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) $ (Lam c (k (c~)))))))"
| "eqvt k \<Longrightarrow> atom c \<sharp> (x, M) \<Longrightarrow> (Lam x M)*k = k (Lam x (Lam 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> (Lam x M)^l = l $ (Lam x (Lam 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 "Lam c (ka (c~)) = Lam 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 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 "Lam c (CPS1_CPS2_sumC (Inr (M, c~))) = Lam 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 "Lam ca (CPS1_CPS2_sumC (Inr (Ma, ca~))) = Lam 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 "Lam a (CPS1_CPS2_sumC (Inr (M, a~))) = Lam c (CPS1_CPS2_sumC (Inr (M, c~)))")
  apply (thin_tac "Lam a (CPS1_CPS2_sumC (Inr (Ma, a~))) = Lam 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 (rotate_tac 1)
  apply (drule_tac q="(atom ca \<rightleftharpoons> atom x)" in eqvt_at_perm)
  apply (simp add: swap_fresh_fresh)
  apply (simp add: eqvt_at_def swap_fresh_fresh)
  apply (thin_tac "eqvt_at CPS1_CPS2_sumC (Inr (M, c~))")
  apply (simp add: eqvt_at_def)
  apply (drule_tac x="(atom ca \<rightleftharpoons> atom c)" in spec)
  apply simp
  apply (metis (no_types) atom_eq_iff fresh_permute_iff permute_swap_cancel swap_atom_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 "Lam c (CPS1_CPS2_sumC (Inr (M, c~))) = Lam 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 "Lam ca (CPS1_CPS2_sumC (Inr (Ma, ca~))) = Lam 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) (* TODO put sum of fs into fs typeclass *)
  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 "Lam a (CPS1_CPS2_sumC (Inr (M, a~))) = Lam c (CPS1_CPS2_sumC (Inr (M, c~)))")
  apply (thin_tac "Lam a (CPS1_CPS2_sumC (Inr (Ma, a~))) = Lam 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 (rotate_tac 1)
  apply (drule_tac q="(atom ca \<rightleftharpoons> atom x)" in eqvt_at_perm)
  apply (simp add: swap_fresh_fresh)
  apply (simp add: eqvt_at_def swap_fresh_fresh)
  apply (thin_tac "eqvt_at CPS1_CPS2_sumC (Inr (M, c~))")
  apply (simp add: eqvt_at_def)
  apply (drule_tac x="(atom ca \<rightleftharpoons> atom c)" in spec)
  apply simp
  apply (metis (no_types) atom_eq_iff fresh_permute_iff permute_swap_cancel swap_atom_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 (eqvt)
  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^(Lam n (k (Var n))))"
  by (cases M rule: lt.exhaust) auto



end



