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)
⊢ ContinuousAt g' x

Tactic:
have W := g_cont.continuousAt (x := 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
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)
W : ContinuousAt (fun x ↦ TotalSpace.mk' (F →L[ℝ] F →L[ℝ] ℝ) x (g x)) x
⊢ ContinuousAt g' x
Module: Mathlib.Geometry.Manifold.Riemannian.Basic

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type u_2
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : RiemannianBundle fun x ↦ TangentSpace I x
inst✝¹ : IsManifold I 1 M
inst✝ : IsContinuousRiemannianBundle E fun x ↦ TangentSpace I x
x : M
c : ℝ≥0∞
hc : 0 < c
C : ℝ≥0
C_pos : C > 0
hC : ∀ᶠ (y : M) in 𝓝 x, riemannianEDist I x y ≤ ↑C * edist (↑(extChartAt I x) x) (↑(extChartAt I x) y)
this : ↑(extChartAt I x) ⁻¹' eball (↑(extChartAt I x) x) (c / ↑C) ∈ 𝓝 x
y : M
hy : y ∈ ↑(extChartAt I x) ⁻¹' eball (↑(extChartAt I x) x) (c / ↑C)
h'y : riemannianEDist I x y ≤ ↑C * edist (↑(extChartAt I x) x) (↑(extChartAt I x) y)
⊢ edist (↑(extChartAt I x) x) (↑(extChartAt I x) y) < c / ↑C

Tactic:
simpa only [mem_preimage, Metric.mem_eball'] using hy

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'
this : HasMFDerivAt% (fun _x ↦ 1) x 0
t : M
ht : t ∈ s
⊢ ψ t = (fun _x ↦ 1) t

Tactic:
simp [ψ, ht]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : IsManifold I 2 M
inst✝ : CompleteSpace E
cov : ((x : M) → TangentSpace I x) → (x : M) → TangentSpace I x →L[𝕜] TangentSpace I x
hcov : IsCovariantDerivativeOn E cov
x : M
Y : (x : M) → TangentSpace I x
f✝ : M → 𝕜
σ✝ : (x : M) → TangentSpace I x
hf : MDiffAt f✝ x
hX : MDiffAt (T% σ✝) x
⊢ torsionAux cov (f✝ • σ✝) Y x = f✝ x • torsionAux cov σ✝ Y x

Tactic:
simp [torsionAux, hcov.leibniz hX hf, VectorField.mlieBracket_smul_left hf hX]
    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
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
hgy : ((g y) v) v = ((g' y) w) w
hgx : ((g x) ((trivializationAt F E x).symm x w)) ((trivializationAt F E x).symm x w) = ((g' x) w) w
⊢ ‖g' x - g' y‖ * ‖w‖ * ‖w‖ + ((g' y) w) w ≤ δ * ‖w‖ ^ 2 + ((g' y) w) w

Tactic:
rw [pow_two, mul_assoc]; gcongr

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

Goal state before:
pp: case H
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
ω : Quotient (orbitRel α β)
⊢ 0 < Fintype.card ↥(stabilizer α (φ ω))

Tactic:
exact Fintype.card_pos_iff.mpr (by infer_instance)

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:
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)

Goal state after:
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
this :
  ∀ (ω : Quotient (orbitRel α β)),
    Fintype.card α / Fintype.card ↥(stabilizer α (φ ω)) = Fintype.card (α ⧸ stabilizer α (φ ω))
⊢ Fintype.card β = ∑ ω, Fintype.card α / Fintype.card ↥(stabilizer α (φ ω))
Module: Mathlib.GroupTheory.SpecificGroups.Dihedral

Goal state before:
pp: n : ℕ
i : ZMod n
k : ℕ
⊢ r i ^ k = r (i * ↑k)

Tactic:
by
  induction k with
  | zero => simp only [pow_zero, Nat.cast_zero, mul_zero, r_zero]
  | succ k IH =>
    rw [pow_add, pow_one, IH, r_mul_r, Nat.cast_add, Nat.cast_one, r.injEq, mul_add, mul_one]

Goal state after:
Module: Mathlib.GroupTheory.CommutingProbability

Goal state before:
pp: G : Type u_2
inst✝¹ : Group G
inst✝ : Finite G
H : Subgroup G
⊢ ↑(Nat.card { p // Commute p.1 p.2 }) ≤
    ↑(Nat.card { p // Commute p.1 p.2 }) / ↑(Nat.card G) ^ 2 * (↑H.index ^ 2 * ↑(Nat.card ↥H) ^ 2)

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

Tactic:
← mul_pow,

Goal state after:
G : Type u_2
inst✝¹ : Group G
inst✝ : Finite G
H : Subgroup G
⊢ ↑(Nat.card { p // Commute p.1 p.2 }) ≤
    ↑(Nat.card { p // Commute p.1 p.2 }) / ↑(Nat.card G) ^ 2 * (↑H.index * ↑(Nat.card ↥H)) ^ 2

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

Goal state before:
pp: case zero
B : Type u_1
i j : B
p : ℕ
h : 0 < 2 * p
⊢ take 0 (alternatingWord i j (2 * p)) = if Even 0 then alternatingWord i j 0 else alternatingWord j i 0

Tactic:
simp only [take_zero, Even.zero, ↓reduceIte, alternatingWord]

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
δ : ℝ
δpos : 0 < δ
h'δ : 1 + δ * C < r' ^ 2
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)
⊢ ContinuousAt g' x

Tactic:
have W := g_cont.continuousAt (x := 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
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
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)
W : ContinuousAt (fun x ↦ TotalSpace.mk' (F →L[ℝ] F →L[ℝ] ℝ) x (g x)) x
⊢ ContinuousAt g' x
Module: Mathlib.GroupTheory.Coxeter.Basic

Goal state before:
pp: case succ
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)
⊢ 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:
by_cases hm : Even m

Goal state after:
case pos
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)

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)
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
hw : w ≠ 1
⊢ ∃ i, cs.IsLeftDescent w i

Tactic:
rcases cs.exists_isReduced w with ⟨ω, h, rfl⟩
  have h₁ : ω ≠ [] := by rintro rfl; simp at hw
  rcases List.exists_cons_of_ne_nil h₁ with ⟨i, ω', rfl⟩
  use i
  rw [IsLeftDescent, h, wordProd_cons, simple_mul_simple_cancel_left]
  calc
    ℓ (π ω') ≤ ω'.length := cs.length_wordProd_le ω'
    _ < (i :: ω').length := by simp

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

Goal state before:
pp: case succ
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)
ω : List B
⊢ cs.rightInvSeq (drop (j + 1) ω) = drop (j + 1) (cs.rightInvSeq ω)

Tactic:
induction ω with
    | nil => simp
    | cons k ω _ => rw [drop_succ_cons, ih₁ ω, rightInvSeq, drop_succ_cons]

Goal state after:
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
t : M
ht : t ∈ {x | IsOfFinOrder x}
this : Finite ↥(Submonoid.powers t)
C : ℝ := ⨆ g, h ↑g
⊢ t ∈ {a | h a ≤ c₀ / (b - 1)}

Tactic:
have hC : ∀ g ∈ Submonoid.powers t, h g ≤ C :=
    fun g hg ↦ Finite.le_ciSup (fun g : Submonoid.powers t ↦ h g) ⟨g, hg⟩

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}
this : Finite ↥(Submonoid.powers t)
C : ℝ := ⨆ g, h ↑g
hC : ∀ g ∈ Submonoid.powers t, h g ≤ C
⊢ t ∈ {a | h a ≤ c₀ / (b - 1)}
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
a b : G
⊢ (setoid ↑⊥ ↑H) a b ↔ (QuotientGroup.leftRel H) a b

Tactic:
rw [rel_iff, QuotientGroup.leftRel_apply]

Goal state after:
G : Type u_1
inst✝ : Group G
H : Subgroup G
a b : G
⊢ (∃ a_1 ∈ ⊥, ∃ b_1 ∈ H, b = a_1 * a * b_1) ↔ a⁻¹ * b ∈ H
Module: Mathlib.GroupTheory.FiniteAbelian.Duality

Goal state before:
pp: G : Type u_1
M : Type u_2
inst✝² : CommGroup G
inst✝¹ : Finite G
inst✝ : CommMonoid M
hM : HasEnoughRootsOfUnity M (Monoid.exponent G)
H : Subgroup G
this✝¹ : Fintype ↥H
this✝ : HasEnoughRootsOfUnity M (Monoid.exponent ↥H)
this : HasEnoughRootsOfUnity M (Monoid.exponent (G ⧸ H))
⊢ Function.Surjective ⇑(domRestrictHom H Mˣ)

Tactic:
refine MonoidHom.surjective_of_card_ker_le_div _ (le_of_eq ?_)

Goal state after:
G : Type u_1
M : Type u_2
inst✝² : CommGroup G
inst✝¹ : Finite G
inst✝ : CommMonoid M
hM : HasEnoughRootsOfUnity M (Monoid.exponent G)
H : Subgroup G
this✝¹ : Fintype ↥H
this✝ : HasEnoughRootsOfUnity M (Monoid.exponent ↥H)
this : HasEnoughRootsOfUnity M (Monoid.exponent (G ⧸ H))
⊢ Nat.card ↥(domRestrictHom H Mˣ).ker = Nat.card (G →* Mˣ) / Nat.card (↥H →* Mˣ)
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
inst✝² : Nontrivial ι
G : Type u_1
inst✝¹ : Group G
a : ι → G
α : Type u_4
inst✝ : MulAction G α
X Y : ι → Set α
hXnonempty : ∀ (i : ι), (X i).Nonempty
hXdisj : Pairwise (Disjoint on X)
hYdisj : Pairwise (Disjoint on Y)
hXYdisj : ∀ (i j : ι), Disjoint (X i) (Y j)
hX : ∀ (i : ι), a i • (Y i)ᶜ ⊆ X i
hY : ∀ (i : ι), a⁻¹ i • (X i)ᶜ ⊆ Y i
H : ι → Type := fun _i ↦ FreeGroup Unit
f : (i : ι) → H i →* G := fun i ↦ FreeGroup.lift fun x ↦ a i
X' : ι → Set α := fun i ↦ X i ∪ Y i
inhabited_h : Inhabited ι
⊢ 3 ≤ #ι ∨ ∃ i, 3 ≤ #(H i)

Tactic:
right

Goal state after:
ι : Type u_1
inst✝² : Nontrivial ι
G : Type u_1
inst✝¹ : Group G
a : ι → G
α : Type u_4
inst✝ : MulAction G α
X Y : ι → Set α
hXnonempty : ∀ (i : ι), (X i).Nonempty
hXdisj : Pairwise (Disjoint on X)
hYdisj : Pairwise (Disjoint on Y)
hXYdisj : ∀ (i j : ι), Disjoint (X i) (Y j)
hX : ∀ (i : ι), a i • (Y i)ᶜ ⊆ X i
hY : ∀ (i : ι), a⁻¹ i • (X i)ᶜ ⊆ Y i
H : ι → Type := ⋯
f : (i : ι) → H i →* G := ⋯
X' : ι → Set α := ⋯
inhabited_h : Inhabited ι
⊢ ∃ i, 3 ≤ #(H i)
Module: Mathlib.GroupTheory.Schreier

Goal state before:
pp: case neg
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this✝ : (center G).FiniteIndex
h1 : (center G).relIndex (_root_.commutator G) ∣ (center G).index
this : ((center G).subgroupOf (_root_.commutator G)).FiniteIndex
⊢ Nat.card ↥((center G).subgroupOf (_root_.commutator G)) ∣
    (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G))

Tactic:
have h2 := rank_le_index_mul_rank ((center G).subgroupOf (_root_.commutator G))

Goal state after:
case neg
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this✝ : (center G).FiniteIndex
h1 : (center G).relIndex (_root_.commutator G) ∣ (center G).index
this : ((center G).subgroupOf (_root_.commutator G)).FiniteIndex
h2 :
  Group.rank ↥((center G).subgroupOf (_root_.commutator G)) ≤
    ((center G).subgroupOf (_root_.commutator G)).index * Group.rank ↥(_root_.commutator G)
⊢ Nat.card ↥((center G).subgroupOf (_root_.commutator G)) ∣
    (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G))
Module: Mathlib.GroupTheory.FinitelyPresentedGroup

Goal state before:
pp: G : Type u_1
inst✝ : Group G
hg : IsFinitelyPresented G
⊢ ∃ n s, s.Finite ∧ Nonempty (G ≃* PresentedGroup s)

Tactic:
obtain ⟨n, φ, hφ, s, hs, hsφ⟩ := hg
  exact ⟨n, s, hs, ⟨(quotientKerEquivOfSurjective φ hφ).symm.trans (quotientMulEquivOfEq hsφ.symm)⟩⟩

Goal state after:
Module: Mathlib.GroupTheory.FixedPointFree

Goal state before:
pp: F : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : FunLike F G G
inst✝¹ : MonoidHomClass F G G
φ : F
inst✝ : Finite G
hφ : FixedPointFree ⇑φ
h2 : Function.Involutive ⇑φ
g : G
hg : orderOf g = 2
⊢ (fun x ↦ x⁻¹) g = g

Tactic:
inv_eq_iff_mul_eq_one,

Goal state after:
F : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : FunLike F G G
inst✝¹ : MonoidHomClass F G G
φ : F
inst✝ : Finite G
hφ : FixedPointFree ⇑φ
h2 : Function.Involutive ⇑φ
g : G
hg : orderOf g = 2
⊢ g * g = 1
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
⊢ (↑P).transferFocal.ker ⊓ ↑P = (↑P).focalSubgroup

Tactic:
rw [← subgroupOf_map_subtype, ← MonoidHom.ker_domRestrict, ← map_focalSubgroupOf]
  exact congr_arg _ (ker_restrict_transferFocal_eq_focalSubgroupOf P)

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
⊢ M ∈ {H_1 | IsCoatom H_1} → ⨅ a ∈ {H | IsCoatom H}, a ≤ comap φ M

Tactic:
hM

Goal state after:
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}
⊢ ⨅ a ∈ {H | IsCoatom H}, a ≤ comap φ M
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: 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 + 1) → ∀ (g : G), ⁅x, g⁆ ∈ H n
x : G
m : ℕ
hx : x ∈ H (n - m)
g : G
hm : m < n
⊢ H (n - (m + 1) + 1) = H (n - m)

Tactic:
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: 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 + 1) → ∀ (g : G), ⁅x, g⁆ ∈ H n
x : G
m : ℕ
hx : x ∈ H (n - m)
g : G
hm : m < n
⊢ ⁅x, g⁆ ∈ (fun m ↦ H (n - m)) (m + 1)

Tactic:
· apply hH
    convert! hx 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
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
⊢ H (n - m) = H (n - (m + 1) + 1)

Tactic:
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.FreeGroup.Orbit

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
w w' : α × Bool
⊢ Disjoint (startsWith w) (startsWith w') ↔ w ≠ w'

Tactic:
simp_all only [ne_eq, startsWith, Set.disjoint_iff_inter_eq_empty, Set.ext_iff, Set.mem_inter_iff,
    Set.mem_ofPred_eq, Set.mem_empty_iff_false, iff_false, not_and, Option.some.injEq]

Goal state after:
α : Type u_1
inst✝ : DecidableEq α
w w' : α × Bool
⊢ (∀ (x : FreeGroup α), x.toWord[0]? = some w → ¬w = w') ↔ ¬w = w'
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case neg
G : Type u_1
inst✝ : Group G
hH : ¬IsNilpotent G
⊢ nilpotencyClass (G ⧸ center G) = nilpotencyClass G - 1

Tactic:
nilpotencyClass_of_not_nilpotent hH,

Goal state after:
case neg
G : Type u_1
inst✝ : Group G
hH : ¬IsNilpotent G
⊢ nilpotencyClass (G ⧸ center G) = 0 - 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 : 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
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' y - g' x‖ ≤ δ
v : E x
w : F := (Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x) v
hgx : ((g x) v) v = ((g' x) w) w
hgy : ((g y) ((trivializationAt F E x).symm y w)) ((trivializationAt F E x).symm y w) = ((g' y) w) w
⊢ ‖g' y - g' x‖ * ‖w‖ * ‖w‖ + ((g' x) w) w ≤ δ * ‖w‖ ^ 2 + ((g' x) w) w

Tactic:
rw [pow_two, mul_assoc]; gcongr

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

Goal state before:
pp: case inr
M : Type u_3
α : Type u_4
inst✝¹ : Group M
inst✝ : MulAction M α
B : Set α
g : M
h : B = univ
⊢ IsTrivialBlock (g • B)

Tactic:
right

Goal state after:
case inr
M : Type u_3
α : Type u_4
inst✝¹ : Group M
inst✝ : MulAction M α
B : Set α
g : M
h : B = univ
⊢ g • B = univ