The Coq Proof Assistant

# Library Coq.Reals.Cauchy.QExtra

Require Import QArith.
Require Import Qpower.
Require Import Qabs.
Require Import Qround.
Require Import Lia.
Require Import Lqa. Require Import PosExtra.

# Power of 2 open and closed upper and lower bounds for q : Q

Fixpoint Pos_log2floor_plus1 (p : positive) : positive :=
match p with
| (p'~1)%positive => Pos.succ (Pos_log2floor_plus1 p')
| (p'~0)%positive => Pos.succ (Pos_log2floor_plus1 p')
| 1%positive => 1
end.
Lemma Pos_log2floor_plus1_spec : forall (p : positive),
(Pos.pow 2 (Pos_log2floor_plus1 p) <= 2 * p < 2 * Pos.pow 2 (Pos_log2floor_plus1 p))%positive.
Fixpoint Pos_log2ceil_plus1 (p : positive) : positive :=
match p with
| (p'~1)%positive => Pos.succ (Pos.succ (Pos_log2floor_plus1 p'))
| (p'~0)%positive => Pos.succ (Pos_log2ceil_plus1 p')
| 1%positive => 1
end.
Lemma Pos_log2ceil_plus1_spec : forall (p : positive),
(Pos.pow 2 (Pos_log2ceil_plus1 p) < 4 * p <= 2 * Pos.pow 2 (Pos_log2ceil_plus1 p))%positive.
Fixpoint Pos_is_pow2 (p : positive) : bool :=
match p with
| (p'~1)%positive => false
| (p'~0)%positive => Pos_is_pow2 p'
| 1%positive => true
end.

## Power of two closed upper bound q <= 2^z

Definition Qbound_le_ZExp2 (q : Q) : Z :=
match Qnum q with
| Z0 => -1000
| Zneg p => 0
| Zpos p => (Z.pos (Pos_log2ceil_plus1 p) - Z.pos (Pos_log2floor_plus1 (Qden q)))%Z
end.
Lemma Qbound_le_ZExp2_spec : forall (q : Q),
(q <= 2^(Qbound_le_ZExp2 q))%Q.
Definition Qbound_leabs_ZExp2 (q : Q) : Z := Qbound_le_ZExp2 (Qabs q).
Lemma Qbound_leabs_ZExp2_spec : forall (q : Q),
(Qabs q <= 2^(Qbound_leabs_ZExp2 q))%Q.

## Power of two open upper bound q < 2^z and Qabs q < 2^z

Compute a z such that q<2^z. z shall be close to as small as possible, but we need a compromise between the tighness of the bound and the computation speed and proof complexity. Looking just at the log2 of the numerator and denominator, this is a tight bound except when the numerator is a power of 2 and the denomintor is not. E.g. this return 4/5 < 2^1 instead of 4/5< 2^0. If q==0, we return -1000, because as binary integer this has just 10 bits but 2^-1000 should be smaller than almost any number in practice. If numbers are much smaller, computations might be inefficient.
Definition Qbound_lt_ZExp2 (q : Q) : Z :=
match Qnum q with
| Z0 => -1000
| Zneg p => 0
| Zpos p => Z.pos_sub (Pos.succ (Pos_log2floor_plus1 p)) (Pos_log2floor_plus1 (Qden q))
end.
Remark Qbound_lt_ZExp2_test_1 : Qbound_lt_ZExp2 (4#4) = 1%Z.
Remark Qbound_lt_ZExp2_test_2 : Qbound_lt_ZExp2 (5#4) = 1%Z.
Remark Qbound_lt_ZExp2_test_3 : Qbound_lt_ZExp2 (4#5) = 1%Z.
Remark Qbound_lt_ZExp2_test_4 : Qbound_lt_ZExp2 (7#5) = 1%Z.
Lemma Qbound_lt_ZExp2_spec : forall (q : Q),
(q < 2^(Qbound_lt_ZExp2 q))%Q.
Definition Qbound_ltabs_ZExp2 (q : Q) : Z := Qbound_lt_ZExp2 (Qabs q).
Lemma Qbound_ltabs_ZExp2_spec : forall (q : Q),
(Qabs q < 2^(Qbound_ltabs_ZExp2 q))%Q.

## Power of 2 open lower bounds for 2^z < q and 2^z < Qabs q

Note: the -2 is required cause of the Qlt limit. In case q is a power of two, the lower and upper bound must be a factor of 4 apart
Definition Qlowbound_ltabs_ZExp2 (q : Q) : Z := -2 + Qbound_ltabs_ZExp2 q.
Lemma Qlowbound_ltabs_ZExp2_inv: forall q : Q,
q > 0
-> Qlowbound_ltabs_ZExp2 q = (- Qbound_ltabs_ZExp2 (/q))%Z.
Lemma Qlowbound_ltabs_ZExp2_opp: forall q : Q,
(Qlowbound_ltabs_ZExp2 q = Qlowbound_ltabs_ZExp2 (-q))%Z.
Lemma Qlowbound_lt_ZExp2_spec : forall (q : Q) (Hqgt0 : q > 0),
(2^(Qlowbound_ltabs_ZExp2 q) < q)%Q.
Lemma Qlowbound_ltabs_ZExp2_spec : forall (q : Q) (Hqgt0 : ~ q == 0),
(2^(Qlowbound_ltabs_ZExp2 q) < Qabs q)%Q.

## Existential formulations of power of 2 lower and upper bounds

Definition QarchimedeanExp2_Z (q : Q)
: {z : Z | (q < 2^z)%Q}
:= exist _ (Qbound_lt_ZExp2 q) (Qbound_lt_ZExp2_spec q).
Definition QarchimedeanAbsExp2_Z (q : Q)
: {z : Z | (Qabs q < 2^z)%Q}
:= exist _ (Qbound_ltabs_ZExp2 q) (Qbound_ltabs_ZExp2_spec q).
Definition QarchimedeanLowExp2_Z (q : Q) (Hqgt0 : q > 0)
: {z : Z | (2^z < q)%Q}
:= exist _ (Qlowbound_ltabs_ZExp2 q) (Qlowbound_lt_ZExp2_spec q Hqgt0).
Definition QarchimedeanLowAbsExp2_Z (q : Q) (Hqgt0 : ~ q == 0)
: {z : Z | (2^z < Qabs q)%Q}
:= exist _ (Qlowbound_ltabs_ZExp2 q) (Qlowbound_ltabs_ZExp2_spec q Hqgt0).