Lots of speedups (#116)

Groups/Cyclic/Definition.agda

@@ -7,13 +7,13 @@ open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
 open import Numbers.Naturals.Semiring
 open import Numbers.Integers.Integers
 open import Groups.Definition
-open import Groups.Lemmas
 
 module Groups.Cyclic.Definition {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} (G : Group S _·_) where
 
 open Setoid S
 open Group G
 open Equivalence eq
+open import Groups.Lemmas G
 
 positiveEltPower : (x : A) (i : ℕ) → A
 positiveEltPower x 0 = Group.0G G
@@ -40,9 +40,10 @@ elementPowerWellDefinedZ' i j i=j {g} = identityOfIndiscernablesRight _∼_ refl
 
 elementPowerWellDefinedG : (g h : A) → (g ∼ h) → {n : ℤ} → (elementPower g n) ∼ (elementPower h n)
 elementPowerWellDefinedG g h g=h {nonneg n} = positiveEltPowerWellDefinedG g h g=h n
-elementPowerWellDefinedG g h g=h {negSucc x} = inverseWellDefined G (+WellDefined g=h (positiveEltPowerWellDefinedG g h g=h x))
+elementPowerWellDefinedG g h g=h {negSucc x} = inverseWellDefined (+WellDefined g=h (positiveEltPowerWellDefinedG g h g=h x))
 
 record CyclicGroup : Set (m ⊔ n) where
   field
     generator : A
     cyclic : {a : A} → (Sg ℤ (λ i → Setoid._∼_ S (elementPower generator i) a))
+

Groups/FreeProduct/Addition.agda

@@ -15,15 +15,18 @@ module Groups.FreeProduct.Addition {i : _} {I : Set i} (decidableIndex : (x y :
 open import Groups.FreeProduct.Definition decidableIndex decidableGroups G
 open import Groups.FreeProduct.Setoid decidableIndex decidableGroups G
 
+prependEqual : (i : I) (g : A i) .(nonZero : (Setoid._∼_ (S i) g (Group.0G (G i))) → False) → (j : I) → i ≡ j → (w : ReducedSequenceBeginningWith j) → ReducedSequence
+prependEqual i g nonzero .i refl (ofEmpty .i h nonZero) with decidableGroups i (_+_ i g h) (Group.0G (G i))
+prependEqual i g nonzero .i refl (ofEmpty .i h nonZero) | inl sumZero = empty
+prependEqual i g nonzero .i refl (ofEmpty .i h nonZero) | inr sumNotZero = nonempty i (ofEmpty i (_+_ i g h) sumNotZero)
+prependEqual i g nonzero .i refl (prependLetter .i h nonZero x x₁) with decidableGroups i (_+_ i g h) (Group.0G (G i))
+prependEqual i g nonzero .i refl (prependLetter .i h nonZero {j} x x₁) | inl sumZero = nonempty j x
+prependEqual i g nonzero .i refl (prependLetter .i h nonZero x pr) | inr sumNotZero = nonempty i (prependLetter i (_+_ i g h) sumNotZero x pr)
+
 prepend : (i : I) → (g : A i) → .(nonZero : (Setoid._∼_ (S i) g (Group.0G (G i))) → False) → ReducedSequence → ReducedSequence
 prepend i g nonzero empty = nonempty i (ofEmpty i g nonzero)
 prepend i g nonzero (nonempty j x) with decidableIndex i j
-prepend i g nonzero (nonempty .i (ofEmpty .i h nonZero)) | inl refl with decidableGroups i (_+_ i g h) (Group.0G (G i))
-prepend i g nonzero (nonempty .i (ofEmpty .i h nonZero)) | inl refl | inl sumZero = empty
-prepend i g nonzero (nonempty .i (ofEmpty .i h nonZero)) | inl refl | inr sumNotZero = nonempty i (ofEmpty i (_+_ i g h) sumNotZero)
-prepend i g nonzero (nonempty .i (prependLetter .i h nonZero x x₁)) | inl refl with decidableGroups i (_+_ i g h) (Group.0G (G i))
-prepend i g nonzero (nonempty .i (prependLetter .i h nonZero {j} x x₁)) | inl refl | inl sumZero = nonempty j x
-prepend i g nonzero (nonempty .i (prependLetter .i h nonZero x pr)) | inl refl | inr sumNotZero = nonempty i (prependLetter i (_+_ i g h) sumNotZero x pr)
+prepend i g nonzero (nonempty j w) | inl i=j = prependEqual i g nonzero j i=j w
 prepend i g nonzero (nonempty j x) | inr i!=j = nonempty i (prependLetter i g nonzero x i!=j)
 
 plus' : {j : _} → (ReducedSequenceBeginningWith j) → ReducedSequence → ReducedSequence

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

Groups/Orders/Archimedean.agda

@@ -0,0 +1,20 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import Numbers.Naturals.Definition
+open import LogicalFormulae
+open import Groups.Definition
+open import Groups.Orders.Partial.Definition
+open import Setoids.Orders.Partial.Definition
+open import Setoids.Setoids
+open import Functions
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Groups.Orders.Archimedean {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {G : Group S _+_} {c : _} {_<_ : A → A → Set c} {pOrder : SetoidPartialOrder S _<_} (p : PartiallyOrderedGroup G pOrder) where
+
+open Setoid S
+open import Groups.Cyclic.Definition G
+open Group G
+
+Archimedean : Set (a ⊔ c)
+Archimedean = (x y : A) → (0G < x) → (0G < y) → Sg ℕ (λ n → y < (positiveEltPower x n))

Groups/Orders/Partial/Definition.agda

@@ -0,0 +1,17 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import Groups.Definition
+open import Setoids.Orders.Partial.Definition
+open import Setoids.Setoids
+open import Functions
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Groups.Orders.Partial.Definition {n m : _} {A : Set n} {S : Setoid {n} {m} A} {_+_ : A → A → A} (G : Group S _+_) where
+
+open Group G
+open Setoid S
+
+record PartiallyOrderedGroup {p : _} {_<_ : Rel {_} {p} A} (pOrder : SetoidPartialOrder S _<_) : Set (lsuc n ⊔ m ⊔ p) where
+  field
+    orderRespectsAddition : {a b : A} → (a < b) → (c : A) → (a + c) < (b + c)