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 (25ms).
| – Problem 2 remains open; application of the following processors failed [SubtermCriterion (2ms), DependencyGraph (1ms), PolynomialLinearRange4iUR (190ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (586ms), DependencyGraph (0ms), ReductionPairSAT (518ms), DependencyGraph (0ms), SizeChangePrinciple (25ms), ForwardNarrowing (0ms), BackwardInstantiation (1ms), ForwardInstantiation (1ms), Propagation (1ms)].
| – Problem 3 was processed with processor SubtermCriterion (2ms).
| – Problem 4 was processed with processor SubtermCriterion (1ms).
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| cond#(true, x, y) | → | cond#(gr(x, y), x, add(x, y)) |
Rewrite Rules
| cond(true, x, y) | → | cond(gr(x, y), x, add(x, 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)) |
Original Signature
Termination of terms over the following signature is verified: 0, s, false, true, add, gr, cond
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| cond#(true, x, y) | → | gr#(x, y) | | cond#(true, x, y) | → | add#(x, y) |
| gr#(s(x), s(y)) | → | gr#(x, y) | | cond#(true, x, y) | → | cond#(gr(x, y), x, add(x, y)) |
| add#(s(x), y) | → | add#(x, y) |
Rewrite Rules
| cond(true, x, y) | → | cond(gr(x, y), x, add(x, 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)) |
Original Signature
Termination of terms over the following signature is verified: 0, s, true, false, gr, add, cond
Strategy
The following SCCs where found
| gr#(s(x), s(y)) → gr#(x, y) |
| cond#(true, x, y) → cond#(gr(x, y), x, add(x, y)) |
| add#(s(x), y) → add#(x, y) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| gr#(s(x), s(y)) | → | gr#(x, y) |
Rewrite Rules
| cond(true, x, y) | → | cond(gr(x, y), x, add(x, 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)) |
Original Signature
Termination of terms over the following signature is verified: 0, s, true, false, gr, add, cond
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| gr#(s(x), s(y)) | → | gr#(x, y) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| add#(s(x), y) | → | add#(x, y) |
Rewrite Rules
| cond(true, x, y) | → | cond(gr(x, y), x, add(x, 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)) |
Original Signature
Termination of terms over the following signature is verified: 0, s, true, false, gr, add, cond
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| add#(s(x), y) | → | add#(x, y) |