--- a/src/HOL/Library/Quotient_List.thy Mon Nov 08 23:02:20 2010 +0100
+++ b/src/HOL/Library/Quotient_List.thy Tue Nov 09 14:02:12 2010 +0100
@@ -82,21 +82,15 @@
apply(rule list_all2_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: fun_eq_iff fun_map_def cons_prs_aux[OF q])
- (simp)
+ 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)
+ by auto
lemma nil_prs[quot_preserve]:
assumes q: "Quotient R Abs Rep"
@@ -120,15 +114,16 @@
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 fun_map_def map_prs_aux[OF a b])
- (simp_all add: Quotient_abs_rep[OF a])
+ 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
+ apply (simp_all add: fun_rel_def)
apply(rule_tac [!] allI)+
apply(rule_tac [!] impI)
apply(rule_tac [!] allI)+
@@ -146,7 +141,8 @@
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: fun_eq_iff fun_map_def foldr_prs_aux[OF a b])
+ apply (simp add: fun_eq_iff)
+ by (simp only: fun_eq_iff foldr_prs_aux[OF a b])
(simp)
lemma foldl_prs_aux:
@@ -160,8 +156,7 @@
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: fun_eq_iff fun_map_def foldl_prs_aux[OF a b])
- (simp)
+ 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"
@@ -172,7 +167,7 @@
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)
+ 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)
@@ -186,7 +181,7 @@
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
+ 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)
@@ -224,7 +219,7 @@
lemma[quot_respect]:
"((R ===> R ===> op =) ===> list_all2 R ===> list_all2 R ===> op =) list_all2 list_all2"
- by (simp add: list_all2_rsp)
+ by (simp add: list_all2_rsp fun_rel_def)
lemma[quot_preserve]:
assumes a: "Quotient R abs1 rep1"