------------------------------------------------------------------------
-- The Agda standard library
--
-- Relations between properties of functions, such as associativity and
-- commutativity, when the underlying relation is a setoid
------------------------------------------------------------------------
{-# OPTIONS --cubical-compatible --safe #-}
open import Relation.Binary.Bundles using (Setoid)
open import Relation.Binary.Definitions
using (Substitutive; Symmetric; Total)
module Algebra.Consequences.Setoid {a ℓ} (S : Setoid a ℓ) where
import Algebra.Consequences.Base as Base
open import Algebra.Core
open import Data.Sum.Base using (inj₁; inj₂)
open import Data.Product.Base using (_,_)
open import Function.Base using (_$_; id; _∘_)
open import Function.Definitions
import Relation.Binary.Consequences as Bin
open import Relation.Binary.Core using (Rel)
open import Relation.Unary using (Pred)
open Setoid S renaming (Carrier to A)
open import Algebra.Definitions _≈_
open import Relation.Binary.Reasoning.Setoid S
------------------------------------------------------------------------
-- Re-exports
-- Export base lemmas that don't require the setoid
open Base public
hiding (module Congruence)
-- Export congruence lemmas using reflexivity
module Congruence {_∙_ : Op₂ A} (cong : Congruent₂ _∙_) where
open Base.Congruence _≈_ cong refl public
------------------------------------------------------------------------
-- MiddleFourExchange
module _ {_∙_ : Op₂ A} (cong : Congruent₂ _∙_) where
open Congruence cong
comm∧assoc⇒middleFour : Commutative _∙_ → Associative _∙_ →
_∙_ MiddleFourExchange _∙_
comm∧assoc⇒middleFour comm assoc w x y z = begin
(w ∙ x) ∙ (y ∙ z) ≈⟨ assoc w x (y ∙ z) ⟩
w ∙ (x ∙ (y ∙ z)) ≈⟨ ∙-congˡ (assoc x y z) ⟨
w ∙ ((x ∙ y) ∙ z) ≈⟨ ∙-congˡ (∙-congʳ (comm x y)) ⟩
w ∙ ((y ∙ x) ∙ z) ≈⟨ ∙-congˡ (assoc y x z) ⟩
w ∙ (y ∙ (x ∙ z)) ≈⟨ assoc w y (x ∙ z) ⟨
(w ∙ y) ∙ (x ∙ z) ∎
identity∧middleFour⇒assoc : {e : A} → Identity e _∙_ →
_∙_ MiddleFourExchange _∙_ →
Associative _∙_
identity∧middleFour⇒assoc {e} (identityˡ , identityʳ) middleFour x y z = begin
(x ∙ y) ∙ z ≈⟨ ∙-congˡ (identityˡ z) ⟨
(x ∙ y) ∙ (e ∙ z) ≈⟨ middleFour x y e z ⟩
(x ∙ e) ∙ (y ∙ z) ≈⟨ ∙-congʳ (identityʳ x) ⟩
x ∙ (y ∙ z) ∎
identity∧middleFour⇒comm : {_+_ : Op₂ A} {e : A} → Identity e _+_ →
_∙_ MiddleFourExchange _+_ →
Commutative _∙_
identity∧middleFour⇒comm {_+_} {e} (identityˡ , identityʳ) middleFour x y
= begin
x ∙ y ≈⟨ cong (identityˡ x) (identityʳ y) ⟨
(e + x) ∙ (y + e) ≈⟨ middleFour e x y e ⟩
(e + y) ∙ (x + e) ≈⟨ cong (identityˡ y) (identityʳ x) ⟩
y ∙ x ∎
------------------------------------------------------------------------
-- SelfInverse
module _ {f : Op₁ A} (self : SelfInverse f) where
selfInverse⇒involutive : Involutive f
selfInverse⇒involutive = reflexive∧selfInverse⇒involutive _≈_ refl self
selfInverse⇒congruent : Congruent _≈_ _≈_ f
selfInverse⇒congruent {x} {y} x≈y = sym (self (begin
f (f x) ≈⟨ selfInverse⇒involutive x ⟩
x ≈⟨ x≈y ⟩
y ∎))
selfInverse⇒inverseᵇ : Inverseᵇ _≈_ _≈_ f f
selfInverse⇒inverseᵇ = self ∘ sym , self ∘ sym
selfInverse⇒surjective : Surjective _≈_ _≈_ f
selfInverse⇒surjective y = f y , self ∘ sym
selfInverse⇒injective : Injective _≈_ _≈_ f
selfInverse⇒injective {x} {y} x≈y = begin
x ≈⟨ self x≈y ⟨
f (f y) ≈⟨ selfInverse⇒involutive y ⟩
y ∎
selfInverse⇒bijective : Bijective _≈_ _≈_ f
selfInverse⇒bijective = selfInverse⇒injective , selfInverse⇒surjective
------------------------------------------------------------------------
-- Magma-like structures
module _ {_∙_ : Op₂ A} (comm : Commutative _∙_) where
comm∧cancelˡ⇒cancelʳ : LeftCancellative _∙_ → RightCancellative _∙_
comm∧cancelˡ⇒cancelʳ cancelˡ x y z eq = cancelˡ x y z $ begin
x ∙ y ≈⟨ comm x y ⟩
y ∙ x ≈⟨ eq ⟩
z ∙ x ≈⟨ comm z x ⟩
x ∙ z ∎
comm∧cancelʳ⇒cancelˡ : RightCancellative _∙_ → LeftCancellative _∙_
comm∧cancelʳ⇒cancelˡ cancelʳ x y z eq = cancelʳ x y z $ begin
y ∙ x ≈⟨ comm y x ⟩
x ∙ y ≈⟨ eq ⟩
x ∙ z ≈⟨ comm x z ⟩
z ∙ x ∎
------------------------------------------------------------------------
-- Monoid-like structures
module _ {_∙_ : Op₂ A} (comm : Commutative _∙_) {e : A} where
comm∧idˡ⇒idʳ : LeftIdentity e _∙_ → RightIdentity e _∙_
comm∧idˡ⇒idʳ idˡ x = begin
x ∙ e ≈⟨ comm x e ⟩
e ∙ x ≈⟨ idˡ x ⟩
x ∎
comm∧idʳ⇒idˡ : RightIdentity e _∙_ → LeftIdentity e _∙_
comm∧idʳ⇒idˡ idʳ x = begin
e ∙ x ≈⟨ comm e x ⟩
x ∙ e ≈⟨ idʳ x ⟩
x ∎
comm∧idˡ⇒id : LeftIdentity e _∙_ → Identity e _∙_
comm∧idˡ⇒id idˡ = idˡ , comm∧idˡ⇒idʳ idˡ
comm∧idʳ⇒id : RightIdentity e _∙_ → Identity e _∙_
comm∧idʳ⇒id idʳ = comm∧idʳ⇒idˡ idʳ , idʳ
comm∧zeˡ⇒zeʳ : LeftZero e _∙_ → RightZero e _∙_
comm∧zeˡ⇒zeʳ zeˡ x = begin
x ∙ e ≈⟨ comm x e ⟩
e ∙ x ≈⟨ zeˡ x ⟩
e ∎
comm∧zeʳ⇒zeˡ : RightZero e _∙_ → LeftZero e _∙_
comm∧zeʳ⇒zeˡ zeʳ x = begin
e ∙ x ≈⟨ comm e x ⟩
x ∙ e ≈⟨ zeʳ x ⟩
e ∎
comm∧zeˡ⇒ze : LeftZero e _∙_ → Zero e _∙_
comm∧zeˡ⇒ze zeˡ = zeˡ , comm∧zeˡ⇒zeʳ zeˡ
comm∧zeʳ⇒ze : RightZero e _∙_ → Zero e _∙_
comm∧zeʳ⇒ze zeʳ = comm∧zeʳ⇒zeˡ zeʳ , zeʳ
comm∧almostCancelˡ⇒almostCancelʳ : AlmostLeftCancellative e _∙_ →
AlmostRightCancellative e _∙_
comm∧almostCancelˡ⇒almostCancelʳ cancelˡ-nonZero x y z x≉e yx≈zx =
cancelˡ-nonZero x y z x≉e $ begin
x ∙ y ≈⟨ comm x y ⟩
y ∙ x ≈⟨ yx≈zx ⟩
z ∙ x ≈⟨ comm z x ⟩
x ∙ z ∎
comm∧almostCancelʳ⇒almostCancelˡ : AlmostRightCancellative e _∙_ →
AlmostLeftCancellative e _∙_
comm∧almostCancelʳ⇒almostCancelˡ cancelʳ-nonZero x y z x≉e xy≈xz =
cancelʳ-nonZero x y z x≉e $ begin
y ∙ x ≈⟨ comm y x ⟩
x ∙ y ≈⟨ xy≈xz ⟩
x ∙ z ≈⟨ comm x z ⟩
z ∙ x ∎
------------------------------------------------------------------------
-- Group-like structures
module _ {_∙_ : Op₂ A} {_⁻¹ : Op₁ A} {e} (comm : Commutative _∙_) where
comm∧invˡ⇒invʳ : LeftInverse e _⁻¹ _∙_ → RightInverse e _⁻¹ _∙_
comm∧invˡ⇒invʳ invˡ x = begin
x ∙ (x ⁻¹) ≈⟨ comm x (x ⁻¹) ⟩
(x ⁻¹) ∙ x ≈⟨ invˡ x ⟩
e ∎
comm∧invˡ⇒inv : LeftInverse e _⁻¹ _∙_ → Inverse e _⁻¹ _∙_
comm∧invˡ⇒inv invˡ = invˡ , comm∧invˡ⇒invʳ invˡ
comm∧invʳ⇒invˡ : RightInverse e _⁻¹ _∙_ → LeftInverse e _⁻¹ _∙_
comm∧invʳ⇒invˡ invʳ x = begin
(x ⁻¹) ∙ x ≈⟨ comm (x ⁻¹) x ⟩
x ∙ (x ⁻¹) ≈⟨ invʳ x ⟩
e ∎
comm∧invʳ⇒inv : RightInverse e _⁻¹ _∙_ → Inverse e _⁻¹ _∙_
comm∧invʳ⇒inv invʳ = comm∧invʳ⇒invˡ invʳ , invʳ
module _ {_∙_ : Op₂ A} {_⁻¹ : Op₁ A} {e} (cong : Congruent₂ _∙_) where
open Congruence cong
assoc∧id∧invʳ⇒invˡ-unique : Associative _∙_ →
Identity e _∙_ → RightInverse e _⁻¹ _∙_ →
∀ x y → (x ∙ y) ≈ e → x ≈ (y ⁻¹)
assoc∧id∧invʳ⇒invˡ-unique assoc (idˡ , idʳ) invʳ x y eq = begin
x ≈⟨ idʳ x ⟨
x ∙ e ≈⟨ ∙-congˡ (invʳ y) ⟨
x ∙ (y ∙ (y ⁻¹)) ≈⟨ assoc x y (y ⁻¹) ⟨
(x ∙ y) ∙ (y ⁻¹) ≈⟨ ∙-congʳ eq ⟩
e ∙ (y ⁻¹) ≈⟨ idˡ (y ⁻¹) ⟩
y ⁻¹ ∎
assoc∧id∧invˡ⇒invʳ-unique : Associative _∙_ →
Identity e _∙_ → LeftInverse e _⁻¹ _∙_ →
∀ x y → (x ∙ y) ≈ e → y ≈ (x ⁻¹)
assoc∧id∧invˡ⇒invʳ-unique assoc (idˡ , idʳ) invˡ x y eq = begin
y ≈⟨ idˡ y ⟨
e ∙ y ≈⟨ ∙-congʳ (invˡ x) ⟨
((x ⁻¹) ∙ x) ∙ y ≈⟨ assoc (x ⁻¹) x y ⟩
(x ⁻¹) ∙ (x ∙ y) ≈⟨ ∙-congˡ eq ⟩
(x ⁻¹) ∙ e ≈⟨ idʳ (x ⁻¹) ⟩
x ⁻¹ ∎
------------------------------------------------------------------------
-- Bisemigroup-like structures
module _ {_∙_ _◦_ : Op₂ A}
(◦-cong : Congruent₂ _◦_)
(∙-comm : Commutative _∙_)
where
open Congruence ◦-cong renaming (∙-congˡ to ◦-congˡ)
comm∧distrˡ⇒distrʳ : _∙_ DistributesOverˡ _◦_ → _∙_ DistributesOverʳ _◦_
comm∧distrˡ⇒distrʳ distrˡ x y z = begin
(y ◦ z) ∙ x ≈⟨ ∙-comm (y ◦ z) x ⟩
x ∙ (y ◦ z) ≈⟨ distrˡ x y z ⟩
(x ∙ y) ◦ (x ∙ z) ≈⟨ ◦-cong (∙-comm x y) (∙-comm x z) ⟩
(y ∙ x) ◦ (z ∙ x) ∎
comm∧distrʳ⇒distrˡ : _∙_ DistributesOverʳ _◦_ → _∙_ DistributesOverˡ _◦_
comm∧distrʳ⇒distrˡ distrˡ x y z = begin
x ∙ (y ◦ z) ≈⟨ ∙-comm x (y ◦ z) ⟩
(y ◦ z) ∙ x ≈⟨ distrˡ x y z ⟩
(y ∙ x) ◦ (z ∙ x) ≈⟨ ◦-cong (∙-comm y x) (∙-comm z x) ⟩
(x ∙ y) ◦ (x ∙ z) ∎
comm∧distrˡ⇒distr : _∙_ DistributesOverˡ _◦_ → _∙_ DistributesOver _◦_
comm∧distrˡ⇒distr distrˡ = distrˡ , comm∧distrˡ⇒distrʳ distrˡ
comm∧distrʳ⇒distr : _∙_ DistributesOverʳ _◦_ → _∙_ DistributesOver _◦_
comm∧distrʳ⇒distr distrʳ = comm∧distrʳ⇒distrˡ distrʳ , distrʳ
comm⇒sym[distribˡ] : ∀ x → Symmetric (λ y z → (x ◦ (y ∙ z)) ≈ ((x ◦ y) ∙ (x ◦ z)))
comm⇒sym[distribˡ] x {y} {z} prf = begin
x ◦ (z ∙ y) ≈⟨ ◦-congˡ (∙-comm z y) ⟩
x ◦ (y ∙ z) ≈⟨ prf ⟩
(x ◦ y) ∙ (x ◦ z) ≈⟨ ∙-comm (x ◦ y) (x ◦ z) ⟩
(x ◦ z) ∙ (x ◦ y) ∎
module _ {_∙_ _◦_ : Op₂ A}
(∙-cong : Congruent₂ _∙_)
(∙-assoc : Associative _∙_)
(◦-comm : Commutative _◦_)
where
open Congruence ∙-cong
distrib∧absorbs⇒distribˡ : _∙_ Absorbs _◦_ →
_◦_ Absorbs _∙_ →
_◦_ DistributesOver _∙_ →
_∙_ DistributesOverˡ _◦_
distrib∧absorbs⇒distribˡ ∙-absorbs-◦ ◦-absorbs-∙ (◦-distribˡ-∙ , ◦-distribʳ-∙) x y z = begin
x ∙ (y ◦ z) ≈⟨ ∙-congʳ (∙-absorbs-◦ _ _) ⟨
(x ∙ (x ◦ y)) ∙ (y ◦ z) ≈⟨ ∙-congʳ (∙-congˡ (◦-comm _ _)) ⟩
(x ∙ (y ◦ x)) ∙ (y ◦ z) ≈⟨ ∙-assoc _ _ _ ⟩
x ∙ ((y ◦ x) ∙ (y ◦ z)) ≈⟨ ∙-congˡ (◦-distribˡ-∙ _ _ _) ⟨
x ∙ (y ◦ (x ∙ z)) ≈⟨ ∙-congʳ (◦-absorbs-∙ _ _) ⟨
(x ◦ (x ∙ z)) ∙ (y ◦ (x ∙ z)) ≈⟨ ◦-distribʳ-∙ _ _ _ ⟨
(x ∙ y) ◦ (x ∙ z) ∎
------------------------------------------------------------------------
-- Ring-like structures
module _ {_+_ _*_ : Op₂ A}
{_⁻¹ : Op₁ A} {0# : A}
(+-cong : Congruent₂ _+_)
(*-cong : Congruent₂ _*_)
where
open Congruence +-cong renaming (∙-congˡ to +-congˡ; ∙-congʳ to +-congʳ)
open Congruence *-cong renaming (∙-congˡ to *-congˡ; ∙-congʳ to *-congʳ)
assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ : Associative _+_ → _*_ DistributesOverʳ _+_ →
RightIdentity 0# _+_ → RightInverse 0# _⁻¹ _+_ →
LeftZero 0# _*_
assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ +-assoc distribʳ idʳ invʳ x = begin
0# * x ≈⟨ idʳ _ ⟨
(0# * x) + 0# ≈⟨ +-congˡ (invʳ _) ⟨
(0# * x) + ((0# * x) + ((0# * x)⁻¹)) ≈⟨ +-assoc _ _ _ ⟨
((0# * x) + (0# * x)) + ((0# * x)⁻¹) ≈⟨ +-congʳ (distribʳ _ _ _) ⟨
((0# + 0#) * x) + ((0# * x)⁻¹) ≈⟨ +-congʳ (*-congʳ (idʳ _)) ⟩
(0# * x) + ((0# * x)⁻¹) ≈⟨ invʳ _ ⟩
0# ∎
assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ : Associative _+_ → _*_ DistributesOverˡ _+_ →
RightIdentity 0# _+_ → RightInverse 0# _⁻¹ _+_ →
RightZero 0# _*_
assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ +-assoc distribˡ idʳ invʳ x = begin
x * 0# ≈⟨ idʳ _ ⟨
(x * 0#) + 0# ≈⟨ +-congˡ (invʳ _) ⟨
(x * 0#) + ((x * 0#) + ((x * 0#)⁻¹)) ≈⟨ +-assoc _ _ _ ⟨
((x * 0#) + (x * 0#)) + ((x * 0#)⁻¹) ≈⟨ +-congʳ (distribˡ _ _ _) ⟨
(x * (0# + 0#)) + ((x * 0#)⁻¹) ≈⟨ +-congʳ (*-congˡ (idʳ _)) ⟩
((x * 0#) + ((x * 0#)⁻¹)) ≈⟨ invʳ _ ⟩
0# ∎
------------------------------------------------------------------------
-- Without Loss of Generality
module _ {p} {f : Op₂ A} {P : Pred A p}
(≈-subst : Substitutive _≈_ p)
(comm : Commutative f)
where
subst∧comm⇒sym : Symmetric (λ a b → P (f a b))
subst∧comm⇒sym = ≈-subst P (comm _ _)
wlog : ∀ {r} {_R_ : Rel _ r} → Total _R_ →
(∀ a b → a R b → P (f a b)) →
∀ a b → P (f a b)
wlog r-total = Bin.wlog r-total subst∧comm⇒sym
------------------------------------------------------------------------
-- DEPRECATED NAMES
------------------------------------------------------------------------
-- Please use the new names as continuing support for the old names is
-- not guaranteed.
-- Version 2.0
comm+assoc⇒middleFour = comm∧assoc⇒middleFour
{-# WARNING_ON_USAGE comm+assoc⇒middleFour
"Warning: comm+assoc⇒middleFour was deprecated in v2.0.
Please use comm∧assoc⇒middleFour instead."
#-}
identity+middleFour⇒assoc = identity∧middleFour⇒assoc
{-# WARNING_ON_USAGE identity+middleFour⇒assoc
"Warning: identity+middleFour⇒assoc was deprecated in v2.0.
Please use identity∧middleFour⇒assoc instead."
#-}
identity+middleFour⇒comm = identity∧middleFour⇒comm
{-# WARNING_ON_USAGE identity+middleFour⇒comm
"Warning: identity+middleFour⇒comm was deprecated in v2.0.
Please use identity∧middleFour⇒comm instead."
#-}
comm+cancelˡ⇒cancelʳ = comm∧cancelˡ⇒cancelʳ
{-# WARNING_ON_USAGE comm+cancelˡ⇒cancelʳ
"Warning: comm+cancelˡ⇒cancelʳ was deprecated in v2.0.
Please use comm∧cancelˡ⇒cancelʳ instead."
#-}
comm+cancelʳ⇒cancelˡ = comm∧cancelʳ⇒cancelˡ
{-# WARNING_ON_USAGE comm+cancelʳ⇒cancelˡ
"Warning: comm+cancelʳ⇒cancelˡ was deprecated in v2.0.
Please use comm∧cancelʳ⇒cancelˡ instead."
#-}
comm+idˡ⇒idʳ = comm∧idˡ⇒idʳ
{-# WARNING_ON_USAGE comm+idˡ⇒idʳ
"Warning: comm+idˡ⇒idʳ was deprecated in v2.0.
Please use comm∧idˡ⇒idʳ instead."
#-}
comm+idʳ⇒idˡ = comm∧idʳ⇒idˡ
{-# WARNING_ON_USAGE comm+idʳ⇒idˡ
"Warning: comm+idʳ⇒idˡ was deprecated in v2.0.
Please use comm∧idʳ⇒idˡ instead."
#-}
comm+zeˡ⇒zeʳ = comm∧zeˡ⇒zeʳ
{-# WARNING_ON_USAGE comm+zeˡ⇒zeʳ
"Warning: comm+zeˡ⇒zeʳ was deprecated in v2.0.
Please use comm∧zeˡ⇒zeʳ instead."
#-}
comm+zeʳ⇒zeˡ = comm∧zeʳ⇒zeˡ
{-# WARNING_ON_USAGE comm+zeʳ⇒zeˡ
"Warning: comm+zeʳ⇒zeˡ was deprecated in v2.0.
Please use comm∧zeʳ⇒zeˡ instead."
#-}
comm+almostCancelˡ⇒almostCancelʳ = comm∧almostCancelˡ⇒almostCancelʳ
{-# WARNING_ON_USAGE comm+almostCancelˡ⇒almostCancelʳ
"Warning: comm+almostCancelˡ⇒almostCancelʳ was deprecated in v2.0.
Please use comm∧almostCancelˡ⇒almostCancelʳ instead."
#-}
comm+almostCancelʳ⇒almostCancelˡ = comm∧almostCancelʳ⇒almostCancelˡ
{-# WARNING_ON_USAGE comm+almostCancelʳ⇒almostCancelˡ
"Warning: comm+almostCancelʳ⇒almostCancelˡ was deprecated in v2.0.
Please use comm∧almostCancelʳ⇒almostCancelˡ instead."
#-}
comm+invˡ⇒invʳ = comm∧invˡ⇒invʳ
{-# WARNING_ON_USAGE comm+invˡ⇒invʳ
"Warning: comm+invˡ⇒invʳ was deprecated in v2.0.
Please use comm∧invˡ⇒invʳ instead."
#-}
comm+invʳ⇒invˡ = comm∧invʳ⇒invˡ
{-# WARNING_ON_USAGE comm+invʳ⇒invˡ
"Warning: comm+invʳ⇒invˡ was deprecated in v2.0.
Please use comm∧invʳ⇒invˡ instead."
#-}
comm+invˡ⇒inv = comm∧invˡ⇒inv
{-# WARNING_ON_USAGE comm+invˡ⇒inv
"Warning: comm+invˡ⇒inv was deprecated in v2.0.
Please use comm∧invˡ⇒inv instead."
#-}
comm+invʳ⇒inv = comm∧invʳ⇒inv
{-# WARNING_ON_USAGE comm+invʳ⇒inv
"Warning: comm+invʳ⇒inv was deprecated in v2.0.
Please use comm∧invʳ⇒inv instead."
#-}
comm+distrˡ⇒distrʳ = comm∧distrˡ⇒distrʳ
{-# WARNING_ON_USAGE comm+distrˡ⇒distrʳ
"Warning: comm+distrˡ⇒distrʳ was deprecated in v2.0.
Please use comm∧distrˡ⇒distrʳ instead."
#-}
comm+distrʳ⇒distrˡ = comm∧distrʳ⇒distrˡ
{-# WARNING_ON_USAGE comm+distrʳ⇒distrˡ
"Warning: comm+distrʳ⇒distrˡ was deprecated in v2.0.
Please use comm∧distrʳ⇒distrˡ instead."
#-}
assoc+distribʳ+idʳ+invʳ⇒zeˡ = assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ
{-# WARNING_ON_USAGE assoc+distribʳ+idʳ+invʳ⇒zeˡ
"Warning: assoc+distribʳ+idʳ+invʳ⇒zeˡ was deprecated in v2.0.
Please use assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ instead."
#-}
assoc+distribˡ+idʳ+invʳ⇒zeʳ = assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ
{-# WARNING_ON_USAGE assoc+distribˡ+idʳ+invʳ⇒zeʳ
"Warning: assoc+distribˡ+idʳ+invʳ⇒zeʳ was deprecated in v2.0.
Please use assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ instead."
#-}
assoc+id+invʳ⇒invˡ-unique = assoc∧id∧invʳ⇒invˡ-unique
{-# WARNING_ON_USAGE assoc+id+invʳ⇒invˡ-unique
"Warning: assoc+id+invʳ⇒invˡ-unique was deprecated in v2.0.
Please use assoc∧id∧invʳ⇒invˡ-unique instead."
#-}
assoc+id+invˡ⇒invʳ-unique = assoc∧id∧invˡ⇒invʳ-unique
{-# WARNING_ON_USAGE assoc+id+invˡ⇒invʳ-unique
"Warning: assoc+id+invˡ⇒invʳ-unique was deprecated in v2.0.
Please use assoc∧id∧invˡ⇒invʳ-unique instead."
#-}
subst+comm⇒sym = subst∧comm⇒sym
{-# WARNING_ON_USAGE subst+comm⇒sym
"Warning: subst+comm⇒sym was deprecated in v2.0.
Please use subst∧comm⇒sym instead."
#-}
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