# HG changeset patch
# User blanchet
# Date 1394114234 -3600
# Node ID f20d1db5aa3c6a4ce641e39351958b6c41c18775
# Parent  18e52e8c63001e79575c336af6e048c6546e1333
renamed 'set_rel' to 'rel_set'

diff -r 18e52e8c6300 -r f20d1db5aa3c NEWS
--- a/NEWS	Thu Mar 06 14:25:55 2014 +0100
+++ b/NEWS	Thu Mar 06 14:57:14 2014 +0100
@@ -195,6 +195,7 @@
     map_pair ~> prod_map
     fset_rel ~> rel_fset
     cset_rel ~> rel_cset
+    set_rel ~> rel_set
 
 * New theory:
     Cardinals/Ordinal_Arithmetic.thy
diff -r 18e52e8c6300 -r f20d1db5aa3c src/HOL/BNF_Examples/Derivation_Trees/DTree.thy
--- a/src/HOL/BNF_Examples/Derivation_Trees/DTree.thy	Thu Mar 06 14:25:55 2014 +0100
+++ b/src/HOL/BNF_Examples/Derivation_Trees/DTree.thy	Thu Mar 06 14:57:14 2014 +0100
@@ -60,17 +60,17 @@
 shows "tr = tr'"
 by (metis Node_root_cont assms)
 
-lemma set_rel_cont:
-"set_rel \<chi> (cont tr1) (cont tr2) = rel_fset \<chi> (ccont tr1) (ccont tr2)"
+lemma rel_set_cont:
+"rel_set \<chi> (cont tr1) (cont tr2) = rel_fset \<chi> (ccont tr1) (ccont tr2)"
 unfolding cont_def comp_def rel_fset_fset ..
 
 lemma dtree_coinduct[elim, consumes 1, case_names Lift, induct pred: "HOL.eq"]:
 assumes phi: "\<phi> tr1 tr2" and
 Lift: "\<And> tr1 tr2. \<phi> tr1 tr2 \<Longrightarrow>
-                  root tr1 = root tr2 \<and> set_rel (sum_rel op = \<phi>) (cont tr1) (cont tr2)"
+                  root tr1 = root tr2 \<and> rel_set (sum_rel op = \<phi>) (cont tr1) (cont tr2)"
 shows "tr1 = tr2"
 using phi apply(elim dtree.coinduct)
-apply(rule Lift[unfolded set_rel_cont]) .
+apply(rule Lift[unfolded rel_set_cont]) .
 
 lemma unfold:
 "root (unfold rt ct b) = rt b"
diff -r 18e52e8c6300 -r f20d1db5aa3c src/HOL/BNF_Examples/Derivation_Trees/Parallel.thy
--- a/src/HOL/BNF_Examples/Derivation_Trees/Parallel.thy	Thu Mar 06 14:25:55 2014 +0100
+++ b/src/HOL/BNF_Examples/Derivation_Trees/Parallel.thy	Thu Mar 06 14:57:14 2014 +0100
@@ -67,10 +67,10 @@
 
 subsection{* Structural Coinduction Proofs *}
 
-lemma set_rel_sum_rel_eq[simp]:
-"set_rel (sum_rel (op =) \<phi>) A1 A2 \<longleftrightarrow>
- Inl -` A1 = Inl -` A2 \<and> set_rel \<phi> (Inr -` A1) (Inr -` A2)"
-unfolding set_rel_sum_rel set_rel_eq ..
+lemma rel_set_sum_rel_eq[simp]:
+"rel_set (sum_rel (op =) \<phi>) A1 A2 \<longleftrightarrow>
+ Inl -` A1 = Inl -` A2 \<and> rel_set \<phi> (Inr -` A1) (Inr -` A2)"
+unfolding rel_set_sum_rel rel_set_eq ..
 
 (* Detailed proofs of commutativity and associativity: *)
 theorem par_com: "tr1 \<parallel> tr2 = tr2 \<parallel> tr1"
@@ -79,7 +79,7 @@
   {fix trA trB
    assume "?\<theta> trA trB" hence "trA = trB"
    apply (induct rule: dtree_coinduct)
-   unfolding set_rel_sum_rel set_rel_eq unfolding set_rel_def proof safe
+   unfolding rel_set_sum_rel rel_set_eq unfolding rel_set_def proof safe
      fix tr1 tr2  show "root (tr1 \<parallel> tr2) = root (tr2 \<parallel> tr1)"
      unfolding root_par by (rule Nplus_comm)
    next
@@ -114,7 +114,7 @@
   {fix trA trB
    assume "?\<theta> trA trB" hence "trA = trB"
    apply (induct rule: dtree_coinduct)
-   unfolding set_rel_sum_rel set_rel_eq unfolding set_rel_def proof safe
+   unfolding rel_set_sum_rel rel_set_eq unfolding rel_set_def proof safe
      fix tr1 tr2 tr3  show "root ((tr1 \<parallel> tr2) \<parallel> tr3) = root (tr1 \<parallel> (tr2 \<parallel> tr3))"
      unfolding root_par by (rule Nplus_assoc)
    next
diff -r 18e52e8c6300 -r f20d1db5aa3c src/HOL/Library/FSet.thy
--- a/src/HOL/Library/FSet.thy	Thu Mar 06 14:25:55 2014 +0100
+++ b/src/HOL/Library/FSet.thy	Thu Mar 06 14:57:14 2014 +0100
@@ -78,14 +78,14 @@
 
 lemma right_total_Inf_fset_transfer:
   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
-  shows "(set_rel (set_rel A) ===> set_rel A) 
+  shows "(rel_set (rel_set A) ===> rel_set A) 
     (\<lambda>S. if finite (Inter S \<inter> Collect (Domainp A)) then Inter S \<inter> Collect (Domainp A) else {}) 
       (\<lambda>S. if finite (Inf S) then Inf S else {})"
     by transfer_prover
 
 lemma Inf_fset_transfer:
   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
-  shows "(set_rel (set_rel A) ===> set_rel A) (\<lambda>A. if finite (Inf A) then Inf A else {}) 
+  shows "(rel_set (rel_set A) ===> rel_set A) (\<lambda>A. if finite (Inf A) then Inf A else {}) 
     (\<lambda>A. if finite (Inf A) then Inf A else {})"
   by transfer_prover
 
@@ -94,7 +94,7 @@
 
 lemma Sup_fset_transfer:
   assumes [transfer_rule]: "bi_unique A"
-  shows "(set_rel (set_rel A) ===> set_rel A) (\<lambda>A. if finite (Sup A) then Sup A else {})
+  shows "(rel_set (rel_set A) ===> rel_set A) (\<lambda>A. if finite (Sup A) then Sup A else {})
   (\<lambda>A. if finite (Sup A) then Sup A else {})" by transfer_prover
 
 lift_definition Sup_fset :: "'a fset set \<Rightarrow> 'a fset" is "\<lambda>A. if finite (Sup A) then Sup A else {}"
@@ -103,7 +103,7 @@
 lemma finite_Sup: "\<exists>z. finite z \<and> (\<forall>a. a \<in> X \<longrightarrow> a \<le> z) \<Longrightarrow> finite (Sup X)"
 by (auto intro: finite_subset)
 
-lemma transfer_bdd_below[transfer_rule]: "(set_rel (pcr_fset op =) ===> op =) bdd_below bdd_below"
+lemma transfer_bdd_below[transfer_rule]: "(rel_set (pcr_fset op =) ===> op =) bdd_below bdd_below"
   by auto
 
 instance
@@ -762,53 +762,53 @@
 
 subsubsection {* Relator and predicator properties *}
 
-lift_definition rel_fset :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'b fset \<Rightarrow> bool" is set_rel
-parametric set_rel_transfer .
+lift_definition rel_fset :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'b fset \<Rightarrow> bool" is rel_set
+parametric rel_set_transfer .
 
 lemma rel_fset_alt_def: "rel_fset R = (\<lambda>A B. (\<forall>x.\<exists>y. x|\<in>|A \<longrightarrow> y|\<in>|B \<and> R x y) 
   \<and> (\<forall>y. \<exists>x. y|\<in>|B \<longrightarrow> x|\<in>|A \<and> R x y))"
 apply (rule ext)+
 apply transfer'
-apply (subst set_rel_def[unfolded fun_eq_iff]) 
+apply (subst rel_set_def[unfolded fun_eq_iff]) 
 by blast
 
 lemma rel_fset_conversep: "rel_fset (conversep R) = conversep (rel_fset R)"
   unfolding rel_fset_alt_def by auto
 
-lemmas rel_fset_eq [relator_eq] = set_rel_eq[Transfer.transferred]
+lemmas rel_fset_eq [relator_eq] = rel_set_eq[Transfer.transferred]
 
 lemma rel_fset_mono[relator_mono]: "A \<le> B \<Longrightarrow> rel_fset A \<le> rel_fset B"
 unfolding rel_fset_alt_def by blast
 
-lemma finite_set_rel:
+lemma finite_rel_set:
   assumes fin: "finite X" "finite Z"
-  assumes R_S: "set_rel (R OO S) X Z"
-  shows "\<exists>Y. finite Y \<and> set_rel R X Y \<and> set_rel S Y Z"
+  assumes R_S: "rel_set (R OO S) X Z"
+  shows "\<exists>Y. finite Y \<and> rel_set R X Y \<and> rel_set S Y Z"
 proof -
   obtain f where f: "\<forall>x\<in>X. R x (f x) \<and> (\<exists>z\<in>Z. S (f x) z)"
   apply atomize_elim
   apply (subst bchoice_iff[symmetric])
-  using R_S[unfolded set_rel_def OO_def] by blast
+  using R_S[unfolded rel_set_def OO_def] by blast
   
   obtain g where g: "\<forall>z\<in>Z. S (g z) z \<and> (\<exists>x\<in>X. R  x (g z))"
   apply atomize_elim
   apply (subst bchoice_iff[symmetric])
-  using R_S[unfolded set_rel_def OO_def] by blast
+  using R_S[unfolded rel_set_def OO_def] by blast
   
   let ?Y = "f ` X \<union> g ` Z"
   have "finite ?Y" by (simp add: fin)
-  moreover have "set_rel R X ?Y"
-    unfolding set_rel_def
+  moreover have "rel_set R X ?Y"
+    unfolding rel_set_def
     using f g by clarsimp blast
-  moreover have "set_rel S ?Y Z"
-    unfolding set_rel_def
+  moreover have "rel_set S ?Y Z"
+    unfolding rel_set_def
     using f g by clarsimp blast
   ultimately show ?thesis by metis
 qed
 
 lemma rel_fset_OO[relator_distr]: "rel_fset R OO rel_fset S = rel_fset (R OO S)"
 apply (rule ext)+
-by transfer (auto intro: finite_set_rel set_rel_OO[unfolded fun_eq_iff, rule_format, THEN iffD1])
+by transfer (auto intro: finite_rel_set rel_set_OO[unfolded fun_eq_iff, rule_format, THEN iffD1])
 
 lemma Domainp_fset[relator_domain]:
   assumes "Domainp T = P"
@@ -832,7 +832,7 @@
 apply (subst(asm) choice_iff)
 apply clarsimp
 apply (rename_tac A f y, rule_tac x = "f ` y" in exI)
-by (auto simp add: set_rel_def)
+by (auto simp add: rel_set_def)
 
 lemma left_total_rel_fset[reflexivity_rule]: "left_total A \<Longrightarrow> left_total (rel_fset A)"
 unfolding left_total_def 
@@ -840,10 +840,10 @@
 apply (subst(asm) choice_iff)
 apply clarsimp
 apply (rename_tac A f y, rule_tac x = "f ` y" in exI)
-by (auto simp add: set_rel_def)
+by (auto simp add: rel_set_def)
 
-lemmas right_unique_rel_fset[transfer_rule] = right_unique_set_rel[Transfer.transferred]
-lemmas left_unique_rel_fset[reflexivity_rule] = left_unique_set_rel[Transfer.transferred]
+lemmas right_unique_rel_fset[transfer_rule] = right_unique_rel_set[Transfer.transferred]
+lemmas left_unique_rel_fset[reflexivity_rule] = left_unique_rel_set[Transfer.transferred]
 
 thm right_unique_rel_fset left_unique_rel_fset
 
@@ -911,7 +911,7 @@
   "((A ===> B ===> op =) ===> rel_fset A ===> rel_fset B ===> op =)
     rel_fset rel_fset"
   unfolding fun_rel_def
-  using set_rel_transfer[unfolded fun_rel_def,rule_format, Transfer.transferred, where A = A and B = B]
+  using rel_set_transfer[unfolded fun_rel_def,rule_format, Transfer.transferred, where A = A and B = B]
   by simp
 
 lemma bind_transfer [transfer_rule]:
@@ -945,7 +945,7 @@
   using subset_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
 
 lemma fSup_transfer [transfer_rule]:
-  "bi_unique A \<Longrightarrow> (set_rel (rel_fset A) ===> rel_fset A) Sup Sup"
+  "bi_unique A \<Longrightarrow> (rel_set (rel_fset A) ===> rel_fset A) Sup Sup"
   using assms unfolding fun_rel_def
   apply clarify
   apply transfer'
@@ -955,7 +955,7 @@
 
 lemma fInf_transfer [transfer_rule]:
   assumes "bi_unique A" and "bi_total A"
-  shows "(set_rel (rel_fset A) ===> rel_fset A) Inf Inf"
+  shows "(rel_set (rel_fset A) ===> rel_fset A) Inf Inf"
   using assms unfolding fun_rel_def
   apply clarify
   apply transfer'
@@ -986,7 +986,7 @@
 
 lemma rel_fset_alt:
   "rel_fset R a b \<longleftrightarrow> (\<forall>t \<in> fset a. \<exists>u \<in> fset b. R t u) \<and> (\<forall>t \<in> fset b. \<exists>u \<in> fset a. R u t)"
-by transfer (simp add: set_rel_def)
+by transfer (simp add: rel_set_def)
 
 lemma fset_to_fset: "finite A \<Longrightarrow> fset (the_inv fset A) = A"
 apply (rule f_the_inv_into_f[unfolded inj_on_def])
@@ -1037,7 +1037,7 @@
 apply transfer apply simp
 done
 
-lemma rel_fset_fset: "set_rel \<chi> (fset A1) (fset A2) = rel_fset \<chi> A1 A2"
+lemma rel_fset_fset: "rel_set \<chi> (fset A1) (fset A2) = rel_fset \<chi> A1 A2"
   by transfer (rule refl)
 
 end
@@ -1049,50 +1049,50 @@
 
 (* Set vs. sum relators: *)
 
-lemma set_rel_sum_rel[simp]: 
-"set_rel (sum_rel \<chi> \<phi>) A1 A2 \<longleftrightarrow> 
- set_rel \<chi> (Inl -` A1) (Inl -` A2) \<and> set_rel \<phi> (Inr -` A1) (Inr -` A2)"
+lemma rel_set_sum_rel[simp]: 
+"rel_set (sum_rel \<chi> \<phi>) A1 A2 \<longleftrightarrow> 
+ rel_set \<chi> (Inl -` A1) (Inl -` A2) \<and> rel_set \<phi> (Inr -` A1) (Inr -` A2)"
 (is "?L \<longleftrightarrow> ?Rl \<and> ?Rr")
 proof safe
   assume L: "?L"
-  show ?Rl unfolding set_rel_def Bex_def vimage_eq proof safe
+  show ?Rl unfolding rel_set_def Bex_def vimage_eq proof safe
     fix l1 assume "Inl l1 \<in> A1"
     then obtain a2 where a2: "a2 \<in> A2" and "sum_rel \<chi> \<phi> (Inl l1) a2"
-    using L unfolding set_rel_def by auto
+    using L unfolding rel_set_def by auto
     then obtain l2 where "a2 = Inl l2 \<and> \<chi> l1 l2" by (cases a2, auto)
     thus "\<exists> l2. Inl l2 \<in> A2 \<and> \<chi> l1 l2" using a2 by auto
   next
     fix l2 assume "Inl l2 \<in> A2"
     then obtain a1 where a1: "a1 \<in> A1" and "sum_rel \<chi> \<phi> a1 (Inl l2)"
-    using L unfolding set_rel_def by auto
+    using L unfolding rel_set_def by auto
     then obtain l1 where "a1 = Inl l1 \<and> \<chi> l1 l2" by (cases a1, auto)
     thus "\<exists> l1. Inl l1 \<in> A1 \<and> \<chi> l1 l2" using a1 by auto
   qed
-  show ?Rr unfolding set_rel_def Bex_def vimage_eq proof safe
+  show ?Rr unfolding rel_set_def Bex_def vimage_eq proof safe
     fix r1 assume "Inr r1 \<in> A1"
     then obtain a2 where a2: "a2 \<in> A2" and "sum_rel \<chi> \<phi> (Inr r1) a2"
-    using L unfolding set_rel_def by auto
+    using L unfolding rel_set_def by auto
     then obtain r2 where "a2 = Inr r2 \<and> \<phi> r1 r2" by (cases a2, auto)
     thus "\<exists> r2. Inr r2 \<in> A2 \<and> \<phi> r1 r2" using a2 by auto
   next
     fix r2 assume "Inr r2 \<in> A2"
     then obtain a1 where a1: "a1 \<in> A1" and "sum_rel \<chi> \<phi> a1 (Inr r2)"
-    using L unfolding set_rel_def by auto
+    using L unfolding rel_set_def by auto
     then obtain r1 where "a1 = Inr r1 \<and> \<phi> r1 r2" by (cases a1, auto)
     thus "\<exists> r1. Inr r1 \<in> A1 \<and> \<phi> r1 r2" using a1 by auto
   qed
 next
   assume Rl: "?Rl" and Rr: "?Rr"
-  show ?L unfolding set_rel_def Bex_def vimage_eq proof safe
+  show ?L unfolding rel_set_def Bex_def vimage_eq proof safe
     fix a1 assume a1: "a1 \<in> A1"
     show "\<exists> a2. a2 \<in> A2 \<and> sum_rel \<chi> \<phi> a1 a2"
     proof(cases a1)
       case (Inl l1) then obtain l2 where "Inl l2 \<in> A2 \<and> \<chi> l1 l2"
-      using Rl a1 unfolding set_rel_def by blast
+      using Rl a1 unfolding rel_set_def by blast
       thus ?thesis unfolding Inl by auto
     next
       case (Inr r1) then obtain r2 where "Inr r2 \<in> A2 \<and> \<phi> r1 r2"
-      using Rr a1 unfolding set_rel_def by blast
+      using Rr a1 unfolding rel_set_def by blast
       thus ?thesis unfolding Inr by auto
     qed
   next
@@ -1100,11 +1100,11 @@
     show "\<exists> a1. a1 \<in> A1 \<and> sum_rel \<chi> \<phi> a1 a2"
     proof(cases a2)
       case (Inl l2) then obtain l1 where "Inl l1 \<in> A1 \<and> \<chi> l1 l2"
-      using Rl a2 unfolding set_rel_def by blast
+      using Rl a2 unfolding rel_set_def by blast
       thus ?thesis unfolding Inl by auto
     next
       case (Inr r2) then obtain r1 where "Inr r1 \<in> A1 \<and> \<phi> r1 r2"
-      using Rr a2 unfolding set_rel_def by blast
+      using Rr a2 unfolding rel_set_def by blast
       thus ?thesis unfolding Inr by auto
     qed
   qed
diff -r 18e52e8c6300 -r f20d1db5aa3c src/HOL/Library/Mapping.thy
--- a/src/HOL/Library/Mapping.thy	Thu Mar 06 14:25:55 2014 +0100
+++ b/src/HOL/Library/Mapping.thy	Thu Mar 06 14:57:14 2014 +0100
@@ -37,7 +37,7 @@
 
 lemma dom_transfer:
   assumes [transfer_rule]: "bi_total A"
-  shows "((A ===> rel_option B) ===> set_rel A) dom dom" 
+  shows "((A ===> rel_option B) ===> rel_set A) dom dom" 
 unfolding dom_def[abs_def] equal_None_def[symmetric] 
 by transfer_prover
 
diff -r 18e52e8c6300 -r f20d1db5aa3c src/HOL/Lifting_Set.thy
--- a/src/HOL/Lifting_Set.thy	Thu Mar 06 14:25:55 2014 +0100
+++ b/src/HOL/Lifting_Set.thy	Thu Mar 06 14:57:14 2014 +0100
@@ -10,90 +10,90 @@
 
 subsection {* Relator and predicator properties *}
 
-definition set_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool"
-  where "set_rel R = (\<lambda>A B. (\<forall>x\<in>A. \<exists>y\<in>B. R x y) \<and> (\<forall>y\<in>B. \<exists>x\<in>A. R x y))"
+definition rel_set :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool"
+  where "rel_set R = (\<lambda>A B. (\<forall>x\<in>A. \<exists>y\<in>B. R x y) \<and> (\<forall>y\<in>B. \<exists>x\<in>A. R x y))"
 
-lemma set_relI:
+lemma rel_setI:
   assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. R x y"
   assumes "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. R x y"
-  shows "set_rel R A B"
-  using assms unfolding set_rel_def by simp
+  shows "rel_set R A B"
+  using assms unfolding rel_set_def by simp
 
-lemma set_relD1: "\<lbrakk> set_rel R A B; x \<in> A \<rbrakk> \<Longrightarrow> \<exists>y \<in> B. R x y"
-  and set_relD2: "\<lbrakk> set_rel R A B; y \<in> B \<rbrakk> \<Longrightarrow> \<exists>x \<in> A. R x y"
-by(simp_all add: set_rel_def)
+lemma rel_setD1: "\<lbrakk> rel_set R A B; x \<in> A \<rbrakk> \<Longrightarrow> \<exists>y \<in> B. R x y"
+  and rel_setD2: "\<lbrakk> rel_set R A B; y \<in> B \<rbrakk> \<Longrightarrow> \<exists>x \<in> A. R x y"
+by(simp_all add: rel_set_def)
 
-lemma set_rel_conversep [simp]: "set_rel A\<inverse>\<inverse> = (set_rel A)\<inverse>\<inverse>"
-  unfolding set_rel_def by auto
+lemma rel_set_conversep [simp]: "rel_set A\<inverse>\<inverse> = (rel_set A)\<inverse>\<inverse>"
+  unfolding rel_set_def by auto
 
-lemma set_rel_eq [relator_eq]: "set_rel (op =) = (op =)"
-  unfolding set_rel_def fun_eq_iff by auto
+lemma rel_set_eq [relator_eq]: "rel_set (op =) = (op =)"
+  unfolding rel_set_def fun_eq_iff by auto
 
-lemma set_rel_mono[relator_mono]:
+lemma rel_set_mono[relator_mono]:
   assumes "A \<le> B"
-  shows "set_rel A \<le> set_rel B"
-using assms unfolding set_rel_def by blast
+  shows "rel_set A \<le> rel_set B"
+using assms unfolding rel_set_def by blast
 
-lemma set_rel_OO[relator_distr]: "set_rel R OO set_rel S = set_rel (R OO S)"
+lemma rel_set_OO[relator_distr]: "rel_set R OO rel_set S = rel_set (R OO S)"
   apply (rule sym)
   apply (intro ext, rename_tac X Z)
   apply (rule iffI)
   apply (rule_tac b="{y. (\<exists>x\<in>X. R x y) \<and> (\<exists>z\<in>Z. S y z)}" in relcomppI)
-  apply (simp add: set_rel_def, fast)
-  apply (simp add: set_rel_def, fast)
-  apply (simp add: set_rel_def, fast)
+  apply (simp add: rel_set_def, fast)
+  apply (simp add: rel_set_def, fast)
+  apply (simp add: rel_set_def, fast)
   done
 
 lemma Domainp_set[relator_domain]:
   assumes "Domainp T = R"
-  shows "Domainp (set_rel T) = (\<lambda>A. Ball A R)"
-using assms unfolding set_rel_def Domainp_iff[abs_def]
+  shows "Domainp (rel_set T) = (\<lambda>A. Ball A R)"
+using assms unfolding rel_set_def Domainp_iff[abs_def]
 apply (intro ext)
 apply (rule iffI) 
 apply blast
 apply (rename_tac A, rule_tac x="{y. \<exists>x\<in>A. T x y}" in exI, fast)
 done
 
-lemma left_total_set_rel[reflexivity_rule]: 
-  "left_total A \<Longrightarrow> left_total (set_rel A)"
-  unfolding left_total_def set_rel_def
+lemma left_total_rel_set[reflexivity_rule]: 
+  "left_total A \<Longrightarrow> left_total (rel_set A)"
+  unfolding left_total_def rel_set_def
   apply safe
   apply (rename_tac X, rule_tac x="{y. \<exists>x\<in>X. A x y}" in exI, fast)
 done
 
-lemma left_unique_set_rel[reflexivity_rule]: 
-  "left_unique A \<Longrightarrow> left_unique (set_rel A)"
-  unfolding left_unique_def set_rel_def
+lemma left_unique_rel_set[reflexivity_rule]: 
+  "left_unique A \<Longrightarrow> left_unique (rel_set A)"
+  unfolding left_unique_def rel_set_def
   by fast
 
-lemma right_total_set_rel [transfer_rule]:
-  "right_total A \<Longrightarrow> right_total (set_rel A)"
-using left_total_set_rel[of "A\<inverse>\<inverse>"] by simp
+lemma right_total_rel_set [transfer_rule]:
+  "right_total A \<Longrightarrow> right_total (rel_set A)"
+using left_total_rel_set[of "A\<inverse>\<inverse>"] by simp
 
-lemma right_unique_set_rel [transfer_rule]:
-  "right_unique A \<Longrightarrow> right_unique (set_rel A)"
-  unfolding right_unique_def set_rel_def by fast
+lemma right_unique_rel_set [transfer_rule]:
+  "right_unique A \<Longrightarrow> right_unique (rel_set A)"
+  unfolding right_unique_def rel_set_def by fast
 
-lemma bi_total_set_rel [transfer_rule]:
-  "bi_total A \<Longrightarrow> bi_total (set_rel A)"
-by(simp add: bi_total_conv_left_right left_total_set_rel right_total_set_rel)
+lemma bi_total_rel_set [transfer_rule]:
+  "bi_total A \<Longrightarrow> bi_total (rel_set A)"
+by(simp add: bi_total_conv_left_right left_total_rel_set right_total_rel_set)
 
-lemma bi_unique_set_rel [transfer_rule]:
-  "bi_unique A \<Longrightarrow> bi_unique (set_rel A)"
-  unfolding bi_unique_def set_rel_def by fast
+lemma bi_unique_rel_set [transfer_rule]:
+  "bi_unique A \<Longrightarrow> bi_unique (rel_set A)"
+  unfolding bi_unique_def rel_set_def by fast
 
 lemma set_invariant_commute [invariant_commute]:
-  "set_rel (Lifting.invariant P) = Lifting.invariant (\<lambda>A. Ball A P)"
-  unfolding fun_eq_iff set_rel_def Lifting.invariant_def Ball_def by fast
+  "rel_set (Lifting.invariant P) = Lifting.invariant (\<lambda>A. Ball A P)"
+  unfolding fun_eq_iff rel_set_def Lifting.invariant_def Ball_def by fast
 
 subsection {* Quotient theorem for the Lifting package *}
 
 lemma Quotient_set[quot_map]:
   assumes "Quotient R Abs Rep T"
-  shows "Quotient (set_rel R) (image Abs) (image Rep) (set_rel T)"
+  shows "Quotient (rel_set R) (image Abs) (image Rep) (rel_set T)"
   using assms unfolding Quotient_alt_def4
-  apply (simp add: set_rel_OO[symmetric])
-  apply (simp add: set_rel_def, fast)
+  apply (simp add: rel_set_OO[symmetric])
+  apply (simp add: rel_set_def, fast)
   done
 
 subsection {* Transfer rules for the Transfer package *}
@@ -104,143 +104,143 @@
 begin
 interpretation lifting_syntax .
 
-lemma empty_transfer [transfer_rule]: "(set_rel A) {} {}"
-  unfolding set_rel_def by simp
+lemma empty_transfer [transfer_rule]: "(rel_set A) {} {}"
+  unfolding rel_set_def by simp
 
 lemma insert_transfer [transfer_rule]:
-  "(A ===> set_rel A ===> set_rel A) insert insert"
-  unfolding fun_rel_def set_rel_def by auto
+  "(A ===> rel_set A ===> rel_set A) insert insert"
+  unfolding fun_rel_def rel_set_def by auto
 
 lemma union_transfer [transfer_rule]:
-  "(set_rel A ===> set_rel A ===> set_rel A) union union"
-  unfolding fun_rel_def set_rel_def by auto
+  "(rel_set A ===> rel_set A ===> rel_set A) union union"
+  unfolding fun_rel_def rel_set_def by auto
 
 lemma Union_transfer [transfer_rule]:
-  "(set_rel (set_rel A) ===> set_rel A) Union Union"
-  unfolding fun_rel_def set_rel_def by simp fast
+  "(rel_set (rel_set A) ===> rel_set A) Union Union"
+  unfolding fun_rel_def rel_set_def by simp fast
 
 lemma image_transfer [transfer_rule]:
-  "((A ===> B) ===> set_rel A ===> set_rel B) image image"
-  unfolding fun_rel_def set_rel_def by simp fast
+  "((A ===> B) ===> rel_set A ===> rel_set B) image image"
+  unfolding fun_rel_def rel_set_def by simp fast
 
 lemma UNION_transfer [transfer_rule]:
-  "(set_rel A ===> (A ===> set_rel B) ===> set_rel B) UNION UNION"
+  "(rel_set A ===> (A ===> rel_set B) ===> rel_set B) UNION UNION"
   unfolding SUP_def [abs_def] by transfer_prover
 
 lemma Ball_transfer [transfer_rule]:
-  "(set_rel A ===> (A ===> op =) ===> op =) Ball Ball"
-  unfolding set_rel_def fun_rel_def by fast
+  "(rel_set A ===> (A ===> op =) ===> op =) Ball Ball"
+  unfolding rel_set_def fun_rel_def by fast
 
 lemma Bex_transfer [transfer_rule]:
-  "(set_rel A ===> (A ===> op =) ===> op =) Bex Bex"
-  unfolding set_rel_def fun_rel_def by fast
+  "(rel_set A ===> (A ===> op =) ===> op =) Bex Bex"
+  unfolding rel_set_def fun_rel_def by fast
 
 lemma Pow_transfer [transfer_rule]:
-  "(set_rel A ===> set_rel (set_rel A)) Pow Pow"
-  apply (rule fun_relI, rename_tac X Y, rule set_relI)
+  "(rel_set A ===> rel_set (rel_set A)) Pow Pow"
+  apply (rule fun_relI, rename_tac X Y, rule rel_setI)
   apply (rename_tac X', rule_tac x="{y\<in>Y. \<exists>x\<in>X'. A x y}" in rev_bexI, clarsimp)
-  apply (simp add: set_rel_def, fast)
+  apply (simp add: rel_set_def, fast)
   apply (rename_tac Y', rule_tac x="{x\<in>X. \<exists>y\<in>Y'. A x y}" in rev_bexI, clarsimp)
-  apply (simp add: set_rel_def, fast)
+  apply (simp add: rel_set_def, fast)
   done
 
-lemma set_rel_transfer [transfer_rule]:
-  "((A ===> B ===> op =) ===> set_rel A ===> set_rel B ===> op =)
-    set_rel set_rel"
-  unfolding fun_rel_def set_rel_def by fast
+lemma rel_set_transfer [transfer_rule]:
+  "((A ===> B ===> op =) ===> rel_set A ===> rel_set B ===> op =)
+    rel_set rel_set"
+  unfolding fun_rel_def rel_set_def by fast
 
 lemma SUPR_parametric [transfer_rule]:
-  "(set_rel R ===> (R ===> op =) ===> op =) SUPR (SUPR :: _ \<Rightarrow> _ \<Rightarrow> _::complete_lattice)"
+  "(rel_set R ===> (R ===> op =) ===> op =) SUPR (SUPR :: _ \<Rightarrow> _ \<Rightarrow> _::complete_lattice)"
 proof(rule fun_relI)+
   fix A B f and g :: "'b \<Rightarrow> 'c"
-  assume AB: "set_rel R A B"
+  assume AB: "rel_set R A B"
     and fg: "(R ===> op =) f g"
   show "SUPR A f = SUPR B g"
-    by(rule SUPR_eq)(auto 4 4 dest: set_relD1[OF AB] set_relD2[OF AB] fun_relD[OF fg] intro: rev_bexI)
+    by(rule SUPR_eq)(auto 4 4 dest: rel_setD1[OF AB] rel_setD2[OF AB] fun_relD[OF fg] intro: rev_bexI)
 qed
 
 lemma bind_transfer [transfer_rule]:
-  "(set_rel A ===> (A ===> set_rel B) ===> set_rel B) Set.bind Set.bind"
+  "(rel_set A ===> (A ===> rel_set B) ===> rel_set B) Set.bind Set.bind"
 unfolding bind_UNION[abs_def] by transfer_prover
 
 subsubsection {* Rules requiring bi-unique, bi-total or right-total relations *}
 
 lemma member_transfer [transfer_rule]:
   assumes "bi_unique A"
-  shows "(A ===> set_rel A ===> op =) (op \<in>) (op \<in>)"
-  using assms unfolding fun_rel_def set_rel_def bi_unique_def by fast
+  shows "(A ===> rel_set A ===> op =) (op \<in>) (op \<in>)"
+  using assms unfolding fun_rel_def rel_set_def bi_unique_def by fast
 
 lemma right_total_Collect_transfer[transfer_rule]:
   assumes "right_total A"
-  shows "((A ===> op =) ===> set_rel A) (\<lambda>P. Collect (\<lambda>x. P x \<and> Domainp A x)) Collect"
-  using assms unfolding right_total_def set_rel_def fun_rel_def Domainp_iff by fast
+  shows "((A ===> op =) ===> rel_set A) (\<lambda>P. Collect (\<lambda>x. P x \<and> Domainp A x)) Collect"
+  using assms unfolding right_total_def rel_set_def fun_rel_def Domainp_iff by fast
 
 lemma Collect_transfer [transfer_rule]:
   assumes "bi_total A"
-  shows "((A ===> op =) ===> set_rel A) Collect Collect"
-  using assms unfolding fun_rel_def set_rel_def bi_total_def by fast
+  shows "((A ===> op =) ===> rel_set A) Collect Collect"
+  using assms unfolding fun_rel_def rel_set_def bi_total_def by fast
 
 lemma inter_transfer [transfer_rule]:
   assumes "bi_unique A"
-  shows "(set_rel A ===> set_rel A ===> set_rel A) inter inter"
-  using assms unfolding fun_rel_def set_rel_def bi_unique_def by fast
+  shows "(rel_set A ===> rel_set A ===> rel_set A) inter inter"
+  using assms unfolding fun_rel_def rel_set_def bi_unique_def by fast
 
 lemma Diff_transfer [transfer_rule]:
   assumes "bi_unique A"
-  shows "(set_rel A ===> set_rel A ===> set_rel A) (op -) (op -)"
-  using assms unfolding fun_rel_def set_rel_def bi_unique_def
+  shows "(rel_set A ===> rel_set A ===> rel_set A) (op -) (op -)"
+  using assms unfolding fun_rel_def rel_set_def bi_unique_def
   unfolding Ball_def Bex_def Diff_eq
   by (safe, simp, metis, simp, metis)
 
 lemma subset_transfer [transfer_rule]:
   assumes [transfer_rule]: "bi_unique A"
-  shows "(set_rel A ===> set_rel A ===> op =) (op \<subseteq>) (op \<subseteq>)"
+  shows "(rel_set A ===> rel_set A ===> op =) (op \<subseteq>) (op \<subseteq>)"
   unfolding subset_eq [abs_def] by transfer_prover
 
 lemma right_total_UNIV_transfer[transfer_rule]: 
   assumes "right_total A"
-  shows "(set_rel A) (Collect (Domainp A)) UNIV"
-  using assms unfolding right_total_def set_rel_def Domainp_iff by blast
+  shows "(rel_set A) (Collect (Domainp A)) UNIV"
+  using assms unfolding right_total_def rel_set_def Domainp_iff by blast
 
 lemma UNIV_transfer [transfer_rule]:
   assumes "bi_total A"
-  shows "(set_rel A) UNIV UNIV"
-  using assms unfolding set_rel_def bi_total_def by simp
+  shows "(rel_set A) UNIV UNIV"
+  using assms unfolding rel_set_def bi_total_def by simp
 
 lemma right_total_Compl_transfer [transfer_rule]:
   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
-  shows "(set_rel A ===> set_rel A) (\<lambda>S. uminus S \<inter> Collect (Domainp A)) uminus"
+  shows "(rel_set A ===> rel_set A) (\<lambda>S. uminus S \<inter> Collect (Domainp A)) uminus"
   unfolding Compl_eq [abs_def]
   by (subst Collect_conj_eq[symmetric]) transfer_prover
 
 lemma Compl_transfer [transfer_rule]:
   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
-  shows "(set_rel A ===> set_rel A) uminus uminus"
+  shows "(rel_set A ===> rel_set A) uminus uminus"
   unfolding Compl_eq [abs_def] by transfer_prover
 
 lemma right_total_Inter_transfer [transfer_rule]:
   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
-  shows "(set_rel (set_rel A) ===> set_rel A) (\<lambda>S. Inter S \<inter> Collect (Domainp A)) Inter"
+  shows "(rel_set (rel_set A) ===> rel_set A) (\<lambda>S. Inter S \<inter> Collect (Domainp A)) Inter"
   unfolding Inter_eq[abs_def]
   by (subst Collect_conj_eq[symmetric]) transfer_prover
 
 lemma Inter_transfer [transfer_rule]:
   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
-  shows "(set_rel (set_rel A) ===> set_rel A) Inter Inter"
+  shows "(rel_set (rel_set A) ===> rel_set A) Inter Inter"
   unfolding Inter_eq [abs_def] by transfer_prover
 
 lemma filter_transfer [transfer_rule]:
   assumes [transfer_rule]: "bi_unique A"
-  shows "((A ===> op=) ===> set_rel A ===> set_rel A) Set.filter Set.filter"
-  unfolding Set.filter_def[abs_def] fun_rel_def set_rel_def by blast
+  shows "((A ===> op=) ===> rel_set A ===> rel_set A) Set.filter Set.filter"
+  unfolding Set.filter_def[abs_def] fun_rel_def rel_set_def by blast
 
-lemma bi_unique_set_rel_lemma:
-  assumes "bi_unique R" and "set_rel R X Y"
+lemma bi_unique_rel_set_lemma:
+  assumes "bi_unique R" and "rel_set R X Y"
   obtains f where "Y = image f X" and "inj_on f X" and "\<forall>x\<in>X. R x (f x)"
 proof
   let ?f = "\<lambda>x. THE y. R x y"
   from assms show f: "\<forall>x\<in>X. R x (?f x)"
-    apply (clarsimp simp add: set_rel_def)
+    apply (clarsimp simp add: rel_set_def)
     apply (drule (1) bspec, clarify)
     apply (rule theI2, assumption)
     apply (simp add: bi_unique_def)
@@ -248,13 +248,13 @@
     done
   from assms show "Y = image ?f X"
     apply safe
-    apply (clarsimp simp add: set_rel_def)
+    apply (clarsimp simp add: rel_set_def)
     apply (drule (1) bspec, clarify)
     apply (rule image_eqI)
     apply (rule the_equality [symmetric], assumption)
     apply (simp add: bi_unique_def)
     apply assumption
-    apply (clarsimp simp add: set_rel_def)
+    apply (clarsimp simp add: rel_set_def)
     apply (frule (1) bspec, clarify)
     apply (rule theI2, assumption)
     apply (clarsimp simp add: bi_unique_def)
@@ -269,41 +269,41 @@
 qed
 
 lemma finite_transfer [transfer_rule]:
-  "bi_unique A \<Longrightarrow> (set_rel A ===> op =) finite finite"
-  by (rule fun_relI, erule (1) bi_unique_set_rel_lemma,
+  "bi_unique A \<Longrightarrow> (rel_set A ===> op =) finite finite"
+  by (rule fun_relI, erule (1) bi_unique_rel_set_lemma,
     auto dest: finite_imageD)
 
 lemma card_transfer [transfer_rule]:
-  "bi_unique A \<Longrightarrow> (set_rel A ===> op =) card card"
-  by (rule fun_relI, erule (1) bi_unique_set_rel_lemma, simp add: card_image)
+  "bi_unique A \<Longrightarrow> (rel_set A ===> op =) card card"
+  by (rule fun_relI, erule (1) bi_unique_rel_set_lemma, simp add: card_image)
 
 lemma vimage_parametric [transfer_rule]:
   assumes [transfer_rule]: "bi_total A" "bi_unique B"
-  shows "((A ===> B) ===> set_rel B ===> set_rel A) vimage vimage"
+  shows "((A ===> B) ===> rel_set B ===> rel_set A) vimage vimage"
 unfolding vimage_def[abs_def] by transfer_prover
 
 lemma setsum_parametric [transfer_rule]:
   assumes "bi_unique A"
-  shows "((A ===> op =) ===> set_rel A ===> op =) setsum setsum"
+  shows "((A ===> op =) ===> rel_set A ===> op =) setsum setsum"
 proof(rule fun_relI)+
   fix f :: "'a \<Rightarrow> 'c" and g S T
   assume fg: "(A ===> op =) f g"
-    and ST: "set_rel A S T"
+    and ST: "rel_set A S T"
   show "setsum f S = setsum g T"
   proof(rule setsum_reindex_cong)
     let ?f = "\<lambda>t. THE s. A s t"
     show "S = ?f ` T"
-      by(blast dest: set_relD1[OF ST] set_relD2[OF ST] bi_uniqueDl[OF assms] 
+      by(blast dest: rel_setD1[OF ST] rel_setD2[OF ST] bi_uniqueDl[OF assms] 
            intro: rev_image_eqI the_equality[symmetric] subst[rotated, where P="\<lambda>x. x \<in> S"])
 
     show "inj_on ?f T"
     proof(rule inj_onI)
       fix t1 t2
       assume "t1 \<in> T" "t2 \<in> T" "?f t1 = ?f t2"
-      from ST `t1 \<in> T` obtain s1 where "A s1 t1" "s1 \<in> S" by(auto dest: set_relD2)
+      from ST `t1 \<in> T` obtain s1 where "A s1 t1" "s1 \<in> S" by(auto dest: rel_setD2)
       hence "?f t1 = s1" by(auto dest: bi_uniqueDl[OF assms])
       moreover
-      from ST `t2 \<in> T` obtain s2 where "A s2 t2" "s2 \<in> S" by(auto dest: set_relD2)
+      from ST `t2 \<in> T` obtain s2 where "A s2 t2" "s2 \<in> S" by(auto dest: rel_setD2)
       hence "?f t2 = s2" by(auto dest: bi_uniqueDl[OF assms])
       ultimately have "s1 = s2" using `?f t1 = ?f t2` by simp
       with `A s1 t1` `A s2 t2` show "t1 = t2" by(auto dest: bi_uniqueDr[OF assms])
@@ -311,7 +311,7 @@
 
     fix t
     assume "t \<in> T"
-    with ST obtain s where "A s t" "s \<in> S" by(auto dest: set_relD2)
+    with ST obtain s where "A s t" "s \<in> S" by(auto dest: rel_setD2)
     hence "?f t = s" by(auto dest: bi_uniqueDl[OF assms])
     moreover from fg `A s t` have "f s = g t" by(rule fun_relD)
     ultimately show "g t = f (?f t)" by simp
diff -r 18e52e8c6300 -r f20d1db5aa3c src/HOL/List.thy
--- a/src/HOL/List.thy	Thu Mar 06 14:25:55 2014 +0100
+++ b/src/HOL/List.thy	Thu Mar 06 14:57:14 2014 +0100
@@ -6734,7 +6734,7 @@
   by (rule fun_relI, erule list_all2_induct, auto)
 
 lemma set_transfer [transfer_rule]:
-  "(list_all2 A ===> set_rel A) set set"
+  "(list_all2 A ===> rel_set A) set set"
   unfolding set_rec[abs_def] by transfer_prover
 
 lemma map_rec: "map f xs = rec_list Nil (%x _ y. Cons (f x) y) xs"
@@ -6864,7 +6864,7 @@
   done
 
 lemma sublist_transfer [transfer_rule]:
-  "(list_all2 A ===> set_rel (op =) ===> list_all2 A) sublist sublist"
+  "(list_all2 A ===> rel_set (op =) ===> list_all2 A) sublist sublist"
   unfolding sublist_def [abs_def] by transfer_prover
 
 lemma partition_transfer [transfer_rule]:
@@ -6873,25 +6873,25 @@
   unfolding partition_def by transfer_prover
 
 lemma lists_transfer [transfer_rule]:
-  "(set_rel A ===> set_rel (list_all2 A)) lists lists"
-  apply (rule fun_relI, rule set_relI)
+  "(rel_set A ===> rel_set (list_all2 A)) lists lists"
+  apply (rule fun_relI, rule rel_setI)
   apply (erule lists.induct, simp)
-  apply (simp only: set_rel_def list_all2_Cons1, metis lists.Cons)
+  apply (simp only: rel_set_def list_all2_Cons1, metis lists.Cons)
   apply (erule lists.induct, simp)
-  apply (simp only: set_rel_def list_all2_Cons2, metis lists.Cons)
+  apply (simp only: rel_set_def list_all2_Cons2, metis lists.Cons)
   done
 
 lemma set_Cons_transfer [transfer_rule]:
-  "(set_rel A ===> set_rel (list_all2 A) ===> set_rel (list_all2 A))
+  "(rel_set A ===> rel_set (list_all2 A) ===> rel_set (list_all2 A))
     set_Cons set_Cons"
-  unfolding fun_rel_def set_rel_def set_Cons_def
+  unfolding fun_rel_def rel_set_def set_Cons_def
   apply safe
   apply (simp add: list_all2_Cons1, fast)
   apply (simp add: list_all2_Cons2, fast)
   done
 
 lemma listset_transfer [transfer_rule]:
-  "(list_all2 (set_rel A) ===> set_rel (list_all2 A)) listset listset"
+  "(list_all2 (rel_set A) ===> rel_set (list_all2 A)) listset listset"
   unfolding listset_def by transfer_prover
 
 lemma null_transfer [transfer_rule]:
diff -r 18e52e8c6300 -r f20d1db5aa3c src/HOL/Topological_Spaces.thy
--- a/src/HOL/Topological_Spaces.thy	Thu Mar 06 14:25:55 2014 +0100
+++ b/src/HOL/Topological_Spaces.thy	Thu Mar 06 14:57:14 2014 +0100
@@ -2497,19 +2497,19 @@
 by(fastforce simp add: filter_rel_eventually[abs_def] eventually_sup dest: fun_relD)
 
 lemma Sup_filter_parametric [transfer_rule]:
-  "(set_rel (filter_rel A) ===> filter_rel A) Sup Sup"
+  "(rel_set (filter_rel A) ===> filter_rel A) Sup Sup"
 proof(rule fun_relI)
   fix S T
-  assume [transfer_rule]: "set_rel (filter_rel A) S T"
+  assume [transfer_rule]: "rel_set (filter_rel A) S T"
   show "filter_rel A (Sup S) (Sup T)"
     by(simp add: filter_rel_eventually eventually_Sup) transfer_prover
 qed
 
 lemma principal_parametric [transfer_rule]:
-  "(set_rel A ===> filter_rel A) principal principal"
+  "(rel_set A ===> filter_rel A) principal principal"
 proof(rule fun_relI)
   fix S S'
-  assume [transfer_rule]: "set_rel A S S'"
+  assume [transfer_rule]: "rel_set A S S'"
   show "filter_rel A (principal S) (principal S')"
     by(simp add: filter_rel_eventually eventually_principal) transfer_prover
 qed
@@ -2532,7 +2532,7 @@
 begin
 
 lemma Inf_filter_parametric [transfer_rule]:
-  "(set_rel (filter_rel A) ===> filter_rel A) Inf Inf"
+  "(rel_set (filter_rel A) ===> filter_rel A) Inf Inf"
 unfolding Inf_filter_def[abs_def] by transfer_prover
 
 lemma inf_filter_parametric [transfer_rule]:
diff -r 18e52e8c6300 -r f20d1db5aa3c src/HOL/ex/Transfer_Int_Nat.thy
--- a/src/HOL/ex/Transfer_Int_Nat.thy	Thu Mar 06 14:25:55 2014 +0100
+++ b/src/HOL/ex/Transfer_Int_Nat.thy	Thu Mar 06 14:57:14 2014 +0100
@@ -96,19 +96,19 @@
   unfolding fun_rel_def ZN_def by (simp add: transfer_int_nat_gcd)
 
 lemma ZN_atMost [transfer_rule]:
-  "(ZN ===> set_rel ZN) (atLeastAtMost 0) atMost"
-  unfolding fun_rel_def ZN_def set_rel_def
+  "(ZN ===> rel_set ZN) (atLeastAtMost 0) atMost"
+  unfolding fun_rel_def ZN_def rel_set_def
   by (clarsimp simp add: Bex_def, arith)
 
 lemma ZN_atLeastAtMost [transfer_rule]:
-  "(ZN ===> ZN ===> set_rel ZN) atLeastAtMost atLeastAtMost"
-  unfolding fun_rel_def ZN_def set_rel_def
+  "(ZN ===> ZN ===> rel_set ZN) atLeastAtMost atLeastAtMost"
+  unfolding fun_rel_def ZN_def rel_set_def
   by (clarsimp simp add: Bex_def, arith)
 
 lemma ZN_setsum [transfer_rule]:
-  "bi_unique A \<Longrightarrow> ((A ===> ZN) ===> set_rel A ===> ZN) setsum setsum"
+  "bi_unique A \<Longrightarrow> ((A ===> ZN) ===> rel_set A ===> ZN) setsum setsum"
   apply (intro fun_relI)
-  apply (erule (1) bi_unique_set_rel_lemma)
+  apply (erule (1) bi_unique_rel_set_lemma)
   apply (simp add: setsum.reindex int_setsum ZN_def fun_rel_def)
   apply (rule setsum_cong2, simp)
   done