YES
The TRS could be proven terminating. The proof took 20 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (5ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| – Problem 4 was processed with processor SubtermCriterion (0ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| f#(x, g(y, z)) | → | f#(x, y) | | rem#(g(x, y), s(z)) | → | rem#(x, z) |
| norm#(g(x, y)) | → | norm#(x) |
Rewrite Rules
| norm(nil) | → | 0 | | norm(g(x, y)) | → | s(norm(x)) |
| f(x, nil) | → | g(nil, x) | | f(x, g(y, z)) | → | g(f(x, y), z) |
| rem(nil, y) | → | nil | | rem(g(x, y), 0) | → | g(x, y) |
| rem(g(x, y), s(z)) | → | rem(x, z) |
Original Signature
Termination of terms over the following signature is verified: f, g, 0, norm, s, rem, nil
Strategy
The following SCCs where found
| f#(x, g(y, z)) → f#(x, y) |
| rem#(g(x, y), s(z)) → rem#(x, z) |
| norm#(g(x, y)) → norm#(x) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| f#(x, g(y, z)) | → | f#(x, y) |
Rewrite Rules
| norm(nil) | → | 0 | | norm(g(x, y)) | → | s(norm(x)) |
| f(x, nil) | → | g(nil, x) | | f(x, g(y, z)) | → | g(f(x, y), z) |
| rem(nil, y) | → | nil | | rem(g(x, y), 0) | → | g(x, y) |
| rem(g(x, y), s(z)) | → | rem(x, z) |
Original Signature
Termination of terms over the following signature is verified: f, g, 0, norm, s, rem, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| f#(x, g(y, z)) | → | f#(x, y) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| rem#(g(x, y), s(z)) | → | rem#(x, z) |
Rewrite Rules
| norm(nil) | → | 0 | | norm(g(x, y)) | → | s(norm(x)) |
| f(x, nil) | → | g(nil, x) | | f(x, g(y, z)) | → | g(f(x, y), z) |
| rem(nil, y) | → | nil | | rem(g(x, y), 0) | → | g(x, y) |
| rem(g(x, y), s(z)) | → | rem(x, z) |
Original Signature
Termination of terms over the following signature is verified: f, g, 0, norm, s, rem, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| rem#(g(x, y), s(z)) | → | rem#(x, z) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| norm#(g(x, y)) | → | norm#(x) |
Rewrite Rules
| norm(nil) | → | 0 | | norm(g(x, y)) | → | s(norm(x)) |
| f(x, nil) | → | g(nil, x) | | f(x, g(y, z)) | → | g(f(x, y), z) |
| rem(nil, y) | → | nil | | rem(g(x, y), 0) | → | g(x, y) |
| rem(g(x, y), s(z)) | → | rem(x, z) |
Original Signature
Termination of terms over the following signature is verified: f, g, 0, norm, s, rem, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| norm#(g(x, y)) | → | norm#(x) |