Term Rewriting System R:
[x]
f(h(x)) -> f(i(x))
g(i(x)) -> g(h(x))
h(a) -> b
i(a) -> b

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

F(h(x)) -> F(i(x))
F(h(x)) -> I(x)
G(i(x)) -> G(h(x))
G(i(x)) -> H(x)

Furthermore, R contains two SCCs.


   R
DPs
       →DP Problem 1
Polynomial Ordering
       →DP Problem 2
Polo


Dependency Pair:

F(h(x)) -> F(i(x))


Rules:


f(h(x)) -> f(i(x))
g(i(x)) -> g(h(x))
h(a) -> b
i(a) -> b


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(h(x)) -> F(i(x))


Additionally, the following usable rule for innermost w.r.t. to the implicit AFS can be oriented:

i(a) -> b


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(i(x1))=  0  
  POL(b)=  0  
  POL(h(x1))=  1  
  POL(a)=  0  
  POL(F(x1))=  x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
           →DP Problem 3
Dependency Graph
       →DP Problem 2
Polo


Dependency Pair:


Rules:


f(h(x)) -> f(i(x))
g(i(x)) -> g(h(x))
h(a) -> b
i(a) -> b


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polynomial Ordering


Dependency Pair:

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


Rules:


f(h(x)) -> f(i(x))
g(i(x)) -> g(h(x))
h(a) -> b
i(a) -> b


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


Additionally, the following usable rule for innermost w.r.t. to the implicit AFS can be oriented:

h(a) -> b


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(i(x1))=  1  
  POL(G(x1))=  x1  
  POL(b)=  0  
  POL(h(x1))=  0  
  POL(a)=  0  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
           →DP Problem 4
Dependency Graph


Dependency Pair:


Rules:


f(h(x)) -> f(i(x))
g(i(x)) -> g(h(x))
h(a) -> b
i(a) -> b


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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