YES

The TRS could be proven terminating. The proof took 26443 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (847ms).
 | – Problem 2 was processed with processor SubtermCriterion (0ms).
 |    | – Problem 9 was processed with processor SubtermCriterion (0ms).
 |    |    | – Problem 13 was processed with processor PolynomialLinearRange4 (28ms).
 |    |    |    | – Problem 15 was processed with processor PolynomialLinearRange4 (110ms).
 | – Problem 3 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 10 was processed with processor SubtermCriterion (1ms).
 | – Problem 4 was processed with processor SubtermCriterion (0ms).
 |    | – Problem 11 was processed with processor SubtermCriterion (0ms).
 | – Problem 5 was processed with processor SubtermCriterion (0ms).
 | – Problem 6 was processed with processor SubtermCriterion (1ms).
 | – Problem 7 was processed with processor PolynomialLinearRange4 (291ms).
 |    | – Problem 14 was processed with processor PolynomialLinearRange4 (299ms).
 |    |    | – Problem 16 was processed with processor PolynomialLinearRange4 (200ms).
 |    |    |    | – Problem 17 was processed with processor PolynomialLinearRange4 (203ms).
 |    |    |    |    | – Problem 18 was processed with processor PolynomialLinearRange4 (197ms).
 |    |    |    |    |    | – Problem 19 was processed with processor PolynomialLinearRange4 (185ms).
 |    |    |    |    |    |    | – Problem 20 was processed with processor PolynomialLinearRange4 (203ms).
 |    |    |    |    |    |    |    | – Problem 21 was processed with processor PolynomialLinearRange4 (208ms).
 |    |    |    |    |    |    |    |    | – Problem 22 was processed with processor PolynomialLinearRange4 (171ms).
 |    |    |    |    |    |    |    |    |    | – Problem 23 was processed with processor PolynomialLinearRange4 (135ms).
 |    |    |    |    |    |    |    |    |    |    | – Problem 24 was processed with processor ReductionPairSAT (2668ms).
 |    |    |    |    |    |    |    |    |    |    |    | – Problem 25 was processed with processor DependencyGraph (2ms).
 |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 26 was processed with processor ReductionPairSAT (52ms).
 | – Problem 8 was processed with processor SubtermCriterion (2ms).
 |    | – Problem 12 was processed with processor SubtermCriterion (0ms).

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

Strategy

The following SCCs where found

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

Problem 2: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 9: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 13: PolynomialLinearRange4

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

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:

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

Problem 15: PolynomialLinearRange4

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

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:

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

Problem 3: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 10: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 4: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

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 11: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 5: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 6: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 7: PolynomialLinearRange4

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

Strategy

Polynomial Interpretation

Standard Usable rules

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

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

mark#(s(X)) → active#(s(mark(X)))

Problem 14: PolynomialLinearRange4

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

Strategy

Polynomial Interpretation

Standard Usable rules

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

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

active#(plus(N, 0)) → mark#(U11(isNat(N), N))

Problem 16: PolynomialLinearRange4

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

Strategy

Polynomial Interpretation

Standard Usable rules

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

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

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

Problem 17: PolynomialLinearRange4

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

Strategy

Polynomial Interpretation

Standard Usable rules

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

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

active#(isNat(plus(V1, V2))) → mark#(and(isNat(V1), isNat(V2)))

Problem 18: PolynomialLinearRange4

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

Strategy

Polynomial Interpretation

Standard Usable rules

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

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

mark#(s(X)) → mark#(X)

Problem 19: PolynomialLinearRange4

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

Strategy

Polynomial Interpretation

Standard Usable rules

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

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

mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2)))active#(plus(N, s(M))) → mark#(U21(and(isNat(M), isNat(N)), M, N))

Problem 20: PolynomialLinearRange4

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

Strategy

Polynomial Interpretation

Standard Usable rules

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

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

active#(U21(tt, M, N)) → mark#(s(plus(N, M)))

Problem 21: PolynomialLinearRange4

Dependency Pair Problem

Dependency Pairs

mark#(and(X1, X2)) → mark#(X1)active#(isNat(s(V1))) → mark#(isNat(V1))
mark#(U11(X1, X2)) → mark#(X1)mark#(isNat(X)) → active#(isNat(X))
mark#(and(X1, X2)) → active#(and(mark(X1), X2))active#(U11(tt, N)) → mark#(N)
mark#(U11(X1, X2)) → active#(U11(mark(X1), X2))mark#(U21(X1, X2, X3)) → active#(U21(mark(X1), X2, X3))
active#(and(tt, X)) → mark#(X)

Rewrite Rules

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

Original Signature

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

Strategy

Polynomial Interpretation

Standard Usable rules

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

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

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

Problem 22: PolynomialLinearRange4

Dependency Pair Problem

Dependency Pairs

active#(isNat(s(V1))) → mark#(isNat(V1))mark#(and(X1, X2)) → mark#(X1)
mark#(U11(X1, X2)) → mark#(X1)mark#(isNat(X)) → active#(isNat(X))
mark#(and(X1, X2)) → active#(and(mark(X1), X2))active#(and(tt, X)) → mark#(X)
mark#(U11(X1, X2)) → active#(U11(mark(X1), X2))active#(U11(tt, N)) → mark#(N)

Rewrite Rules

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

Original Signature

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

Strategy

Polynomial Interpretation

Standard Usable rules

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

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

active#(isNat(s(V1))) → mark#(isNat(V1))

Problem 23: PolynomialLinearRange4

Dependency Pair Problem

Dependency Pairs

mark#(and(X1, X2)) → mark#(X1)mark#(U11(X1, X2)) → mark#(X1)
mark#(isNat(X)) → active#(isNat(X))mark#(and(X1, X2)) → active#(and(mark(X1), X2))
active#(U11(tt, N)) → mark#(N)mark#(U11(X1, X2)) → active#(U11(mark(X1), X2))
active#(and(tt, X)) → mark#(X)

Rewrite Rules

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

Original Signature

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

Strategy

Polynomial Interpretation

Standard Usable rules

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

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

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

Problem 24: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

mark#(and(X1, X2)) → mark#(X1)mark#(isNat(X)) → active#(isNat(X))
mark#(and(X1, X2)) → active#(and(mark(X1), X2))active#(and(tt, X)) → mark#(X)
mark#(U11(X1, X2)) → active#(U11(mark(X1), X2))

Rewrite Rules

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

Original Signature

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

Strategy

Function Precedence

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

Argument Filtering

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

Status

plus: lexicographic with permutation 1 → 1 2 → 2
0: multiset
tt: multiset
U21: lexicographic with permutation 2 → 2 3 → 1
and: lexicographic with permutation 1 → 1 2 → 2

Usable Rules

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

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

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

active#(and(tt, X)) → mark#(X)

Problem 25: DependencyGraph

Dependency Pair Problem

Dependency Pairs

mark#(and(X1, X2)) → mark#(X1)mark#(isNat(X)) → active#(isNat(X))
mark#(and(X1, X2)) → active#(and(mark(X1), X2))mark#(U11(X1, X2)) → active#(U11(mark(X1), X2))

Rewrite Rules

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

Original Signature

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

Strategy

The following SCCs where found

mark#(and(X1, X2)) → mark#(X1)

Problem 26: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

mark#(and(X1, X2)) → mark#(X1)

Rewrite Rules

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

Original Signature

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

Strategy

Function Precedence

isNat = plus = 0 = s = tt = U11 = active = mark = mark# = U21 = and

Argument Filtering

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

Status

isNat: multiset
plus: multiset
0: multiset
tt: multiset
U11: lexicographic with permutation 1 → 2 2 → 1
mark: multiset
mark#: lexicographic with permutation 1 → 1
U21: lexicographic with permutation 1 → 2 3 → 1
and: multiset

Usable Rules

There are no usable rules.

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

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

mark#(and(X1, X2)) → mark#(X1)

Problem 8: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 12: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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