--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+include "nat/times.ma".
+include "nat/orders.ma".
+
+(* plus *)
+theorem monotonic_le_plus_r:
+\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. 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.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. n \le m \to n + p \le m + p
+\def monotonic_le_plus_l.
+
+theorem le_plus: \forall n1,n2,m1,m2:nat. n1 \le n2 \to m1 \le m2
+\to n1 + m1 \le n2 + m2.
+intros.
+(**
+auto.
+*)
+apply (transitive_le (plus n1 m1) (plus n1 m2) (plus n2 m2) ? ?);
+ [apply (monotonic_le_plus_r n1 m1 m2 ?).
+ apply (H1).
+ |apply (monotonic_le_plus_l m2 n1 n2 ?).
+ apply (H).
+ ]
+(* end auto($Revision$) proof: TIME=0.61 SIZE=100 DEPTH=100 *)
+(*
+apply (trans_le ? (n2 + m1)).
+apply le_plus_l.assumption.
+apply le_plus_r.assumption.
+*)
+qed.
+
+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 le_plus_n_r :\forall n,m:nat. m \le m + n.
+intros.rewrite > sym_plus.
+apply le_plus_n.
+qed.
+
+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.
+qed.
+
+theorem le_plus_to_le:
+\forall a,n,m. a + n \le a + m \to n \le m.
+intro.
+elim a
+ [assumption
+ |apply H.
+ apply le_S_S_to_le.assumption
+ ]
+qed.
+
+(* times *)
+theorem monotonic_le_times_r:
+\forall n:nat.monotonic nat le (\lambda m. n * m).
+simplify.intros.elim n.
+simplify.apply le_O_n.
+simplify.apply le_plus.
+assumption.
+assumption.
+qed.
+
+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.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. n \le m \to n*p \le m*p
+\def monotonic_le_times_l.
+
+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 ? (n2*m1)).
+apply le_times_l.assumption.
+apply le_times_r.assumption.
+qed.
+
+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.
+
+theorem le_times_to_le:
+\forall a,n,m. S O \le a \to a * n \le a * m \to n \le m.
+intro.
+apply nat_elim2;intros
+ [apply le_O_n
+ |apply False_ind.
+ rewrite < times_n_O in H1.
+ generalize in match H1.
+ apply (lt_O_n_elim ? H).
+ intros.
+ simplify in H2.
+ apply (le_to_not_lt ? ? H2).
+ apply lt_O_S
+ |apply le_S_S.
+ apply H
+ [assumption
+ |rewrite < times_n_Sm in H2.
+ rewrite < times_n_Sm in H2.
+ apply (le_plus_to_le a).
+ assumption
+ ]
+ ]
+qed.
+
+theorem le_S_times_SSO: \forall n,m.O < m \to
+n \le m \to S n \le (S(S O))*m.
+intros.
+simplify.
+rewrite > plus_n_O.
+simplify.rewrite > plus_n_Sm.
+apply le_plus
+ [assumption
+ |rewrite < plus_n_O.
+ assumption
+ ]
+qed.
+(*0 and times *)
+theorem O_lt_const_to_le_times_const: \forall a,c:nat.
+O \lt c \to a \le a*c.
+intros.
+rewrite > (times_n_SO a) in \vdash (? % ?).
+apply le_times
+[ apply le_n
+| assumption
+]
+qed.