-- Mathlib/GroupTheory/Perm/ClosureSwap.lean
theorem mem_closure_isSwap {S : Set (Perm α)} (hS : ∀ f ∈ S, f.IsSwap) {f : Perm α} :
    f ∈ closure S ↔ (fixedBy α f)ᶜ.Finite ∧ ∀ x, f x ∈ orbit (closure S) x

Proof state:
case refine_2
α : Type u_2
inst✝ : DecidableEq α
S : Set (Equiv.Perm α)
hS : ∀ f ∈ S, f.IsSwap
f : Equiv.Perm α
⊢ ((fixedBy α f)ᶜ.Finite ∧ ∀ (x : α), f x ∈ orbit (↥(closure S)) x) → f ∈ closure S

Tactic:
rintro ⟨fin, hf⟩

Result:
case refine_2.intro
α : Type u_2
inst✝ : DecidableEq α
S : Set (Equiv.Perm α)
hS : ∀ f ∈ S, f.IsSwap
f : Equiv.Perm α
fin : (fixedBy α f)ᶜ.Finite
hf : ∀ (x : α), f x ∈ orbit (↥(closure S)) x
⊢ f ∈ closure S

Proof state:
α : Type u_2
inst✝ : DecidableEq α
S : Set (Equiv.Perm α)
hS : ∀ f ∈ S, f.IsSwap
supp : Set α
fin : supp.Finite
⊢ ∀ {a : α} {s : Set α},
    a ∉ s →
      s.Finite →
        (∀ {f : Equiv.Perm α}, (∀ (x : α), f x ∈ orbit (↥(closure S)) x) → (fixedBy α f)ᶜ ⊆ s → f ∈ closure S) →
          ∀ {f : Equiv.Perm α}, (∀ (x : α), f x ∈ orbit (↥(closure S)) x) → (fixedBy α f)ᶜ ⊆ insert a s → f ∈ closure S

Tactic:
rintro a s - - ih f hf supp_subset

Result:
α : Type u_2
inst✝ : DecidableEq α
S : Set (Equiv.Perm α)
hS : ∀ f ∈ S, f.IsSwap
supp : Set α
fin : supp.Finite
a : α
s : Set α
ih : ∀ {f : Equiv.Perm α}, (∀ (x : α), f x ∈ orbit (↥(closure S)) x) → (fixedBy α f)ᶜ ⊆ s → f ∈ closure S
f : Equiv.Perm α
hf : ∀ (x : α), f x ∈ orbit (↥(closure S)) x
supp_subset : (fixedBy α f)ᶜ ⊆ insert a s
⊢ f ∈ closure S

Proof state:
α : Type u_2
inst✝ : DecidableEq α
S : Set (Equiv.Perm α)
hS : ∀ f ∈ S, f.IsSwap
f : Equiv.Perm α
⊢ f ∈ closure S ↔ (fixedBy α f)ᶜ.Finite ∧ ∀ (x : α), f x ∈ orbit (↥(closure S)) x

Tactic:
refine ⟨fun hf ↦ ⟨?_, fun x ↦ mem_orbit_iff.mpr ⟨⟨f, hf⟩, rfl⟩⟩, ?_⟩

Result:
case refine_1
α : Type u_2
inst✝ : DecidableEq α
S : Set (Equiv.Perm α)
hS : ∀ f ∈ S, f.IsSwap
f : Equiv.Perm α
hf : f ∈ closure S
⊢ (fixedBy α f)ᶜ.Finite

case refine_2
α : Type u_2
inst✝ : DecidableEq α
S : Set (Equiv.Perm α)
hS : ∀ f ∈ S, f.IsSwap
f : Equiv.Perm α
⊢ ((fixedBy α f)ᶜ.Finite ∧ ∀ (x : α), f x ∈ orbit (↥(closure S)) x) → f ∈ closure S

Proof state:
α : Type u_2
inst✝ : DecidableEq α
S : Set (Equiv.Perm α)
hS : ∀ f ∈ S, f.IsSwap
supp : Set α
fin : supp.Finite
a : α
s : Set α
ih : ∀ {f : Equiv.Perm α}, (∀ (x : α), f x ∈ orbit (↥(closure S)) x) → (fixedBy α f)ᶜ ⊆ s → f ∈ closure S
f : Equiv.Perm α
hf : ∀ (x : α), f x ∈ orbit (↥(closure S)) x
supp_subset : (fixedBy α f)ᶜ ⊆ insert a s
⊢ f ∈ closure S

Tactic:
refine (mul_mem_cancel_left ((swap_mem_closure_isSwap hS).2 (hf a))).1
  (ih (fun b ↦ ?_) fun b hb ↦ ?_)

Result:
case refine_1
α : Type u_2
inst✝ : DecidableEq α
S : Set (Equiv.Perm α)
hS : ∀ f ∈ S, f.IsSwap
supp : Set α
fin : supp.Finite
a : α
s : Set α
ih : ∀ {f : Equiv.Perm α}, (∀ (x : α), f x ∈ orbit (↥(closure S)) x) → (fixedBy α f)ᶜ ⊆ s → f ∈ closure S
f : Equiv.Perm α
hf : ∀ (x : α), f x ∈ orbit (↥(closure S)) x
supp_subset : (fixedBy α f)ᶜ ⊆ insert a s
b : α
⊢ (swap (f a) a * f) b ∈ orbit (↥(closure S)) b

case refine_2
α : Type u_2
inst✝ : DecidableEq α
S : Set (Equiv.Perm α)
hS : ∀ f ∈ S, f.IsSwap
supp : Set α
fin : supp.Finite
a : α
s : Set α
ih : ∀ {f : Equiv.Perm α}, (∀ (x : α), f x ∈ orbit (↥(closure S)) x) → (fixedBy α f)ᶜ ⊆ s → f ∈ closure S
f : Equiv.Perm α
hf : ∀ (x : α), f x ∈ orbit (↥(closure S)) x
supp_subset : (fixedBy α f)ᶜ ⊆ insert a s
b : α
hb : b ∈ (fixedBy α (swap (f a) a * f))ᶜ
⊢ b ∈ s

Proof state:
case refine_1
α : Type u_2
inst✝ : DecidableEq α
S : Set (Equiv.Perm α)
hS : ∀ f ∈ S, f.IsSwap
supp : Set α
fin : supp.Finite
a : α
s : Set α
ih : ∀ {f : Equiv.Perm α}, (∀ (x : α), f x ∈ orbit (↥(closure S)) x) → (fixedBy α f)ᶜ ⊆ s → f ∈ closure S
f : Equiv.Perm α
hf : ∀ (x : α), f x ∈ orbit (↥(closure S)) x
supp_subset : (fixedBy α f)ᶜ ⊆ insert a s
b : α
⊢ (swap (f a) a * f) b ∈ orbit (↥(closure S)) b

Tactic:
rw [Perm.mul_apply, swap_apply_def]

Result:
case refine_1
α : Type u_2
inst✝ : DecidableEq α
S : Set (Equiv.Perm α)
hS : ∀ f ∈ S, f.IsSwap
supp : Set α
fin : supp.Finite
a : α
s : Set α
ih : ∀ {f : Equiv.Perm α}, (∀ (x : α), f x ∈ orbit (↥(closure S)) x) → (fixedBy α f)ᶜ ⊆ s → f ∈ closure S
f : Equiv.Perm α
hf : ∀ (x : α), f x ∈ orbit (↥(closure S)) x
supp_subset : (fixedBy α f)ᶜ ⊆ insert a s
b : α
⊢ (if f b = f a then a else if f b = a then f a else f b) ∈ orbit (↥(closure S)) b

Proof state:
case refine_2.intro
α : Type u_2
inst✝ : DecidableEq α
S : Set (Equiv.Perm α)
hS : ∀ f ∈ S, f.IsSwap
f : Equiv.Perm α
hf : ∀ (x : α), f x ∈ orbit (↥(closure S)) x
supp : Set α := (fixedBy α f)ᶜ
fin : supp.Finite
supp_eq : supp = (fixedBy α f)ᶜ
⊢ f ∈ closure S

Tactic:
suffices h : (fixedBy α f)ᶜ ⊆ supp → f ∈ closure S from h supp_eq.symm.subset

Result:
case refine_2.intro
α : Type u_2
inst✝ : DecidableEq α
S : Set (Equiv.Perm α)
hS : ∀ f ∈ S, f.IsSwap
f : Equiv.Perm α
hf : ∀ (x : α), f x ∈ orbit (↥(closure S)) x
supp : Set α := (fixedBy α f)ᶜ
fin : supp.Finite
supp_eq : supp = (fixedBy α f)ᶜ
⊢ (fixedBy α f)ᶜ ⊆ supp → f ∈ closure S

Proof state:
case h.e'_5
α : Type u_2
inst✝ : DecidableEq α
S : Set (Equiv.Perm α)
hS : ∀ f ∈ S, f.IsSwap
supp : Set α
fin : supp.Finite
f : Equiv.Perm α
emp : (fixedBy α f)ᶜ ⊆ ∅
⊢ f = 1

Tactic:
ext

Result:
case h.e'_5.H
α : Type u_2
inst✝ : DecidableEq α
S : Set (Equiv.Perm α)
hS : ∀ f ∈ S, f.IsSwap
supp : Set α
fin : supp.Finite
f : Equiv.Perm α
emp : (fixedBy α f)ᶜ ⊆ ∅
x✝ : α
⊢ f x✝ = 1 x✝

Proof state:
case neg
α : Type u_2
inst✝ : DecidableEq α
S : Set (Equiv.Perm α)
hS : ∀ f ∈ S, f.IsSwap
supp : Set α
fin : supp.Finite
a : α
s : Set α
ih : ∀ {f : Equiv.Perm α}, (∀ (x : α), f x ∈ orbit (↥(closure S)) x) → (fixedBy α f)ᶜ ⊆ s → f ∈ closure S
f : Equiv.Perm α
hf : ∀ (x : α), f x ∈ orbit (↥(closure S)) x
supp_subset : (fixedBy α f)ᶜ ⊆ insert a s
b : α
hb : b ∉ s
hb' : ¬f b = b
⊢ (if b = a then a else if f b = a then f a else f b) = b

Tactic:
simp [show b = a by simpa [hb] using supp_subset hb']

Result:
no goals

Proof state:
case refine_1
α : Type u_2
inst✝ : DecidableEq α
S : Set (Equiv.Perm α)
hS : ∀ f ∈ S, f.IsSwap
supp : Set α
fin : supp.Finite
a : α
s : Set α
ih : ∀ {f : Equiv.Perm α}, (∀ (x : α), f x ∈ orbit (↥(closure S)) x) → (fixedBy α f)ᶜ ⊆ s → f ∈ closure S
f : Equiv.Perm α
hf : ∀ (x : α), f x ∈ orbit (↥(closure S)) x
supp_subset : (fixedBy α f)ᶜ ⊆ insert a s
b : α
⊢ (if f b = f a then a else if f b = a then f a else f b) ∈ orbit (↥(closure S)) b

Tactic:
split_ifs with h1 h2

Result:
case pos
α : Type u_2
inst✝ : DecidableEq α
S : Set (Equiv.Perm α)
hS : ∀ f ∈ S, f.IsSwap
supp : Set α
fin : supp.Finite
a : α
s : Set α
ih : ∀ {f : Equiv.Perm α}, (∀ (x : α), f x ∈ orbit (↥(closure S)) x) → (fixedBy α f)ᶜ ⊆ s → f ∈ closure S
f : Equiv.Perm α
hf : ∀ (x : α), f x ∈ orbit (↥(closure S)) x
supp_subset : (fixedBy α f)ᶜ ⊆ insert a s
b : α
h1 : f b = f a
⊢ a ∈ orbit (↥(closure S)) b

case pos
α : Type u_2
inst✝ : DecidableEq α
S : Set (Equiv.Perm α)
hS : ∀ f ∈ S, f.IsSwap
supp : Set α
fin : supp.Finite
a : α
s : Set α
ih : ∀ {f : Equiv.Perm α}, (∀ (x : α), f x ∈ orbit (↥(closure S)) x) → (fixedBy α f)ᶜ ⊆ s → f ∈ closure S
f : Equiv.Perm α
hf : ∀ (x : α), f x ∈ orbit (↥(closure S)) x
supp_subset : (fixedBy α f)ᶜ ⊆ insert a s
b : α
h1 : ¬f b = f a
h2 : f b = a
⊢ f a ∈ orbit (↥(closure S)) b

case neg
α : Type u_2
inst✝ : DecidableEq α
S : Set (Equiv.Perm α)
hS : ∀ f ∈ S, f.IsSwap
supp : Set α
fin : supp.Finite
a : α
s : Set α
ih : ∀ {f : Equiv.Perm α}, (∀ (x : α), f x ∈ orbit (↥(closure S)) x) → (fixedBy α f)ᶜ ⊆ s → f ∈ closure S
f : Equiv.Perm α
hf : ∀ (x : α), f x ∈ orbit (↥(closure S)) x
supp_subset : (fixedBy α f)ᶜ ⊆ insert a s
b : α
h1 : ¬f b = f a
h2 : ¬f b = a
⊢ f b ∈ orbit (↥(closure S)) b