module A201605.AltArtemov.Try2.Vec where
open import A201605.AltArtemov.Try2.Tm public
data Vec (g k : ℕ) : ℕ → Set where
[] : Vec g k 0
_∷_ : ∀ {n} → Tm g (n + k) → Vec g k n → Vec g k (suc n)
infixr 15 _∷_
[_] : ∀ {g k} → Tm g k → Vec g k 1
[ t ] = t ∷ []
VARs : ∀ {g k n} → Fin g → Vec g k n
VARs {n = zero} i = []
VARs {n = suc n} i = VAR i ∷ VARs i
LAMs : ∀ {g k n} → Vec (suc g) k n → Vec g k n
LAMs [] = []
LAMs (t ∷ ts) = LAM t ∷ LAMs ts
APPs : ∀ {g k n} → Vec g k n → Vec g k n → Vec g k n
APPs [] [] = []
APPs (t₁ ∷ ts₁) (t₂ ∷ ts₂) = APP t₁ t₂ ∷ APPs ts₁ ts₂
PAIRs : ∀ {g k n} → Vec g k n → Vec g k n → Vec g k n
PAIRs [] [] = []
PAIRs (t₁ ∷ ts₁) (t₂ ∷ ts₂) = PAIR t₁ t₂ ∷ PAIRs ts₁ ts₂
FSTs : ∀ {g k n} → Vec g k n → Vec g k n
FSTs [] = []
FSTs (t ∷ ts) = FST t ∷ FSTs ts
SNDs : ∀ {g k n} → Vec g k n → Vec g k n
SNDs [] = []
SNDs (t ∷ ts) = SND t ∷ SNDs ts
UPs : ∀ {g k n} → Vec g k n → Vec g (suc k) n
UPs [] = []
UPs {g} {k} {suc n} (t ∷ ts) = subst (Tm g) (suc-plus k n) (UP t) ∷ UPs ts
DOWNs : ∀ {g k n} → Vec g (suc k) n → Vec g k n
DOWNs [] = []
DOWNs {g} {k} {suc n} (t ∷ ts) = DOWN (subst (Tm g) (sym (suc-plus k n)) t) ∷ DOWNs ts
head : ∀ {g k n} → Vec g k (suc n) → Tm g (n + k)
head (t ∷ ts) = t
head-LAMs : ∀ {g k n} (ts : Vec (suc g) k (suc n)) →
head (LAMs ts) ≡ LAM (head ts)
head-LAMs (t ∷ ts) = refl
head-APPs : ∀ {g k n} (ts₁ ts₂ : Vec g k (suc n)) →
head (APPs ts₁ ts₂) ≡ APP (head ts₁) (head ts₂)
head-APPs (t₁ ∷ ts₁) (t₂ ∷ ts₂) = refl
head-PAIRs : ∀ {g k n} (ts₁ ts₂ : Vec g k (suc n)) →
head (PAIRs ts₁ ts₂) ≡ PAIR (head ts₁) (head ts₂)
head-PAIRs (t₁ ∷ ts₁) (t₂ ∷ ts₂) = refl
head-FSTs : ∀ {g k n} (ts : Vec g k (suc n)) →
head (FSTs ts) ≡ FST (head ts)
head-FSTs (t ∷ ts) = refl
head-SNDs : ∀ {g k n} (ts : Vec g k (suc n)) →
head (SNDs ts) ≡ SND (head ts)
head-SNDs (t ∷ ts) = refl
head-UPs : ∀ {g k n} (ts : Vec g k (suc n)) →
head (UPs ts) ≡ subst (Tm g) (suc-plus k n) (UP (head ts))
head-UPs (t ∷ ts) = refl
head-DOWNs : ∀ {g k n} (ts : Vec g (suc k) (suc n)) →
head (DOWNs ts) ≡ DOWN (subst (Tm g) (sym (suc-plus k n)) (head ts))
head-DOWNs (t ∷ ts) = refl
tail : ∀ {g k n} → Vec g k (suc n) → Vec g k n
tail (t ∷ ts) = ts
tail-LAMs : ∀ {g k n} (ts : Vec (suc g) k (suc n)) →
tail (LAMs ts) ≡ LAMs (tail ts)
tail-LAMs (t ∷ ts) = refl
tail-APPs : ∀ {g k n} (ts₁ ts₂ : Vec g k (suc n)) →
tail (APPs ts₁ ts₂) ≡ APPs (tail ts₁) (tail ts₂)
tail-APPs (t₁ ∷ ts₁) (t₂ ∷ ts₂) = refl
tail-PAIRs : ∀ {g k n} (ts₁ ts₂ : Vec g k (suc n)) →
tail (PAIRs ts₁ ts₂) ≡ PAIRs (tail ts₁) (tail ts₂)
tail-PAIRs (t₁ ∷ ts₁) (t₂ ∷ ts₂) = refl
tail-FSTs : ∀ {g k n} (ts : Vec g k (suc n)) →
tail (FSTs ts) ≡ FSTs (tail ts)
tail-FSTs (t ∷ ts) = refl
tail-SNDs : ∀ {g k n} (ts : Vec g k (suc n)) →
tail (SNDs ts) ≡ SNDs (tail ts)
tail-SNDs (t ∷ ts) = refl
tail-UPs : ∀ {g k n} (ts : Vec g k (suc n)) →
tail (UPs ts) ≡ UPs (tail ts)
tail-UPs (t ∷ ts) = refl
tail-DOWNs : ∀ {g k n} (ts : Vec g (suc k) (suc n)) →
tail (DOWNs ts) ≡ DOWNs (tail ts)
tail-DOWNs (t ∷ ts) = refl
ren-vec : ∀ {g g′ k n} → g′ ≥ g → Vec g k n → Vec g′ k n
ren-vec h [] = []
ren-vec h (t ∷ ts) = ren-tm h t ∷ ren-vec h ts
wk-vec : ∀ {g k n} → Vec g k n → Vec (suc g) k n
wk-vec = ren-vec ≥wk
wk*-vec : ∀ {g k n} → Vec 0 k n → Vec g k n
wk*-vec = ren-vec ≥to
ren-vec-id : ∀ {g k n} (ts : Vec g k n) → ren-vec ≥id ts ≡ ts
ren-vec-id [] = refl
ren-vec-id (t ∷ ts) = cong₂ _∷_ (ren-tm-id t) (ren-vec-id ts)
ren-vec-○ : ∀ {g g′ g″ k n} (h′ : g″ ≥ g′) (h : g′ ≥ g) (ts : Vec g k n) →
ren-vec h′ (ren-vec h ts) ≡ ren-vec (h′ ○ h) ts
ren-vec-○ h′ h [] = refl
ren-vec-○ h′ h (t ∷ ts) = cong₂ _∷_ (ren-tm-○ h′ h t) (ren-vec-○ h′ h ts)
ren-VARs : ∀ {g g′ k n} (h : g′ ≥ g) (i : Fin g) →
ren-vec h (VARs {k = k} {n = n} i) ≡ VARs (ren-fin h i)
ren-VARs {n = zero} h i = refl
ren-VARs {n = suc n} h i = cong₂ _∷_ refl (ren-VARs h i)
ren-LAMs : ∀ {g g′ k n} (h : g′ ≥ g) (ts : Vec (suc g) k n) →
ren-vec h (LAMs ts) ≡ LAMs (ren-vec (lift h) ts)
ren-LAMs h [] = refl
ren-LAMs h (t ∷ ts) = cong₂ _∷_ refl (ren-LAMs h ts)
ren-APPs : ∀ {g g′ k n} (h : g′ ≥ g) (ts₁ ts₂ : Vec g k n) →
ren-vec h (APPs ts₁ ts₂) ≡ APPs (ren-vec h ts₁) (ren-vec h ts₂)
ren-APPs h [] [] = refl
ren-APPs h (t₁ ∷ ts₁) (t₂ ∷ ts₂) = cong₂ _∷_ refl (ren-APPs h ts₁ ts₂)
ren-PAIRs : ∀ {g g′ k n} (h : g′ ≥ g) (ts₁ ts₂ : Vec g k n) →
ren-vec h (PAIRs ts₁ ts₂) ≡ PAIRs (ren-vec h ts₁) (ren-vec h ts₂)
ren-PAIRs h [] [] = refl
ren-PAIRs h (t₁ ∷ ts₁) (t₂ ∷ ts₂) = cong₂ _∷_ refl (ren-PAIRs h ts₁ ts₂)
ren-FSTs : ∀ {g g′ k n} (h : g′ ≥ g) (ts : Vec g k n) →
ren-vec h (FSTs ts) ≡ FSTs (ren-vec h ts)
ren-FSTs h [] = refl
ren-FSTs h (t ∷ ts) = cong₂ _∷_ refl (ren-FSTs h ts)
ren-SNDs : ∀ {g g′ k n} (h : g′ ≥ g) (ts : Vec g k n) →
ren-vec h (SNDs ts) ≡ SNDs (ren-vec h ts)
ren-SNDs h [] = refl
ren-SNDs h (t ∷ ts) = cong₂ _∷_ refl (ren-SNDs h ts)