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

Goal state before:
pp: 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)

Tactic:
intervalIntegrable_iff_integrableOn_Icc_of_le 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
⊢ IntegrableOn (fun x ↦ f' x • g (f x)) (Icc a b) volume ↔ IntervalIntegrable g volume (f a) (f b)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: 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)

Tactic:
rw [intervalIntegrable_iff_integrableOn_Icc_of_le hab,
      integrableOn_Icc_deriv_smul_iff_of_deriv_nonpos,
      IntervalIntegrable.symm_iff, intervalIntegrable_iff_integrableOn_Icc_of_le]
    · apply M left_mem_uIcc right_mem_uIcc hab
    · rwa [uIcc_of_le hab] at hf
    · grind
    · grind
    · exact hab

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
⊢ ‖(f' x - A) z‖ ≤ ↑δ * ‖z‖

Tactic:
by
    have :
      Tendsto (fun ε : ℝ => ((δ : ℝ) + ε) * (‖z‖ + ε) + ‖f' x - A‖ * ε) (𝓝[>] 0)
        (𝓝 ((δ + 0) * (‖z‖ + 0) + ‖f' x - A‖ * 0)) :=
      Tendsto.mono_left (Continuous.tendsto (by fun_prop) 0) nhdsWithin_le_nhds
    simp only [add_zero, mul_zero] at this
    apply le_of_tendsto_of_tendsto tendsto_const_nhds this
    filter_upwards [self_mem_nhdsWithin]
    exact H

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.UniformOn

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝³ : CommMonoid α
f : ι → β → α
g : β → α
s : Set β
inst✝² : UniformSpace α
inst✝¹ : TopologicalSpace β
hs : IsOpen[inst✝¹] s
inst✝ : LocallyCompactSpace β
h : ∀ K ⊆ s, IsCompact K → HasProdUniformlyOn f g K
⊢ HasProdLocallyUniformlyOn f g s

Tactic:
rw [HasProdLocallyUniformlyOn, tendstoLocallyUniformlyOn_iff_forall_isCompact hs]
  simpa [hasProdUniformlyOn_iff_tendstoUniformlyOn] using h

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 K : 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
δ : ℝ
hδ : 0 < δ
hK : IsCompact K
hU : IsOpen U
hKU : cthickening δ K ⊆ U
⊢ TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K

Tactic:
by
  have h1 : ∀ᶠ n in φ, ContinuousOn (F n) (cthickening δ K) := by
    filter_upwards [hF] with n h using h.continuousOn.mono hKU
  have h2 : IsCompact (cthickening δ K) := hK.cthickening
  have h3 : TendstoUniformlyOn F f φ (cthickening δ K) :=
    (tendstoLocallyUniformlyOn_iff_forall_isCompact hU).mp hf (cthickening δ K) hKU h2
  apply (h3.cderiv hδ h1).congr
  filter_upwards [hF] with n h z hz
  exact cderiv_eq_deriv hU h hδ ((closedBall_subset_cthickening hz δ).trans hKU)

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
F : ι → ℂ → E
inst✝ : CompleteSpace E
u : ι → ℝ
hu : Summable u
hf : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hU : IsOpen U
hF_le : ∀ (i : ι), ∀ w ∈ U, ‖F i w‖ ≤ u i
hc : TendstoLocallyUniformlyOn (fun t x ↦ ∑ n ∈ t, F n x) (fun x ↦ ∑' (n : ι), F n x) atTop U
s : Finset ι
⊢ DifferentiableOn ℂ (fun x ↦ ∑ n ∈ s, F n x) U

Tactic:
exact DifferentiableOn.fun_sum fun i _ => hf i

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

Goal state before:
pp: H : Type u
M : Type u_2
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
G : StructureGroupoid H
e e' : OpenPartialHomeomorph M H
he : e ∈ maximalAtlas M G
he' : e' ∈ maximalAtlas M G
x : H
hx : x ∈ (e.symm ≫ₕ e').source
f : OpenPartialHomeomorph M H := chartAt H (↑e.symm x)
s : Set H := e.target ∩ ↑e.symm ⁻¹' f.source
hs : IsOpen[inst✝²] s
xs : x ∈ s
A : e.symm ≫ₕ f ∈ G
B : f.symm ≫ₕ e' ∈ G
⊢ (e.symm ≫ₕ e').restr s ∈ G

Tactic:
have C : (e.symm ≫ₕ f) ≫ₕ f.symm ≫ₕ e' ∈ G := G.trans A B

Goal state after:
H : Type u
M : Type u_2
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
G : StructureGroupoid H
e e' : OpenPartialHomeomorph M H
he : e ∈ maximalAtlas M G
he' : e' ∈ maximalAtlas M G
x : H
hx : x ∈ (e.symm ≫ₕ e').source
f : OpenPartialHomeomorph M H := chartAt H (↑e.symm x)
s : Set H := e.target ∩ ↑e.symm ⁻¹' f.source
hs : IsOpen[inst✝²] s
xs : x ∈ s
A : e.symm ≫ₕ f ∈ G
B : f.symm ≫ₕ e' ∈ G
C : (e.symm ≫ₕ f) ≫ₕ f.symm ≫ₕ e' ∈ G
⊢ (e.symm ≫ₕ e').restr s ∈ G
Module: Mathlib.Geometry.Manifold.HasGroupoid

Goal state before:
pp: H : Type u
M : Type u_2
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
G : StructureGroupoid H
e e' : OpenPartialHomeomorph M H
he : e ∈ maximalAtlas M G
he' : e' ∈ maximalAtlas M G
x : H
hx : x ∈ (e.symm ≫ₕ e').source
f : OpenPartialHomeomorph M H := chartAt H (↑e.symm x)
s : Set H := e.target ∩ ↑e.symm ⁻¹' f.source
hs : IsOpen[inst✝²] s
xs : x ∈ s
A : e.symm ≫ₕ f ∈ G
B : f.symm ≫ₕ e' ∈ G
C : (e.symm ≫ₕ f) ≫ₕ f.symm ≫ₕ e' ∈ G
⊢ (e.symm ≫ₕ f) ≫ₕ f.symm ≫ₕ e' = e.symm ≫ₕ (f ≫ₕ f.symm) ≫ₕ e'

Tactic:
by simp only [trans_assoc]

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}
ee' : e' ≈ e
⊢ PG.property (↑e) e'.source

Tactic:
simp only [ee'.1, he.1]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
s t : Set M
⊢ ↑(f.extend I).symm ⁻¹' (s ∩ t) ∩ range ↑I = ↑(f.extend I).symm ⁻¹' s ∩ range ↑I ∩ ↑(f.extend I).symm ⁻¹' t

Tactic:
by
  mfld_set_tac

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

Goal state before:
pp: 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type w
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type v'
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type w'
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
this : (range fun x ↦ (↑I x.1, ↑I' x.2)) = range (Prod.map ↑I ↑I')
⊢ (interior (range fun x ↦ (↑I x.1, ↑I' x.2))).Nonempty

Tactic:
simp [this, interior_prod_eq, nonempty_interior]

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

Goal state before:
pp: H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
s : Set H
x : H
f : H → H
e' : OpenPartialHomeomorph H H
he'G : e' ∈ G
he'x : x ∈ e'.source
h : G.IsLocalStructomorphWithinAt f s x
hx : ↑e' x ∈ ↑e'.symm ⁻¹' s
⊢ x ∈ s

Tactic:
simpa only [e'.left_inv he'x, mfld_simps] using hx

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

Goal state before:
pp: case refine_2
H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
s : Set H
x : H
f : H → H
e' : OpenPartialHomeomorph H H
he'G : e' ∈ G
he'x : x ∈ e'.source
h : G.IsLocalStructomorphWithinAt f s x
hx : ↑e' x ∈ ↑e'.symm ⁻¹' s
hxs : x ∈ s
e : OpenPartialHomeomorph H H
heG : e ∈ G
hef : EqOn f (↑e.toPartialEquiv) (s ∩ e.source)
hex : x ∈ e.source
⊢ ↑e' x ∈ (e'.symm ≫ₕ e).source

Tactic:
simp only [hex, he'x, mfld_simps]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
t : Set M
inst✝ : ChartedSpace H M
x : M
ht : t ∈ 𝓝 x
⊢ ↑(extChartAt I x).symm ⁻¹' t ∈ 𝓝 (↑(extChartAt I x) x)

Tactic:
apply (continuousAt_extChartAt_symm x).preimage_mem_nhds

Goal state after:
𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
t : Set M
inst✝ : ChartedSpace H M
x : M
ht : t ∈ 𝓝 x
⊢ t ∈ 𝓝 (↑(extChartAt I x).symm (↑(extChartAt I x) x))
Module: Mathlib.Geometry.Manifold.IsManifold.ExtChartAt

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
s t : Set M
inst✝ : ChartedSpace H M
x : M
⊢ ↑(extChartAt I x).symm ⁻¹' (s ∩ t) ∩ range ↑I = ↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I ∩ ↑(extChartAt I x).symm ⁻¹' t

Tactic:
by
  mfld_set_tac

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_16
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
n : WithTop ℕ∞
x : M
C : OpenPartialHomeomorph M H := chartAt H x
hC : C = chartAt H x
⊢ ↑I.symm ⁻¹' C.target ∩ range ↑I ∈ 𝓝[range ↑I] ↑(extChartAt I x) x

Tactic:
rw [← I.image_eq (chartAt H x).target]

Goal state after:
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_16
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
n : WithTop ℕ∞
x : M
C : OpenPartialHomeomorph M H := chartAt H x
hC : C = chartAt H x
⊢ ↑I '' (chartAt H x).target ∈ 𝓝[range ↑I] ↑(extChartAt I x) x
Module: Mathlib.Geometry.Manifold.ContMDiff.Constructions

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_16
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
n : WithTop ℕ∞
x : M'
C : OpenPartialHomeomorph M' H := chartAt H x
hC : C = chartAt H x
⊢ ↑I.symm ⁻¹' (chartAt H x).target ∩ range ↑I ∈ 𝓝[range ↑I] ↑(extChartAt I x) x

Tactic:
rw [← I.image_eq (chartAt H x).target]

Goal state after:
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_16
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
n : WithTop ℕ∞
x : M'
C : OpenPartialHomeomorph M' H := chartAt H x
hC : C = chartAt H x
⊢ ↑I '' (chartAt H x).target ∈ 𝓝[range ↑I] ↑(extChartAt I x) x
Module: Mathlib.Geometry.Manifold.ContMDiff.Defs

Goal state before:
pp: 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'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
f : M → M'
s : Set M
x x' : M
y : M'
n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
hy : f x' ∈ (extChartAt I' y).source
e : PartialEquiv M E := extChartAt I x
hx : x' ∈ e.source
e' : PartialEquiv M' E' := extChartAt I' (f x)
hc : ContinuousWithinAt f s x'
⊢ ↑e.symm ⁻¹' s ∩ range ↑I =ᶠ[𝓝 (↑e x')] e.target ∩ ↑e.symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source)

Tactic:
rw [← nhdsWithin_eq_iff_eventuallyEq, ← e.image_source_inter_eq',
    ← map_extChartAt_nhdsWithin_eq_image' hx,
    ← map_extChartAt_nhdsWithin' hx, inter_comm, nhdsWithin_inter_of_mem]

Goal state after:
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'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
f : M → M'
s : Set M
x x' : M
y : M'
n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
hy : f x' ∈ (extChartAt I' y).source
e : PartialEquiv M E := extChartAt I x
hx : x' ∈ e.source
e' : PartialEquiv M' E' := extChartAt I' (f x)
hc : ContinuousWithinAt f s x'
⊢ f ⁻¹' (extChartAt I' y).source ∈ 𝓝[s] x'
Module: Mathlib.Geometry.Manifold.ContMDiff.Defs

Goal state before:
pp: 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'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
f : M → M'
s : Set M
n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
hcont : ContinuousOn f s
hdiff :
  ∀ (x : M) (y : M'),
    ContDiffOn 𝕜 n (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm)
      ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source))
x : M
hx : x ∈ s
⊢ ↑(extChartAt I x) x ∈ (extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source)

Tactic:
by simp only [hx, mfld_simps]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
s : Set M
x : M
n : ℕ∞
h : ∀ (m : ℕ), ↑m ≤ n → ContMDiffWithinAt I I' (↑m) f s x
⊢ ContMDiffWithinAt I I' (↑n) f s x

Tactic:
obtain - | n := n

Goal state after:
case none
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
s : Set M
x : M
h : ∀ (m : ℕ), ↑m ≤ none → ContMDiffWithinAt I I' (↑m) f s x
⊢ ContMDiffWithinAt I I' (↑none) f s x

case some
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
s : Set M
x : M
n : ℕ
h : ∀ (m : ℕ), ↑m ≤ some n → ContMDiffWithinAt I I' (↑m) f s x
⊢ ContMDiffWithinAt I I' (↑(some n)) f s x
Module: Mathlib.Geometry.Manifold.ContMDiff.Atlas

Goal state before:
pp: 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'
hf : LiftPropOn (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑f) f.source
x : M
hx : x ∈ f.source
c : OpenPartialHomeomorph M H := chartAt H x
c' : OpenPartialHomeomorph M' H := chartAt H (↑f 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)
⊢ ↑(chartAt H x) x ∈ ↑(chartAt H x).symm ⁻¹' f.source

Tactic:
by simp only [hx, mfld_simps]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
x : M
s t : Set M
ht : t ∈ 𝓝[s] x
⊢ MDiffAt[s ∩ t] f x ↔ MDiffAt[s] f x

Tactic:
by
  rw [MDifferentiableWithinAt, MDifferentiableWithinAt,
    differentiableWithinAt_localInvariantProp.liftPropWithinAt_inter' ht]

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

Goal state before:
pp: 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'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
f : M → M'
s : Set M
inst✝¹ : IsManifold I 1 M
inst✝ : IsManifold I' 1 M'
hcont : ContinuousOn f s
hdiff :
  ∀ (x : M) (y : M'),
    DifferentiableOn 𝕜 (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm)
      ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source))
x : M
hx : x ∈ s
⊢ ↑(extChartAt I x) x ∈ (extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source)

Tactic:
by simp only [hx, mfld_simps]

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

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
x : M
s t : Set M
y : M
h : s =ᶠ[𝓝[{y}ᶜ] x] t
this : T1Space M
u : Set M
u_mem : u ∈ 𝓝 (↑(extChartAt I x).symm (↑(extChartAt I x) x))
hu : u ∩ {x}ᶜ ⊆ {y | (y ∈ s) = (y ∈ t)}
B : (extChartAt I x).target ∪ (range ↑I)ᶜ ∈ 𝓝 (↑(extChartAt I x) x)
z : E
h'z : z ∈ {↑(extChartAt I x) x}ᶜ
hIz : z ∈ range ↑I
hz : ↑(extChartAt I x).symm z ∈ u ∧ z ∈ (extChartAt I x).target
⊢ (↑(extChartAt I x).symm z ∈ s) = (↑(extChartAt I x).symm z ∈ t)

Tactic:
apply hu ⟨hz.1, ?_⟩

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
x : M
s t : Set M
y : M
h : s =ᶠ[𝓝[{y}ᶜ] x] t
this : T1Space M
u : Set M
u_mem : u ∈ 𝓝 (↑(extChartAt I x).symm (↑(extChartAt I x) x))
hu : u ∩ {x}ᶜ ⊆ {y | (y ∈ s) = (y ∈ t)}
B : (extChartAt I x).target ∪ (range ↑I)ᶜ ∈ 𝓝 (↑(extChartAt I x) x)
z : E
h'z : z ∈ {↑(extChartAt I x) x}ᶜ
hIz : z ∈ range ↑I
hz : ↑(extChartAt I x).symm z ∈ u ∧ z ∈ (extChartAt I x).target
⊢ ↑(extChartAt I x).symm z ∈ {x}ᶜ
Module: Mathlib.Geometry.Manifold.MFDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : ChartedSpace H' M'
E'' : Type u_8
inst✝⁴ : NormedAddCommGroup E''
inst✝³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝¹ : TopologicalSpace M''
inst✝ : ChartedSpace H'' M''
f : M → M'
x : M
s : Set M
g : M' → M''
u : Set M'
hg : MDiffAt[u] g (f x)
hf : MDiffAt[s] f x
h : f ⁻¹' u ∈ 𝓝[s] x
hxs : UniqueMDiffAt[s] x
A : s ∩ f ⁻¹' u ∈ 𝓝[s] x
B : mfderiv[s] (g ∘ f) x = mfderiv[s ∩ f ⁻¹' u] (g ∘ f) x
C : mfderiv[s] f x = mfderiv[s ∩ f ⁻¹' u] f x
⊢ mfderiv[s] (g ∘ f) x = mfderiv[u] g (f x) ∘SL mfderiv[s] f x

Tactic:
rw [B, C]

Goal state after:
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : ChartedSpace H' M'
E'' : Type u_8
inst✝⁴ : NormedAddCommGroup E''
inst✝³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝¹ : TopologicalSpace M''
inst✝ : ChartedSpace H'' M''
f : M → M'
x : M
s : Set M
g : M' → M''
u : Set M'
hg : MDiffAt[u] g (f x)
hf : MDiffAt[s] f x
h : f ⁻¹' u ∈ 𝓝[s] x
hxs : UniqueMDiffAt[s] x
A : s ∩ f ⁻¹' u ∈ 𝓝[s] x
B : mfderiv[s] (g ∘ f) x = mfderiv[s ∩ f ⁻¹' u] (g ∘ f) x
C : mfderiv[s] f x = mfderiv[s ∩ f ⁻¹' u] f x
⊢ mfderiv[s ∩ f ⁻¹' u] (g ∘ f) x = mfderiv[u] g (f x) ∘SL mfderiv[s ∩ f ⁻¹' u] f x