Term Rewriting System R:
[X, Y, X1, X2]
f(g(X), Y) -> f(X, nf(g(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
activate(nf(X1, X2)) -> f(X1, X2)
activate(X) -> X

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(g(X), Y) -> F(X, nf(g(X), activate(Y)))
F(g(X), Y) -> ACTIVATE(Y)
ACTIVATE(nf(X1, X2)) -> F(X1, X2)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

ACTIVATE(nf(X1, X2)) -> F(X1, X2)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(g(X), activate(Y)))


Rules:


f(g(X), Y) -> f(X, nf(g(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
activate(nf(X1, X2)) -> f(X1, X2)
activate(X) -> X


Strategy:

innermost




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

F(g(X), Y) -> F(X, nf(g(X), activate(Y)))
two new Dependency Pairs are created:

F(g(X), nf(X1', X2')) -> F(X, nf(g(X), f(X1', X2')))
F(g(X), Y') -> F(X, nf(g(X), Y'))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Forward Instantiation Transformation


Dependency Pairs:

F(g(X), Y') -> F(X, nf(g(X), Y'))
F(g(X), nf(X1', X2')) -> F(X, nf(g(X), f(X1', X2')))
F(g(X), Y) -> ACTIVATE(Y)
ACTIVATE(nf(X1, X2)) -> F(X1, X2)


Rules:


f(g(X), Y) -> f(X, nf(g(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
activate(nf(X1, X2)) -> f(X1, X2)
activate(X) -> X


Strategy:

innermost




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

F(g(X), Y) -> ACTIVATE(Y)
one new Dependency Pair is created:

F(g(X), nf(X1'', X2'')) -> ACTIVATE(nf(X1'', X2''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 3
                 ↳Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(nf(X1, X2)) -> F(X1, X2)
F(g(X), nf(X1'', X2'')) -> ACTIVATE(nf(X1'', X2''))
F(g(X), nf(X1', X2')) -> F(X, nf(g(X), f(X1', X2')))
F(g(X), Y') -> F(X, nf(g(X), Y'))


Rules:


f(g(X), Y) -> f(X, nf(g(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
activate(nf(X1, X2)) -> f(X1, X2)
activate(X) -> X


Strategy:

innermost




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

ACTIVATE(nf(X1, X2)) -> F(X1, X2)
three new Dependency Pairs are created:

ACTIVATE(nf(g(X''), nf(X1''', X2'''))) -> F(g(X''), nf(X1''', X2'''))
ACTIVATE(nf(g(X''), X2')) -> F(g(X''), X2')
ACTIVATE(nf(g(X''), nf(X1'''', X2''''))) -> F(g(X''), nf(X1'''', X2''''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 4
                 ↳Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(nf(g(X''), nf(X1'''', X2''''))) -> F(g(X''), nf(X1'''', X2''''))
ACTIVATE(nf(g(X''), X2')) -> F(g(X''), X2')
F(g(X), Y') -> F(X, nf(g(X), Y'))
F(g(X), nf(X1', X2')) -> F(X, nf(g(X), f(X1', X2')))
ACTIVATE(nf(g(X''), nf(X1''', X2'''))) -> F(g(X''), nf(X1''', X2'''))
F(g(X), nf(X1'', X2'')) -> ACTIVATE(nf(X1'', X2''))


Rules:


f(g(X), Y) -> f(X, nf(g(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
activate(nf(X1, X2)) -> f(X1, X2)
activate(X) -> X


Strategy:

innermost




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

F(g(X), Y') -> F(X, nf(g(X), Y'))
two new Dependency Pairs are created:

F(g(g(X'')), Y'') -> F(g(X''), nf(g(g(X'')), Y''))
F(g(g(X'')), Y''') -> F(g(X''), nf(g(g(X'')), Y'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 5
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

ACTIVATE(nf(g(X''), X2')) -> F(g(X''), X2')
F(g(g(X'')), Y''') -> F(g(X''), nf(g(g(X'')), Y'''))
F(g(g(X'')), Y'') -> F(g(X''), nf(g(g(X'')), Y''))
ACTIVATE(nf(g(X''), nf(X1''', X2'''))) -> F(g(X''), nf(X1''', X2'''))
F(g(X), nf(X1'', X2'')) -> ACTIVATE(nf(X1'', X2''))
F(g(X), nf(X1', X2')) -> F(X, nf(g(X), f(X1', X2')))
ACTIVATE(nf(g(X''), nf(X1'''', X2''''))) -> F(g(X''), nf(X1'''', X2''''))


Rules:


f(g(X), Y) -> f(X, nf(g(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
activate(nf(X1, X2)) -> f(X1, X2)
activate(X) -> X


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:01 minutes