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

Goal state before:
pp: G : Type u_1
inst✝ : Group G
K : Subgroup G
H : Subgroup ↥K
Hsn : H.IsSubnormal
Ksn : K.IsSubnormal
⊢ (map K.subtype H).IsSubnormal

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.IsSubnormal

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
HK : H ≤ K
Hsn : (H.subgroupOf K).IsSubnormal
Ksn : K.IsSubnormal
⊢ H.IsSubnormal

Tactic:
by

Goal state after:
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:
apply step _ (map f K) (map_mono h_le) ih
    rw [normal_subgroupOf_iff_le_normalizer h_le] at hN
    exact normal_subgroupOf_of_le_normalizer ((map_mono hN).trans (H.le_normalizer_map f))

Goal state after:
Module: Mathlib.GroupTheory.IsSubnormal

Goal state before:
pp: 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
hS : H.IsSubnormal
⊢ (map f H).IsSubnormal

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.IsSubnormal

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
G' : Type u_2
inst✝ : Group G'
H' : Subgroup G'
f : G →* G'
h : H'.IsSubnormal
⊢ (comap f H').IsSubnormal

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.IsSubnormal

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
hH : H.IsSubnormal
hK : K.IsSubnormal
⊢ (H ⊓ K).IsSubnormal

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.HNNExtension

Goal state before:
pp: G : Type u_1
inst✝² : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
H : Type u_2
inst✝¹ : Group H
M : Type u_3
inst✝ : Monoid M
d : TransversalPair G A B
u : ℤˣ
w : NormalWord d
this : (p : Prop) → Decidable p := Classical.dec
g' : ↥(toSubgroup A B (-u)) × ↑(d.set u) := unitsSMulGroup φ d u w.head
h : w.head ∈ toSubgroup A B u → ∀ (x : G), ¬w.toList.head? = some (-u, x)
u' : ℤˣ
x : G
hx : w.toList.head? = some (u', x)
hmem : ↑(⋯.equiv w.head).2 ∈ toSubgroup A B u
⊢ w.head ∈ toSubgroup A B u

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.IsSubnormal

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Subsingleton ↥H
⊢ H.IsSubnormal

Tactic:
by

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)
⊢ ↑c₂ * a₁ < ↑a₂ * c₁ ↔ ↑d₂ * b₁ < ↑b₂ * d₁

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommCoprod

Goal state before:
pp: M : Type u_1
N : Type u_2
P : Type u_3
inst✝² : Mul M
inst✝¹ : Mul N
inst✝ : Semigroup P
f : M →ₙ* P
g : N →ₙ* P
comm : ∀ (m : M) (n : N), Commute (f m) (g n)
mn mn' : M × N
⊢ f (mn * mn').1 * g (mn * mn').2 = f mn.1 * g mn.2 * (f mn'.1 * g mn'.2)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommCoprod

Goal state before:
pp: M : Type u_1
N : Type u_2
P : Type u_3
inst✝² : Mul M
inst✝¹ : Mul N
inst✝ : Semigroup P
f : M →ₙ* P
g : N →ₙ* P
comm : ∀ (m : M) (n : N), Commute (f m) (g n)
mn : M × N
⊢ (f.noncommCoprod g comm) mn = g mn.2 * f mn.1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommCoprod

Goal state before:
pp: M : Type u_1
N : Type u_2
P : Type u_3
inst✝² : MulOneClass M
inst✝¹ : MulOneClass N
inst✝ : Monoid P
f : M →* P
g : N →* P
comm : ∀ (m : M) (n : N), Commute (f m) (g n)
⊢ f 1.1 * g 1.2 = 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommCoprod

Goal state before:
pp: M : Type u_1
N : Type u_2
P : Type u_3
inst✝² : MulOneClass M
inst✝¹ : MulOneClass N
inst✝ : Monoid P
f : M →* P
g : N →* P
comm : ∀ (m : M) (n : N), Commute (f m) (g n)
mn : M × N
⊢ (f.noncommCoprod g comm) mn = g mn.2 * f mn.1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommCoprod

Goal state before:
pp: M : Type u_1
N : Type u_2
P : Type u_3
inst✝² : MulOneClass M
inst✝¹ : MulOneClass N
inst✝ : Monoid P
f : M →* P
g : N →* P
comm : ∀ (m : M) (n : N), Commute (f m) (g n)
x : M
⊢ ((f.noncommCoprod g comm).comp (inl M N)) x = f x

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommCoprod

Goal state before:
pp: M : Type u_1
N : Type u_2
P : Type u_3
inst✝² : MulOneClass M
inst✝¹ : MulOneClass N
inst✝ : Monoid P
f : M →* P
g : N →* P
comm : ∀ (m : M) (n : N), Commute (f m) (g n)
x : N
⊢ ((f.noncommCoprod g comm).comp (inr M N)) x = g x

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommCoprod

Goal state before:
pp: M : Type u_1
N : Type u_2
P : Type u_3
inst✝² : MulOneClass M
inst✝¹ : MulOneClass N
inst✝ : Monoid P
f : M × N →* P
x : M × N
⊢ ((f.comp (inl M N)).noncommCoprod (f.comp (inr M N)) ⋯) x = f x

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommCoprod

Goal state before:
pp: M : Type u_1
N : Type u_2
P : Type u_3
inst✝³ : MulOneClass M
inst✝² : MulOneClass N
inst✝¹ : Monoid P
f : M →* P
g : N →* P
comm : ∀ (m : M) (n : N), Commute (f m) (g n)
Q : Type u_4
inst✝ : Monoid Q
h : P →* Q
x : M × N
⊢ (h.comp (f.noncommCoprod g comm)) x = ((h.comp f).noncommCoprod (h.comp g) ⋯) x

Tactic:
by

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 : mk a.1 a.2 ≤ mk b.1 b.2
hbc : mk b.1 b.2 ≤ mk c.1 c.2
⊢ mk a.1 a.2 ≤ mk c.1 c.2

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.HNNExtension

Goal state before:
pp: G : Type u_1
inst✝² : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
H : Type u_2
inst✝¹ : Group H
M : Type u_3
inst✝ : Monoid M
d : TransversalPair G A B
u : ℤˣ
w : NormalWord d
this : (p : Prop) → Decidable p := Classical.dec
h : ¬Cancels u w
g' : ↥(toSubgroup A B (-u)) × ↑(d.set u) := unitsSMulGroup φ d u w.head
⊢ ∀ u' ∈ Option.map Prod.fst ((↑g'.2 * w.head⁻¹) • w).toList.head?,
    ((↑g'.2 * w.head⁻¹) • w).head ∈ toSubgroup A B u → u = u'

Tactic:
by

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✝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

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
⊢ a ≤ b → b ≤ a → a = b

Tactic:
by

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
a b : α × ↥s
hab : mk a.1 a.2 ≤ mk b.1 b.2
c✝ : Localization s
c : α × ↥s
⊢ mk a.1 a.2 * mk c.1 c.2 ≤ mk b.1 b.2 * mk c.1 c.2

Tactic:
by

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 : mk a.1 a.2 * mk b.1 b.2 ≤ mk a.1 a.2 * mk c.1 c.2
⊢ mk b.1 b.2 ≤ mk c.1 c.2

Tactic:
by

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 b✝ : ↥s
a b : α
⊢ { toFun := fun a ↦ mk a b✝, inj' := ⋯ } a ≤ { toFun := fun a ↦ mk a b✝, inj' := ⋯ } b ↔ a ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.HNNExtension

Goal state before:
pp: G : Type u_1
inst✝ : Group G
A B : Subgroup G
d : TransversalPair G A B
u : ℤˣ
w : NormalWord d
h2 : ∀ u' ∈ Option.map Prod.fst w.toList.head?, w.head ∈ toSubgroup A B u → u = u'
⊢ ¬Cancels u w

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝² : CommMonoid α
inst✝¹ : LinearOrder α
inst✝ : IsOrderedCancelMonoid α
s : Submonoid α
a b : Localization s
x✝¹ x✝ : α × ↥s
⊢ mk x✝¹.1 x✝¹.2 ≤ mk x✝.1 x✝.2 ∨ mk x✝.1 x✝.2 ≤ mk x✝¹.1 x✝¹.2

Tactic:
by

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

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝¹ : CommMonoidWithZero M
inst✝ : CommMonoidWithZero N
S : Submonoid M
f : S.LocalizationMap N
m : M
prime : Prime m
n0 : f m ≠ 0
nu : ¬IsUnit (f m)
⊢ Prime (f m)

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝ : Fintype α
c : List ℕ
hc : c.sum ≤ Fintype.card α
klift : (n : ℕ) → n < Fintype.card α → Fin (Fintype.card α) := fun n hn ↦ ⟨n, hn⟩
klift' : (l : List ℕ) → (∀ a ∈ l, a < Fintype.card α) → List (Fin (Fintype.card α)) := fun l hl ↦ pmap klift l hl
⊢ ∀ l ∈ c.ranges, ∀ n ∈ l, n < Fintype.card α

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommCoprod

Goal state before:
pp: M : Type u_4
N : Type u_5
P : Type u_6
inst✝² : Group M
inst✝¹ : Group N
inst✝ : Group P
f : M →* P
g : N →* P
comm : ∀ (m : M) (n : N), Commute (f m) (g n)
h : ∀ (a : M) (b : N), f a * g b = 1 → a = 1 ∧ b = 1
x : P
y : M
hy : f y = x
z : N
hz : g z = x
⊢ f y * g z⁻¹ = 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommCoprod

Goal state before:
pp: M : Type u_4
N : Type u_5
P : Type u_6
inst✝² : Group M
inst✝¹ : Group N
inst✝ : Group P
f : M →* P
g : N →* P
comm : ∀ (m : M) (n : N), Commute (f m) (g n)
h : ∀ (a : M) (b : N), f a * g b = 1 → a = 1 ∧ b = 1
x : P
y : M
hy : f y = x
z : N
hz : g z = x
⊢ f y * g z⁻¹ = 1

Tactic:
by rw [map_inv, hy, hz, mul_inv_cancel]

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
u : ℤˣ
w : NormalWord d
⊢ Cancels (-u) (unitsSMul φ u w) ↔ ¬Cancels u w

Tactic:
by

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

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝² : CommMonoidWithZero M
inst✝¹ : CommMonoidWithZero N
S : Submonoid M
inst✝ : WfDvdMonoid M
f : S.LocalizationMap N
i : M
hi : Irreducible i
u m' : M
hu : IsUnit (f u)
hm' : Irreducible m'
ha0 : u * m' ≠ 0
ha : Irreducible (f (u * m')) → ∃ u_1 m'_1, IsUnit (f u_1) ∧ Irreducible m'_1 ∧ u * m' = u_1 * m'_1
hai : Irreducible (f (u * m')) ∧ IsUnit (f i)
⊢ IsUnit (f (u * i))

Tactic:
by

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

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝² : CommMonoidWithZero M
inst✝¹ : CommMonoidWithZero N
S : Submonoid M
inst✝ : WfDvdMonoid M
f : S.LocalizationMap N
i : M
hi : Irreducible i
u m' : M
hu : IsUnit (f u)
hm' : Irreducible m'
ha0 : u * m' ≠ 0
ha : Irreducible (f (u * m')) → ∃ u_1 m'_1, IsUnit (f u_1) ∧ Irreducible m'_1 ∧ u * m' = u_1 * m'_1
hai : Irreducible (f (u * m')) ∧ IsUnit (f i)
⊢ i * (u * m') = u * i * m'

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝ : Fintype α
c : List ℕ
hc : c.sum ≤ Fintype.card α
klift : (n : ℕ) → n < Fintype.card α → Fin (Fintype.card α) := fun n hn ↦ ⟨n, hn⟩
klift' : (l : List ℕ) → (∀ a ∈ l, a < Fintype.card α) → List (Fin (Fintype.card α)) := fun l hl ↦ pmap klift l hl
hc'_lt : ∀ l ∈ c.ranges, ∀ n ∈ l, n < Fintype.card α
l : List (List (Fin (Fintype.card α))) := pmap klift' c.ranges hc'_lt
⊢ ∀ (a : List ℕ) (ha : a ∈ c.ranges), map (⇑Fin.valEmbedding) (klift' a ⋯) = a

Tactic:
by

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

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝² : CommMonoidWithZero M
inst✝¹ : CommMonoidWithZero N
S : Submonoid M
inst✝ : WfDvdMonoid M
f : S.LocalizationMap N
m : M
hm : Irreducible (f m)
⊢ ∃ u m', IsUnit (f u) ∧ Irreducible m' ∧ m = u * m'

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommCoprod

Goal state before:
pp: M : Type u_4
N : Type u_5
P : Type u_6
inst✝² : Group M
inst✝¹ : Group N
inst✝ : Group P
f : M →* P
g : N →* P
comm : ∀ (m : M) (n : N), Commute (f m) (g n)
hf : ∀ (a : M), f a = 1 → a = 1
hg : ∀ (a : N), g a = 1 → a = 1
hp : f.range ⊓ g.range ≤ ⊥
a : M
b : N
h : f a * g b = 1
⊢ f a⁻¹ = g b

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommCoprod

Goal state before:
pp: M : Type u_4
N : Type u_5
P : Type u_6
inst✝² : Group M
inst✝¹ : Group N
inst✝ : Group P
f : M →* P
g : N →* P
comm : ∀ (m : M) (n : N), Commute (f m) (g n)
hf : ∀ (a : M), f a = 1 → a = 1
hg : ∀ (a : N), g a = 1 → a = 1
hp : f.range ⊓ g.range ≤ ⊥
a : M
b : N
h : f a * g b = 1
key : g b ∈ ⊥
⊢ f a = 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommCoprod

Goal state before:
pp: M : Type u_4
N : Type u_5
P : Type u_6
inst✝² : Group M
inst✝¹ : Group N
inst✝ : Group P
f : M →* P
g : N →* P
comm : ∀ (m : M) (n : N), Commute (f m) (g n)
⊢ Function.Injective ⇑(f.noncommCoprod g comm) ↔
    Function.Injective ⇑f ∧ Function.Injective ⇑g ∧ Disjoint f.range g.range

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommCoprod

Goal state before:
pp: M : Type u_4
N : Type u_5
P : Type u_6
inst✝² : Group M
inst✝¹ : Group N
inst✝ : Group P
f : M →* P
g : N →* P
comm : ∀ (m : M) (n : N), Commute (f m) (g n)
a : M
⊢ (f.noncommCoprod g comm) (a, 1) = f a

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommCoprod

Goal state before:
pp: M : Type u_4
N : Type u_5
P : Type u_6
inst✝² : Group M
inst✝¹ : Group N
inst✝ : Group P
f : M →* P
g : N →* P
comm : ∀ (m : M) (n : N), Commute (f m) (g n)
a : N
⊢ (f.noncommCoprod g comm) (1, a) = g a

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommCoprod

Goal state before:
pp: M : Type u_4
N : Type u_5
P : Type u_6
inst✝² : Group M
inst✝¹ : Group N
inst✝ : Group P
f : M →* P
g : N →* P
comm : ∀ (m : M) (n : N), Commute (f m) (g n)
⊢ (f.noncommCoprod g comm).range = f.range ⊔ g.range

Tactic:
by

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

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝² : CommMonoidWithZero M
inst✝¹ : CommMonoidWithZero N
S : Submonoid M
f : S.LocalizationMap N
inst✝ : UniqueFactorizationMonoid M
this : IsCancelMulZero N
n : N
m : M
s : ↥S
hn : ¬f (m, s).1 = 0
eq : n * f ↑(m, s).2 = f (m, s).1
x✝ : m = 0
⊢ f (m, s).1 = 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ι : Type u_2
f : α → ι
g : (Perm α)ᵈᵐᵃ
⊢ g ∈ stabilizer (Perm α)ᵈᵐᵃ f ↔ f ∘ ⇑(mk.symm g) = f

Tactic:
by

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:
by

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
⊢ stabilizerEquiv_invFun (fun i ↦ Equiv.symm (g i)) (stabilizerEquiv_invFun g a) = a

Tactic:
by

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
⊢ stabilizerEquiv_invFun g (stabilizerEquiv_invFun (fun i ↦ Equiv.symm (g i)) a) = a

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ι : Type u_2
f : α → ι
g : (↥(stabilizer (Perm α)ᵈᵐᵃ f))ᵐᵒᵖ
i : ι
a : α
⊢ f ((mk.symm ↑(MulOpposite.unop g)) a) = i ↔ f a = i

Tactic:
by

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 }
⊢ mk (stabilizerEquiv_invFun_aux g) ∈ stabilizer (Perm α)ᵈᵐᵃ f

Tactic:
by

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 }
⊢ (fun g i ↦ (mk.symm ↑(MulOpposite.unop g)).subtypePerm ⋯)
      ((fun g ↦ { unop' := ⟨mk (stabilizerEquiv_invFun_aux g), ⋯⟩ }) g) =
    g

Tactic:
by

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

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝² : CommMonoidWithZero M
inst✝¹ : CommMonoidWithZero N
S : Submonoid M
f : S.LocalizationMap N
inst✝ : UniqueFactorizationMonoid M
this : IsCancelMulZero N
n : N
hn : n ≠ 0
⊢ ∃ f, (∀ b ∈ f, Prime b) ∧ Associated f.prod n

Tactic:
by

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
u : ℤˣ
w : NormalWord d
⊢ unitsSMul φ (-u) (unitsSMul φ u w) = w

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.HNNExtension

Goal state before:
pp: G : Type u_1
inst✝² : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
H : Type u_2
inst✝¹ : Group H
M : Type u_3
inst✝ : Monoid M
d : TransversalPair G A B
x✝ : NormalWord d
⊢ unitsSMul φ (-1) (unitsSMul φ 1 x✝) = x✝

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.HNNExtension

Goal state before:
pp: G : Type u_1
inst✝² : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
H : Type u_2
inst✝¹ : Group H
M : Type u_3
inst✝ : Monoid M
d : TransversalPair G A B
w : NormalWord d
⊢ unitsSMul φ 1 (unitsSMul φ (-1) w) = w

Tactic:
by

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
⊢ Cancels 1 (↑g • w) ↔ Cancels 1 w

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ι : Type u_2
f : α → ι
inst✝³ : Fintype α
inst✝² : DecidableEq α
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
i : ι
x✝ : i ∈ Finset.univ
⊢ Nat.card (Perm { a // f a = i }) = (Fintype.card { a // f a = i })!

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ι : Type u_2
f : α → ι
inst✝³ : Fintype α
inst✝² : DecidableEq α
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
⊢ Fintype.card { g // f ∘ ⇑g = f } = ∏ i, (Fintype.card { a // f a = i })!

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ι : Type u_2
f : α → ι
inst✝¹ : Finite α
inst✝ : Fintype ι
⊢ {g | f ∘ ⇑g = f}.ncard = ∏ i, {a | f a = i}.ncard !

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ι : Type u_2
f : α → ι
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableEq ι
a : α
⊢ f a ∈ ↑(Finset.image f Finset.univ)

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝ : Fintype α
c : List ℕ
hc : c.sum ≤ Fintype.card α
⊢ ∃ o, map length o = c ∧ (∀ s ∈ o, s.Nodup) ∧ Pairwise Disjoint o

Tactic:
by

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

Goal state before:
pp: 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
⊢ m.toList.sum ≤ Fintype.card α

Tactic:
by

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:
by

Goal state after:
Module: Mathlib.GroupTheory.HNNExtension

Goal state before:
pp: G : Type u_1
inst✝² : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
H : Type u_2
inst✝¹ : Group H
M : Type u_3
inst✝ : Monoid M
d : TransversalPair G A B
⊢ ∀ (a : ↥A),
    unitsSMulEquiv φ * (MulAction.toPermHom G (NormalWord d)) ↑a =
      (MulAction.toPermHom G (NormalWord d)) ↑(φ a) * unitsSMulEquiv φ

Tactic:
by

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 : G
w : NormalWord d
⊢ ReducedWord.prod φ (g • w).toReducedWord = of g * ReducedWord.prod φ w.toReducedWord

Tactic:
by

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 : G
w : NormalWord d
⊢ of g • w = g • w

Tactic:
by

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

Goal state before:
pp: 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
⊢ ∀ x ∈ p, 2 ≤ x.length

Tactic:
by

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
w : NormalWord d
⊢ t • w = unitsSMul φ 1 w

Tactic:
by

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
u : ℤˣ
w : NormalWord d
⊢ t ^ ↑u • w = unitsSMul φ u w

Tactic:
by

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 : G
u : ℤˣ
w : NormalWord d
h1 : w.head ∈ d.set u
h2 : ∀ u' ∈ Option.map Prod.fst w.toList.head?, w.head ∈ toSubgroup A B u → u = u'
⊢ ReducedWord.prod φ (cons g u w h1 h2).toReducedWord = of g * (t ^ ↑u * ReducedWord.prod φ w.toReducedWord)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ι : Type u_2
f : α → ι
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableEq ι
φ : α → ↥(Finset.image f Finset.univ) := Set.codRestrict f ↑(Finset.image f Finset.univ) ⋯
this : ∀ (g : Perm α), f ∘ ⇑g = f ↔ φ ∘ ⇑g = φ
⊢ Fintype.card { g // f ∘ ⇑g = f } = ∏ i ∈ Finset.image f Finset.univ, (Fintype.card { a // f a = i })!

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ι : Type u_2
f : α → ι
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableEq ι
φ : α → ↥(Finset.image f Finset.univ) := Set.codRestrict f ↑(Finset.image f Finset.univ) ⋯
this : ∀ (g : Perm α), f ∘ ⇑g = f ↔ φ ∘ ⇑g = φ
⊢ Fintype.card { g // f ∘ ⇑g = f } = ∏ i ∈ Finset.image f Finset.univ, (Fintype.card { a // f a = i })!

Tactic:
by
    simp only [this, stabilizer_card]
    apply Finset.prod_bij (fun g _ => g.val)
    · exact fun g _ => Finset.coe_mem g
    · exact fun g _ g' _ => SetCoe.ext
    · simp
    · intro i _
      apply congr_arg
      apply Fintype.card_congr
      apply Equiv.subtypeEquiv (Equiv.refl α)
      intro a
      rw [refl_apply, ← Subtype.coe_inj]
      simp only [φ, Set.val_codRestrict_apply]

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

Goal state before:
pp: α : Type u_1
ι : Type u_2
f : α → ι
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableEq ι
⊢ Fintype.card { g // f ∘ ⇑g = f } = ∏ i ∈ Finset.image f Finset.univ, (Fintype.card { a // f a = i })!

Tactic:
by

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
u : ℤˣ
w : NormalWord d
⊢ ReducedWord.prod φ (unitsSMul φ u w).toReducedWord = t ^ ↑u * ReducedWord.prod φ w.toReducedWord

Tactic:
by

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
⊢ ReducedWord.prod φ empty.toReducedWord = 1

Tactic:
by

Goal state after: