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compute_sturm: THEORY
BEGIN
IMPORTING sturm,
remainder_sequence,
clear_denominators,
reals@RealInt,
reals@more_polynomial_props,
reals@real_orders,
reals@real_order_ep
a : VAR [nat->int]
n : VAR posnat
I : VAR RealInt
x,y,z: VAR real
%%% Support Lemmas %%%
finite_bij_real_remove_one: LEMMA
FORALL (m:nat,A:set[real],bij:[below(m)->(A)]):
bijective?(bij) AND A(x) IMPLIES
m-1>=0 AND EXISTS (bijec:[below(m-1)->{r:(A)|r/=x}]):
bijective?(bijec)
finite_bij_real_remove_two: LEMMA
FORALL (m:nat,A:set[real],bij:[below(m)->(A)]):
bijective?(bij) AND A(x) AND A(y) AND x/=y IMPLIES
m-2>=0 AND EXISTS (bijec:[below(m-2)->{r:(A)|r/=x AND r/=y}]):
bijective?(bijec)
computed_sturm_spec: LEMMA a(n)/=0 AND (FORALL (i:nat): i>n IMPLIES a(i)=0) IMPLIES
LET sl = sturm_chain(a,n),
P: [nat->[nat->int]] = (LAMBDA (j:nat): IF j<length(sl)
THEN list2array[int](0)(nth(sl,length(sl)-1-j))
ELSE (LAMBDA (i:nat): 0) ENDIF),
N: [nat->nat] = (LAMBDA (j:nat): IF j<length(sl)
THEN max(length(nth(sl,length(sl)-1-j))-1,0) ELSE 0 ENDIF),
M: nat = length(sl)-1
IN constructed_sturm_sequence?(P,N,M) AND P(0)=a AND N(0)=n
Eq_computed_remainder?(a,(n|a(n)/=0))(sl:list[list[int]]): bool =
sl = sturm_chain(LAMBDA (i:nat): IF i<=n THEN a(i) ELSE 0 ENDIF,n)
% Standard but possibly unbounded interval %
lower_bound,upper_bound: VAR bool
% Formalizatin brings in complications with evaluating at infinity
roots_closed_int(a,(n|a(n)/=0),low:real,(high:real|low<high),lower_bound,upper_bound:bool,
sl:(Eq_computed_remainder?(a,n))):
int =
LET newa = (LAMBDA (i:nat): IF i<=n THEN a(i) ELSE 0 ENDIF),
P: [nat->[nat->int]] = (LAMBDA (j:nat): IF j<length(sl)
THEN list2array[int](0)(nth(sl,length(sl)-1-j))
ELSE (LAMBDA (i:nat): 0) ENDIF),
N: [nat->nat] = (LAMBDA (j:nat): IF j<length(sl)
THEN max(length(nth(sl,length(sl)-1-j))-1,0) ELSE 0 ENDIF),
M: nat = length(sl)-1,
nscnorm = LAMBDA (xyz:real): number_sign_changes(
LAMBDA (i:nat): polynomial(P(i),N(i))(xyz),M-1),
nschighlow = LAMBDA (b:bool): IF b THEN number_sign_changes(LAMBDA (i:nat): P(i)(N(i)),M-1)
ELSE number_sign_changes(LAMBDA (i:nat): (-1)^N(i)*P(i)(N(i)),M-1) ENDIF,
newlow = IF ((NOT lower_bound) OR polynomial(P(0),N(0))(low)/=0
OR polynomial(P(1),N(1))(low)/=0) THEN low
ELSE low-poly_rootless_width(a,n,low,TRUE)/2 ENDIF,
newhigh = IF ((NOT upper_bound) OR polynomial(P(0),N(0))(high)/=0
OR polynomial(P(1),N(1))(high)/=0) THEN high
ELSE high+poly_rootless_width(a,n,high,TRUE)/2 ENDIF,
Nroots = IF lower_bound and upper_bound THEN nscnorm(newlow)`num-nscnorm(newhigh)`num
ELSIF upper_bound THEN nschighlow(FALSE)`num-nscnorm(newhigh)`num
ELSIF lower_bound THEN nscnorm(newlow)`num-nschighlow(TRUE)`num
ELSE nschighlow(FALSE)`num-nschighlow(TRUE)`num ENDIF,
adjlow = IF lower_bound AND polynomial(P(0),N(0))(low)=0 AND polynomial(P(1),N(1))(low)/=0
THEN 1 ELSE 0 ENDIF
IN Nroots + adjlow
roots_closed_int_def_truetrue: LEMMA FORALL (sl:(Eq_computed_remainder?(a,n))): a(n)/=0 AND x<y IMPLIES
LET rootnum = roots_closed_int(a,n,x,y,true,true,sl),
Aset = {z:real|x<=z AND z<=y AND polynomial(a,n)(z)=0}
IN rootnum>=0 AND EXISTS (bij: [below(rootnum)->(Aset)]): bijective?(bij)
roots_closed_int_def_falsetrue: LEMMA FORALL (sl:(Eq_computed_remainder?(a,n))): a(n)/=0 AND x<y IMPLIES
LET rootnum = roots_closed_int(a,n,x,y,false,true,sl),
Aset = {z:real|z<=y AND polynomial(a,n)(z)=0}
IN rootnum>=0 AND EXISTS (bij: [below(rootnum)->(Aset)]): bijective?(bij)
roots_closed_int_def_truefalse: LEMMA FORALL (sl:(Eq_computed_remainder?(a,n))): a(n)/=0 AND x<y IMPLIES
LET rootnum = roots_closed_int(a,n,x,y,true,false,sl),
Aset = {z:real|x<=z AND polynomial(a,n)(z)=0}
IN rootnum>=0 AND EXISTS (bij: [below(rootnum)->(Aset)]): bijective?(bij)
roots_closed_int_def_falsefalse: LEMMA FORALL (sl:(Eq_computed_remainder?(a,n))): a(n)/=0 AND x<y IMPLIES
LET rootnum = roots_closed_int(a,n,x,y,false,false,sl),
Aset = {z:real| polynomial(a,n)(z)=0}
IN rootnum>=0 AND EXISTS (bij: [below(rootnum)->(Aset)]): bijective?(bij)
roots_closed_int_def: LEMMA FORALL (sl:(Eq_computed_remainder?(a,n))): a(n)/=0 AND x<y IMPLIES
LET rootnum = roots_closed_int(a,n,x,y,lower_bound,upper_bound,sl),
Aset = {z:real|(lower_bound IMPLIES x<=z)
AND (upper_bound IMPLIES z<=y)
AND polynomial(a,n)(z)=0}
IN rootnum>=0 AND EXISTS (bij: [below(rootnum)->(Aset)]): bijective?(bij)
number_roots_interval(a,(n|a(n)/=0),I,(sl:(Eq_computed_remainder?(a,n)))): nat =
IF I`lb >= I`ub THEN 0
ELSE
LET rootstotal = roots_closed_int(a,n,I`lb,I`ub,I`bounded_below,I`bounded_above,sl),
adjlow = IF I`bounded_below AND (NOT I`closed_below) AND polynomial(a,n)(I`lb)=0
THEN -1 ELSE 0 ENDIF,
adjhigh = IF I`bounded_above AND (NOT I`closed_above) AND polynomial(a,n)(I`ub)=0
THEN -1 ELSE 0 ENDIF
IN rootstotal + adjlow + adjhigh
ENDIF
number_roots_interval_def: LEMMA FORALL (sl:(Eq_computed_remainder?(a,n))): a(n)/=0 AND I`lb<I`ub
IMPLIES
LET rootnum = number_roots_interval(a,n,I,sl),
Aset = {z:real|contains?(I)(z) AND polynomial(a,n)(z)=0}
IN rootnum>=0 AND EXISTS (bij: [below(rootnum)->(Aset)]): bijective?(bij)
meas_fun_alw_nonneg(a,(n|a(n)/=0),x,y:real,sl:(Eq_computed_remainder?(a,n))):real = y-x
always_nonnegative_int(a,(n|a(n)/=0),x,y:real,(sl:(Eq_computed_remainder?(a,n)))): RECURSIVE {bb:bool|
bb IFF FORALL (z:real): x<=z AND z<=y IMPLIES polynomial(a,n)(z)>=0} =
IF x>y THEN TRUE
ELSIF x=y THEN polynomial(a,n)(x)>=0
ELSIF roots_closed_int(a,n,x,y,TRUE,TRUE,sl) <= 1 THEN
polynomial(a,n)(x)>=0 AND polynomial(a,n)(y)>=0
ELSE
always_nonnegative_int(a,n,x,(x+y)/2,sl) AND
always_nonnegative_int(a,n,(x+y)/2,y,sl)
ENDIF
MEASURE meas_fun_alw_nonneg BY real_ord_ep(min_poly_root_dist(a,n))
always_nonnegative(a,(n|a(n)/=0),(I|I`lb<I`ub)): bool =
LET newa = (LAMBDA (i:nat): IF i<=n THEN a(i) ELSE 0 ENDIF),
sl = sturm_chain(newa,n)
IN
IF I`bounded_below AND I`bounded_above
THEN always_nonnegative_int(a,n,I`lb,I`ub,sl)
ELSE
LET M = Knuth_poly_root_strict_bound(a,n) IN
IF I`bounded_above THEN
always_nonnegative_int(a,n,min(-M,I`ub-1),I`ub,sl)
ELSIF I`bounded_below THEN
always_nonnegative_int(a,n,I`lb,max(M,I`lb+1),sl)
ELSE always_nonnegative_int(a,n,-M,M,sl)
ENDIF
ENDIF
always_nonnegative_def: LEMMA a(n)/=0 AND I`lb<I`ub IMPLIES
(always_nonnegative(a,n,I) IFF
(FORALL (x:real): contains?(I)(x) IMPLIES polynomial(a,n)(x)>=0))
R:VAR [[real,real]->bool]
SturmRel?(R): bool = (R = (<)) OR (R = (<=)) OR (R = (>)) OR (R = (>=))
OR (R = (/=[real]))
realord,ro: VAR (SturmRel?)
strict?(realord): bool = (realord = (<)) OR (realord = (>)) OR (realord = (/=[real]))
swap(realord): (SturmRel?) =
IF realord = (<) THEN >
ELSIF realord = (<=) THEN >=
ELSIF realord = (>) THEN <
ELSIF realord = (>=) THEN <=
ELSE (/=[real])
ENDIF
swap_lt : LEMMA
swap(<) = (>)
swap_gt : LEMMA
swap(>) = (<)
swap_le : LEMMA
swap(<=) = (>=)
swap_ge : LEMMA
swap(>=) = (<=)
swap_ne : LEMMA
swap(/=[real]) = (/=[real])
real_order_scal_pos: LEMMA
FORALL (p:posreal,x,y:real): realord(p*x,p*y)=realord(x,y)
compute_poly_sat(a,(n|a(n)/=0),(I|I`lb<I`ub),realord): bool =
IF realord = (>=) THEN always_nonnegative(a,n,I)
ELSIF realord = (<=) THEN always_nonnegative(-a,n,I)
ELSE
LET newa = (LAMBDA (i:nat): IF i<=n THEN a(i) ELSE 0 ENDIF),
sl = sturm_chain(newa,n)
IN number_roots_interval(a,n,I,sl)=0 AND realord(polynomial(a,n)((I`lb+I`ub)/2),0)
ENDIF
compute_poly_sat_def: LEMMA a(n)/=0 AND I`lb<I`ub IMPLIES
(compute_poly_sat(a,n,I,realord) IFF
(FORALL (x:real): contains?(I)(x) IMPLIES realord(polynomial(a,n)(x),0)))
aq: VAR [nat->rat]
compute_poly_sat_rational(aq,(n|aq(n)/=0),(I|I`lb<I`ub),realord): bool =
compute_poly_sat(rat_poly_to_int(aq,n),n,I,realord)
compute_poly_sat_rational_def: LEMMA aq(n)/=0 AND I`lb<I`ub IMPLIES
(compute_poly_sat_rational(aq,n,I,realord) IFF
(FORALL (x:real): contains?(I)(x) IMPLIES realord(polynomial(aq,n)(x),0)))
% Next function assumes x<y
compute_poly_mono_basic(aq,(n|aq(n)/=0),(I|I`lb<I`ub),realord): bool =
IF n = 1 THEN realord(sign(aq(1))*I`lb,sign(aq(1))*I`ub)
ELSIF (realord = (/=[real])) THEN
compute_poly_sat_rational(poly_deriv(aq),n-1,I,>=) OR
compute_poly_sat_rational(poly_deriv(aq),n-1,I,<=)
ELSIF realord(I`lb,I`ub) THEN
compute_poly_sat_rational(poly_deriv(aq),n-1,I,>=)
ELSE
compute_poly_sat_rational(poly_deriv(aq),n-1,I,<=)
ENDIF
compute_poly_mono_basic_def: LEMMA aq(n)/=0 AND I`lb<I`ub IMPLIES
((FORALL (x,y:real): x<y AND contains?(I)(x) AND contains?(I)(y) IMPLIES
realord(polynomial(aq,n)(x),polynomial(aq,n)(y))) IFF
compute_poly_mono_basic(aq,n,I,realord))
mono(aq,(n|aq(n)/=0),(I|I`lb<I`ub),ro,realord): bool =
IF ro(1,1) AND NOT realord(1,1) THEN FALSE
ELSIF (ro=(<)) OR (ro=(<=)) THEN compute_poly_mono_basic(aq,n,I,realord)
ELSIF (ro=(>)) OR (ro=(>=)) THEN compute_poly_mono_basic(aq,n,I,swap(realord))
ELSIF NOT (realord=(/=[real])) THEN FALSE
ELSE compute_poly_mono_basic(aq,n,I,realord)
OR compute_poly_mono_basic(aq,n,I,swap(realord))
ENDIF
poly_non_constant_real_int: LEMMA aq(n)/=0 AND I`lb<I`ub IMPLIES
EXISTS (x,y:real): x<y AND contains?(I)(x) AND contains?(I)(y) AND
polynomial(aq,n)(x)/=polynomial(aq,n)(y)
mono_def: LEMMA aq(n)/=0 AND I`lb<I`ub IMPLIES
((FORALL (x,y:real): ro(x,y) AND contains?(I)(x) AND contains?(I)(y) IMPLIES
realord(polynomial(aq,n)(x),polynomial(aq,n)(y))) IFF
mono(aq,n,I,ro,realord))
END compute_sturm
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