# HG changeset patch
# User Christian Urban <urbanc@in.tum.de>
# Date 1309832613 -7200
# Node ID 70bbd18ad19479c310466be74effcec1183e0c0f
# Parent  8648ae682442e13142747b194287d3d006ded10c# Parent  fac8895b109adfb8cef96c6777a23449ac87bac9
merged

diff -r fac8895b109a -r 70bbd18ad194 Nominal/Ex/Classical.thy
--- a/Nominal/Ex/Classical.thy	Tue Jul 05 10:13:34 2011 +0900
+++ b/Nominal/Ex/Classical.thy	Tue Jul 05 04:23:33 2011 +0200
@@ -47,269 +47,6 @@
 thm trm.supp
 thm trm.supp[simplified]
 
-lemma Abs_set_fcb2:
-  fixes as bs :: "atom set"
-    and x y :: "'b :: fs"
-    and c::"'c::fs"
-  assumes eq: "[as]set. x = [bs]set. y"
-  and fin: "finite as" "finite bs"
-  and fcb1: "as \<sharp>* f as x c"
-  and fresh1: "as \<sharp>* c"
-  and fresh2: "bs \<sharp>* c"
-  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"
-  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"
-  shows "f as x c = f bs y c"
-proof -
-  have "supp (as, x, c) supports (f as x c)"
-    unfolding  supports_def fresh_def[symmetric]
-    by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)
-  then have fin1: "finite (supp (f as x c))"
-    using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair)
-  have "supp (bs, y, c) supports (f bs y c)"
-    unfolding  supports_def fresh_def[symmetric]
-    by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
-  then have fin2: "finite (supp (f bs y c))"
-    using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair)
-  obtain q::"perm" where 
-    fr1: "(q \<bullet> as) \<sharp>* (x, c, f as x c, f bs y c)" and 
-    fr2: "supp q \<sharp>* ([as]set. x)" and 
-    inc: "supp q \<subseteq> as \<union> (q \<bullet> as)"
-    using at_set_avoiding3[where xs="as" and c="(x, c, f as x c, f bs y c)" and x="[as]set. x"]  
-      fin1 fin2 fin
-    by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
-  have "[q \<bullet> as]set. (q \<bullet> x) = q \<bullet> ([as]set. x)" by simp
-  also have "\<dots> = [as]set. x"
-    by (simp only: fr2 perm_supp_eq)
-  finally have "[q \<bullet> as]set. (q \<bullet> x) = [bs]set. y" using eq by simp
-  then obtain r::perm where 
-    qq1: "q \<bullet> x = r \<bullet> y" and 
-    qq2: "q \<bullet> as = r \<bullet> bs" and 
-    qq3: "supp r \<subseteq> (q \<bullet> as) \<union> bs"
-    apply(drule_tac sym)
-    apply(simp only: Abs_eq_iff2 alphas)
-    apply(erule exE)
-    apply(erule conjE)+
-    apply(drule_tac x="p" in meta_spec)
-    apply(simp add: set_eqvt)
-    apply(blast)
-    done
-  have "as \<sharp>* f as x c" by (rule fcb1)
-  then have "q \<bullet> (as \<sharp>* f as x c)"
-    by (simp add: permute_bool_def)
-  then have "(q \<bullet> as) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"
-    apply(simp add: fresh_star_eqvt set_eqvt)
-    apply(subst (asm) perm1)
-    using inc fresh1 fr1
-    apply(auto simp add: fresh_star_def fresh_Pair)
-    done
-  then have "(r \<bullet> bs) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
-  then have "r \<bullet> (bs \<sharp>* f bs y c)"
-    apply(simp add: fresh_star_eqvt set_eqvt)
-    apply(subst (asm) perm2[symmetric])
-    using qq3 fresh2 fr1
-    apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)
-    done
-  then have fcb2: "bs \<sharp>* f bs y c" by (simp add: permute_bool_def)
-  have "f as x c = q \<bullet> (f as x c)"
-    apply(rule perm_supp_eq[symmetric])
-    using inc fcb1 fr1 by (auto simp add: fresh_star_def)
-  also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c" 
-    apply(rule perm1)
-    using inc fresh1 fr1 by (auto simp add: fresh_star_def)
-  also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
-  also have "\<dots> = r \<bullet> (f bs y c)"
-    apply(rule perm2[symmetric])
-    using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)
-  also have "... = f bs y c"
-    apply(rule perm_supp_eq)
-    using qq3 fr1 fcb2 by (auto simp add: fresh_star_def)
-  finally show ?thesis by simp
-qed
-
-lemma Abs_res_fcb2:
-  fixes as bs :: "atom set"
-    and x y :: "'b :: fs"
-    and c::"'c::fs"
-  assumes eq: "[as]res. x = [bs]res. y"
-  and fin: "finite as" "finite bs"
-  and fcb1: "as \<sharp>* f as x c"
-  and fresh1: "as \<sharp>* c"
-  and fresh2: "bs \<sharp>* c"
-  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"
-  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"
-  shows "f as x c = f bs y c"
-proof -
-  have "supp (as, x, c) supports (f as x c)"
-    unfolding  supports_def fresh_def[symmetric]
-    by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)
-  then have fin1: "finite (supp (f as x c))"
-    using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair)
-  have "supp (bs, y, c) supports (f bs y c)"
-    unfolding  supports_def fresh_def[symmetric]
-    by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
-  then have fin2: "finite (supp (f bs y c))"
-    using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair)
-  obtain q::"perm" where 
-    fr1: "(q \<bullet> as) \<sharp>* (x, c, f as x c, f bs y c)" and 
-    fr2: "supp q \<sharp>* ([as]res. x)" and 
-    inc: "supp q \<subseteq> as \<union> (q \<bullet> as)"
-    using at_set_avoiding3[where xs="as" and c="(x, c, f as x c, f bs y c)" and x="[as]res. x"]  
-      fin1 fin2 fin
-    by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
-  have "[q \<bullet> as]res. (q \<bullet> x) = q \<bullet> ([as]res. x)" by simp
-  also have "\<dots> = [as]res. x"
-    by (simp only: fr2 perm_supp_eq)
-  finally have "[q \<bullet> as]res. (q \<bullet> x) = [bs]res. y" using eq by simp
-  then obtain r::perm where 
-    qq1: "q \<bullet> x = r \<bullet> y" and 
-    qq2: "(q \<bullet> as \<inter> supp (q \<bullet> x)) = r \<bullet> (bs \<inter> supp y)" and 
-    qq3: "supp r \<subseteq> bs \<inter> supp y \<union> q \<bullet> as \<inter> supp (q \<bullet> x)"
-    apply(drule_tac sym)
-    apply(subst(asm) Abs_eq_res_set)
-    apply(simp only: Abs_eq_iff2 alphas)
-    apply(erule exE)
-    apply(erule conjE)+
-    apply(drule_tac x="p" in meta_spec)
-    apply(simp add: set_eqvt)
-    done
-  have "(as \<inter> supp x) \<sharp>* f (as \<inter> supp x) x c" sorry (* FCB? *)
-  then have "q \<bullet> ((as \<inter> supp x) \<sharp>* f (as \<inter> supp x) x c)"
-    by (simp add: permute_bool_def)
-  then have "(q \<bullet> (as \<inter> supp x)) \<sharp>* f (q \<bullet> (as \<inter> supp x)) (q \<bullet> x) c"
-    apply(simp add: fresh_star_eqvt set_eqvt)
-    sorry (* perm? *)
-  then have "r \<bullet> (bs \<inter> supp y) \<sharp>* f (r \<bullet> (bs \<inter> supp y)) (r \<bullet> y) c" using qq2 
-    apply (simp add: inter_eqvt)
-    sorry
-  (* rest similar reversing it other way around... *)
-  show ?thesis sorry
-qed
-
-
-
-lemma Abs_lst_fcb2:
-  fixes as bs :: "atom list"
-    and x y :: "'b :: fs"
-    and c::"'c::fs"
-  assumes eq: "[as]lst. x = [bs]lst. y"
-  and fcb1: "(set as) \<sharp>* f as x c"
-  and fresh1: "set as \<sharp>* c"
-  and fresh2: "set bs \<sharp>* c"
-  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"
-  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"
-  shows "f as x c = f bs y c"
-proof -
-  have "supp (as, x, c) supports (f as x c)"
-    unfolding  supports_def fresh_def[symmetric]
-    by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)
-  then have fin1: "finite (supp (f as x c))"
-    by (auto intro: supports_finite simp add: finite_supp)
-  have "supp (bs, y, c) supports (f bs y c)"
-    unfolding  supports_def fresh_def[symmetric]
-    by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
-  then have fin2: "finite (supp (f bs y c))"
-    by (auto intro: supports_finite simp add: finite_supp)
-  obtain q::"perm" where 
-    fr1: "(q \<bullet> (set as)) \<sharp>* (x, c, f as x c, f bs y c)" and 
-    fr2: "supp q \<sharp>* Abs_lst as x" and 
-    inc: "supp q \<subseteq> (set as) \<union> q \<bullet> (set as)"
-    using at_set_avoiding3[where xs="set as" and c="(x, c, f as x c, f bs y c)" and x="[as]lst. x"]  
-      fin1 fin2
-    by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
-  have "Abs_lst (q \<bullet> as) (q \<bullet> x) = q \<bullet> Abs_lst as x" by simp
-  also have "\<dots> = Abs_lst as x"
-    by (simp only: fr2 perm_supp_eq)
-  finally have "Abs_lst (q \<bullet> as) (q \<bullet> x) = Abs_lst bs y" using eq by simp
-  then obtain r::perm where 
-    qq1: "q \<bullet> x = r \<bullet> y" and 
-    qq2: "q \<bullet> as = r \<bullet> bs" and 
-    qq3: "supp r \<subseteq> (q \<bullet> (set as)) \<union> set bs"
-    apply(drule_tac sym)
-    apply(simp only: Abs_eq_iff2 alphas)
-    apply(erule exE)
-    apply(erule conjE)+
-    apply(drule_tac x="p" in meta_spec)
-    apply(simp add: set_eqvt)
-    apply(blast)
-    done
-  have "(set as) \<sharp>* f as x c" by (rule fcb1)
-  then have "q \<bullet> ((set as) \<sharp>* f as x c)"
-    by (simp add: permute_bool_def)
-  then have "set (q \<bullet> as) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"
-    apply(simp add: fresh_star_eqvt set_eqvt)
-    apply(subst (asm) perm1)
-    using inc fresh1 fr1
-    apply(auto simp add: fresh_star_def fresh_Pair)
-    done
-  then have "set (r \<bullet> bs) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
-  then have "r \<bullet> ((set bs) \<sharp>* f bs y c)"
-    apply(simp add: fresh_star_eqvt set_eqvt)
-    apply(subst (asm) perm2[symmetric])
-    using qq3 fresh2 fr1
-    apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)
-    done
-  then have fcb2: "(set bs) \<sharp>* f bs y c" by (simp add: permute_bool_def)
-  have "f as x c = q \<bullet> (f as x c)"
-    apply(rule perm_supp_eq[symmetric])
-    using inc fcb1 fr1 by (auto simp add: fresh_star_def)
-  also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c" 
-    apply(rule perm1)
-    using inc fresh1 fr1 by (auto simp add: fresh_star_def)
-  also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
-  also have "\<dots> = r \<bullet> (f bs y c)"
-    apply(rule perm2[symmetric])
-    using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)
-  also have "... = f bs y c"
-    apply(rule perm_supp_eq)
-    using qq3 fr1 fcb2 by (auto simp add: fresh_star_def)
-  finally show ?thesis by simp
-qed
-
-lemma Abs_lst1_fcb2:
-  fixes a b :: "atom"
-    and x y :: "'b :: fs"
-    and c::"'c :: fs"
-  assumes e: "(Abs_lst [a] x) = (Abs_lst [b] y)"
-  and fcb1: "a \<sharp> f a x c"
-  and fresh: "{a, b} \<sharp>* c"
-  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f a x c) = f (p \<bullet> a) (p \<bullet> x) c"
-  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b y c) = f (p \<bullet> b) (p \<bullet> y) c"
-  shows "f a x c = f b y c"
-using e
-apply(drule_tac Abs_lst_fcb2[where c="c" and f="\<lambda>(as::atom list) . f (hd as)"])
-apply(simp_all)
-using fcb1 fresh perm1 perm2
-apply(simp_all add: fresh_star_def)
-done
-
-lemma supp_zero_perm_zero:
-  shows "supp (p :: perm) = {} \<longleftrightarrow> p = 0"
-  by (metis supp_perm_singleton supp_zero_perm)
-
-lemma permute_atom_list_id:
-  shows "p \<bullet> l = l \<longleftrightarrow> supp p \<inter> set l = {}"
-  by (induct l) (auto simp add: supp_Nil supp_perm)
-
-lemma permute_length_eq:
-  shows "p \<bullet> xs = ys \<Longrightarrow> length xs = length ys"
-  by (auto simp add: length_eqvt[symmetric] permute_pure)
-
-lemma Abs_lst_binder_length:
-  shows "[xs]lst. T = [ys]lst. S \<Longrightarrow> length xs = length ys"
-  by (auto simp add: Abs_eq_iff alphas length_eqvt[symmetric] permute_pure)
-
-lemma Abs_lst_binder_eq:
-  shows "Abs_lst l T = Abs_lst l S \<longleftrightarrow> T = S"
-  by (rule, simp_all add: Abs_eq_iff2 alphas)
-     (metis fresh_star_zero inf_absorb1 permute_atom_list_id supp_perm_eq
-       supp_zero_perm_zero)
-
-lemma in_permute_list:
-  shows "py \<bullet> p \<bullet> xs = px \<bullet> xs \<Longrightarrow>  x \<in> set xs \<Longrightarrow> py \<bullet> p \<bullet> x = px \<bullet> x"
-  by (induct xs) auto
-
-
-
 
 nominal_primrec 
   crename :: "trm \<Rightarrow> coname \<Rightarrow> coname \<Rightarrow> trm"  ("_[_\<turnstile>c>_]" [100,100,100] 100) 
diff -r fac8895b109a -r 70bbd18ad194 Nominal/Ex/Lambda.thy
--- a/Nominal/Ex/Lambda.thy	Tue Jul 05 10:13:34 2011 +0900
+++ b/Nominal/Ex/Lambda.thy	Tue Jul 05 04:23:33 2011 +0200
@@ -2,117 +2,6 @@
 imports "../Nominal2" 
 begin
 
-lemma Abs_lst_fcb2:
-  fixes as bs :: "atom list"
-    and x y :: "'b :: fs"
-    and c::"'c::fs"
-  assumes eq: "[as]lst. x = [bs]lst. y"
-  and fcb1: "(set as) \<sharp>* f as x c"
-  and fresh1: "set as \<sharp>* c"
-  and fresh2: "set bs \<sharp>* c"
-  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"
-  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"
-  shows "f as x c = f bs y c"
-proof -
-  have "supp (as, x, c) supports (f as x c)"
-    unfolding  supports_def fresh_def[symmetric]
-    by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)
-  then have fin1: "finite (supp (f as x c))"
-    by (auto intro: supports_finite simp add: finite_supp)
-  have "supp (bs, y, c) supports (f bs y c)"
-    unfolding  supports_def fresh_def[symmetric]
-    by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
-  then have fin2: "finite (supp (f bs y c))"
-    by (auto intro: supports_finite simp add: finite_supp)
-  obtain q::"perm" where 
-    fr1: "(q \<bullet> (set as)) \<sharp>* (x, c, f as x c, f bs y c)" and 
-    fr2: "supp q \<sharp>* Abs_lst as x" and 
-    inc: "supp q \<subseteq> (set as) \<union> q \<bullet> (set as)"
-    using at_set_avoiding3[where xs="set as" and c="(x, c, f as x c, f bs y c)" and x="[as]lst. x"]  
-      fin1 fin2
-    by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
-  have "Abs_lst (q \<bullet> as) (q \<bullet> x) = q \<bullet> Abs_lst as x" by simp
-  also have "\<dots> = Abs_lst as x"
-    by (simp only: fr2 perm_supp_eq)
-  finally have "Abs_lst (q \<bullet> as) (q \<bullet> x) = Abs_lst bs y" using eq by simp
-  then obtain r::perm where 
-    qq1: "q \<bullet> x = r \<bullet> y" and 
-    qq2: "q \<bullet> as = r \<bullet> bs" and 
-    qq3: "supp r \<subseteq> (q \<bullet> (set as)) \<union> set bs"
-    apply(drule_tac sym)
-    apply(simp only: Abs_eq_iff2 alphas)
-    apply(erule exE)
-    apply(erule conjE)+
-    apply(drule_tac x="p" in meta_spec)
-    apply(simp add: set_eqvt)
-    apply(blast)
-    done
-  have "(set as) \<sharp>* f as x c" by (rule fcb1)
-  then have "q \<bullet> ((set as) \<sharp>* f as x c)"
-    by (simp add: permute_bool_def)
-  then have "set (q \<bullet> as) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"
-    apply(simp add: fresh_star_eqvt set_eqvt)
-    apply(subst (asm) perm1)
-    using inc fresh1 fr1
-    apply(auto simp add: fresh_star_def fresh_Pair)
-    done
-  then have "set (r \<bullet> bs) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
-  then have "r \<bullet> ((set bs) \<sharp>* f bs y c)"
-    apply(simp add: fresh_star_eqvt set_eqvt)
-    apply(subst (asm) perm2[symmetric])
-    using qq3 fresh2 fr1
-    apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)
-    done
-  then have fcb2: "(set bs) \<sharp>* f bs y c" by (simp add: permute_bool_def)
-  have "f as x c = q \<bullet> (f as x c)"
-    apply(rule perm_supp_eq[symmetric])
-    using inc fcb1 fr1 by (auto simp add: fresh_star_def)
-  also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c" 
-    apply(rule perm1)
-    using inc fresh1 fr1 by (auto simp add: fresh_star_def)
-  also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
-  also have "\<dots> = r \<bullet> (f bs y c)"
-    apply(rule perm2[symmetric])
-    using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)
-  also have "... = f bs y c"
-    apply(rule perm_supp_eq)
-    using qq3 fr1 fcb2 by (auto simp add: fresh_star_def)
-  finally show ?thesis by simp
-qed
-
-lemma Abs_lst1_fcb2:
-  fixes a b :: "atom"
-    and x y :: "'b :: fs"
-    and c::"'c :: fs"
-  assumes e: "(Abs_lst [a] x) = (Abs_lst [b] y)"
-  and fcb1: "a \<sharp> f a x c"
-  and fresh: "{a, b} \<sharp>* c"
-  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f a x c) = f (p \<bullet> a) (p \<bullet> x) c"
-  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b y c) = f (p \<bullet> b) (p \<bullet> y) c"
-  shows "f a x c = f b y c"
-using e
-apply(drule_tac Abs_lst_fcb2[where c="c" and f="\<lambda>(as::atom list) . f (hd as)"])
-apply(simp_all)
-using fcb1 fresh perm1 perm2
-apply(simp_all add: fresh_star_def)
-done
-
-lemma Abs_lst1_fcb2':
-  fixes a b :: "'a::at"
-    and x y :: "'b :: fs"
-    and c::"'c :: fs"
-  assumes e: "(Abs_lst [atom a] x) = (Abs_lst [atom b] y)"
-  and fcb1: "atom a \<sharp> f a x c"
-  and fresh: "{atom a, atom b} \<sharp>* c"
-  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f a x c) = f (p \<bullet> a) (p \<bullet> x) c"
-  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b y c) = f (p \<bullet> b) (p \<bullet> y) c"
-  shows "f a x c = f b y c"
-using e
-apply(drule_tac Abs_lst1_fcb2[where c="c" and f="\<lambda>a . f ((inv atom) a)"])
-using  fcb1 fresh perm1 perm2
-apply(simp_all add: fresh_star_def inv_f_f inj_on_def atom_eqvt)
-done
-
 
 atom_decl name
 
diff -r fac8895b109a -r 70bbd18ad194 Nominal/Ex/Let.thy
--- a/Nominal/Ex/Let.thy	Tue Jul 05 10:13:34 2011 +0900
+++ b/Nominal/Ex/Let.thy	Tue Jul 05 04:23:33 2011 +0200
@@ -77,104 +77,6 @@
 apply(simp add: atom_eqvt)
 done
 
-lemma Abs_lst_fcb2:
-  fixes as bs :: "atom list"
-    and x y :: "'b :: fs"
-    and c::"'c::fs"
-  assumes eq: "[as]lst. x = [bs]lst. y"
-  and fcb1: "(set as) \<sharp>* c \<Longrightarrow> (set as) \<sharp>* f as x c"
-  and fresh1: "set as \<sharp>* c"
-  and fresh2: "set bs \<sharp>* c"
-  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"
-  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"
-  shows "f as x c = f bs y c"
-proof -
-  have "supp (as, x, c) supports (f as x c)"
-    unfolding  supports_def fresh_def[symmetric]
-    by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)
-  then have fin1: "finite (supp (f as x c))"
-    by (auto intro: supports_finite simp add: finite_supp)
-  have "supp (bs, y, c) supports (f bs y c)"
-    unfolding  supports_def fresh_def[symmetric]
-    by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
-  then have fin2: "finite (supp (f bs y c))"
-    by (auto intro: supports_finite simp add: finite_supp)
-  obtain q::"perm" where 
-    fr1: "(q \<bullet> (set as)) \<sharp>* (x, c, f as x c, f bs y c)" and 
-    fr2: "supp q \<sharp>* Abs_lst as x" and 
-    inc: "supp q \<subseteq> (set as) \<union> q \<bullet> (set as)"
-    using at_set_avoiding3[where xs="set as" and c="(x, c, f as x c, f bs y c)" and x="[as]lst. x"]  
-      fin1 fin2
-    by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
-  have "Abs_lst (q \<bullet> as) (q \<bullet> x) = q \<bullet> Abs_lst as x" by simp
-  also have "\<dots> = Abs_lst as x"
-    by (simp only: fr2 perm_supp_eq)
-  finally have "Abs_lst (q \<bullet> as) (q \<bullet> x) = Abs_lst bs y" using eq by simp
-  then obtain r::perm where 
-    qq1: "q \<bullet> x = r \<bullet> y" and 
-    qq2: "q \<bullet> as = r \<bullet> bs" and 
-    qq3: "supp r \<subseteq> (q \<bullet> (set as)) \<union> set bs"
-    apply(drule_tac sym)
-    apply(simp only: Abs_eq_iff2 alphas)
-    apply(erule exE)
-    apply(erule conjE)+
-    apply(drule_tac x="p" in meta_spec)
-    apply(simp add: set_eqvt)
-    apply(blast)
-    done
-  have "(set as) \<sharp>* f as x c" 
-    apply(rule fcb1)
-    apply(rule fresh1)
-    done
-  then have "q \<bullet> ((set as) \<sharp>* f as x c)"
-    by (simp add: permute_bool_def)
-  then have "set (q \<bullet> as) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"
-    apply(simp add: fresh_star_eqvt set_eqvt)
-    apply(subst (asm) perm1)
-    using inc fresh1 fr1
-    apply(auto simp add: fresh_star_def fresh_Pair)
-    done
-  then have "set (r \<bullet> bs) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
-  then have "r \<bullet> ((set bs) \<sharp>* f bs y c)"
-    apply(simp add: fresh_star_eqvt set_eqvt)
-    apply(subst (asm) perm2[symmetric])
-    using qq3 fresh2 fr1
-    apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)
-    done
-  then have fcb2: "(set bs) \<sharp>* f bs y c" by (simp add: permute_bool_def)
-  have "f as x c = q \<bullet> (f as x c)"
-    apply(rule perm_supp_eq[symmetric])
-    using inc fcb1[OF fresh1] fr1 by (auto simp add: fresh_star_def)
-  also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c" 
-    apply(rule perm1)
-    using inc fresh1 fr1 by (auto simp add: fresh_star_def)
-  also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
-  also have "\<dots> = r \<bullet> (f bs y c)"
-    apply(rule perm2[symmetric])
-    using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)
-  also have "... = f bs y c"
-    apply(rule perm_supp_eq)
-    using qq3 fr1 fcb2 by (auto simp add: fresh_star_def)
-  finally show ?thesis by simp
-qed
-
-lemma Abs_lst1_fcb2:
-  fixes a b :: "atom"
-    and x y :: "'b :: fs"
-    and c::"'c :: fs"
-  assumes e: "(Abs_lst [a] x) = (Abs_lst [b] y)"
-  and fcb1: "a \<sharp> c \<Longrightarrow> a \<sharp> f a x c"
-  and fresh: "{a, b} \<sharp>* c"
-  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f a x c) = f (p \<bullet> a) (p \<bullet> x) c"
-  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b y c) = f (p \<bullet> b) (p \<bullet> y) c"
-  shows "f a x c = f b y c"
-using e
-apply(drule_tac Abs_lst_fcb2[where c="c" and f="\<lambda>(as::atom list) . f (hd as)"])
-apply(simp_all)
-using fcb1 fresh perm1 perm2
-apply(simp_all add: fresh_star_def)
-done
-
 
 lemma max_eqvt[eqvt]: "p \<bullet> (max (a :: _ :: pure) b) = max (p \<bullet> a) (p \<bullet> b)"
   by (simp add: permute_pure)
diff -r fac8895b109a -r 70bbd18ad194 Nominal/Ex/LetSimple1.thy
--- a/Nominal/Ex/LetSimple1.thy	Tue Jul 05 10:13:34 2011 +0900
+++ b/Nominal/Ex/LetSimple1.thy	Tue Jul 05 04:23:33 2011 +0200
@@ -2,102 +2,6 @@
 imports "../Nominal2" 
 begin
 
-lemma Abs_lst_fcb2:
-  fixes as bs :: "atom list"
-    and x y :: "'b :: fs"
-    and c::"'c::fs"
-  assumes eq: "[as]lst. x = [bs]lst. y"
-  and fcb1: "(set as) \<sharp>* f as x c"
-  and fresh1: "set as \<sharp>* c"
-  and fresh2: "set bs \<sharp>* c"
-  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"
-  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"
-  shows "f as x c = f bs y c"
-proof -
-  have "supp (as, x, c) supports (f as x c)"
-    unfolding  supports_def fresh_def[symmetric]
-    by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)
-  then have fin1: "finite (supp (f as x c))"
-    by (auto intro: supports_finite simp add: finite_supp)
-  have "supp (bs, y, c) supports (f bs y c)"
-    unfolding  supports_def fresh_def[symmetric]
-    by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
-  then have fin2: "finite (supp (f bs y c))"
-    by (auto intro: supports_finite simp add: finite_supp)
-  obtain q::"perm" where 
-    fr1: "(q \<bullet> (set as)) \<sharp>* (x, c, f as x c, f bs y c)" and 
-    fr2: "supp q \<sharp>* Abs_lst as x" and 
-    inc: "supp q \<subseteq> (set as) \<union> q \<bullet> (set as)"
-    using at_set_avoiding3[where xs="set as" and c="(x, c, f as x c, f bs y c)" and x="[as]lst. x"]  
-      fin1 fin2
-    by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
-  have "Abs_lst (q \<bullet> as) (q \<bullet> x) = q \<bullet> Abs_lst as x" by simp
-  also have "\<dots> = Abs_lst as x"
-    by (simp only: fr2 perm_supp_eq)
-  finally have "Abs_lst (q \<bullet> as) (q \<bullet> x) = Abs_lst bs y" using eq by simp
-  then obtain r::perm where 
-    qq1: "q \<bullet> x = r \<bullet> y" and 
-    qq2: "q \<bullet> as = r \<bullet> bs" and 
-    qq3: "supp r \<subseteq> (q \<bullet> (set as)) \<union> set bs"
-    apply(drule_tac sym)
-    apply(simp only: Abs_eq_iff2 alphas)
-    apply(erule exE)
-    apply(erule conjE)+
-    apply(drule_tac x="p" in meta_spec)
-    apply(simp add: set_eqvt)
-    apply(blast)
-    done
-  have "(set as) \<sharp>* f as x c" by (rule fcb1)
-  then have "q \<bullet> ((set as) \<sharp>* f as x c)"
-    by (simp add: permute_bool_def)
-  then have "set (q \<bullet> as) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"
-    apply(simp add: fresh_star_eqvt set_eqvt)
-    apply(subst (asm) perm1)
-    using inc fresh1 fr1
-    apply(auto simp add: fresh_star_def fresh_Pair)
-    done
-  then have "set (r \<bullet> bs) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
-  then have "r \<bullet> ((set bs) \<sharp>* f bs y c)"
-    apply(simp add: fresh_star_eqvt set_eqvt)
-    apply(subst (asm) perm2[symmetric])
-    using qq3 fresh2 fr1
-    apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)
-    done
-  then have fcb2: "(set bs) \<sharp>* f bs y c" by (simp add: permute_bool_def)
-  have "f as x c = q \<bullet> (f as x c)"
-    apply(rule perm_supp_eq[symmetric])
-    using inc fcb1 fr1 by (auto simp add: fresh_star_def)
-  also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c" 
-    apply(rule perm1)
-    using inc fresh1 fr1 by (auto simp add: fresh_star_def)
-  also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
-  also have "\<dots> = r \<bullet> (f bs y c)"
-    apply(rule perm2[symmetric])
-    using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)
-  also have "... = f bs y c"
-    apply(rule perm_supp_eq)
-    using qq3 fr1 fcb2 by (auto simp add: fresh_star_def)
-  finally show ?thesis by simp
-qed
-
-lemma Abs_lst1_fcb2:
-  fixes a b :: "atom"
-    and x y :: "'b :: fs"
-    and c::"'c :: fs"
-  assumes e: "(Abs_lst [a] x) = (Abs_lst [b] y)"
-  and fcb1: "a \<sharp> f a x c"
-  and fresh: "{a, b} \<sharp>* c"
-  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f a x c) = f (p \<bullet> a) (p \<bullet> x) c"
-  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b y c) = f (p \<bullet> b) (p \<bullet> y) c"
-  shows "f a x c = f b y c"
-using e
-apply(drule_tac Abs_lst_fcb2[where c="c" and f="\<lambda>(as::atom list) . f (hd as)"])
-apply(simp_all)
-using fcb1 fresh perm1 perm2
-apply(simp_all add: fresh_star_def)
-done
-
-
 atom_decl name
 
 nominal_datatype trm =
diff -r fac8895b109a -r 70bbd18ad194 Nominal/Ex/LetSimple2.thy
--- a/Nominal/Ex/LetSimple2.thy	Tue Jul 05 10:13:34 2011 +0900
+++ b/Nominal/Ex/LetSimple2.thy	Tue Jul 05 04:23:33 2011 +0200
@@ -2,105 +2,6 @@
 imports "../Nominal2" 
 begin
 
-
-lemma Abs_lst_fcb2:
-  fixes as bs :: "atom list"
-    and x y :: "'b :: fs"
-    and c::"'c::fs"
-  assumes eq: "[as]lst. x = [bs]lst. y"
-  and fcb1: "(set as) \<sharp>* c \<Longrightarrow> (set as) \<sharp>* f as x c"
-  and fresh1: "set as \<sharp>* c"
-  and fresh2: "set bs \<sharp>* c"
-  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"
-  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"
-  shows "f as x c = f bs y c"
-proof -
-  have "supp (as, x, c) supports (f as x c)"
-    unfolding  supports_def fresh_def[symmetric]
-    by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)
-  then have fin1: "finite (supp (f as x c))"
-    by (auto intro: supports_finite simp add: finite_supp)
-  have "supp (bs, y, c) supports (f bs y c)"
-    unfolding  supports_def fresh_def[symmetric]
-    by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
-  then have fin2: "finite (supp (f bs y c))"
-    by (auto intro: supports_finite simp add: finite_supp)
-  obtain q::"perm" where 
-    fr1: "(q \<bullet> (set as)) \<sharp>* (x, c, f as x c, f bs y c)" and 
-    fr2: "supp q \<sharp>* Abs_lst as x" and 
-    inc: "supp q \<subseteq> (set as) \<union> q \<bullet> (set as)"
-    using at_set_avoiding3[where xs="set as" and c="(x, c, f as x c, f bs y c)" and x="[as]lst. x"]  
-      fin1 fin2
-    by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
-  have "Abs_lst (q \<bullet> as) (q \<bullet> x) = q \<bullet> Abs_lst as x" by simp
-  also have "\<dots> = Abs_lst as x"
-    by (simp only: fr2 perm_supp_eq)
-  finally have "Abs_lst (q \<bullet> as) (q \<bullet> x) = Abs_lst bs y" using eq by simp
-  then obtain r::perm where 
-    qq1: "q \<bullet> x = r \<bullet> y" and 
-    qq2: "q \<bullet> as = r \<bullet> bs" and 
-    qq3: "supp r \<subseteq> (q \<bullet> (set as)) \<union> set bs"
-    apply(drule_tac sym)
-    apply(simp only: Abs_eq_iff2 alphas)
-    apply(erule exE)
-    apply(erule conjE)+
-    apply(drule_tac x="p" in meta_spec)
-    apply(simp add: set_eqvt)
-    apply(blast)
-    done
-  have "(set as) \<sharp>* f as x c" 
-    apply(rule fcb1)
-    apply(rule fresh1)
-    done
-  then have "q \<bullet> ((set as) \<sharp>* f as x c)"
-    by (simp add: permute_bool_def)
-  then have "set (q \<bullet> as) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"
-    apply(simp add: fresh_star_eqvt set_eqvt)
-    apply(subst (asm) perm1)
-    using inc fresh1 fr1
-    apply(auto simp add: fresh_star_def fresh_Pair)
-    done
-  then have "set (r \<bullet> bs) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
-  then have "r \<bullet> ((set bs) \<sharp>* f bs y c)"
-    apply(simp add: fresh_star_eqvt set_eqvt)
-    apply(subst (asm) perm2[symmetric])
-    using qq3 fresh2 fr1
-    apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)
-    done
-  then have fcb2: "(set bs) \<sharp>* f bs y c" by (simp add: permute_bool_def)
-  have "f as x c = q \<bullet> (f as x c)"
-    apply(rule perm_supp_eq[symmetric])
-    using inc fcb1[OF fresh1] fr1 by (auto simp add: fresh_star_def)
-  also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c" 
-    apply(rule perm1)
-    using inc fresh1 fr1 by (auto simp add: fresh_star_def)
-  also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
-  also have "\<dots> = r \<bullet> (f bs y c)"
-    apply(rule perm2[symmetric])
-    using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)
-  also have "... = f bs y c"
-    apply(rule perm_supp_eq)
-    using qq3 fr1 fcb2 by (auto simp add: fresh_star_def)
-  finally show ?thesis by simp
-qed
-
-lemma Abs_lst1_fcb2:
-  fixes a b :: "atom"
-    and x y :: "'b :: fs"
-    and c::"'c :: fs"
-  assumes e: "(Abs_lst [a] x) = (Abs_lst [b] y)"
-  and fcb1: "a \<sharp> c \<Longrightarrow> a \<sharp> f a x c"
-  and fresh: "{a, b} \<sharp>* c"
-  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f a x c) = f (p \<bullet> a) (p \<bullet> x) c"
-  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b y c) = f (p \<bullet> b) (p \<bullet> y) c"
-  shows "f a x c = f b y c"
-using e
-apply(drule_tac Abs_lst_fcb2[where c="c" and f="\<lambda>(as::atom list) . f (hd as)"])
-apply(simp_all)
-using fcb1 fresh perm1 perm2
-apply(simp_all add: fresh_star_def)
-done
-
 atom_decl name
 
 nominal_datatype trm =
@@ -116,8 +17,6 @@
 
 print_theorems
 
-
-
 thm bn_raw.simps
 thm permute_bn_raw.simps
 thm trm_assn.perm_bn_alpha
diff -r fac8895b109a -r 70bbd18ad194 Nominal/Nominal2.thy
--- a/Nominal/Nominal2.thy	Tue Jul 05 10:13:34 2011 +0900
+++ b/Nominal/Nominal2.thy	Tue Jul 05 04:23:33 2011 +0200
@@ -1,6 +1,6 @@
 theory Nominal2
 imports 
-  Nominal2_Base Nominal2_Abs
+  Nominal2_Base Nominal2_Abs Nominal2_FCB
 uses ("nominal_dt_rawfuns.ML")
      ("nominal_dt_alpha.ML")
      ("nominal_dt_quot.ML")
diff -r fac8895b109a -r 70bbd18ad194 Nominal/Nominal2_Abs.thy
--- a/Nominal/Nominal2_Abs.thy	Tue Jul 05 10:13:34 2011 +0900
+++ b/Nominal/Nominal2_Abs.thy	Tue Jul 05 04:23:33 2011 +0200
@@ -1013,112 +1013,6 @@
   unfolding prod_alpha_def
   by (auto intro!: ext)
 
-lemma Abs_lst1_fcb:
-  fixes x y :: "'a :: at_base"
-    and S T :: "'b :: fs"
-  assumes e: "(Abs_lst [atom x] T) = (Abs_lst [atom y] S)"
-  and f1: "x \<noteq> y \<Longrightarrow> atom y \<sharp> T \<Longrightarrow> atom x \<sharp> (atom y \<rightleftharpoons> atom x) \<bullet> T \<Longrightarrow> atom x \<sharp> f x T"
-  and f2: "x \<noteq> y \<Longrightarrow> atom y \<sharp> T \<Longrightarrow> atom x \<sharp> (atom y \<rightleftharpoons> atom x) \<bullet> T \<Longrightarrow> atom y \<sharp> f x T"
-  and p: "S = (atom x \<rightleftharpoons> atom y) \<bullet> T \<Longrightarrow> x \<noteq> y \<Longrightarrow> atom y \<sharp> T \<Longrightarrow> atom x \<sharp> S \<Longrightarrow> (atom x \<rightleftharpoons> atom y) \<bullet> (f x T) = f y S"
-  and s: "sort_of (atom x) = sort_of (atom y)"
-  shows "f x T = f y S"
-  using e
-  apply(case_tac "atom x \<sharp> S")
-  apply(simp add: Abs1_eq_iff'[OF s s])
-  apply(elim conjE disjE)
-  apply(simp)
-  apply(rule trans)
-  apply(rule_tac p="(atom x \<rightleftharpoons> atom y)" in supp_perm_eq[symmetric])
-  apply(rule fresh_star_supp_conv)
-  apply(simp add: supp_swap fresh_star_def s f1 f2)
-  apply(simp add: swap_commute p)
-  apply(simp add: Abs1_eq_iff[OF s s])
-  done
-
-lemma Abs_lst_fcb:
-  fixes xs ys :: "'a :: fs"
-    and S T :: "'b :: fs"
-  assumes e: "(Abs_lst (ba xs) T) = (Abs_lst (ba ys) S)"
-    and f1: "\<And>x. x \<in> set (ba xs) \<Longrightarrow> x \<sharp> f xs T"
-    and f2: "\<And>x. supp T - set (ba xs) = supp S - set (ba ys) \<Longrightarrow> x \<in> set (ba ys) \<Longrightarrow> x \<sharp> f xs T"
-    and eqv: "\<And>p. p \<bullet> T = S \<Longrightarrow> p \<bullet> ba xs = ba ys \<Longrightarrow> supp p \<subseteq> set (ba xs) \<union> set (ba ys) \<Longrightarrow> p \<bullet> (f xs T) = f ys S"
-  shows "f xs T = f ys S"
-  using e apply -
-  apply(subst (asm) Abs_eq_iff2)
-  apply(simp add: alphas)
-  apply(elim exE conjE)
-  apply(rule trans)
-  apply(rule_tac p="p" in supp_perm_eq[symmetric])
-  apply(rule fresh_star_supp_conv)
-  apply(drule fresh_star_perm_set_conv)
-  apply(rule finite_Diff)
-  apply(rule finite_supp)
-  apply(subgoal_tac "(set (ba xs) \<union> set (ba ys)) \<sharp>* f xs T")
-  apply(metis Un_absorb2 fresh_star_Un)
-  apply(subst fresh_star_Un)
-  apply(rule conjI)
-  apply(simp add: fresh_star_def f1)
-  apply(simp add: fresh_star_def f2)
-  apply(simp add: eqv)
-  done
-
-lemma Abs_set_fcb:
-  fixes xs ys :: "'a :: fs"
-    and S T :: "'b :: fs"
-  assumes e: "(Abs_set (ba xs) T) = (Abs_set (ba ys) S)"
-    and f1: "\<And>x. x \<in> ba xs \<Longrightarrow> x \<sharp> f xs T"
-    and f2: "\<And>x. supp T - ba xs = supp S - ba ys \<Longrightarrow> x \<in> ba ys \<Longrightarrow> x \<sharp> f xs T"
-    and eqv: "\<And>p. p \<bullet> T = S \<Longrightarrow> p \<bullet> ba xs = ba ys \<Longrightarrow> supp p \<subseteq> ba xs \<union> ba ys \<Longrightarrow> p \<bullet> (f xs T) = f ys S"
-  shows "f xs T = f ys S"
-  using e apply -
-  apply(subst (asm) Abs_eq_iff2)
-  apply(simp add: alphas)
-  apply(elim exE conjE)
-  apply(rule trans)
-  apply(rule_tac p="p" in supp_perm_eq[symmetric])
-  apply(rule fresh_star_supp_conv)
-  apply(drule fresh_star_perm_set_conv)
-  apply(rule finite_Diff)
-  apply(rule finite_supp)
-  apply(subgoal_tac "(ba xs \<union> ba ys) \<sharp>* f xs T")
-  apply(metis Un_absorb2 fresh_star_Un)
-  apply(subst fresh_star_Un)
-  apply(rule conjI)
-  apply(simp add: fresh_star_def f1)
-  apply(simp add: fresh_star_def f2)
-  apply(simp add: eqv)
-  done
-
-lemma Abs_res_fcb:
-  fixes xs ys :: "('a :: at_base) set"
-    and S T :: "'b :: fs"
-  assumes e: "(Abs_res (atom ` xs) T) = (Abs_res (atom ` ys) S)"
-    and f1: "\<And>x. x \<in> atom ` xs \<Longrightarrow> x \<in> supp T \<Longrightarrow> x \<sharp> f xs T"
-    and f2: "\<And>x. supp T - atom ` xs = supp S - atom ` ys \<Longrightarrow> x \<in> atom ` ys \<Longrightarrow> x \<in> supp S \<Longrightarrow> x \<sharp> f xs T"
-    and eqv: "\<And>p. p \<bullet> T = S \<Longrightarrow> supp p \<subseteq> atom ` xs \<inter> supp T \<union> atom ` ys \<inter> supp S
-               \<Longrightarrow> p \<bullet> (atom ` xs \<inter> supp T) = atom ` ys \<inter> supp S \<Longrightarrow> p \<bullet> (f xs T) = f ys S"
-  shows "f xs T = f ys S"
-  using e apply -
-  apply(subst (asm) Abs_eq_res_set)
-  apply(subst (asm) Abs_eq_iff2)
-  apply(simp add: alphas)
-  apply(elim exE conjE)
-  apply(rule trans)
-  apply(rule_tac p="p" in supp_perm_eq[symmetric])
-  apply(rule fresh_star_supp_conv)
-  apply(drule fresh_star_perm_set_conv)
-  apply(rule finite_Diff)
-  apply(rule finite_supp)
-  apply(subgoal_tac "(atom ` xs \<inter> supp T \<union> atom ` ys \<inter> supp S) \<sharp>* f xs T")
-  apply(metis Un_absorb2 fresh_star_Un)
-  apply(subst fresh_star_Un)
-  apply(rule conjI)
-  apply(simp add: fresh_star_def f1)
-  apply(subgoal_tac "supp T - atom ` xs = supp S - atom ` ys")
-  apply(simp add: fresh_star_def f2)
-  apply(blast)
-  apply(simp add: eqv)
-  done
 
 end
 
diff -r fac8895b109a -r 70bbd18ad194 Nominal/Nominal2_FCB.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/Nominal2_FCB.thy	Tue Jul 05 04:23:33 2011 +0200
@@ -0,0 +1,370 @@
+theory Nominal2_FCB
+imports "Nominal2_Abs" 
+begin
+
+
+lemma Abs_lst1_fcb:
+  fixes x y :: "'a :: at_base"
+    and S T :: "'b :: fs"
+  assumes e: "(Abs_lst [atom x] T) = (Abs_lst [atom y] S)"
+  and f1: "\<lbrakk>x \<noteq> y; atom y \<sharp> T; atom x \<sharp> (atom y \<rightleftharpoons> atom x) \<bullet> T\<rbrakk> \<Longrightarrow> atom x \<sharp> f x T"
+  and f2: "\<lbrakk>x \<noteq> y; atom y \<sharp> T; atom x \<sharp> (atom y \<rightleftharpoons> atom x) \<bullet> T\<rbrakk> \<Longrightarrow> atom y \<sharp> f x T"
+  and p: "\<lbrakk>S = (atom x \<rightleftharpoons> atom y) \<bullet> T; x \<noteq> y; atom y \<sharp> T; atom x \<sharp> S\<rbrakk> 
+    \<Longrightarrow> (atom x \<rightleftharpoons> atom y) \<bullet> (f x T) = f y S"
+  and s: "sort_of (atom x) = sort_of (atom y)"
+  shows "f x T = f y S"
+  using e
+  apply(case_tac "atom x \<sharp> S")
+  apply(simp add: Abs1_eq_iff'[OF s s])
+  apply(elim conjE disjE)
+  apply(simp)
+  apply(rule trans)
+  apply(rule_tac p="(atom x \<rightleftharpoons> atom y)" in supp_perm_eq[symmetric])
+  apply(rule fresh_star_supp_conv)
+  apply(simp add: supp_swap fresh_star_def s f1 f2)
+  apply(simp add: swap_commute p)
+  apply(simp add: Abs1_eq_iff[OF s s])
+  done
+
+lemma Abs_lst_fcb:
+  fixes xs ys :: "'a :: fs"
+    and S T :: "'b :: fs"
+  assumes e: "(Abs_lst (ba xs) T) = (Abs_lst (ba ys) S)"
+    and f1: "\<And>x. x \<in> set (ba xs) \<Longrightarrow> x \<sharp> f xs T"
+    and f2: "\<And>x. \<lbrakk>supp T - set (ba xs) = supp S - set (ba ys); x \<in> set (ba ys)\<rbrakk> \<Longrightarrow> x \<sharp> f xs T"
+    and eqv: "\<And>p. \<lbrakk>p \<bullet> T = S; p \<bullet> ba xs = ba ys; supp p \<subseteq> set (ba xs) \<union> set (ba ys)\<rbrakk> 
+      \<Longrightarrow> p \<bullet> (f xs T) = f ys S"
+  shows "f xs T = f ys S"
+  using e apply -
+  apply(subst (asm) Abs_eq_iff2)
+  apply(simp add: alphas)
+  apply(elim exE conjE)
+  apply(rule trans)
+  apply(rule_tac p="p" in supp_perm_eq[symmetric])
+  apply(rule fresh_star_supp_conv)
+  apply(drule fresh_star_perm_set_conv)
+  apply(rule finite_Diff)
+  apply(rule finite_supp)
+  apply(subgoal_tac "(set (ba xs) \<union> set (ba ys)) \<sharp>* f xs T")
+  apply(metis Un_absorb2 fresh_star_Un)
+  apply(subst fresh_star_Un)
+  apply(rule conjI)
+  apply(simp add: fresh_star_def f1)
+  apply(simp add: fresh_star_def f2)
+  apply(simp add: eqv)
+  done
+
+lemma Abs_set_fcb:
+  fixes xs ys :: "'a :: fs"
+    and S T :: "'b :: fs"
+  assumes e: "(Abs_set (ba xs) T) = (Abs_set (ba ys) S)"
+    and f1: "\<And>x. x \<in> ba xs \<Longrightarrow> x \<sharp> f xs T"
+    and f2: "\<And>x. \<lbrakk>supp T - ba xs = supp S - ba ys; x \<in> ba ys\<rbrakk> \<Longrightarrow> x \<sharp> f xs T"
+    and eqv: "\<And>p. \<lbrakk>p \<bullet> T = S; p \<bullet> ba xs = ba ys; supp p \<subseteq> ba xs \<union> ba ys\<rbrakk> \<Longrightarrow> p \<bullet> (f xs T) = f ys S"
+  shows "f xs T = f ys S"
+  using e apply -
+  apply(subst (asm) Abs_eq_iff2)
+  apply(simp add: alphas)
+  apply(elim exE conjE)
+  apply(rule trans)
+  apply(rule_tac p="p" in supp_perm_eq[symmetric])
+  apply(rule fresh_star_supp_conv)
+  apply(drule fresh_star_perm_set_conv)
+  apply(rule finite_Diff)
+  apply(rule finite_supp)
+  apply(subgoal_tac "(ba xs \<union> ba ys) \<sharp>* f xs T")
+  apply(metis Un_absorb2 fresh_star_Un)
+  apply(subst fresh_star_Un)
+  apply(rule conjI)
+  apply(simp add: fresh_star_def f1)
+  apply(simp add: fresh_star_def f2)
+  apply(simp add: eqv)
+  done
+
+lemma Abs_res_fcb:
+  fixes xs ys :: "('a :: at_base) set"
+    and S T :: "'b :: fs"
+  assumes e: "(Abs_res (atom ` xs) T) = (Abs_res (atom ` ys) S)"
+    and f1: "\<And>x. x \<in> atom ` xs \<Longrightarrow> x \<in> supp T \<Longrightarrow> x \<sharp> f xs T"
+    and f2: "\<And>x. \<lbrakk>supp T - atom ` xs = supp S - atom ` ys; x \<in> atom ` ys; x \<in> supp S\<rbrakk> \<Longrightarrow> x \<sharp> f xs T"
+    and eqv: "\<And>p. \<lbrakk>p \<bullet> T = S; supp p \<subseteq> atom ` xs \<inter> supp T \<union> atom ` ys \<inter> supp S;
+      p \<bullet> (atom ` xs \<inter> supp T) = atom ` ys \<inter> supp S\<rbrakk> \<Longrightarrow> p \<bullet> (f xs T) = f ys S"
+  shows "f xs T = f ys S"
+  using e apply -
+  apply(subst (asm) Abs_eq_res_set)
+  apply(subst (asm) Abs_eq_iff2)
+  apply(simp add: alphas)
+  apply(elim exE conjE)
+  apply(rule trans)
+  apply(rule_tac p="p" in supp_perm_eq[symmetric])
+  apply(rule fresh_star_supp_conv)
+  apply(drule fresh_star_perm_set_conv)
+  apply(rule finite_Diff)
+  apply(rule finite_supp)
+  apply(subgoal_tac "(atom ` xs \<inter> supp T \<union> atom ` ys \<inter> supp S) \<sharp>* f xs T")
+  apply(metis Un_absorb2 fresh_star_Un)
+  apply(subst fresh_star_Un)
+  apply(rule conjI)
+  apply(simp add: fresh_star_def f1)
+  apply(subgoal_tac "supp T - atom ` xs = supp S - atom ` ys")
+  apply(simp add: fresh_star_def f2)
+  apply(blast)
+  apply(simp add: eqv)
+  done
+
+
+
+lemma Abs_set_fcb2:
+  fixes as bs :: "atom set"
+    and x y :: "'b :: fs"
+    and c::"'c::fs"
+  assumes eq: "[as]set. x = [bs]set. y"
+  and fin: "finite as" "finite bs"
+  and fcb1: "as \<sharp>* f as x c"
+  and fresh1: "as \<sharp>* c"
+  and fresh2: "bs \<sharp>* c"
+  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"
+  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"
+  shows "f as x c = f bs y c"
+proof -
+  have "supp (as, x, c) supports (f as x c)"
+    unfolding  supports_def fresh_def[symmetric]
+    by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)
+  then have fin1: "finite (supp (f as x c))"
+    using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair)
+  have "supp (bs, y, c) supports (f bs y c)"
+    unfolding  supports_def fresh_def[symmetric]
+    by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
+  then have fin2: "finite (supp (f bs y c))"
+    using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair)
+  obtain q::"perm" where 
+    fr1: "(q \<bullet> as) \<sharp>* (x, c, f as x c, f bs y c)" and 
+    fr2: "supp q \<sharp>* ([as]set. x)" and 
+    inc: "supp q \<subseteq> as \<union> (q \<bullet> as)"
+    using at_set_avoiding3[where xs="as" and c="(x, c, f as x c, f bs y c)" and x="[as]set. x"]  
+      fin1 fin2 fin
+    by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
+  have "[q \<bullet> as]set. (q \<bullet> x) = q \<bullet> ([as]set. x)" by simp
+  also have "\<dots> = [as]set. x"
+    by (simp only: fr2 perm_supp_eq)
+  finally have "[q \<bullet> as]set. (q \<bullet> x) = [bs]set. y" using eq by simp
+  then obtain r::perm where 
+    qq1: "q \<bullet> x = r \<bullet> y" and 
+    qq2: "q \<bullet> as = r \<bullet> bs" and 
+    qq3: "supp r \<subseteq> (q \<bullet> as) \<union> bs"
+    apply(drule_tac sym)
+    apply(simp only: Abs_eq_iff2 alphas)
+    apply(erule exE)
+    apply(erule conjE)+
+    apply(drule_tac x="p" in meta_spec)
+    apply(simp add: set_eqvt)
+    apply(blast)
+    done
+  have "as \<sharp>* f as x c" by (rule fcb1)
+  then have "q \<bullet> (as \<sharp>* f as x c)"
+    by (simp add: permute_bool_def)
+  then have "(q \<bullet> as) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"
+    apply(simp add: fresh_star_eqvt set_eqvt)
+    apply(subst (asm) perm1)
+    using inc fresh1 fr1
+    apply(auto simp add: fresh_star_def fresh_Pair)
+    done
+  then have "(r \<bullet> bs) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
+  then have "r \<bullet> (bs \<sharp>* f bs y c)"
+    apply(simp add: fresh_star_eqvt set_eqvt)
+    apply(subst (asm) perm2[symmetric])
+    using qq3 fresh2 fr1
+    apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)
+    done
+  then have fcb2: "bs \<sharp>* f bs y c" by (simp add: permute_bool_def)
+  have "f as x c = q \<bullet> (f as x c)"
+    apply(rule perm_supp_eq[symmetric])
+    using inc fcb1 fr1 by (auto simp add: fresh_star_def)
+  also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c" 
+    apply(rule perm1)
+    using inc fresh1 fr1 by (auto simp add: fresh_star_def)
+  also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
+  also have "\<dots> = r \<bullet> (f bs y c)"
+    apply(rule perm2[symmetric])
+    using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)
+  also have "... = f bs y c"
+    apply(rule perm_supp_eq)
+    using qq3 fr1 fcb2 by (auto simp add: fresh_star_def)
+  finally show ?thesis by simp
+qed
+
+
+text {* NOT DONE 
+lemma Abs_res_fcb2:
+  fixes as bs :: "atom set"
+    and x y :: "'b :: fs"
+    and c::"'c::fs"
+  assumes eq: "[as]res. x = [bs]res. y"
+  and fin: "finite as" "finite bs"
+  and fcb1: "as \<sharp>* f as x c"
+  and fresh1: "as \<sharp>* c"
+  and fresh2: "bs \<sharp>* c"
+  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"
+  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"
+  shows "f as x c = f bs y c"
+proof -
+  have "supp (as, x, c) supports (f as x c)"
+    unfolding  supports_def fresh_def[symmetric]
+    by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)
+  then have fin1: "finite (supp (f as x c))"
+    using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair)
+  have "supp (bs, y, c) supports (f bs y c)"
+    unfolding  supports_def fresh_def[symmetric]
+    by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
+  then have fin2: "finite (supp (f bs y c))"
+    using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair)
+  obtain q::"perm" where 
+    fr1: "(q \<bullet> as) \<sharp>* (x, c, f as x c, f bs y c)" and 
+    fr2: "supp q \<sharp>* ([as]res. x)" and 
+    inc: "supp q \<subseteq> as \<union> (q \<bullet> as)"
+    using at_set_avoiding3[where xs="as" and c="(x, c, f as x c, f bs y c)" and x="[as]res. x"]  
+      fin1 fin2 fin
+    by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
+  have "[q \<bullet> as]res. (q \<bullet> x) = q \<bullet> ([as]res. x)" by simp
+  also have "\<dots> = [as]res. x"
+    by (simp only: fr2 perm_supp_eq)
+  finally have "[q \<bullet> as]res. (q \<bullet> x) = [bs]res. y" using eq by simp
+  then obtain r::perm where 
+    qq1: "q \<bullet> x = r \<bullet> y" and 
+    qq2: "(q \<bullet> as \<inter> supp (q \<bullet> x)) = r \<bullet> (bs \<inter> supp y)" and 
+    qq3: "supp r \<subseteq> bs \<inter> supp y \<union> q \<bullet> as \<inter> supp (q \<bullet> x)"
+    apply(drule_tac sym)
+    apply(subst(asm) Abs_eq_res_set)
+    apply(simp only: Abs_eq_iff2 alphas)
+    apply(erule exE)
+    apply(erule conjE)+
+    apply(drule_tac x="p" in meta_spec)
+    apply(simp add: set_eqvt)
+    done
+  have "(as \<inter> supp x) \<sharp>* f (as \<inter> supp x) x c" sorry (* FCB? *)
+  then have "q \<bullet> ((as \<inter> supp x) \<sharp>* f (as \<inter> supp x) x c)"
+    by (simp add: permute_bool_def)
+  then have "(q \<bullet> (as \<inter> supp x)) \<sharp>* f (q \<bullet> (as \<inter> supp x)) (q \<bullet> x) c"
+    apply(simp add: fresh_star_eqvt set_eqvt)
+    sorry (* perm? *)
+  then have "r \<bullet> (bs \<inter> supp y) \<sharp>* f (r \<bullet> (bs \<inter> supp y)) (r \<bullet> y) c" using qq2 
+    apply (simp add: inter_eqvt)
+    sorry
+  (* rest similar reversing it other way around... *)
+  show ?thesis sorry
+qed
+*}
+
+
+lemma Abs_lst_fcb2:
+  fixes as bs :: "atom list"
+    and x y :: "'b :: fs"
+    and c::"'c::fs"
+  assumes eq: "[as]lst. x = [bs]lst. y"
+  and fcb1: "(set as) \<sharp>* f as x c"
+  and fresh1: "set as \<sharp>* c"
+  and fresh2: "set bs \<sharp>* c"
+  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"
+  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"
+  shows "f as x c = f bs y c"
+proof -
+  have "supp (as, x, c) supports (f as x c)"
+    unfolding  supports_def fresh_def[symmetric]
+    by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)
+  then have fin1: "finite (supp (f as x c))"
+    by (auto intro: supports_finite simp add: finite_supp)
+  have "supp (bs, y, c) supports (f bs y c)"
+    unfolding  supports_def fresh_def[symmetric]
+    by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
+  then have fin2: "finite (supp (f bs y c))"
+    by (auto intro: supports_finite simp add: finite_supp)
+  obtain q::"perm" where 
+    fr1: "(q \<bullet> (set as)) \<sharp>* (x, c, f as x c, f bs y c)" and 
+    fr2: "supp q \<sharp>* Abs_lst as x" and 
+    inc: "supp q \<subseteq> (set as) \<union> q \<bullet> (set as)"
+    using at_set_avoiding3[where xs="set as" and c="(x, c, f as x c, f bs y c)" and x="[as]lst. x"]  
+      fin1 fin2
+    by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
+  have "Abs_lst (q \<bullet> as) (q \<bullet> x) = q \<bullet> Abs_lst as x" by simp
+  also have "\<dots> = Abs_lst as x"
+    by (simp only: fr2 perm_supp_eq)
+  finally have "Abs_lst (q \<bullet> as) (q \<bullet> x) = Abs_lst bs y" using eq by simp
+  then obtain r::perm where 
+    qq1: "q \<bullet> x = r \<bullet> y" and 
+    qq2: "q \<bullet> as = r \<bullet> bs" and 
+    qq3: "supp r \<subseteq> (q \<bullet> (set as)) \<union> set bs"
+    apply(drule_tac sym)
+    apply(simp only: Abs_eq_iff2 alphas)
+    apply(erule exE)
+    apply(erule conjE)+
+    apply(drule_tac x="p" in meta_spec)
+    apply(simp add: set_eqvt)
+    apply(blast)
+    done
+  have "(set as) \<sharp>* f as x c" by (rule fcb1)
+  then have "q \<bullet> ((set as) \<sharp>* f as x c)"
+    by (simp add: permute_bool_def)
+  then have "set (q \<bullet> as) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"
+    apply(simp add: fresh_star_eqvt set_eqvt)
+    apply(subst (asm) perm1)
+    using inc fresh1 fr1
+    apply(auto simp add: fresh_star_def fresh_Pair)
+    done
+  then have "set (r \<bullet> bs) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
+  then have "r \<bullet> ((set bs) \<sharp>* f bs y c)"
+    apply(simp add: fresh_star_eqvt set_eqvt)
+    apply(subst (asm) perm2[symmetric])
+    using qq3 fresh2 fr1
+    apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)
+    done
+  then have fcb2: "(set bs) \<sharp>* f bs y c" by (simp add: permute_bool_def)
+  have "f as x c = q \<bullet> (f as x c)"
+    apply(rule perm_supp_eq[symmetric])
+    using inc fcb1 fr1 by (auto simp add: fresh_star_def)
+  also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c" 
+    apply(rule perm1)
+    using inc fresh1 fr1 by (auto simp add: fresh_star_def)
+  also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
+  also have "\<dots> = r \<bullet> (f bs y c)"
+    apply(rule perm2[symmetric])
+    using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)
+  also have "... = f bs y c"
+    apply(rule perm_supp_eq)
+    using qq3 fr1 fcb2 by (auto simp add: fresh_star_def)
+  finally show ?thesis by simp
+qed
+
+lemma Abs_lst1_fcb2:
+  fixes a b :: "atom"
+    and x y :: "'b :: fs"
+    and c::"'c :: fs"
+  assumes e: "(Abs_lst [a] x) = (Abs_lst [b] y)"
+  and fcb1: "a \<sharp> f a x c"
+  and fresh: "{a, b} \<sharp>* c"
+  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f a x c) = f (p \<bullet> a) (p \<bullet> x) c"
+  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b y c) = f (p \<bullet> b) (p \<bullet> y) c"
+  shows "f a x c = f b y c"
+using e
+apply(drule_tac Abs_lst_fcb2[where c="c" and f="\<lambda>(as::atom list) . f (hd as)"])
+apply(simp_all)
+using fcb1 fresh perm1 perm2
+apply(simp_all add: fresh_star_def)
+done
+
+lemma Abs_lst1_fcb2':
+  fixes a b :: "'a::at"
+    and x y :: "'b :: fs"
+    and c::"'c :: fs"
+  assumes e: "(Abs_lst [atom a] x) = (Abs_lst [atom b] y)"
+  and fcb1: "atom a \<sharp> f a x c"
+  and fresh: "{atom a, atom b} \<sharp>* c"
+  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f a x c) = f (p \<bullet> a) (p \<bullet> x) c"
+  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b y c) = f (p \<bullet> b) (p \<bullet> y) c"
+  shows "f a x c = f b y c"
+using e
+apply(drule_tac Abs_lst1_fcb2[where c="c" and f="\<lambda>a . f ((inv atom) a)"])
+using  fcb1 fresh perm1 perm2
+apply(simp_all add: fresh_star_def inv_f_f inj_on_def atom_eqvt)
+done
+
+end
\ No newline at end of file