Library Coq.micromega.ZMicromega
Require Import OrderedRing.
Require Import RingMicromega.
Require Import ZCoeff.
Require Import Refl.
Require Import ZArith.
Require Import List.
Require Import Bool.
Ltac flatten_bool :=
repeat match goal with
[
id : (
_ && _)%
bool = true |-
_ ] =>
destruct (
andb_prop _ _ id);
clear id
| [
id : (
_ || _)%
bool = true |-
_ ] =>
destruct (
orb_prop _ _ id);
clear id
end.
Ltac inv H :=
inversion H ;
try subst ;
clear H.
Require Import EnvRing.
Open Scope Z_scope.
Lemma Zsor :
SOR 0 1
Zplus Zmult Zminus Zopp (@
eq Z)
Zle Zlt.
Lemma ZSORaddon :
SORaddon 0 1
Zplus Zmult Zminus Zopp (@
eq Z)
Zle
0%
Z 1%
Z Zplus Zmult Zminus Zopp
Zeq_bool Zle_bool
(
fun x =>
x) (
fun x =>
x) (
pow_N 1
Zmult).
Fixpoint Zeval_expr (
env :
PolEnv Z) (
e:
PExpr Z) :
Z :=
match e with
|
PEc c =>
c
|
PEX x =>
env x
|
PEadd e1 e2 =>
Zeval_expr env e1 + Zeval_expr env e2
|
PEmul e1 e2 =>
Zeval_expr env e1 * Zeval_expr env e2
|
PEpow e1 n =>
Zpower (
Zeval_expr env e1) (
Z_of_N n)
|
PEsub e1 e2 =>
(Zeval_expr env e1) - (Zeval_expr env e2)
|
PEopp e =>
Zopp (
Zeval_expr env e)
end.
Definition eval_expr :=
eval_pexpr Zplus Zmult Zminus Zopp (
fun x =>
x) (
fun x =>
x) (
pow_N 1
Zmult).
Lemma ZNpower :
forall r n,
r ^ Z_of_N n = pow_N 1
Zmult r n.
Lemma Zeval_expr_compat :
forall env e,
Zeval_expr env e = eval_expr env e.
Definition Zeval_op2 (
o :
Op2) :
Z ->
Z ->
Prop :=
match o with
|
OpEq => @
eq Z
|
OpNEq =>
fun x y =>
~ x = y
|
OpLe =>
Zle
|
OpGe =>
Zge
|
OpLt =>
Zlt
|
OpGt =>
Zgt
end.
Definition Zeval_formula (
env :
PolEnv Z) (
f :
Formula Z):=
let (
lhs,
op,
rhs) :=
f in
(
Zeval_op2 op) (
Zeval_expr env lhs) (
Zeval_expr env rhs).
Definition Zeval_formula' :=
eval_formula Zplus Zmult Zminus Zopp (@
eq Z)
Zle Zlt (
fun x =>
x) (
fun x =>
x) (
pow_N 1
Zmult).
Lemma Zeval_formula_compat :
forall env f,
Zeval_formula env f <-> Zeval_formula' env f.
Definition eval_nformula :=
eval_nformula 0
Zplus Zmult (@
eq Z)
Zle Zlt (
fun x =>
x) .
Definition Zeval_op1 (
o :
Op1) :
Z ->
Prop :=
match o with
|
Equal =>
fun x :
Z =>
x = 0
|
NonEqual =>
fun x :
Z =>
x <> 0
|
Strict =>
fun x :
Z => 0
< x
|
NonStrict =>
fun x :
Z => 0
<= x
end.
Lemma Zeval_nformula_dec :
forall env d,
(eval_nformula env d) \/ ~ (eval_nformula env d).
Definition ZWitness :=
Psatz Z.
Definition ZWeakChecker :=
check_normalised_formulas 0 1
Zplus Zmult Zeq_bool Zle_bool.
Lemma ZWeakChecker_sound :
forall (
l :
list (
NFormula Z)) (
cm :
ZWitness),
ZWeakChecker l cm = true ->
forall env,
make_impl (
eval_nformula env)
l False.
Definition psub :=
psub Z0 Zplus Zminus Zopp Zeq_bool.
Definition padd :=
padd Z0 Zplus Zeq_bool.
Definition norm :=
norm 0 1
Zplus Zmult Zminus Zopp Zeq_bool.
Definition eval_pol :=
eval_pol 0
Zplus Zmult (
fun x =>
x).
Lemma eval_pol_sub :
forall env lhs rhs,
eval_pol env (
psub lhs rhs)
= eval_pol env lhs - eval_pol env rhs.
Lemma eval_pol_add :
forall env lhs rhs,
eval_pol env (
padd lhs rhs)
= eval_pol env lhs + eval_pol env rhs.
Lemma eval_pol_norm :
forall env e,
eval_expr env e = eval_pol env (
norm e) .
Definition xnormalise (
t:
Formula Z) :
list (
NFormula Z) :=
let (
lhs,
o,
rhs) :=
t in
let lhs :=
norm lhs in
let rhs :=
norm rhs in
match o with
|
OpEq =>
((psub lhs (
padd rhs (
Pc 1))
),NonStrict)::((psub rhs (
padd lhs (
Pc 1))
),NonStrict)::nil
|
OpNEq =>
(psub lhs rhs,Equal) :: nil
|
OpGt =>
(psub rhs lhs,NonStrict) :: nil
|
OpLt =>
(psub lhs rhs,NonStrict) :: nil
|
OpGe =>
(psub rhs (
padd lhs (
Pc 1))
,NonStrict) :: nil
|
OpLe =>
(psub lhs (
padd rhs (
Pc 1))
,NonStrict) :: nil
end.
Require Import Tauto.
Definition normalise (
t:
Formula Z) :
cnf (
NFormula Z) :=
List.map (
fun x =>
x::nil) (
xnormalise t).
Lemma normalise_correct :
forall env t,
eval_cnf (
eval_nformula env) (
normalise t)
<-> Zeval_formula env t.
Opaque padd.
Transparent padd.
Definition xnegate (
t:
RingMicromega.Formula Z) :
list (
NFormula Z) :=
let (
lhs,
o,
rhs) :=
t in
let lhs :=
norm lhs in
let rhs :=
norm rhs in
match o with
|
OpEq =>
(psub lhs rhs,Equal) :: nil
|
OpNEq =>
((psub lhs (
padd rhs (
Pc 1))
),NonStrict)::((psub rhs (
padd lhs (
Pc 1))
),NonStrict)::nil
|
OpGt =>
(psub lhs (
padd rhs (
Pc 1))
,NonStrict) :: nil
|
OpLt =>
(psub rhs (
padd lhs (
Pc 1))
,NonStrict) :: nil
|
OpGe =>
(psub lhs rhs,NonStrict) :: nil
|
OpLe =>
(psub rhs lhs,NonStrict) :: nil
end.
Definition negate (
t:
RingMicromega.Formula Z) :
cnf (
NFormula Z) :=
List.map (
fun x =>
x::nil) (
xnegate t).
Lemma negate_correct :
forall env t,
eval_cnf (
eval_nformula env) (
negate t)
<-> ~ Zeval_formula env t.
Opaque padd.
Transparent padd.
Definition ZweakTautoChecker (
w:
list ZWitness) (
f :
BFormula (
Formula Z)) :
bool :=
@
tauto_checker (
Formula Z) (
NFormula Z)
normalise negate ZWitness ZWeakChecker f w.
Require Import Zdiv.
Open Scope Z_scope.
Definition ceiling (
a b:
Z) :
Z :=
let (
q,
r) :=
Zdiv_eucl a b in
match r with
|
Z0 =>
q
|
_ =>
q + 1
end.
Lemma narrow_interval_lower_bound :
forall a b x,
a > 0 ->
a * x >= b ->
x >= ceiling b a.
NB: narrow_interval_upper_bound is Zdiv.Zdiv_le_lower_bound
Require Import QArith.
Inductive ZArithProof :
Type :=
|
DoneProof
|
RatProof :
ZWitness ->
ZArithProof ->
ZArithProof
|
CutProof :
ZWitness ->
ZArithProof ->
ZArithProof
|
EnumProof :
ZWitness ->
ZWitness ->
list ZArithProof ->
ZArithProof.
Require Import Znumtheory.
Definition isZ0 (
x:
Z) :=
match x with
|
Z0 =>
true
|
_ =>
false
end.
Lemma isZ0_0 :
forall x,
isZ0 x = true <-> x = 0.
Lemma isZ0_n0 :
forall x,
isZ0 x = false <-> x <> 0.
Definition ZgcdM (
x y :
Z) :=
Zmax (
Zgcd x y) 1.
Fixpoint Zgcd_pol (
p :
PolC Z) : (
Z * Z) :=
match p with
|
Pc c =>
(0
,c)
|
Pinj _ p =>
Zgcd_pol p
|
PX p _ q =>
let (
g1,
c1) :=
Zgcd_pol p in
let (
g2,
c2) :=
Zgcd_pol q in
(ZgcdM (
ZgcdM g1 c1)
g2 , c2)
end.
Fixpoint Zdiv_pol (
p:
PolC Z) (
x:
Z) :
PolC Z :=
match p with
|
Pc c =>
Pc (
Zdiv c x)
|
Pinj j p =>
Pinj j (
Zdiv_pol p x)
|
PX p j q =>
PX (
Zdiv_pol p x)
j (
Zdiv_pol q x)
end.
Inductive Zdivide_pol (
x:
Z):
PolC Z ->
Prop :=
|
Zdiv_Pc :
forall c,
(x | c) ->
Zdivide_pol x (
Pc c)
|
Zdiv_Pinj :
forall p,
Zdivide_pol x p ->
forall j,
Zdivide_pol x (
Pinj j p)
|
Zdiv_PX :
forall p q,
Zdivide_pol x p ->
Zdivide_pol x q ->
forall j,
Zdivide_pol x (
PX p j q).
Lemma Zdiv_pol_correct :
forall a p, 0
< a ->
Zdivide_pol a p ->
forall env,
eval_pol env p = a * eval_pol env (
Zdiv_pol p a).
Lemma Zgcd_pol_ge :
forall p,
fst (
Zgcd_pol p)
>= 0.
Lemma Zdivide_pol_Zdivide :
forall p x y,
Zdivide_pol x p ->
(y | x) ->
Zdivide_pol y p.
Lemma Zdivide_pol_one :
forall p,
Zdivide_pol 1
p.
Lemma Zgcd_minus :
forall a b c,
(a | c - b ) ->
(Zgcd a b | c).
Lemma Zdivide_pol_sub :
forall p a b,
0
< Zgcd a b ->
Zdivide_pol a (
PsubC Zminus p b) ->
Zdivide_pol (
Zgcd a b)
p.
Lemma Zdivide_pol_sub_0 :
forall p a,
Zdivide_pol a (
PsubC Zminus p 0) ->
Zdivide_pol a p.
Lemma Zgcd_pol_div :
forall p g c,
Zgcd_pol p = (g, c) ->
Zdivide_pol g (
PsubC Zminus p c).
Lemma Zgcd_pol_correct_lt :
forall p env g c,
Zgcd_pol p = (g,c) -> 0
< g ->
eval_pol env p = g * (eval_pol env (
Zdiv_pol (
PsubC Zminus p c)
g)
) + c.
Definition makeCuttingPlane (
p :
PolC Z) :
PolC Z * Z :=
let (
g,
c) :=
Zgcd_pol p in
if Zgt_bool g Z0
then (Zdiv_pol (
PsubC Zminus p c)
g , Zopp (
ceiling (
Zopp c)
g)
)
else (p,Z0).
Definition genCuttingPlane (
f :
NFormula Z) :
option (
PolC Z * Z * Op1) :=
let (
e,
op) :=
f in
match op with
|
Equal =>
let (
g,
c) :=
Zgcd_pol e in
if andb (
Zgt_bool g Z0) (
andb (
Zgt_bool c Z0) (
negb (
Zeq_bool (
Zgcd g c)
g)))
then None
else Some (e, Z0,op)
|
NonEqual =>
Some (e,Z0,op)
|
Strict =>
let (
p,
c) :=
makeCuttingPlane (
PsubC Zminus e 1)
in
Some (p,c,NonStrict)
|
NonStrict =>
let (
p,
c) :=
makeCuttingPlane e in
Some (p,c,NonStrict)
end.
Definition nformula_of_cutting_plane (
t :
PolC Z * Z * Op1) :
NFormula Z :=
let (
e_z,
o) :=
t in
let (
e,
z) :=
e_z in
(padd e (
Pc z)
, o).
Definition is_pol_Z0 (
p :
PolC Z) :
bool :=
match p with
|
Pc Z0 =>
true
|
_ =>
false
end.
Lemma is_pol_Z0_eval_pol :
forall p,
is_pol_Z0 p = true ->
forall env,
eval_pol env p = 0.
Definition eval_Psatz :
list (
NFormula Z) ->
ZWitness ->
option (
NFormula Z) :=
eval_Psatz 0 1
Zplus Zmult Zeq_bool Zle_bool.
Definition check_inconsistent :=
check_inconsistent 0
Zeq_bool Zle_bool.
Fixpoint ZChecker (
l:
list (
NFormula Z)) (
pf :
ZArithProof) {
struct pf} :
bool :=
match pf with
|
DoneProof =>
false
|
RatProof w pf =>
match eval_Psatz l w with
|
None =>
false
|
Some f =>
if check_inconsistent f then true
else ZChecker (
f::l)
pf
end
|
CutProof w pf =>
match eval_Psatz l w with
|
None =>
false
|
Some f =>
match genCuttingPlane f with
|
None =>
true
|
Some cp =>
ZChecker (
nformula_of_cutting_plane cp::l)
pf
end
end
|
EnumProof w1 w2 pf =>
match eval_Psatz l w1 ,
eval_Psatz l w2 with
|
Some f1 ,
Some f2 =>
match genCuttingPlane f1 ,
genCuttingPlane f2 with
|
Some (e1,z1,op1) ,
Some (e2,z2,op2) =>
match op1 ,
op2 with
|
NonStrict ,
NonStrict =>
if is_pol_Z0 (
padd e1 e2)
then
(
fix label (
pfs:
list ZArithProof) :=
fun lb ub =>
match pfs with
|
nil =>
if Zgt_bool lb ub then true else false
|
pf::rsr =>
andb (
ZChecker (
(psub e1 (
Pc lb)
, Equal) :: l)
pf) (
label rsr (
Zplus lb 1%
Z)
ub)
end)
pf (
Zopp z1)
z2
else false
|
_ ,
_ =>
false
end
|
_ ,
_ =>
false
end
|
_ ,
_ =>
false
end
end.
Fixpoint bdepth (
pf :
ZArithProof) :
nat :=
match pf with
|
DoneProof =>
O
|
RatProof _ p =>
S (
bdepth p)
|
CutProof _ p =>
S (
bdepth p)
|
EnumProof _ _ l =>
S (
List.fold_right (
fun pf x =>
Max.max (
bdepth pf)
x)
O l)
end.
Require Import Wf_nat.
Lemma in_bdepth :
forall l a b y,
In y l ->
ltof ZArithProof bdepth y (
EnumProof a b l).
Lemma eval_Psatz_sound :
forall env w l f',
make_conj (
eval_nformula env)
l ->
eval_Psatz l w = Some f' ->
eval_nformula env f'.
Lemma makeCuttingPlane_sound :
forall env e e' c,
eval_nformula env (e, NonStrict) ->
makeCuttingPlane e = (e',c) ->
eval_nformula env (
nformula_of_cutting_plane (e', c, NonStrict)).
Lemma cutting_plane_sound :
forall env f p,
eval_nformula env f ->
genCuttingPlane f = Some p ->
eval_nformula env (
nformula_of_cutting_plane p).
Lemma genCuttingPlaneNone :
forall env f,
genCuttingPlane f = None ->
eval_nformula env f ->
False.
Lemma ZChecker_sound :
forall w l,
ZChecker l w = true ->
forall env,
make_impl (
eval_nformula env)
l False.
Definition ZTautoChecker (
f :
BFormula (
Formula Z)) (
w:
list ZArithProof):
bool :=
@
tauto_checker (
Formula Z) (
NFormula Z)
normalise negate ZArithProof ZChecker f w.
Lemma ZTautoChecker_sound :
forall f w,
ZTautoChecker f w = true ->
forall env,
eval_f (
Zeval_formula env)
f.
Fixpoint xhyps_of_pt (
base:
nat) (
acc :
list nat) (
pt:
ZArithProof) :
list nat :=
match pt with
|
DoneProof =>
acc
|
RatProof c pt =>
xhyps_of_pt (
S base ) (
xhyps_of_psatz base acc c)
pt
|
CutProof c pt =>
xhyps_of_pt (
S base ) (
xhyps_of_psatz base acc c)
pt
|
EnumProof c1 c2 l =>
let acc :=
xhyps_of_psatz base (
xhyps_of_psatz base acc c2)
c1 in
List.fold_left (
xhyps_of_pt (
S base))
l acc
end.
Definition hyps_of_pt (
pt :
ZArithProof) :
list nat :=
xhyps_of_pt 0
nil pt.
Open Scope Z_scope.
To ease bindings from ml code