(*  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