------------------------------------------------------------------------
-- The Agda standard library
--
-- A bunch of properties about natural number operations
------------------------------------------------------------------------

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

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

module Data.Nat.Properties where

open import Axiom.UniquenessOfIdentityProofs
open import Algebra.Bundles
open import Algebra.Morphism
open import Algebra.Consequences.Propositional
open import Algebra.Construct.NaturalChoice.Base
import Algebra.Construct.NaturalChoice.MinMaxOp as MinMaxOp
import Algebra.Lattice.Construct.NaturalChoice.MinMaxOp as LatticeMinMaxOp
import Algebra.Properties.CommutativeSemigroup as CommSemigroupProperties
open import Data.Bool.Base using (Bool; false; true; T)
open import Data.Bool.Properties using (T?)
open import Data.Empty using ()
open import Data.Nat.Base
open import Data.Product using (; ; _×_; _,_)
open import Data.Sum.Base as Sum
open import Data.Unit using (tt)
open import Function.Base
open import Function.Bundles using (_↣_)
open import Function.Metric.Nat
open import Level using (0ℓ)
open import Relation.Unary as U using (Pred)
open import Relation.Binary
open import Relation.Binary.Consequences using (flip-Connex)
open import Relation.Binary.PropositionalEquality
open import Relation.Nullary hiding (Irrelevant)
open import Relation.Nullary.Decidable using (True; via-injection; map′)
open import Relation.Nullary.Negation using (contradiction; contradiction₂)
open import Relation.Nullary.Reflects using (fromEquivalence)

open import Algebra.Definitions {A = } _≡_
  hiding (LeftCancellative; RightCancellative; Cancellative)
open import Algebra.Definitions
  using (LeftCancellative; RightCancellative; Cancellative)
open import Algebra.Structures {A = } _≡_

------------------------------------------------------------------------
-- Properties of NonZero
------------------------------------------------------------------------

nonZero? : U.Decidable NonZero
nonZero? zero    = no NonZero.nonZero
nonZero? (suc n) = yes _

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

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

≡ᵇ⇒≡ :  m n  T (m ≡ᵇ n)  m  n
≡ᵇ⇒≡ zero    zero    _  = refl
≡ᵇ⇒≡ (suc m) (suc n) eq = cong suc (≡ᵇ⇒≡ m n eq)

≡⇒≡ᵇ :  m n  m  n  T (m ≡ᵇ n)
≡⇒≡ᵇ zero    zero    eq = _
≡⇒≡ᵇ (suc m) (suc n) eq = ≡⇒≡ᵇ m n (suc-injective eq)

-- NB: we use the builtin function `_≡ᵇ_` here so that the function
-- quickly decides whether to return `yes` or `no`. It still takes
-- a linear amount of time to generate the proof if it is inspected.
-- We expect the main benefit to be visible in compiled code as the
-- backend erases proofs.

infix 4 _≟_
_≟_ : DecidableEquality 
m  n = map′ (≡ᵇ⇒≡ m n) (≡⇒≡ᵇ m n) (T? (m ≡ᵇ n))

≡-irrelevant : Irrelevant {A = } _≡_
≡-irrelevant = Decidable⇒UIP.≡-irrelevant _≟_

≟-diag :  {m n} (eq : m  n)  (m  n)  yes eq
≟-diag = ≡-≟-identity _≟_

≡-isDecEquivalence : IsDecEquivalence (_≡_ {A = })
≡-isDecEquivalence = record
  { isEquivalence = isEquivalence
  ; _≟_           = _≟_
  }

≡-decSetoid : DecSetoid 0ℓ 0ℓ
≡-decSetoid = record
  { Carrier          = 
  ; _≈_              = _≡_
  ; isDecEquivalence = ≡-isDecEquivalence
  }

0≢1+n :  {n}  0  suc n
0≢1+n ()

1+n≢0 :  {n}  suc n  0
1+n≢0 ()

1+n≢n :  {n}  suc n  n
1+n≢n {suc n} = 1+n≢n  suc-injective

------------------------------------------------------------------------
-- Properties of _<ᵇ_
------------------------------------------------------------------------

<ᵇ⇒< :  m n  T (m <ᵇ n)  m < n
<ᵇ⇒< zero    (suc n) m<n = z<s
<ᵇ⇒< (suc m) (suc n) m<n = s<s (<ᵇ⇒< m n m<n)

<⇒<ᵇ :  {m n}  m < n  T (m <ᵇ n)
<⇒<ᵇ z<s               = tt
<⇒<ᵇ (s<s m<n@(s≤s _)) = <⇒<ᵇ m<n

<ᵇ-reflects-< :  m n  Reflects (m < n) (m <ᵇ n)
<ᵇ-reflects-< m n = fromEquivalence (<ᵇ⇒< m n) <⇒<ᵇ

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

≤ᵇ⇒≤ :  m n  T (m ≤ᵇ n)  m  n
≤ᵇ⇒≤ zero    n m≤n = z≤n
≤ᵇ⇒≤ (suc m) n m≤n = <ᵇ⇒< m n m≤n

≤⇒≤ᵇ :  {m n}  m  n  T (m ≤ᵇ n)
≤⇒≤ᵇ z≤n         = tt
≤⇒≤ᵇ m≤n@(s≤s _) = <⇒<ᵇ m≤n

≤ᵇ-reflects-≤ :  m n  Reflects (m  n) (m ≤ᵇ n)
≤ᵇ-reflects-≤ m n = fromEquivalence (≤ᵇ⇒≤ m n) ≤⇒≤ᵇ

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

------------------------------------------------------------------------
-- Relational properties of _≤_

≤-reflexive : _≡_  _≤_
≤-reflexive {zero}  refl = z≤n
≤-reflexive {suc m} refl = s≤s (≤-reflexive refl)

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

≤-antisym : Antisymmetric _≡_ _≤_
≤-antisym z≤n       z≤n       = refl
≤-antisym (s≤s m≤n) (s≤s n≤m) = cong suc (≤-antisym m≤n n≤m)

≤-trans : Transitive _≤_
≤-trans z≤n       _         = z≤n
≤-trans (s≤s m≤n) (s≤s n≤o) = s≤s (≤-trans m≤n n≤o)

≤-total : Total _≤_
≤-total zero    _       = inj₁ z≤n
≤-total _       zero    = inj₂ z≤n
≤-total (suc m) (suc n) with ≤-total m n
... | inj₁ m≤n = inj₁ (s≤s m≤n)
... | inj₂ n≤m = inj₂ (s≤s n≤m)

≤-irrelevant : Irrelevant _≤_
≤-irrelevant z≤n        z≤n        = refl
≤-irrelevant (s≤s m≤n₁) (s≤s m≤n₂) = cong s≤s (≤-irrelevant m≤n₁ m≤n₂)

-- NB: we use the builtin function `_<ᵇ_` here so that the function
-- quickly decides whether to return `yes` or `no`. It still takes
-- a linear amount of time to generate the proof if it is inspected.
-- We expect the main benefit to be visible in compiled code as the
-- backend erases proofs.

infix 4 _≤?_ _≥?_

_≤?_ : Decidable _≤_
m ≤? n = map′ (≤ᵇ⇒≤ m n) ≤⇒≤ᵇ (T? (m ≤ᵇ n))

_≥?_ : Decidable _≥_
_≥?_ = flip _≤?_

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

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

≤-isTotalPreorder : IsTotalPreorder _≡_ _≤_
≤-isTotalPreorder = record
  { isPreorder = ≤-isPreorder
  ; total      = ≤-total
  }

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

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

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

------------------------------------------------------------------------
-- Bundles

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

≤-totalPreorder : TotalPreorder 0ℓ 0ℓ 0ℓ
≤-totalPreorder = record
  { isTotalPreorder = ≤-isTotalPreorder
  }

≤-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
  }

------------------------------------------------------------------------
-- Other properties of _≤_

s≤s-injective :  {m n} {p q : m  n}  s≤s p  s≤s q  p  q
s≤s-injective refl = refl

≤-pred :  {m n}  suc m  suc n  m  n
≤-pred (s≤s m≤n) = m≤n

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

n≤0⇒n≡0 :  {n}  n  0  n  0
n≤0⇒n≡0 z≤n = refl

n≤1⇒n≡0∨n≡1 :  {n : }  n  1  n  0  n  1
n≤1⇒n≡0∨n≡1 z≤n       = inj₁ refl
n≤1⇒n≡0∨n≡1 (s≤s z≤n) = inj₂ refl

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

-- Relationships between the various relations

<⇒≤ : _<_  _≤_
<⇒≤ z<s               = z≤n
<⇒≤ (s<s m<n@(s≤s _)) = s≤s (<⇒≤ m<n)

<⇒≢ : _<_  _≢_
<⇒≢ m<n refl = 1+n≰n m<n

>⇒≢ : _>_  _≢_
>⇒≢ = ≢-sym  <⇒≢

≤⇒≯ : _≤_  _≯_
≤⇒≯ (s≤s m≤n) (s≤s n≤m) = ≤⇒≯ m≤n n≤m

<⇒≱ : _<_  _≱_
<⇒≱ (s≤s m+1≤n) (s≤s n≤m) = <⇒≱ m+1≤n n≤m

<⇒≯ : _<_  _≯_
<⇒≯ (s≤s m<n) (s≤s n<m) = <⇒≯ m<n n<m

≰⇒≮ : _≰_  _≮_
≰⇒≮ m≰n 1+m≤n = m≰n (<⇒≤ 1+m≤n)

≰⇒> : _≰_  _>_
≰⇒> {zero}          z≰n = contradiction z≤n z≰n
≰⇒> {suc m} {zero}  _   = z<s
≰⇒> {suc m} {suc n} m≰n = s<s (≰⇒> (m≰n  s≤s))

≰⇒≥ : _≰_  _≥_
≰⇒≥ = <⇒≤  ≰⇒>

≮⇒≥ : _≮_  _≥_
≮⇒≥ {_}     {zero}  _       = z≤n
≮⇒≥ {zero}  {suc j} 1≮j+1   = contradiction z<s 1≮j+1
≮⇒≥ {suc i} {suc j} i+1≮j+1 = s≤s (≮⇒≥ (i+1≮j+1  s<s))

≤∧≢⇒< :  {m n}  m  n  m  n  m < n
≤∧≢⇒< {_} {zero}  z≤n       m≢n     = contradiction refl m≢n
≤∧≢⇒< {_} {suc n} z≤n       m≢n     = z<s
≤∧≢⇒< {_} {suc n} (s≤s m≤n) 1+m≢1+n =
  s<s (≤∧≢⇒< m≤n (1+m≢1+n  cong suc))

≤∧≮⇒≡ :  {m n}  m  n  m  n  m  n
≤∧≮⇒≡ m≤n m≮n = ≤-antisym m≤n (≮⇒≥ m≮n)

≤-<-connex : Connex _≤_ _<_
≤-<-connex m n with m ≤? n
... | yes m≤n = inj₁ m≤n
... | no ¬m≤n = inj₂ (≰⇒> ¬m≤n)

≥->-connex : Connex _≥_ _>_
≥->-connex = flip ≤-<-connex

<-≤-connex : Connex _<_ _≤_
<-≤-connex = flip-Connex ≤-<-connex

>-≥-connex : Connex _>_ _≥_
>-≥-connex = flip-Connex ≥->-connex

------------------------------------------------------------------------
-- Relational properties of _<_

<-irrefl : Irreflexive _≡_ _<_
<-irrefl refl (s<s n<n) = <-irrefl refl n<n

<-asym : Asymmetric _<_
<-asym (s<s n<m) (s<s m<n) = <-asym n<m m<n

<-trans : Transitive _<_
<-trans (s≤s i≤j) (s≤s j<k) = s≤s (≤-trans i≤j (≤-trans (n≤1+n _) j<k))

<-transʳ : Trans _≤_ _<_ _<_
<-transʳ m≤n (s≤s n≤o) = s≤s (≤-trans m≤n n≤o)

<-transˡ : Trans _<_ _≤_ _<_
<-transˡ (s≤s m≤n) (s≤s n≤o) = s≤s (≤-trans m≤n n≤o)

-- NB: we use the builtin function `_<ᵇ_` here so that the function
-- quickly decides which constructor to return. It still takes a
-- linear amount of time to generate the proof if it is inspected.
-- We expect the main benefit to be visible in compiled code as the
-- backend erases proofs.

<-cmp : Trichotomous _≡_ _<_
<-cmp m n with m  n | T? (m <ᵇ n)
... | yes m≡n | _       = tri≈ (<-irrefl m≡n) m≡n (<-irrefl (sym m≡n))
... | no  m≢n | yes m<n = tri< (<ᵇ⇒< m n m<n) m≢n (<⇒≯ (<ᵇ⇒< m n m<n))
... | no  m≢n | no  m≮n = tri> (m≮n  <⇒<ᵇ)   m≢n (≤∧≢⇒< (≮⇒≥ (m≮n  <⇒<ᵇ)) (m≢n  sym))

infix 4 _<?_ _>?_

_<?_ : Decidable _<_
m <? n = suc m ≤? n

_>?_ : Decidable _>_
_>?_ = flip _<?_

<-irrelevant : Irrelevant _<_
<-irrelevant = ≤-irrelevant

<-resp₂-≡ : _<_ Respects₂ _≡_
<-resp₂-≡ = subst (_ <_) , subst (_< _)

------------------------------------------------------------------------
-- Bundles

<-isStrictPartialOrder : IsStrictPartialOrder _≡_ _<_
<-isStrictPartialOrder = record
  { isEquivalence = isEquivalence
  ; irrefl        = <-irrefl
  ; trans         = <-trans
  ; <-resp-≈      = <-resp₂-≡
  }

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

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

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

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

s<s-injective :  {m n} {p q : m < n}  s<s p  s<s q  p  q
s<s-injective refl = refl

<-pred :  {m n}  suc m < suc n  m < n
<-pred (s<s m<n) = m<n

<-step :  {m n}  m < n  m < 1 + n
<-step z<s               = z<s
<-step (s<s m<n@(s≤s _)) = s<s (<-step m<n)

n≮0 :  {n}  n  0
n≮0 ()

n≮n :  n  n  n
n≮n n = <-irrefl (refl {x = n})

0<1+n :  {n}  0 < suc n
0<1+n = z<s

n<1+n :  n  n < suc n
n<1+n n = ≤-refl

n<1⇒n≡0 :  {n}  n < 1  n  0
n<1⇒n≡0 (s≤s n≤0) = n≤0⇒n≡0 n≤0

n>0⇒n≢0 :  {n}  n > 0  n  0
n>0⇒n≢0 {suc n} _ ()

n≢0⇒n>0 :  {n}  n  0  n > 0
n≢0⇒n>0 {zero}  0≢0 =  contradiction refl 0≢0
n≢0⇒n>0 {suc n} _   =  0<1+n

m<n⇒0<n :  {m n}  m < n  0 < n
m<n⇒0<n = ≤-trans 0<1+n

m<n⇒n≢0 :  {m n}  m < n  n  0
m<n⇒n≢0 (s≤s m≤n) ()

m<n⇒m≤1+n :  {m n}  m < n  m  suc n
m<n⇒m≤1+n = ≤-step  <⇒≤

m<1+n⇒m<n∨m≡n :   {m n}  m < suc n  m < n  m  n
m<1+n⇒m<n∨m≡n {0}     {0}     _          =  inj₂ refl
m<1+n⇒m<n∨m≡n {0}     {suc n} _          =  inj₁ 0<1+n
m<1+n⇒m<n∨m≡n {suc m} {suc n} (s<s m<1+n)  with m<1+n⇒m<n∨m≡n m<1+n
... | inj₂ m≡n = inj₂ (cong suc m≡n)
... | inj₁ m<n = inj₁ (s<s m<n)

m≤n⇒m<n∨m≡n :   {m n}  m  n  m < n  m  n
m≤n⇒m<n∨m≡n m≤n = m<1+n⇒m<n∨m≡n (s≤s m≤n)

m<1+n⇒m≤n :  {m n}  m < suc n  m  n
m<1+n⇒m≤n (s≤s m≤n) = m≤n

∀[m≤n⇒m≢o]⇒n<o :  n o  (∀ {m}  m  n  m  o)  n < o
∀[m≤n⇒m≢o]⇒n<o _       zero    m≤n⇒n≢0 = contradiction refl (m≤n⇒n≢0 z≤n)
∀[m≤n⇒m≢o]⇒n<o zero    (suc o) _       = 0<1+n
∀[m≤n⇒m≢o]⇒n<o (suc n) (suc o) m≤n⇒n≢o = s≤s (∀[m≤n⇒m≢o]⇒n<o n o rec)
  where
  rec :  {m}  m  n  m  o
  rec m≤n refl = m≤n⇒n≢o (s≤s m≤n) refl

∀[m<n⇒m≢o]⇒n≤o :  n o  (∀ {m}  m < n  m  o)  n  o
∀[m<n⇒m≢o]⇒n≤o zero    n       _       = z≤n
∀[m<n⇒m≢o]⇒n≤o (suc n) zero    m<n⇒m≢0 = contradiction refl (m<n⇒m≢0 0<1+n)
∀[m<n⇒m≢o]⇒n≤o (suc n) (suc o) m<n⇒m≢o = s≤s (∀[m<n⇒m≢o]⇒n≤o n o rec)
  where
  rec :  {m}  m < n  m  o
  rec x<m refl = m<n⇒m≢o (s≤s x<m) refl

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

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

open ≤-Reasoning

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

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

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

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

+-identityˡ : LeftIdentity 0 _+_
+-identityˡ _ = refl

+-identityʳ : RightIdentity 0 _+_
+-identityʳ zero    = refl
+-identityʳ (suc n) = cong suc (+-identityʳ n)

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

+-comm : Commutative _+_
+-comm zero    n = sym (+-identityʳ n)
+-comm (suc m) n = begin-equality
  suc m + n   ≡⟨⟩
  suc (m + n) ≡⟨ cong suc (+-comm m n) 
  suc (n + m) ≡⟨ sym (+-suc n m) 
  n + suc m   

+-cancelˡ-≡ : LeftCancellative _≡_ _+_
+-cancelˡ-≡ zero    _ _ eq = eq
+-cancelˡ-≡ (suc m) _ _ eq = +-cancelˡ-≡ m _ _ (cong pred eq)

+-cancelʳ-≡ : RightCancellative _≡_ _+_
+-cancelʳ-≡ = comm+cancelˡ⇒cancelʳ +-comm +-cancelˡ-≡

+-cancel-≡ : Cancellative _≡_ _+_
+-cancel-≡ = +-cancelˡ-≡ , +-cancelʳ-≡

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

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

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

+-isCommutativeSemigroup : IsCommutativeSemigroup _+_
+-isCommutativeSemigroup = record
  { isSemigroup = +-isSemigroup
  ; comm        = +-comm
  }

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

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

------------------------------------------------------------------------
-- Raw bundles

+-rawMagma : RawMagma 0ℓ 0ℓ
+-rawMagma = record
  { _≈_ = _≡_
  ; _∙_ = _+_
  }

+-0-rawMonoid : RawMonoid 0ℓ 0ℓ
+-0-rawMonoid = record
  { _≈_ = _≡_
  ; _∙_ = _+_
  ; ε   = 0
  }

------------------------------------------------------------------------
-- Bundles

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

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

+-commutativeSemigroup : CommutativeSemigroup 0ℓ 0ℓ
+-commutativeSemigroup = record
  { isCommutativeSemigroup = +-isCommutativeSemigroup
  }

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

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

∸-magma : Magma 0ℓ 0ℓ
∸-magma = magma _∸_


------------------------------------------------------------------------
-- Other properties of _+_ and _≡_

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

m≢1+n+m :  m {n}  m  suc (n + m)
m≢1+n+m m m≡1+n+m = m≢1+m+n m (trans m≡1+n+m (cong suc (+-comm _ m)))

m+1+n≢m :  m {n}  m + suc n  m
m+1+n≢m (suc m) = (m+1+n≢m m)  suc-injective

m+1+n≢n :  m {n}  m + suc n  n
m+1+n≢n m {n} rewrite +-suc m n = ≢-sym (m≢1+n+m n)

m+1+n≢0 :  m {n}  m + suc n  0
m+1+n≢0 m {n} rewrite +-suc m n = λ()

m+n≡0⇒m≡0 :  m {n}  m + n  0  m  0
m+n≡0⇒m≡0 zero eq = refl

m+n≡0⇒n≡0 :  m {n}  m + n  0  n  0
m+n≡0⇒n≡0 m {n} m+n≡0 = m+n≡0⇒m≡0 n (trans (+-comm n m) (m+n≡0))

------------------------------------------------------------------------
-- Properties of _+_ and _≤_/_<_

+-cancelˡ-≤ : LeftCancellative _≤_ _+_
+-cancelˡ-≤ zero    _ _ le       = le
+-cancelˡ-≤ (suc m) _ _ (s≤s le) = +-cancelˡ-≤ m _ _ le

+-cancelʳ-≤ : RightCancellative _≤_ _+_
+-cancelʳ-≤ m n o le =
  +-cancelˡ-≤ m _ _ (subst₂ _≤_ (+-comm n m) (+-comm o m) le)

+-cancel-≤ : Cancellative _≤_ _+_
+-cancel-≤ = +-cancelˡ-≤ , +-cancelʳ-≤

+-cancelˡ-< : LeftCancellative _<_ _+_
+-cancelˡ-< m n o = +-cancelˡ-≤ m (suc n) o  subst (_≤ m + o) (sym (+-suc m n))

+-cancelʳ-< : RightCancellative _<_ _+_
+-cancelʳ-< m n o n+m<o+m = +-cancelʳ-≤ m (suc n) o n+m<o+m

+-cancel-< : Cancellative _<_ _+_
+-cancel-< = +-cancelˡ-< , +-cancelʳ-<

≤-stepsˡ :  {m n} o  m  n  m  o + n
≤-stepsˡ zero    m≤n = m≤n
≤-stepsˡ (suc o) m≤n = ≤-step (≤-stepsˡ o m≤n)

≤-stepsʳ :  {m n} o  m  n  m  n + o
≤-stepsʳ {m} o m≤n = subst (m ≤_) (+-comm o _) (≤-stepsˡ o 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≤n+m :  m n  m  n + m
m≤n+m m n = subst (m ≤_) (+-comm m n) (m≤m+n m n)

m+n≤o⇒m≤o :  m {n o}  m + n  o  m  o
m+n≤o⇒m≤o zero    m+n≤o       = z≤n
m+n≤o⇒m≤o (suc m) (s≤s m+n≤o) = s≤s (m+n≤o⇒m≤o m m+n≤o)

m+n≤o⇒n≤o :  m {n o}  m + n  o  n  o
m+n≤o⇒n≤o zero    n≤o   = n≤o
m+n≤o⇒n≤o (suc m) m+n<o = m+n≤o⇒n≤o m (<⇒≤ m+n<o)

+-mono-≤ : _+_ Preserves₂ _≤_  _≤_  _≤_
+-mono-≤ {_} {m} z≤n       o≤p = ≤-trans o≤p (m≤n+m _ m)
+-mono-≤ {_} {_} (s≤s m≤n) o≤p = s≤s (+-mono-≤ m≤n o≤p)

+-monoˡ-≤ :  n  (_+ n) Preserves _≤_  _≤_
+-monoˡ-≤ n m≤o = +-mono-≤ m≤o (≤-refl {n})

+-monoʳ-≤ :  n  (n +_) Preserves _≤_  _≤_
+-monoʳ-≤ n m≤o = +-mono-≤ (≤-refl {n}) m≤o

+-mono-<-≤ : _+_ Preserves₂ _<_  _≤_  _<_
+-mono-<-≤ {_} {suc n} z<s               o≤p = s≤s (≤-stepsˡ n o≤p)
+-mono-<-≤ {_} {_}     (s<s m<n@(s≤s _)) o≤p = s≤s (+-mono-<-≤ m<n o≤p)

+-mono-≤-< : _+_ Preserves₂ _≤_  _<_  _<_
+-mono-≤-< {_} {n} z≤n       o<p = ≤-trans o<p (m≤n+m _ n)
+-mono-≤-< {_} {_} (s≤s m≤n) o<p = s≤s (+-mono-≤-< m≤n o<p)

+-mono-< : _+_ Preserves₂ _<_  _<_  _<_
+-mono-< m≤n = +-mono-≤-< (<⇒≤ m≤n)

+-monoˡ-< :  n  (_+ n) Preserves _<_  _<_
+-monoˡ-< n = +-monoˡ-≤ n

+-monoʳ-< :  n  (n +_) Preserves _<_  _<_
+-monoʳ-< zero    m≤o = m≤o
+-monoʳ-< (suc n) m≤o = s≤s (+-monoʳ-< n m≤o)

m+1+n≰m :  m {n}  m + suc n  m
m+1+n≰m (suc m) (s≤s m+1+n≤m) = m+1+n≰m m m+1+n≤m

m<m+n :  m {n}  n > 0  m < m + n
m<m+n zero    n>0 = n>0
m<m+n (suc m) n>0 = s<s (m<m+n m n>0)

m<n+m :  m {n}  n > 0  m < n + m
m<n+m m {n} n>0 rewrite +-comm n m = m<m+n m n>0

m+n≮n :  m n  m + n  n
m+n≮n zero    n                   = n≮n n
m+n≮n (suc m) (suc n) (s<s m+n<n) = m+n≮n m (suc n) (<-step m+n<n)

m+n≮m :  m n  m + n  m
m+n≮m m n = subst (_≮ m) (+-comm n m) (m+n≮n n m)

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

*-suc :  m n  m * suc n  m + m * n
*-suc zero    n = refl
*-suc (suc m) n = begin-equality
  suc m * suc n         ≡⟨⟩
  suc n + m * suc n     ≡⟨ cong (suc n +_) (*-suc m n) 
  suc n + (m + m * n)   ≡⟨⟩
  suc (n + (m + m * n)) ≡⟨ cong suc (sym (+-assoc n m (m * n))) 
  suc (n + m + m * n)   ≡⟨ cong  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)) ≡⟨⟩
  suc m + suc m * n     

------------------------------------------------------------------------
-- Algebraic properties of _*_

*-identityˡ : LeftIdentity 1 _*_
*-identityˡ n = +-identityʳ n

*-identityʳ : RightIdentity 1 _*_
*-identityʳ zero    = refl
*-identityʳ (suc n) = cong suc (*-identityʳ n)

*-identity : Identity 1 _*_
*-identity = *-identityˡ , *-identityʳ

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

*-zeroʳ : RightZero 0 _*_
*-zeroʳ zero    = refl
*-zeroʳ (suc n) = *-zeroʳ n

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

*-comm : Commutative _*_
*-comm zero    n = sym (*-zeroʳ n)
*-comm (suc m) n = begin-equality
  suc m * n  ≡⟨⟩
  n + m * n  ≡⟨ cong (n +_) (*-comm m n) 
  n + n * m  ≡⟨ sym (*-suc n m) 
  n * suc m  

*-distribʳ-+ : _*_ DistributesOverʳ _+_
*-distribʳ-+ m zero    o = refl
*-distribʳ-+ m (suc n) o = begin-equality
  (suc n + o) * m     ≡⟨⟩
  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   ≡⟨⟩
  suc n * m + o * m   

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

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

*-assoc : Associative _*_
*-assoc zero    n o = refl
*-assoc (suc m) n o = begin-equality
  (suc m * n) * o     ≡⟨⟩
  (n + m * n) * o     ≡⟨ *-distribʳ-+ o n (m * n) 
  n * o + (m * n) * o ≡⟨ cong (n * o +_) (*-assoc m n o) 
  n * o + m * (n * o) ≡⟨⟩
  suc m * (n * o)     

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

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

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

*-isCommutativeSemigroup : IsCommutativeSemigroup _*_
*-isCommutativeSemigroup = record
  { isSemigroup = *-isSemigroup
  ; comm        = *-comm
  }

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

*-1-isCommutativeMonoid : IsCommutativeMonoid _*_ 1
*-1-isCommutativeMonoid = record
  { isMonoid = *-1-isMonoid
  ; comm     = *-comm
  }

+-*-isSemiring : IsSemiring _+_ _*_ 0 1
+-*-isSemiring = record
  { isSemiringWithoutAnnihilatingZero = record
    { +-isCommutativeMonoid = +-0-isCommutativeMonoid
    ; *-cong                = cong₂ _*_
    ; *-assoc               = *-assoc
    ; *-identity            = *-identity
    ; distrib               = *-distrib-+
    }
  ; zero = *-zero
  }

+-*-isCommutativeSemiring : IsCommutativeSemiring _+_ _*_ 0 1
+-*-isCommutativeSemiring = record
  { isSemiring = +-*-isSemiring
  ; *-comm     = *-comm
  }

------------------------------------------------------------------------
-- Bundles

*-rawMagma : RawMagma 0ℓ 0ℓ
*-rawMagma = record
  { _≈_ = _≡_
  ; _∙_ = _*_
  }

*-1-rawMonoid : RawMonoid 0ℓ 0ℓ
*-1-rawMonoid = record
  { _≈_ = _≡_
  ; _∙_ = _*_
  ; ε   = 1
  }

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

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

*-commutativeSemigroup : CommutativeSemigroup 0ℓ 0ℓ
*-commutativeSemigroup = record
  { isCommutativeSemigroup = *-isCommutativeSemigroup
  }

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

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

+-*-semiring : Semiring 0ℓ 0ℓ
+-*-semiring = record
  { isSemiring = +-*-isSemiring
  }

+-*-commutativeSemiring : CommutativeSemiring 0ℓ 0ℓ
+-*-commutativeSemiring = record
  { isCommutativeSemiring = +-*-isCommutativeSemiring
  }

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

*-cancelʳ-≡ :  m n o .{{_ : NonZero o}}  m * o  n * o  m  n
*-cancelʳ-≡ zero    zero    (suc o) eq = refl
*-cancelʳ-≡ (suc m) (suc n) (suc o) eq =
  cong suc (*-cancelʳ-≡ m n (suc o) (+-cancelˡ-≡ (suc o) (m * suc o) (n * suc o) eq))

*-cancelˡ-≡ :  m n o .{{_ : NonZero o}}  o * m  o * n  m  n
*-cancelˡ-≡ m n o rewrite *-comm o m | *-comm o n = *-cancelʳ-≡ m n o

m*n≡0⇒m≡0∨n≡0 :  m {n}  m * n  0  m  0  n  0
m*n≡0⇒m≡0∨n≡0 zero    {n}     eq = inj₁ refl
m*n≡0⇒m≡0∨n≡0 (suc m) {zero}  eq = inj₂ refl

m*n≢0 :  m n  .{{_ : NonZero m}} .{{_ : NonZero n}}  NonZero (m * n)
m*n≢0 (suc m) (suc n) = _

m*n≡0⇒m≡0 :  m n .{{_ : NonZero n}}  m * n  0  m  0
m*n≡0⇒m≡0 zero (suc _) eq = refl

m*n≡1⇒m≡1 :  m n  m * n  1  m  1
m*n≡1⇒m≡1 (suc zero)    n          _  = refl
m*n≡1⇒m≡1 (suc (suc m)) (suc zero) ()
m*n≡1⇒m≡1 (suc (suc m)) zero       eq =
  contradiction (trans (sym $ *-zeroʳ m) eq) λ()

m*n≡1⇒n≡1 :  m n  m * n  1  n  1
m*n≡1⇒n≡1 m n eq = m*n≡1⇒m≡1 n m (trans (*-comm n m) eq)

[m*n]*[o*p]≡[m*o]*[n*p] :  m n o p  (m * n) * (o * p)  (m * o) * (n * p)
[m*n]*[o*p]≡[m*o]*[n*p] m n o p = begin-equality
  (m * n) * (o * p) ≡⟨  *-assoc m n (o * p) 
  m * (n * (o * p)) ≡⟨  cong (m *_) (x∙yz≈y∙xz n o p) 
  m * (o * (n * p)) ≡˘⟨ *-assoc m o (n * p) 
  (m * o) * (n * p) 
  where open CommSemigroupProperties *-commutativeSemigroup

------------------------------------------------------------------------
-- Other properties of _*_ and _≤_/_<_

*-cancelʳ-≤ :  m n o .{{_ : NonZero o}}  m * o  n * o  m  n
*-cancelʳ-≤ zero    _       (suc o) _  = z≤n
*-cancelʳ-≤ (suc m) (suc n) (suc o) le =
  s≤s (*-cancelʳ-≤ m n (suc o) (+-cancelˡ-≤ _ _ _ le))

*-cancelˡ-≤ :  {m n} o .{{_ : NonZero o}}  o * m  o * n  m  n
*-cancelˡ-≤ {m} {n} o rewrite *-comm o m | *-comm o n = *-cancelʳ-≤ m n o

*-mono-≤ : _*_ Preserves₂ _≤_  _≤_  _≤_
*-mono-≤ z≤n       _   = z≤n
*-mono-≤ (s≤s m≤n) u≤v = +-mono-≤ u≤v (*-mono-≤ m≤n u≤v)

*-monoˡ-≤ :  n  (_* n) Preserves _≤_  _≤_
*-monoˡ-≤ n m≤o = *-mono-≤ m≤o (≤-refl {n})

*-monoʳ-≤ :  n  (n *_) Preserves _≤_  _≤_
*-monoʳ-≤ n m≤o = *-mono-≤ (≤-refl {n}) m≤o

*-mono-< : _*_ Preserves₂ _<_  _<_  _<_
*-mono-< z<s               u<v@(s≤s _) = 0<1+n
*-mono-< (s<s m<n@(s≤s _)) u<v@(s≤s _) = +-mono-< u<v (*-mono-< m<n u<v)

*-monoˡ-< :  n .{{_ : NonZero n}}  (_* n) Preserves _<_  _<_
*-monoˡ-< (suc n) z<s       = 0<1+n
*-monoˡ-< (suc n) (s<s m<o@(s≤s _)) =
  +-mono-≤-< (≤-refl {suc n}) (*-monoˡ-< (suc n) m<o)

*-monoʳ-< :  n .{{_ : NonZero n}}  (n *_) Preserves _<_  _<_
*-monoʳ-< (suc zero)    m<o@(s≤s _) = +-mono-≤ m<o z≤n
*-monoʳ-< (suc (suc n)) m<o@(s≤s _) =
  +-mono-≤ m<o (<⇒≤ (*-monoʳ-< (suc n) m<o))

m≤m*n :  m n .{{_ : NonZero n}}  m  m * n
m≤m*n m n@(suc _) = begin
  m     ≡⟨ sym (*-identityʳ m) 
  m * 1 ≤⟨ *-monoʳ-≤ m 0<1+n 
  m * n 

m≤n*m :  m n .{{_ : NonZero n}}  m  n * m
m≤n*m m n@(suc _) = begin
  m     ≤⟨ m≤m*n m n 
  m * n ≡⟨ *-comm m n 
  n * m 

m<m*n :  m n .{{_ : NonZero m}}  1 < n  m < m * n
m<m*n m@(suc m-1) n@(suc (suc n-2)) (s≤s (s≤s _)) = begin-strict
  m           <⟨ s≤s (s≤s (m≤n+m m-1 n-2)) 
  n + m-1     ≤⟨ +-monoʳ-≤ n (m≤m*n m-1 n) 
  n + m-1 * n ≡⟨⟩
  m * n       

m<n⇒m<n*o :  {m n} o .{{_ : NonZero o}}  m < n  m < n * o
m<n⇒m<n*o {m} {n} o m<n = <-transˡ m<n (m≤m*n n o)

m<n⇒m<o*n :  {m n} o .{{_ : NonZero o}}  m < n  m < o * n
m<n⇒m<o*n {m} {n} o m<n = begin-strict
  m     <⟨ m<n⇒m<n*o o m<n 
  n * o ≡⟨ *-comm n o 
  o * n 

*-cancelʳ-< : RightCancellative _<_ _*_
*-cancelʳ-< zero    zero    (suc o) _     = 0<1+n
*-cancelʳ-< (suc m) zero    (suc o) _     = 0<1+n
*-cancelʳ-< m       (suc n) (suc o) nm<om =
  s≤s (*-cancelʳ-< m n o (+-cancelˡ-< m _ _ nm<om))

*-cancelˡ-< : LeftCancellative _<_ _*_
*-cancelˡ-< x y z rewrite *-comm x y | *-comm x z = *-cancelʳ-< x y z

*-cancel-< : Cancellative _<_ _*_
*-cancel-< = *-cancelˡ-< , *-cancelʳ-<

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

^-identityʳ : RightIdentity 1 _^_
^-identityʳ zero    = refl
^-identityʳ (suc n) = cong suc (^-identityʳ n)

^-zeroˡ : LeftZero 1 _^_
^-zeroˡ zero    = refl
^-zeroˡ (suc n) = begin-equality
  1 ^ suc n   ≡⟨⟩
  1 * (1 ^ n) ≡⟨ *-identityˡ (1 ^ n) 
  1 ^ n       ≡⟨ ^-zeroˡ n 
  1           

^-distribˡ-+-* :  m n o  m ^ (n + o)  m ^ n * m ^ o
^-distribˡ-+-* m zero    o = sym (+-identityʳ (m ^ o))
^-distribˡ-+-* m (suc n) o = begin-equality
  m * (m ^ (n + o))       ≡⟨ cong (m *_) (^-distribˡ-+-* m n o) 
  m * ((m ^ n) * (m ^ o)) ≡⟨ sym (*-assoc m _ _) 
  (m * (m ^ n)) * (m ^ o) 

^-semigroup-morphism :  {n}  (n ^_) Is +-semigroup -Semigroup⟶ *-semigroup
^-semigroup-morphism = record
  { ⟦⟧-cong = cong (_ ^_)
  ; ∙-homo  = ^-distribˡ-+-* _
  }

^-monoid-morphism :  {n}  (n ^_) Is +-0-monoid -Monoid⟶ *-1-monoid
^-monoid-morphism = record
  { sm-homo = ^-semigroup-morphism
  ; ε-homo  = refl
  }

^-*-assoc :  m n o  (m ^ n) ^ o  m ^ (n * o)
^-*-assoc m n zero    = cong (m ^_) (sym $ *-zeroʳ n)
^-*-assoc m n (suc o) = begin-equality
  (m ^ n) * ((m ^ n) ^ o) ≡⟨ cong ((m ^ n) *_) (^-*-assoc m n o) 
  (m ^ n) * (m ^ (n * o)) ≡⟨ sym (^-distribˡ-+-* m n (n * o)) 
  m ^ (n + n * o)         ≡⟨ cong (m ^_) (sym (*-suc n o)) 
  m ^ (n * (suc o)) 

m^n≡0⇒m≡0 :  m n  m ^ n  0  m  0
m^n≡0⇒m≡0 m (suc n) eq = [ id , m^n≡0⇒m≡0 m n ]′ (m*n≡0⇒m≡0∨n≡0 m eq)

m^n≡1⇒n≡0∨m≡1 :  m n  m ^ n  1  n  0  m  1
m^n≡1⇒n≡0∨m≡1 m zero    _  = inj₁ refl
m^n≡1⇒n≡0∨m≡1 m (suc n) eq = inj₂ (m*n≡1⇒m≡1 m (m ^ n) eq)

m^n≢0 :  m n .{{_ : NonZero m}}  NonZero (m ^ n)
m^n≢0 m n = ≢-nonZero (≢-nonZero⁻¹ m ∘′ m^n≡0⇒m≡0 m n)

2^n>0 :  (n : )  2 ^ n > 0
2^n>0 zero = s≤s z≤n
2^n>0 (suc n) = ≤-trans (2^n>0 n) (m≤m+n (2 ^ n) ((2 ^ n) + zero))

------------------------------------------------------------------------
-- Properties of _⊓_ and _⊔_
------------------------------------------------------------------------
-- Basic specification in terms of _≤_

m≤n⇒m⊔n≡n :  {m n}  m  n  m  n  n
m≤n⇒m⊔n≡n {zero}  _         = refl
m≤n⇒m⊔n≡n {suc m} (s≤s m≤n) = cong suc (m≤n⇒m⊔n≡n m≤n)

m≥n⇒m⊔n≡m :  {m n}  m  n  m  n  m
m≥n⇒m⊔n≡m {zero}  {zero}  z≤n       = refl
m≥n⇒m⊔n≡m {suc m} {zero}  z≤n       = refl
m≥n⇒m⊔n≡m {suc m} {suc n} (s≤s m≥n) = cong suc (m≥n⇒m⊔n≡m m≥n)

m≤n⇒m⊓n≡m :  {m n}  m  n  m  n  m
m≤n⇒m⊓n≡m {zero}  z≤n       = refl
m≤n⇒m⊓n≡m {suc m} (s≤s m≤n) = cong suc (m≤n⇒m⊓n≡m m≤n)

m≥n⇒m⊓n≡n :  {m n}  m  n  m  n  n
m≥n⇒m⊓n≡n {zero}  {zero}  z≤n       = refl
m≥n⇒m⊓n≡n {suc m} {zero}  z≤n       = refl
m≥n⇒m⊓n≡n {suc m} {suc n} (s≤s m≤n) = cong suc (m≥n⇒m⊓n≡n m≤n)

⊓-operator : MinOperator ≤-totalPreorder
⊓-operator = record
  { x≤y⇒x⊓y≈x = m≤n⇒m⊓n≡m
  ; x≥y⇒x⊓y≈y = m≥n⇒m⊓n≡n
  }

⊔-operator : MaxOperator ≤-totalPreorder
⊔-operator = record
  { x≤y⇒x⊔y≈y = m≤n⇒m⊔n≡n
  ; x≥y⇒x⊔y≈x = m≥n⇒m⊔n≡m
  }

------------------------------------------------------------------------
-- Derived properties of _⊓_ and _⊔_

private
  module ⊓-⊔-properties        = MinMaxOp        ⊓-operator ⊔-operator
  module ⊓-⊔-latticeProperties = LatticeMinMaxOp ⊓-operator ⊔-operator

open ⊓-⊔-properties public
  using
  ( ⊓-idem                    -- : Idempotent _⊓_
  ; ⊓-sel                     -- : Selective _⊓_
  ; ⊓-assoc                   -- : Associative _⊓_
  ; ⊓-comm                    -- : Commutative _⊓_

  ; ⊔-idem                    -- : Idempotent _⊔_
  ; ⊔-sel                     -- : Selective _⊔_
  ; ⊔-assoc                   -- : Associative _⊔_
  ; ⊔-comm                    -- : Commutative _⊔_

  ; ⊓-distribˡ-⊔              -- : _⊓_ DistributesOverˡ _⊔_
  ; ⊓-distribʳ-⊔              -- : _⊓_ DistributesOverʳ _⊔_
  ; ⊓-distrib-⊔               -- : _⊓_ DistributesOver  _⊔_
  ; ⊔-distribˡ-⊓              -- : _⊔_ DistributesOverˡ _⊓_
  ; ⊔-distribʳ-⊓              -- : _⊔_ DistributesOverʳ _⊓_
  ; ⊔-distrib-⊓               -- : _⊔_ DistributesOver  _⊓_
  ; ⊓-absorbs-⊔               -- : _⊓_ Absorbs _⊔_
  ; ⊔-absorbs-⊓               -- : _⊔_ Absorbs _⊓_
  ; ⊔-⊓-absorptive            -- : Absorptive _⊔_ _⊓_
  ; ⊓-⊔-absorptive            -- : Absorptive _⊓_ _⊔_

  ; ⊓-isMagma                 -- : IsMagma _⊓_
  ; ⊓-isSemigroup             -- : IsSemigroup _⊓_
  ; ⊓-isCommutativeSemigroup  -- : IsCommutativeSemigroup _⊓_
  ; ⊓-isBand                  -- : IsBand _⊓_
  ; ⊓-isSelectiveMagma        -- : IsSelectiveMagma _⊓_

  ; ⊔-isMagma                 -- : IsMagma _⊔_
  ; ⊔-isSemigroup             -- : IsSemigroup _⊔_
  ; ⊔-isCommutativeSemigroup  -- : IsCommutativeSemigroup _⊔_
  ; ⊔-isBand                  -- : IsBand _⊔_
  ; ⊔-isSelectiveMagma        -- : IsSelectiveMagma _⊔_

  ; ⊓-magma                   -- : Magma _ _
  ; ⊓-semigroup               -- : Semigroup _ _
  ; ⊓-band                    -- : Band _ _
  ; ⊓-commutativeSemigroup    -- : CommutativeSemigroup _ _
  ; ⊓-selectiveMagma          -- : SelectiveMagma _ _

  ; ⊔-magma                   -- : Magma _ _
  ; ⊔-semigroup               -- : Semigroup _ _
  ; ⊔-band                    -- : Band _ _
  ; ⊔-commutativeSemigroup    -- : CommutativeSemigroup _ _
  ; ⊔-selectiveMagma          -- : SelectiveMagma _ _

  ; ⊓-glb                     -- : ∀ {m n o} → m ≥ o → n ≥ o → m ⊓ n ≥ o
  ; ⊓-triangulate             -- : ∀ m n o → m ⊓ n ⊓ o ≡ (m ⊓ n) ⊓ (n ⊓ o)
  ; ⊓-mono-≤                  -- : _⊓_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_
  ; ⊓-monoˡ-≤                 -- : ∀ n → (_⊓ n) Preserves _≤_ ⟶ _≤_
  ; ⊓-monoʳ-≤                 -- : ∀ n → (n ⊓_) Preserves _≤_ ⟶ _≤_

  ; ⊔-lub                     -- : ∀ {m n o} → m ≤ o → n ≤ o → m ⊔ n ≤ o
  ; ⊔-triangulate             -- : ∀ m n o → m ⊔ n ⊔ o ≡ (m ⊔ n) ⊔ (n ⊔ o)
  ; ⊔-mono-≤                  -- : _⊔_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_
  ; ⊔-monoˡ-≤                 -- : ∀ n → (_⊔ n) Preserves _≤_ ⟶ _≤_
  ; ⊔-monoʳ-≤                 -- : ∀ n → (n ⊔_) Preserves _≤_ ⟶ _≤_
  )
  renaming
  ( x⊓y≈y⇒y≤x to m⊓n≡n⇒n≤m    -- : ∀ {m n} → m ⊓ n ≡ n → n ≤ m
  ; x⊓y≈x⇒x≤y to m⊓n≡m⇒m≤n    -- : ∀ {m n} → m ⊓ n ≡ m → m ≤ n
  ; x⊓y≤x     to m⊓n≤m        -- : ∀ m n → m ⊓ n ≤ m
  ; x⊓y≤y     to m⊓n≤n        -- : ∀ m n → m ⊓ n ≤ n
  ; x≤y⇒x⊓z≤y to m≤n⇒m⊓o≤n    -- : ∀ {m n} o → m ≤ n → m ⊓ o ≤ n
  ; x≤y⇒z⊓x≤y to m≤n⇒o⊓m≤n    -- : ∀ {m n} o → m ≤ n → o ⊓ m ≤ n
  ; x≤y⊓z⇒x≤y to m≤n⊓o⇒m≤n    -- : ∀ {m} n o → m ≤ n ⊓ o → m ≤ n
  ; x≤y⊓z⇒x≤z to m≤n⊓o⇒m≤o    -- : ∀ {m} n o → m ≤ n ⊓ o → m ≤ o

  ; x⊔y≈y⇒x≤y to m⊔n≡n⇒m≤n    -- : ∀ {m n} → m ⊔ n ≡ n → m ≤ n
  ; x⊔y≈x⇒y≤x to m⊔n≡m⇒n≤m    -- : ∀ {m n} → m ⊔ n ≡ m → n ≤ m
  ; x≤x⊔y     to m≤m⊔n        -- : ∀ m n → m ≤ m ⊔ n
  ; x≤y⊔x     to m≤n⊔m        -- : ∀ m n → m ≤ n ⊔ m
  ; x≤y⇒x≤y⊔z to m≤n⇒m≤n⊔o    -- : ∀ {m n} o → m ≤ n → m ≤ n ⊔ o
  ; x≤y⇒x≤z⊔y to m≤n⇒m≤o⊔n    -- : ∀ {m n} o → m ≤ n → m ≤ o ⊔ n
  ; x⊔y≤z⇒x≤z to m⊔n≤o⇒m≤o    -- : ∀ m n {o} → m ⊔ n ≤ o → m ≤ o
  ; x⊔y≤z⇒y≤z to m⊔n≤o⇒n≤o    -- : ∀ m n {o} → m ⊔ n ≤ o → n ≤ o

  ; x⊓y≤x⊔y   to m⊓n≤m⊔n      -- : ∀ m n → m ⊓ n ≤ m ⊔ n
  )

open ⊓-⊔-latticeProperties public
  using
  ( ⊓-isSemilattice           -- : IsSemilattice _⊓_
  ; ⊔-isSemilattice           -- : IsSemilattice _⊔_
  ; ⊔-⊓-isLattice             -- : IsLattice _⊔_ _⊓_
  ; ⊓-⊔-isLattice             -- : IsLattice _⊓_ _⊔_
  ; ⊔-⊓-isDistributiveLattice -- : IsDistributiveLattice _⊔_ _⊓_
  ; ⊓-⊔-isDistributiveLattice -- : IsDistributiveLattice _⊓_ _⊔_

  ; ⊓-semilattice             -- : Semilattice _ _
  ; ⊔-semilattice             -- : Semilattice _ _
  ; ⊔-⊓-lattice               -- : Lattice _ _
  ; ⊓-⊔-lattice               -- : Lattice _ _
  ; ⊔-⊓-distributiveLattice   -- : DistributiveLattice _ _
  ; ⊓-⊔-distributiveLattice   -- : DistributiveLattice _ _
  )

------------------------------------------------------------------------
-- Automatically derived properties of _⊓_ and _⊔_

⊔-identityˡ : LeftIdentity 0 _⊔_
⊔-identityˡ _ = refl

⊔-identityʳ : RightIdentity 0 _⊔_
⊔-identityʳ zero    = refl
⊔-identityʳ (suc n) = refl

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

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

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

⊔-0-isCommutativeMonoid : IsCommutativeMonoid _⊔_ 0
⊔-0-isCommutativeMonoid = record
  { isMonoid = ⊔-0-isMonoid
  ; comm     = ⊔-comm
  }

------------------------------------------------------------------------
-- Bundles

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

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

------------------------------------------------------------------------
-- Other properties of _⊔_ and _≤_/_<_

mono-≤-distrib-⊔ :  {f}  f Preserves _≤_  _≤_ 
                    m n  f (m  n)  f m  f n
mono-≤-distrib-⊔ {f} = ⊓-⊔-properties.mono-≤-distrib-⊔ (cong f)

mono-≤-distrib-⊓ :  {f}  f Preserves _≤_  _≤_ 
                    m n  f (m  n)  f m  f n
mono-≤-distrib-⊓ {f} = ⊓-⊔-properties.mono-≤-distrib-⊓ (cong f)

antimono-≤-distrib-⊓ :  {f}  f Preserves _≤_  _≥_ 
                        m n  f (m  n)  f m  f n
antimono-≤-distrib-⊓ {f} = ⊓-⊔-properties.antimono-≤-distrib-⊓ (cong f)

antimono-≤-distrib-⊔ :  {f}  f Preserves _≤_  _≥_ 
                        m n  f (m  n)  f m  f n
antimono-≤-distrib-⊔ {f} = ⊓-⊔-properties.antimono-≤-distrib-⊔ (cong f)

m<n⇒m<n⊔o :  {m n} o  m < n  m < n  o
m<n⇒m<n⊔o = m≤n⇒m≤n⊔o

m<n⇒m<o⊔n :  {m n} o  m < n  m < o  n
m<n⇒m<o⊔n = m≤n⇒m≤o⊔n

m⊔n<o⇒m<o :  m n {o}  m  n < o  m < o
m⊔n<o⇒m<o m n m⊔n<o = <-transʳ (m≤m⊔n m n) m⊔n<o

m⊔n<o⇒n<o :  m n {o}  m  n < o  n < o
m⊔n<o⇒n<o m n m⊔n<o = <-transʳ (m≤n⊔m m n) m⊔n<o

⊔-mono-< : _⊔_ Preserves₂ _<_  _<_  _<_
⊔-mono-< = ⊔-mono-≤

⊔-pres-<m :  {m n o}  n < m  o < m  n  o < m
⊔-pres-<m {m} n<m o<m = subst (_ <_) (⊔-idem m) (⊔-mono-< n<m o<m)

------------------------------------------------------------------------
-- Other properties of _⊔_ and _+_

+-distribˡ-⊔ : _+_ DistributesOverˡ _⊔_
+-distribˡ-⊔ zero    n o = refl
+-distribˡ-⊔ (suc m) n o = cong suc (+-distribˡ-⊔ m n o)

+-distribʳ-⊔ : _+_ DistributesOverʳ _⊔_
+-distribʳ-⊔ = comm+distrˡ⇒distrʳ +-comm +-distribˡ-⊔

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

m⊔n≤m+n :  m n  m  n  m + n
m⊔n≤m+n m n with ⊔-sel m n
... | inj₁ m⊔n≡m rewrite m⊔n≡m = m≤m+n m n
... | inj₂ m⊔n≡n rewrite m⊔n≡n = m≤n+m n m

------------------------------------------------------------------------
-- Other properties of _⊔_ and _*_

*-distribˡ-⊔ : _*_ DistributesOverˡ _⊔_
*-distribˡ-⊔ m zero o = sym (cong (_⊔ m * o) (*-zeroʳ m))
*-distribˡ-⊔ m (suc n) zero = begin-equality
  m * (suc n  zero)         ≡⟨⟩
  m * suc n                  ≡˘⟨ ⊔-identityʳ (m * suc n) 
  m * suc n  zero           ≡˘⟨ cong (m * suc n ⊔_) (*-zeroʳ m) 
  m * suc n  m * zero       
*-distribˡ-⊔ m (suc n) (suc o) = begin-equality
  m * (suc n  suc o)        ≡⟨⟩
  m * suc (n  o)            ≡⟨ *-suc m (n  o) 
  m + m * (n  o)            ≡⟨ cong (m +_) (*-distribˡ-⊔ m n o) 
  m + (m * n  m * o)        ≡⟨ +-distribˡ-⊔ m (m * n) (m * o) 
  (m + m * n)  (m + m * o)  ≡˘⟨ cong₂ _⊔_ (*-suc m n) (*-suc m o) 
  (m * suc n)  (m * suc o)  

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

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

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

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

⊓-zeroˡ : LeftZero 0 _⊓_
⊓-zeroˡ _ = refl

⊓-zeroʳ : RightZero 0 _⊓_
⊓-zeroʳ zero    = refl
⊓-zeroʳ (suc n) = refl

⊓-zero : Zero 0 _⊓_
⊓-zero = ⊓-zeroˡ , ⊓-zeroʳ

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

⊔-⊓-isSemiringWithoutOne : IsSemiringWithoutOne _⊔_ _⊓_ 0
⊔-⊓-isSemiringWithoutOne = record
  { +-isCommutativeMonoid = ⊔-0-isCommutativeMonoid
  ; *-cong                = cong₂ _⊓_
  ; *-assoc               = ⊓-assoc
  ; distrib               = ⊓-distrib-⊔
  ; zero                  = ⊓-zero
  }

⊔-⊓-isCommutativeSemiringWithoutOne
  : IsCommutativeSemiringWithoutOne _⊔_ _⊓_ 0
⊔-⊓-isCommutativeSemiringWithoutOne = record
  { isSemiringWithoutOne = ⊔-⊓-isSemiringWithoutOne
  ; *-comm               = ⊓-comm
  }

------------------------------------------------------------------------
-- Bundles

⊔-⊓-commutativeSemiringWithoutOne : CommutativeSemiringWithoutOne 0ℓ 0ℓ
⊔-⊓-commutativeSemiringWithoutOne = record
  { isCommutativeSemiringWithoutOne =
      ⊔-⊓-isCommutativeSemiringWithoutOne
  }

------------------------------------------------------------------------
-- Other properties of _⊓_ and _≤_/_<_

m<n⇒m⊓o<n :  {m n} o  m < n  m  o < n
m<n⇒m⊓o<n o m<n = <-transʳ (m⊓n≤m _ o) m<n

m<n⇒o⊓m<n :  {m n} o  m < n  o  m < n
m<n⇒o⊓m<n o m<n = <-transʳ (m⊓n≤n o _) m<n

m<n⊓o⇒m<n :  {m} n o  m < n  o  m < n
m<n⊓o⇒m<n = m≤n⊓o⇒m≤n

m<n⊓o⇒m<o :  {m} n o  m < n  o  m < o
m<n⊓o⇒m<o = m≤n⊓o⇒m≤o

⊓-mono-< : _⊓_ Preserves₂ _<_  _<_  _<_
⊓-mono-< = ⊓-mono-≤

⊓-pres-m< :  {m n o}  m < n  m < o  m < n  o
⊓-pres-m< {m} m<n m<o = subst (_< _) (⊓-idem m) (⊓-mono-< m<n m<o)

------------------------------------------------------------------------
-- Other properties of _⊓_ and _+_

+-distribˡ-⊓ : _+_ DistributesOverˡ _⊓_
+-distribˡ-⊓ zero    n o = refl
+-distribˡ-⊓ (suc m) n o = cong suc (+-distribˡ-⊓ m n o)

+-distribʳ-⊓ : _+_ DistributesOverʳ _⊓_
+-distribʳ-⊓ = comm+distrˡ⇒distrʳ +-comm +-distribˡ-⊓

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

m⊓n≤m+n :  m n  m  n  m + n
m⊓n≤m+n m n with ⊓-sel m n
... | inj₁ m⊓n≡m rewrite m⊓n≡m = m≤m+n m n
... | inj₂ m⊓n≡n rewrite m⊓n≡n = m≤n+m n m

------------------------------------------------------------------------
-- Other properties of _⊓_ and _*_

*-distribˡ-⊓ : _*_ DistributesOverˡ _⊓_
*-distribˡ-⊓ m 0 o = begin-equality
  m * (0  o)               ≡⟨⟩
  m * 0                     ≡⟨ *-zeroʳ m 
  0                         ≡⟨⟩
  0  (m * o)               ≡˘⟨ cong (_⊓ (m * o)) (*-zeroʳ m) 
  (m * 0)  (m * o)         
*-distribˡ-⊓ m (suc n) 0 = begin-equality
  m * (suc n  0)           ≡⟨⟩
  m * 0                     ≡⟨ *-zeroʳ m 
  0                         ≡˘⟨ ⊓-zeroʳ (m * suc n) 
  (m * suc n)  0           ≡˘⟨ cong (m * suc n ⊓_) (*-zeroʳ m) 
  (m * suc n)  (m * 0)     
*-distribˡ-⊓ m (suc n) (suc o) = begin-equality
  m * (suc n  suc o)       ≡⟨⟩
  m * suc (n  o)           ≡⟨ *-suc m (n  o) 
  m + m * (n  o)           ≡⟨ cong (m +_) (*-distribˡ-⊓ m n o) 
  m + (m * n)  (m * o)     ≡⟨ +-distribˡ-⊓ m (m * n) (m * o) 
  (m + m * n)  (m + m * o) ≡˘⟨ cong₂ _⊓_ (*-suc m n) (*-suc m o) 
  (m * suc n)  (m * suc o) 

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

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

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

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

------------------------------------------------------------------------
-- Properties of _∸_ and pred

pred[m∸n]≡m∸[1+n] :  m n  pred (m  n)  m  suc n
pred[m∸n]≡m∸[1+n] zero    zero    = refl
pred[m∸n]≡m∸[1+n] (suc m) zero    = refl
pred[m∸n]≡m∸[1+n] zero (suc n)    = refl
pred[m∸n]≡m∸[1+n] (suc m) (suc n) = pred[m∸n]≡m∸[1+n] m n

------------------------------------------------------------------------
-- Properties of _∸_ and _≤_/_<_

m∸n≤m :  m n  m  n  m
m∸n≤m n       zero    = ≤-refl
m∸n≤m zero    (suc n) = ≤-refl
m∸n≤m (suc m) (suc n) = ≤-trans (m∸n≤m m n) (n≤1+n m)

m≮m∸n :  m n  m  m  n
m≮m∸n m       zero    = n≮n m
m≮m∸n (suc m) (suc n) = m≮m∸n m n  ≤-trans (n≤1+n (suc m))

1+m≢m∸n :  {m} n  suc m  m  n
1+m≢m∸n {m} n eq = m≮m∸n m n (≤-reflexive eq)

∸-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       (z≤n {n = n₁}) = ≤-trans (m∸n≤m _ n₁) m₁≤m₂

∸-monoˡ-≤ :  {m n} o  m  n  m  o  n  o
∸-monoˡ-≤ o m≤n = ∸-mono {u = o} m≤n ≤-refl

∸-monoʳ-≤ :  {m n} o  m  n  o  m  o  n
∸-monoʳ-≤ _ m≤n = ∸-mono ≤-refl m≤n

∸-monoˡ-< :  {m n o}  m < o  n  m  m  n < o  n
∸-monoˡ-< {m}     {zero}  {o}     m<o       n≤m       = m<o
∸-monoˡ-< {suc m} {suc n} {suc o} (s≤s m<o) (s≤s n≤m) = ∸-monoˡ-< m<o n≤m

∸-monoʳ-< :  {m n o}  o < n  n  m  m  n < m  o
∸-monoʳ-< {n = suc n} {zero}  (s≤s o<n) (s≤s n<m) = s≤s (m∸n≤m _ n)
∸-monoʳ-< {n = suc n} {suc o} (s≤s o<n) (s≤s n<m) = ∸-monoʳ-< o<n n<m

∸-cancelʳ-≤ :  {m n o}  m  o  o  n  o  m  m  n
∸-cancelʳ-≤ {_}     {_}     z≤n       _       = z≤n
∸-cancelʳ-≤ {suc m} {zero}  (s≤s _)   o<o∸m   = contradiction o<o∸m (m≮m∸n _ m)
∸-cancelʳ-≤ {suc m} {suc n} (s≤s m≤o) o∸n<o∸m = s≤s (∸-cancelʳ-≤ m≤o o∸n<o∸m)

∸-cancelʳ-< :  {m n o}  o  m < o  n  n < m
∸-cancelʳ-< {zero}  {n}     {o}     o<o∸n   = contradiction o<o∸n (m≮m∸n o n)
∸-cancelʳ-< {suc m} {zero}  {_}     o∸n<o∸m = 0<1+n
∸-cancelʳ-< {suc m} {suc n} {suc o} o∸n<o∸m = s≤s (∸-cancelʳ-< o∸n<o∸m)

∸-cancelˡ-≡ :   {m n o}  n  m  o  m  m  n  m  o  n  o
∸-cancelˡ-≡ {_}         z≤n       z≤n       _  = refl
∸-cancelˡ-≡ {o = suc o} z≤n       (s≤s _)   eq = contradiction eq (1+m≢m∸n o)
∸-cancelˡ-≡ {n = suc n} (s≤s _)   z≤n       eq = contradiction (sym eq) (1+m≢m∸n n)
∸-cancelˡ-≡ {_}         (s≤s n≤m) (s≤s o≤m) eq = cong suc (∸-cancelˡ-≡ n≤m o≤m eq)

∸-cancelʳ-≡ :   {m n o}  o  m  o  n  m  o  n  o  m  n
∸-cancelʳ-≡  z≤n       z≤n      eq = eq
∸-cancelʳ-≡ (s≤s o≤m) (s≤s o≤n) eq = cong suc (∸-cancelʳ-≡ o≤m o≤n eq)

m∸n≡0⇒m≤n :  {m n}  m  n  0  m  n
m∸n≡0⇒m≤n {zero}  {_}    _   = z≤n
m∸n≡0⇒m≤n {suc m} {suc n} eq = s≤s (m∸n≡0⇒m≤n eq)

m≤n⇒m∸n≡0 :  {m n}  m  n  m  n  0
m≤n⇒m∸n≡0 {n = n} z≤n      = 0∸n≡0 n
m≤n⇒m∸n≡0 {_}    (s≤s m≤n) = m≤n⇒m∸n≡0 m≤n

m<n⇒0<n∸m :  {m n}  m < n  0 < n  m
m<n⇒0<n∸m {zero}  {suc n} _         = 0<1+n
m<n⇒0<n∸m {suc m} {suc n} (s≤s m<n) = m<n⇒0<n∸m m<n

m∸n≢0⇒n<m :  {m n}  m  n  0  n < m
m∸n≢0⇒n<m {m} {n} m∸n≢0 with n <? m
... | yes n<m = n<m
... | no  n≮m = contradiction (m≤n⇒m∸n≡0 (≮⇒≥ n≮m)) m∸n≢0

m>n⇒m∸n≢0 :  {m n}  m > n  m  n  0
m>n⇒m∸n≢0 {n = suc n} (s≤s m>n) = m>n⇒m∸n≢0 m>n

m≤n⇒n∸m≤n :  {m n}  m  n  n  m  n
m≤n⇒n∸m≤n z≤n       = ≤-refl
m≤n⇒n∸m≤n (s≤s m≤n) = ≤-step (m≤n⇒n∸m≤n m≤n)

---------------------------------------------------------------
-- Properties of _∸_ and _+_

+-∸-comm :  {m} n {o}  o  m  (m + n)  o  (m  o) + n
+-∸-comm {zero}  _ {zero}  _         = refl
+-∸-comm {suc m} _ {zero}  _         = refl
+-∸-comm {suc m} n {suc o} (s≤s o≤m) = +-∸-comm n o≤m

∸-+-assoc :  m n o  (m  n)  o  m  (n + o)
∸-+-assoc zero zero o = refl
∸-+-assoc zero (suc n) o = 0∸n≡0 o
∸-+-assoc (suc m) zero o = refl
∸-+-assoc (suc m) (suc n) o = ∸-+-assoc m n o

+-∸-assoc :  m {n o}  o  n  (m + n)  o  m + (n  o)
+-∸-assoc m (z≤n {n = n})             = begin-equality m + n 
+-∸-assoc m (s≤s {m = o} {n = n} o≤n) = begin-equality
  (m + suc n)  suc o  ≡⟨ cong (_∸ suc o) (+-suc m n) 
  suc (m + n)  suc o  ≡⟨⟩
  (m + n)  o          ≡⟨ +-∸-assoc m o≤n 
  m + (n  o)          

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

m+n∸n≡m :  m n  m + n  n  m
m+n∸n≡m m n = begin-equality
  (m + n)  n  ≡⟨ +-∸-assoc m (≤-refl {x = n}) 
  m + (n  n)  ≡⟨ cong (m +_) (n∸n≡0 n) 
  m + 0        ≡⟨ +-identityʳ m 
  m            

m+n∸m≡n :  m n  m + n  m  n
m+n∸m≡n m n = trans (cong (_∸ m) (+-comm m n)) (m+n∸n≡m n m)

m+[n∸m]≡n :  {m n}  m  n  m + (n  m)  n
m+[n∸m]≡n {m} {n} m≤n = begin-equality
  m + (n  m)  ≡⟨ sym $ +-∸-assoc m m≤n 
  (m + n)  m  ≡⟨ cong (_∸ m) (+-comm m n) 
  (n + m)  m  ≡⟨ m+n∸n≡m n m 
  n            

m∸n+n≡m :  {m n}  n  m  (m  n) + n  m
m∸n+n≡m {m} {n} n≤m = begin-equality
  (m  n) + n ≡⟨ sym (+-∸-comm n n≤m) 
  (m + n)  n ≡⟨ m+n∸n≡m m n 
  m           

m∸[m∸n]≡n :  {m n}  n  m  m  (m  n)  n
m∸[m∸n]≡n {m}     {_}     z≤n       = n∸n≡0 m
m∸[m∸n]≡n {suc m} {suc n} (s≤s n≤m) = begin-equality
  suc m  (m  n)   ≡⟨ +-∸-assoc 1 (m∸n≤m m n) 
  suc (m  (m  n)) ≡⟨ cong suc (m∸[m∸n]≡n n≤m) 
  suc n             

[m+n]∸[m+o]≡n∸o :  m n o  (m + n)  (m + o)  n  o
[m+n]∸[m+o]≡n∸o zero    n o = refl
[m+n]∸[m+o]≡n∸o (suc m) n o = [m+n]∸[m+o]≡n∸o m n o

------------------------------------------------------------------------
-- Properties of _∸_ and _*_

*-distribʳ-∸ : _*_ DistributesOverʳ _∸_
*-distribʳ-∸ m       zero    zero    = refl
*-distribʳ-∸ zero    zero    (suc o) = sym (0∸n≡0 (o * zero))
*-distribʳ-∸ (suc m) zero    (suc o) = refl
*-distribʳ-∸ m       (suc n) zero    = refl
*-distribʳ-∸ m       (suc n) (suc o) = begin-equality
  (n  o) * m             ≡⟨ *-distribʳ-∸ m n o 
  n * m  o * m           ≡⟨ sym $ [m+n]∸[m+o]≡n∸o m _ _ 
  m + n * m  (m + o * m) 

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

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

even≢odd :   m n  2 * m  suc (2 * n)
even≢odd (suc m) zero    eq = contradiction (suc-injective eq) (m+1+n≢0 m)
even≢odd (suc m) (suc n) eq = even≢odd m n (suc-injective (begin-equality
  suc (2 * m)         ≡⟨ sym (+-suc m _) 
  m + suc (m + 0)     ≡⟨ suc-injective eq 
  suc n + suc (n + 0) ≡⟨ cong suc (+-suc n _) 
  suc (suc (2 * n))   ))

------------------------------------------------------------------------
-- Properties of _∸_ and _⊓_ and _⊔_

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

∸-distribˡ-⊓-⊔ :  m n o  m  (n  o)  (m  n)  (m  o)
∸-distribˡ-⊓-⊔ m n o = antimono-≤-distrib-⊓ (∸-monoʳ-≤ m) n o

∸-distribʳ-⊓ : _∸_ DistributesOverʳ _⊓_
∸-distribʳ-⊓ m n o = mono-≤-distrib-⊓ (∸-monoˡ-≤ m) n o

∸-distribˡ-⊔-⊓ :  m n o  m  (n  o)  (m  n)  (m  o)
∸-distribˡ-⊔-⊓ m n o = antimono-≤-distrib-⊔ (∸-monoʳ-≤ m) n o

∸-distribʳ-⊔ : _∸_ DistributesOverʳ _⊔_
∸-distribʳ-⊔ m n o = mono-≤-distrib-⊔ (∸-monoˡ-≤ m) n o

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

pred-mono : pred Preserves _≤_  _≤_
pred-mono m≤n = ∸-mono m≤n (≤-refl {1})

pred[n]≤n :  {n}  pred n  n
pred[n]≤n {zero}  = z≤n
pred[n]≤n {suc n} = n≤1+n n

≤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}  le = le
≤⇒pred≤ {suc m} le = ≤-trans (n≤1+n m) le

<⇒≤pred :  {m n}  m < n  m  pred n
<⇒≤pred (s≤s le) = le

suc-pred :  n .{{_ : NonZero n}}  suc (pred n)  n
suc-pred (suc n) = refl

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

------------------------------------------------------------------------
-- Basic

m≡n⇒∣m-n∣≡0 :  {m n}  m  n   m - n   0
m≡n⇒∣m-n∣≡0 {zero}  refl = refl
m≡n⇒∣m-n∣≡0 {suc m} refl = m≡n⇒∣m-n∣≡0 {m} refl

∣m-n∣≡0⇒m≡n :  {m n}   m - n   0  m  n
∣m-n∣≡0⇒m≡n {zero}  {zero}  eq = refl
∣m-n∣≡0⇒m≡n {suc m} {suc n} eq = cong suc (∣m-n∣≡0⇒m≡n eq)

m≤n⇒∣n-m∣≡n∸m :  {m n}  m  n   n - m   n  m
m≤n⇒∣n-m∣≡n∸m {_} {zero}  z≤n       = refl
m≤n⇒∣n-m∣≡n∸m {_} {suc m} z≤n       = refl
m≤n⇒∣n-m∣≡n∸m {_} {_}     (s≤s m≤n) = m≤n⇒∣n-m∣≡n∸m m≤n

∣m-n∣≡m∸n⇒n≤m :  {m n}   m - n   m  n  n  m
∣m-n∣≡m∸n⇒n≤m {zero}  {zero}  eq = z≤n
∣m-n∣≡m∸n⇒n≤m {suc m} {zero}  eq = z≤n
∣m-n∣≡m∸n⇒n≤m {suc m} {suc n} eq = s≤s (∣m-n∣≡m∸n⇒n≤m eq)

∣n-n∣≡0 :  n   n - n   0
∣n-n∣≡0 n = m≡n⇒∣m-n∣≡0 {n} refl

∣m-m+n∣≡n :  m n   m - m + n   n
∣m-m+n∣≡n zero    n = refl
∣m-m+n∣≡n (suc m) n = ∣m-m+n∣≡n m n

∣m+n-m+o∣≡∣n-o∣ :  m n o   m + n - m + o    n - o 
∣m+n-m+o∣≡∣n-o∣ zero    n o = refl
∣m+n-m+o∣≡∣n-o∣ (suc m) n o = ∣m+n-m+o∣≡∣n-o∣ m n o

m∸n≤∣m-n∣ :  m n  m  n   m - n 
m∸n≤∣m-n∣ m n with ≤-total m n
... | inj₁ m≤n = subst (_≤  m - n ) (sym (m≤n⇒m∸n≡0 m≤n)) z≤n
... | inj₂ n≤m = subst (m  n ≤_) (sym (m≤n⇒∣n-m∣≡n∸m n≤m)) ≤-refl

∣m-n∣≤m⊔n :  m n   m - n   m  n
∣m-n∣≤m⊔n zero    m       = ≤-refl
∣m-n∣≤m⊔n (suc m) zero    = ≤-refl
∣m-n∣≤m⊔n (suc m) (suc n) = ≤-step (∣m-n∣≤m⊔n m n)

∣-∣-identityˡ : LeftIdentity 0 ∣_-_∣
∣-∣-identityˡ x = refl

∣-∣-identityʳ : RightIdentity 0 ∣_-_∣
∣-∣-identityʳ zero    = refl
∣-∣-identityʳ (suc x) = refl

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

∣-∣-comm : Commutative ∣_-_∣
∣-∣-comm zero    zero    = refl
∣-∣-comm zero    (suc n) = refl
∣-∣-comm (suc m) zero    = refl
∣-∣-comm (suc m) (suc n) = ∣-∣-comm m n

∣m-n∣≡[m∸n]∨[n∸m] :  m n  ( m - n   m  n)  ( m - n   n  m)
∣m-n∣≡[m∸n]∨[n∸m] m n with ≤-total m n
... | inj₂ n≤m = inj₁ $ m≤n⇒∣n-m∣≡n∸m n≤m
... | inj₁ m≤n = inj₂ $ begin-equality
   m - n  ≡⟨ ∣-∣-comm m n 
   n - m  ≡⟨ m≤n⇒∣n-m∣≡n∸m m≤n 
  n  m     

private

  *-distribˡ-∣-∣-aux :  a m n  m  n  a *  n - m    a * n - a * m 
  *-distribˡ-∣-∣-aux a m n m≤n = begin-equality
    a *  n - m      ≡⟨ cong (a *_) (m≤n⇒∣n-m∣≡n∸m m≤n) 
    a * (n  m)       ≡⟨ *-distribˡ-∸ a n m 
    a * n  a * m     ≡⟨ sym $′ m≤n⇒∣n-m∣≡n∸m (*-monoʳ-≤ a m≤n) 
     a * n - a * m  

*-distribˡ-∣-∣ : _*_ DistributesOverˡ ∣_-_∣
*-distribˡ-∣-∣ a m n with ≤-total m n
... | inj₂ n≤m = *-distribˡ-∣-∣-aux a n m n≤m
... | inj₁ m≤n = begin-equality
  a *  m - n      ≡⟨ cong (a *_) (∣-∣-comm m n) 
  a *  n - m      ≡⟨ *-distribˡ-∣-∣-aux a m n m≤n 
   a * n - a * m  ≡⟨ ∣-∣-comm (a * n) (a * m) 
   a * m - a * n  

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

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

m≤n+∣n-m∣ :  m n  m  n +  n - m 
m≤n+∣n-m∣ zero    n       = z≤n
m≤n+∣n-m∣ (suc m) zero    = ≤-refl
m≤n+∣n-m∣ (suc m) (suc n) = s≤s (m≤n+∣n-m∣ m n)

m≤n+∣m-n∣ :  m n  m  n +  m - n 
m≤n+∣m-n∣ m n = subst (m ≤_) (cong (n +_) (∣-∣-comm n m)) (m≤n+∣n-m∣ m n)

m≤∣m-n∣+n :  m n  m   m - n  + n
m≤∣m-n∣+n m n = subst (m ≤_) (+-comm n _) (m≤n+∣m-n∣ m n)

∣-∣-triangle : TriangleInequality ∣_-_∣
∣-∣-triangle zero    y       z       = m≤n+∣n-m∣ z y
∣-∣-triangle x       zero    z       = begin
   x - z      ≤⟨ ∣m-n∣≤m⊔n x z 
  x  z         ≤⟨ m⊔n≤m+n x z 
  x + z         ≡⟨ cong₂ _+_ (sym (∣-∣-identityʳ x)) refl 
   x - 0  + z 
  where open ≤-Reasoning
∣-∣-triangle x       y       zero    = begin
   x - 0              ≡⟨ ∣-∣-identityʳ x 
  x                     ≤⟨ m≤∣m-n∣+n x y 
   x - y  + y         ≡⟨ cong₂ _+_ refl (sym (∣-∣-identityʳ y)) 
   x - y  +  y - 0  
  where open ≤-Reasoning
∣-∣-triangle (suc x) (suc y) (suc z) = ∣-∣-triangle x y z

------------------------------------------------------------------------
-- Metric structures

∣-∣-isProtoMetric : IsProtoMetric _≡_ ∣_-_∣
∣-∣-isProtoMetric = record
  { isPartialOrder  = ≤-isPartialOrder
  ; ≈-isEquivalence = isEquivalence
  ; cong            = cong₂ ∣_-_∣
  ; nonNegative     = z≤n
  }

∣-∣-isPreMetric : IsPreMetric _≡_ ∣_-_∣
∣-∣-isPreMetric = record
  { isProtoMetric = ∣-∣-isProtoMetric
  ; ≈⇒0           = m≡n⇒∣m-n∣≡0
  }

∣-∣-isQuasiSemiMetric : IsQuasiSemiMetric _≡_ ∣_-_∣
∣-∣-isQuasiSemiMetric = record
  { isPreMetric = ∣-∣-isPreMetric
  ; 0⇒≈         = ∣m-n∣≡0⇒m≡n
  }

∣-∣-isSemiMetric : IsSemiMetric _≡_ ∣_-_∣
∣-∣-isSemiMetric = record
  { isQuasiSemiMetric = ∣-∣-isQuasiSemiMetric
  ; sym               = ∣-∣-comm
  }

∣-∣-isMetric : IsMetric _≡_ ∣_-_∣
∣-∣-isMetric = record
  { isSemiMetric = ∣-∣-isSemiMetric
  ; triangle     = ∣-∣-triangle
  }

------------------------------------------------------------------------
-- Metric bundles

∣-∣-quasiSemiMetric : QuasiSemiMetric 0ℓ 0ℓ
∣-∣-quasiSemiMetric = record
  { isQuasiSemiMetric = ∣-∣-isQuasiSemiMetric
  }

∣-∣-semiMetric : SemiMetric 0ℓ 0ℓ
∣-∣-semiMetric = record
  { isSemiMetric = ∣-∣-isSemiMetric
  }

∣-∣-preMetric : PreMetric 0ℓ 0ℓ
∣-∣-preMetric = record
  { isPreMetric = ∣-∣-isPreMetric
  }

∣-∣-metric : Metric 0ℓ 0ℓ
∣-∣-metric = record
  { isMetric = ∣-∣-isMetric
  }

------------------------------------------------------------------------
-- Properties of ⌊_/2⌋ and ⌈_/2⌉
------------------------------------------------------------------------

⌊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)

⌊n/2⌋≤⌈n/2⌉ :  n   n /2⌋   n /2⌉
⌊n/2⌋≤⌈n/2⌉ zero          = z≤n
⌊n/2⌋≤⌈n/2⌉ (suc zero)    = z≤n
⌊n/2⌋≤⌈n/2⌉ (suc (suc n)) = s≤s (⌊n/2⌋≤⌈n/2⌉ n)

⌊n/2⌋+⌈n/2⌉≡n :  n   n /2⌋ +  n /2⌉  n
⌊n/2⌋+⌈n/2⌉≡n zero    = refl
⌊n/2⌋+⌈n/2⌉≡n (suc n) = begin-equality
   suc n /2⌋ + suc  n /2⌋   ≡⟨ +-comm  suc n /2⌋ (suc  n /2⌋) 
  suc  n /2⌋ +  suc n /2⌋   ≡⟨⟩
  suc ( n /2⌋ +  suc n /2⌋) ≡⟨ cong suc (⌊n/2⌋+⌈n/2⌉≡n n) 
  suc n                       

⌊n/2⌋≤n :  n   n /2⌋  n
⌊n/2⌋≤n zero          = z≤n
⌊n/2⌋≤n (suc zero)    = z≤n
⌊n/2⌋≤n (suc (suc n)) = s≤s (≤-step (⌊n/2⌋≤n n))

⌊n/2⌋<n :  n   suc n /2⌋ < suc n
⌊n/2⌋<n zero    = z<s
⌊n/2⌋<n (suc n) = s<s (s≤s (⌊n/2⌋≤n n))

n≡⌊n+n/2⌋ :  n  n   n + n /2⌋
n≡⌊n+n/2⌋ zero          = refl
n≡⌊n+n/2⌋ (suc zero)    = refl
n≡⌊n+n/2⌋ (suc (suc n)) =
  cong suc (trans (n≡⌊n+n/2⌋ _) (cong ⌊_/2⌋ (sym (+-suc n (suc n)))))

⌈n/2⌉≤n :  n   n /2⌉  n
⌈n/2⌉≤n zero    = z≤n
⌈n/2⌉≤n (suc n) = s≤s (⌊n/2⌋≤n n)

⌈n/2⌉<n :  n   suc (suc n) /2⌉ < suc (suc n)
⌈n/2⌉<n n = s<s (⌊n/2⌋<n n)

n≡⌈n+n/2⌉ :  n  n   n + n /2⌉
n≡⌈n+n/2⌉ zero          = refl
n≡⌈n+n/2⌉ (suc zero)    = refl
n≡⌈n+n/2⌉ (suc (suc n)) =
  cong suc (trans (n≡⌈n+n/2⌉ _) (cong ⌈_/2⌉ (sym (+-suc n (suc n)))))

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

1≤n! :  n  1  n !
1≤n! zero    = ≤-refl
1≤n! (suc n) = *-mono-≤ (m≤m+n 1 n) (1≤n! n)

_!≢0 :  n  NonZero (n !)
n !≢0 = >-nonZero (1≤n! n)

_!*_!≢0 :  m n  NonZero (m ! * n !)
m !* n !≢0 = m*n≢0 _ _ {{m !≢0}} {{n !≢0}}

------------------------------------------------------------------------
-- Properties of _≤′_ and _<′_

≤′-trans : Transitive _≤′_
≤′-trans m≤n ≤′-refl       = m≤n
≤′-trans m≤n (≤′-step n≤o) = ≤′-step (≤′-trans m≤n n≤o)

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)

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

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

≤′-step-injective :  {m n} {p q : m ≤′ n}  ≤′-step p  ≤′-step q  p  q
≤′-step-injective refl = refl

------------------------------------------------------------------------
-- Properties of _<′_ and _<_
------------------------------------------------------------------------

z<′s :  {n}  zero <′ suc n
z<′s {zero}  = <′-base
z<′s {suc n} = <′-step (z<′s {n})

s<′s :  {m n}  m <′ n  suc m <′ suc n
s<′s <′-base        = <′-base
s<′s (<′-step m<′n) = <′-step (s<′s m<′n)

<⇒<′ :  {m n}  m < n  m <′ n
<⇒<′ z<s               = z<′s
<⇒<′ (s<s m<n@(s≤s _)) = s<′s (<⇒<′ m<n)

<′⇒< :  {m n}  m <′ n  m < n
<′⇒< <′-base        = n<1+n _
<′⇒< (<′-step m<′n) = <-step (<′⇒< m<′n)

m<1+n⇒m<n∨m≡n′ :  {m n}  m < suc n  m < n  m  n
m<1+n⇒m<n∨m≡n′ m<n with <⇒<′ m<n
... | <′-base      = inj₂ refl
... | <′-step m<′n = inj₁ (<′⇒< m<′n)

------------------------------------------------------------------------
-- Other properties of _≤′_ and _<′_
------------------------------------------------------------------------

infix 4 _≤′?_ _<′?_ _≥′?_ _>′?_

_≤′?_ : Decidable _≤′_
m ≤′? n = map′ ≤⇒≤′ ≤′⇒≤ (m ≤? n)

_<′?_ : Decidable _<′_
m <′? n = suc m ≤′? n

_≥′?_ : Decidable _≥′_
_≥′?_ = flip _≤′?_

_>′?_ : Decidable _>′_
_>′?_ = flip _<′?_

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/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)

------------------------------------------------------------------------
-- Properties of _≤″_ and _<″_
------------------------------------------------------------------------

m<ᵇn⇒1+m+[n-1+m]≡n :  m n  T (m <ᵇ n)  suc m + (n  suc m)  n
m<ᵇn⇒1+m+[n-1+m]≡n m n lt = m+[n∸m]≡n (<ᵇ⇒< m n lt)

m<ᵇ1+m+n :  m {n}  T (m <ᵇ suc (m + n))
m<ᵇ1+m+n m = <⇒<ᵇ (m≤m+n (suc m) _)

<ᵇ⇒<″ :  {m n}  T (m <ᵇ n)  m <″ n
<ᵇ⇒<″ {m} {n} leq = less-than-or-equal (m+[n∸m]≡n (<ᵇ⇒< m n leq))

<″⇒<ᵇ :  {m n}  m <″ n  T (m <ᵇ n)
<″⇒<ᵇ {m} (less-than-or-equal refl) = <⇒<ᵇ (m≤m+n (suc m) _)

-- equivalence to _≤_

≤″⇒≤ : _≤″_  _≤_
≤″⇒≤ {zero}  (less-than-or-equal refl) = z≤n
≤″⇒≤ {suc m} (less-than-or-equal refl) =
  s≤s (≤″⇒≤ (less-than-or-equal refl))

≤⇒≤″ : _≤_  _≤″_
≤⇒≤″ = less-than-or-equal  m+[n∸m]≡n

-- NB: we use the builtin function `_<ᵇ_ : (m n : ℕ) → Bool` here so
-- that the function quickly decides whether to return `yes` or `no`.
-- It still takes a linear amount of time to generate the proof if it
-- is inspected. We expect the main benefit to be visible for compiled
-- code: the backend erases proofs.

infix 4 _<″?_ _≤″?_ _≥″?_ _>″?_

_<″?_ : Decidable _<″_
m <″? n = map′ <ᵇ⇒<″ <″⇒<ᵇ (T? (m <ᵇ n))

_≤″?_ : Decidable _≤″_
zero  ≤″? n = yes (less-than-or-equal refl)
suc m ≤″? n = m <″? n

_≥″?_ : Decidable _≥″_
_≥″?_ = flip _≤″?_

_>″?_ : Decidable _>″_
_>″?_ = flip _<″?_

≤″-irrelevant : Irrelevant _≤″_
≤″-irrelevant {m} (less-than-or-equal eq₁)
                  (less-than-or-equal eq₂)
  with +-cancelˡ-≡ m _ _ (trans eq₁ (sym eq₂))
... | refl = cong less-than-or-equal (≡-irrelevant eq₁ eq₂)

<″-irrelevant : Irrelevant _<″_
<″-irrelevant = ≤″-irrelevant

>″-irrelevant : Irrelevant _>″_
>″-irrelevant = ≤″-irrelevant

≥″-irrelevant : Irrelevant _≥″_
≥″-irrelevant = ≤″-irrelevant

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

≤‴⇒≤″ : ∀{m n}  m ≤‴ n  m ≤″ n
≤‴⇒≤″ {m = m} ≤‴-refl     = less-than-or-equal {k = 0} (+-identityʳ m)
≤‴⇒≤″ {m = m} (≤‴-step x) = less-than-or-equal (trans (+-suc m _) (_≤″_.proof ind)) where
  ind = ≤‴⇒≤″ x

m≤‴m+k : ∀{m n k}  m + k  n  m ≤‴ n
m≤‴m+k {m} {k = zero} refl = subst  z  m ≤‴ z) (sym (+-identityʳ m)) (≤‴-refl {m})
m≤‴m+k {m} {k = suc k} proof
  = ≤‴-step (m≤‴m+k {k = k} (trans (sym (+-suc m _)) proof))

≤″⇒≤‴ : ∀{m n}  m ≤″ n  m ≤‴ n
≤″⇒≤‴ (less-than-or-equal {k} proof) = m≤‴m+k proof

0≤‴n : ∀{n}  0 ≤‴ n
0≤‴n {n} = m≤‴m+k refl

<ᵇ⇒<‴ :  {m n}  T (m <ᵇ n)  m <‴ n
<ᵇ⇒<‴ {m} {n} leq = ≤″⇒≤‴ (<ᵇ⇒<″ leq)

<‴⇒<ᵇ :  {m n}  m <‴ n  T (m <ᵇ n)
<‴⇒<ᵇ leq = <″⇒<ᵇ (≤‴⇒≤″ leq)

infix 4 _<‴?_ _≤‴?_ _≥‴?_ _>‴?_

_<‴?_ : Decidable _<‴_
m <‴? n = map′ <ᵇ⇒<‴ <‴⇒<ᵇ (T? (m <ᵇ n))

_≤‴?_ : Decidable _≤‴_
zero ≤‴? n = yes 0≤‴n
suc m ≤‴? n = m <‴? n

_≥‴?_ : Decidable _≥‴_
_≥‴?_ = flip _≤‴?_

_>‴?_ : Decidable _>‴_
_>‴?_ = flip _<‴?_

≤⇒≤‴ : _≤_  _≤‴_
≤⇒≤‴ = ≤″⇒≤‴  ≤⇒≤″

≤‴⇒≤ : _≤‴_  _≤_
≤‴⇒≤ = ≤″⇒≤  ≤‴⇒≤″

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

-- If there is an injection from a type to ℕ, then the type has
-- decidable equality.

eq? :  {a} {A : Set a}  A    DecidableEquality A
eq? inj = via-injection inj _≟_

-- It's possible to decide existential and universal predicates up to
-- a limit.

module _ {p} {P : Pred  p} (P? : U.Decidable P) where

  anyUpTo? :  v  Dec ( λ n  n < v × P n)
  anyUpTo? zero    = no λ {(_ , () , _)}
  anyUpTo? (suc v) with P? v | anyUpTo? v
  ... | yes Pv | _                  = yes (v , ≤-refl , Pv)
  ... | _      | yes (n , n<v , Pn) = yes (n , ≤-step n<v , Pn)
  ... | no ¬Pv | no ¬Pn<v           = no ¬Pn<1+v
    where
    ¬Pn<1+v :  λ n  n < suc v × P n
    ¬Pn<1+v (n , s≤s n≤v , Pn) with n  v
    ... | yes refl = ¬Pv Pn
    ... | no  n≢v  = ¬Pn<v (n , ≤∧≢⇒< n≤v n≢v , Pn)

  allUpTo? :  v  Dec (∀ {n}  n < v  P n)
  allUpTo? zero    = yes λ()
  allUpTo? (suc v) with P? v | allUpTo? v
  ... | no ¬Pv | _        = no  prf  ¬Pv   (prf ≤-refl))
  ... | _      | no ¬Pn<v = no  prf  ¬Pn<v (prf  ≤-step))
  ... | yes Pn | yes Pn<v = yes Pn<1+v
    where
      Pn<1+v :  {n}  n < suc v  P n
      Pn<1+v {n} (s≤s n≤v) with n  v
      ... | yes refl = Pn
      ... | no  n≢v  = Pn<v (≤∧≢⇒< n≤v n≢v)



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

-- Version 1.3

∀[m≤n⇒m≢o]⇒o<n :  n o  (∀ {m}  m  n  m  o)  n < o
∀[m≤n⇒m≢o]⇒o<n = ∀[m≤n⇒m≢o]⇒n<o
{-# WARNING_ON_USAGE ∀[m≤n⇒m≢o]⇒o<n
"Warning: ∀[m≤n⇒m≢o]⇒o<n was deprecated in v1.3.
Please use ∀[m≤n⇒m≢o]⇒n<o instead."
#-}
∀[m<n⇒m≢o]⇒o≤n :  n o  (∀ {m}  m < n  m  o)  n  o
∀[m<n⇒m≢o]⇒o≤n = ∀[m<n⇒m≢o]⇒n≤o
{-# WARNING_ON_USAGE ∀[m<n⇒m≢o]⇒o≤n
"Warning: ∀[m<n⇒m≢o]⇒o≤n was deprecated in v1.3.
Please use ∀[m<n⇒m≢o]⇒n≤o instead."
#-}

-- Version 1.4

*-+-isSemiring = +-*-isSemiring
{-# WARNING_ON_USAGE *-+-isSemiring
"Warning: *-+-isSemiring was deprecated in v1.4.
Please use +-*-isSemiring instead."
#-}
*-+-isCommutativeSemiring = +-*-isCommutativeSemiring
{-# WARNING_ON_USAGE *-+-isCommutativeSemiring
"Warning: *-+-isCommutativeSemiring was deprecated in v1.4.
Please use +-*-isCommutativeSemiring instead."
#-}
*-+-semiring = +-*-semiring
{-# WARNING_ON_USAGE *-+-semiring
"Warning: *-+-semiring was deprecated in v1.4.
Please use +-*-semiring instead."
#-}
*-+-commutativeSemiring = +-*-commutativeSemiring
{-# WARNING_ON_USAGE *-+-commutativeSemiring
"Warning: *-+-commutativeSemiring was deprecated in v1.4.
Please use +-*-commutativeSemiring instead."
#-}

-- Version 1.6

∣m+n-m+o∣≡∣n-o| = ∣m+n-m+o∣≡∣n-o∣
{-# WARNING_ON_USAGE ∣m+n-m+o∣≡∣n-o|
"Warning: ∣m+n-m+o∣≡∣n-o| was deprecated in v1.6.
Please use ∣m+n-m+o∣≡∣n-o∣ instead. Note the final is a \\| rather than a |"
#-}
m≤n⇒n⊔m≡n = m≥n⇒m⊔n≡m
{-# WARNING_ON_USAGE m≤n⇒n⊔m≡n
"Warning: m≤n⇒n⊔m≡n was deprecated in v1.6. Please use m≥n⇒m⊔n≡m instead."
#-}
m≤n⇒n⊓m≡m = m≥n⇒m⊓n≡n
{-# WARNING_ON_USAGE m≤n⇒n⊓m≡m
"Warning: m≤n⇒n⊓m≡m was deprecated in v1.6. Please use m≥n⇒m⊓n≡n instead."
#-}
n⊔m≡m⇒n≤m = m⊔n≡n⇒m≤n
{-# WARNING_ON_USAGE n⊔m≡m⇒n≤m
"Warning: n⊔m≡m⇒n≤m was deprecated in v1.6. Please use m⊔n≡n⇒m≤n instead."
#-}
n⊔m≡n⇒m≤n = m⊔n≡m⇒n≤m
{-# WARNING_ON_USAGE n⊔m≡n⇒m≤n
"Warning: n⊔m≡n⇒m≤n was deprecated in v1.6. Please use m⊔n≡m⇒n≤m instead."
#-}
n≤m⊔n = m≤n⊔m
{-# WARNING_ON_USAGE n≤m⊔n
"Warning: n≤m⊔n was deprecated in v1.6. Please use m≤n⊔m instead."
#-}
⊔-least = ⊔-lub
{-# WARNING_ON_USAGE ⊔-least
"Warning: ⊔-least was deprecated in v1.6. Please use ⊔-lub instead."
#-}
⊓-greatest = ⊓-glb
{-# WARNING_ON_USAGE ⊓-greatest
"Warning: ⊓-greatest was deprecated in v1.6. Please use ⊓-glb instead."
#-}
⊔-pres-≤m = ⊔-lub
{-# WARNING_ON_USAGE ⊔-pres-≤m
"Warning: ⊔-pres-≤m was deprecated in v1.6. Please use ⊔-lub instead."
#-}
⊓-pres-m≤ = ⊓-glb
{-# WARNING_ON_USAGE ⊓-pres-m≤
"Warning: ⊓-pres-m≤ was deprecated in v1.6. Please use ⊓-glb instead."
#-}
⊔-abs-⊓ = ⊔-absorbs-⊓
{-# WARNING_ON_USAGE ⊔-abs-⊓
"Warning: ⊔-abs-⊓ was deprecated in v1.6. Please use ⊔-absorbs-⊓ instead."
#-}
⊓-abs-⊔ = ⊓-absorbs-⊔
{-# WARNING_ON_USAGE ⊓-abs-⊔
"Warning: ⊓-abs-⊔ was deprecated in v1.6. Please use ⊓-absorbs-⊔ instead."
#-}

-- Version 2.0

suc[pred[n]]≡n :  {n}  n  0  suc (pred n)  n
suc[pred[n]]≡n {zero}  0≢0 = contradiction refl 0≢0
suc[pred[n]]≡n {suc n} _   = refl
{-# WARNING_ON_USAGE suc[pred[n]]≡n
"Warning: suc[pred[n]]≡n was deprecated in v2.0. Please use suc-pred instead. Note that the proof now uses instance arguments"
#-}