Term Rewriting System R:
[x, y, z]
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
+(x, +(y, z)) -> +(+(x, y), z)
f(g(f(x))) -> f(h(s(0), x))
f(g(h(x, y))) -> f(h(s(x), y))
f(h(x, h(y, z))) -> f(h(+(x, y), z))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

+'(x, s(y)) -> +'(x, y)
+'(s(x), y) -> +'(x, y)
+'(x, +(y, z)) -> +'(+(x, y), z)
+'(x, +(y, z)) -> +'(x, y)
F(g(f(x))) -> F(h(s(0), x))
F(g(h(x, y))) -> F(h(s(x), y))
F(h(x, h(y, z))) -> F(h(+(x, y), z))
F(h(x, h(y, z))) -> +'(x, y)

Furthermore, R contains two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation
       →DP Problem 2
         ↳Remaining


Dependency Pairs:

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


Rules:


+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
+(x, +(y, z)) -> +(+(x, y), z)
f(g(f(x))) -> f(h(s(0), x))
f(g(h(x, y))) -> f(h(s(x), y))
f(h(x, h(y, z))) -> f(h(+(x, y), z))


Strategy:

innermost




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

+'(x, +(y, z)) -> +'(+(x, y), z)
three new Dependency Pairs are created:

+'(0, +(y'', z)) -> +'(y'', z)
+'(s(x''), +(y'', z)) -> +'(s(+(x'', y'')), z)
+'(x'', +(+(y'', z''), z)) -> +'(+(+(x'', y''), z''), z)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 3
             ↳Forward Instantiation Transformation
       →DP Problem 2
         ↳Remaining


Dependency Pairs:

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


Rules:


+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
+(x, +(y, z)) -> +(+(x, y), z)
f(g(f(x))) -> f(h(s(0), x))
f(g(h(x, y))) -> f(h(s(x), y))
f(h(x, h(y, z))) -> f(h(+(x, y), z))


Strategy:

innermost




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

+'(x, s(y)) -> +'(x, y)
five new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 3
             ↳FwdInst
             ...
               →DP Problem 4
                 ↳Forward Instantiation Transformation
       →DP Problem 2
         ↳Remaining


Dependency Pairs:

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


Rules:


+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
+(x, +(y, z)) -> +(+(x, y), z)
f(g(f(x))) -> f(h(s(0), x))
f(g(h(x, y))) -> f(h(s(x), y))
f(h(x, h(y, z))) -> f(h(+(x, y), z))


Strategy:

innermost




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

+'(s(x), y) -> +'(x, y)
nine new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 3
             ↳FwdInst
             ...
               →DP Problem 5
                 ↳Forward Instantiation Transformation
       →DP Problem 2
         ↳Remaining


Dependency Pairs:

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


Rules:


+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
+(x, +(y, z)) -> +(+(x, y), z)
f(g(f(x))) -> f(h(s(0), x))
f(g(h(x, y))) -> f(h(s(x), y))
f(h(x, h(y, z))) -> f(h(+(x, y), z))


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)
seven new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



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




The following remains to be proven:


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




The following remains to be proven:

Innermost Termination of R could not be shown.
Duration:
0:25 minutes