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 (48ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (5ms), PolynomialLinearRange4iUR (343ms), DependencyGraph (3ms), PolynomialLinearRange8NegiUR (5346ms), DependencyGraph (4ms), ReductionPairSAT (1939ms), DependencyGraph (5ms), SizeChangePrinciple (127ms), ForwardNarrowing (1ms), BackwardInstantiation (2ms), ForwardInstantiation (1ms), Propagation (3ms)].
| – Problem 4 was processed with processor SubtermCriterion (1ms).
The following open problems remain:
Open Dependency Pair Problem 3
Dependency Pairs
| cond2#(false, x, y) | → | cond1#(neq(x, 0), p(x), y) | | cond1#(true, x, y) | → | cond2#(gr(x, y), x, y) |
| cond2#(true, x, y) | → | cond1#(neq(x, 0), y, y) |
Rewrite Rules
| cond1(true, x, y) | → | cond2(gr(x, y), x, y) | | cond2(true, x, y) | → | cond1(neq(x, 0), y, y) |
| cond2(false, x, y) | → | cond1(neq(x, 0), p(x), y) | | gr(0, x) | → | false |
| gr(s(x), 0) | → | true | | gr(s(x), s(y)) | → | gr(x, y) |
| neq(0, 0) | → | false | | neq(0, s(x)) | → | true |
| neq(s(x), 0) | → | true | | neq(s(x), s(y)) | → | neq(x, y) |
| p(0) | → | 0 | | p(s(x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: cond2, 0, s, p, false, true, neq, gr, cond1
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| cond1#(true, x, y) | → | gr#(x, y) | | cond2#(false, x, y) | → | neq#(x, 0) |
| cond2#(false, x, y) | → | cond1#(neq(x, 0), p(x), y) | | cond2#(false, x, y) | → | p#(x) |
| cond1#(true, x, y) | → | cond2#(gr(x, y), x, y) | | gr#(s(x), s(y)) | → | gr#(x, y) |
| neq#(s(x), s(y)) | → | neq#(x, y) | | cond2#(true, x, y) | → | cond1#(neq(x, 0), y, y) |
| cond2#(true, x, y) | → | neq#(x, 0) |
Rewrite Rules
| cond1(true, x, y) | → | cond2(gr(x, y), x, y) | | cond2(true, x, y) | → | cond1(neq(x, 0), y, y) |
| cond2(false, x, y) | → | cond1(neq(x, 0), p(x), y) | | gr(0, x) | → | false |
| gr(s(x), 0) | → | true | | gr(s(x), s(y)) | → | gr(x, y) |
| neq(0, 0) | → | false | | neq(0, s(x)) | → | true |
| neq(s(x), 0) | → | true | | neq(s(x), s(y)) | → | neq(x, y) |
| p(0) | → | 0 | | p(s(x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: cond2, 0, s, p, true, false, neq, gr, cond1
Strategy
The following SCCs where found
| neq#(s(x), s(y)) → neq#(x, y) |
| gr#(s(x), s(y)) → gr#(x, y) |
| cond2#(false, x, y) → cond1#(neq(x, 0), p(x), y) | cond1#(true, x, y) → cond2#(gr(x, y), x, y) |
| cond2#(true, x, y) → cond1#(neq(x, 0), y, y) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| neq#(s(x), s(y)) | → | neq#(x, y) |
Rewrite Rules
| cond1(true, x, y) | → | cond2(gr(x, y), x, y) | | cond2(true, x, y) | → | cond1(neq(x, 0), y, y) |
| cond2(false, x, y) | → | cond1(neq(x, 0), p(x), y) | | gr(0, x) | → | false |
| gr(s(x), 0) | → | true | | gr(s(x), s(y)) | → | gr(x, y) |
| neq(0, 0) | → | false | | neq(0, s(x)) | → | true |
| neq(s(x), 0) | → | true | | neq(s(x), s(y)) | → | neq(x, y) |
| p(0) | → | 0 | | p(s(x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: cond2, 0, s, p, true, false, neq, gr, cond1
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| neq#(s(x), s(y)) | → | neq#(x, y) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| gr#(s(x), s(y)) | → | gr#(x, y) |
Rewrite Rules
| cond1(true, x, y) | → | cond2(gr(x, y), x, y) | | cond2(true, x, y) | → | cond1(neq(x, 0), y, y) |
| cond2(false, x, y) | → | cond1(neq(x, 0), p(x), y) | | gr(0, x) | → | false |
| gr(s(x), 0) | → | true | | gr(s(x), s(y)) | → | gr(x, y) |
| neq(0, 0) | → | false | | neq(0, s(x)) | → | true |
| neq(s(x), 0) | → | true | | neq(s(x), s(y)) | → | neq(x, y) |
| p(0) | → | 0 | | p(s(x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: cond2, 0, s, p, true, false, neq, gr, cond1
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| gr#(s(x), s(y)) | → | gr#(x, y) |