lean2/library/data/list/basic.lean

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----------------------------------------------------------------------------------------------------
--- Copyright (c) 2014 Parikshit Khanna. All rights reserved.
--- Released under Apache 2.0 license as described in the file LICENSE.
--- Authors: Parikshit Khanna, Jeremy Avigad
----------------------------------------------------------------------------------------------------
-- Theory list
-- ===========
--
-- Basic properties of lists.
import tools.tactic
import data.nat
import logic
-- import if -- for find
using nat
using eq_ops
namespace list
-- Type
-- ----
inductive list (T : Type) : Type :=
nil {} : list T,
cons : T → list T → list T
infix `::` := cons
section
variable {T : Type}
theorem list_induction_on {P : list T → Prop} (l : list T) (Hnil : P nil)
(Hind : forall x : T, forall l : list T, forall H : P l, P (cons x l)) : P l :=
list_rec Hnil Hind l
theorem list_cases_on {P : list T → Prop} (l : list T) (Hnil : P nil)
(Hcons : forall x : T, forall l : list T, P (cons x l)) : P l :=
list_induction_on l Hnil (take x l IH, Hcons x l)
notation `[` l:(foldr `,` (h t, cons h t) nil) `]` := l
-- Concat
-- ------
definition concat (s t : list T) : list T :=
list_rec t (fun x : T, fun l : list T, fun u : list T, cons x u) s
infixl `++` : 65 := concat
theorem nil_concat (t : list T) : nil ++ t = t := refl _
theorem cons_concat (x : T) (s t : list T) : (x :: s) ++ t = x :: (s ++ t) := refl _
theorem concat_nil (t : list T) : t ++ nil = t :=
list_induction_on t (refl _)
(take (x : T) (l : list T) (H : concat l nil = l),
show concat (cons x l) nil = cons x l, from H ▸ refl _)
theorem concat_assoc (s t u : list T) : s ++ t ++ u = s ++ (t ++ u) :=
list_induction_on s (refl _)
(take x l,
assume H : concat (concat l t) u = concat l (concat t u),
calc
concat (concat (cons x l) t) u = cons x (concat (concat l t) u) : refl _
... = cons x (concat l (concat t u)) : { H }
... = concat (cons x l) (concat t u) : refl _)
-- Length
-- ------
definition length : list T → := list_rec 0 (fun x l m, succ m)
theorem length_nil : length (@nil T) = 0 := refl _
theorem length_cons (x : T) (t : list T) : length (x :: t) = succ (length t) := refl _
theorem length_concat (s t : list T) : length (s ++ t) = length s + length t :=
list_induction_on s
(calc
length (concat nil t) = length t : refl _
... = zero + length t : {symm (add_zero_left (length t))}
... = length (@nil T) + length t : refl _)
(take x s,
assume H : length (concat s t) = length s + length t,
calc
length (concat (cons x s) t ) = succ (length (concat s t)) : refl _
... = succ (length s + length t) : { H }
... = succ (length s) + length t : {symm (add_succ_left _ _)}
... = length (cons x s) + length t : refl _)
-- add_rewrite length_nil length_cons
-- Append
-- ------
definition append (x : T) : list T → list T := list_rec [x] (fun y l l', y :: l')
theorem append_nil (x : T) : append x nil = [x] := refl _
theorem append_cons (x : T) (y : T) (l : list T) : append x (y :: l) = y :: (append x l) := refl _
theorem append_eq_concat (x : T) (l : list T) : append x l = l ++ [x] := refl _
-- add_rewrite append_nil append_cons
-- Reverse
-- -------
definition reverse : list T → list T := list_rec nil (fun x l r, r ++ [x])
theorem reverse_nil : reverse (@nil T) = nil := refl _
theorem reverse_cons (x : T) (l : list T) : reverse (x :: l) = append x (reverse l) := refl _
theorem reverse_singleton (x : T) : reverse [x] = [x] := refl _
theorem reverse_concat (s t : list T) : reverse (s ++ t) = (reverse t) ++ (reverse s) :=
list_induction_on s (symm (concat_nil _))
(take x s,
assume IH : reverse (s ++ t) = concat (reverse t) (reverse s),
calc
reverse ((x :: s) ++ t) = reverse (s ++ t) ++ [x] : refl _
... = reverse t ++ reverse s ++ [x] : {IH}
... = reverse t ++ (reverse s ++ [x]) : concat_assoc _ _ _
... = reverse t ++ (reverse (x :: s)) : refl _)
theorem reverse_reverse (l : list T) : reverse (reverse l) = l :=
list_induction_on l (refl _)
(take x l',
assume H: reverse (reverse l') = l',
show reverse (reverse (x :: l')) = x :: l', from
calc
reverse (reverse (x :: l')) = reverse (reverse l' ++ [x]) : refl _
... = reverse [x] ++ reverse (reverse l') : reverse_concat _ _
... = [x] ++ l' : { H }
... = x :: l' : refl _)
theorem append_eq_reverse_cons (x : T) (l : list T) : append x l = reverse (x :: reverse l) :=
list_induction_on l (refl _)
(take y l',
assume H : append x l' = reverse (x :: reverse l'),
calc
append x (y :: l') = (y :: l') ++ [ x ] : append_eq_concat _ _
... = concat (reverse (reverse (y :: l'))) [ x ] : {symm (reverse_reverse _)}
... = reverse (x :: (reverse (y :: l'))) : refl _)
-- Head and tail
-- -------------
definition head (x0 : T) : list T → T := list_rec x0 (fun x l h, x)
theorem head_nil (x0 : T) : head x0 (@nil T) = x0 := refl _
theorem head_cons (x : T) (x0 : T) (t : list T) : head x0 (x :: t) = x := refl _
theorem head_concat (s t : list T) (x0 : T) : s ≠ nil → (head x0 (s ++ t) = head x0 s) :=
list_cases_on s
(take H : nil ≠ nil, absurd_elim (head x0 (concat nil t) = head x0 nil) (refl nil) H)
(take x s,
take H : cons x s ≠ nil,
calc
head x0 (concat (cons x s) t) = head x0 (cons x (concat s t)) : {cons_concat _ _ _}
... = x : {head_cons _ _ _}
... = head x0 (cons x s) : {symm ( head_cons x x0 s)})
definition tail : list T → list T := list_rec nil (fun x l b, l)
theorem tail_nil : tail (@nil T) = nil := refl _
theorem tail_cons (x : T) (l : list T) : tail (cons x l) = l := refl _
theorem cons_head_tail (x0 : T) (l : list T) : l ≠ nil → (head x0 l) :: (tail l) = l :=
list_cases_on l
(assume H : nil ≠ nil, absurd_elim _ (refl _) H)
(take x l, assume H : cons x l ≠ nil, refl _)
-- List membership
-- ---------------
definition mem (x : T) : list T → Prop := list_rec false (fun y l H, x = y H)
infix `∈` := mem
-- TODO: constructively, equality is stronger. Use that?
theorem mem_nil (x : T) : x ∈ nil ↔ false := iff_refl _
theorem mem_cons (x : T) (y : T) (l : list T) : mem x (cons y l) ↔ (x = y mem x l) := iff_refl _
theorem mem_concat_imp_or (x : T) (s t : list T) : x ∈ s ++ t → x ∈ s x ∈ t :=
list_induction_on s (or_intro_right _)
(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_imp_or_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_intro_left _ H2)
(take H2 : x ∈ s, or_intro_right _ (IH (or_intro_left _ H2))))
(assume H1 : x ∈ t, or_intro_right _ (IH (or_intro_right _ 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 _ _ _)
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 x) 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 l (subst H1 (refl _))))
(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 subst H3 (refl (y :: l)),
exists_intro _ (exists_intro _ H4)))
-- Find
-- ----
-- to do this: need decidability of = for nat
-- definition find (x : T) : list T → nat
-- := list_rec 0 (fun y l b, if x = y then 0 else succ b)
-- theorem find_nil (f : T) : find f nil = 0
-- :=refl _
-- theorem find_cons (x y : T) (l : list T) : find x (cons y l) =
-- if x = y then 0 else succ (find x l)
-- := refl _
-- theorem not_mem_find (l : list T) (x : T) : ¬ mem x l → find x l = length l
-- :=
-- @list_induction_on T (λl, ¬ mem x l → find x l = length l) l
-- -- list_induction_on l
-- (assume P1 : ¬ mem x nil,
-- show find x nil = length nil, from
-- calc
-- find x nil = 0 : find_nil _
-- ... = length nil : by simp)
-- (take y l,
-- assume IH : ¬ (mem x l) → find x l = length l,
-- assume P1 : ¬ (mem x (cons y l)),
-- have P2 : ¬ (mem x l (y = x)), from subst P1 (mem_cons _ _ _),
-- have P3 : ¬ (mem x l) ∧ (y ≠ x),from subst P2 (not_or _ _),
-- have P4 : x ≠ y, from ne_symm (and_elim_right P3),
-- calc
-- find x (cons y l) = succ (find x l) :
-- trans (find_cons _ _ _) (not_imp_if_eq P4 _ _)
-- ... = succ (length l) : {IH (and_elim_left P3)}
-- ... = length (cons y l) : symm (length_cons _ _))
-- nth element
-- -----------
definition nth (x0 : T) (l : list T) (n : ) : T :=
nat_rec (λl, head x0 l) (λm f l, f (tail l)) n l
theorem nth_zero (x0 : T) (l : list T) : nth x0 l 0 = head x0 l := refl _
theorem nth_succ (x0 : T) (l : list T) (n : ) : nth x0 l (succ n) = nth x0 (tail l) n := refl _
end
-- declare global notation outside the section
infixl `++` := concat
end list