-- Basic intuitionistic contextual modal logic, without ∨ or ⊥.
-- Gentzen-style formalisation of syntax with context pairs, after Nanevski-Pfenning-Pientka.
-- Normal forms and neutrals.

module A201607.BasicICML.Syntax.DyadicGentzenNormalForm where

open import A201607.BasicICML.Syntax.DyadicGentzen public


-- Derivations.

mutual
  -- Normal forms, or introductions.
  infix 3 _⊢ⁿᶠ_
  data _⊢ⁿᶠ_ : Cx² Ty Box → Ty → Set where
    neⁿᶠ   : ∀ {A Γ Δ}   → Γ ⁏ Δ ⊢ⁿᵉ A → Γ ⁏ Δ ⊢ⁿᶠ A
    lamⁿᶠ  : ∀ {A B Γ Δ} → Γ , A ⁏ Δ ⊢ⁿᶠ B → Γ ⁏ Δ ⊢ⁿᶠ A ▻ B
    boxⁿᶠ  : ∀ {Ψ A Γ Δ} → Ψ ⁏ Δ ⊢ⁿᶠ A → Γ ⁏ Δ ⊢ⁿᶠ [ Ψ ] A
    pairⁿᶠ : ∀ {A B Γ Δ} → Γ ⁏ Δ ⊢ⁿᶠ A → Γ ⁏ Δ ⊢ⁿᶠ B → Γ ⁏ Δ ⊢ⁿᶠ A ∧ B
    unitⁿᶠ : ∀ {Γ Δ}     → Γ ⁏ Δ ⊢ⁿᶠ ⊤

  -- Neutrals, or eliminations.
  infix 3 _⊢ⁿᵉ_
  data _⊢ⁿᵉ_ : Cx² Ty Box → Ty → Set where
    varⁿᵉ   : ∀ {A Γ Δ}     → A ∈ Γ → Γ ⁏ Δ ⊢ⁿᵉ A
    appⁿᵉ   : ∀ {A B Γ Δ}   → Γ ⁏ Δ ⊢ⁿᵉ A ▻ B → Γ ⁏ Δ ⊢ⁿᶠ A → Γ ⁏ Δ ⊢ⁿᵉ B
    mvarⁿᵉ  : ∀ {Ψ A Γ Δ}   → [ Ψ ] A ∈ Δ → Γ ⁏ Δ ⊢⋆ⁿᶠ Ψ → Γ ⁏ Δ ⊢ⁿᵉ A
    unboxⁿᵉ : ∀ {Ψ A C Γ Δ} → Γ ⁏ Δ ⊢ⁿᵉ [ Ψ ] A → Γ ⁏ Δ , [ Ψ ] A ⊢ⁿᶠ C → Γ ⁏ Δ ⊢ⁿᵉ C
    fstⁿᵉ   : ∀ {A B Γ Δ}   → Γ ⁏ Δ ⊢ⁿᵉ A ∧ B → Γ ⁏ Δ ⊢ⁿᵉ A
    sndⁿᵉ   : ∀ {A B Γ Δ}   → Γ ⁏ Δ ⊢ⁿᵉ A ∧ B → Γ ⁏ Δ ⊢ⁿᵉ B

  infix 3 _⊢⋆ⁿᶠ_
  _⊢⋆ⁿᶠ_ : Cx² Ty Box → Cx Ty → Set
  Γ ⁏ Δ ⊢⋆ⁿᶠ ∅     = 𝟙
  Γ ⁏ Δ ⊢⋆ⁿᶠ Ξ , A = Γ ⁏ Δ ⊢⋆ⁿᶠ Ξ × Γ ⁏ Δ ⊢ⁿᶠ A

infix 3 _⊢⋆ⁿᵉ_
_⊢⋆ⁿᵉ_ : Cx² Ty Box → Cx Ty → Set
Γ ⁏ Δ ⊢⋆ⁿᵉ ∅     = 𝟙
Γ ⁏ Δ ⊢⋆ⁿᵉ Ξ , A = Γ ⁏ Δ ⊢⋆ⁿᵉ Ξ × Γ ⁏ Δ ⊢ⁿᵉ A


-- Translation to simple terms.

mutual
  nf→tm : ∀ {A Γ Δ} → Γ ⁏ Δ ⊢ⁿᶠ A → Γ ⁏ Δ ⊢ A
  nf→tm (neⁿᶠ t)     = ne→tm t
  nf→tm (lamⁿᶠ t)    = lam (nf→tm t)
  nf→tm (boxⁿᶠ t)    = box (nf→tm t)
  nf→tm (pairⁿᶠ t u) = pair (nf→tm t) (nf→tm u)
  nf→tm unitⁿᶠ       = unit

  ne→tm : ∀ {A Γ Δ} → Γ ⁏ Δ ⊢ⁿᵉ A → Γ ⁏ Δ ⊢ A
  ne→tm (varⁿᵉ i)         = var i
  ne→tm (appⁿᵉ t u)       = app (ne→tm t) (nf→tm u)
  ne→tm (mvarⁿᵉ i ts)     = mvar i (nf→tm⋆ ts)
  ne→tm (unboxⁿᵉ t u)     = unbox (ne→tm t) (nf→tm u)
  ne→tm (fstⁿᵉ t)         = fst (ne→tm t)
  ne→tm (sndⁿᵉ t)         = snd (ne→tm t)

  nf→tm⋆ : ∀ {Ξ Γ Δ} → Γ ⁏ Δ ⊢⋆ⁿᶠ Ξ → Γ ⁏ Δ ⊢⋆ Ξ
  nf→tm⋆ {∅}     ∙        = ∙
  nf→tm⋆ {Ξ , A} (ts , t) = nf→tm⋆ ts , nf→tm t

ne→tm⋆ : ∀ {Ξ Γ Δ} → Γ ⁏ Δ ⊢⋆ⁿᵉ Ξ → Γ ⁏ Δ ⊢⋆ Ξ
ne→tm⋆ {∅}     ∙        = ∙
ne→tm⋆ {Ξ , A} (ts , t) = ne→tm⋆ ts , ne→tm t


-- Monotonicity with respect to context inclusion.

mutual
  mono⊢ⁿᶠ : ∀ {A Γ Γ′ Δ} → Γ ⊆ Γ′ → Γ ⁏ Δ ⊢ⁿᶠ A → Γ′ ⁏ Δ ⊢ⁿᶠ A
  mono⊢ⁿᶠ η (neⁿᶠ t)     = neⁿᶠ (mono⊢ⁿᵉ η t)
  mono⊢ⁿᶠ η (lamⁿᶠ t)    = lamⁿᶠ (mono⊢ⁿᶠ (keep η) t)
  mono⊢ⁿᶠ η (boxⁿᶠ t)    = boxⁿᶠ t
  mono⊢ⁿᶠ η (pairⁿᶠ t u) = pairⁿᶠ (mono⊢ⁿᶠ η t) (mono⊢ⁿᶠ η u)
  mono⊢ⁿᶠ η unitⁿᶠ       = unitⁿᶠ

  mono⊢ⁿᵉ : ∀ {A Γ Γ′ Δ} → Γ ⊆ Γ′ → Γ ⁏ Δ ⊢ⁿᵉ A → Γ′ ⁏ Δ ⊢ⁿᵉ A
  mono⊢ⁿᵉ η (varⁿᵉ i)         = varⁿᵉ (mono∈ η i)
  mono⊢ⁿᵉ η (appⁿᵉ t u)       = appⁿᵉ (mono⊢ⁿᵉ η t) (mono⊢ⁿᶠ η u)
  mono⊢ⁿᵉ η (mvarⁿᵉ i ts)     = mvarⁿᵉ i (mono⊢⋆ⁿᶠ η ts)
  mono⊢ⁿᵉ η (unboxⁿᵉ t u)     = unboxⁿᵉ (mono⊢ⁿᵉ η t) (mono⊢ⁿᶠ η u)
  mono⊢ⁿᵉ η (fstⁿᵉ t)         = fstⁿᵉ (mono⊢ⁿᵉ η t)
  mono⊢ⁿᵉ η (sndⁿᵉ t)         = sndⁿᵉ (mono⊢ⁿᵉ η t)

  mono⊢⋆ⁿᶠ : ∀ {Ξ Γ Γ′ Δ} → Γ ⊆ Γ′ → Γ ⁏ Δ ⊢⋆ⁿᶠ Ξ → Γ′ ⁏ Δ ⊢⋆ⁿᶠ Ξ
  mono⊢⋆ⁿᶠ {∅}     η ∙        = ∙
  mono⊢⋆ⁿᶠ {Ξ , A} η (ts , t) = mono⊢⋆ⁿᶠ η ts , mono⊢ⁿᶠ η t

mono⊢⋆ⁿᵉ : ∀ {Ξ Γ Γ′ Δ} → Γ ⊆ Γ′ → Γ ⁏ Δ ⊢⋆ⁿᵉ Ξ → Γ′ ⁏ Δ ⊢⋆ⁿᵉ Ξ
mono⊢⋆ⁿᵉ {∅}     η ∙        = ∙
mono⊢⋆ⁿᵉ {Ξ , A} η (ts , t) = mono⊢⋆ⁿᵉ η ts , mono⊢ⁿᵉ η t


-- Monotonicity with respect to modal context inclusion.

mutual
  mmono⊢ⁿᶠ : ∀ {A Γ Δ Δ′} → Δ ⊆ Δ′ → Γ ⁏ Δ ⊢ⁿᶠ A → Γ ⁏ Δ′ ⊢ⁿᶠ A
  mmono⊢ⁿᶠ θ (neⁿᶠ t)     = neⁿᶠ (mmono⊢ⁿᵉ θ t)
  mmono⊢ⁿᶠ θ (lamⁿᶠ t)    = lamⁿᶠ (mmono⊢ⁿᶠ θ t)
  mmono⊢ⁿᶠ θ (boxⁿᶠ t)    = boxⁿᶠ (mmono⊢ⁿᶠ θ t)
  mmono⊢ⁿᶠ θ (pairⁿᶠ t u) = pairⁿᶠ (mmono⊢ⁿᶠ θ t) (mmono⊢ⁿᶠ θ u)
  mmono⊢ⁿᶠ θ unitⁿᶠ       = unitⁿᶠ

  mmono⊢ⁿᵉ : ∀ {A Γ Δ Δ′} → Δ ⊆ Δ′ → Γ ⁏ Δ ⊢ⁿᵉ A → Γ ⁏ Δ′ ⊢ⁿᵉ A
  mmono⊢ⁿᵉ θ (varⁿᵉ i)         = varⁿᵉ i
  mmono⊢ⁿᵉ θ (appⁿᵉ t u)       = appⁿᵉ (mmono⊢ⁿᵉ θ t) (mmono⊢ⁿᶠ θ u)
  mmono⊢ⁿᵉ θ (mvarⁿᵉ i ts)     = mvarⁿᵉ (mono∈ θ i) (mmono⊢⋆ⁿᶠ θ ts)
  mmono⊢ⁿᵉ θ (unboxⁿᵉ t u)     = unboxⁿᵉ (mmono⊢ⁿᵉ θ t) (mmono⊢ⁿᶠ (keep θ) u)
  mmono⊢ⁿᵉ θ (fstⁿᵉ t)         = fstⁿᵉ (mmono⊢ⁿᵉ θ t)
  mmono⊢ⁿᵉ θ (sndⁿᵉ t)         = sndⁿᵉ (mmono⊢ⁿᵉ θ t)

  mmono⊢⋆ⁿᶠ : ∀ {Ξ Γ Δ Δ′} → Δ ⊆ Δ′ → Γ ⁏ Δ ⊢⋆ⁿᶠ Ξ → Γ ⁏ Δ′ ⊢⋆ⁿᶠ Ξ
  mmono⊢⋆ⁿᶠ {∅}     θ ∙        = ∙
  mmono⊢⋆ⁿᶠ {Ξ , A} θ (ts , t) = mmono⊢⋆ⁿᶠ θ ts , mmono⊢ⁿᶠ θ t

mmono⊢⋆ⁿᵉ : ∀ {Ξ Γ Δ Δ′} → Δ ⊆ Δ′ → Γ ⁏ Δ ⊢⋆ⁿᵉ Ξ → Γ ⁏ Δ′ ⊢⋆ⁿᵉ Ξ
mmono⊢⋆ⁿᵉ {∅}     θ ∙        = ∙
mmono⊢⋆ⁿᵉ {Ξ , A} θ (ts , t) = mmono⊢⋆ⁿᵉ θ ts , mmono⊢ⁿᵉ θ t


-- Monotonicity using context pairs.

mono²⊢ⁿᶠ : ∀ {A Π Π′} → Π ⊆² Π′ → Π ⊢ⁿᶠ A → Π′ ⊢ⁿᶠ A
mono²⊢ⁿᶠ (η , θ) = mono⊢ⁿᶠ η ∘ mmono⊢ⁿᶠ θ

mono²⊢ⁿᵉ : ∀ {A Π Π′} → Π ⊆² Π′ → Π ⊢ⁿᵉ A → Π′ ⊢ⁿᵉ A
mono²⊢ⁿᵉ (η , θ) = mono⊢ⁿᵉ η ∘ mmono⊢ⁿᵉ θ


-- Generalised reflexivity.

grefl⊢⋆ⁿᵉ : ∀ {Γ Ψ Δ} → Ψ ⊆ Γ → Γ ⁏ Δ ⊢⋆ⁿᵉ Ψ
grefl⊢⋆ⁿᵉ {∅}     done     = ∙
grefl⊢⋆ⁿᵉ {Γ , A} (skip η) = mono⊢⋆ⁿᵉ weak⊆ (grefl⊢⋆ⁿᵉ η)
grefl⊢⋆ⁿᵉ {Γ , A} (keep η) = mono⊢⋆ⁿᵉ weak⊆ (grefl⊢⋆ⁿᵉ η) , varⁿᵉ top