{-# OPTIONS --guardedness --sized-types #-}
module A201903.4-2-1-Properties-SmallStep-NO₂ where
open import A201903.2-2-Semantics-SmallStep
open NO₂ public
import A201903.4-1-Properties-SmallStep-CBN as CBN
data RF? {n} : Pred₀ (Tm n) where
cbn-yës : ∀ {e} → CBN.RF e → RF? e
yës : ∀ {e} → WHNF e → RF e → RF? e
nö : ∀ {e} → NF e → RF? e
rf? : ∀ {n} (e : Tm n) → RF? e
rf? e with CBN.rf? e
... | CBN.yës (_ , r) = cbn-yës (_ , r)
rf? (var s x) | CBN.nö _ = nö (nf var)
rf? (lam s e) | CBN.nö _ with rf? e
... | cbn-yës (_ , r) = yës lam (_ , cbn-lam (λ p′ → r ↯ CBN.nrf←whnf p′) r)
... | yës p (_ , r) = yës lam (_ , lam p r)
... | nö p = nö (lam p)
rf? (app e₁ e₂) | CBN.nö (whnf (app p₁)) with rf? e₁ | rf? e₂
... | cbn-yës (_ , r₁) | _ = cbn-yës (_ , CBN.app₁ r₁)
... | yës p₁′ (_ , r₁) | _ = yës (whnf (app p₁)) (_ , app₁ p₁ r₁)
... | nö (lam p₁′) | _ = cbn-yës (_ , CBN.applam)
... | nö (nf p₁′) | cbn-yës (_ , r₂) = yës (whnf (app p₁))
(_ , cbn-app₂ p₁′ (λ p₂′ → r₂ ↯ CBN.nrf←whnf p₂′) r₂)
... | nö (nf p₁′) | yës p₂ (_ , r₂) = yës (whnf (app p₁)) (_ , app₂ p₁′ p₂ r₂)
... | nö (nf p₁′) | nö p₂ = nö (nf (app p₁′ p₂))
cbn-rf|nf←nrf : ∀ {n} {e : Tm n} → NRF e → CBN.RF e ⊎ NF e
cbn-rf|nf←nrf p with rf? _
... | cbn-yës (_ , r) = inj₁ (_ , r)
... | yës p′ (_ , r) = r ↯ p
... | nö p′ = inj₂ p′
nrf←cbn-rf : ∀ {n} {e : Tm n} → CBN.RF e → NRF e
nrf←cbn-rf (_ , r) = λ { (cbn-lam ¬p r′) → case r of λ ()
; (lam p r′) → case r of λ ()
; (app₁ p₁ r₁) → case r of
λ { CBN.applam → case p₁ of λ ()
; (CBN.app₁ r₁′) → r₁′ ↯ CBN.nrf←naxnf p₁ }
; (cbn-app₂ p₁ ¬p₂ r₂) → case r of
λ { CBN.applam → case p₁ of λ ()
; (CBN.app₁ r₁′) → r₁′ ↯ CBN.nrf←naxnf (naxnf←nanf p₁) }
; (app₂ p₁ p₂ r₂) → case r of
λ { CBN.applam → case p₁ of λ ()
; (CBN.app₁ r₁′) → r₁′ ↯ CBN.nrf←naxnf (naxnf←nanf p₁) } }
mutual
nrf←nf : ∀ {n} {e : Tm n} → NF e → NRF e
nrf←nf (lam p) = λ { (cbn-lam ¬p r) → whnf←nf p ↯ ¬p
; (lam p′ r) → r ↯ nrf←nf p }
nrf←nf (nf p) = nrf←nanf p
nrf←nanf : ∀ {n} {e : Tm n} → NANF e → NRF e
nrf←nanf var = λ ()
nrf←nanf (app p₁ p₂) = λ { (app₁ p₁′ r₁) → r₁ ↯ nrf←nanf p₁
; (cbn-app₂ p₁′ ¬p₂ r₂) → r₂ ↯ CBN.nrf←whnf (whnf←nf p₂)
; (app₂ p₁′ p₂′ r₂) → r₂ ↯ nrf←nf p₂ }
uniq-⇒ : Unique _⇒_
uniq-⇒ {e = var _ _} () ()
uniq-⇒ {e = lam _ _} (cbn-lam ¬p r) (cbn-lam ¬p′ r′) = cbn-lam & uniq-¬whnf ¬p ¬p′
⊗ CBN.uniq-⇒ r r′
uniq-⇒ {e = lam _ _} (cbn-lam ¬p r) (lam p′ r′) = p′ ↯ ¬p
uniq-⇒ {e = lam _ _} (lam p r) (cbn-lam ¬p′ r′) = p ↯ ¬p′
uniq-⇒ {e = lam _ _} (lam p r) (lam p′ r′) = lam & uniq-whnf p p′
⊗ uniq-⇒ r r′
uniq-⇒ {e = app _ _} (app₁ p₁ r₁) (app₁ p₁′ r₁′) = app₁ & uniq-naxnf p₁ p₁′
⊗ uniq-⇒ r₁ r₁′
uniq-⇒ {e = app _ _} (app₁ p₁ r₁) (cbn-app₂ p₁′ ¬p₂′ r₂′) = r₁ ↯ nrf←nanf p₁′
uniq-⇒ {e = app _ _} (app₁ p₁ r₁) (app₂ p₁′ p₂′ r₂′) = r₁ ↯ nrf←nanf p₁′
uniq-⇒ {e = app _ _} (cbn-app₂ p₁ ¬p₂ r₂) (app₁ p₁′ r₁′) = r₁′ ↯ nrf←nanf p₁
uniq-⇒ {e = app _ _} (cbn-app₂ p₁ ¬p₂ r₂) (cbn-app₂ p₁′ ¬p₂′ r₂′) = cbn-app₂ & uniq-nanf p₁ p₁′
⊗ uniq-¬whnf ¬p₂ ¬p₂′
⊗ CBN.uniq-⇒ r₂ r₂′
uniq-⇒ {e = app _ _} (cbn-app₂ p₁ ¬p₂ r₂) (app₂ p₁′ p₂′ r₂′) = p₂′ ↯ ¬p₂
uniq-⇒ {e = app _ _} (app₂ p₁ p₂ r₂) (app₁ p₁′ r₁′) = r₁′ ↯ nrf←nanf p₁
uniq-⇒ {e = app _ _} (app₂ p₁ p₂ r₂) (cbn-app₂ p₁′ ¬p₂′ r₂′) = p₂ ↯ ¬p₂′
uniq-⇒ {e = app _ _} (app₂ p₁ p₂ r₂) (app₂ p₁′ p₂′ r₂′) = app₂ & uniq-nanf p₁ p₁′
⊗ uniq-whnf p₂ p₂′
⊗ uniq-⇒ r₂ r₂′
det-⇒ : Deterministic _⇒_
det-⇒ (cbn-lam ¬p r) (cbn-lam ¬p′ r′) = lam & refl ⊗ CBN.det-⇒ r r′
det-⇒ (cbn-lam ¬p r) (lam p′ r′) = p′ ↯ ¬p
det-⇒ (lam p r) (cbn-lam ¬p′ r′) = p ↯ ¬p′
det-⇒ (lam p r) (lam p′ r′) = lam & refl ⊗ det-⇒ r r′
det-⇒ (app₁ p₁ r₁) (app₁ p₁′ r₁′) = app & det-⇒ r₁ r₁′ ⊗ refl
det-⇒ (app₁ p₁ r₁) (cbn-app₂ p₁′ ¬p₂′ r₂′) = r₁ ↯ nrf←nanf p₁′
det-⇒ (app₁ p₁ r₁) (app₂ p₁′ p₂′ r₂′) = r₁ ↯ nrf←nanf p₁′
det-⇒ (cbn-app₂ p₁ ¬p₂ r₂) (app₁ p₁′ r₁′) = r₁′ ↯ nrf←nanf p₁
det-⇒ (cbn-app₂ p₁ ¬p₂ r₂) (cbn-app₂ p₁′ ¬p₂′ r₂′) = app & refl ⊗ CBN.det-⇒ r₂ r₂′
det-⇒ (cbn-app₂ p₁ ¬p₂ r₂) (app₂ p₁′ p₂′ r₂′) = p₂′ ↯ ¬p₂
det-⇒ (app₂ p₁ p₂ r₂) (app₁ p₁′ r₁′) = r₁′ ↯ nrf←nanf p₁
det-⇒ (app₂ p₁ p₂ r₂) (cbn-app₂ p₁′ ¬p₂′ r₂′) = p₂ ↯ ¬p₂′
det-⇒ (app₂ p₁ p₂ r₂) (app₂ p₁′ p₂′ r₂′) = app & refl ⊗ det-⇒ r₂ r₂′
conf-⇒ : Confluent _⇒_
conf-⇒ = cor-conf-⇒ det-⇒
det-⇓-nrf : Deterministic _⇓[ NRF ]_
det-⇓-nrf = cor-det-⇓-nrf det-⇒
naxnf-⇒ : ∀ {n} {e : Tm n} {e′} → NAXNF e → e ⇒ e′ → NAXNF e′
naxnf-⇒ var ()
naxnf-⇒ (app _) (app₁ p₁ r₁) = app (naxnf-⇒ p₁ r₁)
naxnf-⇒ (app p₁) (cbn-app₂ p₁′ ¬p₂ r₂) = app p₁
naxnf-⇒ (app p₁) (app₂ p₁′ p₂ r₂) = app p₁
whnf-⇒ : ∀ {n} {e : Tm n} {e′} → e ⇒ e′ → WHNF e′
whnf-⇒ (cbn-lam ¬p r) = lam
whnf-⇒ (lam p r) = lam
whnf-⇒ (app₁ p₁ r₁) = whnf (app (naxnf-⇒ p₁ r₁))
whnf-⇒ (cbn-app₂ p₁ ¬p₂ r₂) = whnf (app (naxnf←nanf p₁))
whnf-⇒ (app₂ p₁ ¬p₂ r₂) = whnf (app (naxnf←nanf p₁))
rev-whnf-⇒ : ∀ {n} {e : Tm n} {e′} → e ⇒ e′ → WHNF e
rev-whnf-⇒ (cbn-lam ¬p r) = lam
rev-whnf-⇒ (lam p r) = lam
rev-whnf-⇒ (app₁ p₁ r₁) = whnf (app p₁)
rev-whnf-⇒ (cbn-app₂ p₁ ¬p₂ r₂) = whnf (app (naxnf←nanf p₁))
rev-whnf-⇒ (app₂ p₁ p₂ r₂) = whnf (app (naxnf←nanf p₁))