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 (80ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| – Problem 4 remains open; application of the following processors failed [SubtermCriterion (0ms), DependencyGraph (5ms), PolynomialLinearRange4iUR (514ms), DependencyGraph (3ms), PolynomialLinearRange8NegiUR (4991ms), DependencyGraph (45ms), ReductionPairSAT (541ms), DependencyGraph (3ms), SizeChangePrinciple (152ms), ForwardNarrowing (1ms), BackwardInstantiation (1ms), ForwardInstantiation (2ms), Propagation (1ms)].
The following open problems remain:
Open Dependency Pair Problem 4
Dependency Pairs
| cond2#(true, x) | → | cond1#(neq(x, 0), div2(x)) | | cond1#(true, x) | → | cond2#(even(x), x) |
| cond2#(false, x) | → | cond1#(neq(x, 0), p(x)) |
Rewrite Rules
| cond1(true, x) | → | cond2(even(x), x) | | cond2(true, x) | → | cond1(neq(x, 0), div2(x)) |
| cond2(false, x) | → | cond1(neq(x, 0), p(x)) | | neq(0, 0) | → | false |
| neq(0, s(x)) | → | true | | neq(s(x), 0) | → | true |
| neq(s(x), s(y)) | → | neq(x, y) | | even(0) | → | true |
| even(s(0)) | → | false | | even(s(s(x))) | → | even(x) |
| div2(0) | → | 0 | | div2(s(0)) | → | 0 |
| div2(s(s(x))) | → | s(div2(x)) | | p(0) | → | 0 |
| p(s(x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: cond2, 0, s, p, false, true, even, neq, y, div2, cond1
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| cond2#(true, x) | → | div2#(x) | | cond2#(true, x) | → | cond1#(neq(x, 0), div2(x)) |
| cond1#(true, x) | → | cond2#(even(x), x) | | cond2#(true, x) | → | neq#(x, 0) |
| even#(s(s(x))) | → | even#(x) | | cond2#(false, x) | → | neq#(x, 0) |
| cond2#(false, x) | → | p#(x) | | cond1#(true, x) | → | even#(x) |
| cond2#(false, x) | → | cond1#(neq(x, 0), p(x)) | | neq#(s(x), s(y)) | → | neq#(x, y) |
| div2#(s(s(x))) | → | div2#(x) |
Rewrite Rules
| cond1(true, x) | → | cond2(even(x), x) | | cond2(true, x) | → | cond1(neq(x, 0), div2(x)) |
| cond2(false, x) | → | cond1(neq(x, 0), p(x)) | | neq(0, 0) | → | false |
| neq(0, s(x)) | → | true | | neq(s(x), 0) | → | true |
| neq(s(x), s(y)) | → | neq(x, y) | | even(0) | → | true |
| even(s(0)) | → | false | | even(s(s(x))) | → | even(x) |
| div2(0) | → | 0 | | div2(s(0)) | → | 0 |
| div2(s(s(x))) | → | s(div2(x)) | | p(0) | → | 0 |
| p(s(x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: cond2, 0, s, p, true, false, even, neq, cond1, div2, y
Strategy
The following SCCs where found
| even#(s(s(x))) → even#(x) |
| cond2#(true, x) → cond1#(neq(x, 0), div2(x)) | cond1#(true, x) → cond2#(even(x), x) |
| cond2#(false, x) → cond1#(neq(x, 0), p(x)) |
| div2#(s(s(x))) → div2#(x) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| even#(s(s(x))) | → | even#(x) |
Rewrite Rules
| cond1(true, x) | → | cond2(even(x), x) | | cond2(true, x) | → | cond1(neq(x, 0), div2(x)) |
| cond2(false, x) | → | cond1(neq(x, 0), p(x)) | | neq(0, 0) | → | false |
| neq(0, s(x)) | → | true | | neq(s(x), 0) | → | true |
| neq(s(x), s(y)) | → | neq(x, y) | | even(0) | → | true |
| even(s(0)) | → | false | | even(s(s(x))) | → | even(x) |
| div2(0) | → | 0 | | div2(s(0)) | → | 0 |
| div2(s(s(x))) | → | s(div2(x)) | | p(0) | → | 0 |
| p(s(x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: cond2, 0, s, p, true, false, even, neq, cond1, div2, y
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| even#(s(s(x))) | → | even#(x) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| div2#(s(s(x))) | → | div2#(x) |
Rewrite Rules
| cond1(true, x) | → | cond2(even(x), x) | | cond2(true, x) | → | cond1(neq(x, 0), div2(x)) |
| cond2(false, x) | → | cond1(neq(x, 0), p(x)) | | neq(0, 0) | → | false |
| neq(0, s(x)) | → | true | | neq(s(x), 0) | → | true |
| neq(s(x), s(y)) | → | neq(x, y) | | even(0) | → | true |
| even(s(0)) | → | false | | even(s(s(x))) | → | even(x) |
| div2(0) | → | 0 | | div2(s(0)) | → | 0 |
| div2(s(s(x))) | → | s(div2(x)) | | p(0) | → | 0 |
| p(s(x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: cond2, 0, s, p, true, false, even, neq, cond1, div2, y
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| div2#(s(s(x))) | → | div2#(x) |