mathlib-initiative/mathlib-tactics huggingface dataset:
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 (sSup (slope f x '' {y | y ∈ S ∧ y < x})) (Iio 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 (sSup (slope f x '' {y | y ∈ S ∧ y < x})) (Iio x) x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: case inl
x y : ℝ
h : x * y < 1
hy : y ≤ 0
⊢ arctan x + arctan y < π / 2 + arctan 0

Tactic:
]

Goal state after:
case inl
x y : ℝ
h : x * y < 1
hy : y ≤ 0
⊢ arctan x + arctan y < π / 2 + arctan 0
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: case inr
x y : ℝ
h : x < y⁻¹
hy : 0 < y
⊢ arctan x + arctan y < π / 2

Tactic:
]

Goal state after:
case inr
x y : ℝ
h : x < y⁻¹
hy : 0 < y
⊢ arctan x + arctan y < π / 2
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: case inr
x y : ℝ
hy : 0 < y
h : arctan x + arctan y < π / 2
⊢ arctan x + arctan y < π / 2

Tactic:
]

Goal state after:
case inr
x y : ℝ
hy : 0 < y
h : arctan x + arctan y < π / 2
⊢ arctan x + arctan y < π / 2
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x y : ℝ
h : x * y < 1
⊢ arctan (tan (arctan x + arctan y)) = arctan ((x + y) / (1 - x * y))

pp: case hx₁
x y : ℝ
h : x * y < 1
⊢ -(π / 2) < arctan x + arctan y

pp: case hx₂
x y : ℝ
h : x * y < 1
⊢ arctan x + arctan y < π / 2

Tactic:
]

Goal state after:
x y : ℝ
h : x * y < 1
⊢ arctan (tan (arctan x + arctan y)) = arctan ((x + y) / (1 - x * y))

case hx₁
x y : ℝ
h : x * y < 1
⊢ -(π / 2) < arctan x + arctan y

case hx₂
x y : ℝ
h : x * y < 1
⊢ arctan x + arctan y < 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 a x ⊆ {y | y ∈ S ∧ y < x}
h_Ioo : Tendsto (slope f x) (𝓝[<] x) (𝓝 (sSup (slope f x '' Ioo a x)))
this : sSup (slope f x '' Ioo a x) = sSup (slope f x '' {y | y ∈ S ∧ y < x})
⊢ Tendsto (slope f x) (𝓝[<] x) (𝓝 (sSup (slope f x '' Ioo a x)))

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 a x ⊆ {y | y ∈ S ∧ y < x}
h_Ioo : Tendsto (slope f x) (𝓝[<] x) (𝓝 (sSup (slope f x '' Ioo a x)))
this : sSup (slope f x '' Ioo a x) = sSup (slope f x '' {y | y ∈ S ∧ y < x})
⊢ Tendsto (slope f x) (𝓝[<] x) (𝓝 (sSup (slope f x '' Ioo a x)))
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: case e'_3
x : ℝ
hx1 : -(3 * π / 2) ≤ x
hx2 : x ≤ 3 * π / 2
hx3 : x ≠ 0
this : ∀ {y : ℝ}, 0 < y → y ≤ 3 * π / 2 → cos y < 1 / √(y ^ 2 + 1)
h✝ : x < 0
⊢ cos x = cos x

Tactic:
]

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

Goal state before:
pp: case hx₁
x y : ℝ
h : x * y < 1
⊢ arctan (-x) + arctan (-y) < π / 2

Tactic:
]

Goal state after:
case hx₁
x y : ℝ
h : x * y < 1
⊢ arctan (-x) + arctan (-y) < π / 2
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: case hx₁
x y : ℝ
h : -x * -y < 1
⊢ arctan (-x) + arctan (-y) < π / 2

Tactic:
]

Goal state after:
case hx₁
x y : ℝ
h : -x * -y < 1
⊢ arctan (-x) + arctan (-y) < π / 2
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: case e'_4
x : ℝ
hx1 : -(3 * π / 2) ≤ x
hx2 : x ≤ 3 * π / 2
hx3 : x ≠ 0
this : ∀ {y : ℝ}, 0 < y → y ≤ 3 * π / 2 → cos y < 1 / √(y ^ 2 + 1)
h✝ : x < 0
⊢ 1 / √(x ^ 2 + 1) = 1 / √(x ^ 2 + 1)

Tactic:
]

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

Goal state before:
pp: x y : ℝ
h : 1 < x * y
hx : 0 < x
hy : 0 < y
k : arctan (-((x⁻¹ + y⁻¹) / (1 - x⁻¹ * y⁻¹))) + π = arctan x + arctan y
⊢ arctan x + arctan y = arctan ((x + y) / (1 - x * y)) + π

Tactic:
]

Goal state after:
x y : ℝ
h : 1 < x * y
hx : 0 < x
hy : 0 < y
k : arctan (-((x⁻¹ + y⁻¹) / (1 - x⁻¹ * y⁻¹))) + π = arctan x + arctan y
⊢ arctan x + arctan y = arctan ((x + y) / (1 - x * y)) + π
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x y : ℝ
h : 1 < -x * -y
hx : x < 0
⊢ arctan x + arctan y = arctan ((x + y) / (1 - x * y)) - π

Tactic:
]

Goal state after:
x y : ℝ
h : 1 < -x * -y
hx : x < 0
⊢ arctan x + arctan y = arctan ((x + y) / (1 - x * y)) - π
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x y : ℝ
h : 1 < -x * -y
hx : x < 0
k : -(arctan (-x) + arctan (-y)) = -(arctan (-((x + y) / (1 - x * y))) + π)
⊢ arctan x + arctan y = arctan ((x + y) / (1 - x * y)) - π

Tactic:
]

Goal state after:
x y : ℝ
h : 1 < -x * -y
hx : x < 0
k : -(arctan (-x) + arctan (-y)) = -(arctan (-((x + y) / (1 - x * y))) + π)
⊢ arctan x + arctan y = arctan ((x + y) / (1 - x * y)) - π
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x : ℝ
h₁ : -1 < x
h₂ : x < 1
⊢ arctan ((x + x) / (1 - x * x)) = arctan (2 * x / (1 - x ^ 2))

Tactic:
]

Goal state after:
x : ℝ
h₁ : -1 < x
h₂ : x < 1
⊢ arctan ((x + x) / (1 - x * x)) = arctan (2 * x / (1 - x ^ 2))
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case inr.refine_1
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
x : ℝ
hxs : x ∈ interior S
y : ℝ
hys : y ∈ interior S
hxy✝ : x ≤ y
hxy : x < y
hys' : ∃ l u, y ∈ Ioo l u ∧ Ioo l u ⊆ S
⊢ (slope f y '' {y_1 | y_1 ∈ S ∧ y < y_1}).Nonempty

Tactic:
]

Goal state after:
case inr.refine_1
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
x : ℝ
hxs : x ∈ interior S
y : ℝ
hys : y ∈ interior S
hxy✝ : x ≤ y
hxy : x < y
hys' : ∃ l u, y ∈ Ioo l u ∧ Ioo l u ⊆ S
⊢ (slope f y '' {y_1 | y_1 ∈ S ∧ y < y_1}).Nonempty
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case inr.refine_1
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
x : ℝ
hxs : x ∈ interior S
y : ℝ
hys : y ∈ interior S
hxy✝ : x ≤ y
hxy : x < y
a b : ℝ
hxab : y ∈ Ioo a b
habs : Ioo a b ⊆ S
⊢ (slope f y '' {y_1 | y_1 ∈ S ∧ y < y_1}).Nonempty

Tactic:
image_nonempty

Goal state after:
case inr.refine_1
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
x : ℝ
hxs : x ∈ interior S
y : ℝ
hys : y ∈ interior S
hxy✝ : x ≤ y
hxy : x < y
a b : ℝ
hxab : y ∈ Ioo a b
habs : Ioo a b ⊆ S
⊢ {y_1 | y_1 ∈ S ∧ y < y_1}.Nonempty
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

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

Tactic:
]

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

Goal state before:
pp: case inr.refine_1
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
x : ℝ
hxs : x ∈ interior S
y : ℝ
hys : y ∈ interior S
hxy✝ : x ≤ y
hxy : x < y
a b : ℝ
hxab : y ∈ Ioo a b
habs : Ioo a b ⊆ S
⊢ {y_1 | y_1 ∈ S ∧ y < y_1}.Nonempty

Tactic:
]

Goal state after:
case inr.refine_1
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
x : ℝ
hxs : x ∈ interior S
y : ℝ
hys : y ∈ interior S
hxy✝ : x ≤ y
hxy : x < y
a b : ℝ
hxab : y ∈ Ioo a b
habs : Ioo a b ⊆ S
⊢ {y_1 | y_1 ∈ S ∧ y < y_1}.Nonempty
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

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

Tactic:
]

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

Goal state before:
pp: case inr.refine_2
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
x : ℝ
hxs : x ∈ interior S
y : ℝ
hys : y ∈ interior S
hxy✝ : x ≤ y
hxy : x < y
z : ℝ
hzs : z ∈ S
hyz : y < z
⊢ slope f y x ≤ slope f y z

Tactic:
]

Goal state after:
case inr.refine_2
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
x : ℝ
hxs : x ∈ interior S
y : ℝ
hys : y ∈ interior S
hxy✝ : x ≤ y
hxy : x < y
z : ℝ
hzs : z ∈ S
hyz : y < z
⊢ slope f y x ≤ slope f y z
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: ⊢ arctan ((2⁻¹ + 3⁻¹) / (1 - 2⁻¹ * 3⁻¹)) = π / 4

pp: ⊢ 2⁻¹ * 3⁻¹ < 1

Tactic:
]

Goal state after:
⊢ arctan ((2⁻¹ + 3⁻¹) / (1 - 2⁻¹ * 3⁻¹)) = π / 4

⊢ 2⁻¹ * 3⁻¹ < 1
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: ⊢ arctan (2 * 2⁻¹ / (1 - 2⁻¹ ^ 2)) = arctan ((1 + 7⁻¹) / (1 - 1 * 7⁻¹))

pp: ⊢ 1 * 7⁻¹ < 1

pp: case h₁
⊢ -1 < 2⁻¹

pp: case h₂
⊢ 2⁻¹ < 1

Tactic:
]

Goal state after:
⊢ arctan (2 * 2⁻¹ / (1 - 2⁻¹ ^ 2)) = arctan ((1 + 7⁻¹) / (1 - 1 * 7⁻¹))

⊢ 1 * 7⁻¹ < 1

case h₁
⊢ -1 < 2⁻¹

case h₂
⊢ 2⁻¹ < 1
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: ⊢ arctan ((2 * 3⁻¹ / (1 - 3⁻¹ ^ 2) + 7⁻¹) / (1 - 2 * 3⁻¹ / (1 - 3⁻¹ ^ 2) * 7⁻¹)) = π / 4

pp: ⊢ 2 * 3⁻¹ / (1 - 3⁻¹ ^ 2) * 7⁻¹ < 1

pp: case h₁
⊢ -1 < 3⁻¹

pp: case h₂
⊢ 3⁻¹ < 1

Tactic:
]

Goal state after:
⊢ arctan ((2 * 3⁻¹ / (1 - 3⁻¹ ^ 2) + 7⁻¹) / (1 - 2 * 3⁻¹ / (1 - 3⁻¹ ^ 2) * 7⁻¹)) = π / 4

⊢ 2 * 3⁻¹ / (1 - 3⁻¹ ^ 2) * 7⁻¹ < 1

case h₁
⊢ -1 < 3⁻¹

case h₂
⊢ 3⁻¹ < 1
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: ⊢ arctan (2 * (2 * 5⁻¹ / (1 - 5⁻¹ ^ 2)) / (1 - (2 * 5⁻¹ / (1 - 5⁻¹ ^ 2)) ^ 2)) = arctan ((1 + 239⁻¹) / (1 - 1 * 239⁻¹))

pp: ⊢ 1 * 239⁻¹ < 1

pp: case h₁
⊢ -1 < 2 * 5⁻¹ / (1 - 5⁻¹ ^ 2)

pp: case h₂
⊢ 2 * 5⁻¹ / (1 - 5⁻¹ ^ 2) < 1

pp: case h₁
⊢ -1 < 5⁻¹

pp: case h₂
⊢ 5⁻¹ < 1

Tactic:
]

Goal state after:
⊢ arctan (2 * (2 * 5⁻¹ / (1 - 5⁻¹ ^ 2)) / (1 - (2 * 5⁻¹ / (1 - 5⁻¹ ^ 2)) ^ 2)) = arctan ((1 + 239⁻¹) / (1 - 1 * 239⁻¹))

⊢ 1 * 239⁻¹ < 1

case h₁
⊢ -1 < 2 * 5⁻¹ / (1 - 5⁻¹ ^ 2)

case h₂
⊢ 2 * 5⁻¹ / (1 - 5⁻¹ ^ 2) < 1

case h₁
⊢ -1 < 5⁻¹

case h₂
⊢ 5⁻¹ < 1
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: case inl
x : ℝ
hx1 : -(3 * π / 2) ≤ x
hx2 : x ≤ 3 * π / 2
hx3 : x ≠ 0
y : ℝ
hy1 : 0 < y
hy2 : y ≤ 3 * π / 2
hy3 : 0 < y ^ 2 + 1
hy2' : y < π / 2
hy4 : 0 < cos y
⊢ y < tan y

Tactic:
]

Goal state after:
case inl
x : ℝ
hx1 : -(3 * π / 2) ≤ x
hx2 : x ≤ 3 * π / 2
hx3 : x ≠ 0
y : ℝ
hy1 : 0 < y
hy2 : y ≤ 3 * π / 2
hy3 : 0 < y ^ 2 + 1
hy2' : y < π / 2
hy4 : 0 < cos y
⊢ y < tan y
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case inr.refine_1
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
x : ℝ
hxs : x ∈ interior S
y : ℝ
hys : y ∈ interior S
hxy✝ : x ≤ y
hxy : x < y
hxs' : ∃ l u, x ∈ Ioo l u ∧ Ioo l u ⊆ S
⊢ (slope f x '' {y | y ∈ S ∧ y < x}).Nonempty

Tactic:
]

Goal state after:
case inr.refine_1
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
x : ℝ
hxs : x ∈ interior S
y : ℝ
hys : y ∈ interior S
hxy✝ : x ≤ y
hxy : x < y
hxs' : ∃ l u, x ∈ Ioo l u ∧ Ioo l u ⊆ S
⊢ (slope f x '' {y | y ∈ S ∧ y < x}).Nonempty
Module: Mathlib.Analysis.Complex.Angle

Goal state before:
pp: x y : ℂ
hx : x ≠ 0
hy : y ≠ 0
⊢ inner ℝ x y / (‖x‖ * ‖y‖) = (x / y).re / ‖x / y‖

Tactic:
]

Goal state after:
x y : ℂ
hx : x ≠ 0
hy : y ≠ 0
⊢ inner ℝ x y / (‖x‖ * ‖y‖) = (x / y).re / ‖x / y‖
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case inr.refine_1
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
x : ℝ
hxs : x ∈ interior S
y : ℝ
hys : y ∈ interior S
hxy✝ : x ≤ y
hxy : x < y
a b : ℝ
hxab : x ∈ Ioo a b
habs : Ioo a b ⊆ S
⊢ (slope f x '' {y | y ∈ S ∧ y < x}).Nonempty

Tactic:
image_nonempty

Goal state after:
case inr.refine_1
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
x : ℝ
hxs : x ∈ interior S
y : ℝ
hys : y ∈ interior S
hxy✝ : x ≤ y
hxy : x < y
a b : ℝ
hxab : x ∈ Ioo a b
habs : Ioo a b ⊆ S
⊢ {y | y ∈ S ∧ y < x}.Nonempty
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case inr.refine_1
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
x : ℝ
hxs : x ∈ interior S
y : ℝ
hys : y ∈ interior S
hxy✝ : x ≤ y
hxy : x < y
a b : ℝ
hxab : x ∈ Ioo a b
habs : Ioo a b ⊆ S
⊢ {y | y ∈ S ∧ y < x}.Nonempty

Tactic:
]

Goal state after:
case inr.refine_1
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
x : ℝ
hxs : x ∈ interior S
y : ℝ
hys : y ∈ interior S
hxy✝ : x ≤ y
hxy : x < y
a b : ℝ
hxab : x ∈ Ioo a b
habs : Ioo a b ⊆ S
⊢ {y | y ∈ S ∧ y < x}.Nonempty
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x : ℝ
⊢ ‖2 * sin (x / 2)‖ ≤ ‖x‖

Tactic:
]

Goal state after:
x : ℝ
⊢ ‖2 * sin (x / 2)‖ ≤ ‖x‖
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x : ℝ
⊢ 2 * (|x| / 2) = |x|

Tactic:
]

Goal state after:
x : ℝ
⊢ 2 * (|x| / 2) = |x|
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x : ℝ
⊢ ENNReal.ofReal ‖Complex.exp (Complex.I * ↑x) - 1‖ ≤ ‖x‖ₑ

Tactic:
]

Goal state after:
x : ℝ
⊢ ENNReal.ofReal ‖Complex.exp (Complex.I * ↑x) - 1‖ ≤ ‖x‖ₑ
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x : ℝ
⊢ ENNReal.ofReal ‖Complex.exp (Complex.I * ↑x) - 1‖ ≤ ENNReal.ofReal ‖x‖

Tactic:
]

Goal state after:
x : ℝ
⊢ ENNReal.ofReal ‖Complex.exp (Complex.I * ↑x) - 1‖ ≤ ENNReal.ofReal ‖x‖
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x : ℝ
⊢ ↑‖Complex.exp (Complex.I * ↑x) - 1‖₊ ≤ ↑‖x‖₊

Tactic:
]

Goal state after:
x : ℝ
⊢ ↑‖Complex.exp (Complex.I * ↑x) - 1‖₊ ≤ ↑‖x‖₊
Module: Mathlib.Analysis.Complex.Angle

Goal state before:
pp: case inr
a x y : ℂ
ha : a ≠ 0
⊢ angle x (y * a) = angle x (y * a)

Tactic:
]

Goal state after:
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
⊢ derivWithin f (Iio x) x ≤ derivWithin f (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
⊢ derivWithin f (Iio x) x ≤ derivWithin f (Ioi x) x
Module: Mathlib.Analysis.Complex.Angle

Goal state before:
pp: a x y : ℂ
⊢ angle (x * a) y = angle (x * a) y

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Complex.Arg

Goal state before:
pp: x y : ℂ
⊢ x = 0 ∨ y = 0 ∨ x.arg = y.arg ↔ ↑(x / y).arg = 0

Tactic:
]

Goal state after:
x y : ℂ
⊢ x = 0 ∨ y = 0 ∨ x.arg = y.arg ↔ ↑(x / y).arg = 0
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
⊢ sSup (slope f x '' {y | y ∈ S ∧ y < x}) ≤ sInf (slope f x '' {y | y ∈ S ∧ x < y})

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
⊢ sSup (slope f x '' {y | y ∈ S ∧ y < x}) ≤ sInf (slope f x '' {y | y ∈ S ∧ x < y})
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
c : ℝ
hc : c ≠ 0
x y : V
this : c * c ≠ 0
⊢ arccos (⟪x, y⟫ / (‖x‖ * ‖y‖)) = arccos (⟪x, y⟫ / (‖x‖ * ‖y‖))

Tactic:
]

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

Goal state before:
pp: case refine_1
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
⊢ (slope f x '' {y | y ∈ S ∧ y < x}).Nonempty

Tactic:
image_nonempty

Goal state after:
case refine_1
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
⊢ {y | y ∈ S ∧ y < x}.Nonempty
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ E
inst✝ : InnerProductSpace ℝ F
f : E →ₗᵢ[ℝ] F
u v : E
⊢ arccos (⟪u, v⟫ / (‖u‖ * ‖v‖)) = arccos (⟪u, v⟫ / (‖u‖ * ‖v‖))

Tactic:
]

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

Goal state before:
pp: case refine_1
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
⊢ {y | y ∈ S ∧ y < x}.Nonempty

Tactic:
]

Goal state after:
case refine_1
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
⊢ {y | y ∈ S ∧ y < x}.Nonempty
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ arccos (⟪y, x⟫ / (‖y‖ * ‖x‖)) = arccos (⟪y, x⟫ / (‖y‖ * ‖x‖))

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ arccos (⟪x, y⟫ / (‖x‖ * ‖y‖)) = arccos (⟪x, y⟫ / (‖x‖ * ‖y‖))

Tactic:
]

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

Goal state before:
pp: case refine_2.refine_1
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
z : ℝ
hzs : z ∈ S
hzx : z < x
⊢ (slope f x '' {y | y ∈ S ∧ x < y}).Nonempty

Tactic:
image_nonempty

Goal state after:
case refine_2.refine_1
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
z : ℝ
hzs : z ∈ S
hzx : z < x
⊢ {y | y ∈ S ∧ x < y}.Nonempty
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case refine_2.refine_1
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
z : ℝ
hzs : z ∈ S
hzx : z < x
⊢ {y | y ∈ S ∧ x < y}.Nonempty

Tactic:
]

Goal state after:
case refine_2.refine_1
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
z : ℝ
hzs : z ∈ S
hzx : z < x
⊢ {y | y ∈ S ∧ x < y}.Nonempty
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ arccos (-(⟪x, y⟫ / (‖x‖ * ‖y‖))) = arccos (-(⟪x, y⟫ / (‖x‖ * ‖y‖)))

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ π - angle x y = π - angle x y

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Complex.Angle

Goal state before:
pp: case inr
x y : ℂ
hx : ‖x‖ = 1
hy : ‖y‖ = 1
this✝ : ∀ {x y : ℂ}, ‖x‖ = 1 → ‖y‖ = 1 → y = 1 → ‖x - y‖ ∈ Icc (2 / π * angle x y) (angle x y)
h : ¬y = 1
this : ‖x - y‖ ∈ Icc (2 / π * angle x y) (angle x y)
⊢ ‖x - y‖ ∈ Icc (2 / π * angle x y) (angle x y)

pp: case inr
x y : ℂ
hx : ‖x‖ = 1
hy : ‖y‖ = 1
this✝ : ∀ {x y : ℂ}, ‖x‖ = 1 → ‖y‖ = 1 → y = 1 → ‖x - y‖ ∈ Icc (2 / π * angle x y) (angle x y)
h : ¬y = 1
this : ‖x / y - 1‖ ∈ Icc (2 / π * angle x (1 * y)) (angle x (1 * y))
⊢ y ≠ 0

Tactic:
]

Goal state after:
case inr
x y : ℂ
hx : ‖x‖ = 1
hy : ‖y‖ = 1
this✝ : ∀ {x y : ℂ}, ‖x‖ = 1 → ‖y‖ = 1 → y = 1 → ‖x - y‖ ∈ Icc (2 / π * angle x y) (angle x y)
h : ¬y = 1
this : ‖x - y‖ ∈ Icc (2 / π * angle x y) (angle x y)
⊢ ‖x - y‖ ∈ Icc (2 / π * angle x y) (angle x y)

case inr
x y : ℂ
hx : ‖x‖ = 1
hy : ‖y‖ = 1
this✝ : ∀ {x y : ℂ}, ‖x‖ = 1 → ‖y‖ = 1 → y = 1 → ‖x - y‖ ∈ Icc (2 / π * angle x y) (angle x y)
h : ¬y = 1
this : ‖x / y - 1‖ ∈ Icc (2 / π * angle x (1 * y)) (angle x (1 * y))
⊢ y ≠ 0
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x : V
⊢ π / 2 = π / 2

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x : V
hx : x ≠ 0
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Complex.Angle

Goal state before:
pp: x✝ y x : ℂ
hx : ∃ θ ∈ Ioc (-π) π, cexp (↑θ * I) = x
hy : ‖1‖ = 1
⊢ ‖x - 1‖ ∈ Icc (2 / π * angle x 1) (angle x 1)

Tactic:
]

Goal state after:
x✝ y x : ℂ
hx : ∃ θ ∈ Ioc (-π) π, cexp (↑θ * I) = x
hy : ‖1‖ = 1
⊢ ‖x - 1‖ ∈ Icc (2 / π * angle x 1) (angle x 1)
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x : V
hx : x ≠ 0
⊢ π = π

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Complex.Angle

Goal state before:
pp: x y : ℂ
hy : ‖1‖ = 1
θ : ℝ
hθ : θ ∈ Ioc (-π) π
⊢ ‖cos ↑θ - 1 + sin ↑θ * I‖ ∈ Icc (2 / π * |θ|) |θ|

pp: x y : ℂ
hy : ‖1‖ = 1
θ : ℝ
hθ : θ ∈ Ioc (-π) π
⊢ θ ∈ Ioc (-π) (-π + 2 * π)

Tactic:
]

Goal state after:
x y : ℂ
hy : ‖1‖ = 1
θ : ℝ
hθ : θ ∈ Ioc (-π) π
⊢ ‖cos ↑θ - 1 + sin ↑θ * I‖ ∈ Icc (2 / π * |θ|) |θ|

x y : ℂ
hy : ‖1‖ = 1
θ : ℝ
hθ : θ ∈ Ioc (-π) π
⊢ θ ∈ Ioc (-π) (-π + 2 * π)
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
r : ℝ
hr : 0 < r
⊢ arccos (⟪x, y⟫ / (‖x‖ * ‖y‖)) = arccos (⟪x, y⟫ / (‖x‖ * ‖y‖))

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
r : ℝ
hr : 0 < r
⊢ angle x y = angle x y

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Complex.Angle

Goal state before:
pp: x y : ℂ
hy : ‖1‖ = 1
θ : ℝ
hθ : θ ∈ Ioc (-π) π
⊢ √((Real.cos θ - 1) ^ 2 + Real.sin θ ^ 2) ∈ Icc (2 / π * |θ|) |θ|

Tactic:
]

Goal state after:
x y : ℂ
hy : ‖1‖ = 1
θ : ℝ
hθ : θ ∈ Ioc (-π) π
⊢ √((Real.cos θ - 1) ^ 2 + Real.sin θ ^ 2) ∈ Icc (2 / π * |θ|) |θ|
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
r : ℝ
hr : r < 0
⊢ π - angle x y = π - angle x y

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
r : ℝ
hr : r < 0
⊢ angle (-x) y = angle (-x) y

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Complex.Angle

Goal state before:
pp: case refine_1
x y : ℂ
hy : ‖1‖ = 1
θ : ℝ
hθ : θ ∈ Ioc (-π) π
⊢ (2 / π) ^ 2 * θ ^ 2 ≤ (Real.cos θ - 1) ^ 2 + Real.sin θ ^ 2

Tactic:
]

Goal state after:
case refine_1
x y : ℂ
hy : ‖1‖ = 1
θ : ℝ
hθ : θ ∈ Ioc (-π) π
⊢ (2 / π) ^ 2 * θ ^ 2 ≤ (Real.cos θ - 1) ^ 2 + Real.sin θ ^ 2
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ ⟪x, y⟫ = ⟪x, y⟫

pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ ‖x‖ * ‖y‖ = 0 → ⟪x, y⟫ = 0

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ ‖x‖ * ‖y‖ = 0 → ⟪x, y⟫ = 0
Module: Mathlib.Analysis.Complex.Angle

Goal state before:
pp: x y : ℂ
hy : ‖1‖ = 1
θ : ℝ
hθ : θ ∈ Ioc (-π) π
⊢ (2 / π) ^ 2 * θ ^ 2 = 2 * (1 - (1 - 2 / π ^ 2 * θ ^ 2))

Tactic:
by ring

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ √(1 - (⟪x, y⟫ / (‖x‖ * ‖y‖)) ^ 2) * (‖x‖ * ‖y‖) = √(⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫)

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ √(1 - (⟪x, y⟫ / (‖x‖ * ‖y‖)) ^ 2) * (‖x‖ * ‖y‖) = √(⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫)
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ √(1 - (⟪x, y⟫ / (‖x‖ * ‖y‖)) ^ 2) * √((‖x‖ * ‖y‖) ^ 2) = √(⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫)

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ √(1 - (⟪x, y⟫ / (‖x‖ * ‖y‖)) ^ 2) * √((‖x‖ * ‖y‖) ^ 2) = √(⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫)
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ √((1 - ⟪x, y⟫ / (‖x‖ * ‖y‖) * (⟪x, y⟫ / (‖x‖ * ‖y‖))) * (‖x‖ * ‖y‖) ^ 2) = √(⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫)

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ √((1 - ⟪x, y⟫ / (‖x‖ * ‖y‖) * (⟪x, y⟫ / (‖x‖ * ‖y‖))) * (‖x‖ * ‖y‖) ^ 2) = √(⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫)