```------------------------------------------------------------------------
-- A bunch of properties about natural number operations
------------------------------------------------------------------------

-- See README.Nat for some examples showing how this module can be
-- used.

module Data.Nat.Properties where

open import Data.Nat as Nat
open ≤-Reasoning
renaming (begin_ to start_; _∎ to _□; _≡⟨_⟩_ to _≡⟨_⟩'_)
open import Relation.Binary
open DecTotalOrder Nat.decTotalOrder using () renaming (refl to ≤-refl)
open import Function
open import Algebra
open import Algebra.Structures
open import Relation.Nullary
open import Relation.Binary.PropositionalEquality as PropEq
using (_≡_; _≢_; refl; sym; cong; cong₂)
open PropEq.≡-Reasoning
import Algebra.FunctionProperties as P; open P _≡_
open import Data.Product

------------------------------------------------------------------------
-- (ℕ, +, *, 0, 1) is a commutative semiring

private

+-assoc : Associative _+_
+-assoc zero    _ _ = refl
+-assoc (suc m) n o = cong suc \$ +-assoc m n o

+-identity : Identity 0 _+_
+-identity = (λ _ → refl) , n+0≡n
where
n+0≡n : RightIdentity 0 _+_
n+0≡n zero    = refl
n+0≡n (suc n) = cong suc \$ n+0≡n n

m+1+n≡1+m+n : ∀ m n → m + suc n ≡ suc (m + n)
m+1+n≡1+m+n zero    n = refl
m+1+n≡1+m+n (suc m) n = cong suc (m+1+n≡1+m+n m n)

+-comm : Commutative _+_
+-comm zero    n = sym \$ proj₂ +-identity n
+-comm (suc m) n =
begin
suc m + n
≡⟨ refl ⟩
suc (m + n)
≡⟨ cong suc (+-comm m n) ⟩
suc (n + m)
≡⟨ sym (m+1+n≡1+m+n n m) ⟩
n + suc m
∎

m*1+n≡m+mn : ∀ m n → m * suc n ≡ m + m * n
m*1+n≡m+mn zero    n = refl
m*1+n≡m+mn (suc m) n =
begin
suc m * suc n
≡⟨ refl ⟩
suc n + m * suc n
≡⟨ cong (λ x → suc n + x) (m*1+n≡m+mn m n) ⟩
suc n + (m + m * n)
≡⟨ refl ⟩
suc (n + (m + m * n))
≡⟨ cong suc (sym \$ +-assoc n m (m * n)) ⟩
suc (n + m + m * n)
≡⟨ cong (λ x → suc (x + m * n)) (+-comm n m) ⟩
suc (m + n + m * n)
≡⟨ cong suc (+-assoc m n (m * n)) ⟩
suc (m + (n + m * n))
≡⟨ refl ⟩
suc m + suc m * n
∎

*-zero : Zero 0 _*_
*-zero = (λ _ → refl) , n*0≡0
where
n*0≡0 : RightZero 0 _*_
n*0≡0 zero    = refl
n*0≡0 (suc n) = n*0≡0 n

*-comm : Commutative _*_
*-comm zero    n = sym \$ proj₂ *-zero n
*-comm (suc m) n =
begin
suc m * n
≡⟨ refl ⟩
n + m * n
≡⟨ cong (λ x → n + x) (*-comm m n) ⟩
n + n * m
≡⟨ sym (m*1+n≡m+mn n m) ⟩
n * suc m
∎

distribʳ-*-+ : _*_ DistributesOverʳ _+_
distribʳ-*-+ m zero    o = refl
distribʳ-*-+ m (suc n) o =
begin
(suc n + o) * m
≡⟨ refl ⟩
m + (n + o) * m
≡⟨ cong (_+_ m) \$ distribʳ-*-+ m n o ⟩
m + (n * m + o * m)
≡⟨ sym \$ +-assoc m (n * m) (o * m) ⟩
m + n * m + o * m
≡⟨ refl ⟩
suc n * m + o * m
∎

*-assoc : Associative _*_
*-assoc zero    n o = refl
*-assoc (suc m) n o =
begin
(suc m * n) * o
≡⟨ refl ⟩
(n + m * n) * o
≡⟨ distribʳ-*-+ o n (m * n) ⟩
n * o + (m * n) * o
≡⟨ cong (λ x → n * o + x) \$ *-assoc m n o ⟩
n * o + m * (n * o)
≡⟨ refl ⟩
suc m * (n * o)
∎

isCommutativeSemiring : IsCommutativeSemiring _≡_ _+_ _*_ 0 1
isCommutativeSemiring = record
{ +-isCommutativeMonoid = record
{ isSemigroup = record
{ isEquivalence = PropEq.isEquivalence
; assoc         = +-assoc
; ∙-cong        = cong₂ _+_
}
; identityˡ = proj₁ +-identity
; comm      = +-comm
}
; *-isCommutativeMonoid = record
{ isSemigroup = record
{ isEquivalence = PropEq.isEquivalence
; assoc         = *-assoc
; ∙-cong        = cong₂ _*_
}
; identityˡ = proj₂ +-identity
; comm      = *-comm
}
; distribʳ = distribʳ-*-+
; zeroˡ    = proj₁ *-zero
}

commutativeSemiring : CommutativeSemiring _ _
commutativeSemiring = record
{ _+_                   = _+_
; _*_                   = _*_
; 0#                    = 0
; 1#                    = 1
; isCommutativeSemiring = isCommutativeSemiring
}

import Algebra.RingSolver.Simple as Solver
import Algebra.RingSolver.AlmostCommutativeRing as ACR
module SemiringSolver =
Solver (ACR.fromCommutativeSemiring commutativeSemiring)

------------------------------------------------------------------------
-- (ℕ, ⊔, ⊓, 0) is a commutative semiring without one

private

⊔-assoc : Associative _⊔_
⊔-assoc zero    _       _       = refl
⊔-assoc (suc m) zero    o       = refl
⊔-assoc (suc m) (suc n) zero    = refl
⊔-assoc (suc m) (suc n) (suc o) = cong suc \$ ⊔-assoc m n o

⊔-identity : Identity 0 _⊔_
⊔-identity = (λ _ → refl) , n⊔0≡n
where
n⊔0≡n : RightIdentity 0 _⊔_
n⊔0≡n zero    = refl
n⊔0≡n (suc n) = refl

⊔-comm : Commutative _⊔_
⊔-comm zero    n       = sym \$ proj₂ ⊔-identity n
⊔-comm (suc m) zero    = refl
⊔-comm (suc m) (suc n) =
begin
suc m ⊔ suc n
≡⟨ refl ⟩
suc (m ⊔ n)
≡⟨ cong suc (⊔-comm m n) ⟩
suc (n ⊔ m)
≡⟨ refl ⟩
suc n ⊔ suc m
∎

⊓-assoc : Associative _⊓_
⊓-assoc zero    _       _       = refl
⊓-assoc (suc m) zero    o       = refl
⊓-assoc (suc m) (suc n) zero    = refl
⊓-assoc (suc m) (suc n) (suc o) = cong suc \$ ⊓-assoc m n o

⊓-zero : Zero 0 _⊓_
⊓-zero = (λ _ → refl) , n⊓0≡0
where
n⊓0≡0 : RightZero 0 _⊓_
n⊓0≡0 zero    = refl
n⊓0≡0 (suc n) = refl

⊓-comm : Commutative _⊓_
⊓-comm zero    n       = sym \$ proj₂ ⊓-zero n
⊓-comm (suc m) zero    = refl
⊓-comm (suc m) (suc n) =
begin
suc m ⊓ suc n
≡⟨ refl ⟩
suc (m ⊓ n)
≡⟨ cong suc (⊓-comm m n) ⟩
suc (n ⊓ m)
≡⟨ refl ⟩
suc n ⊓ suc m
∎

distrib-⊓-⊔ : _⊓_ DistributesOver _⊔_
distrib-⊓-⊔ = (distribˡ-⊓-⊔ , distribʳ-⊓-⊔)
where
distribʳ-⊓-⊔ : _⊓_ DistributesOverʳ _⊔_
distribʳ-⊓-⊔ (suc m) (suc n) (suc o) = cong suc \$ distribʳ-⊓-⊔ m n o
distribʳ-⊓-⊔ (suc m) (suc n) zero    = cong suc \$ refl
distribʳ-⊓-⊔ (suc m) zero    o       = refl
distribʳ-⊓-⊔ zero    n       o       = begin
(n ⊔ o) ⊓ 0    ≡⟨ ⊓-comm (n ⊔ o) 0 ⟩
0 ⊓ (n ⊔ o)    ≡⟨ refl ⟩
0 ⊓ n ⊔ 0 ⊓ o  ≡⟨ ⊓-comm 0 n ⟨ cong₂ _⊔_ ⟩ ⊓-comm 0 o ⟩
n ⊓ 0 ⊔ o ⊓ 0  ∎

distribˡ-⊓-⊔ : _⊓_ DistributesOverˡ _⊔_
distribˡ-⊓-⊔ m n o = begin
m ⊓ (n ⊔ o)    ≡⟨ ⊓-comm m _ ⟩
(n ⊔ o) ⊓ m    ≡⟨ distribʳ-⊓-⊔ m n o ⟩
n ⊓ m ⊔ o ⊓ m  ≡⟨ ⊓-comm n m ⟨ cong₂ _⊔_ ⟩ ⊓-comm o m ⟩
m ⊓ n ⊔ m ⊓ o  ∎

⊔-⊓-0-isCommutativeSemiringWithoutOne
: IsCommutativeSemiringWithoutOne _≡_ _⊔_ _⊓_ 0
⊔-⊓-0-isCommutativeSemiringWithoutOne = record
{ isSemiringWithoutOne = record
{ +-isCommutativeMonoid = record
{ isSemigroup = record
{ isEquivalence = PropEq.isEquivalence
; assoc         = ⊔-assoc
; ∙-cong        = cong₂ _⊔_
}
; identityˡ = proj₁ ⊔-identity
; comm      = ⊔-comm
}
; *-isSemigroup = record
{ isEquivalence = PropEq.isEquivalence
; assoc         = ⊓-assoc
; ∙-cong        = cong₂ _⊓_
}
; distrib = distrib-⊓-⊔
; zero    = ⊓-zero
}
; *-comm = ⊓-comm
}

⊔-⊓-0-commutativeSemiringWithoutOne : CommutativeSemiringWithoutOne _ _
⊔-⊓-0-commutativeSemiringWithoutOne = record
{ _+_                             = _⊔_
; _*_                             = _⊓_
; 0#                              = 0
; isCommutativeSemiringWithoutOne =
⊔-⊓-0-isCommutativeSemiringWithoutOne
}

------------------------------------------------------------------------
-- (ℕ, ⊓, ⊔) is a lattice

private

absorptive-⊓-⊔ : Absorptive _⊓_ _⊔_
absorptive-⊓-⊔ = abs-⊓-⊔ , abs-⊔-⊓
where
abs-⊔-⊓ : _⊔_ Absorbs _⊓_
abs-⊔-⊓ zero    n       = refl
abs-⊔-⊓ (suc m) zero    = refl
abs-⊔-⊓ (suc m) (suc n) = cong suc \$ abs-⊔-⊓ m n

abs-⊓-⊔ : _⊓_ Absorbs _⊔_
abs-⊓-⊔ zero    n       = refl
abs-⊓-⊔ (suc m) (suc n) = cong suc \$ abs-⊓-⊔ m n
abs-⊓-⊔ (suc m) zero    = cong suc \$
begin
m ⊓ m
≡⟨ cong (_⊓_ m) \$ sym \$ proj₂ ⊔-identity m ⟩
m ⊓ (m ⊔ 0)
≡⟨ abs-⊓-⊔ m zero ⟩
m
∎

isDistributiveLattice : IsDistributiveLattice _≡_ _⊓_ _⊔_
isDistributiveLattice = record
{ isLattice = record
{ isEquivalence = PropEq.isEquivalence
; ∨-comm        = ⊓-comm
; ∨-assoc       = ⊓-assoc
; ∨-cong        = cong₂ _⊓_
; ∧-comm        = ⊔-comm
; ∧-assoc       = ⊔-assoc
; ∧-cong        = cong₂ _⊔_
; absorptive    = absorptive-⊓-⊔
}
; ∨-∧-distribʳ = proj₂ distrib-⊓-⊔
}

distributiveLattice : DistributiveLattice _ _
distributiveLattice = record
{ _∨_                   = _⊓_
; _∧_                   = _⊔_
; isDistributiveLattice = isDistributiveLattice
}

------------------------------------------------------------------------
-- Converting between ≤ and ≤′

≤-step : ∀ {m n} → m ≤ n → m ≤ 1 + n
≤-step z≤n       = z≤n
≤-step (s≤s m≤n) = s≤s (≤-step m≤n)

≤′⇒≤ : _≤′_ ⇒ _≤_
≤′⇒≤ ≤′-refl        = ≤-refl
≤′⇒≤ (≤′-step m≤′n) = ≤-step (≤′⇒≤ m≤′n)

z≤′n : ∀ {n} → zero ≤′ n
z≤′n {zero}  = ≤′-refl
z≤′n {suc n} = ≤′-step z≤′n

s≤′s : ∀ {m n} → m ≤′ n → suc m ≤′ suc n
s≤′s ≤′-refl        = ≤′-refl
s≤′s (≤′-step m≤′n) = ≤′-step (s≤′s m≤′n)

≤⇒≤′ : _≤_ ⇒ _≤′_
≤⇒≤′ z≤n       = z≤′n
≤⇒≤′ (s≤s m≤n) = s≤′s (≤⇒≤′ m≤n)

------------------------------------------------------------------------
-- Various order-related properties

≤-steps : ∀ {m n} k → m ≤ n → m ≤ k + n
≤-steps zero    m≤n = m≤n
≤-steps (suc k) m≤n = ≤-step (≤-steps k m≤n)

m≤m+n : ∀ m n → m ≤ m + n
m≤m+n zero    n = z≤n
m≤m+n (suc m) n = s≤s (m≤m+n m n)

m≤′m+n : ∀ m n → m ≤′ m + n
m≤′m+n m n = ≤⇒≤′ (m≤m+n m n)

n≤′m+n : ∀ m n → n ≤′ m + n
n≤′m+n zero    n = ≤′-refl
n≤′m+n (suc m) n = ≤′-step (n≤′m+n m n)

n≤m+n : ∀ m n → n ≤ m + n
n≤m+n m n = ≤′⇒≤ (n≤′m+n m n)

n≤1+n : ∀ n → n ≤ 1 + n
n≤1+n _ = ≤-step ≤-refl

1+n≰n : ∀ {n} → ¬ 1 + n ≤ n
1+n≰n (s≤s le) = 1+n≰n le

≤pred⇒≤ : ∀ m n → m ≤ pred n → m ≤ n
≤pred⇒≤ m zero    le = le
≤pred⇒≤ m (suc n) le = ≤-step le

≤⇒pred≤ : ∀ m n → m ≤ n → pred m ≤ n
≤⇒pred≤ zero    n le = le
≤⇒pred≤ (suc m) n le = start
m     ≤⟨ n≤1+n m ⟩
suc m ≤⟨ le ⟩
n     □

¬i+1+j≤i : ∀ i {j} → ¬ i + suc j ≤ i
¬i+1+j≤i zero    ()
¬i+1+j≤i (suc i) le = ¬i+1+j≤i i (≤-pred le)

n∸m≤n : ∀ m n → n ∸ m ≤ n
n∸m≤n zero    n       = ≤-refl
n∸m≤n (suc m) zero    = ≤-refl
n∸m≤n (suc m) (suc n) = start
n ∸ m  ≤⟨ n∸m≤n m n ⟩
n      ≤⟨ n≤1+n n ⟩
suc n  □

n≤m+n∸m : ∀ m n → n ≤ m + (n ∸ m)
n≤m+n∸m m       zero    = z≤n
n≤m+n∸m zero    (suc n) = ≤-refl
n≤m+n∸m (suc m) (suc n) = s≤s (n≤m+n∸m m n)

m⊓n≤m : ∀ m n → m ⊓ n ≤ m
m⊓n≤m zero    _       = z≤n
m⊓n≤m (suc m) zero    = z≤n
m⊓n≤m (suc m) (suc n) = s≤s \$ m⊓n≤m m n

m≤m⊔n : ∀ m n → m ≤ m ⊔ n
m≤m⊔n zero    _       = z≤n
m≤m⊔n (suc m) zero    = ≤-refl
m≤m⊔n (suc m) (suc n) = s≤s \$ m≤m⊔n m n

⌈n/2⌉≤′n : ∀ n → ⌈ n /2⌉ ≤′ n
⌈n/2⌉≤′n zero          = ≤′-refl
⌈n/2⌉≤′n (suc zero)    = ≤′-refl
⌈n/2⌉≤′n (suc (suc n)) = s≤′s (≤′-step (⌈n/2⌉≤′n n))

⌊n/2⌋≤′n : ∀ n → ⌊ n /2⌋ ≤′ n
⌊n/2⌋≤′n zero    = ≤′-refl
⌊n/2⌋≤′n (suc n) = ≤′-step (⌈n/2⌉≤′n n)

<-trans : Transitive _<_
<-trans {i} {j} {k} i<j j<k = start
1 + i  ≤⟨ i<j ⟩
j      ≤⟨ n≤1+n j ⟩
1 + j  ≤⟨ j<k ⟩
k      □

------------------------------------------------------------------------
-- (ℕ, _≡_, _<_) is a strict total order

m≢1+m+n : ∀ m {n} → m ≢ suc (m + n)
m≢1+m+n zero    ()
m≢1+m+n (suc m) eq = m≢1+m+n m (cong pred eq)

strictTotalOrder : StrictTotalOrder _ _ _
strictTotalOrder = record
{ Carrier            = ℕ
; _≈_                = _≡_
; _<_                = _<_
; isStrictTotalOrder = record
{ isEquivalence = PropEq.isEquivalence
; trans         = <-trans
; compare       = cmp
; <-resp-≈      = PropEq.resp₂ _<_
}
}
where
2+m+n≰m : ∀ {m n} → ¬ 2 + (m + n) ≤ m
2+m+n≰m (s≤s le) = 2+m+n≰m le

cmp : Trichotomous _≡_ _<_
cmp m n with compare m n
cmp .m .(suc (m + k)) | less    m k = tri< (m≤m+n (suc m) k) (m≢1+m+n _) 2+m+n≰m
cmp .n             .n | equal   n   = tri≈ 1+n≰n refl 1+n≰n
cmp .(suc (n + k)) .n | greater n k = tri> 2+m+n≰m (m≢1+m+n _ ∘ sym) (m≤m+n (suc n) k)

------------------------------------------------------------------------
-- Miscellaneous other properties

0∸n≡0 : LeftZero zero _∸_
0∸n≡0 zero    = refl
0∸n≡0 (suc _) = refl

∸-+-assoc : ∀ m n o → (m ∸ n) ∸ o ≡ m ∸ (n + o)
∸-+-assoc m       n       zero    = cong (_∸_ m) (sym \$ proj₂ +-identity n)
∸-+-assoc zero    zero    (suc o) = refl
∸-+-assoc zero    (suc n) (suc o) = refl
∸-+-assoc (suc m) zero    (suc o) = refl
∸-+-assoc (suc m) (suc n) (suc o) = ∸-+-assoc m n (suc o)

m+n∸n≡m : ∀ m n → (m + n) ∸ n ≡ m
m+n∸n≡m m       zero    = proj₂ +-identity m
m+n∸n≡m zero    (suc n) = m+n∸n≡m zero n
m+n∸n≡m (suc m) (suc n) = begin
m + suc n ∸ n
≡⟨ cong (λ x → x ∸ n) (m+1+n≡1+m+n m n) ⟩
suc m + n ∸ n
≡⟨ m+n∸n≡m (suc m) n ⟩
suc m
∎

m⊓n+n∸m≡n : ∀ m n → (m ⊓ n) + (n ∸ m) ≡ n
m⊓n+n∸m≡n zero    n       = refl
m⊓n+n∸m≡n (suc m) zero    = refl
m⊓n+n∸m≡n (suc m) (suc n) = cong suc \$ m⊓n+n∸m≡n m n

[m∸n]⊓[n∸m]≡0 : ∀ m n → (m ∸ n) ⊓ (n ∸ m) ≡ 0
[m∸n]⊓[n∸m]≡0 zero zero       = refl
[m∸n]⊓[n∸m]≡0 zero (suc n)    = refl
[m∸n]⊓[n∸m]≡0 (suc m) zero    = refl
[m∸n]⊓[n∸m]≡0 (suc m) (suc n) = [m∸n]⊓[n∸m]≡0 m n

-- TODO: Can this proof be simplified? An automatic solver which can
-- handle ∸ would be nice...

i∸k∸j+j∸k≡i+j∸k : ∀ i j k → i ∸ (k ∸ j) + (j ∸ k) ≡ i + j ∸ k
i∸k∸j+j∸k≡i+j∸k zero j k = begin
0 ∸ (k ∸ j) + (j ∸ k)
≡⟨ cong (λ x → x + (j ∸ k)) (0∸n≡0 (k ∸ j)) ⟩
0 + (j ∸ k)
≡⟨ refl ⟩
j ∸ k
∎
i∸k∸j+j∸k≡i+j∸k (suc i) j zero = begin
suc i ∸ (0 ∸ j) + j
≡⟨ cong (λ x → suc i ∸ x + j) (0∸n≡0 j) ⟩
suc i ∸ 0 + j
≡⟨ refl ⟩
suc (i + j)
∎
i∸k∸j+j∸k≡i+j∸k (suc i) zero (suc k) = begin
i ∸ k + 0
≡⟨ proj₂ +-identity _ ⟩
i ∸ k
≡⟨ cong (λ x → x ∸ k) (sym (proj₂ +-identity _)) ⟩
i + 0 ∸ k
∎
i∸k∸j+j∸k≡i+j∸k (suc i) (suc j) (suc k) = begin
suc i ∸ (k ∸ j) + (j ∸ k)
≡⟨ i∸k∸j+j∸k≡i+j∸k (suc i) j k ⟩
suc i + j ∸ k
≡⟨ cong (λ x → x ∸ k)
(sym (m+1+n≡1+m+n i j)) ⟩
i + suc j ∸ k
∎

m+n∸m≡n : ∀ {m n} → m ≤ n → m + (n ∸ m) ≡ n
m+n∸m≡n z≤n       = refl
m+n∸m≡n (s≤s m≤n) = cong suc \$ m+n∸m≡n m≤n

i+j≡0⇒i≡0 : ∀ i {j} → i + j ≡ 0 → i ≡ 0
i+j≡0⇒i≡0 zero    eq = refl
i+j≡0⇒i≡0 (suc i) ()

i+j≡0⇒j≡0 : ∀ i {j} → i + j ≡ 0 → j ≡ 0
i+j≡0⇒j≡0 i {j} i+j≡0 = i+j≡0⇒i≡0 j \$ begin
j + i
≡⟨ +-comm j i ⟩
i + j
≡⟨ i+j≡0 ⟩
0
∎

i*j≡1⇒i≡1 : ∀ i j → i * j ≡ 1 → i ≡ 1
i*j≡1⇒i≡1 (suc zero)    j             _  = refl
i*j≡1⇒i≡1 zero          j             ()
i*j≡1⇒i≡1 (suc (suc i)) (suc (suc j)) ()
i*j≡1⇒i≡1 (suc (suc i)) (suc zero)    ()
i*j≡1⇒i≡1 (suc (suc i)) zero          eq with begin
0      ≡⟨ *-comm 0 i ⟩
i * 0  ≡⟨ eq ⟩
1      ∎
... | ()

i*j≡1⇒j≡1 : ∀ i j → i * j ≡ 1 → j ≡ 1
i*j≡1⇒j≡1 i j eq = i*j≡1⇒i≡1 j i (begin
j * i  ≡⟨ *-comm j i ⟩
i * j  ≡⟨ eq ⟩
1      ∎)

cancel-+-left : ∀ i {j k} → i + j ≡ i + k → j ≡ k
cancel-+-left zero    eq = eq
cancel-+-left (suc i) eq = cancel-+-left i (cong pred eq)

cancel-*-right : ∀ i j {k} → i * suc k ≡ j * suc k → i ≡ j
cancel-*-right zero    zero        eq = refl
cancel-*-right zero    (suc j)     ()
cancel-*-right (suc i) zero        ()
cancel-*-right (suc i) (suc j) {k} eq =
cong suc (cancel-*-right i j (cancel-+-left (suc k) eq))

im≡jm+n⇒[i∸j]m≡n
: ∀ i j m n →
i * m ≡ j * m + n → (i ∸ j) * m ≡ n
im≡jm+n⇒[i∸j]m≡n i       zero    m n eq = eq
im≡jm+n⇒[i∸j]m≡n zero    (suc j) m n eq =
sym \$ i+j≡0⇒j≡0 (m + j * m) \$ sym eq
im≡jm+n⇒[i∸j]m≡n (suc i) (suc j) m n eq =
im≡jm+n⇒[i∸j]m≡n i j m n (cancel-+-left m eq')
where
eq' = begin
m + i * m
≡⟨ eq ⟩
m + j * m + n
≡⟨ +-assoc m (j * m) n ⟩
m + (j * m + n)
∎

i+1+j≢i : ∀ i {j} → i + suc j ≢ i
i+1+j≢i i eq = ¬i+1+j≤i i (reflexive eq)
where open DecTotalOrder decTotalOrder

⌊n/2⌋-mono : ⌊_/2⌋ Preserves _≤_ ⟶ _≤_
⌊n/2⌋-mono z≤n             = z≤n
⌊n/2⌋-mono (s≤s z≤n)       = z≤n
⌊n/2⌋-mono (s≤s (s≤s m≤n)) = s≤s (⌊n/2⌋-mono m≤n)

⌈n/2⌉-mono : ⌈_/2⌉ Preserves _≤_ ⟶ _≤_
⌈n/2⌉-mono m≤n = ⌊n/2⌋-mono (s≤s m≤n)

pred-mono : pred Preserves _≤_ ⟶ _≤_
pred-mono z≤n      = z≤n
pred-mono (s≤s le) = le

_+-mono_ : _+_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_
_+-mono_ {zero} {m₂} {n₁} {n₂} z≤n n₁≤n₂ = start
n₁      ≤⟨ n₁≤n₂ ⟩
n₂      ≤⟨ n≤m+n m₂ n₂ ⟩
m₂ + n₂ □
s≤s m₁≤m₂ +-mono n₁≤n₂ = s≤s (m₁≤m₂ +-mono n₁≤n₂)

_*-mono_ : _*_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_
z≤n       *-mono n₁≤n₂ = z≤n
s≤s m₁≤m₂ *-mono n₁≤n₂ = n₁≤n₂ +-mono (m₁≤m₂ *-mono n₁≤n₂)
```