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

module Data.Nat.Base where

import Level using (zero)
open import Function using (_∘_)
open import Relation.Binary
open import Relation.Binary.Core
open import Relation.Binary.PropositionalEquality.Core
import Relation.Binary.PropositionalEquality.TrustMe as TrustMe
open import Relation.Nullary using (¬_; Dec; yes; no)

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

------------------------------------------------------------------------
-- The types

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

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

_<_ : Rel  Level.zero
m < n = suc m  n

_≥_ : Rel  Level.zero
m  n = n  m

_>_ : Rel  Level.zero
m > n = n < m

_≰_ : Rel  Level.zero
a  b = ¬ a  b

_≮_ : Rel  Level.zero
a  b = ¬ a < b

_≱_ : Rel  Level.zero
a  b = ¬ a  b

_≯_ : Rel  Level.zero
a  b = ¬ a > b

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

infix 4 _≤′_ _<′_ _≥′_ _>′_

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

_<′_ : Rel  Level.zero
m <′ n = suc m ≤′ n

_≥′_ : Rel  Level.zero
m ≥′ n = n ≤′ m

_>′_ : Rel  Level.zero
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  Level.zero
m <″ n = suc m ≤″ n

_≥″_ : Rel  Level.zero
m ≥″ n = n ≤″ m

_>″_ : Rel  Level.zero
m >″ n = n <″ m

erase :  {m n}  m ≤″ n  m ≤″ n
erase (less-than-or-equal eq) = less-than-or-equal (TrustMe.erase eq)

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

pred :   
pred zero    = zero
pred (suc n) = n

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

------------------------------------------------------------------------
-- Queries

infix 4 _≟_ _≤?_

_≟_ : Decidable {A = } _≡_
zero   zero   = yes refl
suc m  suc n  with m  n
suc m  suc .m | yes refl = yes refl
suc m  suc n  | no prf   = no (prf   p  subst  x  m  pred x) p refl))
zero   suc n  = no λ()
suc m  zero   = no λ()

≤-pred :  {m n}  suc m  suc n  m  n
≤-pred (s≤s m≤n) = m≤n

_≤?_ : Decidable _≤_
zero  ≤? _     = yes z≤n
suc m ≤? zero  = no λ()
suc m ≤? suc n with m ≤? n
...            | yes m≤n = yes (s≤s m≤n)
...            | no  m≰n = no  (m≰n  ≤-pred)

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

data Ordering : Rel  Level.zero 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
compare (suc .m)           (suc .(suc m + k)) | less    m k = less    (suc m) k
compare (suc .m)           (suc .m)           | equal   m   = equal   (suc m)
compare (suc .(suc m + k)) (suc .m)           | greater m k = greater (suc m) k