------------------------------------------------------------------------
-- The Agda standard library
--
-- Some properties about integers
------------------------------------------------------------------------

{-# OPTIONS --without-K --safe #-}

-- Disabled to prevent warnings from deprecated names
{-# OPTIONS --warn=noUserWarning #-}

module Data.Integer.Properties where

open import Algebra
import Algebra.Morphism as Morphism
import Algebra.Properties.AbelianGroup
open import Data.Integer.Base renaming (suc to sucℤ)
open import Data.Nat as 
  using (; suc; zero; _∸_; s≤s; z≤n)
  hiding (module )
import Data.Nat.Properties as ℕₚ
open import Data.Nat.Solver
open import Data.Product using (proj₁; proj₂; _,_)
open import Data.Sum as Sum using (inj₁; inj₂)
open import Data.Sign as Sign using () renaming (_*_ to _𝕊*_)
import Data.Sign.Properties as 𝕊ₚ
open import Function using (_∘_; _$_)
open import Level using (0ℓ)
open import Relation.Binary
open import Relation.Binary.PropositionalEquality
open import Relation.Nullary using (yes; no)
open import Relation.Nullary.Negation using (contradiction)
import Relation.Nullary.Decidable as Dec

open import Algebra.FunctionProperties {A = } _≡_
open import Algebra.FunctionProperties.Consequences.Propositional
open import Algebra.Structures {A = } _≡_
module ℤtoℕ = Morphism.Definitions   _≡_
module ℕtoℤ = Morphism.Definitions   _≡_
open +-*-Solver

------------------------------------------------------------------------
-- Equality
------------------------------------------------------------------------

+-injective :  {m n}  + m  + n  m  n
+-injective refl = refl

-[1+-injective :  {m n}  -[1+ m ]  -[1+ n ]  m  n
-[1+-injective refl = refl

+[1+-injective :  {m n}  +[1+ m ]  +[1+ n ]  m  n
+[1+-injective refl = refl

infix 4 _≟_
_≟_ : Decidable {A = } _≡_
+ m       + n      = Dec.map′ (cong (+_)) +-injective (m ℕ.≟ n)
+ m       -[1+ n ] = no λ()
-[1+ m ]  + n      = no λ()
-[1+ m ]  -[1+ n ] = Dec.map′ (cong -[1+_]) -[1+-injective (m ℕ.≟ n)

≡-setoid : Setoid 0ℓ 0ℓ
≡-setoid = setoid 

≡-decSetoid : DecSetoid 0ℓ 0ℓ
≡-decSetoid = decSetoid _≟_

------------------------------------------------------------------------
-- Properties of _≤_
------------------------------------------------------------------------

drop‿+≤+ :  {m n}  + m  + n  m ℕ.≤ n
drop‿+≤+ (+≤+ m≤n) = m≤n

drop‿-≤- :  {m n}  -[1+ m ]  -[1+ n ]  n ℕ.≤ m
drop‿-≤- (-≤- n≤m) = n≤m

------------------------------------------------------------------------
-- Relational properties

≤-reflexive : _≡_  _≤_
≤-reflexive { -[1+ n ]} refl = -≤- ℕₚ.≤-refl
≤-reflexive {+ n}       refl = +≤+ ℕₚ.≤-refl

≤-refl : Reflexive _≤_
≤-refl = ≤-reflexive refl

≤-trans : Transitive _≤_
≤-trans -≤+       (+≤+ n≤m) = -≤+
≤-trans (-≤- n≤m) -≤+       = -≤+
≤-trans (-≤- n≤m) (-≤- k≤n) = -≤- (ℕₚ.≤-trans k≤n n≤m)
≤-trans (+≤+ m≤n) (+≤+ n≤k) = +≤+ (ℕₚ.≤-trans m≤n n≤k)

≤-antisym : Antisymmetric _≡_ _≤_
≤-antisym (-≤- n≤m) (-≤- m≤n) = cong -[1+_] $ ℕₚ.≤-antisym m≤n n≤m
≤-antisym (+≤+ m≤n) (+≤+ n≤m) = cong (+_)   $ ℕₚ.≤-antisym m≤n n≤m

≤-total : Total _≤_
≤-total (-[1+ m ]) (-[1+ n ]) = Sum.map -≤- -≤- (ℕₚ.≤-total n m)
≤-total (-[1+ m ]) (+    n  ) = inj₁ -≤+
≤-total (+    m  ) (-[1+ n ]) = inj₂ -≤+
≤-total (+    m  ) (+    n  ) = Sum.map +≤+ +≤+ (ℕₚ.≤-total m n)

infix  4 _≤?_
_≤?_ : Decidable _≤_
-[1+ m ] ≤? -[1+ n ] = Dec.map′ -≤- drop‿-≤- (n ℕ.≤? m)
-[1+ m ] ≤? +    n   = yes -≤+
+    m   ≤? -[1+ n ] = no λ ()
+    m   ≤? +    n   = Dec.map′ +≤+ drop‿+≤+ (m ℕ.≤? n)

≤-irrelevant : Irrelevant _≤_
≤-irrelevant -≤+       -≤+         = refl
≤-irrelevant (-≤- n≤m₁) (-≤- n≤m₂) = cong -≤- (ℕₚ.≤-irrelevant n≤m₁ n≤m₂)
≤-irrelevant (+≤+ n≤m₁) (+≤+ n≤m₂) = cong +≤+ (ℕₚ.≤-irrelevant n≤m₁ n≤m₂)

------------------------------------------------------------------------
-- Structures

≤-isPreorder : IsPreorder _≡_ _≤_
≤-isPreorder = record
  { isEquivalence = isEquivalence
  ; reflexive     = ≤-reflexive
  ; trans         = ≤-trans
  }

≤-isPartialOrder : IsPartialOrder _≡_ _≤_
≤-isPartialOrder = record
  { isPreorder = ≤-isPreorder
  ; antisym    = ≤-antisym
  }

≤-isTotalOrder : IsTotalOrder _≡_ _≤_
≤-isTotalOrder = record
  { isPartialOrder = ≤-isPartialOrder
  ; total          = ≤-total
  }

≤-isDecTotalOrder : IsDecTotalOrder _≡_ _≤_
≤-isDecTotalOrder = record
  { isTotalOrder = ≤-isTotalOrder
  ; _≟_          = _≟_
  ; _≤?_         = _≤?_
  }

------------------------------------------------------------------------
-- Packages

≤-preorder : Preorder 0ℓ 0ℓ 0ℓ
≤-preorder = record
  { isPreorder = ≤-isPreorder
  }

≤-poset : Poset 0ℓ 0ℓ 0ℓ
≤-poset = record
  { isPartialOrder = ≤-isPartialOrder
  }

≤-totalOrder : TotalOrder 0ℓ 0ℓ 0ℓ
≤-totalOrder = record
  { isTotalOrder = ≤-isTotalOrder
  }

≤-decTotalOrder : DecTotalOrder 0ℓ 0ℓ 0ℓ
≤-decTotalOrder = record
  { isDecTotalOrder = ≤-isDecTotalOrder
  }

------------------------------------------------------------------------
-- Properties _<_
------------------------------------------------------------------------

drop‿+<+ :  {m n}  + m < + n  m ℕ.< n
drop‿+<+ (+<+ m<n) = m<n

drop‿-<- :  {m n}  -[1+ m ] < -[1+ n ]  n ℕ.< m
drop‿-<- (-<- n<m) = n<m

+≮0 :  {n}  + n  +0
+≮0 (+<+ ())

+≮- :  {m n}  + m  -[1+ n ]
+≮- ()

------------------------------------------------------------------------
-- Relationship between other operators

<⇒≤ :  {i j}  i < j  i  j
<⇒≤ (-<- i<j) = -≤- (ℕₚ.<⇒≤ i<j)
<⇒≤ -<+       = -≤+
<⇒≤ (+<+ i<j) = +≤+ (ℕₚ.<⇒≤ i<j)

>⇒≰ :  {i j}  i > j  i  j
>⇒≰ (-<- n<m) = ℕₚ.<⇒≱ n<m  drop‿-≤-
>⇒≰ (+<+ m<n) = ℕₚ.<⇒≱ m<n  drop‿+≤+

≰⇒> :  {i j}  i  j  i > j
≰⇒> {+ n}       {+_ n₁}      i≰j = +<+ (ℕₚ.≰⇒> (i≰j  +≤+))
≰⇒> {+ n}       { -[1+_] n₁} i≰j = -<+
≰⇒> { -[1+_] n} {+_ n₁}      i≰j = contradiction -≤+ i≰j
≰⇒> { -[1+_] n} { -[1+_] n₁} i≰j = -<- (ℕₚ.≰⇒> (i≰j  -≤-))

------------------------------------------------------------------------
-- Relational properties

<-irrefl : Irreflexive _≡_ _<_
<-irrefl { -[1+ n ]} refl = ℕₚ.<-irrefl refl  drop‿-<-
<-irrefl { +0}       refl (+<+ ())
<-irrefl { +[1+ n ]} refl = ℕₚ.<-irrefl refl  drop‿+<+

<-asym : Asymmetric _<_
<-asym (-<- n<m) = ℕₚ.<-asym n<m  drop‿-<-
<-asym (+<+ m<n) = ℕₚ.<-asym m<n  drop‿+<+

≤-<-trans : Trans _≤_ _<_ _<_
≤-<-trans (-≤- n≤m) (-<- o<n) = -<- (ℕₚ.<-transˡ o<n n≤m)
≤-<-trans (-≤- n≤m) -<+       = -<+
≤-<-trans -≤+       (+<+ m<o) = -<+
≤-<-trans (+≤+ m≤n) (+<+ n<o) = +<+ (ℕₚ.<-transʳ m≤n n<o)

<-≤-trans : Trans _<_ _≤_ _<_
<-≤-trans (-<- n<m) (-≤- o≤n) = -<- (ℕₚ.<-transʳ o≤n n<m)
<-≤-trans (-<- n<m) -≤+       = -<+
<-≤-trans -<+       (+≤+ m≤n) = -<+
<-≤-trans (+<+ m<n) (+≤+ n≤o) = +<+ (ℕₚ.<-transˡ m<n n≤o)

<-trans : Transitive _<_
<-trans m<n n<p = ≤-<-trans (<⇒≤ m<n) n<p

<-cmp : Trichotomous _≡_ _<_
<-cmp +0       +0       = tri≈ +≮0 refl +≮0
<-cmp +0       +[1+ n ] = tri< (+<+ (s≤s z≤n)) (λ()) +≮0
<-cmp +[1+ n ] +0       = tri> +≮0 (λ()) (+<+ (s≤s z≤n))
<-cmp (+ m)    -[1+ n ] = tri> +≮- (λ()) -<+
<-cmp -[1+ m ] (+ n)    = tri< -<+ (λ()) +≮-
<-cmp -[1+ m ] -[1+ n ] with ℕₚ.<-cmp m n
... | tri< m<n m≢n n≯m = tri> (n≯m  drop‿-<-) (m≢n  -[1+-injective) (-<- m<n)
... | tri≈ m≮n m≡n n≯m = tri≈ (n≯m  drop‿-<-) (cong -[1+_] m≡n) (m≮n  drop‿-<-)
... | tri> m≮n m≢n n>m = tri< (-<- n>m) (m≢n  -[1+-injective) (m≮n  drop‿-<-)
<-cmp +[1+ m ] +[1+ n ] with ℕₚ.<-cmp m n
... | tri< m<n m≢n n≯m = tri< (+<+ (s≤s m<n))              (m≢n  +[1+-injective) (n≯m  ℕₚ.≤-pred  drop‿+<+)
... | tri≈ m≮n m≡n n≯m = tri≈ (m≮n  ℕₚ.≤-pred  drop‿+<+) (cong (+_  suc) m≡n)  (n≯m  ℕₚ.≤-pred  drop‿+<+)
... | tri> m≮n m≢n n>m = tri> (m≮n  ℕₚ.≤-pred  drop‿+<+) (m≢n  +[1+-injective) (+<+ (s≤s n>m))

infix 4 _<?_
_<?_ : Decidable _<_
-[1+ m ] <? -[1+ n ] = Dec.map′ -<- drop‿-<- (n ℕ.<? m)
-[1+ m ] <? + n      = yes -<+
+ m      <? -[1+ n ] = no λ()
+ m      <? + n      = Dec.map′ +<+ drop‿+<+ (m ℕ.<? n)

<-irrelevant : Irrelevant _<_
<-irrelevant (-<- n<m₁) (-<- n<m₂) = cong -<- (ℕₚ.<-irrelevant n<m₁ n<m₂)
<-irrelevant -<+       -<+         = refl
<-irrelevant (+<+ m<n₁) (+<+ m<n₂) = cong +<+ (ℕₚ.<-irrelevant m<n₁ m<n₂)

------------------------------------------------------------------------
-- Structures

<-isStrictPartialOrder : IsStrictPartialOrder _≡_ _<_
<-isStrictPartialOrder = record
  { isEquivalence = isEquivalence
  ; irrefl        = <-irrefl
  ; trans         = <-trans
  ; <-resp-≈      = subst (_ <_) , subst (_< _)
  }

<-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_
<-isStrictTotalOrder = record
  { isEquivalence = isEquivalence
  ; trans         = <-trans
  ; compare       = <-cmp
  }

------------------------------------------------------------------------
-- Packages

<-strictPartialOrder : StrictPartialOrder 0ℓ 0ℓ 0ℓ
<-strictPartialOrder = record
  { isStrictPartialOrder = <-isStrictPartialOrder
  }

<-strictTotalOrder : StrictTotalOrder 0ℓ 0ℓ 0ℓ
<-strictTotalOrder = record
  { isStrictTotalOrder = <-isStrictTotalOrder
  }

------------------------------------------------------------------------
-- Other properties of _<_

n≮n :  {n}  n  n
n≮n {n} = <-irrefl refl

>-irrefl : Irreflexive _≡_ _>_
>-irrefl = <-irrefl  sym

------------------------------------------------------------------------
-- A specialised module for reasoning about the _≤_ and _<_ relations
------------------------------------------------------------------------

module ≤-Reasoning where
  open import Relation.Binary.Reasoning.Base.Triple
    ≤-isPreorder
    <-trans
    (resp₂ _<_)
    <⇒≤
    <-≤-trans
    ≤-<-trans
    public
    hiding (_≈⟨_⟩_; _≈˘⟨_⟩_)

------------------------------------------------------------------------
-- Properties of -_
------------------------------------------------------------------------

neg-involutive :  n  - - n  n
neg-involutive -[1+ n ] = refl
neg-involutive +0       = refl
neg-involutive +[1+ n ] = refl

neg-injective :  {m n}  - m  - n  m  n
neg-injective {m} {n} -m≡-n = begin
  m     ≡⟨ sym (neg-involutive m) 
  - - m ≡⟨ cong -_ -m≡-n 
  - - n ≡⟨ neg-involutive n 
  n      where open ≡-Reasoning

neg-≤-pos :  {m n}  - (+ m)  + n
neg-≤-pos {zero}  = +≤+ z≤n
neg-≤-pos {suc m} = -≤+

neg-mono-<-> : -_ Preserves _<_  _>_
neg-mono-<-> { -[1+ _ ]} { -[1+ _ ]} (-<- n<m) = +<+ (s≤s n<m)
neg-mono-<-> { -[1+ _ ]} { +0}       -<+       = +<+ (s≤s z≤n)
neg-mono-<-> { -[1+ _ ]} { +[1+ n ]} -<+       = -<+
neg-mono-<-> { +0}       { +[1+ n ]} (+<+ _)   = -<+
neg-mono-<-> { +[1+ m ]} { +[1+ n ]} (+<+ m<n) = -<- (ℕₚ.≤-pred m<n)

------------------------------------------------------------------------
-- Properties of ∣_∣
------------------------------------------------------------------------

∣n∣≡0⇒n≡0 :  {n}   n   0  n  + 0
∣n∣≡0⇒n≡0 {+0} refl = refl

∣-n∣≡∣n∣ :  n   - n    n 
∣-n∣≡∣n∣ -[1+ n ] = refl
∣-n∣≡∣n∣ +0       = refl
∣-n∣≡∣n∣ +[1+ n ] = refl

0≤n⇒+∣n∣≡n :  {n}  + 0  n  +  n   n
0≤n⇒+∣n∣≡n (+≤+ 0≤n) = refl

+∣n∣≡n⇒0≤n :  {n}  +  n   n  + 0  n
+∣n∣≡n⇒0≤n {+ n} _ = +≤+ z≤n

------------------------------------------------------------------------
-- Properties of sign and _◃_

◃-inverse :  i  sign i   i   i
◃-inverse -[1+ n ] = refl
◃-inverse +0       = refl
◃-inverse +[1+ n ] = refl

◃-cong :  {i j}  sign i  sign j   i    j   i  j
◃-cong {i} {j} sign-≡ abs-≡ = begin
  i               ≡⟨ sym $ ◃-inverse i 
  sign i   i   ≡⟨ cong₂ _◃_ sign-≡ abs-≡ 
  sign j   j   ≡⟨ ◃-inverse j 
  j                where open ≡-Reasoning

+◃n≡+n :  n  Sign.+  n  + n
+◃n≡+n zero    = refl
+◃n≡+n (suc _) = refl

-◃n≡-n :  n  Sign.-  n  - + n
-◃n≡-n zero    = refl
-◃n≡-n (suc _) = refl

sign-◃ :  s n  sign (s  suc n)  s
sign-◃ Sign.- _ = refl
sign-◃ Sign.+ _ = refl

abs-◃ :  s n   s  n   n
abs-◃ _      zero    = refl
abs-◃ Sign.- (suc n) = refl
abs-◃ Sign.+ (suc n) = refl

signₙ◃∣n∣≡n :  n  sign n   n   n
signₙ◃∣n∣≡n (+ n)    = +◃n≡+n n
signₙ◃∣n∣≡n -[1+ n ] = refl

sign-cong :  {s₁ s₂ n₁ n₂} 
            s₁  suc n₁  s₂  suc n₂  s₁  s₂
sign-cong {s₁} {s₂} {n₁} {n₂} eq = begin
  s₁                  ≡⟨ sym $ sign-◃ s₁ n₁ 
  sign (s₁  suc n₁)  ≡⟨ cong sign eq 
  sign (s₂  suc n₂)  ≡⟨ sign-◃ s₂ n₂ 
  s₂                   where open ≡-Reasoning

abs-cong :  {s₁ s₂ n₁ n₂}  s₁  n₁  s₂  n₂  n₁  n₂
abs-cong {s₁} {s₂} {n₁} {n₂} eq = begin
  n₁           ≡⟨ sym $ abs-◃ s₁ n₁ 
   s₁  n₁   ≡⟨ cong ∣_∣ eq 
   s₂  n₂   ≡⟨ abs-◃ s₂ n₂ 
  n₂            where open ≡-Reasoning

∣s◃m∣*∣t◃n∣≡m*n :  s t m n   s  m  ℕ.*  t  n   m ℕ.* n
∣s◃m∣*∣t◃n∣≡m*n s t m n = cong₂ ℕ._*_ (abs-◃ s m) (abs-◃ t n)

◃-≡ :  {m n}  sign m  sign n   m    n   m  n
◃-≡ {+ m}       {+ n }      ≡-sign refl = refl
◃-≡ { -[1+ m ]} { -[1+ n ]} ≡-sign refl = refl

+◃-mono-< :  {m n}  m ℕ.< n  Sign.+  m < Sign.+  n
+◃-mono-< {zero}  {suc n} m<n = +<+ m<n
+◃-mono-< {suc m} {suc n} m<n = +<+ m<n

+◃-cancel-< :  {m n}  Sign.+  m < Sign.+  n  m ℕ.< n
+◃-cancel-< {zero}  {zero}  (+<+ ())
+◃-cancel-< {suc m} {zero}  (+<+ ())
+◃-cancel-< {zero}  {suc n} (+<+ m<n) = m<n
+◃-cancel-< {suc m} {suc n} (+<+ m<n) = m<n

neg◃-cancel-< :  {m n}  Sign.-  m < Sign.-  n  n ℕ.< m
neg◃-cancel-< {zero}  {suc n} ()
neg◃-cancel-< {zero}  {zero}  (+<+ ())
neg◃-cancel-< {suc m} {zero}  -<+       = s≤s z≤n
neg◃-cancel-< {suc m} {suc n} (-<- n<m) = s≤s n<m

-◃<+◃ :  m n  Sign.-  (suc m) < Sign.+  n
-◃<+◃ m zero    = -<+
-◃<+◃ m (suc n) = -<+

+◃≮-◃ :  {m n}  Sign.+  m  Sign.-  n
+◃≮-◃ {zero}  {zero} (+<+ ())
+◃≮-◃ {suc m} {zero} (+<+ ())

------------------------------------------------------------------------
-- Properties of _⊖_
------------------------------------------------------------------------

n⊖n≡0 :  n  n  n  + 0
n⊖n≡0 zero    = refl
n⊖n≡0 (suc n) = n⊖n≡0 n

⊖-swap :  a b  a  b  - (b  a)
⊖-swap zero    zero    = refl
⊖-swap (suc _) zero    = refl
⊖-swap zero    (suc _) = refl
⊖-swap (suc a) (suc b) = ⊖-swap a b

⊖-≥ :  {m n}  m ℕ.≥ n  m  n  + (m  n)
⊖-≥ z≤n         = refl
⊖-≥ (ℕ.s≤s n≤m) = ⊖-≥ n≤m

⊖-< :  {m n}  m ℕ.< n  m  n  - + (n  m)
⊖-< {zero}  (ℕ.s≤s z≤n) = refl
⊖-< {suc m} (ℕ.s≤s m<n) = ⊖-< m<n

⊖-≰ :  {m n}  n ℕ.≰ m  m  n  - + (n  m)
⊖-≰ = ⊖-<  ℕₚ.≰⇒>

∣⊖∣-< :  {m n}  m ℕ.< n   m  n   n  m
∣⊖∣-< {zero}  (ℕ.s≤s z≤n) = refl
∣⊖∣-< {suc n} (ℕ.s≤s m<n) = ∣⊖∣-< m<n

∣⊖∣-≰ :  {m n}  n ℕ.≰ m   m  n   n  m
∣⊖∣-≰ = ∣⊖∣-<  ℕₚ.≰⇒>

-[n⊖m]≡-m+n :  m n  - (m  n)  (- (+ m)) + (+ n)
-[n⊖m]≡-m+n zero    zero    = refl
-[n⊖m]≡-m+n zero    (suc n) = refl
-[n⊖m]≡-m+n (suc m) zero    = refl
-[n⊖m]≡-m+n (suc m) (suc n) = sym (⊖-swap n m)

∣m⊖n∣≡∣n⊖m∣ :  x y   x  y    y  x 
∣m⊖n∣≡∣n⊖m∣ zero    zero    = refl
∣m⊖n∣≡∣n⊖m∣ zero    (suc _) = refl
∣m⊖n∣≡∣n⊖m∣ (suc _) zero    = refl
∣m⊖n∣≡∣n⊖m∣ (suc x) (suc y) = ∣m⊖n∣≡∣n⊖m∣ x y

+-cancelˡ-⊖ :  a b c  (a ℕ.+ b)  (a ℕ.+ c)  b  c
+-cancelˡ-⊖ zero    b c = refl
+-cancelˡ-⊖ (suc a) b c = +-cancelˡ-⊖ a b c

m⊖n≤m :  m n  m  n  + m
m⊖n≤m m       zero    = ≤-refl
m⊖n≤m zero    (suc n) = -≤+
m⊖n≤m (suc m) (suc n) = ≤-trans (m⊖n≤m m n) (+≤+ (ℕₚ.n≤1+n m))

m⊖n<1+m :  m n  m  n < +[1+ m ]
m⊖n<1+m m       zero    = +<+ ℕₚ.≤-refl
m⊖n<1+m zero    (suc n) = -<+
m⊖n<1+m (suc m) (suc n) = <-trans (m⊖n<1+m m n) (+<+ ℕₚ.≤-refl)

m⊖1+n<m :  m n  m  suc n < + m
m⊖1+n<m zero    n = -<+
m⊖1+n<m (suc m) n = m⊖n<1+m m n

-1+m<n⊖m :  m n  -[1+ m ] < n  m
-1+m<n⊖m zero    n       = -<+
-1+m<n⊖m (suc m) zero    = -<- ℕₚ.≤-refl
-1+m<n⊖m (suc m) (suc n) = <-trans (-<- ℕₚ.≤-refl) (-1+m<n⊖m m n)

-[1+m]≤n⊖m+1 :  m n  -[1+ m ]  n  suc m
-[1+m]≤n⊖m+1 m zero    = ≤-refl
-[1+m]≤n⊖m+1 m (suc n) = <⇒≤ (-1+m<n⊖m m n)

-1+m≤n⊖m :  m n  -[1+ m ]  n  m
-1+m≤n⊖m m n = <⇒≤ (-1+m<n⊖m m n)

0⊖m≤+ :  m {n}  0  m  + n
0⊖m≤+ zero    = +≤+ z≤n
0⊖m≤+ (suc m) = -≤+

sign-⊖-< :  {m n}  m ℕ.< n  sign (m  n)  Sign.-
sign-⊖-< {zero}  (ℕ.s≤s z≤n) = refl
sign-⊖-< {suc n} (ℕ.s≤s m<n) = sign-⊖-< m<n

sign-⊖-≰ :  {m n}  n ℕ.≰ m  sign (m  n)  Sign.-
sign-⊖-≰ = sign-⊖-<  ℕₚ.≰⇒>

⊖-monoʳ-≥-≤ :  p  (p ⊖_) Preserves ℕ._≥_  _≤_
⊖-monoʳ-≥-≤ zero    (z≤n {n})     = 0⊖m≤+ n
⊖-monoʳ-≥-≤ zero    (s≤s m≤n)     = -≤- m≤n
⊖-monoʳ-≥-≤ (suc p) (z≤n {zero})  = ≤-refl
⊖-monoʳ-≥-≤ (suc p) (z≤n {suc n}) = ≤-trans (⊖-monoʳ-≥-≤ p (z≤n {n})) (+≤+ (ℕₚ.n≤1+n p))
⊖-monoʳ-≥-≤ (suc p) (s≤s m≤n)     = ⊖-monoʳ-≥-≤ p m≤n

⊖-monoˡ-≤ :  p  (_⊖ p) Preserves ℕ._≤_  _≤_
⊖-monoˡ-≤ zero    m≤n             = +≤+ m≤n
⊖-monoˡ-≤ (suc p) (z≤n {0})       = ≤-refl
⊖-monoˡ-≤ (suc p) (z≤n {(suc m)}) = ≤-trans (⊖-monoʳ-≥-≤ 0 (ℕₚ.n≤1+n p)) (⊖-monoˡ-≤ p z≤n)
⊖-monoˡ-≤ (suc p) (s≤s m≤n)       = ⊖-monoˡ-≤ p m≤n

⊖-monoʳ->-< :  p  (p ⊖_) Preserves ℕ._>_  _<_
⊖-monoʳ->-< zero    {_}     (s≤s z≤n)       = -<+
⊖-monoʳ->-< zero    {_}     (s≤s (s≤s m≤n)) = -<- (s≤s m≤n)
⊖-monoʳ->-< (suc p) {suc m} (s≤s z≤n)       = m⊖n<1+m p m
⊖-monoʳ->-< (suc p) {_}     (s≤s (s≤s m≤n)) = ⊖-monoʳ->-< p (s≤s m≤n)

⊖-monoˡ-< :  p  (_⊖ p) Preserves ℕ._<_  _<_
⊖-monoˡ-< zero    m<n             = +<+ m<n
⊖-monoˡ-< (suc p) (s≤s z≤n)       = -1+m<n⊖m p _
⊖-monoˡ-< (suc p) (s≤s (s≤s m<n)) = ⊖-monoˡ-< p (s≤s m<n)

------------------------------------------------------------------------
-- Properties of _+_
------------------------------------------------------------------------

------------------------------------------------------------------------
-- Algebraic properties of _+_

+-comm : Commutative _+_
+-comm -[1+ a ] -[1+ b ] = cong (-[1+_]  suc) (ℕₚ.+-comm a b)
+-comm (+   a ) (+   b ) = cong +_ (ℕₚ.+-comm a b)
+-comm -[1+ _ ] (+   _ ) = refl
+-comm (+   _ ) -[1+ _ ] = refl

+-identityˡ : LeftIdentity +0 _+_
+-identityˡ -[1+ _ ] = refl
+-identityˡ (+   _ ) = refl

+-identityʳ : RightIdentity +0 _+_
+-identityʳ = comm+idˡ⇒idʳ +-comm +-identityˡ

+-identity : Identity +0 _+_
+-identity = +-identityˡ , +-identityʳ

distribˡ-⊖-+-pos :  a b c  b  c + + a  b ℕ.+ a  c
distribˡ-⊖-+-pos _ zero    zero    = refl
distribˡ-⊖-+-pos _ zero    (suc _) = refl
distribˡ-⊖-+-pos _ (suc _) zero    = refl
distribˡ-⊖-+-pos a (suc b) (suc c) = distribˡ-⊖-+-pos a b c

distribˡ-⊖-+-neg :  a b c  b  c + -[1+ a ]  b  (suc c ℕ.+ a)
distribˡ-⊖-+-neg _ zero    zero    = refl
distribˡ-⊖-+-neg _ zero    (suc _) = refl
distribˡ-⊖-+-neg _ (suc _) zero    = refl
distribˡ-⊖-+-neg a (suc b) (suc c) = distribˡ-⊖-+-neg a b c

distribʳ-⊖-+-pos :  a b c  + a + (b  c)  a ℕ.+ b  c
distribʳ-⊖-+-pos a b c = begin
  + a + (b  c) ≡⟨ +-comm (+ a) (b  c) 
  (b  c) + + a ≡⟨ distribˡ-⊖-+-pos a b c 
  b ℕ.+ a  c   ≡⟨ cong (_⊖ c) (ℕₚ.+-comm b a) 
  a ℕ.+ b  c    where open ≡-Reasoning

distribʳ-⊖-+-neg :  a b c  -[1+ a ] + (b  c)  b  (suc a ℕ.+ c)
distribʳ-⊖-+-neg a b c = begin
  -[1+ a ] + (b  c) ≡⟨ +-comm -[1+ a ] (b  c) 
  (b  c) + -[1+ a ] ≡⟨ distribˡ-⊖-+-neg a b c 
  b  suc (c ℕ.+ a)  ≡⟨ cong  x  b  suc x) (ℕₚ.+-comm c a) 
  b  suc (a ℕ.+ c)   where open ≡-Reasoning

+-assoc : Associative _+_
+-assoc +0 y z rewrite +-identityˡ      y  | +-identityˡ (y + z) = refl
+-assoc x +0 z rewrite +-identityʳ  x      | +-identityˡ      z  = refl
+-assoc x y +0 rewrite +-identityʳ (x + y) | +-identityʳ  y      = refl
+-assoc -[1+ a ] -[1+ b ] +[1+ c ] = sym (distribʳ-⊖-+-neg a c b)
+-assoc -[1+ a ] +[1+ b ] +[1+ c ] = distribˡ-⊖-+-pos (suc c) b a
+-assoc +[1+ a ] -[1+ b ] -[1+ c ] = distribˡ-⊖-+-neg c a b
+-assoc +[1+ a ] -[1+ b ] +[1+ c ]
  rewrite distribˡ-⊖-+-pos (suc c) a b
        | distribʳ-⊖-+-pos (suc a) c b
        | sym (ℕₚ.+-assoc a 1 c)
        | ℕₚ.+-comm a 1
        = refl
+-assoc +[1+ a ] +[1+ b ] -[1+ c ]
  rewrite distribʳ-⊖-+-pos (suc a) b c
        | sym (ℕₚ.+-assoc a 1 b)
        | ℕₚ.+-comm a 1
        = refl
+-assoc -[1+ a ] -[1+ b ] -[1+ c ]
  rewrite sym (ℕₚ.+-assoc a 1 (b ℕ.+ c))
        | ℕₚ.+-comm a 1
        | ℕₚ.+-assoc a b c
        = refl
+-assoc -[1+ a ] +[1+ b ] -[1+ c ]
  rewrite distribʳ-⊖-+-neg a b c
        | distribˡ-⊖-+-neg c b a
        = refl
+-assoc +[1+ a ] +[1+ b ] +[1+ c ]
  rewrite ℕₚ.+-assoc (suc a) (suc b) (suc c)
        = refl

+-inverseˡ : LeftInverse +0 -_ _+_
+-inverseˡ -[1+ n ] = n⊖n≡0 n
+-inverseˡ +0       = refl
+-inverseˡ +[1+ n ] = n⊖n≡0 n

+-inverseʳ : RightInverse +0 -_ _+_
+-inverseʳ = comm+invˡ⇒invʳ +-comm +-inverseˡ

+-inverse : Inverse +0 -_ _+_
+-inverse = +-inverseˡ , +-inverseʳ

------------------------------------------------------------------------
-- Structures

+-isMagma : IsMagma _+_
+-isMagma = record
  { isEquivalence = isEquivalence
  ; ∙-cong        = cong₂ _+_
  }

+-isSemigroup : IsSemigroup _+_
+-isSemigroup = record
  { isMagma = +-isMagma
  ; assoc   = +-assoc
  }

+-0-isMonoid : IsMonoid _+_ +0
+-0-isMonoid = record
  { isSemigroup = +-isSemigroup
  ; identity    = +-identity
  }

+-0-isCommutativeMonoid : IsCommutativeMonoid _+_ +0
+-0-isCommutativeMonoid = record
  { isSemigroup = +-isSemigroup
  ; identityˡ   = +-identityˡ
  ; comm        = +-comm
  }

+-0-isGroup : IsGroup _+_ +0 (-_)
+-0-isGroup = record
  { isMonoid = +-0-isMonoid
  ; inverse  = +-inverse
  ; ⁻¹-cong  = cong (-_)
  }

+-isAbelianGroup : IsAbelianGroup _+_ +0 (-_)
+-isAbelianGroup = record
  { isGroup = +-0-isGroup
  ; comm    = +-comm
  }

------------------------------------------------------------------------
-- Packages

+-magma : Magma 0ℓ 0ℓ
+-magma = record
  { isMagma = +-isMagma
  }

+-semigroup : Semigroup 0ℓ 0ℓ
+-semigroup = record
  { isSemigroup = +-isSemigroup
  }

+-0-monoid : Monoid 0ℓ 0ℓ
+-0-monoid = record
  { isMonoid = +-0-isMonoid
  }

+-0-commutativeMonoid : CommutativeMonoid 0ℓ 0ℓ
+-0-commutativeMonoid = record
  { isCommutativeMonoid = +-0-isCommutativeMonoid
  }

+-0-abelianGroup : AbelianGroup 0ℓ 0ℓ
+-0-abelianGroup = record
  { isAbelianGroup = +-isAbelianGroup
  }

------------------------------------------------------------------------
-- Properties of _+_ and +_/-_.

pos-+-commute : ℕtoℤ.Homomorphic₂ +_ ℕ._+_ _+_
pos-+-commute zero    n = refl
pos-+-commute (suc m) n = cong sucℤ (pos-+-commute m n)

neg-distrib-+ :  m n  - (m + n)  (- m) + (- n)
neg-distrib-+ +0        +0        = refl
neg-distrib-+ +0        +[1+ n ]  = refl
neg-distrib-+ +[1+ m ]  +0        = cong -[1+_] (ℕₚ.+-identityʳ m)
neg-distrib-+ +[1+ m ]  +[1+ n ]  = cong -[1+_] (ℕₚ.+-suc m n)
neg-distrib-+ -[1+ m ]  -[1+ n ]  = cong  v  + suc v) (sym (ℕₚ.+-suc m n))
neg-distrib-+ (+   m)   -[1+ n ]  = -[n⊖m]≡-m+n m (suc n)
neg-distrib-+ -[1+ m ]  (+   n)   =
  trans (-[n⊖m]≡-m+n n (suc m)) (+-comm (- + n) (+ suc m))

◃-distrib-+ :  s m n  s  (m ℕ.+ n)  (s  m) + (s  n)
◃-distrib-+ Sign.- m n = begin
  Sign.-  (m ℕ.+ n)          ≡⟨ -◃n≡-n (m ℕ.+ n) 
  - (+ (m ℕ.+ n))             ≡⟨⟩
  - ((+ m) + (+ n))           ≡⟨ neg-distrib-+ (+ m) (+ n) 
  (- (+ m)) + (- (+ n))       ≡⟨ sym (cong₂ _+_ (-◃n≡-n m) (-◃n≡-n n)) 
  (Sign.-  m) + (Sign.-  n)  where open ≡-Reasoning
◃-distrib-+ Sign.+ m n = begin
  Sign.+  (m ℕ.+ n)          ≡⟨ +◃n≡+n (m ℕ.+ n) 
  + (m ℕ.+ n)                 ≡⟨⟩
  (+ m) + (+ n)               ≡⟨ sym (cong₂ _+_ (+◃n≡+n m) (+◃n≡+n n)) 
  (Sign.+  m) + (Sign.+  n)  where open ≡-Reasoning

------------------------------------------------------------------------
-- Properties of _+_ and _≤_

+-pos-monoʳ-≤ :  n  (_+_ (+ n)) Preserves _≤_  _≤_
+-pos-monoʳ-≤ n {_}         (-≤- o≤m) = ⊖-monoʳ-≥-≤ n (s≤s o≤m)
+-pos-monoʳ-≤ n { -[1+ m ]} -≤+       = ≤-trans (m⊖n≤m n (suc m)) (+≤+ (ℕₚ.m≤m+n n _))
+-pos-monoʳ-≤ n {_}         (+≤+ m≤o) = +≤+ (ℕₚ.+-monoʳ-≤ n m≤o)

+-neg-monoʳ-≤ :  n  (_+_ (-[1+ n ])) Preserves _≤_  _≤_
+-neg-monoʳ-≤ n {_} {_}   (-≤- n≤m) = -≤- (ℕₚ.+-monoʳ-≤ (suc n) n≤m)
+-neg-monoʳ-≤ n {_} {+ m} -≤+       = ≤-trans (-≤- (ℕₚ.m≤m+n (suc n) _)) (-1+m≤n⊖m (suc n) m)
+-neg-monoʳ-≤ n {_} {_}   (+≤+ m≤n) = ⊖-monoˡ-≤ (suc n) m≤n

+-monoʳ-≤ :  n  (_+_ n) Preserves _≤_  _≤_
+-monoʳ-≤ (+ n)    = +-pos-monoʳ-≤ n
+-monoʳ-≤ -[1+ n ] = +-neg-monoʳ-≤ n

+-monoˡ-≤ :  n  (_+ n) Preserves _≤_  _≤_
+-monoˡ-≤ n {i} {j} i≤j
  rewrite +-comm i n
        | +-comm j n
        = +-monoʳ-≤ n i≤j

+-mono-≤ : _+_ Preserves₂ _≤_  _≤_  _≤_
+-mono-≤ {m} {n} {i} {j} m≤n i≤j = begin
  m + i ≤⟨ +-monoˡ-≤ i m≤n 
  n + i ≤⟨ +-monoʳ-≤ n i≤j 
  n + j 
  where open ≤-Reasoning

≤-steps :  {m n} p  m  n  m  + p + n
≤-steps p m≤n = subst (_≤ _) (+-identityˡ _) (+-mono-≤ (+≤+ z≤n) m≤n)

m≤m+n :  {m} n  m  m + + n
m≤m+n {m} n = begin
  m       ≡⟨ sym (+-identityʳ m) 
  m + + 0 ≤⟨ +-monoʳ-≤ m (+≤+ z≤n) 
  m + + n 
  where open ≤-Reasoning

n≤m+n :  m {n}  n  + m + n
n≤m+n m {n} rewrite +-comm (+ m) n = m≤m+n m

------------------------------------------------------------------------
-- Properties of _+_ and _<_

+-monoʳ-< :  n  (_+_ n) Preserves _<_  _<_
+-monoʳ-< (+ n)    {_} {_}   (-<- o<m) = ⊖-monoʳ->-< n (s≤s o<m)
+-monoʳ-< (+ n)    {_} {_}   -<+       = <-≤-trans (m⊖1+n<m n _) (+≤+ (ℕₚ.m≤m+n n _))
+-monoʳ-< (+ n)    {_} {_}   (+<+ m<o) = +<+ (ℕₚ.+-monoʳ-< n m<o)
+-monoʳ-< -[1+ n ] {_} {_}   (-<- o<m) = -<- (ℕₚ.+-monoʳ-< (suc n) o<m)
+-monoʳ-< -[1+ n ] {_} {+ o} -<+       = <-≤-trans (-<- (ℕₚ.m≤m+n (suc n) _)) (-[1+m]≤n⊖m+1 n o)
+-monoʳ-< -[1+ n ] {_} {_}   (+<+ m<o) = ⊖-monoˡ-< (suc n) m<o

+-monoˡ-< :  n  (_+ n) Preserves _<_  _<_
+-monoˡ-< n {i} {j} i<j
  rewrite +-comm i n
        | +-comm j n
        = +-monoʳ-< n i<j

+-mono-< : _+_ Preserves₂ _<_  _<_  _<_
+-mono-< {m} {n} {i} {j} m<n i<j = begin-strict
  m + i <⟨ +-monoˡ-< i m<n 
  n + i <⟨ +-monoʳ-< n i<j 
  n + j 
  where open ≤-Reasoning

+-mono-≤-< : _+_ Preserves₂ _≤_  _<_  _<_
+-mono-≤-< {m} {n} {i} m≤n i<j = ≤-<-trans (+-monoˡ-≤ i m≤n) (+-monoʳ-< n i<j)

+-mono-<-≤ : _+_ Preserves₂ _<_  _≤_  _<_
+-mono-<-≤ {m} {n} {i} m<n i≤j = <-≤-trans (+-monoˡ-< i m<n) (+-monoʳ-≤ n i≤j)

------------------------------------------------------------------------
-- Properties of _-_
------------------------------------------------------------------------

neg-minus-pos :  x y  -[1+ x ] - (+ y)  -[1+ (y ℕ.+ x) ]
neg-minus-pos x       zero    = refl
neg-minus-pos zero    (suc y) = cong (-[1+_]  suc) (sym (ℕₚ.+-identityʳ y))
neg-minus-pos (suc x) (suc y) = cong (-[1+_]  suc) (ℕₚ.+-comm (suc x) y)

+-minus-telescope :  x y z  (x - y) + (y - z)  x - z
+-minus-telescope x y z = begin
  (x - y) + (y - z)   ≡⟨ +-assoc x (- y) (y - z) 
  x + (- y + (y - z)) ≡⟨ cong  v  x + v) (sym (+-assoc (- y) y _)) 
  x + ((- y + y) - z) ≡⟨ sym (+-assoc x (- y + y) (- z)) 
  x + (- y + y) - z   ≡⟨ cong  a  x + a - z) (+-inverseˡ y) 
  x + +0 - z          ≡⟨ cong (_- z) (+-identityʳ x) 
  x - z                where open ≡-Reasoning

[+m]-[+n]≡m⊖n :  x y  (+ x) - (+ y)  x  y
[+m]-[+n]≡m⊖n zero    zero    = refl
[+m]-[+n]≡m⊖n zero    (suc y) = refl
[+m]-[+n]≡m⊖n (suc x) zero    = cong (+_  suc) (ℕₚ.+-identityʳ x)
[+m]-[+n]≡m⊖n (suc x) (suc y) = refl

∣m-n∣≡∣n-m∣ : (x y : )   x - y    y - x 
∣m-n∣≡∣n-m∣ -[1+ x ] -[1+ y ] = ∣m⊖n∣≡∣n⊖m∣ y x
∣m-n∣≡∣n-m∣ -[1+ x ] (+ y)    = begin
   -[1+ x ] - (+ y)    ≡⟨ cong ∣_∣ (neg-minus-pos x y) 
  suc (y ℕ.+ x)          ≡⟨ sym (ℕₚ.+-suc y x) 
  y ℕ.+ suc x             where open ≡-Reasoning
∣m-n∣≡∣n-m∣ (+ x)    -[1+ y ] = begin
  x ℕ.+ suc y            ≡⟨ ℕₚ.+-suc x y 
  suc (x ℕ.+ y)          ≡⟨ cong ∣_∣ (sym (neg-minus-pos y x)) 
   -[1+ y ] + - (+ x)   where open ≡-Reasoning
∣m-n∣≡∣n-m∣ (+ x)     (+ y) = begin
   (+ x) - (+ y)  ≡⟨ cong ∣_∣ ([+m]-[+n]≡m⊖n x y) 
   x  y          ≡⟨ ∣m⊖n∣≡∣n⊖m∣ x y 
   y  x          ≡⟨ cong ∣_∣ (sym ([+m]-[+n]≡m⊖n y x)) 
   (+ y) - (+ x)   where open ≡-Reasoning

m≡n⇒m-n≡0 :  m n  m  n  m - n  + 0
m≡n⇒m-n≡0 m m refl = +-inverseʳ m

m-n≡0⇒m≡n :  m n  m - n  + 0  m  n
m-n≡0⇒m≡n m n m-n≡0 = begin
  m             ≡⟨ sym (+-identityʳ m) 
  m + + 0       ≡⟨ cong (_+_ m) (sym (+-inverseˡ n)) 
  m + (- n + n) ≡⟨ sym (+-assoc m (- n) n) 
  (m - n) + n   ≡⟨ cong (_+ n) m-n≡0 
  + 0 + n       ≡⟨ +-identityˡ n 
  n              where open ≡-Reasoning

≤-steps-neg :  {m n} p  m  n  m - + p  n
≤-steps-neg {m}         zero    m≤n rewrite +-identityʳ m = m≤n
≤-steps-neg {+ m}       (suc p) m≤n = ≤-trans (m⊖n≤m m (suc p)) m≤n
≤-steps-neg { -[1+ n ]} (suc p) m≤n = ≤-trans (-≤- (ℕₚ.≤-trans (ℕₚ.m≤m+n n p) (ℕₚ.n≤1+n _))) m≤n

neg-mono-≤-≥ : -_ Preserves _≤_  _≥_
neg-mono-≤-≥ -≤+             = neg-≤-pos
neg-mono-≤-≥ (-≤- n≤m)       = +≤+ (s≤s n≤m)
neg-mono-≤-≥ (+≤+ z≤n)       = neg-≤-pos
neg-mono-≤-≥ (+≤+ (s≤s m≤n)) = -≤- m≤n

m-n≤m :  m n  m - + n  m
m-n≤m m n = ≤-steps-neg n ≤-refl

m≤n⇒m-n≤0 :  {m n}  m  n  m - n  + 0
m≤n⇒m-n≤0 (-≤+ {n = n})         = ≤-steps-neg n -≤+
m≤n⇒m-n≤0 (-≤- {n = n} n≤m)     = ≤-trans (⊖-monoʳ-≥-≤ n n≤m) (≤-reflexive (n⊖n≡0 n))
m≤n⇒m-n≤0 {n = + 0}     (+≤+ z≤n) = +≤+ z≤n
m≤n⇒m-n≤0 {n = + suc n} (+≤+ z≤n) = -≤+
m≤n⇒m-n≤0 (+≤+ (s≤s {m} m≤n))   = ≤-trans (⊖-monoʳ-≥-≤ m m≤n) (≤-reflexive (n⊖n≡0 m))

m-n≤0⇒m≤n :  {m n}  m - n  + 0  m  n
m-n≤0⇒m≤n {m} {n} m-n≤0 = begin
  m             ≡⟨ sym (+-identityʳ m) 
  m + + 0       ≡⟨ cong (_+_ m) (sym (+-inverseˡ n)) 
  m + (- n + n) ≡⟨ sym (+-assoc m (- n) n) 
  (m - n) + n   ≤⟨ +-monoˡ-≤ n m-n≤0 
  + 0 + n       ≡⟨ +-identityˡ n 
  n             
  where open ≤-Reasoning

m≤n⇒0≤n-m :  {m n}  m  n  + 0  n - m
m≤n⇒0≤n-m {m} {n} m≤n = begin
  + 0   ≡⟨ sym (+-inverseʳ m) 
  m - m ≤⟨ +-monoˡ-≤ (- m) m≤n 
  n - m 
  where open ≤-Reasoning

0≤n-m⇒m≤n :  {m n}  + 0  n - m  m  n
0≤n-m⇒m≤n {m} {n} 0≤n-m = begin
  m             ≡⟨ sym (+-identityˡ m) 
  + 0 + m       ≤⟨ +-monoˡ-≤ m 0≤n-m 
  n - m + m     ≡⟨ +-assoc n (- m) m 
  n + (- m + m) ≡⟨ cong (_+_ n) (+-inverseˡ m) 
  n + + 0       ≡⟨ +-identityʳ n 
  n             
  where open ≤-Reasoning

------------------------------------------------------------------------
-- Properties of suc
------------------------------------------------------------------------

≤-step :  {n m}  n  m  n  sucℤ m
≤-step = ≤-steps 1

n≤1+n :  n  n  sucℤ n
n≤1+n n = ≤-steps 1 ≤-refl

suc-+ :  m n  + suc m + n  sucℤ (+ m + n)
suc-+ m (+ n)      = refl
suc-+ m (-[1+ n ]) = sym (distribʳ-⊖-+-pos 1 m (suc n))

n≢1+n :  {n}  n  sucℤ n
n≢1+n {+ _}           ()
n≢1+n { -[1+ 0 ]}     ()
n≢1+n { -[1+ suc n ]} ()

1-[1+n]≡-n :  n  sucℤ -[1+ n ]  - (+ n)
1-[1+n]≡-n zero    = refl
1-[1+n]≡-n (suc n) = refl

suc-mono : sucℤ Preserves _≤_  _≤_
suc-mono (-≤+ {m}) = 0⊖m≤+ m
suc-mono (-≤- n≤m) = ⊖-monoʳ-≥-≤ zero n≤m
suc-mono (+≤+ m≤n) = +≤+ (s≤s m≤n)

------------------------------------------------------------------------
-- Properties of pred
------------------------------------------------------------------------

suc-pred :  m  sucℤ (pred m)  m
suc-pred m = begin
  sucℤ (pred m) ≡⟨ sym (+-assoc (+ 1) (- + 1) m) 
  + 0 + m       ≡⟨ +-identityˡ m 
  m              where open ≡-Reasoning

pred-suc :  m  pred (sucℤ m)  m
pred-suc m = begin
  pred (sucℤ m) ≡⟨ sym (+-assoc (- + 1) (+ 1) m) 
  + 0 + m       ≡⟨ +-identityˡ m 
  m              where open ≡-Reasoning

+-pred :  m n  m + pred n  pred (m + n)
+-pred m n = begin
  m + (-[1+ 0 ] + n) ≡⟨ sym (+-assoc m -[1+ 0 ] n) 
  m + -[1+ 0 ] + n   ≡⟨ cong (_+ n) (+-comm m -[1+ 0 ]) 
  -[1+ 0 ] + m + n   ≡⟨ +-assoc -[1+ 0 ] m n 
  -[1+ 0 ] + (m + n)  where open ≡-Reasoning

pred-+ :  m n  pred m + n  pred (m + n)
pred-+ m n = begin
  pred m + n   ≡⟨ +-comm (pred m) n 
  n + pred m   ≡⟨ +-pred n m 
  pred (n + m) ≡⟨ cong pred (+-comm n m) 
  pred (m + n)  where open ≡-Reasoning

neg-suc :  m  - + suc m  pred (- + m)
neg-suc zero    = refl
neg-suc (suc m) = refl

minus-suc :  m n  m - + suc n  pred (m - + n)
minus-suc m n = begin
  m + - + suc n      ≡⟨ cong (_+_ m) (neg-suc n) 
  m + pred (- (+ n)) ≡⟨ +-pred m (- + n) 
  pred (m - + n)      where open ≡-Reasoning

m≤pred[n]⇒m<n :  {m n}  m  pred n  m < n
m≤pred[n]⇒m<n {m} { + n}      m≤predn = ≤-<-trans m≤predn (m⊖1+n<m n 0)
m≤pred[n]⇒m<n {m} { -[1+ n ]} m≤predn = ≤-<-trans m≤predn (-<- ℕₚ.≤-refl)

m<n⇒m≤pred[n] :  {m n}  m < n  m  pred n
m<n⇒m≤pred[n] {_} { +0}       -<+       = -≤- z≤n
m<n⇒m≤pred[n] {_} { +[1+ n ]} -<+       = -≤+
m<n⇒m≤pred[n] {_} { +[1+ n ]} (+<+ m<n) = +≤+ (ℕₚ.≤-pred m<n)
m<n⇒m≤pred[n] {_} { -[1+ n ]} (-<- n<m) = -≤- n<m

≤-step-neg :  {m n}  m  n  pred m  n
≤-step-neg -≤+               = -≤+
≤-step-neg (-≤- n≤m)         = -≤- (ℕₚ.≤-step n≤m)
≤-step-neg (+≤+ z≤n)         = -≤+
≤-step-neg (+≤+ (s≤s m≤n)) = +≤+ (ℕₚ.≤-step m≤n)

pred-mono : pred Preserves _≤_  _≤_
pred-mono (-≤+ {n = 0})     = -≤- z≤n
pred-mono (-≤+ {n = suc n}) = -≤+
pred-mono (-≤- n≤m)         = -≤- (s≤s n≤m)
pred-mono (+≤+ m≤n)         = ⊖-monoˡ-≤ 1 m≤n

------------------------------------------------------------------------
-- Properties of _*_
------------------------------------------------------------------------

------------------------------------------------------------------------
-- Algebraic properties

*-comm : Commutative _*_
*-comm -[1+ a ] -[1+ b ] rewrite ℕₚ.*-comm (suc a) (suc b) = refl
*-comm -[1+ a ] (+   b ) rewrite ℕₚ.*-comm (suc a) b       = refl
*-comm (+   a ) -[1+ b ] rewrite ℕₚ.*-comm a       (suc b) = refl
*-comm (+   a ) (+   b ) rewrite ℕₚ.*-comm a       b       = refl

*-identityˡ : LeftIdentity (+ 1) _*_
*-identityˡ -[1+ n ] rewrite ℕₚ.+-identityʳ n = refl
*-identityˡ +0    = refl
*-identityˡ +[1+ n ] rewrite ℕₚ.+-identityʳ n = refl

*-identityʳ : RightIdentity (+ 1) _*_
*-identityʳ = comm+idˡ⇒idʳ *-comm *-identityˡ

*-identity : Identity (+ 1) _*_
*-identity = *-identityˡ , *-identityʳ

*-zeroˡ : LeftZero +0 _*_
*-zeroˡ n = refl

*-zeroʳ : RightZero +0 _*_
*-zeroʳ = comm+zeˡ⇒zeʳ *-comm *-zeroˡ

*-zero : Zero +0 _*_
*-zero = *-zeroˡ , *-zeroʳ

private
  lemma :  a b c  c ℕ.+ (b ℕ.+ a ℕ.* suc b) ℕ.* suc c
                   c ℕ.+ b ℕ.* suc c ℕ.+ a ℕ.* suc (c ℕ.+ b ℕ.* suc c)
  lemma =
    solve 3  a b c  c :+ (b :+ a :* (con 1 :+ b)) :* (con 1 :+ c)
                    := c :+ b :* (con 1 :+ c) :+
                       a :* (con 1 :+ (c :+ b :* (con 1 :+ c))))
            refl

*-assoc : Associative _*_
*-assoc +0 _ _ = refl
*-assoc x +0 z rewrite ℕₚ.*-zeroʳ  x  = refl
*-assoc x y +0 rewrite
    ℕₚ.*-zeroʳ  y 
  | ℕₚ.*-zeroʳ  x 
  | ℕₚ.*-zeroʳ  sign x 𝕊* sign y   x  ℕ.*  y  
  = refl
*-assoc -[1+ a ] -[1+ b ] +[1+ c ] = cong (+_  suc) (lemma a b c)
*-assoc -[1+ a ] +[1+ b ] -[1+ c ] = cong (+_  suc) (lemma a b c)
*-assoc +[1+ a ] +[1+ b ] +[1+ c ] = cong (+_  suc) (lemma a b c)
*-assoc +[1+ a ] -[1+ b ] -[1+ c ] = cong (+_  suc) (lemma a b c)
*-assoc -[1+ a ] -[1+ b ] -[1+ c ] = cong -[1+_] (lemma a b c)
*-assoc -[1+ a ] +[1+ b ] +[1+ c ] = cong -[1+_] (lemma a b c)
*-assoc +[1+ a ] -[1+ b ] +[1+ c ] = cong -[1+_] (lemma a b c)
*-assoc +[1+ a ] +[1+ b ] -[1+ c ] = cong -[1+_] (lemma a b c)

private

  -- lemma used to prove distributivity.
  distrib-lemma :
     a b c  (c  b) * -[1+ a ]  a ℕ.+ b ℕ.* suc a  (a ℕ.+ c ℕ.* suc a)
  distrib-lemma a b c
    rewrite +-cancelˡ-⊖ a (b ℕ.* suc a) (c ℕ.* suc a)
          | ⊖-swap (b ℕ.* suc a) (c ℕ.* suc a)
    with b ℕ.≤? c
  ... | yes b≤c
    rewrite ⊖-≥ b≤c
          | ⊖-≥ (ℕₚ.*-mono-≤ b≤c (ℕₚ.≤-refl {x = suc a}))
          | -◃n≡-n ((c  b) ℕ.* suc a)
          | ℕₚ.*-distribʳ-∸ (suc a) c b
          = refl
  ... | no b≰c
    rewrite sign-⊖-≰ b≰c
          | ∣⊖∣-≰ b≰c
          | +◃n≡+n ((b  c) ℕ.* suc a)
          | ⊖-≰ (b≰c  ℕₚ.*-cancelʳ-≤ b c a)
          | neg-involutive (+ (b ℕ.* suc a  c ℕ.* suc a))
          | ℕₚ.*-distribʳ-∸ (suc a) b c
          = refl

*-distribʳ-+ : _*_ DistributesOverʳ _+_
*-distribʳ-+ +0 y z
  rewrite ℕₚ.*-zeroʳ  y 
        | ℕₚ.*-zeroʳ  z 
        | ℕₚ.*-zeroʳ  y + z 
        = refl
*-distribʳ-+ x +0 z
  rewrite +-identityˡ z
        | +-identityˡ (sign z 𝕊* sign x   z  ℕ.*  x )
        = refl
*-distribʳ-+ x y +0
  rewrite +-identityʳ y
        | +-identityʳ (sign y 𝕊* sign x   y  ℕ.*  x )
        = refl
*-distribʳ-+ -[1+ a ] -[1+ b ] -[1+ c ] = cong (+_) $
  solve 3  a b c  (con 2 :+ b :+ c) :* (con 1 :+ a)
                  := (con 1 :+ b) :* (con 1 :+ a) :+
                     (con 1 :+ c) :* (con 1 :+ a))
          refl a b c
*-distribʳ-+ (+ suc a) (+ suc b) (+ suc c) = cong (+_) $
  solve 3  a b c  (con 1 :+ b :+ (con 1 :+ c)) :* (con 1 :+ a)
                  := (con 1 :+ b) :* (con 1 :+ a) :+
                     (con 1 :+ c) :* (con 1 :+ a))
        refl a b c
*-distribʳ-+ -[1+ a ] (+ suc b) (+ suc c) = cong -[1+_] $
  solve 3  a b c  a :+ (b :+ (con 1 :+ c)) :* (con 1 :+ a)
                   := (con 1 :+ b) :* (con 1 :+ a) :+
                      (a :+ c :* (con 1 :+ a)))
         refl a b c
*-distribʳ-+ (+ suc a) -[1+ b ] -[1+ c ] = cong -[1+_] $
  solve 3  a b c  a :+ (con 1 :+ a :+ (b :+ c) :* (con 1 :+ a))
                  := (con 1 :+ b) :* (con 1 :+ a) :+
                     (a :+ c :* (con 1 :+ a)))
         refl a b c
*-distribʳ-+ -[1+ a ] -[1+ b ] (+ suc c) = distrib-lemma a b c
*-distribʳ-+ -[1+ a ] (+ suc b) -[1+ c ] = distrib-lemma a c b
*-distribʳ-+ (+ suc a) -[1+ b ] (+ suc c) with b ℕ.≤? c
... | yes b≤c
  rewrite +-cancelˡ-⊖ a (c ℕ.* suc a) (b ℕ.* suc a)
        | ⊖-≥ b≤c
        | +-comm (- (+ (a ℕ.+ b ℕ.* suc a))) (+ (a ℕ.+ c ℕ.* suc a))
        | ⊖-≥ (ℕₚ.*-mono-≤ b≤c (ℕₚ.≤-refl {x = suc a}))
        | ℕₚ.*-distribʳ-∸ (suc a) c b
        | +◃n≡+n (c ℕ.* suc a  b ℕ.* suc a)
        = refl
... | no b≰c
  rewrite +-cancelˡ-⊖ a (c ℕ.* suc a) (b ℕ.* suc a)
        | sign-⊖-≰ b≰c
        | ∣⊖∣-≰ b≰c
        | -◃n≡-n ((b  c) ℕ.* suc a)
        | ⊖-≰ (b≰c  ℕₚ.*-cancelʳ-≤ b c a)
        | ℕₚ.*-distribʳ-∸ (suc a) b c
        = refl
*-distribʳ-+ (+ suc c) (+ suc a) -[1+ b ] with b ℕ.≤? a
... | yes b≤a
  rewrite +-cancelˡ-⊖ c (a ℕ.* suc c) (b ℕ.* suc c)
        | ⊖-≥ b≤a
        | ⊖-≥ (ℕₚ.*-mono-≤ b≤a (ℕₚ.≤-refl {x = suc c}))
        | +◃n≡+n ((a  b) ℕ.* suc c)
        | ℕₚ.*-distribʳ-∸ (suc c) a b
        = refl
... | no b≰a
  rewrite +-cancelˡ-⊖ c (a ℕ.* suc c) (b ℕ.* suc c)
        | sign-⊖-≰ b≰a
        | ∣⊖∣-≰ b≰a
        | ⊖-≰ (b≰a  ℕₚ.*-cancelʳ-≤ b a c)
        | -◃n≡-n ((b  a) ℕ.* suc c)
        | ℕₚ.*-distribʳ-∸ (suc c) b a
        = refl

*-distribˡ-+ : _*_ DistributesOverˡ _+_
*-distribˡ-+ = comm+distrʳ⇒distrˡ *-comm *-distribʳ-+

*-distrib-+ : _*_ DistributesOver _+_
*-distrib-+ = *-distribˡ-+ , *-distribʳ-+

------------------------------------------------------------------------
-- Structures

*-isMagma : IsMagma _*_
*-isMagma = record
  { isEquivalence = isEquivalence
  ; ∙-cong        = cong₂ _*_
  }

*-isSemigroup : IsSemigroup _*_
*-isSemigroup = record
  { isMagma = *-isMagma
  ; assoc   = *-assoc
  }

*-1-isMonoid : IsMonoid _*_ (+ 1)
*-1-isMonoid = record
  { isSemigroup = *-isSemigroup
  ; identity    = *-identity
  }

*-1-isCommutativeMonoid : IsCommutativeMonoid _*_ (+ 1)
*-1-isCommutativeMonoid = record
  { isSemigroup = *-isSemigroup
  ; identityˡ   = *-identityˡ
  ; comm        = *-comm
  }

+-*-isCommutativeSemiring : IsCommutativeSemiring _+_ _*_ +0 (+ 1)
+-*-isCommutativeSemiring = record
  { +-isCommutativeMonoid = +-0-isCommutativeMonoid
  ; *-isCommutativeMonoid = *-1-isCommutativeMonoid
  ; distribʳ              = *-distribʳ-+
  ; zeroˡ                 = *-zeroˡ
  }

+-*-isRing : IsRing _+_ _*_ -_ +0 (+ 1)
+-*-isRing = record
  { +-isAbelianGroup = +-isAbelianGroup
  ; *-isMonoid       = *-1-isMonoid
  ; distrib          = *-distrib-+
  }

+-*-isCommutativeRing : IsCommutativeRing _+_ _*_ -_ +0 (+ 1)
+-*-isCommutativeRing = record
  { isRing = +-*-isRing
  ; *-comm = *-comm
  }

------------------------------------------------------------------------
-- Packages

*-magma : Magma 0ℓ 0ℓ
*-magma = record
  { isMagma = *-isMagma
  }

*-semigroup : Semigroup 0ℓ 0ℓ
*-semigroup = record
  { isSemigroup = *-isSemigroup
  }

*-1-monoid : Monoid 0ℓ 0ℓ
*-1-monoid = record
  { isMonoid = *-1-isMonoid
  }

*-1-commutativeMonoid : CommutativeMonoid 0ℓ 0ℓ
*-1-commutativeMonoid = record
  { isCommutativeMonoid = *-1-isCommutativeMonoid
  }

+-*-ring : Ring 0ℓ 0ℓ
+-*-ring = record
  { isRing = +-*-isRing
  }

+-*-commutativeRing : CommutativeRing 0ℓ 0ℓ
+-*-commutativeRing = record
  { isCommutativeRing = +-*-isCommutativeRing
  }

------------------------------------------------------------------------
-- Other properties of _*_ and _≡_

abs-*-commute : ℤtoℕ.Homomorphic₂ ∣_∣ _*_ ℕ._*_
abs-*-commute i j = abs-◃ _ _

*-cancelʳ-≡ :  i j k  k  + 0  i * k  j * k  i  j
*-cancelʳ-≡ i j k            ≢0 eq with signAbs k
*-cancelʳ-≡ i j .+0       ≢0 eq | s  zero  = contradiction refl ≢0
*-cancelʳ-≡ i j .(s  suc n) ≢0 eq | s  suc n
  with  s  suc n  | abs-◃ s (suc n) | sign (s  suc n) | sign-◃ s n
...  | .(suc n)      | refl            | .s               | refl =
  ◃-cong (sign-i≡sign-j i j eq) $
         ℕₚ.*-cancelʳ-≡  i   j  $ abs-cong eq
  where
  sign-i≡sign-j :  i j 
                  (sign i 𝕊* s)  ( i  ℕ.* suc n) 
                  (sign j 𝕊* s)  ( j  ℕ.* suc n) 
                  sign i  sign j
  sign-i≡sign-j i              j              eq with signAbs i | signAbs j
  sign-i≡sign-j .+0         .+0         eq | s₁  zero   | s₂  zero   = refl
  sign-i≡sign-j .+0         .(s₂  suc n₂) eq | s₁  zero   | s₂  suc n₂
    with  s₂  suc n₂  | abs-◃ s₂ (suc n₂)
  ... | .(suc n₂) | refl
    with abs-cong {s₁} {sign (s₂  suc n₂) 𝕊* s} {0} {suc n₂ ℕ.* suc n} eq
  ...   | ()
  sign-i≡sign-j .(s₁  suc n₁) .+0         eq | s₁  suc n₁ | s₂  zero
    with  s₁  suc n₁  | abs-◃ s₁ (suc n₁)
  ... | .(suc n₁) | refl
    with abs-cong {sign (s₁  suc n₁) 𝕊* s} {s₁} {suc n₁ ℕ.* suc n} {0} eq
  ...   | ()
  sign-i≡sign-j .(s₁  suc n₁) .(s₂  suc n₂) eq | s₁  suc n₁ | s₂  suc n₂
    with  s₁  suc n₁  | abs-◃ s₁ (suc n₁)
       | sign (s₁  suc n₁) | sign-◃ s₁ n₁
       |  s₂  suc n₂  | abs-◃ s₂ (suc n₂)
       | sign (s₂  suc n₂) | sign-◃ s₂ n₂
  ... | .(suc n₁) | refl | .s₁ | refl | .(suc n₂) | refl | .s₂ | refl =
    𝕊ₚ.*-cancelʳ-≡ s₁ s₂ (sign-cong eq)

*-cancelˡ-≡ :  i j k  i  + 0  i * j  i * k  j  k
*-cancelˡ-≡ i j k
  rewrite *-comm i j
        | *-comm i k
        = *-cancelʳ-≡ j k i

suc-* :  m n  sucℤ m * n  n + m * n
suc-* m n = begin
  sucℤ m * n      ≡⟨ *-distribʳ-+ n (+ 1) m 
  + 1 * n + m * n ≡⟨ cong (_+ m * n) (*-identityˡ n) 
  n + m * n       
  where open ≡-Reasoning

*-suc :  m n  m * sucℤ n  m + m * n
*-suc m n = begin
  m * sucℤ n ≡⟨ *-comm m _ 
  sucℤ n * m ≡⟨ suc-* n m 
  m + n * m  ≡⟨ cong  v  m + v) (*-comm n m) 
  m + m * n  
  where open ≡-Reasoning

-1*n≡-n :  n  -[1+ 0 ] * n  - n
-1*n≡-n -[1+ n ] = cong  v  + suc v) (ℕₚ.+-identityʳ n)
-1*n≡-n +0       = refl
-1*n≡-n +[1+ n ] = cong -[1+_] (ℕₚ.+-identityʳ n)

------------------------------------------------------------------------
-- Properties of _*_ and +_/-_

pos-distrib-* :  x y  (+ x) * (+ y)  + (x ℕ.* y)
pos-distrib-* zero    y       = refl
pos-distrib-* (suc x) zero    = pos-distrib-* x zero
pos-distrib-* (suc x) (suc y) = refl

neg-distribˡ-* :  x y  - (x * y)  (- x) * y
neg-distribˡ-* x y = begin
  - (x * y)          ≡⟨ sym (-1*n≡-n (x * y)) 
  -[1+ 0 ] * (x * y) ≡⟨ sym (*-assoc -[1+ 0 ] x y) 
  -[1+ 0 ] * x * y   ≡⟨ cong (_* y) (-1*n≡-n x) 
  - x * y             where open ≡-Reasoning

neg-distribʳ-* :  x y  - (x * y)  x * (- y)
neg-distribʳ-* x y = begin
  - (x * y) ≡⟨ cong -_ (*-comm x y) 
  - (y * x) ≡⟨ neg-distribˡ-* y x 
  - y * x   ≡⟨ *-comm (- y) x 
  x * (- y)  where open ≡-Reasoning

------------------------------------------------------------------------
-- Properties of _*_ and _◃_

◃-distrib-* :   s t m n  (s 𝕊* t)  (m ℕ.* n)  (s  m) * (t  n)
◃-distrib-* s t zero    zero    = refl
◃-distrib-* s t zero    (suc n) = refl
◃-distrib-* s t (suc m) zero    =
  trans
    (cong₂ _◃_ (𝕊ₚ.*-comm s t) (ℕₚ.*-comm m 0))
    (*-comm (t  zero) (s  suc m))
◃-distrib-* s t (suc m) (suc n) =
  sym (cong₂ _◃_
    (cong₂ _𝕊*_ (sign-◃ s m) (sign-◃ t n))
    (∣s◃m∣*∣t◃n∣≡m*n s t (suc m) (suc n)))

------------------------------------------------------------------------
-- Properties of _*_ and _≤_

*-cancelʳ-≤-pos :  m n o  m * + suc o  n * + suc o  m  n
*-cancelʳ-≤-pos (-[1+ m ]) (-[1+ n ]) o (-≤- n≤m) =
  -≤- (ℕₚ.≤-pred (ℕₚ.*-cancelʳ-≤ (suc n) (suc m) o (s≤s n≤m)))
*-cancelʳ-≤-pos -[1+ _ ]   (+ _)      _ _         = -≤+
*-cancelʳ-≤-pos +0      +0      _ _         = +≤+ z≤n
*-cancelʳ-≤-pos +0      (+ suc _)  _ _         = +≤+ z≤n
*-cancelʳ-≤-pos (+ suc _)  +0      _ (+≤+ ())
*-cancelʳ-≤-pos (+ suc m)  (+ suc n)  o (+≤+ m≤n) =
  +≤+ (ℕₚ.*-cancelʳ-≤ (suc m) (suc n) o m≤n)

*-cancelˡ-≤-pos :  m n o  + suc m * n  + suc m * o  n  o
*-cancelˡ-≤-pos m n o
  rewrite *-comm (+ suc m) n
        | *-comm (+ suc m) o
        = *-cancelʳ-≤-pos n o m

*-monoʳ-≤-pos :  n  (_* + suc n) Preserves _≤_  _≤_
*-monoʳ-≤-pos _ (-≤+             {n = 0})         = -≤+
*-monoʳ-≤-pos _ (-≤+             {n = suc _})     = -≤+
*-monoʳ-≤-pos x (-≤-                         n≤m) =
  -≤- (ℕₚ.≤-pred (ℕₚ.*-mono-≤ (s≤s n≤m) (ℕₚ.≤-refl {x = suc x})))
*-monoʳ-≤-pos k {+ 0} {+ 0}     (+≤+ m≤n) = +≤+ m≤n
*-monoʳ-≤-pos k {+ 0} {+ suc _} (+≤+ m≤n) = +≤+ z≤n
*-monoʳ-≤-pos x (+≤+ {m = suc _} {n = suc _} m≤n) =
  +≤+ ((ℕₚ.*-mono-≤ m≤n (ℕₚ.≤-refl {x = suc x})))

*-monoʳ-≤-non-neg :  n  (_* + n) Preserves _≤_  _≤_
*-monoʳ-≤-non-neg (suc n) = *-monoʳ-≤-pos n
*-monoʳ-≤-non-neg zero {i} {j} i≤j
  rewrite *-zeroʳ i
        | *-zeroʳ j
        = +≤+ z≤n

*-monoˡ-≤-non-neg :  n  (+ n *_) Preserves _≤_  _≤_
*-monoˡ-≤-non-neg n {i} {j} i≤j
  rewrite *-comm (+ n) i
        | *-comm (+ n) j
        = *-monoʳ-≤-non-neg n i≤j

*-monoˡ-≤-pos :  n  (+ suc n *_) Preserves _≤_  _≤_
*-monoˡ-≤-pos n = *-monoˡ-≤-non-neg (suc n)

------------------------------------------------------------------------
-- Properties of _*_ and _≤_

*-monoˡ-<-pos :  n  (+[1+ n ] *_) Preserves _<_  _<_
*-monoˡ-<-pos n {+ m}       {+ o}       (+<+ m<o) = +◃-mono-< (ℕₚ.+-mono-<-≤ m<o (ℕₚ.*-monoʳ-≤ n (ℕₚ.<⇒≤ m<o)))
*-monoˡ-<-pos n { -[1+ m ]} {+ o}       leq       = -◃<+◃ _ (suc n ℕ.* o)
*-monoˡ-<-pos n { -[1+ m ]} { -[1+ o ]} (-<- o<m) = -<- (ℕₚ.+-mono-<-≤ o<m (ℕₚ.*-monoʳ-≤ n (ℕₚ.<⇒≤ (s≤s o<m))))

*-monoʳ-<-pos :  n  (_* +[1+ n ]) Preserves _<_  _<_
*-monoʳ-<-pos n {m} {o} rewrite *-comm m +[1+ n ] | *-comm o +[1+ n ]
  = *-monoˡ-<-pos n

*-cancelˡ-<-non-neg :  n {i j}  + n * i < + n * j  i < j
*-cancelˡ-<-non-neg n {+ i}       {+ j}       leq = +<+ (ℕₚ.*-cancelˡ-< n (+◃-cancel-< leq))
*-cancelˡ-<-non-neg n {+ i}       { -[1+ j ]} leq = contradiction leq +◃≮-◃
*-cancelˡ-<-non-neg n { -[1+ i ]} {+ j}       leq = -<+
*-cancelˡ-<-non-neg n { -[1+ i ]} { -[1+ j ]} leq = -<- (ℕₚ.≤-pred (ℕₚ.*-cancelˡ-< n (neg◃-cancel-< leq)))

*-cancelʳ-<-non-neg :  {i j} n  i * + n < j * + n  i < j
*-cancelʳ-<-non-neg {i} {j} n rewrite *-comm i (+ n) | *-comm j (+ n)
  = *-cancelˡ-<-non-neg n

------------------------------------------------------------------------
-- Properties of _⊓_
------------------------------------------------------------------------

------------------------------------------------------------------------
-- Algebraic properties

⊓-comm : Commutative _⊓_
⊓-comm -[1+ m ] -[1+ n ] = cong -[1+_] (ℕₚ.⊔-comm m n)
⊓-comm -[1+ m ] (+ n)    = refl
⊓-comm (+ m)    -[1+ n ] = refl
⊓-comm (+ m)    (+ n)    = cong +_ (ℕₚ.⊓-comm m n)

⊓-assoc : Associative _⊓_
⊓-assoc -[1+ m ] -[1+ n ] -[1+ p ] = cong -[1+_] (ℕₚ.⊔-assoc m n p)
⊓-assoc -[1+ m ] -[1+ n ] (+ p)    = refl
⊓-assoc -[1+ m ] (+ n)    -[1+ p ] = refl
⊓-assoc -[1+ m ] (+ n)    (+ p)    = refl
⊓-assoc (+ m)    -[1+ n ] -[1+ p ] = refl
⊓-assoc (+ m)    -[1+ n ] (+ p)    = refl
⊓-assoc (+ m)    (+ n)    -[1+ p ] = refl
⊓-assoc (+ m)    (+ n)    (+ p)    = cong +_ (ℕₚ.⊓-assoc m n p)

⊓-idem : Idempotent _⊓_
⊓-idem (+ m)    = cong +_ (ℕₚ.⊓-idem m)
⊓-idem -[1+ m ] = cong -[1+_] (ℕₚ.⊔-idem m)

⊓-sel : Selective _⊓_
⊓-sel -[1+ m ] -[1+ n ] = Sum.map (cong -[1+_]) (cong -[1+_]) $ ℕₚ.⊔-sel m n
⊓-sel -[1+ m ] (+ n)    = inj₁ refl
⊓-sel (+ m)    -[1+ n ] = inj₂ refl
⊓-sel (+ m)    (+ n)    = Sum.map (cong ℤ.pos) (cong ℤ.pos) $ ℕₚ.⊓-sel m n

------------------------------------------------------------------------
-- Other properties

m≤n⇒m⊓n≡m :  {m n}  m  n  m  n  m
m≤n⇒m⊓n≡m -≤+       = refl
m≤n⇒m⊓n≡m (-≤- n≤m) = cong -[1+_] (ℕₚ.m≤n⇒n⊔m≡n n≤m)
m≤n⇒m⊓n≡m (+≤+ m≤n) = cong +_ (ℕₚ.m≤n⇒m⊓n≡m m≤n)

m⊓n≡m⇒m≤n :  {m n}  m  n  m  m  n
m⊓n≡m⇒m≤n { -[1+ m ]} { -[1+ n ]} eq = -≤- (ℕₚ.n⊔m≡n⇒m≤n (-[1+-injective eq))
m⊓n≡m⇒m≤n { -[1+ m ]} {+ n}       eq = -≤+
m⊓n≡m⇒m≤n {+ m}       {+ n}       eq = +≤+ (ℕₚ.m⊓n≡m⇒m≤n (+-injective eq))

m≥n⇒m⊓n≡n :  {m n}  m  n  m  n  n
m≥n⇒m⊓n≡n {m} {n} pr rewrite ⊓-comm m n = m≤n⇒m⊓n≡m pr

m⊓n≡n⇒m≥n :  {m n}  m  n  n  m  n
m⊓n≡n⇒m≥n {m} {n} eq rewrite ⊓-comm m n = m⊓n≡m⇒m≤n eq

m⊓n≤n :  m n  m  n  n
m⊓n≤n -[1+ m ] -[1+ n ] = -≤- (ℕₚ.n≤m⊔n m n)
m⊓n≤n -[1+ m ] (+ n)    = -≤+
m⊓n≤n (+ m)    -[1+ n ] = -≤- ℕₚ.≤-refl
m⊓n≤n (+ m)    (+ n)    = +≤+ (ℕₚ.m⊓n≤n m n)

m⊓n≤m :  m n  m  n  m
m⊓n≤m m n rewrite ⊓-comm m n = m⊓n≤n n m

------------------------------------------------------------------------
-- Properties _⊔_
------------------------------------------------------------------------

------------------------------------------------------------------------
-- Algebraic properties

⊔-assoc : Associative _⊔_
⊔-assoc -[1+ m ] -[1+ n ] -[1+ p ] = cong -[1+_] (ℕₚ.⊓-assoc m n p)
⊔-assoc -[1+ m ] -[1+ n ] (+ p)    = refl
⊔-assoc -[1+ m ] (+ n)    -[1+ p ] = refl
⊔-assoc -[1+ m ] (+ n)    (+ p)    = refl
⊔-assoc (+ m)    -[1+ n ] -[1+ p ] = refl
⊔-assoc (+ m)    -[1+ n ] (+ p)    = refl
⊔-assoc (+ m)    (+ n)    -[1+ p ] = refl
⊔-assoc (+ m)    (+ n)    (+ p)    = cong +_ (ℕₚ.⊔-assoc m n p)

⊔-comm : Commutative _⊔_
⊔-comm -[1+ m ] -[1+ n ] = cong -[1+_] (ℕₚ.⊓-comm m n)
⊔-comm -[1+ m ] (+ n)    = refl
⊔-comm (+ m)    -[1+ n ] = refl
⊔-comm (+ m)    (+ n)    = cong +_ (ℕₚ.⊔-comm m n)

⊔-idem : Idempotent _⊔_
⊔-idem (+ m)    = cong +_ (ℕₚ.⊔-idem m)
⊔-idem -[1+ m ] = cong -[1+_] (ℕₚ.⊓-idem m)

⊔-sel : Selective _⊔_
⊔-sel -[1+ m ] -[1+ n ] = Sum.map (cong -[1+_]) (cong -[1+_]) $ ℕₚ.⊓-sel m n
⊔-sel -[1+ m ] (+ n)    = inj₂ refl
⊔-sel (+ m)    -[1+ n ] = inj₁ refl
⊔-sel (+ m)    (+ n)    = Sum.map (cong ℤ.pos) (cong ℤ.pos) $ ℕₚ.⊔-sel m n

------------------------------------------------------------------------
-- Other properties

m≤n⇒m⊔n≡n :  {m n}  m  n  m  n  n
m≤n⇒m⊔n≡n -≤+       = refl
m≤n⇒m⊔n≡n (-≤- n≤m) = cong -[1+_] (ℕₚ.m≤n⇒n⊓m≡m n≤m)
m≤n⇒m⊔n≡n (+≤+ m≤n) = cong +_ (ℕₚ.m≤n⇒m⊔n≡n m≤n)

m⊔n≡n⇒m≤n :  {m n}  m  n  n  m  n
m⊔n≡n⇒m≤n { -[1+ m ]} { -[1+ n ]} eq = -≤- (ℕₚ.m⊓n≡n⇒n≤m (-[1+-injective eq))
m⊔n≡n⇒m≤n { -[1+ m ]} {+ n}       eq = -≤+
m⊔n≡n⇒m≤n {+ m}       {+ n}       eq = +≤+ (ℕₚ.n⊔m≡m⇒n≤m (+-injective eq))

m≥n⇒m⊔n≡m :  {m n}  m  n  m  n  m
m≥n⇒m⊔n≡m {m} {n} pr rewrite ⊔-comm m n = m≤n⇒m⊔n≡n pr

m⊔n≡m⇒m≥n :  {m n}  m  n  m  m  n
m⊔n≡m⇒m≥n {m} {n} eq rewrite ⊔-comm m n = m⊔n≡n⇒m≤n eq

m≤m⊔n :  m n  m  m  n
m≤m⊔n -[1+ m ] -[1+ n ] = -≤- (ℕₚ.m⊓n≤m m n)
m≤m⊔n -[1+ m ] (+ n)    = -≤+
m≤m⊔n (+ m)    -[1+ n ] = +≤+ ℕₚ.≤-refl
m≤m⊔n (+ m)    (+ n)    = +≤+ (ℕₚ.m≤m⊔n m n)

n≤m⊔n :  m n  n  m  n
n≤m⊔n m n rewrite ⊔-comm m n = m≤m⊔n n m

neg-distrib-⊔-⊓ :  m n  - (m  n)  - m  - n
neg-distrib-⊔-⊓ -[1+ m ] -[1+ n ] = refl
neg-distrib-⊔-⊓ -[1+ m ] +0       = refl
neg-distrib-⊔-⊓ -[1+ m ] +[1+ n ] = refl
neg-distrib-⊔-⊓ +0       -[1+ n ] = refl
neg-distrib-⊔-⊓ +0       +0       = refl
neg-distrib-⊔-⊓ +0       +[1+ n ] = refl
neg-distrib-⊔-⊓ +[1+ m ] -[1+ n ] = refl
neg-distrib-⊔-⊓ +[1+ m ] +0       = refl
neg-distrib-⊔-⊓ +[1+ m ] +[1+ n ] = refl

neg-distrib-⊓-⊔ :  m n  - (m  n)  - m  - n
neg-distrib-⊓-⊔ -[1+ m ] -[1+ n ] = refl
neg-distrib-⊓-⊔ -[1+ m ] +0       = refl
neg-distrib-⊓-⊔ -[1+ m ] +[1+ n ] = refl
neg-distrib-⊓-⊔ +0       -[1+ n ] = refl
neg-distrib-⊓-⊔ +0       +0       = refl
neg-distrib-⊓-⊔ +0       +[1+ n ] = refl
neg-distrib-⊓-⊔ +[1+ m ] -[1+ n ] = refl
neg-distrib-⊓-⊔ +[1+ m ] +0       = refl
neg-distrib-⊓-⊔ +[1+ m ] +[1+ n ] = refl


------------------------------------------------------------------------
-- DEPRECATED NAMES
------------------------------------------------------------------------
-- Please use the new names as continuing support for the old names is
-- not guaranteed.

-- Version 0.15

inverseˡ = +-inverseˡ
{-# WARNING_ON_USAGE inverseˡ
"Warning: inverseˡ was deprecated in v0.15.
Please use +-inverseˡ instead."
#-}
inverseʳ = +-inverseʳ
{-# WARNING_ON_USAGE inverseʳ
"Warning: inverseʳ was deprecated in v0.15.
Please use +-inverseʳ instead."
#-}
distribʳ = *-distribʳ-+
{-# WARNING_ON_USAGE distribʳ
"Warning: distribʳ was deprecated in v0.15.
Please use *-distribʳ-+ instead."
#-}
isCommutativeSemiring = +-*-isCommutativeSemiring
{-# WARNING_ON_USAGE isCommutativeSemiring
"Warning: isCommutativeSemiring was deprecated in v0.15.
Please use +-*-isCommutativeSemiring instead."
#-}
commutativeRing = +-*-commutativeRing
{-# WARNING_ON_USAGE commutativeRing
"Warning: commutativeRing was deprecated in v0.15.
Please use +-*-commutativeRing instead."
#-}
*-+-right-mono = *-monoʳ-≤-pos
{-# WARNING_ON_USAGE *-+-right-mono
"Warning: *-+-right-mono was deprecated in v0.15.
Please use *-monoʳ-≤-pos instead."
#-}
cancel-*-+-right-≤ = *-cancelʳ-≤-pos
{-# WARNING_ON_USAGE cancel-*-+-right-≤
"Warning: cancel-*-+-right-≤ was deprecated in v0.15.
Please use *-cancelʳ-≤-pos instead."
#-}
cancel-*-right = *-cancelʳ-≡
{-# WARNING_ON_USAGE cancel-*-right
"Warning: cancel-*-right was deprecated in v0.15.
Please use *-cancelʳ-≡ instead."
#-}
doubleNeg = neg-involutive
{-# WARNING_ON_USAGE doubleNeg
"Warning: doubleNeg was deprecated in v0.15.
Please use neg-involutive instead."
#-}
-‿involutive = neg-involutive
{-# WARNING_ON_USAGE -‿involutive
"Warning: -‿involutive was deprecated in v0.15.
Please use neg-involutive instead."
#-}
+-⊖-left-cancel = +-cancelˡ-⊖
{-# WARNING_ON_USAGE +-⊖-left-cancel
"Warning: +-⊖-left-cancel was deprecated in v0.15.
Please use +-cancelˡ-⊖ instead."
#-}

-- Version 1.0

≰→> = ≰⇒>
{-# WARNING_ON_USAGE ≰→>
"Warning: ≰→> was deprecated in v1.0.
Please use ≰⇒> instead."
#-}
≤-irrelevance = ≤-irrelevant
{-# WARNING_ON_USAGE ≤-irrelevance
"Warning: ≤-irrelevance was deprecated in v1.0.
Please use ≤-irrelevant instead."
#-}
<-irrelevance = <-irrelevant
{-# WARNING_ON_USAGE <-irrelevance
"Warning: <-irrelevance was deprecated in v1.0.
Please use <-irrelevant instead."
#-}


-- Version 1.1

-- Not all of the below have deprecation warnings attached as they are
-- reused by other deprecated results.

-<′+ :  {m n}  -[1+ m ] <′ + n
-<′+ {0}     = +≤+ z≤n
-<′+ {suc _} = -≤+
{-# WARNING_ON_USAGE -<′+
"Warning: _<′_ was deprecated in v1.1.
Please use _<_ instead."
#-}
<′-irrefl : Irreflexive _≡_ _<′_
<′-irrefl { + n}          refl (+≤+ 1+n≤n) = ℕₚ.<-irrefl refl 1+n≤n
<′-irrefl { -[1+ suc n ]} refl (-≤- 1+n≤n) = ℕₚ.<-irrefl refl 1+n≤n
{-# WARNING_ON_USAGE <′-irrefl
"Warning: _<′_ was deprecated in v1.1.
Please use _<_ instead."
#-}
>′-irrefl : Irreflexive _≡_ _>′_
>′-irrefl = <′-irrefl  sym
{-# WARNING_ON_USAGE >′-irrefl
"Warning: _>′_ was deprecated in v1.1.
Please use _>_ instead."
#-}
<′-asym : Asymmetric _<′_
<′-asym {+ n}           {+ m}           (+≤+ n<m) (+≤+ m<n) = ℕₚ.<-asym n<m m<n
<′-asym { -[1+ suc n ]} { -[1+ suc m ]} (-≤- n<m) (-≤- m<n) = ℕₚ.<-asym n<m m<n
{-# WARNING_ON_USAGE <′-asym
"Warning: _<′_ was deprecated in v1.1.
Please use _<_ instead."
#-}
≤-<′-trans : Trans _≤_ _<′_ _<′_
≤-<′-trans { -[1+ m ]} {+ n} {+ p} -≤+ (+≤+ 1+n≤p) = -<′+ {m} {p}
≤-<′-trans {+ m} {+ n} {+ p} (+≤+ m≤n) (+≤+ 1+n≤p) = +≤+ (ℕₚ.≤-trans (ℕ.s≤s m≤n) 1+n≤p)
≤-<′-trans { -[1+ m ]} { -[1+ n ]} (-≤- n≤m) n<p = ≤-trans (⊖-monoʳ-≥-≤ 0 n≤m) n<p
{-# WARNING_ON_USAGE ≤-<′-trans
"Warning: _<′_ was deprecated in v1.1.
Please use _<_ instead."
#-}
<′-≤-trans : Trans _<′_ _≤_ _<′_
<′-≤-trans = ≤-trans
{-# WARNING_ON_USAGE <′-≤-trans
"Warning: _<′_ was deprecated in v1.1.
Please use _<_ instead."
#-}
<′⇒≤ :  {m n}  m <′ n  m  n
<′⇒≤ m<n =  ≤-trans (n≤1+n _) m<n
{-# WARNING_ON_USAGE <′⇒≤
"Warning: _<′_ was deprecated in v1.1.
Please use _<_ instead."
#-}
<′-trans : Transitive _<′_
<′-trans {m} {n} {p} m<n n<p = ≤-<′-trans {m} {n} {p} (<′⇒≤ m<n) n<p
{-# WARNING_ON_USAGE <′-trans
"Warning: _<′_ was deprecated in v1.1.
Please use _<_ instead."
#-}
<′-cmp : Trichotomous _≡_ _<′_
<′-cmp (+ m) (+ n) with ℕₚ.<-cmp m n
... | tri< m<n m≢n m≯n =
  tri< (+≤+ m<n)         (m≢n  +-injective) (m≯n  drop‿+≤+)
... | tri≈ m≮n m≡n m≯n =
  tri≈ (m≮n  drop‿+≤+) (cong (+_) m≡n)     (m≯n  drop‿+≤+)
... | tri> m≮n m≢n m>n =
  tri> (m≮n  drop‿+≤+) (m≢n  +-injective) (+≤+ m>n)
<′-cmp (+_ m)       -[1+ 0 ]     = tri> (λ())     (λ()) (+≤+ z≤n)
<′-cmp (+_ m)       -[1+ suc n ] = tri> (λ())     (λ()) -≤+
<′-cmp -[1+ 0 ]     (+ n)        = tri< (+≤+ z≤n) (λ()) (λ())
<′-cmp -[1+ suc m ] (+ n)        = tri< -≤+       (λ()) (λ())
<′-cmp -[1+ 0 ]     -[1+ 0 ]     = tri≈ (λ())     refl  (λ())
<′-cmp -[1+ 0 ]     -[1+ suc n ] = tri> (λ())     (λ()) (-≤- z≤n)
<′-cmp -[1+ suc m ] -[1+ 0 ]     = tri< (-≤- z≤n) (λ()) (λ())
<′-cmp -[1+ suc m ] -[1+ suc n ] with ℕₚ.<-cmp (suc m) (suc n)
... | tri< m<n m≢n m≯n =
  tri> (m≯n  s≤s  drop‿-≤-) (m≢n  -[1+-injective) (-≤- (ℕₚ.≤-pred m<n))
... | tri≈ m≮n m≡n m≯n =
  tri≈ (m≯n  ℕ.s≤s  drop‿-≤-) (cong -[1+_] m≡n) (m≮n  ℕ.s≤s  drop‿-≤-)
... | tri> m≮n m≢n m>n =
  tri< (-≤- (ℕₚ.≤-pred m>n)) (m≢n  -[1+-injective) (m≮n  s≤s  drop‿-≤-)
{-# WARNING_ON_USAGE <′-cmp
"Warning: _<′_ was deprecated in v1.1.
Please use _<_ instead."
#-}
<′-isStrictPartialOrder : IsStrictPartialOrder _≡_ _<′_
<′-isStrictPartialOrder = record
  { isEquivalence = isEquivalence
  ; irrefl        = <′-irrefl
  ; trans         = λ {i}  <′-trans {i}
  ; <-resp-≈      =  {x}  subst (x <′_)) , subst (_<′ _)
  }
{-# WARNING_ON_USAGE <′-isStrictPartialOrder
"Warning: _<′_ was deprecated in v1.1.
Please use _<_ instead."
#-}
<′-strictPartialOrder : StrictPartialOrder _ _ _
<′-strictPartialOrder = record
  { isStrictPartialOrder = <′-isStrictPartialOrder
  }
{-# WARNING_ON_USAGE <′-strictPartialOrder
"Warning: _<′_ was deprecated in v1.1.
Please use _<_ instead."
#-}
<′-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<′_
<′-isStrictTotalOrder = record
  { isEquivalence = isEquivalence
  ; trans         = λ {i}  <′-trans {i}
  ; compare       = <′-cmp
  }
{-# WARNING_ON_USAGE <′-isStrictTotalOrder
"Warning: _<′_ was deprecated in v1.1.
Please use _<_ instead."
#-}
<′-strictTotalOrder : StrictTotalOrder _ _ _
<′-strictTotalOrder = record
  { isStrictTotalOrder = <′-isStrictTotalOrder
  }
{-# WARNING_ON_USAGE <′-strictTotalOrder
"Warning: _<′_ was deprecated in v1.1.
Please use _<_ instead."
#-}
n≮′n :  {n}  n ≮′ n
n≮′n {+ n}           (+≤+ n<n) =  contradiction n<n ℕₚ.1+n≰n
n≮′n { -[1+ suc n ]} (-≤- n<n) =  contradiction n<n ℕₚ.1+n≰n
{-# WARNING_ON_USAGE n≮′n
"Warning: n≮′n was deprecated in v1.1.
Please use n≮n instead."
#-}
>′⇒≰′ :  {x y}  x >′ y  x  y
>′⇒≰′ {y = y} x>y x≤y = contradiction (<′-≤-trans {i = y} x>y x≤y) n≮′n
{-# WARNING_ON_USAGE >′⇒≰′
"Warning: _<′_ was deprecated in v1.1.
Please use _<_ instead."
#-}
≰⇒>′ :  {x y}  x  y  x >′ y
≰⇒>′ {+ m}           {+ n}           m≰n =  +≤+ (ℕₚ.≰⇒> (m≰n  +≤+))
≰⇒>′ {+ m}           { -[1+ n ]}     _   =  -<′+ {n} {m}
≰⇒>′ { -[1+ m ]}     {+ _}           m≰n =  contradiction -≤+ m≰n
≰⇒>′ { -[1+ 0 ]}     { -[1+ 0 ]}     m≰n =  contradiction ≤-refl m≰n
≰⇒>′ { -[1+ suc _ ]} { -[1+ 0 ]}     m≰n =  contradiction (-≤- z≤n) m≰n
≰⇒>′ { -[1+ m ]}     { -[1+ suc n ]} m≰n with m ℕ.≤? n
... | yes m≤n  = -≤- m≤n
... | no  m≰n' = contradiction (-≤- (ℕₚ.≰⇒> m≰n')) m≰n
{-# WARNING_ON_USAGE ≰⇒>′
"Warning: _<′_ was deprecated in v1.1.
Please use _<_ instead."
#-}
<′-irrelevant : Irrelevant _<′_
<′-irrelevant = ≤-irrelevant
{-# WARNING_ON_USAGE <′-irrelevant
"Warning: _<′_ was deprecated in v1.1.
Please use _<_ instead."
#-}
+-monoˡ-<′ :  n  (_+ n) Preserves _<′_  _<′_
+-monoˡ-<′ n {i} {j} i<j
  rewrite sym (+-assoc (+ 1) i n)
          = +-monoˡ-≤ n i<j
{-# WARNING_ON_USAGE +-monoˡ-<′
"Warning: _<′_ was deprecated in v1.1.
Please use _<_ instead."
#-}
+-monoʳ-<′ :  n  (_+_ n) Preserves _<′_  _<′_
+-monoʳ-<′ n {i} {j} i<j
  rewrite +-comm n i
        | +-comm n j
        = +-monoˡ-<′ n {i} {j} i<j
{-# WARNING_ON_USAGE +-monoʳ-<′
"Warning: _<′_ was deprecated in v1.1.
Please use _<_ instead."
#-}
+-mono-<′ : _+_ Preserves₂ _<′_  _<′_  _<′_
+-mono-<′ {m} {n} {i} {j} m<n i<j = begin
  sucℤ (m + i) ≤⟨ suc-mono {m + i} (<′⇒≤ (+-monoˡ-<′ i {m} {n} m<n)) 
  sucℤ (n + i) ≤⟨ +-monoʳ-<′ n i<j 
  n + j        
  where open ≤-Reasoning
{-# WARNING_ON_USAGE +-mono-<′
"Warning: _<′_ was deprecated in v1.1.
Please use _<_ instead."
#-}
+-mono-≤-<′ : _+_ Preserves₂ _≤_  _<′_  _<′_
+-mono-≤-<′ {m} {n} {i} {j} m≤n i<j = ≤-<′-trans (+-monoˡ-≤ i m≤n) (+-monoʳ-<′ n i<j)
{-# WARNING_ON_USAGE +-mono-≤-<′
"Warning: _<′_ was deprecated in v1.1.
Please use _<_ instead."
#-}
+-mono-<′-≤ : _+_ Preserves₂ _<′_  _≤_  _<′_
+-mono-<′-≤ {m} {n} {i} {j} m<n i≤j =
  <′-≤-trans {m + i} {n + i} {n + j} (+-monoˡ-<′ i {m} {n} m<n) (+-monoʳ-≤ n i≤j)
{-# WARNING_ON_USAGE +-mono-<′-≤
"Warning: _<′_ was deprecated in v1.1.
Please use _<_ instead."
#-}
m≤pred[n]⇒m<′n :  {m n}  m  pred n  m <′ n
m≤pred[n]⇒m<′n {m} {n} m≤predn = begin
  sucℤ m               ≤⟨ +-monoʳ-≤ (+ 1) m≤predn 
  + 1 + pred n         ≡⟨ sym (+-assoc (+ 1) -[1+ 0 ] n) 
  (+ 1 + -[1+ 0 ]) + n ≡⟨ cong (_+ n) (+-inverseʳ (+ 1)) 
  + 0 + n              ≡⟨ +-identityˡ n 
  n                    
  where open ≤-Reasoning
{-# WARNING_ON_USAGE m≤pred[n]⇒m<′n
"Warning: _<′_ was deprecated in v1.1.
Please use _<_ instead."
#-}
m<′n⇒m≤pred[n] :  {m n}  m <′ n  m  pred n
m<′n⇒m≤pred[n] {m} {n} m<n = begin
  m             ≡⟨ sym (pred-suc m) 
  pred (sucℤ m) ≤⟨ pred-mono m<n 
  pred n        
  where open ≤-Reasoning
{-# WARNING_ON_USAGE m<′n⇒m≤pred[n]
"Warning: _<′_ was deprecated in v1.1.
Please use _<_ instead."
#-}

-- Version 1.2

[1+m]*n≡n+m*n = suc-*
{-# WARNING_ON_USAGE [1+m]*n≡n+m*n
"Warning: [1+m]*n≡n+m*n was deprecated in v1.1.
Please use suc-* instead."
#-}