TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60060 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (77ms).
| – Problem 2 was processed with processor ForwardNarrowing (2ms).
| | – Problem 4 was processed with processor ForwardInstantiation (1ms).
| | | – Problem 6 remains open; application of the following processors failed [Propagation (1ms), ForwardNarrowing (1ms), BackwardInstantiation (2ms), ForwardInstantiation (2ms), Propagation (1ms)].
| – Problem 3 was processed with processor ForwardNarrowing (1ms).
| | – Problem 5 was processed with processor ForwardInstantiation (1ms).
| | | – Problem 7 remains open; application of the following processors failed [Propagation (0ms), ForwardNarrowing (0ms), BackwardInstantiation (1ms), ForwardInstantiation (0ms), Propagation (0ms)].
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| a#(f, a(s, x)) | → | a#(f, a(p, a(s, x))) |
Rewrite Rules
| a(f, 0) | → | a(s, 0) | | a(d, 0) | → | 0 |
| a(d, a(s, x)) | → | a(s, a(s, a(d, a(p, a(s, x))))) | | a(f, a(s, x)) | → | a(d, a(f, a(p, a(s, x)))) |
| a(p, a(s, x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: f, d, 0, s, p, a
Open Dependency Pair Problem 3
Dependency Pairs
| a#(d, a(s, x)) | → | a#(d, a(p, a(s, x))) |
Rewrite Rules
| a(f, 0) | → | a(s, 0) | | a(d, 0) | → | 0 |
| a(d, a(s, x)) | → | a(s, a(s, a(d, a(p, a(s, x))))) | | a(f, a(s, x)) | → | a(d, a(f, a(p, a(s, x)))) |
| a(p, a(s, x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: f, d, 0, s, p, a
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| a#(f, 0) | → | a#(s, 0) | | a#(d, a(s, x)) | → | a#(p, a(s, x)) |
| a#(d, a(s, x)) | → | a#(s, a(d, a(p, a(s, x)))) | | a#(d, a(s, x)) | → | a#(d, a(p, a(s, x))) |
| a#(d, a(s, x)) | → | a#(s, a(s, a(d, a(p, a(s, x))))) | | a#(f, a(s, x)) | → | a#(f, a(p, a(s, x))) |
| a#(f, a(s, x)) | → | a#(d, a(f, a(p, a(s, x)))) | | a#(f, a(s, x)) | → | a#(p, a(s, x)) |
| a#(f, a(s, x)) | → | a#(s, x) | | a#(d, a(s, x)) | → | a#(s, x) |
Rewrite Rules
| a(f, 0) | → | a(s, 0) | | a(d, 0) | → | 0 |
| a(d, a(s, x)) | → | a(s, a(s, a(d, a(p, a(s, x))))) | | a(f, a(s, x)) | → | a(d, a(f, a(p, a(s, x)))) |
| a(p, a(s, x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: f, d, 0, s, a, p
Strategy
The following SCCs where found
| a#(d, a(s, x)) → a#(d, a(p, a(s, x))) |
| a#(f, a(s, x)) → a#(f, a(p, a(s, x))) |
Problem 2: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| a#(f, a(s, x)) | → | a#(f, a(p, a(s, x))) |
Rewrite Rules
| a(f, 0) | → | a(s, 0) | | a(d, 0) | → | 0 |
| a(d, a(s, x)) | → | a(s, a(s, a(d, a(p, a(s, x))))) | | a(f, a(s, x)) | → | a(d, a(f, a(p, a(s, x)))) |
| a(p, a(s, x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: f, d, 0, s, a, p
Strategy
The right-hand side of the rule a
#(f, a(s,
x)) → a
#(f, a(p, a(s,
x))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
| Relevant Terms | Irrelevant Terms |
|---|
| a#(f, _x31) | |
Thus, the rule a
#(f, a(s,
x)) → a
#(f, a(p, a(s,
x))) is replaced by the following rules:
| a#(f, a(s, _x31)) → a#(f, _x31) |
Problem 4: ForwardInstantiation
Dependency Pair Problem
Dependency Pairs
| a#(f, a(s, _x31)) | → | a#(f, _x31) |
Rewrite Rules
| a(f, 0) | → | a(s, 0) | | a(d, 0) | → | 0 |
| a(d, a(s, x)) | → | a(s, a(s, a(d, a(p, a(s, x))))) | | a(f, a(s, x)) | → | a(d, a(f, a(p, a(s, x)))) |
| a(p, a(s, x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: f, d, 0, s, p, a
Strategy
Instantiation
For all potential successors l → r of the rule a
#(f, a(s,
_x31)) → a
#(f,
_x31) on dependency pair chains it holds that:
- a#(f, _x31) matches l,
- all variables of a#(f, _x31) are embedded in constructor contexts, i.e., each subterm of a#(f, _x31) containing a variable is rooted by a constructor symbol.
Thus, a
#(f, a(s,
_x31)) → a
#(f,
_x31) is replaced by instances determined through the above matching. These instances are:
| a#(f, a(s, a(s, __x31))) → a#(f, a(s, __x31)) |
Problem 3: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| a#(d, a(s, x)) | → | a#(d, a(p, a(s, x))) |
Rewrite Rules
| a(f, 0) | → | a(s, 0) | | a(d, 0) | → | 0 |
| a(d, a(s, x)) | → | a(s, a(s, a(d, a(p, a(s, x))))) | | a(f, a(s, x)) | → | a(d, a(f, a(p, a(s, x)))) |
| a(p, a(s, x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: f, d, 0, s, a, p
Strategy
The right-hand side of the rule a
#(d, a(s,
x)) → a
#(d, a(p, a(s,
x))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
| Relevant Terms | Irrelevant Terms |
|---|
| a#(d, _x31) | |
Thus, the rule a
#(d, a(s,
x)) → a
#(d, a(p, a(s,
x))) is replaced by the following rules:
| a#(d, a(s, _x31)) → a#(d, _x31) |
Problem 5: ForwardInstantiation
Dependency Pair Problem
Dependency Pairs
| a#(d, a(s, _x31)) | → | a#(d, _x31) |
Rewrite Rules
| a(f, 0) | → | a(s, 0) | | a(d, 0) | → | 0 |
| a(d, a(s, x)) | → | a(s, a(s, a(d, a(p, a(s, x))))) | | a(f, a(s, x)) | → | a(d, a(f, a(p, a(s, x)))) |
| a(p, a(s, x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: f, d, 0, s, p, a
Strategy
Instantiation
For all potential successors l → r of the rule a
#(d, a(s,
_x31)) → a
#(d,
_x31) on dependency pair chains it holds that:
- a#(d, _x31) matches l,
- all variables of a#(d, _x31) are embedded in constructor contexts, i.e., each subterm of a#(d, _x31) containing a variable is rooted by a constructor symbol.
Thus, a
#(d, a(s,
_x31)) → a
#(d,
_x31) is replaced by instances determined through the above matching. These instances are:
| a#(d, a(s, a(s, __x31))) → a#(d, a(s, __x31)) |