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Z is a Euclidean domain (#86)

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Patrick Stevens
2019-12-07 13:00:18 +00:00
committed by GitHub
parent cfd9787bb8
commit e192f0e1f1
38 changed files with 1018 additions and 486 deletions
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27
Groups/Groups.agda
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@@ -42,30 +42,3 @@ fourWay+Associative'' {S = S} {_·_ = _·_} G = transitive +Associative (symmetr
open Group G
open Setoid S
open Equivalence eq
{-
quotientHom : {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 : A → B} → (fHom : GroupHom G H f) → A → A
quotientHom {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {f = f} fHom a = {!!}
quotientInjection : {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 : A → B} → (fHom : GroupHom G H f) → GroupHom (quotientGroup G fHom) G (quotientHom G fHom)
GroupHom.groupHom (quotientInjection {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {f = f} fHom) {x} {y} = {!!}
where
open Setoid S
open Equivalence eq
open Reflexive reflexiveEq
GroupHom.wellDefined (quotientInjection {S = S} {T = T} {_·A_ = _·A_} G {H = H} {f = f} fHom) {x} {y} x~y = {!!}
where
open Group G
open Setoid S
open Setoid T renaming (_∼_ to _∼T_)
open Equivalence (Setoid.eq S)
open Reflexive reflexiveEq
have : f (x ·A inverse y) ∼T Group.0G H
have = x~y
need : x ∼ y
need = {!!}
quotientIsSubgroup : {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 : A → B} → {fHom : GroupHom G H f} → Subgroup G (quotientGroup G fHom) (quotientInjection G fHom)
quotientIsSubgroup = {!!}
-}