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Nominal/Ex/CPS/CPS3_DanvyFilinski.thy
author Cezary Kaliszyk <kaliszyk@in.tum.de>
Mon, 20 Jun 2011 20:09:30 +0900
changeset 2877 3e82c1ced5e4
parent 2866 9f667f6da04f
child 2882 186ec672cc51
permissions -rw-r--r--
function for let-rec

header {* CPS transformation of Danvy and Filinski *}
theory 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 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~)))")
  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))
--"-"
  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 "psi V == V*(\<lambda>x. x)"

section {* Simple consequence of CPS *}

lemma value_eq1 : "isValue V \<Longrightarrow> eqvt k \<Longrightarrow> V*k = k (psi V)"
  apply (cases V rule: lt.exhaust)
  apply (auto simp add: psi_def)
  apply (subst CPS1.simps)
  apply (simp add: eqvt_def eqvt_bound eqvt_lambda)
  apply rule
  apply (rule_tac x="(name, lt)" and ?'a="name" in obtain_fresh)
  apply (subst CPS1.simps(3))
  apply assumption+
  apply (subst CPS1.simps(3))
  apply (simp add: eqvt_def eqvt_bound eqvt_lambda)
  apply assumption
  apply rule
  done

end
nominal2