TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60093 ms.
The following DP Processors were used
Problem 1 was processed with processor PolynomialLinearRange4iUR (7845ms).
| – Problem 2 was processed with processor DependencyGraph (20ms).
| | – Problem 3 remains open; application of the following processors failed [PolynomialLinearRange4iUR (842ms), DependencyGraph (0ms), PolynomialLinearRange4iUR (797ms), DependencyGraph (1ms), PolynomialLinearRange4iUR (851ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (9988ms), DependencyGraph (0ms)].
| | – Problem 4 was processed with processor PolynomialLinearRange4iUR (3505ms).
| | | – Problem 5 was processed with processor PolynomialLinearRange4iUR (463ms).
The following open problems remain:
Open Dependency Pair Problem 3
Dependency Pairs
| f#(c(a, b(b(z, a), y), x)) | → | f#(c(x, b(z, x), y)) |
Rewrite Rules
| b(a, b(c(z, x, y), a)) | → | b(b(z, c(y, z, a)), x) | | f(c(a, b(b(z, a), y), x)) | → | f(c(x, b(z, x), y)) |
| c(f(c(a, y, a)), x, z) | → | f(b(b(z, z), f(b(y, b(x, a))))) |
Original Signature
Termination of terms over the following signature is verified: f, b, c, a
Problem 1: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| c#(f(c(a, y, a)), x, z) | → | b#(y, b(x, a)) | | f#(c(a, b(b(z, a), y), x)) | → | f#(c(x, b(z, x), y)) |
| b#(a, b(c(z, x, y), a)) | → | b#(b(z, c(y, z, a)), x) | | b#(a, b(c(z, x, y), a)) | → | b#(z, c(y, z, a)) |
| c#(f(c(a, y, a)), x, z) | → | b#(z, z) | | f#(c(a, b(b(z, a), y), x)) | → | b#(z, x) |
| c#(f(c(a, y, a)), x, z) | → | f#(b(b(z, z), f(b(y, b(x, a))))) | | c#(f(c(a, y, a)), x, z) | → | b#(x, a) |
| b#(a, b(c(z, x, y), a)) | → | c#(y, z, a) | | c#(f(c(a, y, a)), x, z) | → | f#(b(y, b(x, a))) |
| f#(c(a, b(b(z, a), y), x)) | → | c#(x, b(z, x), y) | | c#(f(c(a, y, a)), x, z) | → | b#(b(z, z), f(b(y, b(x, a)))) |
Rewrite Rules
| b(a, b(c(z, x, y), a)) | → | b(b(z, c(y, z, a)), x) | | f(c(a, b(b(z, a), y), x)) | → | f(c(x, b(z, x), y)) |
| c(f(c(a, y, a)), x, z) | → | f(b(b(z, z), f(b(y, b(x, a))))) |
Original Signature
Termination of terms over the following signature is verified: f, b, c, a
Strategy
Polynomial Interpretation
- a: 1
- b(x,y): 0
- b#(x,y): 0
- c(x,y,z): 2y + 1
- c#(x,y,z): 0
- f(x): x
- f#(x): 2x
Improved Usable rules
| b(a, b(c(z, x, y), a)) | → | b(b(z, c(y, z, a)), x) | | c(f(c(a, y, a)), x, z) | → | f(b(b(z, z), f(b(y, b(x, a))))) |
| f(c(a, b(b(z, a), y), x)) | → | f(c(x, b(z, x), y)) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| f#(c(a, b(b(z, a), y), x)) | → | b#(z, x) | | f#(c(a, b(b(z, a), y), x)) | → | c#(x, b(z, x), y) |
Problem 2: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| c#(f(c(a, y, a)), x, z) | → | b#(y, b(x, a)) | | f#(c(a, b(b(z, a), y), x)) | → | f#(c(x, b(z, x), y)) |
| b#(a, b(c(z, x, y), a)) | → | b#(b(z, c(y, z, a)), x) | | b#(a, b(c(z, x, y), a)) | → | b#(z, c(y, z, a)) |
| c#(f(c(a, y, a)), x, z) | → | b#(z, z) | | c#(f(c(a, y, a)), x, z) | → | f#(b(b(z, z), f(b(y, b(x, a))))) |
| c#(f(c(a, y, a)), x, z) | → | b#(x, a) | | c#(f(c(a, y, a)), x, z) | → | f#(b(y, b(x, a))) |
| b#(a, b(c(z, x, y), a)) | → | c#(y, z, a) | | c#(f(c(a, y, a)), x, z) | → | b#(b(z, z), f(b(y, b(x, a)))) |
Rewrite Rules
| b(a, b(c(z, x, y), a)) | → | b(b(z, c(y, z, a)), x) | | f(c(a, b(b(z, a), y), x)) | → | f(c(x, b(z, x), y)) |
| c(f(c(a, y, a)), x, z) | → | f(b(b(z, z), f(b(y, b(x, a))))) |
Original Signature
Termination of terms over the following signature is verified: f, b, c, a
Strategy
The following SCCs where found
| f#(c(a, b(b(z, a), y), x)) → f#(c(x, b(z, x), y)) |
| c#(f(c(a, y, a)), x, z) → b#(y, b(x, a)) | b#(a, b(c(z, x, y), a)) → b#(b(z, c(y, z, a)), x) |
| b#(a, b(c(z, x, y), a)) → b#(z, c(y, z, a)) | c#(f(c(a, y, a)), x, z) → b#(z, z) |
| c#(f(c(a, y, a)), x, z) → b#(x, a) | b#(a, b(c(z, x, y), a)) → c#(y, z, a) |
| c#(f(c(a, y, a)), x, z) → b#(b(z, z), f(b(y, b(x, a)))) |
Problem 4: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| c#(f(c(a, y, a)), x, z) | → | b#(y, b(x, a)) | | b#(a, b(c(z, x, y), a)) | → | b#(b(z, c(y, z, a)), x) |
| b#(a, b(c(z, x, y), a)) | → | b#(z, c(y, z, a)) | | c#(f(c(a, y, a)), x, z) | → | b#(z, z) |
| c#(f(c(a, y, a)), x, z) | → | b#(x, a) | | b#(a, b(c(z, x, y), a)) | → | c#(y, z, a) |
| c#(f(c(a, y, a)), x, z) | → | b#(b(z, z), f(b(y, b(x, a)))) |
Rewrite Rules
| b(a, b(c(z, x, y), a)) | → | b(b(z, c(y, z, a)), x) | | f(c(a, b(b(z, a), y), x)) | → | f(c(x, b(z, x), y)) |
| c(f(c(a, y, a)), x, z) | → | f(b(b(z, z), f(b(y, b(x, a))))) |
Original Signature
Termination of terms over the following signature is verified: f, b, c, a
Strategy
Polynomial Interpretation
- a: 0
- b(x,y): y + x
- b#(x,y): 2y
- c(x,y,z): 3z + y + 2x + 1
- c#(x,y,z): 2z + 2y + 1
- f(x): 0
Improved Usable rules
| b(a, b(c(z, x, y), a)) | → | b(b(z, c(y, z, a)), x) | | c(f(c(a, y, a)), x, z) | → | f(b(b(z, z), f(b(y, b(x, a))))) |
| f(c(a, b(b(z, a), y), x)) | → | f(c(x, b(z, x), y)) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| c#(f(c(a, y, a)), x, z) | → | b#(y, b(x, a)) | | b#(a, b(c(z, x, y), a)) | → | b#(b(z, c(y, z, a)), x) |
| c#(f(c(a, y, a)), x, z) | → | b#(z, z) | | c#(f(c(a, y, a)), x, z) | → | b#(x, a) |
| b#(a, b(c(z, x, y), a)) | → | c#(y, z, a) | | c#(f(c(a, y, a)), x, z) | → | b#(b(z, z), f(b(y, b(x, a)))) |
Problem 5: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| b#(a, b(c(z, x, y), a)) | → | b#(z, c(y, z, a)) |
Rewrite Rules
| b(a, b(c(z, x, y), a)) | → | b(b(z, c(y, z, a)), x) | | f(c(a, b(b(z, a), y), x)) | → | f(c(x, b(z, x), y)) |
| c(f(c(a, y, a)), x, z) | → | f(b(b(z, z), f(b(y, b(x, a))))) |
Original Signature
Termination of terms over the following signature is verified: f, b, c, a
Strategy
Polynomial Interpretation
- a: 1
- b(x,y): 2
- b#(x,y): y + 1
- c(x,y,z): 1
- f(x): 0
Improved Usable rules
| c(f(c(a, y, a)), x, z) | → | f(b(b(z, z), f(b(y, b(x, a))))) | | f(c(a, b(b(z, a), y), x)) | → | f(c(x, b(z, x), y)) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| b#(a, b(c(z, x, y), a)) | → | b#(z, c(y, z, a)) |