TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60002 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (6942ms).
 | – Problem 2 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 12 was processed with processor ReductionPairSAT (79ms).
 |    |    | – Problem 19 was processed with processor ReductionPairSAT (70ms).
 | – Problem 3 was processed with processor SubtermCriterion (3ms).
 |    | – Problem 13 was processed with processor ReductionPairSAT (52ms).
 |    |    | – Problem 20 was processed with processor ReductionPairSAT (112ms).
 | – Problem 4 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 14 was processed with processor ReductionPairSAT (387ms).
 |    |    | – Problem 21 was processed with processor ReductionPairSAT (297ms).
 |    |    |    | – Problem 24 remains open; application of the following processors failed [DependencyGraph (4ms)].
 | – Problem 5 was processed with processor SubtermCriterion (1ms).
 | – Problem 6 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 15 was processed with processor ReductionPairSAT (408ms).
 |    |    | – Problem 22 was processed with processor ReductionPairSAT (264ms).
 |    |    |    | – Problem 25 remains open; application of the following processors failed [DependencyGraph (3ms)].
 | – Problem 7 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 16 was processed with processor PolynomialLinearRange4iUR (70ms).
 |    |    | – Problem 17 was processed with processor PolynomialLinearRange4iUR (60ms).
 | – Problem 8 was processed with processor SubtermCriterion (4ms).
 | – Problem 9 was processed with processor SubtermCriterion (1ms).
 | – Problem 10 was processed with processor ReductionPairSAT (14450ms).
 |    | – Problem 18 was processed with processor ReductionPairSAT (13111ms).
 |    |    | – Problem 23 remains open; application of the following processors failed [DependencyGraph (733ms), ReductionPairSAT (timeout)].
 | – Problem 11 was processed with processor SubtermCriterion (1ms).

The following open problems remain:

Open Dependency Pair Problem 23

Dependency Pairs

mark#(U11(X1, X2)) → mark#(X1)mark#(isNat(X)) → active#(isNat(X))
mark#(U21(X)) → active#(U21(mark(X)))active#(U21(tt)) → mark#(tt)
mark#(U21(X)) → mark#(X)mark#(U42(X1, X2, X3)) → active#(U42(mark(X1), X2, X3))
mark#(plus(X1, X2)) → mark#(X2)active#(plus(N, s(M))) → mark#(U41(isNat(M), M, N))
mark#(U31(X1, X2)) → mark#(X1)mark#(s(X)) → mark#(X)
active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2))mark#(U11(X1, X2)) → active#(U11(mark(X1), X2))
mark#(U31(X1, X2)) → active#(U31(mark(X1), X2))active#(U41(tt, M, N)) → mark#(U42(isNat(N), M, N))
mark#(U12(X)) → mark#(X)mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2)))
mark#(s(X)) → active#(s(mark(X)))active#(U11(tt, V2)) → mark#(U12(isNat(V2)))
active#(isNat(0)) → mark#(tt)active#(plus(N, 0)) → mark#(U31(isNat(N), N))
mark#(U42(X1, X2, X3)) → mark#(X1)active#(isNat(s(V1))) → mark#(U21(isNat(V1)))
mark#(U12(X)) → active#(U12(mark(X)))mark#(plus(X1, X2)) → mark#(X1)
active#(U12(tt)) → mark#(tt)mark#(U41(X1, X2, X3)) → mark#(X1)
mark#(U41(X1, X2, X3)) → active#(U41(mark(X1), X2, X3))active#(U31(tt, N)) → mark#(N)
active#(U42(tt, M, N)) → mark#(s(plus(N, M)))

Rewrite Rules

active(U11(tt, V2)) → mark(U12(isNat(V2)))active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(0) → active(0)
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)
U12(mark(X)) → U12(X)U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)U21(active(X)) → U21(X)
U31(mark(X1), X2) → U31(X1, X2)U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)U31(X1, active(X2)) → U31(X1, X2)
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)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(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)

Original Signature

Termination of terms over the following signature is verified: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, U12, U31, U21

Open Dependency Pair Problem 25

Dependency Pairs

U42#(X1, X2, active(X3)) → U42#(X1, X2, X3)U42#(X1, X2, mark(X3)) → U42#(X1, X2, X3)

Rewrite Rules

active(U11(tt, V2)) → mark(U12(isNat(V2)))active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(0) → active(0)
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)
U12(mark(X)) → U12(X)U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)U21(active(X)) → U21(X)
U31(mark(X1), X2) → U31(X1, X2)U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)U31(X1, active(X2)) → U31(X1, X2)
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)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(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)

Original Signature

Termination of terms over the following signature is verified: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, U12, U31, U21

Open Dependency Pair Problem 24

Dependency Pairs

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

Rewrite Rules

active(U11(tt, V2)) → mark(U12(isNat(V2)))active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(0) → active(0)
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)
U12(mark(X)) → U12(X)U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)U21(active(X)) → U21(X)
U31(mark(X1), X2) → U31(X1, X2)U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)U31(X1, active(X2)) → U31(X1, X2)
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)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(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)

Original Signature

Termination of terms over the following signature is verified: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, U12, U31, U21

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

U42#(X1, X2, active(X3)) → U42#(X1, X2, X3)mark#(U11(X1, X2)) → mark#(X1)
active#(U41(tt, M, N)) → isNat#(N)mark#(U21(X)) → active#(U21(mark(X)))
active#(U21(tt)) → mark#(tt)U11#(mark(X1), X2) → U11#(X1, X2)
active#(plus(N, 0)) → U31#(isNat(N), N)U12#(mark(X)) → U12#(X)
mark#(s(X)) → s#(mark(X))active#(plus(N, s(M))) → mark#(U41(isNat(M), M, N))
active#(U41(tt, M, N)) → U42#(isNat(N), M, N)plus#(X1, mark(X2)) → plus#(X1, X2)
U31#(active(X1), X2) → U31#(X1, X2)active#(plus(N, s(M))) → U41#(isNat(M), M, N)
mark#(U31(X1, X2)) → mark#(X1)isNat#(active(X)) → isNat#(X)
mark#(s(X)) → mark#(X)mark#(U11(X1, X2)) → active#(U11(mark(X1), X2))
active#(plus(N, 0)) → isNat#(N)active#(U41(tt, M, N)) → mark#(U42(isNat(N), M, N))
mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2)))mark#(U12(X)) → U12#(mark(X))
U11#(active(X1), X2) → U11#(X1, X2)U42#(mark(X1), X2, X3) → U42#(X1, X2, X3)
active#(U11(tt, V2)) → mark#(U12(isNat(V2)))U42#(active(X1), X2, X3) → U42#(X1, X2, X3)
active#(isNat(0)) → mark#(tt)active#(plus(N, 0)) → mark#(U31(isNat(N), N))
mark#(U42(X1, X2, X3)) → mark#(X1)U42#(X1, mark(X2), X3) → U42#(X1, X2, X3)
U42#(X1, active(X2), X3) → U42#(X1, X2, X3)mark#(U12(X)) → active#(U12(mark(X)))
U11#(X1, mark(X2)) → U11#(X1, X2)mark#(plus(X1, X2)) → mark#(X1)
active#(U11(tt, V2)) → U12#(isNat(V2))U11#(X1, active(X2)) → U11#(X1, X2)
active#(U12(tt)) → mark#(tt)U41#(X1, active(X2), X3) → U41#(X1, X2, X3)
mark#(U41(X1, X2, X3)) → mark#(X1)mark#(U42(X1, X2, X3)) → U42#(mark(X1), X2, X3)
U12#(active(X)) → U12#(X)U41#(active(X1), X2, X3) → U41#(X1, X2, X3)
plus#(X1, active(X2)) → plus#(X1, X2)plus#(mark(X1), X2) → plus#(X1, X2)
plus#(active(X1), X2) → plus#(X1, X2)active#(U42(tt, M, N)) → mark#(s(plus(N, M)))
mark#(U41(X1, X2, X3)) → U41#(mark(X1), X2, X3)mark#(isNat(X)) → active#(isNat(X))
mark#(tt) → active#(tt)U31#(X1, active(X2)) → U31#(X1, X2)
mark#(isNat(X)) → isNat#(X)active#(U42(tt, M, N)) → s#(plus(N, M))
active#(isNat(plus(V1, V2))) → isNat#(V1)isNat#(mark(X)) → isNat#(X)
U31#(X1, mark(X2)) → U31#(X1, X2)mark#(U21(X)) → mark#(X)
mark#(U42(X1, X2, X3)) → active#(U42(mark(X1), X2, X3))mark#(plus(X1, X2)) → mark#(X2)
mark#(U11(X1, X2)) → U11#(mark(X1), X2)U41#(mark(X1), X2, X3) → U41#(X1, X2, X3)
active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2))U41#(X1, X2, active(X3)) → U41#(X1, X2, X3)
mark#(U31(X1, X2)) → active#(U31(mark(X1), X2))active#(isNat(s(V1))) → isNat#(V1)
mark#(U12(X)) → mark#(X)mark#(0) → active#(0)
mark#(s(X)) → active#(s(mark(X)))active#(U11(tt, V2)) → isNat#(V2)
U41#(X1, X2, mark(X3)) → U41#(X1, X2, X3)U21#(mark(X)) → U21#(X)
active#(isNat(s(V1))) → U21#(isNat(V1))U21#(active(X)) → U21#(X)
mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2))U41#(X1, mark(X2), X3) → U41#(X1, X2, X3)
active#(isNat(s(V1))) → mark#(U21(isNat(V1)))U31#(mark(X1), X2) → U31#(X1, X2)
active#(U42(tt, M, N)) → plus#(N, M)s#(mark(X)) → s#(X)
active#(isNat(plus(V1, V2))) → U11#(isNat(V1), V2)mark#(U31(X1, X2)) → U31#(mark(X1), X2)
active#(plus(N, s(M))) → isNat#(M)mark#(U41(X1, X2, X3)) → active#(U41(mark(X1), X2, X3))
s#(active(X)) → s#(X)active#(U31(tt, N)) → mark#(N)
U42#(X1, X2, mark(X3)) → U42#(X1, X2, X3)mark#(U21(X)) → U21#(mark(X))

Rewrite Rules

active(U11(tt, V2)) → mark(U12(isNat(V2)))active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(0) → active(0)
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)
U12(mark(X)) → U12(X)U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)U21(active(X)) → U21(X)
U31(mark(X1), X2) → U31(X1, X2)U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)U31(X1, active(X2)) → U31(X1, X2)
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)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(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)

Original Signature

Termination of terms over the following signature is verified: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, U12, U31, U21

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)
U42#(X1, X2, active(X3)) → U42#(X1, X2, X3)U42#(mark(X1), X2, X3) → U42#(X1, X2, X3)
U42#(active(X1), X2, X3) → U42#(X1, X2, X3)U42#(X1, mark(X2), X3) → U42#(X1, X2, X3)
U42#(X1, X2, mark(X3)) → U42#(X1, X2, X3)U42#(X1, active(X2), X3) → U42#(X1, X2, X3)
U12#(active(X)) → U12#(X)U12#(mark(X)) → U12#(X)
s#(mark(X)) → s#(X)s#(active(X)) → s#(X)
U21#(mark(X)) → U21#(X)U21#(active(X)) → U21#(X)
isNat#(active(X)) → isNat#(X)isNat#(mark(X)) → isNat#(X)
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)
U31#(active(X1), X2) → U31#(X1, X2)U31#(X1, active(X2)) → U31#(X1, X2)
U31#(X1, mark(X2)) → U31#(X1, X2)U31#(mark(X1), X2) → U31#(X1, X2)
mark#(U11(X1, X2)) → mark#(X1)mark#(isNat(X)) → active#(isNat(X))
mark#(U21(X)) → active#(U21(mark(X)))active#(U21(tt)) → mark#(tt)
mark#(tt) → active#(tt)mark#(U21(X)) → mark#(X)
mark#(U42(X1, X2, X3)) → active#(U42(mark(X1), X2, X3))mark#(plus(X1, X2)) → mark#(X2)
active#(plus(N, s(M))) → mark#(U41(isNat(M), M, N))mark#(U31(X1, X2)) → mark#(X1)
mark#(s(X)) → mark#(X)mark#(U11(X1, X2)) → active#(U11(mark(X1), X2))
active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2))mark#(U31(X1, X2)) → active#(U31(mark(X1), X2))
active#(U41(tt, M, N)) → mark#(U42(isNat(N), M, N))mark#(0) → active#(0)
mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2)))mark#(U12(X)) → mark#(X)
mark#(s(X)) → active#(s(mark(X)))active#(U11(tt, V2)) → mark#(U12(isNat(V2)))
active#(isNat(0)) → mark#(tt)active#(plus(N, 0)) → mark#(U31(isNat(N), N))
mark#(U42(X1, X2, X3)) → mark#(X1)active#(isNat(s(V1))) → mark#(U21(isNat(V1)))
mark#(U12(X)) → active#(U12(mark(X)))mark#(plus(X1, X2)) → mark#(X1)
active#(U12(tt)) → mark#(tt)mark#(U41(X1, X2, X3)) → mark#(X1)
mark#(U41(X1, X2, X3)) → active#(U41(mark(X1), X2, X3))active#(U31(tt, N)) → mark#(N)
active#(U42(tt, M, N)) → mark#(s(plus(N, M)))

Problem 2: 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, V2)) → mark(U12(isNat(V2)))active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(0) → active(0)
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)
U12(mark(X)) → U12(X)U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)U21(active(X)) → U21(X)
U31(mark(X1), X2) → U31(X1, X2)U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)U31(X1, active(X2)) → U31(X1, X2)
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)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(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)

Original Signature

Termination of terms over the following signature is verified: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, U12, U31, U21

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 12: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(U11(tt, V2)) → mark(U12(isNat(V2)))active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(0) → active(0)
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)
U12(mark(X)) → U12(X)U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)U21(active(X)) → U21(X)
U31(mark(X1), X2) → U31(X1, X2)U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)U31(X1, active(X2)) → U31(X1, X2)
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)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(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)

Original Signature

Termination of terms over the following signature is verified: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, U12, U31, U21

Strategy

Function Precedence

mark < active < plus = isNat = 0 = s = U42 = tt = U41 = U11 = plus# = U12 = U31 = U21

Argument Filtering

plus: all arguments are removed from plus
mark: 1
isNat: collapses to 1
0: all arguments are removed from 0
s: all arguments are removed from s
U42: all arguments are removed from U42
tt: all arguments are removed from tt
U41: all arguments are removed from U41
U11: 1 2
plus#: collapses to 2
active: collapses to 1
U12: all arguments are removed from U12
U31: all arguments are removed from U31
U21: 1

Status

plus: multiset
mark: multiset
0: multiset
s: multiset
U42: multiset
tt: multiset
U41: multiset
U11: lexicographic with permutation 1 → 1 2 → 2
U12: multiset
U31: multiset
U21: lexicographic with permutation 1 → 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.

plus#(X1, mark(X2)) → plus#(X1, X2)

Problem 19: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

plus#(X1, active(X2)) → plus#(X1, X2)

Rewrite Rules

active(U11(tt, V2)) → mark(U12(isNat(V2)))active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(0) → active(0)
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)
U12(mark(X)) → U12(X)U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)U21(active(X)) → U21(X)
U31(mark(X1), X2) → U31(X1, X2)U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)U31(X1, active(X2)) → U31(X1, X2)
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)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(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)

Original Signature

Termination of terms over the following signature is verified: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, U12, U31, U21

Strategy

Function Precedence

active < plus = mark = isNat = 0 = s = U42 = tt = U41 = U11 = plus# = U12 = U31 = U21

Argument Filtering

plus: all arguments are removed from plus
mark: all arguments are removed from mark
isNat: all arguments are removed from isNat
0: all arguments are removed from 0
s: all arguments are removed from s
U42: all arguments are removed from U42
tt: all arguments are removed from tt
U41: all arguments are removed from U41
U11: all arguments are removed from U11
plus#: 1 2
active: 1
U12: all arguments are removed from U12
U31: 1 2
U21: all arguments are removed from U21

Status

plus: multiset
mark: multiset
isNat: multiset
0: multiset
s: multiset
U42: multiset
tt: multiset
U41: multiset
U11: multiset
plus#: lexicographic with permutation 1 → 1 2 → 2
active: multiset
U12: multiset
U31: lexicographic with permutation 1 → 1 2 → 2
U21: 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.

plus#(X1, active(X2)) → plus#(X1, X2)

Problem 3: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

U31#(active(X1), X2) → U31#(X1, X2)U31#(X1, active(X2)) → U31#(X1, X2)
U31#(X1, mark(X2)) → U31#(X1, X2)U31#(mark(X1), X2) → U31#(X1, X2)

Rewrite Rules

active(U11(tt, V2)) → mark(U12(isNat(V2)))active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(0) → active(0)
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)
U12(mark(X)) → U12(X)U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)U21(active(X)) → U21(X)
U31(mark(X1), X2) → U31(X1, X2)U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)U31(X1, active(X2)) → U31(X1, X2)
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)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(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)

Original Signature

Termination of terms over the following signature is verified: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, U12, U31, U21

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

U31#(active(X1), X2) → U31#(X1, X2)U31#(mark(X1), X2) → U31#(X1, X2)

Problem 13: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

U31#(X1, active(X2)) → U31#(X1, X2)U31#(X1, mark(X2)) → U31#(X1, X2)

Rewrite Rules

active(U11(tt, V2)) → mark(U12(isNat(V2)))active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(0) → active(0)
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)
U12(mark(X)) → U12(X)U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)U21(active(X)) → U21(X)
U31(mark(X1), X2) → U31(X1, X2)U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)U31(X1, active(X2)) → U31(X1, X2)
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)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(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)

Original Signature

Termination of terms over the following signature is verified: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, U12, U31, U21

Strategy

Function Precedence

mark < plus = U31# = isNat = 0 = s = U42 = tt = U41 = U11 = active = U12 = U31 = U21

Argument Filtering

plus: all arguments are removed from plus
U31#: collapses to 2
mark: collapses to 1
isNat: all arguments are removed from isNat
0: all arguments are removed from 0
s: all arguments are removed from s
U42: all arguments are removed from U42
tt: all arguments are removed from tt
U41: all arguments are removed from U41
U11: 1 2
active: 1
U12: all arguments are removed from U12
U31: all arguments are removed from U31
U21: all arguments are removed from U21

Status

plus: multiset
isNat: multiset
0: multiset
s: multiset
U42: multiset
tt: multiset
U41: multiset
U11: lexicographic with permutation 1 → 1 2 → 2
active: multiset
U12: multiset
U31: multiset
U21: 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.

U31#(X1, active(X2)) → U31#(X1, X2)

Problem 20: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

U31#(X1, mark(X2)) → U31#(X1, X2)

Rewrite Rules

active(U11(tt, V2)) → mark(U12(isNat(V2)))active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(0) → active(0)
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)
U12(mark(X)) → U12(X)U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)U21(active(X)) → U21(X)
U31(mark(X1), X2) → U31(X1, X2)U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)U31(X1, active(X2)) → U31(X1, X2)
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)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(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)

Original Signature

Termination of terms over the following signature is verified: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, U12, U31, U21

Strategy

Function Precedence

U31# < mark < plus = isNat = 0 = s = U42 = tt = U41 = U11 = active = U12 = U31 = U21

Argument Filtering

plus: all arguments are removed from plus
U31#: collapses to 2
mark: 1
isNat: all arguments are removed from isNat
0: all arguments are removed from 0
s: collapses to 1
U42: collapses to 3
tt: all arguments are removed from tt
U41: all arguments are removed from U41
U11: all arguments are removed from U11
active: all arguments are removed from active
U12: 1
U31: 1 2
U21: all arguments are removed from U21

Status

plus: multiset
mark: multiset
isNat: multiset
0: multiset
tt: multiset
U41: multiset
U11: multiset
active: multiset
U12: lexicographic with permutation 1 → 1
U31: lexicographic with permutation 1 → 2 2 → 1
U21: 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.

U31#(X1, mark(X2)) → U31#(X1, X2)

Problem 4: 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, V2)) → mark(U12(isNat(V2)))active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(0) → active(0)
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)
U12(mark(X)) → U12(X)U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)U21(active(X)) → U21(X)
U31(mark(X1), X2) → U31(X1, X2)U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)U31(X1, active(X2)) → U31(X1, X2)
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)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(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)

Original Signature

Termination of terms over the following signature is verified: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, U12, U31, U21

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 14: ReductionPairSAT

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, V2)) → mark(U12(isNat(V2)))active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(0) → active(0)
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)
U12(mark(X)) → U12(X)U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)U21(active(X)) → U21(X)
U31(mark(X1), X2) → U31(X1, X2)U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)U31(X1, active(X2)) → U31(X1, X2)
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)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(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)

Original Signature

Termination of terms over the following signature is verified: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, U12, U31, U21

Strategy

Function Precedence

mark = active < plus = isNat = U41# = 0 = s = U42 = tt = U41 = U11 = U12 = U31 = U21

Argument Filtering

plus: all arguments are removed from plus
mark: 1
isNat: 1
U41#: 1 3
0: all arguments are removed from 0
s: all arguments are removed from s
U42: 1 2 3
tt: all arguments are removed from tt
U41: 2 3
U11: 1
active: collapses to 1
U12: collapses to 1
U31: all arguments are removed from U31
U21: collapses to 1

Status

plus: multiset
mark: multiset
isNat: lexicographic with permutation 1 → 1
U41#: multiset
0: multiset
s: multiset
U42: lexicographic with permutation 1 → 1 2 → 2 3 → 3
tt: multiset
U41: lexicographic with permutation 2 → 1 3 → 2
U11: lexicographic with permutation 1 → 1
U31: 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.

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

Problem 21: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

U41#(X1, active(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, V2)) → mark(U12(isNat(V2)))active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(0) → active(0)
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)
U12(mark(X)) → U12(X)U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)U21(active(X)) → U21(X)
U31(mark(X1), X2) → U31(X1, X2)U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)U31(X1, active(X2)) → U31(X1, X2)
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)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(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)

Original Signature

Termination of terms over the following signature is verified: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, U12, U31, U21

Strategy

Function Precedence

active < U41# < mark < plus = isNat = 0 = s = U42 = tt = U41 = U11 = U12 = U31 = U21

Argument Filtering

plus: 1
mark: 1
isNat: all arguments are removed from isNat
U41#: collapses to 2
0: all arguments are removed from 0
s: collapses to 1
U42: all arguments are removed from U42
tt: all arguments are removed from tt
U41: all arguments are removed from U41
U11: all arguments are removed from U11
active: 1
U12: 1
U31: 1 2
U21: all arguments are removed from U21

Status

plus: lexicographic with permutation 1 → 1
mark: multiset
isNat: multiset
0: multiset
U42: multiset
tt: multiset
U41: multiset
U11: multiset
active: multiset
U12: lexicographic with permutation 1 → 1
U31: lexicographic with permutation 1 → 2 2 → 1
U21: 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.

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

Problem 5: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

U21#(mark(X)) → U21#(X)U21#(active(X)) → U21#(X)

Rewrite Rules

active(U11(tt, V2)) → mark(U12(isNat(V2)))active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(0) → active(0)
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)
U12(mark(X)) → U12(X)U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)U21(active(X)) → U21(X)
U31(mark(X1), X2) → U31(X1, X2)U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)U31(X1, active(X2)) → U31(X1, X2)
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)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(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)

Original Signature

Termination of terms over the following signature is verified: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, U12, U31, U21

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

U21#(mark(X)) → U21#(X)U21#(active(X)) → U21#(X)

Problem 6: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

U42#(X1, X2, active(X3)) → U42#(X1, X2, X3)U42#(mark(X1), X2, X3) → U42#(X1, X2, X3)
U42#(active(X1), X2, X3) → U42#(X1, X2, X3)U42#(X1, mark(X2), X3) → U42#(X1, X2, X3)
U42#(X1, X2, mark(X3)) → U42#(X1, X2, X3)U42#(X1, active(X2), X3) → U42#(X1, X2, X3)

Rewrite Rules

active(U11(tt, V2)) → mark(U12(isNat(V2)))active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(0) → active(0)
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)
U12(mark(X)) → U12(X)U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)U21(active(X)) → U21(X)
U31(mark(X1), X2) → U31(X1, X2)U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)U31(X1, active(X2)) → U31(X1, X2)
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)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(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)

Original Signature

Termination of terms over the following signature is verified: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, U12, U31, U21

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

U42#(mark(X1), X2, X3) → U42#(X1, X2, X3)U42#(active(X1), X2, X3) → U42#(X1, X2, X3)

Problem 15: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

U42#(X1, X2, active(X3)) → U42#(X1, X2, X3)U42#(X1, mark(X2), X3) → U42#(X1, X2, X3)
U42#(X1, X2, mark(X3)) → U42#(X1, X2, X3)U42#(X1, active(X2), X3) → U42#(X1, X2, X3)

Rewrite Rules

active(U11(tt, V2)) → mark(U12(isNat(V2)))active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(0) → active(0)
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)
U12(mark(X)) → U12(X)U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)U21(active(X)) → U21(X)
U31(mark(X1), X2) → U31(X1, X2)U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)U31(X1, active(X2)) → U31(X1, X2)
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)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(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)

Original Signature

Termination of terms over the following signature is verified: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, U12, U31, U21

Strategy

Function Precedence

mark = active < plus = isNat = U42# = 0 = s = U42 = tt = U41 = U11 = U12 = U31 = U21

Argument Filtering

plus: all arguments are removed from plus
mark: collapses to 1
isNat: collapses to 1
U42#: collapses to 2
0: all arguments are removed from 0
s: collapses to 1
U42: collapses to 2
tt: all arguments are removed from tt
U41: 1 2 3
U11: all arguments are removed from U11
active: 1
U12: collapses to 1
U31: 1 2
U21: collapses to 1

Status

plus: multiset
0: multiset
tt: multiset
U41: lexicographic with permutation 1 → 2 2 → 1 3 → 3
U11: multiset
active: lexicographic with permutation 1 → 1
U31: lexicographic with permutation 1 → 2 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.

U42#(X1, active(X2), X3) → U42#(X1, X2, X3)

Problem 22: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

U42#(X1, X2, active(X3)) → U42#(X1, X2, X3)U42#(X1, mark(X2), X3) → U42#(X1, X2, X3)
U42#(X1, X2, mark(X3)) → U42#(X1, X2, X3)

Rewrite Rules

active(U11(tt, V2)) → mark(U12(isNat(V2)))active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(0) → active(0)
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)
U12(mark(X)) → U12(X)U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)U21(active(X)) → U21(X)
U31(mark(X1), X2) → U31(X1, X2)U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)U31(X1, active(X2)) → U31(X1, X2)
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)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(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)

Original Signature

Termination of terms over the following signature is verified: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, U12, U31, U21

Strategy

Function Precedence

active < plus = mark = isNat = U42# = 0 = s = U42 = tt = U41 = U11 = U12 = U31 = U21

Argument Filtering

plus: 1 2
mark: 1
isNat: all arguments are removed from isNat
U42#: 2
0: all arguments are removed from 0
s: all arguments are removed from s
U42: collapses to 1
tt: all arguments are removed from tt
U41: all arguments are removed from U41
U11: 1 2
active: all arguments are removed from active
U12: 1
U31: all arguments are removed from U31
U21: 1

Status

plus: lexicographic with permutation 1 → 2 2 → 1
mark: multiset
isNat: multiset
U42#: lexicographic with permutation 2 → 1
0: multiset
s: multiset
tt: multiset
U41: multiset
U11: lexicographic with permutation 1 → 1 2 → 2
active: multiset
U12: lexicographic with permutation 1 → 1
U31: multiset
U21: lexicographic with permutation 1 → 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.

U42#(X1, mark(X2), X3) → U42#(X1, X2, X3)

Problem 7: 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, V2)) → mark(U12(isNat(V2)))active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(0) → active(0)
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)
U12(mark(X)) → U12(X)U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)U21(active(X)) → U21(X)
U31(mark(X1), X2) → U31(X1, X2)U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)U31(X1, active(X2)) → U31(X1, X2)
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)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(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)

Original Signature

Termination of terms over the following signature is verified: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, U12, U31, U21

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 16: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(U11(tt, V2)) → mark(U12(isNat(V2)))active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(0) → active(0)
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)
U12(mark(X)) → U12(X)U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)U21(active(X)) → U21(X)
U31(mark(X1), X2) → U31(X1, X2)U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)U31(X1, active(X2)) → U31(X1, X2)
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)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(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)

Original Signature

Termination of terms over the following signature is verified: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, U12, U31, U21

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:

U11#(X1, active(X2)) → U11#(X1, X2)

Problem 17: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

U11#(X1, mark(X2)) → U11#(X1, X2)

Rewrite Rules

active(U11(tt, V2)) → mark(U12(isNat(V2)))active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(0) → active(0)
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)
U12(mark(X)) → U12(X)U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)U21(active(X)) → U21(X)
U31(mark(X1), X2) → U31(X1, X2)U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)U31(X1, active(X2)) → U31(X1, X2)
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)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(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)

Original Signature

Termination of terms over the following signature is verified: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, U12, U31, U21

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:

U11#(X1, mark(X2)) → U11#(X1, X2)

Problem 8: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(U11(tt, V2)) → mark(U12(isNat(V2)))active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(0) → active(0)
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)
U12(mark(X)) → U12(X)U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)U21(active(X)) → U21(X)
U31(mark(X1), X2) → U31(X1, X2)U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)U31(X1, active(X2)) → U31(X1, X2)
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)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(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)

Original Signature

Termination of terms over the following signature is verified: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, U12, U31, U21

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 9: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(U11(tt, V2)) → mark(U12(isNat(V2)))active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(0) → active(0)
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)
U12(mark(X)) → U12(X)U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)U21(active(X)) → U21(X)
U31(mark(X1), X2) → U31(X1, X2)U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)U31(X1, active(X2)) → U31(X1, X2)
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)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(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)

Original Signature

Termination of terms over the following signature is verified: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, U12, U31, U21

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 10: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

mark#(U11(X1, X2)) → mark#(X1)mark#(isNat(X)) → active#(isNat(X))
mark#(U21(X)) → active#(U21(mark(X)))active#(U21(tt)) → mark#(tt)
mark#(tt) → active#(tt)mark#(U21(X)) → mark#(X)
mark#(U42(X1, X2, X3)) → active#(U42(mark(X1), X2, X3))mark#(plus(X1, X2)) → mark#(X2)
active#(plus(N, s(M))) → mark#(U41(isNat(M), M, N))mark#(U31(X1, X2)) → mark#(X1)
mark#(s(X)) → mark#(X)mark#(U11(X1, X2)) → active#(U11(mark(X1), X2))
active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2))mark#(U31(X1, X2)) → active#(U31(mark(X1), X2))
active#(U41(tt, M, N)) → mark#(U42(isNat(N), M, N))mark#(U12(X)) → mark#(X)
mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2)))mark#(0) → active#(0)
mark#(s(X)) → active#(s(mark(X)))active#(U11(tt, V2)) → mark#(U12(isNat(V2)))
active#(isNat(0)) → mark#(tt)active#(plus(N, 0)) → mark#(U31(isNat(N), N))
mark#(U42(X1, X2, X3)) → mark#(X1)active#(isNat(s(V1))) → mark#(U21(isNat(V1)))
mark#(U12(X)) → active#(U12(mark(X)))mark#(plus(X1, X2)) → mark#(X1)
active#(U12(tt)) → mark#(tt)mark#(U41(X1, X2, X3)) → mark#(X1)
mark#(U41(X1, X2, X3)) → active#(U41(mark(X1), X2, X3))active#(U31(tt, N)) → mark#(N)
active#(U42(tt, M, N)) → mark#(s(plus(N, M)))

Rewrite Rules

active(U11(tt, V2)) → mark(U12(isNat(V2)))active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(0) → active(0)
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)
U12(mark(X)) → U12(X)U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)U21(active(X)) → U21(X)
U31(mark(X1), X2) → U31(X1, X2)U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)U31(X1, active(X2)) → U31(X1, X2)
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)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(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)

Original Signature

Termination of terms over the following signature is verified: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, U12, U31, U21

Strategy

Function Precedence

active < plus = mark = mark# = isNat = 0 = s = U42 = tt = U41 = U11 = U12 = U31 = active# = U21

Argument Filtering

plus: all arguments are removed from plus
mark: all arguments are removed from mark
mark#: all arguments are removed from mark#
isNat: all arguments are removed from isNat
0: all arguments are removed from 0
s: all arguments are removed from s
U42: all arguments are removed from U42
tt: all arguments are removed from tt
U41: all arguments are removed from U41
active: collapses to 1
U11: all arguments are removed from U11
U12: all arguments are removed from U12
U31: all arguments are removed from U31
active#: collapses to 1
U21: all arguments are removed from U21

Status

plus: multiset
mark: multiset
mark#: multiset
isNat: multiset
0: multiset
s: multiset
U42: multiset
tt: multiset
U41: multiset
U11: multiset
U12: multiset
U31: multiset
U21: multiset

Usable Rules

mark(isNat(X)) → active(isNat(X))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
active(U21(tt)) → mark(tt)mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
plus(mark(X1), X2) → plus(X1, X2)U11(active(X1), X2) → U11(X1, X2)
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))U11(X1, mark(X2)) → U11(X1, X2)
active(U31(tt, N)) → mark(N)U21(active(X)) → U21(X)
plus(X1, active(X2)) → plus(X1, X2)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
isNat(active(X)) → isNat(X)U21(mark(X)) → U21(X)
mark(U12(X)) → active(U12(mark(X)))U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)mark(U31(X1, X2)) → active(U31(mark(X1), X2))
plus(active(X1), X2) → plus(X1, X2)U12(active(X)) → U12(X)
U31(X1, active(X2)) → U31(X1, X2)active(plus(N, s(M))) → mark(U41(isNat(M), M, N))
mark(U21(X)) → active(U21(mark(X)))active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))U31(mark(X1), X2) → U31(X1, X2)
mark(s(X)) → active(s(mark(X)))U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U11(X1, active(X2)) → U11(X1, X2)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
active(isNat(0)) → mark(tt)active(plus(N, 0)) → mark(U31(isNat(N), N))
mark(0) → active(0)s(active(X)) → s(X)
U12(mark(X)) → U12(X)U11(mark(X1), X2) → U11(X1, X2)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)plus(X1, mark(X2)) → plus(X1, X2)
active(U42(tt, M, N)) → mark(s(plus(N, M)))mark(tt) → active(tt)
active(U12(tt)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
isNat(mark(X)) → isNat(X)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U31(X1, mark(X2)) → U31(X1, X2)s(mark(X)) → s(X)
active(isNat(s(V1))) → mark(U21(isNat(V1)))U31(active(X1), X2) → U31(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)

Problem 18: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

mark#(U11(X1, X2)) → mark#(X1)mark#(isNat(X)) → active#(isNat(X))
mark#(U21(X)) → active#(U21(mark(X)))active#(U21(tt)) → mark#(tt)
mark#(tt) → active#(tt)mark#(U21(X)) → mark#(X)
mark#(U42(X1, X2, X3)) → active#(U42(mark(X1), X2, X3))mark#(plus(X1, X2)) → mark#(X2)
active#(plus(N, s(M))) → mark#(U41(isNat(M), M, N))mark#(U31(X1, X2)) → mark#(X1)
mark#(s(X)) → mark#(X)active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2))
mark#(U11(X1, X2)) → active#(U11(mark(X1), X2))mark#(U31(X1, X2)) → active#(U31(mark(X1), X2))
active#(U41(tt, M, N)) → mark#(U42(isNat(N), M, N))mark#(U12(X)) → mark#(X)
mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2)))mark#(s(X)) → active#(s(mark(X)))
active#(U11(tt, V2)) → mark#(U12(isNat(V2)))active#(isNat(0)) → mark#(tt)
active#(plus(N, 0)) → mark#(U31(isNat(N), N))mark#(U42(X1, X2, X3)) → mark#(X1)
active#(isNat(s(V1))) → mark#(U21(isNat(V1)))mark#(U12(X)) → active#(U12(mark(X)))
mark#(plus(X1, X2)) → mark#(X1)active#(U12(tt)) → mark#(tt)
mark#(U41(X1, X2, X3)) → mark#(X1)mark#(U41(X1, X2, X3)) → active#(U41(mark(X1), X2, X3))
active#(U31(tt, N)) → mark#(N)active#(U42(tt, M, N)) → mark#(s(plus(N, M)))

Rewrite Rules

active(U11(tt, V2)) → mark(U12(isNat(V2)))active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(0) → active(0)
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)
U12(mark(X)) → U12(X)U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)U21(active(X)) → U21(X)
U31(mark(X1), X2) → U31(X1, X2)U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)U31(X1, active(X2)) → U31(X1, X2)
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)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(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)

Original Signature

Termination of terms over the following signature is verified: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, U12, U31, U21

Strategy

Function Precedence

active < active# < plus = mark = mark# = isNat = 0 = s = U42 = tt = U41 = U11 = U12 = U31 = U21

Argument Filtering

plus: all arguments are removed from plus
mark: all arguments are removed from mark
mark#: all arguments are removed from mark#
isNat: all arguments are removed from isNat
0: all arguments are removed from 0
s: all arguments are removed from s
U42: all arguments are removed from U42
tt: all arguments are removed from tt
U41: all arguments are removed from U41
active: collapses to 1
U11: all arguments are removed from U11
U12: all arguments are removed from U12
U31: all arguments are removed from U31
active#: collapses to 1
U21: all arguments are removed from U21

Status

plus: multiset
mark: multiset
mark#: multiset
isNat: multiset
0: multiset
s: multiset
U42: multiset
tt: multiset
U41: multiset
U11: multiset
U12: multiset
U31: multiset
U21: multiset

Usable Rules

mark(isNat(X)) → active(isNat(X))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
active(U21(tt)) → mark(tt)mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
plus(mark(X1), X2) → plus(X1, X2)U11(active(X1), X2) → U11(X1, X2)
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))U11(X1, mark(X2)) → U11(X1, X2)
active(U31(tt, N)) → mark(N)U21(active(X)) → U21(X)
plus(X1, active(X2)) → plus(X1, X2)U41(X1, X2, active(X3)) → U41(X1, X2, X3)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
isNat(active(X)) → isNat(X)U21(mark(X)) → U21(X)
mark(U12(X)) → active(U12(mark(X)))U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)mark(U31(X1, X2)) → active(U31(mark(X1), X2))
plus(active(X1), X2) → plus(X1, X2)U12(active(X)) → U12(X)
U31(X1, active(X2)) → U31(X1, X2)active(plus(N, s(M))) → mark(U41(isNat(M), M, N))
mark(U21(X)) → active(U21(mark(X)))active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))U31(mark(X1), X2) → U31(X1, X2)
mark(s(X)) → active(s(mark(X)))U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U41(active(X1), X2, X3) → U41(X1, X2, X3)
U11(X1, active(X2)) → U11(X1, X2)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
active(isNat(0)) → mark(tt)active(plus(N, 0)) → mark(U31(isNat(N), N))
mark(0) → active(0)s(active(X)) → s(X)
U12(mark(X)) → U12(X)U11(mark(X1), X2) → U11(X1, X2)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)plus(X1, mark(X2)) → plus(X1, X2)
active(U42(tt, M, N)) → mark(s(plus(N, M)))mark(tt) → active(tt)
active(U12(tt)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
isNat(mark(X)) → isNat(X)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U31(X1, mark(X2)) → U31(X1, X2)s(mark(X)) → s(X)
active(isNat(s(V1))) → mark(U21(isNat(V1)))U31(active(X1), X2) → U31(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#(tt) → active#(tt)

Problem 11: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

U12#(active(X)) → U12#(X)U12#(mark(X)) → U12#(X)

Rewrite Rules

active(U11(tt, V2)) → mark(U12(isNat(V2)))active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(0) → active(0)
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)
U12(mark(X)) → U12(X)U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)U21(active(X)) → U21(X)
U31(mark(X1), X2) → U31(X1, X2)U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)U31(X1, active(X2)) → U31(X1, X2)
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)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)U42(X1, X2, active(X3)) → U42(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)

Original Signature

Termination of terms over the following signature is verified: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, U12, U31, U21

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

U12#(active(X)) → U12#(X)U12#(mark(X)) → U12#(X)