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.

source

@[deprecated "Use `[Semifield K] [LinearOrder K] [IsStrictOrderedRing K]` instead." (since := "2025-04-10")]

structure LinearOrderedSemifield (K : Type u_1) extends LinearOrderedCommSemiring K, Semifield K :

Type u_1

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

add : K → K → K
add_assoc (a b c : K) : a + b + c = a + (b + c)
zero : K
zero_add (a : K) : 0 + a = a
add_zero (a : K) : a + 0 = a
nsmul : ℕ → K → K
nsmul_zero (x : K) : AddMonoid.nsmul 0 x = 0
nsmul_succ (n : ℕ) (x : K) : AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
add_comm (a b : K) : a + b = b + a
mul : K → K → K
left_distrib (a b c : K) : a * (b + c) = a * b + a * c
right_distrib (a b c : K) : (a + b) * c = a * c + b * c
zero_mul (a : K) : 0 * a = 0
mul_zero (a : K) : a * 0 = 0
mul_assoc (a b c : K) : a * b * c = a * (b * c)
one : K
one_mul (a : K) : 1 * a = a
mul_one (a : K) : a * 1 = a
natCast : ℕ → K
natCast_zero : ↑0 = 0
natCast_succ (n : ℕ) : ↑(n + 1) = ↑n + 1
npow : ℕ → K → K
npow_zero (x : K) : Semiring.npow 0 x = 1
npow_succ (n : ℕ) (x : K) : Semiring.npow (n + 1) x = Semiring.npow n x * x
le : K → K → Prop
lt : K → K → Prop
le_refl (a : K) : a ≤ a
le_trans (a b c : K) : a ≤ b → b ≤ c → a ≤ c
lt_iff_le_not_le (a b : K) : a < b ↔ a ≤ b ∧ ¬b ≤ a
le_antisymm (a b : K) : a ≤ b → b ≤ a → a = b
add_le_add_left (a b : K) : a ≤ b → ∀ (c : K), c + a ≤ c + b
le_of_add_le_add_left (a b c : K) : a + b ≤ a + c → b ≤ c
exists_pair_ne : ∃ (x : K), ∃ (y : K), x ≠ y
zero_le_one : 0 ≤ 1
mul_lt_mul_of_pos_left (a b c : K) : a < b → 0 < c → c * a < c * b
mul_lt_mul_of_pos_right (a b c : K) : a < b → 0 < c → a * c < b * c
mul_comm (a b : K) : a * b = b * a
min : K → K → K
max : K → K → K
compare : K → K → Ordering
le_total (a b : K) : a ≤ b ∨ b ≤ a
toDecidableLE : DecidableLE K
toDecidableEq : DecidableEq K
toDecidableLT : DecidableLT K
min_def (a b : K) : a ⊓ b = if a ≤ b then a else b
max_def (a b : K) : a ⊔ b = if a ≤ b then b else a
compare_eq_compareOfLessAndEq (a b : K) : compare a b = compareOfLessAndEq a b
inv : K → K
div : K → K → K
div_eq_mul_inv (a b : K) : a / b = a * b⁻¹
zpow : ℤ → K → K
zpow_zero' (a : K) : self.zpow 0 a = 1
zpow_succ' (n : ℕ) (a : K) : self.zpow (↑n.succ) a = self.zpow (↑n) a * a
zpow_neg' (n : ℕ) (a : K) : self.zpow (Int.negSucc n) a = (self.zpow (↑n.succ) a)⁻¹
inv_zero : 0⁻¹ = 0
mul_inv_cancel (a : K) : a ≠ 0 → a * a⁻¹ = 1
nnratCast : ℚ≥0 → K
nnratCast_def (q : ℚ≥0) : ↑q = ↑q.num / ↑q.den
nnqsmul : ℚ≥0 → K → K
nnqsmul_def (q : ℚ≥0) (a : K) : self.nnqsmul q a = ↑q * a

Instances For

source

@[deprecated "Use `[Field K] [LinearOrder K] [IsStrictOrderedRing K]` instead." (since := "2025-04-10")]

structure LinearOrderedField (K : Type u_1) extends LinearOrderedCommRing K, Field K :

Type u_1

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

add : K → K → K
add_assoc (a b c : K) : a + b + c = a + (b + c)
zero : K
zero_add (a : K) : 0 + a = a
add_zero (a : K) : a + 0 = a
nsmul : ℕ → K → K
nsmul_zero (x : K) : AddMonoid.nsmul 0 x = 0
nsmul_succ (n : ℕ) (x : K) : AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
add_comm (a b : K) : a + b = b + a
mul : K → K → K
left_distrib (a b c : K) : a * (b + c) = a * b + a * c
right_distrib (a b c : K) : (a + b) * c = a * c + b * c
zero_mul (a : K) : 0 * a = 0
mul_zero (a : K) : a * 0 = 0
mul_assoc (a b c : K) : a * b * c = a * (b * c)
one : K
one_mul (a : K) : 1 * a = a
mul_one (a : K) : a * 1 = a
natCast : ℕ → K
natCast_zero : ↑0 = 0
natCast_succ (n : ℕ) : ↑(n + 1) = ↑n + 1
npow : ℕ → K → K
npow_zero (x : K) : Semiring.npow 0 x = 1
npow_succ (n : ℕ) (x : K) : Semiring.npow (n + 1) x = Semiring.npow n x * x
neg : K → K
sub : K → K → K
sub_eq_add_neg (a b : K) : a - b = a + -b
zsmul : ℤ → K → K
zsmul_zero' (a : K) : Ring.zsmul 0 a = 0
zsmul_succ' (n : ℕ) (a : K) : Ring.zsmul (↑n.succ) a = Ring.zsmul (↑n) a + a
zsmul_neg' (n : ℕ) (a : K) : Ring.zsmul (Int.negSucc n) a = -Ring.zsmul (↑n.succ) a
neg_add_cancel (a : K) : -a + a = 0
intCast : ℤ → K
intCast_ofNat (n : ℕ) : IntCast.intCast ↑n = ↑n
intCast_negSucc (n : ℕ) : IntCast.intCast (Int.negSucc n) = -↑(n + 1)
le : K → K → Prop
lt : K → K → Prop
le_refl (a : K) : a ≤ a
le_trans (a b c : K) : a ≤ b → b ≤ c → a ≤ c
lt_iff_le_not_le (a b : K) : a < b ↔ a ≤ b ∧ ¬b ≤ a
le_antisymm (a b : K) : a ≤ b → b ≤ a → a = b
add_le_add_left (a b : K) : a ≤ b → ∀ (c : K), c + a ≤ c + b
exists_pair_ne : ∃ (x : K), ∃ (y : K), x ≠ y
zero_le_one : 0 ≤ 1
mul_pos (a b : K) : 0 < a → 0 < b → 0 < a * b
min : K → K → K
max : K → K → K
compare : K → K → Ordering
le_total (a b : K) : a ≤ b ∨ b ≤ a
toDecidableLE : DecidableLE K
toDecidableEq : DecidableEq K
toDecidableLT : DecidableLT K
min_def (a b : K) : a ⊓ b = if a ≤ b then a else b
max_def (a b : K) : a ⊔ b = if a ≤ b then b else a
compare_eq_compareOfLessAndEq (a b : K) : compare a b = compareOfLessAndEq a b
mul_comm (a b : K) : a * b = b * a
inv : K → K
div : K → K → K
div_eq_mul_inv (a b : K) : a / b = a * b⁻¹
zpow : ℤ → K → K
zpow_zero' (a : K) : self.zpow 0 a = 1
zpow_succ' (n : ℕ) (a : K) : self.zpow (↑n.succ) a = self.zpow (↑n) a * a
zpow_neg' (n : ℕ) (a : K) : self.zpow (Int.negSucc n) a = (self.zpow (↑n.succ) a)⁻¹
nnratCast : ℚ≥0 → K
ratCast : ℚ → K
mul_inv_cancel (a : K) : a ≠ 0 → a * a⁻¹ = 1
inv_zero : 0⁻¹ = 0
nnratCast_def (q : ℚ≥0) : ↑q = ↑q.num / ↑q.den
nnqsmul : ℚ≥0 → K → K
nnqsmul_def (q : ℚ≥0) (a : K) : self.nnqsmul q a = ↑q * a
ratCast_def (q : ℚ) : ↑q = ↑q.num / ↑q.den
qsmul : ℚ → K → K
qsmul_def (a : ℚ) (x : K) : self.qsmul a x = ↑a * x

Instances For

Documentation

Mathlib.Algebra.Order.Field.Defs

Linear ordered (semi)fields #

Main Definitions #