--- a/src/HOL/Library/Quotient_List.thy Tue Jun 22 19:46:16 2010 +0200
+++ b/src/HOL/Library/Quotient_List.thy Wed Jun 23 08:42:41 2010 +0200
@@ -8,15 +8,7 @@
imports Main Quotient_Syntax
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)"
-
-declare [[map list = (map, list_rel)]]
+declare [[map list = (map, list_all2)]]
lemma split_list_all:
shows "(\<forall>x. P x) \<longleftrightarrow> P [] \<and> (\<forall>x xs. P (x#xs))"
@@ -33,52 +25,47 @@
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_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:
+lemma list_all2_symp:
assumes a: "equivp R"
- shows "list_rel R xs ys \<Longrightarrow> list_rel R ys xs"
- apply(induct xs ys rule: list_induct2')
+ 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_rel_transp:
+thm list_induct3
+
+lemma list_all2_transp:
assumes a: "equivp R"
- shows "list_rel R xs1 xs2 \<Longrightarrow> list_rel R xs2 xs3 \<Longrightarrow> list_rel R xs1 xs3"
- using a
- apply(induct R xs1 xs2 arbitrary: xs3 rule: list_rel.induct)
- apply(simp)
- apply(simp)
- apply(simp)
- apply(case_tac xs3)
- apply(clarify)
- apply(simp (no_asm_use))
- apply(clarify)
- apply(simp (no_asm_use))
- apply(auto intro: equivp_transp)
+ 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_rel R)"
- apply(rule equivpI)
+ shows "equivp (list_all2 R)"
+ apply (intro 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])
+ 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_rel_rel:
+lemma list_all2_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))"
+ 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]
@@ -87,21 +74,16 @@
lemma list_quotient[quot_thm]:
assumes q: "Quotient R Abs Rep"
- shows "Quotient (list_rel R) (map Abs) (map 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(rule conjI)
- apply(rule allI)
+ apply(intro conjI 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])
+ apply(simp_all add: Quotient_rep_reflp[OF q])
+ 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"
@@ -115,7 +97,7 @@
lemma cons_rsp[quot_respect]:
assumes q: "Quotient R Abs Rep"
- shows "(R ===> list_rel R ===> list_rel R) (op #) (op #)"
+ shows "(R ===> list_all2 R ===> list_all2 R) (op #) (op #)"
by (auto)
lemma nil_prs[quot_preserve]:
@@ -125,7 +107,7 @@
lemma nil_rsp[quot_respect]:
assumes q: "Quotient R Abs Rep"
- shows "list_rel R [] []"
+ shows "list_all2 R [] []"
by simp
lemma map_prs_aux:
@@ -146,8 +128,8 @@
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"
- and "((R1 ===> op =) ===> (list_rel R1) ===> op =) map map"
+ shows "((R1 ===> R2) ===> (list_all2 R1) ===> list_all2 R2) map map"
+ and "((R1 ===> op =) ===> (list_all2 R1) ===> op =) map map"
apply simp_all
apply(rule_tac [!] allI)+
apply(rule_tac [!] impI)
@@ -183,53 +165,45 @@
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"
+lemma list_all2_empty:
+ shows "list_all2 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"
+ shows "((R1 ===> R2 ===> R1) ===> R1 ===> list_all2 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 (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_rel_len)
+ 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_rel R1 ===> R2 ===> R2) foldr foldr"
+ shows "((R1 ===> R2 ===> R2) ===> list_all2 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(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_rel_len)
+ apply (rule list_all2_lengthD)
apply (simp_all)
done
-lemma list_rel_rsp:
+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_rel R x y"
- and l2: "list_rel R a b"
- shows "list_rel S x a = list_rel 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_rel_len[OF l1, symmetric])
- have c: "length a = length b" by (rule list_rel_len[OF l2])
+ 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
@@ -243,20 +217,20 @@
next
case False
have d: "length x \<noteq> length a" by fact
- then have e: "\<not>list_rel S x a" using list_rel_len by auto
+ 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_rel T y b" using list_rel_len by auto
+ 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_rel R ===> list_rel R ===> op =) list_rel list_rel"
- by (simp add: list_rel_rsp)
+ "((R ===> R ===> op =) ===> list_all2 R ===> list_all2 R ===> op =) list_all2 list_all2"
+ by (simp add: list_all2_rsp)
lemma[quot_preserve]:
assumes a: "Quotient R abs1 rep1"
- shows "((abs1 ---> abs1 ---> id) ---> map rep1 ---> map rep1 ---> id) list_rel = list_rel"
+ shows "((abs1 ---> abs1 ---> id) ---> map rep1 ---> map rep1 ---> id) list_all2 = list_all2"
apply (simp add: expand_fun_eq)
apply clarify
apply (induct_tac xa xb rule: list_induct2')
@@ -265,29 +239,29 @@
lemma[quot_preserve]:
assumes a: "Quotient R abs1 rep1"
- shows "(list_rel ((rep1 ---> rep1 ---> id) R) l m) = (l = m)"
+ 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_rel_eq[id_simps]:
- shows "(list_rel (op =)) = (op =)"
+lemma list_all2_eq[id_simps]:
+ shows "(list_all2 (op =)) = (op =)"
unfolding expand_fun_eq
apply(rule allI)+
apply(induct_tac x xa rule: list_induct2')
apply(simp_all)
done
-lemma list_rel_find_element:
+lemma list_all2_find_element:
assumes a: "x \<in> set a"
- and b: "list_rel R a b"
+ 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_rel_len)
+ 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_rel_refl:
+lemma list_all2_refl:
assumes a: "\<And>x y. R x y = (R x = R y)"
- shows "list_rel R x x"
+ shows "list_all2 R x x"
by (induct x) (auto simp add: a)
end