X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2Fnat%2Fsigma_and_pi.ma;h=4f5f6cba008299370b94fb0e95f7b0b6248407a6;hb=97c2d258a5c524eb5c4b85208899d80751a2c82f;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..4f5f6cba0 100644
--- a/helm/matita/library/nat/sigma_and_pi.ma
+++ b/helm/matita/library/nat/sigma_and_pi.ma
@@ -14,14 +14,66 @@
 
 set "baseuri" "cic:/matita/nat/sigma_and_pi".
 
-include "nat/times.ma".
+include "nat/factorial.ma".
+include "nat/lt_arith.ma".
+include "nat/exp.ma".
 
-let rec sigma n f \def
+let rec sigma n f m \def
   match n with 
-  [ O \Rightarrow O
-  | (S p) \Rightarrow (f p)+(sigma p f)].
+  [ O \Rightarrow (f m)
+  | (S p) \Rightarrow (f (S p+m))+(sigma p f m)].
 
-let rec pi n f \def
+let rec pi n f m \def
   match n with 
-  [ O \Rightarrow (S O)
-  | (S p) \Rightarrow (f p)*(pi p f)].
\ No newline at end of file
+  [ O \Rightarrow f m
+  | (S p) \Rightarrow (f (S p+m))*(pi p f m)].
+  
+theorem eq_sigma: \forall f,g:nat \to nat.
+\forall n,m:nat.
+(\forall i:nat. m \le i \to i \le m+n \to f i = g i) \to
+(sigma n f m) = (sigma n g m).
+intros 3.elim n.
+simplify.apply H.apply le_n.rewrite < plus_n_O.apply le_n.
+simplify.
+apply eq_f2.apply H1.
+change with (m \le (S n1)+m).apply le_plus_n.
+rewrite > (sym_plus m).apply le_n.
+apply H.intros.apply H1.assumption.
+rewrite < plus_n_Sm.
+apply le_S.assumption.
+qed.
+
+theorem eq_pi: \forall f,g:nat \to nat.
+\forall n,m:nat.
+(\forall i:nat. m \le i \to i \le m+n \to f i = g i) \to
+(pi n f m) = (pi n g m).
+intros 3.elim n.
+simplify.apply H.apply le_n.rewrite < plus_n_O.apply le_n.
+simplify.
+apply eq_f2.apply H1.
+change with (m \le (S n1)+m).apply le_plus_n.
+rewrite > (sym_plus m).apply le_n.
+apply H.intros.apply H1.assumption.
+rewrite < plus_n_Sm.
+apply le_S.assumption.
+qed.
+
+theorem eq_fact_pi: \forall n. (S n)! = pi n (\lambda m.m) (S O).
+intro.elim n.
+simplify.reflexivity.
+change with ((S(S n1))*(S n1)! = ((S n1)+(S O))*(pi n1 (\lambda m.m) (S O))).
+rewrite < plus_n_Sm.rewrite < plus_n_O.
+apply eq_f.assumption.
+qed.
+
+theorem exp_pi_l: \forall f:nat\to nat.\forall n,m,a:nat.
+(exp a (S n))*pi n f m= pi n (\lambda p.a*(f p)) m.
+intros.elim n.simplify.rewrite < times_n_SO.reflexivity.
+simplify.
+rewrite < H.
+rewrite > assoc_times. 
+rewrite > assoc_times in\vdash (? ?  ? %).
+apply eq_f.rewrite < assoc_times. 
+rewrite < assoc_times. 
+apply eq_f2.apply sym_times.reflexivity.
+qed.