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pingu
2025-12-26 13:30:39 +01:00
parent 03083966a5
commit e1212b208c
1 changed files with 94 additions and 1 deletions
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@@ -115,8 +115,9 @@ pattern _,_ Γ X = X ∷ Γ
infix 9 _++_
_++_ : {A : Type} → List A → List A → List A
[] ++ ys = ys
(xs , x) ++ ys = xs ++ (x ∷ ys)
(x ∷ xs) ++ ys = x ∷ (xs ++ ys)
infix 8 _⊆_
_⊆_ : {A : Type} → (Γ Γ' : List A) → Type
_⊆_ {A} Γ Γ' = ∀ {X} → X ∈ Γ → X ∈ Γ'
@@ -133,10 +134,92 @@ data _/_⊢_ (Δ Γ : Context) : Formula → Type where
⇒ᵢ : Δ / Γ , X ⊢ Y → Δ / Γ ⊢ X ⇒ Y
¬ᵢ : Δ / Γ , X ⊢ ⊥ → Δ / Γ ⊢ ∼ X
¬ₑ : Δ / Γ ⊢ ∼ X → Δ / Γ ⊢ X → Δ / Γ ⊢ ⊥
⊥ₑ : Δ / Γ ⊢ ⊥ → Δ / Γ ⊢ X
□ᵢ : [] / Δ ⊢ X → Δ / Γ ⊢ □ X
□ₑ : Δ / Γ ⊢ □ X → (Δ , X) / Γ ⊢ Y → Δ / Γ ⊢ Y
-- TODO: Maybe make it KT45
inSwap : {A : Type} {X Y Z : A} (Γ Γ' : List A) → Z ∈ (Γ' ++ Γ , X , Y) → Z ∈ (Γ' ++ Γ , Y , X)
inSwap Γ [] (zero x xs) = succ x (Γ , x) (zero x Γ)
inSwap Γ [] (succ x xs (zero x₁ xs₁)) = zero x (Γ , _)
inSwap Γ [] (succ x xs (succ x₁ xs₁ x₂)) = succ x (Γ , _) (succ x Γ x₂)
inSwap Γ (Γ' , x₁) (zero x xs) = zero x₁ (Γ' ++ Γ , _ , _)
inSwap Γ (Γ' , x₁) (succ x xs x₂) = succ x (Γ' ++ Γ , _ , _) (inSwap Γ Γ' x₂)
exchange : (Γ' : Context) → Δ / Γ' ++ (Γ , A , B) ⊢ C → Δ / Γ' ++ (Γ , B , A) ⊢ C
exchange Γ' (var x) = var (inSwap _ Γ' x)
exchange Γ' (mp x x₁) = mp (exchange Γ' x) (exchange Γ' x₁)
exchange Γ' (∧ᵢ x x₁) = ∧ᵢ (exchange Γ' x) (exchange Γ' x₁)
exchange Γ' (∧ₑ₁ x) = ∧ₑ₁ (exchange Γ' x)
exchange Γ' (∧ₑ₂ x) = ∧ₑ₂ (exchange Γ' x)
exchange Γ' (∨ᵢ₁ x) = ∨ᵢ₁ (exchange Γ' x)
exchange Γ' (∨ᵢ₂ x) = ∨ᵢ₂ (exchange Γ' x)
exchange Γ' (∨ₑ x x₁ x₂) = ∨ₑ (exchange Γ' x) (exchange (Γ' , _) x₁) (exchange (Γ' , _) x₂)
exchange Γ' (⇒ᵢ x) = ⇒ᵢ (exchange (Γ' , _) x)
exchange Γ' (¬ᵢ x) = ¬ᵢ (exchange (Γ' , _) x)
exchange Γ' (¬ₑ x x₁) = ¬ₑ (exchange Γ' x) (exchange Γ' x₁)
exchange Γ' (⊥ₑ x) = ⊥ₑ (exchange Γ' x)
exchange Γ' (□ᵢ x) = □ᵢ x
exchange Γ' (□ₑ x x₁) = □ₑ (exchange Γ' x) (exchange Γ' x₁)
exchange-modal : (Δ' : Context) → Δ' ++ Δ , A , B / Γ ⊢ C → Δ' ++ Δ , B , A / Γ ⊢ C
exchange-modal Δ' (var x) = var x
exchange-modal Δ' (mp x x₁) = mp (exchange-modal Δ' x) (exchange-modal Δ' x₁)
exchange-modal Δ' (∧ᵢ x x₁) = ∧ᵢ (exchange-modal Δ' x) (exchange-modal Δ' x₁)
exchange-modal Δ' (∧ₑ₁ x) = ∧ₑ₁ (exchange-modal Δ' x)
exchange-modal Δ' (∧ₑ₂ x) = ∧ₑ₂ (exchange-modal Δ' x)
exchange-modal Δ' (∨ᵢ₁ x) = ∨ᵢ₁ (exchange-modal Δ' x)
exchange-modal Δ' (∨ᵢ₂ x) = ∨ᵢ₂ (exchange-modal Δ' x)
exchange-modal Δ' (∨ₑ x x₁ x₂) = ∨ₑ (exchange-modal Δ' x) (exchange-modal Δ' x₁) (exchange-modal Δ' x₂)
exchange-modal Δ' (⇒ᵢ x) = ⇒ᵢ (exchange-modal Δ' x)
exchange-modal Δ' (¬ᵢ x) = ¬ᵢ (exchange-modal Δ' x)
exchange-modal Δ' (¬ₑ x x₁) = ¬ₑ (exchange-modal Δ' x) (exchange-modal Δ' x₁)
exchange-modal Δ' (⊥ₑ x) = ⊥ₑ (exchange-modal Δ' x)
exchange-modal Δ' (□ᵢ x) = □ᵢ (exchange Δ' x)
exchange-modal Δ' (□ₑ x x₁) = □ₑ (exchange-modal Δ' x) (exchange-modal (Δ' , _) x₁)
inBoth : {A : Type} {Γ Γ' : List A} {X : A} → Γ ⊆ Γ' → (Γ , X) ⊆ (Γ' , X)
inBoth {A} {Γ} {Γ'} {X} x {X₁} (zero x₁ xs) = zero X Γ'
inBoth {A} {Γ} {Γ'} {X} x {X₁} (succ x₁ xs x₂) = succ X₁ Γ' (x x₂)
weak : Δ / Γ ⊢ A → Γ ⊆ Γ' → Δ / Γ' ⊢ A
weak {Δ} {Γ} {A} {Γ'} (var x) x₁ = var (x₁ x)
weak {Δ} {Γ} {A} {Γ'} (mp x x₂) x₁ = mp (weak x x₁) (weak x₂ x₁)
weak {Δ} {Γ} {A} {Γ'} (∧ᵢ x x₂) x₁ = ∧ᵢ (weak x x₁) (weak x₂ x₁)
weak {Δ} {Γ} {A} {Γ'} (∧ₑ₁ x) x₁ = ∧ₑ₁ (weak x x₁)
weak {Δ} {Γ} {A} {Γ'} (∧ₑ₂ x) x₁ = ∧ₑ₂ (weak x x₁)
weak {Δ} {Γ} {A} {Γ'} (∨ᵢ₁ x) x₁ = ∨ᵢ₁ (weak x x₁)
weak {Δ} {Γ} {A} {Γ'} (∨ᵢ₂ x) x₁ = ∨ᵢ₂ (weak x x₁)
weak {Δ} {Γ} {A} {Γ'} (∨ₑ x x₂ x₃) x₁ = ∨ₑ (weak x x₁) (weak x₂ (inBoth x₁)) (weak x₃ (inBoth x₁))
weak {Δ} {Γ} {A} {Γ'} (⇒ᵢ x) x₁ = ⇒ᵢ (weak x (inBoth x₁))
weak {Δ} {Γ} {A} {Γ'} (¬ᵢ x) x₁ = ¬ᵢ (weak x (inBoth x₁))
weak {Δ} {Γ} {A} {Γ'} (¬ₑ x x₂) x₁ = ¬ₑ (weak x x₁) (weak x₂ x₁)
weak {Δ} {Γ} {A} {Γ'} (⊥ₑ x) x₁ = ⊥ₑ (weak x x₁)
weak {Δ} {Γ} {A} {Γ'} (□ᵢ x) x₁ = □ᵢ x
weak {Δ} {Γ} {A} {Γ'} (□ₑ x x₂) x₁ = □ₑ (weak x x₁) (weak x₂ x₁)
weak-modal : Δ / Γ ⊢ A → Δ ⊆ Δ' → Δ' / Γ ⊢ A
weak-modal (var x) x₁ = var x
weak-modal (mp x x₂) x₁ = mp (weak-modal x x₁) (weak-modal x₂ x₁)
weak-modal (∧ᵢ x x₂) x₁ = ∧ᵢ (weak-modal x x₁) (weak-modal x₂ x₁)
weak-modal (∧ₑ₁ x) x₁ = ∧ₑ₁ (weak-modal x x₁)
weak-modal (∧ₑ₂ x) x₁ = ∧ₑ₂ (weak-modal x x₁)
weak-modal (∨ᵢ₁ x) x₁ = ∨ᵢ₁ (weak-modal x x₁)
weak-modal (∨ᵢ₂ x) x₁ = ∨ᵢ₂ (weak-modal x x₁)
weak-modal (∨ₑ x x₂ x₃) x₁ = ∨ₑ (weak-modal x x₁) (weak-modal x₂ x₁) (weak-modal x₃ x₁)
weak-modal (⇒ᵢ x) x₁ = ⇒ᵢ (weak-modal x x₁)
weak-modal (¬ᵢ x) x₁ = ¬ᵢ (weak-modal x x₁)
weak-modal (¬ₑ x x₂) x₁ = ¬ₑ (weak-modal x x₁) (weak-modal x₂ x₁)
weak-modal (⊥ₑ x) x₁ = ⊥ₑ (weak-modal x x₁)
weak-modal (□ᵢ x) x₁ = □ᵢ (weak x x₁)
weak-modal (□ₑ x x₂) x₁ = □ₑ (weak-modal x x₁) (weak-modal x₂ (inBoth x₁))
cut : Δ / Γ ⊢ A → Δ / Γ' ++ (Γ , A) ⊢ B → Δ / Γ' ++ Γ ⊢ B
cut = {!!}
cut-modal : Δ / Γ ⊢ A → Δ' ++ (Δ , A) / Γ ⊢ B → Δ' ++ Δ / Γ ⊢ B
cut-modal = {!!}
KSym : [] / [] ⊢ □ (X ⇒ Y) ⇒ (□ X ⇒ □ Y)
KSym {X} {Y} = ⇒ᵢ (⇒ᵢ (□ₑ (var (zero (□ X) ([] , (□ (X ⇒ Y)))))
(□ₑ (var (succ (□(X ⇒ Y)) ([] , (□ (X ⇒ Y))) (zero (□ (X ⇒ Y)) [])))
@@ -147,6 +230,16 @@ KSym {X} {Y} = ⇒ᵢ (⇒ᵢ (□ₑ (var (zero (□ X) ([] , (□ (X ⇒ Y))))
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))))))))
DedSym : Δ / Γ ⊢ A ⇒ B → Δ / Γ , A ⊢ B
DedSym {Δ} {Γ} {A} {B} (var x) = mp (var (succ (A ⇒ B) Γ x)) (var (zero A Γ))
DedSym {Δ} {Γ} {A} {B} (mp x x₁) = {!!}
DedSym {Δ} {Γ} {A} {B} (∧ₑ₁ x) = {!!}
DedSym {Δ} {Γ} {A} {B} (∧ₑ₂ x) = {!!}
DedSym {Δ} {Γ} {A} {B} (∨ₑ x x₁ x₂) = {!!}
DedSym {Δ} {Γ} {A} {B} (⇒ᵢ x) = x
DedSym {Δ} {Γ} {A} {B} (⊥ₑ x) = weak (⊥ₑ x) λ {X} → succ X Γ
DedSym {Δ} {Γ} {A} {B} (□ₑ x x₁) = {!!}
ExampleSyn : [] / [] ⊢ □ X ⇒ □ (Y ⇒ X)
ExampleSyn {X} {Y} = ⇒ᵢ (□ₑ (var (zero (□ X) [])) (□ᵢ (⇒ᵢ (var (succ X ([] , X) (zero X []))))))