R
↳Dependency Pair Analysis
APP(app(add, app(s, x)), y) -> APP(s, app(app(add, x), y))
APP(app(add, app(s, x)), y) -> APP(app(add, x), y)
APP(app(add, app(s, x)), y) -> APP(add, 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)
R
↳DPs
→DP Problem 1
↳Narrowing Transformation
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(add, app(s, x)), y) -> APP(app(add, x), y)
app(id, x) -> x
app(add, 0) -> id
app(app(add, app(s, x)), y) -> app(s, app(app(add, 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))
innermost
seven new Dependency Pairs are created:
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(app(map, id), app(app(cons, x''), xs)) -> APP(app(cons, x''), app(app(map, id), xs))
APP(app(map, add), app(app(cons, 0), xs)) -> APP(app(cons, id), app(app(map, add), xs))
APP(app(map, app(add, app(s, x''))), app(app(cons, x0), xs)) -> APP(app(cons, app(s, app(app(add, x''), x0))), app(app(map, app(add, 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, 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'')))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Forward Instantiation Transformation
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, app(add, app(s, x''))), app(app(cons, x0), xs)) -> APP(app(cons, app(s, app(app(add, x''), x0))), app(app(map, app(add, app(s, x''))), xs))
APP(app(map, add), app(app(cons, 0), xs)) -> APP(app(cons, id), app(app(map, add), xs))
APP(app(add, app(s, x)), y) -> APP(app(add, x), y)
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(id, x) -> x
app(add, 0) -> id
app(app(add, app(s, x)), y) -> app(s, app(app(add, 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))
innermost
seven new Dependency Pairs are created:
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
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(add, app(s, x''))), app(app(cons, x0), xs)) -> APP(app(add, app(s, x'')), x0)
APP(app(map, app(map, add)), app(app(cons, app(app(cons, 0), xs'')), xs)) -> APP(app(map, add), app(app(cons, 0), xs''))
APP(app(map, app(map, app(add, app(s, x'''')))), app(app(cons, app(app(cons, x0''), xs'')), xs)) -> APP(app(map, app(add, 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, 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'''')))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳FwdInst
...
→DP Problem 3
↳Remaining Obligation(s)
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(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(add, app(s, x'''')))), app(app(cons, app(app(cons, x0''), xs'')), xs)) -> APP(app(map, app(add, app(s, x''''))), app(app(cons, x0''), xs''))
APP(app(map, app(map, add)), app(app(cons, app(app(cons, 0), xs'')), xs)) -> APP(app(map, add), app(app(cons, 0), xs''))
APP(app(map, app(add, app(s, x''))), app(app(cons, x0), xs)) -> APP(app(add, 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(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(add, app(s, x''))), app(app(cons, x0), xs)) -> APP(app(cons, app(s, app(app(add, x''), x0))), app(app(map, app(add, app(s, x''))), xs))
APP(app(map, add), app(app(cons, 0), xs)) -> APP(app(cons, id), app(app(map, add), xs))
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(add, app(s, x)), y) -> APP(app(add, 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'')))
app(id, x) -> x
app(add, 0) -> id
app(app(add, app(s, x)), y) -> app(s, app(app(add, 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))
innermost