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