(* Title: HOL/Library/Quotient_List.thy
Author: Cezary Kaliszyk and Christian Urban
*)
header {* Quotient infrastructure for the list type *}
theory Quotient_List
imports Main Quotient_Syntax
begin
declare [[map list = (map, list_all2)]]
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: fun_eq_iff)
apply(rule allI)
apply(induct_tac x)
apply(simp_all)
done
lemma list_all2_reflp:
shows "equivp R \<Longrightarrow> list_all2 R xs xs"
by (induct xs, simp_all add: equivp_reflp)
lemma list_all2_symp:
assumes a: "equivp R"
and b: "list_all2 R xs ys"
shows "list_all2 R ys xs"
using list_all2_lengthD[OF b] b
apply(induct xs ys rule: list_induct2)
apply(simp_all)
apply(rule equivp_symp[OF a])
apply(simp)
done
lemma list_all2_transp:
assumes a: "equivp R"
and b: "list_all2 R xs1 xs2"
and c: "list_all2 R xs2 xs3"
shows "list_all2 R xs1 xs3"
using list_all2_lengthD[OF b] list_all2_lengthD[OF c] b c
apply(induct rule: list_induct3)
apply(simp_all)
apply(auto intro: equivp_transp[OF a])
done
lemma list_equivp[quot_equiv]:
assumes a: "equivp R"
shows "equivp (list_all2 R)"
apply (intro equivpI)
unfolding reflp_def symp_def transp_def
apply(simp add: list_all2_reflp[OF a])
apply(blast intro: list_all2_symp[OF a])
apply(blast intro: list_all2_transp[OF a])
done
lemma list_all2_rel:
assumes q: "Quotient R Abs Rep"
shows "list_all2 R r s = (list_all2 R r r \<and> list_all2 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_all2 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(intro conjI allI)
apply(induct_tac a)
apply(simp_all add: Quotient_rep_reflp[OF q])
apply(rule list_all2_rel[OF q])
done
lemma cons_prs[quot_preserve]:
assumes q: "Quotient R Abs Rep"
shows "(Rep ---> (map Rep) ---> (map Abs)) (op #) = (op #)"
by (auto simp add: fun_eq_iff comp_def Quotient_abs_rep [OF q])
lemma cons_rsp[quot_respect]:
assumes q: "Quotient R Abs Rep"
shows "(R ===> list_all2 R ===> list_all2 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_all2 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"
and "((abs1 ---> id) ---> map rep1 ---> id) map = map"
by (simp_all only: fun_eq_iff map_prs_aux[OF a b] comp_def)
(simp_all add: Quotient_abs_rep[OF a] Quotient_abs_rep[OF b])
lemma map_rsp[quot_respect]:
assumes q1: "Quotient R1 Abs1 Rep1"
and q2: "Quotient R2 Abs2 Rep2"
shows "((R1 ===> R2) ===> (list_all2 R1) ===> list_all2 R2) map map"
and "((R1 ===> op =) ===> (list_all2 R1) ===> op =) map map"
apply (simp_all add: fun_rel_def)
apply(rule_tac [!] allI)+
apply(rule_tac [!] impI)
apply(rule_tac [!] 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"
apply (simp add: fun_eq_iff)
by (simp only: fun_eq_iff 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 add: fun_eq_iff foldl_prs_aux [OF a b])
lemma list_all2_empty:
shows "list_all2 R [] b \<Longrightarrow> length b = 0"
by (induct b) (simp_all)
(* 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_all2 R2 ===> R1) foldl foldl"
apply(auto simp add: fun_rel_def)
apply (subgoal_tac "R1 xa ya \<longrightarrow> list_all2 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_all2_lengthD)
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_all2 R1 ===> R2 ===> R2) foldr foldr"
apply (auto simp add: fun_rel_def)
apply(subgoal_tac "R2 xb yb \<longrightarrow> list_all2 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_all2_lengthD)
apply (simp_all)
done
lemma list_all2_rsp:
assumes r: "\<forall>x y. R x y \<longrightarrow> (\<forall>a b. R a b \<longrightarrow> S x a = T y b)"
and l1: "list_all2 R x y"
and l2: "list_all2 R a b"
shows "list_all2 S x a = list_all2 T y b"
proof -
have a: "length y = length x" by (rule list_all2_lengthD[OF l1, symmetric])
have c: "length a = length b" by (rule list_all2_lengthD[OF l2])
show ?thesis proof (cases "length x = length a")
case True
have b: "length x = length a" by fact
show ?thesis using a b c r l1 l2 proof (induct rule: list_induct4)
case Nil
show ?case using assms by simp
next
case (Cons h t)
then show ?case by auto
qed
next
case False
have d: "length x \<noteq> length a" by fact
then have e: "\<not>list_all2 S x a" using list_all2_lengthD by auto
have "length y \<noteq> length b" using d a c by simp
then have "\<not>list_all2 T y b" using list_all2_lengthD by auto
then show ?thesis using e by simp
qed
qed
lemma[quot_respect]:
"((R ===> R ===> op =) ===> list_all2 R ===> list_all2 R ===> op =) list_all2 list_all2"
by (simp add: list_all2_rsp fun_rel_def)
lemma[quot_preserve]:
assumes a: "Quotient R abs1 rep1"
shows "((abs1 ---> abs1 ---> id) ---> map rep1 ---> map rep1 ---> id) list_all2 = list_all2"
apply (simp add: fun_eq_iff)
apply clarify
apply (induct_tac xa xb rule: list_induct2')
apply (simp_all add: Quotient_abs_rep[OF a])
done
lemma[quot_preserve]:
assumes a: "Quotient R abs1 rep1"
shows "(list_all2 ((rep1 ---> rep1 ---> id) R) l m) = (l = m)"
by (induct l m rule: list_induct2') (simp_all add: Quotient_rel_rep[OF a])
lemma list_all2_eq[id_simps]:
shows "(list_all2 (op =)) = (op =)"
unfolding fun_eq_iff
apply(rule allI)+
apply(induct_tac x xa rule: list_induct2')
apply(simp_all)
done
lemma list_all2_find_element:
assumes a: "x \<in> set a"
and b: "list_all2 R a b"
shows "\<exists>y. (y \<in> set b \<and> R x y)"
proof -
have "length a = length b" using b by (rule list_all2_lengthD)
then show ?thesis using a b by (induct a b rule: list_induct2) auto
qed
lemma list_all2_refl:
assumes a: "\<And>x y. R x y = (R x = R y)"
shows "list_all2 R x x"
by (induct x) (auto simp add: a)
end