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Direct sums: preparing for vector spaces (#74)

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Patrick Stevens
2019-11-16 15:06:57 +00:00
committed by GitHub
parent 9419587eb0
commit ff6ef4f1a1
10 changed files with 148 additions and 19 deletions
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19
Groups/Abelian/DirectSum.agda Normal file
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@@ -0,0 +1,19 @@
{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Setoids.Setoids
open import Functions
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Numbers.Naturals.Naturals
open import Sets.FinSet
open import Groups.Definition
open import Sets.EquivalenceRelations
open import Groups.Abelian.Definition
module Groups.Abelian.DirectSum {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+1_ : A → A → A} {_+2_ : B → B → B} {G1' : Group S _+1_} {G2' : Group T _+2_} (G1 : AbelianGroup G1') (G2 : AbelianGroup G2') where
open import Groups.DirectSum.Definition G1' G2'
open import Setoids.DirectSum S T
directSumAbGroup : AbelianGroup directSumGroup
AbelianGroup.commutative directSumAbGroup = directSumLift (AbelianGroup.commutative G1) (AbelianGroup.commutative G2)

3
Groups/Abelian/Lemmas.agda
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@@ -2,6 +2,7 @@
open import LogicalFormulae
open import Setoids.Setoids
open import Setoids.DirectSum
open import Functions
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Numbers.Naturals.Naturals
@@ -16,7 +17,7 @@ open import Groups.Isomorphisms.Definition
module Groups.Abelian.Lemmas where
directSumAbelianGroup : {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} {underG : Group S _·A_} {underH : Group T _·B_} (G : AbelianGroup underG) (h : AbelianGroup underH) → (AbelianGroup (directSum underG underH))
directSumAbelianGroup : {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} {underG : Group S _·A_} {underH : Group T _·B_} (G : AbelianGroup underG) (h : AbelianGroup underH) → (AbelianGroup (directSumGroup underG underH))
AbelianGroup.commutative (directSumAbelianGroup {A = A} {B} G H) = AbelianGroup.commutative G ,, AbelianGroup.commutative H
subgroupOfAbelianIsAbelian : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : B → A} {fHom : GroupHom H G f} → Subgroup G H fHom → AbelianGroup G → AbelianGroup H

5
Groups/Cyclic/Lemmas.agda
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@@ -30,8 +30,9 @@ GroupHom.wellDefined (elementPowerHom {S = S} G x) {.y} {y} refl = reflexive
open Equivalence (Setoid.eq S)
subgroupOfCyclicIsCyclic : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : B → A} {fHom : GroupHom H G f} → Subgroup G H fHom → CyclicGroup G → CyclicGroup H
CyclicGroup.generator (subgroupOfCyclicIsCyclic {f = f} subgrp record { generator = generator ; cyclic = cyclic }) = {!f generator!}
CyclicGroup.cyclic (subgroupOfCyclicIsCyclic subgrp gCyclic) = {!!}
CyclicGroup.generator (subgroupOfCyclicIsCyclic {f = f} subgrp record { generator = generator ; cyclic = cyclic }) = {!!}
CyclicGroup.cyclic (subgroupOfCyclicIsCyclic {f = f} subgrp record { generator = generator ; cyclic = cyclic }) {a} with cyclic {f a}
CyclicGroup.cyclic (subgroupOfCyclicIsCyclic {f = f} subgrp record { generator = generator ; cyclic = cyclic }) {a} | N , generator^n=fa = N , {!!}
-- Prefer to prove that subgroup of cyclic is cyclic, then deduce this like with abelian groups
{-

22
Groups/DirectSum/Definition.agda
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@@ -9,14 +9,16 @@ open import Sets.FinSet
open import Groups.Definition
open import Sets.EquivalenceRelations
module Groups.DirectSum.Definition where
module Groups.DirectSum.Definition {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_) where
directSum : {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_) → Group (directSumSetoid S T) (λ x1 y1 → (((_&&_.fst x1) ·A (_&&_.fst y1)) ,, ((_&&_.snd x1) ·B (_&&_.snd y1))))
Group.+WellDefined (directSum {A = A} {B} g h) (s ,, t) (u ,, v) = Group.+WellDefined g s u ,, Group.+WellDefined h t v
Group.0G (directSum {A = A} {B} g h) = (Group.0G g ,, Group.0G h)
Group.inverse (directSum {A = A} {B} g h) (g1 ,, h1) = (Group.inverse g g1) ,, (Group.inverse h h1)
Group.+Associative (directSum {A = A} {B} g h) = Group.+Associative g ,, Group.+Associative h
Group.identRight (directSum {A = A} {B} g h) = Group.identRight g ,, Group.identRight h
Group.identLeft (directSum {A = A} {B} g h) = Group.identLeft g ,, Group.identLeft h
Group.invLeft (directSum {A = A} {B} g h) = Group.invLeft g ,, Group.invLeft h
Group.invRight (directSum {A = A} {B} g h) = Group.invRight g ,, Group.invRight h
open import Setoids.DirectSum S T
directSumGroup : Group directSumSetoid (λ x1 y1 → (((_&&_.fst x1) ·A (_&&_.fst y1)) ,, ((_&&_.snd x1) ·B (_&&_.snd y1))))
Group.+WellDefined directSumGroup (s ,, t) (u ,, v) = Group.+WellDefined G s u ,, Group.+WellDefined H t v
Group.0G directSumGroup = (Group.0G G ,, Group.0G H)
Group.inverse directSumGroup (g1 ,, H1) = (Group.inverse G g1) ,, (Group.inverse H H1)
Group.+Associative directSumGroup = Group.+Associative G ,, Group.+Associative H
Group.identRight directSumGroup = Group.identRight G ,, Group.identRight H
Group.identLeft directSumGroup = Group.identLeft G ,, Group.identLeft H
Group.invLeft directSumGroup = Group.invLeft G ,, Group.invLeft H
Group.invRight directSumGroup = Group.invRight G ,, Group.invRight H