Term Rewriting System R:
[X, Y]
f(s(X)) -> f(X)
g(cons(0, Y)) -> g(Y)
g(cons(s(X), Y)) -> s(X)
h(cons(X, Y)) -> h(g(cons(X, Y)))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(s(X)) -> F(X)
G(cons(0, Y)) -> G(Y)
H(cons(X, Y)) -> H(g(cons(X, Y)))
H(cons(X, Y)) -> G(cons(X, Y))

Furthermore, R contains three SCCs. 


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


Dependency Pair:

F(s(X)) -> F(X)


Rules:


f(s(X)) -> f(X)
g(cons(0, Y)) -> g(Y)
g(cons(s(X), Y)) -> s(X)
h(cons(X, Y)) -> h(g(cons(X, Y)))





The following dependency pair can be strictly oriented:

F(s(X)) -> F(X)


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(s(x1))=  1 + x1  
  POL(F(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 4
             ↳Dependency Graph
       →DP Problem 2
         ↳FwdInst
       →DP Problem 3
         ↳Nar


Dependency Pair:


Rules:


f(s(X)) -> f(X)
g(cons(0, Y)) -> g(Y)
g(cons(s(X), Y)) -> s(X)
h(cons(X, Y)) -> h(g(cons(X, Y)))





Using the Dependency Graph resulted in no new DP problems.


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


Dependency Pair:

G(cons(0, Y)) -> G(Y)


Rules:


f(s(X)) -> f(X)
g(cons(0, Y)) -> g(Y)
g(cons(s(X), Y)) -> s(X)
h(cons(X, Y)) -> h(g(cons(X, Y)))





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

G(cons(0, Y)) -> G(Y)
one new Dependency Pair is created:

G(cons(0, cons(0, Y''))) -> G(cons(0, Y''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳FwdInst
           →DP Problem 5
             ↳Forward Instantiation Transformation
       →DP Problem 3
         ↳Nar


Dependency Pair:

G(cons(0, cons(0, Y''))) -> G(cons(0, Y''))


Rules:


f(s(X)) -> f(X)
g(cons(0, Y)) -> g(Y)
g(cons(s(X), Y)) -> s(X)
h(cons(X, Y)) -> h(g(cons(X, Y)))





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

G(cons(0, cons(0, Y''))) -> G(cons(0, Y''))
one new Dependency Pair is created:

G(cons(0, cons(0, cons(0, Y'''')))) -> G(cons(0, cons(0, Y'''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳FwdInst
           →DP Problem 5
             ↳FwdInst
             ...
               →DP Problem 6
                 ↳Polynomial Ordering
       →DP Problem 3
         ↳Nar


Dependency Pair:

G(cons(0, cons(0, cons(0, Y'''')))) -> G(cons(0, cons(0, Y'''')))


Rules:


f(s(X)) -> f(X)
g(cons(0, Y)) -> g(Y)
g(cons(s(X), Y)) -> s(X)
h(cons(X, Y)) -> h(g(cons(X, Y)))





The following dependency pair can be strictly oriented:

G(cons(0, cons(0, cons(0, Y'''')))) -> G(cons(0, cons(0, Y'''')))


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  0  
  POL(G(x1))=  1 + x1  
  POL(cons(x1, x2))=  1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳FwdInst
           →DP Problem 5
             ↳FwdInst
             ...
               →DP Problem 7
                 ↳Dependency Graph
       →DP Problem 3
         ↳Nar


Dependency Pair:


Rules:


f(s(X)) -> f(X)
g(cons(0, Y)) -> g(Y)
g(cons(s(X), Y)) -> s(X)
h(cons(X, Y)) -> h(g(cons(X, Y)))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳FwdInst
       →DP Problem 3
         ↳Narrowing Transformation


Dependency Pair:

H(cons(X, Y)) -> H(g(cons(X, Y)))


Rules:


f(s(X)) -> f(X)
g(cons(0, Y)) -> g(Y)
g(cons(s(X), Y)) -> s(X)
h(cons(X, Y)) -> h(g(cons(X, Y)))





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

H(cons(X, Y)) -> H(g(cons(X, Y)))
two new Dependency Pairs are created:

H(cons(0, Y'')) -> H(g(Y''))
H(cons(s(X''), Y'')) -> H(s(X''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳FwdInst
       →DP Problem 3
         ↳Nar
           →DP Problem 8
             ↳Narrowing Transformation


Dependency Pair:

H(cons(0, Y'')) -> H(g(Y''))


Rules:


f(s(X)) -> f(X)
g(cons(0, Y)) -> g(Y)
g(cons(s(X), Y)) -> s(X)
h(cons(X, Y)) -> h(g(cons(X, Y)))





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

H(cons(0, Y'')) -> H(g(Y''))
two new Dependency Pairs are created:

H(cons(0, cons(0, Y'))) -> H(g(Y'))
H(cons(0, cons(s(X'), Y'))) -> H(s(X'))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳FwdInst
       →DP Problem 3
         ↳Nar
           →DP Problem 8
             ↳Nar
             ...
               →DP Problem 9
                 ↳Forward Instantiation Transformation


Dependency Pair:

H(cons(0, cons(0, Y'))) -> H(g(Y'))


Rules:


f(s(X)) -> f(X)
g(cons(0, Y)) -> g(Y)
g(cons(s(X), Y)) -> s(X)
h(cons(X, Y)) -> h(g(cons(X, Y)))





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

H(cons(0, cons(0, Y'))) -> H(g(Y'))
no new Dependency Pairs are created.
The transformation is resulting in no new DP problems.


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