TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60013 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (44ms).
| – Problem 2 was processed with processor SubtermCriterion (2ms).
| – Problem 3 remains open; application of the following processors failed [SubtermCriterion (0ms), DependencyGraph (3ms), PolynomialLinearRange4iUR (431ms), DependencyGraph (4ms), PolynomialLinearRange8NegiUR (6848ms), DependencyGraph (27ms), ReductionPairSAT (533ms), DependencyGraph (5ms), SizeChangePrinciple (111ms), ForwardNarrowing (1ms), BackwardInstantiation (3ms), ForwardInstantiation (2ms), Propagation (0ms)].
| – Problem 4 was processed with processor SubtermCriterion (1ms).
The following open problems remain:
Open Dependency Pair Problem 3
Dependency Pairs
| if_gcd#(true, x, y) | → | gcd#(minus(x, y), y) | | if_gcd#(false, x, y) | → | gcd#(minus(y, x), x) |
| gcd#(s(x), s(y)) | → | if_gcd#(le(y, x), s(x), s(y)) |
Rewrite Rules
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) | | minus(x, 0) | → | x |
| minus(0, x) | → | 0 | | minus(s(x), s(y)) | → | minus(x, y) |
| gcd(0, y) | → | y | | gcd(s(x), 0) | → | s(x) |
| gcd(s(x), s(y)) | → | if_gcd(le(y, x), s(x), s(y)) | | if_gcd(true, x, y) | → | gcd(minus(x, y), y) |
| if_gcd(false, x, y) | → | gcd(minus(y, x), x) |
Original Signature
Termination of terms over the following signature is verified: minus, 0, s, le, if_gcd, false, true, gcd
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| gcd#(s(x), s(y)) | → | le#(y, x) | | if_gcd#(true, x, y) | → | gcd#(minus(x, y), y) |
| le#(s(x), s(y)) | → | le#(x, y) | | if_gcd#(false, x, y) | → | minus#(y, x) |
| minus#(s(x), s(y)) | → | minus#(x, y) | | if_gcd#(true, x, y) | → | minus#(x, y) |
| if_gcd#(false, x, y) | → | gcd#(minus(y, x), x) | | gcd#(s(x), s(y)) | → | if_gcd#(le(y, x), s(x), s(y)) |
Rewrite Rules
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) | | minus(x, 0) | → | x |
| minus(0, x) | → | 0 | | minus(s(x), s(y)) | → | minus(x, y) |
| gcd(0, y) | → | y | | gcd(s(x), 0) | → | s(x) |
| gcd(s(x), s(y)) | → | if_gcd(le(y, x), s(x), s(y)) | | if_gcd(true, x, y) | → | gcd(minus(x, y), y) |
| if_gcd(false, x, y) | → | gcd(minus(y, x), x) |
Original Signature
Termination of terms over the following signature is verified: 0, minus, le, s, if_gcd, true, false, gcd
Strategy
The following SCCs where found
| le#(s(x), s(y)) → le#(x, y) |
| minus#(s(x), s(y)) → minus#(x, y) |
| if_gcd#(true, x, y) → gcd#(minus(x, y), y) | if_gcd#(false, x, y) → gcd#(minus(y, x), x) |
| gcd#(s(x), s(y)) → if_gcd#(le(y, x), s(x), s(y)) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| minus#(s(x), s(y)) | → | minus#(x, y) |
Rewrite Rules
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) | | minus(x, 0) | → | x |
| minus(0, x) | → | 0 | | minus(s(x), s(y)) | → | minus(x, y) |
| gcd(0, y) | → | y | | gcd(s(x), 0) | → | s(x) |
| gcd(s(x), s(y)) | → | if_gcd(le(y, x), s(x), s(y)) | | if_gcd(true, x, y) | → | gcd(minus(x, y), y) |
| if_gcd(false, x, y) | → | gcd(minus(y, x), x) |
Original Signature
Termination of terms over the following signature is verified: 0, minus, le, s, if_gcd, true, false, gcd
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| minus#(s(x), s(y)) | → | minus#(x, y) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| le#(s(x), s(y)) | → | le#(x, y) |
Rewrite Rules
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) | | minus(x, 0) | → | x |
| minus(0, x) | → | 0 | | minus(s(x), s(y)) | → | minus(x, y) |
| gcd(0, y) | → | y | | gcd(s(x), 0) | → | s(x) |
| gcd(s(x), s(y)) | → | if_gcd(le(y, x), s(x), s(y)) | | if_gcd(true, x, y) | → | gcd(minus(x, y), y) |
| if_gcd(false, x, y) | → | gcd(minus(y, x), x) |
Original Signature
Termination of terms over the following signature is verified: 0, minus, le, s, if_gcd, true, false, gcd
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| le#(s(x), s(y)) | → | le#(x, y) |