--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Quot/QuotList.thy	Mon Dec 07 14:09:50 2009 +0100
@@ -0,0 +1,174 @@
+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