• 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), conditionally_complete_linear_ordered_field.nsmul 0 x = 0) . "try_refl_tac"
• nsmul_succ' : (∀ (n : ℕ) (x : α), conditionally_complete_linear_ordered_field.nsmul n.succ x = x + conditionally_complete_linear_ordered_field.nsmul n x) . "try_refl_tac"
• neg : α → α
• sub : α → α → α
• sub_eq_add_neg : (∀ (a b : α), a - b = a + -b) . "try_refl_tac"
• zsmul : ℤ → α → α
• zsmul_zero' : (∀ (a : α), conditionally_complete_linear_ordered_field.zsmul 0 a = 0) . "try_refl_tac"
• zsmul_succ' : (∀ (n : ℕ) (a : α), conditionally_complete_linear_ordered_field.zsmul (int.of_nat n.succ) a = a + conditionally_complete_linear_ordered_field.zsmul (int.of_nat n) a) . "try_refl_tac"
• zsmul_neg' : (∀ (n : ℕ) (a : α), conditionally_complete_linear_ordered_field.zsmul -[1+ n] a = -conditionally_complete_linear_ordered_field.zsmul ↑(n.succ) a) . "try_refl_tac"
• add_left_neg : ∀ (a : α), -a + a = 0
• add_comm : ∀ (a b : α), a + b = b + a
• int_cast : ℤ → α
• nat_cast : ℕ → α
• one : α
• nat_cast_zero : conditionally_complete_linear_ordered_field.nat_cast 0 = 0 . "control_laws_tac"
• nat_cast_succ : (∀ (n : ℕ), conditionally_complete_linear_ordered_field.nat_cast (n + 1) = conditionally_complete_linear_ordered_field.nat_cast n + 1) . "control_laws_tac"
• int_cast_of_nat : (∀ (n : ℕ), conditionally_complete_linear_ordered_field.int_cast ↑n = ↑n) . "control_laws_tac"
• int_cast_neg_succ_of_nat : (∀ (n : ℕ), conditionally_complete_linear_ordered_field.int_cast (-↑(n + 1)) = -↑(n + 1)) . "control_laws_tac"
• mul : α → α → α
• mul_assoc : ∀ (a b c : α), a * b * c = a * (b * c)
• one_mul : ∀ (a : α), 1 * a = a
• mul_one : ∀ (a : α), a * 1 = a
• npow : ℕ → α → α
• npow_zero' : (∀ (x : α), conditionally_complete_linear_ordered_field.npow 0 x = 1) . "try_refl_tac"
• npow_succ' : (∀ (n : ℕ) (x : α), conditionally_complete_linear_ordered_field.npow n.succ x = x * conditionally_complete_linear_ordered_field.npow n x) . "try_refl_tac"
• 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
• 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
• exists_pair_ne : ∃ (x y : α), x ≠ y
• zero_le_one : 0 ≤ 1
• mul_pos : ∀ (a b : α), 0 < a → 0 < b → 0 < a * b
• 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
• sup : α → α → α
• max_def : conditionally_complete_linear_ordered_field.sup = max_default . "reflexivity"
• inf : α → α → α
• min_def : conditionally_complete_linear_ordered_field.inf = min_default . "reflexivity"
• 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) . "try_refl_tac"
• zpow : ℤ → α → α
• zpow_zero' : (∀ (a : α), conditionally_complete_linear_ordered_field.zpow 0 a = 1) . "try_refl_tac"
• zpow_succ' : (∀ (n : ℕ) (a : α), conditionally_complete_linear_ordered_field.zpow (int.of_nat n.succ) a = a * conditionally_complete_linear_ordered_field.zpow (int.of_nat n) a) . "try_refl_tac"
• zpow_neg' : (∀ (n : ℕ) (a : α), conditionally_complete_linear_ordered_field.zpow -[1+ n] a = (conditionally_complete_linear_ordered_field.zpow ↑(n.succ) a)⁻¹) . "try_refl_tac"
• rat_cast : ℚ → α
• mul_inv_cancel : ∀ {a : α}, a ≠ 0 → a * a⁻¹ = 1
• inv_zero : 0⁻¹ = 0
• rat_cast_mk : (∀ (a : ℤ) (b : ℕ) (h1 : 0 < b) (h2 : a.nat_abs.coprime b), conditionally_complete_linear_ordered_field.rat_cast {num := a, denom := b, pos := h1, cop := h2} = ↑a * (↑b)⁻¹) . "try_refl_tac"
• qsmul : ℚ → α → α
• qsmul_eq_mul' : (∀ (a : ℚ) (x : α), conditionally_complete_linear_ordered_field.qsmul a x = conditionally_complete_linear_ordered_field.rat_cast a * x) . "try_refl_tac"
• le_sup_left : ∀ (a b : α), a ≤ a ⊔ b
• le_sup_right : ∀ (a b : α), b ≤ a ⊔ b
• sup_le : ∀ (a b c : α), a ≤ c → b ≤ c → a ⊔ b ≤ c
• inf_le_left : ∀ (a b : α), a ⊓ b ≤ a
• inf_le_right : ∀ (a b : α), a ⊓ b ≤ b
• le_inf : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c
• Sup : set α → α
• Inf : set α → α
• le_cSup : ∀ (s : set α) (a : α), bdd_above s → a ∈ s → a ≤ conditionally_complete_linear_ordered_field.Sup s
• cSup_le : ∀ (s : set α) (a : α), s.nonempty → a ∈ upper_bounds s → conditionally_complete_linear_ordered_field.Sup s ≤ a
• cInf_le : ∀ (s : set α) (a : α), bdd_below s → a ∈ s → conditionally_complete_linear_ordered_field.Inf s ≤ a
• le_cInf : ∀ (s : set α) (a : α), s.nonempty → a ∈ lower_bounds s → a ≤ conditionally_complete_linear_ordered_field.Inf s

A field which is both linearly ordered and conditionally complete with respect to the order. This axiomatizes the reals.