Term Rewriting System R:
[x, y]
f(g(x), g(y)) -> f(p(f(g(x), s(y))), g(s(p(x))))
p(0) -> g(0)
g(s(p(x))) -> p(x)

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(g(x), g(y)) -> F(p(f(g(x), s(y))), g(s(p(x))))
F(g(x), g(y)) -> P(f(g(x), s(y)))
F(g(x), g(y)) -> F(g(x), s(y))
F(g(x), g(y)) -> G(s(p(x)))
F(g(x), g(y)) -> P(x)
P(0) -> G(0)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pair:

F(g(x), g(y)) -> F(p(f(g(x), s(y))), g(s(p(x))))


Rules:


f(g(x), g(y)) -> f(p(f(g(x), s(y))), g(s(p(x))))
p(0) -> g(0)
g(s(p(x))) -> p(x)


Strategy:

innermost




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

F(g(x), g(y)) -> F(p(f(g(x), s(y))), g(s(p(x))))
two new Dependency Pairs are created:

F(g(x''), g(y)) -> F(p(f(g(x''), s(y))), p(x''))
F(g(0), g(y)) -> F(p(f(g(0), s(y))), g(s(g(0))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Narrowing Transformation


Dependency Pairs:

F(g(0), g(y)) -> F(p(f(g(0), s(y))), g(s(g(0))))
F(g(x''), g(y)) -> F(p(f(g(x''), s(y))), p(x''))


Rules:


f(g(x), g(y)) -> f(p(f(g(x), s(y))), g(s(p(x))))
p(0) -> g(0)
g(s(p(x))) -> p(x)


Strategy:

innermost




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

F(g(x''), g(y)) -> F(p(f(g(x''), s(y))), p(x''))
one new Dependency Pair is created:

F(g(0), g(y)) -> F(p(f(g(0), s(y))), g(0))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 3
                 ↳Narrowing Transformation


Dependency Pairs:

F(g(0), g(y)) -> F(p(f(g(0), s(y))), g(0))
F(g(0), g(y)) -> F(p(f(g(0), s(y))), g(s(g(0))))


Rules:


f(g(x), g(y)) -> f(p(f(g(x), s(y))), g(s(p(x))))
p(0) -> g(0)
g(s(p(x))) -> p(x)


Strategy:

innermost




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

F(g(0), g(y)) -> F(p(f(g(0), s(y))), g(s(g(0))))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 4
                 ↳Narrowing Transformation


Dependency Pair:

F(g(0), g(y)) -> F(p(f(g(0), s(y))), g(0))


Rules:


f(g(x), g(y)) -> f(p(f(g(x), s(y))), g(s(p(x))))
p(0) -> g(0)
g(s(p(x))) -> p(x)


Strategy:

innermost




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

F(g(0), g(y)) -> F(p(f(g(0), s(y))), g(0))
no new Dependency Pairs are created.
The transformation is resulting in no new DP problems.


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