YES
The TRS could be proven terminating. The proof took 42 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (26ms).
| – Problem 2 was processed with processor SubtermCriterion (3ms).
| – Problem 3 was processed with processor SubtermCriterion (1ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| fib#(s(s(x))) | → | sp#(g(x)) | | g#(s(x)) | → | np#(g(x)) |
| +#(x, s(y)) | → | +#(x, y) | | fib#(s(s(x))) | → | g#(x) |
| np#(pair(x, y)) | → | +#(x, y) | | g#(s(x)) | → | g#(x) |
| sp#(pair(x, y)) | → | +#(x, y) |
Rewrite Rules
| fib(0) | → | 0 | | fib(s(0)) | → | s(0) |
| fib(s(s(0))) | → | s(0) | | fib(s(s(x))) | → | sp(g(x)) |
| g(0) | → | pair(s(0), 0) | | g(s(0)) | → | pair(s(0), s(0)) |
| g(s(x)) | → | np(g(x)) | | sp(pair(x, y)) | → | +(x, y) |
| np(pair(x, y)) | → | pair(+(x, y), x) | | +(x, 0) | → | x |
| +(x, s(y)) | → | s(+(x, y)) |
Original Signature
Termination of terms over the following signature is verified: g, 0, s, sp, pair, np, +, fib
Strategy
The following SCCs where found
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| fib(0) | → | 0 | | fib(s(0)) | → | s(0) |
| fib(s(s(0))) | → | s(0) | | fib(s(s(x))) | → | sp(g(x)) |
| g(0) | → | pair(s(0), 0) | | g(s(0)) | → | pair(s(0), s(0)) |
| g(s(x)) | → | np(g(x)) | | sp(pair(x, y)) | → | +(x, y) |
| np(pair(x, y)) | → | pair(+(x, y), x) | | +(x, 0) | → | x |
| +(x, s(y)) | → | s(+(x, y)) |
Original Signature
Termination of terms over the following signature is verified: g, 0, s, sp, pair, np, +, fib
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| fib(0) | → | 0 | | fib(s(0)) | → | s(0) |
| fib(s(s(0))) | → | s(0) | | fib(s(s(x))) | → | sp(g(x)) |
| g(0) | → | pair(s(0), 0) | | g(s(0)) | → | pair(s(0), s(0)) |
| g(s(x)) | → | np(g(x)) | | sp(pair(x, y)) | → | +(x, y) |
| np(pair(x, y)) | → | pair(+(x, y), x) | | +(x, 0) | → | x |
| +(x, s(y)) | → | s(+(x, y)) |
Original Signature
Termination of terms over the following signature is verified: g, 0, s, sp, pair, np, +, fib
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed: