R
↳Dependency Pair Analysis
AP(ap(ap(g, x), y), ap(s, z)) -> AP(ap(ap(g, x), y), ap(ap(x, y), 0))
AP(ap(ap(g, x), y), ap(s, z)) -> AP(ap(x, y), 0)
AP(ap(ap(g, x), y), ap(s, z)) -> AP(x, y)
R
↳DPs
→DP Problem 1
↳Narrowing Transformation
AP(ap(ap(g, x), y), ap(s, z)) -> AP(x, y)
AP(ap(ap(g, x), y), ap(s, z)) -> AP(ap(ap(g, x), y), ap(ap(x, y), 0))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
two new Dependency Pairs are created:
AP(ap(ap(g, x), y), ap(s, z)) -> AP(ap(ap(g, x), y), ap(ap(x, y), 0))
AP(ap(ap(g, f), y'), ap(s, z)) -> AP(ap(ap(g, f), y'), ap(y', 0))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Narrowing Transformation
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0))
AP(ap(ap(g, f), y'), ap(s, z)) -> AP(ap(ap(g, f), y'), ap(y', 0))
AP(ap(ap(g, x), y), ap(s, z)) -> AP(x, y)
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
two new Dependency Pairs are created:
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0))
AP(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'')), ap(ap(ap(ap(g, f), y'''), ap(y''', 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z''')), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 3
↳Forward Instantiation Transformation
AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z''')), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'')), ap(ap(ap(ap(g, f), y'''), ap(y''', 0)), 0))
AP(ap(ap(g, x), y), ap(s, z)) -> AP(x, y)
AP(ap(ap(g, f), y'), ap(s, z)) -> AP(ap(ap(g, f), y'), ap(y', 0))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
four new Dependency Pairs are created:
AP(ap(ap(g, x), y), ap(s, z)) -> AP(x, y)
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, f), y'''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 29
↳Remaining Obligation(s)
AP(ap(ap(g, f), y'), ap(s, z)) -> AP(ap(ap(g, f), y'), ap(y', 0))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1)), ap(ap(ap(ap(g, f), y''), ap(y'', 0)), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1)), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0)), 0)), 0))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 5
↳Narrowing Transformation
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'')), ap(ap(ap(ap(g, f), y'''), ap(y''', 0)), 0))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z''')), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
two new Dependency Pairs are created:
AP(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'')), ap(ap(ap(ap(g, f), y'''), ap(y''', 0)), 0))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0))
AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(ap(ap(ap(g, f), f), 0), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 6
↳Rewriting Transformation
AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(ap(ap(ap(g, f), f), 0), 0))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z''')), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
one new Dependency Pair is created:
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 7
↳Rewriting Transformation
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z''')), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(ap(ap(ap(g, f), f), 0), 0))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
one new Dependency Pair is created:
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 8
↳Rewriting Transformation
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0))
AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(ap(ap(ap(g, f), f), 0), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z''')), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
one new Dependency Pair is created:
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 9
↳Rewriting Transformation
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z''')), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(ap(ap(ap(g, f), f), 0), 0))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
one new Dependency Pair is created:
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 10
↳Rewriting Transformation
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0))
AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(ap(ap(ap(g, f), f), 0), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z''')), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
one new Dependency Pair is created:
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 11
↳Narrowing Transformation
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z''')), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(ap(ap(ap(g, f), f), 0), 0))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
two new Dependency Pairs are created:
AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z''')), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), y'')), ap(s, z''')), ap(ap(ap(ap(g, f), y''), ap(y'', 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 12
↳Narrowing Transformation
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), y'')), ap(s, z''')), ap(ap(ap(ap(g, f), y''), ap(y'', 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(ap(ap(ap(g, f), f), 0), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
no new Dependency Pairs are created.
AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(ap(ap(ap(g, f), f), 0), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 13
↳Narrowing Transformation
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), y'')), ap(s, z''')), ap(ap(ap(ap(g, f), y''), ap(y'', 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
two new Dependency Pairs are created:
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), y'')), ap(s, z''')), ap(ap(ap(ap(g, f), y''), ap(y'', 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z''')), ap(ap(ap(ap(g, f), f), 0), 0)), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 14
↳Rewriting Transformation
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z''')), ap(ap(ap(ap(g, f), f), 0), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
one new Dependency Pair is created:
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 15
↳Rewriting Transformation
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z''')), ap(ap(ap(ap(g, f), f), 0), 0)), 0))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
one new Dependency Pair is created:
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 16
↳Rewriting Transformation
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z''')), ap(ap(ap(ap(g, f), f), 0), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
one new Dependency Pair is created:
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 17
↳Rewriting Transformation
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z''')), ap(ap(ap(ap(g, f), f), 0), 0)), 0))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
one new Dependency Pair is created:
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 18
↳Rewriting Transformation
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z''')), ap(ap(ap(ap(g, f), f), 0), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
one new Dependency Pair is created:
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 19
↳Narrowing Transformation
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z''')), ap(ap(ap(ap(g, f), f), 0), 0)), 0))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
two new Dependency Pairs are created:
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0)), ap(ap(ap(ap(g, f), y'''), ap(y''', 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1)), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0)), 0)), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 20
↳Narrowing Transformation
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1)), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0)), ap(ap(ap(ap(g, f), y'''), ap(y''', 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z''')), ap(ap(ap(ap(g, f), f), 0), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
no new Dependency Pairs are created.
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z''')), ap(ap(ap(ap(g, f), f), 0), 0)), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 21
↳Narrowing Transformation
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0)), ap(ap(ap(ap(g, f), y'''), ap(y''', 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1)), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0)), 0)), 0))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
two new Dependency Pairs are created:
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0)), ap(ap(ap(ap(g, f), y'''), ap(y''', 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z'0)), ap(ap(ap(ap(g, f), f), 0), 0)), 0)), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 22
↳Rewriting Transformation
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z'0)), ap(ap(ap(ap(g, f), f), 0), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1)), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
one new Dependency Pair is created:
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0)), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 23
↳Rewriting Transformation
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1)), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z'0)), ap(ap(ap(ap(g, f), f), 0), 0)), 0)), 0))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
one new Dependency Pair is created:
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0)), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 24
↳Rewriting Transformation
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z'0)), ap(ap(ap(ap(g, f), f), 0), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1)), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0)), 0)), 0))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
one new Dependency Pair is created:
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0)), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 25
↳Rewriting Transformation
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1)), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z'0)), ap(ap(ap(ap(g, f), f), 0), 0)), 0)), 0))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
one new Dependency Pair is created:
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0)), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 26
↳Rewriting Transformation
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z'0)), ap(ap(ap(ap(g, f), f), 0), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1)), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0)), 0)), 0))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
one new Dependency Pair is created:
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0)), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 27
↳Narrowing Transformation
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1)), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z'0)), ap(ap(ap(ap(g, f), f), 0), 0)), 0)), 0))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
two new Dependency Pairs are created:
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1)), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1)), ap(ap(ap(ap(g, f), y''), ap(y'', 0)), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1)), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0)), 0)), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 28
↳Narrowing Transformation
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1)), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1)), ap(ap(ap(ap(g, f), y''), ap(y'', 0)), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z'0)), ap(ap(ap(ap(g, f), f), 0), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0)), 0))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
no new Dependency Pairs are created.
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z'0)), ap(ap(ap(ap(g, f), f), 0), 0)), 0)), 0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 29
↳Remaining Obligation(s)
AP(ap(ap(g, f), y'), ap(s, z)) -> AP(ap(ap(g, f), y'), ap(y', 0))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1)), ap(ap(ap(ap(g, f), y''), ap(y'', 0)), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1)), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0)), 0)), 0))
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))
innermost