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