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Groups lectures (#25)

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Patrick Stevens
2019-02-19 08:24:38 +00:00
committed by GitHub
parent d2192c72d9
commit 674aae49cd
8 changed files with 593 additions and 71 deletions
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8
Groups/Groups.agda
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@@ -58,14 +58,6 @@ module Groups.Groups where
m : y ∼ z · e
m = transitive l (wellDefined ~refl invRight)
identityIsUnique : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → (e : A) → ((b : A) → (Setoid._∼_ S (b · e) b)) → (Setoid._∼_ S e (Group.identity G))
identityIsUnique {S = S} {_·_} g thing fb = transitive (symmetric multIdentLeft) (fb e)
where
open Group g renaming (inverse to _^-1) renaming (identity to e)
open Setoid S
open Transitive (Equivalence.transitiveEq eq)
open Symmetric (Equivalence.symmetricEq eq)
replaceGroupOp : {l m : _} {A : Set l} {S : Setoid {l} {m} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b c d w x y z : A} → (Setoid._∼_ S a c) → (Setoid._∼_ S b d) → (Setoid._∼_ S w y) → (Setoid._∼_ S x z) → Setoid._∼_ S (a · b) (w · x) → Setoid._∼_ S (c · d) (y · z)
replaceGroupOp {S = S} {_·_} G a~c b~d w~y x~z pr = transitive (symmetric (wellDefined a~c b~d)) (transitive pr (wellDefined w~y x~z))
where