--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/CPS/Lt.thy	Thu Jun 16 13:14:53 2011 +0100
@@ -0,0 +1,131 @@
+header {* The Call-by-Value Lambda Calculus *}
+theory Lt
+imports Nominal2
+begin
+
+atom_decl name
+
+nominal_datatype lt =
+    Var name       ("_~"  [150] 149)
+  | Abs x::"name" t::"lt"  bind  x in t
+  | App lt lt         (infixl "$" 100)
+
+nominal_primrec
+  subst :: "lt \<Rightarrow> lt \<Rightarrow> name \<Rightarrow> lt" ("_[_'/_]" [200,0,0] 190)
+where
+  "(y~)[L/x] = (if y = x then L else y~)"
+| "atom y\<sharp>L \<Longrightarrow> atom y\<sharp>x \<Longrightarrow> (Abs y M)[L/x]  = Abs y (M[L/x])"
+| "(M $ N)[L/x] = M[L/x] $ N[L/x]"
+  unfolding eqvt_def subst_graph_def
+  apply(perm_simp)
+  apply(auto)
+  apply(rule_tac y="a" and c="(aa, b)" in lt.strong_exhaust)
+  apply(simp_all add: fresh_star_def fresh_Pair)
+  apply blast+
+  apply (erule Abs_lst1_fcb)
+  apply (simp_all add: Abs_fresh_iff)[2]
+  apply(drule_tac a="atom (ya)" in fresh_eqvt_at)
+  apply(simp add: finite_supp fresh_Pair)
+  apply(simp_all add: fresh_Pair Abs_fresh_iff)
+  apply(subgoal_tac "(atom y \<rightleftharpoons> atom ya) \<bullet> La = La")
+  apply(subgoal_tac "(atom y \<rightleftharpoons> atom ya) \<bullet> xa = xa")
+  apply(simp add: atom_eqvt eqvt_at_def Abs1_eq_iff swap_commute)
+  apply (simp_all add: swap_fresh_fresh)
+  done
+
+termination
+  by (relation "measure (\<lambda>(t, _, _). size t)")
+     (simp_all add: lt.size)
+
+lemma subst_eqvt[eqvt]:
+  shows "p\<bullet>(M[V/(x::name)]) = (p\<bullet>M)[(p\<bullet>V)/(p\<bullet>x)]"
+  by (induct M V x rule: subst.induct) (simp_all)
+
+lemma forget[simp]:
+  shows "atom x \<sharp> M \<Longrightarrow> M[s/x] = M"
+  by (nominal_induct M avoiding: x s rule: lt.strong_induct)
+     (auto simp add: lt.fresh fresh_at_base)
+
+lemma [simp]: "supp ( M[V/(x::name)] ) <= (supp(M) - {atom x}) Un (supp V)"
+  by (nominal_induct M avoiding: x V rule: lt.strong_induct)
+     (auto simp add: lt.supp supp_at_base, blast, blast)
+
+nominal_primrec 
+  isValue:: "lt => bool"
+where
+  "isValue (Var x) = True"
+| "isValue (Abs y N) = True"
+| "isValue (A $ B) = False"
+  unfolding eqvt_def isValue_graph_def
+  by (perm_simp, auto)
+     (erule lt.exhaust, auto)
+
+termination
+  by (relation "measure size")
+     (simp_all add: lt.size)
+
+lemma is_Value_eqvt[eqvt]:
+  shows "p\<bullet>(isValue (M::lt)) = isValue (p\<bullet>M)"
+  by (induct M rule: lt.induct) (simp_all add: eqvts)
+
+inductive
+  eval :: "[lt, lt] \<Rightarrow> bool" (" _ \<longrightarrow>\<^isub>\<beta> _" [80,80] 80)
+  where
+   evbeta: "\<lbrakk>atom x\<sharp>V; isValue V\<rbrakk> \<Longrightarrow> ((Abs x M) $ V) \<longrightarrow>\<^isub>\<beta> (M[V/x])"
+|  ev1: "\<lbrakk>isValue V; M \<longrightarrow>\<^isub>\<beta> M' \<rbrakk> \<Longrightarrow> (V $ M) \<longrightarrow>\<^isub>\<beta> (V $ M')"
+|  ev2: "M \<longrightarrow>\<^isub>\<beta> M' \<Longrightarrow> (M $ N) \<longrightarrow>\<^isub>\<beta> (M' $ N)"
+
+equivariance eval
+
+nominal_inductive eval
+done
+
+(*lemmas [simp] = lt.supp(2)*)
+
+lemma closedev1: assumes "s \<longrightarrow>\<^isub>\<beta> t"
+  shows "supp t <= supp s"
+  using assms
+    by (induct, auto simp add: lt.supp)
+
+
+lemma [simp]: "~ ((Abs x M) \<longrightarrow>\<^isub>\<beta> N)"
+by (rule, erule eval.cases, simp_all)
+
+lemma [simp]: assumes "M \<longrightarrow>\<^isub>\<beta> N" shows "~ isValue M"
+using assms
+by (cases, auto)
+
+
+inductive
+  eval_star :: "[lt, lt] \<Rightarrow> bool" (" _ \<longrightarrow>\<^isub>\<beta>\<^sup>* _" [80,80] 80)
+  where
+   evs1: "M \<longrightarrow>\<^isub>\<beta>\<^sup>* M"
+|  evs2: "\<lbrakk>M \<longrightarrow>\<^isub>\<beta> M'; M' \<longrightarrow>\<^isub>\<beta>\<^sup>*  M'' \<rbrakk> \<Longrightarrow> M \<longrightarrow>\<^isub>\<beta>\<^sup>*  M''"
+
+lemma eval_evs: assumes *: "M \<longrightarrow>\<^isub>\<beta> M'" shows "M \<longrightarrow>\<^isub>\<beta>\<^sup>* M'"
+by (rule evs2, rule *, rule evs1)
+
+lemma eval_trans[trans]:
+  assumes "M1  \<longrightarrow>\<^isub>\<beta>\<^sup>* M2"
+      and "M2  \<longrightarrow>\<^isub>\<beta>\<^sup>*  M3"
+  shows "M1  \<longrightarrow>\<^isub>\<beta>\<^sup>* M3"
+  using assms
+  by (induct, auto intro: evs2)
+
+lemma evs3[rule_format]: assumes "M1  \<longrightarrow>\<^isub>\<beta>\<^sup>* M2"
+  shows "M2  \<longrightarrow>\<^isub>\<beta> M3 \<longrightarrow> M1  \<longrightarrow>\<^isub>\<beta>\<^sup>* M3"
+  using assms
+    by (induct, auto intro: eval_evs evs2)
+
+equivariance eval_star
+
+lemma evbeta':
+  fixes x :: name
+  assumes "isValue V" and "atom x\<sharp>V" and "N = (M[V/x])"
+  shows "((Abs x M) $ V) \<longrightarrow>\<^isub>\<beta>\<^sup>* N"
+  using assms by simp (rule evs2, rule evbeta, simp_all add: evs1)
+
+end
+
+
+