YES
The TRS could be proven terminating. The proof took 41348 ms.
The following DP Processors were used
Problem 1 was processed with processor PolynomialLinearRange4 (279ms).
| – Problem 2 was processed with processor PolynomialLinearRange4 (203ms).
| | – Problem 3 was processed with processor DependencyGraph (14ms).
| | | – Problem 4 was processed with processor PolynomialLinearRange4 (94ms).
| | | | – Problem 5 was processed with processor DependencyGraph (19ms).
| | | | | – Problem 6 was processed with processor PolynomialLinearRange4 (107ms).
| | | | | | – Problem 7 was processed with processor DependencyGraph (1ms).
Problem 1: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | mark#(isNat(X)) | → | a__isNat#(X) |
| a__isNat#(plus(V1, V2)) | → | a__and#(a__isNat(V1), isNat(V2)) | | a__U11#(tt, N) | → | mark#(N) |
| a__U21#(tt, M, N) | → | mark#(N) | | mark#(plus(X1, X2)) | → | a__plus#(mark(X1), mark(X2)) |
| mark#(U21(X1, X2, X3)) | → | a__U21#(mark(X1), X2, X3) | | a__U21#(tt, M, N) | → | a__plus#(mark(N), mark(M)) |
| a__isNat#(s(V1)) | → | a__isNat#(V1) | | a__plus#(N, 0) | → | a__U11#(a__isNat(N), N) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | mark#(and(X1, X2)) | → | mark#(X1) |
| a__plus#(N, s(M)) | → | a__U21#(a__and(a__isNat(M), isNat(N)), M, N) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(and(X1, X2)) | → | a__and#(mark(X1), X2) | | a__and#(tt, X) | → | mark#(X) |
| a__plus#(N, 0) | → | a__isNat#(N) | | a__isNat#(plus(V1, V2)) | → | a__isNat#(V1) |
| mark#(U11(X1, X2)) | → | a__U11#(mark(X1), X2) | | a__plus#(N, s(M)) | → | a__isNat#(M) |
| mark#(U21(X1, X2, X3)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| a__plus#(N, s(M)) | → | a__and#(a__isNat(M), isNat(N)) | | a__U21#(tt, M, N) | → | mark#(M) |
Rewrite Rules
| a__U11(tt, N) | → | mark(N) | | a__U21(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__and(tt, X) | → | mark(X) | | a__isNat(0) | → | tt |
| a__isNat(plus(V1, V2)) | → | a__and(a__isNat(V1), isNat(V2)) | | a__isNat(s(V1)) | → | a__isNat(V1) |
| a__plus(N, 0) | → | a__U11(a__isNat(N), N) | | a__plus(N, s(M)) | → | a__U21(a__and(a__isNat(M), isNat(N)), M, N) |
| mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) | | mark(U21(X1, X2, X3)) | → | a__U21(mark(X1), X2, X3) |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(isNat(X)) | → | a__isNat(X) | | mark(tt) | → | tt |
| mark(s(X)) | → | s(mark(X)) | | mark(0) | → | 0 |
| a__U11(X1, X2) | → | U11(X1, X2) | | a__U21(X1, X2, X3) | → | U21(X1, X2, X3) |
| a__plus(X1, X2) | → | plus(X1, X2) | | a__and(X1, X2) | → | and(X1, X2) |
| a__isNat(X) | → | isNat(X) |
Original Signature
Termination of terms over the following signature is verified: plus, a__plus, mark, and, a__and, isNat, 0, s, tt, a__isNat, U11, a__U11, U21, a__U21
Strategy
Polynomial Interpretation
- 0: 1
- U11(x,y): y + x
- U21(x,y,z): z + y + x
- a__U11(x,y): y + x
- a__U11#(x,y): 2y
- a__U21(x,y,z): z + y + x
- a__U21#(x,y,z): 2z + 2y
- a__and(x,y): y + x
- a__and#(x,y): 2y
- a__isNat(x): 0
- a__isNat#(x): 0
- a__plus(x,y): y + x
- a__plus#(x,y): 2y + 2x
- and(x,y): y + x
- isNat(x): 0
- mark(x): x
- mark#(x): 2x
- plus(x,y): y + x
- s(x): x
- tt: 0
Standard Usable rules
| mark(tt) | → | tt | | a__U11(X1, X2) | → | U11(X1, X2) |
| mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) | | mark(0) | → | 0 |
| a__plus(X1, X2) | → | plus(X1, X2) | | a__and(X1, X2) | → | and(X1, X2) |
| a__plus(N, s(M)) | → | a__U21(a__and(a__isNat(M), isNat(N)), M, N) | | a__and(tt, X) | → | mark(X) |
| a__U21(tt, M, N) | → | s(a__plus(mark(N), mark(M))) | | mark(isNat(X)) | → | a__isNat(X) |
| a__isNat(0) | → | tt | | a__U11(tt, N) | → | mark(N) |
| a__isNat(X) | → | isNat(X) | | a__plus(N, 0) | → | a__U11(a__isNat(N), N) |
| a__U21(X1, X2, X3) | → | U21(X1, X2, X3) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(s(X)) | → | s(mark(X)) | | a__isNat(s(V1)) | → | a__isNat(V1) |
| a__isNat(plus(V1, V2)) | → | a__and(a__isNat(V1), isNat(V2)) | | mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) |
| mark(U21(X1, X2, X3)) | → | a__U21(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) | → | a__U11#(a__isNat(N), N) | | a__plus#(N, 0) | → | a__isNat#(N) |
Problem 2: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | mark#(isNat(X)) | → | a__isNat#(X) |
| a__isNat#(plus(V1, V2)) | → | a__and#(a__isNat(V1), isNat(V2)) | | a__U11#(tt, N) | → | mark#(N) |
| a__U21#(tt, M, N) | → | mark#(N) | | mark#(plus(X1, X2)) | → | a__plus#(mark(X1), mark(X2)) |
| mark#(U21(X1, X2, X3)) | → | a__U21#(mark(X1), X2, X3) | | a__U21#(tt, M, N) | → | a__plus#(mark(N), mark(M)) |
| a__isNat#(s(V1)) | → | a__isNat#(V1) | | mark#(plus(X1, X2)) | → | mark#(X1) |
| mark#(and(X1, X2)) | → | mark#(X1) | | a__plus#(N, s(M)) | → | a__U21#(a__and(a__isNat(M), isNat(N)), M, N) |
| mark#(and(X1, X2)) | → | a__and#(mark(X1), X2) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| a__and#(tt, X) | → | mark#(X) | | mark#(U11(X1, X2)) | → | a__U11#(mark(X1), X2) |
| a__isNat#(plus(V1, V2)) | → | a__isNat#(V1) | | mark#(U21(X1, X2, X3)) | → | mark#(X1) |
| a__plus#(N, s(M)) | → | a__isNat#(M) | | mark#(s(X)) | → | mark#(X) |
| a__plus#(N, s(M)) | → | a__and#(a__isNat(M), isNat(N)) | | a__U21#(tt, M, N) | → | mark#(M) |
Rewrite Rules
| a__U11(tt, N) | → | mark(N) | | a__U21(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__and(tt, X) | → | mark(X) | | a__isNat(0) | → | tt |
| a__isNat(plus(V1, V2)) | → | a__and(a__isNat(V1), isNat(V2)) | | a__isNat(s(V1)) | → | a__isNat(V1) |
| a__plus(N, 0) | → | a__U11(a__isNat(N), N) | | a__plus(N, s(M)) | → | a__U21(a__and(a__isNat(M), isNat(N)), M, N) |
| mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) | | mark(U21(X1, X2, X3)) | → | a__U21(mark(X1), X2, X3) |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(isNat(X)) | → | a__isNat(X) | | mark(tt) | → | tt |
| mark(s(X)) | → | s(mark(X)) | | mark(0) | → | 0 |
| a__U11(X1, X2) | → | U11(X1, X2) | | a__U21(X1, X2, X3) | → | U21(X1, X2, X3) |
| a__plus(X1, X2) | → | plus(X1, X2) | | a__and(X1, X2) | → | and(X1, X2) |
| a__isNat(X) | → | isNat(X) |
Original Signature
Termination of terms over the following signature is verified: plus, a__plus, mark, and, a__and, isNat, 0, s, tt, a__isNat, U11, a__U11, a__U21, U21
Strategy
Polynomial Interpretation
- 0: 1
- U11(x,y): y + x + 1
- U21(x,y,z): 2z + y + x + 1
- a__U11(x,y): y + x + 1
- a__U11#(x,y): 2y + 2
- a__U21(x,y,z): 2z + y + x + 1
- a__U21#(x,y,z): 2z + 2y + 2
- a__and(x,y): 2y + 2x
- a__and#(x,y): 2y
- a__isNat(x): 0
- a__isNat#(x): 0
- a__plus(x,y): y + 2x
- a__plus#(x,y): 2y + 2x
- and(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
| mark(tt) | → | tt | | a__U11(X1, X2) | → | U11(X1, X2) |
| mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) | | mark(0) | → | 0 |
| a__plus(X1, X2) | → | plus(X1, X2) | | a__and(X1, X2) | → | and(X1, X2) |
| a__plus(N, s(M)) | → | a__U21(a__and(a__isNat(M), isNat(N)), M, N) | | a__and(tt, X) | → | mark(X) |
| a__U21(tt, M, N) | → | s(a__plus(mark(N), mark(M))) | | mark(isNat(X)) | → | a__isNat(X) |
| a__isNat(0) | → | tt | | a__U11(tt, N) | → | mark(N) |
| a__isNat(X) | → | isNat(X) | | a__plus(N, 0) | → | a__U11(a__isNat(N), N) |
| a__U21(X1, X2, X3) | → | U21(X1, X2, X3) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(s(X)) | → | s(mark(X)) | | a__isNat(s(V1)) | → | a__isNat(V1) |
| a__isNat(plus(V1, V2)) | → | a__and(a__isNat(V1), isNat(V2)) | | mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) |
| mark(U21(X1, X2, X3)) | → | a__U21(mark(X1), X2, X3) |
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) | | a__U11#(tt, N) | → | mark#(N) |
| a__U21#(tt, M, N) | → | mark#(N) | | a__U21#(tt, M, N) | → | a__plus#(mark(N), mark(M)) |
| a__plus#(N, s(M)) | → | a__isNat#(M) | | mark#(U21(X1, X2, X3)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | a__plus#(N, s(M)) | → | a__and#(a__isNat(M), isNat(N)) |
| a__U21#(tt, M, N) | → | mark#(M) |
Problem 3: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(isNat(X)) | → | a__isNat#(X) | | a__isNat#(plus(V1, V2)) | → | a__and#(a__isNat(V1), isNat(V2)) |
| mark#(plus(X1, X2)) | → | a__plus#(mark(X1), mark(X2)) | | mark#(U21(X1, X2, X3)) | → | a__U21#(mark(X1), X2, X3) |
| a__isNat#(s(V1)) | → | a__isNat#(V1) | | mark#(plus(X1, X2)) | → | mark#(X1) |
| mark#(and(X1, X2)) | → | mark#(X1) | | a__plus#(N, s(M)) | → | a__U21#(a__and(a__isNat(M), isNat(N)), M, N) |
| mark#(and(X1, X2)) | → | a__and#(mark(X1), X2) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| a__and#(tt, X) | → | mark#(X) | | mark#(U11(X1, X2)) | → | a__U11#(mark(X1), X2) |
| a__isNat#(plus(V1, V2)) | → | a__isNat#(V1) |
Rewrite Rules
| a__U11(tt, N) | → | mark(N) | | a__U21(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__and(tt, X) | → | mark(X) | | a__isNat(0) | → | tt |
| a__isNat(plus(V1, V2)) | → | a__and(a__isNat(V1), isNat(V2)) | | a__isNat(s(V1)) | → | a__isNat(V1) |
| a__plus(N, 0) | → | a__U11(a__isNat(N), N) | | a__plus(N, s(M)) | → | a__U21(a__and(a__isNat(M), isNat(N)), M, N) |
| mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) | | mark(U21(X1, X2, X3)) | → | a__U21(mark(X1), X2, X3) |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(isNat(X)) | → | a__isNat(X) | | mark(tt) | → | tt |
| mark(s(X)) | → | s(mark(X)) | | mark(0) | → | 0 |
| a__U11(X1, X2) | → | U11(X1, X2) | | a__U21(X1, X2, X3) | → | U21(X1, X2, X3) |
| a__plus(X1, X2) | → | plus(X1, X2) | | a__and(X1, X2) | → | and(X1, X2) |
| a__isNat(X) | → | isNat(X) |
Original Signature
Termination of terms over the following signature is verified: plus, a__plus, mark, and, a__and, isNat, 0, s, tt, a__isNat, U11, a__U11, U21, a__U21
Strategy
The following SCCs where found
| mark#(and(X1, X2)) → mark#(X1) | a__isNat#(plus(V1, V2)) → a__and#(a__isNat(V1), isNat(V2)) |
| mark#(isNat(X)) → a__isNat#(X) | mark#(plus(X1, X2)) → mark#(X2) |
| mark#(and(X1, X2)) → a__and#(mark(X1), X2) | a__and#(tt, X) → mark#(X) |
| a__isNat#(plus(V1, V2)) → a__isNat#(V1) | a__isNat#(s(V1)) → a__isNat#(V1) |
| mark#(plus(X1, X2)) → mark#(X1) |
Problem 4: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(and(X1, X2)) | → | mark#(X1) | | a__isNat#(plus(V1, V2)) | → | a__and#(a__isNat(V1), isNat(V2)) |
| mark#(isNat(X)) | → | a__isNat#(X) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(and(X1, X2)) | → | a__and#(mark(X1), X2) | | a__and#(tt, X) | → | mark#(X) |
| a__isNat#(plus(V1, V2)) | → | a__isNat#(V1) | | a__isNat#(s(V1)) | → | a__isNat#(V1) |
| mark#(plus(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| a__U11(tt, N) | → | mark(N) | | a__U21(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__and(tt, X) | → | mark(X) | | a__isNat(0) | → | tt |
| a__isNat(plus(V1, V2)) | → | a__and(a__isNat(V1), isNat(V2)) | | a__isNat(s(V1)) | → | a__isNat(V1) |
| a__plus(N, 0) | → | a__U11(a__isNat(N), N) | | a__plus(N, s(M)) | → | a__U21(a__and(a__isNat(M), isNat(N)), M, N) |
| mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) | | mark(U21(X1, X2, X3)) | → | a__U21(mark(X1), X2, X3) |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(isNat(X)) | → | a__isNat(X) | | mark(tt) | → | tt |
| mark(s(X)) | → | s(mark(X)) | | mark(0) | → | 0 |
| a__U11(X1, X2) | → | U11(X1, X2) | | a__U21(X1, X2, X3) | → | U21(X1, X2, X3) |
| a__plus(X1, X2) | → | plus(X1, X2) | | a__and(X1, X2) | → | and(X1, X2) |
| a__isNat(X) | → | isNat(X) |
Original Signature
Termination of terms over the following signature is verified: plus, a__plus, mark, and, a__and, isNat, 0, s, tt, a__isNat, U11, a__U11, U21, a__U21
Strategy
Polynomial Interpretation
- 0: 1
- U11(x,y): y + 1
- U21(x,y,z): z + 2y + 2
- a__U11(x,y): y + 1
- a__U21(x,y,z): z + 2y + 2
- a__and(x,y): 2y + x
- a__and#(x,y): 2y
- a__isNat(x): 2x + 1
- a__isNat#(x): 2x + 1
- a__plus(x,y): 2y + x + 1
- and(x,y): 2y + x
- isNat(x): 2x + 1
- mark(x): x
- mark#(x): x
- plus(x,y): 2y + x + 1
- s(x): x + 1
- tt: 0
Standard Usable rules
| mark(tt) | → | tt | | a__U11(X1, X2) | → | U11(X1, X2) |
| mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) | | mark(0) | → | 0 |
| a__plus(X1, X2) | → | plus(X1, X2) | | a__and(X1, X2) | → | and(X1, X2) |
| a__and(tt, X) | → | mark(X) | | a__plus(N, s(M)) | → | a__U21(a__and(a__isNat(M), isNat(N)), M, N) |
| a__U21(tt, M, N) | → | s(a__plus(mark(N), mark(M))) | | mark(isNat(X)) | → | a__isNat(X) |
| a__isNat(0) | → | tt | | a__U11(tt, N) | → | mark(N) |
| a__isNat(X) | → | isNat(X) | | a__plus(N, 0) | → | a__U11(a__isNat(N), N) |
| a__U21(X1, X2, X3) | → | U21(X1, X2, X3) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(s(X)) | → | s(mark(X)) | | a__isNat(s(V1)) | → | a__isNat(V1) |
| a__isNat(plus(V1, V2)) | → | a__and(a__isNat(V1), isNat(V2)) | | mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) |
| mark(U21(X1, X2, X3)) | → | a__U21(mark(X1), X2, X3) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| a__isNat#(plus(V1, V2)) | → | a__and#(a__isNat(V1), isNat(V2)) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| a__isNat#(plus(V1, V2)) | → | a__isNat#(V1) | | a__isNat#(s(V1)) | → | a__isNat#(V1) |
| mark#(plus(X1, X2)) | → | mark#(X1) |
Problem 5: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(and(X1, X2)) | → | mark#(X1) | | mark#(isNat(X)) | → | a__isNat#(X) |
| mark#(and(X1, X2)) | → | a__and#(mark(X1), X2) | | a__and#(tt, X) | → | mark#(X) |
Rewrite Rules
| a__U11(tt, N) | → | mark(N) | | a__U21(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__and(tt, X) | → | mark(X) | | a__isNat(0) | → | tt |
| a__isNat(plus(V1, V2)) | → | a__and(a__isNat(V1), isNat(V2)) | | a__isNat(s(V1)) | → | a__isNat(V1) |
| a__plus(N, 0) | → | a__U11(a__isNat(N), N) | | a__plus(N, s(M)) | → | a__U21(a__and(a__isNat(M), isNat(N)), M, N) |
| mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) | | mark(U21(X1, X2, X3)) | → | a__U21(mark(X1), X2, X3) |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(isNat(X)) | → | a__isNat(X) | | mark(tt) | → | tt |
| mark(s(X)) | → | s(mark(X)) | | mark(0) | → | 0 |
| a__U11(X1, X2) | → | U11(X1, X2) | | a__U21(X1, X2, X3) | → | U21(X1, X2, X3) |
| a__plus(X1, X2) | → | plus(X1, X2) | | a__and(X1, X2) | → | and(X1, X2) |
| a__isNat(X) | → | isNat(X) |
Original Signature
Termination of terms over the following signature is verified: plus, a__plus, mark, and, a__and, isNat, 0, s, tt, a__isNat, U11, a__U11, a__U21, U21
Strategy
The following SCCs where found
| mark#(and(X1, X2)) → mark#(X1) | mark#(and(X1, X2)) → a__and#(mark(X1), X2) |
| a__and#(tt, X) → mark#(X) |
Problem 6: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(and(X1, X2)) | → | mark#(X1) | | mark#(and(X1, X2)) | → | a__and#(mark(X1), X2) |
| a__and#(tt, X) | → | mark#(X) |
Rewrite Rules
| a__U11(tt, N) | → | mark(N) | | a__U21(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__and(tt, X) | → | mark(X) | | a__isNat(0) | → | tt |
| a__isNat(plus(V1, V2)) | → | a__and(a__isNat(V1), isNat(V2)) | | a__isNat(s(V1)) | → | a__isNat(V1) |
| a__plus(N, 0) | → | a__U11(a__isNat(N), N) | | a__plus(N, s(M)) | → | a__U21(a__and(a__isNat(M), isNat(N)), M, N) |
| mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) | | mark(U21(X1, X2, X3)) | → | a__U21(mark(X1), X2, X3) |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(isNat(X)) | → | a__isNat(X) | | mark(tt) | → | tt |
| mark(s(X)) | → | s(mark(X)) | | mark(0) | → | 0 |
| a__U11(X1, X2) | → | U11(X1, X2) | | a__U21(X1, X2, X3) | → | U21(X1, X2, X3) |
| a__plus(X1, X2) | → | plus(X1, X2) | | a__and(X1, X2) | → | and(X1, X2) |
| a__isNat(X) | → | isNat(X) |
Original Signature
Termination of terms over the following signature is verified: plus, a__plus, mark, and, a__and, isNat, 0, s, tt, a__isNat, U11, a__U11, a__U21, U21
Strategy
Polynomial Interpretation
- 0: 0
- U11(x,y): y
- U21(x,y,z): 2z + y + 2
- a__U11(x,y): y
- a__U21(x,y,z): 2z + y + 2
- a__and(x,y): y + 2x + 2
- a__and#(x,y): y
- a__isNat(x): x
- a__plus(x,y): y + 2x + 2
- and(x,y): y + 2x + 2
- isNat(x): x
- mark(x): x
- mark#(x): x
- plus(x,y): y + 2x + 2
- s(x): x
- tt: 0
Standard Usable rules
| mark(tt) | → | tt | | a__U11(X1, X2) | → | U11(X1, X2) |
| mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) | | mark(0) | → | 0 |
| a__plus(X1, X2) | → | plus(X1, X2) | | a__and(X1, X2) | → | and(X1, X2) |
| a__and(tt, X) | → | mark(X) | | a__plus(N, s(M)) | → | a__U21(a__and(a__isNat(M), isNat(N)), M, N) |
| a__U21(tt, M, N) | → | s(a__plus(mark(N), mark(M))) | | mark(isNat(X)) | → | a__isNat(X) |
| a__isNat(0) | → | tt | | a__isNat(X) | → | isNat(X) |
| a__U11(tt, N) | → | mark(N) | | a__plus(N, 0) | → | a__U11(a__isNat(N), N) |
| a__U21(X1, X2, X3) | → | U21(X1, X2, X3) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(s(X)) | → | s(mark(X)) | | a__isNat(s(V1)) | → | a__isNat(V1) |
| a__isNat(plus(V1, V2)) | → | a__and(a__isNat(V1), isNat(V2)) | | mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) |
| mark(U21(X1, X2, X3)) | → | a__U21(mark(X1), X2, X3) |
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#(and(X1, X2)) | → | a__and#(mark(X1), X2) |
Problem 7: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| a__and#(tt, X) | → | mark#(X) |
Rewrite Rules
| a__U11(tt, N) | → | mark(N) | | a__U21(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__and(tt, X) | → | mark(X) | | a__isNat(0) | → | tt |
| a__isNat(plus(V1, V2)) | → | a__and(a__isNat(V1), isNat(V2)) | | a__isNat(s(V1)) | → | a__isNat(V1) |
| a__plus(N, 0) | → | a__U11(a__isNat(N), N) | | a__plus(N, s(M)) | → | a__U21(a__and(a__isNat(M), isNat(N)), M, N) |
| mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) | | mark(U21(X1, X2, X3)) | → | a__U21(mark(X1), X2, X3) |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(isNat(X)) | → | a__isNat(X) | | mark(tt) | → | tt |
| mark(s(X)) | → | s(mark(X)) | | mark(0) | → | 0 |
| a__U11(X1, X2) | → | U11(X1, X2) | | a__U21(X1, X2, X3) | → | U21(X1, X2, X3) |
| a__plus(X1, X2) | → | plus(X1, X2) | | a__and(X1, X2) | → | and(X1, X2) |
| a__isNat(X) | → | isNat(X) |
Original Signature
Termination of terms over the following signature is verified: plus, a__plus, mark, and, a__and, isNat, 0, s, tt, a__isNat, U11, a__U11, U21, a__U21
Strategy
There are no SCCs!