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

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

F(1) -> F(g(1))
F(1) -> G(1)
G(0) -> G(f(0))
G(0) -> F(0)

Furthermore, R contains two SCCs.


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


Dependency Pair:

F(1) -> F(g(1))


Rules:


f(1) -> f(g(1))
f(f(x)) -> f(x)
g(0) -> g(f(0))
g(g(x)) -> g(x)


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(1) -> F(g(1))


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

g(0) -> g(f(0))
g(g(x)) -> g(x)
f(1) -> f(g(1))
f(f(x)) -> f(x)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  0  
  POL(g(x1))=  0  
  POL(1)=  1  
  POL(f(x1))=  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(1) -> f(g(1))
f(f(x)) -> f(x)
g(0) -> g(f(0))
g(g(x)) -> g(x)


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(0) -> G(f(0))


Rules:


f(1) -> f(g(1))
f(f(x)) -> f(x)
g(0) -> g(f(0))
g(g(x)) -> g(x)


Strategy:

innermost




The following dependency pair can be strictly oriented:

G(0) -> G(f(0))


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

g(0) -> g(f(0))
g(g(x)) -> g(x)
f(1) -> f(g(1))
f(f(x)) -> f(x)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  1  
  POL(g(x1))=  0  
  POL(G(x1))=  x1  
  POL(1)=  0  
  POL(f(x1))=  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(1) -> f(g(1))
f(f(x)) -> f(x)
g(0) -> g(f(0))
g(g(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