Term Rewriting System R:
[x]
f(a, x) -> f(b, f(c, x))
f(a, f(b, x)) -> f(b, f(a, x))
f(d, f(c, x)) -> f(d, f(a, x))
f(a, f(c, x)) -> f(c, f(a, x))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(a, x) -> F(b, f(c, x))
F(a, x) -> F(c, x)
F(a, f(b, x)) -> F(b, f(a, x))
F(a, f(b, x)) -> F(a, x)
F(d, f(c, x)) -> F(d, f(a, x))
F(d, f(c, x)) -> F(a, x)
F(a, f(c, x)) -> F(c, f(a, x))
F(a, f(c, x)) -> F(a, x)

Furthermore, R contains two SCCs. 


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


Dependency Pairs:

F(a, f(c, x)) -> F(a, x)
F(a, f(b, x)) -> F(a, x)


Rules:


f(a, x) -> f(b, f(c, x))
f(a, f(b, x)) -> f(b, f(a, x))
f(d, f(c, x)) -> f(d, f(a, x))
f(a, f(c, x)) -> f(c, f(a, x))


Strategy:

innermost




As we are in the innermost case, we can delete all 4 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 3
             ↳A-Transformation
       →DP Problem 2
         ↳UsableRules


Dependency Pairs:

F(a, f(c, x)) -> F(a, x)
F(a, f(b, x)) -> F(a, x)


Rule:

none


Strategy:

innermost




We have an applicative DP problem with proper arity. Thus we can use the A-Transformation to obtain one new DP problem which consists of the A-transformed TRSs.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 3
             ↳ATrans
             ...
               →DP Problem 4
                 ↳Size-Change Principle
       →DP Problem 2
         ↳UsableRules


Dependency Pairs:

A(c(x)) -> A(x)
A(b(x)) -> A(x)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. A(c(x)) -> A(x)
  2. A(b(x)) -> A(x)
and get the following Size-Change Graph(s):
{1, 2} , {1, 2}
1>1

which lead(s) to this/these maximal multigraph(s):
{1, 2} , {1, 2}
1>1

DP: empty set
Oriented Rules: none

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

with Argument Filtering System:
c(x1) -> c(x1)
b(x1) -> b(x1)

We obtain no new DP problems. 


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


Dependency Pair:

F(d, f(c, x)) -> F(d, f(a, x))


Rules:


f(a, x) -> f(b, f(c, x))
f(a, f(b, x)) -> f(b, f(a, x))
f(d, f(c, x)) -> f(d, f(a, x))
f(a, f(c, x)) -> f(c, f(a, x))


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
         ↳UsableRules
           →DP Problem 5
             ↳A-Transformation


Dependency Pair:

F(d, f(c, x)) -> F(d, f(a, x))


Rules:


f(a, f(b, x)) -> f(b, f(a, x))
f(a, x) -> f(b, f(c, x))
f(a, f(c, x)) -> f(c, f(a, x))


Strategy:

innermost




We have an applicative DP problem with proper arity. Thus we can use the A-Transformation to obtain one new DP problem which consists of the A-transformed TRSs.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
           →DP Problem 5
             ↳ATrans
             ...
               →DP Problem 6
                 ↳Negative Polynomial Order


Dependency Pair:

D(c(x)) -> D(a(x))


Rules:


a(b(x)) -> b(a(x))
a(x) -> b(c(x))
a(c(x)) -> c(a(x))


Strategy:

innermost




The following Dependency Pair can be strictly oriented using the given order.

D(c(x)) -> D(a(x))


Moreover, the following usable rules (regarding the implicit AFS) are oriented.

a(x) -> b(c(x))
a(b(x)) -> b(a(x))
a(c(x)) -> c(a(x))


Used ordering:
Polynomial Order with Interpretation:

POL( D(x1) ) = x1

POL( c(x1) ) = x1 + 1

POL( a(x1) ) = x1

POL( b(x1) ) = 0


This results in one new DP problem. 


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
           →DP Problem 5
             ↳ATrans
             ...
               →DP Problem 7
                 ↳Dependency Graph


Dependency Pair:


Rules:


a(b(x)) -> b(a(x))
a(x) -> b(c(x))
a(c(x)) -> c(a(x))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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