mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Manifold.PartitionOfUnity

Goal state before:
pp: E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type uH
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : FiniteDimensional ℝ E
inst✝² : IsManifold I ∞ M
n : ℕ∞
inst✝¹ : T2Space M
inst✝ : SigmaCompactSpace M
s t : Set M
hs : IsClosed[inst✝⁵] s
ht : IsClosed[inst✝⁵] t
hd : Disjoint s t
⊢ ∃ f, EqOn (⇑f) 0 s ∧ EqOn (⇑f) 1 t ∧ ∀ (x : M), f x ∈ Icc 0 1

Tactic:
have : ∀ x ∈ t, sᶜ ∈ 𝓝 x := fun x hx => hs.isOpen_compl.mem_nhds (disjoint_right.1 hd hx)

Goal state after:
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type uH
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : FiniteDimensional ℝ E
inst✝² : IsManifold I ∞ M
n : ℕ∞
inst✝¹ : T2Space M
inst✝ : SigmaCompactSpace M
s t : Set M
hs : IsClosed[inst✝⁵] s
ht : IsClosed[inst✝⁵] t
hd : Disjoint s t
this : ∀ x ∈ t, sᶜ ∈ 𝓝 x
⊢ ∃ f, EqOn (⇑f) 0 s ∧ EqOn (⇑f) 1 t ∧ ∀ (x : M), f x ∈ Icc 0 1
Module: Mathlib.Geometry.Manifold.PartitionOfUnity

Goal state before:
pp: E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
H : Type uH
inst✝¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
inst✝ : FiniteDimensional ℝ E
n : ℕ∞
s : Set H
hs : IsOpen[inst✝¹] s
⊢ ∃ f, support f = s ∧ ContMDiff I 𝓘(ℝ, ℝ) (↑n) f ∧ range f ⊆ Icc 0 1

Tactic:
have h's : IsOpen (I.symm ⁻¹' s) := I.continuous_symm.isOpen_preimage _ hs

Goal state after:
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
H : Type uH
inst✝¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
inst✝ : FiniteDimensional ℝ E
n : ℕ∞
s : Set H
hs : IsOpen[inst✝¹] s
h's : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (↑I.symm ⁻¹' s)
⊢ ∃ f, support f = s ∧ ContMDiff I 𝓘(ℝ, ℝ) (↑n) f ∧ range f ⊆ Icc 0 1
Module: Mathlib.Geometry.Manifold.PartitionOfUnity

Goal state before:
pp: case refine_3
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type uH
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : FiniteDimensional ℝ E
inst✝² : IsManifold I ∞ M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
n : ℕ∞
s : Set M
hs : IsOpen[inst✝⁵] s
f : SmoothPartitionOfUnity M I M
hf : f.IsSubordinate fun x ↦ (chartAt H x).source
g : M → H → ℝ
g_supp : ∀ (c : M), support (g c) = (chartAt H c).target ∩ ↑(chartAt H c).symm ⁻¹' s
g_diff : ∀ (c : M), ContMDiff I 𝓘(ℝ, ℝ) (↑n) (g c)
hg : ∀ (c : M), range (g c) ⊆ Icc 0 1
h'g : ∀ (c : M) (x : H), 0 ≤ g c x
h''g : ∀ (c x : M), 0 ≤ (f c) x * g c (↑(chartAt H c) x)
x : M
⊢ 0 ≤ (fun x ↦ ∑ᶠ (c : M), (f c) x * g c (↑(chartAt H c) x)) x

Tactic:
apply finsum_nonneg (fun c ↦ h''g c x)

Goal state after:
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
h'f : HasCompactSupport f
hg : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] g
v : E
K : Set E := cthickening ‖v‖ (tsupport f)
K_compact : IsCompact K
t : ℝ
ht : t ∈ Ioc 0 1
t_pos : 0 < t
x : E
hx : x ∉ K
⊢ f x = 0

Tactic:
← Function.notMem_support

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
h'f : HasCompactSupport f
hg : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] g
v : E
K : Set E := cthickening ‖v‖ (tsupport f)
K_compact : IsCompact K
t : ℝ
ht : t ∈ Ioc 0 1
t_pos : 0 < t
x : E
hx : x ∉ K
⊢ x ∉ Function.support f
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
h'f : HasCompactSupport f
hg : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] g
v : E
K : Set E := cthickening ‖v‖ (tsupport f)
K_compact : IsCompact K
t : ℝ
ht : t ∈ Ioc 0 1
t_pos : 0 < t
x : E
hx : x ∉ K
A : f x = 0
⊢ f (x + t • v) = 0

Tactic:
← Function.notMem_support

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
h'f : HasCompactSupport f
hg : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] g
v : E
K : Set E := cthickening ‖v‖ (tsupport f)
K_compact : IsCompact K
t : ℝ
ht : t ∈ Ioc 0 1
t_pos : 0 < t
x : E
hx : x ∉ K
A : f x = 0
⊢ x + t • v ∉ Function.support f
Module: Mathlib.Analysis.Calculus.TangentCone.Seq

Goal state before:
pp: case mp
R : Type u_1
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : SMul R E
inst✝¹ : TopologicalSpace E
inst✝ : FirstCountableTopology E
s : Set E
x y : E
h : y ∈ tangentConeAt R s x
⊢ ∃ c d, Tendsto d atTop (𝓝 0) ∧ (∀ᶠ (n : ℕ) in atTop, x + d n ∈ s) ∧ Tendsto (fun n ↦ c n • d n) atTop (𝓝 y)

Tactic:
simp only [tangentConeAt_def, Set.mem_ofPred, ← map₂_smul, ← map_prod_eq_map₂, ClusterPt,
      ← neBot_inf_comap_iff_map'] at h

Goal state after:
case mp
R : Type u_1
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : SMul R E
inst✝¹ : TopologicalSpace E
inst✝ : FirstCountableTopology E
s : Set E
x y : E
h : (comap (fun p ↦ p.1 • p.2) (𝓝 y) ⊓ ⊤ ×ˢ 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0).NeBot
⊢ ∃ c d, Tendsto d atTop (𝓝 0) ∧ (∀ᶠ (n : ℕ) in atTop, x + d n ∈ s) ∧ Tendsto (fun n ↦ c n • d n) atTop (𝓝 y)
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: case hf.hcf
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C D : ℝ≥0
f g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
hg : LipschitzWith D g
h'g : HasCompactSupport g
v : E
A :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ) (𝓝[>] 0) (𝓝 (∫ (x : E), lineDeriv ℝ f x v * g x ∂μ))
B :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (g (x + t • -v) - g x) * f x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μ))
t : ℝ
S3 : ∫ (x : E), f (x + t • v) * g x ∂μ = ∫ (x : E), f x * g (x + t • -v) ∂μ
⊢ HasCompactSupport fun x ↦ f x * g (x + t • -v)

Tactic:
exact (h'g.comp_homeomorph (Homeomorph.addRight (t • (-v)))).mul_left

Goal state after:
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: case hf.hf
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C D : ℝ≥0
f g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
hg : LipschitzWith D g
h'g : HasCompactSupport g
v : E
A :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ) (𝓝[>] 0) (𝓝 (∫ (x : E), lineDeriv ℝ f x v * g x ∂μ))
B :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (g (x + t • -v) - g x) * f x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μ))
t : ℝ
S3 : ∫ (x : E), f (x + t • v) * g x ∂μ = ∫ (x : E), f x * g (x + t • -v) ∂μ
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] fun x ↦ f (x + t • v) * g x

pp: case hf.hcf
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C D : ℝ≥0
f g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
hg : LipschitzWith D g
h'g : HasCompactSupport g
v : E
A :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ) (𝓝[>] 0) (𝓝 (∫ (x : E), lineDeriv ℝ f x v * g x ∂μ))
B :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (g (x + t • -v) - g x) * f x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μ))
t : ℝ
S3 : ∫ (x : E), f (x + t • v) * g x ∂μ = ∫ (x : E), f x * g (x + t • -v) ∂μ
⊢ HasCompactSupport fun x ↦ f (x + t • v) * g x

Tactic:
· exact (hf.continuous.comp (continuous_add_const _)).mul hg.continuous

Goal state after:
case hf.hcf
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C D : ℝ≥0
f g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
hg : LipschitzWith D g
h'g : HasCompactSupport g
v : E
A :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ) (𝓝[>] 0) (𝓝 (∫ (x : E), lineDeriv ℝ f x v * g x ∂μ))
B :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (g (x + t • -v) - g x) * f x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μ))
t : ℝ
S3 : ∫ (x : E), f (x + t • v) * g x ∂μ = ∫ (x : E), f x * g (x + t • -v) ∂μ
⊢ HasCompactSupport fun x ↦ f (x + t • v) * g x
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Slope

Goal state before:
pp: f : ℝ → ℝ
a b c : ℝ
hf : MonotoneOn f (uIcc a (b + c))
hab : a ≤ b
hc✝ : 0 ≤ c
hc : 0 < c
hf' : IntervalIntegrable f volume a (b + c)
⊢ ∫ (x : ℝ) in b..b + c, f x ≤ c * f (b + c)

Tactic:
grw [intervalIntegral.integral_mono_on (g := fun _ ↦ f (b + c))
          (by linarith)
          (hf'.mono_set (by grind [uIcc]))
          (by simp)
          (by intros; apply hf <;> grind [uIcc])]

Goal state after:
f : ℝ → ℝ
a b c : ℝ
hf : MonotoneOn f (uIcc a (b + c))
hab : a ≤ b
hc✝ : 0 ≤ c
hc : 0 < c
hf' : IntervalIntegrable f volume a (b + c)
⊢ ∫ (u : ℝ) in b..b + c, f (b + c) ≤ c * f (b + c)
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: case hgf
⊢ (fun x ↦ 0) ≤ fun E ↦ volume (⋃ i ∈ Finset.range E.1, uIoc (E.2 i).1 (E.2 i).2)

Tactic:
intro; simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: case hgf
a b : ℝ
⊢ (fun x ↦ 0) ≤ fun E ↦ (volume.restrict (uIoc a b)) (⋃ i ∈ Finset.range E.1, uIoc (E.2 i).1 (E.2 i).2)

Tactic:
intro; simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: case h₁
F : Type u_2
inst✝³ : SeminormedAddCommGroup F
a b : ℝ
M : Type u_3
inst✝² : SeminormedRing M
inst✝¹ : Module M F
inst✝ : NormSMulClass M F
f : ℝ → M
g : ℝ → F
hf :
  Tendsto (fun E ↦ ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b))
    (𝓝 0)
hg :
  Tendsto (fun E ↦ ∑ i ∈ Finset.range E.1, dist (g (E.2 i).1) (g (E.2 i).2)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b))
    (𝓝 0)
C : ℝ
hC : ∀ x ∈ uIcc a b, ‖f x‖ ≤ C
D : ℝ
hD : ∀ x ∈ uIcc a b, ‖g x‖ ≤ D
n : ℕ
I : ℕ → ℝ × ℝ
hnI :
  (∀ i ∈ Finset.range n, (I i).1 ∈ uIcc a b ∧ (I i).2 ∈ uIcc a b) ∧
    (↑(Finset.range n)).PairwiseDisjoint fun i ↦ uIoc (I i).1 (I i).2
i : ℕ
hi : i ∈ Finset.range n
⊢ dist (f (I i).1 • g (I i).1) (f (I i).1 • g (I i).2) ≤ C * dist (g (I i).1) (g (I i).2)

Tactic:
rw [dist_smul₀]

Goal state after:
case h₁
F : Type u_2
inst✝³ : SeminormedAddCommGroup F
a b : ℝ
M : Type u_3
inst✝² : SeminormedRing M
inst✝¹ : Module M F
inst✝ : NormSMulClass M F
f : ℝ → M
g : ℝ → F
hf :
  Tendsto (fun E ↦ ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b))
    (𝓝 0)
hg :
  Tendsto (fun E ↦ ∑ i ∈ Finset.range E.1, dist (g (E.2 i).1) (g (E.2 i).2)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b))
    (𝓝 0)
C : ℝ
hC : ∀ x ∈ uIcc a b, ‖f x‖ ≤ C
D : ℝ
hD : ∀ x ∈ uIcc a b, ‖g x‖ ≤ D
n : ℕ
I : ℕ → ℝ × ℝ
hnI :
  (∀ i ∈ Finset.range n, (I i).1 ∈ uIcc a b ∧ (I i).2 ∈ uIcc a b) ∧
    (↑(Finset.range n)).PairwiseDisjoint fun i ↦ uIoc (I i).1 (I i).2
i : ℕ
hi : i ∈ Finset.range n
⊢ ‖f (I i).1‖ * dist (g (I i).1) (g (I i).2) ≤ C * dist (g (I i).1) (g (I i).2)
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: case inr.hu'
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℝ → F
x x₀ : ℝ
n : ℕ
hf : ContDiffOn ℝ (↑(n + 1)) f [[x₀, x]]
this✝ : x₀ ≠ x
this : UniqueDiffOn ℝ [[x₀, x]]
k : ℕ
hk : k ≤ n
⊢ IntervalIntegrable (deriv fun t ↦ (x - t) ^ k / ↑k !) MeasureTheory.volume x₀ x

pp: case inr.hv'
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℝ → F
x x₀ : ℝ
n : ℕ
hf : ContDiffOn ℝ (↑(n + 1)) f [[x₀, x]]
this✝ : x₀ ≠ x
this : UniqueDiffOn ℝ [[x₀, x]]
k : ℕ
hk : k ≤ n
⊢ IntervalIntegrable (deriv fun t ↦ iteratedDerivWithin k f [[x₀, x]] t) MeasureTheory.volume x₀ x

Tactic:
· apply ContinuousOn.intervalIntegrable
    fun_prop

Goal state after:
case inr.hv'
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℝ → F
x x₀ : ℝ
n : ℕ
hf : ContDiffOn ℝ (↑(n + 1)) f [[x₀, x]]
this✝ : x₀ ≠ x
this : UniqueDiffOn ℝ [[x₀, x]]
k : ℕ
hk : k ≤ n
⊢ IntervalIntegrable (deriv fun t ↦ iteratedDerivWithin k f [[x₀, x]] t) MeasureTheory.volume x₀ x
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: case inr.x
F : Type u_2
inst✝ : SeminormedAddCommGroup F
a✝ b✝ : ℝ
f : ℝ → F
a b : ℝ
hf :
  ∀ ε > 0,
    ∃ δ > 0,
      ∀ E ∈ disjWithin a b,
        ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ →
          ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < ε
hab₀ : a ≤ b
hab : a < b
δ : ℝ
hδ₁ : δ > 0
hδ₂ :
  ∀ E ∈ disjWithin a b,
    ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ → ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < 1
hab₁ : 0 < b - a
n : ℕ
hn : 1 / (↑n + 1) < δ / (b - a)
δ' : ℝ := (b - a) / (↑n + 1)
hδ₃ : δ' < δ
h_mono : Monotone fun i ↦ a + ↑i * δ'
v_sum : eVariationOn f (Icc a b) = ∑ i ∈ Finset.range (n + 1), eVariationOn f (Icc (a + ↑i * δ') (a + (↑i + 1) * δ'))
v_each : ∀ (x y : ℝ), a ≤ x → x ≤ y → y < x + δ → y ≤ b → eVariationOn f (Icc x y) ≤ 1
i : ℕ
hi : i < n + 1
⊢ a + (↑i * δ' + 1 * δ') < a + ↑i * δ' + δ

Tactic:
← add_assoc

Goal state after:
case inr.x
F : Type u_2
inst✝ : SeminormedAddCommGroup F
a✝ b✝ : ℝ
f : ℝ → F
a b : ℝ
hf :
  ∀ ε > 0,
    ∃ δ > 0,
      ∀ E ∈ disjWithin a b,
        ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ →
          ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < ε
hab₀ : a ≤ b
hab : a < b
δ : ℝ
hδ₁ : δ > 0
hδ₂ :
  ∀ E ∈ disjWithin a b,
    ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ → ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < 1
hab₁ : 0 < b - a
n : ℕ
hn : 1 / (↑n + 1) < δ / (b - a)
δ' : ℝ := (b - a) / (↑n + 1)
hδ₃ : δ' < δ
h_mono : Monotone fun i ↦ a + ↑i * δ'
v_sum : eVariationOn f (Icc a b) = ∑ i ∈ Finset.range (n + 1), eVariationOn f (Icc (a + ↑i * δ') (a + (↑i + 1) * δ'))
v_each : ∀ (x y : ℝ), a ≤ x → x ≤ y → y < x + δ → y ≤ b → eVariationOn f (Icc x y) ≤ 1
i : ℕ
hi : i < n + 1
⊢ a + ↑i * δ' + 1 * δ' < a + ↑i * δ' + δ
Module: Mathlib.Analysis.Complex.AbsMax

Goal state before:
pp: F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : ℂ → F
z w : ℂ
hd : DiffContOnCl ℂ f (ball z (dist w z))
hz : IsMaxOn (norm ∘ f) (closedBall z (dist w z)) z
⊢ ‖f w‖ = ‖f z‖

Tactic:
set e : F →L[ℂ] F̂ := UniformSpace.Completion.toComplL

Goal state after:
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : ℂ → F
z w : ℂ
hd : DiffContOnCl ℂ f (ball z (dist w z))
hz : IsMaxOn (norm ∘ f) (closedBall z (dist w z)) z
e : F →L[ℂ] F̂ := UniformSpace.Completion.toComplL
⊢ ‖f w‖ = ‖f z‖
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: z : ℂ
⊢ tan (2 * z) = 2 * tan z / (1 - tan z ^ 2)

Tactic:
by_cases! h : ∀ k : ℤ, z ≠ (2 * k + 1) * π / 2
  · rw [two_mul, two_mul, sq, tan_add (Or.inl ⟨h, h⟩)]
  · rw [two_mul, two_mul, sq, tan_add (Or.inr ⟨h, h⟩)]

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

Goal state before:
pp: x : ℂ
h : cos x ≠ -1
⊢ cos x = (1 - tan (x / 2) ^ 2) / (1 + tan (x / 2) ^ 2)

Tactic:
by
  conv_lhs => rw [← mul_div_cancel₀ x two_ne_zero, cos_two_mul']
  have : cos (x / 2) ≠ 0 := by grind [cos_ne_zero_iff, cos_eq_neg_one_iff]
  rw [div_eq_mul_inv (1 - tan (x / 2) ^ 2) (1 + tan (x / 2) ^ 2), inv_one_add_tan_sq this,
    ← tan_mul_cos this]
  ring

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

Goal state before:
pp: n : ℕ
a : ℝ
h : ∀ (k : ℤ), a ≠ ↑k * (2 * π)
b : ℝ
⊢ ∑ i ∈ range n, sin (a * ↑i + b) = sin (↑n * a / 2) * sin ((↑n - 1) * a / 2 + b) / sin (a / 2)

Tactic:
have h := Complex.sum_sin n (a := a) (by exact_mod_cast h) b
  exact_mod_cast congr($(h).re)

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

Goal state before:
pp: this : -(π / 2) < π / 2
⊢ Tendsto (fun x ↦ tan ↑x) atBot atBot

Tactic:
by rw [tendsto_comp_coe_Ioo_atBot this]; exact tendsto_tan_neg_pi_div_two

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

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

Tactic:
by
  wlog hx : 0 ≤ x
  · grind [sin_neg]
  · grind [Real.sin_le, abs_of_nonneg, sin_ge_sub_cube]

Goal state after:
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:
rw [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.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:
rw [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.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 (c * (c * ⟪x, y⟫) / (|c| * |c| * (‖x‖ * ‖y‖))) = arccos (⟪x, y⟫ / (‖x‖ * ‖y‖))

Tactic:
abs_mul_abs_self,

Goal state after:
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 (c * (c * ⟪x, y⟫) / (c * c * (‖x‖ * ‖y‖))) = arccos (⟪x, y⟫ / (‖x‖ * ‖y‖))
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:
rw [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.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:
rw [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.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 y (r • x) = angle (-x) y

Tactic:
angle_smul_right_of_neg y x hr,

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
r : ℝ
hr : r < 0
⊢ angle y (-x) = angle (-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
⊢ angle x y = 0 ↔ x ≠ 0 ∧ ∃ r, 0 < r ∧ y = r • x

Tactic:
rw [angle, ← real_inner_div_norm_mul_norm_eq_one_iff, Real.arccos_eq_zero, LE.le.ge_iff_eq',
    eq_comm]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ ⟪x, y⟫ / (‖x‖ * ‖y‖) ≤ 1
Module: Mathlib.Analysis.Complex.BorelCaratheodory

Goal state before:
pp: f : ℂ → ℂ
M R : ℝ
z : ℂ
hM : 0 < M
hf : DifferentiableOn ℂ f (ball 0 R)
hf₁ : Set.MapsTo f (ball 0 R) {z | z.re ≤ M}
hz : z ∈ ball 0 R
hf₂ : f 0 = 0
⊢ DifferentiableOn ℂ (fun {z} ↦ f z / (2 * ↑M - f z)) (ball 0 R)

Tactic:
apply hf.div (hf.const_sub _) fun x hx h ↦ ?_
    have := sub_eq_zero.mp h ▸ hf₁ hx
    aesop

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

Goal state before:
pp: f : ℂ → ℂ
c z : ℂ
R₁ R₂ : ℝ
n : ℕ
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)
hn : (fun x ↦ f x - f c) =o[𝓝 c] fun w ↦ (w - c) ^ n
hz : z ∈ ball c R₁
hR₁ : 0 < R₁
⊢ ‖f z - f c‖ ≤ R₂ * (‖z - c‖ / R₁) ^ (n + 1)

Tactic:
wlog hd' : DifferentiableOn ℂ f (closedBall c R₁) ∧
    MapsTo f (closedBall c R₁) (closedBall (f c) R₂) generalizing R₁

Goal state after:
case inr
f : ℂ → ℂ
c z : ℂ
R₁ R₂ : ℝ
n : ℕ
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)
hn : (fun x ↦ f x - f c) =o[𝓝 c] fun w ↦ (w - c) ^ n
hz : z ∈ ball c R₁
hR₁ : 0 < R₁
this :
  ∀ {R₁ : ℝ},
    DifferentiableOn ℂ f (ball c R₁) →
      MapsTo f (ball c R₁) (closedBall (f c) R₂) →
        z ∈ ball c R₁ →
          0 < R₁ →
            DifferentiableOn ℂ f (closedBall c R₁) ∧ MapsTo f (closedBall c R₁) (closedBall (f c) R₂) →
              ‖f z - f c‖ ≤ R₂ * (‖z - c‖ / R₁) ^ (n + 1)
hd' : ¬(DifferentiableOn ℂ f (closedBall c R₁) ∧ MapsTo f (closedBall c R₁) (closedBall (f c) R₂))
⊢ ‖f z - f c‖ ≤ R₂ * (‖z - c‖ / R₁) ^ (n + 1)

f : ℂ → ℂ
c z : ℂ
R₂ : ℝ
n : ℕ
hn : (fun x ↦ f x - f c) =o[𝓝 c] fun w ↦ (w - c) ^ n
R₁ : ℝ
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)
hz : z ∈ ball c R₁
hR₁ : 0 < R₁
hd' : DifferentiableOn ℂ f (closedBall c R₁) ∧ MapsTo f (closedBall c R₁) (closedBall (f c) R₂)
⊢ ‖f z - f c‖ ≤ R₂ * (‖z - c‖ / R₁) ^ (n + 1)
Module: Mathlib.Analysis.Complex.Schwarz

Goal state before:
pp: case hd
f : ℂ → ℂ
c z : ℂ
R₂ : ℝ
n : ℕ
hn : (fun x ↦ f x - f c) =o[𝓝 c] fun w ↦ (w - c) ^ n
R₁ : ℝ
hz : z ∈ ball c R₁
hR₁ : 0 < R₁
hd : DifferentiableOn ℂ f (closedBall c R₁)
h_maps : MapsTo f (closedBall c R₁) (closedBall (f c) R₂)
hne : z ≠ c
g : ℂ → ℂ := fun w ↦ ((w - c) ^ (n + 1))⁻¹ * (f w - f c)
g' : ℂ → ℂ := update g c ((𝓝[≠] c).limUnder g)
⊢ DifferentiableOn ℂ (fun w ↦ ((w - c) ^ (n + 1))⁻¹) (closedBall c R₁ \ {c})

Tactic:
fun_prop (disch := simp +contextual [sub_eq_zero])

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

Goal state before:
pp: case refine_1
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : ProperSpace 𝕜
n : ℤ
hn : ↑n ≠ 0
this : ∀ (x : 𝕜), x ^ n = 0 ↔ x = 0
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ((fun x ↦ x ^ n) ⁻¹' {0}ᶜ)

Tactic:
convert isClosed_singleton (x := (0 : 𝕜)).isOpen_compl
    ext; simp [this]

Goal state after: