0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 AND
↳5 QDP
↳6 QDPOrderProof (⇔)
↳7 QDP
↳8 PisEmptyProof (⇔)
↳9 TRUE
↳10 QDP
↳11 QDPOrderProof (⇔)
↳12 QDP
↳13 PisEmptyProof (⇔)
↳14 TRUE
↳15 QDP
↳16 QDPOrderProof (⇔)
↳17 QDP
↳18 QDPOrderProof (⇔)
↳19 QDP
↳20 PisEmptyProof (⇔)
↳21 TRUE
↳22 QDP
↳23 QDPOrderProof (⇔)
↳24 QDP
↳25 PisEmptyProof (⇔)
↳26 TRUE
↳27 QDP
↳28 QDPOrderProof (⇔)
↳29 QDP
↳30 QDPOrderProof (⇔)
↳31 QDP
↳32 PisEmptyProof (⇔)
↳33 TRUE
active(f(f(a))) → mark(f(g(f(a))))
active(f(X)) → f(active(X))
f(mark(X)) → mark(f(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
ACTIVE(f(f(a))) → F(g(f(a)))
ACTIVE(f(f(a))) → G(f(a))
ACTIVE(f(X)) → F(active(X))
ACTIVE(f(X)) → ACTIVE(X)
F(mark(X)) → F(X)
PROPER(f(X)) → F(proper(X))
PROPER(f(X)) → PROPER(X)
PROPER(g(X)) → G(proper(X))
PROPER(g(X)) → PROPER(X)
F(ok(X)) → F(X)
G(ok(X)) → G(X)
TOP(mark(X)) → TOP(proper(X))
TOP(mark(X)) → PROPER(X)
TOP(ok(X)) → TOP(active(X))
TOP(ok(X)) → ACTIVE(X)
active(f(f(a))) → mark(f(g(f(a))))
active(f(X)) → f(active(X))
f(mark(X)) → mark(f(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
G(ok(X)) → G(X)
active(f(f(a))) → mark(f(g(f(a))))
active(f(X)) → f(active(X))
f(mark(X)) → mark(f(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
G(ok(X)) → G(X)
proper1 > a > ok1
top > active1 > a > ok1
ok1: [1]
active1: multiset
a: multiset
proper1: [1]
top: []
active(f(f(a))) → mark(f(g(f(a))))
active(f(X)) → f(active(X))
f(mark(X)) → mark(f(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
active(f(f(a))) → mark(f(g(f(a))))
active(f(X)) → f(active(X))
f(mark(X)) → mark(f(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
F(ok(X)) → F(X)
F(mark(X)) → F(X)
active(f(f(a))) → mark(f(g(f(a))))
active(f(X)) → f(active(X))
f(mark(X)) → mark(f(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
F(ok(X)) → F(X)
F(mark(X)) → F(X)
top > active1 > mark1
top > active1 > a
top > proper1 > ok1
top > proper1 > a
ok1: multiset
mark1: [1]
active1: multiset
a: multiset
proper1: [1]
top: multiset
active(f(f(a))) → mark(f(g(f(a))))
active(f(X)) → f(active(X))
f(mark(X)) → mark(f(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
active(f(f(a))) → mark(f(g(f(a))))
active(f(X)) → f(active(X))
f(mark(X)) → mark(f(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
PROPER(g(X)) → PROPER(X)
PROPER(f(X)) → PROPER(X)
active(f(f(a))) → mark(f(g(f(a))))
active(f(X)) → f(active(X))
f(mark(X)) → mark(f(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
PROPER(g(X)) → PROPER(X)
a > g1
top > active1 > g1
g1: multiset
active1: multiset
a: multiset
top: multiset
active(f(f(a))) → mark(f(g(f(a))))
active(f(X)) → f(active(X))
f(mark(X)) → mark(f(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
PROPER(f(X)) → PROPER(X)
active(f(f(a))) → mark(f(g(f(a))))
active(f(X)) → f(active(X))
f(mark(X)) → mark(f(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
PROPER(f(X)) → PROPER(X)
f1 > g
a > g
top > g
f1: multiset
a: multiset
g: multiset
top: []
active(f(f(a))) → mark(f(g(f(a))))
active(f(X)) → f(active(X))
f(mark(X)) → mark(f(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
active(f(f(a))) → mark(f(g(f(a))))
active(f(X)) → f(active(X))
f(mark(X)) → mark(f(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
ACTIVE(f(X)) → ACTIVE(X)
active(f(f(a))) → mark(f(g(f(a))))
active(f(X)) → f(active(X))
f(mark(X)) → mark(f(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
ACTIVE(f(X)) → ACTIVE(X)
f1 > g
a > g
top > g
f1: multiset
a: multiset
g: multiset
top: []
active(f(f(a))) → mark(f(g(f(a))))
active(f(X)) → f(active(X))
f(mark(X)) → mark(f(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
active(f(f(a))) → mark(f(g(f(a))))
active(f(X)) → f(active(X))
f(mark(X)) → mark(f(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
TOP(ok(X)) → TOP(active(X))
TOP(mark(X)) → TOP(proper(X))
active(f(f(a))) → mark(f(g(f(a))))
active(f(X)) → f(active(X))
f(mark(X)) → mark(f(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
TOP(mark(X)) → TOP(proper(X))
TOP1 > top1
f1 > mark1 > top1
f1 > a > top1
f1 > g > top1
TOP1: multiset
mark1: [1]
f1: multiset
a: multiset
g: multiset
top1: [1]
active(f(f(a))) → mark(f(g(f(a))))
active(f(X)) → f(active(X))
f(mark(X)) → mark(f(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
TOP(ok(X)) → TOP(active(X))
active(f(f(a))) → mark(f(g(f(a))))
active(f(X)) → f(active(X))
f(mark(X)) → mark(f(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
TOP(ok(X)) → TOP(active(X))
proper1 > ok > active > mark
proper1 > ok > top > mark
proper1 > a > mark
proper1 > g1 > mark
ok: []
active: multiset
a: multiset
mark: []
g1: multiset
proper1: [1]
top: []
active(f(f(a))) → mark(f(g(f(a))))
active(f(X)) → f(active(X))
f(mark(X)) → mark(f(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
active(f(f(a))) → mark(f(g(f(a))))
active(f(X)) → f(active(X))
f(mark(X)) → mark(f(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))