ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ENNReal.rpow_lt_top_iff_of_pos
lemma rpow_lt_top_iff_of_pos {x : ℝ≥0∞} {y : ℝ} (hy : 0 < y) : x ^ y < ∞ ↔ x < ∞ 
Arguments: {x : ℝ≥0∞} {y : ℝ} (hy : 0 < y)
Goal: x ^ y < ⊤ ↔ x < ⊤
-- ENNReal.rpow_eq_top_of_nonneg
theorem rpow_eq_top_of_nonneg (x : ℝ≥0∞) {y : ℝ} (hy0 : 0 ≤ y) : x ^ y = ⊤ → x = ⊤ 
Arguments: (x : ℝ≥0∞) {y : ℝ} (hy0 : 0 ≤ y)
Goal: x ^ y = ⊤ → x = ⊤
-- ENNReal.rpow_ne_top_of_nonneg
theorem rpow_ne_top_of_nonneg {x : ℝ≥0∞} {y : ℝ} (hy0 : 0 ≤ y) (h : x ≠ ⊤) : x ^ y ≠ ⊤ 
Arguments: {x : ℝ≥0∞} {y : ℝ} (hy0 : 0 ≤ y) (h : x ≠ ⊤)
Goal: x ^ y ≠ ⊤
-- ENNReal.rpow_lt_top_of_nonneg
theorem rpow_lt_top_of_nonneg {x : ℝ≥0∞} {y : ℝ} (hy0 : 0 ≤ y) (h : x ≠ ⊤) : x ^ y < ⊤ 
Arguments: {x : ℝ≥0∞} {y : ℝ} (hy0 : 0 ≤ y) (h : x ≠ ⊤)
Goal: x ^ y < ⊤
-- ENNReal.rpow_add
theorem rpow_add {x : ℝ≥0∞} (y z : ℝ) (hx : x ≠ 0) (h'x : x ≠ ⊤) : x ^ (y + z) = x ^ y * x ^ z 
Arguments: {x : ℝ≥0∞} (y z : ℝ) (hx : x ≠ 0) (h'x : x ≠ ⊤)
Goal: x ^ (y + z) = x ^ y * x ^ z
-- ENNReal.rpow_add_of_nonneg
theorem rpow_add_of_nonneg {x : ℝ≥0∞} (y z : ℝ) (hy : 0 ≤ y) (hz : 0 ≤ z) :
    x ^ (y + z) = x ^ y * x ^ z 
Arguments: {x : ℝ≥0∞} (y z : ℝ) (hy : 0 ≤ y) (hz : 0 ≤ z)
Goal: x ^ (y + z) = x ^ y * x ^ z
-- ENNReal.rpow_neg
theorem rpow_neg (x : ℝ≥0∞) (y : ℝ) : x ^ (-y) = (x ^ y)⁻¹ 
Arguments: (x : ℝ≥0∞) (y : ℝ)
Goal: x ^ (-y) = (x ^ y)⁻¹
-- ENNReal.rpow_sub
theorem rpow_sub {x : ℝ≥0∞} (y z : ℝ) (hx : x ≠ 0) (h'x : x ≠ ⊤) : x ^ (y - z) = x ^ y / x ^ z 
Arguments: {x : ℝ≥0∞} (y z : ℝ) (hx : x ≠ 0) (h'x : x ≠ ⊤)
Goal: x ^ (y - z) = x ^ y / x ^ z
-- ENNReal.rpow_neg_one
theorem rpow_neg_one (x : ℝ≥0∞) : x ^ (-1 : ℝ) = x⁻¹ 
Arguments: (x : ℝ≥0∞)
Goal: x ^ (-1) = x⁻¹
-- ENNReal.rpow_mul
theorem rpow_mul (x : ℝ≥0∞) (y z : ℝ) : x ^ (y * z) = (x ^ y) ^ z 
Arguments: (x : ℝ≥0∞) (y z : ℝ)
Goal: x ^ (y * z) = (x ^ y) ^ z
-- ENNReal.rpow_natCast
@[simp, norm_cast]
theorem rpow_natCast (x : ℝ≥0∞) (n : ℕ) : x ^ (n : ℝ) = x ^ n 
Arguments: (x : ℝ≥0∞) (n : ℕ)
Goal: x ^ ↑n = x ^ n
-- ENNReal.rpow_ofNat
@[simp]
lemma rpow_ofNat (x : ℝ≥0∞) (n : ℕ) [n.AtLeastTwo] :
    x ^ (no_index (OfNat.ofNat n) : ℝ) = x ^ (OfNat.ofNat n) 
Arguments: (x : ℝ≥0∞) (n : ℕ) [n.AtLeastTwo]
Goal: x ^ OfNat.ofNat n = x ^ OfNat.ofNat n
-- ENNReal.rpow_intCast
@[simp, norm_cast]
lemma rpow_intCast (x : ℝ≥0∞) (n : ℤ) : x ^ (n : ℝ) = x ^ n 
Arguments: (x : ℝ≥0∞) (n : ℤ)
Goal: x ^ ↑n = x ^ n
-- ENNReal.rpow_two
theorem rpow_two (x : ℝ≥0∞) : x ^ (2 : ℝ) = x ^ 2 
Arguments: (x : ℝ≥0∞)
Goal: x ^ 2 = x ^ 2
-- ENNReal.mul_rpow_eq_ite
theorem mul_rpow_eq_ite (x y : ℝ≥0∞) (z : ℝ) :
    (x * y) ^ z = if (x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else x ^ z * y ^ z 
Arguments: (x y : ℝ≥0∞) (z : ℝ)
Goal: (x * y) ^ z = if (x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else x ^ z * y ^ z
-- ENNReal.mul_rpow_of_ne_top
theorem mul_rpow_of_ne_top {x y : ℝ≥0∞} (hx : x ≠ ⊤) (hy : y ≠ ⊤) (z : ℝ) :
    (x * y) ^ z = x ^ z * y ^ z 
Arguments: {x y : ℝ≥0∞} (hx : x ≠ ⊤) (hy : y ≠ ⊤) (z : ℝ)
Goal: (x * y) ^ z = x ^ z * y ^ z
-- ENNReal.coe_mul_rpow
@[norm_cast]
theorem coe_mul_rpow (x y : ℝ≥0) (z : ℝ) : ((x : ℝ≥0∞) * y) ^ z = (x : ℝ≥0∞) ^ z * (y : ℝ≥0∞) ^ z 
Arguments: (x y : ℝ≥0) (z : ℝ)
Goal: (↑x * ↑y) ^ z = ↑x ^ z * ↑y ^ z
-- ENNReal.prod_coe_rpow
theorem prod_coe_rpow {ι} (s : Finset ι) (f : ι → ℝ≥0) (r : ℝ) :
    ∏ i ∈ s, (f i : ℝ≥0∞) ^ r = ((∏ i ∈ s, f i : ℝ≥0) : ℝ≥0∞) ^ r 
Arguments: {ι : Type u_1} (s : Finset ι) (f : ι → ℝ≥0) (r : ℝ)
Goal: ∏ i ∈ s, ↑(f i) ^ r = ↑(∏ i ∈ s, f i) ^ r
-- ENNReal.mul_rpow_of_ne_zero
theorem mul_rpow_of_ne_zero {x y : ℝ≥0∞} (hx : x ≠ 0) (hy : y ≠ 0) (z : ℝ) :
    (x * y) ^ z = x ^ z * y ^ z 
Arguments: {x y : ℝ≥0∞} (hx : x ≠ 0) (hy : y ≠ 0) (z : ℝ)
Goal: (x * y) ^ z = x ^ z * y ^ z
-- ENNReal.mul_rpow_of_nonneg
theorem mul_rpow_of_nonneg (x y : ℝ≥0∞) {z : ℝ} (hz : 0 ≤ z) : (x * y) ^ z = x ^ z * y ^ z 
Arguments: (x y : ℝ≥0∞) {z : ℝ} (hz : 0 ≤ z)
Goal: (x * y) ^ z = x ^ z * y ^ z
-- ENNReal.prod_rpow_of_ne_top
theorem prod_rpow_of_ne_top {ι} {s : Finset ι} {f : ι → ℝ≥0∞} (hf : ∀ i ∈ s, f i ≠ ∞) (r : ℝ) :
    ∏ i ∈ s, f i ^ r = (∏ i ∈ s, f i) ^ r 
Arguments: {ι : Type u_1} {s : Finset ι} {f : ι → ℝ≥0∞} (hf : ∀ i ∈ s, f i ≠ ⊤) (r : ℝ)
Goal: ∏ i ∈ s, f i ^ r = (∏ i ∈ s, f i) ^ r
-- ENNReal.prod_rpow_of_nonneg
theorem prod_rpow_of_nonneg {ι} {s : Finset ι} {f : ι → ℝ≥0∞} {r : ℝ} (hr : 0 ≤ r) :
    ∏ i ∈ s, f i ^ r = (∏ i ∈ s, f i) ^ r 
Arguments: {ι : Type u_1} {s : Finset ι} {f : ι → ℝ≥0∞} {r : ℝ} (hr : 0 ≤ r)
Goal: ∏ i ∈ s, f i ^ r = (∏ i ∈ s, f i) ^ r
-- ENNReal.inv_rpow
theorem inv_rpow (x : ℝ≥0∞) (y : ℝ) : x⁻¹ ^ y = (x ^ y)⁻¹ 
Arguments: (x : ℝ≥0∞) (y : ℝ)
Goal: x⁻¹ ^ y = (x ^ y)⁻¹
-- ENNReal.div_rpow_of_nonneg
theorem div_rpow_of_nonneg (x y : ℝ≥0∞) {z : ℝ} (hz : 0 ≤ z) : (x / y) ^ z = x ^ z / y ^ z 
Arguments: (x y : ℝ≥0∞) {z : ℝ} (hz : 0 ≤ z)
Goal: (x / y) ^ z = x ^ z / y ^ z
-- ENNReal.strictMono_rpow_of_pos
theorem strictMono_rpow_of_pos {z : ℝ} (h : 0 < z) : StrictMono fun x : ℝ≥0∞ => x ^ z 
Arguments: {z : ℝ} (h : 0 < z)
Goal: StrictMono fun x => x ^ z
-- ENNReal.monotone_rpow_of_nonneg
theorem monotone_rpow_of_nonneg {z : ℝ} (h : 0 ≤ z) : Monotone fun x : ℝ≥0∞ => x ^ z 
Arguments: {z : ℝ} (h : 0 ≤ z)
Goal: Monotone fun x => x ^ z
-- ENNReal.orderIsoRpow_apply
@[simps! apply]
def orderIsoRpow (y : ℝ) (hy : 0 < y) : ℝ≥0∞ ≃o ℝ≥0∞ 
Arguments: (y : ℝ) (hy : 0 < y) (x : ℝ≥0∞)
Goal: (ENNReal.orderIsoRpow y hy) x = x ^ y
-- ENNReal.orderIsoRpow_symm_apply
theorem orderIsoRpow_symm_apply (y : ℝ) (hy : 0 < y) :
    (orderIsoRpow y hy).symm = orderIsoRpow (1 / y) (one_div_pos.2 hy) 
Arguments: (y : ℝ) (hy : 0 < y)
Goal: (ENNReal.orderIsoRpow y hy).symm = ENNReal.orderIsoRpow (1 / y) ⋯
-- ENNReal.rpow_le_rpow
@[gcongr] theorem rpow_le_rpow {x y : ℝ≥0∞} {z : ℝ} (h₁ : x ≤ y) (h₂ : 0 ≤ z) : x ^ z ≤ y ^ z 
Arguments: {x y : ℝ≥0∞} {z : ℝ} (h₁ : x ≤ y) (h₂ : 0 ≤ z)
Goal: x ^ z ≤ y ^ z
-- ENNReal.rpow_lt_rpow
@[gcongr] theorem rpow_lt_rpow {x y : ℝ≥0∞} {z : ℝ} (h₁ : x < y) (h₂ : 0 < z) : x ^ z < y ^ z 
Arguments: {x y : ℝ≥0∞} {z : ℝ} (h₁ : x < y) (h₂ : 0 < z)
Goal: x ^ z < y ^ z
-- ENNReal.rpow_le_rpow_iff
theorem rpow_le_rpow_iff {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z) : x ^ z ≤ y ^ z ↔ x ≤ y 
Arguments: {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z)
Goal: x ^ z ≤ y ^ z ↔ x ≤ y
-- ENNReal.rpow_lt_rpow_iff
theorem rpow_lt_rpow_iff {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z) : x ^ z < y ^ z ↔ x < y 
Arguments: {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z)
Goal: x ^ z < y ^ z ↔ x < y
-- ENNReal.le_rpow_inv_iff
theorem le_rpow_inv_iff {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z) : x ≤ y ^ z⁻¹ ↔ x ^ z ≤ y 
Arguments: {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z)
Goal: x ≤ y ^ z⁻¹ ↔ x ^ z ≤ y
-- ENNReal.le_rpow_one_div_iff
@[deprecated le_rpow_inv_iff (since := "2024-07-10")]
theorem le_rpow_one_div_iff {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z) : x ≤ y ^ (1 / z) ↔ x ^ z ≤ y 
Arguments: {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z)
Goal: x ≤ y ^ (1 / z) ↔ x ^ z ≤ y
-- ENNReal.rpow_inv_lt_iff
theorem rpow_inv_lt_iff {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z) : x ^ z⁻¹ < y ↔ x < y ^ z 
Arguments: {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z)
Goal: x ^ z⁻¹ < y ↔ x < y ^ z
-- ENNReal.lt_rpow_inv_iff
theorem lt_rpow_inv_iff {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z) : x < y ^ z⁻¹ ↔ x ^ z < y 
Arguments: {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z)
Goal: x < y ^ z⁻¹ ↔ x ^ z < y
-- ENNReal.lt_rpow_one_div_iff
@[deprecated lt_rpow_inv_iff (since := "2024-07-10")]
theorem lt_rpow_one_div_iff {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z) : x < y ^ (1 / z) ↔ x ^ z < y 
Arguments: {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z)
Goal: x < y ^ (1 / z) ↔ x ^ z < y
-- ENNReal.rpow_inv_le_iff
theorem rpow_inv_le_iff {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z) : x ^ z⁻¹ ≤ y ↔ x ≤ y ^ z 
Arguments: {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z)
Goal: x ^ z⁻¹ ≤ y ↔ x ≤ y ^ z
-- ENNReal.rpow_one_div_le_iff
@[deprecated rpow_inv_le_iff (since := "2024-07-10")]
theorem rpow_one_div_le_iff {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z) : x ^ (1 / z) ≤ y ↔ x ≤ y ^ z 
Arguments: {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z)
Goal: x ^ (1 / z) ≤ y ↔ x ≤ y ^ z
-- ENNReal.rpow_lt_rpow_of_exponent_lt
theorem rpow_lt_rpow_of_exponent_lt {x : ℝ≥0∞} {y z : ℝ} (hx : 1 < x) (hx' : x ≠ ⊤) (hyz : y < z) :
    x ^ y < x ^ z 
Arguments: {x : ℝ≥0∞} {y z : ℝ} (hx : 1 < x) (hx' : x ≠ ⊤) (hyz : y < z)
Goal: x ^ y < x ^ z
-- ENNReal.rpow_le_rpow_of_exponent_le
@[gcongr] theorem rpow_le_rpow_of_exponent_le {x : ℝ≥0∞} {y z : ℝ} (hx : 1 ≤ x) (hyz : y ≤ z) :
    x ^ y ≤ x ^ z 
Arguments: {x : ℝ≥0∞} {y z : ℝ} (hx : 1 ≤ x) (hyz : y ≤ z)
Goal: x ^ y ≤ x ^ z
-- ENNReal.rpow_lt_rpow_of_exponent_gt
theorem rpow_lt_rpow_of_exponent_gt {x : ℝ≥0∞} {y z : ℝ} (hx0 : 0 < x) (hx1 : x < 1) (hyz : z < y) :
    x ^ y < x ^ z 
Arguments: {x : ℝ≥0∞} {y z : ℝ} (hx0 : 0 < x) (hx1 : x < 1) (hyz : z < y)
Goal: x ^ y < x ^ z
-- ENNReal.rpow_le_rpow_of_exponent_ge
theorem rpow_le_rpow_of_exponent_ge {x : ℝ≥0∞} {y z : ℝ} (hx1 : x ≤ 1) (hyz : z ≤ y) :
    x ^ y ≤ x ^ z 
Arguments: {x : ℝ≥0∞} {y z : ℝ} (hx1 : x ≤ 1) (hyz : z ≤ y)
Goal: x ^ y ≤ x ^ z
-- ENNReal.rpow_le_self_of_le_one
theorem rpow_le_self_of_le_one {x : ℝ≥0∞} {z : ℝ} (hx : x ≤ 1) (h_one_le : 1 ≤ z) : x ^ z ≤ x 
Arguments: {x : ℝ≥0∞} {z : ℝ} (hx : x ≤ 1) (h_one_le : 1 ≤ z)
Goal: x ^ z ≤ x
-- ENNReal.le_rpow_self_of_one_le
theorem le_rpow_self_of_one_le {x : ℝ≥0∞} {z : ℝ} (hx : 1 ≤ x) (h_one_le : 1 ≤ z) : x ≤ x ^ z 
Arguments: {x : ℝ≥0∞} {z : ℝ} (hx : 1 ≤ x) (h_one_le : 1 ≤ z)
Goal: x ≤ x ^ z
-- ENNReal.rpow_pos_of_nonneg
theorem rpow_pos_of_nonneg {p : ℝ} {x : ℝ≥0∞} (hx_pos : 0 < x) (hp_nonneg : 0 ≤ p) : 0 < x ^ p 
Arguments: {p : ℝ} {x : ℝ≥0∞} (hx_pos : 0 < x) (hp_nonneg : 0 ≤ p)
Goal: 0 < x ^ p
-- ENNReal.rpow_pos
theorem rpow_pos {p : ℝ} {x : ℝ≥0∞} (hx_pos : 0 < x) (hx_ne_top : x ≠ ⊤) : 0 < x ^ p 
Arguments: {p : ℝ} {x : ℝ≥0∞} (hx_pos : 0 < x) (hx_ne_top : x ≠ ⊤)
Goal: 0 < x ^ p
-- ENNReal.rpow_lt_one
theorem rpow_lt_one {x : ℝ≥0∞} {z : ℝ} (hx : x < 1) (hz : 0 < z) : x ^ z < 1 
Arguments: {x : ℝ≥0∞} {z : ℝ} (hx : x < 1) (hz : 0 < z)
Goal: x ^ z < 1
-- ENNReal.rpow_le_one
theorem rpow_le_one {x : ℝ≥0∞} {z : ℝ} (hx : x ≤ 1) (hz : 0 ≤ z) : x ^ z ≤ 1 
Arguments: {x : ℝ≥0∞} {z : ℝ} (hx : x ≤ 1) (hz : 0 ≤ z)
Goal: x ^ z ≤ 1
-- ENNReal.rpow_lt_one_of_one_lt_of_neg
theorem rpow_lt_one_of_one_lt_of_neg {x : ℝ≥0∞} {z : ℝ} (hx : 1 < x) (hz : z < 0) : x ^ z < 1 
Arguments: {x : ℝ≥0∞} {z : ℝ} (hx : 1 < x) (hz : z < 0)
Goal: x ^ z < 1
-- ENNReal.rpow_le_one_of_one_le_of_neg
theorem rpow_le_one_of_one_le_of_neg {x : ℝ≥0∞} {z : ℝ} (hx : 1 ≤ x) (hz : z < 0) : x ^ z ≤ 1 
Arguments: {x : ℝ≥0∞} {z : ℝ} (hx : 1 ≤ x) (hz : z < 0)
Goal: x ^ z ≤ 1
-- ENNReal.one_lt_rpow
theorem one_lt_rpow {x : ℝ≥0∞} {z : ℝ} (hx : 1 < x) (hz : 0 < z) : 1 < x ^ z 
Arguments: {x : ℝ≥0∞} {z : ℝ} (hx : 1 < x) (hz : 0 < z)
Goal: 1 < x ^ z
-- ENNReal.one_le_rpow
theorem one_le_rpow {x : ℝ≥0∞} {z : ℝ} (hx : 1 ≤ x) (hz : 0 < z) : 1 ≤ x ^ z 
Arguments: {x : ℝ≥0∞} {z : ℝ} (hx : 1 ≤ x) (hz : 0 < z)
Goal: 1 ≤ x ^ z
-- ENNReal.one_lt_rpow_of_pos_of_lt_one_of_neg
theorem one_lt_rpow_of_pos_of_lt_one_of_neg {x : ℝ≥0∞} {z : ℝ} (hx1 : 0 < x) (hx2 : x < 1)
    (hz : z < 0) : 1 < x ^ z 
Arguments: {x : ℝ≥0∞} {z : ℝ} (hx1 : 0 < x) (hx2 : x < 1) (hz : z < 0)
Goal: 1 < x ^ z
-- ENNReal.one_le_rpow_of_pos_of_le_one_of_neg
theorem one_le_rpow_of_pos_of_le_one_of_neg {x : ℝ≥0∞} {z : ℝ} (hx1 : 0 < x) (hx2 : x ≤ 1)
    (hz : z < 0) : 1 ≤ x ^ z 
Arguments: {x : ℝ≥0∞} {z : ℝ} (hx1 : 0 < x) (hx2 : x ≤ 1) (hz : z < 0)
Goal: 1 ≤ x ^ z
-- ENNReal.toNNReal_rpow
theorem toNNReal_rpow (x : ℝ≥0∞) (z : ℝ) : x.toNNReal ^ z = (x ^ z).toNNReal 
Arguments: (x : ℝ≥0∞) (z : ℝ)
Goal: x.toNNReal ^ z = (x ^ z).toNNReal
-- ENNReal.toReal_rpow
theorem toReal_rpow (x : ℝ≥0∞) (z : ℝ) : x.toReal ^ z = (x ^ z).toReal 
Arguments: (x : ℝ≥0∞) (z : ℝ)
Goal: x.toReal ^ z = (x ^ z).toReal
-- ENNReal.ofReal_rpow_of_pos
theorem ofReal_rpow_of_pos {x p : ℝ} (hx_pos : 0 < x) :
    ENNReal.ofReal x ^ p = ENNReal.ofReal (x ^ p) 
Arguments: {x p : ℝ} (hx_pos : 0 < x)
Goal: ENNReal.ofReal x ^ p = ENNReal.ofReal (x ^ p)
-- ENNReal.ofReal_rpow_of_nonneg
theorem ofReal_rpow_of_nonneg {x p : ℝ} (hx_nonneg : 0 ≤ x) (hp_nonneg : 0 ≤ p) :
    ENNReal.ofReal x ^ p = ENNReal.ofReal (x ^ p) 
Arguments: {x p : ℝ} (hx_nonneg : 0 ≤ x) (hp_nonneg : 0 ≤ p)
Goal: ENNReal.ofReal x ^ p = ENNReal.ofReal (x ^ p)
-- ENNReal.rpow_rpow_inv
@[simp] lemma rpow_rpow_inv {y : ℝ} (hy : y ≠ 0) (x : ℝ≥0∞) : (x ^ y) ^ y⁻¹ = x 
Arguments: {y : ℝ} (hy : y ≠ 0) (x : ℝ≥0∞)
Goal: (x ^ y) ^ y⁻¹ = x
-- ENNReal.rpow_inv_rpow
@[simp] lemma rpow_inv_rpow {y : ℝ} (hy : y ≠ 0) (x : ℝ≥0∞) : (x ^ y⁻¹) ^ y = x 
Arguments: {y : ℝ} (hy : y ≠ 0) (x : ℝ≥0∞)
Goal: (x ^ y⁻¹) ^ y = x
-- ENNReal.pow_rpow_inv_natCast
lemma pow_rpow_inv_natCast {n : ℕ} (hn : n ≠ 0) (x : ℝ≥0∞) : (x ^ n) ^ (n⁻¹ : ℝ) = x 
Arguments: {n : ℕ} (hn : n ≠ 0) (x : ℝ≥0∞)
Goal: (x ^ n) ^ (↑n)⁻¹ = x
-- ENNReal.rpow_inv_natCast_pow
lemma rpow_inv_natCast_pow {n : ℕ} (hn : n ≠ 0) (x : ℝ≥0∞) : (x ^ (n⁻¹ : ℝ)) ^ n = x 
Arguments: {n : ℕ} (hn : n ≠ 0) (x : ℝ≥0∞)
Goal: (x ^ (↑n)⁻¹) ^ n = x
-- ENNReal.rpow_natCast_mul
lemma rpow_natCast_mul (x : ℝ≥0∞) (n : ℕ) (z : ℝ) : x ^ (n * z) = (x ^ n) ^ z 
Arguments: (x : ℝ≥0∞) (n : ℕ) (z : ℝ)
Goal: x ^ (↑n * z) = (x ^ n) ^ z
-- ENNReal.rpow_mul_natCast
lemma rpow_mul_natCast (x : ℝ≥0∞) (y : ℝ) (n : ℕ) : x ^ (y * n) = (x ^ y) ^ n 
Arguments: (x : ℝ≥0∞) (y : ℝ) (n : ℕ)
Goal: x ^ (y * ↑n) = (x ^ y) ^ n
-- ENNReal.rpow_intCast_mul
lemma rpow_intCast_mul (x : ℝ≥0∞) (n : ℤ) (z : ℝ) : x ^ (n * z) = (x ^ n) ^ z 
Arguments: (x : ℝ≥0∞) (n : ℤ) (z : ℝ)
Goal: x ^ (↑n * z) = (x ^ n) ^ z
-- ENNReal.rpow_mul_intCast
lemma rpow_mul_intCast (x : ℝ≥0∞) (y : ℝ) (n : ℤ) : x ^ (y * n) = (x ^ y) ^ n 
Arguments: (x : ℝ≥0∞) (y : ℝ) (n : ℤ)
Goal: x ^ (y * ↑n) = (x ^ y) ^ n
-- ENNReal.rpow_left_injective
lemma rpow_left_injective {x : ℝ} (hx : x ≠ 0) : Injective fun y : ℝ≥0∞ ↦ y ^ x 
Arguments: {x : ℝ} (hx : x ≠ 0)
Goal: Function.Injective fun y => y ^ x
-- ENNReal.rpow_left_surjective
theorem rpow_left_surjective {x : ℝ} (hx : x ≠ 0) : Function.Surjective fun y : ℝ≥0∞ => y ^ x 
Arguments: {x : ℝ} (hx : x ≠ 0)
Goal: Function.Surjective fun y => y ^ x
-- ENNReal.rpow_left_bijective
theorem rpow_left_bijective {x : ℝ} (hx : x ≠ 0) : Function.Bijective fun y : ℝ≥0∞ => y ^ x 
Arguments: {x : ℝ} (hx : x ≠ 0)
Goal: Function.Bijective fun y => y ^ x
-- tendstoUniformlyOn_tsum
theorem tendstoUniformlyOn_tsum {f : α → β → F} (hu : Summable u) {s : Set β}
    (hfu : ∀ n x, x ∈ s → ‖f n x‖ ≤ u n) :
    TendstoUniformlyOn (fun t : Finset α => fun x => ∑ n ∈ t, f n x) (fun x => ∑' n, f n x) atTop
      s 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_3} [NormedAddCommGroup F] [CompleteSpace F] {u : α → ℝ} {f : α → β → F} (hu : Summable u) {s : Set β} (hfu : ∀ (n : α), ∀ x ∈ s, ‖f n x‖ ≤ u n)
Goal: TendstoUniformlyOn (fun t x => ∑ n ∈ t, f n x) (fun x => ∑' (n : α), f n x) Filter.atTop s
-- tendstoUniformlyOn_tsum_nat
theorem tendstoUniformlyOn_tsum_nat {f : ℕ → β → F} {u : ℕ → ℝ} (hu : Summable u) {s : Set β}
    (hfu : ∀ n x, x ∈ s → ‖f n x‖ ≤ u n) :
    TendstoUniformlyOn (fun N => fun x => ∑ n ∈ Finset.range N, f n x) (fun x => ∑' n, f n x) atTop
      s 
Arguments: {β : Type u_2} {F : Type u_3} [NormedAddCommGroup F] [CompleteSpace F] {f : ℕ → β → F} {u : ℕ → ℝ} (hu : Summable u) {s : Set β} (hfu : ∀ (n : ℕ), ∀ x ∈ s, ‖f n x‖ ≤ u n)
Goal: TendstoUniformlyOn (fun N x => ∑ n ∈ Finset.range N, f n x) (fun x => ∑' (n : ℕ), f n x) Filter.atTop s
-- tendstoUniformly_tsum
theorem tendstoUniformly_tsum {f : α → β → F} (hu : Summable u) (hfu : ∀ n x, ‖f n x‖ ≤ u n) :
    TendstoUniformly (fun t : Finset α => fun x => ∑ n ∈ t, f n x)
      (fun x => ∑' n, f n x) atTop 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_3} [NormedAddCommGroup F] [CompleteSpace F] {u : α → ℝ} {f : α → β → F} (hu : Summable u) (hfu : ∀ (n : α) (x : β), ‖f n x‖ ≤ u n)
Goal: TendstoUniformly (fun t x => ∑ n ∈ t, f n x) (fun x => ∑' (n : α), f n x) Filter.atTop
-- tendstoUniformly_tsum_nat
theorem tendstoUniformly_tsum_nat {f : ℕ → β → F} {u : ℕ → ℝ} (hu : Summable u)
    (hfu : ∀ n x, ‖f n x‖ ≤ u n) :
    TendstoUniformly (fun N => fun x => ∑ n ∈ Finset.range N, f n x) (fun x => ∑' n, f n x)
      atTop 
Arguments: {β : Type u_2} {F : Type u_3} [NormedAddCommGroup F] [CompleteSpace F] {f : ℕ → β → F} {u : ℕ → ℝ} (hu : Summable u) (hfu : ∀ (n : ℕ) (x : β), ‖f n x‖ ≤ u n)
Goal: TendstoUniformly (fun N x => ∑ n ∈ Finset.range N, f n x) (fun x => ∑' (n : ℕ), f n x) Filter.atTop
-- continuousOn_tsum
theorem continuousOn_tsum [TopologicalSpace β] {f : α → β → F} {s : Set β}
    (hf : ∀ i, ContinuousOn (f i) s) (hu : Summable u) (hfu : ∀ n x, x ∈ s → ‖f n x‖ ≤ u n) :
    ContinuousOn (fun x => ∑' n, f n x) s 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_3} [NormedAddCommGroup F] [CompleteSpace F] {u : α → ℝ} [TopologicalSpace β] {f : α → β → F} {s : Set β} (hf : ∀ (i : α), ContinuousOn (f i) s) (hu : Summable u) (hfu : ∀ (n : α), ∀ x ∈ s, ‖f n x‖ ≤ u n)
Goal: ContinuousOn (fun x => ∑' (n : α), f n x) s
-- continuous_tsum
theorem continuous_tsum [TopologicalSpace β] {f : α → β → F} (hf : ∀ i, Continuous (f i))
    (hu : Summable u) (hfu : ∀ n x, ‖f n x‖ ≤ u n) : Continuous fun x => ∑' n, f n x 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_3} [NormedAddCommGroup F] [CompleteSpace F] {u : α → ℝ} [TopologicalSpace β] {f : α → β → F} (hf : ∀ (i : α), Continuous (f i)) (hu : Summable u) (hfu : ∀ (n : α) (x : β), ‖f n x‖ ≤ u n)
Goal: Continuous fun x => ∑' (n : α), f n x
-- Complex.continuousOn_one_add_mul_inv
lemma continuousOn_one_add_mul_inv {z : ℂ} (hz : 1 + z ∈ slitPlane) :
    ContinuousOn (fun t : ℝ ↦ (1 + t • z)⁻¹) (Set.Icc 0 1) 
Arguments: {z : ℂ} (hz : 1 + z ∈ Complex.slitPlane)
Goal: ContinuousOn (fun t => (1 + t • z)⁻¹) (Set.Icc 0 1)
-- Complex.log_eq_integral
lemma log_eq_integral {z : ℂ} (hz : 1 + z ∈ slitPlane) :
    log (1 + z) = z * ∫ (t : ℝ) in (0 : ℝ)..1, (1 + t • z)⁻¹ 
Arguments: {z : ℂ} (hz : 1 + z ∈ Complex.slitPlane)
Goal: Complex.log (1 + z) = z * ∫ (t : ℝ) in 0 ..1, (1 + t • z)⁻¹
-- Complex.log_inv_eq_integral
lemma log_inv_eq_integral {z : ℂ} (hz : 1 - z ∈ slitPlane) :
    log (1 - z)⁻¹ = z * ∫ (t : ℝ) in (0 : ℝ)..1, (1 - t • z)⁻¹ 
Arguments: {z : ℂ} (hz : 1 - z ∈ Complex.slitPlane)
Goal: Complex.log (1 - z)⁻¹ = z * ∫ (t : ℝ) in 0 ..1, (1 - t • z)⁻¹
-- Complex.logTaylor_zero
lemma logTaylor_zero : logTaylor 0 = fun _ ↦ 0 
Goal: Complex.logTaylor 0 = fun x => 0
-- Complex.logTaylor_succ
lemma logTaylor_succ (n : ℕ) :
    logTaylor (n + 1) = logTaylor n + (fun z : ℂ ↦ (-1) ^ (n + 1) * z ^ n / n) 
Arguments: (n : ℕ)
Goal: Complex.logTaylor (n + 1) = Complex.logTaylor n + fun z => (-1) ^ (n + 1) * z ^ n / ↑n
-- Complex.logTaylor_at_zero
lemma logTaylor_at_zero (n : ℕ) : logTaylor n 0 = 0 
Arguments: (n : ℕ)
Goal: Complex.logTaylor n 0 = 0
-- Complex.hasDerivAt_logTaylor
lemma hasDerivAt_logTaylor (n : ℕ) (z : ℂ) :
    HasDerivAt (logTaylor (n + 1)) (∑ j ∈ Finset.range n, (-1) ^ j * z ^ j) z 
Arguments: (n : ℕ) (z : ℂ)
Goal: HasDerivAt (Complex.logTaylor (n + 1)) (∑ j ∈ Finset.range n, (-1) ^ j * z ^ j) z
-- Complex.hasDerivAt_log_sub_logTaylor
lemma hasDerivAt_log_sub_logTaylor (n : ℕ) {z : ℂ} (hz : 1 + z ∈ slitPlane) :
    HasDerivAt (fun z : ℂ ↦ log (1 + z) - logTaylor (n + 1) z) ((-z) ^ n * (1 + z)⁻¹) z 
Arguments: (n : ℕ) {z : ℂ} (hz : 1 + z ∈ Complex.slitPlane)
Goal: HasDerivAt (fun z => Complex.log (1 + z) - Complex.logTaylor (n + 1) z) ((-z) ^ n * (1 + z)⁻¹) z
-- Complex.norm_one_add_mul_inv_le
lemma norm_one_add_mul_inv_le {t : ℝ} (ht : t ∈ Set.Icc 0 1) {z : ℂ} (hz : ‖z‖ < 1) :
    ‖(1 + t * z)⁻¹‖ ≤ (1 - ‖z‖)⁻¹ 
Arguments: {t : ℝ} (ht : t ∈ Set.Icc 0 1) {z : ℂ} (hz : ‖z‖ < 1)
Goal: ‖(1 + ↑t * z)⁻¹‖ ≤ (1 - ‖z‖)⁻¹
-- Complex.integrable_pow_mul_norm_one_add_mul_inv
lemma integrable_pow_mul_norm_one_add_mul_inv (n : ℕ) {z : ℂ} (hz : ‖z‖ < 1) :
    IntervalIntegrable (fun t : ℝ ↦ t ^ n * ‖(1 + t * z)⁻¹‖) MeasureTheory.volume 0 1 
Arguments: (n : ℕ) {z : ℂ} (hz : ‖z‖ < 1)
Goal: IntervalIntegrable (fun t => t ^ n * ‖(1 + ↑t * z)⁻¹‖) MeasureTheory.volume 0 1
-- Complex.norm_log_sub_logTaylor_le
lemma norm_log_sub_logTaylor_le (n : ℕ) {z : ℂ} (hz : ‖z‖ < 1) :
    ‖log (1 + z) - logTaylor (n + 1) z‖ ≤ ‖z‖ ^ (n + 1) * (1 - ‖z‖)⁻¹ / (n + 1) 
Arguments: (n : ℕ) {z : ℂ} (hz : ‖z‖ < 1)
Goal: ‖Complex.log (1 + z) - Complex.logTaylor (n + 1) z‖ ≤ ‖z‖ ^ (n + 1) * (1 - ‖z‖)⁻¹ / (↑n + 1)
-- Complex.norm_log_one_add_sub_self_le
lemma norm_log_one_add_sub_self_le {z : ℂ} (hz : ‖z‖ < 1) :
    ‖log (1 + z) - z‖ ≤ ‖z‖ ^ 2 * (1 - ‖z‖)⁻¹ / 2 
Arguments: {z : ℂ} (hz : ‖z‖ < 1)
Goal: ‖Complex.log (1 + z) - z‖ ≤ ‖z‖ ^ 2 * (1 - ‖z‖)⁻¹ / 2
-- Complex.norm_log_one_sub_inv_add_logTaylor_neg_le
lemma norm_log_one_sub_inv_add_logTaylor_neg_le (n : ℕ) {z : ℂ} (hz : ‖z‖ < 1) :
    ‖log (1 - z)⁻¹ + logTaylor (n + 1) (-z)‖ ≤ ‖z‖ ^ (n + 1) * (1 - ‖z‖)⁻¹ / (n + 1) 
Arguments: (n : ℕ) {z : ℂ} (hz : ‖z‖ < 1)
Goal: ‖Complex.log (1 - z)⁻¹ + Complex.logTaylor (n + 1) (-z)‖ ≤ ‖z‖ ^ (n + 1) * (1 - ‖z‖)⁻¹ / (↑n + 1)
-- Complex.norm_log_one_sub_inv_sub_self_le
lemma norm_log_one_sub_inv_sub_self_le {z : ℂ} (hz : ‖z‖ < 1) :
    ‖log (1 - z)⁻¹ - z‖ ≤ ‖z‖ ^ 2 * (1 - ‖z‖)⁻¹ / 2 
Arguments: {z : ℂ} (hz : ‖z‖ < 1)
Goal: ‖Complex.log (1 - z)⁻¹ - z‖ ≤ ‖z‖ ^ 2 * (1 - ‖z‖)⁻¹ / 2
-- Complex.hasSum_taylorSeries_log
lemma hasSum_taylorSeries_log {z : ℂ} (hz : ‖z‖ < 1) :
    HasSum (fun n : ℕ ↦ (-1) ^ (n + 1) * z ^ n / n) (log (1 + z)) 
Arguments: {z : ℂ} (hz : ‖z‖ < 1)
Goal: HasSum (fun n => (-1) ^ (n + 1) * z ^ n / ↑n) (Complex.log (1 + z))
-- Complex.hasSum_taylorSeries_neg_log
lemma hasSum_taylorSeries_neg_log {z : ℂ} (hz : ‖z‖ < 1) :
    HasSum (fun n : ℕ ↦ z ^ n / n) (-log (1 - z)) 
Arguments: {z : ℂ} (hz : ‖z‖ < 1)
Goal: HasSum (fun n => z ^ n / ↑n) (-Complex.log (1 - z))
-- Real.Gamma_integrand_isLittleO
theorem Gamma_integrand_isLittleO (s : ℝ) :
    (fun x : ℝ => exp (-x) * x ^ s) =o[atTop] fun x : ℝ => exp (-(1 / 2) * x) 
Arguments: (s : ℝ)
Goal: (fun x => rexp (-x) * x ^ s) =o[Filter.atTop] fun x => rexp (-(1 / 2) * x)
-- Real.GammaIntegral_convergent
theorem GammaIntegral_convergent {s : ℝ} (h : 0 < s) :
    IntegrableOn (fun x : ℝ => exp (-x) * x ^ (s - 1)) (Ioi 0) 
Arguments: {s : ℝ} (h : 0 < s)
Goal: MeasureTheory.IntegrableOn (fun x => rexp (-x) * x ^ (s - 1)) (Set.Ioi 0) MeasureTheory.volume
-- Complex.GammaIntegral_convergent
theorem GammaIntegral_convergent {s : ℂ} (hs : 0 < s.re) :
    IntegrableOn (fun x => (-x).exp * x ^ (s - 1) : ℝ → ℂ) (Ioi 0) 
Arguments: {s : ℂ} (hs : 0 < s.re)
Goal: MeasureTheory.IntegrableOn (fun x => ↑(rexp (-x)) * ↑x ^ (s - 1)) (Set.Ioi 0) MeasureTheory.volume
-- Complex.GammaIntegral_conj
theorem GammaIntegral_conj (s : ℂ) : GammaIntegral (conj s) = conj (GammaIntegral s) 
Arguments: (s : ℂ)
Goal: ((starRingEnd ℂ) s).GammaIntegral = (starRingEnd ℂ) s.GammaIntegral
-- Complex.GammaIntegral_ofReal
theorem GammaIntegral_ofReal (s : ℝ) :
    GammaIntegral ↑s = ↑(∫ x : ℝ in Ioi 0, Real.exp (-x) * x ^ (s - 1)) 
Arguments: (s : ℝ)
Goal: (↑s).GammaIntegral = ↑(∫ (x : ℝ) in Set.Ioi 0, rexp (-x) * x ^ (s - 1))
-- Complex.GammaIntegral_one
@[simp]
theorem GammaIntegral_one : GammaIntegral 1 = 1 
Goal: Complex.GammaIntegral 1 = 1
-- Complex.tendsto_partialGamma
theorem tendsto_partialGamma {s : ℂ} (hs : 0 < s.re) :
    Tendsto (fun X : ℝ => partialGamma s X) atTop (𝓝 <| GammaIntegral s) 
Arguments: {s : ℂ} (hs : 0 < s.re)
Goal: Filter.Tendsto (fun X => s.partialGamma X) Filter.atTop (𝓝 s.GammaIntegral)
-- _private.Mathlib.Analysis.SpecialFunctions.Gamma.Basic.0.Complex.Gamma_integrand_intervalIntegrable
private theorem Gamma_integrand_intervalIntegrable (s : ℂ) {X : ℝ} (hs : 0 < s.re) (hX : 0 ≤ X) :
    IntervalIntegrable (fun x => (-x).exp * x ^ (s - 1) : ℝ → ℂ) volume 0 X 
Arguments: (s : ℂ) {X : ℝ} (hs : 0 < s.re) (hX : 0 ≤ X)
Goal: IntervalIntegrable (fun x => ↑(rexp (-x)) * ↑x ^ (s - 1)) MeasureTheory.volume 0 X
-- _private.Mathlib.Analysis.SpecialFunctions.Gamma.Basic.0.Complex.Gamma_integrand_deriv_integrable_A
private theorem Gamma_integrand_deriv_integrable_A {s : ℂ} (hs : 0 < s.re) {X : ℝ} (hX : 0 ≤ X) :
    IntervalIntegrable (fun x => -((-x).exp * x ^ s) : ℝ → ℂ) volume 0 X 
Arguments: {s : ℂ} (hs : 0 < s.re) {X : ℝ} (hX : 0 ≤ X)
Goal: IntervalIntegrable (fun x => -(↑(rexp (-x)) * ↑x ^ s)) MeasureTheory.volume 0 X