Term Rewriting System R:
[X]
f(a, a) -> f(a, b)
f(a, b) -> f(s(a), c)
f(s(X), c) -> f(X, c)
f(c, c) -> f(a, a)

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(a, a) -> F(a, b)
F(a, b) -> F(s(a), c)
F(s(X), c) -> F(X, c)
F(c, c) -> F(a, a)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pairs:

F(c, c) -> F(a, a)
F(s(X), c) -> F(X, c)
F(a, b) -> F(s(a), c)
F(a, a) -> F(a, b)


Rules:


f(a, a) -> f(a, b)
f(a, b) -> f(s(a), c)
f(s(X), c) -> f(X, c)
f(c, c) -> f(a, a)


Strategy:

innermost




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

F(s(X), c) -> F(X, c)
two new Dependency Pairs are created:

F(s(s(X'')), c) -> F(s(X''), c)
F(s(c), c) -> F(c, c)

The transformation is resulting in one new DP problem: 



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


Dependency Pair:

F(s(s(X'')), c) -> F(s(X''), c)


Rules:


f(a, a) -> f(a, b)
f(a, b) -> f(s(a), c)
f(s(X), c) -> f(X, c)
f(c, c) -> f(a, a)


Strategy:

innermost




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

F(s(s(X'')), c) -> F(s(X''), c)
one new Dependency Pair is created:

F(s(s(s(X''''))), c) -> F(s(s(X'''')), c)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 3
                 ↳Argument Filtering and Ordering


Dependency Pair:

F(s(s(s(X''''))), c) -> F(s(s(X'''')), c)


Rules:


f(a, a) -> f(a, b)
f(a, b) -> f(s(a), c)
f(s(X), c) -> f(X, c)
f(c, c) -> f(a, a)


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(s(s(s(X''''))), c) -> F(s(s(X'''')), c)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(c)=  0  
  POL(s(x1))=  1 + x1  
  POL(F(x1, x2))=  1 + x1 + x2  

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


   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 4
                 ↳Dependency Graph


Dependency Pair:


Rules:


f(a, a) -> f(a, b)
f(a, b) -> f(s(a), c)
f(s(X), c) -> f(X, c)
f(c, c) -> f(a, a)


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes