mathlib-initiative/mathlib-tactics huggingface dataset:
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
v : (x : M) → TangentSpace I x
inst✝ : BoundarylessManifold I M
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
x : M
γ : ℝ → ℝ → M
hγx : ∀ (a : ℝ), γ a 0 = x
hγ : ∀ a > 0, IsMIntegralCurveOn (γ a) v (Ioo (-a) a)
t : ℝ
⊢ t ∈ Ioo (-(|t| + 1)) (|t| + 1)

Tactic:
mem_Ioo,

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
v : (x : M) → TangentSpace I x
inst✝ : BoundarylessManifold I M
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
x : M
γ : ℝ → ℝ → M
hγx : ∀ (a : ℝ), γ a 0 = x
hγ : ∀ a > 0, IsMIntegralCurveOn (γ a) v (Ioo (-a) a)
t : ℝ
⊢ -(|t| + 1) < t ∧ t < |t| + 1
Module: Mathlib.Geometry.Manifold.ContMDiffMFDeriv

Goal state before:
pp: case h
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
Is : IsManifold I 1 M
x : M
v : TangentSpace I x
N : ↑I.symm ⁻¹' (chartAt H x).target ∈ 𝓝 (↑I (↑(chartAt H x) x))
A : (MDiffAt fun x ↦ ⟨x, 0⟩) x
⊢ DifferentiableAt 𝕜 (fun x' ↦ (x', 0)) (↑I (↑(chartAt H x) x))

Tactic:
· exact differentiableAt_id.prodMk (differentiableAt_const _)

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

Goal state before:
pp: 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
this : CompleteSpace E'
u : Set M
u_open : IsOpen[inst✝⁹] u
x₀u : x₀ ∈ u
u_mem : u ∈ 𝓝 x₀
s' : Set M := ⋯
ut : s' ⊆ f ⁻¹' t
maps_u : s' ⊆ f ⁻¹' (extChartAt I' (f x₀)).source
u_smooth : ContMDiffOn I I' 2 f s'
hs' : s' = s ∩ u
s'_eq : s' =ᶠ[𝓝 x₀] s
t' : Set M' := ⋯
ht' : t' = t ∩ (extChartAt I' (f x₀)).source
⊢ u ∈ 𝓝 (↑(extChartAt I x₀).symm (↑(extChartAt I x₀) x₀))

Tactic:
apply u_open.mem_nhds (by simpa using x₀u)

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝²⁶ : NontriviallyNormedField 𝕜
EM : Type u_2
inst✝²⁵ : NormedAddCommGroup EM
inst✝²⁴ : NormedSpace 𝕜 EM
HM : Type u_3
inst✝²³ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
E : Type u_4
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace 𝕜 E
H : Type u_5
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_6
inst✝¹⁹ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E H'
M : Type u
inst✝¹⁸ : TopologicalSpace M
inst✝¹⁷ : ChartedSpace HM M
N G A A' R : Type u
inst✝¹⁶ : TopologicalSpace N
inst✝¹⁵ : ChartedSpace H N
inst✝¹⁴ : TopologicalSpace G
inst✝¹³ : ChartedSpace H G
inst✝¹² : TopologicalSpace A
inst✝¹¹ : ChartedSpace H A
inst✝¹⁰ : TopologicalSpace A'
inst✝⁹ : ChartedSpace H' A'
inst✝⁸ : TopologicalSpace R
inst✝⁷ : ChartedSpace H R
EP : Type u_7
inst✝⁶ : NormedAddCommGroup EP
inst✝⁵ : NormedSpace 𝕜 EP
HP : Type u_8
inst✝⁴ : TopologicalSpace HP
IP : ModelWithCorners 𝕜 EP HP
P : Type u
inst✝³ : TopologicalSpace P
inst✝² : ChartedSpace HP P
inst✝¹ : CommRing R
inst✝ : ContMDiffRing I ∞ R
⊢ CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↑(TopCat.of M)) (smoothPresheafCommRing IM I M R)

Tactic:
rw [CategoryTheory.Presheaf.isSheaf_iff_isSheaf_forget _ _
      (CategoryTheory.forget CommRingCat)]

Goal state after:
𝕜 : Type u_1
inst✝²⁶ : NontriviallyNormedField 𝕜
EM : Type u_2
inst✝²⁵ : NormedAddCommGroup EM
inst✝²⁴ : NormedSpace 𝕜 EM
HM : Type u_3
inst✝²³ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
E : Type u_4
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace 𝕜 E
H : Type u_5
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_6
inst✝¹⁹ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E H'
M : Type u
inst✝¹⁸ : TopologicalSpace M
inst✝¹⁷ : ChartedSpace HM M
N G A A' R : Type u
inst✝¹⁶ : TopologicalSpace N
inst✝¹⁵ : ChartedSpace H N
inst✝¹⁴ : TopologicalSpace G
inst✝¹³ : ChartedSpace H G
inst✝¹² : TopologicalSpace A
inst✝¹¹ : ChartedSpace H A
inst✝¹⁰ : TopologicalSpace A'
inst✝⁹ : ChartedSpace H' A'
inst✝⁸ : TopologicalSpace R
inst✝⁷ : ChartedSpace H R
EP : Type u_7
inst✝⁶ : NormedAddCommGroup EP
inst✝⁵ : NormedSpace 𝕜 EP
HP : Type u_8
inst✝⁴ : TopologicalSpace HP
IP : ModelWithCorners 𝕜 EP HP
P : Type u
inst✝³ : TopologicalSpace P
inst✝² : ChartedSpace HP P
inst✝¹ : CommRing R
inst✝ : ContMDiffRing I ∞ R
⊢ CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↑(TopCat.of M))
    (CategoryTheory.Functor.comp (smoothPresheafCommRing IM I M R) (CategoryTheory.forget CommRingCat))
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 : Set M
x : M
n : ℕ∞ω
hs : IsLocalFrameOn I F n s u
hx : x ∈ u
i : ι
⊢ (hs.toBasisAt hx) i = s i x

Tactic:
by
  simpa only [toBasisAt] using Basis.mk_apply (hs.linearIndependent hx) (hs.generating hx) i

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

Goal state before:
pp: case inr
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 : FiberBundle F V
ι : Type u_7
s : ι → (x : M) → V x
u : Set M
n : ℕ∞ω
t : (x : M) → V x
hs : IsLocalFrameOn I F n s u
inst✝¹ : VectorBundle 𝕜 F V
inst✝ : FiniteDimensional 𝕜 F
h : ∀ (i : ι), ContMDiffOn I 𝓘(𝕜, 𝕜) n ((LinearMap.piApply (hs.coeff i)) t) u
x : M
hx : x ∈ u
this : Fintype ι
⊢ ContMDiffOn I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, t x⟩) u

Tactic:
have this (i) : CMDiff[u] n (T% ((LinearMap.piApply (hs.coeff i)) t • s i)) :=
    (h i).smul_section (hs.contMDiffOn i)

Goal state after:
case inr
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 : FiberBundle F V
ι : Type u_7
s : ι → (x : M) → V x
u : Set M
n : ℕ∞ω
t : (x : M) → V x
hs : IsLocalFrameOn I F n s u
inst✝¹ : VectorBundle 𝕜 F V
inst✝ : FiniteDimensional 𝕜 F
h : ∀ (i : ι), ContMDiffOn I 𝓘(𝕜, 𝕜) n ((LinearMap.piApply (hs.coeff i)) t) u
x : M
hx : x ∈ u
this✝ : Fintype ι
this : ∀ (i : ι), ContMDiffOn I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, ((LinearMap.piApply (hs.coeff i)) t • s i) x⟩) u
⊢ ContMDiffOn I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, t x⟩) u
Module: Mathlib.Geometry.Manifold.VectorBundle.LocalFrame

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹³ : TopologicalSpace M
inst✝¹² : ChartedSpace H M
F : Type u_5
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝⁹ : TopologicalSpace (TotalSpace F V)
inst✝⁸ : (x : M) → AddCommGroup (V x)
inst✝⁷ : (x : M) → Module 𝕜 (V x)
inst✝⁶ : (x : M) → TopologicalSpace (V x)
inst✝⁵ : FiberBundle F V
inst✝⁴ : VectorBundle 𝕜 F V
inst✝³ : ContMDiffVectorBundle 1 F V I
e : Trivialization F TotalSpace.proj
inst✝² : MemTrivializationAtlas e
ι : Type u_7
b : Basis ι 𝕜 F
s : (x : M) → V x
x : M
inst✝¹ : FiniteDimensional 𝕜 F
inst✝ : CompleteSpace 𝕜
hxe : x ∈ e.baseSet
hs : MDiffAt (T% s) x
i : ι
aux : M → 𝕜 := fun x ↦ (b.repr (↑e ((fun x ↦ ⟨x, s x⟩) x)).2) i
⊢ (MDiffAt fun x ↦ (b.repr (↑e ⟨x, s x⟩).2) i) x

Tactic:
have h₁ : MDiffAt (fun x ↦ (e ((T% s) x)).2) x := by
    simpa using (e.mdifferentiableAt_section_iff I s hxe).1 hs

Goal state after:
𝕜 : Type u_1
inst✝¹⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹³ : TopologicalSpace M
inst✝¹² : ChartedSpace H M
F : Type u_5
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝⁹ : TopologicalSpace (TotalSpace F V)
inst✝⁸ : (x : M) → AddCommGroup (V x)
inst✝⁷ : (x : M) → Module 𝕜 (V x)
inst✝⁶ : (x : M) → TopologicalSpace (V x)
inst✝⁵ : FiberBundle F V
inst✝⁴ : VectorBundle 𝕜 F V
inst✝³ : ContMDiffVectorBundle 1 F V I
e : Trivialization F TotalSpace.proj
inst✝² : MemTrivializationAtlas e
ι : Type u_7
b : Basis ι 𝕜 F
s : (x : M) → V x
x : M
inst✝¹ : FiniteDimensional 𝕜 F
inst✝ : CompleteSpace 𝕜
hxe : x ∈ e.baseSet
hs : MDiffAt (T% s) x
i : ι
aux : M → 𝕜 := fun x ↦ (b.repr (↑e ((fun x ↦ ⟨x, s x⟩) x)).2) i
h₁ : (MDiffAt fun x ↦ (↑e ((fun x ↦ ⟨x, s x⟩) x)).2) x
⊢ (MDiffAt fun x ↦ (b.repr (↑e ⟨x, s x⟩).2) i) x
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
s : Set M
cov : ((x : M) → V x) → (x : M) → TangentSpace I x →L[𝕜] V x
hcov : IsCovariantDerivativeOn F cov s
cov' : ((x : M) → V x) → (x : M) → TangentSpace I x →L[𝕜] V x
hcov' : IsCovariantDerivativeOn F cov' s
g : M → 𝕜
σ✝ : (x : M) → V x
g✝ : M → 𝕜
x✝ : M
hσ : MDiffAt (T% σ✝) x✝
hφ : MDiffAt g✝ x✝
hx : x✝ ∈ s
⊢ (g • cov (g✝ • σ✝) + (1 - g) • cov' (g✝ • σ✝)) x✝ =
    g✝ x✝ • (g • cov σ✝ + (1 - g) • cov' σ✝) x✝ + (d% g✝ x✝).smulRight (σ✝ x✝)

Tactic:
simp [hcov.leibniz hσ hφ, hcov'.leibniz hσ hφ]
    module

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

Goal state before:
pp: B : Type u_1
inst✝⁷ : TopologicalSpace B
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
E : B → Type u_3
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (x : B) → NormedAddCommGroup (E x)
inst✝² : (x : B) → InnerProductSpace ℝ (E x)
inst✝¹ : FiberBundle F E
inst✝ : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x : B
r : ℝ
hr : 1 < r
r' : ℝ
hr' : 1 < r'
r'r : r' < r
h'x : x ∈ (trivializationAt F E x).baseSet
G : E x ≃L[ℝ] F := Trivialization.continuousLinearEquivAt ℝ (trivializationAt F E x) x h'x
C : ℝ := ‖↑G‖ ^ 2
⊢ ∀ᶠ (y : B) in 𝓝 x,
    ‖Trivialization.symmL ℝ (trivializationAt F E x) y ∘SL
          Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x‖ <
      r

Tactic:
obtain ⟨δ, δpos, h'δ⟩ : ∃ δ, 0 < δ ∧ (1 + δ * C) < r' ^ 2 := by
    have A : ∀ᶠ δ in 𝓝[>] (0 : ℝ), 0 < δ := self_mem_nhdsWithin
    have B : Tendsto (fun δ ↦ 1 + δ * C) (𝓝[>] 0) (𝓝 (1 + 0 * C)) := by
      apply tendsto_inf_left
      exact tendsto_const_nhds.add (tendsto_id.mul tendsto_const_nhds)
    have B' : ∀ᶠ δ in 𝓝[>] 0, 1 + δ * C < r' ^ 2 := by
      apply (tendsto_order.1 B).2
      simpa using hr'.trans_le (le_abs_self _)
    exact (A.and B').exists

Goal state after:
B : Type u_1
inst✝⁷ : TopologicalSpace B
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
E : B → Type u_3
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (x : B) → NormedAddCommGroup (E x)
inst✝² : (x : B) → InnerProductSpace ℝ (E x)
inst✝¹ : FiberBundle F E
inst✝ : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x : B
r : ℝ
hr : 1 < r
r' : ℝ
hr' : 1 < r'
r'r : r' < r
h'x : x ∈ (trivializationAt F E x).baseSet
G : E x ≃L[ℝ] F := Trivialization.continuousLinearEquivAt ℝ (trivializationAt F E x) x h'x
C : ℝ := ‖↑G‖ ^ 2
δ : ℝ
δpos : 0 < δ
h'δ : 1 + δ * C < r' ^ 2
⊢ ∀ᶠ (y : B) in 𝓝 x,
    ‖Trivialization.symmL ℝ (trivializationAt F E x) y ∘SL
          Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x‖ <
      r
Module: Mathlib.GroupTheory.Coxeter.Basic

Goal state before:
pp: case succ
B : Type u_1
m : ℕ
ih : ∀ (i i' : B), alternatingWord i i' (m + 1) = (if Even m then i' else i) :: alternatingWord i i' m
i i' : B
⊢ alternatingWord i i' (m + 1 + 1) = (if Even (m + 1) then i' else i) :: alternatingWord i i' (m + 1)

Tactic:
rw [alternatingWord]

Goal state after:
case succ
B : Type u_1
m : ℕ
ih : ∀ (i i' : B), alternatingWord i i' (m + 1) = (if Even m then i' else i) :: alternatingWord i i' m
i i' : B
⊢ (alternatingWord i' i (m + 1)).concat i' = (if Even (m + 1) then i' else i) :: alternatingWord i i' (m + 1)
Module: Mathlib.GroupTheory.Coxeter.Basic

Goal state before:
pp: case succ
B : Type u_1
m : ℕ
ih : ∀ (i i' : B), alternatingWord i i' (m + 1) = (if Even m then i' else i) :: alternatingWord i i' m
i i' : B
⊢ ((if Even m then i else i') :: alternatingWord i' i m).concat i' =
    (if Even (m + 1) then i' else i) :: alternatingWord i i' (m + 1)

Tactic:
rw [alternatingWord]

Goal state after:
case succ
B : Type u_1
m : ℕ
ih : ∀ (i i' : B), alternatingWord i i' (m + 1) = (if Even m then i' else i) :: alternatingWord i i' m
i i' : B
⊢ ((if Even m then i else i') :: alternatingWord i' i m).concat i' =
    (if Even (m + 1) then i' else i) :: (alternatingWord i' i m).concat i'
Module: Mathlib.GroupTheory.Coxeter.Basic

Goal state before:
pp: case inr
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
k : ℤ
this : ¬Even (2 * k + 1)
⊢ cs.simple i' * (cs.simple i * cs.simple i') ^ ((2 * k + 1) / 2) =
    (if Even (↑(M.M i i') * 2 - (2 * k + 1)) then 1 else cs.simple i) *
      (cs.simple i' * cs.simple i) ^ ((↑(M.M i i') * 2 - (2 * k + 1)) / 2)

Tactic:
have : ¬Even (↑(M i i') * 2 - (2 * k + 1)) :=
      Int.not_even_iff_odd.2 ⟨↑(M i i') - k - 1, by ring⟩

Goal state after:
case inr
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
k : ℤ
this✝ : ¬Even (2 * k + 1)
this : ¬Even (↑(M.M i i') * 2 - (2 * k + 1))
⊢ cs.simple i' * (cs.simple i * cs.simple i') ^ ((2 * k + 1) / 2) =
    (if Even (↑(M.M i i') * 2 - (2 * k + 1)) then 1 else cs.simple i) *
      (cs.simple i' * cs.simple i) ^ ((↑(M.M i i') * 2 - (2 * k + 1)) / 2)
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: case e'_2
M✝ : Type u_1
inst✝¹ : AddCommMonoid M✝
M : Type u_2
inst✝ : AddCommGroup M
a b : ℚ
m : M
s : ℕ+
this : ((a * b).num * ↑a.den * ↑b.den * ↑↑s) • m = (a.num * b.num * ↑(a * b).den * ↑↑s) • m
⊢ ↑1 • ↑(⟨a.den, ⋯⟩ * (⟨b.den, ⋯⟩ * s)) • (a * b).num • m = ((a * b).num * ↑a.den * ↑b.den * ↑↑s) • m

Tactic:
simp [← natCast_zsmul, smul_smul]
      ring_nf

Goal state after:
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: case e'_3
M✝ : Type u_1
inst✝¹ : AddCommMonoid M✝
M : Type u_2
inst✝ : AddCommGroup M
a b : ℚ
m : M
s : ℕ+
this : ((a * b).num * ↑a.den * ↑b.den * ↑↑s) • m = (a.num * b.num * ↑(a * b).den * ↑↑s) • m
⊢ ↑1 • ↑(⟨(a * b).den, ⋯⟩ * s) • a.num • b.num • m = (a.num * b.num * ↑(a * b).den * ↑↑s) • m

Tactic:
simp [← natCast_zsmul, smul_smul]
      ring_nf

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
⊢ ⋃ q, quotToDoubleCoset H K q = Set.univ

Tactic:
by
  ext x
  simp only [Set.mem_iUnion, quotToDoubleCoset, mem_doubleCoset, SetLike.mem_coe, Set.mem_univ,
    iff_true]
  use mk H K x
  obtain ⟨h, k, h3, h4, h5⟩ := mk_out_eq_mul H K x
  refine ⟨h⁻¹, H.inv_mem h3, k⁻¹, K.inv_mem h4, ?_⟩
  simp only [h5, ← mul_assoc, one_mul, inv_mul_cancel, mul_inv_cancel_right]

Goal state after:
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: case mk.mk
M✝ : Type u_1
inst✝³ : AddCommMonoid M✝
M : Type u_2
inst✝² : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
c : DivisibleHull M
ma : M
sa : ℕ+
mb : M
sb : ℕ+
hbc : mk mb sb ≤ c
hab : mk ma sa ≤ mk mb sb
⊢ mk ma sa ≤ c

Tactic:
induction c with | mk mc sc

Goal state after:
case mk.mk.mk
M✝ : Type u_1
inst✝³ : AddCommMonoid M✝
M : Type u_2
inst✝² : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
ma : M
sa : ℕ+
mb : M
sb : ℕ+
hab : mk ma sa ≤ mk mb sb
mc : M
sc : ℕ+
hbc : mk mb sb ≤ mk mc sc
⊢ mk ma sa ≤ mk mc sc
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: case mk.mk
M✝ : Type u_1
inst✝³ : AddCommMonoid M✝
M : Type u_2
inst✝² : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
c : DivisibleHull M
ma : M
sa : ℕ+
mb : M
sb : ℕ+
h : mk mb sb < c
⊢ mk ma sa + mk mb sb < mk ma sa + c

Tactic:
induction c with | mk mc sc

Goal state after:
case mk.mk.mk
M✝ : Type u_1
inst✝³ : AddCommMonoid M✝
M : Type u_2
inst✝² : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
ma : M
sa : ℕ+
mb : M
sb : ℕ+
mc : M
sc : ℕ+
h : mk mb sb < mk mc sc
⊢ mk ma sa + mk mb sb < mk ma sa + mk mc sc
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: case mk
M✝ : Type u_1
inst✝³ : AddCommMonoid M✝
M : Type u_2
inst✝² : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
a : ℚ≥0
ha : 0 < a
c : DivisibleHull M
mb : M
sb : ℕ+
h : mk mb sb < c
⊢ a • mk mb sb < a • c

Tactic:
induction c with | mk mc sc

Goal state after:
case mk.mk
M✝ : Type u_1
inst✝³ : AddCommMonoid M✝
M : Type u_2
inst✝² : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
a : ℚ≥0
ha : 0 < a
mb : M
sb : ℕ+
mc : M
sc : ℕ+
h : mk mb sb < mk mc sc
⊢ a • mk mb sb < a • mk mc sc
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: M✝ : Type u_1
inst✝³ : AddCommMonoid M✝
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedAddMonoid M
x : DivisibleHull M
x✝³ x✝² : M
x✝¹ x✝ : ℕ+
h : mk x✝³ x✝¹ = mk x✝² x✝
⊢ (archimedeanClassOrderHom M) (ArchimedeanClass.mk x✝³) = (archimedeanClassOrderHom M) (ArchimedeanClass.mk x✝²)

Tactic:
apply_fun ArchimedeanClass.mk at h

Goal state after:
M✝ : Type u_1
inst✝³ : AddCommMonoid M✝
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedAddMonoid M
x : DivisibleHull M
x✝³ x✝² : M
x✝¹ x✝ : ℕ+
h : ArchimedeanClass.mk (mk x✝³ x✝¹) = ArchimedeanClass.mk (mk x✝² x✝)
⊢ (archimedeanClassOrderHom M) (ArchimedeanClass.mk x✝³) = (archimedeanClassOrderHom M) (ArchimedeanClass.mk x✝²)
Module: Mathlib.GroupTheory.Transfer

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
g : G
q : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)
k : ZMod (minimalPeriod (fun x ↦ g • x) (Quotient.out q))
⊢ g ^ (1 + ⟨q, -1 + k⟩.snd.cast) * Quotient.out (Quotient.out ⟨q, -1 + k⟩.fst) =
    if k = 0 then g ^ minimalPeriod (fun x ↦ g • x) (Quotient.out q) * Quotient.out (Quotient.out q)
    else g ^ k.cast * Quotient.out (Quotient.out q)

Tactic:
←
    sub_eq_neg_add,

Goal state after:
G : Type u_1
inst✝ : Group G
H : Subgroup G
g : G
q : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)
k : ZMod (minimalPeriod (fun x ↦ g • x) (Quotient.out q))
⊢ g ^ (1 + ⟨q, k - 1⟩.snd.cast) * Quotient.out (Quotient.out ⟨q, k - 1⟩.fst) =
    if k = 0 then g ^ minimalPeriod (fun x ↦ g • x) (Quotient.out q) * Quotient.out (Quotient.out q)
    else g ^ k.cast * Quotient.out (Quotient.out q)
Module: Mathlib.GroupTheory.Focal

Goal state before:
pp: G : Type u_1
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
P : Sylow p G
inst✝ : (↑P).FiniteIndex
g : ↥P
hQ : IsPGroup p (↥↑P ⧸ (↑P).focalSubgroupOf)
⊢ (↑P).transferFocal ↑g = 1 ↔ g ∈ (↑P).focalSubgroupOf

Tactic:
transferFocal_eq_pow

Goal state after:
G : Type u_1
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
P : Sylow p G
inst✝ : (↑P).FiniteIndex
g : ↥P
hQ : IsPGroup p (↥↑P ⧸ (↑P).focalSubgroupOf)
⊢ ↑g ^ (↑P).index = 1 ↔ g ∈ (↑P).focalSubgroupOf
Module: Mathlib.GroupTheory.Frattini

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : Finite G
⊢ Group.IsNilpotent ↥(frattini G)

Tactic:
have q := (Group.isNilpotent_of_finite_tfae (G := frattini G)).out 0 3

Goal state after:
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite G
q : Group.IsNilpotent ↥(frattini G) ↔ ∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (P : Sylow p ↥(frattini G)), (↑P).Normal
⊢ Group.IsNilpotent ↥(frattini G)
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
H : Type u_2
inst✝ : Group H
e : G ≃* H
⊢ (∃ n, upperCentralSeries G n = ⊤) ↔ ∃ n, upperCentralSeries H n = ⊤

Tactic:
refine exists_congr fun n ↦ ⟨fun h ↦ ?_, fun h ↦ ?_⟩

Goal state after:
case refine_1
G : Type u_1
inst✝¹ : Group G
H : Type u_2
inst✝ : Group H
e : G ≃* H
n : ℕ
h : upperCentralSeries G n = ⊤
⊢ upperCentralSeries H n = ⊤

case refine_2
G : Type u_1
inst✝¹ : Group G
H : Type u_2
inst✝ : Group H
e : G ≃* H
n : ℕ
h : upperCentralSeries H n = ⊤
⊢ upperCentralSeries G n = ⊤
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
inst✝² : Group G
N : Subgroup G
inst✝¹ : N.Normal
S : Subgroup G
inst✝ : S.Characteristic
n : ℕ
⊢ (S.lowerCentralSeries n).Characteristic

Tactic:
induction n with

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
S : Subgroup G
K : Type u_2
inst✝ : Group K
f : G →* K
n : ℕ
⊢ map f (S.lowerCentralSeries n) = (map f S).lowerCentralSeries n

Tactic:
induction n with

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
inst✝ : Group G
hG : Group.IsNilpotent G
n : ℕ
⊢ upperCentralSeries G n = ⊤ ↔ nilpotencyClass G ≤ n

Tactic:
by
  classical
  constructor
  · intro h
    rw [nilpotencyClass_def]
    exact Nat.find_le h
  · intro h
    rw [eq_top_iff, ← upperCentralSeries_nilpotencyClass]
    exact upperCentralSeries_mono _ h

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
inst✝ : Group G
n : ℕ
⊢ Monotone fun S ↦ S.lowerCentralSeries n

Tactic:
induction n with

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
inst✝² : Group G
N : Subgroup G
inst✝¹ : N.Normal
S : Subgroup G
inst✝ : S.Normal
n : ℕ
⊢ (S.lowerCentralSeries n).Normal

Tactic:
induction n with

Goal state after: