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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

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


Rule:


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


Strategy:

innermost




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

F(f(f(f(j, a), b), c), d) -> F(f(a, b), f(f(a, d), c))
three new Dependency Pairs are created:

F(f(f(f(j, f(f(f(j, a''), b'''), c'')), b''), c), d) -> F(f(f(a'', b'''), f(f(a'', b''), c'')), f(f(f(f(f(j, a''), b'''), c''), d), c))
F(f(f(f(j, f(f(j, a''), b'')), b), c''), d'') -> F(f(f(f(j, a''), b''), b), f(f(a'', b''), f(f(a'', c''), d'')))
F(f(f(f(j, f(f(f(j, a''), b''), c'')), b), c), d'') -> F(f(f(f(f(j, a''), b''), c''), b), f(f(f(a'', b''), f(f(a'', d''), c'')), c))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(f(f(f(j, f(f(f(j, a''), b''), c'')), b), c), d'') -> F(f(f(f(f(j, a''), b''), c''), b), f(f(f(a'', b''), f(f(a'', d''), c'')), c))
F(f(f(f(j, f(f(j, a''), b'')), b), c''), d'') -> F(f(f(f(j, a''), b''), b), f(f(a'', b''), f(f(a'', c''), d'')))
F(f(f(f(j, f(f(f(j, a''), b'''), c'')), b''), c), d) -> F(f(f(a'', b'''), f(f(a'', b''), c'')), f(f(f(f(f(j, a''), b'''), c''), d), c))
F(f(f(f(j, a), b), c), d) -> F(f(a, d), c)
F(f(f(f(j, a), b), c), d) -> F(a, b)
F(f(f(f(j, a), b), c), d) -> F(a, d)


Rule:


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


Strategy:

innermost




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

F(f(f(f(j, f(f(f(j, a''), b'''), c'')), b''), c), d) -> F(f(f(a'', b'''), f(f(a'', b''), c'')), f(f(f(f(f(j, a''), b'''), c''), d), c))
one new Dependency Pair is created:

F(f(f(f(j, f(f(f(j, a''), b'''), c'')), b''), c), d) -> F(f(f(a'', b'''), f(f(a'', b''), c'')), f(f(f(a'', b'''), f(f(a'', d), c'')), c))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(f(f(f(j, f(f(f(j, a''), b'''), c'')), b''), c), d) -> F(f(f(a'', b'''), f(f(a'', b''), c'')), f(f(f(a'', b'''), f(f(a'', d), c'')), c))
F(f(f(f(j, f(f(j, a''), b'')), b), c''), d'') -> F(f(f(f(j, a''), b''), b), f(f(a'', b''), f(f(a'', c''), d'')))
F(f(f(f(j, a), b), c), d) -> F(a, d)
F(f(f(f(j, a), b), c), d) -> F(f(a, d), c)
F(f(f(f(j, a), b), c), d) -> F(a, b)
F(f(f(f(j, f(f(f(j, a''), b''), c'')), b), c), d'') -> F(f(f(f(f(j, a''), b''), c''), b), f(f(f(a'', b''), f(f(a'', d''), c'')), c))


Rule:


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


Strategy:

innermost




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

F(f(f(f(j, f(f(f(j, a''), b''), c'')), b), c), d'') -> F(f(f(f(f(j, a''), b''), c''), b), f(f(f(a'', b''), f(f(a'', d''), c'')), c))
one new Dependency Pair is created:

F(f(f(f(j, f(f(f(j, a''), b''), c'')), b), c), d'') -> F(f(f(a'', b''), f(f(a'', b), c'')), f(f(f(a'', b''), f(f(a'', d''), c'')), c))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Rw
             ...
               →DP Problem 4
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

F(f(f(f(j, f(f(f(j, a''), b''), c'')), b), c), d'') -> F(f(f(a'', b''), f(f(a'', b), c'')), f(f(f(a'', b''), f(f(a'', d''), c'')), c))
F(f(f(f(j, f(f(j, a''), b'')), b), c''), d'') -> F(f(f(f(j, a''), b''), b), f(f(a'', b''), f(f(a'', c''), d'')))
F(f(f(f(j, a), b), c), d) -> F(a, d)
F(f(f(f(j, a), b), c), d) -> F(f(a, d), c)
F(f(f(f(j, a), b), c), d) -> F(a, b)
F(f(f(f(j, f(f(f(j, a''), b'''), c'')), b''), c), d) -> F(f(f(a'', b'''), f(f(a'', b''), c'')), f(f(f(a'', b'''), f(f(a'', d), c'')), c))


Rule:


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


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:02 minutes