# Theory Standardization

theory Standardization
imports NormalForm
```(*  Title:      HOL/Proofs/Lambda/Standardization.thy
Author:     Stefan Berghofer
*)

section ‹Standardization›

theory Standardization
imports NormalForm
begin

text ‹
Based on lecture notes by Ralph Matthes @{cite "Matthes-ESSLLI2000"},
›

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 (→⇩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"

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 (→⇩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"
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 (→⇩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
```