YES
The TRS could be proven terminating. The proof took 54 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (38ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor SubtermCriterion (0ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| s#(f(x, y)) | → | s#(x) | | f#(g(x, y), g(u, v)) | → | f#(x, u) |
| f#(g(x, y), g(u, v)) | → | f#(y, v) | | s#(f(x, y)) | → | f#(s(y), s(x)) |
| s#(f(x, y)) | → | s#(y) | | s#(g(x, y)) | → | g#(s(x), s(y)) |
| f#(g(x, y), g(u, v)) | → | g#(f(x, u), f(y, v)) | | s#(g(x, y)) | → | s#(y) |
| s#(g(x, y)) | → | s#(x) |
Rewrite Rules
| s(a) | → | a | | s(s(x)) | → | x |
| s(f(x, y)) | → | f(s(y), s(x)) | | s(g(x, y)) | → | g(s(x), s(y)) |
| f(x, a) | → | x | | f(a, y) | → | y |
| f(g(x, y), g(u, v)) | → | g(f(x, u), f(y, v)) | | g(a, a) | → | a |
Original Signature
Termination of terms over the following signature is verified: f, g, s, a
Strategy
The following SCCs where found
| s#(f(x, y)) → s#(x) | s#(f(x, y)) → s#(y) |
| s#(g(x, y)) → s#(y) | s#(g(x, y)) → s#(x) |
| f#(g(x, y), g(u, v)) → f#(x, u) | f#(g(x, y), g(u, v)) → f#(y, v) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| s#(f(x, y)) | → | s#(x) | | s#(f(x, y)) | → | s#(y) |
| s#(g(x, y)) | → | s#(y) | | s#(g(x, y)) | → | s#(x) |
Rewrite Rules
| s(a) | → | a | | s(s(x)) | → | x |
| s(f(x, y)) | → | f(s(y), s(x)) | | s(g(x, y)) | → | g(s(x), s(y)) |
| f(x, a) | → | x | | f(a, y) | → | y |
| f(g(x, y), g(u, v)) | → | g(f(x, u), f(y, v)) | | g(a, a) | → | a |
Original Signature
Termination of terms over the following signature is verified: f, g, s, a
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| s#(f(x, y)) | → | s#(x) | | s#(f(x, y)) | → | s#(y) |
| s#(g(x, y)) | → | s#(y) | | s#(g(x, y)) | → | s#(x) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| f#(g(x, y), g(u, v)) | → | f#(x, u) | | f#(g(x, y), g(u, v)) | → | f#(y, v) |
Rewrite Rules
| s(a) | → | a | | s(s(x)) | → | x |
| s(f(x, y)) | → | f(s(y), s(x)) | | s(g(x, y)) | → | g(s(x), s(y)) |
| f(x, a) | → | x | | f(a, y) | → | y |
| f(g(x, y), g(u, v)) | → | g(f(x, u), f(y, v)) | | g(a, a) | → | a |
Original Signature
Termination of terms over the following signature is verified: f, g, s, a
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| f#(g(x, y), g(u, v)) | → | f#(y, v) | | f#(g(x, y), g(u, v)) | → | f#(x, u) |