MAYBE

The TRS could not be proven terminating. The proof attempt took 1017 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 (133ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (358ms), DependencyGraph (1ms), ReductionPairSAT (340ms), DependencyGraph (1ms), SizeChangePrinciple (6ms)].

The following open problems remain:



Open Dependency Pair Problem 3

Dependency Pairs

cond#(true, x, y)cond#(gr(x, y), y, x)

Rewrite Rules

cond(true, x, y)cond(gr(x, y), y, x)gr(0, x)false
gr(s(x), 0)truegr(s(x), s(y))gr(x, y)

Original Signature

Termination of terms over the following signature is verified: 0, s, false, true, gr, cond


Problem 1: DependencyGraph



Dependency Pair Problem

Dependency Pairs

cond#(true, x, y)gr#(x, y)gr#(s(x), s(y))gr#(x, y)
cond#(true, x, y)cond#(gr(x, y), y, x)

Rewrite Rules

cond(true, x, y)cond(gr(x, y), y, x)gr(0, x)false
gr(s(x), 0)truegr(s(x), s(y))gr(x, y)

Original Signature

Termination of terms over the following signature is verified: 0, s, true, false, gr, cond

Strategy


The following SCCs where found

gr#(s(x), s(y)) → gr#(x, y)

cond#(true, x, y) → cond#(gr(x, y), y, x)

Problem 2: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

gr#(s(x), s(y))gr#(x, y)

Rewrite Rules

cond(true, x, y)cond(gr(x, y), y, x)gr(0, x)false
gr(s(x), 0)truegr(s(x), s(y))gr(x, y)

Original Signature

Termination of terms over the following signature is verified: 0, s, true, false, gr, cond

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

gr#(s(x), s(y))gr#(x, y)