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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(g(x), a) -> F(x, g(a))
F(g(x), g(y)) -> H(g(y), x, g(y))
H(g(x), y, z) -> F(y, h(x, y, z))
H(g(x), y, z) -> H(x, y, z)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering


Dependency Pairs:

H(g(x), y, z) -> H(x, y, z)
H(g(x), y, z) -> F(y, h(x, y, z))
F(g(x), g(y)) -> H(g(y), x, g(y))
F(g(x), a) -> F(x, g(a))


Rules:


f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


The following usable rules for innermost can be oriented:

h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z
f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
a > g

resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2) -> x2
g(x1) -> g
H(x1, x2, x3) -> x3
h(x1, x2, x3) -> x3
f(x1, x2) -> x2


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


Dependency Pairs:

H(g(x), y, z) -> H(x, y, z)
H(g(x), y, z) -> F(y, h(x, y, z))
F(g(x), g(y)) -> H(g(y), x, g(y))


Rules:


f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z


Strategy:

innermost




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

H(g(x), y, z) -> F(y, h(x, y, z))
two new Dependency Pairs are created:

H(g(g(x'')), y'', z'') -> F(y'', f(y'', h(x'', y'', z'')))
H(g(a), y'', z'') -> F(y'', z'')

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

H(g(a), y'', z'') -> F(y'', z'')
F(g(x), g(y)) -> H(g(y), x, g(y))
H(g(g(x'')), y'', z'') -> F(y'', f(y'', h(x'', y'', z'')))
H(g(x), y, z) -> H(x, y, z)


Rules:


f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z


Strategy:

innermost




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

H(g(x), y, z) -> H(x, y, z)
three new Dependency Pairs are created:

H(g(g(x'')), y'', z'') -> H(g(x''), y'', z'')
H(g(g(g(x''''))), y', z') -> H(g(g(x'''')), y', z')
H(g(g(a)), y', z') -> H(g(a), y', z')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 4
                 ↳Forward Instantiation Transformation


Dependency Pairs:

H(g(g(a)), y', z') -> H(g(a), y', z')
H(g(g(g(x''''))), y', z') -> H(g(g(x'''')), y', z')
H(g(g(x'')), y'', z'') -> H(g(x''), y'', z'')
H(g(g(x'')), y'', z'') -> F(y'', f(y'', h(x'', y'', z'')))
F(g(x), g(y)) -> H(g(y), x, g(y))
H(g(a), y'', z'') -> F(y'', z'')


Rules:


f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z


Strategy:

innermost




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

F(g(x), g(y)) -> H(g(y), x, g(y))
four new Dependency Pairs are created:

F(g(x'), g(g(x''''))) -> H(g(g(x'''')), x', g(g(x'''')))
F(g(x'), g(a)) -> H(g(a), x', g(a))
F(g(x'), g(g(g(x'''''')))) -> H(g(g(g(x''''''))), x', g(g(g(x''''''))))
F(g(x'), g(g(a))) -> H(g(g(a)), x', g(g(a)))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 5
                 ↳Forward Instantiation Transformation


Dependency Pairs:

F(g(x'), g(g(a))) -> H(g(g(a)), x', g(g(a)))
H(g(g(g(x''''))), y', z') -> H(g(g(x'''')), y', z')
H(g(g(x'')), y'', z'') -> H(g(x''), y'', z'')
F(g(x'), g(g(g(x'''''')))) -> H(g(g(g(x''''''))), x', g(g(g(x''''''))))
F(g(x'), g(a)) -> H(g(a), x', g(a))
H(g(g(x'')), y'', z'') -> F(y'', f(y'', h(x'', y'', z'')))
F(g(x'), g(g(x''''))) -> H(g(g(x'''')), x', g(g(x'''')))
H(g(a), y'', z'') -> F(y'', z'')
H(g(g(a)), y', z') -> H(g(a), y', z')


Rules:


f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z


Strategy:

innermost




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

H(g(a), y'', z'') -> F(y'', z'')
four new Dependency Pairs are created:

H(g(a), g(x'''), g(g(x''''''))) -> F(g(x'''), g(g(x'''''')))
H(g(a), g(x'''), g(a)) -> F(g(x'''), g(a))
H(g(a), g(x'''), g(g(g(x'''''''')))) -> F(g(x'''), g(g(g(x''''''''))))
H(g(a), g(x'''), g(g(a))) -> F(g(x'''), g(g(a)))

The transformation is resulting in two new DP problems: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 6
                 ↳Argument Filtering and Ordering


Dependency Pairs:

F(g(x'), g(a)) -> H(g(a), x', g(a))
H(g(a), g(x'''), g(a)) -> F(g(x'''), g(a))


Rules:


f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(g(x'), g(a)) -> H(g(a), x', g(a))


There are no usable rules for innermost that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
H(x1, x2, x3) -> x2
g(x1) -> g(x1)
F(x1, x2) -> x1


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 8
                 ↳Dependency Graph


Dependency Pair:

H(g(a), g(x'''), g(a)) -> F(g(x'''), g(a))


Rules:


f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 7
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

H(g(a), g(x'''), g(g(a))) -> F(g(x'''), g(g(a)))
H(g(a), g(x'''), g(g(g(x'''''''')))) -> F(g(x'''), g(g(g(x''''''''))))
F(g(x'), g(g(g(x'''''')))) -> H(g(g(g(x''''''))), x', g(g(g(x''''''))))
H(g(a), g(x'''), g(g(x''''''))) -> F(g(x'''), g(g(x'''''')))
H(g(g(a)), y', z') -> H(g(a), y', z')
H(g(g(g(x''''))), y', z') -> H(g(g(x'''')), y', z')
H(g(g(x'')), y'', z'') -> H(g(x''), y'', z'')
F(g(x'), g(g(x''''))) -> H(g(g(x'''')), x', g(g(x'''')))
H(g(g(x'')), y'', z'') -> F(y'', f(y'', h(x'', y'', z'')))
F(g(x'), g(g(a))) -> H(g(g(a)), x', g(g(a)))


Rules:


f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:03 minutes