eliminated hard tabulators, guessing at each author's individual tab-width;
tuned headers;
(* Title: HOL/Lambda/StrongNorm.thy
Author: Stefan Berghofer
Copyright 2000 TU Muenchen
*)
header {* Strong normalization for simply-typed lambda calculus *}
theory StrongNorm imports Type InductTermi begin
text {*
Formalization by Stefan Berghofer. Partly based on a paper proof by
Felix Joachimski and Ralph Matthes \cite{Matthes-Joachimski-AML}.
*}
subsection {* Properties of @{text IT} *}
lemma lift_IT [intro!]: "IT t \<Longrightarrow> IT (lift t i)"
apply (induct arbitrary: i set: IT)
apply (simp (no_asm))
apply (rule conjI)
apply
(rule impI,
rule IT.Var,
erule listsp.induct,
simp (no_asm),
rule listsp.Nil,
simp (no_asm),
rule listsp.Cons,
blast,
assumption)+
apply auto
done
lemma lifts_IT: "listsp IT ts \<Longrightarrow> listsp IT (map (\<lambda>t. lift t 0) ts)"
by (induct ts) auto
lemma subst_Var_IT: "IT r \<Longrightarrow> IT (r[Var i/j])"
apply (induct arbitrary: i j set: IT)
txt {* Case @{term Var}: *}
apply (simp (no_asm) add: subst_Var)
apply
((rule conjI impI)+,
rule IT.Var,
erule listsp.induct,
simp (no_asm),
rule listsp.Nil,
simp (no_asm),
rule listsp.Cons,
fast,
assumption)+
txt {* Case @{term Lambda}: *}
apply atomize
apply simp
apply (rule IT.Lambda)
apply fast
txt {* Case @{term Beta}: *}
apply atomize
apply (simp (no_asm_use) add: subst_subst [symmetric])
apply (rule IT.Beta)
apply auto
done
lemma Var_IT: "IT (Var n)"
apply (subgoal_tac "IT (Var n \<degree>\<degree> [])")
apply simp
apply (rule IT.Var)
apply (rule listsp.Nil)
done
lemma app_Var_IT: "IT t \<Longrightarrow> IT (t \<degree> Var i)"
apply (induct set: IT)
apply (subst app_last)
apply (rule IT.Var)
apply simp
apply (rule listsp.Cons)
apply (rule Var_IT)
apply (rule listsp.Nil)
apply (rule IT.Beta [where ?ss = "[]", unfolded foldl_Nil [THEN eq_reflection]])
apply (erule subst_Var_IT)
apply (rule Var_IT)
apply (subst app_last)
apply (rule IT.Beta)
apply (subst app_last [symmetric])
apply assumption
apply assumption
done
subsection {* Well-typed substitution preserves termination *}
lemma subst_type_IT:
"\<And>t e T u i. IT t \<Longrightarrow> e\<langle>i:U\<rangle> \<turnstile> t : T \<Longrightarrow>
IT u \<Longrightarrow> e \<turnstile> u : U \<Longrightarrow> IT (t[u/i])"
(is "PROP ?P U" is "\<And>t e T u i. _ \<Longrightarrow> PROP ?Q t e T u i U")
proof (induct U)
fix T t
assume MI1: "\<And>T1 T2. T = T1 \<Rightarrow> T2 \<Longrightarrow> PROP ?P T1"
assume MI2: "\<And>T1 T2. T = T1 \<Rightarrow> T2 \<Longrightarrow> PROP ?P T2"
assume "IT t"
thus "\<And>e T' u i. PROP ?Q t e T' u i T"
proof induct
fix e T' u i
assume uIT: "IT u"
assume uT: "e \<turnstile> u : T"
{
case (Var rs n e_ T'_ u_ i_)
assume nT: "e\<langle>i:T\<rangle> \<turnstile> Var n \<degree>\<degree> rs : T'"
let ?ty = "\<lambda>t. \<exists>T'. e\<langle>i:T\<rangle> \<turnstile> t : T'"
let ?R = "\<lambda>t. \<forall>e T' u i.
e\<langle>i:T\<rangle> \<turnstile> t : T' \<longrightarrow> IT u \<longrightarrow> e \<turnstile> u : T \<longrightarrow> IT (t[u/i])"
show "IT ((Var n \<degree>\<degree> rs)[u/i])"
proof (cases "n = i")
case True
show ?thesis
proof (cases rs)
case Nil
with uIT True show ?thesis by simp
next
case (Cons a as)
with nT have "e\<langle>i:T\<rangle> \<turnstile> Var n \<degree> a \<degree>\<degree> as : T'" by simp
then obtain Ts
where headT: "e\<langle>i:T\<rangle> \<turnstile> Var n \<degree> a : Ts \<Rrightarrow> T'"
and argsT: "e\<langle>i:T\<rangle> \<tturnstile> as : Ts"
by (rule list_app_typeE)
from headT obtain T''
where varT: "e\<langle>i:T\<rangle> \<turnstile> Var n : T'' \<Rightarrow> Ts \<Rrightarrow> T'"
and argT: "e\<langle>i:T\<rangle> \<turnstile> a : T''"
by cases simp_all
from varT True have T: "T = T'' \<Rightarrow> Ts \<Rrightarrow> T'"
by cases auto
with uT have uT': "e \<turnstile> u : T'' \<Rightarrow> Ts \<Rrightarrow> T'" by simp
from T have "IT ((Var 0 \<degree>\<degree> map (\<lambda>t. lift t 0)
(map (\<lambda>t. t[u/i]) as))[(u \<degree> a[u/i])/0])"
proof (rule MI2)
from T have "IT ((lift u 0 \<degree> Var 0)[a[u/i]/0])"
proof (rule MI1)
have "IT (lift u 0)" by (rule lift_IT [OF uIT])
thus "IT (lift u 0 \<degree> Var 0)" by (rule app_Var_IT)
show "e\<langle>0:T''\<rangle> \<turnstile> lift u 0 \<degree> Var 0 : Ts \<Rrightarrow> T'"
proof (rule typing.App)
show "e\<langle>0:T''\<rangle> \<turnstile> lift u 0 : T'' \<Rightarrow> Ts \<Rrightarrow> T'"
by (rule lift_type) (rule uT')
show "e\<langle>0:T''\<rangle> \<turnstile> Var 0 : T''"
by (rule typing.Var) simp
qed
from Var have "?R a" by cases (simp_all add: Cons)
with argT uIT uT show "IT (a[u/i])" by simp
from argT uT show "e \<turnstile> a[u/i] : T''"
by (rule subst_lemma) simp
qed
thus "IT (u \<degree> a[u/i])" by simp
from Var have "listsp ?R as"
by cases (simp_all add: Cons)
moreover from argsT have "listsp ?ty as"
by (rule lists_typings)
ultimately have "listsp (\<lambda>t. ?R t \<and> ?ty t) as"
by simp
hence "listsp IT (map (\<lambda>t. lift t 0) (map (\<lambda>t. t[u/i]) as))"
(is "listsp IT (?ls as)")
proof induct
case Nil
show ?case by fastsimp
next
case (Cons b bs)
hence I: "?R b" by simp
from Cons obtain U where "e\<langle>i:T\<rangle> \<turnstile> b : U" by fast
with uT uIT I have "IT (b[u/i])" by simp
hence "IT (lift (b[u/i]) 0)" by (rule lift_IT)
hence "listsp IT (lift (b[u/i]) 0 # ?ls bs)"
by (rule listsp.Cons) (rule Cons)
thus ?case by simp
qed
thus "IT (Var 0 \<degree>\<degree> ?ls as)" by (rule IT.Var)
have "e\<langle>0:Ts \<Rrightarrow> T'\<rangle> \<turnstile> Var 0 : Ts \<Rrightarrow> T'"
by (rule typing.Var) simp
moreover from uT argsT have "e \<tturnstile> map (\<lambda>t. t[u/i]) as : Ts"
by (rule substs_lemma)
hence "e\<langle>0:Ts \<Rrightarrow> T'\<rangle> \<tturnstile> ?ls as : Ts"
by (rule lift_types)
ultimately show "e\<langle>0:Ts \<Rrightarrow> T'\<rangle> \<turnstile> Var 0 \<degree>\<degree> ?ls as : T'"
by (rule list_app_typeI)
from argT uT have "e \<turnstile> a[u/i] : T''"
by (rule subst_lemma) (rule refl)
with uT' show "e \<turnstile> u \<degree> a[u/i] : Ts \<Rrightarrow> T'"
by (rule typing.App)
qed
with Cons True show ?thesis
by (simp add: map_compose [symmetric] comp_def)
qed
next
case False
from Var have "listsp ?R rs" by simp
moreover from nT obtain Ts where "e\<langle>i:T\<rangle> \<tturnstile> rs : Ts"
by (rule list_app_typeE)
hence "listsp ?ty rs" by (rule lists_typings)
ultimately have "listsp (\<lambda>t. ?R t \<and> ?ty t) rs"
by simp
hence "listsp IT (map (\<lambda>x. x[u/i]) rs)"
proof induct
case Nil
show ?case by fastsimp
next
case (Cons a as)
hence I: "?R a" by simp
from Cons obtain U where "e\<langle>i:T\<rangle> \<turnstile> a : U" by fast
with uT uIT I have "IT (a[u/i])" by simp
hence "listsp IT (a[u/i] # map (\<lambda>t. t[u/i]) as)"
by (rule listsp.Cons) (rule Cons)
thus ?case by simp
qed
with False show ?thesis by (auto simp add: subst_Var)
qed
next
case (Lambda r e_ T'_ u_ i_)
assume "e\<langle>i:T\<rangle> \<turnstile> Abs r : T'"
and "\<And>e T' u i. PROP ?Q r e T' u i T"
with uIT uT show "IT (Abs r[u/i])"
by fastsimp
next
case (Beta r a as e_ T'_ u_ i_)
assume T: "e\<langle>i:T\<rangle> \<turnstile> Abs r \<degree> a \<degree>\<degree> as : T'"
assume SI1: "\<And>e T' u i. PROP ?Q (r[a/0] \<degree>\<degree> as) e T' u i T"
assume SI2: "\<And>e T' u i. PROP ?Q a e T' u i T"
have "IT (Abs (r[lift u 0/Suc i]) \<degree> a[u/i] \<degree>\<degree> map (\<lambda>t. t[u/i]) as)"
proof (rule IT.Beta)
have "Abs r \<degree> a \<degree>\<degree> as \<rightarrow>\<^sub>\<beta> r[a/0] \<degree>\<degree> as"
by (rule apps_preserves_beta) (rule beta.beta)
with T have "e\<langle>i:T\<rangle> \<turnstile> r[a/0] \<degree>\<degree> as : T'"
by (rule subject_reduction)
hence "IT ((r[a/0] \<degree>\<degree> as)[u/i])"
using uIT uT by (rule SI1)
thus "IT (r[lift u 0/Suc i][a[u/i]/0] \<degree>\<degree> map (\<lambda>t. t[u/i]) as)"
by (simp del: subst_map add: subst_subst subst_map [symmetric])
from T obtain U where "e\<langle>i:T\<rangle> \<turnstile> Abs r \<degree> a : U"
by (rule list_app_typeE) fast
then obtain T'' where "e\<langle>i:T\<rangle> \<turnstile> a : T''" by cases simp_all
thus "IT (a[u/i])" using uIT uT by (rule SI2)
qed
thus "IT ((Abs r \<degree> a \<degree>\<degree> as)[u/i])" by simp
}
qed
qed
subsection {* Well-typed terms are strongly normalizing *}
lemma type_implies_IT:
assumes "e \<turnstile> t : T"
shows "IT t"
using assms
proof induct
case Var
show ?case by (rule Var_IT)
next
case Abs
show ?case by (rule IT.Lambda) (rule Abs)
next
case (App e s T U t)
have "IT ((Var 0 \<degree> lift t 0)[s/0])"
proof (rule subst_type_IT)
have "IT (lift t 0)" using `IT t` by (rule lift_IT)
hence "listsp IT [lift t 0]" by (rule listsp.Cons) (rule listsp.Nil)
hence "IT (Var 0 \<degree>\<degree> [lift t 0])" by (rule IT.Var)
also have "Var 0 \<degree>\<degree> [lift t 0] = Var 0 \<degree> lift t 0" by simp
finally show "IT \<dots>" .
have "e\<langle>0:T \<Rightarrow> U\<rangle> \<turnstile> Var 0 : T \<Rightarrow> U"
by (rule typing.Var) simp
moreover have "e\<langle>0:T \<Rightarrow> U\<rangle> \<turnstile> lift t 0 : T"
by (rule lift_type) (rule App.hyps)
ultimately show "e\<langle>0:T \<Rightarrow> U\<rangle> \<turnstile> Var 0 \<degree> lift t 0 : U"
by (rule typing.App)
show "IT s" by fact
show "e \<turnstile> s : T \<Rightarrow> U" by fact
qed
thus ?case by simp
qed
theorem type_implies_termi: "e \<turnstile> t : T \<Longrightarrow> termip beta t"
proof -
assume "e \<turnstile> t : T"
hence "IT t" by (rule type_implies_IT)
thus ?thesis by (rule IT_implies_termi)
qed
end