TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60001 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (25ms).
| – Problem 2 was processed with processor PolynomialLinearRange4iUR (320ms).
| | – Problem 3 was processed with processor DependencyGraph (2ms).
| | | – Problem 4 remains open; application of the following processors failed [PolynomialLinearRange4iUR (121ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (697ms), DependencyGraph (1ms), ReductionPairSAT (837ms), DependencyGraph (1ms), SizeChangePrinciple (29ms), ForwardNarrowing (1ms), BackwardInstantiation (0ms), ForwardInstantiation (2ms), Propagation (1ms)].
The following open problems remain:
Open Dependency Pair Problem 4
Dependency Pairs
| a__f#(a, b, X) | → | a__f#(X, X, mark(X)) |
Rewrite Rules
| a__f(a, b, X) | → | a__f(X, X, mark(X)) | | a__c | → | a |
| a__c | → | b | | mark(f(X1, X2, X3)) | → | a__f(X1, X2, mark(X3)) |
| mark(c) | → | a__c | | mark(a) | → | a |
| 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, a__c, mark, a__f
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(f(X1, X2, X3)) | → | a__f#(X1, X2, mark(X3)) | | a__f#(a, b, X) | → | a__f#(X, X, mark(X)) |
| mark#(f(X1, X2, X3)) | → | mark#(X3) | | mark#(c) | → | a__c# |
| a__f#(a, b, X) | → | mark#(X) |
Rewrite Rules
| a__f(a, b, X) | → | a__f(X, X, mark(X)) | | a__c | → | a |
| a__c | → | b | | mark(f(X1, X2, X3)) | → | a__f(X1, X2, mark(X3)) |
| mark(c) | → | a__c | | mark(a) | → | a |
| 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, a__c, mark, a__f
Strategy
The following SCCs where found
| mark#(f(X1, X2, X3)) → a__f#(X1, X2, mark(X3)) | a__f#(a, b, X) → a__f#(X, X, mark(X)) |
| mark#(f(X1, X2, X3)) → mark#(X3) | a__f#(a, b, X) → mark#(X) |
Problem 2: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(f(X1, X2, X3)) | → | a__f#(X1, X2, mark(X3)) | | a__f#(a, b, X) | → | a__f#(X, X, mark(X)) |
| mark#(f(X1, X2, X3)) | → | mark#(X3) | | a__f#(a, b, X) | → | mark#(X) |
Rewrite Rules
| a__f(a, b, X) | → | a__f(X, X, mark(X)) | | a__c | → | a |
| a__c | → | b | | mark(f(X1, X2, X3)) | → | a__f(X1, X2, mark(X3)) |
| mark(c) | → | a__c | | mark(a) | → | a |
| 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, a__c, mark, a__f
Strategy
Polynomial Interpretation
- a: 0
- a__c: 2
- a__f(x,y,z): 2z + 1
- a__f#(x,y,z): z
- b: 2
- c: 2
- f(x,y,z): 2z + 1
- mark(x): x
- mark#(x): x
Improved Usable rules
| mark(b) | → | b | | mark(c) | → | a__c |
| mark(a) | → | a | | a__f(a, b, X) | → | a__f(X, X, mark(X)) |
| a__c | → | b | | a__c | → | a |
| a__f(X1, X2, X3) | → | f(X1, X2, X3) | | mark(f(X1, X2, X3)) | → | a__f(X1, X2, mark(X3)) |
| a__c | → | c |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(f(X1, X2, X3)) | → | a__f#(X1, X2, mark(X3)) | | mark#(f(X1, X2, X3)) | → | mark#(X3) |
Problem 3: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| a__f#(a, b, X) | → | a__f#(X, X, mark(X)) | | a__f#(a, b, X) | → | mark#(X) |
Rewrite Rules
| a__f(a, b, X) | → | a__f(X, X, mark(X)) | | a__c | → | a |
| a__c | → | b | | mark(f(X1, X2, X3)) | → | a__f(X1, X2, mark(X3)) |
| mark(c) | → | a__c | | mark(a) | → | a |
| 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, a__c, mark, a__f
Strategy
The following SCCs where found
| a__f#(a, b, X) → a__f#(X, X, mark(X)) |