Lemmas for linarith.

Those in the Linarith namespace should stay here.

Those outside the Linarith namespace may be deleted as they are ported to mathlib4.

theorem Linarith.lt_irrefl {α : Type u} [Preorder α] {a : α} : ¬a < a

theorem Linarith.eq_of_eq_of_eq {α : Type u_1} [OrderedSemiring α] {a : α} {b : α} (ha : a = 0) (hb : b = 0) : a + b = 0

theorem Linarith.le_of_eq_of_le {α : Type u_1} [OrderedSemiring α] {a : α} {b : α} (ha : a = 0) (hb : b ≤ 0) : a + b ≤ 0

theorem Linarith.lt_of_eq_of_lt {α : Type u_1} [OrderedSemiring α] {a : α} {b : α} (ha : a = 0) (hb : b < 0) : a + b < 0

theorem Linarith.le_of_le_of_eq {α : Type u_1} [OrderedSemiring α] {a : α} {b : α} (ha : a ≤ 0) (hb : b = 0) : a + b ≤ 0

theorem Linarith.lt_of_lt_of_eq {α : Type u_1} [OrderedSemiring α] {a : α} {b : α} (ha : a < 0) (hb : b = 0) : a + b < 0

theorem Linarith.mul_neg {α : Type u_1} [StrictOrderedRing α] {a : α} {b : α} (ha : a < 0) (hb : 0 < b) : b * a < 0

theorem Linarith.mul_nonpos {α : Type u_1} [OrderedRing α] {a : α} {b : α} (ha : a ≤ 0) (hb : 0 < b) : b * a ≤ 0

theorem Linarith.mul_eq {α : Type u_1} [OrderedSemiring α] {a : α} {b : α} (ha : a = 0) : 0 < b → b * a = 0

theorem Linarith.eq_of_not_lt_of_not_gt {α : Type u_1} [LinearOrder α] (a : α) (b : α) (h1 : ¬a < b) (h2 : ¬b < a) : a = b

theorem Linarith.mul_zero_eq {α : Type u_1} {R : α → α → Prop} [Semiring α] {a : α} {b : α} : R a 0 → b = 0 → a * b = 0

theorem Linarith.zero_mul_eq {α : Type u_1} {R : α → α → Prop} [Semiring α] {a : α} {b : α} (h : a = 0) : R b 0 → a * b = 0

theorem Linarith.natCast_nonneg (α : Type u) [OrderedSemiring α] (n : ℕ) : 0 ≤ ↑n

@[deprecated Linarith.natCast_nonneg]

theorem Linarith.nat_cast_nonneg (α : Type u) [OrderedSemiring α] (n : ℕ) : 0 ≤ ↑n

Alias of Linarith.natCast_nonneg.

theorem lt_zero_of_zero_gt {α : Type u_1} [Zero α] [LT α] {a : α} (h : 0 > a) : a < 0

theorem le_zero_of_zero_ge {α : Type u_1} [Zero α] [LE α] {a : α} (h : 0 ≥ a) : a ≤ 0