Deplacement d'un lemme sur nat de ZArith vers Arith

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@4146 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 11 insertions, 0 deletions

diff --git a/theories/Arith/Mult.v b/theories/Arith/Mult.v
index 761979db3..14dd98dac 100755
--- a/theories/Arith/Mult.v
+++ b/theories/Arith/Mult.v
@@ -129,6 +129,17 @@ Rewrite mult_sym.
 Replace (mult n (S p)) with (mult (S p) n); Auto with arith.
 Qed.
 
+Theorem lt_mult_left :
+ (x,y,z:nat) (lt x y) -> (lt (mult (S z) x) (mult (S z) y)).
+Proof.
+Intros x y z H;Elim z; [
+  Simpl; Do 2 Rewrite <- plus_n_O; Assumption
+| Simpl; Intros n H1; Apply lt_trans with m:=(plus y (plus x (mult n x))); [
+    Rewrite (plus_sym x (plus x (mult n x)));
+    Rewrite (plus_sym y (plus x (mult n x))); Apply lt_reg_l; Assumption
+  | Apply lt_reg_l;Assumption ]].
+Qed.
+
 Lemma mult_le_conv_1 : (m,n,p:nat) (le (mult (S m) n) (mult (S m) p)) -> (le n p).
 Proof.
   Intros. Elim (le_or_lt n p). Trivial.