
diff -r dc003667cd17 -r 09834ba7ce59 Nominal/Ex/Classical.thy
--- a/Nominal/Ex/Classical.thy	Mon Jul 04 23:56:19 2011 +0200
+++ b/Nominal/Ex/Classical.thy	Tue Jul 05 04:18:45 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) 




