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

Goal state before:
pp: case refine_1
h : DifferentiableAt ℝ binEntropy 0
⊢ False

Tactic:
unfold binEntropy at h

Goal state after:
case refine_1
h : DifferentiableAt ℝ (fun p ↦ p * log p⁻¹ + (1 - p) * log (1 - p)⁻¹) 0
⊢ False
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: case refine_2
h : DifferentiableAt ℝ binEntropy 1
⊢ False

Tactic:
unfold binEntropy at h

Goal state after:
case refine_2
h : DifferentiableAt ℝ (fun p ↦ p * log p⁻¹ + (1 - p) * log (1 - p)⁻¹) 1
⊢ False
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: case refine_1
h : DifferentiableAt ℝ (fun p ↦ p * log p⁻¹) 0
⊢ False

pp: case refine_1
h : DifferentiableAt ℝ (fun p ↦ p * log p⁻¹ + (1 - p) * log (1 - p)⁻¹) 0
⊢ DifferentiableAt ℝ (fun p ↦ (1 - p) * log (1 - p)⁻¹) 0

Tactic:
]

Goal state after:
case refine_1
h : DifferentiableAt ℝ (fun p ↦ p * log p⁻¹) 0
⊢ False

case refine_1
h : DifferentiableAt ℝ (fun p ↦ p * log p⁻¹ + (1 - p) * log (1 - p)⁻¹) 0
⊢ DifferentiableAt ℝ (fun p ↦ (1 - p) * log (1 - p)⁻¹) 0
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: case refine_2
h : DifferentiableAt ℝ (fun x ↦ (1 - (1 - x)) * log (1 - (1 - x))⁻¹) (1 - 1)
⊢ False

pp: case refine_2
h : DifferentiableAt ℝ (fun p ↦ p * log p⁻¹ + (1 - p) * log (1 - p)⁻¹) 1
⊢ DifferentiableAt ℝ (fun p ↦ p * log p⁻¹) 1

Tactic:
]

Goal state after:
case refine_2
h : DifferentiableAt ℝ (fun x ↦ (1 - (1 - x)) * log (1 - (1 - x))⁻¹) (1 - 1)
⊢ False

case refine_2
h : DifferentiableAt ℝ (fun p ↦ p * log p⁻¹ + (1 - p) * log (1 - p)⁻¹) 1
⊢ DifferentiableAt ℝ (fun p ↦ p * log p⁻¹) 1
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: case pos
p : ℝ
hp₀ : p ≠ 0
hp₁ : 1 - p ≠ 0
⊢ deriv binEntropy p = log (1 - p) - log p

Tactic:
]

Goal state after:
case pos
p : ℝ
hp₀ : p ≠ 0
hp₁ : 1 - p ≠ 0
⊢ deriv binEntropy p = log (1 - p) - log p
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: case pos
p : ℝ
hp₀ : p ≠ 0
hp₁ : 1 - p ≠ 0
⊢ -log p - 1 + -(-log (1 - p) - 1) = log (1 - p) - log p

pp: case pos.hf
p : ℝ
hp₀ : p ≠ 0
hp₁ : 1 - p ≠ 0
⊢ DifferentiableAt ℝ negMulLog p

pp: case pos.hg
p : ℝ
hp₀ : p ≠ 0
hp₁ : 1 - p ≠ 0
⊢ DifferentiableAt ℝ (fun p ↦ (1 - p).negMulLog) p

Tactic:
]

Goal state after:
case pos
p : ℝ
hp₀ : p ≠ 0
hp₁ : 1 - p ≠ 0
⊢ -log p - 1 + -(-log (1 - p) - 1) = log (1 - p) - log p

case pos.hf
p : ℝ
hp₀ : p ≠ 0
hp₁ : 1 - p ≠ 0
⊢ DifferentiableAt ℝ negMulLog p

case pos.hg
p : ℝ
hp₀ : p ≠ 0
hp₁ : 1 - p ≠ 0
⊢ DifferentiableAt ℝ (fun p ↦ (1 - p).negMulLog) p
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: case neg
p : ℝ
hp : ¬(p ≠ 0 ∧ p ≠ 1)
⊢ 0 = log (1 - p) - log p

Tactic:
]

Goal state after:
case neg
p : ℝ
hp : ¬(p ≠ 0 ∧ p ≠ 1)
⊢ 0 = log (1 - p) - log p
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: q : ℕ
p : ℝ
hp₀ : 0 ≤ p
hp₁ : p ≤ 1
⊢ 0 ≤ qaryEntropy q p

Tactic:
obtain rfl | hp₀ := hp₀.eq_or_lt
  · simp
  obtain rfl | hp₁ := hp₁.eq_or_lt
  · simpa [qaryEntropy, -Int.cast_sub] using log_intCast_nonneg _
  exact (qaryEntropy_pos hp₀ hp₁).le

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: q : ℕ
p : ℝ
hp₀ : p ≠ 0
hp₁ : p ≠ 1
⊢ deriv (fun p ↦ p * log ↑(↑q - 1)) p + deriv binEntropy p = log (↑q - 1) + log (1 - p) - log p

pp: case hf
q : ℕ
p : ℝ
hp₀ : p ≠ 0
hp₁ : p ≠ 1
⊢ DifferentiableAt ℝ (fun p ↦ p * log ↑(↑q - 1)) p

pp: case hg
q : ℕ
p : ℝ
hp₀ : p ≠ 0
hp₁ : p ≠ 1
⊢ DifferentiableAt ℝ binEntropy p

Tactic:
]

Goal state after:
q : ℕ
p : ℝ
hp₀ : p ≠ 0
hp₁ : p ≠ 1
⊢ deriv (fun p ↦ p * log ↑(↑q - 1)) p + deriv binEntropy p = log (↑q - 1) + log (1 - p) - log p

case hf
q : ℕ
p : ℝ
hp₀ : p ≠ 0
hp₁ : p ≠ 1
⊢ DifferentiableAt ℝ (fun p ↦ p * log ↑(↑q - 1)) p

case hg
q : ℕ
p : ℝ
hp₀ : p ≠ 0
hp₁ : p ≠ 1
⊢ DifferentiableAt ℝ binEntropy p
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: q : ℕ
this : (fun p ↦ log (↑q - 1) + log (1 - p) - log p) = fun p ↦ log (↑q - 1) + (log (1 - p) - log p)
⊢ Tendsto (fun p ↦ log (↑q - 1) + (log (1 - p) - log p)) (𝓝[<] 1) atBot

Tactic:
]

Goal state after:
q : ℕ
this : (fun p ↦ log (↑q - 1) + log (1 - p) - log p) = fun p ↦ log (↑q - 1) + (log (1 - p) - log p)
⊢ Tendsto (fun p ↦ log (↑q - 1) + (log (1 - p) - log p)) (𝓝[<] 1) atBot
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: q : ℕ
this : (fun p ↦ log (↑q - 1) + log (1 - p) - log p) = fun p ↦ log (↑q - 1) + (log (1 - p) - log p)
⊢ Tendsto (fun p ↦ log (↑q - 1) + (log (1 - p) - log p)) (𝓝[>] 0) atTop

Tactic:
]

Goal state after:
q : ℕ
this : (fun p ↦ log (↑q - 1) + log (1 - p) - log p) = fun p ↦ log (↑q - 1) + (log (1 - p) - log p)
⊢ Tendsto (fun p ↦ log (↑q - 1) + (log (1 - p) - log p)) (𝓝[>] 0) atTop
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: q : ℕ
p : ℝ
xne0 : p ≠ 0
xne1 : p ≠ 1
this : ∀ᶠ (y : ℝ) in 𝓝 p, deriv (fun p ↦ qaryEntropy q p) y = log (↑q - 1) + log (1 - y) - log y
⊢ deriv (fun x ↦ log (↑q - 1) + log (1 - x)) p - deriv log p = -1 / (p * (1 - p))

pp: q : ℕ
p : ℝ
xne0 : p ≠ 0
xne1 : p ≠ 1
this : ∀ᶠ (y : ℝ) in 𝓝 p, deriv (fun p ↦ qaryEntropy q p) y = log (↑q - 1) + log (1 - y) - log y
⊢ DifferentiableAt ℝ (fun x ↦ log (↑q - 1) + log (1 - x)) p

Tactic:
]

Goal state after:
q : ℕ
p : ℝ
xne0 : p ≠ 0
xne1 : p ≠ 1
this : ∀ᶠ (y : ℝ) in 𝓝 p, deriv (fun p ↦ qaryEntropy q p) y = log (↑q - 1) + log (1 - y) - log y
⊢ deriv (fun x ↦ log (↑q - 1) + log (1 - x)) p - deriv log p = -1 / (p * (1 - p))

q : ℕ
p : ℝ
xne0 : p ≠ 0
xne1 : p ≠ 1
this : ∀ᶠ (y : ℝ) in 𝓝 p, deriv (fun p ↦ qaryEntropy q p) y = log (↑q - 1) + log (1 - y) - log y
⊢ DifferentiableAt ℝ (fun x ↦ log (↑q - 1) + log (1 - x)) p
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: q : ℕ
p : ℝ
xne0 : p ≠ 0
xne1 : p ≠ 1
this : ∀ᶠ (y : ℝ) in 𝓝 p, deriv (fun p ↦ qaryEntropy q p) y = log (↑q - 1) + log (1 - y) - log y
⊢ deriv (fun x ↦ log (↑q - 1) + log (1 - x)) p - deriv (fun x ↦ x) p / p = -1 / (p * (1 - p))

Tactic:
]

Goal state after:
q : ℕ
p : ℝ
xne0 : p ≠ 0
xne1 : p ≠ 1
this : ∀ᶠ (y : ℝ) in 𝓝 p, deriv (fun p ↦ qaryEntropy q p) y = log (↑q - 1) + log (1 - y) - log y
⊢ deriv (fun x ↦ log (↑q - 1) + log (1 - x)) p - deriv (fun x ↦ x) p / p = -1 / (p * (1 - p))
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: case neg
q : ℕ
p : ℝ
is_x_where_nondiff : ¬(p ≠ 0 ∧ p ≠ 1)
this : p = 0 ∨ p = 1
⊢ 0 = -1 / (p * (1 - p))

pp: case neg
q : ℕ
p : ℝ
is_x_where_nondiff : ¬(p ≠ 0 ∧ p ≠ 1)
this : p = 0 ∨ p = 1
⊢ ¬DifferentiableAt ℝ (deriv (qaryEntropy q)) p

Tactic:
]

Goal state after:
case neg
q : ℕ
p : ℝ
is_x_where_nondiff : ¬(p ≠ 0 ∧ p ≠ 1)
this : p = 0 ∨ p = 1
⊢ 0 = -1 / (p * (1 - p))

case neg
q : ℕ
p : ℝ
is_x_where_nondiff : ¬(p ≠ 0 ∧ p ≠ 1)
this : p = 0 ∨ p = 1
⊢ ¬DifferentiableAt ℝ (deriv (qaryEntropy q)) p
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc 0 (1 - 1 / ↑q)
p2 : ℝ
hp2 : p2 ∈ Icc 0 (1 - 1 / ↑q)
p1le2 : p1 < p2
p : ℝ
this✝ : 2 ≤ ↑q
zero_le_qinv : 0 < (↑q)⁻¹
this : 0 < 1 - p
hp : 0 < p ∧ p < 1 - (↑q)⁻¹
⊢ 0 < log (↑q - 1) + log (1 - p) - log p

pp: q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc 0 (1 - 1 / ↑q)
p2 : ℝ
hp2 : p2 ∈ Icc 0 (1 - 1 / ↑q)
p1le2 : p1 < p2
p : ℝ
this✝ : 2 ≤ ↑q
zero_le_qinv : 0 < (↑q)⁻¹
this : 0 < 1 - p
hp : 0 < p ∧ p < 1 - (↑q)⁻¹
⊢ p ≠ 1

Tactic:
]

Goal state after:
q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc 0 (1 - 1 / ↑q)
p2 : ℝ
hp2 : p2 ∈ Icc 0 (1 - 1 / ↑q)
p1le2 : p1 < p2
p : ℝ
this✝ : 2 ≤ ↑q
zero_le_qinv : 0 < (↑q)⁻¹
this : 0 < 1 - p
hp : 0 < p ∧ p < 1 - (↑q)⁻¹
⊢ 0 < log (↑q - 1) + log (1 - p) - log p

q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc 0 (1 - 1 / ↑q)
p2 : ℝ
hp2 : p2 ∈ Icc 0 (1 - 1 / ↑q)
p1le2 : p1 < p2
p : ℝ
this✝ : 2 ≤ ↑q
zero_le_qinv : 0 < (↑q)⁻¹
this : 0 < 1 - p
hp : 0 < p ∧ p < 1 - (↑q)⁻¹
⊢ p ≠ 1
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc 0 (1 - 1 / ↑q)
p2 : ℝ
hp2 : p2 ∈ Icc 0 (1 - 1 / ↑q)
p1le2 : p1 < p2
p : ℝ
this✝ : 2 ≤ ↑q
zero_le_qinv : 0 < (↑q)⁻¹
this : 0 < 1 - p
hp : 0 < p ∧ p < 1 - (↑q)⁻¹
⊢ log p < log ((↑q - 1) * (1 - p))

Tactic:
]

Goal state after:
q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc 0 (1 - 1 / ↑q)
p2 : ℝ
hp2 : p2 ∈ Icc 0 (1 - 1 / ↑q)
p1le2 : p1 < p2
p : ℝ
this✝ : 2 ≤ ↑q
zero_le_qinv : 0 < (↑q)⁻¹
this : 0 < 1 - p
hp : 0 < p ∧ p < 1 - (↑q)⁻¹
⊢ log p < log ((↑q - 1) * (1 - p))
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: θ : ℝ
n : ℕ
⊢ -(2 * 2 * ((↑(n + 1) + 1) * (0 ^ (n + 1) * cos θ))) + 2 * (↑(n + 1) + 1) * (2 * ↑(n + 1) + 1) * I (n + 1) θ -
      4 * (↑(n + 1) + 1) * ↑(n + 1) * I (n + 1 - 1) θ =
    2 * (↑n + 2) * (2 * ↑n + 3) * I (n + 1) θ - 4 * (↑n + 2) * (↑n + 1) * I n θ

Tactic:
]

Goal state after:
θ : ℝ
n : ℕ
⊢ -(2 * 2 * ((↑(n + 1) + 1) * (0 ^ (n + 1) * cos θ))) + 2 * (↑(n + 1) + 1) * (2 * ↑(n + 1) + 1) * I (n + 1) θ -
      4 * (↑(n + 1) + 1) * ↑(n + 1) * I (n + 1 - 1) θ =
    2 * (↑n + 2) * (2 * ↑n + 3) * I (n + 1) θ - 4 * (↑n + 2) * (↑n + 1) * I n θ
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.ConjSqrt

Goal state before:
pp: A : Type u_1
inst✝¹⁰ : PartialOrder A
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : StarOrderedRing A
inst✝⁵ : Algebra ℝ A
inst✝⁴ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : SeparatelyContinuousMul A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
c a : A
hc : IsStrictlyPositive c
hc' : IsSelfAdjoint (sqrt c)
⊢ IsStrictlyPositive (sqrt c * a * sqrt c) ↔ IsStrictlyPositive a

Tactic:
]

Goal state after:
A : Type u_1
inst✝¹⁰ : PartialOrder A
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : StarOrderedRing A
inst✝⁵ : Algebra ℝ A
inst✝⁴ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : SeparatelyContinuousMul A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
c a : A
hc : IsStrictlyPositive c
hc' : IsSelfAdjoint (sqrt c)
⊢ IsStrictlyPositive (sqrt c * a * sqrt c) ↔ IsStrictlyPositive a
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: θ : ℝ
⊢ -(2 * 2 * ((↑0 + 1) * (0 ^ 0 * cos θ))) * θ + 2 * (↑0 + 1) * (2 * ↑0 + 1) * (2 * sin θ) -
      4 * (↑0 + 1) * ↑0 * I (0 - 1) θ * θ =
    4 * sin θ - 4 * θ * cos θ

Tactic:
]

Goal state after:
θ : ℝ
⊢ -(2 * 2 * ((↑0 + 1) * (0 ^ 0 * cos θ))) * θ + 2 * (↑0 + 1) * (2 * ↑0 + 1) * (2 * sin θ) -
      4 * (↑0 + 1) * ↑0 * I (0 - 1) θ * θ =
    4 * sin θ - 4 * θ * cos θ
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: n : ℕ
⊢ ((2 * (2 * ↑n + 3)) • sinPoly (n + 1) + (monomial 2) (-4) * sinPoly n).natDegree ≤ n + 2

Tactic:
]

Goal state after:
n : ℕ
⊢ ((2 * (2 * ↑n + 3)) • sinPoly (n + 1) + (monomial 2) (-4) * sinPoly n).natDegree ≤ n + 2
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: n : ℕ
⊢ ((2 * (2 * ↑n + 3)) • cosPoly (n + 1) + (monomial 2) (-4) * cosPoly n).natDegree ≤ n + 2

Tactic:
]

Goal state after:
n : ℕ
⊢ ((2 * (2 * ↑n + 3)) • cosPoly (n + 1) + (monomial 2) (-4) * cosPoly n).natDegree ≤ n + 2
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.ConjSqrt

Goal state before:
pp: A : Type u_1
inst✝¹⁰ : PartialOrder A
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : StarOrderedRing A
inst✝⁵ : Algebra ℝ A
inst✝⁴ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : SeparatelyContinuousMul A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
c : A
hc : 0 ≤ c
⊢ c = c

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc (1 - 1 / ↑q) 1
p2 : ℝ
hp2 : p2 ∈ Icc (1 - 1 / ↑q) 1
p1le2 : p1 < p2
p : ℝ
this : 2 ≤ ↑q
qinv_lt_1 : (↑q)⁻¹ < 1
zero_lt_1_sub_p : 0 < 1 - p
hp : 1 - (↑q)⁻¹ < p ∧ p < 1
⊢ log (↑q - 1) + log (1 - p) - log p < 0

pp: q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc (1 - 1 / ↑q) 1
p2 : ℝ
hp2 : p2 ∈ Icc (1 - 1 / ↑q) 1
p1le2 : p1 < p2
p : ℝ
this : 2 ≤ ↑q
qinv_lt_1 : (↑q)⁻¹ < 1
zero_lt_1_sub_p : 0 < 1 - p
hp : 1 - (↑q)⁻¹ < p ∧ p < 1
⊢ p ≠ 1

Tactic:
]

Goal state after:
q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc (1 - 1 / ↑q) 1
p2 : ℝ
hp2 : p2 ∈ Icc (1 - 1 / ↑q) 1
p1le2 : p1 < p2
p : ℝ
this : 2 ≤ ↑q
qinv_lt_1 : (↑q)⁻¹ < 1
zero_lt_1_sub_p : 0 < 1 - p
hp : 1 - (↑q)⁻¹ < p ∧ p < 1
⊢ log (↑q - 1) + log (1 - p) - log p < 0

q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc (1 - 1 / ↑q) 1
p2 : ℝ
hp2 : p2 ∈ Icc (1 - 1 / ↑q) 1
p1le2 : p1 < p2
p : ℝ
this : 2 ≤ ↑q
qinv_lt_1 : (↑q)⁻¹ < 1
zero_lt_1_sub_p : 0 < 1 - p
hp : 1 - (↑q)⁻¹ < p ∧ p < 1
⊢ p ≠ 1
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: θ : ℝ
n : ℕ
⊢ I (n + 2) θ * θ ^ (2 * (n + 2) + 1) = I (n + 2) θ * θ ^ 2 * θ ^ (2 * n + 3)

Tactic:
by ring

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc (1 - 1 / ↑q) 1
p2 : ℝ
hp2 : p2 ∈ Icc (1 - 1 / ↑q) 1
p1le2 : p1 < p2
p : ℝ
this : 2 ≤ ↑q
qinv_lt_1 : (↑q)⁻¹ < 1
zero_lt_1_sub_p : 0 < 1 - p
hp : 1 - (↑q)⁻¹ < p ∧ p < 1
⊢ log ((↑q - 1) * (1 - p)) < log p

Tactic:
]

Goal state after:
q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc (1 - 1 / ↑q) 1
p2 : ℝ
hp2 : p2 ∈ Icc (1 - 1 / ↑q) 1
p1le2 : p1 < p2
p : ℝ
this : 2 ≤ ↑q
qinv_lt_1 : (↑q)⁻¹ < 1
zero_lt_1_sub_p : 0 < 1 - p
hp : 1 - (↑q)⁻¹ < p ∧ p < 1
⊢ log ((↑q - 1) * (1 - p)) < log p
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: θ : ℝ
n : ℕ
⊢ (2 * (↑n + 2) * (2 * ↑n + 3) * I (n + 1) θ - 4 * (↑n + 2) * (↑n + 1) * I n θ) * θ ^ (2 * n + 3) =
    2 * (↑n + 2) * (2 * ↑n + 3) * (I (n + 1) θ * θ ^ (2 * (n + 1) + 1)) -
      4 * (↑n + 2) * (↑n + 1) * θ ^ 2 * (I n θ * θ ^ (2 * n + 1))

Tactic:
]

Goal state after:
θ : ℝ
n : ℕ
⊢ (2 * (↑n + 2) * (2 * ↑n + 3) * I (n + 1) θ - 4 * (↑n + 2) * (↑n + 1) * I n θ) * θ ^ (2 * n + 3) =
    2 * (↑n + 2) * (2 * ↑n + 3) * (I (n + 1) θ * θ ^ (2 * (n + 1) + 1)) -
      4 * (↑n + 2) * (↑n + 1) * θ ^ 2 * (I n θ * θ ^ (2 * n + 1))
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc (1 - 1 / ↑q) 1
p2 : ℝ
hp2 : p2 ∈ Icc (1 - 1 / ↑q) 1
p1le2 : p1 < p2
p : ℝ
this : 2 ≤ ↑q
qinv_lt_1 : (↑q)⁻¹ < 1
zero_lt_1_sub_p : 0 < 1 - p
hp : 1 - (↑q)⁻¹ < p ∧ p < 1
qpos : 0 < ↑q
h1 : (1 - (↑q)⁻¹) * ↑q < p * ↑q
⊢ (1 - (↑q)⁻¹) * ↑q = ↑q - (↑q)⁻¹ * ↑q

Tactic:
by ring

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc (1 - 1 / ↑q) 1
p2 : ℝ
hp2 : p2 ∈ Icc (1 - 1 / ↑q) 1
p1le2 : p1 < p2
p : ℝ
this : 2 ≤ ↑q
qinv_lt_1 : (↑q)⁻¹ < 1
zero_lt_1_sub_p : 0 < 1 - p
hp : 1 - (↑q)⁻¹ < p ∧ p < 1
qpos : 0 < ↑q
h1 : ↑q - 1 < p * ↑q
h2 : (1 - (↑q)⁻¹) * ↑q = ↑q - 1
⊢ ↑q - 1 < p * ↑q

Tactic:
]

Goal state after:
q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc (1 - 1 / ↑q) 1
p2 : ℝ
hp2 : p2 ∈ Icc (1 - 1 / ↑q) 1
p1le2 : p1 < p2
p : ℝ
this : 2 ≤ ↑q
qinv_lt_1 : (↑q)⁻¹ < 1
zero_lt_1_sub_p : 0 < 1 - p
hp : 1 - (↑q)⁻¹ < p ∧ p < 1
qpos : 0 < ↑q
h1 : ↑q - 1 < p * ↑q
h2 : (1 - (↑q)⁻¹) * ↑q = ↑q - 1
⊢ ↑q - 1 < p * ↑q
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: ⊢ StrictMonoOn (qaryEntropy 2) (Icc 0 (1 - 1 / 2))

Tactic:
]

Goal state after:
⊢ StrictMonoOn (qaryEntropy 2) (Icc 0 (1 - 1 / 2))
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: ⊢ StrictAntiOn (qaryEntropy 2) (Icc (1 / 2) 1)

Tactic:
]

Goal state after:
⊢ StrictAntiOn (qaryEntropy 2) (Icc (1 / 2) 1)
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: q : ℕ
p : ℝ
hp : p ∈ interior (Icc 0 1)
⊢ -1 / (p * (1 - p)) < 0

Tactic:
]

Goal state after:
q : ℕ
p : ℝ
hp : p ∈ interior (Icc 0 1)
⊢ -1 / (p * (1 - p)) < 0
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order

Goal state before:
pp: case pos
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
hp : p ∈ Ioo 0 1 ∪ {0} ∪ {1}
a b : A
hab : a ≤ b
ha : 0 ≤ a
hb : 0 ≤ b
hIcc : Icc 0 1 = Ioo 0 1 ∪ {0} ∪ {1}
⊢ (fun a ↦ a ^ p) a ≤ (fun a ↦ a ^ p) b

Tactic:
]

Goal state after:
case pos
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
hp : p ∈ Ioo 0 1 ∪ {0} ∪ {1}
a b : A
hab : a ≤ b
ha : 0 ≤ a
hb : 0 ≤ b
hIcc : Icc 0 1 = Ioo 0 1 ∪ {0} ∪ {1}
⊢ (fun a ↦ a ^ p) a ≤ (fun a ↦ a ^ p) b
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: case inr
p : ℤ[X]
a b : ℤ
k : ℕ
hp : p.natDegree ≤ k
hb : b ≠ 0
⊢ ∑ i ∈ p.support, eval₂ (Int.castRingHom ℝ) (↑a / ↑b) ((monomial i) (p.coeff i)) * ↑b ^ k =
    ∑ x ∈ p.support, ↑(p.coeff x * a ^ x * b ^ (k - x))

Tactic:
]

Goal state after:
case inr
p : ℤ[X]
a b : ℤ
k : ℕ
hp : p.natDegree ≤ k
hb : b ≠ 0
⊢ ∑ i ∈ p.support, eval₂ (Int.castRingHom ℝ) (↑a / ↑b) ((monomial i) (p.coeff i)) * ↑b ^ k =
    ∑ x ∈ p.support, ↑(p.coeff x * a ^ x * b ^ (k - x))
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : a ≤ b
⊢ a ^ (1 / 2) ≤ b ^ (1 / 2)

Tactic:
]

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : a ≤ b
⊢ a ^ (1 / 2) ≤ b ^ (1 / 2)
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: case inr
p : ℤ[X]
a b : ℤ
k : ℕ
hp : p.natDegree ≤ k
hb : b ≠ 0
i : ℕ
hi : i ∈ p.support
ik : i ≤ k
⊢ ↑(p.coeff i) * ↑(a ^ i) * ↑(b ^ (k - i)) = ↑(p.coeff i) * ↑(a ^ i) * ↑(b ^ (k - i))

pp: case inr
p : ℤ[X]
a b : ℤ
k : ℕ
hp : p.natDegree ≤ k
hb : b ≠ 0
i : ℕ
hi : i ∈ p.support
ik : i ≤ k
⊢ b ^ i ∣ b ^ (k - i) * b ^ i

Tactic:
]

Goal state after:
case inr
p : ℤ[X]
a b : ℤ
k : ℕ
hp : p.natDegree ≤ k
hb : b ≠ 0
i : ℕ
hi : i ∈ p.support
ik : i ≤ k
⊢ b ^ i ∣ b ^ (k - i) * b ^ i
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: case refine_1
n : ℕ
x : ℝ
hx : x ∈ Ioc (-1) 1
⊢ -1 ≤ x ∧ x ≤ 1

Tactic:
]

Goal state after:
case refine_1
n : ℕ
x : ℝ
hx : x ∈ Ioc (-1) 1
⊢ -1 ≤ x ∧ x ≤ 1
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: n : ℕ
⊢ ‖I n (π / 2)‖ ≤ 2

Tactic:
]

Goal state after:
n : ℕ
⊢ ‖I n (π / 2)‖ ≤ 2
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: n : ℕ
x : ℝ
hx : -1 < x ∧ x ≤ 1
⊢ |1 - x ^ 2| ^ n * |cos (x * (π / 2))| ≤ 1

Tactic:
]

Goal state after:
n : ℕ
x : ℝ
hx : -1 < x ∧ x ≤ 1
⊢ |1 - x ^ 2| ^ n * |cos (x * (π / 2))| ≤ 1
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: n : ℕ
x : ℝ
hx : -1 < x ∧ x ≤ 1
⊢ -1 ≤ 1 - x ^ 2 ∧ 1 - x ^ 2 ≤ 1

Tactic:
]

Goal state after:
n : ℕ
x : ℝ
hx : -1 < x ∧ x ≤ 1
⊢ -1 ≤ 1 - x ^ 2 ∧ 1 - x ^ 2 ≤ 1
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: a : ℝ
⊢ Tendsto (fun n ↦ a ^ (2 * n + 1) / ↑n !) atTop (nhds (a * 0))

Tactic:
]

Goal state after:
a : ℝ
⊢ Tendsto (fun n ↦ a ^ (2 * n + 1) / ↑n !) atTop (nhds (a * 0))
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: a : ℝ
x : ℕ
⊢ a * a ^ (2 * x) / ↑x ! = a * a ^ (2 * x) / ↑x !

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
hp : p ∈ Ioo 0 1 ∪ {0} ∪ {1}
hIcc : Icc 0 1 = Ioo 0 1 ∪ {0} ∪ {1}
⊢ ConcaveOn ℝ (Ici 0) fun a ↦ a ^ p

Tactic:
]

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
hp : p ∈ Ioo 0 1 ∪ {0} ∪ {1}
hIcc : Icc 0 1 = Ioo 0 1 ∪ {0} ∪ {1}
⊢ ConcaveOn ℝ (Ici 0) fun a ↦ a ^ p
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: x : ℝ
⊢ (∃ y, ↑y = x) → ∃ a b, 0 < b ∧ x = ↑a / ↑b

Tactic:
]

Goal state after:
x : ℝ
⊢ (∃ y, ↑y = x) → ∃ a b, 0 < b ∧ x = ↑a / ↑b
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: ⊢ Irrational (π / 2)

Tactic:
]

Goal state after:
⊢ Irrational (π / 2)
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: h' : ¬Irrational (π / 2)
a : ℤ
b : ℕ
hb : 0 < b
h : π / 2 = ↑a / ↑b
⊢ 0 < ↑a

Tactic:
have : 0 < (a : ℝ) / b := h ▸ pi_div_two_pos

Goal state after:
h' : ¬Irrational (π / 2)
a : ℤ
b : ℕ
hb : 0 < b
h : π / 2 = ↑a / ↑b
this : 0 < ↑a / ↑b
⊢ 0 < ↑a
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: h' : ¬Irrational (π / 2)
a : ℤ
b : ℕ
hb : 0 < b
h : π / 2 = ↑a / ↑b
this : 0 < ↑a
⊢ 0 < ↑a

Tactic:
]

Goal state after:
h' : ¬Irrational (π / 2)
a : ℤ
b : ℕ
hb : 0 < b
h : π / 2 = ↑a / ↑b
this : 0 < ↑a
⊢ 0 < ↑a
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
q : ℝ≥0 := ⟨p, ⋯⟩
⊢ Monotone fun a ↦ a ^ ↑q

Tactic:
obtain hq | hq := eq_zero_or_pos q

Goal state after:
case inl
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
q : ℝ≥0 := ⟨p, ⋯⟩
hq : q = 0
⊢ Monotone fun a ↦ a ^ ↑q

case inr
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
q : ℝ≥0 := ⟨p, ⋯⟩
hq : 0 < q
⊢ Monotone fun a ↦ a ^ ↑q
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order

Goal state before:
pp: case inl
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
q : ℝ≥0 := ⟨p, ⋯⟩
hq : q = 0
⊢ Monotone fun a ↦ a ^ ↑0

Tactic:
]

Goal state after:
case inl
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
q : ℝ≥0 := ⟨p, ⋯⟩
hq : q = 0
⊢ Monotone fun a ↦ a ^ ↑0
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: h' : ¬Irrational (π / 2)
a : ℤ
b : ℕ
hb : 0 < b
h : π / 2 = ↑a / ↑b
ha : 0 < ↑a
k : ∀ (n : ℕ), 0 < ↑a ^ (2 * n + 1) / ↑n !
this : ∀ᶠ (a_1 : ℕ) in atTop, ↑a ^ (2 * a_1 + 1) / ↑a_1 ! < 1 / 2
n : ℕ
hn : ↑a ^ (2 * n + 1) / ↑n ! * 2 < 1
⊢ ↑a ^ (2 * n + 1) / ↑n ! * I n (π / 2) < 1

Tactic:
]

Goal state after:
h' : ¬Irrational (π / 2)
a : ℤ
b : ℕ
hb : 0 < b
h : π / 2 = ↑a / ↑b
ha : 0 < ↑a
k : ∀ (n : ℕ), 0 < ↑a ^ (2 * n + 1) / ↑n !
this : ∀ᶠ (a_1 : ℕ) in atTop, ↑a ^ (2 * a_1 + 1) / ↑a_1 ! < 1 / 2
n : ℕ
hn : ↑a ^ (2 * n + 1) / ↑n ! * 2 < 1
⊢ ↑a ^ (2 * n + 1) / ↑n ! * I n (π / 2) < 1
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
q : ℝ≥0 := ⟨p, ⋯⟩
⊢ ConcaveOn ℝ (Ici 0) fun a ↦ a ^ ↑q

Tactic:
obtain hq | hq := eq_zero_or_pos q

Goal state after:
case inl
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
q : ℝ≥0 := ⟨p, ⋯⟩
hq : q = 0
⊢ ConcaveOn ℝ (Ici 0) fun a ↦ a ^ ↑q

case inr
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
q : ℝ≥0 := ⟨p, ⋯⟩
hq : 0 < q
⊢ ConcaveOn ℝ (Ici 0) fun a ↦ a ^ ↑q
Module: Mathlib.Analysis.SpecialFunctions.Log.RpowTendsto

Goal state before:
pp: case e_a.e_a.e_a
p x : ℝ
p_pos : 0 < p
x_pos : 0 < x
hx : ‖p * log x‖ ≤ 1
pinv_nonneg : 0 ≤ p⁻¹
⊢ x ^ p = x ^ p

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.RpowTendsto

Goal state before:
pp: s : Set ℝ
hs : s ⊆ Ioi 0
hs' : IsCompact s
⊢ ∀ (i : ℝ), 0 < i → ∀ᶠ (n : ℝ) in 𝓝[>] 0, ∀ x ∈ s, (log x, n⁻¹ * (x ^ n - 1)) ∈ {p | dist p.1 p.2 ≤ i}

Tactic:
]

Goal state after:
s : Set ℝ
hs : s ⊆ Ioi 0
hs' : IsCompact s
⊢ ∀ (i : ℝ), 0 < i → ∀ᶠ (n : ℝ) in 𝓝[>] 0, ∀ x ∈ s, (log x, n⁻¹ * (x ^ n - 1)) ∈ {p | dist p.1 p.2 ≤ i}