TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60002 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (11990ms).
 | – Problem 2 was processed with processor SubtermCriterion (2ms).
 | – Problem 3 was processed with processor SubtermCriterion (4ms).
 | – Problem 4 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 15 remains open; application of the following processors failed [DependencyGraph (4ms), PolynomialLinearRange4iUR (16ms), DependencyGraph (5ms), PolynomialLinearRange4iUR (11ms), DependencyGraph (4ms), PolynomialLinearRange4iUR (26ms), DependencyGraph (4ms), PolynomialLinearRange8NegiUR (0ms), DependencyGraph (4ms)].
 | – Problem 5 was processed with processor SubtermCriterion (1ms).
 | – Problem 6 was processed with processor SubtermCriterion (2ms).
 | – Problem 7 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 16 was processed with processor PolynomialLinearRange4iUR (40ms).
 |    |    | – Problem 21 was processed with processor PolynomialLinearRange4iUR (88ms).
 | – Problem 8 was processed with processor SubtermCriterion (3ms).
 | – Problem 9 was processed with processor SubtermCriterion (2ms).
 |    | – Problem 17 was processed with processor PolynomialLinearRange4iUR (126ms).
 | – Problem 10 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 18 was processed with processor PolynomialLinearRange4iUR (60ms).
 | – Problem 11 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 19 was processed with processor PolynomialLinearRange4iUR (76ms).
 | – Problem 12 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 20 was processed with processor PolynomialLinearRange4iUR (72ms).
 |    |    | – Problem 22 was processed with processor PolynomialLinearRange4iUR (65ms).
 | – Problem 13 remains open; application of the following processors failed [SubtermCriterion (3ms), DependencyGraph (2191ms), PolynomialLinearRange4iUR (1437ms), DependencyGraph (2269ms), PolynomialLinearRange4iUR (2532ms), DependencyGraph (1897ms), PolynomialLinearRange4iUR (5033ms), DependencyGraph (2004ms), PolynomialLinearRange8NegiUR (15021ms), DependencyGraph (1891ms), ReductionPairSAT (timeout)].
 | – Problem 14 was processed with processor SubtermCriterion (1ms).

The following open problems remain:



Open Dependency Pair Problem 13

Dependency Pairs

mark#(cons(X1, X2))active#(cons(mark(X1), X2))mark#(2ndspos(X1, X2))mark#(X2)
active#(2ndsneg(0, Z))mark#(rnil)mark#(square(X))mark#(X)
mark#(pi(X))mark#(X)mark#(pi(X))active#(pi(mark(X)))
mark#(times(X1, X2))mark#(X2)mark#(negrecip(X))mark#(X)
mark#(2ndsneg(X1, X2))mark#(X1)active#(plus(s(X), Y))mark#(s(plus(X, Y)))
active#(2ndsneg(s(N), cons(X, cons(Y, Z))))mark#(rcons(negrecip(Y), 2ndspos(N, Z)))mark#(plus(X1, X2))mark#(X2)
mark#(times(X1, X2))mark#(X1)mark#(2ndsneg(X1, X2))mark#(X2)
mark#(rnil)active#(rnil)mark#(s(X))mark#(X)
active#(times(0, Y))mark#(0)mark#(square(X))active#(square(mark(X)))
active#(times(s(X), Y))mark#(plus(Y, times(X, Y)))mark#(2ndspos(X1, X2))active#(2ndspos(mark(X1), mark(X2)))
active#(2ndspos(s(N), cons(X, cons(Y, Z))))mark#(rcons(posrecip(Y), 2ndsneg(N, Z)))mark#(0)active#(0)
mark#(plus(X1, X2))active#(plus(mark(X1), mark(X2)))active#(plus(0, Y))mark#(Y)
mark#(s(X))active#(s(mark(X)))mark#(rcons(X1, X2))active#(rcons(mark(X1), mark(X2)))
mark#(negrecip(X))active#(negrecip(mark(X)))mark#(times(X1, X2))active#(times(mark(X1), mark(X2)))
mark#(rcons(X1, X2))mark#(X2)mark#(2ndsneg(X1, X2))active#(2ndsneg(mark(X1), mark(X2)))
mark#(from(X))mark#(X)active#(square(X))mark#(times(X, X))
mark#(posrecip(X))active#(posrecip(mark(X)))mark#(cons(X1, X2))mark#(X1)
active#(from(X))mark#(cons(X, from(s(X))))mark#(plus(X1, X2))mark#(X1)
active#(2ndspos(0, Z))mark#(rnil)mark#(from(X))active#(from(mark(X)))
mark#(posrecip(X))mark#(X)mark#(2ndspos(X1, X2))mark#(X1)
mark#(rcons(X1, X2))mark#(X1)active#(pi(X))mark#(2ndspos(X, from(0)))

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(2ndspos(0, Z))mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z))))mark(rcons(posrecip(Y), 2ndsneg(N, Z)))active(2ndsneg(0, Z))mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z))))mark(rcons(negrecip(Y), 2ndspos(N, Z)))active(pi(X))mark(2ndspos(X, from(0)))
active(plus(0, Y))mark(Y)active(plus(s(X), Y))mark(s(plus(X, Y)))
active(times(0, Y))mark(0)active(times(s(X), Y))mark(plus(Y, times(X, Y)))
active(square(X))mark(times(X, X))mark(from(X))active(from(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(2ndspos(X1, X2))active(2ndspos(mark(X1), mark(X2)))mark(0)active(0)
mark(rnil)active(rnil)mark(rcons(X1, X2))active(rcons(mark(X1), mark(X2)))
mark(posrecip(X))active(posrecip(mark(X)))mark(2ndsneg(X1, X2))active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X))active(negrecip(mark(X)))mark(pi(X))active(pi(mark(X)))
mark(plus(X1, X2))active(plus(mark(X1), mark(X2)))mark(times(X1, X2))active(times(mark(X1), mark(X2)))
mark(square(X))active(square(mark(X)))from(mark(X))from(X)
from(active(X))from(X)cons(mark(X1), X2)cons(X1, X2)
cons(X1, mark(X2))cons(X1, X2)cons(active(X1), X2)cons(X1, X2)
cons(X1, active(X2))cons(X1, X2)s(mark(X))s(X)
s(active(X))s(X)2ndspos(mark(X1), X2)2ndspos(X1, X2)
2ndspos(X1, mark(X2))2ndspos(X1, X2)2ndspos(active(X1), X2)2ndspos(X1, X2)
2ndspos(X1, active(X2))2ndspos(X1, X2)rcons(mark(X1), X2)rcons(X1, X2)
rcons(X1, mark(X2))rcons(X1, X2)rcons(active(X1), X2)rcons(X1, X2)
rcons(X1, active(X2))rcons(X1, X2)posrecip(mark(X))posrecip(X)
posrecip(active(X))posrecip(X)2ndsneg(mark(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, mark(X2))2ndsneg(X1, X2)2ndsneg(active(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, active(X2))2ndsneg(X1, X2)negrecip(mark(X))negrecip(X)
negrecip(active(X))negrecip(X)pi(mark(X))pi(X)
pi(active(X))pi(X)plus(mark(X1), X2)plus(X1, X2)
plus(X1, mark(X2))plus(X1, X2)plus(active(X1), X2)plus(X1, X2)
plus(X1, active(X2))plus(X1, X2)times(mark(X1), X2)times(X1, X2)
times(X1, mark(X2))times(X1, X2)times(active(X1), X2)times(X1, X2)
times(X1, active(X2))times(X1, X2)square(mark(X))square(X)
square(active(X))square(X)

Original Signature

Termination of terms over the following signature is verified: plus, posrecip, negrecip, rnil, mark, from, rcons, 2ndspos, 0, s, times, 2ndsneg, active, square, pi, cons




Open Dependency Pair Problem 15

Dependency Pairs

2ndspos#(X1, mark(X2))2ndspos#(X1, X2)2ndspos#(X1, active(X2))2ndspos#(X1, X2)

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(2ndspos(0, Z))mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z))))mark(rcons(posrecip(Y), 2ndsneg(N, Z)))active(2ndsneg(0, Z))mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z))))mark(rcons(negrecip(Y), 2ndspos(N, Z)))active(pi(X))mark(2ndspos(X, from(0)))
active(plus(0, Y))mark(Y)active(plus(s(X), Y))mark(s(plus(X, Y)))
active(times(0, Y))mark(0)active(times(s(X), Y))mark(plus(Y, times(X, Y)))
active(square(X))mark(times(X, X))mark(from(X))active(from(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(2ndspos(X1, X2))active(2ndspos(mark(X1), mark(X2)))mark(0)active(0)
mark(rnil)active(rnil)mark(rcons(X1, X2))active(rcons(mark(X1), mark(X2)))
mark(posrecip(X))active(posrecip(mark(X)))mark(2ndsneg(X1, X2))active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X))active(negrecip(mark(X)))mark(pi(X))active(pi(mark(X)))
mark(plus(X1, X2))active(plus(mark(X1), mark(X2)))mark(times(X1, X2))active(times(mark(X1), mark(X2)))
mark(square(X))active(square(mark(X)))from(mark(X))from(X)
from(active(X))from(X)cons(mark(X1), X2)cons(X1, X2)
cons(X1, mark(X2))cons(X1, X2)cons(active(X1), X2)cons(X1, X2)
cons(X1, active(X2))cons(X1, X2)s(mark(X))s(X)
s(active(X))s(X)2ndspos(mark(X1), X2)2ndspos(X1, X2)
2ndspos(X1, mark(X2))2ndspos(X1, X2)2ndspos(active(X1), X2)2ndspos(X1, X2)
2ndspos(X1, active(X2))2ndspos(X1, X2)rcons(mark(X1), X2)rcons(X1, X2)
rcons(X1, mark(X2))rcons(X1, X2)rcons(active(X1), X2)rcons(X1, X2)
rcons(X1, active(X2))rcons(X1, X2)posrecip(mark(X))posrecip(X)
posrecip(active(X))posrecip(X)2ndsneg(mark(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, mark(X2))2ndsneg(X1, X2)2ndsneg(active(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, active(X2))2ndsneg(X1, X2)negrecip(mark(X))negrecip(X)
negrecip(active(X))negrecip(X)pi(mark(X))pi(X)
pi(active(X))pi(X)plus(mark(X1), X2)plus(X1, X2)
plus(X1, mark(X2))plus(X1, X2)plus(active(X1), X2)plus(X1, X2)
plus(X1, active(X2))plus(X1, X2)times(mark(X1), X2)times(X1, X2)
times(X1, mark(X2))times(X1, X2)times(active(X1), X2)times(X1, X2)
times(X1, active(X2))times(X1, X2)square(mark(X))square(X)
square(active(X))square(X)

Original Signature

Termination of terms over the following signature is verified: plus, posrecip, negrecip, rnil, mark, from, rcons, 2ndspos, 0, s, times, 2ndsneg, active, square, pi, cons


Problem 1: DependencyGraph



Dependency Pair Problem

Dependency Pairs

mark#(2ndsneg(X1, X2))2ndsneg#(mark(X1), mark(X2))active#(2ndspos(s(N), cons(X, cons(Y, Z))))posrecip#(Y)
active#(2ndsneg(0, Z))mark#(rnil)active#(pi(X))from#(0)
rcons#(X1, active(X2))rcons#(X1, X2)mark#(pi(X))mark#(X)
mark#(negrecip(X))negrecip#(mark(X))mark#(pi(X))active#(pi(mark(X)))
mark#(times(X1, X2))mark#(X2)mark#(negrecip(X))mark#(X)
mark#(2ndsneg(X1, X2))mark#(X1)active#(plus(s(X), Y))mark#(s(plus(X, Y)))
times#(active(X1), X2)times#(X1, X2)mark#(times(X1, X2))mark#(X1)
mark#(s(X))mark#(X)mark#(plus(X1, X2))active#(plus(mark(X1), mark(X2)))
2ndspos#(X1, mark(X2))2ndspos#(X1, X2)mark#(rcons(X1, X2))mark#(X2)
active#(square(X))mark#(times(X, X))mark#(pi(X))pi#(mark(X))
mark#(cons(X1, X2))mark#(X1)rcons#(mark(X1), X2)rcons#(X1, X2)
times#(mark(X1), X2)times#(X1, X2)active#(square(X))times#(X, X)
negrecip#(mark(X))negrecip#(X)active#(from(X))mark#(cons(X, from(s(X))))
2ndspos#(mark(X1), X2)2ndspos#(X1, X2)mark#(2ndspos(X1, X2))2ndspos#(mark(X1), mark(X2))
plus#(mark(X1), X2)plus#(X1, X2)plus#(active(X1), X2)plus#(X1, X2)
cons#(mark(X1), X2)cons#(X1, X2)mark#(rcons(X1, X2))rcons#(mark(X1), mark(X2))
from#(mark(X))from#(X)active#(2ndsneg(s(N), cons(X, cons(Y, Z))))mark#(rcons(negrecip(Y), 2ndspos(N, Z)))
square#(mark(X))square#(X)times#(X1, mark(X2))times#(X1, X2)
cons#(X1, mark(X2))cons#(X1, X2)active#(pi(X))2ndspos#(X, from(0))
mark#(square(X))square#(mark(X))active#(times(0, Y))mark#(0)
mark#(2ndspos(X1, X2))active#(2ndspos(mark(X1), mark(X2)))active#(plus(s(X), Y))plus#(X, Y)
active#(2ndspos(s(N), cons(X, cons(Y, Z))))mark#(rcons(posrecip(Y), 2ndsneg(N, Z)))posrecip#(mark(X))posrecip#(X)
mark#(0)active#(0)mark#(negrecip(X))active#(negrecip(mark(X)))
active#(2ndsneg(s(N), cons(X, cons(Y, Z))))negrecip#(Y)mark#(s(X))active#(s(mark(X)))
active#(times(s(X), Y))plus#(Y, times(X, Y))mark#(posrecip(X))active#(posrecip(mark(X)))
cons#(active(X1), X2)cons#(X1, X2)active#(2ndsneg(s(N), cons(X, cons(Y, Z))))2ndspos#(N, Z)
active#(2ndspos(0, Z))mark#(rnil)2ndsneg#(X1, mark(X2))2ndsneg#(X1, X2)
rcons#(X1, mark(X2))rcons#(X1, X2)active#(from(X))from#(s(X))
mark#(cons(X1, X2))active#(cons(mark(X1), X2))active#(2ndspos(s(N), cons(X, cons(Y, Z))))rcons#(posrecip(Y), 2ndsneg(N, Z))
mark#(2ndspos(X1, X2))mark#(X2)mark#(square(X))mark#(X)
2ndspos#(X1, active(X2))2ndspos#(X1, X2)active#(2ndsneg(s(N), cons(X, cons(Y, Z))))rcons#(negrecip(Y), 2ndspos(N, Z))
mark#(s(X))s#(mark(X))pi#(mark(X))pi#(X)
plus#(X1, mark(X2))plus#(X1, X2)mark#(2ndsneg(X1, X2))mark#(X2)
mark#(rnil)active#(rnil)mark#(posrecip(X))posrecip#(mark(X))
pi#(active(X))pi#(X)active#(times(s(X), Y))mark#(plus(Y, times(X, Y)))
active#(plus(0, Y))mark#(Y)2ndspos#(active(X1), X2)2ndspos#(X1, X2)
mark#(2ndsneg(X1, X2))active#(2ndsneg(mark(X1), mark(X2)))mark#(from(X))mark#(X)
posrecip#(active(X))posrecip#(X)mark#(plus(X1, X2))mark#(X1)
active#(from(X))s#(X)cons#(X1, active(X2))cons#(X1, X2)
mark#(from(X))from#(mark(X))mark#(from(X))active#(from(mark(X)))
plus#(X1, active(X2))plus#(X1, X2)mark#(posrecip(X))mark#(X)
from#(active(X))from#(X)active#(plus(s(X), Y))s#(plus(X, Y))
square#(active(X))square#(X)active#(times(s(X), Y))times#(X, Y)
mark#(plus(X1, X2))mark#(X2)2ndsneg#(active(X1), X2)2ndsneg#(X1, X2)
negrecip#(active(X))negrecip#(X)mark#(times(X1, X2))times#(mark(X1), mark(X2))
active#(from(X))cons#(X, from(s(X)))2ndsneg#(mark(X1), X2)2ndsneg#(X1, X2)
mark#(cons(X1, X2))cons#(mark(X1), X2)mark#(square(X))active#(square(mark(X)))
mark#(rcons(X1, X2))active#(rcons(mark(X1), mark(X2)))mark#(times(X1, X2))active#(times(mark(X1), mark(X2)))
times#(X1, active(X2))times#(X1, X2)mark#(plus(X1, X2))plus#(mark(X1), mark(X2))
2ndsneg#(X1, active(X2))2ndsneg#(X1, X2)s#(mark(X))s#(X)
rcons#(active(X1), X2)rcons#(X1, X2)s#(active(X))s#(X)
mark#(2ndspos(X1, X2))mark#(X1)mark#(rcons(X1, X2))mark#(X1)
active#(pi(X))mark#(2ndspos(X, from(0)))active#(2ndspos(s(N), cons(X, cons(Y, Z))))2ndsneg#(N, Z)

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(2ndspos(0, Z))mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z))))mark(rcons(posrecip(Y), 2ndsneg(N, Z)))active(2ndsneg(0, Z))mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z))))mark(rcons(negrecip(Y), 2ndspos(N, Z)))active(pi(X))mark(2ndspos(X, from(0)))
active(plus(0, Y))mark(Y)active(plus(s(X), Y))mark(s(plus(X, Y)))
active(times(0, Y))mark(0)active(times(s(X), Y))mark(plus(Y, times(X, Y)))
active(square(X))mark(times(X, X))mark(from(X))active(from(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(2ndspos(X1, X2))active(2ndspos(mark(X1), mark(X2)))mark(0)active(0)
mark(rnil)active(rnil)mark(rcons(X1, X2))active(rcons(mark(X1), mark(X2)))
mark(posrecip(X))active(posrecip(mark(X)))mark(2ndsneg(X1, X2))active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X))active(negrecip(mark(X)))mark(pi(X))active(pi(mark(X)))
mark(plus(X1, X2))active(plus(mark(X1), mark(X2)))mark(times(X1, X2))active(times(mark(X1), mark(X2)))
mark(square(X))active(square(mark(X)))from(mark(X))from(X)
from(active(X))from(X)cons(mark(X1), X2)cons(X1, X2)
cons(X1, mark(X2))cons(X1, X2)cons(active(X1), X2)cons(X1, X2)
cons(X1, active(X2))cons(X1, X2)s(mark(X))s(X)
s(active(X))s(X)2ndspos(mark(X1), X2)2ndspos(X1, X2)
2ndspos(X1, mark(X2))2ndspos(X1, X2)2ndspos(active(X1), X2)2ndspos(X1, X2)
2ndspos(X1, active(X2))2ndspos(X1, X2)rcons(mark(X1), X2)rcons(X1, X2)
rcons(X1, mark(X2))rcons(X1, X2)rcons(active(X1), X2)rcons(X1, X2)
rcons(X1, active(X2))rcons(X1, X2)posrecip(mark(X))posrecip(X)
posrecip(active(X))posrecip(X)2ndsneg(mark(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, mark(X2))2ndsneg(X1, X2)2ndsneg(active(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, active(X2))2ndsneg(X1, X2)negrecip(mark(X))negrecip(X)
negrecip(active(X))negrecip(X)pi(mark(X))pi(X)
pi(active(X))pi(X)plus(mark(X1), X2)plus(X1, X2)
plus(X1, mark(X2))plus(X1, X2)plus(active(X1), X2)plus(X1, X2)
plus(X1, active(X2))plus(X1, X2)times(mark(X1), X2)times(X1, X2)
times(X1, mark(X2))times(X1, X2)times(active(X1), X2)times(X1, X2)
times(X1, active(X2))times(X1, X2)square(mark(X))square(X)
square(active(X))square(X)

Original Signature

Termination of terms over the following signature is verified: plus, posrecip, negrecip, rnil, mark, from, rcons, 2ndspos, 0, s, times, 2ndsneg, active, square, pi, cons

Strategy


The following SCCs where found

posrecip#(mark(X)) → posrecip#(X)posrecip#(active(X)) → posrecip#(X)

from#(active(X)) → from#(X)from#(mark(X)) → from#(X)

square#(mark(X)) → square#(X)square#(active(X)) → square#(X)

pi#(mark(X)) → pi#(X)pi#(active(X)) → pi#(X)

times#(active(X1), X2) → times#(X1, X2)times#(X1, active(X2)) → times#(X1, X2)
times#(X1, mark(X2)) → times#(X1, X2)times#(mark(X1), X2) → times#(X1, X2)

2ndspos#(mark(X1), X2) → 2ndspos#(X1, X2)2ndspos#(active(X1), X2) → 2ndspos#(X1, X2)
2ndspos#(X1, mark(X2)) → 2ndspos#(X1, X2)2ndspos#(X1, active(X2)) → 2ndspos#(X1, X2)

2ndsneg#(X1, mark(X2)) → 2ndsneg#(X1, X2)2ndsneg#(active(X1), X2) → 2ndsneg#(X1, X2)
2ndsneg#(mark(X1), X2) → 2ndsneg#(X1, X2)2ndsneg#(X1, active(X2)) → 2ndsneg#(X1, X2)

cons#(X1, active(X2)) → cons#(X1, X2)cons#(mark(X1), X2) → cons#(X1, X2)
cons#(active(X1), X2) → cons#(X1, X2)cons#(X1, mark(X2)) → cons#(X1, X2)

negrecip#(active(X)) → negrecip#(X)negrecip#(mark(X)) → negrecip#(X)

mark#(cons(X1, X2)) → active#(cons(mark(X1), X2))mark#(2ndspos(X1, X2)) → mark#(X2)
active#(2ndsneg(0, Z)) → mark#(rnil)mark#(square(X)) → mark#(X)
mark#(pi(X)) → mark#(X)mark#(pi(X)) → active#(pi(mark(X)))
mark#(negrecip(X)) → mark#(X)mark#(times(X1, X2)) → mark#(X2)
mark#(2ndsneg(X1, X2)) → mark#(X1)active#(2ndsneg(s(N), cons(X, cons(Y, Z)))) → mark#(rcons(negrecip(Y), 2ndspos(N, Z)))
active#(plus(s(X), Y)) → mark#(s(plus(X, Y)))mark#(plus(X1, X2)) → mark#(X2)
mark#(times(X1, X2)) → mark#(X1)mark#(2ndsneg(X1, X2)) → mark#(X2)
mark#(s(X)) → mark#(X)mark#(rnil) → active#(rnil)
active#(times(0, Y)) → mark#(0)active#(times(s(X), Y)) → mark#(plus(Y, times(X, Y)))
mark#(square(X)) → active#(square(mark(X)))mark#(2ndspos(X1, X2)) → active#(2ndspos(mark(X1), mark(X2)))
active#(2ndspos(s(N), cons(X, cons(Y, Z)))) → mark#(rcons(posrecip(Y), 2ndsneg(N, Z)))active#(plus(0, Y)) → mark#(Y)
mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2)))mark#(0) → active#(0)
mark#(negrecip(X)) → active#(negrecip(mark(X)))mark#(rcons(X1, X2)) → active#(rcons(mark(X1), mark(X2)))
mark#(s(X)) → active#(s(mark(X)))mark#(rcons(X1, X2)) → mark#(X2)
mark#(times(X1, X2)) → active#(times(mark(X1), mark(X2)))mark#(2ndsneg(X1, X2)) → active#(2ndsneg(mark(X1), mark(X2)))
active#(square(X)) → mark#(times(X, X))mark#(from(X)) → mark#(X)
mark#(cons(X1, X2)) → mark#(X1)mark#(posrecip(X)) → active#(posrecip(mark(X)))
active#(from(X)) → mark#(cons(X, from(s(X))))mark#(plus(X1, X2)) → mark#(X1)
active#(2ndspos(0, Z)) → mark#(rnil)mark#(from(X)) → active#(from(mark(X)))
mark#(posrecip(X)) → mark#(X)mark#(2ndspos(X1, X2)) → mark#(X1)
mark#(rcons(X1, X2)) → mark#(X1)active#(pi(X)) → mark#(2ndspos(X, from(0)))

rcons#(active(X1), X2) → rcons#(X1, X2)rcons#(X1, active(X2)) → rcons#(X1, X2)
rcons#(mark(X1), X2) → rcons#(X1, X2)rcons#(X1, mark(X2)) → rcons#(X1, X2)

s#(mark(X)) → s#(X)s#(active(X)) → s#(X)

plus#(X1, active(X2)) → plus#(X1, X2)plus#(X1, mark(X2)) → plus#(X1, X2)
plus#(mark(X1), X2) → plus#(X1, X2)plus#(active(X1), X2) → plus#(X1, X2)

Problem 2: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

s#(mark(X))s#(X)s#(active(X))s#(X)

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(2ndspos(0, Z))mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z))))mark(rcons(posrecip(Y), 2ndsneg(N, Z)))active(2ndsneg(0, Z))mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z))))mark(rcons(negrecip(Y), 2ndspos(N, Z)))active(pi(X))mark(2ndspos(X, from(0)))
active(plus(0, Y))mark(Y)active(plus(s(X), Y))mark(s(plus(X, Y)))
active(times(0, Y))mark(0)active(times(s(X), Y))mark(plus(Y, times(X, Y)))
active(square(X))mark(times(X, X))mark(from(X))active(from(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(2ndspos(X1, X2))active(2ndspos(mark(X1), mark(X2)))mark(0)active(0)
mark(rnil)active(rnil)mark(rcons(X1, X2))active(rcons(mark(X1), mark(X2)))
mark(posrecip(X))active(posrecip(mark(X)))mark(2ndsneg(X1, X2))active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X))active(negrecip(mark(X)))mark(pi(X))active(pi(mark(X)))
mark(plus(X1, X2))active(plus(mark(X1), mark(X2)))mark(times(X1, X2))active(times(mark(X1), mark(X2)))
mark(square(X))active(square(mark(X)))from(mark(X))from(X)
from(active(X))from(X)cons(mark(X1), X2)cons(X1, X2)
cons(X1, mark(X2))cons(X1, X2)cons(active(X1), X2)cons(X1, X2)
cons(X1, active(X2))cons(X1, X2)s(mark(X))s(X)
s(active(X))s(X)2ndspos(mark(X1), X2)2ndspos(X1, X2)
2ndspos(X1, mark(X2))2ndspos(X1, X2)2ndspos(active(X1), X2)2ndspos(X1, X2)
2ndspos(X1, active(X2))2ndspos(X1, X2)rcons(mark(X1), X2)rcons(X1, X2)
rcons(X1, mark(X2))rcons(X1, X2)rcons(active(X1), X2)rcons(X1, X2)
rcons(X1, active(X2))rcons(X1, X2)posrecip(mark(X))posrecip(X)
posrecip(active(X))posrecip(X)2ndsneg(mark(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, mark(X2))2ndsneg(X1, X2)2ndsneg(active(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, active(X2))2ndsneg(X1, X2)negrecip(mark(X))negrecip(X)
negrecip(active(X))negrecip(X)pi(mark(X))pi(X)
pi(active(X))pi(X)plus(mark(X1), X2)plus(X1, X2)
plus(X1, mark(X2))plus(X1, X2)plus(active(X1), X2)plus(X1, X2)
plus(X1, active(X2))plus(X1, X2)times(mark(X1), X2)times(X1, X2)
times(X1, mark(X2))times(X1, X2)times(active(X1), X2)times(X1, X2)
times(X1, active(X2))times(X1, X2)square(mark(X))square(X)
square(active(X))square(X)

Original Signature

Termination of terms over the following signature is verified: plus, posrecip, negrecip, rnil, mark, from, rcons, 2ndspos, 0, s, times, 2ndsneg, active, square, pi, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

s#(mark(X))s#(X)s#(active(X))s#(X)

Problem 3: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

posrecip#(mark(X))posrecip#(X)posrecip#(active(X))posrecip#(X)

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(2ndspos(0, Z))mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z))))mark(rcons(posrecip(Y), 2ndsneg(N, Z)))active(2ndsneg(0, Z))mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z))))mark(rcons(negrecip(Y), 2ndspos(N, Z)))active(pi(X))mark(2ndspos(X, from(0)))
active(plus(0, Y))mark(Y)active(plus(s(X), Y))mark(s(plus(X, Y)))
active(times(0, Y))mark(0)active(times(s(X), Y))mark(plus(Y, times(X, Y)))
active(square(X))mark(times(X, X))mark(from(X))active(from(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(2ndspos(X1, X2))active(2ndspos(mark(X1), mark(X2)))mark(0)active(0)
mark(rnil)active(rnil)mark(rcons(X1, X2))active(rcons(mark(X1), mark(X2)))
mark(posrecip(X))active(posrecip(mark(X)))mark(2ndsneg(X1, X2))active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X))active(negrecip(mark(X)))mark(pi(X))active(pi(mark(X)))
mark(plus(X1, X2))active(plus(mark(X1), mark(X2)))mark(times(X1, X2))active(times(mark(X1), mark(X2)))
mark(square(X))active(square(mark(X)))from(mark(X))from(X)
from(active(X))from(X)cons(mark(X1), X2)cons(X1, X2)
cons(X1, mark(X2))cons(X1, X2)cons(active(X1), X2)cons(X1, X2)
cons(X1, active(X2))cons(X1, X2)s(mark(X))s(X)
s(active(X))s(X)2ndspos(mark(X1), X2)2ndspos(X1, X2)
2ndspos(X1, mark(X2))2ndspos(X1, X2)2ndspos(active(X1), X2)2ndspos(X1, X2)
2ndspos(X1, active(X2))2ndspos(X1, X2)rcons(mark(X1), X2)rcons(X1, X2)
rcons(X1, mark(X2))rcons(X1, X2)rcons(active(X1), X2)rcons(X1, X2)
rcons(X1, active(X2))rcons(X1, X2)posrecip(mark(X))posrecip(X)
posrecip(active(X))posrecip(X)2ndsneg(mark(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, mark(X2))2ndsneg(X1, X2)2ndsneg(active(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, active(X2))2ndsneg(X1, X2)negrecip(mark(X))negrecip(X)
negrecip(active(X))negrecip(X)pi(mark(X))pi(X)
pi(active(X))pi(X)plus(mark(X1), X2)plus(X1, X2)
plus(X1, mark(X2))plus(X1, X2)plus(active(X1), X2)plus(X1, X2)
plus(X1, active(X2))plus(X1, X2)times(mark(X1), X2)times(X1, X2)
times(X1, mark(X2))times(X1, X2)times(active(X1), X2)times(X1, X2)
times(X1, active(X2))times(X1, X2)square(mark(X))square(X)
square(active(X))square(X)

Original Signature

Termination of terms over the following signature is verified: plus, posrecip, negrecip, rnil, mark, from, rcons, 2ndspos, 0, s, times, 2ndsneg, active, square, pi, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

posrecip#(mark(X))posrecip#(X)posrecip#(active(X))posrecip#(X)

Problem 4: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

2ndspos#(mark(X1), X2)2ndspos#(X1, X2)2ndspos#(active(X1), X2)2ndspos#(X1, X2)
2ndspos#(X1, mark(X2))2ndspos#(X1, X2)2ndspos#(X1, active(X2))2ndspos#(X1, X2)

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(2ndspos(0, Z))mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z))))mark(rcons(posrecip(Y), 2ndsneg(N, Z)))active(2ndsneg(0, Z))mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z))))mark(rcons(negrecip(Y), 2ndspos(N, Z)))active(pi(X))mark(2ndspos(X, from(0)))
active(plus(0, Y))mark(Y)active(plus(s(X), Y))mark(s(plus(X, Y)))
active(times(0, Y))mark(0)active(times(s(X), Y))mark(plus(Y, times(X, Y)))
active(square(X))mark(times(X, X))mark(from(X))active(from(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(2ndspos(X1, X2))active(2ndspos(mark(X1), mark(X2)))mark(0)active(0)
mark(rnil)active(rnil)mark(rcons(X1, X2))active(rcons(mark(X1), mark(X2)))
mark(posrecip(X))active(posrecip(mark(X)))mark(2ndsneg(X1, X2))active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X))active(negrecip(mark(X)))mark(pi(X))active(pi(mark(X)))
mark(plus(X1, X2))active(plus(mark(X1), mark(X2)))mark(times(X1, X2))active(times(mark(X1), mark(X2)))
mark(square(X))active(square(mark(X)))from(mark(X))from(X)
from(active(X))from(X)cons(mark(X1), X2)cons(X1, X2)
cons(X1, mark(X2))cons(X1, X2)cons(active(X1), X2)cons(X1, X2)
cons(X1, active(X2))cons(X1, X2)s(mark(X))s(X)
s(active(X))s(X)2ndspos(mark(X1), X2)2ndspos(X1, X2)
2ndspos(X1, mark(X2))2ndspos(X1, X2)2ndspos(active(X1), X2)2ndspos(X1, X2)
2ndspos(X1, active(X2))2ndspos(X1, X2)rcons(mark(X1), X2)rcons(X1, X2)
rcons(X1, mark(X2))rcons(X1, X2)rcons(active(X1), X2)rcons(X1, X2)
rcons(X1, active(X2))rcons(X1, X2)posrecip(mark(X))posrecip(X)
posrecip(active(X))posrecip(X)2ndsneg(mark(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, mark(X2))2ndsneg(X1, X2)2ndsneg(active(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, active(X2))2ndsneg(X1, X2)negrecip(mark(X))negrecip(X)
negrecip(active(X))negrecip(X)pi(mark(X))pi(X)
pi(active(X))pi(X)plus(mark(X1), X2)plus(X1, X2)
plus(X1, mark(X2))plus(X1, X2)plus(active(X1), X2)plus(X1, X2)
plus(X1, active(X2))plus(X1, X2)times(mark(X1), X2)times(X1, X2)
times(X1, mark(X2))times(X1, X2)times(active(X1), X2)times(X1, X2)
times(X1, active(X2))times(X1, X2)square(mark(X))square(X)
square(active(X))square(X)

Original Signature

Termination of terms over the following signature is verified: plus, posrecip, negrecip, rnil, mark, from, rcons, 2ndspos, 0, s, times, 2ndsneg, active, square, pi, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

2ndspos#(mark(X1), X2)2ndspos#(X1, X2)2ndspos#(active(X1), X2)2ndspos#(X1, X2)

Problem 5: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

from#(active(X))from#(X)from#(mark(X))from#(X)

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(2ndspos(0, Z))mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z))))mark(rcons(posrecip(Y), 2ndsneg(N, Z)))active(2ndsneg(0, Z))mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z))))mark(rcons(negrecip(Y), 2ndspos(N, Z)))active(pi(X))mark(2ndspos(X, from(0)))
active(plus(0, Y))mark(Y)active(plus(s(X), Y))mark(s(plus(X, Y)))
active(times(0, Y))mark(0)active(times(s(X), Y))mark(plus(Y, times(X, Y)))
active(square(X))mark(times(X, X))mark(from(X))active(from(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(2ndspos(X1, X2))active(2ndspos(mark(X1), mark(X2)))mark(0)active(0)
mark(rnil)active(rnil)mark(rcons(X1, X2))active(rcons(mark(X1), mark(X2)))
mark(posrecip(X))active(posrecip(mark(X)))mark(2ndsneg(X1, X2))active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X))active(negrecip(mark(X)))mark(pi(X))active(pi(mark(X)))
mark(plus(X1, X2))active(plus(mark(X1), mark(X2)))mark(times(X1, X2))active(times(mark(X1), mark(X2)))
mark(square(X))active(square(mark(X)))from(mark(X))from(X)
from(active(X))from(X)cons(mark(X1), X2)cons(X1, X2)
cons(X1, mark(X2))cons(X1, X2)cons(active(X1), X2)cons(X1, X2)
cons(X1, active(X2))cons(X1, X2)s(mark(X))s(X)
s(active(X))s(X)2ndspos(mark(X1), X2)2ndspos(X1, X2)
2ndspos(X1, mark(X2))2ndspos(X1, X2)2ndspos(active(X1), X2)2ndspos(X1, X2)
2ndspos(X1, active(X2))2ndspos(X1, X2)rcons(mark(X1), X2)rcons(X1, X2)
rcons(X1, mark(X2))rcons(X1, X2)rcons(active(X1), X2)rcons(X1, X2)
rcons(X1, active(X2))rcons(X1, X2)posrecip(mark(X))posrecip(X)
posrecip(active(X))posrecip(X)2ndsneg(mark(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, mark(X2))2ndsneg(X1, X2)2ndsneg(active(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, active(X2))2ndsneg(X1, X2)negrecip(mark(X))negrecip(X)
negrecip(active(X))negrecip(X)pi(mark(X))pi(X)
pi(active(X))pi(X)plus(mark(X1), X2)plus(X1, X2)
plus(X1, mark(X2))plus(X1, X2)plus(active(X1), X2)plus(X1, X2)
plus(X1, active(X2))plus(X1, X2)times(mark(X1), X2)times(X1, X2)
times(X1, mark(X2))times(X1, X2)times(active(X1), X2)times(X1, X2)
times(X1, active(X2))times(X1, X2)square(mark(X))square(X)
square(active(X))square(X)

Original Signature

Termination of terms over the following signature is verified: plus, posrecip, negrecip, rnil, mark, from, rcons, 2ndspos, 0, s, times, 2ndsneg, active, square, pi, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

from#(active(X))from#(X)from#(mark(X))from#(X)

Problem 6: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

square#(mark(X))square#(X)square#(active(X))square#(X)

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(2ndspos(0, Z))mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z))))mark(rcons(posrecip(Y), 2ndsneg(N, Z)))active(2ndsneg(0, Z))mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z))))mark(rcons(negrecip(Y), 2ndspos(N, Z)))active(pi(X))mark(2ndspos(X, from(0)))
active(plus(0, Y))mark(Y)active(plus(s(X), Y))mark(s(plus(X, Y)))
active(times(0, Y))mark(0)active(times(s(X), Y))mark(plus(Y, times(X, Y)))
active(square(X))mark(times(X, X))mark(from(X))active(from(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(2ndspos(X1, X2))active(2ndspos(mark(X1), mark(X2)))mark(0)active(0)
mark(rnil)active(rnil)mark(rcons(X1, X2))active(rcons(mark(X1), mark(X2)))
mark(posrecip(X))active(posrecip(mark(X)))mark(2ndsneg(X1, X2))active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X))active(negrecip(mark(X)))mark(pi(X))active(pi(mark(X)))
mark(plus(X1, X2))active(plus(mark(X1), mark(X2)))mark(times(X1, X2))active(times(mark(X1), mark(X2)))
mark(square(X))active(square(mark(X)))from(mark(X))from(X)
from(active(X))from(X)cons(mark(X1), X2)cons(X1, X2)
cons(X1, mark(X2))cons(X1, X2)cons(active(X1), X2)cons(X1, X2)
cons(X1, active(X2))cons(X1, X2)s(mark(X))s(X)
s(active(X))s(X)2ndspos(mark(X1), X2)2ndspos(X1, X2)
2ndspos(X1, mark(X2))2ndspos(X1, X2)2ndspos(active(X1), X2)2ndspos(X1, X2)
2ndspos(X1, active(X2))2ndspos(X1, X2)rcons(mark(X1), X2)rcons(X1, X2)
rcons(X1, mark(X2))rcons(X1, X2)rcons(active(X1), X2)rcons(X1, X2)
rcons(X1, active(X2))rcons(X1, X2)posrecip(mark(X))posrecip(X)
posrecip(active(X))posrecip(X)2ndsneg(mark(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, mark(X2))2ndsneg(X1, X2)2ndsneg(active(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, active(X2))2ndsneg(X1, X2)negrecip(mark(X))negrecip(X)
negrecip(active(X))negrecip(X)pi(mark(X))pi(X)
pi(active(X))pi(X)plus(mark(X1), X2)plus(X1, X2)
plus(X1, mark(X2))plus(X1, X2)plus(active(X1), X2)plus(X1, X2)
plus(X1, active(X2))plus(X1, X2)times(mark(X1), X2)times(X1, X2)
times(X1, mark(X2))times(X1, X2)times(active(X1), X2)times(X1, X2)
times(X1, active(X2))times(X1, X2)square(mark(X))square(X)
square(active(X))square(X)

Original Signature

Termination of terms over the following signature is verified: plus, posrecip, negrecip, rnil, mark, from, rcons, 2ndspos, 0, s, times, 2ndsneg, active, square, pi, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

square#(mark(X))square#(X)square#(active(X))square#(X)

Problem 7: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

rcons#(active(X1), X2)rcons#(X1, X2)rcons#(X1, active(X2))rcons#(X1, X2)
rcons#(mark(X1), X2)rcons#(X1, X2)rcons#(X1, mark(X2))rcons#(X1, X2)

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(2ndspos(0, Z))mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z))))mark(rcons(posrecip(Y), 2ndsneg(N, Z)))active(2ndsneg(0, Z))mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z))))mark(rcons(negrecip(Y), 2ndspos(N, Z)))active(pi(X))mark(2ndspos(X, from(0)))
active(plus(0, Y))mark(Y)active(plus(s(X), Y))mark(s(plus(X, Y)))
active(times(0, Y))mark(0)active(times(s(X), Y))mark(plus(Y, times(X, Y)))
active(square(X))mark(times(X, X))mark(from(X))active(from(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(2ndspos(X1, X2))active(2ndspos(mark(X1), mark(X2)))mark(0)active(0)
mark(rnil)active(rnil)mark(rcons(X1, X2))active(rcons(mark(X1), mark(X2)))
mark(posrecip(X))active(posrecip(mark(X)))mark(2ndsneg(X1, X2))active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X))active(negrecip(mark(X)))mark(pi(X))active(pi(mark(X)))
mark(plus(X1, X2))active(plus(mark(X1), mark(X2)))mark(times(X1, X2))active(times(mark(X1), mark(X2)))
mark(square(X))active(square(mark(X)))from(mark(X))from(X)
from(active(X))from(X)cons(mark(X1), X2)cons(X1, X2)
cons(X1, mark(X2))cons(X1, X2)cons(active(X1), X2)cons(X1, X2)
cons(X1, active(X2))cons(X1, X2)s(mark(X))s(X)
s(active(X))s(X)2ndspos(mark(X1), X2)2ndspos(X1, X2)
2ndspos(X1, mark(X2))2ndspos(X1, X2)2ndspos(active(X1), X2)2ndspos(X1, X2)
2ndspos(X1, active(X2))2ndspos(X1, X2)rcons(mark(X1), X2)rcons(X1, X2)
rcons(X1, mark(X2))rcons(X1, X2)rcons(active(X1), X2)rcons(X1, X2)
rcons(X1, active(X2))rcons(X1, X2)posrecip(mark(X))posrecip(X)
posrecip(active(X))posrecip(X)2ndsneg(mark(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, mark(X2))2ndsneg(X1, X2)2ndsneg(active(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, active(X2))2ndsneg(X1, X2)negrecip(mark(X))negrecip(X)
negrecip(active(X))negrecip(X)pi(mark(X))pi(X)
pi(active(X))pi(X)plus(mark(X1), X2)plus(X1, X2)
plus(X1, mark(X2))plus(X1, X2)plus(active(X1), X2)plus(X1, X2)
plus(X1, active(X2))plus(X1, X2)times(mark(X1), X2)times(X1, X2)
times(X1, mark(X2))times(X1, X2)times(active(X1), X2)times(X1, X2)
times(X1, active(X2))times(X1, X2)square(mark(X))square(X)
square(active(X))square(X)

Original Signature

Termination of terms over the following signature is verified: plus, posrecip, negrecip, rnil, mark, from, rcons, 2ndspos, 0, s, times, 2ndsneg, active, square, pi, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

rcons#(active(X1), X2)rcons#(X1, X2)rcons#(mark(X1), X2)rcons#(X1, X2)

Problem 16: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

rcons#(X1, active(X2))rcons#(X1, X2)rcons#(X1, mark(X2))rcons#(X1, X2)

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(2ndspos(0, Z))mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z))))mark(rcons(posrecip(Y), 2ndsneg(N, Z)))active(2ndsneg(0, Z))mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z))))mark(rcons(negrecip(Y), 2ndspos(N, Z)))active(pi(X))mark(2ndspos(X, from(0)))
active(plus(0, Y))mark(Y)active(plus(s(X), Y))mark(s(plus(X, Y)))
active(times(0, Y))mark(0)active(times(s(X), Y))mark(plus(Y, times(X, Y)))
active(square(X))mark(times(X, X))mark(from(X))active(from(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(2ndspos(X1, X2))active(2ndspos(mark(X1), mark(X2)))mark(0)active(0)
mark(rnil)active(rnil)mark(rcons(X1, X2))active(rcons(mark(X1), mark(X2)))
mark(posrecip(X))active(posrecip(mark(X)))mark(2ndsneg(X1, X2))active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X))active(negrecip(mark(X)))mark(pi(X))active(pi(mark(X)))
mark(plus(X1, X2))active(plus(mark(X1), mark(X2)))mark(times(X1, X2))active(times(mark(X1), mark(X2)))
mark(square(X))active(square(mark(X)))from(mark(X))from(X)
from(active(X))from(X)cons(mark(X1), X2)cons(X1, X2)
cons(X1, mark(X2))cons(X1, X2)cons(active(X1), X2)cons(X1, X2)
cons(X1, active(X2))cons(X1, X2)s(mark(X))s(X)
s(active(X))s(X)2ndspos(mark(X1), X2)2ndspos(X1, X2)
2ndspos(X1, mark(X2))2ndspos(X1, X2)2ndspos(active(X1), X2)2ndspos(X1, X2)
2ndspos(X1, active(X2))2ndspos(X1, X2)rcons(mark(X1), X2)rcons(X1, X2)
rcons(X1, mark(X2))rcons(X1, X2)rcons(active(X1), X2)rcons(X1, X2)
rcons(X1, active(X2))rcons(X1, X2)posrecip(mark(X))posrecip(X)
posrecip(active(X))posrecip(X)2ndsneg(mark(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, mark(X2))2ndsneg(X1, X2)2ndsneg(active(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, active(X2))2ndsneg(X1, X2)negrecip(mark(X))negrecip(X)
negrecip(active(X))negrecip(X)pi(mark(X))pi(X)
pi(active(X))pi(X)plus(mark(X1), X2)plus(X1, X2)
plus(X1, mark(X2))plus(X1, X2)plus(active(X1), X2)plus(X1, X2)
plus(X1, active(X2))plus(X1, X2)times(mark(X1), X2)times(X1, X2)
times(X1, mark(X2))times(X1, X2)times(active(X1), X2)times(X1, X2)
times(X1, active(X2))times(X1, X2)square(mark(X))square(X)
square(active(X))square(X)

Original Signature

Termination of terms over the following signature is verified: plus, posrecip, negrecip, rnil, mark, from, rcons, 2ndspos, 0, s, times, 2ndsneg, active, square, pi, cons

Strategy


Polynomial Interpretation

There are no usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

rcons#(X1, mark(X2))rcons#(X1, X2)

Problem 21: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

rcons#(X1, active(X2))rcons#(X1, X2)

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(2ndspos(0, Z))mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z))))mark(rcons(posrecip(Y), 2ndsneg(N, Z)))active(2ndsneg(0, Z))mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z))))mark(rcons(negrecip(Y), 2ndspos(N, Z)))active(pi(X))mark(2ndspos(X, from(0)))
active(plus(0, Y))mark(Y)active(plus(s(X), Y))mark(s(plus(X, Y)))
active(times(0, Y))mark(0)active(times(s(X), Y))mark(plus(Y, times(X, Y)))
active(square(X))mark(times(X, X))mark(from(X))active(from(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(2ndspos(X1, X2))active(2ndspos(mark(X1), mark(X2)))mark(0)active(0)
mark(rnil)active(rnil)mark(rcons(X1, X2))active(rcons(mark(X1), mark(X2)))
mark(posrecip(X))active(posrecip(mark(X)))mark(2ndsneg(X1, X2))active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X))active(negrecip(mark(X)))mark(pi(X))active(pi(mark(X)))
mark(plus(X1, X2))active(plus(mark(X1), mark(X2)))mark(times(X1, X2))active(times(mark(X1), mark(X2)))
mark(square(X))active(square(mark(X)))from(mark(X))from(X)
from(active(X))from(X)cons(mark(X1), X2)cons(X1, X2)
cons(X1, mark(X2))cons(X1, X2)cons(active(X1), X2)cons(X1, X2)
cons(X1, active(X2))cons(X1, X2)s(mark(X))s(X)
s(active(X))s(X)2ndspos(mark(X1), X2)2ndspos(X1, X2)
2ndspos(X1, mark(X2))2ndspos(X1, X2)2ndspos(active(X1), X2)2ndspos(X1, X2)
2ndspos(X1, active(X2))2ndspos(X1, X2)rcons(mark(X1), X2)rcons(X1, X2)
rcons(X1, mark(X2))rcons(X1, X2)rcons(active(X1), X2)rcons(X1, X2)
rcons(X1, active(X2))rcons(X1, X2)posrecip(mark(X))posrecip(X)
posrecip(active(X))posrecip(X)2ndsneg(mark(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, mark(X2))2ndsneg(X1, X2)2ndsneg(active(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, active(X2))2ndsneg(X1, X2)negrecip(mark(X))negrecip(X)
negrecip(active(X))negrecip(X)pi(mark(X))pi(X)
pi(active(X))pi(X)plus(mark(X1), X2)plus(X1, X2)
plus(X1, mark(X2))plus(X1, X2)plus(active(X1), X2)plus(X1, X2)
plus(X1, active(X2))plus(X1, X2)times(mark(X1), X2)times(X1, X2)
times(X1, mark(X2))times(X1, X2)times(active(X1), X2)times(X1, X2)
times(X1, active(X2))times(X1, X2)square(mark(X))square(X)
square(active(X))square(X)

Original Signature

Termination of terms over the following signature is verified: plus, posrecip, negrecip, rnil, mark, from, rcons, 2ndspos, 0, s, times, 2ndsneg, active, square, pi, cons

Strategy


Polynomial Interpretation

There are no usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

rcons#(X1, active(X2))rcons#(X1, X2)

Problem 8: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

pi#(mark(X))pi#(X)pi#(active(X))pi#(X)

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(2ndspos(0, Z))mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z))))mark(rcons(posrecip(Y), 2ndsneg(N, Z)))active(2ndsneg(0, Z))mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z))))mark(rcons(negrecip(Y), 2ndspos(N, Z)))active(pi(X))mark(2ndspos(X, from(0)))
active(plus(0, Y))mark(Y)active(plus(s(X), Y))mark(s(plus(X, Y)))
active(times(0, Y))mark(0)active(times(s(X), Y))mark(plus(Y, times(X, Y)))
active(square(X))mark(times(X, X))mark(from(X))active(from(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(2ndspos(X1, X2))active(2ndspos(mark(X1), mark(X2)))mark(0)active(0)
mark(rnil)active(rnil)mark(rcons(X1, X2))active(rcons(mark(X1), mark(X2)))
mark(posrecip(X))active(posrecip(mark(X)))mark(2ndsneg(X1, X2))active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X))active(negrecip(mark(X)))mark(pi(X))active(pi(mark(X)))
mark(plus(X1, X2))active(plus(mark(X1), mark(X2)))mark(times(X1, X2))active(times(mark(X1), mark(X2)))
mark(square(X))active(square(mark(X)))from(mark(X))from(X)
from(active(X))from(X)cons(mark(X1), X2)cons(X1, X2)
cons(X1, mark(X2))cons(X1, X2)cons(active(X1), X2)cons(X1, X2)
cons(X1, active(X2))cons(X1, X2)s(mark(X))s(X)
s(active(X))s(X)2ndspos(mark(X1), X2)2ndspos(X1, X2)
2ndspos(X1, mark(X2))2ndspos(X1, X2)2ndspos(active(X1), X2)2ndspos(X1, X2)
2ndspos(X1, active(X2))2ndspos(X1, X2)rcons(mark(X1), X2)rcons(X1, X2)
rcons(X1, mark(X2))rcons(X1, X2)rcons(active(X1), X2)rcons(X1, X2)
rcons(X1, active(X2))rcons(X1, X2)posrecip(mark(X))posrecip(X)
posrecip(active(X))posrecip(X)2ndsneg(mark(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, mark(X2))2ndsneg(X1, X2)2ndsneg(active(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, active(X2))2ndsneg(X1, X2)negrecip(mark(X))negrecip(X)
negrecip(active(X))negrecip(X)pi(mark(X))pi(X)
pi(active(X))pi(X)plus(mark(X1), X2)plus(X1, X2)
plus(X1, mark(X2))plus(X1, X2)plus(active(X1), X2)plus(X1, X2)
plus(X1, active(X2))plus(X1, X2)times(mark(X1), X2)times(X1, X2)
times(X1, mark(X2))times(X1, X2)times(active(X1), X2)times(X1, X2)
times(X1, active(X2))times(X1, X2)square(mark(X))square(X)
square(active(X))square(X)

Original Signature

Termination of terms over the following signature is verified: plus, posrecip, negrecip, rnil, mark, from, rcons, 2ndspos, 0, s, times, 2ndsneg, active, square, pi, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

pi#(mark(X))pi#(X)pi#(active(X))pi#(X)

Problem 9: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

plus#(X1, active(X2))plus#(X1, X2)plus#(X1, mark(X2))plus#(X1, X2)
plus#(mark(X1), X2)plus#(X1, X2)plus#(active(X1), X2)plus#(X1, X2)

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(2ndspos(0, Z))mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z))))mark(rcons(posrecip(Y), 2ndsneg(N, Z)))active(2ndsneg(0, Z))mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z))))mark(rcons(negrecip(Y), 2ndspos(N, Z)))active(pi(X))mark(2ndspos(X, from(0)))
active(plus(0, Y))mark(Y)active(plus(s(X), Y))mark(s(plus(X, Y)))
active(times(0, Y))mark(0)active(times(s(X), Y))mark(plus(Y, times(X, Y)))
active(square(X))mark(times(X, X))mark(from(X))active(from(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(2ndspos(X1, X2))active(2ndspos(mark(X1), mark(X2)))mark(0)active(0)
mark(rnil)active(rnil)mark(rcons(X1, X2))active(rcons(mark(X1), mark(X2)))
mark(posrecip(X))active(posrecip(mark(X)))mark(2ndsneg(X1, X2))active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X))active(negrecip(mark(X)))mark(pi(X))active(pi(mark(X)))
mark(plus(X1, X2))active(plus(mark(X1), mark(X2)))mark(times(X1, X2))active(times(mark(X1), mark(X2)))
mark(square(X))active(square(mark(X)))from(mark(X))from(X)
from(active(X))from(X)cons(mark(X1), X2)cons(X1, X2)
cons(X1, mark(X2))cons(X1, X2)cons(active(X1), X2)cons(X1, X2)
cons(X1, active(X2))cons(X1, X2)s(mark(X))s(X)
s(active(X))s(X)2ndspos(mark(X1), X2)2ndspos(X1, X2)
2ndspos(X1, mark(X2))2ndspos(X1, X2)2ndspos(active(X1), X2)2ndspos(X1, X2)
2ndspos(X1, active(X2))2ndspos(X1, X2)rcons(mark(X1), X2)rcons(X1, X2)
rcons(X1, mark(X2))rcons(X1, X2)rcons(active(X1), X2)rcons(X1, X2)
rcons(X1, active(X2))rcons(X1, X2)posrecip(mark(X))posrecip(X)
posrecip(active(X))posrecip(X)2ndsneg(mark(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, mark(X2))2ndsneg(X1, X2)2ndsneg(active(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, active(X2))2ndsneg(X1, X2)negrecip(mark(X))negrecip(X)
negrecip(active(X))negrecip(X)pi(mark(X))pi(X)
pi(active(X))pi(X)plus(mark(X1), X2)plus(X1, X2)
plus(X1, mark(X2))plus(X1, X2)plus(active(X1), X2)plus(X1, X2)
plus(X1, active(X2))plus(X1, X2)times(mark(X1), X2)times(X1, X2)
times(X1, mark(X2))times(X1, X2)times(active(X1), X2)times(X1, X2)
times(X1, active(X2))times(X1, X2)square(mark(X))square(X)
square(active(X))square(X)

Original Signature

Termination of terms over the following signature is verified: plus, posrecip, negrecip, rnil, mark, from, rcons, 2ndspos, 0, s, times, 2ndsneg, active, square, pi, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

plus#(mark(X1), X2)plus#(X1, X2)plus#(active(X1), X2)plus#(X1, X2)

Problem 17: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

plus#(X1, mark(X2))plus#(X1, X2)plus#(X1, active(X2))plus#(X1, X2)

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(2ndspos(0, Z))mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z))))mark(rcons(posrecip(Y), 2ndsneg(N, Z)))active(2ndsneg(0, Z))mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z))))mark(rcons(negrecip(Y), 2ndspos(N, Z)))active(pi(X))mark(2ndspos(X, from(0)))
active(plus(0, Y))mark(Y)active(plus(s(X), Y))mark(s(plus(X, Y)))
active(times(0, Y))mark(0)active(times(s(X), Y))mark(plus(Y, times(X, Y)))
active(square(X))mark(times(X, X))mark(from(X))active(from(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(2ndspos(X1, X2))active(2ndspos(mark(X1), mark(X2)))mark(0)active(0)
mark(rnil)active(rnil)mark(rcons(X1, X2))active(rcons(mark(X1), mark(X2)))
mark(posrecip(X))active(posrecip(mark(X)))mark(2ndsneg(X1, X2))active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X))active(negrecip(mark(X)))mark(pi(X))active(pi(mark(X)))
mark(plus(X1, X2))active(plus(mark(X1), mark(X2)))mark(times(X1, X2))active(times(mark(X1), mark(X2)))
mark(square(X))active(square(mark(X)))from(mark(X))from(X)
from(active(X))from(X)cons(mark(X1), X2)cons(X1, X2)
cons(X1, mark(X2))cons(X1, X2)cons(active(X1), X2)cons(X1, X2)
cons(X1, active(X2))cons(X1, X2)s(mark(X))s(X)
s(active(X))s(X)2ndspos(mark(X1), X2)2ndspos(X1, X2)
2ndspos(X1, mark(X2))2ndspos(X1, X2)2ndspos(active(X1), X2)2ndspos(X1, X2)
2ndspos(X1, active(X2))2ndspos(X1, X2)rcons(mark(X1), X2)rcons(X1, X2)
rcons(X1, mark(X2))rcons(X1, X2)rcons(active(X1), X2)rcons(X1, X2)
rcons(X1, active(X2))rcons(X1, X2)posrecip(mark(X))posrecip(X)
posrecip(active(X))posrecip(X)2ndsneg(mark(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, mark(X2))2ndsneg(X1, X2)2ndsneg(active(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, active(X2))2ndsneg(X1, X2)negrecip(mark(X))negrecip(X)
negrecip(active(X))negrecip(X)pi(mark(X))pi(X)
pi(active(X))pi(X)plus(mark(X1), X2)plus(X1, X2)
plus(X1, mark(X2))plus(X1, X2)plus(active(X1), X2)plus(X1, X2)
plus(X1, active(X2))plus(X1, X2)times(mark(X1), X2)times(X1, X2)
times(X1, mark(X2))times(X1, X2)times(active(X1), X2)times(X1, X2)
times(X1, active(X2))times(X1, X2)square(mark(X))square(X)
square(active(X))square(X)

Original Signature

Termination of terms over the following signature is verified: plus, posrecip, negrecip, rnil, mark, from, rcons, 2ndspos, 0, s, times, 2ndsneg, active, square, pi, cons

Strategy


Polynomial Interpretation

There are no usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

plus#(X1, active(X2))plus#(X1, X2)plus#(X1, mark(X2))plus#(X1, X2)

Problem 10: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

times#(active(X1), X2)times#(X1, X2)times#(X1, active(X2))times#(X1, X2)
times#(X1, mark(X2))times#(X1, X2)times#(mark(X1), X2)times#(X1, X2)

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(2ndspos(0, Z))mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z))))mark(rcons(posrecip(Y), 2ndsneg(N, Z)))active(2ndsneg(0, Z))mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z))))mark(rcons(negrecip(Y), 2ndspos(N, Z)))active(pi(X))mark(2ndspos(X, from(0)))
active(plus(0, Y))mark(Y)active(plus(s(X), Y))mark(s(plus(X, Y)))
active(times(0, Y))mark(0)active(times(s(X), Y))mark(plus(Y, times(X, Y)))
active(square(X))mark(times(X, X))mark(from(X))active(from(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(2ndspos(X1, X2))active(2ndspos(mark(X1), mark(X2)))mark(0)active(0)
mark(rnil)active(rnil)mark(rcons(X1, X2))active(rcons(mark(X1), mark(X2)))
mark(posrecip(X))active(posrecip(mark(X)))mark(2ndsneg(X1, X2))active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X))active(negrecip(mark(X)))mark(pi(X))active(pi(mark(X)))
mark(plus(X1, X2))active(plus(mark(X1), mark(X2)))mark(times(X1, X2))active(times(mark(X1), mark(X2)))
mark(square(X))active(square(mark(X)))from(mark(X))from(X)
from(active(X))from(X)cons(mark(X1), X2)cons(X1, X2)
cons(X1, mark(X2))cons(X1, X2)cons(active(X1), X2)cons(X1, X2)
cons(X1, active(X2))cons(X1, X2)s(mark(X))s(X)
s(active(X))s(X)2ndspos(mark(X1), X2)2ndspos(X1, X2)
2ndspos(X1, mark(X2))2ndspos(X1, X2)2ndspos(active(X1), X2)2ndspos(X1, X2)
2ndspos(X1, active(X2))2ndspos(X1, X2)rcons(mark(X1), X2)rcons(X1, X2)
rcons(X1, mark(X2))rcons(X1, X2)rcons(active(X1), X2)rcons(X1, X2)
rcons(X1, active(X2))rcons(X1, X2)posrecip(mark(X))posrecip(X)
posrecip(active(X))posrecip(X)2ndsneg(mark(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, mark(X2))2ndsneg(X1, X2)2ndsneg(active(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, active(X2))2ndsneg(X1, X2)negrecip(mark(X))negrecip(X)
negrecip(active(X))negrecip(X)pi(mark(X))pi(X)
pi(active(X))pi(X)plus(mark(X1), X2)plus(X1, X2)
plus(X1, mark(X2))plus(X1, X2)plus(active(X1), X2)plus(X1, X2)
plus(X1, active(X2))plus(X1, X2)times(mark(X1), X2)times(X1, X2)
times(X1, mark(X2))times(X1, X2)times(active(X1), X2)times(X1, X2)
times(X1, active(X2))times(X1, X2)square(mark(X))square(X)
square(active(X))square(X)

Original Signature

Termination of terms over the following signature is verified: plus, posrecip, negrecip, rnil, mark, from, rcons, 2ndspos, 0, s, times, 2ndsneg, active, square, pi, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

times#(active(X1), X2)times#(X1, X2)times#(mark(X1), X2)times#(X1, X2)

Problem 18: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

times#(X1, active(X2))times#(X1, X2)times#(X1, mark(X2))times#(X1, X2)

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(2ndspos(0, Z))mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z))))mark(rcons(posrecip(Y), 2ndsneg(N, Z)))active(2ndsneg(0, Z))mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z))))mark(rcons(negrecip(Y), 2ndspos(N, Z)))active(pi(X))mark(2ndspos(X, from(0)))
active(plus(0, Y))mark(Y)active(plus(s(X), Y))mark(s(plus(X, Y)))
active(times(0, Y))mark(0)active(times(s(X), Y))mark(plus(Y, times(X, Y)))
active(square(X))mark(times(X, X))mark(from(X))active(from(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(2ndspos(X1, X2))active(2ndspos(mark(X1), mark(X2)))mark(0)active(0)
mark(rnil)active(rnil)mark(rcons(X1, X2))active(rcons(mark(X1), mark(X2)))
mark(posrecip(X))active(posrecip(mark(X)))mark(2ndsneg(X1, X2))active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X))active(negrecip(mark(X)))mark(pi(X))active(pi(mark(X)))
mark(plus(X1, X2))active(plus(mark(X1), mark(X2)))mark(times(X1, X2))active(times(mark(X1), mark(X2)))
mark(square(X))active(square(mark(X)))from(mark(X))from(X)
from(active(X))from(X)cons(mark(X1), X2)cons(X1, X2)
cons(X1, mark(X2))cons(X1, X2)cons(active(X1), X2)cons(X1, X2)
cons(X1, active(X2))cons(X1, X2)s(mark(X))s(X)
s(active(X))s(X)2ndspos(mark(X1), X2)2ndspos(X1, X2)
2ndspos(X1, mark(X2))2ndspos(X1, X2)2ndspos(active(X1), X2)2ndspos(X1, X2)
2ndspos(X1, active(X2))2ndspos(X1, X2)rcons(mark(X1), X2)rcons(X1, X2)
rcons(X1, mark(X2))rcons(X1, X2)rcons(active(X1), X2)rcons(X1, X2)
rcons(X1, active(X2))rcons(X1, X2)posrecip(mark(X))posrecip(X)
posrecip(active(X))posrecip(X)2ndsneg(mark(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, mark(X2))2ndsneg(X1, X2)2ndsneg(active(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, active(X2))2ndsneg(X1, X2)negrecip(mark(X))negrecip(X)
negrecip(active(X))negrecip(X)pi(mark(X))pi(X)
pi(active(X))pi(X)plus(mark(X1), X2)plus(X1, X2)
plus(X1, mark(X2))plus(X1, X2)plus(active(X1), X2)plus(X1, X2)
plus(X1, active(X2))plus(X1, X2)times(mark(X1), X2)times(X1, X2)
times(X1, mark(X2))times(X1, X2)times(active(X1), X2)times(X1, X2)
times(X1, active(X2))times(X1, X2)square(mark(X))square(X)
square(active(X))square(X)

Original Signature

Termination of terms over the following signature is verified: plus, posrecip, negrecip, rnil, mark, from, rcons, 2ndspos, 0, s, times, 2ndsneg, active, square, pi, cons

Strategy


Polynomial Interpretation

There are no usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

times#(X1, active(X2))times#(X1, X2)times#(X1, mark(X2))times#(X1, X2)

Problem 11: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

2ndsneg#(X1, mark(X2))2ndsneg#(X1, X2)2ndsneg#(active(X1), X2)2ndsneg#(X1, X2)
2ndsneg#(mark(X1), X2)2ndsneg#(X1, X2)2ndsneg#(X1, active(X2))2ndsneg#(X1, X2)

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(2ndspos(0, Z))mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z))))mark(rcons(posrecip(Y), 2ndsneg(N, Z)))active(2ndsneg(0, Z))mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z))))mark(rcons(negrecip(Y), 2ndspos(N, Z)))active(pi(X))mark(2ndspos(X, from(0)))
active(plus(0, Y))mark(Y)active(plus(s(X), Y))mark(s(plus(X, Y)))
active(times(0, Y))mark(0)active(times(s(X), Y))mark(plus(Y, times(X, Y)))
active(square(X))mark(times(X, X))mark(from(X))active(from(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(2ndspos(X1, X2))active(2ndspos(mark(X1), mark(X2)))mark(0)active(0)
mark(rnil)active(rnil)mark(rcons(X1, X2))active(rcons(mark(X1), mark(X2)))
mark(posrecip(X))active(posrecip(mark(X)))mark(2ndsneg(X1, X2))active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X))active(negrecip(mark(X)))mark(pi(X))active(pi(mark(X)))
mark(plus(X1, X2))active(plus(mark(X1), mark(X2)))mark(times(X1, X2))active(times(mark(X1), mark(X2)))
mark(square(X))active(square(mark(X)))from(mark(X))from(X)
from(active(X))from(X)cons(mark(X1), X2)cons(X1, X2)
cons(X1, mark(X2))cons(X1, X2)cons(active(X1), X2)cons(X1, X2)
cons(X1, active(X2))cons(X1, X2)s(mark(X))s(X)
s(active(X))s(X)2ndspos(mark(X1), X2)2ndspos(X1, X2)
2ndspos(X1, mark(X2))2ndspos(X1, X2)2ndspos(active(X1), X2)2ndspos(X1, X2)
2ndspos(X1, active(X2))2ndspos(X1, X2)rcons(mark(X1), X2)rcons(X1, X2)
rcons(X1, mark(X2))rcons(X1, X2)rcons(active(X1), X2)rcons(X1, X2)
rcons(X1, active(X2))rcons(X1, X2)posrecip(mark(X))posrecip(X)
posrecip(active(X))posrecip(X)2ndsneg(mark(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, mark(X2))2ndsneg(X1, X2)2ndsneg(active(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, active(X2))2ndsneg(X1, X2)negrecip(mark(X))negrecip(X)
negrecip(active(X))negrecip(X)pi(mark(X))pi(X)
pi(active(X))pi(X)plus(mark(X1), X2)plus(X1, X2)
plus(X1, mark(X2))plus(X1, X2)plus(active(X1), X2)plus(X1, X2)
plus(X1, active(X2))plus(X1, X2)times(mark(X1), X2)times(X1, X2)
times(X1, mark(X2))times(X1, X2)times(active(X1), X2)times(X1, X2)
times(X1, active(X2))times(X1, X2)square(mark(X))square(X)
square(active(X))square(X)

Original Signature

Termination of terms over the following signature is verified: plus, posrecip, negrecip, rnil, mark, from, rcons, 2ndspos, 0, s, times, 2ndsneg, active, square, pi, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

2ndsneg#(active(X1), X2)2ndsneg#(X1, X2)2ndsneg#(mark(X1), X2)2ndsneg#(X1, X2)

Problem 19: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

2ndsneg#(X1, mark(X2))2ndsneg#(X1, X2)2ndsneg#(X1, active(X2))2ndsneg#(X1, X2)

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(2ndspos(0, Z))mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z))))mark(rcons(posrecip(Y), 2ndsneg(N, Z)))active(2ndsneg(0, Z))mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z))))mark(rcons(negrecip(Y), 2ndspos(N, Z)))active(pi(X))mark(2ndspos(X, from(0)))
active(plus(0, Y))mark(Y)active(plus(s(X), Y))mark(s(plus(X, Y)))
active(times(0, Y))mark(0)active(times(s(X), Y))mark(plus(Y, times(X, Y)))
active(square(X))mark(times(X, X))mark(from(X))active(from(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(2ndspos(X1, X2))active(2ndspos(mark(X1), mark(X2)))mark(0)active(0)
mark(rnil)active(rnil)mark(rcons(X1, X2))active(rcons(mark(X1), mark(X2)))
mark(posrecip(X))active(posrecip(mark(X)))mark(2ndsneg(X1, X2))active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X))active(negrecip(mark(X)))mark(pi(X))active(pi(mark(X)))
mark(plus(X1, X2))active(plus(mark(X1), mark(X2)))mark(times(X1, X2))active(times(mark(X1), mark(X2)))
mark(square(X))active(square(mark(X)))from(mark(X))from(X)
from(active(X))from(X)cons(mark(X1), X2)cons(X1, X2)
cons(X1, mark(X2))cons(X1, X2)cons(active(X1), X2)cons(X1, X2)
cons(X1, active(X2))cons(X1, X2)s(mark(X))s(X)
s(active(X))s(X)2ndspos(mark(X1), X2)2ndspos(X1, X2)
2ndspos(X1, mark(X2))2ndspos(X1, X2)2ndspos(active(X1), X2)2ndspos(X1, X2)
2ndspos(X1, active(X2))2ndspos(X1, X2)rcons(mark(X1), X2)rcons(X1, X2)
rcons(X1, mark(X2))rcons(X1, X2)rcons(active(X1), X2)rcons(X1, X2)
rcons(X1, active(X2))rcons(X1, X2)posrecip(mark(X))posrecip(X)
posrecip(active(X))posrecip(X)2ndsneg(mark(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, mark(X2))2ndsneg(X1, X2)2ndsneg(active(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, active(X2))2ndsneg(X1, X2)negrecip(mark(X))negrecip(X)
negrecip(active(X))negrecip(X)pi(mark(X))pi(X)
pi(active(X))pi(X)plus(mark(X1), X2)plus(X1, X2)
plus(X1, mark(X2))plus(X1, X2)plus(active(X1), X2)plus(X1, X2)
plus(X1, active(X2))plus(X1, X2)times(mark(X1), X2)times(X1, X2)
times(X1, mark(X2))times(X1, X2)times(active(X1), X2)times(X1, X2)
times(X1, active(X2))times(X1, X2)square(mark(X))square(X)
square(active(X))square(X)

Original Signature

Termination of terms over the following signature is verified: plus, posrecip, negrecip, rnil, mark, from, rcons, 2ndspos, 0, s, times, 2ndsneg, active, square, pi, cons

Strategy


Polynomial Interpretation

There are no usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

2ndsneg#(X1, mark(X2))2ndsneg#(X1, X2)2ndsneg#(X1, active(X2))2ndsneg#(X1, X2)

Problem 12: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

cons#(X1, active(X2))cons#(X1, X2)cons#(mark(X1), X2)cons#(X1, X2)
cons#(active(X1), X2)cons#(X1, X2)cons#(X1, mark(X2))cons#(X1, X2)

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(2ndspos(0, Z))mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z))))mark(rcons(posrecip(Y), 2ndsneg(N, Z)))active(2ndsneg(0, Z))mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z))))mark(rcons(negrecip(Y), 2ndspos(N, Z)))active(pi(X))mark(2ndspos(X, from(0)))
active(plus(0, Y))mark(Y)active(plus(s(X), Y))mark(s(plus(X, Y)))
active(times(0, Y))mark(0)active(times(s(X), Y))mark(plus(Y, times(X, Y)))
active(square(X))mark(times(X, X))mark(from(X))active(from(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(2ndspos(X1, X2))active(2ndspos(mark(X1), mark(X2)))mark(0)active(0)
mark(rnil)active(rnil)mark(rcons(X1, X2))active(rcons(mark(X1), mark(X2)))
mark(posrecip(X))active(posrecip(mark(X)))mark(2ndsneg(X1, X2))active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X))active(negrecip(mark(X)))mark(pi(X))active(pi(mark(X)))
mark(plus(X1, X2))active(plus(mark(X1), mark(X2)))mark(times(X1, X2))active(times(mark(X1), mark(X2)))
mark(square(X))active(square(mark(X)))from(mark(X))from(X)
from(active(X))from(X)cons(mark(X1), X2)cons(X1, X2)
cons(X1, mark(X2))cons(X1, X2)cons(active(X1), X2)cons(X1, X2)
cons(X1, active(X2))cons(X1, X2)s(mark(X))s(X)
s(active(X))s(X)2ndspos(mark(X1), X2)2ndspos(X1, X2)
2ndspos(X1, mark(X2))2ndspos(X1, X2)2ndspos(active(X1), X2)2ndspos(X1, X2)
2ndspos(X1, active(X2))2ndspos(X1, X2)rcons(mark(X1), X2)rcons(X1, X2)
rcons(X1, mark(X2))rcons(X1, X2)rcons(active(X1), X2)rcons(X1, X2)
rcons(X1, active(X2))rcons(X1, X2)posrecip(mark(X))posrecip(X)
posrecip(active(X))posrecip(X)2ndsneg(mark(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, mark(X2))2ndsneg(X1, X2)2ndsneg(active(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, active(X2))2ndsneg(X1, X2)negrecip(mark(X))negrecip(X)
negrecip(active(X))negrecip(X)pi(mark(X))pi(X)
pi(active(X))pi(X)plus(mark(X1), X2)plus(X1, X2)
plus(X1, mark(X2))plus(X1, X2)plus(active(X1), X2)plus(X1, X2)
plus(X1, active(X2))plus(X1, X2)times(mark(X1), X2)times(X1, X2)
times(X1, mark(X2))times(X1, X2)times(active(X1), X2)times(X1, X2)
times(X1, active(X2))times(X1, X2)square(mark(X))square(X)
square(active(X))square(X)

Original Signature

Termination of terms over the following signature is verified: plus, posrecip, negrecip, rnil, mark, from, rcons, 2ndspos, 0, s, times, 2ndsneg, active, square, pi, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

cons#(mark(X1), X2)cons#(X1, X2)cons#(active(X1), X2)cons#(X1, X2)

Problem 20: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

cons#(X1, active(X2))cons#(X1, X2)cons#(X1, mark(X2))cons#(X1, X2)

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(2ndspos(0, Z))mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z))))mark(rcons(posrecip(Y), 2ndsneg(N, Z)))active(2ndsneg(0, Z))mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z))))mark(rcons(negrecip(Y), 2ndspos(N, Z)))active(pi(X))mark(2ndspos(X, from(0)))
active(plus(0, Y))mark(Y)active(plus(s(X), Y))mark(s(plus(X, Y)))
active(times(0, Y))mark(0)active(times(s(X), Y))mark(plus(Y, times(X, Y)))
active(square(X))mark(times(X, X))mark(from(X))active(from(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(2ndspos(X1, X2))active(2ndspos(mark(X1), mark(X2)))mark(0)active(0)
mark(rnil)active(rnil)mark(rcons(X1, X2))active(rcons(mark(X1), mark(X2)))
mark(posrecip(X))active(posrecip(mark(X)))mark(2ndsneg(X1, X2))active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X))active(negrecip(mark(X)))mark(pi(X))active(pi(mark(X)))
mark(plus(X1, X2))active(plus(mark(X1), mark(X2)))mark(times(X1, X2))active(times(mark(X1), mark(X2)))
mark(square(X))active(square(mark(X)))from(mark(X))from(X)
from(active(X))from(X)cons(mark(X1), X2)cons(X1, X2)
cons(X1, mark(X2))cons(X1, X2)cons(active(X1), X2)cons(X1, X2)
cons(X1, active(X2))cons(X1, X2)s(mark(X))s(X)
s(active(X))s(X)2ndspos(mark(X1), X2)2ndspos(X1, X2)
2ndspos(X1, mark(X2))2ndspos(X1, X2)2ndspos(active(X1), X2)2ndspos(X1, X2)
2ndspos(X1, active(X2))2ndspos(X1, X2)rcons(mark(X1), X2)rcons(X1, X2)
rcons(X1, mark(X2))rcons(X1, X2)rcons(active(X1), X2)rcons(X1, X2)
rcons(X1, active(X2))rcons(X1, X2)posrecip(mark(X))posrecip(X)
posrecip(active(X))posrecip(X)2ndsneg(mark(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, mark(X2))2ndsneg(X1, X2)2ndsneg(active(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, active(X2))2ndsneg(X1, X2)negrecip(mark(X))negrecip(X)
negrecip(active(X))negrecip(X)pi(mark(X))pi(X)
pi(active(X))pi(X)plus(mark(X1), X2)plus(X1, X2)
plus(X1, mark(X2))plus(X1, X2)plus(active(X1), X2)plus(X1, X2)
plus(X1, active(X2))plus(X1, X2)times(mark(X1), X2)times(X1, X2)
times(X1, mark(X2))times(X1, X2)times(active(X1), X2)times(X1, X2)
times(X1, active(X2))times(X1, X2)square(mark(X))square(X)
square(active(X))square(X)

Original Signature

Termination of terms over the following signature is verified: plus, posrecip, negrecip, rnil, mark, from, rcons, 2ndspos, 0, s, times, 2ndsneg, active, square, pi, cons

Strategy


Polynomial Interpretation

There are no usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

cons#(X1, mark(X2))cons#(X1, X2)

Problem 22: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

cons#(X1, active(X2))cons#(X1, X2)

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(2ndspos(0, Z))mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z))))mark(rcons(posrecip(Y), 2ndsneg(N, Z)))active(2ndsneg(0, Z))mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z))))mark(rcons(negrecip(Y), 2ndspos(N, Z)))active(pi(X))mark(2ndspos(X, from(0)))
active(plus(0, Y))mark(Y)active(plus(s(X), Y))mark(s(plus(X, Y)))
active(times(0, Y))mark(0)active(times(s(X), Y))mark(plus(Y, times(X, Y)))
active(square(X))mark(times(X, X))mark(from(X))active(from(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(2ndspos(X1, X2))active(2ndspos(mark(X1), mark(X2)))mark(0)active(0)
mark(rnil)active(rnil)mark(rcons(X1, X2))active(rcons(mark(X1), mark(X2)))
mark(posrecip(X))active(posrecip(mark(X)))mark(2ndsneg(X1, X2))active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X))active(negrecip(mark(X)))mark(pi(X))active(pi(mark(X)))
mark(plus(X1, X2))active(plus(mark(X1), mark(X2)))mark(times(X1, X2))active(times(mark(X1), mark(X2)))
mark(square(X))active(square(mark(X)))from(mark(X))from(X)
from(active(X))from(X)cons(mark(X1), X2)cons(X1, X2)
cons(X1, mark(X2))cons(X1, X2)cons(active(X1), X2)cons(X1, X2)
cons(X1, active(X2))cons(X1, X2)s(mark(X))s(X)
s(active(X))s(X)2ndspos(mark(X1), X2)2ndspos(X1, X2)
2ndspos(X1, mark(X2))2ndspos(X1, X2)2ndspos(active(X1), X2)2ndspos(X1, X2)
2ndspos(X1, active(X2))2ndspos(X1, X2)rcons(mark(X1), X2)rcons(X1, X2)
rcons(X1, mark(X2))rcons(X1, X2)rcons(active(X1), X2)rcons(X1, X2)
rcons(X1, active(X2))rcons(X1, X2)posrecip(mark(X))posrecip(X)
posrecip(active(X))posrecip(X)2ndsneg(mark(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, mark(X2))2ndsneg(X1, X2)2ndsneg(active(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, active(X2))2ndsneg(X1, X2)negrecip(mark(X))negrecip(X)
negrecip(active(X))negrecip(X)pi(mark(X))pi(X)
pi(active(X))pi(X)plus(mark(X1), X2)plus(X1, X2)
plus(X1, mark(X2))plus(X1, X2)plus(active(X1), X2)plus(X1, X2)
plus(X1, active(X2))plus(X1, X2)times(mark(X1), X2)times(X1, X2)
times(X1, mark(X2))times(X1, X2)times(active(X1), X2)times(X1, X2)
times(X1, active(X2))times(X1, X2)square(mark(X))square(X)
square(active(X))square(X)

Original Signature

Termination of terms over the following signature is verified: plus, posrecip, negrecip, rnil, mark, from, rcons, 2ndspos, 0, s, times, 2ndsneg, active, square, pi, cons

Strategy


Polynomial Interpretation

There are no usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

cons#(X1, active(X2))cons#(X1, X2)

Problem 14: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

negrecip#(active(X))negrecip#(X)negrecip#(mark(X))negrecip#(X)

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(2ndspos(0, Z))mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z))))mark(rcons(posrecip(Y), 2ndsneg(N, Z)))active(2ndsneg(0, Z))mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z))))mark(rcons(negrecip(Y), 2ndspos(N, Z)))active(pi(X))mark(2ndspos(X, from(0)))
active(plus(0, Y))mark(Y)active(plus(s(X), Y))mark(s(plus(X, Y)))
active(times(0, Y))mark(0)active(times(s(X), Y))mark(plus(Y, times(X, Y)))
active(square(X))mark(times(X, X))mark(from(X))active(from(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(2ndspos(X1, X2))active(2ndspos(mark(X1), mark(X2)))mark(0)active(0)
mark(rnil)active(rnil)mark(rcons(X1, X2))active(rcons(mark(X1), mark(X2)))
mark(posrecip(X))active(posrecip(mark(X)))mark(2ndsneg(X1, X2))active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X))active(negrecip(mark(X)))mark(pi(X))active(pi(mark(X)))
mark(plus(X1, X2))active(plus(mark(X1), mark(X2)))mark(times(X1, X2))active(times(mark(X1), mark(X2)))
mark(square(X))active(square(mark(X)))from(mark(X))from(X)
from(active(X))from(X)cons(mark(X1), X2)cons(X1, X2)
cons(X1, mark(X2))cons(X1, X2)cons(active(X1), X2)cons(X1, X2)
cons(X1, active(X2))cons(X1, X2)s(mark(X))s(X)
s(active(X))s(X)2ndspos(mark(X1), X2)2ndspos(X1, X2)
2ndspos(X1, mark(X2))2ndspos(X1, X2)2ndspos(active(X1), X2)2ndspos(X1, X2)
2ndspos(X1, active(X2))2ndspos(X1, X2)rcons(mark(X1), X2)rcons(X1, X2)
rcons(X1, mark(X2))rcons(X1, X2)rcons(active(X1), X2)rcons(X1, X2)
rcons(X1, active(X2))rcons(X1, X2)posrecip(mark(X))posrecip(X)
posrecip(active(X))posrecip(X)2ndsneg(mark(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, mark(X2))2ndsneg(X1, X2)2ndsneg(active(X1), X2)2ndsneg(X1, X2)
2ndsneg(X1, active(X2))2ndsneg(X1, X2)negrecip(mark(X))negrecip(X)
negrecip(active(X))negrecip(X)pi(mark(X))pi(X)
pi(active(X))pi(X)plus(mark(X1), X2)plus(X1, X2)
plus(X1, mark(X2))plus(X1, X2)plus(active(X1), X2)plus(X1, X2)
plus(X1, active(X2))plus(X1, X2)times(mark(X1), X2)times(X1, X2)
times(X1, mark(X2))times(X1, X2)times(active(X1), X2)times(X1, X2)
times(X1, active(X2))times(X1, X2)square(mark(X))square(X)
square(active(X))square(X)

Original Signature

Termination of terms over the following signature is verified: plus, posrecip, negrecip, rnil, mark, from, rcons, 2ndspos, 0, s, times, 2ndsneg, active, square, pi, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

negrecip#(active(X))negrecip#(X)negrecip#(mark(X))negrecip#(X)