{-# OPTIONS --sized-types #-}

module A201607.BasicIS4.Metatheory.Gentzen-TarskiClosedOvergluedImplicit where

open import A201607.BasicIS4.Syntax.Gentzen public
open import A201607.BasicIS4.Semantics.TarskiClosedOvergluedImplicit public

open ImplicitSyntax (∅ ⊢_) public


-- Completeness with respect to a particular model.

module _ {{_ : Model}} where
  reify : ∀ {A} → ⊩ A → ∅ ⊢ A
  reify {α P}   s = syn s
  reify {A ▻ B} s = syn s
  reify {□ A}   s = syn s
  reify {A ∧ B} s = pair (reify (π₁ s)) (reify (π₂ s))
  reify {⊤}    s = unit

  reify⋆ : ∀ {Ξ} → ⊩⋆ Ξ → ∅ ⊢⋆ Ξ
  reify⋆ {∅}     ∙        = ∙
  reify⋆ {Ξ , A} (ts , t) = reify⋆ ts , reify t


-- Soundness with respect to all models, or evaluation.

mutual
  eval : ∀ {A Γ} → Γ ⊢ A → Γ ⊨ A
  eval (var i)         γ = lookup i γ
  eval (lam t)         γ = multicut (reify⋆ γ) (lam t) ⅋ λ a →
                             eval t (γ , a)
  eval (app t u)       γ = eval t γ ⟪$⟫ eval u γ
  eval (multibox ts u) γ = multicut (reify⋆ γ) (multibox ts u) ⅋
                             eval u (eval⋆ ts γ)
  eval (down t)        γ = ⟪↓⟫ (eval t γ)
  eval (pair t u)      γ = eval t γ , eval u γ
  eval (fst t)         γ = π₁ (eval t γ)
  eval (snd t)         γ = π₂ (eval t γ)
  eval unit            γ = ∙

  eval⋆ : ∀ {Ξ Γ} → Γ ⊢⋆ Ξ → Γ ⊨⋆ Ξ
  eval⋆ {∅}     ∙        γ = ∙
  eval⋆ {Ξ , A} (ts , t) γ = eval⋆ ts γ , eval t γ


-- TODO: Correctness of evaluation with respect to conversion.


-- The canonical model.

private
  instance
    canon : Model
    canon = record
      { ⊩ᵅ_ = λ P → ∅ ⊢ α P
      }


-- Completeness with respect to all models, or quotation, for closed terms only.

quot₀ : ∀ {A} → ∅ ⊨ A → ∅ ⊢ A
quot₀ t = reify (t ∙)


-- Normalisation by evaluation, for closed terms only.

norm₀ : ∀ {A} → ∅ ⊢ A → ∅ ⊢ A
norm₀ = quot₀ ∘ eval


-- TODO: Correctness of normalisation with respect to conversion.