_induct2')+
−
  apply(simp_all)+
−
  apply(rule equivp_symp[OF a])+
−
  apply(simp)+
−
  done+
−
+
−
lemma list_rel_transp:+
−
  assumes a: "equivp R"+
−
  shows "list_rel R xs1 xs2 \<Longrightarrow> list_rel R xs2 xs3 \<Longrightarrow> list_rel R xs1 xs3"+
−
  apply(induct xs1 xs2 arbitrary: xs3 rule: list_induct2')+
−
  apply(simp_all)+
−
  apply(case_tac xs3)+
−
  apply(simp_all)+
−
  apply(rule equivp_transp[OF a])+
−
  apply(auto)+
−
  done+
−
+
−
lemma list_equivp[quot_equiv]:+
−
  assumes a: "equivp R"+
−
  shows "equivp (list_rel R)"+
−
  apply(rule equivpI)+
−
  unfolding reflp_def symp_def transp_def+
−
  apply(subst split_list_all)+
−
  apply(simp add: equivp_reflp[OF a] list_rel_reflp[OF a])+
−
  apply(blast intro: list_rel_symp[OF a])+
−
  apply(blast intro: list_rel_transp[OF a])+
−
  done+
−
+
−
lemma list_rel_rel:+
−
  assumes q: "Quotient R Abs Rep"+
−
  shows "list_rel R r s = (list_rel R r r \<and> list_rel R s s \<and> (map Abs r = map Abs s))"+
−
  apply(induct r s rule: list_induct2')+
−
  apply(simp_all)+
−
  using Quotient_rel[OF q]+
−
  apply(metis)+
−
  done+
−
+
−
lemma list_quotient[quot_thm]:+
−
  assumes q: "Quotient R Abs Rep"+
−
  shows "Quotient (list_rel R) (map Abs) (map Rep)"+
−
  unfolding Quotient_def+
−
  apply(subst split_list_all)+
−
  apply(simp add: Quotient_abs_rep[OF q] abs_o_rep[OF q] map_id)+
−
  apply(rule conjI)+
−
  apply(rule allI)+
−
  apply(induct_tac a)+
−
  apply(simp)+
−
  apply(simp)+
−
  apply(simp add: Quotient_rep_reflp[OF q])+
−
  apply(rule allI)++
−
  apply(rule list_rel_rel[OF q])+
−
  done+
−
+
−
+
−
lemma cons_prs_aux:+
−
  assumes q: "Quotient R Abs Rep"+
−
  shows "(map Abs) ((Rep h) # (map Rep t)) = h # t"+
−
  by (induct t) (simp_all add: Quotient_abs_rep[OF q])+
−
+
−
lemma cons_prs[quot_preserve]:+
−
  assumes q: "Quotient R Abs Rep"+
−
  shows "(Rep ---> (map Rep) ---> (map Abs)) (op #) = (op #)"+
−
  by (simp only: expand_fun_eq fun_map_def cons_prs_aux[OF q])+
−
     (simp)+
−
+
−
lemma cons_rsp[quot_respect]:+
−
  assumes q: "Quotient R Abs Rep"+
−
  shows "(R ===> list_rel R ===> list_rel R) (op #) (op #)"+
−
  by (auto)+
−
+
−
lemma nil_prs[quot_preserve]:+
−
  assumes q: "Quotient R Abs Rep"+
−
  shows "map Abs [] = []"+
−
  by simp+
−
+
−
lemma nil_rsp[quot_respect]:+
−
  assumes q: "Quotient R Abs Rep"+
−
  shows "list_rel R [] []"+
−
  by simp+
−
+
−
lemma map_prs_aux:+
−
  assumes a: "Quotient R1 abs1 rep1"+
−
  and     b: "Quotient R2 abs2 rep2"+
−
  shows "(map abs2) (map ((abs1 ---> rep2) f) (map rep1 l)) = map f l"+
−
  by (induct l)+
−
     (simp_all add: Quotient_abs_rep[OF a] Quotient_abs_rep[OF b])+
−
+
−
+
−
lemma map_prs[quot_preserve]:+
−
  assumes a: "Quotient R1 abs1 rep1"+
−
  and     b: "Quotient R2 abs2 rep2"+
−
  shows "((abs1 ---> rep2) ---> (map rep1) ---> (map abs2)) map = map"+
−
  by (simp only: expand_fun_eq fun_map_def map_prs_aux[OF a b])+
−
     (simp)+
−
+
−
+
−
lemma map_rsp[quot_respect]:+
−
  assumes q1: "Quotient R1 Abs1 Rep1"+
−
  and     q2: "Quotient R2 Abs2 Rep2"+
−
  shows "((R1 ===> R2) ===> (list_rel R1) ===> list_rel R2) map map"+
−
  apply(simp)+
−
  apply(rule allI)++
−
  apply(rule impI)+
−
  apply(rule allI)++
−
  apply (induct_tac xa ya rule: list_induct2')+
−
  apply simp_all+
−
  done+
−
+
−
lemma foldr_prs_aux:+
−
  assumes a: "Quotient R1 abs1 rep1"+
−
  and     b: "Quotient R2 abs2 rep2"+
−
  shows "abs2 (foldr ((abs1 ---> abs2 ---> rep2) f) (map rep1 l) (rep2 e)) = foldr f l e"+
−
  by (induct l) (simp_all add: Quotient_abs_rep[OF a] Quotient_abs_rep[OF b])+
−
+
−
lemma foldr_prs[quot_preserve]:+
−
  assumes a: "Quotient R1 abs1 rep1"+
−
  and     b: "Quotient R2 abs2 rep2"+
−
  shows "((abs1 ---> abs2 ---> rep2) ---> (map rep1) ---> rep2 ---> abs2) foldr = foldr"+
−
  by (simp only: expand_fun_eq fun_map_def foldr_prs_aux[OF a b])+
−
     (simp)+
−
+
−
lemma foldl_prs_aux:+
−
  assumes a: "Quotient R1 abs1 rep1"+
−
  and     b: "Quotient R2 abs2 rep2"+
−
  shows "abs1 (foldl ((abs1 ---> abs2 ---> rep1) f) (rep1 e) (map rep2 l)) = foldl f e l"+
−
  by (induct l arbitrary:e) (simp_all add: Quotient_abs_rep[OF a] Quotient_abs_rep[OF b])+
−
+
−
+
−
lemma foldl_prs[quot_preserve]:+
−
  assumes a: "Quotient R1 abs1 rep1"+
−
  and     b: "Quotient R2 abs2 rep2"+
−
  shows "((abs1 ---> abs2 ---> rep1) ---> rep1 ---> (map rep2) ---> abs1) foldl = foldl"+
−
  by (simp only: expand_fun_eq fun_map_def foldl_prs_aux[OF a b])+
−
     (simp)+
−
+
−
lemma list_rel_empty:+
−
  shows "list_rel R [] b \<Longrightarrow> length b = 0"+
−
  by (induct b) (simp_all)+
−
+
−
lemma list_rel_len:+
−
  shows "list_rel R a b \<Longrightarrow> length a = length b"+
−
  apply (induct a arbitrary: b)+
−
  apply (simp add: list_rel_empty)+
−
  apply (case_tac b)+
−
  apply simp_all+
−
  done+
−
+
−
(* induct_tac doesn't accept 'arbitrary', so we manually 'spec' *)+
−
lemma foldl_rsp[quot_respect]:+
−
  assumes q1: "Quotient R1 Abs1 Rep1"+
−
  and     q2: "Quotient R2 Abs2 Rep2"+
−
  shows "((R1 ===> R2 ===> R1) ===> R1 ===> list_rel R2 ===> R1) foldl foldl"+
−
  apply(auto)+
−
  apply (subgoal_tac "R1 xa ya \<longrightarrow> list_rel R2 xb yb \<longrightarrow> R1 (foldl x xa xb) (foldl y ya yb)")+
−
  apply simp+
−
  apply (rule_tac x="xa" in spec)+
−
  apply (rule_tac x="ya" in spec)+
−
  apply (rule_tac xs="xb" and ys="yb" in list_induct2)+
−
  apply (rule list_rel_len)+
−
  apply (simp_all)+
−
  done+
−
+
−
lemma foldr_rsp[quot_respect]:+
−
  assumes q1: "Quotient R1 Abs1 Rep1"+
−
  and     q2: "Quotient R2 Abs2 Rep2"+
−
  shows "((R1 ===> R2 ===> R2) ===> list_rel R1 ===> R2 ===> R2) foldr foldr"+
−
  apply auto+
−
  apply(subgoal_tac "R2 xb yb \<longrightarrow> list_rel R1 xa ya \<longrightarrow> R2 (foldr x xa xb) (foldr y ya yb)")+
−
  apply simp+
−
  apply (rule_tac xs="xa" and ys="ya" in list_induct2)+
−
  apply (rule list_rel_len)+
−
  apply (simp_all)+
−
  done+
−
+
−
lemma list_rel_eq[id_simps]:+
−
  shows "(list_rel (op =)) = (op =)"+
−
  unfolding expand_fun_eq+
−
  apply(rule allI)++
−
  apply(induct_tac x xa rule: list_induct2')+
−
  apply(simp_all)+
−
  done+
−
+
−
lemma list_rel_refl:+
−
  assumes a: "\<And>x y. R x y = (R x = R y)"+
−
  shows "list_rel R x x"+
−
  by (induct x) (auto simp add: a)+
−
+
−
end+
−