a(a(a(x1))) → b(b(x1))
b(b(b(x1))) → c(x1)
c(x1) → d(d(x1))
d(x1) → a(a(x1))
↳ QTRS
↳ DependencyPairsProof
a(a(a(x1))) → b(b(x1))
b(b(b(x1))) → c(x1)
c(x1) → d(d(x1))
d(x1) → a(a(x1))
D(x1) → A(x1)
A(a(a(x1))) → B(x1)
B(b(b(x1))) → C(x1)
C(x1) → D(x1)
A(a(a(x1))) → B(b(x1))
D(x1) → A(a(x1))
C(x1) → D(d(x1))
a(a(a(x1))) → b(b(x1))
b(b(b(x1))) → c(x1)
c(x1) → d(d(x1))
d(x1) → a(a(x1))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
D(x1) → A(x1)
A(a(a(x1))) → B(x1)
B(b(b(x1))) → C(x1)
C(x1) → D(x1)
A(a(a(x1))) → B(b(x1))
D(x1) → A(a(x1))
C(x1) → D(d(x1))
a(a(a(x1))) → b(b(x1))
b(b(b(x1))) → c(x1)
c(x1) → d(d(x1))
d(x1) → a(a(x1))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
D(x1) → A(x1)
A(a(a(x1))) → B(x1)
C(x1) → D(x1)
A(a(a(x1))) → B(b(x1))
Used ordering: Polynomial interpretation [25,35]:
B(b(b(x1))) → C(x1)
D(x1) → A(a(x1))
C(x1) → D(d(x1))
The value of delta used in the strict ordering is 1/8.
POL(C(x1)) = 3/4 + (1/4)x_1
POL(c(x1)) = 4 + x_1
POL(D(x1)) = 1/4 + (1/4)x_1
POL(B(x1)) = (1/4)x_1
POL(a(x1)) = 1 + x_1
POL(A(x1)) = (1/4)x_1
POL(b(x1)) = 3/2 + x_1
POL(d(x1)) = 2 + x_1
c(x1) → d(d(x1))
b(b(b(x1))) → c(x1)
a(a(a(x1))) → b(b(x1))
d(x1) → a(a(x1))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
B(b(b(x1))) → C(x1)
D(x1) → A(a(x1))
C(x1) → D(d(x1))
a(a(a(x1))) → b(b(x1))
b(b(b(x1))) → c(x1)
c(x1) → d(d(x1))
d(x1) → a(a(x1))