(* *)
(**************************************************************************)
-include "Ground-2/arith.ma".
-include "Ground-2/notation.ma".
+include "Ground_2/arith.ma".
(* LISTS ********************************************************************)
inductive list (A:Type[0]) : Type[0] :=
| nil : list A
- | cons: A -> list A -> list A.
+ | cons: A → list A → list A.
interpretation "nil (list)" 'Nil = (nil ?).
interpretation "cons (list)" 'Cons hd tl = (cons ? hd tl).
-let rec append A (l1: list A) l2 on l1 ≝
- match l1 with
- [ nil ⇒ l2
- | cons hd tl ⇒ hd :: append A tl l2
+let rec all A (R:predicate A) (l:list A) on l ≝
+ match l with
+ [ nil ⇒ True
+ | cons hd tl ⇒ R hd ∧ all A R tl
].
-interpretation "append (list)" 'Append l1 l2 = (append ? l1 l2).
+inductive list2 (A1,A2:Type[0]) : Type[0] :=
+ | nil2 : list2 A1 A2
+ | cons2: A1 → A2 → list2 A1 A2 → list2 A1 A2.
+
+interpretation "nil (list of pairs)" 'Nil2 = (nil2 ? ?).
+
+interpretation "cons (list of pairs)" 'Cons hd1 hd2 tl = (cons2 ? ? hd1 hd2 tl).
+
+let rec append2 (A1,A2:Type[0]) (l1,l2:list2 A1 A2) on l1 ≝ match l1 with
+[ nil2 ⇒ l2
+| cons2 a1 a2 tl ⇒ {a1, a2} :: append2 A1 A2 tl l2
+].
+
+interpretation "append (list of pairs)"
+ 'Append l1 l2 = (append2 ? ? l1 l2).
+
+let rec length2 (A1,A2:Type[0]) (l:list2 A1 A2) on l ≝ match l with
+[ nil2 ⇒ 0
+| cons2 _ _ l ⇒ length2 A1 A2 l + 1
+].
+
+interpretation "length (list of pairs)"
+ 'card l = (length2 ? ? l).