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