Termination Proof using AProVE

Term Rewriting System R:
[y, x, xs, f]
app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(times, 0), y) -> 0
app(app(times, app(s, x)), y) -> app(app(plus, app(app(times, x), y)), y)
app(inc, xs) -> app(app(map, app(plus, app(s, 0))), xs)
app(double, xs) -> app(app(map, app(times, app(s, app(s, 0)))), xs)
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

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(times, app(s, x)), y) -> APP(app(plus, app(app(times, x), y)), y)
APP(app(times, app(s, x)), y) -> APP(plus, app(app(times, x), y))
APP(app(times, app(s, x)), y) -> APP(app(times, x), y)
APP(app(times, app(s, x)), y) -> APP(times, x)
APP(inc, xs) -> APP(app(map, app(plus, app(s, 0))), xs)
APP(inc, xs) -> APP(map, app(plus, app(s, 0)))
APP(inc, xs) -> APP(plus, app(s, 0))
APP(inc, xs) -> APP(s, 0)
APP(double, xs) -> APP(app(map, app(times, app(s, app(s, 0)))), xs)
APP(double, xs) -> APP(map, app(times, app(s, app(s, 0))))
APP(double, xs) -> APP(times, app(s, app(s, 0)))
APP(double, xs) -> APP(s, app(s, 0))
APP(double, xs) -> APP(s, 0)
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(app(map, f), app(app(cons, x), xs)) -> APP(cons, app(f, x))
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(double, xs) -> APP(app(map, app(times, app(s, app(s, 0)))), xs)
APP(inc, xs) -> APP(app(map, app(plus, app(s, 0))), xs)
APP(app(times, app(s, x)), y) -> APP(app(times, x), y)
APP(app(times, app(s, x)), y) -> APP(app(plus, app(app(times, x), y)), y)
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)

Rules:

app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(times, 0), y) -> 0
app(app(times, app(s, x)), y) -> app(app(plus, app(app(times, x), y)), y)
app(inc, xs) -> app(app(map, app(plus, app(s, 0))), xs)
app(double, xs) -> app(app(map, app(times, app(s, app(s, 0)))), xs)
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))

Strategy:

innermost

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

APP(app(times, app(s, x)), y) -> APP(app(plus, app(app(times, x), y)), y)

two new Dependency Pairs are created:

APP(app(times, app(s, 0)), y'') -> APP(app(plus, 0), y'')
APP(app(times, app(s, app(s, x''))), y'') -> APP(app(plus, app(app(plus, app(app(times, x''), y'')), y'')), y'')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

APP(app(times, app(s, app(s, x''))), y'') -> APP(app(plus, app(app(plus, app(app(times, x''), y'')), y'')), y'')
APP(app(times, app(s, 0)), y'') -> APP(app(plus, 0), y'')
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(double, xs) -> APP(app(map, app(times, app(s, app(s, 0)))), xs)
APP(inc, xs) -> APP(app(map, app(plus, app(s, 0))), xs)
APP(app(times, app(s, x)), y) -> APP(app(times, x), y)
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)

Rules:

app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(times, 0), y) -> 0
app(app(times, app(s, x)), y) -> app(app(plus, app(app(times, x), y)), y)
app(inc, xs) -> app(app(map, app(plus, app(s, 0))), xs)
app(double, xs) -> app(app(map, app(times, app(s, app(s, 0)))), xs)
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))

Strategy:

innermost

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

APP(app(map, f), app(app(cons, x), xs)) -> APP(app(cons, app(f, x)), app(app(map, f), xs))

10 new Dependency Pairs are created:

APP(app(map, app(plus, 0)), app(app(cons, x'), xs)) -> APP(app(cons, x'), app(app(map, app(plus, 0)), xs))
APP(app(map, app(plus, app(s, x''))), app(app(cons, x0), xs)) -> APP(app(cons, app(s, app(app(plus, x''), x0))), app(app(map, app(plus, app(s, x''))), xs))
APP(app(map, app(times, 0)), app(app(cons, x'), xs)) -> APP(app(cons, 0), app(app(map, app(times, 0)), xs))
APP(app(map, app(times, app(s, x''))), app(app(cons, x0), xs)) -> APP(app(cons, app(app(plus, app(app(times, x''), x0)), x0)), app(app(map, app(times, app(s, x''))), xs))
APP(app(map, inc), app(app(cons, x'), xs)) -> APP(app(cons, app(app(map, app(plus, app(s, 0))), x')), app(app(map, inc), xs))
APP(app(map, double), app(app(cons, x'), xs)) -> APP(app(cons, app(app(map, app(times, app(s, app(s, 0)))), x')), app(app(map, double), xs))
APP(app(map, app(map, f'')), app(app(cons, nil), xs)) -> APP(app(cons, nil), app(app(map, app(map, f'')), xs))
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(map, f''), xs''))), app(app(map, app(map, f'')), xs))
APP(app(map, f''), app(app(cons, x), nil)) -> APP(app(cons, app(f'', x)), nil)
APP(app(map, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(f'', x)), app(app(cons, app(f'', x'')), app(app(map, f''), xs'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 3

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(map, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(f'', x)), app(app(cons, app(f'', x'')), app(app(map, f''), xs'')))
APP(app(map, f''), app(app(cons, x), nil)) -> APP(app(cons, app(f'', x)), nil)
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(map, f''), xs''))), app(app(map, app(map, f'')), xs))
APP(app(map, double), app(app(cons, x'), xs)) -> APP(app(cons, app(app(map, app(times, app(s, app(s, 0)))), x')), app(app(map, double), xs))
APP(app(map, inc), app(app(cons, x'), xs)) -> APP(app(cons, app(app(map, app(plus, app(s, 0))), x')), app(app(map, inc), xs))
APP(app(map, app(times, app(s, x''))), app(app(cons, x0), xs)) -> APP(app(cons, app(app(plus, app(app(times, x''), x0)), x0)), app(app(map, app(times, app(s, x''))), xs))
APP(app(map, app(times, 0)), app(app(cons, x'), xs)) -> APP(app(cons, 0), app(app(map, app(times, 0)), xs))
APP(app(times, app(s, 0)), y'') -> APP(app(plus, 0), y'')
APP(app(map, app(plus, app(s, x''))), app(app(cons, x0), xs)) -> APP(app(cons, app(s, app(app(plus, x''), x0))), app(app(map, app(plus, app(s, x''))), xs))
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(double, xs) -> APP(app(map, app(times, app(s, app(s, 0)))), xs)
APP(inc, xs) -> APP(app(map, app(plus, app(s, 0))), xs)
APP(app(times, app(s, x)), y) -> APP(app(times, x), y)
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)
APP(app(times, app(s, app(s, x''))), y'') -> APP(app(plus, app(app(plus, app(app(times, x''), y'')), y'')), y'')

Rules:

app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(times, 0), y) -> 0
app(app(times, app(s, x)), y) -> app(app(plus, app(app(times, x), y)), y)
app(inc, xs) -> app(app(map, app(plus, app(s, 0))), xs)
app(double, xs) -> app(app(map, app(times, app(s, app(s, 0)))), xs)
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))

Strategy:

innermost

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

APP(app(times, app(s, 0)), y'') -> APP(app(plus, 0), y'')

no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 4

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

APP(app(map, f''), app(app(cons, x), nil)) -> APP(app(cons, app(f'', x)), nil)
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(map, f''), xs''))), app(app(map, app(map, f'')), xs))
APP(app(map, double), app(app(cons, x'), xs)) -> APP(app(cons, app(app(map, app(times, app(s, app(s, 0)))), x')), app(app(map, double), xs))
APP(app(map, inc), app(app(cons, x'), xs)) -> APP(app(cons, app(app(map, app(plus, app(s, 0))), x')), app(app(map, inc), xs))
APP(app(map, app(times, app(s, x''))), app(app(cons, x0), xs)) -> APP(app(cons, app(app(plus, app(app(times, x''), x0)), x0)), app(app(map, app(times, app(s, x''))), xs))
APP(app(map, app(times, 0)), app(app(cons, x'), xs)) -> APP(app(cons, 0), app(app(map, app(times, 0)), xs))
APP(app(times, app(s, app(s, x''))), y'') -> APP(app(plus, app(app(plus, app(app(times, x''), y'')), y'')), y'')
APP(app(map, app(plus, app(s, x''))), app(app(cons, x0), xs)) -> APP(app(cons, app(s, app(app(plus, x''), x0))), app(app(map, app(plus, app(s, x''))), xs))
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(double, xs) -> APP(app(map, app(times, app(s, app(s, 0)))), xs)
APP(inc, xs) -> APP(app(map, app(plus, app(s, 0))), xs)
APP(app(times, app(s, x)), y) -> APP(app(times, x), y)
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)
APP(app(map, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(f'', x)), app(app(cons, app(f'', x'')), app(app(map, f''), xs'')))

Rules:

app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(times, 0), y) -> 0
app(app(times, app(s, x)), y) -> app(app(plus, app(app(times, x), y)), y)
app(inc, xs) -> app(app(map, app(plus, app(s, 0))), xs)
app(double, xs) -> app(app(map, app(times, app(s, app(s, 0)))), xs)
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:20 minutes