module A201607.BasicIPC.Metatheory.Gentzen-TarskiConcreteGluedImplicit where

open import A201607.BasicIPC.Syntax.Gentzen public
open import A201607.BasicIPC.Semantics.TarskiConcreteGluedImplicit public

open ImplicitSyntax (_⊢_) public


-- Completeness with respect to a particular model.

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

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


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

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 (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 = ∙


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


-- The canonical model.

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


-- Soundness with respect to the canonical model.

reflectᶜ : ∀ {A w} → unwrap w ⊢ A → w ⊩ A
reflectᶜ {α P}   t = t ⅋ t
reflectᶜ {A ▻ B} t = t ⅋ λ a → reflectᶜ (app t (reify a))
reflectᶜ {A ∧ B} t = reflectᶜ (fst t) , reflectᶜ (snd t)
reflectᶜ {⊤}    t = ∙

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


-- Reflexivity and transitivity.

refl⊩⋆ : ∀ {w} → w ⊩⋆ unwrap w
refl⊩⋆ = reflectᶜ⋆ refl⊢⋆

trans⊩⋆ : ∀ {w w′ w″} → w ⊩⋆ unwrap w′ → w′ ⊩⋆ unwrap w″ → w ⊩⋆ unwrap w″
trans⊩⋆ ts us = reflectᶜ⋆ (trans⊢⋆ (reify⋆ ts) (reify⋆ us))


-- Completeness with respect to all models, or quotation.

quot : ∀ {A Γ} → Γ ⊨ A → Γ ⊢ A
quot s = reify (s refl⊩⋆)


-- Normalisation by evaluation.

norm : ∀ {A Γ} → Γ ⊢ A → Γ ⊢ A
norm = quot ∘ eval


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