Digit

# Library Digit

`Require Export ZArithRing.
Require Export Omega.
Require Export Faux.
Section Pdigit.
Variable n : Z.
Hypothesis nMoreThan1 : (1 < n)%Z.
Let nMoreThanOne := Zlt_1_O _ (Zlt_le_weak _ _ nMoreThan1).
Hint Resolve nMoreThanOne: zarith.
Theorem Zpower_nat_less : forall q : nat, (0 < Zpower_nat n q)%Z.
Hint Resolve Zpower_nat_less: zarith.
Theorem Zpower_nat_monotone_S :
forall p : nat, (Zpower_nat n p < Zpower_nat n (S p))%Z.
Theorem Zpower_nat_monotone_lt :
forall p q : nat, p < q -> (Zpower_nat n p < Zpower_nat n q)%Z.
Hint Resolve Zpower_nat_monotone_lt: zarith.
Theorem Zpower_nat_anti_monotone_lt :
forall p q : nat, (Zpower_nat n p < Zpower_nat n q)%Z -> p < q.
Theorem Zpower_nat_monotone_le :
forall p q : nat, p <= q -> (Zpower_nat n p <= Zpower_nat n q)%Z.
Theorem Zpower_nat_anti_monotone_le :
forall p q : nat, (Zpower_nat n p <= Zpower_nat n q)%Z -> p <= q.
Theorem Zpower_nat_anti_eq :
forall p q : nat, Zpower_nat n p = Zpower_nat n q -> p = q.
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.
Definition digit (q : Z) :=
match q with
| Z0 => 0
| Zpos q' => digitAux (Zabs q) 1 (xO q')
| Zneg q' => digitAux (Zabs q) 1 (xO q')
end.
Hint Unfold digit.
Theorem digitAux1 :
forall p r, (Zpower_nat n (S p) * r)%Z = (Zpower_nat n p * (n * r))%Z.
Theorem Zcompare_correct :
forall p q : Z,
match (p ?= q)%Z with
| Datatypes.Gt => (q < p)%Z
| Datatypes.Lt => (p < q)%Z
| Datatypes.Eq => p = q
end.
Theorem digitAuxLess :
forall (v r : Z) (q : positive),
match digitAux v r q with
| S r' => (Zpower_nat n r' * r <= v)%Z
| O => True
end.
Theorem digitLess :
forall q : Z, q <> 0%Z -> (Zpower_nat n (pred (digit q)) <= Zabs q)%Z.
Hint Resolve digitLess: zarith.
Hint Resolve Zmult_gt_0_lt_compat_r Zmult_gt_0_lt_compat_l: zarith.
Fixpoint pos_length (p : positive) : nat :=
match p with
| xH => 0
| xO p' => S (pos_length p')
| xI p' => S (pos_length p')
end.
Theorem digitAuxMore :
forall (v r : Z) (q : positive),
(0 < r)%Z ->
(v < Zpower_nat n (pos_length q) * r)%Z ->
(v < Zpower_nat n (digitAux v r q) * r)%Z.
Theorem pos_length_pow :
forall p : positive, (Zpos p < Zpower_nat n (S (pos_length p)))%Z.
Theorem digitMore : forall q : Z, (Zabs q < Zpower_nat n (digit q))%Z.
Hint Resolve digitMore: zarith.
Theorem digitInv :
forall (q : Z) (r : nat),
(Zpower_nat n (pred r) <= Zabs q)%Z ->
(Zabs q < Zpower_nat n r)%Z -> digit q = r.
Theorem digitO : digit 0 = 0.
Theorem digit1 : digit 1 = 1.
Theorem digit_monotone :
forall p q : Z, (Zabs p <= Zabs q)%Z -> digit p <= digit q.
Hint Resolve digit_monotone: arith.
Theorem digitNotZero : forall q : Z, q <> 0%Z -> 0 < digit q.
Hint Resolve Zlt_gt: zarith.
Theorem digitAdd :
forall (q : Z) (r : nat),
q <> 0%Z -> digit (q * Zpower_nat n r) = digit q + r.
Theorem digit_minus1 : forall p : nat, digit (Zpower_nat n p - 1) = p.
Theorem digit_bound :
forall (x y z : Z) (n : nat),
(Zabs x <= Zabs y)%Z ->
(Zabs y <= Zabs z)%Z -> digit x = n -> digit z = n -> digit y = n.
Theorem digit_abs : forall p : Z, digit (Zabs p) = digit p.
Theorem digit_anti_monotone_lt :
(1 < n)%Z -> forall p q : Z, digit p < digit q -> (Zabs p < Zabs q)%Z.
End Pdigit.
Hint Resolve Zpower_nat_less: zarith.
Hint Resolve Zpower_nat_monotone_lt: zarith.
Hint Resolve Zpower_nat_monotone_le: zarith.
Hint Unfold digit.
Hint Resolve digitLess: zarith.
Hint Resolve digitMore: zarith.
Hint Resolve digit_monotone: arith.
`