mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: case inl
M : Type u_1
α : Type u_2
inst✝² : Group M
inst✝¹ : MulAction M α
s B : Set α
G : Subgroup M
inst✝ : IsPreprimitive ↥(stabilizer (↥G) s) ↑s
hB : IsBlock (↥G) B
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → B.Subsingleton
hG : stabilizer M s < G
hBs : B ⊆ s
hB' : (Subtype.val ⁻¹' B).Subsingleton
⊢ (s ∩ B).Subsingleton

Tactic:
← Subtype.image_preimage_val

Goal state after:
case inl
M : Type u_1
α : Type u_2
inst✝² : Group M
inst✝¹ : MulAction M α
s B : Set α
G : Subgroup M
inst✝ : IsPreprimitive ↥(stabilizer (↥G) s) ↑s
hB : IsBlock (↥G) B
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → B.Subsingleton
hG : stabilizer M s < G
hBs : B ⊆ s
hB' : (Subtype.val ⁻¹' B).Subsingleton
⊢ (Subtype.val '' Subtype.val ⁻¹' B).Subsingleton
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝² : Group M
inst✝¹ : MulAction M α
s B : Set α
G : Subgroup M
inst✝ : IsPreprimitive ↥(stabilizer (↥G) s) ↑s
hB : IsBlock (↥G) B
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → B.Subsingleton
hG : stabilizer M s < G
hBs : B ⊆ s
this : IsPreprimitive ↥(stabilizer (↥G) s) ↑s
φ' : ↥(stabilizer (↥G) s) → ↥G := Subtype.val
f' : ↑s →ₑ[φ'] α := { toFun := Subtype.val, map_smul' := ⋯ }
⊢ IsBlock (↥(stabilizer (↥G) s)) (Subtype.val ⁻¹' B)

Tactic:
apply MulAction.IsBlock.preimage f' hB

Goal state after:
Module: Mathlib.GroupTheory.GroupExtension.Basic

Goal state before:
pp: case h
N : Type u_1
G : Type u_2
inst✝² : Group N
inst✝¹ : Group G
E : Type u_3
inst✝ : Group E
S : GroupExtension N E G
σ : S.Section
g₁ g₂ : G
n : N
hn : S.inl n = (σ (g₁ * g₂))⁻¹ * σ g₁ * σ g₂
⊢ σ (g₁ * g₂) = σ g₁ * σ g₂ * S.inl n⁻¹

Tactic:
rw [map_inv, eq_mul_inv_iff_mul_eq, ← eq_inv_mul_iff_mul_eq, ← mul_assoc, hn]

Goal state after:
Module: Mathlib.GroupTheory.GroupExtension.Basic

Goal state before:
pp: N : Type u_1
G : Type u_2
inst✝³ : Group N
inst✝² : Group G
E : Type u_3
inst✝¹ : Group E
S : GroupExtension N E G
E' : Type u_4
inst✝ : Group E'
S' : GroupExtension N E' G
f : E →* E'
comp_inl : f.comp S.inl = S'.inl
rightHom_comp : S'.rightHom.comp f = S.rightHom
e' : E'
⊢ (↑f).toFun
      ((fun e' ↦
          let e := Function.surjInv ⋯ (S'.rightHom e');
          e * S.inl (Function.invFun (⇑S'.inl) ((f e)⁻¹ * e')))
        e') =
    e'

Tactic:
simp only [OneHom.toFun_eq_coe, MonoidHom.toOneHom_coe, map_mul]
    rw [← eq_inv_mul_iff_mul_eq, ← MonoidHom.comp_apply, comp_inl]
    apply Function.invFun_eq
    rw [← MonoidHom.mem_range, S'.range_inl_eq_ker_rightHom, MonoidHom.mem_ker, map_mul, map_inv,
      ← MonoidHom.comp_apply, rightHom_comp]
    simpa only [Function.surjInv_eq] using inv_mul_cancel (S'.rightHom e')

Goal state after:
Module: Mathlib.GroupTheory.GroupExtension.Basic

Goal state before:
pp: N : Type u_1
G : Type u_2
inst✝² : Group N
inst✝¹ : Group G
E : Type u_3
inst✝ : Group E
S : GroupExtension N E G
s : S.Splitting
e : E
⊢ (fun x ↦
        match x with
        | ⟨n, g⟩ => S.inl n * s g)
      ((fun e ↦ ⟨Function.invFun (⇑S.inl) (e * (s (S.rightHom e))⁻¹), S.rightHom e⟩) e) =
    e

Tactic:
simp only [← eq_mul_inv_iff_mul_eq]
    apply Function.invFun_eq
    rw [← MonoidHom.mem_range, S.range_inl_eq_ker_rightHom, MonoidHom.mem_ker, map_mul, map_inv,
      rightHom_splitting, mul_inv_cancel]

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

Goal state before:
pp: case hα
α : Type u_2
inst✝ : DecidableEq α
n : ℕ
h_one_le : 1 ≤ n
hn : n < Nat.card α
hα : Nat.card α ≠ 2 * n
this✝² : Finite α
this✝¹ : Fintype α
this✝ : IsPretransitive (Perm α) ↑(powersetCard α n)
this : Nontrivial ↑(powersetCard α n)
s : ↑(powersetCard α n)
⊢ Nat.card α ≠ 2 * (↑s).ncard

Tactic:
rwa [ncard_eq]

Goal state after:
Module: Mathlib.GroupTheory.IsSubnormal

Goal state before:
pp: case step
G : Type u_1
inst✝ : Group G
H✝ : Subgroup G
hG : IsSimpleGroup G
H K : Subgroup G
h_le : H ≤ K
hSubn : K.IsSubnormal
hN : (H.subgroupOf K).Normal
Knorm : K.Normal
⊢ H.Normal

Tactic:
obtain rfl | rfl := Knorm.eq_bot_or_eq_top

Goal state after:
case step.inl
G : Type u_1
inst✝ : Group G
H✝ : Subgroup G
hG : IsSimpleGroup G
H : Subgroup G
h_le : H ≤ ⊥
hSubn : ⊥.IsSubnormal
hN : (H.subgroupOf ⊥).Normal
Knorm : ⊥.Normal
⊢ H.Normal

case step.inr
G : Type u_1
inst✝ : Group G
H✝ : Subgroup G
hG : IsSimpleGroup G
H : Subgroup G
h_le : H ≤ ⊤
hSubn : ⊤.IsSubnormal
hN : (H.subgroupOf ⊤).Normal
Knorm : ⊤.Normal
⊢ H.Normal
Module: Mathlib.GroupTheory.IsSubnormal

Goal state before:
pp: case step
G : Type u_1
inst✝ : Group G
H✝ H K : Subgroup G
h_le : H ≤ K
hSubn : K.IsSubnormal
hN : (H.subgroupOf K).Normal
ih : K ≠ ⊤ → ∃ K_1, K_1.Normal ∧ K ≤ K_1 ∧ K_1 < ⊤
ne_top : H ≠ ⊤
⊢ ∃ K, K.Normal ∧ H ≤ K ∧ K < ⊤

Tactic:
| step H K h_le hSubn hN ih =>

Goal state after:
Module: Mathlib.GroupTheory.IsSubnormal

Goal state before:
pp: case step
G : Type u_1
inst✝ : Group G
H✝ H K : Subgroup G
h_le : H ≤ K
hSubn : K.IsSubnormal
hN : (H.subgroupOf K).Normal
ih : ∃ n f, Monotone f ∧ (∀ (i : ℕ), ((f i).subgroupOf (f (i + 1))).Normal) ∧ f 0 = K ∧ f n = ⊤
⊢ ∃ n f, Monotone f ∧ (∀ (i : ℕ), ((f i).subgroupOf (f (i + 1))).Normal) ∧ f 0 = H ∧ f n = ⊤

Tactic:
| step H K h_le hSubn hN ih =>

Goal state after:
Module: Mathlib.GroupTheory.MonoidLocalization.Order

Goal state before:
pp: α : Type u_1
inst✝² : CommMonoid α
inst✝¹ : PartialOrder α
inst✝ : IsOrderedCancelMonoid α
s : Submonoid α
a₁✝ b₁✝ : α
a₂✝ b₂✝ : ↥s
a b : Localization s
a₁ b₁ : α
a₂ b₂ : ↥s
c₁ d₁ : α
c₂ d₂ : ↥s
hab : (r s) (a₁, a₂) (b₁, b₂)
hcd : (r s) (c₁, c₂) (d₁, d₂)
e f : ↥s
he : ↑b₂ * a₁ = ↑a₂ * b₁
hf : ↑d₂ * c₁ = ↑c₂ * d₁
⊢ ↑c₂ * a₁ * ?a ≤ ↑a₂ * c₁ * ?a ↔ ↑d₂ * b₁ ≤ ↑b₂ * d₁

pp: case a
α : Type u_1
inst✝² : CommMonoid α
inst✝¹ : PartialOrder α
inst✝ : IsOrderedCancelMonoid α
s : Submonoid α
a₁✝ b₁✝ : α
a₂✝ b₂✝ : ↥s
a b : Localization s
a₁ b₁ : α
a₂ b₂ : ↥s
c₁ d₁ : α
c₂ d₂ : ↥s
hab : (r s) (a₁, a₂) (b₁, b₂)
hcd : (r s) (c₁, c₂) (d₁, d₂)
e f : ↥s
he : ↑b₂ * a₁ = ↑a₂ * b₁
hf : ↑d₂ * c₁ = ↑c₂ * d₁
⊢ α

Tactic:
mul_right_comm,

Goal state after:
α : Type u_1
inst✝² : CommMonoid α
inst✝¹ : PartialOrder α
inst✝ : IsOrderedCancelMonoid α
s : Submonoid α
a₁✝ b₁✝ : α
a₂✝ b₂✝ : ↥s
a b : Localization s
a₁ b₁ : α
a₂ b₂ : ↥s
c₁ d₁ : α
c₂ d₂ : ↥s
hab : (r s) (a₁, a₂) (b₁, b₂)
hcd : (r s) (c₁, c₂) (d₁, d₂)
e f : ↥s
he : ↑b₂ * a₁ = ↑a₂ * b₁
hf : ↑d₂ * c₁ = ↑c₂ * d₁
⊢ ↑c₂ * ?c * a₁ ≤ ↑a₂ * c₁ * ?c ↔ ↑d₂ * b₁ ≤ ↑b₂ * d₁

case c
α : Type u_1
inst✝² : CommMonoid α
inst✝¹ : PartialOrder α
inst✝ : IsOrderedCancelMonoid α
s : Submonoid α
a₁✝ b₁✝ : α
a₂✝ b₂✝ : ↥s
a b : Localization s
a₁ b₁ : α
a₂ b₂ : ↥s
c₁ d₁ : α
c₂ d₂ : ↥s
hab : (r s) (a₁, a₂) (b₁, b₂)
hcd : (r s) (c₁, c₂) (d₁, d₂)
e f : ↥s
he : ↑b₂ * a₁ = ↑a₂ * b₁
hf : ↑d₂ * c₁ = ↑c₂ * d₁
⊢ α
Module: Mathlib.GroupTheory.MonoidLocalization.Order

Goal state before:
pp: α : Type u_1
inst✝² : CommMonoid α
inst✝¹ : PartialOrder α
inst✝ : IsOrderedCancelMonoid α
s : Submonoid α
a₁✝ b₁✝ : α
a₂✝ b₂✝ : ↥s
a b : Localization s
a₁ b₁ : α
a₂ b₂ : ↥s
c₁ d₁ : α
c₂ d₂ : ↥s
hab : (r s) (a₁, a₂) (b₁, b₂)
hcd : (r s) (c₁, c₂) (d₁, d₂)
e f : ↥s
he : ↑b₂ * a₁ = ↑a₂ * b₁
hf : ↑d₂ * c₁ = ↑c₂ * d₁
⊢ ↑d₂ * c₁ * a₁ ≤ ↑a₂ * c₁ * d₁ ↔ ↑d₂ * b₁ ≤ ↑b₂ * d₁

Tactic:
mul_right_comm,

Goal state after:
α : Type u_1
inst✝² : CommMonoid α
inst✝¹ : PartialOrder α
inst✝ : IsOrderedCancelMonoid α
s : Submonoid α
a₁✝ b₁✝ : α
a₂✝ b₂✝ : ↥s
a b : Localization s
a₁ b₁ : α
a₂ b₂ : ↥s
c₁ d₁ : α
c₂ d₂ : ↥s
hab : (r s) (a₁, a₂) (b₁, b₂)
hcd : (r s) (c₁, c₂) (d₁, d₂)
e f : ↥s
he : ↑b₂ * a₁ = ↑a₂ * b₁
hf : ↑d₂ * c₁ = ↑c₂ * d₁
⊢ ↑d₂ * a₁ * c₁ ≤ ↑a₂ * c₁ * d₁ ↔ ↑d₂ * b₁ ≤ ↑b₂ * d₁
Module: Mathlib.GroupTheory.MonoidLocalization.Order

Goal state before:
pp: α : Type u_1
inst✝² : CommMonoid α
inst✝¹ : PartialOrder α
inst✝ : IsOrderedCancelMonoid α
s : Submonoid α
a₁✝ b₁✝ : α
a₂✝ b₂✝ : ↥s
a b : Localization s
a₁ b₁ : α
a₂ b₂ : ↥s
c₁ d₁ : α
c₂ d₂ : ↥s
hab : (r s) (a₁, a₂) (b₁, b₂)
hcd : (r s) (c₁, c₂) (d₁, d₂)
e f : ↥s
he : ↑b₂ * a₁ = ↑a₂ * b₁
hf : ↑d₂ * c₁ = ↑c₂ * d₁
⊢ ↑c₂ * a₁ * ?a < ↑a₂ * c₁ * ?a ↔ ↑d₂ * b₁ < ↑b₂ * d₁

pp: case a
α : Type u_1
inst✝² : CommMonoid α
inst✝¹ : PartialOrder α
inst✝ : IsOrderedCancelMonoid α
s : Submonoid α
a₁✝ b₁✝ : α
a₂✝ b₂✝ : ↥s
a b : Localization s
a₁ b₁ : α
a₂ b₂ : ↥s
c₁ d₁ : α
c₂ d₂ : ↥s
hab : (r s) (a₁, a₂) (b₁, b₂)
hcd : (r s) (c₁, c₂) (d₁, d₂)
e f : ↥s
he : ↑b₂ * a₁ = ↑a₂ * b₁
hf : ↑d₂ * c₁ = ↑c₂ * d₁
⊢ α

Tactic:
mul_right_comm,

Goal state after:
α : Type u_1
inst✝² : CommMonoid α
inst✝¹ : PartialOrder α
inst✝ : IsOrderedCancelMonoid α
s : Submonoid α
a₁✝ b₁✝ : α
a₂✝ b₂✝ : ↥s
a b : Localization s
a₁ b₁ : α
a₂ b₂ : ↥s
c₁ d₁ : α
c₂ d₂ : ↥s
hab : (r s) (a₁, a₂) (b₁, b₂)
hcd : (r s) (c₁, c₂) (d₁, d₂)
e f : ↥s
he : ↑b₂ * a₁ = ↑a₂ * b₁
hf : ↑d₂ * c₁ = ↑c₂ * d₁
⊢ ↑c₂ * ?c * a₁ < ↑a₂ * c₁ * ?c ↔ ↑d₂ * b₁ < ↑b₂ * d₁

case c
α : Type u_1
inst✝² : CommMonoid α
inst✝¹ : PartialOrder α
inst✝ : IsOrderedCancelMonoid α
s : Submonoid α
a₁✝ b₁✝ : α
a₂✝ b₂✝ : ↥s
a b : Localization s
a₁ b₁ : α
a₂ b₂ : ↥s
c₁ d₁ : α
c₂ d₂ : ↥s
hab : (r s) (a₁, a₂) (b₁, b₂)
hcd : (r s) (c₁, c₂) (d₁, d₂)
e f : ↥s
he : ↑b₂ * a₁ = ↑a₂ * b₁
hf : ↑d₂ * c₁ = ↑c₂ * d₁
⊢ α
Module: Mathlib.GroupTheory.MonoidLocalization.Order

Goal state before:
pp: α : Type u_1
inst✝² : CommMonoid α
inst✝¹ : PartialOrder α
inst✝ : IsOrderedCancelMonoid α
s : Submonoid α
a₁✝ b₁✝ : α
a₂✝ b₂✝ : ↥s
a b : Localization s
a₁ b₁ : α
a₂ b₂ : ↥s
c₁ d₁ : α
c₂ d₂ : ↥s
hab : (r s) (a₁, a₂) (b₁, b₂)
hcd : (r s) (c₁, c₂) (d₁, d₂)
e f : ↥s
he : ↑b₂ * a₁ = ↑a₂ * b₁
hf : ↑d₂ * c₁ = ↑c₂ * d₁
⊢ ↑d₂ * c₁ * a₁ < ↑a₂ * c₁ * d₁ ↔ ↑d₂ * b₁ < ↑b₂ * d₁

Tactic:
mul_right_comm,

Goal state after:
α : Type u_1
inst✝² : CommMonoid α
inst✝¹ : PartialOrder α
inst✝ : IsOrderedCancelMonoid α
s : Submonoid α
a₁✝ b₁✝ : α
a₂✝ b₂✝ : ↥s
a b : Localization s
a₁ b₁ : α
a₂ b₂ : ↥s
c₁ d₁ : α
c₂ d₂ : ↥s
hab : (r s) (a₁, a₂) (b₁, b₂)
hcd : (r s) (c₁, c₂) (d₁, d₂)
e f : ↥s
he : ↑b₂ * a₁ = ↑a₂ * b₁
hf : ↑d₂ * c₁ = ↑c₂ * d₁
⊢ ↑d₂ * a₁ * c₁ < ↑a₂ * c₁ * d₁ ↔ ↑d₂ * b₁ < ↑b₂ * d₁
Module: Mathlib.GroupTheory.IsSubnormal

Goal state before:
pp: case step
G : Type u_1
inst✝¹ : Group G
H✝ : Subgroup G
G' : Type u_2
inst✝ : Group G'
f : G →* G'
hf : Function.Surjective ⇑f
H K : Subgroup G
h_le : H ≤ K
hSubn : K.IsSubnormal
hN : (H.subgroupOf K).Normal
ih : (map f K).IsSubnormal
⊢ (map f H).IsSubnormal

Tactic:
| step H K h_le hSubn hN ih =>

Goal state after:
Module: Mathlib.GroupTheory.IsSubnormal

Goal state before:
pp: case step
G : Type u_1
inst✝¹ : Group G
G' : Type u_2
inst✝ : Group G'
H' : Subgroup G'
f : G →* G'
H K : Subgroup G'
h_le : H ≤ K
hSubn : K.IsSubnormal
hN : (H.subgroupOf K).Normal
ih : (comap f K).IsSubnormal
⊢ (comap f H).IsSubnormal

Tactic:
| step H K h_le hSubn hN ih =>

Goal state after:
Module: Mathlib.GroupTheory.MonoidLocalization.Order

Goal state before:
pp: α : Type u_1
inst✝² : CommMonoid α
inst✝¹ : PartialOrder α
inst✝ : IsOrderedCancelMonoid α
s : Submonoid α
a₁ b₁ : α
a₂ b₂ : ↥s
a✝ b✝ c✝ : Localization s
a b c : α × ↥s
hab : ↑b.2 * a.1 ≤ ↑a.2 * b.1
hbc : ↑c.2 * b.1 ≤ ↑b.2 * c.1
⊢ α

Tactic:
exact ↑b.2

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

Goal state before:
pp: α : Type u_1
ι : Type u_2
f : α → ι
g : (i : ι) → Perm { a // f a = i }
a : α
i : ι
h : f a = i
⊢ stabilizerEquiv_invFun g a = ↑((g i) ⟨a, h⟩)

Tactic:
subst h; rfl

Goal state after:
Module: Mathlib.GroupTheory.Perm.Cycle.PossibleTypes

Goal state before:
pp: case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
m : Multiset ℕ
hc : m.sum ≤ Fintype.card α
h2c : ∀ a ∈ m, 2 ≤ a
hc' : m.toList.sum ≤ Fintype.card α
p : List (List α)
hp_length : List.map List.length p = m.toList
hp_nodup : ∀ s ∈ p, s.Nodup
hp_disj : List.Pairwise List.Disjoint p
hp2 : ∀ x ∈ p, 2 ≤ x.length
x : List α
hx : x ∈ p
⊢ x.toFinset.card = x.length

Tactic:
rw [List.toFinset_card_of_nodup (hp_nodup x hx)]

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
this : ∀ (c : ↥g.cycleFactorsFinset), (↑c).support.Nonempty
⊢ Nonempty g.Basis

Tactic:
exact ⟨fun c ↦ (this c).choose, fun c ↦ (this c).choose_spec⟩

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
a : g.Basis
c : ↥g.cycleFactorsFinset
τ✝ τ : ↥(range_toPermHom' g)
x : α
⊢ a.ofPermHomFun τ⁻¹ (a.ofPermHomFun τ x) = x

Tactic:
← ofPermHomFun_mul,

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
a : g.Basis
c : ↥g.cycleFactorsFinset
τ✝ τ : ↥(range_toPermHom' g)
x : α
⊢ a.ofPermHomFun (τ⁻¹ * τ) x = x
Module: Mathlib.GroupTheory.Perm.Centralizer

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
a : g.Basis
c : ↥g.cycleFactorsFinset
τ✝ τ : ↥(range_toPermHom' g)
x : α
⊢ a.ofPermHomFun τ (a.ofPermHomFun τ⁻¹ x) = x

Tactic:
← ofPermHomFun_mul,

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
a : g.Basis
c : ↥g.cycleFactorsFinset
τ✝ τ : ↥(range_toPermHom' g)
x : α
⊢ a.ofPermHomFun (τ * τ⁻¹) x = x
Module: Mathlib.GroupTheory.SchurZassenhaus

Goal state before:
pp: G : Type u_1
inst✝² : Group G
H : Subgroup G
inst✝¹ : IsMulCommutative ↥H
inst✝ : H.FiniteIndex
α β : H.LeftTransversal
hH : H.Normal
g : Gᵐᵒᵖ
this : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
h₁ h₂ : ↥H
⊢ ⟨(unop g)⁻¹ * ↑(h₁ * h₂) * unop g, ⋯⟩ = ⟨(unop g)⁻¹ * ↑h₁ * unop g, ⋯⟩ * ⟨(unop g)⁻¹ * ↑h₂ * unop g, ⋯⟩

Tactic:
simp only [Subtype.ext_iff, coe_mul, mul_assoc, mul_inv_cancel_left]

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
sc : ↥g.cycleFactorsFinset → ℕ := ⋯
hsc : sc = fun c ↦ #(↑c).support
x✝ : Perm ↥g.cycleFactorsFinset
⊢ x✝ ∈ (toPermHom g).range ↔ x✝ ∈ {k | sc ∘ ⇑k = sc}

Tactic:
mem_range_toPermHom_iff',

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
sc : ↥g.cycleFactorsFinset → ℕ := fun c ↦ #(↑c).support
hsc : sc = fun c ↦ #(↑c).support
x✝ : Perm ↥g.cycleFactorsFinset
⊢ ((fun c ↦ #(↑c).support) ∘ ⇑x✝ = fun c ↦ #(↑c).support) ↔ x✝ ∈ {k | sc ∘ ⇑k = sc}
Module: Mathlib.GroupTheory.Perm.Centralizer

Goal state before:
pp: case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
u : Perm ↑(Function.fixedPoints ⇑g)
v : (c : ↥g.cycleFactorsFinset) → ↥(zpowers ↑c)
x : α
hx : g.cycleOf x ∈ g.cycleFactorsFinset
hx' : x ∉ Function.fixedPoints ⇑g
⊢ ↑((u, v).2 ⟨g.cycleOf x, hx⟩) x ∉
    ((Finset.univ.erase ⟨g.cycleOf x, hx⟩).noncommProd (fun i ↦ ↑((u, v).2 i)) ⋯).support

Tactic:
contrapose hx'

Goal state after:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
u : Perm ↑(Function.fixedPoints ⇑g)
v : (c : ↥g.cycleFactorsFinset) → ↥(zpowers ↑c)
x : α
hx : g.cycleOf x ∈ g.cycleFactorsFinset
hx' :
  ↑((u, v).2 ⟨g.cycleOf x, hx⟩) x ∈
    ((Finset.univ.erase ⟨g.cycleOf x, hx⟩).noncommProd (fun i ↦ ↑((u, v).2 i)) ⋯).support
⊢ x ∈ Function.fixedPoints ⇑g
Module: Mathlib.GroupTheory.SchurZassenhaus

Goal state before:
pp: G : Type u
inst✝² : Group G
N : Subgroup G
inst✝¹ : N.Normal
h1 : (Nat.card ↥N).Coprime N.index
h2 :
  ∀ (G' : Type u) [inst : Group G'] [Finite G'],
    Nat.card G' < Nat.card G →
      ∀ {N' : Subgroup G'} [N'.Normal], (Nat.card ↥N').Coprime N'.index → ∃ H', N'.IsComplement' H'
inst✝ : Finite G
K : Subgroup G
hK : K ⊔ N = ⊤
h3 : K ≠ ⊤
⊢ (comap K.subtype N).index = N.relIndex ⊤

Tactic:
← hK

Goal state after:
G : Type u
inst✝² : Group G
N : Subgroup G
inst✝¹ : N.Normal
h1 : (Nat.card ↥N).Coprime N.index
h2 :
  ∀ (G' : Type u) [inst : Group G'] [Finite G'],
    Nat.card G' < Nat.card G →
      ∀ {N' : Subgroup G'} [N'.Normal], (Nat.card ↥N').Coprime N'.index → ∃ H', N'.IsComplement' H'
inst✝ : Finite G
K : Subgroup G
hK : K ⊔ N = ⊤
h3 : K ≠ ⊤
⊢ (comap K.subtype N).index = N.relIndex (K ⊔ N)
Module: Mathlib.GroupTheory.SchurZassenhaus

Goal state before:
pp: G : Type u
inst✝² : Group G
N : Subgroup G
inst✝¹ : N.Normal
h1 : (Nat.card ↥N).Coprime N.index
h2 :
  ∀ (G' : Type u) [inst : Group G'] [Finite G'],
    Nat.card G' < Nat.card G →
      ∀ {N' : Subgroup G'} [N'.Normal], (Nat.card ↥N').Coprime N'.index → ∃ H', N'.IsComplement' H'
inst✝ : Finite G
K : Subgroup G
hK : K ⊔ N = ⊤
h3 : K ≠ ⊤
h4 : (comap K.subtype N).index = N.index
⊢ ∃ H, N.IsComplement' H

Tactic:
have h5 : Nat.card K < Nat.card G := by
    rw [← K.index_mul_card]
    exact lt_mul_of_one_lt_left Nat.card_pos (one_lt_index_of_ne_top h3)

Goal state after:
G : Type u
inst✝² : Group G
N : Subgroup G
inst✝¹ : N.Normal
h1 : (Nat.card ↥N).Coprime N.index
h2 :
  ∀ (G' : Type u) [inst : Group G'] [Finite G'],
    Nat.card G' < Nat.card G →
      ∀ {N' : Subgroup G'} [N'.Normal], (Nat.card ↥N').Coprime N'.index → ∃ H', N'.IsComplement' H'
inst✝ : Finite G
K : Subgroup G
hK : K ⊔ N = ⊤
h3 : K ≠ ⊤
h4 : (comap K.subtype N).index = N.index
h5 : Nat.card ↥K < Nat.card G
⊢ ∃ H, N.IsComplement' H
Module: Mathlib.GroupTheory.Perm.Centralizer

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
⊢ Nat.card ↑{h | IsConj g h} =
    Nat.card ↑{h | IsConj g h} *
        ((Fintype.card α - g.cycleType.sum)! * g.cycleType.prod *
          ∏ n ∈ g.cycleType.toFinset, (Multiset.count n g.cycleType)!) /
      ((Fintype.card α - g.cycleType.sum)! * g.cycleType.prod *
        ∏ n ∈ g.cycleType.toFinset, (Multiset.count n g.cycleType)!)

Tactic:
Nat.div_eq_of_eq_mul_left _

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
⊢ Nat.card ↑{h | IsConj g h} = ?m.55

α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
⊢ Nat.card ↑{h | IsConj g h} *
      ((Fintype.card α - g.cycleType.sum)! * g.cycleType.prod *
        ∏ n ∈ g.cycleType.toFinset, (Multiset.count n g.cycleType)!) =
    ?m.55 *
      ((Fintype.card α - g.cycleType.sum)! * g.cycleType.prod *
        ∏ n ∈ g.cycleType.toFinset, (Multiset.count n g.cycleType)!)

α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
⊢ ℕ

α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
⊢ 0 <
    (Fintype.card α - g.cycleType.sum)! * g.cycleType.prod * ∏ n ∈ g.cycleType.toFinset, (Multiset.count n g.cycleType)!
Module: Mathlib.GroupTheory.Perm.Centralizer

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
⊢ 0 <
    (Fintype.card α - g.cycleType.sum)! * g.cycleType.prod * ∏ n ∈ g.cycleType.toFinset, (Multiset.count n g.cycleType)!

Tactic:
rw [← nat_card_centralizer g]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
⊢ 0 < Nat.card ↥(centralizer {g})
Module: Mathlib.GroupTheory.PushoutI

Goal state before:
pp: case neg
ι : Type u_1
G : ι → Type u_2
H : Type u_3
inst✝³ : (i : ι) → Group (G i)
inst✝² : Group H
φ : (i : ι) → H →* G i
d : Transversal φ
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (G i)
w : Word G
i : ι
h : H
hw✝ : ∀ (i : ι) (g : G i), ⟨i, g⟩ ∈ w.toList → ↑(⋯.equiv g).2 = g
hφw : ∀ (j : ι) (g : G j), ⟨j, g⟩ ∈ (CoprodI.of ((φ i) h) • w).toList → ↑(⋯.equiv g).2 = g
hhead : ↑(⋯.equiv ((Word.equivPair i) w).head).2 = ((Word.equivPair i) w).head
hh1 : ¬h = 1
hep : ∀ (h : ¬w.toList = []), (w.toList.head h).fst ≠ i
hw : ¬w.toList = []
⊢ ⟨i, (φ i) h * 1⟩ ∈ w.toList.tail ∨ w.toList.head? = some ⟨i, 1⟩ ∨ ¬w.fstIdx = some i ∧ 1 = 1

Tactic:
· simp [head?_eq_some_head hw, Word.fstIdx, hep hw]

Goal state after:
Module: Mathlib.GroupTheory.PushoutI

Goal state before:
pp: ι : Type u_1
G : ι → Type u_2
H : Type u_3
inst✝³ : (i : ι) → Group (G i)
inst✝² : Group H
φ : (i : ι) → H →* G i
d : Transversal φ
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (G i)
i : ι
head₁ : G i
tail₁ : Word G
fstIdx_ne✝¹ : tail₁.fstIdx ≠ some i
normalized✝¹ :
  ∀ (i_1 : ι) (g : G i_1),
    ⟨i_1, g⟩ ∈ { head := head₁, tail := tail₁, fstIdx_ne := fstIdx_ne✝¹ }.tail.toList → g ∈ d.set i_1
head₂ : G i
h₁ : ↑(⋯.equiv head₁).2 = ↑(⋯.equiv head₂).2
fstIdx_ne✝ : tail₁.fstIdx ≠ some i
normalized✝ :
  ∀ (i_1 : ι) (g : G i_1),
    ⟨i_1, g⟩ ∈ { head := head₂, tail := tail₁, fstIdx_ne := fstIdx_ne✝ }.tail.toList → g ∈ d.set i_1
h₃ : (⋯.equiv head₁).1 = (⋯.equiv head₂).1
⊢ ↑(⋯.equiv head₁).1 * ↑(⋯.equiv head₁).2 = ↑(⋯.equiv head₂).1 * ↑(⋯.equiv head₂).2 ∧ tail₁ = tail₁

Tactic:
h₁,

Goal state after:
ι : Type u_1
G : ι → Type u_2
H : Type u_3
inst✝³ : (i : ι) → Group (G i)
inst✝² : Group H
φ : (i : ι) → H →* G i
d : Transversal φ
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (G i)
i : ι
head₁ : G i
tail₁ : Word G
fstIdx_ne✝¹ : tail₁.fstIdx ≠ some i
normalized✝¹ :
  ∀ (i_1 : ι) (g : G i_1),
    ⟨i_1, g⟩ ∈ { head := head₁, tail := tail₁, fstIdx_ne := fstIdx_ne✝¹ }.tail.toList → g ∈ d.set i_1
head₂ : G i
h₁ : ↑(⋯.equiv head₁).2 = ↑(⋯.equiv head₂).2
fstIdx_ne✝ : tail₁.fstIdx ≠ some i
normalized✝ :
  ∀ (i_1 : ι) (g : G i_1),
    ⟨i_1, g⟩ ∈ { head := head₂, tail := tail₁, fstIdx_ne := fstIdx_ne✝ }.tail.toList → g ∈ d.set i_1
h₃ : (⋯.equiv head₁).1 = (⋯.equiv head₂).1
⊢ ↑(⋯.equiv head₁).1 * ↑(⋯.equiv head₂).2 = ↑(⋯.equiv head₂).1 * ↑(⋯.equiv head₂).2 ∧ tail₁ = tail₁
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Nat.card α = 4
g : Perm α
hg : g ∈ alternatingGroup α
hg' : ⟨g, hg⟩ ∈ ↑(kleinFour α)
⊢ g ^ 2 = 1

Tactic:
coe_kleinFour_of_card_eq_four hα4

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Nat.card α = 4
g : Perm α
hg : g ∈ alternatingGroup α
hg' : ⟨g, hg⟩ ∈ {1} ∪ {g | (↑g).cycleType = {2, 2}}
⊢ g ^ 2 = 1
Module: Mathlib.GroupTheory.SpecificGroups.Quaternion

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
i : ZMod (2 * n)
⊢ orderOf (a 1) / (orderOf (a 1)).gcd i.val = 2 * n / (2 * n).gcd i.val

Tactic:
orderOf_a_one

Goal state after:
n : ℕ
inst✝ : NeZero n
i : ZMod (2 * n)
⊢ 2 * n / (2 * n).gcd i.val = 2 * n / (2 * n).gcd i.val
Module: Mathlib.GroupTheory.SpecificGroups.Quaternion

Goal state before:
pp: case inr.a.a
n : ℕ
hn : NeZero n
m : ZMod (2 * n)
⊢ orderOf (a m) ∣ lcm (2 * n) 4

Tactic:
orderOf_a

Goal state after:
case inr.a.a
n : ℕ
hn : NeZero n
m : ZMod (2 * n)
⊢ 2 * n / (2 * n).gcd m.val ∣ lcm (2 * n) 4
Module: Mathlib.GroupTheory.SpecificGroups.ZGroup

Goal state before:
pp: G : Type u_1
G' : Type u_2
inst✝² : Group G
inst✝¹ : Group G'
f : G →* G'
inst✝ : Finite G
hG : IsZGroup G
hf : Function.Surjective ⇑f
⊢ IsZGroup G'

Tactic:
rw [isZGroup_iff] at hG ⊢

Goal state after:
G : Type u_1
G' : Type u_2
inst✝² : Group G
inst✝¹ : Group G'
f : G →* G'
inst✝ : Finite G
hG : ∀ (p : ℕ), Nat.Prime p → ∀ (P : Sylow p G), IsCyclic ↥↑P
hf : Function.Surjective ⇑f
⊢ ∀ (p : ℕ), Nat.Prime p → ∀ (P : Sylow p G'), IsCyclic ↥↑P
Module: Mathlib.GroupTheory.SpecificGroups.ZGroup

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : Finite G
inst✝ : IsZGroup G
p : ℕ
hp : Nat.Prime p
this : Fact (Nat.Prime p)
P : Sylow p G := default
⊢ Monoid.exponent ↥↑P ∣ Monoid.exponent G

Tactic:
exact Monoid.exponent_dvd_of_monoidHom P.1.subtype P.1.subtype_injective

Goal state after:
Module: Mathlib.Probability.Kernel.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
s : Set β
κ : Kernel α β
hs : MeasurableSet s
a : α
f : β → ℝ≥0∞
⊢ ∫⁻ (b : β), f b ∂(κ.restrict hs) a = ∫⁻ (b : β) in s, f b ∂κ a

Tactic:
rw [restrict_apply]

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.Comp

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
⊢ swap α α ∘ₖ copy α = copy α

Tactic:
by
  ext a s hs
  rw [comp_apply, copy_apply, Measure.dirac_bind (Kernel.measurable _), swap_apply' _ hs,
    Measure.dirac_apply' _ hs]
  congr

Goal state after:
Module: Mathlib.Probability.Kernel.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ : Measure α
h : ∀ᵐ (a : α) ∂μ, IsProbabilityMeasure (κ a)
h' : μ ≠ 0
s : Set α
s_meas : MeasurableSet s
μs : μ s = 0
hs : ∀ a ∉ s, IsProbabilityMeasure (κ a)
⊢ sᶜ.Nonempty

Tactic:
contrapose! h'

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ : Measure α
h : ∀ᵐ (a : α) ∂μ, IsProbabilityMeasure (κ a)
s : Set α
s_meas : MeasurableSet s
μs : μ s = 0
hs : ∀ a ∉ s, IsProbabilityMeasure (κ a)
h' : sᶜ = ∅
⊢ μ = 0