mathlib-initiative/mathlib-tactics huggingface dataset:
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
n : ℕ
s : Simplex ℝ P (n + 2)
i₁ : Fin (n + 3)
p : P
h : ∀ (i₂ : Fin (n + 3)), i₁ ≠ i₂ → p ∈ s.mongePlane i₁ i₂
i₂ : Fin (n + 3)
hne : i₁ ≠ i₂
⊢ p -ᵥ s.mongePoint ∈ (ℝ ∙ (s.points i₁ -ᵥ s.points i₂))ᗮ ⊓ vectorSpan ℝ (Set.range s.points)

Tactic:
rw [← s.direction_mongePlane, vsub_right_mem_direction_iff_mem s.mongePoint_mem_mongePlane]

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
n : ℕ
s : Simplex ℝ P (n + 2)
i₁ : Fin (n + 3)
p : P
h : ∀ (i₂ : Fin (n + 3)), i₁ ≠ i₂ → p ∈ s.mongePlane i₁ i₂
i₂ : Fin (n + 3)
hne : i₁ ≠ i₂
⊢ p ∈ s.mongePlane i₁ i₂
Module: Mathlib.Geometry.Group.Growth.LinearLowerBound

Goal state before:
pp: case inl
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)}
hmn : 0 ≤ n
hm : ↑X ^ (0 - 1) = ↑(closure ↑X)
⊢ ↑X ^ (n - 1) = ↑(closure ↑X)

Tactic:
simp [eq_comm (a := (1 : Set _)), coe_set_eq_one, -Set.subset_singleton_iff,
      hX.coe.not_subset_singleton] at hm

Goal state after:
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
⊢ ↑(e'.extend I) x ∈ interior (range ↑I)

Tactic:
have hφx : φ.source ∈ 𝓝 (e.extend I x) := by
    simp_rw [φ, extendCoordChange, PartialEquiv.trans_source, PartialEquiv.symm_source,
      Filter.inter_mem_iff, mem_interior_iff_mem_nhds.1 hx, true_and, e'.extend_source]
    exact e.extend_preimage_mem_nhds hex <| e'.open_source.mem_nhds hex'

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
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)
Module: Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : NormedAddCommGroup F
inst✝¹¹ : NormedSpace 𝕜 F
inst✝¹⁰ : TopologicalSpace (TotalSpace F E)
inst✝⁹ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝⁸ : NormedAddCommGroup EB
inst✝⁷ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝⁶ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝⁵ : TopologicalSpace B
inst✝⁴ : ChartedSpace HB B
inst✝³ : FiberBundle F E
inst✝² : (x : B) → AddCommMonoid (E x)
inst✝¹ : (x : B) → Module 𝕜 (E x)
inst✝ : VectorBundle 𝕜 F E
⊢ MDiff (zeroSection F E)

Tactic:
intro x
  unfold zeroSection
  rw [mdifferentiableAt_section]
  apply (mdifferentiableAt_const (c := 0)).congr_of_eventuallyEq
  filter_upwards [(trivializationAt F E x).open_baseSet.mem_nhds
    (mem_baseSet_trivializationAt F E x)] with y hy
    using congr_arg Prod.snd <| (trivializationAt F E x).zeroSection 𝕜 hy

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

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
s t : (x : B) → E x
x₀ : B
hs : MDiffAt (T% s) x₀
ht : MDiffAt (T% t) x₀
⊢ (MDiffAt fun x ↦ ⟨x, (s + -t) x⟩) x₀

Tactic:
apply mdifferentiableAt_add_section hs <| mdifferentiableAt_neg_section ht

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁶ : TopologicalSpace H
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝³ : TopologicalSpace G
inst✝² : ChartedSpace H G
inst✝¹ : Group G
inst✝ : LieGroup I (minSmoothness 𝕜 3) G
g : G
v : GroupLieAlgebra I G
M : minSmoothness 𝕜 3 ≠ 0
h x : G
⊢ h * x = ((fun b ↦ g⁻¹ * b) ∘ fun x ↦ g * h * x) x

Tactic:
simp only [comp_apply]

Goal state after:
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁶ : TopologicalSpace H
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝³ : TopologicalSpace G
inst✝² : ChartedSpace H G
inst✝¹ : Group G
inst✝ : LieGroup I (minSmoothness 𝕜 3) G
g : G
v : GroupLieAlgebra I G
M : minSmoothness 𝕜 3 ≠ 0
h x : G
⊢ h * x = g⁻¹ * (g * h * x)
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: case hV
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁶ : TopologicalSpace H
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
x : M
V W V₁ : (x : M) → TangentSpace I x
inst✝¹ : IsManifold I 2 M
inst✝ : CompleteSpace E
hV : MDiffAt[s] (T% V) x
hV₁ : MDiffAt[s] (T% V₁) x
hs : UniqueMDiffAt[s] x
⊢ DifferentiableWithinAt 𝕜 (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V (range ↑I))
    (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x)

pp: case hV₁
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁶ : TopologicalSpace H
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
x : M
V W V₁ : (x : M) → TangentSpace I x
inst✝¹ : IsManifold I 2 M
inst✝ : CompleteSpace E
hV : MDiffAt[s] (T% V) x
hV₁ : MDiffAt[s] (T% V₁) x
hs : UniqueMDiffAt[s] x
⊢ DifferentiableWithinAt 𝕜 (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V₁ (range ↑I))
    (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x)

pp: case hs
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁶ : TopologicalSpace H
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
x : M
V W V₁ : (x : M) → TangentSpace I x
inst✝¹ : IsManifold I 2 M
inst✝ : CompleteSpace E
hV : MDiffAt[s] (T% V) x
hV₁ : MDiffAt[s] (T% V₁) x
hs : UniqueMDiffAt[s] x
⊢ UniqueDiffWithinAt 𝕜 (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x)

Tactic:
· exact hV.differentiableWithinAt_mpullbackWithin_vectorField

Goal state after:
case hV₁
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁶ : TopologicalSpace H
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
x : M
V W V₁ : (x : M) → TangentSpace I x
inst✝¹ : IsManifold I 2 M
inst✝ : CompleteSpace E
hV : MDiffAt[s] (T% V) x
hV₁ : MDiffAt[s] (T% V₁) x
hs : UniqueMDiffAt[s] x
⊢ DifferentiableWithinAt 𝕜 (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V₁ (range ↑I))
    (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x)

case hs
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁶ : TopologicalSpace H
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
x : M
V W V₁ : (x : M) → TangentSpace I x
inst✝¹ : IsManifold I 2 M
inst✝ : CompleteSpace E
hV : MDiffAt[s] (T% V) x
hV₁ : MDiffAt[s] (T% V₁) x
hs : UniqueMDiffAt[s] x
⊢ UniqueDiffWithinAt 𝕜 (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x)
Module: Mathlib.Topology.Homeomorph.TransferInstance

Goal state before:
pp: case e'_4
R : Type u_1
α : Type u_2
β : Type u_3
inst✝ : TopologicalSpace β
e : α ≃ β
this : TopologicalSpace α := e.topologicalSpace
⊢ e.topologicalSpace = TopologicalSpace.coinduced (⇑e.symm) inst✝

Tactic:
e.coinduced_symm

Goal state after:
case e'_4
R : Type u_1
α : Type u_2
β : Type u_3
inst✝ : TopologicalSpace β
e : α ≃ β
this : TopologicalSpace α := e.topologicalSpace
⊢ e.topologicalSpace = TopologicalSpace.induced (⇑e) inst✝
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: case e_6.hV
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁶ : TopologicalSpace H
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s t : Set M
x : M
V W : (x : M) → TangentSpace I x
inst✝¹ : IsManifold I 2 M
inst✝ : CompleteSpace E
st : t ∈ 𝓝[s] x
hs : UniqueMDiffAt[s] x
hV : MDiffAt[t] (T% V) x
hW : MDiffAt[t] (T% W) x
⊢ DifferentiableWithinAt 𝕜 (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V (range ↑I))
    (↑(extChartAt I x).symm ⁻¹' t ∩ range ↑I) (↑(extChartAt I x) x)

pp: case e_6.hW
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁶ : TopologicalSpace H
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s t : Set M
x : M
V W : (x : M) → TangentSpace I x
inst✝¹ : IsManifold I 2 M
inst✝ : CompleteSpace E
st : t ∈ 𝓝[s] x
hs : UniqueMDiffAt[s] x
hV : MDiffAt[t] (T% V) x
hW : MDiffAt[t] (T% W) x
⊢ DifferentiableWithinAt 𝕜 (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) W (range ↑I))
    (↑(extChartAt I x).symm ⁻¹' t ∩ range ↑I) (↑(extChartAt I x) x)

Tactic:
· exact hV.differentiableWithinAt_mpullbackWithin_vectorField

Goal state after:
case e_6.hW
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁶ : TopologicalSpace H
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s t : Set M
x : M
V W : (x : M) → TangentSpace I x
inst✝¹ : IsManifold I 2 M
inst✝ : CompleteSpace E
st : t ∈ 𝓝[s] x
hs : UniqueMDiffAt[s] x
hV : MDiffAt[t] (T% V) x
hW : MDiffAt[t] (T% W) x
⊢ DifferentiableWithinAt 𝕜 (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) W (range ↑I))
    (↑(extChartAt I x).symm ⁻¹' t ∩ range ↑I) (↑(extChartAt I x) x)
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
⊢ if h : IsRCLikeNormedField ℝ then Convex ℝ (range Subtype.val) else range Subtype.val = univ

Tactic:
simp only [instIsRCLikeNormedField, ↓reduceDIte]
    apply Convex.convex_isRCLikeNormedField
    rw [range_euclideanQuadrant]
    exact EuclideanQuadrant.convex

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

Goal state before:
pp: x y : ℝ
hxy : Fact (x < y)
⊢ ↑((IccLeftChart x y).extend (𝓡∂ 1)) ⊥ ∈ frontier (range ↑(𝓡∂ 1))

Tactic:
IccLeftChart_extend_bot,

Goal state after:
x y : ℝ
hxy : Fact (x < y)
⊢ 0 ∈ frontier (range ↑(𝓡∂ 1))
Module: Mathlib.Geometry.Manifold.Instances.Real

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

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

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

Goal state before:
pp: case inl
x✝ y✝ : ℝ
hxy : Fact (x✝ < y✝)
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M✝ : Type u_3
inst✝² : TopologicalSpace M✝
inst✝¹ : ChartedSpace H M✝
x y : ℝ
inst✝ : Fact (x < y)
n : ℕ∞ω
M :
  ContDiff ℝ n
    (have this := fun z ↦ toLp 2 fun i ↦ -z.ofLp i + (y - x);
    this)
e' : OpenPartialHomeomorph (↑(Icc x y)) (EuclideanHalfSpace 1)
he' : e' ∈ ChartedSpace.atlas
⊢ ContDiffOn ℝ n (↑(𝓡∂ 1) ∘ ↑((IccLeftChart x y).symm ≫ₕ e') ∘ ↑(𝓡∂ 1).symm)
    (↑(𝓡∂ 1).symm ⁻¹' ((IccLeftChart x y).symm ≫ₕ e').source ∩ range ↑(𝓡∂ 1))

Tactic:
rcases he' with (rfl | rfl)

Goal state after:
case inl.inl
x✝ y✝ : ℝ
hxy : Fact (x✝ < y✝)
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M✝ : Type u_3
inst✝² : TopologicalSpace M✝
inst✝¹ : ChartedSpace H M✝
x y : ℝ
inst✝ : Fact (x < y)
n : ℕ∞ω
M :
  ContDiff ℝ n
    (have this := fun z ↦ toLp 2 fun i ↦ -z.ofLp i + (y - x);
    this)
⊢ ContDiffOn ℝ n (↑(𝓡∂ 1) ∘ ↑((IccLeftChart x y).symm ≫ₕ IccLeftChart x y) ∘ ↑(𝓡∂ 1).symm)
    (↑(𝓡∂ 1).symm ⁻¹' ((IccLeftChart x y).symm ≫ₕ IccLeftChart x y).source ∩ range ↑(𝓡∂ 1))

case inl.inr
x✝ y✝ : ℝ
hxy : Fact (x✝ < y✝)
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M✝ : Type u_3
inst✝² : TopologicalSpace M✝
inst✝¹ : ChartedSpace H M✝
x y : ℝ
inst✝ : Fact (x < y)
n : ℕ∞ω
M :
  ContDiff ℝ n
    (have this := fun z ↦ toLp 2 fun i ↦ -z.ofLp i + (y - x);
    this)
⊢ ContDiffOn ℝ n (↑(𝓡∂ 1) ∘ ↑((IccLeftChart x y).symm ≫ₕ IccRightChart x y) ∘ ↑(𝓡∂ 1).symm)
    (↑(𝓡∂ 1).symm ⁻¹' ((IccLeftChart x y).symm ≫ₕ IccRightChart x y).source ∩ range ↑(𝓡∂ 1))
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: case inr
x✝ y✝ : ℝ
hxy : Fact (x✝ < y✝)
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M✝ : Type u_3
inst✝² : TopologicalSpace M✝
inst✝¹ : ChartedSpace H M✝
x y : ℝ
inst✝ : Fact (x < y)
n : ℕ∞ω
M :
  ContDiff ℝ n
    (have this := fun z ↦ toLp 2 fun i ↦ -z.ofLp i + (y - x);
    this)
e' : OpenPartialHomeomorph (↑(Icc x y)) (EuclideanHalfSpace 1)
he' : e' ∈ ChartedSpace.atlas
⊢ ContDiffOn ℝ n (↑(𝓡∂ 1) ∘ ↑((IccRightChart x y).symm ≫ₕ e') ∘ ↑(𝓡∂ 1).symm)
    (↑(𝓡∂ 1).symm ⁻¹' ((IccRightChart x y).symm ≫ₕ e').source ∩ range ↑(𝓡∂ 1))

Tactic:
rcases he' with (rfl | rfl)

Goal state after:
case inr.inl
x✝ y✝ : ℝ
hxy : Fact (x✝ < y✝)
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M✝ : Type u_3
inst✝² : TopologicalSpace M✝
inst✝¹ : ChartedSpace H M✝
x y : ℝ
inst✝ : Fact (x < y)
n : ℕ∞ω
M :
  ContDiff ℝ n
    (have this := fun z ↦ toLp 2 fun i ↦ -z.ofLp i + (y - x);
    this)
⊢ ContDiffOn ℝ n (↑(𝓡∂ 1) ∘ ↑((IccRightChart x y).symm ≫ₕ IccLeftChart x y) ∘ ↑(𝓡∂ 1).symm)
    (↑(𝓡∂ 1).symm ⁻¹' ((IccRightChart x y).symm ≫ₕ IccLeftChart x y).source ∩ range ↑(𝓡∂ 1))

case inr.inr
x✝ y✝ : ℝ
hxy : Fact (x✝ < y✝)
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M✝ : Type u_3
inst✝² : TopologicalSpace M✝
inst✝¹ : ChartedSpace H M✝
x y : ℝ
inst✝ : Fact (x < y)
n : ℕ∞ω
M :
  ContDiff ℝ n
    (have this := fun z ↦ toLp 2 fun i ↦ -z.ofLp i + (y - x);
    this)
⊢ ContDiffOn ℝ n (↑(𝓡∂ 1) ∘ ↑((IccRightChart x y).symm ≫ₕ IccRightChart x y) ∘ ↑(𝓡∂ 1).symm)
    (↑(𝓡∂ 1).symm ⁻¹' ((IccRightChart x y).symm ≫ₕ IccRightChart x y).source ∩ range ↑(𝓡∂ 1))
Module: Mathlib.Geometry.Manifold.Immersion

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁷ : NontriviallyNormedField 𝕜
E : Type u_2
E''' : Type u_4
E'' : Type u
F : Type u_5
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace 𝕜 E
inst✝¹⁴ : NormedAddCommGroup E''
inst✝¹³ : NormedSpace 𝕜 E''
inst✝¹² : NormedAddCommGroup E'''
inst✝¹¹ : NormedSpace 𝕜 E'''
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
H : Type u_7
inst✝⁸ : TopologicalSpace H
G : Type u_9
inst✝⁷ : TopologicalSpace G
G' : Type u_10
inst✝⁶ : TopologicalSpace G'
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 E'' G
J' : ModelWithCorners 𝕜 E''' G'
M : Type u_11
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
N : Type u_13
inst✝³ : TopologicalSpace N
inst✝² : ChartedSpace G N
N' : Type u_14
inst✝¹ : TopologicalSpace N'
inst✝ : ChartedSpace G' N'
n : ℕ∞ω
x : M
f : M → N
φ : N → N'
hφ : IsImmersionAtOfComplement F J J' n φ (f x)
x✝ : ContinuousAt f x ∧ ContMDiffAt I J' n (φ ∘ f) x
hf : ContinuousAt f x
h' : ContMDiffAt I J' n (φ ∘ f) x
⊢ ContMDiffAt I J n f x

Tactic:
have : hφ.domChart.source ∈ 𝓝 (f x) := hφ.domChart.open_source.mem_nhds hφ.mem_domChart_source

Goal state after:
𝕜 : Type u_1
inst✝¹⁷ : NontriviallyNormedField 𝕜
E : Type u_2
E''' : Type u_4
E'' : Type u
F : Type u_5
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace 𝕜 E
inst✝¹⁴ : NormedAddCommGroup E''
inst✝¹³ : NormedSpace 𝕜 E''
inst✝¹² : NormedAddCommGroup E'''
inst✝¹¹ : NormedSpace 𝕜 E'''
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
H : Type u_7
inst✝⁸ : TopologicalSpace H
G : Type u_9
inst✝⁷ : TopologicalSpace G
G' : Type u_10
inst✝⁶ : TopologicalSpace G'
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 E'' G
J' : ModelWithCorners 𝕜 E''' G'
M : Type u_11
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
N : Type u_13
inst✝³ : TopologicalSpace N
inst✝² : ChartedSpace G N
N' : Type u_14
inst✝¹ : TopologicalSpace N'
inst✝ : ChartedSpace G' N'
n : ℕ∞ω
x : M
f : M → N
φ : N → N'
hφ : IsImmersionAtOfComplement F J J' n φ (f x)
x✝ : ContinuousAt f x ∧ ContMDiffAt I J' n (φ ∘ f) x
hf : ContinuousAt f x
h' : ContMDiffAt I J' n (φ ∘ f) x
this : hφ.domChart.source ∈ 𝓝 (f x)
⊢ ContMDiffAt I J n f x
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
w : E
hw : w ∈ (ℝ ∙ v)ᗮ
h₁ : 0 < ‖w‖ ^ 2 + 4
⊢ stereoInvFunAux v w ∈ sphere 0 1

Tactic:
suffices ‖(4 : ℝ) • w + (‖w‖ ^ 2 - 4) • v‖ = ‖w‖ ^ 2 + 4 by
    simp only [mem_sphere_zero_iff_norm, norm_smul, Real.norm_eq_abs, abs_inv, this,
      abs_of_pos h₁, stereoInvFunAux_apply, inv_mul_cancel₀ h₁.ne']

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
v : E
hv : ‖v‖ = 1
w : E
hw : w ∈ (ℝ ∙ v)ᗮ
h₁ : 0 < ‖w‖ ^ 2 + 4
⊢ ‖4 • w + (‖w‖ ^ 2 - 4) • v‖ = ‖w‖ ^ 2 + 4
Module: Mathlib.Geometry.Manifold.IntegralCurve.Basic

Goal state before:
pp: case mp
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
⊢ IsMIntegralCurveAt γ v t₀ → ∃ s ∈ 𝓝 t₀, IsMIntegralCurveOn γ v s

Tactic:
intro h
    rw [IsMIntegralCurveAt, Filter.eventually_iff_exists_mem] at h
    obtain ⟨s, hs, h⟩ := h
    exact ⟨s, hs, fun t ht ↦ (h t ht).hasMFDerivWithinAt⟩

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
⊢ Injective ⇑(d% Subtype.val v)

Tactic:
rw [((contMDiff_coe_sphere v).mdifferentiableAt one_ne_zero).mvfderiv]
  simp only [chartAt, fderivWithin_univ, mfld_simps]
  let U := (OrthonormalBasis.fromOrthogonalSpanSingleton
      (𝕜 := ℝ) n (ne_zero_of_mem_unit_sphere (-v))).repr
  suffices Injective (fderiv ℝ ((stereoInvFunAux (-v : E) ∘ (↑)) ∘ U.symm) 0) by
    convert! this using 3
    congr 2
    apply stereographic'_neg (v := v)
  have : HasFDerivAt (stereoInvFunAux (-v : E) ∘ (Subtype.val : (ℝ ∙ (↑(-v) : E))ᗮ → E))
      (ℝ ∙ (↑(-v) : E))ᗮ.subtypeL (U.symm 0) := by
    convert! hasFDerivAt_stereoInvFunAux_comp_coe (-v : E)
    -- Otherwise, the lemma `EmbeddingLike.map_eq_zero_iff` is not applied.
    set_option backward.isDefEq.respectTransparency false in
    simp
  have := congr_arg DFunLike.coe <| (this.comp 0 U.symm.toContinuousLinearEquiv.hasFDerivAt).fderiv
  refine Eq.subst this.symm ?_
  rw [ContinuousLinearMap.coe_comp, ContinuousLinearEquiv.coe_coe]
  set_option backward.isDefEq.respectTransparency false in
  simpa [-Subtype.val_injective] using Subtype.val_injective

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

Goal state before:
pp: case neg
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
γ γ' : ℝ → M
v : (x : M) → TangentSpace I x
inst✝ : BoundarylessManifold I M
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
a b a' b' : ℝ
hγ : IsMIntegralCurveOn γ v (Ioo a b)
hγ' : IsMIntegralCurveOn γ' v (Ioo a' b')
t₀ : ℝ
ht₀ : t₀ ∈ Ioo a b ∩ Ioo a' b'
h : γ t₀ = γ' t₀
t : ℝ
ht : t ∈ Ioo a b ∪ Ioo a' b'
hmem : t ∉ Ioo a b
⊢ HasMFDerivAt[Ioo a b ∪ Ioo a' b'] ((Ioo a b).piecewise γ γ') t
    (ContinuousLinearMap.smulRight 1 (v ((Ioo a b).piecewise γ γ' t)))

Tactic:
have ht' := ht
    rw [mem_union, or_iff_not_imp_left] at ht
    rw [piecewise, if_neg hmem]
    apply hγ' t (ht hmem) |>.hasMFDerivAt (Ioo_mem_nhds (ht hmem).1 (ht hmem).2)
      |>.hasMFDerivWithinAt (s := Ioo a b ∪ Ioo a' b')
      |>.congr_of_eventuallyEq _ (by rw [piecewise, if_neg hmem])
    rw [Filter.eventuallyEq_iff_exists_mem]
    refine ⟨Ioo a' b', ?_,
      eqOn_piecewise_of_isMIntegralCurveOn_Ioo hv hγ hγ' ht₀ h⟩
    rw [(isOpen_Ioo.union isOpen_Ioo).nhdsWithin_eq ht']
    exact Ioo_mem_nhds (ht hmem).1 (ht hmem).2

Goal state after: