191 lines
6.4 KiB
Text
191 lines
6.4 KiB
Text
----------------------------------------------------------------------------------------------------
|
||
--- 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 logic data.nat
|
||
-- import congr
|
||
|
||
open nat
|
||
-- open congr
|
||
open eq_ops eq
|
||
|
||
inductive list (T : Type) : Type :=
|
||
nil {} : list T,
|
||
cons : T → list T → list T
|
||
|
||
definition refl := @eq.refl
|
||
|
||
namespace list
|
||
|
||
-- Type
|
||
-- ----
|
||
|
||
infix `::` : 65 := 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),
|
||
H ▸ (refl (cons x (concat l nil))))
|
||
|
||
theorem concat_nil2 (t : list T) : t ++ nil = t :=
|
||
list_induction_on t (refl _)
|
||
(take (x : T) (l : list T) (H : concat l nil = l),
|
||
-- H ▸ (refl (cons x (concat l nil))))
|
||
H ▸ (refl (concat (cons x l) nil)))
|
||
|
||
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),
|
||
H ▸ refl _)
|
||
|
||
theorem concat_assoc2 (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 _
|
||
... = concat (cons x l) (concat t u) : { H })
|
||
|
||
theorem concat_assoc3 (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 l (concat t u)) : { H }
|
||
... = concat (cons x l) (concat t u) : refl _)
|
||
|
||
theorem concat_assoc4 (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 → ℕ := list.rec 0 (fun x l m, succ m)
|
||
|
||
-- TODO: cannot replace zero by 0
|
||
theorem length_nil : length (@nil T) = zero := 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 (@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 _)
|
||
|
||
-- 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) = (reverse l) ++ (cons x nil) := refl _
|
||
|
||
-- opaque_hint (hiding reverse)
|
||
|
||
theorem reverse_concat (s t : list T) : reverse (s ++ t) = (reverse t) ++ (reverse s) :=
|
||
list_induction_on s
|
||
(calc
|
||
reverse (concat nil t) = reverse t : { nil_concat _ }
|
||
... = concat (reverse t) nil : symm (concat_nil _)
|
||
... = concat (reverse t) (reverse nil) : {symm (reverse_nil)})
|
||
(take x l,
|
||
assume H : reverse (concat l t) = concat (reverse t) (reverse l),
|
||
calc
|
||
reverse (concat (cons x l) t) = concat (reverse (concat l t)) (cons x nil) : refl _
|
||
... = concat (concat (reverse t) (reverse l)) (cons x nil) : { H }
|
||
... = concat (reverse t) (concat (reverse l) (cons x nil)) : concat_assoc _ _ _
|
||
... = concat (reverse t) (reverse (cons x l)) : refl _)
|
||
|
||
|
||
-- -- add_rewrite length_nil length_cons
|
||
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 (cons x l')) = cons x l', from
|
||
calc
|
||
reverse (reverse (cons x l')) =
|
||
concat (reverse (cons x nil)) (reverse (reverse l')) : {reverse_concat _ _}
|
||
... = cons x l' : {H})
|
||
-- Append
|
||
-- ------
|
||
|
||
-- TODO: define reverse from append
|
||
|
||
definition append (x : T) : list T → list T := list.rec (x :: nil) (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] :=
|
||
list_induction_on l (refl _)
|
||
(take y l,
|
||
assume P : append x l = concat l [x],
|
||
P ▸ refl _)
|
||
|
||
theorem append_eq_reverse_cons (x : T) (l : list T) : append x l = reverse (x :: reverse l) :=
|
||
list_induction_on l
|
||
(calc
|
||
append x nil = [x] : (refl _)
|
||
... = concat nil [x] : {symm (nil_concat _)}
|
||
... = concat (reverse nil) [x] : {symm (reverse_nil)}
|
||
... = reverse [x] : {symm (reverse_cons _ _)}
|
||
... = reverse (x :: (reverse nil)) : {symm (reverse_nil)})
|
||
(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 _)
|
||
end
|
||
end list
|