- renamed ocaml/ to components/

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

diff --git a/helm/matita/library/nat/le_arith.ma b/helm/matita/library/nat/le_arith.ma
deleted file mode 100644 (file)
index a761830..0000000
--- a/helm/matita/library/nat/le_arith.ma
+++ /dev/null
@@ -1,95 +0,0 @@
-(**************************************************************************)
-(*       ___                                                               *)
-(*      ||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.