--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Quot/Quotient_List.thy	Thu Feb 11 10:06:02 2010 +0100
@@ -0,0 +1,232 @@
+(*  Title:      Quotient_List.thy
+    Author:     Cezary Kaliszyk and Christian Urban
+*)
+theory Quotient_List
+imports Quotient 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