TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60001 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (28ms).
 | – Problem 2 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (2ms), PolynomialLinearRange4iUR (125ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (237ms), DependencyGraph (1ms), ReductionPairSAT (725ms), DependencyGraph (1ms), SizeChangePrinciple (34ms), ForwardNarrowing (1ms), BackwardInstantiation (1ms), ForwardInstantiation (0ms), Propagation (0ms)].
 | – Problem 3 was processed with processor SubtermCriterion (1ms).

The following open problems remain:



Open Dependency Pair Problem 2

Dependency Pairs

f#(0, s(0), X)f#(X, double(X), X)

Rewrite Rules

+(X, 0)X+(X, s(Y))s(+(X, Y))
double(X)+(X, X)f(0, s(0), X)f(X, double(X), X)
g(X, Y)Xg(X, Y)Y

Original Signature

Termination of terms over the following signature is verified: f, g, 0, s, +, double


Problem 1: DependencyGraph



Dependency Pair Problem

Dependency Pairs

double#(X)+#(X, X)f#(0, s(0), X)double#(X)
f#(0, s(0), X)f#(X, double(X), X)+#(X, s(Y))+#(X, Y)

Rewrite Rules

+(X, 0)X+(X, s(Y))s(+(X, Y))
double(X)+(X, X)f(0, s(0), X)f(X, double(X), X)
g(X, Y)Xg(X, Y)Y

Original Signature

Termination of terms over the following signature is verified: f, g, 0, s, +, double

Strategy


The following SCCs where found

f#(0, s(0), X) → f#(X, double(X), X)

+#(X, s(Y)) → +#(X, Y)

Problem 3: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

+#(X, s(Y))+#(X, Y)

Rewrite Rules

+(X, 0)X+(X, s(Y))s(+(X, Y))
double(X)+(X, X)f(0, s(0), X)f(X, double(X), X)
g(X, Y)Xg(X, Y)Y

Original Signature

Termination of terms over the following signature is verified: f, g, 0, s, +, double

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

+#(X, s(Y))+#(X, Y)