YES
The TRS could be proven terminating. The proof took 11441 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (233ms).
| – Problem 2 was processed with processor PolynomialLinearRange4iUR (3739ms).
| | – Problem 8 was processed with processor PolynomialLinearRange4iUR (3125ms).
| | | – Problem 9 was processed with processor PolynomialLinearRange4iUR (2691ms).
| | | | – Problem 10 was processed with processor PolynomialLinearRange4iUR (1191ms).
| | | | | – Problem 11 was processed with processor PolynomialLinearRange4iUR (106ms).
| | | | | | – Problem 12 was processed with processor PolynomialLinearRange4iUR (86ms).
| | | | | | | – Problem 13 was processed with processor PolynomialLinearRange4iUR (63ms).
| | | | | | | | – Problem 14 was processed with processor DependencyGraph (0ms).
| – Problem 3 was processed with processor SubtermCriterion (0ms).
| – Problem 4 was processed with processor SubtermCriterion (0ms).
| – Problem 5 was processed with processor SubtermCriterion (1ms).
| – Problem 6 was processed with processor SubtermCriterion (0ms).
| – Problem 7 was processed with processor SubtermCriterion (27ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| h#(mark(X)) | → | h#(X) | | mark#(g(X)) | → | g#(X) |
| g#(mark(X)) | → | g#(X) | | active#(f(f(X))) | → | c#(f(g(f(X)))) |
| mark#(c(X)) | → | active#(c(X)) | | f#(active(X)) | → | f#(X) |
| active#(h(X)) | → | mark#(c(d(X))) | | mark#(g(X)) | → | active#(g(X)) |
| d#(active(X)) | → | d#(X) | | mark#(h(X)) | → | active#(h(mark(X))) |
| active#(f(f(X))) | → | g#(f(X)) | | active#(h(X)) | → | c#(d(X)) |
| active#(h(X)) | → | d#(X) | | h#(active(X)) | → | h#(X) |
| mark#(f(X)) | → | f#(mark(X)) | | f#(mark(X)) | → | f#(X) |
| c#(active(X)) | → | c#(X) | | active#(c(X)) | → | mark#(d(X)) |
| d#(mark(X)) | → | d#(X) | | active#(c(X)) | → | d#(X) |
| mark#(d(X)) | → | d#(X) | | mark#(f(X)) | → | mark#(X) |
| mark#(f(X)) | → | active#(f(mark(X))) | | active#(f(f(X))) | → | f#(X) |
| g#(active(X)) | → | g#(X) | | mark#(d(X)) | → | active#(d(X)) |
| active#(f(f(X))) | → | mark#(c(f(g(f(X))))) | | active#(f(f(X))) | → | f#(g(f(X))) |
| c#(mark(X)) | → | c#(X) | | mark#(c(X)) | → | c#(X) |
| mark#(h(X)) | → | mark#(X) | | mark#(h(X)) | → | h#(mark(X)) |
Rewrite Rules
| active(f(f(X))) | → | mark(c(f(g(f(X))))) | | active(c(X)) | → | mark(d(X)) |
| active(h(X)) | → | mark(c(d(X))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(c(X)) | → | active(c(X)) | | mark(g(X)) | → | active(g(X)) |
| mark(d(X)) | → | active(d(X)) | | mark(h(X)) | → | active(h(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| c(mark(X)) | → | c(X) | | c(active(X)) | → | c(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
| d(mark(X)) | → | d(X) | | d(active(X)) | → | d(X) |
| h(mark(X)) | → | h(X) | | h(active(X)) | → | h(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, d, c, active, mark, h
Strategy
The following SCCs where found
| d#(mark(X)) → d#(X) | d#(active(X)) → d#(X) |
| f#(active(X)) → f#(X) | f#(mark(X)) → f#(X) |
| active#(f(f(X))) → mark#(c(f(g(f(X))))) | mark#(d(X)) → active#(d(X)) |
| mark#(h(X)) → active#(h(mark(X))) | mark#(c(X)) → active#(c(X)) |
| active#(c(X)) → mark#(d(X)) | mark#(f(X)) → mark#(X) |
| active#(h(X)) → mark#(c(d(X))) | mark#(h(X)) → mark#(X) |
| mark#(f(X)) → active#(f(mark(X))) | mark#(g(X)) → active#(g(X)) |
| c#(active(X)) → c#(X) | c#(mark(X)) → c#(X) |
| g#(active(X)) → g#(X) | g#(mark(X)) → g#(X) |
| h#(mark(X)) → h#(X) | h#(active(X)) → h#(X) |
Problem 2: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| active#(f(f(X))) | → | mark#(c(f(g(f(X))))) | | mark#(d(X)) | → | active#(d(X)) |
| mark#(h(X)) | → | active#(h(mark(X))) | | mark#(c(X)) | → | active#(c(X)) |
| active#(c(X)) | → | mark#(d(X)) | | mark#(f(X)) | → | mark#(X) |
| active#(h(X)) | → | mark#(c(d(X))) | | mark#(h(X)) | → | mark#(X) |
| mark#(f(X)) | → | active#(f(mark(X))) | | mark#(g(X)) | → | active#(g(X)) |
Rewrite Rules
| active(f(f(X))) | → | mark(c(f(g(f(X))))) | | active(c(X)) | → | mark(d(X)) |
| active(h(X)) | → | mark(c(d(X))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(c(X)) | → | active(c(X)) | | mark(g(X)) | → | active(g(X)) |
| mark(d(X)) | → | active(d(X)) | | mark(h(X)) | → | active(h(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| c(mark(X)) | → | c(X) | | c(active(X)) | → | c(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
| d(mark(X)) | → | d(X) | | d(active(X)) | → | d(X) |
| h(mark(X)) | → | h(X) | | h(active(X)) | → | h(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, d, c, active, mark, h
Strategy
Polynomial Interpretation
- active(x): 3
- active#(x): x + 1
- c(x): 2
- d(x): 0
- f(x): 2
- g(x): 2
- h(x): 2
- mark(x): 3
- mark#(x): 3
Improved Usable rules
| d(active(X)) | → | d(X) | | c(mark(X)) | → | c(X) |
| g(active(X)) | → | g(X) | | h(mark(X)) | → | h(X) |
| c(active(X)) | → | c(X) | | f(active(X)) | → | f(X) |
| g(mark(X)) | → | g(X) | | f(mark(X)) | → | f(X) |
| h(active(X)) | → | h(X) | | d(mark(X)) | → | d(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(d(X)) | → | active#(d(X)) |
Problem 8: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| active#(f(f(X))) | → | mark#(c(f(g(f(X))))) | | mark#(h(X)) | → | active#(h(mark(X))) |
| mark#(c(X)) | → | active#(c(X)) | | active#(c(X)) | → | mark#(d(X)) |
| active#(h(X)) | → | mark#(c(d(X))) | | mark#(f(X)) | → | mark#(X) |
| mark#(h(X)) | → | mark#(X) | | mark#(f(X)) | → | active#(f(mark(X))) |
| mark#(g(X)) | → | active#(g(X)) |
Rewrite Rules
| active(f(f(X))) | → | mark(c(f(g(f(X))))) | | active(c(X)) | → | mark(d(X)) |
| active(h(X)) | → | mark(c(d(X))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(c(X)) | → | active(c(X)) | | mark(g(X)) | → | active(g(X)) |
| mark(d(X)) | → | active(d(X)) | | mark(h(X)) | → | active(h(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| c(mark(X)) | → | c(X) | | c(active(X)) | → | c(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
| d(mark(X)) | → | d(X) | | d(active(X)) | → | d(X) |
| h(mark(X)) | → | h(X) | | h(active(X)) | → | h(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, d, c, active, mark, h
Strategy
Polynomial Interpretation
- active(x): 1
- active#(x): 0
- c(x): 0
- d(x): 0
- f(x): 2x
- g(x): x + 1
- h(x): 2x
- mark(x): 1
- mark#(x): x
Improved Usable rules
| d(active(X)) | → | d(X) | | c(mark(X)) | → | c(X) |
| c(active(X)) | → | c(X) | | d(mark(X)) | → | d(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(g(X)) | → | active#(g(X)) |
Problem 9: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| active#(f(f(X))) | → | mark#(c(f(g(f(X))))) | | mark#(h(X)) | → | active#(h(mark(X))) |
| mark#(c(X)) | → | active#(c(X)) | | active#(c(X)) | → | mark#(d(X)) |
| mark#(f(X)) | → | mark#(X) | | active#(h(X)) | → | mark#(c(d(X))) |
| mark#(h(X)) | → | mark#(X) | | mark#(f(X)) | → | active#(f(mark(X))) |
Rewrite Rules
| active(f(f(X))) | → | mark(c(f(g(f(X))))) | | active(c(X)) | → | mark(d(X)) |
| active(h(X)) | → | mark(c(d(X))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(c(X)) | → | active(c(X)) | | mark(g(X)) | → | active(g(X)) |
| mark(d(X)) | → | active(d(X)) | | mark(h(X)) | → | active(h(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| c(mark(X)) | → | c(X) | | c(active(X)) | → | c(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
| d(mark(X)) | → | d(X) | | d(active(X)) | → | d(X) |
| h(mark(X)) | → | h(X) | | h(active(X)) | → | h(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, d, c, active, mark, h
Strategy
Polynomial Interpretation
- active(x): 0
- active#(x): 0
- c(x): 0
- d(x): 0
- f(x): 2x
- g(x): 0
- h(x): 2x + 1
- mark(x): 0
- mark#(x): 2x
Improved Usable rules
| d(active(X)) | → | d(X) | | c(mark(X)) | → | c(X) |
| c(active(X)) | → | c(X) | | d(mark(X)) | → | d(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(h(X)) | → | active#(h(mark(X))) | | mark#(h(X)) | → | mark#(X) |
Problem 10: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| active#(f(f(X))) | → | mark#(c(f(g(f(X))))) | | mark#(c(X)) | → | active#(c(X)) |
| active#(c(X)) | → | mark#(d(X)) | | active#(h(X)) | → | mark#(c(d(X))) |
| mark#(f(X)) | → | mark#(X) | | mark#(f(X)) | → | active#(f(mark(X))) |
Rewrite Rules
| active(f(f(X))) | → | mark(c(f(g(f(X))))) | | active(c(X)) | → | mark(d(X)) |
| active(h(X)) | → | mark(c(d(X))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(c(X)) | → | active(c(X)) | | mark(g(X)) | → | active(g(X)) |
| mark(d(X)) | → | active(d(X)) | | mark(h(X)) | → | active(h(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| c(mark(X)) | → | c(X) | | c(active(X)) | → | c(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
| d(mark(X)) | → | d(X) | | d(active(X)) | → | d(X) |
| h(mark(X)) | → | h(X) | | h(active(X)) | → | h(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, d, c, active, mark, h
Strategy
Polynomial Interpretation
- active(x): 0
- active#(x): 0
- c(x): 0
- d(x): 0
- f(x): x + 1
- g(x): 0
- h(x): 3x + 3
- mark(x): 0
- mark#(x): 2x
Improved Usable rules
| d(active(X)) | → | d(X) | | c(mark(X)) | → | c(X) |
| c(active(X)) | → | c(X) | | d(mark(X)) | → | d(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(f(X)) | → | mark#(X) | | mark#(f(X)) | → | active#(f(mark(X))) |
Problem 11: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| active#(f(f(X))) | → | mark#(c(f(g(f(X))))) | | mark#(c(X)) | → | active#(c(X)) |
| active#(c(X)) | → | mark#(d(X)) | | active#(h(X)) | → | mark#(c(d(X))) |
Rewrite Rules
| active(f(f(X))) | → | mark(c(f(g(f(X))))) | | active(c(X)) | → | mark(d(X)) |
| active(h(X)) | → | mark(c(d(X))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(c(X)) | → | active(c(X)) | | mark(g(X)) | → | active(g(X)) |
| mark(d(X)) | → | active(d(X)) | | mark(h(X)) | → | active(h(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| c(mark(X)) | → | c(X) | | c(active(X)) | → | c(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
| d(mark(X)) | → | d(X) | | d(active(X)) | → | d(X) |
| h(mark(X)) | → | h(X) | | h(active(X)) | → | h(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, d, c, active, mark, h
Strategy
Polynomial Interpretation
- active(x): x
- active#(x): x + 1
- c(x): 0
- d(x): 0
- f(x): 1
- g(x): x
- h(x): 0
- mark(x): 3
- mark#(x): 1
Improved Usable rules
| c(mark(X)) | → | c(X) | | c(active(X)) | → | c(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| active#(f(f(X))) | → | mark#(c(f(g(f(X))))) |
Problem 12: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(c(X)) | → | active#(c(X)) | | active#(c(X)) | → | mark#(d(X)) |
| active#(h(X)) | → | mark#(c(d(X))) |
Rewrite Rules
| active(f(f(X))) | → | mark(c(f(g(f(X))))) | | active(c(X)) | → | mark(d(X)) |
| active(h(X)) | → | mark(c(d(X))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(c(X)) | → | active(c(X)) | | mark(g(X)) | → | active(g(X)) |
| mark(d(X)) | → | active(d(X)) | | mark(h(X)) | → | active(h(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| c(mark(X)) | → | c(X) | | c(active(X)) | → | c(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
| d(mark(X)) | → | d(X) | | d(active(X)) | → | d(X) |
| h(mark(X)) | → | h(X) | | h(active(X)) | → | h(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, d, c, active, mark, h
Strategy
Polynomial Interpretation
- active(x): x
- active#(x): x + 1
- c(x): 2x
- d(x): 0
- f(x): 0
- g(x): 0
- h(x): 1
- mark(x): x
- mark#(x): 2x + 1
Improved Usable rules
| d(active(X)) | → | d(X) | | c(mark(X)) | → | c(X) |
| c(active(X)) | → | c(X) | | d(mark(X)) | → | d(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| active#(h(X)) | → | mark#(c(d(X))) |
Problem 13: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(c(X)) | → | active#(c(X)) | | active#(c(X)) | → | mark#(d(X)) |
Rewrite Rules
| active(f(f(X))) | → | mark(c(f(g(f(X))))) | | active(c(X)) | → | mark(d(X)) |
| active(h(X)) | → | mark(c(d(X))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(c(X)) | → | active(c(X)) | | mark(g(X)) | → | active(g(X)) |
| mark(d(X)) | → | active(d(X)) | | mark(h(X)) | → | active(h(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| c(mark(X)) | → | c(X) | | c(active(X)) | → | c(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
| d(mark(X)) | → | d(X) | | d(active(X)) | → | d(X) |
| h(mark(X)) | → | h(X) | | h(active(X)) | → | h(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, d, c, active, mark, h
Strategy
Polynomial Interpretation
- active(x): 3x + 3
- active#(x): 1
- c(x): 1
- d(x): 0
- f(x): 0
- g(x): 0
- h(x): 0
- mark(x): 3
- mark#(x): x
Improved Usable rules
| d(active(X)) | → | d(X) | | d(mark(X)) | → | d(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| active#(c(X)) | → | mark#(d(X)) |
Problem 14: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(c(X)) | → | active#(c(X)) |
Rewrite Rules
| active(f(f(X))) | → | mark(c(f(g(f(X))))) | | active(c(X)) | → | mark(d(X)) |
| active(h(X)) | → | mark(c(d(X))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(c(X)) | → | active(c(X)) | | mark(g(X)) | → | active(g(X)) |
| mark(d(X)) | → | active(d(X)) | | mark(h(X)) | → | active(h(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| c(mark(X)) | → | c(X) | | c(active(X)) | → | c(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
| d(mark(X)) | → | d(X) | | d(active(X)) | → | d(X) |
| h(mark(X)) | → | h(X) | | h(active(X)) | → | h(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, d, c, active, mark, h
Strategy
There are no SCCs!
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| c#(active(X)) | → | c#(X) | | c#(mark(X)) | → | c#(X) |
Rewrite Rules
| active(f(f(X))) | → | mark(c(f(g(f(X))))) | | active(c(X)) | → | mark(d(X)) |
| active(h(X)) | → | mark(c(d(X))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(c(X)) | → | active(c(X)) | | mark(g(X)) | → | active(g(X)) |
| mark(d(X)) | → | active(d(X)) | | mark(h(X)) | → | active(h(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| c(mark(X)) | → | c(X) | | c(active(X)) | → | c(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
| d(mark(X)) | → | d(X) | | d(active(X)) | → | d(X) |
| h(mark(X)) | → | h(X) | | h(active(X)) | → | h(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, d, c, active, mark, h
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| c#(active(X)) | → | c#(X) | | c#(mark(X)) | → | c#(X) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| d#(mark(X)) | → | d#(X) | | d#(active(X)) | → | d#(X) |
Rewrite Rules
| active(f(f(X))) | → | mark(c(f(g(f(X))))) | | active(c(X)) | → | mark(d(X)) |
| active(h(X)) | → | mark(c(d(X))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(c(X)) | → | active(c(X)) | | mark(g(X)) | → | active(g(X)) |
| mark(d(X)) | → | active(d(X)) | | mark(h(X)) | → | active(h(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| c(mark(X)) | → | c(X) | | c(active(X)) | → | c(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
| d(mark(X)) | → | d(X) | | d(active(X)) | → | d(X) |
| h(mark(X)) | → | h(X) | | h(active(X)) | → | h(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, d, c, active, mark, h
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| d#(mark(X)) | → | d#(X) | | d#(active(X)) | → | d#(X) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| f#(active(X)) | → | f#(X) | | f#(mark(X)) | → | f#(X) |
Rewrite Rules
| active(f(f(X))) | → | mark(c(f(g(f(X))))) | | active(c(X)) | → | mark(d(X)) |
| active(h(X)) | → | mark(c(d(X))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(c(X)) | → | active(c(X)) | | mark(g(X)) | → | active(g(X)) |
| mark(d(X)) | → | active(d(X)) | | mark(h(X)) | → | active(h(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| c(mark(X)) | → | c(X) | | c(active(X)) | → | c(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
| d(mark(X)) | → | d(X) | | d(active(X)) | → | d(X) |
| h(mark(X)) | → | h(X) | | h(active(X)) | → | h(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, d, c, active, mark, h
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| f#(active(X)) | → | f#(X) | | f#(mark(X)) | → | f#(X) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| g#(active(X)) | → | g#(X) | | g#(mark(X)) | → | g#(X) |
Rewrite Rules
| active(f(f(X))) | → | mark(c(f(g(f(X))))) | | active(c(X)) | → | mark(d(X)) |
| active(h(X)) | → | mark(c(d(X))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(c(X)) | → | active(c(X)) | | mark(g(X)) | → | active(g(X)) |
| mark(d(X)) | → | active(d(X)) | | mark(h(X)) | → | active(h(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| c(mark(X)) | → | c(X) | | c(active(X)) | → | c(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
| d(mark(X)) | → | d(X) | | d(active(X)) | → | d(X) |
| h(mark(X)) | → | h(X) | | h(active(X)) | → | h(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, d, c, active, mark, h
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| g#(active(X)) | → | g#(X) | | g#(mark(X)) | → | g#(X) |
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| h#(mark(X)) | → | h#(X) | | h#(active(X)) | → | h#(X) |
Rewrite Rules
| active(f(f(X))) | → | mark(c(f(g(f(X))))) | | active(c(X)) | → | mark(d(X)) |
| active(h(X)) | → | mark(c(d(X))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(c(X)) | → | active(c(X)) | | mark(g(X)) | → | active(g(X)) |
| mark(d(X)) | → | active(d(X)) | | mark(h(X)) | → | active(h(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| c(mark(X)) | → | c(X) | | c(active(X)) | → | c(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
| d(mark(X)) | → | d(X) | | d(active(X)) | → | d(X) |
| h(mark(X)) | → | h(X) | | h(active(X)) | → | h(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, d, c, active, mark, h
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| h#(mark(X)) | → | h#(X) | | h#(active(X)) | → | h#(X) |