Canonically ordered rings and semirings.

@[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

@[instance 100]

instance CanonicallyOrderedAdd.toMulRightMono {R : Type u} [NonUnitalNonAssocSemiring R] [LE R] [CanonicallyOrderedAdd R] : MulRightMono 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] [IsReduced 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] [Std.Total 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] [Std.Total 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] [Std.Total 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] [Std.Total 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] [Std.Total 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] [Std.Total 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] [Std.Total 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] [Std.Total 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] [Std.Total fun (x1 x2 : R) => x1 ≤ x2] [AddLeftReflectLE R] (a : R) : a * a - 1 = (a + 1) * (a - 1)