feat: Properties of lists
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HepLean/Mathematics/List.lean
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HepLean/Mathematics/List.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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/-!
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# List lemmas
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-/
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namespace HepLean.List
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open Fin
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variable {n : Nat}
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def Fin.equivCons {n m : ℕ} (e : Fin n ≃ Fin m) : Fin n.succ ≃ Fin m.succ where
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toFun := Fin.cons 0 (Fin.succ ∘ e.toFun)
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invFun := Fin.cons 0 (Fin.succ ∘ e.invFun)
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left_inv i := by
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rcases Fin.eq_zero_or_eq_succ i with hi | hi
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· subst hi
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simp
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· obtain ⟨j, hj⟩ := hi
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subst hj
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simp
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right_inv i := by
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rcases Fin.eq_zero_or_eq_succ i with hi | hi
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· subst hi
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simp
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· obtain ⟨j, hj⟩ := hi
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subst hj
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simp
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@[simp]
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lemma Fin.equivCons_trans {n m k : ℕ} (e : Fin n ≃ Fin m) (f : Fin m ≃ Fin k) :
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Fin.equivCons (e.trans f) = (Fin.equivCons e).trans (Fin.equivCons f) := by
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refine Equiv.ext_iff.mpr ?_
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intro x
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simp [Fin.equivCons]
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match x with
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| ⟨0, h⟩ => rfl
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| ⟨i + 1, h⟩ => rfl
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@[simp]
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lemma Fin.equivCons_castOrderIso {n m : ℕ} (h : n = m) :
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(Fin.equivCons (Fin.castOrderIso h).toEquiv) = (Fin.castOrderIso (by simp [h])).toEquiv := by
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refine Equiv.ext_iff.mpr ?_
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intro x
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simp [Fin.equivCons]
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match x with
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| ⟨0, h⟩ => rfl
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| ⟨i + 1, h⟩ => rfl
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@[simp]
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lemma Fin.equivCons_symm_succ {n m : ℕ} (e : Fin n ≃ Fin m) (i : ℕ) (hi : i + 1 < m.succ) :
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(Fin.equivCons e).symm ⟨i + 1, hi⟩ = (e.symm ⟨i , Nat.succ_lt_succ_iff.mp hi⟩).succ := by
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simp [Fin.equivCons]
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have hi : ⟨i + 1, hi⟩ = Fin.succ ⟨i, Nat.succ_lt_succ_iff.mp hi⟩ := by rfl
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rw [hi]
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rw [Fin.cons_succ]
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simp
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def insertEquiv {α : Type} (r : α → α → Prop) [DecidableRel r] (a : α) : (l : List α) →
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Fin (a :: l).length ≃ Fin (List.orderedInsert r a l).length
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| [] => Equiv.refl _
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| b :: l => by
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if r a b then
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exact (Fin.castOrderIso (List.orderedInsert_length r (b :: l) a).symm).toEquiv
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else
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let e := insertEquiv (r := r) a l
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let e2 : Fin (a :: b :: l).length ≃ Fin (b :: a :: l).length :=
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Equiv.swap ⟨0, Nat.zero_lt_succ (b :: l).length⟩ ⟨1, Nat.one_lt_succ_succ l.length⟩
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let e3 : Fin (b :: a :: l).length ≃ Fin (b :: List.orderedInsert r a l).length :=
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Fin.equivCons e
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let e4 : Fin (b :: List.orderedInsert r a l).length ≃ Fin (List.orderedInsert r a (b :: l)).length :=
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(Fin.castOrderIso (by
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rw [List.orderedInsert_length]
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simpa using List.orderedInsert_length r l a
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)).toEquiv
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exact e2.trans (e3.trans e4)
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lemma insertEquiv_congr {α : Type} {r : α → α → Prop} [DecidableRel r] (a : α) (l l' : List α)
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(h : l = l') : insertEquiv r a l = (Fin.castOrderIso (by simp [h])).toEquiv.trans
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((insertEquiv r a l').trans (Fin.castOrderIso (by simp [h])).toEquiv) := by
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subst h
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rfl
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lemma insertEquiv_cons_pos {α : Type} {r : α → α → Prop} [DecidableRel r] (a b : α) (hab : r a b)
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(l : List α) : insertEquiv r a (b :: l) =
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(Fin.castOrderIso (List.orderedInsert_length r (b :: l) a).symm).toEquiv := by
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simp [insertEquiv, hab]
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lemma insertEquiv_cons_neg {α : Type} {r : α → α → Prop} [DecidableRel r] (a b : α) (hab : ¬ r a b)
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(l : List α) : insertEquiv r a (b :: l) =
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let e := insertEquiv r a l
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let e2 : Fin (a :: b :: l).length ≃ Fin (b :: a :: l).length :=
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Equiv.swap ⟨0, Nat.zero_lt_succ (b :: l).length⟩ ⟨1, Nat.one_lt_succ_succ l.length⟩
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let e3 : Fin (b :: a :: l).length ≃ Fin (b :: List.orderedInsert r a l).length :=
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Fin.equivCons e
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let e4 : Fin (b :: List.orderedInsert r a l).length ≃ Fin (List.orderedInsert r a (b :: l)).length :=
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(Fin.castOrderIso (by
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rw [List.orderedInsert_length]
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simpa using List.orderedInsert_length r l a
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)).toEquiv
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e2.trans (e3.trans e4) := by
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simp [insertEquiv, hab]
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lemma insertEquiv_get {α : Type} {r : α → α → Prop} [DecidableRel r] (a : α) : (l : List α) →
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(a :: l).get ∘ (insertEquiv r a l).symm = (List.orderedInsert r a l).get
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| [] => by
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simp [insertEquiv]
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| b :: l => by
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by_cases hr : r a b
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· rw [insertEquiv_cons_pos a b hr l]
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simp_all only [List.orderedInsert.eq_2, List.length_cons, OrderIso.toEquiv_symm, symm_castOrderIso,
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RelIso.coe_fn_toEquiv]
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ext x : 1
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simp_all only [Function.comp_apply, castOrderIso_apply, List.get_eq_getElem, List.length_cons, coe_cast,
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↓reduceIte]
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· rw [insertEquiv_cons_neg a b hr l]
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trans (b :: List.orderedInsert r a l).get ∘ Fin.cast (by
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rw [List.orderedInsert_length]
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simp [List.orderedInsert_length])
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· simp
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ext x
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match x with
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| ⟨0, h⟩ => rfl
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| ⟨Nat.succ x, h⟩ =>
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simp
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have hswap (n : Fin (b :: a :: l).length) :
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(a :: b :: l).get (Equiv.swap ⟨0, by simp⟩ ⟨1, by simp⟩ n) = (b :: a :: l).get n := by
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match n with
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| ⟨0, h⟩ => rfl
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| ⟨1, h⟩ => rfl
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| ⟨Nat.succ (Nat.succ x), h⟩ => rfl
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trans (a :: b :: l).get (Equiv.swap ⟨0, by simp⟩ ⟨1, by simp⟩ ((insertEquiv r a l).symm ⟨x, by simpa [List.orderedInsert_length, hr] using h⟩).succ)
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· simp
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· rw [hswap]
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simp
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change _ = (List.orderedInsert r a l).get _
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rw [← insertEquiv_get (r := r) a l]
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simp
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· simp_all only [List.orderedInsert.eq_2, List.length_cons]
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ext x : 1
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simp_all only [Function.comp_apply, List.get_eq_getElem, List.length_cons, coe_cast, ↓reduceIte]
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def insertionSortEquiv {α : Type} (r : α → α → Prop) [DecidableRel r] : (l : List α) →
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Fin l.length ≃ Fin (List.insertionSort r l).length
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| [] => Equiv.refl _
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| a :: l =>
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(Fin.equivCons (insertionSortEquiv r l)).trans (insertEquiv r a (List.insertionSort r l))
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lemma insertionSortEquiv_get {α : Type} {r : α → α → Prop} [DecidableRel r] : (l : List α) →
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l.get ∘ (insertionSortEquiv r l).symm = (List.insertionSort r l).get
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| [] => by
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simp [insertionSortEquiv]
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| a :: l => by
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rw [insertionSortEquiv]
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change ((a :: l).get ∘ ((Fin.equivCons (insertionSortEquiv r l))).symm) ∘ (insertEquiv r a (List.insertionSort r l)).symm = _
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have hl : (a :: l).get ∘ ((Fin.equivCons (insertionSortEquiv r l))).symm = (a :: List.insertionSort r l).get := by
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ext x
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match x with
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| ⟨0, h⟩ => rfl
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| ⟨Nat.succ x, h⟩ =>
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change _ = (List.insertionSort r l).get _
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rw [← insertionSortEquiv_get (r := r) l]
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rfl
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rw [hl]
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rw [insertEquiv_get (r := r) a (List.insertionSort r l)]
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rfl
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lemma insertionSort_eq_ofFn {α : Type} {r : α → α → Prop} [DecidableRel r] (l : List α) :
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List.insertionSort r l = List.ofFn (l.get ∘ (insertionSortEquiv r l).symm) := by
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rw [insertionSortEquiv_get (r := r)]
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exact Eq.symm (List.ofFn_get (List.insertionSort r l))
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end HepLean.List
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