X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fnat%2Fiteration2.ma;h=211df69d0fa0d9941a9dcf14bebca51e1812ffda;hb=10f29fdd78ee089a9a94446207b543d33d6c851c;hp=4f0238498c500d0409c25e3d73196e6d3704dd31;hpb=3ed7d56cf4fab7401f8b400c45b2e35579ba71dd;p=helm.git

diff --git a/helm/software/matita/library/nat/iteration2.ma b/helm/software/matita/library/nat/iteration2.ma
index 4f0238498..211df69d0 100644
--- a/helm/software/matita/library/nat/iteration2.ma
+++ b/helm/software/matita/library/nat/iteration2.ma
@@ -16,13 +16,13 @@ set "baseuri" "cic:/matita/nat/iteration2".
 
 include "nat/primes.ma".
 include "nat/ord.ma".
-include "nat/generic_sigma_p.ma".
+include "nat/generic_iter_p.ma".
 include "nat/count.ma".(*necessary just to use bool_to_nat and bool_to_nat_andb*)
 
 
 (* 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).
+\lambda n, p, g. (iter_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)).
@@ -40,7 +40,7 @@ p n = true \to sigma_p (S n) p g =
 (g n)+(sigma_p n p g).
 intros.
 unfold sigma_p.
-apply true_to_sigma_p_Sn_gen.
+apply true_to_iter_p_gen_Sn.
 assumption.
 qed.
    
@@ -49,10 +49,9 @@ theorem false_to_sigma_p_Sn:
 p n = false \to sigma_p (S n) p g = sigma_p n p g.
 intros.
 unfold sigma_p.
-apply false_to_sigma_p_Sn_gen.
+apply false_to_iter_p_gen_Sn.
 assumption.
-
-qed.
+qed.  
 
 theorem eq_sigma_p: \forall p1,p2:nat \to bool.
 \forall g1,g2: nat \to nat.\forall n.
@@ -61,7 +60,7 @@ theorem eq_sigma_p: \forall p1,p2:nat \to bool.
 sigma_p n p1 g1 = sigma_p n p2 g2.
 intros.
 unfold sigma_p.
-apply eq_sigma_p_gen;
+apply eq_iter_p_gen;
   assumption.
 qed.
 
@@ -72,7 +71,7 @@ theorem eq_sigma_p1: \forall p1,p2:nat \to bool.
 sigma_p n p1 g1 = sigma_p n p2 g2.
 intros.
 unfold sigma_p.
-apply eq_sigma_p1_gen;
+apply eq_iter_p_gen1;
   assumption.
 qed.
 
@@ -80,7 +79,7 @@ theorem sigma_p_false:
 \forall g: nat \to nat.\forall n.sigma_p n (\lambda x.false) g = O.
 intros.
 unfold sigma_p.
-apply sigma_p_false_gen.
+apply iter_p_gen_false.
 qed.
 
 theorem sigma_p_plus: \forall n,k:nat.\forall p:nat \to bool.
@@ -89,7 +88,7 @@ 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.
 unfold sigma_p.
-apply (sigma_p_plusA_gen nat n k p g O plus)
+apply (iter_p_gen_plusA nat n k p g O plus)
 [ apply symmetricIntPlus.
 | intros.
   apply sym_eq.
@@ -104,10 +103,88 @@ theorem false_to_eq_sigma_p: \forall n,m:nat.n \le m \to
 p i = false) \to sigma_p m p g = sigma_p n p g.
 intros.
 unfold sigma_p.
-apply (false_to_eq_sigma_p_gen);
+apply (false_to_eq_iter_p_gen);
   assumption.
 qed.
 
+theorem or_false_to_eq_sigma_p:
+\forall n,m:nat.\forall p:nat \to bool.
+\forall g: nat \to nat. 
+n \le m \to (\forall i:nat. n \le i \to i < m \to p i = false \lor g i = O)
+\to sigma_p m p g = sigma_p n p g.
+intros.
+unfold sigma_p.
+apply or_false_eq_baseA_to_eq_iter_p_gen
+  [intros.reflexivity
+  |assumption
+  |assumption
+  ]
+qed.
+
+theorem bool_to_nat_to_eq_sigma_p:
+\forall n:nat.\forall p1,p2:nat \to bool.
+\forall g1,g2: nat \to nat. 
+(\forall i:nat. 
+bool_to_nat (p1 i)*(g1 i) = bool_to_nat (p2 i)*(g2 i)) 
+\to sigma_p n p1 g1 = sigma_p n p2 g2.
+intros.elim n
+  [reflexivity
+  |generalize in match (H n1).
+   apply (bool_elim ? (p1 n1));intro
+    [apply (bool_elim ? (p2 n1));intros
+      [rewrite > true_to_sigma_p_Sn
+        [rewrite > true_to_sigma_p_Sn
+          [apply eq_f2
+            [simplify in H4.
+             rewrite > plus_n_O.
+             rewrite > plus_n_O in ⊢ (? ? ? %).
+             assumption
+            |assumption
+            ]
+          |assumption
+          ]
+        |assumption
+        ]
+      |rewrite > true_to_sigma_p_Sn
+        [rewrite > false_to_sigma_p_Sn
+          [change in ⊢ (? ? ? %) with (O + sigma_p n1 p2 g2).
+           apply eq_f2
+            [simplify in H4.
+             rewrite > plus_n_O.
+             assumption
+            |assumption
+            ]
+          |assumption
+          ]
+        |assumption
+        ]
+      ]
+    |apply (bool_elim ? (p2 n1));intros
+      [rewrite > false_to_sigma_p_Sn
+        [rewrite > true_to_sigma_p_Sn
+          [change in ⊢ (? ? % ?) with (O + sigma_p n1 p1 g1).
+           apply eq_f2
+            [simplify in H4.
+             rewrite < plus_n_O in H4.
+             assumption
+            |assumption
+            ]
+          |assumption
+          ]
+        |assumption
+        ]
+      |rewrite > false_to_sigma_p_Sn
+        [rewrite > false_to_sigma_p_Sn
+          [assumption
+          |assumption
+          ]
+        |assumption
+        ]
+      ]
+    ]
+  ]
+qed.
+            
 theorem sigma_p2 : 
 \forall n,m:nat.
 \forall p1,p2:nat \to bool.
@@ -119,7 +196,7 @@ sigma_p n p1
   (\lambda x.sigma_p m p2 (g x)).
 intros.
 unfold sigma_p.
-apply (sigma_p2_gen n m p1 p2 nat g O plus)
+apply (iter_p_gen2 n m p1 p2 nat g O plus)
 [ apply symmetricIntPlus
 | apply associative_plus
 | intros.
@@ -134,13 +211,13 @@ theorem sigma_p2' :
 \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.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 (iter_p_gen2' n m p1 p2 nat g O plus)
 [ apply symmetricIntPlus
 | apply associative_plus
 | intros.
@@ -154,7 +231,7 @@ lemma sigma_p_gi: \forall g: nat \to nat.
 sigma_p n p g = g i + sigma_p n (\lambda x. andb (p x) (notb (eqb x i))) g.
 intros.
 unfold sigma_p.
-apply (sigma_p_gi_gen)
+apply (iter_p_gen_gi)
 [ apply symmetricIntPlus
 | apply associative_plus
 | intros.
@@ -174,10 +251,10 @@ theorem eq_sigma_p_gh:
 (\forall j. j < n1 \to p2 j = true \to p1 (h1 j) = true) \to
 (\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.
+sigma_p n p1 (\lambda x.g(h x)) = sigma_p n1 p2 g.
 intros.
 unfold sigma_p.
-apply (eq_sigma_p_gh_gen nat O plus ? ? ? g h h1 n n1 p1 p2)
+apply (eq_iter_p_gen_gh nat O plus ? ? ? g h h1 n n1 p1 p2)
 [ apply symmetricIntPlus
 | apply associative_plus
 | intros.
@@ -192,7 +269,249 @@ apply (eq_sigma_p_gh_gen nat O plus ? ? ? g h h1 n n1 p1 p2)
 ]
 qed.
 
+theorem eq_sigma_p_pred: 
+\forall n,p,g. p O = true \to
+sigma_p (S n) (\lambda i.p (pred i)) (\lambda i.g(pred i)) = 
+plus (sigma_p n p g) (g O).
+intros.
+unfold sigma_p.
+apply eq_iter_p_gen_pred
+  [assumption
+  |apply symmetricIntPlus
+  |apply associative_plus
+  |intros.apply sym_eq.apply plus_n_O
+  ]
+qed.
+
+(* monotonicity *)
+theorem le_sigma_p: 
+\forall n:nat. \forall p:nat \to bool. \forall g1,g2:nat \to nat.
+(\forall i. i < n \to p i = true \to g1 i \le g2 i ) \to 
+sigma_p n p g1 \le sigma_p n p g2.
+intros.
+generalize in match H.
+elim n
+  [apply le_n.
+  |apply (bool_elim ? (p n1));intros
+    [rewrite > true_to_sigma_p_Sn
+      [rewrite > true_to_sigma_p_Sn in ⊢ (? ? %)
+        [apply le_plus
+          [apply H2[apply le_n|assumption]
+          |apply H1.
+           intros.
+           apply H2[apply le_S.assumption|assumption]
+          ]
+        |assumption
+        ]
+      |assumption
+      ]
+    |rewrite > false_to_sigma_p_Sn
+      [rewrite > false_to_sigma_p_Sn in ⊢ (? ? %)
+        [apply H1.
+         intros.
+         apply H2[apply le_S.assumption|assumption]
+        |assumption
+        ]
+      |assumption
+      ]
+    ]
+  ]
+qed.
 
+(* a slightly more general result *)
+theorem le_sigma_p1: 
+\forall n:nat. \forall p1,p2:nat \to bool. \forall g1,g2:nat \to nat.
+(\forall i. i < n \to 
+bool_to_nat (p1 i)*(g1 i) \le bool_to_nat (p2 i)*g2 i) \to 
+sigma_p n p1 g1 \le sigma_p n p2 g2.
+intros.
+generalize in match H.
+elim n
+  [apply le_n.
+  |apply (bool_elim ? (p1 n1));intros
+    [apply (bool_elim ? (p2 n1));intros
+      [rewrite > true_to_sigma_p_Sn
+        [rewrite > true_to_sigma_p_Sn in ⊢ (? ? %)
+          [apply le_plus
+            [lapply (H2 n1) as H5
+              [rewrite > H3 in H5.
+               rewrite > H4 in H5.
+               simplify in H5.
+               rewrite < plus_n_O in H5.
+               rewrite < plus_n_O in H5.
+               assumption
+              |apply le_S_S.apply le_n
+              ]
+            |apply H1.intros.
+             apply H2.apply le_S.assumption
+            ]
+          |assumption
+          ]
+        |assumption
+        ]
+      |rewrite > true_to_sigma_p_Sn
+        [rewrite > false_to_sigma_p_Sn in ⊢ (? ? %)
+          [change in ⊢ (? ? %) with (O + sigma_p n1 p2 g2).
+           apply le_plus
+            [lapply (H2 n1) as H5
+              [rewrite > H3 in H5.
+               rewrite > H4 in H5.
+               simplify in H5.
+               rewrite < plus_n_O in H5.
+               assumption
+              |apply le_S_S.apply le_n
+              ]
+            |apply H1.intros.
+             apply H2.apply le_S.assumption
+            ]
+          |assumption
+          ]
+        |assumption
+        ]
+      ]
+    |apply (bool_elim ? (p2 n1));intros
+      [rewrite > false_to_sigma_p_Sn
+        [rewrite > true_to_sigma_p_Sn in ⊢ (? ? %)
+          [change in ⊢ (? % ?) with (O + sigma_p n1 p1 g1).
+           apply le_plus
+            [lapply (H2 n1) as H5
+              [rewrite > H3 in H5.
+               rewrite > H4 in H5.
+               simplify in H5.
+               rewrite < plus_n_O in H5.
+               assumption
+              |apply le_S_S.apply le_n
+              ]
+            |apply H1.intros.
+             apply H2.apply le_S.assumption
+            ]
+          |assumption
+          ]
+        |assumption
+        ]
+      |rewrite > false_to_sigma_p_Sn
+        [rewrite > false_to_sigma_p_Sn in ⊢ (? ? %)
+          [apply H1.intros.
+           apply H2.apply le_S.assumption
+          |assumption
+          ]
+        |assumption
+        ]
+      ]
+    ]
+  ]
+qed.    
+
+theorem lt_sigma_p: 
+\forall n:nat. \forall p:nat \to bool. \forall g1,g2:nat \to nat.
+(\forall i. i < n \to p i = true \to g1 i \le g2 i ) \to
+(\exists i. i < n \and (p i = true) \and (g1 i < g2 i)) \to
+sigma_p n p g1 < sigma_p n p g2.
+intros 4.
+elim n
+  [elim H1.clear H1.
+   elim H2.clear H2.
+   elim H1.clear H1.
+   apply False_ind.
+   apply (lt_to_not_le ? ? H2).
+   apply le_O_n
+  |apply (bool_elim ? (p n1));intros
+    [apply (bool_elim ? (leb (S (g1 n1)) (g2 n1)));intros
+      [rewrite > true_to_sigma_p_Sn
+        [rewrite > true_to_sigma_p_Sn in ⊢ (? ? %)
+          [change with 
+           (S (g1 n1)+sigma_p n1 p g1 \le g2 n1+sigma_p n1 p g2).
+           apply le_plus
+            [apply leb_true_to_le.assumption
+            |apply le_sigma_p.intros.
+             apply H1
+              [apply lt_to_le.apply le_S_S.assumption
+              |assumption
+              ]
+            ]
+          |assumption
+          ]
+        |assumption
+        ]
+      |rewrite > true_to_sigma_p_Sn
+        [rewrite > true_to_sigma_p_Sn in ⊢ (? ? %)
+          [unfold lt.
+           rewrite > plus_n_Sm.
+           apply le_plus
+            [apply H1
+              [apply le_n
+              |assumption
+              ]
+            |apply H
+              [intros.apply H1
+                [apply lt_to_le.apply le_S_S.assumption
+                |assumption
+                ]
+              |elim H2.clear H2.
+               elim H5.clear H5.
+               elim H2.clear H2.
+               apply (ex_intro ? ? a).
+               split
+                [split
+                  [elim (le_to_or_lt_eq a n1)
+                    [assumption
+                    |absurd (g1 a < g2 a)
+                      [assumption
+                      |apply leb_false_to_not_le.
+                       rewrite > H2.
+                       assumption
+                      ]
+                    |apply le_S_S_to_le.
+                     assumption
+                    ]
+                  |assumption
+                  ]
+                |assumption
+                ]
+              ]
+            ]
+          |assumption
+          ]
+        |assumption
+        ]
+      ]
+    |rewrite > false_to_sigma_p_Sn
+      [rewrite > false_to_sigma_p_Sn in ⊢ (? ? %)
+        [apply H
+          [intros.apply H1
+            [apply lt_to_le.apply le_S_S.assumption
+            |assumption
+            ]
+          |elim H2.clear H2.
+           elim H4.clear H4.
+           elim H2.clear H2.
+           apply (ex_intro ? ? a).
+           split
+            [split
+              [elim (le_to_or_lt_eq a n1)
+                [assumption
+                |apply False_ind.
+                 apply not_eq_true_false.
+                 rewrite < H6.
+                 rewrite < H3.
+                 rewrite < H2. 
+                 reflexivity
+                |apply le_S_S_to_le.
+                 assumption
+                ]
+              |assumption
+              ]
+            |assumption
+            ]
+          ]
+        |assumption
+        ]
+      |assumption
+      ]
+    ]
+  ]
+qed.
+          
 theorem sigma_p_divides: 
 \forall n,m,p:nat.O < n \to prime p \to Not (divides p n) \to 
 \forall g: nat \to nat.
@@ -201,7 +520,7 @@ 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 nat O plus n m p ? ? ? g)
+apply (iter_p_gen_divides nat O plus n m p ? ? ? g)
 [ assumption
 | assumption
 | assumption
@@ -216,7 +535,7 @@ 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 (distributive_times_plus_iter_p_gen nat plus O times n k p g)
 [ apply symmetricIntPlus
 | apply associative_plus
 | intros.
@@ -310,32 +629,33 @@ elim n
 qed.
 
 theorem sigma_p_plus_1: \forall n:nat. \forall f,g:nat \to nat.
-sigma_p n (\lambda b:nat. true) (\lambda a:nat.(f a) + (g a)) = 
-sigma_p n (\lambda b:nat. true) f + sigma_p n (\lambda b:nat. true) g.
+\forall p.
+sigma_p n p (\lambda a:nat.(f a) + (g a)) = 
+sigma_p n p f + sigma_p n p g.
 intros.
 elim n
 [ simplify.
   reflexivity
-| rewrite > true_to_sigma_p_Sn
-  [ rewrite > (true_to_sigma_p_Sn n1 (\lambda c:nat.true) f)
-    [ rewrite > (true_to_sigma_p_Sn n1 (\lambda c:nat.true) g)
-      [ rewrite > assoc_plus in \vdash (? ? ? %).
-        rewrite < assoc_plus in \vdash (? ? ? (? ? %)).
-        rewrite < sym_plus in \vdash (? ? ? (? ? (? % ?))).
-        rewrite > assoc_plus in \vdash (? ? ? (? ? %)).
-        rewrite < assoc_plus in \vdash (? ? ? %).
-        apply eq_f.
-        assumption
-      | reflexivity
-      ]
-    | reflexivity
-    ]
-  | reflexivity
-  ]
-]
+| apply (bool_elim ? (p n1)); intro;
+  [ rewrite > true_to_sigma_p_Sn
+    [ rewrite > (true_to_sigma_p_Sn n1 p f)
+      [ rewrite > (true_to_sigma_p_Sn n1 p g)
+        [ rewrite > assoc_plus in \vdash (? ? ? %).
+          rewrite < assoc_plus in \vdash (? ? ? (? ? %)).
+          rewrite < sym_plus in \vdash (? ? ? (? ? (? % ?))).
+          rewrite > assoc_plus in \vdash (? ? ? (? ? %)).
+          rewrite < assoc_plus in \vdash (? ? ? %).
+          apply eq_f.
+          assumption]]]
+   assumption
+ | rewrite > false_to_sigma_p_Sn
+    [ rewrite > (false_to_sigma_p_Sn n1 p f)
+      [ rewrite > (false_to_sigma_p_Sn n1 p g)
+        [assumption]]]
+   assumption
+]]
 qed.
 
-
 theorem eq_sigma_p_sigma_p_times1 : \forall n,m:nat.\forall f:nat \to nat.
 sigma_p (n*m) (\lambda x:nat.true) f =
 sigma_p m (\lambda x:nat.true) 
@@ -382,7 +702,6 @@ rewrite > sym_times.
 apply eq_sigma_p_sigma_p_times1.
 qed.
 
-
 theorem sigma_p_times:\forall n,m:nat. 
 \forall f,f1,f2:nat \to bool.
 \forall g:nat \to nat \to nat. 
@@ -581,3 +900,84 @@ rewrite > (S_pred ((S n)*(S m))) in \vdash (? ? (? % ? ?) ?)
 | apply lt_O_times_S_S
 ]
 qed.
+
+theorem sigma_p_knm: 
+\forall g: nat \to nat.
+\forall h2:nat \to nat \to nat.
+\forall h11,h12:nat \to nat. 
+\forall k,n,m.
+\forall p1,p21:nat \to bool.
+\forall p22:nat \to nat \to bool.
+(\forall x. x < k \to p1 x = true \to 
+p21 (h11 x) = true \land p22 (h11 x) (h12 x) = true
+\land h2 (h11 x) (h12 x) = x 
+\land (h11 x) < n \land (h12 x) < m) \to
+(\forall i,j. i < n \to j < m \to p21 i = true \to p22 i j = true \to 
+p1 (h2 i j) = true \land 
+h11 (h2 i j) = i \land h12 (h2 i j) = j
+\land h2 i j < k) \to
+sigma_p k p1 g=
+sigma_p n p21 (\lambda x:nat.sigma_p m (p22 x) (\lambda y. g (h2 x y))).
+intros.
+unfold sigma_p.
+unfold sigma_p in \vdash (? ? ? (? ? ? ? (\lambda x:?.%) ? ?)).
+apply iter_p_gen_knm
+  [apply symmetricIntPlus
+  |apply associative_plus
+  |intro.rewrite < plus_n_O.reflexivity
+  |exact h11
+  |exact h12
+  |assumption
+  |assumption
+  ]
+qed.
+  
+  
+theorem sigma_p2_eq: 
+\forall g: nat \to nat \to nat.
+\forall h11,h12,h21,h22: nat \to nat \to nat. 
+\forall n1,m1,n2,m2.
+\forall p11,p21:nat \to bool.
+\forall p12,p22:nat \to nat \to bool.
+(\forall i,j. i < n2 \to j < m2 \to p21 i = true \to p22 i j = true \to 
+p11 (h11 i j) = true \land p12 (h11 i j) (h12 i j) = true
+\land h21 (h11 i j) (h12 i j) = i \land h22 (h11 i j) (h12 i j) = j
+\land h11 i j < n1 \land h12 i j < m1) \to
+(\forall i,j. i < n1 \to j < m1 \to p11 i = true \to p12 i j = true \to 
+p21 (h21 i j) = true \land p22 (h21 i j) (h22 i j) = true
+\land h11 (h21 i j) (h22 i j) = i \land h12 (h21 i j) (h22 i j) = j
+\land (h21 i j) < n2 \land (h22 i j) < m2) \to
+sigma_p n1 p11 (\lambda x:nat .sigma_p m1 (p12 x) (\lambda y. g x y)) =
+sigma_p n2 p21 (\lambda x:nat .sigma_p m2 (p22 x) (\lambda y. g (h11 x y) (h12 x y))).
+intros.
+unfold sigma_p.
+unfold sigma_p in \vdash (? ? (? ? ? ? (\lambda x:?.%) ? ?) ?).
+unfold sigma_p in \vdash (? ? ? (? ? ? ? (\lambda x:?.%) ? ?)).
+
+apply(iter_p_gen_2_eq nat O plus ? ? ? g h11 h12 h21 h22 n1 m1 n2 m2 p11 p21 p12 p22)
+[ apply symmetricIntPlus
+| apply associative_plus
+| intro.
+  rewrite < (plus_n_O).
+  reflexivity
+| assumption
+| assumption
+]
+qed.
+
+theorem sigma_p_sigma_p: 
+\forall g: nat \to nat \to nat.
+\forall n,m.
+\forall p11,p21:nat \to bool.
+\forall p12,p22:nat \to nat \to bool.
+(\forall x,y. x < n \to y < m \to 
+ (p11 x \land p12 x y) = (p21 y \land p22 y x)) \to
+sigma_p n p11 (\lambda x:nat.sigma_p m (p12 x) (\lambda y. g x y)) =
+sigma_p m p21 (\lambda y:nat.sigma_p n (p22 y) (\lambda x. g x y)).
+intros.
+unfold sigma_p.unfold sigma_p.
+apply (iter_p_gen_iter_p_gen ? ? ? sym_plus assoc_plus)
+  [intros.apply sym_eq.apply plus_n_O.
+  |assumption
+  ]
+qed.
\ No newline at end of file