Documentation

Mathlib.Algebra.Order.Ring.Canonical

Canonically ordered rings and semirings. #

TODO #

We're still missing some typeclasses, like

source
class CanonicallyOrderedCommSemiring (α : Type u_2) extends CanonicallyOrderedAddCommMonoid , Mul , One , NatCast :
Type u_2

A canonically ordered commutative semiring is an ordered, commutative semiring in which a ≤ b iff there exists c with b = a + c. This is satisfied by the natural numbers, for example, but not the integers or other ordered groups.

  • 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 = x + AddMonoid.nsmul n x
  • add_comm : ∀ (a b : α), a + b = b + a
  • le : α → α → Prop
  • lt : α → α → Prop
  • le_refl : ∀ (a : α), a ≤ a
  • le_trans : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c
  • lt_iff_le_not_le : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a
  • le_antisymm : ∀ (a b : α), a ≤ b → b ≤ a → a = b
  • add_le_add_left : ∀ (a b : α), a ≤ b → ∀ (c : α), c + a ≤ c + b
  • bot : α
  • bot_le : ∀ (a : α), ⊥ ≤ a
  • exists_add_of_le : ∀ {a b : α}, a ≤ b → ∃ (c : α), b = a + c
  • le_self_add : ∀ (a b : α), a ≤ a + b
  • 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

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

    Multiplication is associative

  • 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), CanonicallyOrderedCommSemiring.npow 0 x = 1

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

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

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

  • mul_comm : ∀ (a b : α), a * b = b * a

    Multiplication is commutative in a commutative multiplicative magma.

  • eq_zero_or_eq_zero_of_mul_eq_zero : ∀ {a b : α}, a * b = 0 → a = 0 ∨ b = 0

    No zero divisors.

Instances
    source
    instance CanonicallyOrderedCommSemiring.toNoZeroDivisors {α : Type u} [CanonicallyOrderedCommSemiring α] :
    NoZeroDivisors α
    Equations
    • ⋯ = ⋯
    source
    instance CanonicallyOrderedCommSemiring.toCovariantClassMulLE {α : Type u} [CanonicallyOrderedCommSemiring α] :
    CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1
    Equations
    • ⋯ = ⋯
    source
    instance CanonicallyOrderedCommSemiring.toOrderedCommMonoid {α : Type u} [CanonicallyOrderedCommSemiring α] :
    OrderedCommMonoid α
    Equations
    source
    instance CanonicallyOrderedCommSemiring.toOrderedCommSemiring {α : Type u} [CanonicallyOrderedCommSemiring α] :
    OrderedCommSemiring α
    Equations
    source
    @[simp]
    theorem CanonicallyOrderedCommSemiring.mul_pos {α : Type u} [CanonicallyOrderedCommSemiring α] {a : α} {b : α} :
    0 < a * b ↔ 0 < a ∧ 0 < b
    source
    theorem CanonicallyOrderedCommSemiring.mul_lt_mul_of_lt_of_lt {α : Type u} [CanonicallyOrderedCommSemiring α] {a : α} {b : α} {c : α} {d : α} [PosMulStrictMono α] (hab : a < b) (hcd : c < d) :
    a * c < b * d
    source
    theorem AddLECancellable.mul_tsub {α : Type u} [CanonicallyOrderedCommSemiring α] {a : α} {b : α} {c : α} [Sub α] [OrderedSub α] [IsTotal α fun (x x_1 : α) => x ≤ x_1] (h : AddLECancellable (a * c)) :
    a * (b - c) = a * b - a * c
    source
    theorem AddLECancellable.tsub_mul {α : Type u} [CanonicallyOrderedCommSemiring α] {a : α} {b : α} {c : α} [Sub α] [OrderedSub α] [IsTotal α fun (x x_1 : α) => x ≤ x_1] (h : AddLECancellable (b * c)) :
    (a - b) * c = a * c - b * c
    source
    theorem mul_tsub {α : Type u} [CanonicallyOrderedCommSemiring α] [Sub α] [OrderedSub α] [IsTotal α fun (x x_1 : α) => x ≤ x_1] [ContravariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (a : α) (b : α) (c : α) :
    a * (b - c) = a * b - a * c
    source
    theorem tsub_mul {α : Type u} [CanonicallyOrderedCommSemiring α] [Sub α] [OrderedSub α] [IsTotal α fun (x x_1 : α) => x ≤ x_1] [ContravariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (a : α) (b : α) (c : α) :
    (a - b) * c = a * c - b * c