mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Euclidean.Sphere.SecondInter

Goal state before:
pp: case refine_1
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p : P
hp : p ∈ s
v : V
p' : P
hp' : p' ∈ AffineSubspace.mk' p (ℝ ∙ v)
h : p' = p ∨ p' = s.secondInter p v
⊢ p' ∈ s

pp: case refine_2
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p : P
hp : p ∈ s
v : V
p' : P
hp' : p' ∈ AffineSubspace.mk' p (ℝ ∙ v)
h : p' ∈ s
⊢ p' = p ∨ p' = s.secondInter p v

Tactic:
· rcases h with (h | h)
    · rwa [h]
    · rwa [h, Sphere.secondInter_mem]

Goal state after:
case refine_2
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p : P
hp : p ∈ s
v : V
p' : P
hp' : p' ∈ AffineSubspace.mk' p (ℝ ∙ v)
h : p' ∈ s
⊢ p' = p ∨ p' = s.secondInter p v
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
i₁ i₂ : Fin 3
p : P
h₁₂ : i₁ ≠ i₂
i₃ : Fin 3
h₂ : p ∈ mongePlane t i₃ i₁
h₁ : p ∈ mongePlane t i₃ i₂
h₂₃ : i₂ ≠ i₃
h₁₃ : i₁ ≠ i₃
i : Fin 3
hi : i₃ ≠ i
⊢ i₁ = i ∨ i₂ = i

Tactic:
lia

Goal state after:
Module: Mathlib.Geometry.Group.Growth.LinearLowerBound

Goal state before:
pp: case e_a
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
X : Finset G
hX₁ : 1 ∈ X
hX : X.Nontrivial
n : ℕ
hn : n ∈ {n | ↑X ^ (n - 1) ≠ ↑(closure ↑X)}
m : ℕ
hmn : m ≤ n
hm : ↑X ^ (m - 1) = ↑(closure ↑X)
hm₀ : m > 0
⊢ n - 1 = n - m + (m - 1)

Tactic:
lia

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
x : M
h : I.IsInteriorPoint x
⊢ range ↑I ∈ 𝓝 (↑(extChartAt I x) x)

Tactic:
rw [mem_nhds_iff]

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
x : M
h : I.IsInteriorPoint x
⊢ ∃ t ⊆ range ↑I, IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t ∧ ↑(extChartAt I x) x ∈ t
Module: Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary

Goal state before:
pp: 𝕜✝ : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜✝
E✝ : Type u_2
inst✝¹¹ : NormedAddCommGroup E✝
inst✝¹⁰ : NormedSpace 𝕜✝ E✝
H✝ : Type u_3
inst✝⁹ : TopologicalSpace H✝
I✝ : ModelWithCorners 𝕜✝ E✝ H✝
M✝ : Type u_4
inst✝⁸ : TopologicalSpace M✝
inst✝⁷ : ChartedSpace H✝ M✝
n✝ : WithTop ℕ∞
e✝ e'✝ : OpenPartialHomeomorph M✝ H✝
x✝² : M✝
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
n : WithTop ℕ∞
inst✝ : IsManifold I n M
e e' : OpenPartialHomeomorph M H
x : M
hn : n ≠ 0
he : e ∈ atlas H M
he' : e' ∈ atlas H M
hex : x ∈ e.source
hex' : x ∈ e'.source
hx : ↑(e.extend I) x ∈ interior (e.extend I).target
φ : PartialEquiv E E := extendCoordChange e e'
hφ : ContDiffOn 𝕜 n (↑φ) φ.source
hφx' : Surjective ⇑(fderivWithin 𝕜 (↑φ) φ.source (↑(e.extend I) x))
h✝ : IsRCLikeNormedField 𝕜
x✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
x✝ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
hφx : φ.source ∈ 𝓝 (↑(e.extend I) x)
⊢ ↑(e'.extend I) x ∈ interior (range ↑I)

Tactic:
← ContinuousLinearMap.coe_restrictScalars' (R := ℝ),

Goal state after:
𝕜✝ : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜✝
E✝ : Type u_2
inst✝¹¹ : NormedAddCommGroup E✝
inst✝¹⁰ : NormedSpace 𝕜✝ E✝
H✝ : Type u_3
inst✝⁹ : TopologicalSpace H✝
I✝ : ModelWithCorners 𝕜✝ E✝ H✝
M✝ : Type u_4
inst✝⁸ : TopologicalSpace M✝
inst✝⁷ : ChartedSpace H✝ M✝
n✝ : WithTop ℕ∞
e✝ e'✝ : OpenPartialHomeomorph M✝ H✝
x✝² : M✝
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
n : WithTop ℕ∞
inst✝ : IsManifold I n M
e e' : OpenPartialHomeomorph M H
x : M
hn : n ≠ 0
he : e ∈ atlas H M
he' : e' ∈ atlas H M
hex : x ∈ e.source
hex' : x ∈ e'.source
hx : ↑(e.extend I) x ∈ interior (e.extend I).target
φ : PartialEquiv E E := extendCoordChange e e'
hφ : ContDiffOn 𝕜 n (↑φ) φ.source
h✝ : IsRCLikeNormedField 𝕜
x✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
x✝ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
hφx' : Surjective ⇑(ContinuousLinearMap.restrictScalars ℝ (fderivWithin 𝕜 (↑φ) φ.source (↑(e.extend I) x)))
hφx : φ.source ∈ 𝓝 (↑(e.extend I) x)
⊢ ↑(e'.extend I) x ∈ interior (range ↑I)
Module: Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary

Goal state before:
pp: 𝕜✝ : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜✝
E✝ : Type u_2
inst✝¹¹ : NormedAddCommGroup E✝
inst✝¹⁰ : NormedSpace 𝕜✝ E✝
H✝ : Type u_3
inst✝⁹ : TopologicalSpace H✝
I✝ : ModelWithCorners 𝕜✝ E✝ H✝
M✝ : Type u_4
inst✝⁸ : TopologicalSpace M✝
inst✝⁷ : ChartedSpace H✝ M✝
n✝ : WithTop ℕ∞
e✝ e'✝ : OpenPartialHomeomorph M✝ H✝
x✝² : M✝
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
n : WithTop ℕ∞
inst✝ : IsManifold I n M
e e' : OpenPartialHomeomorph M H
x : M
hn : n ≠ 0
he : e ∈ atlas H M
he' : e' ∈ atlas H M
hex : x ∈ e.source
hex' : x ∈ e'.source
hx : ↑(e.extend I) x ∈ interior (e.extend I).target
φ : PartialEquiv E E := extendCoordChange e e'
hφ : ContDiffOn 𝕜 n (↑φ) φ.source
hφx' : Surjective ⇑(fderivWithin 𝕜 (↑φ) φ.source (↑(e.extend I) x))
h✝ : IsRCLikeNormedField 𝕜
x✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
x✝ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
hφx : φ.source ∈ 𝓝 (↑(e.extend I) x)
⊢ ↑(e'.extend I) x ∈ interior (range ↑I)

Tactic:
rw [← ContinuousLinearMap.coe_restrictScalars' (R := ℝ),
    (hφ.differentiableOn hn _ (by simp [φ, hex, hex'])).restrictScalars_fderivWithin (𝕜 := ℝ)
      (uniqueDiffWithinAt_of_mem_nhds hφx), fderivWithin_of_mem_nhds <| hφx] at hφx'

Goal state after:
𝕜✝ : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜✝
E✝ : Type u_2
inst✝¹¹ : NormedAddCommGroup E✝
inst✝¹⁰ : NormedSpace 𝕜✝ E✝
H✝ : Type u_3
inst✝⁹ : TopologicalSpace H✝
I✝ : ModelWithCorners 𝕜✝ E✝ H✝
M✝ : Type u_4
inst✝⁸ : TopologicalSpace M✝
inst✝⁷ : ChartedSpace H✝ M✝
n✝ : WithTop ℕ∞
e✝ e'✝ : OpenPartialHomeomorph M✝ H✝
x✝² : M✝
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
n : WithTop ℕ∞
inst✝ : IsManifold I n M
e e' : OpenPartialHomeomorph M H
x : M
hn : n ≠ 0
he : e ∈ atlas H M
he' : e' ∈ atlas H M
hex : x ∈ e.source
hex' : x ∈ e'.source
hx : ↑(e.extend I) x ∈ interior (e.extend I).target
φ : PartialEquiv E E := extendCoordChange e e'
hφ : ContDiffOn 𝕜 n (↑φ) φ.source
h✝ : IsRCLikeNormedField 𝕜
x✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
x✝ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
hφx' : Surjective ⇑(fderiv ℝ (↑φ) (↑(e.extend I) x))
hφx : φ.source ∈ 𝓝 (↑(e.extend I) x)
⊢ ↑(e'.extend I) x ∈ interior (range ↑I)
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Set P
t t₀ : Triangle ℝ P
ht : Set.range t.points ⊆ insert t₀.orthocenter (Set.range t₀.points)
ht₀o : t₀.orthocenter ∉ Set.range t₀.points
ht₀s : s = insert t₀.orthocenter (Set.range t₀.points)
i₁ i₂ i₃ j₂ j₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h₁ : t.points i₁ = t₀.orthocenter
hj₂₃ : j₂ ≠ j₃
h₂ : t₀.points j₂ = t.points i₂
h₃ : t₀.points j₃ = t.points i₃
j₁ : Fin 3
hj₁₂ : j₁ ≠ j₂
hj₁₃ : j₁ ≠ j₃
hj₁₂₃ : ∀ (j : Fin 3), j = j₁ ∨ j = j₂ ∨ j = j₃
h : t₀.points j₁ = t.orthocenter
hui : Set.univ = {i₁, i₂, i₃}
huj : Set.univ = {j₁, j₂, j₃}
⊢ insert t₀.orthocenter (t₀.points '' {j₁, j₂, j₃}) = insert (t₀.points j₁) (t.points '' {i₁, i₂, i₃})

Tactic:
simp_rw [Set.image_insert_eq, Set.image_singleton, h₁, ← h₂, ← h₃]

Goal state after:
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Set P
t t₀ : Triangle ℝ P
ht : Set.range t.points ⊆ insert t₀.orthocenter (Set.range t₀.points)
ht₀o : t₀.orthocenter ∉ Set.range t₀.points
ht₀s : s = insert t₀.orthocenter (Set.range t₀.points)
i₁ i₂ i₃ j₂ j₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h₁ : t.points i₁ = t₀.orthocenter
hj₂₃ : j₂ ≠ j₃
h₂ : t₀.points j₂ = t.points i₂
h₃ : t₀.points j₃ = t.points i₃
j₁ : Fin 3
hj₁₂ : j₁ ≠ j₂
hj₁₃ : j₁ ≠ j₃
hj₁₂₃ : ∀ (j : Fin 3), j = j₁ ∨ j = j₂ ∨ j = j₃
h : t₀.points j₁ = t.orthocenter
hui : Set.univ = {i₁, i₂, i₃}
huj : Set.univ = {j₁, j₂, j₃}
⊢ {t₀.orthocenter, t₀.points j₁, t₀.points j₂, t₀.points j₃} =
    {t₀.points j₁, t₀.orthocenter, t₀.points j₂, t₀.points j₃}
Module: Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable

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
F : Type u_5
inst✝⁵ : NormedAddCommGroup F
V : M → Type u_6
inst✝⁴ : TopologicalSpace (TotalSpace F V)
inst✝³ : (x : M) → AddCommGroup (V x)
inst✝² : (x : M) → TopologicalSpace (V x)
inst✝¹ : FiberBundle F V
inst✝ : NormedSpace 𝕜 F
k : WithTop ℕ∞
x : M
σ₀ : V x
t : Trivialization F TotalSpace.proj := trivializationAt F V x
w : F := (↑t ⟨x, σ₀⟩).2
this : ContMDiffAt I 𝓘(𝕜, F) k (fun _x ↦ w) x
⊢ (fun x_1 ↦ (↑t ⟨x_1, extend F σ₀ x_1⟩).2) =ᶠ[𝓝 x] fun _x ↦ w

Tactic:
apply eventually_nhds_iff.mpr

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
F : Type u_5
inst✝⁵ : NormedAddCommGroup F
V : M → Type u_6
inst✝⁴ : TopologicalSpace (TotalSpace F V)
inst✝³ : (x : M) → AddCommGroup (V x)
inst✝² : (x : M) → TopologicalSpace (V x)
inst✝¹ : FiberBundle F V
inst✝ : NormedSpace 𝕜 F
k : WithTop ℕ∞
x : M
σ₀ : V x
t : Trivialization F TotalSpace.proj := ⋯
w : F := ⋯
this : ContMDiffAt I 𝓘(𝕜, F) k (fun _x ↦ w) x
⊢ ∃ t_1, (∀ y ∈ t_1, (fun x_1 ↦ (↑t ⟨x_1, extend F σ₀ x_1⟩).2) y = (fun _x ↦ w) y) ∧ IsOpen[inst✝⁷] t_1 ∧ x ∈ t_1
Module: Mathlib.Geometry.Manifold.ContMDiffMFDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
m n : WithTop ℕ∞
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : ChartedSpace H' M'
F : Type u_8
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_9
inst✝² : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_10
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
Js : IsManifold J 1 N
Is : IsManifold I 1 M
I's : IsManifold I' 1 M'
x₀ : N
f : N → M → M'
g : N → M
t : Set N
u : Set M
hf : ContMDiffWithinAt (J.prod I) I' n (uncurry f) (t ×ˢ u) (x₀, g x₀)
hg : ContMDiffWithinAt J I m g t x₀
hx₀ : x₀ ∈ t
hu : MapsTo g t u
hmn : m + 1 ≤ n
h'u : UniqueMDiff[u]
t' : Set N := t ∩ g ⁻¹' (extChartAt I (g x₀)).source
ht't : t' ⊆ t
hx₀gx₀ : (x₀, g x₀) ∈ t ×ˢ u
h4f✝ : ContinuousWithinAt (fun x ↦ f x (g x)) t x₀
h4f : (fun x ↦ f x (g x)) ⁻¹' (extChartAt I' (f x₀ (g x₀))).source ∈ 𝓝[t] x₀
h3f :
  ∀ᶠ (x' : N × M) in 𝓝[insert (x₀, g x₀) (t ×ˢ u)] (x₀, g x₀), ContMDiffWithinAt (J.prod I) I' 1 (uncurry f) (t ×ˢ u) x'
⊢ ContMDiffWithinAt J 𝓘(𝕜, E →L[𝕜] E') m
    (inTangentCoordinates I I' g (fun x ↦ f x (g x)) (fun x ↦ mfderiv[u] (f x) (g x)) x₀) t' x₀

Tactic:
simp only [hx₀gx₀, insert_eq_of_mem] at h3f

Goal state after:
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
m n : WithTop ℕ∞
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : ChartedSpace H' M'
F : Type u_8
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_9
inst✝² : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_10
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
Js : IsManifold J 1 N
Is : IsManifold I 1 M
I's : IsManifold I' 1 M'
x₀ : N
f : N → M → M'
g : N → M
t : Set N
u : Set M
hf : ContMDiffWithinAt (J.prod I) I' n (uncurry f) (t ×ˢ u) (x₀, g x₀)
hg : ContMDiffWithinAt J I m g t x₀
hx₀ : x₀ ∈ t
hu : MapsTo g t u
hmn : m + 1 ≤ n
h'u : UniqueMDiff[u]
t' : Set N := t ∩ g ⁻¹' (extChartAt I (g x₀)).source
ht't : t' ⊆ t
hx₀gx₀ : (x₀, g x₀) ∈ t ×ˢ u
h4f✝ : ContinuousWithinAt (fun x ↦ f x (g x)) t x₀
h4f : (fun x ↦ f x (g x)) ⁻¹' (extChartAt I' (f x₀ (g x₀))).source ∈ 𝓝[t] x₀
h3f : ∀ᶠ (x' : N × M) in 𝓝[t ×ˢ u] (x₀, g x₀), ContMDiffWithinAt (J.prod I) I' 1 (uncurry f) (t ×ˢ u) x'
⊢ ContMDiffWithinAt J 𝓘(𝕜, E →L[𝕜] E') m
    (inTangentCoordinates I I' g (fun x ↦ f x (g x)) (fun x ↦ mfderiv[u] (f x) (g x)) x₀) t' x₀
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁴ : TopologicalSpace H
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
s : Set M
x : M
V W : (x : M) → TangentSpace I x
⊢ mlieBracketWithin I V W s x = -mlieBracketWithin I W V s x

Tactic:
rw [mlieBracketWithin, lieBracketWithin_swap, mpullback_neg]
  rfl

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

Goal state before:
pp: case hf
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁷ : TopologicalSpace H
E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝⁴ : TopologicalSpace G
inst✝³ : ChartedSpace H G
inst✝² : Group G
inst✝¹ : LieGroup I (minSmoothness 𝕜 3) G
inst✝ : CompleteSpace E
v w : GroupLieAlgebra I G
g : G
⊢ ContMDiffAt I I (minSmoothness 𝕜 3) (fun x ↦ g⁻¹ * x) g

pp: case hn
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁷ : TopologicalSpace H
E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝⁴ : TopologicalSpace G
inst✝³ : ChartedSpace H G
inst✝² : Group G
inst✝¹ : LieGroup I (minSmoothness 𝕜 3) G
inst✝ : CompleteSpace E
v w : GroupLieAlgebra I G
g : G
⊢ minSmoothness 𝕜 2 ≤ minSmoothness 𝕜 3

Tactic:
· exact contMDiffAt_mul_left

Goal state after:
case hn
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁷ : TopologicalSpace H
E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝⁴ : TopologicalSpace G
inst✝³ : ChartedSpace H G
inst✝² : Group G
inst✝¹ : LieGroup I (minSmoothness 𝕜 3) G
inst✝ : CompleteSpace E
v w : GroupLieAlgebra I G
g : G
⊢ minSmoothness 𝕜 2 ≤ minSmoothness 𝕜 3
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: x y : ℝ
h : Fact (x < y)
⊢ ∀ ⦃x_1 : EuclideanHalfSpace 1⦄,
    x_1 ∈ {z | (↑z).ofLp 0 < y - x} → ⟨toLp 2 fun x_2 ↦ ↑⟨min ((↑x_1).ofLp 0 + x) y, ⋯⟩ - x, ⋯⟩ = x_1

Tactic:
rintro ⟨z, hz⟩ h'z
    rw [Subtype.mk_eq_mk]
    ext i
    dsimp at hz h'z
    have A : x + z 0 ≤ y := by linarith
    rw [Subsingleton.elim i 0]
    simp only [Fin.isValue, add_comm, A, inf_of_le_left, add_sub_cancel_left]

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: x✝ y✝ : ℝ
hxy : Fact (x✝ < y✝)
x y : ℝ
h : Fact (x < y)
z : ↑(Icc x y)
⊢ (if ↑z < y then IccLeftChart x y else IccRightChart x y) ∈ {IccLeftChart x y, IccRightChart x y}

Tactic:
by_cases h' : (z : ℝ) < y

Goal state after:
case pos
x✝ y✝ : ℝ
hxy : Fact (x✝ < y✝)
x y : ℝ
h : Fact (x < y)
z : ↑(Icc x y)
h' : ↑z < y
⊢ (if ↑z < y then IccLeftChart x y else IccRightChart x y) ∈ {IccLeftChart x y, IccRightChart x y}

case neg
x✝ y✝ : ℝ
hxy : Fact (x✝ < y✝)
x y : ℝ
h : Fact (x < y)
z : ↑(Icc x y)
h' : ¬↑z < y
⊢ (if ↑z < y then IccLeftChart x y else IccRightChart x y) ∈ {IccLeftChart x y, IccRightChart x y}
Module: Mathlib.Geometry.Manifold.Immersion

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
E'' : Type u
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup E''
inst✝⁶ : NormedSpace 𝕜 E''
H : Type u_7
inst✝⁵ : TopologicalSpace H
G : Type u_9
inst✝⁴ : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 E'' G
M : Type u_11
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
N : Type u_13
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n : ℕ∞ω
f : M → N
x : M
hx : x ∈ {x | IsImmersionAt I J n f x}
⊢ x ∈ {x_1 | IsImmersionAtOfComplement (complement hx) I J n f x_1}

Tactic:
by simp [hx.isImmersionAtOfComplement_complement]

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

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_7
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_11
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
n : ℕ∞ω
x : M
inst✝ : IsManifold I n M
s : TopologicalSpace.Opens M
hx : x ∈ s
⊢ IsImmersionAt I I n Subtype.val ⟨x, hx⟩

Tactic:
use PUnit, by infer_instance, by infer_instance
  apply Manifold.IsImmersionAtOfComplement.of_opens

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

Goal state before:
pp: case mpr
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
H : Type u_2
inst✝² : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
γ : ℝ → M
v : (x : M) → TangentSpace I x
t₀ : ℝ
s : Set ℝ
hs : s ∈ 𝓝 t₀
h : IsMIntegralCurveOn γ v s
s' : Set ℝ
h1 : s' ⊆ s
h2 : IsOpen s'
h3 : t₀ ∈ s'
t : ℝ
ht : t ∈ s'
⊢ s ∈ 𝓝 t

Tactic:
rw [mem_nhds_iff]

Goal state after:
case mpr
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
H : Type u_2
inst✝² : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
γ : ℝ → M
v : (x : M) → TangentSpace I x
t₀ : ℝ
s : Set ℝ
hs : s ∈ 𝓝 t₀
h : IsMIntegralCurveOn γ v s
s' : Set ℝ
h1 : s' ⊆ s
h2 : IsOpen s'
h3 : t₀ ∈ s'
t : ℝ
ht : t ∈ s'
⊢ ∃ t_1 ⊆ s, IsOpen t_1 ∧ t ∈ t_1
Module: Mathlib.Geometry.Manifold.Instances.Sphere

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
v : E
hv : ‖v‖ = 1
x : ↑(sphere 0 1)
hx : ↑x ≠ v
⊢ stereoInvFun hv (stereoToFun v ↑x) = x

Tactic:
by
  ext
  simp only [stereoToFun_apply, stereoInvFun_apply, smul_add]
  -- name two frequently-occurring quantities and write down their basic properties
  set a : ℝ := innerSL _ v x
  set y := (ℝ ∙ v)ᗮ.orthogonalProjectionOnto x
  have split : ↑x = a • v + ↑y := by
    rw [← ((ℝ ∙ v).starProjection_add_starProjection_orthogonal x),
      Submodule.starProjection_unit_singleton ℝ hv x]
    rfl
  have hvy : ⟪v, y⟫_ℝ = 0 := Submodule.mem_orthogonal_singleton_iff_inner_right.mp y.2
  have pythag : 1 = a ^ 2 + ‖y‖ ^ 2 := by
    have hvy' : ⟪a • v, y⟫_ℝ = 0 := by simp only [inner_smul_left, hvy, mul_zero]
    convert! norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero _ _ hvy' using 2
    · simp [← split]
    · simp [norm_smul, hv, ← sq, sq_abs]
    · exact sq _
  -- a fact which will be helpful for clearing denominators in the main calculation
  have ha : 0 < 1 - a := by
    have : a < 1 := (inner_lt_one_iff_real_of_norm_eq_one hv (by simp)).mpr hx.symm
    linarith
  rw [split, Submodule.coe_smul_of_tower]
  simp only [norm_smul, norm_div, Real.norm_eq_abs, abs_of_nonneg ha.le]
  match_scalars
  · field_simp
    linear_combination 4 * pythag
  · field_simp
    linear_combination 4 * (a - 1) * pythag

Goal state after:
Module: Mathlib.Geometry.Manifold.IntegralCurve.Transform

Goal state before:
pp: case e'_12
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
H : Type u_2
inst✝² : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
γ : ℝ → M
v : (x : M) → TangentSpace I x
s : Set ℝ
a : ℝ
ha : a ≠ 0
hγ : IsMIntegralCurveOn (γ ∘ fun x ↦ x * a) (a • v) {t | t * a ∈ s}
⊢ s = {t | t * a⁻¹ ∈ {t | t * a ∈ s}}

Tactic:
simp only [mem_ofPred_eq, mul_assoc, inv_mul_eq_div, div_self ha, mul_one, ofPred_mem_eq]

Goal state after:
Module: Mathlib.Geometry.Manifold.IntegralCurve.UniformTime

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type u_2
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I 1 M
inst✝¹ : T2Space M
inst✝ : BoundarylessManifold I M
v : (x : M) → TangentSpace I x
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
ε : ℝ
hε : 0 < ε
h : ∀ (x : M), ∃ γ, γ 0 = x ∧ IsMIntegralCurveOn γ v (Ioo (-ε) ε)
x : M
s : Set ℝ := {a | ∃ γ, γ 0 = x ∧ IsMIntegralCurveOn γ v (Ioo (-a) a)}
hbdd : BddAbove s
asup : ℝ := sSup s
hasup : asup = sSup s
a : ℝ
ha : ∃ γ, γ 0 = x ∧ IsMIntegralCurveOn γ v (Ioo (-a) a)
hlt : sSup s + -(ε / 2) < a
⊢ False

Tactic:
rw [← hasup, ← sub_eq_add_neg] at hlt

Goal state after:
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type u_2
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I 1 M
inst✝¹ : T2Space M
inst✝ : BoundarylessManifold I M
v : (x : M) → TangentSpace I x
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
ε : ℝ
hε : 0 < ε
h : ∀ (x : M), ∃ γ, γ 0 = x ∧ IsMIntegralCurveOn γ v (Ioo (-ε) ε)
x : M
s : Set ℝ := {a | ∃ γ, γ 0 = x ∧ IsMIntegralCurveOn γ v (Ioo (-a) a)}
hbdd : BddAbove s
asup : ℝ := sSup s
hasup : asup = sSup s
a : ℝ
ha : ∃ γ, γ 0 = x ∧ IsMIntegralCurveOn γ v (Ioo (-a) a)
hlt : asup - ε / 2 < a
⊢ False
Module: Mathlib.Geometry.Manifold.IntegralCurve.ExistUnique

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type u_2
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I 1 M
γ : ℝ → M
v : (x : M) → TangentSpace I x
inst✝¹ : T2Space M
inst✝ : BoundarylessManifold I M
hγ : IsMIntegralCurve γ v
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
⊢ Xor (∃ T > 0, Periodic γ T) (Injective γ)

Tactic:
rw [xor_iff_iff_not]

Goal state after:
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type u_2
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I 1 M
γ : ℝ → M
v : (x : M) → TangentSpace I x
inst✝¹ : T2Space M
inst✝ : BoundarylessManifold I M
hγ : IsMIntegralCurve γ v
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
⊢ (∃ T > 0, Periodic γ T) ↔ ¬Injective γ