TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60015 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (7375ms).
 | – Problem 2 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (8ms), PolynomialLinearRange4iUR (3333ms), DependencyGraph (7ms), PolynomialLinearRange4iUR (5019ms), DependencyGraph (6ms), PolynomialLinearRange8NegiUR (15000ms), DependencyGraph (7ms), ReductionPairSAT (18294ms), DependencyGraph (5ms), ReductionPairSAT (timeout)].
 | – Problem 3 was processed with processor SubtermCriterion (5ms).
 | – Problem 4 was processed with processor SubtermCriterion (2ms).
 | – Problem 5 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 14 was processed with processor ReductionPairSAT (102ms).
 | – Problem 6 was processed with processor SubtermCriterion (2ms).
 | – Problem 7 was processed with processor SubtermCriterion (1ms).
 | – Problem 8 was processed with processor SubtermCriterion (3ms).
 | – Problem 9 was processed with processor SubtermCriterion (1ms).
 | – Problem 10 was processed with processor SubtermCriterion (1ms).
 | – Problem 11 was processed with processor SubtermCriterion (2ms).
 |    | – Problem 15 was processed with processor PolynomialLinearRange4iUR (76ms).
 | – Problem 12 was processed with processor SubtermCriterion (3ms).
 | – Problem 13 was processed with processor SubtermCriterion (1ms).

The following open problems remain:

Open Dependency Pair Problem 2

Dependency Pairs

top#(mark(X)) → top#(proper(X))top#(ok(X)) → top#(active(X))

Rewrite Rules

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

Original Signature

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

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

Strategy

The following SCCs where found

plus#(ok(X1), ok(X2)) → plus#(X1, X2)plus#(X1, mark(X2)) → plus#(X1, X2)
plus#(mark(X1), X2) → plus#(X1, X2)
active#(plus(X1, X2)) → active#(X1)active#(U31(X)) → active#(X)
active#(s(X)) → active#(X)active#(x(X1, X2)) → active#(X1)
active#(x(X1, X2)) → active#(X2)active#(plus(X1, X2)) → active#(X2)
active#(and(X1, X2)) → active#(X1)active#(U11(X1, X2)) → active#(X1)
active#(U41(X1, X2, X3)) → active#(X1)active#(U21(X1, X2, X3)) → active#(X1)
U21#(ok(X1), ok(X2), ok(X3)) → U21#(X1, X2, X3)U21#(mark(X1), X2, X3) → U21#(X1, X2, X3)
x#(mark(X1), X2) → x#(X1, X2)x#(ok(X1), ok(X2)) → x#(X1, X2)
x#(X1, mark(X2)) → x#(X1, X2)
proper#(isNat(X)) → proper#(X)proper#(U11(X1, X2)) → proper#(X1)
proper#(U41(X1, X2, X3)) → proper#(X3)proper#(U21(X1, X2, X3)) → proper#(X3)
proper#(x(X1, X2)) → proper#(X1)proper#(and(X1, X2)) → proper#(X1)
proper#(x(X1, X2)) → proper#(X2)proper#(U41(X1, X2, X3)) → proper#(X2)
proper#(s(X)) → proper#(X)proper#(U11(X1, X2)) → proper#(X2)
proper#(and(X1, X2)) → proper#(X2)proper#(U21(X1, X2, X3)) → proper#(X1)
proper#(U41(X1, X2, X3)) → proper#(X1)proper#(plus(X1, X2)) → proper#(X1)
proper#(U21(X1, X2, X3)) → proper#(X2)proper#(plus(X1, X2)) → proper#(X2)
proper#(U31(X)) → proper#(X)
U31#(ok(X)) → U31#(X)U31#(mark(X)) → U31#(X)
U11#(ok(X1), ok(X2)) → U11#(X1, X2)U11#(mark(X1), X2) → U11#(X1, X2)
isNat#(ok(X)) → isNat#(X)
and#(ok(X1), ok(X2)) → and#(X1, X2)and#(mark(X1), X2) → and#(X1, X2)
s#(mark(X)) → s#(X)s#(ok(X)) → s#(X)
U41#(mark(X1), X2, X3) → U41#(X1, X2, X3)U41#(ok(X1), ok(X2), ok(X3)) → U41#(X1, X2, X3)
top#(mark(X)) → top#(proper(X))top#(ok(X)) → top#(active(X))

Problem 3: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

proper#(isNat(X)) → proper#(X)proper#(U11(X1, X2)) → proper#(X1)
proper#(U41(X1, X2, X3)) → proper#(X3)proper#(U21(X1, X2, X3)) → proper#(X3)
proper#(x(X1, X2)) → proper#(X1)proper#(and(X1, X2)) → proper#(X1)
proper#(U41(X1, X2, X3)) → proper#(X2)proper#(x(X1, X2)) → proper#(X2)
proper#(s(X)) → proper#(X)proper#(U11(X1, X2)) → proper#(X2)
proper#(and(X1, X2)) → proper#(X2)proper#(U21(X1, X2, X3)) → proper#(X1)
proper#(U41(X1, X2, X3)) → proper#(X1)proper#(plus(X1, X2)) → proper#(X1)
proper#(U21(X1, X2, X3)) → proper#(X2)proper#(plus(X1, X2)) → proper#(X2)
proper#(U31(X)) → proper#(X)

Rewrite Rules

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

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

proper#(isNat(X)) → proper#(X)proper#(U11(X1, X2)) → proper#(X1)
proper#(U41(X1, X2, X3)) → proper#(X3)proper#(U21(X1, X2, X3)) → proper#(X3)
proper#(x(X1, X2)) → proper#(X1)proper#(and(X1, X2)) → proper#(X1)
proper#(x(X1, X2)) → proper#(X2)proper#(U41(X1, X2, X3)) → proper#(X2)
proper#(s(X)) → proper#(X)proper#(U11(X1, X2)) → proper#(X2)
proper#(and(X1, X2)) → proper#(X2)proper#(U21(X1, X2, X3)) → proper#(X1)
proper#(U41(X1, X2, X3)) → proper#(X1)proper#(plus(X1, X2)) → proper#(X1)
proper#(U21(X1, X2, X3)) → proper#(X2)proper#(plus(X1, X2)) → proper#(X2)
proper#(U31(X)) → proper#(X)

Problem 4: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

active#(plus(X1, X2)) → active#(X1)active#(U31(X)) → active#(X)
active#(s(X)) → active#(X)active#(x(X1, X2)) → active#(X1)
active#(x(X1, X2)) → active#(X2)active#(plus(X1, X2)) → active#(X2)
active#(and(X1, X2)) → active#(X1)active#(U11(X1, X2)) → active#(X1)
active#(U41(X1, X2, X3)) → active#(X1)active#(U21(X1, X2, X3)) → active#(X1)

Rewrite Rules

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

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

active#(U31(X)) → active#(X)active#(plus(X1, X2)) → active#(X1)
active#(s(X)) → active#(X)active#(x(X1, X2)) → active#(X1)
active#(x(X1, X2)) → active#(X2)active#(plus(X1, X2)) → active#(X2)
active#(and(X1, X2)) → active#(X1)active#(U11(X1, X2)) → active#(X1)
active#(U21(X1, X2, X3)) → active#(X1)active#(U41(X1, X2, X3)) → active#(X1)

Problem 5: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 14: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

plus#(X1, mark(X2)) → plus#(X1, X2)

Rewrite Rules

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

Original Signature

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

Strategy

Function Precedence

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

Argument Filtering

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

Status

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

Usable Rules

There are no usable rules.

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

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

plus#(X1, mark(X2)) → plus#(X1, X2)

Problem 6: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

isNat#(ok(X)) → isNat#(X)

Rewrite Rules

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

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

isNat#(ok(X)) → isNat#(X)

Problem 7: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

U31#(ok(X)) → U31#(X)U31#(mark(X)) → U31#(X)

Rewrite Rules

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

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

U31#(ok(X)) → U31#(X)U31#(mark(X)) → U31#(X)

Problem 8: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

s#(mark(X)) → s#(X)s#(ok(X)) → s#(X)

Rewrite Rules

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

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

s#(mark(X)) → s#(X)s#(ok(X)) → s#(X)

Problem 9: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 10: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 11: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 15: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

x#(X1, mark(X2)) → x#(X1, X2)

Rewrite Rules

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

Original Signature

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

Strategy

Polynomial Interpretation

There are no usable rules

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

x#(X1, mark(X2)) → x#(X1, X2)

Problem 12: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 13: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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