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

-- Basic intuitionistic propositional calculus, without ∨ or ⊥.
-- Kripke-style semantics with contexts as concrete worlds.

module A201607.BasicIPC.Semantics.KripkeConcrete where

open import A201607.BasicIPC.Syntax.Common public
open import A201607.Common.Semantics public

open ConcreteWorlds (Ty) public


-- Partial intuitionistic Kripke models.

record Model : Set₁ where
  infix 3 _⊩ᵅ_
  field
    -- Forcing for atomic propositions; monotonic.
    _⊩ᵅ_   : World → Atom → Set
    mono⊩ᵅ : ∀ {P w w′} → w ≤ w′ → w ⊩ᵅ P → w′ ⊩ᵅ P

open Model {{…}} public


-- Forcing in a particular world of a particular model.

module _ {{_ : Model}} where
  infix 3 _⊩_
  _⊩_ : World → Ty → Set
  w ⊩ α P   = w ⊩ᵅ P
  w ⊩ A ▻ B = ∀ {w′} → w ≤ w′ → w′ ⊩ A → w′ ⊩ B
  w ⊩ A ∧ B = w ⊩ A × w ⊩ B
  w ⊩ ⊤    = 𝟙

  infix 3 _⊩⋆_
  _⊩⋆_ : World → Cx Ty → Set
  w ⊩⋆ ∅     = 𝟙
  w ⊩⋆ Ξ , A = w ⊩⋆ Ξ × w ⊩ A


-- Monotonicity with respect to context inclusion.

module _ {{_ : Model}} where
  mono⊩ : ∀ {A w w′} → w ≤ w′ → w ⊩ A → w′ ⊩ A
  mono⊩ {α P}   ξ s = mono⊩ᵅ ξ s
  mono⊩ {A ▻ B} ξ s = λ ξ′ → s (trans≤ ξ ξ′)
  mono⊩ {A ∧ B} ξ s = mono⊩ {A} ξ (π₁ s) , mono⊩ {B} ξ (π₂ s)
  mono⊩ {⊤}    ξ s = ∙

  mono⊩⋆ : ∀ {Ξ w w′} → w ≤ w′ → w ⊩⋆ Ξ → w′ ⊩⋆ Ξ
  mono⊩⋆ {∅}     ξ ∙        = ∙
  mono⊩⋆ {Ξ , A} ξ (ts , t) = mono⊩⋆ {Ξ} ξ ts , mono⊩ {A} ξ t


-- Additional useful equipment.

module _ {{_ : Model}} where
  _⟪$⟫_ : ∀ {A B w} → w ⊩ A ▻ B → w ⊩ A → w ⊩ B
  f ⟪$⟫ a = f refl≤ a

  ⟪K⟫ : ∀ {A B w} → w ⊩ A → w ⊩ B ▻ A
  ⟪K⟫ {A} a ξ = K (mono⊩ {A} ξ a)

  ⟪S⟫ : ∀ {A B C w} → w ⊩ A ▻ B ▻ C → w ⊩ A ▻ B → w ⊩ A → w ⊩ C
  ⟪S⟫ {A} {B} {C} s₁ s₂ a = _⟪$⟫_ {B} {C} (_⟪$⟫_ {A} {B ▻ C} s₁ a) (_⟪$⟫_ {A} {B} s₂ a)

  ⟪S⟫′ : ∀ {A B C w} → w ⊩ A ▻ B ▻ C → w ⊩ (A ▻ B) ▻ A ▻ C
  ⟪S⟫′ {A} {B} {C} s₁ ξ s₂ ξ′ a = let s₁′ = mono⊩ {A ▻ B ▻ C} (trans≤ ξ ξ′) s₁
                                      s₂′ = mono⊩ {A ▻ B} ξ′ s₂
                                  in  ⟪S⟫ {A} {B} {C} s₁′ s₂′ a

  _⟪,⟫′_ : ∀ {A B w} → w ⊩ A → w ⊩ B ▻ A ∧ B
  _⟪,⟫′_ {A} {B} a ξ = _,_ (mono⊩ {A} ξ a)


-- Forcing in a particular world of a particular model, for sequents.

module _ {{_ : Model}} where
  infix 3 _⊩_⇒_
  _⊩_⇒_ : World → Cx Ty → Ty → Set
  w ⊩ Γ ⇒ A = w ⊩⋆ Γ → w ⊩ A

  infix 3 _⊩_⇒⋆_
  _⊩_⇒⋆_ : World → Cx Ty → Cx Ty → Set
  w ⊩ Γ ⇒⋆ Ξ = w ⊩⋆ Γ → w ⊩⋆ Ξ


-- Entailment, or forcing in all worlds of all models, for sequents.

infix 3 _⊨_
_⊨_ : Cx Ty → Ty → Set₁
Γ ⊨ A = ∀ {{_ : Model}} {w : World} → w ⊩ Γ ⇒ A

infix 3 _⊨⋆_
_⊨⋆_ : Cx Ty → Cx Ty → Set₁
Γ ⊨⋆ Ξ = ∀ {{_ : Model}} {w : World} → w ⊩ Γ ⇒⋆ Ξ


-- Additional useful equipment, for sequents.

module _ {{_ : Model}} where
  lookup : ∀ {A Γ w} → A ∈ Γ → w ⊩ Γ ⇒ A
  lookup top     (γ , a) = a
  lookup (pop i) (γ , b) = lookup i γ

  ⟦λ⟧ : ∀ {A B Γ w} → (∀ {w′} → w′ ⊩ Γ , A ⇒ B) → w ⊩ Γ ⇒ A ▻ B
  ⟦λ⟧ s γ = λ ξ a → s (mono⊩⋆ ξ γ , a)

  _⟦$⟧_ : ∀ {A B Γ w} → w ⊩ Γ ⇒ A ▻ B → w ⊩ Γ ⇒ A → w ⊩ Γ ⇒ B
  _⟦$⟧_ {A} {B} s₁ s₂ γ = _⟪$⟫_ {A} {B} (s₁ γ) (s₂ γ)

  ⟦K⟧ : ∀ {A B Γ w} → w ⊩ Γ ⇒ A → w ⊩ Γ ⇒ B ▻ A
  ⟦K⟧ {A} {B} a γ = ⟪K⟫ {A} {B} (a γ)

  ⟦S⟧ : ∀ {A B C Γ w} → w ⊩ Γ ⇒ A ▻ B ▻ C → w ⊩ Γ ⇒ A ▻ B → w ⊩ Γ ⇒ A → w ⊩ Γ ⇒ C
  ⟦S⟧ {A} {B} {C} s₁ s₂ a γ = ⟪S⟫ {A} {B} {C} (s₁ γ) (s₂ γ) (a γ)

  _⟦,⟧_ : ∀ {A B Γ w} → w ⊩ Γ ⇒ A → w ⊩ Γ ⇒ B → w ⊩ Γ ⇒ A ∧ B
  (a ⟦,⟧ b) γ = a γ , b γ

  ⟦π₁⟧ : ∀ {A B Γ w} → w ⊩ Γ ⇒ A ∧ B → w ⊩ Γ ⇒ A
  ⟦π₁⟧ s γ = π₁ (s γ)

  ⟦π₂⟧ : ∀ {A B Γ w} → w ⊩ Γ ⇒ A ∧ B → w ⊩ Γ ⇒ B
  ⟦π₂⟧ s γ = π₂ (s γ)