TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60055 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (23ms).
| – Problem 2 remains open; application of the following processors failed [SubtermCriterion (2ms), DependencyGraph (1ms), PolynomialLinearRange4iUR (74ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (192ms), DependencyGraph (1ms), ReductionPairSAT (279ms), DependencyGraph (3ms), SizeChangePrinciple (4ms), ForwardNarrowing (0ms), BackwardInstantiation (0ms), ForwardInstantiation (1ms), Propagation (1ms)].
| – Problem 3 was processed with processor SubtermCriterion (0ms).
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| a__f#(b, X, c) | → | a__f#(X, a__c, X) |
Rewrite Rules
| a__f(b, X, c) | → | a__f(X, a__c, X) | | a__c | → | b |
| mark(f(X1, X2, X3)) | → | a__f(X1, mark(X2), X3) | | mark(c) | → | a__c |
| mark(b) | → | b | | a__f(X1, X2, X3) | → | f(X1, X2, X3) |
| a__c | → | c |
Original Signature
Termination of terms over the following signature is verified: f, b, c, a__c, mark, a__f
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| a__f#(b, X, c) | → | a__c# | | a__f#(b, X, c) | → | a__f#(X, a__c, X) |
| mark#(f(X1, X2, X3)) | → | a__f#(X1, mark(X2), X3) | | mark#(c) | → | a__c# |
| mark#(f(X1, X2, X3)) | → | mark#(X2) |
Rewrite Rules
| a__f(b, X, c) | → | a__f(X, a__c, X) | | a__c | → | b |
| mark(f(X1, X2, X3)) | → | a__f(X1, mark(X2), X3) | | mark(c) | → | a__c |
| mark(b) | → | b | | a__f(X1, X2, X3) | → | f(X1, X2, X3) |
| a__c | → | c |
Original Signature
Termination of terms over the following signature is verified: f, b, c, a__c, mark, a__f
Strategy
The following SCCs where found
| a__f#(b, X, c) → a__f#(X, a__c, X) |
| mark#(f(X1, X2, X3)) → mark#(X2) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| mark#(f(X1, X2, X3)) | → | mark#(X2) |
Rewrite Rules
| a__f(b, X, c) | → | a__f(X, a__c, X) | | a__c | → | b |
| mark(f(X1, X2, X3)) | → | a__f(X1, mark(X2), X3) | | mark(c) | → | a__c |
| mark(b) | → | b | | a__f(X1, X2, X3) | → | f(X1, X2, X3) |
| a__c | → | c |
Original Signature
Termination of terms over the following signature is verified: f, b, c, a__c, mark, a__f
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| mark#(f(X1, X2, X3)) | → | mark#(X2) |