TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60013 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (18458ms).
 | – Problem 2 was processed with processor SubtermCriterion (2ms).
 | – Problem 3 was processed with processor SubtermCriterion (2ms).
 |    | – Problem 15 remains open; application of the following processors failed [DependencyGraph (5ms), PolynomialLinearRange4iUR (18ms), DependencyGraph (5ms), PolynomialLinearRange4iUR (26ms), DependencyGraph (6ms), PolynomialLinearRange4iUR (8ms), DependencyGraph (5ms), PolynomialLinearRange4iUR (13ms), DependencyGraph (5ms), PolynomialLinearRange4iUR (0ms), DependencyGraph (5ms), PolynomialLinearRange4iUR (11ms), DependencyGraph (5ms), PolynomialLinearRange8NegiUR (8ms)].
 | – Problem 4 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 16 was processed with processor PolynomialLinearRange4iUR (68ms).
 |    |    | – Problem 22 was processed with processor PolynomialLinearRange4iUR (72ms).
 |    |    |    | – Problem 26 was processed with processor PolynomialLinearRange4iUR (64ms).
 |    |    |    |    | – Problem 29 remains open; application of the following processors failed [DependencyGraph (2ms), PolynomialLinearRange4iUR (0ms), DependencyGraph (2ms), PolynomialLinearRange4iUR (18ms), DependencyGraph (2ms), PolynomialLinearRange8NegiUR (19ms)].
 | – 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 17 was processed with processor PolynomialLinearRange4iUR (89ms).
 | – Problem 8 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 18 was processed with processor PolynomialLinearRange4iUR (49ms).
 | – Problem 9 was processed with processor SubtermCriterion (3ms).
 | – Problem 10 was processed with processor SubtermCriterion (2ms).
 |    | – Problem 19 was processed with processor PolynomialLinearRange4iUR (112ms).
 |    |    | – Problem 23 was processed with processor PolynomialLinearRange4iUR (19ms).
 | – Problem 11 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (2286ms), PolynomialLinearRange4iUR (1251ms), DependencyGraph (2198ms), PolynomialLinearRange4iUR (1666ms), DependencyGraph (2209ms), PolynomialLinearRange4iUR (2503ms), DependencyGraph (2294ms), PolynomialLinearRange4iUR (2534ms), DependencyGraph (2205ms), PolynomialLinearRange4iUR (2500ms), DependencyGraph (2203ms), PolynomialLinearRange4iUR (3369ms), DependencyGraph (2299ms), PolynomialLinearRange8NegiUR (10000ms), DependencyGraph (timeout)].
 | – Problem 12 was processed with processor SubtermCriterion (2ms).
 |    | – Problem 20 was processed with processor PolynomialLinearRange4iUR (49ms).
 |    |    | – Problem 25 was processed with processor PolynomialLinearRange4iUR (27ms).
 |    |    |    | – Problem 27 was processed with processor PolynomialLinearRange4iUR (50ms).
 |    |    |    |    | – Problem 28 was processed with processor PolynomialLinearRange4iUR (17ms).
 | – Problem 13 was processed with processor SubtermCriterion (4ms).
 |    | – Problem 21 was processed with processor PolynomialLinearRange4iUR (26ms).
 |    |    | – Problem 24 was processed with processor PolynomialLinearRange4iUR (24ms).
 | – Problem 14 was processed with processor SubtermCriterion (1ms).

The following open problems remain:

Open Dependency Pair Problem 11

Dependency Pairs

active#(isNatKind(0)) → mark#(tt)active#(U21(tt, V1)) → mark#(U22(isNat(V1)))
mark#(isNat(X)) → active#(isNat(X))mark#(isNatKind(X)) → active#(isNatKind(X))
mark#(tt) → active#(tt)mark#(U21(X1, X2)) → mark#(X1)
active#(U41(tt, M, N)) → mark#(s(plus(N, M)))mark#(U11(X1, X2, X3)) → active#(U11(mark(X1), X2, X3))
active#(U22(tt)) → mark#(tt)mark#(plus(X1, X2)) → mark#(X2)
mark#(U21(X1, X2)) → active#(U21(mark(X1), X2))mark#(and(X1, X2)) → active#(and(mark(X1), X2))
mark#(U31(X1, X2)) → mark#(X1)mark#(s(X)) → mark#(X)
mark#(U13(X)) → active#(U13(mark(X)))active#(isNatKind(s(V1))) → mark#(isNatKind(V1))
active#(U13(tt)) → mark#(tt)mark#(U31(X1, X2)) → active#(U31(mark(X1), X2))
mark#(U12(X1, X2)) → mark#(X1)active#(plus(N, 0)) → mark#(U31(and(isNat(N), isNatKind(N)), N))
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)
active#(isNatKind(plus(V1, V2))) → mark#(and(isNatKind(V1), isNatKind(V2)))active#(U11(tt, V1, V2)) → mark#(U12(isNat(V1), V2))
active#(and(tt, X)) → mark#(X)active#(isNat(s(V1))) → mark#(U21(isNatKind(V1), V1))
mark#(plus(X1, X2)) → mark#(X1)mark#(and(X1, X2)) → mark#(X1)
mark#(U22(X)) → mark#(X)active#(plus(N, s(M))) → mark#(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark#(U41(X1, X2, X3)) → mark#(X1)mark#(U41(X1, X2, X3)) → active#(U41(mark(X1), X2, X3))
mark#(U12(X1, X2)) → active#(U12(mark(X1), X2))mark#(U22(X)) → active#(U22(mark(X)))
active#(U31(tt, N)) → mark#(N)active#(U12(tt, V2)) → mark#(U13(isNat(V2)))
active#(isNat(plus(V1, V2))) → mark#(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))mark#(U11(X1, X2, X3)) → mark#(X1)
mark#(U13(X)) → mark#(X)

Rewrite Rules

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U13, U31, U21, U22

Open Dependency Pair Problem 29

Dependency Pairs

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

Rewrite Rules

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U13, U31, U21, U22

Open Dependency Pair Problem 15

Dependency Pairs

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

Rewrite Rules

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U31, U13, U21, U22

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

active#(isNatKind(0)) → mark#(tt)U12#(active(X1), X2) → U12#(X1, X2)
active#(isNat(plus(V1, V2))) → U11#(and(isNatKind(V1), isNatKind(V2)), V1, V2)active#(isNatKind(plus(V1, V2))) → isNatKind#(V1)
mark#(U11(X1, X2, X3)) → U11#(mark(X1), X2, X3)active#(U22(tt)) → mark#(tt)
active#(plus(N, 0)) → and#(isNat(N), isNatKind(N))active#(U12(tt, V2)) → isNat#(V2)
isNat#(active(X)) → isNat#(X)U21#(X1, active(X2)) → U21#(X1, X2)
mark#(s(X)) → mark#(X)active#(plus(N, 0)) → isNat#(N)
mark#(U12(X1, X2)) → mark#(X1)mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2)))
U11#(X1, active(X2), X3) → U11#(X1, X2, X3)active#(plus(N, s(M))) → and#(isNat(N), isNatKind(N))
active#(U11(tt, V1, V2)) → mark#(U12(isNat(V1), V2))and#(mark(X1), X2) → and#(X1, X2)
active#(U21(tt, V1)) → isNat#(V1)active#(isNat(s(V1))) → isNatKind#(V1)
mark#(and(X1, X2)) → mark#(X1)U41#(X1, active(X2), X3) → U41#(X1, X2, X3)
active#(plus(N, s(M))) → isNatKind#(M)mark#(U41(X1, X2, X3)) → mark#(X1)
mark#(U22(X)) → U22#(mark(X))mark#(U22(X)) → active#(U22(mark(X)))
active#(U12(tt, V2)) → mark#(U13(isNat(V2)))plus#(mark(X1), X2) → plus#(X1, X2)
mark#(U11(X1, X2, X3)) → mark#(X1)plus#(active(X1), X2) → plus#(X1, X2)
mark#(U13(X)) → U13#(mark(X))U12#(X1, active(X2)) → U12#(X1, X2)
mark#(tt) → active#(tt)U31#(X1, active(X2)) → U31#(X1, X2)
active#(U11(tt, V1, V2)) → isNat#(V1)U21#(X1, mark(X2)) → U21#(X1, X2)
mark#(U21(X1, X2)) → active#(U21(mark(X1), X2))U11#(active(X1), X2, X3) → U11#(X1, X2, X3)
U13#(mark(X)) → U13#(X)mark#(U13(X)) → active#(U13(mark(X)))
active#(isNatKind(s(V1))) → mark#(isNatKind(V1))U41#(X1, X2, active(X3)) → U41#(X1, X2, X3)
mark#(U31(X1, X2)) → active#(U31(mark(X1), X2))mark#(isNatKind(X)) → isNatKind#(X)
isNatKind#(mark(X)) → isNatKind#(X)active#(U11(tt, V1, V2)) → U12#(isNat(V1), V2)
active#(plus(N, 0)) → mark#(U31(and(isNat(N), isNatKind(N)), N))mark#(0) → active#(0)
mark#(s(X)) → active#(s(mark(X)))U41#(X1, X2, mark(X3)) → U41#(X1, X2, X3)
U21#(active(X1), X2) → U21#(X1, X2)active#(U41(tt, M, N)) → s#(plus(N, M))
active#(plus(N, s(M))) → and#(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))mark#(U22(X)) → mark#(X)
active#(plus(N, s(M))) → mark#(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))active#(isNat(s(V1))) → U21#(isNatKind(V1), V1)
active#(plus(N, s(M))) → isNat#(M)mark#(U12(X1, X2)) → active#(U12(mark(X1), X2))
mark#(U41(X1, X2, X3)) → active#(U41(mark(X1), X2, X3))U13#(active(X)) → U13#(X)
U11#(X1, X2, active(X3)) → U11#(X1, X2, X3)active#(plus(N, s(M))) → and#(isNat(M), isNatKind(M))
active#(U21(tt, V1)) → mark#(U22(isNat(V1)))mark#(isNatKind(X)) → active#(isNatKind(X))
mark#(s(X)) → s#(mark(X))mark#(U11(X1, X2, X3)) → active#(U11(mark(X1), X2, X3))
plus#(X1, mark(X2)) → plus#(X1, X2)U31#(active(X1), X2) → U31#(X1, X2)
U22#(mark(X)) → U22#(X)mark#(U31(X1, X2)) → mark#(X1)
U12#(mark(X1), X2) → U12#(X1, X2)active#(isNatKind(plus(V1, V2))) → and#(isNatKind(V1), isNatKind(V2))
active#(U41(tt, M, N)) → plus#(N, M)active#(isNat(0)) → mark#(tt)
mark#(and(X1, X2)) → and#(mark(X1), X2)active#(isNatKind(plus(V1, V2))) → mark#(and(isNatKind(V1), isNatKind(V2)))
U11#(mark(X1), X2, X3) → U11#(X1, X2, X3)active#(U12(tt, V2)) → U13#(isNat(V2))
active#(U21(tt, V1)) → U22#(isNat(V1))mark#(plus(X1, X2)) → mark#(X1)
isNatKind#(active(X)) → isNatKind#(X)U41#(active(X1), X2, X3) → U41#(X1, X2, X3)
plus#(X1, active(X2)) → plus#(X1, X2)U22#(active(X)) → U22#(X)
active#(plus(N, s(M))) → U41#(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)and#(active(X1), X2) → and#(X1, X2)
mark#(U41(X1, X2, X3)) → U41#(mark(X1), X2, X3)mark#(isNat(X)) → active#(isNat(X))
active#(plus(N, s(M))) → isNat#(N)and#(X1, active(X2)) → and#(X1, X2)
active#(plus(N, s(M))) → isNatKind#(N)mark#(U21(X1, X2)) → U21#(mark(X1), X2)
mark#(U21(X1, X2)) → mark#(X1)mark#(isNat(X)) → isNat#(X)
isNat#(mark(X)) → isNat#(X)U31#(X1, mark(X2)) → U31#(X1, X2)
active#(U41(tt, M, N)) → mark#(s(plus(N, M)))active#(isNat(plus(V1, V2))) → isNatKind#(V2)
active#(isNat(plus(V1, V2))) → isNatKind#(V1)active#(plus(N, 0)) → isNatKind#(N)
U11#(X1, X2, mark(X3)) → U11#(X1, X2, X3)mark#(plus(X1, X2)) → mark#(X2)
mark#(and(X1, X2)) → active#(and(mark(X1), X2))U41#(mark(X1), X2, X3) → U41#(X1, X2, X3)
active#(U13(tt)) → mark#(tt)active#(plus(N, 0)) → U31#(and(isNat(N), isNatKind(N)), N)
active#(isNat(plus(V1, V2))) → and#(isNatKind(V1), isNatKind(V2))U12#(X1, mark(X2)) → U12#(X1, X2)
active#(isNatKind(s(V1))) → isNatKind#(V1)and#(X1, mark(X2)) → and#(X1, X2)
active#(isNatKind(plus(V1, V2))) → isNatKind#(V2)mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2))
active#(isNat(s(V1))) → mark#(U21(isNatKind(V1), V1))active#(and(tt, X)) → mark#(X)
U41#(X1, mark(X2), X3) → U41#(X1, X2, X3)U31#(mark(X1), X2) → U31#(X1, X2)
mark#(U12(X1, X2)) → U12#(mark(X1), X2)s#(mark(X)) → s#(X)
mark#(U31(X1, X2)) → U31#(mark(X1), X2)U21#(mark(X1), X2) → U21#(X1, X2)
U11#(X1, mark(X2), X3) → U11#(X1, X2, X3)s#(active(X)) → s#(X)
active#(U31(tt, N)) → mark#(N)active#(isNat(plus(V1, V2))) → mark#(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
mark#(U13(X)) → mark#(X)

Rewrite Rules

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U31, U13, U21, U22

Strategy

The following SCCs where found

U21#(X1, mark(X2)) → U21#(X1, X2)U21#(mark(X1), X2) → U21#(X1, X2)
U21#(active(X1), X2) → U21#(X1, X2)U21#(X1, active(X2)) → U21#(X1, X2)
U12#(active(X1), X2) → U12#(X1, X2)U12#(X1, active(X2)) → U12#(X1, X2)
U12#(mark(X1), X2) → U12#(X1, X2)U12#(X1, mark(X2)) → U12#(X1, X2)
U13#(mark(X)) → U13#(X)U13#(active(X)) → U13#(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)
isNat#(active(X)) → isNat#(X)isNat#(mark(X)) → isNat#(X)
s#(mark(X)) → s#(X)s#(active(X)) → s#(X)
active#(isNatKind(0)) → mark#(tt)mark#(isNat(X)) → active#(isNat(X))
active#(U21(tt, V1)) → mark#(U22(isNat(V1)))mark#(isNatKind(X)) → active#(isNatKind(X))
mark#(tt) → active#(tt)mark#(U21(X1, X2)) → mark#(X1)
active#(U41(tt, M, N)) → mark#(s(plus(N, M)))mark#(U11(X1, X2, X3)) → active#(U11(mark(X1), X2, X3))
active#(U22(tt)) → mark#(tt)mark#(U21(X1, X2)) → active#(U21(mark(X1), X2))
mark#(plus(X1, X2)) → mark#(X2)mark#(and(X1, X2)) → active#(and(mark(X1), X2))
mark#(U31(X1, X2)) → mark#(X1)mark#(U13(X)) → active#(U13(mark(X)))
mark#(s(X)) → mark#(X)active#(isNatKind(s(V1))) → mark#(isNatKind(V1))
mark#(U31(X1, X2)) → active#(U31(mark(X1), X2))active#(U13(tt)) → mark#(tt)
mark#(U12(X1, X2)) → mark#(X1)active#(plus(N, 0)) → mark#(U31(and(isNat(N), isNatKind(N)), N))
mark#(0) → active#(0)mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2)))
mark#(s(X)) → active#(s(mark(X)))active#(isNat(0)) → mark#(tt)
active#(isNatKind(plus(V1, V2))) → mark#(and(isNatKind(V1), isNatKind(V2)))active#(U11(tt, V1, V2)) → mark#(U12(isNat(V1), V2))
active#(isNat(s(V1))) → mark#(U21(isNatKind(V1), V1))active#(and(tt, X)) → mark#(X)
mark#(plus(X1, X2)) → mark#(X1)mark#(and(X1, X2)) → mark#(X1)
mark#(U22(X)) → mark#(X)mark#(U41(X1, X2, X3)) → mark#(X1)
active#(plus(N, s(M))) → mark#(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))mark#(U12(X1, X2)) → active#(U12(mark(X1), X2))
mark#(U41(X1, X2, X3)) → active#(U41(mark(X1), X2, X3))active#(U31(tt, N)) → mark#(N)
mark#(U22(X)) → active#(U22(mark(X)))active#(isNat(plus(V1, V2))) → mark#(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active#(U12(tt, V2)) → mark#(U13(isNat(V2)))mark#(U11(X1, X2, X3)) → mark#(X1)
mark#(U13(X)) → mark#(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)
isNatKind#(active(X)) → isNatKind#(X)isNatKind#(mark(X)) → isNatKind#(X)
U22#(mark(X)) → U22#(X)U22#(active(X)) → U22#(X)
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)
U11#(X1, X2, mark(X3)) → U11#(X1, X2, X3)U11#(X1, active(X2), X3) → U11#(X1, X2, X3)
U11#(active(X1), X2, X3) → U11#(X1, X2, X3)U11#(X1, mark(X2), X3) → U11#(X1, X2, X3)
U11#(mark(X1), X2, X3) → U11#(X1, X2, X3)U11#(X1, X2, active(X3)) → U11#(X1, X2, X3)
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)

Problem 2: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

U13#(mark(X)) → U13#(X)U13#(active(X)) → U13#(X)

Rewrite Rules

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U31, U13, U21, U22

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

U13#(mark(X)) → U13#(X)U13#(active(X)) → U13#(X)

Problem 3: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U31, U13, U21, U22

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

U21#(mark(X1), X2) → U21#(X1, X2)U21#(active(X1), X2) → U21#(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, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U31, U13, U21, U22

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 16: 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, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U13, U31, U21, U22

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 22: 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, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U31, U13, U21, U22

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 26: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U13, U31, U21, U22

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 5: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

U22#(mark(X)) → U22#(X)U22#(active(X)) → U22#(X)

Rewrite Rules

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U31, U13, U21, U22

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

U22#(mark(X)) → U22#(X)U22#(active(X)) → U22#(X)

Problem 6: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

isNatKind#(active(X)) → isNatKind#(X)isNatKind#(mark(X)) → isNatKind#(X)

Rewrite Rules

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U31, U13, U21, U22

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

isNatKind#(active(X)) → isNatKind#(X)isNatKind#(mark(X)) → isNatKind#(X)

Problem 7: 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, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U31, U13, U21, U22

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 17: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U13, U31, U21, U22

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:

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

Problem 8: 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, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U31, U13, U21, U22

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 18: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U13, U31, U21, U22

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:

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

Problem 9: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U31, U13, U21, U22

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: 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, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U31, U13, U21, U22

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 19: 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, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U13, U31, U21, U22

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 23: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

plus#(X1, mark(X2)) → plus#(X1, X2)

Rewrite Rules

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U31, U13, U21, U22

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 12: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

U11#(X1, X2, mark(X3)) → U11#(X1, X2, X3)U11#(X1, active(X2), X3) → U11#(X1, X2, X3)
U11#(active(X1), X2, X3) → U11#(X1, X2, X3)U11#(X1, mark(X2), X3) → U11#(X1, X2, X3)
U11#(mark(X1), X2, X3) → U11#(X1, X2, X3)U11#(X1, X2, active(X3)) → U11#(X1, X2, X3)

Rewrite Rules

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U31, U13, U21, U22

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

U11#(active(X1), X2, X3) → U11#(X1, X2, X3)U11#(mark(X1), X2, X3) → U11#(X1, X2, X3)

Problem 20: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

U11#(X1, X2, mark(X3)) → U11#(X1, X2, X3)U11#(X1, active(X2), X3) → U11#(X1, X2, X3)
U11#(X1, mark(X2), X3) → U11#(X1, X2, X3)U11#(X1, X2, active(X3)) → U11#(X1, X2, X3)

Rewrite Rules

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U13, U31, U21, U22

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, X2, mark(X3)) → U11#(X1, X2, X3)

Problem 25: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

U11#(X1, active(X2), X3) → U11#(X1, X2, X3)U11#(X1, mark(X2), X3) → U11#(X1, X2, X3)
U11#(X1, X2, active(X3)) → U11#(X1, X2, X3)

Rewrite Rules

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U31, U13, U21, U22

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, X2, active(X3)) → U11#(X1, X2, X3)

Problem 27: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

U11#(X1, active(X2), X3) → U11#(X1, X2, X3)U11#(X1, mark(X2), X3) → U11#(X1, X2, X3)

Rewrite Rules

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U13, U31, U21, U22

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), X3) → U11#(X1, X2, X3)

Problem 28: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

U11#(X1, mark(X2), X3) → U11#(X1, X2, X3)

Rewrite Rules

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U31, U13, U21, U22

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), X3) → U11#(X1, X2, X3)

Problem 13: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

U12#(active(X1), X2) → U12#(X1, X2)U12#(X1, active(X2)) → U12#(X1, X2)
U12#(mark(X1), X2) → U12#(X1, X2)U12#(X1, mark(X2)) → U12#(X1, X2)

Rewrite Rules

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U31, U13, U21, U22

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

U12#(active(X1), X2) → U12#(X1, X2)U12#(mark(X1), X2) → U12#(X1, X2)

Problem 21: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

U12#(X1, active(X2)) → U12#(X1, X2)U12#(X1, mark(X2)) → U12#(X1, X2)

Rewrite Rules

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U13, U31, U21, U22

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:

U12#(X1, active(X2)) → U12#(X1, X2)

Problem 24: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

U12#(X1, mark(X2)) → U12#(X1, X2)

Rewrite Rules

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U31, U13, U21, U22

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:

U12#(X1, mark(X2)) → U12#(X1, X2)

Problem 14: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)U22(active(X)) → U22(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)
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)
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)
isNatKind(mark(X)) → isNatKind(X)isNatKind(active(X)) → isNatKind(X)

Original Signature

Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, U12, U31, U13, U21, U22

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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