src/HOL/Library/Quotient_List.thy
author wenzelm
Mon Dec 28 17:43:30 2015 +0100 (2015-12-28)
changeset 61952 546958347e05
parent 60500 903bb1495239
child 62954 c5d0fdc260fa
permissions -rw-r--r--
prefer symbols for "Union", "Inter";
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(*  Title:      HOL/Library/Quotient_List.thy
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    Author:     Cezary Kaliszyk and Christian Urban
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*)
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section \<open>Quotient infrastructure for the list type\<close>
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theory Quotient_List
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imports Main Quotient_Set Quotient_Product Quotient_Option
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begin
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subsection \<open>Rules for the Quotient package\<close>
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lemma map_id [id_simps]:
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  "map id = id"
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  by (fact List.map.id)
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lemma list_all2_eq [id_simps]:
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  "list_all2 (op =) = (op =)"
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proof (rule ext)+
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  fix xs ys
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  show "list_all2 (op =) xs ys \<longleftrightarrow> xs = ys"
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    by (induct xs ys rule: list_induct2') simp_all
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qed
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lemma reflp_list_all2:
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  assumes "reflp R"
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  shows "reflp (list_all2 R)"
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proof (rule reflpI)
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  from assms have *: "\<And>xs. R xs xs" by (rule reflpE)
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  fix xs
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  show "list_all2 R xs xs"
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    by (induct xs) (simp_all add: *)
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qed
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lemma list_symp:
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  assumes "symp R"
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  shows "symp (list_all2 R)"
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proof (rule sympI)
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  from assms have *: "\<And>xs ys. R xs ys \<Longrightarrow> R ys xs" by (rule sympE)
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  fix xs ys
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  assume "list_all2 R xs ys"
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  then show "list_all2 R ys xs"
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    by (induct xs ys rule: list_induct2') (simp_all add: *)
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qed
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lemma list_transp:
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  assumes "transp R"
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  shows "transp (list_all2 R)"
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proof (rule transpI)
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  from assms have *: "\<And>xs ys zs. R xs ys \<Longrightarrow> R ys zs \<Longrightarrow> R xs zs" by (rule transpE)
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  fix xs ys zs
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  assume "list_all2 R xs ys" and "list_all2 R ys zs"
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  then show "list_all2 R xs zs"
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    by (induct arbitrary: zs) (auto simp: list_all2_Cons1 intro: *)
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qed
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lemma list_equivp [quot_equiv]:
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  "equivp R \<Longrightarrow> equivp (list_all2 R)"
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  by (blast intro: equivpI reflp_list_all2 list_symp list_transp elim: equivpE)
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lemma list_quotient3 [quot_thm]:
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  assumes "Quotient3 R Abs Rep"
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  shows "Quotient3 (list_all2 R) (map Abs) (map Rep)"
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proof (rule Quotient3I)
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  from assms have "\<And>x. Abs (Rep x) = x" by (rule Quotient3_abs_rep)
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  then show "\<And>xs. map Abs (map Rep xs) = xs" by (simp add: comp_def)
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next
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  from assms have "\<And>x y. R (Rep x) (Rep y) \<longleftrightarrow> x = y" by (rule Quotient3_rel_rep)
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  then show "\<And>xs. list_all2 R (map Rep xs) (map Rep xs)"
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    by (simp add: list_all2_map1 list_all2_map2 list_all2_eq)
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next
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  fix xs ys
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  from assms have "\<And>x y. R x x \<and> R y y \<and> Abs x = Abs y \<longleftrightarrow> R x y" by (rule Quotient3_rel)
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  then show "list_all2 R xs ys \<longleftrightarrow> list_all2 R xs xs \<and> list_all2 R ys ys \<and> map Abs xs = map Abs ys"
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    by (induct xs ys rule: list_induct2') auto
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qed
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declare [[mapQ3 list = (list_all2, list_quotient3)]]
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lemma cons_prs [quot_preserve]:
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  assumes q: "Quotient3 R Abs Rep"
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  shows "(Rep ---> (map Rep) ---> (map Abs)) (op #) = (op #)"
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  by (auto simp add: fun_eq_iff comp_def Quotient3_abs_rep [OF q])
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lemma cons_rsp [quot_respect]:
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  assumes q: "Quotient3 R Abs Rep"
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  shows "(R ===> list_all2 R ===> list_all2 R) (op #) (op #)"
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  by auto
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lemma nil_prs [quot_preserve]:
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  assumes q: "Quotient3 R Abs Rep"
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  shows "map Abs [] = []"
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  by simp
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lemma nil_rsp [quot_respect]:
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  assumes q: "Quotient3 R Abs Rep"
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  shows "list_all2 R [] []"
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  by simp
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lemma map_prs_aux:
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  assumes a: "Quotient3 R1 abs1 rep1"
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  and     b: "Quotient3 R2 abs2 rep2"
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  shows "(map abs2) (map ((abs1 ---> rep2) f) (map rep1 l)) = map f l"
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  by (induct l)
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     (simp_all add: Quotient3_abs_rep[OF a] Quotient3_abs_rep[OF b])
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lemma map_prs [quot_preserve]:
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  assumes a: "Quotient3 R1 abs1 rep1"
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  and     b: "Quotient3 R2 abs2 rep2"
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  shows "((abs1 ---> rep2) ---> (map rep1) ---> (map abs2)) map = map"
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  and   "((abs1 ---> id) ---> map rep1 ---> id) map = map"
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  by (simp_all only: fun_eq_iff map_prs_aux[OF a b] comp_def)
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    (simp_all add: Quotient3_abs_rep[OF a] Quotient3_abs_rep[OF b])
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lemma map_rsp [quot_respect]:
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  assumes q1: "Quotient3 R1 Abs1 Rep1"
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  and     q2: "Quotient3 R2 Abs2 Rep2"
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  shows "((R1 ===> R2) ===> (list_all2 R1) ===> list_all2 R2) map map"
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  and   "((R1 ===> op =) ===> (list_all2 R1) ===> op =) map map"
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  unfolding list_all2_eq [symmetric] by (rule list.map_transfer)+
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lemma foldr_prs_aux:
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  assumes a: "Quotient3 R1 abs1 rep1"
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  and     b: "Quotient3 R2 abs2 rep2"
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  shows "abs2 (foldr ((abs1 ---> abs2 ---> rep2) f) (map rep1 l) (rep2 e)) = foldr f l e"
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  by (induct l) (simp_all add: Quotient3_abs_rep[OF a] Quotient3_abs_rep[OF b])
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lemma foldr_prs [quot_preserve]:
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  assumes a: "Quotient3 R1 abs1 rep1"
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  and     b: "Quotient3 R2 abs2 rep2"
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  shows "((abs1 ---> abs2 ---> rep2) ---> (map rep1) ---> rep2 ---> abs2) foldr = foldr"
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  apply (simp add: fun_eq_iff)
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  by (simp only: fun_eq_iff foldr_prs_aux[OF a b])
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     (simp)
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lemma foldl_prs_aux:
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  assumes a: "Quotient3 R1 abs1 rep1"
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  and     b: "Quotient3 R2 abs2 rep2"
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  shows "abs1 (foldl ((abs1 ---> abs2 ---> rep1) f) (rep1 e) (map rep2 l)) = foldl f e l"
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  by (induct l arbitrary:e) (simp_all add: Quotient3_abs_rep[OF a] Quotient3_abs_rep[OF b])
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lemma foldl_prs [quot_preserve]:
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  assumes a: "Quotient3 R1 abs1 rep1"
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  and     b: "Quotient3 R2 abs2 rep2"
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  shows "((abs1 ---> abs2 ---> rep1) ---> rep1 ---> (map rep2) ---> abs1) foldl = foldl"
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  by (simp add: fun_eq_iff foldl_prs_aux [OF a b])
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(* induct_tac doesn't accept 'arbitrary', so we manually 'spec' *)
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lemma foldl_rsp[quot_respect]:
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  assumes q1: "Quotient3 R1 Abs1 Rep1"
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  and     q2: "Quotient3 R2 Abs2 Rep2"
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  shows "((R1 ===> R2 ===> R1) ===> R1 ===> list_all2 R2 ===> R1) foldl foldl"
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  by (rule foldl_transfer)
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lemma foldr_rsp[quot_respect]:
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  assumes q1: "Quotient3 R1 Abs1 Rep1"
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  and     q2: "Quotient3 R2 Abs2 Rep2"
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  shows "((R1 ===> R2 ===> R2) ===> list_all2 R1 ===> R2 ===> R2) foldr foldr"
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  by (rule foldr_transfer)
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lemma list_all2_rsp:
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  assumes r: "\<forall>x y. R x y \<longrightarrow> (\<forall>a b. R a b \<longrightarrow> S x a = T y b)"
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  and l1: "list_all2 R x y"
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  and l2: "list_all2 R a b"
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  shows "list_all2 S x a = list_all2 T y b"
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  using l1 l2
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  by (induct arbitrary: a b rule: list_all2_induct,
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    auto simp: list_all2_Cons1 list_all2_Cons2 r)
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lemma [quot_respect]:
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  "((R ===> R ===> op =) ===> list_all2 R ===> list_all2 R ===> op =) list_all2 list_all2"
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  by (rule list.rel_transfer)
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lemma [quot_preserve]:
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  assumes a: "Quotient3 R abs1 rep1"
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  shows "((abs1 ---> abs1 ---> id) ---> map rep1 ---> map rep1 ---> id) list_all2 = list_all2"
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  apply (simp add: fun_eq_iff)
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  apply clarify
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  apply (induct_tac xa xb rule: list_induct2')
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  apply (simp_all add: Quotient3_abs_rep[OF a])
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  done
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lemma [quot_preserve]:
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  assumes a: "Quotient3 R abs1 rep1"
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  shows "(list_all2 ((rep1 ---> rep1 ---> id) R) l m) = (l = m)"
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  by (induct l m rule: list_induct2') (simp_all add: Quotient3_rel_rep[OF a])
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lemma list_all2_find_element:
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  assumes a: "x \<in> set a"
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  and b: "list_all2 R a b"
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  shows "\<exists>y. (y \<in> set b \<and> R x y)"
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  using b a by induct auto
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lemma list_all2_refl:
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  assumes a: "\<And>x y. R x y = (R x = R y)"
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  shows "list_all2 R x x"
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  by (induct x) (auto simp add: a)
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end