mathlib-initiative/mathlib-tactics huggingface dataset:
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
δ : ℝ
δpos : 0 < δ
hδ : (r' ^ 2)⁻¹ < 1 - δ * C
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_cont :
  Continuous[inst✝⁷, topologicalSpaceTotalSpace (RingHom.id ℝ) F E (F →L[ℝ] ℝ) fun x ↦ E x →L[ℝ] ℝ] fun x ↦
    TotalSpace.mk' (F →L[ℝ] F →L[ℝ] ℝ) x (g x)
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = ((g x) v) w
g' : B → F →L[ℝ] F →L[ℝ] ℝ := fun y ↦ inCoordinates F E (F →L[ℝ] ℝ) (fun x ↦ E x →L[ℝ] ℝ) x y x y (g y)
hg' : ContinuousAt g' x
this✝ : ∀ᶠ (y : B) in 𝓝 x, dist (g' y) (g' x) < δ
y : B
hy : dist (g' y) (g' x) < δ
h'y : y ∈ (trivializationAt F E x).baseSet
this : ‖g' x - g' y‖ ≤ δ
v : E y
⊢ ‖(Trivialization.symmL ℝ (trivializationAt F E x) x ∘SL
          Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) y)
        v‖ ≤
    r' * ‖v‖

Tactic:
let w := (trivializationAt F E x).continuousLinearMapAt ℝ y v

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
hδ : (r' ^ 2)⁻¹ < 1 - δ * C
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_cont :
  Continuous[inst✝⁷, topologicalSpaceTotalSpace (RingHom.id ℝ) F E (F →L[ℝ] ℝ) fun x ↦ E x →L[ℝ] ℝ] fun x ↦
    TotalSpace.mk' (F →L[ℝ] F →L[ℝ] ℝ) x (g x)
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = ((g x) v) w
g' : B → F →L[ℝ] F →L[ℝ] ℝ := fun y ↦ inCoordinates F E (F →L[ℝ] ℝ) (fun x ↦ E x →L[ℝ] ℝ) x y x y (g y)
hg' : ContinuousAt g' x
this✝ : ∀ᶠ (y : B) in 𝓝 x, dist (g' y) (g' x) < δ
y : B
hy : dist (g' y) (g' x) < δ
h'y : y ∈ (trivializationAt F E x).baseSet
this : ‖g' x - g' y‖ ≤ δ
v : E y
w : F := (Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) y) v
⊢ ‖(Trivialization.symmL ℝ (trivializationAt F E x) x ∘SL
          Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) y)
        v‖ ≤
    r' * ‖v‖
Module: Mathlib.GroupTheory.SpecificGroups.Dihedral

Goal state before:
pp: n : ℕ
x✝¹ : n + 3 ≠ 1
x✝ : n + 3 ≠ 2
h' : IsMulCommutative (DihedralGroup (n + 3))
this : ZMod.val 2 = 0
⊢ False

Tactic:
ZMod.val_two_eq_two_mod

Goal state after:
n : ℕ
x✝¹ : n + 3 ≠ 1
x✝ : n + 3 ≠ 2
h' : IsMulCommutative (DihedralGroup (n + 3))
this : 2 % (n + 3) = 0
⊢ False
Module: Mathlib.GroupTheory.CommutingProbability

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

Tactic:
div_one

Goal state after:
case pos
n : ℕ
h0 : ¬n = 0
h1 : n = 1
⊢ 1 = 1
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 y : B
hy :
  ‖Trivialization.symmL ℝ (trivializationAt F E x) x ∘SL
        Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) y‖ <
    2
v : F
⊢ (Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x ∘SL
        Trivialization.symmL ℝ (trivializationAt F E x) x)
      v =
    (ContinuousLinearMap.id ℝ F) v

Tactic:
have h'x : x ∈ (trivializationAt F E x).baseSet := FiberBundle.mem_baseSet_trivializationAt' x

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 y : B
hy :
  ‖Trivialization.symmL ℝ (trivializationAt F E x) x ∘SL
        Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) y‖ <
    2
v : F
h'x : x ∈ (trivializationAt F E x).baseSet
⊢ (Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x ∘SL
        Trivialization.symmL ℝ (trivializationAt F E x) x)
      v =
    (ContinuousLinearMap.id ℝ F) v
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
⊢ ∀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:
have h'x : x ∈ (trivializationAt F E x).baseSet := FiberBundle.mem_baseSet_trivializationAt' x

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
⊢ ∀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 neg
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
ih :
  cs.wordProd (alternatingWord i i' m) = (if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)
hm : ¬Even m
⊢ cs.simple (if Even m then i' else i) *
      ((if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)) =
    (if Even (m + 1) then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ ((m + 1) / 2)

Tactic:
have h₁ : Even (m + 1) := by simp [hm, parity_simps]
      have h₂ : (m + 1) / 2 = m / 2 + 1 := Nat.succ_div_of_dvd h₁.two_dvd
      simp [hm, h₁, h₂, ← pow_succ', ← mul_assoc]

Goal state after:
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
i i' : B
m : ℕ
k : ℤ
⊢ Even (↑(M.M i i') * 2 - 2 * k)

Tactic:
use -k + (M i i')

Goal state after:
case h
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
⊢ ↑(M.M i i') * 2 - 2 * k = -k + ↑(M.M i i') + (-k + ↑(M.M i i'))
Module: Mathlib.GroupTheory.Coxeter.Inversion

Goal state before:
pp: B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
w t : W
ht : cs.IsReflection t
⊢ cs.length (w⁻¹ * t) < cs.length w⁻¹ ↔ cs.length (t * w) < cs.length w

Tactic:
← length_inv,

Goal state after:
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
w t : W
ht : cs.IsReflection t
⊢ cs.length (w⁻¹ * t)⁻¹ < cs.length w⁻¹ ↔ cs.length (t * w) < cs.length w
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
i j : ι
w : NeWord M i j
⊢ w.toList ≠ []

Tactic:
by
  induction w
  · rintro ⟨rfl⟩
  · apply List.append_ne_nil_of_left_ne_nil
    assumption

Goal state after:
Module: Mathlib.GroupTheory.Coxeter.Inversion

Goal state before:
pp: case succ.cons
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
j : ℕ
ih₁ : ∀ (ω : List B), cs.rightInvSeq (drop j ω) = drop j (cs.rightInvSeq ω)
k : B
ω : List B
tail_ih✝ : cs.rightInvSeq (drop (j + 1) ω) = drop (j + 1) (cs.rightInvSeq ω)
⊢ cs.rightInvSeq (drop (j + 1) (k :: ω)) = drop (j + 1) (cs.rightInvSeq (k :: ω))

Tactic:
drop_succ_cons,

Goal state after:
case succ.cons
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
j : ℕ
ih₁ : ∀ (ω : List B), cs.rightInvSeq (drop j ω) = drop j (cs.rightInvSeq ω)
k : B
ω : List B
tail_ih✝ : cs.rightInvSeq (drop (j + 1) ω) = drop (j + 1) (cs.rightInvSeq ω)
⊢ cs.rightInvSeq (drop j ω) = drop (j + 1) (cs.rightInvSeq (k :: ω))
Module: Mathlib.GroupTheory.Descent

Goal state before:
pp: M : Type u_1
inst✝¹ : Monoid M
n : ℕ
h : M → ℝ
b c₀ : ℝ
hb : 1 < b
H : ∀ (x : M), b * h x - c₀ ≤ h (x ^ n)
inst✝ : Northcott h
⊢ Finite ↑{x | IsOfFinOrder x}

Tactic:
refine (Northcott.finite_le (h := h) (c₀ / (b - 1))).subset fun t ht ↦ ?_

Goal state after:
M : Type u_1
inst✝¹ : Monoid M
n : ℕ
h : M → ℝ
b c₀ : ℝ
hb : 1 < b
H : ∀ (x : M), b * h x - c₀ ≤ h (x ^ n)
inst✝ : Northcott h
t : M
ht : t ∈ {x | IsOfFinOrder x}
⊢ t ∈ {a | h a ≤ c₀ / (b - 1)}
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: M✝ : Type u_1
inst✝¹ : AddCommMonoid M✝
M : Type u_2
inst✝ : AddCommGroup M
a b : ℚ
x : DivisibleHull M
⊢ (a * b) • x = a • b • x

Tactic:
by
    induction x with | mk m s
    simp_rw [qsmul_mk, mk_eq_mk]
    use 1
    suffices ((a * b).num * a.den * b.den * s) • m = (a.num * b.num * (a * b).den * s) • m by
      convert! this using 1
      all_goals
      simp [← natCast_zsmul, smul_smul]
      ring_nf
    rw [Rat.mul_num_den']

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
cover : ⋃ q, quotToDoubleCoset H K q = Set.univ
x : Quot ⇑(leftRel K)
y : G
hy : ⟦y⟧ = x
⊢ ∃ i, x ∈ Quot.mk ⇑(leftRel K) '' doubleCoset (out i) ↑H ↑K

Tactic:
have ⟨i, hi⟩ : ∃ i : Quotient H K, y ∈ doubleCoset (out i) H K := by
    contrapose cover
    exact (Set.ne_univ_iff_exists_notMem _).mpr ⟨y, by simpa using! cover⟩

Goal state after:
G : Type u_1
inst✝ : Group G
H K : Subgroup G
cover : ⋃ q, quotToDoubleCoset H K q = Set.univ
x : Quot ⇑(leftRel K)
y : G
hy : ⟦y⟧ = x
i : Quotient ↑H ↑K
hi : y ∈ doubleCoset (out i) ↑H ↑K
⊢ ∃ i, x ∈ Quot.mk ⇑(leftRel K) '' doubleCoset (out i) ↑H ↑K
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
cover : ⋃ q, quotToDoubleCoset H K q = Set.univ
x : Quot ⇑(rightRel H)
y : G
hy : ⟦y⟧ = x
⊢ ∃ i, x ∈ Quot.mk ⇑(rightRel H) '' doubleCoset (out i) ↑H ↑K

Tactic:
have ⟨i, hi⟩ : ∃ i : Quotient H K, y ∈ doubleCoset (out i) H K := by
    contrapose cover
    exact (Set.ne_univ_iff_exists_notMem _).mpr ⟨y, by simpa using! cover⟩

Goal state after:
G : Type u_1
inst✝ : Group G
H K : Subgroup G
cover : ⋃ q, quotToDoubleCoset H K q = Set.univ
x : Quot ⇑(rightRel H)
y : G
hy : ⟦y⟧ = x
i : Quotient ↑H ↑K
hi : y ∈ doubleCoset (out i) ↑H ↑K
⊢ ∃ i, x ∈ Quot.mk ⇑(rightRel H) '' doubleCoset (out i) ↑H ↑K
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: case h₁
M✝ : Type u_1
inst✝³ : AddCommMonoid M✝
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedAddMonoid M
a : ArchimedeanClass (DivisibleHull M)
⊢ ((archimedeanClassOrderHom M).comp (archimedeanClassOrderHomInv M)) a = OrderHom.id a

Tactic:
induction a with | mk a

Goal state after:
case h₁.mk
M✝ : Type u_1
inst✝³ : AddCommMonoid M✝
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedAddMonoid M
a : DivisibleHull M
⊢ ((archimedeanClassOrderHom M).comp (archimedeanClassOrderHomInv M)) (ArchimedeanClass.mk a) =
    OrderHom.id (ArchimedeanClass.mk a)
Module: Mathlib.GroupTheory.Transfer

Goal state before:
pp: G : Type u_1
inst✝² : Group G
H : Subgroup G
A : Type u_2
inst✝¹ : CommGroup A
ϕ : ↥H →* A
inst✝ : H.FiniteIndex
T : H.LeftTransversal := default
g h : G
⊢ diff ϕ T ((g * h) • T) = diff ϕ T (g • T) * diff ϕ T (h • T)

Tactic:
rw [mul_smul, ← diff_mul_diff, smul_diff_smul]

Goal state after:
Module: Mathlib.GroupTheory.FinitelyPresentedGroup

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
N : Subgroup H
f : G →* H
hf : Surjective ⇑f
S : Set H
hS_fin : S.Finite
hS : normalClosure S = N
T : Set G
hT_fin : T.Finite
hT : normalClosure T = f.ker
⊢ ⇑f '' surjInv hf '' S = S

Tactic:
by rw [← image_comp, comp_surjInv, image_id]

Goal state after:
Module: Mathlib.GroupTheory.Frattini

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
φ : G →* H
hφ : Function.Surjective ⇑φ
M : Subgroup H
hM : M ∈ {H_1 | IsCoatom H_1}
⊢ comap φ M ∈ {H | IsCoatom H}

Tactic:
exact isCoatom_comap_of_surjective hφ hM

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : ℕ → Subgroup G
hH : IsAscendingCentralSeries H
n : ℕ
⊢ H (n + 1) ≤ upperCentralSeries G (n + 1)

Tactic:
by
    intro x hx
    rw [mem_upperCentralSeries_succ_iff]
    exact fun y => ascending_central_series_le_upper H hH n (hH.2 x n hx y)

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case refine_2
G : Type u_1
inst✝ : Group G
S : Subgroup G
⊢ (∃ n, S.lowerCentralSeries n = ⊥) → ∃ n H, IsDescendingCentralSeries H ∧ H n = ⊥

Tactic:
rintro ⟨n, hn⟩

Goal state after:
case refine_2
G : Type u_1
inst✝ : Group G
S : Subgroup G
n : ℕ
hn : S.lowerCentralSeries n = ⊥
⊢ ∃ n H, IsDescendingCentralSeries H ∧ H n = ⊥
Module: Mathlib.GroupTheory.Goursat

Goal state before:
pp: case mp
G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
I : Subgroup (G × H)
G' : Subgroup G := ⋯
H' : Subgroup H := ⋯
P : ↥I →* ↥G' := ⋯
Q : ↥I →* ↥H' := ⋯
I' : Subgroup (↥G' × ↥H') := ⋯
hI₁' : Surjective (Prod.fst ∘ ⇑I'.subtype)
hI₂' : Surjective (Prod.snd ∘ ⇑I'.subtype)
this✝ : I'.goursatFst.Normal
this : I'.goursatSnd.Normal
e : ↥G' ⧸ I'.goursatFst ≃* ↥H' ⧸ I'.goursatSnd
he :
  (((QuotientGroup.mk' I'.goursatFst).prodMap (QuotientGroup.mk' I'.goursatSnd)).comp I'.subtype).range =
    e.toMonoidHom.graph
g : G
h : H
⊢ (g, h) ∈ I →
    (g, h) ∈ map ((map (MonoidHom.fst G H) I).subtype.prodMap (map (MonoidHom.snd G H) I).subtype) (P.prod Q).range

Tactic:
intro hgh

Goal state after:
case mp
G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
I : Subgroup (G × H)
G' : Subgroup G := map (MonoidHom.fst G H) I
H' : Subgroup H := map (MonoidHom.snd G H) I
P : ↥I →* ↥G' := (MonoidHom.fst G H).subgroupMap I
Q : ↥I →* ↥H' := (MonoidHom.snd G H).subgroupMap I
I' : Subgroup (↥G' × ↥H') := (P.prod Q).range
hI₁' : Surjective (Prod.fst ∘ ⇑I'.subtype)
hI₂' : Surjective (Prod.snd ∘ ⇑I'.subtype)
this✝ : I'.goursatFst.Normal
this : I'.goursatSnd.Normal
e : ↥G' ⧸ I'.goursatFst ≃* ↥H' ⧸ I'.goursatSnd
he :
  (((QuotientGroup.mk' I'.goursatFst).prodMap (QuotientGroup.mk' I'.goursatSnd)).comp I'.subtype).range =
    e.toMonoidHom.graph
g : G
h : H
hgh : (g, h) ∈ I
⊢ (g, h) ∈ map ((map (MonoidHom.fst G H) I).subtype.prodMap (map (MonoidHom.snd G H) I).subtype) (P.prod Q).range
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
hH : IsNilpotent G
inst✝ : Nontrivial G
⊢ nilpotencyClass G = nilpotencyClass (G ⧸ center G) + 1

Tactic:
rw [nilpotencyClass_quotient_center]
  rcases h : Group.nilpotencyClass G with ⟨⟩
  · exfalso
    rw [nilpotencyClass_zero_iff_subsingleton] at h
    apply false_of_nontrivial_of_subsingleton G
  · simp

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
B₁ B₂ : Set X
h₁ : ∀ ⦃g₁ g₂ : G⦄, (g₁ • B₁ ∩ g₂ • B₁).Nonempty → g₁ • B₁ = g₂ • B₁
h₂ : ∀ ⦃g₁ g₂ : G⦄, (g₁ • B₂ ∩ g₂ • B₂).Nonempty → g₁ • B₂ = g₂ • B₂
⊢ ∀ ⦃g₁ g₂ : G⦄, (g₁ • B₁ ∩ g₁ • B₂ ∩ (g₂ • B₁ ∩ g₂ • B₂)).Nonempty → g₁ • B₁ ∩ g₁ • B₂ = g₂ • B₁ ∩ g₂ • B₂

Tactic:
rintro g₁ g₂ ⟨a, ha₁, ha₂⟩

Goal state after:
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
B₁ B₂ : Set X
h₁ : ∀ ⦃g₁ g₂ : G⦄, (g₁ • B₁ ∩ g₂ • B₁).Nonempty → g₁ • B₁ = g₂ • B₁
h₂ : ∀ ⦃g₁ g₂ : G⦄, (g₁ • B₂ ∩ g₂ • B₂).Nonempty → g₁ • B₂ = g₂ • B₂
g₁ g₂ : G
a : X
ha₁ : a ∈ g₁ • B₁ ∩ g₁ • B₂
ha₂ : a ∈ g₂ • B₁ ∩ g₂ • B₂
⊢ g₁ • B₁ ∩ g₁ • B₂ = g₂ • B₁ ∩ g₂ • B₂
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
ι : Sort u_3
B : ι → Set X
hB : ∀ (i : ι) ⦃g₁ g₂ : G⦄, (g₁ • B i ∩ g₂ • B i).Nonempty → g₁ • B i = g₂ • B i
⊢ ∀ ⦃g₁ g₂ : G⦄, ((⋂ i, g₁ • B i) ∩ ⋂ i, g₂ • B i).Nonempty → ⋂ i, g₁ • B i = ⋂ i, g₂ • B i

Tactic:
rintro g₁ g₂ ⟨a, ha₁, ha₂⟩

Goal state after:
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
ι : Sort u_3
B : ι → Set X
hB : ∀ (i : ι) ⦃g₁ g₂ : G⦄, (g₁ • B i ∩ g₂ • B i).Nonempty → g₁ • B i = g₂ • B i
g₁ g₂ : G
a : X
ha₁ : a ∈ ⋂ i, g₁ • B i
ha₂ : a ∈ ⋂ i, g₂ • B i
⊢ ⋂ i, g₁ • B i = ⋂ i, g₂ • B i
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
C : SubMulAction G X
B : Set ↥C
g₁ g₂ : G
⊢ ⇑C.inclusion '' (g₁ • B) ≠ ⇑C.inclusion '' (g₂ • B) → Disjoint (⇑C.inclusion '' (g₁ • B)) (⇑C.inclusion '' (g₂ • B)) ↔
    g₁ • B ≠ g₂ • B → Disjoint (g₁ • B) (g₂ • B)

Tactic:
ne_eq,

Goal state after:
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
C : SubMulAction G X
B : Set ↥C
g₁ g₂ : G
⊢ ¬⇑C.inclusion '' (g₁ • B) = ⇑C.inclusion '' (g₂ • B) →
      Disjoint (⇑C.inclusion '' (g₁ • B)) (⇑C.inclusion '' (g₂ • B)) ↔
    g₁ • B ≠ g₂ • B → Disjoint (g₁ • B) (g₂ • 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
B : Set X
hGX : IsPretransitive G X
hB : IsBlock G B
hBe : B.Nonempty
⊢ IsBlockSystem G (range fun g ↦ g • B)

Tactic:
refine ⟨⟨?nonempty, ?cover⟩, ?mem_blocks⟩

Goal state after:
case nonempty
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
B : Set X
hGX : IsPretransitive G X
hB : IsBlock G B
hBe : B.Nonempty
⊢ ∅ ∉ range fun g ↦ g • B

case cover
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
B : Set X
hGX : IsPretransitive G X
hB : IsBlock G B
hBe : B.Nonempty
⊢ ∀ (a : X), ∃! b, (b ∈ range fun g ↦ g • B) ∧ a ∈ b

case mem_blocks
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
B : Set X
hGX : IsPretransitive G X
hB : IsBlock G B
hBe : B.Nonempty
⊢ ∀ ⦃B_1 : Set X⦄, (B_1 ∈ range fun g ↦ g • B) → IsBlock G B_1
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
N : Subgroup G
nN : N.Normal
a : X
g : G
⊢ g • orbit (↥N) a = orbit (↥N) (g • a)

Tactic:
simp only [orbit, Set.smul_set_range]

Goal state after:
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : N.Normal
a : X
g : G
⊢ (range fun i ↦ g • i • a) = range fun m ↦ m • g • a
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: case mpr
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g ∈ H
⊢ g • orbit (↥H) a = orbit (↥H) a

Tactic:
← Subgroup.coe_mk H g hg,

Goal state after:
case mpr
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g ∈ H
⊢ ↑⟨g, hg⟩ • orbit (↥H) a = orbit (↥H) a
Module: Mathlib.GroupTheory.SpecificGroups.Alternating

Goal state before:
pp: case inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
hσ✝ : σ ∈ alternatingGroup α
hσ : (↑⟨σ, hσ✝⟩).support.card + 2 ≤ card α
τ : Perm α
hτ : τ ∈ alternatingGroup α
hc : IsConj ↑⟨σ, hσ✝⟩ ↑⟨τ, hτ⟩
π : Perm α
hπ : π * σ * π⁻¹ = τ
h : sign π = 1
⊢ IsConj ⟨σ, hσ✝⟩ ⟨τ, hτ⟩

Tactic:
isConj_iff

Goal state after:
case inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
hσ✝ : σ ∈ alternatingGroup α
hσ : (↑⟨σ, hσ✝⟩).support.card + 2 ≤ card α
τ : Perm α
hτ : τ ∈ alternatingGroup α
hc : IsConj ↑⟨σ, hσ✝⟩ ↑⟨τ, hτ⟩
π : Perm α
hπ : π * σ * π⁻¹ = τ
h : sign π = 1
⊢ ∃ c, c * ⟨σ, hσ✝⟩ * c⁻¹ = ⟨τ, hτ⟩
Module: Mathlib.GroupTheory.SpecificGroups.Alternating

Goal state before:
pp: case inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
hσ✝ : σ ∈ alternatingGroup α
hσ : (↑⟨σ, hσ✝⟩).support.card + 2 ≤ card α
τ : Perm α
hτ : τ ∈ alternatingGroup α
hc : IsConj ↑⟨σ, hσ✝⟩ ↑⟨τ, hτ⟩
π : Perm α
hπ : π * σ * π⁻¹ = τ
h : sign π = -1
h2 : 2 ≤ σ.supportᶜ.card
⊢ IsConj ⟨σ, hσ✝⟩ ⟨τ, hτ⟩

Tactic:
obtain ⟨a, ha, b, hb, ab⟩ := Finset.one_lt_card.1 h2

Goal state after:
case inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
hσ✝ : σ ∈ alternatingGroup α
hσ : (↑⟨σ, hσ✝⟩).support.card + 2 ≤ card α
τ : Perm α
hτ : τ ∈ alternatingGroup α
hc : IsConj ↑⟨σ, hσ✝⟩ ↑⟨τ, hτ⟩
π : Perm α
hπ : π * σ * π⁻¹ = τ
h : sign π = -1
h2 : 2 ≤ σ.supportᶜ.card
a : α
ha : a ∈ σ.supportᶜ
b : α
hb : b ∈ σ.supportᶜ
ab : a ≠ b
⊢ IsConj ⟨σ, hσ✝⟩ ⟨τ, hτ⟩
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 y : B
hy :
  ‖Trivialization.symmL ℝ (trivializationAt F E x) y ∘SL
        Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x‖ <
    2
v : F
⊢ (Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x ∘SL
        Trivialization.symmL ℝ (trivializationAt F E x) x)
      v =
    (ContinuousLinearMap.id ℝ F) v

Tactic:
have h'x : x ∈ (trivializationAt F E x).baseSet := FiberBundle.mem_baseSet_trivializationAt' x

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 y : B
hy :
  ‖Trivialization.symmL ℝ (trivializationAt F E x) y ∘SL
        Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x‖ <
    2
v : F
h'x : x ∈ (trivializationAt F E x).baseSet
⊢ (Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x ∘SL
        Trivialization.symmL ℝ (trivializationAt F E x) x)
      v =
    (ContinuousLinearMap.id ℝ F) v
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
    suffices G = ⊤ by rw [this, Subgroup.index_top] at hG; cases hG
    rwa [eq_top_iff, ← closure_isSwap, G.closure_le]

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

Goal state before:
pp: case refine_2
G : Type u_1
α : Type u_2
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : IsPretransitive G α
n : ℕ
a : α
hn : IsMultiplyPretransitive (↥(stabilizer G a)) (↥(ofStabilizer G a)) n
x y : Fin n.succ ↪ α
⊢ ∃ g, g • x = y

Tactic:
obtain ⟨gx, x1, hgx⟩ := exists_smul_of_last_eq G a x
    -- gy • y = y1 :: a
    obtain ⟨gy, y1, hgy⟩ := exists_smul_of_last_eq G a y
    -- g • x1 = y1,
    obtain ⟨g, hg⟩ := hn.exists_smul_eq x1 y1
    use gy⁻¹ * g * gx
    ext i
    simp only [mul_smul, smul_apply, inv_smul_eq_iff]
    simp only [← smul_apply _ _ i, hgy, hgx]
    simp only [smul_apply]
    rcases Fin.eq_castSucc_or_eq_last i with ⟨i, rfl⟩ | ⟨rfl⟩
    · simp [ofStabilizer.snoc_castSucc, ← hg, SetLike.val_smul, subgroup_smul_def]
    · simp only [ofStabilizer.snoc_last, ← hg]
      exact g.prop

Goal state after: