(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/list/".
include "logic/equality.ma".
+include "datatypes/bool.ma".
include "higher_order_defs/functions.ma".
+include "nat/plus.ma".
+include "nat/orders.ma".
-inductive list (A:Set) : Set :=
+inductive list (A:Type) : Type :=
| nil: list A
| cons: A -> list A -> list A.
notation "hvbox(hd break :: tl)"
- right associative with precedence 46
+ right associative with precedence 47
for @{'cons $hd $tl}.
notation "[ list0 x sep ; ]"
right associative with precedence 47
for @{'append $l1 $l2 }.
-interpretation "nil" 'nil = (cic:/matita/list/list.ind#xpointer(1/1/1) _).
-interpretation "cons" 'cons hd tl =
- (cic:/matita/list/list.ind#xpointer(1/1/2) _ hd tl).
+interpretation "nil" 'nil = (nil ?).
+interpretation "cons" 'cons hd tl = (cons ? hd tl).
(* theorem test_notation: [O; S O; S (S O)] = O :: S O :: S (S O) :: []. *)
theorem nil_cons:
- \forall A:Set.\forall l:list A.\forall a:A.
- a::l <> [].
+ \forall A:Type.\forall l:list A.\forall a:A. a::l ≠ [].
intros;
unfold Not;
intros;
- discriminate H.
+ destruct H.
qed.
let rec id_list A (l: list A) on l :=
[ nil => l2
| (cons hd tl) => hd :: append A tl l2 ].
-definition tail := \lambda A:Set. \lambda l: list A.
+definition tail := \lambda A:Type. \lambda l: list A.
match l with
[ nil => []
| (cons hd tl) => tl].
-interpretation "append" 'append l1 l2 = (cic:/matita/list/append.con _ l1 l2).
+interpretation "append" 'append l1 l2 = (append ? l1 l2).
-theorem append_nil: \forall A:Set.\forall l:list A.l @ [] = l.
+theorem append_nil: \forall A:Type.\forall l:list A.l @ [] = l.
intros;
elim l;
[ reflexivity;
]
qed.
-theorem associative_append: \forall A:Set.associative (list A) (append A).
+theorem associative_append: \forall A:Type.associative (list A) (append A).
intros; unfold; intros;
elim x;
[ simplify;
qed.
theorem cons_append_commute:
- \forall A:Set.\forall l1,l2:list A.\forall a:A.
+ \forall A:Type.\forall l1,l2:list A.\forall a:A.
a :: (l1 @ l2) = (a :: l1) @ l2.
intros;
reflexivity;
qed.
-inductive permutation (A:Set) : list A -> list A -> Prop \def
+lemma append_cons:\forall A.\forall a:A.\forall l,l1.
+l@(a::l1)=(l@[a])@l1.
+intros.
+rewrite > associative_append.
+reflexivity.
+qed.
+
+inductive permutation (A:Type) : list A -> list A -> Prop \def
| refl : \forall l:list A. permutation ? l l
| swap : \forall l:list A. \forall x,y:A.
permutation ? (x :: y :: l) (y :: x :: l)
| step : \forall l1,l2:list A. \forall x,y:A.
permut1 ? (l1 @ (x :: y :: l2)) (l1 @ (y :: x :: l2)).
-include "nat/nat.ma".
-
+(*
+
definition x1 \def S O.
definition x2 \def S x1.
definition x3 \def S x2.
apply (step ? (x1::[]) [] x2 x3).
qed.
-
-(*
theorem nil_append_nil_both:
- \forall A:Set.\forall l1,l2:list A.
+ \forall A:Type.\forall l1,l2:list A.
l1 @ l2 = [] \to l1 = [] \land l2 = [].
-*)
-
-(*
-include "nat/nat.ma".
theorem test_notation: [O; S O; S (S O)] = O :: S O :: S (S O) :: [].
reflexivity.
simplify.
reflexivity.
qed.
+
*)
+
+definition nth ≝
+ λA:Type.
+ let rec nth l d n on n ≝
+ match n with
+ [ O ⇒
+ match l with
+ [ nil ⇒ d
+ | cons (x : A) _ ⇒ x
+ ]
+ | S n' ⇒ nth (tail ? l) d n']
+ in nth.
+
+definition map ≝
+ λA,B:Type.λf:A→B.
+ let rec map (l : list A) on l : list B ≝
+ match l with [ nil ⇒ nil ? | cons x tl ⇒ f x :: (map tl)]
+ in map.
+
+definition foldr ≝
+ λA,B:Type.λf:A→B→B.λb:B.
+ let rec foldr (l : list A) on l : B :=
+ match l with [ nil ⇒ b | (cons a l) ⇒ f a (foldr l)]
+ in foldr.
+
+definition length ≝ λT:Type.λl:list T.foldr T nat (λx,c.S c) O l.
+
+definition filter \def
+ \lambda T:Type.\lambda l:list T.\lambda p:T \to bool.
+ foldr T (list T)
+ (\lambda x,l0.match (p x) with [ true => x::l0 | false => l0]) [] l.
+
+definition iota : nat → nat → list nat ≝
+ λn,m. nat_rect (λ_.list ?) (nil ?) (λx,acc.cons ? (n+x) acc) m.
+
+(* ### induction principle for functions visiting 2 lists in parallel *)
+lemma list_ind2 :
+ ∀T1,T2:Type.∀l1:list T1.∀l2:list T2.∀P:list T1 → list T2 → Prop.
+ length ? l1 = length ? l2 →
+ (P (nil ?) (nil ?)) →
+ (∀tl1,tl2,hd1,hd2. P tl1 tl2 → P (hd1::tl1) (hd2::tl2)) →
+ P l1 l2.
+intros (T1 T2 l1 l2 P Hl Pnil Pcons);
+elim l1 in Hl l2 ⊢ % 1 (l2 x1); [ cases l2; intros (Hl); [assumption| simplify in Hl; destruct Hl]]
+intros 3 (tl1 IH l2); cases l2; [1: simplify; intros 1 (Hl); destruct Hl]
+intros 1 (Hl); apply Pcons; apply IH; simplify in Hl; destruct Hl; assumption;
+qed.
+
+lemma eq_map : ∀A,B,f,g,l. (∀x.f x = g x) → map A B f l = map A B g l.
+intros (A B f g l Efg); elim l; simplify; [1: reflexivity ];
+rewrite > (Efg a); rewrite > H; reflexivity;
+qed.
+
+lemma le_length_filter : \forall A,l,p.length A (filter A l p) \leq length A l.
+intros;elim l
+ [simplify;apply le_n
+ |simplify;apply (bool_elim ? (p a));intro
+ [simplify;apply le_S_S;assumption
+ |simplify;apply le_S;assumption]]
+qed.
+
+lemma length_append : ∀A,l,m.length A (l@m) = length A l + length A m.
+intros;elim l
+[reflexivity
+|simplify;rewrite < H;reflexivity]
+qed.