
diff -r db158e995bfc -r 9df6144e281b Quot/Quotient_List.thy
--- a/Quot/Quotient_List.thy	Thu Feb 25 07:48:57 2010 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,232 +0,0 @@
-(*  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




