0 QTRS
↳1 Overlay + Local Confluence (⇔)
↳2 QTRS
↳3 DependencyPairsProof (⇔)
↳4 QDP
↳5 DependencyGraphProof (⇔)
↳6 AND
↳7 QDP
↳8 UsableRulesProof (⇔)
↳9 QDP
↳10 QReductionProof (⇔)
↳11 QDP
↳12 QDPSizeChangeProof (⇔)
↳13 TRUE
↳14 QDP
↳15 UsableRulesProof (⇔)
↳16 QDP
↳17 QReductionProof (⇔)
↳18 QDP
↳19 QDPOrderProof (⇔)
↳20 QDP
↳21 PisEmptyProof (⇔)
↳22 TRUE
↳23 QDP
↳24 UsableRulesProof (⇔)
↳25 QDP
↳26 QReductionProof (⇔)
↳27 QDP
↳28 QDPSizeChangeProof (⇔)
↳29 TRUE
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(a(x, y, h), l))
a(h, h, x) → s(x)
a(x, s(y), h) → a(x, y, s(h))
a(x, s(y), s(z)) → a(x, y, a(x, s(y), z))
a(s(x), h, z) → a(x, z, z)
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(a(x, y, h), l))
a(h, h, x) → s(x)
a(x, s(y), h) → a(x, y, s(h))
a(x, s(y), s(z)) → a(x, y, a(x, s(y), z))
a(s(x), h, z) → a(x, z, z)
app(nil, x0)
app(x0, nil)
app(cons(x0, x1), x2)
sum(cons(x0, nil))
sum(cons(x0, cons(x1, x2)))
a(h, h, x0)
a(x0, s(x1), h)
a(x0, s(x1), s(x2))
a(s(x0), h, x1)
APP(cons(x, l), k) → APP(l, k)
SUM(cons(x, cons(y, l))) → SUM(cons(a(x, y, h), l))
SUM(cons(x, cons(y, l))) → A(x, y, h)
A(x, s(y), h) → A(x, y, s(h))
A(x, s(y), s(z)) → A(x, y, a(x, s(y), z))
A(x, s(y), s(z)) → A(x, s(y), z)
A(s(x), h, z) → A(x, z, z)
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(a(x, y, h), l))
a(h, h, x) → s(x)
a(x, s(y), h) → a(x, y, s(h))
a(x, s(y), s(z)) → a(x, y, a(x, s(y), z))
a(s(x), h, z) → a(x, z, z)
app(nil, x0)
app(x0, nil)
app(cons(x0, x1), x2)
sum(cons(x0, nil))
sum(cons(x0, cons(x1, x2)))
a(h, h, x0)
a(x0, s(x1), h)
a(x0, s(x1), s(x2))
a(s(x0), h, x1)
A(x, s(y), s(z)) → A(x, y, a(x, s(y), z))
A(x, s(y), s(z)) → A(x, s(y), z)
A(x, s(y), h) → A(x, y, s(h))
A(s(x), h, z) → A(x, z, z)
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(a(x, y, h), l))
a(h, h, x) → s(x)
a(x, s(y), h) → a(x, y, s(h))
a(x, s(y), s(z)) → a(x, y, a(x, s(y), z))
a(s(x), h, z) → a(x, z, z)
app(nil, x0)
app(x0, nil)
app(cons(x0, x1), x2)
sum(cons(x0, nil))
sum(cons(x0, cons(x1, x2)))
a(h, h, x0)
a(x0, s(x1), h)
a(x0, s(x1), s(x2))
a(s(x0), h, x1)
A(x, s(y), s(z)) → A(x, y, a(x, s(y), z))
A(x, s(y), s(z)) → A(x, s(y), z)
A(x, s(y), h) → A(x, y, s(h))
A(s(x), h, z) → A(x, z, z)
a(x, s(y), s(z)) → a(x, y, a(x, s(y), z))
a(s(x), h, z) → a(x, z, z)
a(x, s(y), h) → a(x, y, s(h))
a(h, h, x) → s(x)
app(nil, x0)
app(x0, nil)
app(cons(x0, x1), x2)
sum(cons(x0, nil))
sum(cons(x0, cons(x1, x2)))
a(h, h, x0)
a(x0, s(x1), h)
a(x0, s(x1), s(x2))
a(s(x0), h, x1)
app(nil, x0)
app(x0, nil)
app(cons(x0, x1), x2)
sum(cons(x0, nil))
sum(cons(x0, cons(x1, x2)))
A(x, s(y), s(z)) → A(x, y, a(x, s(y), z))
A(x, s(y), s(z)) → A(x, s(y), z)
A(x, s(y), h) → A(x, y, s(h))
A(s(x), h, z) → A(x, z, z)
a(x, s(y), s(z)) → a(x, y, a(x, s(y), z))
a(s(x), h, z) → a(x, z, z)
a(x, s(y), h) → a(x, y, s(h))
a(h, h, x) → s(x)
a(h, h, x0)
a(x0, s(x1), h)
a(x0, s(x1), s(x2))
a(s(x0), h, x1)
From the DPs we obtained the following set of size-change graphs:
SUM(cons(x, cons(y, l))) → SUM(cons(a(x, y, h), l))
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(a(x, y, h), l))
a(h, h, x) → s(x)
a(x, s(y), h) → a(x, y, s(h))
a(x, s(y), s(z)) → a(x, y, a(x, s(y), z))
a(s(x), h, z) → a(x, z, z)
app(nil, x0)
app(x0, nil)
app(cons(x0, x1), x2)
sum(cons(x0, nil))
sum(cons(x0, cons(x1, x2)))
a(h, h, x0)
a(x0, s(x1), h)
a(x0, s(x1), s(x2))
a(s(x0), h, x1)
SUM(cons(x, cons(y, l))) → SUM(cons(a(x, y, h), l))
a(h, h, x) → s(x)
a(x, s(y), s(z)) → a(x, y, a(x, s(y), z))
a(s(x), h, z) → a(x, z, z)
a(x, s(y), h) → a(x, y, s(h))
app(nil, x0)
app(x0, nil)
app(cons(x0, x1), x2)
sum(cons(x0, nil))
sum(cons(x0, cons(x1, x2)))
a(h, h, x0)
a(x0, s(x1), h)
a(x0, s(x1), s(x2))
a(s(x0), h, x1)
app(nil, x0)
app(x0, nil)
app(cons(x0, x1), x2)
sum(cons(x0, nil))
sum(cons(x0, cons(x1, x2)))
SUM(cons(x, cons(y, l))) → SUM(cons(a(x, y, h), l))
a(h, h, x) → s(x)
a(x, s(y), s(z)) → a(x, y, a(x, s(y), z))
a(s(x), h, z) → a(x, z, z)
a(x, s(y), h) → a(x, y, s(h))
a(h, h, x0)
a(x0, s(x1), h)
a(x0, s(x1), s(x2))
a(s(x0), h, x1)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
SUM(cons(x, cons(y, l))) → SUM(cons(a(x, y, h), l))
POL(SUM(x1)) = x1
POL(a(x1, x2, x3)) = 0
POL(cons(x1, x2)) = 1 + x2
POL(h) = 0
POL(s(x1)) = 0
a(h, h, x) → s(x)
a(x, s(y), s(z)) → a(x, y, a(x, s(y), z))
a(s(x), h, z) → a(x, z, z)
a(x, s(y), h) → a(x, y, s(h))
a(h, h, x0)
a(x0, s(x1), h)
a(x0, s(x1), s(x2))
a(s(x0), h, x1)
APP(cons(x, l), k) → APP(l, k)
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(a(x, y, h), l))
a(h, h, x) → s(x)
a(x, s(y), h) → a(x, y, s(h))
a(x, s(y), s(z)) → a(x, y, a(x, s(y), z))
a(s(x), h, z) → a(x, z, z)
app(nil, x0)
app(x0, nil)
app(cons(x0, x1), x2)
sum(cons(x0, nil))
sum(cons(x0, cons(x1, x2)))
a(h, h, x0)
a(x0, s(x1), h)
a(x0, s(x1), s(x2))
a(s(x0), h, x1)
APP(cons(x, l), k) → APP(l, k)
app(nil, x0)
app(x0, nil)
app(cons(x0, x1), x2)
sum(cons(x0, nil))
sum(cons(x0, cons(x1, x2)))
a(h, h, x0)
a(x0, s(x1), h)
a(x0, s(x1), s(x2))
a(s(x0), h, x1)
app(nil, x0)
app(x0, nil)
app(cons(x0, x1), x2)
sum(cons(x0, nil))
sum(cons(x0, cons(x1, x2)))
a(h, h, x0)
a(x0, s(x1), h)
a(x0, s(x1), s(x2))
a(s(x0), h, x1)
APP(cons(x, l), k) → APP(l, k)
From the DPs we obtained the following set of size-change graphs: