theory QuotList

imports QuotScript List

begin

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)"

lemma list_equivp:

  assumes a: "equivp R"

  shows "equivp (list_rel R)"

  unfolding equivp_def

  apply(rule allI)+

  apply(induct_tac x y rule: list_induct2')

  apply(simp_all add: expand_fun_eq)

  apply(metis list_rel.simps(1) list_rel.simps(2))

  apply(metis list_rel.simps(1) list_rel.simps(2))

  apply(rule iffI)

  apply(rule allI)

  apply(case_tac x)

  apply(simp_all)

  using a

  apply(unfold equivp_def)

  apply(auto)[1]

  apply(metis list_rel.simps(4))

  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:

  assumes q: "Quotient R Abs Rep"

  shows "Quotient (list_rel R) (map Abs) (map Rep)"

  unfolding Quotient_def

  apply(rule conjI)

  apply(rule allI)

  apply(induct_tac a)

  apply(simp)

  apply(simp add: Quotient_abs_rep[OF q])

  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:

  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_rsp:

  assumes q: "Quotient R Abs Rep"

  shows "(R ===> list_rel R ===> list_rel R) op # op #"

by (auto)

lemma nil_prs:

  assumes q: "Quotient R Abs Rep"

  shows "map Abs [] \<equiv> []"

by (simp)

lemma nil_rsp:

  assumes q: "Quotient R Abs Rep"

  shows "list_rel R [] []"

by simp

lemma map_prs:

  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_rsp:

  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

(* TODO: if the above is correct, we can remove this one *)

lemma map_rsp_lo:

  assumes q1: "Quotient R1 Abs1 Rep1"

  and     q2: "Quotient R2 Abs2 Rep2"

  and     a: "(R1 ===> R2) f1 f2"

  and     b: "list_rel R1 l1 l2"

  shows "list_rel R2 (map f1 l1) (map f2 l2)"

using b a

by (induct l1 l2 rule: list_induct2') (simp_all)

lemma foldr_prs:

  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 foldl_prs:

  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 list_rel_empty: "list_rel R [] b \<Longrightarrow> length b = 0"

by (induct b) (simp_all)

lemma list_rel_len: "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

(* TODO: induct_tac doesn't accept 'arbitrary'.

         induct     doesn't accept 'rule'.

   that's why the proof uses manual generalisation and needs assumptions

   both in conclusion for induction and in assumptions. *)

lemma foldl_rsp:

  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

(* TODO: foldr_rsp should be similar *)

(* TODO: Rest are unused *)

lemma list_map_id:

  shows "map (\<lambda>x. x) = (\<lambda>x. x)"

  by simp

lemma list_rel_eq:

  shows "list_rel (op =) \<equiv> (op =)"

apply(rule eq_reflection)

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