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

Goal state before:
pp: M : Type u_1
inst✝² : Monoid M
ι : Type u_2
inst✝¹ : Fintype ι
N : ι → Type u_3
inst✝ : (i : ι) → Monoid (N i)
ϕ : (i : ι) → N i →* M
hcomm : Pairwise fun i j ↦ ∀ (x : N i) (y : N j), Commute ((ϕ i) x) ((ϕ j) y)
h : (i : ι) → N i
⊢ (noncommPiCoprod ϕ hcomm) h = Finset.univ.noncommProd (fun i ↦ (ϕ i) (h i)) ⋯

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommPiCoprod

Goal state before:
pp: M : Type u_1
inst✝³ : Monoid M
ι : Type u_2
inst✝² : Fintype ι
N : ι → Type u_3
inst✝¹ : (i : ι) → Monoid (N i)
ϕ : (i : ι) → N i →* M
hcomm : Pairwise fun i j ↦ ∀ (x : N i) (y : N j), Commute ((ϕ i) x) ((ϕ j) y)
P : Type u_4
inst✝ : Monoid P
f : M →* P
i j : ι
x : N i
h : ∀ (y : N j), Commute ((ϕ i) x) ((ϕ j) y)
y : N j
⊢ Commute ((f.comp (ϕ i)) x) ((f.comp (ϕ j)) y)

Tactic:
by

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

Goal state before:
pp: α : Type u_2
s : Set α
hs : s.Countable
⊢ ∃ f, f.IsCycleOn s ∧ {x | f x ≠ x} ⊆ s

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommPiCoprod

Goal state before:
pp: M : Type u_1
inst✝³ : Monoid M
ι : Type u_2
inst✝² : Fintype ι
N : ι → Type u_3
inst✝¹ : (i : ι) → Monoid (N i)
ϕ : (i : ι) → N i →* M
hcomm : Pairwise fun i j ↦ ∀ (x : N i) (y : N j), Commute ((ϕ i) x) ((ϕ j) y)
P : Type u_4
inst✝ : Monoid P
f : M →* P
hcomm' : Pairwise fun i j ↦ ∀ (x : N i) (y : N j), Commute ((f.comp (ϕ i)) x) ((f.comp (ϕ j)) y)
x✝ : (i : ι) → N i
⊢ (f.comp (noncommPiCoprod ϕ hcomm)) x✝ = (noncommPiCoprod (fun i ↦ f.comp (ϕ i)) hcomm') x✝

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f : Perm α
s : Set α
hf : f.IsCycleOn s
⊢ s ×ˢ s = ⋃ n, (fun a ↦ (a, (f ^ n) a)) '' s

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f : Perm α
inst✝ : DecidableRel f.SameCycle
x y : α
⊢ (f.cycleOf x) y = if f.SameCycle x y then f y else y

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommPiCoprod

Goal state before:
pp: G : Type u_1
inst✝² : Group G
ι : Type u_2
H : ι → Type u_3
inst✝¹ : (i : ι) → Group (H i)
ϕ : (i : ι) → H i →* G
inst✝ : Fintype ι
hcomm : Pairwise fun i j ↦ ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)
⊢ (noncommPiCoprod ϕ hcomm).range = ⨆ i, (ϕ i).range

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommPiCoprod

Goal state before:
pp: G : Type u_1
inst✝² : Group G
ι : Type u_2
H : ι → Type u_3
inst✝¹ : (i : ι) → Group (H i)
ϕ : (i : ι) → H i →* G
inst✝ : Fintype ι
hcomm : Pairwise fun i j ↦ ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)
hind : iSupIndep fun i ↦ (ϕ i).range
hinj : ∀ (i : ι), Function.Injective ⇑(ϕ i)
⊢ Function.Injective ⇑(noncommPiCoprod ϕ hcomm)

Tactic:
apply (MonoidHom.ker_eq_bot_iff _).mp

Goal state after:
G : Type u_1
inst✝² : Group G
ι : Type u_2
H : ι → Type u_3
inst✝¹ : (i : ι) → Group (H i)
ϕ : (i : ι) → H i →* G
inst✝ : Fintype ι
hcomm : Pairwise fun i j ↦ ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)
hind : iSupIndep fun i ↦ (ϕ i).range
hinj : ∀ (i : ι), Function.Injective ⇑(ϕ i)
⊢ (noncommPiCoprod ϕ hcomm).ker = ⊥
Module: Mathlib.GroupTheory.Perm.Cycle.Factors

Goal state before:
pp: α : Type u_2
f : Perm α
inst✝ : DecidableRel f.SameCycle
x y : α
⊢ (f.cycleOf x)⁻¹ y = (f⁻¹.cycleOf x) y

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f : Perm α
s : Finset α
hf : f.IsCycleOn ↑s
⊢ (↑(range #s)).PairwiseDisjoint fun k ↦ map { toFun := fun i ↦ (i, (f ^ k) i), inj' := ⋯ } s

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommPiCoprod

Goal state before:
pp: G : Type u_1
inst✝² : Group G
ι : Type u_2
H : ι → Type u_3
inst✝¹ : (i : ι) → Group (H i)
ϕ : (i : ι) → H i →* G
inst✝ : Fintype ι
hcomm : Pairwise fun i j ↦ ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)
hind : iSupIndep fun i ↦ (ϕ i).range
hinj : ∀ (i : ι), Function.Injective ⇑(ϕ i)
f : (i : ι) → H i
heq1 : f ∈ (noncommPiCoprod ϕ hcomm).ker
⊢ ∀ x ∈ Finset.univ, (ϕ x) (f x) ∈ (ϕ x).range

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f : Perm α
inst✝ : DecidableRel f.SameCycle
x : α
⊢ ∀ (n : ℕ), (f.cycleOf x ^ n) x = (f ^ n) x

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f : Perm α
s : Finset α
hf : f.IsCycleOn ↑s
a : α
n : ℕ
hn : n < #s
hx : a ∈ s ∧ (f ^ n) a ∈ s
⊢ (⇑f)^[n] a = (f ^ n) a

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f : Perm α
s : Finset α
hf : f.IsCycleOn ↑s
⊢ s ×ˢ s = (range #s).disjiUnion (fun k ↦ map { toFun := fun i ↦ (i, (f ^ k) i), inj' := ⋯ } s) ⋯

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommPiCoprod

Goal state before:
pp: G : Type u_1
inst✝² : Group G
ι : Type u_2
H : ι → Type u_3
inst✝¹ : (i : ι) → Group (H i)
ϕ : (i : ι) → H i →* G
inst✝ : Fintype ι
hcomm : Pairwise fun i j ↦ ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)
hind : iSupIndep fun i ↦ (ϕ i).range
hinj : ∀ (i : ι), Function.Injective ⇑(ϕ i)
⊢ Function.Injective ⇑(noncommPiCoprod ϕ hcomm)

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f : Perm α
inst✝ : DecidableRel f.SameCycle
x : α
⊢ ∀ (n : ℤ), (f.cycleOf x ^ n) x = (f ^ n) x

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
s : Finset (Perm α)
h1 : ∀ f ∈ s, f.IsCycle
h2 : (↑s).Pairwise Disjoint
h0 : s.noncommProd id ⋯ = σ
⊢ σ.cycleType = Multiset.map (Finset.card ∘ support) s.val

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f : Perm α
x y : α
inst✝ : DecidableRel f.SameCycle
h : f.SameCycle x y
⊢ f.cycleOf x = f.cycleOf y

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f : Perm α
inst✝ : DecidableRel f.SameCycle
x : α
k : ℤ
⊢ (f.cycleOf x) ((f ^ k) x) = (f ^ (k + 1)) x

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f : Perm α
inst✝ : DecidableRel f.SameCycle
x : α
k : ℕ
⊢ (f.cycleOf x) ((f ^ k) x) = (f ^ (k + 1)) x

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
α : Type u_2
β : Type u_3
inst✝² : Semiring α
inst✝¹ : AddCommMonoid β
inst✝ : Module α β
s : Finset ι
σ : Perm ι
hσ : σ.IsCycleOn ↑s
f : ι → α
g : ι → β
⊢ (∑ i ∈ s, f i) • ∑ i ∈ s, g i = ∑ k ∈ range #s, ∑ i ∈ s, f i • g ((σ ^ k) i)

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f : Perm α
inst✝ : DecidableRel f.SameCycle
x : α
⊢ (f.cycleOf x) (f x) = f (f x)

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f : Perm α
x : α
inst✝ : DecidableRel f.SameCycle
hf : f.IsCycle
hx : f x ≠ x
y : α
h : f.SameCycle x y
⊢ (f.cycleOf x) y = f y

Tactic:
by

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

Goal state before:
pp: α : Type u_2
g : Perm α
s : Finset α
hs : ∀ (x : α), g x ∈ s ↔ x ∈ s
n : ℕ
x : α
hx : x ∈ s
⊢ ↑((g.subtypePerm hs ^ n) ⟨x, hx⟩) = (g ^ n) x

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f : Perm α
x : α
inst✝ : DecidableRel f.SameCycle
hf : f.IsCycle
hx : f x ≠ x
y : α
h : ¬f.SameCycle x y
⊢ (f.cycleOf x) y = f y

Tactic:
by

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

Goal state before:
pp: α : Type u_2
g : Perm α
s : Finset α
hs : ∀ (x : α), g x ∈ s ↔ x ∈ s
i : ℤ
x : α
hx : x ∈ s
⊢ ↑((g.subtypePerm hs ^ i) ⟨x, hx⟩) = (g ^ i) x

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
hg : g.IsCycle
⊢ g.support.Nonempty

Tactic:
by

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

Goal state before:
pp: α : Type u_2
x : α
f : Perm α
inst✝ : DecidableRel f.SameCycle
h : f x = x
y : α
hy : ¬f y = y
⊢ ¬(f x = x ↔ f y = y)

Tactic:
by

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

Goal state before:
pp: α : Type u_2
x : α
f : Perm α
inst✝ : DecidableRel f.SameCycle
⊢ f.cycleOf x = 1 ↔ f x = x

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f : Perm α
x : α
inst✝¹ : DecidableRel f.SameCycle
inst✝ : DecidableEq α
hf : f.IsCycle
⊢ f.cycleOf x = if f x = x then 1 else f

Tactic:
by

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

Goal state before:
pp: α : Type u_2
x : α
f : Perm α
inst✝ : DecidableRel f.SameCycle
hx : f x ≠ x
⊢ (f.cycleOf x) x ≠ x

Tactic:
by

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

Goal state before:
pp: α : Type u_2
x : α
f : Perm α
inst✝ : DecidableRel f.SameCycle
hx : f x ≠ x
this : (f.cycleOf x) x ≠ x
y : α
h : (f.cycleOf x) y ≠ y
hxy : f.SameCycle x y
i : ℤ
hi : (f ^ i) x = y
⊢ (f.cycleOf x ^ i) x = y

Tactic:
by

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

Goal state before:
pp: α : Type u_2
x : α
f : Perm α
inst✝ : DecidableRel f.SameCycle
hx : f x ≠ x
this : (f.cycleOf x) x ≠ x
y : α
h : (f.cycleOf x) y ≠ y
hxy : ¬f.SameCycle x y
⊢ (f.cycleOf x).SameCycle x y

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
l : List (Perm α)
h0 : l.prod = σ
h1 : ∀ σ ∈ l, σ.IsCycle
h2 : List.Pairwise Disjoint l
⊢ σ.cycleType = ↑(List.map (Finset.card ∘ support) l)

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f : Perm α
inst✝ : DecidableRel f.SameCycle
n : ℕ
x : α
⊢ (f ^ (n % orderOf (f.cycleOf x))) x = (f ^ n) x

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f g : Perm α
inst✝¹ : DecidableRel f.SameCycle
inst✝ : DecidableRel (f * g).SameCycle
h : Commute f g
x : α
hx : g x = x
⊢ (f * g).cycleOf x = f.cycleOf x

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f g : Perm α
inst✝² : DecidableRel f.SameCycle
inst✝¹ : DecidableRel g.SameCycle
inst✝ : DecidableRel (f * g).SameCycle
h : f.Disjoint g
x : α
⊢ (f * g).cycleOf x = f.cycleOf x * g.cycleOf x

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
n : ℕ
⊢ Multiset.count n σ.cycleType = Fintype.card { c // #(↑c).support = n }

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
⊢ σ.cycleType = 0 ↔ σ = 1

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
⊢ σ.cycleType.card = 0 ↔ σ = 1

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
n : ℕ
h : n ∈ σ.cycleType
⊢ 2 ≤ n

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommPiCoprod

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
ι : Type u_2
H : ι → Type u_3
inst✝² : (i : ι) → Group (H i)
ϕ : (i : ι) → H i →* G
hcomm : Pairwise fun i j ↦ ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)
inst✝¹ : Finite ι
inst✝ : (i : ι) → Fintype (H i)
hcoprime : Pairwise fun i j ↦ (Fintype.card (H i)).Coprime (Fintype.card (H j))
val✝ : Fintype ι
this : DecidableEq ι := Classical.decEq ι
i : ι
f : G
hxp : f ∈ ↑(⨆ x, (ϕ ↑x).range)
g : (i_1 : { j // ¬j = i }) → H ↑i_1
hgf : (noncommPiCoprod (fun x ↦ ϕ ↑x) ⋯) g = f
g' : H i
hg'f : (ϕ i) g' = f
⊢ orderOf f ∣ Fintype.card (H i)

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g c : Perm α
hc : c.IsCycle
hc' : ∀ (x : α), g x ∈ c.support ↔ x ∈ c.support
i : ℤ
hi : (fun x ↦ c.subtypePermOfSupport ^ x) i = g.subtypePerm hc'
x : α
hx : x ∈ c.support
hi' : ∀ x ∈ c.support, g x = (c ^ i) x
⊢ g (c x) = c (g x)

Tactic:
by

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

Goal state before:
pp: case neg
α : Type u_2
f : Perm α
x y : α
inst✝² : DecidableRel f.SameCycle
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hx : ¬f x = x
hy : ¬f.SameCycle x y
⊢ y ≠ y ↔ f.SameCycle x y ∧ x ∈ f.support

Tactic:
simpa [hx] using hy

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
⊢ σ.cycleType.card = 1 ↔ σ.IsCycle

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f : Perm α
x y : α
inst✝² : DecidableRel f.SameCycle
inst✝¹ : DecidableEq α
inst✝ : Fintype α
⊢ y ∈ (f.cycleOf x).support ↔ f.SameCycle x y ∧ x ∈ f.support

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f : Perm α
x y : α
hx : f x ≠ x
inst✝² : DecidableRel f.SameCycle
inst✝¹ : DecidableEq α
inst✝ : Fintype α
⊢ y ∈ (f.cycleOf x).support ↔ f.SameCycle x y

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f : Perm α
x y : α
hx : f x ≠ x
inst✝² : DecidableRel f.SameCycle
inst✝¹ : DecidableEq α
inst✝ : Fintype α
⊢ y ∈ (f.cycleOf x).support ↔ f.SameCycle x y

Tactic:
by
  rw [mem_support_cycleOf_iff_aux, and_iff_left (mem_support.2 hx)]

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

Goal state before:
pp: case neg
α : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g c : Perm α
hc : c.IsCycle
hc' : ∀ (x : α), g x ∈ c.support ↔ x ∈ c.support
i : ℤ
hi : (fun x ↦ c.subtypePermOfSupport ^ x) i = g.subtypePerm hc'
x : α
hx : x ∉ c.support
⊢ g x ∉ c.support

Tactic:
contrapose hx

Goal state after:
case neg
α : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g c : Perm α
hc : c.IsCycle
hc' : ∀ (x : α), g x ∈ c.support ↔ x ∈ c.support
i : ℤ
hi : (fun x ↦ c.subtypePermOfSupport ^ x) i = g.subtypePerm hc'
x : α
hx : g x ∈ c.support
⊢ x ∈ c.support
Module: Mathlib.GroupTheory.Perm.Cycle.Factors

Goal state before:
pp: α : Type u_2
x : α
f : Perm α
inst✝ : DecidableRel f.SameCycle
⊢ (f.cycleOf x).IsCycle ↔ f x ≠ x

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ τ : Perm α
h : σ.Disjoint τ
⊢ (σ * τ).cycleType = σ.cycleType + τ.cycleType

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g c : Perm α
hc : c.IsCycle
⊢ Commute g c ↔
    ∃ (hc' : ∀ (x : α), g x ∈ c.support ↔ x ∈ c.support), g.subtypePerm hc' ∈ Subgroup.zpowers c.subtypePermOfSupport

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ✝ σ : Perm α
hσ : σ.IsCycle
⊢ σ⁻¹.cycleType = σ.cycleType

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommPiCoprod

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
ι : Type u_2
H : ι → Type u_3
inst✝² : (i : ι) → Group (H i)
ϕ : (i : ι) → H i →* G
hcomm : Pairwise fun i j ↦ ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)
inst✝¹ : Finite ι
inst✝ : (i : ι) → Fintype (H i)
hcoprime : Pairwise fun i j ↦ (Fintype.card (H i)).Coprime (Fintype.card (H j))
val✝ : Fintype ι
this : DecidableEq ι := Classical.decEq ι
i : ι
f : G
hxp : f ∈ ↑(⨆ x, (ϕ ↑x).range)
g : (i_1 : { j // ¬j = i }) → H ↑i_1
hgf : (noncommPiCoprod (fun x ↦ ϕ ↑x) ⋯) g = f
g' : H i
hg'f : (ϕ i) g' = f
hxi : orderOf f ∣ Fintype.card (H i)
⊢ orderOf f ∣ ∏ j, Fintype.card (H ↑j)

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝² : DecidableEq α
inst✝¹ : Fintype α
f : Perm α
inst✝ : DecidableRel f.SameCycle
x : α
⊢ ∀ (a : α), f a ∈ ↑(f.cycleOf x).support ↔ a ∈ ↑(f.cycleOf x).support

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ✝ σ τ : Perm α
hστ : σ.Disjoint τ
x✝ : σ.IsCycle
hσ : σ⁻¹.cycleType = σ.cycleType
hτ : τ⁻¹.cycleType = τ.cycleType
⊢ (σ * τ)⁻¹.cycleType = (σ * τ).cycleType

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝² : DecidableEq α
inst✝¹ : Fintype α
f : Perm α
inst✝ : DecidableRel f.SameCycle
x a : α
ha : a ∈ ↑(f.cycleOf x).support
b : α
hb : b ∈ ↑(f.cycleOf x).support
⊢ f.SameCycle a b

Tactic:
by

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

Goal state before:
pp: α : Type u_2
x y : α
f : Perm α
inst✝² : DecidableEq α
inst✝¹ : Fintype α
inst✝ : DecidableRel f.SameCycle
h : f.SameCycle x y
hx : f x ≠ x
⊢ x ∈ (f.cycleOf x).support

Tactic:
by

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

Goal state before:
pp: α : Type u_2
x y : α
f : Perm α
inst✝² : DecidableEq α
inst✝¹ : Fintype α
inst✝ : DecidableRel f.SameCycle
h : f.SameCycle x y
hx : f x ≠ x
⊢ y ∈ (f.cycleOf x).support

Tactic:
by

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

Goal state before:
pp: α : Type u_2
x y : α
f : Perm α
inst✝² : DecidableEq α
inst✝¹ : Fintype α
inst✝ : DecidableRel f.SameCycle
h : f.SameCycle x y
hx : x ∈ f.support
⊢ ∃ i < #(f.cycleOf x).support, (f ^ i) x = y

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ τ : Perm α
⊢ (τ * σ * τ⁻¹).cycleType = σ.cycleType

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
α : Type u_2
β : Type u_3
f✝ g : Perm α
x✝ y✝ : α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f : Perm α
x y : α
⊢ y ∈ List.iterate (⇑f) x (Fintype.card α) ↔ f.SameCycle x y

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f : Perm α
x : α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
h : f x ≠ x
⊢ 2 ≤ #(f.cycleOf x).support

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
⊢ σ.cycleType.sum = #σ.support

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
⊢ σ.cycleType.sum = #σ.support

Tactic:
by
  induction σ using cycle_induction_on with
  | base_one => simp
  | base_cycles σ hσ => rw [hσ.cycleType, Multiset.sum_singleton]
  | induction_disjoint σ τ hd _ hσ hτ => rw [hd.cycleType_mul, sum_add, hσ, hτ, hd.card_support_mul]

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

Goal state before:
pp: α : Type u_2
f : Perm α
x : α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
⊢ 2 ≤ #(f.cycleOf x).support ↔ f x ≠ x

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
⊢ Fintype.card ↑(Function.fixedPoints ⇑σ) = Fintype.card α - σ.cycleType.sum

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f : Perm α
x : α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
⊢ (f.cycleOf x).support.Nonempty ↔ f x ≠ x

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
⊢ sign σ = (Multiset.map (fun n ↦ -(-1) ^ n) σ.cycleType).prod

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g c : Perm α
hc : c.IsCycle
⊢ Commute g c ↔ ∃ (hc' : ∀ (x : α), g x ∈ c.support ↔ x ∈ c.support), ofSubtype (g.subtypePerm hc') ∈ Subgroup.zpowers c

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
x y : α
hx : x ∈ g.support
hy : y ∈ g.support
⊢ g.SameCycle x y ↔ g.cycleOf x = g.cycleOf y

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f : Perm α
x : α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
⊢ (f.cycleOf x).support = ∅ ↔ x ∉ f.support

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
⊢ sign f = (-1) ^ (f.cycleType.sum + f.cycleType.card)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
⊢ σ.cycleType.lcm = orderOf σ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
n : ℕ
h : n ∈ σ.cycleType
⊢ n ∣ orderOf σ

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f : Perm α
x : α
⊢ (f.cycleOf x).support ⊆ f.support

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f : Perm α
n : ℕ
x : α
⊢ (f ^ (n % #(f.cycleOf x).support)) x = (f ^ n) x

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommPiCoprod

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
ι : Type u_2
H : ι → Type u_3
inst✝² : (i : ι) → Group (H i)
ϕ : (i : ι) → H i →* G
hcomm : Pairwise fun i j ↦ ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)
inst✝¹ : Finite ι
inst✝ : (i : ι) → Fintype (H i)
hcoprime : Pairwise fun i j ↦ (Fintype.card (H i)).Coprime (Fintype.card (H j))
⊢ iSupIndep fun i ↦ (ϕ i).range

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x : α
⊢ orderOf (f.cycleOf x) ∣ orderOf f

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
hσ : σ ^ 2 = 1
n : ℕ
hn : n ∈ σ.cycleType
⊢ 2 ∣ n

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommPiCoprod

Goal state before:
pp: G : Type u_1
inst✝ : Group G
ι : Type u_2
H : ι → Subgroup G
hcomm : Pairwise fun i j ↦ ∀ (x y : G), x ∈ H i → y ∈ H j → Commute x y
i j : ι
hne : i ≠ j
⊢ ∀ (x : ↥(H i)) (y : ↥(H j)), Commute ((H i).subtype x) ((H j).subtype y)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommPiCoprod

Goal state before:
pp: G : Type u_1
inst✝² : Group G
ι : Type u_2
H : ι → Subgroup G
hcomm : Pairwise fun i j ↦ ∀ (x y : G), x ∈ H i → y ∈ H j → Commute x y
inst✝¹ : Finite ι
inst✝ : (i : ι) → Fintype ↥(H i)
hcoprime : Pairwise fun i j ↦ (Fintype.card ↥(H i)).Coprime (Fintype.card ↥(H j))
⊢ iSupIndep H

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommPiCoprod

Goal state before:
pp: G : Type u_1
inst✝² : Group G
ι : Type u_2
H : ι → Subgroup G
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
hcomm : Pairwise fun i j ↦ ∀ (x y : G), x ∈ H i → y ∈ H j → Commute x y
i : ι
y : ↥(H i)
⊢ (noncommPiCoprod hcomm) (Pi.mulSingle i y) = ↑y

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.NoncommPiCoprod

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
ι : Type u_2
H : ι → Subgroup G
inst✝ : Fintype ι
hcomm : Pairwise fun i j ↦ ∀ (x y : G), x ∈ H i → y ∈ H j → Commute x y
⊢ (noncommPiCoprod hcomm).range = ⨆ i, H i

Tactic:
by

Goal state after: