]
qed.
+theorem divides_div: \forall d,n. divides d n \to divides (n/d) n.
+intros.
+apply (witness ? ? d).
+apply sym_eq.
+apply divides_to_div.
+assumption.
+qed.
+
theorem div_div: \forall n,d:nat. O < n \to divides d n \to
n/(n/d) = d.
intros.
[apply (nat_case m)
[intro.apply divides_n_n
|simplify.intros.apply False_ind.
- apply not_eq_true_false.apply sym_eq.assumption
+ apply not_eq_true_false.apply sym_eq.
+ assumption
]
|intros.
apply divides_b_true_to_divides1
absurd (n \divides m).assumption.assumption.
qed.
+theorem divides_to_divides_b_true1 : \forall n,m:nat.
+O < m \to n \divides m \to divides_b n m = true.
+intro.
+elim (le_to_or_lt_eq O n (le_O_n n))
+ [apply divides_to_divides_b_true
+ [assumption|assumption]
+ |apply False_ind.
+ rewrite < H in H2.
+ elim H2.
+ simplify in H3.
+ apply (not_le_Sn_O O).
+ rewrite > H3 in H1.
+ assumption
+ ]
+qed.
+
theorem not_divides_to_divides_b_false: \forall n,m:nat. O < n \to
\lnot(n \divides m) \to (divides_b n m) = false.
intros.
reflexivity.
qed.
+theorem divides_b_div_true:
+\forall d,n. O < n \to
+ divides_b d n = true \to divides_b (n/d) n = true.
+intros.
+apply divides_to_divides_b_true1
+ [assumption
+ |apply divides_div.
+ apply divides_b_true_to_divides.
+ assumption
+ ]
+qed.
+
theorem divides_b_true_to_lt_O: \forall n,m. O < n \to divides_b m n = true \to O < m.
intros.
elim (le_to_or_lt_eq ? ? (le_O_n m))