--- a/Nominal/Ex/CPS/CPS3_DanvyFilinski.thy Sat Dec 17 17:03:01 2011 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,253 +0,0 @@
-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
-
-
-