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Define the free product (#104)

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Patrick Stevens
2020-03-28 21:34:14 +00:00
committed by GitHub
parent 162a1c7a40
commit a27375db4e
21 changed files with 1326 additions and 18 deletions
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11
Groups/Lemmas.agda
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@@ -44,8 +44,8 @@ groupsHaveRightCancellation x y z pr = transitive m identRight
m : y ∼ z · 0G
m = transitive l (+WellDefined ~refl invRight)
rightInversesAreUnique : (x : A) → (y : A) → (y · x) ∼ 0G → y ∼ (inverse x)
rightInversesAreUnique x y f = transitive i (transitive j (transitive k (transitive l m)))
rightInversesAreUnique : {x : A} → {y : A} → (y · x) ∼ 0G → y ∼ (inverse x)
rightInversesAreUnique {x} {y} f = transitive i (transitive j (transitive k (transitive l m)))
where
_^-1 = inverse
i : y ∼ y · 0G
@@ -60,7 +60,7 @@ rightInversesAreUnique x y f = transitive i (transitive j (transitive k (transit
m = identLeft
leftInversesAreUnique : {x : A} → {y : A} → (x · y) ∼ 0G → y ∼ (inverse x)
leftInversesAreUnique {x} {y} f = rightInversesAreUnique x y l
leftInversesAreUnique {x} {y} f = rightInversesAreUnique {x} {y} l
where
_^-1 = inverse
i : y · (x · y) ∼ y · 0G
@@ -76,7 +76,7 @@ leftInversesAreUnique {x} {y} f = rightInversesAreUnique x y l
invTwice : (x : A) → (inverse (inverse x)) ∼ x
invTwice x = symmetric (rightInversesAreUnique (x ^-1) x invRight)
invTwice x = symmetric (rightInversesAreUnique {x ^-1} {x} invRight)
where
_^-1 = inverse
@@ -139,3 +139,6 @@ invContravariant {x} {y} = ans
equalsDoubleImpliesZero : {x : A} → (x · x) ∼ x → x ∼ 0G
equalsDoubleImpliesZero 2x=x = transitive (symmetric identLeft) (transitive (+WellDefined (symmetric invLeft) reflexive) (transitive (symmetric +Associative) (transitive (+WellDefined reflexive 2x=x) invLeft)))
invZeroImpliesZero : {x : A} → (inverse x) ∼ 0G → x ∼ 0G
invZeroImpliesZero {x} pr = transitive (symmetric (invTwice x)) (transitive (inverseWellDefined pr) invIdent)