mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case pos
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
⊢ ∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x)

pp: case neg
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
h : ∃ w ∈ Ioo x y, deriv f w = 0
⊢ ∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x)

Tactic:
·

Goal state after:
case pos
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
⊢ ∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x)
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case neg
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
h : ∃ w ∈ Ioo x y, deriv f w = 0
⊢ ∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x)

Tactic:
·

Goal state after:
case neg
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
h : ∃ w ∈ Ioo x y, deriv f w = 0
⊢ ∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x)
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case neg
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
h : ∃ w ∈ Ioo x y, deriv f w = 0
⊢ ∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x)

Tactic:
rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩

Goal state after:
case neg
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
⊢ ∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x)
Module: Mathlib.Analysis.Complex.AbsMax

Goal state before:
pp: case inr
E : Type u
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℂ F
inst✝¹ : Nontrivial E
inst✝ : FiniteDimensional ℂ E
f : E → F
U : Set E
hb : Bornology.IsBounded U
hne : U.Nonempty
hd : DiffContOnCl ℂ f U
hc : IsCompact (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U)
w : E
hle : IsMaxOn (norm ∘ f) (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U) w
hwU : w ∈ frontier U
⊢ ∃ z ∈ frontier U, IsMaxOn (norm ∘ f) (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U) z

pp: case inl
E : Type u
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℂ F
inst✝¹ : Nontrivial E
inst✝ : FiniteDimensional ℂ E
f : E → F
U : Set E
hb : Bornology.IsBounded U
hne : U.Nonempty
hd : DiffContOnCl ℂ f U
hc : IsCompact (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U)
w : E
hle : IsMaxOn (norm ∘ f) (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U) w
hwU : w ∈ interior U
⊢ ∃ z ∈ frontier U, IsMaxOn (norm ∘ f) (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U) z

Tactic:
·

Goal state after:
case inr
E : Type u
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℂ F
inst✝¹ : Nontrivial E
inst✝ : FiniteDimensional ℂ E
f : E → F
U : Set E
hb : Bornology.IsBounded U
hne : U.Nonempty
hd : DiffContOnCl ℂ f U
hc : IsCompact (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U)
w : E
hle : IsMaxOn (norm ∘ f) (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U) w
hwU : w ∈ frontier U
⊢ ∃ z ∈ frontier U, IsMaxOn (norm ∘ f) (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U) z
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: case pos
x : ℂ
h : cos (x / 2) = 0
⊢ 2 * sin (x / 2) * cos (x / 2) = 2 * tan (x / 2) / (1 + tan (x / 2) ^ 2)

pp: case neg
x : ℂ
h : ¬cos (x / 2) = 0
⊢ 2 * sin (x / 2) * cos (x / 2) = 2 * tan (x / 2) / (1 + tan (x / 2) ^ 2)

Tactic:
·

Goal state after:
case pos
x : ℂ
h : cos (x / 2) = 0
⊢ 2 * sin (x / 2) * cos (x / 2) = 2 * tan (x / 2) / (1 + tan (x / 2) ^ 2)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: case neg
x : ℂ
h : ¬cos (x / 2) = 0
⊢ 2 * sin (x / 2) * cos (x / 2) = 2 * tan (x / 2) / (1 + tan (x / 2) ^ 2)

Tactic:
·

Goal state after:
case neg
x : ℂ
h : ¬cos (x / 2) = 0
⊢ 2 * sin (x / 2) * cos (x / 2) = 2 * tan (x / 2) / (1 + tan (x / 2) ^ 2)
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
ha : deriv f a < (f w - f x) / (w - x)
hxa : x < a
haw : a < w
⊢ ContinuousOn f (Icc w y)

pp: x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
ha : deriv f a < (f w - f x) / (w - x)
hxa : x < a
haw : a < w
⊢ StrictMonoOn (deriv f) (Ioo w y)

pp: x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
ha : deriv f a < (f w - f x) / (w - x)
hxa : x < a
haw : a < w
⊢ ∀ w_1 ∈ Ioo w y, deriv f w_1 ≠ 0

Tactic:
·

Goal state after:
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
ha : deriv f a < (f w - f x) / (w - x)
hxa : x < a
haw : a < w
⊢ ContinuousOn f (Icc w y)
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
ha : deriv f a < (f w - f x) / (w - x)
hxa : x < a
haw : a < w
⊢ StrictMonoOn (deriv f) (Ioo w y)

pp: x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
ha : deriv f a < (f w - f x) / (w - x)
hxa : x < a
haw : a < w
⊢ ∀ w_1 ∈ Ioo w y, deriv f w_1 ≠ 0

Tactic:
·

Goal state after:
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
ha : deriv f a < (f w - f x) / (w - x)
hxa : x < a
haw : a < w
⊢ StrictMonoOn (deriv f) (Ioo w y)
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
ha : deriv f a < (f w - f x) / (w - x)
hxa : x < a
haw : a < w
⊢ ∀ w_1 ∈ Ioo w y, deriv f w_1 ≠ 0

Tactic:
·

Goal state after:
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
ha : deriv f a < (f w - f x) / (w - x)
hxa : x < a
haw : a < w
⊢ ∀ w_1 ∈ Ioo w y, deriv f w_1 ≠ 0
Module: Mathlib.Analysis.Complex.AbsMax

Goal state before:
pp: case inl
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : Nontrivial E
f : E → F
U : Set E
hU : Bornology.IsBounded U
hd : DiffContOnCl ℂ f U
C : ℝ
hC : ∀ z ∈ frontier U, ‖f z‖ ≤ C
z : E
hz : z ∈ frontier U
⊢ ‖f z‖ ≤ C

pp: case inr
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : Nontrivial E
f : E → F
U : Set E
hU : Bornology.IsBounded U
hd : DiffContOnCl ℂ f U
C : ℝ
hC : ∀ z ∈ frontier U, ‖f z‖ ≤ C
z : E
hz : z ∈ U
⊢ ‖f z‖ ≤ C

Tactic:
·

Goal state after:
case inl
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : Nontrivial E
f : E → F
U : Set E
hU : Bornology.IsBounded U
hd : DiffContOnCl ℂ f U
C : ℝ
hC : ∀ z ∈ frontier U, ‖f z‖ ≤ C
z : E
hz : z ∈ frontier U
⊢ ‖f z‖ ≤ C
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
hxa : x < a
haw : a < w
b : ℝ
hwb : w < b
hby : b < y
ha : deriv f a * (w - x) < f w - f x
hb : deriv f b * (y - w) < f y - f w
⊢ deriv f a < deriv f b

pp: x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
hxa : x < a
haw : a < w
b : ℝ
hwb : w < b
hby : b < y
ha : deriv f a * (w - x) < f w - f x
hb : deriv f b * (y - w) < f y - f w
⊢ 0 ≤ deriv f b

Tactic:
·

Goal state after:
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
hxa : x < a
haw : a < w
b : ℝ
hwb : w < b
hby : b < y
ha : deriv f a * (w - x) < f w - f x
hb : deriv f b * (y - w) < f y - f w
⊢ deriv f a < deriv f b
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
hxa : x < a
haw : a < w
b : ℝ
hwb : w < b
hby : b < y
ha : deriv f a * (w - x) < f w - f x
hb : deriv f b * (y - w) < f y - f w
⊢ 0 ≤ deriv f b

Tactic:
·

Goal state after:
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
hxa : x < a
haw : a < w
b : ℝ
hwb : w < b
hby : b < y
ha : deriv f a * (w - x) < f w - f x
hb : deriv f b * (y - w) < f y - f w
⊢ 0 ≤ deriv f b
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: case mp
x : ℝ
⊢ ((∃ n, ↑n * (2 * π) = x) ∨ ∃ k, π + ↑k * (2 * π) = x) → ∃ k, ↑k * π = x

pp: case mpr
x : ℝ
⊢ (∃ k, ↑k * π = x) → (∃ n, ↑n * (2 * π) = x) ∨ ∃ k, π + ↑k * (2 * π) = x

Tactic:
·

Goal state after:
case mp
x : ℝ
⊢ ((∃ n, ↑n * (2 * π) = x) ∨ ∃ k, π + ↑k * (2 * π) = x) → ∃ k, ↑k * π = x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: case mp.inl
x : ℝ
n : ℤ
h : ↑n * (2 * π) = x
⊢ ∃ k, ↑k * π = x

pp: case mp.inr
x : ℝ
n : ℤ
h : π + ↑n * (2 * π) = x
⊢ ∃ k, ↑k * π = x

Tactic:
·

Goal state after:
case mp.inl
x : ℝ
n : ℤ
h : ↑n * (2 * π) = x
⊢ ∃ k, ↑k * π = x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: case mp.inr
x : ℝ
n : ℤ
h : π + ↑n * (2 * π) = x
⊢ ∃ k, ↑k * π = x

Tactic:
·

Goal state after:
case mp.inr
x : ℝ
n : ℤ
h : π + ↑n * (2 * π) = x
⊢ ∃ k, ↑k * π = x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: case mpr
x : ℝ
⊢ (∃ k, ↑k * π = x) → (∃ n, ↑n * (2 * π) = x) ∨ ∃ k, π + ↑k * (2 * π) = x

Tactic:
·

Goal state after:
case mpr
x : ℝ
⊢ (∃ k, ↑k * π = x) → (∃ n, ↑n * (2 * π) = x) ∨ ∃ k, π + ↑k * (2 * π) = x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: case mpr.inl
x : ℝ
n : ℤ
h : ↑(n + n) * π = x
⊢ (∃ n, ↑n * (2 * π) = x) ∨ ∃ k, π + ↑k * (2 * π) = x

pp: case mpr.inr
x : ℝ
n : ℤ
h : ↑(2 * n + 1) * π = x
⊢ (∃ n, ↑n * (2 * π) = x) ∨ ∃ k, π + ↑k * (2 * π) = x

Tactic:
·

Goal state after:
case mpr.inl
x : ℝ
n : ℤ
h : ↑(n + n) * π = x
⊢ (∃ n, ↑n * (2 * π) = x) ∨ ∃ k, π + ↑k * (2 * π) = x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: case mpr.inr
x : ℝ
n : ℤ
h : ↑(2 * n + 1) * π = x
⊢ (∃ n, ↑n * (2 * π) = x) ∨ ∃ k, π + ↑k * (2 * π) = x

Tactic:
·

Goal state after:
case mpr.inr
x : ℝ
n : ℤ
h : ↑(2 * n + 1) * π = x
⊢ (∃ n, ↑n * (2 * π) = x) ∨ ∃ k, π + ↑k * (2 * π) = x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: case mp
x : ℝ
⊢ ((∃ k, π / 2 + ↑k * (2 * π) = x) ∨ ∃ k, -(π / 2) + ↑k * (2 * π) = x) → ∃ k, π / 2 + ↑k * π = x

pp: case mpr
x : ℝ
⊢ (∃ k, π / 2 + ↑k * π = x) → (∃ k, π / 2 + ↑k * (2 * π) = x) ∨ ∃ k, -(π / 2) + ↑k * (2 * π) = x

Tactic:
·

Goal state after:
case mp
x : ℝ
⊢ ((∃ k, π / 2 + ↑k * (2 * π) = x) ∨ ∃ k, -(π / 2) + ↑k * (2 * π) = x) → ∃ k, π / 2 + ↑k * π = x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: case mp.inl
x : ℝ
n : ℤ
h : π / 2 + ↑n * (2 * π) = x
⊢ ∃ k, π / 2 + ↑k * π = x

pp: case mp.inr
x : ℝ
n : ℤ
h : -(π / 2) + ↑n * (2 * π) = x
⊢ ∃ k, π / 2 + ↑k * π = x

Tactic:
·

Goal state after:
case mp.inl
x : ℝ
n : ℤ
h : π / 2 + ↑n * (2 * π) = x
⊢ ∃ k, π / 2 + ↑k * π = x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: case mp.inr
x : ℝ
n : ℤ
h : -(π / 2) + ↑n * (2 * π) = x
⊢ ∃ k, π / 2 + ↑k * π = x

Tactic:
·

Goal state after:
case mp.inr
x : ℝ
n : ℤ
h : -(π / 2) + ↑n * (2 * π) = x
⊢ ∃ k, π / 2 + ↑k * π = x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: case mpr
x : ℝ
⊢ (∃ k, π / 2 + ↑k * π = x) → (∃ k, π / 2 + ↑k * (2 * π) = x) ∨ ∃ k, -(π / 2) + ↑k * (2 * π) = x

Tactic:
·

Goal state after:
case mpr
x : ℝ
⊢ (∃ k, π / 2 + ↑k * π = x) → (∃ k, π / 2 + ↑k * (2 * π) = x) ∨ ∃ k, -(π / 2) + ↑k * (2 * π) = x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: case mpr.inl
x : ℝ
n : ℤ
h : π / 2 + ↑(n + n) * π = x
⊢ (∃ k, π / 2 + ↑k * (2 * π) = x) ∨ ∃ k, -(π / 2) + ↑k * (2 * π) = x

pp: case mpr.inr
x : ℝ
n : ℤ
h : π / 2 + ↑(2 * n + 1) * π = x
⊢ (∃ k, π / 2 + ↑k * (2 * π) = x) ∨ ∃ k, -(π / 2) + ↑k * (2 * π) = x

Tactic:
·

Goal state after:
case mpr.inl
x : ℝ
n : ℤ
h : π / 2 + ↑(n + n) * π = x
⊢ (∃ k, π / 2 + ↑k * (2 * π) = x) ∨ ∃ k, -(π / 2) + ↑k * (2 * π) = x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: case mpr.inr
x : ℝ
n : ℤ
h : π / 2 + ↑(2 * n + 1) * π = x
⊢ (∃ k, π / 2 + ↑k * (2 * π) = x) ∨ ∃ k, -(π / 2) + ↑k * (2 * π) = x

Tactic:
·

Goal state after:
case mpr.inr
x : ℝ
n : ℤ
h : π / 2 + ↑(2 * n + 1) * π = x
⊢ (∃ k, π / 2 + ↑k * (2 * π) = x) ∨ ∃ k, -(π / 2) + ↑k * (2 * π) = x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

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

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

Tactic:
·

Goal state after:
case inl
x : ℝ
hx_gt : -(π / 2) < x
hx_lt : x < π / 2
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 inr
x : ℝ
hx_gt : -(π / 2) < x
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 : -(π / 2) < x
hx_lt : x < π / 2
r : ℤ
hxr_eq : x = (2 * ↑r + 1) * π / 2
h : r < 0
⊢ False
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 (deriv f) (interior D)

pp: case refine_2
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 ∈ interior D, 0 ≤ deriv^[2] f x

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 (deriv f) (interior D)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: case inl
x : ℝ
h : 0 < x
h' : 1 < x
⊢ sin x < x

pp: case inr
x : ℝ
h : 0 < x
h' : x ≤ 1
⊢ sin x < x

Tactic:
·

Goal state after:
case inl
x : ℝ
h : 0 < x
h' : 1 < x
⊢ sin x < x
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case refine_2
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 ∈ interior D, 0 ≤ deriv^[2] f x

Tactic:
·

Goal state after:
case refine_2
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 ∈ interior D, 0 ≤ deriv^[2] f x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: case inl
hx : 0 ≤ 0
⊢ sin 0 ≤ 0

pp: case inr
x : ℝ
hx✝ : 0 ≤ x
hx : 0 < x
⊢ sin x ≤ x

Tactic:
·

Goal state after:
case inl
hx : 0 ≤ 0
⊢ sin 0 ≤ 0
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: case inr
x : ℝ
hx✝ : 0 ≤ x
hx : 0 < x
⊢ sin x ≤ x

Tactic:
·

Goal state after:
case inr
x : ℝ
hx✝ : 0 ≤ x
hx : 0 < x
⊢ sin x ≤ x
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
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
⊢ 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
⊢ 0 < 2
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 (deriv f) (interior D)

pp: case refine_2
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 ∈ interior D, deriv^[2] f x ≤ 0

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 (deriv f) (interior D)
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case refine_2
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 ∈ interior D, deriv^[2] f x ≤ 0

Tactic:
·

Goal state after:
case refine_2
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 ∈ interior D, deriv^[2] f x ≤ 0
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

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

Tactic:
·

Goal state after:
case hx₁
⊢ -(π / 2) < π / 3
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: case hx₂
⊢ π / 3 < π / 2

Tactic:
·

Goal state after:
case hx₂
⊢ π / 3 < π / 2
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

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

Tactic:
·

Goal state after:
case hx₁
⊢ -(π / 2) < π / 6
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: case hx₂
⊢ π / 6 < π / 2

Tactic:
·

Goal state after:
case hx₂
⊢ π / 6 < π / 2
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: case inl
hx : 0 ≤ 0
⊢ 0 - 0 ^ 3 / 6 ≤ sin 0

pp: case inr
x : ℝ
hx✝ : 0 ≤ x
hx : 0 < x
⊢ x - x ^ 3 / 6 ≤ sin x

Tactic:
·

Goal state after:
case inl
hx : 0 ≤ 0
⊢ 0 - 0 ^ 3 / 6 ≤ sin 0
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: case inr
x : ℝ
this : ∀ (x : ℝ), 0 ≤ x → |x - sin x| ≤ |x| ^ 3 / 6
hx : ¬0 ≤ x
⊢ |x - sin x| ≤ |x| ^ 3 / 6

pp: x : ℝ
hx : 0 ≤ x
⊢ |x - sin x| ≤ |x| ^ 3 / 6

Tactic:
·

Goal state after:
case inr
x : ℝ
this : ∀ (x : ℝ), 0 ≤ x → |x - sin x| ≤ |x| ^ 3 / 6
hx : ¬0 ≤ x
⊢ |x - sin x| ≤ |x| ^ 3 / 6
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x : ℝ
hx : 0 ≤ x
⊢ |x - sin x| ≤ |x| ^ 3 / 6

Tactic:
·

Goal state after:
x : ℝ
hx : 0 ≤ x
⊢ |x - sin x| ≤ |x| ^ 3 / 6
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

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

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

Tactic:
·

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

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

pp: x : ℝ
h : 0 ≤ x
⊢ 1 + x ^ 2 ≠ 0

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

Tactic:
add_sub_cancel_left

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

x : ℝ
h : 0 ≤ x
⊢ 1 + x ^ 2 ≠ 0

x : ℝ
h : 0 ≤ x
⊢ 0 ≤ (1 + x ^ 2)⁻¹
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case refine_1
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
x : 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
hfc : ConvexOn 𝕜 s f
hxs : x ∈ interior s
y : 𝕜
hyx : y < x
hys : y ∈ s
y' : 𝕜
x✝ : y' ∈ slope f x '' {y | y ∈ s ∧ x < y}
z : 𝕜
hz : z ∈ {y | y ∈ s ∧ x < y}
hz' : slope f x z = y'
⊢ y ∈ s \ {x}

pp: case refine_2
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
x : 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
hfc : ConvexOn 𝕜 s f
hxs : x ∈ interior s
y : 𝕜
hyx : y < x
hys : y ∈ s
y' : 𝕜
x✝ : y' ∈ slope f x '' {y | y ∈ s ∧ x < y}
z : 𝕜
hz : z ∈ {y | y ∈ s ∧ x < y}
hz' : slope f x z = y'
⊢ z ∈ s \ {x}

Tactic:
·

Goal state after:
case refine_1
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
x : 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
hfc : ConvexOn 𝕜 s f
hxs : x ∈ interior s
y : 𝕜
hyx : y < x
hys : y ∈ s
y' : 𝕜
x✝ : y' ∈ slope f x '' {y | y ∈ s ∧ x < y}
z : 𝕜
hz : z ∈ {y | y ∈ s ∧ x < y}
hz' : slope f x z = y'
⊢ y ∈ s \ {x}
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
x : 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
hfc : ConvexOn 𝕜 s f
hxs : x ∈ interior s
y : 𝕜
hyx : y < x
hys : y ∈ s
y' : 𝕜
x✝ : y' ∈ slope f x '' {y | y ∈ s ∧ x < y}
z : 𝕜
hz : z ∈ {y | y ∈ s ∧ x < y}
hz' : slope f x z = y'
⊢ z ∈ s \ {x}

Tactic:
·

Goal state after:
case refine_2
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
x : 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
hfc : ConvexOn 𝕜 s f
hxs : x ∈ interior s
y : 𝕜
hyx : y < x
hys : y ∈ s
y' : 𝕜
x✝ : y' ∈ slope f x '' {y | y ∈ s ∧ x < y}
z : 𝕜
hz : z ∈ {y | y ∈ s ∧ x < y}
hz' : slope f x z = y'
⊢ z ∈ s \ {x}