(* Title: HOL/Proofs/Lambda/Standardization.thy
Author: Stefan Berghofer
Copyright 2005 TU Muenchen
*)
section \<open>Standardization\<close>
theory Standardization
imports NormalForm
begin
text \<open>
Based on lecture notes by Ralph Matthes @{cite "Matthes-ESSLLI2000"},
original proof idea due to Ralph Loader @{cite Loader1998}.
\<close>
subsection \<open>Standard reduction relation\<close>
declare listrel_mono [mono_set]
inductive
sred :: "dB \<Rightarrow> dB \<Rightarrow> bool" (infixl "\<rightarrow>\<^sub>s" 50)
and sredlist :: "dB list \<Rightarrow> dB list \<Rightarrow> bool" (infixl "[\<rightarrow>\<^sub>s]" 50)
where
"s [\<rightarrow>\<^sub>s] t \<equiv> listrelp (\<rightarrow>\<^sub>s) s t"
| Var: "rs [\<rightarrow>\<^sub>s] rs' \<Longrightarrow> Var x \<degree>\<degree> rs \<rightarrow>\<^sub>s Var x \<degree>\<degree> rs'"
| Abs: "r \<rightarrow>\<^sub>s r' \<Longrightarrow> ss [\<rightarrow>\<^sub>s] ss' \<Longrightarrow> Abs r \<degree>\<degree> ss \<rightarrow>\<^sub>s Abs r' \<degree>\<degree> ss'"
| Beta: "r[s/0] \<degree>\<degree> ss \<rightarrow>\<^sub>s t \<Longrightarrow> Abs r \<degree> s \<degree>\<degree> ss \<rightarrow>\<^sub>s t"
lemma refl_listrelp: "\<forall>x\<in>set xs. R x x \<Longrightarrow> listrelp R xs xs"
by (induct xs) (auto intro: listrelp.intros)
lemma refl_sred: "t \<rightarrow>\<^sub>s t"
by (induct t rule: Apps_dB_induct) (auto intro: refl_listrelp sred.intros)
lemma refl_sreds: "ts [\<rightarrow>\<^sub>s] ts"
by (simp add: refl_sred refl_listrelp)
lemma listrelp_conj1: "listrelp (\<lambda>x y. R x y \<and> S x y) x y \<Longrightarrow> listrelp R x y"
by (erule listrelp.induct) (auto intro: listrelp.intros)
lemma listrelp_conj2: "listrelp (\<lambda>x y. R x y \<and> S x y) x y \<Longrightarrow> listrelp S x y"
by (erule listrelp.induct) (auto intro: listrelp.intros)
lemma listrelp_app:
assumes xsys: "listrelp R xs ys"
shows "listrelp R xs' ys' \<Longrightarrow> listrelp R (xs @ xs') (ys @ ys')" using xsys
by (induct arbitrary: xs' ys') (auto intro: listrelp.intros)
lemma lemma1:
assumes r: "r \<rightarrow>\<^sub>s r'" and s: "s \<rightarrow>\<^sub>s s'"
shows "r \<degree> s \<rightarrow>\<^sub>s r' \<degree> s'" using r
proof induct
case (Var rs rs' x)
then have "rs [\<rightarrow>\<^sub>s] rs'" by (rule listrelp_conj1)
moreover have "[s] [\<rightarrow>\<^sub>s] [s']" by (iprover intro: s listrelp.intros)
ultimately have "rs @ [s] [\<rightarrow>\<^sub>s] rs' @ [s']" by (rule listrelp_app)
hence "Var x \<degree>\<degree> (rs @ [s]) \<rightarrow>\<^sub>s Var x \<degree>\<degree> (rs' @ [s'])" by (rule sred.Var)
thus ?case by (simp only: app_last)
next
case (Abs r r' ss ss')
from Abs(3) have "ss [\<rightarrow>\<^sub>s] ss'" by (rule listrelp_conj1)
moreover have "[s] [\<rightarrow>\<^sub>s] [s']" by (iprover intro: s listrelp.intros)
ultimately have "ss @ [s] [\<rightarrow>\<^sub>s] ss' @ [s']" by (rule listrelp_app)
with \<open>r \<rightarrow>\<^sub>s r'\<close> have "Abs r \<degree>\<degree> (ss @ [s]) \<rightarrow>\<^sub>s Abs r' \<degree>\<degree> (ss' @ [s'])"
by (rule sred.Abs)
thus ?case by (simp only: app_last)
next
case (Beta r u ss t)
hence "r[u/0] \<degree>\<degree> (ss @ [s]) \<rightarrow>\<^sub>s t \<degree> s'" by (simp only: app_last)
hence "Abs r \<degree> u \<degree>\<degree> (ss @ [s]) \<rightarrow>\<^sub>s t \<degree> s'" by (rule sred.Beta)
thus ?case by (simp only: app_last)
qed
lemma lemma1':
assumes ts: "ts [\<rightarrow>\<^sub>s] ts'"
shows "r \<rightarrow>\<^sub>s r' \<Longrightarrow> r \<degree>\<degree> ts \<rightarrow>\<^sub>s r' \<degree>\<degree> ts'" using ts
by (induct arbitrary: r r') (auto intro: lemma1)
lemma lemma2_1:
assumes beta: "t \<rightarrow>\<^sub>\<beta> u"
shows "t \<rightarrow>\<^sub>s u" using beta
proof induct
case (beta s t)
have "Abs s \<degree> t \<degree>\<degree> [] \<rightarrow>\<^sub>s s[t/0] \<degree>\<degree> []" by (iprover intro: sred.Beta refl_sred)
thus ?case by simp
next
case (appL s t u)
thus ?case by (iprover intro: lemma1 refl_sred)
next
case (appR s t u)
thus ?case by (iprover intro: lemma1 refl_sred)
next
case (abs s t)
hence "Abs s \<degree>\<degree> [] \<rightarrow>\<^sub>s Abs t \<degree>\<degree> []" by (iprover intro: sred.Abs listrelp.Nil)
thus ?case by simp
qed
lemma listrelp_betas:
assumes ts: "listrelp (\<rightarrow>\<^sub>\<beta>\<^sup>*) ts ts'"
shows "\<And>t t'. t \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<Longrightarrow> t \<degree>\<degree> ts \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<degree>\<degree> ts'" using ts
by induct auto
lemma lemma2_2:
assumes t: "t \<rightarrow>\<^sub>s u"
shows "t \<rightarrow>\<^sub>\<beta>\<^sup>* u" using t
by induct (auto dest: listrelp_conj2
intro: listrelp_betas apps_preserves_beta converse_rtranclp_into_rtranclp)
lemma sred_lift:
assumes s: "s \<rightarrow>\<^sub>s t"
shows "lift s i \<rightarrow>\<^sub>s lift t i" using s
proof (induct arbitrary: i)
case (Var rs rs' x)
hence "map (\<lambda>t. lift t i) rs [\<rightarrow>\<^sub>s] map (\<lambda>t. lift t i) rs'"
by induct (auto intro: listrelp.intros)
thus ?case by (cases "x < i") (auto intro: sred.Var)
next
case (Abs r r' ss ss')
from Abs(3) have "map (\<lambda>t. lift t i) ss [\<rightarrow>\<^sub>s] map (\<lambda>t. lift t i) ss'"
by induct (auto intro: listrelp.intros)
thus ?case by (auto intro: sred.Abs Abs)
next
case (Beta r s ss t)
thus ?case by (auto intro: sred.Beta)
qed
lemma lemma3:
assumes r: "r \<rightarrow>\<^sub>s r'"
shows "s \<rightarrow>\<^sub>s s' \<Longrightarrow> r[s/x] \<rightarrow>\<^sub>s r'[s'/x]" using r
proof (induct arbitrary: s s' x)
case (Var rs rs' y)
hence "map (\<lambda>t. t[s/x]) rs [\<rightarrow>\<^sub>s] map (\<lambda>t. t[s'/x]) rs'"
by induct (auto intro: listrelp.intros Var)
moreover have "Var y[s/x] \<rightarrow>\<^sub>s Var y[s'/x]"
proof (cases "y < x")
case True thus ?thesis by simp (rule refl_sred)
next
case False
thus ?thesis
by (cases "y = x") (auto simp add: Var intro: refl_sred)
qed
ultimately show ?case by simp (rule lemma1')
next
case (Abs r r' ss ss')
from Abs(4) have "lift s 0 \<rightarrow>\<^sub>s lift s' 0" by (rule sred_lift)
hence "r[lift s 0/Suc x] \<rightarrow>\<^sub>s r'[lift s' 0/Suc x]" by (fast intro: Abs.hyps)
moreover from Abs(3) have "map (\<lambda>t. t[s/x]) ss [\<rightarrow>\<^sub>s] map (\<lambda>t. t[s'/x]) ss'"
by induct (auto intro: listrelp.intros Abs)
ultimately show ?case by simp (rule sred.Abs)
next
case (Beta r u ss t)
thus ?case by (auto simp add: subst_subst intro: sred.Beta)
qed
lemma lemma4_aux:
assumes rs: "listrelp (\<lambda>t u. t \<rightarrow>\<^sub>s u \<and> (\<forall>r. u \<rightarrow>\<^sub>\<beta> r \<longrightarrow> t \<rightarrow>\<^sub>s r)) rs rs'"
shows "rs' => ss \<Longrightarrow> rs [\<rightarrow>\<^sub>s] ss" using rs
proof (induct arbitrary: ss)
case Nil
thus ?case by cases (auto intro: listrelp.Nil)
next
case (Cons x y xs ys)
note Cons' = Cons
show ?case
proof (cases ss)
case Nil with Cons show ?thesis by simp
next
case (Cons y' ys')
hence ss: "ss = y' # ys'" by simp
from Cons Cons' have "y \<rightarrow>\<^sub>\<beta> y' \<and> ys' = ys \<or> y' = y \<and> ys => ys'" by simp
hence "x # xs [\<rightarrow>\<^sub>s] y' # ys'"
proof
assume H: "y \<rightarrow>\<^sub>\<beta> y' \<and> ys' = ys"
with Cons' have "x \<rightarrow>\<^sub>s y'" by blast
moreover from Cons' have "xs [\<rightarrow>\<^sub>s] ys" by (iprover dest: listrelp_conj1)
ultimately have "x # xs [\<rightarrow>\<^sub>s] y' # ys" by (rule listrelp.Cons)
with H show ?thesis by simp
next
assume H: "y' = y \<and> ys => ys'"
with Cons' have "x \<rightarrow>\<^sub>s y'" by blast
moreover from H have "xs [\<rightarrow>\<^sub>s] ys'" by (blast intro: Cons')
ultimately show ?thesis by (rule listrelp.Cons)
qed
with ss show ?thesis by simp
qed
qed
lemma lemma4:
assumes r: "r \<rightarrow>\<^sub>s r'"
shows "r' \<rightarrow>\<^sub>\<beta> r'' \<Longrightarrow> r \<rightarrow>\<^sub>s r''" using r
proof (induct arbitrary: r'')
case (Var rs rs' x)
then obtain ss where rs: "rs' => ss" and r'': "r'' = Var x \<degree>\<degree> ss"
by (blast dest: head_Var_reduction)
from Var(1) rs have "rs [\<rightarrow>\<^sub>s] ss" by (rule lemma4_aux)
hence "Var x \<degree>\<degree> rs \<rightarrow>\<^sub>s Var x \<degree>\<degree> ss" by (rule sred.Var)
with r'' show ?case by simp
next
case (Abs r r' ss ss')
from \<open>Abs r' \<degree>\<degree> ss' \<rightarrow>\<^sub>\<beta> r''\<close> show ?case
proof
fix s
assume r'': "r'' = s \<degree>\<degree> ss'"
assume "Abs r' \<rightarrow>\<^sub>\<beta> s"
then obtain r''' where s: "s = Abs r'''" and r''': "r' \<rightarrow>\<^sub>\<beta> r'''" by cases auto
from r''' have "r \<rightarrow>\<^sub>s r'''" by (blast intro: Abs)
moreover from Abs have "ss [\<rightarrow>\<^sub>s] ss'" by (iprover dest: listrelp_conj1)
ultimately have "Abs r \<degree>\<degree> ss \<rightarrow>\<^sub>s Abs r''' \<degree>\<degree> ss'" by (rule sred.Abs)
with r'' s show "Abs r \<degree>\<degree> ss \<rightarrow>\<^sub>s r''" by simp
next
fix rs'
assume "ss' => rs'"
with Abs(3) have "ss [\<rightarrow>\<^sub>s] rs'" by (rule lemma4_aux)
with \<open>r \<rightarrow>\<^sub>s r'\<close> have "Abs r \<degree>\<degree> ss \<rightarrow>\<^sub>s Abs r' \<degree>\<degree> rs'" by (rule sred.Abs)
moreover assume "r'' = Abs r' \<degree>\<degree> rs'"
ultimately show "Abs r \<degree>\<degree> ss \<rightarrow>\<^sub>s r''" by simp
next
fix t u' us'
assume "ss' = u' # us'"
with Abs(3) obtain u us where
ss: "ss = u # us" and u: "u \<rightarrow>\<^sub>s u'" and us: "us [\<rightarrow>\<^sub>s] us'"
by cases (auto dest!: listrelp_conj1)
have "r[u/0] \<rightarrow>\<^sub>s r'[u'/0]" using Abs(1) and u by (rule lemma3)
with us have "r[u/0] \<degree>\<degree> us \<rightarrow>\<^sub>s r'[u'/0] \<degree>\<degree> us'" by (rule lemma1')
hence "Abs r \<degree> u \<degree>\<degree> us \<rightarrow>\<^sub>s r'[u'/0] \<degree>\<degree> us'" by (rule sred.Beta)
moreover assume "Abs r' = Abs t" and "r'' = t[u'/0] \<degree>\<degree> us'"
ultimately show "Abs r \<degree>\<degree> ss \<rightarrow>\<^sub>s r''" using ss by simp
qed
next
case (Beta r s ss t)
show ?case
by (rule sred.Beta) (rule Beta)+
qed
lemma rtrancl_beta_sred:
assumes r: "r \<rightarrow>\<^sub>\<beta>\<^sup>* r'"
shows "r \<rightarrow>\<^sub>s r'" using r
by induct (iprover intro: refl_sred lemma4)+
subsection \<open>Leftmost reduction and weakly normalizing terms\<close>
inductive
lred :: "dB \<Rightarrow> dB \<Rightarrow> bool" (infixl "\<rightarrow>\<^sub>l" 50)
and lredlist :: "dB list \<Rightarrow> dB list \<Rightarrow> bool" (infixl "[\<rightarrow>\<^sub>l]" 50)
where
"s [\<rightarrow>\<^sub>l] t \<equiv> listrelp (\<rightarrow>\<^sub>l) s t"
| Var: "rs [\<rightarrow>\<^sub>l] rs' \<Longrightarrow> Var x \<degree>\<degree> rs \<rightarrow>\<^sub>l Var x \<degree>\<degree> rs'"
| Abs: "r \<rightarrow>\<^sub>l r' \<Longrightarrow> Abs r \<rightarrow>\<^sub>l Abs r'"
| Beta: "r[s/0] \<degree>\<degree> ss \<rightarrow>\<^sub>l t \<Longrightarrow> Abs r \<degree> s \<degree>\<degree> ss \<rightarrow>\<^sub>l t"
lemma lred_imp_sred:
assumes lred: "s \<rightarrow>\<^sub>l t"
shows "s \<rightarrow>\<^sub>s t" using lred
proof induct
case (Var rs rs' x)
then have "rs [\<rightarrow>\<^sub>s] rs'"
by induct (iprover intro: listrelp.intros)+
then show ?case by (rule sred.Var)
next
case (Abs r r')
from \<open>r \<rightarrow>\<^sub>s r'\<close>
have "Abs r \<degree>\<degree> [] \<rightarrow>\<^sub>s Abs r' \<degree>\<degree> []" using listrelp.Nil
by (rule sred.Abs)
then show ?case by simp
next
case (Beta r s ss t)
from \<open>r[s/0] \<degree>\<degree> ss \<rightarrow>\<^sub>s t\<close>
show ?case by (rule sred.Beta)
qed
inductive WN :: "dB => bool"
where
Var: "listsp WN rs \<Longrightarrow> WN (Var n \<degree>\<degree> rs)"
| Lambda: "WN r \<Longrightarrow> WN (Abs r)"
| Beta: "WN ((r[s/0]) \<degree>\<degree> ss) \<Longrightarrow> WN ((Abs r \<degree> s) \<degree>\<degree> ss)"
lemma listrelp_imp_listsp1:
assumes H: "listrelp (\<lambda>x y. P x) xs ys"
shows "listsp P xs" using H
by induct auto
lemma listrelp_imp_listsp2:
assumes H: "listrelp (\<lambda>x y. P y) xs ys"
shows "listsp P ys" using H
by induct auto
lemma lemma5:
assumes lred: "r \<rightarrow>\<^sub>l r'"
shows "WN r" and "NF r'" using lred
by induct
(iprover dest: listrelp_conj1 listrelp_conj2
listrelp_imp_listsp1 listrelp_imp_listsp2 intro: WN.intros
NF.intros [simplified listall_listsp_eq])+
lemma lemma6:
assumes wn: "WN r"
shows "\<exists>r'. r \<rightarrow>\<^sub>l r'" using wn
proof induct
case (Var rs n)
then have "\<exists>rs'. rs [\<rightarrow>\<^sub>l] rs'"
by induct (iprover intro: listrelp.intros)+
then show ?case by (iprover intro: lred.Var)
qed (iprover intro: lred.intros)+
lemma lemma7:
assumes r: "r \<rightarrow>\<^sub>s r'"
shows "NF r' \<Longrightarrow> r \<rightarrow>\<^sub>l r'" using r
proof induct
case (Var rs rs' x)
from \<open>NF (Var x \<degree>\<degree> rs')\<close> have "listall NF rs'"
by cases simp_all
with Var(1) have "rs [\<rightarrow>\<^sub>l] rs'"
proof induct
case Nil
show ?case by (rule listrelp.Nil)
next
case (Cons x y xs ys)
hence "x \<rightarrow>\<^sub>l y" and "xs [\<rightarrow>\<^sub>l] ys" by simp_all
thus ?case by (rule listrelp.Cons)
qed
thus ?case by (rule lred.Var)
next
case (Abs r r' ss ss')
from \<open>NF (Abs r' \<degree>\<degree> ss')\<close>
have ss': "ss' = []" by (rule Abs_NF)
from Abs(3) have ss: "ss = []" using ss'
by cases simp_all
from ss' Abs have "NF (Abs r')" by simp
hence "NF r'" by cases simp_all
with Abs have "r \<rightarrow>\<^sub>l r'" by simp
hence "Abs r \<rightarrow>\<^sub>l Abs r'" by (rule lred.Abs)
with ss ss' show ?case by simp
next
case (Beta r s ss t)
hence "r[s/0] \<degree>\<degree> ss \<rightarrow>\<^sub>l t" by simp
thus ?case by (rule lred.Beta)
qed
lemma WN_eq: "WN t = (\<exists>t'. t \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t')"
proof
assume "WN t"
then have "\<exists>t'. t \<rightarrow>\<^sub>l t'" by (rule lemma6)
then obtain t' where t': "t \<rightarrow>\<^sub>l t'" ..
then have NF: "NF t'" by (rule lemma5)
from t' have "t \<rightarrow>\<^sub>s t'" by (rule lred_imp_sred)
then have "t \<rightarrow>\<^sub>\<beta>\<^sup>* t'" by (rule lemma2_2)
with NF show "\<exists>t'. t \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t'" by iprover
next
assume "\<exists>t'. t \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t'"
then obtain t' where t': "t \<rightarrow>\<^sub>\<beta>\<^sup>* t'" and NF: "NF t'"
by iprover
from t' have "t \<rightarrow>\<^sub>s t'" by (rule rtrancl_beta_sred)
then have "t \<rightarrow>\<^sub>l t'" using NF by (rule lemma7)
then show "WN t" by (rule lemma5)
qed
end