changes

[helm.git] / matita / matita / lib / arithmetics / nat.ma

diff --git a/matita/matita/lib/arithmetics/nat.ma b/matita/matita/lib/arithmetics/nat.ma
index edb99b8c0c9dac85517e253e9282d7f34597bd89..67b7de50b7dfb5711e9679b502720862e49c6d03 100644 (file)
--- a/matita/matita/lib/arithmetics/nat.ma
+++ b/matita/matita/lib/arithmetics/nat.ma
@@ -305,6 +305,10 @@ lemma lt_to_le: ∀x,y. x < y → x ≤ y.
 lemma inv_eq_minus_O: ∀x,y. x - y = 0 → x ≤ y.
 // qed-.
 
+lemma le_x_times_x: ∀x. x ≤ x * x.
+#x elim x -x //
+qed.
+
 (* lt *)
 
 theorem transitive_lt: transitive nat lt.
@@ -491,6 +495,13 @@ cut (∀q:nat. q ≤ n → P q) /2/
  ]
 qed.
 
+lemma f_ind: ∀A. ∀f:A→ℕ. ∀P:predicate A.
+             (∀n. (∀a. f a < n → P a) → ∀a. f a = n → P a) → ∀a. P a.
+#A #f #P #H #a
+cut (∀n,a. f a = n → P a) /2 width=3/ -a
+#n @(nat_elim1 … n) -n #n /3 width=3/ (**) (* auto very slow (274s) without #n *)
+qed-.
+
 (* More negated equalities **************************************************)
 
 theorem lt_to_not_eq : ∀n,m:nat. n < m → n ≠ m.