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

Goal state before:
pp: α : Type u
inst✝ : Finite α
σ τ π ρ : Perm α
hc1 : IsConj σ π
hc2 : IsConj τ ρ
hd1 : σ.Disjoint τ
hd2 : π.Disjoint ρ
⊢ IsConj (σ * τ) (π * ρ)

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
σ : Perm α
h1 : σ.IsCycle
h2 : σ.support = univ
x : α
H : Subgroup (Perm α) := closure {σ, swap x (σ x)}
h3 : σ ∈ H
h4 : swap x (σ x) ∈ H
step1 : ∀ (n : ℕ), swap ((σ ^ n) x) ((σ ^ (n + 1)) x) ∈ H
step2 : ∀ (n : ℕ), swap x ((σ ^ n) x) ∈ H
⊢ ∀ (y : α), swap x y ∈ H

Tactic:
by

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

Goal state before:
pp: α : Type u_4
inst✝ : DecidableEq α
⊢ ∀ (n : ℕ) {b x : α} {f : Perm α},
    (swap x (f x) * f) b ≠ b → (f ^ n) (f x) = b → ∃ i, ((swap x (f x) * f) ^ i) (f x) = b

Tactic:
by

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

Goal state before:
pp: α : Type u
g p : Perm α
hp : p ∈ Subgroup.centralizer {g}
x : α
⊢ p x ∈ fixedPoints ⇑g ↔ x ∈ fixedPoints ⇑g

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : DecidableEq α
g : Perm α
u : Perm ↑(fixedPoints ⇑g)
⊢ (ofSubtype u).Disjoint g

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
σ : Perm α
n : ℕ
h : n.Coprime (orderOf σ)
⊢ (σ ^ n).support = σ.support

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
σ : Perm α
x : Perm ↑(fixedPoints ⇑σ)
⊢ _root_.Disjoint (ofSubtype x).support σ.support

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
H K : Subgroup (Perm α)
h : ∀ a ∈ H, ∀ b ∈ K, _root_.Disjoint a.support b.support
⊢ _root_.Disjoint H K

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
σ : Perm α
h1 : σ.IsCycle
h2 : σ.support = univ
x : α
H : Subgroup (Perm α) := closure {σ, swap x (σ x)}
h3 : σ ∈ H
h4 : swap x (σ x) ∈ H
step1 : ∀ (n : ℕ), swap ((σ ^ n) x) ((σ ^ (n + 1)) x) ∈ H
step2 : ∀ (n : ℕ), swap x ((σ ^ n) x) ∈ H
step3 : ∀ (y : α), swap x y ∈ H
⊢ ∀ (y z : α), swap y z ∈ H

Tactic:
by

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

Goal state before:
pp: α : Type u_4
inst✝ : DecidableEq α
n : ℕ
b x : α
f : Perm α
hb : (swap x (f x) * f) b ≠ b
h : (f ^ Int.negSucc n) (f x) = b
hfxb : f x ≠ b
hfb : f b ≠ b
hbx : b ≠ x
⊢ (swap x ((Equiv.symm f) x) * f⁻¹) ((Equiv.symm f) b) ≠ (Equiv.symm f) b

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
σ : Perm α
h1 : σ.IsCycle
h2 : σ.support = univ
x : α
⊢ closure {σ, swap x (σ x)} = ⊤

Tactic:
by

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

Goal state before:
pp: α : Type u_4
inst✝ : DecidableEq α
n : ℕ
b x : α
f : Perm α
h : (f ^ Int.negSucc n) (f x) = b
hfxb : f x ≠ b
hfb : f b ≠ b
hbx : b ≠ x
hb : (swap x ((Equiv.symm f) x) * f⁻¹) ((Equiv.symm f) b) ≠ (Equiv.symm f) b
⊢ (Equiv.symm f ^ n) ((Equiv.symm f) x) = (Equiv.symm f) b

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
S : Set (Perm α)
⊢ ∀ a ∈ closure S, ↑a.support ⊆ ⋃ b ∈ S, ↑b.support

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
n : ℕ
σ : Perm α
h0 : n.Coprime (orderOf σ)
h1 : σ.IsCycle
h2 : σ.support = univ
x : α
m : ℕ
hm : (σ ^ n) ^ m = σ
h2' : (σ ^ n).support = univ
⊢ ((σ ^ n) ^ ↑m).IsCycle

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : DecidableEq α
n : ℕ
x : α
l : List α
f : Perm α
e : α ≃ Fin n
h : ∀ (x_1 : α), f x_1 ≠ x_1 → x_1 ∈ x :: l
hfx : ¬x = f x
hy : ∀ (y : α), (swap x (f x) * f) y ≠ y → y ∈ l
⊢ (e.symm.trans (swap x (f x) * f)).trans e = swap (e x) (e (f x)) * (e.symm.trans f).trans e

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
n : ℕ
σ : Perm α
h0 : n.Coprime (Fintype.card α)
h1 : σ.IsCycle
h2 : σ.support = univ
x : α
⊢ closure {σ, swap x ((σ ^ n) x)} = ⊤

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
S T : Set (Perm α)
h : ∀ a ∈ S, ∀ b ∈ T, _root_.Disjoint a.support b.support
⊢ ∀ a ∈ closure S, ∀ b ∈ closure T, _root_.Disjoint a.support b.support

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
S T : Set (Perm α)
h : ∀ a ∈ S, ∀ b ∈ T, _root_.Disjoint a.support b.support
⊢ _root_.Disjoint (closure S) (closure T)

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : DecidableEq α
n : ℕ
x : α
l : List α
f : Perm α
e : α ≃ Fin n
h : ∀ (x_1 : α), f x_1 ≠ x_1 → x_1 ∈ x :: l
⊢ signAux ((e.symm.trans f).trans e) = signAux2 (x :: l) f

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
σ τ : Perm α
h0 : Nat.Prime (Fintype.card α)
h1 : σ.IsCycle
h2 : σ.support = univ
h3 : τ.IsSwap
⊢ closure {σ, τ} = ⊤

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝² : DecidableEq α
inst✝¹ : Fintype α
p : α → Prop
inst✝ : DecidablePred p
g : Perm α
⊢ g ∈ ofSubtype.range ↔ ↑g.support ⊆ Set.ofPred p

Tactic:
by

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

Goal state before:
pp: α : Type u_4
inst✝ : DecidableEq α
n : ℕ
b x : α
f : Perm α
h : (f ^ Int.negSucc n) (f x) = b
hfxb : f x ≠ b
hfb : f b ≠ b
hbx : b ≠ x
hb : (swap x ((Equiv.symm f) x) * f⁻¹) ((Equiv.symm f) b) ≠ (Equiv.symm f) b
i : ℤ
hi : ((swap x ((Equiv.symm f) x) * Equiv.symm f) ^ i) ((Equiv.symm f) x) = (Equiv.symm f) b
⊢ f⁻¹ b ≠ x

Tactic:
by

Goal state after:
Module: Mathlib.Data.List.Iterate

Goal state before:
pp: α : Type u_1
f : α → α
a : α
n : ℕ
⊢ (iterate f a n).length = n

Tactic:
by

Goal state after:
Module: Mathlib.Data.List.Iterate

Goal state before:
pp: α : Type u_1
f : α → α
a : α
n : ℕ
⊢ iterate f a n = [] ↔ n = 0

Tactic:
by

Goal state after:
Module: Mathlib.Data.List.Iterate

Goal state before:
pp: α : Type u_1
f : α → α
a : α
n : ℕ
x✝ : 0 < n + 1
⊢ (iterate f a (n + 1))[0]? = some (f^[0] a)

Tactic:
by

Goal state after:
Module: Mathlib.Data.List.Iterate

Goal state before:
pp: α : Type u_1
f : α → α
a : α
n i : ℕ
h : i + 1 < n + 1
⊢ i < n

Tactic:
by

Goal state after:
Module: Mathlib.Data.List.Iterate

Goal state before:
pp: α : Type u_1
f : α → α
a : α
n i : ℕ
h : i + 1 < n + 1
⊢ (iterate f a (n + 1))[i + 1]? = some (f^[i + 1] a)

Tactic:
by

Goal state after:
Module: Mathlib.Data.List.Iterate

Goal state before:
pp: α : Type u_1
f : α → α
a : α
n i : ℕ
h : i < (iterate f a n).length
⊢ i < n

Tactic:
by

Goal state after:
Module: Mathlib.Data.List.Iterate

Goal state before:
pp: α : Type u_1
f : α → α
a : α
n : ℕ
b : α
⊢ b ∈ iterate f a n ↔ ∃ m, m < n ∧ b = f^[m] a

Tactic:
by

Goal state after:
Module: Mathlib.Data.List.Iterate

Goal state before:
pp: α : Type u_1
n : ℕ
f : α → α
a : α
⊢ map (fun x ↦ f^[x] a) (range n) = iterate f a n

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝¹ : DecidableEq α
β : Type v
inst✝ : Finite α
f : Perm α
s : Multiset α
l₁ l₂ : List α
h : l₁ ≈ l₂
⊢ (fun x ↦ signAux2 l₁ f) ≍ fun x ↦ signAux2 l₂ f

Tactic:
by

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

Goal state before:
pp: α : Type u_4
inst✝ : DecidableEq α
n : ℕ
b x : α
f : Perm α
h : (f ^ Int.negSucc n) (f x) = b
hfxb : f x ≠ b
hfb : f b ≠ b
hbx : b ≠ x
hb : (swap x ((Equiv.symm f) x) * f⁻¹) ((Equiv.symm f) b) ≠ (Equiv.symm f) b
i : ℤ
hi : ((swap x ((Equiv.symm f) x) * Equiv.symm f) ^ i) ((Equiv.symm f) x) = (Equiv.symm f) b
⊢ f⁻¹ b ≠ f⁻¹ x

Tactic:
by

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

Goal state before:
pp: α : Type u_4
inst✝ : DecidableEq α
n : ℕ
b x : α
f : Perm α
hb : (swap x (f x) * f) b ≠ b
h : (f ^ Int.negSucc n) (f x) = b
⊢ ∃ i, ((swap x (f x) * f) ^ i) (f x) = b

Tactic:
by

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

Goal state before:
pp: α : Type u_4
inst✝ : DecidableEq α
f : Perm α
hf : f.IsCycle
x : α
hfx : f x ≠ x
hffx : f (f x) = x
y z : α
hz : f z ≠ z ∧ ∀ ⦃y : α⦄, f y ≠ y → f.SameCycle z y
i : ℤ
hi : (f ^ i) z = x
hyx : y = x
⊢ f y = (swap x (f x)) y

Tactic:
by

Goal state after:
Module: Mathlib.Data.Set.Pairwise.List

Goal state before:
pp: α : Type u_1
r : α → α → Prop
l : List α
inst✝ : Std.Symm r
hl : l.Nodup
⊢ {a | a ∈ l}.Pairwise r ↔ Pairwise r l

Tactic:
by

Goal state after:
Module: Mathlib.Data.Nat.GCD.BigOperators

Goal state before:
pp: l : List ℕ
k : ℕ
⊢ l.prod.Coprime k ↔ ∀ n ∈ l, n.Coprime k

Tactic:
by

Goal state after:
Module: Mathlib.Data.Nat.GCD.BigOperators

Goal state before:
pp: k : ℕ
l : List ℕ
⊢ k.Coprime l.prod ↔ ∀ n ∈ l, k.Coprime n

Tactic:
by

Goal state after:
Module: Mathlib.Data.Nat.GCD.BigOperators

Goal state before:
pp: m : Multiset ℕ
k : ℕ
⊢ m.prod.Coprime k ↔ ∀ n ∈ m, n.Coprime k

Tactic:
by

Goal state after:
Module: Mathlib.Data.Nat.GCD.BigOperators

Goal state before:
pp: k : ℕ
m : Multiset ℕ
⊢ k.Coprime m.prod ↔ ∀ n ∈ m, k.Coprime n

Tactic:
by

Goal state after:
Module: Mathlib.Data.Nat.GCD.BigOperators

Goal state before:
pp: ι : Type u_1
t : Finset ι
s : ι → ℕ
x : ℕ
⊢ (∏ i ∈ t, s i).Coprime x ↔ ∀ i ∈ t, (s i).Coprime x

Tactic:
by

Goal state after:
Module: Mathlib.Data.Nat.GCD.BigOperators

Goal state before:
pp: ι : Type u_1
x : ℕ
t : Finset ι
s : ι → ℕ
⊢ x.Coprime (∏ i ∈ t, s i) ↔ ∀ i ∈ t, x.Coprime (s i)

Tactic:
by

Goal state after:
Module: Mathlib.Data.Nat.GCD.BigOperators

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
s : ι → ℕ
x : ℕ
⊢ (∏ i, s i).Coprime x ↔ ∀ (i : ι), (s i).Coprime x

Tactic:
by

Goal state after:
Module: Mathlib.Data.Nat.GCD.BigOperators

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
x : ℕ
s : ι → ℕ
⊢ x.Coprime (∏ i, s i) ↔ ∀ (i : ι), x.Coprime (s i)

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝¹ : DecidableEq α
inst✝ : Finite α
f g : Perm α
n : ℕ
e : α ≃ Fin n
l : List α
hs : ∀ (x : α), x ∈ ⟦l⟧
h : Fin n
⊢ ((e.symm.trans (f * g)).trans e) h = ((e.symm.trans f).trans e * (e.symm.trans g).trans e) h

Tactic:
by

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

Goal state before:
pp: α : Type u_4
inst✝ : DecidableEq α
f : Perm α
hf : f.IsCycle
x : α
hfx : f x ≠ x
hffx : f (f x) = x
y z : α
hz : f z ≠ z ∧ ∀ ⦃y : α⦄, f y ≠ y → f.SameCycle z y
i : ℤ
hi : (f ^ i) z = x
hyx : ¬y = x
hfyx : y = f x
⊢ f y = (swap x (f x)) y

Tactic:
by

Goal state after:
Module: Mathlib.Data.List.Iterate

Goal state before:
pp: case succ
α : Type u_1
f : α → α
n✝ n : ℕ
ih : ∀ (a : α), iterate f a (n + n✝) = iterate f a n ++ iterate f (f^[n] a) n✝
a : α
⊢ iterate f a (n + 1 + n✝) = iterate f a (n + 1) ++ iterate f (f^[n + 1] a) n✝

Tactic:
rw [iterate, add_right_comm, iterate, ih, Nat.iterate, cons_append]

Goal state after:
Module: Mathlib.Data.List.Iterate

Goal state before:
pp: α : Type u_1
f : α → α
a : α
m n : ℕ
⊢ iterate f a (m + n) = iterate f a m ++ iterate f (f^[m] a) n

Tactic:
by

Goal state after:
Module: Mathlib.Data.List.Iterate

Goal state before:
pp: α : Type u_1
f : α → α
a : α
m n : ℕ
⊢ take m (iterate f a n) = iterate f a (min m n)

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝¹ : DecidableEq α
inst✝ : Finite α
f g : Perm α
s : Multiset α
hs : ∀ (x : α), x ∈ s
⊢ (f * g).signAux3 hs = f.signAux3 hs * g.signAux3 hs ∧ Pairwise fun x y ↦ (swap x y).signAux3 hs = -1

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝³ : DecidableEq α
β : Type v
inst✝² : Finite α
inst✝¹ : DecidableEq β
inst✝ : Finite β
f : Perm α
e : α ≃ β
s : Multiset α
t : Multiset β
hs : ∀ (x : α), x ∈ s
ht : ∀ (x : β), x ∈ t
⊢ signAux3 ((e.symm.trans f).trans e) ht = f.signAux3 hs

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
⊢ sign (Equiv.trans f g) = sign g * sign f

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f : Perm α
⊢ sign f⁻¹ = sign f

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
x y : α
H : x = y
⊢ sign (swap x y) = if x = y then 1 else -1

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
x y : α
H : ¬x = y
⊢ sign (swap x y) = if x = y then 1 else -1

Tactic:
by

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

Goal state before:
pp: α : Type u_4
inst✝ : DecidableEq α
f : Perm α
hf : f.IsCycle
x : α
hfx : f x ≠ x
hffx : f (f x) = x
y z : α
hz : f z ≠ z ∧ ∀ ⦃y : α⦄, f y ≠ y → f.SameCycle z y
i : ℤ
hi : (f ^ i) z = x
hyx : ¬y = x
hfyx : ¬y = f x
⊢ f y = (swap x (f x)) y

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
l : List (Perm α)
hl : ∀ g ∈ l, g.IsSwap
⊢ (List.map (⇑sign) l).length = l.length

Tactic:
by

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

Goal state before:
pp: α : Type u_4
inst✝ : DecidableEq α
f : Perm α
hf : f.IsCycle
x : α
hx : f x ≠ x
hffx : f (f x) ≠ x
i : ℤ
hy : (swap x (f x) * f) ((f ^ i) x) ≠ (f ^ i) x
⊢ (f ^ (i - 1)) (f x) = (f ^ i) x

Tactic:
by

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

Goal state before:
pp: α : Type u_4
inst✝ : DecidableEq α
f : Perm α
hf : f.IsCycle
x : α
hx : f x ≠ x
hffx : f (f x) ≠ x
⊢ (swap x (f x) * f).IsCycle

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f : Perm α
hf : f.IsCycle
x : α
hx : f x ≠ x ∧ ∀ ⦃y : α⦄, f y ≠ y → f.SameCycle x y
⊢ Perm.sign f = Perm.sign (swap x (f x) * (swap x (f x) * f))

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
l : List (Perm α)
hl : ∀ g ∈ l, g.IsSwap
⊢ sign l.prod = (-1) ^ l.length

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f : Perm α
hf : f.IsCycle
x : α
hx : f x ≠ x ∧ ∀ ⦃y : α⦄, f y ≠ y → f.SameCycle x y
h1 : f (f x) = x
⊢ swap x (f x) * f = 1

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f : Perm α
⊢ |↑(sign f)| = 1

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝² : DecidableEq α
inst✝¹ : Fintype α
inst✝ : Nontrivial α
a : ℤˣ
h : a = 1
⊢ sign 1 = a

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝² : DecidableEq α
inst✝¹ : Fintype α
inst✝ : Nontrivial α
a : ℤˣ
h : a = -1
x y : α
hxy : x ≠ y
⊢ sign (swap x y) = a

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Perm α →* ℤˣ
hs : Surjective ⇑s
f✝ : Perm α
x✝¹ : f✝.IsSwap
x y : α
hxy : x ≠ y
hxy' : f✝ = swap x y
h : ¬s (swap x y) = -1
f : Perm α
x✝ : f.IsSwap
a b : α
hab : a ≠ b
hab' : f = swap a b
⊢ s f = 1

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f : Perm α
hf : f.IsCycle
x : α
hx : f x ≠ x ∧ ∀ ⦃y : α⦄, f y ≠ y → f.SameCycle x y
h1 : f (f x) = x
⊢ Perm.sign (swap x (f x) * (swap x (f x) * f)) = -(-1) ^ #f.support

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f : Perm α
hf : f.IsCycle
x : α
hx : f x ≠ x ∧ ∀ ⦃y : α⦄, f y ≠ y → f.SameCycle x y
h1 : ¬f (f x) = x
⊢ #(swap x (f x) * f).support + 1 = #f.support

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f : Perm α
hf : f.IsCycle
x : α
hx : f x ≠ x ∧ ∀ ⦃y : α⦄, f y ≠ y → f.SameCycle x y
h1 : ¬f (f x) = x
⊢ Perm.sign (swap x (f x) * (swap x (f x) * f)) = -(-1) ^ #f.support

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f : Perm α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
n : ℕ
h1 : (f ^ n).IsCycle
h2 : f.support ⊆ (f ^ n).support
⊢ ∀ (x : α), (f ^ n) x ≠ x ↔ f x ≠ x

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Perm α →* ℤˣ
hs : Surjective ⇑s
f : Perm α
x✝ : f.IsSwap
x y : α
hxy : x ≠ y
hxy' : f = swap x y
h : ¬s (swap x y) = -1
this✝ : ∀ (f : Perm α), f.IsSwap → s f = 1
g : Perm α
hg : s g = -1
l : List (Perm α)
hl : l.prod = g ∧ ∀ g ∈ l, g.IsSwap
this : ∀ a ∈ List.map (⇑s) l, a = 1
⊢ s l.prod = 1

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f : Perm α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
n : ℕ
h2 : f.support ⊆ (f ^ n).support
key : ∀ (x : α), (f ^ n) x ≠ x ↔ f x ≠ x
x : α
hx1 : (f ^ n) x ≠ x
hx2 : ∀ ⦃y : α⦄, (f ^ n) y ≠ y → (f ^ n).SameCycle x y
y : α
hy : f y ≠ y
i : ℤ
h✝ : ((f ^ n) ^ i) x = y
⊢ (f ^ (↑n * i)) x = y

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f : Perm α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
n : ℕ
h1 : (f ^ n).IsCycle
h2 : f.support ⊆ (f ^ n).support
⊢ f.IsCycle

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Perm α →* ℤˣ
hs : Surjective ⇑s
f : Perm α
x✝ : f.IsSwap
x y : α
hxy : x ≠ y
hxy' : f = swap x y
h : ¬s (swap x y) = -1
this✝¹ : ∀ (f : Perm α), f.IsSwap → s f = 1
g : Perm α
hg : s g = -1
l : List (Perm α)
hl : l.prod = g ∧ ∀ g ∈ l, g.IsSwap
this✝ : ∀ a ∈ List.map (⇑s) l, a = 1
this : -1 = 1
⊢ ¬-1 = 1

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Perm α →* ℤˣ
hs : Surjective ⇑s
f : Perm α
x✝ : f.IsSwap
x y : α
hxy : x ≠ y
hxy' : f = swap x y
h : ¬s (swap x y) = -1
⊢ False

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f : Perm α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
n : ℤ
h1 : (f ^ n).IsCycle
h2 : f.support ⊆ (f ^ n).support
⊢ f.IsCycle

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Perm α →* ℤˣ
hs : Surjective ⇑s
this : ∀ {f : Perm α}, f.IsSwap → s f = -1
f : Perm α
⊢ s f = sign f

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f g : Perm α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hf : f.IsCycle
hg : g.IsCycle
h : f.support ⊆ g.support
h' : ∀ x ∈ f.support, f x = g x
z : α
hz : z ∈ g.support
x : α
hx : f x ≠ x
right✝ : ∀ ⦃y : α⦄, f y ≠ y → f.SameCycle x y
⊢ g x ≠ x

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f g : Perm α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hf : f.IsCycle
hg : g.IsCycle
h : f.support ⊆ g.support
h' : ∀ x ∈ f.support, f x = g x
z : α
hz : z ∈ g.support
x : α
hx : f x ≠ x
right✝ : ∀ ⦃y : α⦄, f y ≠ y → f.SameCycle x y
hx' : g x ≠ x
m : ℕ
hm : (g ^ m) x = z
⊢ ∀ x ∈ f.support ∩ g.support, f x = g x

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝² : DecidableEq α
inst✝¹ : Fintype α
f : Perm α
p : α → Prop
inst✝ : DecidablePred p
h₁ : ∀ (x : α), p (f x) ↔ p x
h₂ : ∀ (x : α), f x ≠ x → p x
l : { l // l.prod = f.subtypePerm h₁ ∧ ∀ g ∈ l, g.IsSwap } := (f.subtypePerm h₁).truncSwapFactors.out
hl' : ∀ g' ∈ List.map ⇑ofSubtype ↑l, g'.IsSwap
⊢ (List.map ⇑ofSubtype ↑l).prod = f

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f g : Perm α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hf : f.IsCycle
hg : g.IsCycle
h : f.support ⊆ g.support
h' : ∀ x ∈ f.support, f x = g x
⊢ f.support = g.support

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f g : Perm α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hf : f.IsCycle
hg : g.IsCycle
h : f.support ⊆ g.support
h' : ∀ x ∈ f.support, f x = g x
⊢ f = g

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f g : Perm α
x : α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hf : f.IsCycle
hg : g.IsCycle
h : ∀ x ∈ f.support ∩ g.support, f x = g x
hx : f x = g x
hx' : x ∈ f.support
⊢ x ∈ g.support

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f g : Perm α
x : α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hf : f.IsCycle
hg : g.IsCycle
h : ∀ x ∈ f.support ∩ g.support, f x = g x
hx : f x = g x
hx' : x ∈ f.support
hx'' : x ∈ g.support
⊢ f.support ⊆ g.support

Tactic:
by

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

Goal state before:
pp: α : Type u_2
f g : Perm α
x : α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hf : f.IsCycle
hg : g.IsCycle
h : ∀ x ∈ f.support ∩ g.support, f x = g x
hx : f x = g x
hx' : x ∈ f.support
⊢ f = g

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝² : DecidableEq α
inst✝¹ : Fintype α
f : Perm α
p : α → Prop
inst✝ : DecidablePred p
h₁ : ∀ (x : α), p (f x) ↔ p x
h₂ : ∀ (x : α), f x ≠ x → p x
⊢ sign (f.subtypePerm h₁) = sign f

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝³ : DecidableEq α
β : Type v
inst✝² : Fintype α
inst✝¹ : DecidableEq β
inst✝ : Fintype β
f : Perm α
g : Perm β
e : α ≃ β
h : ∀ (x : α), e (f x) = g (e x)
⊢ ∀ (x : β), g x = ((e.symm.trans f).trans e) x

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝³ : DecidableEq α
β : Type v
inst✝² : Fintype α
inst✝¹ : DecidableEq β
inst✝ : Fintype β
f : Perm α
g : Perm β
e : α ≃ β
h : ∀ (x : α), e (f x) = g (e x)
⊢ sign f = sign g

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝³ : DecidableEq α
β : Type v
inst✝² : Fintype α
inst✝¹ : DecidableEq β
inst✝ : Fintype β
f : Perm α
g : Perm β
i : (x : α) → f x ≠ x → β
h : ∀ (x : α) (hx : f x ≠ x) (hx' : f (f x) ≠ f x), i (f x) hx' = g (i x hx)
hi : ∀ (x₁ x₂ : α) (hx₁ : f x₁ ≠ x₁) (hx₂ : f x₂ ≠ x₂), i x₁ hx₁ = i x₂ hx₂ → x₁ = x₂
hg : ∀ (y : β), g y ≠ y → ∃ x, ∃ (hx : f x ≠ x), i x hx = y
⊢ ∀ (x : α), f (f x) ≠ f x ↔ f x ≠ x

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝³ : DecidableEq α
β : Type v
inst✝² : Fintype α
inst✝¹ : DecidableEq β
inst✝ : Fintype β
f : Perm α
g : Perm β
i : (x : α) → f x ≠ x → β
h : ∀ (x : α) (hx : f x ≠ x) (hx' : f (f x) ≠ f x), i (f x) hx' = g (i x hx)
hi : ∀ (x₁ x₂ : α) (hx₁ : f x₁ ≠ x₁) (hx₂ : f x₂ ≠ x₂), i x₁ hx₁ = i x₂ hx₂ → x₁ = x₂
hg : ∀ (y : β), g y ≠ y → ∃ x, ∃ (hx : f x ≠ x), i x hx = y
⊢ ∀ (x : β), g (g x) ≠ g x ↔ g x ≠ x

Tactic:
by

Goal state after: