a(a(x1)) → b(c(x1))
b(b(x1)) → c(d(x1))
c(c(x1)) → d(d(d(x1)))
d(d(d(x1))) → a(c(x1))
↳ QTRS
↳ DependencyPairsProof
a(a(x1)) → b(c(x1))
b(b(x1)) → c(d(x1))
c(c(x1)) → d(d(d(x1)))
d(d(d(x1))) → a(c(x1))
A(a(x1)) → B(c(x1))
C(c(x1)) → D(d(x1))
C(c(x1)) → D(d(d(x1)))
B(b(x1)) → D(x1)
A(a(x1)) → C(x1)
D(d(d(x1))) → C(x1)
C(c(x1)) → D(x1)
D(d(d(x1))) → A(c(x1))
B(b(x1)) → C(d(x1))
a(a(x1)) → b(c(x1))
b(b(x1)) → c(d(x1))
c(c(x1)) → d(d(d(x1)))
d(d(d(x1))) → a(c(x1))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
A(a(x1)) → B(c(x1))
C(c(x1)) → D(d(x1))
C(c(x1)) → D(d(d(x1)))
B(b(x1)) → D(x1)
A(a(x1)) → C(x1)
D(d(d(x1))) → C(x1)
C(c(x1)) → D(x1)
D(d(d(x1))) → A(c(x1))
B(b(x1)) → C(d(x1))
a(a(x1)) → b(c(x1))
b(b(x1)) → c(d(x1))
c(c(x1)) → d(d(d(x1)))
d(d(d(x1))) → a(c(x1))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
C(c(x1)) → D(d(x1))
B(b(x1)) → D(x1)
A(a(x1)) → C(x1)
D(d(d(x1))) → C(x1)
C(c(x1)) → D(x1)
B(b(x1)) → C(d(x1))
Used ordering: Polynomial interpretation [25,35]:
A(a(x1)) → B(c(x1))
C(c(x1)) → D(d(d(x1)))
D(d(d(x1))) → A(c(x1))
The value of delta used in the strict ordering is 4.
POL(C(x1)) = 4 + (4)x_1
POL(c(x1)) = 3 + x_1
POL(B(x1)) = 4 + (4)x_1
POL(D(x1)) = (4)x_1
POL(a(x1)) = 3 + x_1
POL(A(x1)) = 4 + (4)x_1
POL(d(x1)) = 2 + x_1
POL(b(x1)) = 3 + x_1
d(d(d(x1))) → a(c(x1))
b(b(x1)) → c(d(x1))
a(a(x1)) → b(c(x1))
c(c(x1)) → d(d(d(x1)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
A(a(x1)) → B(c(x1))
C(c(x1)) → D(d(d(x1)))
D(d(d(x1))) → A(c(x1))
a(a(x1)) → b(c(x1))
b(b(x1)) → c(d(x1))
c(c(x1)) → d(d(d(x1)))
d(d(d(x1))) → a(c(x1))