YES
The TRS could be proven terminating. The proof took 1735 ms.
The following DP Processors were used
Problem 1 was processed with processor PolynomialLinearRange4iUR (763ms).
| – Problem 2 was processed with processor DependencyGraph (9ms).
| | – Problem 3 was processed with processor PolynomialLinearRange4iUR (235ms).
| | | – Problem 4 was processed with processor PolynomialLinearRange4iUR (317ms).
| | | | – Problem 5 was processed with processor DependencyGraph (1ms).
| | | | | – Problem 6 was processed with processor PolynomialLinearRange4iUR (150ms).
| | | | | – Problem 7 was processed with processor PolynomialLinearRange4iUR (6ms).
Problem 1: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(and(X1, X2)) | → | mark#(X1) | | a__plus#(N, s(M)) | → | a__plus#(mark(N), mark(M)) |
| mark#(and(X1, X2)) | → | a__and#(mark(X1), X2) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(plus(X1, X2)) | → | a__plus#(mark(X1), mark(X2)) | | a__and#(tt, X) | → | mark#(X) |
| a__plus#(N, s(M)) | → | mark#(N) | | mark#(s(X)) | → | mark#(X) |
| a__plus#(N, 0) | → | mark#(N) | | a__plus#(N, s(M)) | → | mark#(M) |
| mark#(plus(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| a__and(tt, X) | → | mark(X) | | a__plus(N, 0) | → | mark(N) |
| a__plus(N, s(M)) | → | s(a__plus(mark(N), mark(M))) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | | mark(tt) | → | tt |
| mark(0) | → | 0 | | mark(s(X)) | → | s(mark(X)) |
| a__and(X1, X2) | → | and(X1, X2) | | a__plus(X1, X2) | → | plus(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, a__plus, mark, a__and, and
Strategy
Polynomial Interpretation
- 0: 0
- a__and(x,y): y + 2x
- a__and#(x,y): y + 2x
- a__plus(x,y): 2y + x
- a__plus#(x,y): y + x
- and(x,y): y + 2x
- mark(x): x
- mark#(x): x
- plus(x,y): 2y + x
- s(x): x
- tt: 1
Improved Usable rules
| mark(tt) | → | tt | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(0) | → | 0 | | a__plus(N, 0) | → | mark(N) |
| a__plus(N, s(M)) | → | s(a__plus(mark(N), mark(M))) | | mark(s(X)) | → | s(mark(X)) |
| a__plus(X1, X2) | → | plus(X1, X2) | | a__and(X1, X2) | → | and(X1, X2) |
| a__and(tt, X) | → | mark(X) | | mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| a__and#(tt, X) | → | mark#(X) |
Problem 2: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(and(X1, X2)) | → | mark#(X1) | | a__plus#(N, s(M)) | → | a__plus#(mark(N), mark(M)) |
| mark#(plus(X1, X2)) | → | mark#(X2) | | mark#(and(X1, X2)) | → | a__and#(mark(X1), X2) |
| mark#(plus(X1, X2)) | → | a__plus#(mark(X1), mark(X2)) | | a__plus#(N, s(M)) | → | mark#(N) |
| mark#(s(X)) | → | mark#(X) | | a__plus#(N, 0) | → | mark#(N) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | a__plus#(N, s(M)) | → | mark#(M) |
Rewrite Rules
| a__and(tt, X) | → | mark(X) | | a__plus(N, 0) | → | mark(N) |
| a__plus(N, s(M)) | → | s(a__plus(mark(N), mark(M))) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | | mark(tt) | → | tt |
| mark(0) | → | 0 | | mark(s(X)) | → | s(mark(X)) |
| a__and(X1, X2) | → | and(X1, X2) | | a__plus(X1, X2) | → | plus(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, a__plus, mark, and, a__and
Strategy
The following SCCs where found
| mark#(and(X1, X2)) → mark#(X1) | a__plus#(N, s(M)) → a__plus#(mark(N), mark(M)) |
| mark#(plus(X1, X2)) → mark#(X2) | mark#(plus(X1, X2)) → a__plus#(mark(X1), mark(X2)) |
| a__plus#(N, s(M)) → mark#(N) | mark#(s(X)) → mark#(X) |
| a__plus#(N, 0) → mark#(N) | mark#(plus(X1, X2)) → mark#(X1) |
| a__plus#(N, s(M)) → mark#(M) |
Problem 3: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(and(X1, X2)) | → | mark#(X1) | | a__plus#(N, s(M)) | → | a__plus#(mark(N), mark(M)) |
| mark#(plus(X1, X2)) | → | mark#(X2) | | mark#(plus(X1, X2)) | → | a__plus#(mark(X1), mark(X2)) |
| a__plus#(N, s(M)) | → | mark#(N) | | mark#(s(X)) | → | mark#(X) |
| a__plus#(N, 0) | → | mark#(N) | | mark#(plus(X1, X2)) | → | mark#(X1) |
| a__plus#(N, s(M)) | → | mark#(M) |
Rewrite Rules
| a__and(tt, X) | → | mark(X) | | a__plus(N, 0) | → | mark(N) |
| a__plus(N, s(M)) | → | s(a__plus(mark(N), mark(M))) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | | mark(tt) | → | tt |
| mark(0) | → | 0 | | mark(s(X)) | → | s(mark(X)) |
| a__and(X1, X2) | → | and(X1, X2) | | a__plus(X1, X2) | → | plus(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, a__plus, mark, and, a__and
Strategy
Polynomial Interpretation
- 0: 1
- a__and(x,y): 2y + 2x
- a__plus(x,y): 3y + 2x
- a__plus#(x,y): 3y + 3x
- and(x,y): 2y + 2x
- mark(x): x
- mark#(x): 3x
- plus(x,y): 3y + 2x
- s(x): x
- tt: 0
Improved Usable rules
| mark(tt) | → | tt | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(0) | → | 0 | | a__plus(N, 0) | → | mark(N) |
| a__plus(N, s(M)) | → | s(a__plus(mark(N), mark(M))) | | mark(s(X)) | → | s(mark(X)) |
| a__plus(X1, X2) | → | plus(X1, X2) | | a__and(X1, X2) | → | and(X1, X2) |
| a__and(tt, X) | → | mark(X) | | mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| a__plus#(N, 0) | → | mark#(N) |
Problem 4: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(and(X1, X2)) | → | mark#(X1) | | a__plus#(N, s(M)) | → | a__plus#(mark(N), mark(M)) |
| mark#(plus(X1, X2)) | → | mark#(X2) | | mark#(plus(X1, X2)) | → | a__plus#(mark(X1), mark(X2)) |
| a__plus#(N, s(M)) | → | mark#(N) | | mark#(s(X)) | → | mark#(X) |
| a__plus#(N, s(M)) | → | mark#(M) | | mark#(plus(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| a__and(tt, X) | → | mark(X) | | a__plus(N, 0) | → | mark(N) |
| a__plus(N, s(M)) | → | s(a__plus(mark(N), mark(M))) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | | mark(tt) | → | tt |
| mark(0) | → | 0 | | mark(s(X)) | → | s(mark(X)) |
| a__and(X1, X2) | → | and(X1, X2) | | a__plus(X1, X2) | → | plus(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, a__plus, mark, a__and, and
Strategy
Polynomial Interpretation
- 0: 0
- a__and(x,y): y + x + 2
- a__plus(x,y): 2y + 2x + 2
- a__plus#(x,y): 2y + 2x
- and(x,y): y + x + 2
- mark(x): x
- mark#(x): x
- plus(x,y): 2y + 2x + 2
- s(x): x
- tt: 0
Improved Usable rules
| mark(tt) | → | tt | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(0) | → | 0 | | a__plus(N, 0) | → | mark(N) |
| a__plus(N, s(M)) | → | s(a__plus(mark(N), mark(M))) | | mark(s(X)) | → | s(mark(X)) |
| a__plus(X1, X2) | → | plus(X1, X2) | | a__and(X1, X2) | → | and(X1, X2) |
| a__and(tt, X) | → | mark(X) | | mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(and(X1, X2)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(plus(X1, X2)) | → | a__plus#(mark(X1), mark(X2)) | | mark#(plus(X1, X2)) | → | mark#(X1) |
Problem 5: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| a__plus#(N, s(M)) | → | a__plus#(mark(N), mark(M)) | | a__plus#(N, s(M)) | → | mark#(N) |
| mark#(s(X)) | → | mark#(X) | | a__plus#(N, s(M)) | → | mark#(M) |
Rewrite Rules
| a__and(tt, X) | → | mark(X) | | a__plus(N, 0) | → | mark(N) |
| a__plus(N, s(M)) | → | s(a__plus(mark(N), mark(M))) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | | mark(tt) | → | tt |
| mark(0) | → | 0 | | mark(s(X)) | → | s(mark(X)) |
| a__and(X1, X2) | → | and(X1, X2) | | a__plus(X1, X2) | → | plus(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, a__plus, mark, and, a__and
Strategy
The following SCCs where found
| a__plus#(N, s(M)) → a__plus#(mark(N), mark(M)) |
Problem 6: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| a__plus#(N, s(M)) | → | a__plus#(mark(N), mark(M)) |
Rewrite Rules
| a__and(tt, X) | → | mark(X) | | a__plus(N, 0) | → | mark(N) |
| a__plus(N, s(M)) | → | s(a__plus(mark(N), mark(M))) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | | mark(tt) | → | tt |
| mark(0) | → | 0 | | mark(s(X)) | → | s(mark(X)) |
| a__and(X1, X2) | → | and(X1, X2) | | a__plus(X1, X2) | → | plus(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, a__plus, mark, and, a__and
Strategy
Polynomial Interpretation
- 0: 0
- a__and(x,y): 2y
- a__plus(x,y): 2y + x
- a__plus#(x,y): 2y
- and(x,y): 2y
- mark(x): x
- plus(x,y): 2y + x
- s(x): x + 1
- tt: 0
Improved Usable rules
| mark(tt) | → | tt | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(0) | → | 0 | | a__plus(N, 0) | → | mark(N) |
| a__plus(N, s(M)) | → | s(a__plus(mark(N), mark(M))) | | mark(s(X)) | → | s(mark(X)) |
| a__plus(X1, X2) | → | plus(X1, X2) | | a__and(X1, X2) | → | and(X1, X2) |
| a__and(tt, X) | → | mark(X) | | mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| a__plus#(N, s(M)) | → | a__plus#(mark(N), mark(M)) |
Problem 7: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| a__and(tt, X) | → | mark(X) | | a__plus(N, 0) | → | mark(N) |
| a__plus(N, s(M)) | → | s(a__plus(mark(N), mark(M))) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | | mark(tt) | → | tt |
| mark(0) | → | 0 | | mark(s(X)) | → | s(mark(X)) |
| a__and(X1, X2) | → | and(X1, X2) | | a__plus(X1, X2) | → | plus(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, a__plus, mark, and, a__and
Strategy
Polynomial Interpretation
- 0: 0
- a__and(x,y): 0
- a__plus(x,y): 0
- and(x,y): 0
- mark(x): 0
- mark#(x): x
- plus(x,y): 0
- s(x): x + 2
- tt: 0
There are no usable rules
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed: