feat: Cardinality of involutions and refactor
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572
HepLean/Mathematics/Fin/Involutions.lean
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572
HepLean/Mathematics/Fin/Involutions.lean
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/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import Mathlib.LinearAlgebra.PiTensorProduct
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import Mathlib.Tactic.Polyrith
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import Mathlib.Tactic.Linarith
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import Mathlib.Data.Nat.Factorial.DoubleFactorial
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/-!
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# Fin involutions
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Some properties of involutions of `Fin n`.
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These involutions are used in e.g. proving results about Wick contractions.
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-/
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namespace HepLean.Fin
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open Nat
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def involutionCons (n : ℕ) : {f : Fin n.succ → Fin n.succ // Function.Involutive f } ≃
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(f : {f : Fin n → Fin n // Function.Involutive f}) × {i : Option (Fin n) //
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∀ (h : i.isSome), f.1 (Option.get i h) = (Option.get i h)} where
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toFun f := ⟨⟨
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fun i =>
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if h : f.1 i.succ = 0 then i
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else Fin.pred (f.1 i.succ) h , by
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intro i
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by_cases h : f.1 i.succ = 0
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· simp [h]
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· simp [h]
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simp [f.2 i.succ]
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intro h
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exact False.elim (Fin.succ_ne_zero i h)⟩,
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⟨if h : f.1 0 = 0 then none else Fin.pred (f.1 0) h , by
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by_cases h0 : f.1 0 = 0
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· simp [h0]
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· simp [h0]
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refine fun h => False.elim (h (f.2 0))⟩⟩
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invFun f := ⟨
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if h : (f.2.1).isSome then
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Fin.cons (f.2.1.get h).succ (Function.update (Fin.succ ∘ f.1.1) (f.2.1.get h) 0)
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else
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Fin.cons 0 (Fin.succ ∘ f.1.1)
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, by
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by_cases hs : (f.2.1).isSome
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· simp only [Nat.succ_eq_add_one, hs, ↓reduceDIte, Fin.coe_eq_castSucc]
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let a := f.2.1.get hs
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change Function.Involutive (Fin.cons a.succ (Function.update (Fin.succ ∘ ↑f.fst) a 0))
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intro i
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rcases Fin.eq_zero_or_eq_succ i with hi | ⟨j, hj⟩
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· subst hi
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rw [Fin.cons_zero, Fin.cons_succ]
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simp
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· subst hj
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rw [Fin.cons_succ]
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by_cases hja : j = a
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· subst hja
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simp
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· rw [Function.update_apply ]
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rw [if_neg hja]
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simp
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have hf2 := f.2.2 hs
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change f.1.1 a = a at hf2
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have hjf1 : f.1.1 j ≠ a := by
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by_contra hn
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have haj : j = f.1.1 a := by
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rw [← hn]
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rw [f.1.2]
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rw [hf2] at haj
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exact hja haj
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rw [Function.update_apply, if_neg hjf1]
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simp
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rw [f.1.2]
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· simp [hs]
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intro i
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rcases Fin.eq_zero_or_eq_succ i with hi | ⟨j, hj⟩
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· subst hi
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simp
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· subst hj
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simp
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rw [f.1.2]⟩
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left_inv f := by
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match f with
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| ⟨f, hf⟩ =>
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simp
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ext i
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by_cases h0 : f 0 = 0
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· simp [h0]
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rcases Fin.eq_zero_or_eq_succ i with hi | ⟨j, hj⟩
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· subst hi
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simp [h0]
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· subst hj
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simp [h0]
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by_cases hj : f j.succ =0
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· rw [← h0] at hj
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have hn := Function.Involutive.injective hf hj
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exact False.elim (Fin.succ_ne_zero j hn)
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· simp [hj]
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rw [Fin.ext_iff] at hj
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simp at hj
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omega
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· rw [if_neg h0]
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by_cases hf' : i = f 0
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· subst hf'
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simp
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rw [hf]
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simp
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· rw [Function.update_apply, if_neg hf']
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rcases Fin.eq_zero_or_eq_succ i with hi | ⟨j, hj⟩
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· subst hi
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simp
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· subst hj
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simp
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by_cases hj : f j.succ =0
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· rw [← hj] at hf'
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rw [hf] at hf'
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simp at hf'
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· simp [hj]
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rw [Fin.ext_iff] at hj
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simp at hj
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omega
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right_inv f := by
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match f with
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| ⟨⟨f, hf⟩, ⟨f0, hf0⟩⟩ =>
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ext i
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· simp
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by_cases hs : f0.isSome
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· simp [hs]
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by_cases hi : i = f0.get hs
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· simp [hi, Function.update_apply]
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exact Eq.symm (Fin.val_eq_of_eq (hf0 hs))
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· simp [hi]
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split
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· rename_i h
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exact False.elim (Fin.succ_ne_zero (f i) h)
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· rfl
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· simp [hs]
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split
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· rename_i h
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exact False.elim (Fin.succ_ne_zero (f i) h)
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· rfl
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· simp only [Nat.succ_eq_add_one, Option.mem_def,
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Option.dite_none_left_eq_some, Option.some.injEq]
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by_cases hs : f0.isSome
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· simp only [hs, ↓reduceDIte]
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simp [Fin.cons_zero]
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have hx : ¬ (f0.get hs).succ = 0 := (Fin.succ_ne_zero (f0.get hs))
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simp [hx]
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refine Iff.intro (fun hi => ?_) (fun hi => ?_)
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· rw [← hi]
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exact
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Option.eq_some_of_isSome
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(Eq.mpr_prop (Eq.refl (f0.isSome = true))
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(of_eq_true (Eq.trans (congrArg (fun x => x = true) hs) (eq_self true))))
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· subst hi
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exact rfl
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· simp [hs]
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simp at hs
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subst hs
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exact ne_of_beq_false rfl
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lemma involutionCons_ext {n : ℕ} {f1 f2 : (f : {f : Fin n → Fin n // Function.Involutive f}) ×
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{i : Option (Fin n) // ∀ (h : i.isSome), f.1 (Option.get i h) = (Option.get i h)}}
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(h1 : f1.1 = f2.1) (h2 : f1.2 = Equiv.subtypeEquivRight (by rw [h1]; simp) f2.2) : f1 = f2 := by
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cases f1
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cases f2
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simp at h1 h2
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subst h1
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rename_i fst snd snd_1
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simp_all only [Sigma.mk.inj_iff, heq_eq_eq, true_and]
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obtain ⟨val, property⟩ := fst
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obtain ⟨val_1, property_1⟩ := snd
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obtain ⟨val_2, property_2⟩ := snd_1
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simp_all only
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rfl
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def involutionAddEquiv {n : ℕ} (f : {f : Fin n → Fin n // Function.Involutive f}) :
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{i : Option (Fin n) // ∀ (h : i.isSome), f.1 (Option.get i h) = (Option.get i h)} ≃
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Option (Fin (Finset.univ.filter fun i => f.1 i = i).card) := by
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let e1 : {i : Option (Fin n) // ∀ (h : i.isSome), f.1 (Option.get i h) = (Option.get i h)}
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≃ Option {i : Fin n // f.1 i = i} :=
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{ toFun := fun i => match i with
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| ⟨some i, h⟩ => some ⟨i, by simpa using h⟩
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| ⟨none, h⟩ => none
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invFun := fun i => match i with
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| some ⟨i, h⟩ => ⟨some i, by simpa using h⟩
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| none => ⟨none, by simp⟩
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left_inv := by
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intro a
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cases a
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aesop
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right_inv := by
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intro a
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cases a
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rfl
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simp_all only [Subtype.coe_eta] }
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let s : Finset (Fin n) := Finset.univ.filter fun i => f.1 i = i
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let e2' : { i : Fin n // f.1 i = i} ≃ {i // i ∈ s} := by
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refine Equiv.subtypeEquivProp ?h
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funext i
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simp [s]
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let e2 : {i // i ∈ s} ≃ Fin (Finset.card s) := by
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refine (Finset.orderIsoOfFin _ ?_).symm.toEquiv
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simp [s]
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refine e1.trans (Equiv.optionCongr (e2'.trans (e2)))
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lemma involutionAddEquiv_none_image_zero {n : ℕ} :
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{f : {f : Fin n.succ → Fin n.succ // Function.Involutive f}}
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→ involutionAddEquiv (involutionCons n f).1 (involutionCons n f).2 = none
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→ f.1 ⟨0, Nat.zero_lt_succ n⟩ = ⟨0, Nat.zero_lt_succ n⟩ := by
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intro f h
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simp only [Nat.succ_eq_add_one, involutionCons, Equiv.coe_fn_mk, involutionAddEquiv,
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Option.isSome_some, Option.get_some, Option.isSome_none, Equiv.trans_apply,
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Equiv.optionCongr_apply, Equiv.coe_trans, RelIso.coe_fn_toEquiv, Option.map_eq_none'] at h
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simp_all only [List.length_cons, Fin.zero_eta]
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obtain ⟨val, property⟩ := f
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simp_all only [List.length_cons]
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split at h
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next i i_1 h_1 heq =>
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split at heq
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next h_2 => simp_all only [reduceCtorEq]
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next h_2 => simp_all only [reduceCtorEq]
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next i h_1 heq =>
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split at heq
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next h_2 => simp_all only
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next h_2 => simp_all only [Subtype.mk.injEq, reduceCtorEq]
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lemma involutionAddEquiv_cast {n : ℕ} {f1 f2 : {f : Fin n → Fin n // Function.Involutive f}}
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(hf : f1 = f2):
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involutionAddEquiv f1 = (Equiv.subtypeEquivRight (by rw [hf]; simp)).trans
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((involutionAddEquiv f2).trans (Equiv.optionCongr (finCongr (by rw [hf])))):= by
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subst hf
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simp
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rfl
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lemma involutionAddEquiv_cast' {m : ℕ} {f1 f2 : {f : Fin m → Fin m // Function.Involutive f}}
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{N : ℕ} (hf : f1 = f2) (n : Option (Fin N)) (hn1 : N = (Finset.filter (fun i => f1.1 i = i) Finset.univ).card)
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(hn2 : N = (Finset.filter (fun i => f2.1 i = i) Finset.univ).card):
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HEq ((involutionAddEquiv f1).symm (Option.map (finCongr hn1) n)) ((involutionAddEquiv f2).symm (Option.map (finCongr hn2) n)) := by
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subst hf
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rfl
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lemma involutionAddEquiv_none_succ {n : ℕ}
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{f : {f : Fin n.succ → Fin n.succ // Function.Involutive f}}
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(h : involutionAddEquiv (involutionCons n f).1 (involutionCons n f).2 = none)
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(x : Fin n) : f.1 x.succ = x.succ ↔ (involutionCons n f).1.1 x = x := by
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simp [involutionCons]
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have hn' := involutionAddEquiv_none_image_zero h
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have hx : ¬ f.1 x.succ = ⟨0, Nat.zero_lt_succ n⟩:= by
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rw [← hn']
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exact fun hn => Fin.succ_ne_zero x (Function.Involutive.injective f.2 hn)
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apply Iff.intro
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· intro h2 h3
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rw [Fin.ext_iff]
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simp [h2]
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· intro h2
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have h2' := h2 hx
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conv_rhs => rw [← h2']
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simp
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lemma involutionAddEquiv_isSome_image_zero {n : ℕ} :
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{f : {f : Fin n.succ → Fin n.succ // Function.Involutive f}}
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→ (involutionAddEquiv (involutionCons n f).1 (involutionCons n f).2).isSome
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→ ¬ f.1 ⟨0, Nat.zero_lt_succ n⟩ = ⟨0, Nat.zero_lt_succ n⟩ := by
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intro f hf
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simp [involutionAddEquiv, involutionCons] at hf
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simp_all only [List.length_cons, Fin.zero_eta]
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obtain ⟨val, property⟩ := f
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simp_all only [List.length_cons]
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apply Aesop.BuiltinRules.not_intro
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intro a
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simp_all only [↓reduceDIte, Option.isSome_none, Bool.false_eq_true]
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def involutionNoFixedEquivSum {n : ℕ} :
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{f : Fin (2 * n.succ) → Fin (2 * n.succ) // Function.Involutive f
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∧ ∀ i, f i ≠ i} ≃ Σ (k : Fin (2 * n + 1)),
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{f : Fin (2 * n.succ) → Fin (2 * n.succ) // Function.Involutive f
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∧ (∀ i, f i ≠ i) ∧ f 0 = k.succ} where
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toFun f := ⟨(f.1 0).pred (f.2.2 0), ⟨f.1, f.2.1, by simpa using f.2.2⟩⟩
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invFun f := ⟨f.2.1, ⟨f.2.2.1, f.2.2.2.1⟩⟩
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left_inv f := by
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rfl
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right_inv f := by
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simp
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ext
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· simp
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rw [f.2.2.2.2]
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simp
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· simp
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/-!
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## Equivalences of involutions with no fixed points.
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The main aim of thes equivalences is to define `involutionNoFixedZeroEquivProd`.
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-/
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def involutionNoFixedZeroSetEquivEquiv {n : ℕ}
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(k : Fin (2 * n + 1)) (e : Fin (2 * n.succ) ≃ Fin (2 * n.succ)) :
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{f : Fin (2 * n.succ) → Fin (2 * n.succ) // Function.Involutive f
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∧ (∀ i, f i ≠ i) ∧ f 0 = k.succ} ≃
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{f : Fin (2 * n.succ) → Fin (2 * n.succ) // Function.Involutive (e.symm ∘ f ∘ e) ∧
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(∀ i, (e.symm ∘ f ∘ e) i ≠ i) ∧ (e.symm ∘ f ∘ e) 0 = k.succ} where
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toFun f := ⟨e ∘ f.1 ∘ e.symm, by
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intro i
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simp
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rw [f.2.1], by
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simpa using f.2.2.1, by simpa using f.2.2.2⟩
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invFun f := ⟨e.symm ∘ f.1 ∘ e, by
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intro i
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simp
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have hf2 := f.2.1 i
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simpa using hf2, by
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simpa using f.2.2.1, by simpa using f.2.2.2⟩
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left_inv f := by
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ext i
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simp
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right_inv f := by
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ext i
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simp
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def involutionNoFixedZeroSetEquivSetEquiv {n : ℕ} (k : Fin (2 * n + 1)) (e : Fin (2 * n.succ) ≃ Fin (2 * n.succ)) :
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{f : Fin (2 * n.succ) → Fin (2 * n.succ) // Function.Involutive (e.symm ∘ f ∘ e) ∧
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(∀ i, (e.symm ∘ f ∘ e) i ≠ i) ∧ (e.symm ∘ f ∘ e) 0 = k.succ} ≃
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{f : Fin (2 * n.succ) → Fin (2 * n.succ) // Function.Involutive f ∧
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(∀ i, f i ≠ i) ∧ (e.symm ∘ f ∘ e) 0 = k.succ} := by
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refine Equiv.subtypeEquivRight ?_
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intro f
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have h1 : Function.Involutive (⇑e.symm ∘ f ∘ ⇑e) ↔ Function.Involutive f := by
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apply Iff.intro
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· intro h i
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have hi := h (e.symm i)
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simpa using hi
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· intro h i
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have hi := h (e i)
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simp [hi]
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rw [h1]
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simp
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intro h1 h2
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apply Iff.intro
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· intro h i
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have hi := h (e.symm i)
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simpa using hi
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· intro h i
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have hi := h (e i)
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by_contra hn
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nth_rewrite 2 [← hn] at hi
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simp at hi
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def involutionNoFixedZeroSetEquivEquiv' {n : ℕ} (k : Fin (2 * n + 1)) (e : Fin (2 * n.succ) ≃ Fin (2 * n.succ)) :
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{f : Fin (2 * n.succ) → Fin (2 * n.succ) // Function.Involutive f ∧
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(∀ i, f i ≠ i) ∧ (e.symm ∘ f ∘ e) 0 = k.succ}
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≃ {f : Fin (2 * n.succ) → Fin (2 * n.succ) // Function.Involutive f ∧
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(∀ i, f i ≠ i) ∧ f (e 0) = e k.succ} := by
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refine Equiv.subtypeEquivRight ?_
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simp
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intro f hi h1
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exact Equiv.symm_apply_eq e
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def involutionNoFixedZeroSetEquivSetOne {n : ℕ} (k : Fin (2 * n + 1)) :
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{f : Fin (2 * n.succ) → Fin (2 * n.succ) // Function.Involutive f ∧
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(∀ i, f i ≠ i) ∧ f 0 = k.succ}
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≃ {f : Fin (2 * n.succ) → Fin (2 * n.succ) // Function.Involutive f ∧
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(∀ i, f i ≠ i) ∧ f 0 = 1} := by
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refine Equiv.trans (involutionNoFixedZeroSetEquivEquiv k (Equiv.swap k.succ 1)) ?_
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refine Equiv.trans (involutionNoFixedZeroSetEquivSetEquiv k (Equiv.swap k.succ 1)) ?_
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refine Equiv.trans (involutionNoFixedZeroSetEquivEquiv' k (Equiv.swap k.succ 1)) ?_
|
||||
refine Equiv.subtypeEquivRight ?_
|
||||
simp
|
||||
intro f hi h1
|
||||
rw [Equiv.swap_apply_of_ne_of_ne]
|
||||
· exact Ne.symm (Fin.succ_ne_zero k)
|
||||
· exact Fin.zero_ne_one
|
||||
|
||||
def involutionNoFixedSetOne {n : ℕ} :
|
||||
{f : Fin (2 * n.succ) → Fin (2 * n.succ) // Function.Involutive f ∧
|
||||
(∀ i, f i ≠ i) ∧ f 0 = 1} ≃
|
||||
{f : Fin (2 * n) → Fin (2 * n) // Function.Involutive f ∧
|
||||
(∀ i, f i ≠ i)} where
|
||||
toFun f := by
|
||||
have hf1 : f.1 1 = 0 := by
|
||||
have hf := f.2.2.2
|
||||
simp [← hf]
|
||||
rw [f.2.1]
|
||||
let f' := f.1 ∘ Fin.succ ∘ Fin.succ
|
||||
have hf' (i : Fin (2 * n)) : f' i ≠ 0 := by
|
||||
simp [f']
|
||||
simp [← hf1]
|
||||
by_contra hn
|
||||
have hn' := Function.Involutive.injective f.2.1 hn
|
||||
simp [Fin.ext_iff] at hn'
|
||||
let f'' := fun i => (f' i).pred (hf' i)
|
||||
have hf'' (i : Fin (2 * n)) : f'' i ≠ 0 := by
|
||||
simp [f'']
|
||||
rw [@Fin.pred_eq_iff_eq_succ]
|
||||
simp [f']
|
||||
simp [← f.2.2.2 ]
|
||||
by_contra hn
|
||||
have hn' := Function.Involutive.injective f.2.1 hn
|
||||
simp [Fin.ext_iff] at hn'
|
||||
let f''' := fun i => (f'' i).pred (hf'' i)
|
||||
refine ⟨f''', ?_, ?_⟩
|
||||
· intro i
|
||||
simp [f''', f'', f']
|
||||
simp [f.2.1 i.succ.succ]
|
||||
· intro i
|
||||
simp [f''', f'', f']
|
||||
rw [@Fin.pred_eq_iff_eq_succ]
|
||||
rw [@Fin.pred_eq_iff_eq_succ]
|
||||
exact f.2.2.1 i.succ.succ
|
||||
invFun f := by
|
||||
let f' := fun (i : Fin (2 * n.succ))=>
|
||||
match i with
|
||||
| ⟨0, h⟩ => 1
|
||||
| ⟨1, h⟩ => 0
|
||||
| ⟨(Nat.succ (Nat.succ n)), h⟩ => (f.1 ⟨n, by omega⟩).succ.succ
|
||||
refine ⟨f', ?_, ?_, ?_⟩
|
||||
· intro i
|
||||
match i with
|
||||
| ⟨0, h⟩ => rfl
|
||||
| ⟨1, h⟩ => rfl
|
||||
| ⟨(Nat.succ (Nat.succ m)), h⟩ =>
|
||||
simp [f']
|
||||
split
|
||||
· rename_i h
|
||||
simp at h
|
||||
exact False.elim (Fin.succ_ne_zero (f.1 ⟨m, _⟩).succ h)
|
||||
· rename_i h
|
||||
simp [Fin.ext_iff] at h
|
||||
· rename_i h
|
||||
rename_i x r
|
||||
simp_all [Fin.ext_iff]
|
||||
have hfn {a b : ℕ} {ha : a < 2 * n} {hb : b < 2 * n}
|
||||
(hab : ↑(f.1 ⟨a, ha⟩) = b): ↑(f.1 ⟨b, hb⟩) = a := by
|
||||
have ht : f.1 ⟨a, ha⟩ = ⟨b, hb⟩ := by
|
||||
simp [hab, Fin.ext_iff]
|
||||
rw [← ht, f.2.1]
|
||||
exact hfn h
|
||||
· intro i
|
||||
match i with
|
||||
| ⟨0, h⟩ =>
|
||||
simp [f']
|
||||
split
|
||||
· rename_i h
|
||||
simp
|
||||
· rename_i h
|
||||
simp [Fin.ext_iff] at h
|
||||
· rename_i h
|
||||
simp [Fin.ext_iff] at h
|
||||
| ⟨1, h⟩ =>
|
||||
simp [f']
|
||||
split
|
||||
· rename_i h
|
||||
simp at h
|
||||
· rename_i h
|
||||
simp
|
||||
· rename_i h
|
||||
simp [Fin.ext_iff] at h
|
||||
| ⟨(Nat.succ (Nat.succ m)), h⟩ =>
|
||||
simp [f', Fin.ext_iff]
|
||||
have hf:= f.2.2 ⟨m, by exact Nat.add_lt_add_iff_right.mp h⟩
|
||||
simp [Fin.ext_iff] at hf
|
||||
omega
|
||||
· simp [f']
|
||||
split
|
||||
· rename_i h
|
||||
simp
|
||||
· rename_i h
|
||||
simp at h
|
||||
· rename_i h
|
||||
simp [Fin.ext_iff] at h
|
||||
left_inv f := by
|
||||
have hf1 : f.1 1 = 0 := by
|
||||
have hf := f.2.2.2
|
||||
simp [← hf]
|
||||
rw [f.2.1]
|
||||
simp
|
||||
ext i
|
||||
simp
|
||||
split
|
||||
· simp
|
||||
rw [f.2.2.2]
|
||||
simp
|
||||
· simp
|
||||
rw [hf1]
|
||||
simp
|
||||
· rfl
|
||||
right_inv f := by
|
||||
simp
|
||||
ext i
|
||||
simp
|
||||
split
|
||||
· rename_i h
|
||||
simp [Fin.ext_iff] at h
|
||||
· rename_i h
|
||||
simp [Fin.ext_iff] at h
|
||||
· rename_i h
|
||||
simp
|
||||
congr
|
||||
apply congrArg
|
||||
simp_all [Fin.ext_iff]
|
||||
|
||||
|
||||
def involutionNoFixedZeroSetEquiv {n : ℕ} (k : Fin (2 * n + 1)) :
|
||||
{f : Fin (2 * n.succ) → Fin (2 * n.succ) // Function.Involutive f ∧
|
||||
(∀ i, f i ≠ i) ∧ f 0 = k.succ}
|
||||
≃ {f : Fin (2 * n) → Fin (2 * n) // Function.Involutive f ∧ (∀ i, f i ≠ i)} := by
|
||||
refine Equiv.trans (involutionNoFixedZeroSetEquivSetOne k) involutionNoFixedSetOne
|
||||
|
||||
def involutionNoFixedEquivSumSame {n : ℕ} :
|
||||
{f : Fin (2 * n.succ) → Fin (2 * n.succ) // Function.Involutive f ∧ (∀ i, f i ≠ i)}
|
||||
≃ Σ (_ : Fin (2 * n + 1)), {f : Fin (2 * n) → Fin (2 * n) // Function.Involutive f ∧ (∀ i, f i ≠ i)} := by
|
||||
refine Equiv.trans involutionNoFixedEquivSum ?_
|
||||
refine Equiv.sigmaCongrRight involutionNoFixedZeroSetEquiv
|
||||
|
||||
def involutionNoFixedZeroEquivProd {n : ℕ} :
|
||||
{f : Fin (2 * n.succ) → Fin (2 * n.succ) // Function.Involutive f ∧ (∀ i, f i ≠ i)}
|
||||
≃ Fin (2 * n + 1) × {f : Fin (2 * n) → Fin (2 * n) // Function.Involutive f ∧ (∀ i, f i ≠ i)} := by
|
||||
refine Equiv.trans involutionNoFixedEquivSumSame ?_
|
||||
exact Equiv.sigmaEquivProd (Fin (2 * n + 1))
|
||||
{ f // Function.Involutive f ∧ ∀ (i : Fin (2 * n)), f i ≠ i}
|
||||
|
||||
/-!
|
||||
|
||||
## Cardinality
|
||||
|
||||
-/
|
||||
|
||||
|
||||
instance {n : ℕ} : Fintype { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i } := by
|
||||
haveI : DecidablePred fun x => Function.Involutive x := by
|
||||
intro f
|
||||
apply Fintype.decidableForallFintype (α := Fin n)
|
||||
haveI : DecidablePred fun x => Function.Involutive x ∧ ∀ (i : Fin n), x i ≠ i := by
|
||||
intro x
|
||||
apply instDecidableAnd
|
||||
apply Subtype.fintype
|
||||
|
||||
lemma involutionNoFixed_card_succ {n : ℕ} :
|
||||
Fintype.card {f : Fin (2 * n.succ) → Fin (2 * n.succ) // Function.Involutive f ∧ (∀ i, f i ≠ i)}
|
||||
= (2 * n + 1) * Fintype.card {f : Fin (2 * n) → Fin (2 * n) // Function.Involutive f ∧ (∀ i, f i ≠ i)} := by
|
||||
rw [Fintype.card_congr (involutionNoFixedZeroEquivProd)]
|
||||
rw [Fintype.card_prod ]
|
||||
congr
|
||||
exact Fintype.card_fin (2 * n + 1)
|
||||
|
||||
|
||||
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 => by
|
||||
rw [involutionNoFixed_card_succ]
|
||||
rw [involutionNoFixed_card_mul_two n]
|
||||
exact Eq.symm (Nat.doubleFactorial_add_one (Nat.mul 2 n))
|
||||
|
||||
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
|
||||
intro n he
|
||||
obtain ⟨r, hr⟩ := he
|
||||
have hr' : n = 2 * r := by omega
|
||||
subst hr'
|
||||
exact involutionNoFixed_card_mul_two r
|
||||
|
||||
end HepLean.Fin
|
||||
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