module A201903.5-2-Equivalence-NO where
open import A201903.1-2-Syntax-Predicates
import A201903.2-1-Semantics-BigStep as BS
import A201903.2-2-Semantics-SmallStep as SS
import A201903.3-1-Properties-BigStep-CBN as BS-CBN
import A201903.3-2-Properties-BigStep-NO as BS-NO
import A201903.3-2-1-Properties-BigStep-NO₂ as BS-NO₂
import A201903.4-1-Properties-SmallStep-CBN as SS-CBN
import A201903.4-2-Properties-SmallStep-NO as SS-NO
import A201903.4-2-1-Properties-SmallStep-NO₂ as SS-NO₂
open import A201903.5-1-Equivalence-CBN
module Lem-5-2-1 where
open SS
cbn←no : ∀ {n} {e : Tm n} {e′} → ¬ WHNF e → e NO.⇒ e′ → e CBN.⇒ e′
cbn←no ¬p NO.applam = CBN.applam
cbn←no ¬p (NO.lam r) = lam ↯ ¬p
cbn←no ¬p (NO.app₁ p₁ r₁) with whnf? _
... | no ¬p₁′ = CBN.app₁ (cbn←no ¬p₁′ r₁)
... | yes p₁′ = whnf (app (naxnf←whnf p₁′ p₁)) ↯ ¬p
cbn←no ¬p (NO.app₂ p₁ r₂) = whnf (app (naxnf←nanf p₁)) ↯ ¬p
no₂←no : ∀ {n} {e : Tm n} {e′} → WHNF e → e NO.⇒ e′ → e NO₂.⇒ e′
no₂←no lam (NO.lam r) with whnf? _
... | no ¬p = NO₂.cbn-lam ¬p (cbn←no ¬p r)
... | yes p = NO₂.lam p (no₂←no p r)
no₂←no (whnf var) ()
no₂←no (whnf (app ())) NO.applam
no₂←no (whnf (app p₁)) (NO.app₁ p₁′ r₁) = NO₂.app₁ p₁ (no₂←no (whnf p₁) r₁)
no₂←no (whnf (app _)) (NO.app₂ p₁ r₂) with whnf? _
... | no ¬p₂ = NO₂.cbn-app₂ p₁ ¬p₂ (cbn←no ¬p₂ r₂)
... | yes p₂ = NO₂.app₂ p₁ p₂ (no₂←no p₂ r₂)
cbn|no₂←no : ∀ {n} {e : Tm n} {e′} → e NO.⇒ e′ → (e CBN.⇒ e′) ⊎ (e NO₂.⇒ e′)
cbn|no₂←no r with whnf? _
... | no ¬p = inj₁ (cbn←no ¬p r)
... | yes p = inj₂ (no₂←no p r)
no₂←no* : ∀ {n} {e : Tm n} {e′} → WHNF e → e NO.⇒* e′ → e NO₂.⇒* e′
no₂←no* p ε = ε
no₂←no* p (r ◅ rs) = no₂←no p r ◅ no₂←no* (SS-NO₂.whnf-⇒ (no₂←no p r)) rs
cbn|no₂←no* : ∀ {n} {e : Tm n} {e′} → e NO.⇒* e′ → NF e′ →
(∃ λ e″ →
e CBN.⇒* e″ × WHNF e″ × e″ NO₂.⇒* e′)
cbn|no₂←no* ε p′ = _ , ε , whnf←nf p′ , ε
cbn|no₂←no* (r ◅ rs) p′ with cbn|no₂←no r
... | inj₂ r′ = _ , ε , SS-NO₂.rev-whnf-⇒ r′ , r′ ◅ no₂←no* (SS-NO₂.whnf-⇒ r′) rs
... | inj₁ r′ with cbn|no₂←no* rs p′
... | _ , rs′ , p″ , rs″ = _ , r′ ◅ rs′ , p″ , rs″
module Lem-5-2-2 where
open BS-NO₂
open SS-NO₂
rev-lam* : ∀ {n i s} {e : Tm (suc n)} {e′} → WHNF e → lam s e ⇒*⟨ i ⟩ lam s e′ → e ⇒*⟨ i ⟩ e′
rev-lam* p ε = ε
rev-lam* p (cbn-lam ¬p r ◅ rs) = p ↯ ¬p
rev-lam* p (lam p′ r ◅ rs) = r ◅ rev-lam* (whnf-⇒ r) rs
rev-cbn-lam* : ∀ {n i s} {e : Tm (suc n)} {e′} →
lam s e ⇒*⟨ i ⟩ lam s e′ → NF e′ →
(∃ λ e″ →
e SS.CBN.⇒*⟨ i ⟩ e″ × WHNF e″ × e″ ⇒* e′)
rev-cbn-lam* ε p′ = _ , ε , whnf←nf p′ , ε
rev-cbn-lam* (cbn-lam ¬p r ◅ rs) p′ with rev-cbn-lam* rs p′
... | _ , rs′ , p″ , rs″ = _ , r ◅ rs′ , p″ , rs″
rev-cbn-lam* (lam p r ◅ rs) p′ = _ , ε , p , r ◅ rev-lam* (whnf-⇒ r) rs
¬lam⇒*var : ∀ {n s} {e : Tm (suc n)} {s′ x} → ¬ (lam s e ⇒* var s′ x)
¬lam⇒*var = λ { (cbn-lam ¬p r ◅ rs) → rs ↯ ¬lam⇒*var
; (lam p r ◅ rs) → rs ↯ ¬lam⇒*var }
¬lam-s⇒*lam-s′ : ∀ {n s} {e : Tm (suc n)} {s′ e′} → s ≢ s′ → ¬ (lam s e ⇒* lam s′ e′)
¬lam-s⇒*lam-s′ s≢s′ = λ { ε → refl ↯ s≢s′
; (cbn-lam ¬p r ◅ rs) → rs ↯ ¬lam-s⇒*lam-s′ s≢s′
; (lam p r ◅ rs) → rs ↯ ¬lam-s⇒*lam-s′ s≢s′ }
¬lam⇒*app : ∀ {n s} {e : Tm (suc n)} {e₁ e₂} → ¬ (lam s e ⇒* app e₁ e₂)
¬lam⇒*app = λ { (cbn-lam ¬p r ◅ rs) → rs ↯ ¬lam⇒*app
; (lam p r ◅ rs) → rs ↯ ¬lam⇒*app }
rev-app₂* : ∀ {n i} {e₁ e₂ : Tm n} {e′} →
NANF e₁ → WHNF e₂ → app e₁ e₂ ⇒*⟨ i ⟩ e′ → NF e′ →
(∃ λ e₂′ →
e₂ ⇒*⟨ i ⟩ e₂′ × NF e₂′ × app e₁ e₂′ ≡ e′)
rev-app₂* p₁ p₂ ε (nf (app p₁′ p₂′)) = _ , ε , p₂′ , refl
rev-app₂* p₁ p₂ (app₁ p₁′ r₁ ◅ rs) p′ = r₁ ↯ nrf←nanf p₁
rev-app₂* p₁ p₂ (cbn-app₂ p₁′ ¬p₂ r₂ ◅ rs) p′ = p₂ ↯ ¬p₂
rev-app₂* p₁ p₂ (app₂ p₁′ p₂′ r₂ ◅ rs) p′ with rev-app₂* p₁′ (whnf-⇒ r₂) rs p′
... | _ , rs₂ , p₂″ , refl = _ , r₂ ◅ rs₂ , p₂″ , refl
rev-cbn-app₂* : ∀ {n i} {e₁ e₂ : Tm n} {e′} →
NANF e₁ → app e₁ e₂ ⇒*⟨ i ⟩ e′ → NF e′ →
(∃² λ e₂′ e₂″ →
e₂ SS.CBN.⇒*⟨ i ⟩ e₂′ × WHNF e₂′ × e₂′ ⇒*⟨ i ⟩ e₂″ × NF e₂″ × app e₁ e₂″ ≡ e′)
rev-cbn-app₂* p₁ ε (nf (app p₁′ p₂′)) = _ , ε , whnf←nf p₂′ , ε , p₂′ , refl
rev-cbn-app₂* p₁ (app₁ p₁′ r₁ ◅ rs) p′ = r₁ ↯ nrf←nanf p₁
rev-cbn-app₂* p₁ (cbn-app₂ p₁′ ¬p₂ r₂ ◅ rs) p′ with rev-cbn-app₂* p₁′ rs p′
... | _ , rs₂ , p₂ , rs₂′ , p₂′ , refl = _ , r₂ ◅ rs₂ , p₂ , rs₂′ , p₂′ , refl
rev-cbn-app₂* p₁ (app₂ p₁′ p₂ r₂ ◅ rs) p′ with rev-app₂* p₁′ (whnf-⇒ r₂) rs p′
... | _ , rs₂ , p₂′ , refl = _ , ε , p₂ , r₂ ◅ rs₂ , p₂′ , refl
rev-app₁* : ∀ {n i} {e₁ e₂ : Tm n} {e′} →
app e₁ e₂ ⇒*⟨ i ⟩ e′ → NF e′ →
(∃³ λ e₁′ e₂′ e₂″ →
e₁ ⇒*⟨ i ⟩ e₁′ × NANF e₁′ × e₂ SS.CBN.⇒*⟨ i ⟩ e₂′ × WHNF e₂′ ×
e₂′ ⇒*⟨ i ⟩ e₂″ × NF e₂″ × app e₁′ e₂″ ≡ e′)
rev-app₁* ε (nf (app p₁′ p₂′)) = _ , ε , p₁′ , ε , whnf←nf p₂′ , ε , p₂′ , refl
rev-app₁* (app₁ p₁ r₁ ◅ rs) p′ with rev-app₁* rs p′
... | _ , rs₁ , p₁′ , rs₂ , p₂ , rs₂′ , p₂′ , refl = _ , r₁ ◅ rs₁ , p₁′ , rs₂ , p₂ , rs₂′ , p₂′ , refl
rev-app₁* (cbn-app₂ p₁ ¬p₂ r₂ ◅ rs) p′ with rev-cbn-app₂* p₁ rs p′
... | _ , rs₂ , p₂ , rs₂′ , p₂′ , refl = _ , ε , p₁ , r₂ ◅ rs₂ , p₂ , rs₂′ , p₂′ , refl
rev-app₁* (app₂ p₁ p₂ r₂ ◅ rs) p′ with rev-app₂* p₁ (whnf-⇒ r₂) rs p′
... | _ , rs₂ , p₂′ , refl = _ , ε , p₁ , ε , p₂ , r₂ ◅ rs₂ , p₂′ , refl
mutual
bs←ss : ∀ {n i} {e : Tm n} {e′} → e ⇒*⟨ i ⟩ e′ → NF e′ → e ⟱ e′
bs←ss ε p′ = refl-⟱ p′
bs←ss (r ◅ rs) p′ = bs←ss′ r rs p′
bs←ss′ : ∀ {n i} {e : Tm n} {e′ e″} → e ⇒ e′ → e′ ⇒*⟨ i ⟩ e″ → NF e″ → e ⟱ e″
bs←ss′ (cbn-lam {s = s} ¬p r) rs (lam {s = s′} p″) with s ≟ s′
... | no s≢s′ = rs ↯ ¬lam-s⇒*lam-s′ s≢s′
... | yes refl with rev-cbn-lam* rs p″
... | _ , rs′ , p′ , rs″ = lam (Lem-5-1-1.bs←ss′ r rs′ p′)
(bs←ss rs″ p″)
bs←ss′ (cbn-lam ¬p r) rs (nf var) = rs ↯ ¬lam⇒*var
bs←ss′ (cbn-lam ¬p r) rs (nf (app _ _)) = rs ↯ ¬lam⇒*app
bs←ss′ (lam {s = s} p r) rs (lam {s = s′} p″) with s ≟ s′
... | no s≢s′ = rs ↯ ¬lam-s⇒*lam-s′ s≢s′
... | yes refl = lam (BS-CBN.refl-⟱ p)
(bs←ss′ r (rev-lam* (whnf-⇒ r) rs) p″)
bs←ss′ (lam p r) rs (nf var) = rs ↯ ¬lam⇒*var
bs←ss′ (lam p r) rs (nf (app _ _)) = rs ↯ ¬lam⇒*app
bs←ss′ (app₁ p₁ r₁) rs p″ with rev-app₁* rs p″
... | _ , rs₁ , p₁′ , rs₂ , p₂ , rs₂′ , p₂′ , refl = app p₁ (bs←ss′ r₁ rs₁ (nf p₁′))
(Lem-5-1-1.bs←ss rs₂ p₂)
(bs←ss rs₂′ p₂′)
bs←ss′ (cbn-app₂ p₁ ¬p₂ r₂) rs p″ with rev-cbn-app₂* p₁ rs p″
... | _ , rs₂ , p₂ , rs₂′ , p₂′ , refl = app (naxnf←nanf p₁) (refl-⟱′ p₁)
(Lem-5-1-1.bs←ss′ r₂ rs₂ p₂)
(bs←ss rs₂′ p₂′)
bs←ss′ (app₂ p₁ p₂ r₂) rs p″ with rev-app₂* p₁ (whnf-⇒ r₂) rs p″
... | _ , rs₂ , p₂′ , refl = app (naxnf←nanf p₁) (refl-⟱′ p₁)
(BS-CBN.refl-⟱ p₂)
(bs←ss′ r₂ rs₂ p₂′)
module Lem-5-2-3 where
open BS
no←cbn|no₂ : ∀ {n} {e : Tm n} {e′ e″} → e CBN.⟱ e′ → e′ NO₂.⟱ e″ → e NO.⟱ e″
no←cbn|no₂ (CBN.applam r₁ r) r′ = NO.applam r₁ (no←cbn|no₂ r r′)
no←cbn|no₂ CBN.var NO₂.var = NO.var
no←cbn|no₂ CBN.lam (NO₂.lam r r′) = NO.lam (no←cbn|no₂ r r′)
no←cbn|no₂ (CBN.app r₁ p₁′) (NO₂.app p₁ r₁′ r₂ r₂′) = NO.app r₁ p₁′ r₁″ (no←cbn|no₂ r₂ r₂′)
where
r₁″ = no←cbn|no₂ (BS-CBN.refl-⟱ (BS-CBN.whnf-⟱ r₁)) r₁′
module Cor-5-2-4 where
open SS-NO
open BS-NO
bs←ss : ∀ {n} {e : Tm n} {e′} → e ⇒* e′ → NF e′ → e ⟱ e′
bs←ss rs p′ with Lem-5-2-1.cbn|no₂←no* rs p′
... | _ , rs′ , p″ , rs″ = Lem-5-2-3.no←cbn|no₂ (Lem-5-1-1.bs←ss rs′ p″)
(Lem-5-2-2.bs←ss rs″ p′)
module Lem-5-2-5 where
open SS-NO
open BS-NO
applam* : ∀ {n s} {e₁ : Tm (suc n)} {e₂ : Tm n} → app (lam s e₁) e₂ ⇒* e₁ [ e₂ ]
applam* = applam ◅ ε
lam* : ∀ {n s} {e : Tm (suc n)} {e′} → e ⇒* e′ → lam s e ⇒* lam s e′
lam* = map* lam
cbn-app₁* : ∀ {n} {e₁ e₂ : Tm n} {e₁′} → e₁ SS.CBN.⇒* e₁′ → app e₁ e₂ ⇒* app e₁′ e₂
cbn-app₁* = map* cbn-app₁
app₁* : ∀ {n} {e₁ e₂ : Tm n} {e₁′} → NAXNF e₁ → e₁ ⇒* e₁′ → app e₁ e₂ ⇒* app e₁′ e₂
app₁* p₁ ε = ε
app₁* p₁ (r₁ ◅ rs₁) = app₁ (na←naxnf p₁) r₁ ◅ app₁* (naxnf-⇒ p₁ r₁) rs₁
app₂* : ∀ {n} {e₁ e₂ : Tm n} {e₂′} → NANF e₁ → e₂ ⇒* e₂′ → app e₁ e₂ ⇒* app e₁ e₂′
app₂* p₁ = map* (app₂ p₁)
bs-applam : ∀ {n s} {e₁ e₂ : Tm n} {e₁′ e′} →
e₁ SS.CBN.⇒* lam s e₁′ → e₁′ [ e₂ ] ⇒* e′ →
app e₁ e₂ ⇒* e′
bs-applam rs₁ rs = cbn-app₁* rs₁ ◅◅ applam* ◅◅ rs
bs-lam : ∀ {n s} {e : Tm (suc n)} {e′} → e ⇒* e′ → lam s e ⇒* lam s e′
bs-lam = lam*
bs-app : ∀ {n} {e₁ e₂ : Tm n} {e₁′ e₁″ e₂′} →
e₁ SS.CBN.⇒* e₁′ → NAXNF e₁′ → e₁′ ⇒* e₁″ → NANF e₁″ → e₂ ⇒* e₂′ →
app e₁ e₂ ⇒* app e₁″ e₂′
bs-app rs₁ p₁′ rs₁′ p₁″ rs₂ = cbn-app₁* rs₁ ◅◅ app₁* p₁′ rs₁′ ◅◅ app₂* p₁″ rs₂
ss←bs : ∀ {n} {e : Tm n} {e′} → e ⟱ e′ → e ⇒* e′
ss←bs (applam r₁ r) = bs-applam (Lem-5-1-2.ss←bs r₁) (ss←bs r)
ss←bs var = ε
ss←bs (lam r) = bs-lam (ss←bs r)
ss←bs (app r₁ p₁′ r₁′ r₂) = bs-app (Lem-5-1-2.ss←bs r₁) p₁″ (ss←bs r₁′) p₁‴ (ss←bs r₂)
where
p₁″ = naxnf←whnf (BS-CBN.whnf-⟱ r₁) p₁′
p₁‴ = nanf←nf (nf-⟱ r₁′) (na←whnf-⟱ (BS-CBN.whnf-⟱ r₁) p₁′ r₁′)
module Thm-5-2-6 where
ss↔bs : ∀ {n} {e : Tm n} {e′} → (e SS.NO.⇒* e′ × NF e′) ↔ e BS.NO.⟱ e′
ss↔bs = uncurry Cor-5-2-4.bs←ss , λ r → Lem-5-2-5.ss←bs r , BS-NO.nf-⟱ r