Semirings and rings

This file defines semirings, rings and domains. This is analogous to Algebra.Group.Defs and Algebra.Group.Basic, the difference being that the former is about + and * separately, while the present file is about their interaction.

Main definitions

• Distrib: Typeclass for distributivity of multiplication over addition.
• HasDistribNeg: Typeclass for commutativity of negation and multiplication. This is useful when dealing with multiplicative submonoids which are closed under negation without being closed under addition, for example Units.
• (NonUnital)(NonAssoc)(Semi)Ring: Typeclasses for possibly non-unital or non-associative rings and semirings. Some combinations are not defined yet because they haven't found use.

Tags

Semiring, CommSemiring, Ring, CommRing, domain, IsDomain, nonzero, units

Distrib class

class Distrib (R : Type u_1) extends Mul , Add : Type u_1

A typeclass stating that multiplication is left and right distributive over addition.

• mul : R → R → R
• add : R → R → R
• left_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c

  Multiplication is left distributive over addition

• right_distrib : ∀ (a b c : R), (a + b) * c = a * c + b * c

  Multiplication is right distributive over addition

class LeftDistribClass (R : Type u_1) [Mul R] [Add R] : Prop

A typeclass stating that multiplication is left distributive over addition.

• left_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c

  Multiplication is left distributive over addition

class RightDistribClass (R : Type u_1) [Mul R] [Add R] : Prop

A typeclass stating that multiplication is right distributive over addition.

• right_distrib : ∀ (a b c : R), (a + b) * c = a * c + b * c

  Multiplication is right distributive over addition

instance Distrib.leftDistribClass (R : Type u_1) [Distrib R] : LeftDistribClass R

Equations

• ⋯ = ⋯

instance Distrib.rightDistribClass (R : Type u_1) [Distrib R] : RightDistribClass R

Equations

• ⋯ = ⋯

theorem left_distrib {R : Type x} [Mul R] [Add R] [LeftDistribClass R] (a : R) (b : R) (c : R) : a * (b + c) = a * b + a * c

theorem mul_add {R : Type x} [Mul R] [Add R] [LeftDistribClass R] (a : R) (b : R) (c : R) : a * (b + c) = a * b + a * c

Alias of left_distrib.

theorem right_distrib {R : Type x} [Mul R] [Add R] [RightDistribClass R] (a : R) (b : R) (c : R) : (a + b) * c = a * c + b * c

theorem add_mul {R : Type x} [Mul R] [Add R] [RightDistribClass R] (a : R) (b : R) (c : R) : (a + b) * c = a * c + b * c

Alias of right_distrib.

theorem distrib_three_right {R : Type x} [Mul R] [Add R] [RightDistribClass R] (a : R) (b : R) (c : R) (d : R) : (a + b + c) * d = a * d + b * d + c * d

Classes of semirings and rings

We make sure that the canonical path from NonAssocSemiring to Ring passes through Semiring, as this is a path which is followed all the time in linear algebra where the defining semilinear map σ : R →+* S depends on the NonAssocSemiring structure of R and S while the module definition depends on the Semiring structure.

It is not currently possible to adjust priorities by hand (see lean4#2115). Instead, the last declared instance is used, so we make sure that Semiring is declared after NonAssocRing, so that Semiring -> NonAssocSemiring is tried before NonAssocRing -> NonAssocSemiring. TODO: clean this once lean4#2115 is fixed

class NonUnitalNonAssocSemiring (α : Type u) extends AddCommMonoid , Mul : Type u

A not-necessarily-unital, not-necessarily-associative semiring.

• add : α → α → α
• add_assoc : ∀ (a b c : α), a + b + c = a + (b + c)
• zero : α
• zero_add : ∀ (a : α), 0 + a = a
• add_zero : ∀ (a : α), a + 0 = a
• nsmul : ℕ → α → α
• nsmul_zero : ∀ (x : α), AddMonoid.nsmul 0 x = 0
• nsmul_succ : ∀ (n : ℕ) (x : α), AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• add_comm : ∀ (a b : α), a + b = b + a
• mul : α → α → α
• left_distrib : ∀ (a b c : α), a * (b + c) = a * b + a * c

  Multiplication is left distributive over addition

• right_distrib : ∀ (a b c : α), (a + b) * c = a * c + b * c

  Multiplication is right distributive over addition

• zero_mul : ∀ (a : α), 0 * a = 0

  Zero is a left absorbing element for multiplication

• mul_zero : ∀ (a : α), a * 0 = 0

  Zero is a right absorbing element for multiplication

class NonUnitalSemiring (α : Type u) extends NonUnitalNonAssocSemiring : Type u

An associative but not-necessarily unital semiring.

• add : α → α → α
• add_assoc : ∀ (a b c : α), a + b + c = a + (b + c)
• zero : α
• zero_add : ∀ (a : α), 0 + a = a
• add_zero : ∀ (a : α), a + 0 = a
• nsmul : ℕ → α → α
• nsmul_zero : ∀ (x : α), AddMonoid.nsmul 0 x = 0
• nsmul_succ : ∀ (n : ℕ) (x : α), AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• add_comm : ∀ (a b : α), a + b = b + a
• mul : α → α → α
• left_distrib : ∀ (a b c : α), a * (b + c) = a * b + a * c
• right_distrib : ∀ (a b c : α), (a + b) * c = a * c + b * c
• zero_mul : ∀ (a : α), 0 * a = 0
• mul_zero : ∀ (a : α), a * 0 = 0
• mul_assoc : ∀ (a b c : α), a * b * c = a * (b * c)

  Multiplication is associative

class NonAssocSemiring (α : Type u) extends NonUnitalNonAssocSemiring , One , NatCast : Type u

A unital but not-necessarily-associative semiring.

• add : α → α → α
• add_assoc : ∀ (a b c : α), a + b + c = a + (b + c)
• zero : α
• zero_add : ∀ (a : α), 0 + a = a
• add_zero : ∀ (a : α), a + 0 = a
• nsmul : ℕ → α → α
• nsmul_zero : ∀ (x : α), AddMonoid.nsmul 0 x = 0
• nsmul_succ : ∀ (n : ℕ) (x : α), AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• add_comm : ∀ (a b : α), a + b = b + a
• mul : α → α → α
• left_distrib : ∀ (a b c : α), a * (b + c) = a * b + a * c
• right_distrib : ∀ (a b c : α), (a + b) * c = a * c + b * c
• zero_mul : ∀ (a : α), 0 * a = 0
• mul_zero : ∀ (a : α), a * 0 = 0
• one : α
• one_mul : ∀ (a : α), 1 * a = a

  One is a left neutral element for multiplication

• mul_one : ∀ (a : α), a * 1 = a

  One is a right neutral element for multiplication

• natCast : ℕ → α
• natCast_zero : NatCast.natCast 0 = 0

  The canonical map ℕ → R sends 0 : ℕ to 0 : R.

• natCast_succ : ∀ (n : ℕ), NatCast.natCast (n + 1) = NatCast.natCast n + 1

  The canonical map ℕ → R is a homomorphism.

class NonUnitalNonAssocRing (α : Type u) extends AddCommGroup , Mul : Type u

A not-necessarily-unital, not-necessarily-associative ring.

• add : α → α → α
• add_assoc : ∀ (a b c : α), a + b + c = a + (b + c)
• zero : α
• zero_add : ∀ (a : α), 0 + a = a
• add_zero : ∀ (a : α), a + 0 = a
• nsmul : ℕ → α → α
• nsmul_zero : ∀ (x : α), AddMonoid.nsmul 0 x = 0
• nsmul_succ : ∀ (n : ℕ) (x : α), AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• neg : α → α
• sub : α → α → α
• sub_eq_add_neg : ∀ (a b : α), a - b = a + -b
• zsmul : ℤ → α → α
• zsmul_zero' : ∀ (a : α), SubNegMonoid.zsmul 0 a = 0
• zsmul_succ' : ∀ (n : ℕ) (a : α), SubNegMonoid.zsmul (Int.ofNat (Nat.succ n)) a = SubNegMonoid.zsmul (Int.ofNat n) a + a
• zsmul_neg' : ∀ (n : ℕ) (a : α), SubNegMonoid.zsmul (Int.negSucc n) a = -SubNegMonoid.zsmul (↑(Nat.succ n)) a
• add_left_neg : ∀ (a : α), -a + a = 0
• add_comm : ∀ (a b : α), a + b = b + a
• mul : α → α → α
• left_distrib : ∀ (a b c : α), a * (b + c) = a * b + a * c

  Multiplication is left distributive over addition

• right_distrib : ∀ (a b c : α), (a + b) * c = a * c + b * c

  Multiplication is right distributive over addition

• zero_mul : ∀ (a : α), 0 * a = 0

  Zero is a left absorbing element for multiplication

• mul_zero : ∀ (a : α), a * 0 = 0

  Zero is a right absorbing element for multiplication

class NonUnitalRing (α : Type u_1) extends NonUnitalNonAssocRing : Type u_1

An associative but not-necessarily unital ring.

• add : α → α → α
• add_assoc : ∀ (a b c : α), a + b + c = a + (b + c)
• zero : α
• zero_add : ∀ (a : α), 0 + a = a
• add_zero : ∀ (a : α), a + 0 = a
• nsmul : ℕ → α → α
• nsmul_zero : ∀ (x : α), AddMonoid.nsmul 0 x = 0
• nsmul_succ : ∀ (n : ℕ) (x : α), AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• neg : α → α
• sub : α → α → α
• sub_eq_add_neg : ∀ (a b : α), a - b = a + -b
• zsmul : ℤ → α → α
• zsmul_zero' : ∀ (a : α), SubNegMonoid.zsmul 0 a = 0
• zsmul_succ' : ∀ (n : ℕ) (a : α), SubNegMonoid.zsmul (Int.ofNat (Nat.succ n)) a = SubNegMonoid.zsmul (Int.ofNat n) a + a
• zsmul_neg' : ∀ (n : ℕ) (a : α), SubNegMonoid.zsmul (Int.negSucc n) a = -SubNegMonoid.zsmul (↑(Nat.succ n)) a
• add_left_neg : ∀ (a : α), -a + a = 0
• add_comm : ∀ (a b : α), a + b = b + a
• mul : α → α → α
• left_distrib : ∀ (a b c : α), a * (b + c) = a * b + a * c
• right_distrib : ∀ (a b c : α), (a + b) * c = a * c + b * c
• zero_mul : ∀ (a : α), 0 * a = 0
• mul_zero : ∀ (a : α), a * 0 = 0
• mul_assoc : ∀ (a b c : α), a * b * c = a * (b * c)

  Multiplication is associative

class NonAssocRing (α : Type u_1) extends NonUnitalNonAssocRing , One , NatCast , IntCast : Type u_1

A unital but not-necessarily-associative ring.

• add : α → α → α
• add_assoc : ∀ (a b c : α), a + b + c = a + (b + c)
• zero : α
• zero_add : ∀ (a : α), 0 + a = a
• add_zero : ∀ (a : α), a + 0 = a
• nsmul : ℕ → α → α
• nsmul_zero : ∀ (x : α), AddMonoid.nsmul 0 x = 0
• nsmul_succ : ∀ (n : ℕ) (x : α), AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• neg : α → α
• sub : α → α → α
• sub_eq_add_neg : ∀ (a b : α), a - b = a + -b
• zsmul : ℤ → α → α
• zsmul_zero' : ∀ (a : α), SubNegMonoid.zsmul 0 a = 0
• zsmul_succ' : ∀ (n : ℕ) (a : α), SubNegMonoid.zsmul (Int.ofNat (Nat.succ n)) a = SubNegMonoid.zsmul (Int.ofNat n) a + a
• zsmul_neg' : ∀ (n : ℕ) (a : α), SubNegMonoid.zsmul (Int.negSucc n) a = -SubNegMonoid.zsmul (↑(Nat.succ n)) a
• add_left_neg : ∀ (a : α), -a + a = 0
• add_comm : ∀ (a b : α), a + b = b + a
• mul : α → α → α
• left_distrib : ∀ (a b c : α), a * (b + c) = a * b + a * c
• right_distrib : ∀ (a b c : α), (a + b) * c = a * c + b * c
• zero_mul : ∀ (a : α), 0 * a = 0
• mul_zero : ∀ (a : α), a * 0 = 0
• one : α
• one_mul : ∀ (a : α), 1 * a = a

  One is a left neutral element for multiplication

• mul_one : ∀ (a : α), a * 1 = a

  One is a right neutral element for multiplication

• natCast : ℕ → α
• natCast_zero : NatCast.natCast 0 = 0

  The canonical map ℕ → R sends 0 : ℕ to 0 : R.

• natCast_succ : ∀ (n : ℕ), NatCast.natCast (n + 1) = NatCast.natCast n + 1

  The canonical map ℕ → R is a homomorphism.

• intCast : ℤ → α
• intCast_ofNat : ∀ (n : ℕ), IntCast.intCast ↑n = ↑n

  The canonical homomorphism ℤ → R agrees with the one from ℕ → R on ℕ.

• intCast_negSucc : ∀ (n : ℕ), IntCast.intCast (Int.negSucc n) = -↑(n + 1)

  The canonical homomorphism ℤ → R for negative values is just the negation of the values of the canonical homomorphism ℕ → R.

class Semiring (α : Type u) extends NonUnitalSemiring , One , NatCast : Type u

A Semiring is a type with addition, multiplication, a 0 and a 1 where addition is commutative and associative, multiplication is associative and left and right distributive over addition, and 0 and 1 are additive and multiplicative identities.

• add : α → α → α
• add_assoc : ∀ (a b c : α), a + b + c = a + (b + c)
• zero : α
• zero_add : ∀ (a : α), 0 + a = a
• add_zero : ∀ (a : α), a + 0 = a
• nsmul : ℕ → α → α
• nsmul_zero : ∀ (x : α), AddMonoid.nsmul 0 x = 0
• nsmul_succ : ∀ (n : ℕ) (x : α), AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• add_comm : ∀ (a b : α), a + b = b + a
• mul : α → α → α
• left_distrib : ∀ (a b c : α), a * (b + c) = a * b + a * c
• right_distrib : ∀ (a b c : α), (a + b) * c = a * c + b * c
• zero_mul : ∀ (a : α), 0 * a = 0
• mul_zero : ∀ (a : α), a * 0 = 0
• mul_assoc : ∀ (a b c : α), a * b * c = a * (b * c)
• one : α
• one_mul : ∀ (a : α), 1 * a = a

  One is a left neutral element for multiplication

• mul_one : ∀ (a : α), a * 1 = a

  One is a right neutral element for multiplication

• natCast : ℕ → α
• natCast_zero : NatCast.natCast 0 = 0

  The canonical map ℕ → R sends 0 : ℕ to 0 : R.

• natCast_succ : ∀ (n : ℕ), NatCast.natCast (n + 1) = NatCast.natCast n + 1

  The canonical map ℕ → R is a homomorphism.

• npow : ℕ → α → α

  Raising to the power of a natural number.

• npow_zero : ∀ (x : α), Semiring.npow 0 x = 1

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

• npow_succ : ∀ (n : ℕ) (x : α), Semiring.npow (n + 1) x = Semiring.npow n x * x

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

class Ring (R : Type u) extends Semiring , Neg , Sub , IntCast : Type u

A Ring is a Semiring with negation making it an additive group.

• add : R → R → R
• add_assoc : ∀ (a b c : R), a + b + c = a + (b + c)
• zero : R
• zero_add : ∀ (a : R), 0 + a = a
• add_zero : ∀ (a : R), a + 0 = a
• nsmul : ℕ → R → R
• nsmul_zero : ∀ (x : R), AddMonoid.nsmul 0 x = 0
• nsmul_succ : ∀ (n : ℕ) (x : R), AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• add_comm : ∀ (a b : R), a + b = b + a
• mul : R → R → R
• left_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c
• right_distrib : ∀ (a b c : R), (a + b) * c = a * c + b * c
• zero_mul : ∀ (a : R), 0 * a = 0
• mul_zero : ∀ (a : R), a * 0 = 0
• mul_assoc : ∀ (a b c : R), a * b * c = a * (b * c)
• one : R
• one_mul : ∀ (a : R), 1 * a = a
• mul_one : ∀ (a : R), a * 1 = a
• natCast : ℕ → R
• natCast_zero : NatCast.natCast 0 = 0
• natCast_succ : ∀ (n : ℕ), NatCast.natCast (n + 1) = NatCast.natCast n + 1
• npow : ℕ → R → R
• npow_zero : ∀ (x : R), Semiring.npow 0 x = 1
• npow_succ : ∀ (n : ℕ) (x : R), Semiring.npow (n + 1) x = Semiring.npow n x * x
• neg : R → R
• sub : R → R → R
• sub_eq_add_neg : ∀ (a b : R), a - b = a + -b
• zsmul : ℤ → R → R

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

• zsmul_zero' : ∀ (a : R), Ring.zsmul 0 a = 0
• zsmul_succ' : ∀ (n : ℕ) (a : R), Ring.zsmul (Int.ofNat (Nat.succ n)) a = Ring.zsmul (Int.ofNat n) a + a
• zsmul_neg' : ∀ (n : ℕ) (a : R), Ring.zsmul (Int.negSucc n) a = -Ring.zsmul (↑(Nat.succ n)) a
• add_left_neg : ∀ (a : R), -a + a = 0
• intCast : ℤ → R
• intCast_ofNat : ∀ (n : ℕ), IntCast.intCast ↑n = ↑n

  The canonical homomorphism ℤ → R agrees with the one from ℕ → R on ℕ.

• intCast_negSucc : ∀ (n : ℕ), IntCast.intCast (Int.negSucc n) = -↑(n + 1)

  The canonical homomorphism ℤ → R for negative values is just the negation of the values of the canonical homomorphism ℕ → R.

Semirings

theorem add_one_mul {α : Type u} [Add α] [MulOneClass α] [RightDistribClass α] (a : α) (b : α) : (a + 1) * b = a * b + b

theorem mul_add_one {α : Type u} [Add α] [MulOneClass α] [LeftDistribClass α] (a : α) (b : α) : a * (b + 1) = a * b + a

theorem one_add_mul {α : Type u} [Add α] [MulOneClass α] [RightDistribClass α] (a : α) (b : α) : (1 + a) * b = b + a * b

theorem mul_one_add {α : Type u} [Add α] [MulOneClass α] [LeftDistribClass α] (a : α) (b : α) : a * (1 + b) = a + a * b

theorem two_mul {α : Type u} [NonAssocSemiring α] (n : α) : 2 * n = n + n

theorem bit0_eq_two_mul {α : Type u} [NonAssocSemiring α] (n : α) : bit0 n = 2 * n

theorem mul_two {α : Type u} [NonAssocSemiring α] (n : α) : n * 2 = n + n

theorem add_ite {α : Type u_1} [Add α] (P : Prop) [Decidable P] (a : α) (b : α) (c : α) : (a + if P then b else c) = if P then a + b else a + c

@[simp]

theorem mul_ite {α : Type u_1} [Mul α] (P : Prop) [Decidable P] (a : α) (b : α) (c : α) : (a * if P then b else c) = if P then a * b else a * c

theorem ite_add {α : Type u_1} [Add α] (P : Prop) [Decidable P] (a : α) (b : α) (c : α) : (if P then a else b) + c = if P then a + c else b + c

@[simp]

theorem ite_mul {α : Type u_1} [Mul α] (P : Prop) [Decidable P] (a : α) (b : α) (c : α) : (if P then a else b) * c = if P then a * c else b * c

theorem ite_sub_ite {α : Type u_1} [Sub α] (P : Prop) [Decidable P] (a : α) (b : α) (c : α) (d : α) : ((if P then a else b) - if P then c else d) = if P then a - c else b - d

theorem ite_add_ite {α : Type u_1} [Add α] (P : Prop) [Decidable P] (a : α) (b : α) (c : α) (d : α) : ((if P then a else b) + if P then c else d) = if P then a + c else b + d

theorem ite_zero_mul {α : Type u} [MulZeroClass α] (P : Prop) [Decidable P] (a : α) (b : α) : (if P then a else 0) * b = if P then a * b else 0

theorem mul_ite_zero {α : Type u} [MulZeroClass α] (P : Prop) [Decidable P] (a : α) (b : α) : (a * if P then b else 0) = if P then a * b else 0

theorem ite_zero_mul_ite_zero {α : Type u} [MulZeroClass α] (P : Prop) (Q : Prop) [Decidable P] [Decidable Q] (a : α) (b : α) : ((if P then a else 0) * if Q then b else 0) = if P ∧ Q then a * b else 0

theorem mul_boole {α : Type u_1} [MulZeroOneClass α] (P : Prop) [Decidable P] (a : α) : (a * if P then 1 else 0) = if P then a else 0

theorem boole_mul {α : Type u_1} [MulZeroOneClass α] (P : Prop) [Decidable P] (a : α) : (if P then 1 else 0) * a = if P then a else 0

class NonUnitalNonAssocCommSemiring (α : Type u) extends NonUnitalNonAssocSemiring : Type u

A not-necessarily-unital, not-necessarily-associative, but commutative semiring.

• add : α → α → α
• add_assoc : ∀ (a b c : α), a + b + c = a + (b + c)
• zero : α
• zero_add : ∀ (a : α), 0 + a = a
• add_zero : ∀ (a : α), a + 0 = a
• nsmul : ℕ → α → α
• nsmul_zero : ∀ (x : α), AddMonoid.nsmul 0 x = 0
• nsmul_succ : ∀ (n : ℕ) (x : α), AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• add_comm : ∀ (a b : α), a + b = b + a
• mul : α → α → α
• left_distrib : ∀ (a b c : α), a * (b + c) = a * b + a * c
• right_distrib : ∀ (a b c : α), (a + b) * c = a * c + b * c
• zero_mul : ∀ (a : α), 0 * a = 0
• mul_zero : ∀ (a : α), a * 0 = 0
• mul_comm : ∀ (a b : α), a * b = b * a

  Multiplication is commutative in a commutative multiplicative magma.

class NonUnitalCommSemiring (α : Type u) extends NonUnitalSemiring : Type u

A non-unital commutative semiring is a NonUnitalSemiring with commutative multiplication. In other words, it is a type with the following structures: additive commutative monoid (AddCommMonoid), commutative semigroup (CommSemigroup), distributive laws (Distrib), and multiplication by zero law (MulZeroClass).

• add : α → α → α
• add_assoc : ∀ (a b c : α), a + b + c = a + (b + c)
• zero : α
• zero_add : ∀ (a : α), 0 + a = a
• add_zero : ∀ (a : α), a + 0 = a
• nsmul : ℕ → α → α
• nsmul_zero : ∀ (x : α), AddMonoid.nsmul 0 x = 0
• nsmul_succ : ∀ (n : ℕ) (x : α), AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• add_comm : ∀ (a b : α), a + b = b + a
• mul : α → α → α
• left_distrib : ∀ (a b c : α), a * (b + c) = a * b + a * c
• right_distrib : ∀ (a b c : α), (a + b) * c = a * c + b * c
• zero_mul : ∀ (a : α), 0 * a = 0
• mul_zero : ∀ (a : α), a * 0 = 0
• mul_assoc : ∀ (a b c : α), a * b * c = a * (b * c)
• mul_comm : ∀ (a b : α), a * b = b * a

  Multiplication is commutative in a commutative multiplicative magma.

class CommSemiring (R : Type u) extends Semiring : Type u

• add : R → R → R
• add_assoc : ∀ (a b c : R), a + b + c = a + (b + c)
• zero : R
• zero_add : ∀ (a : R), 0 + a = a
• add_zero : ∀ (a : R), a + 0 = a
• nsmul : ℕ → R → R
• nsmul_zero : ∀ (x : R), AddMonoid.nsmul 0 x = 0
• nsmul_succ : ∀ (n : ℕ) (x : R), AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• add_comm : ∀ (a b : R), a + b = b + a
• mul : R → R → R
• left_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c
• right_distrib : ∀ (a b c : R), (a + b) * c = a * c + b * c
• zero_mul : ∀ (a : R), 0 * a = 0
• mul_zero : ∀ (a : R), a * 0 = 0
• mul_assoc : ∀ (a b c : R), a * b * c = a * (b * c)
• one : R
• one_mul : ∀ (a : R), 1 * a = a
• mul_one : ∀ (a : R), a * 1 = a
• natCast : ℕ → R
• natCast_zero : NatCast.natCast 0 = 0
• natCast_succ : ∀ (n : ℕ), NatCast.natCast (n + 1) = NatCast.natCast n + 1
• npow : ℕ → R → R
• npow_zero : ∀ (x : R), Semiring.npow 0 x = 1
• npow_succ : ∀ (n : ℕ) (x : R), Semiring.npow (n + 1) x = Semiring.npow n x * x
• mul_comm : ∀ (a b : R), a * b = b * a

  Multiplication is commutative in a commutative multiplicative magma.

instance CommSemiring.toNonUnitalCommSemiring {α : Type u} [CommSemiring α] : NonUnitalCommSemiring α

Equations

• CommSemiring.toNonUnitalCommSemiring = let __src := inferInstanceAs (CommMonoid α); let __src_1 := inferInstanceAs (CommSemiring α); NonUnitalCommSemiring.mk ⋯

instance CommSemiring.toCommMonoidWithZero {α : Type u} [CommSemiring α] : CommMonoidWithZero α

Equations

• CommSemiring.toCommMonoidWithZero = let __src := inferInstanceAs (CommMonoid α); let __src_1 := inferInstanceAs (CommSemiring α); CommMonoidWithZero.mk ⋯ ⋯

theorem add_mul_self_eq {α : Type u} [CommSemiring α] (a : α) (b : α) : (a + b) * (a + b) = a * a + 2 * a * b + b * b

class HasDistribNeg (α : Type u_1) [Mul α] extends InvolutiveNeg : Type u_1

Typeclass for a negation operator that distributes across multiplication.

This is useful for dealing with submonoids of a ring that contain -1 without having to duplicate lemmas.

• neg : α → α
• neg_neg : ∀ (x : α), - -x = x
• neg_mul : ∀ (x y : α), -x * y = -(x * y)

  Negation is left distributive over multiplication

• mul_neg : ∀ (x y : α), x * -y = -(x * y)

  Negation is right distributive over multiplication

@[simp]

theorem neg_mul {α : Type u} [Mul α] [HasDistribNeg α] (a : α) (b : α) : -a * b = -(a * b)

@[simp]

theorem mul_neg {α : Type u} [Mul α] [HasDistribNeg α] (a : α) (b : α) : a * -b = -(a * b)

theorem neg_mul_neg {α : Type u} [Mul α] [HasDistribNeg α] (a : α) (b : α) : -a * -b = a * b

theorem neg_mul_eq_neg_mul {α : Type u} [Mul α] [HasDistribNeg α] (a : α) (b : α) : -(a * b) = -a * b

theorem neg_mul_eq_mul_neg {α : Type u} [Mul α] [HasDistribNeg α] (a : α) (b : α) : -(a * b) = a * -b

theorem neg_mul_comm {α : Type u} [Mul α] [HasDistribNeg α] (a : α) (b : α) : -a * b = a * -b

theorem neg_eq_neg_one_mul {α : Type u} [MulOneClass α] [HasDistribNeg α] (a : α) : -a = -1 * a

theorem mul_neg_one {α : Type u} [MulOneClass α] [HasDistribNeg α] (a : α) : a * -1 = -a

An element of a ring multiplied by the additive inverse of one is the element's additive inverse.

theorem neg_one_mul {α : Type u} [MulOneClass α] [HasDistribNeg α] (a : α) : -1 * a = -a

The additive inverse of one multiplied by an element of a ring is the element's additive inverse.

instance MulZeroClass.negZeroClass {α : Type u} [MulZeroClass α] [HasDistribNeg α] : NegZeroClass α

Equations

• MulZeroClass.negZeroClass = let __spread.0 := inferInstanceAs (Zero α); let __spread.1 := inferInstanceAs (InvolutiveNeg α); NegZeroClass.mk ⋯

Rings

instance NonUnitalNonAssocRing.toHasDistribNeg {α : Type u} [NonUnitalNonAssocRing α] : HasDistribNeg α

Equations

• NonUnitalNonAssocRing.toHasDistribNeg = HasDistribNeg.mk ⋯ ⋯

theorem mul_sub_left_distrib {α : Type u} [NonUnitalNonAssocRing α] (a : α) (b : α) (c : α) : a * (b - c) = a * b - a * c

theorem mul_sub {α : Type u} [NonUnitalNonAssocRing α] (a : α) (b : α) (c : α) : a * (b - c) = a * b - a * c

Alias of mul_sub_left_distrib.

theorem mul_sub_right_distrib {α : Type u} [NonUnitalNonAssocRing α] (a : α) (b : α) (c : α) : (a - b) * c = a * c - b * c

theorem sub_mul {α : Type u} [NonUnitalNonAssocRing α] (a : α) (b : α) (c : α) : (a - b) * c = a * c - b * c

Alias of mul_sub_right_distrib.

theorem sub_one_mul {α : Type u} [NonAssocRing α] (a : α) (b : α) : (a - 1) * b = a * b - b

theorem mul_sub_one {α : Type u} [NonAssocRing α] (a : α) (b : α) : a * (b - 1) = a * b - a

theorem one_sub_mul {α : Type u} [NonAssocRing α] (a : α) (b : α) : (1 - a) * b = b - a * b

theorem mul_one_sub {α : Type u} [NonAssocRing α] (a : α) (b : α) : a * (1 - b) = a - a * b

instance Ring.toNonUnitalRing {α : Type u} [Ring α] : NonUnitalRing α

Equations

• Ring.toNonUnitalRing = let __src := inst; NonUnitalRing.mk ⋯

instance Ring.toNonAssocRing {α : Type u} [Ring α] : NonAssocRing α

Equations

• Ring.toNonAssocRing = let __src := inst; NonAssocRing.mk ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

instance instSemiring {α : Type u} [Ring α] : Semiring α

Equations

• instSemiring = let __src := inst; Semiring.mk ⋯ ⋯ ⋯ ⋯ Semiring.npow ⋯ ⋯

class NonUnitalNonAssocCommRing (α : Type u) extends NonUnitalNonAssocRing : Type u

A non-unital non-associative commutative ring is a NonUnitalNonAssocRing with commutative multiplication.

• add : α → α → α
• add_assoc : ∀ (a b c : α), a + b + c = a + (b + c)
• zero : α
• zero_add : ∀ (a : α), 0 + a = a
• add_zero : ∀ (a : α), a + 0 = a
• nsmul : ℕ → α → α
• nsmul_zero : ∀ (x : α), AddMonoid.nsmul 0 x = 0
• nsmul_succ : ∀ (n : ℕ) (x : α), AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• neg : α → α
• sub : α → α → α
• sub_eq_add_neg : ∀ (a b : α), a - b = a + -b
• zsmul : ℤ → α → α
• zsmul_zero' : ∀ (a : α), SubNegMonoid.zsmul 0 a = 0
• zsmul_succ' : ∀ (n : ℕ) (a : α), SubNegMonoid.zsmul (Int.ofNat (Nat.succ n)) a = SubNegMonoid.zsmul (Int.ofNat n) a + a
• zsmul_neg' : ∀ (n : ℕ) (a : α), SubNegMonoid.zsmul (Int.negSucc n) a = -SubNegMonoid.zsmul (↑(Nat.succ n)) a
• add_left_neg : ∀ (a : α), -a + a = 0
• add_comm : ∀ (a b : α), a + b = b + a
• mul : α → α → α
• left_distrib : ∀ (a b c : α), a * (b + c) = a * b + a * c
• right_distrib : ∀ (a b c : α), (a + b) * c = a * c + b * c
• zero_mul : ∀ (a : α), 0 * a = 0
• mul_zero : ∀ (a : α), a * 0 = 0
• mul_comm : ∀ (a b : α), a * b = b * a

  Multiplication is commutative in a commutative multiplicative magma.

class NonUnitalCommRing (α : Type u) extends NonUnitalRing : Type u

A non-unital commutative ring is a NonUnitalRing with commutative multiplication.

• add : α → α → α
• add_assoc : ∀ (a b c : α), a + b + c = a + (b + c)
• zero : α
• zero_add : ∀ (a : α), 0 + a = a
• add_zero : ∀ (a : α), a + 0 = a
• nsmul : ℕ → α → α
• nsmul_zero : ∀ (x : α), AddMonoid.nsmul 0 x = 0
• nsmul_succ : ∀ (n : ℕ) (x : α), AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• neg : α → α
• sub : α → α → α
• sub_eq_add_neg : ∀ (a b : α), a - b = a + -b
• zsmul : ℤ → α → α
• zsmul_zero' : ∀ (a : α), SubNegMonoid.zsmul 0 a = 0
• zsmul_succ' : ∀ (n : ℕ) (a : α), SubNegMonoid.zsmul (Int.ofNat (Nat.succ n)) a = SubNegMonoid.zsmul (Int.ofNat n) a + a
• zsmul_neg' : ∀ (n : ℕ) (a : α), SubNegMonoid.zsmul (Int.negSucc n) a = -SubNegMonoid.zsmul (↑(Nat.succ n)) a
• add_left_neg : ∀ (a : α), -a + a = 0
• add_comm : ∀ (a b : α), a + b = b + a
• mul : α → α → α
• left_distrib : ∀ (a b c : α), a * (b + c) = a * b + a * c
• right_distrib : ∀ (a b c : α), (a + b) * c = a * c + b * c
• zero_mul : ∀ (a : α), 0 * a = 0
• mul_zero : ∀ (a : α), a * 0 = 0
• mul_assoc : ∀ (a b c : α), a * b * c = a * (b * c)
• mul_comm : ∀ (a b : α), a * b = b * a

  Multiplication is commutative in a commutative multiplicative magma.

instance NonUnitalCommRing.toNonUnitalCommSemiring {α : Type u} [s : NonUnitalCommRing α] : NonUnitalCommSemiring α

Equations

• NonUnitalCommRing.toNonUnitalCommSemiring = NonUnitalCommSemiring.mk ⋯

class CommRing (α : Type u) extends Ring : Type u

• add : α → α → α
• add_assoc : ∀ (a b c : α), a + b + c = a + (b + c)
• zero : α
• zero_add : ∀ (a : α), 0 + a = a
• add_zero : ∀ (a : α), a + 0 = a
• nsmul : ℕ → α → α
• nsmul_zero : ∀ (x : α), AddMonoid.nsmul 0 x = 0
• nsmul_succ : ∀ (n : ℕ) (x : α), AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• add_comm : ∀ (a b : α), a + b = b + a
• mul : α → α → α
• left_distrib : ∀ (a b c : α), a * (b + c) = a * b + a * c
• right_distrib : ∀ (a b c : α), (a + b) * c = a * c + b * c
• zero_mul : ∀ (a : α), 0 * a = 0
• mul_zero : ∀ (a : α), a * 0 = 0
• mul_assoc : ∀ (a b c : α), a * b * c = a * (b * c)
• one : α
• one_mul : ∀ (a : α), 1 * a = a
• mul_one : ∀ (a : α), a * 1 = a
• natCast : ℕ → α
• natCast_zero : NatCast.natCast 0 = 0
• natCast_succ : ∀ (n : ℕ), NatCast.natCast (n + 1) = NatCast.natCast n + 1
• npow : ℕ → α → α
• npow_zero : ∀ (x : α), Semiring.npow 0 x = 1
• npow_succ : ∀ (n : ℕ) (x : α), Semiring.npow (n + 1) x = Semiring.npow n x * x
• neg : α → α
• sub : α → α → α
• sub_eq_add_neg : ∀ (a b : α), a - b = a + -b
• zsmul : ℤ → α → α
• zsmul_zero' : ∀ (a : α), Ring.zsmul 0 a = 0
• zsmul_succ' : ∀ (n : ℕ) (a : α), Ring.zsmul (Int.ofNat (Nat.succ n)) a = Ring.zsmul (Int.ofNat n) a + a
• zsmul_neg' : ∀ (n : ℕ) (a : α), Ring.zsmul (Int.negSucc n) a = -Ring.zsmul (↑(Nat.succ n)) a
• add_left_neg : ∀ (a : α), -a + a = 0
• intCast : ℤ → α
• intCast_ofNat : ∀ (n : ℕ), IntCast.intCast ↑n = ↑n
• intCast_negSucc : ∀ (n : ℕ), IntCast.intCast (Int.negSucc n) = -↑(n + 1)
• mul_comm : ∀ (a b : α), a * b = b * a

  Multiplication is commutative in a commutative multiplicative magma.

instance CommRing.toCommSemiring {α : Type u} [s : CommRing α] : CommSemiring α

Equations

• CommRing.toCommSemiring = CommSemiring.mk ⋯

instance CommRing.toNonUnitalCommRing {α : Type u} [s : CommRing α] : NonUnitalCommRing α

Equations

• CommRing.toNonUnitalCommRing = NonUnitalCommRing.mk ⋯

instance CommRing.toAddCommGroupWithOne {α : Type u} [s : CommRing α] : AddCommGroupWithOne α

Equations

• CommRing.toAddCommGroupWithOne = AddCommGroupWithOne.mk ⋯ ⋯ ⋯ ⋯

class IsDomain (α : Type u) [Semiring α] extends IsCancelMulZero , Nontrivial : Prop

A domain is a nontrivial semiring such multiplication by a non zero element is cancellative, on both sides. In other words, a nontrivial semiring R satisfying ∀ {a b c : R}, a ≠ 0 → a * b = a * c → b = c and ∀ {a b c : R}, b ≠ 0 → a * b = c * b → a = c.

This is implemented as a mixin for Semiring α. To obtain an integral domain use [CommRing α] [IsDomain α].

Previously an import dependency on Mathlib.Algebra.Group.Basic had crept in. In general, the .Defs files in the basic algebraic hierarchy should only depend on earlier .Defs files, without importing .Basic theory development.

These assert_not_exists statements guard against this returning.