# HG changeset patch
# User haftmann
# Date 1291129101 -3600
# Node ID 9f32d7b8b24fc6b078ecd8421e5312cace912994
# Parent  eeaa59fb5ad82346bdb9e9f88223b52e40080920# Parent  fd9c98ead9a98c1638426b1d54d1ddcfab2c5012
merged

diff -r eeaa59fb5ad8 -r 9f32d7b8b24f NEWS
--- a/NEWS	Tue Nov 30 00:12:29 2010 +0100
+++ b/NEWS	Tue Nov 30 15:58:21 2010 +0100
@@ -92,6 +92,9 @@
 
 *** HOL ***
 
+* Abandoned locale equiv for equivalence relations.  INCOMPATIBILITY: use
+equivI rather than equiv_intro.
+
 * Code generator: globbing constant expressions "*" and "Theory.*" have been
 replaced by the more idiomatic "_" and "Theory._".  INCOMPATIBILITY.
 
diff -r eeaa59fb5ad8 -r 9f32d7b8b24f src/HOL/Algebra/Coset.thy
--- a/src/HOL/Algebra/Coset.thy	Tue Nov 30 00:12:29 2010 +0100
+++ b/src/HOL/Algebra/Coset.thy	Tue Nov 30 15:58:21 2010 +0100
@@ -606,7 +606,7 @@
 proof -
   interpret group G by fact
   show ?thesis
-  proof (intro equiv.intro)
+  proof (intro equivI)
     show "refl_on (carrier G) (rcong H)"
       by (auto simp add: r_congruent_def refl_on_def) 
   next
diff -r eeaa59fb5ad8 -r 9f32d7b8b24f src/HOL/Equiv_Relations.thy
--- a/src/HOL/Equiv_Relations.thy	Tue Nov 30 00:12:29 2010 +0100
+++ b/src/HOL/Equiv_Relations.thy	Tue Nov 30 15:58:21 2010 +0100
@@ -8,13 +8,19 @@
 imports Big_Operators Relation Plain
 begin
 
-subsection {* Equivalence relations *}
+subsection {* Equivalence relations -- set version *}
+
+definition equiv :: "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool" where
+  "equiv A r \<longleftrightarrow> refl_on A r \<and> sym r \<and> trans r"
 
-locale equiv =
-  fixes A and r
-  assumes refl_on: "refl_on A r"
-    and sym: "sym r"
-    and trans: "trans r"
+lemma equivI:
+  "refl_on A r \<Longrightarrow> sym r \<Longrightarrow> trans r \<Longrightarrow> equiv A r"
+  by (simp add: equiv_def)
+
+lemma equivE:
+  assumes "equiv A r"
+  obtains "refl_on A r" and "sym r" and "trans r"
+  using assms by (simp add: equiv_def)
 
 text {*
   Suppes, Theorem 70: @{text r} is an equiv relation iff @{text "r\<inverse> O
@@ -157,9 +163,17 @@
 subsection {* Defining unary operations upon equivalence classes *}
 
 text{*A congruence-preserving function*}
-locale congruent =
-  fixes r and f
-  assumes congruent: "(y,z) \<in> r ==> f y = f z"
+
+definition congruent :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"  where
+  "congruent r f \<longleftrightarrow> (\<forall>(y, z) \<in> r. f y = f z)"
+
+lemma congruentI:
+  "(\<And>y z. (y, z) \<in> r \<Longrightarrow> f y = f z) \<Longrightarrow> congruent r f"
+  by (auto simp add: congruent_def)
+
+lemma congruentD:
+  "congruent r f \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> f y = f z"
+  by (auto simp add: congruent_def)
 
 abbreviation
   RESPECTS :: "('a => 'b) => ('a * 'a) set => bool"
@@ -214,10 +228,18 @@
 subsection {* Defining binary operations upon equivalence classes *}
 
 text{*A congruence-preserving function of two arguments*}
-locale congruent2 =
-  fixes r1 and r2 and f
-  assumes congruent2:
-    "(y1,z1) \<in> r1 ==> (y2,z2) \<in> r2 ==> f y1 y2 = f z1 z2"
+
+definition congruent2 :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> ('b \<times> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> bool" where
+  "congruent2 r1 r2 f \<longleftrightarrow> (\<forall>(y1, z1) \<in> r1. \<forall>(y2, z2) \<in> r2. f y1 y2 = f z1 z2)"
+
+lemma congruent2I':
+  assumes "\<And>y1 z1 y2 z2. (y1, z1) \<in> r1 \<Longrightarrow> (y2, z2) \<in> r2 \<Longrightarrow> f y1 y2 = f z1 z2"
+  shows "congruent2 r1 r2 f"
+  using assms by (auto simp add: congruent2_def)
+
+lemma congruent2D:
+  "congruent2 r1 r2 f \<Longrightarrow> (y1, z1) \<in> r1 \<Longrightarrow> (y2, z2) \<in> r2 \<Longrightarrow> f y1 y2 = f z1 z2"
+  using assms by (auto simp add: congruent2_def)
 
 text{*Abbreviation for the common case where the relations are identical*}
 abbreviation
@@ -331,4 +353,99 @@
 apply simp
 done
 
+
+subsection {* Equivalence relations -- predicate version *}
+
+text {* Partial equivalences *}
+
+definition part_equivp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
+  "part_equivp R \<longleftrightarrow> (\<exists>x. R x x) \<and> (\<forall>x y. R x y \<longleftrightarrow> R x x \<and> R y y \<and> R x = R y)"
+    -- {* John-Harrison-style characterization *}
+
+lemma part_equivpI:
+  "(\<exists>x. R x x) \<Longrightarrow> symp R \<Longrightarrow> transp R \<Longrightarrow> part_equivp R"
+  by (auto simp add: part_equivp_def mem_def) (auto elim: sympE transpE)
+
+lemma part_equivpE:
+  assumes "part_equivp R"
+  obtains x where "R x x" and "symp R" and "transp R"
+proof -
+  from assms have 1: "\<exists>x. R x x"
+    and 2: "\<And>x y. R x y \<longleftrightarrow> R x x \<and> R y y \<and> R x = R y"
+    by (unfold part_equivp_def) blast+
+  from 1 obtain x where "R x x" ..
+  moreover have "symp R"
+  proof (rule sympI)
+    fix x y
+    assume "R x y"
+    with 2 [of x y] show "R y x" by auto
+  qed
+  moreover have "transp R"
+  proof (rule transpI)
+    fix x y z
+    assume "R x y" and "R y z"
+    with 2 [of x y] 2 [of y z] show "R x z" by auto
+  qed
+  ultimately show thesis by (rule that)
+qed
+
+lemma part_equivp_refl_symp_transp:
+  "part_equivp R \<longleftrightarrow> (\<exists>x. R x x) \<and> symp R \<and> transp R"
+  by (auto intro: part_equivpI elim: part_equivpE)
+
+lemma part_equivp_symp:
+  "part_equivp R \<Longrightarrow> R x y \<Longrightarrow> R y x"
+  by (erule part_equivpE, erule sympE)
+
+lemma part_equivp_transp:
+  "part_equivp R \<Longrightarrow> R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
+  by (erule part_equivpE, erule transpE)
+
+lemma part_equivp_typedef:
+  "part_equivp R \<Longrightarrow> \<exists>d. d \<in> (\<lambda>c. \<exists>x. R x x \<and> c = R x)"
+  by (auto elim: part_equivpE simp add: mem_def)
+
+
+text {* Total equivalences *}
+
+definition equivp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
+  "equivp R \<longleftrightarrow> (\<forall>x y. R x y = (R x = R y))" -- {* John-Harrison-style characterization *}
+
+lemma equivpI:
+  "reflp R \<Longrightarrow> symp R \<Longrightarrow> transp R \<Longrightarrow> equivp R"
+  by (auto elim: reflpE sympE transpE simp add: equivp_def mem_def)
+
+lemma equivpE:
+  assumes "equivp R"
+  obtains "reflp R" and "symp R" and "transp R"
+  using assms by (auto intro!: that reflpI sympI transpI simp add: equivp_def)
+
+lemma equivp_implies_part_equivp:
+  "equivp R \<Longrightarrow> part_equivp R"
+  by (auto intro: part_equivpI elim: equivpE reflpE)
+
+lemma equivp_equiv:
+  "equiv UNIV A \<longleftrightarrow> equivp (\<lambda>x y. (x, y) \<in> A)"
+  by (auto intro: equivpI elim: equivpE simp add: equiv_def reflp_def symp_def transp_def)
+
+lemma equivp_reflp_symp_transp:
+  shows "equivp R \<longleftrightarrow> reflp R \<and> symp R \<and> transp R"
+  by (auto intro: equivpI elim: equivpE)
+
+lemma identity_equivp:
+  "equivp (op =)"
+  by (auto intro: equivpI reflpI sympI transpI)
+
+lemma equivp_reflp:
+  "equivp R \<Longrightarrow> R x x"
+  by (erule equivpE, erule reflpE)
+
+lemma equivp_symp:
+  "equivp R \<Longrightarrow> R x y \<Longrightarrow> R y x"
+  by (erule equivpE, erule sympE)
+
+lemma equivp_transp:
+  "equivp R \<Longrightarrow> R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
+  by (erule equivpE, erule transpE)
+
 end
diff -r eeaa59fb5ad8 -r 9f32d7b8b24f src/HOL/Int.thy
--- a/src/HOL/Int.thy	Tue Nov 30 00:12:29 2010 +0100
+++ b/src/HOL/Int.thy	Tue Nov 30 15:58:21 2010 +0100
@@ -102,7 +102,7 @@
 lemma minus: "- Abs_Integ(intrel``{(x,y)}) = Abs_Integ(intrel `` {(y,x)})"
 proof -
   have "(\<lambda>(x,y). intrel``{(y,x)}) respects intrel"
-    by (simp add: congruent_def) 
+    by (auto simp add: congruent_def)
   thus ?thesis
     by (simp add: minus_int_def UN_equiv_class [OF equiv_intrel])
 qed
@@ -113,7 +113,7 @@
 proof -
   have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). intrel `` {(x+u, y+v)}) w) z) 
         respects2 intrel"
-    by (simp add: congruent2_def)
+    by (auto simp add: congruent2_def)
   thus ?thesis
     by (simp add: add_int_def UN_UN_split_split_eq
                   UN_equiv_class2 [OF equiv_intrel equiv_intrel])
@@ -288,7 +288,7 @@
 lemma of_int: "of_int (Abs_Integ (intrel `` {(i,j)})) = of_nat i - of_nat j"
 proof -
   have "(\<lambda>(i,j). { of_nat i - (of_nat j :: 'a) }) respects intrel"
-    by (simp add: congruent_def algebra_simps of_nat_add [symmetric]
+    by (auto simp add: congruent_def) (simp add: algebra_simps of_nat_add [symmetric]
             del: of_nat_add) 
   thus ?thesis
     by (simp add: of_int_def UN_equiv_class [OF equiv_intrel])
@@ -394,7 +394,7 @@
 lemma nat: "nat (Abs_Integ (intrel``{(x,y)})) = x-y"
 proof -
   have "(\<lambda>(x,y). {x-y}) respects intrel"
-    by (simp add: congruent_def) arith
+    by (auto simp add: congruent_def)
   thus ?thesis
     by (simp add: nat_def UN_equiv_class [OF equiv_intrel])
 qed
diff -r eeaa59fb5ad8 -r 9f32d7b8b24f src/HOL/Library/Fraction_Field.thy
--- a/src/HOL/Library/Fraction_Field.thy	Tue Nov 30 00:12:29 2010 +0100
+++ b/src/HOL/Library/Fraction_Field.thy	Tue Nov 30 15:58:21 2010 +0100
@@ -43,7 +43,7 @@
 qed
   
 lemma equiv_fractrel: "equiv {x. snd x \<noteq> 0} fractrel"
-  by (rule equiv.intro [OF refl_fractrel sym_fractrel trans_fractrel])
+  by (rule equivI [OF refl_fractrel sym_fractrel trans_fractrel])
 
 lemmas UN_fractrel = UN_equiv_class [OF equiv_fractrel]
 lemmas UN_fractrel2 = UN_equiv_class2 [OF equiv_fractrel equiv_fractrel]
diff -r eeaa59fb5ad8 -r 9f32d7b8b24f src/HOL/Library/Quotient_List.thy
--- a/src/HOL/Library/Quotient_List.thy	Tue Nov 30 00:12:29 2010 +0100
+++ b/src/HOL/Library/Quotient_List.thy	Tue Nov 30 15:58:21 2010 +0100
@@ -10,94 +10,96 @@
 
 declare [[map list = (map, list_all2)]]
 
-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]:
+  "map id = id"
+  by (simp add: id_def fun_eq_iff map.identity)
 
-lemma map_id[id_simps]:
-  shows "map id = id"
-  apply(simp add: fun_eq_iff)
-  apply(rule allI)
-  apply(induct_tac x)
-  apply(simp_all)
-  done
+lemma list_all2_map1:
+  "list_all2 R (map f xs) ys \<longleftrightarrow> list_all2 (\<lambda>x. R (f x)) xs ys"
+  by (induct xs ys rule: list_induct2') simp_all
+
+lemma list_all2_map2:
+  "list_all2 R xs (map f ys) \<longleftrightarrow> list_all2 (\<lambda>x y. R x (f y)) xs ys"
+  by (induct xs ys rule: list_induct2') simp_all
 
-lemma list_all2_reflp:
-  shows "equivp R \<Longrightarrow> list_all2 R xs xs"
-  by (induct xs, simp_all add: equivp_reflp)
+lemma list_all2_eq [id_simps]:
+  "list_all2 (op =) = (op =)"
+proof (rule ext)+
+  fix xs ys
+  show "list_all2 (op =) xs ys \<longleftrightarrow> xs = ys"
+    by (induct xs ys rule: list_induct2') simp_all
+qed
 
-lemma list_all2_symp:
-  assumes a: "equivp R"
-  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_reflp:
+  assumes "reflp R"
+  shows "reflp (list_all2 R)"
+proof (rule reflpI)
+  from assms have *: "\<And>xs. R xs xs" by (rule reflpE)
+  fix xs
+  show "list_all2 R xs xs"
+    by (induct xs) (simp_all add: *)
+qed
 
-lemma list_all2_transp:
-  assumes a: "equivp R"
-  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_symp:
+  assumes "symp R"
+  shows "symp (list_all2 R)"
+proof (rule sympI)
+  from assms have *: "\<And>xs ys. R xs ys \<Longrightarrow> R ys xs" by (rule sympE)
+  fix xs ys
+  assume "list_all2 R xs ys"
+  then show "list_all2 R ys xs"
+    by (induct xs ys rule: list_induct2') (simp_all add: *)
+qed
 
-lemma list_equivp[quot_equiv]:
-  assumes a: "equivp R"
-  shows "equivp (list_all2 R)"
-  apply (intro equivpI)
-  unfolding reflp_def symp_def transp_def
-  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_transp:
+  assumes "transp R"
+  shows "transp (list_all2 R)"
+proof (rule transpI)
+  from assms have *: "\<And>xs ys zs. R xs ys \<Longrightarrow> R ys zs \<Longrightarrow> R xs zs" by (rule transpE)
+  fix xs ys zs
+  assume A: "list_all2 R xs ys" "list_all2 R ys zs"
+  then have "length xs = length ys" "length ys = length zs" by (blast dest: list_all2_lengthD)+
+  then show "list_all2 R xs zs" using A
+    by (induct xs ys zs rule: list_induct3) (auto intro: *)
+qed
 
-lemma list_all2_rel:
-  assumes q: "Quotient R Abs Rep"
-  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]
-  apply(metis)
-  done
+lemma list_equivp [quot_equiv]:
+  "equivp R \<Longrightarrow> equivp (list_all2 R)"
+  by (blast intro: equivpI list_reflp list_symp list_transp elim: equivpE)
 
-lemma list_quotient[quot_thm]:
-  assumes q: "Quotient R Abs Rep"
+lemma list_quotient [quot_thm]:
+  assumes "Quotient R Abs 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(intro conjI allI)
-  apply(induct_tac a)
-  apply(simp_all add: Quotient_rep_reflp[OF q])
-  apply(rule list_all2_rel[OF q])
-  done
+proof (rule QuotientI)
+  from assms have "\<And>x. Abs (Rep x) = x" by (rule Quotient_abs_rep)
+  then show "\<And>xs. map Abs (map Rep xs) = xs" by (simp add: comp_def)
+next
+  from assms have "\<And>x y. R (Rep x) (Rep y) \<longleftrightarrow> x = y" by (rule Quotient_rel_rep)
+  then show "\<And>xs. list_all2 R (map Rep xs) (map Rep xs)"
+    by (simp add: list_all2_map1 list_all2_map2 list_all2_eq)
+next
+  fix xs ys
+  from assms have "\<And>x y. R x x \<and> R y y \<and> Abs x = Abs y \<longleftrightarrow> R x y" by (rule Quotient_rel)
+  then show "list_all2 R xs ys \<longleftrightarrow> list_all2 R xs xs \<and> list_all2 R ys ys \<and> map Abs xs = map Abs ys"
+    by (induct xs ys rule: list_induct2') auto
+qed
 
-lemma cons_prs[quot_preserve]:
+lemma cons_prs [quot_preserve]:
   assumes q: "Quotient R Abs Rep"
   shows "(Rep ---> (map Rep) ---> (map Abs)) (op #) = (op #)"
   by (auto simp add: fun_eq_iff comp_def Quotient_abs_rep [OF q])
 
-lemma cons_rsp[quot_respect]:
+lemma cons_rsp [quot_respect]:
   assumes q: "Quotient R Abs Rep"
   shows "(R ===> list_all2 R ===> list_all2 R) (op #) (op #)"
   by auto
 
-lemma nil_prs[quot_preserve]:
+lemma nil_prs [quot_preserve]:
   assumes q: "Quotient R Abs Rep"
   shows "map Abs [] = []"
   by simp
 
-lemma nil_rsp[quot_respect]:
+lemma nil_rsp [quot_respect]:
   assumes q: "Quotient R Abs Rep"
   shows "list_all2 R [] []"
   by simp
@@ -109,7 +111,7 @@
   by (induct l)
      (simp_all add: Quotient_abs_rep[OF a] Quotient_abs_rep[OF b])
 
-lemma map_prs[quot_preserve]:
+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"
@@ -117,8 +119,7 @@
   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]:
+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"
@@ -137,7 +138,7 @@
   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]:
+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"
@@ -151,8 +152,7 @@
   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]:
+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"
@@ -217,11 +217,11 @@
     qed
   qed
 
-lemma[quot_respect]:
+lemma [quot_respect]:
   "((R ===> R ===> op =) ===> list_all2 R ===> list_all2 R ===> op =) list_all2 list_all2"
   by (simp add: list_all2_rsp fun_rel_def)
 
-lemma[quot_preserve]:
+lemma [quot_preserve]:
   assumes a: "Quotient R abs1 rep1"
   shows "((abs1 ---> abs1 ---> id) ---> map rep1 ---> map rep1 ---> id) list_all2 = list_all2"
   apply (simp add: fun_eq_iff)
@@ -230,19 +230,11 @@
   apply (simp_all add: Quotient_abs_rep[OF a])
   done
 
-lemma[quot_preserve]:
+lemma [quot_preserve]:
   assumes a: "Quotient R abs1 rep1"
   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_all2_eq[id_simps]:
-  shows "(list_all2 (op =)) = (op =)"
-  unfolding fun_eq_iff
-  apply(rule allI)+
-  apply(induct_tac x xa rule: list_induct2')
-  apply(simp_all)
-  done
-
 lemma list_all2_find_element:
   assumes a: "x \<in> set a"
   and b: "list_all2 R a b"
diff -r eeaa59fb5ad8 -r 9f32d7b8b24f src/HOL/Library/Quotient_Option.thy
--- a/src/HOL/Library/Quotient_Option.thy	Tue Nov 30 00:12:29 2010 +0100
+++ b/src/HOL/Library/Quotient_Option.thy	Tue Nov 30 15:58:21 2010 +0100
@@ -18,64 +18,73 @@
 
 declare [[map option = (Option.map, option_rel)]]
 
-text {* should probably be in Option.thy *}
-lemma split_option_all:
-  shows "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>a. P (Some a))"
-  apply(auto)
-  apply(case_tac x)
-  apply(simp_all)
+lemma option_rel_unfold:
+  "option_rel R x y = (case (x, y) of (None, None) \<Rightarrow> True
+    | (Some x, Some y) \<Rightarrow> R x y
+    | _ \<Rightarrow> False)"
+  by (cases x) (cases y, simp_all)+
+
+lemma option_rel_map1:
+  "option_rel R (Option.map f x) y \<longleftrightarrow> option_rel (\<lambda>x. R (f x)) x y"
+  by (simp add: option_rel_unfold split: option.split)
+
+lemma option_rel_map2:
+  "option_rel R x (Option.map f y) \<longleftrightarrow> option_rel (\<lambda>x y. R x (f y)) x y"
+  by (simp add: option_rel_unfold split: option.split)
+
+lemma option_map_id [id_simps]:
+  "Option.map id = id"
+  by (simp add: id_def Option.map.identity fun_eq_iff)
+
+lemma option_rel_eq [id_simps]:
+  "option_rel (op =) = (op =)"
+  by (simp add: option_rel_unfold fun_eq_iff split: option.split)
+
+lemma option_reflp:
+  "reflp R \<Longrightarrow> reflp (option_rel R)"
+  by (auto simp add: option_rel_unfold split: option.splits intro!: reflpI elim: reflpE)
+
+lemma option_symp:
+  "symp R \<Longrightarrow> symp (option_rel R)"
+  by (auto simp add: option_rel_unfold split: option.splits intro!: sympI elim: sympE)
+
+lemma option_transp:
+  "transp R \<Longrightarrow> transp (option_rel R)"
+  by (auto simp add: option_rel_unfold split: option.splits intro!: transpI elim: transpE)
+
+lemma option_equivp [quot_equiv]:
+  "equivp R \<Longrightarrow> equivp (option_rel R)"
+  by (blast intro: equivpI option_reflp option_symp option_transp elim: equivpE)
+
+lemma option_quotient [quot_thm]:
+  assumes "Quotient R Abs Rep"
+  shows "Quotient (option_rel R) (Option.map Abs) (Option.map Rep)"
+  apply (rule QuotientI)
+  apply (simp_all add: Option.map.compositionality Option.map.identity option_rel_eq option_rel_map1 option_rel_map2 Quotient_abs_rep [OF assms] Quotient_rel_rep [OF assms])
+  using Quotient_rel [OF assms]
+  apply (simp add: option_rel_unfold split: option.split)
   done
 
-lemma option_quotient[quot_thm]:
-  assumes q: "Quotient R Abs Rep"
-  shows "Quotient (option_rel R) (Option.map Abs) (Option.map Rep)"
-  unfolding Quotient_def
-  apply(simp add: split_option_all)
-  apply(simp add: Quotient_abs_rep[OF q] Quotient_rel_rep[OF q])
-  using q
-  unfolding Quotient_def
-  apply(blast)
-  done
-
-lemma option_equivp[quot_equiv]:
-  assumes a: "equivp R"
-  shows "equivp (option_rel R)"
-  apply(rule equivpI)
-  unfolding reflp_def symp_def transp_def
-  apply(simp_all add: split_option_all)
-  apply(blast intro: equivp_reflp[OF a])
-  apply(blast intro: equivp_symp[OF a])
-  apply(blast intro: equivp_transp[OF a])
-  done
-
-lemma option_None_rsp[quot_respect]:
+lemma option_None_rsp [quot_respect]:
   assumes q: "Quotient R Abs Rep"
   shows "option_rel R None None"
   by simp
 
-lemma option_Some_rsp[quot_respect]:
+lemma option_Some_rsp [quot_respect]:
   assumes q: "Quotient R Abs Rep"
   shows "(R ===> option_rel R) Some Some"
   by auto
 
-lemma option_None_prs[quot_preserve]:
+lemma option_None_prs [quot_preserve]:
   assumes q: "Quotient R Abs Rep"
   shows "Option.map Abs None = None"
   by simp
 
-lemma option_Some_prs[quot_preserve]:
+lemma option_Some_prs [quot_preserve]:
   assumes q: "Quotient R Abs Rep"
   shows "(Rep ---> Option.map Abs) Some = Some"
   apply(simp add: fun_eq_iff)
   apply(simp add: Quotient_abs_rep[OF q])
   done
 
-lemma option_map_id[id_simps]:
-  shows "Option.map id = id"
-  by (simp add: fun_eq_iff split_option_all)
-
-lemma option_rel_eq[id_simps]:
-  shows "option_rel (op =) = (op =)"
-  by (simp add: fun_eq_iff split_option_all)
-
 end
diff -r eeaa59fb5ad8 -r 9f32d7b8b24f src/HOL/Library/Quotient_Product.thy
--- a/src/HOL/Library/Quotient_Product.thy	Tue Nov 30 00:12:29 2010 +0100
+++ b/src/HOL/Library/Quotient_Product.thy	Tue Nov 30 15:58:21 2010 +0100
@@ -19,38 +19,39 @@
   "prod_rel R1 R2 (a, b) (c, d) \<longleftrightarrow> R1 a c \<and> R2 b d"
   by (simp add: prod_rel_def)
 
-lemma prod_equivp[quot_equiv]:
-  assumes a: "equivp R1"
-  assumes b: "equivp R2"
+lemma map_pair_id [id_simps]:
+  shows "map_pair id id = id"
+  by (simp add: fun_eq_iff)
+
+lemma prod_rel_eq [id_simps]:
+  shows "prod_rel (op =) (op =) = (op =)"
+  by (simp add: fun_eq_iff)
+
+lemma prod_equivp [quot_equiv]:
+  assumes "equivp R1"
+  assumes "equivp R2"
   shows "equivp (prod_rel R1 R2)"
-  apply(rule equivpI)
-  unfolding reflp_def symp_def transp_def
-  apply(simp_all add: split_paired_all prod_rel_def)
-  apply(blast intro: equivp_reflp[OF a] equivp_reflp[OF b])
-  apply(blast intro: equivp_symp[OF a] equivp_symp[OF b])
-  apply(blast intro: equivp_transp[OF a] equivp_transp[OF b])
+  using assms by (auto intro!: equivpI reflpI sympI transpI elim!: equivpE elim: reflpE sympE transpE)
+
+lemma prod_quotient [quot_thm]:
+  assumes "Quotient R1 Abs1 Rep1"
+  assumes "Quotient R2 Abs2 Rep2"
+  shows "Quotient (prod_rel R1 R2) (map_pair Abs1 Abs2) (map_pair Rep1 Rep2)"
+  apply (rule QuotientI)
+  apply (simp add: map_pair.compositionality map_pair.identity
+     Quotient_abs_rep [OF assms(1)] Quotient_abs_rep [OF assms(2)])
+  apply (simp add: split_paired_all Quotient_rel_rep [OF assms(1)] Quotient_rel_rep [OF assms(2)])
+  using Quotient_rel [OF assms(1)] Quotient_rel [OF assms(2)]
+  apply (auto simp add: split_paired_all)
   done
 
-lemma prod_quotient[quot_thm]:
-  assumes q1: "Quotient R1 Abs1 Rep1"
-  assumes q2: "Quotient R2 Abs2 Rep2"
-  shows "Quotient (prod_rel R1 R2) (map_pair Abs1 Abs2) (map_pair Rep1 Rep2)"
-  unfolding Quotient_def
-  apply(simp add: split_paired_all)
-  apply(simp add: Quotient_abs_rep[OF q1] Quotient_rel_rep[OF q1])
-  apply(simp add: Quotient_abs_rep[OF q2] Quotient_rel_rep[OF q2])
-  using q1 q2
-  unfolding Quotient_def
-  apply(blast)
-  done
-
-lemma Pair_rsp[quot_respect]:
+lemma Pair_rsp [quot_respect]:
   assumes q1: "Quotient R1 Abs1 Rep1"
   assumes q2: "Quotient R2 Abs2 Rep2"
   shows "(R1 ===> R2 ===> prod_rel R1 R2) Pair Pair"
   by (auto simp add: prod_rel_def)
 
-lemma Pair_prs[quot_preserve]:
+lemma Pair_prs [quot_preserve]:
   assumes q1: "Quotient R1 Abs1 Rep1"
   assumes q2: "Quotient R2 Abs2 Rep2"
   shows "(Rep1 ---> Rep2 ---> (map_pair Abs1 Abs2)) Pair = Pair"
@@ -58,35 +59,35 @@
   apply(simp add: Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])
   done
 
-lemma fst_rsp[quot_respect]:
+lemma fst_rsp [quot_respect]:
   assumes "Quotient R1 Abs1 Rep1"
   assumes "Quotient R2 Abs2 Rep2"
   shows "(prod_rel R1 R2 ===> R1) fst fst"
   by auto
 
-lemma fst_prs[quot_preserve]:
+lemma fst_prs [quot_preserve]:
   assumes q1: "Quotient R1 Abs1 Rep1"
   assumes q2: "Quotient R2 Abs2 Rep2"
   shows "(map_pair Rep1 Rep2 ---> Abs1) fst = fst"
   by (simp add: fun_eq_iff Quotient_abs_rep[OF q1])
 
-lemma snd_rsp[quot_respect]:
+lemma snd_rsp [quot_respect]:
   assumes "Quotient R1 Abs1 Rep1"
   assumes "Quotient R2 Abs2 Rep2"
   shows "(prod_rel R1 R2 ===> R2) snd snd"
   by auto
 
-lemma snd_prs[quot_preserve]:
+lemma snd_prs [quot_preserve]:
   assumes q1: "Quotient R1 Abs1 Rep1"
   assumes q2: "Quotient R2 Abs2 Rep2"
   shows "(map_pair Rep1 Rep2 ---> Abs2) snd = snd"
   by (simp add: fun_eq_iff Quotient_abs_rep[OF q2])
 
-lemma split_rsp[quot_respect]:
+lemma split_rsp [quot_respect]:
   shows "((R1 ===> R2 ===> (op =)) ===> (prod_rel R1 R2) ===> (op =)) split split"
   by (auto intro!: fun_relI elim!: fun_relE)
 
-lemma split_prs[quot_preserve]:
+lemma split_prs [quot_preserve]:
   assumes q1: "Quotient R1 Abs1 Rep1"
   and     q2: "Quotient R2 Abs2 Rep2"
   shows "(((Abs1 ---> Abs2 ---> id) ---> map_pair Rep1 Rep2 ---> id) split) = split"
@@ -111,12 +112,4 @@
 
 declare Pair_eq[quot_preserve]
 
-lemma map_pair_id[id_simps]:
-  shows "map_pair id id = id"
-  by (simp add: fun_eq_iff)
-
-lemma prod_rel_eq[id_simps]:
-  shows "prod_rel (op =) (op =) = (op =)"
-  by (simp add: fun_eq_iff)
-
 end
diff -r eeaa59fb5ad8 -r 9f32d7b8b24f src/HOL/Library/Quotient_Sum.thy
--- a/src/HOL/Library/Quotient_Sum.thy	Tue Nov 30 00:12:29 2010 +0100
+++ b/src/HOL/Library/Quotient_Sum.thy	Tue Nov 30 15:58:21 2010 +0100
@@ -18,53 +18,68 @@
 
 declare [[map sum = (sum_map, sum_rel)]]
 
+lemma sum_rel_unfold:
+  "sum_rel R1 R2 x y = (case (x, y) of (Inl x, Inl y) \<Rightarrow> R1 x y
+    | (Inr x, Inr y) \<Rightarrow> R2 x y
+    | _ \<Rightarrow> False)"
+  by (cases x) (cases y, simp_all)+
 
-text {* should probably be in @{theory Sum_Type} *}
-lemma split_sum_all:
-  shows "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (Inl x)) \<and> (\<forall>x. P (Inr x))"
-  apply(auto)
-  apply(case_tac x)
-  apply(simp_all)
-  done
+lemma sum_rel_map1:
+  "sum_rel R1 R2 (sum_map f1 f2 x) y \<longleftrightarrow> sum_rel (\<lambda>x. R1 (f1 x)) (\<lambda>x. R2 (f2 x)) x y"
+  by (simp add: sum_rel_unfold split: sum.split)
+
+lemma sum_rel_map2:
+  "sum_rel R1 R2 x (sum_map f1 f2 y) \<longleftrightarrow> sum_rel (\<lambda>x y. R1 x (f1 y)) (\<lambda>x y. R2 x (f2 y)) x y"
+  by (simp add: sum_rel_unfold split: sum.split)
+
+lemma sum_map_id [id_simps]:
+  "sum_map id id = id"
+  by (simp add: id_def sum_map.identity fun_eq_iff)
 
-lemma sum_equivp[quot_equiv]:
-  assumes a: "equivp R1"
-  assumes b: "equivp R2"
-  shows "equivp (sum_rel R1 R2)"
-  apply(rule equivpI)
-  unfolding reflp_def symp_def transp_def
-  apply(simp_all add: split_sum_all)
-  apply(blast intro: equivp_reflp[OF a] equivp_reflp[OF b])
-  apply(blast intro: equivp_symp[OF a] equivp_symp[OF b])
-  apply(blast intro: equivp_transp[OF a] equivp_transp[OF b])
-  done
+lemma sum_rel_eq [id_simps]:
+  "sum_rel (op =) (op =) = (op =)"
+  by (simp add: sum_rel_unfold fun_eq_iff split: sum.split)
+
+lemma sum_reflp:
+  "reflp R1 \<Longrightarrow> reflp R2 \<Longrightarrow> reflp (sum_rel R1 R2)"
+  by (auto simp add: sum_rel_unfold split: sum.splits intro!: reflpI elim: reflpE)
 
-lemma sum_quotient[quot_thm]:
+lemma sum_symp:
+  "symp R1 \<Longrightarrow> symp R2 \<Longrightarrow> symp (sum_rel R1 R2)"
+  by (auto simp add: sum_rel_unfold split: sum.splits intro!: sympI elim: sympE)
+
+lemma sum_transp:
+  "transp R1 \<Longrightarrow> transp R2 \<Longrightarrow> transp (sum_rel R1 R2)"
+  by (auto simp add: sum_rel_unfold split: sum.splits intro!: transpI elim: transpE)
+
+lemma sum_equivp [quot_equiv]:
+  "equivp R1 \<Longrightarrow> equivp R2 \<Longrightarrow> equivp (sum_rel R1 R2)"
+  by (blast intro: equivpI sum_reflp sum_symp sum_transp elim: equivpE)
+  
+lemma sum_quotient [quot_thm]:
   assumes q1: "Quotient R1 Abs1 Rep1"
   assumes q2: "Quotient R2 Abs2 Rep2"
   shows "Quotient (sum_rel R1 R2) (sum_map Abs1 Abs2) (sum_map Rep1 Rep2)"
-  unfolding Quotient_def
-  apply(simp add: split_sum_all)
-  apply(simp_all add: Quotient_abs_rep[OF q1] Quotient_rel_rep[OF q1])
-  apply(simp_all add: Quotient_abs_rep[OF q2] Quotient_rel_rep[OF q2])
-  using q1 q2
-  unfolding Quotient_def
-  apply(blast)+
+  apply (rule QuotientI)
+  apply (simp_all add: sum_map.compositionality sum_map.identity sum_rel_eq sum_rel_map1 sum_rel_map2
+    Quotient_abs_rep [OF q1] Quotient_rel_rep [OF q1] Quotient_abs_rep [OF q2] Quotient_rel_rep [OF q2])
+  using Quotient_rel [OF q1] Quotient_rel [OF q2]
+  apply (simp add: sum_rel_unfold split: sum.split)
   done
 
-lemma sum_Inl_rsp[quot_respect]:
+lemma sum_Inl_rsp [quot_respect]:
   assumes q1: "Quotient R1 Abs1 Rep1"
   assumes q2: "Quotient R2 Abs2 Rep2"
   shows "(R1 ===> sum_rel R1 R2) Inl Inl"
   by auto
 
-lemma sum_Inr_rsp[quot_respect]:
+lemma sum_Inr_rsp [quot_respect]:
   assumes q1: "Quotient R1 Abs1 Rep1"
   assumes q2: "Quotient R2 Abs2 Rep2"
   shows "(R2 ===> sum_rel R1 R2) Inr Inr"
   by auto
 
-lemma sum_Inl_prs[quot_preserve]:
+lemma sum_Inl_prs [quot_preserve]:
   assumes q1: "Quotient R1 Abs1 Rep1"
   assumes q2: "Quotient R2 Abs2 Rep2"
   shows "(Rep1 ---> sum_map Abs1 Abs2) Inl = Inl"
@@ -72,7 +87,7 @@
   apply(simp add: Quotient_abs_rep[OF q1])
   done
 
-lemma sum_Inr_prs[quot_preserve]:
+lemma sum_Inr_prs [quot_preserve]:
   assumes q1: "Quotient R1 Abs1 Rep1"
   assumes q2: "Quotient R2 Abs2 Rep2"
   shows "(Rep2 ---> sum_map Abs1 Abs2) Inr = Inr"
@@ -80,12 +95,4 @@
   apply(simp add: Quotient_abs_rep[OF q2])
   done
 
-lemma sum_map_id[id_simps]:
-  shows "sum_map id id = id"
-  by (simp add: fun_eq_iff split_sum_all)
-
-lemma sum_rel_eq[id_simps]:
-  shows "sum_rel (op =) (op =) = (op =)"
-  by (simp add: fun_eq_iff split_sum_all)
-
 end
diff -r eeaa59fb5ad8 -r 9f32d7b8b24f src/HOL/NSA/StarDef.thy
--- a/src/HOL/NSA/StarDef.thy	Tue Nov 30 00:12:29 2010 +0100
+++ b/src/HOL/NSA/StarDef.thy	Tue Nov 30 15:58:21 2010 +0100
@@ -62,7 +62,7 @@
 by (simp add: starrel_def)
 
 lemma equiv_starrel: "equiv UNIV starrel"
-proof (rule equiv.intro)
+proof (rule equivI)
   show "refl starrel" by (simp add: refl_on_def)
   show "sym starrel" by (simp add: sym_def eq_commute)
   show "trans starrel" by (auto intro: transI elim!: ultra)
diff -r eeaa59fb5ad8 -r 9f32d7b8b24f src/HOL/Predicate.thy
--- a/src/HOL/Predicate.thy	Tue Nov 30 00:12:29 2010 +0100
+++ b/src/HOL/Predicate.thy	Tue Nov 30 15:58:21 2010 +0100
@@ -363,6 +363,44 @@
 abbreviation single_valuedP :: "('a => 'b => bool) => bool" where
   "single_valuedP r == single_valued {(x, y). r x y}"
 
+(*FIXME inconsistencies: abbreviations vs. definitions, suffix `P` vs. suffix `p`*)
+
+definition reflp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
+  "reflp r \<longleftrightarrow> refl {(x, y). r x y}"
+
+definition symp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
+  "symp r \<longleftrightarrow> sym {(x, y). r x y}"
+
+definition transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
+  "transp r \<longleftrightarrow> trans {(x, y). r x y}"
+
+lemma reflpI:
+  "(\<And>x. r x x) \<Longrightarrow> reflp r"
+  by (auto intro: refl_onI simp add: reflp_def)
+
+lemma reflpE:
+  assumes "reflp r"
+  obtains "r x x"
+  using assms by (auto dest: refl_onD simp add: reflp_def)
+
+lemma sympI:
+  "(\<And>x y. r x y \<Longrightarrow> r y x) \<Longrightarrow> symp r"
+  by (auto intro: symI simp add: symp_def)
+
+lemma sympE:
+  assumes "symp r" and "r x y"
+  obtains "r y x"
+  using assms by (auto dest: symD simp add: symp_def)
+
+lemma transpI:
+  "(\<And>x y z. r x y \<Longrightarrow> r y z \<Longrightarrow> r x z) \<Longrightarrow> transp r"
+  by (auto intro: transI simp add: transp_def)
+  
+lemma transpE:
+  assumes "transp r" and "r x y" and "r y z"
+  obtains "r x z"
+  using assms by (auto dest: transD simp add: transp_def)
+
 
 subsection {* Predicates as enumerations *}
 
diff -r eeaa59fb5ad8 -r 9f32d7b8b24f src/HOL/Quotient.thy
--- a/src/HOL/Quotient.thy	Tue Nov 30 00:12:29 2010 +0100
+++ b/src/HOL/Quotient.thy	Tue Nov 30 15:58:21 2010 +0100
@@ -14,131 +14,15 @@
   ("Tools/Quotient/quotient_tacs.ML")
 begin
 
-
 text {*
   Basic definition for equivalence relations
   that are represented by predicates.
 *}
 
-definition
-  "reflp E \<longleftrightarrow> (\<forall>x. E x x)"
-
-lemma refl_reflp:
-  "refl A \<longleftrightarrow> reflp (\<lambda>x y. (x, y) \<in> A)"
-  by (simp add: refl_on_def reflp_def)
-
-definition
-  "symp E \<longleftrightarrow> (\<forall>x y. E x y \<longrightarrow> E y x)"
-
-lemma sym_symp:
-  "sym A \<longleftrightarrow> symp (\<lambda>x y. (x, y) \<in> A)"
-  by (simp add: sym_def symp_def)
-
-definition
-  "transp E \<longleftrightarrow> (\<forall>x y z. E x y \<and> E y z \<longrightarrow> E x z)"
-
-lemma trans_transp:
-  "trans A \<longleftrightarrow> transp (\<lambda>x y. (x, y) \<in> A)"
-  by (auto simp add: trans_def transp_def)
-
-definition
-  "equivp E \<longleftrightarrow> (\<forall>x y. E x y = (E x = E y))"
-
-lemma equivp_reflp_symp_transp:
-  shows "equivp E = (reflp E \<and> symp E \<and> transp E)"
-  unfolding equivp_def reflp_def symp_def transp_def fun_eq_iff
-  by blast
-
-lemma equiv_equivp:
-  "equiv UNIV A \<longleftrightarrow> equivp (\<lambda>x y. (x, y) \<in> A)"
-  by (simp add: equiv_def equivp_reflp_symp_transp refl_reflp sym_symp trans_transp)
-
-lemma equivp_reflp:
-  shows "equivp E \<Longrightarrow> E x x"
-  by (simp only: equivp_reflp_symp_transp reflp_def)
-
-lemma equivp_symp:
-  shows "equivp E \<Longrightarrow> E x y \<Longrightarrow> E y x"
-  by (simp add: equivp_def)
-
-lemma equivp_transp:
-  shows "equivp E \<Longrightarrow> E x y \<Longrightarrow> E y z \<Longrightarrow> E x z"
-  by (simp add: equivp_def)
-
-lemma equivpI:
-  assumes "reflp R" "symp R" "transp R"
-  shows "equivp R"
-  using assms by (simp add: equivp_reflp_symp_transp)
-
-lemma identity_equivp:
-  shows "equivp (op =)"
-  unfolding equivp_def
-  by auto
-
-text {* Partial equivalences *}
-
-definition
-  "part_equivp E \<longleftrightarrow> (\<exists>x. E x x) \<and> (\<forall>x y. E x y = (E x x \<and> E y y \<and> (E x = E y)))"
-
-lemma equivp_implies_part_equivp:
-  assumes a: "equivp E"
-  shows "part_equivp E"
-  using a
-  unfolding equivp_def part_equivp_def
-  by auto
-
-lemma part_equivp_symp:
-  assumes e: "part_equivp R"
-  and a: "R x y"
-  shows "R y x"
-  using e[simplified part_equivp_def] a
-  by (metis)
-
-lemma part_equivp_typedef:
-  shows "part_equivp R \<Longrightarrow> \<exists>d. d \<in> (\<lambda>c. \<exists>x. R x x \<and> c = R x)"
-  unfolding part_equivp_def mem_def
-  apply clarify
-  apply (intro exI)
-  apply (rule conjI)
-  apply assumption
-  apply (rule refl)
-  done
-
-lemma part_equivp_refl_symp_transp:
-  shows "part_equivp E \<longleftrightarrow> ((\<exists>x. E x x) \<and> symp E \<and> transp E)"
-proof
-  assume "part_equivp E"
-  then show "(\<exists>x. E x x) \<and> symp E \<and> transp E"
-  unfolding part_equivp_def symp_def transp_def
-  by metis
-next
-  assume a: "(\<exists>x. E x x) \<and> symp E \<and> transp E"
-  then have b: "(\<forall>x y. E x y \<longrightarrow> E y x)" and c: "(\<forall>x y z. E x y \<and> E y z \<longrightarrow> E x z)"
-    unfolding symp_def transp_def by (metis, metis)
-  have "(\<forall>x y. E x y = (E x x \<and> E y y \<and> E x = E y))"
-  proof (intro allI iffI conjI)
-    fix x y
-    assume d: "E x y"
-    then show "E x x" using b c by metis
-    show "E y y" using b c d by metis
-    show "E x = E y" unfolding fun_eq_iff using b c d by metis
-  next
-    fix x y
-    assume "E x x \<and> E y y \<and> E x = E y"
-    then show "E x y" using b c by metis
-  qed
-  then show "part_equivp E" unfolding part_equivp_def using a by metis
-qed
-
-lemma part_equivpI:
-  assumes "\<exists>x. R x x" "symp R" "transp R"
-  shows "part_equivp R"
-  using assms by (simp add: part_equivp_refl_symp_transp)
-
 text {* Composition of Relations *}
 
 abbreviation
-  rel_conj (infixr "OOO" 75)
+  rel_conj :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" (infixr "OOO" 75)
 where
   "r1 OOO r2 \<equiv> r1 OO r2 OO r1"
 
@@ -169,16 +53,16 @@
 definition
   fun_rel :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool" (infixr "===>" 55)
 where
-  "fun_rel E1 E2 = (\<lambda>f g. \<forall>x y. E1 x y \<longrightarrow> E2 (f x) (g y))"
+  "fun_rel R1 R2 = (\<lambda>f g. \<forall>x y. R1 x y \<longrightarrow> R2 (f x) (g y))"
 
 lemma fun_relI [intro]:
-  assumes "\<And>x y. E1 x y \<Longrightarrow> E2 (f x) (g y)"
-  shows "(E1 ===> E2) f g"
+  assumes "\<And>x y. R1 x y \<Longrightarrow> R2 (f x) (g y)"
+  shows "(R1 ===> R2) f g"
   using assms by (simp add: fun_rel_def)
 
 lemma fun_relE:
-  assumes "(E1 ===> E2) f g" and "E1 x y"
-  obtains "E2 (f x) (g y)"
+  assumes "(R1 ===> R2) f g" and "R1 x y"
+  obtains "R2 (f x) (g y)"
   using assms by (simp add: fun_rel_def)
 
 lemma fun_rel_eq:
@@ -189,34 +73,41 @@
 subsection {* Quotient Predicate *}
 
 definition
-  "Quotient E Abs Rep \<longleftrightarrow>
-     (\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. E (Rep a) (Rep a)) \<and>
-     (\<forall>r s. E r s = (E r r \<and> E s s \<and> (Abs r = Abs s)))"
+  "Quotient R Abs Rep \<longleftrightarrow>
+     (\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. R (Rep a) (Rep a)) \<and>
+     (\<forall>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s)"
+
+lemma QuotientI:
+  assumes "\<And>a. Abs (Rep a) = a"
+    and "\<And>a. R (Rep a) (Rep a)"
+    and "\<And>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s"
+  shows "Quotient R Abs Rep"
+  using assms unfolding Quotient_def by blast
 
 lemma Quotient_abs_rep:
-  assumes a: "Quotient E Abs Rep"
+  assumes a: "Quotient R Abs Rep"
   shows "Abs (Rep a) = a"
   using a
   unfolding Quotient_def
   by simp
 
 lemma Quotient_rep_reflp:
-  assumes a: "Quotient E Abs Rep"
-  shows "E (Rep a) (Rep a)"
+  assumes a: "Quotient R Abs Rep"
+  shows "R (Rep a) (Rep a)"
   using a
   unfolding Quotient_def
   by blast
 
 lemma Quotient_rel:
-  assumes a: "Quotient E Abs Rep"
-  shows " E r s = (E r r \<and> E s s \<and> (Abs r = Abs s))"
+  assumes a: "Quotient R Abs Rep"
+  shows "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" -- {* orientation does not loop on rewriting *}
   using a
   unfolding Quotient_def
   by blast
 
 lemma Quotient_rel_rep:
   assumes a: "Quotient R Abs Rep"
-  shows "R (Rep a) (Rep b) = (a = b)"
+  shows "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
   using a
   unfolding Quotient_def
   by metis
@@ -228,22 +119,20 @@
   by blast
 
 lemma Quotient_rel_abs:
-  assumes a: "Quotient E Abs Rep"
-  shows "E r s \<Longrightarrow> Abs r = Abs s"
+  assumes a: "Quotient R Abs Rep"
+  shows "R r s \<Longrightarrow> Abs r = Abs s"
   using a unfolding Quotient_def
   by blast
 
 lemma Quotient_symp:
-  assumes a: "Quotient E Abs Rep"
-  shows "symp E"
-  using a unfolding Quotient_def symp_def
-  by metis
+  assumes a: "Quotient R Abs Rep"
+  shows "symp R"
+  using a unfolding Quotient_def using sympI by metis
 
 lemma Quotient_transp:
-  assumes a: "Quotient E Abs Rep"
-  shows "transp E"
-  using a unfolding Quotient_def transp_def
-  by metis
+  assumes a: "Quotient R Abs Rep"
+  shows "transp R"
+  using a unfolding Quotient_def using transpI by metis
 
 lemma identity_quotient:
   shows "Quotient (op =) id id"
@@ -291,8 +180,7 @@
   and     a: "R xa xb" "R ya yb"
   shows "R xa ya = R xb yb"
   using a Quotient_symp[OF q] Quotient_transp[OF q]
-  unfolding symp_def transp_def
-  by blast
+  by (blast elim: sympE transpE)
 
 lemma lambda_prs:
   assumes q1: "Quotient R1 Abs1 Rep1"
@@ -300,7 +188,7 @@
   shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"
   unfolding fun_eq_iff
   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
-  by (simp add:)
+  by simp
 
 lemma lambda_prs1:
   assumes q1: "Quotient R1 Abs1 Rep1"
@@ -308,7 +196,7 @@
   shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)"
   unfolding fun_eq_iff
   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
-  by (simp add:)
+  by simp
 
 lemma rep_abs_rsp:
   assumes q: "Quotient R Abs Rep"
@@ -392,9 +280,7 @@
   apply(simp add: in_respects fun_rel_def)
   apply(rule impI)
   using a equivp_reflp_symp_transp[of "R2"]
-  apply(simp add: reflp_def)
-  apply(simp)
-  apply(simp)
+  apply (auto elim: equivpE reflpE)
   done
 
 lemma bex_reg_eqv_range:
@@ -406,7 +292,7 @@
   apply(simp add: Respects_def in_respects fun_rel_def)
   apply(rule impI)
   using a equivp_reflp_symp_transp[of "R2"]
-  apply(simp add: reflp_def)
+  apply (auto elim: equivpE reflpE)
   done
 
 (* Next four lemmas are unused *)
diff -r eeaa59fb5ad8 -r 9f32d7b8b24f src/HOL/Rat.thy
--- a/src/HOL/Rat.thy	Tue Nov 30 00:12:29 2010 +0100
+++ b/src/HOL/Rat.thy	Tue Nov 30 15:58:21 2010 +0100
@@ -44,7 +44,7 @@
 qed
   
 lemma equiv_ratrel: "equiv {x. snd x \<noteq> 0} ratrel"
-  by (rule equiv.intro [OF refl_on_ratrel sym_ratrel trans_ratrel])
+  by (rule equivI [OF refl_on_ratrel sym_ratrel trans_ratrel])
 
 lemmas UN_ratrel = UN_equiv_class [OF equiv_ratrel]
 lemmas UN_ratrel2 = UN_equiv_class2 [OF equiv_ratrel equiv_ratrel]
@@ -146,7 +146,7 @@
 lemma minus_rat [simp]: "- Fract a b = Fract (- a) b"
 proof -
   have "(\<lambda>x. ratrel `` {(- fst x, snd x)}) respects ratrel"
-    by (simp add: congruent_def)
+    by (simp add: congruent_def split_paired_all)
   then show ?thesis by (simp add: Fract_def minus_rat_def UN_ratrel)
 qed
 
@@ -781,7 +781,7 @@
 
 lemma of_rat_congruent:
   "(\<lambda>(a, b). {of_int a / of_int b :: 'a::field_char_0}) respects ratrel"
-apply (rule congruent.intro)
+apply (rule congruentI)
 apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
 apply (simp only: of_int_mult [symmetric])
 done
diff -r eeaa59fb5ad8 -r 9f32d7b8b24f src/HOL/RealDef.thy
--- a/src/HOL/RealDef.thy	Tue Nov 30 00:12:29 2010 +0100
+++ b/src/HOL/RealDef.thy	Tue Nov 30 15:58:21 2010 +0100
@@ -324,7 +324,7 @@
 
 lemma equiv_realrel: "equiv {X. cauchy X} realrel"
   using refl_realrel sym_realrel trans_realrel
-  by (rule equiv.intro)
+  by (rule equivI)
 
 subsection {* The field of real numbers *}
 
@@ -358,7 +358,7 @@
   apply (simp add: quotientI X)
   apply (rule the_equality)
   apply clarsimp
-  apply (erule congruent.congruent [OF f])
+  apply (erule congruentD [OF f])
   apply (erule bspec)
   apply simp
   apply (rule refl_onD [OF refl_realrel])
@@ -370,14 +370,14 @@
   assumes X: "cauchy X" and Y: "cauchy Y"
   shows "real_case (\<lambda>X. real_case (\<lambda>Y. f X Y) (Real Y)) (Real X) = f X Y"
  apply (subst real_case_1 [OF _ X])
-  apply (rule congruent.intro)
+  apply (rule congruentI)
   apply (subst real_case_1 [OF _ Y])
    apply (rule congruent2_implies_congruent [OF equiv_realrel f])
    apply (simp add: realrel_def)
   apply (subst real_case_1 [OF _ Y])
    apply (rule congruent2_implies_congruent [OF equiv_realrel f])
    apply (simp add: realrel_def)
-  apply (erule congruent2.congruent2 [OF f])
+  apply (erule congruent2D [OF f])
   apply (rule refl_onD [OF refl_realrel])
   apply (simp add: Y)
   apply (rule real_case_1 [OF _ Y])
@@ -416,7 +416,7 @@
 
 lemma minus_respects_realrel:
   "(\<lambda>X. Real (\<lambda>n. - X n)) respects realrel"
-proof (rule congruent.intro)
+proof (rule congruentI)
   fix X Y assume "(X, Y) \<in> realrel"
   hence X: "cauchy X" and Y: "cauchy Y" and XY: "vanishes (\<lambda>n. X n - Y n)"
     unfolding realrel_def by simp_all
@@ -492,7 +492,7 @@
 lemma inverse_respects_realrel:
   "(\<lambda>X. if vanishes X then c else Real (\<lambda>n. inverse (X n))) respects realrel"
     (is "?inv respects realrel")
-proof (rule congruent.intro)
+proof (rule congruentI)
   fix X Y assume "(X, Y) \<in> realrel"
   hence X: "cauchy X" and Y: "cauchy Y" and XY: "vanishes (\<lambda>n. X n - Y n)"
     unfolding realrel_def by simp_all
@@ -622,7 +622,7 @@
   assumes sym: "sym r"
   assumes P: "\<And>x y. (x, y) \<in> r \<Longrightarrow> P x \<Longrightarrow> P y"
   shows "P respects r"
-apply (rule congruent.intro)
+apply (rule congruentI)
 apply (rule iffI)
 apply (erule (1) P)
 apply (erule (1) P [OF symD [OF sym]])