Termination Proof using AProVE

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

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

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

Furthermore, R contains two SCCs.

     ↳DPs

       →DP Problem 1

         ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Remaining

Dependency Pair:

F(+(x, 0)) -> F(x)

Rules:

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

Strategy:

innermost

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

F(+(x, 0)) -> F(x)

one new Dependency Pair is created:

F(+(+(x'', 0), 0)) -> F(+(x'', 0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
F(+(+(x'', 0), 0)) -> F(+(x'', 0))

Rules:

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

Strategy:
innermost
Dependency Pair:
+'(x, +(y, z)) -> +'(x, y)

Rules:

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

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
F(+(+(x'', 0), 0)) -> F(+(x'', 0))

Rules:

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

Strategy:
innermost
Dependency Pair:
+'(x, +(y, z)) -> +'(x, y)

Rules:

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

Strategy:
innermost

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