Towards chebyshev.

[helm.git] / matita / library / nat / div_and_mod.ma

diff --git a/matita/library/nat/div_and_mod.ma b/matita/library/nat/div_and_mod.ma
index 65ba0e9e5127a77ed8c69bb13f50300a058e3213..0323b18fb17849949b5ce4a49e3b33476f14fcec 100644 (file)
--- a/matita/library/nat/div_and_mod.ma
+++ b/matita/library/nat/div_and_mod.ma
@@ -17,6 +17,7 @@ set "baseuri" "cic:/matita/nat/div_and_mod".
 include "datatypes/constructors.ma".
 include "nat/minus.ma".
 
+
 let rec mod_aux p m n: nat \def
 match (leb m n) with
 [ true \Rightarrow m
@@ -100,6 +101,16 @@ simplify.
 apply div_aux_mod_aux.
 qed.
 
+theorem eq_times_div_minus_mod:
+\forall a,b:nat. O \lt b \to
+(a /b)*b = a - (a \mod b).
+intros.
+rewrite > (div_mod a b) in \vdash (? ? ? (? % ?))
+[ apply (minus_plus_m_m (times (div a b) b) (mod a b))
+| assumption
+]
+qed.
+
 inductive div_mod_spec (n,m,q,r:nat) : Prop \def
 div_mod_spec_intro: r < m \to n=q*m+r \to (div_mod_spec n m q r).
 
@@ -215,11 +226,21 @@ qed.
 (* some properties of div and mod *)
 theorem div_times: \forall n,m:nat. ((S n)*m) / (S n) = m.
 intros.
-apply (div_mod_spec_to_eq ((S n)*m) (S n) ? ? ? O).
-goal 15. (* ?11 is closed with the following tactics *)
-apply div_mod_spec_div_mod.
-unfold lt.apply le_S_S.apply le_O_n.
-apply div_mod_spec_times.
+apply (div_mod_spec_to_eq ((S n)*m) (S n) ? ? ? O);
+[2: apply div_mod_spec_div_mod.
+    unfold lt.apply le_S_S.apply le_O_n.
+|   skip
+|   apply div_mod_spec_times
+]
+qed.
+
+(*a simple variant of div_times theorem *)
+theorem lt_O_to_div_times: \forall a,b:nat. O \lt b \to
+a*b/b = a.
+intros.
+rewrite > sym_times.
+rewrite > (S_pred b H).
+apply div_times.
 qed.
 
 theorem div_n_n: \forall n:nat. O < n \to n / n = S O.
@@ -272,6 +293,63 @@ constructor 1.
 assumption.reflexivity.
 qed.
 
+theorem mod_SO: \forall n:nat. mod n (S O) = O.
+intro.
+apply sym_eq.
+apply le_n_O_to_eq.
+apply le_S_S_to_le.
+apply lt_mod_m_m.
+apply le_n.
+qed.
+
+theorem div_SO: \forall n:nat. div n (S O) = n.
+intro.
+rewrite > (div_mod ? (S O)) in \vdash (? ? ? %)
+  [rewrite > mod_SO.
+   rewrite < plus_n_O.
+   apply times_n_SO
+  |apply le_n
+  ]
+qed.
+
+theorem le_div: \forall n,m. O < n \to m/n \le m.
+intros.
+rewrite > (div_mod m n) in \vdash (? ? %)
+  [apply (trans_le ? (m/n*n))
+    [rewrite > times_n_SO in \vdash (? % ?).
+     apply le_times
+      [apply le_n|assumption]
+    |apply le_plus_n_r
+    ]
+  |assumption
+  ]
+qed.
+
+theorem or_div_mod: \forall n,q. O < q \to
+((S (n \mod q)=q) \land S n = (S (div n q)) * q \lor
+((S (n \mod q)<q) \land S n= (div n q) * q + S (n\mod q))).
+intros.
+elim (le_to_or_lt_eq ? ? (lt_mod_m_m n q H))
+  [right.split
+    [assumption
+    |rewrite < plus_n_Sm.
+     apply eq_f.
+     apply div_mod.
+     assumption
+    ]
+  |left.split
+    [assumption
+    |simplify.
+     rewrite > sym_plus.
+     rewrite < H1 in ⊢ (? ? ? (? ? %)).
+     rewrite < plus_n_Sm.
+     apply eq_f.
+     apply div_mod.
+     assumption
+    ]
+  ]
+qed.
+
 (* injectivity *)
 theorem injective_times_r: \forall n:nat.injective nat nat (\lambda m:nat.(S n)*m).
 change with (\forall n,p,q:nat.(S n)*p = (S n)*q \to p=q).
@@ -330,3 +408,4 @@ let rec n_divides_aux p n m acc \def
 
 (* n_divides n m = <q,r> if m divides n q times, with remainder r *)
 definition n_divides \def \lambda n,m:nat.n_divides_aux n n m O.
+