notational update in lambdadelta completed

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / lib / list.ma

diff --git a/matita/matita/contribs/lambdadelta/ground_2/lib/list.ma b/matita/matita/contribs/lambdadelta/ground_2/lib/list.ma
index 9355f9c56185175fa263fba844548e432a7672f9..ffce3128641951c7429d9aaeef7ae488dd000305 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/lib/list.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/lib/list.ma
@@ -13,9 +13,7 @@
 (**************************************************************************)
 
 include "ground_2/notation/constructors/nil_0.ma".
-include "ground_2/notation/constructors/cons_2.ma".
-include "ground_2/notation/constructors/cons_3.ma".
-include "ground_2/notation/functions/append_2.ma".
+include "ground_2/notation/constructors/oplusright_3.ma".
 include "ground_2/lib/arith.ma".
 
 (* LISTS ********************************************************************)
@@ -26,34 +24,45 @@ inductive list (A:Type[0]) : Type[0] :=
 
 interpretation "nil (list)" 'Nil = (nil ?).
 
-interpretation "cons (list)" 'Cons hd tl = (cons ? hd tl).
+interpretation "cons (list)" 'OPlusRight A hd tl = (cons A hd tl).
 
-let rec all A (R:predicate A) (l:list A) on l ≝
+rec definition length A (l:list A) on l ≝ match l with
+[ nil      ⇒ 0
+| cons _ l ⇒ ↑(length A l)
+].
+
+interpretation "length (list)"
+   'card l = (length ? l).
+
+rec definition all A (R:predicate A) (l:list A) on l ≝
   match l with
   [ nil        ⇒ ⊤
   | cons hd tl ⇒ R hd ∧ all A R tl
   ].
 
-inductive list2 (A1,A2:Type[0]) : Type[0] :=
-  | nil2 : list2 A1 A2
-  | cons2: A1 → A2 → list2 A1 A2 → list2 A1 A2.
+(* Basic properties on length ***********************************************)
 
-interpretation "nil (list of pairs)" 'Nil = (nil2 ? ?).
+lemma length_nil (A:Type[0]): |nil A| = 0.
+// qed.
 
-interpretation "cons (list of pairs)" 'Cons hd1 hd2 tl = (cons2 ? ? hd1 hd2 tl).
+lemma length_cons (A:Type[0]) (l:list A) (a:A): |a⨮l| = ↑|l|.
+// qed.
 
-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
-].
+(* Basic inversion lemmas on length *****************************************)
 
-interpretation "append (list of pairs)"
-   'Append l1 l2 = (append2 ? ? l1 l2).
+lemma length_inv_zero_dx (A:Type[0]) (l:list A): |l| = 0 → l = ◊.
+#A * // #a #l >length_cons #H destruct
+qed-.
 
-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
-].
+lemma length_inv_zero_sn (A:Type[0]) (l:list A): 0 = |l| → l = ◊.
+/2 width=1 by length_inv_zero_dx/ qed-.
+
+lemma length_inv_succ_dx (A:Type[0]) (l:list A) (x): |l| = ↑x →
+                         ∃∃tl,a. x = |tl| & l = a ⨮ tl.
+#A * /2 width=4 by ex2_2_intro/
+>length_nil #x #H destruct
+qed-.
 
-interpretation "length (list of pairs)"
-   'card l = (length2 ? ? l).
+lemma length_inv_succ_sn (A:Type[0]) (l:list A) (x): ↑x = |l| →
+                         ∃∃tl,a. x = |tl| & l = a ⨮ tl.
+/2 width=1 by length_inv_succ_dx/ qed.