YES
The TRS could be proven terminating. The proof took 7064 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (158ms).
| – Problem 2 was processed with processor PolynomialLinearRange4iUR (2813ms).
| | – Problem 3 was processed with processor PolynomialLinearRange4iUR (1841ms).
| | | – Problem 4 was processed with processor PolynomialLinearRange4iUR (1184ms).
| | | | – Problem 5 was processed with processor PolynomialLinearRange4iUR (861ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(U12(X)) | → | mark#(X) | | a____#(__(X, Y), Z) | → | mark#(Y) |
| a____#(__(X, Y), Z) | → | a____#(mark(Y), mark(Z)) | | mark#(__(X1, X2)) | → | a____#(mark(X1), mark(X2)) |
| mark#(U12(X)) | → | a__U12#(mark(X)) | | mark#(U11(X)) | → | mark#(X) |
| a____#(nil, X) | → | mark#(X) | | mark#(isNePal(X)) | → | a__isNePal#(mark(X)) |
| a____#(__(X, Y), Z) | → | mark#(X) | | a____#(X, nil) | → | mark#(X) |
| a____#(__(X, Y), Z) | → | mark#(Z) | | a____#(__(X, Y), Z) | → | a____#(mark(X), a____(mark(Y), mark(Z))) |
| mark#(U11(X)) | → | a__U11#(mark(X)) | | a__U11#(tt) | → | a__U12#(tt) |
| mark#(__(X1, X2)) | → | mark#(X1) | | a__isNePal#(__(I, __(P, I))) | → | a__U11#(tt) |
| mark#(isNePal(X)) | → | mark#(X) | | mark#(__(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| a____(__(X, Y), Z) | → | a____(mark(X), a____(mark(Y), mark(Z))) | | a____(X, nil) | → | mark(X) |
| a____(nil, X) | → | mark(X) | | a__U11(tt) | → | a__U12(tt) |
| a__U12(tt) | → | tt | | a__isNePal(__(I, __(P, I))) | → | a__U11(tt) |
| mark(__(X1, X2)) | → | a____(mark(X1), mark(X2)) | | mark(U11(X)) | → | a__U11(mark(X)) |
| mark(U12(X)) | → | a__U12(mark(X)) | | mark(isNePal(X)) | → | a__isNePal(mark(X)) |
| mark(nil) | → | nil | | mark(tt) | → | tt |
| a____(X1, X2) | → | __(X1, X2) | | a__U11(X) | → | U11(X) |
| a__U12(X) | → | U12(X) | | a__isNePal(X) | → | isNePal(X) |
Original Signature
Termination of terms over the following signature is verified: a____, tt, isNePal, __, U11, mark, a__U12, U12, a__U11, a__isNePal, nil
Strategy
The following SCCs where found
| mark#(U12(X)) → mark#(X) | a____#(__(X, Y), Z) → mark#(Y) |
| a____#(__(X, Y), Z) → a____#(mark(Y), mark(Z)) | mark#(__(X1, X2)) → a____#(mark(X1), mark(X2)) |
| mark#(U11(X)) → mark#(X) | a____#(nil, X) → mark#(X) |
| a____#(__(X, Y), Z) → mark#(X) | a____#(X, nil) → mark#(X) |
| a____#(__(X, Y), Z) → mark#(Z) | a____#(__(X, Y), Z) → a____#(mark(X), a____(mark(Y), mark(Z))) |
| mark#(__(X1, X2)) → mark#(X1) | mark#(isNePal(X)) → mark#(X) |
| mark#(__(X1, X2)) → mark#(X2) |
Problem 2: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| a____#(__(X, Y), Z) | → | mark#(Y) | | mark#(U12(X)) | → | mark#(X) |
| mark#(__(X1, X2)) | → | a____#(mark(X1), mark(X2)) | | a____#(__(X, Y), Z) | → | a____#(mark(Y), mark(Z)) |
| mark#(U11(X)) | → | mark#(X) | | a____#(nil, X) | → | mark#(X) |
| a____#(__(X, Y), Z) | → | mark#(X) | | a____#(X, nil) | → | mark#(X) |
| a____#(__(X, Y), Z) | → | mark#(Z) | | a____#(__(X, Y), Z) | → | a____#(mark(X), a____(mark(Y), mark(Z))) |
| mark#(__(X1, X2)) | → | mark#(X1) | | mark#(isNePal(X)) | → | mark#(X) |
| mark#(__(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| a____(__(X, Y), Z) | → | a____(mark(X), a____(mark(Y), mark(Z))) | | a____(X, nil) | → | mark(X) |
| a____(nil, X) | → | mark(X) | | a__U11(tt) | → | a__U12(tt) |
| a__U12(tt) | → | tt | | a__isNePal(__(I, __(P, I))) | → | a__U11(tt) |
| mark(__(X1, X2)) | → | a____(mark(X1), mark(X2)) | | mark(U11(X)) | → | a__U11(mark(X)) |
| mark(U12(X)) | → | a__U12(mark(X)) | | mark(isNePal(X)) | → | a__isNePal(mark(X)) |
| mark(nil) | → | nil | | mark(tt) | → | tt |
| a____(X1, X2) | → | __(X1, X2) | | a__U11(X) | → | U11(X) |
| a__U12(X) | → | U12(X) | | a__isNePal(X) | → | isNePal(X) |
Original Signature
Termination of terms over the following signature is verified: a____, tt, isNePal, __, U11, mark, a__U12, U12, a__U11, a__isNePal, nil
Strategy
Polynomial Interpretation
- U11(x): x + 2
- U12(x): x
- __(x,y): y + x
- a__U11(x): x + 2
- a__U12(x): x
- a____(x,y): y + x
- a____#(x,y): y + x
- a__isNePal(x): 2x + 2
- isNePal(x): 2x + 2
- mark(x): x
- mark#(x): x
- nil: 2
- tt: 0
Improved Usable rules
| mark(tt) | → | tt | | a__isNePal(__(I, __(P, I))) | → | a__U11(tt) |
| mark(U12(X)) | → | a__U12(mark(X)) | | a____(X, nil) | → | mark(X) |
| mark(isNePal(X)) | → | a__isNePal(mark(X)) | | a____(X1, X2) | → | __(X1, X2) |
| a____(__(X, Y), Z) | → | a____(mark(X), a____(mark(Y), mark(Z))) | | a__isNePal(X) | → | isNePal(X) |
| a____(nil, X) | → | mark(X) | | mark(U11(X)) | → | a__U11(mark(X)) |
| a__U12(X) | → | U12(X) | | a__U11(tt) | → | a__U12(tt) |
| mark(nil) | → | nil | | mark(__(X1, X2)) | → | a____(mark(X1), mark(X2)) |
| a__U11(X) | → | U11(X) | | a__U12(tt) | → | tt |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(U11(X)) | → | mark#(X) | | a____#(nil, X) | → | mark#(X) |
| a____#(X, nil) | → | mark#(X) | | mark#(isNePal(X)) | → | mark#(X) |
Problem 3: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| a____#(__(X, Y), Z) | → | mark#(Z) | | mark#(U12(X)) | → | mark#(X) |
| a____#(__(X, Y), Z) | → | mark#(Y) | | a____#(__(X, Y), Z) | → | a____#(mark(X), a____(mark(Y), mark(Z))) |
| a____#(__(X, Y), Z) | → | a____#(mark(Y), mark(Z)) | | mark#(__(X1, X2)) | → | a____#(mark(X1), mark(X2)) |
| mark#(__(X1, X2)) | → | mark#(X1) | | a____#(__(X, Y), Z) | → | mark#(X) |
| mark#(__(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| a____(__(X, Y), Z) | → | a____(mark(X), a____(mark(Y), mark(Z))) | | a____(X, nil) | → | mark(X) |
| a____(nil, X) | → | mark(X) | | a__U11(tt) | → | a__U12(tt) |
| a__U12(tt) | → | tt | | a__isNePal(__(I, __(P, I))) | → | a__U11(tt) |
| mark(__(X1, X2)) | → | a____(mark(X1), mark(X2)) | | mark(U11(X)) | → | a__U11(mark(X)) |
| mark(U12(X)) | → | a__U12(mark(X)) | | mark(isNePal(X)) | → | a__isNePal(mark(X)) |
| mark(nil) | → | nil | | mark(tt) | → | tt |
| a____(X1, X2) | → | __(X1, X2) | | a__U11(X) | → | U11(X) |
| a__U12(X) | → | U12(X) | | a__isNePal(X) | → | isNePal(X) |
Original Signature
Termination of terms over the following signature is verified: a____, tt, isNePal, U11, __, U12, a__U12, mark, a__isNePal, a__U11, nil
Strategy
Polynomial Interpretation
- U11(x): 2
- U12(x): 2x + 2
- __(x,y): y + x
- a__U11(x): 2
- a__U12(x): 2x + 2
- a____(x,y): y + x
- a____#(x,y): y + x
- a__isNePal(x): 2
- isNePal(x): 2
- mark(x): x
- mark#(x): x
- nil: 0
- tt: 0
Improved Usable rules
| mark(tt) | → | tt | | a__isNePal(__(I, __(P, I))) | → | a__U11(tt) |
| mark(U12(X)) | → | a__U12(mark(X)) | | a____(X, nil) | → | mark(X) |
| mark(isNePal(X)) | → | a__isNePal(mark(X)) | | a____(X1, X2) | → | __(X1, X2) |
| a____(__(X, Y), Z) | → | a____(mark(X), a____(mark(Y), mark(Z))) | | a__isNePal(X) | → | isNePal(X) |
| a____(nil, X) | → | mark(X) | | mark(U11(X)) | → | a__U11(mark(X)) |
| a__U12(X) | → | U12(X) | | a__U11(tt) | → | a__U12(tt) |
| mark(nil) | → | nil | | mark(__(X1, X2)) | → | a____(mark(X1), mark(X2)) |
| a__U11(X) | → | U11(X) | | a__U12(tt) | → | tt |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
Problem 4: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| a____#(__(X, Y), Z) | → | mark#(Z) | | a____#(__(X, Y), Z) | → | mark#(Y) |
| a____#(__(X, Y), Z) | → | a____#(mark(X), a____(mark(Y), mark(Z))) | | mark#(__(X1, X2)) | → | a____#(mark(X1), mark(X2)) |
| a____#(__(X, Y), Z) | → | a____#(mark(Y), mark(Z)) | | mark#(__(X1, X2)) | → | mark#(X1) |
| a____#(__(X, Y), Z) | → | mark#(X) | | mark#(__(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| a____(__(X, Y), Z) | → | a____(mark(X), a____(mark(Y), mark(Z))) | | a____(X, nil) | → | mark(X) |
| a____(nil, X) | → | mark(X) | | a__U11(tt) | → | a__U12(tt) |
| a__U12(tt) | → | tt | | a__isNePal(__(I, __(P, I))) | → | a__U11(tt) |
| mark(__(X1, X2)) | → | a____(mark(X1), mark(X2)) | | mark(U11(X)) | → | a__U11(mark(X)) |
| mark(U12(X)) | → | a__U12(mark(X)) | | mark(isNePal(X)) | → | a__isNePal(mark(X)) |
| mark(nil) | → | nil | | mark(tt) | → | tt |
| a____(X1, X2) | → | __(X1, X2) | | a__U11(X) | → | U11(X) |
| a__U12(X) | → | U12(X) | | a__isNePal(X) | → | isNePal(X) |
Original Signature
Termination of terms over the following signature is verified: a____, tt, isNePal, __, U11, mark, a__U12, U12, a__U11, a__isNePal, nil
Strategy
Polynomial Interpretation
- U11(x): 0
- U12(x): 0
- __(x,y): y + x + 1
- a__U11(x): 0
- a__U12(x): 0
- a____(x,y): y + x + 1
- a____#(x,y): y + x + 1
- a__isNePal(x): 0
- isNePal(x): 0
- mark(x): x
- mark#(x): x + 1
- nil: 0
- tt: 0
Improved Usable rules
| mark(tt) | → | tt | | a__isNePal(__(I, __(P, I))) | → | a__U11(tt) |
| mark(U12(X)) | → | a__U12(mark(X)) | | a____(X, nil) | → | mark(X) |
| mark(isNePal(X)) | → | a__isNePal(mark(X)) | | a____(X1, X2) | → | __(X1, X2) |
| a____(__(X, Y), Z) | → | a____(mark(X), a____(mark(Y), mark(Z))) | | a__isNePal(X) | → | isNePal(X) |
| a____(nil, X) | → | mark(X) | | mark(U11(X)) | → | a__U11(mark(X)) |
| a__U12(X) | → | U12(X) | | a__U11(tt) | → | a__U12(tt) |
| mark(nil) | → | nil | | mark(__(X1, X2)) | → | a____(mark(X1), mark(X2)) |
| a__U11(X) | → | U11(X) | | a__U12(tt) | → | tt |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| a____#(__(X, Y), Z) | → | mark#(Z) | | a____#(__(X, Y), Z) | → | mark#(Y) |
| a____#(__(X, Y), Z) | → | a____#(mark(Y), mark(Z)) | | mark#(__(X1, X2)) | → | a____#(mark(X1), mark(X2)) |
| mark#(__(X1, X2)) | → | mark#(X1) | | a____#(__(X, Y), Z) | → | mark#(X) |
| mark#(__(X1, X2)) | → | mark#(X2) |
Problem 5: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| a____#(__(X, Y), Z) | → | a____#(mark(X), a____(mark(Y), mark(Z))) |
Rewrite Rules
| a____(__(X, Y), Z) | → | a____(mark(X), a____(mark(Y), mark(Z))) | | a____(X, nil) | → | mark(X) |
| a____(nil, X) | → | mark(X) | | a__U11(tt) | → | a__U12(tt) |
| a__U12(tt) | → | tt | | a__isNePal(__(I, __(P, I))) | → | a__U11(tt) |
| mark(__(X1, X2)) | → | a____(mark(X1), mark(X2)) | | mark(U11(X)) | → | a__U11(mark(X)) |
| mark(U12(X)) | → | a__U12(mark(X)) | | mark(isNePal(X)) | → | a__isNePal(mark(X)) |
| mark(nil) | → | nil | | mark(tt) | → | tt |
| a____(X1, X2) | → | __(X1, X2) | | a__U11(X) | → | U11(X) |
| a__U12(X) | → | U12(X) | | a__isNePal(X) | → | isNePal(X) |
Original Signature
Termination of terms over the following signature is verified: a____, tt, isNePal, U11, __, U12, a__U12, mark, a__isNePal, a__U11, nil
Strategy
Polynomial Interpretation
- U11(x): 0
- U12(x): 0
- __(x,y): y + x + 1
- a__U11(x): 0
- a__U12(x): 0
- a____(x,y): y + x + 1
- a____#(x,y): 2x
- a__isNePal(x): 0
- isNePal(x): 0
- mark(x): x
- nil: 0
- tt: 0
Improved Usable rules
| mark(tt) | → | tt | | a__isNePal(__(I, __(P, I))) | → | a__U11(tt) |
| mark(U12(X)) | → | a__U12(mark(X)) | | a____(X, nil) | → | mark(X) |
| mark(isNePal(X)) | → | a__isNePal(mark(X)) | | a____(X1, X2) | → | __(X1, X2) |
| a____(__(X, Y), Z) | → | a____(mark(X), a____(mark(Y), mark(Z))) | | a__isNePal(X) | → | isNePal(X) |
| a____(nil, X) | → | mark(X) | | mark(U11(X)) | → | a__U11(mark(X)) |
| a__U12(X) | → | U12(X) | | a__U11(tt) | → | a__U12(tt) |
| mark(nil) | → | nil | | mark(__(X1, X2)) | → | a____(mark(X1), mark(X2)) |
| a__U11(X) | → | U11(X) | | a__U12(tt) | → | tt |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| a____#(__(X, Y), Z) | → | a____#(mark(X), a____(mark(Y), mark(Z))) |