Term Rewriting System R:
[y, n, x]
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(add(n, x), y) -> APP(x, y)
REVERSE(add(n, x)) -> APP(reverse(x), add(n, nil))
REVERSE(add(n, x)) -> REVERSE(x)
SHUFFLE(add(n, x)) -> SHUFFLE(reverse(x))
SHUFFLE(add(n, x)) -> REVERSE(x)

Furthermore, R contains three SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳Nar


Dependency Pair:

APP(add(n, x), y) -> APP(x, y)


Rules:


app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))





The following dependency pair can be strictly oriented:

APP(add(n, x), y) -> APP(x, y)


There are no usable rules using the Ce-refinement 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:
APP(x1, x2) -> APP(x1, x2)
add(x1, x2) -> add(x1, x2)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 4
             ↳Dependency Graph
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳Nar


Dependency Pair:


Rules:


app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Argument Filtering and Ordering
       →DP Problem 3
         ↳Nar


Dependency Pair:

REVERSE(add(n, x)) -> REVERSE(x)


Rules:


app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))





The following dependency pair can be strictly oriented:

REVERSE(add(n, x)) -> REVERSE(x)


There are no usable rules using the Ce-refinement 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:
REVERSE(x1) -> REVERSE(x1)
add(x1, x2) -> add(x1, x2)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
           →DP Problem 5
             ↳Dependency Graph
       →DP Problem 3
         ↳Nar


Dependency Pair:


Rules:


app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳Narrowing Transformation


Dependency Pair:

SHUFFLE(add(n, x)) -> SHUFFLE(reverse(x))


Rules:


app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))





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

SHUFFLE(add(n, x)) -> SHUFFLE(reverse(x))
two new Dependency Pairs are created:

SHUFFLE(add(n, nil)) -> SHUFFLE(nil)
SHUFFLE(add(n, add(n'', x''))) -> SHUFFLE(app(reverse(x''), add(n'', nil)))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳Nar
           →DP Problem 6
             ↳Narrowing Transformation


Dependency Pair:

SHUFFLE(add(n, add(n'', x''))) -> SHUFFLE(app(reverse(x''), add(n'', nil)))


Rules:


app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))





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

SHUFFLE(add(n, add(n'', x''))) -> SHUFFLE(app(reverse(x''), add(n'', nil)))
two new Dependency Pairs are created:

SHUFFLE(add(n, add(n'', nil))) -> SHUFFLE(app(nil, add(n'', nil)))
SHUFFLE(add(n, add(n'', add(n''', x')))) -> SHUFFLE(app(app(reverse(x'), add(n''', nil)), add(n'', nil)))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳Nar
           →DP Problem 6
             ↳Nar
             ...
               →DP Problem 7
                 ↳Narrowing Transformation


Dependency Pairs:

SHUFFLE(add(n, add(n'', add(n''', x')))) -> SHUFFLE(app(app(reverse(x'), add(n''', nil)), add(n'', nil)))
SHUFFLE(add(n, add(n'', nil))) -> SHUFFLE(app(nil, add(n'', nil)))


Rules:


app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))





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

SHUFFLE(add(n, add(n'', nil))) -> SHUFFLE(app(nil, add(n'', nil)))
one new Dependency Pair is created:

SHUFFLE(add(n, add(n''', nil))) -> SHUFFLE(add(n''', nil))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳Nar
           →DP Problem 6
             ↳Nar
             ...
               →DP Problem 8
                 ↳Narrowing Transformation


Dependency Pair:

SHUFFLE(add(n, add(n'', add(n''', x')))) -> SHUFFLE(app(app(reverse(x'), add(n''', nil)), add(n'', nil)))


Rules:


app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))





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

SHUFFLE(add(n, add(n'', add(n''', x')))) -> SHUFFLE(app(app(reverse(x'), add(n''', nil)), add(n'', nil)))
two new Dependency Pairs are created:

SHUFFLE(add(n, add(n'', add(n''', nil)))) -> SHUFFLE(app(app(nil, add(n''', nil)), add(n'', nil)))
SHUFFLE(add(n, add(n'', add(n''', add(n'0, x''))))) -> SHUFFLE(app(app(app(reverse(x''), add(n'0, nil)), add(n''', nil)), add(n'', nil)))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳Nar
           →DP Problem 6
             ↳Nar
             ...
               →DP Problem 9
                 ↳Narrowing Transformation


Dependency Pairs:

SHUFFLE(add(n, add(n'', add(n''', add(n'0, x''))))) -> SHUFFLE(app(app(app(reverse(x''), add(n'0, nil)), add(n''', nil)), add(n'', nil)))
SHUFFLE(add(n, add(n'', add(n''', nil)))) -> SHUFFLE(app(app(nil, add(n''', nil)), add(n'', nil)))


Rules:


app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))





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

SHUFFLE(add(n, add(n'', add(n''', nil)))) -> SHUFFLE(app(app(nil, add(n''', nil)), add(n'', nil)))
one new Dependency Pair is created:

SHUFFLE(add(n, add(n'', add(n'''', nil)))) -> SHUFFLE(app(add(n'''', nil), add(n'', nil)))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳Nar
           →DP Problem 6
             ↳Nar
             ...
               →DP Problem 10
                 ↳Narrowing Transformation


Dependency Pairs:

SHUFFLE(add(n, add(n'', add(n'''', nil)))) -> SHUFFLE(app(add(n'''', nil), add(n'', nil)))
SHUFFLE(add(n, add(n'', add(n''', add(n'0, x''))))) -> SHUFFLE(app(app(app(reverse(x''), add(n'0, nil)), add(n''', nil)), add(n'', nil)))


Rules:


app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))





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

SHUFFLE(add(n, add(n'', add(n''', add(n'0, x''))))) -> SHUFFLE(app(app(app(reverse(x''), add(n'0, nil)), add(n''', nil)), add(n'', nil)))
two new Dependency Pairs are created:

SHUFFLE(add(n, add(n'', add(n''', add(n'0, nil))))) -> SHUFFLE(app(app(app(nil, add(n'0, nil)), add(n''', nil)), add(n'', nil)))
SHUFFLE(add(n, add(n'', add(n''', add(n'0, add(n'1, x')))))) -> SHUFFLE(app(app(app(app(reverse(x'), add(n'1, nil)), add(n'0, nil)), add(n''', nil)), add(n'', nil)))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳Nar
           →DP Problem 6
             ↳Nar
             ...
               →DP Problem 11
                 ↳Narrowing Transformation


Dependency Pairs:

SHUFFLE(add(n, add(n'', add(n''', add(n'0, add(n'1, x')))))) -> SHUFFLE(app(app(app(app(reverse(x'), add(n'1, nil)), add(n'0, nil)), add(n''', nil)), add(n'', nil)))
SHUFFLE(add(n, add(n'', add(n''', add(n'0, nil))))) -> SHUFFLE(app(app(app(nil, add(n'0, nil)), add(n''', nil)), add(n'', nil)))
SHUFFLE(add(n, add(n'', add(n'''', nil)))) -> SHUFFLE(app(add(n'''', nil), add(n'', nil)))


Rules:


app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))





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

SHUFFLE(add(n, add(n'', add(n'''', nil)))) -> SHUFFLE(app(add(n'''', nil), add(n'', nil)))
one new Dependency Pair is created:

SHUFFLE(add(n, add(n''', add(n''''', nil)))) -> SHUFFLE(add(n''''', app(nil, add(n''', nil))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳Nar
           →DP Problem 6
             ↳Nar
             ...
               →DP Problem 12
                 ↳Narrowing Transformation


Dependency Pairs:

SHUFFLE(add(n, add(n''', add(n''''', nil)))) -> SHUFFLE(add(n''''', app(nil, add(n''', nil))))
SHUFFLE(add(n, add(n'', add(n''', add(n'0, nil))))) -> SHUFFLE(app(app(app(nil, add(n'0, nil)), add(n''', nil)), add(n'', nil)))
SHUFFLE(add(n, add(n'', add(n''', add(n'0, add(n'1, x')))))) -> SHUFFLE(app(app(app(app(reverse(x'), add(n'1, nil)), add(n'0, nil)), add(n''', nil)), add(n'', nil)))


Rules:


app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))





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

SHUFFLE(add(n, add(n'', add(n''', add(n'0, nil))))) -> SHUFFLE(app(app(app(nil, add(n'0, nil)), add(n''', nil)), add(n'', nil)))
one new Dependency Pair is created:

SHUFFLE(add(n, add(n'', add(n''', add(n'0', nil))))) -> SHUFFLE(app(app(add(n'0', nil), add(n''', nil)), add(n'', nil)))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳Nar
           →DP Problem 6
             ↳Nar
             ...
               →DP Problem 13
                 ↳Narrowing Transformation


Dependency Pairs:

SHUFFLE(add(n, add(n'', add(n''', add(n'0', nil))))) -> SHUFFLE(app(app(add(n'0', nil), add(n''', nil)), add(n'', nil)))
SHUFFLE(add(n, add(n'', add(n''', add(n'0, add(n'1, x')))))) -> SHUFFLE(app(app(app(app(reverse(x'), add(n'1, nil)), add(n'0, nil)), add(n''', nil)), add(n'', nil)))
SHUFFLE(add(n, add(n''', add(n''''', nil)))) -> SHUFFLE(add(n''''', app(nil, add(n''', nil))))


Rules:


app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))





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

SHUFFLE(add(n, add(n'', add(n''', add(n'0, add(n'1, x')))))) -> SHUFFLE(app(app(app(app(reverse(x'), add(n'1, nil)), add(n'0, nil)), add(n''', nil)), add(n'', nil)))
two new Dependency Pairs are created:

SHUFFLE(add(n, add(n'', add(n''', add(n'0, add(n'1, nil)))))) -> SHUFFLE(app(app(app(app(nil, add(n'1, nil)), add(n'0, nil)), add(n''', nil)), add(n'', nil)))
SHUFFLE(add(n, add(n'', add(n''', add(n'0, add(n'1, add(n'2, x''))))))) -> SHUFFLE(app(app(app(app(app(reverse(x''), add(n'2, nil)), add(n'1, nil)), add(n'0, nil)), add(n''', nil)), add(n'', nil)))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳Nar
           →DP Problem 6
             ↳Nar
             ...
               →DP Problem 14
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

SHUFFLE(add(n, add(n'', add(n''', add(n'0, add(n'1, add(n'2, x''))))))) -> SHUFFLE(app(app(app(app(app(reverse(x''), add(n'2, nil)), add(n'1, nil)), add(n'0, nil)), add(n''', nil)), add(n'', nil)))
SHUFFLE(add(n, add(n'', add(n''', add(n'0, add(n'1, nil)))))) -> SHUFFLE(app(app(app(app(nil, add(n'1, nil)), add(n'0, nil)), add(n''', nil)), add(n'', nil)))
SHUFFLE(add(n, add(n''', add(n''''', nil)))) -> SHUFFLE(add(n''''', app(nil, add(n''', nil))))
SHUFFLE(add(n, add(n'', add(n''', add(n'0', nil))))) -> SHUFFLE(app(app(add(n'0', nil), add(n''', nil)), add(n'', nil)))


Rules:


app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))




Termination of R could not be shown.
Duration:
0:33 minutes