mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.VectorBundle.Hom

Goal state before:
pp: 𝕜 : Type u_8
F₁ : Type u_9
F₂ : Type u_10
B₁ : Type u_11
B₂ : Type u_12
M : Type u_13
E₁ : B₁ → Type u_14
E₂ : B₂ → Type u_15
inst✝¹⁹ : NontriviallyNormedField 𝕜
inst✝¹⁸ : (x : B₁) → AddCommGroup (E₁ x)
inst✝¹⁷ : (x : B₁) → Module 𝕜 (E₁ x)
inst✝¹⁶ : NormedAddCommGroup F₁
inst✝¹⁵ : NormedSpace 𝕜 F₁
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝¹³ : (x : B₁) → TopologicalSpace (E₁ x)
inst✝¹² : (x : B₂) → AddCommGroup (E₂ x)
inst✝¹¹ : (x : B₂) → Module 𝕜 (E₂ x)
inst✝¹⁰ : NormedAddCommGroup F₂
inst✝⁹ : NormedSpace 𝕜 F₂
inst✝⁸ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁷ : (x : B₂) → TopologicalSpace (E₂ x)
inst✝⁶ : TopologicalSpace B₁
inst✝⁵ : TopologicalSpace B₂
inst✝⁴ : TopologicalSpace M
inst✝³ : FiberBundle F₁ E₁
inst✝² : VectorBundle 𝕜 F₁ E₁
inst✝¹ : FiberBundle F₂ E₂
inst✝ : VectorBundle 𝕜 F₂ E₂
b₁ : M → B₁
b₂ : M → B₂
m₀ : M
ϕ : (m : M) → E₁ (b₁ m) →L[𝕜] E₂ (b₂ m)
v : (m : M) → E₁ (b₁ m)
s : Set M
hϕ : ContinuousWithinAt (fun m ↦ inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) s m₀
hv : ContinuousWithinAt (fun m ↦ { proj := b₁ m, snd := v m }) s m₀
hb₂ : ContinuousWithinAt b₂ s m₀
⊢ ContinuousWithinAt (fun m ↦ { proj := b₂ m, snd := (ϕ m) (v m) }) s m₀

Tactic:
by
  rw [← continuousWithinAt_insert_self] at hϕ hv hb₂ ⊢
  rw [FiberBundle.continuousWithinAt_totalSpace] at hv ⊢
  refine ⟨hb₂, ?_⟩
  apply (ContinuousWithinAt.clm_apply hϕ hv.2).congr_of_eventuallyEq_of_mem ?_ (mem_insert m₀ s)
  have A : ∀ᶠ m in 𝓝[insert m₀ s] m₀, b₁ m ∈ (trivializationAt F₁ E₁ (b₁ m₀)).baseSet := by
    apply hv.1
    apply (trivializationAt F₁ E₁ (b₁ m₀)).open_baseSet.mem_nhds
    exact FiberBundle.mem_baseSet_trivializationAt' (b₁ m₀)
  have A' : ∀ᶠ m in 𝓝[insert m₀ s] m₀, b₂ m ∈ (trivializationAt F₂ E₂ (b₂ m₀)).baseSet := by
    apply hb₂
    apply (trivializationAt F₂ E₂ (b₂ m₀)).open_baseSet.mem_nhds
    exact FiberBundle.mem_baseSet_trivializationAt' (b₂ m₀)
  filter_upwards [A, A'] with m hm h'm using by simp [inCoordinates_eq hm h'm, hm]

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

Goal state before:
pp: case rt
𝕜 : Type u_1
E : Type u_2
F : Type u_4
H : Type u_6
G : Type u_8
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : TopologicalSpace H
inst✝⁴ : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 F G
M : Type u_10
N : Type u_12
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n : ℕ∞ω
f g : M → N
x : M
P : (M → N) → OpenPartialHomeomorph M H → OpenPartialHomeomorph N G → Prop
hP : IsLocalSourceTargetProperty P
hf : LiftSourceTargetPropertyAt I J n f x P
h' : f =ᶠ[𝓝 x] g
s' : Set M
hxs' : s' ∈ 𝓝 x
hfg : EqOn f g s'
s : Set M
hss' : s ⊆ s'
hs : IsOpen[inst✝³] s
hxs : x ∈ s
⊢ (hf.domChart.restr s).source ⊆ f ⁻¹' hf.codChart.source

Tactic:
· exact Subset.trans (by simp) hf.source_subset_preimage_source

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

Goal state before:
pp: n : ℕ
x✝⁶ : EuclideanSpace ℝ (Fin n)
hx : x✝⁶ ∈ {x | ∀ (i : Fin n), 0 ≤ x.ofLp i}
x✝⁵ : EuclideanSpace ℝ (Fin n)
hy : x✝⁵ ∈ {x | ∀ (i : Fin n), 0 ≤ x.ofLp i}
x✝⁴ x✝³ : ℝ
x✝² : 0 ≤ x✝⁴
x✝¹ : 0 ≤ x✝³
x✝ : x✝⁴ + x✝³ = 1
i : Fin n
⊢ 0 ≤ (x✝⁴ • x✝⁶ + x✝³ • x✝⁵).ofLp i

Tactic:
dsimp at hx hy ⊢; specialize hx i; specialize hy i; positivity

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

Goal state before:
pp: x y : ℝ
hxy : Fact (x < y)
z : EuclideanHalfSpace 1
hz : (↑z).ofLp 0 ≤ y - x
⊢ y - (↑z).ofLp 0 ∈ Icc x y

Tactic:
simp [z.prop, sub_eq_add_neg, add_le_of_le_sub_left hz]

Goal state after:
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:
simp only [stereoToFun_apply, stereoInvFun_apply, smul_add]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
v : E
hv : ‖v‖ = 1
x : ↑(sphere 0 1)
hx : ↑x ≠ v
⊢ (‖(2 / (1 - ((innerSL ℝ) v) ↑x)) • (ℝ ∙ v)ᗮ.orthogonalProjectionOnto ↑x‖ ^ 2 + 4)⁻¹ •
        4 • ↑((2 / (1 - ((innerSL ℝ) v) ↑x)) • (ℝ ∙ v)ᗮ.orthogonalProjectionOnto ↑x) +
      (‖(2 / (1 - ((innerSL ℝ) v) ↑x)) • (ℝ ∙ v)ᗮ.orthogonalProjectionOnto ↑x‖ ^ 2 + 4)⁻¹ •
        (‖(2 / (1 - ((innerSL ℝ) v) ↑x)) • (ℝ ∙ v)ᗮ.orthogonalProjectionOnto ↑x‖ ^ 2 - 4) • v =
    ↑x
Module: Mathlib.Geometry.Manifold.ContMDiffMFDeriv

Goal state before:
pp: case hf
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'
h2f : ∀ᶠ (x₂ : N) in 𝓝[t] x₀, ContMDiffWithinAt I I' 1 (f x₂) u (g x₂)
h2g : g ⁻¹' (extChartAt I (g x₀)).source ∈ 𝓝[t] x₀
this✝ :
  ContDiffWithinAt 𝕜 m
    (fun x ↦
      fderivWithin 𝕜 (↑(extChartAt I' (f x₀ (g x₀))) ∘ f (↑(extChartAt J x₀).symm x) ∘ ↑(extChartAt I (g x₀)).symm)
        ((extChartAt I (g x₀)).target ∩ ↑(extChartAt I (g x₀)).symm ⁻¹' u)
        (↑(extChartAt I (g x₀)) (g (↑(extChartAt J x₀).symm x))))
    (↑(extChartAt J x₀).symm ⁻¹' t' ∩ range ↑J) (↑(extChartAt J x₀) x₀)
this :
  ContMDiffWithinAt J 𝓘(𝕜, E →L[𝕜] E') m
    (fun x ↦
      fderivWithin 𝕜 (↑(extChartAt I' (f x₀ (g x₀))) ∘ f x ∘ ↑(extChartAt I (g x₀)).symm)
        ((extChartAt I (g x₀)).target ∩ ↑(extChartAt I (g x₀)).symm ⁻¹' u) (↑(extChartAt I (g x₀)) (g x)))
    t' x₀
x : N
hx : ContMDiffWithinAt I I' 1 (f x) u (g x)
h'x : x ∈ (fun x ↦ f x (g x)) ⁻¹' (extChartAt I' (f x₀ (g x₀))).source
h2 : x ∈ g ⁻¹' (extChartAt I (g x₀)).source
hxt : x ∈ t
h1 : g x ∈ u
h3 : UniqueMDiffAt[(extChartAt I (g x₀)).target ∩ ↑(extChartAt I (g x₀)).symm ⁻¹' u] (↑(extChartAt I (g x₀)) (g x))
A :
  mfderiv[range ↑I] ↑(extChartAt I (g x₀)).symm (↑(extChartAt I (g x₀)) (g x)) =
    mfderiv[(extChartAt I (g x₀)).target ∩ ↑(extChartAt I (g x₀)).symm ⁻¹' u] ↑(extChartAt I (g x₀)).symm
      (↑(extChartAt I (g x₀)) (g x))
⊢ MDiffAt[(extChartAt I (g x₀)).target ∩ ↑(extChartAt I (g x₀)).symm ⁻¹' u] ↑(extChartAt I (g x₀)).symm
    (↑(extChartAt I (g x₀)) (g x))

Tactic:
apply (mdifferentiableWithinAt_extChartAt_symm _).mono
    · exact inter_subset_left.trans (extChartAt_target_subset_range (g x₀))
    · exact PartialEquiv.map_source (extChartAt I (g x₀)) h2

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

Goal state before:
pp: 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 dt ε : ℝ
hε : ε > 0
h : IsMIntegralCurveOn γ v (Metric.ball t₀ ε)
⊢ {y | |y - t₀ + dt| < ε} = {t | |t + dt - t₀| < ε}

Tactic:
add_sub_right_comm

Goal state after:
Module: Mathlib.Geometry.Manifold.Riemannian.PathELength

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : (x : M) → ENorm (TangentSpace I x)
inst✝ : ∀ (x : M), ENormSMulClass ℝ (TangentSpace I x)
x y : M
a b : ℝ
γ : ℝ → M
hγ : ContMDiffOn 𝓘(ℝ, ℝ) I 1 γ (Icc a b)
ha : γ a = x
hb : γ b = y
hab : a ≤ b
η : ℝ →ᴬ[ℝ] ℝ := ContinuousAffineMap.lineMap a b
hη : ContMDiffOn 𝓘(ℝ, ℝ) I 1 (γ ∘ ⇑η) (Icc 0 1)
f : ↑unitInterval → M := fun t ↦ (γ ∘ ⇑η) ↑t
hf : ContMDiff (𝓡∂ 1) I 1 f
g : Path x y := { toFun := f, continuous_toFun := ⋯, source' := ⋯, target' := ⋯ }
⊢ riemannianEDist I x y ≤ ∫⁻ (x_1 : ↑unitInterval), ‖(mfderiv% ⇑g x_1) 1‖ₑ

Tactic:
riemannianEDist

Goal state after:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : (x : M) → ENorm (TangentSpace I x)
inst✝ : ∀ (x : M), ENormSMulClass ℝ (TangentSpace I x)
x y : M
a b : ℝ
γ : ℝ → M
hγ : ContMDiffOn 𝓘(ℝ, ℝ) I 1 γ (Icc a b)
ha : γ a = x
hb : γ b = y
hab : a ≤ b
η : ℝ →ᴬ[ℝ] ℝ := ContinuousAffineMap.lineMap a b
hη : ContMDiffOn 𝓘(ℝ, ℝ) I 1 (γ ∘ ⇑η) (Icc 0 1)
f : ↑unitInterval → M := fun t ↦ (γ ∘ ⇑η) ↑t
hf : ContMDiff (𝓡∂ 1) I 1 f
g : Path x y := { toFun := f, continuous_toFun := ⋯, source' := ⋯, target' := ⋯ }
⊢ ⨅ γ, ⨅ (_ : ContMDiff (𝓡∂ 1) I 1 ⇑γ), ∫⁻ (x_2 : ↑unitInterval), ‖(mfderiv% ⇑γ x_2) 1‖ₑ ≤
    ∫⁻ (x_1 : ↑unitInterval), ‖(mfderiv% ⇑g x_1) 1‖ₑ
Module: Mathlib.Geometry.Manifold.Riemannian.PathELength

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : (x : M) → ENorm (TangentSpace I x)
inst✝ : ∀ (x : M), ENormSMulClass ℝ (TangentSpace I x)
x y : M
r : ℝ≥0∞
a b : ℝ
hr : riemannianEDist I x y < r
hab : a < b
γ : ℝ → M
hγx : γ 0 = x
hγy : γ 1 = y
γ_smooth : ContMDiffOn 𝓘(ℝ, ℝ) I 1 γ (Icc 0 1)
hγ : pathELength I γ 0 1 < r
a' : ℝ
haa' : a < a'
ha'b : a' < b
b' : ℝ
ha'b' : a' < b'
hb'b : b' < b
⊢ ∃ γ,
    γ a = x ∧
      γ b = y ∧ ContMDiff 𝓘(ℝ, ℝ) I 1 γ ∧ pathELength I γ a b < r ∧ (γ =ᶠ[𝓝 a] fun x_1 ↦ x) ∧ γ =ᶠ[𝓝 b] fun x ↦ y

Tactic:
let η (t : ℝ) : ℝ := Real.smoothTransition ((b' - a')⁻¹ * (t - a'))

Goal state after:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : (x : M) → ENorm (TangentSpace I x)
inst✝ : ∀ (x : M), ENormSMulClass ℝ (TangentSpace I x)
x y : M
r : ℝ≥0∞
a b : ℝ
hr : riemannianEDist I x y < r
hab : a < b
γ : ℝ → M
hγx : γ 0 = x
hγy : γ 1 = y
γ_smooth : ContMDiffOn 𝓘(ℝ, ℝ) I 1 γ (Icc 0 1)
hγ : pathELength I γ 0 1 < r
a' : ℝ
haa' : a < a'
ha'b : a' < b
b' : ℝ
ha'b' : a' < b'
hb'b : b' < b
η : ℝ → ℝ := fun t ↦ ((b' - a')⁻¹ * (t - a')).smoothTransition
⊢ ∃ γ,
    γ a = x ∧
      γ b = y ∧ ContMDiff 𝓘(ℝ, ℝ) I 1 γ ∧ pathELength I γ a b < r ∧ (γ =ᶠ[𝓝 a] fun x_1 ↦ x) ∧ γ =ᶠ[𝓝 b] fun x ↦ y
Module: Mathlib.Geometry.Manifold.Riemannian.PathELength

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : (x : M) → ENorm (TangentSpace I x)
inst✝ : ∀ (x : M), ENormSMulClass ℝ (TangentSpace I x)
x✝ y✝ x y : M
r : ℝ≥0∞
hr : riemannianEDist I x y < r
γ : ℝ → M
γ0 : γ 0 = x
γ1 : γ 1 = y
γ_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
hγ : pathELength I γ 0 1 < r
η : ℝ → ℝ := fun t ↦ -t
h_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 (γ ∘ η)
this : riemannianEDist I y x ≤ pathELength I ((γ ∘ η) ∘ η) 0 1
t : ℝ
⊢ ((γ ∘ η) ∘ η) t = γ t

Tactic:
simp [η]

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

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
n : ℕ∞ω
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
F : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x : F
⊢ IsVonNBounded ℝ {v | ((innerSL ℝ) v) v < 1}

Tactic:
change IsVonNBounded ℝ {v : F | ⟪v, v⟫ < 1}
    have : Metric.ball (0 : F) 1 = {v : F | ⟪v, v⟫ < 1} := by
      ext v
      simp only [Metric.mem_ball, dist_zero_right, norm_eq_sqrt_re_inner (𝕜 := ℝ),
        RCLike.re_to_real, Set.mem_ofPred_eq]
      conv_lhs => rw [show (1 : ℝ) = √1 by simp]
      rw [Real.sqrt_lt_sqrt_iff]
      exact real_inner_self_nonneg
    rw [← this]
    exact NormedSpace.isVonNBounded_ball ℝ F 1

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

Goal state before:
pp: B : Type u_1
inst✝⁹ : TopologicalSpace B
F : Type u_2
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
E : B → Type u_3
inst✝⁶ : TopologicalSpace (TotalSpace F E)
inst✝⁵ : (x : B) → NormedAddCommGroup (E x)
inst✝⁴ : (x : B) → InnerProductSpace ℝ (E x)
inst✝³ : FiberBundle F E
inst✝² : VectorBundle ℝ F E
F₁ : Type u_4
inst✝¹ : NormedAddCommGroup F₁
inst✝ : InnerProductSpace ℝ F₁
x : B
⊢ ContinuousAt (fun x ↦ TotalSpace.mk' (F₁ →L[ℝ] F₁ →L[ℝ] ℝ) x ((fun x ↦ innerSL ℝ) x)) x

Tactic:
rw [FiberBundle.continuousAt_totalSpace]

Goal state after:
B : Type u_1
inst✝⁹ : TopologicalSpace B
F : Type u_2
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
E : B → Type u_3
inst✝⁶ : TopologicalSpace (TotalSpace F E)
inst✝⁵ : (x : B) → NormedAddCommGroup (E x)
inst✝⁴ : (x : B) → InnerProductSpace ℝ (E x)
inst✝³ : FiberBundle F E
inst✝² : VectorBundle ℝ F E
F₁ : Type u_4
inst✝¹ : NormedAddCommGroup F₁
inst✝ : InnerProductSpace ℝ F₁
x : B
⊢ ContinuousAt (fun x ↦ (TotalSpace.mk' (F₁ →L[ℝ] F₁ →L[ℝ] ℝ) x ((fun x ↦ innerSL ℝ) x)).proj) x ∧
    ContinuousAt
      (fun x_1 ↦
        (↑(trivializationAt (F₁ →L[ℝ] F₁ →L[ℝ] ℝ) (fun x ↦ Trivial B F₁ x →L[ℝ] Trivial B F₁ x →L[ℝ] ℝ)
                (TotalSpace.mk' (F₁ →L[ℝ] F₁ →L[ℝ] ℝ) x ((fun x ↦ innerSL ℝ) x)).proj)
            (TotalSpace.mk' (F₁ →L[ℝ] F₁ →L[ℝ] ℝ) x_1 ((fun x ↦ innerSL ℝ) x_1))).2)
      x
Module: Mathlib.Geometry.Manifold.Riemannian.Basic

Goal state before:
pp: E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : RiemannianBundle fun x ↦ TangentSpace I x
inst✝² : IsManifold I 1 M
inst✝¹ : IsContinuousRiemannianBundle E fun x ↦ TangentSpace I x
inst✝ : RegularSpace M
x : M
s : Set M
hs : s ∈ 𝓝 x
C : ℝ≥0
C_pos : C > 0
hC : ∀ᶠ (y : M) in 𝓝 x, ‖mfderiv% ↑(extChartAt I x) y‖ₑ < ↑C
u : Set M
u_mem : u ∈ 𝓝 x
u_closed : IsClosed[inst✝⁵] u
us : u ⊆ s
hu : u ⊆ {y | ‖mfderiv% ↑(extChartAt I x) y‖ₑ < ↑C}
uc : u ⊆ (extChartAt I x).source
uc' : u ⊆ (chartAt H x).source
⊢ ∃ c > 0, {y | riemannianEDist I x y < ↑c} ⊆ s

Tactic:
obtain ⟨v, ⟨v_mem, v_open⟩, hv⟩ : ∃ v, (v ∈ 𝓝 x ∧ IsOpen v) ∧ v ⊆ u :=
    (nhds_basis_opens' x).mem_iff.1 u_mem

Goal state after:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : RiemannianBundle fun x ↦ TangentSpace I x
inst✝² : IsManifold I 1 M
inst✝¹ : IsContinuousRiemannianBundle E fun x ↦ TangentSpace I x
inst✝ : RegularSpace M
x : M
s : Set M
hs : s ∈ 𝓝 x
C : ℝ≥0
C_pos : C > 0
hC : ∀ᶠ (y : M) in 𝓝 x, ‖mfderiv% ↑(extChartAt I x) y‖ₑ < ↑C
u : Set M
u_mem : u ∈ 𝓝 x
u_closed : IsClosed[inst✝⁵] u
us : u ⊆ s
hu : u ⊆ {y | ‖mfderiv% ↑(extChartAt I x) y‖ₑ < ↑C}
uc : u ⊆ (extChartAt I x).source
uc' : u ⊆ (chartAt H x).source
v : Set M
hv : v ⊆ u
v_mem : v ∈ 𝓝 x
v_open : IsOpen[inst✝⁵] v
⊢ ∃ c > 0, {y | riemannianEDist I x y < ↑c} ⊆ s
Module: Mathlib.Geometry.Manifold.Submersion

Goal state before:
pp: 𝕜 : Type u_1
E'' : Type u_3
F : Type u_5
H : Type u_7
G : Type u_9
E : Type u
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedAddCommGroup E''
inst✝⁸ : NormedSpace 𝕜 E''
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : TopologicalSpace H
inst✝⁴ : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 E'' G
M : Type u_11
N : Type u_13
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n : ℕ∞ω
f : M → N
x : M
h : IsSubmersionAtOfComplement F I J n f x
⊢ IsSubmersionAt I J n f x

Tactic:
by
  use h.smallComplement, by infer_instance, by infer_instance
  exact (IsSubmersionAtOfComplement.congr_F h.smallEquiv).mp h

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

Goal state before:
pp: E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
n : ℕ∞ω
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : RiemannianBundle fun x ↦ TangentSpace I x
inst✝² : IsManifold I 1 M
inst✝¹ : IsContinuousRiemannianBundle E fun x ↦ TangentSpace I x
inst✝ : RegularSpace M
⊢ IsRiemannianManifold I M

Tactic:
let : PseudoEMetricSpace M := .ofRiemannianMetric I M
  exact ⟨fun x y ↦ rfl⟩

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝²⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝²⁷ : NormedAddCommGroup E
inst✝²⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝²⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝²⁴ : TopologicalSpace M
inst✝²³ : ChartedSpace H M
F : Type u_5
inst✝²² : NormedAddCommGroup F
inst✝²¹ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝²⁰ : TopologicalSpace (TotalSpace F V)
inst✝¹⁹ : (x : M) → AddCommGroup (V x)
inst✝¹⁸ : (x : M) → Module 𝕜 (V x)
inst✝¹⁷ : (x : M) → TopologicalSpace (V x)
inst✝¹⁶ : FiberBundle F V
F' : Type u_7
inst✝¹⁵ : NormedAddCommGroup F'
inst✝¹⁴ : NormedSpace 𝕜 F'
V' : M → Type u_8
inst✝¹³ : TopologicalSpace (TotalSpace F' V')
inst✝¹² : (x : M) → AddCommGroup (V' x)
inst✝¹¹ : (x : M) → Module 𝕜 (V' x)
inst✝¹⁰ : (x : M) → TopologicalSpace (V' x)
inst✝⁹ : FiberBundle F' V'
A : Type u_9
inst✝⁸ : AddCommGroup A
inst✝⁷ : Module 𝕜 A
inst✝⁶ : VectorBundle 𝕜 F V
inst✝⁵ : VectorBundle 𝕜 F' V'
inst✝⁴ : CompleteSpace 𝕜
inst✝³ : FiniteDimensional 𝕜 F
inst✝² : FiniteDimensional 𝕜 F'
inst✝¹ : ContMDiffVectorBundle 1 F V I
inst✝ : ContMDiffVectorBundle 1 F' V' I
Φ : ((x : M) → V x) → ((x : M) → V' x) → A
x : M
hΦ₁ : ∀ (τ : (x : M) → V' x), MDiffAt (T% τ) x → TensorialAt I F (fun x ↦ Φ x τ) x
hΦ₂ : ∀ (σ : (x : M) → V x), MDiffAt (T% σ) x → TensorialAt I F' (fun x ↦ Φ σ x) x
σ σ' : (x : M) → V x
τ τ' : (x : M) → V' x
hσ : MDiffAt (T% σ) x
hσ' : MDiffAt (T% σ') x
hτ : MDiffAt (T% τ) x
hτ' : MDiffAt (T% τ') x
hσσ' : σ x = σ' x
hττ' : τ x = τ' x
⊢ Φ σ τ = Φ σ' τ'

Tactic:
by
  trans Φ σ' τ
  · exact (hΦ₁ _ hτ).pointwise hσ hσ' hσσ'
  · exact (hΦ₂ _ hσ').pointwise hτ hτ' hττ'

Goal state after:
Module: Mathlib.GroupTheory.Commensurable

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
x✝¹ : ConjAct G
x✝ : x✝¹ ∈ {g | (g • H).Commensurable H}
⊢ x✝¹⁻¹ ∈ {g | (g • H).Commensurable H}

Tactic:
rwa [Set.mem_ofPred_eq, comm, ← commensurable_inv]

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.KleinFour

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ #{1} + 1 + 1 + 1 = 4

pp: G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ y ∉ {1}

pp: G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ x ∉ {y, 1}

Tactic:
all_goals simp_all

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.KleinFour

Goal state before:
pp: case neg
G : Type u_1
inst✝⁴ : Group G
inst✝³ : IsKleinFour G
G₁ : Type u_2
G₂ : Type u_3
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : IsKleinFour G₁
e : G₁ ≃ G₂
he : e 1 = 1
h : Monoid.exponent G₂ = 2
_inst₁ : Fintype G₁ := Fintype.ofFinite G₁
_inst₂ : Fintype G₂ := Fintype.ofEquiv G₁ e
x y : G₁
hx : ¬x = 1
hy : ¬y = 1
hxy : ¬x = y
univ₂ : {e (x * y), e x, e y, 1} = univ
⊢ e (x * y) = e x * e y

Tactic:
rw [← Ne, ← e.injective.ne_iff] at hx hy hxy

Goal state after:
case neg
G : Type u_1
inst✝⁴ : Group G
inst✝³ : IsKleinFour G
G₁ : Type u_2
G₂ : Type u_3
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : IsKleinFour G₁
e : G₁ ≃ G₂
he : e 1 = 1
h : Monoid.exponent G₂ = 2
_inst₁ : Fintype G₁ := Fintype.ofFinite G₁
_inst₂ : Fintype G₂ := Fintype.ofEquiv G₁ e
x y : G₁
hx : e x ≠ e 1
hy : e y ≠ e 1
hxy : e x ≠ e y
univ₂ : {e (x * y), e x, e y, 1} = univ
⊢ e (x * y) = e x * e y
Module: Mathlib.GroupTheory.SpecificGroups.Dihedral

Goal state before:
pp: case succ
n : ℕ
i : ZMod n
k : ℕ
IH : r i ^ k = r (i * ↑k)
⊢ r (i * ↑k + i) = r (i * (↑k + ↑1))

Tactic:
Nat.cast_one,

Goal state after:
case succ
n : ℕ
i : ZMod n
k : ℕ
IH : r i ^ k = r (i * ↑k)
⊢ r (i * ↑k + i) = r (i * (↑k + 1))
Module: Mathlib.GroupTheory.SpecificGroups.Dihedral

Goal state before:
pp: case e'_3
n : ℕ
hn : NeZero n
⊢ 2 = orderOf (sr 0)

Tactic:
exact (orderOf_sr 0).symm

Goal state after:
Module: Mathlib.GroupTheory.CommutingProbability

Goal state before:
pp: n : ℕ
h1 : n ≠ 1
h2 : Odd n
⊢ reciprocalFactors n = n % 4 * n :: reciprocalFactors (n / 4 + 1)

Tactic:
have h0 : n ≠ 0 := by
    rintro rfl
    norm_num [← Nat.not_even_iff_odd] at h2

Goal state after:
n : ℕ
h1 : n ≠ 1
h2 : Odd n
h0 : n ≠ 0
⊢ reciprocalFactors n = n % 4 * n :: reciprocalFactors (n / 4 + 1)
Module: Mathlib.GroupTheory.CommutingProbability

Goal state before:
pp: case pos
n : ℕ
h0 : ¬n = 0
h1 : n = 1
⊢ 1 = 1 / ↑1

Tactic:
Nat.cast_one,

Goal state after:
case pos
n : ℕ
h0 : ¬n = 0
h1 : n = 1
⊢ 1 = 1 / 1
Module: Mathlib.GroupTheory.FreeGroup.IsFreeGroup

Goal state before:
pp: ι : Type u_1
G : Type u_3
inst✝ : Group G
b : FreeGroupBasis ι G
i : ι
⊢ b.repr (b i) = FreeGroup.of i

Tactic:
change b.repr (b.repr.symm (FreeGroup.of i)) = FreeGroup.of i
  simp

Goal state after:
Module: Mathlib.GroupTheory.Coprod.Basic

Goal state before:
pp: case of_mul
G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
x : G ⊕ H
xs : FreeMonoid (G ⊕ H)
ihx : mk (ofList (List.map (Sum.map Inv.inv Inv.inv) (toList xs)).reverse) * mk xs = 1
⊢ mk (ofList (List.map (Sum.map Inv.inv Inv.inv) (toList xs)).reverse) * mk (of (Sum.map Inv.inv Inv.inv x)) *
      (mk (of x) * mk xs) =
    1

Tactic:
rwa [mul_assoc, ← mul_assoc (mk (of _)), mk_of_inv_mul, one_mul]

Goal state after:
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
M : ι → Type u_2
inst✝² : (i : ι) → Monoid (M i)
N : Type u_3
inst✝¹ : Monoid N
G : ι → Type u_4
inst✝ : (i : ι) → Group (G i)
m : CoprodI G
⊢ m⁻¹ * m = 1

Tactic:
inv_def

Goal state after:
ι : Type u_1
M : ι → Type u_2
inst✝² : (i : ι) → Monoid (M i)
N : Type u_3
inst✝¹ : Monoid N
G : ι → Type u_4
inst✝ : (i : ι) → Group (G i)
m : CoprodI G
⊢ MulOpposite.unop ((lift fun i ↦ (MonoidHom.op of).comp (MulEquiv.inv' (G i)).toMonoidHom) m) * m = 1
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
M : ι → Type u_2
inst✝¹ : (i : ι) → Monoid (M i)
inst✝ : (i : ι) → DecidableEq (M i)
i : ι
p : Pair M i
hm : p.head = 1
⊢ (if h : p.head = 1 then p.tail else cons p.head p.tail ⋯ h).prod = of p.head * p.tail.prod

Tactic:
dif_pos hm,

Goal state after:
ι : Type u_1
M : ι → Type u_2
inst✝¹ : (i : ι) → Monoid (M i)
inst✝ : (i : ι) → DecidableEq (M i)
i : ι
p : Pair M i
hm : p.head = 1
⊢ p.tail.prod = of p.head * p.tail.prod