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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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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