Structures · Algebra
AddCancelMonoid
An additive monoid in which addition is cancellative on both sides.
Main examples are ℕ and groups. This is the right typeclass for many sum lemmas, as having a zero
is useful to define the sum over the empty set, so AddRightCancelMonoid is not enough.
- Defined in
- Mathlib.Algebra.Group.Defs
- Shape
- One type argument
Extends2
Extended by2
Forgetful instances
Provided automatically by
Concrete types that are instances10
- Matrix
- DomAddAct
- FreeAddMonoid
- DyckWord
- Prod
- OrderDual
- ULift
- Lex
- AddOpposite
- Colex
How is a type an instance?
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Assumed by34
- Finset.HasAntidiagonal.antidiagonal_congr
- MeasureTheory.FinMeasAdditive.map_empty_eq_zero
- Finset.Nonempty.card_nsmul_mono
- Finset.card_nsmul_mono
- Finset.HasAntidiagonal.antidiagonal_subtype_ext
- Measurable.add_stronglyMeasurable
- Measurable.add_simpleFunc
- Measurable.simpleFunc_add
- Finset.Nontrivial.nsmul
- Finset.card_le_card_nsmul
- Set.Nontrivial.nsmul
- Colex.instAddCancelMonoid
- Prod.instAddCancelMonoid
- AddCancelMonoid.toAddRightCancelMonoid
- Function.Injective.addCancelMonoid
- eq_zero_of_add_left'
- AddMonoidHom.instAddMonoidHomClassAddHom_2
- eq_zero_of_add_right'
- AddCancelMonoid.toAddLeftCancelMonoid
- OrderDual.instAddCancelMonoid
- Matrix.instAddCancelMonoid
- AddCancelMonoid.toIsRightCancelAdd
- FunLike.addCancelMonoid
- Measurable.stronglyMeasurable_add
- Finset.HasAntidiagonal.antidiagonal_subtype_ext_iff
- DomAddAct.instAddCancelMonoidOfAddOpposite
- ULift.addCancelMonoid
- add_ne_zero'
- Lex.instAddCancelMonoid
- AddOpposite.instAddCancelMonoid
- Pi.addCancelMonoid
- AddCancelMonoid.toIsCancelAdd
- instIsCancelVAdd
- add_eq_zero'