- renamed ocaml/ to components/

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

diff --git a/helm/matita/library/nat/sigma_and_pi.ma b/helm/matita/library/nat/sigma_and_pi.ma
deleted file mode 100644 (file)
index 4f5f6cb..0000000
--- a/helm/matita/library/nat/sigma_and_pi.ma
+++ /dev/null
@@ -1,79 +0,0 @@
-(**************************************************************************)
-(*       ___                                                               *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
-(*      ||A||       E.Tassi, S.Zacchiroli                                 *)
-(*      \   /                                                             *)
-(*       \ /        Matita is distributed under the terms of the          *)
-(*        v         GNU Lesser General Public License Version 2.1         *)
-(*                                                                        *)
-(**************************************************************************)
-
-set "baseuri" "cic:/matita/nat/sigma_and_pi".
-
-include "nat/factorial.ma".
-include "nat/lt_arith.ma".
-include "nat/exp.ma".
-
-let rec sigma n f m \def
-  match n with 
-  [ O \Rightarrow (f m)
-  | (S p) \Rightarrow (f (S p+m))+(sigma p f m)].
-
-let rec pi n f m \def
-  match n with 
-  [ 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.