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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(f(x)) -> F(g(f(x), x))
F(f(x)) -> G(f(x), x)
F(f(x)) -> F(h(f(x), f(x)))
F(f(x)) -> H(f(x), f(x))
H(x, x) -> G(x, 0)

Furthermore, R contains one SCC. 


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


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




As we are in the innermost case, we can delete all 2 non-usable-rules.


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


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

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

F(f(x)) -> F(g(f(x), 0))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(f(x)) -> F(g(f(x), 0))
F(f(x)) -> F(g(f(x), x))


Rules:


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


Strategy:

innermost




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

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

F(f(x)) -> F(x)

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(f(x)) -> F(x)
F(f(x)) -> F(g(f(x), 0))


Rules:


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


Strategy:

innermost




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

F(f(x)) -> F(g(f(x), 0))
one new Dependency Pair is created:

F(f(x)) -> F(0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 2
             ↳Rw
             ...
               →DP Problem 5
                 ↳Usable Rules (Innermost)


Dependency Pair:

F(f(x)) -> F(x)


Rules:


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


Strategy:

innermost




As we are in the innermost case, we can delete all 2 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 2
             ↳Rw
             ...
               →DP Problem 6
                 ↳Size-Change Principle


Dependency Pair:

F(f(x)) -> F(x)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. F(f(x)) -> F(x)
and get the following Size-Change Graph(s):
{1} , {1}
1>1

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
f(x1) -> f(x1)

We obtain no new DP problems.

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