268 lines
7.2 KiB
Text
268 lines
7.2 KiB
Text
/-
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Copyright (c) 2016 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Leonardo de Moura
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-/
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prelude
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import init.logic init.data.nat.basic init.data.bool.basic
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open decidable list
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universes u v w
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instance (α : Type u) : inhabited (list α) :=
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⟨list.nil⟩
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variables {α : Type u} {β : Type v} {γ : Type w}
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namespace list
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@[simp] protected def append : list α → list α → list α
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| [] l := l
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| (h :: s) t := h :: (append s t)
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instance : has_append (list α) :=
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⟨list.append⟩
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protected def mem : α → list α → Prop
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| a [] := false
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| a (b :: l) := a = b ∨ mem a l
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instance : has_mem α (list α) :=
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⟨list.mem⟩
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instance decidable_mem [decidable_eq α] (a : α) : ∀ (l : list α), decidable (a ∈ l)
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| [] := is_false not_false
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| (b::l) :=
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if h₁ : a = b then is_true (or.inl h₁)
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else match decidable_mem l with
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| is_true h₂ := is_true (or.inr h₂)
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| is_false h₂ := is_false (not_or h₁ h₂)
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end
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instance : has_emptyc (list α) :=
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⟨list.nil⟩
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protected def erase {α} [decidable_eq α] : list α → α → list α
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| [] b := []
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| (a::l) b := if a = b then l else a :: erase l b
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protected def bag_inter {α} [decidable_eq α] : list α → list α → list α
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| [] _ := []
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| _ [] := []
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| (a::l₁) l₂ := if a ∈ l₂ then a :: bag_inter l₁ (l₂.erase a) else bag_inter l₁ l₂
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protected def diff {α} [decidable_eq α] : list α → list α → list α
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| l [] := l
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| l₁ (a::l₂) := if a ∈ l₁ then diff (l₁.erase a) l₂ else diff l₁ l₂
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@[simp] def length : list α → nat
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| [] := 0
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| (a :: l) := length l + 1
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def empty : list α → bool
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| [] := tt
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| (_ :: _) := ff
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open option nat
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@[simp] def nth : list α → nat → option α
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| [] n := none
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| (a :: l) 0 := some a
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| (a :: l) (n+1) := nth l n
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@[simp] def nth_le : Π (l : list α) (n), n < l.length → α
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| [] n h := absurd h (not_lt_zero n)
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| (a :: l) 0 h := a
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| (a :: l) (n+1) h := nth_le l n (le_of_succ_le_succ h)
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@[simp] def head [inhabited α] : list α → α
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| [] := default α
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| (a :: l) := a
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@[simp] def tail : list α → list α
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| [] := []
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| (a :: l) := l
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def reverse_core : list α → list α → list α
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| [] r := r
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| (a::l) r := reverse_core l (a::r)
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def reverse : list α → list α :=
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λ l, reverse_core l []
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@[simp] def map (f : α → β) : list α → list β
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| [] := []
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| (a :: l) := f a :: map l
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@[simp] def map₂ (f : α → β → γ) : list α → list β → list γ
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| [] _ := []
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| _ [] := []
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| (x::xs) (y::ys) := f x y :: map₂ xs ys
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def join : list (list α) → list α
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| [] := []
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| (l :: ls) := l ++ join ls
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def filter_map (f : α → option β) : list α → list β
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| [] := []
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| (a::l) :=
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match f a with
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| none := filter_map l
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| some b := b :: filter_map l
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end
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def filter (p : α → Prop) [decidable_pred p] : list α → list α
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| [] := []
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| (a::l) := if p a then a :: filter l else filter l
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def partition (p : α → Prop) [decidable_pred p] : list α → list α × list α
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| [] := ([], [])
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| (a::l) := let (l₁, l₂) := partition l in if p a then (a :: l₁, l₂) else (l₁, a :: l₂)
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def drop_while (p : α → Prop) [decidable_pred p] : list α → list α
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| [] := []
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| (a::l) := if p a then drop_while l else a::l
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def span (p : α → Prop) [decidable_pred p] : list α → list α × list α
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| [] := ([], [])
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| (a::xs) := if p a then let (l, r) := span xs in (a :: l, r) else ([], a::xs)
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def find_index (p : α → Prop) [decidable_pred p] : list α → nat
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| [] := 0
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| (a::l) := if p a then 0 else succ (find_index l)
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def index_of [decidable_eq α] (a : α) : list α → nat := find_index (eq a)
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def remove_all [decidable_eq α] (xs ys : list α) : list α :=
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filter (∉ ys) xs
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def update_nth : list α → ℕ → α → list α
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| (x::xs) 0 a := a :: xs
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| (x::xs) (i+1) a := x :: update_nth xs i a
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| [] _ _ := []
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def remove_nth : list α → ℕ → list α
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| [] _ := []
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| (x::xs) 0 := xs
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| (x::xs) (i+1) := x :: remove_nth xs i
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@[simp] def drop : ℕ → list α → list α
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| 0 a := a
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| (succ n) [] := []
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| (succ n) (x::r) := drop n r
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@[simp] def take : ℕ → list α → list α
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| 0 a := []
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| (succ n) [] := []
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| (succ n) (x :: r) := x :: take n r
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@[simp] def foldl (f : α → β → α) : α → list β → α
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| a [] := a
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| a (b :: l) := foldl (f a b) l
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@[simp] def foldr (f : α → β → β) (b : β) : list α → β
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| [] := b
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| (a :: l) := f a (foldr l)
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def any (l : list α) (p : α → bool) : bool :=
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foldr (λ a r, p a || r) ff l
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def all (l : list α) (p : α → bool) : bool :=
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foldr (λ a r, p a && r) tt l
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def bor (l : list bool) : bool := any l id
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def band (l : list bool) : bool := all l id
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def zip_with (f : α → β → γ) : list α → list β → list γ
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| (x::xs) (y::ys) := f x y :: zip_with xs ys
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| _ _ := []
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def zip : list α → list β → list (prod α β) :=
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zip_with prod.mk
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def unzip : list (α × β) → list α × list β
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| [] := ([], [])
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| ((a, b) :: t) := match unzip t with (al, bl) := (a::al, b::bl) end
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protected def insert [decidable_eq α] (a : α) (l : list α) : list α :=
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if a ∈ l then l else a :: l
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instance [decidable_eq α] : has_insert α (list α) :=
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⟨list.insert⟩
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protected def union [decidable_eq α] (l₁ l₂ : list α) : list α :=
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foldr insert l₂ l₁
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instance [decidable_eq α] : has_union (list α) :=
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⟨list.union⟩
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protected def inter [decidable_eq α] (l₁ l₂ : list α) : list α :=
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filter (∈ l₂) l₁
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instance [decidable_eq α] : has_inter (list α) :=
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⟨list.inter⟩
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@[simp] def repeat (a : α) : ℕ → list α
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| 0 := []
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| (succ n) := a :: repeat n
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def range_core : ℕ → list ℕ → list ℕ
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| 0 l := l
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| (succ n) l := range_core n (n :: l)
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def range (n : ℕ) : list ℕ :=
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range_core n []
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def iota : ℕ → list ℕ
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| 0 := []
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| (succ n) := succ n :: iota n
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def enum_from : ℕ → list α → list (ℕ × α)
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| n [] := nil
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| n (x :: xs) := (n, x) :: enum_from (n + 1) xs
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def enum : list α → list (ℕ × α) := enum_from 0
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@[simp] def last : Π l : list α, l ≠ [] → α
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| [] h := absurd rfl h
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| [a] h := a
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| (a::b::l) h := last (b::l) (λ h, list.no_confusion h)
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def ilast [inhabited α] : list α → α
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| [] := arbitrary α
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| [a] := a
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| [a, b] := b
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| (a::b::l) := ilast l
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def init : list α → list α
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| [] := []
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| [a] := []
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| (a::l) := a::init l
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def intersperse (sep : α) : list α → list α
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| [] := []
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| [x] := [x]
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| (x::xs) := x::sep::intersperse xs
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def intercalate (sep : list α) (xs : list (list α)) : list α :=
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join (intersperse sep xs)
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@[inline] def bind {α : Type u} {β : Type v} (a : list α) (b : α → list β) : list β :=
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join (map b a)
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@[inline] def ret {α : Type u} (a : α) : list α :=
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[a]
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end list
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namespace bin_tree
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private def to_list_aux : bin_tree α → list α → list α
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| empty as := as
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| (leaf a) as := a::as
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| (node l r) as := to_list_aux l (to_list_aux r as)
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def to_list (t : bin_tree α) : list α :=
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to_list_aux t []
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end bin_tree
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