-- Basic intuitionistic modal logic S4, without ∨, ⊥, or ◇.
-- Basic Kripke-style semantics with abstract worlds, for soundness only.
-- Alechina-style conditions.

module A201607.BasicIS4.Semantics.BasicKripkeAlechina where

open import A201607.BasicIS4.Syntax.Common public


-- Intuitionistic modal Kripke models, with Alechina-Mendler-de Paiva-Ritter frame conditions.

record Model : Set₁ where
  infix  _⊩ᵅ_
  field
    World : Set

    -- Intuitionistic accessibility; preorder.
    _≤_    : World → World → Set
    refl≤  : ∀ {w} → w ≤ w
    trans≤ : ∀ {w w′ w″} → w ≤ w′ → w′ ≤ w″ → w ≤ w″

    -- Modal accessibility; preorder.
    _R_    : World → World → Set
    reflR  : ∀ {w} → w R w
    transR : ∀ {w w′ w″} → w R w′ → w′ R w″ → w R w″

    -- Forcing for atomic propositions; monotonic.
    _⊩ᵅ_   : World → Atom → Set
    mono⊩ᵅ : ∀ {P w w′} → w ≤ w′ → w ⊩ᵅ P → w′ ⊩ᵅ P


  -- Composition of accessibility.

  _≤⨾R_ : World → World → Set
  _≤⨾R_ = _≤_ ⨾ _R_

  _R⨾≤_ : World → World → Set
  _R⨾≤_ = _R_ ⨾ _≤_


  -- Minor brilliance condition.
  --
  --           v′  →   w′      v′
  --           ●   →   ◌───R───●
  --           │   →   │
  --      ζ,ξ  ≤   →   ≤
  --           │   →   │
  --   ●───R───◌   →   ●
  --   w       v   →   w
  --
  --           v′      w″  →   v″              w″
  --           ◌───R───●   →   ◌───────R───────●
  --           │           →   │
  --           ≤  ξ′,ζ′    →   │
  --   v       │           →   │
  --   ◌───R───●           →   ≤
  --   │       w′          →   │
  --   ≤  ξ,ζ              →   │
  --   │                   →   │
  --   ●                   →   ●
  --   w                   →   w

  field
    R⨾≤→≤⨾R : ∀ {w v′} → w R⨾≤ v′ → w ≤⨾R v′

  refl≤⨾R : ∀ {w} → w ≤⨾R w
  refl≤⨾R {w} = w , (refl≤ , reflR)

  trans≤⨾R : ∀ {w′ w w″} → w ≤⨾R w′ → w′ ≤⨾R w″ → w ≤⨾R w″
  trans≤⨾R {w′} (v , (ξ , ζ)) (v′ , (ξ′ , ζ′)) = let v″ , (ξ″ , ζ″) = R⨾≤→≤⨾R (w′ , (ζ , ξ′))
                                                 in  v″ , (trans≤ ξ ξ″ , transR ζ″ ζ′)

open Model {{…}} public


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

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

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


-- Monotonicity with respect to intuitionistic accessibility.

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

  mono⊩⋆ : ∀ {Γ} {w w′ : World} → w ≤ w′ → w ⊩⋆ Γ → w′ ⊩⋆ Γ
  mono⊩⋆ {∅}     ξ ∙       = ∙
  mono⊩⋆ {Γ , A} ξ (γ , a) = mono⊩⋆ {Γ} ξ γ , mono⊩ {A} ξ a


-- Additional useful equipment.

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

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

  ⟪S⟫ : ∀ {A B C} {w : World} → 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 : World} → 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

  _⟪D⟫_ : ∀ {A B} {w : World} → w ⊩ □ (A ▻ B) → w ⊩ □ A → w ⊩ □ B
  _⟪D⟫_ {A} {B} s₁ s₂ ξ ζ = let s₁′ = s₁ ξ ζ
                                s₂′ = s₂ ξ ζ
                            in  _⟪$⟫_ {A} {B} s₁′ s₂′

  _⟪D⟫′_ : ∀ {A B} {w : World} → w ⊩ □ (A ▻ B) → w ⊩ □ A ▻ □ B
  _⟪D⟫′_ {A} {B} s₁ ξ = _⟪D⟫_ {A} {B} (mono⊩ {□ (A ▻ B)} ξ s₁)

  ⟪↑⟫ : ∀ {A} {w : World} → w ⊩ □ A → w ⊩ □ □ A
  ⟪↑⟫ s ξ ζ ξ′ ζ′ = let _ , (ξ″ , ζ″) = trans≤⨾R (_ , (ξ , ζ)) (_ , (ξ′ , ζ′))
                    in  s ξ″ ζ″

  ⟪↓⟫ : ∀ {A} {w : World} → w ⊩ □ A → w ⊩ A
  ⟪↓⟫ s = s refl≤ reflR

  _⟪,⟫′_ : ∀ {A B} {w : World} → 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  _⊩_⇒_
  _⊩_⇒_ : World → Cx Ty → Ty → Set
  w ⊩ Γ ⇒ A = w ⊩⋆ Γ → w ⊩ A

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

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

infix  _⁏_⊨_
_⁏_⊨_ : Cx Ty → Cx Ty → Ty → Set₁
Γ ⁏ Δ ⊨ A = ∀ {{_ : Model}} {w : World}
             → w ⊩⋆ Γ → (∀ {w′ : World} → w ≤ w′ → ∀ {v′ : World} → w′ R v′ → v′ ⊩⋆ Δ) → w ⊩ A


-- Additional useful equipment, for sequents.

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