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pingu
2025-12-25 14:38:00 +01:00
parent 1e949cb9ad
commit 0e3754808e
1 changed files with 4 additions and 1 deletions
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@@ -122,7 +122,7 @@ data _/_⊢_ (Δ Γ : Context) : Formula → Type where
∨ₑ : Δ / Γ ⊢ X ∨ Y → Δ / Γ , X ⊢ Z → Δ / Γ , Y ⊢ Z → Δ / Γ ⊢ Z ∨ₑ : Δ / Γ ⊢ X ∨ Y → Δ / Γ , X ⊢ Z → Δ / Γ , Y ⊢ Z → Δ / Γ ⊢ Z
⇒ᵢ : Δ / Γ , X ⊢ Y → Δ / Γ ⊢ X ⇒ Y ⇒ᵢ : Δ / Γ , X ⊢ Y → Δ / Γ ⊢ X ⇒ Y
¬ᵢ : Δ / Γ , X ⊢ ⊥ → Δ / Γ ⊢ ∼ X ¬ᵢ : Δ / Γ , X ⊢ ⊥ → Δ / Γ ⊢ ∼ X
¬ₑ : Δ / Γ ⊢ X → Δ / Γ ⊢ ∼ X → Δ / Γ ⊢ ⊥ ¬ₑ : Δ / Γ ⊢ ∼ X → Δ / Γ ⊢ X → Δ / Γ ⊢ ⊥
□ᵢ : [] / Δ ⊢ X → Δ / Γ ⊢ □ X □ᵢ : [] / Δ ⊢ X → Δ / Γ ⊢ □ X
□ₑ : Δ / Γ ⊢ □ X → (Δ , X) / Γ ⊢ Y → Δ / Γ ⊢ Y □ₑ : Δ / Γ ⊢ □ X → (Δ , X) / Γ ⊢ Y → Δ / Γ ⊢ Y
-- TODO: Maybe make it KT45 -- TODO: Maybe make it KT45
@@ -134,6 +134,9 @@ KSym {X} {Y} = ⇒ᵢ (⇒ᵢ (□ₑ (var (zero (□ X) ([] , (□ (X ⇒ Y))))
(var (zero (X ⇒ Y) ([] , X))) (var (zero (X ⇒ Y) ([] , X)))
(var (succ X ([] , X) (zero X [])))))))) (var (succ X ([] , X) (zero X []))))))))
MTSym : [] / [] ⊢ (A ⇒ B) ⇒ ∼ B ⇒ ∼ A
MTSym {A} {B} = ⇒ᵢ (⇒ᵢ (¬ᵢ (¬ₑ (var (succ (∼ B) ([] , (A ⇒ B) , (∼ B)) (zero (∼ B) ([] , (A ⇒ B))))) (mp (var (succ (A ⇒ B) ([] , (A ⇒ B) , (∼ B)) (succ (A ⇒ B) ([] , (A ⇒ B)) (zero (A ⇒ B) [])))) (var (zero A ([] , (A ⇒ B) , (∼ B))))))))
ExampleSyn : [] / [] ⊢ □ X ⇒ □ (Y ⇒ X) ExampleSyn : [] / [] ⊢ □ X ⇒ □ (Y ⇒ X)
ExampleSyn {X} {Y} = ⇒ᵢ (□ₑ (var (zero (□ X) [])) (□ᵢ (⇒ᵢ (var (succ X ([] , X) (zero X [])))))) ExampleSyn {X} {Y} = ⇒ᵢ (□ₑ (var (zero (□ X) [])) (□ᵢ (⇒ᵢ (var (succ X ([] , X) (zero X []))))))