{-# OPTIONS --no-exact-split --safe #-}
module Cubical.Data.Nat.Properties where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Function
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Transport
open import Cubical.Data.Nat.Base
open import Cubical.Data.Empty as 
open import Cubical.Data.Sigma
open import Cubical.Data.Sum.Base
open import Cubical.Relation.Nullary
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)
max :     
max zero m = m
max (suc n) zero = suc n
max (suc n) (suc m) = suc (max n m)
maxComm : (n m : )  max n m  max m n
maxComm zero zero = refl
maxComm zero (suc m) = refl
maxComm (suc n) zero = refl
maxComm (suc n) (suc m) = cong suc (maxComm 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
-- encode decode caracterisation of equality
codeℕ : (n m : )  Type ℓ-zero
codeℕ zero zero = Unit
codeℕ zero (suc m) = 
codeℕ (suc n) zero = 
codeℕ (suc n) (suc m) = codeℕ n m
encodeℕ : (n m : )  (n  m)  codeℕ n m
encodeℕ n m p = subst  m  codeℕ n m) p (reflCode n)
 where
 reflCode : (n : )  codeℕ n n
 reflCode zero = tt
 reflCode (suc n) = reflCode n
compute-eqℕ : (n m : )  (n  m )  codeℕ n m
compute-eqℕ zero zero p = tt
compute-eqℕ zero (suc m) p = znots p
compute-eqℕ (suc n) zero p = snotz p
compute-eqℕ (suc n) (suc m) p = compute-eqℕ n m (injSuc p)
decodeℕ : (n m : )  codeℕ n m  (n  m)
decodeℕ zero zero = λ _  refl
decodeℕ zero (suc m) = ⊥.rec
decodeℕ (suc n) zero = ⊥.rec
decodeℕ (suc n) (suc m) = λ r  cong suc (decodeℕ n m r)
≡ℕ≃Codeℕ : (n m : )  (n  m)  codeℕ n m
≡ℕ≃Codeℕ n m = isoToEquiv is
  where
  is : Iso (n  m) (codeℕ n m)
  Iso.fun is = encodeℕ n m
  Iso.inv is = decodeℕ n m
  Iso.rightInv is = sect n m
    where
    sect : (n m : )  (r : codeℕ n m)  (encodeℕ n m (decodeℕ n m r)  r)
    sect zero zero tt = refl
    sect zero (suc m) r = ⊥.rec r
    sect (suc n) zero r = ⊥.rec r
    sect (suc n) (suc m) r = sect n m r
  Iso.leftInv is = retr n m
    where
    reflRetr : (n : )  decodeℕ n n (encodeℕ n n refl)  refl
    reflRetr zero = refl
    reflRetr (suc n) i = cong suc (reflRetr n i)
    retr : (n m : )  (p : n  m)  (decodeℕ n m (encodeℕ n m p)  p)
    retr n m p = J  m p  decodeℕ n m (encodeℕ n m p)  p) (reflRetr n) p

≡ℕ≃Codeℕ' : (n m : )  (n  m)  codeℕ n m
≡ℕ≃Codeℕ' n m = isoToEquiv is
  where
  is : Iso (n  m) (codeℕ n m)
  Iso.fun is = compute-eqℕ n m
  Iso.inv is = decodeℕ n m
  Iso.rightInv is = sect n m
    where
    sect : (n m : )  (r : codeℕ n m)  compute-eqℕ n m (decodeℕ n m r)  r
    sect zero zero tt = refl
    sect (suc n) (suc m) r = sect n m r
  Iso.leftInv is = retr n m
    where
    reflRetr : (n : )  decodeℕ n n (compute-eqℕ n n refl)  refl
    reflRetr zero = refl
    reflRetr (suc n) i = cong suc (reflRetr n i)
    retr : (n m : )  (p : n  m)  decodeℕ n m (compute-eqℕ n m p)  p
    retr n m p = J  m p  decodeℕ n m (compute-eqℕ n m p)  p) (reflRetr n) 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))
separatedℕ : Separated 
separatedℕ = Discrete→Separated discreteℕ
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)
integral-domain-· : {k l : }  (k  0  )  (l  0  )  (k · l  0  )
integral-domain-· {zero} {l} ¬p ¬q r = ¬p refl
integral-domain-· {suc k} {zero} ¬p ¬q r = ¬q refl
integral-domain-· {suc k} {suc l} ¬p ¬q r = snotz r
-- Arithmetic facts about ∸
zero∸ :  n  zero  n  zero
zero∸ zero = refl
zero∸ (suc _) = refl

n∸n : (n : )  n  n  0
n∸n zero = refl
n∸n (suc n) = n∸n n
∸-cancelˡ :  k m n  (k + m)  (k + n)  m  n
∸-cancelˡ zero    = λ _ _  refl
∸-cancelˡ (suc k) = ∸-cancelˡ k
+∸ :  k n  (k + n)  n  k
+∸ zero n = n∸n n
+∸ (suc k) zero = cong suc (+-comm k zero)
+∸ (suc k) (suc n) = cong (_∸ n) (+-suc k n)  +∸ (suc k) n
∸+ :  k n  (n + k)  n  k
∸+ k n = cong  X  X  n) (+-comm n k)  +∸ k n
∸-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))

-- factorial:
_! :   
zero ! = 1
suc n ! = (suc n) · (n !)
--binomial coefficient:
_choose_ :     
n choose zero = 1
zero choose suc k = 0
suc n choose suc k = n choose (suc k) + n choose k
evenOrOdd : (n : )  isEvenT n  isOddT n
evenOrOdd zero = inl tt
evenOrOdd (suc zero) = inr tt
evenOrOdd (suc (suc n)) = evenOrOdd n
¬evenAndOdd : (n : )  ¬ isEvenT n × isOddT n
¬evenAndOdd zero (p , ())
¬evenAndOdd (suc zero) ()
¬evenAndOdd (suc (suc n)) = ¬evenAndOdd n
isPropIsEvenT : (n : )  isProp (isEvenT n)
isPropIsEvenT zero x y = refl
isPropIsEvenT (suc zero) = isProp⊥
isPropIsEvenT (suc (suc n)) = isPropIsEvenT n
isPropIsOddT : (n : )  isProp (isOddT n)
isPropIsOddT n = isPropIsEvenT (suc n)
isPropEvenOrOdd : (n : )  isProp (isEvenT n  isOddT n)
isPropEvenOrOdd n (inl x) (inl x₁) = cong inl (isPropIsEvenT n x x₁)
isPropEvenOrOdd n (inl x) (inr x₁) = ⊥.rec (¬evenAndOdd n (x , x₁))
isPropEvenOrOdd n (inr x) (inl x₁) = ⊥.rec (¬evenAndOdd (suc n) (x , x₁))
isPropEvenOrOdd n (inr x) (inr x₁) = cong inr (isPropIsEvenT (suc n) x x₁)
module PlusBis where
  _+'_ :     
  zero +' b = b
  suc a +' zero = suc a
  suc a +' suc b = 2 + (a + b)
  +'≡+ : (n m : )  n +' m  n + m
  +'≡+ zero m = refl
  +'≡+ (suc n) zero = cong suc (sym (+-comm n zero))
  +'≡+ (suc n) (suc m) = cong suc (sym (+-suc n m))
  +'-comm : (n m : )  n +' m  m +' n
  +'-comm n m = +'≡+ n m ∙∙ +-comm n m ∙∙ sym (+'≡+ m n)
  +'-assoc : (n m l : )  (n +' (m +' l))  ((n +' m) +' l)
  +'-assoc n m l =
        i  +'≡+ n (+'≡+ m l i) i)
    ∙∙ +-assoc n m l
    ∙∙  i  +'≡+ (+'≡+ n m (~ i)) l (~ i))
  +'-rid : (n : )  n +' 0  n
  +'-rid zero = refl
  +'-rid (suc n) = refl
  +'-lid : (n : )  0 +' n  n
  +'-lid n = refl
  +'-suc : (n m : )  suc (n +' m)  suc n +' m
  +'-suc zero zero = refl
  +'-suc zero (suc m) = refl
  +'-suc (suc n) zero = refl
  +'-suc (suc n) (suc m) = refl
-- Neat transport lemma for ℕ
compSubstℕ :  {} {A :   Type } {n m l : }
   (p : n  m) (q : m  l) (r : n  l)
    {x : _}
    subst A q (subst A p x)
    subst A r x
compSubstℕ {A = A} p q r {x = x} =
  sym (substComposite A p q x)
   λ i  subst A (isSetℕ _ _ (p  q) r i) x