Typeclasses for (semi)groups and monoids

In this file we define typeclasses for algebraic structures with one binary operation. The classes are named (Add)?(Comm)?(Semigroup|Monoid|Group), where Add means that the class uses additive notation and Comm means that the class assumes that the binary operation is commutative.

The file does not contain any lemmas except for

• axioms of typeclasses restated in the root namespace;
• lemmas required for instances.

For basic lemmas about these classes see Algebra.Group.Basic.

We register the following instances:

• Pow M ℕ, for monoids M, and Pow G ℤ for groups G;
• SMul ℕ M for additive monoids M, and SMul ℤ G for additive groups G.

Notation

• +, -, *, /, ^ : the usual arithmetic operations; the underlying functions are Add.add, Neg.neg/Sub.sub, Mul.mul, Div.div, and HPow.hPow.

def leftMul {G : Type u_1} [Mul G] : G → G → G

leftMul g denotes left multiplication by g

Equations

• leftMul g x = g * x

def leftAdd {G : Type u_1} [Add G] : G → G → G

leftAdd g denotes left addition by g

Equations

• leftAdd g x = g + x

def rightMul {G : Type u_1} [Mul G] : G → G → G

rightMul g denotes right multiplication by g

Equations

• rightMul g x = x * g

def rightAdd {G : Type u_1} [Add G] : G → G → G

rightAdd g denotes right addition by g

Equations

• rightAdd g x = x + g

class IsLeftCancelMul (G : Type u) [Mul G] : Prop

A mixin for left cancellative multiplication.

• mul_left_cancel (a b c : G) : a * b = a * c → b = c

  Multiplication is left cancellative.

class IsRightCancelMul (G : Type u) [Mul G] : Prop

A mixin for right cancellative multiplication.

• mul_right_cancel (a b c : G) : a * b = c * b → a = c

  Multiplication is right cancellative.

class IsCancelMul (G : Type u) [Mul G] extends IsLeftCancelMul G, IsRightCancelMul G : Prop

A mixin for cancellative multiplication.

• mul_left_cancel (a b c : G) : a * b = a * c → b = c
• mul_right_cancel (a b c : G) : a * b = c * b → a = c

class IsLeftCancelAdd (G : Type u) [Add G] : Prop

A mixin for left cancellative addition.

• add_left_cancel (a b c : G) : a + b = a + c → b = c

  Addition is left cancellative.

class IsRightCancelAdd (G : Type u) [Add G] : Prop

A mixin for right cancellative addition.

• add_right_cancel (a b c : G) : a + b = c + b → a = c

  Addition is right cancellative.

class IsCancelAdd (G : Type u) [Add G] extends IsLeftCancelAdd G, IsRightCancelAdd G : Prop

A mixin for cancellative addition.

• add_left_cancel (a b c : G) : a + b = a + c → b = c
• add_right_cancel (a b c : G) : a + b = c + b → a = c

theorem mul_left_cancel {G : Type u_1} [Mul G] [IsLeftCancelMul G] {a b c : G} : a * b = a * c → b = c

theorem add_left_cancel {G : Type u_1} [Add G] [IsLeftCancelAdd G] {a b c : G} : a + b = a + c → b = c

theorem mul_left_cancel_iff {G : Type u_1} [Mul G] [IsLeftCancelMul G] {a b c : G} : a * b = a * c ↔ b = c

theorem add_left_cancel_iff {G : Type u_1} [Add G] [IsLeftCancelAdd G] {a b c : G} : a + b = a + c ↔ b = c

theorem mul_right_cancel {G : Type u_1} [Mul G] [IsRightCancelMul G] {a b c : G} : a * b = c * b → a = c

theorem add_right_cancel {G : Type u_1} [Add G] [IsRightCancelAdd G] {a b c : G} : a + b = c + b → a = c

theorem mul_right_cancel_iff {G : Type u_1} [Mul G] [IsRightCancelMul G] {a b c : G} : b * a = c * a ↔ b = c

theorem add_right_cancel_iff {G : Type u_1} [Add G] [IsRightCancelAdd G] {a b c : G} : b + a = c + a ↔ b = c

class Semigroup (G : Type u) extends Mul G : Type u

A semigroup is a type with an associative (*).

• mul : G → G → G
• mul_assoc (a b c : G) : a * b * c = a * (b * c)

  Multiplication is associative

theorem Semigroup.ext {G : Type u} {x y : Semigroup G} (mul : Mul.mul = Mul.mul) : x = y

class AddSemigroup (G : Type u) extends Add G : Type u

An additive semigroup is a type with an associative (+).

• add : G → G → G
• add_assoc (a b c : G) : a + b + c = a + (b + c)

  Addition is associative

theorem AddSemigroup.ext {G : Type u} {x y : AddSemigroup G} (add : Add.add = Add.add) : x = y

theorem mul_assoc {G : Type u_1} [Semigroup G] (a b c : G) : a * b * c = a * (b * c)

theorem add_assoc {G : Type u_1} [AddSemigroup G] (a b c : G) : a + b + c = a + (b + c)

class AddCommMagma (G : Type u) extends Add G : Type u

A commutative additive magma is a type with an addition which commutes.

• add : G → G → G
• add_comm (a b : G) : a + b = b + a

  Addition is commutative in an commutative additive magma.

theorem AddCommMagma.ext {G : Type u} {x y : AddCommMagma G} (add : Add.add = Add.add) : x = y

class CommMagma (G : Type u) extends Mul G : Type u

A commutative multiplicative magma is a type with a multiplication which commutes.

• mul : G → G → G
• mul_comm (a b : G) : a * b = b * a

  Multiplication is commutative in a commutative multiplicative magma.

theorem CommMagma.ext {G : Type u} {x y : CommMagma G} (mul : Mul.mul = Mul.mul) : x = y

class CommSemigroup (G : Type u) extends Semigroup G, CommMagma G : Type u

A commutative semigroup is a type with an associative commutative (*).

• mul : G → G → G
• mul_assoc (a b c : G) : a * b * c = a * (b * c)
• mul_comm (a b : G) : a * b = b * a

theorem CommSemigroup.ext {G : Type u} {x y : CommSemigroup G} (mul : Mul.mul = Mul.mul) : x = y

class AddCommSemigroup (G : Type u) extends AddSemigroup G, AddCommMagma G : Type u

A commutative additive semigroup is a type with an associative commutative (+).

• add : G → G → G
• add_assoc (a b c : G) : a + b + c = a + (b + c)
• add_comm (a b : G) : a + b = b + a

theorem AddCommSemigroup.ext {G : Type u} {x y : AddCommSemigroup G} (add : Add.add = Add.add) : x = y

theorem mul_comm {G : Type u_1} [CommMagma G] (a b : G) : a * b = b * a

theorem add_comm {G : Type u_1} [AddCommMagma G] (a b : G) : a + b = b + a

theorem CommMagma.IsRightCancelMul.toIsLeftCancelMul (G : Type u) [CommMagma G] [IsRightCancelMul G] : IsLeftCancelMul G

Any CommMagma G that satisfies IsRightCancelMul G also satisfies IsLeftCancelMul G.

theorem AddCommMagma.IsRightCancelAdd.toIsLeftCancelAdd (G : Type u) [AddCommMagma G] [IsRightCancelAdd G] : IsLeftCancelAdd G

Any AddCommMagma G that satisfies IsRightCancelAdd G also satisfies IsLeftCancelAdd G.

theorem CommMagma.IsLeftCancelMul.toIsRightCancelMul (G : Type u) [CommMagma G] [IsLeftCancelMul G] : IsRightCancelMul G

Any CommMagma G that satisfies IsLeftCancelMul G also satisfies IsRightCancelMul G.

theorem AddCommMagma.IsLeftCancelAdd.toIsRightCancelAdd (G : Type u) [AddCommMagma G] [IsLeftCancelAdd G] : IsRightCancelAdd G

Any AddCommMagma G that satisfies IsLeftCancelAdd G also satisfies IsRightCancelAdd G.

theorem CommMagma.IsLeftCancelMul.toIsCancelMul (G : Type u) [CommMagma G] [IsLeftCancelMul G] : IsCancelMul G

Any CommMagma G that satisfies IsLeftCancelMul G also satisfies IsCancelMul G.

theorem AddCommMagma.IsLeftCancelAdd.toIsCancelAdd (G : Type u) [AddCommMagma G] [IsLeftCancelAdd G] : IsCancelAdd G

Any AddCommMagma G that satisfies IsLeftCancelAdd G also satisfies IsCancelAdd G.

theorem CommMagma.IsRightCancelMul.toIsCancelMul (G : Type u) [CommMagma G] [IsRightCancelMul G] : IsCancelMul G

Any CommMagma G that satisfies IsRightCancelMul G also satisfies IsCancelMul G.

theorem AddCommMagma.IsRightCancelAdd.toIsCancelAdd (G : Type u) [AddCommMagma G] [IsRightCancelAdd G] : IsCancelAdd G

Any AddCommMagma G that satisfies IsRightCancelAdd G also satisfies IsCancelAdd G.

class LeftCancelSemigroup (G : Type u) extends Semigroup G : Type u

A LeftCancelSemigroup is a semigroup such that a * b = a * c implies b = c.

• mul : G → G → G
• mul_assoc (a b c : G) : a * b * c = a * (b * c)
• mul_left_cancel (a b c : G) : a * b = a * c → b = c

theorem LeftCancelSemigroup.ext {G : Type u} {x y : LeftCancelSemigroup G} (mul : Mul.mul = Mul.mul) : x = y

class AddLeftCancelSemigroup (G : Type u) extends AddSemigroup G : Type u

An AddLeftCancelSemigroup is an additive semigroup such that a + b = a + c implies b = c.

• add : G → G → G
• add_assoc (a b c : G) : a + b + c = a + (b + c)
• add_left_cancel (a b c : G) : a + b = a + c → b = c

theorem AddLeftCancelSemigroup.ext {G : Type u} {x y : AddLeftCancelSemigroup G} (add : Add.add = Add.add) : x = y

@[instance 100]

instance LeftCancelSemigroup.toIsLeftCancelMul (G : Type u) [LeftCancelSemigroup G] : IsLeftCancelMul G

Any LeftCancelSemigroup satisfies IsLeftCancelMul.

@[instance 100]

instance AddLeftCancelSemigroup.toIsLeftCancelAdd (G : Type u) [AddLeftCancelSemigroup G] : IsLeftCancelAdd G

Any AddLeftCancelSemigroup satisfies IsLeftCancelAdd.

class RightCancelSemigroup (G : Type u) extends Semigroup G : Type u

A RightCancelSemigroup is a semigroup such that a * b = c * b implies a = c.

• mul : G → G → G
• mul_assoc (a b c : G) : a * b * c = a * (b * c)
• mul_right_cancel (a b c : G) : a * b = c * b → a = c

theorem RightCancelSemigroup.ext {G : Type u} {x y : RightCancelSemigroup G} (mul : Mul.mul = Mul.mul) : x = y

class AddRightCancelSemigroup (G : Type u) extends AddSemigroup G : Type u

An AddRightCancelSemigroup is an additive semigroup such that a + b = c + b implies a = c.

• add : G → G → G
• add_assoc (a b c : G) : a + b + c = a + (b + c)
• add_right_cancel (a b c : G) : a + b = c + b → a = c

theorem AddRightCancelSemigroup.ext {G : Type u} {x y : AddRightCancelSemigroup G} (add : Add.add = Add.add) : x = y

@[instance 100]

instance RightCancelSemigroup.toIsRightCancelMul (G : Type u) [RightCancelSemigroup G] : IsRightCancelMul G

Any RightCancelSemigroup satisfies IsRightCancelMul.

@[instance 100]

instance AddRightCancelSemigroup.toIsRightCancelAdd (G : Type u) [AddRightCancelSemigroup G] : IsRightCancelAdd G

Any AddRightCancelSemigroup satisfies IsRightCancelAdd.

class MulOneClass (M : Type u) extends One M, Mul M : Type u

Typeclass for expressing that a type M with multiplication and a one satisfies 1 * a = a and a * 1 = a for all a : M.

• one : M
• mul : M → M → M
• one_mul (a : M) : 1 * a = a

  One is a left neutral element for multiplication

• mul_one (a : M) : a * 1 = a

  One is a right neutral element for multiplication

class AddZeroClass (M : Type u) extends Zero M, Add M : Type u

Typeclass for expressing that a type M with addition and a zero satisfies 0 + a = a and a + 0 = a for all a : M.

• zero : M
• add : M → M → M
• zero_add (a : M) : 0 + a = a

  Zero is a left neutral element for addition

• add_zero (a : M) : a + 0 = a

  Zero is a right neutral element for addition

theorem AddZeroClass.ext {M : Type u} ⦃m₁ m₂ : AddZeroClass M⦄ : Add.add = Add.add → m₁ = m₂

theorem MulOneClass.ext {M : Type u} ⦃m₁ m₂ : MulOneClass M⦄ : Mul.mul = Mul.mul → m₁ = m₂

@[simp]

theorem one_mul {M : Type u} [MulOneClass M] (a : M) : 1 * a = a

@[simp]

theorem zero_add {M : Type u} [AddZeroClass M] (a : M) : 0 + a = a

@[simp]

theorem mul_one {M : Type u} [MulOneClass M] (a : M) : a * 1 = a

@[simp]

theorem add_zero {M : Type u} [AddZeroClass M] (a : M) : a + 0 = a

def npowRec {M : Type u} [One M] [Mul M] : ℕ → M → M

The fundamental power operation in a monoid. npowRec n a = a*a*...*a n times. Use instead a ^ n, which has better definitional behavior.

Equations

• npowRec 0 x✝ = 1
• npowRec n.succ x✝ = npowRec n x✝ * x✝

def nsmulRec {M : Type u} [Zero M] [Add M] : ℕ → M → M

The fundamental scalar multiplication in an additive monoid. nsmulRec n a = a+a+...+a n times. Use instead n • a, which has better definitional behavior.

Equations

• nsmulRec 0 x✝ = 0
• nsmulRec n.succ x✝ = nsmulRec n x✝ + x✝

theorem npowRec_add {M : Type u} [One M] [Semigroup M] (m n : ℕ) (hn : n ≠ 0) (a : M) (ha : 1 * a = a) : npowRec (m + n) a = npowRec m a * npowRec n a

theorem nsmulRec_add {M : Type u} [Zero M] [AddSemigroup M] (m n : ℕ) (hn : n ≠ 0) (a : M) (ha : 0 + a = a) : nsmulRec (m + n) a = nsmulRec m a + nsmulRec n a

theorem npowRec_succ {M : Type u} [One M] [Semigroup M] (n : ℕ) (hn : n ≠ 0) (a : M) (ha : 1 * a = a) : npowRec (n + 1) a = a * npowRec n a

theorem nsmulRec_succ {M : Type u} [Zero M] [AddSemigroup M] (n : ℕ) (hn : n ≠ 0) (a : M) (ha : 0 + a = a) : nsmulRec (n + 1) a = a + nsmulRec n a

Design note on AddMonoid and Monoid

An AddMonoid has a natural ℕ-action, defined by n • a = a + ... + a, that we want to declare as an instance as it makes it possible to use the language of linear algebra. However, there are often other natural ℕ-actions. For instance, for any semiring R, the space of polynomials Polynomial R has a natural R-action defined by multiplication on the coefficients. This means that Polynomial ℕ would have two natural ℕ-actions, which are equal but not defeq. The same goes for linear maps, tensor products, and so on (and even for ℕ itself).

To solve this issue, we embed an ℕ-action in the definition of an AddMonoid (which is by default equal to the naive action a + ... + a, but can be adjusted when needed), and declare a SMul ℕ α instance using this action. See Note [forgetful inheritance] for more explanations on this pattern.

For example, when we define Polynomial R, then we declare the ℕ-action to be by multiplication on each coefficient (using the ℕ-action on R that comes from the fact that R is an AddMonoid). In this way, the two natural SMul ℕ (Polynomial ℕ) instances are defeq.

The tactic to_additive transfers definitions and results from multiplicative monoids to additive monoids. To work, it has to map fields to fields. This means that we should also add corresponding fields to the multiplicative structure Monoid, which could solve defeq problems for powers if needed. These problems do not come up in practice, so most of the time we will not need to adjust the npow field when defining multiplicative objects.

def npowBinRec {M : Type u_2} [One M] [Mul M] (k : ℕ) : M → M

Exponentiation by repeated squaring.

Equations

• npowBinRec k = npowBinRec.go k 1

def nsmulBinRec {M : Type u_2} [Zero M] [Add M] (k : ℕ) : M → M

Scalar multiplication by repeated self-addition, the additive version of exponentation by repeated squaring.

Equations

• nsmulBinRec k = nsmulBinRec.go k 0

def nsmulBinRec.go {M : Type u_2} [Add M] (k : ℕ) : M → M → M

Auxiliary tail-recursive implementation for nsmulBinRec.

Equations

• nsmulBinRec.go k = Nat.binaryRec (motive := fun (x : ℕ) => M → M → M) (fun (y x : M) => y) (fun (bn : Bool) (_n : ℕ) (fn : M → M → M) (y x : M) => fn (bif bn then y + x else y) (x + x)) k

def npowBinRec.go {M : Type u_2} [Mul M] (k : ℕ) : M → M → M

Auxiliary tail-recursive implementation for npowBinRec.

Equations

• npowBinRec.go k = Nat.binaryRec (motive := fun (x : ℕ) => M → M → M) (fun (y x : M) => y) (fun (bn : Bool) (_n : ℕ) (fn : M → M → M) (y x : M) => fn (bif bn then y * x else y) (x * x)) k

def npowRec' {M : Type u_2} [One M] [Mul M] : ℕ → M → M

A variant of npowRec which is a semigroup homomorphisms from ℕ₊ to M.

Equations

• npowRec' 0 x✝ = 1
• npowRec' 1 x✝ = x✝
• npowRec' k.succ.succ x✝ = npowRec' (k + 1) x✝ * x✝

def nsmulRec' {M : Type u_2} [Zero M] [Add M] : ℕ → M → M

A variant of nsmulRec which is a semigroup homomorphisms from ℕ₊ to M.

Equations

• nsmulRec' 0 x✝ = 0
• nsmulRec' 1 x✝ = x✝
• nsmulRec' k.succ.succ x✝ = nsmulRec' (k + 1) x✝ + x✝

theorem npowRec'_succ {M : Type u_2} [Semigroup M] [One M] {k : ℕ} : k ≠ 0 → ∀ (m : M), npowRec' (k + 1) m = npowRec' k m * m

theorem nsmulRec'_succ {M : Type u_2} [AddSemigroup M] [Zero M] {k : ℕ} : k ≠ 0 → ∀ (m : M), nsmulRec' (k + 1) m = nsmulRec' k m + m

theorem npowRec'_two_mul {M : Type u_2} [Semigroup M] [One M] (k : ℕ) (m : M) : npowRec' (2 * k) m = npowRec' k (m * m)

theorem nsmulRec'_two_add {M : Type u_2} [AddSemigroup M] [Zero M] (k : ℕ) (m : M) : nsmulRec' (2 * k) m = nsmulRec' k (m + m)

theorem npowRec'_mul_comm {M : Type u_2} [Semigroup M] [One M] {k : ℕ} (k0 : k ≠ 0) (m : M) : m * npowRec' k m = npowRec' k m * m

theorem nsmulRec'_add_comm {M : Type u_2} [AddSemigroup M] [Zero M] {k : ℕ} (k0 : k ≠ 0) (m : M) : m + nsmulRec' k m = nsmulRec' k m + m

theorem npowRec_eq {M : Type u_2} [Semigroup M] [One M] (k : ℕ) (m : M) : npowRec (k + 1) m = 1 * npowRec' (k + 1) m

theorem nsmulRec_eq {M : Type u_2} [AddSemigroup M] [Zero M] (k : ℕ) (m : M) : nsmulRec (k + 1) m = 0 + nsmulRec' (k + 1) m

theorem npowBinRec.go_spec {M : Type u_2} [Semigroup M] [One M] (k : ℕ) (m n : M) : npowBinRec.go (k + 1) m n = m * npowRec' (k + 1) n

theorem nsmulBinRec.go_spec {M : Type u_2} [AddSemigroup M] [Zero M] (k : ℕ) (m n : M) : nsmulBinRec.go (k + 1) m n = m + nsmulRec' (k + 1) n

@[reducible, inline]

abbrev npowRecAuto {M : Type u_2} [Semigroup M] [One M] (k : ℕ) (m : M) : M

An abbreviation for npowRec with an additional typeclass assumption on associativity so that we can use @[csimp] to replace it with an implementation by repeated squaring in compiled code.

Equations

• npowRecAuto k m = npowRec k m

@[reducible, inline]

abbrev nsmulRecAuto {M : Type u_2} [AddSemigroup M] [Zero M] (k : ℕ) (m : M) : M

An abbreviation for nsmulRec with an additional typeclass assumptions on associativity so that we can use @[csimp] to replace it with an implementation by repeated doubling in compiled code as an automatic parameter.

Equations

• nsmulRecAuto k m = nsmulRec k m

@[reducible, inline]

abbrev npowBinRecAuto {M : Type u_2} [Semigroup M] [One M] (k : ℕ) (m : M) : M

An abbreviation for npowBinRec with an additional typeclass assumption on associativity so that we can use it in @[csimp] for more performant code generation.

Equations

• npowBinRecAuto k m = npowBinRec k m

@[reducible, inline]

abbrev nsmulBinRecAuto {M : Type u_2} [AddSemigroup M] [Zero M] (k : ℕ) (m : M) : M

An abbreviation for nsmulBinRec with an additional typeclass assumption on associativity so that we can use it in @[csimp] for more performant code generation as an automatic parameter.

Equations

• nsmulBinRecAuto k m = nsmulBinRec k m

@[csimp]

theorem npowRec_eq_npowBinRec : @npowRecAuto = @npowBinRecAuto

@[csimp]

theorem nsmulRec_eq_nsmulBinRec : @nsmulRecAuto = @nsmulBinRecAuto

class AddMonoid (M : Type u) extends AddSemigroup M, AddZeroClass M : Type u

An AddMonoid is an AddSemigroup with an element 0 such that 0 + a = a + 0 = a.

• add : M → M → M
• add_assoc (a b c : M) : a + b + c = a + (b + c)
• zero : M
• zero_add (a : M) : 0 + a = a
• add_zero (a : M) : a + 0 = a
• nsmul : ℕ → M → M

  Multiplication by a natural number. Set this to nsmulRec unless Module diamonds are possible.

• nsmul_zero (x : M) : AddMonoid.nsmul 0 x = 0

  Multiplication by (0 : ℕ) gives 0.

• nsmul_succ (n : ℕ) (x : M) : AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x

  Multiplication by (n + 1 : ℕ) behaves as expected.

class Monoid (M : Type u) extends Semigroup M, MulOneClass M : Type u

A Monoid is a Semigroup with an element 1 such that 1 * a = a * 1 = a.

• mul : M → M → M
• mul_assoc (a b c : M) : a * b * c = a * (b * c)
• one : M
• one_mul (a : M) : 1 * a = a
• mul_one (a : M) : a * 1 = a
• npow : ℕ → M → M

  Raising to the power of a natural number.

• npow_zero (x : M) : Monoid.npow 0 x = 1

  Raising to the power (0 : ℕ) gives 1.

• npow_succ (n : ℕ) (x : M) : Monoid.npow (n + 1) x = Monoid.npow n x * x

  Raising to the power (n + 1 : ℕ) behaves as expected.

@[defaultInstance 10000]

instance Monoid.toNatPow {M : Type u_2} [Monoid M] : Pow M ℕ

Equations

• Monoid.toNatPow = { pow := fun (x : M) (n : ℕ) => Monoid.npow n x }

instance AddMonoid.toNatSMul {M : Type u_2} [AddMonoid M] : SMul ℕ M

Equations

• AddMonoid.toNatSMul = { smul := AddMonoid.nsmul }

@[simp]

theorem npow_eq_pow {M : Type u_2} [Monoid M] (n : ℕ) (x : M) : Monoid.npow n x = x ^ n

@[simp]

theorem nsmul_eq_smul {M : Type u_2} [AddMonoid M] (n : ℕ) (x : M) : AddMonoid.nsmul n x = n • x

theorem left_inv_eq_right_inv {M : Type u_2} [Monoid M] {a b c : M} (hba : b * a = 1) (hac : a * c = 1) : b = c

theorem left_neg_eq_right_neg {M : Type u_2} [AddMonoid M] {a b c : M} (hba : b + a = 0) (hac : a + c = 0) : b = c

@[simp]

theorem pow_zero {M : Type u_2} [Monoid M] (a : M) : a ^ 0 = 1

theorem zero_nsmul {M : Type u_2} [AddMonoid M] (a : M) : 0 • a = 0

theorem pow_succ {M : Type u_2} [Monoid M] (a : M) (n : ℕ) : a ^ (n + 1) = a ^ n * a

theorem succ_nsmul {M : Type u_2} [AddMonoid M] (a : M) (n : ℕ) : (n + 1) • a = n • a + a

@[simp]

theorem pow_one {M : Type u_2} [Monoid M] (a : M) : a ^ 1 = a

@[simp]

theorem one_nsmul {M : Type u_2} [AddMonoid M] (a : M) : 1 • a = a

theorem pow_succ' {M : Type u_2} [Monoid M] (a : M) (n : ℕ) : a ^ (n + 1) = a * a ^ n

theorem succ_nsmul' {M : Type u_2} [AddMonoid M] (a : M) (n : ℕ) : (n + 1) • a = a + n • a

theorem pow_mul_comm' {M : Type u_2} [Monoid M] (a : M) (n : ℕ) : a ^ n * a = a * a ^ n

theorem nsmul_add_comm' {M : Type u_2} [AddMonoid M] (a : M) (n : ℕ) : n • a + a = a + n • a

theorem pow_two {M : Type u_2} [Monoid M] (a : M) : a ^ 2 = a * a

Note that most of the lemmas about powers of two refer to it as sq.

theorem two_nsmul {M : Type u_2} [AddMonoid M] (a : M) : 2 • a = a + a

theorem sq {M : Type u_2} [Monoid M] (a : M) : a ^ 2 = a * a

Alias of pow_two.

---

Note that most of the lemmas about powers of two refer to it as sq.

theorem pow_three' {M : Type u_2} [Monoid M] (a : M) : a ^ 3 = a * a * a

theorem three'_nsmul {M : Type u_2} [AddMonoid M] (a : M) : 3 • a = a + a + a

theorem pow_three {M : Type u_2} [Monoid M] (a : M) : a ^ 3 = a * (a * a)

theorem three_nsmul {M : Type u_2} [AddMonoid M] (a : M) : 3 • a = a + (a + a)

@[simp]

theorem one_pow {M : Type u_2} [Monoid M] (n : ℕ) : 1 ^ n = 1

theorem nsmul_zero {M : Type u_2} [AddMonoid M] (n : ℕ) : n • 0 = 0

theorem pow_add {M : Type u_2} [Monoid M] (a : M) (m n : ℕ) : a ^ (m + n) = a ^ m * a ^ n

theorem add_nsmul {M : Type u_2} [AddMonoid M] (a : M) (m n : ℕ) : (m + n) • a = m • a + n • a

theorem pow_mul_comm {M : Type u_2} [Monoid M] (a : M) (m n : ℕ) : a ^ m * a ^ n = a ^ n * a ^ m

theorem nsmul_add_comm {M : Type u_2} [AddMonoid M] (a : M) (m n : ℕ) : m • a + n • a = n • a + m • a

theorem pow_mul {M : Type u_2} [Monoid M] (a : M) (m n : ℕ) : a ^ (m * n) = (a ^ m) ^ n

theorem mul_nsmul {M : Type u_2} [AddMonoid M] (a : M) (m n : ℕ) : (m * n) • a = n • m • a

theorem pow_mul' {M : Type u_2} [Monoid M] (a : M) (m n : ℕ) : a ^ (m * n) = (a ^ n) ^ m

theorem mul_nsmul' {M : Type u_2} [AddMonoid M] (a : M) (m n : ℕ) : (m * n) • a = m • n • a

theorem pow_right_comm {M : Type u_2} [Monoid M] (a : M) (m n : ℕ) : (a ^ m) ^ n = (a ^ n) ^ m

theorem nsmul_left_comm {M : Type u_2} [AddMonoid M] (a : M) (m n : ℕ) : n • m • a = m • n • a

class AddCommMonoid (M : Type u) extends AddMonoid M, AddCommSemigroup M : Type u

An additive commutative monoid is an additive monoid with commutative (+).

• add : M → M → M
• add_assoc (a b c : M) : a + b + c = a + (b + c)
• zero : M
• zero_add (a : M) : 0 + a = a
• add_zero (a : M) : a + 0 = a
• nsmul : ℕ → M → M
• nsmul_zero (x : M) : AddMonoid.nsmul 0 x = 0
• nsmul_succ (n : ℕ) (x : M) : AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• add_comm (a b : M) : a + b = b + a

class CommMonoid (M : Type u) extends Monoid M, CommSemigroup M : Type u

A commutative monoid is a monoid with commutative (*).

• mul : M → M → M
• mul_assoc (a b c : M) : a * b * c = a * (b * c)
• one : M
• one_mul (a : M) : 1 * a = a
• mul_one (a : M) : a * 1 = a
• npow : ℕ → M → M
• npow_zero (x : M) : Monoid.npow 0 x = 1
• npow_succ (n : ℕ) (x : M) : Monoid.npow (n + 1) x = Monoid.npow n x * x
• mul_comm (a b : M) : a * b = b * a

class AddLeftCancelMonoid (M : Type u) extends AddMonoid M, AddLeftCancelSemigroup M : Type u

An additive monoid in which addition is left-cancellative. 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 AddLeftCancelSemigroup is not enough.

• add : M → M → M
• add_assoc (a b c : M) : a + b + c = a + (b + c)
• zero : M
• zero_add (a : M) : 0 + a = a
• add_zero (a : M) : a + 0 = a
• nsmul : ℕ → M → M
• nsmul_zero (x : M) : AddMonoid.nsmul 0 x = 0
• nsmul_succ (n : ℕ) (x : M) : AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• add_left_cancel (a b c : M) : a + b = a + c → b = c

class LeftCancelMonoid (M : Type u) extends Monoid M, LeftCancelSemigroup M : Type u

A monoid in which multiplication is left-cancellative.

• mul : M → M → M
• mul_assoc (a b c : M) : a * b * c = a * (b * c)
• one : M
• one_mul (a : M) : 1 * a = a
• mul_one (a : M) : a * 1 = a
• npow : ℕ → M → M
• npow_zero (x : M) : Monoid.npow 0 x = 1
• npow_succ (n : ℕ) (x : M) : Monoid.npow (n + 1) x = Monoid.npow n x * x
• mul_left_cancel (a b c : M) : a * b = a * c → b = c

class AddRightCancelMonoid (M : Type u) extends AddMonoid M, AddRightCancelSemigroup M : Type u

An additive monoid in which addition is right-cancellative. 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 AddRightCancelSemigroup is not enough.

• add : M → M → M
• add_assoc (a b c : M) : a + b + c = a + (b + c)
• zero : M
• zero_add (a : M) : 0 + a = a
• add_zero (a : M) : a + 0 = a
• nsmul : ℕ → M → M
• nsmul_zero (x : M) : AddMonoid.nsmul 0 x = 0
• nsmul_succ (n : ℕ) (x : M) : AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• add_right_cancel (a b c : M) : a + b = c + b → a = c

class RightCancelMonoid (M : Type u) extends Monoid M, RightCancelSemigroup M : Type u

A monoid in which multiplication is right-cancellative.

• mul : M → M → M
• mul_assoc (a b c : M) : a * b * c = a * (b * c)
• one : M
• one_mul (a : M) : 1 * a = a
• mul_one (a : M) : a * 1 = a
• npow : ℕ → M → M
• npow_zero (x : M) : Monoid.npow 0 x = 1
• npow_succ (n : ℕ) (x : M) : Monoid.npow (n + 1) x = Monoid.npow n x * x
• mul_right_cancel (a b c : M) : a * b = c * b → a = c

class AddCancelMonoid (M : Type u) extends AddLeftCancelMonoid M, AddRightCancelMonoid M : Type u

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.

• add : M → M → M
• add_assoc (a b c : M) : a + b + c = a + (b + c)
• zero : M
• zero_add (a : M) : 0 + a = a
• add_zero (a : M) : a + 0 = a
• nsmul : ℕ → M → M
• nsmul_zero (x : M) : AddMonoid.nsmul 0 x = 0
• nsmul_succ (n : ℕ) (x : M) : AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• add_left_cancel (a b c : M) : a + b = a + c → b = c
• add_right_cancel (a b c : M) : a + b = c + b → a = c

class CancelMonoid (M : Type u) extends LeftCancelMonoid M, RightCancelMonoid M : Type u

A monoid in which multiplication is cancellative.

• mul : M → M → M
• mul_assoc (a b c : M) : a * b * c = a * (b * c)
• one : M
• one_mul (a : M) : 1 * a = a
• mul_one (a : M) : a * 1 = a
• npow : ℕ → M → M
• npow_zero (x : M) : Monoid.npow 0 x = 1
• npow_succ (n : ℕ) (x : M) : Monoid.npow (n + 1) x = Monoid.npow n x * x
• mul_left_cancel (a b c : M) : a * b = a * c → b = c
• mul_right_cancel (a b c : M) : a * b = c * b → a = c

class AddCancelCommMonoid (M : Type u) extends AddCommMonoid M, AddLeftCancelMonoid M : Type u

Commutative version of AddCancelMonoid.

• add : M → M → M
• add_assoc (a b c : M) : a + b + c = a + (b + c)
• zero : M
• zero_add (a : M) : 0 + a = a
• add_zero (a : M) : a + 0 = a
• nsmul : ℕ → M → M
• nsmul_zero (x : M) : AddMonoid.nsmul 0 x = 0
• nsmul_succ (n : ℕ) (x : M) : AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• add_comm (a b : M) : a + b = b + a
• add_left_cancel (a b c : M) : a + b = a + c → b = c

class CancelCommMonoid (M : Type u) extends CommMonoid M, LeftCancelMonoid M : Type u

Commutative version of CancelMonoid.

• mul : M → M → M
• mul_assoc (a b c : M) : a * b * c = a * (b * c)
• one : M
• one_mul (a : M) : 1 * a = a
• mul_one (a : M) : a * 1 = a
• npow : ℕ → M → M
• npow_zero (x : M) : Monoid.npow 0 x = 1
• npow_succ (n : ℕ) (x : M) : Monoid.npow (n + 1) x = Monoid.npow n x * x
• mul_comm (a b : M) : a * b = b * a
• mul_left_cancel (a b c : M) : a * b = a * c → b = c

@[instance 100]

instance CancelCommMonoid.toCancelMonoid (M : Type u) [CancelCommMonoid M] : CancelMonoid M

Equations

• CancelCommMonoid.toCancelMonoid M = CancelMonoid.mk ⋯

@[instance 100]

instance AddCancelCommMonoid.toAddCancelMonoid (M : Type u) [AddCancelCommMonoid M] : AddCancelMonoid M

Equations

• AddCancelCommMonoid.toAddCancelMonoid M = AddCancelMonoid.mk ⋯

@[instance 100]

instance CancelMonoid.toIsCancelMul (M : Type u) [CancelMonoid M] : IsCancelMul M

Any CancelMonoid G satisfies IsCancelMul G.

@[instance 100]

instance AddCancelMonoid.toIsCancelAdd (M : Type u) [AddCancelMonoid M] : IsCancelAdd M

Any AddCancelMonoid G satisfies IsCancelAdd G.

def zpowRec {G : Type u_1} [One G] [Mul G] [Inv G] (npow : ℕ → G → G := npowRec) : ℤ → G → G

The fundamental power operation in a group. zpowRec n a = a*a*...*a n times, for integer n. Use instead a ^ n, which has better definitional behavior.

Equations

• zpowRec npow (Int.ofNat n) x✝ = npow n x✝
• zpowRec npow (Int.negSucc n) x✝ = (npow n.succ x✝)⁻¹

def zsmulRec {G : Type u_1} [Zero G] [Add G] [Neg G] (nsmul : ℕ → G → G := nsmulRec) : ℤ → G → G

The fundamental scalar multiplication in an additive group. zpowRec n a = a+a+...+a n times, for integer n. Use instead n • a, which has better definitional behavior.

Equations

• zsmulRec nsmul (Int.ofNat n) x✝ = nsmul n x✝
• zsmulRec nsmul (Int.negSucc n) x✝ = -nsmul n.succ x✝

class InvolutiveNeg (A : Type u_2) extends Neg A : Type u_2

Auxiliary typeclass for types with an involutive Neg.

• neg : A → A
• neg_neg (x : A) : - -x = x

class InvolutiveInv (G : Type u_2) extends Inv G : Type u_2

Auxiliary typeclass for types with an involutive Inv.

• inv : G → G
• inv_inv (x : G) : x⁻¹⁻¹ = x

@[simp]

theorem inv_inv {G : Type u_1} [InvolutiveInv G] (a : G) : a⁻¹⁻¹ = a

@[simp]

theorem neg_neg {G : Type u_1} [InvolutiveNeg G] (a : G) : - -a = a

Design note on DivInvMonoid/SubNegMonoid and DivisionMonoid/SubtractionMonoid

Those two pairs of made-up classes fulfill slightly different roles.

DivInvMonoid/SubNegMonoid provides the minimum amount of information to define the ℤ action (zpow or zsmul). Further, it provides a div field, matching the forgetful inheritance pattern. This is useful to shorten extension clauses of stronger structures (Group, GroupWithZero, DivisionRing, Field) and for a few structures with a rather weak pseudo-inverse (Matrix).

DivisionMonoid/SubtractionMonoid is targeted at structures with stronger pseudo-inverses. It is an ad hoc collection of axioms that are mainly respected by three things:

• Groups
• Groups with zero
• The pointwise monoids Set α, Finset α, Filter α

It acts as a middle ground for structures with an inversion operator that plays well with multiplication, except for the fact that it might not be a true inverse (a / a ≠ 1 in general). The axioms are pretty arbitrary (many other combinations are equivalent to it), but they are independent:

• Without DivisionMonoid.div_eq_mul_inv, you can define / arbitrarily.
• Without DivisionMonoid.inv_inv, you can consider WithTop Unit with a⁻¹ = ⊤ for all a.
• Without DivisionMonoid.mul_inv_rev, you can consider WithTop α with a⁻¹ = a for all a where α non commutative.
• Without DivisionMonoid.inv_eq_of_mul, you can consider any CommMonoid with a⁻¹ = a for all a.

As a consequence, a few natural structures do not fit in this framework. For example, ENNReal respects everything except for the fact that (0 * ∞)⁻¹ = 0⁻¹ = ∞ while ∞⁻¹ * 0⁻¹ = 0 * ∞ = 0.

def DivInvMonoid.div' {G : Type u} [Monoid G] [Inv G] (a b : G) : G

In a class equipped with instances of both Monoid and Inv, this definition records what the default definition for Div would be: a * b⁻¹. This is later provided as the default value for the Div instance in DivInvMonoid.

We keep it as a separate definition rather than inlining it in DivInvMonoid so that the Div field of individual DivInvMonoids constructed using that default value will not be unfolded at .instance transparency.

Equations

• DivInvMonoid.div' a b = a * b⁻¹

class DivInvMonoid (G : Type u) extends Monoid G, Inv G, Div G : Type u

A DivInvMonoid is a Monoid with operations / and ⁻¹ satisfying div_eq_mul_inv : ∀ a b, a / b = a * b⁻¹.

This deduplicates the name div_eq_mul_inv. The default for div is such that a / b = a * b⁻¹ holds by definition.

Adding div as a field rather than defining a / b := a * b⁻¹ allows us to avoid certain classes of unification failures, for example: Let Foo X be a type with a ∀ X, Div (Foo X) instance but no ∀ X, Inv (Foo X), e.g. when Foo X is a EuclideanDomain. Suppose we also have an instance ∀ X [Cromulent X], GroupWithZero (Foo X). Then the (/) coming from GroupWithZero.div cannot be definitionally equal to the (/) coming from Foo.Div.

In the same way, adding a zpow field makes it possible to avoid definitional failures in diamonds. See the definition of Monoid and Note [forgetful inheritance] for more explanations on this.

• mul : G → G → G
• mul_assoc (a b c : G) : a * b * c = a * (b * c)
• one : G
• one_mul (a : G) : 1 * a = a
• mul_one (a : G) : a * 1 = a
• npow : ℕ → G → G
• npow_zero (x : G) : Monoid.npow 0 x = 1
• npow_succ (n : ℕ) (x : G) : Monoid.npow (n + 1) x = Monoid.npow n x * x
• inv : G → G
• div : G → G → G
• div_eq_mul_inv (a b : G) : a / b = a * b⁻¹

  a / b := a * b⁻¹

• zpow : ℤ → G → G

  The power operation: a ^ n = a * ··· * a; a ^ (-n) = a⁻¹ * ··· a⁻¹ (n times)

• zpow_zero' (a : G) : DivInvMonoid.zpow 0 a = 1

  a ^ 0 = 1

• zpow_succ' (n : ℕ) (a : G) : DivInvMonoid.zpow (↑n.succ) a = DivInvMonoid.zpow (↑n) a * a

  a ^ (n + 1) = a ^ n * a

• zpow_neg' (n : ℕ) (a : G) : DivInvMonoid.zpow (Int.negSucc n) a = (DivInvMonoid.zpow (↑n.succ) a)⁻¹

  a ^ -(n + 1) = (a ^ (n + 1))⁻¹

def SubNegMonoid.sub' {G : Type u} [AddMonoid G] [Neg G] (a b : G) : G

In a class equipped with instances of both AddMonoid and Neg, this definition records what the default definition for Sub would be: a + -b. This is later provided as the default value for the Sub instance in SubNegMonoid.

We keep it as a separate definition rather than inlining it in SubNegMonoid so that the Sub field of individual SubNegMonoids constructed using that default value will not be unfolded at .instance transparency.

Equations

• SubNegMonoid.sub' a b = a + -b

class SubNegMonoid (G : Type u) extends AddMonoid G, Neg G, Sub G : Type u

A SubNegMonoid is an AddMonoid with unary - and binary - operations satisfying sub_eq_add_neg : ∀ a b, a - b = a + -b.

The default for sub is such that a - b = a + -b holds by definition.

Adding sub as a field rather than defining a - b := a + -b allows us to avoid certain classes of unification failures, for example: Let foo X be a type with a ∀ X, Sub (Foo X) instance but no ∀ X, Neg (Foo X). Suppose we also have an instance ∀ X [Cromulent X], AddGroup (Foo X). Then the (-) coming from AddGroup.sub cannot be definitionally equal to the (-) coming from Foo.Sub.

In the same way, adding a zsmul field makes it possible to avoid definitional failures in diamonds. See the definition of AddMonoid and Note [forgetful inheritance] for more explanations on this.

• add : G → G → G
• add_assoc (a b c : G) : a + b + c = a + (b + c)
• zero : G
• zero_add (a : G) : 0 + a = a
• add_zero (a : G) : a + 0 = a
• nsmul : ℕ → G → G
• nsmul_zero (x : G) : AddMonoid.nsmul 0 x = 0
• nsmul_succ (n : ℕ) (x : G) : AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• neg : G → G
• sub : G → G → G
• sub_eq_add_neg (a b : G) : a - b = a + -b
• zsmul : ℤ → G → G

  Multiplication by an integer. Set this to zsmulRec unless Module diamonds are possible.

• zsmul_zero' (a : G) : SubNegMonoid.zsmul 0 a = 0
• zsmul_succ' (n : ℕ) (a : G) : SubNegMonoid.zsmul (↑n.succ) a = SubNegMonoid.zsmul (↑n) a + a
• zsmul_neg' (n : ℕ) (a : G) : SubNegMonoid.zsmul (Int.negSucc n) a = -SubNegMonoid.zsmul (↑n.succ) a

instance DivInvMonoid.toZPow {M : Type u_2} [DivInvMonoid M] : Pow M ℤ

Equations

• DivInvMonoid.toZPow = { pow := fun (x : M) (n : ℤ) => DivInvMonoid.zpow n x }

instance SubNegMonoid.toZSMul {M : Type u_2} [SubNegMonoid M] : SMul ℤ M

Equations

• SubNegMonoid.toZSMul = { smul := SubNegMonoid.zsmul }

class IsAddCyclic (G : Type u) [SMul ℤ G] : Prop

A group is called cyclic if it is generated by a single element.

• exists_zsmul_surjective : ∃ (g : G), Function.Surjective fun (x : ℤ) => x • g

class IsCyclic (G : Type u) [Pow G ℤ] : Prop

A group is called cyclic if it is generated by a single element.

• exists_zpow_surjective : ∃ (g : G), Function.Surjective fun (x : ℤ) => g ^ x

theorem exists_zpow_surjective (G : Type u_2) [Pow G ℤ] [IsCyclic G] : ∃ (g : G), Function.Surjective fun (x : ℤ) => g ^ x

theorem exists_zsmul_surjective (G : Type u_2) [SMul ℤ G] [IsAddCyclic G] : ∃ (g : G), Function.Surjective fun (x : ℤ) => x • g

@[simp]

theorem zpow_eq_pow {G : Type u_1} [DivInvMonoid G] (n : ℤ) (x : G) : DivInvMonoid.zpow n x = x ^ n

@[simp]

theorem zsmul_eq_smul {G : Type u_1} [SubNegMonoid G] (n : ℤ) (x : G) : SubNegMonoid.zsmul n x = n • x

@[simp]

theorem zpow_zero {G : Type u_1} [DivInvMonoid G] (a : G) : a ^ 0 = 1

@[simp]

theorem zero_zsmul {G : Type u_1} [SubNegMonoid G] (a : G) : 0 • a = 0

@[simp]

theorem zpow_natCast {G : Type u_1} [DivInvMonoid G] (a : G) (n : ℕ) : a ^ ↑n = a ^ n

@[simp]

theorem natCast_zsmul {G : Type u_1} [SubNegMonoid G] (a : G) (n : ℕ) : ↑n • a = n • a

@[deprecated zpow_natCast]

theorem zpow_coe_nat {G : Type u_1} [DivInvMonoid G] (a : G) (n : ℕ) : a ^ ↑n = a ^ n

Alias of zpow_natCast.

@[deprecated natCast_zsmul]

theorem coe_nat_zsmul {G : Type u_1} [SubNegMonoid G] (a : G) (n : ℕ) : ↑n • a = n • a

Alias of natCast_zsmul.

theorem zpow_ofNat {G : Type u_1} [DivInvMonoid G] (a : G) (n : ℕ) : a ^ OfNat.ofNat n = a ^ OfNat.ofNat n

theorem ofNat_zsmul {G : Type u_1} [SubNegMonoid G] (a : G) (n : ℕ) : OfNat.ofNat n • a = OfNat.ofNat n • a

@[simp]

theorem zpow_negSucc {G : Type u_1} [DivInvMonoid G] (a : G) (n : ℕ) : a ^ Int.negSucc n = (a ^ (n + 1))⁻¹

@[simp]

theorem negSucc_zsmul {G : Type u_2} [SubNegMonoid G] (a : G) (n : ℕ) : Int.negSucc n • a = -((n + 1) • a)

theorem div_eq_mul_inv {G : Type u_1} [DivInvMonoid G] (a b : G) : a / b = a * b⁻¹

Dividing by an element is the same as multiplying by its inverse.

This is a duplicate of DivInvMonoid.div_eq_mul_inv ensuring that the types unfold better.

theorem sub_eq_add_neg {G : Type u_1} [SubNegMonoid G] (a b : G) : a - b = a + -b

Subtracting an element is the same as adding by its negative. This is a duplicate of SubNegMonoid.sub_eq_mul_neg ensuring that the types unfold better.

theorem division_def {G : Type u_1} [DivInvMonoid G] (a b : G) : a / b = a * b⁻¹

Alias of div_eq_mul_inv.

---

Dividing by an element is the same as multiplying by its inverse.

This is a duplicate of DivInvMonoid.div_eq_mul_inv ensuring that the types unfold better.

@[simp]

theorem zpow_one {G : Type u_1} [DivInvMonoid G] (a : G) : a ^ 1 = a

@[simp]

theorem one_zsmul {G : Type u_1} [SubNegMonoid G] (a : G) : 1 • a = a

theorem zpow_two {G : Type u_1} [DivInvMonoid G] (a : G) : a ^ 2 = a * a

theorem two_zsmul {G : Type u_1} [SubNegMonoid G] (a : G) : 2 • a = a + a

theorem zpow_neg_one {G : Type u_1} [DivInvMonoid G] (x : G) : x ^ (-1) = x⁻¹

theorem neg_one_zsmul {G : Type u_1} [SubNegMonoid G] (x : G) : -1 • x = -x

theorem zpow_neg_coe_of_pos {G : Type u_1} [DivInvMonoid G] (a : G) {n : ℕ} : 0 < n → a ^ (-↑n) = (a ^ n)⁻¹

theorem zsmul_neg_coe_of_pos {G : Type u_1} [SubNegMonoid G] (a : G) {n : ℕ} : 0 < n → -↑n • a = -(n • a)

class NegZeroClass (G : Type u_2) extends Zero G, Neg G : Type u_2

Typeclass for expressing that -0 = 0.

• zero : G
• neg : G → G
• neg_zero : -0 = 0

class SubNegZeroMonoid (G : Type u_2) extends SubNegMonoid G, NegZeroClass G : Type u_2

A SubNegMonoid where -0 = 0.

• add : G → G → G
• add_assoc (a b c : G) : a + b + c = a + (b + c)
• zero : G
• zero_add (a : G) : 0 + a = a
• add_zero (a : G) : a + 0 = a
• nsmul : ℕ → G → G
• nsmul_zero (x : G) : AddMonoid.nsmul 0 x = 0
• nsmul_succ (n : ℕ) (x : G) : AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• neg : G → G
• sub : G → G → G
• sub_eq_add_neg (a b : G) : a - b = a + -b
• zsmul : ℤ → G → G
• zsmul_zero' (a : G) : SubNegMonoid.zsmul 0 a = 0
• zsmul_succ' (n : ℕ) (a : G) : SubNegMonoid.zsmul (↑n.succ) a = SubNegMonoid.zsmul (↑n) a + a
• zsmul_neg' (n : ℕ) (a : G) : SubNegMonoid.zsmul (Int.negSucc n) a = -SubNegMonoid.zsmul (↑n.succ) a
• neg_zero : -0 = 0

class InvOneClass (G : Type u_2) extends One G, Inv G : Type u_2

Typeclass for expressing that 1⁻¹ = 1.

• one : G
• inv : G → G
• inv_one : 1⁻¹ = 1

class DivInvOneMonoid (G : Type u_2) extends DivInvMonoid G, InvOneClass G : Type u_2

A DivInvMonoid where 1⁻¹ = 1.

• mul : G → G → G
• mul_assoc (a b c : G) : a * b * c = a * (b * c)
• one : G
• one_mul (a : G) : 1 * a = a
• mul_one (a : G) : a * 1 = a
• npow : ℕ → G → G
• npow_zero (x : G) : Monoid.npow 0 x = 1
• npow_succ (n : ℕ) (x : G) : Monoid.npow (n + 1) x = Monoid.npow n x * x
• inv : G → G
• div : G → G → G
• div_eq_mul_inv (a b : G) : a / b = a * b⁻¹
• zpow : ℤ → G → G
• zpow_zero' (a : G) : DivInvMonoid.zpow 0 a = 1
• zpow_succ' (n : ℕ) (a : G) : DivInvMonoid.zpow (↑n.succ) a = DivInvMonoid.zpow (↑n) a * a
• zpow_neg' (n : ℕ) (a : G) : DivInvMonoid.zpow (Int.negSucc n) a = (DivInvMonoid.zpow (↑n.succ) a)⁻¹
• inv_one : 1⁻¹ = 1

@[simp]

theorem inv_one {G : Type u_1} [InvOneClass G] : 1⁻¹ = 1

@[simp]

theorem neg_zero {G : Type u_1} [NegZeroClass G] : -0 = 0

class SubtractionMonoid (G : Type u) extends SubNegMonoid G, InvolutiveNeg G : Type u

A SubtractionMonoid is a SubNegMonoid with involutive negation and such that -(a + b) = -b + -a and a + b = 0 → -a = b.

• add : G → G → G
• add_assoc (a b c : G) : a + b + c = a + (b + c)
• zero : G
• zero_add (a : G) : 0 + a = a
• add_zero (a : G) : a + 0 = a
• nsmul : ℕ → G → G
• nsmul_zero (x : G) : AddMonoid.nsmul 0 x = 0
• nsmul_succ (n : ℕ) (x : G) : AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• neg : G → G
• sub : G → G → G
• sub_eq_add_neg (a b : G) : a - b = a + -b
• zsmul : ℤ → G → G
• zsmul_zero' (a : G) : SubNegMonoid.zsmul 0 a = 0
• zsmul_succ' (n : ℕ) (a : G) : SubNegMonoid.zsmul (↑n.succ) a = SubNegMonoid.zsmul (↑n) a + a
• zsmul_neg' (n : ℕ) (a : G) : SubNegMonoid.zsmul (Int.negSucc n) a = -SubNegMonoid.zsmul (↑n.succ) a
• neg_neg (x : G) : - -x = x
• neg_add_rev (a b : G) : -(a + b) = -b + -a
• neg_eq_of_add (a b : G) : a + b = 0 → -a = b

  Despite the asymmetry of neg_eq_of_add, the symmetric version is true thanks to the involutivity of negation.

class DivisionMonoid (G : Type u) extends DivInvMonoid G, InvolutiveInv G : Type u

A DivisionMonoid is a DivInvMonoid with involutive inversion and such that (a * b)⁻¹ = b⁻¹ * a⁻¹ and a * b = 1 → a⁻¹ = b.

This is the immediate common ancestor of Group and GroupWithZero.

• mul : G → G → G
• mul_assoc (a b c : G) : a * b * c = a * (b * c)
• one : G
• one_mul (a : G) : 1 * a = a
• mul_one (a : G) : a * 1 = a
• npow : ℕ → G → G
• npow_zero (x : G) : Monoid.npow 0 x = 1
• npow_succ (n : ℕ) (x : G) : Monoid.npow (n + 1) x = Monoid.npow n x * x
• inv : G → G
• div : G → G → G
• div_eq_mul_inv (a b : G) : a / b = a * b⁻¹
• zpow : ℤ → G → G
• zpow_zero' (a : G) : DivInvMonoid.zpow 0 a = 1
• zpow_succ' (n : ℕ) (a : G) : DivInvMonoid.zpow (↑n.succ) a = DivInvMonoid.zpow (↑n) a * a
• zpow_neg' (n : ℕ) (a : G) : DivInvMonoid.zpow (Int.negSucc n) a = (DivInvMonoid.zpow (↑n.succ) a)⁻¹
• inv_inv (x : G) : x⁻¹⁻¹ = x
• mul_inv_rev (a b : G) : (a * b)⁻¹ = b⁻¹ * a⁻¹
• inv_eq_of_mul (a b : G) : a * b = 1 → a⁻¹ = b

  Despite the asymmetry of inv_eq_of_mul, the symmetric version is true thanks to the involutivity of inversion.

@[simp]

theorem mul_inv_rev {G : Type u_1} [DivisionMonoid G] (a b : G) : (a * b)⁻¹ = b⁻¹ * a⁻¹

@[simp]

theorem neg_add_rev {G : Type u_1} [SubtractionMonoid G] (a b : G) : -(a + b) = -b + -a

theorem inv_eq_of_mul_eq_one_right {G : Type u_1} [DivisionMonoid G] {a b : G} : a * b = 1 → a⁻¹ = b

theorem neg_eq_of_add_eq_zero_right {G : Type u_1} [SubtractionMonoid G] {a b : G} : a + b = 0 → -a = b

theorem inv_eq_of_mul_eq_one_left {G : Type u_1} [DivisionMonoid G] {a b : G} (h : a * b = 1) : b⁻¹ = a

theorem neg_eq_of_add_eq_zero_left {G : Type u_1} [SubtractionMonoid G] {a b : G} (h : a + b = 0) : -b = a

theorem eq_inv_of_mul_eq_one_left {G : Type u_1} [DivisionMonoid G] {a b : G} (h : a * b = 1) : a = b⁻¹

theorem eq_neg_of_add_eq_zero_left {G : Type u_1} [SubtractionMonoid G] {a b : G} (h : a + b = 0) : a = -b

class SubtractionCommMonoid (G : Type u) extends SubtractionMonoid G, AddCommMonoid G : Type u

Commutative SubtractionMonoid.

• add : G → G → G
• add_assoc (a b c : G) : a + b + c = a + (b + c)
• zero : G
• zero_add (a : G) : 0 + a = a
• add_zero (a : G) : a + 0 = a
• nsmul : ℕ → G → G
• nsmul_zero (x : G) : AddMonoid.nsmul 0 x = 0
• nsmul_succ (n : ℕ) (x : G) : AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• neg : G → G
• sub : G → G → G
• sub_eq_add_neg (a b : G) : a - b = a + -b
• zsmul : ℤ → G → G
• zsmul_zero' (a : G) : SubNegMonoid.zsmul 0 a = 0
• zsmul_succ' (n : ℕ) (a : G) : SubNegMonoid.zsmul (↑n.succ) a = SubNegMonoid.zsmul (↑n) a + a
• zsmul_neg' (n : ℕ) (a : G) : SubNegMonoid.zsmul (Int.negSucc n) a = -SubNegMonoid.zsmul (↑n.succ) a
• neg_neg (x : G) : - -x = x
• neg_add_rev (a b : G) : -(a + b) = -b + -a
• neg_eq_of_add (a b : G) : a + b = 0 → -a = b
• add_comm (a b : G) : a + b = b + a

class DivisionCommMonoid (G : Type u) extends DivisionMonoid G, CommMonoid G : Type u

Commutative DivisionMonoid.

This is the immediate common ancestor of CommGroup and CommGroupWithZero.

• mul : G → G → G
• mul_assoc (a b c : G) : a * b * c = a * (b * c)
• one : G
• one_mul (a : G) : 1 * a = a
• mul_one (a : G) : a * 1 = a
• npow : ℕ → G → G
• npow_zero (x : G) : Monoid.npow 0 x = 1
• npow_succ (n : ℕ) (x : G) : Monoid.npow (n + 1) x = Monoid.npow n x * x
• inv : G → G
• div : G → G → G
• div_eq_mul_inv (a b : G) : a / b = a * b⁻¹
• zpow : ℤ → G → G
• zpow_zero' (a : G) : DivInvMonoid.zpow 0 a = 1
• zpow_succ' (n : ℕ) (a : G) : DivInvMonoid.zpow (↑n.succ) a = DivInvMonoid.zpow (↑n) a * a
• zpow_neg' (n : ℕ) (a : G) : DivInvMonoid.zpow (Int.negSucc n) a = (DivInvMonoid.zpow (↑n.succ) a)⁻¹
• inv_inv (x : G) : x⁻¹⁻¹ = x
• mul_inv_rev (a b : G) : (a * b)⁻¹ = b⁻¹ * a⁻¹
• inv_eq_of_mul (a b : G) : a * b = 1 → a⁻¹ = b
• mul_comm (a b : G) : a * b = b * a

class Group (G : Type u) extends DivInvMonoid G : Type u

A Group is a Monoid with an operation ⁻¹ satisfying a⁻¹ * a = 1.

There is also a division operation / such that a / b = a * b⁻¹, with a default so that a / b = a * b⁻¹ holds by definition.

Use Group.ofLeftAxioms or Group.ofRightAxioms to define a group structure on a type with the minimum proof obligations.

• mul : G → G → G
• mul_assoc (a b c : G) : a * b * c = a * (b * c)
• one : G
• one_mul (a : G) : 1 * a = a
• mul_one (a : G) : a * 1 = a
• npow : ℕ → G → G
• npow_zero (x : G) : Monoid.npow 0 x = 1
• npow_succ (n : ℕ) (x : G) : Monoid.npow (n + 1) x = Monoid.npow n x * x
• inv : G → G
• div : G → G → G
• div_eq_mul_inv (a b : G) : a / b = a * b⁻¹
• zpow : ℤ → G → G
• zpow_zero' (a : G) : DivInvMonoid.zpow 0 a = 1
• zpow_succ' (n : ℕ) (a : G) : DivInvMonoid.zpow (↑n.succ) a = DivInvMonoid.zpow (↑n) a * a
• zpow_neg' (n : ℕ) (a : G) : DivInvMonoid.zpow (Int.negSucc n) a = (DivInvMonoid.zpow (↑n.succ) a)⁻¹
• inv_mul_cancel (a : G) : a⁻¹ * a = 1

class AddGroup (A : Type u) extends SubNegMonoid A : Type u

An AddGroup is an AddMonoid with a unary - satisfying -a + a = 0.

There is also a binary operation - such that a - b = a + -b, with a default so that a - b = a + -b holds by definition.

Use AddGroup.ofLeftAxioms or AddGroup.ofRightAxioms to define an additive group structure on a type with the minimum proof obligations.

• add : A → A → A
• add_assoc (a b c : A) : a + b + c = a + (b + c)
• zero : A
• zero_add (a : A) : 0 + a = a
• add_zero (a : A) : a + 0 = a
• nsmul : ℕ → A → A
• nsmul_zero (x : A) : AddMonoid.nsmul 0 x = 0
• nsmul_succ (n : ℕ) (x : A) : AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• neg : A → A
• sub : A → A → A
• sub_eq_add_neg (a b : A) : a - b = a + -b
• zsmul : ℤ → A → A
• zsmul_zero' (a : A) : SubNegMonoid.zsmul 0 a = 0
• zsmul_succ' (n : ℕ) (a : A) : SubNegMonoid.zsmul (↑n.succ) a = SubNegMonoid.zsmul (↑n) a + a
• zsmul_neg' (n : ℕ) (a : A) : SubNegMonoid.zsmul (Int.negSucc n) a = -SubNegMonoid.zsmul (↑n.succ) a
• neg_add_cancel (a : A) : -a + a = 0

@[simp]

theorem inv_mul_cancel {G : Type u_1} [Group G] (a : G) : a⁻¹ * a = 1

@[simp]

theorem neg_add_cancel {G : Type u_1} [AddGroup G] (a : G) : -a + a = 0

@[simp]

theorem mul_inv_cancel {G : Type u_1} [Group G] (a : G) : a * a⁻¹ = 1

@[simp]

theorem add_neg_cancel {G : Type u_1} [AddGroup G] (a : G) : a + -a = 0

@[deprecated inv_mul_cancel]

theorem mul_left_inv {G : Type u_1} [Group G] (a : G) : a⁻¹ * a = 1

Alias of inv_mul_cancel.

@[deprecated mul_inv_cancel]

theorem mul_right_inv {G : Type u_1} [Group G] (a : G) : a * a⁻¹ = 1

Alias of mul_inv_cancel.

@[deprecated neg_add_cancel]

theorem add_left_neg {G : Type u_1} [AddGroup G] (a : G) : -a + a = 0

Alias of neg_add_cancel.

@[deprecated add_neg_cancel]

theorem add_right_neg {G : Type u_1} [AddGroup G] (a : G) : a + -a = 0

Alias of add_neg_cancel.

@[deprecated inv_mul_cancel]

theorem inv_mul_self {G : Type u_1} [Group G] (a : G) : a⁻¹ * a = 1

Alias of inv_mul_cancel.

@[deprecated mul_inv_cancel]

theorem mul_inv_self {G : Type u_1} [Group G] (a : G) : a * a⁻¹ = 1

Alias of mul_inv_cancel.

@[deprecated neg_add_cancel]

theorem neg_add_self {G : Type u_1} [AddGroup G] (a : G) : -a + a = 0

Alias of neg_add_cancel.

@[deprecated add_neg_cancel]

theorem add_right_self {G : Type u_1} [AddGroup G] (a : G) : a + -a = 0

Alias of add_neg_cancel.

@[simp]

theorem inv_mul_cancel_left {G : Type u_1} [Group G] (a b : G) : a⁻¹ * (a * b) = b

@[simp]

theorem neg_add_cancel_left {G : Type u_1} [AddGroup G] (a b : G) : -a + (a + b) = b

@[simp]

theorem mul_inv_cancel_left {G : Type u_1} [Group G] (a b : G) : a * (a⁻¹ * b) = b

@[simp]

theorem add_neg_cancel_left {G : Type u_1} [AddGroup G] (a b : G) : a + (-a + b) = b

@[simp]

theorem mul_inv_cancel_right {G : Type u_1} [Group G] (a b : G) : a * b * b⁻¹ = a

@[simp]

theorem add_neg_cancel_right {G : Type u_1} [AddGroup G] (a b : G) : a + b + -b = a

@[simp]

theorem inv_mul_cancel_right {G : Type u_1} [Group G] (a b : G) : a * b⁻¹ * b = a

@[simp]

theorem neg_add_cancel_right {G : Type u_1} [AddGroup G] (a b : G) : a + -b + b = a

@[instance 100]

instance Group.toDivisionMonoid {G : Type u_1} [Group G] : DivisionMonoid G

Equations

• Group.toDivisionMonoid = DivisionMonoid.mk ⋯ ⋯ ⋯

@[instance 100]

instance AddGroup.toSubtractionMonoid {G : Type u_1} [AddGroup G] : SubtractionMonoid G

Equations

• AddGroup.toSubtractionMonoid = SubtractionMonoid.mk ⋯ ⋯ ⋯

@[instance 100]

instance Group.toCancelMonoid {G : Type u_1} [Group G] : CancelMonoid G

Equations

• Group.toCancelMonoid = CancelMonoid.mk ⋯

@[instance 100]

instance AddGroup.toAddCancelMonoid {G : Type u_1} [AddGroup G] : AddCancelMonoid G

Equations

• AddGroup.toAddCancelMonoid = AddCancelMonoid.mk ⋯

class AddCommGroup (G : Type u) extends AddGroup G, AddCommMonoid G : Type u

An additive commutative group is an additive group with commutative (+).

• add : G → G → G
• add_assoc (a b c : G) : a + b + c = a + (b + c)
• zero : G
• zero_add (a : G) : 0 + a = a
• add_zero (a : G) : a + 0 = a
• nsmul : ℕ → G → G
• nsmul_zero (x : G) : AddMonoid.nsmul 0 x = 0
• nsmul_succ (n : ℕ) (x : G) : AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• neg : G → G
• sub : G → G → G
• sub_eq_add_neg (a b : G) : a - b = a + -b
• zsmul : ℤ → G → G
• zsmul_zero' (a : G) : SubNegMonoid.zsmul 0 a = 0
• zsmul_succ' (n : ℕ) (a : G) : SubNegMonoid.zsmul (↑n.succ) a = SubNegMonoid.zsmul (↑n) a + a
• zsmul_neg' (n : ℕ) (a : G) : SubNegMonoid.zsmul (Int.negSucc n) a = -SubNegMonoid.zsmul (↑n.succ) a
• neg_add_cancel (a : G) : -a + a = 0
• add_comm (a b : G) : a + b = b + a

class CommGroup (G : Type u) extends Group G, CommMonoid G : Type u

A commutative group is a group with commutative (*).

• mul : G → G → G
• mul_assoc (a b c : G) : a * b * c = a * (b * c)
• one : G
• one_mul (a : G) : 1 * a = a
• mul_one (a : G) : a * 1 = a
• npow : ℕ → G → G
• npow_zero (x : G) : Monoid.npow 0 x = 1
• npow_succ (n : ℕ) (x : G) : Monoid.npow (n + 1) x = Monoid.npow n x * x
• inv : G → G
• div : G → G → G
• div_eq_mul_inv (a b : G) : a / b = a * b⁻¹
• zpow : ℤ → G → G
• zpow_zero' (a : G) : DivInvMonoid.zpow 0 a = 1
• zpow_succ' (n : ℕ) (a : G) : DivInvMonoid.zpow (↑n.succ) a = DivInvMonoid.zpow (↑n) a * a
• zpow_neg' (n : ℕ) (a : G) : DivInvMonoid.zpow (Int.negSucc n) a = (DivInvMonoid.zpow (↑n.succ) a)⁻¹
• inv_mul_cancel (a : G) : a⁻¹ * a = 1
• mul_comm (a b : G) : a * b = b * a

@[instance 100]

instance CommGroup.toCancelCommMonoid {G : Type u_1} [CommGroup G] : CancelCommMonoid G

Equations

• CommGroup.toCancelCommMonoid = CancelCommMonoid.mk ⋯

@[instance 100]

instance AddCommGroup.toAddCancelCommMonoid {G : Type u_1} [AddCommGroup G] : AddCancelCommMonoid G

Equations

• AddCommGroup.toAddCancelCommMonoid = AddCancelCommMonoid.mk ⋯

@[instance 100]

instance CommGroup.toDivisionCommMonoid {G : Type u_1} [CommGroup G] : DivisionCommMonoid G

Equations

• CommGroup.toDivisionCommMonoid = DivisionCommMonoid.mk ⋯

@[instance 100]

instance AddCommGroup.toDivisionAddCommMonoid {G : Type u_1} [AddCommGroup G] : SubtractionCommMonoid G

Equations

• AddCommGroup.toDivisionAddCommMonoid = SubtractionCommMonoid.mk ⋯

@[simp]

theorem inv_mul_cancel_comm {G : Type u_1} [CommGroup G] (a b : G) : a⁻¹ * b * a = b

@[simp]

theorem neg_add_cancel_comm {G : Type u_1} [AddCommGroup G] (a b : G) : -a + b + a = b

@[simp]

theorem mul_inv_cancel_comm {G : Type u_1} [CommGroup G] (a b : G) : a * b * a⁻¹ = b

@[simp]

theorem add_neg_cancel_comm {G : Type u_1} [AddCommGroup G] (a b : G) : a + b + -a = b

@[simp]

theorem inv_mul_cancel_comm_assoc {G : Type u_1} [CommGroup G] (a b : G) : a⁻¹ * (b * a) = b

@[simp]

theorem neg_add_cancel_comm_assoc {G : Type u_1} [AddCommGroup G] (a b : G) : -a + (b + a) = b

@[simp]

theorem mul_inv_cancel_comm_assoc {G : Type u_1} [CommGroup G] (a b : G) : a * (b * a⁻¹) = b

@[simp]

theorem add_neg_cancel_comm_assoc {G : Type u_1} [AddCommGroup G] (a b : G) : a + (b + -a) = b

We initialize all projections for @[simps] here, so that we don't have to do it in later files.

Note: the lemmas generated for the npow/zpow projections will not apply to x ^ y, since the argument order of these projections doesn't match the argument order of ^. The nsmul/zsmul lemmas will be correct.