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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(x, h(y)) -> F(h(x), y)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Instantiation Transformation


Dependency Pair:

F(x, h(y)) -> F(h(x), y)


Rules:


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


Strategy:

innermost




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

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

F(h(x''), h(y'')) -> F(h(h(x'')), y'')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Instantiation Transformation


Dependency Pair:

F(h(x''), h(y'')) -> F(h(h(x'')), y'')


Rules:


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


Strategy:

innermost




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

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

F(h(h(x'''')), h(y'''')) -> F(h(h(h(x''''))), y'''')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 3
                 ↳Argument Filtering and Ordering


Dependency Pair:

F(h(h(x'''')), h(y'''')) -> F(h(h(h(x''''))), y'''')


Rules:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(h(h(x'''')), h(y'''')) -> F(h(h(h(x''''))), y'''')


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:
F(x1, x2) -> x2
h(x1) -> h(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 4
                 ↳Dependency Graph


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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