X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2Fnat%2Fle_arith.ma;h=a76183063b1e6d211d5c03a75cd1845c844c73aa;hb=4167cea65ca58897d1a3dbb81ff95de5074700cc;hp=a36f96bd451caba6a4091f0e7ecd9ea530957ed1;hpb=7bbce6bc163892cfd99cfcda65db42001b86789f;p=helm.git

diff --git a/helm/matita/library/nat/le_arith.ma b/helm/matita/library/nat/le_arith.ma
index a36f96bd4..a76183063 100644
--- a/helm/matita/library/nat/le_arith.ma
+++ b/helm/matita/library/nat/le_arith.ma
@@ -14,47 +14,44 @@
 
 set "baseuri" "cic:/matita/nat/le_arith".
 
-include "higher_order_defs/functions.ma".
 include "nat/times.ma".
 include "nat/orders.ma".
-include "nat/compare.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 \le m \to p + n \le 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.
+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 \le m \to n + p \le 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 \le n2  \to m1 \le m2 
+\to n1 + m1 \le 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 \le n + m.
+intros.change with (O+m \le n+m).
 apply le_plus_l.apply le_O_n.
 qed.
 
-theorem eq_plus_to_le: \forall n,m,p:nat.eq nat n (plus m p) 
-\to le m n.
+theorem eq_plus_to_le: \forall n,m,p:nat.n=m+p \to m \le n.
 intros.rewrite > H.
 rewrite < sym_plus.
 apply le_plus_n.
@@ -62,7 +59,7 @@ 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.
@@ -70,30 +67,29 @@ 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 \le m \to p*n \le 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.
+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 \le m \to n*p \le 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 \le n2  \to m1 \le m2 
+\to n1*m1 \le 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) \le n \to m \le n*m.
 intros.elim H.simplify.
 elim (plus_n_O ?).apply le_n.
 simplify.rewrite < sym_plus.apply le_plus_n.
 qed.
-