mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: case hx
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁶ : TopologicalSpace H
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : IsManifold I (minSmoothness 𝕜 3) M
inst✝ : CompleteSpace E
U V W : (x : M) → TangentSpace I x
s : Set M
x : M
hs : UniqueMDiff[s]
h's : x ∈ closure[inst✝³] (interior s)
hx : x ∈ s
hU : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, U x⟩) s x
hV : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, V x⟩) s x
hW : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, W x⟩) s x
A : minSmoothness 𝕜 2 + 1 ≤ minSmoothness 𝕜 3
s_inter_mem : s ∩ (extChartAt I x).source ∈ 𝓝[s] x
pre_mem : ↑(extChartAt I x) ⁻¹' ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) ∈ 𝓝[s] x
U' : (x : E) → TangentSpace 𝓘(𝕜, E) x := ⋯
V' : (x : E) → TangentSpace 𝓘(𝕜, E) x := ⋯
W' : (x : E) → TangentSpace 𝓘(𝕜, E) x := ⋯
J0U :
  ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, U' x⟩)
    ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) x)
J0V :
  ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, V' x⟩)
    ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) x)
J0W :
  ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, W' x⟩)
    ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) x)
J1U :
  ∀ᶠ (y : M) in 𝓝[s] x,
    ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, U' x⟩)
      ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) y)
J1V :
  ∀ᶠ (y : M) in 𝓝[s] x,
    ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, V' x⟩)
      ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) y)
J1W :
  ∀ᶠ (y : M) in 𝓝[s] x,
    ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, W' x⟩)
      ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) y)
JU : U =ᶠ[𝓝[s] x] mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) U'
JV : V =ᶠ[𝓝[s] x] mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) V'
JW : W =ᶠ[𝓝[s] x] mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) W'
this :
  ∀ᶠ (y : M) in 𝓝[s] x,
    mlieBracketWithin I (mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) V') (mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) W') s y =
      mpullback I 𝓘(𝕜, E) (↑(extChartAt I x))
        (mlieBracketWithin 𝓘(𝕜, E) V' W' ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s)) y
⊢ ↑(extChartAt I x) x ∈ (extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s

Tactic:
exact ⟨mem_extChartAt_target x, by simp [hx]⟩

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

Goal state before:
pp: case hx
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁶ : TopologicalSpace H
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : IsManifold I (minSmoothness 𝕜 3) M
inst✝ : CompleteSpace E
U V W : (x : M) → TangentSpace I x
s : Set M
x : M
hs : UniqueMDiff[s]
h's : x ∈ closure[inst✝³] (interior s)
hx : x ∈ s
hU : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, U x⟩) s x
hV : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, V x⟩) s x
hW : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, W x⟩) s x
A : minSmoothness 𝕜 2 + 1 ≤ minSmoothness 𝕜 3
s_inter_mem : s ∩ (extChartAt I x).source ∈ 𝓝[s] x
pre_mem : ↑(extChartAt I x) ⁻¹' ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) ∈ 𝓝[s] x
U' : (x : E) → TangentSpace 𝓘(𝕜, E) x := ⋯
V' : (x : E) → TangentSpace 𝓘(𝕜, E) x := ⋯
W' : (x : E) → TangentSpace 𝓘(𝕜, E) x := ⋯
J0U :
  ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, U' x⟩)
    ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) x)
J0V :
  ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, V' x⟩)
    ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) x)
J0W :
  ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, W' x⟩)
    ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) x)
J1U :
  ∀ᶠ (y : M) in 𝓝[s] x,
    ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, U' x⟩)
      ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) y)
J1V :
  ∀ᶠ (y : M) in 𝓝[s] x,
    ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, V' x⟩)
      ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) y)
J1W :
  ∀ᶠ (y : M) in 𝓝[s] x,
    ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, W' x⟩)
      ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) y)
JU : U =ᶠ[𝓝[s] x] mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) U'
JV : V =ᶠ[𝓝[s] x] mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) V'
JW : W =ᶠ[𝓝[s] x] mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) W'
this✝ :
  ∀ᶠ (y : M) in 𝓝[s] x,
    mlieBracketWithin I (mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) V') (mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) W') s y =
      mpullback I 𝓘(𝕜, E) (↑(extChartAt I x))
        (mlieBracketWithin 𝓘(𝕜, E) V' W' ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s)) y
this :
  ∀ᶠ (y : M) in 𝓝[s] x,
    mlieBracketWithin I (mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) U') (mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) V') s y =
      mpullback I 𝓘(𝕜, E) (↑(extChartAt I x))
        (mlieBracketWithin 𝓘(𝕜, E) U' V' ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s)) y
⊢ ↑(extChartAt I x) x ∈ (extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s

Tactic:
exact ⟨mem_extChartAt_target x, by simp [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
hxs' : x ∈ s
ψ : M → 𝕜 := fun x' ↦ if x' ∈ s then 1 else 0
hψx : ψ x = 1
H : ∀ (x' : M), (ψ • σ) x' = (ψ • σ') x'
hψ' : HasMFDerivAt% ψ x 0
⊢ cov σ x = cov σ' x

Tactic:
have := hcov.leibniz hσ hψ'.mdifferentiableAt

Goal state after:
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
hxs' : x ∈ s
ψ : M → 𝕜 := fun x' ↦ if x' ∈ s then 1 else 0
hψx : ψ x = 1
H : ∀ (x' : M), (ψ • σ) x' = (ψ • σ') x'
hψ' : HasMFDerivAt% ψ x 0
this : cov (ψ • σ) x = ψ x • cov σ x + (d% ψ x).smulRight (σ x)
⊢ cov σ x = cov σ' x
Module: Mathlib.GroupTheory.Congruence.Star

Goal state before:
pp: A : Type u_1
inst✝¹ : AddMonoid A
inst✝ : StarAddMonoid A
r : A → A → Prop
hr : ∀ (a b : A), r a b → r (Star.star a) (Star.star b)
a b w✝ x✝ y✝ z✝ : A
h1 : Rel r w✝ x✝
h2 : Rel r y✝ z✝
⊢ Rel r (Star.star w✝ + Star.star y✝) (Star.star (x✝ + z✝))

Tactic:
star_add

Goal state after:
A : Type u_1
inst✝¹ : AddMonoid A
inst✝ : StarAddMonoid A
r : A → A → Prop
hr : ∀ (a b : A), r a b → r (Star.star a) (Star.star b)
a b w✝ x✝ y✝ z✝ : A
h1 : Rel r w✝ x✝
h2 : Rel r y✝ z✝
⊢ Rel r (Star.star w✝ + Star.star y✝) (Star.star x✝ + Star.star z✝)
Module: Mathlib.GroupTheory.CommutingProbability

Goal state before:
pp: case neg.inl
n : ℕ
h0 : ¬n = 0
h1 : ¬n = 1
h2 : Even n
⊢ commProb (Product (reciprocalFactors n)) = 1 / ↑n

Tactic:
rw [reciprocalFactors_even h0 h2, commProb_cons, commProb_reciprocal (n / 2),
        commProb_odd (by decide)]
    simp [field, h2.two_dvd]
    norm_num

Goal state after:
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: case mul
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)
x y : CoprodI G
ihx : MulOpposite.unop ((lift fun i ↦ (MonoidHom.op of).comp (MulEquiv.inv' (G i)).toMonoidHom) x) * x = 1
ihy : MulOpposite.unop ((lift fun i ↦ (MonoidHom.op of).comp (MulEquiv.inv' (G i)).toMonoidHom) y) * y = 1
⊢ MulOpposite.unop ((lift fun i ↦ (MonoidHom.op of).comp (MulEquiv.inv' (G i)).toMonoidHom) (x * y)) * (x * y) = 1

Tactic:
| mul x y ihx ihy =>

Goal state after:
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
M : ι → Type u_2
inst✝³ : (i : ι) → Monoid (M i)
N : Type u_3
inst✝² : Monoid N
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
i : ι
w : Word M
⊢ rcons { head := 1, tail := empty, fstIdx_ne := ⋯ } = empty

Tactic:
by simp [rcons]

Goal state after:
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
M : ι → Type u_2
inst✝³ : (i : ι) → Monoid (M i)
N : Type u_3
inst✝² : Monoid N
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
i : ι
w✝ : Word M
j : ι
m : M j
w : Word M
h1 : w.fstIdx ≠ some j
h2 : m ≠ 1
x✝ : { p // rcons p = w }
ij : ¬i = j
⊢ rcons { head := 1, tail := cons m w h1 h2, fstIdx_ne := ⋯ } = cons m w h1 h2

Tactic:
by simp [rcons]

Goal state after:
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: case neg
ι : Type u_1
M : ι → Type u_2
inst✝² : (i : ι) → Monoid (M i)
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
j : ι
m₂ : M j
w : Word M
m₁ : M j
hw : ⟨j, m₁⟩ ∉ w.toList.tail
⊢ (⟨j, m₁⟩ ∈ w.toList.tail ∨
      ¬m₁ = 1 ∧
        m₁ = m₂ * if h : ∃ (h : ¬w.toList = []), (w.toList.head h).fst = j then ⋯ ▸ (w.toList.head ⋯).snd else 1) ↔
    ¬m₁ = 1 ∧ (⟨j, m₁⟩ ∈ w.toList.tail ∨ (∃ m', ⟨j, m'⟩ ∈ w.toList.head? ∧ m₁ = m₂ * m') ∨ ¬w.fstIdx = some j ∧ m₁ = m₂)

Tactic:
simp only [hw, false_or, Option.mem_def, and_congr_right_iff]

Goal state after:
case neg
ι : Type u_1
M : ι → Type u_2
inst✝² : (i : ι) → Monoid (M i)
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
j : ι
m₂ : M j
w : Word M
m₁ : M j
hw : ⟨j, m₁⟩ ∉ w.toList.tail
⊢ ¬m₁ = 1 →
    ((m₁ = m₂ * if h : ∃ (h : ¬w.toList = []), (w.toList.head h).fst = j then ⋯ ▸ (w.toList.head ⋯).snd else 1) ↔
      (∃ m', w.toList.head? = some ⟨j, m'⟩ ∧ m₁ = m₂ * m') ∨ ¬w.fstIdx = some j ∧ m₁ = m₂)
Module: Mathlib.GroupTheory.Coxeter.Length

Goal state before:
pp: B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
w : W
i : B
this : cs.length (w⁻¹ * cs.simple i) = cs.length w⁻¹ + 1 ∨ cs.length (w⁻¹ * cs.simple i) + 1 = cs.length w⁻¹
⊢ cs.length (cs.simple i * w) = cs.length w + 1 ∨ cs.length (cs.simple i * w) + 1 = cs.length w

Tactic:
(by simp : w⁻¹ * (s i) = ((s i) * w)⁻¹),

Goal state after:
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
w : W
i : B
this : cs.length (cs.simple i * w)⁻¹ = cs.length w⁻¹ + 1 ∨ cs.length (cs.simple i * w)⁻¹ + 1 = cs.length w⁻¹
⊢ cs.length (cs.simple i * w) = cs.length w + 1 ∨ cs.length (cs.simple i * w) + 1 = cs.length w
Module: Mathlib.GroupTheory.Descent

Goal state before:
pp: G : Type u_1
inst✝¹ : CommGroup G
n : ℕ
h : G → ℝ
a b c₀ : ℝ
c : G → ℝ
ha : 0 ≤ a
H₀ : a < b
H₁ : (powMonoidHom n).range.FiniteIndex
H₂ : ∀ (g x : G), h x ≤ a * h (g * x) + c g
H₃ : ∀ (x : G), b * h x - c₀ ≤ h (x ^ n)
inst✝ : Northcott h
f : G →* G := powMonoidHom n
q : G → G ⧸ f.range := QuotientGroup.mk
qi : G ⧸ f.range → G := Function.surjInv ⋯
s : Set G := Set.range qi
g : G
hg₁ : g ∈ s
hg₂ : ∀ b ∈ s, c b ≤ c g
H₁' : s * ↑f.range = Set.univ
c' : ℝ := max c₀ (c g)
⊢ Group.FG G

Tactic:
have H₃' x : b * h x - c' ≤ h (f x) := by grind [powMonoidHom_apply]

Goal state after:
G : Type u_1
inst✝¹ : CommGroup G
n : ℕ
h : G → ℝ
a b c₀ : ℝ
c : G → ℝ
ha : 0 ≤ a
H₀ : a < b
H₁ : (powMonoidHom n).range.FiniteIndex
H₂ : ∀ (g x : G), h x ≤ a * h (g * x) + c g
H₃ : ∀ (x : G), b * h x - c₀ ≤ h (x ^ n)
inst✝ : Northcott h
f : G →* G := powMonoidHom n
q : G → G ⧸ f.range := QuotientGroup.mk
qi : G ⧸ f.range → G := Function.surjInv ⋯
s : Set G := Set.range qi
g : G
hg₁ : g ∈ s
hg₂ : ∀ b ∈ s, c b ≤ c g
H₁' : s * ↑f.range = Set.univ
c' : ℝ := max c₀ (c g)
H₃' : ∀ (x : G), b * h x - c' ≤ h (f x)
⊢ Group.FG G
Module: Mathlib.GroupTheory.Schreier

Goal state before:
pp: case refine_1
G : Type u_1
inst✝ : Group G
H : Subgroup G
R S : Set G
hR : IsComplement (↑H) R
hR1 : 1 ∈ R
hS : closure S = ⊤
f : G → ↑R := hR.toRightFun
U : Set G := (fun g ↦ g * (↑(f g))⁻¹) '' (R * S)
g : G
x✝ : g ∈ ⊤
⊢ 1 ∈ ↑(closure U) * R

Tactic:
exact ⟨1, (closure U).one_mem, 1, hR1, one_mul 1⟩

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case neg
G : Type u_1
inst✝ : Group G
H : ℕ → Subgroup G
n : ℕ
hn : H n = ⊥
h0 : H 0 = ⊤
hH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), ⁅x, g⁆ ∈ H (n + 1)
x : G
m : ℕ
hx : x ∈ H (n - (m + 1))
g : G
hm : m < n
⊢ ⁅x, g⁆ ∈ H (n - m)

Tactic:
convert! hH x _ hx g using 1
    rw [tsub_add_eq_add_tsub (Nat.succ_le_of_lt hm), Nat.succ_eq_add_one, Nat.add_sub_add_right]

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
inst✝ : Group G
hG : Group.IsNilpotent G
⊢ Nat.find ⋯ = nilpotencyClass G

Tactic:
rw [← least_ascending_central_series_length_eq_nilpotencyClass]

Goal state after:
G : Type u_1
inst✝ : Group G
hG : Group.IsNilpotent G
⊢ Nat.find ⋯ = Nat.find ⋯
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
inst✝ : Group G
S : Subgroup G
n : ℕ
⊢ S.lowerCentralSeries n ≤ S

Tactic:
simpa using S.lowerCentralSeries_antitone (Nat.zero_le n)

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case succ
G : Type u_1
inst✝ : Group G
i : ℕ
ih : derivedSeries G i ≤ ⊤.lowerCentralSeries i
⊢ derivedSeries G (i + 1) ≤ ⊤.lowerCentralSeries (i + 1)

Tactic:
| succ i ih =>

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
inst✝⁵ : Group G
N : Subgroup G
inst✝⁴ : N.Normal
G₁ : Type u_2
G₂ : Type u_3
inst✝³ : Group G₁
inst✝² : Group G₂
inst✝¹ : IsNilpotent G₁
inst✝ : IsNilpotent G₂
⊢ ⊥.prod ⊥ = ⊥

Tactic:
bot_prod_bot

Goal state after:
G : Type u_1
inst✝⁵ : Group G
N : Subgroup G
inst✝⁴ : N.Normal
G₁ : Type u_2
G₂ : Type u_3
inst✝³ : Group G₁
inst✝² : Group G₂
inst✝¹ : IsNilpotent G₁
inst✝ : IsNilpotent G₂
⊢ ⊥ = ⊥
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case succ
η : Type u_2
Gs : η → Type u_3
inst✝ : (i : η) → Group (Gs i)
Ss : (i : η) → Subgroup (Gs i)
n : ℕ
ih : (pi Set.univ Ss).lowerCentralSeries n ≤ pi Set.univ fun i ↦ (Ss i).lowerCentralSeries n
⊢ (pi Set.univ Ss).lowerCentralSeries (n + 1) ≤ pi Set.univ fun i ↦ (Ss i).lowerCentralSeries (n + 1)

Tactic:
simp_rw [lowerCentralSeries_succ]

Goal state after:
case succ
η : Type u_2
Gs : η → Type u_3
inst✝ : (i : η) → Group (Gs i)
Ss : (i : η) → Subgroup (Gs i)
n : ℕ
ih : (pi Set.univ Ss).lowerCentralSeries n ≤ pi Set.univ fun i ↦ (Ss i).lowerCentralSeries n
⊢ ⁅(pi Set.univ Ss).lowerCentralSeries n, pi Set.univ Ss⁆ ≤ pi Set.univ fun i ↦ ⁅(Ss i).lowerCentralSeries n, Ss i⁆
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: η : Type u_2
Gs : η → Type u_3
inst✝² : (i : η) → Group (Gs i)
inst✝¹ : Fintype η
inst✝ : ∀ (i : η), IsNilpotent (Gs i)
⊢ nilpotencyClass ((i : η) → Gs i) = Finset.univ.sup fun i ↦ nilpotencyClass (Gs i)

Tactic:
by
  apply eq_of_forall_ge_iff
  intro k
  simp only [Finset.sup_le_iff, ← lowerCentralSeries_eq_bot_iff_nilpotencyClass_le,
    top_lowerCentralSeries_pi_of_finite, pi_eq_bot_iff, Finset.mem_univ, true_imp_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
a b : X
⊢ orbit G a = orbit G b ∨ Disjoint (orbit G a) (orbit G b)

Tactic:
apply (em (Disjoint (orbit G a) (orbit G b))).symm.imp _ id
  simp +contextual
    only [Set.not_disjoint_iff, ← orbit_eq_iff, forall_exists_index, eq_comm, implies_true]

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
hσ : σ ∈ alternatingGroup α
hind : ∀ (n : ℕ) (l : List (Perm α)), (∀ g ∈ l, g.IsSwap) → l.length = 2 * n → l.prod ∈ closure {σ | σ.IsThreeCycle}
⊢ σ ∈ closure {σ | σ.IsThreeCycle}

Tactic:
by
    obtain ⟨l, rfl, hl⟩ := truncSwapFactors σ
    obtain ⟨n, hn⟩ := (prod_list_swap_mem_alternatingGroup_iff_even_length hl).1 hσ
    rw [← two_mul] at hn
    exact hind n l hl hn

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

Goal state before:
pp: case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Nat.card α
⊢ {σ | σ.IsThreeCycle} ⊆ ↑(closure {g | g.cycleType = {2, 2}})

Tactic:
intro g hg3

Goal state after:
case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Nat.card α
g : Perm α
hg3 : g ∈ {σ | σ.IsThreeCycle}
⊢ g ∈ ↑(closure {g | g.cycleType = {2, 2}})
Module: Mathlib.GroupTheory.SpecificGroups.Alternating

Goal state before:
pp: g : Perm (Fin 5)
ha : (-1) ^ g.cycleType.card = 1
h1 : g ≠ 1
h2 : g.cycleType = Multiset.replicate g.cycleType.card 2
h✝ : g.cycleType.card * 2 ≤ card (Fin 5)
h : g.cycleType.card ≤ 3
⊢ -1 ≠ 1

pp: g : Perm (Fin 5)
ha : Even g.cycleType.card
h1 : g ≠ 1
h2 : g.cycleType = Multiset.replicate g.cycleType.card 2
h✝ : g.cycleType.card * 2 ≤ card (Fin 5)
h : g.cycleType.card ≤ 3
⊢ IsConj (swap 0 4 * swap 1 3) g

Tactic:
· decide

Goal state after:
g : Perm (Fin 5)
ha : Even g.cycleType.card
h1 : g ≠ 1
h2 : g.cycleType = Multiset.replicate g.cycleType.card 2
h✝ : g.cycleType.card * 2 ≤ card (Fin 5)
h : g.cycleType.card ≤ 3
⊢ IsConj (swap 0 4 * swap 1 3) g
Module: Mathlib.GroupTheory.SpecificGroups.Alternating

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Perm α)
hG : G.index = 2
a✝ : Nontrivial α
⊢ ∃ g, g.IsSwap ∧ g ∉ G

Tactic:
by_contra! h

Goal state after:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Perm α)
hG : G.index = 2
a✝ : Nontrivial α
h : ∀ (g : Perm α), g.IsSwap → g ∈ G
⊢ False
Module: Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s t : Set α
g : M
⊢ g ∈ fixingSubgroup M (s ∪ t) ↔ g ∈ fixingSubgroup M s ∧ g ∈ fixingSubgroup M t

Tactic:
simp [fixingSubgroup_union, Subgroup.mem_inf]

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

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝³ : Group M
inst✝² : MulAction M α
m n : ℕ
inst✝¹ : IsMultiplyPreprimitive M α n
s : Set α
inst✝ : Finite ↑s
hs : s.ncard + m = n
t : Set ↥(ofFixingSubgroup M s)
ht : t.encard + 1 = ↑m
⊢ IsPreprimitive ↥(fixingSubgroup (↥(fixingSubgroup M s)) t) ↥(ofFixingSubgroup (↥(fixingSubgroup M s)) t)

Tactic:
let t' : Set α := Subtype.val '' t

Goal state after:
M : Type u_1
α : Type u_2
inst✝³ : Group M
inst✝² : MulAction M α
m n : ℕ
inst✝¹ : IsMultiplyPreprimitive M α n
s : Set α
inst✝ : Finite ↑s
hs : s.ncard + m = n
t : Set ↥(ofFixingSubgroup M s)
ht : t.encard + 1 = ↑m
t' : Set α := Subtype.val '' t
⊢ IsPreprimitive ↥(fixingSubgroup (↥(fixingSubgroup M s)) t) ↥(ofFixingSubgroup (↥(fixingSubgroup M s)) t)
Module: Mathlib.GroupTheory.GroupAction.MultipleTransitivity

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h2 : 2 ≤ Nat.card α
h2le : Nat.card α - 2 ≤ Nat.card α
this✝ : IsMultiplyPretransitive (Equiv.Perm α) α (Nat.card α)
this : IsMultiplyPretransitive (Equiv.Perm α) α (Nat.card α - 2)
x y : Fin (Nat.card α - 2) ↪ α
g : Equiv.Perm α
hg : g • x = y
h : Equiv.Perm.sign g = -1
⊢ Nat.card α - (Nat.card α - 2) = 2

Tactic:
tsub_tsub_cancel_of_le h2

Goal state after:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h2 : 2 ≤ Nat.card α
h2le : Nat.card α - 2 ≤ Nat.card α
this✝ : IsMultiplyPretransitive (Equiv.Perm α) α (Nat.card α)
this : IsMultiplyPretransitive (Equiv.Perm α) α (Nat.card α - 2)
x y : Fin (Nat.card α - 2) ↪ α
g : Equiv.Perm α
hg : g • x = y
h : Equiv.Perm.sign g = -1
⊢ 2 = 2
Module: Mathlib.GroupTheory.GroupAction.MultipleTransitivity

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Nat.card α
h4 : 3 < Nat.card α
this : IsMultiplyPretransitive (↥(alternatingGroup α)) α (Nat.card α - 2) := isMultiplyPretransitive α
⊢ 2 + 2 ≤ Nat.card α - 2 + 2

Tactic:
Nat.sub_add_cancel (le_of_lt h2)

Goal state after:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Nat.card α
h4 : 3 < Nat.card α
this : IsMultiplyPretransitive (↥(alternatingGroup α)) α (Nat.card α - 2) := isMultiplyPretransitive α
⊢ 2 + 2 ≤ Nat.card α
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: α : Type u_2
inst✝ : DecidableEq α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < t.encard → ∃ g, g.IsSwap ∧ g ∈ stabilizer (Perm α) t
h1' : s.encard ≤ 1
h1c' : sᶜ.encard ≤ 1
⊢ univ.encard = 2

Tactic:
rw [← Set.encard_add_encard_compl s]

Goal state after:
α : Type u_2
inst✝ : DecidableEq α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < t.encard → ∃ g, g.IsSwap ∧ g ∈ stabilizer (Perm α) t
h1' : s.encard ≤ 1
h1c' : sᶜ.encard ≤ 1
⊢ s.encard + sᶜ.encard = 2
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s B : Set α
G : Subgroup ↥(alternatingGroup α)
hs : s.Nontrivial
hB_ss_sc : B ⊂ sᶜ
hG : stabilizer (↥(alternatingGroup α)) s ≤ G
hB : IsBlock (↥G) B
⊢ B.Subsingleton

Tactic:
by
  apply hB.subsingleton_of_ssubset_of_stabilizer_le hB_ss_sc
  intro g
  obtain ⟨⟨k, hk⟩, rfl⟩ := alternatingGroup.stabilizer.surjective_toPerm (by rwa [compl_compl]) g
  rw [stabilizer_compl] at hk
  exact ⟨⟨⟨k, hG hk⟩, by aesop⟩, rfl⟩

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

Goal state before:
pp: case inl
α : Type u_1
G : Subgroup (Perm α)
inst✝¹ : DecidableEq α
inst✝ : Finite α
hG : IsPreprimitive (↥G) α
g : Perm α
h2g : g.IsSwap
hg : g ∈ G
this✝ : Fintype α
hα3 : Nat.card α ≤ 2
this : Nonempty ↥G
⊢ orderOf g ∣ Fintype.card ↥G

Tactic:
orderOf_submonoid ⟨g, hg⟩

Goal state after:
case inl
α : Type u_1
G : Subgroup (Perm α)
inst✝¹ : DecidableEq α
inst✝ : Finite α
hG : IsPreprimitive (↥G) α
g : Perm α
h2g : g.IsSwap
hg : g ∈ G
this✝ : Fintype α
hα3 : Nat.card α ≤ 2
this : Nonempty ↥G
⊢ orderOf ⟨g, hg⟩ ∣ Fintype.card ↥G