module A201607.BasicIPC.Metatheory.Gentzen-KripkeMcKinseyTarski where

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


-- 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 = λ ξ a  eval t (mono⊩⋆ ξ γ , a)
eval (app {A} {B} t u) γ = _⟪$⟫_ {A} {B} (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
      { World   = Cx Ty
      ; _≤_     = _⊆_
      ; refl≤   = refl⊆
      ; trans≤  = trans⊆
      ; _⊩ᵅ_   = λ Γ P  Γ  α P
      ; mono⊩ᵅ = mono⊢
      }


-- Soundness and completeness with respect to the canonical model.

mutual
  reflectᶜ :  {A Γ}  Γ  A  Γ  A
  reflectᶜ {α P}   t = t
  reflectᶜ {A  B} t = λ η a  reflectᶜ (app (mono⊢ η t) (reifyᶜ a))
  reflectᶜ {A  B} t = reflectᶜ (fst t) , reflectᶜ (snd t)
  reflectᶜ {}    t = 

  reifyᶜ :  {A Γ}  Γ  A  Γ  A
  reifyᶜ {α P}   s = s
  reifyᶜ {A  B} s = lam (reifyᶜ (s weak⊆ (reflectᶜ {A} v₀)))
  reifyᶜ {A  B} s = pair (reifyᶜ (π₁ s)) (reifyᶜ (π₂ s))
  reifyᶜ {}    s = unit

reflectᶜ⋆ :  {Ξ Γ}  Γ ⊢⋆ Ξ  Γ ⊩⋆ Ξ
reflectᶜ⋆ {}             = 
reflectᶜ⋆ {Ξ , A} (ts , t) = reflectᶜ⋆ ts , reflectᶜ t

reifyᶜ⋆ :  {Ξ Γ}  Γ ⊩⋆ Ξ  Γ ⊢⋆ Ξ
reifyᶜ⋆ {}             = 
reifyᶜ⋆ {Ξ , A} (ts , t) = reifyᶜ⋆ ts , reifyᶜ t


-- Reflexivity and transitivity.

refl⊩⋆ :  {Γ}  Γ ⊩⋆ Γ
refl⊩⋆ = reflectᶜ⋆ refl⊢⋆

trans⊩⋆ :  {Γ Γ′ Γ″}  Γ ⊩⋆ Γ′  Γ′ ⊩⋆ Γ″  Γ ⊩⋆ Γ″
trans⊩⋆ ts us = reflectᶜ⋆ (trans⊢⋆ (reifyᶜ⋆ ts) (reifyᶜ⋆ us))


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

quot :  {A Γ}  Γ  A  Γ  A
quot s = reifyᶜ (s refl⊩⋆)


-- Normalisation by evaluation.

norm :  {A Γ}  Γ  A  Γ  A
norm = quot  eval


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