• 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), canonically_linear_ordered_semifield.nsmul 0 x = 0) . "try_refl_tac"
• nsmul_succ' : (∀ (n : ℕ) (x : α), canonically_linear_ordered_semifield.nsmul n.succ x = x + canonically_linear_ordered_semifield.nsmul n x) . "try_refl_tac"
• add_comm : ∀ (a b : α), a + b = b + a
• 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) . "order_laws_tac"
• 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
• bot : α
• bot_le : ∀ (x : α), ⊥ ≤ x
• exists_add_of_le : ∀ {a b : α}, a ≤ b → (∃ (c : α), b = a + c)
• le_self_add : ∀ (a b : α), a ≤ a + b
• 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
• nat_cast : ℕ → α
• nat_cast_zero : canonically_linear_ordered_semifield.nat_cast 0 = 0 . "control_laws_tac"
• nat_cast_succ : (∀ (n : ℕ), canonically_linear_ordered_semifield.nat_cast (n + 1) = canonically_linear_ordered_semifield.nat_cast n + 1) . "control_laws_tac"
• npow : ℕ → α → α
• npow_zero' : (∀ (x : α), canonically_linear_ordered_semifield.npow 0 x = 1) . "try_refl_tac"
• npow_succ' : (∀ (n : ℕ) (x : α), canonically_linear_ordered_semifield.npow n.succ x = x * canonically_linear_ordered_semifield.npow n x) . "try_refl_tac"
• mul_comm : ∀ (a b : α), a * b = b * a
• eq_zero_or_eq_zero_of_mul_eq_zero : ∀ {a b : α}, a * b = 0 → a = 0 ∨ b = 0
• le_of_add_le_add_left : ∀ (a b c : α), a + b ≤ a + c → b ≤ c
• exists_pair_ne : ∃ (x 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
• le_total : ∀ (a b : α), a ≤ b ∨ b ≤ a
• decidable_le : decidable_rel has_le.le
• decidable_eq : decidable_eq α
• decidable_lt : decidable_rel has_lt.lt
• max : α → α → α
• max_def : canonically_linear_ordered_semifield.max = max_default . "reflexivity"
• min : α → α → α
• min_def : canonically_linear_ordered_semifield.min = min_default . "reflexivity"
• inv : α → α
• div : α → α → α
• div_eq_mul_inv : (∀ (a b : α), a / b = a * b⁻¹) . "try_refl_tac"
• zpow : ℤ → α → α
• zpow_zero' : (∀ (a : α), canonically_linear_ordered_semifield.zpow 0 a = 1) . "try_refl_tac"
• zpow_succ' : (∀ (n : ℕ) (a : α), canonically_linear_ordered_semifield.zpow (int.of_nat n.succ) a = a * canonically_linear_ordered_semifield.zpow (int.of_nat n) a) . "try_refl_tac"
• zpow_neg' : (∀ (n : ℕ) (a : α), canonically_linear_ordered_semifield.zpow -[1+ n] a = (canonically_linear_ordered_semifield.zpow ↑(n.succ) a)⁻¹) . "try_refl_tac"
• inv_zero : 0⁻¹ = 0
• mul_inv_cancel : ∀ (a : α), a ≠ 0 → a * a⁻¹ = 1

A canonically linear ordered field is a linear ordered field in which a ≤ b iff there exists c with b = a + c.

Instances of this typeclass

• nonneg.canonically_linear_ordered_semifield
• nnreal.canonically_linear_ordered_semifield
• nnrat.canonically_linear_ordered_semifield

Instances of other typeclasses for canonically_linear_ordered_semifield

• canonically_linear_ordered_semifield.has_sizeof_inst