Canonically ordered rings and semirings.

@[deprecated "Use `[OrderedCommSemiring R] [CanonicallyOrderedAdd R] [NoZeroDivisors R]` instead." (since := "2025-01-13")]

structure CanonicallyOrderedCommSemiring (R : Type u_1) extends CanonicallyOrderedAddCommMonoid R, CommSemiring R : 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 : 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
• le : R → R → Prop
• lt : R → R → Prop
• le_refl (a : R) : a ≤ a
• le_trans (a b c : R) : a ≤ b → b ≤ c → a ≤ c
• lt_iff_le_not_le (a b : R) : a < b ↔ a ≤ b ∧ ¬b ≤ a
• le_antisymm (a b : R) : a ≤ b → b ≤ a → a = b
• add_le_add_left (a b : R) : a ≤ b → ∀ (c : R), c + a ≤ c + b
• bot : R
• bot_le (a : R) : ⊥ ≤ a
• exists_add_of_le {a b : R} : a ≤ b → ∃ (c : R), b = a + c
• le_self_add (a b : R) : a ≤ a + b
• 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 : ↑0 = 0
• natCast_succ (n : ℕ) : ↑(n + 1) = ↑n + 1
• npow : ℕ → R → R
• npow_zero (x : R) : self.npow 0 x = 1
• npow_succ (n : ℕ) (x : R) : self.npow (n + 1) x = self.npow n x * x
• mul_comm (a b : R) : a * b = b * a
• eq_zero_or_eq_zero_of_mul_eq_zero {a b : R} : a * b = 0 → a = 0 ∨ b = 0

  No zero divisors.

@[instance 10]

instance CanonicallyOrderedAdd.toZeroLEOneClass {R : Type u} [AddZeroClass R] [One R] [LE R] [CanonicallyOrderedAdd R] : ZeroLEOneClass R

theorem Odd.pos {R : Type u} [Semiring R] [PartialOrder R] [CanonicallyOrderedAdd R] [Nontrivial R] {a : R} : Odd a → 0 < a

@[instance 100]

instance CanonicallyOrderedAdd.toMulLeftMono {R : Type u} [NonUnitalNonAssocSemiring R] [LE R] [CanonicallyOrderedAdd R] : MulLeftMono R

theorem CanonicallyOrderedAdd.toIsOrderedMonoid {R : Type u} [CommSemiring R] [PartialOrder R] [CanonicallyOrderedAdd R] : IsOrderedMonoid R

theorem CanonicallyOrderedAdd.toIsOrderedRing {R : Type u} [CommSemiring R] [PartialOrder R] [CanonicallyOrderedAdd R] : IsOrderedRing R

@[simp]

theorem CanonicallyOrderedAdd.mul_pos {R : Type u} [CommSemiring R] [PartialOrder R] [CanonicallyOrderedAdd R] [NoZeroDivisors R] {a b : R} : 0 < a * b ↔ 0 < a ∧ 0 < b

theorem CanonicallyOrderedAdd.pow_pos {R : Type u} [CommSemiring R] [PartialOrder R] [CanonicallyOrderedAdd R] [NoZeroDivisors R] {a : R} (ha : 0 < a) (n : ℕ) : 0 < a ^ n

theorem CanonicallyOrderedAdd.mul_lt_mul_of_lt_of_lt {R : Type u} [CommSemiring R] [PartialOrder R] [CanonicallyOrderedAdd R] [PosMulStrictMono R] {a b c d : R} (hab : a < b) (hcd : c < d) : a * c < b * d

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

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

theorem mul_tsub {R : Type u} [NonUnitalNonAssocSemiring R] [PartialOrder R] [CanonicallyOrderedAdd R] [Sub R] [OrderedSub R] [IsTotal R fun (x1 x2 : R) => x1 ≤ x2] [AddLeftReflectLE R] (a b c : R) : a * (b - c) = a * b - a * c

theorem tsub_mul {R : Type u} [NonUnitalNonAssocSemiring R] [PartialOrder R] [CanonicallyOrderedAdd R] [Sub R] [OrderedSub R] [IsTotal R fun (x1 x2 : R) => x1 ≤ x2] [AddLeftReflectLE R] [MulRightMono R] (a b c : R) : (a - b) * c = a * c - b * c

theorem mul_tsub_one {R : Type u} [NonAssocSemiring R] [PartialOrder R] [CanonicallyOrderedAdd R] [Sub R] [OrderedSub R] [IsTotal R fun (x1 x2 : R) => x1 ≤ x2] [AddLeftReflectLE R] (a b : R) : a * (b - 1) = a * b - a

theorem tsub_one_mul {R : Type u} [NonAssocSemiring R] [PartialOrder R] [CanonicallyOrderedAdd R] [Sub R] [OrderedSub R] [IsTotal R fun (x1 x2 : R) => x1 ≤ x2] [MulRightMono R] [AddLeftReflectLE R] (a b : R) : (a - 1) * b = a * b - b

theorem mul_self_tsub_mul_self {R : Type u} [CommSemiring R] [PartialOrder R] [CanonicallyOrderedAdd R] [Sub R] [OrderedSub R] [IsTotal R fun (x1 x2 : R) => x1 ≤ x2] [AddLeftReflectLE R] (a b : R) : 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 {R : Type u} [CommSemiring R] [PartialOrder R] [CanonicallyOrderedAdd R] [Sub R] [OrderedSub R] [IsTotal R fun (x1 x2 : R) => x1 ≤ x2] [AddLeftReflectLE R] (a b : R) : 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 {R : Type u} [CommSemiring R] [PartialOrder R] [CanonicallyOrderedAdd R] [Sub R] [OrderedSub R] [IsTotal R fun (x1 x2 : R) => x1 ≤ x2] [AddLeftReflectLE R] (a : R) : a * a - 1 = (a + 1) * (a - 1)