1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| A.Asperti, C.Sacerdoti Coen, *)
8 (* ||A|| E.Tassi, S.Zacchiroli *)
10 (* \ / Matita is distributed under the terms of the *)
11 (* v GNU Lesser General Public License Version 2.1 *)
13 (**************************************************************************)
15 set "baseuri" "cic:/matita/nat/sigma_and_pi".
17 include "nat/lt_arith.ma".
19 let rec sigma n f \def
22 | (S p) \Rightarrow (f p)+(sigma p f)].
27 | (S p) \Rightarrow (f p)*(pi p f)].
29 theorem eq_sigma: \forall f,g:nat \to nat.
30 \forall n:nat. (\forall m:nat. m < n \to f m = g m) \to
31 (sigma n f) = (sigma n g).
35 apply eq_f2.apply H1.simplify. apply le_n.
36 apply H.intros.apply H1.
37 apply trans_lt ? n1.assumption.simplify.apply le_n.
40 theorem eq_pi: \forall f,g:nat \to nat.
41 \forall n:nat. (\forall m:nat. m < n \to f m = g m) \to
46 apply eq_f2.apply H1.simplify. apply le_n.
47 apply H.intros.apply H1.
48 apply trans_lt ? n1.assumption.simplify.apply le_n.