X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2Fnat%2Fsigma_and_pi.ma;h=6bd6af38c52eae0f4163938fd245f3bfa4e1b657;hb=7273c698dd60c1a8a0f35b44376acb548c6a4a33;hp=29df3a1dfdab2b76f726748897621bd7758dd3d1;hpb=5a702cea95883f7095c16b450e065ccb1714fc5a;p=helm.git

diff --git a/helm/matita/library/nat/sigma_and_pi.ma b/helm/matita/library/nat/sigma_and_pi.ma
index 29df3a1df..6bd6af38c 100644
--- a/helm/matita/library/nat/sigma_and_pi.ma
+++ b/helm/matita/library/nat/sigma_and_pi.ma
@@ -14,7 +14,8 @@
 
 set "baseuri" "cic:/matita/nat/sigma_and_pi".
 
-include "nat/times.ma".
+include "nat/factorial.ma".
+include "nat/lt_arith.ma".
 
 let rec sigma n f \def
   match n with 
@@ -24,4 +25,33 @@ let rec sigma n f \def
 let rec pi n f \def
   match n with 
   [ O \Rightarrow (S O)
-  | (S p) \Rightarrow (f p)*(pi p f)].
\ No newline at end of file
+  | (S p) \Rightarrow (f p)*(pi p f)].
+  
+theorem eq_sigma: \forall f,g:nat \to nat.
+\forall n:nat. (\forall m:nat. m < n \to f m = g m) \to
+(sigma n f) = (sigma n g).
+intros 3.elim n.
+simplify.reflexivity.
+simplify.
+apply eq_f2.apply H1.simplify. apply le_n.
+apply H.intros.apply H1.
+apply trans_lt ? n1.assumption.simplify.apply le_n.
+qed.
+
+theorem eq_pi: \forall f,g:nat \to nat.
+\forall n:nat. (\forall m:nat. m < n \to f m = g m) \to
+(pi n f) = (pi n g).
+intros 3.elim n.
+simplify.reflexivity.
+simplify.
+apply eq_f2.apply H1.simplify. apply le_n.
+apply H.intros.apply H1.
+apply trans_lt ? n1.assumption.simplify.apply le_n.
+qed.
+
+theorem eq_fact_pi: \forall n. n! = pi n S.
+intro.elim n.
+simplify.reflexivity.
+change with (S n1)*n1! = (S n1)*(pi n1 S).
+apply eq_f.assumption.
+qed.
\ No newline at end of file