Term Rewriting System R:
[x]
f(f(a, x), a) -> f(f(f(a, a), f(x, a)), a)

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(f(a, x), a) -> F(f(f(a, a), f(x, a)), a)
F(f(a, x), a) -> F(f(a, a), f(x, a))
F(f(a, x), a) -> F(a, a)
F(f(a, x), a) -> F(x, a)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pairs:

F(f(a, x), a) -> F(x, a)
F(f(a, x), a) -> F(f(a, a), f(x, a))
F(f(a, x), a) -> F(f(f(a, a), f(x, a)), a)


Rule:


f(f(a, x), a) -> f(f(f(a, a), f(x, a)), a)


Strategy:

innermost




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

F(f(a, x), a) -> F(f(a, a), f(x, a))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(f(a, x), a) -> F(f(f(a, a), f(x, a)), a)
F(f(a, x), a) -> F(x, a)


Rule:


f(f(a, x), a) -> f(f(f(a, a), f(x, a)), a)


Strategy:

innermost




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

F(f(a, x), a) -> F(f(f(a, a), f(x, a)), a)
one new Dependency Pair is created:

F(f(a, f(a, x'')), a) -> F(f(f(a, a), f(f(f(a, a), f(x'', a)), a)), a)

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(f(a, f(a, x'')), a) -> F(f(f(a, a), f(f(f(a, a), f(x'', a)), a)), a)
F(f(a, x), a) -> F(x, a)


Rule:


f(f(a, x), a) -> f(f(f(a, a), f(x, a)), a)


Strategy:

innermost




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

F(f(a, x), a) -> F(x, a)
two new Dependency Pairs are created:

F(f(a, f(a, x'')), a) -> F(f(a, x''), a)
F(f(a, f(a, f(a, x''''))), a) -> F(f(a, f(a, x'''')), a)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 4
                 ↳Argument Filtering and Ordering


Dependency Pairs:

F(f(a, f(a, f(a, x''''))), a) -> F(f(a, f(a, x'''')), a)
F(f(a, f(a, x'')), a) -> F(f(a, x''), a)
F(f(a, f(a, x'')), a) -> F(f(f(a, a), f(f(f(a, a), f(x'', a)), a)), a)


Rule:


f(f(a, x), a) -> f(f(f(a, a), f(x, a)), a)


Strategy:

innermost




The following dependency pairs can be strictly oriented:

F(f(a, f(a, f(a, x''''))), a) -> F(f(a, f(a, x'''')), a)
F(f(a, f(a, x'')), a) -> F(f(a, x''), a)


The following usable rule for innermost can be oriented:

f(f(a, x), a) -> f(f(f(a, a), f(x, a)), a)


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
F > a
f > a

resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2) -> F(x1, x2)
f(x1, x2) -> f(x2)


   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 5
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

F(f(a, f(a, x'')), a) -> F(f(f(a, a), f(f(f(a, a), f(x'', a)), a)), a)


Rule:


f(f(a, x), a) -> f(f(f(a, a), f(x, a)), a)


Strategy:

innermost



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