--- a/Quot/Nominal/Abs.thy	Mon Feb 01 16:23:47 2010 +0100
+++ b/Quot/Nominal/Abs.thy	Mon Feb 01 16:46:07 2010 +0100
@@ -80,34 +80,34 @@
 notation
   alpha_abs ("_ \<approx>abs _")
 
-lemma test1:
+lemma alpha_abs_swap:
   assumes a1: "a \<notin> (supp x) - bs"
   and     a2: "b \<notin> (supp x) - bs"
   shows "(bs, x) \<approx>abs ((a \<rightleftharpoons> b) \<bullet> bs, (a \<rightleftharpoons> b) \<bullet> x)"
-apply(simp)
-apply(rule_tac x="(a \<rightleftharpoons> b)" in exI)
-apply(simp add: alpha_gen)
-apply(simp add: supp_eqvt[symmetric] Diff_eqvt[symmetric])
-apply(simp add: swap_set_fresh[OF a1 a2])
-apply(subgoal_tac "supp (a \<rightleftharpoons> b) \<subseteq> {a, b}")
-using a1 a2
-apply(simp add: fresh_star_def fresh_def)
-apply(blast)
-apply(simp add: supp_swap)
-done
+  apply(simp)
+  apply(rule_tac x="(a \<rightleftharpoons> b)" in exI)
+  apply(simp add: alpha_gen)
+  apply(simp add: supp_eqvt[symmetric] Diff_eqvt[symmetric])
+  apply(simp add: swap_set_not_in[OF a1 a2])
+  apply(subgoal_tac "supp (a \<rightleftharpoons> b) \<subseteq> {a, b}")
+  using a1 a2
+  apply(simp add: fresh_star_def fresh_def)
+  apply(blast)
+  apply(simp add: supp_swap)
+  done
 
 fun
-  s_test
+  supp_abs_fun
 where
-  "s_test (bs, x) = (supp x) - bs"
+  "supp_abs_fun (bs, x) = (supp x) - bs"
 
-lemma s_test_lemma:
+lemma supp_abs_fun_lemma:
   assumes a: "x \<approx>abs y" 
-  shows "s_test x = s_test y"
-using a
-apply(induct rule: alpha_abs.induct)
-apply(simp add: alpha_gen)
-done
+  shows "supp_abs_fun x = supp_abs_fun y"
+  using a
+  apply(induct rule: alpha_abs.induct)
+  apply(simp add: alpha_gen)
+  done
   
 quotient_type 'a abs = "(atom set \<times> 'a::pt)" / "alpha_abs"
   apply(rule equivpI)
@@ -137,30 +137,30 @@
 
 lemma [quot_respect]:
   shows "((op =) ===> (op =) ===> alpha_abs) Pair Pair"
-apply(clarsimp)
-apply(rule exI)
-apply(rule alpha_gen_refl)
-apply(simp)
-done
+  apply(clarsimp)
+  apply(rule exI)
+  apply(rule alpha_gen_refl)
+  apply(simp)
+  done
 
 lemma [quot_respect]:
   shows "((op =) ===> alpha_abs ===> alpha_abs) permute permute"
-apply(clarsimp)
-apply(rule exI)
-apply(rule alpha_gen_eqvt)
-apply(assumption)
-apply(simp_all add: supp_eqvt)
-done
+  apply(clarsimp)
+  apply(rule exI)
+  apply(rule alpha_gen_eqvt)
+  apply(assumption)
+  apply(simp_all add: supp_eqvt)
+  done
 
 lemma [quot_respect]:
-  shows "(alpha_abs ===> (op =)) s_test s_test"
-apply(simp add: s_test_lemma)
-done
+  shows "(alpha_abs ===> (op =)) supp_abs_fun supp_abs_fun"
+  apply(simp add: supp_abs_fun_lemma)
+  done
 
 lemma abs_induct:
   "\<lbrakk>\<And>as (x::'a::pt). P (Abs as x)\<rbrakk> \<Longrightarrow> P t"
-apply(lifting prod.induct[where 'a="atom set" and 'b="'a"])
-done
+  apply(lifting prod.induct[where 'a="atom set" and 'b="'a"])
+  done
 
 instantiation abs :: (pt) pt
 begin
@@ -173,8 +173,7 @@
 lemma permute_ABS [simp]:
   fixes x::"'a::pt"  (* ??? has to be 'a \<dots> 'b does not work *)
   shows "(p \<bullet> (Abs as x)) = Abs (p \<bullet> as) (p \<bullet> x)"
-apply(lifting permute_prod.simps(1)[where 'a="atom set" and 'b="'a"])
-done
+  by (lifting permute_prod.simps(1)[where 'a="atom set" and 'b="'a"])
 
 instance
   apply(default)
@@ -184,120 +183,91 @@
 
 end
 
-lemma test1_lifted:
+quotient_definition
+  "supp_Abs_fun :: ('a::pt) abs \<Rightarrow> atom \<Rightarrow> bool"
+as
+  "supp_abs_fun"
+
+lemma supp_Abs_fun_simp:
+  shows "supp_Abs_fun (Abs bs x) = (supp x) - bs"
+  by (lifting supp_abs_fun.simps(1))
+
+lemma supp_Abs_fun_eqvt:
+  shows "(p \<bullet> supp_Abs_fun) = supp_Abs_fun"
+  apply(subst permute_fun_def)
+  apply(subst expand_fun_eq)
+  apply(rule allI)
+  apply(induct_tac x rule: abs_induct)
+  apply(simp add: supp_Abs_fun_simp supp_eqvt Diff_eqvt)
+  done
+
+lemma supp_Abs_fun_fresh:
+  shows "a \<sharp> Abs bs x \<Longrightarrow> a \<sharp> supp_Abs_fun (Abs bs x)"
+  apply(rule fresh_fun_eqvt_app)
+  apply(simp add: supp_Abs_fun_eqvt)
+  apply(simp)
+  done
+
+lemma Abs_swap:
   assumes a1: "a \<notin> (supp x) - bs"
   and     a2: "b \<notin> (supp x) - bs"
   shows "(Abs bs x) = (Abs ((a \<rightleftharpoons> b) \<bullet> bs) ((a \<rightleftharpoons> b) \<bullet> x))"
-using a1 a2
-apply(lifting test1)
-done
+  using a1 a2 by (lifting alpha_abs_swap)
 
 lemma Abs_supports:
   shows "((supp x) - as) supports (Abs as x)"
-unfolding supports_def
-apply(clarify)
-apply(simp (no_asm))
-apply(subst test1_lifted[symmetric])
-apply(simp_all)
-done
-
-quotient_definition
-  "s_test_lifted :: ('a::pt) abs \<Rightarrow> atom \<Rightarrow> bool"
-as
-  "s_test"
-
-lemma s_test_lifted_simp:
-  shows "s_test_lifted (Abs bs x) = (supp x) - bs"
-apply(lifting s_test.simps(1))
-done
-
-lemma s_test_lifted_eqvt:
-  shows "(p \<bullet> (s_test_lifted ab)) = s_test_lifted (p \<bullet> ab)"
-apply(induct ab rule: abs_induct)
-apply(simp add: s_test_lifted_simp supp_eqvt Diff_eqvt)
-done
-
-lemma fresh_f_empty_supp:
-  assumes a: "\<forall>p. p \<bullet> f = f"
-  shows "a \<sharp> x \<Longrightarrow> a \<sharp> (f x)"
-apply(simp add: fresh_def)
-apply(simp add: supp_def)
-apply(simp add: permute_fun_app_eq)
-apply(simp add: a)
-apply(rule finite_subset)
-prefer 2
-apply(assumption)
-apply(auto)
-done
-
+  unfolding supports_def
+  apply(clarify)
+  apply(simp (no_asm))
+  apply(subst Abs_swap[symmetric])
+  apply(simp_all)
+  done
 
-lemma s_test_fresh_lemma:
-  shows "(a \<sharp> Abs bs x) \<Longrightarrow> (a \<sharp> s_test_lifted (Abs bs x))"
-apply(rule fresh_f_empty_supp)
-apply(rule allI)
-apply(subst permute_fun_def)
-apply(simp add: s_test_lifted_eqvt)
-apply(simp)
-done
-
+lemma supp_Abs_subset1:
+  fixes x::"'a::fs"
+  shows "(supp x) - as \<subseteq> supp (Abs as x)"
+  apply(simp add: supp_conv_fresh)
+  apply(auto)
+  apply(drule_tac supp_Abs_fun_fresh)
+  apply(simp only: supp_Abs_fun_simp)
+  apply(simp add: fresh_def)
+  apply(simp add: supp_finite_atom_set finite_supp)
+  done
 
-lemma supp_finite_set:
-  fixes S::"atom set"
-  assumes "finite S"
-  shows "supp S = S"
-  apply(rule finite_supp_unique)
-  apply(simp add: supports_def)
-  apply(auto simp add: permute_set_eq swap_atom)[1]
-  apply(metis)
-  apply(rule assms)
-  apply(auto simp add: permute_set_eq swap_atom)[1]
-done
-
-lemma s_test_subset:
+lemma supp_Abs_subset2:
   fixes x::"'a::fs"
-  shows "((supp x) - as) \<subseteq> (supp (Abs as x))"
-apply(rule subsetI)
-apply(rule contrapos_pp)
-apply(assumption)
-unfolding fresh_def[symmetric]
-thm s_test_fresh_lemma
-apply(drule_tac s_test_fresh_lemma)
-apply(simp only: s_test_lifted_simp)
-apply(simp add: fresh_def)
-apply(subgoal_tac "finite (supp x - as)")
-apply(simp add: supp_finite_set)
-apply(simp add: finite_supp)
-done
+  shows "supp (Abs as x) \<subseteq> (supp x) - as"
+  apply(rule supp_is_subset)
+  apply(rule Abs_supports)
+  apply(simp add: finite_supp)
+  done
 
 lemma supp_Abs:
   fixes x::"'a::fs"
   shows "supp (Abs as x) = (supp x) - as"
-apply(rule subset_antisym)
-apply(rule supp_is_subset)
-apply(rule Abs_supports)
-apply(simp add: finite_supp)
-apply(rule s_test_subset)
-done
+  apply(rule_tac subset_antisym)
+  apply(rule supp_Abs_subset2)
+  apply(rule supp_Abs_subset1)
+  done
 
 instance abs :: (fs) fs
-apply(default)
-apply(induct_tac x rule: abs_induct)
-apply(simp add: supp_Abs)
-apply(simp add: finite_supp)
-done
+  apply(default)
+  apply(induct_tac x rule: abs_induct)
+  apply(simp add: supp_Abs)
+  apply(simp add: finite_supp)
+  done
 
-lemma fresh_abs:
+lemma Abs_fresh_iff:
   fixes x::"'a::fs"
   shows "a \<sharp> Abs bs x = (a \<in> bs \<or> (a \<notin> bs \<and> a \<sharp> x))"
-apply(simp add: fresh_def)
-apply(simp add: supp_Abs)
-apply(auto)
-done
+  apply(simp add: fresh_def)
+  apply(simp add: supp_Abs)
+  apply(auto)
+  done
 
-lemma abs_eq:
+lemma Abs_eq_iff:
   shows "(Abs bs x) = (Abs cs y) \<longleftrightarrow> (\<exists>p. (bs, x) \<approx>gen (op =) supp p (cs, y))"
-apply(lifting alpha_abs.simps(1))
-done
+  by (lifting alpha_abs.simps(1))
 
 end