TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60043 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (114ms).
| – Problem 2 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (15ms), PolynomialLinearRange4iUR (859ms), DependencyGraph (13ms), PolynomialLinearRange8NegiUR (9901ms), DependencyGraph (9ms), ReductionPairSAT (timeout)].
| – Problem 3 was processed with processor SubtermCriterion (1ms).
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| if#(true, b1, b2, b3, x, y) | → | if2#(b1, b2, b3, x, y) | | if4#(false, x, y) | → | average#(s(x), p(p(y))) |
| if2#(false, b2, b3, x, y) | → | if3#(b2, b3, x, y) | | if3#(false, b3, x, y) | → | if4#(b3, x, y) |
| average#(x, y) | → | if#(le(x, 0), le(y, 0), le(y, s(0)), le(y, s(s(0))), x, y) | | if#(false, b1, b2, b3, x, y) | → | average#(p(x), s(y)) |
Rewrite Rules
| p(s(x)) | → | x | | p(0) | → | 0 |
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) | | average(x, y) | → | if(le(x, 0), le(y, 0), le(y, s(0)), le(y, s(s(0))), x, y) |
| if(true, b1, b2, b3, x, y) | → | if2(b1, b2, b3, x, y) | | if(false, b1, b2, b3, x, y) | → | average(p(x), s(y)) |
| if2(true, b2, b3, x, y) | → | 0 | | if2(false, b2, b3, x, y) | → | if3(b2, b3, x, y) |
| if3(true, b3, x, y) | → | 0 | | if3(false, b3, x, y) | → | if4(b3, x, y) |
| if4(true, x, y) | → | s(0) | | if4(false, x, y) | → | average(s(x), p(p(y))) |
Original Signature
Termination of terms over the following signature is verified: 0, le, s, if, p, if3, if4, false, true, if2, average
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| if4#(false, x, y) | → | p#(y) | | if#(true, b1, b2, b3, x, y) | → | if2#(b1, b2, b3, x, y) |
| if2#(false, b2, b3, x, y) | → | if3#(b2, b3, x, y) | | if4#(false, x, y) | → | average#(s(x), p(p(y))) |
| if3#(false, b3, x, y) | → | if4#(b3, x, y) | | if#(false, b1, b2, b3, x, y) | → | p#(x) |
| le#(s(x), s(y)) | → | le#(x, y) | | average#(x, y) | → | le#(x, 0) |
| average#(x, y) | → | le#(y, s(0)) | | average#(x, y) | → | if#(le(x, 0), le(y, 0), le(y, s(0)), le(y, s(s(0))), x, y) |
| average#(x, y) | → | le#(y, 0) | | if#(false, b1, b2, b3, x, y) | → | average#(p(x), s(y)) |
| if4#(false, x, y) | → | p#(p(y)) | | average#(x, y) | → | le#(y, s(s(0))) |
Rewrite Rules
| p(s(x)) | → | x | | p(0) | → | 0 |
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) | | average(x, y) | → | if(le(x, 0), le(y, 0), le(y, s(0)), le(y, s(s(0))), x, y) |
| if(true, b1, b2, b3, x, y) | → | if2(b1, b2, b3, x, y) | | if(false, b1, b2, b3, x, y) | → | average(p(x), s(y)) |
| if2(true, b2, b3, x, y) | → | 0 | | if2(false, b2, b3, x, y) | → | if3(b2, b3, x, y) |
| if3(true, b3, x, y) | → | 0 | | if3(false, b3, x, y) | → | if4(b3, x, y) |
| if4(true, x, y) | → | s(0) | | if4(false, x, y) | → | average(s(x), p(p(y))) |
Original Signature
Termination of terms over the following signature is verified: 0, s, le, if, p, if3, true, false, if4, if2, average
Strategy
The following SCCs where found
| if2#(false, b2, b3, x, y) → if3#(b2, b3, x, y) | if4#(false, x, y) → average#(s(x), p(p(y))) |
| if#(true, b1, b2, b3, x, y) → if2#(b1, b2, b3, x, y) | if3#(false, b3, x, y) → if4#(b3, x, y) |
| average#(x, y) → if#(le(x, 0), le(y, 0), le(y, s(0)), le(y, s(s(0))), x, y) | if#(false, b1, b2, b3, x, y) → average#(p(x), s(y)) |
| le#(s(x), s(y)) → le#(x, y) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| le#(s(x), s(y)) | → | le#(x, y) |
Rewrite Rules
| p(s(x)) | → | x | | p(0) | → | 0 |
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) | | average(x, y) | → | if(le(x, 0), le(y, 0), le(y, s(0)), le(y, s(s(0))), x, y) |
| if(true, b1, b2, b3, x, y) | → | if2(b1, b2, b3, x, y) | | if(false, b1, b2, b3, x, y) | → | average(p(x), s(y)) |
| if2(true, b2, b3, x, y) | → | 0 | | if2(false, b2, b3, x, y) | → | if3(b2, b3, x, y) |
| if3(true, b3, x, y) | → | 0 | | if3(false, b3, x, y) | → | if4(b3, x, y) |
| if4(true, x, y) | → | s(0) | | if4(false, x, y) | → | average(s(x), p(p(y))) |
Original Signature
Termination of terms over the following signature is verified: 0, s, le, if, p, if3, true, false, if4, if2, average
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| le#(s(x), s(y)) | → | le#(x, y) |