Term Rewriting System R:
[x]
f(f(x)) -> f(c(f(x)))
f(f(x)) -> f(d(f(x)))
g(c(x)) -> x
g(d(x)) -> x
g(c(h(0))) -> g(d(1))
g(c(1)) -> g(d(h(0)))
g(h(x)) -> g(x)

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(f(x)) -> F(c(f(x)))
F(f(x)) -> F(d(f(x)))
G(c(h(0))) -> G(d(1))
G(c(1)) -> G(d(h(0)))
G(h(x)) -> G(x)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pair:

G(h(x)) -> G(x)


Rules:


f(f(x)) -> f(c(f(x)))
f(f(x)) -> f(d(f(x)))
g(c(x)) -> x
g(d(x)) -> x
g(c(h(0))) -> g(d(1))
g(c(1)) -> g(d(h(0)))
g(h(x)) -> g(x)


Strategy:

innermost




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

G(h(x)) -> G(x)
one new Dependency Pair is created:

G(h(h(x''))) -> G(h(x''))

The transformation is resulting in one new DP problem: 



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


Dependency Pair:

G(h(h(x''))) -> G(h(x''))


Rules:


f(f(x)) -> f(c(f(x)))
f(f(x)) -> f(d(f(x)))
g(c(x)) -> x
g(d(x)) -> x
g(c(h(0))) -> g(d(1))
g(c(1)) -> g(d(h(0)))
g(h(x)) -> g(x)


Strategy:

innermost




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

G(h(h(x''))) -> G(h(x''))
one new Dependency Pair is created:

G(h(h(h(x'''')))) -> G(h(h(x'''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 3
                 ↳Polynomial Ordering


Dependency Pair:

G(h(h(h(x'''')))) -> G(h(h(x'''')))


Rules:


f(f(x)) -> f(c(f(x)))
f(f(x)) -> f(d(f(x)))
g(c(x)) -> x
g(d(x)) -> x
g(c(h(0))) -> g(d(1))
g(c(1)) -> g(d(h(0)))
g(h(x)) -> g(x)


Strategy:

innermost




The following dependency pair can be strictly oriented:

G(h(h(h(x'''')))) -> G(h(h(x'''')))


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(G(x1))=  1 + x1  
  POL(h(x1))=  1 + x1  

resulting in one new DP problem. 



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


Dependency Pair:


Rules:


f(f(x)) -> f(c(f(x)))
f(f(x)) -> f(d(f(x)))
g(c(x)) -> x
g(d(x)) -> x
g(c(h(0))) -> g(d(1))
g(c(1)) -> g(d(h(0)))
g(h(x)) -> g(x)


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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