X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambda_delta%2Fground_2%2Farith.ma;h=f7f5e35c4ce10e08a684f7c6f8955530c9899f73;hb=8b1bc0a74dbc6c5854cbce31240ae829dfe7e8bf;hp=c65a47bcc70b31c7649709157b5e7ac81493c280;hpb=78d4844bcccb3deb58a3179151c3045298782b18;p=helm.git

diff --git a/matita/matita/contribs/lambda_delta/ground_2/arith.ma b/matita/matita/contribs/lambda_delta/ground_2/arith.ma
index c65a47bcc..f7f5e35c4 100644
--- a/matita/matita/contribs/lambda_delta/ground_2/arith.ma
+++ b/matita/matita/contribs/lambda_delta/ground_2/arith.ma
@@ -19,6 +19,9 @@ include "ground_2/star.ma".
 
 (* Equations ****************************************************************)
 
+lemma plus_n_2: ∀n. n + 2 = n + 1 + 1.
+// qed.
+
 lemma le_plus_minus: ∀m,n,p. p ≤ n → m + n - p = m + (n - p).
 /2 by plus_minus/ qed.
 
@@ -66,11 +69,28 @@ lemma false_lt_to_le: ∀x,y. (x < y → ⊥) → y ≤ x.
 #Hxy elim (H Hxy)
 qed-.
 
-(*
-lemma pippo: ∀x,y,z. x < z → y < z - x → x + y < z.
-/3 width=2/
-
-lemma le_or_ge: ∀m,n. m ≤ n ∨ n ≤ m.
-#m #n elim (lt_or_ge m n) /2 width=1/ /3 width=2/
+lemma plus_xySz_x_false: ∀z,x,y. x + y + S z = x → ⊥.
+#z #x elim x -x normalize
+[ #y <plus_n_Sm #H destruct
+| /3 width=2/
+]
 qed-.
-*)
+
+(* iterators ****************************************************************)
+
+(* Note: see also: lib/arithemetcs/bigops.ma *)
+let rec iter (n:nat) (B:Type[0]) (op: B → B) (nil: B) ≝
+  match n with
+   [ O   ⇒ nil
+   | S k ⇒ op (iter k B op nil)
+   ].
+
+interpretation "iterated function" 'exp op n = (iter n ? op).
+
+lemma iter_SO: ∀B:Type[0]. ∀f:B→B. ∀b,l. f^(l+1) b = f (f^l b).
+#B #f #b #l >commutative_plus //
+qed.
+
+lemma iter_n_Sm: ∀B:Type[0]. ∀f:B→B. ∀b,l. f^l (f b) = f (f^l b).
+#B #f #b #l elim l -l normalize //
+qed.