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Restructure towards ideals

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Smaug123
2019-11-20 21:20:03 +00:00
parent f2f4e867fc
commit b03a5279bc
22 changed files with 417 additions and 296 deletions
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18
Groups/Subgroups/Definition.agda
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@@ -11,7 +11,21 @@ open import Groups.Homomorphisms.Definition
module Groups.Subgroups.Definition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} (G : Group S _+_) where
open import Setoids.Subset S
open Group G
subgroup : {c : _} {pred : A → Set c} → (wd : {x y : A} → (Setoid._∼_ S x y) → (pred x → pred y)) → Set (a ⊔ c)
subgroup {pred = pred} wd = ({g h : A} → (pred g) → (pred h) → pred (g + h)) & pred 0G & ({g : A} → (pred g) → (pred (inverse g)))
subgroup : {c : _} (pred : A → Set c) → Set (a ⊔ b ⊔ c)
subgroup pred = subset pred && (({g h : A} → (pred g) → (pred h) → pred (g + h)) & pred 0G & ({g : A} → (pred g) → (pred (inverse g))))
subgroupOp : {c : _} {pred : A → Set c} → (s : subgroup pred) → Sg A pred → Sg A pred → Sg A pred
subgroupOp {pred = pred} (_ ,, record { one = one ; two = two ; three = three }) (a , prA) (b , prB) = (a + b) , one prA prB
subgroupIsGroup : {c : _} {pred : A → Set c} → (subs : subset pred) → (s : subgroup pred) → Group (subsetSetoid subs) (subgroupOp s)
Group.+WellDefined (subgroupIsGroup _ s) {m , prM} {n , prN} {x , prX} {y , prY} m=x n=y = +WellDefined m=x n=y
Group.0G (subgroupIsGroup _ (_ ,, record { two = two })) = 0G , two
Group.inverse (subgroupIsGroup _ (_ ,, record { three = three })) (a , prA) = (inverse a) , three prA
Group.+Associative (subgroupIsGroup _ s) {a , prA} {b , prB} {c , prC} = +Associative
Group.identRight (subgroupIsGroup _ s) {a , prA} = identRight
Group.identLeft (subgroupIsGroup _ s) {a , prA} = identLeft
Group.invLeft (subgroupIsGroup _ s) {a , prA} = invLeft
Group.invRight (subgroupIsGroup _ s) {a , prA} = invRight

5
Groups/Subgroups/Normal/Definition.agda
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@@ -16,11 +16,10 @@ open import Groups.Isomorphisms.Definition
open import Groups.Subgroups.Definition
open import Groups.Lemmas
open import Groups.Abelian.Definition
open import Groups.QuotientGroup.Definition
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Groups.Subgroups.Normal.Definition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} (G : Group S _+_) where
normalSubgroup : {c : _} {pred : A → Set c} (wd : {x y : A} → (Setoid._∼_ S x y) → (pred x → pred y)) (sub : subgroup G wd) → Set (a ⊔ c)
normalSubgroup {pred = pred} wd sub = {g k : A} → pred k → pred (g + (k + Group.inverse G g))
normalSubgroup : {c : _} {pred : A → Set c} (sub : subgroup G pred) → Set (a ⊔ c)
normalSubgroup {pred = pred} sub = {g k : A} → pred k → pred (g + (k + Group.inverse G g))

15
Groups/Subgroups/Normal/Lemmas.agda
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@@ -64,12 +64,13 @@ groupKernelGroupPred {T = T} G {H = H} {f = f} hom a = Setoid._∼_ T (f a) (Gro
groupKernelGroupPredWd : {a b c d : _} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} {f : A → B} (hom : GroupHom G H f) → {x y : A} → (Setoid._∼_ S x y) → (groupKernelGroupPred G hom x → groupKernelGroupPred G hom y)
groupKernelGroupPredWd {S = S} {T = T} G hom {x} {y} x=y fx=0 = Equivalence.transitive (Setoid.eq T) (GroupHom.wellDefined hom (Equivalence.symmetric (Setoid.eq S) x=y)) fx=0
groupKernelGroupIsSubgroup : {a b c d : _} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} {f : A → B} (hom : GroupHom G H f) → subgroup G (groupKernelGroupPredWd G hom)
_&_&_.one (groupKernelGroupIsSubgroup {S = S} {T = T} G {H = H} hom) g=0 h=0 = Equivalence.transitive (Setoid.eq T) (GroupHom.groupHom hom) (Equivalence.transitive (Setoid.eq T) (Group.+WellDefined H g=0 h=0) (Group.identLeft H))
_&_&_.two (groupKernelGroupIsSubgroup G hom) = imageOfIdentityIsIdentity hom
_&_&_.three (groupKernelGroupIsSubgroup {S = S} {T = T} G {H = H} hom) g=0 = Equivalence.transitive (Setoid.eq T) (homRespectsInverse hom) (Equivalence.transitive (Setoid.eq T) (inverseWellDefined H g=0) (invIdent H))
groupKernelGroupIsSubgroup : {a b c d : _} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} {f : A → B} (hom : GroupHom G H f) → subgroup G (groupKernelGroupPred G hom)
_&_&_.one (_&&_.snd (groupKernelGroupIsSubgroup {S = S} {T = T} G {H = H} hom)) g=0 h=0 = Equivalence.transitive (Setoid.eq T) (GroupHom.groupHom hom) (Equivalence.transitive (Setoid.eq T) (Group.+WellDefined H g=0 h=0) (Group.identLeft H))
_&_&_.two (_&&_.snd (groupKernelGroupIsSubgroup G hom)) = imageOfIdentityIsIdentity hom
_&_&_.three (_&&_.snd (groupKernelGroupIsSubgroup {S = S} {T = T} G {H = H} hom)) g=0 = Equivalence.transitive (Setoid.eq T) (homRespectsInverse hom) (Equivalence.transitive (Setoid.eq T) (inverseWellDefined H g=0) (invIdent H))
_&&_.fst (groupKernelGroupIsSubgroup G hom) = groupKernelGroupPredWd G hom
groupKernelGroupIsNormalSubgroup : {a b c d : _} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} {f : A → B} (hom : GroupHom G H f) → normalSubgroup G (groupKernelGroupPredWd G hom) (groupKernelGroupIsSubgroup G hom)
groupKernelGroupIsNormalSubgroup : {a b c d : _} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} {f : A → B} (hom : GroupHom G H f) → normalSubgroup G (groupKernelGroupIsSubgroup G hom)
groupKernelGroupIsNormalSubgroup {T = T} G {H = H} hom k=0 = transitive groupHom (transitive (+WellDefined reflexive groupHom) (transitive (+WellDefined reflexive (transitive (+WellDefined k=0 reflexive) identLeft)) (transitive (symmetric groupHom) (transitive (wellDefined (Group.invRight G)) (imageOfIdentityIsIdentity hom)))))
where
open Setoid T
@@ -77,8 +78,8 @@ groupKernelGroupIsNormalSubgroup {T = T} G {H = H} hom k=0 = transitive groupHom
open Equivalence eq
open GroupHom hom
abelianGroupSubgroupIsNormal : {a b c : _} {A : Set a} {B : Set b} {S : Setoid {a} {b} A} {_+_ : A → A → A} {G : Group S _+_} {pred : A → Set c} (predWd : {x y : A} → (Setoid._∼_ S x y) → (pred x → pred y)) → (s : subgroup G predWd) → AbelianGroup G → normalSubgroup G predWd s
abelianGroupSubgroupIsNormal {S = S} {_+_ = _+_} {G = G} predWd record { one = respectsPlus ; two = respectsId ; three = respectsInv } abelian {k} {l} prK = predWd (transitive (transitive (transitive (symmetric identLeft) (+WellDefined (symmetric invRight) reflexive)) (symmetric +Associative)) (+WellDefined reflexive commutative)) prK
abelianGroupSubgroupIsNormal : {a b c : _} {A : Set a} {B : Set b} {S : Setoid {a} {b} A} {_+_ : A → A → A} {G : Group S _+_} {pred : A → Set c} → (s : subgroup G pred) → AbelianGroup G → normalSubgroup G s
abelianGroupSubgroupIsNormal {S = S} {_+_ = _+_} {G = G} (predWd ,, record { one = respectsPlus ; two = respectsId ; three = respectsInv }) abelian {k} {l} prK = predWd (transitive (transitive (transitive (symmetric identLeft) (+WellDefined (symmetric invRight) reflexive)) (symmetric +Associative)) (+WellDefined reflexive commutative)) prK
where
open Group G
open AbelianGroup abelian