Term Rewriting System R:
[x, y, z]
+(+(x, y), z) -> +(x, +(y, z))
+(f(x), f(y)) -> f(+(x, y))
+(f(x), +(f(y), z)) -> +(f(+(x, y)), z)
Innermost Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
+'(+(x, y), z) -> +'(x, +(y, z))
+'(+(x, y), z) -> +'(y, z)
+'(f(x), f(y)) -> +'(x, y)
+'(f(x), +(f(y), z)) -> +'(f(+(x, y)), z)
+'(f(x), +(f(y), z)) -> +'(x, y)
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Size-Change Principle
Dependency Pairs:
+'(f(x), +(f(y), z)) -> +'(x, y)
+'(+(x, y), z) -> +'(y, z)
+'(f(x), f(y)) -> +'(x, y)
+'(+(x, y), z) -> +'(x, +(y, z))
Rules:
+(+(x, y), z) -> +(x, +(y, z))
+(f(x), f(y)) -> f(+(x, y))
+(f(x), +(f(y), z)) -> +(f(+(x, y)), z)
Strategy:
innermost
We number the DPs as follows:
- +'(f(x), +(f(y), z)) -> +'(x, y)
- +'(+(x, y), z) -> +'(y, z)
- +'(f(x), f(y)) -> +'(x, y)
- +'(+(x, y), z) -> +'(x, +(y, z))
and get the following Size-Change Graph(s):
which lead(s) to this/these maximal multigraph(s):
DP: empty set
Oriented Rules: none
We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial
with Argument Filtering System:
f(x1) -> f(x1)
We obtain no new DP problems.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes