------------------------------------------------------------------------
-- The algorithmic equality is closed under weakening.
------------------------------------------------------------------------

open import Definition.Typed.Restrictions
open import Graded.Modality

module Definition.Conversion.Weakening
  {a} {M : Set a}
  {𝕄 : Modality M}
  (R : Type-restrictions 𝕄)
  where

open Type-restrictions R

open import Definition.Untyped M as U hiding (wk)
open import Definition.Untyped.Neutral M type-variant
open import Definition.Untyped.Properties M
open import Definition.Typed R
open import Definition.Typed.Weakening R
open import Definition.Typed.Consequences.Syntactic R
open import Definition.Conversion R
open import Definition.Conversion.Soundness R

open import Tools.Fin
open import Tools.Function
open import Tools.Nat
import Tools.PropositionalEquality as PE
open import Tools.Product

private
  variable
    m n : Nat
    ρ : Wk m n
    p r : M

mutual
  -- Weakening of algorithmic equality of neutrals.
  wk~↑ : ∀ {t u A Γ Δ} ([ρ] : ρ ∷ Δ ⊇ Γ) → ⊢ Δ
      → Γ ⊢ t ~ u ↑ A
      → Δ ⊢ U.wk ρ t ~ U.wk ρ u ↑ U.wk ρ A
  wk~↑ {ρ = ρ} [ρ] ⊢Δ (var-refl x₁ x≡y) = var-refl (wkTerm [ρ] ⊢Δ x₁) (PE.cong (wkVar ρ) x≡y)
  wk~↑ ρ ⊢Δ (app-cong {G = G} t~u x) =
    PE.subst (λ x → _ ⊢ _ ~ _ ↑ x) (PE.sym (wk-β G))
             (app-cong (wk~↓ ρ ⊢Δ t~u) (wkConv↑Term ρ ⊢Δ x))
  wk~↑ ρ ⊢Δ (fst-cong p~r) =
    fst-cong (wk~↓ ρ ⊢Δ p~r)
  wk~↑ ρ ⊢Δ (snd-cong {G = G} p~r) =
    PE.subst (λ x → _ ⊢ _ ~ _ ↑ x)
             (PE.sym (wk-β G))
             (snd-cong (wk~↓ ρ ⊢Δ p~r))
  wk~↑ {ρ = ρ} {Δ = Δ} [ρ] ⊢Δ
    (natrec-cong
       {F = F} {G} {a₀} {b₀} {h} {g} {k} {l} {p} {q = q} {r = r}
       x x₁ x₂ t~u) =
    let ⊢Δℕ = ⊢Δ ∙ (ℕⱼ ⊢Δ)
        Δℕ⊢F = wk (lift [ρ]) ⊢Δℕ (proj₁ (syntacticEq (soundnessConv↑ x)))
    in  PE.subst (λ x → _ ⊢ U.wk ρ (natrec p q r F a₀ h k) ~ _ ↑ x) (PE.sym (wk-β F))
                 (natrec-cong (wkConv↑ (lift [ρ]) (⊢Δ ∙ ℕⱼ ⊢Δ) x)
                              (PE.subst (λ x → _ ⊢ _ [conv↑] _ ∷ x) (wk-β F)
                                        (wkConv↑Term [ρ] ⊢Δ x₁))
                              (PE.subst (λ x → (Δ ∙ ℕ ∙ U.wk (lift ρ) F) ⊢ U.wk (lift (lift ρ)) h
                                             [conv↑] U.wk (lift (lift ρ)) g ∷ x)
                              (wk-β-natrec _ F) (wkConv↑Term (lift (lift [ρ]))
                                                             (⊢Δℕ ∙ Δℕ⊢F) x₂))
                              (wk~↓ [ρ] ⊢Δ t~u))
  wk~↑
    {ρ = ρ} {Δ = Δ} [ρ] ⊢Δ
    (prodrec-cong {C = C} {E} {g} {h} {u} {v} x g~h x₁) =
    let ρg~ρh = wk~↓ [ρ] ⊢Δ g~h
        ⊢ρΣ , _ , _ = syntacticEqTerm (soundness~↓ ρg~ρh)
        ⊢ρF , ⊢ρG = syntacticΣ ⊢ρΣ
        u↓v = PE.subst (λ x → _ ⊢ U.wk (liftn ρ ) u [conv↑] U.wk (liftn ρ ) v ∷ x)
                       (wk-β-prodrec ρ C)
                       (wkConv↑Term (lift (lift [ρ])) (⊢Δ ∙ ⊢ρF ∙ ⊢ρG) x₁)
    in  PE.subst  (λ x → _ ⊢ U.wk ρ (prodrec _ _ _ C g u) ~
                           U.wk ρ (prodrec _ _ _ E h v) ↑ x)
                  (PE.sym (wk-β C))
                  (prodrec-cong (wkConv↑ (lift [ρ]) (⊢Δ ∙ ⊢ρΣ) x)
                     ρg~ρh u↓v)
  wk~↑ [ρ] ⊢Δ (emptyrec-cong x t~u) =
    emptyrec-cong (wkConv↑ [ρ] ⊢Δ x) (wk~↓ [ρ] ⊢Δ t~u)
  wk~↑ {ρ = ρ} {Δ = Δ} [ρ] ⊢Δ (unitrec-cong {F = F} x x₁ x₂ no-η) =
    let k~l = wk~↓ [ρ] ⊢Δ x₁
        ⊢Unit , _ = syntacticEqTerm (soundness~↓ k~l)
        u↑v = PE.subst (λ x → _ ⊢ _ [conv↑] _ ∷ x)
                       (wk-β F)
                       (wkConv↑Term [ρ] ⊢Δ x₂)
    in  PE.subst (λ x → _ ⊢ U.wk ρ (unitrec _ _ _ _ _) ~ _ ↑ x)
                 (PE.sym (wk-β F))
                 (unitrec-cong (wkConv↑ (lift [ρ]) (⊢Δ ∙ ⊢Unit) x)
                               k~l u↑v no-η)
  wk~↑
    {ρ} {Δ} [ρ] ⊢Δ
    (J-cong {A₁} {B₁} {B₂} A₁≡A₂ t₁≡t₂ B₁≡B₂ u₁≡u₂ v₁≡v₂ w₁~w₂ ≡Id) =
    case syntacticEq (soundnessConv↑ A₁≡A₂) .proj₁ of λ {
      ⊢A₁ →
    case syntacticEqTerm (soundnessConv↑Term t₁≡t₂) .proj₂ .proj₁ of λ {
      ⊢t₁ →
    case ⊢Δ ∙ wk [ρ] ⊢Δ ⊢A₁ of λ {
      ⊢Δ∙wk-ρ-A₁ →
    PE.subst (_ ⊢ J _ _ _ _ _ _ _ _ ~ _ ↑_)
      (PE.sym $ wk-β-doubleSubst _ B₁ _ _) $
    J-cong (wkConv↑ [ρ] ⊢Δ A₁≡A₂) (wkConv↑Term [ρ] ⊢Δ t₁≡t₂)
      (PE.subst
         (λ Id →
            Δ ∙ U.wk ρ A₁ ∙ Id ⊢
              U.wk (lift (lift ρ)) B₁ [conv↑] U.wk (lift (lift ρ)) B₂)
         (PE.cong₂ (λ A t → Id A t (var x0))
            (PE.sym $ wk1-wk≡lift-wk1 _ _)
            (PE.sym $ wk1-wk≡lift-wk1 _ _)) $
       wkConv↑ (lift (lift [ρ]))
         (⊢Δ∙wk-ρ-A₁ ∙
          Idⱼ
            (PE.subst₂ (_⊢_∷_ _)
               (PE.sym $ lift-wk1 _ _)
               (PE.sym $ lift-wk1 _ _) $
             wkTerm (step [ρ]) ⊢Δ∙wk-ρ-A₁ ⊢t₁)
            (PE.subst (_⊢_∷_ _ _) (wk1-wk≡lift-wk1 _ _) $
             var ⊢Δ∙wk-ρ-A₁ here))
         B₁≡B₂)
      (PE.subst (_⊢_[conv↑]_∷_ _ _ _) (wk-β-doubleSubst _ B₁ _ _) $
       wkConv↑Term [ρ] ⊢Δ u₁≡u₂)
      (wkConv↑Term [ρ] ⊢Δ v₁≡v₂) (wk~↓ [ρ] ⊢Δ w₁~w₂)
      (wkEq [ρ] ⊢Δ ≡Id) }}}
  wk~↑ [ρ] ⊢Δ (K-cong {B₁} A₁≡A₂ t₁≡t₂ B₁≡B₂ u₁≡u₂ v₁~v₂ ≡Id ok) =
    case syntacticEqTerm (soundnessConv↑Term t₁≡t₂) .proj₂ .proj₁ of λ {
      ⊢t₁ →
    PE.subst (_ ⊢ K _ _ _ _ _ _ ~ _ ↑_)
      (PE.sym $ wk-β B₁) $
    K-cong (wkConv↑ [ρ] ⊢Δ A₁≡A₂) (wkConv↑Term [ρ] ⊢Δ t₁≡t₂)
      (wkConv↑ (lift [ρ]) (⊢Δ ∙ wk [ρ] ⊢Δ (Idⱼ ⊢t₁ ⊢t₁)) B₁≡B₂)
      (PE.subst (_⊢_[conv↑]_∷_ _ _ _) (wk-β B₁) $
       wkConv↑Term [ρ] ⊢Δ u₁≡u₂)
      (wk~↓ [ρ] ⊢Δ v₁~v₂) (wkEq [ρ] ⊢Δ ≡Id) ok }
  wk~↑ [ρ] ⊢Δ ([]-cong-cong A₁≡A₂ t₁≡t₂ u₁≡u₂ v₁~v₂ ≡Id ok) =
    []-cong-cong (wkConv↑ [ρ] ⊢Δ A₁≡A₂) (wkConv↑Term [ρ] ⊢Δ t₁≡t₂)
      (wkConv↑Term [ρ] ⊢Δ u₁≡u₂) (wk~↓ [ρ] ⊢Δ v₁~v₂) (wkEq [ρ] ⊢Δ ≡Id)
      ok

  -- Weakening of algorithmic equality of neutrals in WHNF.
  wk~↓ : ∀ {t u A Γ Δ} ([ρ] : ρ ∷ Δ ⊇ Γ) → ⊢ Δ
      → Γ ⊢ t ~ u ↓ A
      → Δ ⊢ U.wk ρ t ~ U.wk ρ u ↓ U.wk ρ A
  wk~↓ {ρ} [ρ] ⊢Δ ([~] A₁ D k~l) =
    [~] (U.wk ρ A₁) (wkRed↘ [ρ] ⊢Δ D) (wk~↑ [ρ] ⊢Δ k~l)

  -- Weakening of algorithmic equality of types.
  wkConv↑ : ∀ {A B Γ Δ} ([ρ] : ρ ∷ Δ ⊇ Γ) → ⊢ Δ
          → Γ ⊢ A [conv↑] B
          → Δ ⊢ U.wk ρ A [conv↑] U.wk ρ B
  wkConv↑ {ρ} [ρ] ⊢Δ ([↑] A′ B′ D D′ A′<>B′) =
    [↑] (U.wk ρ A′) (U.wk ρ B′) (wkRed↘ [ρ] ⊢Δ D) (wkRed↘ [ρ] ⊢Δ D′)
      (wkConv↓ [ρ] ⊢Δ A′<>B′)

  -- Weakening of algorithmic equality of types in WHNF.
  wkConv↓ : ∀ {A B Γ Δ} ([ρ] : ρ ∷ Δ ⊇ Γ) → ⊢ Δ
         → Γ ⊢ A [conv↓] B
         → Δ ⊢ U.wk ρ A [conv↓] U.wk ρ B
  wkConv↓ ρ ⊢Δ (U-refl x) = U-refl ⊢Δ
  wkConv↓ ρ ⊢Δ (ℕ-refl x) = ℕ-refl ⊢Δ
  wkConv↓ ρ ⊢Δ (Empty-refl x) = Empty-refl ⊢Δ
  wkConv↓ ρ ⊢Δ (Unit-refl x ok) = Unit-refl ⊢Δ ok
  wkConv↓ ρ ⊢Δ (ne x) = ne (wk~↓ ρ ⊢Δ x)
  wkConv↓ ρ ⊢Δ (ΠΣ-cong x A<>B A<>B₁ ok) =
    let ⊢ρF = wk ρ ⊢Δ x
    in  ΠΣ-cong ⊢ρF (wkConv↑ ρ ⊢Δ A<>B)
          (wkConv↑ (lift ρ) (⊢Δ ∙ ⊢ρF) A<>B₁) ok
  wkConv↓ ρ ⊢Δ (Id-cong A₁≡A₂ t₁≡t₂ u₁≡u₂) =
    Id-cong (wkConv↑ ρ ⊢Δ A₁≡A₂) (wkConv↑Term ρ ⊢Δ t₁≡t₂)
      (wkConv↑Term ρ ⊢Δ u₁≡u₂)

  -- Weakening of algorithmic equality of terms.
  wkConv↑Term : ∀ {t u A Γ Δ} ([ρ] : ρ ∷ Δ ⊇ Γ) → ⊢ Δ
             → Γ ⊢ t [conv↑] u ∷ A
             → Δ ⊢ U.wk ρ t [conv↑] U.wk ρ u ∷ U.wk ρ A
  wkConv↑Term {ρ} [ρ] ⊢Δ ([↑]ₜ B t′ u′ D d d′ t<>u) =
    [↑]ₜ (U.wk ρ B) (U.wk ρ t′) (U.wk ρ u′)
         (wkRed↘ [ρ] ⊢Δ D) (wkRed↘Term [ρ] ⊢Δ d) (wkRed↘Term [ρ] ⊢Δ d′)
         (wkConv↓Term [ρ] ⊢Δ t<>u)

  -- Weakening of algorithmic equality of terms in WHNF.
  wkConv↓Term : ∀ {t u A Γ Δ} ([ρ] : ρ ∷ Δ ⊇ Γ) → ⊢ Δ
             → Γ ⊢ t [conv↓] u ∷ A
             → Δ ⊢ U.wk ρ t [conv↓] U.wk ρ u ∷ U.wk ρ A
  wkConv↓Term ρ ⊢Δ (ℕ-ins x) =
    ℕ-ins (wk~↓ ρ ⊢Δ x)
  wkConv↓Term ρ ⊢Δ (Empty-ins x) =
    Empty-ins (wk~↓ ρ ⊢Δ x)
  wkConv↓Term ρ ⊢Δ (Unit-ins x) =
    Unit-ins (wk~↓ ρ ⊢Δ x)
  wkConv↓Term ρ ⊢Δ (Σʷ-ins t u x) =
    Σʷ-ins (wkTerm ρ ⊢Δ t) (wkTerm ρ ⊢Δ u) (wk~↓ ρ ⊢Δ x)
  wkConv↓Term {ρ = ρ} [ρ] ⊢Δ (ne-ins t u x x₁) =
    ne-ins (wkTerm [ρ] ⊢Δ t) (wkTerm [ρ] ⊢Δ u) (wkNeutral ρ x) (wk~↓ [ρ] ⊢Δ x₁)
  wkConv↓Term ρ ⊢Δ (univ x x₁ x₂) =
    univ (wkTerm ρ ⊢Δ x) (wkTerm ρ ⊢Δ x₁) (wkConv↓ ρ ⊢Δ x₂)
  wkConv↓Term ρ ⊢Δ (zero-refl x) = zero-refl ⊢Δ
  wkConv↓Term ρ ⊢Δ (starʷ-refl _ ok no-η) = starʷ-refl ⊢Δ ok no-η
  wkConv↓Term ρ ⊢Δ (suc-cong t<>u) = suc-cong (wkConv↑Term ρ ⊢Δ t<>u)
  wkConv↓Term ρ ⊢Δ (prod-cong {G = G} x x₁ x₂ x₃ ok) =
    let ⊢ρF = wk ρ ⊢Δ x
        ⊢ρG = wk (lift ρ) (⊢Δ ∙ ⊢ρF) x₁
    in  prod-cong ⊢ρF ⊢ρG (wkConv↑Term ρ ⊢Δ x₂)
          (PE.subst (λ x → _ ⊢ _ [conv↑] _ ∷ x) (wk-β G)
             (wkConv↑Term ρ ⊢Δ x₃))
          ok
  wkConv↓Term {ρ = ρ} {Δ = Δ} [ρ] ⊢Δ (η-eq {F = F} {G = G} x₁ x₂ y y₁ t<>u) =
    let ⊢F , _ = syntacticΠ (syntacticTerm x₁)
        ⊢ρF = wk [ρ] ⊢Δ ⊢F
    in
    η-eq (wkTerm [ρ] ⊢Δ x₁) (wkTerm [ρ] ⊢Δ x₂)
      (wkFunction ρ y) (wkFunction ρ y₁) $
    PE.subst₃ (λ x y z → Δ ∙ U.wk ρ F ⊢ x [conv↑] y ∷ z)
      (PE.cong₃ _∘⟨_⟩_ (PE.sym (wk1-wk≡lift-wk1 _ _)) PE.refl PE.refl)
      (PE.cong₃ _∘⟨_⟩_ (PE.sym (wk1-wk≡lift-wk1 _ _)) PE.refl PE.refl)
      PE.refl $
    wkConv↑Term (lift [ρ]) (⊢Δ ∙ ⊢ρF) t<>u
  wkConv↓Term {ρ = ρ} [ρ] ⊢Δ (Σ-η {G = G} ⊢p ⊢r pProd rProd fstConv sndConv) =
    Σ-η (wkTerm [ρ] ⊢Δ ⊢p)
        (wkTerm [ρ] ⊢Δ ⊢r)
        (wkProduct ρ pProd)
        (wkProduct ρ rProd)
        (wkConv↑Term [ρ] ⊢Δ fstConv)
        (PE.subst (λ x → _ ⊢ _ [conv↑] _ ∷ x)
                  (wk-β G)
                  (wkConv↑Term [ρ] ⊢Δ sndConv))
  wkConv↓Term {ρ = ρ} [ρ] ⊢Δ (η-unit [t] [u] tWhnf uWhnf ok) =
    η-unit (wkTerm [ρ] ⊢Δ [t]) (wkTerm [ρ] ⊢Δ [u])
           (wkWhnf ρ tWhnf) (wkWhnf ρ uWhnf) ok
  wkConv↓Term ρ ⊢Δ (Id-ins ⊢v₁ v₁~v₂) =
    Id-ins (wkTerm ρ ⊢Δ ⊢v₁) (wk~↓ ρ ⊢Δ v₁~v₂)
  wkConv↓Term ρ ⊢Δ (rfl-refl t≡u) =
    rfl-refl (wkEqTerm ρ ⊢Δ t≡u)