Classical.Theories.Group¶

The equational theory of groups¶

This is the Classical.Theories.Group module of the Agda Universal Algebra Library.

Th-Group has five equations: the three monoid equations — associativity, left identity, right identity — composed from the same builders Th-Monoid uses, plus the two inverse laws LeftInverse and RightInverse from Classical.Equations applied to Sig-Group's symbols. The inverse laws are the first equations in the Classical/ tree to mention a unary operation symbol (⁻¹-Op); their arity-conformance evidence is the triple refl refl refl for the binary ∙-Op, the unary ⁻¹-Op, and the nullary ε-Op respectively. The variable carrier is uniformly Fin 3 as for Th-Monoid (per ADR-002 v2 §2); the identity and inverse equations use 0F and ignore 1F, 2F.

{-# OPTIONS --cubical-compatible --exact-split --safe #-}

module Classical.Theories.Group where

-- Imports from Agda and the Agda Standard Library ----------------------------
open import Agda.Primitive                         using () renaming ( Set to Type )
open import Data.Fin.Base                          using ( Fin )
open import Data.Fin.Patterns                      using ( 0F ; 1F ; 2F )
open import Data.Product                           using ( _×_ )
open import Relation.Binary.PropositionalEquality  using ( refl )

-- Imports from the Agda Universal Algebra Library ----------------------------
open import Classical.Signatures.Group             using ( Sig-Group ; ∙-Op ; ε-Op ; ⁻¹-Op )
open import Classical.Equations                    using ( Associative ; LeftIdentity ; RightIdentity
                                                         ; LeftInverse ; RightInverse )
open import Overture.Terms {𝑆 = Sig-Group}         using ( Term )

data Eq-Group : Type where
  assoc idˡ idʳ invˡ invʳ : Eq-Group

Th-Group : Eq-Group → Term (Fin 3) × Term (Fin 3)
Th-Group assoc = Associative   ∙-Op             refl           0F 1F 2F
Th-Group idˡ   = LeftIdentity  ∙-Op ε-Op        refl refl      0F
Th-Group idʳ   = RightIdentity ∙-Op ε-Op        refl refl      0F
Th-Group invˡ  = LeftInverse   ∙-Op ⁻¹-Op ε-Op  refl refl refl 0F
Th-Group invʳ  = RightInverse  ∙-Op ⁻¹-Op ε-Op  refl refl refl 0F