R
↳Dependency Pair Analysis
APP(app(h, z), app(e, x)) -> APP(app(h, app(c, z)), app(app(d, z), x))
APP(app(h, z), app(e, x)) -> APP(h, app(c, z))
APP(app(h, z), app(e, x)) -> APP(c, z)
APP(app(h, z), app(e, x)) -> APP(app(d, z), x)
APP(app(h, z), app(e, x)) -> APP(d, z)
APP(app(d, z), app(app(g, 0), 0)) -> APP(e, 0)
APP(app(d, z), app(app(g, x), y)) -> APP(app(g, app(e, x)), app(app(d, z), y))
APP(app(d, z), app(app(g, x), y)) -> APP(g, app(e, x))
APP(app(d, z), app(app(g, x), y)) -> APP(e, x)
APP(app(d, z), app(app(g, x), y)) -> APP(app(d, z), y)
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(g, app(app(d, app(c, z)), app(app(g, x), y)))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, app(c, z)), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, z), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(d, z)
APP(app(g, app(e, x)), app(e, y)) -> APP(e, app(app(g, x), y))
APP(app(g, app(e, x)), app(e, y)) -> APP(app(g, x), y)
APP(app(g, app(e, x)), app(e, y)) -> APP(g, x)
R
↳DPs
→DP Problem 1
↳Polynomial Ordering
APP(app(g, app(e, x)), app(e, y)) -> APP(app(g, x), y)
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, z), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, app(c, z)), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
APP(app(d, z), app(app(g, x), y)) -> APP(app(d, z), y)
APP(app(d, z), app(app(g, x), y)) -> APP(app(g, app(e, x)), app(app(d, z), y))
APP(app(h, z), app(e, x)) -> APP(app(d, z), x)
APP(app(h, z), app(e, x)) -> APP(app(h, app(c, z)), app(app(d, z), x))
app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))
innermost
APP(app(h, z), app(e, x)) -> APP(app(d, z), x)
app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))
POL(c) = 1 POL(0) = 0 POL(g) = 0 POL(e) = 0 POL(d) = 0 POL(h) = 1 POL(app(x1, x2)) = x1 POL(APP(x1, x2)) = x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polynomial Ordering
APP(app(g, app(e, x)), app(e, y)) -> APP(app(g, x), y)
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, z), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, app(c, z)), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
APP(app(d, z), app(app(g, x), y)) -> APP(app(d, z), y)
APP(app(d, z), app(app(g, x), y)) -> APP(app(g, app(e, x)), app(app(d, z), y))
APP(app(h, z), app(e, x)) -> APP(app(h, app(c, z)), app(app(d, z), x))
app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))
innermost
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
APP(app(d, z), app(app(g, x), y)) -> APP(app(g, app(e, x)), app(app(d, z), y))
app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))
POL(c) = 1 POL(0) = 0 POL(g) = 0 POL(e) = 0 POL(d) = 1 POL(h) = 1 POL(app(x1, x2)) = x1 POL(APP(x1, x2)) = x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
...
→DP Problem 3
↳Dependency Graph
APP(app(g, app(e, x)), app(e, y)) -> APP(app(g, x), y)
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, z), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, app(c, z)), app(app(g, x), y))
APP(app(d, z), app(app(g, x), y)) -> APP(app(d, z), y)
APP(app(h, z), app(e, x)) -> APP(app(h, app(c, z)), app(app(d, z), x))
app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))
innermost
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
...
→DP Problem 7
↳Remaining Obligation(s)
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, z), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, app(c, z)), app(app(g, x), y))
APP(app(d, z), app(app(g, x), y)) -> APP(app(d, z), y)
app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))
innermost
APP(app(h, z''), app(e, app(app(g, x''), y'))) -> APP(app(h, app(c, z'')), app(app(g, app(e, x'')), app(app(d, z''), y')))
APP(app(g, app(e, x)), app(e, y)) -> APP(app(g, x), y)
APP(app(h, app(c, z'')), app(e, app(app(g, app(app(g, x''), y')), 0))) -> APP(app(h, app(c, app(c, z''))), app(app(g, app(app(d, app(c, z'')), app(app(g, x''), y'))), app(app(d, z''), app(app(g, x''), y'))))
app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))
innermost
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
...
→DP Problem 5
↳Narrowing Transformation
APP(app(h, z), app(e, x)) -> APP(app(h, app(c, z)), app(app(d, z), x))
APP(app(g, app(e, x)), app(e, y)) -> APP(app(g, x), y)
app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))
innermost
three new Dependency Pairs are created:
APP(app(h, z), app(e, x)) -> APP(app(h, app(c, z)), app(app(d, z), x))
APP(app(h, z''), app(e, app(app(g, 0), 0))) -> APP(app(h, app(c, z'')), app(e, 0))
APP(app(h, z''), app(e, app(app(g, x''), y'))) -> APP(app(h, app(c, z'')), app(app(g, app(e, x'')), app(app(d, z''), y')))
APP(app(h, app(c, z'')), app(e, app(app(g, app(app(g, x''), y')), 0))) -> APP(app(h, app(c, app(c, z''))), app(app(g, app(app(d, app(c, z'')), app(app(g, x''), y'))), app(app(d, z''), app(app(g, x''), y'))))
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
...
→DP Problem 6
↳Narrowing Transformation
APP(app(h, app(c, z'')), app(e, app(app(g, app(app(g, x''), y')), 0))) -> APP(app(h, app(c, app(c, z''))), app(app(g, app(app(d, app(c, z'')), app(app(g, x''), y'))), app(app(d, z''), app(app(g, x''), y'))))
APP(app(h, z''), app(e, app(app(g, x''), y'))) -> APP(app(h, app(c, z'')), app(app(g, app(e, x'')), app(app(d, z''), y')))
APP(app(h, z''), app(e, app(app(g, 0), 0))) -> APP(app(h, app(c, z'')), app(e, 0))
APP(app(g, app(e, x)), app(e, y)) -> APP(app(g, x), y)
app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))
innermost
no new Dependency Pairs are created.
APP(app(h, z''), app(e, app(app(g, 0), 0))) -> APP(app(h, app(c, z'')), app(e, 0))
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
...
→DP Problem 7
↳Remaining Obligation(s)
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, z), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, app(c, z)), app(app(g, x), y))
APP(app(d, z), app(app(g, x), y)) -> APP(app(d, z), y)
app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))
innermost
APP(app(h, z''), app(e, app(app(g, x''), y'))) -> APP(app(h, app(c, z'')), app(app(g, app(e, x'')), app(app(d, z''), y')))
APP(app(g, app(e, x)), app(e, y)) -> APP(app(g, x), y)
APP(app(h, app(c, z'')), app(e, app(app(g, app(app(g, x''), y')), 0))) -> APP(app(h, app(c, app(c, z''))), app(app(g, app(app(d, app(c, z'')), app(app(g, x''), y'))), app(app(d, z''), app(app(g, x''), y'))))
app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))
innermost