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 (133ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| – Problem 4 was processed with processor SubtermCriterion (0ms).
| – Problem 5 remains open; application of the following processors failed [SubtermCriterion (2ms), DependencyGraph (10ms), PolynomialLinearRange4iUR (1541ms), DependencyGraph (9ms), PolynomialLinearRange8NegiUR (29997ms), DependencyGraph (9ms), ReductionPairSAT (5875ms), DependencyGraph (47ms), SizeChangePrinciple (805ms), ForwardNarrowing (3ms), BackwardInstantiation (2ms), ForwardInstantiation (4ms), Propagation (1ms)].
The following open problems remain:
Open Dependency Pair Problem 5
Dependency Pairs
| cond3#(false, x, y) | → | cond1#(gr(add(x, y), 0), x, p(y)) | | cond2#(true, x, y) | → | cond1#(gr(add(x, y), 0), p(x), y) |
| cond3#(true, x, y) | → | cond1#(gr(add(x, y), 0), p(x), y) | | cond1#(true, x, y) | → | cond2#(gr(x, y), x, y) |
| cond2#(false, x, y) | → | cond3#(eq(x, y), x, y) |
Rewrite Rules
| cond1(true, x, y) | → | cond2(gr(x, y), x, y) | | cond2(true, x, y) | → | cond1(gr(add(x, y), 0), p(x), y) |
| cond2(false, x, y) | → | cond3(eq(x, y), x, y) | | cond3(true, x, y) | → | cond1(gr(add(x, y), 0), p(x), y) |
| cond3(false, x, y) | → | cond1(gr(add(x, y), 0), x, p(y)) | | gr(0, x) | → | false |
| gr(s(x), 0) | → | true | | gr(s(x), s(y)) | → | gr(x, y) |
| add(0, x) | → | x | | add(s(x), y) | → | s(add(x, y)) |
| eq(0, 0) | → | true | | eq(0, s(x)) | → | false |
| eq(s(x), 0) | → | false | | eq(s(x), s(y)) | → | eq(x, y) |
| p(0) | → | 0 | | p(s(x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: cond2, cond3, 0, s, p, false, true, add, gr, cond1, eq
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| cond1#(true, x, y) | → | gr#(x, y) | | cond3#(true, x, y) | → | add#(x, y) |
| cond2#(true, x, y) | → | gr#(add(x, y), 0) | | cond3#(true, x, y) | → | cond1#(gr(add(x, y), 0), p(x), y) |
| cond2#(false, x, y) | → | cond3#(eq(x, y), x, y) | | cond3#(true, x, y) | → | p#(x) |
| cond2#(true, x, y) | → | add#(x, y) | | cond3#(true, x, y) | → | gr#(add(x, y), 0) |
| cond2#(false, x, y) | → | eq#(x, y) | | cond3#(false, x, y) | → | cond1#(gr(add(x, y), 0), x, p(y)) |
| cond2#(true, x, y) | → | cond1#(gr(add(x, y), 0), p(x), y) | | cond3#(false, x, y) | → | p#(y) |
| cond2#(true, x, y) | → | p#(x) | | cond1#(true, x, y) | → | cond2#(gr(x, y), x, y) |
| cond3#(false, x, y) | → | add#(x, y) | | gr#(s(x), s(y)) | → | gr#(x, y) |
| add#(s(x), y) | → | add#(x, y) | | eq#(s(x), s(y)) | → | eq#(x, y) |
| cond3#(false, x, y) | → | gr#(add(x, y), 0) |
Rewrite Rules
| cond1(true, x, y) | → | cond2(gr(x, y), x, y) | | cond2(true, x, y) | → | cond1(gr(add(x, y), 0), p(x), y) |
| cond2(false, x, y) | → | cond3(eq(x, y), x, y) | | cond3(true, x, y) | → | cond1(gr(add(x, y), 0), p(x), y) |
| cond3(false, x, y) | → | cond1(gr(add(x, y), 0), x, p(y)) | | gr(0, x) | → | false |
| gr(s(x), 0) | → | true | | gr(s(x), s(y)) | → | gr(x, y) |
| add(0, x) | → | x | | add(s(x), y) | → | s(add(x, y)) |
| eq(0, 0) | → | true | | eq(0, s(x)) | → | false |
| eq(s(x), 0) | → | false | | eq(s(x), s(y)) | → | eq(x, y) |
| p(0) | → | 0 | | p(s(x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: cond2, cond3, 0, s, p, true, false, gr, add, cond1, eq
Strategy
The following SCCs where found
| cond3#(false, x, y) → cond1#(gr(add(x, y), 0), x, p(y)) | cond2#(true, x, y) → cond1#(gr(add(x, y), 0), p(x), y) |
| cond3#(true, x, y) → cond1#(gr(add(x, y), 0), p(x), y) | cond1#(true, x, y) → cond2#(gr(x, y), x, y) |
| cond2#(false, x, y) → cond3#(eq(x, y), x, y) |
| gr#(s(x), s(y)) → gr#(x, y) |
| add#(s(x), y) → add#(x, y) |
| eq#(s(x), s(y)) → eq#(x, y) |
Problem 2: 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(gr(add(x, y), 0), p(x), y) |
| cond2(false, x, y) | → | cond3(eq(x, y), x, y) | | cond3(true, x, y) | → | cond1(gr(add(x, y), 0), p(x), y) |
| cond3(false, x, y) | → | cond1(gr(add(x, y), 0), x, p(y)) | | gr(0, x) | → | false |
| gr(s(x), 0) | → | true | | gr(s(x), s(y)) | → | gr(x, y) |
| add(0, x) | → | x | | add(s(x), y) | → | s(add(x, y)) |
| eq(0, 0) | → | true | | eq(0, s(x)) | → | false |
| eq(s(x), 0) | → | false | | eq(s(x), s(y)) | → | eq(x, y) |
| p(0) | → | 0 | | p(s(x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: cond2, cond3, 0, s, p, true, false, gr, add, cond1, eq
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| gr#(s(x), s(y)) | → | gr#(x, y) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| add#(s(x), y) | → | add#(x, y) |
Rewrite Rules
| cond1(true, x, y) | → | cond2(gr(x, y), x, y) | | cond2(true, x, y) | → | cond1(gr(add(x, y), 0), p(x), y) |
| cond2(false, x, y) | → | cond3(eq(x, y), x, y) | | cond3(true, x, y) | → | cond1(gr(add(x, y), 0), p(x), y) |
| cond3(false, x, y) | → | cond1(gr(add(x, y), 0), x, p(y)) | | gr(0, x) | → | false |
| gr(s(x), 0) | → | true | | gr(s(x), s(y)) | → | gr(x, y) |
| add(0, x) | → | x | | add(s(x), y) | → | s(add(x, y)) |
| eq(0, 0) | → | true | | eq(0, s(x)) | → | false |
| eq(s(x), 0) | → | false | | eq(s(x), s(y)) | → | eq(x, y) |
| p(0) | → | 0 | | p(s(x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: cond2, cond3, 0, s, p, true, false, gr, add, cond1, eq
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| add#(s(x), y) | → | add#(x, y) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| eq#(s(x), s(y)) | → | eq#(x, y) |
Rewrite Rules
| cond1(true, x, y) | → | cond2(gr(x, y), x, y) | | cond2(true, x, y) | → | cond1(gr(add(x, y), 0), p(x), y) |
| cond2(false, x, y) | → | cond3(eq(x, y), x, y) | | cond3(true, x, y) | → | cond1(gr(add(x, y), 0), p(x), y) |
| cond3(false, x, y) | → | cond1(gr(add(x, y), 0), x, p(y)) | | gr(0, x) | → | false |
| gr(s(x), 0) | → | true | | gr(s(x), s(y)) | → | gr(x, y) |
| add(0, x) | → | x | | add(s(x), y) | → | s(add(x, y)) |
| eq(0, 0) | → | true | | eq(0, s(x)) | → | false |
| eq(s(x), 0) | → | false | | eq(s(x), s(y)) | → | eq(x, y) |
| p(0) | → | 0 | | p(s(x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: cond2, cond3, 0, s, p, true, false, gr, add, cond1, eq
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| eq#(s(x), s(y)) | → | eq#(x, y) |