YES
The TRS could be proven terminating. The proof took 42372 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (499ms).
| – Problem 2 was processed with processor PolynomialLinearRange4 (592ms).
| | – Problem 3 was processed with processor DependencyGraph (81ms).
| | | – Problem 4 was processed with processor PolynomialLinearRange4 (536ms).
| | | | – Problem 5 was processed with processor DependencyGraph (9ms).
| | | | | – Problem 6 was processed with processor PolynomialLinearRange4 (136ms).
| | | | | | – Problem 7 was processed with processor PolynomialLinearRange4 (107ms).
| | | | | | | – Problem 8 was processed with processor DependencyGraph (1ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| and#(tt, X) | → | activate#(X) | | isNat#(n__plus(V1, V2)) | → | isNatKind#(activate(V1)) |
| plus#(N, s(M)) | → | isNat#(N) | | U41#(tt, M, N) | → | activate#(M) |
| plus#(N, 0) | → | and#(isNat(N), n__isNatKind(N)) | | U41#(tt, M, N) | → | plus#(activate(N), activate(M)) |
| isNatKind#(n__s(V1)) | → | activate#(V1) | | plus#(N, s(M)) | → | and#(isNat(M), n__isNatKind(M)) |
| U12#(tt, V2) | → | activate#(V2) | | isNat#(n__plus(V1, V2)) | → | activate#(V2) |
| U21#(tt, V1) | → | U22#(isNat(activate(V1))) | | plus#(N, 0) | → | isNat#(N) |
| U41#(tt, M, N) | → | s#(plus(activate(N), activate(M))) | | U11#(tt, V1, V2) | → | activate#(V2) |
| isNatKind#(n__plus(V1, V2)) | → | and#(isNatKind(activate(V1)), n__isNatKind(activate(V2))) | | isNatKind#(n__plus(V1, V2)) | → | isNatKind#(activate(V1)) |
| plus#(N, 0) | → | U31#(and(isNat(N), n__isNatKind(N)), N) | | isNatKind#(n__plus(V1, V2)) | → | activate#(V2) |
| U11#(tt, V1, V2) | → | isNat#(activate(V1)) | | isNat#(n__plus(V1, V2)) | → | U11#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) |
| isNat#(n__s(V1)) | → | activate#(V1) | | U12#(tt, V2) | → | isNat#(activate(V2)) |
| isNatKind#(n__plus(V1, V2)) | → | activate#(V1) | | U21#(tt, V1) | → | activate#(V1) |
| plus#(N, s(M)) | → | and#(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))) | | U31#(tt, N) | → | activate#(N) |
| U41#(tt, M, N) | → | activate#(N) | | U11#(tt, V1, V2) | → | activate#(V1) |
| activate#(n__s(X)) | → | s#(X) | | plus#(N, s(M)) | → | U41#(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) |
| plus#(N, s(M)) | → | isNat#(M) | | activate#(n__isNatKind(X)) | → | isNatKind#(X) |
| isNat#(n__plus(V1, V2)) | → | and#(isNatKind(activate(V1)), n__isNatKind(activate(V2))) | | U11#(tt, V1, V2) | → | U12#(isNat(activate(V1)), activate(V2)) |
| activate#(n__0) | → | 0# | | isNat#(n__plus(V1, V2)) | → | activate#(V1) |
| U21#(tt, V1) | → | isNat#(activate(V1)) | | isNatKind#(n__s(V1)) | → | isNatKind#(activate(V1)) |
| isNat#(n__s(V1)) | → | U21#(isNatKind(activate(V1)), activate(V1)) | | activate#(n__and(X1, X2)) | → | and#(X1, X2) |
| isNat#(n__s(V1)) | → | isNatKind#(activate(V1)) | | U12#(tt, V2) | → | U13#(isNat(activate(V2))) |
| activate#(n__plus(X1, X2)) | → | plus#(X1, X2) |
Rewrite Rules
| U11(tt, V1, V2) | → | U12(isNat(activate(V1)), activate(V2)) | | U12(tt, V2) | → | U13(isNat(activate(V2))) |
| U13(tt) | → | tt | | U21(tt, V1) | → | U22(isNat(activate(V1))) |
| U22(tt) | → | tt | | U31(tt, N) | → | activate(N) |
| U41(tt, M, N) | → | s(plus(activate(N), activate(M))) | | and(tt, X) | → | activate(X) |
| isNat(n__0) | → | tt | | isNat(n__plus(V1, V2)) | → | U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) |
| isNat(n__s(V1)) | → | U21(isNatKind(activate(V1)), activate(V1)) | | isNatKind(n__0) | → | tt |
| isNatKind(n__plus(V1, V2)) | → | and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) | | isNatKind(n__s(V1)) | → | isNatKind(activate(V1)) |
| plus(N, 0) | → | U31(and(isNat(N), n__isNatKind(N)), N) | | plus(N, s(M)) | → | U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) |
| 0 | → | n__0 | | plus(X1, X2) | → | n__plus(X1, X2) |
| isNatKind(X) | → | n__isNatKind(X) | | s(X) | → | n__s(X) |
| and(X1, X2) | → | n__and(X1, X2) | | activate(n__0) | → | 0 |
| activate(n__plus(X1, X2)) | → | plus(X1, X2) | | activate(n__isNatKind(X)) | → | isNatKind(X) |
| activate(n__s(X)) | → | s(X) | | activate(n__and(X1, X2)) | → | and(X1, X2) |
| activate(X) | → | X |
Original Signature
Termination of terms over the following signature is verified: plus, isNatKind, n__plus, and, n__s, activate, isNat, n__and, 0, n__0, s, tt, U41, U11, U12, U31, U13, U21, U22, n__isNatKind
Strategy
The following SCCs where found
| and#(tt, X) → activate#(X) | plus#(N, s(M)) → isNat#(N) |
| isNat#(n__plus(V1, V2)) → isNatKind#(activate(V1)) | plus#(N, 0) → and#(isNat(N), n__isNatKind(N)) |
| U41#(tt, M, N) → activate#(M) | U41#(tt, M, N) → plus#(activate(N), activate(M)) |
| isNatKind#(n__s(V1)) → activate#(V1) | plus#(N, s(M)) → and#(isNat(M), n__isNatKind(M)) |
| U12#(tt, V2) → activate#(V2) | isNat#(n__plus(V1, V2)) → activate#(V2) |
| plus#(N, 0) → isNat#(N) | U11#(tt, V1, V2) → activate#(V2) |
| isNatKind#(n__plus(V1, V2)) → and#(isNatKind(activate(V1)), n__isNatKind(activate(V2))) | isNatKind#(n__plus(V1, V2)) → isNatKind#(activate(V1)) |
| isNatKind#(n__plus(V1, V2)) → activate#(V2) | U11#(tt, V1, V2) → isNat#(activate(V1)) |
| plus#(N, 0) → U31#(and(isNat(N), n__isNatKind(N)), N) | isNat#(n__plus(V1, V2)) → U11#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) |
| isNat#(n__s(V1)) → activate#(V1) | U12#(tt, V2) → isNat#(activate(V2)) |
| isNatKind#(n__plus(V1, V2)) → activate#(V1) | U21#(tt, V1) → activate#(V1) |
| plus#(N, s(M)) → and#(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))) | U31#(tt, N) → activate#(N) |
| U41#(tt, M, N) → activate#(N) | U11#(tt, V1, V2) → activate#(V1) |
| plus#(N, s(M)) → U41#(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) | activate#(n__isNatKind(X)) → isNatKind#(X) |
| plus#(N, s(M)) → isNat#(M) | isNat#(n__plus(V1, V2)) → and#(isNatKind(activate(V1)), n__isNatKind(activate(V2))) |
| U11#(tt, V1, V2) → U12#(isNat(activate(V1)), activate(V2)) | isNat#(n__plus(V1, V2)) → activate#(V1) |
| isNatKind#(n__s(V1)) → isNatKind#(activate(V1)) | U21#(tt, V1) → isNat#(activate(V1)) |
| activate#(n__and(X1, X2)) → and#(X1, X2) | isNat#(n__s(V1)) → U21#(isNatKind(activate(V1)), activate(V1)) |
| activate#(n__plus(X1, X2)) → plus#(X1, X2) | isNat#(n__s(V1)) → isNatKind#(activate(V1)) |
Problem 2: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| and#(tt, X) | → | activate#(X) | | plus#(N, s(M)) | → | isNat#(N) |
| isNat#(n__plus(V1, V2)) | → | isNatKind#(activate(V1)) | | plus#(N, 0) | → | and#(isNat(N), n__isNatKind(N)) |
| U41#(tt, M, N) | → | activate#(M) | | U41#(tt, M, N) | → | plus#(activate(N), activate(M)) |
| isNatKind#(n__s(V1)) | → | activate#(V1) | | plus#(N, s(M)) | → | and#(isNat(M), n__isNatKind(M)) |
| U12#(tt, V2) | → | activate#(V2) | | isNat#(n__plus(V1, V2)) | → | activate#(V2) |
| plus#(N, 0) | → | isNat#(N) | | U11#(tt, V1, V2) | → | activate#(V2) |
| isNatKind#(n__plus(V1, V2)) | → | and#(isNatKind(activate(V1)), n__isNatKind(activate(V2))) | | isNatKind#(n__plus(V1, V2)) | → | isNatKind#(activate(V1)) |
| plus#(N, 0) | → | U31#(and(isNat(N), n__isNatKind(N)), N) | | U11#(tt, V1, V2) | → | isNat#(activate(V1)) |
| isNatKind#(n__plus(V1, V2)) | → | activate#(V2) | | isNat#(n__s(V1)) | → | activate#(V1) |
| isNat#(n__plus(V1, V2)) | → | U11#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) | | U12#(tt, V2) | → | isNat#(activate(V2)) |
| isNatKind#(n__plus(V1, V2)) | → | activate#(V1) | | U21#(tt, V1) | → | activate#(V1) |
| plus#(N, s(M)) | → | and#(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))) | | U31#(tt, N) | → | activate#(N) |
| U41#(tt, M, N) | → | activate#(N) | | U11#(tt, V1, V2) | → | activate#(V1) |
| plus#(N, s(M)) | → | U41#(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) | | activate#(n__isNatKind(X)) | → | isNatKind#(X) |
| plus#(N, s(M)) | → | isNat#(M) | | isNat#(n__plus(V1, V2)) | → | and#(isNatKind(activate(V1)), n__isNatKind(activate(V2))) |
| U11#(tt, V1, V2) | → | U12#(isNat(activate(V1)), activate(V2)) | | isNat#(n__plus(V1, V2)) | → | activate#(V1) |
| isNatKind#(n__s(V1)) | → | isNatKind#(activate(V1)) | | U21#(tt, V1) | → | isNat#(activate(V1)) |
| activate#(n__and(X1, X2)) | → | and#(X1, X2) | | isNat#(n__s(V1)) | → | U21#(isNatKind(activate(V1)), activate(V1)) |
| activate#(n__plus(X1, X2)) | → | plus#(X1, X2) | | isNat#(n__s(V1)) | → | isNatKind#(activate(V1)) |
Rewrite Rules
| U11(tt, V1, V2) | → | U12(isNat(activate(V1)), activate(V2)) | | U12(tt, V2) | → | U13(isNat(activate(V2))) |
| U13(tt) | → | tt | | U21(tt, V1) | → | U22(isNat(activate(V1))) |
| U22(tt) | → | tt | | U31(tt, N) | → | activate(N) |
| U41(tt, M, N) | → | s(plus(activate(N), activate(M))) | | and(tt, X) | → | activate(X) |
| isNat(n__0) | → | tt | | isNat(n__plus(V1, V2)) | → | U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) |
| isNat(n__s(V1)) | → | U21(isNatKind(activate(V1)), activate(V1)) | | isNatKind(n__0) | → | tt |
| isNatKind(n__plus(V1, V2)) | → | and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) | | isNatKind(n__s(V1)) | → | isNatKind(activate(V1)) |
| plus(N, 0) | → | U31(and(isNat(N), n__isNatKind(N)), N) | | plus(N, s(M)) | → | U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) |
| 0 | → | n__0 | | plus(X1, X2) | → | n__plus(X1, X2) |
| isNatKind(X) | → | n__isNatKind(X) | | s(X) | → | n__s(X) |
| and(X1, X2) | → | n__and(X1, X2) | | activate(n__0) | → | 0 |
| activate(n__plus(X1, X2)) | → | plus(X1, X2) | | activate(n__isNatKind(X)) | → | isNatKind(X) |
| activate(n__s(X)) | → | s(X) | | activate(n__and(X1, X2)) | → | and(X1, X2) |
| activate(X) | → | X |
Original Signature
Termination of terms over the following signature is verified: plus, isNatKind, n__plus, and, n__s, activate, isNat, n__and, 0, n__0, s, tt, U41, U11, U12, U31, U13, U21, U22, n__isNatKind
Strategy
Polynomial Interpretation
- 0: 1
- U11(x,y,z): z + 3x
- U11#(x,y,z): z + y + 2x
- U12(x,y): y
- U12#(x,y): y
- U13(x): 0
- U21(x,y): 2x
- U21#(x,y): y
- U22(x): 0
- U31(x,y): y
- U31#(x,y): y
- U41(x,y,z): 2z + 3y
- U41#(x,y,z): z + 2y
- activate(x): x
- activate#(x): x
- and(x,y): y
- and#(x,y): y
- isNat(x): 2x
- isNat#(x): x
- isNatKind(x): x
- isNatKind#(x): x
- n__0: 1
- n__and(x,y): y
- n__isNatKind(x): x
- n__plus(x,y): 3y + 2x
- n__s(x): x
- plus(x,y): 3y + 2x
- plus#(x,y): 2y + x
- s(x): x
- tt: 0
Standard Usable rules
| isNat(n__plus(V1, V2)) | → | U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) | | U31(tt, N) | → | activate(N) |
| activate(n__s(X)) | → | s(X) | | U11(tt, V1, V2) | → | U12(isNat(activate(V1)), activate(V2)) |
| activate(n__0) | → | 0 | | isNat(n__0) | → | tt |
| U13(tt) | → | tt | | plus(N, s(M)) | → | U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) |
| s(X) | → | n__s(X) | | isNatKind(n__s(V1)) | → | isNatKind(activate(V1)) |
| U21(tt, V1) | → | U22(isNat(activate(V1))) | | isNatKind(n__plus(V1, V2)) | → | and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) |
| U22(tt) | → | tt | | plus(N, 0) | → | U31(and(isNat(N), n__isNatKind(N)), N) |
| plus(X1, X2) | → | n__plus(X1, X2) | | isNat(n__s(V1)) | → | U21(isNatKind(activate(V1)), activate(V1)) |
| U41(tt, M, N) | → | s(plus(activate(N), activate(M))) | | activate(n__plus(X1, X2)) | → | plus(X1, X2) |
| activate(n__isNatKind(X)) | → | isNatKind(X) | | and(X1, X2) | → | n__and(X1, X2) |
| U12(tt, V2) | → | U13(isNat(activate(V2))) | | isNatKind(X) | → | n__isNatKind(X) |
| 0 | → | n__0 | | isNatKind(n__0) | → | tt |
| and(tt, X) | → | activate(X) | | activate(X) | → | X |
| activate(n__and(X1, X2)) | → | and(X1, X2) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| plus#(N, 0) | → | and#(isNat(N), n__isNatKind(N)) | | plus#(N, 0) | → | isNat#(N) |
| plus#(N, 0) | → | U31#(and(isNat(N), n__isNatKind(N)), N) |
Problem 3: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| and#(tt, X) | → | activate#(X) | | isNat#(n__plus(V1, V2)) | → | isNatKind#(activate(V1)) |
| plus#(N, s(M)) | → | isNat#(N) | | U41#(tt, M, N) | → | activate#(M) |
| isNatKind#(n__s(V1)) | → | activate#(V1) | | U41#(tt, M, N) | → | plus#(activate(N), activate(M)) |
| plus#(N, s(M)) | → | and#(isNat(M), n__isNatKind(M)) | | U12#(tt, V2) | → | activate#(V2) |
| isNat#(n__plus(V1, V2)) | → | activate#(V2) | | U11#(tt, V1, V2) | → | activate#(V2) |
| isNatKind#(n__plus(V1, V2)) | → | and#(isNatKind(activate(V1)), n__isNatKind(activate(V2))) | | isNatKind#(n__plus(V1, V2)) | → | isNatKind#(activate(V1)) |
| U11#(tt, V1, V2) | → | isNat#(activate(V1)) | | isNatKind#(n__plus(V1, V2)) | → | activate#(V2) |
| isNat#(n__s(V1)) | → | activate#(V1) | | isNat#(n__plus(V1, V2)) | → | U11#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) |
| U12#(tt, V2) | → | isNat#(activate(V2)) | | isNatKind#(n__plus(V1, V2)) | → | activate#(V1) |
| U21#(tt, V1) | → | activate#(V1) | | plus#(N, s(M)) | → | and#(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))) |
| U31#(tt, N) | → | activate#(N) | | U41#(tt, M, N) | → | activate#(N) |
| U11#(tt, V1, V2) | → | activate#(V1) | | plus#(N, s(M)) | → | U41#(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) |
| plus#(N, s(M)) | → | isNat#(M) | | activate#(n__isNatKind(X)) | → | isNatKind#(X) |
| U11#(tt, V1, V2) | → | U12#(isNat(activate(V1)), activate(V2)) | | isNat#(n__plus(V1, V2)) | → | and#(isNatKind(activate(V1)), n__isNatKind(activate(V2))) |
| isNat#(n__plus(V1, V2)) | → | activate#(V1) | | U21#(tt, V1) | → | isNat#(activate(V1)) |
| isNatKind#(n__s(V1)) | → | isNatKind#(activate(V1)) | | isNat#(n__s(V1)) | → | U21#(isNatKind(activate(V1)), activate(V1)) |
| activate#(n__and(X1, X2)) | → | and#(X1, X2) | | isNat#(n__s(V1)) | → | isNatKind#(activate(V1)) |
| activate#(n__plus(X1, X2)) | → | plus#(X1, X2) |
Rewrite Rules
| U11(tt, V1, V2) | → | U12(isNat(activate(V1)), activate(V2)) | | U12(tt, V2) | → | U13(isNat(activate(V2))) |
| U13(tt) | → | tt | | U21(tt, V1) | → | U22(isNat(activate(V1))) |
| U22(tt) | → | tt | | U31(tt, N) | → | activate(N) |
| U41(tt, M, N) | → | s(plus(activate(N), activate(M))) | | and(tt, X) | → | activate(X) |
| isNat(n__0) | → | tt | | isNat(n__plus(V1, V2)) | → | U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) |
| isNat(n__s(V1)) | → | U21(isNatKind(activate(V1)), activate(V1)) | | isNatKind(n__0) | → | tt |
| isNatKind(n__plus(V1, V2)) | → | and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) | | isNatKind(n__s(V1)) | → | isNatKind(activate(V1)) |
| plus(N, 0) | → | U31(and(isNat(N), n__isNatKind(N)), N) | | plus(N, s(M)) | → | U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) |
| 0 | → | n__0 | | plus(X1, X2) | → | n__plus(X1, X2) |
| isNatKind(X) | → | n__isNatKind(X) | | s(X) | → | n__s(X) |
| and(X1, X2) | → | n__and(X1, X2) | | activate(n__0) | → | 0 |
| activate(n__plus(X1, X2)) | → | plus(X1, X2) | | activate(n__isNatKind(X)) | → | isNatKind(X) |
| activate(n__s(X)) | → | s(X) | | activate(n__and(X1, X2)) | → | and(X1, X2) |
| activate(X) | → | X |
Original Signature
Termination of terms over the following signature is verified: plus, isNatKind, n__plus, and, n__s, activate, isNat, n__and, 0, n__0, s, tt, U41, U11, U12, U13, U31, U21, n__isNatKind, U22
Strategy
The following SCCs where found
| and#(tt, X) → activate#(X) | plus#(N, s(M)) → isNat#(N) |
| isNat#(n__plus(V1, V2)) → isNatKind#(activate(V1)) | U41#(tt, M, N) → activate#(M) |
| U41#(tt, M, N) → plus#(activate(N), activate(M)) | isNatKind#(n__s(V1)) → activate#(V1) |
| plus#(N, s(M)) → and#(isNat(M), n__isNatKind(M)) | U12#(tt, V2) → activate#(V2) |
| isNat#(n__plus(V1, V2)) → activate#(V2) | U11#(tt, V1, V2) → activate#(V2) |
| isNatKind#(n__plus(V1, V2)) → and#(isNatKind(activate(V1)), n__isNatKind(activate(V2))) | isNatKind#(n__plus(V1, V2)) → isNatKind#(activate(V1)) |
| isNatKind#(n__plus(V1, V2)) → activate#(V2) | U11#(tt, V1, V2) → isNat#(activate(V1)) |
| isNat#(n__plus(V1, V2)) → U11#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) | isNat#(n__s(V1)) → activate#(V1) |
| U12#(tt, V2) → isNat#(activate(V2)) | isNatKind#(n__plus(V1, V2)) → activate#(V1) |
| U21#(tt, V1) → activate#(V1) | plus#(N, s(M)) → and#(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))) |
| U41#(tt, M, N) → activate#(N) | U11#(tt, V1, V2) → activate#(V1) |
| plus#(N, s(M)) → U41#(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) | activate#(n__isNatKind(X)) → isNatKind#(X) |
| plus#(N, s(M)) → isNat#(M) | U11#(tt, V1, V2) → U12#(isNat(activate(V1)), activate(V2)) |
| isNat#(n__plus(V1, V2)) → and#(isNatKind(activate(V1)), n__isNatKind(activate(V2))) | isNat#(n__plus(V1, V2)) → activate#(V1) |
| U21#(tt, V1) → isNat#(activate(V1)) | isNatKind#(n__s(V1)) → isNatKind#(activate(V1)) |
| activate#(n__and(X1, X2)) → and#(X1, X2) | isNat#(n__s(V1)) → U21#(isNatKind(activate(V1)), activate(V1)) |
| activate#(n__plus(X1, X2)) → plus#(X1, X2) | isNat#(n__s(V1)) → isNatKind#(activate(V1)) |
Problem 4: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| and#(tt, X) | → | activate#(X) | | plus#(N, s(M)) | → | isNat#(N) |
| isNat#(n__plus(V1, V2)) | → | isNatKind#(activate(V1)) | | U41#(tt, M, N) | → | activate#(M) |
| U41#(tt, M, N) | → | plus#(activate(N), activate(M)) | | isNatKind#(n__s(V1)) | → | activate#(V1) |
| plus#(N, s(M)) | → | and#(isNat(M), n__isNatKind(M)) | | U12#(tt, V2) | → | activate#(V2) |
| isNat#(n__plus(V1, V2)) | → | activate#(V2) | | U11#(tt, V1, V2) | → | activate#(V2) |
| isNatKind#(n__plus(V1, V2)) | → | and#(isNatKind(activate(V1)), n__isNatKind(activate(V2))) | | isNatKind#(n__plus(V1, V2)) | → | isNatKind#(activate(V1)) |
| U11#(tt, V1, V2) | → | isNat#(activate(V1)) | | isNatKind#(n__plus(V1, V2)) | → | activate#(V2) |
| isNat#(n__s(V1)) | → | activate#(V1) | | isNat#(n__plus(V1, V2)) | → | U11#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) |
| U12#(tt, V2) | → | isNat#(activate(V2)) | | isNatKind#(n__plus(V1, V2)) | → | activate#(V1) |
| U21#(tt, V1) | → | activate#(V1) | | plus#(N, s(M)) | → | and#(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))) |
| U41#(tt, M, N) | → | activate#(N) | | U11#(tt, V1, V2) | → | activate#(V1) |
| plus#(N, s(M)) | → | U41#(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) | | plus#(N, s(M)) | → | isNat#(M) |
| activate#(n__isNatKind(X)) | → | isNatKind#(X) | | U11#(tt, V1, V2) | → | U12#(isNat(activate(V1)), activate(V2)) |
| isNat#(n__plus(V1, V2)) | → | and#(isNatKind(activate(V1)), n__isNatKind(activate(V2))) | | isNat#(n__plus(V1, V2)) | → | activate#(V1) |
| U21#(tt, V1) | → | isNat#(activate(V1)) | | isNatKind#(n__s(V1)) | → | isNatKind#(activate(V1)) |
| activate#(n__and(X1, X2)) | → | and#(X1, X2) | | isNat#(n__s(V1)) | → | U21#(isNatKind(activate(V1)), activate(V1)) |
| activate#(n__plus(X1, X2)) | → | plus#(X1, X2) | | isNat#(n__s(V1)) | → | isNatKind#(activate(V1)) |
Rewrite Rules
| U11(tt, V1, V2) | → | U12(isNat(activate(V1)), activate(V2)) | | U12(tt, V2) | → | U13(isNat(activate(V2))) |
| U13(tt) | → | tt | | U21(tt, V1) | → | U22(isNat(activate(V1))) |
| U22(tt) | → | tt | | U31(tt, N) | → | activate(N) |
| U41(tt, M, N) | → | s(plus(activate(N), activate(M))) | | and(tt, X) | → | activate(X) |
| isNat(n__0) | → | tt | | isNat(n__plus(V1, V2)) | → | U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) |
| isNat(n__s(V1)) | → | U21(isNatKind(activate(V1)), activate(V1)) | | isNatKind(n__0) | → | tt |
| isNatKind(n__plus(V1, V2)) | → | and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) | | isNatKind(n__s(V1)) | → | isNatKind(activate(V1)) |
| plus(N, 0) | → | U31(and(isNat(N), n__isNatKind(N)), N) | | plus(N, s(M)) | → | U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) |
| 0 | → | n__0 | | plus(X1, X2) | → | n__plus(X1, X2) |
| isNatKind(X) | → | n__isNatKind(X) | | s(X) | → | n__s(X) |
| and(X1, X2) | → | n__and(X1, X2) | | activate(n__0) | → | 0 |
| activate(n__plus(X1, X2)) | → | plus(X1, X2) | | activate(n__isNatKind(X)) | → | isNatKind(X) |
| activate(n__s(X)) | → | s(X) | | activate(n__and(X1, X2)) | → | and(X1, X2) |
| activate(X) | → | X |
Original Signature
Termination of terms over the following signature is verified: plus, isNatKind, n__plus, and, n__s, activate, isNat, n__and, 0, n__0, s, tt, U41, U11, U12, U13, U31, U21, n__isNatKind, U22
Strategy
Polynomial Interpretation
- 0: 1
- U11(x,y,z): 0
- U11#(x,y,z): 2z + 2y + 2
- U12(x,y): 0
- U12#(x,y): 2y + 2
- U13(x): 0
- U21(x,y): 0
- U21#(x,y): y + 3
- U22(x): 0
- U31(x,y): y
- U41(x,y,z): 3z + 2y + 2
- U41#(x,y,z): 3z + 2y + 1
- activate(x): x
- activate#(x): x + 1
- and(x,y): y + 2x
- and#(x,y): y + x + 1
- isNat(x): 0
- isNat#(x): x + 2
- isNatKind(x): 2x + 1
- isNatKind#(x): x + 2
- n__0: 1
- n__and(x,y): y + 2x
- n__isNatKind(x): 2x + 1
- n__plus(x,y): 2y + 3x + 1
- n__s(x): x + 1
- plus(x,y): 2y + 3x + 1
- plus#(x,y): 2y + 3x + 1
- s(x): x + 1
- tt: 0
Standard Usable rules
| isNat(n__plus(V1, V2)) | → | U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) | | U31(tt, N) | → | activate(N) |
| activate(n__s(X)) | → | s(X) | | U11(tt, V1, V2) | → | U12(isNat(activate(V1)), activate(V2)) |
| activate(n__0) | → | 0 | | isNat(n__0) | → | tt |
| U13(tt) | → | tt | | plus(N, s(M)) | → | U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) |
| s(X) | → | n__s(X) | | isNatKind(n__s(V1)) | → | isNatKind(activate(V1)) |
| U21(tt, V1) | → | U22(isNat(activate(V1))) | | isNatKind(n__plus(V1, V2)) | → | and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) |
| U22(tt) | → | tt | | plus(N, 0) | → | U31(and(isNat(N), n__isNatKind(N)), N) |
| plus(X1, X2) | → | n__plus(X1, X2) | | isNat(n__s(V1)) | → | U21(isNatKind(activate(V1)), activate(V1)) |
| U41(tt, M, N) | → | s(plus(activate(N), activate(M))) | | activate(n__plus(X1, X2)) | → | plus(X1, X2) |
| activate(n__isNatKind(X)) | → | isNatKind(X) | | and(X1, X2) | → | n__and(X1, X2) |
| U12(tt, V2) | → | U13(isNat(activate(V2))) | | isNatKind(X) | → | n__isNatKind(X) |
| 0 | → | n__0 | | isNatKind(n__0) | → | tt |
| and(tt, X) | → | activate(X) | | activate(X) | → | X |
| activate(n__and(X1, X2)) | → | and(X1, X2) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| isNat#(n__plus(V1, V2)) | → | isNatKind#(activate(V1)) | | plus#(N, s(M)) | → | isNat#(N) |
| isNatKind#(n__s(V1)) | → | activate#(V1) | | plus#(N, s(M)) | → | and#(isNat(M), n__isNatKind(M)) |
| U12#(tt, V2) | → | activate#(V2) | | isNat#(n__plus(V1, V2)) | → | activate#(V2) |
| U11#(tt, V1, V2) | → | activate#(V2) | | isNatKind#(n__plus(V1, V2)) | → | isNatKind#(activate(V1)) |
| isNatKind#(n__plus(V1, V2)) | → | activate#(V2) | | isNat#(n__s(V1)) | → | activate#(V1) |
| isNat#(n__plus(V1, V2)) | → | U11#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) | | isNatKind#(n__plus(V1, V2)) | → | activate#(V1) |
| U21#(tt, V1) | → | activate#(V1) | | U11#(tt, V1, V2) | → | activate#(V1) |
| plus#(N, s(M)) | → | U41#(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) | | plus#(N, s(M)) | → | isNat#(M) |
| isNat#(n__plus(V1, V2)) | → | activate#(V1) | | isNatKind#(n__s(V1)) | → | isNatKind#(activate(V1)) |
| U21#(tt, V1) | → | isNat#(activate(V1)) | | isNat#(n__s(V1)) | → | isNatKind#(activate(V1)) |
| activate#(n__plus(X1, X2)) | → | plus#(X1, X2) |
Problem 5: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| and#(tt, X) | → | activate#(X) | | plus#(N, s(M)) | → | and#(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))) |
| U41#(tt, M, N) | → | activate#(M) | | U41#(tt, M, N) | → | plus#(activate(N), activate(M)) |
| U41#(tt, M, N) | → | activate#(N) | | isNatKind#(n__plus(V1, V2)) | → | and#(isNatKind(activate(V1)), n__isNatKind(activate(V2))) |
| activate#(n__isNatKind(X)) | → | isNatKind#(X) | | U11#(tt, V1, V2) | → | isNat#(activate(V1)) |
| U11#(tt, V1, V2) | → | U12#(isNat(activate(V1)), activate(V2)) | | isNat#(n__plus(V1, V2)) | → | and#(isNatKind(activate(V1)), n__isNatKind(activate(V2))) |
| activate#(n__and(X1, X2)) | → | and#(X1, X2) | | isNat#(n__s(V1)) | → | U21#(isNatKind(activate(V1)), activate(V1)) |
| U12#(tt, V2) | → | isNat#(activate(V2)) |
Rewrite Rules
| U11(tt, V1, V2) | → | U12(isNat(activate(V1)), activate(V2)) | | U12(tt, V2) | → | U13(isNat(activate(V2))) |
| U13(tt) | → | tt | | U21(tt, V1) | → | U22(isNat(activate(V1))) |
| U22(tt) | → | tt | | U31(tt, N) | → | activate(N) |
| U41(tt, M, N) | → | s(plus(activate(N), activate(M))) | | and(tt, X) | → | activate(X) |
| isNat(n__0) | → | tt | | isNat(n__plus(V1, V2)) | → | U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) |
| isNat(n__s(V1)) | → | U21(isNatKind(activate(V1)), activate(V1)) | | isNatKind(n__0) | → | tt |
| isNatKind(n__plus(V1, V2)) | → | and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) | | isNatKind(n__s(V1)) | → | isNatKind(activate(V1)) |
| plus(N, 0) | → | U31(and(isNat(N), n__isNatKind(N)), N) | | plus(N, s(M)) | → | U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) |
| 0 | → | n__0 | | plus(X1, X2) | → | n__plus(X1, X2) |
| isNatKind(X) | → | n__isNatKind(X) | | s(X) | → | n__s(X) |
| and(X1, X2) | → | n__and(X1, X2) | | activate(n__0) | → | 0 |
| activate(n__plus(X1, X2)) | → | plus(X1, X2) | | activate(n__isNatKind(X)) | → | isNatKind(X) |
| activate(n__s(X)) | → | s(X) | | activate(n__and(X1, X2)) | → | and(X1, X2) |
| activate(X) | → | X |
Original Signature
Termination of terms over the following signature is verified: plus, isNatKind, n__plus, and, n__s, activate, isNat, n__and, 0, n__0, s, tt, U41, U11, U12, U31, U13, U21, U22, n__isNatKind
Strategy
The following SCCs where found
| activate#(n__isNatKind(X)) → isNatKind#(X) | isNatKind#(n__plus(V1, V2)) → and#(isNatKind(activate(V1)), n__isNatKind(activate(V2))) |
| and#(tt, X) → activate#(X) | activate#(n__and(X1, X2)) → and#(X1, X2) |
Problem 6: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| activate#(n__isNatKind(X)) | → | isNatKind#(X) | | isNatKind#(n__plus(V1, V2)) | → | and#(isNatKind(activate(V1)), n__isNatKind(activate(V2))) |
| and#(tt, X) | → | activate#(X) | | activate#(n__and(X1, X2)) | → | and#(X1, X2) |
Rewrite Rules
| U11(tt, V1, V2) | → | U12(isNat(activate(V1)), activate(V2)) | | U12(tt, V2) | → | U13(isNat(activate(V2))) |
| U13(tt) | → | tt | | U21(tt, V1) | → | U22(isNat(activate(V1))) |
| U22(tt) | → | tt | | U31(tt, N) | → | activate(N) |
| U41(tt, M, N) | → | s(plus(activate(N), activate(M))) | | and(tt, X) | → | activate(X) |
| isNat(n__0) | → | tt | | isNat(n__plus(V1, V2)) | → | U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) |
| isNat(n__s(V1)) | → | U21(isNatKind(activate(V1)), activate(V1)) | | isNatKind(n__0) | → | tt |
| isNatKind(n__plus(V1, V2)) | → | and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) | | isNatKind(n__s(V1)) | → | isNatKind(activate(V1)) |
| plus(N, 0) | → | U31(and(isNat(N), n__isNatKind(N)), N) | | plus(N, s(M)) | → | U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) |
| 0 | → | n__0 | | plus(X1, X2) | → | n__plus(X1, X2) |
| isNatKind(X) | → | n__isNatKind(X) | | s(X) | → | n__s(X) |
| and(X1, X2) | → | n__and(X1, X2) | | activate(n__0) | → | 0 |
| activate(n__plus(X1, X2)) | → | plus(X1, X2) | | activate(n__isNatKind(X)) | → | isNatKind(X) |
| activate(n__s(X)) | → | s(X) | | activate(n__and(X1, X2)) | → | and(X1, X2) |
| activate(X) | → | X |
Original Signature
Termination of terms over the following signature is verified: plus, isNatKind, n__plus, and, n__s, activate, isNat, n__and, 0, n__0, s, tt, U41, U11, U12, U31, U13, U21, U22, n__isNatKind
Strategy
Polynomial Interpretation
- 0: 1
- U11(x,y,z): 2x
- U12(x,y): 1
- U13(x): 1
- U21(x,y): 2
- U22(x): 2
- U31(x,y): y + 1
- U41(x,y,z): 0
- activate(x): x + 1
- activate#(x): 2x
- and(x,y): y + 1
- and#(x,y): 2y
- isNat(x): 2
- isNatKind(x): 1
- isNatKind#(x): 0
- n__0: 1
- n__and(x,y): y + 1
- n__isNatKind(x): 0
- n__plus(x,y): 2y + x
- n__s(x): 0
- plus(x,y): 2y + x
- s(x): 0
- tt: 1
Standard Usable rules
| isNat(n__plus(V1, V2)) | → | U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) | | U31(tt, N) | → | activate(N) |
| activate(n__s(X)) | → | s(X) | | U11(tt, V1, V2) | → | U12(isNat(activate(V1)), activate(V2)) |
| activate(n__0) | → | 0 | | isNat(n__0) | → | tt |
| plus(N, s(M)) | → | U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) | | U13(tt) | → | tt |
| s(X) | → | n__s(X) | | isNatKind(n__s(V1)) | → | isNatKind(activate(V1)) |
| U21(tt, V1) | → | U22(isNat(activate(V1))) | | isNatKind(n__plus(V1, V2)) | → | and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) |
| U22(tt) | → | tt | | plus(N, 0) | → | U31(and(isNat(N), n__isNatKind(N)), N) |
| plus(X1, X2) | → | n__plus(X1, X2) | | isNat(n__s(V1)) | → | U21(isNatKind(activate(V1)), activate(V1)) |
| U41(tt, M, N) | → | s(plus(activate(N), activate(M))) | | activate(n__plus(X1, X2)) | → | plus(X1, X2) |
| activate(n__isNatKind(X)) | → | isNatKind(X) | | and(X1, X2) | → | n__and(X1, X2) |
| U12(tt, V2) | → | U13(isNat(activate(V2))) | | isNatKind(X) | → | n__isNatKind(X) |
| 0 | → | n__0 | | isNatKind(n__0) | → | tt |
| and(tt, X) | → | activate(X) | | activate(X) | → | X |
| activate(n__and(X1, X2)) | → | and(X1, X2) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| activate#(n__and(X1, X2)) | → | and#(X1, X2) |
Problem 7: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| and#(tt, X) | → | activate#(X) | | isNatKind#(n__plus(V1, V2)) | → | and#(isNatKind(activate(V1)), n__isNatKind(activate(V2))) |
| activate#(n__isNatKind(X)) | → | isNatKind#(X) |
Rewrite Rules
| U11(tt, V1, V2) | → | U12(isNat(activate(V1)), activate(V2)) | | U12(tt, V2) | → | U13(isNat(activate(V2))) |
| U13(tt) | → | tt | | U21(tt, V1) | → | U22(isNat(activate(V1))) |
| U22(tt) | → | tt | | U31(tt, N) | → | activate(N) |
| U41(tt, M, N) | → | s(plus(activate(N), activate(M))) | | and(tt, X) | → | activate(X) |
| isNat(n__0) | → | tt | | isNat(n__plus(V1, V2)) | → | U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) |
| isNat(n__s(V1)) | → | U21(isNatKind(activate(V1)), activate(V1)) | | isNatKind(n__0) | → | tt |
| isNatKind(n__plus(V1, V2)) | → | and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) | | isNatKind(n__s(V1)) | → | isNatKind(activate(V1)) |
| plus(N, 0) | → | U31(and(isNat(N), n__isNatKind(N)), N) | | plus(N, s(M)) | → | U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) |
| 0 | → | n__0 | | plus(X1, X2) | → | n__plus(X1, X2) |
| isNatKind(X) | → | n__isNatKind(X) | | s(X) | → | n__s(X) |
| and(X1, X2) | → | n__and(X1, X2) | | activate(n__0) | → | 0 |
| activate(n__plus(X1, X2)) | → | plus(X1, X2) | | activate(n__isNatKind(X)) | → | isNatKind(X) |
| activate(n__s(X)) | → | s(X) | | activate(n__and(X1, X2)) | → | and(X1, X2) |
| activate(X) | → | X |
Original Signature
Termination of terms over the following signature is verified: plus, isNatKind, n__plus, and, n__s, activate, isNat, n__and, 0, n__0, s, tt, U41, U11, U12, U13, U31, U21, n__isNatKind, U22
Strategy
Polynomial Interpretation
- 0: 0
- U11(x,y,z): 0
- U12(x,y): 0
- U13(x): 0
- U21(x,y): 1
- U22(x): 1
- U31(x,y): 2y + 1
- U41(x,y,z): 3z + 2y + 1
- activate(x): x
- activate#(x): x
- and(x,y): 2y
- and#(x,y): 2y
- isNat(x): 2
- isNatKind(x): 2x
- isNatKind#(x): 2x
- n__0: 0
- n__and(x,y): 2y
- n__isNatKind(x): 2x
- n__plus(x,y): 2y + 3x + 1
- n__s(x): x
- plus(x,y): 2y + 3x + 1
- s(x): x
- tt: 0
Standard Usable rules
| isNat(n__plus(V1, V2)) | → | U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) | | U31(tt, N) | → | activate(N) |
| activate(n__s(X)) | → | s(X) | | U11(tt, V1, V2) | → | U12(isNat(activate(V1)), activate(V2)) |
| activate(n__0) | → | 0 | | isNat(n__0) | → | tt |
| plus(N, s(M)) | → | U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) | | U13(tt) | → | tt |
| s(X) | → | n__s(X) | | isNatKind(n__s(V1)) | → | isNatKind(activate(V1)) |
| U21(tt, V1) | → | U22(isNat(activate(V1))) | | isNatKind(n__plus(V1, V2)) | → | and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) |
| U22(tt) | → | tt | | plus(N, 0) | → | U31(and(isNat(N), n__isNatKind(N)), N) |
| plus(X1, X2) | → | n__plus(X1, X2) | | isNat(n__s(V1)) | → | U21(isNatKind(activate(V1)), activate(V1)) |
| U41(tt, M, N) | → | s(plus(activate(N), activate(M))) | | activate(n__plus(X1, X2)) | → | plus(X1, X2) |
| activate(n__isNatKind(X)) | → | isNatKind(X) | | and(X1, X2) | → | n__and(X1, X2) |
| U12(tt, V2) | → | U13(isNat(activate(V2))) | | isNatKind(X) | → | n__isNatKind(X) |
| 0 | → | n__0 | | isNatKind(n__0) | → | tt |
| and(tt, X) | → | activate(X) | | activate(X) | → | X |
| activate(n__and(X1, X2)) | → | and(X1, X2) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| isNatKind#(n__plus(V1, V2)) | → | and#(isNatKind(activate(V1)), n__isNatKind(activate(V2))) |
Problem 8: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| activate#(n__isNatKind(X)) | → | isNatKind#(X) | | and#(tt, X) | → | activate#(X) |
Rewrite Rules
| U11(tt, V1, V2) | → | U12(isNat(activate(V1)), activate(V2)) | | U12(tt, V2) | → | U13(isNat(activate(V2))) |
| U13(tt) | → | tt | | U21(tt, V1) | → | U22(isNat(activate(V1))) |
| U22(tt) | → | tt | | U31(tt, N) | → | activate(N) |
| U41(tt, M, N) | → | s(plus(activate(N), activate(M))) | | and(tt, X) | → | activate(X) |
| isNat(n__0) | → | tt | | isNat(n__plus(V1, V2)) | → | U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) |
| isNat(n__s(V1)) | → | U21(isNatKind(activate(V1)), activate(V1)) | | isNatKind(n__0) | → | tt |
| isNatKind(n__plus(V1, V2)) | → | and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) | | isNatKind(n__s(V1)) | → | isNatKind(activate(V1)) |
| plus(N, 0) | → | U31(and(isNat(N), n__isNatKind(N)), N) | | plus(N, s(M)) | → | U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) |
| 0 | → | n__0 | | plus(X1, X2) | → | n__plus(X1, X2) |
| isNatKind(X) | → | n__isNatKind(X) | | s(X) | → | n__s(X) |
| and(X1, X2) | → | n__and(X1, X2) | | activate(n__0) | → | 0 |
| activate(n__plus(X1, X2)) | → | plus(X1, X2) | | activate(n__isNatKind(X)) | → | isNatKind(X) |
| activate(n__s(X)) | → | s(X) | | activate(n__and(X1, X2)) | → | and(X1, X2) |
| activate(X) | → | X |
Original Signature
Termination of terms over the following signature is verified: plus, isNatKind, n__plus, and, n__s, activate, isNat, n__and, 0, n__0, s, tt, U41, U11, U12, U31, U13, U21, U22, n__isNatKind
Strategy
There are no SCCs!