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
         ↳Argument Filtering and Ordering


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




The following dependency pairs can be strictly oriented:

+'(f(x), +(f(y), z)) -> +'(x, y)
+'(+(x, y), z) -> +'(y, z)
+'(f(x), f(y)) -> +'(x, y)
+'(+(x, y), z) -> +'(x, +(y, z))


There are no usable rules for innermost w.r.t. to the AFS 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:
+'(x1, x2) -> x1
+(x1, x2) -> +(x1, x2)
f(x1) -> f(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Dependency Graph


Dependency Pair:


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




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes