mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: x✝ : ℝ≥0
w y✝ z y : ℝ
hy : 0 < y
x : ℝ≥0
⊢ x = x

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx : 0 < x
hy : 0 < y
hz : z < 0
⊢ y ^ z ≤ x ↔ y ^ z ≤ x

pp: x y z : ℝ
hx : 0 < x
hy : 0 < y
hz : z < 0
⊢ 0 ≤ x

pp: x y z : ℝ
hx : 0 < x
hy : 0 < y
hz : z < 0
⊢ 0 < x ^ z⁻¹

Tactic:
]

Goal state after:
x y z : ℝ
hx : 0 < x
hy : 0 < y
hz : z < 0
⊢ 0 ≤ x

x y z : ℝ
hx : 0 < x
hy : 0 < y
hz : z < 0
⊢ 0 < x ^ z⁻¹
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx : 1 < x
hyz : y < z
⊢ rexp (log x * y) < x ^ z

Tactic:
]

Goal state after:
x y z : ℝ
hx : 1 < x
hyz : y < z
⊢ rexp (log x * y) < x ^ z
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: x : ℝ≥0∞
⊢ 0 < 1

Tactic:
]

Goal state after:
x : ℝ≥0∞
⊢ 0 < 1
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx : 1 < x
hyz : y < z
⊢ rexp (log x * y) < rexp (log x * z)

Tactic:
]

Goal state after:
x y z : ℝ
hx : 1 < x
hyz : y < z
⊢ rexp (log x * y) < rexp (log x * z)
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: y : ℝ
h : 0 < y
⊢ some 0 ^ y = some 0

Tactic:
]

Goal state after:
y : ℝ
h : 0 < y
⊢ some 0 ^ y = some 0
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: y : ℝ
h : y < 0
⊢ some 0 ^ y = ∞

Tactic:
]

Goal state after:
y : ℝ
h : y < 0
⊢ some 0 ^ y = ∞
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx : 1 < x
hyz : y < z
⊢ log x * y < log x * z

Tactic:
]

Goal state after:
x y z : ℝ
hx : 1 < x
hyz : y < z
⊢ log x * y < log x * z
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx : 1 ≤ x
hyz : y ≤ z
⊢ rexp (log x * y) ≤ x ^ z

Tactic:
]

Goal state after:
x y z : ℝ
hx : 1 ≤ x
hyz : y ≤ z
⊢ rexp (log x * y) ≤ x ^ z
Module: Mathlib.Analysis.SpecialFunctions.Pow.Continuity

Goal state before:
pp: case refine_2
z✝ : ℂ
hz✝ : 0 < z✝.re
hz₀ : z✝ ≠ 0
C : ℝ
hC : |z✝.im| < C
z : ℂ × ℂ
hz : |(im ∘ Prod.snd) z| < C
⊢ |z.1.arg| * |z.2.im| ≤ π * C

Tactic:
]

Goal state after:
case refine_2
z✝ : ℂ
hz✝ : 0 < z✝.re
hz₀ : z✝ ≠ 0
C : ℝ
hC : |z✝.im| < C
z : ℂ × ℂ
hz : |(im ∘ Prod.snd) z| < C
⊢ |z.1.arg| * |z.2.im| ≤ π * C
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx : 1 ≤ x
hyz : y ≤ z
⊢ rexp (log x * y) ≤ rexp (log x * z)

Tactic:
]

Goal state after:
x y z : ℝ
hx : 1 ≤ x
hyz : y ≤ z
⊢ rexp (log x * y) ≤ rexp (log x * z)
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: y : ℝ
⊢ ((if 0 < y then 0 else if y = 0 then 1 else ∞) * if 0 < y then 0 else if y = 0 then 1 else ∞) =
    if 0 < y then 0 else if y = 0 then 1 else ∞

Tactic:
]

Goal state after:
y : ℝ
⊢ ((if 0 < y then 0 else if y = 0 then 1 else ∞) * if 0 < y then 0 else if y = 0 then 1 else ∞) =
    if 0 < y then 0 else if y = 0 then 1 else ∞
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx : 1 ≤ x
hyz : y ≤ z
⊢ log x * y ≤ log x * z

Tactic:
]

Goal state after:
x y z : ℝ
hx : 1 ≤ x
hyz : y ≤ z
⊢ log x * y ≤ log x * z
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: x : ℝ≥0
h : x ≠ 0
y : ℝ
⊢ some (x ^ y) = ↑x ^ y

Tactic:
]

Goal state after:
x : ℝ≥0
h : x ≠ 0
y : ℝ
⊢ some (x ^ y) = ↑x ^ y
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx : 1 < x
x_pos : 0 < x
⊢ y ≤ z ↔ y ≤ z

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Continuity

Goal state before:
pp: z w : ℂ
h₁ : (0 < (z, w).1.re ∨ (z, w).1.im ≠ 0) ∨ (z, w).1 = 0
h₂ : 0 < (z, w).2.re
⊢ ContinuousAt (fun x ↦ x.1 ^ x.2) (z, w)

Tactic:
]

Goal state after:
z w : ℂ
h₁ : (0 < (z, w).1.re ∨ (z, w).1.im ≠ 0) ∨ (z, w).1 = 0
h₂ : 0 < (z, w).2.re
⊢ ContinuousAt (fun x ↦ x.1 ^ x.2) (z, w)
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx : 1 < x
⊢ ¬z ≤ y ↔ ¬z ≤ y

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx0 : 0 < x
hx1 : x < 1
hyz : z < y
⊢ rexp (log x * y) < x ^ z

Tactic:
]

Goal state after:
x y z : ℝ
hx0 : 0 < x
hx1 : x < 1
hyz : z < y
⊢ rexp (log x * y) < x ^ z
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: M : ℝ≥0
P : ℝ
hP : 0 ≤ P
⊢ (↑M).rpow P = (↑M).rpow P

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx0 : 0 < x
hx1 : x < 1
hyz : z < y
⊢ rexp (log x * y) < rexp (log x * z)

Tactic:
]

Goal state after:
x y z : ℝ
hx0 : 0 < x
hx1 : x < 1
hyz : z < y
⊢ rexp (log x * y) < rexp (log x * z)
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: x : ℝ
⊢ ↑(1 ^ x) = ↑1

Tactic:
]

Goal state after:
x : ℝ
⊢ ↑(1 ^ x) = ↑1
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx0 : 0 < x
hx1 : x < 1
hyz : z < y
⊢ log x * y < log x * z

Tactic:
]

Goal state after:
x y z : ℝ
hx0 : 0 < x
hx1 : x < 1
hyz : z < y
⊢ log x * y < log x * z
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx0 : 0 < x
hx1 : x ≤ 1
hyz : z ≤ y
⊢ rexp (log x * y) ≤ x ^ z

Tactic:
]

Goal state after:
x y z : ℝ
hx0 : 0 < x
hx1 : x ≤ 1
hyz : z ≤ y
⊢ rexp (log x * y) ≤ x ^ z
Module: Mathlib.Analysis.SpecialFunctions.Pow.Continuity

Goal state before:
pp: case inl
x : ℝ
y : ℂ
h : 0 < y.re ∨ x ≠ 0
hx : 0 < x
this : ContinuousAt (fun p ↦ (↑p.1, p.2)) (x, y)
⊢ 0 < (x, y).1

Tactic:
]

Goal state after:
case inl
x : ℝ
y : ℂ
h : 0 < y.re ∨ x ≠ 0
hx : 0 < x
this : ContinuousAt (fun p ↦ (↑p.1, p.2)) (x, y)
⊢ 0 < (x, y).1
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx0 : 0 < x
hx1 : x ≤ 1
hyz : z ≤ y
⊢ rexp (log x * y) ≤ rexp (log x * z)

Tactic:
]

Goal state after:
x y z : ℝ
hx0 : 0 < x
hx1 : x ≤ 1
hyz : z ≤ y
⊢ rexp (log x * y) ≤ rexp (log x * z)
Module: Mathlib.Analysis.SpecialFunctions.Pow.Continuity

Goal state before:
pp: y : ℂ
h : 0 < y.re ∨ 0 ≠ 0
⊢ ContinuousAt (fun p ↦ p.1 ^ p.2) (0, y)

Tactic:
]

Goal state after:
y : ℂ
h : 0 < y.re ∨ 0 ≠ 0
⊢ ContinuousAt (fun p ↦ p.1 ^ p.2) (0, y)
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx0 : 0 < x
hx1 : x ≤ 1
hyz : z ≤ y
⊢ log x * y ≤ log x * z

Tactic:
]

Goal state after:
x y z : ℝ
hx0 : 0 < x
hx1 : x ≤ 1
hyz : z ≤ y
⊢ log x * y ≤ log x * z
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx0 : 0 < x
hx1 : x < 1
⊢ z ≤ y ↔ z ≤ y

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx0 : 0 < x
hx1 : x < 1
⊢ ¬y ≤ z ↔ ¬y ≤ z

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x z : ℝ
hx1 : 0 ≤ x
hx2 : x < 1
hz : 0 < z
⊢ x ^ z < 1 ^ z

Tactic:
]

Goal state after:
x z : ℝ
hx1 : 0 ≤ x
hx2 : x < 1
hz : 0 < z
⊢ x ^ z < 1 ^ z
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x z : ℝ
hx1 : 0 ≤ x
hx2 : x ≤ 1
hz : 0 ≤ z
⊢ x ^ z ≤ 1 ^ z

Tactic:
]

Goal state after:
x z : ℝ
hx1 : 0 ≤ x
hx2 : x ≤ 1
hz : 0 ≤ z
⊢ x ^ z ≤ 1 ^ z
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x z : ℝ
hx : 1 < x
hz : 0 < z
⊢ 1 ^ z < x ^ z

Tactic:
]

Goal state after:
x z : ℝ
hx : 1 < x
hz : 0 < z
⊢ 1 ^ z < x ^ z
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x z : ℝ
hx : 1 ≤ x
hz : 0 ≤ z
⊢ 1 ^ z ≤ x ^ z

Tactic:
]

Goal state after:
x z : ℝ
hx : 1 ≤ x
hz : 0 ≤ z
⊢ 1 ^ z ≤ x ^ z
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y : ℝ
hx : 0 < x
⊢ 1 < x ∧ y < 0 ∨ x < 1 ∧ 0 < y ↔ 1 < x ∧ y < 0 ∨ x < 1 ∧ 0 < y

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: case inl
y : ℝ
hx : 0 ≤ 0
⊢ 0 ^ y < 1 ↔ 0 = 0 ∧ y ≠ 0 ∨ 1 < 0 ∧ y < 0 ∨ 0 < 1 ∧ 0 < y

Tactic:
rcases _root_.em (y = 0) with (rfl | hy) <;> simp [*, zero_lt_one]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Continuity

Goal state before:
pp: case inr.inr.hf
x : ℝ
y : ℂ
h : 0 < y.re ∨ x ≠ 0
hx : x < 0
A : ContinuousAt (fun p ↦ (-↑p.1, p.2)) (x, y)
⊢ (x, y).1 < 0

Tactic:
]

Goal state after:
case inr.inr.hf
x : ℝ
y : ℂ
h : 0 < y.re ∨ x ≠ 0
hx : x < 0
A : ContinuousAt (fun p ↦ (-↑p.1, p.2)) (x, y)
⊢ (x, y).1 < 0
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y : ℝ
hx : 0 ≤ x
hy : 0 < y
⊢ x ^ y < 1 ↔ x ^ y < 1

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y : ℝ
hx : 0 < x
⊢ 1 < x ∧ 0 < y ∨ x < 1 ∧ y < 0 ↔ 1 < x ∧ 0 < y ∨ x < 1 ∧ y < 0

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Continuity

Goal state before:
pp: x : ℝ≥0
y : ℝ
h : x ≠ 0 ∨ 0 < y
this : (fun p ↦ p.1 ^ p.2) = toNNReal ∘ (fun p ↦ p.1 ^ p.2) ∘ fun p ↦ (↑p.1, p.2)
⊢ ContinuousAt (toNNReal ∘ (fun p ↦ p.1 ^ p.2) ∘ fun p ↦ (↑p.1, p.2)) (x, y)

Tactic:
]

Goal state after:
x : ℝ≥0
y : ℝ
h : x ≠ 0 ∨ 0 < y
this : (fun p ↦ p.1 ^ p.2) = toNNReal ∘ (fun p ↦ p.1 ^ p.2) ∘ fun p ↦ (↑p.1, p.2)
⊢ ContinuousAt (toNNReal ∘ (fun p ↦ p.1 ^ p.2) ∘ fun p ↦ (↑p.1, p.2)) (x, y)
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: case top.inl
z : ℝ
hz : 0 ≤ z
hy : 0 ≤ 0
⊢ ∞ ^ z = ∞ ^ z

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: case inl.inl
z : ℝ
hx0 : 0 ≤ 0
hx1 : 0 ≤ 1
hyz : z ≤ 0
h : 0 = 0 → 0 = 0 → z = 0
⊢ 0 ^ 0 ≤ 0 ^ 0

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: case inl.inr
y z : ℝ
hyz : z ≤ y
hx0 : 0 ≤ 0
hx1 : 0 ≤ 1
h : 0 = 0 → y = 0 → z = 0
hy0 : y ≠ 0
⊢ 0 ≤ 0 ^ z

Tactic:
]

Goal state after:
case inl.inr
y z : ℝ
hyz : z ≤ y
hx0 : 0 ≤ 0
hx1 : 0 ≤ 1
h : 0 = 0 → y = 0 → z = 0
hy0 : y ≠ 0
⊢ 0 ≤ 0 ^ z
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: case top.inr.inl
y : ℝ
hy✝ : 0 ≤ y
hy : 0 < y
hz : 0 ≤ 0
⊢ ∞ ^ y = ∞ ^ y

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: case inl
x y p : ℝ
hx : 0 ≤ x
hy : 0 ≤ y
hp : 0 ≤ p
hxy : x ≤ y
⊢ y ^ p = y ^ p

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Continuity

Goal state before:
pp: x y : ℝ≥0
hy : 1 < y
m : ℕ
hm : x < y ^ m
⊢ ∀ᶠ (n : ℕ) in atTop, x ^ (1 / ↑n) ≤ y

Tactic:
]

Goal state after:
x y : ℝ≥0
hy : 1 < y
m : ℕ
hm : x < y ^ m
⊢ ∀ᶠ (n : ℕ) in atTop, x ^ (1 / ↑n) ≤ y
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: case inr
x : ℝ≥0∞
hx : x ≠ ∞
y z : ℝ
hyz : 0 < y + z
hx' : x ≠ 0
⊢ x ^ y * x ^ z = x ^ y * x ^ z

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: case inr
x y p : ℝ
hx : 0 ≤ x
hy : 0 ≤ y
hp : 0 ≤ p
hxy : y ≤ x
⊢ x ^ p = x ^ p

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: x : ℝ≥0∞
y z : ℝ
hx : x ≠ 0
h'x : x ≠ ∞
⊢ x ^ y * (x ^ z)⁻¹ = x ^ y * (x ^ z)⁻¹

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y : ℝ
h₁ : 1 ≤ x
h₂ : 1 ≤ y
⊢ x ≤ x ^ y

Tactic:
by
  simpa only [rpow_one] using rpow_le_rpow_of_exponent_le h₁ h₂

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y : ℝ
h₁ : 1 ≤ x
h₂ : y ≤ 1
⊢ x ^ y ≤ x

Tactic:
by
  simpa only [rpow_one] using rpow_le_rpow_of_exponent_le h₁ h₂

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Continuity

Goal state before:
pp: case neg
x y : ℝ≥0
hy : 1 < ↑y
this : ∀ᶠ (n : ℕ) in atTop, x ^ (1 / ↑n) ≤ y
n : ℕ
⊢ x ^ (1 / ↑n) ≤ y ↔ x ^ (1 / ↑n) ≤ y

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x : ℝ
hx : x ≠ 0
y : ℝ
hy : y ∈ {y | 0 ≤ y}
z : ℝ
hz : z ∈ {y | 0 ≤ y}
hyz : y ^ x = z ^ x
⊢ (z ^ x) ^ x⁻¹ = (z ^ x) ^ x⁻¹

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx : 0 ≤ x
hy : 0 ≤ y
hz : z ≠ 0
⊢ x = y ^ z ↔ x = y ^ z

pp: x y z : ℝ
hx : 0 ≤ x
hy : 0 ≤ y
hz : z ≠ 0
⊢ 0 ≤ x ^ z⁻¹

Tactic:
]

Goal state after:
x y z : ℝ
hx : 0 ≤ x
hy : 0 ≤ y
hz : z ≠ 0
⊢ 0 ≤ x ^ z⁻¹
Module: Mathlib.Analysis.SpecialFunctions.Pow.Continuity

Goal state before:
pp: case pos
x : ℝ≥0∞
y : ℝ
h : 0 < y
hx : x = ∞
⊢ Tendsto (fun a ↦ a ^ y) (𝓝 ∞) (𝓝 (∞ ^ y))

Tactic:
]

Goal state after:
case pos
x : ℝ≥0∞
y : ℝ
h : 0 < y
hx : x = ∞
⊢ Tendsto (fun a ↦ a ^ y) (𝓝 ∞) (𝓝 (∞ ^ y))
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: case e'_2
x y : ℝ≥0∞
z : ℝ
hz : z < 0 ∨ 0 < z
this :
  ∀ (x y : ℝ≥0∞) (z : ℝ),
    z < 0 ∨ 0 < z → x ≤ y → (x * y) ^ z = if (x = 0 ∧ y = ∞ ∨ x = ∞ ∧ y = 0) ∧ z < 0 then ∞ else x ^ z * y ^ z
hxy : ¬x ≤ y
⊢ x * y = y * x

Tactic:
simp only [mul_comm, and_comm, or_comm]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: case e'_3.h₁
x y : ℝ≥0∞
z : ℝ
hz : z < 0 ∨ 0 < z
this :
  ∀ (x y : ℝ≥0∞) (z : ℝ),
    z < 0 ∨ 0 < z → x ≤ y → (x * y) ^ z = if (x = 0 ∧ y = ∞ ∨ x = ∞ ∧ y = 0) ∧ z < 0 then ∞ else x ^ z * y ^ z
hxy : ¬x ≤ y
⊢ (x = 0 ∧ y = ∞ ∨ x = ∞ ∧ y = 0) ∧ z < 0 ↔ (y = 0 ∧ x = ∞ ∨ y = ∞ ∧ x = 0) ∧ z < 0

Tactic:
simp only [mul_comm, and_comm, or_comm]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: case e'_3.h₃
x y : ℝ≥0∞
z : ℝ
hz : z < 0 ∨ 0 < z
this :
  ∀ (x y : ℝ≥0∞) (z : ℝ),
    z < 0 ∨ 0 < z → x ≤ y → (x * y) ^ z = if (x = 0 ∧ y = ∞ ∨ x = ∞ ∧ y = 0) ∧ z < 0 then ∞ else x ^ z * y ^ z
hxy : ¬x ≤ y
a✝ : ¬((y = 0 ∧ x = ∞ ∨ y = ∞ ∧ x = 0) ∧ z < 0)
⊢ x ^ z * y ^ z = y ^ z * x ^ z

Tactic:
simp only [mul_comm, and_comm, or_comm]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx : 0 ≤ x
hy : 0 ≤ y
hz : z ≠ 0
⊢ x ^ z = y ↔ x ^ z = y

pp: x y z : ℝ
hx : 0 ≤ x
hy : 0 ≤ y
hz : z ≠ 0
⊢ 0 ≤ y ^ z⁻¹

Tactic:
]

Goal state after:
x y z : ℝ
hx : 0 ≤ x
hy : 0 ≤ y
hz : z ≠ 0
⊢ 0 ≤ y ^ z⁻¹
Module: Mathlib.Analysis.SpecialFunctions.Pow.Continuity

Goal state before:
pp: case neg
y : ℝ
h : 0 < y
x : ℝ≥0
⊢ ContinuousAt ((fun a ↦ a ^ y) ∘ ofNNReal) x

Tactic:
]

Goal state after:
case neg
y : ℝ
h : 0 < y
x : ℝ≥0
⊢ ContinuousAt ((fun a ↦ a ^ y) ∘ ofNNReal) x
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx : 0 < x
hy : 0 < y
⊢ log x ≤ z * log y ↔ log x ≤ z * log y

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx : 0 < x
hy : 0 < y
⊢ log x < z * log y ↔ log x < z * log y

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: case inr.inr.coe.top
z : ℝ
hz : z < 0 ∨ 0 < z
x✝ : ℝ≥0
hx0 : ¬x✝ = 0
hy0 : ∞ ≠ 0
hxy : ↑x✝ ≤ ∞
⊢ (↑x✝ * ∞) ^ z = if (↑x✝ = 0 ∧ ∞ = ∞ ∨ ↑x✝ = ∞ ∧ ∞ = 0) ∧ z < 0 then ∞ else ↑x✝ ^ z * ∞ ^ z

Tactic:
]

Goal state after:
case inr.inr.coe.top
z : ℝ
hz : z < 0 ∨ 0 < z
x✝ : ℝ≥0
hx0 : ¬x✝ = 0
hy0 : ∞ ≠ 0
hxy : ↑x✝ ≤ ∞
⊢ (↑x✝ * ∞) ^ z = if (↑x✝ = 0 ∧ ∞ = ∞ ∨ ↑x✝ = ∞ ∧ ∞ = 0) ∧ z < 0 then ∞ else ↑x✝ ^ z * ∞ ^ z
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx : 0 < x
hy : 0 < y
⊢ z * log x ≤ log y ↔ z * log x ≤ log y

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx : 0 < x
hy : 0 < y
⊢ x ^ z ≤ y ↔ z * log x ≤ log y

Tactic:
by
  rw [← log_le_log_iff (rpow_pos_of_pos hx _) hy, log_rpow hx]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y : ℝ
n : ℕ
hx : 0 < x
hy : 0 < y
⊢ x ^ n ≤ y ↔ x ^ n ≤ y

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y : ℝ
n : ℤ
hx : 0 < x
hy : 0 < y
⊢ x ^ n ≤ y ↔ x ^ n ≤ y

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: case inr.inr.coe.coe
z : ℝ
x✝¹ x✝ : ℝ≥0
hz : z < 0 ∨ 0 < z
hx0 : ¬x✝¹ = 0
hy0 : ¬x✝ = 0
hxy : x✝¹ ≤ x✝
⊢ ↑(x✝¹ ^ z * x✝ ^ z) = if (False ∧ ↑x✝ = ∞ ∨ ↑x✝¹ = ∞ ∧ False) ∧ z < 0 then ∞ else ↑x✝¹ ^ z * ↑x✝ ^ z

Tactic:
]

Goal state after:
case inr.inr.coe.coe
z : ℝ
x✝¹ x✝ : ℝ≥0
hz : z < 0 ∨ 0 < z
hx0 : ¬x✝¹ = 0
hy0 : ¬x✝ = 0
hxy : x✝¹ ≤ x✝
⊢ ↑(x✝¹ ^ z * x✝ ^ z) = if (False ∧ ↑x✝ = ∞ ∨ ↑x✝¹ = ∞ ∧ False) ∧ z < 0 then ∞ else ↑x✝¹ ^ z * ↑x✝ ^ z
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx : 0 < x
hy : 0 < y
⊢ z * log x < log y ↔ z * log x < log y

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y : ℝ
n : ℕ
hx : 0 < x
hy : 0 < y
⊢ x ^ n < y ↔ x ^ n < y

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: case insert
ι : Type u_1
f : ι → ℝ≥0∞
r : ℝ
i : ι
s : Finset ι
hi : i ∉ s
ih : (∀ i ∈ s, f i ≠ ∞) → ∏ i ∈ s, f i ^ r = (∏ i ∈ s, f i) ^ r
hf : ∀ i_1 ∈ insert i s, f i_1 ≠ ∞
h2f : ∀ i ∈ s, f i ≠ ∞
⊢ (f i * ∏ i ∈ s, f i) ^ r = (f i * ∏ x ∈ s, f x) ^ r

pp: case insert.hy
ι : Type u_1
f : ι → ℝ≥0∞
r : ℝ
i : ι
s : Finset ι
hi : i ∉ s
ih : (∀ i ∈ s, f i ≠ ∞) → ∏ i ∈ s, f i ^ r = (∏ i ∈ s, f i) ^ r
hf : ∀ i_1 ∈ insert i s, f i_1 ≠ ∞
h2f : ∀ i ∈ s, f i ≠ ∞
⊢ ∏ i ∈ s, f i ≠ ∞

Tactic:
]

Goal state after:
case insert.hy
ι : Type u_1
f : ι → ℝ≥0∞
r : ℝ
i : ι
s : Finset ι
hi : i ∉ s
ih : (∀ i ∈ s, f i ≠ ∞) → ∏ i ∈ s, f i ^ r = (∏ i ∈ s, f i) ^ r
hf : ∀ i_1 ∈ insert i s, f i_1 ≠ ∞
h2f : ∀ i ∈ s, f i ≠ ∞
⊢ ∏ i ∈ s, f i ≠ ∞
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y : ℝ
n : ℤ
hx : 0 < x
hy : 0 < y
⊢ x ^ n < y ↔ x ^ n < y

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: case inr.inr.inr
x : ℝ≥0∞
y : ℝ
hy : y < 0 ∨ 0 < y
h0 : x ≠ 0
h_top : x ≠ ∞
⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: x y : ℝ≥0∞
z : ℝ
hz : 0 ≤ z
⊢ x ^ z * (y ^ z)⁻¹ = x ^ z * (y ^ z)⁻¹

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y : ℝ
hx : 0 < x
⊢ 1 ≤ x ∧ y ≤ 0 ∨ x ≤ 1 ∧ 0 ≤ y ↔ 1 ≤ x ∧ y ≤ 0 ∨ x ≤ 1 ∧ 0 ≤ y

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x t : ℝ
h1 : 0 < x
h2 : x ≤ 1
ht : 0 < t
⊢ |log x * x ^ t| < 1 / t

Tactic:
lt_div_iff₀ ht

Goal state after:
x t : ℝ
h1 : 0 < x
h2 : x ≤ 1
ht : 0 < t
⊢ |log x * x ^ t| * t < 1
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x t : ℝ
h1 : 0 < x
h2 : x ≤ 1
ht : 0 < t
⊢ |log x * x ^ t| * t < 1

Tactic:
]

Goal state after:
x t : ℝ
h1 : 0 < x
h2 : x ≤ 1
ht : 0 < t
⊢ |log x * x ^ t| * t < 1
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: y : ℝ
hy : 0 < y
x : ℝ≥0∞
⊢ x = x

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: case inl
x y : ℝ≥0∞
p : ℝ
hp : 0 ≤ p
hxy : x ≤ y
⊢ y ^ p = y ^ p

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: case inr
x y : ℝ≥0∞
p : ℝ
hp : 0 ≤ p
hxy : y ≤ x
⊢ x ^ p = x ^ p

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: x y : ℝ≥0∞
z : ℝ
hz : 0 < z
⊢ x ^ 1 ≤ y ^ z⁻¹ ↔ x ^ z ≤ y

Tactic:
]

Goal state after:
x y : ℝ≥0∞
z : ℝ
hz : 0 < z
⊢ x ^ 1 ≤ y ^ z⁻¹ ↔ x ^ z ≤ y
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: x y : ℝ≥0∞
z : ℝ
hz : 0 < z
⊢ x ^ (z * z⁻¹) ≤ y ^ z⁻¹ ↔ x ^ z ≤ y

Tactic:
]

Goal state after:
x y : ℝ≥0∞
z : ℝ
hz : 0 < z
⊢ x ^ (z * z⁻¹) ≤ y ^ z⁻¹ ↔ x ^ z ≤ y
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x t : ℝ
h1 : 0 < x
h2 : x ≤ 1
ht : 0 < t
this : |log x * x ^ t| * t < 1
⊢ |log x * x ^ t| * t < 1

Tactic:
]

Goal state after:
x t : ℝ
h1 : 0 < x
h2 : x ≤ 1
ht : 0 < t
this : |log x * x ^ t| * t < 1
⊢ |log x * x ^ t| * t < 1
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: x y : ℝ≥0∞
z : ℝ
hz : 0 < z
⊢ x ^ z ≤ y ↔ x ^ z ≤ y

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: x y : ℝ≥0∞
z : ℝ
hz : 0 < z
⊢ x ^ 1 < y ^ z⁻¹ ↔ x ^ z < y

Tactic:
]

Goal state after:
x y : ℝ≥0∞
z : ℝ
hz : 0 < z
⊢ x ^ 1 < y ^ z⁻¹ ↔ x ^ z < y
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: x y : ℝ≥0∞
z : ℝ
hz : 0 < z
⊢ x ^ (z * z⁻¹) < y ^ z⁻¹ ↔ x ^ z < y

Tactic:
]

Goal state after:
x y : ℝ≥0∞
z : ℝ
hz : 0 < z
⊢ x ^ (z * z⁻¹) < y ^ z⁻¹ ↔ x ^ z < y
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: x y : ℝ≥0∞
z : ℝ
hz : 0 < z
⊢ x ^ z < y ↔ x ^ z < y

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: case inl
ε : ℝ
hε : 0 < ε
hx : 0 ≤ 0
⊢ 0 ≤ 0

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: x y : ℝ≥0∞
z : ℝ
hz : 0 < z
⊢ x ^ z⁻¹ ≤ y ^ 1 ↔ x ≤ y ^ z

Tactic:
]

Goal state after:
x y : ℝ≥0∞
z : ℝ
hz : 0 < z
⊢ x ^ z⁻¹ ≤ y ^ 1 ↔ x ≤ y ^ z
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: x y : ℝ≥0∞
z : ℝ
hz : 0 < z
⊢ x ^ z⁻¹ ≤ y ^ (z * z⁻¹) ↔ x ≤ y ^ z

Tactic:
]

Goal state after:
x y : ℝ≥0∞
z : ℝ
hz : 0 < z
⊢ x ^ z⁻¹ ≤ y ^ (z * z⁻¹) ↔ x ≤ y ^ z
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: x y : ℝ≥0∞
z : ℝ
hz : 0 < z
⊢ x ≤ y ^ z ↔ x ≤ y ^ z

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: case inr
x ε : ℝ
hx : 0 ≤ x
hε : 0 < ε
h : 0 < x
⊢ ε * log x ≤ x ^ ε

Tactic:
]

Goal state after:
case inr
x ε : ℝ
hx : 0 ≤ x
hε : 0 < ε
h : 0 < x
⊢ ε * log x ≤ x ^ ε
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: y z : ℝ
hyz : y < z
x : ℝ≥0
hx : 1 < x
⊢ ↑x ^ y < ↑x ^ z

Tactic:
]

Goal state after:
y z : ℝ
hyz : y < z
x : ℝ≥0
hx : 1 < x
⊢ ↑x ^ y < ↑x ^ z
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx₀ : 0 < x
hx₁ : x ≠ 1
H : y = z
⊢ x ^ z = x ^ z

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: p : Nat.Primes
s : ℂ
hs : 1 < s.re
⊢ -s.re ≠ 0

Tactic:
]

Goal state after:
p : Nat.Primes
s : ℂ
hs : 1 < s.re
⊢ -s.re ≠ 0
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: p : Nat.Primes
s : ℂ
hs : 1 < s.re
⊢ ↑↑p ^ (-s).re ≤ 1 / 2

Tactic:
]

Goal state after:
p : Nat.Primes
s : ℂ
hs : 1 < s.re
⊢ ↑↑p ^ (-s).re ≤ 1 / 2
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: p : Nat.Primes
s : ℂ
hs : 1 < s.re
⊢ -s.re ≤ 0

Tactic:
]

Goal state after:
p : Nat.Primes
s : ℂ
hs : 1 < s.re
⊢ -s.re ≤ 0
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: p : Nat.Primes
s : ℂ
hs : 1 < s.re
⊢ 2 ^ (-s).re ≤ 2 ^ (-1)

Tactic:
]

Goal state after:
p : Nat.Primes
s : ℂ
hs : 1 < s.re
⊢ 2 ^ (-s).re ≤ 2 ^ (-1)
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: p : ℕ
hp : Nat.Prime p
s : ℂ
hs : 1 < s.re
H : 1 = ↑p ^ (-s)
this : 1 ≤ 1 / 2
⊢ False

Tactic:
]

Goal state after:
p : ℕ
hp : Nat.Prime p
s : ℂ
hs : 1 < s.re
H : 1 = ↑p ^ (-s)
this : 1 ≤ 1 / 2
⊢ False
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: case inl
ε : ℝ
hε : 0 < ε
⊢ 0 ≤ 0

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: case inr
n : ℕ
ε : ℝ
hε : 0 < ε
h : n > 0
⊢ Real.log ↑n ≤ ↑n ^ ε / ε

Tactic:
]

Goal state after:
case inr
n : ℕ
ε : ℝ
hε : 0 < ε
h : n > 0
⊢ Real.log ↑n ≤ ↑n ^ ε / ε
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: case inl
x : ℝ
h : 0 ≤ x
⊢ x = x ^ (1 / 2 * ↑2)

Tactic:
]

Goal state after:
case inl
x : ℝ
h : 0 ≤ x
⊢ x = x ^ (1 / 2 * ↑2)
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: case inr
x : ℝ
h : x < 0
this : 1 / 2 * π = π / 2
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x r : ℝ
hx : 0 ≤ x
⊢ x ^ (r / 2) = x ^ (1 / 2 * r)

Tactic:
]

Goal state after:
x r : ℝ
hx : 0 ≤ x
⊢ x ^ (r / 2) = x ^ (1 / 2 * r)