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

Goal state before:
pp: E : Type u_1
F : Type u_2
G : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
L : E →L[ℝ] F →L[ℝ] G
u : ℝ → E
v : ℝ → F
u' : ℝ → E
v' : ℝ → F
hu : ∀ x ∈ tsupport v, HasDerivAt u (u' x) x
hv : ∀ x ∈ tsupport u, HasDerivAt v (v' x) x
huv' : Integrable (fun x ↦ (L (u x)) (v' x)) volume
hu'v : Integrable (fun x ↦ (L (u' x)) (v x)) volume
huv : Integrable (fun x ↦ (L (u x)) (v x)) volume
⊢ ∫ (x : ℝ), (L (u x)) (v' x) = -∫ (x : ℝ), (L (u' x)) (v x)

Tactic:
by_cases hG : CompleteSpace G

Goal state after:
case pos
E : Type u_1
F : Type u_2
G : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
L : E →L[ℝ] F →L[ℝ] G
u : ℝ → E
v : ℝ → F
u' : ℝ → E
v' : ℝ → F
hu : ∀ x ∈ tsupport v, HasDerivAt u (u' x) x
hv : ∀ x ∈ tsupport u, HasDerivAt v (v' x) x
huv' : Integrable (fun x ↦ (L (u x)) (v' x)) volume
hu'v : Integrable (fun x ↦ (L (u' x)) (v x)) volume
huv : Integrable (fun x ↦ (L (u x)) (v x)) volume
hG : CompleteSpace G
⊢ ∫ (x : ℝ), (L (u x)) (v' x) = -∫ (x : ℝ), (L (u' x)) (v x)

case neg
E : Type u_1
F : Type u_2
G : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
L : E →L[ℝ] F →L[ℝ] G
u : ℝ → E
v : ℝ → F
u' : ℝ → E
v' : ℝ → F
hu : ∀ x ∈ tsupport v, HasDerivAt u (u' x) x
hv : ∀ x ∈ tsupport u, HasDerivAt v (v' x) x
huv' : Integrable (fun x ↦ (L (u x)) (v' x)) volume
hu'v : Integrable (fun x ↦ (L (u' x)) (v x)) volume
huv : Integrable (fun x ↦ (L (u x)) (v x)) volume
hG : ¬CompleteSpace G
⊢ ∫ (x : ℝ), (L (u x)) (v' x) = -∫ (x : ℝ), (L (u' x)) (v x)
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
h1 : ∀ᶠ (n : ι) in φ, ContinuousOn (F n) (cthickening δ K)
h2 : IsCompact (cthickening δ K)
h3 : TendstoUniformlyOn F f φ (cthickening δ K)
n : ι
h : DifferentiableOn ℂ (F n) U
z : ℂ
hz : z ∈ K
⊢ (cderiv δ ∘ F) n z = (deriv ∘ F) n z

Tactic:
exact cderiv_eq_deriv hU h hδ ((closedBall_subset_cthickening hz δ).trans hKU)

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

Goal state before:
pp: case inl
H : Type u_5
M : Type u_6
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsEmpty H
hM : IsEmpty M
⊢ IsEmpty M

Tactic:
exact hM

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

Goal state before:
pp: H : Type u
M : Type u_2
M' : Type u_3
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : TopologicalSpace M'
cm : ChartedSpace H M
cm' : ChartedSpace H M'
x' : M'
⊢ chartAt H (Sum.inr x') = (chartAt H x').lift_openEmbedding ⋯

Tactic:
by
  simp +instances only [chartAt, sum, nonempty_of_chartedSpace x', ↓reduceDIte]
  rfl

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
⊢ IsOpen[inst✝²] (e.source ∩ s)

Tactic:
exact e.open_source.inter hs

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.invFun e.target

Tactic:
let : TopologicalSpace M := c.toTopologicalSpace
      apply continuousOn_isOpen_of_generateFrom
      intro t ht
      simp only [exists_prop, mem_iUnion, mem_singleton_iff] at ht
      rcases ht with ⟨e', e'_atlas, s, s_open, ts⟩
      rw [ts]
      let f := e.symm.trans e'
      have : IsOpen (f ⁻¹' s ∩ f.source) := by
        simpa [f, inter_comm] using (continuousOn_open_iff (c.open_source e e' he e'_atlas)).1
          (c.continuousOn_toFun e e' he e'_atlas) s s_open
      have A : e' ∘ e.symm ⁻¹' s ∩ (e.target ∩ e.symm ⁻¹' e'.source) =
          e.target ∩ (e' ∘ e.symm ⁻¹' s ∩ e.symm ⁻¹' e'.source) := by
        rw [← inter_assoc, ← inter_assoc]
        congr 1
        exact inter_comm _ _
      simpa [f, PartialEquiv.trans_source, preimage_inter, preimage_comp.symm, A] using this

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 : Set M
hG : G.LocalInvariantProp G' P
h : ∀ x ∈ s, ∃ u, IsOpen[inst✝⁴] u ∧ x ∈ u ∧ LiftPropOn P g (s ∩ u)
x : M
hx : x ∈ s
u : Set M
u_open : IsOpen[inst✝⁴] u
xu : x ∈ u
hu : LiftPropOn P g (s ∩ u)
this : LiftPropWithinAt P g (s ∩ u) x
⊢ LiftPropWithinAt P g s x

Tactic:
hG.liftPropWithinAt_inter

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'
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
hG : G.LocalInvariantProp G' P
h : ∀ x ∈ s, ∃ u, IsOpen[inst✝⁴] u ∧ x ∈ u ∧ LiftPropOn P g (s ∩ u)
x : M
hx : x ∈ s
u : Set M
u_open : IsOpen[inst✝⁴] u
xu : x ∈ u
hu : LiftPropOn P g (s ∩ u)
this : LiftPropWithinAt P g s x
⊢ LiftPropWithinAt P g s x

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 : Set M
hG : G.LocalInvariantProp G' P
h : ∀ x ∈ s, ∃ u, IsOpen[inst✝⁴] u ∧ x ∈ u ∧ LiftPropOn P g (s ∩ u)
x : M
hx : x ∈ s
u : Set M
u_open : IsOpen[inst✝⁴] u
xu : x ∈ u
hu : LiftPropOn P g (s ∩ u)
this : LiftPropWithinAt P g (s ∩ u) x
⊢ u ∈ 𝓝 x
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'
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
mono : ∀ ⦃s : Set H⦄ ⦃x : H⦄ ⦃t : Set H⦄ ⦃f : H → H'⦄, t ⊆ s → P f s x → P f t x
h : LiftProp P g
⊢ LiftPropOn P g s

Tactic:
rw [← liftPropOn_univ] at h
  exact liftPropOn_mono mono h (subset_univ _)

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

Goal state before:
pp: case left
n : ℕ∞ω
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
E' : Type u_5
H' : Type u_6
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
inst✝ : TopologicalSpace H'
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
e : OpenPartialHomeomorph H H
e' : OpenPartialHomeomorph H' H'
he : (contDiffPregroupoid n I).property (↑e) e.source
he_symm : (contDiffPregroupoid n I).property (↑e.symm) e.target
he' : (contDiffPregroupoid n I').property (↑e') e'.source
he'_symm : (contDiffPregroupoid n I').property (↑e'.symm) e'.target
h3 :
  ContDiffOn 𝕜 n (Prod.map (↑I ∘ ↑e ∘ ↑I.symm) (↑I' ∘ ↑e' ∘ ↑I'.symm))
    ((↑I.symm ⁻¹' e.source ∩ range ↑I) ×ˢ (↑I'.symm ⁻¹' e'.source ∩ range ↑I'))
⊢ ContDiffOn 𝕜 n (↑(I.prod I') ∘ ↑(e.prod e') ∘ ↑(I.prod I').symm)
    (↑(I.prod I').symm ⁻¹' (e.prod e'.toPartialEquiv).source ∩ range ↑(I.prod I'))

Tactic:
rw [← I.image_eq, ← I'.image_eq, prod_image_image_eq] at h3

Goal state after:
case left
n : ℕ∞ω
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
E' : Type u_5
H' : Type u_6
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
inst✝ : TopologicalSpace H'
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
e : OpenPartialHomeomorph H H
e' : OpenPartialHomeomorph H' H'
he : (contDiffPregroupoid n I).property (↑e) e.source
he_symm : (contDiffPregroupoid n I).property (↑e.symm) e.target
he' : (contDiffPregroupoid n I').property (↑e') e'.source
he'_symm : (contDiffPregroupoid n I').property (↑e'.symm) e'.target
h3 :
  ContDiffOn 𝕜 n (Prod.map (↑I ∘ ↑e ∘ ↑I.symm) (↑I' ∘ ↑e' ∘ ↑I'.symm))
    ((fun p ↦ (↑I p.1, ↑I' p.2)) '' e.source ×ˢ e'.source)
⊢ ContDiffOn 𝕜 n (↑(I.prod I') ∘ ↑(e.prod e') ∘ ↑(I.prod I').symm)
    (↑(I.prod I').symm ⁻¹' (e.prod e'.toPartialEquiv).source ∩ range ↑(I.prod I'))
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: case right
n : ℕ∞ω
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
E' : Type u_5
H' : Type u_6
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
inst✝ : TopologicalSpace H'
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
e : OpenPartialHomeomorph H H
e' : OpenPartialHomeomorph H' H'
he : (contDiffPregroupoid n I).property (↑e) e.source
he_symm : (contDiffPregroupoid n I).property (↑e.symm) e.target
he' : (contDiffPregroupoid n I').property (↑e') e'.source
he'_symm : (contDiffPregroupoid n I').property (↑e'.symm) e'.target
h3 :
  ContDiffOn 𝕜 n (Prod.map (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I' ∘ ↑e'.symm ∘ ↑I'.symm))
    ((↑I.symm ⁻¹' e.target ∩ range ↑I) ×ˢ (↑I'.symm ⁻¹' e'.target ∩ range ↑I'))
⊢ ContDiffOn 𝕜 n (↑(I.prod I') ∘ ↑(e.prod e').symm ∘ ↑(I.prod I').symm)
    (↑(I.prod I').symm ⁻¹' (e.prod e'.toPartialEquiv).target ∩ range ↑(I.prod I'))

Tactic:
rw [← I.image_eq, ← I'.image_eq, prod_image_image_eq] at h3

Goal state after:
case right
n : ℕ∞ω
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
E' : Type u_5
H' : Type u_6
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
inst✝ : TopologicalSpace H'
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
e : OpenPartialHomeomorph H H
e' : OpenPartialHomeomorph H' H'
he : (contDiffPregroupoid n I).property (↑e) e.source
he_symm : (contDiffPregroupoid n I).property (↑e.symm) e.target
he' : (contDiffPregroupoid n I').property (↑e') e'.source
he'_symm : (contDiffPregroupoid n I').property (↑e'.symm) e'.target
h3 :
  ContDiffOn 𝕜 n (Prod.map (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I' ∘ ↑e'.symm ∘ ↑I'.symm))
    ((fun p ↦ (↑I p.1, ↑I' p.2)) '' e.target ×ˢ e'.target)
⊢ ContDiffOn 𝕜 n (↑(I.prod I') ∘ ↑(e.prod e').symm ∘ ↑(I.prod I').symm)
    (↑(I.prod I').symm ⁻¹' (e.prod e'.toPartialEquiv).target ∩ range ↑(I.prod I'))
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},
    e' ∈ G →
      s ⊆ f ⁻¹' e'.source →
        f x ∈ e'.source → G.IsLocalStructomorphWithinAt f s x → G.IsLocalStructomorphWithinAt (↑e' ∘ f) s x

Tactic:
by
      intro s x f e' he'G _ hfx h hx
      rcases h hx with ⟨e, heG, hef, hex⟩
      refine ⟨e.trans e', G.trans heG he'G, ?_, ?_⟩
      · intro y hy
        simp only [mfld_simps] at hy
        simp only [hef ⟨hy.1, hy.2.1⟩, mfld_simps]
      · simpa only [hex, hef ⟨hx, hex⟩, mfld_simps] using hfx

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
n : ℕ∞ω
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsEmpty M
e e' : OpenPartialHomeomorph M H
a✝¹ : e ∈ atlas H M
a✝ : e' ∈ atlas H M
x : E
t : Set E := ↑I.symm ⁻¹' (e.symm ≫ₕ e').source ∩ range ↑I
hx : x ∈ t
⊢ e.symm.source ∩ ↑e.symm ⁻¹' e'.source = e.symm.source ∩ ↑e.symm ⁻¹' ∅

Tactic:
by rw [eq_empty_of_isEmpty (e'.source)]

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[s] f x
⊢ HasMFDerivAt[s] f x (mfderiv[s] f x)

Tactic:
by
  refine ⟨h.1, ?_⟩
  simp only [mfderivWithin, h, mfld_simps]
  exact DifferentiableWithinAt.hasFDerivWithinAt h.2

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
f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)
h : HasMFDerivAt[s] f x f'
hxs : UniqueMDiffAt[s] x
x✝ : TangentSpace I x
⊢ (mfderiv[s] f x) x✝ = f' x✝

Tactic:
hxs.eq h h.mdifferentiableWithinAt.hasMFDerivWithinAt

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
f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)
h : HasMFDerivAt[s] f x f'
hxs : UniqueMDiffAt[s] x
x✝ : TangentSpace I x
⊢ (mfderiv[s] f x) x✝ = (mfderiv[s] f x) x✝
Module: Mathlib.Topology.VectorBundle.Basic

Goal state before:
pp: case pos.refine_2
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁹ : Semiring R
inst✝⁸ : TopologicalSpace F
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace (TotalSpace F E)
e✝ : Trivialization F TotalSpace.proj
x : TotalSpace F E
b✝ : B
y : E b✝
inst✝⁵ : AddCommMonoid F
inst✝⁴ : Module R F
inst✝³ : (x : B) → AddCommMonoid (E x)
inst✝² : (x : B) → Module R (E x)
e e' : Trivialization F TotalSpace.proj
inst✝¹ : Trivialization.IsLinear R e
inst✝ : Trivialization.IsLinear R e'
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
⊢ ∀ (x : F), { proj := b, snd := ↑(linearEquivAt R e b ⋯).symm x } ∈ e'.source

Tactic:
· exact fun y => e'.mem_source.mpr hb.2

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

Goal state before:
pp: case refine_1
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
f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)
y : M
h : s =ᶠ[𝓝[{y}ᶜ] x] t
this : T1Space M
⊢ ContinuousWithinAt f s x ↔ ContinuousWithinAt f t x

Tactic:
exact continuousWithinAt_congr_set' _ h

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

Goal state before:
pp: B : Type u_2
F : Type u_3
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : TopologicalSpace B
𝕜₁ : Type u_5
𝕜₂ : Type u_6
inst✝⁵ : NontriviallyNormedField 𝕜₁
inst✝⁴ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
B' : Type u_7
inst✝³ : TopologicalSpace B'
inst✝² : NormedSpace 𝕜₁ F
F' : Type u_8
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace 𝕜₂ F'
ι : Type u_10
ι' : Type u_11
Z : VectorBundleCore 𝕜₁ B F ι
Z' : VectorBundleCore 𝕜₂ B' F' ι'
x₀ x : B
y₀ y : B'
ϕ : F →SL[σ] F'
hx : x ∈ Z.baseSet (Z.indexAt x₀)
hy : y ∈ Z'.baseSet (Z'.indexAt y₀)
⊢ Trivialization.continuousLinearMapAt 𝕜₂ (trivializationAt F' Z'.Fiber y₀) y ∘SL
      ϕ ∘SL Trivialization.symmL 𝕜₁ (trivializationAt F Z.Fiber x₀) x =
    Z'.coordChange (Z'.indexAt y) (Z'.indexAt y₀) y ∘SL ϕ ∘SL Z.coordChange (Z.indexAt x₀) (Z.indexAt x) x

Tactic:
Z'.trivializationAt_continuousLinearMapAt hy,

Goal state after:
B : Type u_2
F : Type u_3
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : TopologicalSpace B
𝕜₁ : Type u_5
𝕜₂ : Type u_6
inst✝⁵ : NontriviallyNormedField 𝕜₁
inst✝⁴ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
B' : Type u_7
inst✝³ : TopologicalSpace B'
inst✝² : NormedSpace 𝕜₁ F
F' : Type u_8
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace 𝕜₂ F'
ι : Type u_10
ι' : Type u_11
Z : VectorBundleCore 𝕜₁ B F ι
Z' : VectorBundleCore 𝕜₂ B' F' ι'
x₀ x : B
y₀ y : B'
ϕ : F →SL[σ] F'
hx : x ∈ Z.baseSet (Z.indexAt x₀)
hy : y ∈ Z'.baseSet (Z'.indexAt y₀)
⊢ Z'.coordChange (Z'.indexAt y) (Z'.indexAt y₀) y ∘SL ϕ ∘SL Trivialization.symmL 𝕜₁ (trivializationAt F Z.Fiber x₀) x =
    Z'.coordChange (Z'.indexAt y) (Z'.indexAt y₀) y ∘SL ϕ ∘SL Z.coordChange (Z.indexAt x₀) (Z.indexAt x) x
Module: Mathlib.Geometry.Manifold.VectorBundle.Basic

Goal state before:
pp: n : ℕ∞ω
𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹⁵ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁴ : NormedAddCommGroup EB
inst✝¹³ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹² : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹¹ : TopologicalSpace B
inst✝¹⁰ : ChartedSpace HB B
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
s : (x : B) → E x
a : Set B
x₀ : B
e : Trivialization F TotalSpace.proj
inst✝ : MemTrivializationAtlas e
hx₀ : x₀ ∈ e.baseSet
⊢ TotalSpace.mk' F x₀ (s x₀) ∈ e.source

Tactic:
rwa [mem_source]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsManifold I 1 M
x : M
y : E
hy : y ∈ (extChartAt I x).target
h'y : ↑(extChartAt I x).symm y ∈ (extChartAt I x).source
h''y : ↑(extChartAt I x).symm y ∈ (chartAt H x).source
⊢ mfderiv[range ↑I] ↑(extChartAt I x).symm y ∘SL mfderiv% ↑(extChartAt I x) (↑(extChartAt I x).symm y) =
    ContinuousLinearMap.id 𝕜 (TangentSpace I (↑(extChartAt I x).symm y))

Tactic:
have U' : UniqueMDiffAt[(extChartAt I x).source] ((extChartAt I x).symm y) :=
    (isOpen_extChartAt_source x).uniqueMDiffWithinAt h'y

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsManifold I 1 M
x : M
y : E
hy : y ∈ (extChartAt I x).target
h'y : ↑(extChartAt I x).symm y ∈ (extChartAt I x).source
h''y : ↑(extChartAt I x).symm y ∈ (chartAt H x).source
U' : UniqueMDiffAt[(extChartAt I x).source] (↑(extChartAt I x).symm y)
⊢ mfderiv[range ↑I] ↑(extChartAt I x).symm y ∘SL mfderiv% ↑(extChartAt I x) (↑(extChartAt I x).symm y) =
    ContinuousLinearMap.id 𝕜 (TangentSpace I (↑(extChartAt I x).symm y))
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
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'
s : Set M
x : M
hs : UniqueMDiffAt[s] x
f : M → M'
f' : E →L[𝕜] E'
hf : HasMFDerivAt[s] f x f'
hd : DenseRange ⇑f'
⊢ UniqueMDiffAt[f '' s] (f x)

Tactic:
have := hs.inter' <| hf.1 (extChartAt_source_mem_nhds (I := I') (f x))
  refine (((hf.2.mono ?sub1).uniqueDiffWithinAt this hd).mono ?sub2).congr_pt ?pt
  case pt => simp only [mfld_simps]
  case sub1 => mfld_set_tac
  case sub2 =>
    rintro _ ⟨y, ⟨⟨hys, hfy⟩, -⟩, rfl⟩
    exact ⟨⟨_, hys, ((extChartAt I' (f x)).left_inv hfy).symm⟩, mem_range_self _⟩

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³² : NormedAddCommGroup E
inst✝³¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝³⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝²⁹ : TopologicalSpace M
inst✝²⁸ : ChartedSpace H M
E' : Type u_5
inst✝²⁷ : NormedAddCommGroup E'
inst✝²⁶ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝²⁵ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝²⁴ : TopologicalSpace M'
inst✝²³ : ChartedSpace H' M'
E'' : Type u_8
inst✝²² : NormedAddCommGroup E''
inst✝²¹ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝²⁰ : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝¹⁹ : TopologicalSpace M''
inst✝¹⁸ : ChartedSpace H'' M''
F : Type u_11
inst✝¹⁷ : NormedAddCommGroup F
inst✝¹⁶ : NormedSpace 𝕜 F
G : Type u_12
inst✝¹⁵ : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_13
inst✝¹⁴ : TopologicalSpace N
inst✝¹³ : ChartedSpace G N
F' : Type u_14
inst✝¹² : NormedAddCommGroup F'
inst✝¹¹ : NormedSpace 𝕜 F'
G' : Type u_15
inst✝¹⁰ : TopologicalSpace G'
J' : ModelWithCorners 𝕜 F' G'
N' : Type u_16
inst✝⁹ : TopologicalSpace N'
inst✝⁸ : ChartedSpace G' N'
F₁ : Type u_17
inst✝⁷ : NormedAddCommGroup F₁
inst✝⁶ : NormedSpace 𝕜 F₁
F₂ : Type u_18
inst✝⁵ : NormedAddCommGroup F₂
inst✝⁴ : NormedSpace 𝕜 F₂
F₃ : Type u_19
inst✝³ : NormedAddCommGroup F₃
inst✝² : NormedSpace 𝕜 F₃
F₄ : Type u_20
inst✝¹ : NormedAddCommGroup F₄
inst✝ : NormedSpace 𝕜 F₄
s : Set M
x z : M
f g : M → E'
f' g' : TangentSpace I z →L[𝕜] E'
hf : MDiffAt[s] f z
hg : MDiffAt[s] g z
hs : UniqueMDiffAt[s] z
⊢ TangentSpace I z →L[𝕜] TangentSpace 𝓘(𝕜, E') ((f + g) z)

Tactic:
by exact mfderiv[s] f z

Goal state after: