-- module Demo where

open import Relation.Binary.PropositionalEquality 

-- NATURAL NUMBERS
data ℕ : Set where
  zero : ℕ
  succ : ℕ → ℕ


-- Addition
_+_ : ℕ → ℕ → ℕ
zero + b = b
succ a + b = succ (a + b)


-- BOOLEANS
data Bool : Set where
  true : Bool
  false : Bool

-- Negation
¬ : Bool → Bool
¬ true = false
¬ false = true

-- Proof
not-not : (b : Bool) → ¬(¬ b) ≡ b
not-not true = refl
not-not false = refl



-- ******* -- 

-- LIST 
data List (A : Set) : Set where
  [] : List A
  _::_ : A → List A → List A



{-# BUILTIN NATURAL ℕ #-}

infixr 5 _::_

example = 1 :: 2 :: []



-- Concatenation
_++_ : {A : Set} → List A → List A → List A
[] ++ ys = ys
(x :: xs) ++ ys = x :: (xs ++ ys)


-- Define length function for lists
length : {A : Set} → List A → ℕ
length [] = 0
length (x :: xs) = 1 + length xs



-- Proof

-- length (xs ++ ys) ≡ length xs + length ys
length-concat : {A : Set} (xs ys : List A) → length (xs ++ ys) ≡ length xs + length ys
length-concat [] ys = {!   !}
length-concat (x :: xs) ys = cong succ (length-concat xs ys) 



-- VECTOR
data Vec (A : Set) : ℕ → Set where
  [] : Vec A 0
  _::_ : {n : ℕ} → A → Vec A n → Vec A (1 + n)

-- Head
head : {A : Set} {n : ℕ} -> Vec A (1 + n) -> A
head (x :: xs) = x