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compute_sturm_tarski: THEORY
BEGIN
IMPORTING sturmtarski,
Sturm@remainder_sequence,
Sturm@clear_denominators,
reals@more_polynomial_props
a,g,g1,g2 : VAR [nat->int]
n : VAR posnat
k,k1,k2 : VAR nat
x,y,z: VAR real
aq: VAR [nat->rat]
gq: VAR [nat->rat]
%%% Support Lemmas %%%
der_prod(a,n,g,k)(i:nat): int =
polynomial_prod(g,k,poly_deriv(a),n-1)(i)
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 g(k)/=0 AND
(FORALL (i:nat): i>n IMPLIES a(i)=0) IMPLIES
LET sl = remainder_seq(a,n,der_prod(a,n,g,k),n-1+k),
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,g,k,M) AND P(0)=a AND N(0)=n
Eq_computed_remainder?(a,(n|a(n)/=0),g,(k|g(k)/=0))(sl:list[list[int]]): bool =
sl = remainder_seq((LAMBDA (i:nat): IF i<=n THEN a(i) ELSE 0 ENDIF),
n,der_prod((LAMBDA (i:nat): IF i<=n THEN a(i) ELSE 0 ENDIF),n,g,k),n-1+k)
% Standard but possibly unbounded interval %
lower_bound,upper_bound: VAR bool
% Formalizatin brings in complications with evaluating at infinity
compute_TQ_param(a,(n|a(n)/=0),g,(k|g(k)/=0),
sl:(Eq_computed_remainder?(a,n,g,k))):
int =
LET 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,
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):
(IF even?(N(i)) THEN 1 ELSE -1 ENDIF)*P(i)(N(i)),M-1) ENDIF,
Nroots = nschighlow(FALSE)`num-nschighlow(TRUE)`num
IN Nroots
TQ(a,(n|a(n)/=0),g,(k|g(k)/=0)): int =
LET newa = (LAMBDA (i:nat): IF i<=n THEN a(i) ELSE 0 ENDIF),
sl = remainder_seq(newa,n,
der_prod(newa,n,g,k),n-1+k)
IN compute_TQ_param(newa,n,g,k,sl)
TQ_def: LEMMA
a(n)/=0 AND g(k)/=0 IMPLIES
TQ(a,n,g,k) = NSol(a,n,g,k,>)-NSol(a,n,g,k,<)
TQ_eq_g: LEMMA a(n)/=0 AND g1(k)/=0 AND g2(k)/=0 AND
(FORALL (i:upto(k)): g1(i)=g2(i))
IMPLIES
TQ(a,n,g1,k)=TQ(a,n,g2,k)
END compute_sturm_tarski
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