TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60003 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (24ms).
| – Problem 2 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (4ms), PolynomialLinearRange4iUR (136ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (366ms), DependencyGraph (2ms), ReductionPairSAT (2083ms), DependencyGraph (1ms), SizeChangePrinciple (12ms), ForwardNarrowing (1ms), BackwardInstantiation (0ms), ForwardInstantiation (1ms), Propagation (1ms)].
| – Problem 3 remains open; application of the following processors failed [SubtermCriterion (2ms), DependencyGraph (2ms), PolynomialLinearRange4iUR (52ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (77ms), DependencyGraph (1ms), ReductionPairSAT (195ms), DependencyGraph (0ms), SizeChangePrinciple (4ms), ForwardNarrowing (0ms), BackwardInstantiation (0ms), ForwardInstantiation (1ms), Propagation (0ms)].
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| g#(f(s(x), s(y), z)) | → | g#(f(x, y, z)) |
Rewrite Rules
| f(s(a), s(b), x) | → | f(x, x, x) | | g(f(s(x), s(y), z)) | → | g(f(x, y, z)) |
| cons(x, y) | → | x | | cons(x, y) | → | y |
Original Signature
Termination of terms over the following signature is verified: f, g, b, s, a, cons
Open Dependency Pair Problem 3
Dependency Pairs
| f#(s(a), s(b), x) | → | f#(x, x, x) |
Rewrite Rules
| f(s(a), s(b), x) | → | f(x, x, x) | | g(f(s(x), s(y), z)) | → | g(f(x, y, z)) |
| cons(x, y) | → | x | | cons(x, y) | → | y |
Original Signature
Termination of terms over the following signature is verified: f, g, b, s, a, cons
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| g#(f(s(x), s(y), z)) | → | f#(x, y, z) | | f#(s(a), s(b), x) | → | f#(x, x, x) |
| g#(f(s(x), s(y), z)) | → | g#(f(x, y, z)) |
Rewrite Rules
| f(s(a), s(b), x) | → | f(x, x, x) | | g(f(s(x), s(y), z)) | → | g(f(x, y, z)) |
| cons(x, y) | → | x | | cons(x, y) | → | y |
Original Signature
Termination of terms over the following signature is verified: f, g, s, b, a, cons
Strategy
The following SCCs where found
| f#(s(a), s(b), x) → f#(x, x, x) |
| g#(f(s(x), s(y), z)) → g#(f(x, y, z)) |