73 lines
2.4 KiB
Agda
73 lines
2.4 KiB
Agda
{-# OPTIONS --prop #-}
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module PropositionalKripke (PV : Set) where
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open import ListUtil
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open import PropUtil
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open import PropositionalLogic PV
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private
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variable
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x : PV
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y : PV
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F : Form
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G : Form
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Γ : Con
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Γ' : Con
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Η : Con
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Η' : Con
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record Kripke : Set₁ where
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field
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Worlds : Set₀
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_≤_ : Worlds → Worlds → Prop
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refl≤ : {w : Worlds} → w ≤ w
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tran≤ : {a b c : Worlds} → a ≤ b → b ≤ c → a ≤ c
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_⊩_ : Worlds → PV → Prop
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mon⊩ : {a b : Worlds} → a ≤ b → {p : PV} → a ⊩ p → b ⊩ p
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private
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variable
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w : Worlds
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w' : Worlds
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w₁ : Worlds
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w₂ : Worlds
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w₃ : Worlds
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{- Extending ⊩ to Formulas and Contexts -}
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_⊩ᶠ_ : Worlds → Form → Prop
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w ⊩ᶠ Var x = w ⊩ x
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w ⊩ᶠ (fp ⇒ fq) = {w' : Worlds} → w ≤ w' → w' ⊩ᶠ fp → w' ⊩ᶠ fq
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w ⊩ᶠ (fp ∧∧ fq) = w ⊩ᶠ fp ∧ w ⊩ᶠ fq
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w ⊩ᶠ ⊤⊤ = ⊤
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_⊩ᶜ_ : Worlds → Con → Prop
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w ⊩ᶜ [] = ⊤
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w ⊩ᶜ (p ∷ c) = (w ⊩ᶠ p) ∧ (w ⊩ᶜ c)
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-- The extensions are monotonous
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mon⊩ᶠ : w ≤ w' → w ⊩ᶠ F → w' ⊩ᶠ F
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mon⊩ᶠ {F = Var x} ww' wF = mon⊩ ww' wF
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mon⊩ᶠ {F = F ⇒ G} ww' wF w'w'' w''F = wF (tran≤ ww' w'w'') w''F
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mon⊩ᶠ {F = F ∧∧ G} ww' ⟨ wF , wG ⟩ = ⟨ mon⊩ᶠ {F = F} ww' wF , mon⊩ᶠ {F = G} ww' wG ⟩
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mon⊩ᶠ {F = ⊤⊤} ww' wF = tt
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mon⊩ᶜ : w ≤ w' → w ⊩ᶜ Γ → w' ⊩ᶜ Γ
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mon⊩ᶜ {Γ = []} ww' wΓ = wΓ
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mon⊩ᶜ {Γ = F ∷ Γ} ww' wΓ = ⟨ mon⊩ᶠ {F = F} ww' (proj₁ wΓ) , mon⊩ᶜ ww' (proj₂ wΓ) ⟩
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{- General operator matching the shape of ⊢ -}
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_⊫_ : Con → Form → Prop
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Γ ⊫ F = {w : Worlds} → w ⊩ᶜ Γ → w ⊩ᶠ F
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{- Soundness -}
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⟦_⟧ : Γ ⊢ F → Γ ⊫ F
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⟦ zero zero∈ ⟧ wΓ = proj₁ wΓ
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⟦ zero (next∈ h) ⟧ wΓ = ⟦ zero h ⟧ (proj₂ wΓ)
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⟦ lam p ⟧ = λ wΓ w≤ w'A → ⟦ p ⟧ ⟨ w'A , mon⊩ᶜ w≤ wΓ ⟩
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⟦ app p p₁ ⟧ wΓ = ⟦ p ⟧ wΓ refl≤ (⟦ p₁ ⟧ wΓ)
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⟦ andi p₁ p₂ ⟧ wΓ = ⟨ (⟦ p₁ ⟧ wΓ) , (⟦ p₂ ⟧ wΓ) ⟩
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⟦ ande₁ p ⟧ wΓ = proj₁ $ ⟦ p ⟧ wΓ
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⟦ ande₂ p ⟧ wΓ = proj₂ $ ⟦ p ⟧ wΓ
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⟦ true ⟧ wΓ = tt
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