Term Rewriting System R:
[x, y]
average(s(x), y) -> average(x, s(y))
average(x, s(s(s(y)))) -> s(average(s(x), y))
average(0, 0) -> 0
average(0, s(0)) -> 0
average(0, s(s(0))) -> s(0)

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

AVERAGE(s(x), y) -> AVERAGE(x, s(y))
AVERAGE(x, s(s(s(y)))) -> AVERAGE(s(x), y)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pairs:

AVERAGE(x, s(s(s(y)))) -> AVERAGE(s(x), y)
AVERAGE(s(x), y) -> AVERAGE(x, s(y))


Rules:


average(s(x), y) -> average(x, s(y))
average(x, s(s(s(y)))) -> s(average(s(x), y))
average(0, 0) -> 0
average(0, s(0)) -> 0
average(0, s(s(0))) -> s(0)


Strategy:

innermost




On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

AVERAGE(s(x), y) -> AVERAGE(x, s(y))
two new Dependency Pairs are created:

AVERAGE(s(s(x'')), y'') -> AVERAGE(s(x''), s(y''))
AVERAGE(s(x''), s(s(y''))) -> AVERAGE(x'', s(s(s(y''))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳Forward Instantiation Transformation


Dependency Pairs:

AVERAGE(s(x''), s(s(y''))) -> AVERAGE(x'', s(s(s(y''))))
AVERAGE(s(s(x'')), y'') -> AVERAGE(s(x''), s(y''))
AVERAGE(x, s(s(s(y)))) -> AVERAGE(s(x), y)


Rules:


average(s(x), y) -> average(x, s(y))
average(x, s(s(s(y)))) -> s(average(s(x), y))
average(0, 0) -> 0
average(0, s(0)) -> 0
average(0, s(s(0))) -> s(0)


Strategy:

innermost




On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

AVERAGE(x, s(s(s(y)))) -> AVERAGE(s(x), y)
three new Dependency Pairs are created:

AVERAGE(x'', s(s(s(s(s(s(y''))))))) -> AVERAGE(s(x''), s(s(s(y''))))
AVERAGE(s(x''''), s(s(s(y')))) -> AVERAGE(s(s(x'''')), y')
AVERAGE(x', s(s(s(s(s(y'''')))))) -> AVERAGE(s(x'), s(s(y'''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 3
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

AVERAGE(x', s(s(s(s(s(y'''')))))) -> AVERAGE(s(x'), s(s(y'''')))
AVERAGE(s(x''''), s(s(s(y')))) -> AVERAGE(s(s(x'''')), y')
AVERAGE(x'', s(s(s(s(s(s(y''))))))) -> AVERAGE(s(x''), s(s(s(y''))))
AVERAGE(s(s(x'')), y'') -> AVERAGE(s(x''), s(y''))
AVERAGE(s(x''), s(s(y''))) -> AVERAGE(x'', s(s(s(y''))))


Rules:


average(s(x), y) -> average(x, s(y))
average(x, s(s(s(y)))) -> s(average(s(x), y))
average(0, 0) -> 0
average(0, s(0)) -> 0
average(0, s(s(0))) -> s(0)


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:01 minutes