TIMEOUT

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

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (4426ms).
 | – Problem 2 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (7ms), PolynomialLinearRange4iUR (3333ms), DependencyGraph (6ms), PolynomialLinearRange4iUR (5000ms), DependencyGraph (27ms), PolynomialLinearRange8NegiUR (15000ms), DependencyGraph (7ms), ReductionPairSAT (12899ms), DependencyGraph (5ms), ReductionPairSAT (12773ms), DependencyGraph (5ms), SizeChangePrinciple (timeout)].
 | – Problem 3 was processed with processor SubtermCriterion (2ms).
 | – Problem 4 was processed with processor SubtermCriterion (3ms).
 |    | – Problem 12 was processed with processor ReductionPairSAT (90ms).
 | – Problem 5 was processed with processor SubtermCriterion (1ms).
 | – Problem 6 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 13 was processed with processor PolynomialLinearRange4iUR (23ms).
 | – Problem 7 was processed with processor SubtermCriterion (1ms).
 | – Problem 8 was processed with processor SubtermCriterion (1ms).
 | – Problem 9 was processed with processor SubtermCriterion (4ms).
 | – Problem 10 was processed with processor SubtermCriterion (1ms).
 | – Problem 11 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, M, N))mark(U12(tt, M, N))active(U12(tt, M, N))mark(s(plus(N, M)))
active(U21(tt, M, N))mark(U22(tt, M, N))active(U22(tt, M, N))mark(plus(x(N, M), N))
active(plus(N, 0))mark(N)active(plus(N, s(M)))mark(U11(tt, M, N))
active(x(N, 0))mark(0)active(x(N, s(M)))mark(U21(tt, M, N))
active(U11(X1, X2, X3))U11(active(X1), X2, X3)active(U12(X1, X2, X3))U12(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(U21(X1, X2, X3))U21(active(X1), X2, X3)
active(U22(X1, X2, X3))U22(active(X1), X2, X3)active(x(X1, X2))x(active(X1), X2)
active(x(X1, X2))x(X1, active(X2))U11(mark(X1), X2, X3)mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3)mark(U12(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))
U21(mark(X1), X2, X3)mark(U21(X1, X2, X3))U22(mark(X1), X2, X3)mark(U22(X1, X2, X3))
x(mark(X1), X2)mark(x(X1, X2))x(X1, mark(X2))mark(x(X1, X2))
proper(U11(X1, X2, X3))U11(proper(X1), proper(X2), proper(X3))proper(tt)ok(tt)
proper(U12(X1, X2, X3))U12(proper(X1), proper(X2), proper(X3))proper(s(X))s(proper(X))
proper(plus(X1, X2))plus(proper(X1), proper(X2))proper(U21(X1, X2, X3))U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3))U22(proper(X1), proper(X2), proper(X3))proper(x(X1, X2))x(proper(X1), proper(X2))
proper(0)ok(0)U11(ok(X1), ok(X2), ok(X3))ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3))ok(U12(X1, X2, X3))s(ok(X))ok(s(X))
plus(ok(X1), ok(X2))ok(plus(X1, X2))U21(ok(X1), ok(X2), ok(X3))ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2), ok(X3))ok(U22(X1, X2, X3))x(ok(X1), ok(X2))ok(x(X1, X2))
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, 0, s, tt, active, U11, U12, ok, proper, U21, top, U22, x


Problem 1: DependencyGraph



Dependency Pair Problem

Dependency Pairs

active#(x(N, s(M)))U21#(tt, M, N)top#(ok(X))top#(active(X))
U21#(ok(X1), ok(X2), ok(X3))U21#(X1, X2, X3)proper#(U11(X1, X2, X3))proper#(X3)
U22#(mark(X1), X2, X3)U22#(X1, X2, X3)proper#(U22(X1, X2, X3))U22#(proper(X1), proper(X2), proper(X3))
proper#(U11(X1, X2, X3))proper#(X2)plus#(X1, mark(X2))plus#(X1, X2)
proper#(plus(X1, X2))proper#(X1)x#(ok(X1), ok(X2))x#(X1, X2)
x#(X1, mark(X2))x#(X1, X2)top#(mark(X))proper#(X)
proper#(plus(X1, X2))plus#(proper(X1), proper(X2))top#(mark(X))top#(proper(X))
U12#(mark(X1), X2, X3)U12#(X1, X2, X3)proper#(U22(X1, X2, X3))proper#(X3)
U21#(mark(X1), X2, X3)U21#(X1, X2, X3)U11#(mark(X1), X2, X3)U11#(X1, X2, X3)
active#(U21(X1, X2, X3))U21#(active(X1), X2, X3)active#(U22(X1, X2, X3))U22#(active(X1), X2, X3)
proper#(U12(X1, X2, X3))proper#(X3)active#(x(X1, X2))x#(X1, active(X2))
active#(U21(X1, X2, X3))active#(X1)active#(U11(X1, X2, X3))active#(X1)
active#(U12(X1, X2, X3))active#(X1)proper#(s(X))proper#(X)
active#(plus(X1, X2))active#(X1)U12#(ok(X1), ok(X2), ok(X3))U12#(X1, X2, X3)
active#(U12(tt, M, N))s#(plus(N, M))U22#(ok(X1), ok(X2), ok(X3))U22#(X1, X2, X3)
active#(plus(X1, X2))active#(X2)U11#(ok(X1), ok(X2), ok(X3))U11#(X1, X2, X3)
plus#(mark(X1), X2)plus#(X1, X2)proper#(U12(X1, X2, X3))proper#(X1)
proper#(U21(X1, X2, X3))proper#(X3)proper#(x(X1, X2))proper#(X1)
active#(U11(X1, X2, X3))U11#(active(X1), X2, X3)active#(x(X1, X2))x#(active(X1), X2)
top#(ok(X))active#(X)active#(U12(X1, X2, X3))U12#(active(X1), X2, X3)
x#(mark(X1), X2)x#(X1, X2)proper#(U21(X1, X2, X3))proper#(X1)
proper#(U22(X1, X2, X3))proper#(X1)proper#(x(X1, X2))x#(proper(X1), proper(X2))
proper#(plus(X1, X2))proper#(X2)plus#(ok(X1), ok(X2))plus#(X1, X2)
active#(U21(tt, M, N))U22#(tt, M, N)proper#(U12(X1, X2, X3))proper#(X2)
proper#(U12(X1, X2, X3))U12#(proper(X1), proper(X2), proper(X3))proper#(U22(X1, X2, X3))proper#(X2)
active#(U22(tt, M, N))x#(N, M)active#(x(X1, X2))active#(X1)
active#(U12(tt, M, N))plus#(N, M)proper#(U21(X1, X2, X3))U21#(proper(X1), proper(X2), proper(X3))
active#(s(X))s#(active(X))proper#(x(X1, X2))proper#(X2)
s#(ok(X))s#(X)proper#(U11(X1, X2, X3))U11#(proper(X1), proper(X2), proper(X3))
proper#(U11(X1, X2, X3))proper#(X1)s#(mark(X))s#(X)
active#(U22(tt, M, N))plus#(x(N, M), N)active#(plus(X1, X2))plus#(X1, active(X2))
active#(U22(X1, X2, X3))active#(X1)active#(U11(tt, M, N))U12#(tt, M, N)
active#(s(X))active#(X)active#(plus(X1, X2))plus#(active(X1), X2)
proper#(s(X))s#(proper(X))active#(x(X1, X2))active#(X2)
proper#(U21(X1, X2, X3))proper#(X2)active#(plus(N, s(M)))U11#(tt, M, N)

Rewrite Rules

active(U11(tt, M, N))mark(U12(tt, M, N))active(U12(tt, M, N))mark(s(plus(N, M)))
active(U21(tt, M, N))mark(U22(tt, M, N))active(U22(tt, M, N))mark(plus(x(N, M), N))
active(plus(N, 0))mark(N)active(plus(N, s(M)))mark(U11(tt, M, N))
active(x(N, 0))mark(0)active(x(N, s(M)))mark(U21(tt, M, N))
active(U11(X1, X2, X3))U11(active(X1), X2, X3)active(U12(X1, X2, X3))U12(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(U21(X1, X2, X3))U21(active(X1), X2, X3)
active(U22(X1, X2, X3))U22(active(X1), X2, X3)active(x(X1, X2))x(active(X1), X2)
active(x(X1, X2))x(X1, active(X2))U11(mark(X1), X2, X3)mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3)mark(U12(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))
U21(mark(X1), X2, X3)mark(U21(X1, X2, X3))U22(mark(X1), X2, X3)mark(U22(X1, X2, X3))
x(mark(X1), X2)mark(x(X1, X2))x(X1, mark(X2))mark(x(X1, X2))
proper(U11(X1, X2, X3))U11(proper(X1), proper(X2), proper(X3))proper(tt)ok(tt)
proper(U12(X1, X2, X3))U12(proper(X1), proper(X2), proper(X3))proper(s(X))s(proper(X))
proper(plus(X1, X2))plus(proper(X1), proper(X2))proper(U21(X1, X2, X3))U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3))U22(proper(X1), proper(X2), proper(X3))proper(x(X1, X2))x(proper(X1), proper(X2))
proper(0)ok(0)U11(ok(X1), ok(X2), ok(X3))ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3))ok(U12(X1, X2, X3))s(ok(X))ok(s(X))
plus(ok(X1), ok(X2))ok(plus(X1, X2))U21(ok(X1), ok(X2), ok(X3))ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2), ok(X3))ok(U22(X1, X2, X3))x(ok(X1), ok(X2))ok(x(X1, X2))
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, 0, s, tt, active, U11, U12, ok, proper, U21, x, top, U22

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)

proper#(U11(X1, X2, X3)) → proper#(X3)proper#(U12(X1, X2, X3)) → proper#(X2)
proper#(U22(X1, X2, X3)) → proper#(X2)proper#(U21(X1, X2, X3)) → proper#(X3)
proper#(U22(X1, X2, X3)) → proper#(X3)proper#(x(X1, X2)) → proper#(X1)
proper#(x(X1, X2)) → proper#(X2)proper#(U12(X1, X2, X3)) → proper#(X3)
proper#(U11(X1, X2, X3)) → proper#(X1)proper#(s(X)) → proper#(X)
proper#(U11(X1, X2, X3)) → proper#(X2)proper#(U21(X1, X2, X3)) → proper#(X1)
proper#(U22(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#(U12(X1, X2, X3)) → proper#(X1)

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

U22#(mark(X1), X2, X3) → U22#(X1, X2, X3)U22#(ok(X1), ok(X2), ok(X3)) → U22#(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)

active#(U11(X1, X2, X3)) → active#(X1)active#(U22(X1, X2, X3)) → active#(X1)
active#(U12(X1, X2, X3)) → active#(X1)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#(U21(X1, X2, X3)) → active#(X1)

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

s#(mark(X)) → s#(X)s#(ok(X)) → s#(X)

U12#(ok(X1), ok(X2), ok(X3)) → U12#(X1, X2, X3)U12#(mark(X1), X2, X3) → U12#(X1, X2, X3)

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

Problem 3: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

s#(mark(X))s#(X)s#(ok(X))s#(X)

Rewrite Rules

active(U11(tt, M, N))mark(U12(tt, M, N))active(U12(tt, M, N))mark(s(plus(N, M)))
active(U21(tt, M, N))mark(U22(tt, M, N))active(U22(tt, M, N))mark(plus(x(N, M), N))
active(plus(N, 0))mark(N)active(plus(N, s(M)))mark(U11(tt, M, N))
active(x(N, 0))mark(0)active(x(N, s(M)))mark(U21(tt, M, N))
active(U11(X1, X2, X3))U11(active(X1), X2, X3)active(U12(X1, X2, X3))U12(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(U21(X1, X2, X3))U21(active(X1), X2, X3)
active(U22(X1, X2, X3))U22(active(X1), X2, X3)active(x(X1, X2))x(active(X1), X2)
active(x(X1, X2))x(X1, active(X2))U11(mark(X1), X2, X3)mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3)mark(U12(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))
U21(mark(X1), X2, X3)mark(U21(X1, X2, X3))U22(mark(X1), X2, X3)mark(U22(X1, X2, X3))
x(mark(X1), X2)mark(x(X1, X2))x(X1, mark(X2))mark(x(X1, X2))
proper(U11(X1, X2, X3))U11(proper(X1), proper(X2), proper(X3))proper(tt)ok(tt)
proper(U12(X1, X2, X3))U12(proper(X1), proper(X2), proper(X3))proper(s(X))s(proper(X))
proper(plus(X1, X2))plus(proper(X1), proper(X2))proper(U21(X1, X2, X3))U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3))U22(proper(X1), proper(X2), proper(X3))proper(x(X1, X2))x(proper(X1), proper(X2))
proper(0)ok(0)U11(ok(X1), ok(X2), ok(X3))ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3))ok(U12(X1, X2, X3))s(ok(X))ok(s(X))
plus(ok(X1), ok(X2))ok(plus(X1, X2))U21(ok(X1), ok(X2), ok(X3))ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2), ok(X3))ok(U22(X1, X2, X3))x(ok(X1), ok(X2))ok(x(X1, X2))
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, 0, s, tt, active, U11, U12, ok, proper, U21, x, top, U22

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 4: 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, M, N))mark(U12(tt, M, N))active(U12(tt, M, N))mark(s(plus(N, M)))
active(U21(tt, M, N))mark(U22(tt, M, N))active(U22(tt, M, N))mark(plus(x(N, M), N))
active(plus(N, 0))mark(N)active(plus(N, s(M)))mark(U11(tt, M, N))
active(x(N, 0))mark(0)active(x(N, s(M)))mark(U21(tt, M, N))
active(U11(X1, X2, X3))U11(active(X1), X2, X3)active(U12(X1, X2, X3))U12(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(U21(X1, X2, X3))U21(active(X1), X2, X3)
active(U22(X1, X2, X3))U22(active(X1), X2, X3)active(x(X1, X2))x(active(X1), X2)
active(x(X1, X2))x(X1, active(X2))U11(mark(X1), X2, X3)mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3)mark(U12(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))
U21(mark(X1), X2, X3)mark(U21(X1, X2, X3))U22(mark(X1), X2, X3)mark(U22(X1, X2, X3))
x(mark(X1), X2)mark(x(X1, X2))x(X1, mark(X2))mark(x(X1, X2))
proper(U11(X1, X2, X3))U11(proper(X1), proper(X2), proper(X3))proper(tt)ok(tt)
proper(U12(X1, X2, X3))U12(proper(X1), proper(X2), proper(X3))proper(s(X))s(proper(X))
proper(plus(X1, X2))plus(proper(X1), proper(X2))proper(U21(X1, X2, X3))U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3))U22(proper(X1), proper(X2), proper(X3))proper(x(X1, X2))x(proper(X1), proper(X2))
proper(0)ok(0)U11(ok(X1), ok(X2), ok(X3))ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3))ok(U12(X1, X2, X3))s(ok(X))ok(s(X))
plus(ok(X1), ok(X2))ok(plus(X1, X2))U21(ok(X1), ok(X2), ok(X3))ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2), ok(X3))ok(U22(X1, X2, X3))x(ok(X1), ok(X2))ok(x(X1, X2))
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, 0, s, tt, active, U11, U12, ok, proper, U21, x, top, U22

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 12: ReductionPairSAT



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(U11(tt, M, N))mark(U12(tt, M, N))active(U12(tt, M, N))mark(s(plus(N, M)))
active(U21(tt, M, N))mark(U22(tt, M, N))active(U22(tt, M, N))mark(plus(x(N, M), N))
active(plus(N, 0))mark(N)active(plus(N, s(M)))mark(U11(tt, M, N))
active(x(N, 0))mark(0)active(x(N, s(M)))mark(U21(tt, M, N))
active(U11(X1, X2, X3))U11(active(X1), X2, X3)active(U12(X1, X2, X3))U12(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(U21(X1, X2, X3))U21(active(X1), X2, X3)
active(U22(X1, X2, X3))U22(active(X1), X2, X3)active(x(X1, X2))x(active(X1), X2)
active(x(X1, X2))x(X1, active(X2))U11(mark(X1), X2, X3)mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3)mark(U12(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))
U21(mark(X1), X2, X3)mark(U21(X1, X2, X3))U22(mark(X1), X2, X3)mark(U22(X1, X2, X3))
x(mark(X1), X2)mark(x(X1, X2))x(X1, mark(X2))mark(x(X1, X2))
proper(U11(X1, X2, X3))U11(proper(X1), proper(X2), proper(X3))proper(tt)ok(tt)
proper(U12(X1, X2, X3))U12(proper(X1), proper(X2), proper(X3))proper(s(X))s(proper(X))
proper(plus(X1, X2))plus(proper(X1), proper(X2))proper(U21(X1, X2, X3))U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3))U22(proper(X1), proper(X2), proper(X3))proper(x(X1, X2))x(proper(X1), proper(X2))
proper(0)ok(0)U11(ok(X1), ok(X2), ok(X3))ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3))ok(U12(X1, X2, X3))s(ok(X))ok(s(X))
plus(ok(X1), ok(X2))ok(plus(X1, X2))U21(ok(X1), ok(X2), ok(X3))ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2), ok(X3))ok(U22(X1, X2, X3))x(ok(X1), ok(X2))ok(x(X1, X2))
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, 0, s, tt, active, U11, U12, ok, proper, U21, U22, x, top

Strategy


Function Precedence

plus = mark = 0 = s = tt = U11 = active = U12 = ok = proper = x# = U21 = x = top = U22

Argument Filtering

plus: collapses to 1
mark: 1
0: all arguments are removed from 0
s: all arguments are removed from s
tt: all arguments are removed from tt
U11: 2 3
active: collapses to 1
U12: all arguments are removed from U12
ok: collapses to 1
proper: collapses to 1
x#: collapses to 2
U21: 2 3
x: 1 2
top: all arguments are removed from top
U22: all arguments are removed from U22

Status

mark: multiset
0: multiset
s: multiset
tt: multiset
U11: lexicographic with permutation 2 → 2 3 → 1
U12: multiset
U21: lexicographic with permutation 2 → 2 3 → 1
x: lexicographic with permutation 1 → 2 2 → 1
top: multiset
U22: multiset

Usable Rules

There are no usable rules.

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

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

x#(X1, mark(X2)) → x#(X1, X2)

Problem 5: 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, M, N))mark(U12(tt, M, N))active(U12(tt, M, N))mark(s(plus(N, M)))
active(U21(tt, M, N))mark(U22(tt, M, N))active(U22(tt, M, N))mark(plus(x(N, M), N))
active(plus(N, 0))mark(N)active(plus(N, s(M)))mark(U11(tt, M, N))
active(x(N, 0))mark(0)active(x(N, s(M)))mark(U21(tt, M, N))
active(U11(X1, X2, X3))U11(active(X1), X2, X3)active(U12(X1, X2, X3))U12(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(U21(X1, X2, X3))U21(active(X1), X2, X3)
active(U22(X1, X2, X3))U22(active(X1), X2, X3)active(x(X1, X2))x(active(X1), X2)
active(x(X1, X2))x(X1, active(X2))U11(mark(X1), X2, X3)mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3)mark(U12(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))
U21(mark(X1), X2, X3)mark(U21(X1, X2, X3))U22(mark(X1), X2, X3)mark(U22(X1, X2, X3))
x(mark(X1), X2)mark(x(X1, X2))x(X1, mark(X2))mark(x(X1, X2))
proper(U11(X1, X2, X3))U11(proper(X1), proper(X2), proper(X3))proper(tt)ok(tt)
proper(U12(X1, X2, X3))U12(proper(X1), proper(X2), proper(X3))proper(s(X))s(proper(X))
proper(plus(X1, X2))plus(proper(X1), proper(X2))proper(U21(X1, X2, X3))U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3))U22(proper(X1), proper(X2), proper(X3))proper(x(X1, X2))x(proper(X1), proper(X2))
proper(0)ok(0)U11(ok(X1), ok(X2), ok(X3))ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3))ok(U12(X1, X2, X3))s(ok(X))ok(s(X))
plus(ok(X1), ok(X2))ok(plus(X1, X2))U21(ok(X1), ok(X2), ok(X3))ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2), ok(X3))ok(U22(X1, X2, X3))x(ok(X1), ok(X2))ok(x(X1, X2))
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, 0, s, tt, active, U11, U12, ok, proper, U21, x, top, U22

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 6: 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, M, N))mark(U12(tt, M, N))active(U12(tt, M, N))mark(s(plus(N, M)))
active(U21(tt, M, N))mark(U22(tt, M, N))active(U22(tt, M, N))mark(plus(x(N, M), N))
active(plus(N, 0))mark(N)active(plus(N, s(M)))mark(U11(tt, M, N))
active(x(N, 0))mark(0)active(x(N, s(M)))mark(U21(tt, M, N))
active(U11(X1, X2, X3))U11(active(X1), X2, X3)active(U12(X1, X2, X3))U12(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(U21(X1, X2, X3))U21(active(X1), X2, X3)
active(U22(X1, X2, X3))U22(active(X1), X2, X3)active(x(X1, X2))x(active(X1), X2)
active(x(X1, X2))x(X1, active(X2))U11(mark(X1), X2, X3)mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3)mark(U12(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))
U21(mark(X1), X2, X3)mark(U21(X1, X2, X3))U22(mark(X1), X2, X3)mark(U22(X1, X2, X3))
x(mark(X1), X2)mark(x(X1, X2))x(X1, mark(X2))mark(x(X1, X2))
proper(U11(X1, X2, X3))U11(proper(X1), proper(X2), proper(X3))proper(tt)ok(tt)
proper(U12(X1, X2, X3))U12(proper(X1), proper(X2), proper(X3))proper(s(X))s(proper(X))
proper(plus(X1, X2))plus(proper(X1), proper(X2))proper(U21(X1, X2, X3))U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3))U22(proper(X1), proper(X2), proper(X3))proper(x(X1, X2))x(proper(X1), proper(X2))
proper(0)ok(0)U11(ok(X1), ok(X2), ok(X3))ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3))ok(U12(X1, X2, X3))s(ok(X))ok(s(X))
plus(ok(X1), ok(X2))ok(plus(X1, X2))U21(ok(X1), ok(X2), ok(X3))ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2), ok(X3))ok(U22(X1, X2, X3))x(ok(X1), ok(X2))ok(x(X1, X2))
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, 0, s, tt, active, U11, U12, ok, proper, U21, x, top, U22

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 13: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(U11(tt, M, N))mark(U12(tt, M, N))active(U12(tt, M, N))mark(s(plus(N, M)))
active(U21(tt, M, N))mark(U22(tt, M, N))active(U22(tt, M, N))mark(plus(x(N, M), N))
active(plus(N, 0))mark(N)active(plus(N, s(M)))mark(U11(tt, M, N))
active(x(N, 0))mark(0)active(x(N, s(M)))mark(U21(tt, M, N))
active(U11(X1, X2, X3))U11(active(X1), X2, X3)active(U12(X1, X2, X3))U12(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(U21(X1, X2, X3))U21(active(X1), X2, X3)
active(U22(X1, X2, X3))U22(active(X1), X2, X3)active(x(X1, X2))x(active(X1), X2)
active(x(X1, X2))x(X1, active(X2))U11(mark(X1), X2, X3)mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3)mark(U12(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))
U21(mark(X1), X2, X3)mark(U21(X1, X2, X3))U22(mark(X1), X2, X3)mark(U22(X1, X2, X3))
x(mark(X1), X2)mark(x(X1, X2))x(X1, mark(X2))mark(x(X1, X2))
proper(U11(X1, X2, X3))U11(proper(X1), proper(X2), proper(X3))proper(tt)ok(tt)
proper(U12(X1, X2, X3))U12(proper(X1), proper(X2), proper(X3))proper(s(X))s(proper(X))
proper(plus(X1, X2))plus(proper(X1), proper(X2))proper(U21(X1, X2, X3))U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3))U22(proper(X1), proper(X2), proper(X3))proper(x(X1, X2))x(proper(X1), proper(X2))
proper(0)ok(0)U11(ok(X1), ok(X2), ok(X3))ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3))ok(U12(X1, X2, X3))s(ok(X))ok(s(X))
plus(ok(X1), ok(X2))ok(plus(X1, X2))U21(ok(X1), ok(X2), ok(X3))ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2), ok(X3))ok(U22(X1, X2, X3))x(ok(X1), ok(X2))ok(x(X1, X2))
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, 0, s, tt, active, U11, U12, ok, proper, U21, U22, x, top

Strategy


Polynomial Interpretation

There are no usable rules

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

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

Problem 7: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

U22#(mark(X1), X2, X3)U22#(X1, X2, X3)U22#(ok(X1), ok(X2), ok(X3))U22#(X1, X2, X3)

Rewrite Rules

active(U11(tt, M, N))mark(U12(tt, M, N))active(U12(tt, M, N))mark(s(plus(N, M)))
active(U21(tt, M, N))mark(U22(tt, M, N))active(U22(tt, M, N))mark(plus(x(N, M), N))
active(plus(N, 0))mark(N)active(plus(N, s(M)))mark(U11(tt, M, N))
active(x(N, 0))mark(0)active(x(N, s(M)))mark(U21(tt, M, N))
active(U11(X1, X2, X3))U11(active(X1), X2, X3)active(U12(X1, X2, X3))U12(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(U21(X1, X2, X3))U21(active(X1), X2, X3)
active(U22(X1, X2, X3))U22(active(X1), X2, X3)active(x(X1, X2))x(active(X1), X2)
active(x(X1, X2))x(X1, active(X2))U11(mark(X1), X2, X3)mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3)mark(U12(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))
U21(mark(X1), X2, X3)mark(U21(X1, X2, X3))U22(mark(X1), X2, X3)mark(U22(X1, X2, X3))
x(mark(X1), X2)mark(x(X1, X2))x(X1, mark(X2))mark(x(X1, X2))
proper(U11(X1, X2, X3))U11(proper(X1), proper(X2), proper(X3))proper(tt)ok(tt)
proper(U12(X1, X2, X3))U12(proper(X1), proper(X2), proper(X3))proper(s(X))s(proper(X))
proper(plus(X1, X2))plus(proper(X1), proper(X2))proper(U21(X1, X2, X3))U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3))U22(proper(X1), proper(X2), proper(X3))proper(x(X1, X2))x(proper(X1), proper(X2))
proper(0)ok(0)U11(ok(X1), ok(X2), ok(X3))ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3))ok(U12(X1, X2, X3))s(ok(X))ok(s(X))
plus(ok(X1), ok(X2))ok(plus(X1, X2))U21(ok(X1), ok(X2), ok(X3))ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2), ok(X3))ok(U22(X1, X2, X3))x(ok(X1), ok(X2))ok(x(X1, X2))
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, 0, s, tt, active, U11, U12, ok, proper, U21, x, top, U22

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

U22#(ok(X1), ok(X2), ok(X3))U22#(X1, X2, X3)U22#(mark(X1), X2, X3)U22#(X1, X2, X3)

Problem 8: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

active#(U11(X1, X2, X3))active#(X1)active#(U22(X1, X2, X3))active#(X1)
active#(U12(X1, X2, X3))active#(X1)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#(U21(X1, X2, X3))active#(X1)

Rewrite Rules

active(U11(tt, M, N))mark(U12(tt, M, N))active(U12(tt, M, N))mark(s(plus(N, M)))
active(U21(tt, M, N))mark(U22(tt, M, N))active(U22(tt, M, N))mark(plus(x(N, M), N))
active(plus(N, 0))mark(N)active(plus(N, s(M)))mark(U11(tt, M, N))
active(x(N, 0))mark(0)active(x(N, s(M)))mark(U21(tt, M, N))
active(U11(X1, X2, X3))U11(active(X1), X2, X3)active(U12(X1, X2, X3))U12(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(U21(X1, X2, X3))U21(active(X1), X2, X3)
active(U22(X1, X2, X3))U22(active(X1), X2, X3)active(x(X1, X2))x(active(X1), X2)
active(x(X1, X2))x(X1, active(X2))U11(mark(X1), X2, X3)mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3)mark(U12(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))
U21(mark(X1), X2, X3)mark(U21(X1, X2, X3))U22(mark(X1), X2, X3)mark(U22(X1, X2, X3))
x(mark(X1), X2)mark(x(X1, X2))x(X1, mark(X2))mark(x(X1, X2))
proper(U11(X1, X2, X3))U11(proper(X1), proper(X2), proper(X3))proper(tt)ok(tt)
proper(U12(X1, X2, X3))U12(proper(X1), proper(X2), proper(X3))proper(s(X))s(proper(X))
proper(plus(X1, X2))plus(proper(X1), proper(X2))proper(U21(X1, X2, X3))U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3))U22(proper(X1), proper(X2), proper(X3))proper(x(X1, X2))x(proper(X1), proper(X2))
proper(0)ok(0)U11(ok(X1), ok(X2), ok(X3))ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3))ok(U12(X1, X2, X3))s(ok(X))ok(s(X))
plus(ok(X1), ok(X2))ok(plus(X1, X2))U21(ok(X1), ok(X2), ok(X3))ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2), ok(X3))ok(U22(X1, X2, X3))x(ok(X1), ok(X2))ok(x(X1, X2))
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, 0, s, tt, active, U11, U12, ok, proper, U21, x, top, U22

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

active#(U11(X1, X2, X3))active#(X1)active#(U22(X1, X2, X3))active#(X1)
active#(U12(X1, X2, X3))active#(X1)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#(U21(X1, X2, X3))active#(X1)

Problem 9: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

proper#(U11(X1, X2, X3))proper#(X3)proper#(U12(X1, X2, X3))proper#(X2)
proper#(U22(X1, X2, X3))proper#(X2)proper#(U21(X1, X2, X3))proper#(X3)
proper#(U22(X1, X2, X3))proper#(X3)proper#(x(X1, X2))proper#(X1)
proper#(U12(X1, X2, X3))proper#(X3)proper#(x(X1, X2))proper#(X2)
proper#(U11(X1, X2, X3))proper#(X1)proper#(s(X))proper#(X)
proper#(U11(X1, X2, X3))proper#(X2)proper#(U21(X1, X2, X3))proper#(X1)
proper#(U22(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#(U12(X1, X2, X3))proper#(X1)

Rewrite Rules

active(U11(tt, M, N))mark(U12(tt, M, N))active(U12(tt, M, N))mark(s(plus(N, M)))
active(U21(tt, M, N))mark(U22(tt, M, N))active(U22(tt, M, N))mark(plus(x(N, M), N))
active(plus(N, 0))mark(N)active(plus(N, s(M)))mark(U11(tt, M, N))
active(x(N, 0))mark(0)active(x(N, s(M)))mark(U21(tt, M, N))
active(U11(X1, X2, X3))U11(active(X1), X2, X3)active(U12(X1, X2, X3))U12(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(U21(X1, X2, X3))U21(active(X1), X2, X3)
active(U22(X1, X2, X3))U22(active(X1), X2, X3)active(x(X1, X2))x(active(X1), X2)
active(x(X1, X2))x(X1, active(X2))U11(mark(X1), X2, X3)mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3)mark(U12(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))
U21(mark(X1), X2, X3)mark(U21(X1, X2, X3))U22(mark(X1), X2, X3)mark(U22(X1, X2, X3))
x(mark(X1), X2)mark(x(X1, X2))x(X1, mark(X2))mark(x(X1, X2))
proper(U11(X1, X2, X3))U11(proper(X1), proper(X2), proper(X3))proper(tt)ok(tt)
proper(U12(X1, X2, X3))U12(proper(X1), proper(X2), proper(X3))proper(s(X))s(proper(X))
proper(plus(X1, X2))plus(proper(X1), proper(X2))proper(U21(X1, X2, X3))U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3))U22(proper(X1), proper(X2), proper(X3))proper(x(X1, X2))x(proper(X1), proper(X2))
proper(0)ok(0)U11(ok(X1), ok(X2), ok(X3))ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3))ok(U12(X1, X2, X3))s(ok(X))ok(s(X))
plus(ok(X1), ok(X2))ok(plus(X1, X2))U21(ok(X1), ok(X2), ok(X3))ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2), ok(X3))ok(U22(X1, X2, X3))x(ok(X1), ok(X2))ok(x(X1, X2))
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, 0, s, tt, active, U11, U12, ok, proper, U21, x, top, U22

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

proper#(U11(X1, X2, X3))proper#(X3)proper#(U12(X1, X2, X3))proper#(X2)
proper#(U22(X1, X2, X3))proper#(X2)proper#(U21(X1, X2, X3))proper#(X3)
proper#(U22(X1, X2, X3))proper#(X3)proper#(x(X1, X2))proper#(X1)
proper#(x(X1, X2))proper#(X2)proper#(U12(X1, X2, X3))proper#(X3)
proper#(U11(X1, X2, X3))proper#(X1)proper#(s(X))proper#(X)
proper#(U11(X1, X2, X3))proper#(X2)proper#(U21(X1, X2, X3))proper#(X1)
proper#(U22(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#(U12(X1, X2, X3))proper#(X1)

Problem 10: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(U11(tt, M, N))mark(U12(tt, M, N))active(U12(tt, M, N))mark(s(plus(N, M)))
active(U21(tt, M, N))mark(U22(tt, M, N))active(U22(tt, M, N))mark(plus(x(N, M), N))
active(plus(N, 0))mark(N)active(plus(N, s(M)))mark(U11(tt, M, N))
active(x(N, 0))mark(0)active(x(N, s(M)))mark(U21(tt, M, N))
active(U11(X1, X2, X3))U11(active(X1), X2, X3)active(U12(X1, X2, X3))U12(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(U21(X1, X2, X3))U21(active(X1), X2, X3)
active(U22(X1, X2, X3))U22(active(X1), X2, X3)active(x(X1, X2))x(active(X1), X2)
active(x(X1, X2))x(X1, active(X2))U11(mark(X1), X2, X3)mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3)mark(U12(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))
U21(mark(X1), X2, X3)mark(U21(X1, X2, X3))U22(mark(X1), X2, X3)mark(U22(X1, X2, X3))
x(mark(X1), X2)mark(x(X1, X2))x(X1, mark(X2))mark(x(X1, X2))
proper(U11(X1, X2, X3))U11(proper(X1), proper(X2), proper(X3))proper(tt)ok(tt)
proper(U12(X1, X2, X3))U12(proper(X1), proper(X2), proper(X3))proper(s(X))s(proper(X))
proper(plus(X1, X2))plus(proper(X1), proper(X2))proper(U21(X1, X2, X3))U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3))U22(proper(X1), proper(X2), proper(X3))proper(x(X1, X2))x(proper(X1), proper(X2))
proper(0)ok(0)U11(ok(X1), ok(X2), ok(X3))ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3))ok(U12(X1, X2, X3))s(ok(X))ok(s(X))
plus(ok(X1), ok(X2))ok(plus(X1, X2))U21(ok(X1), ok(X2), ok(X3))ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2), ok(X3))ok(U22(X1, X2, X3))x(ok(X1), ok(X2))ok(x(X1, X2))
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, 0, s, tt, active, U11, U12, ok, proper, U21, x, top, U22

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 11: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

U12#(ok(X1), ok(X2), ok(X3))U12#(X1, X2, X3)U12#(mark(X1), X2, X3)U12#(X1, X2, X3)

Rewrite Rules

active(U11(tt, M, N))mark(U12(tt, M, N))active(U12(tt, M, N))mark(s(plus(N, M)))
active(U21(tt, M, N))mark(U22(tt, M, N))active(U22(tt, M, N))mark(plus(x(N, M), N))
active(plus(N, 0))mark(N)active(plus(N, s(M)))mark(U11(tt, M, N))
active(x(N, 0))mark(0)active(x(N, s(M)))mark(U21(tt, M, N))
active(U11(X1, X2, X3))U11(active(X1), X2, X3)active(U12(X1, X2, X3))U12(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(U21(X1, X2, X3))U21(active(X1), X2, X3)
active(U22(X1, X2, X3))U22(active(X1), X2, X3)active(x(X1, X2))x(active(X1), X2)
active(x(X1, X2))x(X1, active(X2))U11(mark(X1), X2, X3)mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3)mark(U12(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))
U21(mark(X1), X2, X3)mark(U21(X1, X2, X3))U22(mark(X1), X2, X3)mark(U22(X1, X2, X3))
x(mark(X1), X2)mark(x(X1, X2))x(X1, mark(X2))mark(x(X1, X2))
proper(U11(X1, X2, X3))U11(proper(X1), proper(X2), proper(X3))proper(tt)ok(tt)
proper(U12(X1, X2, X3))U12(proper(X1), proper(X2), proper(X3))proper(s(X))s(proper(X))
proper(plus(X1, X2))plus(proper(X1), proper(X2))proper(U21(X1, X2, X3))U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3))U22(proper(X1), proper(X2), proper(X3))proper(x(X1, X2))x(proper(X1), proper(X2))
proper(0)ok(0)U11(ok(X1), ok(X2), ok(X3))ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3))ok(U12(X1, X2, X3))s(ok(X))ok(s(X))
plus(ok(X1), ok(X2))ok(plus(X1, X2))U21(ok(X1), ok(X2), ok(X3))ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2), ok(X3))ok(U22(X1, X2, X3))x(ok(X1), ok(X2))ok(x(X1, X2))
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, 0, s, tt, active, U11, U12, ok, proper, U21, x, top, U22

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

U12#(ok(X1), ok(X2), ok(X3))U12#(X1, X2, X3)U12#(mark(X1), X2, X3)U12#(X1, X2, X3)