Section Pdigit.
(* n is the base *)
Variable n : Z.
(* and it is greater or equal to 2 *)
Hypothesis nMoreThan1 : (1 < n)%Z.
Let nMoreThanOne := Zlt_1_O _ (Zlt_le_weak _ _ nMoreThan1).
Hint Resolve nMoreThanOne: zarith.
intros q; elim q; simpl in |- *;
auto with zarith.
Qed.
Hint Resolve Zpower_nat_less: zarith.
intros p; rewrite <- (Zmult_1_l (Zpower_nat n p)); replace (S p) with (1 + p);
[ rewrite Zpower_nat_is_exp | auto with zarith ].
rewrite Zpower_nat_1; auto with zarith.
apply Zmult_gt_0_lt_compat_r; auto with zarith.
apply Z.lt_gt; auto with zarith.
Qed.
intros p q H'; elim H'; simpl in |- *; auto.
apply Zpower_nat_monotone_S.
intros m H H0; apply Z.lt_trans with (1 := H0).
apply Zpower_nat_monotone_S.
Qed.
Hint Resolve Zpower_nat_monotone_lt: zarith.
intros p q H'.
case (le_or_lt q p); auto; (intros H'1; generalize H'; case H'1).
intros H'0; Contradict H'0; auto with zarith.
intros m H'0 H'2; Contradict H'2; auto with zarith.
Qed.
intros p q H'; case (le_lt_or_eq _ _ H'); auto with zarith.
intros H1; rewrite H1; auto with zarith.
Qed.
intros p q H'; case (le_or_lt p q); intros H'0; auto with arith.
Contradict H'; auto with zarith.
Qed.
intros p q H'; apply le_antisym; apply Zpower_nat_anti_monotone_le;
rewrite H'; auto with zarith.
Qed.
(* To compute the number of digits structurally, we suppose that we know already an upper bound q. So we start from q down to 0 to find the bigger exponent r such that n^(r-1) < v *)
Fixpoint digitAux (v r : Z) (q : positive) {struct q} : nat :=
match q with
| xH => 0
| xI q' =>
match (n * r)%Z with
| r' =>
match (r ?= v)%Z with
| Datatypes.Gt => 0
| _ => S (digitAux v r' q')
end
end
| xO q' =>
match (n * r)%Z with
| r' =>
match (r ?= v)%Z with
| Datatypes.Gt => 0
| _ => S (digitAux v r' q')
end
end
end.
(* As we know that log_n q < log_2 q we can define our function digit*)
Definition digit (q : Z) :=
match q with
| Z0 => 0
| Zpos q' => digitAux (Z.abs q) 1 (xO q')
| Zneg q' => digitAux (Z.abs q) 1 (xO q')
end.
Hint Unfold digit.
intros p r; replace (S p) with (1 + p);
[ rewrite Zpower_nat_is_exp | auto with arith ].
rewrite Zpower_nat_1; ring.
Qed.
intros p q; unfold Z.lt in |- *; generalize (Zcompare_EGAL p q);
(CaseEq (p ?= q)%Z; simpl in |- *; auto).
intros H H0; case (Zcompare_Gt_Lt_antisym p q); auto.
Qed.
intros v r q; generalize r; elim q; clear r q; simpl in |- *; auto.
intros q' Rec r; generalize (Zcompare_correct r v); case (r ?= v)%Z; auto.
intros H1; generalize (Rec (n * r)%Z); case (digitAux v (n * r) q').
intros; rewrite H1; rewrite Zpower_nat_O; auto with zarith.
intros r'; rewrite digitAux1; auto.
intros H1; generalize (Rec (n * r)%Z); case (digitAux v (n * r) q').
intros; rewrite Zpower_nat_O; auto with zarith.
apply Z.le_trans with (m := r); auto with zarith.
intros r'; rewrite digitAux1; auto.
intros q' Rec r; generalize (Zcompare_correct r v); case (r ?= v)%Z; auto.
intros H1; generalize (Rec (n * r)%Z); case (digitAux v (n * r) q').
intros; rewrite H1; rewrite Zpower_nat_O; auto with zarith.
intros r'; rewrite digitAux1; auto.
intros H1; generalize (Rec (n * r)%Z); case (digitAux v (n * r) q').
intros; rewrite Zpower_nat_O; auto with zarith.
apply Z.le_trans with (m := r); auto with zarith.
intros r'; rewrite digitAux1; auto.
Qed.
(* digit is correct (first part) *)

Theorem digitLess :
forall q : Z, q <> 0%Z -> (Zpower_nat n (pred (digit q)) <= Z.abs q)%Z.