X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2FZ%2Fsigma_p.ma;h=c2bdb0901852f732e64cf4c026c82f2c71f78735;hb=8ae1653eb75d2b57c50e077c49cb9d078313ea9d;hp=dbd5666c9ff9feea1b5efd23eb8859b86d509a11;hpb=c883f2899a5f0604b65287f913292d049d9081a5;p=helm.git

diff --git a/helm/software/matita/library/Z/sigma_p.ma b/helm/software/matita/library/Z/sigma_p.ma
index dbd5666c9..c2bdb0901 100644
--- a/helm/software/matita/library/Z/sigma_p.ma
+++ b/helm/software/matita/library/Z/sigma_p.ma
@@ -12,16 +12,14 @@
 (*                                                                        *)
 (**************************************************************************)
 
-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".
+include "nat/generic_iter_p.ma".
 
-(* sigma_p in Z is a specialization of sigma_p_gen *)
+(* sigma_p in Z is a specialization of iter_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).
+\lambda n, p, g. (iter_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)).
@@ -36,7 +34,7 @@ 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.
    
@@ -45,7 +43,7 @@ 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.
 
@@ -56,7 +54,7 @@ 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.
 
@@ -67,7 +65,7 @@ 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.
 
@@ -75,7 +73,7 @@ theorem sigma_p_false:
 \forall g: nat \to Z.\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.
@@ -84,10 +82,10 @@ 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 Z n k p g OZ Zplus)
+apply (iter_p_gen_plusA Z n k p g OZ Zplus)
 [ apply symmetricZPlus.
 | intros.
-  apply cic:/matita/Z/plus/Zplus_z_OZ.con
+  apply Zplus_z_OZ.
 | apply associative_Zplus
 ]
 qed.
@@ -98,7 +96,7 @@ 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.
 
@@ -113,7 +111,7 @@ sigma_p n p1
   (\lambda x.sigma_p m p2 (g x)).
 intros.
 unfold sigma_p.
-apply (sigma_p2_gen n m p1 p2 Z g OZ Zplus)
+apply (iter_p_gen2 n m p1 p2 Z g OZ Zplus)
 [ apply symmetricZPlus
 | apply associative_Zplus
 | intros.
@@ -135,7 +133,7 @@ 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 Z g OZ Zplus)
+apply (iter_p_gen2' n m p1 p2 Z g OZ Zplus)
 [ apply symmetricZPlus
 | apply associative_Zplus
 | intros.
@@ -148,7 +146,7 @@ lemma sigma_p_gi: \forall g: nat \to Z.
 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 symmetricZPlus
 | apply associative_Zplus
 | intros.
@@ -168,10 +166,10 @@ 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 Z OZ Zplus ? ? ? g h h1 n n1 p1 p2)
+apply (eq_iter_p_gen_gh Z OZ Zplus ? ? ? g h h1 n n1 p1 p2)
 [ apply symmetricZPlus
 | apply associative_Zplus
 | intros.
@@ -273,7 +271,7 @@ 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 Z OZ Zplus n m p ? ? ? g)
+apply (iter_p_gen_divides Z OZ Zplus n m p ? ? ? g)
 [ assumption
 | assumption
 | assumption
@@ -289,7 +287,7 @@ qed.
 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.
-apply (distributive_times_plus_sigma_p_generic Z Zplus OZ Ztimes n z p f)
+apply (distributive_times_plus_iter_p_gen Z Zplus OZ Ztimes n z p f)
 [ apply symmetricZPlus
 | apply associative_Zplus
 | intros.
@@ -311,4 +309,513 @@ apply eq_sigma_p
   [intros.reflexivity
   |intros.apply sym_Ztimes
   ]
-qed.
\ No newline at end of file
+qed.
+
+
+theorem sigma_p_knm: 
+\forall g: nat \to Z.
+\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 symmetricZPlus
+  |apply associative_Zplus
+  | intro.
+    apply (Zplus_z_OZ a)
+  | exact h11
+  | exact h12
+  | assumption
+  | assumption
+  ]
+qed.
+
+
+theorem sigma_p2_eq: 
+\forall g: nat \to nat \to Z.
+\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 Z OZ Zplus ? ? ? g h11 h12 h21 h22 n1 m1 n2 m2 p11 p21 p12 p22)
+[ apply symmetricZPlus
+| apply associative_Zplus
+| intro.
+  apply (Zplus_z_OZ a)
+| assumption
+| assumption
+]
+qed.
+
+
+
+
+(*
+
+
+
+
+
+rewrite < sigma_p2'.
+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 ? ? 
+(sigma_p n2 p21 (\lambda x:nat. sigma_p m2 (p22 x)
+ (\lambda y:nat.(g (((h11 x y)*m1+(h12 x y))/m1) (((h11 x y)*m1+(h12 x y))\mod m1)) ) ) ))
+[
+  apply (sigma_p_knm (\lambda e. (g (e/m1) (e \mod m1))) ha ha12 ha22);intros
+  [ elim (and_true ? ? H3).
+    cut(O \lt m1)
+    [ cut(x/m1 < n1)
+      [ cut((x \mod m1) < m1)
+        [ elim (H1 ? ? Hcut1 Hcut2 H4 H5).
+          elim H6.clear H6.
+          elim H8.clear H8.
+          elim H6.clear H6.
+          elim H8.clear H8.
+          split
+          [ split
+            [ split
+              [ split
+                [ assumption
+                | assumption
+                ]
+              | rewrite > H11.
+                rewrite > H10.
+                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 H2.
+      apply (le_n_O_elim ? H6).
+      rewrite < times_n_O.
+      apply le_to_not_lt.
+      apply le_O_n.              
+    ]
+  | elim (H ? ? H2 H3 H4 H5).
+    elim H6.clear H6.
+    elim H8.clear H8.
+    elim H6.clear H6.
+    elim H8.clear H8.
+    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
+              ] 
+            | rewrite > Hcut1.
+              rewrite > Hcut.
+              assumption
+            ]
+          | rewrite > Hcut1.
+            rewrite > Hcut.
+            assumption            
+          ]
+        | cut(O \lt m1)
+          [ cut(O \lt n1)      
+            [ apply (lt_to_le_to_lt ? ((h11 i j)*m1 + m1) )
+              [ 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 H9.
+              apply (le_n_O_elim ? H8).       
+              apply le_to_not_lt.
+              apply le_O_n
+            ]
+          | apply not_le_to_lt.unfold.intro.
+            generalize in match H7.
+            apply (le_n_O_elim ? H8).       
+            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_sigma_p1)
+  [ intros. reflexivity
+  | intros.
+    apply (eq_sigma_p1)
+    [ intros. reflexivity
+    | intros.
+      rewrite > (div_plus_times)
+      [ rewrite > (mod_plus_times)
+        [ reflexivity
+        | elim (H x x1 H2 H4 H3 H5).
+          assumption
+        ]
+      | elim (H x x1 H2 H4 H3 H5).       
+        assumption
+      ]
+    ]
+  ]
+]
+qed.
+
+rewrite < sigma_p2' in \vdash (? ? ? %).
+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 ? ? 
+  (sigma_p (n2*m2) (\lambda x:nat.p21 (x/m2)\land p22 (x/m2) (x\mod m2))
+  (\lambda x:nat.g ((h x)/m1) ((h x)\mod m1))))
+  [clear h.clear h1.
+   apply eq_sigma_p1
+    [intros.reflexivity
+    |intros.
+     cut (O < m2)
+      [cut (x/m2 < n2)
+        [cut (x \mod m2 < m2)
+          [elim (and_true ? ? H3).
+           elim (H ? ? Hcut1 Hcut2 H4 H5).
+           elim H6.clear H6.
+           elim H8.clear H8.
+           elim H6.clear H6.
+           elim H8.clear H8.
+           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 H2.
+       apply (le_n_O_elim ? H4).
+       rewrite < times_n_O.
+       apply le_to_not_lt.
+       apply le_O_n  
+      ]      
+    ]
+  |apply (eq_sigma_p_gh ? h h1);intros
+    [cut (O < m2)
+      [cut (i/m2 < n2)
+        [cut (i \mod m2 < m2)
+          [elim (and_true ? ? H3).
+           elim (H ? ? Hcut1 Hcut2 H4 H5).
+           elim H6.clear H6.
+           elim H8.clear H8.
+           elim H6.clear H6.
+           elim H8.clear H8.
+           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 > H6.
+               rewrite > H12.
+               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 H2.
+       apply (le_n_O_elim ? H4).
+       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 ? ? H3).
+           elim (H ? ? Hcut1 Hcut2 H4 H5).
+           elim H6.clear H6.
+           elim H8.clear H8.
+           elim H6.clear H6.
+           elim H8.clear H8.
+           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 > H10.
+               rewrite > H11.
+               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 H2.
+       apply (le_n_O_elim ? H4).
+       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 ? ? H3).
+           elim (H ? ? Hcut1 Hcut2 H4 H5).
+           elim H6.clear H6.
+           elim H8.clear H8.
+           elim H6.clear H6.
+           elim H8.clear H8.
+           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 H2.
+       apply (le_n_O_elim ? H4).
+       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 ? ? H3).
+           elim (H1 ? ? Hcut1 Hcut2 H4 H5).
+           elim H6.clear H6.
+           elim H8.clear H8.
+           elim H6.clear H6.
+           elim H8.clear H8.
+           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 > H6.
+               rewrite > H12.
+               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 H2.
+       apply (le_n_O_elim ? H4).
+       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 ? ? H3).
+           elim (H1 ? ? Hcut1 Hcut2 H4 H5).
+           elim H6.clear H6.
+           elim H8.clear H8.
+           elim H6.clear H6.
+           elim H8.clear H8.
+           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 > H10.
+               rewrite > H11.
+               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 H2.
+       apply (le_n_O_elim ? H4).
+       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 ? ? H3).
+           elim (H1 ? ? Hcut1 Hcut2 H4 H5).
+           elim H6.clear H6.
+           elim H8.clear H8.
+           elim H6.clear H6.
+           elim H8.clear H8.
+           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 H2.
+       apply (le_n_O_elim ? H4).
+       rewrite < times_n_O.
+       apply le_to_not_lt.
+       apply le_O_n  
+      ]
+    ]
+  ]
+qed.
+*)
+
+