------------------------------------------------------------------------
-- The Agda standard library
--
-- Natural numbers, basic types and operations
------------------------------------------------------------------------

-- See README.Data.Nat for examples of how to use and reason about
-- naturals.

{-# OPTIONS --without-K --safe #-}

module Data.Nat.Base where

open import Data.Bool.Base using (Bool; true; false; T; not)
open import Level using (0ℓ)
open import Relation.Binary.Core using (Rel)
open import Relation.Binary.PropositionalEquality.Core
  using (_≡_; _≢_; refl)
open import Relation.Nullary using (¬_)
open import Relation.Nullary.Negation.Core using (contradiction)
open import Relation.Unary using (Pred)

------------------------------------------------------------------------
-- Types

open import Agda.Builtin.Nat public
  using (zero; suc) renaming (Nat to )

------------------------------------------------------------------------
-- Boolean equality relation

open import Agda.Builtin.Nat public
  using () renaming (_==_ to _≡ᵇ_)

------------------------------------------------------------------------
-- Boolean ordering relation

open import Agda.Builtin.Nat public
  using () renaming (_<_ to _<ᵇ_)

infix 4 _≤ᵇ_
_≤ᵇ_ : (m n : )  Bool
zero  ≤ᵇ n = true
suc m ≤ᵇ n = m <ᵇ n

------------------------------------------------------------------------
-- Standard ordering relations

infix 4 _≤_ _<_ _≥_ _>_ _≰_ _≮_ _≱_ _≯_

data _≤_ : Rel  0ℓ where
  z≤n :  {n}                  zero   n
  s≤s :  {m n} (m≤n : m  n)  suc m  suc n

_<_ : Rel  0ℓ
m < n = suc m  n

_≥_ : Rel  0ℓ
m  n = n  m

_>_ : Rel  0ℓ
m > n = n < m

_≰_ : Rel  0ℓ
a  b = ¬ a  b

_≮_ : Rel  0ℓ
a  b = ¬ a < b

_≱_ : Rel  0ℓ
a  b = ¬ a  b

_≯_ : Rel  0ℓ
a  b = ¬ a > b

------------------------------------------------------------------------
-- Simple predicates

-- Defining `NonZero` in terms of `T` and therefore ultimately `⊤` and
-- `⊥` allows Agda to automatically infer nonZero-ness for any natural
-- of the form `suc n`. Consequently in many circumstances this
-- eliminates the need to explicitly pass a proof when the NonZero
-- argument is either an implicit or an instance argument.
--
-- See `Data.Nat.DivMod` for an example.

record NonZero (n : ) : Set where
  field
    nonZero : T (not (n ≡ᵇ 0))

-- Instances

instance
  nonZero :  {n}  NonZero (suc n)
  nonZero = _

-- Constructors

≢-nonZero :  {n}  n  0  NonZero n
≢-nonZero {zero}  0≢0 = contradiction refl 0≢0
≢-nonZero {suc n} n≢0 = _

>-nonZero :  {n}  n > 0  NonZero n
>-nonZero (s≤s 0<n) = _

-- Destructors

≢-nonZero⁻¹ :  {n}  .(NonZero n)  n  0
≢-nonZero⁻¹ {suc n} _ ()

>-nonZero⁻¹ :  {n}  .(NonZero n)  n > 0
>-nonZero⁻¹ {suc n} _ = s≤s z≤n

------------------------------------------------------------------------
-- Arithmetic

open import Agda.Builtin.Nat public
  using (_+_; _*_) renaming (_-_ to _∸_)

pred :   
pred n = n  1

infixl 7 _⊓_
infixl 6 _+⋎_ _⊔_

-- Argument-swapping addition. Used by Data.Vec._⋎_.

_+⋎_ :     
zero  +⋎ n = n
suc m +⋎ n = suc (n +⋎ m)

-- Max.

_⊔_ :     
zero   n     = n
suc m  zero  = suc m
suc m  suc n = suc (m  n)

-- Min.

_⊓_ :     
zero   n     = zero
suc m  zero  = zero
suc m  suc n = suc (m  n)

-- Division by 2, rounded downwards.

⌊_/2⌋ :   
 0 /2⌋           = 0
 1 /2⌋           = 0
 suc (suc n) /2⌋ = suc  n /2⌋

-- Division by 2, rounded upwards.

⌈_/2⌉ :   
 n /2⌉ =  suc n /2⌋

-- Naïve exponentiation

_^_ :     
x ^ zero  = 1
x ^ suc n = x * x ^ n

-- Distance

∣_-_∣ :     
 zero  - y      = y
 x     - zero   = x
 suc x - suc y  =  x - y 

------------------------------------------------------------------------
-- Alternative definition of _≤_

-- The following definition of _≤_ is more suitable for well-founded
-- induction (see Data.Nat.Induction)

infix 4 _≤′_ _<′_ _≥′_ _>′_

data _≤′_ (m : ) :   Set where
  ≤′-refl :                         m ≤′ m
  ≤′-step :  {n} (m≤′n : m ≤′ n)  m ≤′ suc n

_<′_ : Rel  0ℓ
m <′ n = suc m ≤′ n

_≥′_ : Rel  0ℓ
m ≥′ n = n ≤′ m

_>′_ : Rel  0ℓ
m >′ n = n <′ m

------------------------------------------------------------------------
-- Another alternative definition of _≤_

record _≤″_ (m n : ) : Set where
  constructor less-than-or-equal
  field
    {k}   : 
    proof : m + k  n

infix 4 _≤″_ _<″_ _≥″_ _>″_

_<″_ : Rel  0ℓ
m <″ n = suc m ≤″ n

_≥″_ : Rel  0ℓ
m ≥″ n = n ≤″ m

_>″_ : Rel  0ℓ
m >″ n = n <″ m

------------------------------------------------------------------------
-- Another alternative definition of _≤_

-- Useful for induction when you have an upper bound.

data _≤‴_ :     Set where
  ≤‴-refl : ∀{m}  m ≤‴ m
  ≤‴-step : ∀{m n}  suc m ≤‴ n  m ≤‴ n

infix 4 _≤‴_ _<‴_ _≥‴_ _>‴_

_<‴_ : Rel  0ℓ
m <‴ n = suc m ≤‴ n

_≥‴_ : Rel  0ℓ
m ≥‴ n = n ≤‴ m

_>‴_ : Rel  0ℓ
m >‴ n = n <‴ m

------------------------------------------------------------------------
-- A comparison view. Taken from "View from the left"
-- (McBride/McKinna); details may differ.

data Ordering : Rel  0ℓ where
  less    :  m k  Ordering m (suc (m + k))
  equal   :  m    Ordering m m
  greater :  m k  Ordering (suc (m + k)) m

compare :  m n  Ordering m n
compare zero    zero    = equal   zero
compare (suc m) zero    = greater zero m
compare zero    (suc n) = less    zero n
compare (suc m) (suc n) with compare m n
... | less    m k = less (suc m) k
... | equal   m   = equal (suc m)
... | greater n k = greater (suc n) k