/
basic.lean
364 lines (278 loc) · 10.8 KB
/
basic.lean
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/-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura
-/
prelude
import init.logic init.data.nat.basic init.data.bool.basic init.propext
open decidable list
universes u v w
instance (α : Type u) : inhabited (list α) :=
⟨list.nil⟩
variables {α : Type u} {β : Type v} {γ : Type w}
namespace list
protected def has_dec_eq [s : decidable_eq α] : decidable_eq (list α)
| [] [] := is_true rfl
| (a::as) [] := is_false (λ h, list.no_confusion h)
| [] (b::bs) := is_false (λ h, list.no_confusion h)
| (a::as) (b::bs) :=
match s a b with
| is_true hab :=
match has_dec_eq as bs with
| is_true habs := is_true (eq.subst hab (eq.subst habs rfl))
| is_false nabs := is_false (λ h, list.no_confusion h (λ _ habs, absurd habs nabs))
end
| is_false nab := is_false (λ h, list.no_confusion h (λ hab _, absurd hab nab))
end
instance [decidable_eq α] : decidable_eq (list α) :=
list.has_dec_eq
@[simp] protected def append : list α → list α → list α
| [] l := l
| (h :: s) t := h :: (append s t)
instance : has_append (list α) :=
⟨list.append⟩
protected def mem : α → list α → Prop
| a [] := false
| a (b :: l) := a = b ∨ mem a l
instance : has_mem α (list α) :=
⟨list.mem⟩
instance decidable_mem [decidable_eq α] (a : α) : ∀ (l : list α), decidable (a ∈ l)
| [] := is_false not_false
| (b::l) :=
if h₁ : a = b then is_true (or.inl h₁)
else match decidable_mem l with
| is_true h₂ := is_true (or.inr h₂)
| is_false h₂ := is_false (not_or h₁ h₂)
end
instance : has_emptyc (list α) :=
⟨list.nil⟩
protected def erase {α} [decidable_eq α] : list α → α → list α
| [] b := []
| (a::l) b := if a = b then l else a :: erase l b
protected def bag_inter {α} [decidable_eq α] : list α → list α → list α
| [] _ := []
| _ [] := []
| (a::l₁) l₂ := if a ∈ l₂ then a :: bag_inter l₁ (l₂.erase a) else bag_inter l₁ l₂
protected def diff {α} [decidable_eq α] : list α → list α → list α
| l [] := l
| l₁ (a::l₂) := if a ∈ l₁ then diff (l₁.erase a) l₂ else diff l₁ l₂
@[simp] def length : list α → nat
| [] := 0
| (a :: l) := length l + 1
def empty : list α → bool
| [] := tt
| (_ :: _) := ff
open option nat
@[simp] def nth : list α → nat → option α
| [] n := none
| (a :: l) 0 := some a
| (a :: l) (n+1) := nth l n
@[simp] def nth_le : Π (l : list α) (n), n < l.length → α
| [] n h := absurd h (not_lt_zero n)
| (a :: l) 0 h := a
| (a :: l) (n+1) h := nth_le l n (le_of_succ_le_succ h)
@[simp] def head [inhabited α] : list α → α
| [] := default α
| (a :: l) := a
@[simp] def tail : list α → list α
| [] := []
| (a :: l) := l
def reverse_core : list α → list α → list α
| [] r := r
| (a::l) r := reverse_core l (a::r)
def reverse : list α → list α :=
λ l, reverse_core l []
@[simp] def map (f : α → β) : list α → list β
| [] := []
| (a :: l) := f a :: map l
@[simp] def map₂ (f : α → β → γ) : list α → list β → list γ
| [] _ := []
| _ [] := []
| (x::xs) (y::ys) := f x y :: map₂ xs ys
def map_with_index_core (f : ℕ → α → β) : ℕ → list α → list β
| k [] := []
| k (a::as) := f k a::(map_with_index_core (k+1) as)
/-- Given a function `f : ℕ → α → β` and `as : list α`, `as = [a₀, a₁, ...]`, returns the list
`[f 0 a₀, f 1 a₁, ...]`. -/
def map_with_index (f : ℕ → α → β) (as : list α) : list β :=
map_with_index_core f 0 as
def join : list (list α) → list α
| [] := []
| (l :: ls) := l ++ join ls
def filter_map (f : α → option β) : list α → list β
| [] := []
| (a::l) :=
match f a with
| none := filter_map l
| some b := b :: filter_map l
end
def filter (p : α → Prop) [decidable_pred p] : list α → list α
| [] := []
| (a::l) := if p a then a :: filter l else filter l
def partition (p : α → Prop) [decidable_pred p] : list α → list α × list α
| [] := ([], [])
| (a::l) := let (l₁, l₂) := partition l in if p a then (a :: l₁, l₂) else (l₁, a :: l₂)
def drop_while (p : α → Prop) [decidable_pred p] : list α → list α
| [] := []
| (a::l) := if p a then drop_while l else a::l
/-- `after p xs` is the suffix of `xs` after the first element that satisfies
`p`, not including that element.
```lean
after (eq 1) [0, 1, 2, 3] = [2, 3]
drop_while (not ∘ eq 1) [0, 1, 2, 3] = [1, 2, 3]
```
-/
def after (p : α → Prop) [decidable_pred p] : list α → list α
| [] := []
| (x :: xs) := if p x then xs else after xs
def span (p : α → Prop) [decidable_pred p] : list α → list α × list α
| [] := ([], [])
| (a::xs) := if p a then let (l, r) := span xs in (a :: l, r) else ([], a::xs)
def find_index (p : α → Prop) [decidable_pred p] : list α → nat
| [] := 0
| (a::l) := if p a then 0 else succ (find_index l)
def index_of [decidable_eq α] (a : α) : list α → nat := find_index (eq a)
def remove_all [decidable_eq α] (xs ys : list α) : list α :=
filter (∉ ys) xs
def update_nth : list α → ℕ → α → list α
| (x::xs) 0 a := a :: xs
| (x::xs) (i+1) a := x :: update_nth xs i a
| [] _ _ := []
def remove_nth : list α → ℕ → list α
| [] _ := []
| (x::xs) 0 := xs
| (x::xs) (i+1) := x :: remove_nth xs i
@[simp] def drop : ℕ → list α → list α
| 0 a := a
| (succ n) [] := []
| (succ n) (x::r) := drop n r
@[simp] def take : ℕ → list α → list α
| 0 a := []
| (succ n) [] := []
| (succ n) (x :: r) := x :: take n r
@[simp] def foldl (f : α → β → α) : α → list β → α
| a [] := a
| a (b :: l) := foldl (f a b) l
@[simp] def foldr (f : α → β → β) (b : β) : list α → β
| [] := b
| (a :: l) := f a (foldr l)
def any (l : list α) (p : α → bool) : bool :=
foldr (λ a r, p a || r) ff l
def all (l : list α) (p : α → bool) : bool :=
foldr (λ a r, p a && r) tt l
def bor (l : list bool) : bool := any l id
def band (l : list bool) : bool := all l id
def zip_with (f : α → β → γ) : list α → list β → list γ
| (x::xs) (y::ys) := f x y :: zip_with xs ys
| _ _ := []
def zip : list α → list β → list (prod α β) :=
zip_with prod.mk
def unzip : list (α × β) → list α × list β
| [] := ([], [])
| ((a, b) :: t) := match unzip t with (al, bl) := (a::al, b::bl) end
protected def insert [decidable_eq α] (a : α) (l : list α) : list α :=
if a ∈ l then l else a :: l
instance [decidable_eq α] : has_insert α (list α) :=
⟨list.insert⟩
instance : has_singleton α (list α) := ⟨λ x, [x]⟩
instance [decidable_eq α] : is_lawful_singleton α (list α) :=
⟨λ x, show (if x ∈ ([] : list α) then [] else [x]) = [x], from if_neg not_false⟩
protected def union [decidable_eq α] (l₁ l₂ : list α) : list α :=
foldr insert l₂ l₁
instance [decidable_eq α] : has_union (list α) :=
⟨list.union⟩
protected def inter [decidable_eq α] (l₁ l₂ : list α) : list α :=
filter (∈ l₂) l₁
instance [decidable_eq α] : has_inter (list α) :=
⟨list.inter⟩
@[simp] def repeat (a : α) : ℕ → list α
| 0 := []
| (succ n) := a :: repeat n
def range_core : ℕ → list ℕ → list ℕ
| 0 l := l
| (succ n) l := range_core n (n :: l)
def range (n : ℕ) : list ℕ :=
range_core n []
def iota : ℕ → list ℕ
| 0 := []
| (succ n) := succ n :: iota n
def enum_from : ℕ → list α → list (ℕ × α)
| n [] := nil
| n (x :: xs) := (n, x) :: enum_from (n + 1) xs
def enum : list α → list (ℕ × α) := enum_from 0
@[simp] def last : Π l : list α, l ≠ [] → α
| [] h := absurd rfl h
| [a] h := a
| (a::b::l) h := last (b::l) (λ h, list.no_confusion h)
def ilast [inhabited α] : list α → α
| [] := arbitrary α
| [a] := a
| [a, b] := b
| (a::b::l) := ilast l
def init : list α → list α
| [] := []
| [a] := []
| (a::l) := a::init l
def intersperse (sep : α) : list α → list α
| [] := []
| [x] := [x]
| (x::xs) := x::sep::intersperse xs
def intercalate (sep : list α) (xs : list (list α)) : list α :=
join (intersperse sep xs)
@[inline] protected def bind {α : Type u} {β : Type v} (a : list α) (b : α → list β) : list β :=
join (map b a)
@[inline] protected def ret {α : Type u} (a : α) : list α :=
[a]
protected def lt [has_lt α] : list α → list α → Prop
| [] [] := false
| [] (b::bs) := true
| (a::as) [] := false
| (a::as) (b::bs) := a < b ∨ (¬ b < a ∧ lt as bs)
instance [has_lt α] : has_lt (list α) :=
⟨list.lt⟩
instance has_decidable_lt [has_lt α] [h : decidable_rel ((<) : α → α → Prop)] : Π l₁ l₂ : list α, decidable (l₁ < l₂)
| [] [] := is_false not_false
| [] (b::bs) := is_true trivial
| (a::as) [] := is_false not_false
| (a::as) (b::bs) :=
match h a b with
| is_true h₁ := is_true (or.inl h₁)
| is_false h₁ :=
match h b a with
| is_true h₂ := is_false (λ h, or.elim h (λ h, absurd h h₁) (λ ⟨h, _⟩, absurd h₂ h))
| is_false h₂ :=
match has_decidable_lt as bs with
| is_true h₃ := is_true (or.inr ⟨h₂, h₃⟩)
| is_false h₃ := is_false (λ h, or.elim h (λ h, absurd h h₁) (λ ⟨_, h⟩, absurd h h₃))
end
end
end
@[reducible] protected def le [has_lt α] (a b : list α) : Prop :=
¬ b < a
instance [has_lt α] : has_le (list α) :=
⟨list.le⟩
instance has_decidable_le [has_lt α] [h : decidable_rel ((<) : α → α → Prop)] : Π l₁ l₂ : list α, decidable (l₁ ≤ l₂) :=
λ a b, not.decidable
lemma le_eq_not_gt [has_lt α] : ∀ l₁ l₂ : list α, (l₁ ≤ l₂) = ¬ (l₂ < l₁) :=
λ l₁ l₂, rfl
lemma lt_eq_not_ge [has_lt α] [decidable_rel ((<) : α → α → Prop)] : ∀ l₁ l₂ : list α, (l₁ < l₂) = ¬ (l₂ ≤ l₁) :=
λ l₁ l₂,
show (l₁ < l₂) = ¬ ¬ (l₁ < l₂), from
eq.subst (propext (not_not_iff (l₁ < l₂))).symm rfl
/-- `is_prefix_of l₁ l₂` returns `tt` iff `l₁` is a prefix of `l₂`. -/
def is_prefix_of [decidable_eq α] : list α → list α → bool
| [] _ := tt
| _ [] := ff
| (a::as) (b::bs) := to_bool (a = b) && is_prefix_of as bs
/-- `is_suffix_of l₁ l₂` returns `tt` iff `l₁` is a suffix of `l₂`. -/
def is_suffix_of [decidable_eq α] (l₁ l₂ : list α) : bool :=
is_prefix_of l₁.reverse l₂.reverse
end list
namespace bin_tree
private def to_list_aux : bin_tree α → list α → list α
| empty as := as
| (leaf a) as := a::as
| (node l r) as := to_list_aux l (to_list_aux r as)
def to_list (t : bin_tree α) : list α :=
to_list_aux t []
end bin_tree