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
Instantiation Transformation


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




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

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

F(i(i(x'''''')), i(i(i(x'''''')))) -> F(i(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 6
Instantiation Transformation


Dependency Pairs:

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

F(i(i(x'''''''')), i(i(x''''''''))) -> F(i(i(i(x''''''''))), i(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 7
Instantiation Transformation


Dependency Pairs:

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

F(i(i(i(x''''''''''))), i(i(i(i(x''''''''''))))) -> F(i(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 8
Instantiation Transformation


Dependency Pairs:

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

F(i(i(i(x''''''''''''))), i(i(i(x'''''''''''')))) -> F(i(i(i(i(x'''''''''''')))), i(i(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 9
Instantiation Transformation


Dependency Pairs:

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

F(i(i(i(i(x'''''''''''''')))), i(i(i(i(i(x'''''''''''''')))))) -> F(i(i(i(i(x'''''''''''''')))), i(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 10
Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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



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