module A201605.AltArtemov.Old.GN.True2 where

open import A201605.AltArtemov.Old.GN.Core public


mutual
  data True (Γ : Cx) :  {n}  Ty n  Set where
    var  :  {n} {A : Ty n}  Var Γ A  True Γ A
    lam  :  {n} {A B : Ty n}  True (Γ , A) B  True Γ (A  B)
    app  :  {n} {A B : Ty n}  True Γ (A  B)  True Γ A  True Γ B
    pair :  {n} {A B : Ty n}  True Γ A  True Γ B  True Γ (A  B)
    fst  :  {n} {A B : Ty n}  True Γ (A  B)  True Γ A
    snd  :  {n} {A B : Ty n}  True Γ (A  B)  True Γ B
    up   :  {n} {A : Ty n} (j : True  A)  True Γ (ᵗ⌊ j   A)
    down :  {n} {t : Tm 0 n} {A : Ty n}  True Γ (t  A)  True Γ A

  ᵗ⌊_⌋ :  {Γ n} {A : Ty n}  True Γ A  Tm ᵍ⌊ Γ  n
  ᵗ⌊ var x       = VAR ⁱ⌊ x 
  ᵗ⌊ lam j       = LAM ᵗ⌊ j 
  ᵗ⌊ app j₁ j₂   = APP ᵗ⌊ j₁  ᵗ⌊ j₂ 
  ᵗ⌊ pair j₁ j₂  = PAIR ᵗ⌊ j₁  ᵗ⌊ j₂ 
  ᵗ⌊ fst j       = FST ᵗ⌊ j 
  ᵗ⌊ snd j       = SND ᵗ⌊ j 
  ᵗ⌊ up j        = UP (wk*-tm ᵗ⌊ j )
  ᵗ⌊ down j      = DOWN ᵗ⌊ j 

ren-true :  {Γ Γ′ n} {A : Ty n}  Γ′  Γ  True Γ A  True Γ′ A
ren-true η (var x)      = var (ren-var η x)
ren-true η (lam j)      = lam (ren-true (lift η) j)
ren-true η (app j₁ j₂)  = app (ren-true η j₁) (ren-true η j₂)
ren-true η (pair j₁ j₂) = pair (ren-true η j₁) (ren-true η j₂)
ren-true η (fst j)      = fst (ren-true η j)
ren-true η (snd j)      = snd (ren-true η j)
ren-true η (up j)       = up j
ren-true η (down j)     = down (ren-true η j)

wk-true :  {Γ n n′} {A : Ty n} {C : Ty n′}  True Γ C  True (Γ , A) C
wk-true = ren-true ⊇wk

wk*-true :  {Γ n} {C : Ty n}  True  C  True Γ C
wk*-true = ren-true ⊇to

ren-true-id :  {Γ n} {A : Ty n} (j : True Γ A)  ren-true ⊇id j  j
ren-true-id (var x)      = cong var (ren-var-id x)
ren-true-id (lam j)      = cong lam (ren-true-id j)
ren-true-id (app j₁ j₂)  = cong₂ app (ren-true-id j₁) (ren-true-id j₂)
ren-true-id (pair j₁ j₂) = cong₂ pair (ren-true-id j₁) (ren-true-id j₂)
ren-true-id (fst j)      = cong fst (ren-true-id j)
ren-true-id (snd j)      = cong snd (ren-true-id j)
ren-true-id (up j)       = refl
ren-true-id (down j)     = cong down (ren-true-id j)

ren-true-● :  {Γ Γ′ Γ″ n} {A : Ty n} (η′ : Γ″  Γ′) (η : Γ′  Γ) (j : True Γ A) 
               ren-true η′ (ren-true η j)  ren-true (η′  η) j
ren-true-● η′ η (var x)      = cong var (ren-var-● η′ η x)
ren-true-● η′ η (lam j)      = cong lam (ren-true-● (lift η′) (lift η) j)
ren-true-● η′ η (app j₁ j₂)  = cong₂ app (ren-true-● η′ η j₁) (ren-true-● η′ η j₂)
ren-true-● η′ η (pair j₁ j₂) = cong₂ pair (ren-true-● η′ η j₁) (ren-true-● η′ η j₂)
ren-true-● η′ η (fst j)      = cong fst (ren-true-● η′ η j)
ren-true-● η′ η (snd j)      = cong snd (ren-true-● η′ η j)
ren-true-● η′ η (up j)       = refl
ren-true-● η′ η (down j)     = cong down (ren-true-● η′ η j)

v₀ :  {Γ n} {A : Ty n}  True (Γ , A) A
v₀ = var x₀

v₁ :  {Γ n n′} {A : Ty n} {B : Ty n′}  True ((Γ , A) , B) A
v₁ = var x₁

v₂ :  {Γ n n′ n″} {A : Ty n} {B : Ty n′} {C : Ty n″}  True (((Γ , A) , B) , C) A
v₂ = var x₂

I :  {Γ n} {A : Ty n}  True Γ (A  A)
I = lam v₀

K :  {Γ n} {A B : Ty n}  True Γ (A  B  A)
K = lam (lam v₁)

S :  {Γ n} {A B C : Ty n}  True Γ ((A  B  C)  (A  B)  A  C)
S = lam (lam (lam
      (app (app v₂ v₀)
           (app v₁ v₀))))