MAYBE
The TRS could not be proven terminating. The proof attempt took 1109 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 (3ms), PolynomialLinearRange4iUR (215ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (465ms), DependencyGraph (2ms), ReductionPairSAT (167ms), DependencyGraph (1ms), SizeChangePrinciple (15ms)].
The following open problems remain:
Open Dependency Pair Problem 3
Dependency Pairs
| d#(x) | → | if#(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x) | | if#(true, x) | → | d#(s(x)) |
Rewrite Rules
| digits | → | d(0) | | d(x) | → | if(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x) |
| if(true, x) | → | cons(x, d(s(x))) | | if(false, x) | → | nil |
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) |
Original Signature
Termination of terms over the following signature is verified: d, 0, le, s, if, false, true, digits, nil, cons
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| le#(s(x), s(y)) | → | le#(x, y) | | d#(x) | → | if#(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x) |
| d#(x) | → | le#(x, s(s(s(s(s(s(s(s(s(0)))))))))) | | if#(true, x) | → | d#(s(x)) |
| digits# | → | d#(0) |
Rewrite Rules
| digits | → | d(0) | | d(x) | → | if(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x) |
| if(true, x) | → | cons(x, d(s(x))) | | if(false, x) | → | nil |
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) |
Original Signature
Termination of terms over the following signature is verified: d, 0, s, le, if, true, false, digits, cons, nil
Strategy
The following SCCs where found
| le#(s(x), s(y)) → le#(x, y) |
| d#(x) → if#(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x) | if#(true, x) → d#(s(x)) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| le#(s(x), s(y)) | → | le#(x, y) |
Rewrite Rules
| digits | → | d(0) | | d(x) | → | if(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x) |
| if(true, x) | → | cons(x, d(s(x))) | | if(false, x) | → | nil |
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) |
Original Signature
Termination of terms over the following signature is verified: d, 0, s, le, if, true, false, digits, cons, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| le#(s(x), s(y)) | → | le#(x, y) |