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

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+(**************************************************************************)
+(*       ___                                                               *)
+(*      ||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         *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/nat/le_arith".
+
+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.
+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 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.
+
+(* 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.