TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60000 ms.
The following DP Processors were used
Problem 1 was processed with processor PolynomialLinearRange4iUR (423ms).
| – Problem 2 was processed with processor ForwardNarrowing (2ms).
| | – Problem 3 was processed with processor ForwardNarrowing (0ms).
| | | – Problem 4 was processed with processor ForwardNarrowing (28ms).
| | | | – Problem 5 was processed with processor ForwardNarrowing (2ms).
| | | | | – Problem 6 was processed with processor ForwardNarrowing (2ms).
| | | | | | – Problem 7 was processed with processor ForwardNarrowing (2ms).
| | | | | | | – Problem 8 was processed with processor ForwardNarrowing (5ms).
| | | | | | | | – Problem 9 was processed with processor ForwardNarrowing (3ms).
| | | | | | | | | – Problem 10 was processed with processor ForwardNarrowing (3ms).
| | | | | | | | | | – Problem 11 was processed with processor ForwardNarrowing (3ms).
| | | | | | | | | | | – Problem 12 was processed with processor ForwardNarrowing (4ms).
| | | | | | | | | | | | – Problem 13 was processed with processor ForwardNarrowing (4ms).
| | | | | | | | | | | | | – Problem 14 was processed with processor ForwardNarrowing (9ms).
| | | | | | | | | | | | | | – Problem 15 was processed with processor ForwardNarrowing (5ms).
| | | | | | | | | | | | | | | – Problem 16 was processed with processor ForwardNarrowing (5ms).
| | | | | | | | | | | | | | | | – Problem 17 was processed with processor ForwardNarrowing (7ms).
| | | | | | | | | | | | | | | | | – Problem 18 was processed with processor ForwardNarrowing (7ms).
| | | | | | | | | | | | | | | | | | – Problem 19 was processed with processor ForwardNarrowing (7ms).
| | | | | | | | | | | | | | | | | | | – Problem 20 was processed with processor ForwardNarrowing (6ms).
| | | | | | | | | | | | | | | | | | | | – Problem 21 was processed with processor ForwardNarrowing (7ms).
| | | | | | | | | | | | | | | | | | | | | – Problem 22 was processed with processor ForwardNarrowing (26ms).
| | | | | | | | | | | | | | | | | | | | | | – Problem 23 was processed with processor ForwardNarrowing (104ms).
| | | | | | | | | | | | | | | | | | | | | | | – Problem 24 was processed with processor ForwardNarrowing (132ms).
| | | | | | | | | | | | | | | | | | | | | | | | – Problem 25 was processed with processor ForwardNarrowing (222ms).
| | | | | | | | | | | | | | | | | | | | | | | | | – Problem 26 was processed with processor ForwardNarrowing (336ms).
| | | | | | | | | | | | | | | | | | | | | | | | | | – Problem 27 was processed with processor ForwardNarrowing (484ms).
| | | | | | | | | | | | | | | | | | | | | | | | | | | – Problem 28 remains open; application of the following processors failed [ForwardNarrowing (timeout)].
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| f#(f(a, x), a) | → | f#(f(f(x, a), f(a, a)), a) | | f#(f(a, x), a) | → | f#(a, a) |
| f#(f(a, x), a) | → | f#(x, a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Problem 1: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| f#(f(a, x), a) | → | f#(f(f(x, a), f(a, a)), a) | | f#(f(a, x), a) | → | f#(f(x, a), f(a, a)) |
| f#(f(a, x), a) | → | f#(a, a) | | f#(f(a, x), a) | → | f#(x, a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
Polynomial Interpretation
- a: 1
- f(x,y): 0
- f#(x,y): y
Improved Usable rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| f#(f(a, x), a) | → | f#(f(x, a), f(a, a)) |
Problem 2: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| f#(f(a, x), a) | → | f#(f(f(x, a), f(a, a)), a) | | f#(f(a, x), a) | → | f#(a, a) |
| f#(f(a, x), a) | → | f#(x, a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
#(f(a,
x), a) → f
#(f(f(
x, a), f(a, a)), a) 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 |
|---|
| f#(f(f(f(f(_x31, a), f(a, a)), a), f(a, a)), a) | |
Thus, the rule f
#(f(a,
x), a) → f
#(f(f(
x, a), f(a, a)), a) is replaced by the following rules:
| f#(f(a, f(a, _x31)), a) → f#(f(f(f(f(_x31, a), f(a, a)), a), f(a, a)), a) |
Problem 3: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| f#(f(a, x), a) | → | f#(a, a) | | f#(f(a, f(a, _x31)), a) | → | f#(f(f(f(f(_x31, a), f(a, a)), a), f(a, a)), a) |
| f#(f(a, x), a) | → | f#(x, a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
#(f(a,
x), a) → f
#(a, a) 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 |
|---|
Thus, the rule f
#(f(a,
x), a) → f
#(a, a) is deleted.
Problem 4: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| f#(f(a, f(a, _x31)), a) | → | f#(f(f(f(f(_x31, a), f(a, a)), a), f(a, a)), a) | | f#(f(a, x), a) | → | f#(x, a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
#(f(a, f(a,
_x31)), a) → f
#(f(f(f(f(
_x31, a), f(a, a)), a), f(a, a)), a) 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 |
|---|
| f#(f(f(f(f(f(f(_x61, a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | |
Thus, the rule f
#(f(a, f(a,
_x31)), a) → f
#(f(f(f(f(
_x31, a), f(a, a)), a), f(a, a)), a) is replaced by the following rules:
| f#(f(a, f(a, f(a, _x61))), a) → f#(f(f(f(f(f(f(_x61, a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Problem 5: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| f#(f(a, f(a, f(a, _x61))), a) | → | f#(f(f(f(f(f(f(_x61, a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | | f#(f(a, x), a) | → | f#(x, a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
#(f(a, f(a, f(a,
_x61))), a) → f
#(f(f(f(f(f(f(
_x61, a), f(a, a)), a), f(a, a)), a), f(a, a)), a) 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 |
|---|
| f#(f(f(f(f(f(f(f(f(_x81, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | |
Thus, the rule f
#(f(a, f(a, f(a,
_x61))), a) → f
#(f(f(f(f(f(f(
_x61, a), f(a, a)), a), f(a, a)), a), f(a, a)), a) is replaced by the following rules:
| f#(f(a, f(a, f(a, f(a, _x81)))), a) → f#(f(f(f(f(f(f(f(f(_x81, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Problem 6: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| f#(f(a, f(a, f(a, f(a, _x81)))), a) | → | f#(f(f(f(f(f(f(f(f(_x81, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | | f#(f(a, x), a) | → | f#(x, a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
#(f(a, f(a, f(a, f(a,
_x81)))), a) → f
#(f(f(f(f(f(f(f(f(
_x81, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) 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 |
|---|
| f#(f(f(f(f(f(f(f(f(f(f(_x101, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | |
Thus, the rule f
#(f(a, f(a, f(a, f(a,
_x81)))), a) → f
#(f(f(f(f(f(f(f(f(
_x81, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) is replaced by the following rules:
| f#(f(a, f(a, f(a, f(a, f(a, _x101))))), a) → f#(f(f(f(f(f(f(f(f(f(f(_x101, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Problem 7: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| f#(f(a, f(a, f(a, f(a, f(a, _x101))))), a) | → | f#(f(f(f(f(f(f(f(f(f(f(_x101, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | | f#(f(a, x), a) | → | f#(x, a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
#(f(a, f(a, f(a, f(a, f(a,
_x101))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(
_x101, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) 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 |
|---|
| f#(f(f(f(f(f(f(f(f(f(f(f(f(_x121, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | |
Thus, the rule f
#(f(a, f(a, f(a, f(a, f(a,
_x101))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(
_x101, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) is replaced by the following rules:
| f#(f(a, f(a, f(a, f(a, f(a, f(a, _x121)))))), a) → f#(f(f(f(f(f(f(f(f(f(f(f(f(_x121, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Problem 8: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| f#(f(a, x), a) | → | f#(x, a) | | f#(f(a, f(a, f(a, f(a, f(a, f(a, _x121)))))), a) | → | f#(f(f(f(f(f(f(f(f(f(f(f(f(_x121, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a,
_x121)))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(
_x121, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) 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 |
|---|
| f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x141, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | |
Thus, the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a,
_x121)))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(
_x121, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) is replaced by the following rules:
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x141))))))), a) → f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x141, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Problem 9: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x141))))))), a) | → | f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x141, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | | f#(f(a, x), a) | → | f#(x, a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x141))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x141, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) 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 |
|---|
| f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x161, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | |
Thus, the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x141))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x141, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) is replaced by the following rules:
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x161)))))))), a) → f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x161, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Problem 10: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| f#(f(a, x), a) | → | f#(x, a) | | f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x161)))))))), a) | → | f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x161, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x161)))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x161, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) 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 |
|---|
| f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x181, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | |
Thus, the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x161)))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x161, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) is replaced by the following rules:
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x181))))))))), a) → f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x181, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Problem 11: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| f#(f(a, x), a) | → | f#(x, a) | | f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x181))))))))), a) | → | f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x181, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x181))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x181, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) 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 |
|---|
| f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x201, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | |
Thus, the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x181))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x181, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) is replaced by the following rules:
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x201)))))))))), a) → f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x201, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Problem 12: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x201)))))))))), a) | → | f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x201, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | | f#(f(a, x), a) | → | f#(x, a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x201)))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x201, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) 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 |
|---|
| f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x231, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | |
Thus, the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x201)))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x201, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) is replaced by the following rules:
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x231))))))))))), a) → f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x231, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Problem 13: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x231))))))))))), a) | → | f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x231, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | | f#(f(a, x), a) | → | f#(x, a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x231))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x231, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) 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 |
|---|
| f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x251, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | |
Thus, the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x231))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x231, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) is replaced by the following rules:
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x251)))))))))))), a) → f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x251, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Problem 14: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x251)))))))))))), a) | → | f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x251, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | | f#(f(a, x), a) | → | f#(x, a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x251)))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x251, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) 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 |
|---|
| f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x271, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | |
Thus, the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x251)))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x251, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) is replaced by the following rules:
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x271))))))))))))), a) → f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x271, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Problem 15: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x271))))))))))))), a) | → | f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x271, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | | f#(f(a, x), a) | → | f#(x, a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x271))))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x271, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) 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 |
|---|
| f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x291, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | |
Thus, the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x271))))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x271, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) is replaced by the following rules:
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x291)))))))))))))), a) → f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x291, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Problem 16: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x291)))))))))))))), a) | → | f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x291, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | | f#(f(a, x), a) | → | f#(x, a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x291)))))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x291, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) 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 |
|---|
| f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x311, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | |
Thus, the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x291)))))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x291, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) is replaced by the following rules:
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x311))))))))))))))), a) → f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x311, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Problem 17: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x311))))))))))))))), a) | → | f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x311, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | | f#(f(a, x), a) | → | f#(x, a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x311))))))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x311, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) 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 |
|---|
| f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x331, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | |
Thus, the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x311))))))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x311, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) is replaced by the following rules:
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x331)))))))))))))))), a) → f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x331, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Problem 18: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x331)))))))))))))))), a) | → | f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x331, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | | f#(f(a, x), a) | → | f#(x, a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x331)))))))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x331, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) 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 |
|---|
| f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x351, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | |
Thus, the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x331)))))))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x331, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) is replaced by the following rules:
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x351))))))))))))))))), a) → f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x351, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Problem 19: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x351))))))))))))))))), a) | → | f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x351, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | | f#(f(a, x), a) | → | f#(x, a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x351))))))))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x351, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) 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 |
|---|
| f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x371, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | |
Thus, the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x351))))))))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x351, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) is replaced by the following rules:
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x371)))))))))))))))))), a) → f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x371, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Problem 20: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x371)))))))))))))))))), a) | → | f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x371, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | | f#(f(a, x), a) | → | f#(x, a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x371)))))))))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x371, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) 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 |
|---|
| f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x391, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | |
Thus, the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x371)))))))))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x371, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) is replaced by the following rules:
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x391))))))))))))))))))), a) → f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x391, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Problem 21: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| f#(f(a, x), a) | → | f#(x, a) | | f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x391))))))))))))))))))), a) | → | f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x391, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x391))))))))))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x391, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) 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 |
|---|
| f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x411, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | |
Thus, the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x391))))))))))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x391, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) is replaced by the following rules:
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x411)))))))))))))))))))), a) → f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x411, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Problem 22: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x1011)))))))))))))))))))))))))))))))))))))))))))))))))), a) | → | f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x1011, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | | f#(f(a, x), a) | → | f#(x, a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x1011)))))))))))))))))))))))))))))))))))))))))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x1011, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) 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 |
|---|
| f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x1031, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | |
Thus, the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x1011)))))))))))))))))))))))))))))))))))))))))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x1011, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) is replaced by the following rules:
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x1031))))))))))))))))))))))))))))))))))))))))))))))))))), a) → f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x1031, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Problem 23: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x2011)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), a) | → | f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x2011, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | | f#(f(a, x), a) | → | f#(x, a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x2011)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x2011, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) 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 |
|---|
| f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x2041, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | |
Thus, the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x2011)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x2011, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) is replaced by the following rules:
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x2041))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), a) → f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x2041, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Problem 24: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x3021)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), a) | → | f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x3021, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | | f#(f(a, x), a) | → | f#(x, a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x3021)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x3021, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) 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 |
|---|
| f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x3041, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | |
Thus, the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x3021)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x3021, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) is replaced by the following rules:
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x3041))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), a) → f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x3041, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Problem 25: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| f#(f(a, x), a) | → | f#(x, a) | | f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x4021)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), a) | → | f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x4021, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
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_x4021)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
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| Relevant Terms | Irrelevant Terms |
|---|
| f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x4041, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | |
Thus, the rule f
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_x4021)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x4021, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) is replaced by the following rules:
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x4041))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), a) → f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x4041, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Problem 26: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x5021)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), a) | → | f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x5021, 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a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | | f#(f(a, x), a) | → | f#(x, a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
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| Relevant Terms | Irrelevant Terms |
|---|
| f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x5041, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | |
Thus, the rule f
#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a,
_x5021)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), a) → f
#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(
_x5021, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) is replaced by the following rules:
| f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x5041))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), a) → f#(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(_x5041, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) |
Problem 27: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
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f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | | f#(f(a, x), a) | → | f#(x, a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, a), f(a, a)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The right-hand side of the rule f
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| Relevant Terms | Irrelevant Terms |
|---|
| 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a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a), f(a, a)), a) | |
Thus, the rule f
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_x6021)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), a) → f
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