MAYBE

The TRS could not be proven terminating. The proof attempt took 372 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (0ms).
 | – Problem 2 was processed with processor SubtermCriterion (0ms).
 | – Problem 3 was processed with processor SubtermCriterion (0ms).
 |    | – Problem 6 was processed with processor DependencyGraph (0ms).
 | – Problem 4 was processed with processor SubtermCriterion (0ms).
 | – Problem 5 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (1ms), PolynomialLinearRange4iUR (66ms), DependencyGraph (2ms), PolynomialLinearRange8NegiUR (36ms), DependencyGraph (1ms), ReductionPairSAT (26ms), DependencyGraph (1ms), SizeChangePrinciple (5ms)].

The following open problems remain:



Open Dependency Pair Problem 5

Dependency Pairs

from#(X)from#(s(X))

Rewrite Rules

from(X)cons(X, from(s(X)))2ndspos(0, Z)rnil
2ndspos(s(N), cons(X, Z))2ndspos(s(N), cons2(X, Z))2ndspos(s(N), cons2(X, cons(Y, Z)))rcons(posrecip(Y), 2ndsneg(N, Z))
2ndsneg(0, Z)rnil2ndsneg(s(N), cons(X, Z))2ndsneg(s(N), cons2(X, Z))
2ndsneg(s(N), cons2(X, cons(Y, Z)))rcons(negrecip(Y), 2ndspos(N, Z))pi(X)2ndspos(X, from(0))
plus(0, Y)Yplus(s(X), Y)s(plus(X, Y))
times(0, Y)0times(s(X), Y)plus(Y, times(X, Y))
square(X)times(X, X)

Original Signature

Termination of terms over the following signature is verified: posrecip, negrecip, plus, cons2, rnil, from, rcons, 2ndspos, 0, s, times, 2ndsneg, square, pi, cons


Problem 1: DependencyGraph



Dependency Pair Problem

Dependency Pairs

2ndspos#(s(N), cons2(X, cons(Y, Z)))2ndsneg#(N, Z)square#(X)times#(X, X)
from#(X)from#(s(X))2ndsneg#(s(N), cons(X, Z))2ndsneg#(s(N), cons2(X, Z))
plus#(s(X), Y)plus#(X, Y)2ndspos#(s(N), cons(X, Z))2ndspos#(s(N), cons2(X, Z))
pi#(X)2ndspos#(X, from(0))pi#(X)from#(0)
times#(s(X), Y)plus#(Y, times(X, Y))2ndsneg#(s(N), cons2(X, cons(Y, Z)))2ndspos#(N, Z)
times#(s(X), Y)times#(X, Y)

Rewrite Rules

from(X)cons(X, from(s(X)))2ndspos(0, Z)rnil
2ndspos(s(N), cons(X, Z))2ndspos(s(N), cons2(X, Z))2ndspos(s(N), cons2(X, cons(Y, Z)))rcons(posrecip(Y), 2ndsneg(N, Z))
2ndsneg(0, Z)rnil2ndsneg(s(N), cons(X, Z))2ndsneg(s(N), cons2(X, Z))
2ndsneg(s(N), cons2(X, cons(Y, Z)))rcons(negrecip(Y), 2ndspos(N, Z))pi(X)2ndspos(X, from(0))
plus(0, Y)Yplus(s(X), Y)s(plus(X, Y))
times(0, Y)0times(s(X), Y)plus(Y, times(X, Y))
square(X)times(X, X)

Original Signature

Termination of terms over the following signature is verified: posrecip, negrecip, plus, cons2, rnil, from, rcons, 2ndspos, 0, s, times, 2ndsneg, square, pi, cons

Strategy


The following SCCs where found

2ndspos#(s(N), cons2(X, cons(Y, Z))) → 2ndsneg#(N, Z)2ndsneg#(s(N), cons(X, Z)) → 2ndsneg#(s(N), cons2(X, Z))
2ndspos#(s(N), cons(X, Z)) → 2ndspos#(s(N), cons2(X, Z))2ndsneg#(s(N), cons2(X, cons(Y, Z))) → 2ndspos#(N, Z)

from#(X) → from#(s(X))

plus#(s(X), Y) → plus#(X, Y)

times#(s(X), Y) → times#(X, Y)

Problem 2: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

times#(s(X), Y)times#(X, Y)

Rewrite Rules

from(X)cons(X, from(s(X)))2ndspos(0, Z)rnil
2ndspos(s(N), cons(X, Z))2ndspos(s(N), cons2(X, Z))2ndspos(s(N), cons2(X, cons(Y, Z)))rcons(posrecip(Y), 2ndsneg(N, Z))
2ndsneg(0, Z)rnil2ndsneg(s(N), cons(X, Z))2ndsneg(s(N), cons2(X, Z))
2ndsneg(s(N), cons2(X, cons(Y, Z)))rcons(negrecip(Y), 2ndspos(N, Z))pi(X)2ndspos(X, from(0))
plus(0, Y)Yplus(s(X), Y)s(plus(X, Y))
times(0, Y)0times(s(X), Y)plus(Y, times(X, Y))
square(X)times(X, X)

Original Signature

Termination of terms over the following signature is verified: posrecip, negrecip, plus, cons2, rnil, from, rcons, 2ndspos, 0, s, times, 2ndsneg, square, pi, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

times#(s(X), Y)times#(X, Y)

Problem 3: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

2ndspos#(s(N), cons2(X, cons(Y, Z)))2ndsneg#(N, Z)2ndsneg#(s(N), cons(X, Z))2ndsneg#(s(N), cons2(X, Z))
2ndspos#(s(N), cons(X, Z))2ndspos#(s(N), cons2(X, Z))2ndsneg#(s(N), cons2(X, cons(Y, Z)))2ndspos#(N, Z)

Rewrite Rules

from(X)cons(X, from(s(X)))2ndspos(0, Z)rnil
2ndspos(s(N), cons(X, Z))2ndspos(s(N), cons2(X, Z))2ndspos(s(N), cons2(X, cons(Y, Z)))rcons(posrecip(Y), 2ndsneg(N, Z))
2ndsneg(0, Z)rnil2ndsneg(s(N), cons(X, Z))2ndsneg(s(N), cons2(X, Z))
2ndsneg(s(N), cons2(X, cons(Y, Z)))rcons(negrecip(Y), 2ndspos(N, Z))pi(X)2ndspos(X, from(0))
plus(0, Y)Yplus(s(X), Y)s(plus(X, Y))
times(0, Y)0times(s(X), Y)plus(Y, times(X, Y))
square(X)times(X, X)

Original Signature

Termination of terms over the following signature is verified: posrecip, negrecip, plus, cons2, rnil, from, rcons, 2ndspos, 0, s, times, 2ndsneg, square, pi, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

2ndspos#(s(N), cons2(X, cons(Y, Z)))2ndsneg#(N, Z)2ndsneg#(s(N), cons2(X, cons(Y, Z)))2ndspos#(N, Z)

Problem 6: DependencyGraph



Dependency Pair Problem

Dependency Pairs

2ndsneg#(s(N), cons(X, Z))2ndsneg#(s(N), cons2(X, Z))2ndspos#(s(N), cons(X, Z))2ndspos#(s(N), cons2(X, Z))

Rewrite Rules

from(X)cons(X, from(s(X)))2ndspos(0, Z)rnil
2ndspos(s(N), cons(X, Z))2ndspos(s(N), cons2(X, Z))2ndspos(s(N), cons2(X, cons(Y, Z)))rcons(posrecip(Y), 2ndsneg(N, Z))
2ndsneg(0, Z)rnil2ndsneg(s(N), cons(X, Z))2ndsneg(s(N), cons2(X, Z))
2ndsneg(s(N), cons2(X, cons(Y, Z)))rcons(negrecip(Y), 2ndspos(N, Z))pi(X)2ndspos(X, from(0))
plus(0, Y)Yplus(s(X), Y)s(plus(X, Y))
times(0, Y)0times(s(X), Y)plus(Y, times(X, Y))
square(X)times(X, X)

Original Signature

Termination of terms over the following signature is verified: posrecip, negrecip, plus, cons2, rnil, from, rcons, 2ndspos, 0, s, times, 2ndsneg, square, pi, cons

Strategy


There are no SCCs!

Problem 4: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

plus#(s(X), Y)plus#(X, Y)

Rewrite Rules

from(X)cons(X, from(s(X)))2ndspos(0, Z)rnil
2ndspos(s(N), cons(X, Z))2ndspos(s(N), cons2(X, Z))2ndspos(s(N), cons2(X, cons(Y, Z)))rcons(posrecip(Y), 2ndsneg(N, Z))
2ndsneg(0, Z)rnil2ndsneg(s(N), cons(X, Z))2ndsneg(s(N), cons2(X, Z))
2ndsneg(s(N), cons2(X, cons(Y, Z)))rcons(negrecip(Y), 2ndspos(N, Z))pi(X)2ndspos(X, from(0))
plus(0, Y)Yplus(s(X), Y)s(plus(X, Y))
times(0, Y)0times(s(X), Y)plus(Y, times(X, Y))
square(X)times(X, X)

Original Signature

Termination of terms over the following signature is verified: posrecip, negrecip, plus, cons2, rnil, from, rcons, 2ndspos, 0, s, times, 2ndsneg, square, pi, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

plus#(s(X), Y)plus#(X, Y)