Physlib.Mathematics.Fin.Involutions

Physlib
Mathematics
Fin
Involutions

Imports

/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
module

public import Mathlib.Data.Finset.Sort
public import Mathlib.Data.Fintype.Pi
public import Mathlib.Data.Fintype.Prod
public import Mathlib.Data.Nat.Factorial.DoubleFactorial

Fin involutions

Some properties of involutions of Fin n.

These involutions are used in e.g. proving results about Wick contractions.

@[expose] public section

There is an equivalence between involutions of Fin n.succ and involutions of Fin n and an optional valid choice of an element in Fin n (which is where 0 in Fin n.succ will be sent).


def involutionCons (n : ℕ) : {f : Fin n.succ → Fin n.succ // Function.Involutive f } ≃
    (f : {f : Fin n → Fin n // Function.Involutive f}) × {i : Option (Fin n) //
      ∀ (h : i.isSome), f.1 (Option.get i h) = (Option.get i h)} where
  toFun f := ⟨⟨
    fun i =>
    if h : f.1 i.succ = 0 then i
    else Fin.pred (f.1 i.succ) h, byn:ℕf:{ f // Function.Involutive f }⊢ Function.Involutive fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h
    intro in:ℕf:{ f // Function.Involutive f }i:Fin n⊢ (fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h)
    ((fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h) i) =
  i
    by_cases h : f.1 i.succ = 0posn:ℕf:{ f // Function.Involutive f }i:Fin nh:↑f i.succ = 0⊢ (fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h)
    ((fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h) i) =
  inegn:ℕf:{ f // Function.Involutive f }i:Fin nh:¬↑f i.succ = 0⊢ (fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h)
    ((fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h) i) =
  i
    ·posn:ℕf:{ f // Function.Involutive f }i:Fin nh:↑f i.succ = 0⊢ (fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h)
    ((fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h) i) =
  i simp [h]All goals completed! 🐙
    ·negn:ℕf:{ f // Function.Involutive f }i:Fin nh:¬↑f i.succ = 0⊢ (fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h)
    ((fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h) i) =
  i simp only [succ_eq_add_one, h, ↓reduceDIte, Fin.succ_pred, f.2 i.succ, Fin.pred_succ,
        dite_eq_ite, ite_eq_right_iff]negn:ℕf:{ f // Function.Involutive f }i:Fin nh:¬↑f i.succ = 0⊢ i.succ = 0 → (↑f i.succ).pred ⋯ = i
      exact fun h => False.elim (Fin.succ_ne_zero i h)All goals completed! 🐙⟩,
    ⟨if h : f.1 0 = 0 then none else Fin.pred (f.1 0) h, byn:ℕf:{ f // Function.Involutive f }⊢ ∀ (h : (if h : ↑f 0 = 0 then none else some ((↑f 0).pred h)).isSome = true),
  ↑⟨fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h, ⋯⟩
      ((if h : ↑f 0 = 0 then none else some ((↑f 0).pred h)).get h) =
    (if h : ↑f 0 = 0 then none else some ((↑f 0).pred h)).get h
    by_cases h0 : f.1 0 = 0posn:ℕf:{ f // Function.Involutive f }h0:↑f 0 = 0⊢ ∀ (h : (if h : ↑f 0 = 0 then none else some ((↑f 0).pred h)).isSome = true),
  ↑⟨fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h, ⋯⟩
      ((if h : ↑f 0 = 0 then none else some ((↑f 0).pred h)).get h) =
    (if h : ↑f 0 = 0 then none else some ((↑f 0).pred h)).get hnegn:ℕf:{ f // Function.Involutive f }h0:¬↑f 0 = 0⊢ ∀ (h : (if h : ↑f 0 = 0 then none else some ((↑f 0).pred h)).isSome = true),
  ↑⟨fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h, ⋯⟩
      ((if h : ↑f 0 = 0 then none else some ((↑f 0).pred h)).get h) =
    (if h : ↑f 0 = 0 then none else some ((↑f 0).pred h)).get h
    ·posn:ℕf:{ f // Function.Involutive f }h0:↑f 0 = 0⊢ ∀ (h : (if h : ↑f 0 = 0 then none else some ((↑f 0).pred h)).isSome = true),
  ↑⟨fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h, ⋯⟩
      ((if h : ↑f 0 = 0 then none else some ((↑f 0).pred h)).get h) =
    (if h : ↑f 0 = 0 then none else some ((↑f 0).pred h)).get h simp [h0]All goals completed! 🐙
    ·negn:ℕf:{ f // Function.Involutive f }h0:¬↑f 0 = 0⊢ ∀ (h : (if h : ↑f 0 = 0 then none else some ((↑f 0).pred h)).isSome = true),
  ↑⟨fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h, ⋯⟩
      ((if h : ↑f 0 = 0 then none else some ((↑f 0).pred h)).get h) =
    (if h : ↑f 0 = 0 then none else some ((↑f 0).pred h)).get h simp only [succ_eq_add_one, h0, ↓reduceDIte, Option.isSome_some, Option.get_some,
      Fin.succ_pred, dite_eq_left_iff, Fin.pred_inj, forall_const]negn:ℕf:{ f // Function.Involutive f }h0:¬↑f 0 = 0⊢ ¬↑f (↑f 0) = 0 → ↑f (↑f 0) = ↑f 0
      refine fun h => False.elim (h (f.2 0))All goals completed! 🐙⟩⟩
  invFun f := ⟨
      if h : (f.2.1).isSome then
        Fin.cons (f.2.1.get h).succ (Function.update (Fin.succ ∘ f.1.1) (f.2.1.get h) 0)
      else
        Fin.cons 0 (Fin.succ ∘ f.1.1), byn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }⊢ Function.Involutive
  (if h : (↑f.snd).isSome = true then
    Fin.cons ((↑f.snd).get h).succ (Function.update (Fin.succ ∘ ↑f.fst) ((↑f.snd).get h) 0)
  else Fin.cons 0 (Fin.succ ∘ ↑f.fst))
    by_cases hs : (f.2.1).isSomeposn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = true⊢ Function.Involutive
  (if h : (↑f.snd).isSome = true then
    Fin.cons ((↑f.snd).get h).succ (Function.update (Fin.succ ∘ ↑f.fst) ((↑f.snd).get h) 0)
  else Fin.cons 0 (Fin.succ ∘ ↑f.fst))negn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:¬(↑f.snd).isSome = true⊢ Function.Involutive
  (if h : (↑f.snd).isSome = true then
    Fin.cons ((↑f.snd).get h).succ (Function.update (Fin.succ ∘ ↑f.fst) ((↑f.snd).get h) 0)
  else Fin.cons 0 (Fin.succ ∘ ↑f.fst))
    ·posn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = true⊢ Function.Involutive
  (if h : (↑f.snd).isSome = true then
    Fin.cons ((↑f.snd).get h).succ (Function.update (Fin.succ ∘ ↑f.fst) ((↑f.snd).get h) 0)
  else Fin.cons 0 (Fin.succ ∘ ↑f.fst)) simp only [Nat.succ_eq_add_one, hs, ↓reduceDIte]posn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = true⊢ Function.Involutive (Fin.cons ((↑f.snd).get ⋯).succ (Function.update (Fin.succ ∘ ↑f.fst) ((↑f.snd).get ⋯) 0))
      let a := f.2.1.get hsposn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hs⊢ Function.Involutive (Fin.cons ((↑f.snd).get ⋯).succ (Function.update (Fin.succ ∘ ↑f.fst) ((↑f.snd).get ⋯) 0))
      change Function.Involutive (Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0))posn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hs⊢ Function.Involutive (Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0))
      intro iposn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsi:Fin (n + 1)⊢ Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0)
    (Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0) i) =
  i
      rcases Fin.eq_zero_or_eq_succ i with hi | ⟨j, hj⟩pos.inln:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsi:Fin (n + 1)hi:i = 0⊢ Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0)
    (Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0) i) =
  ipos.inrn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsi:Fin (n + 1)j:Fin nhj:i = j.succ⊢ Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0)
    (Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0) i) =
  i
      ·pos.inln:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsi:Fin (n + 1)hi:i = 0⊢ Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0)
    (Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0) i) =
  i subst hipos.inln:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hs⊢ Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0)
    (Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0) 0) =
  0
        rw [Fin.cons_zero,pos.inln:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hs⊢ Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0) a.succ = 0pos.inln:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hs⊢ Function.update (Fin.succ ∘ ↑f.fst) a 0 a = 0 Fin.cons_succpos.inln:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hs⊢ Function.update (Fin.succ ∘ ↑f.fst) a 0 a = 0pos.inln:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hs⊢ Function.update (Fin.succ ∘ ↑f.fst) a 0 a = 0]pos.inln:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hs⊢ Function.update (Fin.succ ∘ ↑f.fst) a 0 a = 0
        simpAll goals completed! 🐙
      ·pos.inrn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsi:Fin (n + 1)j:Fin nhj:i = j.succ⊢ Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0)
    (Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0) i) =
  i subst hjpos.inrn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin n⊢ Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0)
    (Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0) j.succ) =
  j.succ
        rw [Fin.cons_succpos.inrn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin n⊢ Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0) (Function.update (Fin.succ ∘ ↑f.fst) a 0 j) = j.succpos.inrn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin n⊢ Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0) (Function.update (Fin.succ ∘ ↑f.fst) a 0 j) = j.succ]pos.inrn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin n⊢ Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0) (Function.update (Fin.succ ∘ ↑f.fst) a 0 j) = j.succ
        by_cases hja : j = aposn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:j = a⊢ Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0) (Function.update (Fin.succ ∘ ↑f.fst) a 0 j) = j.succnegn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = a⊢ Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0) (Function.update (Fin.succ ∘ ↑f.fst) a 0 j) = j.succ
        ·posn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:j = a⊢ Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0) (Function.update (Fin.succ ∘ ↑f.fst) a 0 j) = j.succ subst hjaposn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hs⊢ Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0) (Function.update (Fin.succ ∘ ↑f.fst) a 0 a) = a.succ
          simpAll goals completed! 🐙
        ·negn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = a⊢ Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0) (Function.update (Fin.succ ∘ ↑f.fst) a 0 j) = j.succ rw [Function.update_apply,negn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = a⊢ Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0) (if j = a then 0 else (Fin.succ ∘ ↑f.fst) j) = j.succnegn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = a⊢ Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0) ((Fin.succ ∘ ↑f.fst) j) = j.succ if_neg hjanegn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = a⊢ Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0) ((Fin.succ ∘ ↑f.fst) j) = j.succnegn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = a⊢ Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0) ((Fin.succ ∘ ↑f.fst) j) = j.succ]negn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = a⊢ Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0) ((Fin.succ ∘ ↑f.fst) j) = j.succ
          simp only [Function.comp_apply, Fin.cons_succ]negn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = a⊢ Function.update (Fin.succ ∘ ↑f.fst) a 0 (↑f.fst j) = j.succ
          have hf2 := f.2.2 hsnegn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = ahf2:↑f.fst ((↑f.snd).get hs) = (↑f.snd).get hs⊢ Function.update (Fin.succ ∘ ↑f.fst) a 0 (↑f.fst j) = j.succ
          change f.1.1 a = a at hf2negn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = ahf2:↑f.fst a = a⊢ Function.update (Fin.succ ∘ ↑f.fst) a 0 (↑f.fst j) = j.succ
          have hjf1 : f.1.1 j ≠ a := fun hn => hja (byn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = ahf2:↑f.fst a = ahn:↑f.fst j = a⊢ j = anegn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = ahf2:↑f.fst a = ahjf1:↑f.fst j ≠ a⊢ Function.update (Fin.succ ∘ ↑f.fst) a 0 (↑f.fst j) = j.succ rw [← f.1.2 j,n:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = ahf2:↑f.fst a = ahn:↑f.fst j = a⊢ ↑f.fst (↑f.fst j) = aAll goals completed! 🐙negn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = ahf2:↑f.fst a = ahjf1:↑f.fst j ≠ a⊢ Function.update (Fin.succ ∘ ↑f.fst) a 0 (↑f.fst j) = j.succ hn,n:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = ahf2:↑f.fst a = ahn:↑f.fst j = a⊢ ↑f.fst a = aAll goals completed! 🐙negn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = ahf2:↑f.fst a = ahjf1:↑f.fst j ≠ a⊢ Function.update (Fin.succ ∘ ↑f.fst) a 0 (↑f.fst j) = j.succ hf2n:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = ahf2:↑f.fst a = ahn:↑f.fst j = a⊢ a = aAll goals completed! 🐙negn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = ahf2:↑f.fst a = ahjf1:↑f.fst j ≠ a⊢ Function.update (Fin.succ ∘ ↑f.fst) a 0 (↑f.fst j) = j.succ]All goals completed! 🐙negn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = ahf2:↑f.fst a = ahjf1:↑f.fst j ≠ a⊢ Function.update (Fin.succ ∘ ↑f.fst) a 0 (↑f.fst j) = j.succ)negn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = ahf2:↑f.fst a = ahjf1:↑f.fst j ≠ a⊢ Function.update (Fin.succ ∘ ↑f.fst) a 0 (↑f.fst j) = j.succ
          rw [Function.update_apply,negn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = ahf2:↑f.fst a = ahjf1:↑f.fst j ≠ a⊢ (if ↑f.fst j = a then 0 else (Fin.succ ∘ ↑f.fst) (↑f.fst j)) = j.succnegn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = ahf2:↑f.fst a = ahjf1:↑f.fst j ≠ a⊢ (Fin.succ ∘ ↑f.fst) (↑f.fst j) = j.succ if_neg hjf1negn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = ahf2:↑f.fst a = ahjf1:↑f.fst j ≠ a⊢ (Fin.succ ∘ ↑f.fst) (↑f.fst j) = j.succnegn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = ahf2:↑f.fst a = ahjf1:↑f.fst j ≠ a⊢ (Fin.succ ∘ ↑f.fst) (↑f.fst j) = j.succ]negn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = ahf2:↑f.fst a = ahjf1:↑f.fst j ≠ a⊢ (Fin.succ ∘ ↑f.fst) (↑f.fst j) = j.succ
          simp only [Function.comp_apply, Fin.succ_inj]negn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = ahf2:↑f.fst a = ahjf1:↑f.fst j ≠ a⊢ ↑f.fst (↑f.fst j) = j
          rw [f.1.2negn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:(↑f.snd).isSome = truea:Fin n := (↑f.snd).get hsj:Fin nhja:¬j = ahf2:↑f.fst a = ahjf1:↑f.fst j ≠ a⊢ j = jAll goals completed! 🐙]All goals completed! 🐙
    ·negn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:¬(↑f.snd).isSome = true⊢ Function.Involutive
  (if h : (↑f.snd).isSome = true then
    Fin.cons ((↑f.snd).get h).succ (Function.update (Fin.succ ∘ ↑f.fst) ((↑f.snd).get h) 0)
  else Fin.cons 0 (Fin.succ ∘ ↑f.fst)) simp only [succ_eq_add_one, hs, Bool.false_eq_true, ↓reduceDIte]negn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:¬(↑f.snd).isSome = true⊢ Function.Involutive (Fin.cons 0 (Fin.succ ∘ ↑f.fst))
      intro inegn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:¬(↑f.snd).isSome = truei:Fin (n + 1)⊢ Fin.cons 0 (Fin.succ ∘ ↑f.fst) (Fin.cons 0 (Fin.succ ∘ ↑f.fst) i) = i
      rcases Fin.eq_zero_or_eq_succ i with hi | ⟨j, hj⟩neg.inln:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:¬(↑f.snd).isSome = truei:Fin (n + 1)hi:i = 0⊢ Fin.cons 0 (Fin.succ ∘ ↑f.fst) (Fin.cons 0 (Fin.succ ∘ ↑f.fst) i) = ineg.inrn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:¬(↑f.snd).isSome = truei:Fin (n + 1)j:Fin nhj:i = j.succ⊢ Fin.cons 0 (Fin.succ ∘ ↑f.fst) (Fin.cons 0 (Fin.succ ∘ ↑f.fst) i) = i
      ·neg.inln:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:¬(↑f.snd).isSome = truei:Fin (n + 1)hi:i = 0⊢ Fin.cons 0 (Fin.succ ∘ ↑f.fst) (Fin.cons 0 (Fin.succ ∘ ↑f.fst) i) = i subst hineg.inln:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:¬(↑f.snd).isSome = true⊢ Fin.cons 0 (Fin.succ ∘ ↑f.fst) (Fin.cons 0 (Fin.succ ∘ ↑f.fst) 0) = 0
        simpAll goals completed! 🐙
      ·neg.inrn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:¬(↑f.snd).isSome = truei:Fin (n + 1)j:Fin nhj:i = j.succ⊢ Fin.cons 0 (Fin.succ ∘ ↑f.fst) (Fin.cons 0 (Fin.succ ∘ ↑f.fst) i) = i subst hjneg.inrn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:¬(↑f.snd).isSome = truej:Fin n⊢ Fin.cons 0 (Fin.succ ∘ ↑f.fst) (Fin.cons 0 (Fin.succ ∘ ↑f.fst) j.succ) = j.succ
        simp only [Fin.cons_succ, Function.comp_apply, Fin.succ_inj]neg.inrn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:¬(↑f.snd).isSome = truej:Fin n⊢ ↑f.fst (↑f.fst j) = j
        rw [f.1.2neg.inrn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }hs:¬(↑f.snd).isSome = truej:Fin n⊢ j = jAll goals completed! 🐙]All goals completed! 🐙⟩
  left_inv f := byn:ℕf:{ f // Function.Involutive f }⊢ (fun f =>
      ⟨if h : (↑f.snd).isSome = true then
          Fin.cons ((↑f.snd).get h).succ (Function.update (Fin.succ ∘ ↑f.fst) ((↑f.snd).get h) 0)
        else Fin.cons 0 (Fin.succ ∘ ↑f.fst),
        ⋯⟩)
    ((fun f =>
        ⟨⟨fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h, ⋯⟩,
          ⟨if h : ↑f 0 = 0 then none else some ((↑f 0).pred h), ⋯⟩⟩)
      f) =
  f
    match f with
    | ⟨f, hf⟩ =>n:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive f⊢ (fun f =>
      ⟨if h : (↑f.snd).isSome = true then
          Fin.cons ((↑f.snd).get h).succ (Function.update (Fin.succ ∘ ↑f.fst) ((↑f.snd).get h) 0)
        else Fin.cons 0 (Fin.succ ∘ ↑f.fst),
        ⋯⟩)
    ((fun f =>
        ⟨⟨fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h, ⋯⟩,
          ⟨if h : ↑f 0 = 0 then none else some ((↑f 0).pred h), ⋯⟩⟩)
      ⟨f, hf⟩) =
  ⟨f, hf⟩
    have hpred (j : Fin n) (hj : f j.succ ≠ 0) :
        (↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 : ℕ) = ↑(f j.succ) := byn:ℕf:{ f // Function.Involutive f }⊢ (fun f =>
      ⟨if h : (↑f.snd).isSome = true then
          Fin.cons ((↑f.snd).get h).succ (Function.update (Fin.succ ∘ ↑f.fst) ((↑f.snd).get h) 0)
        else Fin.cons 0 (Fin.succ ∘ ↑f.fst),
        ⋯⟩)
    ((fun f =>
        ⟨⟨fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h, ⋯⟩,
          ⟨if h : ↑f 0 = 0 then none else some ((↑f 0).pred h), ⋯⟩⟩)
      f) =
  fn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)⊢ (fun f =>
      ⟨if h : (↑f.snd).isSome = true then
          Fin.cons ((↑f.snd).get h).succ (Function.update (Fin.succ ∘ ↑f.fst) ((↑f.snd).get h) 0)
        else Fin.cons 0 (Fin.succ ∘ ↑f.fst),
        ⋯⟩)
    ((fun f =>
        ⟨⟨fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h, ⋯⟩,
          ⟨if h : ↑f 0 = 0 then none else some ((↑f 0).pred h), ⋯⟩⟩)
      ⟨f, hf⟩) =
  ⟨f, hf⟩
      rw [dif_neg hj,n:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fj:Fin nhj:f j.succ ≠ 0⊢ ↑((f j.succ).pred hj) + 1 = ↑(f j.succ)n:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fj:Fin nhj:f j.succ ≠ 0⊢ ↑(f j.succ) - 1 + 1 = ↑(f j.succ)n:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)⊢ (fun f =>
      ⟨if h : (↑f.snd).isSome = true then
          Fin.cons ((↑f.snd).get h).succ (Function.update (Fin.succ ∘ ↑f.fst) ((↑f.snd).get h) 0)
        else Fin.cons 0 (Fin.succ ∘ ↑f.fst),
        ⋯⟩)
    ((fun f =>
        ⟨⟨fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h, ⋯⟩,
          ⟨if h : ↑f 0 = 0 then none else some ((↑f 0).pred h), ⋯⟩⟩)
      ⟨f, hf⟩) =
  ⟨f, hf⟩ Fin.val_predn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fj:Fin nhj:f j.succ ≠ 0⊢ ↑(f j.succ) - 1 + 1 = ↑(f j.succ)n:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fj:Fin nhj:f j.succ ≠ 0⊢ ↑(f j.succ) - 1 + 1 = ↑(f j.succ)n:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)⊢ (fun f =>
      ⟨if h : (↑f.snd).isSome = true then
          Fin.cons ((↑f.snd).get h).succ (Function.update (Fin.succ ∘ ↑f.fst) ((↑f.snd).get h) 0)
        else Fin.cons 0 (Fin.succ ∘ ↑f.fst),
        ⋯⟩)
    ((fun f =>
        ⟨⟨fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h, ⋯⟩,
          ⟨if h : ↑f 0 = 0 then none else some ((↑f 0).pred h), ⋯⟩⟩)
      ⟨f, hf⟩) =
  ⟨f, hf⟩]n:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fj:Fin nhj:f j.succ ≠ 0⊢ ↑(f j.succ) - 1 + 1 = ↑(f j.succ)n:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)⊢ (fun f =>
      ⟨if h : (↑f.snd).isSome = true then
          Fin.cons ((↑f.snd).get h).succ (Function.update (Fin.succ ∘ ↑f.fst) ((↑f.snd).get h) 0)
        else Fin.cons 0 (Fin.succ ∘ ↑f.fst),
        ⋯⟩)
    ((fun f =>
        ⟨⟨fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h, ⋯⟩,
          ⟨if h : ↑f 0 = 0 then none else some ((↑f 0).pred h), ⋯⟩⟩)
      ⟨f, hf⟩) =
  ⟨f, hf⟩
      have hv : (f j.succ).val ≠ 0 := fun h => hj (Fin.ext (byn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fj:Fin nhj:f j.succ ≠ 0h:↑(f j.succ) = 0⊢ ↑(f j.succ) = ↑0n:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fj:Fin nhj:f j.succ ≠ 0hv:↑(f j.succ) ≠ 0⊢ ↑(f j.succ) - 1 + 1 = ↑(f j.succ)n:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)⊢ (fun f =>
      ⟨if h : (↑f.snd).isSome = true then
          Fin.cons ((↑f.snd).get h).succ (Function.update (Fin.succ ∘ ↑f.fst) ((↑f.snd).get h) 0)
        else Fin.cons 0 (Fin.succ ∘ ↑f.fst),
        ⋯⟩)
    ((fun f =>
        ⟨⟨fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h, ⋯⟩,
          ⟨if h : ↑f 0 = 0 then none else some ((↑f 0).pred h), ⋯⟩⟩)
      ⟨f, hf⟩) =
  ⟨f, hf⟩ simpa using hAll goals completed! 🐙n:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fj:Fin nhj:f j.succ ≠ 0hv:↑(f j.succ) ≠ 0⊢ ↑(f j.succ) - 1 + 1 = ↑(f j.succ)n:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)⊢ (fun f =>
      ⟨if h : (↑f.snd).isSome = true then
          Fin.cons ((↑f.snd).get h).succ (Function.update (Fin.succ ∘ ↑f.fst) ((↑f.snd).get h) 0)
        else Fin.cons 0 (Fin.succ ∘ ↑f.fst),
        ⋯⟩)
    ((fun f =>
        ⟨⟨fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h, ⋯⟩,
          ⟨if h : ↑f 0 = 0 then none else some ((↑f 0).pred h), ⋯⟩⟩)
      ⟨f, hf⟩) =
  ⟨f, hf⟩))n:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fj:Fin nhj:f j.succ ≠ 0hv:↑(f j.succ) ≠ 0⊢ ↑(f j.succ) - 1 + 1 = ↑(f j.succ)n:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)⊢ (fun f =>
      ⟨if h : (↑f.snd).isSome = true then
          Fin.cons ((↑f.snd).get h).succ (Function.update (Fin.succ ∘ ↑f.fst) ((↑f.snd).get h) 0)
        else Fin.cons 0 (Fin.succ ∘ ↑f.fst),
        ⋯⟩)
    ((fun f =>
        ⟨⟨fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h, ⋯⟩,
          ⟨if h : ↑f 0 = 0 then none else some ((↑f 0).pred h), ⋯⟩⟩)
      ⟨f, hf⟩) =
  ⟨f, hf⟩
      omegan:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)⊢ (fun f =>
      ⟨if h : (↑f.snd).isSome = true then
          Fin.cons ((↑f.snd).get h).succ (Function.update (Fin.succ ∘ ↑f.fst) ((↑f.snd).get h) 0)
        else Fin.cons 0 (Fin.succ ∘ ↑f.fst),
        ⋯⟩)
    ((fun f =>
        ⟨⟨fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h, ⋯⟩,
          ⟨if h : ↑f 0 = 0 then none else some ((↑f 0).pred h), ⋯⟩⟩)
      ⟨f, hf⟩) =
  ⟨f, hf⟩n:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)⊢ (fun f =>
      ⟨if h : (↑f.snd).isSome = true then
          Fin.cons ((↑f.snd).get h).succ (Function.update (Fin.succ ∘ ↑f.fst) ((↑f.snd).get h) 0)
        else Fin.cons 0 (Fin.succ ∘ ↑f.fst),
        ⋯⟩)
    ((fun f =>
        ⟨⟨fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h, ⋯⟩,
          ⟨if h : ↑f 0 = 0 then none else some ((↑f 0).pred h), ⋯⟩⟩)
      ⟨f, hf⟩) =
  ⟨f, hf⟩
    simp only [succ_eq_add_one, Option.isSome_dite', Option.get_dite', Fin.succ_pred,
      Fin.cons_update, dite_eq_ite, ite_not, Subtype.mk.injEq]n:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)⊢ (if f 0 = 0 then Fin.cons 0 (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯)
  else
    Function.update (Fin.cons (f 0) (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯)) (f 0) 0) =
  f
    ext in:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)⊢ ↑((if f 0 = 0 then Fin.cons 0 (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯)
      else
        Function.update (Fin.cons (f 0) (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯)) (f 0)
          0)
      i) =
  ↑(f i)
    by_cases h0 : f 0 = 0posn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)h0:f 0 = 0⊢ ↑((if f 0 = 0 then Fin.cons 0 (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯)
      else
        Function.update (Fin.cons (f 0) (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯)) (f 0)
          0)
      i) =
  ↑(f i)negn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)h0:¬f 0 = 0⊢ ↑((if f 0 = 0 then Fin.cons 0 (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯)
      else
        Function.update (Fin.cons (f 0) (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯)) (f 0)
          0)
      i) =
  ↑(f i)
    ·posn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)h0:f 0 = 0⊢ ↑((if f 0 = 0 then Fin.cons 0 (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯)
      else
        Function.update (Fin.cons (f 0) (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯)) (f 0)
          0)
      i) =
  ↑(f i) simp only [h0, ↓reduceIte]posn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)h0:f 0 = 0⊢ ↑(Fin.cons 0 (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯) i) = ↑(f i)
      rcases Fin.eq_zero_or_eq_succ i with hi | ⟨j, hj⟩pos.inln:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)h0:f 0 = 0hi:i = 0⊢ ↑(Fin.cons 0 (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯) i) = ↑(f i)pos.inrn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)h0:f 0 = 0j:Fin nhj:i = j.succ⊢ ↑(Fin.cons 0 (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯) i) = ↑(f i)
      ·pos.inln:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)h0:f 0 = 0hi:i = 0⊢ ↑(Fin.cons 0 (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯) i) = ↑(f i) subst hipos.inln:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)h0:f 0 = 0⊢ ↑(Fin.cons 0 (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯) 0) = ↑(f 0)
        simp [h0]All goals completed! 🐙
      ·pos.inrn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)h0:f 0 = 0j:Fin nhj:i = j.succ⊢ ↑(Fin.cons 0 (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯) i) = ↑(f i) subst hjpos.inrn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)h0:f 0 = 0j:Fin n⊢ ↑(Fin.cons 0 (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯) j.succ) = ↑(f j.succ)
        simp only [Fin.cons_succ, Function.comp_apply, Fin.val_succ]pos.inrn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)h0:f 0 = 0j:Fin n⊢ ↑(if h : f j.succ = 0 then j else (f j.succ).pred ⋯) + 1 = ↑(f j.succ)
        refine hpred j fun hj => Fin.succ_ne_zero j (hf.injective (hj.trans h0.symm))All goals completed! 🐙
    ·negn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)h0:¬f 0 = 0⊢ ↑((if f 0 = 0 then Fin.cons 0 (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯)
      else
        Function.update (Fin.cons (f 0) (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯)) (f 0)
          0)
      i) =
  ↑(f i) rw [if_neg h0negn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)h0:¬f 0 = 0⊢ ↑(Function.update (Fin.cons (f 0) (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯)) (f 0) 0 i) =
  ↑(f i)negn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)h0:¬f 0 = 0⊢ ↑(Function.update (Fin.cons (f 0) (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯)) (f 0) 0 i) =
  ↑(f i)]negn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)h0:¬f 0 = 0⊢ ↑(Function.update (Fin.cons (f 0) (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯)) (f 0) 0 i) =
  ↑(f i)
      by_cases hf' : i = f 0posn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)h0:¬f 0 = 0hf':i = f 0⊢ ↑(Function.update (Fin.cons (f 0) (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯)) (f 0) 0 i) =
  ↑(f i)negn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)h0:¬f 0 = 0hf':¬i = f 0⊢ ↑(Function.update (Fin.cons (f 0) (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯)) (f 0) 0 i) =
  ↑(f i)
      ·posn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)h0:¬f 0 = 0hf':i = f 0⊢ ↑(Function.update (Fin.cons (f 0) (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯)) (f 0) 0 i) =
  ↑(f i) subst hf'posn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)h0:¬f 0 = 0⊢ ↑(Function.update (Fin.cons (f 0) (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯)) (f 0) 0
      (f 0)) =
  ↑(f (f 0))
        simp only [Function.update_self, Fin.val_zero]posn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)h0:¬f 0 = 0⊢ 0 = ↑(f (f 0))
        rw [hfposn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)h0:¬f 0 = 0⊢ 0 = ↑0posn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)h0:¬f 0 = 0⊢ 0 = ↑0]posn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)h0:¬f 0 = 0⊢ 0 = ↑0
        simpAll goals completed! 🐙
      ·negn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)h0:¬f 0 = 0hf':¬i = f 0⊢ ↑(Function.update (Fin.cons (f 0) (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯)) (f 0) 0 i) =
  ↑(f i) rw [Function.update_apply,negn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)h0:¬f 0 = 0hf':¬i = f 0⊢ ↑(if i = f 0 then 0 else Fin.cons (f 0) (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯) i) =
  ↑(f i)negn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)h0:¬f 0 = 0hf':¬i = f 0⊢ ↑(Fin.cons (f 0) (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯) i) = ↑(f i) if_neg hf'negn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)h0:¬f 0 = 0hf':¬i = f 0⊢ ↑(Fin.cons (f 0) (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯) i) = ↑(f i)negn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)h0:¬f 0 = 0hf':¬i = f 0⊢ ↑(Fin.cons (f 0) (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯) i) = ↑(f i)]negn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)h0:¬f 0 = 0hf':¬i = f 0⊢ ↑(Fin.cons (f 0) (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯) i) = ↑(f i)
        rcases Fin.eq_zero_or_eq_succ i with hi | ⟨j, hj⟩neg.inln:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)h0:¬f 0 = 0hf':¬i = f 0hi:i = 0⊢ ↑(Fin.cons (f 0) (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯) i) = ↑(f i)neg.inrn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)h0:¬f 0 = 0hf':¬i = f 0j:Fin nhj:i = j.succ⊢ ↑(Fin.cons (f 0) (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯) i) = ↑(f i)
        ·neg.inln:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)h0:¬f 0 = 0hf':¬i = f 0hi:i = 0⊢ ↑(Fin.cons (f 0) (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯) i) = ↑(f i) subst hineg.inln:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)h0:¬f 0 = 0hf':¬0 = f 0⊢ ↑(Fin.cons (f 0) (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯) 0) = ↑(f 0)
          simpAll goals completed! 🐙
        ·neg.inrn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)i:Fin (n + 1)h0:¬f 0 = 0hf':¬i = f 0j:Fin nhj:i = j.succ⊢ ↑(Fin.cons (f 0) (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯) i) = ↑(f i) subst hjneg.inrn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)h0:¬f 0 = 0j:Fin nhf':¬j.succ = f 0⊢ ↑(Fin.cons (f 0) (Fin.succ ∘ fun i => if h : f i.succ = 0 then i else (f i.succ).pred ⋯) j.succ) = ↑(f j.succ)
          simp only [Fin.cons_succ, Function.comp_apply, Fin.val_succ]neg.inrn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)h0:¬f 0 = 0j:Fin nhf':¬j.succ = f 0⊢ ↑(if h : f j.succ = 0 then j else (f j.succ).pred ⋯) + 1 = ↑(f j.succ)
          refine hpred j fun hj => hf' ?_neg.inrn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)h0:¬f 0 = 0j:Fin nhf':¬j.succ = f 0hj:f j.succ = 0⊢ j.succ = f 0
          rw [← hj,neg.inrn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)h0:¬f 0 = 0j:Fin nhf':¬j.succ = f 0hj:f j.succ = 0⊢ j.succ = f (f j.succ)All goals completed! 🐙 hfneg.inrn:ℕf✝:{ f // Function.Involutive f }f:Fin n.succ → Fin n.succhf:Function.Involutive fhpred:∀ (j : Fin n), f j.succ ≠ 0 → ↑(if h : f j.succ = 0 then j else (f j.succ).pred h) + 1 = ↑(f j.succ)h0:¬f 0 = 0j:Fin nhf':¬j.succ = f 0hj:f j.succ = 0⊢ j.succ = j.succAll goals completed! 🐙]All goals completed! 🐙
  right_inv f := byn:ℕf:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }⊢ (fun f =>
      ⟨⟨fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h, ⋯⟩,
        ⟨if h : ↑f 0 = 0 then none else some ((↑f 0).pred h), ⋯⟩⟩)
    ((fun f =>
        ⟨if h : (↑f.snd).isSome = true then
            Fin.cons ((↑f.snd).get h).succ (Function.update (Fin.succ ∘ ↑f.fst) ((↑f.snd).get h) 0)
          else Fin.cons 0 (Fin.succ ∘ ↑f.fst),
          ⋯⟩)
      f) =
  f
    match f with
    | ⟨⟨f, hf⟩, ⟨f0, hf0⟩⟩ =>n:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get h⊢ (fun f =>
      ⟨⟨fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h, ⋯⟩,
        ⟨if h : ↑f 0 = 0 then none else some ((↑f 0).pred h), ⋯⟩⟩)
    ((fun f =>
        ⟨if h : (↑f.snd).isSome = true then
            Fin.cons ((↑f.snd).get h).succ (Function.update (Fin.succ ∘ ↑f.fst) ((↑f.snd).get h) 0)
          else Fin.cons 0 (Fin.succ ∘ ↑f.fst),
          ⋯⟩)
      ⟨⟨f, hf⟩, ⟨f0, hf0⟩⟩) =
  ⟨⟨f, hf⟩, ⟨f0, hf0⟩⟩
    ext ian:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin n⊢ ↑(↑((fun f =>
              ⟨⟨fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h, ⋯⟩,
                ⟨if h : ↑f 0 = 0 then none else some ((↑f 0).pred h), ⋯⟩⟩)
            ((fun f =>
                ⟨if h : (↑f.snd).isSome = true then
                    Fin.cons ((↑f.snd).get h).succ (Function.update (Fin.succ ∘ ↑f.fst) ((↑f.snd).get h) 0)
                  else Fin.cons 0 (Fin.succ ∘ ↑f.fst),
                  ⋯⟩)
              ⟨⟨f, hf⟩, ⟨f0, hf0⟩⟩)).fst
      i) =
  ↑(↑⟨⟨f, hf⟩, ⟨f0, hf0⟩⟩.fst i)an:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin n⊢ ↑((fun f =>
            ⟨⟨fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h, ⋯⟩,
              ⟨if h : ↑f 0 = 0 then none else some ((↑f 0).pred h), ⋯⟩⟩)
          ((fun f =>
              ⟨if h : (↑f.snd).isSome = true then
                  Fin.cons ((↑f.snd).get h).succ (Function.update (Fin.succ ∘ ↑f.fst) ((↑f.snd).get h) 0)
                else Fin.cons 0 (Fin.succ ∘ ↑f.fst),
                ⋯⟩)
            ⟨⟨f, hf⟩, ⟨f0, hf0⟩⟩)).snd =
    some i ↔
  ↑⟨⟨f, hf⟩, ⟨f0, hf0⟩⟩.snd = some i
    ·an:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin n⊢ ↑(↑((fun f =>
              ⟨⟨fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h, ⋯⟩,
                ⟨if h : ↑f 0 = 0 then none else some ((↑f 0).pred h), ⋯⟩⟩)
            ((fun f =>
                ⟨if h : (↑f.snd).isSome = true then
                    Fin.cons ((↑f.snd).get h).succ (Function.update (Fin.succ ∘ ↑f.fst) ((↑f.snd).get h) 0)
                  else Fin.cons 0 (Fin.succ ∘ ↑f.fst),
                  ⋯⟩)
              ⟨⟨f, hf⟩, ⟨f0, hf0⟩⟩)).fst
      i) =
  ↑(↑⟨⟨f, hf⟩, ⟨f0, hf0⟩⟩.fst i) simp only [succ_eq_add_one, Fin.cons_update]an:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin n⊢ ↑(if h :
        (if h : f0.isSome = true then Function.update (Fin.cons (f0.get ⋯).succ (Fin.succ ∘ f)) (f0.get ⋯).succ 0
            else Fin.cons 0 (Fin.succ ∘ f))
            i.succ =
          0 then
      i
    else
      ((if h : f0.isSome = true then Function.update (Fin.cons (f0.get ⋯).succ (Fin.succ ∘ f)) (f0.get ⋯).succ 0
            else Fin.cons 0 (Fin.succ ∘ f))
            i.succ).pred
        ⋯) =
  ↑(f i)
      by_cases hs : f0.isSomeposn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:f0.isSome = true⊢ ↑(if h :
        (if h : f0.isSome = true then Function.update (Fin.cons (f0.get ⋯).succ (Fin.succ ∘ f)) (f0.get ⋯).succ 0
            else Fin.cons 0 (Fin.succ ∘ f))
            i.succ =
          0 then
      i
    else
      ((if h : f0.isSome = true then Function.update (Fin.cons (f0.get ⋯).succ (Fin.succ ∘ f)) (f0.get ⋯).succ 0
            else Fin.cons 0 (Fin.succ ∘ f))
            i.succ).pred
        ⋯) =
  ↑(f i)negn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:¬f0.isSome = true⊢ ↑(if h :
        (if h : f0.isSome = true then Function.update (Fin.cons (f0.get ⋯).succ (Fin.succ ∘ f)) (f0.get ⋯).succ 0
            else Fin.cons 0 (Fin.succ ∘ f))
            i.succ =
          0 then
      i
    else
      ((if h : f0.isSome = true then Function.update (Fin.cons (f0.get ⋯).succ (Fin.succ ∘ f)) (f0.get ⋯).succ 0
            else Fin.cons 0 (Fin.succ ∘ f))
            i.succ).pred
        ⋯) =
  ↑(f i)
      ·posn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:f0.isSome = true⊢ ↑(if h :
        (if h : f0.isSome = true then Function.update (Fin.cons (f0.get ⋯).succ (Fin.succ ∘ f)) (f0.get ⋯).succ 0
            else Fin.cons 0 (Fin.succ ∘ f))
            i.succ =
          0 then
      i
    else
      ((if h : f0.isSome = true then Function.update (Fin.cons (f0.get ⋯).succ (Fin.succ ∘ f)) (f0.get ⋯).succ 0
            else Fin.cons 0 (Fin.succ ∘ f))
            i.succ).pred
        ⋯) =
  ↑(f i) simp only [hs, ↓reduceDIte]posn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:f0.isSome = true⊢ ↑(if h : Function.update (Fin.cons (f0.get ⋯).succ (Fin.succ ∘ f)) (f0.get ⋯).succ 0 i.succ = 0 then i
    else (Function.update (Fin.cons (f0.get ⋯).succ (Fin.succ ∘ f)) (f0.get ⋯).succ 0 i.succ).pred ⋯) =
  ↑(f i)
        by_cases hi : i = f0.get hsposn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:f0.isSome = truehi:i = f0.get hs⊢ ↑(if h : Function.update (Fin.cons (f0.get ⋯).succ (Fin.succ ∘ f)) (f0.get ⋯).succ 0 i.succ = 0 then i
    else (Function.update (Fin.cons (f0.get ⋯).succ (Fin.succ ∘ f)) (f0.get ⋯).succ 0 i.succ).pred ⋯) =
  ↑(f i)negn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:f0.isSome = truehi:¬i = f0.get hs⊢ ↑(if h : Function.update (Fin.cons (f0.get ⋯).succ (Fin.succ ∘ f)) (f0.get ⋯).succ 0 i.succ = 0 then i
    else (Function.update (Fin.cons (f0.get ⋯).succ (Fin.succ ∘ f)) (f0.get ⋯).succ 0 i.succ).pred ⋯) =
  ↑(f i)
        ·posn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:f0.isSome = truehi:i = f0.get hs⊢ ↑(if h : Function.update (Fin.cons (f0.get ⋯).succ (Fin.succ ∘ f)) (f0.get ⋯).succ 0 i.succ = 0 then i
    else (Function.update (Fin.cons (f0.get ⋯).succ (Fin.succ ∘ f)) (f0.get ⋯).succ 0 i.succ).pred ⋯) =
  ↑(f i) simp only [Function.update_apply, hi, ↓reduceIte, ↓reduceDIte]posn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:f0.isSome = truehi:i = f0.get hs⊢ ↑(f0.get hs) = ↑(f (f0.get hs))
          exact Eq.symm (Fin.val_eq_of_eq (hf0 hs))All goals completed! 🐙
        ·negn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:f0.isSome = truehi:¬i = f0.get hs⊢ ↑(if h : Function.update (Fin.cons (f0.get ⋯).succ (Fin.succ ∘ f)) (f0.get ⋯).succ 0 i.succ = 0 then i
    else (Function.update (Fin.cons (f0.get ⋯).succ (Fin.succ ∘ f)) (f0.get ⋯).succ 0 i.succ).pred ⋯) =
  ↑(f i) simp only [ne_eq, Fin.succ_inj, hi, not_false_eq_true, Function.update_of_ne,
          Fin.cons_succ, Function.comp_apply, Fin.pred_succ, dite_eq_ite]negn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:f0.isSome = truehi:¬i = f0.get hs⊢ ↑(if (f i).succ = 0 then i else f i) = ↑(f i)
          rw [if_neg (Fin.succ_ne_zero (f i))negn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:f0.isSome = truehi:¬i = f0.get hs⊢ ↑(f i) = ↑(f i)All goals completed! 🐙]All goals completed! 🐙
      ·negn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:¬f0.isSome = true⊢ ↑(if h :
        (if h : f0.isSome = true then Function.update (Fin.cons (f0.get ⋯).succ (Fin.succ ∘ f)) (f0.get ⋯).succ 0
            else Fin.cons 0 (Fin.succ ∘ f))
            i.succ =
          0 then
      i
    else
      ((if h : f0.isSome = true then Function.update (Fin.cons (f0.get ⋯).succ (Fin.succ ∘ f)) (f0.get ⋯).succ 0
            else Fin.cons 0 (Fin.succ ∘ f))
            i.succ).pred
        ⋯) =
  ↑(f i) simp only [hs, Bool.false_eq_true, ↓reduceDIte, Fin.cons_succ, Function.comp_apply,
        Fin.pred_succ, dite_eq_ite]negn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:¬f0.isSome = true⊢ ↑(if (f i).succ = 0 then i else f i) = ↑(f i)
        rw [if_neg (Fin.succ_ne_zero (f i))negn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:¬f0.isSome = true⊢ ↑(f i) = ↑(f i)All goals completed! 🐙]All goals completed! 🐙
    ·an:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin n⊢ ↑((fun f =>
            ⟨⟨fun i => if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h, ⋯⟩,
              ⟨if h : ↑f 0 = 0 then none else some ((↑f 0).pred h), ⋯⟩⟩)
          ((fun f =>
              ⟨if h : (↑f.snd).isSome = true then
                  Fin.cons ((↑f.snd).get h).succ (Function.update (Fin.succ ∘ ↑f.fst) ((↑f.snd).get h) 0)
                else Fin.cons 0 (Fin.succ ∘ ↑f.fst),
                ⋯⟩)
            ⟨⟨f, hf⟩, ⟨f0, hf0⟩⟩)).snd =
    some i ↔
  ↑⟨⟨f, hf⟩, ⟨f0, hf0⟩⟩.snd = some i simp only [Nat.succ_eq_add_one, Option.dite_none_left_eq_some, Option.some.injEq]an:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin n⊢ (∃ (h :
    ¬(if h : f0.isSome = true then Fin.cons (f0.get ⋯).succ (Function.update (Fin.succ ∘ f) (f0.get ⋯) 0)
          else Fin.cons 0 (Fin.succ ∘ f))
          0 =
        0),
    ((if h : f0.isSome = true then Fin.cons (f0.get ⋯).succ (Function.update (Fin.succ ∘ f) (f0.get ⋯) 0)
            else Fin.cons 0 (Fin.succ ∘ f))
            0).pred
        ⋯ =
      i) ↔
  f0 = some i
      by_cases hs : f0.isSomeposn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:f0.isSome = true⊢ (∃ (h :
    ¬(if h : f0.isSome = true then Fin.cons (f0.get ⋯).succ (Function.update (Fin.succ ∘ f) (f0.get ⋯) 0)
          else Fin.cons 0 (Fin.succ ∘ f))
          0 =
        0),
    ((if h : f0.isSome = true then Fin.cons (f0.get ⋯).succ (Function.update (Fin.succ ∘ f) (f0.get ⋯) 0)
            else Fin.cons 0 (Fin.succ ∘ f))
            0).pred
        ⋯ =
      i) ↔
  f0 = some inegn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:¬f0.isSome = true⊢ (∃ (h :
    ¬(if h : f0.isSome = true then Fin.cons (f0.get ⋯).succ (Function.update (Fin.succ ∘ f) (f0.get ⋯) 0)
          else Fin.cons 0 (Fin.succ ∘ f))
          0 =
        0),
    ((if h : f0.isSome = true then Fin.cons (f0.get ⋯).succ (Function.update (Fin.succ ∘ f) (f0.get ⋯) 0)
            else Fin.cons 0 (Fin.succ ∘ f))
            0).pred
        ⋯ =
      i) ↔
  f0 = some i
      ·posn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:f0.isSome = true⊢ (∃ (h :
    ¬(if h : f0.isSome = true then Fin.cons (f0.get ⋯).succ (Function.update (Fin.succ ∘ f) (f0.get ⋯) 0)
          else Fin.cons 0 (Fin.succ ∘ f))
          0 =
        0),
    ((if h : f0.isSome = true then Fin.cons (f0.get ⋯).succ (Function.update (Fin.succ ∘ f) (f0.get ⋯) 0)
            else Fin.cons 0 (Fin.succ ∘ f))
            0).pred
        ⋯ =
      i) ↔
  f0 = some i simp only [hs, ↓reduceDIte]posn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:f0.isSome = true⊢ (∃ (h : ¬Fin.cons (f0.get ⋯).succ (Function.update (Fin.succ ∘ f) (f0.get ⋯) 0) 0 = 0),
    (Fin.cons (f0.get ⋯).succ (Function.update (Fin.succ ∘ f) (f0.get ⋯) 0) 0).pred ⋯ = i) ↔
  f0 = some i
        simp only [Fin.cons_zero, Fin.pred_succ, exists_prop]posn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:f0.isSome = true⊢ ¬(f0.get ⋯).succ = 0 ∧ f0.get ⋯ = i ↔ f0 = some i
        have hx : ¬ (f0.get hs).succ = 0 := (Fin.succ_ne_zero (f0.get hs))posn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:f0.isSome = truehx:¬(f0.get hs).succ = 0⊢ ¬(f0.get ⋯).succ = 0 ∧ f0.get ⋯ = i ↔ f0 = some i
        simp only [hx, not_false_eq_true, true_and]posn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:f0.isSome = truehx:¬(f0.get hs).succ = 0⊢ f0.get ⋯ = i ↔ f0 = some i
        rw [Option.eq_some_iff_get_eqposn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:f0.isSome = truehx:¬(f0.get hs).succ = 0⊢ f0.get ⋯ = i ↔ ∃ (h : f0.isSome = true), f0.get h = iposn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:f0.isSome = truehx:¬(f0.get hs).succ = 0⊢ f0.get ⋯ = i ↔ ∃ (h : f0.isSome = true), f0.get h = i]posn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:f0.isSome = truehx:¬(f0.get hs).succ = 0⊢ f0.get ⋯ = i ↔ ∃ (h : f0.isSome = true), f0.get h = i
        simp [hs]All goals completed! 🐙
      ·negn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:¬f0.isSome = true⊢ (∃ (h :
    ¬(if h : f0.isSome = true then Fin.cons (f0.get ⋯).succ (Function.update (Fin.succ ∘ f) (f0.get ⋯) 0)
          else Fin.cons 0 (Fin.succ ∘ f))
          0 =
        0),
    ((if h : f0.isSome = true then Fin.cons (f0.get ⋯).succ (Function.update (Fin.succ ∘ f) (f0.get ⋯) 0)
            else Fin.cons 0 (Fin.succ ∘ f))
            0).pred
        ⋯ =
      i) ↔
  f0 = some i simp only [hs, Bool.false_eq_true, ↓reduceDIte, Fin.cons_zero, not_true_eq_false,
        IsEmpty.exists_iff, false_iff]negn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:¬f0.isSome = true⊢ ¬f0 = some i
        simp only [Bool.not_eq_true, Option.isSome_eq_false_iff, Option.isNone_iff_eq_none] at hsnegn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive ff0:Option (Fin n)hf0:∀ (h : f0.isSome = true), ↑⟨f, hf⟩ (f0.get h) = f0.get hi:Fin nhs:f0 = none⊢ ¬f0 = some i
        subst hsnegn:ℕf✝:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f:Fin n → Fin nhf:Function.Involutive fi:Fin nhf0:∀ (h : none.isSome = true), ↑⟨f, hf⟩ (none.get h) = none.get h⊢ ¬none = some i
        exact ne_of_beq_false rflAll goals completed! 🐙



lemma involutionCons_ext {n : ℕ} {f1 f2 : (f : {f : Fin n → Fin n // Function.Involutive f}) ×
    {i : Option (Fin n) // ∀ (h : i.isSome), f.1 (Option.get i h) = (Option.get i h)}}
    (h1 : f1.1 = f2.1) (h2 : f1.2 = Equiv.subtypeEquivRight (byn:ℕf1:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f2:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }h1:f1.fst = f2.fst⊢ ∀ (x : Option (Fin n)),
  (∀ (h : x.isSome = true), ↑f2.fst (x.get h) = x.get h) ↔ ∀ (h : x.isSome = true), ↑f1.fst (x.get h) = x.get h rw [h1n:ℕf1:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f2:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }h1:f1.fst = f2.fst⊢ ∀ (x : Option (Fin n)),
  (∀ (h : x.isSome = true), ↑f2.fst (x.get h) = x.get h) ↔ ∀ (h : x.isSome = true), ↑f2.fst (x.get h) = x.get hn:ℕf1:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f2:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }h1:f1.fst = f2.fst⊢ ∀ (x : Option (Fin n)),
  (∀ (h : x.isSome = true), ↑f2.fst (x.get h) = x.get h) ↔ ∀ (h : x.isSome = true), ↑f2.fst (x.get h) = x.get h]n:ℕf1:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f2:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }h1:f1.fst = f2.fst⊢ ∀ (x : Option (Fin n)),
  (∀ (h : x.isSome = true), ↑f2.fst (x.get h) = x.get h) ↔ ∀ (h : x.isSome = true), ↑f2.fst (x.get h) = x.get h; simpAll goals completed! 🐙) f2.2) : f1 = f2 := byn:ℕf1:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }f2:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }h1:f1.fst = f2.fsth2:f1.snd = (Equiv.subtypeEquivRight ⋯) f2.snd⊢ f1 = f2
  cases f1mkn:ℕf2:(f : { f // Function.Involutive f }) × { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }fst✝:{ f // Function.Involutive f }snd✝:{ i // ∀ (h : i.isSome = true), ↑fst✝ (i.get h) = i.get h }h1:⟨fst✝, snd✝⟩.fst = f2.fsth2:⟨fst✝, snd✝⟩.snd = (Equiv.subtypeEquivRight ⋯) f2.snd⊢ ⟨fst✝, snd✝⟩ = f2
  cases f2mk.mkn:ℕfst✝¹:{ f // Function.Involutive f }snd✝¹:{ i // ∀ (h : i.isSome = true), ↑fst✝ (i.get h) = i.get h }fst✝:{ f // Function.Involutive f }snd✝:{ i // ∀ (h : i.isSome = true), ↑fst✝ (i.get h) = i.get h }h1:⟨fst✝¹, snd✝¹⟩.fst = ⟨fst✝, snd✝⟩.fsth2:⟨fst✝¹, snd✝¹⟩.snd = (Equiv.subtypeEquivRight ⋯) ⟨fst✝, snd✝⟩.snd⊢ ⟨fst✝¹, snd✝¹⟩ = ⟨fst✝, snd✝⟩
  simp only at h1 h2mk.mkn:ℕfst✝¹:{ f // Function.Involutive f }snd✝¹:{ i // ∀ (h : i.isSome = true), ↑fst✝ (i.get h) = i.get h }fst✝:{ f // Function.Involutive f }snd✝:{ i // ∀ (h : i.isSome = true), ↑fst✝ (i.get h) = i.get h }h1:fst✝¹ = fst✝h2:snd✝¹ = (Equiv.subtypeEquivRight ⋯) snd✝⊢ ⟨fst✝¹, snd✝¹⟩ = ⟨fst✝, snd✝⟩
  subst h1mk.mkn:ℕfst✝:{ f // Function.Involutive f }snd✝¹:{ i // ∀ (h : i.isSome = true), ↑fst✝ (i.get h) = i.get h }snd✝:{ i // ∀ (h : i.isSome = true), ↑fst✝ (i.get h) = i.get h }h2:snd✝¹ = (Equiv.subtypeEquivRight ⋯) snd✝⊢ ⟨fst✝, snd✝¹⟩ = ⟨fst✝, snd✝⟩
  rename_i fst snd snd_1mk.mkn:ℕfst:{ f // Function.Involutive f }snd:{ i // ∀ (h : i.isSome = true), ↑fst✝ (i.get h) = i.get h }snd_1:{ i // ∀ (h : i.isSome = true), ↑fst✝ (i.get h) = i.get h }h2:snd✝¹ = (Equiv.subtypeEquivRight ⋯) snd✝⊢ ⟨fst✝, snd✝¹⟩ = ⟨fst✝, snd✝⟩
  simp_all only [Sigma.mk.inj_iff, heq_eq_eq, true_and]mk.mkn:ℕfst:{ f // Function.Involutive f }snd:{ i // ∀ (h : i.isSome = true), ↑fst✝ (i.get h) = i.get h }snd_1:{ i // ∀ (h : i.isSome = true), ↑fst✝ (i.get h) = i.get h }h2:snd✝¹ = (Equiv.subtypeEquivRight ⋯) snd✝⊢ (Equiv.subtypeEquivRight ⋯) snd_1 = snd_1
  obtain ⟨val, property⟩ := fstmk.mkn:ℕval:Fin n → Fin nproperty:Function.Involutive valsnd:{ i // ∀ (h : i.isSome = true), ↑⟨val, property⟩ (i.get h) = i.get h }snd_1:{ i // ∀ (h : i.isSome = true), ↑⟨val, property⟩ (i.get h) = i.get h }h2:snd = (Equiv.subtypeEquivRight ⋯) snd_1⊢ (Equiv.subtypeEquivRight ⋯) snd_1 = snd_1
  obtain ⟨val_1, property_1⟩ := sndmk.mkn:ℕval:Fin n → Fin nproperty:Function.Involutive valsnd_1:{ i // ∀ (h : i.isSome = true), ↑⟨val, property⟩ (i.get h) = i.get h }val_1:Option (Fin n)property_1:∀ (h : val_1.isSome = true), ↑⟨val, property⟩ (val_1.get h) = val_1.get hh2:⟨val_1, property_1⟩ = (Equiv.subtypeEquivRight ⋯) snd_1⊢ (Equiv.subtypeEquivRight ⋯) snd_1 = snd_1
  obtain ⟨val_2, property_2⟩ := snd_1mk.mkn:ℕval:Fin n → Fin nproperty:Function.Involutive valval_1:Option (Fin n)property_1:∀ (h : val_1.isSome = true), ↑⟨val, property⟩ (val_1.get h) = val_1.get hval_2:Option (Fin n)property_2:∀ (h : val_2.isSome = true), ↑⟨val, property⟩ (val_2.get h) = val_2.get hh2:⟨val_1, property_1⟩ = (Equiv.subtypeEquivRight ⋯) ⟨val_2, property_2⟩⊢ (Equiv.subtypeEquivRight ⋯) ⟨val_2, property_2⟩ = ⟨val_2, property_2⟩
  simp_all onlymk.mkn:ℕval:Fin n → Fin nproperty:Function.Involutive valval_1:Option (Fin n)property_1:∀ (h : val_1.isSome = true), ↑⟨val, property⟩ (val_1.get h) = val_1.get hval_2:Option (Fin n)property_2:∀ (h : val_2.isSome = true), ↑⟨val, property⟩ (val_2.get h) = val_2.get hh2:⟨val_1, property_1⟩ = (Equiv.subtypeEquivRight ⋯) ⟨val_2, property_2⟩⊢ (Equiv.subtypeEquivRight ⋯) ⟨val_2, property_2⟩ = ⟨val_2, property_2⟩
  rflAll goals completed! 🐙

Given an involution of Fin n, the optional choice of an element in Fin n which maps to itself is equivalent to the optional choice of an element in Fin (Finset.univ.filter fun i => f.1 i = i).card.


def involutionAddEquiv {n : ℕ} (f : {f : Fin n → Fin n // Function.Involutive f}) :
    {i : Option (Fin n) // ∀ (h : i.isSome), f.1 (Option.get i h) = (Option.get i h)} ≃
    Option (Fin (Finset.univ.filter fun i => f.1 i = i).card) := byn:ℕf:{ f // Function.Involutive f }⊢ { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option (Fin {i | ↑f i = i}.card)
  let e1 : {i : Option (Fin n) // ∀ (h : i.isSome), f.1 (Option.get i h) = (Option.get i h)}
        ≃ Option {i : Fin n // f.1 i = i} :=
    { toFun := fun i => match i with
        | ⟨some i, h⟩ => some ⟨i, byn:ℕf:{ f // Function.Involutive f }i✝:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h }i:Fin nh:∀ (h : (some i).isSome = true), ↑f ((some i).get h) = (some i).get h⊢ ↑f i = in:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }⊢ { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option (Fin {i | ↑f i = i}.card) simpa using hAll goals completed! 🐙n:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }⊢ { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option (Fin {i | ↑f i = i}.card)⟩
        | ⟨none, h⟩ => none
      invFun := fun i => match i with
        | some ⟨i, h⟩ => ⟨some i, byn:ℕf:{ f // Function.Involutive f }i✝:Option { i // ↑f i = i }i:Fin nh:↑f i = i⊢ ∀ (h : (some i).isSome = true), ↑f ((some i).get h) = (some i).get hn:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }⊢ { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option (Fin {i | ↑f i = i}.card) simpa using hAll goals completed! 🐙n:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }⊢ { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option (Fin {i | ↑f i = i}.card)⟩
        | none => ⟨none, byn:ℕf:{ f // Function.Involutive f }i:Option { i // ↑f i = i }⊢ ∀ (h : none.isSome = true), ↑f (none.get h) = none.get hn:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }⊢ { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option (Fin {i | ↑f i = i}.card) simpAll goals completed! 🐙n:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }⊢ { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option (Fin {i | ↑f i = i}.card)⟩
      left_inv := byn:ℕf:{ f // Function.Involutive f }⊢ Function.LeftInverse
  (fun i =>
    match i with
    | some ⟨i, h⟩ => ⟨some i, ⋯⟩
    | none => ⟨none, ⋯⟩)
  fun i =>
  match i with
  | ⟨some i, h⟩ => some ⟨i, ⋯⟩
  | ⟨none, h⟩ => nonen:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }⊢ { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option (Fin {i | ↑f i = i}.card)
        rintro ⟨_ | i, h⟩nonen:ℕf:{ f // Function.Involutive f }h:∀ (h : none.isSome = true), ↑f (none.get h) = none.get h⊢ (fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩)
    ((fun i =>
        match i with
        | ⟨some i, h⟩ => some ⟨i, ⋯⟩
        | ⟨none, h⟩ => none)
      ⟨none, h⟩) =
  ⟨none, h⟩somen:ℕf:{ f // Function.Involutive f }i:Fin nh:∀ (h : (some i).isSome = true), ↑f ((some i).get h) = (some i).get h⊢ (fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩)
    ((fun i =>
        match i with
        | ⟨some i, h⟩ => some ⟨i, ⋯⟩
        | ⟨none, h⟩ => none)
      ⟨some i, h⟩) =
  ⟨some i, h⟩n:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }⊢ { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option (Fin {i | ↑f i = i}.card) <;>nonen:ℕf:{ f // Function.Involutive f }h:∀ (h : none.isSome = true), ↑f (none.get h) = none.get h⊢ (fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩)
    ((fun i =>
        match i with
        | ⟨some i, h⟩ => some ⟨i, ⋯⟩
        | ⟨none, h⟩ => none)
      ⟨none, h⟩) =
  ⟨none, h⟩somen:ℕf:{ f // Function.Involutive f }i:Fin nh:∀ (h : (some i).isSome = true), ↑f ((some i).get h) = (some i).get h⊢ (fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩)
    ((fun i =>
        match i with
        | ⟨some i, h⟩ => some ⟨i, ⋯⟩
        | ⟨none, h⟩ => none)
      ⟨some i, h⟩) =
  ⟨some i, h⟩n:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }⊢ { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option (Fin {i | ↑f i = i}.card) rflAll goals completed! 🐙n:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }⊢ { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option (Fin {i | ↑f i = i}.card)
      right_inv := byn:ℕf:{ f // Function.Involutive f }⊢ Function.RightInverse
  (fun i =>
    match i with
    | some ⟨i, h⟩ => ⟨some i, ⋯⟩
    | none => ⟨none, ⋯⟩)
  fun i =>
  match i with
  | ⟨some i, h⟩ => some ⟨i, ⋯⟩
  | ⟨none, h⟩ => nonen:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }⊢ { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option (Fin {i | ↑f i = i}.card)
        rintro (_ | ⟨i, h⟩)nonen:ℕf:{ f // Function.Involutive f }⊢ (fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none)
    ((fun i =>
        match i with
        | some ⟨i, h⟩ => ⟨some i, ⋯⟩
        | none => ⟨none, ⋯⟩)
      none) =
  nonesomen:ℕf:{ f // Function.Involutive f }i:Fin nh:↑f i = i⊢ (fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none)
    ((fun i =>
        match i with
        | some ⟨i, h⟩ => ⟨some i, ⋯⟩
        | none => ⟨none, ⋯⟩)
      (some ⟨i, h⟩)) =
  some ⟨i, h⟩n:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }⊢ { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option (Fin {i | ↑f i = i}.card) <;>nonen:ℕf:{ f // Function.Involutive f }⊢ (fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none)
    ((fun i =>
        match i with
        | some ⟨i, h⟩ => ⟨some i, ⋯⟩
        | none => ⟨none, ⋯⟩)
      none) =
  nonesomen:ℕf:{ f // Function.Involutive f }i:Fin nh:↑f i = i⊢ (fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none)
    ((fun i =>
        match i with
        | some ⟨i, h⟩ => ⟨some i, ⋯⟩
        | none => ⟨none, ⋯⟩)
      (some ⟨i, h⟩)) =
  some ⟨i, h⟩n:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }⊢ { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option (Fin {i | ↑f i = i}.card) rflAll goals completed! 🐙n:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }⊢ { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option (Fin {i | ↑f i = i}.card) }n:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }⊢ { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option (Fin {i | ↑f i = i}.card)
  let s : Finset (Fin n) := Finset.univ.filter fun i => f.1 i = in:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }s:Finset (Fin n) := {i | ↑f i = i}⊢ { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option (Fin {i | ↑f i = i}.card)
  let e2' : { i : Fin n // f.1 i = i} ≃ {i // i ∈ s} := byn:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }s:Finset (Fin n) := {i | ↑f i = i}⊢ { i // ↑f i = i } ≃ ↥sn:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }s:Finset (Fin n) := {i | ↑f i = i}e2':{ i // ↑f i = i } ≃ ↥s := Equiv.subtypeEquivProp ⋯⊢ { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option (Fin {i | ↑f i = i}.card)
    apply Equiv.subtypeEquivPropn:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }s:Finset (Fin n) := {i | ↑f i = i}⊢ (fun i => ↑f i = i) = fun i => i ∈ sn:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }s:Finset (Fin n) := {i | ↑f i = i}e2':{ i // ↑f i = i } ≃ ↥s := Equiv.subtypeEquivProp ⋯⊢ { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option (Fin {i | ↑f i = i}.card)
    simp [s]n:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }s:Finset (Fin n) := {i | ↑f i = i}e2':{ i // ↑f i = i } ≃ ↥s := Equiv.subtypeEquivProp ⋯⊢ { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option (Fin {i | ↑f i = i}.card)n:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }s:Finset (Fin n) := {i | ↑f i = i}e2':{ i // ↑f i = i } ≃ ↥s := Equiv.subtypeEquivProp ⋯⊢ { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option (Fin {i | ↑f i = i}.card)
  let e2 : {i // i ∈ s} ≃ Fin (Finset.card s) := byn:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }s:Finset (Fin n) := {i | ↑f i = i}e2':{ i // ↑f i = i } ≃ ↥s := Equiv.subtypeEquivProp ⋯⊢ ↥s ≃ Fin s.cardn:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }s:Finset (Fin n) := {i | ↑f i = i}e2':{ i // ↑f i = i } ≃ ↥s := Equiv.subtypeEquivProp ⋯e2:↥s ≃ Fin s.card := (s.orderIsoOfFin ⋯).symm.toEquiv⊢ { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option (Fin {i | ↑f i = i}.card)
    refine (Finset.orderIsoOfFin _ ?_).symm.toEquivn:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }s:Finset (Fin n) := {i | ↑f i = i}e2':{ i // ↑f i = i } ≃ ↥s := Equiv.subtypeEquivProp ⋯⊢ s.card = s.cardn:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }s:Finset (Fin n) := {i | ↑f i = i}e2':{ i // ↑f i = i } ≃ ↥s := Equiv.subtypeEquivProp ⋯e2:↥s ≃ Fin s.card := (s.orderIsoOfFin ⋯).symm.toEquiv⊢ { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option (Fin {i | ↑f i = i}.card)
    simp [s]n:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }s:Finset (Fin n) := {i | ↑f i = i}e2':{ i // ↑f i = i } ≃ ↥s := Equiv.subtypeEquivProp ⋯e2:↥s ≃ Fin s.card := (s.orderIsoOfFin ⋯).symm.toEquiv⊢ { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option (Fin {i | ↑f i = i}.card)n:ℕf:{ f // Function.Involutive f }e1:{ i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option { i // ↑f i = i } := 
  {
    toFun := fun i =>
      match i with
      | ⟨some i, h⟩ => some ⟨i, ⋯⟩
      | ⟨none, h⟩ => none,
    invFun := fun i =>
      match i with
      | some ⟨i, h⟩ => ⟨some i, ⋯⟩
      | none => ⟨none, ⋯⟩,
    left_inv := ⋯, right_inv := ⋯ }s:Finset (Fin n) := {i | ↑f i = i}e2':{ i // ↑f i = i } ≃ ↥s := Equiv.subtypeEquivProp ⋯e2:↥s ≃ Fin s.card := (s.orderIsoOfFin ⋯).symm.toEquiv⊢ { i // ∀ (h : i.isSome = true), ↑f (i.get h) = i.get h } ≃ Option (Fin {i | ↑f i = i}.card)
  refine e1.trans (Equiv.optionCongr (e2'.trans (e2)))All goals completed! 🐙

              lemma involutionAddEquiv_none_image_zero {n : ℕ} :
    {f : {f : Fin n.succ → Fin n.succ // Function.Involutive f}}
    → involutionAddEquiv (involutionCons n f).1 (involutionCons n f).2 = none
    → f.1 ⟨0, Nat.zero_lt_succ n⟩ = ⟨0, Nat.zero_lt_succ n⟩ := byn:ℕ⊢ ∀ {f : { f // Function.Involutive f }},
  (involutionAddEquiv ((involutionCons n) f).fst) ((involutionCons n) f).snd = none → ↑f ⟨0, ⋯⟩ = ⟨0, ⋯⟩
  intro f hn:ℕf:{ f // Function.Involutive f }h:(involutionAddEquiv ((involutionCons n) f).fst) ((involutionCons n) f).snd = none⊢ ↑f ⟨0, ⋯⟩ = ⟨0, ⋯⟩
  by_contra hf0n:ℕf:{ f // Function.Involutive f }h:(involutionAddEquiv ((involutionCons n) f).fst) ((involutionCons n) f).snd = nonehf0:¬↑f ⟨0, ⋯⟩ = ⟨0, ⋯⟩⊢ False
  simp only [Fin.zero_eta] at hf0n:ℕf:{ f // Function.Involutive f }h:(involutionAddEquiv ((involutionCons n) f).fst) ((involutionCons n) f).snd = nonehf0:¬↑f 0 = 0⊢ False
  simp only [succ_eq_add_one, involutionCons, Equiv.coe_fn_mk, involutionAddEquiv,
    Option.isSome_some, Option.get_some, Option.isSome_none, dif_neg hf0] at hn:ℕf:{ f // Function.Involutive f }hf0:¬↑f 0 = 0h:({
          toFun := fun i =>
            match i with
            | ⟨some i, h⟩ => some ⟨i, ⋯⟩
            | ⟨none, h⟩ => none,
          invFun := fun i =>
            match i with
            | some ⟨i, h⟩ => ⟨some i, ⋯⟩
            | none => ⟨none, ⋯⟩,
          left_inv := ⋯, right_inv := ⋯ }.trans
      ((Equiv.subtypeEquivProp ⋯).trans
          ({i | (if h : ↑f i.succ = 0 then i else (↑f i.succ).pred h) = i}.orderIsoOfFin ⋯).symm.toEquiv).optionCongr)
    ⟨some ((↑f 0).pred hf0), ⋯⟩ =
  none⊢ False
  exact absurd h (Option.some_ne_none _)All goals completed! 🐙

lemma involutionAddEquiv_cast {n : ℕ} {f1 f2 : {f : Fin n → Fin n // Function.Involutive f}}
    (hf : f1 = f2) :
    involutionAddEquiv f1 = (Equiv.subtypeEquivRight (byn:ℕf1:{ f // Function.Involutive f }f2:{ f // Function.Involutive f }hf:f1 = f2⊢ ∀ (x : Option (Fin n)),
  (∀ (h : x.isSome = true), ↑f1 (x.get h) = x.get h) ↔ ∀ (h : x.isSome = true), ↑f2 (x.get h) = x.get h rw [hfn:ℕf1:{ f // Function.Involutive f }f2:{ f // Function.Involutive f }hf:f1 = f2⊢ ∀ (x : Option (Fin n)),
  (∀ (h : x.isSome = true), ↑f2 (x.get h) = x.get h) ↔ ∀ (h : x.isSome = true), ↑f2 (x.get h) = x.get hn:ℕf1:{ f // Function.Involutive f }f2:{ f // Function.Involutive f }hf:f1 = f2⊢ ∀ (x : Option (Fin n)),
  (∀ (h : x.isSome = true), ↑f2 (x.get h) = x.get h) ↔ ∀ (h : x.isSome = true), ↑f2 (x.get h) = x.get h]n:ℕf1:{ f // Function.Involutive f }f2:{ f // Function.Involutive f }hf:f1 = f2⊢ ∀ (x : Option (Fin n)),
  (∀ (h : x.isSome = true), ↑f2 (x.get h) = x.get h) ↔ ∀ (h : x.isSome = true), ↑f2 (x.get h) = x.get h; simpAll goals completed! 🐙)).trans
      ((involutionAddEquiv f2).trans (Equiv.optionCongr (finCongr (byn:ℕf1:{ f // Function.Involutive f }f2:{ f // Function.Involutive f }hf:f1 = f2⊢ {i | ↑f2 i = i}.card = {i | ↑f1 i = i}.card rw [hfn:ℕf1:{ f // Function.Involutive f }f2:{ f // Function.Involutive f }hf:f1 = f2⊢ {i | ↑f2 i = i}.card = {i | ↑f2 i = i}.cardAll goals completed! 🐙]All goals completed! 🐙)))) := byn:ℕf1:{ f // Function.Involutive f }f2:{ f // Function.Involutive f }hf:f1 = f2⊢ involutionAddEquiv f1 = (Equiv.subtypeEquivRight ⋯).trans ((involutionAddEquiv f2).trans (finCongr ⋯).optionCongr)
  subst hfn:ℕf1:{ f // Function.Involutive f }⊢ involutionAddEquiv f1 = (Equiv.subtypeEquivRight ⋯).trans ((involutionAddEquiv f1).trans (finCongr ⋯).optionCongr)
  rw [finCongr_refl,n:ℕf1:{ f // Function.Involutive f }⊢ involutionAddEquiv f1 =
  (Equiv.subtypeEquivRight ⋯).trans ((involutionAddEquiv f1).trans (Equiv.refl (Fin {i | ↑f1 i = i}.card)).optionCongr)n:ℕf1:{ f // Function.Involutive f }⊢ involutionAddEquiv f1 =
  (Equiv.subtypeEquivRight ⋯).trans ((involutionAddEquiv f1).trans (Equiv.refl (Option (Fin {i | ↑f1 i = i}.card)))) Equiv.optionCongr_refln:ℕf1:{ f // Function.Involutive f }⊢ involutionAddEquiv f1 =
  (Equiv.subtypeEquivRight ⋯).trans ((involutionAddEquiv f1).trans (Equiv.refl (Option (Fin {i | ↑f1 i = i}.card))))n:ℕf1:{ f // Function.Involutive f }⊢ involutionAddEquiv f1 =
  (Equiv.subtypeEquivRight ⋯).trans ((involutionAddEquiv f1).trans (Equiv.refl (Option (Fin {i | ↑f1 i = i}.card))))]n:ℕf1:{ f // Function.Involutive f }⊢ involutionAddEquiv f1 =
  (Equiv.subtypeEquivRight ⋯).trans ((involutionAddEquiv f1).trans (Equiv.refl (Option (Fin {i | ↑f1 i = i}.card))))
  rflAll goals completed! 🐙lemma involutionAddEquiv_cast' {m : ℕ} {f1 f2 : {f : Fin m → Fin m // Function.Involutive f}}
    {N : ℕ} (hf : f1 = f2) (n : Option (Fin N))
    (hn1 : N = (Finset.filter (fun i => f1.1 i = i) Finset.univ).card)
    (hn2 : N = (Finset.filter (fun i => f2.1 i = i) Finset.univ).card) :
    HEq ((involutionAddEquiv f1).symm (Option.map (finCongr hn1) n))
    ((involutionAddEquiv f2).symm (Option.map (finCongr hn2) n)) := bym:ℕf1:{ f // Function.Involutive f }f2:{ f // Function.Involutive f }N:ℕhf:f1 = f2n:Option (Fin N)hn1:N = {i | ↑f1 i = i}.cardhn2:N = {i | ↑f2 i = i}.card⊢ (involutionAddEquiv f1).symm (Option.map (⇑(finCongr hn1)) n) ≍
  (involutionAddEquiv f2).symm (Option.map (⇑(finCongr hn2)) n)
  subst hfm:ℕf1:{ f // Function.Involutive f }N:ℕn:Option (Fin N)hn1:N = {i | ↑f1 i = i}.cardhn2:N = {i | ↑f1 i = i}.card⊢ (involutionAddEquiv f1).symm (Option.map (⇑(finCongr hn1)) n) ≍
  (involutionAddEquiv f1).symm (Option.map (⇑(finCongr hn2)) n)
  rflAll goals completed! 🐙lemma involutionAddEquiv_none_succ {n : ℕ}
    {f : {f : Fin n.succ → Fin n.succ // Function.Involutive f}}
    (h : involutionAddEquiv (involutionCons n f).1 (involutionCons n f).2 = none)
    (x : Fin n) : f.1 x.succ = x.succ ↔ (involutionCons n f).1.1 x = x := byn:ℕf:{ f // Function.Involutive f }h:(involutionAddEquiv ((involutionCons n) f).fst) ((involutionCons n) f).snd = nonex:Fin n⊢ ↑f x.succ = x.succ ↔ ↑((involutionCons n) f).fst x = x
  simp only [succ_eq_add_one, involutionCons, Fin.cons_update, Equiv.coe_fn_mk, dite_eq_left_iff]n:ℕf:{ f // Function.Involutive f }h:(involutionAddEquiv ((involutionCons n) f).fst) ((involutionCons n) f).snd = nonex:Fin n⊢ ↑f x.succ = x.succ ↔ ∀ (h : ¬↑f x.succ = 0), (↑f x.succ).pred h = x
  have hx : ¬ f.1 x.succ = ⟨0, Nat.zero_lt_succ n⟩:=
    involutionAddEquiv_none_image_zero h ▸
      fun hn => Fin.succ_ne_zero x (Function.Involutive.injective f.2 hn)n:ℕf:{ f // Function.Involutive f }h:(involutionAddEquiv ((involutionCons n) f).fst) ((involutionCons n) f).snd = nonex:Fin nhx:¬↑f x.succ = ⟨0, ⋯⟩⊢ ↑f x.succ = x.succ ↔ ∀ (h : ¬↑f x.succ = 0), (↑f x.succ).pred h = x
  exact Iff.intro (fun h2 ↦ byn:ℕf:{ f // Function.Involutive f }h:(involutionAddEquiv ((involutionCons n) f).fst) ((involutionCons n) f).snd = nonex:Fin nhx:¬↑f x.succ = ⟨0, ⋯⟩h2:↑f x.succ = x.succ⊢ ∀ (h : ¬↑f x.succ = 0), (↑f x.succ).pred h = x simp [h2]All goals completed! 🐙) (fun h2 ↦ (Fin.pred_eq_iff_eq_succ hx).mp (h2 hx))

Equivalences of involutions with no fixed points.

The main aim of these equivalences is to define involutionNoFixedZeroEquivProd.

Fixed point free involutions of Fin n.succ can be separated based on where they sent 0.

def involutionNoFixedEquivSum {n : ℕ} :
    {f : Fin n.succ → Fin n.succ // Function.Involutive f
    ∧ ∀ i, f i ≠ i} ≃ Σ (k : Fin n), {f : Fin n.succ → Fin n.succ // Function.Involutive f
    ∧ (∀ i, f i ≠ i) ∧ f 0 = k.succ} where
  toFun f := ⟨(f.1 0).pred (f.2.2 0), ⟨f.1, f.2.1, byn:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n.succ), f i ≠ i }⊢ (∀ (i : Fin n.succ), ↑f i ≠ i) ∧ ↑f 0 = ((↑f 0).pred ⋯).succ simpa using f.2.2All goals completed! 🐙⟩⟩
  invFun f := ⟨f.2.1, ⟨f.2.2.1, f.2.2.2.1⟩⟩
  left_inv f := rfl
  right_inv f := byn:ℕf:(k : Fin n) × { f // Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ f 0 = k.succ }⊢ (fun f => ⟨(↑f 0).pred ⋯, ⟨↑f, ⋯⟩⟩) ((fun f => ⟨↑f.snd, ⋯⟩) f) = f extan:ℕf:(k : Fin n) × { f // Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ f 0 = k.succ }⊢ ↑((fun f => ⟨(↑f 0).pred ⋯, ⟨↑f, ⋯⟩⟩) ((fun f => ⟨↑f.snd, ⋯⟩) f)).fst = ↑f.fstan:ℕf:(k : Fin n) × { f // Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ f 0 = k.succ }x✝:Fin n.succ⊢ ↑(↑((fun f => ⟨(↑f 0).pred ⋯, ⟨↑f, ⋯⟩⟩) ((fun f => ⟨↑f.snd, ⋯⟩) f)).snd x✝) = ↑(↑f.snd x✝) <;>an:ℕf:(k : Fin n) × { f // Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ f 0 = k.succ }⊢ ↑((fun f => ⟨(↑f 0).pred ⋯, ⟨↑f, ⋯⟩⟩) ((fun f => ⟨↑f.snd, ⋯⟩) f)).fst = ↑f.fstan:ℕf:(k : Fin n) × { f // Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ f 0 = k.succ }x✝:Fin n.succ⊢ ↑(↑((fun f => ⟨(↑f 0).pred ⋯, ⟨↑f, ⋯⟩⟩) ((fun f => ⟨↑f.snd, ⋯⟩) f)).snd x✝) = ↑(↑f.snd x✝) try aesopAll goals completed! 🐙

The condition on fixed point free involutions of Fin n.succ for a fixed value of f 0, can be modified by conjugation with an equivalence.


def involutionNoFixedZeroSetEquivEquiv {n : ℕ}
    (k : Fin n) (e : Fin n.succ ≃ Fin n.succ) :
    {f : Fin n.succ → Fin n.succ // Function.Involutive f ∧ (∀ i, f i ≠ i) ∧ f 0 = k.succ} ≃
    {f : Fin n.succ → Fin n.succ // Function.Involutive (e.symm ∘ f ∘ e) ∧
      (∀ i, (e.symm ∘ f ∘ e) i ≠ i) ∧ (e.symm ∘ f ∘ e) 0 = k.succ} where
  toFun f := ⟨e ∘ f.1 ∘ e.symm, byn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ f 0 = k.succ }⊢ Function.Involutive (⇑e.symm ∘ (⇑e ∘ ↑f ∘ ⇑e.symm) ∘ ⇑e)
    intro in:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ f 0 = k.succ }i:Fin n.succ⊢ (⇑e.symm ∘ (⇑e ∘ ↑f ∘ ⇑e.symm) ∘ ⇑e) ((⇑e.symm ∘ (⇑e ∘ ↑f ∘ ⇑e.symm) ∘ ⇑e) i) = i
    simp only [succ_eq_add_one, ne_eq, Function.comp_apply, Equiv.symm_apply_apply]n:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ f 0 = k.succ }i:Fin n.succ⊢ ↑f (↑f i) = i
    rw [f.2.1n:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ f 0 = k.succ }i:Fin n.succ⊢ i = iAll goals completed! 🐙]All goals completed! 🐙, byn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ f 0 = k.succ }⊢ ∀ (i : Fin n.succ), (⇑e.symm ∘ (⇑e ∘ ↑f ∘ ⇑e.symm) ∘ ⇑e) i ≠ i simpa using f.2.2.1All goals completed! 🐙, byn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ f 0 = k.succ }⊢ (⇑e.symm ∘ (⇑e ∘ ↑f ∘ ⇑e.symm) ∘ ⇑e) 0 = k.succ simpa using f.2.2.2All goals completed! 🐙⟩
  invFun f := ⟨e.symm ∘ f.1 ∘ e, byn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:{ f //
  Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ∧
    (∀ (i : Fin n.succ), (⇑e.symm ∘ f ∘ ⇑e) i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ }⊢ Function.Involutive (⇑e.symm ∘ ↑f ∘ ⇑e)
    intro in:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:{ f //
  Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ∧
    (∀ (i : Fin n.succ), (⇑e.symm ∘ f ∘ ⇑e) i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ }i:Fin n.succ⊢ (⇑e.symm ∘ ↑f ∘ ⇑e) ((⇑e.symm ∘ ↑f ∘ ⇑e) i) = i
    simpa using f.2.1 iAll goals completed! 🐙, byn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:{ f //
  Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ∧
    (∀ (i : Fin n.succ), (⇑e.symm ∘ f ∘ ⇑e) i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ }⊢ ∀ (i : Fin n.succ), (⇑e.symm ∘ ↑f ∘ ⇑e) i ≠ i simpa using f.2.2.1All goals completed! 🐙, byn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:{ f //
  Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ∧
    (∀ (i : Fin n.succ), (⇑e.symm ∘ f ∘ ⇑e) i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ }⊢ (⇑e.symm ∘ ↑f ∘ ⇑e) 0 = k.succ simpa using f.2.2.2All goals completed! 🐙⟩
  left_inv f := byn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ f 0 = k.succ }⊢ (fun f => ⟨⇑e.symm ∘ ↑f ∘ ⇑e, ⋯⟩) ((fun f => ⟨⇑e ∘ ↑f ∘ ⇑e.symm, ⋯⟩) f) = f ext in:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ f 0 = k.succ }i:Fin n.succ⊢ ↑(↑((fun f => ⟨⇑e.symm ∘ ↑f ∘ ⇑e, ⋯⟩) ((fun f => ⟨⇑e ∘ ↑f ∘ ⇑e.symm, ⋯⟩) f)) i) = ↑(↑f i); simpAll goals completed! 🐙
  right_inv f := byn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:{ f //
  Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ∧
    (∀ (i : Fin n.succ), (⇑e.symm ∘ f ∘ ⇑e) i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ }⊢ (fun f => ⟨⇑e ∘ ↑f ∘ ⇑e.symm, ⋯⟩) ((fun f => ⟨⇑e.symm ∘ ↑f ∘ ⇑e, ⋯⟩) f) = f ext in:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:{ f //
  Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ∧
    (∀ (i : Fin n.succ), (⇑e.symm ∘ f ∘ ⇑e) i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ }i:Fin n.succ⊢ ↑(↑((fun f => ⟨⇑e ∘ ↑f ∘ ⇑e.symm, ⋯⟩) ((fun f => ⟨⇑e.symm ∘ ↑f ∘ ⇑e, ⋯⟩) f)) i) = ↑(↑f i); simpAll goals completed! 🐙

The condition on fixed point free involutions of Fin n.succ for a fixed value of f 0 given an equivalence e, can be modified so that only the condition on f 0 is up-to the equivalence e.


def involutionNoFixedZeroSetEquivSetEquiv {n : ℕ} (k : Fin n)
    (e : Fin n.succ ≃ Fin n.succ) :
    {f : Fin n.succ → Fin n.succ // Function.Involutive (e.symm ∘ f ∘ e) ∧
    (∀ i, (e.symm ∘ f ∘ e) i ≠ i) ∧ (e.symm ∘ f ∘ e) 0 = k.succ} ≃
    {f : Fin n.succ → Fin n.succ // Function.Involutive f ∧
      (∀ i, f i ≠ i) ∧ (e.symm ∘ f ∘ e) 0 = k.succ} := byn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succ⊢ { f //
    Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ∧
      (∀ (i : Fin n.succ), (⇑e.symm ∘ f ∘ ⇑e) i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ } ≃
  { f // Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ }
  refine Equiv.subtypeEquivRight fun f ↦ ?_n:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succ⊢ Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ∧
    (∀ (i : Fin n.succ), (⇑e.symm ∘ f ∘ ⇑e) i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ ↔
  Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ
  have h1 : Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ↔ Function.Involutive f := byn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succ⊢ { f //
    Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ∧
      (∀ (i : Fin n.succ), (⇑e.symm ∘ f ∘ ⇑e) i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ } ≃
  { f // Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ }n:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh1:Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ↔ Function.Involutive f⊢ Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ∧
    (∀ (i : Fin n.succ), (⇑e.symm ∘ f ∘ ⇑e) i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ ↔
  Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ
    apply Iff.intrompn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succ⊢ Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) → Function.Involutive fmprn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succ⊢ Function.Involutive f → Function.Involutive (⇑e.symm ∘ f ∘ ⇑e)n:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh1:Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ↔ Function.Involutive f⊢ Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ∧
    (∀ (i : Fin n.succ), (⇑e.symm ∘ f ∘ ⇑e) i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ ↔
  Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ <;>mpn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succ⊢ Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) → Function.Involutive fmprn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succ⊢ Function.Involutive f → Function.Involutive (⇑e.symm ∘ f ∘ ⇑e)n:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh1:Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ↔ Function.Involutive f⊢ Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ∧
    (∀ (i : Fin n.succ), (⇑e.symm ∘ f ∘ ⇑e) i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ ↔
  Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ intro h imprn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh:Function.Involutive fi:Fin n.succ⊢ (⇑e.symm ∘ f ∘ ⇑e) ((⇑e.symm ∘ f ∘ ⇑e) i) = in:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh1:Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ↔ Function.Involutive f⊢ Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ∧
    (∀ (i : Fin n.succ), (⇑e.symm ∘ f ∘ ⇑e) i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ ↔
  Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ
    ·mpn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh:Function.Involutive (⇑e.symm ∘ f ∘ ⇑e)i:Fin n.succ⊢ f (f i) = in:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh1:Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ↔ Function.Involutive f⊢ Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ∧
    (∀ (i : Fin n.succ), (⇑e.symm ∘ f ∘ ⇑e) i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ ↔
  Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ simpa using h (e.symm i)All goals completed! 🐙n:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh1:Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ↔ Function.Involutive f⊢ Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ∧
    (∀ (i : Fin n.succ), (⇑e.symm ∘ f ∘ ⇑e) i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ ↔
  Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ
    ·mprn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh:Function.Involutive fi:Fin n.succ⊢ (⇑e.symm ∘ f ∘ ⇑e) ((⇑e.symm ∘ f ∘ ⇑e) i) = in:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh1:Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ↔ Function.Involutive f⊢ Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ∧
    (∀ (i : Fin n.succ), (⇑e.symm ∘ f ∘ ⇑e) i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ ↔
  Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ simp [h (e i)]n:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh1:Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ↔ Function.Involutive f⊢ Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ∧
    (∀ (i : Fin n.succ), (⇑e.symm ∘ f ∘ ⇑e) i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ ↔
  Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh1:Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ↔ Function.Involutive f⊢ Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ∧
    (∀ (i : Fin n.succ), (⇑e.symm ∘ f ∘ ⇑e) i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ ↔
  Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ
  rw [h1n:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh1:Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ↔ Function.Involutive f⊢ Function.Involutive f ∧ (∀ (i : Fin n.succ), (⇑e.symm ∘ f ∘ ⇑e) i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ ↔
  Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh1:Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ↔ Function.Involutive f⊢ Function.Involutive f ∧ (∀ (i : Fin n.succ), (⇑e.symm ∘ f ∘ ⇑e) i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ ↔
  Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ]n:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh1:Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ↔ Function.Involutive f⊢ Function.Involutive f ∧ (∀ (i : Fin n.succ), (⇑e.symm ∘ f ∘ ⇑e) i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ ↔
  Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ
  simp only [succ_eq_add_one, Function.comp_apply, ne_eq, and_congr_right_iff, and_congr_left_iff]n:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh1:Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ↔ Function.Involutive f⊢ Function.Involutive f →
  e.symm (f (e 0)) = k.succ → ((∀ (i : Fin (n + 1)), ¬e.symm (f (e i)) = i) ↔ ∀ (i : Fin (n + 1)), ¬f i = i)
  intro h1 h2n:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh1✝:Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ↔ Function.Involutive fh1:Function.Involutive fh2:e.symm (f (e 0)) = k.succ⊢ (∀ (i : Fin (n + 1)), ¬e.symm (f (e i)) = i) ↔ ∀ (i : Fin (n + 1)), ¬f i = i
  apply Iff.intrompn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh1✝:Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ↔ Function.Involutive fh1:Function.Involutive fh2:e.symm (f (e 0)) = k.succ⊢ (∀ (i : Fin (n + 1)), ¬e.symm (f (e i)) = i) → ∀ (i : Fin (n + 1)), ¬f i = imprn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh1✝:Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ↔ Function.Involutive fh1:Function.Involutive fh2:e.symm (f (e 0)) = k.succ⊢ (∀ (i : Fin (n + 1)), ¬f i = i) → ∀ (i : Fin (n + 1)), ¬e.symm (f (e i)) = i
  ·mpn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh1✝:Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ↔ Function.Involutive fh1:Function.Involutive fh2:e.symm (f (e 0)) = k.succ⊢ (∀ (i : Fin (n + 1)), ¬e.symm (f (e i)) = i) → ∀ (i : Fin (n + 1)), ¬f i = i intro h impn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh1✝:Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ↔ Function.Involutive fh1:Function.Involutive fh2:e.symm (f (e 0)) = k.succh:∀ (i : Fin (n + 1)), ¬e.symm (f (e i)) = ii:Fin (n + 1)⊢ ¬f i = i
    simpa using h (e.symm i)All goals completed! 🐙
  ·mprn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh1✝:Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ↔ Function.Involutive fh1:Function.Involutive fh2:e.symm (f (e 0)) = k.succ⊢ (∀ (i : Fin (n + 1)), ¬f i = i) → ∀ (i : Fin (n + 1)), ¬e.symm (f (e i)) = i intro h imprn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh1✝:Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ↔ Function.Involutive fh1:Function.Involutive fh2:e.symm (f (e 0)) = k.succh:∀ (i : Fin (n + 1)), ¬f i = ii:Fin (n + 1)⊢ ¬e.symm (f (e i)) = i
    have hi := h (e i)mprn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh1✝:Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ↔ Function.Involutive fh1:Function.Involutive fh2:e.symm (f (e 0)) = k.succh:∀ (i : Fin (n + 1)), ¬f i = ii:Fin (n + 1)hi:¬f (e i) = e i⊢ ¬e.symm (f (e i)) = i
    by_contra hnmprn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh1✝:Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ↔ Function.Involutive fh1:Function.Involutive fh2:e.symm (f (e 0)) = k.succh:∀ (i : Fin (n + 1)), ¬f i = ii:Fin (n + 1)hi:¬f (e i) = e ihn:e.symm (f (e i)) = i⊢ False
    nth_rewrite 2 [← hn] at himprn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin n.succ → Fin n.succh1✝:Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ↔ Function.Involutive fh1:Function.Involutive fh2:e.symm (f (e 0)) = k.succh:∀ (i : Fin (n + 1)), ¬f i = ii:Fin (n + 1)hi:¬f (e i) = e (e.symm (f (e i)))hn:e.symm (f (e i)) = i⊢ False
    simp at hiAll goals completed! 🐙

Fixed point free involutions of Fin n.succ fixing (e.symm ∘ f ∘ e) = k.succ for a given e are equivalent to fixing f (e 0) = e k.succ.

def involutionNoFixedZeroSetEquivEquiv' {n : ℕ} (k : Fin n) (e : Fin n.succ ≃ Fin n.succ) :
    {f : Fin n.succ → Fin n.succ // Function.Involutive f ∧
    (∀ i, f i ≠ i) ∧ (e.symm ∘ f ∘ e) 0 = k.succ} ≃
    {f : Fin n.succ → Fin n.succ // Function.Involutive f ∧
    (∀ i, f i ≠ i) ∧ f (e 0) = e k.succ} := byn:ℕk:Fin ne:Fin n.succ ≃ Fin n.succ⊢ { f // Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ (⇑e.symm ∘ f ∘ ⇑e) 0 = k.succ } ≃
  { f // Function.Involutive f ∧ (∀ (i : Fin n.succ), f i ≠ i) ∧ f (e 0) = e k.succ }
  refine Equiv.subtypeEquivRight ?_n:ℕk:Fin ne:Fin n.succ ≃ Fin n.succ⊢ ∀ (x : Fin n.succ → Fin n.succ),
  Function.Involutive x ∧ (∀ (i : Fin n.succ), x i ≠ i) ∧ (⇑e.symm ∘ x ∘ ⇑e) 0 = k.succ ↔
    Function.Involutive x ∧ (∀ (i : Fin n.succ), x i ≠ i) ∧ x (e 0) = e k.succ
  simp only [succ_eq_add_one, ne_eq, Function.comp_apply, and_congr_right_iff]n:ℕk:Fin ne:Fin n.succ ≃ Fin n.succ⊢ ∀ (x : Fin (n + 1) → Fin (n + 1)),
  Function.Involutive x → (∀ (i : Fin (n + 1)), ¬x i = i) → (e.symm (x (e 0)) = k.succ ↔ x (e 0) = e k.succ)
  intro f hi h1n:ℕk:Fin ne:Fin n.succ ≃ Fin n.succf:Fin (n + 1) → Fin (n + 1)hi:Function.Involutive fh1:∀ (i : Fin (n + 1)), ¬f i = i⊢ e.symm (f (e 0)) = k.succ ↔ f (e 0) = e k.succ
  exact Equiv.symm_apply_eq eAll goals completed! 🐙

Fixed point involutions of Fin n.succ.succ with f 0 = k.succ are equivalent to fixed point involutions with f 0 = 1.


def involutionNoFixedZeroSetEquivSetOne {n : ℕ} (k : Fin n.succ) :
    {f : Fin n.succ.succ → Fin n.succ.succ // Function.Involutive f ∧
      (∀ i, f i ≠ i) ∧ f 0 = k.succ}
      ≃ {f : Fin n.succ.succ → Fin n.succ.succ // Function.Involutive f ∧
      (∀ i, f i ≠ i) ∧ f 0 = 1} := byn:ℕk:Fin n.succ⊢ { f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = k.succ } ≃
  { f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }
  refine (involutionNoFixedZeroSetEquivEquiv k (Equiv.swap k.succ 1)).trans ?_n:ℕk:Fin n.succ⊢ { f //
    Function.Involutive (⇑(Equiv.symm (Equiv.swap k.succ 1)) ∘ f ∘ ⇑(Equiv.swap k.succ 1)) ∧
      (∀ (i : Fin n.succ.succ), (⇑(Equiv.symm (Equiv.swap k.succ 1)) ∘ f ∘ ⇑(Equiv.swap k.succ 1)) i ≠ i) ∧
        (⇑(Equiv.symm (Equiv.swap k.succ 1)) ∘ f ∘ ⇑(Equiv.swap k.succ 1)) 0 = k.succ } ≃
  { f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }
  refine (involutionNoFixedZeroSetEquivSetEquiv k (Equiv.swap k.succ 1)).trans ?_n:ℕk:Fin n.succ⊢ { f //
    Function.Involutive f ∧
      (∀ (i : Fin n.succ.succ), f i ≠ i) ∧
        (⇑(Equiv.symm (Equiv.swap k.succ 1)) ∘ f ∘ ⇑(Equiv.swap k.succ 1)) 0 = k.succ } ≃
  { f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }
  refine (involutionNoFixedZeroSetEquivEquiv' k (Equiv.swap k.succ 1)).trans ?_n:ℕk:Fin n.succ⊢ { f //
    Function.Involutive f ∧
      (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f ((Equiv.swap k.succ 1) 0) = (Equiv.swap k.succ 1) k.succ } ≃
  { f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }
  refine Equiv.subtypeEquivRight ?_n:ℕk:Fin n.succ⊢ ∀ (x : Fin n.succ.succ → Fin n.succ.succ),
  Function.Involutive x ∧
      (∀ (i : Fin n.succ.succ), x i ≠ i) ∧ x ((Equiv.swap k.succ 1) 0) = (Equiv.swap k.succ 1) k.succ ↔
    Function.Involutive x ∧ (∀ (i : Fin n.succ.succ), x i ≠ i) ∧ x 0 = 1
  simp only [succ_eq_add_one, ne_eq, Equiv.swap_apply_left, and_congr_right_iff]n:ℕk:Fin n.succ⊢ ∀ (x : Fin (n + 1 + 1) → Fin (n + 1 + 1)),
  Function.Involutive x → (∀ (i : Fin (n + 1 + 1)), ¬x i = i) → (x ((Equiv.swap k.succ 1) 0) = 1 ↔ x 0 = 1)
  intro f hi h1n:ℕk:Fin n.succf:Fin (n + 1 + 1) → Fin (n + 1 + 1)hi:Function.Involutive fh1:∀ (i : Fin (n + 1 + 1)), ¬f i = i⊢ f ((Equiv.swap k.succ 1) 0) = 1 ↔ f 0 = 1
  rw [Equiv.swap_apply_of_ne_of_nen:ℕk:Fin n.succf:Fin (n + 1 + 1) → Fin (n + 1 + 1)hi:Function.Involutive fh1:∀ (i : Fin (n + 1 + 1)), ¬f i = i⊢ f 0 = 1 ↔ f 0 = 1an:ℕk:Fin n.succf:Fin (n + 1 + 1) → Fin (n + 1 + 1)hi:Function.Involutive fh1:∀ (i : Fin (n + 1 + 1)), ¬f i = i⊢ 0 ≠ k.succan:ℕk:Fin n.succf:Fin (n + 1 + 1) → Fin (n + 1 + 1)hi:Function.Involutive fh1:∀ (i : Fin (n + 1 + 1)), ¬f i = i⊢ 0 ≠ 1an:ℕk:Fin n.succf:Fin (n + 1 + 1) → Fin (n + 1 + 1)hi:Function.Involutive fh1:∀ (i : Fin (n + 1 + 1)), ¬f i = i⊢ 0 ≠ k.succan:ℕk:Fin n.succf:Fin (n + 1 + 1) → Fin (n + 1 + 1)hi:Function.Involutive fh1:∀ (i : Fin (n + 1 + 1)), ¬f i = i⊢ 0 ≠ 1]an:ℕk:Fin n.succf:Fin (n + 1 + 1) → Fin (n + 1 + 1)hi:Function.Involutive fh1:∀ (i : Fin (n + 1 + 1)), ¬f i = i⊢ 0 ≠ k.succan:ℕk:Fin n.succf:Fin (n + 1 + 1) → Fin (n + 1 + 1)hi:Function.Involutive fh1:∀ (i : Fin (n + 1 + 1)), ¬f i = i⊢ 0 ≠ 1
  ·an:ℕk:Fin n.succf:Fin (n + 1 + 1) → Fin (n + 1 + 1)hi:Function.Involutive fh1:∀ (i : Fin (n + 1 + 1)), ¬f i = i⊢ 0 ≠ k.succ exact Ne.symm (Fin.succ_ne_zero k)All goals completed! 🐙
  ·an:ℕk:Fin n.succf:Fin (n + 1 + 1) → Fin (n + 1 + 1)hi:Function.Involutive fh1:∀ (i : Fin (n + 1 + 1)), ¬f i = i⊢ 0 ≠ 1 exact Fin.zero_ne_oneAll goals completed! 🐙

Fixed point involutions of Fin n.succ.succ fixing f 0 = 1 are equivalent to fixed point involutions of Fin n.


def involutionNoFixedSetOne {n : ℕ} :
    {f : Fin n.succ.succ → Fin n.succ.succ // Function.Involutive f ∧
    (∀ i, f i ≠ i) ∧ f 0 = 1} ≃ {f : Fin n → Fin n // Function.Involutive f ∧
    (∀ i, f i ≠ i)} where
  toFun f := byn:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }⊢ { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }
    have f_succ_succ_ne_zero_one (i : Fin n) : f.1 i.succ.succ ≠ 0 ∧ f.1 i.succ.succ ≠ 1 := by
      have hf1 : f.1 1 = 0 := byn:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }⊢ { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }i:Fin nhf1:↑f 1 = 0⊢ ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1⊢ { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }
        simp only [← f.2.2.2]n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }i:Fin n⊢ ↑f (↑f 0) = 0n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }i:Fin nhf1:↑f 1 = 0⊢ ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1⊢ { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }
        rw [f.2.1n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }i:Fin n⊢ 0 = 0n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }i:Fin nhf1:↑f 1 = 0⊢ ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1⊢ { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }]n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }i:Fin nhf1:↑f 1 = 0⊢ ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1⊢ { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }i:Fin nhf1:↑f 1 = 0⊢ ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1⊢ { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }
      refine ⟨fun hn => ?_, fun hn => ?_⟩refine_1n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }i:Fin nhf1:↑f 1 = 0hn:↑f i.succ.succ = 0⊢ Falserefine_2n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }i:Fin nhf1:↑f 1 = 0hn:↑f i.succ.succ = 1⊢ Falsen:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1⊢ { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }
      ·refine_1n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }i:Fin nhf1:↑f 1 = 0hn:↑f i.succ.succ = 0⊢ Falsen:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1⊢ { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i } simpa [Fin.ext_iff] using f.2.1.injective (hn.trans hf1.symm)All goals completed! 🐙n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1⊢ { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }
      ·refine_2n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }i:Fin nhf1:↑f 1 = 0hn:↑f i.succ.succ = 1⊢ Falsen:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1⊢ { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i } simpa [Fin.ext_iff] using f.2.1.injective (hn.trans f.2.2.2.symm)n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1⊢ { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1⊢ { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }
    let f' := f.1 ∘ Fin.succ ∘ Fin.succn:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succ⊢ { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }
    have hf' (i : Fin n) : f' i ≠ 0 := (f_succ_succ_ne_zero_one i).1n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0⊢ { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }
    let f'' := fun i => (f' i).pred (hf' i)n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯⊢ { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }
    have hf'' (i : Fin n) : f'' i ≠ 0 := by
      rw [ne_eq,n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯i:Fin n⊢ ¬f'' i = 0n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯i:Fin n⊢ ¬f' i = 1n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯hf'':∀ (i : Fin n), f'' i ≠ 0⊢ { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i } Fin.pred_eq_iff_eq_succ,n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯i:Fin n⊢ ¬f' i = Fin.succ 0n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯i:Fin n⊢ ¬f' i = 1n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯hf'':∀ (i : Fin n), f'' i ≠ 0⊢ { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i } Fin.succ_zero_eq_onen:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯i:Fin n⊢ ¬f' i = 1n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯i:Fin n⊢ ¬f' i = 1n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯hf'':∀ (i : Fin n), f'' i ≠ 0⊢ { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }]n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯i:Fin n⊢ ¬f' i = 1n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯hf'':∀ (i : Fin n), f'' i ≠ 0⊢ { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }
      exact (f_succ_succ_ne_zero_one i).2n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯hf'':∀ (i : Fin n), f'' i ≠ 0⊢ { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯hf'':∀ (i : Fin n), f'' i ≠ 0⊢ { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }
    let f''' := fun i => (f'' i).pred (hf'' i)n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯hf'':∀ (i : Fin n), f'' i ≠ 0f''':Fin n → Fin n := fun i => (f'' i).pred ⋯⊢ { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }
    refine ⟨f''', ?_, ?_⟩refine_1n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯hf'':∀ (i : Fin n), f'' i ≠ 0f''':Fin n → Fin n := fun i => (f'' i).pred ⋯⊢ Function.Involutive f'''refine_2n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯hf'':∀ (i : Fin n), f'' i ≠ 0f''':Fin n → Fin n := fun i => (f'' i).pred ⋯⊢ ∀ (i : Fin n), f''' i ≠ i
    ·refine_1n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯hf'':∀ (i : Fin n), f'' i ≠ 0f''':Fin n → Fin n := fun i => (f'' i).pred ⋯⊢ Function.Involutive f''' intro irefine_1n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯hf'':∀ (i : Fin n), f'' i ≠ 0f''':Fin n → Fin n := fun i => (f'' i).pred ⋯i:Fin n⊢ f''' (f''' i) = i
      simp only [succ_eq_add_one, ne_eq, Function.comp_apply, Fin.succ_pred, f''', f'', f']refine_1n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯hf'':∀ (i : Fin n), f'' i ≠ 0f''':Fin n → Fin n := fun i => (f'' i).pred ⋯i:Fin n⊢ ((↑f (↑f i.succ.succ)).pred ⋯).pred ⋯ = i
      simp [f.2.1 i.succ.succ]All goals completed! 🐙
    ·refine_2n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯hf'':∀ (i : Fin n), f'' i ≠ 0f''':Fin n → Fin n := fun i => (f'' i).pred ⋯⊢ ∀ (i : Fin n), f''' i ≠ i intro irefine_2n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯hf'':∀ (i : Fin n), f'' i ≠ 0f''':Fin n → Fin n := fun i => (f'' i).pred ⋯i:Fin n⊢ f''' i ≠ i
      simp only [succ_eq_add_one, ne_eq, Function.comp_apply, f''', f'', f']refine_2n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯hf'':∀ (i : Fin n), f'' i ≠ 0f''':Fin n → Fin n := fun i => (f'' i).pred ⋯i:Fin n⊢ ¬((↑f i.succ.succ).pred ⋯).pred ⋯ = i
      rw [Fin.pred_eq_iff_eq_succ,refine_2n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯hf'':∀ (i : Fin n), f'' i ≠ 0f''':Fin n → Fin n := fun i => (f'' i).pred ⋯i:Fin n⊢ ¬(↑f i.succ.succ).pred ⋯ = i.succrefine_2n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯hf'':∀ (i : Fin n), f'' i ≠ 0f''':Fin n → Fin n := fun i => (f'' i).pred ⋯i:Fin n⊢ ¬↑f i.succ.succ = i.succ.succ Fin.pred_eq_iff_eq_succrefine_2n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯hf'':∀ (i : Fin n), f'' i ≠ 0f''':Fin n → Fin n := fun i => (f'' i).pred ⋯i:Fin n⊢ ¬↑f i.succ.succ = i.succ.succrefine_2n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯hf'':∀ (i : Fin n), f'' i ≠ 0f''':Fin n → Fin n := fun i => (f'' i).pred ⋯i:Fin n⊢ ¬↑f i.succ.succ = i.succ.succ]refine_2n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }f_succ_succ_ne_zero_one:∀ (i : Fin n), ↑f i.succ.succ ≠ 0 ∧ ↑f i.succ.succ ≠ 1f':Fin n → Fin n.succ.succ := ↑f ∘ Fin.succ ∘ Fin.succhf':∀ (i : Fin n), f' i ≠ 0f'':Fin n → Fin (n + 1) := fun i => (f' i).pred ⋯hf'':∀ (i : Fin n), f'' i ≠ 0f''':Fin n → Fin n := fun i => (f'' i).pred ⋯i:Fin n⊢ ¬↑f i.succ.succ = i.succ.succ
      exact f.2.2.1 i.succ.succAll goals completed! 🐙
  invFun f := byn:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }⊢ { f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }
    let f' := fun (i : Fin n.succ.succ)=>
      match i with
      | ⟨0, h⟩ => 1
      | ⟨1, h⟩ => 0
      | ⟨(Nat.succ (Nat.succ n)), h⟩ => (f.1 ⟨n, byn✝:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }i:Fin n✝.succ.succn:ℕh:n.succ.succ < n✝.succ.succ⊢ n < n✝n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succ⊢ { f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 } omegaAll goals completed! 🐙n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succ⊢ { f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }⟩).succ.succn:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succ⊢ { f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }
    refine ⟨f', ?_, ?_, ?_⟩refine_1n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succ⊢ Function.Involutive f'refine_2n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succ⊢ ∀ (i : Fin n.succ.succ), f' i ≠ irefine_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succ⊢ f' 0 = 1
    ·refine_1n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succ⊢ Function.Involutive f' intro irefine_1n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succ⊢ f' (f' i) = i
      match i with
      | ⟨0, h⟩ =>n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succh:0 < n.succ.succ⊢ f' (f' ⟨0, h⟩) = ⟨0, h⟩ rflAll goals completed! 🐙
      | ⟨1, h⟩ =>n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succh:1 < n.succ.succ⊢ f' (f' ⟨1, h⟩) = ⟨1, h⟩ rflAll goals completed! 🐙
      | ⟨(Nat.succ (Nat.succ m)), h⟩ =>n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh:m.succ.succ < n.succ.succ⊢ f' (f' ⟨m.succ.succ, h⟩) = ⟨m.succ.succ, h⟩
        simp only [succ_eq_add_one, ne_eq, f']n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh:m.succ.succ < n.succ.succ⊢ (match (↑f ⟨m, ⋯⟩).succ.succ with
  | ⟨0, h⟩ => 1
  | ⟨1, h⟩ => 0
  | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succ) =
  ⟨m + 1 + 1, h⟩
        splith_1n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh:m.succ.succ < n.succ.succi✝:Fin n✝.succ.succh✝:0 < n.succ.succheq✝:(↑f ⟨m, ⋯⟩).succ.succ = ⟨0, h✝⟩⊢ 1 = ⟨m + 1 + 1, h⟩h_2n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh:m.succ.succ < n.succ.succi✝:Fin n✝.succ.succh✝:1 < n.succ.succheq✝:(↑f ⟨m, ⋯⟩).succ.succ = ⟨1, h✝⟩⊢ 0 = ⟨m + 1 + 1, h⟩h_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh:m.succ.succ < n.succ.succi✝:Fin n✝.succ.succn✝:ℕh✝:n✝.succ.succ < n.succ.succheq✝:(↑f ⟨m, ⋯⟩).succ.succ = ⟨n✝.succ.succ, h✝⟩⊢ (↑f ⟨n✝, ⋯⟩).succ.succ = ⟨m + 1 + 1, h⟩
        ·h_1n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh:m.succ.succ < n.succ.succi✝:Fin n✝.succ.succh✝:0 < n.succ.succheq✝:(↑f ⟨m, ⋯⟩).succ.succ = ⟨0, h✝⟩⊢ 1 = ⟨m + 1 + 1, h⟩ rename_i hh_1n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh✝¹:m.succ.succ < n.succ.succi✝:Fin n✝.succ.succh✝:0 < n.succ.succh:(↑f ⟨m, ⋯⟩).succ.succ = ⟨0, h✝⟩⊢ 1 = ⟨m + 1 + 1, h⟩
          simp [Fin.ext_iff] at hAll goals completed! 🐙
        ·h_2n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh:m.succ.succ < n.succ.succi✝:Fin n✝.succ.succh✝:1 < n.succ.succheq✝:(↑f ⟨m, ⋯⟩).succ.succ = ⟨1, h✝⟩⊢ 0 = ⟨m + 1 + 1, h⟩ rename_i hh_2n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh✝¹:m.succ.succ < n.succ.succi✝:Fin n✝.succ.succh✝:1 < n.succ.succh:(↑f ⟨m, ⋯⟩).succ.succ = ⟨1, h✝⟩⊢ 0 = ⟨m + 1 + 1, h⟩
          simp [Fin.ext_iff] at hAll goals completed! 🐙
        ·h_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh:m.succ.succ < n.succ.succi✝:Fin n✝.succ.succn✝:ℕh✝:n✝.succ.succ < n.succ.succheq✝:(↑f ⟨m, ⋯⟩).succ.succ = ⟨n✝.succ.succ, h✝⟩⊢ (↑f ⟨n✝, ⋯⟩).succ.succ = ⟨m + 1 + 1, h⟩ rename_i hh_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh✝¹:m.succ.succ < n.succ.succi✝:Fin n✝.succ.succn✝:ℕh✝:n✝.succ.succ < n.succ.succh:(↑f ⟨m, ⋯⟩).succ.succ = ⟨n✝.succ.succ, h✝⟩⊢ (↑f ⟨n✝, ⋯⟩).succ.succ = ⟨m + 1 + 1, h⟩
          rename_i x rh_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh✝:m.succ.succ < n.succ.succi✝:Fin n✝.succ.succx:ℕr:n✝.succ.succ < n.succ.succh:(↑f ⟨m, ⋯⟩).succ.succ = ⟨n✝.succ.succ, h✝⟩⊢ (↑f ⟨n✝, ⋯⟩).succ.succ = ⟨m + 1 + 1, h⟩
          simp_all only [succ_eq_add_one, Fin.ext_iff, Fin.val_succ, add_left_inj]h_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh✝:m.succ.succ < n.succ.succi✝:Fin n✝.succ.succx:ℕr:n✝.succ.succ < n.succ.succh:↑(↑f ⟨m, ⋯⟩) = x⊢ ↑(↑f ⟨x, ⋯⟩) = m
          have ht : f.1 ⟨m, byn:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh✝:m.succ.succ < n.succ.succi✝:Fin n✝.succ.succx:ℕr:n✝.succ.succ < n.succ.succh:↑(↑f ⟨m, ⋯⟩) = x⊢ m < nh_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh✝:m.succ.succ < n.succ.succi✝:Fin n✝.succ.succx:ℕr:n✝.succ.succ < n.succ.succh:↑(↑f ⟨m, ⋯⟩) = xht:↑f ⟨m, ⋯⟩ = ⟨x, ⋯⟩⊢ ↑(↑f ⟨x, ⋯⟩) = m omegaAll goals completed! 🐙h_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh✝:m.succ.succ < n.succ.succi✝:Fin n✝.succ.succx:ℕr:n✝.succ.succ < n.succ.succh:↑(↑f ⟨m, ⋯⟩) = xht:↑f ⟨m, ⋯⟩ = ⟨x, ⋯⟩⊢ ↑(↑f ⟨x, ⋯⟩) = m⟩ = ⟨x, byn:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh✝:m.succ.succ < n.succ.succi✝:Fin n✝.succ.succx:ℕr:n✝.succ.succ < n.succ.succh:↑(↑f ⟨m, ⋯⟩) = x⊢ x < nh_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh✝:m.succ.succ < n.succ.succi✝:Fin n✝.succ.succx:ℕr:n✝.succ.succ < n.succ.succh:↑(↑f ⟨m, ⋯⟩) = xht:↑f ⟨m, ⋯⟩ = ⟨x, ⋯⟩⊢ ↑(↑f ⟨x, ⋯⟩) = m omegaAll goals completed! 🐙h_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh✝:m.succ.succ < n.succ.succi✝:Fin n✝.succ.succx:ℕr:n✝.succ.succ < n.succ.succh:↑(↑f ⟨m, ⋯⟩) = xht:↑f ⟨m, ⋯⟩ = ⟨x, ⋯⟩⊢ ↑(↑f ⟨x, ⋯⟩) = m⟩ := Fin.ext hh_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh✝:m.succ.succ < n.succ.succi✝:Fin n✝.succ.succx:ℕr:n✝.succ.succ < n.succ.succh:↑(↑f ⟨m, ⋯⟩) = xht:↑f ⟨m, ⋯⟩ = ⟨x, ⋯⟩⊢ ↑(↑f ⟨x, ⋯⟩) = m
          rw [← ht,h_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh✝:m.succ.succ < n.succ.succi✝:Fin n✝.succ.succx:ℕr:n✝.succ.succ < n.succ.succh:↑(↑f ⟨m, ⋯⟩) = xht:↑f ⟨m, ⋯⟩ = ⟨x, ⋯⟩⊢ ↑(↑f (↑f ⟨m, ⋯⟩)) = mAll goals completed! 🐙 f.2.1h_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh✝:m.succ.succ < n.succ.succi✝:Fin n✝.succ.succx:ℕr:n✝.succ.succ < n.succ.succh:↑(↑f ⟨m, ⋯⟩) = xht:↑f ⟨m, ⋯⟩ = ⟨x, ⋯⟩⊢ ↑⟨m, ⋯⟩ = mAll goals completed! 🐙]All goals completed! 🐙
    ·refine_2n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succ⊢ ∀ (i : Fin n.succ.succ), f' i ≠ i intro irefine_2n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succ⊢ f' i ≠ i
      match i with
      | ⟨0, h⟩ =>n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succh:0 < n.succ.succ⊢ f' ⟨0, h⟩ ≠ ⟨0, h⟩
        simp only [succ_eq_add_one, ne_eq, Fin.zero_eta, f']n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succh:0 < n.succ.succ⊢ ¬(match 0 with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succ) =
    0
        splith_1n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succh:0 < n.succ.succi✝:Fin n✝.succ.succh✝:0 < n.succ.succheq✝:0 = ⟨0, h✝⟩⊢ ¬1 = 0h_2n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succh:0 < n.succ.succi✝:Fin n✝.succ.succh✝:1 < n.succ.succheq✝:0 = ⟨1, h✝⟩⊢ ¬0 = 0h_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succh:0 < n.succ.succi✝:Fin n✝.succ.succn✝:ℕh✝:n✝.succ.succ < n.succ.succheq✝:0 = ⟨n✝.succ.succ, h✝⟩⊢ ¬(↑f ⟨n✝, ⋯⟩).succ.succ = 0 <;>h_1n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succh:0 < n.succ.succi✝:Fin n✝.succ.succh✝:0 < n.succ.succheq✝:0 = ⟨0, h✝⟩⊢ ¬1 = 0h_2n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succh:0 < n.succ.succi✝:Fin n✝.succ.succh✝:1 < n.succ.succheq✝:0 = ⟨1, h✝⟩⊢ ¬0 = 0h_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succh:0 < n.succ.succi✝:Fin n✝.succ.succn✝:ℕh✝:n✝.succ.succ < n.succ.succheq✝:0 = ⟨n✝.succ.succ, h✝⟩⊢ ¬(↑f ⟨n✝, ⋯⟩).succ.succ = 0 try simp_allAll goals completed! 🐙
      | ⟨1, h⟩ =>n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succh:1 < n.succ.succ⊢ f' ⟨1, h⟩ ≠ ⟨1, h⟩
        simp only [succ_eq_add_one, ne_eq, Fin.mk_one, f']n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succh:1 < n.succ.succ⊢ ¬(match 1 with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succ) =
    1
        splith_1n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succh:1 < n.succ.succi✝:Fin n✝.succ.succh✝:0 < n.succ.succheq✝:1 = ⟨0, h✝⟩⊢ ¬1 = 1h_2n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succh:1 < n.succ.succi✝:Fin n✝.succ.succh✝:1 < n.succ.succheq✝:1 = ⟨1, h✝⟩⊢ ¬0 = 1h_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succh:1 < n.succ.succi✝:Fin n✝.succ.succn✝:ℕh✝:n✝.succ.succ < n.succ.succheq✝:1 = ⟨n✝.succ.succ, h✝⟩⊢ ¬(↑f ⟨n✝, ⋯⟩).succ.succ = 1 <;>h_1n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succh:1 < n.succ.succi✝:Fin n✝.succ.succh✝:0 < n.succ.succheq✝:1 = ⟨0, h✝⟩⊢ ¬1 = 1h_2n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succh:1 < n.succ.succi✝:Fin n✝.succ.succh✝:1 < n.succ.succheq✝:1 = ⟨1, h✝⟩⊢ ¬0 = 1h_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succh:1 < n.succ.succi✝:Fin n✝.succ.succn✝:ℕh✝:n✝.succ.succ < n.succ.succheq✝:1 = ⟨n✝.succ.succ, h✝⟩⊢ ¬(↑f ⟨n✝, ⋯⟩).succ.succ = 1 try simp_allAll goals completed! 🐙
      | ⟨(Nat.succ (Nat.succ m)), h⟩ =>n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh:m.succ.succ < n.succ.succ⊢ f' ⟨m.succ.succ, h⟩ ≠ ⟨m.succ.succ, h⟩
        simp only [succ_eq_add_one, ne_eq, Fin.ext_iff, Fin.val_succ, add_left_inj, f']n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh:m.succ.succ < n.succ.succ⊢ ¬↑(↑f ⟨m, ⋯⟩) = m
        have hf := f.2.2 ⟨m, Nat.add_lt_add_iff_right.mp h⟩n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh:m.succ.succ < n.succ.succhf:↑f ⟨m, ⋯⟩ ≠ ⟨m, ⋯⟩⊢ ¬↑(↑f ⟨m, ⋯⟩) = m
        simp only [ne_eq, Fin.ext_iff] at hfn:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi:Fin n✝.succ.succm:ℕh:m.succ.succ < n.succ.succhf:¬↑(↑f ⟨m, ⋯⟩) = m⊢ ¬↑(↑f ⟨m, ⋯⟩) = m
        omegaAll goals completed! 🐙
    ·refine_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succ⊢ f' 0 = 1 simp only [succ_eq_add_one, ne_eq, f']refine_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succ⊢ (match 0 with
  | ⟨0, h⟩ => 1
  | ⟨1, h⟩ => 0
  | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succ) =
  1
      splith_1n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi✝:Fin n✝.succ.succh✝:0 < n.succ.succheq✝:0 = ⟨0, h✝⟩⊢ 1 = 1h_2n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi✝:Fin n✝.succ.succh✝:1 < n.succ.succheq✝:0 = ⟨1, h✝⟩⊢ 0 = 1h_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi✝:Fin n✝.succ.succn✝:ℕh✝:n✝.succ.succ < n.succ.succheq✝:0 = ⟨n✝.succ.succ, h✝⟩⊢ (↑f ⟨n✝, ⋯⟩).succ.succ = 1 <;>h_1n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi✝:Fin n✝.succ.succh✝:0 < n.succ.succheq✝:0 = ⟨0, h✝⟩⊢ 1 = 1h_2n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi✝:Fin n✝.succ.succh✝:1 < n.succ.succheq✝:0 = ⟨1, h✝⟩⊢ 0 = 1h_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }f':Fin n.succ.succ → Fin (n + 1 + 1) := 
  fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succi✝:Fin n✝.succ.succn✝:ℕh✝:n✝.succ.succ < n.succ.succheq✝:0 = ⟨n✝.succ.succ, h✝⟩⊢ (↑f ⟨n✝, ⋯⟩).succ.succ = 1 try simp_allAll goals completed! 🐙
  left_inv f := byn:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }⊢ (fun f =>
      let f' := fun i =>
        match i with
        | ⟨0, h⟩ => 1
        | ⟨1, h⟩ => 0
        | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succ;
      ⟨f', ⋯⟩)
    ((fun f =>
        have f_succ_succ_ne_zero_one := ⋯;
        let f' := ↑f ∘ Fin.succ ∘ Fin.succ;
        have hf' := ⋯;
        let f'' := fun i => (f' i).pred ⋯;
        have hf'' := ⋯;
        let f''' := fun i => (f'' i).pred ⋯;
        ⟨f''', ⋯⟩)
      f) =
  f
    have hf1 : f.1 1 = 0 := by
      simp only [succ_eq_add_one, ne_eq, ← f.2.2.2]n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }⊢ ↑f (↑f 0) = 0n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }hf1:↑f 1 = 0⊢ (fun f =>
      let f' := fun i =>
        match i with
        | ⟨0, h⟩ => 1
        | ⟨1, h⟩ => 0
        | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succ;
      ⟨f', ⋯⟩)
    ((fun f =>
        have f_succ_succ_ne_zero_one := ⋯;
        let f' := ↑f ∘ Fin.succ ∘ Fin.succ;
        have hf' := ⋯;
        let f'' := fun i => (f' i).pred ⋯;
        have hf'' := ⋯;
        let f''' := fun i => (f'' i).pred ⋯;
        ⟨f''', ⋯⟩)
      f) =
  f
      rw [f.2.1n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }⊢ 0 = 0n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }hf1:↑f 1 = 0⊢ (fun f =>
      let f' := fun i =>
        match i with
        | ⟨0, h⟩ => 1
        | ⟨1, h⟩ => 0
        | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succ;
      ⟨f', ⋯⟩)
    ((fun f =>
        have f_succ_succ_ne_zero_one := ⋯;
        let f' := ↑f ∘ Fin.succ ∘ Fin.succ;
        have hf' := ⋯;
        let f'' := fun i => (f' i).pred ⋯;
        have hf'' := ⋯;
        let f''' := fun i => (f'' i).pred ⋯;
        ⟨f''', ⋯⟩)
      f) =
  f]n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }hf1:↑f 1 = 0⊢ (fun f =>
      let f' := fun i =>
        match i with
        | ⟨0, h⟩ => 1
        | ⟨1, h⟩ => 0
        | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succ;
      ⟨f', ⋯⟩)
    ((fun f =>
        have f_succ_succ_ne_zero_one := ⋯;
        let f' := ↑f ∘ Fin.succ ∘ Fin.succ;
        have hf' := ⋯;
        let f'' := fun i => (f' i).pred ⋯;
        have hf'' := ⋯;
        let f''' := fun i => (f'' i).pred ⋯;
        ⟨f''', ⋯⟩)
      f) =
  fn:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }hf1:↑f 1 = 0⊢ (fun f =>
      let f' := fun i =>
        match i with
        | ⟨0, h⟩ => 1
        | ⟨1, h⟩ => 0
        | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succ;
      ⟨f', ⋯⟩)
    ((fun f =>
        have f_succ_succ_ne_zero_one := ⋯;
        let f' := ↑f ∘ Fin.succ ∘ Fin.succ;
        have hf' := ⋯;
        let f'' := fun i => (f' i).pred ⋯;
        have hf'' := ⋯;
        let f''' := fun i => (f'' i).pred ⋯;
        ⟨f''', ⋯⟩)
      f) =
  f
    simp only [succ_eq_add_one, ne_eq, Function.comp_apply, Fin.succ_mk, Fin.succ_pred]n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }hf1:↑f 1 = 0⊢ ⟨fun i =>
    match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => ↑f ⟨n_1 + 1 + 1, ⋯⟩,
    ⋯⟩ =
  f
    ext in:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }hf1:↑f 1 = 0i:Fin (n + 1 + 1)⊢ ↑(↑⟨fun i =>
          match i with
          | ⟨0, h⟩ => 1
          | ⟨1, h⟩ => 0
          | ⟨n_1.succ.succ, h⟩ => ↑f ⟨n_1 + 1 + 1, ⋯⟩,
          ⋯⟩
      i) =
  ↑(↑f i)
    simp onlyn:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }hf1:↑f 1 = 0i:Fin (n + 1 + 1)⊢ ↑(match i with
    | ⟨0, h⟩ => 1
    | ⟨1, h⟩ => 0
    | ⟨n_1.succ.succ, h⟩ => ↑f ⟨n_1 + 1 + 1, ⋯⟩) =
  ↑(↑f i)
    splith_1n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }hf1:↑f 1 = 0i✝:Fin n✝.succ.succh✝:0 < n.succ.succ⊢ ↑1 = ↑(↑f ⟨0, h✝⟩)h_2n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }hf1:↑f 1 = 0i✝:Fin n✝.succ.succh✝:1 < n.succ.succ⊢ ↑0 = ↑(↑f ⟨1, h✝⟩)h_3n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }hf1:↑f 1 = 0i✝:Fin n✝.succ.succn✝:ℕh✝:n✝.succ.succ < n.succ.succ⊢ ↑(↑f ⟨n✝ + 1 + 1, ⋯⟩) = ↑(↑f ⟨n✝.succ.succ, h✝⟩)
    ·h_1n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }hf1:↑f 1 = 0i✝:Fin n✝.succ.succh✝:0 < n.succ.succ⊢ ↑1 = ↑(↑f ⟨0, h✝⟩) simp [succ_eq_add_one, Fin.zero_eta, f.2.2.2]All goals completed! 🐙
    ·h_2n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }hf1:↑f 1 = 0i✝:Fin n✝.succ.succh✝:1 < n.succ.succ⊢ ↑0 = ↑(↑f ⟨1, h✝⟩) exact congrArg Fin.val hf1.symmAll goals completed! 🐙
    ·h_3n:ℕf:{ f // Function.Involutive f ∧ (∀ (i : Fin n.succ.succ), f i ≠ i) ∧ f 0 = 1 }hf1:↑f 1 = 0i✝:Fin n✝.succ.succn✝:ℕh✝:n✝.succ.succ < n.succ.succ⊢ ↑(↑f ⟨n✝ + 1 + 1, ⋯⟩) = ↑(↑f ⟨n✝.succ.succ, h✝⟩) exact rflAll goals completed! 🐙
  right_inv f := byn:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }⊢ (fun f =>
      have f_succ_succ_ne_zero_one := ⋯;
      let f' := ↑f ∘ Fin.succ ∘ Fin.succ;
      have hf' := ⋯;
      let f'' := fun i => (f' i).pred ⋯;
      have hf'' := ⋯;
      let f''' := fun i => (f'' i).pred ⋯;
      ⟨f''', ⋯⟩)
    ((fun f =>
        let f' := fun i =>
          match i with
          | ⟨0, h⟩ => 1
          | ⟨1, h⟩ => 0
          | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succ;
        ⟨f', ⋯⟩)
      f) =
  f
    simp only [ne_eq, succ_eq_add_one, Function.comp_apply]n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }⊢ ⟨fun i =>
    ((match i.succ.succ with
            | ⟨0, h⟩ => 1
            | ⟨1, h⟩ => 0
            | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succ).pred
          ⋯).pred
      ⋯,
    ⋯⟩ =
  f
    ext in:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }i:Fin n⊢ ↑(↑⟨fun i =>
          ((match i.succ.succ with
                  | ⟨0, h⟩ => 1
                  | ⟨1, h⟩ => 0
                  | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succ).pred
                ⋯).pred
            ⋯,
          ⋯⟩
      i) =
  ↑(↑f i)
    simp only [Fin.val_pred]n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }i:Fin n⊢ ↑(match i.succ.succ with
        | ⟨0, h⟩ => 1
        | ⟨1, h⟩ => 0
        | ⟨n_1.succ.succ, h⟩ => (↑f ⟨n_1, ⋯⟩).succ.succ) -
      1 -
    1 =
  ↑(↑f i)
    splith_1n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }i:Fin ni✝:Fin n✝.succ.succh✝:0 < n.succ.succheq✝:i.succ.succ = ⟨0, h✝⟩⊢ ↑1 - 1 - 1 = ↑(↑f i)h_2n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }i:Fin ni✝:Fin n✝.succ.succh✝:1 < n.succ.succheq✝:i.succ.succ = ⟨1, h✝⟩⊢ ↑0 - 1 - 1 = ↑(↑f i)h_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }i:Fin ni✝:Fin n✝.succ.succn✝:ℕh✝:n✝.succ.succ < n.succ.succheq✝:i.succ.succ = ⟨n✝.succ.succ, h✝⟩⊢ ↑(↑f ⟨n✝, ⋯⟩).succ.succ - 1 - 1 = ↑(↑f i)
    ·h_1n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }i:Fin ni✝:Fin n✝.succ.succh✝:0 < n.succ.succheq✝:i.succ.succ = ⟨0, h✝⟩⊢ ↑1 - 1 - 1 = ↑(↑f i) rename_i hh_1n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }i:Fin ni✝:Fin n✝.succ.succh✝:0 < n.succ.succh:i.succ.succ = ⟨0, h✝⟩⊢ ↑1 - 1 - 1 = ↑(↑f i)
      simp [Fin.ext_iff] at hAll goals completed! 🐙
    ·h_2n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }i:Fin ni✝:Fin n✝.succ.succh✝:1 < n.succ.succheq✝:i.succ.succ = ⟨1, h✝⟩⊢ ↑0 - 1 - 1 = ↑(↑f i) rename_i hh_2n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }i:Fin ni✝:Fin n✝.succ.succh✝:1 < n.succ.succh:i.succ.succ = ⟨1, h✝⟩⊢ ↑0 - 1 - 1 = ↑(↑f i)
      simp [Fin.ext_iff] at hAll goals completed! 🐙
    ·h_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }i:Fin ni✝:Fin n✝.succ.succn✝:ℕh✝:n✝.succ.succ < n.succ.succheq✝:i.succ.succ = ⟨n✝.succ.succ, h✝⟩⊢ ↑(↑f ⟨n✝, ⋯⟩).succ.succ - 1 - 1 = ↑(↑f i) simp only [Fin.val_succ, add_tsub_cancel_right]h_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }i:Fin ni✝:Fin n✝.succ.succn✝:ℕh✝:n✝.succ.succ < n.succ.succheq✝:i.succ.succ = ⟨n✝.succ.succ, h✝⟩⊢ ↑(↑f ⟨n✝, ⋯⟩) = ↑(↑f i)
      congrh_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }i:Fin ni✝:Fin n✝.succ.succn✝:ℕh✝:n✝.succ.succ < n.succ.succheq✝:i.succ.succ = ⟨n✝.succ.succ, h✝⟩⊢ ↑f ⟨n✝, ⋯⟩ = ↑f i
      apply congrArgh_3n:ℕf:{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }i:Fin ni✝:Fin n✝.succ.succn✝:ℕh✝:n✝.succ.succ < n.succ.succheq✝:i.succ.succ = ⟨n✝.succ.succ, h✝⟩⊢ ⟨n✝, ⋯⟩ = i
      simp_all [Fin.ext_iff]All goals completed! 🐙

Fixed point involutions of Fin n.succ.succ for fixed f 0 = k.succ are equivalent to fixed point involutions of Fin n.

def involutionNoFixedZeroSetEquiv {n : ℕ} (k : Fin n.succ) :
    {f : Fin n.succ.succ → Fin n.succ.succ // Function.Involutive f ∧
      (∀ i, f i ≠ i) ∧ f 0 = k.succ}
        ≃ {f : Fin n → Fin n // Function.Involutive f ∧ (∀ i, f i ≠ i)} :=
  (involutionNoFixedZeroSetEquivSetOne k).trans involutionNoFixedSetOne

The type of fixed point free involutions of Fin n.succ.succ is equivalent to the sum of Fin n.succ copies of fixed point involutions of Fin n.

def involutionNoFixedEquivSumSame {n : ℕ} :
    {f : Fin n.succ.succ → Fin n.succ.succ // Function.Involutive f ∧ (∀ i, f i ≠ i)}
    ≃ Σ (_ : Fin n.succ), {f : Fin n → Fin n // Function.Involutive f ∧ (∀ i, f i ≠ i)} :=
    involutionNoFixedEquivSum.trans <| .sigmaCongrRight involutionNoFixedZeroSetEquiv

Ever fixed-point free involutions of Fin n.succ.succ can be decomposed into a element of Fin n.succ (where 0 is sent) and a fixed-point free involution of Fin n.

def involutionNoFixedZeroEquivProd {n : ℕ} :
    {f : Fin n.succ.succ → Fin n.succ.succ // Function.Involutive f ∧ (∀ i, f i ≠ i)}
    ≃ Fin n.succ × {f : Fin n → Fin n // Function.Involutive f ∧ (∀ i, f i ≠ i)} :=
  involutionNoFixedEquivSumSame.trans <| .sigmaEquivProd ..

Cardinality

The type of fixed-point free involutions of Fin n is finite.

instance {n : ℕ} : Fintype { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i } := byn:ℕ⊢ Fintype { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }
  have : DecidablePred fun x ↦ Function.Involutive x :=
    fun f ↦ Fintype.decidableForallFintype (α := Fin n)n:ℕthis:DecidablePred fun x => Function.Involutive x⊢ Fintype { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }
  exact Subtype.fintype ..All goals completed! 🐙

              lemma involutionNoFixed_card_succ {n : ℕ} :
    Fintype.card
    {f : Fin n.succ.succ → Fin n.succ.succ // Function.Involutive f ∧ (∀ i, f i ≠ i)}
    = n.succ *
    Fintype.card {f : Fin n → Fin n // Function.Involutive f ∧ (∀ i, f i ≠ i)} := byn:ℕ⊢ Fintype.card { f // Function.Involutive f ∧ ∀ (i : Fin n.succ.succ), f i ≠ i } =
  n.succ * Fintype.card { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i }
  simp [Fintype.card_congr involutionNoFixedZeroEquivProd]All goals completed! 🐙lemma involutionNoFixed_card_mul_two : (n : ℕ) →
    Fintype.card {f : Fin (2 * n) → Fin (2 * n) // Function.Involutive f ∧ (∀ i, f i ≠ i)}
    = (2 * n - 1)‼
  | 0 => rfl
  | Nat.succ n =>n:ℕ⊢ Fintype.card { f // Function.Involutive f ∧ ∀ (i : Fin (2 * n.succ)), f i ≠ i } = (2 * n.succ - 1)‼ byn:ℕ⊢ Fintype.card { f // Function.Involutive f ∧ ∀ (i : Fin (2 * n.succ)), f i ≠ i } = (2 * n.succ - 1)‼
    erw [involutionNoFixed_card_succ,n:ℕ⊢ (Nat.mul 2 n).succ * Fintype.card { f // Function.Involutive f ∧ ∀ (i : Fin (Nat.mul 2 n)), f i ≠ i } =
  (2 * n.succ - 1)‼ involutionNoFixed_card_mul_two nn:ℕ⊢ (Nat.mul 2 n).succ * (2 * n - 1)‼ = (2 * n.succ - 1)‼]n:ℕ⊢ (Nat.mul 2 n).succ * (2 * n - 1)‼ = (2 * n.succ - 1)‼
    exact (Nat.doubleFactorial_add_one (Nat.mul 2 n)).symmAll goals completed! 🐙lemma involutionNoFixed_card_mul_two_plus_one : (n : ℕ) →
    Fintype.card {f : Fin (2 * n + 1) → Fin (2 * n + 1) // Function.Involutive f ∧ (∀ i, f i ≠ i)}
    = 0
  | 0 => rfl
  | Nat.succ n =>n:ℕ⊢ Fintype.card { f // Function.Involutive f ∧ ∀ (i : Fin (2 * n.succ + 1)), f i ≠ i } = 0 byn:ℕ⊢ Fintype.card { f // Function.Involutive f ∧ ∀ (i : Fin (2 * n.succ + 1)), f i ≠ i } = 0
    erw [involutionNoFixed_card_succ,n:ℕ⊢ (Nat.mul 2 n + 1).succ * Fintype.card { f // Function.Involutive f ∧ ∀ (i : Fin (Nat.mul 2 n + 1)), f i ≠ i } = 0 involutionNoFixed_card_mul_two_plus_one nn:ℕ⊢ (Nat.mul 2 n + 1).succ * 0 = 0]n:ℕ⊢ (Nat.mul 2 n + 1).succ * 0 = 0
    ringAll goals completed! 🐙

lemma involutionNoFixed_card_even : (n : ℕ) → (he : Even n) →
    Fintype.card {f : Fin n → Fin n // Function.Involutive f ∧ (∀ i, f i ≠ i)} = (n - 1)‼ := by⊢ ∀ (n : ℕ), Even n → Fintype.card { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i } = (n - 1)‼
  intro n hen:ℕhe:Even n⊢ Fintype.card { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i } = (n - 1)‼
  obtain ⟨r, hr⟩ := hen:ℕr:ℕhr:n = r + r⊢ Fintype.card { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i } = (n - 1)‼
  have hr' : n = 2 * r := by⊢ ∀ (n : ℕ), Even n → Fintype.card { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i } = (n - 1)‼n:ℕr:ℕhr:n = r + rhr':n = 2 * r⊢ Fintype.card { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i } = (n - 1)‼ omegan:ℕr:ℕhr:n = r + rhr':n = 2 * r⊢ Fintype.card { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i } = (n - 1)‼n:ℕr:ℕhr:n = r + rhr':n = 2 * r⊢ Fintype.card { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i } = (n - 1)‼
  subst hr'r:ℕhr:2 * r = r + r⊢ Fintype.card { f // Function.Involutive f ∧ ∀ (i : Fin (2 * r)), f i ≠ i } = (2 * r - 1)‼
  exact involutionNoFixed_card_mul_two rAll goals completed! 🐙lemma involutionNoFixed_card_odd : (n : ℕ) → (ho : Odd n) →
    Fintype.card {f : Fin n → Fin n // Function.Involutive f ∧ (∀ i, f i ≠ i)} = 0 := by⊢ ∀ (n : ℕ), Odd n → Fintype.card { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i } = 0
  intro n hon:ℕho:Odd n⊢ Fintype.card { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i } = 0
  obtain ⟨r, hr⟩ := hon:ℕr:ℕhr:n = 2 * r + 1⊢ Fintype.card { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i } = 0
  subst hrr:ℕ⊢ Fintype.card { f // Function.Involutive f ∧ ∀ (i : Fin (2 * r + 1)), f i ≠ i } = 0
  exact involutionNoFixed_card_mul_two_plus_one rAll goals completed! 🐙