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namespace List
theorem listFilterSmallerOrEqualList (l : List α) (p : α → Bool) : l.length ≥ (l.filter p).length := by
induction l with
| nil => simp[List.filter]
| cons a as hi =>
simp[List.length, List.filter]
cases (p a) with
| false =>
simp
constructor
assumption
| true => simp_arith[hi]
def quicksortBy {α : Type} (pred : α → α → Bool): List α → List α
| [] => []
| a :: as =>
let smallerPred := λ b ↦ pred b a
let largerEqualPred := not ∘ smallerPred
have : List.length (List.filter smallerPred as) < Nat.succ (List.length as) := by simp_arith[listFilterSmallerOrEqualList]
have : List.length (List.filter largerEqualPred as) < Nat.succ (List.length as) := by simp_arith[listFilterSmallerOrEqualList]
let smallers := as.filter smallerPred
let biggers := as.filter largerEqualPred
(quicksortBy pred smallers) ++ [a] ++ (quicksortBy pred biggers)
termination_by l => l.length
def quicksort {α : Type} [Ord α] : List α → List α := quicksortBy λ a b ↦ Ord.compare a b == Ordering.lt
/-- Maps a List to another list, but keeps state.
The output list is a list of the state, and **has** the initial state
prepended!
-/
def scan {α σ : Type} (step : σ → α → σ) (init : σ): List α → List σ
| [] => [init]
| a :: as =>
let next := step init a
init :: scan step next as
def compare {α : Type} [Ord α] (a b : List α) := match a, b with
| _ :: _, [] => Ordering.gt
| [], _ :: _ => Ordering.lt
| [], [] => Ordering.eq
| a :: as, b :: bs =>
let ab := Ord.compare a b
if ab != Ordering.eq then
ab
else
compare as bs
instance {α : Type} [Ord α] : Ord (List α) where
compare := compare
def not_empty_iff_size_gt_zero {list : List α} : list.isEmpty = false ↔ list.length > 0 :=
match list with
| [] => ⟨nofun, nofun⟩
| l :: ls => ⟨λ_ ↦ (List.length_cons l ls).substr (Nat.succ_pos ls.length), λ_ => rfl⟩
def ne_nil_of_not_empty {list : List α} : list.isEmpty = false ↔ list ≠ [] :=
match list with
| [] => ⟨nofun, nofun⟩
| _ :: _ => ⟨nofun, λ_ ↦ List.isEmpty_cons⟩
def drop_while_length_le (list : List α) (f : α → Bool) : (list.dropWhile f).length ≤ list.length := by
unfold dropWhile
split
case h_1 => exact Nat.le_refl _
case h_2 =>
split
case h_2 => exact Nat.le_refl _
case h_1 a as _ _ =>
have := drop_while_length_le as f
exact Nat.le_trans this (Nat.le_succ as.length)
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