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It kinda works :D

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pingu
2025-12-25 13:58:52 +01:00
parent b322bac402
commit 45b888cc73
1 changed files with 38 additions and 25 deletions
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@@ -36,9 +36,10 @@ data Formula : Type where
_⇒_ : Formula → Formula → Formula _⇒_ : Formula → Formula → Formula
infixr 4 _⇒_ infixr 4 _⇒_
infix 19 _∧_
variable variable
X Y Z : Formula A B C X Y Z : Formula
_⇔_ : Formula → Formula → Formula _⇔_ : Formula → Formula → Formula
X ⇔ Y = (X ⇒ Y) ∧ (Y ⇒ X) X ⇔ Y = (X ⇒ Y) ∧ (Y ⇒ X)
@@ -83,12 +84,12 @@ ExampleModel .M.L 1 = 1 ∷ 2 ∷ []
ExampleModel .M.L 2 = 0 ∷ 2 ∷ [] ExampleModel .M.L 2 = 0 ∷ 2 ∷ []
ExampleModel .M.L _ = [] ExampleModel .M.L _ = []
Example : ExampleModel , 0 ⊩ □ (atom 2) ExampleSem : ExampleModel , 0 ⊩ □ (atom 2)
Example y zeroone = succ 2 (2 ∷ []) (zero 2 []) ExampleSem y zeroone = succ 2 (2 ∷ []) (zero 2 [])
Example y zerotwo = succ 2 (2 ∷ []) (zero 2 []) ExampleSem y zerotwo = succ 2 (2 ∷ []) (zero 2 [])
Example2 : ExampleModel , 0 ⊩ ◇ (atom 1) Example2Sem : ExampleModel , 0 ⊩ ◇ (atom 1)
Example2 = pair 1 (λ _ → zero 1 (2 ∷ [])) Example2Sem = pair 1 (λ _ → zero 1 (2 ∷ []))
Context : Type Context : Type
Context = List Formula Context = List Formula
@@ -104,23 +105,35 @@ _++_ : {A : Type} → List A → List A → List A
[] ++ ys = ys [] ++ ys = ys
(xs , x) ++ ys = xs ++ (x ∷ ys) (xs , x) ++ ys = xs ++ (x ∷ ys)
boxed : Context → Context infixr 2 _/_⊢_
boxed [] = [] data _/_⊢_ (Δ Γ : Context) : Formula → Type where
boxed (xs , x) = boxed xs , □ x var : X ∈ Γ → Δ / Γ ⊢ X
mp : Δ / Γ ⊢ X ⇒ Y → Δ / Γ ⊢ X → Δ / Γ ⊢ Y
infixr 2 _⊢_ ∧ᵢ : Δ / Γ ⊢ X → Δ / Γ ⊢ Y → Δ / Γ ⊢ X ∧ Y
data _⊢_ : Context → Formula → Type where ∧ₑ₁ : Δ / Γ ⊢ X ∧ Y → Δ / Γ ⊢ X
var : X ∈ Γ → Γ ⊢ X ∧ₑ₂ : Δ / Γ ⊢ X ∧ Y → Δ / Γ ⊢ Y
∧ᵢ : Γ ⊢ X → Γ ⊢ Y → Γ ⊢ X ∧ Y ∨ᵢ₁ : Δ / Γ ⊢ X → Δ / Γ ⊢ X ∨ Y
∧ₑ₁ : Γ ⊢ X ∧ Y → Γ ⊢ X ∨ᵢ₂ : Δ / Γ ⊢ Y → Δ / Γ ⊢ X ∨ Y
∧ₑ₂ : Γ ⊢ X ∧ Y → Γ ⊢ Y ∨ₑ : Δ / Γ ⊢ X ∨ Y → Δ / Γ , X ⊢ Z → Δ / Γ , Y ⊢ Z → Δ / Γ ⊢ Z
∨ᵢ₁ : Γ ⊢ X → Γ ⊢ X ∨ Y ⇒ᵢ : Δ / Γ , X ⊢ Y → Δ / Γ ⊢ X ⇒ Y
∨ᵢ₂ : Γ ⊢ Y → Γ ⊢ X ∨ Y ¬ᵢ : Δ / Γ , X ⊢ ⊥ → Δ / Γ ⊢ ∼ X
∨ₑ : Γ ⊢ X ∨ Y → Γ , X ⊢ Z → Γ , Y ⊢ Z → Γ ⊢ Z ¬ₑ : Δ / Γ ⊢ X → Δ / Γ ⊢ ∼ X → Δ / Γ ⊢ ⊥
mp : Γ ⊢ X ⇒ Y → Γ ⊢ X → Γ ⊢ Y □ᵢ : [] / Δ ⊢ X → Δ / Γ ⊢ □ X
⇒ᵢ : Γ , X ⊢ Y → Γ ⊢ X ⇒ Y □ₑ : Δ / Γ ⊢ □ X → (Δ , X) / Γ ⊢ Y → Δ / Γ ⊢ Y
¬ᵢ : Γ , X ⊢ ⊥ → Γ ⊢ ∼ X
¬ₑ : Γ ⊢ X → Γ ⊢ ∼ X → Γ ⊢ ⊥
□ₑ : Γ ⊢ □ X → X ∈ Δ → Γ ++ boxed Δ ⊢ Y
□ᵢ : X ∈ Δ → Γ ++ boxed Δ ⊢ □ X
-- TODO: Maybe make it KT45 -- TODO: Maybe make it KT45
K : [] / [] ⊢ □ (X ⇒ Y) ⇒ (□ X ⇒ □ Y)
K {X} {Y} = ⇒ᵢ (⇒ᵢ (□ₑ (var (zero (□ X) ([] , (□ (X ⇒ Y)))))
(□ₑ (var (succ (□(X ⇒ Y)) ([] , (□ (X ⇒ Y))) (zero (□ (X ⇒ Y)) [])))
(□ᵢ (mp
(var (zero (X ⇒ Y) ([] , X)))
(var (succ X ([] , X) (zero X []))))))))
ExampleSyn : [] / [] ⊢ □ X ⇒ □ (Y ⇒ X)
ExampleSyn {X} {Y} = ⇒ᵢ (□ₑ (var (zero (□ X) [])) (□ᵢ (⇒ᵢ (var (succ X ([] , X) (zero X []))))))
ExampleSyn2 : [] / [] ⊢ □(A ∧ B) ⇒ (□ A ∧ □ B)
ExampleSyn2 {A} {B} = ⇒ᵢ (∧ᵢ (□ₑ (var (zero (□ (A ∧ B)) [])) (□ᵢ (∧ₑ₁ (var (zero (A ∧ B) []))))) (□ₑ (var (zero (□ (A ∧ B)) [])) (□ᵢ (∧ₑ₂ (var (zero (A ∧ B) []))))))
ExampleSyn3 : [] / [] ⊢ (□ A ∧ □ B) ⇒ □(A ∧ B)
ExampleSyn3 {A} {B} = ⇒ᵢ (□ₑ (∧ₑ₁ (var (zero ((□ A) ∧ (□ B)) []))) (□ₑ (∧ₑ₂ (var (zero ((□ A) ∧ (□ B)) []))) (□ᵢ (∧ᵢ (var (succ A ([] , A) (zero A []))) (var (zero B ([] , A)))))))