X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fnat%2Fgeneric_iter_p.ma;h=28ef391eb4e6790124601cba297cf7a7ea16c467;hb=063523ae5f8da7e6458232f4afb6744ec86dc8bd;hp=c424f82e0318064b1a31ee6c406d4f8b55089327;hpb=6d49a181a1b771f797d37b02661b5743aee86ac1;p=helm.git

diff --git a/helm/software/matita/library/nat/generic_iter_p.ma b/helm/software/matita/library/nat/generic_iter_p.ma
index c424f82e0..28ef391eb 100644
--- a/helm/software/matita/library/nat/generic_iter_p.ma
+++ b/helm/software/matita/library/nat/generic_iter_p.ma
@@ -12,8 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/nat/generic_iter_p".
-
+include "nat/div_and_mod_diseq.ma".
 include "nat/primes.ma".
 include "nat/ord.ma".
 
@@ -47,8 +46,6 @@ rewrite > H.
 reflexivity.
 qed.
 
-
-
 theorem false_to_iter_p_gen_Sn: 
 \forall n:nat. \forall p:nat \to bool. \forall A:Type. \forall g:nat \to A.
 \forall baseA:A. \forall plusA: A \to A \to A.
@@ -59,7 +56,6 @@ rewrite > H.
 reflexivity.
 qed.
 
-
 theorem eq_iter_p_gen: \forall p1,p2:nat \to bool. \forall A:Type.
 \forall g1,g2: nat \to A. \forall baseA: A. 
 \forall plusA: A \to A \to A. \forall n:nat.
@@ -162,8 +158,7 @@ iter_p_gen (k + n) p A g baseA plusA
 intros.
 
 elim k
-[ rewrite < (plus_n_O n).
-  simplify.
+[ simplify.
   rewrite > H in \vdash (? ? ? %).
   rewrite > (H1 ?).
   reflexivity
@@ -205,6 +200,47 @@ elim H
 ]
 qed.
 
+(* a therem slightly more general than the previous one *)
+theorem or_false_eq_baseA_to_eq_iter_p_gen: \forall A:Type. \forall n,m:nat.\forall p:nat \to bool.
+\forall g: nat \to A. \forall baseA:A. \forall plusA: A \to A \to A.
+(\forall a. plusA baseA a = a) \to
+n \le m \to (\forall i:nat. n \le i \to i < m \to p i = false \lor g i = baseA)
+\to iter_p_gen m p A g baseA plusA = iter_p_gen n p A g baseA plusA.
+intros 9.
+elim H1
+[reflexivity
+|apply (bool_elim ? (p n1));intro
+  [elim (H4 n1)
+    [apply False_ind.
+     apply not_eq_true_false.
+     rewrite < H5.
+     rewrite < H6.
+     reflexivity
+    |rewrite > true_to_iter_p_gen_Sn
+      [rewrite > H6.
+       rewrite > H.
+       apply H3.intros.
+       apply H4
+        [assumption
+        |apply le_S.assumption
+        ]
+      |assumption
+      ]
+    |assumption
+    |apply le_n
+    ]
+  |rewrite > false_to_iter_p_gen_Sn
+    [apply H3.intros.
+     apply H4
+      [assumption
+      |apply le_S.assumption
+      ]
+    |assumption
+    ]
+  ]
+]
+qed.
+    
 theorem iter_p_gen2 : 
 \forall n,m:nat.
 \forall p1,p2:nat \to bool.
@@ -246,8 +282,7 @@ elim n
           rewrite > sym_plus.
           rewrite > (div_plus_times ? ? ? H5).
           rewrite > (mod_plus_times ? ? ? H5).
-          rewrite > H4.
-          simplify.reflexivity.   
+          reflexivity.   
         ]
       | reflexivity
       ]
@@ -329,8 +364,7 @@ elim n
           rewrite > sym_plus.
           rewrite > (div_plus_times ? ? ? H5).
           rewrite > (mod_plus_times ? ? ? H5).
-          rewrite > H4.
-          simplify.reflexivity.   
+          reflexivity.   
         ]
       | reflexivity
       ]
@@ -667,9 +701,38 @@ elim n
 ]
 qed.
 
-
-(* da spostare *)
-
+theorem eq_iter_p_gen_pred: 
+\forall A:Type.
+\forall baseA: A.
+\forall plusA: A \to A \to A.
+\forall n,p,g.
+p O = true \to
+(symmetric A plusA) \to (associative A plusA) \to (\forall a:A.(plusA a  baseA) = a) \to
+iter_p_gen (S n) (\lambda i.p (pred i)) A (\lambda i.g(pred i)) baseA plusA = 
+plusA (iter_p_gen n p A g baseA plusA) (g O).
+intros.
+elim n
+  [rewrite > true_to_iter_p_gen_Sn
+    [simplify.apply H1
+    |assumption
+    ]
+  |apply (bool_elim ? (p n1));intro
+    [rewrite > true_to_iter_p_gen_Sn
+      [rewrite > true_to_iter_p_gen_Sn in ⊢ (? ? ? %)
+        [rewrite > H2 in ⊢ (? ? ? %).
+         apply eq_f.assumption
+        |assumption
+        ]
+      |assumption
+      ]
+    |rewrite > false_to_iter_p_gen_Sn
+      [rewrite > false_to_iter_p_gen_Sn in ⊢ (? ? ? %);assumption
+      |assumption
+      ]
+    ]
+  ]
+qed.
+    
 definition p_ord_times \def
 \lambda p,m,x.
   match p_ord x p with
@@ -708,23 +771,6 @@ elim (le_to_or_lt_eq O ? (le_O_n m))
   ]
 qed.
 
-(* lemmino da postare *)
-theorem lt_times_to_lt_div: \forall i,n,m. i < n*m \to i/m < n.
-intros.
-cut (O < m)
-  [apply (lt_times_n_to_lt m)
-    [assumption
-    |apply (le_to_lt_to_lt ? i)
-      [rewrite > (div_mod i m) in \vdash (? ? %).
-       apply le_plus_n_r.
-       assumption
-      |assumption
-      ]
-    ]
-  |apply (lt_times_to_lt_O ? ? ? H)
-  ]
-qed.
-
 theorem iter_p_gen_knm:
 \forall A:Type.
 \forall baseA: A.
@@ -1142,4 +1188,576 @@ elim n
 ]
 qed.
 
+(* old version - proved without theorem iter_p_gen_knm
+theorem iter_p_gen_2_eq: 
+\forall A:Type.
+\forall baseA: A.
+\forall plusA: A \to A \to A. 
+(symmetric A plusA) \to 
+(associative A plusA) \to 
+(\forall a:A.(plusA a  baseA) = a)\to
+\forall g: nat \to nat \to A.
+\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
+iter_p_gen n1 p11 A 
+     (\lambda x:nat .iter_p_gen m1 (p12 x) A (\lambda y. g x y) baseA plusA) 
+     baseA plusA =
+iter_p_gen n2 p21 A 
+    (\lambda x:nat .iter_p_gen m2 (p22 x) A  (\lambda y. g (h11 x y) (h12 x y)) baseA plusA )
+    baseA plusA.
+intros.
+rewrite < (iter_p_gen2' ? ? ? ? ? ? ? ? H H1 H2).
+rewrite < (iter_p_gen2' ? ? ? ? ? ? ? ? H H1 H2).
+apply sym_eq.
+letin h := (\lambda x.(h11 (x/m2) (x\mod m2))*m1 + (h12 (x/m2) (x\mod m2))).
+letin h1 := (\lambda x.(h21 (x/m1) (x\mod m1))*m2 + (h22 (x/m1) (x\mod m1))).
+apply (trans_eq ? ? 
+  (iter_p_gen (n2*m2) (\lambda x:nat.p21 (x/m2)\land p22 (x/m2) (x\mod m2)) A
+  (\lambda x:nat.g ((h x)/m1) ((h x)\mod m1)) baseA plusA ))
+  [clear h.clear h1.
+   apply eq_iter_p_gen1
+    [intros.reflexivity
+    |intros.
+     cut (O < m2)
+      [cut (x/m2 < n2)
+        [cut (x \mod m2 < m2)
+          [elim (and_true ? ? H6).
+           elim (H3 ? ? Hcut1 Hcut2 H7 H8).
+           elim H9.clear H9.
+           elim H11.clear H11.
+           elim H9.clear H9.
+           elim H11.clear H11.
+           apply eq_f2
+            [apply sym_eq.
+             apply div_plus_times.
+             assumption
+            | apply sym_eq.
+              apply mod_plus_times.
+              assumption
+            ]
+          |apply lt_mod_m_m.
+           assumption
+          ]
+        |apply (lt_times_n_to_lt m2)
+          [assumption
+          |apply (le_to_lt_to_lt ? x)
+            [apply (eq_plus_to_le ? ? (x \mod m2)).
+             apply div_mod.
+             assumption
+            |assumption
+            ]
+          ]
+        ]
+      |apply not_le_to_lt.unfold.intro.
+       generalize in match H5.
+       apply (le_n_O_elim ? H7).
+       rewrite < times_n_O.
+       apply le_to_not_lt.
+       apply le_O_n  
+      ]      
+    ]
+  |apply (eq_iter_p_gen_gh ? ? ? H H1 H2 ? h h1);intros
+    [cut (O < m2)
+      [cut (i/m2 < n2)
+        [cut (i \mod m2 < m2)
+          [elim (and_true ? ? H6).
+           elim (H3 ? ? Hcut1 Hcut2 H7 H8).
+           elim H9.clear H9.
+           elim H11.clear H11.
+           elim H9.clear H9.
+           elim H11.clear H11.
+           cut ((h11 (i/m2) (i\mod m2)*m1+h12 (i/m2) (i\mod m2))/m1 = 
+                 h11 (i/m2) (i\mod m2))
+            [cut ((h11 (i/m2) (i\mod m2)*m1+h12 (i/m2) (i\mod m2))\mod m1 =
+                  h12 (i/m2) (i\mod m2))
+              [rewrite > Hcut3.
+               rewrite > Hcut4.
+               rewrite > H9.
+               rewrite > H15.
+               reflexivity
+              |apply mod_plus_times. 
+               assumption
+              ]
+            |apply div_plus_times.
+             assumption
+            ]
+          |apply lt_mod_m_m.
+           assumption
+          ]
+        |apply (lt_times_n_to_lt m2)
+          [assumption
+          |apply (le_to_lt_to_lt ? i)
+            [apply (eq_plus_to_le ? ? (i \mod m2)).
+             apply div_mod.
+             assumption
+            |assumption
+            ]
+          ]
+        ]
+      |apply not_le_to_lt.unfold.intro.
+       generalize in match H5.
+       apply (le_n_O_elim ? H7).
+       rewrite < times_n_O.
+       apply le_to_not_lt.
+       apply le_O_n  
+      ]      
+    |cut (O < m2)
+      [cut (i/m2 < n2)
+        [cut (i \mod m2 < m2)
+          [elim (and_true ? ? H6).
+           elim (H3 ? ? Hcut1 Hcut2 H7 H8).
+           elim H9.clear H9.
+           elim H11.clear H11.
+           elim H9.clear H9.
+           elim H11.clear H11.
+           cut ((h11 (i/m2) (i\mod m2)*m1+h12 (i/m2) (i\mod m2))/m1 = 
+                 h11 (i/m2) (i\mod m2))
+            [cut ((h11 (i/m2) (i\mod m2)*m1+h12 (i/m2) (i\mod m2))\mod m1 =
+                  h12 (i/m2) (i\mod m2))
+              [rewrite > Hcut3.
+               rewrite > Hcut4.
+               rewrite > H13.
+               rewrite > H14.
+               apply sym_eq.
+               apply div_mod.
+               assumption
+              |apply mod_plus_times. 
+               assumption
+              ]
+            |apply div_plus_times.
+             assumption
+            ]
+          |apply lt_mod_m_m.
+           assumption
+          ]
+        |apply (lt_times_n_to_lt m2)
+          [assumption
+          |apply (le_to_lt_to_lt ? i)
+            [apply (eq_plus_to_le ? ? (i \mod m2)).
+             apply div_mod.
+             assumption
+            |assumption
+            ]
+          ]
+        ]
+      |apply not_le_to_lt.unfold.intro.
+       generalize in match H5.
+       apply (le_n_O_elim ? H7).
+       rewrite < times_n_O.
+       apply le_to_not_lt.
+       apply le_O_n  
+      ]      
+    |cut (O < m2)
+      [cut (i/m2 < n2)
+        [cut (i \mod m2 < m2)
+          [elim (and_true ? ? H6).
+           elim (H3 ? ? Hcut1 Hcut2 H7 H8).
+           elim H9.clear H9.
+           elim H11.clear H11.
+           elim H9.clear H9.
+           elim H11.clear H11.
+           apply lt_times_plus_times
+            [assumption|assumption]
+          |apply lt_mod_m_m.
+           assumption
+          ]
+        |apply (lt_times_n_to_lt m2)
+          [assumption
+          |apply (le_to_lt_to_lt ? i)
+            [apply (eq_plus_to_le ? ? (i \mod m2)).
+             apply div_mod.
+             assumption
+            |assumption
+            ]
+          ]
+        ]
+      |apply not_le_to_lt.unfold.intro.
+       generalize in match H5.
+       apply (le_n_O_elim ? H7).
+       rewrite < times_n_O.
+       apply le_to_not_lt.
+       apply le_O_n  
+      ]
+    |cut (O < m1)
+      [cut (j/m1 < n1)
+        [cut (j \mod m1 < m1)
+          [elim (and_true ? ? H6).
+           elim (H4 ? ? Hcut1 Hcut2 H7 H8).
+           elim H9.clear H9.
+           elim H11.clear H11.
+           elim H9.clear H9.
+           elim H11.clear H11.
+           cut ((h21 (j/m1) (j\mod m1)*m2+h22 (j/m1) (j\mod m1))/m2 = 
+                 h21 (j/m1) (j\mod m1))
+            [cut ((h21 (j/m1) (j\mod m1)*m2+h22 (j/m1) (j\mod m1))\mod m2 =
+                  h22 (j/m1) (j\mod m1))
+              [rewrite > Hcut3.
+               rewrite > Hcut4.
+               rewrite > H9.
+               rewrite > H15.
+               reflexivity
+              |apply mod_plus_times. 
+               assumption
+              ]
+            |apply div_plus_times.
+             assumption
+            ]
+          |apply lt_mod_m_m.
+           assumption
+          ] 
+        |apply (lt_times_n_to_lt m1)
+          [assumption
+          |apply (le_to_lt_to_lt ? j)
+            [apply (eq_plus_to_le ? ? (j \mod m1)).
+             apply div_mod.
+             assumption
+            |assumption
+            ]
+          ]
+        ]
+      |apply not_le_to_lt.unfold.intro.
+       generalize in match H5.
+       apply (le_n_O_elim ? H7).
+       rewrite < times_n_O.
+       apply le_to_not_lt.
+       apply le_O_n  
+      ] 
+    |cut (O < m1)
+      [cut (j/m1 < n1)
+        [cut (j \mod m1 < m1)
+          [elim (and_true ? ? H6).
+           elim (H4 ? ? Hcut1 Hcut2 H7 H8).
+           elim H9.clear H9.
+           elim H11.clear H11.
+           elim H9.clear H9.
+           elim H11.clear H11.
+           cut ((h21 (j/m1) (j\mod m1)*m2+h22 (j/m1) (j\mod m1))/m2 = 
+                 h21 (j/m1) (j\mod m1))
+            [cut ((h21 (j/m1) (j\mod m1)*m2+h22 (j/m1) (j\mod m1))\mod m2 =
+                  h22 (j/m1) (j\mod m1))
+              [rewrite > Hcut3.
+               rewrite > Hcut4.               
+               rewrite > H13.
+               rewrite > H14.
+               apply sym_eq.
+               apply div_mod.
+               assumption
+              |apply mod_plus_times. 
+               assumption
+              ]
+            |apply div_plus_times.
+             assumption
+            ]
+          |apply lt_mod_m_m.
+           assumption
+          ] 
+        |apply (lt_times_n_to_lt m1)
+          [assumption
+          |apply (le_to_lt_to_lt ? j)
+            [apply (eq_plus_to_le ? ? (j \mod m1)).
+             apply div_mod.
+             assumption
+            |assumption
+            ]
+          ]
+        ]
+      |apply not_le_to_lt.unfold.intro.
+       generalize in match H5.
+       apply (le_n_O_elim ? H7).
+       rewrite < times_n_O.
+       apply le_to_not_lt.
+       apply le_O_n  
+      ] 
+    |cut (O < m1)
+      [cut (j/m1 < n1)
+        [cut (j \mod m1 < m1)
+          [elim (and_true ? ? H6).
+           elim (H4 ? ? Hcut1 Hcut2 H7 H8).
+           elim H9.clear H9.
+           elim H11.clear H11.
+           elim H9.clear H9.
+           elim H11.clear H11.
+           apply (lt_times_plus_times ? ? ? m2)
+            [assumption|assumption]
+          |apply lt_mod_m_m.
+           assumption
+          ] 
+        |apply (lt_times_n_to_lt m1)
+          [assumption
+          |apply (le_to_lt_to_lt ? j)
+            [apply (eq_plus_to_le ? ? (j \mod m1)).
+             apply div_mod.
+             assumption
+            |assumption
+            ]
+          ]
+        ]
+      |apply not_le_to_lt.unfold.intro.
+       generalize in match H5.
+       apply (le_n_O_elim ? H7).
+       rewrite < times_n_O.
+       apply le_to_not_lt.
+       apply le_O_n  
+      ]
+    ]
+  ]
+qed.*)
+
 
+theorem iter_p_gen_2_eq: 
+\forall A:Type.
+\forall baseA: A.
+\forall plusA: A \to A \to A. 
+(symmetric A plusA) \to 
+(associative A plusA) \to 
+(\forall a:A.(plusA a  baseA) = a)\to
+\forall g: nat \to nat \to A.
+\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
+iter_p_gen n1 p11 A 
+     (\lambda x:nat .iter_p_gen m1 (p12 x) A (\lambda y. g x y) baseA plusA) 
+     baseA plusA =
+iter_p_gen n2 p21 A 
+    (\lambda x:nat .iter_p_gen m2 (p22 x) A  (\lambda y. g (h11 x y) (h12 x y)) baseA plusA )
+    baseA plusA.
+
+intros.
+rewrite < (iter_p_gen2' ? ? ? ? ? ? ? ? H H1 H2).
+letin ha:= (\lambda x,y.(((h11 x y)*m1) + (h12 x y))).
+letin ha12:= (\lambda x.(h21 (x/m1) (x \mod m1))).
+letin ha22:= (\lambda x.(h22 (x/m1) (x \mod m1))).
+
+apply (trans_eq ? ? 
+(iter_p_gen n2 p21 A (\lambda x:nat. iter_p_gen m2 (p22 x) A
+ (\lambda y:nat.(g (((h11 x y)*m1+(h12 x y))/m1) (((h11 x y)*m1+(h12 x y))\mod m1))) baseA plusA ) baseA plusA))
+[
+  apply (iter_p_gen_knm A baseA plusA H H1 H2 (\lambda e. (g (e/m1) (e \mod m1))) ha ha12 ha22);intros
+  [ elim (and_true ? ? H6).
+    cut(O \lt m1)
+    [ cut(x/m1 < n1)
+      [ cut((x \mod m1) < m1)
+        [ elim (H4 ? ? Hcut1 Hcut2 H7 H8).
+          elim H9.clear H9.
+          elim H11.clear H11.
+          elim H9.clear H9.
+          elim H11.clear H11.
+          split
+          [ split
+            [ split
+              [ split
+                [ assumption
+                | assumption
+                ]
+              | unfold ha.
+                unfold ha12.
+                unfold ha22.
+                rewrite > H14.
+                rewrite > H13.
+                apply sym_eq.
+                apply div_mod.
+                assumption
+              ]
+            | assumption
+            ]
+          | assumption
+          ]
+        | apply lt_mod_m_m.
+          assumption
+        ]
+      | apply (lt_times_n_to_lt m1)
+        [ assumption
+        | apply (le_to_lt_to_lt ? x)
+          [ apply (eq_plus_to_le ? ? (x \mod m1)).
+            apply div_mod.
+            assumption
+          | assumption
+        ]
+      ]  
+    ]
+    | apply not_le_to_lt.unfold.intro.
+      generalize in match H5.
+      apply (le_n_O_elim ? H9).
+      rewrite < times_n_O.
+      apply le_to_not_lt.
+      apply le_O_n.              
+    ]
+  | elim (H3 ? ? H5 H6 H7 H8).
+    elim H9.clear H9.
+    elim H11.clear H11.
+    elim H9.clear H9.
+    elim H11.clear H11.
+    cut(((h11 i j)*m1 + (h12 i j))/m1 = (h11 i j))
+    [ cut(((h11 i j)*m1 + (h12 i j)) \mod m1 = (h12 i j))
+      [ split
+        [ split
+          [ split
+            [ apply true_to_true_to_andb_true
+              [ rewrite > Hcut.
+                assumption
+              | rewrite > Hcut1.
+                rewrite > Hcut.
+                assumption
+              ] 
+            | unfold ha.
+              unfold ha12.
+              rewrite > Hcut1.
+              rewrite > Hcut.
+              assumption
+            ]
+          | unfold ha.
+            unfold ha22.
+            rewrite > Hcut1.
+            rewrite > Hcut.
+            assumption            
+          ]
+        | cut(O \lt m1)
+          [ cut(O \lt n1)      
+            [ apply (lt_to_le_to_lt ? ((h11 i j)*m1 + m1) )
+              [ unfold ha.
+                apply (lt_plus_r).
+                assumption
+              | rewrite > sym_plus.
+                rewrite > (sym_times (h11 i j) m1).
+                rewrite > times_n_Sm.
+                rewrite > sym_times.
+                apply (le_times_l).
+                assumption  
+              ]
+            | apply not_le_to_lt.unfold.intro.
+              generalize in match H12.
+              apply (le_n_O_elim ? H11).       
+              apply le_to_not_lt.
+              apply le_O_n
+            ]
+          | apply not_le_to_lt.unfold.intro.
+            generalize in match H10.
+            apply (le_n_O_elim ? H11).       
+            apply le_to_not_lt.
+            apply le_O_n
+          ]  
+        ]
+      | rewrite > (mod_plus_times m1 (h11 i j) (h12 i j)).
+        reflexivity.
+        assumption
+      ]     
+    | rewrite > (div_plus_times m1 (h11 i j) (h12 i j)).
+      reflexivity.
+      assumption
+    ]
+  ]
+| apply (eq_iter_p_gen1)
+  [ intros. reflexivity 
+  | intros.
+    apply (eq_iter_p_gen1)
+    [ intros. reflexivity
+    | intros.
+      rewrite > (div_plus_times)
+      [ rewrite > (mod_plus_times)
+        [ reflexivity
+        | elim (H3 x x1 H5 H7 H6 H8).
+          assumption
+        ]
+      | elim (H3 x x1 H5 H7 H6 H8).       
+        assumption
+      ]
+    ]
+  ]
+]
+qed.
+
+theorem iter_p_gen_iter_p_gen: 
+\forall A:Type.
+\forall baseA: A.
+\forall plusA: A \to A \to A. 
+(symmetric A plusA) \to 
+(associative A plusA) \to 
+(\forall a:A.(plusA a  baseA) = a)\to
+\forall g: nat \to nat \to A.
+\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
+iter_p_gen n p11 A 
+  (\lambda x:nat.iter_p_gen m (p12 x) A (\lambda y. g x y) baseA plusA) 
+  baseA plusA =
+iter_p_gen m p21 A 
+  (\lambda y:nat.iter_p_gen n (p22 y) A  (\lambda x. g x y) baseA plusA )
+  baseA plusA.
+intros.
+apply (iter_p_gen_2_eq A baseA plusA H H1 H2 (\lambda x,y. g x y) (\lambda x,y.y) (\lambda x,y.x) (\lambda x,y.y) (\lambda x,y.x)
+       n m m n p11 p21 p12 p22)
+  [intros.split
+    [split
+      [split
+        [split
+          [split
+            [apply (andb_true_true ? (p12 j i)).
+             rewrite > H3
+              [rewrite > H6.rewrite > H7.reflexivity
+              |assumption
+              |assumption
+              ]
+            |apply (andb_true_true_r (p11 j)).
+             rewrite > H3
+              [rewrite > H6.rewrite > H7.reflexivity
+              |assumption
+              |assumption
+              ]
+            ]
+          |reflexivity
+          ]
+        |reflexivity
+        ]
+      |assumption
+      ]
+    |assumption
+    ]
+  |intros.split
+    [split
+      [split
+        [split
+          [split
+            [apply (andb_true_true ? (p22 j i)).
+             rewrite < H3
+              [rewrite > H6.rewrite > H7.reflexivity
+              |assumption
+              |assumption
+              ]
+            |apply (andb_true_true_r (p21 j)).
+             rewrite < H3
+              [rewrite > H6.rewrite > H7.reflexivity
+              |assumption
+              |assumption
+              ]
+            ]
+          |reflexivity
+          ]
+        |reflexivity
+        ]
+      |assumption
+      ]
+    |assumption
+    ]
+  ]
+qed.
\ No newline at end of file