Not working, yet

mambo.agda

@@ -8,13 +8,14 @@ data Bottom : Type where
 ¬ : Type → Type
 ¬ A = A → Bottom
 
-Rel : Type → Type₁
-Rel A = A → A → Type
+Rel : Type → Type → Type₁
+Rel A B = A → B → Type
 
-record _×_ (A B : Type) : Type where
-  field
-    fst : A
-    snd : B
+data Σ (A : Type) (B : A -> Type) : Type where
+ pair : (a : A) -> (b : B a) -> Σ A B
+
+∃ : {A : Type} (B : A → Type) → Type
+∃ {A} B = Σ A B
 
 data Formula : Type where
   ⊥ : Formula
@@ -50,6 +51,7 @@ data _∈_ {A : Type} : A → List A → Type where
   zero : (x : A) (xs : List A) → x ∈ (x ∷ xs)
   succ : {y : A} (x : A) (xs : List A) → (x ∈ xs) → (x ∈ (y ∷ xs))
 
+
 infixr 2 _⊢_
 data _⊢_ : Context → Formula → Type where
   var : X ∈ Γ → Γ ⊢ X
@@ -63,17 +65,22 @@ data _⊢_ : Context → Formula → Type where
   ⇒ᵢ : Γ , X ⊢ Y → Γ ⊢ X ⇒ Y
   ¬ᵢ : Γ , X ⊢ ⊥ → Γ ⊢ ∼ X
   ¬ₑ : Γ ⊢ X → Γ ⊢ ∼ X → Γ ⊢ ⊥
-  -- ¬¬ₑ : Γ ⊢ ¬ ¬ X → Γ ⊢ X
-  -- Unsure if i want classical logic
   -- TODO: entailments for ◇ and □
 
 record M (W : Type) : Type₁ where
   field
-    R : Rel (W × W)
+    R : Rel W W
     L : W → List Nat
 
 infixr 2 _,_⊩_
-data _,_⊩_ {W : Type} (Model : M W) (x : W) : Formula → Type where
+data _,_⊩_ {W : Type} (Model : M W) (x : W) : (p : Formula) → Type where
   top : Model , x ⊩ ⊤
   atom : {p : Nat} → p ∈ (M.L Model x) →  Model , x ⊩ atom p
-  -- TODO: The rest :D
+  neg : {p : Formula} → ¬ (Model , x ⊩ p) → Model , x ⊩ ∼ p
+  and : {p q : Formula} → Model , x ⊩ p → Model , x ⊩ q → Model , x ⊩ p ∧ q
+  orl : {p q : Formula} → Model , x ⊩ p → Model , x ⊩ p ∨ q
+  orr : {p q : Formula} → Model , x ⊩ q → Model , x ⊩ p ∨ q
+  impl : {p q : Formula} → Model , x ⊩ p → Model , x ⊩ q → Model , x ⊩ p ⇒ q 
+  -- impln : {p q : Formula} → ¬ (Model , x ⊩ p) → Model , x ⊩ p ⇒ q 
+  box : {p : Formula} → (∀ (y : W) → M.R Model x y → Model , y ⊩ p) → Model , x ⊩ □ p
+  dia : {p : Formula} → (∃ λ (y : W) → M.R Model x y → Model , y ⊩ p) → Model , x ⊩ ◇ p