--- a/Nominal/Ex/CPS/CPS3_DanvyFilinski_FCB2.thy	Mon Jul 11 23:42:22 2011 +0900
+++ b/Nominal/Ex/CPS/CPS3_DanvyFilinski_FCB2.thy	Mon Jul 11 23:42:52 2011 +0900
@@ -1,105 +1,6 @@
 header {* CPS transformation of Danvy and Filinski *}
 theory CPS3_DanvyFilinski imports Lt 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
-
 nominal_primrec
   CPS1 :: "lt \<Rightarrow> (lt \<Rightarrow> lt) \<Rightarrow> lt" ("_*_"  [100,100] 100)
 and
@@ -145,9 +46,6 @@
 --"-"
   apply (rule arg_cong)
   back
-  apply simp
-  apply (erule Abs_lst1_fcb2)
-  apply simp
   apply (thin_tac "eqvt ka")
   apply (rule_tac x="(c, ca, x, xa, M, Ma)" and ?'a="name" in obtain_fresh)
   apply (subgoal_tac "Abs c (CPS1_CPS2_sumC (Inr (M, c~))) = Abs a (CPS1_CPS2_sumC (Inr (M, a~)))")
@@ -173,182 +71,12 @@
   apply (simp add: supp_Inr finite_supp)
   apply (simp add: fresh_Inr fresh_Pair lt.fresh fresh_at_base)
   apply (simp only: )
-  apply (erule Abs_lst1_fcb2)
-  apply (simp add: Abs_fresh_iff)
+  apply (erule_tac c="a" in Abs_lst1_fcb2')
+  apply (simp add: Abs_fresh_iff lt.fresh)
   apply (simp add: fresh_star_def fresh_Pair_elim lt.fresh fresh_at_base)
   apply (subgoal_tac "p \<bullet> CPS1_CPS2_sumC (Inr (M, a~)) = CPS1_CPS2_sumC (p \<bullet> (Inr (M, a~)))")
   apply (simp add: perm_supp_eq fresh_star_def lt.fresh)
-  apply (drule sym)
-  apply (simp only: )
-  apply (simp only: permute_Abs_lst)
-  apply simp
-  apply (simp add: eqvt_at_def)
-  apply (drule sym)
-  apply (simp only:)
-  apply (simp add: Abs_fresh_iff lt.fresh)
-  apply clarify
-  apply (erule fresh_eqvt_at)
-  apply (simp add: supp_Inr finite_supp)
-  apply (simp add: fresh_Inr fresh_Pair lt.fresh fresh_at_base)
-  apply (drule sym)
-  apply (drule sym)
-  apply (drule sym)
-  apply (simp only:)
-  apply (thin_tac "Abs a (CPS1_CPS2_sumC (Inr (M, a~))) = Abs c (CPS1_CPS2_sumC (Inr (M, c~)))")
-  apply (thin_tac "Abs a (CPS1_CPS2_sumC (Inr (Ma, a~))) = Abs ca (CPS1_CPS2_sumC (Inr (Ma, ca~)))")
-  apply (thin_tac "atom a \<sharp> (c, ca, x, xa, M, Ma)")
-  apply (simp add: fresh_Pair_elim)
-  apply (subst iffD1[OF meta_eq_to_obj_eq[OF eqvt_at_def]])
-  back
-  back
-  back
-  apply assumption
-  apply (simp add: Abs1_eq_iff' fresh_Pair_elim fresh_at_base swap_fresh_fresh lt.fresh)
-  apply (case_tac "(atom x \<rightleftharpoons> atom xa) \<bullet> c = ca")
-  apply simp_all[3]
-  apply rule
-  apply (case_tac "c = xa")
-  apply simp_all[2]
-  apply (simp add: eqvt_at_def)
-  apply clarify
-  apply (smt flip_def permute_flip_at permute_swap_cancel swap_fresh_fresh)
-  apply (simp add: eqvt_at_def)
-  apply clarify
-  apply (smt atom_eq_iff atom_eqvt flip_def fresh_eqvt permute_flip_at permute_swap_cancel swap_at_base_simps(3) swap_fresh_fresh)
-  apply (case_tac "c = xa")
-  apply simp
-  apply (subgoal_tac "((ca \<leftrightarrow> x) \<bullet> (atom x)) \<sharp> (ca \<leftrightarrow> x) \<bullet> CPS1_CPS2_sumC (Inr (Ma, ca~))")
-  apply (simp add: atom_eqvt eqvt_at_def)
-  apply (simp add: flip_fresh_fresh)
-  apply (subst fresh_permute_iff)
-  apply (erule fresh_eqvt_at)
-  apply (simp add: supp_Inr finite_supp)
-  apply (simp add: fresh_Inr lt.fresh fresh_at_base fresh_Pair)
-  apply simp
-  apply clarify
-  apply (subgoal_tac "atom ca \<sharp> (atom x \<rightleftharpoons> atom xa) \<bullet> CPS1_CPS2_sumC (Inr (M, c~))")
-  apply (simp add: eqvt_at_def)
-  apply (subgoal_tac "(atom x \<rightleftharpoons> atom xa) \<bullet> atom ca \<sharp> CPS1_CPS2_sumC (Inr (M, c~))")
-  apply (metis Nominal2_Base.swap_commute fresh_permute_iff permute_swap_cancel2)
-  apply (erule fresh_eqvt_at)
-  apply (simp add: finite_supp supp_Inr)
-  apply (simp add: fresh_Inr fresh_Pair lt.fresh)
-  apply rule
-  apply (metis Nominal2_Base.swap_commute fresh_permute_iff permute_swap_cancel2)
-  apply (simp add: fresh_def supp_at_base)
-  apply (metis atom_eq_iff permute_swap_cancel2 swap_atom_simps(3))
---"-"
-  apply (rule_tac x="(c, ca, x, xa, M, Ma)" and ?'a="name" in obtain_fresh)
-  apply (subgoal_tac "Abs c (CPS1_CPS2_sumC (Inr (M, c~))) = Abs a (CPS1_CPS2_sumC (Inr (M, a~)))")
-  prefer 2
-  apply (simp add: Abs1_eq_iff')
-  apply (case_tac "c = a")
-  apply simp_all[2]
-  apply rule
-  apply (simp add: eqvt_at_def)
-  apply (simp add: swap_fresh_fresh fresh_Pair_elim)
-  apply (erule fresh_eqvt_at)
-  apply (simp add: supp_Inr finite_supp)
-  apply (simp add: fresh_Inr fresh_Pair lt.fresh fresh_at_base)
-  apply (subgoal_tac "Abs ca (CPS1_CPS2_sumC (Inr (Ma, ca~))) = Abs a (CPS1_CPS2_sumC (Inr (Ma, a~)))")
-  prefer 2
-  apply (simp add: Abs1_eq_iff')
-  apply (case_tac "ca = a")
-  apply simp_all[2]
-  apply rule
-  apply (simp add: eqvt_at_def)
-  apply (simp add: swap_fresh_fresh fresh_Pair_elim)
-  apply (erule fresh_eqvt_at)
-  apply (simp add: supp_Inr finite_supp)
-  apply (simp add: fresh_Inr fresh_Pair lt.fresh fresh_at_base)
-  apply (simp only: )
-  apply (erule Abs_lst1_fcb)
-  apply (simp add: Abs_fresh_iff)
-  apply (drule sym)
-  apply (simp only:)
-  apply (simp add: Abs_fresh_iff lt.fresh)
-  apply clarify
-  apply (erule fresh_eqvt_at)
-  apply (simp add: supp_Inr finite_supp)
-  apply (simp add: fresh_Inr fresh_Pair lt.fresh fresh_at_base)
-  apply (drule sym)
-  apply (drule sym)
-  apply (drule sym)
-  apply (simp only:)
-  apply (thin_tac "Abs a (CPS1_CPS2_sumC (Inr (M, a~))) = Abs c (CPS1_CPS2_sumC (Inr (M, c~)))")
-  apply (thin_tac "Abs a (CPS1_CPS2_sumC (Inr (Ma, a~))) = Abs ca (CPS1_CPS2_sumC (Inr (Ma, ca~)))")
-  apply (thin_tac "atom a \<sharp> (c, ca, x, xa, M, Ma)")
-  apply (simp add: fresh_Pair_elim)
-  apply (subst iffD1[OF meta_eq_to_obj_eq[OF eqvt_at_def]])
-  back
-  back
-  back
-  apply assumption
-  apply (simp add: Abs1_eq_iff' fresh_Pair_elim fresh_at_base swap_fresh_fresh lt.fresh)
-  apply (case_tac "(atom x \<rightleftharpoons> atom xa) \<bullet> c = ca")
-  apply simp_all[3]
-  apply rule
-  apply (case_tac "c = xa")
-  apply simp_all[2]
-  apply (simp add: eqvt_at_def)
-  apply clarify
-  apply (smt flip_def permute_flip_at permute_swap_cancel swap_fresh_fresh)
-  apply (simp add: eqvt_at_def)
-  apply clarify
-  apply (smt atom_eq_iff atom_eqvt flip_def fresh_eqvt permute_flip_at permute_swap_cancel swap_at_base_simps(3) swap_fresh_fresh)
-  apply (case_tac "c = xa")
-  apply simp
-  apply (subgoal_tac "((ca \<leftrightarrow> x) \<bullet> (atom x)) \<sharp> (ca \<leftrightarrow> x) \<bullet> CPS1_CPS2_sumC (Inr (Ma, ca~))")
-  apply (simp add: atom_eqvt eqvt_at_def)
-  apply (simp add: flip_fresh_fresh)
-  apply (subst fresh_permute_iff)
-  apply (erule fresh_eqvt_at)
-  apply (simp add: supp_Inr finite_supp)
-  apply (simp add: fresh_Inr lt.fresh fresh_at_base fresh_Pair)
-  apply simp
-  apply clarify
-  apply (subgoal_tac "atom ca \<sharp> (atom x \<rightleftharpoons> atom xa) \<bullet> CPS1_CPS2_sumC (Inr (M, c~))")
-  apply (simp add: eqvt_at_def)
-  apply (subgoal_tac "(atom x \<rightleftharpoons> atom xa) \<bullet> atom ca \<sharp> CPS1_CPS2_sumC (Inr (M, c~))")
-  apply (metis Nominal2_Base.swap_commute fresh_permute_iff permute_swap_cancel2)
-  apply (erule fresh_eqvt_at)
-  apply (simp add: finite_supp supp_Inr)
-  apply (simp add: fresh_Inr fresh_Pair lt.fresh)
-  apply rule
-  apply (metis Nominal2_Base.swap_commute fresh_permute_iff permute_swap_cancel2)
-  apply (simp add: fresh_def supp_at_base)
-  apply (metis atom_eq_iff permute_swap_cancel2 swap_atom_simps(3))
-  done
-
-termination
-  by lexicographic_order
-
-definition psi:: "lt => lt"
-  where [simp]: "psi V == V*(\<lambda>x. x)"
-
-section {* Simple consequence of CPS *}
-
-lemma [simp]: "eqvt (\<lambda>x\<Colon>lt. x)"
-  by (simp add: eqvt_def eqvt_bound eqvt_lambda)
-
-lemma value_eq1 : "isValue V \<Longrightarrow> eqvt k \<Longrightarrow> V*k = k (psi V)"
-  apply (cases V rule: lt.exhaust)
-  apply simp_all
-  apply (rule_tac x="(name, lt)" and ?'a="name" in obtain_fresh)
-  apply simp
-  done
-
-lemma value_eq2 : "isValue V \<Longrightarrow> V^K = K $ (psi V)"
-  apply (cases V rule: lt.exhaust)
-  apply simp_all
-  apply (rule_tac x="(name, lt)" and ?'a="name" in obtain_fresh)
-  apply simp
-  done
-
-lemma value_eq3' : "~isValue M \<Longrightarrow> eqvt k \<Longrightarrow> M*k = (M^(Abs n (k (Var n))))"
-  by (cases M rule: lt.exhaust) auto
-
-
+  oops
 
 end