mathlib-initiative/mathlib-tactics huggingface dataset:
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 ω) = drop (j + 1) (cs.rightInvSeq (k :: ω))

Tactic:
ih₁ ω,

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)
⊢ drop j (cs.rightInvSeq ω) = drop (j + 1) (cs.rightInvSeq (k :: ω))
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)
H₃' : ∀ (x : G), b * h x - c' ≤ h (f x)
⊢ Group.FG G

Tactic:
refine Group.fg_of_descent (fun U u hu ↦ ?_) ha H₀ s.toFinite H₁' (fun g' hg' x ↦ ?_) H₃'

Goal state after:
case refine_1
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)
U : Subgroup G
u : G
hu : u ∈ Subgroup.map f U
⊢ u ∈ U

case refine_2
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)
g' : G
hg' : g' ∈ s
x : G
⊢ h x ≤ a * h (g' * x) + c'
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: M : Type u_1
inst✝ : AddCommMonoid M
m1 m2 : M
⊢ ↑(m1 + m2) = ↑m1 + ↑m2

Tactic:
simp [mk_add_mk_left]

Goal state after:
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: M : Type u_2
inst✝ : AddCommGroup M
m : M
s : ℕ+
⊢ mk (-m) s + mk m s = 0

Tactic:
simp [mk_add_mk_left]

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
a b : G
h : a⁻¹ * b ∈ H
⊢ b = 1 * a * (a⁻¹ * b)

Tactic:
rw [one_mul, mul_inv_cancel_left]

Goal state after:
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: case inl
ι : Type u_1
G : Type u_4
inst✝³ : Group G
H : ι → Type u_5
inst✝² : (i : ι) → Group (H i)
f : (i : ι) → H i →* G
α : Type u_6
inst✝¹ : MulAction G α
X : ι → Set α
hXnonempty : ∀ (i : ι), (X i).Nonempty
hXdisj : Pairwise (Disjoint on X)
hpp : Pairwise fun i j ↦ ∀ (h : H i), h ≠ 1 → (f i) h • X j ⊆ X i
inst✝ : Nontrivial ι
i✝ j : ι
w : NeWord H i✝ j
hcard : 3 ≤ #ι
i : ι
h1 : i ≠ i✝
h2 : i ≠ j
⊢ (lift f) w.prod ≠ 1

Tactic:
exact lift_word_prod_nontrivial_of_other_i f X hXnonempty hXdisj hpp w h1 h2

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
i : Quotient ↑H ↑K
a : G
hg : mk H K a = i
⊢ a ∈ quotToDoubleCoset H K i

Tactic:
← out_eq' _ _ i

Goal state after:
G : Type u_1
inst✝ : Group G
H K : Subgroup G
i : Quotient ↑H ↑K
a : G
hg : mk H K a = mk H K (Quotient.out i)
⊢ a ∈ quotToDoubleCoset H K i
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
H : ∀ (n : ℕ), n ∣ Monoid.exponent G → ∀ (a : ZMod n), a ≠ 0 → ∃ φ, φ (Multiplicative.ofAdd a) ≠ 1
a : G
ha : a ≠ 1
ι : Type
w✝ : Fintype ι
n : ι → ℕ
left✝ : ∀ (i : ι), 1 < n i
h : Nonempty (G ≃* ((i : ι) → Multiplicative (ZMod (n i))))
e : G ≃* ((i : ι) → Multiplicative (ZMod (n i))) := h.some
⊢ ∃ φ, φ a ≠ 1

Tactic:
obtain ⟨i, hi⟩ : ∃ i : ι, e a i ≠ 1 := by
    contrapose! ha
    exact (MulEquiv.map_eq_one_iff e).mp <| funext ha

Goal state after:
G : Type u_1
M : Type u_2
inst✝² : CommGroup G
inst✝¹ : Finite G
inst✝ : CommMonoid M
H : ∀ (n : ℕ), n ∣ Monoid.exponent G → ∀ (a : ZMod n), a ≠ 0 → ∃ φ, φ (Multiplicative.ofAdd a) ≠ 1
a : G
ha : a ≠ 1
ι : Type
w✝ : Fintype ι
n : ι → ℕ
left✝ : ∀ (i : ι), 1 < n i
h : Nonempty (G ≃* ((i : ι) → Multiplicative (ZMod (n i))))
e : G ≃* ((i : ι) → Multiplicative (ZMod (n i))) := h.some
i : ι
hi : e a i ≠ 1
⊢ ∃ φ, φ a ≠ 1
Module: Mathlib.GroupTheory.Focal

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
⊢ _root_.commutator ↥H ≤ H.focalSubgroupOf

Tactic:
by
  rw [focalSubgroupOf_eq_closure]
  apply closure_mono
  rintro - ⟨a, -, b, -, -, rfl⟩
  exact ⟨a, a.2, b, rfl⟩

Goal state after:
Module: Mathlib.Order.Radical

Goal state before:
pp: α : Type u_1
inst✝¹ : CompleteLattice α
inst✝ : IsCoatomic α
a : α
h : a ⊔ radical α = ⊤
⊢ a = ⊤

Tactic:
obtain (rfl | w) := eq_top_or_exists_le_coatom a

Goal state after:
case inl
α : Type u_1
inst✝¹ : CompleteLattice α
inst✝ : IsCoatomic α
h : ⊤ ⊔ radical α = ⊤
⊢ ⊤ = ⊤

case inr
α : Type u_1
inst✝¹ : CompleteLattice α
inst✝ : IsCoatomic α
a : α
h : a ⊔ radical α = ⊤
w : ∃ a_1, IsCoatom a_1 ∧ a ≤ a_1
⊢ a = ⊤
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case succ
G : Type u_1
inst✝² : Group G
N : Subgroup G
inst✝¹ : N.Normal
S : Subgroup G
inst✝ : S.Normal
n : ℕ
a✝ : (S.lowerCentralSeries n).Normal
⊢ (S.lowerCentralSeries (n + 1)).Normal

Tactic:
| succ n _ =>

Goal state after:
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)
hI₁ : Surjective (Prod.fst ∘ ⇑I.subtype)
hI₂ : Surjective (Prod.snd ∘ ⇑I.subtype)
x y : G × H
hx : x ∈ I
hy : y ∈ I
this✝ : I.goursatFst.Normal
this : I.goursatSnd.Normal
h : (y.1 / x.1, 1) ∈ I
⊢ (1, x.2 / y.2) ∈ I

Tactic:
simpa [Prod.mul_def, Prod.div_def] using div_mem (mul_mem h hx) hy

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: P : (G : Type u_2) → [inst : Group G] → [IsNilpotent G] → Prop
G : Type u_2
inst✝¹ : Group G
inst✝ : IsNilpotent G
hbase : ∀ (G : Type u_2) [inst : Group G] [inst_1 : Subsingleton G], P G
hstep : ∀ (G : Type u_2) [inst : Group G] [inst_1 : IsNilpotent G], P (G ⧸ center G) → P G
⊢ P G

Tactic:
obtain ⟨n, h⟩ : ∃ n, Group.nilpotencyClass G = n := ⟨_, rfl⟩

Goal state after:
P : (G : Type u_2) → [inst : Group G] → [IsNilpotent G] → Prop
G : Type u_2
inst✝¹ : Group G
inst✝ : IsNilpotent G
hbase : ∀ (G : Type u_2) [inst : Group G] [inst_1 : Subsingleton G], P G
hstep : ∀ (G : Type u_2) [inst : Group G] [inst_1 : IsNilpotent G], P (G ⧸ center G) → P G
n : ℕ
h : nilpotencyClass G = n
⊢ P G
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: case inr
M : Type u_3
α : Type u_4
N : Type u_5
β : Type u_6
inst✝³ : Monoid M
inst✝² : MulAction M α
inst✝¹ : Monoid N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : B = univ
⊢ IsTrivialBlock (⇑f '' B)

Tactic:
apply Or.intro_right

Goal state after:
case inr
M : Type u_3
α : Type u_4
N : Type u_5
β : Type u_6
inst✝³ : Monoid M
inst✝² : MulAction M α
inst✝¹ : Monoid N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : B = univ
⊢ ⇑f '' B = univ
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: case inr
M : Type u_3
α : Type u_4
N : Type u_5
β : Type u_6
inst✝³ : Monoid M
inst✝² : MulAction M α
inst✝¹ : Monoid N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : B = univ
⊢ IsTrivialBlock (⇑f ⁻¹' B)

Tactic:
apply Or.intro_right

Goal state after:
case inr
M : Type u_3
α : Type u_4
N : Type u_5
β : Type u_6
inst✝³ : Monoid M
inst✝² : MulAction M α
inst✝¹ : Monoid N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : B = univ
⊢ ⇑f ⁻¹' B = univ
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: M : Type u_3
α : Type u_4
inst✝¹ : Group M
inst✝ : MulAction M α
B : Set α
hB : IsTrivialBlock B
g : M
⊢ IsTrivialBlock (g • B)

Tactic:
by
  cases hB with
  | inl h =>
    left
    exact (Function.Injective.subsingleton_image_iff (MulAction.injective g)).mpr h
  | inr h =>
    right
    rw [h, ← Set.image_smul, Set.image_univ_of_surjective (MulAction.surjective g)]

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
C : SubMulAction G X
B : Set ↥C
⊢ IsBlock G (Subtype.val '' B) ↔ IsBlock G B

Tactic:
refine forall₂_congr fun g₁ g₂ ↦ ?_
  rw [← SubMulAction.inclusion.coe_eq, ← image_smul_set, ← image_smul_set, ne_eq,
    Set.image_eq_image C.inclusion_injective, disjoint_image_iff C.inclusion_injective]

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

Goal state before:
pp: G : Type u_1
inst✝⁵ : Group G
S : Type u_3
H : Type u_4
inst✝⁴ : Group H
inst✝³ : SetLike S H
inst✝² : SubgroupClass S H
s : S
inst✝¹ : MulAction G H
inst✝ : IsScalarTower G Hᵐᵒᵖ H
a b : G
hab : a • ↑s ≠ b • ↑s
c : H
hc : c ∈ ↑s
d : H
hd : d ∈ ↑s
hcd : b • d = a • c
⊢ (a • op c) • ↑s = b • op c • ↑s

Tactic:
← op_smul,

Goal state after:
G : Type u_1
inst✝⁵ : Group G
S : Type u_3
H : Type u_4
inst✝⁴ : Group H
inst✝³ : SetLike S H
inst✝² : SubgroupClass S H
s : S
inst✝¹ : MulAction G H
inst✝ : IsScalarTower G Hᵐᵒᵖ H
a b : G
hab : a • ↑s ≠ b • ↑s
c : H
hc : c ∈ ↑s
d : H
hd : d ∈ ↑s
hcd : b • d = a • c
⊢ op (a • c) • ↑s = b • op c • ↑s
Module: Mathlib.GroupTheory.IndexNormal

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
hHp : H.index = (Nat.card G).minFac
hG0 : ¬Nat.card G = 0
hG1 : ¬Nat.card G = 1
this : Finite G
index_ne_zero : H.index ≠ 0
hp : Nat.Prime H.index
h : H.normalCore.index ∣ H.index !
hr1 : H.normalCore.index ≠ 1
⊢ H.index - 1 < H.normalCore.index.minFac

Tactic:
exact lt_of_lt_of_le (Nat.sub_one_lt hp.ne_zero) <|
    hHp ▸ minFac_le_of_dvd (Nat.minFac_prime hr1).two_le
      (dvd_trans (minFac_dvd H.normalCore.index) (H.normalCore.index_dvd_card))

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

Goal state before:
pp: G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
inst✝ : IsPretransitive G X
B : Set X
a : X
hfB : B.Finite
B' : Set X := ⋂ k, ⋂ (_ : a ∈ k • B), k • B
hB'₀ : ∀ (k : G), a ∈ k • B → B' ⊆ k • B
b : X
hb : b ∈ B
k : G
hk : k • b = a
⊢ B'.Finite

Tactic:
apply Set.Finite.subset (Set.Finite.map _ hfB) (hB'₀ k ⟨b, hb, hk⟩)

Goal state after:
Module: Mathlib.GroupTheory.GroupAction.SubMulAction.OfStabilizer

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
s : Set α
g : ↥(stabilizer G s)
x : ↑s
⊢ g • ↑x ∈ s

Tactic:
by
    convert! Set.smul_mem_smul_set x.prop
    exact (mem_stabilizer_iff.mp g.prop).symm

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

Goal state before:
pp: c g' : Perm (Fin 5)
h : 3 ∈ (c * g').cycleType
hd : c.Disjoint g'
left✝ : c.IsCycle
h3 : c.support.card = 3
⊢ (c * (g' * (c * g'))).IsThreeCycle

Tactic:
rw [hd.commute.eq, ← mul_assoc g']

Goal state after:
c g' : Perm (Fin 5)
h : 3 ∈ (c * g').cycleType
hd : c.Disjoint g'
left✝ : c.IsCycle
h3 : c.support.card = 3
⊢ (c * (g' * g' * c)).IsThreeCycle
Module: Mathlib.GroupTheory.SpecificGroups.Alternating

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Finset α
⊢ Function.Injective ⇑(ofSubtype s)

Tactic:
by
  rw [← Function.Injective.of_comp_iff (alternatingGroup α).subtype_injective]
  exact Perm.ofSubtype_injective.comp (alternatingGroup s).subtype_injective

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

Goal state before:
pp: G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
h2 : IsMultiplyPretransitive G α 2
⊢ IsPreprimitive G α

Tactic:
have : IsPretransitive G α := isPretransitive_of_is_two_pretransitive
  apply IsPreprimitive.mk
  intro B hB
  rcases B.subsingleton_or_nontrivial with h | h
  · left
    exact h
  · right
    obtain ⟨a, ha, b, hb, h⟩ := h
    rw [← top_eq_univ, eq_top_iff]
    intro c _
    by_cases h' : a = c
    · rw [← h']; exact ha
    · rw [is_two_pretransitive_iff] at h2
      obtain ⟨g, hga, hgb⟩ := h2 h h'
      rw [MulAction.isBlock_iff_smul_eq_of_mem] at hB
      rw [← hB (g := g) ha (by rw [hga]; exact ha), ← hgb]
      exact smul_mem_smul_set hb

Goal state after:
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
a b : α
hab : a ≠ b
g✝ g : Perm α
x y : α
hxy : x ≠ y
ih : g ∈ G ↔ g ∈ alternatingGroup α
habG : swap x y ∈ G
⊢ swap a b ∈ G

Tactic:
← (isConj_iff.mp <| isConj_swap hxy hab).choose_spec

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
a b : α
hab : a ≠ b
g✝ g : Perm α
x y : α
hxy : x ≠ y
ih : g ∈ G ↔ g ∈ alternatingGroup α
habG : swap x y ∈ G
⊢ ⋯.choose * swap x y * ⋯.choose⁻¹ ∈ G
Module: Mathlib.GroupTheory.GroupAction.MultiplePrimitivity

Goal state before:
pp: case mp
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
H1 : IsMultiplyPreprimitive M α 1
⊢ IsPreprimitive M α

Tactic:
← isPreprimitive_of_fixingSubgroup_empty_iff

Goal state after:
case mp
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
H1 : IsMultiplyPreprimitive M α 1
⊢ IsPreprimitive ↥(fixingSubgroup M ∅) ↥(ofFixingSubgroup M ∅)
Module: Mathlib.GroupTheory.GroupAction.Jordan

Goal state before:
pp: n : ℕ
hrec :
  ∀ m < n,
    ∀ {G : Type u_1} {α : Type u_2} [inst : Group G] [inst_1 : MulAction G α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          s.ncard = m + 1 →
            m + 2 < Nat.card α →
              (IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2) ∧
                (IsPreprimitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPreprimitive G α 2)
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hG : IsPreprimitive G α
s : Set α
hsn : s.ncard = n + 1
hsn' : n + 2 < Nat.card α
this : Finite α
hs_ne_univ : s ≠ Set.univ
hs_nonempty : s.Nonempty
hn : n + 1 ≥ 2
hn1 : 2 * (n + 1) < Nat.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := ⋯
ht : t.Finite
htm : t.ncard = t.ncard - 1 + 1
⊢ t ⊂ s

Tactic:
exact ⟨Set.inter_subset_left, fun h ↦ hgb (Set.inter_subset_right (h hb))⟩

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

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝² : Group M
inst✝¹ : MulAction M α
n : ℕ
hα : ↑n.succ ≤ ENat.card α
inst✝ : IsMultiplyPretransitive M α n.succ
⊢ IsMultiplyPreprimitive M α n

Tactic:
rcases Nat.eq_zero_or_pos n with hn | hn

Goal state after:
case inl
M : Type u_1
α : Type u_2
inst✝² : Group M
inst✝¹ : MulAction M α
n : ℕ
hα : ↑n.succ ≤ ENat.card α
inst✝ : IsMultiplyPretransitive M α n.succ
hn : n = 0
⊢ IsMultiplyPreprimitive M α n

case inr
M : Type u_1
α : Type u_2
inst✝² : Group M
inst✝¹ : MulAction M α
n : ℕ
hα : ↑n.succ ≤ ENat.card α
inst✝ : IsMultiplyPretransitive M α n.succ
hn : n > 0
⊢ IsMultiplyPreprimitive M α n
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
N : Type u_3
β : Type u_4
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Bijective ⇑f
n : ℕ
h : IsMultiplyPreprimitive M α n
t : Set β
ht : t.encard + 1 = ↑n
s : Set α := ⋯
hs' : ⇑f '' s = t
x✝¹ : ↥(fixingSubgroup M s)
m : M
hm : m ∈ fixingSubgroup M s
x✝ : ↑t
y : β
hy : y ∈ t
x : α
hx : x ∈ s
hx' : f x = y
⊢ m • x = x

Tactic:
rw [mem_fixingSubgroup_iff] at hm

Goal state after:
M : Type u_1
α : Type u_2
inst✝³ : Group M
inst✝² : MulAction M α
N : Type u_3
β : Type u_4
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Bijective ⇑f
n : ℕ
h : IsMultiplyPreprimitive M α n
t : Set β
ht : t.encard + 1 = ↑n
s : Set α := ⇑f ⁻¹' t
hs' : ⇑f '' s = t
x✝¹ : ↥(fixingSubgroup M s)
m : M
hm : ∀ y ∈ s, m • y = y
x✝ : ↑t
y : β
hy : y ∈ t
x : α
hx : x ∈ s
hx' : f x = y
⊢ m • x = x
Module: Mathlib.GroupTheory.GroupAction.Jordan

Goal state before:
pp: n : ℕ
hrec :
  ∀ m < n,
    ∀ {G : Type u_1} {α : Type u_2} [inst : Group G] [inst_1 : MulAction G α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          s.ncard = m + 1 →
            m + 2 < Nat.card α →
              (IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2) ∧
                (IsPreprimitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPreprimitive G α 2)
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hG : IsPreprimitive G α
s : Set α
hsn : s.ncard = n + 1
hsn' : n + 2 < Nat.card α
this : Finite α
hs_ne_univ : s ≠ Set.univ
hs_nonempty : s.Nonempty
hn : n + 1 ≥ 2
hn1 : 2 * (n + 1) < Nat.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := ⋯
ht : t.Finite
htm : t.ncard = t.ncard - 1 + 1
hmn : t.ncard - 1 < n
htm' : t.ncard - 1 + 2 < Nat.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s)
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(ofFixingSubgroup G t)
⊢ t.ncard - 1 + 2 < n + 2

Tactic:
exact Nat.add_lt_add_right hmn 2

Goal state after:
Module: Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup

Goal state before:
pp: case inr
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup M s) ↥(ofFixingSubgroup M s)
g : M
a : α
ha : a ∈ sᶜ ∩ (g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ ofFixingSubgroup M (s ∩ g • s)
hx : x ∉ g • s
⊢ ∃ g_1, g_1 • ⟨a, ha'⟩ = ⟨x, hx✝⟩

Tactic:
have hg'x : g⁻¹ • x ∈ ofFixingSubgroup M s := mt Set.mem_smul_set_iff_inv_smul_mem.mpr hx
    have hg'a : g⁻¹ • a ∈ ofFixingSubgroup M s := mt Set.mem_smul_set_iff_inv_smul_mem.mpr ha.2
    obtain ⟨⟨k, hk⟩, hkax⟩ := hs.exists_smul_eq ⟨g⁻¹ • a, hg'a⟩ ⟨g⁻¹ • x, hg'x⟩
    use ⟨g * k * g⁻¹, ?_⟩
    · simp only [← SetLike.coe_eq_coe] at hkax ⊢
      rwa [SetLike.val_smul, Subgroup.mk_smul, eq_inv_smul_iff, smul_smul, smul_smul] at hkax
    · rw [mem_fixingSubgroup_iff] at hk ⊢
      intro y hy
      rw [mul_smul, mul_smul, smul_eq_iff_eq_inv_smul g]
      exact hk _ (Set.mem_smul_set_iff_inv_smul_mem.mp hy.2)

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

Goal state before:
pp: α : Type u_2
inst✝ : DecidableEq α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
this : swap a b • s ⊆ s
⊢ swap a b • s = s

Tactic:
apply Set.Subset.antisymm this

Goal state after:
α : Type u_2
inst✝ : DecidableEq α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
this : swap a b • s ⊆ s
⊢ s ⊆ swap a b • s
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: case inr
M : Type u_1
α : Type u_2
inst✝² : Group M
inst✝¹ : MulAction M α
B : Set α
G : Subgroup M
hB : IsBlock (↥G) B
inst✝ : IsPreprimitive ↥(stabilizer (↥G) B) ↑B
hB_not_le_sc : ∀ (B_1 : Set α), IsBlock (↥G) B_1 → B_1 ⊆ Bᶜ → B_1.Subsingleton
hG : stabilizer M B < G
hBs : B ⊆ B
hB' : Subtype.val ⁻¹' B = Set.univ
g' : M
hg' : g' ∈ G
hg's : g' ∉ stabilizer M B
h : _root_.Disjoint (⟨g', hg'⟩ • B) B
⊢ B.Subsingleton

Tactic:
suffices (g' • B).Subsingleton by
        exact subsingleton_of_image (MulAction.injective g') B this

Goal state after:
case inr
M : Type u_1
α : Type u_2
inst✝² : Group M
inst✝¹ : MulAction M α
B : Set α
G : Subgroup M
hB : IsBlock (↥G) B
inst✝ : IsPreprimitive ↥(stabilizer (↥G) B) ↑B
hB_not_le_sc : ∀ (B_1 : Set α), IsBlock (↥G) B_1 → B_1 ⊆ Bᶜ → B_1.Subsingleton
hG : stabilizer M B < G
hBs : B ⊆ B
hB' : Subtype.val ⁻¹' B = Set.univ
g' : M
hg' : g' ∈ G
hg's : g' ∉ stabilizer M B
h : _root_.Disjoint (⟨g', hg'⟩ • B) B
⊢ (g' • B).Subsingleton
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝³ : Group M
inst✝² : MulAction M α
s : Set α
inst✝¹ : Finite α
inst✝ : IsMultiplyPretransitive M α (s.ncard + 1)
G : Subgroup M
hG : stabilizer M s ≤ G
B : Set α
hBs : ¬B ⊆ s
hBsc : ¬B ⊆ sᶜ
hB : IsBlock (↥G) B
a : α
ha : a ∈ B
ha' : a ∈ s
b : α
hb : b ∈ B
hb' : b ∈ sᶜ
x : α
hx' : x ∈ sᶜ
this : ∃ k, k • b = x
⊢ x ∈ B

Tactic:
obtain ⟨⟨k, hk⟩, rfl⟩ := this

Goal state after:
M : Type u_1
α : Type u_2
inst✝³ : Group M
inst✝² : MulAction M α
s : Set α
inst✝¹ : Finite α
inst✝ : IsMultiplyPretransitive M α (s.ncard + 1)
G : Subgroup M
hG : stabilizer M s ≤ G
B : Set α
hBs : ¬B ⊆ s
hBsc : ¬B ⊆ sᶜ
hB : IsBlock (↥G) B
a : α
ha : a ∈ B
ha' : a ∈ s
b : α
hb : b ∈ B
hb' : b ∈ sᶜ
k : M
hk : k ∈ fixingSubgroup M s
hx' : ⟨k, hk⟩ • b ∈ sᶜ
⊢ ⟨k, hk⟩ • b ∈ B
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝³ : Group M
inst✝² : MulAction M α
s : Set α
inst✝¹ : Finite α
inst✝ : IsMultiplyPretransitive M α (s.ncard + 1)
G : Subgroup M
hG : stabilizer M s ≤ G
B : Set α
hBs : ¬B ⊆ s
hBsc : ¬B ⊆ sᶜ
hB : IsBlock (↥G) B
a : α
ha : a ∈ B
ha' : a ∈ s
b : α
hb : b ∈ B
hb' : b ∈ sᶜ
k : M
hk : k ∈ fixingSubgroup M s
hx' : ⟨k, hk⟩ • b ∈ sᶜ
⊢ k • B = B

Tactic:
apply isBlock_iff_smul_eq_of_nonempty.mp hB (g := ⟨k, ?_⟩)

Goal state after:
M : Type u_1
α : Type u_2
inst✝³ : Group M
inst✝² : MulAction M α
s : Set α
inst✝¹ : Finite α
inst✝ : IsMultiplyPretransitive M α (s.ncard + 1)
G : Subgroup M
hG : stabilizer M s ≤ G
B : Set α
hBs : ¬B ⊆ s
hBsc : ¬B ⊆ sᶜ
hB : IsBlock (↥G) B
a : α
ha : a ∈ B
ha' : a ∈ s
b : α
hb : b ∈ B
hb' : b ∈ sᶜ
k : M
hk : k ∈ fixingSubgroup M s
hx' : ⟨k, hk⟩ • b ∈ sᶜ
⊢ (⟨k, ?m.152⟩ • B ∩ B).Nonempty

M : Type u_1
α : Type u_2
inst✝³ : Group M
inst✝² : MulAction M α
s : Set α
inst✝¹ : Finite α
inst✝ : IsMultiplyPretransitive M α (s.ncard + 1)
G : Subgroup M
hG : stabilizer M s ≤ G
B : Set α
hBs : ¬B ⊆ s
hBsc : ¬B ⊆ sᶜ
hB : IsBlock (↥G) B
a : α
ha : a ∈ B
ha' : a ∈ s
b : α
hb : b ∈ B
hb' : b ∈ sᶜ
k : M
hk : k ∈ fixingSubgroup M s
hx' : ⟨k, hk⟩ • b ∈ sᶜ
⊢ k ∈ G
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : s.Nonempty
h1 : sᶜ.Nonempty
hs : Nat.card α ≠ 2 * s.ncard
⊢ IsCoatom (stabilizer (↥(alternatingGroup α)) s)

Tactic:
rw [← ncard_add_ncard_compl s, two_mul, ne_eq, Nat.add_left_cancel_iff] at hs
  wlog hs' : ncard s < ncard sᶜ
  · rw [← stabilizer_compl]
    apply this h1 <;> rw [compl_compl] <;> grind
  · by_cases h0' : s.Nontrivial
    · apply isCoatom_stabilizer_of_ncard_lt_ncard_compl h0' hs'
    · simp only [not_nontrivial_iff] at h0'
      apply isCoatom_stabilizer_singleton _ h0 h0'
      rw [← ncard_add_ncard_compl s]
      rw [← ncard_pos] at h0 h1
      grind

Goal state after: