# HG changeset patch
# User berghofe
# Date 1326236731 -3600
# Node ID eda2c0aeb1f2077fd528d42fb6af9d1ece42b803
# Parent  1c5c88f6feb53e8a22e4f8b6e2513fb2726c7841# Parent  b4aa5e39f9443f0ce57c83a4d8cc8de1d8885dcd
merged

diff -r 1c5c88f6feb5 -r eda2c0aeb1f2 src/HOL/Nominal/Examples/Class2.thy
--- a/src/HOL/Nominal/Examples/Class2.thy	Tue Jan 10 23:26:27 2012 +0100
+++ b/src/HOL/Nominal/Examples/Class2.thy	Wed Jan 11 00:05:31 2012 +0100
@@ -2123,7 +2123,7 @@
 lemma NOTRIGHT_eqvt_name:
   fixes pi::"name prm"
   shows "(pi\<bullet>(NOTRIGHT (NOT B) X)) = NOTRIGHT (NOT B) (pi\<bullet>X)"
-apply(auto simp add: perm_set_eq)
+apply(auto simp add: perm_set_def)
 apply(rule_tac x="pi\<bullet>a" in exI) 
 apply(rule_tac x="pi\<bullet>xb" in exI) 
 apply(rule_tac x="pi\<bullet>M" in exI)
@@ -2149,7 +2149,7 @@
 lemma NOTRIGHT_eqvt_coname:
   fixes pi::"coname prm"
   shows "(pi\<bullet>(NOTRIGHT (NOT B) X)) = NOTRIGHT (NOT B) (pi\<bullet>X)"
-apply(auto simp add: perm_set_eq)
+apply(auto simp add: perm_set_def)
 apply(rule_tac x="pi\<bullet>a" in exI) 
 apply(rule_tac x="pi\<bullet>xb" in exI) 
 apply(rule_tac x="pi\<bullet>M" in exI)
@@ -2182,7 +2182,7 @@
 lemma NOTLEFT_eqvt_name:
   fixes pi::"name prm"
   shows "(pi\<bullet>(NOTLEFT (NOT B) X)) = NOTLEFT (NOT B) (pi\<bullet>X)"
-apply(auto simp add: perm_set_eq)
+apply(auto simp add: perm_set_def)
 apply(rule_tac x="pi\<bullet>a" in exI) 
 apply(rule_tac x="pi\<bullet>xb" in exI) 
 apply(rule_tac x="pi\<bullet>M" in exI)
@@ -2208,7 +2208,7 @@
 lemma NOTLEFT_eqvt_coname:
   fixes pi::"coname prm"
   shows "(pi\<bullet>(NOTLEFT (NOT B) X)) = NOTLEFT (NOT B) (pi\<bullet>X)"
-apply(auto simp add: perm_set_eq)
+apply(auto simp add: perm_set_def)
 apply(rule_tac x="pi\<bullet>a" in exI) 
 apply(rule_tac x="pi\<bullet>xb" in exI) 
 apply(rule_tac x="pi\<bullet>M" in exI)
@@ -2242,7 +2242,7 @@
 lemma ANDRIGHT_eqvt_name:
   fixes pi::"name prm"
   shows "(pi\<bullet>(ANDRIGHT (A AND B) X Y)) = ANDRIGHT (A AND B) (pi\<bullet>X) (pi\<bullet>Y)"
-apply(auto simp add: perm_set_eq)
+apply(auto simp add: perm_set_def)
 apply(rule_tac x="pi\<bullet>c" in exI)
 apply(rule_tac x="pi\<bullet>a" in exI)
 apply(rule_tac x="pi\<bullet>b" in exI)
@@ -2277,7 +2277,7 @@
 lemma ANDRIGHT_eqvt_coname:
   fixes pi::"coname prm"
   shows "(pi\<bullet>(ANDRIGHT (A AND B) X Y)) = ANDRIGHT (A AND B) (pi\<bullet>X) (pi\<bullet>Y)"
-apply(auto simp add: perm_set_eq)
+apply(auto simp add: perm_set_def)
 apply(rule_tac x="pi\<bullet>c" in exI)
 apply(rule_tac x="pi\<bullet>a" in exI)
 apply(rule_tac x="pi\<bullet>b" in exI)
@@ -2319,7 +2319,7 @@
 lemma ANDLEFT1_eqvt_name:
   fixes pi::"name prm"
   shows "(pi\<bullet>(ANDLEFT1 (A AND B) X)) = ANDLEFT1 (A AND B) (pi\<bullet>X)"
-apply(auto simp add: perm_set_eq)
+apply(auto simp add: perm_set_def)
 apply(rule_tac x="pi\<bullet>xb" in exI) 
 apply(rule_tac x="pi\<bullet>y" in exI) 
 apply(rule_tac x="pi\<bullet>M" in exI)
@@ -2345,7 +2345,7 @@
 lemma ANDLEFT1_eqvt_coname:
   fixes pi::"coname prm"
   shows "(pi\<bullet>(ANDLEFT1 (A AND B) X)) = ANDLEFT1 (A AND B) (pi\<bullet>X)"
-apply(auto simp add: perm_set_eq)
+apply(auto simp add: perm_set_def)
 apply(rule_tac x="pi\<bullet>xb" in exI) 
 apply(rule_tac x="pi\<bullet>y" in exI) 
 apply(rule_tac x="pi\<bullet>M" in exI)
@@ -2378,7 +2378,7 @@
 lemma ANDLEFT2_eqvt_name:
   fixes pi::"name prm"
   shows "(pi\<bullet>(ANDLEFT2 (A AND B) X)) = ANDLEFT2 (A AND B) (pi\<bullet>X)"
-apply(auto simp add: perm_set_eq)
+apply(auto simp add: perm_set_def)
 apply(rule_tac x="pi\<bullet>xb" in exI) 
 apply(rule_tac x="pi\<bullet>y" in exI) 
 apply(rule_tac x="pi\<bullet>M" in exI)
@@ -2404,7 +2404,7 @@
 lemma ANDLEFT2_eqvt_coname:
   fixes pi::"coname prm"
   shows "(pi\<bullet>(ANDLEFT2 (A AND B) X)) = ANDLEFT2 (A AND B) (pi\<bullet>X)"
-apply(auto simp add: perm_set_eq)
+apply(auto simp add: perm_set_def)
 apply(rule_tac x="pi\<bullet>xb" in exI) 
 apply(rule_tac x="pi\<bullet>y" in exI) 
 apply(rule_tac x="pi\<bullet>M" in exI)
@@ -2438,7 +2438,7 @@
 lemma ORLEFT_eqvt_name:
   fixes pi::"name prm"
   shows "(pi\<bullet>(ORLEFT (A OR B) X Y)) = ORLEFT (A OR B) (pi\<bullet>X) (pi\<bullet>Y)"
-apply(auto simp add: perm_set_eq)
+apply(auto simp add: perm_set_def)
 apply(rule_tac x="pi\<bullet>xb" in exI)
 apply(rule_tac x="pi\<bullet>y" in exI)
 apply(rule_tac x="pi\<bullet>z" in exI)
@@ -2473,7 +2473,7 @@
 lemma ORLEFT_eqvt_coname:
   fixes pi::"coname prm"
   shows "(pi\<bullet>(ORLEFT (A OR B) X Y)) = ORLEFT (A OR B) (pi\<bullet>X) (pi\<bullet>Y)"
-apply(auto simp add: perm_set_eq)
+apply(auto simp add: perm_set_def)
 apply(rule_tac x="pi\<bullet>xb" in exI)
 apply(rule_tac x="pi\<bullet>y" in exI)
 apply(rule_tac x="pi\<bullet>z" in exI)
@@ -2515,7 +2515,7 @@
 lemma ORRIGHT1_eqvt_name:
   fixes pi::"name prm"
   shows "(pi\<bullet>(ORRIGHT1 (A OR B) X)) = ORRIGHT1 (A OR B) (pi\<bullet>X)"
-apply(auto simp add: perm_set_eq)
+apply(auto simp add: perm_set_def)
 apply(rule_tac x="pi\<bullet>a" in exI) 
 apply(rule_tac x="pi\<bullet>b" in exI) 
 apply(rule_tac x="pi\<bullet>M" in exI)
@@ -2541,7 +2541,7 @@
 lemma ORRIGHT1_eqvt_coname:
   fixes pi::"coname prm"
   shows "(pi\<bullet>(ORRIGHT1 (A OR B) X)) = ORRIGHT1 (A OR B) (pi\<bullet>X)"
-apply(auto simp add: perm_set_eq)
+apply(auto simp add: perm_set_def)
 apply(rule_tac x="pi\<bullet>a" in exI) 
 apply(rule_tac x="pi\<bullet>b" in exI) 
 apply(rule_tac x="pi\<bullet>M" in exI)
@@ -2574,7 +2574,7 @@
 lemma ORRIGHT2_eqvt_name:
   fixes pi::"name prm"
   shows "(pi\<bullet>(ORRIGHT2 (A OR B) X)) = ORRIGHT2 (A OR B) (pi\<bullet>X)"
-apply(auto simp add: perm_set_eq)
+apply(auto simp add: perm_set_def)
 apply(rule_tac x="pi\<bullet>a" in exI) 
 apply(rule_tac x="pi\<bullet>b" in exI) 
 apply(rule_tac x="pi\<bullet>M" in exI)
@@ -2600,7 +2600,7 @@
 lemma ORRIGHT2_eqvt_coname:
   fixes pi::"coname prm"
   shows "(pi\<bullet>(ORRIGHT2 (A OR B) X)) = ORRIGHT2 (A OR B) (pi\<bullet>X)"
-apply(auto simp add: perm_set_eq)
+apply(auto simp add: perm_set_def)
 apply(rule_tac x="pi\<bullet>a" in exI) 
 apply(rule_tac x="pi\<bullet>b" in exI) 
 apply(rule_tac x="pi\<bullet>M" in exI)
@@ -2636,7 +2636,7 @@
 lemma IMPRIGHT_eqvt_name:
   fixes pi::"name prm"
   shows "(pi\<bullet>(IMPRIGHT (A IMP B) X Y Z U)) = IMPRIGHT (A IMP B) (pi\<bullet>X) (pi\<bullet>Y) (pi\<bullet>Z) (pi\<bullet>U)"
-apply(auto simp add: perm_set_eq)
+apply(auto simp add: perm_set_def)
 apply(rule_tac x="pi\<bullet>xb" in exI)
 apply(rule_tac x="pi\<bullet>a" in exI)
 apply(rule_tac x="pi\<bullet>b" in exI)
@@ -2696,7 +2696,7 @@
 lemma IMPRIGHT_eqvt_coname:
   fixes pi::"coname prm"
   shows "(pi\<bullet>(IMPRIGHT (A IMP B) X Y Z U)) = IMPRIGHT (A IMP B) (pi\<bullet>X) (pi\<bullet>Y) (pi\<bullet>Z) (pi\<bullet>U)"
-apply(auto simp add: perm_set_eq)
+apply(auto simp add: perm_set_def)
 apply(rule_tac x="pi\<bullet>xb" in exI)
 apply(rule_tac x="pi\<bullet>a" in exI)
 apply(rule_tac x="pi\<bullet>b" in exI)
@@ -2763,7 +2763,7 @@
 lemma IMPLEFT_eqvt_name:
   fixes pi::"name prm"
   shows "(pi\<bullet>(IMPLEFT (A IMP B) X Y)) = IMPLEFT (A IMP B) (pi\<bullet>X) (pi\<bullet>Y)"
-apply(auto simp add: perm_set_eq)
+apply(auto simp add: perm_set_def)
 apply(rule_tac x="pi\<bullet>xb" in exI) 
 apply(rule_tac x="pi\<bullet>a" in exI)
 apply(rule_tac x="pi\<bullet>y" in exI) 
@@ -2798,7 +2798,7 @@
 lemma IMPLEFT_eqvt_coname:
   fixes pi::"coname prm"
   shows "(pi\<bullet>(IMPLEFT (A IMP B) X Y)) = IMPLEFT (A IMP B) (pi\<bullet>X) (pi\<bullet>Y)"
-apply(auto simp add: perm_set_eq)
+apply(auto simp add: perm_set_def)
 apply(rule_tac x="pi\<bullet>xb" in exI) 
 apply(rule_tac x="pi\<bullet>a" in exI)
 apply(rule_tac x="pi\<bullet>y" in exI) 
@@ -2902,7 +2902,7 @@
   and   Y::"('b::pt_coname) set set"
   shows "(pi1\<bullet>(\<Inter> X)) = \<Inter> (pi1\<bullet>X)"
   and   "(pi2\<bullet>(\<Inter> Y)) = \<Inter> (pi2\<bullet>Y)"
-apply(auto simp add: perm_set_eq)
+apply(auto simp add: perm_set_def)
 apply(rule_tac x="(rev pi1)\<bullet>x" in exI)
 apply(perm_simp)
 apply(rule ballI)
@@ -3130,7 +3130,7 @@
   fixes pi::"name prm"
   shows "(pi\<bullet>(BINDINGn B X)) = BINDINGn B (pi\<bullet>X)" 
   and   "(pi\<bullet>(BINDINGc B Y)) = BINDINGc B (pi\<bullet>Y)" 
-apply(auto simp add: BINDINGn_def BINDINGc_def perm_set_eq)
+apply(auto simp add: BINDINGn_def BINDINGc_def perm_set_def)
 apply(rule_tac x="pi\<bullet>xb" in exI)
 apply(rule_tac x="pi\<bullet>M" in exI)
 apply(simp)
@@ -3185,7 +3185,7 @@
   fixes pi::"coname prm"
   shows "(pi\<bullet>(BINDINGn B X)) = BINDINGn B (pi\<bullet>X)" 
   and   "(pi\<bullet>(BINDINGc B Y)) = BINDINGc B (pi\<bullet>Y)" 
-apply(auto simp add: BINDINGn_def BINDINGc_def perm_set_eq)
+apply(auto simp add: BINDINGn_def BINDINGc_def perm_set_def)
 apply(rule_tac x="pi\<bullet>xb" in exI)
 apply(rule_tac x="pi\<bullet>M" in exI)
 apply(simp)
diff -r 1c5c88f6feb5 -r eda2c0aeb1f2 src/HOL/Nominal/Examples/Fsub.thy
--- a/src/HOL/Nominal/Examples/Fsub.thy	Tue Jan 10 23:26:27 2012 +0100
+++ b/src/HOL/Nominal/Examples/Fsub.thy	Wed Jan 11 00:05:31 2012 +0100
@@ -270,7 +270,7 @@
   fixes x::"vrs"
   shows "x\<sharp>vrs_of b = x\<sharp>b"
   by (nominal_induct b rule: binding.strong_induct)
-    (simp_all add: fresh_singleton [OF pt_vrs_inst at_vrs_inst] fresh_set_empty ty_vrs_fresh fresh_atm)
+    (simp_all add: fresh_singleton fresh_set_empty ty_vrs_fresh fresh_atm)
 
 lemma fresh_trm_dom: 
   fixes x::"vrs"
@@ -292,27 +292,7 @@
 | valid_consT[simp]: "\<lbrakk>\<turnstile> \<Gamma> ok; X\<sharp>(ty_dom  \<Gamma>); T closed_in \<Gamma>\<rbrakk>  \<Longrightarrow>  \<turnstile> (TVarB X T#\<Gamma>) ok"
 | valid_cons [simp]: "\<lbrakk>\<turnstile> \<Gamma> ok; x\<sharp>(trm_dom \<Gamma>); T closed_in \<Gamma>\<rbrakk>  \<Longrightarrow>  \<turnstile> (VarB  x T#\<Gamma>) ok"
 
-(* FIXME The two following theorems should be subsumed by:
 equivariance valid_rel
-*)
-
-lemma valid_eqvt[eqvt]:
-  fixes pi::"tyvrs prm"
-  assumes a: "\<turnstile> \<Gamma> ok" shows "\<turnstile> (pi\<bullet>\<Gamma>) ok"
-  using a
-apply(induct \<Gamma>)
-apply auto
-apply (metis closed_in_eqvt doms_eqvt(1) fresh_bij(1) valid_consT)
-by (metis closed_in_eqvt fresh_aux(3) fresh_trm_dom perm_dj(1) valid_cons)
-
-lemma valid_eqvt'[eqvt]:
-  fixes pi::"vrs prm"
-  assumes a: "\<turnstile> \<Gamma> ok" shows "\<turnstile> (pi\<bullet>\<Gamma>) ok"
-  using a
-apply(induct \<Gamma>)
-apply auto
-apply (metis closed_in_eqvt' perm_dj(2) ty_dom_vrs_prm_simp valid_consT)
-by (metis closed_in_eqvt' fresh_bij(2) fresh_trm_dom valid_cons)
 
 declare binding.inject [simp add]
 declare trm.inject     [simp add]
@@ -372,7 +352,7 @@
         by (simp add:  fresh_ty_dom_cons 
                        fresh_fin_union[OF pt_tyvrs_inst  at_tyvrs_inst fs_tyvrs_inst]  
                        finite_vrs finite_doms, 
-            auto simp add: fresh_atm fresh_singleton [OF pt_tyvrs_inst at_tyvrs_inst])
+            auto simp add: fresh_atm fresh_singleton)
     qed (simp)
   }
   ultimately show "T=S" by (auto simp add: binding.inject)
@@ -396,7 +376,7 @@
       thus "\<not> (\<exists>T.(VarB x' T)\<in>set(y#\<Gamma>'))" 
         by (simp add:  fresh_fin_union[OF pt_vrs_inst  at_vrs_inst fs_vrs_inst]  
                        finite_vrs finite_doms, 
-            auto simp add: fresh_atm fresh_singleton [OF pt_vrs_inst at_vrs_inst])
+            auto simp add: fresh_atm fresh_singleton)
     qed (simp)
   }
   ultimately show "T=S" by (auto simp add: binding.inject)
diff -r 1c5c88f6feb5 -r eda2c0aeb1f2 src/HOL/Nominal/Examples/Support.thy
--- a/src/HOL/Nominal/Examples/Support.thy	Tue Jan 10 23:26:27 2012 +0100
+++ b/src/HOL/Nominal/Examples/Support.thy	Wed Jan 11 00:05:31 2012 +0100
@@ -98,14 +98,14 @@
     proof (cases "x\<in>S")
       case True
       have "x\<in>S" by fact
-      hence "\<forall>b\<in>(UNIV-S). [(x,b)]\<bullet>S\<noteq>S" by (auto simp add: perm_set_eq calc_atm)
+      hence "\<forall>b\<in>(UNIV-S). [(x,b)]\<bullet>S\<noteq>S" by (auto simp add: perm_set_def calc_atm)
       with asm2 have "infinite {b\<in>(UNIV-S). [(x,b)]\<bullet>S\<noteq>S}" by (rule infinite_Collection)
       hence "infinite {b. [(x,b)]\<bullet>S\<noteq>S}" by (rule_tac infinite_super, auto)
       then show "x\<in>(supp S)" by (simp add: supp_def)
     next
       case False
       have "x\<notin>S" by fact
-      hence "\<forall>b\<in>S. [(x,b)]\<bullet>S\<noteq>S" by (auto simp add: perm_set_eq calc_atm)
+      hence "\<forall>b\<in>S. [(x,b)]\<bullet>S\<noteq>S" by (auto simp add: perm_set_def calc_atm)
       with asm1 have "infinite {b\<in>S. [(x,b)]\<bullet>S\<noteq>S}" by (rule infinite_Collection)
       hence "infinite {b. [(x,b)]\<bullet>S\<noteq>S}" by (rule_tac infinite_super, auto)
       then show "x\<in>(supp S)" by (simp add: supp_def)
@@ -129,7 +129,7 @@
   shows "supp (UNIV::atom set) = ({}::atom set)"
 proof -
   have "\<forall>(x::atom) (y::atom). [(x,y)]\<bullet>UNIV = (UNIV::atom set)"
-    by (auto simp add: perm_set_eq calc_atm)
+    by (auto simp add: perm_set_def calc_atm)
   then show "supp (UNIV::atom set) = ({}::atom set)" by (simp add: supp_def)
 qed
 
diff -r 1c5c88f6feb5 -r eda2c0aeb1f2 src/HOL/Nominal/Nominal.thy
--- a/src/HOL/Nominal/Nominal.thy	Tue Jan 10 23:26:27 2012 +0100
+++ b/src/HOL/Nominal/Nominal.thy	Wed Jan 11 00:05:31 2012 +0100
@@ -58,7 +58,7 @@
   "perm_bool pi b = b"
 
 definition perm_set :: "'x prm \<Rightarrow> 'a set \<Rightarrow> 'a set" where
-  "perm_set pi A = {x. rev pi \<bullet> x \<in> A}"
+  "perm_set pi X = {pi \<bullet> x | x. x \<in> X}"
 
 primrec perm_unit :: "'x prm \<Rightarrow> unit \<Rightarrow> unit"  where 
   "perm_unit pi () = ()"
@@ -137,11 +137,15 @@
 (* permutation on sets *)
 lemma empty_eqvt:
   shows "pi\<bullet>{} = {}"
-  by (simp add: perm_set_def fun_eq_iff)
+  by (simp add: perm_set_def)
 
 lemma union_eqvt:
   shows "(pi\<bullet>(X\<union>Y)) = (pi\<bullet>X) \<union> (pi\<bullet>Y)"
-  by (simp add: perm_set_def fun_eq_iff Un_def)
+  by (auto simp add: perm_set_def)
+
+lemma insert_eqvt:
+  shows "pi\<bullet>(insert x X) = insert (pi\<bullet>x) (pi\<bullet>X)"
+  by (auto simp add: perm_set_def)
 
 (* permutations on products *)
 lemma fst_eqvt:
@@ -166,6 +170,12 @@
   shows "pi\<bullet>(rev l) = rev (pi\<bullet>l)"
   by (induct l) (simp_all add: append_eqvt)
 
+lemma set_eqvt:
+  fixes pi :: "'x prm"
+  and   xs :: "'a list"
+  shows "pi\<bullet>(set xs) = set (pi\<bullet>xs)"
+by (induct xs) (auto simp add: empty_eqvt insert_eqvt)
+
 (* permutation on characters and strings *)
 lemma perm_string:
   fixes s::"string"
@@ -1018,16 +1028,13 @@
   by (simp add: pt_def perm_bool_def)
 
 lemma pt_set_inst:
-  assumes pta: "pt TYPE('a) TYPE('x)"
-  and     at:  "at TYPE('x)"
-  shows "pt TYPE('a set) TYPE('x)"
-proof -
-  have *: "\<And>pi A. pi \<bullet> A = {x. (pi \<bullet> (\<lambda>x. x \<in> A)) x}"
-    by (simp add: perm_fun_def perm_bool_def perm_set_def)
-  from pta pt_bool_inst at
-    have "pt TYPE('a \<Rightarrow> bool) TYPE('x)" by (rule pt_fun_inst)
-  then show ?thesis by (simp add: pt_def *)
-qed
+  assumes pt: "pt TYPE('a) TYPE('x)"
+  shows  "pt TYPE('a set) TYPE('x)"
+apply(simp add: pt_def)
+apply(simp_all add: perm_set_def)
+apply(simp add: pt1[OF pt])
+apply(force simp add: pt2[OF pt] pt3[OF pt])
+done
 
 lemma pt_unit_inst:
   shows "pt TYPE(unit) TYPE('x)"
@@ -1245,43 +1252,13 @@
 apply(rule pt1[OF pt])
 done
 
-lemma perm_set_eq:
-  assumes pt: "pt TYPE('a) TYPE('x)"
-  and at: "at TYPE('x)" 
-  shows "(pi::'x prm)\<bullet>(X::'a set) = {pi\<bullet>x | x. x\<in>X}"
-  apply (auto simp add: perm_fun_def perm_bool perm_set_def)
-  apply (rule_tac x="rev pi \<bullet> x" in exI)
-  apply (simp add: pt_pi_rev [OF pt at])
-  apply (simp add: pt_rev_pi [OF pt at])
-  done
-
-lemma pt_insert_eqvt:
-  fixes pi::"'x prm"
-  and   x::"'a"
-  assumes pt: "pt TYPE('a) TYPE('x)"
-  and at: "at TYPE('x)" 
-  shows "(pi\<bullet>(insert x X)) = insert (pi\<bullet>x) (pi\<bullet>X)"
-  by (auto simp add: perm_set_eq [OF pt at])
-
-lemma pt_set_eqvt:
-  fixes pi :: "'x prm"
-  and   xs :: "'a list"
-  assumes pt: "pt TYPE('a) TYPE('x)"
-  and at: "at TYPE('x)" 
-  shows "pi\<bullet>(set xs) = set (pi\<bullet>xs)"
-by (induct xs) (auto simp add: empty_eqvt pt_insert_eqvt [OF pt at])
-
 lemma supp_singleton:
-  assumes pt: "pt TYPE('a) TYPE('x)"
-  and at: "at TYPE('x)" 
-  shows "(supp {x::'a} :: 'x set) = supp x"
-  by (force simp add: supp_def perm_set_eq [OF pt at])
+  shows "supp {x} = supp x"
+  by (force simp add: supp_def perm_set_def)
 
 lemma fresh_singleton:
-  assumes pt: "pt TYPE('a) TYPE('x)"
-  and at: "at TYPE('x)" 
-  shows "(a::'x)\<sharp>{x::'a} = a\<sharp>x"
-  by (simp add: fresh_def supp_singleton [OF pt at])
+  shows "a\<sharp>{x} = a\<sharp>x"
+  by (simp add: fresh_def supp_singleton)
 
 lemma pt_set_bij1:
   fixes pi :: "'x prm"
@@ -1290,7 +1267,7 @@
   assumes pt: "pt TYPE('a) TYPE('x)"
   and     at: "at TYPE('x)"
   shows "((pi\<bullet>x)\<in>X) = (x\<in>((rev pi)\<bullet>X))"
-  by (force simp add: perm_set_eq [OF pt at] pt_rev_pi[OF pt, OF at] pt_pi_rev[OF pt, OF at])
+  by (force simp add: perm_set_def pt_rev_pi[OF pt, OF at] pt_pi_rev[OF pt, OF at])
 
 lemma pt_set_bij1a:
   fixes pi :: "'x prm"
@@ -1299,7 +1276,7 @@
   assumes pt: "pt TYPE('a) TYPE('x)"
   and     at: "at TYPE('x)"
   shows "(x\<in>(pi\<bullet>X)) = (((rev pi)\<bullet>x)\<in>X)"
-  by (force simp add: perm_set_eq [OF pt at] pt_rev_pi[OF pt, OF at] pt_pi_rev[OF pt, OF at])
+  by (force simp add: perm_set_def pt_rev_pi[OF pt, OF at] pt_pi_rev[OF pt, OF at])
 
 lemma pt_set_bij:
   fixes pi :: "'x prm"
@@ -1308,7 +1285,7 @@
   assumes pt: "pt TYPE('a) TYPE('x)"
   and     at: "at TYPE('x)"
   shows "((pi\<bullet>x)\<in>(pi\<bullet>X)) = (x\<in>X)"
-  by (simp add: perm_set_eq [OF pt at] pt_bij[OF pt, OF at])
+  by (simp add: perm_set_def pt_bij[OF pt, OF at])
 
 lemma pt_in_eqvt:
   fixes pi :: "'x prm"
@@ -1355,7 +1332,7 @@
   assumes pt: "pt TYPE('a) TYPE('x)"
   and     at: "at TYPE('x)"
   shows "(pi\<bullet>(X\<subseteq>Y)) = ((pi\<bullet>X)\<subseteq>(pi\<bullet>Y))"
-by (auto simp add: perm_set_eq [OF pt at] perm_bool pt_bij[OF pt, OF at])
+by (auto simp add: perm_set_def perm_bool pt_bij[OF pt, OF at])
 
 lemma pt_set_diff_eqvt:
   fixes X::"'a set"
@@ -1364,14 +1341,14 @@
   assumes pt: "pt TYPE('a) TYPE('x)"
   and     at: "at TYPE('x)"
   shows "pi\<bullet>(X - Y) = (pi\<bullet>X) - (pi\<bullet>Y)"
-  by (auto simp add: perm_set_eq [OF pt at] pt_bij[OF pt, OF at])
+  by (auto simp add: perm_set_def pt_bij[OF pt, OF at])
 
 lemma pt_Collect_eqvt:
   fixes pi::"'x prm"
   assumes pt: "pt TYPE('a) TYPE('x)"
   and     at: "at TYPE('x)"
   shows "pi\<bullet>{x::'a. P x} = {x. P ((rev pi)\<bullet>x)}"
-apply(auto simp add: perm_set_eq [OF pt at] pt_rev_pi[OF pt, OF at])
+apply(auto simp add: perm_set_def pt_rev_pi[OF pt, OF at])
 apply(rule_tac x="(rev pi)\<bullet>x" in exI)
 apply(simp add: pt_pi_rev[OF pt, OF at])
 done
@@ -1415,7 +1392,7 @@
   and     at: "at TYPE('y)"
   shows "finite (pi\<bullet>X) = finite X"
 proof -
-  have image: "(pi\<bullet>X) = (perm pi ` X)" by (force simp only: perm_set_eq [OF pt at])
+  have image: "(pi\<bullet>X) = (perm pi ` X)" by (force simp only: perm_set_def)
   show ?thesis
   proof (rule iffI)
     assume "finite (pi\<bullet>X)"
@@ -1445,17 +1422,17 @@
   and     cp:  "cp TYPE('a) TYPE('x) TYPE('y)"
   shows "(pi\<bullet>((supp x)::'y set)) = supp (pi\<bullet>x)" (is "?LHS = ?RHS")
 proof -
-  have "?LHS = {pi\<bullet>a | a. infinite {b. [(a,b)]\<bullet>x \<noteq> x}}" by (simp add: supp_def perm_set_eq [OF ptb at])
+  have "?LHS = {pi\<bullet>a | a. infinite {b. [(a,b)]\<bullet>x \<noteq> x}}" by (simp add: supp_def perm_set_def)
   also have "\<dots> = {pi\<bullet>a | a. infinite {pi\<bullet>b | b. [(a,b)]\<bullet>x \<noteq> x}}" 
   proof (rule Collect_permI, rule allI, rule iffI)
     fix a
     assume "infinite {b::'y. [(a,b)]\<bullet>x  \<noteq> x}"
     hence "infinite (pi\<bullet>{b::'y. [(a,b)]\<bullet>x \<noteq> x})" by (simp add: pt_set_infinite_ineq[OF ptb, OF at])
-    thus "infinite {pi\<bullet>b |b::'y. [(a,b)]\<bullet>x  \<noteq> x}" by (simp add: perm_set_eq [OF ptb at])
+    thus "infinite {pi\<bullet>b |b::'y. [(a,b)]\<bullet>x  \<noteq> x}" by (simp add: perm_set_def)
   next
     fix a
     assume "infinite {pi\<bullet>b |b::'y. [(a,b)]\<bullet>x \<noteq> x}"
-    hence "infinite (pi\<bullet>{b::'y. [(a,b)]\<bullet>x \<noteq> x})" by (simp add: perm_set_eq [OF ptb at])
+    hence "infinite (pi\<bullet>{b::'y. [(a,b)]\<bullet>x \<noteq> x})" by (simp add: perm_set_def)
     thus "infinite {b::'y. [(a,b)]\<bullet>x  \<noteq> x}" 
       by (simp add: pt_set_infinite_ineq[OF ptb, OF at])
   qed
@@ -1981,7 +1958,7 @@
   shows "X supports X"
 proof -
   have "\<forall>a b. a\<notin>X \<and> b\<notin>X \<longrightarrow> [(a,b)]\<bullet>X = X"
-    by (auto simp add: perm_set_eq [OF at_pt_inst [OF at] at] at_calc[OF at])
+    by (auto simp add: perm_set_def at_calc[OF at])
   then show ?thesis by (simp add: supports_def)
 qed
 
@@ -2008,7 +1985,7 @@
   { fix a::"'x"
     assume asm: "a\<in>X"
     hence "\<forall>b\<in>(UNIV-X). [(a,b)]\<bullet>X\<noteq>X"
-      by (auto simp add: perm_set_eq [OF at_pt_inst [OF at] at] at_calc[OF at])
+      by (auto simp add: perm_set_def at_calc[OF at])
     with inf have "infinite {b\<in>(UNIV-X). [(a,b)]\<bullet>X\<noteq>X}" by (rule infinite_Collection)
     hence "infinite {b. [(a,b)]\<bullet>X\<noteq>X}" by (rule_tac infinite_super, auto)
     hence "a\<in>(supp X)" by (simp add: supp_def)
@@ -2229,7 +2206,7 @@
   proof (rule equalityI)
     case goal1
     show "pi\<bullet>(\<Union>x\<in>X. f x) \<subseteq> (\<Union>x\<in>(pi\<bullet>X). (pi\<bullet>f) x)"
-      apply(auto simp add: perm_set_eq [OF pt at] perm_set_eq [OF at_pt_inst [OF at] at])
+      apply(auto simp add: perm_set_def)
       apply(rule_tac x="pi\<bullet>xb" in exI)
       apply(rule conjI)
       apply(rule_tac x="xb" in exI)
@@ -2245,7 +2222,7 @@
   next
     case goal2
     show "(\<Union>x\<in>(pi\<bullet>X). (pi\<bullet>f) x) \<subseteq> pi\<bullet>(\<Union>x\<in>X. f x)"
-      apply(auto simp add: perm_set_eq [OF pt at] perm_set_eq [OF at_pt_inst [OF at] at])
+      apply(auto simp add: perm_set_def)
       apply(rule_tac x="(rev pi)\<bullet>x" in exI)
       apply(rule conjI)
       apply(simp add: pt_pi_rev[OF pt_x, OF at])
@@ -2278,7 +2255,7 @@
   apply(rule allI)+
   apply(rule impI)
   apply(erule conjE)
-  apply(simp add: perm_set_eq [OF pt at])
+  apply(simp add: perm_set_def)
   apply(auto)
   apply(subgoal_tac "[(a,b)]\<bullet>xa = xa")(*A*)
   apply(simp)
@@ -2362,7 +2339,7 @@
   also have "\<dots> = (supp {x})\<union>(supp X)"
     by (rule supp_fin_union[OF pt, OF at, OF fs], simp_all add: f)
   finally show "(supp (insert x X)) = (supp x)\<union>((supp X)::'x set)" 
-    by (simp add: supp_singleton [OF pt at])
+    by (simp add: supp_singleton)
 qed
 
 lemma fresh_fin_union:
@@ -2513,10 +2490,10 @@
 apply(simp add: pt_fresh_left_ineq[OF pta, OF ptb, OF at, OF cp])
 apply(drule_tac x="pi\<bullet>xa" in bspec)
 apply(simp add: pt_set_bij1[OF ptb, OF at])
-apply(simp add: pt_set_eqvt [OF ptb at] pt_rev_pi[OF pt_list_inst[OF ptb], OF at])
+apply(simp add: set_eqvt pt_rev_pi[OF pt_list_inst[OF ptb], OF at])
 apply(simp add: pt_fresh_bij_ineq[OF pta, OF ptb, OF at, OF cp])
 apply(drule_tac x="(rev pi)\<bullet>xa" in bspec)
-apply(simp add: pt_set_bij1[OF ptb, OF at] pt_set_eqvt [OF ptb at])
+apply(simp add: pt_set_bij1[OF ptb, OF at] set_eqvt)
 apply(simp add: pt_fresh_left_ineq[OF pta, OF ptb, OF at, OF cp])
 done
 
@@ -2638,10 +2615,7 @@
     have "({}::'a set) \<inter> As = {}" by simp
     moreover
     have "set ([]::'a prm) \<subseteq> {} \<times> {}" by simp
-    moreover
-    have "([]::'a prm)\<bullet>{} = ({}::'a set)" 
-      by (rule pt1 [OF pt_set_inst, OF at_pt_inst [OF at] at])
-    ultimately show ?case by simp
+    ultimately show ?case by (simp add: empty_eqvt)
   next
     case (insert x Xs)
     then have ih: "\<exists>pi. (pi\<bullet>Xs)\<sharp>*c \<and> (pi\<bullet>Xs) \<inter> As = {} \<and> set pi \<subseteq> Xs \<times> (pi\<bullet>Xs)" by simp
@@ -2655,7 +2629,7 @@
     obtain y::"'a" where fr: "y\<sharp>(c,pi\<bullet>Xs,As)" 
       apply(rule_tac at_exists_fresh[OF at, where x="(c,pi\<bullet>Xs,As)"])
       apply(auto simp add: supp_prod at_supp[OF at] at_fin_set_supp[OF at]
-        pt_supp_finite_pi[OF pt_set_inst[OF at_pt_inst[OF at] at] at])
+        pt_supp_finite_pi[OF pt_set_inst[OF at_pt_inst[OF at]] at])
       done
     have "({y}\<union>(pi\<bullet>Xs))\<sharp>*c" using a1 fr by (simp add: fresh_star_def)
     moreover
@@ -2671,20 +2645,18 @@
       have eq: "[(pi\<bullet>x,y)]\<bullet>(pi\<bullet>Xs) = (pi\<bullet>Xs)" 
       proof -
         have "(pi\<bullet>x)\<sharp>(pi\<bullet>Xs)" using b d2 
-          by (simp add: pt_fresh_bij [OF pt_set_inst, OF at_pt_inst [OF at], 
-            OF at, OF at]
+          by (simp add: pt_fresh_bij [OF pt_set_inst [OF at_pt_inst [OF at]], OF at]
             at_fin_set_fresh [OF at])
         moreover
         have "y\<sharp>(pi\<bullet>Xs)" using fr by simp
         ultimately show "[(pi\<bullet>x,y)]\<bullet>(pi\<bullet>Xs) = (pi\<bullet>Xs)" 
-          by (simp add: pt_fresh_fresh[OF pt_set_inst, 
-            OF at_pt_inst[OF at], OF at, OF at])
+          by (simp add: pt_fresh_fresh[OF pt_set_inst
+            [OF at_pt_inst[OF at]], OF at])
       qed
       have "(((pi\<bullet>x,y)#pi)\<bullet>({x}\<union>Xs)) = ([(pi\<bullet>x,y)]\<bullet>(pi\<bullet>({x}\<union>Xs)))"
-        by (simp add: pt2[symmetric, OF pt_set_inst, OF at_pt_inst[OF at], 
-          OF at])
+        by (simp add: pt2[symmetric, OF pt_set_inst [OF at_pt_inst[OF at]]])
       also have "\<dots> = {y}\<union>([(pi\<bullet>x,y)]\<bullet>(pi\<bullet>Xs))" 
-        by (simp only: union_eqvt perm_set_eq[OF at_pt_inst[OF at], OF at] at_calc[OF at])(auto)
+        by (simp only: union_eqvt perm_set_def at_calc[OF at])(auto)
       finally show "(((pi\<bullet>x,y)#pi)\<bullet>({x} \<union> Xs)) = {y}\<union>(pi\<bullet>Xs)" using eq by simp
     qed
     ultimately 
@@ -2714,6 +2686,17 @@
 apply(auto)
 done
 
+lemma cp_set_inst:
+  assumes c1: "cp TYPE ('a) TYPE('x) TYPE('y)"
+  shows "cp TYPE ('a set) TYPE('x) TYPE('y)"
+using c1
+apply(simp add: cp_def)
+apply(auto)
+apply(auto simp add: perm_set_def)
+apply(rule_tac x="pi2\<bullet>xc" in exI)
+apply(auto)
+done
+
 lemma cp_option_inst:
   assumes c1: "cp TYPE ('a) TYPE('x) TYPE('y)"
   shows "cp TYPE ('a option) TYPE('x) TYPE('y)"
@@ -3596,7 +3579,7 @@
   plus_int_eqvt minus_int_eqvt mult_int_eqvt div_int_eqvt
   
   (* sets *)
-  union_eqvt empty_eqvt
+  union_eqvt empty_eqvt insert_eqvt set_eqvt
   
  
 (* the lemmas numeral_nat_eqvt numeral_int_eqvt do not conform with the *)
diff -r 1c5c88f6feb5 -r eda2c0aeb1f2 src/HOL/Nominal/nominal_atoms.ML
--- a/src/HOL/Nominal/nominal_atoms.ML	Tue Jan 10 23:26:27 2012 +0100
+++ b/src/HOL/Nominal/nominal_atoms.ML	Wed Jan 11 00:05:31 2012 +0100
@@ -542,7 +542,7 @@
                      rtac (thm RS pt1) 1, rtac (thm RS pt2) 1, rtac (thm RS pt3) 1, atac 1];
 
           val pt_thm_fun   = at_thm RS (pt_inst RS (pt_inst RS pt_fun_inst));
-          val pt_thm_set   = at_thm RS (pt_inst RS pt_set_inst);
+          val pt_thm_set   = pt_inst RS pt_set_inst;
           val pt_thm_noptn = pt_inst RS pt_noptn_inst; 
           val pt_thm_optn  = pt_inst RS pt_optn_inst; 
           val pt_thm_list  = pt_inst RS pt_list_inst;
@@ -635,6 +635,7 @@
        val cp_fun_inst     = @{thm "cp_fun_inst"};
        val cp_option_inst  = @{thm "cp_option_inst"};
        val cp_noption_inst = @{thm "cp_noption_inst"};
+       val cp_set_inst     = @{thm "cp_set_inst"};
        val pt_perm_compose = @{thm "pt_perm_compose"};
 
        val dj_pp_forget    = @{thm "dj_perm_perm_forget"};
@@ -696,6 +697,7 @@
             val cp_thm_fun  = at_inst RS (pt_inst RS (cp_inst RS (cp_inst RS cp_fun_inst)));
             val cp_thm_optn = cp_inst RS cp_option_inst;
             val cp_thm_noptn = cp_inst RS cp_noption_inst;
+            val cp_thm_set = cp_inst RS cp_set_inst;
         in
          thy'
          |> AxClass.prove_arity ("Product_Type.unit",[],[cls_name]) (cp_proof cp_thm_unit)
@@ -704,6 +706,7 @@
          |> AxClass.prove_arity ("fun",[[cls_name],[cls_name]],[cls_name]) (cp_proof cp_thm_fun)
          |> AxClass.prove_arity ("Option.option",[[cls_name]],[cls_name]) (cp_proof cp_thm_optn)
          |> AxClass.prove_arity ("Nominal.noption",[[cls_name]],[cls_name]) (cp_proof cp_thm_noptn)
+         |> AxClass.prove_arity (@{type_name Set.set},[[cls_name]],[cls_name]) (cp_proof cp_thm_set)
         end) ak_names thy) ak_names thy25;
 
      (* show that discrete nominal types are permutation types, finitely     *)
@@ -817,9 +820,6 @@
        val pt_perm_supp_ineq   = @{thm "Nominal.pt_perm_supp_ineq"};
        val pt_perm_supp        = @{thm "Nominal.pt_perm_supp"};
        val subseteq_eqvt       = @{thm "Nominal.pt_subseteq_eqvt"};
-       val insert_eqvt         = @{thm "Nominal.pt_insert_eqvt"};
-       val set_eqvt            = @{thm "Nominal.pt_set_eqvt"};
-       val perm_set_eq         = @{thm "Nominal.perm_set_eq"};
 
        (* Now we collect and instantiate some lemmas w.r.t. all atom      *)
        (* types; this allows for example to use abs_perm (which is a      *)
@@ -985,9 +985,6 @@
             ||>> add_thmss_string
               let val thms1 = inst_pt_at [subseteq_eqvt]
               in [(("subseteq_eqvt", thms1),[NominalThmDecls.eqvt_add])] end
-            ||>> add_thmss_string [(("insert_eqvt", inst_pt_at [insert_eqvt]), [NominalThmDecls.eqvt_add])]
-            ||>> add_thmss_string [(("set_eqvt", inst_pt_at [set_eqvt]), [NominalThmDecls.eqvt_add])]
-            ||>> add_thmss_string [(("perm_set_eq", inst_pt_at [perm_set_eq]), [])]
             ||>> add_thmss_string
               let val thms1 = inst_pt_at [fresh_aux]
                   and thms2 = inst_pt_pt_at_cp_dj [fresh_perm_app_ineq]