YES
The TRS could be proven terminating. The proof took 1985 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (409ms).
| – Problem 2 was processed with processor SubtermCriterion (2ms).
| – Problem 3 was processed with processor PolynomialLinearRange4iUR (821ms).
| | – Problem 10 was processed with processor PolynomialLinearRange4iUR (476ms).
| – Problem 4 was processed with processor SubtermCriterion (2ms).
| – Problem 5 was processed with processor SubtermCriterion (2ms).
| – Problem 6 was processed with processor SubtermCriterion (29ms).
| – Problem 7 was processed with processor SubtermCriterion (2ms).
| | – Problem 9 was processed with processor SubtermCriterion (3ms).
| – Problem 8 was processed with processor SubtermCriterion (21ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| active#(U11(X)) | → | active#(X) | | top#(ok(X)) | → | top#(active(X)) |
| active#(__(__(X, Y), Z)) | → | __#(X, __(Y, Z)) | | active#(U12(X)) | → | active#(X) |
| proper#(U11(X)) | → | proper#(X) | | top#(ok(X)) | → | active#(X) |
| U12#(mark(X)) | → | U12#(X) | | proper#(__(X1, X2)) | → | __#(proper(X1), proper(X2)) |
| active#(isNePal(__(I, __(P, I)))) | → | U11#(tt) | | active#(U12(X)) | → | U12#(active(X)) |
| active#(__(X1, X2)) | → | active#(X2) | | proper#(U11(X)) | → | U11#(proper(X)) |
| active#(__(X1, X2)) | → | __#(X1, active(X2)) | | active#(isNePal(X)) | → | isNePal#(active(X)) |
| top#(mark(X)) | → | proper#(X) | | U11#(mark(X)) | → | U11#(X) |
| active#(__(X1, X2)) | → | __#(active(X1), X2) | | proper#(__(X1, X2)) | → | proper#(X1) |
| __#(mark(X1), X2) | → | __#(X1, X2) | | active#(isNePal(X)) | → | active#(X) |
| top#(mark(X)) | → | top#(proper(X)) | | active#(__(X1, X2)) | → | active#(X1) |
| proper#(U12(X)) | → | proper#(X) | | active#(U11(X)) | → | U11#(active(X)) |
| __#(ok(X1), ok(X2)) | → | __#(X1, X2) | | active#(U11(tt)) | → | U12#(tt) |
| proper#(isNePal(X)) | → | proper#(X) | | proper#(U12(X)) | → | U12#(proper(X)) |
| __#(X1, mark(X2)) | → | __#(X1, X2) | | active#(__(__(X, Y), Z)) | → | __#(Y, Z) |
| isNePal#(mark(X)) | → | isNePal#(X) | | proper#(__(X1, X2)) | → | proper#(X2) |
| isNePal#(ok(X)) | → | isNePal#(X) | | proper#(isNePal(X)) | → | isNePal#(proper(X)) |
| U12#(ok(X)) | → | U12#(X) | | U11#(ok(X)) | → | U11#(X) |
Rewrite Rules
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | active(__(X, nil)) | → | mark(X) |
| active(__(nil, X)) | → | mark(X) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) |
| active(__(X1, X2)) | → | __(active(X1), X2) | | active(__(X1, X2)) | → | __(X1, active(X2)) |
| active(U11(X)) | → | U11(active(X)) | | active(U12(X)) | → | U12(active(X)) |
| active(isNePal(X)) | → | isNePal(active(X)) | | __(mark(X1), X2) | → | mark(__(X1, X2)) |
| __(X1, mark(X2)) | → | mark(__(X1, X2)) | | U11(mark(X)) | → | mark(U11(X)) |
| U12(mark(X)) | → | mark(U12(X)) | | isNePal(mark(X)) | → | mark(isNePal(X)) |
| proper(__(X1, X2)) | → | __(proper(X1), proper(X2)) | | proper(nil) | → | ok(nil) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNePal(X)) | → | isNePal(proper(X)) |
| __(ok(X1), ok(X2)) | → | ok(__(X1, X2)) | | U11(ok(X)) | → | ok(U11(X)) |
| U12(ok(X)) | → | ok(U12(X)) | | isNePal(ok(X)) | → | ok(isNePal(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: tt, isNePal, active, __, U11, mark, U12, ok, proper, nil, top
Strategy
The following SCCs where found
| U12#(ok(X)) → U12#(X) | U12#(mark(X)) → U12#(X) |
| U11#(ok(X)) → U11#(X) | U11#(mark(X)) → U11#(X) |
| active#(isNePal(X)) → active#(X) | active#(U11(X)) → active#(X) |
| active#(__(X1, X2)) → active#(X1) | active#(__(X1, X2)) → active#(X2) |
| active#(U12(X)) → active#(X) |
| proper#(__(X1, X2)) → proper#(X1) | proper#(U12(X)) → proper#(X) |
| proper#(isNePal(X)) → proper#(X) | proper#(U11(X)) → proper#(X) |
| proper#(__(X1, X2)) → proper#(X2) |
| __#(mark(X1), X2) → __#(X1, X2) | __#(ok(X1), ok(X2)) → __#(X1, X2) |
| __#(X1, mark(X2)) → __#(X1, X2) |
| isNePal#(mark(X)) → isNePal#(X) | isNePal#(ok(X)) → isNePal#(X) |
| top#(mark(X)) → top#(proper(X)) | top#(ok(X)) → top#(active(X)) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| isNePal#(mark(X)) | → | isNePal#(X) | | isNePal#(ok(X)) | → | isNePal#(X) |
Rewrite Rules
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | active(__(X, nil)) | → | mark(X) |
| active(__(nil, X)) | → | mark(X) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) |
| active(__(X1, X2)) | → | __(active(X1), X2) | | active(__(X1, X2)) | → | __(X1, active(X2)) |
| active(U11(X)) | → | U11(active(X)) | | active(U12(X)) | → | U12(active(X)) |
| active(isNePal(X)) | → | isNePal(active(X)) | | __(mark(X1), X2) | → | mark(__(X1, X2)) |
| __(X1, mark(X2)) | → | mark(__(X1, X2)) | | U11(mark(X)) | → | mark(U11(X)) |
| U12(mark(X)) | → | mark(U12(X)) | | isNePal(mark(X)) | → | mark(isNePal(X)) |
| proper(__(X1, X2)) | → | __(proper(X1), proper(X2)) | | proper(nil) | → | ok(nil) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNePal(X)) | → | isNePal(proper(X)) |
| __(ok(X1), ok(X2)) | → | ok(__(X1, X2)) | | U11(ok(X)) | → | ok(U11(X)) |
| U12(ok(X)) | → | ok(U12(X)) | | isNePal(ok(X)) | → | ok(isNePal(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: tt, isNePal, active, __, U11, mark, U12, ok, proper, nil, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| isNePal#(mark(X)) | → | isNePal#(X) | | isNePal#(ok(X)) | → | isNePal#(X) |
Problem 3: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| top#(mark(X)) | → | top#(proper(X)) | | top#(ok(X)) | → | top#(active(X)) |
Rewrite Rules
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | active(__(X, nil)) | → | mark(X) |
| active(__(nil, X)) | → | mark(X) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) |
| active(__(X1, X2)) | → | __(active(X1), X2) | | active(__(X1, X2)) | → | __(X1, active(X2)) |
| active(U11(X)) | → | U11(active(X)) | | active(U12(X)) | → | U12(active(X)) |
| active(isNePal(X)) | → | isNePal(active(X)) | | __(mark(X1), X2) | → | mark(__(X1, X2)) |
| __(X1, mark(X2)) | → | mark(__(X1, X2)) | | U11(mark(X)) | → | mark(U11(X)) |
| U12(mark(X)) | → | mark(U12(X)) | | isNePal(mark(X)) | → | mark(isNePal(X)) |
| proper(__(X1, X2)) | → | __(proper(X1), proper(X2)) | | proper(nil) | → | ok(nil) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNePal(X)) | → | isNePal(proper(X)) |
| __(ok(X1), ok(X2)) | → | ok(__(X1, X2)) | | U11(ok(X)) | → | ok(U11(X)) |
| U12(ok(X)) | → | ok(U12(X)) | | isNePal(ok(X)) | → | ok(isNePal(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: tt, isNePal, active, __, U11, mark, U12, ok, proper, nil, top
Strategy
Polynomial Interpretation
- U11(x): x + 2
- U12(x): x + 1
- __(x,y): y + 2x + 1
- active(x): x
- isNePal(x): x + 1
- mark(x): x + 1
- nil: 1
- ok(x): x
- proper(x): x
- top(x): 0
- top#(x): 2x
- tt: 0
Improved Usable rules
| active(__(X1, X2)) | → | __(active(X1), X2) | | active(__(X, nil)) | → | mark(X) |
| U12(mark(X)) | → | mark(U12(X)) | | U11(mark(X)) | → | mark(U11(X)) |
| __(mark(X1), X2) | → | mark(__(X1, X2)) | | active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) |
| active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) | | isNePal(ok(X)) | → | ok(isNePal(X)) |
| active(U12(X)) | → | U12(active(X)) | | __(ok(X1), ok(X2)) | → | ok(__(X1, X2)) |
| active(__(nil, X)) | → | mark(X) | | U11(ok(X)) | → | ok(U11(X)) |
| proper(__(X1, X2)) | → | __(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| active(U12(tt)) | → | mark(tt) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(isNePal(X)) | → | isNePal(active(X)) | | active(U11(X)) | → | U11(active(X)) |
| proper(isNePal(X)) | → | isNePal(proper(X)) | | U12(ok(X)) | → | ok(U12(X)) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(U12(X)) | → | U12(proper(X)) |
| proper(nil) | → | ok(nil) | | active(__(X1, X2)) | → | __(X1, active(X2)) |
| isNePal(mark(X)) | → | mark(isNePal(X)) | | __(X1, mark(X2)) | → | mark(__(X1, X2)) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| top#(mark(X)) | → | top#(proper(X)) |
Problem 10: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| top#(ok(X)) | → | top#(active(X)) |
Rewrite Rules
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | active(__(X, nil)) | → | mark(X) |
| active(__(nil, X)) | → | mark(X) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) |
| active(__(X1, X2)) | → | __(active(X1), X2) | | active(__(X1, X2)) | → | __(X1, active(X2)) |
| active(U11(X)) | → | U11(active(X)) | | active(U12(X)) | → | U12(active(X)) |
| active(isNePal(X)) | → | isNePal(active(X)) | | __(mark(X1), X2) | → | mark(__(X1, X2)) |
| __(X1, mark(X2)) | → | mark(__(X1, X2)) | | U11(mark(X)) | → | mark(U11(X)) |
| U12(mark(X)) | → | mark(U12(X)) | | isNePal(mark(X)) | → | mark(isNePal(X)) |
| proper(__(X1, X2)) | → | __(proper(X1), proper(X2)) | | proper(nil) | → | ok(nil) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNePal(X)) | → | isNePal(proper(X)) |
| __(ok(X1), ok(X2)) | → | ok(__(X1, X2)) | | U11(ok(X)) | → | ok(U11(X)) |
| U12(ok(X)) | → | ok(U12(X)) | | isNePal(ok(X)) | → | ok(isNePal(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: tt, isNePal, U11, __, active, ok, U12, mark, proper, top, nil
Strategy
Polynomial Interpretation
- U11(x): x
- U12(x): x
- __(x,y): 2x
- active(x): x
- isNePal(x): 2x
- mark(x): 0
- nil: 0
- ok(x): x + 1
- proper(x): 0
- top(x): 0
- top#(x): 2x + 1
- tt: 2
Improved Usable rules
| active(__(X1, X2)) | → | __(active(X1), X2) | | __(ok(X1), ok(X2)) | → | ok(__(X1, X2)) |
| active(__(nil, X)) | → | mark(X) | | U11(ok(X)) | → | ok(U11(X)) |
| active(__(X, nil)) | → | mark(X) | | U12(mark(X)) | → | mark(U12(X)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(X)) | → | isNePal(active(X)) |
| active(U11(tt)) | → | mark(U12(tt)) | | U11(mark(X)) | → | mark(U11(X)) |
| active(U11(X)) | → | U11(active(X)) | | __(mark(X1), X2) | → | mark(__(X1, X2)) |
| U12(ok(X)) | → | ok(U12(X)) | | active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) |
| active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) | | active(__(X1, X2)) | → | __(X1, active(X2)) |
| isNePal(ok(X)) | → | ok(isNePal(X)) | | isNePal(mark(X)) | → | mark(isNePal(X)) |
| __(X1, mark(X2)) | → | mark(__(X1, X2)) | | active(U12(X)) | → | U12(active(X)) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| top#(ok(X)) | → | top#(active(X)) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| proper#(__(X1, X2)) | → | proper#(X1) | | proper#(U12(X)) | → | proper#(X) |
| proper#(isNePal(X)) | → | proper#(X) | | proper#(U11(X)) | → | proper#(X) |
| proper#(__(X1, X2)) | → | proper#(X2) |
Rewrite Rules
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | active(__(X, nil)) | → | mark(X) |
| active(__(nil, X)) | → | mark(X) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) |
| active(__(X1, X2)) | → | __(active(X1), X2) | | active(__(X1, X2)) | → | __(X1, active(X2)) |
| active(U11(X)) | → | U11(active(X)) | | active(U12(X)) | → | U12(active(X)) |
| active(isNePal(X)) | → | isNePal(active(X)) | | __(mark(X1), X2) | → | mark(__(X1, X2)) |
| __(X1, mark(X2)) | → | mark(__(X1, X2)) | | U11(mark(X)) | → | mark(U11(X)) |
| U12(mark(X)) | → | mark(U12(X)) | | isNePal(mark(X)) | → | mark(isNePal(X)) |
| proper(__(X1, X2)) | → | __(proper(X1), proper(X2)) | | proper(nil) | → | ok(nil) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNePal(X)) | → | isNePal(proper(X)) |
| __(ok(X1), ok(X2)) | → | ok(__(X1, X2)) | | U11(ok(X)) | → | ok(U11(X)) |
| U12(ok(X)) | → | ok(U12(X)) | | isNePal(ok(X)) | → | ok(isNePal(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: tt, isNePal, active, __, U11, mark, U12, ok, proper, nil, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| proper#(__(X1, X2)) | → | proper#(X1) | | proper#(U12(X)) | → | proper#(X) |
| proper#(isNePal(X)) | → | proper#(X) | | proper#(U11(X)) | → | proper#(X) |
| proper#(__(X1, X2)) | → | proper#(X2) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U11#(ok(X)) | → | U11#(X) | | U11#(mark(X)) | → | U11#(X) |
Rewrite Rules
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | active(__(X, nil)) | → | mark(X) |
| active(__(nil, X)) | → | mark(X) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) |
| active(__(X1, X2)) | → | __(active(X1), X2) | | active(__(X1, X2)) | → | __(X1, active(X2)) |
| active(U11(X)) | → | U11(active(X)) | | active(U12(X)) | → | U12(active(X)) |
| active(isNePal(X)) | → | isNePal(active(X)) | | __(mark(X1), X2) | → | mark(__(X1, X2)) |
| __(X1, mark(X2)) | → | mark(__(X1, X2)) | | U11(mark(X)) | → | mark(U11(X)) |
| U12(mark(X)) | → | mark(U12(X)) | | isNePal(mark(X)) | → | mark(isNePal(X)) |
| proper(__(X1, X2)) | → | __(proper(X1), proper(X2)) | | proper(nil) | → | ok(nil) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNePal(X)) | → | isNePal(proper(X)) |
| __(ok(X1), ok(X2)) | → | ok(__(X1, X2)) | | U11(ok(X)) | → | ok(U11(X)) |
| U12(ok(X)) | → | ok(U12(X)) | | isNePal(ok(X)) | → | ok(isNePal(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: tt, isNePal, active, __, U11, mark, U12, ok, proper, nil, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U11#(ok(X)) | → | U11#(X) | | U11#(mark(X)) | → | U11#(X) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| active#(isNePal(X)) | → | active#(X) | | active#(U11(X)) | → | active#(X) |
| active#(__(X1, X2)) | → | active#(X1) | | active#(__(X1, X2)) | → | active#(X2) |
| active#(U12(X)) | → | active#(X) |
Rewrite Rules
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | active(__(X, nil)) | → | mark(X) |
| active(__(nil, X)) | → | mark(X) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) |
| active(__(X1, X2)) | → | __(active(X1), X2) | | active(__(X1, X2)) | → | __(X1, active(X2)) |
| active(U11(X)) | → | U11(active(X)) | | active(U12(X)) | → | U12(active(X)) |
| active(isNePal(X)) | → | isNePal(active(X)) | | __(mark(X1), X2) | → | mark(__(X1, X2)) |
| __(X1, mark(X2)) | → | mark(__(X1, X2)) | | U11(mark(X)) | → | mark(U11(X)) |
| U12(mark(X)) | → | mark(U12(X)) | | isNePal(mark(X)) | → | mark(isNePal(X)) |
| proper(__(X1, X2)) | → | __(proper(X1), proper(X2)) | | proper(nil) | → | ok(nil) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNePal(X)) | → | isNePal(proper(X)) |
| __(ok(X1), ok(X2)) | → | ok(__(X1, X2)) | | U11(ok(X)) | → | ok(U11(X)) |
| U12(ok(X)) | → | ok(U12(X)) | | isNePal(ok(X)) | → | ok(isNePal(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: tt, isNePal, active, __, U11, mark, U12, ok, proper, nil, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| active#(U11(X)) | → | active#(X) | | active#(isNePal(X)) | → | active#(X) |
| active#(__(X1, X2)) | → | active#(X1) | | active#(__(X1, X2)) | → | active#(X2) |
| active#(U12(X)) | → | active#(X) |
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| __#(mark(X1), X2) | → | __#(X1, X2) | | __#(ok(X1), ok(X2)) | → | __#(X1, X2) |
| __#(X1, mark(X2)) | → | __#(X1, X2) |
Rewrite Rules
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | active(__(X, nil)) | → | mark(X) |
| active(__(nil, X)) | → | mark(X) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) |
| active(__(X1, X2)) | → | __(active(X1), X2) | | active(__(X1, X2)) | → | __(X1, active(X2)) |
| active(U11(X)) | → | U11(active(X)) | | active(U12(X)) | → | U12(active(X)) |
| active(isNePal(X)) | → | isNePal(active(X)) | | __(mark(X1), X2) | → | mark(__(X1, X2)) |
| __(X1, mark(X2)) | → | mark(__(X1, X2)) | | U11(mark(X)) | → | mark(U11(X)) |
| U12(mark(X)) | → | mark(U12(X)) | | isNePal(mark(X)) | → | mark(isNePal(X)) |
| proper(__(X1, X2)) | → | __(proper(X1), proper(X2)) | | proper(nil) | → | ok(nil) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNePal(X)) | → | isNePal(proper(X)) |
| __(ok(X1), ok(X2)) | → | ok(__(X1, X2)) | | U11(ok(X)) | → | ok(U11(X)) |
| U12(ok(X)) | → | ok(U12(X)) | | isNePal(ok(X)) | → | ok(isNePal(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: tt, isNePal, active, __, U11, mark, U12, ok, proper, nil, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| __#(mark(X1), X2) | → | __#(X1, X2) | | __#(ok(X1), ok(X2)) | → | __#(X1, X2) |
Problem 9: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| __#(X1, mark(X2)) | → | __#(X1, X2) |
Rewrite Rules
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | active(__(X, nil)) | → | mark(X) |
| active(__(nil, X)) | → | mark(X) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) |
| active(__(X1, X2)) | → | __(active(X1), X2) | | active(__(X1, X2)) | → | __(X1, active(X2)) |
| active(U11(X)) | → | U11(active(X)) | | active(U12(X)) | → | U12(active(X)) |
| active(isNePal(X)) | → | isNePal(active(X)) | | __(mark(X1), X2) | → | mark(__(X1, X2)) |
| __(X1, mark(X2)) | → | mark(__(X1, X2)) | | U11(mark(X)) | → | mark(U11(X)) |
| U12(mark(X)) | → | mark(U12(X)) | | isNePal(mark(X)) | → | mark(isNePal(X)) |
| proper(__(X1, X2)) | → | __(proper(X1), proper(X2)) | | proper(nil) | → | ok(nil) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNePal(X)) | → | isNePal(proper(X)) |
| __(ok(X1), ok(X2)) | → | ok(__(X1, X2)) | | U11(ok(X)) | → | ok(U11(X)) |
| U12(ok(X)) | → | ok(U12(X)) | | isNePal(ok(X)) | → | ok(isNePal(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: tt, isNePal, U11, __, active, ok, U12, mark, proper, top, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| __#(X1, mark(X2)) | → | __#(X1, X2) |
Problem 8: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U12#(ok(X)) | → | U12#(X) | | U12#(mark(X)) | → | U12#(X) |
Rewrite Rules
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | active(__(X, nil)) | → | mark(X) |
| active(__(nil, X)) | → | mark(X) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) |
| active(__(X1, X2)) | → | __(active(X1), X2) | | active(__(X1, X2)) | → | __(X1, active(X2)) |
| active(U11(X)) | → | U11(active(X)) | | active(U12(X)) | → | U12(active(X)) |
| active(isNePal(X)) | → | isNePal(active(X)) | | __(mark(X1), X2) | → | mark(__(X1, X2)) |
| __(X1, mark(X2)) | → | mark(__(X1, X2)) | | U11(mark(X)) | → | mark(U11(X)) |
| U12(mark(X)) | → | mark(U12(X)) | | isNePal(mark(X)) | → | mark(isNePal(X)) |
| proper(__(X1, X2)) | → | __(proper(X1), proper(X2)) | | proper(nil) | → | ok(nil) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNePal(X)) | → | isNePal(proper(X)) |
| __(ok(X1), ok(X2)) | → | ok(__(X1, X2)) | | U11(ok(X)) | → | ok(U11(X)) |
| U12(ok(X)) | → | ok(U12(X)) | | isNePal(ok(X)) | → | ok(isNePal(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: tt, isNePal, active, __, U11, mark, U12, ok, proper, nil, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U12#(ok(X)) | → | U12#(X) | | U12#(mark(X)) | → | U12#(X) |