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:
  1. +'(f(x), +(f(y), z)) -> +'(x, y)
  2. +'(+(x, y), z) -> +'(y, z)
  3. +'(f(x), f(y)) -> +'(x, y)
  4. +'(+(x, y), z) -> +'(x, +(y, z))
and get the following Size-Change Graph(s):
{3, 1} , {3, 1}
1>1
2>2
{2} , {2}
1>1
2=2
{4} , {4}
1>1

which lead(s) to this/these maximal multigraph(s):
{2} , {2}
1>1
2=2
{3, 1} , {3, 1}
1>1
2>2
{3, 1} , {2}
1>1
2>2
{4} , {3, 1}
1>1
{3, 1} , {4}
1>1
{2} , {3, 1}
1>1
2>2
{4} , {2}
1>1
{2} , {3, 1}
1>1
{3, 1} , {3, 1}
1>1
{2} , {2}
1>1
2>2
{2} , {2}
1>1
{3, 1} , {2}
1>1

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