module A202401.STLC-Naturals-SWNF-NBE where
open import A202401.STLC-Naturals-SWNF public
open import A202401.Kit4 public
infix 3 _⊩_
_⊩_ : Ctx → Ty → Set
W ⊩ A ⌜⊃⌝ B = ∀ {W′} → W ⊑ W′ → W′ ⊩ A → W′ ⊩ B
W ⊩ ⌜ℕ⌝ = Σ (W ⊢ ⌜ℕ⌝) NF
vren : ∀ {A W W′} → W ⊑ W′ → W ⊩ A → W′ ⊩ A
vren {A ⌜⊃⌝ B} ϱ v = λ ϱ′ → v (trans⊑ ϱ ϱ′)
vren {⌜ℕ⌝} ϱ (_ , p) = _ , renNF ϱ p
open ValKit (kit _⊩_ vren) public
mutual
↑ : ∀ {A Γ} → Σ (Γ ⊢ A) NNF → Γ ⊩ A
↑ {A ⌜⊃⌝ B} (_ , p₁) = λ ϱ v₂ → let _ , p₂ = ↓ v₂
in ↑ (_ , renNNF ϱ p₁ ⌜$⌝ p₂)
↑ {⌜ℕ⌝} (_ , p) = _ , nnf p
↓ : ∀ {A Γ} → Γ ⊩ A → Σ (Γ ⊢ A) NF
↓ {A ⌜⊃⌝ B} v = let t , p = ↓ (v (wk⊑ id⊑) (↑ (var zero , var-)))
in ⌜λ⌝ t , ⌜λ⌝-
↓ {⌜ℕ⌝} v = v
vid§ : ∀ {Γ} → Γ ⊩§ Γ
vid§ {∙} = ∙
vid§ {Γ , A} = vren§ (wk⊑ id⊑) vid§ , ↑ (var zero , var-)
⟦_⟧⁻¹ : ∀ {Γ A} → Γ ⊨ A → Σ (Γ ⊢ A) NF
⟦ v ⟧⁻¹ = ↓ (v vid§)
⟦zero⟧ : ∀ {Γ} → Γ ⊩ ⌜ℕ⌝
⟦zero⟧ = _ , ⌜zero⌝
⟦suc⟧ : ∀ {Γ} → Γ ⊩ ⌜ℕ⌝ → Γ ⊩ ⌜ℕ⌝
⟦suc⟧ (_ , p) = _ , ⌜suc⌝ p
⟦rec⟧ : ∀ {Γ A} → Γ ⊩ ⌜ℕ⌝ → Γ ⊩ A → Γ ⊩ ⌜ℕ⌝ ⌜⊃⌝ A ⌜⊃⌝ A → Γ ⊩ A
⟦rec⟧ (_ , ⌜zero⌝) v₀ vₛ = v₀
⟦rec⟧ (_ , ⌜suc⌝ pₙ) v₀ vₛ = vₛ id⊑ (_ , pₙ) id⊑ v₀
⟦rec⟧ (_ , nnf pₙ) v₀ vₛ = let _ , p₀ = ↓ v₀
_ , pₛ = ↓ (vₛ (wk⊑ (wk⊑ id⊑)) (↑ (var (wk∋ zero) , var-))
id⊑ (↑ (var zero , var-)))
in ↑ (_ , ⌜rec⌝ pₙ p₀ pₛ)
⟦_⟧ : ∀ {Γ A} → Γ ⊢ A → Γ ⊨ A
⟦ var i ⟧ γ = ⟦ i ⟧∋ γ
⟦ ⌜λ⌝ t ⟧ γ = λ ϱ v → ⟦ t ⟧ (vren§ ϱ γ , v)
⟦ t₁ ⌜$⌝ t₂ ⟧ γ = ⟦ t₁ ⟧ γ id⊑ $ ⟦ t₂ ⟧ γ
⟦ ⌜zero⌝ ⟧ γ = ⟦zero⟧
⟦ ⌜suc⌝ t ⟧ γ = ⟦suc⟧ (⟦ t ⟧ γ)
⟦ ⌜rec⌝ tₙ t₀ tₛ ⟧ γ = ⟦rec⟧ (⟦ tₙ ⟧ γ) (⟦ t₀ ⟧ γ) λ { ϱ (tₙ′ , pₙ′) ϱ′ vₐ →
⟦ tₛ ⟧ ((vren§ (trans⊑ ϱ ϱ′) γ , (_ , renNF ϱ′ pₙ′)) , vₐ) }
nbe : ∀ {Γ A} → Γ ⊢ A → Σ (Γ ⊢ A) NF
nbe t = ⟦ ⟦ t ⟧ ⟧⁻¹