YES
The TRS could be proven terminating. The proof took 3221 ms.
The following DP Processors were used
Problem 1 was processed with processor PolynomialLinearRange4iUR (1710ms).
| – Problem 2 was processed with processor PolynomialLinearRange4iUR (1199ms).
| | – Problem 3 was processed with processor DependencyGraph (2ms).
| | | – Problem 4 was processed with processor PolynomialLinearRange4iUR (15ms).
| | | | – Problem 5 was processed with processor PolynomialLinearRange4iUR (10ms).
Problem 1: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(plus(X1, X2)) | → | a__plus#(mark(X1), mark(X2)) | | a__plus#(N, s(M)) | → | a__U11#(tt, M, N) |
| a__U12#(tt, M, N) | → | mark#(M) | | mark#(plus(X1, X2)) | → | mark#(X1) |
| a__U11#(tt, M, N) | → | a__U12#(tt, M, N) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| a__U12#(tt, M, N) | → | mark#(N) | | mark#(U12(X1, X2, X3)) | → | a__U12#(mark(X1), X2, X3) |
| mark#(s(X)) | → | mark#(X) | | a__plus#(N, 0) | → | mark#(N) |
| mark#(U12(X1, X2, X3)) | → | mark#(X1) | | mark#(U11(X1, X2, X3)) | → | a__U11#(mark(X1), X2, X3) |
| mark#(U11(X1, X2, X3)) | → | mark#(X1) | | a__U12#(tt, M, N) | → | a__plus#(mark(N), mark(M)) |
Rewrite Rules
| a__U11(tt, M, N) | → | a__U12(tt, M, N) | | a__U12(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__plus(N, 0) | → | mark(N) | | a__plus(N, s(M)) | → | a__U11(tt, M, N) |
| mark(U11(X1, X2, X3)) | → | a__U11(mark(X1), X2, X3) | | mark(U12(X1, X2, X3)) | → | a__U12(mark(X1), X2, X3) |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | | mark(tt) | → | tt |
| mark(s(X)) | → | s(mark(X)) | | mark(0) | → | 0 |
| a__U11(X1, X2, X3) | → | U11(X1, X2, X3) | | a__U12(X1, X2, X3) | → | U12(X1, X2, X3) |
| a__plus(X1, X2) | → | plus(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, a__plus, U11, a__U12, mark, U12, a__U11
Strategy
Polynomial Interpretation
- 0: 1
- U11(x,y,z): z + y + x
- U12(x,y,z): z + y + x
- a__U11(x,y,z): z + y + x
- a__U11#(x,y,z): 2z + 2y
- a__U12(x,y,z): z + y + x
- a__U12#(x,y,z): 2z + 2y
- a__plus(x,y): y + x
- a__plus#(x,y): 2y + 2x
- mark(x): x
- mark#(x): 2x
- plus(x,y): y + x
- s(x): x
- tt: 0
Improved Usable rules
| mark(tt) | → | tt | | a__U11(X1, X2, X3) | → | U11(X1, X2, X3) |
| mark(0) | → | 0 | | a__plus(N, 0) | → | mark(N) |
| a__plus(X1, X2) | → | plus(X1, X2) | | a__U12(X1, X2, X3) | → | U12(X1, X2, X3) |
| mark(U12(X1, X2, X3)) | → | a__U12(mark(X1), X2, X3) | | a__plus(N, s(M)) | → | a__U11(tt, M, N) |
| a__U11(tt, M, N) | → | a__U12(tt, M, N) | | mark(s(X)) | → | s(mark(X)) |
| a__U12(tt, M, N) | → | s(a__plus(mark(N), mark(M))) | | mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) |
| mark(U11(X1, X2, X3)) | → | a__U11(mark(X1), X2, X3) |
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 2: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(plus(X1, X2)) | → | a__plus#(mark(X1), mark(X2)) | | a__plus#(N, s(M)) | → | a__U11#(tt, M, N) |
| a__U12#(tt, M, N) | → | mark#(M) | | mark#(plus(X1, X2)) | → | mark#(X1) |
| a__U11#(tt, M, N) | → | a__U12#(tt, M, N) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| a__U12#(tt, M, N) | → | mark#(N) | | mark#(U12(X1, X2, X3)) | → | a__U12#(mark(X1), X2, X3) |
| mark#(s(X)) | → | mark#(X) | | mark#(U12(X1, X2, X3)) | → | mark#(X1) |
| mark#(U11(X1, X2, X3)) | → | a__U11#(mark(X1), X2, X3) | | mark#(U11(X1, X2, X3)) | → | mark#(X1) |
| a__U12#(tt, M, N) | → | a__plus#(mark(N), mark(M)) |
Rewrite Rules
| a__U11(tt, M, N) | → | a__U12(tt, M, N) | | a__U12(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__plus(N, 0) | → | mark(N) | | a__plus(N, s(M)) | → | a__U11(tt, M, N) |
| mark(U11(X1, X2, X3)) | → | a__U11(mark(X1), X2, X3) | | mark(U12(X1, X2, X3)) | → | a__U12(mark(X1), X2, X3) |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | | mark(tt) | → | tt |
| mark(s(X)) | → | s(mark(X)) | | mark(0) | → | 0 |
| a__U11(X1, X2, X3) | → | U11(X1, X2, X3) | | a__U12(X1, X2, X3) | → | U12(X1, X2, X3) |
| a__plus(X1, X2) | → | plus(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, a__plus, U11, U12, mark, a__U12, a__U11
Strategy
Polynomial Interpretation
- 0: 0
- U11(x,y,z): z + y + x
- U12(x,y,z): z + y + x
- a__U11(x,y,z): z + y + x
- a__U11#(x,y,z): 2z + 2y + x
- a__U12(x,y,z): z + y + x
- a__U12#(x,y,z): 2z + 2y + x
- a__plus(x,y): y + x + 1
- a__plus#(x,y): 2y + 2x
- mark(x): x
- mark#(x): 2x
- plus(x,y): y + x + 1
- s(x): x + 1
- tt: 2
Improved Usable rules
| mark(tt) | → | tt | | a__U11(X1, X2, X3) | → | U11(X1, X2, X3) |
| mark(0) | → | 0 | | a__plus(N, 0) | → | mark(N) |
| a__plus(X1, X2) | → | plus(X1, X2) | | a__U12(X1, X2, X3) | → | U12(X1, X2, X3) |
| mark(U12(X1, X2, X3)) | → | a__U12(mark(X1), X2, X3) | | a__plus(N, s(M)) | → | a__U11(tt, M, N) |
| a__U11(tt, M, N) | → | a__U12(tt, M, N) | | mark(s(X)) | → | s(mark(X)) |
| a__U12(tt, M, N) | → | s(a__plus(mark(N), mark(M))) | | mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) |
| mark(U11(X1, X2, X3)) | → | a__U11(mark(X1), X2, X3) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(plus(X1, X2)) | → | a__plus#(mark(X1), mark(X2)) | | a__U12#(tt, M, N) | → | mark#(M) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| a__U12#(tt, M, N) | → | mark#(N) | | mark#(s(X)) | → | mark#(X) |
| a__U12#(tt, M, N) | → | a__plus#(mark(N), mark(M)) |
Problem 3: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| a__U11#(tt, M, N) | → | a__U12#(tt, M, N) | | mark#(U12(X1, X2, X3)) | → | a__U12#(mark(X1), X2, X3) |
| mark#(U12(X1, X2, X3)) | → | mark#(X1) | | a__plus#(N, s(M)) | → | a__U11#(tt, M, N) |
| mark#(U11(X1, X2, X3)) | → | a__U11#(mark(X1), X2, X3) | | mark#(U11(X1, X2, X3)) | → | mark#(X1) |
Rewrite Rules
| a__U11(tt, M, N) | → | a__U12(tt, M, N) | | a__U12(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__plus(N, 0) | → | mark(N) | | a__plus(N, s(M)) | → | a__U11(tt, M, N) |
| mark(U11(X1, X2, X3)) | → | a__U11(mark(X1), X2, X3) | | mark(U12(X1, X2, X3)) | → | a__U12(mark(X1), X2, X3) |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | | mark(tt) | → | tt |
| mark(s(X)) | → | s(mark(X)) | | mark(0) | → | 0 |
| a__U11(X1, X2, X3) | → | U11(X1, X2, X3) | | a__U12(X1, X2, X3) | → | U12(X1, X2, X3) |
| a__plus(X1, X2) | → | plus(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, a__plus, U11, a__U12, mark, U12, a__U11
Strategy
The following SCCs where found
| mark#(U12(X1, X2, X3)) → mark#(X1) | mark#(U11(X1, X2, X3)) → mark#(X1) |
Problem 4: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(U12(X1, X2, X3)) | → | mark#(X1) | | mark#(U11(X1, X2, X3)) | → | mark#(X1) |
Rewrite Rules
| a__U11(tt, M, N) | → | a__U12(tt, M, N) | | a__U12(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__plus(N, 0) | → | mark(N) | | a__plus(N, s(M)) | → | a__U11(tt, M, N) |
| mark(U11(X1, X2, X3)) | → | a__U11(mark(X1), X2, X3) | | mark(U12(X1, X2, X3)) | → | a__U12(mark(X1), X2, X3) |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | | mark(tt) | → | tt |
| mark(s(X)) | → | s(mark(X)) | | mark(0) | → | 0 |
| a__U11(X1, X2, X3) | → | U11(X1, X2, X3) | | a__U12(X1, X2, X3) | → | U12(X1, X2, X3) |
| a__plus(X1, X2) | → | plus(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, a__plus, U11, a__U12, mark, U12, a__U11
Strategy
Polynomial Interpretation
- 0: 0
- U11(x,y,z): 3z + 2y + x + 1
- U12(x,y,z): 2y + 2x
- a__U11(x,y,z): 0
- a__U12(x,y,z): 0
- a__plus(x,y): 0
- mark(x): 0
- mark#(x): x
- plus(x,y): 0
- s(x): 0
- 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:
| mark#(U11(X1, X2, X3)) | → | mark#(X1) |
Problem 5: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(U12(X1, X2, X3)) | → | mark#(X1) |
Rewrite Rules
| a__U11(tt, M, N) | → | a__U12(tt, M, N) | | a__U12(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__plus(N, 0) | → | mark(N) | | a__plus(N, s(M)) | → | a__U11(tt, M, N) |
| mark(U11(X1, X2, X3)) | → | a__U11(mark(X1), X2, X3) | | mark(U12(X1, X2, X3)) | → | a__U12(mark(X1), X2, X3) |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | | mark(tt) | → | tt |
| mark(s(X)) | → | s(mark(X)) | | mark(0) | → | 0 |
| a__U11(X1, X2, X3) | → | U11(X1, X2, X3) | | a__U12(X1, X2, X3) | → | U12(X1, X2, X3) |
| a__plus(X1, X2) | → | plus(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, a__plus, U11, U12, mark, a__U12, a__U11
Strategy
Polynomial Interpretation
- 0: 0
- U11(x,y,z): 0
- U12(x,y,z): 3y + 2x + 1
- a__U11(x,y,z): 0
- a__U12(x,y,z): 0
- a__plus(x,y): 0
- mark(x): 0
- mark#(x): 3x
- plus(x,y): 0
- s(x): 0
- 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:
| mark#(U12(X1, X2, X3)) | → | mark#(X1) |