refactor: Lint and update involutions of Fin n
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jstoobysmith
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4 changed files with 103 additions and 67 deletions
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@ -19,6 +19,9 @@ namespace HepLean.Fin
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open Nat
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/-- There is an equivalence between involutions of `Fin n.succ` and involutions of
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`Fin n` and an optional valid choice of an element in `Fin n` (which is where `0`
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in `Fin n.succ` will be sent). -/
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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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@ -180,6 +183,9 @@ lemma involutionCons_ext {n : ℕ} {f1 f2 : (f : {f : Fin n → Fin n // Functio
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simp_all only
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rfl
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/-- Given an involution of `Fin n`, the optional choice of an element in `Fin n` which
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maps to itself is equivalent to the optional choice of an element in
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`Fin (Finset.univ.filter fun i => f.1 i = i).card`. -/
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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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@ -240,9 +246,9 @@ lemma involutionAddEquiv_cast {n : ℕ} {f1 f2 : {f : Fin n → Fin n // Functio
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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))
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(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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{N : ℕ} (hf : f1 = f2) (n : Option (Fin N))
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(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))
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((involutionAddEquiv f2).symm (Option.map (finCongr hn2) n)) := by
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subst hf
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@ -281,10 +287,19 @@ lemma involutionAddEquiv_isSome_image_zero {n : ℕ} :
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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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/-!
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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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/-- Fixed point free involutions of `Fin n.succ` can be seperated based on where they sent
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`0`. -/
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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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{f : Fin n.succ → Fin n.succ // Function.Involutive f
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∧ ∀ i, f i ≠ i} ≃ Σ (k : Fin n), {f : Fin n.succ → Fin 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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@ -298,19 +313,12 @@ def involutionNoFixedEquivSum {n : ℕ} :
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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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/-- The condition on fixed point free involutions of `Fin n.succ` for a fixed value of `f 0`,
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can be modified by conjugation with an equivalence. -/
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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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(k : Fin n) (e : Fin n.succ ≃ Fin n.succ) :
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{f : Fin n.succ → Fin n.succ // Function.Involutive f ∧ (∀ i, f i ≠ i) ∧ f 0 = k.succ} ≃
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{f : Fin n.succ → Fin 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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@ -330,11 +338,14 @@ def involutionNoFixedZeroSetEquivEquiv {n : ℕ}
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ext i
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simp
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def involutionNoFixedZeroSetEquivSetEquiv {n : ℕ} (k : Fin (2 * n + 1))
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(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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/-- The condition on fixed point free involutions of `Fin n.succ` for a fixed value of `f 0`
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given an equivalence `e`,
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can be modified so that only the condition on `f 0` is up-to the equivalence `e`. -/
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def involutionNoFixedZeroSetEquivSetEquiv {n : ℕ} (k : Fin n)
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(e : Fin n.succ ≃ Fin n.succ) :
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{f : Fin n.succ → Fin 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 n.succ → Fin 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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@ -359,21 +370,24 @@ def involutionNoFixedZeroSetEquivSetEquiv {n : ℕ} (k : Fin (2 * n + 1))
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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))
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(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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/-- Fixed point free involutions of `Fin n.succ` fixing `(e.symm ∘ f ∘ e) = k.succ` for a given `e`
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are equivalent to fixing `f (e 0) = e k.succ`. -/
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def involutionNoFixedZeroSetEquivEquiv' {n : ℕ} (k : Fin n) (e : Fin n.succ ≃ Fin n.succ) :
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{f : Fin n.succ → Fin 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 n.succ → Fin 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 only [succ_eq_add_one, ne_eq, Function.comp_apply, and_congr_right_iff]
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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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/-- Fixed point involutions of `Fin n.succ.succ` with `f 0 = k.succ` are equivalent
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to fixed point involutions with `f 0 = 1`. -/
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def involutionNoFixedZeroSetEquivSetOne {n : ℕ} (k : Fin n.succ) :
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{f : Fin n.succ.succ → Fin n.succ.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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≃ {f : Fin n.succ.succ → Fin n.succ.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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@ -385,9 +399,11 @@ def involutionNoFixedZeroSetEquivSetOne {n : ℕ} (k : Fin (2 * n + 1)) :
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· exact Ne.symm (Fin.succ_ne_zero k)
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· exact Fin.zero_ne_one
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/-- Fixed point involutions of `Fin n.succ.succ` fixing `f 0 = 1` are equivalent to
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fixed point involutions of `Fin n`. -/
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def involutionNoFixedSetOne {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) ∧ f 0 = 1} ≃ {f : Fin (2 * n) → Fin (2 * n) // Function.Involutive f ∧
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{f : Fin n.succ.succ → Fin n.succ.succ // Function.Involutive f ∧
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(∀ i, f i ≠ i) ∧ f 0 = 1} ≃ {f : Fin n → Fin n // Function.Involutive f ∧
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(∀ i, f i ≠ i)} where
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toFun f := by
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have hf1 : f.1 1 = 0 := by
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@ -395,14 +411,14 @@ def involutionNoFixedSetOne {n : ℕ} :
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simp only [succ_eq_add_one, ne_eq, ← hf]
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rw [f.2.1]
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let f' := f.1 ∘ Fin.succ ∘ Fin.succ
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have hf' (i : Fin (2 * n)) : f' i ≠ 0 := by
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have hf' (i : Fin n) : f' i ≠ 0 := by
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simp only [succ_eq_add_one, mul_eq, ne_eq, Function.comp_apply, f']
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simp only [← hf1, succ_eq_add_one, ne_eq]
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by_contra hn
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have hn' := Function.Involutive.injective f.2.1 hn
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simp [Fin.ext_iff] at hn'
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let f'' := fun i => (f' i).pred (hf' i)
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have hf'' (i : Fin (2 * n)) : f'' i ≠ 0 := by
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have hf'' (i : Fin n) : f'' i ≠ 0 := by
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simp only [mul_eq, ne_eq, f'']
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rw [@Fin.pred_eq_iff_eq_succ]
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simp only [mul_eq, succ_eq_add_one, ne_eq, Function.comp_apply, Fin.succ_zero_eq_one, f']
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@ -420,7 +436,7 @@ def involutionNoFixedSetOne {n : ℕ} :
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rw [Fin.pred_eq_iff_eq_succ, Fin.pred_eq_iff_eq_succ]
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exact f.2.2.1 i.succ.succ
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invFun f := by
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let f' := fun (i : Fin (2 * n.succ))=>
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let f' := fun (i : Fin n.succ.succ)=>
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match i with
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| ⟨0, h⟩ => 1
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| ⟨1, h⟩ => 0
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@ -441,7 +457,7 @@ def involutionNoFixedSetOne {n : ℕ} :
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· rename_i h
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rename_i x r
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simp_all only [succ_eq_add_one, Fin.ext_iff, Fin.val_succ, add_left_inj]
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have hfn {a b : ℕ} {ha : a < 2 * n} {hb : b < 2 * n}
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have hfn {a b : ℕ} {ha : a < n} {hb : b < n}
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(hab : ↑(f.1 ⟨a, ha⟩) = b) : ↑(f.1 ⟨b, hb⟩) = a := by
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have ht : f.1 ⟨a, ha⟩ = ⟨b, hb⟩ := by
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simp [hab, Fin.ext_iff]
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@ -511,26 +527,33 @@ def involutionNoFixedSetOne {n : ℕ} :
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apply congrArg
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simp_all [Fin.ext_iff]
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def involutionNoFixedZeroSetEquiv {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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/-- Fixed point involutions of `Fin n.succ.succ` for fixed `f 0 = k.succ` are
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equivalent to fixed point involutions of `Fin n`. -/
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def involutionNoFixedZeroSetEquiv {n : ℕ} (k : Fin n.succ) :
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{f : Fin n.succ.succ → Fin n.succ.succ // Function.Involutive f ∧
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(∀ i, f i ≠ i) ∧ f 0 = k.succ}
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≃ {f : Fin (2 * n) → Fin (2 * n) // Function.Involutive f ∧ (∀ i, f i ≠ i)} := by
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≃ {f : Fin n → Fin n // Function.Involutive f ∧ (∀ i, f i ≠ i)} := by
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refine Equiv.trans (involutionNoFixedZeroSetEquivSetOne k) involutionNoFixedSetOne
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/-- The type of fixed point free involutions of `Fin n.succ.succ` is equivalent to the sum
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of `Fin n.succ` copies of fixed point involutions of `Fin n`. -/
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def involutionNoFixedEquivSumSame {n : ℕ} :
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{f : Fin (2 * n.succ) → Fin (2 * n.succ) // Function.Involutive f ∧ (∀ i, f i ≠ i)}
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≃ Σ (_ : Fin (2 * n + 1)),
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{f : Fin (2 * n) → Fin (2 * n) // Function.Involutive f ∧ (∀ i, f i ≠ i)} := by
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{f : Fin n.succ.succ → Fin n.succ.succ // Function.Involutive f ∧ (∀ i, f i ≠ i)}
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≃ Σ (_ : Fin n.succ),
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{f : Fin n → Fin n // Function.Involutive f ∧ (∀ i, f i ≠ i)} := by
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refine Equiv.trans involutionNoFixedEquivSum ?_
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refine Equiv.sigmaCongrRight involutionNoFixedZeroSetEquiv
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/-- Ever fixed-point free involutions of `Fin n.succ.succ` can be decomponsed into a
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element of `Fin n.succ` (where `0` is sent) and a fixed-point free involution of
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`Fin n`. -/
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def involutionNoFixedZeroEquivProd {n : ℕ} :
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{f : Fin (2 * n.succ) → Fin (2 * n.succ) // Function.Involutive f ∧ (∀ i, f i ≠ i)}
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≃ Fin (2 * n + 1) ×
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{f : Fin (2 * n) → Fin (2 * n) // Function.Involutive f ∧ (∀ i, f i ≠ i)} := by
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{f : Fin n.succ.succ → Fin n.succ.succ // Function.Involutive f ∧ (∀ i, f i ≠ i)}
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≃ Fin n.succ ×
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{f : Fin n → Fin n // Function.Involutive f ∧ (∀ i, f i ≠ i)} := by
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refine Equiv.trans involutionNoFixedEquivSumSame ?_
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exact Equiv.sigmaEquivProd (Fin (2 * n + 1))
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{ f // Function.Involutive f ∧ ∀ (i : Fin (2 * n)), f i ≠ i}
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exact Equiv.sigmaEquivProd (Fin n.succ)
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{ f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i}
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/-!
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@ -538,6 +561,7 @@ def involutionNoFixedZeroEquivProd {n : ℕ} :
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-/
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/-- The type of fixed-point free involutions of `Fin n` is finite. -/
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instance {n : ℕ} : Fintype { f // Function.Involutive f ∧ ∀ (i : Fin n), f i ≠ i } := by
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haveI : DecidablePred fun x => Function.Involutive x :=
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fun f => Fintype.decidableForallFintype (α := Fin n)
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@ -547,22 +571,31 @@ instance {n : ℕ} : Fintype { f // Function.Involutive f ∧ ∀ (i : Fin n), f
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lemma involutionNoFixed_card_succ {n : ℕ} :
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Fintype.card
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{f : Fin (2 * n.succ) → Fin (2 * n.succ) // Function.Involutive f ∧ (∀ i, f i ≠ i)}
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= (2 * n + 1) *
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Fintype.card {f : Fin (2 * n) → Fin (2 * n) // Function.Involutive f ∧ (∀ i, f i ≠ i)} := by
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{f : Fin n.succ.succ → Fin n.succ.succ // Function.Involutive f ∧ (∀ i, f i ≠ i)}
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= n.succ *
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Fintype.card {f : Fin n → Fin n // Function.Involutive f ∧ (∀ i, f i ≠ i)} := by
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rw [Fintype.card_congr (involutionNoFixedZeroEquivProd), Fintype.card_prod]
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congr
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exact Fintype.card_fin (2 * n + 1)
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exact Fintype.card_fin n.succ
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lemma involutionNoFixed_card_mul_two : (n : ℕ) →
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Fintype.card {f : Fin (2 * n) → Fin (2 * n) // Function.Involutive f ∧ (∀ i, f i ≠ i)}
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= (2 * n - 1)‼
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| 0 => rfl
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| Nat.succ n => by
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rw [involutionNoFixed_card_succ]
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rw [involutionNoFixed_card_mul_two n]
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erw [involutionNoFixed_card_succ]
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erw [involutionNoFixed_card_mul_two n]
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exact Eq.symm (Nat.doubleFactorial_add_one (Nat.mul 2 n))
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lemma involutionNoFixed_card_mul_two_plus_one : (n : ℕ) →
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Fintype.card {f : Fin (2 * n + 1) → Fin (2 * n + 1) // Function.Involutive f ∧ (∀ i, f i ≠ i)}
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= 0
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| 0 => rfl
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| Nat.succ n => by
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erw [involutionNoFixed_card_succ]
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erw [involutionNoFixed_card_mul_two_plus_one n]
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exact rfl
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lemma involutionNoFixed_card_even : (n : ℕ) → (he : Even n) →
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Fintype.card {f : Fin n → Fin n // Function.Involutive f ∧ (∀ i, f i ≠ i)} = (n - 1)‼ := by
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intro n he
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@ -571,4 +604,11 @@ lemma involutionNoFixed_card_even : (n : ℕ) → (he : Even n) →
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subst hr'
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exact involutionNoFixed_card_mul_two r
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lemma involutionNoFixed_card_odd : (n : ℕ) → (ho : Odd n) →
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Fintype.card {f : Fin n → Fin n // Function.Involutive f ∧ (∀ i, f i ≠ i)} = 0 := by
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intro n ho
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obtain ⟨r, hr⟩ := ho
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subst hr
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exact involutionNoFixed_card_mul_two_plus_one r
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end HepLean.Fin
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