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-- β-short weak normal forms

module A202401.FOR-STLC-Base-WNF where

open import A202401.FOR-STLC-Base public


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mutual
  data NF {Γ} : ∀ {A} → Γ ⊢ A → Set where
    ⌜λ⌝- : ∀ {A B} {t : Γ , A ⊢ B} → NF (⌜λ⌝ t)
    nnf  : ∀ {A} {t : Γ ⊢ A} (p : NNF t) → NF t

  data NNF {Γ} : ∀ {A} → Γ ⊢ A → Set where
    var-  : ∀ {A} {i : Γ ∋ A} → NNF (var i)
    _⌜$⌝_ : ∀ {A B} {t₁ : Γ ⊢ A ⌜⊃⌝ B} {t₂ : Γ ⊢ A} (p₁ : NNF t₁) (p₂ : NF t₂) → NNF (t₁ ⌜$⌝ t₂)

-- TODO: kit
data NNF§ {Γ} : ∀ {Δ} → Γ ⊢§ Δ → Set where
  ∙   : NNF§ ∙
  _,_ : ∀ {Δ A} {τ : Γ ⊢§ Δ} {t : Γ ⊢ A} → NNF§ τ → NNF t → NNF§ (τ , t)

mutual
  uniNF : ∀ {Γ A} {t : Γ ⊢ A} (p p′ : NF t) → p ≡ p′
  uniNF ⌜λ⌝-    ⌜λ⌝-     = refl
  uniNF (nnf p) (nnf p′) = nnf & uniNNF p p′

  uniNNF : ∀ {Γ A} {t : Γ ⊢ A} (p p′ : NNF t) → p ≡ p′
  uniNNF var-        var-          = refl
  uniNNF (p₁ ⌜$⌝ p₂) (p₁′ ⌜$⌝ p₂′) = _⌜$⌝_ & uniNNF p₁ p₁′ ⊗ uniNF p₂ p₂′

mutual
  NF? : ∀ {Γ A} (t : Γ ⊢ A) → Dec (NF t)
  NF? (var i)           = yes (nnf var-)
  NF? (⌜λ⌝ t)           = yes ⌜λ⌝-
  NF? (t₁ ⌜$⌝ t₂)       with NNF? t₁ | NF? t₂
  ... | yes p₁ | yes p₂   = yes (nnf (p₁ ⌜$⌝ p₂))
  ... | yes p₁ | no ¬p₂   = no λ { (nnf (p₁ ⌜$⌝ p₂)) → p₂ ↯ ¬p₂ }
  ... | no ¬p₁ | _        = no λ { (nnf (p₁ ⌜$⌝ p₂)) → p₁ ↯ ¬p₁ }

  NNF? : ∀ {Γ A} (t : Γ ⊢ A) → Dec (NNF t)
  NNF? (var i)          = yes var-
  NNF? (⌜λ⌝ t)          = no λ ()
  NNF? (t₁ ⌜$⌝ t₂)      with NNF? t₁ | NF? t₂
  ... | yes p₁ | yes p₂   = yes (p₁ ⌜$⌝ p₂)
  ... | yes p₁ | no ¬p₂   = no λ { (p₁ ⌜$⌝ p₂) → p₂ ↯ ¬p₂ }
  ... | no ¬p₁ | _        = no λ { (p₁ ⌜$⌝ p₂) → p₁ ↯ ¬p₁ }


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mutual
  renNF : ∀ {Γ Γ′ A} {t : Γ ⊢ A} (ϱ : Γ ⊑ Γ′) → NF t → NF (ren ϱ t)
  renNF ϱ ⌜λ⌝-    = ⌜λ⌝-
  renNF ϱ (nnf p) = nnf (renNNF ϱ p)

  renNNF : ∀ {Γ Γ′ A} {t : Γ ⊢ A} (ϱ : Γ ⊑ Γ′) → NNF t → NNF (ren ϱ t)
  renNNF ϱ var-        = var-
  renNNF ϱ (p₁ ⌜$⌝ p₂) = renNNF ϱ p₁ ⌜$⌝ renNF ϱ p₂

sub∋NNF : ∀ {Γ Ξ A} {σ : Ξ ⊢§ Γ} {i : Γ ∋ A} → NNF§ σ → NNF (sub∋ σ i)
sub∋NNF {i = zero}  (ψ , p) = p
sub∋NNF {i = wk∋ i} (ψ , p) = sub∋NNF ψ

mutual
  subNF : ∀ {Γ Ξ A} {σ : Ξ ⊢§ Γ} {t : Γ ⊢ A} → NNF§ σ → NF t → NF (sub σ t)
  subNF ψ ⌜λ⌝-    = ⌜λ⌝-
  subNF ψ (nnf p) = nnf (subNNF ψ p)

  subNNF : ∀ {Γ Ξ A} {σ : Ξ ⊢§ Γ} {t : Γ ⊢ A} → NNF§ σ → NNF t → NNF (sub σ t)
  subNNF ψ var-        = sub∋NNF ψ
  subNNF ψ (p₁ ⌜$⌝ p₂) = subNNF ψ p₁ ⌜$⌝ subNF ψ p₂


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