mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup

Goal state before:
pp: case right
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s t : Set α
g : M
hst✝ : g • s = t
hst : g⁻¹ • t = s
x : α
hx : x ∈ ofFixingSubgroup M t
⊢ ∃ a, (conjMap_ofFixingSubgroup hst✝) a = ⟨x, hx⟩

Tactic:
use (SubMulAction.conjMap_ofFixingSubgroup hst) ⟨x, hx⟩

Goal state after:
case h
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s t : Set α
g : M
hst✝ : g • s = t
hst : g⁻¹ • t = s
x : α
hx : x ∈ ofFixingSubgroup M t
⊢ (conjMap_ofFixingSubgroup hst✝) ((conjMap_ofFixingSubgroup hst) ⟨x, hx⟩) = ⟨x, hx⟩
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 t : Set α
hst : s = t
x✝ : ↥(ofFixingSubgroup M t)
x : α
hxt : x ∈ ofFixingSubgroup M t
⊢ (ofFixingSubgroup_of_eq M hst) ⟨x, ⋯⟩ = ⟨x, hxt⟩

Tactic:
by simp [← SetLike.coe_eq_coe]

Goal state after:
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 := s ∩ g • s
ht : t.Finite
htm : t.ncard = t.ncard - 1 + 1
hmn : t.ncard - 1 < n
htm' : t.ncard - 1 + 2 < Nat.card α
this : IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
hs_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s)
⊢ IsMultiplyPreprimitive G α 2

Tactic:
have ht_prim : IsPreprimitive (fixingSubgroup G t) (ofFixingSubgroup G t) := by
        apply IsPreprimitive.isPreprimitive_ofFixingSubgroup_inter hs_prim
        apply Set.union_ne_univ_of_ncard_add_ncard_lt
        rwa [Set.ncard_smul_set, hsn, ← two_mul]

Goal state after:
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 := s ∩ g • s
ht : t.Finite
htm : t.ncard = t.ncard - 1 + 1
hmn : t.ncard - 1 < n
htm' : t.ncard - 1 + 2 < Nat.card α
this : IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
hs_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s)
ht_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(ofFixingSubgroup G t)
⊢ IsMultiplyPreprimitive G α 2
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
hφ : Function.Surjective φ
f : α →ₑ[φ] β
hf : Function.Bijective ⇑f
n : ℕ
H : IsMultiplyPreprimitive N β n
s : Set α
hs : s.encard + 1 = ↑n
t : Set β := ⇑f '' s
x✝ : ↥(fixingSubgroup M s)
g : M
hg : ∀ y ∈ s, g • y = y
y : β
hy : y ∈ t
⊢ ∃ x ∈ s, y = f x

Tactic:
obtain ⟨x, rfl⟩ := hf.surjective y

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
hφ : Function.Surjective φ
f : α →ₑ[φ] β
hf : Function.Bijective ⇑f
n : ℕ
H : IsMultiplyPreprimitive N β n
s : Set α
hs : s.encard + 1 = ↑n
t : Set β := ⇑f '' s
x✝ : ↥(fixingSubgroup M s)
g : M
hg : ∀ y ∈ s, g • y = y
x : α
hy : f x ∈ t
⊢ ∃ x_1 ∈ s, f x = f x_1
Module: Mathlib.GroupTheory.GroupAction.MultiplePrimitivity

Goal state before:
pp: case right
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
hφ : Function.Surjective φ
f : α →ₑ[φ] β
hf : Function.Bijective ⇑f
n : ℕ
H : IsMultiplyPreprimitive N β n
s : Set α
hs : s.encard + 1 = ↑n
t : Set β := ⇑f '' s
ψ : ↥(fixingSubgroup M s) → ↥(fixingSubgroup N t) :=
  fun x ↦
    match x with
    | ⟨g, hg⟩ => ⟨φ g, ⋯⟩
g : ↥(ofFixingSubgroup M s) →ₑ[ψ] ↥(ofFixingSubgroup N t) := { toFun := fun x ↦ ⟨f ↑x, ⋯⟩, map_smul' := ⋯ }
⊢ IsPreprimitive ↥(fixingSubgroup M s) ↥(ofFixingSubgroup M s)

Tactic:
rw [isPreprimitive_congr (f := g)]

Goal state after:
case right
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
hφ : Function.Surjective φ
f : α →ₑ[φ] β
hf : Function.Bijective ⇑f
n : ℕ
H : IsMultiplyPreprimitive N β n
s : Set α
hs : s.encard + 1 = ↑n
t : Set β := ⇑f '' s
ψ : ↥(fixingSubgroup M s) → ↥(fixingSubgroup N t) :=
  fun x ↦
    match x with
    | ⟨g, hg⟩ => ⟨φ g, ⋯⟩
g : ↥(ofFixingSubgroup M s) →ₑ[ψ] ↥(ofFixingSubgroup N t) := { toFun := fun x ↦ ⟨f ↑x, ⋯⟩, map_smul' := ⋯ }
⊢ IsPreprimitive ↥(fixingSubgroup N t) ↥(ofFixingSubgroup N t)

case right.hφ
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
hφ : Function.Surjective φ
f : α →ₑ[φ] β
hf : Function.Bijective ⇑f
n : ℕ
H : IsMultiplyPreprimitive N β n
s : Set α
hs : s.encard + 1 = ↑n
t : Set β := ⇑f '' s
ψ : ↥(fixingSubgroup M s) → ↥(fixingSubgroup N t) :=
  fun x ↦
    match x with
    | ⟨g, hg⟩ => ⟨φ g, ⋯⟩
g : ↥(ofFixingSubgroup M s) →ₑ[ψ] ↥(ofFixingSubgroup N t) := { toFun := fun x ↦ ⟨f ↑x, ⋯⟩, map_smul' := ⋯ }
⊢ Function.Surjective ψ

case right.hf
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
hφ : Function.Surjective φ
f : α →ₑ[φ] β
hf : Function.Bijective ⇑f
n : ℕ
H : IsMultiplyPreprimitive N β n
s : Set α
hs : s.encard + 1 = ↑n
t : Set β := ⇑f '' s
ψ : ↥(fixingSubgroup M s) → ↥(fixingSubgroup N t) :=
  fun x ↦
    match x with
    | ⟨g, hg⟩ => ⟨φ g, ⋯⟩
g : ↥(ofFixingSubgroup M s) →ₑ[ψ] ↥(ofFixingSubgroup N t) := { toFun := fun x ↦ ⟨f ↑x, ⋯⟩, map_smul' := ⋯ }
⊢ Function.Bijective ⇑g
Module: Mathlib.GroupTheory.GroupAction.SubMulAction.Combination

Goal state before:
pp: case refine_2
α : Type u_2
inst✝² : DecidableEq α
G : Type u_3
inst✝¹ : AddGroup G
inst✝ : AddAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑n < ENat.card α
g : G
h : AddAction.toPerm g = 1
⊢ AddAction.toPerm g = 1

Tactic:
· simp only [Equiv.ext_iff, AddAction.toPerm_apply] at h ⊢
    simp [Subtype.ext_iff, Finset.ext_iff, mem_vadd_finset, h]

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 * σ g₂ * (σ (g₁ * g₂))⁻¹
⊢ σ (g₁ * g₂) = (S.inl n)⁻¹ * (σ g₁ * σ g₂)

Tactic:
eq_inv_mul_iff_mul_eq,

Goal state after:
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 * σ g₂ * (σ (g₁ * g₂))⁻¹
⊢ S.inl n * σ (g₁ * g₂) = σ g₁ * σ g₂
Module: Mathlib.GroupTheory.GroupAction.Jordan

Goal state before:
pp: case inr.hsn
α : Type u_1
G : Subgroup (Perm α)
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Perm α
h3g : g.IsThreeCycle
hg : g ∈ G
hα4 : Nat.card α ≥ 4
n : ℕ
hn : Nat.card α = n + 4
⊢ (↑g.support)ᶜ.ncard = n + 1

pp: case inr.hsn'
α : Type u_1
G : Subgroup (Perm α)
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Perm α
h3g : g.IsThreeCycle
hg : g ∈ G
hα4 : Nat.card α ≥ 4
n : ℕ
hn : Nat.card α = n + 4
⊢ n + 2 < Nat.card α

pp: case inr.hprim
α : Type u_1
G : Subgroup (Perm α)
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Perm α
h3g : g.IsThreeCycle
hg : g ∈ G
hα4 : Nat.card α ≥ 4
n : ℕ
hn : Nat.card α = n + 4
⊢ IsPreprimitive ↥(fixingSubgroup (↥G) (↑g.support)ᶜ) ↥(ofFixingSubgroup (↥G) (↑g.support)ᶜ)

Tactic:
· apply Nat.add_left_cancel
    rw [Set.ncard_add_ncard_compl, Set.ncard_coe_finset,
      h3g.card_support, add_comm, hn]

Goal state after:
case inr.hsn'
α : Type u_1
G : Subgroup (Perm α)
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Perm α
h3g : g.IsThreeCycle
hg : g ∈ G
hα4 : Nat.card α ≥ 4
n : ℕ
hn : Nat.card α = n + 4
⊢ n + 2 < Nat.card α

case inr.hprim
α : Type u_1
G : Subgroup (Perm α)
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Perm α
h3g : g.IsThreeCycle
hg : g ∈ G
hα4 : Nat.card α ≥ 4
n : ℕ
hn : Nat.card α = n + 4
⊢ IsPreprimitive ↥(fixingSubgroup (↥G) (↑g.support)ᶜ) ↥(ofFixingSubgroup (↥G) (↑g.support)ᶜ)
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✝¹ : α
b✝¹ : ↥s
a✝ : α
b✝ : ↥s
hab : ↑b✝ * a✝¹ ≤ ↑b✝¹ * a✝
hba : ↑b✝¹ * a✝ ≤ ↑b✝ * a✝¹
⊢ ↑1 * (↑b✝ * a✝¹) = ↑1 * (↑b✝¹ * a✝)

Tactic:
by rw [hab.antisymm hba]

Goal state after:
Module: Mathlib.GroupTheory.HNNExtension

Goal state before:
pp: G : Type u_1
inst✝ : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
d : TransversalPair G A B
g : ↥A
w : NormalWord d
⊢ unitsSMul φ 1 (↑g • w) = ↑(φ g) • unitsSMul φ 1 w

Tactic:
unfold unitsSMul
  have : Cancels 1 ((g : G) • w) ↔ Cancels 1 w := by
    simp [Cancels, Subgroup.mul_mem_cancel_left]
  by_cases hcan : Cancels 1 w
  · simp only [unitsSMulWithCancel, toSubgroup_one, id_eq, toSubgroup_neg_one, toSubgroupEquiv_one,
      group_smul_head, mul_inv_rev, dif_pos (this.2 hcan), dif_pos hcan]
    cases w using consRecOn
    · simp [Cancels] at hcan
    · simp only [smul_cons, consRecOn_cons]
      rw [← mul_smul, ← Subgroup.coe_mul, ← map_mul φ]
      rfl
  · rw [dif_neg (mt this.1 hcan), dif_neg hcan]
    -- Before https://github.com/leanprover/lean4/pull/2644, all this was just
    -- `simp [← mul_smul, mul_assoc, unitsSMulGroup]`
    simp +instances only [toSubgroup_neg_one, unitsSMulGroup, toSubgroup_one, toSubgroupEquiv_one,
      SetLike.coe_sort_coe, group_smul_head, mul_inv_rev, ← mul_smul, mul_assoc, inv_mul_cancel,
      mul_one, smul_cons]
    -- This used to be the end of the proof before https://github.com/leanprover/lean4/pull/2644
    congr 1
    · conv_lhs => erw [IsComplement.equiv_mul_left]
      simp_rw [toSubgroup_one]
      simp only [SetLike.coe_sort_coe, map_mul, Subgroup.coe_mul]
    conv_lhs => erw [IsComplement.equiv_mul_left]
    rfl

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
c : ↥g.cycleFactorsFinset
⊢ 1 • c = c

Tactic:
rw [← Subtype.coe_inj]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
c : ↥g.cycleFactorsFinset
⊢ ↑(1 • c) = ↑c
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 d : ↥g.cycleFactorsFinset
h : a c = a d
⊢ c = d

Tactic:
rw [← Subtype.coe_inj]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
a : g.Basis
c d : ↥g.cycleFactorsFinset
h : a c = a d
⊢ ↑c = ↑d
Module: Mathlib.GroupTheory.Perm.Centralizer

Goal state before:
pp: case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
a : g.Basis
τ : ↥(range_toPermHom' g)
x : α
c : ↥g.cycleFactorsFinset
hx : x ∈ Function.fixedPoints ⇑g
⊢ ∀ (d : ↥g.cycleFactorsFinset), x ∉ (↑d).support

Tactic:
intro d

Goal state after:
case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
a : g.Basis
τ : ↥(range_toPermHom' g)
x : α
c : ↥g.cycleFactorsFinset
hx : x ∈ Function.fixedPoints ⇑g
d : ↥g.cycleFactorsFinset
⊢ x ∉ (↑d).support
Module: Mathlib.GroupTheory.Perm.Centralizer

Goal state before:
pp: case inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
a : g.Basis
x : α
c : ↥g.cycleFactorsFinset
hc : x ∈ (↑c).support
m : ℤ
hm : (g ^ m) (a c) = x
⊢ a.ofPermHomFun 1 x = x

Tactic:
ofPermHomFun_apply_of_cycleOf_mem a _ hc hm,

Goal state after:
case inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
a : g.Basis
x : α
c : ↥g.cycleFactorsFinset
hc : x ∈ (↑c).support
m : ℤ
hm : (g ^ m) (a c) = x
⊢ (g ^ m) (a (↑1 c)) = x
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
a : g.Basis
τ : ↥(range_toPermHom' g)
x : α
c : ↥g.cycleFactorsFinset
hc : x ∈ (↑c).support
m : ℤ
hm : (g ^ m) (a c) = x
H : ↑τ c = c
⊢ ∀ (a : ↥g.cycleFactorsFinset), ↑τ a ≠ a → ↑a x = x

Tactic:
intro d

Goal state after:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
a : g.Basis
τ : ↥(range_toPermHom' g)
x : α
c : ↥g.cycleFactorsFinset
hc : x ∈ (↑c).support
m : ℤ
hm : (g ^ m) (a c) = x
H : ↑τ c = c
d : ↥g.cycleFactorsFinset
⊢ ↑τ d ≠ d → ↑d 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
τ : ↥(range_toPermHom' g)
x : α
⊢ (a.ofPermHom τ) (g x) = g ((a.ofPermHom τ) x)

Tactic:
exact ofPermHomFun_commute_zpow_apply a τ x 1

Goal state after:
Module: Mathlib.GroupTheory.PushoutI

Goal state before:
pp: ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Monoid (G i)
inst✝ : Monoid H
φ : (i : ι) → H →* G i
f : (i : ι) → G i →* K
k : H →* K
hf : ∀ (i : ι), (f i).comp (φ i) = k
g : H
⊢ (lift f k hf) ((base φ) g) = k g

Tactic:
delta PushoutI lift base
  simp only [MonoidHom.coe_comp, Con.coe_mk', comp_apply, Con.lift_coe, lift_apply_inr]

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

Goal state before:
pp: case e'_2.e'_6
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h : Subgroup.centralizer {g} ≤ alternatingGroup α
c : Perm α
hc : c ∈ g.cycleFactorsFinset
d : Perm α
hd : d ∈ g.cycleFactorsFinset
hm : #c.support = #d.support
hm' : c ≠ d
τ : Perm ↥g.cycleFactorsFinset := swap ⟨c, hc⟩ ⟨d, hd⟩
a : g.Basis
hτ : τ ∈ range_toPermHom' g
k : ↥(Subgroup.centralizer {g}) := a.toCentralizer ⟨τ, hτ⟩
hk : k = a.toCentralizer ⟨τ, hτ⟩
⊢ ⟨τ, hτ⟩ ^ 2 = 1

Tactic:
rw [← Subtype.coe_inj]

Goal state after:
case e'_2.e'_6
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h : Subgroup.centralizer {g} ≤ alternatingGroup α
c : Perm α
hc : c ∈ g.cycleFactorsFinset
d : Perm α
hd : d ∈ g.cycleFactorsFinset
hm : #c.support = #d.support
hm' : c ≠ d
τ : Perm ↥g.cycleFactorsFinset := swap ⟨c, hc⟩ ⟨d, hd⟩
a : g.Basis
hτ : τ ∈ range_toPermHom' g
k : ↥(Subgroup.centralizer {g}) := a.toCentralizer ⟨τ, hτ⟩
hk : k = a.toCentralizer ⟨τ, hτ⟩
⊢ ↑(⟨τ, hτ⟩ ^ 2) = ↑1
Module: Mathlib.GroupTheory.Perm.Centralizer

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hm : g.cycleType.sum ≤ Fintype.card α ∧ ∀ a ∈ g.cycleType, 2 ≤ a
⊢ Fintype.card { x // True ∧ x.cycleType = g.cycleType } = Fintype.card { h // IsConj g h }

Tactic:
true_and,

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hm : g.cycleType.sum ≤ Fintype.card α ∧ ∀ a ∈ g.cycleType, 2 ≤ a
⊢ Fintype.card { x // x.cycleType = g.cycleType } = Fintype.card { h // IsConj g h }
Module: Mathlib.GroupTheory.SchurZassenhaus

Goal state before:
pp: case pos
G : Type u
inst✝¹ : Group G
N : Subgroup G
inst✝ : N.Normal
hN : (Nat.card ↥N).Coprime N.index
hN1 : Nat.card ↥N = 0
⊢ ∃ H, N.IsComplement' H

pp: case neg
G : Type u
inst✝¹ : Group G
N : Subgroup G
inst✝ : N.Normal
hN : (Nat.card ↥N).Coprime N.index
hN1 : ¬Nat.card ↥N = 0
⊢ ∃ H, N.IsComplement' H

Tactic:
· rw [hN1, Nat.coprime_zero_left, index_eq_one] at hN
    rw [hN]
    exact ⟨⊥, isComplement'_top_bot⟩

Goal state after:
case neg
G : Type u
inst✝¹ : Group G
N : Subgroup G
inst✝ : N.Normal
hN : (Nat.card ↥N).Coprime N.index
hN1 : ¬Nat.card ↥N = 0
⊢ ∃ H, N.IsComplement' H
Module: Mathlib.GroupTheory.PushoutI

Goal state before:
pp: case cons
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝⁴ : Monoid K
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)
motive : NormalWord d → Sort ?u.126
empty : motive NormalWord.empty
cons :
  (i : ι) →
    (g : G i) →
      (w : NormalWord d) →
        (hmw : w.fstIdx ≠ some i) →
          g ∈ d.set i → (hgr : g ∉ (φ i).range) → w.head = 1 → motive w → motive (NormalWord.cons g w hmw hgr)
base : (h : H) → (w : NormalWord d) → w.head = 1 → motive w → motive ((PushoutI.base φ) h • w)
head : H
i : ι
g : G i
w : Word G
h1 : w.fstIdx ≠ some i
hg1 : g ≠ 1
ih : (h3 : ∀ (i : ι) (g : G i), ⟨i, g⟩ ∈ w.toList → g ∈ d.set i) → motive { toWord := w, head := 1, normalized := h3 }
h3 : ∀ (i_1 : ι) (g_1 : G i_1), ⟨i_1, g_1⟩ ∈ (Word.cons g w h1 hg1).toList → g_1 ∈ d.set i_1
⊢ g ∉ (φ i).range

Tactic:
· rwa [← SetLike.mem_coe,
          ← coe_equiv_snd_eq_one_iff_mem (d.compl i) (d.one_mem _),
          (equiv_snd_eq_self_iff_mem (d.compl i) (one_mem _)).2
          (h3 _ _ List.mem_cons_self)]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
κ : Kernel α β
f : β → γ
hf : ¬Measurable f
⊢ κ.map f = 0

Tactic:
by
  simp [map, hf]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
κ : Kernel α β
f : β → γ
hf : Measurable f
⊢ κ.mapOfMeasurable f hf = κ.map f

Tactic:
by
  simp [map, hf]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
f : β → γ
κ : Kernel α β
inst✝ : IsMarkovKernel κ
hf : Measurable f
a : α
⊢ (κ a) Set.univ = 1

Tactic:
measure_univ

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
f : β → γ
κ : Kernel α β
inst✝ : IsMarkovKernel κ
hf : Measurable f
a : α
⊢ 1 = 1
Module: Mathlib.Probability.Kernel.Composition.MeasureCompProd

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
inst✝¹ : SFinite μ
inst✝ : IsSFiniteKernel κ
h : μ ⊗ₘ κ = 0
⊢ ∀ᵐ (a : α) ∂μ, κ a = 0

Tactic:
simp_rw [← measure_univ_eq_zero]
    refine (lintegral_eq_zero_iff (Kernel.measurable_coe _ .univ)).mp ?_
    rw [← setLIntegral_univ, ← compProd_apply_prod .univ .univ, h]
    simp

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
s : Set (α × α)
hs : MeasurableSet s
⊢ (μ ⊗ₘ Kernel.id) s = (map Function.diag μ) s

Tactic:
rw [compProd_apply hs, map_apply (measurable_id.prod measurable_id) hs]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
s : Set (α × α)
hs : MeasurableSet s
⊢ ∫⁻ (a : α), (Kernel.id a) (Prod.mk a ⁻¹' s) ∂μ = μ (Function.diag ⁻¹' s)
Module: Mathlib.Probability.Kernel.Composition.MapComap

Goal state before:
pp: α : Type u_1
β : Type u_2
γ✝ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ✝ : MeasurableSpace γ✝
γ : Type u_4
δ : Type u_5
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
f : β → γ
g : γ → α
κ : Kernel α β
inst✝ : IsMarkovKernel κ
hg : Measurable g
a : γ
⊢ (κ (g a)) Set.univ = 1

Tactic:
measure_univ

Goal state after:
α : Type u_1
β : Type u_2
γ✝ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ✝ : MeasurableSpace γ✝
γ : Type u_4
δ : Type u_5
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
f : β → γ
g : γ → α
κ : Kernel α β
inst✝ : IsMarkovKernel κ
hg : Measurable g
a : γ
⊢ 1 = 1
Module: Mathlib.Probability.Kernel.Composition.ParallelComp

Goal state before:
pp: case h₂
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
κ : Kernel α β
η : Kernel γ δ
x : α × γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α × γ
⊢ (η a.2) Set.univ ≤ η.bound

Tactic:
exact measure_le_bound η a.2 Set.univ

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

Goal state before:

Tactic:
(κ ∥ₖ η) a Set.univ
  _ = κ a.1 Set.univ * η a.2 Set.univ := parallelComp_apply_univ
  _ ≤ κ.bound * η.bound := by
    gcongr
    · exact measure_le_bound κ a.1 Set.univ
    · exact measure_le_bound η a.2 Set.univ

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
⊢ deterministic Prod.swap ⋯ ∘ₖ deterministic Prod.swap ⋯ = Kernel.id

Tactic:
Kernel.deterministic_comp_deterministic,

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
⊢ deterministic (Prod.swap ∘ Prod.swap) ⋯ = Kernel.id
Module: Mathlib.Probability.Kernel.Composition.MapComap

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α (β × γ)
a : α
g : γ → ℝ≥0∞
hg : Measurable g
⊢ ∫⁻ (c : γ), g c ∂κ.snd a = ∫⁻ (bc : β × γ), g bc.2 ∂κ a

Tactic:
rw [snd_eq, lintegral_map _ measurable_snd a hg]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ✝ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ✝ : MeasurableSpace γ✝
γ : Type u_4
δ : Type u_5
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
κ : Kernel (α × β) γ
inst✝ : ∀ (b : β), IsMarkovKernel (κ.sectL b)
x✝ : α × β
⊢ ((κ.sectL x✝.2) x✝.1) Set.univ = 1

Tactic:
measure_univ

Goal state after:
α : Type u_1
β : Type u_2
γ✝ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ✝ : MeasurableSpace γ✝
γ : Type u_4
δ : Type u_5
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
κ : Kernel (α × β) γ
inst✝ : ∀ (b : β), IsMarkovKernel (κ.sectL b)
x✝ : α × β
⊢ 1 = 1
Module: Mathlib.Probability.Kernel.Composition.MapComap

Goal state before:
pp: α : Type u_1
β : Type u_2
γ✝ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ✝ : MeasurableSpace γ✝
γ : Type u_4
δ : Type u_5
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
κ : Kernel (α × β) γ
inst✝ : ∀ (b : α), IsMarkovKernel (κ.sectR b)
x✝ : α × β
⊢ ((κ.sectR x✝.1) x✝.2) Set.univ = 1

Tactic:
measure_univ

Goal state after:
α : Type u_1
β : Type u_2
γ✝ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ✝ : MeasurableSpace γ✝
γ : Type u_4
δ : Type u_5
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
κ : Kernel (α × β) γ
inst✝ : ∀ (b : α), IsMarkovKernel (κ.sectR b)
x✝ : α × β
⊢ 1 = 1