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) | → | X | | g(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) | → | X | | g(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) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
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) | → | X | | g(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: