Term Rewriting System R:
[x, y, z]
i(0) -> 0
i(i(x)) -> x
i(+(x, y)) -> +(i(x), i(y))
+(0, y) -> y
+(x, 0) -> x
+(i(x), x) -> 0
+(x, i(x)) -> 0
+(x, +(y, z)) -> +(+(x, y), z)
+(+(x, i(y)), y) -> x
+(+(x, y), i(y)) -> x

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

I(+(x, y)) -> +'(i(x), i(y))
I(+(x, y)) -> I(x)
I(+(x, y)) -> I(y)
+'(x, +(y, z)) -> +'(+(x, y), z)
+'(x, +(y, z)) -> +'(x, y)

Furthermore, R contains two SCCs. 


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


Dependency Pair:

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


Rules:


i(0) -> 0
i(i(x)) -> x
i(+(x, y)) -> +(i(x), i(y))
+(0, y) -> y
+(x, 0) -> x
+(i(x), x) -> 0
+(x, i(x)) -> 0
+(x, +(y, z)) -> +(+(x, y), z)
+(+(x, i(y)), y) -> x
+(+(x, y), i(y)) -> x


Strategy:

innermost




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

+'(x, +(y, z)) -> +'(x, y)
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem: 



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




The following remains to be proven:


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




The following remains to be proven:

Innermost Termination of R could not be shown.
Duration:
0:00 minutes