YES
The TRS could be proven terminating. The proof took 22 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (8ms).
| – Problem 2 was processed with processor SubtermCriterion (0ms).
| | – Problem 4 was processed with processor SubtermCriterion (0ms).
| – Problem 3 was processed with processor SubtermCriterion (1ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| times#(x, s(y)) | → | plus#(times(x, y), x) | | plus#(x, s(y)) | → | plus#(x, y) |
| plus#(s(x), y) | → | plus#(x, y) | | times#(x, s(y)) | → | times#(x, y) |
Rewrite Rules
| times(x, 0) | → | 0 | | times(x, s(y)) | → | plus(times(x, y), x) |
| plus(x, 0) | → | x | | plus(0, x) | → | x |
| plus(x, s(y)) | → | s(plus(x, y)) | | plus(s(x), y) | → | s(plus(x, y)) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, times
Strategy
The following SCCs where found
| times#(x, s(y)) → times#(x, y) |
| plus#(s(x), y) → plus#(x, y) | plus#(x, s(y)) → plus#(x, y) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| plus#(s(x), y) | → | plus#(x, y) | | plus#(x, s(y)) | → | plus#(x, y) |
Rewrite Rules
| times(x, 0) | → | 0 | | times(x, s(y)) | → | plus(times(x, y), x) |
| plus(x, 0) | → | x | | plus(0, x) | → | x |
| plus(x, s(y)) | → | s(plus(x, y)) | | plus(s(x), y) | → | s(plus(x, y)) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, times
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| plus#(s(x), y) | → | plus#(x, y) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| plus#(x, s(y)) | → | plus#(x, y) |
Rewrite Rules
| times(x, 0) | → | 0 | | times(x, s(y)) | → | plus(times(x, y), x) |
| plus(x, 0) | → | x | | plus(0, x) | → | x |
| plus(x, s(y)) | → | s(plus(x, y)) | | plus(s(x), y) | → | s(plus(x, y)) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, times
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| plus#(x, s(y)) | → | plus#(x, y) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| times#(x, s(y)) | → | times#(x, y) |
Rewrite Rules
| times(x, 0) | → | 0 | | times(x, s(y)) | → | plus(times(x, y), x) |
| plus(x, 0) | → | x | | plus(0, x) | → | x |
| plus(x, s(y)) | → | s(plus(x, y)) | | plus(s(x), y) | → | s(plus(x, y)) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, times
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| times#(x, s(y)) | → | times#(x, y) |