mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Manifold.Riemannian.PathELength

Goal state before:
pp: case hxs
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)
a b : ℝ
γ : ℝ → M
inst✝ : ∀ (x : M), ENormSMulClass ℝ (TangentSpace I x)
f : ℝ → ℝ
h✝ : a ≤ b
hf : AntitoneOn f (Icc a b)
h'f : DifferentiableOn ℝ f (Icc a b)
hγ : MDiff[f '' Icc a b] γ
h : a < b
f_im : f '' Icc a b = Icc (f b) (f a)
B : ∀ t ∈ Icc a b, HasDerivWithinAt f (derivWithin f (Icc a b) t) (Icc a b) t
t : ℝ
ht : t ∈ Icc a b
⊢ UniqueDiffWithinAt ℝ (Icc a b) t

Tactic:
exact uniqueDiffOn_Icc h _ ht

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 := s ∩ u
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' := t ∩ (extChartAt I' (f x₀)).source
ht' : t' = t ∩ (extChartAt I' (f x₀)).source
⊢ mpullbackWithin I I' f (mlieBracketWithin I' V W t) s x₀ = mpullbackWithin I I' f (mlieBracketWithin I' V W t) s' x₀

Tactic:
simp only [mpullbackWithin, hs', mfderivWithin_inter u_mem]

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✝ 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 := ⋯
h_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 (γ ∘ η)
this : riemannianEDist I y x ≤ pathELength I ((γ ∘ η) ∘ η) 0 1
⊢ pathELength I ((γ ∘ η) ∘ η) 0 1 < r

Tactic:
convert! hγ

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✝ 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
⊢ (γ ∘ η) ∘ η = γ
Module: Mathlib.Geometry.Manifold.Sheaf.LocallyRingedSpace

Goal state before:
pp: 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
f : ↑((smoothSheafCommRing IM 𝓘(𝕜, 𝕜) M 𝕜).presheaf.stalk x)
⊢ IsUnit f ↔ f ∉ ↑(RingHom.ker (eval IM 𝓘(𝕜, 𝕜) M 𝕜 x))

Tactic:
apply smoothSheafCommRing.isUnit_stalk_iff

Goal state after:
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
⊢ ↑(h.codChart.extend J) '' f '' h.domChart.source ⊆ (h.codChart.extend J).target

Tactic:
← PartialEquiv.image_source_eq_target

Goal state after:
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
⊢ ↑(h.codChart.extend J) '' f '' h.domChart.source ⊆ ↑(h.codChart.extend J) '' (h.codChart.extend J).source
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
equiv : E ≃L[𝕜] E'' × F
domChart : OpenPartialHomeomorph M H
codChart : OpenPartialHomeomorph N G
hx : x ∈ domChart.source
hfx : f x ∈ codChart.source
hdomChart : domChart ∈ IsManifold.maximalAtlas I n M
hcodChart : codChart ∈ IsManifold.maximalAtlas J n N
hsource : domChart.source ⊆ f ⁻¹' codChart.source
hwrittenInExtend :
  EqOn (↑(codChart.extend J) ∘ f ∘ ↑(domChart.extend I).symm) (Prod.fst ∘ ⇑equiv) (domChart.extend I).target
⊢ IsSubmersionAt I J n f x

Tactic:
have aux : IsSubmersionAtOfComplement F I J n f x := by
    apply IsSubmersionAtOfComplement.mk_of_charts <;> assumption

Goal state after:
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
equiv : E ≃L[𝕜] E'' × F
domChart : OpenPartialHomeomorph M H
codChart : OpenPartialHomeomorph N G
hx : x ∈ domChart.source
hfx : f x ∈ codChart.source
hdomChart : domChart ∈ IsManifold.maximalAtlas I n M
hcodChart : codChart ∈ IsManifold.maximalAtlas J n N
hsource : domChart.source ⊆ f ⁻¹' codChart.source
hwrittenInExtend :
  EqOn (↑(codChart.extend J) ∘ f ∘ ↑(domChart.extend I).symm) (Prod.fst ∘ ⇑equiv) (domChart.extend I).target
aux : IsSubmersionAtOfComplement F I J n f x
⊢ IsSubmersionAt I J n f x
Module: Mathlib.Geometry.Manifold.Submersion

Goal state before:
pp: case h
𝕜 : Type u_1
E'' : Type u_3
H : Type u_7
G : Type u_9
E : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup E''
inst✝⁶ : NormedSpace 𝕜 E''
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
h : IsSubmersion I J n f
x : M
⊢ IsSubmersionAtOfComplement h.complement I J n f x

Tactic:
exact h.isSubmersionOfComplement_complement x

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

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 : FiberBundle F V
ι : Type u_7
s : ι → (x : M) → V x
u : Set M
n : ℕ∞ω
x : M
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), ContMDiffAt I 𝓘(𝕜, 𝕜) n ((LinearMap.piApply (hs.coeff i)) t) x
hu : IsOpen[inst✝¹⁰] u
hx : x ∈ u
⊢ ContMDiffAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, t x⟩) x

Tactic:
by
  have := fintypeOfFiniteDimensional hs hx
  exact hs.contMDiffAt_of_coeff h (hu.mem_nhds hx)

Goal state after:
Module: Mathlib.GroupTheory.GroupAction.CardCommute

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : Group α
inst✝⁴ : MulAction α β
inst✝³ : Fintype α
inst✝² : Fintype β
inst✝¹ : Fintype (Quotient (orbitRel α β))
inst✝ : (b : β) → Fintype ↥(stabilizer α b)
φ : Quotient (orbitRel α β) → β
hφ : LeftInverse Quotient.mk'' φ
⊢ Fintype.card β = ∑ ω, Fintype.card α / Fintype.card ↥(stabilizer α (φ ω))

Tactic:
by
  classical
    have : ∀ ω : Ω, Fintype.card α / Fintype.card (stabilizer α (φ ω)) =
        Fintype.card (α ⧸ stabilizer α (φ ω)) := by
      intro ω
      rw [Fintype.card_congr (@Subgroup.groupEquivQuotientProdSubgroup α _ (stabilizer α <| φ ω)),
        Fintype.card_prod, Nat.mul_div_cancel]
      exact Fintype.card_pos_iff.mpr (by infer_instance)
    simp_rw [this, ← Fintype.card_sigma,
      Fintype.card_congr (selfEquivSigmaOrbitsQuotientStabilizer' α β hφ)]

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

Goal state before:
pp: n : ℕ
⊢ Function.LeftInverse
    (fun x ↦
      match x with
      | Sum.inl j => r j
      | Sum.inr j => sr j)
    fun x ↦
    match x with
    | r j => Sum.inl j
    | sr j => Sum.inr j

Tactic:
rintro (x | x) <;> rfl

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

Goal state before:
pp: n : ℕ
⊢ Function.RightInverse
    (fun x ↦
      match x with
      | Sum.inl j => r j
      | Sum.inr j => sr j)
    fun x ↦
    match x with
    | r j => Sum.inl j
    | sr j => Sum.inr j

Tactic:
rintro (x | x) <;> rfl

Goal state after:
Module: Mathlib.GroupTheory.CommutingProbability

Goal state before:
pp: case h
G : Type u_2
inst✝¹ : Group G
inst✝ : Finite G
H : Subgroup G
⊢ ↑(Nat.card G) ^ 2 ≠ 0

pp: G : Type u_2
inst✝¹ : Group G
inst✝ : Finite G
H : Subgroup G
⊢ 0 < ↑(Nat.card ↥H) ^ 2

Tactic:
· exact pow_ne_zero 2 (Nat.cast_ne_zero.mpr Finite.card_pos.ne')

Goal state after:
G : Type u_2
inst✝¹ : Group G
inst✝ : Finite G
H : Subgroup G
⊢ 0 < ↑(Nat.card ↥H) ^ 2
Module: Mathlib.GroupTheory.Coprod.Basic

Goal state before:
pp: M : Type u_1
N : Type u_2
P : Type u_3
inst✝³ : MulOneClass M
inst✝² : MulOneClass N
inst✝¹ : Monoid P
P' : Type u_4
inst✝ : Monoid P'
f : P →* P'
g₁ : M →* P
g₂ : N →* P
⊢ f.comp g₂ = (lift (f.comp g₁) (f.comp g₂)).comp inr

Tactic:
lift_comp_inr

Goal state after:
M : Type u_1
N : Type u_2
P : Type u_3
inst✝³ : MulOneClass M
inst✝² : MulOneClass N
inst✝¹ : Monoid P
P' : Type u_4
inst✝ : Monoid P'
f : P →* P'
g₁ : M →* P
g₂ : N →* P
⊢ f.comp g₂ = f.comp g₂
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: M : Type u_2
inst✝² : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
m n m' n' : M
s t s' t' : ℕ+
h : ↑s' • m = ↑s • m'
h' : ↑t' • n = ↑t • n'
⊢ (↑s' * ↑t') • ↑t • m ≤ (↑s' * ↑t') • ↑s • n ↔ ↑t' • m' ≤ ↑s' • n'

Tactic:
convert! (nsmul_le_nsmul_iff_right (M := M) (mul_ne_zero s.ne_zero t.ne_zero)) using 2

Goal state after:
case e'_1.e'_3
M : Type u_2
inst✝² : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
m n m' n' : M
s t s' t' : ℕ+
h : ↑s' • m = ↑s • m'
h' : ↑t' • n = ↑t • n'
⊢ (↑s' * ↑t') • ↑t • m = (↑s * ↑t) • ↑t' • m'

case e'_1.e'_4
M : Type u_2
inst✝² : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
m n m' n' : M
s t s' t' : ℕ+
h : ↑s' • m = ↑s • m'
h' : ↑t' • n = ↑t • n'
⊢ (↑s' * ↑t') • ↑s • n = (↑s * ↑t) • ↑s' • n'
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
g : G
this : mk H K g = mk H K g ↔ ∃ h ∈ H, ∃ k ∈ K, g = h * Quotient.out (mk H K g) * k
h : G
h_h : h ∈ H
k : G
hk : k ∈ K
T : g = h * Quotient.out (mk H K g) * k
⊢ h * Quotient.out (mk H K g) * k = g

Tactic:
← T

Goal state after:
G : Type u_1
inst✝ : Group G
H K : Subgroup G
g : G
this : mk H K g = mk H K g ↔ ∃ h ∈ H, ∃ k ∈ K, g = h * Quotient.out (mk H K g) * k
h : G
h_h : h ∈ H
k : G
hk : k ∈ K
T : g = h * Quotient.out (mk H K g) * k
⊢ g = g
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: case h
G : Type u_1
inst✝ : Group G
H K : Subgroup G
x : G
⊢ ∃ x_1 ∈ H, ∃ y ∈ K, x = x_1 * Quotient.out (mk H K x) * y

Tactic:
obtain ⟨h, k, h3, h4, h5⟩ := mk_out_eq_mul H K x

Goal state after:
case h
G : Type u_1
inst✝ : Group G
H K : Subgroup G
x h k : G
h3 : h ∈ H
h4 : k ∈ K
h5 : Quotient.out (mk H K x) = h * x * k
⊢ ∃ x_1 ∈ H, ∃ y ∈ K, x = x_1 * Quotient.out (mk H K x) * y
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
a : ℚ
ha : 0 < a
b c : DivisibleHull M
h : b < c
⊢ a • b < a • c

Tactic:
simp_rw [qsmul_of_nonneg ha.le]
    apply smul_lt_smul_of_pos_left h (by simpa using! ha)

Goal state after:
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.cast⟩.snd.cast) * Quotient.out (Quotient.out ⟨q, -1 + k.cast⟩.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:
cast_id',

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, -1 + id k⟩.snd.cast) * Quotient.out (Quotient.out ⟨q, -1 + id 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)
Module: Mathlib.GroupTheory.Schreier

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
R S : Set G
hR : IsComplement (↑H) R
hR1 : 1 ∈ R
hS : closure S = ⊤
hU : closure ((fun g ↦ g * (↑(hR.toRightFun g))⁻¹) '' (R * S)) ≤ H
g : G
hg : g ∈ ↑(closure ((fun g ↦ g * (↑(hR.toRightFun g))⁻¹) '' (R * S)))
r : G
hr : r ∈ R
hh : (fun x1 x2 ↦ x1 * x2) g r ∈ H
⊢ (fun x1 x2 ↦ x1 * x2) g r ∈ closure ((fun g ↦ g * (↑(hR.toRightFun g))⁻¹) '' (R * S))

Tactic:
suffices (⟨r, hr⟩ : R) = (⟨1, hR1⟩ : R) by
    simpa only [show r = 1 from Subtype.ext_iff.mp this, mul_one]

Goal state after:
G : Type u_1
inst✝ : Group G
H : Subgroup G
R S : Set G
hR : IsComplement (↑H) R
hR1 : 1 ∈ R
hS : closure S = ⊤
hU : closure ((fun g ↦ g * (↑(hR.toRightFun g))⁻¹) '' (R * S)) ≤ H
g : G
hg : g ∈ ↑(closure ((fun g ↦ g * (↑(hR.toRightFun g))⁻¹) '' (R * S)))
r : G
hr : r ∈ R
hh : (fun x1 x2 ↦ x1 * x2) g r ∈ H
⊢ ⟨r, hr⟩ = ⟨1, hR1⟩
Module: Mathlib.GroupTheory.Transfer

Goal state before:
pp: case neg
G : Type u_3
inst✝¹ : Group G
inst✝ : Finite G
P : Sylow (Nat.card G).minFac G
hP : IsCyclic ↥↑P
hn : ¬Nat.card G = 1
this : Fact (Nat.Prime (Nat.card G).minFac)
key : Nat.card (↥(normalizer ↑↑P) ⧸ (↑P).normalizerMonoidHom.ker) ∣ Nat.card (MulAut ↥↑P)
⊢ normalizer ↑P ≤ centralizer ↑P

Tactic:
rw [normalizerMonoidHom_ker, ← index, ← relIndex] at key

Goal state after:
case neg
G : Type u_3
inst✝¹ : Group G
inst✝ : Finite G
P : Sylow (Nat.card G).minFac G
hP : IsCyclic ↥↑P
hn : ¬Nat.card G = 1
this : Fact (Nat.Prime (Nat.card G).minFac)
key : (centralizer ↑↑P).relIndex (normalizer ↑↑P) ∣ Nat.card (MulAut ↥↑P)
⊢ normalizer ↑P ≤ centralizer ↑P
Module: Mathlib.GroupTheory.GroupAction.Transitive

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
⊢ (∀ (x : X), ∃ g, g • a = x) ↔ ∀ (x : X), x ∈ orbit G a ↔ x ∈ Set.univ

Tactic:
apply forall_congr'

Goal state after:
G : Type u_1
X : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
⊢ ∀ (a_1 : X), (∃ g, g • a = a_1) ↔ (a_1 ∈ orbit G a ↔ a_1 ∈ Set.univ)
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝ : SMul G X
B : Set X
s : Set G
g : G
hB : IsBlock G B
hgs : ¬g • B ⊆ s • B
⊢ Disjoint (g • B) (⋃ a ∈ s, a • B)

Tactic:
disjoint_iUnion₂_right

Goal state after:
G : Type u_1
X : Type u_2
inst✝ : SMul G X
B : Set X
s : Set G
g : G
hB : IsBlock G B
hgs : ¬g • B ⊆ s • B
⊢ ∀ i ∈ s, Disjoint (g • B) (i • B)
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
hG : Group G
inst✝ : Finite G
e : ((p : ↥(Nat.card G).primeFactors) → (P : Sylow (↑p) G) → ↥↑P) ≃* G
ps : Finset ℕ := (Nat.card G).primeFactors
⊢ IsNilpotent G

Tactic:
have : ∀ (p : ps) (P : Sylow p G), IsNilpotent (↑P : Subgroup G) := by
    intro p P
    have : Fact (Nat.Prime ↑p) := Fact.mk <| Nat.prime_of_mem_primeFactors p.2
    exact P.isPGroup'.isNilpotent

Goal state after:
G : Type u_1
hG : Group G
inst✝ : Finite G
e : ((p : ↥(Nat.card G).primeFactors) → (P : Sylow (↑p) G) → ↥↑P) ≃* G
ps : Finset ℕ := (Nat.card G).primeFactors
this : ∀ (p : ↥ps) (P : Sylow (↑p) G), IsNilpotent ↥↑P
⊢ IsNilpotent G
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
B : Set X
H : Type u_3
Y : Type u_4
inst✝ : SMul H Y
φ : G → H
j : X →ₑ[φ] Y
hφ : Function.Surjective φ
hj : Function.Injective ⇑j
hB : IsBlock G B
⊢ IsBlock H (⇑j '' B)

Tactic:
simp only [IsBlock, hφ.forall, ← image_smul_setₛₗ]

Goal state after:
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
B : Set X
H : Type u_3
Y : Type u_4
inst✝ : SMul H Y
φ : G → H
j : X →ₑ[φ] Y
hφ : Function.Surjective φ
hj : Function.Injective ⇑j
hB : IsBlock G B
⊢ ∀ (x x_1 : G), ⇑j '' (x • B) ≠ ⇑j '' (x_1 • B) → Disjoint (⇑j '' (x • B)) (⇑j '' (x_1 • B))
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: case right
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
N : Subgroup G
inst✝ : N.Normal
⊢ ∀ ⦃B : Set X⦄, (B ∈ range fun a ↦ orbit (↥N) a) → IsBlock G B

Tactic:
intro b

Goal state after:
case right
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
N : Subgroup G
inst✝ : N.Normal
b : Set X
⊢ (b ∈ range fun a ↦ orbit (↥N) a) → IsBlock G b
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
⊢ IsBlock G (orbit (↥H) a)

Tactic:
rw [isBlock_iff_smul_eq_of_nonempty]
  rintro g ⟨-, ⟨-, ⟨h₁, rfl⟩, h⟩, h₂, rfl⟩
  suffices g ∈ H by
    rw [← Subgroup.coe_mk H g this, ← H.toSubmonoid.smul_def, smul_orbit (⟨g, this⟩ : H) a]
  rw [← mul_mem_cancel_left h₂⁻¹.2, ← mul_mem_cancel_right h₁.2]
  apply hH
  simpa only [mem_stabilizer_iff, InvMemClass.coe_inv, mul_smul, inv_smul_eq_iff]

Goal state after:
Module: Mathlib.GroupTheory.IndexNormal

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : H.index = 2
x : G
hxH : x ∈ H
g : G
⊢ g * x * g⁻¹ ∈ H

Tactic:
simp_rw [mul_mem_iff_of_index_two hH, hxH, iff_true, inv_mem_iff]

Goal state after:
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
B✝ : Set X
a : X
B : Set X
ha : a ∈ B
hB : IsBlock G B
⊢ ⟨B, ⋯⟩ ≤ ⟨univ, ⋯⟩

Tactic:
simp only [Subtype.mk_le_mk, subset_univ]

Goal state after:
Module: Mathlib.GroupTheory.Perm.ConjAct

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
⊢ c ∈ (k * g * k⁻¹).cycleFactorsFinset ↔ c ∈ Finset.map (MulAut.conj k).toEmbedding g.cycleFactorsFinset

Tactic:
Finset.mem_map_equiv,

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
⊢ c ∈ (k * g * k⁻¹).cycleFactorsFinset ↔ (MulAut.conj k).symm c ∈ g.cycleFactorsFinset
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
inst✝ : IsPretransitive G X
B : Set X
hX : Finite X
hB : IsBlock G B
hB' : Nat.card X < B.ncard * 2
⊢ B = Set.univ

Tactic:
rcases Set.eq_empty_or_nonempty B with rfl | hB_ne

Goal state after:
case inl
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
inst✝ : IsPretransitive G X
hX : Finite X
hB : IsBlock G ∅
hB' : Nat.card X < ∅.ncard * 2
⊢ ∅ = Set.univ

case inr
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
inst✝ : IsPretransitive G X
B : Set X
hX : Finite X
hB : IsBlock G B
hB' : Nat.card X < B.ncard * 2
hB_ne : B.Nonempty
⊢ B = Set.univ
Module: Mathlib.GroupTheory.GroupAction.Jordan

Goal state before:
pp: G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hsn' : 2 < ENat.card α
hG' : IsMultiplyPretransitive G α 2
a : α
this✝ : IsPretransitive G α
this : Nontrivial α
⊢ IsCoatom (stabilizer G a)

Tactic:
rw [isCoatom_stabilizer_iff_preprimitive]

Goal state after:
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hsn' : 2 < ENat.card α
hG' : IsMultiplyPretransitive G α 2
a : α
this✝ : IsPretransitive G α
this : Nontrivial α
⊢ IsPreprimitive G α
Module: Mathlib.GroupTheory.GroupAction.MultiplePrimitivity

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝² : Group M
inst✝¹ : MulAction M α
n : ℕ
hα : ↑n.succ ≤ ENat.card α
inst✝ : IsMultiplyPretransitive M α n.succ
hn : n > 0
s : Set α
m : ℕ
hm : n = Nat.succ 0 + m
hs : s.encard + 1 = ↑(m + 1)
⊢ s.encard = ↑m

Tactic:
exact ENat.add_left_injective_of_ne_top ENat.one_ne_top hs

Goal state after:
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s : Set α
hs : ∀ a ∈ s, ∀ b ∈ s, ∃ g, g • a = b
hs' : ∀ a ∈ sᶜ, ∀ b ∈ sᶜ, ∃ g, g • a = b
hM : stabilizer M s ≠ ⊤
this : ∃ a b g, a ∈ s ∧ b ∈ sᶜ ∧ g • a = b
⊢ IsPretransitive M α

Tactic:
by
    obtain ⟨a, b, g, ha, hb, hgab⟩ := this
    rw [isPretransitive_iff_base a]
    intro x
    by_cases hx : x ∈ s
    · exact hs a ha x hx
    · rw [← Set.mem_compl_iff] at hx
      obtain ⟨k, hk⟩ := hs' b hb x hx
      use k * g
      rw [mul_smul, hgab, hk]

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
a b c : α
ha : a ∈ s
hb : b ∈ s
hc : c ∈ s
hab : a ≠ b
hac : a ≠ c
hbc : b ≠ c
⊢ swap a b * swap a c ∈ stabilizer (Perm α) s

Tactic:
rw [mem_stabilizer_set_iff_subset_smul_set s.toFinite, subset_smul_set_iff]

Goal state after:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
a b c : α
ha : a ∈ s
hb : b ∈ s
hc : c ∈ s
hab : a ≠ b
hac : a ≠ c
hbc : b ≠ c
⊢ (swap a b * swap a c)⁻¹ • s ⊆ s