X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fnat%2Flt_arith.ma;h=683ec262773cbe38541ba1499b6cf1eeb3775440;hb=68dbcd02022874a025a9444aa1125b0458816fbb;hp=e7151472610319dd7a69c2e15ead3e3104f40d6c;hpb=a9f72dea3b74e3c0c33daf5be8f4d5d75611492c;p=helm.git

diff --git a/helm/software/matita/library/nat/lt_arith.ma b/helm/software/matita/library/nat/lt_arith.ma
index e71514726..683ec2627 100644
--- a/helm/software/matita/library/nat/lt_arith.ma
+++ b/helm/software/matita/library/nat/lt_arith.ma
@@ -12,8 +12,6 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/nat/lt_arith".
-
 include "nat/div_and_mod.ma".
 
 (* plus *)
@@ -118,6 +116,17 @@ apply (nat_case1 a)
 ]
 qed.
 
+lemma lt_times_to_lt_O: \forall i,n,m:nat. i < n*m \to O < m.
+intros.
+elim (le_to_or_lt_eq O ? (le_O_n m))
+  [assumption
+  |apply False_ind.
+   rewrite < H1 in H.
+   rewrite < times_n_O in H.
+   apply (not_le_Sn_O ? H)
+  ]
+qed.
+
 (* times *)
 theorem monotonic_lt_times_r: 
 \forall n:nat.monotonic nat lt (\lambda m.(S n)*m).
@@ -239,6 +248,8 @@ apply (nat_case n)
   ]
 qed.
 
+
+
 theorem nat_compare_times_l : \forall n,p,q:nat. 
 nat_compare p q = nat_compare ((S n) * p) ((S n) * q).
 intros.apply nat_compare_elim.intro.
@@ -262,6 +273,26 @@ apply lt_to_not_eq.assumption.
 intro.reflexivity.
 qed.
 
+(* times and plus *)
+theorem lt_times_plus_times: \forall a,b,n,m:nat. 
+a < n \to b < m \to a*m + b < n*m.
+intros 3.
+apply (nat_case n)
+  [intros.apply False_ind.apply (not_le_Sn_O ? H)
+  |intros.simplify.
+   rewrite < sym_plus.
+   unfold.
+   change with (S b+a*m1 \leq m1+m*m1).
+   apply le_plus
+    [assumption
+    |apply le_times
+      [apply le_S_S_to_le.assumption
+      |apply le_n
+      ]
+    ]
+  ]
+qed.
+
 (* div *) 
 
 theorem eq_mod_O_to_lt_O_div: \forall n,m:nat. O < m \to O < n\to n \mod m = O \to O < n / m. 
@@ -300,6 +331,236 @@ apply (trans_lt ? (S O)).
 unfold lt. apply le_n.assumption.
 qed.
 
+theorem eq_div_div_div_times: \forall n,m,q. O < n \to O < m \to
+q/n/m = q/(n*m).
+intros.
+apply (div_mod_spec_to_eq q (n*m) ? (q\mod n+n*(q/n\mod m)) ? (mod q (n*m)))
+  [apply div_mod_spec_intro
+    [apply (lt_to_le_to_lt ? (n*(S (q/n\mod m))))
+      [rewrite < times_n_Sm.
+       apply lt_plus_l.
+       apply lt_mod_m_m.
+       assumption
+      |apply le_times_r.
+       apply lt_mod_m_m.
+       assumption
+      ]
+    |rewrite > sym_times in ⊢ (? ? ? (? (? ? %) ?)).
+     rewrite < assoc_times.
+     rewrite > (eq_times_div_minus_mod ? ? H1).
+     rewrite > sym_times.
+     rewrite > distributive_times_minus.
+     rewrite > sym_times.
+     rewrite > (eq_times_div_minus_mod ? ? H).
+     rewrite < sym_plus in ⊢ (? ? ? (? ? %)).
+     rewrite < assoc_plus.
+     rewrite < plus_minus_m_m
+      [rewrite < plus_minus_m_m
+        [reflexivity
+        |apply (eq_plus_to_le ? ? ((q/n)*n)).
+         rewrite < sym_plus.
+         apply div_mod.
+         assumption
+        ]
+      |apply (trans_le ? (n*(q/n)))
+        [apply le_times_r.
+         apply (eq_plus_to_le ? ? ((q/n)/m*m)).
+         rewrite < sym_plus.
+         apply div_mod.
+         assumption
+        |rewrite > sym_times.
+         rewrite > eq_times_div_minus_mod
+          [apply le_n
+          |assumption
+          ]
+        ]
+      ]
+    ]
+  |apply div_mod_spec_div_mod.
+   rewrite > (times_n_O O).
+   apply lt_times;assumption
+  ]
+qed.
+
+theorem eq_div_div_div_div: \forall n,m,q. O < n \to O < m \to
+q/n/m = q/m/n.
+intros.
+apply (trans_eq ? ? (q/(n*m)))
+  [apply eq_div_div_div_times;assumption
+  |rewrite > sym_times.
+   apply sym_eq.
+   apply eq_div_div_div_times;assumption
+  ]
+qed.
+
+theorem SSO_mod: \forall n,m. O < m \to (S(S O))*n/m = (n/m)*(S(S O)) + mod ((S(S O))*n/m) (S(S O)).
+intros.
+rewrite < (lt_O_to_div_times n (S(S O))) in ⊢ (? ? ? (? (? (? % ?) ?) ?))
+  [rewrite > eq_div_div_div_div
+    [rewrite > sym_times in ⊢ (? ? ? (? (? (? (? % ?) ?) ?) ?)).
+     apply div_mod.
+     apply lt_O_S
+    |apply lt_O_S
+    |assumption
+    ]
+  |apply lt_O_S
+  ]
+qed.
+(* Forall a,b : N. 0 < b \to b * (a/b) <= a < b * (a/b +1) *)
+(* The theorem is shown in two different parts: *)
+
+theorem lt_to_div_to_and_le_times_lt_S: \forall a,b,c:nat.
+O \lt b \to a/b = c \to (b*c \le a \land a \lt b*(S c)).
+intros.
+split
+[ rewrite < H1.
+  rewrite > sym_times.
+  rewrite > eq_times_div_minus_mod
+  [ apply (le_minus_m a (a \mod b))
+  | assumption
+  ]
+| rewrite < (times_n_Sm b c).
+  rewrite < H1.
+  rewrite > sym_times.
+  rewrite > (div_mod a b) in \vdash (? % ?)
+  [ rewrite > (sym_plus b ((a/b)*b)).
+    apply lt_plus_r.
+    apply lt_mod_m_m.
+    assumption
+  | assumption
+  ]
+]
+qed.
+
+theorem lt_to_le_times_to_lt_S_to_div: \forall a,c,b:nat.
+O \lt b \to (b*c) \le a \to a \lt (b*(S c)) \to a/b = c.
+intros.
+apply (le_to_le_to_eq)
+[ apply (leb_elim (a/b) c);intros
+  [ assumption
+  | cut (c \lt (a/b))
+    [ apply False_ind.
+      apply (lt_to_not_le (a \mod b) O)
+      [ apply (lt_plus_to_lt_l ((a/b)*b)).
+        simplify.
+        rewrite < sym_plus.
+        rewrite < div_mod
+        [ apply (lt_to_le_to_lt ? (b*(S c)) ?)
+          [ assumption
+          | rewrite > (sym_times (a/b) b).
+            apply le_times_r.
+            assumption
+          ]
+        | assumption
+        ]
+      | apply le_O_n
+      ]
+    | apply not_le_to_lt.
+      assumption
+    ]
+  ]
+| apply (leb_elim c (a/b));intros
+  [ assumption
+  | cut((a/b) \lt c) 
+    [ apply False_ind.
+      apply (lt_to_not_le (a \mod b) b)
+      [ apply (lt_mod_m_m).
+        assumption
+      | apply (le_plus_to_le ((a/b)*b)).
+        rewrite < (div_mod a b)
+        [ apply (trans_le ? (b*c) ?)
+          [ rewrite > (sym_times (a/b) b).
+            rewrite > (times_n_SO b) in \vdash (? (? ? %) ?).
+            rewrite < distr_times_plus.
+            rewrite > sym_plus.
+            simplify in \vdash (? (? ? %) ?).
+            apply le_times_r.
+            assumption
+          | assumption
+          ]
+        | assumption
+        ]
+      ]
+    | apply not_le_to_lt. 
+      assumption
+    ]
+  ]
+]
+qed.
+
+
+theorem lt_to_lt_to_eq_div_div_times_times: \forall a,b,c:nat. 
+O \lt c \to O \lt b \to (a/b) = (a*c)/(b*c).
+intros.
+apply sym_eq.
+cut (b*(a/b) \le a \land a \lt b*(S (a/b)))
+[ elim Hcut.
+  apply lt_to_le_times_to_lt_S_to_div
+  [ rewrite > (S_pred b)
+    [ rewrite > (S_pred c)
+      [ apply (lt_O_times_S_S)
+      | assumption
+      ]
+    | assumption
+    ]
+  | rewrite > assoc_times.
+    rewrite > (sym_times c (a/b)).
+    rewrite < assoc_times.
+    rewrite > (sym_times (b*(a/b)) c).
+    rewrite > (sym_times a c).
+    apply (le_times_r c (b*(a/b)) a).
+    assumption
+  | rewrite > (sym_times a c).
+    rewrite > (assoc_times ).
+    rewrite > (sym_times c (S (a/b))).
+    rewrite < (assoc_times).
+    rewrite > (sym_times (b*(S (a/b))) c).
+    apply (lt_times_r1 c a (b*(S (a/b))));
+      assumption    
+  ]
+| apply (lt_to_div_to_and_le_times_lt_S)
+  [ assumption
+  | reflexivity
+  ]
+]
+qed.
+
+theorem times_mod: \forall a,b,c:nat.
+O \lt c \to O \lt b \to ((a*c) \mod (b*c)) = c*(a\mod b).
+intros.
+apply (div_mod_spec_to_eq2 (a*c) (b*c) (a/b) ((a*c) \mod (b*c)) (a/b) (c*(a \mod b)))
+[ rewrite > (lt_to_lt_to_eq_div_div_times_times a b c)
+  [ apply div_mod_spec_div_mod.
+    rewrite > (S_pred b)
+    [ rewrite > (S_pred c)
+      [ apply lt_O_times_S_S
+      | assumption
+      ]
+    | assumption
+    ]
+  | assumption
+  | assumption
+  ]
+| apply div_mod_spec_intro
+  [ rewrite > (sym_times b c).
+    apply (lt_times_r1 c)
+    [ assumption
+    | apply (lt_mod_m_m).
+      assumption
+    ]
+  | rewrite < (assoc_times (a/b) b c).
+    rewrite > (sym_times a c).
+    rewrite > (sym_times ((a/b)*b) c).
+    rewrite < (distr_times_plus c ? ?).
+    apply eq_f.
+    apply (div_mod a b).
+    assumption
+  ]
+]
+qed.
+
+
+
 
 (* general properties of functions *)
 theorem monotonic_to_injective: \forall f:nat\to nat.
@@ -322,3 +583,4 @@ increasing f \to injective nat nat f.
 intros.apply monotonic_to_injective.
 apply increasing_to_monotonic.assumption.
 qed.
+