header {* The Call-by-Value Lambda Calculus *}

theory Lt

imports "../../Nominal2"

begin

atom_decl name

nominal_datatype lt =

    Var name       ("_~"  [150] 149)

  | App lt lt         (infixl "$$" 100)

  | Lam x::"name" t::"lt"  binds  x in t

  subst :: "lt \<Rightarrow> name \<Rightarrow> lt \<Rightarrow> lt"  ("_ [_ ::= _]" [90, 90, 90] 90)

where

  "(Var x)[y ::= s] = (if x = y then s else (Var x))"

| "(App t1 t2)[y ::= s] = App (t1[y ::= s]) (t2[y ::= s])"

| "atom x \<sharp> (y, s) \<Longrightarrow> (Lam x t)[y ::= s] = Lam x (t[y ::= s])"

  unfolding eqvt_def subst_graph_aux_def

  apply (simp)

  apply(rule TrueI)

  using [[simproc del: alpha_lst]]

  apply(auto simp add: lt.distinct lt.eq_iff)

  apply(rule_tac y="a" and c="(aa, b)" in lt.strong_exhaust)

  apply blast

  apply(simp_all add: fresh_star_def fresh_Pair_elim)

  apply (erule_tac c="(ya,sa)" in Abs_lst1_fcb2)

  apply(simp add: Abs_fresh_iff)

  apply(simp add: fresh_star_def fresh_Pair)

  apply(simp add: eqvt_at_def)

  apply(simp add: perm_supp_eq fresh_star_Pair)

  apply(simp add: eqvt_at_def)

  apply(simp add: perm_supp_eq fresh_star_Pair)

done

termination (eqvt) by lexicographic_order

lemma forget[simp]:

  shows "atom x \<sharp> M \<Longrightarrow> M[x ::= s] = M"

  by (nominal_induct M avoiding: x s rule: lt.strong_induct)

     (auto simp add: lt.fresh fresh_at_base)

lemma [simp]: "supp (M[x ::= V]) <= (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)

  isValue:: "lt => bool"

where

  "isValue (Var x) = True"

| "isValue (Lam y N) = True"

| "isValue (A $$ B) = False"

  unfolding eqvt_def isValue_graph_aux_def

  by (auto)

     (erule lt.exhaust, auto)

termination (eqvt)

  by (relation "measure size") (simp_all)

inductive

  eval :: "[lt, lt] \<Rightarrow> bool" (" _ \<longrightarrow>\<^isub>\<beta> _" [80,80] 80)

  where

   evbeta: "\<lbrakk>atom x \<sharp> V; isValue V\<rbrakk> \<Longrightarrow> ((Lam x M) $$ V) \<longrightarrow>\<^isub>\<beta> (M[x ::= V])"

|  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]: "~ ((Lam 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[x ::= V])"

  shows "((Lam x M) $$ V) \<longrightarrow>\<^isub>\<beta>\<^sup>* N"

  using assms by simp (rule evs2, rule evbeta, simp_all add: evs1)

end