MAYBE
The TRS could not be proven terminating. The proof attempt took 526 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 (2ms), PolynomialLinearRange4iUR (106ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (50ms), DependencyGraph (1ms), ReductionPairSAT (197ms), DependencyGraph (3ms), SizeChangePrinciple (4ms)].
| – Problem 3 was processed with processor SubtermCriterion (0ms).
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| a__f#(X, g(X), Y) | → | a__f#(Y, Y, Y) |
Rewrite Rules
| a__f(X, g(X), Y) | → | a__f(Y, Y, Y) | | a__g(b) | → | c |
| a__b | → | c | | mark(f(X1, X2, X3)) | → | a__f(X1, X2, X3) |
| mark(g(X)) | → | a__g(mark(X)) | | mark(b) | → | a__b |
| mark(c) | → | c | | a__f(X1, X2, X3) | → | f(X1, X2, X3) |
| a__g(X) | → | g(X) | | a__b | → | b |
Original Signature
Termination of terms over the following signature is verified: f, g, b, c, mark, a__b, a__g, a__f
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| a__f#(X, g(X), Y) | → | a__f#(Y, Y, Y) | | mark#(g(X)) | → | mark#(X) |
| mark#(f(X1, X2, X3)) | → | a__f#(X1, X2, X3) | | mark#(g(X)) | → | a__g#(mark(X)) |
| mark#(b) | → | a__b# |
Rewrite Rules
| a__f(X, g(X), Y) | → | a__f(Y, Y, Y) | | a__g(b) | → | c |
| a__b | → | c | | mark(f(X1, X2, X3)) | → | a__f(X1, X2, X3) |
| mark(g(X)) | → | a__g(mark(X)) | | mark(b) | → | a__b |
| mark(c) | → | c | | a__f(X1, X2, X3) | → | f(X1, X2, X3) |
| a__g(X) | → | g(X) | | a__b | → | b |
Original Signature
Termination of terms over the following signature is verified: f, g, b, c, a__b, mark, a__g, a__f
Strategy
The following SCCs where found
| a__f#(X, g(X), Y) → a__f#(Y, Y, Y) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| a__f(X, g(X), Y) | → | a__f(Y, Y, Y) | | a__g(b) | → | c |
| a__b | → | c | | mark(f(X1, X2, X3)) | → | a__f(X1, X2, X3) |
| mark(g(X)) | → | a__g(mark(X)) | | mark(b) | → | a__b |
| mark(c) | → | c | | a__f(X1, X2, X3) | → | f(X1, X2, X3) |
| a__g(X) | → | g(X) | | a__b | → | b |
Original Signature
Termination of terms over the following signature is verified: f, g, b, c, a__b, mark, a__g, a__f
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed: