YES
The TRS could be proven terminating. The proof took 36 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (16ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor SubtermCriterion (0ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| g#(s(x), y) | → | g#(x, +(y, s(x))) | | +#(x, s(y)) | → | +#(x, y) |
| f#(s(x)) | → | g#(x, s(x)) | | g#(s(x), y) | → | g#(x, s(+(y, x))) |
| g#(s(x), y) | → | +#(y, s(x)) | | g#(s(x), y) | → | +#(y, x) |
Rewrite Rules
| f(0) | → | 1 | | f(s(x)) | → | g(x, s(x)) |
| g(0, y) | → | y | | g(s(x), y) | → | g(x, +(y, s(x))) |
| +(x, 0) | → | x | | +(x, s(y)) | → | s(+(x, y)) |
| g(s(x), y) | → | g(x, s(+(y, x))) |
Original Signature
Termination of terms over the following signature is verified: f, g, 1, 0, s, +
Strategy
The following SCCs where found
| g#(s(x), y) → g#(x, +(y, s(x))) | g#(s(x), y) → g#(x, s(+(y, x))) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| g#(s(x), y) | → | g#(x, +(y, s(x))) | | g#(s(x), y) | → | g#(x, s(+(y, x))) |
Rewrite Rules
| f(0) | → | 1 | | f(s(x)) | → | g(x, s(x)) |
| g(0, y) | → | y | | g(s(x), y) | → | g(x, +(y, s(x))) |
| +(x, 0) | → | x | | +(x, s(y)) | → | s(+(x, y)) |
| g(s(x), y) | → | g(x, s(+(y, x))) |
Original Signature
Termination of terms over the following signature is verified: f, g, 1, 0, s, +
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| g#(s(x), y) | → | g#(x, +(y, s(x))) | | g#(s(x), y) | → | g#(x, s(+(y, x))) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| f(0) | → | 1 | | f(s(x)) | → | g(x, s(x)) |
| g(0, y) | → | y | | g(s(x), y) | → | g(x, +(y, s(x))) |
| +(x, 0) | → | x | | +(x, s(y)) | → | s(+(x, y)) |
| g(s(x), y) | → | g(x, s(+(y, x))) |
Original Signature
Termination of terms over the following signature is verified: f, g, 1, 0, s, +
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed: