--- a/Nominal/Ex/CPS/CPS1_Plotkin.thy Fri Aug 19 11:07:17 2011 +0900
+++ b/Nominal/Ex/CPS/CPS1_Plotkin.thy Fri Aug 19 12:49:38 2011 +0900
@@ -3,111 +3,13 @@
imports Lt
begin
-lemma Abs_lst_fcb2:
- fixes as bs :: "atom list"
- and x y :: "'b :: fs"
- and c::"'c::fs"
- assumes eq: "[as]lst. x = [bs]lst. y"
- and fcb1: "(set as) \<sharp>* c \<Longrightarrow> (set as) \<sharp>* f as x c"
- and fresh1: "set as \<sharp>* c"
- and fresh2: "set bs \<sharp>* c"
- and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"
- and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"
- shows "f as x c = f bs y c"
-proof -
- have "supp (as, x, c) supports (f as x c)"
- unfolding supports_def fresh_def[symmetric]
- by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)
- then have fin1: "finite (supp (f as x c))"
- by (auto intro: supports_finite simp add: finite_supp)
- have "supp (bs, y, c) supports (f bs y c)"
- unfolding supports_def fresh_def[symmetric]
- by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
- then have fin2: "finite (supp (f bs y c))"
- by (auto intro: supports_finite simp add: finite_supp)
- obtain q::"perm" where
- fr1: "(q \<bullet> (set as)) \<sharp>* (x, c, f as x c, f bs y c)" and
- fr2: "supp q \<sharp>* Abs_lst as x" and
- inc: "supp q \<subseteq> (set as) \<union> q \<bullet> (set as)"
- using at_set_avoiding3[where xs="set as" and c="(x, c, f as x c, f bs y c)" and x="[as]lst. x"]
- fin1 fin2
- by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
- have "Abs_lst (q \<bullet> as) (q \<bullet> x) = q \<bullet> Abs_lst as x" by simp
- also have "\<dots> = Abs_lst as x"
- by (simp only: fr2 perm_supp_eq)
- finally have "Abs_lst (q \<bullet> as) (q \<bullet> x) = Abs_lst bs y" using eq by simp
- then obtain r::perm where
- qq1: "q \<bullet> x = r \<bullet> y" and
- qq2: "q \<bullet> as = r \<bullet> bs" and
- qq3: "supp r \<subseteq> (q \<bullet> (set as)) \<union> set bs"
- apply(drule_tac sym)
- apply(simp only: Abs_eq_iff2 alphas)
- apply(erule exE)
- apply(erule conjE)+
- apply(drule_tac x="p" in meta_spec)
- apply(simp add: set_eqvt)
- apply(blast)
- done
- have "(set as) \<sharp>* f as x c"
- apply(rule fcb1)
- apply(rule fresh1)
- done
- then have "q \<bullet> ((set as) \<sharp>* f as x c)"
- by (simp add: permute_bool_def)
- then have "set (q \<bullet> as) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"
- apply(simp add: fresh_star_eqvt set_eqvt)
- apply(subst (asm) perm1)
- using inc fresh1 fr1
- apply(auto simp add: fresh_star_def fresh_Pair)
- done
- then have "set (r \<bullet> bs) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
- then have "r \<bullet> ((set bs) \<sharp>* f bs y c)"
- apply(simp add: fresh_star_eqvt set_eqvt)
- apply(subst (asm) perm2[symmetric])
- using qq3 fresh2 fr1
- apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)
- done
- then have fcb2: "(set bs) \<sharp>* f bs y c" by (simp add: permute_bool_def)
- have "f as x c = q \<bullet> (f as x c)"
- apply(rule perm_supp_eq[symmetric])
- using inc fcb1[OF fresh1] fr1 by (auto simp add: fresh_star_def)
- also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c"
- apply(rule perm1)
- using inc fresh1 fr1 by (auto simp add: fresh_star_def)
- also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
- also have "\<dots> = r \<bullet> (f bs y c)"
- apply(rule perm2[symmetric])
- using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)
- also have "... = f bs y c"
- apply(rule perm_supp_eq)
- using qq3 fr1 fcb2 by (auto simp add: fresh_star_def)
- finally show ?thesis by simp
-qed
-
-lemma Abs_lst1_fcb2:
- fixes a b :: "atom"
- and x y :: "'b :: fs"
- and c::"'c :: fs"
- assumes e: "(Abs_lst [a] x) = (Abs_lst [b] y)"
- and fcb1: "a \<sharp> c \<Longrightarrow> a \<sharp> f a x c"
- and fresh: "{a, b} \<sharp>* c"
- and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f a x c) = f (p \<bullet> a) (p \<bullet> x) c"
- and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b y c) = f (p \<bullet> b) (p \<bullet> y) c"
- shows "f a x c = f b y c"
-using e
-apply(drule_tac Abs_lst_fcb2[where c="c" and f="\<lambda>(as::atom list) . f (hd as)"])
-apply(simp_all)
-using fcb1 fresh perm1 perm2
-apply(simp_all add: fresh_star_def)
-done
-
nominal_primrec
CPS :: "lt \<Rightarrow> lt" ("_*" [250] 250)
where
- "atom k \<sharp> x \<Longrightarrow> (x~)* = (Abs k ((k~) $ (x~)))"
-| "atom k \<sharp> (x, M) \<Longrightarrow> (Abs x M)* = Abs k (k~ $ Abs x (M*))"
+ "atom k \<sharp> x \<Longrightarrow> (x~)* = (Lam k ((k~) $ (x~)))"
+| "atom k \<sharp> (x, M) \<Longrightarrow> (Lam x M)* = Lam k (k~ $ Lam x (M*))"
| "atom k \<sharp> (M, N) \<Longrightarrow> atom m \<sharp> (N, k) \<Longrightarrow> atom n \<sharp> (k, m) \<Longrightarrow>
- (M $ N)* = Abs k (M* $ Abs m (N* $ Abs n (m~ $ n~ $ k~)))"
+ (M $ N)* = Lam k (M* $ Lam m (N* $ Lam n (m~ $ n~ $ k~)))"
unfolding eqvt_def CPS_graph_def
apply (rule, perm_simp, rule, rule)
apply (simp_all add: fresh_Pair_elim)
@@ -123,7 +25,7 @@
apply (simp add: fresh_Pair_elim fresh_at_base)
apply (simp add: Abs1_eq_iff lt.fresh fresh_at_base)
--"-"
-apply(rule_tac s="[[atom ka]]lst. ka~ $ Abs x (CPS_sumC M)" in trans)
+apply(rule_tac s="[[atom ka]]lst. ka~ $ Lam x (CPS_sumC M)" in trans)
apply (case_tac "k = ka")
apply simp
apply(simp (no_asm) add: Abs1_eq_iff del:eqvts)
@@ -155,7 +57,7 @@
apply (simp_all add: Abs1_eq_iff lt.fresh flip_def[symmetric] fresh_at_base flip_fresh_fresh permute_eq_iff)
by (metis flip_at_base_simps(3) flip_at_simps(2) flip_commute permute_flip_at)+
-termination (eqvt) by (relation "measure size") (simp_all)
+termination (eqvt) by lexicographic_order
lemmas [simp] = fresh_Pair_elim CPS.simps(2,3)[simplified fresh_Pair_elim]
@@ -170,7 +72,7 @@
convert:: "lt => lt" ("_+" [250] 250)
where
"(Var x)+ = Var x"
-| "(Abs x M)+ = Abs x (M*)"
+| "(Lam x M)+ = Lam x (M*)"
| "(M $ N)+ = M $ N"
unfolding convert_graph_def eqvt_def
apply (rule, perm_simp, rule, rule)
@@ -195,20 +97,16 @@
shows "isValue (p \<bullet> (M::lt)) = isValue M"
by (nominal_induct M rule: lt.strong_induct) auto
-lemma [eqvt]:
- shows "p \<bullet> isValue M = isValue (p \<bullet> M)"
- by (induct M rule: lt.induct) (perm_simp, rule refl)+
-
nominal_primrec
Kapply :: "lt \<Rightarrow> lt \<Rightarrow> lt" (infixl ";" 100)
where
- "Kapply (Abs x M) K = K $ (Abs x M)+"
+ "Kapply (Lam x M) K = K $ (Lam x M)+"
| "Kapply (Var x) K = K $ Var x"
| "isValue M \<Longrightarrow> isValue N \<Longrightarrow> Kapply (M $ N) K = M+ $ N+ $ K"
| "isValue M \<Longrightarrow> \<not>isValue N \<Longrightarrow> atom n \<sharp> M \<Longrightarrow> atom n \<sharp> K \<Longrightarrow>
- Kapply (M $ N) K = N; (Abs n (M+ $ Var n $ K))"
+ Kapply (M $ N) K = N; (Lam n (M+ $ Var n $ K))"
| "\<not>isValue M \<Longrightarrow> atom m \<sharp> N \<Longrightarrow> atom m \<sharp> K \<Longrightarrow> atom n \<sharp> m \<Longrightarrow> atom n \<sharp> K \<Longrightarrow>
- Kapply (M $ N) K = M; (Abs m (N* $ (Abs n (Var m $ Var n $ K))))"
+ Kapply (M $ N) K = M; (Lam m (N* $ (Lam n (Var m $ Var n $ K))))"
unfolding Kapply_graph_def eqvt_def
apply (rule, perm_simp, rule, rule)
apply (simp_all)
@@ -225,10 +123,10 @@
apply (simp add: fresh_Pair_elim fresh_at_base)
apply (auto simp add: Abs1_eq_iff eqvts)[1]
apply (rename_tac M N u K)
-apply (subgoal_tac "Abs n (M+ $ n~ $ K) = Abs u (M+ $ u~ $ K)")
+apply (subgoal_tac "Lam n (M+ $ n~ $ K) = Lam u (M+ $ u~ $ K)")
apply (simp only:)
apply (auto simp add: Abs1_eq_iff flip_def[symmetric] lt.fresh fresh_at_base flip_fresh_fresh[symmetric])[1]
-apply (subgoal_tac "Abs m (Na* $ Abs n (m~ $ n~ $ Ka)) = Abs ma (Na* $ Abs na (ma~ $ na~ $ Ka))")
+apply (subgoal_tac "Lam m (Na* $ Lam n (m~ $ n~ $ Ka)) = Lam ma (Na* $ Lam na (ma~ $ na~ $ Ka))")
apply (simp only:)
apply (simp add: Abs1_eq_iff flip_def[symmetric] lt.fresh fresh_at_base)
apply (subgoal_tac "Ka = (n \<leftrightarrow> na) \<bullet> Ka")
@@ -254,14 +152,14 @@
lemma value_CPS:
assumes "isValue V"
and "atom a \<sharp> V"
- shows "V* = Abs a (a~ $ V+)"
+ shows "V* = Lam a (a~ $ V+)"
using assms
proof (nominal_induct V avoiding: a rule: lt.strong_induct, simp_all add: lt.fresh)
fix name :: name and lt aa
- assume a: "atom name \<sharp> aa" "\<And>b. \<lbrakk>isValue lt; atom b \<sharp> lt\<rbrakk> \<Longrightarrow> lt* = Abs b (b~ $ lt+)"
+ assume a: "atom name \<sharp> aa" "\<And>b. \<lbrakk>isValue lt; atom b \<sharp> lt\<rbrakk> \<Longrightarrow> lt* = Lam b (b~ $ lt+)"
"atom aa \<sharp> lt \<or> aa = name"
obtain ab :: name where b: "atom ab \<sharp> (name, lt, a)" using obtain_fresh by blast
- show "Abs name lt* = Abs aa (aa~ $ Abs name (lt*))" using a b
+ show "Lam name lt* = Lam aa (aa~ $ Lam name (lt*))" using a b
by (simp add: Abs1_eq_iff fresh_at_base lt.fresh)
qed
@@ -269,28 +167,28 @@
lemma CPS_subst_fv:
assumes *:"isValue V"
- shows "((M[V/x])* = (M*)[V+/x])"
+ shows "((M[x ::= V])* = (M*)[x ::= V+])"
using * proof (nominal_induct M avoiding: V x rule: lt.strong_induct)
case (Var name)
assume *: "isValue V"
obtain a :: name where a: "atom a \<sharp> (x, name, V)" using obtain_fresh by blast
- show "((name~)[V/x])* = (name~)*[V+/x]" using a
+ show "((name~)[x ::= V])* = (name~)*[x ::= V+]" using a
by (simp add: fresh_at_base * value_CPS)
next
- case (Abs name lt V x)
- assume *: "atom name \<sharp> V" "atom name \<sharp> x" "\<And>b ba. isValue b \<Longrightarrow> (lt[b/ba])* = lt*[b+/ba]"
+ case (Lam name lt V x)
+ assume *: "atom name \<sharp> V" "atom name \<sharp> x" "\<And>b ba. isValue b \<Longrightarrow> (lt[ba ::= b])* = lt*[ba ::= b+]"
"isValue V"
- obtain a :: name where a: "atom a \<sharp> (name, lt, lt[V/x], x, V)" using obtain_fresh by blast
- show "(Abs name lt[V/x])* = Abs name lt*[V+/x]" using * a
+ obtain a :: name where a: "atom a \<sharp> (name, lt, lt[x ::= V], x, V)" using obtain_fresh by blast
+ show "(Lam name lt[x ::= V])* = Lam name lt*[x ::= V+]" using * a
by (simp add: fresh_at_base)
next
case (App lt1 lt2 V x)
- assume *: "\<And>b ba. isValue b \<Longrightarrow> (lt1[b/ba])* = lt1*[b+/ba]" "\<And>b ba. isValue b \<Longrightarrow> (lt2[b/ba])* = lt2*[b+/ba]"
+ assume *: "\<And>b ba. isValue b \<Longrightarrow> (lt1[ba ::= b])* = lt1*[ba ::= b+]" "\<And>b ba. isValue b \<Longrightarrow> (lt2[ba ::= b])* = lt2*[ba ::= b+]"
"isValue V"
- obtain a :: name where a: "atom a \<sharp> (lt1[V/x], lt1, lt2[V/x], lt2, V, x)" using obtain_fresh by blast
- obtain b :: name where b: "atom b \<sharp> (lt2[V/x], lt2, a, V, x)" using obtain_fresh by blast
+ obtain a :: name where a: "atom a \<sharp> (lt1[x ::= V], lt1, lt2[x ::= V], lt2, V, x)" using obtain_fresh by blast
+ obtain b :: name where b: "atom b \<sharp> (lt2[x ::= V], lt2, a, V, x)" using obtain_fresh by blast
obtain c :: name where c: "atom c \<sharp> (a, b, V, x)" using obtain_fresh by blast
- show "((lt1 $ lt2)[V/x])* = (lt1 $ lt2)*[V+/x]" using * a b c
+ show "((lt1 $ lt2)[x ::= V])* = (lt1 $ lt2)*[x ::= V+]" using * a b c
by (simp add: fresh_at_base)
qed
@@ -312,7 +210,7 @@
assume *: "atom name \<sharp> K" "\<And>b. isValue b \<Longrightarrow> lt* $ b \<longrightarrow>\<^isub>\<beta>\<^sup>* lt ; b" "isValue K"
obtain a :: name where a: "atom a \<sharp> (name, K, lt)" using obtain_fresh by blast
then have b: "atom name \<sharp> a" using fresh_PairD(1) fresh_at_base atom_eq_iff by metis
- show "Abs name lt* $ K \<longrightarrow>\<^isub>\<beta>\<^sup>* K $ Abs name (lt*)" using * a b
+ show "Lam name lt* $ K \<longrightarrow>\<^isub>\<beta>\<^sup>* K $ Lam name (lt*)" using * a b
by simp (rule evbeta', simp_all)
next
fix lt1 lt2 K
@@ -322,24 +220,24 @@
obtain c :: name where c: "atom c \<sharp> (lt1, lt2, K, a, b)" using obtain_fresh by blast
have d: "atom a \<sharp> lt1" "atom a \<sharp> lt2" "atom a \<sharp> K" "atom b \<sharp> lt1" "atom b \<sharp> lt2" "atom b \<sharp> K" "atom b \<sharp> a"
"atom c \<sharp> lt1" "atom c \<sharp> lt2" "atom c \<sharp> K" "atom c \<sharp> a" "atom c \<sharp> b" using fresh_Pair a b c by simp_all
- have "(lt1 $ lt2)* $ K \<longrightarrow>\<^isub>\<beta>\<^sup>* lt1* $ Abs b (lt2* $ Abs c (b~ $ c~ $ K))" using * d
+ have "(lt1 $ lt2)* $ K \<longrightarrow>\<^isub>\<beta>\<^sup>* lt1* $ Lam b (lt2* $ Lam c (b~ $ c~ $ K))" using * d
by (simp add: fresh_at_base) (rule evbeta', simp_all add: fresh_at_base)
also have "... \<longrightarrow>\<^isub>\<beta>\<^sup>* lt1 $ lt2 ; K" proof (cases "isValue lt1")
assume e: "isValue lt1"
- have "lt1* $ Abs b (lt2* $ Abs c (b~ $ c~ $ K)) \<longrightarrow>\<^isub>\<beta>\<^sup>* Abs b (lt2* $ Abs c (b~ $ c~ $ K)) $ lt1+"
+ have "lt1* $ Lam b (lt2* $ Lam c (b~ $ c~ $ K)) \<longrightarrow>\<^isub>\<beta>\<^sup>* Lam b (lt2* $ Lam c (b~ $ c~ $ K)) $ lt1+"
using * d e by simp
- also have "... \<longrightarrow>\<^isub>\<beta>\<^sup>* lt2* $ Abs c (lt1+ $ c~ $ K)"
+ also have "... \<longrightarrow>\<^isub>\<beta>\<^sup>* lt2* $ Lam c (lt1+ $ c~ $ K)"
by (rule evbeta', simp_all add: * d e, metis d(12) fresh_at_base)
also have "... \<longrightarrow>\<^isub>\<beta>\<^sup>* lt1 $ lt2 ; K" proof (cases "isValue lt2")
assume f: "isValue lt2"
- have "lt2* $ Abs c (lt1+ $ c~ $ K) \<longrightarrow>\<^isub>\<beta>\<^sup>* Abs c (lt1+ $ c~ $ K) $ lt2+" using * d e f by simp
+ have "lt2* $ Lam c (lt1+ $ c~ $ K) \<longrightarrow>\<^isub>\<beta>\<^sup>* Lam c (lt1+ $ c~ $ K) $ lt2+" using * d e f by simp
also have "... \<longrightarrow>\<^isub>\<beta>\<^sup>* lt1+ $ lt2+ $ K"
by (rule evbeta', simp_all add: d e f)
finally show ?thesis using * d e f by simp
next
assume f: "\<not> isValue lt2"
- have "lt2* $ Abs c (lt1+ $ c~ $ K) \<longrightarrow>\<^isub>\<beta>\<^sup>* lt2 ; Abs c (lt1+ $ c~ $ K)" using * d e f by simp
- also have "... \<longrightarrow>\<^isub>\<beta>\<^sup>* lt2 ; Abs a (lt1+ $ a~ $ K)" using Kapply.simps(4) d e evs1 f by metis
+ have "lt2* $ Lam c (lt1+ $ c~ $ K) \<longrightarrow>\<^isub>\<beta>\<^sup>* lt2 ; Lam c (lt1+ $ c~ $ K)" using * d e f by simp
+ also have "... \<longrightarrow>\<^isub>\<beta>\<^sup>* lt2 ; Lam a (lt1+ $ a~ $ K)" using Kapply.simps(4) d e evs1 f by metis
finally show ?thesis using * d e f by simp
qed
finally show ?thesis .
@@ -355,11 +253,11 @@
case (evbeta x V M)
fix K
assume a: "isValue K" "isValue V" "atom x \<sharp> V"
- have "Abs x (M*) $ V+ $ K \<longrightarrow>\<^isub>\<beta>\<^sup>* ((M*)[V+/x] $ K)"
+ have "Lam x (M*) $ V+ $ K \<longrightarrow>\<^isub>\<beta>\<^sup>* (((M*)[x ::= V+]) $ K)"
by (rule evs2,rule ev2,rule Lt.evbeta) (simp_all add: fresh_def a[simplified fresh_def] evs1)
- also have "... = ((M[V/x])* $ K)" by (simp add: CPS_subst_fv a)
- also have "... \<longrightarrow>\<^isub>\<beta>\<^sup>* (M[V/x] ; K)" by (rule CPS_eval_Kapply, simp_all add: a)
- finally show "(Abs x M $ V ; K) \<longrightarrow>\<^isub>\<beta>\<^sup>* (M[V/x] ; K)" using a by simp
+ also have "... = ((M[x ::= V])* $ K)" by (simp add: CPS_subst_fv a)
+ also have "... \<longrightarrow>\<^isub>\<beta>\<^sup>* ((M[x ::= V]) ; K)" by (rule CPS_eval_Kapply, simp_all add: a)
+ finally show "(Lam x M $ V ; K) \<longrightarrow>\<^isub>\<beta>\<^sup>* ((M[x ::= V]) ; K)" using a by simp
next
case (ev1 V M N)
fix V M N K
@@ -370,7 +268,7 @@
then show "V $ M ; K \<longrightarrow>\<^isub>\<beta>\<^sup>* V $ N ; K" using a b by simp
next
assume n: "isValue N"
- have c: "M; Abs a (V+ $ a~ $ K) \<longrightarrow>\<^isub>\<beta>\<^sup>* Abs a (V+ $ a~ $ K) $ N+" using a b by (simp add: n)
+ have c: "M; Lam a (V+ $ a~ $ K) \<longrightarrow>\<^isub>\<beta>\<^sup>* Lam a (V+ $ a~ $ K) $ N+" using a b by (simp add: n)
also have d: "... \<longrightarrow>\<^isub>\<beta>\<^sup>* V+ $ N+ $ K" by (rule evbeta') (simp_all add: a b n)
finally show "V $ M ; K \<longrightarrow>\<^isub>\<beta>\<^sup>* V $ N ; K" using a b by (simp add: n)
qed
@@ -381,19 +279,19 @@
obtain b :: name where b: "atom b \<sharp> (a, K, M, N, M', N+)" using obtain_fresh by blast
have d: "atom a \<sharp> K" "atom a \<sharp> M" "atom a \<sharp> N" "atom a \<sharp> M'" "atom b \<sharp> a" "atom b \<sharp> K"
"atom b \<sharp> M" "atom b \<sharp> N" "atom b \<sharp> M'" using a b fresh_Pair by simp_all
- have "M $ N ; K \<longrightarrow>\<^isub>\<beta>\<^sup>* M' ; Abs a (N* $ Abs b (a~ $ b~ $ K))" using * d by simp
+ have "M $ N ; K \<longrightarrow>\<^isub>\<beta>\<^sup>* M' ; Lam a (N* $ Lam b (a~ $ b~ $ K))" using * d by simp
also have "... \<longrightarrow>\<^isub>\<beta>\<^sup>* M' $ N ; K" proof (cases "isValue M'")
assume "\<not> isValue M'"
then show ?thesis using * d by (simp_all add: evs1)
next
assume e: "isValue M'"
- then have "M' ; Abs a (N* $ Abs b (a~ $ b~ $ K)) = Abs a (N* $ Abs b (a~ $ b~ $ K)) $ M'+" by simp
- also have "... \<longrightarrow>\<^isub>\<beta>\<^sup>* (N* $ Abs b (a~ $ b~ $ K))[M'+/a]"
+ then have "M' ; Lam a (N* $ Lam b (a~ $ b~ $ K)) = Lam a (N* $ Lam b (a~ $ b~ $ K)) $ M'+" by simp
+ also have "... \<longrightarrow>\<^isub>\<beta>\<^sup>* (N* $ Lam b (a~ $ b~ $ K))[a ::= M'+]"
by (rule evbeta') (simp_all add: fresh_at_base e d)
- also have "... = N* $ Abs b (M'+ $ b~ $ K)" using * d by (simp add: fresh_at_base)
+ also have "... = N* $ Lam b (M'+ $ b~ $ K)" using * d by (simp add: fresh_at_base)
also have "... \<longrightarrow>\<^isub>\<beta>\<^sup>* M' $ N ; K" proof (cases "isValue N")
assume f: "isValue N"
- have "N* $ Abs b (M'+ $ b~ $ K) \<longrightarrow>\<^isub>\<beta>\<^sup>* Abs b (M'+ $ b~ $ K) $ N+"
+ have "N* $ Lam b (M'+ $ b~ $ K) \<longrightarrow>\<^isub>\<beta>\<^sup>* Lam b (M'+ $ b~ $ K) $ N+"
by (rule eval_trans, rule CPS_eval_Kapply) (simp_all add: d e f * evs1)
also have "... \<longrightarrow>\<^isub>\<beta>\<^sup>* M' $ N ; K" by (rule evbeta') (simp_all add: d e f evs1)
finally show ?thesis .
@@ -415,17 +313,17 @@
lemma
assumes "isValue V" "M \<longrightarrow>\<^isub>\<beta>\<^sup>* V"
- shows "M* $ (Abs x (x~)) \<longrightarrow>\<^isub>\<beta>\<^sup>* V+"
+ shows "M* $ (Lam x (x~)) \<longrightarrow>\<^isub>\<beta>\<^sup>* V+"
proof-
obtain y::name where *: "atom y \<sharp> V" using obtain_fresh by blast
- have e: "Abs x (x~) = Abs y (y~)"
+ have e: "Lam x (x~) = Lam y (y~)"
by (simp add: Abs1_eq_iff lt.fresh fresh_at_base)
- have "M* $ Abs x (x~) \<longrightarrow>\<^isub>\<beta>\<^sup>* M ; Abs x (x~)"
+ have "M* $ Lam x (x~) \<longrightarrow>\<^isub>\<beta>\<^sup>* M ; Lam x (x~)"
by(rule CPS_eval_Kapply,simp_all add: assms)
- also have "... \<longrightarrow>\<^isub>\<beta>\<^sup>* (V ; Abs x (x~))" by (rule Kapply_eval_rtrancl, simp_all add: assms)
- also have "... = V ; Abs y (y~)" using e by (simp only:)
+ also have "... \<longrightarrow>\<^isub>\<beta>\<^sup>* (V ; Lam x (x~))" by (rule Kapply_eval_rtrancl, simp_all add: assms)
+ also have "... = V ; Lam y (y~)" using e by (simp only:)
also have "... \<longrightarrow>\<^isub>\<beta>\<^sup>* (V+)" by (simp add: assms, rule evbeta') (simp_all add: assms *)
- finally show "M* $ (Abs x (x~)) \<longrightarrow>\<^isub>\<beta>\<^sup>* (V+)" .
+ finally show "M* $ (Lam x (x~)) \<longrightarrow>\<^isub>\<beta>\<^sup>* (V+)" .
qed
end