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https://github.com/Smaug123/agdaproofs
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Comparison on the reals (#55)
This commit is contained in:
Patrick Stevens
GitHub
@@ -11,422 +11,422 @@ open import Sets.EquivalenceRelations
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module Groups.Groups where
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reflGroupWellDefined : {lvl : _} {A : Set lvl} {m n x y : A} {op : A → A → A} → m ≡ x → n ≡ y → (op m n) ≡ (op x y)
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reflGroupWellDefined {lvl} {A} {m} {n} {.m} {.n} {op} refl refl = refl
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reflGroupWellDefined : {lvl : _} {A : Set lvl} {m n x y : A} {op : A → A → A} → m ≡ x → n ≡ y → (op m n) ≡ (op x y)
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reflGroupWellDefined {lvl} {A} {m} {n} {.m} {.n} {op} refl refl = refl
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record AbelianGroup {a} {b} {A : Set a} {S : Setoid {a} {b} A} {_·_ : A → A → A} (G : Group S _·_) : Set (lsuc a ⊔ b) where
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open Setoid S
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field
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commutative : {a b : A} → (a · b) ∼ (b · a)
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record AbelianGroup {a} {b} {A : Set a} {S : Setoid {a} {b} A} {_·_ : A → A → A} (G : Group S _·_) : Set (lsuc a ⊔ b) where
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open Setoid S
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field
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commutative : {a b : A} → (a · b) ∼ (b · a)
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groupsHaveLeftCancellation : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → (x y z : A) → (Setoid._∼_ S (x · y) (x · z)) → (Setoid._∼_ S y z)
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groupsHaveLeftCancellation {_·_ = _·_} {S} g x y z pr = o
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groupsHaveLeftCancellation : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → (x y z : A) → (Setoid._∼_ S (x · y) (x · z)) → (Setoid._∼_ S y z)
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groupsHaveLeftCancellation {_·_ = _·_} {S} g x y z pr = o
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where
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open Group g renaming (inverse to _^-1)
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open Setoid S
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transitive = Equivalence.transitive (Setoid.eq S)
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symmetric = Equivalence.symmetric (Setoid.eq S)
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j : ((x ^-1) · x) · y ∼ (x ^-1) · (x · z)
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j = Equivalence.transitive eq (symmetric (+Associative {x ^-1} {x} {y})) (+WellDefined ~refl pr)
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k : ((x ^-1) · x) · y ∼ ((x ^-1) · x) · z
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k = transitive j +Associative
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l : 0G · y ∼ ((x ^-1) · x) · z
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l = transitive (+WellDefined (symmetric invLeft) ~refl) k
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m : 0G · y ∼ 0G · z
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m = transitive l (+WellDefined invLeft ~refl)
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n : y ∼ 0G · z
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n = transitive (symmetric identLeft) m
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o : y ∼ z
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o = transitive n identLeft
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groupsHaveRightCancellation : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → (x y z : A) → (Setoid._∼_ S (y · x) (z · x)) → (Setoid._∼_ S y z)
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groupsHaveRightCancellation {_·_ = _·_} {S} g x y z pr = transitive m identRight
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where
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open Group g renaming (inverse to _^-1)
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open Setoid S
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transitive = Equivalence.transitive (Setoid.eq S)
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symmetric = Equivalence.symmetric (Setoid.eq S)
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i : (y · x) · (x ^-1) ∼ (z · x) · (x ^-1)
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i = +WellDefined pr ~refl
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j : y · (x · (x ^-1)) ∼ (z · x) · (x ^-1)
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j = transitive +Associative i
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j' : y · 0G ∼ (z · x) · (x ^-1)
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j' = transitive (+WellDefined ~refl (symmetric invRight)) j
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k : y ∼ (z · x) · (x ^-1)
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k = transitive (symmetric identRight) j'
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l : y ∼ z · (x · (x ^-1))
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l = transitive k (symmetric +Associative)
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m : y ∼ z · 0G
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m = transitive l (+WellDefined ~refl invRight)
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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)
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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))
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where
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open Group G
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open Setoid S
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open Equivalence eq
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replaceGroupOpRight : {l m : _} {A : Set l} {S : Setoid {l} {m} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b c x : A} → (Setoid._∼_ S a (b · c)) → (Setoid._∼_ S c x) → (Setoid._∼_ S a (b · x))
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replaceGroupOpRight {S = S} {_·_} G a~bc c~x = transitive a~bc (+WellDefined reflexive c~x)
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where
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open Group G
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open Setoid S
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open Equivalence eq
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inverseWellDefined : {l m : _} {A : Set l} {S : Setoid {l} {m} A} {_·_ : A → A → A} (G : Group S _·_) → {x y : A} → (Setoid._∼_ S x y) → Setoid._∼_ S (Group.inverse G x) (Group.inverse G y)
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inverseWellDefined {S = S} {_·_} G {x} {y} x~y = groupsHaveRightCancellation G x (inverse x) (inverse y) q
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where
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open Group G
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open Setoid S
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open Equivalence eq
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p : inverse x · x ∼ inverse y · y
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p = transitive invLeft (symmetric invLeft)
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q : inverse x · x ∼ inverse y · x
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q = replaceGroupOpRight G {_·_ (inverse x) x} {inverse y} {y} {x} p (symmetric x~y)
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rightInversesAreUnique : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → (x : A) → (y : A) → Setoid._∼_ S (y · x) (Group.0G G) → Setoid._∼_ S y (Group.inverse G x)
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rightInversesAreUnique {S = S} {_·_} G x y f = transitive i (transitive j (transitive k (transitive l m)))
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where
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open Group G renaming (inverse to _^-1)
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open Setoid S
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open Equivalence eq
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i : y ∼ y · 0G
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j : y · 0G ∼ y · (x · (x ^-1))
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k : y · (x · (x ^-1)) ∼ (y · x) · (x ^-1)
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l : (y · x) · (x ^-1) ∼ 0G · (x ^-1)
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m : 0G · (x ^-1) ∼ x ^-1
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i = symmetric identRight
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j = +WellDefined ~refl (symmetric invRight)
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k = +Associative
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l = +WellDefined f ~refl
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m = identLeft
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leftInversesAreUnique : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → {x : A} → {y : A} → Setoid._∼_ S (x · y) (Group.0G G) → Setoid._∼_ S y (Group.inverse G x)
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leftInversesAreUnique {S = S} {_·_} G {x} {y} f = rightInversesAreUnique G x y l
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where
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open Group G renaming (inverse to _^-1)
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open Setoid S
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open Equivalence eq
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i : y · (x · y) ∼ y · 0G
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i' : y · (x · y) ∼ y
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j : (y · x) · y ∼ y
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k : (y · x) · y ∼ 0G · y
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l : y · x ∼ 0G
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i = +WellDefined ~refl f
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i' = transitive i identRight
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j = transitive (symmetric +Associative) i'
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k = transitive j (symmetric identLeft)
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l = groupsHaveRightCancellation G y (y · x) 0G k
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invTwice : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → (x : A) → Setoid._∼_ S (Group.inverse G (Group.inverse G x)) x
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invTwice {S = S} {_·_} G x = symmetric (rightInversesAreUnique G (x ^-1) x invRight)
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where
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open Group G renaming (inverse to _^-1)
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open Setoid S
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open Equivalence eq
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fourWay+Associative : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → {r s t u : A} → (Setoid._∼_ S) (r · ((s · t) · u)) ((r · s) · (t · u))
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fourWay+Associative {S = S} {_·_} G {r} {s} {t} {u} = transitive p1 (transitive p2 p3)
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where
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open Group G renaming (inverse to _^-1)
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open Setoid S
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open Equivalence eq
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p1 : r · ((s · t) · u) ∼ (r · (s · t)) · u
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p2 : (r · (s · t)) · u ∼ ((r · s) · t) · u
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p3 : ((r · s) · t) · u ∼ (r · s) · (t · u)
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p1 = Group.+Associative G
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p2 = Group.+WellDefined G (Group.+Associative G) reflexive
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p3 = symmetric (Group.+Associative G)
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fourWay+Associative' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} (G : Group S _·_) {a b c d : A} → (Setoid._∼_ S (((a · b) · c) · d) (a · ((b · c) · d)))
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fourWay+Associative' {S = S} G = transitive (symmetric +Associative) (symmetric (fourWay+Associative G))
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where
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open Group G
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open Setoid S
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open Equivalence eq
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fourWay+Associative'' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} (G : Group S _·_) {a b c d : A} → (Setoid._∼_ S (a · (b · (c · d))) (a · ((b · c) · d)))
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fourWay+Associative'' {S = S} {_·_ = _·_} G = transitive +Associative (symmetric (fourWay+Associative G))
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where
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open Group G
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open Setoid S
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open Equivalence eq
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invContravariant : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → {x y : A} → (Setoid._∼_ S (Group.inverse G (x · y)) ((Group.inverse G y) · (Group.inverse G x)))
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invContravariant {S = S} {_·_} G {x} {y} = ans
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where
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open Group G renaming (inverse to _^-1)
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open Setoid S
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open Equivalence eq
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otherInv = (y ^-1) · (x ^-1)
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many+Associatives : x · ((y · (y ^-1)) · (x ^-1)) ∼ (x · y) · ((y ^-1) · (x ^-1))
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oneMult : (x · y) · otherInv ∼ x · (x ^-1)
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many+Associatives = fourWay+Associative G
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oneMult = symmetric (transitive (Group.+WellDefined G reflexive (transitive (symmetric (Group.identLeft G)) (Group.+WellDefined G (symmetric (Group.invRight G)) reflexive))) many+Associatives)
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otherInvIsInverse : (x · y) · otherInv ∼ 0G
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otherInvIsInverse = transitive oneMult (Group.invRight G)
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ans : (x · y) ^-1 ∼ (y ^-1) · (x ^-1)
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ans = symmetric (leftInversesAreUnique G otherInvIsInverse)
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invIdentity : {l m : _} → {A : Set l} → {S : Setoid {l} {m} A} → {_·_ : A → A → A} → (G : Group S _·_) → Setoid._∼_ S ((Group.inverse G) (Group.0G G)) (Group.0G G)
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invIdentity {S = S} G = transitive (symmetric identLeft) invRight
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where
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open Group G
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open Setoid S
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open Equivalence eq
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transferToRight : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b : A} → Setoid._∼_ S (a · (Group.inverse G b)) (Group.0G G) → Setoid._∼_ S a b
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transferToRight {S = S} {_·_ = _·_} G {a} {b} ab-1 = transitive (symmetric (invTwice G a)) (transitive u (invTwice G b))
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where
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open Setoid S
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open Group G
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open Equivalence eq
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t : inverse a ∼ inverse b
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t = symmetric (leftInversesAreUnique G ab-1)
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u : inverse (inverse a) ∼ inverse (inverse b)
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u = inverseWellDefined G t
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transferToRight' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b : A} → Setoid._∼_ S (a · b) (Group.0G G) → Setoid._∼_ S a (Group.inverse G b)
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transferToRight' {S = S} {_·_ = _·_} G {a} {b} ab-1 = transferToRight G lemma
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where
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open Setoid S
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open Group G
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open Equivalence eq
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lemma : a · (inverse (inverse b)) ∼ 0G
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lemma = transitive (+WellDefined reflexive (invTwice G b)) ab-1
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transferToRight'' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b : A} → Setoid._∼_ S a b → Setoid._∼_ S (a · (Group.inverse G b)) (Group.0G G)
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transferToRight'' {S = S} {_·_ = _·_} G {a} {b} a~b = transitive (Group.+WellDefined G a~b reflexive) invRight
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where
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open Group G
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open Setoid S
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open Equivalence eq
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directSumGroup : {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))))
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Group.+WellDefined (directSumGroup {A = A} {B} g h) (s ,, t) (u ,, v) = Group.+WellDefined g s u ,, Group.+WellDefined h t v
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Group.0G (directSumGroup {A = A} {B} g h) = (Group.0G g ,, Group.0G h)
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Group.inverse (directSumGroup {A = A} {B} g h) (g1 ,, h1) = (Group.inverse g g1) ,, (Group.inverse h h1)
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Group.+Associative (directSumGroup {A = A} {B} g h) = Group.+Associative g ,, Group.+Associative h
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Group.identRight (directSumGroup {A = A} {B} g h) = Group.identRight g ,, Group.identRight h
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Group.identLeft (directSumGroup {A = A} {B} g h) = Group.identLeft g ,, Group.identLeft h
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Group.invLeft (directSumGroup {A = A} {B} g h) = Group.invLeft g ,, Group.invLeft h
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Group.invRight (directSumGroup {A = A} {B} g h) = Group.invRight g ,, Group.invRight h
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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))
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AbelianGroup.commutative (directSumAbelianGroup {A = A} {B} G H) = AbelianGroup.commutative G ,, AbelianGroup.commutative H
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record GroupHom {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_) (f : A → B) : Set (m ⊔ n ⊔ o ⊔ p) where
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open Group H
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open Setoid T
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field
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groupHom : {x y : A} → f (x ·A y) ∼ (f x) ·B (f y)
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wellDefined : {x y : A} → Setoid._∼_ S x y → f x ∼ f y
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record InjectiveGroupHom {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_} {underf : A → B} (f : GroupHom G H underf) : Set (m ⊔ n ⊔ o ⊔ p) where
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open Setoid S renaming (_∼_ to _∼A_)
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open Setoid T renaming (_∼_ to _∼B_)
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field
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injective : SetoidInjection S T underf
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imageOfIdentityIsIdentity : {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {B : Set n} {T : Setoid {n} {p} 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) → Setoid._∼_ T (f (Group.0G G)) (Group.0G H)
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imageOfIdentityIsIdentity {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} {G = G} {H = H} {f = f} hom = Equivalence.symmetric (Setoid.eq T) t
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where
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open Group H
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open Setoid T
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id2 : Setoid._∼_ S (Group.0G G) ((Group.0G G) ·A (Group.0G G))
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id2 = Equivalence.symmetric (Setoid.eq S) (Group.identRight G)
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r : f (Group.0G G) ∼ f (Group.0G G) ·B f (Group.0G G)
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s : 0G ·B f (Group.0G G) ∼ f (Group.0G G) ·B f (Group.0G G)
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t : 0G ∼ f (Group.0G G)
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t = groupsHaveRightCancellation H (f (Group.0G G)) 0G (f (Group.0G G)) s
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s = Equivalence.transitive (Setoid.eq T) identLeft r
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r = Equivalence.transitive (Setoid.eq T) (GroupHom.wellDefined hom id2) (GroupHom.groupHom hom)
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groupHomsCompose : {m n o r s t : _} {A : Set m} {S : Setoid {m} {r} A} {_+A_ : A → A → A} {B : Set n} {T : Setoid {n} {s} B} {_+B_ : B → B → B} {C : Set o} {U : Setoid {o} {t} C} {_+C_ : C → C → C} {G : Group S _+A_} {H : Group T _+B_} {I : Group U _+C_} {f : A → B} {g : B → C} (fHom : GroupHom G H f) (gHom : GroupHom H I g) → GroupHom G I (g ∘ f)
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GroupHom.wellDefined (groupHomsCompose {G = G} {H} {I} {f} {g} fHom gHom) {x} {y} pr = GroupHom.wellDefined gHom (GroupHom.wellDefined fHom pr)
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GroupHom.groupHom (groupHomsCompose {S = S} {_+A_ = _·A_} {T = T} {U = U} {_+C_ = _·C_} {G = G} {H} {I} {f} {g} fHom gHom) {x} {y} = answer
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where
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open Group I
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answer : (Setoid._∼_ U) ((g ∘ f) (x ·A y)) ((g ∘ f) x ·C (g ∘ f) y)
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answer = (Equivalence.transitive (Setoid.eq U)) (GroupHom.wellDefined gHom (GroupHom.groupHom fHom {x} {y}) ) (GroupHom.groupHom gHom {f x} {f y})
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record GroupIso {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_) (f : A → B) : Set (m ⊔ n ⊔ o ⊔ p) where
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open Setoid S renaming (_∼_ to _∼G_)
|
||||
open Setoid T renaming (_∼_ to _∼H_)
|
||||
field
|
||||
groupHom : GroupHom G H f
|
||||
bij : SetoidBijection S T f
|
||||
|
||||
record GroupsIsomorphic {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_) : Set (m ⊔ n ⊔ o ⊔ p) where
|
||||
field
|
||||
isomorphism : A → B
|
||||
proof : GroupIso G H isomorphism
|
||||
|
||||
groupIsosCompose : {m n o r s t : _} {A : Set m} {S : Setoid {m} {r} A} {_+A_ : A → A → A} {B : Set n} {T : Setoid {n} {s} B} {_+B_ : B → B → B} {C : Set o} {U : Setoid {o} {t} C} {_+C_ : C → C → C} {G : Group S _+A_} {H : Group T _+B_} {I : Group U _+C_} {f : A → B} {g : B → C} (fHom : GroupIso G H f) (gHom : GroupIso H I g) → GroupIso G I (g ∘ f)
|
||||
GroupIso.groupHom (groupIsosCompose fHom gHom) = groupHomsCompose (GroupIso.groupHom fHom) (GroupIso.groupHom gHom)
|
||||
GroupIso.bij (groupIsosCompose {A = A} {S = S} {T = T} {C = C} {U = U} {f = f} {g = g} fHom gHom) = record { inj = record { injective = λ pr → (SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij fHom))) (SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij gHom)) pr) ; wellDefined = +WellDefined } ; surj = record { surjective = surj ; wellDefined = +WellDefined } }
|
||||
where
|
||||
open Setoid S renaming (_∼_ to _∼A_)
|
||||
open Setoid U renaming (_∼_ to _∼C_)
|
||||
+WellDefined : {x y : A} → (x ∼A y) → (g (f x) ∼C g (f y))
|
||||
+WellDefined = GroupHom.wellDefined (groupHomsCompose (GroupIso.groupHom fHom) (GroupIso.groupHom gHom))
|
||||
surj : {x : C} → Sg A (λ a → (g (f a) ∼C x))
|
||||
surj {x} with SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij gHom)) {x}
|
||||
surj {x} | a , prA with SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij fHom)) {a}
|
||||
... | b , prB = b , transitive (GroupHom.wellDefined (GroupIso.groupHom gHom) prB) prA
|
||||
where
|
||||
open Group g renaming (inverse to _^-1)
|
||||
open Setoid S
|
||||
transitive = Equivalence.transitive (Setoid.eq S)
|
||||
symmetric = Equivalence.symmetric (Setoid.eq S)
|
||||
j : ((x ^-1) · x) · y ∼ (x ^-1) · (x · z)
|
||||
j = Equivalence.transitive eq (symmetric (+Associative {x ^-1} {x} {y})) (+WellDefined ~refl pr)
|
||||
k : ((x ^-1) · x) · y ∼ ((x ^-1) · x) · z
|
||||
k = transitive j +Associative
|
||||
l : 0G · y ∼ ((x ^-1) · x) · z
|
||||
l = transitive (+WellDefined (symmetric invLeft) ~refl) k
|
||||
m : 0G · y ∼ 0G · z
|
||||
m = transitive l (+WellDefined invLeft ~refl)
|
||||
n : y ∼ 0G · z
|
||||
n = transitive (symmetric identLeft) m
|
||||
o : y ∼ z
|
||||
o = transitive n identLeft
|
||||
open Equivalence (Setoid.eq U)
|
||||
|
||||
groupsHaveRightCancellation : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → (x y z : A) → (Setoid._∼_ S (y · x) (z · x)) → (Setoid._∼_ S y z)
|
||||
groupsHaveRightCancellation {_·_ = _·_} {S} g x y z pr = transitive m identRight
|
||||
record FiniteGroup {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_·_ : A → A → A} (G : Group S _·_) {c : _} (quotientSet : Set c) : Set (a ⊔ b ⊔ c) where
|
||||
field
|
||||
toSet : SetoidToSet S quotientSet
|
||||
finite : FiniteSet quotientSet
|
||||
|
||||
groupOrder : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_·_ : A → A → A} {underG : Group S _·_} {c : _} {quotientSet : Set c} (G : FiniteGroup underG quotientSet) → ℕ
|
||||
groupOrder record { toSet = toSet ; finite = record { size = size ; mapping = mapping ; bij = bij } } = size
|
||||
|
||||
--groupIsoInvertible : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d}} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : A → B} → (iso : GroupIso G H f) → GroupIso H G (Invertible.inverse (bijectionImpliesInvertible (GroupIso.bijective iso)))
|
||||
--GroupHom.groupHom (GroupIso.groupHom (groupIsoInvertible {G = G} {H} {f} iso)) {x} {y} = {!!}
|
||||
-- where
|
||||
-- open Group G renaming (_·_ to _+G_)
|
||||
-- open Group H renaming (_·_ to _+H_)
|
||||
--GroupHom.wellDefined (GroupIso.groupHom (groupIsoInvertible {G = G} {H} {f} iso)) {x} {y} x∼y = {!GroupHom.wellDefined x∼y!}
|
||||
--GroupIso.bijective (groupIsoInvertible {G = G} {H} {f} iso) = invertibleImpliesBijection (inverseIsInvertible (bijectionImpliesInvertible (GroupIso.bijective iso)))
|
||||
|
||||
homRespectsInverse : {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_} {underF : A → B} → (f : GroupHom G H underF) → {x : A} → Setoid._∼_ T (underF (Group.inverse G x)) (Group.inverse H (underF x))
|
||||
homRespectsInverse {T = T} {_·A_ = _·A_} {_·B_ = _·B_} {G = G} {H = H} {underF = f} fHom {x} = rightInversesAreUnique H (f x) (f (Group.inverse G x)) (transitive (symmetric (GroupHom.groupHom fHom)) (transitive (GroupHom.wellDefined fHom (Group.invLeft G)) (imageOfIdentityIsIdentity fHom)))
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
|
||||
quotientGroupSetoid : {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_} → {underf : A → B} → (f : GroupHom G H underf) → (Setoid {a} {d} A)
|
||||
quotientGroupSetoid {A = A} {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {H} {f} fHom = ansSetoid
|
||||
where
|
||||
open Setoid T
|
||||
open Group H
|
||||
open Equivalence eq
|
||||
ansSetoid : Setoid A
|
||||
Setoid._∼_ ansSetoid r s = (f (r ·A (Group.inverse G s))) ∼ 0G
|
||||
Equivalence.reflexive (Setoid.eq ansSetoid) {b} = transitive (GroupHom.wellDefined fHom (Group.invRight G)) (imageOfIdentityIsIdentity fHom)
|
||||
Equivalence.symmetric (Setoid.eq ansSetoid) {m} {n} pr = i
|
||||
where
|
||||
open Group g renaming (inverse to _^-1)
|
||||
open Setoid S
|
||||
transitive = Equivalence.transitive (Setoid.eq S)
|
||||
symmetric = Equivalence.symmetric (Setoid.eq S)
|
||||
i : (y · x) · (x ^-1) ∼ (z · x) · (x ^-1)
|
||||
i = +WellDefined pr ~refl
|
||||
j : y · (x · (x ^-1)) ∼ (z · x) · (x ^-1)
|
||||
j = transitive +Associative i
|
||||
j' : y · 0G ∼ (z · x) · (x ^-1)
|
||||
j' = transitive (+WellDefined ~refl (symmetric invRight)) j
|
||||
k : y ∼ (z · x) · (x ^-1)
|
||||
k = transitive (symmetric identRight) j'
|
||||
l : y ∼ z · (x · (x ^-1))
|
||||
l = transitive k (symmetric +Associative)
|
||||
m : y ∼ z · 0G
|
||||
m = transitive l (+WellDefined ~refl invRight)
|
||||
|
||||
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))
|
||||
g : f (Group.inverse G (m ·A Group.inverse G n)) ∼ 0G
|
||||
g = transitive (homRespectsInverse fHom {m ·A Group.inverse G n}) (transitive (inverseWellDefined H pr) (invIdentity H))
|
||||
h : f (Group.inverse G (Group.inverse G n) ·A Group.inverse G m) ∼ 0G
|
||||
h = transitive (GroupHom.wellDefined fHom (Equivalence.symmetric (Setoid.eq S) (invContravariant G))) g
|
||||
i : f (n ·A Group.inverse G m) ∼ 0G
|
||||
i = transitive (GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.symmetric (Setoid.eq S) (invTwice G n)) (Equivalence.reflexive (Setoid.eq S)))) h
|
||||
Equivalence.transitive (Setoid.eq ansSetoid) {m} {n} {o} prmn prno = transitive (GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.reflexive (Setoid.eq S)) (Equivalence.symmetric (Setoid.eq S) (Group.identLeft G)))) k
|
||||
where
|
||||
open Group G
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
g : f (m ·A Group.inverse G n) ·B f (n ·A Group.inverse G o) ∼ 0G ·B 0G
|
||||
g = replaceGroupOp H reflexive reflexive prmn prno reflexive
|
||||
h : f (m ·A Group.inverse G n) ·B f (n ·A Group.inverse G o) ∼ 0G
|
||||
h = transitive g identLeft
|
||||
i : f ((m ·A Group.inverse G n) ·A (n ·A Group.inverse G o)) ∼ 0G
|
||||
i = transitive (GroupHom.groupHom fHom) h
|
||||
j : f (m ·A (((Group.inverse G n) ·A n) ·A Group.inverse G o)) ∼ 0G
|
||||
j = transitive (GroupHom.wellDefined fHom (fourWay+Associative G)) i
|
||||
k : f (m ·A ((Group.0G G) ·A Group.inverse G o)) ∼ 0G
|
||||
k = transitive (GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.reflexive (Setoid.eq S)) (Group.+WellDefined G (Equivalence.symmetric (Setoid.eq S) (Group.invLeft G)) (Equivalence.reflexive (Setoid.eq S))))) j
|
||||
|
||||
replaceGroupOpRight : {l m : _} {A : Set l} {S : Setoid {l} {m} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b c x : A} → (Setoid._∼_ S a (b · c)) → (Setoid._∼_ S c x) → (Setoid._∼_ S a (b · x))
|
||||
replaceGroupOpRight {S = S} {_·_} G a~bc c~x = transitive a~bc (+WellDefined reflexive c~x)
|
||||
where
|
||||
open Group G
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
quotientGroup : {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_} → {underf : A → B} → (f : GroupHom G H underf) → Group (quotientGroupSetoid G f) _·A_
|
||||
Group.+WellDefined (quotientGroup {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {H = H} {underf = f} fHom) {x} {y} {m} {n} x~m y~n = ans
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence (Setoid.eq T)
|
||||
p1 : f ((x ·A y) ·A (Group.inverse G (m ·A n))) ∼ f ((x ·A y) ·A ((Group.inverse G n) ·A (Group.inverse G m)))
|
||||
p2 : f ((x ·A y) ·A ((Group.inverse G n) ·A (Group.inverse G m))) ∼ f (x ·A ((y ·A (Group.inverse G n)) ·A (Group.inverse G m)))
|
||||
p3 : f (x ·A ((y ·A (Group.inverse G n)) ·A (Group.inverse G m))) ∼ (f x) ·B f ((y ·A (Group.inverse G n)) ·A (Group.inverse G m))
|
||||
p4 : (f x) ·B f ((y ·A (Group.inverse G n)) ·A (Group.inverse G m)) ∼ (f x) ·B (f (y ·A (Group.inverse G n)) ·B f (Group.inverse G m))
|
||||
p5 : (f x) ·B (f (y ·A (Group.inverse G n)) ·B f (Group.inverse G m)) ∼ (f x) ·B (Group.0G H ·B f (Group.inverse G m))
|
||||
p6 : (f x) ·B (Group.0G H ·B f (Group.inverse G m)) ∼ (f x) ·B f (Group.inverse G m)
|
||||
p7 : (f x) ·B f (Group.inverse G m) ∼ f (x ·A (Group.inverse G m))
|
||||
p8 : f (x ·A (Group.inverse G m)) ∼ Group.0G H
|
||||
p1 = GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.reflexive (Setoid.eq S)) (invContravariant G))
|
||||
p2 = GroupHom.wellDefined fHom (Equivalence.symmetric (Setoid.eq S) (fourWay+Associative G))
|
||||
p3 = GroupHom.groupHom fHom
|
||||
p4 = Group.+WellDefined H reflexive (GroupHom.groupHom fHom)
|
||||
p5 = Group.+WellDefined H reflexive (replaceGroupOp H (symmetric y~n) reflexive reflexive reflexive reflexive)
|
||||
p6 = Group.+WellDefined H reflexive (Group.identLeft H)
|
||||
p7 = symmetric (GroupHom.groupHom fHom)
|
||||
p8 = x~m
|
||||
ans : f ((x ·A y) ·A (Group.inverse G (m ·A n))) ∼ Group.0G H
|
||||
ans = transitive p1 (transitive p2 (transitive p3 (transitive p4 (transitive p5 (transitive p6 (transitive p7 p8))))))
|
||||
Group.0G (quotientGroup G fHom) = Group.0G G
|
||||
Group.inverse (quotientGroup G fHom) = Group.inverse G
|
||||
Group.+Associative (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} {b} {c} = ans
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence (Setoid.eq T)
|
||||
ans : f ((a ·A (b ·A c)) ·A (Group.inverse G ((a ·A b) ·A c))) ∼ Group.0G H
|
||||
ans = transitive (GroupHom.wellDefined fHom (transferToRight'' G (Group.+Associative G))) (imageOfIdentityIsIdentity fHom)
|
||||
Group.identRight (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
|
||||
where
|
||||
open Group G
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
transitiveG = Equivalence.transitive (Setoid.eq S)
|
||||
ans : f ((a ·A 0G) ·A inverse a) ∼ Group.0G H
|
||||
ans = transitive (GroupHom.wellDefined fHom (transitiveG (Group.+WellDefined G (Group.identRight G) (Equivalence.reflexive (Setoid.eq S))) (Group.invRight G))) (imageOfIdentityIsIdentity fHom)
|
||||
Group.identLeft (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
|
||||
where
|
||||
open Group G
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
transitiveG = Equivalence.transitive (Setoid.eq S)
|
||||
ans : f ((0G ·A a) ·A (inverse a)) ∼ Group.0G H
|
||||
ans = transitive (GroupHom.wellDefined fHom (transitiveG (Group.+WellDefined G (Group.identLeft G) (Equivalence.reflexive (Setoid.eq S))) (Group.invRight G))) (imageOfIdentityIsIdentity fHom)
|
||||
Group.invLeft (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {x} = ans
|
||||
where
|
||||
open Group G
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
ans : f ((inverse x ·A x) ·A (inverse 0G)) ∼ (Group.0G H)
|
||||
ans = transitive (GroupHom.wellDefined fHom (Equivalence.transitive (Setoid.eq S) (replaceGroupOp G (Equivalence.symmetric (Setoid.eq S) (Group.invLeft G)) (Equivalence.symmetric (Setoid.eq S) (invIdentity G)) (Equivalence.reflexive (Setoid.eq S)) ((Equivalence.reflexive (Setoid.eq S))) ((Equivalence.reflexive (Setoid.eq S)))) (identRight {0G}))) (imageOfIdentityIsIdentity fHom)
|
||||
Group.invRight (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {x} = ans
|
||||
where
|
||||
open Group G
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
ans : f ((x ·A inverse x) ·A (inverse 0G)) ∼ (Group.0G H)
|
||||
ans = transitive (GroupHom.wellDefined fHom (Equivalence.transitive (Setoid.eq S) (replaceGroupOp G (Equivalence.symmetric (Setoid.eq S) (Group.invRight G)) (Equivalence.symmetric (Setoid.eq S) (invIdentity G)) (Equivalence.reflexive (Setoid.eq S)) (Equivalence.reflexive (Setoid.eq S)) (Equivalence.reflexive (Setoid.eq S))) (identRight {0G}))) (imageOfIdentityIsIdentity fHom)
|
||||
|
||||
inverseWellDefined : {l m : _} {A : Set l} {S : Setoid {l} {m} A} {_·_ : A → A → A} (G : Group S _·_) → {x y : A} → (Setoid._∼_ S x y) → Setoid._∼_ S (Group.inverse G x) (Group.inverse G y)
|
||||
inverseWellDefined {S = S} {_·_} G {x} {y} x~y = groupsHaveRightCancellation G x (inverse x) (inverse y) q
|
||||
where
|
||||
open Group G
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
p : inverse x · x ∼ inverse y · y
|
||||
p = transitive invLeft (symmetric invLeft)
|
||||
q : inverse x · x ∼ inverse y · x
|
||||
q = replaceGroupOpRight G {_·_ (inverse x) x} {inverse y} {y} {x} p (symmetric x~y)
|
||||
|
||||
rightInversesAreUnique : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → (x : A) → (y : A) → Setoid._∼_ S (y · x) (Group.0G G) → Setoid._∼_ S y (Group.inverse G x)
|
||||
rightInversesAreUnique {S = S} {_·_} G x y f = transitive i (transitive j (transitive k (transitive l m)))
|
||||
where
|
||||
open Group G renaming (inverse to _^-1)
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
i : y ∼ y · 0G
|
||||
j : y · 0G ∼ y · (x · (x ^-1))
|
||||
k : y · (x · (x ^-1)) ∼ (y · x) · (x ^-1)
|
||||
l : (y · x) · (x ^-1) ∼ 0G · (x ^-1)
|
||||
m : 0G · (x ^-1) ∼ x ^-1
|
||||
i = symmetric identRight
|
||||
j = +WellDefined ~refl (symmetric invRight)
|
||||
k = +Associative
|
||||
l = +WellDefined f ~refl
|
||||
m = identLeft
|
||||
|
||||
leftInversesAreUnique : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → {x : A} → {y : A} → Setoid._∼_ S (x · y) (Group.0G G) → Setoid._∼_ S y (Group.inverse G x)
|
||||
leftInversesAreUnique {S = S} {_·_} G {x} {y} f = rightInversesAreUnique G x y l
|
||||
where
|
||||
open Group G renaming (inverse to _^-1)
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
i : y · (x · y) ∼ y · 0G
|
||||
i' : y · (x · y) ∼ y
|
||||
j : (y · x) · y ∼ y
|
||||
k : (y · x) · y ∼ 0G · y
|
||||
l : y · x ∼ 0G
|
||||
i = +WellDefined ~refl f
|
||||
i' = transitive i identRight
|
||||
j = transitive (symmetric +Associative) i'
|
||||
k = transitive j (symmetric identLeft)
|
||||
l = groupsHaveRightCancellation G y (y · x) 0G k
|
||||
|
||||
invTwice : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → (x : A) → Setoid._∼_ S (Group.inverse G (Group.inverse G x)) x
|
||||
invTwice {S = S} {_·_} G x = symmetric (rightInversesAreUnique G (x ^-1) x invRight)
|
||||
where
|
||||
open Group G renaming (inverse to _^-1)
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
|
||||
fourWay+Associative : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → {r s t u : A} → (Setoid._∼_ S) (r · ((s · t) · u)) ((r · s) · (t · u))
|
||||
fourWay+Associative {S = S} {_·_} G {r} {s} {t} {u} = transitive p1 (transitive p2 p3)
|
||||
where
|
||||
open Group G renaming (inverse to _^-1)
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
p1 : r · ((s · t) · u) ∼ (r · (s · t)) · u
|
||||
p2 : (r · (s · t)) · u ∼ ((r · s) · t) · u
|
||||
p3 : ((r · s) · t) · u ∼ (r · s) · (t · u)
|
||||
p1 = Group.+Associative G
|
||||
p2 = Group.+WellDefined G (Group.+Associative G) reflexive
|
||||
p3 = symmetric (Group.+Associative G)
|
||||
|
||||
fourWay+Associative' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} (G : Group S _·_) {a b c d : A} → (Setoid._∼_ S (((a · b) · c) · d) (a · ((b · c) · d)))
|
||||
fourWay+Associative' {S = S} G = transitive (symmetric +Associative) (symmetric (fourWay+Associative G))
|
||||
where
|
||||
open Group G
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
|
||||
fourWay+Associative'' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} (G : Group S _·_) {a b c d : A} → (Setoid._∼_ S (a · (b · (c · d))) (a · ((b · c) · d)))
|
||||
fourWay+Associative'' {S = S} {_·_ = _·_} G = transitive +Associative (symmetric (fourWay+Associative G))
|
||||
where
|
||||
open Group G
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
|
||||
invContravariant : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → {x y : A} → (Setoid._∼_ S (Group.inverse G (x · y)) ((Group.inverse G y) · (Group.inverse G x)))
|
||||
invContravariant {S = S} {_·_} G {x} {y} = ans
|
||||
where
|
||||
open Group G renaming (inverse to _^-1)
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
otherInv = (y ^-1) · (x ^-1)
|
||||
many+Associatives : x · ((y · (y ^-1)) · (x ^-1)) ∼ (x · y) · ((y ^-1) · (x ^-1))
|
||||
oneMult : (x · y) · otherInv ∼ x · (x ^-1)
|
||||
many+Associatives = fourWay+Associative G
|
||||
oneMult = symmetric (transitive (Group.+WellDefined G reflexive (transitive (symmetric (Group.identLeft G)) (Group.+WellDefined G (symmetric (Group.invRight G)) reflexive))) many+Associatives)
|
||||
otherInvIsInverse : (x · y) · otherInv ∼ 0G
|
||||
otherInvIsInverse = transitive oneMult (Group.invRight G)
|
||||
ans : (x · y) ^-1 ∼ (y ^-1) · (x ^-1)
|
||||
ans = symmetric (leftInversesAreUnique G otherInvIsInverse)
|
||||
|
||||
invIdentity : {l m : _} → {A : Set l} → {S : Setoid {l} {m} A} → {_·_ : A → A → A} → (G : Group S _·_) → Setoid._∼_ S ((Group.inverse G) (Group.0G G)) (Group.0G G)
|
||||
invIdentity {S = S} G = transitive (symmetric identLeft) invRight
|
||||
where
|
||||
open Group G
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
|
||||
transferToRight : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b : A} → Setoid._∼_ S (a · (Group.inverse G b)) (Group.0G G) → Setoid._∼_ S a b
|
||||
transferToRight {S = S} {_·_ = _·_} G {a} {b} ab-1 = transitive (symmetric (invTwice G a)) (transitive u (invTwice G b))
|
||||
where
|
||||
open Setoid S
|
||||
open Group G
|
||||
open Equivalence eq
|
||||
t : inverse a ∼ inverse b
|
||||
t = symmetric (leftInversesAreUnique G ab-1)
|
||||
u : inverse (inverse a) ∼ inverse (inverse b)
|
||||
u = inverseWellDefined G t
|
||||
|
||||
transferToRight' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b : A} → Setoid._∼_ S (a · b) (Group.0G G) → Setoid._∼_ S a (Group.inverse G b)
|
||||
transferToRight' {S = S} {_·_ = _·_} G {a} {b} ab-1 = transferToRight G lemma
|
||||
where
|
||||
open Setoid S
|
||||
open Group G
|
||||
open Equivalence eq
|
||||
lemma : a · (inverse (inverse b)) ∼ 0G
|
||||
lemma = transitive (+WellDefined reflexive (invTwice G b)) ab-1
|
||||
|
||||
transferToRight'' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b : A} → Setoid._∼_ S a b → Setoid._∼_ S (a · (Group.inverse G b)) (Group.0G G)
|
||||
transferToRight'' {S = S} {_·_ = _·_} G {a} {b} a~b = transitive (Group.+WellDefined G a~b reflexive) invRight
|
||||
where
|
||||
open Group G
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
|
||||
directSumGroup : {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 (directSumGroup {A = A} {B} g h) (s ,, t) (u ,, v) = Group.+WellDefined g s u ,, Group.+WellDefined h t v
|
||||
Group.0G (directSumGroup {A = A} {B} g h) = (Group.0G g ,, Group.0G h)
|
||||
Group.inverse (directSumGroup {A = A} {B} g h) (g1 ,, h1) = (Group.inverse g g1) ,, (Group.inverse h h1)
|
||||
Group.+Associative (directSumGroup {A = A} {B} g h) = Group.+Associative g ,, Group.+Associative h
|
||||
Group.identRight (directSumGroup {A = A} {B} g h) = Group.identRight g ,, Group.identRight h
|
||||
Group.identLeft (directSumGroup {A = A} {B} g h) = Group.identLeft g ,, Group.identLeft h
|
||||
Group.invLeft (directSumGroup {A = A} {B} g h) = Group.invLeft g ,, Group.invLeft h
|
||||
Group.invRight (directSumGroup {A = A} {B} g h) = Group.invRight g ,, Group.invRight h
|
||||
|
||||
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
|
||||
|
||||
record GroupHom {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_) (f : A → B) : Set (m ⊔ n ⊔ o ⊔ p) where
|
||||
open Group H
|
||||
open Setoid T
|
||||
field
|
||||
groupHom : {x y : A} → f (x ·A y) ∼ (f x) ·B (f y)
|
||||
wellDefined : {x y : A} → Setoid._∼_ S x y → f x ∼ f y
|
||||
|
||||
record InjectiveGroupHom {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_} {underf : A → B} (f : GroupHom G H underf) : Set (m ⊔ n ⊔ o ⊔ p) where
|
||||
open Setoid S renaming (_∼_ to _∼A_)
|
||||
open Setoid T renaming (_∼_ to _∼B_)
|
||||
field
|
||||
injective : SetoidInjection S T underf
|
||||
|
||||
imageOfIdentityIsIdentity : {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {B : Set n} {T : Setoid {n} {p} 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) → Setoid._∼_ T (f (Group.0G G)) (Group.0G H)
|
||||
imageOfIdentityIsIdentity {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} {G = G} {H = H} {f = f} hom = Equivalence.symmetric (Setoid.eq T) t
|
||||
where
|
||||
open Group H
|
||||
open Setoid T
|
||||
id2 : Setoid._∼_ S (Group.0G G) ((Group.0G G) ·A (Group.0G G))
|
||||
id2 = Equivalence.symmetric (Setoid.eq S) (Group.identRight G)
|
||||
r : f (Group.0G G) ∼ f (Group.0G G) ·B f (Group.0G G)
|
||||
s : 0G ·B f (Group.0G G) ∼ f (Group.0G G) ·B f (Group.0G G)
|
||||
t : 0G ∼ f (Group.0G G)
|
||||
t = groupsHaveRightCancellation H (f (Group.0G G)) 0G (f (Group.0G G)) s
|
||||
s = Equivalence.transitive (Setoid.eq T) identLeft r
|
||||
r = Equivalence.transitive (Setoid.eq T) (GroupHom.wellDefined hom id2) (GroupHom.groupHom hom)
|
||||
|
||||
groupHomsCompose : {m n o r s t : _} {A : Set m} {S : Setoid {m} {r} A} {_+A_ : A → A → A} {B : Set n} {T : Setoid {n} {s} B} {_+B_ : B → B → B} {C : Set o} {U : Setoid {o} {t} C} {_+C_ : C → C → C} {G : Group S _+A_} {H : Group T _+B_} {I : Group U _+C_} {f : A → B} {g : B → C} (fHom : GroupHom G H f) (gHom : GroupHom H I g) → GroupHom G I (g ∘ f)
|
||||
GroupHom.wellDefined (groupHomsCompose {G = G} {H} {I} {f} {g} fHom gHom) {x} {y} pr = GroupHom.wellDefined gHom (GroupHom.wellDefined fHom pr)
|
||||
GroupHom.groupHom (groupHomsCompose {S = S} {_+A_ = _·A_} {T = T} {U = U} {_+C_ = _·C_} {G = G} {H} {I} {f} {g} fHom gHom) {x} {y} = answer
|
||||
where
|
||||
open Group I
|
||||
answer : (Setoid._∼_ U) ((g ∘ f) (x ·A y)) ((g ∘ f) x ·C (g ∘ f) y)
|
||||
answer = (Equivalence.transitive (Setoid.eq U)) (GroupHom.wellDefined gHom (GroupHom.groupHom fHom {x} {y}) ) (GroupHom.groupHom gHom {f x} {f y})
|
||||
|
||||
record GroupIso {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_) (f : A → B) : Set (m ⊔ n ⊔ o ⊔ p) where
|
||||
open Setoid S renaming (_∼_ to _∼G_)
|
||||
open Setoid T renaming (_∼_ to _∼H_)
|
||||
field
|
||||
groupHom : GroupHom G H f
|
||||
bij : SetoidBijection S T f
|
||||
|
||||
record GroupsIsomorphic {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_) : Set (m ⊔ n ⊔ o ⊔ p) where
|
||||
field
|
||||
isomorphism : A → B
|
||||
proof : GroupIso G H isomorphism
|
||||
|
||||
groupIsosCompose : {m n o r s t : _} {A : Set m} {S : Setoid {m} {r} A} {_+A_ : A → A → A} {B : Set n} {T : Setoid {n} {s} B} {_+B_ : B → B → B} {C : Set o} {U : Setoid {o} {t} C} {_+C_ : C → C → C} {G : Group S _+A_} {H : Group T _+B_} {I : Group U _+C_} {f : A → B} {g : B → C} (fHom : GroupIso G H f) (gHom : GroupIso H I g) → GroupIso G I (g ∘ f)
|
||||
GroupIso.groupHom (groupIsosCompose fHom gHom) = groupHomsCompose (GroupIso.groupHom fHom) (GroupIso.groupHom gHom)
|
||||
GroupIso.bij (groupIsosCompose {A = A} {S = S} {T = T} {C = C} {U = U} {f = f} {g = g} fHom gHom) = record { inj = record { injective = λ pr → (SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij fHom))) (SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij gHom)) pr) ; wellDefined = +WellDefined } ; surj = record { surjective = surj ; wellDefined = +WellDefined } }
|
||||
where
|
||||
open Setoid S renaming (_∼_ to _∼A_)
|
||||
open Setoid U renaming (_∼_ to _∼C_)
|
||||
+WellDefined : {x y : A} → (x ∼A y) → (g (f x) ∼C g (f y))
|
||||
+WellDefined = GroupHom.wellDefined (groupHomsCompose (GroupIso.groupHom fHom) (GroupIso.groupHom gHom))
|
||||
surj : {x : C} → Sg A (λ a → (g (f a) ∼C x))
|
||||
surj {x} with SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij gHom)) {x}
|
||||
surj {x} | a , prA with SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij fHom)) {a}
|
||||
... | b , prB = b , transitive (GroupHom.wellDefined (GroupIso.groupHom gHom) prB) prA
|
||||
where
|
||||
open Equivalence (Setoid.eq U)
|
||||
|
||||
record FiniteGroup {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_·_ : A → A → A} (G : Group S _·_) {c : _} (quotientSet : Set c) : Set (a ⊔ b ⊔ c) where
|
||||
field
|
||||
toSet : SetoidToSet S quotientSet
|
||||
finite : FiniteSet quotientSet
|
||||
|
||||
groupOrder : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_·_ : A → A → A} {underG : Group S _·_} {c : _} {quotientSet : Set c} (G : FiniteGroup underG quotientSet) → ℕ
|
||||
groupOrder record { toSet = toSet ; finite = record { size = size ; mapping = mapping ; bij = bij } } = size
|
||||
|
||||
--groupIsoInvertible : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d}} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : A → B} → (iso : GroupIso G H f) → GroupIso H G (Invertible.inverse (bijectionImpliesInvertible (GroupIso.bijective iso)))
|
||||
--GroupHom.groupHom (GroupIso.groupHom (groupIsoInvertible {G = G} {H} {f} iso)) {x} {y} = {!!}
|
||||
-- where
|
||||
-- open Group G renaming (_·_ to _+G_)
|
||||
-- open Group H renaming (_·_ to _+H_)
|
||||
--GroupHom.wellDefined (GroupIso.groupHom (groupIsoInvertible {G = G} {H} {f} iso)) {x} {y} x∼y = {!GroupHom.wellDefined x∼y!}
|
||||
--GroupIso.bijective (groupIsoInvertible {G = G} {H} {f} iso) = invertibleImpliesBijection (inverseIsInvertible (bijectionImpliesInvertible (GroupIso.bijective iso)))
|
||||
|
||||
homRespectsInverse : {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_} {underF : A → B} → (f : GroupHom G H underF) → {x : A} → Setoid._∼_ T (underF (Group.inverse G x)) (Group.inverse H (underF x))
|
||||
homRespectsInverse {T = T} {_·A_ = _·A_} {_·B_ = _·B_} {G = G} {H = H} {underF = f} fHom {x} = rightInversesAreUnique H (f x) (f (Group.inverse G x)) (transitive (symmetric (GroupHom.groupHom fHom)) (transitive (GroupHom.wellDefined fHom (Group.invLeft G)) (imageOfIdentityIsIdentity fHom)))
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
|
||||
quotientGroupSetoid : {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_} → {underf : A → B} → (f : GroupHom G H underf) → (Setoid {a} {d} A)
|
||||
quotientGroupSetoid {A = A} {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {H} {f} fHom = ansSetoid
|
||||
where
|
||||
open Setoid T
|
||||
open Group H
|
||||
open Equivalence eq
|
||||
ansSetoid : Setoid A
|
||||
Setoid._∼_ ansSetoid r s = (f (r ·A (Group.inverse G s))) ∼ 0G
|
||||
Equivalence.reflexive (Setoid.eq ansSetoid) {b} = transitive (GroupHom.wellDefined fHom (Group.invRight G)) (imageOfIdentityIsIdentity fHom)
|
||||
Equivalence.symmetric (Setoid.eq ansSetoid) {m} {n} pr = i
|
||||
where
|
||||
g : f (Group.inverse G (m ·A Group.inverse G n)) ∼ 0G
|
||||
g = transitive (homRespectsInverse fHom {m ·A Group.inverse G n}) (transitive (inverseWellDefined H pr) (invIdentity H))
|
||||
h : f (Group.inverse G (Group.inverse G n) ·A Group.inverse G m) ∼ 0G
|
||||
h = transitive (GroupHom.wellDefined fHom (Equivalence.symmetric (Setoid.eq S) (invContravariant G))) g
|
||||
i : f (n ·A Group.inverse G m) ∼ 0G
|
||||
i = transitive (GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.symmetric (Setoid.eq S) (invTwice G n)) (Equivalence.reflexive (Setoid.eq S)))) h
|
||||
Equivalence.transitive (Setoid.eq ansSetoid) {m} {n} {o} prmn prno = transitive (GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.reflexive (Setoid.eq S)) (Equivalence.symmetric (Setoid.eq S) (Group.identLeft G)))) k
|
||||
where
|
||||
g : f (m ·A Group.inverse G n) ·B f (n ·A Group.inverse G o) ∼ 0G ·B 0G
|
||||
g = replaceGroupOp H reflexive reflexive prmn prno reflexive
|
||||
h : f (m ·A Group.inverse G n) ·B f (n ·A Group.inverse G o) ∼ 0G
|
||||
h = transitive g identLeft
|
||||
i : f ((m ·A Group.inverse G n) ·A (n ·A Group.inverse G o)) ∼ 0G
|
||||
i = transitive (GroupHom.groupHom fHom) h
|
||||
j : f (m ·A (((Group.inverse G n) ·A n) ·A Group.inverse G o)) ∼ 0G
|
||||
j = transitive (GroupHom.wellDefined fHom (fourWay+Associative G)) i
|
||||
k : f (m ·A ((Group.0G G) ·A Group.inverse G o)) ∼ 0G
|
||||
k = transitive (GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.reflexive (Setoid.eq S)) (Group.+WellDefined G (Equivalence.symmetric (Setoid.eq S) (Group.invLeft G)) (Equivalence.reflexive (Setoid.eq S))))) j
|
||||
|
||||
quotientGroup : {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_} → {underf : A → B} → (f : GroupHom G H underf) → Group (quotientGroupSetoid G f) _·A_
|
||||
Group.+WellDefined (quotientGroup {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {H = H} {underf = f} fHom) {x} {y} {m} {n} x~m y~n = ans
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence (Setoid.eq T)
|
||||
p1 : f ((x ·A y) ·A (Group.inverse G (m ·A n))) ∼ f ((x ·A y) ·A ((Group.inverse G n) ·A (Group.inverse G m)))
|
||||
p2 : f ((x ·A y) ·A ((Group.inverse G n) ·A (Group.inverse G m))) ∼ f (x ·A ((y ·A (Group.inverse G n)) ·A (Group.inverse G m)))
|
||||
p3 : f (x ·A ((y ·A (Group.inverse G n)) ·A (Group.inverse G m))) ∼ (f x) ·B f ((y ·A (Group.inverse G n)) ·A (Group.inverse G m))
|
||||
p4 : (f x) ·B f ((y ·A (Group.inverse G n)) ·A (Group.inverse G m)) ∼ (f x) ·B (f (y ·A (Group.inverse G n)) ·B f (Group.inverse G m))
|
||||
p5 : (f x) ·B (f (y ·A (Group.inverse G n)) ·B f (Group.inverse G m)) ∼ (f x) ·B (Group.0G H ·B f (Group.inverse G m))
|
||||
p6 : (f x) ·B (Group.0G H ·B f (Group.inverse G m)) ∼ (f x) ·B f (Group.inverse G m)
|
||||
p7 : (f x) ·B f (Group.inverse G m) ∼ f (x ·A (Group.inverse G m))
|
||||
p8 : f (x ·A (Group.inverse G m)) ∼ Group.0G H
|
||||
p1 = GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.reflexive (Setoid.eq S)) (invContravariant G))
|
||||
p2 = GroupHom.wellDefined fHom (Equivalence.symmetric (Setoid.eq S) (fourWay+Associative G))
|
||||
p3 = GroupHom.groupHom fHom
|
||||
p4 = Group.+WellDefined H reflexive (GroupHom.groupHom fHom)
|
||||
p5 = Group.+WellDefined H reflexive (replaceGroupOp H (symmetric y~n) reflexive reflexive reflexive reflexive)
|
||||
p6 = Group.+WellDefined H reflexive (Group.identLeft H)
|
||||
p7 = symmetric (GroupHom.groupHom fHom)
|
||||
p8 = x~m
|
||||
ans : f ((x ·A y) ·A (Group.inverse G (m ·A n))) ∼ Group.0G H
|
||||
ans = transitive p1 (transitive p2 (transitive p3 (transitive p4 (transitive p5 (transitive p6 (transitive p7 p8))))))
|
||||
Group.0G (quotientGroup G fHom) = Group.0G G
|
||||
Group.inverse (quotientGroup G fHom) = Group.inverse G
|
||||
Group.+Associative (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} {b} {c} = ans
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence (Setoid.eq T)
|
||||
ans : f ((a ·A (b ·A c)) ·A (Group.inverse G ((a ·A b) ·A c))) ∼ Group.0G H
|
||||
ans = transitive (GroupHom.wellDefined fHom (transferToRight'' G (Group.+Associative G))) (imageOfIdentityIsIdentity fHom)
|
||||
Group.identRight (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
|
||||
where
|
||||
open Group G
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
transitiveG = Equivalence.transitive (Setoid.eq S)
|
||||
ans : f ((a ·A 0G) ·A inverse a) ∼ Group.0G H
|
||||
ans = transitive (GroupHom.wellDefined fHom (transitiveG (Group.+WellDefined G (Group.identRight G) (Equivalence.reflexive (Setoid.eq S))) (Group.invRight G))) (imageOfIdentityIsIdentity fHom)
|
||||
Group.identLeft (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
|
||||
where
|
||||
open Group G
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
transitiveG = Equivalence.transitive (Setoid.eq S)
|
||||
ans : f ((0G ·A a) ·A (inverse a)) ∼ Group.0G H
|
||||
ans = transitive (GroupHom.wellDefined fHom (transitiveG (Group.+WellDefined G (Group.identLeft G) (Equivalence.reflexive (Setoid.eq S))) (Group.invRight G))) (imageOfIdentityIsIdentity fHom)
|
||||
Group.invLeft (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {x} = ans
|
||||
where
|
||||
open Group G
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
ans : f ((inverse x ·A x) ·A (inverse 0G)) ∼ (Group.0G H)
|
||||
ans = transitive (GroupHom.wellDefined fHom (Equivalence.transitive (Setoid.eq S) (replaceGroupOp G (Equivalence.symmetric (Setoid.eq S) (Group.invLeft G)) (Equivalence.symmetric (Setoid.eq S) (invIdentity G)) (Equivalence.reflexive (Setoid.eq S)) ((Equivalence.reflexive (Setoid.eq S))) ((Equivalence.reflexive (Setoid.eq S)))) (identRight {0G}))) (imageOfIdentityIsIdentity fHom)
|
||||
Group.invRight (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {x} = ans
|
||||
where
|
||||
open Group G
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
ans : f ((x ·A inverse x) ·A (inverse 0G)) ∼ (Group.0G H)
|
||||
ans = transitive (GroupHom.wellDefined fHom (Equivalence.transitive (Setoid.eq S) (replaceGroupOp G (Equivalence.symmetric (Setoid.eq S) (Group.invRight G)) (Equivalence.symmetric (Setoid.eq S) (invIdentity G)) (Equivalence.reflexive (Setoid.eq S)) (Equivalence.reflexive (Setoid.eq S)) (Equivalence.reflexive (Setoid.eq S))) (identRight {0G}))) (imageOfIdentityIsIdentity fHom)
|
||||
|
||||
record Subgroup {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 : B → A} (hom : GroupHom H G f) : Set (a ⊔ b ⊔ c ⊔ d) where
|
||||
open Setoid T renaming (_∼_ to _∼G_)
|
||||
open Setoid S renaming (_∼_ to _∼H_)
|
||||
field
|
||||
fInj : SetoidInjection T S f
|
||||
record Subgroup {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 : B → A} (hom : GroupHom H G f) : Set (a ⊔ b ⊔ c ⊔ d) where
|
||||
open Setoid T renaming (_∼_ to _∼G_)
|
||||
open Setoid S renaming (_∼_ to _∼H_)
|
||||
field
|
||||
fInj : SetoidInjection T S f
|
||||
{-
|
||||
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 = {!!}
|
||||
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 = {!!}
|
||||
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 = {!!}
|
||||
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 = {!!}
|
||||
|
||||
-}
|
||||
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
|
||||
AbelianGroup.commutative (subgroupOfAbelianIsAbelian {S = S} {_+B_ = _+B_} {fHom = fHom} record { fInj = fInj } record { commutative = commutative }) {x} {y} = SetoidInjection.injective fInj (transitive (GroupHom.groupHom fHom) (transitive commutative (symmetric (GroupHom.groupHom fHom))))
|
||||
where
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
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
|
||||
AbelianGroup.commutative (subgroupOfAbelianIsAbelian {S = S} {_+B_ = _+B_} {fHom = fHom} record { fInj = fInj } record { commutative = commutative }) {x} {y} = SetoidInjection.injective fInj (transitive (GroupHom.groupHom fHom) (transitive commutative (symmetric (GroupHom.groupHom fHom))))
|
||||
where
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
|
||||
abelianIsGroupProperty : {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_} → GroupsIsomorphic G H → AbelianGroup H → AbelianGroup G
|
||||
abelianIsGroupProperty iso abH = subgroupOfAbelianIsAbelian {fHom = GroupIso.groupHom (GroupsIsomorphic.proof iso)} (record { fInj = SetoidBijection.inj (GroupIso.bij (GroupsIsomorphic.proof iso)) }) abH
|
||||
abelianIsGroupProperty : {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_} → GroupsIsomorphic G H → AbelianGroup H → AbelianGroup G
|
||||
abelianIsGroupProperty iso abH = subgroupOfAbelianIsAbelian {fHom = GroupIso.groupHom (GroupsIsomorphic.proof iso)} (record { fInj = SetoidBijection.inj (GroupIso.bij (GroupsIsomorphic.proof iso)) }) abH
|
||||
|
||||
Reference in New Issue
Block a user