{-# OPTIONS --cubical --no-import-sorts --no-exact-split --safe #-}
module Cubical.Data.Nat.Properties where

open import Cubical.Core.Everything

open import Cubical.Foundations.Prelude

open import Cubical.Data.Nat.Base
open import Cubical.Data.Empty as 
open import Cubical.Data.Sigma

open import Cubical.Relation.Nullary
open import Cubical.Relation.Nullary.DecidableEq

private
  variable
    l m n : 

min :     
min zero m = zero
min (suc n) zero = zero
min (suc n) (suc m) = suc (min n m)

minComm : (n m : )  min n m  min m n
minComm zero zero = refl
minComm zero (suc m) = refl
minComm (suc n) zero = refl
minComm (suc n) (suc m) = cong suc (minComm n m)

znots : ¬ (0  suc n)
znots eq = subst (caseNat  ) eq 0

snotz : ¬ (suc n  0)
snotz eq = subst (caseNat  ) eq 0

injSuc : suc m  suc n  m  n
injSuc p = cong predℕ p

discreteℕ : Discrete 
discreteℕ zero zero = yes refl
discreteℕ zero (suc n) = no znots
discreteℕ (suc m) zero = no snotz
discreteℕ (suc m) (suc n) with discreteℕ m n
... | yes p = yes (cong suc p)
... | no p = no  x  p (injSuc x))

isSetℕ : isSet 
isSetℕ = Discrete→isSet discreteℕ

-- Arithmetic facts about predℕ

suc-predℕ :  n  ¬ n  0  n  suc (predℕ n)
suc-predℕ zero p = ⊥.rec (p refl)
suc-predℕ (suc n) p = refl

-- Arithmetic facts about +

+-zero :  m  m + 0  m
+-zero zero = refl
+-zero (suc m) = cong suc (+-zero m)

+-suc :  m n  m + suc n  suc (m + n)
+-suc zero    n = refl
+-suc (suc m) n = cong suc (+-suc m n)

+-comm :  m n  m + n  n + m
+-comm m zero = +-zero m
+-comm m (suc n) = (+-suc m n)  (cong suc (+-comm m n))

-- Addition is associative
+-assoc :  m n o  m + (n + o)  (m + n) + o
+-assoc zero _ _    = refl
+-assoc (suc m) n o = cong suc (+-assoc m n o)

inj-m+ : m + l  m + n  l  n
inj-m+ {zero} p = p
inj-m+ {suc m} p = inj-m+ (injSuc p)

inj-+m : l + m  n + m  l  n
inj-+m {l} {m} {n} p = inj-m+ ((+-comm m l)  (p  (+-comm n m)))

m+n≡n→m≡0 : m + n  n  m  0
m+n≡n→m≡0 {n = zero} = λ p  (sym (+-zero _))  p
m+n≡n→m≡0 {n = suc n} p = m+n≡n→m≡0 (injSuc ((sym (+-suc _ n))  p))

m+n≡0→m≡0×n≡0 : m + n  0  (m  0) × (n  0)
m+n≡0→m≡0×n≡0 {zero} = refl ,_
m+n≡0→m≡0×n≡0 {suc m} p = ⊥.rec (snotz p)

-- Arithmetic facts about ·

0≡m·0 :  m  0  m · 0
0≡m·0 zero = refl
0≡m·0 (suc m) = 0≡m·0 m

·-suc :  m n  m · suc n  m + m · n
·-suc zero n = refl
·-suc (suc m) n
  = cong suc
  ( n + m · suc n ≡⟨ cong (n +_) (·-suc m n) 
    n + (m + m · n) ≡⟨ +-assoc n m (m · n) 
    (n + m) + m · n ≡⟨ cong (_+ m · n) (+-comm n m) 
    (m + n) + m · n ≡⟨ sym (+-assoc m n (m · n)) 
    m + (n + m · n) 
  )

·-comm :  m n  m · n  n · m
·-comm zero n = 0≡m·0 n
·-comm (suc m) n = cong (n +_) (·-comm m n)  sym (·-suc n m)

·-distribʳ :  m n o  (m · o) + (n · o)  (m + n) · o
·-distribʳ zero _ _ = refl
·-distribʳ (suc m) n o = sym (+-assoc o (m · o) (n · o))  cong (o +_) (·-distribʳ m n o)

·-distribˡ :  o m n  (o · m) + (o · n)  o · (m + n)
·-distribˡ o m n =  i  ·-comm o m i + ·-comm o n i)  ·-distribʳ m n o  ·-comm (m + n) o

·-assoc :  m n o  m · (n · o)  (m · n) · o
·-assoc zero _ _ = refl
·-assoc (suc m) n o = cong (n · o +_) (·-assoc m n o)  ·-distribʳ n (m · n) o

·-identityˡ :  m  1 · m  m
·-identityˡ m = +-zero m

·-identityʳ :  m  m · 1  m
·-identityʳ zero = refl
·-identityʳ (suc m) = cong suc (·-identityʳ m)

0≡n·sm→0≡n : 0  n · suc m  0  n
0≡n·sm→0≡n {n = zero} p = refl
0≡n·sm→0≡n {n = suc n} p = ⊥.rec (znots p)

inj-·sm : l · suc m  n · suc m  l  n
inj-·sm {zero} {m} {n} p = 0≡n·sm→0≡n p
inj-·sm {l} {m} {zero} p = sym (0≡n·sm→0≡n (sym p))
inj-·sm {suc l} {m} {suc n} p = cong suc (inj-·sm (inj-m+ {m = suc m} p))

inj-sm· : suc m · l  suc m · n  l  n
inj-sm· {m} {l} {n} p = inj-·sm (·-comm l (suc m)  p  ·-comm (suc m) n)

-- Arithmetic facts about ∸

zero∸ :  n  zero  n  zero
zero∸ zero = refl
zero∸ (suc _) = refl

∸-cancelˡ :  k m n  (k + m)  (k + n)  m  n
∸-cancelˡ zero    = λ _ _  refl
∸-cancelˡ (suc k) = ∸-cancelˡ k

∸-cancelʳ :  m n k  (m + k)  (n + k)  m  n
∸-cancelʳ m n k =  i  +-comm m k i  +-comm n k i)  ∸-cancelˡ k m n

∸-distribʳ :  m n k  (m  n) · k  m · k  n · k
∸-distribʳ m       zero    k = refl
∸-distribʳ zero    (suc n) k = sym (zero∸ (k + n · k))
∸-distribʳ (suc m) (suc n) k = ∸-distribʳ m n k  sym (∸-cancelˡ k (m · k) (n · k))