mathlib-initiative/mathlib-tactics huggingface dataset:
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:
rw [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.Instances.Icc

Goal state before:
pp: x y : ℝ
h : Fact (x < y)
z : ℝ
hz : z ∈ Icc x y
⊢ (mfderiv[Icc x y] (projIcc x y ⋯) z) 1 = 1

Tactic:
change _ = oneTangentSpaceIcc (Set.projIcc x y h.out.le z)
  simp only [oneTangentSpaceIcc]
  congr
  simp [projIcc_of_mem h.out.le hz]

Goal state after:
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
h : IsImmersion I J n f
x : M
⊢ IsImmersionAt I J n f x

Tactic:
IsImmersionAt

Goal state after:
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
h : IsImmersion I J n f
x : M
⊢ ∃ F x_1 x_2, IsImmersionAtOfComplement F 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
this : ‖4 • w + (‖w‖ ^ 2 - 4) • v‖ = ‖w‖ ^ 2 + 4
⊢ stereoInvFunAux v w ∈ sphere 0 1

Tactic:
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:
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
a : ℝ
ha : a ≠ 0
⊢ IsMIntegralCurve (γ ∘ fun x ↦ x * a) (a • v) ↔ IsMIntegralCurve γ v

Tactic:
by
  refine ⟨fun hγ ↦ ?_, fun hγ ↦ hγ.comp_mul _⟩
  convert! hγ.comp_mul a⁻¹
  · ext t
    simp only [Function.comp_apply, mul_assoc, inv_mul_eq_div, div_self ha, mul_one]
  · simp only [smul_smul, inv_mul_eq_div, div_self ha, one_smul]

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)
⊢ (↑(mfderiv% Subtype.val v)).range = (ℝ ∙ ↑v)ᗮ

Tactic:
convert! range_mvfderiv_subtypeVal v

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
n : WithTop ℕ∞
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
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
inst✝¹ : IsManifold I 1 M
inst✝ : IsManifold I' 1 M'
a : TangentBundle I M
b : TangentBundle I' M'
U : ∀ (w : H) (w' : H'), UniqueDiffWithinAt 𝕜 (range (Prod.map ↑I ↑I')) (↑I w, ↑I' w')
smooth_pM : ContMDiffAt (I.tangent.prod I'.tangent) I.tangent n Prod.fst (a, b)
p : TangentBundle I M × TangentBundle I' M'
hp : p.1.proj ∈ (chartAt H a.proj).source ∧ p.2.proj ∈ (chartAt H' b.proj).source
φ : E → E := ⋯
D0 : DifferentiableWithinAt 𝕜 φ (range ↑I) (↑I (↑(chartAt H p.1.proj) p.1.proj))
D :
  ∀ (w : TangentBundle I' M'),
    DifferentiableWithinAt 𝕜 (φ ∘ Prod.fst) (range (Prod.map ↑I ↑I'))
      (↑I (↑(chartAt H p.1.proj) p.1.proj), ↑I' (↑(chartAt H' w.proj) w.proj))
φ' : E' → E' := ⋯
⊢ ↑I' (↑(chartAt H' p.2.proj) p.2.proj) ∈ ((extChartAt I' p.2.proj).symm ≫ extChartAt I' b.proj).source

Tactic:
simp [hp.2]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
n : WithTop ℕ∞
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
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
inst✝¹ : IsManifold I 1 M
inst✝ : IsManifold I' 1 M'
a : TangentBundle I M
b : TangentBundle I' M'
U : ∀ (w : H) (w' : H'), UniqueDiffWithinAt 𝕜 (range (Prod.map ↑I ↑I')) (↑I w, ↑I' w')
smooth_pM' : ContMDiffAt (I.tangent.prod I'.tangent) I'.tangent n Prod.snd (a, b)
p : TangentBundle I M × TangentBundle I' M'
hp : p.1.proj ∈ (chartAt H a.proj).source ∧ p.2.proj ∈ (chartAt H' b.proj).source
φ : E' → E' := ⋯
⊢ ↑I' (↑(chartAt H' p.2.proj) p.2.proj) ∈ ((extChartAt I' p.2.proj).symm ≫ extChartAt I' b.proj).source

Tactic:
simp [hp.2]

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: case neg.hV
𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹² : TopologicalSpace H
E : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
H' : Type u_5
inst✝⁷ : TopologicalSpace H'
E' : Type u_6
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁴ : TopologicalSpace M'
inst✝³ : ChartedSpace H' M'
inst✝² : IsManifold I 2 M
inst✝¹ : IsManifold I' 2 M'
inst✝ : CompleteSpace E
f : M → M'
V W : (x : M') → TangentSpace I' x
x₀ : M
s : Set M
t : Set M'
hV : MDiffAt[t] (T% V) (f x₀)
hW : MDiffAt[t] (T% W) (f x₀)
hu : UniqueMDiff[s]
hf : ContMDiffWithinAt I I' 2 f s x₀
hx₀ : x₀ ∈ s
hst : f ⁻¹' t ∈ 𝓝[s] x₀
hsymm :
  IsSymmSndFDerivWithinAt 𝕜 (↑(extChartAt I' (f x₀)) ∘ f ∘ ↑(extChartAt I x₀).symm)
    (↑(extChartAt I x₀).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x₀) x₀)
A : ↑(extChartAt I x₀).symm (↑(extChartAt I x₀) x₀) = x₀
hfi : ¬(mfderiv[s] f x₀).IsInvertible
⊢ mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x₀).symm) (mpullbackWithin I I' f V s) (range ↑I) (↑(extChartAt I x₀) x₀) =
    0

Tactic:
simp only [mpullbackWithin_apply]
      rw [A, ContinuousLinearMap.inverse_of_not_isInvertible hfi]
      simp [-extChartAt]

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: case neg.hW
𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹² : TopologicalSpace H
E : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
H' : Type u_5
inst✝⁷ : TopologicalSpace H'
E' : Type u_6
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁴ : TopologicalSpace M'
inst✝³ : ChartedSpace H' M'
inst✝² : IsManifold I 2 M
inst✝¹ : IsManifold I' 2 M'
inst✝ : CompleteSpace E
f : M → M'
V W : (x : M') → TangentSpace I' x
x₀ : M
s : Set M
t : Set M'
hV : MDiffAt[t] (T% V) (f x₀)
hW : MDiffAt[t] (T% W) (f x₀)
hu : UniqueMDiff[s]
hf : ContMDiffWithinAt I I' 2 f s x₀
hx₀ : x₀ ∈ s
hst : f ⁻¹' t ∈ 𝓝[s] x₀
hsymm :
  IsSymmSndFDerivWithinAt 𝕜 (↑(extChartAt I' (f x₀)) ∘ f ∘ ↑(extChartAt I x₀).symm)
    (↑(extChartAt I x₀).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x₀) x₀)
A : ↑(extChartAt I x₀).symm (↑(extChartAt I x₀) x₀) = x₀
hfi : ¬(mfderiv[s] f x₀).IsInvertible
⊢ mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x₀).symm) (mpullbackWithin I I' f W s) (range ↑I) (↑(extChartAt I x₀) x₀) =
    0

Tactic:
simp only [mpullbackWithin_apply]
      rw [A, ContinuousLinearMap.inverse_of_not_isInvertible hfi]
      simp [-extChartAt]

Goal state after:
Module: Mathlib.Geometry.Manifold.Sheaf.LocallyRingedSpace

Goal state before:
pp: case mpr
𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
EM : Type u_1
inst✝⁴ : NormedAddCommGroup EM
inst✝³ : NormedSpace 𝕜 EM
HM : Type u_2
inst✝² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
M : Type u
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace HM M
x : M
S : TopCat.Presheaf CommRingCat (TopCat.of M) := (smoothSheafCommRing IM 𝓘(𝕜, 𝕜) M 𝕜).presheaf
U : Opens M
hxU : x ∈ U
f : C^∞⟮IM, ↥U; 𝓘(𝕜, 𝕜), 𝕜⟯
hf : (eval IM 𝓘(𝕜, 𝕜) M 𝕜 x) ((CategoryTheory.ConcreteCategory.hom (S.germ U x hxU)) f) ≠ 0
hf' : f ⟨x, hxU⟩ ≠ 0
V₀ : Set ↥U
hV₀f : ∀ y ∈ V₀, f y ≠ 0
hV₀ : IsOpen[instTopologicalSpaceSubtype] V₀
hxV₀ : ⟨x, hxU⟩ ∈ V₀
V : Opens M := { carrier := Subtype.val '' V₀, is_open' := ⋯ }
hUV : V ≤ U
⊢ IsUnit ((CategoryTheory.ConcreteCategory.hom (S.germ U x hxU)) f)

Tactic:
have hV : V₀ = Set.range (Set.inclusion hUV) := by
      convert! (Set.range_inclusion hUV).symm
      ext y
      change _ ↔ y ∈ Subtype.val ⁻¹' Subtype.val '' V₀
      rw [Set.preimage_image_eq _ Subtype.coe_injective]

Goal state after:
case mpr
𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
EM : Type u_1
inst✝⁴ : NormedAddCommGroup EM
inst✝³ : NormedSpace 𝕜 EM
HM : Type u_2
inst✝² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
M : Type u
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace HM M
x : M
S : TopCat.Presheaf CommRingCat (TopCat.of M) := (smoothSheafCommRing IM 𝓘(𝕜, 𝕜) M 𝕜).presheaf
U : Opens M
hxU : x ∈ U
f : C^∞⟮IM, ↥U; 𝓘(𝕜, 𝕜), 𝕜⟯
hf : (eval IM 𝓘(𝕜, 𝕜) M 𝕜 x) ((CategoryTheory.ConcreteCategory.hom (S.germ U x hxU)) f) ≠ 0
hf' : f ⟨x, hxU⟩ ≠ 0
V₀ : Set ↥U
hV₀f : ∀ y ∈ V₀, f y ≠ 0
hV₀ : IsOpen[instTopologicalSpaceSubtype] V₀
hxV₀ : ⟨x, hxU⟩ ∈ V₀
V : Opens M := { carrier := Subtype.val '' V₀, is_open' := ⋯ }
hUV : V ≤ U
hV : V₀ = Set.range (Set.inclusion hUV)
⊢ IsUnit ((CategoryTheory.ConcreteCategory.hom (S.germ U x hxU)) f)
Module: Mathlib.Geometry.Manifold.Riemannian.Basic

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 : RiemannianBundle fun x ↦ TangentSpace I x
inst✝¹ : IsManifold I 1 M
inst✝ : IsContinuousRiemannianBundle E fun x ↦ TangentSpace I x
x : M
⊢ ∃ C > 0, ∀ᶠ (y : E) in 𝓝[range ↑I] ↑(extChartAt I x) x, ‖mfderiv[range ↑I] ↑(extChartAt I x).symm y‖ₑ < ↑C

Tactic:
rcases eventually_norm_mfderivWithin_symm_extChartAt_lt I x with ⟨C, C_pos, hC⟩
  lift C to ℝ≥0 using C_pos.le
  simp only [gt_iff_lt, NNReal.coe_pos] at C_pos
  refine ⟨C, C_pos, ?_⟩
  filter_upwards [hC] with y hy
  simp only [enorm, nnnorm]
  exact_mod_cast hy

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

Goal state before:
pp: case h₂
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 : RiemannianBundle fun x ↦ TangentSpace I x
inst✝¹ : IsManifold I 1 M
inst✝ : IsContinuousRiemannianBundle E fun x ↦ TangentSpace I x
x : M
C : ℝ≥0
C_pos : C > 0
hC : ∀ᶠ (y : E) in 𝓝[range ↑I] ↑(extChartAt I x) x, ‖mfderiv[range ↑I] ↑(extChartAt I x).symm y‖ₑ < ↑C
r : ℝ
r_pos : r > 0
hr :
  ball (↑(extChartAt I x) x) r ∩ range ↑I ⊆
    (extChartAt I x).target ∩ {y | ‖mfderiv[range ↑I] ↑(extChartAt I x).symm y‖ₑ < ↑C}
A : ↑(extChartAt I x) ⁻¹' (ball (↑(extChartAt I x) x) r ∩ range ↑I) ∈ 𝓝 x
y : M
hy : y ∈ ↑(extChartAt I x) ⁻¹' (ball (↑(extChartAt I x) x) r ∩ range ↑I)
h'y : y ∈ (chartAt H x).source
η : ℝ →ᴬ[ℝ] E := ⋯
γ : ℝ → M := ⋯
hη :
  Icc 0 1 ⊆ ⇑η ⁻¹' (extChartAt I x).target ∧ Icc 0 1 ⊆ ⇑η ⁻¹' {y | ‖mfderiv[range ↑I] ↑(extChartAt I x).symm y‖ₑ < ↑C}
η_smooth : ContMDiffOn 𝓘(ℝ, ℝ) 𝓘(ℝ, E) 1 (⇑η) (Icc 0 1)
this✝¹ : riemannianEDist I x y ≤ pathELength I γ 0 1
t : ℝ
ht : t ∈ Icc 0 1
this✝ : mfderiv[Icc 0 1] γ t = mfderiv[range ↑I] ↑(extChartAt I x).symm (η t) ∘SL mfderiv[Icc 0 1] ⇑η t
this : (mfderiv[Icc 0 1] γ t) 1 = (mfderiv[range ↑I] ↑(extChartAt I x).symm (η t)) ((mfderiv[Icc 0 1] ⇑η t) 1)
⊢ ‖(mfderiv[Icc 0 1] ⇑η t) 1‖ₑ ≤
    ‖(fderivWithin ℝ (⇑(ContinuousAffineMap.lineMap (↑(extChartAt I x) x) (↑(extChartAt I x) y))) (Icc 0 1) t) 1‖ₑ

Tactic:
· simp only [mfderivWithin_eq_fderivWithin]
    exact le_of_eq rfl

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

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
F : Type u_5
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝⁴ : TopologicalSpace (TotalSpace F V)
inst✝³ : (x : M) → AddCommGroup (V x)
inst✝² : (x : M) → Module 𝕜 (V x)
inst✝¹ : (x : M) → TopologicalSpace (V x)
inst✝ : FiberBundle F V
ι : Type u_7
s : ι → (x : M) → V x
u u' : Set M
n : ℕ∞ω
hs : IsLocalFrameOn I F n s u
hu'u : u' ⊆ u
⊢ ∀ {x : M}, x ∈ u' → LinearIndependent 𝕜 fun x_1 ↦ s x_1 x

Tactic:
by
    intro x hx
    exact hs.linearIndependent (hu'u hx)

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
F : Type u_5
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝⁶ : TopologicalSpace (TotalSpace F V)
inst✝⁵ : (x : M) → AddCommGroup (V x)
inst✝⁴ : (x : M) → Module 𝕜 (V x)
inst✝³ : (x : M) → TopologicalSpace (V x)
inst✝² : ∀ (x : M), IsTopologicalAddGroup (V x)
inst✝¹ : ∀ (x : M), ContinuousSMul 𝕜 (V x)
inst✝ : FiberBundle F V
cov : ((x : M) → V x) → (x : M) → TangentSpace I x →L[𝕜] V x
s : Set M
hcov : IsCovariantDerivativeOn F cov s
σ σ' : (x : M) → V x
x : M
hσ : MDiffAt (T% σ) x
hσ' : MDiffAt (T% σ') x
hxs : s ∈ 𝓝 x
hσσ' : ∀ x ∈ s, σ x = σ' x
⊢ cov σ x = cov σ' x

Tactic:
have hxs' : x ∈ s := mem_of_mem_nhds hxs
  let ψ (x' : M) : 𝕜 := if x' ∈ s then 1 else 0
  have hψx : ψ x = 1 := by simp [ψ, hxs']
  -- Observe that `ψ • σ = ψ • σ'` as dependent functions.
  have H (x' : M) : ((ψ : M → 𝕜) • σ) x' = ((ψ : M → 𝕜) • σ') x' := by
    dsimp [ψ]
    split_ifs with hx's
    · simpa using hσσ' _ hx's
    · simp
  have hψ' : HasMFDerivAt I 𝓘(𝕜) ψ x 0 := by
    have : HasMFDerivAt I 𝓘(𝕜, 𝕜) (fun (_x : M) ↦ (1 : 𝕜)) x 0 := hasMFDerivAt_const ..
    refine this.congr_of_eventuallyEq ?_
    apply Filter.eventuallyEq_of_mem hxs
    intro t ht
    simp [ψ, ht]
  have := hcov.leibniz hσ hψ'.mdifferentiableAt
  -- Then, it's a chain of (dependent) equalities.
  calc cov σ x
    _ = cov ((ψ : M → 𝕜) • σ) x := by
      simp [hcov.leibniz hσ hψ'.mdifferentiableAt, hψx, mvfderiv, hψ'.mfderiv]
    _ = cov ((ψ : M → 𝕜) • σ') x := by rw [funext H]
    _ = cov σ' x := by
      simp [hcov.leibniz hσ' hψ'.mdifferentiableAt, hψx, mvfderiv, hψ'.mfderiv]

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Metric

Goal state before:
pp: E : Type u_1
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace ℝ E
H : Type u_2
inst✝¹² : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
F : Type u_4
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
V : M → Type u_5
inst✝⁷ : TopologicalSpace (TotalSpace F V)
inst✝⁶ : (x : M) → NormedAddCommGroup (V x)
inst✝⁵ : (x : M) → InnerProductSpace ℝ (V x)
inst✝⁴ : FiberBundle F V
cov : CovariantDerivative I F V
inst✝³ : VectorBundle ℝ F V
inst✝² : IsContMDiffRiemannianBundle I 1 F V
x : M
inst✝¹ : ContMDiffVectorBundle 1 F V I
inst✝ : FiniteDimensional ℝ F
X₀ : TangentSpace I x
σ₀ τ₀ : V x
⊢ (((cov.derivMetricTensor x) σ₀) τ₀) X₀ =
    ((d% fun x_1 ↦ inner ℝ (FiberBundle.extend F σ₀ x_1) (FiberBundle.extend F τ₀ x_1)) x) X₀ -
        inner ℝ ((↑cov (FiberBundle.extend F σ₀) x) X₀) τ₀ -
      inner ℝ σ₀ ((↑cov (FiberBundle.extend F τ₀) x) X₀)

Tactic:
by
  simp [derivMetricTensor, TensorialAt.mkHom₂_apply_eq_extend]

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
F : Type u_5
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝⁶ : TopologicalSpace (TotalSpace F V)
inst✝⁵ : (x : M) → AddCommGroup (V x)
inst✝⁴ : (x : M) → Module 𝕜 (V x)
inst✝³ : (x : M) → TopologicalSpace (V x)
inst✝² : ∀ (x : M), IsTopologicalAddGroup (V x)
inst✝¹ : ∀ (x : M), ContinuousSMul 𝕜 (V x)
inst✝ : FiberBundle F V
cov : ((x : M) → V x) → (x : M) → TangentSpace I x →L[𝕜] V x
s : Set M
hcov : IsCovariantDerivativeOn F cov s
σ : (x : M) → V x
x : M
a : 𝕜
hσ : MDiffAt (T% σ) x
hx : x ∈ s
⊢ cov (a • σ) x = a • cov σ x

Tactic:
by
  simpa [mvfderiv] using! hcov.leibniz (g := fun _ ↦ a) hσ mdifferentiableAt_const

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

Goal state before:
pp: case convert_2
ι : Type uι
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I ∞ M
inst✝¹ : T2Space M
inst✝ : Fintype ι
s : Set M
f : SmoothBumpCovering ι I M s
x : M
hx : x ∈ s
L : (ι → E × ℝ) →L[ℝ] E := ContinuousLinearMap.fst ℝ E ℝ ∘SL ContinuousLinearMap.proj (f.ind x hx)
this : HasMFDerivAt% (⇑L ∘ ⇑f.embeddingPiTangent) x (L ∘SL mfderiv% ⇑f.embeddingPiTangent x)
⊢ HasMFDerivAt% (⇑L ∘ ⇑f.embeddingPiTangent) x (mfderiv% ↑(chartAt H (f.c (f.ind x hx))) x)

Tactic:
refine (hasMFDerivAt_extChartAt (f.mem_chartAt_ind_source x hx)).congr_of_eventuallyEq ?_

Goal state after:
case convert_2
ι : Type uι
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I ∞ M
inst✝¹ : T2Space M
inst✝ : Fintype ι
s : Set M
f : SmoothBumpCovering ι I M s
x : M
hx : x ∈ s
L : (ι → E × ℝ) →L[ℝ] E := ContinuousLinearMap.fst ℝ E ℝ ∘SL ContinuousLinearMap.proj (f.ind x hx)
this : HasMFDerivAt% (⇑L ∘ ⇑f.embeddingPiTangent) x (L ∘SL mfderiv% ⇑f.embeddingPiTangent x)
⊢ ⇑L ∘ ⇑f.embeddingPiTangent =ᶠ[𝓝 x] ↑(extChartAt I (f.c (f.ind x hx)))
Module: Mathlib.GroupTheory.FreeGroup.IsFreeGroup

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
G✝ : Type u_3
H : Type u_4
inst✝² : Group G✝
inst✝¹ : Group H
G : Type u
inst✝ : Group G
X : Type u
of : X → G
lift : {H : Type u} → [inst : Group H] → (X → H) ≃ (G →* H)
lift_of : ∀ {H : Type u} [inst : Group H] (f : X → H) (a : X), (lift f) (of a) = f a
⊢ (FreeGroup.lift of).comp (lift FreeGroup.of) = MonoidHom.id G

Tactic:
let lift_symm_of : ∀ {H : Type u} [Group H], ∀ (f : G →* H) (a),
          lift.symm f a = f (of a) := by intro H _ f a; simp [← lift_of (lift.symm f)]

Goal state after:
ι : Type u_1
ι' : Type u_2
G✝ : Type u_3
H : Type u_4
inst✝² : Group G✝
inst✝¹ : Group H
G : Type u
inst✝ : Group G
X : Type u
of : X → G
lift : {H : Type u} → [inst : Group H] → (X → H) ≃ (G →* H)
lift_of : ∀ {H : Type u} [inst : Group H] (f : X → H) (a : X), (lift f) (of a) = f a
lift_symm_of : ∀ {H : Type u} [inst : Group H] (f : G →* H) (a : X), lift.symm f a = f (of a) :=
  fun {H} [Group H] f a ↦
    of_eq_true
      (Eq.trans
        (congrFun'
          (congrArg Eq
            (Eq.trans ((fun a ↦ Eq.symm (lift_of (lift.symm f) a)) a)
              (congrFun' (congrArg DFunLike.coe (Equiv.apply_symm_apply lift f)) (of a))))
          (f (of a)))
        (eq_self (f (of a))))
⊢ (FreeGroup.lift of).comp (lift FreeGroup.of) = MonoidHom.id G
Module: Mathlib.GroupTheory.FreeGroup.IsFreeGroup

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
G✝ : Type u_3
H : Type u_4
inst✝² : Group G✝
inst✝¹ : Group H
G : Type u
inst✝ : Group G
X : Type u
of : X → G
lift : {H : Type u} → [inst : Group H] → (X → H) ≃ (G →* H)
lift_of : ∀ {H : Type u} [inst : Group H] (f : X → H) (a : X), (lift f) (of a) = f a
⊢ (FreeGroup.lift of).comp (lift FreeGroup.of) = MonoidHom.id G

Tactic:
let lift_symm_of : ∀ {H : Type u} [Group H], ∀ (f : G →* H) (a),
          lift.symm f a = f (of a) := by intro H _ f a; simp [← lift_of (lift.symm f)]
        apply lift.symm.injective; ext x
        simp only [MonoidHom.coe_comp, Function.comp_apply, MonoidHom.id_apply,
          FreeGroup.lift_apply_of, lift_of, lift_symm_of]

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

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝¹ : MulOneClass M
inst✝ : MulOneClass N
⊢ MonoidHom.mrange inl ⊔ MonoidHom.mrange inr =
    Submonoid.closure ↑(MonoidHom.mrange inl) ⊔ Submonoid.closure ↑(MonoidHom.mrange inr)

Tactic:
Submonoid.closure_eq,

Goal state after:
M : Type u_1
N : Type u_2
inst✝¹ : MulOneClass M
inst✝ : MulOneClass N
⊢ MonoidHom.mrange inl ⊔ MonoidHom.mrange inr = MonoidHom.mrange inl ⊔ Submonoid.closure ↑(MonoidHom.mrange inr)
Module: Mathlib.GroupTheory.Coxeter.Basic

Goal state before:
pp: B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
p : W → Prop
w : W
one : p 1
mul_simple_left : ∀ (w : W) (i : B), p w → p (cs.simple i * w)
⊢ p w

Tactic:
by
  let p' : (w : W) → w ∈ Submonoid.closure (Set.range cs.simple) → Prop :=
    fun w _ ↦ p w
  have := cs.submonoid_closure_range_simple.symm ▸ Submonoid.mem_top w
  induction this using Submonoid.closure_induction_left with
  | one => exact one
  | mul_left i mi y my ih =>
    rw [Set.mem_range] at mi
    exact mi.choose_spec ▸ mul_simple_left _ _ ih

Goal state after:
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
motive : CoprodI M → Prop
m : CoprodI M
one : motive 1
mul : ∀ {i : ι} (m : M i) (x : CoprodI M), motive x → motive (of m * x)
⊢ motive m

Tactic:
induction m using Submonoid.induction_of_closure_eq_top_left mclosure_iUnion_range_of with
  | one => exact one
  | mul_left x hx y ihy =>
    obtain ⟨i, m, rfl⟩ : ∃ (i : ι) (m : M i), of m = x := by simpa using hx
    exact mul m y ihy

Goal state after: