lean2/library/data/list/basic.lean
Leonardo de Moura 2d78387541 refactor(library/logic/basic): rename absurd_elim to absurd, delete contrapos and trivial_not_true theorems
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
2014-08-27 18:34:09 -07:00

291 lines
9.5 KiB
Text
Raw Blame History

This file contains ambiguous Unicode characters

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

----------------------------------------------------------------------------------------------------
--- 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 (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 (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