refactor: style lint
This commit is contained in:
jstoobysmith
parent
7026d8ce66
commit
408a676bbd
10 changed files with 111 additions and 113 deletions
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@ -25,7 +25,7 @@ def involutionCons (n : ℕ) : {f : Fin n.succ → Fin n.succ // Function.Involu
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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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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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@ -33,7 +33,7 @@ def involutionCons (n : ℕ) : {f : Fin n.succ → Fin n.succ // Function.Involu
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simp only [f.2 i.succ, Fin.pred_succ, dite_eq_ite, ite_eq_right_iff]
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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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⟨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 only [succ_eq_add_one, h0, ↓reduceDIte, Option.isSome_some, Option.get_some,
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@ -43,8 +43,7 @@ def involutionCons (n : ℕ) : {f : Fin n.succ → Fin n.succ // Function.Involu
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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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Fin.cons 0 (Fin.succ ∘ f.1.1), 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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@ -59,7 +58,7 @@ def involutionCons (n : ℕ) : {f : Fin n.succ → Fin n.succ // Function.Involu
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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 [Function.update_apply]
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rw [if_neg hja]
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simp only [Function.comp_apply, Fin.cons_succ]
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have hf2 := f.2.2 hs
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@ -145,12 +144,12 @@ def involutionCons (n : ℕ) : {f : Fin n.succ → Fin n.succ // Function.Involu
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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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· 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 only [Fin.cons_zero, Fin.pred_succ, exists_prop]
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have hx : ¬ (f0.get hs).succ = 0 := (Fin.succ_ne_zero (f0.get hs))
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have hx : ¬ (f0.get hs).succ = 0 := (Fin.succ_ne_zero (f0.get hs))
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simp only [hx, not_false_eq_true, true_and]
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refine Iff.intro (fun hi => ?_) (fun hi => ?_)
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· rw [← hi]
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@ -166,7 +165,7 @@ def involutionCons (n : ℕ) : {f : Fin n.succ → Fin n.succ // Function.Involu
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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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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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@ -181,10 +180,9 @@ lemma involutionCons_ext {n : ℕ} {f1 f2 : (f : {f : Fin n → Fin n // Functi
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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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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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@ -208,14 +206,13 @@ def involutionAddEquiv {n : ℕ} (f : {f : Fin n → Fin n // Function.Involutiv
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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 (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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→ 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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@ -235,28 +232,29 @@ lemma involutionAddEquiv_none_image_zero {n : ℕ} :
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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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(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 only [finCongr_refl, Equiv.optionCongr_refl, Equiv.trans_refl]
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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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{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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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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(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 only [succ_eq_add_one, involutionCons, Fin.cons_update, Equiv.coe_fn_mk, dite_eq_left_iff]
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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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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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@ -286,7 +284,7 @@ lemma involutionAddEquiv_isSome_image_zero {n : ℕ} :
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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 (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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@ -310,7 +308,7 @@ The main aim of thes equivalences is to define `involutionNoFixedZeroEquivProd`.
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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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{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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@ -332,11 +330,12 @@ 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)) (e : Fin (2 * n.succ) ≃ Fin (2 * n.succ)) :
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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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(∀ i, f i ≠ i) ∧ (e.symm ∘ f ∘ e) 0 = k.succ} := by
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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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@ -360,12 +359,12 @@ def involutionNoFixedZeroSetEquivSetEquiv {n : ℕ} (k : Fin (2 * n + 1)) (e : F
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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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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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(∀ 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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(∀ 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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@ -373,13 +372,13 @@ def involutionNoFixedZeroSetEquivEquiv' {n : ℕ} (k : Fin (2 * n + 1)) (e : Fin
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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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(∀ 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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(∀ 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)) ?_
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refine Equiv.subtypeEquivRight ?_
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refine Equiv.subtypeEquivRight ?_
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simp only [succ_eq_add_one, ne_eq, Equiv.swap_apply_left, and_congr_right_iff]
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intro f hi h1
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rw [Equiv.swap_apply_of_ne_of_ne]
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@ -387,10 +386,9 @@ def involutionNoFixedZeroSetEquivSetOne {n : ℕ} (k : Fin (2 * n + 1)) :
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· exact Fin.zero_ne_one
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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} ≃
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{f : Fin (2 * n) → Fin (2 * n) // Function.Involutive f ∧
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(∀ i, f i ≠ i)} where
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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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(∀ 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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have hf := f.2.2.2
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@ -444,7 +442,7 @@ def involutionNoFixedSetOne {n : ℕ} :
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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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(hab : ↑(f.1 ⟨a, ha⟩) = b): ↑(f.1 ⟨b, hb⟩) = a := by
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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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rw [← ht, f.2.1]
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@ -504,7 +502,7 @@ def involutionNoFixedSetOne {n : ℕ} :
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simp only [Fin.coe_pred]
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split
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· rename_i h
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simp [Fin.ext_iff] at h
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simp [Fin.ext_iff] at h
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· rename_i h
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simp [Fin.ext_iff] at h
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· rename_i h
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@ -513,22 +511,23 @@ 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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(∀ i, f i ≠ i) ∧ f 0 = k.succ}
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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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refine Equiv.trans (involutionNoFixedZeroSetEquivSetOne k) involutionNoFixedSetOne
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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)), {f : Fin (2 * n) → Fin (2 * n) // Function.Involutive f ∧ (∀ i, f i ≠ i)} := by
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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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refine Equiv.trans involutionNoFixedEquivSum ?_
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refine Equiv.sigmaCongrRight involutionNoFixedZeroSetEquiv
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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) × {f : Fin (2 * n) → Fin (2 * n) // Function.Involutive f ∧ (∀ i, f i ≠ i)} := by
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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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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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@ -539,27 +538,24 @@ def involutionNoFixedZeroEquivProd {n : ℕ} :
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-/
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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 := by
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intro f
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apply Fintype.decidableForallFintype (α := Fin n)
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haveI : DecidablePred fun x => Function.Involutive x ∧ ∀ (i : Fin n), x i ≠ i := by
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intro x
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apply instDecidableAnd
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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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haveI : DecidablePred fun x => Function.Involutive x ∧ ∀ (i : Fin n), x i ≠ i :=
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fun x => instDecidableAnd
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apply Subtype.fintype
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lemma involutionNoFixed_card_succ {n : ℕ} :
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Fintype.card {f : Fin (2 * n.succ) → Fin (2 * n.succ) // Function.Involutive f ∧ (∀ i, f i ≠ i)}
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= (2 * n + 1) * Fintype.card {f : Fin (2 * n) → Fin (2 * n) // Function.Involutive f ∧ (∀ i, f i ≠ i)} := by
|
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rw [Fintype.card_congr (involutionNoFixedZeroEquivProd)]
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rw [Fintype.card_prod ]
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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)}
|
||||
= (2 * n + 1) *
|
||||
Fintype.card {f : Fin (2 * n) → Fin (2 * 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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|
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lemma involutionNoFixed_card_mul_two : (n : ℕ) →
|
||||
Fintype.card {f : Fin (2 * n) → Fin (2 * n) // Function.Involutive f ∧ (∀ i, f i ≠ i)}
|
||||
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
|
||||
|
|
@ -568,7 +564,7 @@ lemma involutionNoFixed_card_mul_two : (n : 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
|
||||
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
|
||||
|
|
|
|||
|
|
@ -662,7 +662,6 @@ def optionErase {I : Type} (l : List I) (i : Option (Fin l.length)) : List I :=
|
|||
| none => l
|
||||
| some i => List.eraseIdx l i
|
||||
|
||||
|
||||
lemma eraseIdx_length {I : Type} (l : List I) (i : Fin l.length) :
|
||||
(List.eraseIdx l i).length + 1 = l.length := by
|
||||
simp only [List.length_eraseIdx, Fin.is_lt, ↓reduceIte]
|
||||
|
|
@ -744,7 +743,7 @@ lemma optionEraseZ_some_length {I : Type} (l : List I) (a : I) (i : (Fin l.lengt
|
|||
lemma optionEraseZ_ext {I : Type} {l l' : List I} {a a' : I} {i : Option (Fin l.length)}
|
||||
{i' : Option (Fin l'.length)} (hl : l = l') (ha : a = a')
|
||||
(hi : Option.map (Fin.cast (by rw [hl])) i = i') :
|
||||
optionEraseZ l a i = optionEraseZ l' a' i' := by
|
||||
optionEraseZ l a i = optionEraseZ l' a' i' := by
|
||||
subst hl
|
||||
subst ha
|
||||
cases hi
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue