YES

The TRS could be proven terminating. The proof took 12539 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (435ms).
 | – Problem 2 was processed with processor PolynomialOrderingProcessor (8865ms).
 |    | – Problem 10 was processed with processor PolynomialOrderingProcessor (1808ms).
 | – Problem 3 was processed with processor SubtermCriterion (0ms).
 | – Problem 4 was processed with processor SubtermCriterion (0ms).
 | – Problem 5 was processed with processor SubtermCriterion (1ms).
 | – Problem 6 was processed with processor SubtermCriterion (0ms).
 |    | – Problem 9 was processed with processor SubtermCriterion (0ms).
 | – Problem 7 was processed with processor SubtermCriterion (3ms).
 | – Problem 8 was processed with processor SubtermCriterion (0ms).

Problem 1: DependencyGraph



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(U11(tt, M, N))mark(U12(tt, M, N))active(U12(tt, M, N))mark(s(plus(N, M)))
active(plus(N, 0))mark(N)active(plus(N, s(M)))mark(U11(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))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))
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(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))
top(mark(X))top(proper(X))top(ok(X))top(active(X))

Original Signature

Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12, ok, proper, 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#(U11(X1, X2, X3)) → active#(X1)active#(U12(X1, X2, X3)) → active#(X1)
active#(plus(X1, X2)) → active#(X1)active#(s(X)) → active#(X)
active#(plus(X1, X2)) → active#(X2)

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

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

s#(mark(X)) → s#(X)s#(ok(X)) → s#(X)

proper#(s(X)) → proper#(X)proper#(U11(X1, X2, X3)) → proper#(X3)
proper#(U11(X1, X2, X3)) → proper#(X2)proper#(U12(X1, X2, X3)) → proper#(X2)
proper#(plus(X1, X2)) → proper#(X1)proper#(plus(X1, X2)) → proper#(X2)
proper#(U12(X1, X2, X3)) → proper#(X3)proper#(U12(X1, X2, X3)) → proper#(X1)
proper#(U11(X1, X2, X3)) → proper#(X1)

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

Problem 2: PolynomialOrderingProcessor



Dependency Pair Problem

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(plus(N, 0))mark(N)active(plus(N, s(M)))mark(U11(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))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))
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(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))
top(mark(X))top(proper(X))top(ok(X))top(active(X))

Original Signature

Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12, ok, proper, top

Strategy


Polynomial Interpretation

Improved Usable rules

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

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

top#(mark(X))top#(proper(X))

Problem 10: PolynomialOrderingProcessor



Dependency Pair Problem

Dependency Pairs

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(plus(N, 0))mark(N)active(plus(N, s(M)))mark(U11(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))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))
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(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))
top(mark(X))top(proper(X))top(ok(X))top(active(X))

Original Signature

Termination of terms over the following signature is verified: plus, 0, s, tt, U11, active, ok, U12, mark, proper, top

Strategy


Polynomial Interpretation

Improved Usable rules

active(plus(X1, X2))plus(X1, active(X2))plus(X1, mark(X2))mark(plus(X1, X2))
active(plus(X1, X2))plus(active(X1), X2)active(s(X))s(active(X))
s(ok(X))ok(s(X))plus(mark(X1), X2)mark(plus(X1, X2))
active(U12(tt, M, N))mark(s(plus(N, M)))active(U11(tt, M, N))mark(U12(tt, M, N))
U11(mark(X1), X2, X3)mark(U11(X1, X2, X3))active(U12(X1, X2, X3))U12(active(X1), X2, X3)
U12(mark(X1), X2, X3)mark(U12(X1, X2, X3))U12(ok(X1), ok(X2), ok(X3))ok(U12(X1, X2, X3))
active(U11(X1, X2, X3))U11(active(X1), X2, X3)U11(ok(X1), ok(X2), ok(X3))ok(U11(X1, X2, X3))
plus(ok(X1), ok(X2))ok(plus(X1, X2))s(mark(X))mark(s(X))
active(plus(N, 0))mark(N)active(plus(N, s(M)))mark(U11(tt, M, N))

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

top#(ok(X))top#(active(X))

Problem 3: 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(plus(N, 0))mark(N)active(plus(N, s(M)))mark(U11(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))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))
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(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))
top(mark(X))top(proper(X))top(ok(X))top(active(X))

Original Signature

Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12, ok, proper, top

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 4: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

active#(U11(X1, X2, X3))active#(X1)active#(U12(X1, X2, X3))active#(X1)
active#(plus(X1, X2))active#(X1)active#(s(X))active#(X)
active#(plus(X1, X2))active#(X2)

Rewrite Rules

active(U11(tt, M, N))mark(U12(tt, M, N))active(U12(tt, M, N))mark(s(plus(N, M)))
active(plus(N, 0))mark(N)active(plus(N, s(M)))mark(U11(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))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))
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(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))
top(mark(X))top(proper(X))top(ok(X))top(active(X))

Original Signature

Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12, ok, proper, top

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

active#(U11(X1, X2, X3))active#(X1)active#(U12(X1, X2, X3))active#(X1)
active#(plus(X1, X2))active#(X1)active#(s(X))active#(X)
active#(plus(X1, X2))active#(X2)

Problem 5: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

proper#(s(X))proper#(X)proper#(U11(X1, X2, X3))proper#(X3)
proper#(U11(X1, X2, X3))proper#(X2)proper#(U12(X1, X2, X3))proper#(X2)
proper#(plus(X1, X2))proper#(X1)proper#(plus(X1, X2))proper#(X2)
proper#(U12(X1, X2, X3))proper#(X3)proper#(U12(X1, X2, X3))proper#(X1)
proper#(U11(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(plus(N, 0))mark(N)active(plus(N, s(M)))mark(U11(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))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))
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(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))
top(mark(X))top(proper(X))top(ok(X))top(active(X))

Original Signature

Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12, ok, proper, top

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

proper#(s(X))proper#(X)proper#(U11(X1, X2, X3))proper#(X3)
proper#(U12(X1, X2, X3))proper#(X2)proper#(U11(X1, X2, X3))proper#(X2)
proper#(plus(X1, X2))proper#(X1)proper#(plus(X1, X2))proper#(X2)
proper#(U12(X1, X2, X3))proper#(X3)proper#(U12(X1, X2, X3))proper#(X1)
proper#(U11(X1, X2, X3))proper#(X1)

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(plus(N, 0))mark(N)active(plus(N, s(M)))mark(U11(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))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))
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(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))
top(mark(X))top(proper(X))top(ok(X))top(active(X))

Original Signature

Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12, ok, proper, 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 9: SubtermCriterion



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(plus(N, 0))mark(N)active(plus(N, s(M)))mark(U11(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))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))
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(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))
top(mark(X))top(proper(X))top(ok(X))top(active(X))

Original Signature

Termination of terms over the following signature is verified: plus, 0, s, tt, U11, active, ok, U12, mark, proper, top

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 7: 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(plus(N, 0))mark(N)active(plus(N, s(M)))mark(U11(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))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))
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(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))
top(mark(X))top(proper(X))top(ok(X))top(active(X))

Original Signature

Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12, ok, proper, 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 8: 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(plus(N, 0))mark(N)active(plus(N, s(M)))mark(U11(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))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))
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(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))
top(mark(X))top(proper(X))top(ok(X))top(active(X))

Original Signature

Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12, ok, proper, top

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)