MAYBE
The TRS could not be proven terminating. The proof attempt took 6443 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (0ms).
| – Problem 2 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (3ms), PolynomialLinearRange4iUR (237ms), DependencyGraph (2ms), PolynomialLinearRange8NegiUR (831ms), DependencyGraph (2ms), ReductionPairSAT (5177ms), DependencyGraph (1ms), SizeChangePrinciple (78ms)].
| – Problem 3 was processed with processor SubtermCriterion (0ms).
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| f#(x, c(x), c(y)) | → | f#(y, x, y) | | f#(x, c(x), c(y)) | → | f#(y, y, f(y, x, y)) |
Rewrite Rules
| f(x, c(x), c(y)) | → | f(y, y, f(y, x, y)) | | f(s(x), y, z) | → | f(x, s(c(y)), c(z)) |
| f(c(x), x, y) | → | c(y) | | g(x, y) | → | x |
| g(x, y) | → | y |
Original Signature
Termination of terms over the following signature is verified: f, g, s, c
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| f#(x, c(x), c(y)) | → | f#(y, x, y) | | f#(x, c(x), c(y)) | → | f#(y, y, f(y, x, y)) |
| f#(s(x), y, z) | → | f#(x, s(c(y)), c(z)) |
Rewrite Rules
| f(x, c(x), c(y)) | → | f(y, y, f(y, x, y)) | | f(s(x), y, z) | → | f(x, s(c(y)), c(z)) |
| f(c(x), x, y) | → | c(y) | | g(x, y) | → | x |
| g(x, y) | → | y |
Original Signature
Termination of terms over the following signature is verified: f, g, s, c
Strategy
The following SCCs where found
| f#(s(x), y, z) → f#(x, s(c(y)), c(z)) |
| f#(x, c(x), c(y)) → f#(y, x, y) | f#(x, c(x), c(y)) → f#(y, y, f(y, x, y)) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| f#(s(x), y, z) | → | f#(x, s(c(y)), c(z)) |
Rewrite Rules
| f(x, c(x), c(y)) | → | f(y, y, f(y, x, y)) | | f(s(x), y, z) | → | f(x, s(c(y)), c(z)) |
| f(c(x), x, y) | → | c(y) | | g(x, y) | → | x |
| g(x, y) | → | y |
Original Signature
Termination of terms over the following signature is verified: f, g, s, c
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| f#(s(x), y, z) | → | f#(x, s(c(y)), c(z)) |