- inverseDistributesOverGroupOp : VerifiedAbelianGroup t =>
(l : t) ->
(r : t) ->
inverse (l <+>
r) =
inverse l <+>
inverse r
A proof that -(x + y) = -x - y in any verified abelian group.
- inverseNeutralIsNeutral : VerifiedGroup t =>
inverse (the t
neutral) =
neutral
A prof that -0 = 0 in any verified group.
- inverseSquaredIsIdentity : VerifiedGroup t =>
(x : t) ->
inverse (inverse x) =
x
A proof that -(-x) = x in any verified group.
- multInverseInversesL : VerifiedRingWithUnity t =>
(l : t) ->
(r : t) ->
inverse l <.>
r =
inverse (l <.>
r)
A proof that inverse operator can be extracted before multiplication
(-x)y = -(xy)
- multInverseInversesR : VerifiedRingWithUnity t =>
(l : t) ->
(r : t) ->
l <.>
inverse r =
inverse (l <.>
r)
A proof that inverse operator can be extracted before multiplication
x(-y) = -(xy)
- multNegativeByNegativeIsPositive : VerifiedRingWithUnity t =>
(l : t) ->
(r : t) ->
inverse l <.>
inverse r =
l <.>
r
A proof that multiplication of inverses is the same as multiplication of original
elements. (-x)(-y) = xy
- multNeutralAbsorbingL : VerifiedRingWithUnity t =>
(r : t) ->
neutral <.>
r =
neutral
A proof that anything multiplied by zero yields zero back.
- multNeutralAbsorbingR : VerifiedRingWithUnity t =>
(l : t) ->
l <.>
neutral =
neutral
A proof that anything multiplied by zero yields zero back.