lean2/library/data/list/basic.lean

/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.

Module: data.list.basic
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura

Basic properties of lists.
-/

import logic tools.helper_tactics data.nat.basic

open eq.ops helper_tactics nat

inductive list (T : Type) : Type :=
nil {} : list T,
cons   : T → list T → list T

namespace list
notation h :: t  := cons h t
notation `[` l:(foldr `,` (h t, cons h t) nil) `]` := l

variable {T : Type}

/- append -/

definition append : list T → list T → list T,
append nil l      := l,
append (h :: s) t := h :: (append s t)

notation l₁ ++ l₂ := append l₁ l₂

theorem append_nil_left (t : list T) : nil ++ t = t

theorem append_cons (x : T) (s t : list T) : (x::s) ++ t = x::(s ++ t)

theorem append_nil_right : ∀ (t : list T), t ++ nil = t,
append_nil_right nil      := rfl,
append_nil_right (a :: l) := calc
  (a :: l) ++ nil = a :: (l ++ nil) : rfl
              ... = a :: l          : append_nil_right l


theorem append.assoc : ∀ (s t u : list T), s ++ t ++ u = s ++ (t ++ u),
append.assoc nil t u      := rfl,
append.assoc (a :: l) t u := calc
  (a :: l) ++ t ++ u = a :: (l ++ t ++ u)   : rfl
                 ... = a :: (l ++ (t ++ u)) : append.assoc
                 ... = (a :: l) ++ (t ++ u) : rfl

/- length -/

definition length : list T → nat,
length nil      := 0,
length (a :: l) := length l + 1

theorem length_nil : length (@nil T) = 0

theorem length_cons (x : T) (t : list T) : length (x::t) = length t + 1

theorem length_append : ∀ (s t : list T), length (s ++ t) = length s + length t,
length_append nil t      := calc
  length (nil ++ t)  = length t : rfl
                 ... = length nil + length t : zero_add,
length_append (a :: s) t := calc
  length (a :: s ++ t) = length (s ++ t) + 1        : rfl
                  ...  = length s + length t + 1    : length_append
                  ...  = (length s + 1) + length t  : add.succ_left
                  ...  = length (a :: s) + length t : rfl

-- add_rewrite length_nil length_cons

/- concat -/

definition concat : Π (x : T), list T → list T,
concat a nil      := [a],
concat a (b :: l) := b :: concat a l

theorem concat_nil (x : T) : concat x nil = [x]

theorem concat_cons (x y : T) (l : list T) : concat x (y::l)  = y::(concat x l)

theorem concat_eq_append (a : T) : ∀ (l : list T), concat a l = l ++ [a],
concat_eq_append nil      := rfl,
concat_eq_append (b :: l) := calc
  concat a (b :: l) = b :: (concat a l) : rfl
               ...  = b :: (l ++ [a])   : concat_eq_append
               ...  = (b :: l) ++ [a]   : rfl

-- add_rewrite append_nil append_cons

/- reverse -/

definition reverse : list T → list T,
reverse nil      := nil,
reverse (a :: l) := concat a (reverse l)

theorem reverse_nil : reverse (@nil T) = nil

theorem reverse_cons (x : T) (l : list T) : reverse (x::l) = concat x (reverse l)

theorem reverse_singleton (x : T) : reverse [x] = [x]

theorem reverse_append : ∀ (s t : list T), reverse (s ++ t) = (reverse t) ++ (reverse s),
reverse_append nil t2      := calc
  reverse (nil ++ t2) = reverse t2                    : rfl
               ...    = (reverse t2) ++ nil           : append_nil_right
               ...    = (reverse t2) ++ (reverse nil) : {reverse_nil⁻¹},
reverse_append (a2 :: s2) t2 := calc
  reverse ((a2 :: s2) ++ t2) = concat a2 (reverse (s2 ++ t2))         : rfl
                      ...    = concat a2 (reverse t2 ++ reverse s2)   : reverse_append
                      ...    = (reverse t2 ++ reverse s2) ++ [a2]     : concat_eq_append
                      ...    = reverse t2 ++ (reverse s2 ++ [a2])     : append.assoc
                      ...    = reverse t2 ++ concat a2 (reverse s2)   : concat_eq_append
                      ...    = reverse t2 ++ reverse (a2 :: s2)       : rfl

theorem reverse_reverse : ∀ (l : list T), reverse (reverse l) = l,
reverse_reverse nil      := rfl,
reverse_reverse (a :: l) := calc
  reverse (reverse (a :: l)) = reverse (concat a (reverse l))     : rfl
                        ...  = reverse (reverse l ++ [a])         : concat_eq_append
                        ...  = reverse [a] ++ reverse (reverse l) : reverse_append
                        ...  = reverse [a] ++ l                   : reverse_reverse
                        ...  = a :: l                             : rfl

theorem concat_eq_reverse_cons (x : T) (l : list T) : concat x l = reverse (x :: reverse l) :=
calc
  concat x l = concat x (reverse (reverse l)) : reverse_reverse
         ... = reverse (x :: reverse l)       : rfl

/- head and tail -/

definition head [h : inhabited T] : list T → T,
head nil      := arbitrary T,
head (a :: l) := a

theorem head_cons [h : inhabited T] (a : T) (l : list T) : head (a::l) = a

theorem head_concat [h : inhabited T] {s : list T} (t : list T) : s ≠ nil → head (s ++ t) = head s :=
list.cases_on s
  (take H : nil ≠ nil, absurd rfl H)
  (take (x : T) (s : list T), take H : x::s ≠ nil,
    calc
      head ((x::s) ++ t) = head (x::(s ++ t)) : rfl
                   ... = x                    : head_cons
                   ... = head (x::s)          : rfl)

definition tail : list T → list T,
tail nil      := nil,
tail (a :: l) := l

theorem tail_nil : tail (@nil T) = nil

theorem tail_cons (a : T) (l : list T) : tail (a::l) = l

theorem cons_head_tail [h : inhabited T] {l : list T} : l ≠ nil → (head l)::(tail l) = l :=
list.cases_on l
  (assume H : nil ≠ nil, absurd rfl H)
  (take x l, assume H : x::l ≠ nil, rfl)

/- list membership -/

definition mem : T → list T → Prop,
mem a nil      := false,
mem a (b :: l) := a = b ∨ mem a l

notation e ∈ s := mem e s

theorem mem_nil (x : T) : x ∈ nil ↔ false :=
iff.rfl

theorem mem_cons (x y : T) (l : list T) : x ∈ y::l ↔ (x = y ∨ x ∈ l) :=
iff.rfl

theorem mem_concat_imp_or {x : T} {s t : list T} : x ∈ s ++ t → x ∈ s ∨ x ∈ t :=
list.induction_on s or.inr
  (take y s,
    assume IH : x ∈ s ++ t → x ∈ s ∨ x ∈ t,
    assume H1 : x ∈ y::s ++ t,
    have H2 : x = y ∨ x ∈ s ++ t, from H1,
    have H3 : x = y ∨ x ∈ s ∨ x ∈ t, from or_of_or_of_imp_right H2 IH,
    iff.elim_right or.assoc H3)

theorem mem_or_imp_concat {x : T} {s t : list T} : x ∈ s ∨ x ∈ t → x ∈ s ++ t :=
list.induction_on s
  (take H, or.elim H false.elim (assume H, H))
  (take y s,
    assume IH : x ∈ s ∨ x ∈ t → x ∈ s ++ t,
    assume H : x ∈ y::s ∨ x ∈ t,
      or.elim H
        (assume H1,
          or.elim H1
            (take H2 : x = y, or.inl H2)
            (take H2 : x ∈ s, or.inr (IH (or.inl H2))))
        (assume H1 : x ∈ t, or.inr (IH (or.inr H1))))

theorem mem_concat (x : T) (s t : list T) : x ∈ s ++ t ↔ x ∈ s ∨ x ∈ t :=
iff.intro mem_concat_imp_or mem_or_imp_concat

local attribute mem [reducible]
local attribute append [reducible]
theorem mem_split {x : T} {l : list T} : x ∈ l → ∃s t : list T, l = s ++ (x::t) :=
list.induction_on l
  (take H : x ∈ nil, false.elim (iff.elim_left !mem_nil H))
  (take y l,
    assume IH : x ∈ l → ∃s t : list T, l = s ++ (x::t),
    assume H : x ∈ y::l,
    or.elim H
      (assume H1 : x = y,
        exists.intro nil (!exists.intro (H1 ▸ rfl)))
      (assume H1 : x ∈ l,
        obtain s (H2 : ∃t : list T, l = s ++ (x::t)), from IH H1,
        obtain t (H3 : l = s ++ (x::t)), from H2,
        have H4 : y :: l = (y::s) ++ (x::t),
          from H3 ▸ rfl,
        !exists.intro (!exists.intro H4)))

definition mem.is_decidable [instance] (H : decidable_eq T) (x : T) (l : list T) : decidable (x ∈ l) :=
list.rec_on l
  (decidable.inr (not_of_iff_false !mem_nil))
  (take (h : T) (l : list T) (iH : decidable (x ∈ l)),
    show decidable (x ∈ h::l), from
    decidable.rec_on iH
      (assume Hp : x ∈ l,
        decidable.rec_on (H x h)
          (assume Heq : x = h,
            decidable.inl (or.inl Heq))
          (assume Hne : x ≠ h,
            decidable.inl (or.inr Hp)))
      (assume Hn : ¬x ∈ l,
        decidable.rec_on (H x h)
          (assume Heq : x = h,
            decidable.inl (or.inl Heq))
          (assume Hne : x ≠ h,
            have H1 : ¬(x = h ∨ x ∈ l), from
              assume H2 : x = h ∨ x ∈ l, or.elim H2
                (assume Heq, absurd Heq Hne)
                (assume Hp,  absurd Hp Hn),
            have H2 : ¬x ∈ h::l, from
              iff.elim_right (not_iff_not_of_iff !mem_cons) H1,
            decidable.inr H2)))

/- find -/

section
variable [H : decidable_eq T]
include H

definition find : T → list T → nat,
find a nil      := 0,
find a (b :: l) := if a = b then 0 else succ (find a l)

theorem find_nil (x : T) : find x nil = 0

theorem find_cons (x y : T) (l : list T) : find x (y::l) = if x = y then 0 else succ (find x l)

theorem find.not_mem {l : list T} {x : T} : ¬x ∈ l → find x l = length l :=
list.rec_on l
   (assume P₁ : ¬x ∈ nil, _)
   (take y l,
      assume iH : ¬x ∈ l → find x l = length l,
      assume P₁ : ¬x ∈ y::l,
      have P₂ : ¬(x = y ∨ x ∈ l), from iff.elim_right (not_iff_not_of_iff !mem_cons) P₁,
      have P₃ : ¬x = y ∧ ¬x ∈ l, from (iff.elim_left not_or_iff_not_and_not P₂),
      calc
        find x (y::l) = if x = y then 0 else succ (find x l) : !find_cons
                  ... = succ (find x l)                      : if_neg (and.elim_left P₃)
                  ... = succ (length l)                      : {iH (and.elim_right P₃)}
                  ... = length (y::l)                        : !length_cons⁻¹)
end

/- nth element -/

definition nth [h : inhabited T] : list T → nat → T,
nth nil      n     := arbitrary T,
nth (a :: l) 0     := a,
nth (a :: l) (n+1) := nth l n

theorem nth_zero [h : inhabited T] (a : T) (l : list T) : nth (a :: l) 0 = a

theorem nth_succ [h : inhabited T] (a : T) (l : list T) (n : nat) : nth (a::l) (n+1) = nth l n

end list