Linear ordered (semi)fields

A linear ordered (semi)field is a (semi)field equipped with a linear order such that

• addition respects the order: a ≤ b → c + a ≤ c + b;
• multiplication of positives is positive: 0 < a → 0 < b → 0 < a * b;
• 0 < 1.

Main Definitions

• LinearOrderedSemifield: Typeclass for linear order semifields.
• LinearOrderedField: Typeclass for linear ordered fields.

class LinearOrderedSemifield (α : Type u_2) extends LinearOrderedCommSemiring , Inv , Div , NNRatCast : Type u_2

A linear ordered semifield is a field with a linear order respecting the operations.

• 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
• mul : α → α → α
• left_distrib : ∀ (a b c : α), a * (b + c) = a * b + a * c
• right_distrib : ∀ (a b c : α), (a + b) * c = a * c + b * c
• zero_mul : ∀ (a : α), 0 * a = 0
• mul_zero : ∀ (a : α), a * 0 = 0
• mul_assoc : ∀ (a b c : α), a * b * c = a * (b * c)
• one : α
• one_mul : ∀ (a : α), 1 * a = a
• mul_one : ∀ (a : α), a * 1 = a
• natCast : ℕ → α
• natCast_zero : NatCast.natCast 0 = 0
• natCast_succ : ∀ (n : ℕ), NatCast.natCast (n + 1) = NatCast.natCast n + 1
• npow : ℕ → α → α
• npow_zero : ∀ (x : α), Semiring.npow 0 x = 1
• npow_succ : ∀ (n : ℕ) (x : α), Semiring.npow (n + 1) x = Semiring.npow n x * x
• 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
• le_of_add_le_add_left : ∀ (a b c : α), a + b ≤ a + c → b ≤ c
• exists_pair_ne : ∃ (x : α), ∃ (y : α), x ≠ y
• zero_le_one : 0 ≤ 1
• mul_lt_mul_of_pos_left : ∀ (a b c : α), a < b → 0 < c → c * a < c * b
• mul_lt_mul_of_pos_right : ∀ (a b c : α), a < b → 0 < c → a * c < b * c
• mul_comm : ∀ (a b : α), a * b = b * a
• min : α → α → α
• max : α → α → α
• compare : α → α → Ordering
• le_total : ∀ (a b : α), a ≤ b ∨ b ≤ a
• decidableLE : DecidableRel fun (x1 x2 : α) => x1 ≤ x2
• decidableEq : DecidableEq α
• decidableLT : DecidableRel fun (x1 x2 : α) => x1 < x2
• min_def : ∀ (a b : α), min a b = if a ≤ b then a else b
• max_def : ∀ (a b : α), max a b = if a ≤ b then b else a
• compare_eq_compareOfLessAndEq : ∀ (a b : α), compare a b = compareOfLessAndEq a b
• inv : α → α
• div : α → α → α
• div_eq_mul_inv : ∀ (a b : α), a / b = a * b⁻¹

  a / b := a * b⁻¹

• zpow : ℤ → α → α

  The power operation: a ^ n = a * ··· * a; a ^ (-n) = a⁻¹ * ··· a⁻¹ (n times)

• zpow_zero' : ∀ (a : α), LinearOrderedSemifield.zpow 0 a = 1

  a ^ 0 = 1

• zpow_succ' : ∀ (n : ℕ) (a : α), LinearOrderedSemifield.zpow (Int.ofNat n.succ) a = LinearOrderedSemifield.zpow (Int.ofNat n) a * a

  a ^ (n + 1) = a ^ n * a

• zpow_neg' : ∀ (n : ℕ) (a : α), LinearOrderedSemifield.zpow (Int.negSucc n) a = (LinearOrderedSemifield.zpow (↑n.succ) a)⁻¹

  a ^ -(n + 1) = (a ^ (n + 1))⁻¹

• inv_zero : 0⁻¹ = 0

  The inverse of 0 in a group with zero is 0.

• mul_inv_cancel : ∀ (a : α), a ≠ 0 → a * a⁻¹ = 1

  Every nonzero element of a group with zero is invertible.

• nnratCast : ℚ≥0 → α
• nnratCast_def : ∀ (q : ℚ≥0), ↑q = ↑q.num / ↑q.den

  However NNRat.cast is defined, it must be propositionally equal to a / b.

  Do not use this lemma directly. Use NNRat.cast_def instead.

• nnqsmul : ℚ≥0 → α → α

  Scalar multiplication by a nonnegative rational number.

  Unless there is a risk of a Module ℚ≥0 _ instance diamond, write nnqsmul := _. This will set nnqsmul to (NNRat.cast · * ·) thanks to unification in the default proof of nnqsmul_def.

  Do not use directly. Instead use the • notation.

• nnqsmul_def : ∀ (q : ℚ≥0) (a : α), LinearOrderedSemifield.nnqsmul q a = ↑q * a

  However qsmul is defined, it must be propositionally equal to multiplication by Rat.cast.

  Do not use this lemma directly. Use NNRat.smul_def instead.

class LinearOrderedField (α : Type u_2) extends LinearOrderedCommRing , Inv , Div , NNRatCast , RatCast : Type u_2

A linear ordered field is a field with a linear order respecting the operations.

• 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
• mul : α → α → α
• left_distrib : ∀ (a b c : α), a * (b + c) = a * b + a * c
• right_distrib : ∀ (a b c : α), (a + b) * c = a * c + b * c
• zero_mul : ∀ (a : α), 0 * a = 0
• mul_zero : ∀ (a : α), a * 0 = 0
• mul_assoc : ∀ (a b c : α), a * b * c = a * (b * c)
• one : α
• one_mul : ∀ (a : α), 1 * a = a
• mul_one : ∀ (a : α), a * 1 = a
• natCast : ℕ → α
• natCast_zero : NatCast.natCast 0 = 0
• natCast_succ : ∀ (n : ℕ), NatCast.natCast (n + 1) = NatCast.natCast n + 1
• npow : ℕ → α → α
• npow_zero : ∀ (x : α), Semiring.npow 0 x = 1
• npow_succ : ∀ (n : ℕ) (x : α), Semiring.npow (n + 1) x = Semiring.npow n x * x
• neg : α → α
• sub : α → α → α
• sub_eq_add_neg : ∀ (a b : α), a - b = a + -b
• zsmul : ℤ → α → α
• zsmul_zero' : ∀ (a : α), Ring.zsmul 0 a = 0
• zsmul_succ' : ∀ (n : ℕ) (a : α), Ring.zsmul (Int.ofNat n.succ) a = Ring.zsmul (Int.ofNat n) a + a
• zsmul_neg' : ∀ (n : ℕ) (a : α), Ring.zsmul (Int.negSucc n) a = -Ring.zsmul (↑n.succ) a
• neg_add_cancel : ∀ (a : α), -a + a = 0
• intCast : ℤ → α
• intCast_ofNat : ∀ (n : ℕ), IntCast.intCast ↑n = ↑n
• intCast_negSucc : ∀ (n : ℕ), IntCast.intCast (Int.negSucc n) = -↑(n + 1)
• 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
• exists_pair_ne : ∃ (x : α), ∃ (y : α), x ≠ y
• zero_le_one : 0 ≤ 1
• mul_pos : ∀ (a b : α), 0 < a → 0 < b → 0 < a * b
• min : α → α → α
• max : α → α → α
• compare : α → α → Ordering
• le_total : ∀ (a b : α), a ≤ b ∨ b ≤ a
• decidableLE : DecidableRel fun (x1 x2 : α) => x1 ≤ x2
• decidableEq : DecidableEq α
• decidableLT : DecidableRel fun (x1 x2 : α) => x1 < x2
• min_def : ∀ (a b : α), min a b = if a ≤ b then a else b
• max_def : ∀ (a b : α), max a b = if a ≤ b then b else a
• compare_eq_compareOfLessAndEq : ∀ (a b : α), compare a b = compareOfLessAndEq a b
• mul_comm : ∀ (a b : α), a * b = b * a
• inv : α → α
• div : α → α → α
• div_eq_mul_inv : ∀ (a b : α), a / b = a * b⁻¹

  a / b := a * b⁻¹

• zpow : ℤ → α → α

  The power operation: a ^ n = a * ··· * a; a ^ (-n) = a⁻¹ * ··· a⁻¹ (n times)

• zpow_zero' : ∀ (a : α), LinearOrderedField.zpow 0 a = 1

  a ^ 0 = 1

• zpow_succ' : ∀ (n : ℕ) (a : α), LinearOrderedField.zpow (Int.ofNat n.succ) a = LinearOrderedField.zpow (Int.ofNat n) a * a

  a ^ (n + 1) = a ^ n * a

• zpow_neg' : ∀ (n : ℕ) (a : α), LinearOrderedField.zpow (Int.negSucc n) a = (LinearOrderedField.zpow (↑n.succ) a)⁻¹

  a ^ -(n + 1) = (a ^ (n + 1))⁻¹

• nnratCast : ℚ≥0 → α
• ratCast : ℚ → α
• mul_inv_cancel : ∀ (a : α), a ≠ 0 → a * a⁻¹ = 1

  For a nonzero a, a⁻¹ is a right multiplicative inverse.

• inv_zero : 0⁻¹ = 0

  The inverse of 0 is 0 by convention.

• nnratCast_def : ∀ (q : ℚ≥0), ↑q = ↑q.num / ↑q.den

  However NNRat.cast is defined, it must be equal to a / b.

  Do not use this lemma directly. Use NNRat.cast_def instead.

• nnqsmul : ℚ≥0 → α → α

  Scalar multiplication by a nonnegative rational number.

  Unless there is a risk of a Module ℚ≥0 _ instance diamond, write nnqsmul := _. This will set nnqsmul to (NNRat.cast · * ·) thanks to unification in the default proof of nnqsmul_def.

  Do not use directly. Instead use the • notation.

• nnqsmul_def : ∀ (q : ℚ≥0) (a : α), LinearOrderedField.nnqsmul q a = ↑q * a

  However qsmul is defined, it must be propositionally equal to multiplication by Rat.cast.

  Do not use this lemma directly. Use NNRat.smul_def instead.

• ratCast_def : ∀ (q : ℚ), ↑q = ↑q.num / ↑q.den

  However Rat.cast q is defined, it must be propositionally equal to q.num / q.den.

  Do not use this lemma directly. Use Rat.cast_def instead.

• qsmul : ℚ → α → α

  Scalar multiplication by a rational number.

  Unless there is a risk of a Module ℚ _ instance diamond, write qsmul := _. This will set qsmul to (Rat.cast · * ·) thanks to unification in the default proof of qsmul_def.

  Do not use directly. Instead use the • notation.

• qsmul_def : ∀ (a : ℚ) (x : α), LinearOrderedField.qsmul a x = ↑a * x

  However qsmul is defined, it must be propositionally equal to multiplication by Rat.cast.

  Do not use this lemma directly. Use Rat.cast_def instead.

@[instance 100]

instance LinearOrderedField.toLinearOrderedSemifield {α : Type u_1} [LinearOrderedField α] : LinearOrderedSemifield α

Equations

• LinearOrderedField.toLinearOrderedSemifield = LinearOrderedSemifield.mk ⋯ LinearOrderedField.zpow ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ LinearOrderedField.nnqsmul ⋯

theorem mul_inv_le_one {α : Type u_1} [LinearOrderedSemifield α] {a : α} : a * a⁻¹ ≤ 1

Equality holds when a ≠ 0. See mul_inv_cancel.

theorem inv_mul_le_one {α : Type u_1} [LinearOrderedSemifield α] {a : α} : a⁻¹ * a ≤ 1

Equality holds when a ≠ 0. See inv_mul_cancel.

theorem mul_inv_left_le {α : Type u_1} [LinearOrderedSemifield α] {a : α} {b : α} (hb : 0 ≤ b) : a * (a⁻¹ * b) ≤ b

Equality holds when a ≠ 0. See mul_inv_cancel_left.

theorem le_mul_inv_left {α : Type u_1} [LinearOrderedSemifield α] {a : α} {b : α} (hb : b ≤ 0) : b ≤ a * (a⁻¹ * b)

Equality holds when a ≠ 0. See mul_inv_cancel_left.

theorem inv_mul_left_le {α : Type u_1} [LinearOrderedSemifield α] {a : α} {b : α} (hb : 0 ≤ b) : a⁻¹ * (a * b) ≤ b

Equality holds when a ≠ 0. See inv_mul_cancel_left.

theorem le_inv_mul_left {α : Type u_1} [LinearOrderedSemifield α] {a : α} {b : α} (hb : b ≤ 0) : b ≤ a⁻¹ * (a * b)

Equality holds when a ≠ 0. See inv_mul_cancel_left.

theorem mul_inv_right_le {α : Type u_1} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 ≤ a) : a * b * b⁻¹ ≤ a

Equality holds when b ≠ 0. See mul_inv_cancel_right.

theorem le_mul_inv_right {α : Type u_1} [LinearOrderedSemifield α] {a : α} {b : α} (ha : a ≤ 0) : a ≤ a * b * b⁻¹

Equality holds when b ≠ 0. See mul_inv_cancel_right.

theorem inv_mul_right_le {α : Type u_1} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 ≤ a) : a * b⁻¹ * b ≤ a

Equality holds when b ≠ 0. See inv_mul_cancel_right.

theorem le_inv_mul_right {α : Type u_1} [LinearOrderedSemifield α] {a : α} {b : α} (ha : a ≤ 0) : a ≤ a * b⁻¹ * b

Equality holds when b ≠ 0. See inv_mul_cancel_right.