YES
The TRS could be proven terminating. The proof took 41549 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (545ms).
| – Problem 2 was processed with processor SubtermCriterion (5ms).
| | – Problem 4 was processed with processor DependencyGraph (7ms).
| – Problem 3 was processed with processor PolynomialLinearRange4 (225ms).
| | – Problem 5 was processed with processor DependencyGraph (20ms).
| | | – Problem 6 was processed with processor PolynomialLinearRange4 (115ms).
| | | | – Problem 7 was processed with processor DependencyGraph (1ms).
| | | | | – Problem 8 was processed with processor PolynomialLinearRange4 (29ms).
| | | | | | – Problem 9 was processed with processor PolynomialLinearRange4 (20ms).
| | | | | | | – Problem 10 was processed with processor PolynomialLinearRange4 (68ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | mark#(isNat(X)) | → | a__isNat#(X) |
| a__plus#(N, 0) | → | a__U31#(a__isNat(N), N) | | a__U31#(tt, N) | → | mark#(N) |
| mark#(U31(X1, X2)) | → | a__U31#(mark(X1), X2) | | mark#(U41(X1, X2, X3)) | → | a__U41#(mark(X1), X2, X3) |
| a__isNat#(s(V1)) | → | a__isNat#(V1) | | mark#(U21(X)) | → | mark#(X) |
| a__U41#(tt, M, N) | → | a__isNat#(N) | | a__isNat#(plus(V1, V2)) | → | a__U11#(a__isNat(V1), V2) |
| a__isNat#(s(V1)) | → | a__U21#(a__isNat(V1)) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| a__U11#(tt, V2) | → | a__isNat#(V2) | | a__plus#(N, s(M)) | → | a__U41#(a__isNat(M), M, N) |
| mark#(U21(X)) | → | a__U21#(mark(X)) | | a__U41#(tt, M, N) | → | a__U42#(a__isNat(N), M, N) |
| a__isNat#(plus(V1, V2)) | → | a__isNat#(V1) | | a__plus#(N, s(M)) | → | a__isNat#(M) |
| mark#(U31(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| a__U42#(tt, M, N) | → | mark#(N) | | a__U42#(tt, M, N) | → | a__plus#(mark(N), mark(M)) |
| mark#(U12(X)) | → | mark#(X) | | mark#(plus(X1, X2)) | → | a__plus#(mark(X1), mark(X2)) |
| mark#(U12(X)) | → | a__U12#(mark(X)) | | mark#(U42(X1, X2, X3)) | → | mark#(X1) |
| a__U11#(tt, V2) | → | a__U12#(a__isNat(V2)) | | mark#(plus(X1, X2)) | → | mark#(X1) |
| a__U42#(tt, M, N) | → | mark#(M) | | mark#(U41(X1, X2, X3)) | → | mark#(X1) |
| mark#(U42(X1, X2, X3)) | → | a__U42#(mark(X1), X2, X3) | | a__plus#(N, 0) | → | a__isNat#(N) |
| mark#(U11(X1, X2)) | → | a__U11#(mark(X1), X2) |
Rewrite Rules
| a__U11(tt, V2) | → | a__U12(a__isNat(V2)) | | a__U12(tt) | → | tt |
| a__U21(tt) | → | tt | | a__U31(tt, N) | → | mark(N) |
| a__U41(tt, M, N) | → | a__U42(a__isNat(N), M, N) | | a__U42(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__isNat(0) | → | tt | | a__isNat(plus(V1, V2)) | → | a__U11(a__isNat(V1), V2) |
| a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) | | a__plus(N, 0) | → | a__U31(a__isNat(N), N) |
| a__plus(N, s(M)) | → | a__U41(a__isNat(M), M, N) | | mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) |
| mark(U12(X)) | → | a__U12(mark(X)) | | mark(isNat(X)) | → | a__isNat(X) |
| mark(U21(X)) | → | a__U21(mark(X)) | | mark(U31(X1, X2)) | → | a__U31(mark(X1), X2) |
| mark(U41(X1, X2, X3)) | → | a__U41(mark(X1), X2, X3) | | mark(U42(X1, X2, X3)) | → | a__U42(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) | → | U11(X1, X2) | | a__U12(X) | → | U12(X) |
| a__isNat(X) | → | isNat(X) | | a__U21(X) | → | U21(X) |
| a__U31(X1, X2) | → | U31(X1, X2) | | a__U41(X1, X2, X3) | → | U41(X1, X2, X3) |
| a__U42(X1, X2, X3) | → | U42(X1, X2, X3) | | a__plus(X1, X2) | → | plus(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, a__plus, mark, isNat, 0, U42, s, U41, tt, a__isNat, a__U41, U11, a__U31, a__U12, U12, a__U42, a__U11, U31, a__U21, U21
Strategy
The following SCCs where found
| mark#(U11(X1, X2)) → mark#(X1) | mark#(U12(X)) → mark#(X) |
| mark#(plus(X1, X2)) → a__plus#(mark(X1), mark(X2)) | a__plus#(N, 0) → a__U31#(a__isNat(N), N) |
| a__U31#(tt, N) → mark#(N) | mark#(U42(X1, X2, X3)) → mark#(X1) |
| mark#(U31(X1, X2)) → a__U31#(mark(X1), X2) | mark#(U41(X1, X2, X3)) → a__U41#(mark(X1), X2, X3) |
| mark#(U21(X)) → mark#(X) | mark#(plus(X1, X2)) → mark#(X1) |
| a__U42#(tt, M, N) → mark#(M) | mark#(U41(X1, X2, X3)) → mark#(X1) |
| mark#(plus(X1, X2)) → mark#(X2) | a__plus#(N, s(M)) → a__U41#(a__isNat(M), M, N) |
| mark#(U42(X1, X2, X3)) → a__U42#(mark(X1), X2, X3) | a__U41#(tt, M, N) → a__U42#(a__isNat(N), M, N) |
| mark#(U31(X1, X2)) → mark#(X1) | mark#(s(X)) → mark#(X) |
| a__U42#(tt, M, N) → mark#(N) | a__U42#(tt, M, N) → a__plus#(mark(N), mark(M)) |
| a__isNat#(plus(V1, V2)) → a__U11#(a__isNat(V1), V2) | a__U11#(tt, V2) → a__isNat#(V2) |
| a__isNat#(plus(V1, V2)) → a__isNat#(V1) | a__isNat#(s(V1)) → a__isNat#(V1) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| a__isNat#(plus(V1, V2)) | → | a__U11#(a__isNat(V1), V2) | | a__U11#(tt, V2) | → | a__isNat#(V2) |
| a__isNat#(plus(V1, V2)) | → | a__isNat#(V1) | | a__isNat#(s(V1)) | → | a__isNat#(V1) |
Rewrite Rules
| a__U11(tt, V2) | → | a__U12(a__isNat(V2)) | | a__U12(tt) | → | tt |
| a__U21(tt) | → | tt | | a__U31(tt, N) | → | mark(N) |
| a__U41(tt, M, N) | → | a__U42(a__isNat(N), M, N) | | a__U42(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__isNat(0) | → | tt | | a__isNat(plus(V1, V2)) | → | a__U11(a__isNat(V1), V2) |
| a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) | | a__plus(N, 0) | → | a__U31(a__isNat(N), N) |
| a__plus(N, s(M)) | → | a__U41(a__isNat(M), M, N) | | mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) |
| mark(U12(X)) | → | a__U12(mark(X)) | | mark(isNat(X)) | → | a__isNat(X) |
| mark(U21(X)) | → | a__U21(mark(X)) | | mark(U31(X1, X2)) | → | a__U31(mark(X1), X2) |
| mark(U41(X1, X2, X3)) | → | a__U41(mark(X1), X2, X3) | | mark(U42(X1, X2, X3)) | → | a__U42(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) | → | U11(X1, X2) | | a__U12(X) | → | U12(X) |
| a__isNat(X) | → | isNat(X) | | a__U21(X) | → | U21(X) |
| a__U31(X1, X2) | → | U31(X1, X2) | | a__U41(X1, X2, X3) | → | U41(X1, X2, X3) |
| a__U42(X1, X2, X3) | → | U42(X1, X2, X3) | | a__plus(X1, X2) | → | plus(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, a__plus, mark, isNat, 0, U42, s, U41, tt, a__isNat, a__U41, U11, a__U31, a__U12, U12, a__U42, a__U11, U31, a__U21, U21
Strategy
Projection
The following projection was used:
- π (a__U11#): 2
- π (a__isNat#): 1
Thus, the following dependency pairs are removed:
| a__isNat#(plus(V1, V2)) | → | a__U11#(a__isNat(V1), V2) | | a__isNat#(plus(V1, V2)) | → | a__isNat#(V1) |
| a__isNat#(s(V1)) | → | a__isNat#(V1) |
Problem 4: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| a__U11#(tt, V2) | → | a__isNat#(V2) |
Rewrite Rules
| a__U11(tt, V2) | → | a__U12(a__isNat(V2)) | | a__U12(tt) | → | tt |
| a__U21(tt) | → | tt | | a__U31(tt, N) | → | mark(N) |
| a__U41(tt, M, N) | → | a__U42(a__isNat(N), M, N) | | a__U42(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__isNat(0) | → | tt | | a__isNat(plus(V1, V2)) | → | a__U11(a__isNat(V1), V2) |
| a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) | | a__plus(N, 0) | → | a__U31(a__isNat(N), N) |
| a__plus(N, s(M)) | → | a__U41(a__isNat(M), M, N) | | mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) |
| mark(U12(X)) | → | a__U12(mark(X)) | | mark(isNat(X)) | → | a__isNat(X) |
| mark(U21(X)) | → | a__U21(mark(X)) | | mark(U31(X1, X2)) | → | a__U31(mark(X1), X2) |
| mark(U41(X1, X2, X3)) | → | a__U41(mark(X1), X2, X3) | | mark(U42(X1, X2, X3)) | → | a__U42(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) | → | U11(X1, X2) | | a__U12(X) | → | U12(X) |
| a__isNat(X) | → | isNat(X) | | a__U21(X) | → | U21(X) |
| a__U31(X1, X2) | → | U31(X1, X2) | | a__U41(X1, X2, X3) | → | U41(X1, X2, X3) |
| a__U42(X1, X2, X3) | → | U42(X1, X2, X3) | | a__plus(X1, X2) | → | plus(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, a__plus, mark, isNat, 0, U42, s, U41, tt, a__isNat, a__U41, U11, a__U31, U12, a__U12, U31, a__U11, a__U42, U21, a__U21
Strategy
There are no SCCs!
Problem 3: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(U12(X)) | → | mark#(X) | | mark#(U11(X1, X2)) | → | mark#(X1) |
| a__plus#(N, 0) | → | a__U31#(a__isNat(N), N) | | mark#(plus(X1, X2)) | → | a__plus#(mark(X1), mark(X2)) |
| a__U31#(tt, N) | → | mark#(N) | | mark#(U42(X1, X2, X3)) | → | mark#(X1) |
| mark#(U41(X1, X2, X3)) | → | a__U41#(mark(X1), X2, X3) | | mark#(U31(X1, X2)) | → | a__U31#(mark(X1), X2) |
| mark#(U21(X)) | → | mark#(X) | | mark#(plus(X1, X2)) | → | mark#(X1) |
| a__U42#(tt, M, N) | → | mark#(M) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(U41(X1, X2, X3)) | → | mark#(X1) | | a__plus#(N, s(M)) | → | a__U41#(a__isNat(M), M, N) |
| mark#(U42(X1, X2, X3)) | → | a__U42#(mark(X1), X2, X3) | | a__U41#(tt, M, N) | → | a__U42#(a__isNat(N), M, N) |
| mark#(U31(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| a__U42#(tt, M, N) | → | mark#(N) | | a__U42#(tt, M, N) | → | a__plus#(mark(N), mark(M)) |
Rewrite Rules
| a__U11(tt, V2) | → | a__U12(a__isNat(V2)) | | a__U12(tt) | → | tt |
| a__U21(tt) | → | tt | | a__U31(tt, N) | → | mark(N) |
| a__U41(tt, M, N) | → | a__U42(a__isNat(N), M, N) | | a__U42(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__isNat(0) | → | tt | | a__isNat(plus(V1, V2)) | → | a__U11(a__isNat(V1), V2) |
| a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) | | a__plus(N, 0) | → | a__U31(a__isNat(N), N) |
| a__plus(N, s(M)) | → | a__U41(a__isNat(M), M, N) | | mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) |
| mark(U12(X)) | → | a__U12(mark(X)) | | mark(isNat(X)) | → | a__isNat(X) |
| mark(U21(X)) | → | a__U21(mark(X)) | | mark(U31(X1, X2)) | → | a__U31(mark(X1), X2) |
| mark(U41(X1, X2, X3)) | → | a__U41(mark(X1), X2, X3) | | mark(U42(X1, X2, X3)) | → | a__U42(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) | → | U11(X1, X2) | | a__U12(X) | → | U12(X) |
| a__isNat(X) | → | isNat(X) | | a__U21(X) | → | U21(X) |
| a__U31(X1, X2) | → | U31(X1, X2) | | a__U41(X1, X2, X3) | → | U41(X1, X2, X3) |
| a__U42(X1, X2, X3) | → | U42(X1, X2, X3) | | a__plus(X1, X2) | → | plus(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, a__plus, mark, isNat, 0, U42, s, U41, tt, a__isNat, a__U41, U11, a__U31, a__U12, U12, a__U42, a__U11, U31, a__U21, U21
Strategy
Polynomial Interpretation
- 0: 0
- U11(x,y): x
- U12(x): x
- U21(x): x
- U31(x,y): y + x
- U41(x,y,z): 2z + y + x + 1
- U42(x,y,z): 2z + y + x + 1
- a__U11(x,y): x
- a__U12(x): x
- a__U21(x): x
- a__U31(x,y): y + x
- a__U31#(x,y): 2y
- a__U41(x,y,z): 2z + y + x + 1
- a__U41#(x,y,z): 2z + 2y + 2x
- a__U42(x,y,z): 2z + y + x + 1
- a__U42#(x,y,z): 2z + 2y
- a__isNat(x): 0
- a__plus(x,y): y + 2x
- a__plus#(x,y): 2y + 2x
- isNat(x): 0
- mark(x): x
- mark#(x): 2x
- plus(x,y): y + 2x
- s(x): x + 1
- tt: 0
Standard Usable rules
| a__U11(tt, V2) | → | a__U12(a__isNat(V2)) | | mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) |
| a__plus(N, s(M)) | → | a__U41(a__isNat(M), M, N) | | mark(U12(X)) | → | a__U12(mark(X)) |
| a__U41(tt, M, N) | → | a__U42(a__isNat(N), M, N) | | a__isNat(0) | → | tt |
| a__plus(N, 0) | → | a__U31(a__isNat(N), N) | | a__isNat(X) | → | isNat(X) |
| mark(U42(X1, X2, X3)) | → | a__U42(mark(X1), X2, X3) | | a__U12(X) | → | U12(X) |
| a__U31(tt, N) | → | mark(N) | | a__U42(X1, X2, X3) | → | U42(X1, X2, X3) |
| a__U21(tt) | → | tt | | a__isNat(plus(V1, V2)) | → | a__U11(a__isNat(V1), V2) |
| mark(U41(X1, X2, X3)) | → | a__U41(mark(X1), X2, X3) | | a__U12(tt) | → | tt |
| mark(tt) | → | tt | | a__U11(X1, X2) | → | U11(X1, X2) |
| mark(0) | → | 0 | | a__plus(X1, X2) | → | plus(X1, X2) |
| a__U21(X) | → | U21(X) | | mark(U21(X)) | → | a__U21(mark(X)) |
| mark(U31(X1, X2)) | → | a__U31(mark(X1), X2) | | mark(isNat(X)) | → | a__isNat(X) |
| a__U41(X1, X2, X3) | → | U41(X1, X2, X3) | | mark(s(X)) | → | s(mark(X)) |
| a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) | | a__U31(X1, X2) | → | U31(X1, X2) |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | | a__U42(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(U42(X1, X2, X3)) | → | mark#(X1) | | mark#(U41(X1, X2, X3)) | → | a__U41#(mark(X1), X2, X3) |
| mark#(U41(X1, X2, X3)) | → | mark#(X1) | | mark#(U42(X1, X2, X3)) | → | a__U42#(mark(X1), X2, X3) |
| a__plus#(N, s(M)) | → | a__U41#(a__isNat(M), M, N) | | mark#(s(X)) | → | mark#(X) |
Problem 5: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(U12(X)) | → | mark#(X) | | mark#(U11(X1, X2)) | → | mark#(X1) |
| a__plus#(N, 0) | → | a__U31#(a__isNat(N), N) | | mark#(plus(X1, X2)) | → | a__plus#(mark(X1), mark(X2)) |
| a__U31#(tt, N) | → | mark#(N) | | mark#(U31(X1, X2)) | → | a__U31#(mark(X1), X2) |
| mark#(U21(X)) | → | mark#(X) | | mark#(plus(X1, X2)) | → | mark#(X1) |
| a__U42#(tt, M, N) | → | mark#(M) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| a__U41#(tt, M, N) | → | a__U42#(a__isNat(N), M, N) | | mark#(U31(X1, X2)) | → | mark#(X1) |
| a__U42#(tt, M, N) | → | mark#(N) | | a__U42#(tt, M, N) | → | a__plus#(mark(N), mark(M)) |
Rewrite Rules
| a__U11(tt, V2) | → | a__U12(a__isNat(V2)) | | a__U12(tt) | → | tt |
| a__U21(tt) | → | tt | | a__U31(tt, N) | → | mark(N) |
| a__U41(tt, M, N) | → | a__U42(a__isNat(N), M, N) | | a__U42(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__isNat(0) | → | tt | | a__isNat(plus(V1, V2)) | → | a__U11(a__isNat(V1), V2) |
| a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) | | a__plus(N, 0) | → | a__U31(a__isNat(N), N) |
| a__plus(N, s(M)) | → | a__U41(a__isNat(M), M, N) | | mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) |
| mark(U12(X)) | → | a__U12(mark(X)) | | mark(isNat(X)) | → | a__isNat(X) |
| mark(U21(X)) | → | a__U21(mark(X)) | | mark(U31(X1, X2)) | → | a__U31(mark(X1), X2) |
| mark(U41(X1, X2, X3)) | → | a__U41(mark(X1), X2, X3) | | mark(U42(X1, X2, X3)) | → | a__U42(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) | → | U11(X1, X2) | | a__U12(X) | → | U12(X) |
| a__isNat(X) | → | isNat(X) | | a__U21(X) | → | U21(X) |
| a__U31(X1, X2) | → | U31(X1, X2) | | a__U41(X1, X2, X3) | → | U41(X1, X2, X3) |
| a__U42(X1, X2, X3) | → | U42(X1, X2, X3) | | a__plus(X1, X2) | → | plus(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, a__plus, mark, isNat, 0, U42, s, U41, tt, a__isNat, a__U41, U11, a__U31, U12, a__U12, U31, a__U11, a__U42, U21, a__U21
Strategy
The following SCCs where found
| mark#(U11(X1, X2)) → mark#(X1) | mark#(U12(X)) → mark#(X) |
| mark#(plus(X1, X2)) → mark#(X2) | mark#(plus(X1, X2)) → a__plus#(mark(X1), mark(X2)) |
| a__plus#(N, 0) → a__U31#(a__isNat(N), N) | a__U31#(tt, N) → mark#(N) |
| mark#(U31(X1, X2)) → mark#(X1) | mark#(U31(X1, X2)) → a__U31#(mark(X1), X2) |
| mark#(U21(X)) → mark#(X) | mark#(plus(X1, X2)) → mark#(X1) |
Problem 6: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | mark#(U12(X)) | → | mark#(X) |
| mark#(plus(X1, X2)) | → | mark#(X2) | | mark#(plus(X1, X2)) | → | a__plus#(mark(X1), mark(X2)) |
| a__plus#(N, 0) | → | a__U31#(a__isNat(N), N) | | a__U31#(tt, N) | → | mark#(N) |
| mark#(U31(X1, X2)) | → | mark#(X1) | | mark#(U31(X1, X2)) | → | a__U31#(mark(X1), X2) |
| mark#(U21(X)) | → | mark#(X) | | mark#(plus(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| a__U11(tt, V2) | → | a__U12(a__isNat(V2)) | | a__U12(tt) | → | tt |
| a__U21(tt) | → | tt | | a__U31(tt, N) | → | mark(N) |
| a__U41(tt, M, N) | → | a__U42(a__isNat(N), M, N) | | a__U42(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__isNat(0) | → | tt | | a__isNat(plus(V1, V2)) | → | a__U11(a__isNat(V1), V2) |
| a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) | | a__plus(N, 0) | → | a__U31(a__isNat(N), N) |
| a__plus(N, s(M)) | → | a__U41(a__isNat(M), M, N) | | mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) |
| mark(U12(X)) | → | a__U12(mark(X)) | | mark(isNat(X)) | → | a__isNat(X) |
| mark(U21(X)) | → | a__U21(mark(X)) | | mark(U31(X1, X2)) | → | a__U31(mark(X1), X2) |
| mark(U41(X1, X2, X3)) | → | a__U41(mark(X1), X2, X3) | | mark(U42(X1, X2, X3)) | → | a__U42(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) | → | U11(X1, X2) | | a__U12(X) | → | U12(X) |
| a__isNat(X) | → | isNat(X) | | a__U21(X) | → | U21(X) |
| a__U31(X1, X2) | → | U31(X1, X2) | | a__U41(X1, X2, X3) | → | U41(X1, X2, X3) |
| a__U42(X1, X2, X3) | → | U42(X1, X2, X3) | | a__plus(X1, X2) | → | plus(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, a__plus, mark, isNat, 0, U42, s, U41, tt, a__isNat, a__U41, U11, a__U31, U12, a__U12, U31, a__U11, a__U42, U21, a__U21
Strategy
Polynomial Interpretation
- 0: 2
- U11(x,y): x
- U12(x): x
- U21(x): x
- U31(x,y): y + x + 1
- U41(x,y,z): x
- U42(x,y,z): 1
- a__U11(x,y): x
- a__U12(x): x
- a__U21(x): x
- a__U31(x,y): y + x + 1
- a__U31#(x,y): 2y
- a__U41(x,y,z): x
- a__U42(x,y,z): 1
- a__isNat(x): 2
- a__plus(x,y): y + x + 1
- a__plus#(x,y): 2x + 1
- isNat(x): 2
- mark(x): x
- mark#(x): 2x
- plus(x,y): y + x + 1
- s(x): 1
- tt: 2
Standard Usable rules
| a__U11(tt, V2) | → | a__U12(a__isNat(V2)) | | mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) |
| a__plus(N, s(M)) | → | a__U41(a__isNat(M), M, N) | | mark(U12(X)) | → | a__U12(mark(X)) |
| a__U41(tt, M, N) | → | a__U42(a__isNat(N), M, N) | | a__isNat(0) | → | tt |
| a__plus(N, 0) | → | a__U31(a__isNat(N), N) | | a__isNat(X) | → | isNat(X) |
| mark(U42(X1, X2, X3)) | → | a__U42(mark(X1), X2, X3) | | a__U12(X) | → | U12(X) |
| a__U31(tt, N) | → | mark(N) | | a__U42(X1, X2, X3) | → | U42(X1, X2, X3) |
| a__U21(tt) | → | tt | | a__isNat(plus(V1, V2)) | → | a__U11(a__isNat(V1), V2) |
| mark(U41(X1, X2, X3)) | → | a__U41(mark(X1), X2, X3) | | a__U12(tt) | → | tt |
| mark(tt) | → | tt | | a__U11(X1, X2) | → | U11(X1, X2) |
| mark(0) | → | 0 | | a__plus(X1, X2) | → | plus(X1, X2) |
| a__U21(X) | → | U21(X) | | mark(U21(X)) | → | a__U21(mark(X)) |
| mark(U31(X1, X2)) | → | a__U31(mark(X1), X2) | | mark(isNat(X)) | → | a__isNat(X) |
| a__U41(X1, X2, X3) | → | U41(X1, X2, X3) | | mark(s(X)) | → | s(mark(X)) |
| a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) | | a__U31(X1, X2) | → | U31(X1, X2) |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | | a__U42(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(plus(X1, X2)) | → | mark#(X2) | | a__plus#(N, 0) | → | a__U31#(a__isNat(N), N) |
| mark#(plus(X1, X2)) | → | a__plus#(mark(X1), mark(X2)) | | mark#(U31(X1, X2)) | → | mark#(X1) |
| mark#(U31(X1, X2)) | → | a__U31#(mark(X1), X2) | | mark#(plus(X1, X2)) | → | mark#(X1) |
Problem 7: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(U12(X)) | → | mark#(X) | | mark#(U11(X1, X2)) | → | mark#(X1) |
| a__U31#(tt, N) | → | mark#(N) | | mark#(U21(X)) | → | mark#(X) |
Rewrite Rules
| a__U11(tt, V2) | → | a__U12(a__isNat(V2)) | | a__U12(tt) | → | tt |
| a__U21(tt) | → | tt | | a__U31(tt, N) | → | mark(N) |
| a__U41(tt, M, N) | → | a__U42(a__isNat(N), M, N) | | a__U42(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__isNat(0) | → | tt | | a__isNat(plus(V1, V2)) | → | a__U11(a__isNat(V1), V2) |
| a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) | | a__plus(N, 0) | → | a__U31(a__isNat(N), N) |
| a__plus(N, s(M)) | → | a__U41(a__isNat(M), M, N) | | mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) |
| mark(U12(X)) | → | a__U12(mark(X)) | | mark(isNat(X)) | → | a__isNat(X) |
| mark(U21(X)) | → | a__U21(mark(X)) | | mark(U31(X1, X2)) | → | a__U31(mark(X1), X2) |
| mark(U41(X1, X2, X3)) | → | a__U41(mark(X1), X2, X3) | | mark(U42(X1, X2, X3)) | → | a__U42(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) | → | U11(X1, X2) | | a__U12(X) | → | U12(X) |
| a__isNat(X) | → | isNat(X) | | a__U21(X) | → | U21(X) |
| a__U31(X1, X2) | → | U31(X1, X2) | | a__U41(X1, X2, X3) | → | U41(X1, X2, X3) |
| a__U42(X1, X2, X3) | → | U42(X1, X2, X3) | | a__plus(X1, X2) | → | plus(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, a__plus, mark, isNat, 0, U42, s, U41, tt, a__isNat, a__U41, U11, a__U31, a__U12, U12, a__U42, a__U11, U31, a__U21, U21
Strategy
The following SCCs where found
| mark#(U12(X)) → mark#(X) | mark#(U11(X1, X2)) → mark#(X1) |
| mark#(U21(X)) → mark#(X) |
Problem 8: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(U12(X)) | → | mark#(X) | | mark#(U11(X1, X2)) | → | mark#(X1) |
| mark#(U21(X)) | → | mark#(X) |
Rewrite Rules
| a__U11(tt, V2) | → | a__U12(a__isNat(V2)) | | a__U12(tt) | → | tt |
| a__U21(tt) | → | tt | | a__U31(tt, N) | → | mark(N) |
| a__U41(tt, M, N) | → | a__U42(a__isNat(N), M, N) | | a__U42(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__isNat(0) | → | tt | | a__isNat(plus(V1, V2)) | → | a__U11(a__isNat(V1), V2) |
| a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) | | a__plus(N, 0) | → | a__U31(a__isNat(N), N) |
| a__plus(N, s(M)) | → | a__U41(a__isNat(M), M, N) | | mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) |
| mark(U12(X)) | → | a__U12(mark(X)) | | mark(isNat(X)) | → | a__isNat(X) |
| mark(U21(X)) | → | a__U21(mark(X)) | | mark(U31(X1, X2)) | → | a__U31(mark(X1), X2) |
| mark(U41(X1, X2, X3)) | → | a__U41(mark(X1), X2, X3) | | mark(U42(X1, X2, X3)) | → | a__U42(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) | → | U11(X1, X2) | | a__U12(X) | → | U12(X) |
| a__isNat(X) | → | isNat(X) | | a__U21(X) | → | U21(X) |
| a__U31(X1, X2) | → | U31(X1, X2) | | a__U41(X1, X2, X3) | → | U41(X1, X2, X3) |
| a__U42(X1, X2, X3) | → | U42(X1, X2, X3) | | a__plus(X1, X2) | → | plus(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, a__plus, mark, isNat, 0, U42, s, U41, tt, a__isNat, a__U41, U11, a__U31, a__U12, U12, a__U42, a__U11, U31, a__U21, U21
Strategy
Polynomial Interpretation
- 0: 0
- U11(x,y): y + 2x
- U12(x): 2x
- U21(x): 2x + 2
- U31(x,y): 0
- U41(x,y,z): 0
- U42(x,y,z): 0
- a__U11(x,y): 0
- a__U12(x): 0
- a__U21(x): 0
- a__U31(x,y): 0
- a__U41(x,y,z): 0
- a__U42(x,y,z): 0
- a__isNat(x): 0
- a__plus(x,y): 0
- isNat(x): 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:
Problem 9: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | mark#(U12(X)) | → | mark#(X) |
Rewrite Rules
| a__U11(tt, V2) | → | a__U12(a__isNat(V2)) | | a__U12(tt) | → | tt |
| a__U21(tt) | → | tt | | a__U31(tt, N) | → | mark(N) |
| a__U41(tt, M, N) | → | a__U42(a__isNat(N), M, N) | | a__U42(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__isNat(0) | → | tt | | a__isNat(plus(V1, V2)) | → | a__U11(a__isNat(V1), V2) |
| a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) | | a__plus(N, 0) | → | a__U31(a__isNat(N), N) |
| a__plus(N, s(M)) | → | a__U41(a__isNat(M), M, N) | | mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) |
| mark(U12(X)) | → | a__U12(mark(X)) | | mark(isNat(X)) | → | a__isNat(X) |
| mark(U21(X)) | → | a__U21(mark(X)) | | mark(U31(X1, X2)) | → | a__U31(mark(X1), X2) |
| mark(U41(X1, X2, X3)) | → | a__U41(mark(X1), X2, X3) | | mark(U42(X1, X2, X3)) | → | a__U42(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) | → | U11(X1, X2) | | a__U12(X) | → | U12(X) |
| a__isNat(X) | → | isNat(X) | | a__U21(X) | → | U21(X) |
| a__U31(X1, X2) | → | U31(X1, X2) | | a__U41(X1, X2, X3) | → | U41(X1, X2, X3) |
| a__U42(X1, X2, X3) | → | U42(X1, X2, X3) | | a__plus(X1, X2) | → | plus(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, a__plus, mark, isNat, 0, U42, s, U41, tt, a__isNat, a__U41, U11, a__U31, U12, a__U12, U31, a__U11, a__U42, U21, a__U21
Strategy
Polynomial Interpretation
- 0: 0
- U11(x,y): 2x + 1
- U12(x): 3x
- U21(x): 0
- U31(x,y): 0
- U41(x,y,z): 0
- U42(x,y,z): 0
- a__U11(x,y): 0
- a__U12(x): 0
- a__U21(x): 0
- a__U31(x,y): 0
- a__U41(x,y,z): 0
- a__U42(x,y,z): 0
- a__isNat(x): 0
- a__plus(x,y): 0
- isNat(x): 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#(U11(X1, X2)) | → | mark#(X1) |
Problem 10: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| a__U11(tt, V2) | → | a__U12(a__isNat(V2)) | | a__U12(tt) | → | tt |
| a__U21(tt) | → | tt | | a__U31(tt, N) | → | mark(N) |
| a__U41(tt, M, N) | → | a__U42(a__isNat(N), M, N) | | a__U42(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__isNat(0) | → | tt | | a__isNat(plus(V1, V2)) | → | a__U11(a__isNat(V1), V2) |
| a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) | | a__plus(N, 0) | → | a__U31(a__isNat(N), N) |
| a__plus(N, s(M)) | → | a__U41(a__isNat(M), M, N) | | mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) |
| mark(U12(X)) | → | a__U12(mark(X)) | | mark(isNat(X)) | → | a__isNat(X) |
| mark(U21(X)) | → | a__U21(mark(X)) | | mark(U31(X1, X2)) | → | a__U31(mark(X1), X2) |
| mark(U41(X1, X2, X3)) | → | a__U41(mark(X1), X2, X3) | | mark(U42(X1, X2, X3)) | → | a__U42(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) | → | U11(X1, X2) | | a__U12(X) | → | U12(X) |
| a__isNat(X) | → | isNat(X) | | a__U21(X) | → | U21(X) |
| a__U31(X1, X2) | → | U31(X1, X2) | | a__U41(X1, X2, X3) | → | U41(X1, X2, X3) |
| a__U42(X1, X2, X3) | → | U42(X1, X2, X3) | | a__plus(X1, X2) | → | plus(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, a__plus, mark, isNat, 0, U42, s, U41, tt, a__isNat, a__U41, U11, a__U31, a__U12, U12, a__U42, a__U11, U31, a__U21, U21
Strategy
Polynomial Interpretation
- 0: 0
- U11(x,y): 0
- U12(x): x + 1
- U21(x): 0
- U31(x,y): 0
- U41(x,y,z): 0
- U42(x,y,z): 0
- a__U11(x,y): 0
- a__U12(x): 0
- a__U21(x): 0
- a__U31(x,y): 0
- a__U41(x,y,z): 0
- a__U42(x,y,z): 0
- a__isNat(x): 0
- a__plus(x,y): 0
- isNat(x): 0
- mark(x): 0
- mark#(x): 2x + 1
- 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: