MAYBE
The TRS could not be proven terminating. The proof attempt took 7200 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 (10ms), PolynomialLinearRange4iUR (437ms), DependencyGraph (9ms), PolynomialLinearRange8NegiUR (5845ms), DependencyGraph (6ms), ReductionPairSAT (555ms), DependencyGraph (5ms), SizeChangePrinciple (47ms)].
| – Problem 3 was processed with processor SubtermCriterion (0ms).
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| f#(s(x)) | → | f#(id_inc(c(x, x))) | | g#(c(s(x), y)) | → | g#(c(y, x)) |
| f#(c(s(x), y)) | → | g#(c(x, y)) | | g#(c(x, s(y))) | → | g#(c(y, x)) |
| g#(c(x, x)) | → | f#(x) |
Rewrite Rules
| f(s(x)) | → | f(id_inc(c(x, x))) | | f(c(s(x), y)) | → | g(c(x, y)) |
| g(c(s(x), y)) | → | g(c(y, x)) | | g(c(x, s(y))) | → | g(c(y, x)) |
| g(c(x, x)) | → | f(x) | | id_inc(c(x, y)) | → | c(id_inc(x), id_inc(y)) |
| id_inc(s(x)) | → | s(id_inc(x)) | | id_inc(0) | → | 0 |
| id_inc(0) | → | s(0) |
Original Signature
Termination of terms over the following signature is verified: f, g, 0, id_inc, s, c
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| f#(s(x)) | → | f#(id_inc(c(x, x))) | | f#(s(x)) | → | id_inc#(c(x, x)) |
| id_inc#(c(x, y)) | → | id_inc#(y) | | f#(c(s(x), y)) | → | g#(c(x, y)) |
| g#(c(s(x), y)) | → | g#(c(y, x)) | | g#(c(x, s(y))) | → | g#(c(y, x)) |
| g#(c(x, x)) | → | f#(x) | | id_inc#(s(x)) | → | id_inc#(x) |
| id_inc#(c(x, y)) | → | id_inc#(x) |
Rewrite Rules
| f(s(x)) | → | f(id_inc(c(x, x))) | | f(c(s(x), y)) | → | g(c(x, y)) |
| g(c(s(x), y)) | → | g(c(y, x)) | | g(c(x, s(y))) | → | g(c(y, x)) |
| g(c(x, x)) | → | f(x) | | id_inc(c(x, y)) | → | c(id_inc(x), id_inc(y)) |
| id_inc(s(x)) | → | s(id_inc(x)) | | id_inc(0) | → | 0 |
| id_inc(0) | → | s(0) |
Original Signature
Termination of terms over the following signature is verified: f, g, id_inc, 0, s, c
Strategy
The following SCCs where found
| id_inc#(c(x, y)) → id_inc#(y) | id_inc#(s(x)) → id_inc#(x) |
| id_inc#(c(x, y)) → id_inc#(x) |
| f#(s(x)) → f#(id_inc(c(x, x))) | f#(c(s(x), y)) → g#(c(x, y)) |
| g#(c(s(x), y)) → g#(c(y, x)) | g#(c(x, s(y))) → g#(c(y, x)) |
| g#(c(x, x)) → f#(x) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| id_inc#(c(x, y)) | → | id_inc#(y) | | id_inc#(s(x)) | → | id_inc#(x) |
| id_inc#(c(x, y)) | → | id_inc#(x) |
Rewrite Rules
| f(s(x)) | → | f(id_inc(c(x, x))) | | f(c(s(x), y)) | → | g(c(x, y)) |
| g(c(s(x), y)) | → | g(c(y, x)) | | g(c(x, s(y))) | → | g(c(y, x)) |
| g(c(x, x)) | → | f(x) | | id_inc(c(x, y)) | → | c(id_inc(x), id_inc(y)) |
| id_inc(s(x)) | → | s(id_inc(x)) | | id_inc(0) | → | 0 |
| id_inc(0) | → | s(0) |
Original Signature
Termination of terms over the following signature is verified: f, g, id_inc, 0, s, c
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| id_inc#(c(x, y)) | → | id_inc#(y) | | id_inc#(s(x)) | → | id_inc#(x) |
| id_inc#(c(x, y)) | → | id_inc#(x) |