Term Rewriting System R:
[x, y, z]
f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)
Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(+(x, 0)) -> F(x)
+'(x, +(y, z)) -> +'(+(x, y), z)
+'(x, +(y, z)) -> +'(x, y)
Furthermore, R contains two SCCs.
R
↳DPs
→DP Problem 1
↳Argument Filtering and Ordering
→DP Problem 2
↳Remaining
Dependency Pair:
F(+(x, 0)) -> F(x)
Rules:
f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)
The following dependency pair can be strictly oriented:
F(+(x, 0)) -> F(x)
There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial
resulting in one new DP problem.
Used Argument Filtering System: F(x1) -> F(x1)
+(x1, x2) -> +(x1, x2)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 3
↳Dependency Graph
→DP Problem 2
↳Remaining
Dependency Pair:
Rules:
f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)
Using the Dependency Graph resulted in no new DP problems.
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Remaining Obligation(s)
The following remains to be proven:
Dependency Pairs:
+'(x, +(y, z)) -> +'(x, y)
+'(x, +(y, z)) -> +'(+(x, y), z)
Rules:
f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)
Termination of R could not be shown.
Duration:
0:00 minutes