{-# OPTIONS --without-K --safe #-}
module Data.Nat.Properties where
open import Axiom.UniquenessOfIdentityProofs
open import Algebra
open import Algebra.Morphism
open import Algebra.FunctionProperties.Consequences.Propositional
open import Data.Bool.Base using (Bool; false; true; T)
open import Data.Bool.Properties using (T?)
open import Data.Empty
open import Data.Nat.Base
open import Data.Product
open import Data.Sum
open import Data.Unit using (tt)
open import Function.Core
open import Function.Injection using (_↣_)
open import Level using (0ℓ)
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)
open import Algebra.FunctionProperties {A = ℕ} _≡_
hiding (LeftCancellative; RightCancellative; Cancellative)
open import Algebra.FunctionProperties
using (LeftCancellative; RightCancellative; Cancellative)
open import Algebra.Structures {A = ℕ} _≡_
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)
infix 4 _≟_
_≟_ : Decidable {A = ℕ} _≡_
m ≟ n = map′ (≡ᵇ⇒≡ m n) (≡⇒≡ᵇ m n) (T? (m ≡ᵇ n))
≡-irrelevant : Irrelevant {A = ℕ} _≡_
≡-irrelevant = Decidable⇒UIP.≡-irrelevant _≟_
≟-diag : ∀ {m n} (eq : m ≡ n) → (m ≟ n) ≡ yes eq
≟-diag = ≡-≟-identity _≟_
≡-isDecEquivalence : IsDecEquivalence (_≡_ {A = ℕ})
≡-isDecEquivalence = record
{ isEquivalence = isEquivalence
; _≟_ = _≟_
}
≡-decSetoid : DecSetoid 0ℓ 0ℓ
≡-decSetoid = record
{ Carrier = ℕ
; _≈_ = _≡_
; 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
<ᵇ⇒< : ∀ m n → T (m <ᵇ n) → m < n
<ᵇ⇒< zero (suc n) m<n = s≤s z≤n
<ᵇ⇒< (suc m) (suc n) m<n = s≤s (<ᵇ⇒< m n m<n)
<⇒<ᵇ : ∀ {m n} → m < n → T (m <ᵇ n)
<⇒<ᵇ (s≤s z≤n) = tt
<⇒<ᵇ (s≤s (s≤s m<n)) = <⇒<ᵇ (s≤s m<n)
-------------
- Propertie o
≤-pred : ∀ {m n} → suc m ≤ suc n → m ≤ n
≤-pred (s≤s m≤n) = m≤n
≤-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)
infix 4 _≤?_ _≥?_
_≤?_ : Decidable _≤_
zero ≤? _ = yes z≤n
suc m ≤? n with T? (m <ᵇ n)
... | yes m<n = yes (<ᵇ⇒< m n m<n)
... | no m≮n = no (m≮n ∘ <⇒<ᵇ)
_≥?_ : Decidable _≥_
_≥?_ = flip _≤?_
≤-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
; _≟_ = _≟_
; _≤?_ = _≤?_
}
≤-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
}
s≤s-injective : ∀ {m n} {p q : m ≤ n} → s≤s p ≡ s≤s q → p ≡ q
s≤s-injective refl = refl
≤-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 le) = 1+n≰n le
n≤0⇒n≡0 : ∀ {n} → n ≤ 0 → n ≡ 0
n≤0⇒n≡0 z≤n = refl
<⇒≤ : _<_ ⇒ _≤_
<⇒≤ (s≤s m≤n) = ≤-trans m≤n (≤-step ≤-refl)
<⇒≢ : _<_ ⇒ _≢_
<⇒≢ m<n refl = 1+n≰n m<n
≤⇒≯ : _≤_ ⇒ _≯_
≤⇒≯ (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} _ = s≤s z≤n
≰⇒> {suc m} {suc n} m≰n = s≤s (≰⇒> (m≰n ∘ s≤s))
≰⇒≥ : _≰_ ⇒ _≥_
≰⇒≥ = <⇒≤ ∘ ≰⇒>
≮⇒≥ : _≮_ ⇒ _≥_
≮⇒≥ {_} {zero} _ = z≤n
≮⇒≥ {zero} {suc j} 1≮j+1 = contradiction (s≤s z≤n) 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 = s≤s z≤n
≤∧≢⇒< {_} {suc n} (s≤s m≤n) 1+m≢1+n =
s≤s (≤∧≢⇒< m≤n (1+m≢1+n ∘ cong suc))
≤-<-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
<-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)
<-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 (_< _)
<-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
}
n≮n : ∀ n → n ≮ n
n≮n n = <-irrefl (refl {x = n})
0<1+n : ∀ {n} → 0 < suc n
0<1+n = s≤s z≤n
n<1+n : ∀ n → n < suc n
n<1+n n = ≤-refl
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⇒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 (s≤s z≤n) = z≤n
m<n⇒m≤1+n (s≤s (s≤s m<n)) = s≤s (m<n⇒m≤1+n (s≤s m<n))
∀[m≤n⇒m≢o]⇒o<n : ∀ n o → (∀ {m} → m ≤ n → m ≢ o) → n < o
∀[m≤n⇒m≢o]⇒o<n _ zero m≤n⇒n≢0 = contradiction refl (m≤n⇒n≢0 z≤n)
∀[m≤n⇒m≢o]⇒o<n zero (suc o) _ = 0<1+n
∀[m≤n⇒m≢o]⇒o<n (suc n) (suc o) m≤n⇒n≢o = s≤s (∀[m≤n⇒m≢o]⇒o<n 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]⇒o≤n : ∀ n o → (∀ {m} → m < n → m ≢ o) → n ≤ o
∀[m<n⇒m≢o]⇒o≤n zero n _ = z≤n
∀[m<n⇒m≢o]⇒o≤n (suc n) zero m<n⇒m≢0 = contradiction refl (m<n⇒m≢0 0<1+n)
∀[m<n⇒m≢o]⇒o≤n (suc n) (suc o) m<n⇒m≢o = s≤s (∀[m<n⇒m≢o]⇒o≤n n o rec)
where
rec : ∀ {m} → m < n → m ≢ o
rec x<m refl = m<n⇒m≢o (s≤s x<m) refl
module ≤-Reasoning where
open import Relation.Binary.Reasoning.Base.Triple
≤-isPreorder
<-trans
(resp₂ _<_)
<⇒≤
<-transˡ
<-transʳ
public
hiding (_≈⟨_⟩_)
open ≤-Reasoning
pred-mono : pred Preserves _≤_ ⟶ _≤_
pred-mono z≤n = z≤n
pred-mono (s≤s le) = le
≤pred⇒≤ : ∀ {m n} → m ≤ pred n → m ≤ n
≤pred⇒≤ {m} {zero} le = le
≤pred⇒≤ {m} {suc n} le = ≤-step le
≤⇒pred≤ : ∀ {m n} → m ≤ n → pred m ≤ n
≤⇒pred≤ {zero} 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]]≡n : ∀ {n} → n ≢ 0 → suc (pred n) ≡ n
suc[pred[n]]≡n {zero} n≢0 = contradiction refl n≢0
suc[pred[n]]≡n {suc n} n≢0 = refl
+-suc : ∀ m n → m + suc n ≡ suc (m + n)
+-suc zero n = refl
+-suc (suc m) n = cong suc (+-suc m n)
+-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-≡
+-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
}
+-rawMagma : RawMagma 0ℓ 0ℓ
+-rawMagma = record
{ _≈_ = _≡_
; _∙_ = _+_
}
+-0-rawMonoid : RawMonoid 0ℓ 0ℓ
+-0-rawMonoid = record
{ _≈_ = _≡_
; _∙_ = _+_
; ε = 0
}
+-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
}
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≢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))
+-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 ∘ subst (_≤ m + o) (sym (+-suc m n))
+-cancelʳ-< : RightCancellative _<_ _+_
+-cancelʳ-< n o n+m<o+m = +-cancelʳ-≤ (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⇒m<n∨m≡n : ∀ {m n} → m ≤ n → m < n ⊎ m ≡ n
m≤n⇒m<n∨m≡n {0} {0} _ = inj₂ refl
m≤n⇒m<n∨m≡n {0} {suc n} _ = inj₁ 0<1+n
m≤n⇒m<n∨m≡n {suc m} {suc n} (s≤s m≤n) with m≤n⇒m<n∨m≡n m≤n
... | inj₂ m≡n = inj₂ (cong suc m≡n)
... | inj₁ m<n = inj₁ (s≤s 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} (s≤s z≤n) o≤p = s≤s (≤-stepsˡ n o≤p)
+-mono-<-≤ {_} {_} (s≤s (s≤s m<n)) o≤p = s≤s (+-mono-<-≤ (s≤s 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) le = m+1+n≰m m (≤-pred le)
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≮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)
*-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 (λ x → suc (x + m * n)) (+-comm n m) ⟩
suc (m + n + m * n) ≡⟨ cong suc (+-assoc m n (m * n)) ⟩
suc (m + (n + m * n)) ≡⟨⟩
suc m + suc m * n ∎
*-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) ∎
*-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
}
*-+-isSemiring : IsSemiring _+_ _*_ 0 1
*-+-isSemiring = record
{ isSemiringWithoutAnnihilatingZero = record
{ +-isCommutativeMonoid = +-0-isCommutativeMonoid
; *-isMonoid = *-1-isMonoid
; distrib = *-distrib-+
}
; zero = *-zero
}
*-+-isCommutativeSemiring : IsCommutativeSemiring _+_ _*_ 0 1
*-+-isCommutativeSemiring = record
{ +-isCommutativeMonoid = +-0-isCommutativeMonoid
; *-isCommutativeMonoid = *-1-isCommutativeMonoid
; distribʳ = *-distribʳ-+
; zeroˡ = *-zeroˡ
}
*-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
}
*-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
}
*-cancelʳ-≡ : ∀ m n {o} → m * suc o ≡ n * suc o → m ≡ n
*-cancelʳ-≡ zero zero eq = refl
*-cancelʳ-≡ (suc m) (suc n) {o} eq =
cong suc (*-cancelʳ-≡ m n (+-cancelˡ-≡ (suc o) eq))
*-cancelˡ-≡ : ∀ {m n} o → suc o * m ≡ suc o * n → m ≡ n
*-cancelˡ-≡ {m} {n} o eq = *-cancelʳ-≡ m n
(subst₂ _≡_ (*-comm (suc o) m) (*-comm (suc o) n) eq)
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≡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)
*-cancelʳ-≤ : ∀ m n o → m * suc o ≤ n * suc o → m ≤ n
*-cancelʳ-≤ zero _ _ _ = z≤n
*-cancelʳ-≤ (suc m) (suc n) o le =
s≤s (*-cancelʳ-≤ m n o (+-cancelˡ-≤ (suc o) le))
*-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-< (s≤s z≤n) (s≤s u≤v) = s≤s z≤n
*-mono-< (s≤s (s≤s m≤n)) (s≤s u≤v) =
+-mono-< (s≤s u≤v) (*-mono-< (s≤s m≤n) (s≤s u≤v))
*-monoˡ-< : ∀ n → (_* suc n) Preserves _<_ ⟶ _<_
*-monoˡ-< n (s≤s z≤n) = s≤s z≤n
*-monoˡ-< n (s≤s (s≤s m≤o)) =
+-mono-≤-< (≤-refl {suc n}) (*-monoˡ-< n (s≤s m≤o))
*-monoʳ-< : ∀ n → (suc n *_) Preserves _<_ ⟶ _<_
*-monoʳ-< zero (s≤s m≤o) = +-mono-≤ (s≤s m≤o) z≤n
*-monoʳ-< (suc n) (s≤s m≤o) =
+-mono-≤ (s≤s m≤o) (<⇒≤ (*-monoʳ-< n (s≤s m≤o)))
m≤m*n : ∀ m {n} → 0 < n → m ≤ m * n
m≤m*n m {n} 0<n = begin
m ≡⟨ sym (*-identityʳ m) ⟩
m * 1 ≤⟨ *-monoʳ-≤ m 0<n ⟩
m * n ∎
m<m*n : ∀ {m n} → 0 < m → 1 < n → m < m * n
m<m*n {m@(suc m-1)} {n@(suc (suc n-2))} (s≤s _) (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 0<1+n) ⟩
n + m-1 * n ≡⟨⟩
m * 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-< n o (+-cancelˡ-< m nm<om))
*-cancelˡ-< : LeftCancellative _<_ _*_
*-cancelˡ-< x {y} {z} rewrite *-comm x y | *-comm x z = *-cancelʳ-< y z
*-cancel-< : Cancellative _<_ _*_
*-cancel-< = *-cancelˡ-< , *-cancelʳ-<
^-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)
⊔-assoc : Associative _⊔_
⊔-assoc zero _ _ = refl
⊔-assoc (suc m) zero o = refl
⊔-assoc (suc m) (suc n) zero = refl
⊔-assoc (suc m) (suc n) (suc o) = cong suc $ ⊔-assoc m n o
⊔-identityˡ : LeftIdentity 0 _⊔_
⊔-identityˡ _ = refl
⊔-identityʳ : RightIdentity 0 _⊔_
⊔-identityʳ zero = refl
⊔-identityʳ (suc n) = refl
⊔-identity : Identity 0 _⊔_
⊔-identity = ⊔-identityˡ , ⊔-identityʳ
⊔-comm : Commutative _⊔_
⊔-comm zero n = sym $ ⊔-identityʳ n
⊔-comm (suc m) zero = refl
⊔-comm (suc m) (suc n) = cong suc (⊔-comm m n)
⊔-sel : Selective _⊔_
⊔-sel zero _ = inj₂ refl
⊔-sel (suc m) zero = inj₁ refl
⊔-sel (suc m) (suc n) with ⊔-sel m n
... | inj₁ m⊔n≡m = inj₁ (cong suc m⊔n≡m)
... | inj₂ m⊔n≡n = inj₂ (cong suc m⊔n≡n)
⊔-idem : Idempotent _⊔_
⊔-idem = sel⇒idem ⊔-sel
⊔-least : ∀ {m n o} → m ≤ o → n ≤ o → m ⊔ n ≤ o
⊔-least {m} {n} m≤o n≤o with ⊔-sel m n
... | inj₁ m⊔n≡m rewrite m⊔n≡m = m≤o
... | inj₂ m⊔n≡n rewrite m⊔n≡n = n≤o
⊔-isMagma : IsMagma _⊔_
⊔-isMagma = record
{ isEquivalence = isEquivalence
; ∙-cong = cong₂ _⊔_
}
⊔-isSemigroup : IsSemigroup _⊔_
⊔-isSemigroup = record
{ isMagma = ⊔-isMagma
; assoc = ⊔-assoc
}
⊔-isBand : IsBand _⊔_
⊔-isBand = record
{ isSemigroup = ⊔-isSemigroup
; idem = ⊔-idem
}
⊔-isSemilattice : IsSemilattice _⊔_
⊔-isSemilattice = record
{ isBand = ⊔-isBand
; comm = ⊔-comm
}
⊔-0-isCommutativeMonoid : IsCommutativeMonoid _⊔_ 0
⊔-0-isCommutativeMonoid = record
{ isSemigroup = ⊔-isSemigroup
; identityˡ = ⊔-identityˡ
; comm = ⊔-comm
}
⊔-magma : Magma 0ℓ 0ℓ
⊔-magma = record
{ isMagma = ⊔-isMagma
}
⊔-semigroup : Semigroup 0ℓ 0ℓ
⊔-semigroup = record
{ isSemigroup = ⊔-isSemigroup
}
⊔-band : Band 0ℓ 0ℓ
⊔-band = record
{ isBand = ⊔-isBand
}
⊔-semilattice : Semilattice 0ℓ 0ℓ
⊔-semilattice = record
{ isSemilattice = ⊔-isSemilattice
}
⊔-0-commutativeMonoid : CommutativeMonoid 0ℓ 0ℓ
⊔-0-commutativeMonoid = record
{ isCommutativeMonoid = ⊔-0-isCommutativeMonoid
}
⊔-triangulate : ∀ m n o → m ⊔ n ⊔ o ≡ (m ⊔ n) ⊔ (n ⊔ o)
⊔-triangulate m n o = begin-equality
m ⊔ n ⊔ o ≡⟨ cong (λ v → m ⊔ v ⊔ o) (sym (⊔-idem n)) ⟩
m ⊔ (n ⊔ n) ⊔ o ≡⟨ ⊔-assoc m _ _ ⟩
m ⊔ ((n ⊔ n) ⊔ o) ≡⟨ cong (m ⊔_) (⊔-assoc n _ _) ⟩
m ⊔ (n ⊔ (n ⊔ o)) ≡⟨ sym (⊔-assoc m _ _) ⟩
(m ⊔ n) ⊔ (n ⊔ o) ∎
m≤m⊔n : ∀ m n → m ≤ m ⊔ n
m≤m⊔n zero _ = z≤n
m≤m⊔n (suc m) zero = ≤-refl
m≤m⊔n (suc m) (suc n) = s≤s $ m≤m⊔n m n
n≤m⊔n : ∀ m n → n ≤ m ⊔ n
n≤m⊔n m n = subst (n ≤_) (⊔-comm n m) (m≤m⊔n n m)
m≤n⇒n⊔m≡n : ∀ {m n} → m ≤ n → n ⊔ m ≡ n
m≤n⇒n⊔m≡n z≤n = ⊔-identityʳ _
m≤n⇒n⊔m≡n (s≤s m≤n) = cong suc (m≤n⇒n⊔m≡n m≤n)
m≤n⇒m⊔n≡n : ∀ {m n} → m ≤ n → m ⊔ n ≡ n
m≤n⇒m⊔n≡n {m} m≤n = trans (⊔-comm m _) (m≤n⇒n⊔m≡n m≤n)
n⊔m≡m⇒n≤m : ∀ {m n} → n ⊔ m ≡ m → n ≤ m
n⊔m≡m⇒n≤m n⊔m≡m = subst (_ ≤_) n⊔m≡m (m≤m⊔n _ _)
n⊔m≡n⇒m≤n : ∀ {m n} → n ⊔ m ≡ n → m ≤ n
n⊔m≡n⇒m≤n n⊔m≡n = subst (_ ≤_) n⊔m≡n (n≤m⊔n _ _)
m≤n⇒m≤n⊔o : ∀ {m n} o → m ≤ n → m ≤ n ⊔ o
m≤n⇒m≤n⊔o o m≤n = ≤-trans m≤n (m≤m⊔n _ o)
m≤n⇒m≤o⊔n : ∀ {m n} o → m ≤ n → m ≤ o ⊔ n
m≤n⇒m≤o⊔n n m≤n = ≤-trans m≤n (n≤m⊔n n _)
m⊔n≤o⇒m≤o : ∀ m n {o} → m ⊔ n ≤ o → m ≤ o
m⊔n≤o⇒m≤o m n m⊔n≤o = ≤-trans (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 = ≤-trans (n≤m⊔n 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<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 (n≤m⊔n m n) m⊔n<o
⊔-mono-≤ : _⊔_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_
⊔-mono-≤ {m} {n} {u} {v} m≤n u≤v with ⊔-sel m u
... | inj₁ m⊔u≡m rewrite m⊔u≡m = ≤-trans m≤n (m≤m⊔n n v)
... | inj₂ m⊔u≡u rewrite m⊔u≡u = ≤-trans u≤v (n≤m⊔n n 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-< = ⊔-mono-≤
+-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
⊓-assoc : Associative _⊓_
⊓-assoc zero _ _ = refl
⊓-assoc (suc m) zero o = refl
⊓-assoc (suc m) (suc n) zero = refl
⊓-assoc (suc m) (suc n) (suc o) = cong suc $ ⊓-assoc m n o
⊓-zeroˡ : LeftZero 0 _⊓_
⊓-zeroˡ _ = refl
⊓-zeroʳ : RightZero 0 _⊓_
⊓-zeroʳ zero = refl
⊓-zeroʳ (suc n) = refl
⊓-zero : Zero 0 _⊓_
⊓-zero = ⊓-zeroˡ , ⊓-zeroʳ
⊓-comm : Commutative _⊓_
⊓-comm zero n = sym $ ⊓-zeroʳ n
⊓-comm (suc m) zero = refl
⊓-comm (suc m) (suc n) = cong suc (⊓-comm m n)
⊓-sel : Selective _⊓_
⊓-sel zero _ = inj₁ refl
⊓-sel (suc m) zero = inj₂ refl
⊓-sel (suc m) (suc n) with ⊓-sel m n
... | inj₁ m⊓n≡m = inj₁ (cong suc m⊓n≡m)
... | inj₂ m⊓n≡n = inj₂ (cong suc m⊓n≡n)
⊓-idem : Idempotent _⊓_
⊓-idem = sel⇒idem ⊓-sel
⊓-greatest : ∀ {m n o} → m ≥ o → n ≥ o → m ⊓ n ≥ o
⊓-greatest {m} {n} m≥o n≥o with ⊓-sel m n
... | inj₁ m⊓n≡m rewrite m⊓n≡m = m≥o
... | inj₂ m⊓n≡n rewrite m⊓n≡n = n≥o
⊓-distribʳ-⊔ : _⊓_ DistributesOverʳ _⊔_
⊓-distribʳ-⊔ (suc m) (suc n) (suc o) = cong suc $ ⊓-distribʳ-⊔ m n o
⊓-distribʳ-⊔ (suc m) (suc n) zero = cong suc $ refl
⊓-distribʳ-⊔ (suc m) zero o = refl
⊓-distribʳ-⊔ zero n o = begin-equality
(n ⊔ o) ⊓ 0 ≡⟨ ⊓-comm (n ⊔ o) 0 ⟩
0 ⊓ (n ⊔ o) ≡⟨⟩
0 ⊓ n ⊔ 0 ⊓ o ≡⟨ ⊓-comm 0 n ⟨ cong₂ _⊔_ ⟩ ⊓-comm 0 o ⟩
n ⊓ 0 ⊔ o ⊓ 0 ∎
⊓-distribˡ-⊔ : _⊓_ DistributesOverˡ _⊔_
⊓-distribˡ-⊔ = comm+distrʳ⇒distrˡ ⊓-comm ⊓-distribʳ-⊔
⊓-distrib-⊔ : _⊓_ DistributesOver _⊔_
⊓-distrib-⊔ = ⊓-distribˡ-⊔ , ⊓-distribʳ-⊔
⊔-abs-⊓ : _⊔_ Absorbs _⊓_
⊔-abs-⊓ zero n = refl
⊔-abs-⊓ (suc m) zero = refl
⊔-abs-⊓ (suc m) (suc n) = cong suc $ ⊔-abs-⊓ m n
⊓-abs-⊔ : _⊓_ Absorbs _⊔_
⊓-abs-⊔ zero n = refl
⊓-abs-⊔ (suc m) (suc n) = cong suc $ ⊓-abs-⊔ m n
⊓-abs-⊔ (suc m) zero = cong suc $ begin-equality
m ⊓ m ≡⟨ cong (m ⊓_) $ sym $ ⊔-identityʳ m ⟩
m ⊓ (m ⊔ 0) ≡⟨ ⊓-abs-⊔ m zero ⟩
m ∎
⊓-⊔-absorptive : Absorptive _⊓_ _⊔_
⊓-⊔-absorptive = ⊓-abs-⊔ , ⊔-abs-⊓
⊓-isMagma : IsMagma _⊓_
⊓-isMagma = record
{ isEquivalence = isEquivalence
; ∙-cong = cong₂ _⊓_
}
⊓-isSemigroup : IsSemigroup _⊓_
⊓-isSemigroup = record
{ isMagma = ⊓-isMagma
; assoc = ⊓-assoc
}
⊓-isBand : IsBand _⊓_
⊓-isBand = record
{ isSemigroup = ⊓-isSemigroup
; idem = ⊓-idem
}
⊓-isSemilattice : IsSemilattice _⊓_
⊓-isSemilattice = record
{ isBand = ⊓-isBand
; comm = ⊓-comm
}
⊔-⊓-isSemiringWithoutOne : IsSemiringWithoutOne _⊔_ _⊓_ 0
⊔-⊓-isSemiringWithoutOne = record
{ +-isCommutativeMonoid = ⊔-0-isCommutativeMonoid
; *-isSemigroup = ⊓-isSemigroup
; distrib = ⊓-distrib-⊔
; zero = ⊓-zero
}
⊔-⊓-isCommutativeSemiringWithoutOne
: IsCommutativeSemiringWithoutOne _⊔_ _⊓_ 0
⊔-⊓-isCommutativeSemiringWithoutOne = record
{ isSemiringWithoutOne = ⊔-⊓-isSemiringWithoutOne
; *-comm = ⊓-comm
}
⊓-⊔-isLattice : IsLattice _⊓_ _⊔_
⊓-⊔-isLattice = record
{ isEquivalence = isEquivalence
; ∨-comm = ⊓-comm
; ∨-assoc = ⊓-assoc
; ∨-cong = cong₂ _⊓_
; ∧-comm = ⊔-comm
; ∧-assoc = ⊔-assoc
; ∧-cong = cong₂ _⊔_
; absorptive = ⊓-⊔-absorptive
}
⊓-⊔-isDistributiveLattice : IsDistributiveLattice _⊓_ _⊔_
⊓-⊔-isDistributiveLattice = record
{ isLattice = ⊓-⊔-isLattice
; ∨-distribʳ-∧ = ⊓-distribʳ-⊔
}
⊓-magma : Magma 0ℓ 0ℓ
⊓-magma = record
{ isMagma = ⊓-isMagma
}
⊓-semigroup : Semigroup 0ℓ 0ℓ
⊓-semigroup = record
{ isSemigroup = ⊔-isSemigroup
}
⊓-band : Band 0ℓ 0ℓ
⊓-band = record
{ isBand = ⊓-isBand
}
⊓-semilattice : Semilattice 0ℓ 0ℓ
⊓-semilattice = record
{ isSemilattice = ⊓-isSemilattice
}
⊔-⊓-commutativeSemiringWithoutOne : CommutativeSemiringWithoutOne 0ℓ 0ℓ
⊔-⊓-commutativeSemiringWithoutOne = record
{ isCommutativeSemiringWithoutOne =
⊔-⊓-isCommutativeSemiringWithoutOne
}
⊓-⊔-lattice : Lattice 0ℓ 0ℓ
⊓-⊔-lattice = record
{ isLattice = ⊓-⊔-isLattice
}
⊓-⊔-distributiveLattice : DistributiveLattice 0ℓ 0ℓ
⊓-⊔-distributiveLattice = record
{ isDistributiveLattice = ⊓-⊔-isDistributiveLattice
}
⊓-triangulate : ∀ m n o → m ⊓ n ⊓ o ≡ (m ⊓ n) ⊓ (n ⊓ o)
⊓-triangulate m n o = begin-equality
m ⊓ n ⊓ o ≡⟨ sym (cong (λ v → m ⊓ v ⊓ o) (⊓-idem n)) ⟩
m ⊓ (n ⊓ n) ⊓ o ≡⟨ ⊓-assoc m _ _ ⟩
m ⊓ ((n ⊓ n) ⊓ o) ≡⟨ cong (m ⊓_) (⊓-assoc n _ _) ⟩
m ⊓ (n ⊓ (n ⊓ o)) ≡⟨ sym (⊓-assoc m _ _) ⟩
(m ⊓ n) ⊓ (n ⊓ o) ∎
m⊓n≤m : ∀ m n → m ⊓ n ≤ m
m⊓n≤m zero _ = z≤n
m⊓n≤m (suc m) zero = z≤n
m⊓n≤m (suc m) (suc n) = s≤s $ m⊓n≤m m n
m⊓n≤n : ∀ m n → m ⊓ n ≤ n
m⊓n≤n m n = subst (_≤ n) (⊓-comm n m) (m⊓n≤m n m)
m≤n⇒m⊓n≡m : ∀ {m n} → m ≤ n → m ⊓ n ≡ m
m≤n⇒m⊓n≡m z≤n = refl
m≤n⇒m⊓n≡m (s≤s m≤n) = cong suc (m≤n⇒m⊓n≡m m≤n)
m≤n⇒n⊓m≡m : ∀ {m n} → m ≤ n → n ⊓ m ≡ m
m≤n⇒n⊓m≡m {m} m≤n = trans (⊓-comm _ m) (m≤n⇒m⊓n≡m m≤n)
m⊓n≡m⇒m≤n : ∀ {m n} → m ⊓ n ≡ m → m ≤ n
m⊓n≡m⇒m≤n m⊓n≡m = subst (_≤ _) m⊓n≡m (m⊓n≤n _ _)
m⊓n≡n⇒n≤m : ∀ {m n} → m ⊓ n ≡ n → n ≤ m
m⊓n≡n⇒n≤m m⊓n≡n = subst (_≤ _) m⊓n≡n (m⊓n≤m _ _)
m≤n⇒m⊓o≤n : ∀ {m n} o → m ≤ n → m ⊓ o ≤ n
m≤n⇒m⊓o≤n o m≤n = ≤-trans (m⊓n≤m _ o) m≤n
m≤n⇒o⊓m≤n : ∀ {m n} o → m ≤ n → o ⊓ m ≤ n
m≤n⇒o⊓m≤n n m≤n = ≤-trans (m⊓n≤n n _) m≤n
m≤n⊓o⇒m≤n : ∀ {m} n o → m ≤ n ⊓ o → m ≤ n
m≤n⊓o⇒m≤n n o m≤n⊓o = ≤-trans m≤n⊓o (m⊓n≤m n o)
m≤n⊓o⇒m≤o : ∀ {m} n o → m ≤ n ⊓ o → m ≤ o
m≤n⊓o⇒m≤o n o m≤n⊓o = ≤-trans m≤n⊓o (m⊓n≤n n o)
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-≤ {m} {n} {u} {v} m≤n u≤v with ⊓-sel n v
... | inj₁ n⊓v≡n rewrite n⊓v≡n = ≤-trans (m⊓n≤m m u) m≤n
... | inj₂ n⊓v≡v rewrite n⊓v≡v = ≤-trans (m⊓n≤n m u) 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-< = ⊓-mono-≤
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 = z≤n
m⊓n≤m⊔n (suc m) (suc n) = s≤s (m⊓n≤m⊔n m n)
+-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
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
n∸m≤n : ∀ m n → n ∸ m ≤ n
n∸m≤n zero n = ≤-refl
n∸m≤n (suc m) zero = ≤-refl
n∸m≤n (suc m) (suc n) = ≤-trans (n∸m≤n m n) (n≤1+n n)
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 (n∸m≤n 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
∸-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 (n∸m≤n 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)
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
+-∸-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 m n zero = cong (m ∸_) (sym $ +-identityʳ n)
∸-+-assoc zero zero (suc o) = refl
∸-+-assoc zero (suc n) (suc o) = refl
∸-+-assoc (suc m) zero (suc o) = refl
∸-+-assoc (suc m) (suc n) (suc o) = ∸-+-assoc m n (suc o)
+-∸-assoc : ∀ m {n o} → o ≤ n → (m + n) ∸ o ≡ m + (n ∸ o)
+-∸-assoc m (z≤n {n = n}) = begin-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 (n∸m≤n n m) ⟩
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
*-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)) ∎))
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 zero zero = sym (⊔-idem m)
∸-distribˡ-⊓-⊔ zero zero (suc o) = refl
∸-distribˡ-⊓-⊔ zero (suc n) zero = refl
∸-distribˡ-⊓-⊔ zero (suc n) (suc o) = refl
∸-distribˡ-⊓-⊔ (suc m) (suc n) zero = sym (m≤n⇒m⊔n≡n (≤-step (n∸m≤n n m)))
∸-distribˡ-⊓-⊔ (suc m) zero (suc o) = sym (m≤n⇒n⊔m≡n (≤-step (n∸m≤n o m)))
∸-distribˡ-⊓-⊔ (suc m) (suc n) (suc o) = ∸-distribˡ-⊓-⊔ m n o
∸-distribʳ-⊓ : _∸_ DistributesOverʳ _⊓_
∸-distribʳ-⊓ zero n o = refl
∸-distribʳ-⊓ (suc m) zero o = refl
∸-distribʳ-⊓ (suc m) (suc n) zero = sym (⊓-zeroʳ (n ∸ m))
∸-distribʳ-⊓ (suc m) (suc n) (suc o) = ∸-distribʳ-⊓ m n o
∸-distribˡ-⊔-⊓ : ∀ m n o → m ∸ (n ⊔ o) ≡ (m ∸ n) ⊓ (m ∸ o)
∸-distribˡ-⊔-⊓ m zero zero = sym (⊓-idem m)
∸-distribˡ-⊔-⊓ zero zero o = 0∸n≡0 o
∸-distribˡ-⊔-⊓ zero (suc n) o = 0∸n≡0 (suc n ⊔ o)
∸-distribˡ-⊔-⊓ (suc m) (suc n) zero = sym (m≤n⇒m⊓n≡m (≤-step (n∸m≤n n m)))
∸-distribˡ-⊔-⊓ (suc m) zero (suc o) = sym (m≤n⇒n⊓m≡m (≤-step (n∸m≤n o m)))
∸-distribˡ-⊔-⊓ (suc m) (suc n) (suc o) = ∸-distribˡ-⊔-⊓ m n o
∸-distribʳ-⊔ : _∸_ DistributesOverʳ _⊔_
∸-distribʳ-⊔ zero n o = refl
∸-distribʳ-⊔ (suc m) zero o = refl
∸-distribʳ-⊔ (suc m) (suc n) zero = sym (⊔-identityʳ (n ∸ m))
∸-distribʳ-⊔ (suc m) (suc n) (suc o) = ∸-distribʳ-⊔ m n o
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
... | 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 ∣ ∎
... | inj₂ n≤m = *-distribˡ-∣-∣-aux a n m n≤m
*-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)
⌊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)
≤′-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
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)
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) _)
≤″⇒≤ : _≤″_ ⇒ _≤_
≤″⇒≤ {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
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
≤‴⇒≤″ : ∀{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 (λ 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
eq? : ∀ {a} {A : Set a} → A ↣ ℕ → Decidable {A = A} _≡_
eq? inj = via-injection inj _≟_
_*-mono_ = *-mono-≤
{-# WARNING_ON_USAGE _*-mono_
"Warning: _*-mono_ was deprecated in v0.14.
Please use *-mono-≤ instead."
#-}
_+-mono_ = +-mono-≤
{-# WARNING_ON_USAGE _+-mono_
"Warning: _+-mono_ was deprecated in v0.14.
Please use +-mono-≤ instead."
#-}
+-right-identity = +-identityʳ
{-# WARNING_ON_USAGE +-right-identity
"Warning: +-right-identity was deprecated in v0.14.
Please use +-identityʳ instead."
#-}
*-right-zero = *-zeroʳ
{-# WARNING_ON_USAGE *-right-zero
"Warning: *-right-zero was deprecated in v0.14.
Please use *-zeroʳ instead."
#-}
distribʳ-*-+ = *-distribʳ-+
{-# WARNING_ON_USAGE distribʳ-*-+
"Warning: distribʳ-*-+ was deprecated in v0.14.
Please use *-distribʳ-+ instead."
#-}
*-distrib-∸ʳ = *-distribʳ-∸
{-# WARNING_ON_USAGE *-distrib-∸ʳ
"Warning: *-distrib-∸ʳ was deprecated in v0.14.
Please use *-distribʳ-∸ instead."
#-}
cancel-+-left = +-cancelˡ-≡
{-# WARNING_ON_USAGE cancel-+-left
"Warning: cancel-+-left was deprecated in v0.14.
Please use +-cancelˡ-≡ instead."
#-}
cancel-+-left-≤ = +-cancelˡ-≤
{-# WARNING_ON_USAGE cancel-+-left-≤
"Warning: cancel-+-left-≤ was deprecated in v0.14.
Please use +-cancelˡ-≤ instead."
#-}
cancel-*-right = *-cancelʳ-≡
{-# WARNING_ON_USAGE cancel-*-right
"Warning: cancel-*-right was deprecated in v0.14.
Please use *-cancelʳ-≡ instead."
#-}
cancel-*-right-≤ = *-cancelʳ-≤
{-# WARNING_ON_USAGE cancel-*-right-≤
"Warning: cancel-*-right-≤ was deprecated in v0.14.
Please use *-cancelʳ-≤ instead."
#-}
strictTotalOrder = <-strictTotalOrder
{-# WARNING_ON_USAGE strictTotalOrder
"Warning: strictTotalOrder was deprecated in v0.14.
Please use <-strictTotalOrder instead."
#-}
isCommutativeSemiring = *-+-isCommutativeSemiring
{-# WARNING_ON_USAGE isCommutativeSemiring
"Warning: isCommutativeSemiring was deprecated in v0.14.
Please use *-+-isCommutativeSemiring instead."
#-}
commutativeSemiring = *-+-commutativeSemiring
{-# WARNING_ON_USAGE commutativeSemiring
"Warning: commutativeSemiring was deprecated in v0.14.
Please use *-+-commutativeSemiring instead."
#-}
isDistributiveLattice = ⊓-⊔-isDistributiveLattice
{-# WARNING_ON_USAGE isDistributiveLattice
"Warning: isDistributiveLattice was deprecated in v0.14.
Please use ⊓-⊔-isDistributiveLattice instead."
#-}
distributiveLattice = ⊓-⊔-distributiveLattice
{-# WARNING_ON_USAGE distributiveLattice
"Warning: distributiveLattice was deprecated in v0.14.
Please use ⊓-⊔-distributiveLattice instead."
#-}
⊔-⊓-0-isSemiringWithoutOne = ⊔-⊓-isSemiringWithoutOne
{-# WARNING_ON_USAGE ⊔-⊓-0-isSemiringWithoutOne
"Warning: ⊔-⊓-0-isSemiringWithoutOne was deprecated in v0.14.
Please use ⊔-⊓-isSemiringWithoutOne instead."
#-}
⊔-⊓-0-isCommutativeSemiringWithoutOne = ⊔-⊓-isCommutativeSemiringWithoutOne
{-# WARNING_ON_USAGE ⊔-⊓-0-isCommutativeSemiringWithoutOne
"Warning: ⊔-⊓-0-isCommutativeSemiringWithoutOne was deprecated in v0.14.
Please use ⊔-⊓-isCommutativeSemiringWithoutOne instead."
#-}
⊔-⊓-0-commutativeSemiringWithoutOne = ⊔-⊓-commutativeSemiringWithoutOne
{-# WARNING_ON_USAGE ⊔-⊓-0-commutativeSemiringWithoutOne
"Warning: ⊔-⊓-0-commutativeSemiringWithoutOne was deprecated in v0.14.
Please use ⊔-⊓-commutativeSemiringWithoutOne instead."
#-}
¬i+1+j≤i = m+1+n≰m
{-# WARNING_ON_USAGE ¬i+1+j≤i
"Warning: ¬i+1+j≤i was deprecated in v0.15.
Please use m+1+n≰m instead."
#-}
≤-steps = ≤-stepsˡ
{-# WARNING_ON_USAGE ≤-steps
"Warning: ≤-steps was deprecated in v0.15.
Please use ≤-stepsˡ instead."
#-}
i∸k∸j+j∸k≡i+j∸k : ∀ i j k → i ∸ (k ∸ j) + (j ∸ k) ≡ i + j ∸ k
i∸k∸j+j∸k≡i+j∸k zero j k = cong (_+ (j ∸ k)) (0∸n≡0 (k ∸ j))
i∸k∸j+j∸k≡i+j∸k (suc i) j zero = cong (λ x → suc i ∸ x + j) (0∸n≡0 j)
i∸k∸j+j∸k≡i+j∸k (suc i) zero (suc k) = begin-equality
i ∸ k + 0 ≡⟨ +-identityʳ _ ⟩
i ∸ k ≡⟨ cong (_∸ k) (sym (+-identityʳ _)) ⟩
i + 0 ∸ k ∎
i∸k∸j+j∸k≡i+j∸k (suc i) (suc j) (suc k) = begin-equality
suc i ∸ (k ∸ j) + (j ∸ k) ≡⟨ i∸k∸j+j∸k≡i+j∸k (suc i) j k ⟩
suc i + j ∸ k ≡⟨ cong (_∸ k) (sym (+-suc i j)) ⟩
i + suc j ∸ k ∎
{-# WARNING_ON_USAGE i∸k∸j+j∸k≡i+j∸k
"Warning: i∸k∸j+j∸k≡i+j∸k was deprecated in v0.17."
#-}
im≡jm+n⇒[i∸j]m≡n : ∀ i j m n → i * m ≡ j * m + n → (i ∸ j) * m ≡ n
im≡jm+n⇒[i∸j]m≡n i j m n eq = begin-equality
(i ∸ j) * m ≡⟨ *-distribʳ-∸ m i j ⟩
(i * m) ∸ (j * m) ≡⟨ cong (_∸ j * m) eq ⟩
(j * m + n) ∸ (j * m) ≡⟨ cong (_∸ j * m) (+-comm (j * m) n) ⟩
(n + j * m) ∸ (j * m) ≡⟨ m+n∸n≡m n (j * m) ⟩
n ∎
{-# WARNING_ON_USAGE im≡jm+n⇒[i∸j]m≡n
"Warning: im≡jm+n⇒[i∸j]m≡n was deprecated in v0.17."
#-}
≤+≢⇒< = ≤∧≢⇒<
{-# WARNING_ON_USAGE ≤+≢⇒<
"Warning: ≤+≢⇒< was deprecated in v0.17.
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."
#-}
i+1+j≢i = m+1+n≢m
{-# WARNING_ON_USAGE i+1+j≢i
"Warning: i+1+j≢i was deprecated in v1.1.
Please use m+1+n≢m instead."
#-}
i+j≡0⇒i≡0 = m+n≡0⇒m≡0
{-# WARNING_ON_USAGE i+j≡0⇒i≡0
"Warning: i+j≡0⇒i≡0 was deprecated in v1.1.
Please use m+n≡0⇒m≡0 instead."
#-}
i+j≡0⇒j≡0 = m+n≡0⇒n≡0
{-# WARNING_ON_USAGE i+j≡0⇒j≡0
"Warning: i+j≡0⇒j≡0 was deprecated in v1.1.
Please use m+n≡0⇒n≡0 instead."
#-}
i+1+j≰i = m+1+n≰m
{-# WARNING_ON_USAGE i+1+j≰i
"Warning: i+1+j≰i was deprecated in v1.1.
Please use m+1+n≰m instead."
#-}
i*j≡0⇒i≡0∨j≡0 = m*n≡0⇒m≡0∨n≡0
{-# WARNING_ON_USAGE i*j≡0⇒i≡0∨j≡0
"Warning: i*j≡0⇒i≡0∨j≡0 was deprecated in v1.1.
Please use m*n≡0⇒m≡0∨n≡0 instead."
#-}
i*j≡1⇒i≡1 = m*n≡1⇒m≡1
{-# WARNING_ON_USAGE i*j≡1⇒i≡1
"Warning: i*j≡1⇒i≡1 was deprecated in v1.1.
Please use m*n≡1⇒m≡1 instead."
#-}
i*j≡1⇒j≡1 = m*n≡1⇒n≡1
{-# WARNING_ON_USAGE i*j≡1⇒j≡1
"Warning: i*j≡1⇒j≡1 was deprecated in v1.1.
Please use m*n≡1⇒n≡1 instead."
#-}
i^j≡0⇒i≡0 = m^n≡0⇒m≡0
{-# WARNING_ON_USAGE i^j≡0⇒i≡0
"Warning: i^j≡0⇒i≡0 was deprecated in v1.1.
Please use m^n≡0⇒m≡0 instead."
#-}
i^j≡1⇒j≡0∨i≡1 = m^n≡1⇒n≡0∨m≡1
{-# WARNING_ON_USAGE i^j≡1⇒j≡0∨i≡1
"Warning: i^j≡1⇒j≡0∨i≡1 was deprecated in v1.1.
Please use m^n≡1⇒n≡0∨m≡1 instead."
#-}
[i+j]∸[i+k]≡j∸k = [m+n]∸[m+o]≡n∸o
{-# WARNING_ON_USAGE [i+j]∸[i+k]≡j∸k
"Warning: [i+j]∸[i+k]≡j∸k was deprecated in v1.1.
Please use [m+n]∸[m+o]≡n∸o instead."
#-}
m≢0⇒suc[pred[m]]≡m = suc[pred[n]]≡n
{-# WARNING_ON_USAGE m≢0⇒suc[pred[m]]≡m
"Warning: m≢0⇒suc[pred[m]]≡m was deprecated in v1.1.
Please use suc[pred[n]]≡n instead."
#-}
n≡m⇒∣n-m∣≡0 = m≡n⇒∣m-n∣≡0
{-# WARNING_ON_USAGE n≡m⇒∣n-m∣≡0
"Warning: n≡m⇒∣n-m∣≡0 was deprecated in v1.1.
Please use m≡n⇒∣m-n∣≡0 instead."
#-}
∣n-m∣≡0⇒n≡m = ∣m-n∣≡0⇒m≡n
{-# WARNING_ON_USAGE ∣n-m∣≡0⇒n≡m
"Warning: ∣n-m∣≡0⇒n≡m was deprecated in v1.1.
Please use ∣m-n∣≡0⇒m≡n instead."
#-}
∣n-m∣≡n∸m⇒m≤n = ∣m-n∣≡m∸n⇒n≤m
{-# WARNING_ON_USAGE ∣n-m∣≡n∸m⇒m≤n
"Warning: ∣n-m∣≡n∸m⇒m≤n was deprecated in v1.1.
Please use ∣m-n∣≡m∸n⇒n≤m instead."
#-}
∣n-n+m∣≡m = ∣m-m+n∣≡n
{-# WARNING_ON_USAGE ∣n-n+m∣≡m
"Warning: ∣n-n+m∣≡m was deprecated in v1.1.
Please use ∣m-m+n∣≡n instead."
#-}
∣n+m-n+o∣≡∣m-o| = ∣m+n-m+o∣≡∣n-o|
{-# WARNING_ON_USAGE ∣n+m-n+o∣≡∣m-o|
"Warning: ∣n+m-n+o∣≡∣m-o| was deprecated in v1.1.
Please use ∣m+n-m+o∣≡∣n-o| instead."
#-}
n∸m≤∣n-m∣ = m∸n≤∣m-n∣
{-# WARNING_ON_USAGE n∸m≤∣n-m∣
"Warning: n∸m≤∣n-m∣ was deprecated in v1.1.
Please use m∸n≤∣m-n∣ instead."
#-}
∣n-m∣≤n⊔m = ∣m-n∣≤m⊔n
{-# WARNING_ON_USAGE ∣n-m∣≤n⊔m
"Warning: ∣n-m∣≤n⊔m was deprecated in v1.1.
Please use ∣m-n∣≤m⊔n instead."
#-}
n≤m+n : ∀ m n → n ≤ m + n
n≤m+n m n = subst (n ≤_) (+-comm n m) (m≤m+n n m)
{-# WARNING_ON_USAGE n≤m+n
"Warning: n≤m+n was deprecated in v1.1.
Please use m≤n+m instead (note, you will need to switch the argument order)."
#-}
n≤m+n∸m : ∀ m n → n ≤ m + (n ∸ m)
n≤m+n∸m m zero = z≤n
n≤m+n∸m zero (suc n) = ≤-refl
n≤m+n∸m (suc m) (suc n) = s≤s (n≤m+n∸m m n)
{-# WARNING_ON_USAGE n≤m+n∸m
"Warning: n≤m+n∸m was deprecated in v1.1.
Please use m≤n+m∸n instead (note, you will need to switch the argument order)."
#-}
∣n-m∣≡[n∸m]∨[m∸n] : ∀ m n → (∣ n - m ∣ ≡ n ∸ m) ⊎ (∣ n - m ∣ ≡ m ∸ n)
∣n-m∣≡[n∸m]∨[m∸n] m n with ≤-total m n
... | inj₁ m≤n = inj₁ $ m≤n⇒∣n-m∣≡n∸m m≤n
... | inj₂ n≤m = inj₂ $ begin-equality
∣ n - m ∣ ≡⟨ ∣-∣-comm n m ⟩
∣ m - n ∣ ≡⟨ m≤n⇒∣n-m∣≡n∸m n≤m ⟩
m ∸ n ∎
{-# WARNING_ON_USAGE ∣n-m∣≡[n∸m]∨[m∸n]
"Warning: ∣n-m∣≡[n∸m]∨[m∸n] was deprecated in v1.1.
Please use ∣m-n∣≡[m∸n]∨[n∸m] instead (note, you will need to switch the argument order)."
#-}
+-*-suc = *-suc
{-# WARNING_ON_USAGE +-*-suc
"Warning: +-*-suc was deprecated in v1.2.
Please use *-suc instead."
#-}