# HG changeset patch
# User Cezary Kaliszyk <kaliszyk@in.tum.de>
# Date 1277275361 -7200
# Node ID ab36b1a50ca8f5e087f156cb7c1303b41d8f8fdb
# Parent  b5989aa323582bda264a081cd8c503611ef1ddfb
Replace 'list_rel' by 'list_all2'; they are equivalent.

diff -r b5989aa32358 -r ab36b1a50ca8 src/HOL/Library/Quotient_List.thy
--- 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
diff -r b5989aa32358 -r ab36b1a50ca8 src/HOL/Quotient_Examples/FSet.thy
--- a/src/HOL/Quotient_Examples/FSet.thy	Tue Jun 22 19:46:16 2010 +0200
+++ b/src/HOL/Quotient_Examples/FSet.thy	Wed Jun 23 08:42:41 2010 +0200
@@ -80,20 +80,20 @@
 
 text {* Composition Quotient *}
 
-lemma list_rel_refl:
-  shows "(list_rel op \<approx>) r r"
-  by (rule list_rel_refl) (metis equivp_def fset_equivp)
+lemma list_all2_refl:
+  shows "(list_all2 op \<approx>) r r"
+  by (rule list_all2_refl) (metis equivp_def fset_equivp)
 
 lemma compose_list_refl:
-  shows "(list_rel op \<approx> OOO op \<approx>) r r"
+  shows "(list_all2 op \<approx> OOO op \<approx>) r r"
 proof
   have *: "r \<approx> r" by (rule equivp_reflp[OF fset_equivp])
-  show "list_rel op \<approx> r r" by (rule list_rel_refl)
-  with * show "(op \<approx> OO list_rel op \<approx>) r r" ..
+  show "list_all2 op \<approx> r r" by (rule list_all2_refl)
+  with * show "(op \<approx> OO list_all2 op \<approx>) r r" ..
 qed
 
 lemma Quotient_fset_list:
-  shows "Quotient (list_rel op \<approx>) (map abs_fset) (map rep_fset)"
+  shows "Quotient (list_all2 op \<approx>) (map abs_fset) (map rep_fset)"
   by (fact list_quotient[OF Quotient_fset])
 
 lemma set_in_eq: "(\<forall>e. ((e \<in> xs) \<longleftrightarrow> (e \<in> ys))) \<equiv> xs = ys"
@@ -104,32 +104,32 @@
   by (simp only: set_map set_in_eq)
 
 lemma quotient_compose_list[quot_thm]:
-  shows  "Quotient ((list_rel op \<approx>) OOO (op \<approx>))
+  shows  "Quotient ((list_all2 op \<approx>) OOO (op \<approx>))
     (abs_fset \<circ> (map abs_fset)) ((map rep_fset) \<circ> rep_fset)"
   unfolding Quotient_def comp_def
 proof (intro conjI allI)
   fix a r s
   show "abs_fset (map abs_fset (map rep_fset (rep_fset a))) = a"
     by (simp add: abs_o_rep[OF Quotient_fset] Quotient_abs_rep[OF Quotient_fset] map_id)
-  have b: "list_rel op \<approx> (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"
-    by (rule list_rel_refl)
-  have c: "(op \<approx> OO list_rel op \<approx>) (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"
+  have b: "list_all2 op \<approx> (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"
+    by (rule list_all2_refl)
+  have c: "(op \<approx> OO list_all2 op \<approx>) (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"
     by (rule, rule equivp_reflp[OF fset_equivp]) (rule b)
-  show "(list_rel op \<approx> OOO op \<approx>) (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"
-    by (rule, rule list_rel_refl) (rule c)
-  show "(list_rel op \<approx> OOO op \<approx>) r s = ((list_rel op \<approx> OOO op \<approx>) r r \<and>
-        (list_rel op \<approx> OOO op \<approx>) s s \<and> abs_fset (map abs_fset r) = abs_fset (map abs_fset s))"
+  show "(list_all2 op \<approx> OOO op \<approx>) (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"
+    by (rule, rule list_all2_refl) (rule c)
+  show "(list_all2 op \<approx> OOO op \<approx>) r s = ((list_all2 op \<approx> OOO op \<approx>) r r \<and>
+        (list_all2 op \<approx> OOO op \<approx>) s s \<and> abs_fset (map abs_fset r) = abs_fset (map abs_fset s))"
   proof (intro iffI conjI)
-    show "(list_rel op \<approx> OOO op \<approx>) r r" by (rule compose_list_refl)
-    show "(list_rel op \<approx> OOO op \<approx>) s s" by (rule compose_list_refl)
+    show "(list_all2 op \<approx> OOO op \<approx>) r r" by (rule compose_list_refl)
+    show "(list_all2 op \<approx> OOO op \<approx>) s s" by (rule compose_list_refl)
   next
-    assume a: "(list_rel op \<approx> OOO op \<approx>) r s"
+    assume a: "(list_all2 op \<approx> OOO op \<approx>) r s"
     then have b: "map abs_fset r \<approx> map abs_fset s"
     proof (elim pred_compE)
       fix b ba
-      assume c: "list_rel op \<approx> r b"
+      assume c: "list_all2 op \<approx> r b"
       assume d: "b \<approx> ba"
-      assume e: "list_rel op \<approx> ba s"
+      assume e: "list_all2 op \<approx> ba s"
       have f: "map abs_fset r = map abs_fset b"
         using Quotient_rel[OF Quotient_fset_list] c by blast
       have "map abs_fset ba = map abs_fset s"
@@ -140,20 +140,20 @@
     then show "abs_fset (map abs_fset r) = abs_fset (map abs_fset s)"
       using Quotient_rel[OF Quotient_fset] by blast
   next
-    assume a: "(list_rel op \<approx> OOO op \<approx>) r r \<and> (list_rel op \<approx> OOO op \<approx>) s s
+    assume a: "(list_all2 op \<approx> OOO op \<approx>) r r \<and> (list_all2 op \<approx> OOO op \<approx>) s s
       \<and> abs_fset (map abs_fset r) = abs_fset (map abs_fset s)"
-    then have s: "(list_rel op \<approx> OOO op \<approx>) s s" by simp
+    then have s: "(list_all2 op \<approx> OOO op \<approx>) s s" by simp
     have d: "map abs_fset r \<approx> map abs_fset s"
       by (subst Quotient_rel[OF Quotient_fset]) (simp add: a)
     have b: "map rep_fset (map abs_fset r) \<approx> map rep_fset (map abs_fset s)"
       by (rule map_rel_cong[OF d])
-    have y: "list_rel op \<approx> (map rep_fset (map abs_fset s)) s"
-      by (fact rep_abs_rsp_left[OF Quotient_fset_list, OF list_rel_refl[of s]])
-    have c: "(op \<approx> OO list_rel op \<approx>) (map rep_fset (map abs_fset r)) s"
+    have y: "list_all2 op \<approx> (map rep_fset (map abs_fset s)) s"
+      by (fact rep_abs_rsp_left[OF Quotient_fset_list, OF list_all2_refl[of s]])
+    have c: "(op \<approx> OO list_all2 op \<approx>) (map rep_fset (map abs_fset r)) s"
       by (rule pred_compI) (rule b, rule y)
-    have z: "list_rel op \<approx> r (map rep_fset (map abs_fset r))"
-      by (fact rep_abs_rsp[OF Quotient_fset_list, OF list_rel_refl[of r]])
-    then show "(list_rel op \<approx> OOO op \<approx>) r s"
+    have z: "list_all2 op \<approx> r (map rep_fset (map abs_fset r))"
+      by (fact rep_abs_rsp[OF Quotient_fset_list, OF list_all2_refl[of r]])
+    then show "(list_all2 op \<approx> OOO op \<approx>) r s"
       using a c pred_compI by simp
   qed
 qed
@@ -336,27 +336,27 @@
   by (simp add: memb_def[symmetric] ffold_raw_rsp_pre)
 
 lemma concat_rsp_pre:
-  assumes a: "list_rel op \<approx> x x'"
+  assumes a: "list_all2 op \<approx> x x'"
   and     b: "x' \<approx> y'"
-  and     c: "list_rel op \<approx> y' y"
+  and     c: "list_all2 op \<approx> y' y"
   and     d: "\<exists>x\<in>set x. xa \<in> set x"
   shows "\<exists>x\<in>set y. xa \<in> set x"
 proof -
   obtain xb where e: "xb \<in> set x" and f: "xa \<in> set xb" using d by auto
-  have "\<exists>y. y \<in> set x' \<and> xb \<approx> y" by (rule list_rel_find_element[OF e a])
+  have "\<exists>y. y \<in> set x' \<and> xb \<approx> y" by (rule list_all2_find_element[OF e a])
   then obtain ya where h: "ya \<in> set x'" and i: "xb \<approx> ya" by auto
   have "ya \<in> set y'" using b h by simp
-  then have "\<exists>yb. yb \<in> set y \<and> ya \<approx> yb" using c by (rule list_rel_find_element)
+  then have "\<exists>yb. yb \<in> set y \<and> ya \<approx> yb" using c by (rule list_all2_find_element)
   then show ?thesis using f i by auto
 qed
 
 lemma [quot_respect]:
-  shows "(list_rel op \<approx> OOO op \<approx> ===> op \<approx>) concat concat"
+  shows "(list_all2 op \<approx> OOO op \<approx> ===> op \<approx>) concat concat"
 proof (rule fun_relI, elim pred_compE)
   fix a b ba bb
-  assume a: "list_rel op \<approx> a ba"
+  assume a: "list_all2 op \<approx> a ba"
   assume b: "ba \<approx> bb"
-  assume c: "list_rel op \<approx> bb b"
+  assume c: "list_all2 op \<approx> bb b"
   have "\<forall>x. (\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)" proof
     fix x
     show "(\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)" proof
@@ -364,9 +364,9 @@
       show "\<exists>xa\<in>set b. x \<in> set xa" by (rule concat_rsp_pre[OF a b c d])
     next
       assume e: "\<exists>xa\<in>set b. x \<in> set xa"
-      have a': "list_rel op \<approx> ba a" by (rule list_rel_symp[OF list_eq_equivp, OF a])
+      have a': "list_all2 op \<approx> ba a" by (rule list_all2_symp[OF list_eq_equivp, OF a])
       have b': "bb \<approx> ba" by (rule equivp_symp[OF list_eq_equivp, OF b])
-      have c': "list_rel op \<approx> b bb" by (rule list_rel_symp[OF list_eq_equivp, OF c])
+      have c': "list_all2 op \<approx> b bb" by (rule list_all2_symp[OF list_eq_equivp, OF c])
       show "\<exists>xa\<in>set a. x \<in> set xa" by (rule concat_rsp_pre[OF c' b' a' e])
     qed
   qed
@@ -581,14 +581,14 @@
 
 text {* Compositional Respectfullness and Preservation *}
 
-lemma [quot_respect]: "(list_rel op \<approx> OOO op \<approx>) [] []"
+lemma [quot_respect]: "(list_all2 op \<approx> OOO op \<approx>) [] []"
   by (fact compose_list_refl)
 
 lemma [quot_preserve]: "(abs_fset \<circ> map f) [] = abs_fset []"
   by simp
 
 lemma [quot_respect]:
-  "(op \<approx> ===> list_rel op \<approx> OOO op \<approx> ===> list_rel op \<approx> OOO op \<approx>) op # op #"
+  "(op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) op # op #"
   apply auto
   apply (simp add: set_in_eq)
   apply (rule_tac b="x # b" in pred_compI)
@@ -607,59 +607,59 @@
   by (simp add: expand_fun_eq Quotient_abs_rep[OF Quotient_fset]
       abs_o_rep[OF Quotient_fset] map_id sup_fset_def)
 
-lemma list_rel_app_l:
+lemma list_all2_app_l:
   assumes a: "reflp R"
-  and b: "list_rel R l r"
-  shows "list_rel R (z @ l) (z @ r)"
+  and b: "list_all2 R l r"
+  shows "list_all2 R (z @ l) (z @ r)"
   by (induct z) (simp_all add: b rev_iffD1[OF a meta_eq_to_obj_eq[OF reflp_def]])
 
 lemma append_rsp2_pre0:
-  assumes a:"list_rel op \<approx> x x'"
-  shows "list_rel op \<approx> (x @ z) (x' @ z)"
+  assumes a:"list_all2 op \<approx> x x'"
+  shows "list_all2 op \<approx> (x @ z) (x' @ z)"
   using a apply (induct x x' rule: list_induct2')
-  by simp_all (rule list_rel_refl)
+  by simp_all (rule list_all2_refl)
 
 lemma append_rsp2_pre1:
-  assumes a:"list_rel op \<approx> x x'"
-  shows "list_rel op \<approx> (z @ x) (z @ x')"
+  assumes a:"list_all2 op \<approx> x x'"
+  shows "list_all2 op \<approx> (z @ x) (z @ x')"
   using a apply (induct x x' arbitrary: z rule: list_induct2')
-  apply (rule list_rel_refl)
+  apply (rule list_all2_refl)
   apply (simp_all del: list_eq.simps)
-  apply (rule list_rel_app_l)
+  apply (rule list_all2_app_l)
   apply (simp_all add: reflp_def)
   done
 
 lemma append_rsp2_pre:
-  assumes a:"list_rel op \<approx> x x'"
-  and     b: "list_rel op \<approx> z z'"
-  shows "list_rel op \<approx> (x @ z) (x' @ z')"
-  apply (rule list_rel_transp[OF fset_equivp])
+  assumes a:"list_all2 op \<approx> x x'"
+  and     b: "list_all2 op \<approx> z z'"
+  shows "list_all2 op \<approx> (x @ z) (x' @ z')"
+  apply (rule list_all2_transp[OF fset_equivp])
   apply (rule append_rsp2_pre0)
   apply (rule a)
   using b apply (induct z z' rule: list_induct2')
   apply (simp_all only: append_Nil2)
-  apply (rule list_rel_refl)
+  apply (rule list_all2_refl)
   apply simp_all
   apply (rule append_rsp2_pre1)
   apply simp
   done
 
 lemma [quot_respect]:
-  "(list_rel op \<approx> OOO op \<approx> ===> list_rel op \<approx> OOO op \<approx> ===> list_rel op \<approx> OOO op \<approx>) op @ op @"
+  "(list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) op @ op @"
 proof (intro fun_relI, elim pred_compE)
   fix x y z w x' z' y' w' :: "'a list list"
-  assume a:"list_rel op \<approx> x x'"
+  assume a:"list_all2 op \<approx> x x'"
   and b:    "x' \<approx> y'"
-  and c:    "list_rel op \<approx> y' y"
-  assume aa: "list_rel op \<approx> z z'"
+  and c:    "list_all2 op \<approx> y' y"
+  assume aa: "list_all2 op \<approx> z z'"
   and bb:   "z' \<approx> w'"
-  and cc:   "list_rel op \<approx> w' w"
-  have a': "list_rel op \<approx> (x @ z) (x' @ z')" using a aa append_rsp2_pre by auto
+  and cc:   "list_all2 op \<approx> w' w"
+  have a': "list_all2 op \<approx> (x @ z) (x' @ z')" using a aa append_rsp2_pre by auto
   have b': "x' @ z' \<approx> y' @ w'" using b bb by simp
-  have c': "list_rel op \<approx> (y' @ w') (y @ w)" using c cc append_rsp2_pre by auto
-  have d': "(op \<approx> OO list_rel op \<approx>) (x' @ z') (y @ w)"
+  have c': "list_all2 op \<approx> (y' @ w') (y @ w)" using c cc append_rsp2_pre by auto
+  have d': "(op \<approx> OO list_all2 op \<approx>) (x' @ z') (y @ w)"
     by (rule pred_compI) (rule b', rule c')
-  show "(list_rel op \<approx> OOO op \<approx>) (x @ z) (y @ w)"
+  show "(list_all2 op \<approx> OOO op \<approx>) (x @ z) (y @ w)"
     by (rule pred_compI) (rule a', rule d')
 qed