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

Goal state before:
pp: ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.NeBot
inst✝¹ : l.IsCountablyGenerated
inst✝ : NormedAddCommGroup E
b : ι → ℝ
f : ℝ → E
I a : ℝ
hfi : ∀ (i : ι), IntegrableOn f (Ioc a (b i)) μ
hb : Tendsto b l atTop
h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in a..b i, ‖f x‖ ∂μ ≤ I
⊢ IntegrableOn f (Ioi a) μ

Tactic:
by
  have hφ : AECover (μ.restrict <| Ioi a) l _ := aecover_Iic hb
  have hfi : ∀ i, IntegrableOn f (Iic (b i)) (μ.restrict <| Ioi a) := by
    intro i
    rw [IntegrableOn, Measure.restrict_restrict (hφ.measurableSet i), inter_comm]
    exact hfi i
  refine hφ.integrable_of_integral_norm_bounded I hfi (h.mp ?_)
  filter_upwards [hb.eventually (eventually_ge_atTop a)] with i hbi
  rw [intervalIntegral.integral_of_le hbi, Measure.restrict_restrict (hφ.measurableSet i),
    inter_comm]
  exact id

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.IsCountablyGenerated
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
b : ι → ℝ
f : ℝ → E
hb : Tendsto b l atTop
⊢ Tendsto (fun i ↦ ∫ (x : ℝ) in Ioi (b i), f x ∂μ) l (𝓝 0)

Tactic:
by
  by_cases! h : ∀ a, ¬ IntegrableOn f (Ioi a) μ
  · exact tendsto_const_nhds.congr (fun i => (integral_undef (h (b i))).symm)
  obtain ⟨a, ha⟩ := h
  have : ∀ᶠ i in l, ∫ x in Ioi a, f x ∂μ - ∫ x in a..b i, f x ∂μ = ∫ x in Ioi (b i), f x ∂μ := by
    filter_upwards [hb.eventually_mem (Ici_mem_atTop a)] with i hi
    rw [sub_eq_iff_eq_add',
      intervalIntegral.integral_interval_add_Ioi ha (ha.mono_set (Ioi_subset_Ioi hi))]
  rw [← sub_self (∫ x in Ioi a, f x ∂μ)]
  exact Tendsto.congr' this (Tendsto.const_sub _ <| intervalIntegral_tendsto_integral_Ioi a ha hb)

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

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
g : ℝ → F
hf : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo a b, f' x ≤ 0
hab : a ≤ b
⊢ IntegrableOn (fun x ↦ f' x • g (f x)) (Icc a b) volume ↔ IntegrableOn g (Icc (f b) (f a)) volume

Tactic:
have M : AntitoneOn f (Icc a b) := by
    apply antitoneOn_of_deriv_nonpos (convex_Icc a b) hf
    · rw [interior_Icc]
      exact fun z hz ↦ (hff' z hz).differentiableAt.differentiableWithinAt
    · rw [interior_Icc]
      intro z hz
      simpa [(hff' z hz).deriv] using hf' z hz
  have A : IntegrableOn g (Icc (f b) (f a)) ↔ IntegrableOn g (f '' (Ioo a b)) := by
    apply integrableOn_congr_set_ae
    rw [← hf.image_Icc_of_antitoneOn hab M]
    refine ae_eq_set.2 ⟨?_, by simp [show f '' Ioo a b \ f '' Icc a b = ∅ by grind]⟩
    have : f '' (Icc a b) \ f '' Ioo a b ⊆ {f a, f b} := by grind
    apply measure_mono_null this
    apply Finite.measure_zero (by simp)
  rw [A, integrableOn_Icc_iff_integrableOn_Ioo,
    integrableOn_image_iff_integrableOn_deriv_smul_of_antitoneOn measurableSet_Ioo (f' := f')]
  · simp
  · exact fun z hz ↦ (hff' z hz).hasDerivWithinAt
  · exact M.mono Ioo_subset_Icc_self

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
⊢ (Measure.map (⇑f.symm) (volume.withDensity fun x ↦ ENNReal.ofReal (g x))) s =
    ENNReal.ofReal (∫ (x : ℝ) in s, |f' x| * g (f x))

Tactic:
rw [MeasurableEquiv.withDensity_ofReal_map_symm_apply_eq_integral_abs_det_fderiv_mul volume hs
      f (by filter_upwards [hg] with x hx using fun _ ↦ hx) hg_int.integrableOn
      (fun x _ => (hf' x).hasDerivWithinAt)]
  simp only [det_toSpanSingleton]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: E : Type u_1
f f' : ℝ → E
m n : E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hderiv : ∀ (x : ℝ), HasDerivAt f (f' x) x
hf' : Integrable f' volume
hbot : Tendsto f atBot (𝓝 m)
htop : Tendsto f atTop (𝓝 n)
⊢ ∫ (x : ℝ), f' x = n - m

Tactic:
← setIntegral_univ,

Goal state after:
E : Type u_1
f f' : ℝ → E
m n : E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hderiv : ∀ (x : ℝ), HasDerivAt f (f' x) x
hf' : Integrable f' volume
hbot : Tendsto f atBot (𝓝 m)
htop : Tendsto f atTop (𝓝 n)
⊢ ∫ (x : ℝ) in univ, f' x = n - m
Module: Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts

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

Tactic:
simp [ν, Measure.map_map L.symm.continuous.measurable L.continuous.measurable]

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) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z = ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • (f z - f w₀)

Tactic:
have h1 : ContinuousOn (fun z : ℂ => ((z - w₀) ^ 2)⁻¹) (sphere c R) := by
      refine ((continuous_id'.sub continuous_const).pow 2).continuousOn.inv₀ fun w hw h => ?_
      exact sphere_disjoint_ball.ne_of_mem hw hw₀ (sub_eq_zero.mp (sq_eq_zero_iff.mp h))

Goal state after:
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
h1 : ContinuousOn (fun z ↦ ((z - w₀) ^ 2)⁻¹) (sphere c R)
⊢ ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z = ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • (f z - f w₀)
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
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
h'f' : ∀ x ∈ s, (f' x).det = 0
R : ℝ
⊢ μ (f '' (s ∩ closedBall 0 R)) = 0

Tactic:
have A : ∀ (ε : ℝ≥0), 0 < ε → μ (f '' (s ∩ closedBall 0 R)) ≤ ε * μ (closedBall 0 R) :=
    fun ε εpos =>
    addHaar_image_eq_zero_of_det_fderivWithin_eq_zero_aux μ
      (fun x hx => (hf' x hx.1).mono inter_subset_left) R inter_subset_right ε εpos
      fun x hx => h'f' x hx.1

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
h'f' : ∀ x ∈ s, (f' x).det = 0
R : ℝ
A : ∀ (ε : ℝ≥0), 0 < ε → μ (f '' (s ∩ closedBall 0 R)) ≤ ↑ε * μ (closedBall 0 R)
⊢ μ (f '' (s ∩ closedBall 0 R)) = 0
Module: Mathlib.Analysis.Complex.LocallyUniformLimit

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
U : Set ℂ
φ : Filter ι
F : ι → ℂ → E
f : ℂ → E
inst✝¹ : CompleteSpace E
inst✝ : φ.NeBot
hf : TendstoLocallyUniformlyOn F f φ U
hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U
hU : IsOpen U
⊢ DifferentiableOn ℂ f U

Tactic:
rintro x hx

Goal state after:
E : Type u_1
ι : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
U : Set ℂ
φ : Filter ι
F : ι → ℂ → E
f : ℂ → E
inst✝¹ : CompleteSpace E
inst✝ : φ.NeBot
hf : TendstoLocallyUniformlyOn F f φ U
hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U
hU : IsOpen U
x : ℂ
hx : x ∈ U
⊢ DifferentiableWithinAt ℂ f U x
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: H : Type u_1
inst✝ : TopologicalSpace H
S : Set (StructureGroupoid H)
⊢ ∀ (e : OpenPartialHomeomorph H H),
    (∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ ⋂ s ∈ S, s.members) → e ∈ ⋂ s ∈ S, s.members

Tactic:
simp only [mem_iInter]
      intro e he i hi
      refine i.locality' e ?_
      intro x hex
      rcases he x hex with ⟨s, hs⟩
      exact ⟨s, ⟨hs.left, ⟨hs.right.left, hs.right.right i hi⟩⟩⟩

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

Goal state before:
pp: H : Type u_1
inst✝ : TopologicalSpace H
S : Set (StructureGroupoid H)
⊢ ∀ (e e' : OpenPartialHomeomorph H H), e ∈ ⋂ s ∈ S, s.members → e' ≈ e → e' ∈ ⋂ s ∈ S, s.members

Tactic:
simp only [mem_iInter]
      intro e e' he he'e
      exact fun i hi => i.mem_of_eqOnSource' e e' (he i hi) he'e

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

Goal state before:
pp: case mk.mk
H : Type u_1
inst✝ : TopologicalSpace H
members✝¹ : Set (OpenPartialHomeomorph H H)
trans'✝¹ : ∀ (e e' : OpenPartialHomeomorph H H), e ∈ members✝¹ → e' ∈ members✝¹ → e ≫ₕ e' ∈ members✝¹
symm'✝¹ : ∀ e ∈ members✝¹, e.symm ∈ members✝¹
id_mem'✝¹ : OpenPartialHomeomorph.refl H ∈ members✝¹
locality'✝¹ :
  ∀ (e : OpenPartialHomeomorph H H),
    (∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ members✝¹) → e ∈ members✝¹
mem_of_eqOnSource'✝¹ : ∀ (e e' : OpenPartialHomeomorph H H), e ∈ members✝¹ → e' ≈ e → e' ∈ members✝¹
members✝ : Set (OpenPartialHomeomorph H H)
trans'✝ : ∀ (e e' : OpenPartialHomeomorph H H), e ∈ members✝ → e' ∈ members✝ → e ≫ₕ e' ∈ members✝
symm'✝ : ∀ e ∈ members✝, e.symm ∈ members✝
id_mem'✝ : OpenPartialHomeomorph.refl H ∈ members✝
locality'✝ :
  ∀ (e : OpenPartialHomeomorph H H),
    (∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ members✝) → e ∈ members✝
mem_of_eqOnSource'✝ : ∀ (e e' : OpenPartialHomeomorph H H), e ∈ members✝ → e' ≈ e → e' ∈ members✝
h : members✝¹ = members✝
⊢ { members := members✝¹, trans' := trans'✝¹, symm' := symm'✝¹, id_mem' := id_mem'✝¹, locality' := locality'✝¹,
      mem_of_eqOnSource' := mem_of_eqOnSource'✝¹ } =
    { members := members✝, trans' := trans'✝, symm' := symm'✝, id_mem' := id_mem'✝, locality' := locality'✝,
      mem_of_eqOnSource' := mem_of_eqOnSource'✝ }

Tactic:
induction h

Goal state after:
case mk.mk.refl
H : Type u_1
inst✝ : TopologicalSpace H
members✝¹ : Set (OpenPartialHomeomorph H H)
trans'✝¹ : ∀ (e e' : OpenPartialHomeomorph H H), e ∈ members✝¹ → e' ∈ members✝¹ → e ≫ₕ e' ∈ members✝¹
symm'✝¹ : ∀ e ∈ members✝¹, e.symm ∈ members✝¹
id_mem'✝¹ : OpenPartialHomeomorph.refl H ∈ members✝¹
locality'✝¹ :
  ∀ (e : OpenPartialHomeomorph H H),
    (∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ members✝¹) → e ∈ members✝¹
mem_of_eqOnSource'✝¹ : ∀ (e e' : OpenPartialHomeomorph H H), e ∈ members✝¹ → e' ≈ e → e' ∈ members✝¹
members✝ : Set (OpenPartialHomeomorph H H)
trans'✝ : ∀ (e e' : OpenPartialHomeomorph H H), e ∈ members✝¹ → e' ∈ members✝¹ → e ≫ₕ e' ∈ members✝¹
symm'✝ : ∀ e ∈ members✝¹, e.symm ∈ members✝¹
id_mem'✝ : OpenPartialHomeomorph.refl H ∈ members✝¹
locality'✝ :
  ∀ (e : OpenPartialHomeomorph H H),
    (∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ members✝¹) → e ∈ members✝¹
mem_of_eqOnSource'✝ : ∀ (e e' : OpenPartialHomeomorph H H), e ∈ members✝¹ → e' ≈ e → e' ∈ members✝¹
⊢ { members := members✝¹, trans' := trans'✝¹, symm' := symm'✝¹, id_mem' := id_mem'✝¹, locality' := locality'✝¹,
      mem_of_eqOnSource' := mem_of_eqOnSource'✝¹ } =
    { members := members✝¹, trans' := trans'✝, symm' := symm'✝, id_mem' := id_mem'✝, locality' := locality'✝,
      mem_of_eqOnSource' := mem_of_eqOnSource'✝ }
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
⊢ ∃ s, IsOpen[inst✝²] s ∧ x ∈ s ∧ (e.symm ≫ₕ e').restr s ∈ G

Tactic:
have hs : IsOpen s := by
    apply e.symm.continuousOn_toFun.isOpen_inter_preimage <;> apply open_source

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
⊢ ∃ s, IsOpen[inst✝²] s ∧ x ∈ s ∧ (e.symm ≫ₕ e').restr s ∈ G
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✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : TopologicalSpace M'
cm : ChartedSpace H M
cm' : ChartedSpace H M'
inst✝ : Nonempty H
p : M ⊕ M'
⊢ p ∈
    (Sum.elim (fun x ↦ (ChartedSpace.chartAt x).lift_openEmbedding ⋯)
        (fun x ↦ (ChartedSpace.chartAt x).lift_openEmbedding ⋯) p).source

Tactic:
cases p with

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✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : TopologicalSpace M'
cm : ChartedSpace H M
cm' : ChartedSpace H M'
inst✝ : Nonempty H
p : M ⊕ M'
⊢ Sum.elim (fun x ↦ (ChartedSpace.chartAt x).lift_openEmbedding ⋯)
      (fun x ↦ (ChartedSpace.chartAt x).lift_openEmbedding ⋯) p ∈
    (fun e ↦ e.lift_openEmbedding ⋯) '' ChartedSpace.atlas ∪ (fun e ↦ e.lift_openEmbedding ⋯) '' ChartedSpace.atlas

Tactic:
cases p with

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

Goal state before:
pp: case inr.inl
H✝ : Type u_1
inst✝¹ : TopologicalSpace H✝
H : Type u_2
inst✝ : TopologicalSpace H
e : OpenPartialHomeomorph H H
he : ∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}
x : H
hx : x ∈ e.source
s : Set H
open_s : IsOpen[inst✝] s
xs : x ∈ s
x's : x ∈ (e.restr s).source
hs : e.restr s = OpenPartialHomeomorph.refl H
this✝ : (e.restr s).source = univ
this : e.source ∩ interior s = univ
⊢ e ∈ {OpenPartialHomeomorph.refl H}

Tactic:
have : univ ⊆ interior s := by
          rw [← this]
          exact inter_subset_right

Goal state after:
case inr.inl
H✝ : Type u_1
inst✝¹ : TopologicalSpace H✝
H : Type u_2
inst✝ : TopologicalSpace H
e : OpenPartialHomeomorph H H
he : ∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}
x : H
hx : x ∈ e.source
s : Set H
open_s : IsOpen[inst✝] s
xs : x ∈ s
x's : x ∈ (e.restr s).source
hs : e.restr s = OpenPartialHomeomorph.refl H
this✝¹ : (e.restr s).source = univ
this✝ : e.source ∩ interior s = univ
this : univ ⊆ interior s
⊢ e ∈ {OpenPartialHomeomorph.refl H}
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: H : Type u_1
inst✝ : TopologicalSpace H
PG : Pregroupoid H
e : OpenPartialHomeomorph H H
he :
  ∀ x ∈ e.source,
    ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
⊢ e ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}

Tactic:
constructor
    · refine PG.locality e.open_source fun x xu ↦ ?_
      rcases he x xu with ⟨s, s_open, xs, hs⟩
      refine ⟨s, s_open, xs, ?_⟩
      convert! hs.1 using 1
      dsimp [OpenPartialHomeomorph.restr]
      rw [s_open.interior_eq]
    · refine PG.locality e.open_target fun x xu ↦ ?_
      rcases he (e.symm x) (e.map_target xu) with ⟨s, s_open, xs, hs⟩
      refine ⟨e.target ∩ e.symm ⁻¹' s, ?_, ⟨xu, xs⟩, ?_⟩
      · exact ContinuousOn.isOpen_inter_preimage e.continuousOn_invFun e.open_target s_open
      · rw [← inter_assoc, inter_self]
        convert! hs.2 using 1
        dsimp [OpenPartialHomeomorph.restr]
        rw [s_open.interior_eq]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace ℝ E
h : IsRCLikeNormedField 𝕜
s : Set E
hs : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ s
this✝ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
u : E
hu : u ∈ s
⊢ ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • u + b • y ∈ s

Tactic:
v

Goal state after:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace ℝ E
h : IsRCLikeNormedField 𝕜
s : Set E
hs : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ s
this✝ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
u : E
hu : u ∈ s
v : E
⊢ v ∈ s → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • u + b • v ∈ s
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: case pos
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
inst✝ : NormedSpace ℝ E
h : IsRCLikeNormedField 𝕜
this : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
W :
  ∀ ⦃x : E⦄, x ∈ range ↑I → ∀ ⦃y : E⦄, y ∈ range ↑I → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ range ↑I
u : E
hu : u ∈ range ↑I
⊢ ∀ ⦃y : E⦄, y ∈ range ↑I → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • u + b • y ∈ range ↑I

Tactic:
v

Goal state after:
case pos
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
inst✝ : NormedSpace ℝ E
h : IsRCLikeNormedField 𝕜
this : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
W :
  ∀ ⦃x : E⦄, x ∈ range ↑I → ∀ ⦃y : E⦄, y ∈ range ↑I → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ range ↑I
u : E
hu : u ∈ range ↑I
v : E
⊢ v ∈ range ↑I → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • u + b • v ∈ range ↑I
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: case pos
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
inst✝ : NormedSpace ℝ E
h : IsRCLikeNormedField 𝕜
⊢ Convex ℝ (range ↑I)

Tactic:
let : RCLike 𝕜 := h.rclike
    have W := I.convex_range'
    simp only [h, ↓reduceDIte, toPartialEquiv_coe] at W
    simp only [Convex, StarConvex] at W ⊢
    intro u hu v hv a b ha hb hab
    convert! W hu hv ha hb hab using 2
    · rw [← @algebraMap_smul (R := ℝ) (A := 𝕜)]
      rfl
    · rw [← @algebraMap_smul (R := ℝ) (A := 𝕜)]
      rfl

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 : Set E
x : M
hx : x ∈ f.source
hs : s ∈ 𝓝[range ↑I] ↑(f.extend I) x
⊢ ↑(f.extend I) ⁻¹' s ∈ 𝓝 x

Tactic:
← map_extend_nhds (I := I) f hx

Goal state after:
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 : Set E
x : M
hx : x ∈ f.source
hs : s ∈ map (↑(f.extend I)) (𝓝 x)
⊢ ↑(f.extend I) ⁻¹' s ∈ 𝓝 x
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: 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
inst✝ : LocallyCompactSpace E
I : ModelWithCorners 𝕜 E H
x : H
⊢ (𝓝 x).HasBasis (fun s ↦ s ∈ 𝓝 (↑I x) ∧ IsCompact s) fun s ↦ ↑I.symm '' (s ∩ range ↑I)

Tactic:
← I.symm_map_nhdsWithin_range

Goal state after:
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
inst✝ : LocallyCompactSpace E
I : ModelWithCorners 𝕜 E H
x : H
⊢ (map (↑I.symm) (𝓝[range ↑I] ↑I x)).HasBasis (fun s ↦ s ∈ 𝓝 (↑I x) ∧ IsCompact s) fun s ↦ ↑I.symm '' (s ∩ range ↑I)
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: case mp.refine_1
H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
f : OpenPartialHomeomorph H H
s : Set H
x : H
hx : x ∈ f.source ∪ sᶜ
hf : G.IsLocalStructomorphWithinAt (↑f) s x
h2x : x ∈ s
e : OpenPartialHomeomorph H H
he : e ∈ G
hfe : EqOn (↑f) (↑e.toPartialEquiv) (s ∩ e.source)
hxe : x ∈ e.source
⊢ e.source ∩ interior f.source ⊆ f.source

Tactic:
exact inter_subset_right.trans interior_subset

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

Goal state before:
pp: case mp.refine_2
H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
f : OpenPartialHomeomorph H H
s : Set H
x : H
hx : x ∈ f.source ∪ sᶜ
hf : G.IsLocalStructomorphWithinAt (↑f) s x
h2x : x ∈ s
e : OpenPartialHomeomorph H H
he : e ∈ G
hfe : EqOn (↑f) (↑e.toPartialEquiv) (s ∩ e.source)
hxe : x ∈ e.source
⊢ EqOn (↑f) (↑(e.restr f.source)) (s ∩ (e.restr f.source).source)

Tactic:
intro x'

Goal state after:
case mp.refine_2
H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
f : OpenPartialHomeomorph H H
s : Set H
x : H
hx : x ∈ f.source ∪ sᶜ
hf : G.IsLocalStructomorphWithinAt (↑f) s x
h2x : x ∈ s
e : OpenPartialHomeomorph H H
he : e ∈ G
hfe : EqOn (↑f) (↑e.toPartialEquiv) (s ∩ e.source)
hxe : x ∈ e.source
x' : H
⊢ x' ∈ s ∩ (e.restr f.source).source → ↑f x' = ↑(e.restr f.source) x'