Lots of speedups (#116)

Groups/Homomorphisms/Definition.agda

@@ -1,5 +1,6 @@
 {-# OPTIONS --safe --warning=error --without-K #-}
 
+open import Sets.EquivalenceRelations
 open import Setoids.Setoids
 open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
 open import Groups.Definition
@@ -12,6 +13,8 @@ record GroupHom {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A →
   field
     groupHom : {x y : A} → f (x ·A y) ∼ (f x) ·B (f y)
     wellDefined : {x y : A} → Setoid._∼_ S x y → f x ∼ f y
+  groupHom' : {x y : A} → (f x) ·B (f y) ∼ f (x ·A y)
+  groupHom' = Equivalence.symmetric eq groupHom
 
 record InjectiveGroupHom {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} {G : Group S _·A_} {H : Group T _·B_} {underf : A → B} (f : GroupHom G H underf) : Set (m ⊔ n ⊔ o ⊔ p) where
   open Setoid S renaming (_∼_ to _∼A_)

Groups/Homomorphisms/Lemmas.agda

@@ -42,3 +42,9 @@ zeroImpliesInverseZero {x} fx=0 = transitive homRespectsInverse (transitive (inv
   where
     open Setoid T
     open Equivalence eq
+
+homRespectsInverse' : {a b : A} → Setoid._∼_ T (f (Group.inverse G a) +B f (Group.inverse G b)) (Group.inverse H (f (b +A a)))
+homRespectsInverse' {a} {b} = transitive (symmetric (GroupHom.groupHom hom)) (transitive (GroupHom.wellDefined hom (Equivalence.symmetric (Setoid.eq S) (invContravariant G))) (homRespectsInverse))
+  where
+    open Setoid T
+    open Equivalence eq

Groups/Homomorphisms/Lemmas2.agda

@@ -5,7 +5,6 @@ open import Groups.Groups
 open import Groups.Homomorphisms.Definition
 open import Groups.Definition
 open import Numbers.Naturals.Naturals
-open import Setoids.Orders
 open import Setoids.Setoids
 open import Functions
 open import Sets.EquivalenceRelations