Theory Standardization

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theory Standardization
imports NormalForm

(*  Title:      HOL/Lambda/Standardization.thy
Author: Stefan Berghofer
Copyright 2005 TU Muenchen
*)


header {* Standardization *}

theory Standardization
imports NormalForm
begin


text {*
Based on lecture notes by Ralph Matthes \cite{Matthes-ESSLLI2000},
original proof idea due to Ralph Loader \cite{Loader1998}.
*}



subsection {* Standard reduction relation *}

declare listrel_mono [mono_set]

inductive
sred :: "dB => dB => bool" (infixl "->s" 50)
and sredlist :: "dB list => dB list => bool" (infixl "[->s]" 50)
where
"s [->s] t ≡ listrelp op ->s s t"
| Var: "rs [->s] rs' ==> Var x °° rs ->s Var x °° rs'"
| Abs: "r ->s r' ==> ss [->s] ss' ==> Abs r °° ss ->s Abs r' °° ss'"
| Beta: "r[s/0] °° ss ->s t ==> Abs r ° s °° ss ->s t"


lemma refl_listrelp: "∀x∈set xs. R x x ==> listrelp R xs xs"
by (induct xs) (auto intro: listrelp.intros)

lemma refl_sred: "t ->s t"
by (induct t rule: Apps_dB_induct) (auto intro: refl_listrelp sred.intros)

lemma refl_sreds: "ts [->s] ts"
by (simp add: refl_sred refl_listrelp)

lemma listrelp_conj1: "listrelp (λx y. R x y ∧ S x y) x y ==> listrelp R x y"
by (erule listrelp.induct) (auto intro: listrelp.intros)

lemma listrelp_conj2: "listrelp (λx y. R x y ∧ S x y) x y ==> 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' ==> listrelp R (xs @ xs') (ys @ ys')"
using xsys
by (induct arbitrary: xs' ys') (auto intro: listrelp.intros)

lemma lemma1:
assumes r: "r ->s r'" and s: "s ->s s'"
shows "r ° s ->s r' ° s'"
using r
proof induct
case (Var rs rs' x)
then have "rs [->s] rs'" by (rule listrelp_conj1)
moreover have "[s] [->s] [s']" by (iprover intro: s listrelp.intros)
ultimately have "rs @ [s] [->s] rs' @ [s']" by (rule listrelp_app)
hence "Var x °° (rs @ [s]) ->s Var x °° (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 [->s] ss'" by (rule listrelp_conj1)
moreover have "[s] [->s] [s']" by (iprover intro: s listrelp.intros)
ultimately have "ss @ [s] [->s] ss' @ [s']" by (rule listrelp_app)
with `r ->s r'` have "Abs r °° (ss @ [s]) ->s Abs r' °° (ss' @ [s'])"
by (rule sred.Abs)
thus ?case by (simp only: app_last)
next
case (Beta r u ss t)
hence "r[u/0] °° (ss @ [s]) ->s t ° s'" by (simp only: app_last)
hence "Abs r ° u °° (ss @ [s]) ->s t ° s'" by (rule sred.Beta)
thus ?case by (simp only: app_last)
qed

lemma lemma1':
assumes ts: "ts [->s] ts'"
shows "r ->s r' ==> r °° ts ->s r' °° ts'"
using ts
by (induct arbitrary: r r') (auto intro: lemma1)

lemma lemma2_1:
assumes beta: "t ->β u"
shows "t ->s u"
using beta
proof induct
case (beta s t)
have "Abs s ° t °° [] ->s s[t/0] °° []" 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 °° [] ->s Abs t °° []" by (iprover intro: sred.Abs listrelp.Nil)
thus ?case by simp
qed

lemma listrelp_betas:
assumes ts: "listrelp op ->β* ts ts'"
shows "!!t t'. t ->β* t' ==> t °° ts ->β* t' °° ts'"
using ts
by induct auto

lemma lemma2_2:
assumes t: "t ->s u"
shows "t ->β* 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 ->s t"
shows "lift s i ->s lift t i"
using s
proof (induct arbitrary: i)
case (Var rs rs' x)
hence "map (λt. lift t i) rs [->s] map (λt. 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 (λt. lift t i) ss [->s] map (λt. 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 ->s r'"
shows "s ->s s' ==> r[s/x] ->s r'[s'/x]"
using r
proof (induct arbitrary: s s' x)
case (Var rs rs' y)
hence "map (λt. t[s/x]) rs [->s] map (λt. t[s'/x]) rs'"
by induct (auto intro: listrelp.intros Var)
moreover have "Var y[s/x] ->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 ->s lift s' 0" by (rule sred_lift)
hence "r[lift s 0/Suc x] ->s r'[lift s' 0/Suc x]" by (fast intro: Abs.hyps)
moreover from Abs(3) have "map (λt. t[s/x]) ss [->s] map (λt. 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 (λt u. t ->s u ∧ (∀r. u ->β r --> t ->s r)) rs rs'"
shows "rs' => ss ==> rs [->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 ->β y' ∧ ys' = ys ∨ y' = y ∧ ys => ys'" by simp
hence "x # xs [->s] y' # ys'"
proof
assume H: "y ->β y' ∧ ys' = ys"
with Cons' have "x ->s y'" by blast
moreover from Cons' have "xs [->s] ys" by (iprover dest: listrelp_conj1)
ultimately have "x # xs [->s] y' # ys" by (rule listrelp.Cons)
with H show ?thesis by simp
next
assume H: "y' = y ∧ ys => ys'"
with Cons' have "x ->s y'" by blast
moreover from H have "xs [->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 ->s r'"
shows "r' ->β r'' ==> r ->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 °° ss"
by (blast dest: head_Var_reduction)
from Var(1) rs have "rs [->s] ss" by (rule lemma4_aux)
hence "Var x °° rs ->s Var x °° ss" by (rule sred.Var)
with r'' show ?case by simp
next
case (Abs r r' ss ss')
from `Abs r' °° ss' ->β r''` show ?case
proof
fix s
assume r'': "r'' = s °° ss'"
assume "Abs r' ->β s"
then obtain r''' where s: "s = Abs r'''" and r''': "r' ->β r'''" by cases auto
from r''' have "r ->s r'''" by (blast intro: Abs)
moreover from Abs have "ss [->s] ss'" by (iprover dest: listrelp_conj1)
ultimately have "Abs r °° ss ->s Abs r''' °° ss'" by (rule sred.Abs)
with r'' s show "Abs r °° ss ->s r''" by simp
next
fix rs'
assume "ss' => rs'"
with Abs(3) have "ss [->s] rs'" by (rule lemma4_aux)
with `r ->s r'` have "Abs r °° ss ->s Abs r' °° rs'" by (rule sred.Abs)
moreover assume "r'' = Abs r' °° rs'"
ultimately show "Abs r °° ss ->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 ->s u'" and us: "us [->s] us'"

by cases (auto dest!: listrelp_conj1)
have "r[u/0] ->s r'[u'/0]" using Abs(1) and u by (rule lemma3)
with us have "r[u/0] °° us ->s r'[u'/0] °° us'" by (rule lemma1')
hence "Abs r ° u °° us ->s r'[u'/0] °° us'" by (rule sred.Beta)
moreover assume "Abs r' = Abs t" and "r'' = t[u'/0] °° us'"
ultimately show "Abs r °° ss ->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 ->β* r'"
shows "r ->s r'"
using r
by induct (iprover intro: refl_sred lemma4)+


subsection {* Leftmost reduction and weakly normalizing terms *}

inductive
lred :: "dB => dB => bool" (infixl "->l" 50)
and lredlist :: "dB list => dB list => bool" (infixl "[->l]" 50)
where
"s [->l] t ≡ listrelp op ->l s t"
| Var: "rs [->l] rs' ==> Var x °° rs ->l Var x °° rs'"
| Abs: "r ->l r' ==> Abs r ->l Abs r'"
| Beta: "r[s/0] °° ss ->l t ==> Abs r ° s °° ss ->l t"


lemma lred_imp_sred:
assumes lred: "s ->l t"
shows "s ->s t"
using lred
proof induct
case (Var rs rs' x)
then have "rs [->s] rs'"
by induct (iprover intro: listrelp.intros)+
then show ?case by (rule sred.Var)
next
case (Abs r r')
from `r ->s r'`
have "Abs r °° [] ->s Abs r' °° []" using listrelp.Nil
by (rule sred.Abs)
then show ?case by simp
next
case (Beta r s ss t)
from `r[s/0] °° ss ->s t`
show ?case by (rule sred.Beta)
qed

inductive WN :: "dB => bool"
where
Var: "listsp WN rs ==> WN (Var n °° rs)"
| Lambda: "WN r ==> WN (Abs r)"
| Beta: "WN ((r[s/0]) °° ss) ==> WN ((Abs r ° s) °° ss)"


lemma listrelp_imp_listsp1:
assumes H: "listrelp (λx y. P x) xs ys"
shows "listsp P xs"
using H
by induct auto

lemma listrelp_imp_listsp2:
assumes H: "listrelp (λx y. P y) xs ys"
shows "listsp P ys"
using H
by induct auto

lemma lemma5:
assumes lred: "r ->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 "∃r'. r ->l r'"
using wn
proof induct
case (Var rs n)
then have "∃rs'. rs [->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 ->s r'"
shows "NF r' ==> r ->l r'"
using r
proof induct
case (Var rs rs' x)
from `NF (Var x °° rs')` have "listall NF rs'"
by cases simp_all
with Var(1) have "rs [->l] rs'"
proof induct
case Nil
show ?case by (rule listrelp.Nil)
next
case (Cons x y xs ys)
hence "x ->l y" and "xs [->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 `NF (Abs r' °° ss')`
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 ->l r'" by simp
hence "Abs r ->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] °° ss ->l t" by simp
thus ?case by (rule lred.Beta)
qed

lemma WN_eq: "WN t = (∃t'. t ->β* t' ∧ NF t')"
proof
assume "WN t"
then have "∃t'. t ->l t'" by (rule lemma6)
then obtain t' where t': "t ->l t'" ..
then have NF: "NF t'" by (rule lemma5)
from t' have "t ->s t'" by (rule lred_imp_sred)
then have "t ->β* t'" by (rule lemma2_2)
with NF show "∃t'. t ->β* t' ∧ NF t'" by iprover
next
assume "∃t'. t ->β* t' ∧ NF t'"
then obtain t' where t': "t ->β* t'" and NF: "NF t'"
by iprover
from t' have "t ->s t'" by (rule rtrancl_beta_sred)
then have "t ->l t'" using NF by (rule lemma7)
then show "WN t" by (rule lemma5)
qed

end