mathlib-initiative/mathlib-tactics huggingface dataset:
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 g : H → H
u v : Set H
hf : ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' u ∩ range ↑I)
hg : ContDiffOn 𝕜 n (↑I ∘ g ∘ ↑I.symm) (↑I.symm ⁻¹' v ∩ range ↑I)
x✝² : IsOpen[inst✝¹] u
x✝¹ : IsOpen[inst✝¹] v
x✝ : IsOpen[inst✝¹] (u ∩ f ⁻¹' v)
this : ↑I ∘ (g ∘ f) ∘ ↑I.symm = (↑I ∘ g ∘ ↑I.symm) ∘ ↑I ∘ f ∘ ↑I.symm
⊢ MapsTo (↑I ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' (u ∩ f ⁻¹' v) ∩ range ↑I) (↑I.symm ⁻¹' v ∩ range ↑I)

Tactic:
rintro x ⟨hx1, _⟩

Goal state after:
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 g : H → H
u v : Set H
hf : ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' u ∩ range ↑I)
hg : ContDiffOn 𝕜 n (↑I ∘ g ∘ ↑I.symm) (↑I.symm ⁻¹' v ∩ range ↑I)
x✝² : IsOpen[inst✝¹] u
x✝¹ : IsOpen[inst✝¹] v
x✝ : IsOpen[inst✝¹] (u ∩ f ⁻¹' v)
this : ↑I ∘ (g ∘ f) ∘ ↑I.symm = (↑I ∘ g ∘ ↑I.symm) ∘ ↑I ∘ f ∘ ↑I.symm
x : E
hx1 : x ∈ ↑I.symm ⁻¹' (u ∩ f ⁻¹' v)
right✝ : x ∈ range ↑I
⊢ (↑I ∘ f ∘ ↑I.symm) x ∈ ↑I.symm ⁻¹' v ∩ range ↑I
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: case refine_1
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
hG : G.LocalInvariantProp G' P
U V : Opens M
hUV : U ≤ V
f : ↥V → M'
x : ↥U
⊢ ContinuousWithinAt f univ (inclusion hUV x) ↔ ContinuousWithinAt (f ∘ inclusion hUV) univ x

Tactic:
simp_rw [continuousWithinAt_univ,
      (TopologicalSpace.Opens.isOpenEmbedding_of_le hUV).continuousAt_iff]

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
E' : Type u_5
M' : Type u_6
H' : Type u_7
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : TopologicalSpace H
inst✝⁶ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
inst✝³ : TopologicalSpace H'
inst✝² : TopologicalSpace M'
I' : ModelWithCorners 𝕜 E' H'
s : Set M
inst✝¹ : ChartedSpace H M
inst✝ : ChartedSpace H' M'
f : M → M'
x : M
hc : ContinuousWithinAt f s x
⊢ ↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I =ᶠ[𝓝 (↑(extChartAt I x) x)]
    (extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source)

Tactic:
by
  rw [← nhdsWithin_eq_iff_eventuallyEq]
  exact hc.nhdsWithin_extChartAt_symm_preimage_inter_range

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
E' : Type u_5
M' : Type u_6
H' : Type u_7
inst✝²⁰ : NontriviallyNormedField 𝕜
inst✝¹⁹ : NormedAddCommGroup E
inst✝¹⁸ : NormedSpace 𝕜 E
inst✝¹⁷ : TopologicalSpace H
inst✝¹⁶ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
inst✝¹⁵ : NormedAddCommGroup E'
inst✝¹⁴ : NormedSpace 𝕜 E'
inst✝¹³ : TopologicalSpace H'
inst✝¹² : TopologicalSpace M'
I' : ModelWithCorners 𝕜 E' H'
inst✝¹¹ : ChartedSpace H M
inst✝¹⁰ : ChartedSpace H' M'
G : Type u_8
G' : Type u_9
F : Type u_10
F' : Type u_11
N : Type u_12
N' : Type u_13
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : TopologicalSpace G
inst✝⁴ : TopologicalSpace N
inst✝³ : TopologicalSpace G'
inst✝² : TopologicalSpace N'
J : ModelWithCorners 𝕜 F G
J' : ModelWithCorners 𝕜 F' G'
inst✝¹ : ChartedSpace G N
inst✝ : ChartedSpace G' N'
f : M → N
g : M' → N'
x : M
x' : M'
⊢ writtenInExtChartAt (I.prod I') (J.prod J') (x, x') (Prod.map f g) =
    Prod.map (writtenInExtChartAt I J x f) (writtenInExtChartAt I' J' x' g)

Tactic:
ext p <;>
  simp [writtenInExtChartAt, I.toPartialEquiv.prod_symm, (chartAt H x).toPartialEquiv.prod_symm]

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

Goal state before:
pp: case refine_1
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
n : ℕ∞ω
x : H
this : ContDiffWithinAt 𝕜 n id (range ↑I) (↑I x)
y : E
hy : y ∈ range ↑I
⊢ (↑I ∘ ↑I.symm) y = id y

pp: case refine_2
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
n : ℕ∞ω
x : H
this : ContDiffWithinAt 𝕜 n id (range ↑I) (↑I x)
⊢ (↑I ∘ ↑I.symm) (↑I x) = id (↑I x)

Tactic:
· simp only [ModelWithCorners.right_inv I hy, mfld_simps]

Goal state after:
case refine_2
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
n : ℕ∞ω
x : H
this : ContDiffWithinAt 𝕜 n id (range ↑I) (↑I x)
⊢ (↑I ∘ ↑I.symm) (↑I x) = id (↑I x)
Module: Mathlib.Geometry.Manifold.ContMDiff.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
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H M
inst✝ : ChartedSpace H' M'
U : TopologicalSpace.Opens M'
f : M → ↥U
x : M
⊢ ContMDiffAt I I' ∞ (Subtype.val ∘ f) x ↔ ContMDiffAt I I' ∞ f x

Tactic:
by
  rw [ContMDiffAt, ContMDiffAt, ContMDiffWithinAt.subtypeVal_comp_iff]

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
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : NullMeasurableSet s μ
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
⊢ ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ = μ (f '' s)

Tactic:
by
  rcases hs.exists_measurable_subset_ae_eq with ⟨t, ts, ht, t_eq_s⟩
  have A : μ (f '' s) = μ (f '' t) := by
    apply measure_congr
    have : s = t ∪ (s \ t) := by simp [union_eq_self_of_subset_left ts]
    rw [this, image_union]
    refine union_ae_eq_left_of_ae_eq_empty (ae_eq_empty.mpr ?_)
    apply addHaar_image_eq_zero_of_differentiableOn_of_addHaar_eq_zero _
      (fun x hx ↦ ?_) (ae_eq_set.1 t_eq_s).2
    exact (hf' x hx.1).differentiableWithinAt.mono sdiff_subset
  have B : (∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ)
      = (∫⁻ x in t, ENNReal.ofReal |(f' x).det| ∂μ) :=
    setLIntegral_congr t_eq_s.symm
  rw [A, B, lintegral_abs_det_fderiv_eq_addHaar_image _ ht _ (hf.mono ts)]
  intro x hx
  exact (hf' x (ts hx)).mono ts

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'
e' : OpenPartialHomeomorph M' H'
f : M → M'
s : Set M
x : M
n : ℕ∞ω
he' : e' ∈ maximalAtlas I' n M'
hy : f x ∈ e'.source
⊢ ContMDiffWithinAt I I' n f s x ↔
    ContinuousWithinAt f s x ∧
      ContDiffWithinAt 𝕜 n (↑(e'.extend I') ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I)
        (↑(extChartAt I x) x)

Tactic:
by
  rw [contMDiffWithinAt_iff_source,
    contMDiffWithinAt_iff_target_of_mem_maximalAtlas he' (by simpa)]
  apply and_congr continuousWithinAt_iff_source.symm
  -- TODO: this is `contMDiffWithinAt_iff_contDiffWithinAt` copied,
  -- which is not put here for import reasons
  simp +contextual only [ContMDiffWithinAt, liftPropWithinAt_iff',
    ContDiffWithinAtProp, iff_def, mfld_simps]
  exact ContDiffWithinAt.continuousWithinAt

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 : Set M
h : MDiffAt[univ] f x
hs : s ∈ 𝓝 x
this : s = univ ∩ s
⊢ MDiffAt f x

Tactic:
mdifferentiableWithinAt_univ

Goal state after:
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 : Set M
h : MDiffAt f x
hs : s ∈ 𝓝 x
this : s = univ ∩ s
⊢ MDiffAt f 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'
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))
h1 : EqOn (↑(c.symm ≫ₕ f ≫ₕ c').symm) (↑e.symm) e.target
h2 : e.target ⊆ (c.symm ≫ₕ f ≫ₕ c').target
⊢ e.target ⊆ ↑c'.symm ⁻¹' f.target

Tactic:
refine h2.trans ?_

Goal state after:
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))
h1 : EqOn (↑(c.symm ≫ₕ f ≫ₕ c').symm) (↑e.symm) e.target
h2 : e.target ⊆ (c.symm ≫ₕ f ≫ₕ c').target
⊢ (c.symm ≫ₕ f ≫ₕ c').target ⊆ ↑c'.symm ⁻¹' f.target
Module: Mathlib.Geometry.Manifold.MFDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H' M'
f : M → M'
s : Set M
inst✝ : IsManifold I' 1 M'
x : M
y : M'
hy : f x ∈ (chartAt H' y).source
hf : ContinuousWithinAt f s x
⊢ LiftPropWithinAt (DifferentiableWithinAtProp I I') (↑(chartAt H' y) ∘ f) s x ↔
    LiftPropWithinAt (DifferentiableWithinAtProp I 𝓘(𝕜, E')) (↑(extChartAt I' y) ∘ f) s x

Tactic:
simp_rw [StructureGroupoid.liftPropWithinAt_self_target]

Goal state after:
𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H' M'
f : M → M'
s : Set M
inst✝ : IsManifold I' 1 M'
x : M
y : M'
hy : f x ∈ (chartAt H' y).source
hf : ContinuousWithinAt f s x
⊢ ContinuousWithinAt (↑(chartAt H' y) ∘ f) s x ∧
      DifferentiableWithinAtProp I I' ((↑(chartAt H' y) ∘ f) ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s)
        (↑(chartAt H x) x) ↔
    ContinuousWithinAt (↑(extChartAt I' y) ∘ f) s x ∧
      DifferentiableWithinAtProp I 𝓘(𝕜, E') ((↑(extChartAt I' y) ∘ f) ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s)
        (↑(chartAt H x) x)
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
⊢ (if MDiffAt[univ] f x then
      fderivWithin 𝕜 ((↑I' ∘ ↑(chartAt H' (f x))) ∘ f ∘ ↑(chartAt H x).symm ∘ ↑I.symm) (range ↑I)
        (↑I (↑(chartAt H x) x))
    else 0) =
    if MDiffAt f x then
      fderivWithin 𝕜 ((↑I' ∘ ↑(chartAt H' (f x))) ∘ f ∘ ↑(chartAt H x).symm ∘ ↑I.symm) (range ↑I)
        (↑I (↑(chartAt H x) x))
    else 0

Tactic:
mdifferentiableWithinAt_univ

Goal state after:
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
⊢ (if MDiffAt f x then
      fderivWithin 𝕜 ((↑I' ∘ ↑(chartAt H' (f x))) ∘ f ∘ ↑(chartAt H x).symm ∘ ↑I.symm) (range ↑I)
        (↑I (↑(chartAt H x) x))
    else 0) =
    if MDiffAt f x then
      fderivWithin 𝕜 ((↑I' ∘ ↑(chartAt H' (f x))) ∘ f ∘ ↑(chartAt H x).symm ∘ ↑I.symm) (range ↑I)
        (↑I (↑(chartAt H x) x))
    else 0
Module: Mathlib.Topology.VectorBundle.Basic

Goal state before:
pp: R : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField R
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace R F
inst✝ : TopologicalSpace B
ι : Type u_5
Z : VectorBundleCore R B F ι
i : ι
b : B
hb : b ∈ (Z.localTriv i).baseSet
⊢ Trivialization.symmL R (Z.localTriv i) b = Z.coordChange i (Z.indexAt b) b

Tactic:
by
  ext1 v
  rw [(Z.localTriv i).symmL_apply hb, (Z.localTriv i).symm_apply]
  exacts [rfl, hb]

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

Goal state before:
pp: case refine_2
n : ℕ∞ω
𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝²⁰ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁹ : NormedAddCommGroup EB
inst✝¹⁸ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹⁷ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹⁶ : TopologicalSpace B
inst✝¹⁵ : ChartedSpace HB B
EM : Type u_9
inst✝¹⁴ : NormedAddCommGroup EM
inst✝¹³ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝¹² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace HM M
inst✝⁹ : (x : B) → AddCommMonoid (E x)
inst✝⁸ : (x : B) → Module 𝕜 (E x)
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : TopologicalSpace (TotalSpace F E)
inst✝⁴ : (x : B) → TopologicalSpace (E x)
inst✝³ : FiberBundle F E
inst✝² : VectorBundle 𝕜 F E
inst✝¹ : ContMDiffVectorBundle n F E IB
inst✝ : IsManifold IB n B
e : OpenPartialHomeomorph (B × F) (B × F)
φ : B → F ≃L[𝕜] F
U : Set B
hU : IsOpen[inst✝¹⁶] U
hφ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ x)) U
h2φ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ x).symm) U
heφ : e.EqOnSource (FiberwiseLinear.openPartialHomeomorph φ hU ⋯ ⋯)
⊢ ContMDiffOn (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) n (↑(FiberwiseLinear.openPartialHomeomorph φ hU ⋯ ⋯).symm) e.target

Tactic:
· rw [EqOnSource.target_eq heφ]
    apply contMDiffOn_fst.prodMk
    exact (h2φ.comp contMDiffOn_fst <| prod_subset_preimage_fst _ _).clm_apply contMDiffOn_snd

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

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
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
H : Type u_5
inst✝⁵ : TopologicalSpace H
G : Type u_7
inst✝⁴ : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 F G
M : Type u_9
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
N : Type u_11
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n : ℕ∞ω
α : Sort u_13
h : M ≃ₘ^n⟮I, J⟯ N
f : α → M
⊢ ⇑h '' range f = ⇑h.symm ⁻¹' range f

Tactic:
image_eq_preimage_symm

Goal state after:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
H : Type u_5
inst✝⁵ : TopologicalSpace H
G : Type u_7
inst✝⁴ : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 F G
M : Type u_9
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
N : Type u_11
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n : ℕ∞ω
α : Sort u_13
h : M ≃ₘ^n⟮I, J⟯ N
f : α → M
⊢ ⇑h.symm ⁻¹' range f = ⇑h.symm ⁻¹' range f
Module: Mathlib.Geometry.Manifold.MFDeriv.UniqueDifferential

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
s : Set M
F : Type u_9
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
Z : M → Type u_10
inst✝² : TopologicalSpace (TotalSpace F Z)
inst✝¹ : (b : M) → TopologicalSpace (Z b)
inst✝ : FiberBundle F Z
p : TotalSpace F Z
hs : UniqueMDiffAt[s] p.proj
⊢ UniqueMDiffAt[TotalSpace.proj ⁻¹' s] p

Tactic:
suffices UniqueMDiffAt[π F Z ⁻¹' (s ∩ (trivializationAt F Z p.proj).baseSet)] p from
    this.mono (by simp)
  apply UniqueMDiffWithinAt.bundle_preimage_aux (hs.inter _) inter_subset_right
  exact (trivializationAt F Z p.proj).open_baseSet.mem_nhds
    (FiberBundle.mem_baseSet_trivializationAt' p.proj)

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

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
s : Set M
z : M
F' : Type u_21
inst✝² : NormedCommRing F'
inst✝¹ : NormedAlgebra 𝕜 F'
ι : Type
t : Finset ι
f : ι → M → F'
f' : ι → TangentSpace I z →L[𝕜] F'
inst✝ : DecidableEq ι
hf : ∀ i ∈ t, HasMFDerivAt[s] (f i) z (f' i)
⊢ HasMFDerivAt[s] (∏ i ∈ t, f i) z (∑ i ∈ t, (∏ j ∈ t.erase i, f j z) • f' i)

Tactic:
by
  induction t using Finset.induction_on with
  | empty => simpa using! hasMFDerivWithinAt_const ..
  | insert i t hi IH =>
    rw [t.sum_insert hi, t.erase_insert hi, t.prod_insert hi, add_comm]
    rw [t.forall_mem_insert] at hf
    convert! hf.1.mul (IH hf.2) using 2
    · simp only [t.smul_sum, ← mul_smul]
      refine t.sum_congr rfl (fun j hj ↦ ?_)
      rw [t.erase_insert_of_ne (by grind), Finset.prod_insert (by grind)]
    · simp

Goal state after:
Module: Mathlib.Topology.PartitionOfUnity

Goal state before:
pp: ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
⊢ {i | x₀ ∈ tsupport ⇑(ρ i)}.Finite

Tactic:
by
  rcases ρ.locallyFinite x₀ with ⟨t, t_in, ht⟩
  apply ht.subset
  rintro i hi
  simp only [inter_comm]
  exact mem_closure_iff_nhds.mp hi t t_in

Goal state after:
Module: Mathlib.Topology.ShrinkingLemma

Goal state before:
pp: case neg.left
ι : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
u : ι → Set X
s : Set X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
v : PartialRefinement u s fun w ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier
si : Set X := ⋯
hsi : si = s ∩ ⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ
hsic : IsCompact si
this : si ⊆ v.toFun i
vi : Set X
hvi : IsOpen vi ∧ si ⊆ vi ∧ closure vi ⊆ v.toFun i ∧ IsCompact (closure vi)
j : ι
hj : j = i ∨ j ∈ v.carrier
h : ¬j = i
⊢ j = i → IsCompact (closure (update v.toFun i vi j))

pp: case neg.right
ι : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
u : ι → Set X
s : Set X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
v : PartialRefinement u s fun w ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier
si : Set X := ⋯
hsi : si = s ∩ ⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ
hsic : IsCompact si
this : si ⊆ v.toFun i
vi : Set X
hvi : IsOpen vi ∧ si ⊆ vi ∧ closure vi ⊆ v.toFun i ∧ IsCompact (closure vi)
j : ι
hj : j = i ∨ j ∈ v.carrier
h : ¬j = i
⊢ j ∈ v.carrier → IsCompact (closure (update v.toFun i vi j))

Tactic:
· intro hji
        exact False.elim (h hji)

Goal state after:
case neg.right
ι : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
u : ι → Set X
s : Set X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
v : PartialRefinement u s fun w ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier
si : Set X := ⋯
hsi : si = s ∩ ⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ
hsic : IsCompact si
this : si ⊆ v.toFun i
vi : Set X
hvi : IsOpen vi ∧ si ⊆ vi ∧ closure vi ⊆ v.toFun i ∧ IsCompact (closure vi)
j : ι
hj : j = i ∨ j ∈ v.carrier
h : ¬j = i
⊢ j ∈ v.carrier → IsCompact (closure (update v.toFun i vi j))