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 (_≡_ {A = ℕ})
open import Data.Product
open import Data.Sum
open import Data.Nat.Properties.Simple
isCommutativeSemiring : IsCommutativeSemiring _≡_ _+_ _*_ 0 1
isCommutativeSemiring = record
{ +-isCommutativeMonoid = record
{ isSemigroup = record
{ isEquivalence = PropEq.isEquivalence
; assoc = +-assoc
; ∙-cong = cong₂ _+_
}
; identityˡ = λ _ → refl
; comm = +-comm
}
; *-isCommutativeMonoid = record
{ isSemigroup = record
{ isEquivalence = PropEq.isEquivalence
; assoc = *-assoc
; ∙-cong = cong₂ _*_
}
; identityˡ = +-right-identity
; comm = *-comm
}
; distribʳ = distribʳ-*-+
; zeroˡ = λ _ → refl
}
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) _≟_
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
}
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
}
≤-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)
≤-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 □
≰⇒> : _≰_ ⇒ _>_
≰⇒> {zero} z≰n with z≰n z≤n
... | ()
≰⇒> {suc m} {zero} _ = s≤s z≤n
≰⇒> {suc m} {suc n} m≰n = s≤s (≰⇒> (m≰n ∘ s≤s))
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)
0∸n≡0 : LeftZero zero _∸_
0∸n≡0 zero = refl
0∸n≡0 (suc _) = refl
n∸n≡0 : ∀ n → n ∸ n ≡ 0
n∸n≡0 zero = refl
n∸n≡0 (suc n) = n∸n≡0 n
∸-+-assoc : ∀ m n o → (m ∸ n) ∸ o ≡ m ∸ (n + o)
∸-+-assoc m n zero = cong (_∸_ m) (sym $ +-right-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)
+-∸-assoc : ∀ m {n o} → o ≤ n → (m + n) ∸ o ≡ m + (n ∸ o)
+-∸-assoc m (z≤n {n = n}) = begin m + n ∎
+-∸-assoc m (s≤s {m = o} {n = n} o≤n) = begin
(m + suc n) ∸ suc o ≡⟨ cong (λ n → n ∸ suc o) (+-suc m n) ⟩
suc (m + n) ∸ suc o ≡⟨ refl ⟩
(m + n) ∸ o ≡⟨ +-∸-assoc m o≤n ⟩
m + (n ∸ o) ∎
m+n∸n≡m : ∀ m n → (m + n) ∸ n ≡ m
m+n∸n≡m m n = begin
(m + n) ∸ n ≡⟨ +-∸-assoc m (≤-refl {x = n}) ⟩
m + (n ∸ n) ≡⟨ cong (_+_ m) (n∸n≡0 n) ⟩
m + 0 ≡⟨ +-right-identity m ⟩
m ∎
m+n∸m≡n : ∀ {m n} → m ≤ n → m + (n ∸ m) ≡ n
m+n∸m≡n {m} {n} m≤n = begin
m + (n ∸ m) ≡⟨ sym $ +-∸-assoc m m≤n ⟩
(m + n) ∸ m ≡⟨ cong (λ n → n ∸ m) (+-comm m n) ⟩
(n + m) ∸ m ≡⟨ m+n∸n≡m n m ⟩
n ∎
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
[i+j]∸[i+k]≡j∸k : ∀ i j k → (i + j) ∸ (i + k) ≡ j ∸ k
[i+j]∸[i+k]≡j∸k zero j k = refl
[i+j]∸[i+k]≡j∸k (suc i) j k = [i+j]∸[i+k]≡j∸k i j k
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
≡⟨ +-right-identity _ ⟩
i ∸ k
≡⟨ cong (λ x → x ∸ k) (sym (+-right-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 (+-suc i j)) ⟩
i + suc j ∸ k
∎
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≡0⇒i≡0∨j≡0 : ∀ i {j} → i * j ≡ 0 → i ≡ 0 ⊎ j ≡ 0
i*j≡0⇒i≡0∨j≡0 zero {j} eq = inj₁ refl
i*j≡0⇒i≡0∨j≡0 (suc i) {zero} eq = inj₂ refl
i*j≡0⇒i≡0∨j≡0 (suc i) {suc j} ()
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-+-left-≤ : ∀ i {j k} → i + j ≤ i + k → j ≤ k
cancel-+-left-≤ zero le = le
cancel-+-left-≤ (suc i) (s≤s le) = cancel-+-left-≤ i le
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))
cancel-*-right-≤ : ∀ i j k → i * suc k ≤ j * suc k → i ≤ j
cancel-*-right-≤ zero _ _ _ = z≤n
cancel-*-right-≤ (suc i) zero _ ()
cancel-*-right-≤ (suc i) (suc j) k le =
s≤s (cancel-*-right-≤ i j k (cancel-+-left-≤ (suc k) le))
*-distrib-∸ʳ : _*_ DistributesOverʳ _∸_
*-distrib-∸ʳ i zero k = begin
(0 ∸ k) * i ≡⟨ cong₂ _*_ (0∸n≡0 k) refl ⟩
0 ≡⟨ sym $ 0∸n≡0 (k * i) ⟩
0 ∸ k * i ∎
*-distrib-∸ʳ i (suc j) zero = begin i + j * i ∎
*-distrib-∸ʳ i (suc j) (suc k) = begin
(j ∸ k) * i ≡⟨ *-distrib-∸ʳ i j k ⟩
j * i ∸ k * i ≡⟨ sym $ [i+j]∸[i+k]≡j∸k i _ _ ⟩
i + j * i ∸ (i + k * i) ∎
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 j m n eq = begin
(i ∸ j) * m ≡⟨ *-distrib-∸ʳ m i j ⟩
(i * m) ∸ (j * m) ≡⟨ cong₂ _∸_ eq (refl {x = j * m}) ⟩
(j * m + n) ∸ (j * m) ≡⟨ cong₂ _∸_ (+-comm (j * m) n) (refl {x = j * m}) ⟩
(n + j * m) ∸ (j * m) ≡⟨ m+n∸n≡m n (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)
∸-mono : _∸_ Preserves₂ _≤_ ⟶ _≥_ ⟶ _≤_
∸-mono z≤n (s≤s n₁≥n₂) = z≤n
∸-mono (s≤s m₁≤m₂) (s≤s n₁≥n₂) = ∸-mono m₁≤m₂ n₁≥n₂
∸-mono {m₁} {m₂} m₁≤m₂ (z≤n {n = n₁}) = start
m₁ ∸ n₁ ≤⟨ n∸m≤n n₁ m₁ ⟩
m₁ ≤⟨ m₁≤m₂ ⟩
m₂ □