Cast of integers

This file defines the canonical homomorphism from the integers into an additive group with a one (typically a Ring). In additive groups with a one element, there exists a unique such homomorphism and we store it in the intCast : ℤ → R field.

Preferentially, the homomorphism is written as a coercion.

Main declarations

• Int.cast: Canonical homomorphism ℤ → R.
• AddGroupWithOne: Type class for Int.cast.

def Int.castDef {R : Type u} [NatCast R] [Neg R] : ℤ → R

Default value for IntCast.intCast in an AddGroupWithOne.

Equations

• (Int.ofNat n).castDef = ↑n
• (Int.negSucc n).castDef = -↑(n + 1)

Additive groups with one

class AddGroupWithOne (R : Type u) extends IntCast , AddMonoidWithOne , Neg , Sub : Type u

An AddGroupWithOne is an AddGroup with a 1. It also contains data for the unique homomorphisms ℕ → R and ℤ → R.

• intCast : ℤ → R
• natCast : ℕ → R
• 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
• one : R
• natCast_zero : NatCast.natCast 0 = 0
• natCast_succ : ∀ (n : ℕ), NatCast.natCast (n + 1) = NatCast.natCast n + 1
• 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), AddGroupWithOne.zsmul 0 a = 0
• zsmul_succ' : ∀ (n : ℕ) (a : R), AddGroupWithOne.zsmul (↑n.succ) a = AddGroupWithOne.zsmul (↑n) a + a
• zsmul_neg' : ∀ (n : ℕ) (a : R), AddGroupWithOne.zsmul (Int.negSucc n) a = -AddGroupWithOne.zsmul (↑n.succ) a
• neg_add_cancel : ∀ (a : R), -a + a = 0
• 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.

theorem AddGroupWithOne.intCast_ofNat {R : Type u} [self : AddGroupWithOne R] (n : ℕ) : IntCast.intCast ↑n = ↑n

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

theorem AddGroupWithOne.intCast_negSucc {R : Type u} [self : AddGroupWithOne R] (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 AddCommGroupWithOne (R : Type u) extends AddCommGroup , IntCast , NatCast , One : Type u

An AddCommGroupWithOne is an AddGroupWithOne satisfying a + b = b + a.

• 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
• neg : R → R
• sub : R → R → R
• sub_eq_add_neg : ∀ (a b : R), a - b = a + -b
• zsmul : ℤ → R → R
• zsmul_zero' : ∀ (a : R), SubNegMonoid.zsmul 0 a = 0
• zsmul_succ' : ∀ (n : ℕ) (a : R), SubNegMonoid.zsmul (↑n.succ) a = SubNegMonoid.zsmul (↑n) a + a
• zsmul_neg' : ∀ (n : ℕ) (a : R), SubNegMonoid.zsmul (Int.negSucc n) a = -SubNegMonoid.zsmul (↑n.succ) a
• neg_add_cancel : ∀ (a : R), -a + a = 0
• add_comm : ∀ (a b : R), a + b = b + a
• intCast : ℤ → R
• natCast : ℕ → R
• one : R
• 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_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.