MAYBE

The TRS could not be proven terminating. The proof attempt took 1014 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 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (1ms), PolynomialLinearRange4iUR (210ms), DependencyGraph (2ms), PolynomialLinearRange8NegiUR (301ms), DependencyGraph (1ms), ReductionPairSAT (355ms), DependencyGraph (1ms), SizeChangePrinciple (9ms)].

The following open problems remain:



Open Dependency Pair Problem 3

Dependency Pairs

f#(s(x), x)f#(s(x), round(s(x)))

Rewrite Rules

f(s(x), x)f(s(x), round(s(x)))round(0)0
round(0)s(0)round(s(0))s(0)
round(s(s(x)))s(s(round(x)))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, round


Problem 1: DependencyGraph



Dependency Pair Problem

Dependency Pairs

f#(s(x), x)f#(s(x), round(s(x)))f#(s(x), x)round#(s(x))
round#(s(s(x)))round#(x)

Rewrite Rules

f(s(x), x)f(s(x), round(s(x)))round(0)0
round(0)s(0)round(s(0))s(0)
round(s(s(x)))s(s(round(x)))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, round

Strategy


The following SCCs where found

f#(s(x), x) → f#(s(x), round(s(x)))

round#(s(s(x))) → round#(x)

Problem 2: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

round#(s(s(x)))round#(x)

Rewrite Rules

f(s(x), x)f(s(x), round(s(x)))round(0)0
round(0)s(0)round(s(0))s(0)
round(s(s(x)))s(s(round(x)))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, round

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

round#(s(s(x)))round#(x)