Started adding conjunction, made modification for PropositionalLogic
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@ -38,6 +38,7 @@ module PropositionalKripke (PV : Set) where
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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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_⊩ᶜ_ : Worlds → Con → Prop
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w ⊩ᶜ [] = ⊤
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@ -47,6 +48,7 @@ module PropositionalKripke (PV : Set) where
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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⊩ᶜ : w ≤ w' → w ⊩ᶜ Γ → w' ⊩ᶜ Γ
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mon⊩ᶜ {Γ = []} ww' wΓ = wΓ
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@ -87,8 +89,10 @@ module PropositionalKripke (PV : Set) where
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⊢→⊩ᶠ : {F : Form} → {Γ : Con} → Γ ⊢ F → Γ ⊩ᶠ F
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⊢→⊩ᶠ {Var x} h = h
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⊢→⊩ᶠ {F ⇒ F₁} h {Γ'} iq hF = ⊢→⊩ᶠ {F₁} (app {Γ'} {F} {F₁} (lam (app (halftran⊢⁺ (addhyp⊢⁺ iq) h) zero)) (⊩ᶠ→⊢ hF))
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⊢→⊩ᶠ {F ∧∧ G} h = {!!}
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⊩ᶠ→⊢ {Var x} h = h
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⊩ᶠ→⊢ {F ⇒ F₁} {Γ} h = lam (⊩ᶠ→⊢ (h (addhyp⊢⁺ refl⊢⁺) (⊢→⊩ᶠ {F} {F ∷ Γ} zero)))
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⊩ᶠ→⊢ {F ∧∧ G} ⟨ hF , hG ⟩ = {!!}
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-- Finally, we can deduce completeness of the Kripke model
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completeness : {F : Form} → [] ⊫ F → [] ⊢ F
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@ -8,7 +8,9 @@ module PropositionalLogic (PV : Set) where
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data Form : Set where
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Var : PV → Form
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_⇒_ : Form → Form → Form
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_∧∧_ : Form → Form → Form
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infixr 10 _∧∧_
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infixr 8 _⇒_
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{- Contexts -}
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@ -36,6 +38,9 @@ module PropositionalLogic (PV : Set) where
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next : Γ ⊢ A → (B ∷ Γ) ⊢ A
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lam : (A ∷ Γ) ⊢ B → Γ ⊢ (A ⇒ B)
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app : Γ ⊢ (A ⇒ B) → Γ ⊢ A → Γ ⊢ B
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andi : Γ ⊢ A → Γ ⊢ B → Γ ⊢ A ∧∧ B
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ande₁ : Γ ⊢ A ∧∧ B → Γ ⊢ A
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ande₂ : Γ ⊢ A ∧∧ B → Γ ⊢ B
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infix 5 _⊢_
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@ -67,6 +72,9 @@ module PropositionalLogic (PV : Set) where
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halftran⊢⁺ h⁺ (next h) = halftran⊢⁺ (proj₂ h⁺) h
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halftran⊢⁺ h⁺ (lam h) = lam (halftran⊢⁺ ⟨ zero , addhyp⊢⁺ h⁺ ⟩ h)
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halftran⊢⁺ h⁺ (app h h') = app (halftran⊢⁺ h⁺ h) (halftran⊢⁺ h⁺ h')
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halftran⊢⁺ h⁺ (andi hf hg) = andi (halftran⊢⁺ h⁺ hf) (halftran⊢⁺ h⁺ hg)
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halftran⊢⁺ h⁺ (ande₁ hfg) = ande₁ (halftran⊢⁺ h⁺ hfg)
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halftran⊢⁺ h⁺ (ande₂ hfg) = ande₂ (halftran⊢⁺ h⁺ hfg)
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-- The relation is transitive
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tran⊢⁺ : {Γ Γ' Γ'' : Con} → Γ ⊢⁺ Γ' → Γ' ⊢⁺ Γ'' → Γ ⊢⁺ Γ''
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@ -146,6 +154,7 @@ module PropositionalLogic (PV : Set) where
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⟦_⟧ᶠ : Form → Env → Prop
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⟦ Var x ⟧ᶠ ρ = ρ x
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⟦ A ⇒ B ⟧ᶠ ρ = (⟦ A ⟧ᶠ ρ) → (⟦ B ⟧ᶠ ρ)
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⟦ A ∧∧ B ⟧ᶠ ρ = (⟦ A ⟧ᶠ ρ) ∧ (⟦ B ⟧ᶠ ρ)
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⟦_⟧ᶜ : Con → Env → Prop
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⟦ [] ⟧ᶜ ρ = ⊤
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@ -156,6 +165,9 @@ module PropositionalLogic (PV : Set) where
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⟦ next th ⟧ᵈ p = ⟦ th ⟧ᵈ (proj₂ p)
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⟦ lam th ⟧ᵈ = λ pₐ p₀ → ⟦ th ⟧ᵈ ⟨ p₀ , pₐ ⟩
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⟦ app th₁ th₂ ⟧ᵈ = λ p → ⟦ th₁ ⟧ᵈ p (⟦ th₂ ⟧ᵈ p)
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⟦ andi hf hg ⟧ᵈ = λ p → ⟨ ⟦ hf ⟧ᵈ p , ⟦ hg ⟧ᵈ p ⟩
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⟦ ande₁ hfg ⟧ᵈ = λ p → proj₁ (⟦ hfg ⟧ᵈ p)
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⟦ ande₂ hfg ⟧ᵈ = λ p → proj₂ (⟦ hfg ⟧ᵈ p)
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@ -166,6 +178,9 @@ module PropositionalLogic (PV : Set) where
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data ⊢sk : Form → Prop where
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SS : ⊢sk ((A ⇒ B ⇒ C) ⇒ (A ⇒ B) ⇒ A ⇒ C)
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KK : ⊢sk (A ⇒ B ⇒ A)
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ANDi : ⊢sk (A ⇒ B ⇒ (A ∧∧ B))
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ANDe₁ : ⊢sk ((A ∧∧ B) ⇒ A)
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ANDe₂ : ⊢sk ((A ∧∧ B) ⇒ B)
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app : ⊢sk (A ⇒ B) → ⊢sk A → ⊢sk B
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data _⊢skC_ : Con → Form → Prop where
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@ -173,20 +188,28 @@ module PropositionalLogic (PV : Set) where
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next : Γ ⊢skC A → (B ∷ Γ) ⊢skC A
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SS : Γ ⊢skC ((A ⇒ B ⇒ C) ⇒ (A ⇒ B) ⇒ A ⇒ C)
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KK : Γ ⊢skC (A ⇒ B ⇒ A)
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ANDi : Γ ⊢skC (A ⇒ B ⇒ (A ∧∧ B))
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ANDe₁ : Γ ⊢skC ((A ∧∧ B) ⇒ A)
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ANDe₂ : Γ ⊢skC ((A ∧∧ B) ⇒ B)
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app : Γ ⊢skC (A ⇒ B) → Γ ⊢skC A → Γ ⊢skC B
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-- combinatory gives the same proofs as classic
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⊢⇔⊢sk : ([] ⊢ A) ⇔ ⊢sk A
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⊢sk→⊢ : ⊢sk A → ([] ⊢ A)
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⊢sk→⊢ SS = lam (lam (lam ( app (app (next (next zero)) zero) (app (next zero) zero))))
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⊢sk→⊢ KK = lam (lam (next zero))
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⊢sk→⊢ ANDi = lam (lam (andi (next zero) zero))
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⊢sk→⊢ ANDe₁ = lam (ande₁ zero)
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⊢sk→⊢ ANDe₂ = lam (ande₂ zero)
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⊢sk→⊢ (app x x₁) = app (⊢sk→⊢ x) (⊢sk→⊢ x₁)
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skC→sk : [] ⊢skC A → ⊢sk A
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skC→sk SS = SS
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skC→sk KK = KK
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skC→sk ANDi = ANDi
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skC→sk ANDe₁ = ANDe₁
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skC→sk ANDe₂ = ANDe₂
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skC→sk (app d e) = app (skC→sk d) (skC→sk e)
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@ -197,12 +220,19 @@ module PropositionalLogic (PV : Set) where
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lam-sk (next x) = app KK x
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lam-sk SS = app KK SS
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lam-sk KK = app KK KK
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lam-sk ANDi = app KK (app (app SS (app (app SS (app KK SS)) (app (app SS (app KK KK)) (app (app SS (app KK ANDi)) (lam-sk-zero))))) (app KK lam-sk-zero))
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lam-sk ANDe₁ = app KK (app (app SS (app KK ANDe₁)) lam-sk-zero)
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lam-sk ANDe₂ = app KK (app (app SS (app KK ANDe₂)) lam-sk-zero)
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lam-sk (app x₁ x₂) = app (app SS (lam-sk x₁)) (lam-sk x₂)
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⊢→⊢skC : Γ ⊢ A → Γ ⊢skC A
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⊢→⊢skC zero = zero
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⊢→⊢skC (next x) = next (⊢→⊢skC x)
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⊢→⊢skC (lam x) = lam-sk (⊢→⊢skC x)
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⊢→⊢skC (app x x₁) = app (⊢→⊢skC x) (⊢→⊢skC x₁)
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⊢→⊢skC (andi x y) = app (app ANDi (⊢→⊢skC x)) (⊢→⊢skC y)
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⊢→⊢skC (ande₁ x) = app ANDe₁ (⊢→⊢skC x)
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⊢→⊢skC (ande₂ x) = app ANDe₂ (⊢→⊢skC x)
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⊢⇔⊢sk = ⟨ (λ x → skC→sk (⊢→⊢skC x)) , ⊢sk→⊢ ⟩
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