YES
The TRS could be proven terminating. The proof took 174 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (9ms).
| – Problem 2 was processed with processor PolynomialLinearRange4 (125ms).
| | – Problem 3 was processed with processor DependencyGraph (0ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| plus#(N, s(M)) | → | U11#(tt, M, N) | | U12#(tt, M, N) | → | T(M) |
| U12#(tt, M, N) | → | plus#(N, M) | | U11#(tt, M, N) | → | U12#(tt, M, N) |
| U12#(tt, M, N) | → | T(N) |
Rewrite Rules
| U11(tt, M, N) | → | U12(tt, M, N) | | U12(tt, M, N) | → | s(plus(N, M)) |
| plus(N, 0) | → | N | | plus(N, s(M)) | → | U11(tt, M, N) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, U11, U12
Strategy
Context-sensitive strategy:
μ(T) = μ(0) = μ(tt) = ∅
μ(U11#) = μ(U12#) = μ(s) = μ(U11) = μ(U12) = {1}
μ(plus) = μ(plus#) = {1, 2}
The following SCCs where found
| plus#(N, s(M)) → U11#(tt, M, N) | U12#(tt, M, N) → plus#(N, M) |
| U11#(tt, M, N) → U12#(tt, M, N) |
Problem 2: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| plus#(N, s(M)) | → | U11#(tt, M, N) | | U12#(tt, M, N) | → | plus#(N, M) |
| U11#(tt, M, N) | → | U12#(tt, M, N) |
Rewrite Rules
| U11(tt, M, N) | → | U12(tt, M, N) | | U12(tt, M, N) | → | s(plus(N, M)) |
| plus(N, 0) | → | N | | plus(N, s(M)) | → | U11(tt, M, N) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, U11, U12
Strategy
Context-sensitive strategy:
μ(T) = μ(0) = μ(tt) = ∅
μ(U11#) = μ(U12#) = μ(s) = μ(U11) = μ(U12) = {1}
μ(plus) = μ(plus#) = {1, 2}
Polynomial Interpretation
- 0: 0
- U11(x,y,z): z + 2y + 2
- U11#(x,y,z): y
- U12(x,y,z): z + 2y + 1
- U12#(x,y,z): y
- plus(x,y): 2y + x
- plus#(x,y): y
- s(x): x + 1
- tt: 0
Standard Usable rules
| plus(N, s(M)) | → | U11(tt, M, N) | | plus(N, 0) | → | N |
| U11(tt, M, N) | → | U12(tt, M, N) | | U12(tt, M, N) | → | s(plus(N, M)) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| plus#(N, s(M)) | → | U11#(tt, M, N) |
Problem 3: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| U12#(tt, M, N) | → | plus#(N, M) | | U11#(tt, M, N) | → | U12#(tt, M, N) |
Rewrite Rules
| U11(tt, M, N) | → | U12(tt, M, N) | | U12(tt, M, N) | → | s(plus(N, M)) |
| plus(N, 0) | → | N | | plus(N, s(M)) | → | U11(tt, M, N) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, U11, U12
Strategy
Context-sensitive strategy:
μ(T) = μ(0) = μ(tt) = ∅
μ(U11#) = μ(U12#) = μ(s) = μ(U11) = μ(U12) = {1}
μ(plus) = μ(plus#) = {1, 2}
There are no SCCs!