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Tidy up cyclic groups (#71)

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Patrick Stevens
2019-11-13 19:56:59 +00:00
committed by GitHub
parent 8128733722
commit d0f8d5ba39
4 changed files with 45 additions and 25 deletions
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1
Everything/Safe.agda
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@@ -28,6 +28,7 @@ open import Groups.Actions.Orbit
open import Groups.SymmetricGroups.Lemmas
open import Groups.ActionIsSymmetry
open import Groups.Cyclic.Definition
open import Groups.Cyclic.DefinitionLemmas
open import Fields.Fields
open import Fields.Orders.Partial.Definition

41
Groups/Cyclic/DefinitionLemmas.agda Normal file
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@@ -0,0 +1,41 @@
{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Setoids.Setoids
open import Sets.EquivalenceRelations
open import Functions
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Numbers.Naturals.Naturals
open import Numbers.Integers.Integers
open import Numbers.Integers.Addition
open import Sets.FinSet
open import Groups.Homomorphisms.Definition
open import Groups.Groups
open import Groups.Lemmas
open import Groups.Subgroups.Definition
open import Groups.Abelian.Definition
open import Groups.Definition
open import Groups.Cyclic.Definition
open import Semirings.Definition
module Groups.Cyclic.DefinitionLemmas {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
elementPowerCommutes : (x : A) (n : ℕ) → (x + positiveEltPower G x n) ∼ ((positiveEltPower G x n) + x)
elementPowerCommutes x zero = transitive identRight (symmetric identLeft)
elementPowerCommutes x (succ n) = transitive (+WellDefined reflexive (elementPowerCommutes x n)) +Associative
elementPowerCollapse : (x : A) (i j : ℤ) → Setoid._∼_ S ((elementPower G x i) + (elementPower G x j)) (elementPower G x (i +Z j))
elementPowerCollapse x (nonneg a) (nonneg b) rewrite equalityCommutative (+Zinherits a b) = positiveEltPowerCollapse G x a b
elementPowerCollapse x (nonneg zero) (negSucc b) = identLeft
elementPowerCollapse x (nonneg (succ a)) (negSucc zero) = transitive (+WellDefined (elementPowerCommutes x a) reflexive) (transitive (symmetric +Associative) (transitive (+WellDefined reflexive (transitive (+WellDefined reflexive (invContravariant G)) (transitive (+WellDefined reflexive (transitive (+WellDefined (invIdent G) reflexive) identLeft)) invRight))) identRight))
elementPowerCollapse x (nonneg (succ a)) (negSucc (succ b)) = transitive (transitive (+WellDefined (elementPowerCommutes x a) reflexive) (transitive (symmetric +Associative) (+WellDefined reflexive (transitive (transitive (+WellDefined reflexive (inverseWellDefined G (transitive (+WellDefined reflexive (elementPowerCommutes x b)) +Associative))) (transitive (+WellDefined reflexive (invContravariant G)) (transitive +Associative (transitive (+WellDefined invRight (invContravariant G)) identLeft)))) (symmetric (invContravariant G)))))) (elementPowerCollapse x (nonneg a) (negSucc b))
elementPowerCollapse x (negSucc zero) (nonneg zero) = identRight
elementPowerCollapse x (negSucc zero) (nonneg (succ b)) = transitive (transitive +Associative (+WellDefined (transitive (+WellDefined (inverseWellDefined G identRight) reflexive) invLeft) reflexive)) identLeft
elementPowerCollapse x (negSucc (succ a)) (nonneg zero) = identRight
elementPowerCollapse x (negSucc (succ a)) (nonneg (succ b)) = transitive (transitive +Associative (+WellDefined (transitive (+WellDefined (invContravariant G) reflexive) (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invLeft) identRight))) reflexive)) (elementPowerCollapse x (negSucc a) (nonneg b))
elementPowerCollapse x (negSucc a) (negSucc b) = transitive (+WellDefined reflexive (invContravariant G)) (transitive (transitive (transitive +Associative (+WellDefined (transitive (symmetric (invContravariant G)) (transitive (inverseWellDefined G +Associative) (transitive (inverseWellDefined G (+WellDefined (symmetric (elementPowerCommutes x b)) reflexive)) (transitive (inverseWellDefined G (symmetric +Associative)) (transitive (invContravariant G) (+WellDefined (inverseWellDefined G (transitive (positiveEltPowerCollapse G x b a) (identityOfIndiscernablesRight _∼_ reflexive (applyEquality (positiveEltPower G x) {b +N a} {a +N b} (Semiring.commutative ℕSemiring b a))))) reflexive)))))) reflexive)) (+WellDefined (symmetric (invContravariant G)) reflexive)) (symmetric (invContravariant G)))

9
Groups/Cyclic/Lemmas.agda
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@@ -11,19 +11,16 @@ open import Numbers.Integers.Addition
open import Sets.FinSet
open import Groups.Homomorphisms.Definition
open import Groups.Groups
open import Groups.Lemmas
open import Groups.Subgroups.Definition
open import Groups.Abelian.Definition
open import Groups.Definition
open import Groups.Cyclic.Definition
open import Groups.Cyclic.DefinitionLemmas
open import Semirings.Definition
module Groups.Cyclic.Lemmas where
elementPowerCollapse : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} (G : Group S _·_) (x : A) (i j : ℤ) → Setoid._∼_ S ((elementPower G x i) · (elementPower G x j)) (elementPower G x (i +Z j))
elementPowerCollapse G x (nonneg a) (nonneg b) rewrite equalityCommutative (+Zinherits a b) = positiveEltPowerCollapse G x a b
elementPowerCollapse G x (nonneg a) (negSucc b) = {!!}
elementPowerCollapse G x (negSucc a) (nonneg b) = {!!}
elementPowerCollapse G x (negSucc a) (negSucc b) = {!!}
elementPowerHom : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} (G : Group S _·_) (x : A) → GroupHom ℤGroup G (λ i → elementPower G x i)
GroupHom.groupHom (elementPowerHom {S = S} G x) {a} {b} = symmetric (elementPowerCollapse G x a b)
where

19
Groups/CyclicGroups.agda
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@@ -1,19 +0,0 @@
{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Setoids.Setoids
open import Sets.EquivalenceRelations
open import Functions
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Numbers.Naturals.Naturals
open import Numbers.Integers.Integers
open import Numbers.Integers.Addition
open import Sets.FinSet
open import Groups.Homomorphisms.Definition
open import Groups.Groups
open import Groups.Subgroups.Definition
open import Groups.Abelian.Definition
open import Groups.Definition
module Groups.CyclicGroups {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} (G : Group S _·_) where