ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- pow_lt_pow_right_of_lt_one
theorem pow_lt_pow_right_of_lt_one (h₀ : 0 < a) (h₁ : a < 1) (hmn : m < n) : a ^ n < a ^ m 
Arguments: {R : Type u_3} [StrictOrderedSemiring R] {a : R} {n m : ℕ} (h₀ : 0 < a) (h₁ : a < 1) (hmn : m < n)
Goal: a ^ n < a ^ m
-- pow_lt_self_of_lt_one
theorem pow_lt_self_of_lt_one (h₀ : 0 < a) (h₁ : a < 1) (hn : 1 < n) : a ^ n < a 
Arguments: {R : Type u_3} [StrictOrderedSemiring R] {a : R} {n : ℕ} (h₀ : 0 < a) (h₁ : a < 1) (hn : 1 < n)
Goal: a ^ n < a
-- sq_pos_of_pos
theorem sq_pos_of_pos (ha : 0 < a) : 0 < a ^ 2 
Arguments: {R : Type u_3} [StrictOrderedSemiring R] {a : R} (ha : 0 < a)
Goal: 0 < a ^ 2
-- sq_pos_of_neg
lemma sq_pos_of_neg (ha : a < 0) : 0 < a ^ 2 
Arguments: {R : Type u_3} [StrictOrderedRing R] {a : R} (ha : a < 0)
Goal: 0 < a ^ 2
-- pow_le_pow_iff_left
lemma pow_le_pow_iff_left (ha : 0 ≤ a) (hb : 0 ≤ b) (hn : n ≠ 0) : a ^ n ≤ b ^ n ↔ a ≤ b 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a b : R} {n : ℕ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hn : n ≠ 0)
Goal: a ^ n ≤ b ^ n ↔ a ≤ b
-- pow_lt_pow_iff_left
lemma pow_lt_pow_iff_left (ha : 0 ≤ a) (hb : 0 ≤ b) (hn : n ≠ 0) : a ^ n < b ^ n ↔ a < b 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a b : R} {n : ℕ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hn : n ≠ 0)
Goal: a ^ n < b ^ n ↔ a < b
-- pow_left_inj
@[simp]
lemma pow_left_inj (ha : 0 ≤ a) (hb : 0 ≤ b) (hn : n ≠ 0) : a ^ n = b ^ n ↔ a = b 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a b : R} {n : ℕ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hn : n ≠ 0)
Goal: a ^ n = b ^ n ↔ a = b
-- pow_right_injective
lemma pow_right_injective (ha₀ : 0 < a) (ha₁ : a ≠ 1) : Injective (a ^ ·) 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a : R} (ha₀ : 0 < a) (ha₁ : a ≠ 1)
Goal: Function.Injective fun x => a ^ x
-- pow_right_inj
@[simp]
lemma pow_right_inj (ha₀ : 0 < a) (ha₁ : a ≠ 1) : a ^ m = a ^ n ↔ m = n 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a : R} {m n : ℕ} (ha₀ : 0 < a) (ha₁ : a ≠ 1)
Goal: a ^ m = a ^ n ↔ m = n
-- pow_le_one_iff_of_nonneg
theorem pow_le_one_iff_of_nonneg (ha : 0 ≤ a) (hn : n ≠ 0) : a ^ n ≤ 1 ↔ a ≤ 1 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a : R} {n : ℕ} (ha : 0 ≤ a) (hn : n ≠ 0)
Goal: a ^ n ≤ 1 ↔ a ≤ 1
-- one_le_pow_iff_of_nonneg
theorem one_le_pow_iff_of_nonneg (ha : 0 ≤ a) (hn : n ≠ 0) : 1 ≤ a ^ n ↔ 1 ≤ a 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a : R} {n : ℕ} (ha : 0 ≤ a) (hn : n ≠ 0)
Goal: 1 ≤ a ^ n ↔ 1 ≤ a
-- pow_lt_one_iff_of_nonneg
theorem pow_lt_one_iff_of_nonneg (ha : 0 ≤ a) (hn : n ≠ 0) : a ^ n < 1 ↔ a < 1 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a : R} {n : ℕ} (ha : 0 ≤ a) (hn : n ≠ 0)
Goal: a ^ n < 1 ↔ a < 1
-- one_lt_pow_iff_of_nonneg
theorem one_lt_pow_iff_of_nonneg (ha : 0 ≤ a) (hn : n ≠ 0) : 1 < a ^ n ↔ 1 < a 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a : R} {n : ℕ} (ha : 0 ≤ a) (hn : n ≠ 0)
Goal: 1 < a ^ n ↔ 1 < a
-- pow_eq_one_iff_of_nonneg
lemma pow_eq_one_iff_of_nonneg (ha : 0 ≤ a) (hn : n ≠ 0) : a ^ n = 1 ↔ a = 1 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a : R} {n : ℕ} (ha : 0 ≤ a) (hn : n ≠ 0)
Goal: a ^ n = 1 ↔ a = 1
-- sq_le_one_iff
theorem sq_le_one_iff {a : R} (ha : 0 ≤ a) : a ^ 2 ≤ 1 ↔ a ≤ 1 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a : R} (ha : 0 ≤ a)
Goal: a ^ 2 ≤ 1 ↔ a ≤ 1
-- sq_lt_one_iff
theorem sq_lt_one_iff {a : R} (ha : 0 ≤ a) : a ^ 2 < 1 ↔ a < 1 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a : R} (ha : 0 ≤ a)
Goal: a ^ 2 < 1 ↔ a < 1
-- one_le_sq_iff
theorem one_le_sq_iff {a : R} (ha : 0 ≤ a) : 1 ≤ a ^ 2 ↔ 1 ≤ a 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a : R} (ha : 0 ≤ a)
Goal: 1 ≤ a ^ 2 ↔ 1 ≤ a
-- one_lt_sq_iff
theorem one_lt_sq_iff {a : R} (ha : 0 ≤ a) : 1 < a ^ 2 ↔ 1 < a 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a : R} (ha : 0 ≤ a)
Goal: 1 < a ^ 2 ↔ 1 < a
-- lt_of_pow_lt_pow_left
theorem lt_of_pow_lt_pow_left (n : ℕ) (hb : 0 ≤ b) (h : a ^ n < b ^ n) : a < b 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a b : R} (n : ℕ) (hb : 0 ≤ b) (h : a ^ n < b ^ n)
Goal: a < b
-- le_of_pow_le_pow_left
theorem le_of_pow_le_pow_left (hn : n ≠ 0) (hb : 0 ≤ b) (h : a ^ n ≤ b ^ n) : a ≤ b 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a b : R} {n : ℕ} (hn : n ≠ 0) (hb : 0 ≤ b) (h : a ^ n ≤ b ^ n)
Goal: a ≤ b
-- sq_eq_sq
@[simp]
theorem sq_eq_sq {a b : R} (ha : 0 ≤ a) (hb : 0 ≤ b) : a ^ 2 = b ^ 2 ↔ a = b 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a b : R} (ha : 0 ≤ a) (hb : 0 ≤ b)
Goal: a ^ 2 = b ^ 2 ↔ a = b
-- lt_of_mul_self_lt_mul_self
theorem lt_of_mul_self_lt_mul_self (hb : 0 ≤ b) : a * a < b * b → a < b 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a b : R} (hb : 0 ≤ b)
Goal: a * a < b * b → a < b
-- add_sq_le
lemma add_sq_le : (a + b) ^ 2 ≤ 2 * (a ^ 2 + b ^ 2) 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a b : R} [ExistsAddOfLE R]
Goal: (a + b) ^ 2 ≤ 2 * (a ^ 2 + b ^ 2)
-- add_pow_le
lemma add_pow_le (ha : 0 ≤ a) (hb : 0 ≤ b) : ∀ n, (a + b) ^ n ≤ 2 ^ (n - 1) * (a ^ n + b ^ n)
  
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a b : R} [ExistsAddOfLE R] (ha : 0 ≤ a) (hb : 0 ≤ b) (n : ℕ)
Goal: (a + b) ^ n ≤ 2 ^ (n - 1) * (a ^ n + b ^ n)
-- Even.add_pow_le
protected lemma Even.add_pow_le (hn : Even n) :
    (a + b) ^ n ≤ 2 ^ (n - 1) * (a ^ n + b ^ n) 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a b : R} {n : ℕ} [ExistsAddOfLE R] (hn : Even n)
Goal: (a + b) ^ n ≤ 2 ^ (n - 1) * (a ^ n + b ^ n)
-- Even.pow_nonneg
lemma Even.pow_nonneg (hn : Even n) (a : R) : 0 ≤ a ^ n 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {n : ℕ} [ExistsAddOfLE R] (hn : Even n) (a : R)
Goal: 0 ≤ a ^ n
-- Even.pow_pos
lemma Even.pow_pos (hn : Even n) (ha : a ≠ 0) : 0 < a ^ n 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a : R} {n : ℕ} [ExistsAddOfLE R] (hn : Even n) (ha : a ≠ 0)
Goal: 0 < a ^ n
-- Even.pow_pos_iff
lemma Even.pow_pos_iff (hn : Even n) (h₀ : n ≠ 0) : 0 < a ^ n ↔ a ≠ 0 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a : R} {n : ℕ} [ExistsAddOfLE R] (hn : Even n) (h₀ : n ≠ 0)
Goal: 0 < a ^ n ↔ a ≠ 0
-- Odd.pow_neg_iff
lemma Odd.pow_neg_iff (hn : Odd n) : a ^ n < 0 ↔ a < 0 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a : R} {n : ℕ} [ExistsAddOfLE R] (hn : Odd n)
Goal: a ^ n < 0 ↔ a < 0
-- Odd.pow_nonneg_iff
lemma Odd.pow_nonneg_iff (hn : Odd n) : 0 ≤ a ^ n ↔ 0 ≤ a 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a : R} {n : ℕ} [ExistsAddOfLE R] (hn : Odd n)
Goal: 0 ≤ a ^ n ↔ 0 ≤ a
-- Odd.pow_nonpos_iff
lemma Odd.pow_nonpos_iff (hn : Odd n) : a ^ n ≤ 0 ↔ a ≤ 0 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a : R} {n : ℕ} [ExistsAddOfLE R] (hn : Odd n)
Goal: a ^ n ≤ 0 ↔ a ≤ 0
-- Odd.pow_pos_iff
lemma Odd.pow_pos_iff (hn : Odd n) : 0 < a ^ n ↔ 0 < a 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a : R} {n : ℕ} [ExistsAddOfLE R] (hn : Odd n)
Goal: 0 < a ^ n ↔ 0 < a
-- Odd.strictMono_pow
lemma Odd.strictMono_pow (hn : Odd n) : StrictMono fun a : R => a ^ n 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {n : ℕ} [ExistsAddOfLE R] (hn : Odd n)
Goal: StrictMono fun a => a ^ n
-- sq_pos_iff
lemma sq_pos_iff {a : R} : 0 < a ^ 2 ↔ a ≠ 0 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] [ExistsAddOfLE R] {a : R}
Goal: 0 < a ^ 2 ↔ a ≠ 0
-- pow_four_le_pow_two_of_pow_two_le
lemma pow_four_le_pow_two_of_pow_two_le (h : a ^ 2 ≤ b) : a ^ 4 ≤ b ^ 2 
Arguments: {R : Type u_3} [LinearOrderedSemiring R] {a b : R} [ExistsAddOfLE R] (h : a ^ 2 ≤ b)
Goal: a ^ 4 ≤ b ^ 2
-- mabs_zpow
@[to_additive] lemma mabs_zpow (n : ℤ) (a : α) : |a ^ n|ₘ = |a|ₘ ^ |n| 
Arguments: {α : Type u_1} [LinearOrderedCommGroup α] (n : ℤ) (a : α)
Goal: mabs (a ^ n) = mabs a ^ |n|
-- abs_zsmul
@[to_additive] lemma mabs_zpow (n : ℤ) (a : α) : |a ^ n|ₘ = |a|ₘ ^ |n| 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (n : ℤ) (a : α)
Goal: |n • a| = |n| • |a|
-- odd_abs
lemma odd_abs [LinearOrder α] [Ring α] {a : α} : Odd (abs a) ↔ Odd a 
Arguments: {α : Type u_1} [LinearOrder α] [Ring α] {a : α}
Goal: Odd |a| ↔ Odd a
-- abs_one
@[simp] lemma abs_one : |(1 : α)| = 1 
Arguments: {α : Type u_1} [LinearOrderedRing α]
Goal: |1| = 1
-- abs_two
lemma abs_two : |(2 : α)| = 2 
Arguments: {α : Type u_1} [LinearOrderedRing α]
Goal: |2| = 2
-- abs_mul
lemma abs_mul (a b : α) : |a * b| = |a| * |b| 
Arguments: {α : Type u_1} [LinearOrderedRing α] (a b : α)
Goal: |a * b| = |a| * |b|
-- abs_pow
@[simp]
lemma abs_pow (a : α) (n : ℕ) : |a ^ n| = |a| ^ n 
Arguments: {α : Type u_1} [LinearOrderedRing α] (a : α) (n : ℕ)
Goal: |a ^ n| = |a| ^ n
-- pow_abs
lemma pow_abs (a : α) (n : ℕ) : |a| ^ n = |a ^ n| 
Arguments: {α : Type u_1} [LinearOrderedRing α] (a : α) (n : ℕ)
Goal: |a| ^ n = |a ^ n|
-- Even.pow_abs
lemma Even.pow_abs (hn : Even n) (a : α) : |a| ^ n = a ^ n 
Arguments: {α : Type u_1} [LinearOrderedRing α] {n : ℕ} (hn : Even n) (a : α)
Goal: |a| ^ n = a ^ n
-- abs_neg_one_pow
lemma abs_neg_one_pow (n : ℕ) : |(-1 : α) ^ n| = 1 
Arguments: {α : Type u_1} [LinearOrderedRing α] (n : ℕ)
Goal: |(-1) ^ n| = 1
-- abs_pow_eq_one
lemma abs_pow_eq_one (a : α) (h : n ≠ 0) : |a ^ n| = 1 ↔ |a| = 1 
Arguments: {α : Type u_1} [LinearOrderedRing α] {n : ℕ} (a : α) (h : n ≠ 0)
Goal: |a ^ n| = 1 ↔ |a| = 1
-- abs_mul_abs_self
@[simp] lemma abs_mul_abs_self (a : α) : |a| * |a| = a * a 
Arguments: {α : Type u_1} [LinearOrderedRing α] (a : α)
Goal: |a| * |a| = a * a
-- abs_mul_self
@[simp]
lemma abs_mul_self (a : α) : |a * a| = a * a 
Arguments: {α : Type u_1} [LinearOrderedRing α] (a : α)
Goal: |a * a| = a * a
-- abs_eq_iff_mul_self_eq
lemma abs_eq_iff_mul_self_eq : |a| = |b| ↔ a * a = b * b 
Arguments: {α : Type u_1} [LinearOrderedRing α] {a b : α}
Goal: |a| = |b| ↔ a * a = b * b
-- abs_lt_iff_mul_self_lt
lemma abs_lt_iff_mul_self_lt : |a| < |b| ↔ a * a < b * b 
Arguments: {α : Type u_1} [LinearOrderedRing α] {a b : α}
Goal: |a| < |b| ↔ a * a < b * b
-- abs_le_iff_mul_self_le
lemma abs_le_iff_mul_self_le : |a| ≤ |b| ↔ a * a ≤ b * b 
Arguments: {α : Type u_1} [LinearOrderedRing α] {a b : α}
Goal: |a| ≤ |b| ↔ a * a ≤ b * b
-- abs_le_one_iff_mul_self_le_one
lemma abs_le_one_iff_mul_self_le_one : |a| ≤ 1 ↔ a * a ≤ 1 
Arguments: {α : Type u_1} [LinearOrderedRing α] {a : α}
Goal: |a| ≤ 1 ↔ a * a ≤ 1
-- sq_abs
@[simp] lemma sq_abs (a : α) : |a| ^ 2 = a ^ 2 
Arguments: {α : Type u_1} [LinearOrderedRing α] (a : α)
Goal: |a| ^ 2 = a ^ 2
-- abs_sq
lemma abs_sq (x : α) : |x ^ 2| = x ^ 2 
Arguments: {α : Type u_1} [LinearOrderedRing α] (x : α)
Goal: |x ^ 2| = x ^ 2
-- sq_lt_sq
lemma sq_lt_sq : a ^ 2 < b ^ 2 ↔ |a| < |b| 
Arguments: {α : Type u_1} [LinearOrderedRing α] {a b : α}
Goal: a ^ 2 < b ^ 2 ↔ |a| < |b|
-- sq_lt_sq'
lemma sq_lt_sq' (h1 : -b < a) (h2 : a < b) : a ^ 2 < b ^ 2 
Arguments: {α : Type u_1} [LinearOrderedRing α] {a b : α} (h1 : -b < a) (h2 : a < b)
Goal: a ^ 2 < b ^ 2
-- sq_le_sq
lemma sq_le_sq : a ^ 2 ≤ b ^ 2 ↔ |a| ≤ |b| 
Arguments: {α : Type u_1} [LinearOrderedRing α] {a b : α}
Goal: a ^ 2 ≤ b ^ 2 ↔ |a| ≤ |b|
-- sq_le_sq'
lemma sq_le_sq' (h1 : -b ≤ a) (h2 : a ≤ b) : a ^ 2 ≤ b ^ 2 
Arguments: {α : Type u_1} [LinearOrderedRing α] {a b : α} (h1 : -b ≤ a) (h2 : a ≤ b)
Goal: a ^ 2 ≤ b ^ 2
-- abs_lt_of_sq_lt_sq
lemma abs_lt_of_sq_lt_sq (h : a ^ 2 < b ^ 2) (hb : 0 ≤ b) : |a| < b 
Arguments: {α : Type u_1} [LinearOrderedRing α] {a b : α} (h : a ^ 2 < b ^ 2) (hb : 0 ≤ b)
Goal: |a| < b
-- abs_lt_of_sq_lt_sq'
lemma abs_lt_of_sq_lt_sq' (h : a ^ 2 < b ^ 2) (hb : 0 ≤ b) : -b < a ∧ a < b 
Arguments: {α : Type u_1} [LinearOrderedRing α] {a b : α} (h : a ^ 2 < b ^ 2) (hb : 0 ≤ b)
Goal: -b < a ∧ a < b
-- abs_le_of_sq_le_sq
lemma abs_le_of_sq_le_sq (h : a ^ 2 ≤ b ^ 2) (hb : 0 ≤ b) : |a| ≤ b 
Arguments: {α : Type u_1} [LinearOrderedRing α] {a b : α} (h : a ^ 2 ≤ b ^ 2) (hb : 0 ≤ b)
Goal: |a| ≤ b
-- abs_le_of_sq_le_sq'
lemma abs_le_of_sq_le_sq' (h : a ^ 2 ≤ b ^ 2) (hb : 0 ≤ b) : -b ≤ a ∧ a ≤ b 
Arguments: {α : Type u_1} [LinearOrderedRing α] {a b : α} (h : a ^ 2 ≤ b ^ 2) (hb : 0 ≤ b)
Goal: -b ≤ a ∧ a ≤ b
-- sq_eq_sq_iff_abs_eq_abs
lemma sq_eq_sq_iff_abs_eq_abs (a b : α) : a ^ 2 = b ^ 2 ↔ |a| = |b| 
Arguments: {α : Type u_1} [LinearOrderedRing α] (a b : α)
Goal: a ^ 2 = b ^ 2 ↔ |a| = |b|
-- sq_le_one_iff_abs_le_one
@[simp] lemma sq_le_one_iff_abs_le_one (a : α) : a ^ 2 ≤ 1 ↔ |a| ≤ 1 
Arguments: {α : Type u_1} [LinearOrderedRing α] (a : α)
Goal: a ^ 2 ≤ 1 ↔ |a| ≤ 1
-- sq_lt_one_iff_abs_lt_one
@[simp] lemma sq_lt_one_iff_abs_lt_one (a : α) : a ^ 2 < 1 ↔ |a| < 1 
Arguments: {α : Type u_1} [LinearOrderedRing α] (a : α)
Goal: a ^ 2 < 1 ↔ |a| < 1
-- one_le_sq_iff_one_le_abs
@[simp] lemma one_le_sq_iff_one_le_abs (a : α) : 1 ≤ a ^ 2 ↔ 1 ≤ |a| 
Arguments: {α : Type u_1} [LinearOrderedRing α] (a : α)
Goal: 1 ≤ a ^ 2 ↔ 1 ≤ |a|
-- one_lt_sq_iff_one_lt_abs
@[simp] lemma one_lt_sq_iff_one_lt_abs (a : α) : 1 < a ^ 2 ↔ 1 < |a| 
Arguments: {α : Type u_1} [LinearOrderedRing α] (a : α)
Goal: 1 < a ^ 2 ↔ 1 < |a|
-- exists_abs_lt
lemma exists_abs_lt {α : Type*} [LinearOrderedRing α] (a : α) : ∃ b > 0, |a| < b 
Arguments: {α : Type u_2} [LinearOrderedRing α] (a : α)
Goal: ∃ b > 0, |a| < b
-- abs_sub_sq
theorem abs_sub_sq (a b : α) : |a - b| * |a - b| = a * a + b * b - (1 + 1) * a * b 
Arguments: {α : Type u_1} [LinearOrderedCommRing α] (a b : α)
Goal: |a - b| * |a - b| = a * a + b * b - (1 + 1) * a * b
-- abs_dvd
@[simp]
theorem abs_dvd (a b : α) : |a| ∣ b ↔ a ∣ b 
Arguments: {α : Type u_1} [Ring α] [LinearOrder α] (a b : α)
Goal: |a| ∣ b ↔ a ∣ b
-- abs_dvd_self
theorem abs_dvd_self (a : α) : |a| ∣ a 
Arguments: {α : Type u_1} [Ring α] [LinearOrder α] (a : α)
Goal: |a| ∣ a
-- dvd_abs
@[simp]
theorem dvd_abs (a b : α) : a ∣ |b| ↔ a ∣ b 
Arguments: {α : Type u_1} [Ring α] [LinearOrder α] (a b : α)
Goal: a ∣ |b| ↔ a ∣ b
-- self_dvd_abs
theorem self_dvd_abs (a : α) : a ∣ |a| 
Arguments: {α : Type u_1} [Ring α] [LinearOrder α] (a : α)
Goal: a ∣ |a|
-- abs_dvd_abs
theorem abs_dvd_abs (a b : α) : |a| ∣ |b| ↔ a ∣ b 
Arguments: {α : Type u_1} [Ring α] [LinearOrder α] (a b : α)
Goal: |a| ∣ |b| ↔ a ∣ b
-- pow_eq_pow_iff_of_ne_zero
lemma pow_eq_pow_iff_of_ne_zero (hn : n ≠ 0) : a ^ n = b ^ n ↔ a = b ∨ a = -b ∧ Even n 
Arguments: {R : Type u_2} [LinearOrderedRing R] {a b : R} {n : ℕ} (hn : n ≠ 0)
Goal: a ^ n = b ^ n ↔ a = b ∨ a = -b ∧ Even n
-- pow_eq_pow_iff_cases
lemma pow_eq_pow_iff_cases : a ^ n = b ^ n ↔ n = 0 ∨ a = b ∨ a = -b ∧ Even n 
Arguments: {R : Type u_2} [LinearOrderedRing R] {a b : R} {n : ℕ}
Goal: a ^ n = b ^ n ↔ n = 0 ∨ a = b ∨ a = -b ∧ Even n
-- pow_eq_one_iff_of_ne_zero
lemma pow_eq_one_iff_of_ne_zero (hn : n ≠ 0) : a ^ n = 1 ↔ a = 1 ∨ a = -1 ∧ Even n 
Arguments: {R : Type u_2} [LinearOrderedRing R] {a : R} {n : ℕ} (hn : n ≠ 0)
Goal: a ^ n = 1 ↔ a = 1 ∨ a = -1 ∧ Even n
-- pow_eq_one_iff_cases
lemma pow_eq_one_iff_cases : a ^ n = 1 ↔ n = 0 ∨ a = 1 ∨ a = -1 ∧ Even n 
Arguments: {R : Type u_2} [LinearOrderedRing R] {a : R} {n : ℕ}
Goal: a ^ n = 1 ↔ n = 0 ∨ a = 1 ∨ a = -1 ∧ Even n
-- pow_eq_neg_pow_iff
lemma pow_eq_neg_pow_iff (hb : b ≠ 0) : a ^ n = -b ^ n ↔ a = -b ∧ Odd n 
Arguments: {R : Type u_2} [LinearOrderedRing R] {a b : R} {n : ℕ} (hb : b ≠ 0)
Goal: a ^ n = -b ^ n ↔ a = -b ∧ Odd n
-- pow_eq_neg_one_iff
lemma pow_eq_neg_one_iff : a ^ n = -1 ↔ a = -1 ∧ Odd n 
Arguments: {R : Type u_2} [LinearOrderedRing R] {a : R} {n : ℕ}
Goal: a ^ n = -1 ↔ a = -1 ∧ Odd n
-- Odd.mod_even_iff
lemma Odd.mod_even_iff (ha : Even a) : Odd (n % a) ↔ Odd n 
Arguments: {n a : ℕ} (ha : Even a)
Goal: Odd (n % a) ↔ Odd n
-- Even.mod_even_iff
lemma Even.mod_even_iff (ha : Even a) : Even (n % a) ↔ Even n 
Arguments: {n a : ℕ} (ha : Even a)
Goal: Even (n % a) ↔ Even n
-- Odd.mod_even
lemma Odd.mod_even (hn : Odd n) (ha : Even a) : Odd (n % a) 
Arguments: {n a : ℕ} (hn : Odd n) (ha : Even a)
Goal: Odd (n % a)
-- Even.mod_even
lemma Even.mod_even (hn : Even n) (ha : Even a) : Even (n % a) 
Arguments: {n a : ℕ} (hn : Even n) (ha : Even a)
Goal: Even (n % a)
-- Odd.of_dvd_nat
lemma Odd.of_dvd_nat (hn : Odd n) (hm : m ∣ n) : Odd m 
Arguments: {m n : ℕ} (hn : Odd n) (hm : m ∣ n)
Goal: Odd m
-- Odd.ne_two_of_dvd_nat
lemma Odd.ne_two_of_dvd_nat {m n : ℕ} (hn : Odd n) (hm : m ∣ n) : m ≠ 2 
Arguments: {m n : ℕ} (hn : Odd n) (hm : m ∣ n)
Goal: m ≠ 2
-- Set.Icc.coe_zero
@[simp, norm_cast]
theorem coe_zero : ↑(0 : Icc (0 : α) 1) = (0 : α) 
Arguments: {α : Type u_1} [OrderedSemiring α]
Goal: ↑0 = 0
-- Set.Icc.coe_one
@[simp, norm_cast]
theorem coe_one : ↑(1 : Icc (0 : α) 1) = (1 : α) 
Arguments: {α : Type u_1} [OrderedSemiring α]
Goal: ↑1 = 1
-- Set.Icc.mk_zero
@[simp]
theorem mk_zero (h : (0 : α) ∈ Icc (0 : α) 1) : (⟨0, h⟩ : Icc (0 : α) 1) = 0 
Arguments: {α : Type u_1} [OrderedSemiring α] (h : 0 ∈ Set.Icc 0 1)
Goal: ⟨0, h⟩ = 0
-- Set.Icc.mk_one
@[simp]
theorem mk_one (h : (1 : α) ∈ Icc (0 : α) 1) : (⟨1, h⟩ : Icc (0 : α) 1) = 1 
Arguments: {α : Type u_1} [OrderedSemiring α] (h : 1 ∈ Set.Icc 0 1)
Goal: ⟨1, h⟩ = 1
-- Set.Icc.coe_eq_zero
@[simp, norm_cast]
theorem coe_eq_zero {x : Icc (0 : α) 1} : (x : α) = 0 ↔ x = 0 
Arguments: {α : Type u_1} [OrderedSemiring α] {x : ↑(Set.Icc 0 1)}
Goal: ↑x = 0 ↔ x = 0
-- Set.Icc.coe_ne_zero
theorem coe_ne_zero {x : Icc (0 : α) 1} : (x : α) ≠ 0 ↔ x ≠ 0 
Arguments: {α : Type u_1} [OrderedSemiring α] {x : ↑(Set.Icc 0 1)}
Goal: ↑x ≠ 0 ↔ x ≠ 0
-- Set.Icc.coe_eq_one
@[simp, norm_cast]
theorem coe_eq_one {x : Icc (0 : α) 1} : (x : α) = 1 ↔ x = 1 
Arguments: {α : Type u_1} [OrderedSemiring α] {x : ↑(Set.Icc 0 1)}
Goal: ↑x = 1 ↔ x = 1
-- Set.Icc.coe_ne_one
theorem coe_ne_one {x : Icc (0 : α) 1} : (x : α) ≠ 1 ↔ x ≠ 1 
Arguments: {α : Type u_1} [OrderedSemiring α] {x : ↑(Set.Icc 0 1)}
Goal: ↑x ≠ 1 ↔ x ≠ 1
-- Set.Icc.coe_nonneg
theorem coe_nonneg (x : Icc (0 : α) 1) : 0 ≤ (x : α) 
Arguments: {α : Type u_1} [OrderedSemiring α] (x : ↑(Set.Icc 0 1))
Goal: 0 ≤ ↑x
-- Set.Icc.coe_le_one
theorem coe_le_one (x : Icc (0 : α) 1) : (x : α) ≤ 1 
Arguments: {α : Type u_1} [OrderedSemiring α] (x : ↑(Set.Icc 0 1))
Goal: ↑x ≤ 1
-- Set.Icc.nonneg
theorem nonneg {t : Icc (0 : α) 1} : 0 ≤ t 
Arguments: {α : Type u_1} [OrderedSemiring α] {t : ↑(Set.Icc 0 1)}
Goal: 0 ≤ t
-- Set.Icc.le_one
theorem le_one {t : Icc (0 : α) 1} : t ≤ 1 
Arguments: {α : Type u_1} [OrderedSemiring α] {t : ↑(Set.Icc 0 1)}
Goal: t ≤ 1
-- Set.Icc.coe_mul
@[simp, norm_cast]
theorem coe_mul (x y : Icc (0 : α) 1) : ↑(x * y) = (x * y : α) 
Arguments: {α : Type u_1} [OrderedSemiring α] (x y : ↑(Set.Icc 0 1))
Goal: ↑(x * y) = ↑x * ↑y
-- Set.Icc.coe_pow
@[simp, norm_cast]
theorem coe_pow (x : Icc (0 : α) 1) (n : ℕ) : ↑(x ^ n) = ((x : α) ^ n) 
Arguments: {α : Type u_1} [OrderedSemiring α] (x : ↑(Set.Icc 0 1)) (n : ℕ)
Goal: ↑(x ^ n) = ↑x ^ n
-- Set.Icc.mul_le_left
theorem mul_le_left {x y : Icc (0 : α) 1} : x * y ≤ x 
Arguments: {α : Type u_1} [OrderedSemiring α] {x y : ↑(Set.Icc 0 1)}
Goal: x * y ≤ x
-- Set.Icc.mul_le_right
theorem mul_le_right {x y : Icc (0 : α) 1} : x * y ≤ y 
Arguments: {α : Type u_1} [OrderedSemiring α] {x y : ↑(Set.Icc 0 1)}
Goal: x * y ≤ y
-- Set.Icc.one_sub_mem
theorem one_sub_mem {t : β} (ht : t ∈ Icc (0 : β) 1) : 1 - t ∈ Icc (0 : β) 1 
Arguments: {β : Type u_2} [OrderedRing β] {t : β} (ht : t ∈ Set.Icc 0 1)
Goal: 1 - t ∈ Set.Icc 0 1
-- Set.Icc.mem_iff_one_sub_mem
theorem mem_iff_one_sub_mem {t : β} : t ∈ Icc (0 : β) 1 ↔ 1 - t ∈ Icc (0 : β) 1 
Arguments: {β : Type u_2} [OrderedRing β] {t : β}
Goal: t ∈ Set.Icc 0 1 ↔ 1 - t ∈ Set.Icc 0 1
-- Set.Icc.one_sub_nonneg
theorem one_sub_nonneg (x : Icc (0 : β) 1) : 0 ≤ 1 - (x : β) 
Arguments: {β : Type u_2} [OrderedRing β] (x : ↑(Set.Icc 0 1))
Goal: 0 ≤ 1 - ↑x
-- Set.Icc.one_sub_le_one
theorem one_sub_le_one (x : Icc (0 : β) 1) : 1 - (x : β) ≤ 1 
Arguments: {β : Type u_2} [OrderedRing β] (x : ↑(Set.Icc 0 1))
Goal: 1 - ↑x ≤ 1
-- Set.Ico.coe_zero
@[simp, norm_cast]
theorem coe_zero [Nontrivial α] : ↑(0 : Ico (0 : α) 1) = (0 : α) 
Arguments: {α : Type u_1} [OrderedSemiring α] [Nontrivial α]
Goal: ↑0 = 0
-- Set.Ico.mk_zero
@[simp]
theorem mk_zero [Nontrivial α] (h : (0 : α) ∈ Ico (0 : α) 1) : (⟨0, h⟩ : Ico (0 : α) 1) = 0 
Arguments: {α : Type u_1} [OrderedSemiring α] [Nontrivial α] (h : 0 ∈ Set.Ico 0 1)
Goal: ⟨0, h⟩ = 0
-- Set.Ico.coe_eq_zero
@[simp, norm_cast]
theorem coe_eq_zero [Nontrivial α] {x : Ico (0 : α) 1} : (x : α) = 0 ↔ x = 0 
Arguments: {α : Type u_1} [OrderedSemiring α] [Nontrivial α] {x : ↑(Set.Ico 0 1)}
Goal: ↑x = 0 ↔ x = 0
-- Set.Ico.coe_ne_zero
theorem coe_ne_zero [Nontrivial α] {x : Ico (0 : α) 1} : (x : α) ≠ 0 ↔ x ≠ 0 
Arguments: {α : Type u_1} [OrderedSemiring α] [Nontrivial α] {x : ↑(Set.Ico 0 1)}
Goal: ↑x ≠ 0 ↔ x ≠ 0
-- Set.Ico.coe_nonneg
theorem coe_nonneg (x : Ico (0 : α) 1) : 0 ≤ (x : α) 
Arguments: {α : Type u_1} [OrderedSemiring α] (x : ↑(Set.Ico 0 1))
Goal: 0 ≤ ↑x
-- Set.Ico.coe_lt_one
theorem coe_lt_one (x : Ico (0 : α) 1) : (x : α) < 1 
Arguments: {α : Type u_1} [OrderedSemiring α] (x : ↑(Set.Ico 0 1))
Goal: ↑x < 1
-- Set.Ico.nonneg
theorem nonneg [Nontrivial α] {t : Ico (0 : α) 1} : 0 ≤ t 
Arguments: {α : Type u_1} [OrderedSemiring α] [Nontrivial α] {t : ↑(Set.Ico 0 1)}
Goal: 0 ≤ t
-- Set.Ico.coe_mul
@[simp, norm_cast]
theorem coe_mul (x y : Ico (0 : α) 1) : ↑(x * y) = (x * y : α) 
Arguments: {α : Type u_1} [OrderedSemiring α] (x y : ↑(Set.Ico 0 1))
Goal: ↑(x * y) = ↑x * ↑y
-- Set.Ioc.coe_one
@[simp, norm_cast]
theorem coe_one : ↑(1 : Ioc (0 : α) 1) = (1 : α) 
Arguments: {α : Type u_1} [StrictOrderedSemiring α]
Goal: ↑1 = 1
-- Set.Ioc.mk_one
@[simp]
theorem mk_one (h : (1 : α) ∈ Ioc (0 : α) 1) : (⟨1, h⟩ : Ioc (0 : α) 1) = 1 
Arguments: {α : Type u_1} [StrictOrderedSemiring α] (h : 1 ∈ Set.Ioc 0 1)
Goal: ⟨1, h⟩ = 1
-- Set.Ioc.coe_eq_one
@[simp, norm_cast]
theorem coe_eq_one {x : Ioc (0 : α) 1} : (x : α) = 1 ↔ x = 1 
Arguments: {α : Type u_1} [StrictOrderedSemiring α] {x : ↑(Set.Ioc 0 1)}
Goal: ↑x = 1 ↔ x = 1
-- Set.Ioc.coe_ne_one
theorem coe_ne_one {x : Ioc (0 : α) 1} : (x : α) ≠ 1 ↔ x ≠ 1 
Arguments: {α : Type u_1} [StrictOrderedSemiring α] {x : ↑(Set.Ioc 0 1)}
Goal: ↑x ≠ 1 ↔ x ≠ 1
-- Set.Ioc.coe_pos
theorem coe_pos (x : Ioc (0 : α) 1) : 0 < (x : α) 
Arguments: {α : Type u_1} [StrictOrderedSemiring α] (x : ↑(Set.Ioc 0 1))
Goal: 0 < ↑x
-- Set.Ioc.coe_le_one
theorem coe_le_one (x : Ioc (0 : α) 1) : (x : α) ≤ 1 
Arguments: {α : Type u_1} [StrictOrderedSemiring α] (x : ↑(Set.Ioc 0 1))
Goal: ↑x ≤ 1
-- Set.Ioc.le_one
theorem le_one {t : Ioc (0 : α) 1} : t ≤ 1 
Arguments: {α : Type u_1} [StrictOrderedSemiring α] {t : ↑(Set.Ioc 0 1)}
Goal: t ≤ 1
-- Set.Ioc.coe_mul
@[simp, norm_cast]
theorem coe_mul (x y : Ioc (0 : α) 1) : ↑(x * y) = (x * y : α) 
Arguments: {α : Type u_1} [StrictOrderedSemiring α] (x y : ↑(Set.Ioc 0 1))
Goal: ↑(x * y) = ↑x * ↑y
-- Set.Ioc.coe_pow
@[simp, norm_cast]
theorem coe_pow (x : Ioc (0 : α) 1) (n : ℕ) : ↑(x ^ n) = ((x : α) ^ n) 
Arguments: {α : Type u_1} [StrictOrderedSemiring α] (x : ↑(Set.Ioc 0 1)) (n : ℕ)
Goal: ↑(x ^ n) = ↑x ^ n
-- Set.Ioo.pos
theorem pos (x : Ioo (0 : α) 1) : 0 < (x : α) 
Arguments: {α : Type u_1} [StrictOrderedSemiring α] (x : ↑(Set.Ioo 0 1))
Goal: 0 < ↑x
-- Set.Ioo.lt_one
theorem lt_one (x : Ioo (0 : α) 1) : (x : α) < 1 
Arguments: {α : Type u_1} [StrictOrderedSemiring α] (x : ↑(Set.Ioo 0 1))
Goal: ↑x < 1
-- Set.Ioo.coe_mul
@[simp, norm_cast]
theorem coe_mul (x y : Ioo (0 : α) 1) : ↑(x * y) = (x * y : α) 
Arguments: {α : Type u_1} [StrictOrderedSemiring α] (x y : ↑(Set.Ioo 0 1))
Goal: ↑(x * y) = ↑x * ↑y
-- Set.Ioo.one_sub_mem
theorem one_sub_mem {t : β} (ht : t ∈ Ioo (0 : β) 1) : 1 - t ∈ Ioo (0 : β) 1 
Arguments: {β : Type u_2} [OrderedRing β] {t : β} (ht : t ∈ Set.Ioo 0 1)
Goal: 1 - t ∈ Set.Ioo 0 1
-- Set.Ioo.mem_iff_one_sub_mem
theorem mem_iff_one_sub_mem {t : β} : t ∈ Ioo (0 : β) 1 ↔ 1 - t ∈ Ioo (0 : β) 1 
Arguments: {β : Type u_2} [OrderedRing β] {t : β}
Goal: t ∈ Set.Ioo 0 1 ↔ 1 - t ∈ Set.Ioo 0 1
-- Set.Ioo.one_minus_pos
theorem one_minus_pos (x : Ioo (0 : β) 1) : 0 < 1 - (x : β) 
Arguments: {β : Type u_2} [OrderedRing β] (x : ↑(Set.Ioo 0 1))
Goal: 0 < 1 - ↑x
-- Set.Ioo.one_minus_lt_one
theorem one_minus_lt_one (x : Ioo (0 : β) 1) : 1 - (x : β) < 1 
Arguments: {β : Type u_2} [OrderedRing β] (x : ↑(Set.Ioo 0 1))
Goal: 1 - ↑x < 1
-- List.prod_pos
lemma prod_pos [StrictOrderedSemiring R] (l : List R) (h : ∀ a ∈ l, (0 : R) < a) :
    0 < l.prod 
Arguments: {R : Type u_1} [StrictOrderedSemiring R] (l : List R) (h : ∀ a ∈ l, 0 < a)
Goal: 0 < l.prod