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