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

Goal state before:
pp: this : ContinuousOn tan {x | cos x ≠ 0}
h_eq : (fun x ↦ sin x / cos x) = tan
⊢ ContinuousOn tan {x | cos x ≠ 0}

Tactic:
]

Goal state after:
this : ContinuousOn tan {x | cos x ≠ 0}
h_eq : (fun x ↦ sin x / cos x) = tan
⊢ ContinuousOn tan {x | cos x ≠ 0}
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : DifferentiableOn ℝ f (interior D)
hf'' : DifferentiableOn ℝ (deriv f) (interior D)
hf''_nonpos : ∀ x ∈ interior D, deriv^[2] f x ≤ 0
⊢ DifferentiableOn ℝ (deriv f) (interior D)

Tactic:
]

Goal state after:
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : DifferentiableOn ℝ f (interior D)
hf'' : DifferentiableOn ℝ (deriv f) (interior D)
hf''_nonpos : ∀ x ∈ interior D, deriv^[2] f x ≤ 0
⊢ DifferentiableOn ℝ (deriv f) (interior D)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x : ℝ
⊢ -(π / 2) < x → x < π / 2 → ¬∃ k, x = (2 * ↑k + 1) * π / 2

Tactic:
]

Goal state after:
x : ℝ
⊢ -(π / 2) < x → x < π / 2 → ¬∃ k, x = (2 * ↑k + 1) * π / 2
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : DifferentiableOn ℝ f (interior D)
hf'' : DifferentiableOn ℝ (deriv f) (interior D)
hf''_nonpos : ∀ x ∈ interior D, deriv^[2] f x ≤ 0
⊢ ∀ x ∈ interior D, deriv (deriv f) x ≤ 0

Tactic:
]

Goal state after:
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : DifferentiableOn ℝ f (interior D)
hf'' : DifferentiableOn ℝ (deriv f) (interior D)
hf''_nonpos : ∀ x ∈ interior D, deriv^[2] f x ≤ 0
⊢ ∀ x ∈ interior D, deriv (deriv f) x ≤ 0
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: case inl
x : ℝ
hx_gt : -(π / 2) < x
hx_lt : ¬π / 2 ≤ x
r : ℤ
hxr_eq : x = (2 * ↑r + 1) * π / 2
h : 0 ≤ r
⊢ False

Tactic:
]

Goal state after:
case inl
x : ℝ
hx_gt : -(π / 2) < x
hx_lt : ¬π / 2 ≤ x
r : ℤ
hxr_eq : x = (2 * ↑r + 1) * π / 2
h : 0 ≤ r
⊢ False
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: case inl
x : ℝ
hx_gt : -(π / 2) < x
hx_lt : ¬π / 2 ≤ x
r : ℤ
hxr_eq : x = (2 * ↑r + 1) * π / 2
h : 0 ≤ r
⊢ 1 ≤ 2 * ↑r + 1

Tactic:
]

Goal state after:
case inl
x : ℝ
hx_gt : -(π / 2) < x
hx_lt : ¬π / 2 ≤ x
r : ℤ
hxr_eq : x = (2 * ↑r + 1) * π / 2
h : 0 ≤ r
⊢ 1 ≤ 2 * ↑r + 1
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: case inr
x : ℝ
hx_gt : ¬x ≤ -(π / 2)
hx_lt : x < π / 2
r : ℤ
hxr_eq : x = (2 * ↑r + 1) * π / 2
h : r < 0
⊢ False

Tactic:
]

Goal state after:
case inr
x : ℝ
hx_gt : ¬x ≤ -(π / 2)
hx_lt : x < π / 2
r : ℤ
hxr_eq : x = (2 * ↑r + 1) * π / 2
h : r < 0
⊢ False
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x : ℝ
h : 0 < x
h' : x ≤ 1
hx : |x| = x
⊢ x ≤ 1

Tactic:
]

Goal state after:
x : ℝ
h : 0 < x
h' : x ≤ 1
hx : |x| = x
⊢ x ≤ 1
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case refine_1
D : Set ℝ
hD : Convex ℝ D
f f' f'' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x
hf''₀ : ∀ x ∈ interior D, 0 ≤ f'' x
this : EqOn (deriv f) f' (interior D)
⊢ DifferentiableOn ℝ f' (interior D)

Tactic:
]

Goal state after:
case refine_1
D : Set ℝ
hD : Convex ℝ D
f f' f'' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x
hf''₀ : ∀ x ∈ interior D, 0 ≤ f'' x
this : EqOn (deriv f) f' (interior D)
⊢ DifferentiableOn ℝ f' (interior D)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: case inr
x : ℝ
hx_gt : ¬x ≤ -(π / 2)
hx_lt : x < π / 2
r : ℤ
hxr_eq : x = (2 * ↑r + 1) * π / 2
h : r < 0
⊢ 2 * ↑r + 1 ≤ -1

Tactic:
]

Goal state after:
case inr
x : ℝ
hx_gt : ¬x ≤ -(π / 2)
hx_lt : x < π / 2
r : ℤ
hxr_eq : x = (2 * ↑r + 1) * π / 2
h : r < 0
⊢ 2 * ↑r + 1 ≤ -1
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: case inr
x : ℝ
h : 0 < x
h' : x ≤ 1
hx : |x| = x
this : sin x ≤ x - x ^ 3 / 6 + x ^ 5 / 100
⊢ sin x < x

Tactic:
]

Goal state after:
case inr
x : ℝ
h : 0 < x
h' : x ≤ 1
hx : |x| = x
this : sin x ≤ x - x ^ 3 / 6 + x ^ 5 / 100
⊢ sin x < x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: case inr
x : ℝ
h : 0 < x
h' : x ≤ 1
hx : |x| = x
this : sin x ≤ x - x ^ 3 / 6 + x ^ 5 / 100
⊢ x ^ 5 / 100 < x ^ 3 * 6⁻¹

Tactic:
]

Goal state after:
case inr
x : ℝ
h : 0 < x
h' : x ≤ 1
hx : |x| = x
this : sin x ≤ x - x ^ 3 / 6 + x ^ 5 / 100
⊢ x ^ 5 / 100 < x ^ 3 * 6⁻¹
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x : ℝ
hx_gt : ¬x ≤ -(π / 2)
hx_lt : x < π / 2
r : ℤ
hxr_eq : x = (2 * ↑r + 1) * π / 2
h : r ≤ -1
⊢ r ≤ -1

Tactic:
]

Goal state after:
x : ℝ
hx_gt : ¬x ≤ -(π / 2)
hx_lt : x < π / 2
r : ℤ
hxr_eq : x = (2 * ↑r + 1) * π / 2
h : r ≤ -1
⊢ r ≤ -1
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: case inr
x : ℝ
hx_gt : ¬x ≤ -(π / 2)
hx_lt : x < π / 2
r : ℤ
hxr_eq : x = (2 * ↑r + 1) * π / 2
h : r < 0
hr_le : r ≤ -1
⊢ ↑r ≤ (-1 - 1) / 2

pp: case inr
x : ℝ
hx_gt : ¬x ≤ -(π / 2)
hx_lt : x < π / 2
r : ℤ
hxr_eq : x = (2 * ↑r + 1) * π / 2
h : r < 0
hr_le : r ≤ -1
⊢ 0 < 2

Tactic:
]

Goal state after:
case inr
x : ℝ
hx_gt : ¬x ≤ -(π / 2)
hx_lt : x < π / 2
r : ℤ
hxr_eq : x = (2 * ↑r + 1) * π / 2
h : r < 0
hr_le : r ≤ -1
⊢ ↑r ≤ (-1 - 1) / 2

case inr
x : ℝ
hx_gt : ¬x ≤ -(π / 2)
hx_lt : x < π / 2
r : ℤ
hxr_eq : x = (2 * ↑r + 1) * π / 2
h : r < 0
hr_le : r ≤ -1
⊢ 0 < 2
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: D : Set ℝ
hD : Convex ℝ D
f f' f'' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x
hf''₀ : ∀ x ∈ interior D, 0 ≤ f'' x
this : EqOn (deriv f) f' (interior D)
x : ℝ
hx : x ∈ interior D
⊢ ?m.194 x = f'' x

pp: D : Set ℝ
hD : Convex ℝ D
f f' f'' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x
hf''₀ : ∀ x ∈ interior D, 0 ≤ f'' x
this : EqOn (deriv f) f' (interior D)
x : ℝ
hx : x ∈ interior D
⊢ ℝ → ℝ

pp: D : Set ℝ
hD : Convex ℝ D
f f' f'' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x
hf''₀ : ∀ x ∈ interior D, 0 ≤ f'' x
this : EqOn (deriv f) f' (interior D)
x : ℝ
hx : x ∈ interior D
y : ℝ
hy : y ∈ interior D
⊢ HasDerivWithinAt (deriv f) (?m.194 y) (interior D) y

Tactic:
]

Goal state after:
D : Set ℝ
hD : Convex ℝ D
f f' f'' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x
hf''₀ : ∀ x ∈ interior D, 0 ≤ f'' x
this : EqOn (deriv f) f' (interior D)
x : ℝ
hx : x ∈ interior D
y : ℝ
hy : y ∈ interior D
⊢ HasDerivWithinAt (deriv f) (f'' y) (interior D) y
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x : ℝ
hx : 0 < x
hx' : x < π / 2
⊢ (π / 2)⁻¹ * x < sin x

Tactic:
]

Goal state after:
x : ℝ
hx : 0 < x
hx' : x < π / 2
⊢ (π / 2)⁻¹ * x < sin x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x : ℝ
hx : 0 < x
hx' : x < π / 2
⊢ x < π / 2

Tactic:
]

Goal state after:
x : ℝ
hx : 0 < x
hx' : x < π / 2
⊢ x < π / 2
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: D : Set ℝ
hD : Convex ℝ D
f f' f'' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x
hf''₀ : ∀ x ∈ interior D, 0 ≤ f'' x
this : EqOn (deriv f) f' (interior D)
x : ℝ
hx : x ∈ interior D
y : ℝ
hy : y ∈ interior D
⊢ f' y = f' y

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x : ℝ
hx : 0 ≤ x
hx' : x ≤ π / 2
⊢ (π / 2)⁻¹ * x ≤ sin x

Tactic:
]

Goal state after:
x : ℝ
hx : 0 ≤ x
hx' : x ≤ π / 2
⊢ (π / 2)⁻¹ * x ≤ sin x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x : ℝ
hx : 0 ≤ x
hx' : x ≤ π / 2
⊢ x ≤ π / 2

Tactic:
]

Goal state after:
x : ℝ
hx : 0 ≤ x
hx' : x ≤ π / 2
⊢ x ≤ π / 2
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: this : -(π / 2) < π / 2
⊢ Tendsto tan (𝓝[>] (-(π / 2))) atBot

Tactic:
]

Goal state after:
this : -(π / 2) < π / 2
⊢ Tendsto tan (𝓝[>] (-(π / 2))) atBot
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: this : -(π / 2) < π / 2
⊢ Tendsto tan (𝓝[<] (π / 2)) atTop

Tactic:
]

Goal state after:
this : -(π / 2) < π / 2
⊢ Tendsto tan (𝓝[<] (π / 2)) atTop
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x : ℝ
hx : |x| ≤ π / 2
this : ∀ {x : ℝ}, |x| ≤ π / 2 → 0 ≤ x → 2 / π * |x| ≤ |sin x|
hx₀ : ¬0 ≤ x
⊢ |x| ≤ π / 2

Tactic:
]

Goal state after:
x : ℝ
hx : |x| ≤ π / 2
this : ∀ {x : ℝ}, |x| ≤ π / 2 → 0 ≤ x → 2 / π * |x| ≤ |sin x|
hx₀ : ¬0 ≤ x
⊢ |x| ≤ π / 2
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x : ℝ
⊢ x ^ 2 / (1 + x ^ 2) = x ^ 2 / (1 + x ^ 2)

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case refine_1
D : Set ℝ
hD : Convex ℝ D
f f' f'' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x
hf''₀ : ∀ x ∈ interior D, f'' x ≤ 0
this : EqOn (deriv f) f' (interior D)
⊢ DifferentiableOn ℝ f' (interior D)

Tactic:
]

Goal state after:
case refine_1
D : Set ℝ
hD : Convex ℝ D
f f' f'' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x
hf''₀ : ∀ x ∈ interior D, f'' x ≤ 0
this : EqOn (deriv f) f' (interior D)
⊢ DifferentiableOn ℝ f' (interior D)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x : ℝ
⊢ (1 + x ^ 2)⁻¹ = (1 + x ^ 2)⁻¹

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x : ℝ
⊢ x / √(1 + x ^ 2) = x / √(1 + x ^ 2)

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x : ℝ
⊢ (√(1 + x ^ 2))⁻¹ = (√(1 + x ^ 2))⁻¹

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x✝ x : ℝ
hx : x ≤ π / 2
hx₀ : 0 ≤ x
⊢ 2 / π * x ≤ |sin x|

Tactic:
]

Goal state after:
x✝ x : ℝ
hx : x ≤ π / 2
hx₀ : 0 ≤ x
⊢ 2 / π * x ≤ |sin x|
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x✝ x : ℝ
hx : x ≠ 0
hx₀ : 0 < x
hxπ : 1 < x
⊢ 1 < x

Tactic:
]

Goal state after:
x✝ x : ℝ
hx : x ≠ 0
hx₀ : 0 < x
hxπ : 1 < x
⊢ 1 < x
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: D : Set ℝ
hD : Convex ℝ D
f f' f'' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x
hf''₀ : ∀ x ∈ interior D, f'' x ≤ 0
this : EqOn (deriv f) f' (interior D)
x : ℝ
hx : x ∈ interior D
⊢ ?m.194 x = f'' x

pp: D : Set ℝ
hD : Convex ℝ D
f f' f'' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x
hf''₀ : ∀ x ∈ interior D, f'' x ≤ 0
this : EqOn (deriv f) f' (interior D)
x : ℝ
hx : x ∈ interior D
⊢ ℝ → ℝ

pp: D : Set ℝ
hD : Convex ℝ D
f f' f'' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x
hf''₀ : ∀ x ∈ interior D, f'' x ≤ 0
this : EqOn (deriv f) f' (interior D)
x : ℝ
hx : x ∈ interior D
y : ℝ
hy : y ∈ interior D
⊢ HasDerivWithinAt (deriv f) (?m.194 y) (interior D) y

Tactic:
]

Goal state after:
D : Set ℝ
hD : Convex ℝ D
f f' f'' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x
hf''₀ : ∀ x ∈ interior D, f'' x ≤ 0
this : EqOn (deriv f) f' (interior D)
x : ℝ
hx : x ∈ interior D
y : ℝ
hy : y ∈ interior D
⊢ HasDerivWithinAt (deriv f) (f'' y) (interior D) y
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x : ℝ
hx : |x| ≤ π
this : ∀ {x : ℝ}, |x| ≤ π → 0 ≤ x → cos x ≤ 1 - 2 / π ^ 2 * x ^ 2
hx₀ : ¬0 ≤ x
⊢ |x| ≤ π

Tactic:
]

Goal state after:
x : ℝ
hx : |x| ≤ π
this : ∀ {x : ℝ}, |x| ≤ π → 0 ≤ x → cos x ≤ 1 - 2 / π ^ 2 * x ^ 2
hx₀ : ¬0 ≤ x
⊢ |x| ≤ π
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: D : Set ℝ
hD : Convex ℝ D
f f' f'' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x
hf''₀ : ∀ x ∈ interior D, f'' x ≤ 0
this : EqOn (deriv f) f' (interior D)
x : ℝ
hx : x ∈ interior D
y : ℝ
hy : y ∈ interior D
⊢ f' y = f' y

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x✝ x : ℝ
hx : x ≤ π
hx₀ : 0 ≤ x
⊢ cos x ≤ 1 - 2 / π ^ 2 * x ^ 2

Tactic:
]

Goal state after:
x✝ x : ℝ
hx : x ≤ π
hx₀ : 0 ≤ x
⊢ cos x ≤ 1 - 2 / π ^ 2 * x ^ 2
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x : ℝ
⊢ arctan x = 0 ↔ arctan x = 0

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x y : ℝ
h : tan x = y
hx : x ∈ Ioo (-(π / 2)) (π / 2)
⊢ y = y

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: ⊢ π / 3 = π / 3

pp: case hx₁
⊢ -(π / 2) < π / 3

pp: case hx₂
⊢ π / 3 < π / 2

Tactic:
]

Goal state after:
case hx₁
⊢ -(π / 2) < π / 3

case hx₂
⊢ π / 3 < π / 2
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf'' : ∀ x ∈ interior D, 0 < deriv^[2] f x
⊢ ∀ x ∈ interior D, 0 < deriv (deriv f) x

Tactic:
]

Goal state after:
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf'' : ∀ x ∈ interior D, 0 < deriv^[2] f x
⊢ ∀ x ∈ interior D, 0 < deriv (deriv f) x
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf'' : ∀ x ∈ interior D, deriv^[2] f x < 0
⊢ ∀ x ∈ interior D, deriv (deriv f) x < 0

Tactic:
]

Goal state after:
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf'' : ∀ x ∈ interior D, deriv^[2] f x < 0
⊢ ∀ x ∈ interior D, deriv (deriv f) x < 0
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: ⊢ π / 6 = π / 6

pp: case hx₁
⊢ -(π / 2) < π / 6

pp: case hx₂
⊢ π / 6 < π / 2

Tactic:
]

Goal state after:
case hx₁
⊢ -(π / 2) < π / 6

case hx₂
⊢ π / 6 < π / 2
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
x : 𝕜
hfc : ConcaveOn 𝕜 s f
hx : x ∈ s
⊢ AntitoneOn ((-slope (-f)) x) (s \ {x})

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
x : 𝕜
hfc : ConcaveOn 𝕜 s f
hx : x ∈ s
⊢ AntitoneOn ((-slope (-f)) x) (s \ {x})
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x : ℝ
⊢ -(π / 4) = -(π / 4)

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y
y : ℝ
hy : y ∈ Ioo 0 (π / 2)
⊢ 0 < sin y

Tactic:
]

Goal state after:
x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y
y : ℝ
hy : y ∈ Ioo 0 (π / 2)
⊢ 0 < sin y
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x : ℝ
h : 0 ≤ x
⊢ arcsin (x / √(1 + x ^ 2)) = arcsin √(1 - (√(1 + x ^ 2))⁻¹ ^ 2)

pp: x : ℝ
h : 0 ≤ x
⊢ 0 ≤ (√(1 + x ^ 2))⁻¹

Tactic:
]

Goal state after:
x : ℝ
h : 0 ≤ x
⊢ arcsin (x / √(1 + x ^ 2)) = arcsin √(1 - (√(1 + x ^ 2))⁻¹ ^ 2)

x : ℝ
h : 0 ≤ x
⊢ 0 ≤ (√(1 + x ^ 2))⁻¹
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y
sin_pos : ∀ {y : ℝ}, y ∈ interior U → 0 < sin y
tan_cts_U : ContinuousOn tan U
tan_minus_id_cts : ContinuousOn (fun y ↦ tan y - y) U
y : ℝ
hy : y ∈ interior U
this : 0 < cos y
⊢ 1 - sin y ^ 2 ≠ 1

Tactic:
]

Goal state after:
x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y
sin_pos : ∀ {y : ℝ}, y ∈ interior U → 0 < sin y
tan_cts_U : ContinuousOn tan U
tan_minus_id_cts : ContinuousOn (fun y ↦ tan y - y) U
y : ℝ
hy : y ∈ interior U
this : 0 < cos y
⊢ 1 - sin y ^ 2 ≠ 1
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x : ℝ
h : 0 < x
⊢ arctan (√(1 - x ^ 2) / x) = π / 2 - arcsin x

Tactic:
]

Goal state after:
x : ℝ
h : 0 < x
⊢ arctan (√(1 - x ^ 2) / x) = π / 2 - arcsin x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x : ℝ
h : 0 < x
⊢ arctan x⁻¹ = arctan x⁻¹

pp: case hx₁
x : ℝ
h : 0 < x
⊢ -(π / 2) < π / 2 - arctan x

pp: case hx₂
x : ℝ
h : 0 < x
⊢ π / 2 - arctan x < π / 2

Tactic:
]

Goal state after:
case hx₁
x : ℝ
h : 0 < x
⊢ -(π / 2) < π / 2 - arctan x

case hx₂
x : ℝ
h : 0 < x
⊢ π / 2 - arctan x < π / 2
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y
sin_pos : ∀ {y : ℝ}, y ∈ interior U → 0 < sin y
tan_cts_U : ContinuousOn tan U
tan_minus_id_cts : ContinuousOn (fun y ↦ tan y - y) U
y : ℝ
hy : y ∈ interior U
this : 0 < cos y
bd2 : cos y ^ 2 < 1
⊢ cos y ^ 2 < 1

pp: case ha
x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y
sin_pos : ∀ {y : ℝ}, y ∈ interior U → 0 < sin y
tan_cts_U : ContinuousOn tan U
tan_minus_id_cts : ContinuousOn (fun y ↦ tan y - y) U
y : ℝ
hy : y ∈ interior U
this : 0 < cos y
bd2 : cos y ^ 2 < 1
⊢ 0 < 1

pp: case hb
x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y
sin_pos : ∀ {y : ℝ}, y ∈ interior U → 0 < sin y
tan_cts_U : ContinuousOn tan U
tan_minus_id_cts : ContinuousOn (fun y ↦ tan y - y) U
y : ℝ
hy : y ∈ interior U
this : 0 < cos y
bd2 : cos y ^ 2 < 1
⊢ 0 < cos y ^ 2

Tactic:
]

Goal state after:
x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y
sin_pos : ∀ {y : ℝ}, y ∈ interior U → 0 < sin y
tan_cts_U : ContinuousOn tan U
tan_minus_id_cts : ContinuousOn (fun y ↦ tan y - y) U
y : ℝ
hy : y ∈ interior U
this : 0 < cos y
bd2 : cos y ^ 2 < 1
⊢ cos y ^ 2 < 1

case ha
x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y
sin_pos : ∀ {y : ℝ}, y ∈ interior U → 0 < sin y
tan_cts_U : ContinuousOn tan U
tan_minus_id_cts : ContinuousOn (fun y ↦ tan y - y) U
y : ℝ
hy : y ∈ interior U
this : 0 < cos y
bd2 : cos y ^ 2 < 1
⊢ 0 < 1

case hb
x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y
sin_pos : ∀ {y : ℝ}, y ∈ interior U → 0 < sin y
tan_cts_U : ContinuousOn tan U
tan_minus_id_cts : ContinuousOn (fun y ↦ tan y - y) U
y : ℝ
hy : y ∈ interior U
this : 0 < cos y
bd2 : cos y ^ 2 < 1
⊢ 0 < cos y ^ 2
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: case hx₂
x : ℝ
h : 0 < x
⊢ arctan 0 < arctan x

Tactic:
]

Goal state after:
case hx₂
x : ℝ
h : 0 < x
⊢ arctan 0 < arctan x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x : ℝ
h : x < 0
this : arctan x⁻¹ = -(π / 2) - arctan x
⊢ arctan x⁻¹ = -(π / 2) - arctan x

Tactic:
]

Goal state after:
x : ℝ
h : x < 0
this : arctan x⁻¹ = -(π / 2) - arctan x
⊢ arctan x⁻¹ = -(π / 2) - arctan x
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: S : Set ℝ
f : ℝ → ℝ
x : ℝ
hfc : ConvexOn ℝ S f
hxs : x ∈ interior S
hxs' : ∃ l u, x ∈ Ioo l u ∧ Ioo l u ⊆ S
⊢ HasDerivWithinAt f (sInf (slope f x '' {y | y ∈ S ∧ x < y})) (Ioi x) x

Tactic:
]

Goal state after:
S : Set ℝ
f : ℝ → ℝ
x : ℝ
hfc : ConvexOn ℝ S f
hxs : x ∈ interior S
hxs' : ∃ l u, x ∈ Ioo l u ∧ Ioo l u ⊆ S
⊢ HasDerivWithinAt f (sInf (slope f x '' {y | y ∈ S ∧ x < y})) (Ioi x) x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x : ℝ
k : ℤ
h : arctan x = (2 * ↑k + 1) * π / 2
lb : -1 < 2 * ↑k + 1
ub : arctan x < π / 2
⊢ False

Tactic:
]

Goal state after:
x : ℝ
k : ℤ
h : arctan x = (2 * ↑k + 1) * π / 2
lb : -1 < 2 * ↑k + 1
ub : arctan x < π / 2
⊢ False
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y
sin_pos : ∀ {y : ℝ}, y ∈ interior U → 0 < sin y
tan_cts_U : ContinuousOn tan U
tan_minus_id_cts : ContinuousOn (fun y ↦ tan y - y) U
deriv_pos : ∀ y ∈ interior U, 0 < deriv (fun y' ↦ tan y' - y') y
mono : StrictMonoOn (fun y ↦ tan y - y) (Ico 0 (π / 2))
⊢ 0 < π / 2

Tactic:
]

Goal state after:
x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y
sin_pos : ∀ {y : ℝ}, y ∈ interior U → 0 < sin y
tan_cts_U : ContinuousOn tan U
tan_minus_id_cts : ContinuousOn (fun y ↦ tan y - y) U
deriv_pos : ∀ y ∈ interior U, 0 < deriv (fun y' ↦ tan y' - y') y
mono : StrictMonoOn (fun y ↦ tan y - y) (Ico 0 (π / 2))
⊢ 0 < π / 2
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: S : Set ℝ
f : ℝ → ℝ
x : ℝ
hfc : ConvexOn ℝ S f
hxs : x ∈ interior S
a b : ℝ
hxab : x ∈ Ioo a b
habs : Ioo a b ⊆ S
h : Ioo x b ⊆ {y | y ∈ S ∧ x < y}
h_Ioo : Tendsto (slope f x) (𝓝[>] x) (𝓝 (sInf (slope f x '' Ioo x b)))
this : sInf (slope f x '' Ioo x b) = sInf (slope f x '' {y | y ∈ S ∧ x < y})
⊢ Tendsto (slope f x) (𝓝[>] x) (𝓝 (sInf (slope f x '' Ioo x b)))

Tactic:
]

Goal state after:
S : Set ℝ
f : ℝ → ℝ
x : ℝ
hfc : ConvexOn ℝ S f
hxs : x ∈ interior S
a b : ℝ
hxab : x ∈ Ioo a b
habs : Ioo a b ⊆ S
h : Ioo x b ⊆ {y | y ∈ S ∧ x < y}
h_Ioo : Tendsto (slope f x) (𝓝[>] x) (𝓝 (sInf (slope f x '' Ioo x b)))
this : sInf (slope f x '' Ioo x b) = sInf (slope f x '' {y | y ∈ S ∧ x < y})
⊢ Tendsto (slope f x) (𝓝[>] x) (𝓝 (sInf (slope f x '' Ioo x b)))
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x : ℝ
k : ℤ
h : arctan x = (2 * ↑k + 1) * π / 2
lb : -1 < 2 * ↑k + 1
ub : 2 * ↑k + 1 < 1
⊢ False

Tactic:
]

Goal state after:
x : ℝ
k : ℤ
h : arctan x = (2 * ↑k + 1) * π / 2
lb : -1 < 2 * ↑k + 1
ub : 2 * ↑k + 1 < 1
⊢ False
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: case inl
h1 : 0 ≤ 0
h2 : 0 < π / 2
⊢ 0 ≤ 0

Tactic:
]

Goal state after: