header {* CPS transformation of Danvy and Filinski *}
theory CPS3_DanvyFilinski imports Lt begin

nominal_primrec
  CPS1 :: "lt \<Rightarrow> (lt \<Rightarrow> lt) \<Rightarrow> lt" ("_*_"  [100,100] 100)
and
  CPS2 :: "lt \<Rightarrow> lt \<Rightarrow> lt" ("_^_" [100,100] 100)
where
  "eqvt k \<Longrightarrow> (x~)*k = k (x~)"
| "eqvt k \<Longrightarrow> (M$N)*k = M*(%m. (N*(%n.((m $ n) $ (Abs c (k (c~)))))))"
| "eqvt k \<Longrightarrow> atom c \<sharp> (x, M) \<Longrightarrow> (Abs x M)*k = k (Abs x (Abs c (M^(c~))))"
| "\<not>eqvt k \<Longrightarrow> (CPS1 t k) = t"
| "(x~)^l = l $ (x~)"
| "(M$N)^l = M*(%m. (N*(%n.((m $ n) $ l))))"
| "atom c \<sharp> (x, M) \<Longrightarrow> (Abs x M)^l = l $ (Abs x (Abs c (M^(c~))))"
  apply (simp only: eqvt_def CPS1_CPS2_graph_def)
  apply (rule, perm_simp, rule)
  apply auto
  apply (case_tac x)
  apply (case_tac a)
  apply (case_tac "eqvt b")
  apply (rule_tac y="aa" in lt.strong_exhaust)
  apply auto[4]
  apply (rule_tac x="(name, lt)" and ?'a="name" in obtain_fresh)
  apply (simp add: fresh_at_base Abs1_eq_iff)
  apply (case_tac b)
  apply (rule_tac y="a" in lt.strong_exhaust)
  apply auto[3]
  apply blast
  apply (rule_tac x="(name, lt)" and ?'a="name" in obtain_fresh) 
  apply (simp add: fresh_at_base Abs1_eq_iff)
  apply blast
--"-"
  apply (subgoal_tac "Abs c (ka (c~)) = Abs ca (ka (ca~))")
  apply (simp only:)
  apply (simp add: Abs1_eq_iff)
  apply (case_tac "c=ca")
  apply simp_all[2]
  apply rule
  apply (perm_simp)
  apply (simp add: eqvt_def)
  apply (simp add: fresh_def)
  apply (rule contra_subsetD[OF supp_fun_app])
  back
  apply (simp add: supp_fun_eqvt lt.supp supp_at_base)
--"-"
  apply (rule arg_cong)
  back
  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~)))")
  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 (simp (no_asm))
  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)
  oops

end