Added ⊤⊤
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@ -39,6 +39,7 @@ module PropositionalKripke (PV : Set) where
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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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@ -49,6 +50,7 @@ module PropositionalKripke (PV : Set) where
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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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@ -67,3 +69,4 @@ module PropositionalKripke (PV : Set) where
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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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@ -4,7 +4,7 @@ module PropositionalKripkeGeneral (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 using (Form; Var; _⇒_; _∧∧_; Con)
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open import PropositionalLogic PV using (Form; Var; _⇒_; _∧∧_; ⊤⊤; Con)
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open import PropositionalKripke PV using (Kripke)
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@ -29,6 +29,7 @@ module PropositionalKripkeGeneral (PV : Set) where
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neu⁰ : {Γ : Con} → {x : PV} → Γ ⊢⁰ Var x → Γ ⊢* Var x
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lam : {Γ : Con} → {F G : Form} → (F ∷ Γ) ⊢* G → Γ ⊢* (F ⇒ G)
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andi : {Γ : Con} → {F G : Form} → Γ ⊢* F → Γ ⊢* G → Γ ⊢* (F ∧∧ G)
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true : {Γ : Con} → Γ ⊢* ⊤⊤
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record NormalizationFrame : Set₁ where
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field
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@ -62,11 +63,12 @@ module PropositionalKripkeGeneral (PV : Set) where
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u {Var x} h = h
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u {F ⇒ F₁} h {Γ'} iq hF = u {F₁} (app {Γ'} {F} {F₁} (⊢tran iq h) (q hF))
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u {F ∧∧ G} h = ⟨ (u {F} (ande₁ h)) , (u {G} (ande₂ h)) ⟩
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u {⊤⊤} h = tt
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q {Var x} h = neu⁰ h
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q {F ⇒ F₁} {Γ} h = lam (q (h (retro (Preorder.refl≤ o)) (u {F} {F ∷ Γ} zero)))
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q {F ∧∧ G} ⟨ hF , hG ⟩ = andi (q {F} hF) (q {G} hG)
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q {⊤⊤} h = true
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@ -87,7 +89,8 @@ module PropositionalKripkeGeneral (PV : Set) where
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ande₂ = ande₂ ;
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neu⁰ = neu⁰ ;
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lam = lam;
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andi = andi
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andi = andi;
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true = true
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}
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BiggestNN : NormalAndNeutral
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@ -101,7 +104,8 @@ module PropositionalKripkeGeneral (PV : Set) where
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ande₂ = ande₂ ;
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neu⁰ = λ x → x ;
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lam = lam ;
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andi = andi
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andi = andi ;
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true = true
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}
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PO⊢⁺ = [ order {Con} _⊢⁺_ refl⊢⁺ tran⊢⁺ ]ᵒᵖ
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@ -9,6 +9,7 @@ module PropositionalLogic (PV : 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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⊤⊤ : Form
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infixr 10 _∧∧_
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infixr 8 _⇒_
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@ -41,6 +42,7 @@ module PropositionalLogic (PV : Set) where
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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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true : Γ ⊢ ⊤⊤
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infix 5 _⊢_
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@ -57,6 +59,7 @@ module PropositionalLogic (PV : Set) where
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addhyp⊢ s (andi h₁ h₂) = andi (addhyp⊢ s h₁) (addhyp⊢ s h₂)
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addhyp⊢ s (ande₁ h) = ande₁ (addhyp⊢ s h)
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addhyp⊢ s (ande₂ h) = ande₂ (addhyp⊢ s h)
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addhyp⊢ s (true) = true
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-- Extension of ⊢ to contexts
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_⊢⁺_ : Con → Con → Prop
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@ -93,6 +96,7 @@ module PropositionalLogic (PV : Set) where
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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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halftran⊢⁺ h⁺ (true) = true
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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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@ -115,6 +119,7 @@ module PropositionalLogic (PV : Set) where
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neu⁰ : Γ ⊢⁰ Var x → Γ ⊢* Var x
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lam : (A ∷ Γ) ⊢* B → Γ ⊢* (A ⇒ B)
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andi : Γ ⊢* A → Γ ⊢* B → Γ ⊢* (A ∧∧ B)
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true : Γ ⊢* ⊤⊤
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infix 5 _⊢⁰_
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infix 5 _⊢*_
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@ -128,6 +133,7 @@ module PropositionalLogic (PV : Set) where
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⊢*→⊢ (neu⁰ x) = ⊢⁰→⊢ x
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⊢*→⊢ (lam h) = lam (⊢*→⊢ h)
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⊢*→⊢ (andi h₁ h₂) = andi (⊢*→⊢ h₁) (⊢*→⊢ h₂)
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⊢*→⊢ true = true
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-- We can add hypotheses to a proof
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addhyp⊢⁰ : Γ ∈* Γ' → Γ ⊢⁰ A → Γ' ⊢⁰ A
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@ -139,6 +145,7 @@ module PropositionalLogic (PV : Set) where
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addhyp⊢* s (neu⁰ x) = neu⁰ (addhyp⊢⁰ s x)
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addhyp⊢* s (lam h) = lam (addhyp⊢* (both∈* s) h)
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addhyp⊢* s (andi h₁ h₂) = andi (addhyp⊢* s h₁) (addhyp⊢* s h₂)
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addhyp⊢* s true = true
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-- Extension of ⊢⁰ to contexts
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-- i.e. there is a neutral proof for any element
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@ -173,6 +180,7 @@ module PropositionalLogic (PV : Set) where
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halftran⊢⁰⁺* h⁺ (neu⁰ x) = neu⁰ (halftran⊢⁰⁺⁰ h⁺ x)
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halftran⊢⁰⁺* h⁺ (lam h) = lam (halftran⊢⁰⁺* ⟨ zero zero∈ , addhyp⊢⁰⁺ (right∈* refl∈*) h⁺ ⟩ h)
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halftran⊢⁰⁺* h⁺ (andi h₁ h₂) = andi (halftran⊢⁰⁺* h⁺ h₁) (halftran⊢⁰⁺* h⁺ h₂)
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halftran⊢⁰⁺* h⁺ true = true
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halftran⊢⁰⁺⁰ {Γ' = x ∷ Γ'} h⁺ (zero zero∈) = proj₁ h⁺
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halftran⊢⁰⁺⁰ {Γ' = x ∷ Γ'} h⁺ (zero (next∈ h)) = halftran⊢⁰⁺⁰ (proj₂ h⁺) (zero h)
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halftran⊢⁰⁺⁰ h⁺ (app h h') = app (halftran⊢⁰⁺⁰ h⁺ h) (halftran⊢⁰⁺* h⁺ h')
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@ -195,6 +203,7 @@ module PropositionalLogic (PV : Set) where
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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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⟦ ⊤⊤ ⟧ᶠ ρ = ⊤
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⟦_⟧ᶜ : Con → Env → Prop
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⟦ [] ⟧ᶜ ρ = ⊤
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@ -208,6 +217,7 @@ module PropositionalLogic (PV : Set) where
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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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⟦ true ⟧ᵈ ρ = tt
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@ -222,6 +232,7 @@ module PropositionalLogic (PV : Set) where
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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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true : ⊢sk ⊤⊤
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data _⊢skC_ : Con → Form → Prop where
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zero : A ∈ Γ → Γ ⊢skC A
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@ -231,6 +242,7 @@ module PropositionalLogic (PV : Set) where
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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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true : Γ ⊢skC ⊤⊤
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-- combinatory gives the same proofs as classic
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⊢⇔⊢sk : ([] ⊢ A) ⇔ ⊢sk A
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@ -242,6 +254,7 @@ module PropositionalLogic (PV : Set) where
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⊢sk→⊢ ANDe₁ = lam (ande₁ (zero zero∈))
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⊢sk→⊢ ANDe₂ = lam (ande₂ (zero zero∈))
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⊢sk→⊢ (app x x₁) = app (⊢sk→⊢ x) (⊢sk→⊢ x₁)
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⊢sk→⊢ true = true
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skC→sk : [] ⊢skC A → ⊢sk A
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skC→sk SS = SS
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@ -250,6 +263,7 @@ module PropositionalLogic (PV : Set) where
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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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skC→sk true = true
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lam-sk : (A ∷ Γ) ⊢skC B → Γ ⊢skC (A ⇒ B)
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@ -263,6 +277,7 @@ module PropositionalLogic (PV : Set) where
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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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lam-sk true = app KK true
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⊢→⊢skC : Γ ⊢ A → Γ ⊢skC A
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@ -272,5 +287,6 @@ module PropositionalLogic (PV : Set) where
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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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⊢→⊢skC (true) = true
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⊢⇔⊢sk = ⟨ (λ x → skC→sk (⊢→⊢skC x)) , ⊢sk→⊢ ⟩
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