From: Cristian Armentano <??>
Date: Fri, 29 Jun 2007 14:12:07 +0000 (+0000)
Subject: theorems about sigma_p proved using sigma_p_gen
X-Git-Tag: 0.4.95@7852~386
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=45d665041eae44ef5527e2c5a65329493d742ef3;p=helm.git

theorems about sigma_p proved using sigma_p_gen
---

diff --git a/matita/library/Z/sigma_p.ma b/matita/library/Z/sigma_p.ma
index 24cb89395..5d85bc653 100644
--- a/matita/library/Z/sigma_p.ma
+++ b/matita/library/Z/sigma_p.ma
@@ -17,29 +17,36 @@ set "baseuri" "cic:/matita/Z/sigma_p.ma".
 include "Z/times.ma".
 include "nat/primes.ma".
 include "nat/ord.ma".
+include "nat/generic_sigma_p.ma".
 
-let rec sigma_p n p (g:nat \to Z) \def
-  match n with
-   [ O \Rightarrow OZ
-   | (S k) \Rightarrow 
-      match p k with
-      [true \Rightarrow (g k)+(sigma_p k p g)
-      |false \Rightarrow sigma_p k p g]
-   ].
+(* sigma_p in Z is a specialization of sigma_p_gen *)
+definition sigma_p: nat \to (nat \to bool) \to (nat \to Z) \to Z \def
+\lambda n, p, g. (sigma_p_gen n p Z g OZ Zplus).
 
+theorem symmetricZPlus: symmetric Z Zplus.
+change with (\forall a,b:Z. (Zplus a b) = (Zplus b a)).
+intros.
+rewrite > sym_Zplus.
+reflexivity.
+qed.
+   
 theorem true_to_sigma_p_Sn: 
 \forall n:nat. \forall p:nat \to bool. \forall g:nat \to Z.
 p n = true \to sigma_p (S n) p g = 
 (g n)+(sigma_p n p g).
-intros.simplify.
-rewrite > H.reflexivity.
+intros.
+unfold sigma_p.
+apply true_to_sigma_p_Sn_gen.
+assumption.
 qed.
    
 theorem false_to_sigma_p_Sn: 
 \forall n:nat. \forall p:nat \to bool. \forall g:nat \to Z.
 p n = false \to sigma_p (S n) p g = sigma_p n p g.
-intros.simplify.
-rewrite > H.reflexivity.
+intros.
+unfold sigma_p.
+apply false_to_sigma_p_Sn_gen.
+assumption.
 qed.
 
 theorem eq_sigma_p: \forall p1,p2:nat \to bool.
@@ -47,32 +54,10 @@ theorem eq_sigma_p: \forall p1,p2:nat \to bool.
 (\forall x.  x < n \to p1 x = p2 x) \to
 (\forall x.  x < n \to g1 x = g2 x) \to
 sigma_p n p1 g1 = sigma_p n p2 g2.
-intros 5.elim n
-  [reflexivity
-  |apply (bool_elim ? (p1 n1))
-    [intro.
-     rewrite > (true_to_sigma_p_Sn ? ? ? H3).
-     rewrite > true_to_sigma_p_Sn
-      [apply eq_f2
-        [apply H2.apply le_n.
-        |apply H
-          [intros.apply H1.apply le_S.assumption
-          |intros.apply H2.apply le_S.assumption
-          ]
-        ]
-      |rewrite < H3.apply sym_eq.apply H1.apply le_n
-      ]
-    |intro.
-     rewrite > (false_to_sigma_p_Sn ? ? ? H3).
-     rewrite > false_to_sigma_p_Sn
-      [apply H
-        [intros.apply H1.apply le_S.assumption
-        |intros.apply H2.apply le_S.assumption
-        ]
-      |rewrite < H3.apply sym_eq.apply H1.apply le_n
-      ]
-    ]
-  ]
+intros.
+unfold sigma_p.
+apply eq_sigma_p_gen;
+  assumption.
 qed.
 
 theorem eq_sigma_p1: \forall p1,p2:nat \to bool.
@@ -80,42 +65,17 @@ theorem eq_sigma_p1: \forall p1,p2:nat \to bool.
 (\forall x.  x < n \to p1 x = p2 x) \to
 (\forall x.  x < n \to p1 x = true \to g1 x = g2 x) \to
 sigma_p n p1 g1 = sigma_p n p2 g2.
-intros 5.
-elim n
-  [reflexivity
-  |apply (bool_elim ? (p1 n1))
-    [intro.
-     rewrite > (true_to_sigma_p_Sn ? ? ? H3).
-     rewrite > true_to_sigma_p_Sn
-      [apply eq_f2
-        [apply H2
-          [apply le_n|assumption]
-        |apply H
-          [intros.apply H1.apply le_S.assumption
-          |intros.apply H2
-            [apply le_S.assumption|assumption]
-          ]
-        ]
-      |rewrite < H3.apply sym_eq.apply H1.apply le_n
-      ]
-    |intro.
-     rewrite > (false_to_sigma_p_Sn ? ? ? H3).
-     rewrite > false_to_sigma_p_Sn
-      [apply H
-        [intros.apply H1.apply le_S.assumption
-        |intros.apply H2
-          [apply le_S.assumption|assumption]
-        ]
-      |rewrite < H3.apply sym_eq.apply H1.apply le_n
-      ]
-    ]
-  ]
+intros.
+unfold sigma_p.
+apply eq_sigma_p1_gen;
+  assumption.
 qed.
 
 theorem sigma_p_false: 
 \forall g: nat \to Z.\forall n.sigma_p n (\lambda x.false) g = O.
 intros.
-elim n[reflexivity|simplify.assumption]
+unfold sigma_p.
+apply sigma_p_false_gen.
 qed.
 
 theorem sigma_p_plus: \forall n,k:nat.\forall p:nat \to bool.
@@ -123,41 +83,23 @@ theorem sigma_p_plus: \forall n,k:nat.\forall p:nat \to bool.
 sigma_p (k+n) p g 
 = sigma_p k (\lambda x.p (x+n)) (\lambda x.g (x+n)) + sigma_p n p g.
 intros.
-elim k
-  [reflexivity
-  |apply (bool_elim ? (p (n1+n)))
-    [intro.
-     simplify in \vdash (? ? (? % ? ?) ?).    
-     rewrite > (true_to_sigma_p_Sn ? ? ? H1).
-     rewrite > (true_to_sigma_p_Sn n1 (\lambda x.p (x+n)) ? H1).
-     rewrite > assoc_Zplus.
-     rewrite < H.reflexivity
-    |intro.
-     simplify in \vdash (? ? (? % ? ?) ?).    
-     rewrite > (false_to_sigma_p_Sn ? ? ? H1).
-     rewrite > (false_to_sigma_p_Sn n1 (\lambda x.p (x+n)) ? H1).
-     assumption.
-    ]
-  ]
+unfold sigma_p.
+apply (sigma_p_plusA_gen Z n k p g OZ Zplus)
+[ apply symmetricZPlus.
+| intros.
+  apply cic:/matita/Z/plus/Zplus_z_OZ.con
+| apply associative_Zplus
+]
 qed.
 
 theorem false_to_eq_sigma_p: \forall n,m:nat.n \le m \to
 \forall p:nat \to bool.
 \forall g: nat \to Z. (\forall i:nat. n \le i \to i < m \to
 p i = false) \to sigma_p m p g = sigma_p n p g.
-intros 5.
-elim H
-  [reflexivity
-  |simplify.
-   rewrite > H3
-    [simplify.
-     apply H2.
-     intros.
-     apply H3[apply H4|apply le_S.assumption]
-    |assumption
-    |apply le_n
-    ]
-  ]
+intros.
+unfold sigma_p.
+apply (false_to_eq_sigma_p_gen);
+  assumption.
 qed.
 
 theorem sigma_p2 : 
@@ -170,57 +112,17 @@ sigma_p (n*m)
 sigma_p n p1 
   (\lambda x.sigma_p m p2 (g x)).
 intros.
-elim n
-  [simplify.reflexivity
-  |apply (bool_elim ? (p1 n1))
-    [intro.
-     rewrite > (true_to_sigma_p_Sn ? ? ? H1).
-     simplify in \vdash (? ? (? % ? ?) ?);
-     rewrite > sigma_p_plus.
-     rewrite < H.
-     apply eq_f2
-      [apply eq_sigma_p
-        [intros.
-         rewrite > sym_plus.
-         rewrite > (div_plus_times ? ? ? H2).
-         rewrite > (mod_plus_times ? ? ? H2).
-         rewrite > H1.
-         simplify.reflexivity
-        |intros.
-         rewrite > sym_plus.
-         rewrite > (div_plus_times ? ? ? H2).
-         rewrite > (mod_plus_times ? ? ? H2).
-         rewrite > H1.
-         simplify.reflexivity.   
-        ]
-      |reflexivity
-      ]
-    |intro.
-     rewrite > (false_to_sigma_p_Sn ? ? ? H1).
-     simplify in \vdash (? ? (? % ? ?) ?);
-     rewrite > sigma_p_plus.
-     rewrite > H.
-     apply (trans_eq ? ? (O+(sigma_p n1 p1 (\lambda x:nat.sigma_p m p2 (g x)))))
-      [apply eq_f2
-        [rewrite > (eq_sigma_p ? (\lambda x.false) ? (\lambda x:nat.g ((x+n1*m)/m) ((x+n1*m)\mod m)))
-          [apply sigma_p_false
-          |intros.
-           rewrite > sym_plus.
-           rewrite > (div_plus_times ? ? ? H2).
-           rewrite > (mod_plus_times ? ? ? H2).
-           rewrite > H1.
-           simplify.reflexivity
-          |intros.reflexivity.
-          ]
-        |reflexivity
-        ]
-      |reflexivity   
-      ]
-    ]
-  ]
+unfold sigma_p.
+apply (sigma_p2_gen n m p1 p2 Z g OZ Zplus)
+[ apply symmetricZPlus
+| apply associative_Zplus
+| intros.
+  apply Zplus_z_OZ
+]
 qed.
 
 (* a stronger, dependent version, required e.g. for dirichlet product *)
+
 theorem sigma_p2' : 
 \forall n,m:nat.
 \forall p1:nat \to bool.
@@ -232,129 +134,28 @@ sigma_p (n*m)
 sigma_p n p1 
   (\lambda x.sigma_p m (p2 x) (g x)).
 intros.
-elim n
-  [simplify.reflexivity
-  |apply (bool_elim ? (p1 n1))
-    [intro.
-     rewrite > (true_to_sigma_p_Sn ? ? ? H1).
-     simplify in \vdash (? ? (? % ? ?) ?);
-     rewrite > sigma_p_plus.
-     rewrite < H.
-     apply eq_f2
-      [apply eq_sigma_p
-        [intros.
-         rewrite > sym_plus.
-         rewrite > (div_plus_times ? ? ? H2).
-         rewrite > (mod_plus_times ? ? ? H2).
-         rewrite > H1.
-         simplify.reflexivity
-        |intros.
-         rewrite > sym_plus.
-         rewrite > (div_plus_times ? ? ? H2).
-         rewrite > (mod_plus_times ? ? ? H2).
-         rewrite > H1.
-         simplify.reflexivity.   
-        ]
-      |reflexivity
-      ]
-    |intro.
-     rewrite > (false_to_sigma_p_Sn ? ? ? H1).
-     simplify in \vdash (? ? (? % ? ?) ?);
-     rewrite > sigma_p_plus.
-     rewrite > H.
-     apply (trans_eq ? ? (O+(sigma_p n1 p1 (\lambda x:nat.sigma_p m (p2 x) (g x)))))
-      [apply eq_f2
-        [rewrite > (eq_sigma_p ? (\lambda x.false) ? (\lambda x:nat.g ((x+n1*m)/m) ((x+n1*m)\mod m)))
-          [apply sigma_p_false
-          |intros.
-           rewrite > sym_plus.
-           rewrite > (div_plus_times ? ? ? H2).
-           rewrite > (mod_plus_times ? ? ? H2).
-           rewrite > H1.
-           simplify.reflexivity
-          |intros.reflexivity.
-          ]
-        |reflexivity
-        ]
-      |reflexivity   
-      ]
-    ]
-  ]
+unfold sigma_p.
+apply (sigma_p2_gen' n m p1 p2 Z g OZ Zplus)
+[ apply symmetricZPlus
+| apply associative_Zplus
+| intros.
+  apply Zplus_z_OZ
+]
 qed.
 
 lemma sigma_p_gi: \forall g: nat \to Z.
 \forall n,i.\forall p:nat \to bool.i < n \to p i = true \to 
 sigma_p n p g = g i + sigma_p n (\lambda x. andb (p x) (notb (eqb x i))) g.
-intros 2.
-elim n
-  [apply False_ind.
-   apply (not_le_Sn_O i).
-   assumption
-  |apply (bool_elim ? (p n1));intro
-    [elim (le_to_or_lt_eq i n1)
-      [rewrite > true_to_sigma_p_Sn
-        [rewrite > true_to_sigma_p_Sn
-          [rewrite < assoc_Zplus.
-           rewrite < sym_Zplus in \vdash (? ? ? (? % ?)).
-           rewrite > assoc_Zplus.
-           apply eq_f2
-            [reflexivity
-            |apply H[assumption|assumption]
-            ]
-          |rewrite > H3.simplify.
-           change with (notb (eqb n1 i) = notb false).
-           apply eq_f.
-           apply not_eq_to_eqb_false.
-           unfold Not.intro.
-           apply (lt_to_not_eq ? ? H4).
-           apply sym_eq.assumption
-          ]
-        |assumption
-        ]
-      |rewrite > true_to_sigma_p_Sn
-        [rewrite > H4.
-         apply eq_f2
-          [reflexivity
-          |rewrite > false_to_sigma_p_Sn
-            [apply eq_sigma_p
-              [intros.
-               elim (p x)
-                [simplify.
-                 change with (notb false = notb (eqb x n1)).
-                 apply eq_f.
-                 apply sym_eq. 
-                 apply not_eq_to_eqb_false.
-                 apply (lt_to_not_eq ? ? H5)
-                |reflexivity
-                ]
-              |intros.reflexivity
-              ]
-            |rewrite > H3.
-             rewrite > (eq_to_eqb_true ? ? (refl_eq ? n1)).
-             reflexivity
-            ]
-          ]
-        |assumption
-        ]
-      |apply le_S_S_to_le.assumption
-      ]
-    |rewrite > false_to_sigma_p_Sn
-      [elim (le_to_or_lt_eq i n1)
-        [rewrite > false_to_sigma_p_Sn
-          [apply H[assumption|assumption]
-          |rewrite > H3.reflexivity
-          ]
-        |apply False_ind. 
-         apply not_eq_true_false.
-         rewrite < H2.
-         rewrite > H4.
-         assumption
-        |apply le_S_S_to_le.assumption
-        ]
-      |assumption
-      ]
-    ] 
-  ] 
+intros.
+unfold sigma_p.
+apply (sigma_p_gi_gen)
+[ apply symmetricZPlus
+| apply associative_Zplus
+| intros.
+  apply Zplus_z_OZ
+| assumption
+| assumption
+]
 qed.
 
 theorem eq_sigma_p_gh: 
@@ -368,155 +169,58 @@ theorem eq_sigma_p_gh:
 (\forall j. j < n1 \to p2 j = true \to h (h1 j) = j) \to 
 (\forall j. j < n1 \to p2 j = true \to h1 j < n) \to 
 sigma_p n p1 (\lambda x.g(h x)) = sigma_p n1 (\lambda x.p2 x) g.
-intros 4.
-elim n
-  [generalize in match H5.
-   elim n1
-    [reflexivity
-    |apply (bool_elim ? (p2 n2));intro
-      [apply False_ind.
-       apply (not_le_Sn_O (h1 n2)).
-       apply H7
-        [apply le_n|assumption]
-      |rewrite > false_to_sigma_p_Sn
-        [apply H6.
-         intros.  
-            apply H7[apply le_S.apply H9|assumption]
-        |assumption
-        ]
-      ]
-    ]
-  |apply (bool_elim ? (p1 n1));intro
-    [rewrite > true_to_sigma_p_Sn
-      [rewrite > (sigma_p_gi g n2 (h n1))
-        [apply eq_f2
-          [reflexivity
-          |apply H
-            [intros.
-             rewrite > H1
-              [simplify.
-               change with ((\not eqb (h i) (h n1))= \not false).
-               apply eq_f.
-               apply not_eq_to_eqb_false.
-               unfold Not.intro.
-               apply (lt_to_not_eq ? ? H8).
-               rewrite < H2
-                [rewrite < (H2 n1)
-                  [apply eq_f.assumption|apply le_n|assumption]
-                |apply le_S.assumption
-                |assumption
-                ]
-              |apply le_S.assumption
-              |assumption
-              ]
-            |intros.
-             apply H2[apply le_S.assumption|assumption]
-            |intros.
-             apply H3[apply le_S.assumption|assumption]
-            |intros.
-             apply H4
-              [assumption
-              |generalize in match H9.
-               elim (p2 j)
-                [reflexivity|assumption]
-              ]
-            |intros.
-             apply H5
-              [assumption
-              |generalize in match H9.
-               elim (p2 j)
-                [reflexivity|assumption]
-              ]
-            |intros.
-             elim (le_to_or_lt_eq (h1 j) n1)
-              [assumption
-              |generalize in match H9.
-               elim (p2 j)
-                [simplify in H11.
-                 absurd (j = (h n1))
-                  [rewrite < H10.
-                   rewrite > H5
-                    [reflexivity|assumption|autobatch]
-                  |apply eqb_false_to_not_eq.
-                   generalize in match H11.
-                   elim (eqb j (h n1))
-                    [apply sym_eq.assumption|reflexivity]
-                  ]
-                |simplify in H11.
-                 apply False_ind.
-                 apply not_eq_true_false.
-                 apply sym_eq.assumption
-                ]
-              |apply le_S_S_to_le.
-               apply H6
-                [assumption
-                |generalize in match H9.
-                 elim (p2 j)
-                  [reflexivity|assumption]
-                ]
-              ]
-            ]
-          ]
-        |apply H3[apply le_n|assumption]
-        |apply H1[apply le_n|assumption]
-        ]
-      |assumption
-      ]
-    |rewrite > false_to_sigma_p_Sn
-     [apply H
-       [intros.apply H1[apply le_S.assumption|assumption]
-       |intros.apply H2[apply le_S.assumption|assumption]
-       |intros.apply H3[apply le_S.assumption|assumption]
-       |intros.apply H4[assumption|assumption]
-       |intros.apply H5[assumption|assumption]
-       |intros.
-        elim (le_to_or_lt_eq (h1 j) n1)
-          [assumption
-          |absurd (j = (h n1))
-            [rewrite < H10.
-             rewrite > H5
-               [reflexivity|assumption|assumption]
-            |unfold Not.intro.
-             apply not_eq_true_false.
-             rewrite < H7.
-             rewrite < H10.
-             rewrite > H4
-              [reflexivity|assumption|assumption]
-            ]
-          |apply le_S_S_to_le.
-           apply H6[assumption|assumption]
-          ]
-        ]
-      |assumption
-      ]
-    ]
-  ]
+intros.
+unfold sigma_p.
+apply (eq_sigma_p_gh_gen Z OZ Zplus ? ? ? g h h1 n n1 p1 p2)
+[ apply symmetricZPlus
+| apply associative_Zplus
+| intros.
+  apply Zplus_z_OZ
+| assumption
+| assumption
+| assumption
+| assumption
+| assumption
+| assumption
+]
 qed.
 
+
+theorem sigma_p_divides_b: 
+\forall n,m,p:nat.O < n \to prime p \to Not (divides p n) \to 
+\forall g: nat \to Z.
+sigma_p (S (n*(exp p m))) (\lambda x.divides_b x (n*(exp p m))) g =
+sigma_p (S n) (\lambda x.divides_b x n)
+  (\lambda x.sigma_p (S m) (\lambda y.true) (\lambda y.g (x*(exp p y)))).
+intros.
+unfold sigma_p.
+apply (sigma_p_divides_gen Z OZ Zplus n m p ? ? ? g)
+[ assumption
+| assumption
+| assumption
+| apply symmetricZPlus
+| apply associative_Zplus
+| intros.
+  apply Zplus_z_OZ
+]
+qed.
+
+    
 (* sigma_p and Ztimes *)
 lemma Ztimes_sigma_pl: \forall z:Z.\forall n:nat.\forall p. \forall f.
 z * (sigma_p n p f) = sigma_p n p (\lambda i.z*(f i)).
 intros.
-elim n
-  [rewrite > Ztimes_z_OZ.reflexivity
-  |apply (bool_elim ? (p n1)); intro
-    [rewrite > true_to_sigma_p_Sn
-      [rewrite > true_to_sigma_p_Sn 
-        [rewrite < H.
-         apply distr_Ztimes_Zplus
-        |assumption
-        ]
-      |assumption
-      ]
-    |rewrite > false_to_sigma_p_Sn
-      [rewrite > false_to_sigma_p_Sn
-        [assumption
-        |assumption
-        ]
-      |assumption
-      ] 
-    ]
-  ]
+apply (distributive_times_plus_sigma_p_generic Z Zplus OZ Ztimes n z p f)
+[ apply symmetricZPlus
+| apply associative_Zplus
+| intros.
+  apply Zplus_z_OZ
+| apply symmetric_Ztimes
+| apply distributive_Ztimes_Zplus
+| intros.
+  rewrite > (Ztimes_z_OZ a).
+  reflexivity
+]
 qed.
 
 lemma Ztimes_sigma_pr: \forall z:Z.\forall n:nat.\forall p. \forall f.
@@ -528,240 +232,4 @@ apply eq_sigma_p
   [intros.reflexivity
   |intros.apply sym_Ztimes
   ]
-qed.
-
-theorem divides_exp_to_lt_ord:\forall n,m,j,p. O < n \to prime p \to
-p \ndivides n \to j \divides n*(exp p m) \to ord j p < S m.
-intros.
-cut (m = ord (n*(exp p m)) p)
-  [apply le_S_S.
-   rewrite > Hcut.
-   apply divides_to_le_ord
-    [elim (le_to_or_lt_eq ? ? (le_O_n j))
-      [assumption
-      |apply False_ind.
-       apply (lt_to_not_eq ? ? H).
-       elim H3.
-       rewrite < H4 in H5.simplify in H5.
-       elim (times_O_to_O ? ? H5)
-        [apply sym_eq.assumption
-        |apply False_ind.
-         apply (not_le_Sn_n O).
-         rewrite < H6 in \vdash (? ? %).
-         apply lt_O_exp.
-         elim H1.apply lt_to_le.assumption
-        ]
-      ]
-    |rewrite > (times_n_O O).
-     apply lt_times
-      [assumption|apply lt_O_exp.apply (prime_to_lt_O ? H1)]
-    |assumption
-    |assumption
-    ]
-  |unfold ord.
-   rewrite > (p_ord_exp1 p ? m n)
-    [reflexivity
-    |apply (prime_to_lt_O ? H1)
-    |assumption
-    |apply sym_times
-    ]
-  ]
-qed.
-
-theorem divides_exp_to_divides_ord_rem:\forall n,m,j,p. O < n \to prime p \to
-p \ndivides n \to j \divides n*(exp p m) \to ord_rem j p \divides n.
-intros.
-cut (O < j)
-  [cut (n = ord_rem (n*(exp p m)) p)
-    [rewrite > Hcut1.
-     apply divides_to_divides_ord_rem
-      [assumption   
-      |rewrite > (times_n_O O).
-       apply lt_times
-        [assumption|apply lt_O_exp.apply (prime_to_lt_O ? H1)]
-      |assumption
-      |assumption
-      ]
-    |unfold ord_rem.
-     rewrite > (p_ord_exp1 p ? m n)
-      [reflexivity
-      |apply (prime_to_lt_O ? H1)
-      |assumption
-      |apply sym_times
-      ]
-    ]
-  |elim (le_to_or_lt_eq ? ? (le_O_n j))
-    [assumption
-    |apply False_ind.
-     apply (lt_to_not_eq ? ? H).
-     elim H3.
-     rewrite < H4 in H5.simplify in H5.
-     elim (times_O_to_O ? ? H5)
-      [apply sym_eq.assumption
-      |apply False_ind.
-       apply (not_le_Sn_n O).
-       rewrite < H6 in \vdash (? ? %).
-       apply lt_O_exp.
-       elim H1.apply lt_to_le.assumption
-      ]
-    ]
-  ] 
-qed.
-
-theorem sigma_p_divides_b: 
-\forall n,m,p:nat.O < n \to prime p \to Not (divides p n) \to 
-\forall g: nat \to Z.
-sigma_p (S (n*(exp p m))) (\lambda x.divides_b x (n*(exp p m))) g =
-sigma_p (S n) (\lambda x.divides_b x n)
-  (\lambda x.sigma_p (S m) (\lambda y.true) (\lambda y.g (x*(exp p y)))).
-intros.
-cut (O < p)
-  [rewrite < sigma_p2.
-   apply (trans_eq ? ? 
-    (sigma_p (S n*S m) (\lambda x:nat.divides_b (x/S m) n)
-       (\lambda x:nat.g (x/S m*(p)\sup(x\mod S m)))))
-    [apply sym_eq.
-     apply (eq_sigma_p_gh g ? (p_ord_inv p (S m)))
-      [intros.
-       lapply (divides_b_true_to_lt_O ? ? H H4).
-       apply divides_to_divides_b_true
-        [rewrite > (times_n_O O).
-         apply lt_times
-          [assumption
-          |apply lt_O_exp.assumption
-          ]
-        |apply divides_times
-          [apply divides_b_true_to_divides.assumption
-          |apply (witness ? ? (p \sup (m-i \mod (S m)))).
-           rewrite < exp_plus_times.
-           apply eq_f.
-           rewrite > sym_plus.
-           apply plus_minus_m_m.
-           autobatch
-          ]
-        ]
-      |intros.
-       lapply (divides_b_true_to_lt_O ? ? H H4).
-       unfold p_ord_inv.
-       rewrite > (p_ord_exp1 p ? (i \mod (S m)) (i/S m))
-        [change with ((i/S m)*S m+i \mod S m=i).
-         apply sym_eq.
-         apply div_mod.
-         apply lt_O_S
-        |assumption
-        |unfold Not.intro.
-         apply H2.
-         apply (trans_divides ? (i/ S m))
-          [assumption|
-           apply divides_b_true_to_divides;assumption]
-        |apply sym_times.
-        ]
-      |intros.
-       apply le_S_S.
-       apply le_times
-        [apply le_S_S_to_le.
-         change with ((i/S m) < S n).
-         apply (lt_times_to_lt_l m).
-         apply (le_to_lt_to_lt ? i)
-          [autobatch|assumption]
-        |apply le_exp
-          [assumption
-          |apply le_S_S_to_le.
-           apply lt_mod_m_m.
-           apply lt_O_S
-          ]
-        ]
-      |intros.
-       cut (ord j p < S m)
-        [rewrite > div_p_ord_inv
-          [apply divides_to_divides_b_true
-            [apply lt_O_ord_rem
-              [elim H1.assumption
-              |apply (divides_b_true_to_lt_O ? ? ? H4).
-                rewrite > (times_n_O O).
-                apply lt_times
-                [assumption|apply lt_O_exp.assumption]
-              ]
-            |apply (divides_exp_to_divides_ord_rem ? m ? ? H H1 H2).
-             apply divides_b_true_to_divides.
-             assumption
-            ]
-            |assumption
-          ]
-        |apply (divides_exp_to_lt_ord ? ? ? ? H H1 H2).
-         apply (divides_b_true_to_divides ? ? H4).
-         apply (divides_b_true_to_lt_O ? ? H4)
-        ]
-      |intros.
-       cut (ord j p < S m)
-        [rewrite > div_p_ord_inv
-          [rewrite > mod_p_ord_inv
-            [rewrite > sym_times.
-             apply sym_eq.
-             apply exp_ord
-              [elim H1.assumption
-              |apply (divides_b_true_to_lt_O ? ? ? H4).
-               rewrite > (times_n_O O).
-               apply lt_times
-                [assumption|apply lt_O_exp.assumption]
-              ]
-            |apply (divides_exp_to_lt_ord ? ? ? ? H H1 H2).
-             apply (divides_b_true_to_divides ? ? H4).
-             apply (divides_b_true_to_lt_O ? ? H4)
-            ]
-          |assumption
-          ]
-        |apply (divides_exp_to_lt_ord ? ? ? ? H H1 H2).
-         apply (divides_b_true_to_divides ? ? H4).
-         apply (divides_b_true_to_lt_O ? ? H4).
-        ]
-      |intros.
-       rewrite > eq_p_ord_inv.
-       rewrite > sym_plus.
-       apply (lt_to_le_to_lt ? (S m +ord_rem j p*S m))
-        [apply lt_plus_l.
-         apply le_S_S.
-         cut (m = ord (n*(p \sup m)) p)
-          [rewrite > Hcut1.
-           apply divides_to_le_ord
-            [apply (divides_b_true_to_lt_O ? ? ? H4).
-             rewrite > (times_n_O O).
-             apply lt_times
-              [assumption|apply lt_O_exp.assumption]
-            |rewrite > (times_n_O O).
-             apply lt_times
-              [assumption|apply lt_O_exp.assumption]
-            |assumption
-            |apply divides_b_true_to_divides.
-             assumption
-            ]
-          |unfold ord.
-           rewrite > sym_times.
-           rewrite > (p_ord_exp1 p ? m n)
-            [reflexivity
-            |assumption
-            |assumption
-            |reflexivity
-            ]
-          ]
-        |change with (S (ord_rem j p)*S m \le S n*S m).
-         apply le_times_l.
-         apply le_S_S.
-         apply divides_to_le
-          [assumption
-          |apply (divides_exp_to_divides_ord_rem ? m ? ? H H1 H2).
-           apply divides_b_true_to_divides.
-           assumption
-          ]
-        ]
-      ]
-    |apply eq_sigma_p
-      [intros.
-       elim (divides_b (x/S m) n);reflexivity
-      |intros.reflexivity
-      ]
-    ]
-  |elim H1.apply lt_to_le.assumption
-  ]
-qed.
-    
+qed.
\ No newline at end of file
diff --git a/matita/library/nat/iteration2.ma b/matita/library/nat/iteration2.ma
index 14e14b263..e1cd09a20 100644
--- a/matita/library/nat/iteration2.ma
+++ b/matita/library/nat/iteration2.ma
@@ -16,29 +16,41 @@ set "baseuri" "cic:/matita/nat/iteration2.ma".
 
 include "nat/primes.ma".
 include "nat/ord.ma".
+include "nat/generic_sigma_p.ma".
 
-let rec sigma_p n p (g:nat \to nat) \def
-  match n with
-   [ O \Rightarrow O
-   | (S k) \Rightarrow 
-      match p k with
-      [true \Rightarrow (g k)+(sigma_p k p g)
-      |false \Rightarrow sigma_p k p g]
-   ].
+
+(* sigma_p on nautral numbers is a specialization of sigma_p_gen *)
+definition sigma_p: nat \to (nat \to bool) \to (nat \to nat) \to nat \def
+\lambda n, p, g. (sigma_p_gen n p nat g O plus).
+
+theorem symmetricIntPlus: symmetric nat plus.
+change with (\forall a,b:nat. (plus a b) = (plus b a)).
+intros.
+rewrite > sym_plus.
+reflexivity.
+qed.
    
+(*the following theorems on sigma_p in N are obtained by the more general ones
+ * in sigma_p_gen.ma
+ *)
 theorem true_to_sigma_p_Sn: 
 \forall n:nat. \forall p:nat \to bool. \forall g:nat \to nat.
 p n = true \to sigma_p (S n) p g = 
 (g n)+(sigma_p n p g).
-intros.simplify.
-rewrite > H.reflexivity.
+intros.
+unfold sigma_p.
+apply true_to_sigma_p_Sn_gen.
+assumption.
 qed.
    
 theorem false_to_sigma_p_Sn: 
 \forall n:nat. \forall p:nat \to bool. \forall g:nat \to nat.
 p n = false \to sigma_p (S n) p g = sigma_p n p g.
-intros.simplify.
-rewrite > H.reflexivity.
+intros.
+unfold sigma_p.
+apply false_to_sigma_p_Sn_gen.
+assumption.
+
 qed.
 
 theorem eq_sigma_p: \forall p1,p2:nat \to bool.
@@ -46,38 +58,28 @@ theorem eq_sigma_p: \forall p1,p2:nat \to bool.
 (\forall x.  x < n \to p1 x = p2 x) \to
 (\forall x.  x < n \to g1 x = g2 x) \to
 sigma_p n p1 g1 = sigma_p n p2 g2.
-intros 5.elim n
-  [reflexivity
-  |apply (bool_elim ? (p1 n1))
-    [intro.
-     rewrite > (true_to_sigma_p_Sn ? ? ? H3).
-     rewrite > true_to_sigma_p_Sn
-      [apply eq_f2
-        [apply H2.apply le_n.
-        |apply H
-          [intros.apply H1.apply le_S.assumption
-          |intros.apply H2.apply le_S.assumption
-          ]
-        ]
-      |rewrite < H3.apply sym_eq.apply H1.apply le_n
-      ]
-    |intro.
-     rewrite > (false_to_sigma_p_Sn ? ? ? H3).
-     rewrite > false_to_sigma_p_Sn
-      [apply H
-        [intros.apply H1.apply le_S.assumption
-        |intros.apply H2.apply le_S.assumption
-        ]
-      |rewrite < H3.apply sym_eq.apply H1.apply le_n
-      ]
-    ]
-  ]
+intros.
+unfold sigma_p.
+apply eq_sigma_p_gen;
+  assumption.
+qed.
+
+theorem eq_sigma_p1: \forall p1,p2:nat \to bool.
+\forall g1,g2: nat \to nat.\forall n.
+(\forall x.  x < n \to p1 x = p2 x) \to
+(\forall x.  x < n \to p1 x = true \to g1 x = g2 x) \to
+sigma_p n p1 g1 = sigma_p n p2 g2.
+intros.
+unfold sigma_p.
+apply eq_sigma_p1_gen;
+  assumption.
 qed.
 
 theorem sigma_p_false: 
 \forall g: nat \to nat.\forall n.sigma_p n (\lambda x.false) g = O.
 intros.
-elim n[reflexivity|simplify.assumption]
+unfold sigma_p.
+apply sigma_p_false_gen.
 qed.
 
 theorem sigma_p_plus: \forall n,k:nat.\forall p:nat \to bool.
@@ -85,41 +87,24 @@ theorem sigma_p_plus: \forall n,k:nat.\forall p:nat \to bool.
 sigma_p (k+n) p g 
 = sigma_p k (\lambda x.p (x+n)) (\lambda x.g (x+n)) + sigma_p n p g.
 intros.
-elim k
-  [reflexivity
-  |apply (bool_elim ? (p (n1+n)))
-    [intro.
-     simplify in \vdash (? ? (? % ? ?) ?).    
-     rewrite > (true_to_sigma_p_Sn ? ? ? H1).
-     rewrite > (true_to_sigma_p_Sn n1 (\lambda x.p (x+n)) ? H1).
-     rewrite > assoc_plus.
-     rewrite < H.reflexivity
-    |intro.
-     simplify in \vdash (? ? (? % ? ?) ?).    
-     rewrite > (false_to_sigma_p_Sn ? ? ? H1).
-     rewrite > (false_to_sigma_p_Sn n1 (\lambda x.p (x+n)) ? H1).
-     assumption.
-    ]
-  ]
+unfold sigma_p.
+apply (sigma_p_plusA_gen nat n k p g O plus)
+[ apply symmetricIntPlus.
+| intros.
+  apply sym_eq.
+  apply plus_n_O
+| apply associative_plus
+]
 qed.
 
 theorem false_to_eq_sigma_p: \forall n,m:nat.n \le m \to
 \forall p:nat \to bool.
 \forall g: nat \to nat. (\forall i:nat. n \le i \to i < m \to
 p i = false) \to sigma_p m p g = sigma_p n p g.
-intros 5.
-elim H
-  [reflexivity
-  |simplify.
-   rewrite > H3
-    [simplify.
-     apply H2.
-     intros.
-     apply H3[apply H4|apply le_S.assumption]
-    |assumption
-    |apply le_n
-    ]
-  ]
+intros.
+unfold sigma_p.
+apply (false_to_eq_sigma_p_gen);
+  assumption.
 qed.
 
 theorem sigma_p2 : 
@@ -132,129 +117,51 @@ sigma_p (n*m)
 sigma_p n p1 
   (\lambda x.sigma_p m p2 (g x)).
 intros.
-elim n
-  [simplify.reflexivity
-  |apply (bool_elim ? (p1 n1))
-    [intro.
-     rewrite > (true_to_sigma_p_Sn ? ? ? H1).
-     simplify in \vdash (? ? (? % ? ?) ?);
-     rewrite > sigma_p_plus.
-     rewrite < H.
-     apply eq_f2
-      [apply eq_sigma_p
-        [intros.
-         rewrite > sym_plus.
-         rewrite > (div_plus_times ? ? ? H2).
-         rewrite > (mod_plus_times ? ? ? H2).
-         rewrite > H1.
-         simplify.reflexivity
-        |intros.
-         rewrite > sym_plus.
-         rewrite > (div_plus_times ? ? ? H2).
-         rewrite > (mod_plus_times ? ? ? H2).
-         rewrite > H1.
-         simplify.reflexivity.   
-        ]
-      |reflexivity
-      ]
-    |intro.
-     rewrite > (false_to_sigma_p_Sn ? ? ? H1).
-     simplify in \vdash (? ? (? % ? ?) ?);
-     rewrite > sigma_p_plus.
-     rewrite > H.
-     apply (trans_eq ? ? (O+(sigma_p n1 p1 (\lambda x:nat.sigma_p m p2 (g x)))))
-      [apply eq_f2
-        [rewrite > (eq_sigma_p ? (\lambda x.false) ? (\lambda x:nat.g ((x+n1*m)/m) ((x+n1*m)\mod m)))
-          [apply sigma_p_false
-          |intros.
-           rewrite > sym_plus.
-           rewrite > (div_plus_times ? ? ? H2).
-           rewrite > (mod_plus_times ? ? ? H2).
-           rewrite > H1.
-           simplify.reflexivity
-          |intros.reflexivity.
-          ]
-        |reflexivity
-        ]
-      |reflexivity   
-      ]
-    ]
-  ]
+unfold sigma_p.
+apply (sigma_p2_gen n m p1 p2 nat g O plus)
+[ apply symmetricIntPlus
+| apply associative_plus
+| intros.
+  apply sym_eq.
+  apply plus_n_O
+]
+qed.
+
+theorem sigma_p2' : 
+\forall n,m:nat.
+\forall p1:nat \to bool.
+\forall p2:nat \to nat \to bool.
+\forall g: nat \to nat \to nat.
+sigma_p (n*m) 
+  (\lambda x.andb (p1 (div x m)) (p2 (div x m) (mod x m))) 
+  (\lambda x.g (div x m) (mod x m)) =
+sigma_p n p1 
+  (\lambda x.sigma_p m (p2 x) (g x)).
+intros.
+unfold sigma_p.
+apply (sigma_p2_gen' n m p1 p2 nat g O plus)
+[ apply symmetricIntPlus
+| apply associative_plus
+| intros.
+  apply sym_eq.
+  apply plus_n_O
+]
 qed.
 
 lemma sigma_p_gi: \forall g: nat \to nat.
 \forall n,i.\forall p:nat \to bool.i < n \to p i = true \to 
 sigma_p n p g = g i + sigma_p n (\lambda x. andb (p x) (notb (eqb x i))) g.
-intros 2.
-elim n
-  [apply False_ind.
-   apply (not_le_Sn_O i).
-   assumption
-  |apply (bool_elim ? (p n1));intro
-    [elim (le_to_or_lt_eq i n1)
-      [rewrite > true_to_sigma_p_Sn
-        [rewrite > true_to_sigma_p_Sn
-          [rewrite < assoc_plus.
-           rewrite < sym_plus in \vdash (? ? ? (? % ?)).
-           rewrite > assoc_plus.
-           apply eq_f2
-            [reflexivity
-            |apply H[assumption|assumption]
-            ]
-          |rewrite > H3.simplify.
-           change with (notb (eqb n1 i) = notb false).
-           apply eq_f.
-           apply not_eq_to_eqb_false.
-           unfold Not.intro.
-           apply (lt_to_not_eq ? ? H4).
-           apply sym_eq.assumption
-          ]
-        |assumption
-        ]
-      |rewrite > true_to_sigma_p_Sn
-        [rewrite > H4.
-         apply eq_f2
-          [reflexivity
-          |rewrite > false_to_sigma_p_Sn
-            [apply eq_sigma_p
-              [intros.
-               elim (p x)
-                [simplify.
-                 change with (notb false = notb (eqb x n1)).
-                 apply eq_f.
-                 apply sym_eq. 
-                 apply not_eq_to_eqb_false.
-                 apply (lt_to_not_eq ? ? H5)
-                |reflexivity
-                ]
-              |intros.reflexivity
-              ]
-            |rewrite > H3.
-             rewrite > (eq_to_eqb_true ? ? (refl_eq ? n1)).
-             reflexivity
-            ]
-          ]
-        |assumption
-        ]
-      |apply le_S_S_to_le.assumption
-      ]
-    |rewrite > false_to_sigma_p_Sn
-      [elim (le_to_or_lt_eq i n1)
-        [rewrite > false_to_sigma_p_Sn
-          [apply H[assumption|assumption]
-          |rewrite > H3.reflexivity
-          ]
-        |apply False_ind. 
-         apply not_eq_true_false.
-         rewrite < H2.
-         rewrite > H4.
-         assumption
-        |apply le_S_S_to_le.assumption
-        ]
-      |assumption
-      ]
-    ] 
-  ] 
+intros.
+unfold sigma_p.
+apply (sigma_p_gi_gen)
+[ apply symmetricIntPlus
+| apply associative_plus
+| intros.
+  apply sym_eq.
+  apply plus_n_O
+| assumption
+| assumption
+]
 qed.
 
 theorem eq_sigma_p_gh: 
@@ -267,157 +174,23 @@ theorem eq_sigma_p_gh:
 (\forall j. j < n1 \to p2 j = true \to h (h1 j) = j) \to 
 (\forall j. j < n1 \to p2 j = true \to h1 j < n) \to 
 sigma_p n p1 (\lambda x.g(h x)) = sigma_p n1 (\lambda x.p2 x) g.
-intros 4.
-elim n
-  [generalize in match H5.
-   elim n1
-    [reflexivity
-    |apply (bool_elim ? (p2 n2));intro
-      [apply False_ind.
-       apply (not_le_Sn_O (h1 n2)).
-       apply H7
-        [apply le_n|assumption]
-      |rewrite > false_to_sigma_p_Sn
-        [apply H6.
-         intros.  
-            apply H7[apply le_S.apply H9|assumption]
-        |assumption
-        ]
-      ]
-    ]
-  |apply (bool_elim ? (p1 n1));intro
-    [rewrite > true_to_sigma_p_Sn
-      [rewrite > (sigma_p_gi g n2 (h n1))
-        [apply eq_f2
-          [reflexivity
-          |apply H
-            [intros.
-             rewrite > H1
-              [simplify.
-               change with ((\not eqb (h i) (h n1))= \not false).
-               apply eq_f.
-               apply not_eq_to_eqb_false.
-               unfold Not.intro.
-               apply (lt_to_not_eq ? ? H8).
-               rewrite < H2
-                [rewrite < (H2 n1)
-                  [apply eq_f.assumption|apply le_n|assumption]
-                |apply le_S.assumption
-                |assumption
-                ]
-              |apply le_S.assumption
-              |assumption
-              ]
-            |intros.
-             apply H2[apply le_S.assumption|assumption]
-            |intros.
-             apply H3[apply le_S.assumption|assumption]
-            |intros.
-             apply H4
-              [assumption
-              |generalize in match H9.
-               elim (p2 j)
-                [reflexivity|assumption]
-              ]
-            |intros.
-             apply H5
-              [assumption
-              |generalize in match H9.
-               elim (p2 j)
-                [reflexivity|assumption]
-              ]
-            |intros.
-             elim (le_to_or_lt_eq (h1 j) n1)
-              [assumption
-              |generalize in match H9.
-               elim (p2 j)
-                [simplify in H11.
-                 absurd (j = (h n1))
-                  [rewrite < H10.
-                   rewrite > H5
-                    [reflexivity|assumption|autobatch]
-                  |apply eqb_false_to_not_eq.
-                   generalize in match H11.
-                   elim (eqb j (h n1))
-                    [apply sym_eq.assumption|reflexivity]
-                  ]
-                |simplify in H11.
-                 apply False_ind.
-                 apply not_eq_true_false.
-                 apply sym_eq.assumption
-                ]
-              |apply le_S_S_to_le.
-               apply H6
-                [assumption
-                |generalize in match H9.
-                 elim (p2 j)
-                  [reflexivity|assumption]
-                ]
-              ]
-            ]
-          ]
-        |apply H3[apply le_n|assumption]
-        |apply H1[apply le_n|assumption]
-        ]
-      |assumption
-      ]
-    |rewrite > false_to_sigma_p_Sn
-     [apply H
-       [intros.apply H1[apply le_S.assumption|assumption]
-       |intros.apply H2[apply le_S.assumption|assumption]
-       |intros.apply H3[apply le_S.assumption|assumption]
-       |intros.apply H4[assumption|assumption]
-       |intros.apply H5[assumption|assumption]
-       |intros.
-        elim (le_to_or_lt_eq (h1 j) n1)
-          [assumption
-          |absurd (j = (h n1))
-            [rewrite < H10.
-             rewrite > H5
-               [reflexivity|assumption|assumption]
-            |unfold Not.intro.
-             apply not_eq_true_false.
-             rewrite < H7.
-             rewrite < H10.
-             rewrite > H4
-              [reflexivity|assumption|assumption]
-            ]
-          |apply le_S_S_to_le.
-           apply H6[assumption|assumption]
-          ]
-        ]
-      |assumption
-      ]
-    ]
-  ]
-qed.
-
-definition p_ord_times \def
-\lambda p,m,x.
-  match p_ord x p with
-  [pair q r \Rightarrow r*m+q].
-  
-theorem  eq_p_ord_times: \forall p,m,x.
-p_ord_times p m x = (ord_rem x p)*m+(ord x p).
-intros.unfold p_ord_times. unfold ord_rem.
-unfold ord.
-elim (p_ord x p).
-reflexivity.
+intros.
+unfold sigma_p.
+apply (eq_sigma_p_gh_gen nat O plus ? ? ? g h h1 n n1 p1 p2)
+[ apply symmetricIntPlus
+| apply associative_plus
+| intros.
+  apply sym_eq.
+  apply plus_n_O
+| assumption
+| assumption
+| assumption
+| assumption
+| assumption
+| assumption
+]
 qed.
 
-theorem div_p_ord_times: 
-\forall p,m,x. ord x p < m \to p_ord_times p m x / m = ord_rem x p.
-intros.rewrite > eq_p_ord_times.
-apply div_plus_times.
-assumption.
-qed.
-
-theorem mod_p_ord_times: 
-\forall p,m,x. ord x p < m \to p_ord_times p m x \mod m = ord x p.
-intros.rewrite > eq_p_ord_times.
-apply mod_plus_times.
-assumption.
-qed.
 
 theorem sigma_p_divides: 
 \forall n,m,p:nat.O < n \to prime p \to Not (divides p n) \to 
@@ -426,258 +199,33 @@ sigma_p (S (n*(exp p m))) (\lambda x.divides_b x (n*(exp p m))) g =
 sigma_p (S n) (\lambda x.divides_b x n)
   (\lambda x.sigma_p (S m) (\lambda y.true) (\lambda y.g (x*(exp p y)))).
 intros.
-cut (O < p)
-  [rewrite < sigma_p2.
-   apply (trans_eq ? ? 
-    (sigma_p (S n*S m) (\lambda x:nat.divides_b (x/S m) n)
-       (\lambda x:nat.g (x/S m*(p)\sup(x\mod S m)))))
-    [apply sym_eq.
-     apply (eq_sigma_p_gh g ? (p_ord_times p (S m)))
-      [intros.
-       lapply (divides_b_true_to_lt_O ? ? H H4).
-       apply divides_to_divides_b_true
-        [rewrite > (times_n_O O).
-         apply lt_times
-          [assumption
-          |apply lt_O_exp.assumption
-          ]
-        |apply divides_times
-          [apply divides_b_true_to_divides.assumption
-          |apply (witness ? ? (p \sup (m-i \mod (S m)))).
-           rewrite < exp_plus_times.
-           apply eq_f.
-           rewrite > sym_plus.
-           apply plus_minus_m_m.
-           autobatch
-          ]
-        ]
-      |intros.
-       lapply (divides_b_true_to_lt_O ? ? H H4).
-       unfold p_ord_times.
-       rewrite > (p_ord_exp1 p ? (i \mod (S m)) (i/S m))
-        [change with ((i/S m)*S m+i \mod S m=i).
-         apply sym_eq.
-         apply div_mod.
-         apply lt_O_S
-        |assumption
-        |unfold Not.intro.
-         apply H2.
-         apply (trans_divides ? (i/ S m))
-          [assumption|
-           apply divides_b_true_to_divides;assumption]
-        |apply sym_times.
-        ]
-      |intros.
-       apply le_S_S.
-       apply le_times
-        [apply le_S_S_to_le.
-         change with ((i/S m) < S n).
-         apply (lt_times_to_lt_l m).
-         apply (le_to_lt_to_lt ? i)
-          [autobatch|assumption]
-        |apply le_exp
-          [assumption
-          |apply le_S_S_to_le.
-           apply lt_mod_m_m.
-           apply lt_O_S
-          ]
-        ]
-      |intros.
-       cut (ord j p < S m)
-        [rewrite > div_p_ord_times
-          [apply divides_to_divides_b_true
-            [apply lt_O_ord_rem
-             [elim H1.assumption
-             |apply (divides_b_true_to_lt_O ? ? ? H4).
-               rewrite > (times_n_O O).
-               apply lt_times
-               [assumption|apply lt_O_exp.assumption]
-             ]
-           |cut (n = ord_rem (n*(exp p m)) p)
-              [rewrite > Hcut2.
-               apply divides_to_divides_ord_rem
-                [apply (divides_b_true_to_lt_O ? ? ? H4).
-                 rewrite > (times_n_O O).
-                 apply lt_times
-                 [assumption|apply lt_O_exp.assumption]
-                 |rewrite > (times_n_O O).
-                   apply lt_times
-                  [assumption|apply lt_O_exp.assumption]
-               |assumption
-               |apply divides_b_true_to_divides.
-                assumption
-               ]
-              |unfold ord_rem.
-              rewrite > (p_ord_exp1 p ? m n)
-                [reflexivity
-               |assumption
-                |assumption
-               |apply sym_times
-               ]
-             ]
-            ]
-          |assumption
-          ]
-        |cut (m = ord (n*(exp p m)) p)
-          [apply le_S_S.
-           rewrite > Hcut1.
-           apply divides_to_le_ord
-            [apply (divides_b_true_to_lt_O ? ? ? H4).
-             rewrite > (times_n_O O).
-             apply lt_times
-              [assumption|apply lt_O_exp.assumption]
-            |rewrite > (times_n_O O).
-             apply lt_times
-              [assumption|apply lt_O_exp.assumption]
-            |assumption
-            |apply divides_b_true_to_divides.
-             assumption
-            ]
-          |unfold ord.
-           rewrite > (p_ord_exp1 p ? m n)
-            [reflexivity
-            |assumption
-            |assumption
-            |apply sym_times
-            ]
-          ]
-        ]
-      |intros.
-       cut (ord j p < S m)
-        [rewrite > div_p_ord_times
-          [rewrite > mod_p_ord_times
-            [rewrite > sym_times.
-             apply sym_eq.
-             apply exp_ord
-              [elim H1.assumption
-              |apply (divides_b_true_to_lt_O ? ? ? H4).
-               rewrite > (times_n_O O).
-               apply lt_times
-                [assumption|apply lt_O_exp.assumption]
-              ]
-           |cut (m = ord (n*(exp p m)) p)
-             [apply le_S_S.
-              rewrite > Hcut2.
-              apply divides_to_le_ord
-               [apply (divides_b_true_to_lt_O ? ? ? H4).
-                rewrite > (times_n_O O).
-                apply lt_times
-                 [assumption|apply lt_O_exp.assumption]
-               |rewrite > (times_n_O O).
-                apply lt_times
-                 [assumption|apply lt_O_exp.assumption]
-               |assumption
-               |apply divides_b_true_to_divides.
-                assumption
-               ]
-             |unfold ord.
-              rewrite > (p_ord_exp1 p ? m n)
-                [reflexivity
-                |assumption
-                |assumption
-                |apply sym_times
-                ]
-              ]
-            ]
-          |assumption
-          ]
-        |cut (m = ord (n*(exp p m)) p)
-          [apply le_S_S.
-           rewrite > Hcut1.
-           apply divides_to_le_ord
-            [apply (divides_b_true_to_lt_O ? ? ? H4).
-             rewrite > (times_n_O O).
-             apply lt_times
-              [assumption|apply lt_O_exp.assumption]
-            |rewrite > (times_n_O O).
-             apply lt_times
-              [assumption|apply lt_O_exp.assumption]
-            |assumption
-            |apply divides_b_true_to_divides.
-             assumption
-            ]
-          |unfold ord.
-           rewrite > (p_ord_exp1 p ? m n)
-            [reflexivity
-            |assumption
-            |assumption
-            |apply sym_times
-            ]
-          ]
-        ]
-      |intros.
-       rewrite > eq_p_ord_times.
-       rewrite > sym_plus.
-       apply (lt_to_le_to_lt ? (S m +ord_rem j p*S m))
-        [apply lt_plus_l.
-         apply le_S_S.
-         cut (m = ord (n*(p \sup m)) p)
-          [rewrite > Hcut1.
-           apply divides_to_le_ord
-            [apply (divides_b_true_to_lt_O ? ? ? H4).
-             rewrite > (times_n_O O).
-             apply lt_times
-              [assumption|apply lt_O_exp.assumption]
-            |rewrite > (times_n_O O).
-             apply lt_times
-              [assumption|apply lt_O_exp.assumption]
-            |assumption
-            |apply divides_b_true_to_divides.
-             assumption
-            ]
-          |unfold ord.
-           rewrite > sym_times.
-           rewrite > (p_ord_exp1 p ? m n)
-            [reflexivity
-            |assumption
-            |assumption
-            |reflexivity
-            ]
-          ]
-        |change with (S (ord_rem j p)*S m \le S n*S m).
-         apply le_times_l.
-         apply le_S_S.
-         cut (n = ord_rem (n*(p \sup m)) p)
-          [rewrite > Hcut1.
-           apply divides_to_le
-            [apply lt_O_ord_rem
-              [elim H1.assumption
-              |rewrite > (times_n_O O).
-               apply lt_times
-                [assumption|apply lt_O_exp.assumption]
-              ]
-            |apply divides_to_divides_ord_rem
-              [apply (divides_b_true_to_lt_O ? ? ? H4).
-               rewrite > (times_n_O O).
-               apply lt_times
-                [assumption|apply lt_O_exp.assumption]
-              |rewrite > (times_n_O O).
-               apply lt_times
-                [assumption|apply lt_O_exp.assumption]
-              |assumption
-              |apply divides_b_true_to_divides.
-               assumption
-              ]
-            ]
-        |unfold ord_rem.
-         rewrite > sym_times.
-         rewrite > (p_ord_exp1 p ? m n)
-          [reflexivity
-          |assumption
-          |assumption
-          |reflexivity
-          ]
-        ]
-      ]
-    ]
-  |apply eq_sigma_p
-    [intros.
-     elim (divides_b (x/S m) n);reflexivity
-    |intros.reflexivity
-    ]
-  ]
-|elim H1.apply lt_to_le.assumption
+unfold sigma_p.
+apply (sigma_p_divides_gen nat O plus n m p ? ? ? g)
+[ assumption
+| assumption
+| assumption
+| apply symmetricIntPlus
+| apply associative_plus
+| intros.
+  apply sym_eq.
+  apply plus_n_O
+]
+qed.
+
+theorem distributive_times_plus_sigma_p: \forall n,k:nat. \forall p:nat \to bool. \forall g:nat \to nat.
+k*(sigma_p n p g) = sigma_p n p (\lambda i:nat.k * (g i)).
+intros.
+apply (distributive_times_plus_sigma_p_generic nat plus O times n k p g)
+[ apply symmetricIntPlus
+| apply associative_plus
+| intros.
+  apply sym_eq.
+  apply plus_n_O
+| apply symmetric_times
+| apply distributive_times_plus
+| intros.
+  rewrite < (times_n_O a).
+  reflexivity
 ]
 qed.
-