X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fnat%2Flt_arith.ma;h=a568ca408be94412a0203a82abf170709edfaa05;hb=071bb57c99e38a52d2606f7aca47bf56cdeaff53;hp=f0fdbdbd6d6cf9aeecba4b23c6b96d9ea270a735;hpb=06a19bec47845ecffe3bf9d9a95d3d4dadf76861;p=helm.git

diff --git a/helm/software/matita/library/nat/lt_arith.ma b/helm/software/matita/library/nat/lt_arith.ma
index f0fdbdbd6..a568ca408 100644
--- a/helm/software/matita/library/nat/lt_arith.ma
+++ b/helm/software/matita/library/nat/lt_arith.ma
@@ -63,11 +63,61 @@ rewrite > sym_plus.
 rewrite > (sym_plus q).assumption.
 qed.
 
+theorem le_to_lt_to_plus_lt: \forall a,b,c,d:nat.
+a \le c \to b \lt d \to (a + b) \lt (c+d).
+intros.
+cut (a \lt c \lor a = c)
+[ elim Hcut
+  [ apply (lt_plus );
+      assumption
+  | rewrite > H2.
+    apply (lt_plus_r c b d).
+    assumption
+  ]
+| apply le_to_or_lt_eq.
+  assumption
+]
+qed.
+
+
 (* times and zero *)
 theorem lt_O_times_S_S: \forall n,m:nat.O < (S n)*(S m).
 intros.simplify.unfold lt.apply le_S_S.apply le_O_n.
 qed.
 
+theorem lt_times_eq_O: \forall a,b:nat.
+O \lt a \to (a * b) = O \to b = O.
+intros.
+apply (nat_case1 b)
+[ intros.
+  reflexivity
+| intros.
+  rewrite > H2 in H1.
+  rewrite > (S_pred a) in H1
+  [ apply False_ind.
+    apply (eq_to_not_lt O ((S (pred a))*(S m)))
+    [ apply sym_eq.
+      assumption
+    | apply lt_O_times_S_S
+    ]
+  | assumption
+  ]
+]
+qed.
+
+theorem O_lt_times_to_O_lt: \forall a,c:nat.
+O \lt (a * c) \to O \lt a.
+intros.
+apply (nat_case1 a)
+[ intros.
+  rewrite > H1 in H.
+  simplify in H.
+  assumption
+| intros.
+  apply lt_O_S
+]
+qed.
+
 (* times *)
 theorem monotonic_lt_times_r: 
 \forall n:nat.monotonic nat lt (\lambda m.(S n)*m).
@@ -77,6 +127,20 @@ simplify.rewrite < plus_n_O.rewrite < plus_n_O.assumption.
 apply lt_plus.assumption.assumption.
 qed.
 
+(* a simple variant of the previus monotionic_lt_times *)
+theorem monotonic_lt_times_variant: \forall c:nat.
+O \lt c \to monotonic nat lt (\lambda t.(t*c)).
+intros.
+apply (increasing_to_monotonic).
+unfold increasing.
+intros.
+simplify.
+rewrite > sym_plus.
+rewrite > plus_n_O in \vdash (? % ?).
+apply lt_plus_r.
+assumption.
+qed.
+
 theorem lt_times_r: \forall n,p,q:nat. p < q \to (S n) * p < (S n) * q
 \def monotonic_lt_times_r.
 
@@ -198,6 +262,26 @@ 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. 
@@ -236,6 +320,163 @@ apply (trans_lt ? (S O)).
 unfold lt. apply le_n.assumption.
 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.
 monotonic nat lt f \to injective nat nat f.
@@ -257,3 +498,4 @@ increasing f \to injective nat nat f.
 intros.apply monotonic_to_injective.
 apply increasing_to_monotonic.assumption.
 qed.
+