mathlib-initiative/mathlib-tactics huggingface dataset:
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
y : M
hy : mfderiv[s'] f y = mfderiv[s] f y
⊢ mpullbackWithin I I' f V s' y = mpullbackWithin I I' f V s y

Tactic:
simp [mpullbackWithin, hy]

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
y : M
hy : mfderiv[s'] f y = mfderiv[s] f y
⊢ mpullbackWithin I I' f W s' y = mpullbackWithin I I' f W s y

Tactic:
simp [mpullbackWithin, hy]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³⁰ : NormedAddCommGroup E
inst✝²⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝²⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝²⁷ : TopologicalSpace M
inst✝²⁶ : ChartedSpace H M
F : Type u_5
inst✝²⁵ : NormedAddCommGroup F
inst✝²⁴ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝²³ : TopologicalSpace (TotalSpace F V)
inst✝²² : (x : M) → AddCommGroup (V x)
inst✝²¹ : (x : M) → Module 𝕜 (V x)
inst✝²⁰ : (x : M) → TopologicalSpace (V x)
inst✝¹⁹ : FiberBundle F V
F' : Type u_7
inst✝¹⁸ : NormedAddCommGroup F'
inst✝¹⁷ : NormedSpace 𝕜 F'
V' : M → Type u_8
inst✝¹⁶ : TopologicalSpace (TotalSpace F' V')
inst✝¹⁵ : (x : M) → AddCommGroup (V' x)
inst✝¹⁴ : (x : M) → Module 𝕜 (V' x)
inst✝¹³ : (x : M) → TopologicalSpace (V' x)
inst✝¹² : FiberBundle F' V'
A : Type u_9
inst✝¹¹ : AddCommGroup A
inst✝¹⁰ : Module 𝕜 A
Φ✝ : ((x : M) → V x) → A
x✝ : M
inst✝⁹ : VectorBundle 𝕜 F V
inst✝⁸ : VectorBundle 𝕜 F' V'
inst✝⁷ : CompleteSpace 𝕜
inst✝⁶ : FiniteDimensional 𝕜 F
inst✝⁵ : FiniteDimensional 𝕜 F'
inst✝⁴ : ContMDiffVectorBundle 1 F V I
inst✝³ : ContMDiffVectorBundle 1 F' V' I
inst✝² : TopologicalSpace A
inst✝¹ : IsTopologicalAddGroup A
inst✝ : ContinuousSMul 𝕜 A
Φ : ((x : M) → V x) → ((x : M) → V' x) → A
x : M
hΦ₁ : ∀ (τ : (x : M) → V' x), MDiffAt (T% τ) x → TensorialAt I F (fun x ↦ Φ x τ) x
hΦ₂ : ∀ (σ : (x : M) → V x), MDiffAt (T% σ) x → TensorialAt I F' (Φ σ) x
this✝⁶ : T2Space (V x)
this✝⁵ : FiniteDimensional 𝕜 (V x)
this✝⁴ : T2Space (V' x)
this✝³ : FiniteDimensional 𝕜 (V' x)
this✝² : IsTopologicalAddGroup (V x)
this✝¹ : IsTopologicalAddGroup (V' x)
this✝ : ∀ (x : M), ContinuousSMul 𝕜 (V x)
this : ∀ (x : M), ContinuousSMul 𝕜 (V' x)
v₁ v₂ : V x
w : V' x
⊢ Φ (extend F (v₁ + v₂)) (extend F' w) = Φ (extend F v₁) (extend F' w) + Φ (extend F v₂) (extend F' w)

Tactic:
rw [← (hΦ₁ _ (mdifferentiableAt_extend ..)).add (mdifferentiableAt_extend ..)
        (mdifferentiableAt_extend ..)]
      apply TensorialAt.pointwise₂ hΦ₁ hΦ₂ (mdifferentiableAt_extend ..) _
        (mdifferentiableAt_extend ..) (mdifferentiableAt_extend ..) _ rfl
      · exact mdifferentiableAt_add_section (mdifferentiableAt_extend ..)
          (mdifferentiableAt_extend ..)
      · simp

Goal state after:
Module: Mathlib.GroupTheory.Commensurable

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
x✝¹ : ConjAct G
x✝ : x✝¹ ∈ {g | (g • H).Commensurable H}
⊢ (x✝¹⁻¹ • H).Commensurable H

Tactic:
comm,

Goal state after:
G : Type u_1
inst✝ : Group G
H : Subgroup G
x✝¹ : ConjAct G
x✝ : x✝¹ ∈ {g | (g • H).Commensurable H}
⊢ H.Commensurable (x✝¹⁻¹ • H)
Module: Mathlib.GroupTheory.Commutator.Finite

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : Finite ↑(commutatorSet G)
⊢ Finite ↑(commutatorSet ↥(closureCommutatorRepresentatives G))

Tactic:
apply Nat.finite_of_card_ne_zero

Goal state after:
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite ↑(commutatorSet G)
⊢ Nat.card ↑(commutatorSet ↥(closureCommutatorRepresentatives G)) ≠ 0
Module: Mathlib.Geometry.Manifold.WhitneyEmbedding

Goal state before:
pp: E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I ∞ M
inst✝¹ : T2Space M
inst✝ : CompactSpace M
⊢ ∃ n e, ContMDiff I (𝓡 n) ∞ e ∧ IsClosedEmbedding e ∧ ∀ (x : M), Injective ⇑(mfderiv% e x)

Tactic:
rcases SmoothBumpCovering.exists_isSubordinate I isClosed_univ fun (x : M) _ => univ_mem with
    ⟨ι, f, -⟩
  have := f.fintype
  rcases f.exists_immersion_euclidean with ⟨n, e, hsmooth, hinj, hinj_mfderiv⟩
  exact ⟨n, e, hsmooth, hsmooth.continuous.isClosedEmbedding hinj, hinj_mfderiv⟩

Goal state after:
Module: Mathlib.GroupTheory.CommutingProbability

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

Tactic:
by_cases h1 : n = 1

Goal state after:
case pos
n : ℕ
h0 : ¬n = 0
h1 : n = 1
⊢ commProb (Product (reciprocalFactors n)) = 1 / ↑n

case neg
n : ℕ
h0 : ¬n = 0
h1 : ¬n = 1
⊢ commProb (Product (reciprocalFactors n)) = 1 / ↑n
Module: Mathlib.GroupTheory.FreeGroup.IsFreeGroup

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
G : Type u_3
H : Type u_4
inst✝¹ : Group G
inst✝ : Group H
b : FreeGroupBasis ι G
f : FreeGroup ι →* H
⊢ (fun f ↦ f.comp b.repr.symm.toMonoidHom) ((fun f ↦ f.comp b.repr.toMonoidHom) f) = f

Tactic:
ext
        simp

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.IsFreeGroup

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
G : Type u_3
H : Type u_4
inst✝¹ : Group G
inst✝ : Group H
b : FreeGroupBasis ι G
f : G →* H
⊢ (fun f ↦ f.comp b.repr.toMonoidHom) ((fun f ↦ f.comp b.repr.symm.toMonoidHom) f) = f

Tactic:
ext
        simp

Goal state after:
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
⊢ cs.length w = 0 ↔ w = 1

Tactic:
constructor
  · intro h
    rcases cs.exists_isReduced w with ⟨ω, hω, rfl⟩
    have : ω = [] := eq_nil_of_length_eq_zero (hω.symm.trans h)
    rw [this, wordProd_nil]
  · rintro rfl
    exact cs.length_one

Goal state after:
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
w₁ w₂ : W
⊢ cs.length (w₁ * w₂) % 2 = (cs.length w₁ + cs.length w₂) % 2

Tactic:
← ZMod.natCast_eq_natCast_iff',

Goal state after:
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
w₁ w₂ : W
⊢ ↑(cs.length (w₁ * w₂)) = ↑(cs.length w₁ + cs.length w₂)
Module: Mathlib.GroupTheory.Coxeter.Inversion

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

Tactic:
· exact (lt_of_le_of_ne' · (ht.length_mul_left_ne w))

Goal state after:
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: case hpp
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
⊢ Pairwise fun i j ↦ ∀ (h : H i), h ≠ 1 → (f i) h • X' j ⊆ X' i

Tactic:
rintro i j hij

Goal state after:
case hpp
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
i j : ι
hij : i ≠ j
⊢ ∀ (h : H i), h ≠ 1 → (f i) h • X' j ⊆ X' i
Module: Mathlib.GroupTheory.FiniteAbelian.Duality

Goal state before:
pp: case refine_2
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)
this : HasEnoughRootsOfUnity M (Monoid.exponent (G →* Mˣ))
⊢ Nat.card G = Nat.card ((G →* Mˣ) →* Mˣ)

Tactic:
rw [card_monoidHom_of_hasEnoughRootsOfUnity, card_monoidHom_of_hasEnoughRootsOfUnity]

Goal state after:
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
⊢ ⟨r, hr⟩ = ⟨1, hR1⟩

Tactic:
apply (isComplement_iff_existsUnique_mul_inv_mem.mp hR r).unique

Goal state after:
case py₁
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 * (↑⟨r, hr⟩)⁻¹ ∈ ↑H

case py₂
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 * (↑⟨1, hR1⟩)⁻¹ ∈ ↑H
Module: Mathlib.GroupTheory.FreeGroup.Orbit

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
w : α × Bool
g : FreeGroup α
h : g ∈ startsWith w
h1 : g = 1
⊢ False

Tactic:
by simp [h1, startsWith, FreeGroup.toWord_one] at h

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

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

Tactic:
induction n with
  | zero => simp
  | succ d hd =>
    rw [lowerCentralSeries_succ, lowerCentralSeries_succ, Subgroup.map_commutator, hd]

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

Goal state before:
pp: case h
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 k : G
hk : k ∈ N
⊢ ⟨g * k * g⁻¹, ⋯⟩ • g • a = g • ⟨k, hk⟩ • a

Tactic:
simp only [Subgroup.mk_smul]

Goal state after:
case h
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 k : G
hk : k ∈ N
⊢ (g * k * g⁻¹) • g • a = g • k • a
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: case h
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 k : G
hk : k ∈ N
⊢ g • ⟨g⁻¹ * k * g, ⋯⟩ • a = ⟨k, hk⟩ • g • a

Tactic:
simp only [Subgroup.mk_smul]

Goal state after:
case h
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 k : G
hk : k ∈ N
⊢ g • (g⁻¹ * k * g) • a = k • g • a
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: case mp
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
B : Set X
inst✝ : IsPretransitive G X
hB : IsBlock G B
a : X
ha : a ∈ B
k : G
k_mem : k ∈ stabilizer G B
⊢ (fun m ↦ m • a) ⟨k, k_mem⟩ ∈ B

Tactic:
simp only [Subgroup.mk_smul]

Goal state after:
case mp
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
B : Set X
inst✝ : IsPretransitive G X
hB : IsBlock G B
a : X
ha : a ∈ B
k : G
k_mem : k ∈ stabilizer G B
⊢ k • a ∈ 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
htGX : IsPretransitive G X
a : X
B : Set X
ha : a ∈ B
hB : IsBlock G B
⊢ ∀ {b : { B // a ∈ B ∧ IsBlock G B }},
    {
            toFun := fun x ↦
              match x with
              | ⟨B, ⋯⟩ => ⟨stabilizer G B, ⋯⟩,
            invFun := fun x ↦
              match x with
              | ⟨H, hH⟩ => ⟨orbit (↥H) a, ⋯⟩,
            left_inv := ⋯, right_inv := ⋯ }
          ⟨B, ⋯⟩ ≤
        {
            toFun := fun x ↦
              match x with
              | ⟨B, ⋯⟩ => ⟨stabilizer G B, ⋯⟩,
            invFun := fun x ↦
              match x with
              | ⟨H, hH⟩ => ⟨orbit (↥H) a, ⋯⟩,
            left_inv := ⋯, right_inv := ⋯ }
          b ↔
      ⟨B, ⋯⟩ ≤ b

Tactic:
rintro ⟨B', ha', hB'⟩

Goal state after:
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
B✝ : Set X
htGX : IsPretransitive G X
a : X
B : Set X
ha : a ∈ B
hB : IsBlock G B
B' : Set X
ha' : a ∈ B'
hB' : IsBlock G B'
⊢ {
          toFun := fun x ↦
            match x with
            | ⟨B, ⋯⟩ => ⟨stabilizer G B, ⋯⟩,
          invFun := fun x ↦
            match x with
            | ⟨H, hH⟩ => ⟨orbit (↥H) a, ⋯⟩,
          left_inv := ⋯, right_inv := ⋯ }
        ⟨B, ⋯⟩ ≤
      {
          toFun := fun x ↦
            match x with
            | ⟨B, ⋯⟩ => ⟨stabilizer G B, ⋯⟩,
          invFun := fun x ↦
            match x with
            | ⟨H, hH⟩ => ⟨orbit (↥H) a, ⋯⟩,
          left_inv := ⋯, right_inv := ⋯ }
        ⟨B', ⋯⟩ ↔
    ⟨B, ⋯⟩ ≤ ⟨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
a : X
⊢ ∀ (a_1 : BlockMem G a), a_1 ≤ ⟨univ, ⋯⟩

Tactic:
by
    rintro ⟨B, ha, hB⟩
    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
k g : Perm α
⊢ ConjAct.ofConjAct (ConjAct.toConjAct k) • g.support = k • g.support

Tactic:
ConjAct.ofConjAct_toConjAct

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
k g : Perm α
⊢ k • g.support = k • g.support
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: 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
a : X
hfB : B.Finite
B' : Set X := ⋂ k, ⋂ (_ : a ∈ k • B), k • B
hfB_ne : B.Nonempty
hB'₀ : ∀ (k : G), a ∈ k • B → B' ⊆ k • B
hfB' : B'.Finite
hag : ∀ (g : G), a ∈ g • B' → B' ⊆ g • B'
⊢ IsBlock G (⋂ k, ⋂ (_ : a ∈ k • B), k • B)

Tactic:
have hag' (g : G) (hg : a ∈ g • B') : B' = g • B' := by
    rw [eq_comm, ← mem_stabilizer_iff, mem_stabilizer_set_iff_subset_smul_set hfB']
    exact hag g hg

Goal state after:
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
a : X
hfB : B.Finite
B' : Set X := ⋂ k, ⋂ (_ : a ∈ k • B), k • B
hfB_ne : B.Nonempty
hB'₀ : ∀ (k : G), a ∈ k • B → B' ⊆ k • B
hfB' : B'.Finite
hag : ∀ (g : G), a ∈ g • B' → B' ⊆ g • B'
hag' : ∀ (g : G), a ∈ g • B' → B' = g • B'
⊢ IsBlock G (⋂ k, ⋂ (_ : a ∈ k • B), k • B)
Module: Mathlib.GroupTheory.SpecificGroups.Alternating

Goal state before:
pp: α : Type u_1
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Nontrivial α
⊢ 2 * Nat.card ↥(alternatingGroup α) = Nat.card (Perm α)

Tactic:
simp only [← alternatingGroup.index_eq_two (α := α), index_mul_card]

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

Goal state before:
pp: k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
hGX : IsPretransitive G α
a : α
has : a ∈ s
t : Set ↥(ofStabilizer G a) := Subtype.val ⁻¹' s
⊢ Subtype.val '' t = s \ {a}

Tactic:
rw [Set.image_preimage_eq_inter_range]

Goal state after:
k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
hGX : IsPretransitive G α
a : α
has : a ∈ s
t : Set ↥(ofStabilizer G a) := Subtype.val ⁻¹' s
⊢ s ∩ range Subtype.val = s \ {a}
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 : Set α
⊢ Injective ⇑(ofFixingSubgroup_equivariantMap M s)

Tactic:
by
  rintro ⟨x, hx⟩ ⟨y, hy⟩ hxy
  simpa [Subtype.mk.injEq] using! hxy

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

Goal state before:
pp: case e'_2
k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
hGX : IsPretransitive G α
a : α
has : a ∈ s
t : Set ↥(ofStabilizer G a) := Subtype.val ⁻¹' s
hat : Subtype.val '' t = s \ {a}
hat' : s = insert a (Subtype.val '' t)
hfs : fixingSubgroup G s = Subgroup.map (stabilizer G a).subtype (fixingSubgroup (↥(stabilizer G a)) t)
htcard : t.ncard = k
⊢ Nat.card α - 1 = Nat.card ↥(ofStabilizer G a)

pp: case e'_3
k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
hGX : IsPretransitive G α
a : α
has : a ∈ s
t : Set ↥(ofStabilizer G a) := Subtype.val ⁻¹' s
hat : Subtype.val '' t = s \ {a}
hat' : s = insert a (Subtype.val '' t)
hfs : fixingSubgroup G s = Subgroup.map (stabilizer G a).subtype (fixingSubgroup (↥(stabilizer G a)) t)
htcard : t.ncard = k
⊢ Nat.card α - 1 = Nat.card ↥(ofStabilizer G a)

Tactic:
all_goals { rw [nat_card_ofStabilizer_eq G a] }

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

Goal state before:
pp: G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
s : Set α
hMk : IsMultiplyPretransitive G α s.ncard
⊢ (fixingSubgroup G s).index = (Nat.card α).choose s.ncard * s.ncard !

Tactic:
by
  apply Nat.eq_of_mul_eq_mul_right (Nat.factorial_pos _)
  rw [hMk.index_of_fixingSubgroup_mul rfl, Nat.choose_mul_factorial_mul_factorial]
  rw [← ncard_univ]
  exact ncard_le_ncard (subset_univ s)

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

Goal state before:
pp: case mpr.right
M : Type u_1
α : Type u_2
inst✝² : Group M
inst✝¹ : MulAction M α
inst✝ : IsPretransitive M α
n : ℕ
hn : 1 ≤ n
a : α
H : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(ofStabilizer M a)) n
s : Set α
hs : s.encard + 1 = ↑n.succ
b : α
hb : b ∈ s
g : M
hg : g • b = a
s' : Set α := g • s
hs' : s' = g • s
t : Set ↥(ofStabilizer M a) := Subtype.val ⁻¹' s'
hst : s' = insert a (Subtype.val '' t)
⊢ IsPreprimitive ↥(fixingSubgroup M s') ↥(ofFixingSubgroup M s')

Tactic:
rw [hst, isPreprimitive_fixingSubgroup_insert_iff]

Goal state after:
case mpr.right
M : Type u_1
α : Type u_2
inst✝² : Group M
inst✝¹ : MulAction M α
inst✝ : IsPretransitive M α
n : ℕ
hn : 1 ≤ n
a : α
H : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(ofStabilizer M a)) n
s : Set α
hs : s.encard + 1 = ↑n.succ
b : α
hb : b ∈ s
g : M
hg : g • b = a
s' : Set α := g • s
hs' : s' = g • s
t : Set ↥(ofStabilizer M a) := Subtype.val ⁻¹' s'
hst : s' = insert a (Subtype.val '' t)
⊢ IsPreprimitive ↥(fixingSubgroup (↥(stabilizer M a)) t) ↥(ofFixingSubgroup (↥(stabilizer M a)) t)
Module: Mathlib.GroupTheory.GroupAction.MultipleTransitivity

Goal state before:
pp: α : Type u_1
inst✝ : Finite α
G : Subgroup (Perm α)
this : Fintype α
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 1)
j : Fin (Fintype.card α - 1) ↪ Fin (Fintype.card α) := castLEEmb ⋯
k : Perm α
a✝ : k ∈ ⊤
x : Fin (Fintype.card α) ↪ α := (Fintype.equivFinOfCardEq ⋯).symm.toEmbedding
x' : Fin (Fintype.card α - 1) ↪ α := j.trans x
g : ↥G
hg' : g • x' = k • x'
⊢ k = ↑g

Tactic:
have hx (x : Fin (Fintype.card α) ↪ α) : Function.Surjective x.toFun := by
    apply Function.Bijective.surjective
    rw [Fintype.bijective_iff_injective_and_card]
    exact ⟨EmbeddingLike.injective x, Fintype.card_fin (Fintype.card α)⟩

Goal state after:
α : Type u_1
inst✝ : Finite α
G : Subgroup (Perm α)
this : Fintype α
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 1)
j : Fin (Fintype.card α - 1) ↪ Fin (Fintype.card α) := castLEEmb ⋯
k : Perm α
a✝ : k ∈ ⊤
x : Fin (Fintype.card α) ↪ α := (Fintype.equivFinOfCardEq ⋯).symm.toEmbedding
x' : Fin (Fintype.card α - 1) ↪ α := j.trans x
g : ↥G
hg' : g • x' = k • x'
hx : ∀ (x : Fin (Fintype.card α) ↪ α), Function.Surjective x.toFun
⊢ k = ↑g
Module: Mathlib.GroupTheory.GroupAction.MultiplePrimitivity

Goal state before:
pp: case zero
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
m : ℕ
hn : IsMultiplyPreprimitive M α 0
hmn : m ≤ 0
hα : ↑0 ≤ ENat.card α
⊢ IsMultiplyPreprimitive M α m

Tactic:
Nat.eq_zero_of_le_zero hmn

Goal state after:
case zero
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
m : ℕ
hn : IsMultiplyPreprimitive M α 0
hmn : m ≤ 0
hα : ↑0 ≤ ENat.card α
⊢ IsMultiplyPreprimitive M α 0