Termination Proof using AProVE

Term Rewriting System R:
[y, x, f, xs, g]
app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
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))
app(app(app(curry, g), x), y) -> app(app(g, x), y)
inc -> app(map, app(app(curry, plus), app(s, 0)))

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(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)
APP(app(app(curry, g), x), y) -> APP(app(g, x), y)
APP(app(app(curry, g), x), y) -> APP(g, x)
INC -> APP(map, app(app(curry, plus), app(s, 0)))
INC -> APP(app(curry, plus), app(s, 0))
INC -> APP(curry, plus)
INC -> APP(s, 0)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

APP(app(app(curry, g), x), y) -> APP(g, x)
APP(app(app(curry, g), x), y) -> APP(app(g, x), y)
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(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(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(app(curry, g), x), y) -> app(app(g, x), y)
inc -> app(map, app(app(curry, plus), app(s, 0)))

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))

seven 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(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, app(app(curry, g'), x'')), app(app(cons, x0), xs)) -> APP(app(cons, app(app(g', x''), x0)), app(app(map, app(app(curry, g'), x'')), 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

             ↳Forward Instantiation 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(app(curry, g'), x'')), app(app(cons, x0), xs)) -> APP(app(cons, app(app(g', x''), x0)), app(app(map, app(app(curry, g'), x'')), 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(app(curry, g), x), y) -> APP(app(g, x), 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(app(plus, app(s, x)), y) -> APP(app(plus, x), y)
APP(app(app(curry, g), x), y) -> APP(g, x)

Rules:

app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
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))
app(app(app(curry, g), x), y) -> app(app(g, x), y)
inc -> app(map, app(app(curry, plus), app(s, 0)))

Strategy:

innermost

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

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

eight new Dependency Pairs are created:

APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(map, f''), app(app(cons, x''), xs''))
APP(app(map, app(plus, app(s, x''))), app(app(cons, x0), xs)) -> APP(app(plus, app(s, x'')), x0)
APP(app(map, app(app(curry, g''), x'')), app(app(cons, x0), xs)) -> APP(app(app(curry, g''), x''), x0)
APP(app(map, app(map, app(plus, app(s, x'''')))), app(app(cons, app(app(cons, x0''), xs'')), xs)) -> APP(app(map, app(plus, app(s, x''''))), app(app(cons, x0''), xs''))
APP(app(map, app(map, app(map, f''''))), app(app(cons, app(app(cons, app(app(cons, x''''), xs'''')), xs'')), xs)) -> APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, x''''), xs'''')), xs''))
APP(app(map, app(map, app(app(curry, g'''), x''''))), app(app(cons, app(app(cons, x0''), xs'')), xs)) -> APP(app(map, app(app(curry, g'''), x'''')), app(app(cons, x0''), xs''))
APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, x''), nil)), xs)) -> APP(app(map, f''''), app(app(cons, x''), nil))
APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, x''), app(app(cons, x''''), xs''''))), xs)) -> APP(app(map, f''''), app(app(cons, x''), app(app(cons, x''''), xs'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 3

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Strategy:

innermost

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

APP(app(app(curry, g), x), y) -> APP(g, x)

14 new Dependency Pairs are created:

APP(app(app(curry, app(app(curry, g''), x'')), x0), y) -> APP(app(app(curry, g''), x''), x0)
APP(app(app(curry, app(plus, app(s, x''))), x0), y) -> APP(app(plus, app(s, x'')), x0)
APP(app(app(curry, app(map, f'')), app(app(cons, x''), xs'')), y) -> APP(app(map, f''), app(app(cons, x''), xs''))
APP(app(app(curry, app(map, app(plus, app(s, x'''')))), app(app(cons, x0''), xs'')), y) -> APP(app(map, app(plus, app(s, x''''))), app(app(cons, x0''), xs''))
APP(app(app(curry, app(map, app(map, f''''))), app(app(cons, app(app(cons, x''''), xs'''')), xs'')), y) -> APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, x''''), xs'''')), xs''))
APP(app(app(curry, app(map, app(app(curry, g'''), x''''))), app(app(cons, x0''), xs'')), y) -> APP(app(map, app(app(curry, g'''), x'''')), app(app(cons, x0''), xs''))
APP(app(app(curry, app(map, f'''')), app(app(cons, x''), nil)), y) -> APP(app(map, f''''), app(app(cons, x''), nil))
APP(app(app(curry, app(map, f'''')), app(app(cons, x''), app(app(cons, x''''), xs''''))), y) -> APP(app(map, f''''), app(app(cons, x''), app(app(cons, x''''), xs'''')))
APP(app(app(curry, app(map, app(app(curry, g''''), x''''))), app(app(cons, x0''), xs'')), y) -> APP(app(map, app(app(curry, g''''), x'''')), app(app(cons, x0''), xs''))
APP(app(app(curry, app(map, app(map, app(plus, app(s, x''''''))))), app(app(cons, app(app(cons, x0''''), xs'''')), xs'')), y) -> APP(app(map, app(map, app(plus, app(s, x'''''')))), app(app(cons, app(app(cons, x0''''), xs'''')), xs''))
APP(app(app(curry, app(map, app(map, app(map, f'''''')))), app(app(cons, app(app(cons, app(app(cons, x''''''), xs'''''')), xs'''')), xs'')), y) -> APP(app(map, app(map, app(map, f''''''))), app(app(cons, app(app(cons, app(app(cons, x''''''), xs'''''')), xs'''')), xs''))
APP(app(app(curry, app(map, app(map, app(app(curry, g'''''), x'''''')))), app(app(cons, app(app(cons, x0''''), xs'''')), xs'')), y) -> APP(app(map, app(map, app(app(curry, g'''''), x''''''))), app(app(cons, app(app(cons, x0''''), xs'''')), xs''))
APP(app(app(curry, app(map, app(map, f''''''))), app(app(cons, app(app(cons, x''''), nil)), xs'')), y) -> APP(app(map, app(map, f'''''')), app(app(cons, app(app(cons, x''''), nil)), xs''))
APP(app(app(curry, app(map, app(map, f''''''))), app(app(cons, app(app(cons, x''''), app(app(cons, x''''''), xs''''''))), xs'')), y) -> APP(app(map, app(map, f'''''')), app(app(cons, app(app(cons, x''''), app(app(cons, x''''''), xs''''''))), xs''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 4

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

APP(app(app(curry, app(map, app(map, f''''''))), app(app(cons, app(app(cons, x''''), app(app(cons, x''''''), xs''''''))), xs'')), y) -> APP(app(map, app(map, f'''''')), app(app(cons, app(app(cons, x''''), app(app(cons, x''''''), xs''''''))), xs''))
APP(app(app(curry, app(map, app(map, f''''''))), app(app(cons, app(app(cons, x''''), nil)), xs'')), y) -> APP(app(map, app(map, f'''''')), app(app(cons, app(app(cons, x''''), nil)), xs''))
APP(app(app(curry, app(map, app(map, app(app(curry, g'''''), x'''''')))), app(app(cons, app(app(cons, x0''''), xs'''')), xs'')), y) -> APP(app(map, app(map, app(app(curry, g'''''), x''''''))), app(app(cons, app(app(cons, x0''''), xs'''')), xs''))
APP(app(app(curry, app(map, app(map, app(map, f'''''')))), app(app(cons, app(app(cons, app(app(cons, x''''''), xs'''''')), xs'''')), xs'')), y) -> APP(app(map, app(map, app(map, f''''''))), app(app(cons, app(app(cons, app(app(cons, x''''''), xs'''''')), xs'''')), xs''))
APP(app(app(curry, app(map, app(map, app(plus, app(s, x''''''))))), app(app(cons, app(app(cons, x0''''), xs'''')), xs'')), y) -> APP(app(map, app(map, app(plus, app(s, x'''''')))), app(app(cons, app(app(cons, x0''''), xs'''')), xs''))
APP(app(app(curry, app(map, app(app(curry, g''''), x''''))), app(app(cons, x0''), xs'')), y) -> APP(app(map, app(app(curry, g''''), x'''')), app(app(cons, x0''), xs''))
APP(app(app(curry, app(map, f'''')), app(app(cons, x''), app(app(cons, x''''), xs''''))), y) -> APP(app(map, f''''), app(app(cons, x''), app(app(cons, x''''), xs'''')))
APP(app(app(curry, app(map, f'''')), app(app(cons, x''), nil)), y) -> APP(app(map, f''''), app(app(cons, x''), nil))
APP(app(app(curry, app(map, app(app(curry, g'''), x''''))), app(app(cons, x0''), xs'')), y) -> APP(app(map, app(app(curry, g'''), x'''')), app(app(cons, x0''), xs''))
APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, x''), nil)), xs)) -> APP(app(map, f''''), app(app(cons, x''), nil))
APP(app(map, app(map, app(app(curry, g'''), x''''))), app(app(cons, app(app(cons, x0''), xs'')), xs)) -> APP(app(map, app(app(curry, g'''), x'''')), app(app(cons, x0''), xs''))
APP(app(map, app(map, app(map, f''''))), app(app(cons, app(app(cons, app(app(cons, x''''), xs'''')), xs'')), xs)) -> APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, x''''), xs'''')), xs''))
APP(app(map, app(map, app(plus, app(s, x'''')))), app(app(cons, app(app(cons, x0''), xs'')), xs)) -> APP(app(map, app(plus, app(s, x''''))), app(app(cons, x0''), xs''))
APP(app(app(curry, app(map, app(map, f''''))), app(app(cons, app(app(cons, x''''), xs'''')), xs'')), y) -> APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, x''''), xs'''')), xs''))
APP(app(app(curry, app(map, app(plus, app(s, x'''')))), app(app(cons, x0''), xs'')), y) -> APP(app(map, app(plus, app(s, x''''))), app(app(cons, x0''), xs''))
APP(app(map, app(app(curry, g''), x'')), app(app(cons, x0), xs)) -> APP(app(app(curry, g''), x''), x0)
APP(app(map, app(plus, app(s, x''))), app(app(cons, x0), xs)) -> APP(app(plus, app(s, x'')), x0)
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(map, f''), app(app(cons, x''), xs''))
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(app(curry, app(map, f'')), app(app(cons, x''), xs'')), y) -> APP(app(map, f''), app(app(cons, x''), xs''))
APP(app(app(curry, app(plus, app(s, x''))), x0), y) -> APP(app(plus, app(s, x'')), x0)
APP(app(app(curry, app(app(curry, g''), x'')), x0), y) -> APP(app(app(curry, g''), x''), x0)
APP(app(map, f''), app(app(cons, x), nil)) -> APP(app(cons, app(f'', x)), nil)
APP(app(map, app(app(curry, g'), x'')), app(app(cons, x0), xs)) -> APP(app(cons, app(app(g', x''), x0)), app(app(map, app(app(curry, g'), x'')), 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(app(curry, g), x), y) -> APP(app(g, x), y)
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), 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, app(map, f'''')), app(app(cons, app(app(cons, x''), app(app(cons, x''''), xs''''))), xs)) -> APP(app(map, f''''), app(app(cons, x''), app(app(cons, x''''), xs'''')))

Rules:

app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
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))
app(app(app(curry, g), x), y) -> app(app(g, x), y)
inc -> app(map, app(app(curry, plus), app(s, 0)))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:46 minutes