a__U11(tt, M, N) → a__U12(tt, M, N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
↳ QTRS
↳ DependencyPairsProof
a__U11(tt, M, N) → a__U12(tt, M, N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__U12(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, s(M)) → A__U11(tt, M, N)
MARK(plus(X1, X2)) → MARK(X2)
A__U11(tt, M, N) → A__U12(tt, M, N)
A__PLUS(N, 0) → MARK(N)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__U12(tt, M, N) → MARK(N)
A__U12(tt, M, N) → MARK(M)
MARK(U12(X1, X2, X3)) → A__U12(mark(X1), X2, X3)
a__U11(tt, M, N) → a__U12(tt, M, N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__U12(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, s(M)) → A__U11(tt, M, N)
MARK(plus(X1, X2)) → MARK(X2)
A__U11(tt, M, N) → A__U12(tt, M, N)
A__PLUS(N, 0) → MARK(N)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__U12(tt, M, N) → MARK(N)
A__U12(tt, M, N) → MARK(M)
MARK(U12(X1, X2, X3)) → A__U12(mark(X1), X2, X3)
a__U11(tt, M, N) → a__U12(tt, M, N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MARK(s(X)) → MARK(X)
A__PLUS(N, s(M)) → A__U11(tt, M, N)
Used ordering: Polynomial interpretation [25,35]:
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
A__U12(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
A__U11(tt, M, N) → A__U12(tt, M, N)
A__PLUS(N, 0) → MARK(N)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__U12(tt, M, N) → MARK(N)
A__U12(tt, M, N) → MARK(M)
MARK(U12(X1, X2, X3)) → A__U12(mark(X1), X2, X3)
The value of delta used in the strict ordering is 16.
POL(plus(x1, x2)) = x_1 + (2)x_2
POL(A__PLUS(x1, x2)) = (4)x_1 + (4)x_2
POL(U11(x1, x2, x3)) = (2)x_1 + (2)x_2 + x_3
POL(mark(x1)) = x_1
POL(a__U12(x1, x2, x3)) = (2)x_1 + (2)x_2 + x_3
POL(0) = 0
POL(a__plus(x1, x2)) = x_1 + (2)x_2
POL(A__U12(x1, x2, x3)) = (4)x_2 + (4)x_3
POL(MARK(x1)) = (4)x_1
POL(A__U11(x1, x2, x3)) = (4)x_2 + (4)x_3
POL(tt) = 2
POL(a__U11(x1, x2, x3)) = (2)x_1 + (2)x_2 + x_3
POL(s(x1)) = 4 + x_1
POL(U12(x1, x2, x3)) = (2)x_1 + (2)x_2 + x_3
a__U11(tt, M, N) → a__U12(tt, M, N)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(N, 0) → mark(N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
a__plus(N, s(M)) → a__U11(tt, M, N)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__U11(X1, X2, X3) → U11(X1, X2, X3)
mark(0) → 0
a__plus(X1, X2) → plus(X1, X2)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
A__U11(tt, M, N) → A__U12(tt, M, N)
A__U12(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → MARK(N)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__U12(tt, M, N) → MARK(M)
A__U12(tt, M, N) → MARK(N)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(X1, X2, X3)) → A__U12(mark(X1), X2, X3)
a__U11(tt, M, N) → a__U12(tt, M, N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
A__U11(tt, M, N) → A__U12(tt, M, N)
A__U12(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__U12(tt, M, N) → MARK(M)
A__U12(tt, M, N) → MARK(N)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(X1, X2, X3)) → A__U12(mark(X1), X2, X3)
Used ordering: Polynomial interpretation [25,35]:
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
A__PLUS(N, 0) → MARK(N)
The value of delta used in the strict ordering is 1/4.
POL(plus(x1, x2)) = 4 + (2)x_1 + (4)x_2
POL(A__PLUS(x1, x2)) = 1/4 + (2)x_1
POL(U11(x1, x2, x3)) = (7/2)x_1 + x_2 + (2)x_3
POL(mark(x1)) = x_1
POL(a__U12(x1, x2, x3)) = (7/4)x_1 + x_2 + (2)x_3
POL(0) = 0
POL(a__plus(x1, x2)) = 4 + (2)x_1 + (4)x_2
POL(A__U12(x1, x2, x3)) = (1/4)x_1 + (2)x_2 + (4)x_3
POL(MARK(x1)) = 1/4 + (2)x_1
POL(A__U11(x1, x2, x3)) = (1/2)x_1 + (2)x_2 + (4)x_3
POL(tt) = 5/2
POL(a__U11(x1, x2, x3)) = (7/2)x_1 + x_2 + (2)x_3
POL(s(x1)) = 3 + (1/4)x_1
POL(U12(x1, x2, x3)) = (7/4)x_1 + x_2 + (2)x_3
a__U11(tt, M, N) → a__U12(tt, M, N)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(N, 0) → mark(N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
a__plus(N, s(M)) → a__U11(tt, M, N)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__U11(X1, X2, X3) → U11(X1, X2, X3)
mark(0) → 0
a__plus(X1, X2) → plus(X1, X2)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
A__PLUS(N, 0) → MARK(N)
a__U11(tt, M, N) → a__U12(tt, M, N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
a__U11(tt, M, N) → a__U12(tt, M, N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
The value of delta used in the strict ordering is 1/4.
POL(MARK(x1)) = (1/2)x_1
POL(U11(x1, x2, x3)) = 9/4 + x_1
POL(U12(x1, x2, x3)) = 1/2 + (3/2)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
a__U11(tt, M, N) → a__U12(tt, M, N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)