Term Rewriting System R:
[x, y]
app(app(minus, x), 0) -> x
app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, x), y)
app(double, 0) -> 0
app(double, app(s, x)) -> app(s, app(s, app(double, x)))
app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(plus, app(s, x)), y) -> app(app(plus, x), app(s, y))
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, app(app(minus, x), y)), app(double, y)))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(minus, app(s, x)), app(s, y)) -> APP(app(minus, x), y)
APP(app(minus, app(s, x)), app(s, y)) -> APP(minus, x)
APP(double, app(s, x)) -> APP(s, app(s, app(double, x)))
APP(double, app(s, x)) -> APP(s, app(double, x))
APP(double, app(s, x)) -> APP(double, x)
APP(app(plus, app(s, x)), y) -> APP(s, app(app(plus, x), y))
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)
APP(app(plus, app(s, x)), y) -> APP(plus, x)
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), app(s, y))
APP(app(plus, app(s, x)), y) -> APP(s, y)
APP(app(plus, app(s, x)), y) -> APP(s, app(app(plus, app(app(minus, x), y)), app(double, y)))
APP(app(plus, app(s, x)), y) -> APP(app(plus, app(app(minus, x), y)), app(double, y))
APP(app(plus, app(s, x)), y) -> APP(plus, app(app(minus, x), y))
APP(app(plus, app(s, x)), y) -> APP(app(minus, x), y)
APP(app(plus, app(s, x)), y) -> APP(minus, x)
APP(app(plus, app(s, x)), y) -> APP(double, y)

Furthermore, R contains two SCCs. 


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


Dependency Pair:

APP(double, app(s, x)) -> APP(double, x)


Rules:


app(app(minus, x), 0) -> x
app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, x), y)
app(double, 0) -> 0
app(double, app(s, x)) -> app(s, app(s, app(double, x)))
app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(plus, app(s, x)), y) -> app(app(plus, x), app(s, y))
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, app(app(minus, x), y)), app(double, y)))


Strategy:

innermost




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

APP(double, app(s, x)) -> APP(double, x)
one new Dependency Pair is created:

APP(double, app(s, app(s, x''))) -> APP(double, app(s, x''))

The transformation is resulting in one new DP problem: 



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




The following remains to be proven:


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




The following remains to be proven:

Innermost Termination of R could not be shown.
Duration:
0:03 minutes