mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
g : E → F
f : 𝕜 → E
s : Set 𝕜
t : Set E
x : 𝕜
hg : ContDiffWithinAt 𝕜 2 g t (f x)
hf : ContDiffWithinAt 𝕜 2 f s x
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
hst : MapsTo f s t
⊢ (((iteratedFDerivWithin 𝕜 ((OrderedFinpartition.extendEquiv 1) ⟨OrderedFinpartition.atomic 1, none⟩).length g t (f x))
        fun j ↦
        iteratedDerivWithin (((OrderedFinpartition.extendEquiv 1) ⟨OrderedFinpartition.atomic 1, none⟩).partSize j) f s
          x) +
      ∑ x_1,
        (iteratedFDerivWithin 𝕜 ((OrderedFinpartition.extendEquiv 1) ⟨OrderedFinpartition.atomic 1, some x_1⟩).length g
            t (f x))
          fun j ↦
          iteratedDerivWithin
            (((OrderedFinpartition.extendEquiv 1) ⟨OrderedFinpartition.atomic 1, some x_1⟩).partSize j) f s x) =
    ((iteratedFDerivWithin 𝕜 2 g t (f x)) fun x_1 ↦ derivWithin f s x) +
      (fderivWithin 𝕜 g t (f x)) (iteratedDerivWithin 2 f s x)

Tactic:
have : (Fin.cons 1 (fun _ ↦ 1) : Fin 2 → ℕ) = fun _ ↦ 1 :=
    funext <| Fin.forall_fin_two.mpr ⟨rfl, rfl⟩

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
g : E → F
f : 𝕜 → E
s : Set 𝕜
t : Set E
x : 𝕜
hg : ContDiffWithinAt 𝕜 2 g t (f x)
hf : ContDiffWithinAt 𝕜 2 f s x
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
hst : MapsTo f s t
this : (Fin.cons 1 fun x ↦ 1) = fun x ↦ 1
⊢ (((iteratedFDerivWithin 𝕜 ((OrderedFinpartition.extendEquiv 1) ⟨OrderedFinpartition.atomic 1, none⟩).length g t (f x))
        fun j ↦
        iteratedDerivWithin (((OrderedFinpartition.extendEquiv 1) ⟨OrderedFinpartition.atomic 1, none⟩).partSize j) f s
          x) +
      ∑ x_1,
        (iteratedFDerivWithin 𝕜 ((OrderedFinpartition.extendEquiv 1) ⟨OrderedFinpartition.atomic 1, some x_1⟩).length g
            t (f x))
          fun j ↦
          iteratedDerivWithin
            (((OrderedFinpartition.extendEquiv 1) ⟨OrderedFinpartition.atomic 1, some x_1⟩).partSize j) f s x) =
    ((iteratedFDerivWithin 𝕜 2 g t (f x)) fun x_1 ↦ derivWithin f s x) +
      (fderivWithin 𝕜 g t (f x)) (iteratedDerivWithin 2 f s x)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: a b : ℝ
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra ℝ 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : IsScalarTower ℝ 𝕜 E
u u' : ℝ → 𝕜
v v' : ℝ → E
hu : ContinuousOn u [[a, b]]
hv : ContinuousOn v [[a, b]]
huu' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt u (u' x) (Ioi x) x
hvv' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt v (v' x) (Ioi x) x
hu' : IntervalIntegrable u' volume a b
hv' : IntervalIntegrable v' volume a b
⊢ ∫ (x : ℝ) in a..b, u x • v' x = u b • v b - u a • v a - ∫ (x : ℝ) in a..b, u' x • v x

Tactic:
rw [← integral_deriv_smul_eq_sub_of_hasDeriv_right hu hv huu' hvv' hu' hv', ← integral_sub]

Goal state after:
a b : ℝ
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra ℝ 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : IsScalarTower ℝ 𝕜 E
u u' : ℝ → 𝕜
v v' : ℝ → E
hu : ContinuousOn u [[a, b]]
hv : ContinuousOn v [[a, b]]
huu' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt u (u' x) (Ioi x) x
hvv' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt v (v' x) (Ioi x) x
hu' : IntervalIntegrable u' volume a b
hv' : IntervalIntegrable v' volume a b
⊢ ∫ (x : ℝ) in a..b, u x • v' x = ∫ (x : ℝ) in a..b, u' x • v x + u x • v' x - u' x • v x

case hf
a b : ℝ
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra ℝ 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : IsScalarTower ℝ 𝕜 E
u u' : ℝ → 𝕜
v v' : ℝ → E
hu : ContinuousOn u [[a, b]]
hv : ContinuousOn v [[a, b]]
huu' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt u (u' x) (Ioi x) x
hvv' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt v (v' x) (Ioi x) x
hu' : IntervalIntegrable u' volume a b
hv' : IntervalIntegrable v' volume a b
⊢ IntervalIntegrable (fun x ↦ u' x • v x + u x • v' x) volume a b

case hg
a b : ℝ
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra ℝ 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : IsScalarTower ℝ 𝕜 E
u u' : ℝ → 𝕜
v v' : ℝ → E
hu : ContinuousOn u [[a, b]]
hv : ContinuousOn v [[a, b]]
huu' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt u (u' x) (Ioi x) x
hvv' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt v (v' x) (Ioi x) x
hu' : IntervalIntegrable u' volume a b
hv' : IntervalIntegrable v' volume a b
⊢ IntervalIntegrable (fun x ↦ u' x • v x) volume a b
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : SecondCountableTopology F
f : E → F
s : Set E
f' : E → E →L[ℝ] F
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
r : (E →L[ℝ] F) → ℝ≥0
rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0
hs : s.Nonempty
T : Set ↑s
T_count : T.Countable
hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))
u : ℕ → ℝ
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
M : ℕ → ↑T → Set E :=
  fun n z ↦ {x | x ∈ s ∧ ∀ y ∈ s ∩ ball x (u n), ‖f y - f x - (f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}
x : E
xs : x ∈ s
⊢ f' x ∈ ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))

Tactic:
refine mem_iUnion.2 ⟨⟨x, xs⟩, ?_⟩

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : SecondCountableTopology F
f : E → F
s : Set E
f' : E → E →L[ℝ] F
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
r : (E →L[ℝ] F) → ℝ≥0
rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0
hs : s.Nonempty
T : Set ↑s
T_count : T.Countable
hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))
u : ℕ → ℝ
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
M : ℕ → ↑T → Set E :=
  fun n z ↦ {x | x ∈ s ∧ ∀ y ∈ s ∩ ball x (u n), ‖f y - f x - (f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}
x : E
xs : x ∈ s
⊢ f' x ∈ ball (f' ↑⟨x, xs⟩) ↑(r (f' ↑⟨x, xs⟩))
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) [[a, b]] volume
hE : CompleteSpace E
h_cont : ContinuousOn (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
x : ℝ
hx : x ∈ Ioo (min a b) (max a b)
c : ℝ
hc : c ∈ Ioo (min a b) x
⊢ HasDerivWithinAt (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) (f' x • (g ∘ f) x) (Ioi x) x

Tactic:
obtain ⟨d, hd⟩ := nonempty_Ioo.mpr hx.2

Goal state after:
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), HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) [[a, b]] volume
hE : CompleteSpace E
h_cont : ContinuousOn (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
x : ℝ
hx : x ∈ Ioo (min a b) (max a b)
c : ℝ
hc : c ∈ Ioo (min a b) x
d : ℝ
hd : d ∈ Ioo x (max a b)
⊢ HasDerivWithinAt (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) (f' x • (g ∘ f) x) (Ioi x) x
Module: Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts

Goal state before:
pp: case e'_12
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
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
L_emb : MeasurableEmbedding ⇑L
x : E' × ℝ
hf : HasLineDerivAt ℝ ((f ∘ ⇑L.symm) ∘ ⇑↑↑L) (f' (L.symm x)) (L.symm x) v
hx : x ∈ tsupport fun x ↦ g (L.symm x)
this : f = (f ∘ ⇑L.symm) ∘ ⇑↑↑L
h2x : L.symm x ∈ tsupport g
⊢ (0, 1) = ↑↑L v

Tactic:
simp [← hL]

Goal state after:
Module: Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts

Goal state before:
pp: case e'_12
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
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
L_emb : MeasurableEmbedding ⇑L
x : E' × ℝ
hg : HasLineDerivAt ℝ ((g ∘ ⇑L.symm) ∘ ⇑↑↑L) (g' (L.symm x)) (L.symm x) v
hx : x ∈ tsupport fun x ↦ f (L.symm x)
this : g = (g ∘ ⇑L.symm) ∘ ⇑↑↑L
h2x : L.symm x ∈ tsupport f
⊢ (0, 1) = ↑↑L v

Tactic:
simp [← hL]

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

Goal state before:
pp: A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ tsupport v, HasDerivAt u (u' x) x
hv : ∀ x ∈ tsupport u, HasDerivAt v (v' x) x
huv : Integrable (u' * v + u * v') volume
h_bot : Tendsto (u * v) atBot (𝓝 a')
h_top : Tendsto (u * v) atTop (𝓝 b')
⊢ ∫ (x : ℝ), u' x * v x + u x * v' x = b' - a'

Tactic:
refine integral_of_hasDerivAt_of_tendsto (fun x ↦ ?_) huv h_bot h_top
  simpa [add_comm] using! (ContinuousLinearMap.mul ℝ A).hasDerivAt_of_bilinear (hu x) (hv x)

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

Goal state before:
pp: case h
α : Type u_1
β : Type u_2
ι : Type u_3
inst✝² : CommMonoid α
s : Set β
inst✝¹ : UniformSpace α
inst✝ : TopologicalSpace β
f f' : ι → β → α
h : ∀ (i : ι), Set.EqOn (f i) (f' i) s
h2 : MultipliableLocallyUniformlyOn f s
⊢ HasProdLocallyUniformlyOn f' ?g s

pp: case g
α : Type u_1
β : Type u_2
ι : Type u_3
inst✝² : CommMonoid α
s : Set β
inst✝¹ : UniformSpace α
inst✝ : TopologicalSpace β
f f' : ι → β → α
h : ∀ (i : ι), Set.EqOn (f i) (f' i) s
h2 : MultipliableLocallyUniformlyOn f s
⊢ β → α

Tactic:
exact (h2.hasProdLocallyUniformlyOn).congr fun v ↦ eqOn_fun_finsetProd h v

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
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
h'f' : ∀ x ∈ s, (f' x).det = 0
H : ∀ (R : ℝ), μ (f '' (s ∩ closedBall 0 R)) = 0
⊢ ∑' (n : ℕ), μ (f '' (s ∩ closedBall 0 ↑n)) ≤ 0

Tactic:
simp only [H, tsum_zero, nonpos_iff_eq_zero]

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

Goal state before:
pp: ι : Type u_1
s : Set ℂ
hs : IsOpen s
x : ℂ
hx : x ∈ s
f : ι → ℂ → ℂ
hf : ∀ (i : ι), f i x ≠ 0
hd : ∀ (i : ι), DifferentiableOn ℂ (f i) s
hm : Summable fun i ↦ logDeriv (f i) x
htend : MultipliableLocallyUniformlyOn f s
hnez : ∏' (i : ι), f i x ≠ 0
⊢ logDeriv (fun x ↦ ∏' (i : ι), f i x) x = ∑' (i : ι), logDeriv (f i) x

Tactic:
by
  rw [Eq.comm, ← hm.hasSum_iff]
  refine logDeriv_tendsto hs hx htend.hasProdLocallyUniformlyOn (.of_forall <| by fun_prop) hnez
    |>.congr fun b ↦ ?_
  rw [logDeriv_prod (fun i _ ↦ hf i) (fun i _ ↦ (hd i x hx).differentiableAt (hs.mem_nhds hx))]

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

Goal state before:
pp: H : Type u
M : Type u_2
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SecondCountableTopology H
s : Set M
hs : ⋃ x ∈ s, (chartAt H x).source = univ
hsc : s.Countable
⊢ SecondCountableTopology M

Tactic:
have : ∀ x : M, SecondCountableTopology (chartAt H x).source :=
    fun x ↦ (chartAt (H := H) x).secondCountableTopology_source

Goal state after:
H : Type u
M : Type u_2
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SecondCountableTopology H
s : Set M
hs : ⋃ x ∈ s, (chartAt H x).source = univ
hsc : s.Countable
this : ∀ (x : M), SecondCountableTopology ↑(chartAt H x).source
⊢ SecondCountableTopology M
Module: Mathlib.Geometry.Manifold.ChartedSpace

Goal state before:
pp: case inl
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'
h : Nonempty H
e : OpenPartialHomeomorph (M ⊕ M') H
x : OpenPartialHomeomorph M H
hx : x ∈ ChartedSpace.atlas
hxe : (fun e ↦ e.lift_openEmbedding ⋯) x = e
⊢ ∃ f ∈ atlas H M, (fun e ↦ e.lift_openEmbedding ⋯) x = f.lift_openEmbedding ⋯

Tactic:
use x

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

Goal state before:
pp: case inr
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'
h : Nonempty H
e : OpenPartialHomeomorph (M ⊕ M') H
x : OpenPartialHomeomorph M' H
hx : x ∈ ChartedSpace.atlas
hxe : (fun e ↦ e.lift_openEmbedding ⋯) x = e
⊢ ∃ f' ∈ atlas H M', (fun e ↦ e.lift_openEmbedding ⋯) x = f'.lift_openEmbedding ⋯

Tactic:
use x

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
hs'' : IsOpen[inst✝³] (↑e' '' (e'.source ∩ s))
⊢ IsOpen[inst✝³] (↑e' '' (e'.source ∩ s))

Tactic:
exact isOpen_image_source_inter e' hs

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

Goal state before:
pp: H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
G : StructureGroupoid H
inst✝ : HasGroupoid M G
s : Opens M
x : ↥s
⊢ ∃ i, (chartAt H ↑x).subtypeRestr ⋯ = (chartAt H ↑i).subtypeRestr ⋯

Tactic:
use x

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

Goal state before:
pp: case inr
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 = ∅}
h : e.source.Nonempty
⊢ e ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}

Tactic:
left
      rcases h with ⟨x, hx⟩
      rcases he x hx with ⟨s, open_s, xs, hs⟩
      have x's : x ∈ (e.restr s).source := by
        rw [restr_source, open_s.interior_eq]
        exact ⟨hx, xs⟩
      rcases hs with hs | hs
      · replace hs : OpenPartialHomeomorph.restr e s = OpenPartialHomeomorph.refl H := by
          simpa only using! hs
        have : (e.restr s).source = univ := by
          rw [hs]
          simp
        have : e.toPartialEquiv.source ∩ interior s = univ := this
        have : univ ⊆ interior s := by
          rw [← this]
          exact inter_subset_right
        have : s = univ := by rwa [open_s.interior_eq, univ_subset_iff] at this
        simpa only [this, restr_univ] using! hs
      · exfalso
        rw [mem_ofPred_eq] at hs
        rwa [hs] at x's

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

Goal state before:
pp: H : Type u_1
H' : Type u_3
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
hG : G.LocalInvariantProp G' P
s t : Set H
x : H
f : H → H'
hu : s =ᶠ[𝓝 x] t
⊢ P f s x ↔ P f t x

Tactic:
obtain ⟨o, host, ho, hxo⟩ := mem_nhds_iff.mp hu.mem_iff
  simp_rw [subset_def, mem_ofPred, ← and_congr_left_iff, ← mem_inter_iff, ← Set.ext_iff] at host
  rw [hG.is_local ho hxo, host, ← hG.is_local ho hxo]

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
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
inst✝ : I.Boundaryless
x : M
hx : x ∈ f.source
⊢ map (↑(f.extend I)) (𝓝 x) = 𝓝 (↑(f.extend I) x)

Tactic:
by
  rw [f.map_extend_nhds hx, I.range_eq_univ, nhdsWithin_univ]

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

Goal state before:
pp: case e'_5
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
hv : v ∈ range ↑I
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • u = a • u

Tactic:
rw [← @algebraMap_smul (R := ℝ) (A := 𝕜)]

Goal state after:
case e'_5
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
hv : v ∈ range ↑I
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ (algebraMap ℝ 𝕜) a • u = a • u
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: case e'_6
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
hv : v ∈ range ↑I
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ b • v = b • v

Tactic:
rw [← @algebraMap_smul (R := ℝ) (A := 𝕜)]

Goal state after:
case e'_6
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
hv : v ∈ range ↑I
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ (algebraMap ℝ 𝕜) b • v = b • v
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: case refine_1
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
X : Type u_4
inst✝ : TopologicalSpace X
f : H → X
s : Set H
x : H
h : ContinuousWithinAt (f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)
this : ContinuousWithinAt ((f ∘ ↑I.symm) ∘ ↑I) (↑I ⁻¹' (↑I.symm ⁻¹' s ∩ range ↑I)) x
⊢ ContinuousWithinAt f s x

Tactic:
simp_rw [preimage_inter, preimage_preimage, I.left_inv, preimage_id', preimage_range,
      inter_univ] at this

Goal state after:
case refine_1
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
X : Type u_4
inst✝ : TopologicalSpace X
f : H → X
s : Set H
x : H
h : ContinuousWithinAt (f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)
this : ContinuousWithinAt ((f ∘ ↑I.symm) ∘ ↑I) s x
⊢ ContinuousWithinAt f s x
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: 𝕜 : Type u
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
H : Type w
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
inst✝⁴ : I.Boundaryless
E' : Type v'
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace 𝕜 E'
H' : Type w'
inst✝¹ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
inst✝ : I'.Boundaryless
⊢ (I.prod I').Boundaryless

Tactic:
constructor
  dsimp
  rw [Set.range_prodMap, ModelWithCorners.Boundaryless.range_eq_univ,
    ModelWithCorners.Boundaryless.range_eq_univ, univ_prod_univ]

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

Goal state before:
pp: case refine_2
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
⊢ P (↑(chartAt H' (f (inclusion hUV x))) ∘ f ∘ ↑(chartAt H (inclusion hUV x)).symm)
      (↑(chartAt H (inclusion hUV x)).symm ⁻¹' univ) (↑(chartAt H (inclusion hUV x)) (inclusion hUV x)) ↔
    P (↑(chartAt H' ((f ∘ inclusion hUV) x)) ∘ (f ∘ inclusion hUV) ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' univ)
      (↑(chartAt H x) x)

Tactic:
· apply hG.congr_iff
    exact (TopologicalSpace.Opens.chartAt_inclusion_symm_eventuallyEq hUV).fun_comp
      (chartAt H' (f (Set.inclusion hUV x)) ∘ f)

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

Goal state before:
pp: case mp
H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
s : Set H
x : H
u : Set H
f : H → H
hu : IsOpen[inst✝¹] u
hux : x ∈ u
h : G.IsLocalStructomorphWithinAt f s x
hx : x ∈ s ∩ u
e : OpenPartialHomeomorph H H
heG : e ∈ G
hef : EqOn f (↑e.toPartialEquiv) (s ∩ e.source)
hex : x ∈ e.source
⊢ ∃ e ∈ G, EqOn f (↑e.toPartialEquiv) (s ∩ u ∩ e.source) ∧ x ∈ e.source

Tactic:
have : s ∩ u ∩ e.source ⊆ s ∩ e.source := by mfld_set_tac

Goal state after:
case mp
H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
s : Set H
x : H
u : Set H
f : H → H
hu : IsOpen[inst✝¹] u
hux : x ∈ u
h : G.IsLocalStructomorphWithinAt f s x
hx : x ∈ s ∩ u
e : OpenPartialHomeomorph H H
heG : e ∈ G
hef : EqOn f (↑e.toPartialEquiv) (s ∩ e.source)
hex : x ∈ e.source
this : s ∩ u ∩ e.source ⊆ s ∩ e.source
⊢ ∃ e ∈ G, EqOn f (↑e.toPartialEquiv) (s ∩ u ∩ e.source) ∧ x ∈ e.source