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Mjau

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pingu
2025-12-26 01:59:38 +01:00
parent 4ddf204ae8
commit 03083966a5
1 changed files with 19 additions and 14 deletions
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mambo.agda
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@@ -26,14 +26,21 @@ data _∪_ (A B : Type) : Type where
inl : A → A ∪ B inl : A → A ∪ B
inr : B → A ∪ B inr : B → A ∪ B
orₑ : {A B C : Type} → A ∪ B → (A → C) → (B → C) → C
orₑ (inl x) f g = f x
orₑ (inr x) f g = g x
absurd : {A : Type} → Bottom → A
absurd ()
data Formula : Type where data Formula : Type where
⊥ : Formula ⊥ : Formula
atom : Nat → Formula atom : Nat → Formula
∼_ : Formula → Formula ∼_ : (p : Formula) → Formula
□_ : Formula → Formula □_ : (p : Formula) → Formula
_∧_ : Formula → Formula → Formula _∧_ : (p q : Formula) → Formula
_∨_ : Formula → Formula → Formula _∨_ : (p q : Formula) → Formula
_⇒_ : Formula → Formula → Formula _⇒_ : (p q : Formula) → Formula
infixr 4 _⇒_ infixr 4 _⇒_
infix 19 _∧_ infix 19 _∧_
@@ -100,16 +107,19 @@ Context : Type
Context = List Formula Context = List Formula
variable variable
Γ Δ : Context Γ Δ Γ' Δ' : Context
infixl 10 _,_ infixl 10 _,_
pattern _,_ Γ X = X ∷ Γ pattern _,_ Γ X = X ∷ Γ
infix 10 _++_ infix 9 _++_
_++_ : {A : Type} → List A → List A → List A _++_ : {A : Type} → List A → List A → List A
[] ++ ys = ys [] ++ ys = ys
(xs , x) ++ ys = xs ++ (x ∷ ys) (xs , x) ++ ys = xs ++ (x ∷ ys)
_⊆_ : {A : Type} → (Γ Γ' : List A) → Type
_⊆_ {A} Γ Γ' = ∀ {X} → X ∈ Γ → X ∈ Γ'
infixr 2 _/_⊢_ infixr 2 _/_⊢_
data _/_⊢_ (Δ Γ : Context) : Formula → Type where data _/_⊢_ (Δ Γ : Context) : Formula → Type where
var : X ∈ Γ → Δ / Γ ⊢ X var : X ∈ Γ → Δ / Γ ⊢ X
@@ -147,10 +157,5 @@ 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))))))) 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)))))))
ExampleSyn4 : [] / [] ⊢ □ (A ⇒ B) ⇒ (◇ A ⇒ ◇ B) ExampleSyn4 : [] / [] ⊢ □ (A ⇒ B) ⇒ (◇ A ⇒ ◇ B)
ExampleSyn4 {A} {B} = {!!} ExampleSyn4 {A} {B} = ⇒ᵢ (⇒ᵢ (¬ᵢ (□ₑ (var (zero (□ (∼ B)) ([] , (□ (A ⇒ B)) , (◇ A)))) (□ₑ (var (succ (□ (A ⇒ B)) ([] , (□ (A ⇒ B)) , (◇ A)) (succ (□ (A ⇒ B)) ([] , (□ (A ⇒ B))) (zero (□ (A ⇒ B)) [])))) (¬ₑ (var (succ (∼ (□ (∼ A))) ([] , (□ (A ⇒ B)) , (◇ A))
(zero (∼ (□ (∼ A))) ([] , (□ (A ⇒ B)))))) (□ᵢ (¬ᵢ (¬ₑ (var (succ ( ∼ B) ([] , (∼ B) , (A ⇒ B)) (succ (∼ B) ([] , (∼ B)) (zero (∼ B) [])))) (mp (var (succ (A ⇒ B) ([] , (∼ B) , (A ⇒ B)) (zero (A ⇒ B) ([] , (∼ B))))) (var (zero A ([] , (∼ B) , (A ⇒ B)))))))))))))
Soundness : {W : Type} → (p : Formula) → [] / [] ⊢ p → (∀ (m : M W) (x : W) → m , x ⊩ p)
Soundness p x m w = {!!}
Completeness : {W : Type} → (p : Formula) → (∀ (m : M W) (x : W) → m , x ⊩ p) → [] / [] ⊢ p
Completeness p x = {!!}