Term Rewriting System R:
[x, y]
f(0, 1, x) -> f(h(x), h(x), x)
h(0) -> 0
h(g(x, y)) -> y

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(0, 1, x) -> F(h(x), h(x), x)
F(0, 1, x) -> H(x)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Usable Rules (Innermost)


Dependency Pair:

F(0, 1, x) -> F(h(x), h(x), x)


Rules:


f(0, 1, x) -> f(h(x), h(x), x)
h(0) -> 0
h(g(x, y)) -> y


Strategy:

innermost




As we are in the innermost case, we can delete all 1 non-usable-rules.


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


Dependency Pair:

F(0, 1, x) -> F(h(x), h(x), x)


Rules:


h(0) -> 0
h(g(x, y)) -> y


Strategy:

innermost




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

F(0, 1, x) -> F(h(x), h(x), x)
four new Dependency Pairs are created:

F(0, 1, 0) -> F(0, h(0), 0)
F(0, 1, g(x'', y')) -> F(y', h(g(x'', y')), g(x'', y'))
F(0, 1, 0) -> F(h(0), 0, 0)
F(0, 1, g(x'', y')) -> F(h(g(x'', y')), y', g(x'', y'))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(0, 1, g(x'', y')) -> F(h(g(x'', y')), y', g(x'', y'))
F(0, 1, g(x'', y')) -> F(y', h(g(x'', y')), g(x'', y'))


Rules:


h(0) -> 0
h(g(x, y)) -> y


Strategy:

innermost




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

F(0, 1, g(x'', y')) -> F(y', h(g(x'', y')), g(x'', y'))
one new Dependency Pair is created:

F(0, 1, g(x'', y')) -> F(y', y', g(x'', y'))

The transformation is resulting in one new DP problem: 



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


Dependency Pair:

F(0, 1, g(x'', y')) -> F(h(g(x'', y')), y', g(x'', y'))


Rules:


h(0) -> 0
h(g(x, y)) -> y


Strategy:

innermost




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

F(0, 1, g(x'', y')) -> F(h(g(x'', y')), y', g(x'', y'))
one new Dependency Pair is created:

F(0, 1, g(x'', y')) -> F(y', y', g(x'', y'))

The transformation is resulting in no new DP problems.


Innermost Termination of R successfully shown.
Duration:
0:20 minutes