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Modules

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Smaug123
2019-11-16 12:18:03 +00:00
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16
Groups/Cyclic/Definition.agda
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@@ -14,6 +14,7 @@ open import Groups.Groups
open import Groups.Subgroups.Definition
open import Groups.Abelian.Definition
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
@@ -25,6 +26,10 @@ positiveEltPower : (x : A) (i : ℕ) → A
positiveEltPower x 0 = Group.0G G
positiveEltPower x (succ i) = x · (positiveEltPower x i)
positiveEltPowerWellDefinedG : (x y : A) → (x ∼ y) → (i : ℕ) → (positiveEltPower x i) ∼ (positiveEltPower y i)
positiveEltPowerWellDefinedG x y x=y 0 = reflexive
positiveEltPowerWellDefinedG x y x=y (succ i) = +WellDefined x=y (positiveEltPowerWellDefinedG x y x=y i)
positiveEltPowerCollapse : (x : A) (i j : ℕ) → Setoid._∼_ S ((positiveEltPower x i) · (positiveEltPower x j)) (positiveEltPower x (i +N j))
positiveEltPowerCollapse x zero j = Group.identLeft G
positiveEltPowerCollapse x (succ i) j = transitive (symmetric +Associative) (+WellDefined reflexive (positiveEltPowerCollapse x i j))
@@ -33,6 +38,17 @@ elementPower : (x : A) (i : ℤ) → A
elementPower x (nonneg i) = positiveEltPower x i
elementPower x (negSucc i) = Group.inverse G (positiveEltPower x (succ i))
-- TODO: move this to lemmas
elementPowerWellDefinedZ : (i j : ℤ) → (i ≡ j) → {g : A} → elementPower g i ≡ elementPower g j
elementPowerWellDefinedZ i j i=j {g} = applyEquality (elementPower g) i=j
elementPowerWellDefinedZ' : (i j : ℤ) → (i ≡ j) → {g : A} → (elementPower g i) ∼ (elementPower g j)
elementPowerWellDefinedZ' i j i=j {g} = identityOfIndiscernablesRight _∼_ reflexive (elementPowerWellDefinedZ i j i=j)
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))
record CyclicGroup : Set (m ⊔ n) where
field
generator : A

27
Groups/Cyclic/DefinitionLemmas.agda
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@@ -39,3 +39,30 @@ 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)))
positiveEltPowerInverse : (m : ℕ) → (x : A) → (positiveEltPower G (inverse x) m) ∼ inverse (positiveEltPower G x m)
positiveEltPowerInverse zero x = symmetric (invIdent G)
positiveEltPowerInverse (succ m) x = transitive (+WellDefined reflexive (positiveEltPowerInverse m x)) (transitive (symmetric (invContravariant G)) (inverseWellDefined G (symmetric (elementPowerCommutes x m))))
positiveEltPowerMultiplies : (m n : ℕ) → (x : A) → (positiveEltPower G x (m *N n)) ∼ (positiveEltPower G (positiveEltPower G x n) m)
positiveEltPowerMultiplies zero n x = reflexive
positiveEltPowerMultiplies (succ m) n x = transitive (symmetric (positiveEltPowerCollapse G x n (m *N n))) (+WellDefined reflexive (positiveEltPowerMultiplies m n x))
powerZero : (m : ℕ) → positiveEltPower G 0G m ∼ 0G
powerZero zero = reflexive
powerZero (succ m) = transitive identLeft (powerZero m)
elementPowerMultiplies : (m n : ℤ) → (x : A) → (elementPower G x (m *Z n)) ∼ (elementPower G (elementPower G x n) m)
elementPowerMultiplies (nonneg m) (nonneg n) x = positiveEltPowerMultiplies m n x
elementPowerMultiplies (nonneg zero) (negSucc n) x = reflexive
elementPowerMultiplies (nonneg (succ m)) (negSucc n) x = transitive (transitive (inverseWellDefined G (transitive (transitive (elementPowerWellDefinedZ' G (nonneg (succ (n +N m *N n) +N m)) (nonneg (m *N succ n +N succ n)) (applyEquality nonneg (transitivity (applyEquality succ (transitivity (equalityCommutative (Semiring.+Associative ℕSemiring n (m *N n) m)) (applyEquality (n +N_) (transitivity (transitivity (Semiring.commutative ℕSemiring _ m) (applyEquality (m +N_) (multiplicationNIsCommutative m n))) (multiplicationNIsCommutative (succ n) m))))) (Semiring.commutative ℕSemiring (succ n) _))) {x}) (symmetric (positiveEltPowerCollapse G x (m *N succ n) (succ n)))) (+WellDefined (positiveEltPowerMultiplies m (succ n) x) reflexive))) (invContravariant G)) (+WellDefined reflexive (symmetric (positiveEltPowerInverse m (x + positiveEltPower G x n))))
elementPowerMultiplies (negSucc m) (nonneg zero) r = transitive (symmetric (invIdent G)) (inverseWellDefined G (transitive (symmetric identLeft) (+WellDefined reflexive (symmetric (powerZero m)))))
elementPowerMultiplies (negSucc m) (nonneg (succ n)) x = inverseWellDefined G (transitive (transitive (+WellDefined reflexive (transitive (elementPowerWellDefinedZ' G (nonneg ((m +N n *N m) +N n)) (nonneg (n +N m *N succ n)) (applyEquality nonneg (transitivity (Semiring.commutative ℕSemiring _ n) (applyEquality (n +N_) (multiplicationNIsCommutative _ m)))) {x}) (symmetric (positiveEltPowerCollapse G x n (m *N succ n))))) +Associative) (+WellDefined reflexive (positiveEltPowerMultiplies m (succ n) x)))
elementPowerMultiplies (negSucc m) (negSucc n) r = transitive (transitive (transitive (transitive (elementPowerWellDefinedZ' G (nonneg (succ m *N succ n)) (nonneg (m *N succ n +N succ n)) (applyEquality nonneg (Semiring.commutative ℕSemiring (succ n) _)) {r}) (symmetric (positiveEltPowerCollapse G r (m *N succ n) (succ n)))) (transitive (elementPowerCollapse r (nonneg (m *N succ n)) (nonneg (succ n))) (transitive (transitive (elementPowerWellDefinedZ' G (nonneg (m *N succ n) +Z nonneg (succ n)) (nonneg (m *N succ n +N succ n)) (equalityCommutative (+Zinherits (m *N succ n) (succ n))) {r}) (symmetric (positiveEltPowerCollapse G r (m *N succ n) (succ n)))) (+WellDefined (positiveEltPowerMultiplies m (succ n) r) reflexive)))) (+WellDefined (transitive (symmetric (invTwice G _)) (inverseWellDefined G (symmetric (positiveEltPowerInverse m (r + positiveEltPower G r n))))) (symmetric (invTwice G _)))) (symmetric (invContravariant G))
positiveEltPowerHomAbelian : ({x y : A} → (x + y) ∼ (y + x)) → (x y : A) → (r : ℕ) → (positiveEltPower G (x + y) r) ∼ (positiveEltPower G x r + positiveEltPower G y r)
positiveEltPowerHomAbelian ab x y zero = symmetric identRight
positiveEltPowerHomAbelian ab x y (succ r) = transitive (symmetric +Associative) (transitive (+WellDefined reflexive (transitive (+WellDefined reflexive (positiveEltPowerHomAbelian ab x y r)) (transitive (+WellDefined reflexive ab) (transitive +Associative ab)))) +Associative)
elementPowerHomAbelian : ({x y : A} → (x + y) ∼ (y + x)) → (x y : A) → (r : ℤ) → (elementPower G (x + y) r) ∼ (elementPower G x r + elementPower G y r)
elementPowerHomAbelian ab x y (nonneg n) = positiveEltPowerHomAbelian ab x y n
elementPowerHomAbelian ab x y (negSucc n) = transitive (transitive (inverseWellDefined G (positiveEltPowerHomAbelian ab x y (succ n))) (inverseWellDefined G ab)) (invContravariant G)

3
Groups/Lemmas.agda
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@@ -140,3 +140,6 @@ invContravariant {x} {y} = ans
otherInvIsInverse = transitive oneMult invRight
ans : (x · y) ^-1 ∼ (y ^-1) · (x ^-1)
ans = symmetric (leftInversesAreUnique otherInvIsInverse)
equalsDoubleImpliesZero : {x : A} → (x · x) ∼ x → x ∼ 0G
equalsDoubleImpliesZero 2x=x = transitive (symmetric identLeft) (transitive (+WellDefined (symmetric invLeft) reflexive) (transitive (symmetric +Associative) (transitive (+WellDefined reflexive 2x=x) invLeft)))

25
Modules/Definition.agda Normal file
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@@ -0,0 +1,25 @@
{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Groups.Groups
open import Groups.Homomorphisms.Definition
open import Groups.Definition
open import Groups.Abelian.Definition
open import Numbers.Naturals.Naturals
open import Setoids.Orders
open import Setoids.Setoids
open import Functions
open import Sets.EquivalenceRelations
open import Rings.Definition
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Modules.Definition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+R_ : A → A → A} {_*_ : A → A → A} (R : Ring S _+R_ _*_) {m n : _} {M : Set m} {T : Setoid {m} {n} M} {_+_ : M → M → M} {G' : Group T _+_} (G : AbelianGroup G') (_·_ : A → M → M) where
record Module : Set (a ⊔ b ⊔ m ⊔ n) where
field
dotWellDefined : {r s : A} {t u : M} → Setoid._∼_ S r s → Setoid._∼_ T t u → Setoid._∼_ T (r · t) (s · u)
dotDistributesLeft : {r : A} {x y : M} → Setoid._∼_ T (r · (x + y)) ((r · x) + (r · y))
dotDistributesRight : {r s : A} {x : M} → Setoid._∼_ T ((r +R s) · x) ((r · x) + (s · x))
dotAssociative : {r s : A} {x : M} → Setoid._∼_ T ((r * s) · x) (r · (s · x))
dotIdentity : {x : M} → Setoid._∼_ T ((Ring.1R R) · x) x

34
Modules/Examples.agda Normal file
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@@ -0,0 +1,34 @@
{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Groups.Groups
open import Groups.Abelian.Definition
open import Groups.Homomorphisms.Definition
open import Groups.Definition
open import Groups.Abelian.Definition
open import Numbers.Naturals.Naturals
open import Numbers.Integers.Integers
open import Setoids.Orders
open import Setoids.Setoids
open import Functions
open import Sets.EquivalenceRelations
open import Rings.Definition
open import Modules.Definition
open import Groups.Cyclic.Definition
open import Groups.Cyclic.DefinitionLemmas
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Modules.Examples where
abGrpModule : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {G' : Group S _+_} (G : AbelianGroup G') → Module ℤRing G (λ x i → elementPower G' i x)
Module.dotWellDefined (abGrpModule {S = S} {G' = G'} G) {m} {n} {g} {h} m=n g=h = transitive (elementPowerWellDefinedG G' g h g=h {m}) (elementPowerWellDefinedZ' G' m n m=n {h})
where
open Setoid S
open Equivalence eq
Module.dotDistributesLeft (abGrpModule {G' = G'} G) {n} {x} {y} = elementPowerHomAbelian G' (AbelianGroup.commutative G) x y n
Module.dotDistributesRight (abGrpModule {S = S} {G' = G'} G) {r} {s} {x} = symmetric (elementPowerCollapse G' x r s)
where
open Equivalence (Setoid.eq S)
Module.dotAssociative (abGrpModule {G' = G'} G) {r} {s} {x} = elementPowerMultiplies G' r s x
Module.dotIdentity (abGrpModule {G' = G'} G) = Group.identRight G'

39
Modules/Lemmas.agda Normal file
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@@ -0,0 +1,39 @@
{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Groups.Groups
open import Groups.Homomorphisms.Definition
open import Groups.Definition
open import Groups.Lemmas
open import Groups.Abelian.Definition
open import Numbers.Naturals.Naturals
open import Setoids.Orders
open import Setoids.Setoids
open import Functions
open import Sets.EquivalenceRelations
open import Rings.Definition
open import Modules.Definition
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Modules.Lemmas {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+R_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+R_ _*_} {m n : _} {M : Set m} {T : Setoid {m} {n} M} {_+_ : M → M → M} {G' : Group T _+_} {G : AbelianGroup G'} {_·_ : A → M → M} (mod : Module R G _·_) where
open Group G'
open Ring R
open Setoid T
open Equivalence eq
open Module mod
moduleTimesZero : {x : M} → (0R · x) ∼ 0G
moduleTimesZero {x} = equalsDoubleImpliesZero G' (symmetric x=2x)
where
x=2x : (0R · x) ∼ (0R · x) + (0R · x)
x=2x = transitive (dotWellDefined (Equivalence.symmetric (Setoid.eq S) (Group.identLeft additiveGroup)) reflexive) dotDistributesRight
moduleTimes-1 : {x : M} → ((Group.inverse additiveGroup 1R) · x) ∼ inverse x
moduleTimes-1 {x} = transitive (transferToRight' G' j) (inverseWellDefined G' dotIdentity)
where
i : ((1R · x) + ((Group.inverse additiveGroup 1R) · x)) ∼ 0G
i = transitive (symmetric (transitive (dotWellDefined (Equivalence.symmetric (Setoid.eq S) (Group.invRight additiveGroup {1R})) reflexive) dotDistributesRight)) (moduleTimesZero)
j : (((Group.inverse additiveGroup 1R) · x) + (1R · x)) ∼ 0G
j = transitive (AbelianGroup.commutative G) i