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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(x, x) -> F(i(x), g(g(x)))
F(x, x) -> G(g(x))
F(x, x) -> G(x)
F(x, i(x)) -> F(x, x)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Rewriting Transformation


Dependency Pairs:

F(x, i(x)) -> F(x, x)
F(x, x) -> F(i(x), g(g(x)))


Rules:


f(x, x) -> f(i(x), g(g(x)))
f(x, y) -> x
f(x, i(x)) -> f(x, x)
f(i(x), i(g(x))) -> a
g(x) -> i(x)


Strategy:

innermost




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

F(x, x) -> F(i(x), g(g(x)))
one new Dependency Pair is created:

F(x, x) -> F(i(x), i(g(x)))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Rw
           →DP Problem 2
             ↳Rewriting Transformation


Dependency Pairs:

F(x, x) -> F(i(x), i(g(x)))
F(x, i(x)) -> F(x, x)


Rules:


f(x, x) -> f(i(x), g(g(x)))
f(x, y) -> x
f(x, i(x)) -> f(x, x)
f(i(x), i(g(x))) -> a
g(x) -> i(x)


Strategy:

innermost




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

F(x, x) -> F(i(x), i(g(x)))
one new Dependency Pair is created:

F(x, x) -> F(i(x), i(i(x)))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Rw
           →DP Problem 2
             ↳Rw
             ...
               →DP Problem 3
                 ↳Instantiation Transformation


Dependency Pairs:

F(x, x) -> F(i(x), i(i(x)))
F(x, i(x)) -> F(x, x)


Rules:


f(x, x) -> f(i(x), g(g(x)))
f(x, y) -> x
f(x, i(x)) -> f(x, x)
f(i(x), i(g(x))) -> a
g(x) -> i(x)


Strategy:

innermost




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

F(x, i(x)) -> F(x, x)
one new Dependency Pair is created:

F(i(x''), i(i(x''))) -> F(i(x''), i(x''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Rw
           →DP Problem 2
             ↳Rw
             ...
               →DP Problem 4
                 ↳Instantiation Transformation


Dependency Pairs:

F(i(x''), i(i(x''))) -> F(i(x''), i(x''))
F(x, x) -> F(i(x), i(i(x)))


Rules:


f(x, x) -> f(i(x), g(g(x)))
f(x, y) -> x
f(x, i(x)) -> f(x, x)
f(i(x), i(g(x))) -> a
g(x) -> i(x)


Strategy:

innermost




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

F(x, x) -> F(i(x), i(i(x)))
one new Dependency Pair is created:

F(i(x''''), i(x'''')) -> F(i(i(x'''')), i(i(i(x''''))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Rw
           →DP Problem 2
             ↳Rw
             ...
               →DP Problem 5
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

F(i(x''''), i(x'''')) -> F(i(i(x'''')), i(i(i(x''''))))
F(i(x''), i(i(x''))) -> F(i(x''), i(x''))


Rules:


f(x, x) -> f(i(x), g(g(x)))
f(x, y) -> x
f(x, i(x)) -> f(x, x)
f(i(x), i(g(x))) -> a
g(x) -> i(x)


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:00 minutes