mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: case inr.hab
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ f' x
M : MonotoneOn f [[a, b]]
hab : b < a
⊢ b ≤ a

Tactic:
exact hab.le

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ f' x
M : MonotoneOn f [[a, b]]
⊢ IntervalIntegrable (fun x ↦ f' x • g (f x)) volume a b ↔ IntervalIntegrable g volume (f a) (f b)

Tactic:
rcases le_or_gt a b with hab | hab

Goal state after:
case inl
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ f' x
M : MonotoneOn f [[a, b]]
hab : a ≤ b
⊢ IntervalIntegrable (fun x ↦ f' x • g (f x)) volume a b ↔ IntervalIntegrable g volume (f a) (f b)

case inr
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ f' x
M : MonotoneOn f [[a, b]]
hab : b < a
⊢ IntervalIntegrable (fun x ↦ f' x • g (f x)) volume a b ↔ IntervalIntegrable g volume (f a) (f b)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
M : AntitoneOn f [[a, b]]
⊢ ∫ (x : ℝ) in a..b, f' x • g (f x) = ∫ (u : ℝ) in f a..f b, g u

Tactic:
rcases le_or_gt a b with hab | hab

Goal state after:
case inl
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
M : AntitoneOn f [[a, b]]
hab : a ≤ b
⊢ ∫ (x : ℝ) in a..b, f' x • g (f x) = ∫ (u : ℝ) in f a..f b, g u

case inr
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
M : AntitoneOn f [[a, b]]
hab : b < a
⊢ ∫ (x : ℝ) in a..b, f' x • g (f x) = ∫ (u : ℝ) in f a..f b, g u
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
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ENNReal.ofReal |A.det| < ↑m
d : ℝ≥0∞ := ENNReal.ofReal |A.det|
HC : IsCompact (⇑A '' closedBall 0 1)
L0 : Tendsto (fun ε ↦ μ (cthickening ε (⇑A '' closedBall 0 1))) (𝓝[>] 0) (𝓝 (μ (⇑A '' closedBall 0 1)))
L1 : Tendsto (fun ε ↦ μ (closedBall 0 ε + ⇑A '' closedBall 0 1)) (𝓝[>] 0) (𝓝 (μ (⇑A '' closedBall 0 1)))
L2 : Tendsto (fun ε ↦ μ (closedBall 0 ε + ⇑A '' closedBall 0 1)) (𝓝[>] 0) (𝓝 (d * μ (closedBall 0 1)))
I : d * μ (closedBall 0 1) < ↑m * μ (closedBall 0 1)
⊢ ∃ ε, μ (closedBall 0 ε + ⇑A '' closedBall 0 1) < ↑m * μ (closedBall 0 1) ∧ 0 < ε

Tactic:
have H :
      ∀ᶠ b : ℝ in 𝓝[>] 0, μ (closedBall 0 b + A '' closedBall 0 1) < m * μ (closedBall 0 1) :=
      (tendsto_order.1 L2).2 _ I

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ENNReal.ofReal |A.det| < ↑m
d : ℝ≥0∞ := ENNReal.ofReal |A.det|
HC : IsCompact (⇑A '' closedBall 0 1)
L0 : Tendsto (fun ε ↦ μ (cthickening ε (⇑A '' closedBall 0 1))) (𝓝[>] 0) (𝓝 (μ (⇑A '' closedBall 0 1)))
L1 : Tendsto (fun ε ↦ μ (closedBall 0 ε + ⇑A '' closedBall 0 1)) (𝓝[>] 0) (𝓝 (μ (⇑A '' closedBall 0 1)))
L2 : Tendsto (fun ε ↦ μ (closedBall 0 ε + ⇑A '' closedBall 0 1)) (𝓝[>] 0) (𝓝 (d * μ (closedBall 0 1)))
I : d * μ (closedBall 0 1) < ↑m * μ (closedBall 0 1)
H : ∀ᶠ (b : ℝ) in 𝓝[>] 0, μ (closedBall 0 b + ⇑A '' closedBall 0 1) < ↑m * μ (closedBall 0 1)
⊢ ∃ ε, μ (closedBall 0 ε + ⇑A '' closedBall 0 1) < ↑m * μ (closedBall 0 1) ∧ 0 < ε
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: a : ℝ
l : Filter ℝ
f g : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l
⊢ Tendsto (fun x ↦ f x / g x) (𝓝[≠] a) l

Tactic:
apply lhopital_zero_nhdsNE <;>
    (first | apply eventually_nhdsWithin_of_eventually_nhds |
      apply tendsto_nhdsWithin_of_tendsto_nhds) <;> assumption

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: case inr.hab
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
M : AntitoneOn f [[a, b]]
hab : b < a
⊢ b ≤ a

Tactic:
exact hab.le

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
M : AntitoneOn f [[a, b]]
⊢ IntervalIntegrable (fun x ↦ f' x • g (f x)) volume a b ↔ IntervalIntegrable g volume (f a) (f b)

Tactic:
rcases le_or_gt a b with hab | hab

Goal state after:
case inl
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
M : AntitoneOn f [[a, b]]
hab : a ≤ b
⊢ IntervalIntegrable (fun x ↦ f' x • g (f x)) volume a b ↔ IntervalIntegrable g volume (f a) (f b)

case inr
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
M : AntitoneOn f [[a, b]]
hab : b < a
⊢ IntervalIntegrable (fun x ↦ f' x • g (f x)) volume a b ↔ IntervalIntegrable g volume (f a) (f b)
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: f : ℝ → ℝ
hf : MeasurableEmbedding f
s : Set ℝ
hs : MeasurableSet s
f' : ℝ → ℝ
hf' : ∀ (x : ℝ), HasDerivAt f (f' x) x
g : ℝ → ℝ
hg : 0 ≤ᵐ[volume] g
hg_int : Integrable g volume
⊢ ∀ᵐ (x : ℝ), x ∈ f '' s → 0 ≤ g x

Tactic:
filter_upwards [hg] with x hx using fun _ ↦ hx

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

Goal state before:
pp: f : ℝ ≃ᵐ ℝ
s : Set ℝ
hs : MeasurableSet s
f' : ℝ → ℝ
hf' : ∀ (x : ℝ), HasDerivAt (⇑f) (f' x) x
g : ℝ → ℝ
hg : 0 ≤ᵐ[volume] g
hg_int : Integrable g volume
⊢ ∀ᵐ (x : ℝ), x ∈ ⇑f '' s → 0 ≤ g x

Tactic:
filter_upwards [hg] with x hx using fun _ ↦ hx

Goal state after:
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
z : ℂ
hz : z ∈ sphere c R
⊢ ((z - w₀) ^ 2)⁻¹ • (f z - f w₀) = (z - w₀)⁻¹ • dslope f w₀ z

Tactic:
simp only [dslope_of_ne, Metric.sphere_disjoint_ball.ne_of_mem hz hw₀, slope, ← smul_assoc, sq,
      mul_inv, Ne, not_false_iff, vsub_eq_sub, smul_eq_mul]

Goal state after:
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)
w : ℂ
hw : w ∈ sphere z r
h : (w - z) ^ 2 = 0
⊢ 0 = r

Tactic:
rwa [mem_sphere_iff_norm, sq_eq_zero_iff.mp h, norm_zero] at hw

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
a b : ℝ
bound : ℝ → ℝ
F F' : 𝕜 → ℝ → E
x₀ : 𝕜
s : Set 𝕜
hs : s ∈ 𝓝 x₀
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
hF_int : IntervalIntegrable (F x₀) μ a b
hF'_meas : AEStronglyMeasurable (F' x₀) (μ.restrict (Ι a b))
h_bound : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → ∀ x ∈ s, ‖F' x t‖ ≤ bound t
bound_integrable : IntervalIntegrable bound μ a b
h_diff : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → ∀ x ∈ s, HasDerivAt (fun x ↦ F x t) (F' x t) x
⊢ IntervalIntegrable (F' x₀) μ a b ∧ HasDerivAt (fun x ↦ ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' x₀ t ∂μ) x₀

Tactic:
rw [← ae_restrict_iff' measurableSet_uIoc] at h_bound h_diff
  simp only [intervalIntegrable_iff] at hF_int bound_integrable ⊢
  simp only [intervalIntegral_eq_integral_uIoc]
  have := hasDerivAt_integral_of_dominated_loc_of_deriv_le hs hF_meas hF_int hF'_meas h_bound
    bound_integrable h_diff
  exact ⟨this.1, this.2.const_smul _⟩

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
⊢ ContinuousOn (↑e) e.source

Tactic:
let : TopologicalSpace M := c.toTopologicalSpace

Goal state after:
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
⊢ ContinuousOn (↑e) e.source
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
⊢ ContinuousOn e.invFun e.target

Tactic:
let : TopologicalSpace M := c.toTopologicalSpace

Goal state after:
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
⊢ ContinuousOn e.invFun e.target
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: case inl
H : Type u_1
inst✝ : TopologicalSpace H
u : StructureGroupoid H
f : OpenPartialHomeomorph H H
hf✝ : f ∈ (idGroupoid H).members
hf : f = OpenPartialHomeomorph.refl H
⊢ OpenPartialHomeomorph.refl H ∈ u.members

Tactic:
apply u.id_mem

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

Goal state before:
pp: case left
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}
he' : e' ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
⊢ IsOpen[inst✝] (e.source ∩ ↑e ⁻¹' e'.source)

Tactic:
apply e.continuousOn_toFun.isOpen_inter_preimage e.open_source e'.open_source

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

Goal state before:
pp: H : Type u
H' : Type u_1
M✝ : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁹ : TopologicalSpace H
inst✝⁸ : TopologicalSpace M✝
inst✝⁷ : ChartedSpace H M✝
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : TopologicalSpace M''
G : StructureGroupoid H
inst✝⁴ : ChartedSpace H M'
inst✝³ : ChartedSpace H M''
M : Type u_5
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : HasGroupoid M G
c c' : OpenPartialHomeomorph M H
hc : c ∈ atlas H M
hc' : c' ∈ atlas H M
⊢ c.symm ≫ₕ (Homeomorph.refl M).toOpenPartialHomeomorph ≫ₕ c' ∈ G

Tactic:
by
      change OpenPartialHomeomorph.symm c ≫ₕ OpenPartialHomeomorph.refl M ≫ₕ c' ∈ G
      rw [OpenPartialHomeomorph.refl_trans]
      exact G.compatible hc hc'

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

Goal state before:
pp: H : Type u_1
inst✝ : TopologicalSpace H
⊢ ∀ (e : OpenPartialHomeomorph H H),
    (∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {e | ∃ s, ∃ (h : IsOpen[inst✝] s), e ≈ ofSet s h}) →
      e ∈ {e | ∃ s, ∃ (h : IsOpen[inst✝] s), e ≈ ofSet s h}

Tactic:
by
    intro e h
    refine ⟨e.source, e.open_source, by simp only [mfld_simps], ?_⟩
    intro x hx
    rcases h x hx with ⟨s, hs, hxs, s', hs', hes'⟩
    have hes : x ∈ (e.restr s).source := by
      rw [e.restr_source]
      refine ⟨hx, ?_⟩
      rw [hs.interior_eq]
      exact hxs
    simpa only [mfld_simps] using OpenPartialHomeomorph.EqOnSource.eqOn hes' hes

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'
f : OpenPartialHomeomorph M' H'
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
x : M
hG : G.LocalInvariantProp G' P
hf : f ∈ maximalAtlas M' G'
xf : g x ∈ f.source
hg : ContinuousWithinAt g s x
⊢ P (↑(chartAt H' (g x)) ∘ g ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x) ↔
    ContinuousWithinAt (↑f ∘ g) s x ∧ P ((↑f ∘ g) ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x)

Tactic:
(f.continuousAt xf).comp_continuousWithinAt hg,

Goal state after:
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'
f : OpenPartialHomeomorph M' H'
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
x : M
hG : G.LocalInvariantProp G' P
hf : f ∈ maximalAtlas M' G'
xf : g x ∈ f.source
hg : ContinuousWithinAt g s x
⊢ P (↑(chartAt H' (g x)) ∘ g ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x) ↔
    True ∧ P ((↑f ∘ g) ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x)
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: H : Type u_1
M : Type u_2
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
G : StructureGroupoid H
e : OpenPartialHomeomorph M H
Q : (H → H) → Set H → H → Prop
x : H
hG : G.LocalInvariantProp G Q
hQ : ∀ (y : H), Q id univ y
he : e ∈ maximalAtlas M G
hx : x ∈ e.target
⊢ LiftPropAt Q (↑e.symm) x

Tactic:
by
  suffices h : Q (e ∘ e.symm) univ x by
    have : e.symm x ∈ e.source := by simp only [hx, mfld_simps]
    rw [LiftPropAt, hG.liftPropWithinAt_indep_chart G.id_mem_maximalAtlas (mem_univ _) he this]
    refine ⟨(e.symm.continuousAt hx).continuousWithinAt, ?_⟩
    simp only [h, mfld_simps]
  exact hG.congr' (e.eventually_right_inverse hx) (hQ x)

Goal state after:
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: case left
𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
n : ℕ∞ω
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
e : OpenPartialHomeomorph M H
he : e ∈ maximalAtlas I n M
e' : OpenPartialHomeomorph M' H'
he' : e' ∈ maximalAtlas I' n M'
f : OpenPartialHomeomorph M H
hf : f ∈ atlas H M
f' : OpenPartialHomeomorph M' H'
hf' : f' ∈ atlas H' M'
⊢ (e.symm ≫ₕ f).prod (e'.symm ≫ₕ f') ∈ contDiffGroupoid n (I.prod I')

Tactic:
apply contDiffGroupoid_prod

Goal state after:
case left.he
𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
n : ℕ∞ω
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
e : OpenPartialHomeomorph M H
he : e ∈ maximalAtlas I n M
e' : OpenPartialHomeomorph M' H'
he' : e' ∈ maximalAtlas I' n M'
f : OpenPartialHomeomorph M H
hf : f ∈ atlas H M
f' : OpenPartialHomeomorph M' H'
hf' : f' ∈ atlas H' M'
⊢ e.symm ≫ₕ f ∈ contDiffGroupoid n I

case left.he'
𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
n : ℕ∞ω
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
e : OpenPartialHomeomorph M H
he : e ∈ maximalAtlas I n M
e' : OpenPartialHomeomorph M' H'
he' : e' ∈ maximalAtlas I' n M'
f : OpenPartialHomeomorph M H
hf : f ∈ atlas H M
f' : OpenPartialHomeomorph M' H'
hf' : f' ∈ atlas H' M'
⊢ e'.symm ≫ₕ f' ∈ contDiffGroupoid n I'
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: case right
𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
n : ℕ∞ω
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
e : OpenPartialHomeomorph M H
he : e ∈ maximalAtlas I n M
e' : OpenPartialHomeomorph M' H'
he' : e' ∈ maximalAtlas I' n M'
f : OpenPartialHomeomorph M H
hf : f ∈ atlas H M
f' : OpenPartialHomeomorph M' H'
hf' : f' ∈ atlas H' M'
⊢ (f.symm ≫ₕ e).prod (f'.symm ≫ₕ e') ∈ contDiffGroupoid n (I.prod I')

Tactic:
apply contDiffGroupoid_prod

Goal state after:
case right.he
𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
n : ℕ∞ω
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
e : OpenPartialHomeomorph M H
he : e ∈ maximalAtlas I n M
e' : OpenPartialHomeomorph M' H'
he' : e' ∈ maximalAtlas I' n M'
f : OpenPartialHomeomorph M H
hf : f ∈ atlas H M
f' : OpenPartialHomeomorph M' H'
hf' : f' ∈ atlas H' M'
⊢ f.symm ≫ₕ e ∈ contDiffGroupoid n I

case right.he'
𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
n : ℕ∞ω
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
e : OpenPartialHomeomorph M H
he : e ∈ maximalAtlas I n M
e' : OpenPartialHomeomorph M' H'
he' : e' ∈ maximalAtlas I' n M'
f : OpenPartialHomeomorph M H
hf : f ∈ atlas H M
f' : OpenPartialHomeomorph M' H'
hf' : f' ∈ atlas H' M'
⊢ f'.symm ≫ₕ e' ∈ contDiffGroupoid n I'
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
u : E → E
u_meas : Measurable u
uf : EqOn u f s
u' : ∀ x ∈ s, HasFDerivWithinAt u (f' x) s x
F : ↑s → E := u ∘ Subtype.val
hF : F = u ∘ Subtype.val
A : Measure.map F (Measure.comap Subtype.val (μ.withDensity fun x ↦ ENNReal.ofReal |(f' x).det|)) = μ.restrict (f '' s)
x : ↑s
⊢ F x = s.domRestrict f x

Tactic:
exact uf x.2

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.Atlas

Goal state before:
pp: case mp
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
n : ℕ∞ω
M' : Type u_5
inst✝² : TopologicalSpace M'
inst✝¹ : IsManifold I n M
inst✝ : ChartedSpace H M'
IsM' : IsManifold I n M'
f : OpenPartialHomeomorph M M'
h : LiftPropOn (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑f) f.source
X : M'
hX : X ∈ f.symm.source
x : M := ↑f.symm X
hx : x ∈ f.source
c : OpenPartialHomeomorph M H := chartAt H x
c' : OpenPartialHomeomorph M' H := chartAt H X
hxf :
  (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑(chartAt H (↑f x)) ∘ ↑f ∘ ↑(chartAt H x).symm)
    (↑(chartAt H x).symm ⁻¹' f.source) (↑(chartAt H x) x)
h2x : ↑(chartAt H X) X ∈ ↑(chartAt H X).symm ⁻¹' f.symm.source
e : OpenPartialHomeomorph H H
he : e ∈ contDiffGroupoid n I
h2e : e.source ⊆ (c.symm ≫ₕ f ≫ₕ c').source
hef : EqOn (↑c' ∘ ↑f ∘ ↑c.symm) (↑e) e.source
hex : ↑c x ∈ e.source
h2X : ↑c' X = ↑e (↑c (↑f.symm X))
h3e : EqOn (↑c ∘ ↑f.symm ∘ ↑c'.symm) (↑e.symm) (↑c'.symm ⁻¹' f.target ∩ e.target)
⊢ ↑e (↑c (↑f.symm X)) ∈ e.symm.source

Tactic:
exact e.mapsTo hex

Goal state after:
Module: Mathlib.Topology.VectorBundle.Basic

Goal state before:
pp: R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁹ : NontriviallyNormedField R
inst✝⁸ : (x : B) → AddCommMonoid (E x)
inst✝⁷ : (x : B) → Module R (E x)
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace R F
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace (TotalSpace F E)
inst✝² : (x : B) → TopologicalSpace (E x)
inst✝¹ : FiberBundle F E
e : Trivialization F TotalSpace.proj
inst✝ : Trivialization.IsLinear R e
b : B
⊢ Continuous[inst✝² b, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun y ↦
    if b ∈ e.baseSet then (↑e { proj := b, snd := y }).2 else 0

Tactic:
refine continuous_if_const _ (fun hb => ?_) fun _ => continuous_zero
      exact (e.continuousOn.comp_continuous (FiberBundle.totalSpaceMk_isInducing F E b).continuous
        fun x => e.mem_source.mpr hb).snd

Goal state after: