sq(0(x1)) → p(s(p(s(p(p(p(p(s(s(s(s(0(p(s(p(s(x1)))))))))))))))))
sq(s(x1)) → s(p(s(p(s(p(p(s(s(twice(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1)))))))))))))))))))))))))))))))))
twice(0(x1)) → p(p(p(p(s(s(p(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1)))))))))))))))))))))))))))
twice(s(x1)) → p(p(s(s(s(p(p(s(s(s(twice(p(s(p(s(x1)))))))))))))))
p(p(s(x1))) → p(x1)
p(s(x1)) → x1
p(0(x1)) → 0(s(s(s(s(s(s(s(s(s(s(s(x1))))))))))))
0(x1) → x1
↳ QTRS
↳ DependencyPairsProof
sq(0(x1)) → p(s(p(s(p(p(p(p(s(s(s(s(0(p(s(p(s(x1)))))))))))))))))
sq(s(x1)) → s(p(s(p(s(p(p(s(s(twice(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1)))))))))))))))))))))))))))))))))
twice(0(x1)) → p(p(p(p(s(s(p(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1)))))))))))))))))))))))))))
twice(s(x1)) → p(p(s(s(s(p(p(s(s(s(twice(p(s(p(s(x1)))))))))))))))
p(p(s(x1))) → p(x1)
p(s(x1)) → x1
p(0(x1)) → 0(s(s(s(s(s(s(s(s(s(s(s(x1))))))))))))
0(x1) → x1
SQ(0(x1)) → P(p(p(s(s(s(s(0(p(s(p(s(x1))))))))))))
TWICE(0(x1)) → P(s(p(s(p(s(x1))))))
SQ(0(x1)) → P(s(p(s(x1))))
SQ(s(x1)) → P(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1)))))))))))))))))))))
SQ(s(x1)) → TWICE(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1))))))))))))))))))))))))
SQ(0(x1)) → 01(p(s(p(s(x1)))))
TWICE(0(x1)) → P(p(p(p(s(s(p(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1)))))))))))))))))))))))))))
SQ(0(x1)) → P(s(s(s(s(0(p(s(p(s(x1))))))))))
SQ(s(x1)) → P(p(s(s(twice(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1))))))))))))))))))))))))))))
TWICE(0(x1)) → P(s(s(p(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1))))))))))))))))))))))))
TWICE(0(x1)) → 01(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1)))))))))))))))))
SQ(0(x1)) → P(p(s(s(s(s(0(p(s(p(s(x1)))))))))))
SQ(s(x1)) → P(p(p(p(p(s(s(s(s(s(s(x1)))))))))))
TWICE(0(x1)) → P(p(s(s(p(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1)))))))))))))))))))))))))
P(0(x1)) → 01(s(s(s(s(s(s(s(s(s(s(s(x1))))))))))))
TWICE(s(x1)) → P(s(x1))
P(p(s(x1))) → P(x1)
SQ(s(x1)) → P(p(p(p(s(s(s(s(s(s(x1))))))))))
TWICE(s(x1)) → TWICE(p(s(p(s(x1)))))
SQ(s(x1)) → P(s(s(s(s(s(s(x1)))))))
TWICE(0(x1)) → P(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1))))))))))))))))
TWICE(0(x1)) → P(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1)))))))))))))))
SQ(s(x1)) → P(s(s(twice(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1)))))))))))))))))))))))))))
SQ(s(x1)) → P(s(p(p(s(s(twice(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1))))))))))))))))))))))))))))))
TWICE(s(x1)) → P(p(s(s(s(twice(p(s(p(s(x1))))))))))
TWICE(s(x1)) → P(s(p(s(x1))))
SQ(s(x1)) → P(p(p(p(p(p(s(s(s(s(s(s(x1))))))))))))
TWICE(0(x1)) → P(p(p(s(s(p(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1))))))))))))))))))))))))))
TWICE(0(x1)) → P(p(s(s(p(s(p(s(p(s(x1))))))))))
SQ(s(x1)) → P(s(p(s(p(p(s(s(twice(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1))))))))))))))))))))))))))))))))
SQ(s(x1)) → P(p(p(s(s(s(s(s(s(x1)))))))))
TWICE(0(x1)) → P(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1)))))))))))))))))))))
SQ(s(x1)) → P(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1)))))))))))))))))))))))
TWICE(s(x1)) → P(s(s(s(twice(p(s(p(s(x1)))))))))
SQ(0(x1)) → P(s(p(p(p(p(s(s(s(s(0(p(s(p(s(x1)))))))))))))))
TWICE(0(x1)) → P(s(s(p(s(p(s(p(s(x1)))))))))
SQ(s(x1)) → P(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1)))))))))))))))))))
TWICE(s(x1)) → P(s(s(s(p(p(s(s(s(twice(p(s(p(s(x1))))))))))))))
SQ(0(x1)) → P(p(p(p(s(s(s(s(0(p(s(p(s(x1)))))))))))))
TWICE(0(x1)) → P(s(x1))
SQ(s(x1)) → P(p(s(s(s(s(s(s(x1))))))))
SQ(0(x1)) → P(s(x1))
TWICE(s(x1)) → P(p(s(s(s(p(p(s(s(s(twice(p(s(p(s(x1)))))))))))))))
TWICE(0(x1)) → P(s(p(s(x1))))
SQ(s(x1)) → SQ(p(p(p(p(p(p(s(s(s(s(s(s(x1)))))))))))))
TWICE(0(x1)) → P(s(s(s(p(p(s(s(p(s(p(s(p(s(x1))))))))))))))
SQ(s(x1)) → P(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1)))))))))))))))))
SQ(s(x1)) → P(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1))))))))))))))))))
SQ(0(x1)) → P(s(p(s(p(p(p(p(s(s(s(s(0(p(s(p(s(x1)))))))))))))))))
sq(0(x1)) → p(s(p(s(p(p(p(p(s(s(s(s(0(p(s(p(s(x1)))))))))))))))))
sq(s(x1)) → s(p(s(p(s(p(p(s(s(twice(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1)))))))))))))))))))))))))))))))))
twice(0(x1)) → p(p(p(p(s(s(p(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1)))))))))))))))))))))))))))
twice(s(x1)) → p(p(s(s(s(p(p(s(s(s(twice(p(s(p(s(x1)))))))))))))))
p(p(s(x1))) → p(x1)
p(s(x1)) → x1
p(0(x1)) → 0(s(s(s(s(s(s(s(s(s(s(s(x1))))))))))))
0(x1) → x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
SQ(0(x1)) → P(p(p(s(s(s(s(0(p(s(p(s(x1))))))))))))
TWICE(0(x1)) → P(s(p(s(p(s(x1))))))
SQ(0(x1)) → P(s(p(s(x1))))
SQ(s(x1)) → P(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1)))))))))))))))))))))
SQ(s(x1)) → TWICE(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1))))))))))))))))))))))))
SQ(0(x1)) → 01(p(s(p(s(x1)))))
TWICE(0(x1)) → P(p(p(p(s(s(p(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1)))))))))))))))))))))))))))
SQ(0(x1)) → P(s(s(s(s(0(p(s(p(s(x1))))))))))
SQ(s(x1)) → P(p(s(s(twice(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1))))))))))))))))))))))))))))
TWICE(0(x1)) → P(s(s(p(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1))))))))))))))))))))))))
TWICE(0(x1)) → 01(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1)))))))))))))))))
SQ(0(x1)) → P(p(s(s(s(s(0(p(s(p(s(x1)))))))))))
SQ(s(x1)) → P(p(p(p(p(s(s(s(s(s(s(x1)))))))))))
TWICE(0(x1)) → P(p(s(s(p(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1)))))))))))))))))))))))))
P(0(x1)) → 01(s(s(s(s(s(s(s(s(s(s(s(x1))))))))))))
TWICE(s(x1)) → P(s(x1))
P(p(s(x1))) → P(x1)
SQ(s(x1)) → P(p(p(p(s(s(s(s(s(s(x1))))))))))
TWICE(s(x1)) → TWICE(p(s(p(s(x1)))))
SQ(s(x1)) → P(s(s(s(s(s(s(x1)))))))
TWICE(0(x1)) → P(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1))))))))))))))))
TWICE(0(x1)) → P(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1)))))))))))))))
SQ(s(x1)) → P(s(s(twice(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1)))))))))))))))))))))))))))
SQ(s(x1)) → P(s(p(p(s(s(twice(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1))))))))))))))))))))))))))))))
TWICE(s(x1)) → P(p(s(s(s(twice(p(s(p(s(x1))))))))))
TWICE(s(x1)) → P(s(p(s(x1))))
SQ(s(x1)) → P(p(p(p(p(p(s(s(s(s(s(s(x1))))))))))))
TWICE(0(x1)) → P(p(p(s(s(p(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1))))))))))))))))))))))))))
TWICE(0(x1)) → P(p(s(s(p(s(p(s(p(s(x1))))))))))
SQ(s(x1)) → P(s(p(s(p(p(s(s(twice(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1))))))))))))))))))))))))))))))))
SQ(s(x1)) → P(p(p(s(s(s(s(s(s(x1)))))))))
TWICE(0(x1)) → P(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1)))))))))))))))))))))
SQ(s(x1)) → P(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1)))))))))))))))))))))))
TWICE(s(x1)) → P(s(s(s(twice(p(s(p(s(x1)))))))))
SQ(0(x1)) → P(s(p(p(p(p(s(s(s(s(0(p(s(p(s(x1)))))))))))))))
TWICE(0(x1)) → P(s(s(p(s(p(s(p(s(x1)))))))))
SQ(s(x1)) → P(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1)))))))))))))))))))
TWICE(s(x1)) → P(s(s(s(p(p(s(s(s(twice(p(s(p(s(x1))))))))))))))
SQ(0(x1)) → P(p(p(p(s(s(s(s(0(p(s(p(s(x1)))))))))))))
TWICE(0(x1)) → P(s(x1))
SQ(s(x1)) → P(p(s(s(s(s(s(s(x1))))))))
SQ(0(x1)) → P(s(x1))
TWICE(s(x1)) → P(p(s(s(s(p(p(s(s(s(twice(p(s(p(s(x1)))))))))))))))
TWICE(0(x1)) → P(s(p(s(x1))))
SQ(s(x1)) → SQ(p(p(p(p(p(p(s(s(s(s(s(s(x1)))))))))))))
TWICE(0(x1)) → P(s(s(s(p(p(s(s(p(s(p(s(p(s(x1))))))))))))))
SQ(s(x1)) → P(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1)))))))))))))))))
SQ(s(x1)) → P(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1))))))))))))))))))
SQ(0(x1)) → P(s(p(s(p(p(p(p(s(s(s(s(0(p(s(p(s(x1)))))))))))))))))
sq(0(x1)) → p(s(p(s(p(p(p(p(s(s(s(s(0(p(s(p(s(x1)))))))))))))))))
sq(s(x1)) → s(p(s(p(s(p(p(s(s(twice(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1)))))))))))))))))))))))))))))))))
twice(0(x1)) → p(p(p(p(s(s(p(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1)))))))))))))))))))))))))))
twice(s(x1)) → p(p(s(s(s(p(p(s(s(s(twice(p(s(p(s(x1)))))))))))))))
p(p(s(x1))) → p(x1)
p(s(x1)) → x1
p(0(x1)) → 0(s(s(s(s(s(s(s(s(s(s(s(x1))))))))))))
0(x1) → x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
P(p(s(x1))) → P(x1)
sq(0(x1)) → p(s(p(s(p(p(p(p(s(s(s(s(0(p(s(p(s(x1)))))))))))))))))
sq(s(x1)) → s(p(s(p(s(p(p(s(s(twice(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1)))))))))))))))))))))))))))))))))
twice(0(x1)) → p(p(p(p(s(s(p(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1)))))))))))))))))))))))))))
twice(s(x1)) → p(p(s(s(s(p(p(s(s(s(twice(p(s(p(s(x1)))))))))))))))
p(p(s(x1))) → p(x1)
p(s(x1)) → x1
p(0(x1)) → 0(s(s(s(s(s(s(s(s(s(s(s(x1))))))))))))
0(x1) → x1
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
P(p(s(x1))) → P(x1)
The value of delta used in the strict ordering is 2.
POL(P(x1)) = (2)x_1
POL(p(x1)) = 1 + x_1
POL(s(x1)) = x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
sq(0(x1)) → p(s(p(s(p(p(p(p(s(s(s(s(0(p(s(p(s(x1)))))))))))))))))
sq(s(x1)) → s(p(s(p(s(p(p(s(s(twice(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1)))))))))))))))))))))))))))))))))
twice(0(x1)) → p(p(p(p(s(s(p(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1)))))))))))))))))))))))))))
twice(s(x1)) → p(p(s(s(s(p(p(s(s(s(twice(p(s(p(s(x1)))))))))))))))
p(p(s(x1))) → p(x1)
p(s(x1)) → x1
p(0(x1)) → 0(s(s(s(s(s(s(s(s(s(s(s(x1))))))))))))
0(x1) → x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
TWICE(s(x1)) → TWICE(p(s(p(s(x1)))))
sq(0(x1)) → p(s(p(s(p(p(p(p(s(s(s(s(0(p(s(p(s(x1)))))))))))))))))
sq(s(x1)) → s(p(s(p(s(p(p(s(s(twice(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1)))))))))))))))))))))))))))))))))
twice(0(x1)) → p(p(p(p(s(s(p(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1)))))))))))))))))))))))))))
twice(s(x1)) → p(p(s(s(s(p(p(s(s(s(twice(p(s(p(s(x1)))))))))))))))
p(p(s(x1))) → p(x1)
p(s(x1)) → x1
p(0(x1)) → 0(s(s(s(s(s(s(s(s(s(s(s(x1))))))))))))
0(x1) → x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
SQ(s(x1)) → SQ(p(p(p(p(p(p(s(s(s(s(s(s(x1)))))))))))))
sq(0(x1)) → p(s(p(s(p(p(p(p(s(s(s(s(0(p(s(p(s(x1)))))))))))))))))
sq(s(x1)) → s(p(s(p(s(p(p(s(s(twice(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1)))))))))))))))))))))))))))))))))
twice(0(x1)) → p(p(p(p(s(s(p(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1)))))))))))))))))))))))))))
twice(s(x1)) → p(p(s(s(s(p(p(s(s(s(twice(p(s(p(s(x1)))))))))))))))
p(p(s(x1))) → p(x1)
p(s(x1)) → x1
p(0(x1)) → 0(s(s(s(s(s(s(s(s(s(s(s(x1))))))))))))
0(x1) → x1