Canonically ordered rings and semirings.

• CanonicallyOrderedCommSemiring
  • CanonicallyOrderedAddCommMonoid & multiplication & * respects ≤ & no zero divisors
  • CommSemiring & a ≤ b ↔ ∃ c, b = a + c & no zero divisors

TODO

We're still missing some typeclasses, like

• CanonicallyOrderedSemiring They have yet to come up in practice.

class CanonicallyOrderedCommSemiring (α : Type u_1) extends CanonicallyOrderedAddCommMonoid , Mul , One , NatCast : Type u_1

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 = AddMonoid.nsmul n x + 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 = CanonicallyOrderedCommSemiring.npow n x * 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.

theorem CanonicallyOrderedCommSemiring.eq_zero_or_eq_zero_of_mul_eq_zero {α : Type u_1} [self : CanonicallyOrderedCommSemiring α] {a : α} {b : α} : a * b = 0 → a = 0 ∨ b = 0

No zero divisors.

theorem Odd.pos {α : Type u} [CanonicallyOrderedCommSemiring α] {a : α} [Nontrivial α] : Odd a → 0 < a

@[instance 100]

instance CanonicallyOrderedCommSemiring.toNoZeroDivisors {α : Type u} [CanonicallyOrderedCommSemiring α] : NoZeroDivisors α

Equations

• ⋯ = ⋯

@[instance 100]

instance CanonicallyOrderedCommSemiring.toCovariantClassMulLE {α : Type u} [CanonicallyOrderedCommSemiring α] : CovariantClass α α (fun (x1 x2 : α) => x1 * x2) fun (x1 x2 : α) => x1 ≤ x2

Equations

• ⋯ = ⋯

@[instance 100]

instance CanonicallyOrderedCommSemiring.toOrderedCommMonoid {α : Type u} [CanonicallyOrderedCommSemiring α] : OrderedCommMonoid α

Equations

• CanonicallyOrderedCommSemiring.toOrderedCommMonoid = OrderedCommMonoid.mk ⋯

@[instance 100]

instance CanonicallyOrderedCommSemiring.toOrderedCommSemiring {α : Type u} [CanonicallyOrderedCommSemiring α] : OrderedCommSemiring α

Equations

• CanonicallyOrderedCommSemiring.toOrderedCommSemiring = OrderedCommSemiring.mk ⋯

@[simp]

theorem CanonicallyOrderedCommSemiring.mul_pos {α : Type u} [CanonicallyOrderedCommSemiring α] {a : α} {b : α} : 0 < a * b ↔ 0 < a ∧ 0 < b

theorem CanonicallyOrderedCommSemiring.pow_pos {α : Type u} [CanonicallyOrderedCommSemiring α] {a : α} (ha : 0 < a) (n : ℕ) : 0 < a ^ n

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

theorem AddLECancellable.mul_tsub {α : Type u} [CanonicallyOrderedCommSemiring α] {a : α} {b : α} {c : α} [Sub α] [OrderedSub α] [IsTotal α fun (x1 x2 : α) => x1 ≤ x2] (h : AddLECancellable (a * c)) : a * (b - c) = a * b - a * c

theorem AddLECancellable.tsub_mul {α : Type u} [CanonicallyOrderedCommSemiring α] {a : α} {b : α} {c : α} [Sub α] [OrderedSub α] [IsTotal α fun (x1 x2 : α) => x1 ≤ x2] (h : AddLECancellable (b * c)) : (a - b) * c = a * c - b * c

theorem mul_tsub {α : Type u} [CanonicallyOrderedCommSemiring α] [Sub α] [OrderedSub α] [IsTotal α fun (x1 x2 : α) => x1 ≤ x2] [ContravariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] (a : α) (b : α) (c : α) : a * (b - c) = a * b - a * c

theorem tsub_mul {α : Type u} [CanonicallyOrderedCommSemiring α] [Sub α] [OrderedSub α] [IsTotal α fun (x1 x2 : α) => x1 ≤ x2] [ContravariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] (a : α) (b : α) (c : α) : (a - b) * c = a * c - b * c

theorem mul_tsub_one {α : Type u} [CanonicallyOrderedCommSemiring α] [Sub α] [OrderedSub α] [IsTotal α fun (x1 x2 : α) => x1 ≤ x2] [ContravariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] (a : α) (b : α) : a * (b - 1) = a * b - a

theorem tsub_one_mul {α : Type u} [CanonicallyOrderedCommSemiring α] [Sub α] [OrderedSub α] [IsTotal α fun (x1 x2 : α) => x1 ≤ x2] [ContravariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] (a : α) (b : α) : (a - 1) * b = a * b - b

theorem mul_self_tsub_mul_self {α : Type u} [CanonicallyOrderedCommSemiring α] [Sub α] [OrderedSub α] [IsTotal α fun (x1 x2 : α) => x1 ≤ x2] [ContravariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] (a : α) (b : α) : a * a - b * b = (a + b) * (a - b)

The tsub version of mul_self_sub_mul_self. Notably, this holds for Nat and NNReal.

theorem sq_tsub_sq {α : Type u} [CanonicallyOrderedCommSemiring α] [Sub α] [OrderedSub α] [IsTotal α fun (x1 x2 : α) => x1 ≤ x2] [ContravariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] (a : α) (b : α) : a ^ 2 - b ^ 2 = (a + b) * (a - b)

The tsub version of sq_sub_sq. Notably, this holds for Nat and NNReal.

theorem mul_self_tsub_one {α : Type u} [CanonicallyOrderedCommSemiring α] [Sub α] [OrderedSub α] [IsTotal α fun (x1 x2 : α) => x1 ≤ x2] [ContravariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] (a : α) : a * a - 1 = (a + 1) * (a - 1)