X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Flibrary%2FZ%2Fsigma_p.ma;fp=matita%2Flibrary%2FZ%2Fsigma_p.ma;h=71340ac7c21a84523f1aa0fcf4ca5410341b7329;hp=0000000000000000000000000000000000000000;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1

diff --git a/matita/library/Z/sigma_p.ma b/matita/library/Z/sigma_p.ma
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+(**************************************************************************)
+(*       ___	                                                          *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
+(*      ||A||       E.Tassi, S.Zacchiroli                                 *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU Lesser General Public License Version 2.1         *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/Z/sigma_p".
+
+include "Z/times.ma".
+include "nat/primes.ma".
+include "nat/ord.ma".
+include "nat/generic_iter_p.ma".
+
+(* 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. (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)).
+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.
+unfold sigma_p.
+apply true_to_iter_p_gen_Sn.
+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.
+unfold sigma_p.
+apply false_to_iter_p_gen_Sn.
+assumption.
+qed.
+
+theorem eq_sigma_p: \forall p1,p2:nat \to bool.
+\forall g1,g2: nat \to Z.\forall n.
+(\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.
+unfold sigma_p.
+apply eq_iter_p_gen;
+  assumption.
+qed.
+
+theorem eq_sigma_p1: \forall p1,p2:nat \to bool.
+\forall g1,g2: nat \to Z.\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_iter_p_gen1;
+  assumption.
+qed.
+
+theorem sigma_p_false: 
+\forall g: nat \to Z.\forall n.sigma_p n (\lambda x.false) g = O.
+intros.
+unfold sigma_p.
+apply iter_p_gen_false.
+qed.
+
+theorem sigma_p_plus: \forall n,k:nat.\forall p:nat \to bool.
+\forall g: nat \to Z.
+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 (iter_p_gen_plusA 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.
+unfold sigma_p.
+apply (false_to_eq_iter_p_gen);
+  assumption.
+qed.
+
+theorem sigma_p2 : 
+\forall n,m:nat.
+\forall p1,p2:nat \to bool.
+\forall g: nat \to nat \to Z.
+sigma_p (n*m) 
+  (\lambda x.andb (p1 (div x m)) (p2 (mod x m))) 
+  (\lambda x.g (div x m) (mod x m)) =
+sigma_p n p1 
+  (\lambda x.sigma_p m p2 (g x)).
+intros.
+unfold sigma_p.
+apply (iter_p_gen2 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.
+\forall p2:nat \to nat \to bool.
+\forall g: nat \to nat \to Z.
+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 (iter_p_gen2' 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.
+unfold sigma_p.
+apply (iter_p_gen_gi)
+[ apply symmetricZPlus
+| apply associative_Zplus
+| intros.
+  apply Zplus_z_OZ
+| assumption
+| assumption
+]
+qed.
+
+theorem eq_sigma_p_gh: 
+\forall g: nat \to Z.
+\forall h,h1: nat \to nat.\forall n,n1.
+\forall p1,p2:nat \to bool.
+(\forall i. i < n \to p1 i = true \to p2 (h i) = true) \to
+(\forall i. i < n \to p1 i = true \to h1 (h i) = i) \to 
+(\forall i. i < n \to p1 i = true \to h i < n1) \to 
+(\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 p2 g.
+intros.
+unfold sigma_p.
+apply (eq_iter_p_gen_gh 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 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.
+unfold sigma_p.
+apply (iter_p_gen_divides 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.
+apply (distributive_times_plus_iter_p_gen 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.
+(sigma_p n p f) * z = sigma_p n p (\lambda i.(f i)*z).
+intros.
+rewrite < sym_Ztimes.
+rewrite > Ztimes_sigma_pl.
+apply eq_sigma_p
+  [intros.reflexivity
+  |intros.apply sym_Ztimes
+  ]
+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.
+*)
+
+