--- a/Quot/Nominal/Abs.thy	Mon Feb 01 12:48:18 2010 +0100
+++ b/Quot/Nominal/Abs.thy	Mon Feb 01 13:00:01 2010 +0100
@@ -1,5 +1,5 @@
 theory Abs
-imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "../QuotMain" 
+imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "../QuotMain" "../QuotProd"
 begin
 
 (* lemmas that should be in Nominal \<dots>\<dots>must be cleaned *)
@@ -73,17 +73,17 @@
   done
 
 fun
-  alpha_abst 
+  alpha_abs 
 where
-  "alpha_abst (bs, x) (cs, y) = (\<exists>p. (bs, x) \<approx>gen (op=) supp p (cs, y))"
+  "alpha_abs (bs, x) (cs, y) = (\<exists>p. (bs, x) \<approx>gen (op=) supp p (cs, y))"
 
 notation
-  alpha_abst ("_ \<approx>abst _")
+  alpha_abs ("_ \<approx>abs _")
 
 lemma test1:
   assumes a1: "a \<notin> (supp x) - bs"
   and     a2: "b \<notin> (supp x) - bs"
-  shows "(bs, x) \<approx>abst ((a \<rightleftharpoons> b) \<bullet> bs, (a \<rightleftharpoons> b) \<bullet> x)"
+  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)
@@ -102,14 +102,14 @@
   "s_test (bs, x) = (supp x) - bs"
 
 lemma s_test_lemma:
-  assumes a: "x \<approx>abst y" 
+  assumes a: "x \<approx>abs y" 
   shows "s_test x = s_test y"
 using a
-apply(induct rule: alpha_abst.induct)
+apply(induct rule: alpha_abs.induct)
 apply(simp add: alpha_gen)
 done
   
-quotient_type 'a abst = "(atom set \<times> 'a::pt)" / "alpha_abst"
+quotient_type 'a abs = "(atom set \<times> 'a::pt)" / "alpha_abs"
   apply(rule equivpI)
   unfolding reflp_def symp_def transp_def
   apply(simp_all)
@@ -131,12 +131,12 @@
   done
 
 quotient_definition
-   "Abst::atom set \<Rightarrow> ('a::pt) \<Rightarrow> 'a abst"
+   "Abs::atom set \<Rightarrow> ('a::pt) \<Rightarrow> 'a abs"
 as
    "Pair::atom set \<Rightarrow> ('a::pt) \<Rightarrow> (atom set \<times> 'a)"
 
 lemma [quot_respect]:
-  shows "((op =) ===> (op =) ===> alpha_abst) Pair Pair"
+  shows "((op =) ===> (op =) ===> alpha_abs) Pair Pair"
 apply(clarsimp)
 apply(rule exI)
 apply(rule alpha_gen_refl)
@@ -144,7 +144,7 @@
 done
 
 lemma [quot_respect]:
-  shows "((op =) ===> alpha_abst ===> alpha_abst) permute permute"
+  shows "((op =) ===> alpha_abs ===> alpha_abs) permute permute"
 apply(clarsimp)
 apply(rule exI)
 apply(rule alpha_gen_eqvt)
@@ -153,32 +153,32 @@
 done
 
 lemma [quot_respect]:
-  shows "(alpha_abst ===> (op =)) s_test s_test"
+  shows "(alpha_abs ===> (op =)) s_test s_test"
 apply(simp add: s_test_lemma)
 done
 
-lemma abst_induct:
-  "\<lbrakk>\<And>as (x::'a::pt). P (Abst as x)\<rbrakk> \<Longrightarrow> P t"
+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
 
-instantiation abst :: (pt) pt
+instantiation abs :: (pt) pt
 begin
 
 quotient_definition
-  "permute_abst::perm \<Rightarrow> ('a::pt abst) \<Rightarrow> 'a abst"
+  "permute_abs::perm \<Rightarrow> ('a::pt abs) \<Rightarrow> 'a abs"
 as
   "permute:: perm \<Rightarrow> (atom set \<times> 'a::pt) \<Rightarrow> (atom set \<times> 'a::pt)"
 
 lemma permute_ABS [simp]:
   fixes x::"'a::pt"  (* ??? has to be 'a \<dots> 'b does not work *)
-  shows "(p \<bullet> (Abst as x)) = Abst (p \<bullet> as) (p \<bullet> x)"
+  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
 
 instance
   apply(default)
-  apply(induct_tac [!] x rule: abst_induct)
+  apply(induct_tac [!] x rule: abs_induct)
   apply(simp_all)
   done
 
@@ -187,13 +187,13 @@
 lemma test1_lifted:
   assumes a1: "a \<notin> (supp x) - bs"
   and     a2: "b \<notin> (supp x) - bs"
-  shows "(Abst bs x) = (Abst ((a \<rightleftharpoons> b) \<bullet> bs) ((a \<rightleftharpoons> b) \<bullet> x))"
+  shows "(Abs bs x) = (Abs ((a \<rightleftharpoons> b) \<bullet> bs) ((a \<rightleftharpoons> b) \<bullet> x))"
 using a1 a2
 apply(lifting test1)
 done
 
-lemma Abst_supports:
-  shows "((supp x) - as) supports (Abst as x)"
+lemma Abs_supports:
+  shows "((supp x) - as) supports (Abs as x)"
 unfolding supports_def
 apply(clarify)
 apply(simp (no_asm))
@@ -202,18 +202,18 @@
 done
 
 quotient_definition
-  "s_test_lifted :: ('a::pt) abst \<Rightarrow> atom \<Rightarrow> bool"
+  "s_test_lifted :: ('a::pt) abs \<Rightarrow> atom \<Rightarrow> bool"
 as
   "s_test"
 
 lemma s_test_lifted_simp:
-  shows "s_test_lifted (Abst bs x) = (supp x) - bs"
+  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: abst_induct)
+apply(induct ab rule: abs_induct)
 apply(simp add: s_test_lifted_simp supp_eqvt Diff_eqvt)
 done
 
@@ -232,7 +232,7 @@
 
 
 lemma s_test_fresh_lemma:
-  shows "(a \<sharp> Abst bs x) \<Longrightarrow> (a \<sharp> s_test_lifted (Abst bs x))"
+  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)
@@ -255,7 +255,7 @@
 
 lemma s_test_subset:
   fixes x::"'a::fs"
-  shows "((supp x) - as) \<subseteq> (supp (Abst as x))"
+  shows "((supp x) - as) \<subseteq> (supp (Abs as x))"
 apply(rule subsetI)
 apply(rule contrapos_pp)
 apply(assumption)
@@ -269,34 +269,34 @@
 apply(simp add: finite_supp)
 done
 
-lemma supp_Abst:
+lemma supp_Abs:
   fixes x::"'a::fs"
-  shows "supp (Abst as x) = (supp x) - as"
+  shows "supp (Abs as x) = (supp x) - as"
 apply(rule subset_antisym)
 apply(rule supp_is_subset)
-apply(rule Abst_supports)
+apply(rule Abs_supports)
 apply(simp add: finite_supp)
 apply(rule s_test_subset)
 done
 
-instance abst :: (fs) fs
+instance abs :: (fs) fs
 apply(default)
-apply(induct_tac x rule: abst_induct)
-apply(simp add: supp_Abst)
+apply(induct_tac x rule: abs_induct)
+apply(simp add: supp_Abs)
 apply(simp add: finite_supp)
 done
 
 lemma fresh_abs:
   fixes x::"'a::fs"
-  shows "a \<sharp> Abst bs x = (a \<in> bs \<or> (a \<notin> bs \<and> a \<sharp> x))"
+  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_Abst)
+apply(simp add: supp_Abs)
 apply(auto)
 done
 
 lemma abs_eq:
-  shows "(Abst bs x) = (Abst cs y) \<longleftrightarrow> (\<exists>p. (bs, x) \<approx>gen (op =) supp p (cs, y))"
-apply(lifting alpha_abst.simps(1))
+  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
 
 end