TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60000 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (21ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (1ms), PolynomialLinearRange4iUR (147ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (468ms), DependencyGraph (1ms), ReductionPairSAT (73ms), DependencyGraph (0ms), SizeChangePrinciple (11ms), ForwardNarrowing (1ms), BackwardInstantiation (1ms), ForwardInstantiation (0ms), Propagation (1ms)].
The following open problems remain:
Open Dependency Pair Problem 3
Dependency Pairs
| cond#(true, x) | → | cond#(odd(x), p(x)) |
Rewrite Rules
| cond(true, x) | → | cond(odd(x), p(x)) | | odd(0) | → | false |
| odd(s(0)) | → | true | | odd(s(s(x))) | → | odd(x) |
| p(0) | → | 0 | | p(s(x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: 0, s, p, false, true, odd, cond
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| odd#(s(s(x))) | → | odd#(x) | | cond#(true, x) | → | odd#(x) |
| cond#(true, x) | → | p#(x) | | cond#(true, x) | → | cond#(odd(x), p(x)) |
Rewrite Rules
| cond(true, x) | → | cond(odd(x), p(x)) | | odd(0) | → | false |
| odd(s(0)) | → | true | | odd(s(s(x))) | → | odd(x) |
| p(0) | → | 0 | | p(s(x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: 0, s, p, true, false, odd, cond
Strategy
The following SCCs where found
| cond#(true, x) → cond#(odd(x), p(x)) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| cond(true, x) | → | cond(odd(x), p(x)) | | odd(0) | → | false |
| odd(s(0)) | → | true | | odd(s(s(x))) | → | odd(x) |
| p(0) | → | 0 | | p(s(x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: 0, s, p, true, false, odd, cond
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed: