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Consequences.agda
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Consequences.agda
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Relations between properties of functions, such as associativity and
-- commutativity
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
open import Relation.Binary using (Rel; Setoid; Substitutive; Symmetric; Total)
module Algebra.FunctionProperties.Consequences
{a ℓ} (S : Setoid a ℓ) where
open Setoid S renaming (Carrier to A)
open import Algebra.FunctionProperties _≈_
open import Data.Sum using (inj₁; inj₂)
open import Data.Product using (_,_)
import Relation.Binary.Consequences as Bin
open import Relation.Binary.EqReasoning S
open import Relation.Unary using (Pred)
------------------------------------------------------------------------
-- Magma-like structures
module _ {_•_ : Op₂ A} (comm : Commutative _•_) where
comm+cancelˡ⇒cancelʳ : LeftCancellative _•_ → RightCancellative _•_
comm+cancelˡ⇒cancelʳ cancelˡ {x} y z eq = cancelˡ x (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 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+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 ∎
------------------------------------------------------------------------
-- 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+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 ∎
module _ {_•_ : Op₂ A} {_⁻¹ : Op₁ A} {e} (cong : Congruent₂ _•_) where
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 ≈⟨ sym (idʳ x) ⟩
x • e ≈⟨ cong refl (sym (invʳ y)) ⟩
x • (y • (y ⁻¹)) ≈⟨ sym (assoc x y (y ⁻¹)) ⟩
(x • y) • (y ⁻¹) ≈⟨ cong eq refl ⟩
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 ≈⟨ sym (idˡ y) ⟩
e • y ≈⟨ cong (sym (invˡ x)) refl ⟩
((x ⁻¹) • x) • y ≈⟨ assoc (x ⁻¹) x y ⟩
(x ⁻¹) • (x • y) ≈⟨ cong refl eq ⟩
(x ⁻¹) • e ≈⟨ idʳ (x ⁻¹) ⟩
x ⁻¹ ∎
----------------------------------------------------------------------
-- Bisemigroup-like structures
module _ {_•_ _◦_ : Op₂ A}
(◦-cong : Congruent₂ _◦_)
(•-comm : Commutative _•_)
where
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⇒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) ≈⟨ ◦-cong refl (•-comm z y) ⟩
x ◦ (y • z) ≈⟨ prf ⟩
(x ◦ y) • (x ◦ z) ≈⟨ •-comm (x ◦ y) (x ◦ z) ⟩
(x ◦ z) • (x ◦ y) ∎
----------------------------------------------------------------------
-- Ring-like structures
module _ {_+_ _*_ : Op₂ A}
{_⁻¹ : Op₁ A} {0# : A}
(+-cong : Congruent₂ _+_)
(*-cong : Congruent₂ _*_)
where
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 ≈⟨ sym (idʳ _) ⟩
(0# * x) + 0# ≈⟨ +-cong refl (sym (invʳ _)) ⟩
(0# * x) + ((0# * x) + ((0# * x)⁻¹)) ≈⟨ sym (+-assoc _ _ _) ⟩
((0# * x) + (0# * x)) + ((0# * x)⁻¹) ≈⟨ +-cong (sym (distribʳ _ _ _)) refl ⟩
((0# + 0#) * x) + ((0# * x)⁻¹) ≈⟨ +-cong (*-cong (idʳ _) refl) refl ⟩
(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# ≈⟨ sym (idʳ _) ⟩
(x * 0#) + 0# ≈⟨ +-cong refl (sym (invʳ _)) ⟩
(x * 0#) + ((x * 0#) + ((x * 0#)⁻¹)) ≈⟨ sym (+-assoc _ _ _) ⟩
((x * 0#) + (x * 0#)) + ((x * 0#)⁻¹) ≈⟨ +-cong (sym (distribˡ _ _ _)) refl ⟩
(x * (0# + 0#)) + ((x * 0#)⁻¹) ≈⟨ +-cong (*-cong refl (idʳ _)) refl ⟩
((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