YES
The TRS could be proven terminating. The proof took 349 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (11ms).
| – Problem 2 was processed with processor SubtermCriterion (0ms).
| – Problem 3 was processed with processor PolynomialLinearRange4iUR (180ms).
| – Problem 4 was processed with processor SubtermCriterion (0ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| h#(cons(X, Y)) | → | h#(g(cons(X, Y))) | | f#(s(X)) | → | f#(X) |
| h#(cons(X, Y)) | → | g#(cons(X, Y)) | | g#(cons(0, Y)) | → | g#(Y) |
Rewrite Rules
| f(s(X)) | → | f(X) | | g(cons(0, Y)) | → | g(Y) |
| g(cons(s(X), Y)) | → | s(X) | | h(cons(X, Y)) | → | h(g(cons(X, Y))) |
Original Signature
Termination of terms over the following signature is verified: f, g, 0, s, h, cons
Strategy
The following SCCs where found
| h#(cons(X, Y)) → h#(g(cons(X, Y))) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| f(s(X)) | → | f(X) | | g(cons(0, Y)) | → | g(Y) |
| g(cons(s(X), Y)) | → | s(X) | | h(cons(X, Y)) | → | h(g(cons(X, Y))) |
Original Signature
Termination of terms over the following signature is verified: f, g, 0, s, h, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
Problem 3: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| h#(cons(X, Y)) | → | h#(g(cons(X, Y))) |
Rewrite Rules
| f(s(X)) | → | f(X) | | g(cons(0, Y)) | → | g(Y) |
| g(cons(s(X), Y)) | → | s(X) | | h(cons(X, Y)) | → | h(g(cons(X, Y))) |
Original Signature
Termination of terms over the following signature is verified: f, g, 0, s, h, cons
Strategy
Polynomial Interpretation
- 0: 1
- cons(x,y): x + 1
- f(x): 0
- g(x): 0
- h(x): 0
- h#(x): x + 1
- s(x): 0
Improved Usable rules
| g(cons(s(X), Y)) | → | s(X) | | g(cons(0, Y)) | → | g(Y) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| h#(cons(X, Y)) | → | h#(g(cons(X, Y))) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| f(s(X)) | → | f(X) | | g(cons(0, Y)) | → | g(Y) |
| g(cons(s(X), Y)) | → | s(X) | | h(cons(X, Y)) | → | h(g(cons(X, Y))) |
Original Signature
Termination of terms over the following signature is verified: f, g, 0, s, h, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed: