- renamed ocaml/ to components/

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

diff --git a/helm/matita/library/nat/times.ma b/helm/matita/library/nat/times.ma
deleted file mode 100644 (file)
index 2ae5ffd..0000000
--- a/helm/matita/library/nat/times.ma
+++ /dev/null
@@ -1,87 +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/times".
-
-include "nat/plus.ma".
-
-let rec times n m \def 
- match n with 
- [ O \Rightarrow O
- | (S p) \Rightarrow m+(times p m) ].
-
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "natural times" 'times x y = (cic:/matita/nat/times/times.con x y).
-
-theorem times_n_O: \forall n:nat. O = n*O.
-intros.elim n.
-simplify.reflexivity.
-simplify.assumption.
-qed.
-
-theorem times_n_Sm : 
-\forall n,m:nat. n+(n*m) = n*(S m).
-intros.elim n.
-simplify.reflexivity.
-simplify.apply eq_f.rewrite < H.
-transitivity ((n1+m)+n1*m).symmetry.apply assoc_plus.
-transitivity ((m+n1)+n1*m).
-apply eq_f2.
-apply sym_plus.
-reflexivity.
-apply assoc_plus.
-qed.
-
-theorem times_n_SO : \forall n:nat. n = n * S O.
-intros.
-rewrite < times_n_Sm.
-rewrite < times_n_O.
-rewrite < plus_n_O.
-reflexivity.
-qed.
-
-theorem symmetric_times : symmetric nat times. 
-unfold symmetric.
-intros.elim x.
-simplify.apply times_n_O.
-simplify.rewrite > H.apply times_n_Sm.
-qed.
-
-variant sym_times : \forall n,m:nat. n*m = m*n \def
-symmetric_times.
-
-theorem distributive_times_plus : distributive nat times plus.
-unfold distributive.
-intros.elim x.
-simplify.reflexivity.
-simplify.rewrite > H. rewrite > assoc_plus.rewrite > assoc_plus.
-apply eq_f.rewrite < assoc_plus. rewrite < (sym_plus ? z).
-rewrite > assoc_plus.reflexivity.
-qed.
-
-variant distr_times_plus: \forall n,m,p:nat. n*(m+p) = n*m + n*p
-\def distributive_times_plus.
-
-theorem associative_times: associative nat times.
-unfold associative.intros.
-elim x.simplify.apply refl_eq.
-simplify.rewrite < sym_times.
-rewrite > distr_times_plus.
-rewrite < sym_times.
-rewrite < (sym_times (times n y) z).
-rewrite < H.apply refl_eq.
-qed.
-
-variant assoc_times: \forall n,m,p:nat. (n*m)*p = n*(m*p) \def
-associative_times.