ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- add_le_mul
theorem add_le_mul [ZeroLEOneClass α] [NeZero (R := α) 1]
    [MulPosStrictMono α] [PosMulStrictMono α] [CovariantClass α α (· + ·) (· ≤ ·)]
    (a2 : 2 ≤ a) (b2 : 2 ≤ b) : a + b ≤ a * b 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a b : α} [ZeroLEOneClass α] [NeZero 1] [MulPosStrictMono α] [PosMulStrictMono α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a2 : 2 ≤ a) (b2 : 2 ≤ b)
Goal: a + b ≤ a * b
-- add_le_mul'
theorem add_le_mul' [ZeroLEOneClass α] [NeZero (R := α) 1]
    [MulPosStrictMono α] [PosMulStrictMono α] [CovariantClass α α (· + ·) (· ≤ ·)]
    (a2 : 2 ≤ a) (b2 : 2 ≤ b) : a + b ≤ b * a 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a b : α} [ZeroLEOneClass α] [NeZero 1] [MulPosStrictMono α] [PosMulStrictMono α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a2 : 2 ≤ a) (b2 : 2 ≤ b)
Goal: a + b ≤ b * a
-- mul_nonneg_iff_right_nonneg_of_pos
theorem mul_nonneg_iff_right_nonneg_of_pos [PosMulStrictMono α]
    (ha : 0 < a) : 0 ≤ a * b ↔ 0 ≤ b 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a b : α} [PosMulStrictMono α] (ha : 0 < a)
Goal: 0 ≤ a * b ↔ 0 ≤ b
-- mul_nonneg_iff_left_nonneg_of_pos
theorem mul_nonneg_iff_left_nonneg_of_pos [PosMulStrictMono α] [MulPosStrictMono α]
    (hb : 0 < b) : 0 ≤ a * b ↔ 0 ≤ a 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a b : α} [PosMulStrictMono α] [MulPosStrictMono α] (hb : 0 < b)
Goal: 0 ≤ a * b ↔ 0 ≤ a
-- nonpos_of_mul_nonneg_left
theorem nonpos_of_mul_nonneg_left [PosMulStrictMono α]
    (h : 0 ≤ a * b) (hb : b < 0) : a ≤ 0 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a b : α} [PosMulStrictMono α] (h : 0 ≤ a * b) (hb : b < 0)
Goal: a ≤ 0
-- nonpos_of_mul_nonneg_right
theorem nonpos_of_mul_nonneg_right [MulPosStrictMono α]
    (h : 0 ≤ a * b) (ha : a < 0) : b ≤ 0 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a b : α} [MulPosStrictMono α] (h : 0 ≤ a * b) (ha : a < 0)
Goal: b ≤ 0
-- Units.inv_pos
@[simp]
theorem Units.inv_pos
    [ZeroLEOneClass α] [NeZero (R := α) 1] [PosMulStrictMono α]
    {u : αˣ} : (0 : α) < ↑u⁻¹ ↔ (0 : α) < u 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] [ZeroLEOneClass α] [NeZero 1] [PosMulStrictMono α] {u : αˣ}
Goal: 0 < ↑u⁻¹ ↔ 0 < ↑u
-- Units.inv_neg
@[simp]
theorem Units.inv_neg
    [ZeroLEOneClass α] [NeZero (R := α) 1] [MulPosMono α] [PosMulMono α]
    {u : αˣ} : ↑u⁻¹ < (0 : α) ↔ ↑u < (0 : α) 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] [ZeroLEOneClass α] [NeZero 1] [MulPosMono α] [PosMulMono α] {u : αˣ}
Goal: ↑u⁻¹ < 0 ↔ ↑u < 0
-- cmp_mul_pos_left
theorem cmp_mul_pos_left [PosMulStrictMono α]
    (ha : 0 < a) (b c : α) : cmp (a * b) (a * c) = cmp b c 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a : α} [PosMulStrictMono α] (ha : 0 < a) (b c : α)
Goal: cmp (a * b) (a * c) = cmp b c
-- cmp_mul_pos_right
theorem cmp_mul_pos_right [MulPosStrictMono α]
    (ha : 0 < a) (b c : α) : cmp (b * a) (c * a) = cmp b c 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a : α} [MulPosStrictMono α] (ha : 0 < a) (b c : α)
Goal: cmp (b * a) (c * a) = cmp b c
-- mul_max_of_nonneg
theorem mul_max_of_nonneg [PosMulMono α]
    (b c : α) (ha : 0 ≤ a) : a * max b c = max (a * b) (a * c) 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a : α} [PosMulMono α] (b c : α) (ha : 0 ≤ a)
Goal: a * max b c = max (a * b) (a * c)
-- mul_min_of_nonneg
theorem mul_min_of_nonneg [PosMulMono α]
    (b c : α) (ha : 0 ≤ a) : a * min b c = min (a * b) (a * c) 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a : α} [PosMulMono α] (b c : α) (ha : 0 ≤ a)
Goal: a * min b c = min (a * b) (a * c)
-- max_mul_of_nonneg
theorem max_mul_of_nonneg [MulPosMono α]
    (a b : α) (hc : 0 ≤ c) : max a b * c = max (a * c) (b * c) 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {c : α} [MulPosMono α] (a b : α) (hc : 0 ≤ c)
Goal: max a b * c = max (a * c) (b * c)
-- min_mul_of_nonneg
theorem min_mul_of_nonneg [MulPosMono α]
    (a b : α) (hc : 0 ≤ c) : min a b * c = min (a * c) (b * c) 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {c : α} [MulPosMono α] (a b : α) (hc : 0 ≤ c)
Goal: min a b * c = min (a * c) (b * c)
-- le_of_mul_le_of_one_le
theorem le_of_mul_le_of_one_le
    [ZeroLEOneClass α] [NeZero (R := α) 1] [MulPosStrictMono α] [PosMulMono α]
    {a b c : α} (h : a * c ≤ b) (hb : 0 ≤ b) (hc : 1 ≤ c) : a ≤ b 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] [ZeroLEOneClass α] [NeZero 1] [MulPosStrictMono α] [PosMulMono α] {a b c : α} (h : a * c ≤ b) (hb : 0 ≤ b) (hc : 1 ≤ c)
Goal: a ≤ b
-- nonneg_le_nonneg_of_sq_le_sq
theorem nonneg_le_nonneg_of_sq_le_sq [PosMulStrictMono α] [MulPosMono α]
    {a b : α} (hb : 0 ≤ b) (h : a * a ≤ b * b) : a ≤ b 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] [PosMulStrictMono α] [MulPosMono α] {a b : α} (hb : 0 ≤ b) (h : a * a ≤ b * b)
Goal: a ≤ b
-- mul_self_le_mul_self_iff
theorem mul_self_le_mul_self_iff [PosMulStrictMono α] [MulPosMono α]
    {a b : α} (h1 : 0 ≤ a) (h2 : 0 ≤ b) : a ≤ b ↔ a * a ≤ b * b 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] [PosMulStrictMono α] [MulPosMono α] {a b : α} (h1 : 0 ≤ a) (h2 : 0 ≤ b)
Goal: a ≤ b ↔ a * a ≤ b * b
-- mul_self_lt_mul_self_iff
theorem mul_self_lt_mul_self_iff [PosMulStrictMono α] [MulPosMono α]
    {a b : α} (h1 : 0 ≤ a) (h2 : 0 ≤ b) : a < b ↔ a * a < b * b 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] [PosMulStrictMono α] [MulPosMono α] {a b : α} (h1 : 0 ≤ a) (h2 : 0 ≤ b)
Goal: a < b ↔ a * a < b * b
-- mul_self_inj
theorem mul_self_inj [PosMulStrictMono α] [MulPosMono α]
    {a b : α} (h1 : 0 ≤ a) (h2 : 0 ≤ b) : a * a = b * b ↔ a = b 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] [PosMulStrictMono α] [MulPosMono α] {a b : α} (h1 : 0 ≤ a) (h2 : 0 ≤ b)
Goal: a * a = b * b ↔ a = b
-- sign_cases_of_C_mul_pow_nonneg
lemma sign_cases_of_C_mul_pow_nonneg [PosMulStrictMono α]
    (h : ∀ n, 0 ≤ a * b ^ n) : a = 0 ∨ 0 < a ∧ 0 ≤ b 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a b : α} [PosMulStrictMono α] (h : ∀ (n : ℕ), 0 ≤ a * b ^ n)
Goal: a = 0 ∨ 0 < a ∧ 0 ≤ b
-- mul_pos_iff
theorem mul_pos_iff [ExistsAddOfLE α] [PosMulStrictMono α] [MulPosStrictMono α]
    [CovariantClass α α (· + ·) (· < ·)] [ContravariantClass α α (· + ·) (· < ·)] :
    0 < a * b ↔ 0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a b : α} [ExistsAddOfLE α] [PosMulStrictMono α] [MulPosStrictMono α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1]
Goal: 0 < a * b ↔ 0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0
-- mul_nonneg_iff
theorem mul_nonneg_iff [ExistsAddOfLE α] [MulPosStrictMono α] [PosMulStrictMono α]
    [ContravariantClass α α (· + ·) (· ≤ ·)] [CovariantClass α α (· + ·) (· ≤ ·)]:
    0 ≤ a * b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a b : α} [ExistsAddOfLE α] [MulPosStrictMono α] [PosMulStrictMono α] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: 0 ≤ a * b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0
-- mul_nonneg_of_three
theorem mul_nonneg_of_three [ExistsAddOfLE α] [MulPosStrictMono α] [PosMulStrictMono α]
    [CovariantClass α α (· + ·) (· ≤ ·)] [ContravariantClass α α (· + ·) (· ≤ ·)]
    (a b c : α) : 0 ≤ a * b ∨ 0 ≤ b * c ∨ 0 ≤ c * a 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] [ExistsAddOfLE α] [MulPosStrictMono α] [PosMulStrictMono α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a b c : α)
Goal: 0 ≤ a * b ∨ 0 ≤ b * c ∨ 0 ≤ c * a
-- mul_nonneg_iff_pos_imp_nonneg
lemma mul_nonneg_iff_pos_imp_nonneg [ExistsAddOfLE α] [PosMulStrictMono α] [MulPosStrictMono α]
    [CovariantClass α α (· + ·) (· ≤ ·)] [ContravariantClass α α (· + ·) (· ≤ ·)] :
    0 ≤ a * b ↔ (0 < a → 0 ≤ b) ∧ (0 < b → 0 ≤ a) 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a b : α} [ExistsAddOfLE α] [PosMulStrictMono α] [MulPosStrictMono α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: 0 ≤ a * b ↔ (0 < a → 0 ≤ b) ∧ (0 < b → 0 ≤ a)
-- mul_le_mul_left_of_neg
@[simp]
theorem mul_le_mul_left_of_neg [ExistsAddOfLE α] [PosMulStrictMono α]
    [CovariantClass α α (swap (· + ·)) (· ≤ ·)] [ContravariantClass α α (swap (· + ·)) (· ≤ ·)]
    {a b c : α} (h : c < 0) : c * a ≤ c * b ↔ b ≤ a 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] [ExistsAddOfLE α] [PosMulStrictMono α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α} (h : c < 0)
Goal: c * a ≤ c * b ↔ b ≤ a
-- mul_le_mul_right_of_neg
@[simp]
theorem mul_le_mul_right_of_neg [ExistsAddOfLE α] [MulPosStrictMono α]
    [CovariantClass α α (swap (· + ·)) (· ≤ ·)] [ContravariantClass α α (swap (· + ·)) (· ≤ ·)]
    {a b c : α} (h : c < 0) : a * c ≤ b * c ↔ b ≤ a 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] [ExistsAddOfLE α] [MulPosStrictMono α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α} (h : c < 0)
Goal: a * c ≤ b * c ↔ b ≤ a
-- mul_lt_mul_left_of_neg
@[simp]
theorem mul_lt_mul_left_of_neg [ExistsAddOfLE α] [PosMulStrictMono α]
    [CovariantClass α α (swap (· + ·)) (· < ·)] [ContravariantClass α α (swap (· + ·)) (· < ·)]
    {a b c : α} (h : c < 0) : c * a < c * b ↔ b < a 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] [ExistsAddOfLE α] [PosMulStrictMono α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] [ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : α} (h : c < 0)
Goal: c * a < c * b ↔ b < a
-- mul_lt_mul_right_of_neg
@[simp]
theorem mul_lt_mul_right_of_neg [ExistsAddOfLE α] [MulPosStrictMono α]
    [CovariantClass α α (swap (· + ·)) (· < ·)] [ContravariantClass α α (swap (· + ·)) (· < ·)]
    {a b c : α} (h : c < 0) : a * c < b * c ↔ b < a 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] [ExistsAddOfLE α] [MulPosStrictMono α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] [ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : α} (h : c < 0)
Goal: a * c < b * c ↔ b < a
-- lt_of_mul_lt_mul_of_nonpos_left
theorem lt_of_mul_lt_mul_of_nonpos_left [ExistsAddOfLE α] [PosMulMono α]
    [CovariantClass α α (swap (· + ·)) (· ≤ ·)] [ContravariantClass α α (swap (· + ·)) (· ≤ ·)]
    (h : c * a < c * b) (hc : c ≤ 0) : b < a 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a b c : α} [ExistsAddOfLE α] [PosMulMono α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : c * a < c * b) (hc : c ≤ 0)
Goal: b < a
-- lt_of_mul_lt_mul_of_nonpos_right
theorem lt_of_mul_lt_mul_of_nonpos_right [ExistsAddOfLE α] [MulPosMono α]
    [CovariantClass α α (swap (· + ·)) (· ≤ ·)] [ContravariantClass α α (swap (· + ·)) (· ≤ ·)]
    (h : a * c < b * c) (hc : c ≤ 0) : b < a 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a b c : α} [ExistsAddOfLE α] [MulPosMono α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : a * c < b * c) (hc : c ≤ 0)
Goal: b < a
-- cmp_mul_neg_left
theorem cmp_mul_neg_left [ExistsAddOfLE α] [PosMulStrictMono α]
    [ContravariantClass α α (swap (· + ·)) (· < ·)] [CovariantClass α α (swap (· + ·)) (· < ·)]
    {a : α} (ha : a < 0) (b c : α) : cmp (a * b) (a * c) = cmp c b 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] [ExistsAddOfLE α] [PosMulStrictMono α] [ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a : α} (ha : a < 0) (b c : α)
Goal: cmp (a * b) (a * c) = cmp c b
-- cmp_mul_neg_right
theorem cmp_mul_neg_right [ExistsAddOfLE α] [MulPosStrictMono α]
    [ContravariantClass α α (swap (· + ·)) (· < ·)] [CovariantClass α α (swap (· + ·)) (· < ·)]
    {a : α} (ha : a < 0) (b c : α) : cmp (b * a) (c * a) = cmp c b 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] [ExistsAddOfLE α] [MulPosStrictMono α] [ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a : α} (ha : a < 0) (b c : α)
Goal: cmp (b * a) (c * a) = cmp c b
-- mul_self_pos
@[simp]
theorem mul_self_pos [ExistsAddOfLE α] [PosMulStrictMono α] [MulPosStrictMono α]
    [CovariantClass α α (· + ·) (· < ·)] [ContravariantClass α α (· + ·) (· < ·)]
    {a : α} : 0 < a * a ↔ a ≠ 0 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] [ExistsAddOfLE α] [PosMulStrictMono α] [MulPosStrictMono α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a : α}
Goal: 0 < a * a ↔ a ≠ 0
-- nonneg_of_mul_nonpos_left
theorem nonneg_of_mul_nonpos_left [ExistsAddOfLE α] [MulPosStrictMono α]
    [CovariantClass α α (swap (· + ·)) (· ≤ ·)] [ContravariantClass α α (swap (· + ·)) (· ≤ ·)]
    {a b : α} (h : a * b ≤ 0) (hb : b < 0) : 0 ≤ a 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] [ExistsAddOfLE α] [MulPosStrictMono α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (h : a * b ≤ 0) (hb : b < 0)
Goal: 0 ≤ a
-- nonneg_of_mul_nonpos_right
theorem nonneg_of_mul_nonpos_right [ExistsAddOfLE α] [MulPosStrictMono α]
    [CovariantClass α α (swap (· + ·)) (· ≤ ·)] [ContravariantClass α α (swap (· + ·)) (· ≤ ·)]
    {a b : α} (h : a * b ≤ 0) (ha : a < 0) : 0 ≤ b 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] [ExistsAddOfLE α] [MulPosStrictMono α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (h : a * b ≤ 0) (ha : a < 0)
Goal: 0 ≤ b
-- pos_of_mul_neg_left
theorem pos_of_mul_neg_left [ExistsAddOfLE α] [MulPosMono α]
    [CovariantClass α α (swap (· + ·)) (· ≤ ·)] [ContravariantClass α α (swap (· + ·)) (· ≤ ·)]
    {a b : α} (h : a * b < 0) (hb : b ≤ 0) : 0 < a 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] [ExistsAddOfLE α] [MulPosMono α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (h : a * b < 0) (hb : b ≤ 0)
Goal: 0 < a
-- pos_of_mul_neg_right
theorem pos_of_mul_neg_right [ExistsAddOfLE α] [MulPosMono α]
    [CovariantClass α α (swap (· + ·)) (· ≤ ·)] [ContravariantClass α α (swap (· + ·)) (· ≤ ·)]
    {a b : α} (h : a * b < 0) (ha : a ≤ 0) : 0 < b 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] [ExistsAddOfLE α] [MulPosMono α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (h : a * b < 0) (ha : a ≤ 0)
Goal: 0 < b
-- neg_iff_pos_of_mul_neg
theorem neg_iff_pos_of_mul_neg [ExistsAddOfLE α] [PosMulMono α] [MulPosMono α]
    [CovariantClass α α (swap (· + ·)) (· ≤ ·)] [ContravariantClass α α (swap (· + ·)) (· ≤ ·)]
    (hab : a * b < 0) : a < 0 ↔ 0 < b 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a b : α} [ExistsAddOfLE α] [PosMulMono α] [MulPosMono α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (hab : a * b < 0)
Goal: a < 0 ↔ 0 < b
-- pos_iff_neg_of_mul_neg
theorem pos_iff_neg_of_mul_neg [ExistsAddOfLE α] [PosMulMono α] [MulPosMono α]
    [CovariantClass α α (swap (· + ·)) (· ≤ ·)] [ContravariantClass α α (swap (· + ·)) (· ≤ ·)]
    (hab : a * b < 0) : 0 < a ↔ b < 0 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a b : α} [ExistsAddOfLE α] [PosMulMono α] [MulPosMono α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (hab : a * b < 0)
Goal: 0 < a ↔ b < 0
-- sq_nonneg
lemma sq_nonneg [IsRightCancelAdd α]
    [ZeroLEOneClass α] [ExistsAddOfLE α] [PosMulMono α] [CovariantClass α α (· + ·) (· < ·)]
    (a : α) : 0 ≤ a ^ 2 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] [IsRightCancelAdd α] [ZeroLEOneClass α] [ExistsAddOfLE α] [PosMulMono α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] (a : α)
Goal: 0 ≤ a ^ 2
-- mul_self_nonneg
lemma mul_self_nonneg [IsRightCancelAdd α]
    [ZeroLEOneClass α] [ExistsAddOfLE α] [PosMulMono α] [CovariantClass α α (· + ·) (· < ·)]
    (a : α) : 0 ≤ a * a 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] [IsRightCancelAdd α] [ZeroLEOneClass α] [ExistsAddOfLE α] [PosMulMono α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] (a : α)
Goal: 0 ≤ a * a
-- mul_self_add_mul_self_eq_zero
lemma mul_self_add_mul_self_eq_zero [IsRightCancelAdd α] [NoZeroDivisors α]
    [ZeroLEOneClass α] [ExistsAddOfLE α] [PosMulMono α]
    [CovariantClass α α (· + ·) (· ≤ ·)] [CovariantClass α α (· + ·) (· < ·)] :
    a * a + b * b = 0 ↔ a = 0 ∧ b = 0 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a b : α} [IsRightCancelAdd α] [NoZeroDivisors α] [ZeroLEOneClass α] [ExistsAddOfLE α] [PosMulMono α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1]
Goal: a * a + b * b = 0 ↔ a = 0 ∧ b = 0
-- eq_zero_of_mul_self_add_mul_self_eq_zero
lemma eq_zero_of_mul_self_add_mul_self_eq_zero [IsRightCancelAdd α] [NoZeroDivisors α]
    [ZeroLEOneClass α] [ExistsAddOfLE α] [PosMulMono α]
    [CovariantClass α α (· + ·) (· ≤ ·)] [CovariantClass α α (· + ·) (· < ·)]
    (h : a * a + b * b = 0) : a = 0 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a b : α} [IsRightCancelAdd α] [NoZeroDivisors α] [ZeroLEOneClass α] [ExistsAddOfLE α] [PosMulMono α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] (h : a * a + b * b = 0)
Goal: a = 0
-- max_mul_mul_le_max_mul_max
lemma max_mul_mul_le_max_mul_max [PosMulMono α] [MulPosMono α] (b c : α) (ha : 0 ≤ a) (hd : 0 ≤ d) :
    max (a * b) (d * c) ≤ max a c * max d b 
Arguments: {α : Type u} [CommSemiring α] [LinearOrder α] {a d : α} [PosMulMono α] [MulPosMono α] (b c : α) (ha : 0 ≤ a) (hd : 0 ≤ d)
Goal: max (a * b) (d * c) ≤ max a c * max d b
-- two_mul_le_add_sq
lemma two_mul_le_add_sq [ExistsAddOfLE α] [MulPosStrictMono α]
    [ContravariantClass α α (· + ·) (· ≤ ·)] [CovariantClass α α (· + ·) (· ≤ ·)]
    (a b : α) : 2 * a * b ≤ a ^ 2 + b ^ 2 
Arguments: {α : Type u} [CommSemiring α] [LinearOrder α] [ExistsAddOfLE α] [MulPosStrictMono α] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: 2 * a * b ≤ a ^ 2 + b ^ 2
-- mul_neg_iff
lemma mul_neg_iff [PosMulStrictMono α] [MulPosStrictMono α]
    [ContravariantClass α α (· + ·) (· < ·)] [CovariantClass α α (· + ·) (· < ·)] :
    a * b < 0 ↔ 0 < a ∧ b < 0 ∨ a < 0 ∧ 0 < b 
Arguments: {α : Type u} [Ring α] [LinearOrder α] {a b : α} [PosMulStrictMono α] [MulPosStrictMono α] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1]
Goal: a * b < 0 ↔ 0 < a ∧ b < 0 ∨ a < 0 ∧ 0 < b
-- mul_nonpos_iff
lemma mul_nonpos_iff [MulPosStrictMono α] [PosMulStrictMono α]
    [ContravariantClass α α (· + ·) (· ≤ ·)] [CovariantClass α α (· + ·) (· ≤ ·)] :
    a * b ≤ 0 ↔ 0 ≤ a ∧ b ≤ 0 ∨ a ≤ 0 ∧ 0 ≤ b 
Arguments: {α : Type u} [Ring α] [LinearOrder α] {a b : α} [MulPosStrictMono α] [PosMulStrictMono α] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: a * b ≤ 0 ↔ 0 ≤ a ∧ b ≤ 0 ∨ a ≤ 0 ∧ 0 ≤ b
-- mul_nonneg_iff_neg_imp_nonpos
lemma mul_nonneg_iff_neg_imp_nonpos [PosMulStrictMono α] [MulPosStrictMono α]
    [CovariantClass α α (· + ·) (· ≤ ·)] [ContravariantClass α α (· + ·) (· ≤ ·)] :
    0 ≤ a * b ↔ (a < 0 → b ≤ 0) ∧ (b < 0 → a ≤ 0) 
Arguments: {α : Type u} [Ring α] [LinearOrder α] {a b : α} [PosMulStrictMono α] [MulPosStrictMono α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: 0 ≤ a * b ↔ (a < 0 → b ≤ 0) ∧ (b < 0 → a ≤ 0)
-- mul_nonpos_iff_pos_imp_nonpos
lemma mul_nonpos_iff_pos_imp_nonpos [PosMulStrictMono α] [MulPosStrictMono α]
    [CovariantClass α α (· + ·) (· ≤ ·)] [ContravariantClass α α (· + ·) (· ≤ ·)] :
    a * b ≤ 0 ↔ (0 < a → b ≤ 0) ∧ (b < 0 → 0 ≤ a) 
Arguments: {α : Type u} [Ring α] [LinearOrder α] {a b : α} [PosMulStrictMono α] [MulPosStrictMono α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: a * b ≤ 0 ↔ (0 < a → b ≤ 0) ∧ (b < 0 → 0 ≤ a)
-- mul_nonpos_iff_neg_imp_nonneg
lemma mul_nonpos_iff_neg_imp_nonneg [PosMulStrictMono α] [MulPosStrictMono α]
    [CovariantClass α α (· + ·) (· ≤ ·)] [ContravariantClass α α (· + ·) (· ≤ ·)] :
    a * b ≤ 0 ↔ (a < 0 → 0 ≤ b) ∧ (0 < b → a ≤ 0) 
Arguments: {α : Type u} [Ring α] [LinearOrder α] {a b : α} [PosMulStrictMono α] [MulPosStrictMono α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: a * b ≤ 0 ↔ (a < 0 → 0 ≤ b) ∧ (0 < b → a ≤ 0)
-- neg_one_lt_zero
lemma neg_one_lt_zero
    [ZeroLEOneClass α] [NeZero (R := α) 1] [CovariantClass α α (· + ·) (· < ·)] :
    -1 < (0 : α) 
Arguments: {α : Type u} [Ring α] [LinearOrder α] [ZeroLEOneClass α] [NeZero 1] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1]
Goal: -1 < 0
-- sub_one_lt
lemma sub_one_lt [ZeroLEOneClass α] [NeZero (R := α) 1]
    [CovariantClass α α (· + ·) (· < ·)]
    (a : α) : a - 1 < a 
Arguments: {α : Type u} [Ring α] [LinearOrder α] [ZeroLEOneClass α] [NeZero 1] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] (a : α)
Goal: a - 1 < a
-- mul_self_le_mul_self_of_le_of_neg_le
lemma mul_self_le_mul_self_of_le_of_neg_le
    [MulPosMono α] [PosMulMono α] [CovariantClass α α (· + ·) (· ≤ ·)]
    (h₁ : a ≤ b) (h₂ : -a ≤ b) : a * a ≤ b * b 
Arguments: {α : Type u} [Ring α] [LinearOrder α] {a b : α} [MulPosMono α] [PosMulMono α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h₁ : a ≤ b) (h₂ : -a ≤ b)
Goal: a * a ≤ b * b
-- HVertexOperator.ext
@[ext]
theorem ext (A B : HVertexOperator Γ R V W) (h : ∀ v : V, A v = B v) :
    A = B 
Arguments: {Γ : Type u_1} [PartialOrder Γ] {R : Type u_2} {V : Type u_3} {W : Type u_4} [CommRing R] [AddCommGroup V] [Module R V] [AddCommGroup W] [Module R W] (A B : HVertexOperator Γ R V W) (h : ∀ (v : V), A v = B v)
Goal: A = B
-- HVertexOperator.coeff_apply
@[simps]
def coeff (A : HVertexOperator Γ R V W) (n : Γ) : V →ₗ[R] W 
Arguments: {Γ : Type u_1} [PartialOrder Γ] {R : Type u_2} {V : Type u_3} {W : Type u_4} [CommRing R] [AddCommGroup V] [Module R V] [AddCommGroup W] [Module R W] (A : HVertexOperator Γ R V W) (n : Γ) (v : V)
Goal: (A.coeff n) v = ((HahnModule.of R).symm (A v)).coeff n
-- HVertexOperator.coeff_isPWOsupport
theorem coeff_isPWOsupport (A : HVertexOperator Γ R V W) (v : V) :
    ((of R).symm (A v)).coeff.support.IsPWO 
Arguments: {Γ : Type u_1} [PartialOrder Γ] {R : Type u_2} {V : Type u_3} {W : Type u_4} [CommRing R] [AddCommGroup V] [Module R V] [AddCommGroup W] [Module R W] (A : HVertexOperator Γ R V W) (v : V)
Goal: (Function.support ((HahnModule.of R).symm (A v)).coeff).IsPWO
-- HVertexOperator.coeff_inj
@[ext]
theorem coeff_inj : Function.Injective (coeff : HVertexOperator Γ R V W → Γ → (V →ₗ[R] W)) 
Arguments: {Γ : Type u_1} [PartialOrder Γ] {R : Type u_2} {V : Type u_3} {W : Type u_4} [CommRing R] [AddCommGroup V] [Module R V] [AddCommGroup W] [Module R W]
Goal: Function.Injective HVertexOperator.coeff