mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts

Goal state before:
pp: E : Type u_1
F : Type u_2
G : Type u_3
W : Type u_4
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace ℝ G
inst✝³ : NormedAddCommGroup W
inst✝² : NormedSpace ℝ W
inst✝¹ : MeasurableSpace E
μ : Measure E
inst✝ : SigmaFinite μ
f f' : E × ℝ → F
g g' : E × ℝ → G
B : F →L[ℝ] G →L[ℝ] W
hf'g : Integrable (fun x ↦ (B (f' x)) (g x)) (μ.prod volume)
hfg' : Integrable (fun x ↦ (B (f x)) (g' x)) (μ.prod volume)
hfg : Integrable (fun x ↦ (B (f x)) (g x)) (μ.prod volume)
hf : ∀ x ∈ tsupport g, HasLineDerivAt ℝ f (f' x) x (0, 1)
hg : ∀ x ∈ tsupport f, HasLineDerivAt ℝ g (g' x) x (0, 1)
x : E
hf'gx : Integrable (fun y ↦ (B (f' (x, y))) (g (x, y))) volume
hfg'x : Integrable (fun y ↦ (B (f (x, y))) (g' (x, y))) volume
hfgx : Integrable (fun y ↦ (B (f (x, y))) (g (x, y))) volume
t : ℝ
ht : t ∈ tsupport fun x_1 ↦ f (x, x_1)
⊢ Continuous[_, instTopologicalSpaceProd] fun y ↦ (x, y)

Tactic:
by fun_prop

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
H : ∀ (ε : ℝ), 0 < ε → ‖(f' x - A) z‖ ≤ (↑δ + ε) * (‖z‖ + ε) + ‖f' x - A‖ * ε
⊢ Continuous fun ε ↦ (↑δ + ε) * (‖z‖ + ε) + ‖f' x - A‖ * ε

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.LineDeriv

Goal state before:
pp: E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
f : E → ℝ
a b : E
f' : ℝ
h : IsLocalExtr f a
hd : HasLineDerivAt ℝ f f' a b
⊢ Continuous[_, inst✝²] fun t ↦ a + t • b

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.LineDeriv

Goal state before:
pp: E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
f : E → ℝ
a : E
h : IsLocalExtr f a
b : E
⊢ Continuous[_, inst✝²] fun t ↦ a + t • b

Tactic:
by fun_prop

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
ε : ℝ
εpos : 0 < ε
B₁ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • closedBall z ε)).Nonempty
ρ : ℝ
ρpos : ρ > 0
hρ : ball x ρ ∩ s ⊆ {y | ‖f y - f x - (f' x) (y - x)‖ ≤ ε * ‖y - x‖}
B₂ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, {x} + r • closedBall z ε ⊆ ball x ρ
r : ℝ
y : E
ys : y ∈ s
hy : y ∈ {x} + r • closedBall z ε
rρ : {x} + r • closedBall z ε ⊆ ball x ρ
rpos : 0 < r
a : E
az : a ∈ closedBall z ε
ya : y = x + r • a
norm_a : ‖a‖ ≤ ‖z‖ + ε
I : r * ‖(f' x - A) a‖ ≤ r * (↑δ + ε) * (‖z‖ + ε)
⊢ ‖(f' x - A) z‖ = ‖(f' x - A) a + (f' x - A) (z - a)‖

Tactic:
congr 1

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
ε : ℝ
εpos : 0 < ε
B₁ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • closedBall z ε)).Nonempty
ρ : ℝ
ρpos : ρ > 0
hρ : ball x ρ ∩ s ⊆ {y | ‖f y - f x - (f' x) (y - x)‖ ≤ ε * ‖y - x‖}
B₂ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, {x} + r • closedBall z ε ⊆ ball x ρ
r : ℝ
y : E
ys : y ∈ s
hy : y ∈ {x} + r • closedBall z ε
rρ : {x} + r • closedBall z ε ⊆ ball x ρ
rpos : 0 < r
a : E
az : a ∈ closedBall z ε
ya : y = x + r • a
norm_a : ‖a‖ ≤ ‖z‖ + ε
I : r * ‖(f' x - A) a‖ ≤ r * (↑δ + ε) * (‖z‖ + ε)
⊢ (f' x - A) z = (f' x - A) a + (f' x - A) (z - a)
Module: Mathlib.Analysis.Complex.RemovableSingularity

Goal state before:
pp: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
U : Set ℂ
hU : IsOpen U
c w₀ : ℂ
R : ℝ
f : ℂ → E
hc : closedBall c R ⊆ U
hf : DifferentiableOn ℂ f U
hw₀ : w₀ ∈ ball c R
hf' : DifferentiableOn ℂ (dslope f w₀) U
h0 : (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z = dslope f w₀ w₀
⊢ (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z =
    (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z

Tactic:
congr 1

Goal state after:
case e_a
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
U : Set ℂ
hU : IsOpen U
c w₀ : ℂ
R : ℝ
f : ℂ → E
hc : closedBall c R ⊆ U
hf : DifferentiableOn ℂ f U
hw₀ : w₀ ∈ ball c R
hf' : DifferentiableOn ℂ (dslope f w₀) U
h0 : (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z = dslope f w₀ w₀
⊢ ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z = ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z
Module: Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts

Goal state before:
pp: case hg
E : Type u_1
F : Type u_2
G : Type u_3
W : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace ℝ G
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace ℝ W
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
f f' : E → F
g g' : E → G
v : E
B : F →L[ℝ] G →L[ℝ] W
hf'g : Integrable (fun x ↦ (B (f' x)) (g x)) μ
hfg' : Integrable (fun x ↦ (B (f x)) (g' x)) μ
hfg : Integrable (fun x ↦ (B (f x)) (g x)) μ
hf : ∀ x ∈ tsupport g, HasLineDerivAt ℝ f (f' x) x v
hW : CompleteSpace W
hv : v ≠ 0
this✝ : Nontrivial E
n : ℕ := finrank ℝ E
E' : Type := Fin (n - 1) → ℝ
L : E ≃L[ℝ] E' × ℝ
hL : L v = (0, 1)
ν : Measure (E' × ℝ) := map (⇑L) μ
L_emb : MeasurableEmbedding ⇑L
x : E' × ℝ
hg : HasLineDerivAt ℝ g (g' (L.symm x)) (L.symm x) v
hx : x ∈ tsupport fun x ↦ f (L.symm x)
this : g = (g ∘ ⇑L.symm) ∘ ⇑↑↑L
h2x : L.symm x ∈ tsupport f
⊢ HasLineDerivAt ℝ (fun x ↦ g (L.symm x)) (g' (L.symm x)) x (0, 1)

Tactic:
rw [this] at hg

Goal state after:
case hg
E : Type u_1
F : Type u_2
G : Type u_3
W : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace ℝ G
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace ℝ W
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
f f' : E → F
g g' : E → G
v : E
B : F →L[ℝ] G →L[ℝ] W
hf'g : Integrable (fun x ↦ (B (f' x)) (g x)) μ
hfg' : Integrable (fun x ↦ (B (f x)) (g' x)) μ
hfg : Integrable (fun x ↦ (B (f x)) (g x)) μ
hf : ∀ x ∈ tsupport g, HasLineDerivAt ℝ f (f' x) x v
hW : CompleteSpace W
hv : v ≠ 0
this✝ : Nontrivial E
n : ℕ := finrank ℝ E
E' : Type := Fin (n - 1) → ℝ
L : E ≃L[ℝ] E' × ℝ
hL : L v = (0, 1)
ν : Measure (E' × ℝ) := map (⇑L) μ
L_emb : MeasurableEmbedding ⇑L
x : E' × ℝ
hg : HasLineDerivAt ℝ ((g ∘ ⇑L.symm) ∘ ⇑↑↑L) (g' (L.symm x)) (L.symm x) v
hx : x ∈ tsupport fun x ↦ f (L.symm x)
this : g = (g ∘ ⇑L.symm) ∘ ⇑↑↑L
h2x : L.symm x ∈ tsupport f
⊢ HasLineDerivAt ℝ (fun x ↦ g (L.symm x)) (g' (L.symm x)) x (0, 1)
Module: Mathlib.Analysis.Complex.LocallyUniformLimit

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
z : ℂ
r : ℝ
f g : ℂ → E
hr : 0 < r
hf : ContinuousOn f (sphere z r)
hg : ContinuousOn g (sphere z r)
h1 : ContinuousOn (fun w ↦ ((w - z) ^ 2)⁻¹) (sphere z r)
⊢ (2 * ↑π * I)⁻¹ • ∮ (w : ℂ) in C(z, r), ((w - z) ^ 2)⁻¹ • (f - g) w =
    (2 * ↑π * I)⁻¹ • ((∮ (w : ℂ) in C(z, r), ((w - z) ^ 2)⁻¹ • f w) - ∮ (w : ℂ) in C(z, r), ((w - z) ^ 2)⁻¹ • g w)

Tactic:
congr 1

Goal state after:
case e_a
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
z : ℂ
r : ℝ
f g : ℂ → E
hr : 0 < r
hf : ContinuousOn f (sphere z r)
hg : ContinuousOn g (sphere z r)
h1 : ContinuousOn (fun w ↦ ((w - z) ^ 2)⁻¹) (sphere z r)
⊢ ∮ (w : ℂ) in C(z, r), ((w - z) ^ 2)⁻¹ • (f - g) w =
    (∮ (w : ℂ) in C(z, r), ((w - z) ^ 2)⁻¹ • f w) - ∮ (w : ℂ) in C(z, r), ((w - z) ^ 2)⁻¹ • g w
Module: Mathlib.Analysis.Complex.LocallyUniformLimit

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
z : ℂ
M r : ℝ
f : ℂ → E
hr : 0 < r
hfM : ∀ w ∈ sphere z r, ‖f w‖ < M
hf : ContinuousOn f (sphere z r)
e1 : (sphere z r).Nonempty
e2 : ContinuousOn (fun w ↦ ‖f w‖) (sphere z r)
⊢ ∃ L < M, ∀ w ∈ sphere z r, ‖f w‖ ≤ L

Tactic:
obtain ⟨x, hx, hx'⟩ := (isCompact_sphere z r).exists_isMaxOn e1 e2

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
z : ℂ
M r : ℝ
f : ℂ → E
hr : 0 < r
hfM : ∀ w ∈ sphere z r, ‖f w‖ < M
hf : ContinuousOn f (sphere z r)
e1 : (sphere z r).Nonempty
e2 : ContinuousOn (fun w ↦ ‖f w‖) (sphere z r)
x : ℂ
hx : x ∈ sphere z r
hx' : IsMaxOn (fun w ↦ ‖f w‖) (sphere z r) x
⊢ ∃ L < M, ∀ w ∈ sphere z r, ‖f w‖ ≤ L
Module: Mathlib.Analysis.Complex.LocallyUniformLimit

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
K : Set ℂ
δ : ℝ
φ : Filter ι
F : ι → ℂ → E
f : ℂ → E
hF : TendstoUniformlyOn F f φ (cthickening δ K)
hδ : 0 < δ
hFn : ∀ᶠ (n : ι) in φ, ContinuousOn (F n) (cthickening δ K)
⊢ TendstoUniformlyOn (Complex.cderiv δ ∘ F) (Complex.cderiv δ f) φ K

Tactic:
rcases φ.eq_or_neBot with rfl | hne

Goal state after:
case inl
E : Type u_1
ι : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
K : Set ℂ
δ : ℝ
F : ι → ℂ → E
f : ℂ → E
hδ : 0 < δ
hF : TendstoUniformlyOn F f ⊥ (cthickening δ K)
hFn : ∀ᶠ (n : ι) in ⊥, ContinuousOn (F n) (cthickening δ K)
⊢ TendstoUniformlyOn (cderiv δ ∘ F) (cderiv δ f) ⊥ K

case inr
E : Type u_1
ι : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
K : Set ℂ
δ : ℝ
φ : Filter ι
F : ι → ℂ → E
f : ℂ → E
hF : TendstoUniformlyOn F f φ (cthickening δ K)
hδ : 0 < δ
hFn : ∀ᶠ (n : ι) in φ, ContinuousOn (F n) (cthickening δ K)
hne : φ.NeBot
⊢ TendstoUniformlyOn (Complex.cderiv δ ∘ F) (Complex.cderiv δ f) φ K
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[>] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
⊢ ∫ (x : ℝ) in Ioi a, u' x * v x + u x * v' x = b' - a'

Tactic:
rw [← Ici_sdiff_left] at h_zero
  let f := Function.update (u * v) a a'
  have hderiv : ∀ x ∈ Ioi a, HasDerivAt f (u' x * v x + u x * v' x) x := by
    intro x (hx : a < x)
    apply ((hu x hx).mul (hv x hx)).congr_of_eventuallyEq
    filter_upwards [eventually_ne_nhds hx.ne.symm] with y hy
    exact Function.update_of_ne hy a' (u * v)
  have htendsto : Tendsto f atTop (𝓝 b') := by
    apply h_infty.congr'
    filter_upwards [eventually_ne_atTop a] with x hx
    exact (Function.update_of_ne hx a' (u * v)).symm
  simpa using integral_Ioi_of_hasDerivAt_of_tendsto
    (continuousWithinAt_update_same.mpr h_zero) hderiv huv htendsto

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

Goal state before:
pp: ι : Type u_1
s : Set ℂ
hs : IsOpen s
x : ℂ
hx : x ∈ s
f : ι → ℂ → ℂ
hf : ∀ (i : ι), f i x ≠ 0
hd : ∀ (i : ι), DifferentiableOn ℂ (f i) s
hm : Summable fun i ↦ logDeriv (f i) x
htend : MultipliableLocallyUniformlyOn f s
hnez : ∏' (i : ι), f i x ≠ 0
⊢ ∀ (x : Finset ι), DifferentiableOn ℂ (fun x2 ↦ ∏ i ∈ x, f i x2) s

Tactic:
by fun_prop

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

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
U : Set ℂ
φ : Filter ι
F : ι → ℂ → E
f : ℂ → E
inst✝ : CompleteSpace E
hf : TendstoLocallyUniformlyOn F f φ U
hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U
hU : IsOpen U
⊢ ∀ K ⊆ U, IsCompact K → TendstoUniformlyOn (_root_.deriv ∘ F) (_root_.deriv f) φ K

Tactic:
rcases φ.eq_or_neBot with rfl | hne

Goal state after:
case inl
E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
U : Set ℂ
F : ι → ℂ → E
f : ℂ → E
inst✝ : CompleteSpace E
hU : IsOpen U
hf : TendstoLocallyUniformlyOn F f ⊥ U
hF : ∀ᶠ (n : ι) in ⊥, DifferentiableOn ℂ (F n) U
⊢ ∀ K ⊆ U, IsCompact K → TendstoUniformlyOn (deriv ∘ F) (deriv f) ⊥ K

case inr
E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
U : Set ℂ
φ : Filter ι
F : ι → ℂ → E
f : ℂ → E
inst✝ : CompleteSpace E
hf : TendstoLocallyUniformlyOn F f φ U
hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U
hU : IsOpen U
hne : φ.NeBot
⊢ ∀ K ⊆ U, IsCompact K → TendstoUniformlyOn (_root_.deriv ∘ F) (_root_.deriv f) φ K
Module: Mathlib.Analysis.RCLike.TangentCone

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
h𝕜 : IsRCLikeNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace ℝ E
s : Set E
x : E
hs : UniqueDiffWithinAt ℝ s x
this : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
⊢ UniqueDiffWithinAt 𝕜 s x

Tactic:
exact hs.mono_field

Goal state after:
Module: Mathlib.Geometry.Manifold.HasGroupoid

Goal state before:
pp: H : Type u
M : Type u_2
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
e e' : OpenPartialHomeomorph M H
he : e ∈ StructureGroupoid.maximalAtlas M G
he' : e' ∈ atlas H M
s : Set M
hs : IsOpen[inst✝²] s
hs'' : IsOpen[inst✝³] (↑e '' (e.source ∩ s))
this : (e.restr (e.source ∩ s)).symm ≫ₕ e' ∈ G
⊢ (e.restr s).symm ≫ₕ e' ≈ (e.restr (e.source ∩ s)).symm ≫ₕ e'

Tactic:
exact EqOnSource.trans' (Setoid.symm e.restr_inter_source).symm' (eqOnSource_refl e')

Goal state after:
Module: Mathlib.Geometry.Manifold.ChartedSpace

Goal state before:
pp: H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝ : TopologicalSpace H
c : ChartedSpaceCore H M
e✝ e : PartialEquiv M H
he : e ∈ c.atlas
this✝ : TopologicalSpace M := c.toTopologicalSpace
t : Set M
e' : PartialEquiv M H
e'_atlas : e' ∈ c.atlas
s : Set H
s_open : IsOpen[inst✝] s
ts : t = ↑e' ⁻¹' s ∩ e'.source
f : PartialEquiv H H := e.symm ≫ e'
this : IsOpen[inst✝] (↑f ⁻¹' s ∩ f.source)
⊢ ↑e' ∘ ↑e.symm ⁻¹' s ∩ e.target ∩ ↑e.symm ⁻¹' e'.source = e.target ∩ ↑e' ∘ ↑e.symm ⁻¹' s ∩ ↑e.symm ⁻¹' e'.source

Tactic:
congr 1

Goal state after:
case e_a
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝ : TopologicalSpace H
c : ChartedSpaceCore H M
e✝ e : PartialEquiv M H
he : e ∈ c.atlas
this✝ : TopologicalSpace M := c.toTopologicalSpace
t : Set M
e' : PartialEquiv M H
e'_atlas : e' ∈ c.atlas
s : Set H
s_open : IsOpen[inst✝] s
ts : t = ↑e' ⁻¹' s ∩ e'.source
f : PartialEquiv H H := e.symm ≫ e'
this : IsOpen[inst✝] (↑f ⁻¹' s ∩ f.source)
⊢ ↑e' ∘ ↑e.symm ⁻¹' s ∩ e.target = e.target ∩ ↑e' ∘ ↑e.symm ⁻¹' s
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: H : Type u_1
inst✝ : TopologicalSpace H
u : StructureGroupoid H
f : OpenPartialHomeomorph H H
hf✝ : f ∈ (idGroupoid H).members
hf : f ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}
⊢ f ∈ u.members

Tactic:
simp only [singleton_union, mem_ofPred_eq, mem_insert_iff] at hf

Goal state after:
H : Type u_1
inst✝ : TopologicalSpace H
u : StructureGroupoid H
f : OpenPartialHomeomorph H H
hf✝ : f ∈ (idGroupoid H).members
hf : f = OpenPartialHomeomorph.refl H ∨ f.source = ∅
⊢ f ∈ u.members
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
ε : ℝ≥0
εpos : 0 < ε
δ : (E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |B.det - A.det| ≤ ↑ε) ∧
        ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → μ (g '' t) ≤ (ENNReal.ofReal |A.det| + ↑ε) * μ t
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ n, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
n : ℕ
x : E
hx : ‖f' x - A n‖₊ ≤ δ (A n)
⊢ |(A n).det| = |(f' x).det - ((f' x).det - (A n).det)|

Tactic:
congr 1

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
ε : ℝ≥0
εpos : 0 < ε
δ : (E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |B.det - A.det| ≤ ↑ε) ∧
        ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → μ (g '' t) ≤ (ENNReal.ofReal |A.det| + ↑ε) * μ t
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ n, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
n : ℕ
x : E
hx : ‖f' x - A n‖₊ ≤ δ (A n)
⊢ (A n).det = (f' x).det - ((f' x).det - (A n).det)
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: case right
H : Type u_1
inst✝ : TopologicalSpace H
PG : Pregroupoid H
e e' : OpenPartialHomeomorph H H
he : e ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
ee' : e' ≈ e
A : e'.symm ≈ e.symm
⊢ PG.property (↑e'.symm) e'.target

Tactic:
apply PG.congr e'.symm.open_source A.2

Goal state after:
case right
H : Type u_1
inst✝ : TopologicalSpace H
PG : Pregroupoid H
e e' : OpenPartialHomeomorph H H
he : e ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
ee' : e' ≈ e
A : e'.symm ≈ e.symm
⊢ PG.property (↑e.symm) e'.symm.source
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
h : IsRCLikeNormedField 𝕜
⊢ UniqueDiffOn 𝕜 (range ↑I)

Tactic:
let := h.rclike 𝕜
    let := NormedSpace.restrictScalars ℝ 𝕜 E
    apply uniqueDiffOn_convex_of_isRCLikeNormedField _ I.nonempty_interior
    simpa [h] using I.convex_range

Goal state after:
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: H : Type u_1
M : Type u_2
H' : Type u_3
M' : Type u_4
inst✝⁵ : TopologicalSpace H
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : TopologicalSpace H'
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
g : M → M'
s t : Set M
x : M
hG : G.LocalInvariantProp G' P
hu : s =ᶠ[𝓝 x] t
⊢ LiftPropWithinAt P g (s ∩ {x | (fun x ↦ s x = t x) x}) x = LiftPropWithinAt P g (t ∩ {x | (fun x ↦ s x = t x) x}) x

Tactic:
congr 1

Goal state after:
case e_s
H : Type u_1
M : Type u_2
H' : Type u_3
M' : Type u_4
inst✝⁵ : TopologicalSpace H
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : TopologicalSpace H'
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
g : M → M'
s t : Set M
x : M
hG : G.LocalInvariantProp G' P
hu : s =ᶠ[𝓝 x] t
⊢ s ∩ {x | (fun x ↦ s x = t x) x} = t ∩ {x | (fun x ↦ s x = t x) x}
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: m n : ℕ∞ω
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
H✝ : Type u_3
inst✝¹ : TopologicalSpace H✝
I : ModelWithCorners 𝕜 E H✝
M : Type u_4
inst✝ : TopologicalSpace M
f : H✝ → H✝
u : Set H✝
x✝ : IsOpen[inst✝¹] u
H : ∀ x ∈ u, ∃ v, IsOpen[inst✝¹] v ∧ x ∈ v ∧ ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' (u ∩ v) ∩ range ↑I)
y : E
hy2 : y ∈ range ↑I
x : H✝
hx : ↑I x = y
hy1 : x ∈ u
v : Set H✝
v_open : IsOpen[inst✝¹] v
xv : x ∈ v
hv : ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' (u ∩ v) ∩ range ↑I)
⊢ ↑I.symm ⁻¹' u ∩ (↑I.symm ⁻¹' v ∩ range ↑I) = ↑I.symm ⁻¹' u ∩ (range ↑I ∩ ↑I.symm ⁻¹' v)

Tactic:
congr 1

Goal state after:
case e_a
m n : ℕ∞ω
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
H✝ : Type u_3
inst✝¹ : TopologicalSpace H✝
I : ModelWithCorners 𝕜 E H✝
M : Type u_4
inst✝ : TopologicalSpace M
f : H✝ → H✝
u : Set H✝
x✝ : IsOpen[inst✝¹] u
H : ∀ x ∈ u, ∃ v, IsOpen[inst✝¹] v ∧ x ∈ v ∧ ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' (u ∩ v) ∩ range ↑I)
y : E
hy2 : y ∈ range ↑I
x : H✝
hx : ↑I x = y
hy1 : x ∈ u
v : Set H✝
v_open : IsOpen[inst✝¹] v
xv : x ∈ v
hv : ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' (u ∩ v) ∩ range ↑I)
⊢ ↑I.symm ⁻¹' v ∩ range ↑I = range ↑I ∩ ↑I.symm ⁻¹' v
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
δ : (E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |B.det - A.det| ≤ ↑ε) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) → ENNReal.ofReal |A.det| * μ t ≤ μ (g '' t) + ↑ε * μ t
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ n, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
s_eq : s = ⋃ n, s ∩ t n
n : ℕ
x : E
hx : ‖f' x - A n‖₊ ≤ δ (A n)
⊢ |(f' x).det| = |(A n).det + ((f' x).det - (A n).det)|

Tactic:
congr 1

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
δ : (E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |B.det - A.det| ≤ ↑ε) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) → ENNReal.ofReal |A.det| * μ t ≤ μ (g '' t) + ↑ε * μ t
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ n, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
s_eq : s = ⋃ n, s ∩ t n
n : ℕ
x : E
hx : ‖f' x - A n‖₊ ≤ δ (A n)
⊢ (f' x).det = (A n).det + ((f' x).det - (A n).det)