A little bit more of notation here and there.

[helm.git] / helm / matita / library / nat / orders_op.ma

diff --git a/helm/matita/library/nat/orders_op.ma b/helm/matita/library/nat/orders_op.ma
index fbfad715c82a6cc155b0cfe24a131a153c58be7d..bfb3668c7b49984a106327cc46171b08f4ac15a5 100644 (file)
--- a/helm/matita/library/nat/orders_op.ma
+++ b/helm/matita/library/nat/orders_op.ma
@@ -20,40 +20,39 @@ include "nat/orders.ma".
 
 (* plus *)
 theorem monotonic_le_plus_r: 
-\forall n:nat.monotonic nat le (\lambda m.plus n m).
+\forall n:nat.monotonic nat le (\lambda m.n+m).
 simplify.intros.elim n.
 simplify.assumption.
 simplify.apply le_S_S.assumption.
 qed.
 
-theorem le_plus_r: \forall p,n,m:nat. le n m \to le (plus p n) (plus p m)
+theorem le_plus_r: \forall p,n,m:nat. n \leq m \to p+n \leq p+m
 \def monotonic_le_plus_r.
 
 theorem monotonic_le_plus_l: 
-\forall m:nat.monotonic nat le (\lambda n.plus n m).
+\forall m:nat.monotonic nat le (\lambda n.n+m).
 simplify.intros.
 rewrite < sym_plus.rewrite < sym_plus m.
 apply le_plus_r.assumption.
 qed.
 
-theorem le_plus_l: \forall p,n,m:nat. le n m \to le (plus n p) (plus m p)
+theorem le_plus_l: \forall p,n,m:nat. n \leq m \to n+p \leq m+p
 \def monotonic_le_plus_l.
 
-theorem le_plus: \forall n1,n2,m1,m2:nat. le n1 n2  \to le m1 m2 
-\to le (plus n1 m1) (plus n2 m2).
+theorem le_plus: \forall n1,n2,m1,m2:nat. n1 \leq n2  \to m1 \leq m2 
+\to n1+m1 \leq n2+m2.
 intros.
-apply trans_le ? (plus n2 m1).
+apply trans_le ? (n2+m1).
 apply le_plus_l.assumption.
 apply le_plus_r.assumption.
 qed.
 
-theorem le_plus_n :\forall n,m:nat. le m (plus n m).
-intros.change with le (plus O m) (plus n m).
+theorem le_plus_n :\forall n,m:nat. m \leq n+m.
+intros.change with O+m \leq n+m.
 apply le_plus_l.apply le_O_n.
 qed.
 
-theorem eq_plus_to_le: \forall n,m,p:nat.n=plus m p
-\to le m n.
+theorem eq_plus_to_le: \forall n,m,p:nat.n=m+p \to m \leq n.
 intros.rewrite > H.
 rewrite < sym_plus.
 apply le_plus_n.
@@ -61,7 +60,7 @@ qed.
 
 (* times *)
 theorem monotonic_le_times_r: 
-\forall n:nat.monotonic nat le (\lambda m.times n m).
+\forall n:nat.monotonic nat le (\lambda m.n*m).
 simplify.intros.elim n.
 simplify.apply le_O_n.
 simplify.apply le_plus.
@@ -69,28 +68,28 @@ assumption.
 assumption.
 qed.
 
-theorem le_times_r: \forall p,n,m:nat. le n m \to le (times p n) (times p m)
+theorem le_times_r: \forall p,n,m:nat. n \leq m \to p*n \leq p*m
 \def monotonic_le_times_r.
 
 theorem monotonic_le_times_l: 
-\forall m:nat.monotonic nat le (\lambda n.times n m).
+\forall m:nat.monotonic nat le (\lambda n.n*m).
 simplify.intros.
 rewrite < sym_times.rewrite < sym_times m.
 apply le_times_r.assumption.
 qed.
 
-theorem le_times_l: \forall p,n,m:nat. le n m \to le (times n p) (times m p)
+theorem le_times_l: \forall p,n,m:nat. n \leq m \to n*p \leq m*p
 \def monotonic_le_times_l.
 
-theorem le_times: \forall n1,n2,m1,m2:nat. le n1 n2  \to le m1 m2 
-\to le (times n1 m1) (times n2 m2).
+theorem le_times: \forall n1,n2,m1,m2:nat. n1 \leq n2  \to m1 \leq m2 
+\to n1*m1 \leq n2*m2.
 intros.
-apply trans_le ? (times n2 m1).
+apply trans_le ? (n2*m1).
 apply le_times_l.assumption.
 apply le_times_r.assumption.
 qed.
 
-theorem le_times_n: \forall n,m:nat.le (S O) n \to le m (times n m).
+theorem le_times_n: \forall n,m:nat.S O \leq n \to m \leq n*m.
 intros.elim H.simplify.
 elim (plus_n_O ?).apply le_n.
 simplify.rewrite < sym_plus.apply le_plus_n.