Term Rewriting System R:
[X, Y]
f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

F(X) -> IF(X, c, nf(ntrue))
IF(false, X, Y) -> ACTIVATE(Y)
ACTIVATE(nf(X)) -> F(activate(X))
ACTIVATE(nf(X)) -> ACTIVATE(X)
ACTIVATE(ntrue) -> TRUE

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Narrowing Transformation


Dependency Pairs:

ACTIVATE(nf(X)) -> ACTIVATE(X)
ACTIVATE(nf(X)) -> F(activate(X))
IF(false, X, Y) -> ACTIVATE(Y)
F(X) -> IF(X, c, nf(ntrue))


Rules:


f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X


Strategy:

innermost




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

ACTIVATE(nf(X)) -> F(activate(X))
three new Dependency Pairs are created:

ACTIVATE(nf(nf(X''))) -> F(f(activate(X'')))
ACTIVATE(nf(ntrue)) -> F(true)
ACTIVATE(nf(X'')) -> F(X'')

The transformation is resulting in one new DP problem:



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


Dependency Pairs:

ACTIVATE(nf(X'')) -> F(X'')
ACTIVATE(nf(ntrue)) -> F(true)
IF(false, X, Y) -> ACTIVATE(Y)
F(X) -> IF(X, c, nf(ntrue))
ACTIVATE(nf(nf(X''))) -> F(f(activate(X'')))
ACTIVATE(nf(X)) -> ACTIVATE(X)


Rules:


f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X


Strategy:

innermost




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

ACTIVATE(nf(ntrue)) -> F(true)
one new Dependency Pair is created:

ACTIVATE(nf(ntrue)) -> F(ntrue)

The transformation is resulting in one new DP problem:



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


Dependency Pairs:

ACTIVATE(nf(ntrue)) -> F(ntrue)
ACTIVATE(nf(nf(X''))) -> F(f(activate(X'')))
ACTIVATE(nf(X)) -> ACTIVATE(X)
IF(false, X, Y) -> ACTIVATE(Y)
F(X) -> IF(X, c, nf(ntrue))
ACTIVATE(nf(X'')) -> F(X'')


Rules:


f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X


Strategy:

innermost




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

IF(false, X, Y) -> ACTIVATE(Y)
one new Dependency Pair is created:

IF(false, c, nf(ntrue)) -> ACTIVATE(nf(ntrue))

The transformation is resulting in one new DP problem:



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


Dependency Pairs:

ACTIVATE(nf(X'')) -> F(X'')
ACTIVATE(nf(nf(X''))) -> F(f(activate(X'')))
ACTIVATE(nf(X)) -> ACTIVATE(X)
IF(false, c, nf(ntrue)) -> ACTIVATE(nf(ntrue))
F(X) -> IF(X, c, nf(ntrue))
ACTIVATE(nf(ntrue)) -> F(ntrue)


Rules:


f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
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(X) -> IF(X, c, nf(ntrue))
one new Dependency Pair is created:

F(false) -> IF(false, c, nf(ntrue))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Rw
             ...
               →DP Problem 5
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(nf(nf(X''))) -> F(f(activate(X'')))
ACTIVATE(nf(X)) -> ACTIVATE(X)
IF(false, c, nf(ntrue)) -> ACTIVATE(nf(ntrue))
F(false) -> IF(false, c, nf(ntrue))
ACTIVATE(nf(X'')) -> F(X'')


Rules:


f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
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(X)) -> ACTIVATE(X)
three new Dependency Pairs are created:

ACTIVATE(nf(nf(X''))) -> ACTIVATE(nf(X''))
ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(nf(nf(X''''))) -> ACTIVATE(nf(X''''))

The transformation is resulting in two new DP problems:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Rw
             ...
               →DP Problem 6
Instantiation Transformation


Dependency Pairs:

ACTIVATE(nf(X'')) -> F(X'')
IF(false, c, nf(ntrue)) -> ACTIVATE(nf(ntrue))
F(false) -> IF(false, c, nf(ntrue))


Rules:


f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X


Strategy:

innermost




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

ACTIVATE(nf(X'')) -> F(X'')
one new Dependency Pair is created:

ACTIVATE(nf(ntrue)) -> F(ntrue)

The transformation is resulting in no new DP problems.



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Rw
             ...
               →DP Problem 7
Argument Filtering and Ordering


Dependency Pairs:

ACTIVATE(nf(nf(X''''))) -> ACTIVATE(nf(X''''))
ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(nf(nf(X''))) -> ACTIVATE(nf(X''))


Rules:


f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X


Strategy:

innermost




The following dependency pairs can be strictly oriented:

ACTIVATE(nf(nf(X''''))) -> ACTIVATE(nf(X''''))
ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(nf(nf(X''))) -> ACTIVATE(nf(X''))


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
ACTIVATE(x1) -> ACTIVATE(x1)
nf(x1) -> nf(x1)


   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Rw
             ...
               →DP Problem 8
Dependency Graph


Dependency Pair:


Rules:


f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:02 minutes