| 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 ; ]"
(* theorem test_notation: [O; S O; S (S O)] = O :: S O :: S (S O) :: []. *)
theorem nil_cons:
- \forall A:Type.\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;
qed.
*)
-let rec nth (A:Type) l d n on n ≝
- match n with
- [ O ⇒
- match l with
- [ nil ⇒ d
- | cons (x : A) _ ⇒ x
- ]
- | S n' ⇒ nth A (tail ? l) d n'
- ].
+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.
-let rec map (A,B:Type) (f: A → B) (l : list A) on l : list B ≝
- match l with [ nil ⇒ nil ? | cons x tl ⇒ f x :: (map A B f tl)].
+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.
-let rec foldr (A,B:Type) (f : A → B → B) (b : B) (l : list A) on l : B :=
- match l with [ nil ⇒ b | (cons a l) ⇒ f a (foldr ? ? f b l)].
+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.