TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60001 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (9567ms).
 | – Problem 2 was processed with processor SubtermCriterion (2ms).
 |    | – Problem 12 was processed with processor ReductionPairSAT (65ms).
 |    |    | – Problem 23 was processed with processor ReductionPairSAT (32ms).
 | – Problem 3 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 13 was processed with processor ReductionPairSAT (97ms).
 |    |    | – Problem 24 remains open; application of the following processors failed [DependencyGraph (1ms)].
 | – Problem 4 was processed with processor SubtermCriterion (2ms).
 |    | – Problem 14 was processed with processor ReductionPairSAT (415ms).
 |    |    | – Problem 25 remains open; application of the following processors failed [DependencyGraph (7ms)].
 | – Problem 5 was processed with processor SubtermCriterion (4ms).
 | – Problem 6 was processed with processor SubtermCriterion (1ms).
 | – Problem 7 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 15 was processed with processor ReductionPairSAT (86ms).
 |    |    | – Problem 26 remains open; application of the following processors failed [DependencyGraph (1ms)].
 | – Problem 8 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 16 was processed with processor PolynomialLinearRange4iUR (40ms).
 |    |    | – Problem 19 was processed with processor PolynomialLinearRange4iUR (50ms).
 | – Problem 9 was processed with processor SubtermCriterion (1ms).
 | – Problem 10 was processed with processor SubtermCriterion (4ms).
 |    | – Problem 17 was processed with processor PolynomialLinearRange4iUR (107ms).
 |    |    | – Problem 18 was processed with processor PolynomialLinearRange4iUR (80ms).
 |    |    |    | – Problem 20 was processed with processor PolynomialLinearRange4iUR (35ms).
 |    |    |    |    | – Problem 21 was processed with processor PolynomialLinearRange4iUR (15ms).
 | – Problem 11 was processed with processor ReductionPairSAT (17951ms).
 |    | – Problem 22 remains open; application of the following processors failed [DependencyGraph (989ms), ReductionPairSAT (timeout)].

The following open problems remain:

Open Dependency Pair Problem 22

Dependency Pairs

mark#(U11(X1, X2)) → mark#(X1)mark#(isNat(X)) → active#(isNat(X))
active#(U31(tt)) → mark#(0)active#(isNat(s(V1))) → mark#(isNat(V1))
active#(x(N, s(M))) → mark#(U41(and(isNat(M), isNat(N)), M, N))active#(plus(N, s(M))) → mark#(U21(and(isNat(M), isNat(N)), M, N))
mark#(plus(X1, X2)) → mark#(X2)mark#(and(X1, X2)) → active#(and(mark(X1), X2))
mark#(s(X)) → mark#(X)active#(U11(tt, N)) → mark#(N)
mark#(U11(X1, X2)) → active#(U11(mark(X1), X2))mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2)))
mark#(s(X)) → active#(s(mark(X)))active#(isNat(0)) → mark#(tt)
mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2)))active#(x(N, 0)) → mark#(U31(isNat(N)))
active#(isNat(plus(V1, V2))) → mark#(and(isNat(V1), isNat(V2)))mark#(U31(X)) → active#(U31(mark(X)))
active#(and(tt, X)) → mark#(X)active#(U41(tt, M, N)) → mark#(plus(x(N, M), N))
active#(isNat(x(V1, V2))) → mark#(and(isNat(V1), isNat(V2)))mark#(plus(X1, X2)) → mark#(X1)
active#(U21(tt, M, N)) → mark#(s(plus(N, M)))mark#(and(X1, X2)) → mark#(X1)
mark#(x(X1, X2)) → mark#(X2)mark#(U41(X1, X2, X3)) → mark#(X1)
active#(plus(N, 0)) → mark#(U11(isNat(N), N))mark#(U41(X1, X2, X3)) → active#(U41(mark(X1), X2, X3))
mark#(U31(X)) → mark#(X)mark#(U21(X1, X2, X3)) → mark#(X1)
mark#(U21(X1, X2, X3)) → active#(U21(mark(X1), X2, X3))mark#(x(X1, X2)) → mark#(X1)

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Open Dependency Pair Problem 25

Dependency Pairs

U21#(X1, active(X2), X3) → U21#(X1, X2, X3)U21#(X1, X2, mark(X3)) → U21#(X1, X2, X3)
U21#(X1, X2, active(X3)) → U21#(X1, X2, X3)

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Open Dependency Pair Problem 24

Dependency Pairs

U11#(X1, active(X2)) → U11#(X1, X2)

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Open Dependency Pair Problem 26

Dependency Pairs

x#(X1, active(X2)) → x#(X1, X2)

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

active#(plus(N, s(M))) → and#(isNat(M), isNat(N))mark#(U11(X1, X2)) → mark#(X1)
U21#(X1, X2, active(X3)) → U21#(X1, X2, X3)U11#(mark(X1), X2) → U11#(X1, X2)
x#(active(X1), X2) → x#(X1, X2)active#(U21(tt, M, N)) → s#(plus(N, M))
x#(X1, active(X2)) → x#(X1, X2)active#(plus(N, s(M))) → mark#(U21(and(isNat(M), isNat(N)), M, N))
active#(isNat(x(V1, V2))) → and#(isNat(V1), isNat(V2))isNat#(active(X)) → isNat#(X)
mark#(s(X)) → mark#(X)x#(X1, mark(X2)) → x#(X1, X2)
mark#(U11(X1, X2)) → active#(U11(mark(X1), X2))active#(plus(N, 0)) → isNat#(N)
active#(isNat(plus(V1, V2))) → isNat#(V2)mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2)))
U11#(active(X1), X2) → U11#(X1, X2)active#(x(N, 0)) → mark#(U31(isNat(N)))
active#(isNat(plus(V1, V2))) → mark#(and(isNat(V1), isNat(V2)))U21#(mark(X1), X2, X3) → U21#(X1, X2, X3)
and#(mark(X1), X2) → and#(X1, X2)active#(isNat(x(V1, V2))) → mark#(and(isNat(V1), isNat(V2)))
U11#(X1, mark(X2)) → U11#(X1, X2)mark#(and(X1, X2)) → mark#(X1)
U11#(X1, active(X2)) → U11#(X1, X2)U41#(X1, active(X2), X3) → U41#(X1, X2, X3)
mark#(U41(X1, X2, X3)) → mark#(X1)U21#(active(X1), X2, X3) → U21#(X1, X2, X3)
mark#(U31(X)) → mark#(X)plus#(mark(X1), X2) → plus#(X1, X2)
plus#(active(X1), X2) → plus#(X1, X2)active#(x(N, 0)) → isNat#(N)
mark#(tt) → active#(tt)active#(x(N, s(M))) → U41#(and(isNat(M), isNat(N)), M, N)
active#(U31(tt)) → mark#(0)x#(mark(X1), X2) → x#(X1, X2)
active#(x(N, s(M))) → mark#(U41(and(isNat(M), isNat(N)), M, N))U21#(X1, X2, mark(X3)) → U21#(X1, X2, X3)
U41#(X1, X2, active(X3)) → U41#(X1, X2, X3)active#(isNat(s(V1))) → isNat#(V1)
active#(plus(N, s(M))) → U21#(and(isNat(M), isNat(N)), M, N)mark#(0) → active#(0)
mark#(s(X)) → active#(s(mark(X)))U41#(X1, X2, mark(X3)) → U41#(X1, X2, X3)
mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2)))active#(U21(tt, M, N)) → plus#(N, M)
active#(x(N, s(M))) → isNat#(M)active#(plus(N, s(M))) → isNat#(M)
active#(x(N, s(M))) → isNat#(N)active#(plus(N, 0)) → mark#(U11(isNat(N), N))
mark#(U31(X)) → U31#(mark(X))mark#(U41(X1, X2, X3)) → active#(U41(mark(X1), X2, X3))
mark#(x(X1, X2)) → x#(mark(X1), mark(X2))active#(plus(N, 0)) → U11#(isNat(N), N)
mark#(x(X1, X2)) → mark#(X1)active#(isNat(x(V1, V2))) → isNat#(V1)
mark#(s(X)) → s#(mark(X))active#(x(N, 0)) → U31#(isNat(N))
active#(isNat(s(V1))) → mark#(isNat(V1))active#(U41(tt, M, N)) → x#(N, M)
plus#(X1, mark(X2)) → plus#(X1, X2)active#(U11(tt, N)) → mark#(N)
active#(isNat(plus(V1, V2))) → and#(isNat(V1), isNat(V2))U21#(X1, mark(X2), X3) → U21#(X1, X2, X3)
active#(isNat(0)) → mark#(tt)mark#(and(X1, X2)) → and#(mark(X1), X2)
mark#(plus(X1, X2)) → mark#(X1)active#(U21(tt, M, N)) → mark#(s(plus(N, M)))
mark#(x(X1, X2)) → mark#(X2)U21#(X1, active(X2), X3) → U21#(X1, X2, X3)
U41#(active(X1), X2, X3) → U41#(X1, X2, X3)plus#(X1, active(X2)) → plus#(X1, X2)
mark#(U21(X1, X2, X3)) → mark#(X1)U31#(mark(X)) → U31#(X)
and#(active(X1), X2) → and#(X1, X2)mark#(U41(X1, X2, X3)) → U41#(mark(X1), X2, X3)
active#(x(N, s(M))) → and#(isNat(M), isNat(N))mark#(isNat(X)) → active#(isNat(X))
active#(plus(N, s(M))) → isNat#(N)and#(X1, active(X2)) → and#(X1, X2)
isNat#(mark(X)) → isNat#(X)active#(isNat(plus(V1, V2))) → isNat#(V1)
mark#(isNat(X)) → isNat#(X)active#(isNat(x(V1, V2))) → isNat#(V2)
mark#(plus(X1, X2)) → mark#(X2)mark#(U11(X1, X2)) → U11#(mark(X1), X2)
mark#(and(X1, X2)) → active#(and(mark(X1), X2))U41#(mark(X1), X2, X3) → U41#(X1, X2, X3)
active#(U41(tt, M, N)) → plus#(x(N, M), N)and#(X1, mark(X2)) → and#(X1, X2)
U31#(active(X)) → U31#(X)mark#(U31(X)) → active#(U31(mark(X)))
mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2))active#(and(tt, X)) → mark#(X)
U41#(X1, mark(X2), X3) → U41#(X1, X2, X3)active#(U41(tt, M, N)) → mark#(plus(x(N, M), N))
mark#(U21(X1, X2, X3)) → U21#(mark(X1), X2, X3)s#(mark(X)) → s#(X)
s#(active(X)) → s#(X)mark#(U21(X1, X2, X3)) → active#(U21(mark(X1), X2, X3))

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Strategy

The following SCCs where found

U11#(X1, active(X2)) → U11#(X1, X2)U11#(active(X1), X2) → U11#(X1, X2)
U11#(mark(X1), X2) → U11#(X1, X2)U11#(X1, mark(X2)) → U11#(X1, X2)
x#(mark(X1), X2) → x#(X1, X2)x#(X1, active(X2)) → x#(X1, X2)
x#(X1, mark(X2)) → x#(X1, X2)x#(active(X1), X2) → x#(X1, X2)
U31#(active(X)) → U31#(X)U31#(mark(X)) → U31#(X)
s#(mark(X)) → s#(X)s#(active(X)) → s#(X)
mark#(U11(X1, X2)) → mark#(X1)mark#(isNat(X)) → active#(isNat(X))
mark#(tt) → active#(tt)active#(U31(tt)) → mark#(0)
active#(isNat(s(V1))) → mark#(isNat(V1))active#(x(N, s(M))) → mark#(U41(and(isNat(M), isNat(N)), M, N))
active#(plus(N, s(M))) → mark#(U21(and(isNat(M), isNat(N)), M, N))mark#(plus(X1, X2)) → mark#(X2)
mark#(and(X1, X2)) → active#(and(mark(X1), X2))mark#(s(X)) → mark#(X)
active#(U11(tt, N)) → mark#(N)mark#(U11(X1, X2)) → active#(U11(mark(X1), X2))
mark#(0) → active#(0)mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2)))
mark#(s(X)) → active#(s(mark(X)))mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2)))
active#(isNat(0)) → mark#(tt)active#(x(N, 0)) → mark#(U31(isNat(N)))
active#(isNat(plus(V1, V2))) → mark#(and(isNat(V1), isNat(V2)))mark#(U31(X)) → active#(U31(mark(X)))
active#(and(tt, X)) → mark#(X)active#(U41(tt, M, N)) → mark#(plus(x(N, M), N))
mark#(plus(X1, X2)) → mark#(X1)active#(isNat(x(V1, V2))) → mark#(and(isNat(V1), isNat(V2)))
active#(U21(tt, M, N)) → mark#(s(plus(N, M)))mark#(and(X1, X2)) → mark#(X1)
mark#(x(X1, X2)) → mark#(X2)mark#(U41(X1, X2, X3)) → mark#(X1)
active#(plus(N, 0)) → mark#(U11(isNat(N), N))mark#(U41(X1, X2, X3)) → active#(U41(mark(X1), X2, X3))
mark#(U31(X)) → mark#(X)mark#(U21(X1, X2, X3)) → mark#(X1)
mark#(U21(X1, X2, X3)) → active#(U21(mark(X1), X2, X3))mark#(x(X1, X2)) → mark#(X1)
isNat#(active(X)) → isNat#(X)isNat#(mark(X)) → isNat#(X)
and#(active(X1), X2) → and#(X1, X2)and#(X1, active(X2)) → and#(X1, X2)
and#(mark(X1), X2) → and#(X1, X2)and#(X1, mark(X2)) → and#(X1, X2)
U41#(X1, active(X2), X3) → U41#(X1, X2, X3)U41#(X1, X2, mark(X3)) → U41#(X1, X2, X3)
U41#(active(X1), X2, X3) → U41#(X1, X2, X3)U41#(mark(X1), X2, X3) → U41#(X1, X2, X3)
U41#(X1, X2, active(X3)) → U41#(X1, X2, X3)U41#(X1, mark(X2), X3) → U41#(X1, X2, X3)
plus#(X1, mark(X2)) → plus#(X1, X2)plus#(X1, active(X2)) → plus#(X1, X2)
plus#(mark(X1), X2) → plus#(X1, X2)plus#(active(X1), X2) → plus#(X1, X2)
U21#(X1, active(X2), X3) → U21#(X1, X2, X3)U21#(X1, mark(X2), X3) → U21#(X1, X2, X3)
U21#(active(X1), X2, X3) → U21#(X1, X2, X3)U21#(X1, X2, mark(X3)) → U21#(X1, X2, X3)
U21#(mark(X1), X2, X3) → U21#(X1, X2, X3)U21#(X1, X2, active(X3)) → U21#(X1, X2, X3)

Problem 2: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

and#(active(X1), X2) → and#(X1, X2)and#(X1, active(X2)) → and#(X1, X2)
and#(mark(X1), X2) → and#(X1, X2)and#(X1, mark(X2)) → and#(X1, X2)

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

and#(active(X1), X2) → and#(X1, X2)and#(mark(X1), X2) → and#(X1, X2)

Problem 12: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

and#(X1, active(X2)) → and#(X1, X2)and#(X1, mark(X2)) → and#(X1, X2)

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Strategy

Function Precedence

mark < active < plus = and# = and = isNat = 0 = s = tt = U41 = U11 = U31 = U21 = x

Argument Filtering

plus: all arguments are removed from plus
mark: 1
and#: collapses to 2
and: collapses to 1
isNat: all arguments are removed from isNat
0: all arguments are removed from 0
s: all arguments are removed from s
tt: all arguments are removed from tt
U41: collapses to 1
U11: all arguments are removed from U11
active: collapses to 1
U31: all arguments are removed from U31
U21: all arguments are removed from U21
x: 1 2

Status

plus: multiset
mark: multiset
isNat: multiset
0: multiset
s: multiset
tt: multiset
U11: multiset
U31: multiset
U21: multiset
x: lexicographic with permutation 1 → 1 2 → 2

Usable Rules

There are no usable rules.

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

The following dependency pairs (at least) can be eliminated according to the given precedence.

and#(X1, mark(X2)) → and#(X1, X2)

Problem 23: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

and#(X1, active(X2)) → and#(X1, X2)

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Strategy

Function Precedence

active < and# < plus = mark = and = isNat = 0 = s = tt = U41 = U11 = U31 = U21 = x

Argument Filtering

plus: 1 2
mark: all arguments are removed from mark
and#: collapses to 2
and: all arguments are removed from and
isNat: all arguments are removed from isNat
0: all arguments are removed from 0
s: all arguments are removed from s
tt: all arguments are removed from tt
U41: all arguments are removed from U41
U11: all arguments are removed from U11
active: 1
U31: all arguments are removed from U31
U21: 1 2
x: 2

Status

plus: lexicographic with permutation 1 → 2 2 → 1
mark: multiset
and: multiset
isNat: multiset
0: multiset
s: multiset
tt: multiset
U41: multiset
U11: multiset
active: multiset
U31: multiset
U21: lexicographic with permutation 1 → 1 2 → 2
x: lexicographic with permutation 2 → 1

Usable Rules

There are no usable rules.

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

The following dependency pairs (at least) can be eliminated according to the given precedence.

and#(X1, active(X2)) → and#(X1, X2)

Problem 3: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

U11#(X1, active(X2)) → U11#(X1, X2)U11#(active(X1), X2) → U11#(X1, X2)
U11#(mark(X1), X2) → U11#(X1, X2)U11#(X1, mark(X2)) → U11#(X1, X2)

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

U11#(active(X1), X2) → U11#(X1, X2)U11#(mark(X1), X2) → U11#(X1, X2)

Problem 13: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

U11#(X1, active(X2)) → U11#(X1, X2)U11#(X1, mark(X2)) → U11#(X1, X2)

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Strategy

Function Precedence

mark < active < plus = U11# = and = isNat = 0 = s = tt = U41 = U11 = U31 = U21 = x

Argument Filtering

plus: all arguments are removed from plus
U11#: collapses to 2
mark: 1
and: 1
isNat: 1
0: all arguments are removed from 0
s: collapses to 1
tt: all arguments are removed from tt
U41: all arguments are removed from U41
U11: 1 2
active: collapses to 1
U31: collapses to 1
U21: all arguments are removed from U21
x: all arguments are removed from x

Status

plus: multiset
mark: multiset
and: lexicographic with permutation 1 → 1
isNat: lexicographic with permutation 1 → 1
0: multiset
tt: multiset
U41: multiset
U11: lexicographic with permutation 1 → 2 2 → 1
U21: multiset
x: multiset

Usable Rules

There are no usable rules.

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

The following dependency pairs (at least) can be eliminated according to the given precedence.

U11#(X1, mark(X2)) → U11#(X1, X2)

Problem 4: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

U21#(X1, active(X2), X3) → U21#(X1, X2, X3)U21#(X1, mark(X2), X3) → U21#(X1, X2, X3)
U21#(active(X1), X2, X3) → U21#(X1, X2, X3)U21#(X1, X2, mark(X3)) → U21#(X1, X2, X3)
U21#(mark(X1), X2, X3) → U21#(X1, X2, X3)U21#(X1, X2, active(X3)) → U21#(X1, X2, X3)

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

U21#(active(X1), X2, X3) → U21#(X1, X2, X3)U21#(mark(X1), X2, X3) → U21#(X1, X2, X3)

Problem 14: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

U21#(X1, mark(X2), X3) → U21#(X1, X2, X3)U21#(X1, active(X2), X3) → U21#(X1, X2, X3)
U21#(X1, X2, mark(X3)) → U21#(X1, X2, X3)U21#(X1, X2, active(X3)) → U21#(X1, X2, X3)

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Strategy

Function Precedence

active < mark < plus = U21# = and = isNat = 0 = s = tt = U41 = U11 = U31 = U21 = x

Argument Filtering

plus: all arguments are removed from plus
U21#: collapses to 2
mark: 1
and: 1 2
isNat: all arguments are removed from isNat
0: all arguments are removed from 0
s: all arguments are removed from s
tt: all arguments are removed from tt
U41: all arguments are removed from U41
U11: 1 2
active: collapses to 1
U31: all arguments are removed from U31
U21: collapses to 3
x: all arguments are removed from x

Status

plus: multiset
mark: multiset
and: lexicographic with permutation 1 → 2 2 → 1
isNat: multiset
0: multiset
s: multiset
tt: multiset
U41: multiset
U11: lexicographic with permutation 1 → 1 2 → 2
U31: multiset
x: multiset

Usable Rules

There are no usable rules.

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

The following dependency pairs (at least) can be eliminated according to the given precedence.

U21#(X1, mark(X2), X3) → U21#(X1, X2, X3)

Problem 5: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

s#(mark(X)) → s#(X)s#(active(X)) → s#(X)

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

s#(mark(X)) → s#(X)s#(active(X)) → s#(X)

Problem 6: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

U31#(active(X)) → U31#(X)U31#(mark(X)) → U31#(X)

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

U31#(active(X)) → U31#(X)U31#(mark(X)) → U31#(X)

Problem 7: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

x#(mark(X1), X2) → x#(X1, X2)x#(X1, active(X2)) → x#(X1, X2)
x#(X1, mark(X2)) → x#(X1, X2)x#(active(X1), X2) → x#(X1, X2)

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

x#(mark(X1), X2) → x#(X1, X2)x#(active(X1), X2) → x#(X1, X2)

Problem 15: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

x#(X1, active(X2)) → x#(X1, X2)x#(X1, mark(X2)) → x#(X1, X2)

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Strategy

Function Precedence

mark < active < plus = and = isNat = 0 = s = tt = U41 = U11 = U31 = x# = U21 = x

Argument Filtering

plus: all arguments are removed from plus
mark: 1
and: 1 2
isNat: all arguments are removed from isNat
0: all arguments are removed from 0
s: collapses to 1
tt: all arguments are removed from tt
U41: 2
U11: 1 2
active: collapses to 1
U31: collapses to 1
x#: 1 2
U21: all arguments are removed from U21
x: all arguments are removed from x

Status

plus: multiset
mark: multiset
and: lexicographic with permutation 1 → 1 2 → 2
isNat: multiset
0: multiset
tt: multiset
U41: lexicographic with permutation 2 → 1
U11: lexicographic with permutation 1 → 2 2 → 1
x#: lexicographic with permutation 1 → 1 2 → 2
U21: multiset
x: multiset

Usable Rules

There are no usable rules.

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

The following dependency pairs (at least) can be eliminated according to the given precedence.

x#(X1, mark(X2)) → x#(X1, X2)

Problem 8: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

plus#(X1, mark(X2)) → plus#(X1, X2)plus#(X1, active(X2)) → plus#(X1, X2)
plus#(mark(X1), X2) → plus#(X1, X2)plus#(active(X1), X2) → plus#(X1, X2)

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

plus#(mark(X1), X2) → plus#(X1, X2)plus#(active(X1), X2) → plus#(X1, X2)

Problem 16: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

plus#(X1, active(X2)) → plus#(X1, X2)plus#(X1, mark(X2)) → plus#(X1, X2)

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Strategy

Polynomial Interpretation

There are no usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

plus#(X1, active(X2)) → plus#(X1, X2)

Problem 19: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

plus#(X1, mark(X2)) → plus#(X1, X2)

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Strategy

Polynomial Interpretation

There are no usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

plus#(X1, mark(X2)) → plus#(X1, X2)

Problem 9: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

isNat#(active(X)) → isNat#(X)isNat#(mark(X)) → isNat#(X)

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

isNat#(active(X)) → isNat#(X)isNat#(mark(X)) → isNat#(X)

Problem 10: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

U41#(X1, active(X2), X3) → U41#(X1, X2, X3)U41#(X1, X2, mark(X3)) → U41#(X1, X2, X3)
U41#(active(X1), X2, X3) → U41#(X1, X2, X3)U41#(mark(X1), X2, X3) → U41#(X1, X2, X3)
U41#(X1, X2, active(X3)) → U41#(X1, X2, X3)U41#(X1, mark(X2), X3) → U41#(X1, X2, X3)

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

U41#(active(X1), X2, X3) → U41#(X1, X2, X3)U41#(mark(X1), X2, X3) → U41#(X1, X2, X3)

Problem 17: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

U41#(X1, active(X2), X3) → U41#(X1, X2, X3)U41#(X1, X2, mark(X3)) → U41#(X1, X2, X3)
U41#(X1, mark(X2), X3) → U41#(X1, X2, X3)U41#(X1, X2, active(X3)) → U41#(X1, X2, X3)

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Strategy

Polynomial Interpretation

There are no usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

U41#(X1, X2, active(X3)) → U41#(X1, X2, X3)

Problem 18: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

U41#(X1, active(X2), X3) → U41#(X1, X2, X3)U41#(X1, X2, mark(X3)) → U41#(X1, X2, X3)
U41#(X1, mark(X2), X3) → U41#(X1, X2, X3)

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Strategy

Polynomial Interpretation

There are no usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

U41#(X1, mark(X2), X3) → U41#(X1, X2, X3)

Problem 20: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

U41#(X1, active(X2), X3) → U41#(X1, X2, X3)U41#(X1, X2, mark(X3)) → U41#(X1, X2, X3)

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Strategy

Polynomial Interpretation

There are no usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

U41#(X1, X2, mark(X3)) → U41#(X1, X2, X3)

Problem 21: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

U41#(X1, active(X2), X3) → U41#(X1, X2, X3)

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Strategy

Polynomial Interpretation

There are no usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

U41#(X1, active(X2), X3) → U41#(X1, X2, X3)

Problem 11: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

mark#(U11(X1, X2)) → mark#(X1)mark#(isNat(X)) → active#(isNat(X))
mark#(tt) → active#(tt)active#(U31(tt)) → mark#(0)
active#(isNat(s(V1))) → mark#(isNat(V1))active#(x(N, s(M))) → mark#(U41(and(isNat(M), isNat(N)), M, N))
active#(plus(N, s(M))) → mark#(U21(and(isNat(M), isNat(N)), M, N))mark#(plus(X1, X2)) → mark#(X2)
mark#(and(X1, X2)) → active#(and(mark(X1), X2))mark#(s(X)) → mark#(X)
active#(U11(tt, N)) → mark#(N)mark#(U11(X1, X2)) → active#(U11(mark(X1), X2))
mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2)))mark#(0) → active#(0)
mark#(s(X)) → active#(s(mark(X)))active#(isNat(0)) → mark#(tt)
mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2)))active#(x(N, 0)) → mark#(U31(isNat(N)))
active#(isNat(plus(V1, V2))) → mark#(and(isNat(V1), isNat(V2)))mark#(U31(X)) → active#(U31(mark(X)))
active#(and(tt, X)) → mark#(X)active#(U41(tt, M, N)) → mark#(plus(x(N, M), N))
active#(isNat(x(V1, V2))) → mark#(and(isNat(V1), isNat(V2)))mark#(plus(X1, X2)) → mark#(X1)
active#(U21(tt, M, N)) → mark#(s(plus(N, M)))mark#(and(X1, X2)) → mark#(X1)
mark#(x(X1, X2)) → mark#(X2)mark#(U41(X1, X2, X3)) → mark#(X1)
active#(plus(N, 0)) → mark#(U11(isNat(N), N))mark#(U41(X1, X2, X3)) → active#(U41(mark(X1), X2, X3))
mark#(U31(X)) → mark#(X)mark#(U21(X1, X2, X3)) → mark#(X1)
mark#(U21(X1, X2, X3)) → active#(U21(mark(X1), X2, X3))mark#(x(X1, X2)) → mark#(X1)

Rewrite Rules

active(U11(tt, N)) → mark(N)active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U31(X)) → active(U31(mark(X)))mark(0) → active(0)
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
U11(mark(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)plus(X1, active(X2)) → plus(X1, X2)
U31(mark(X)) → U31(X)U31(active(X)) → U31(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)

Original Signature

Termination of terms over the following signature is verified: plus, mark, and, isNat, 0, s, tt, U41, active, U11, U31, U21, x

Strategy

Function Precedence

active < active# < plus = mark = mark# = and = isNat = 0 = s = tt = U41 = U11 = U31 = U21 = x

Argument Filtering

plus: all arguments are removed from plus
mark: all arguments are removed from mark
mark#: all arguments are removed from mark#
and: all arguments are removed from and
isNat: all arguments are removed from isNat
0: all arguments are removed from 0
s: all arguments are removed from s
tt: all arguments are removed from tt
U41: all arguments are removed from U41
U11: all arguments are removed from U11
active: collapses to 1
U31: all arguments are removed from U31
active#: collapses to 1
U21: all arguments are removed from U21
x: all arguments are removed from x

Status

plus: multiset
mark: multiset
mark#: multiset
and: multiset
isNat: multiset
0: multiset
s: multiset
tt: multiset
U41: multiset
U11: multiset
U31: multiset
U21: multiset
x: multiset

Usable Rules

mark(isNat(X)) → active(isNat(X))active(plus(N, 0)) → mark(U11(isNat(N), N))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
and(active(X1), X2) → and(X1, X2)U21(mark(X1), X2, X3) → U21(X1, X2, X3)
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))plus(mark(X1), X2) → plus(X1, X2)
U11(active(X1), X2) → U11(X1, X2)U11(X1, mark(X2)) → U11(X1, X2)
active(isNat(s(V1))) → mark(isNat(V1))plus(X1, active(X2)) → plus(X1, X2)
U31(active(X)) → U31(X)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)isNat(active(X)) → isNat(X)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)active(U11(tt, N)) → mark(N)
active(and(tt, X)) → mark(X)mark(U31(X)) → active(U31(mark(X)))
plus(active(X1), X2) → plus(X1, X2)and(mark(X1), X2) → and(X1, X2)
x(mark(X1), X2) → x(X1, X2)U21(active(X1), X2, X3) → U21(X1, X2, X3)
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))active(U31(tt)) → mark(0)
U31(mark(X)) → U31(X)mark(s(X)) → active(s(mark(X)))
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))
active(U41(tt, M, N)) → mark(plus(x(N, M), N))and(X1, mark(X2)) → and(X1, X2)
mark(and(X1, X2)) → active(and(mark(X1), X2))x(active(X1), X2) → x(X1, X2)
U41(active(X1), X2, X3) → U41(X1, X2, X3)active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))
U11(X1, active(X2)) → U11(X1, X2)x(X1, active(X2)) → x(X1, X2)
active(isNat(0)) → mark(tt)s(active(X)) → s(X)
mark(0) → active(0)active(U21(tt, M, N)) → mark(s(plus(N, M)))
U11(mark(X1), X2) → U11(X1, X2)mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
plus(X1, mark(X2)) → plus(X1, X2)U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)mark(tt) → active(tt)
active(x(N, 0)) → mark(U31(isNat(N)))U21(X1, active(X2), X3) → U21(X1, X2, X3)
x(X1, mark(X2)) → x(X1, X2)isNat(mark(X)) → isNat(X)
s(mark(X)) → s(X)and(X1, active(X2)) → and(X1, X2)

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

The following dependency pairs (at least) can be eliminated according to the given precedence.

mark#(0) → active#(0)mark#(tt) → active#(tt)