include "logic/equality.ma".
include "higher_order_defs/functions.ma".
+inductive list (A:Set) : Set :=
+ | nil: list A
+ | cons: A -> list A -> list A.
+
notation "hvbox(hd break :: tl)"
right associative with precedence 46
for @{'cons $hd $tl}.
right associative with precedence 47
for @{'append $l1 $l2 }.
-inductive list (A:Set) : Set \def
- | nil: list A
- | cons: A \to list A \to list A.
-
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).
+(* 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 \neq [].
- intros.
- unfold; intros.
+ a::l <> [].
+ intros;
+ unfold Not;
+ intros;
discriminate H.
qed.
-let rec id_list A (l: list A) on l \def
+let rec id_list A (l: list A) on l :=
match l with
- [ nil \Rightarrow []
- | (cons hd tl) \Rightarrow hd :: id_list A tl ].
+ [ nil => []
+ | (cons hd tl) => hd :: id_list A tl ].
-let rec append A (l1: list A) l2 on l1 \def
+let rec append A (l1: list A) l2 on l1 :=
match l1 with
- [ nil \Rightarrow l2
- | (cons hd tl) \Rightarrow hd :: append A tl l2 ].
+ [ nil => l2
+ | (cons hd tl) => hd :: append A tl l2 ].
+
+definition tail := \lambda A:Set. \lambda l: list A.
+ match l with
+ [ nil => []
+ | (cons hd tl) => tl].
interpretation "append" 'append l1 l2 = (cic:/matita/list/append.con _ l1 l2).
theorem append_nil: \forall A:Set.\forall l:list A.l @ [] = l.
- intros.
- elim l.
- reflexivity.
- simplify.
- rewrite > H.
- reflexivity.
+ intros;
+ elim l;
+ [ reflexivity;
+ | simplify;
+ rewrite > H;
+ reflexivity;
+ ]
qed.
theorem associative_append: \forall A:Set.associative (list A) (append A).
- intros; unfold; intros.
- elim x.
- simplify; reflexivity.
- simplify.
- rewrite > H.
- reflexivity.
+ intros; unfold; intros;
+ elim x;
+ [ simplify;
+ reflexivity;
+ | simplify;
+ rewrite > H;
+ reflexivity;
+ ]
qed.
theorem cons_append_commute:
\forall A:Set.\forall l1,l2:list A.\forall a:A.
a :: (l1 @ l2) = (a :: l1) @ l2.
- intros.
- reflexivity.
+ intros;
+ reflexivity;
qed.
(*