YES

The TRS could be proven terminating. The proof took 52621 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (433ms).
 | – Problem 2 was processed with processor SubtermCriterion (37ms).
 |    | – Problem 7 was processed with processor SubtermCriterion (0ms).
 | – Problem 3 was processed with processor PolynomialLinearRange4 (187ms).
 |    | – Problem 10 was processed with processor ReductionPairSAT (2742ms).
 |    |    | – Problem 11 was processed with processor ReductionPairSAT (1390ms).
 |    |    |    | – Problem 12 was processed with processor ReductionPairSAT (2660ms).
 |    |    |    |    | – Problem 13 was processed with processor ReductionPairSAT (2403ms).
 |    |    |    |    |    | – Problem 14 was processed with processor ReductionPairSAT (905ms).
 |    |    |    |    |    |    | – Problem 15 was processed with processor ReductionPairSAT (1249ms).
 |    |    |    |    |    |    |    | – Problem 16 was processed with processor DependencyGraph (4ms).
 |    |    |    |    |    |    |    |    | – Problem 17 was processed with processor ReductionPairSAT (19ms).
 |    |    |    |    |    |    |    |    |    | – Problem 18 was processed with processor ReductionPairSAT (10ms).
 |    |    |    |    |    |    |    |    |    |    | – Problem 19 was processed with processor ReductionPairSAT (7ms).
 | – Problem 4 was processed with processor SubtermCriterion (0ms).
 | – Problem 5 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 8 was processed with processor SubtermCriterion (1ms).
 | – Problem 6 was processed with processor SubtermCriterion (0ms).
 |    | – Problem 9 was processed with processor SubtermCriterion (0ms).

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

and#(active(X1), X2) → and#(X1, X2)active#(x(N, s(M))) → plus#(x(N, M), N)
and#(X1, active(X2)) → and#(X1, X2)mark#(tt) → active#(tt)
x#(active(X1), X2) → x#(X1, X2)mark#(s(X)) → s#(mark(X))
active#(x(N, s(M))) → mark#(plus(x(N, M), N))x#(mark(X1), X2) → x#(X1, X2)
x#(X1, active(X2)) → x#(X1, X2)mark#(plus(X1, X2)) → mark#(X2)
plus#(X1, mark(X2)) → plus#(X1, X2)mark#(and(X1, X2)) → active#(and(mark(X1), X2))
mark#(s(X)) → mark#(X)x#(X1, mark(X2)) → x#(X1, X2)
and#(X1, mark(X2)) → and#(X1, X2)active#(plus(N, s(M))) → plus#(N, M)
mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2)))mark#(0) → active#(0)
mark#(s(X)) → active#(s(mark(X)))active#(x(N, s(M))) → x#(N, M)
mark#(and(X1, X2)) → and#(mark(X1), X2)mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2)))
active#(plus(N, 0)) → mark#(N)mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2))
and#(mark(X1), X2) → and#(X1, X2)active#(and(tt, X)) → mark#(X)
mark#(plus(X1, X2)) → mark#(X1)active#(x(N, 0)) → mark#(0)
active#(plus(N, s(M))) → s#(plus(N, M))mark#(and(X1, X2)) → mark#(X1)
s#(mark(X)) → s#(X)mark#(x(X1, X2)) → mark#(X2)
active#(plus(N, s(M))) → mark#(s(plus(N, M)))plus#(X1, active(X2)) → plus#(X1, X2)
s#(active(X)) → s#(X)mark#(x(X1, X2)) → x#(mark(X1), mark(X2))
plus#(mark(X1), X2) → plus#(X1, X2)mark#(x(X1, X2)) → mark#(X1)
plus#(active(X1), X2) → plus#(X1, X2)

Rewrite Rules

active(and(tt, X)) → mark(X)active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(tt) → active(tt)mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)mark(s(X)) → active(s(mark(X)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

Original Signature

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

Strategy

The following SCCs where found

x#(X1, active(X2)) → x#(X1, X2)x#(mark(X1), X2) → x#(X1, X2)
x#(X1, mark(X2)) → x#(X1, X2)x#(active(X1), X2) → x#(X1, X2)
mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2)))mark#(s(X)) → active#(s(mark(X)))
mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2)))active#(plus(N, 0)) → mark#(N)
active#(and(tt, X)) → mark#(X)active#(x(N, s(M))) → mark#(plus(x(N, M), N))
mark#(plus(X1, X2)) → mark#(X1)mark#(and(X1, X2)) → mark#(X1)
mark#(x(X1, X2)) → mark#(X2)mark#(plus(X1, X2)) → mark#(X2)
active#(plus(N, s(M))) → mark#(s(plus(N, M)))mark#(and(X1, X2)) → active#(and(mark(X1), X2))
mark#(s(X)) → mark#(X)mark#(x(X1, X2)) → mark#(X1)
s#(mark(X)) → s#(X)s#(active(X)) → s#(X)
and#(active(X1), X2) → and#(X1, X2)and#(X1, active(X2)) → and#(X1, X2)
and#(mark(X1), X2) → and#(X1, X2)and#(X1, mark(X2)) → and#(X1, X2)
plus#(X1, active(X2)) → plus#(X1, X2)plus#(X1, mark(X2)) → plus#(X1, X2)
plus#(mark(X1), X2) → plus#(X1, X2)plus#(active(X1), X2) → plus#(X1, X2)

Problem 2: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(and(tt, X)) → mark(X)active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(tt) → active(tt)mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)mark(s(X)) → active(s(mark(X)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 7: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(and(tt, X)) → mark(X)active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(tt) → active(tt)mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)mark(s(X)) → active(s(mark(X)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 3: PolynomialLinearRange4

Dependency Pair Problem

Dependency Pairs

mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2)))mark#(s(X)) → active#(s(mark(X)))
mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2)))active#(plus(N, 0)) → mark#(N)
active#(and(tt, X)) → mark#(X)active#(x(N, s(M))) → mark#(plus(x(N, M), N))
mark#(plus(X1, X2)) → mark#(X1)mark#(and(X1, X2)) → mark#(X1)
mark#(x(X1, X2)) → mark#(X2)mark#(plus(X1, X2)) → mark#(X2)
active#(plus(N, s(M))) → mark#(s(plus(N, M)))mark#(and(X1, X2)) → active#(and(mark(X1), X2))
mark#(s(X)) → mark#(X)mark#(x(X1, X2)) → mark#(X1)

Rewrite Rules

active(and(tt, X)) → mark(X)active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(tt) → active(tt)mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)mark(s(X)) → active(s(mark(X)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

Original Signature

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

Strategy

Polynomial Interpretation

Standard Usable rules

active(x(N, 0)) → mark(0)mark(s(X)) → active(s(mark(X)))
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
and(active(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
plus(mark(X1), X2) → plus(X1, X2)mark(and(X1, X2)) → active(and(mark(X1), X2))
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
mark(0) → active(0)s(active(X)) → s(X)
plus(X1, active(X2)) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
mark(tt) → active(tt)active(plus(N, s(M))) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)plus(active(X1), X2) → plus(X1, X2)
and(mark(X1), X2) → and(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(mark(X1), X2) → x(X1, X2)active(plus(N, 0)) → mark(N)
s(mark(X)) → s(X)mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
and(X1, active(X2)) → and(X1, X2)

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

mark#(s(X)) → active#(s(mark(X)))

Problem 10: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2)))mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2)))
active#(plus(N, 0)) → mark#(N)active#(and(tt, X)) → mark#(X)
active#(x(N, s(M))) → mark#(plus(x(N, M), N))mark#(plus(X1, X2)) → mark#(X1)
mark#(and(X1, X2)) → mark#(X1)mark#(x(X1, X2)) → mark#(X2)
mark#(plus(X1, X2)) → mark#(X2)active#(plus(N, s(M))) → mark#(s(plus(N, M)))
mark#(and(X1, X2)) → active#(and(mark(X1), X2))mark#(s(X)) → mark#(X)
mark#(x(X1, X2)) → mark#(X1)

Rewrite Rules

active(and(tt, X)) → mark(X)active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(tt) → active(tt)mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)mark(s(X)) → active(s(mark(X)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

Original Signature

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

Strategy

Function Precedence

0 = mark = active# < mark# = and < x < plus < active < s = tt

Argument Filtering

plus: 1 2
0: all arguments are removed from 0
s: 1
tt: all arguments are removed from tt
active: collapses to 1
mark: collapses to 1
active#: collapses to 1
mark#: collapses to 1
x: 1 2
and: 1 2

Status

plus: lexicographic with permutation 1 → 1 2 → 2
0: multiset
s: lexicographic with permutation 1 → 1
tt: multiset
x: lexicographic with permutation 1 → 2 2 → 1
and: lexicographic with permutation 1 → 1 2 → 2

Usable Rules

active(x(N, 0)) → mark(0)mark(s(X)) → active(s(mark(X)))
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
and(active(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
plus(mark(X1), X2) → plus(X1, X2)mark(and(X1, X2)) → active(and(mark(X1), X2))
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
mark(0) → active(0)s(active(X)) → s(X)
plus(X1, active(X2)) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
mark(tt) → active(tt)active(plus(N, s(M))) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)plus(active(X1), X2) → plus(X1, X2)
and(mark(X1), X2) → and(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(mark(X1), X2) → x(X1, X2)active(plus(N, 0)) → mark(N)
s(mark(X)) → s(X)mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
and(X1, active(X2)) → and(X1, X2)

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

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

active#(and(tt, X)) → mark#(X)

Problem 11: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

mark#(and(X1, X2)) → mark#(X1)mark#(x(X1, X2)) → mark#(X2)
mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2)))mark#(plus(X1, X2)) → mark#(X2)
mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2)))active#(plus(N, s(M))) → mark#(s(plus(N, M)))
active#(plus(N, 0)) → mark#(N)mark#(and(X1, X2)) → active#(and(mark(X1), X2))
mark#(s(X)) → mark#(X)mark#(x(X1, X2)) → mark#(X1)
active#(x(N, s(M))) → mark#(plus(x(N, M), N))mark#(plus(X1, X2)) → mark#(X1)

Rewrite Rules

active(and(tt, X)) → mark(X)active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(tt) → active(tt)mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)mark(s(X)) → active(s(mark(X)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

Original Signature

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

Strategy

Function Precedence

s < plus = 0 = active = mark = active# = mark# = x < tt < and

Argument Filtering

plus: all arguments are removed from plus
0: all arguments are removed from 0
s: all arguments are removed from s
tt: all arguments are removed from tt
active: all arguments are removed from active
mark: all arguments are removed from mark
active#: collapses to 1
mark#: all arguments are removed from mark#
x: all arguments are removed from x
and: all arguments are removed from and

Status

plus: multiset
0: multiset
s: multiset
tt: multiset
active: multiset
mark: multiset
mark#: multiset
x: multiset
and: multiset

Usable Rules

active(x(N, 0)) → mark(0)mark(s(X)) → active(s(mark(X)))
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
and(active(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
plus(mark(X1), X2) → plus(X1, X2)mark(and(X1, X2)) → active(and(mark(X1), X2))
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
mark(0) → active(0)s(active(X)) → s(X)
plus(X1, active(X2)) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
mark(tt) → active(tt)active(plus(N, s(M))) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)plus(active(X1), X2) → plus(X1, X2)
and(mark(X1), X2) → and(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(mark(X1), X2) → x(X1, X2)active(plus(N, 0)) → mark(N)
s(mark(X)) → s(X)mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
and(X1, active(X2)) → and(X1, X2)

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

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

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

Problem 12: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

mark#(and(X1, X2)) → mark#(X1)mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2)))
mark#(x(X1, X2)) → mark#(X2)mark#(plus(X1, X2)) → mark#(X2)
mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2)))active#(plus(N, 0)) → mark#(N)
active#(plus(N, s(M))) → mark#(s(plus(N, M)))mark#(s(X)) → mark#(X)
active#(x(N, s(M))) → mark#(plus(x(N, M), N))mark#(x(X1, X2)) → mark#(X1)
mark#(plus(X1, X2)) → mark#(X1)

Rewrite Rules

active(and(tt, X)) → mark(X)active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(tt) → active(tt)mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)mark(s(X)) → active(s(mark(X)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

Original Signature

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

Strategy

Function Precedence

0 = mark = and < active < x < plus = tt = active# = mark# < s

Argument Filtering

plus: 1 2
0: all arguments are removed from 0
s: 1
tt: all arguments are removed from tt
active: collapses to 1
mark: collapses to 1
active#: 1
mark#: 1
x: 1 2
and: 1 2

Status

plus: lexicographic with permutation 1 → 1 2 → 2
0: multiset
s: lexicographic with permutation 1 → 1
tt: multiset
active#: multiset
mark#: multiset
x: lexicographic with permutation 1 → 1 2 → 2
and: lexicographic with permutation 1 → 1 2 → 2

Usable Rules

active(x(N, 0)) → mark(0)mark(s(X)) → active(s(mark(X)))
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
and(active(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
plus(mark(X1), X2) → plus(X1, X2)mark(and(X1, X2)) → active(and(mark(X1), X2))
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
s(active(X)) → s(X)mark(0) → active(0)
plus(X1, active(X2)) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
mark(tt) → active(tt)active(plus(N, s(M))) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)plus(active(X1), X2) → plus(X1, X2)
x(X1, mark(X2)) → x(X1, X2)and(mark(X1), X2) → and(X1, X2)
x(mark(X1), X2) → x(X1, X2)active(plus(N, 0)) → mark(N)
s(mark(X)) → s(X)mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
and(X1, active(X2)) → and(X1, X2)

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

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

active#(plus(N, 0)) → mark#(N)

Problem 13: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

mark#(and(X1, X2)) → mark#(X1)mark#(x(X1, X2)) → mark#(X2)
mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2)))mark#(plus(X1, X2)) → mark#(X2)
mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2)))active#(plus(N, s(M))) → mark#(s(plus(N, M)))
mark#(s(X)) → mark#(X)mark#(x(X1, X2)) → mark#(X1)
active#(x(N, s(M))) → mark#(plus(x(N, M), N))mark#(plus(X1, X2)) → mark#(X1)

Rewrite Rules

active(and(tt, X)) → mark(X)active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(tt) → active(tt)mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)mark(s(X)) → active(s(mark(X)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

Original Signature

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

Strategy

Function Precedence

0 < x < mark# < plus = mark < tt < s = active = active# = and

Argument Filtering

plus: 1 2
0: all arguments are removed from 0
s: 1
tt: all arguments are removed from tt
active: collapses to 1
mark: collapses to 1
active#: collapses to 1
mark#: collapses to 1
x: 1 2
and: 1 2

Status

plus: lexicographic with permutation 1 → 2 2 → 1
0: multiset
s: lexicographic with permutation 1 → 1
tt: multiset
x: lexicographic with permutation 1 → 2 2 → 1
and: lexicographic with permutation 1 → 2 2 → 1

Usable Rules

active(x(N, 0)) → mark(0)mark(s(X)) → active(s(mark(X)))
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
and(active(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
plus(mark(X1), X2) → plus(X1, X2)mark(and(X1, X2)) → active(and(mark(X1), X2))
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
s(active(X)) → s(X)mark(0) → active(0)
plus(X1, active(X2)) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
mark(tt) → active(tt)active(plus(N, s(M))) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)plus(active(X1), X2) → plus(X1, X2)
x(X1, mark(X2)) → x(X1, X2)and(mark(X1), X2) → and(X1, X2)
x(mark(X1), X2) → x(X1, X2)active(plus(N, 0)) → mark(N)
s(mark(X)) → s(X)mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
and(X1, active(X2)) → and(X1, X2)

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

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

active#(x(N, s(M))) → mark#(plus(x(N, M), N))

Problem 14: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

mark#(and(X1, X2)) → mark#(X1)mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2)))
mark#(x(X1, X2)) → mark#(X2)mark#(plus(X1, X2)) → mark#(X2)
mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2)))active#(plus(N, s(M))) → mark#(s(plus(N, M)))
mark#(s(X)) → mark#(X)mark#(x(X1, X2)) → mark#(X1)
mark#(plus(X1, X2)) → mark#(X1)

Rewrite Rules

active(and(tt, X)) → mark(X)active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(tt) → active(tt)mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)mark(s(X)) → active(s(mark(X)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

Original Signature

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

Strategy

Function Precedence

s < plus = active# = mark# < 0 = tt = active = mark = x = and

Argument Filtering

plus: all arguments are removed from plus
0: all arguments are removed from 0
s: all arguments are removed from s
tt: all arguments are removed from tt
active: all arguments are removed from active
mark: all arguments are removed from mark
active#: collapses to 1
mark#: all arguments are removed from mark#
x: all arguments are removed from x
and: all arguments are removed from and

Status

plus: multiset
0: multiset
s: multiset
tt: multiset
active: multiset
mark: multiset
mark#: multiset
x: multiset
and: multiset

Usable Rules

active(x(N, 0)) → mark(0)mark(s(X)) → active(s(mark(X)))
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
and(active(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
plus(mark(X1), X2) → plus(X1, X2)mark(and(X1, X2)) → active(and(mark(X1), X2))
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
s(active(X)) → s(X)mark(0) → active(0)
plus(X1, active(X2)) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
mark(tt) → active(tt)active(plus(N, s(M))) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)plus(active(X1), X2) → plus(X1, X2)
x(X1, mark(X2)) → x(X1, X2)and(mark(X1), X2) → and(X1, X2)
x(mark(X1), X2) → x(X1, X2)active(plus(N, 0)) → mark(N)
s(mark(X)) → s(X)mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
and(X1, active(X2)) → and(X1, X2)

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

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

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

Problem 15: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

mark#(and(X1, X2)) → mark#(X1)mark#(x(X1, X2)) → mark#(X2)
mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2)))mark#(plus(X1, X2)) → mark#(X2)
active#(plus(N, s(M))) → mark#(s(plus(N, M)))mark#(s(X)) → mark#(X)
mark#(x(X1, X2)) → mark#(X1)mark#(plus(X1, X2)) → mark#(X1)

Rewrite Rules

active(and(tt, X)) → mark(X)active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(tt) → active(tt)mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)mark(s(X)) → active(s(mark(X)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

Original Signature

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

Strategy

Function Precedence

0 < x < plus = mark# < mark < s = tt = active = active# = and

Argument Filtering

plus: 1 2
0: all arguments are removed from 0
s: 1
tt: all arguments are removed from tt
active: collapses to 1
mark: collapses to 1
active#: collapses to 1
mark#: collapses to 1
x: 1 2
and: 1 2

Status

plus: lexicographic with permutation 1 → 1 2 → 2
0: multiset
s: lexicographic with permutation 1 → 1
tt: multiset
x: lexicographic with permutation 1 → 2 2 → 1
and: lexicographic with permutation 1 → 2 2 → 1

Usable Rules

active(x(N, 0)) → mark(0)mark(s(X)) → active(s(mark(X)))
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
and(active(X1), X2) → and(X1, X2)and(X1, mark(X2)) → and(X1, X2)
plus(mark(X1), X2) → plus(X1, X2)mark(and(X1, X2)) → active(and(mark(X1), X2))
x(active(X1), X2) → x(X1, X2)x(X1, active(X2)) → x(X1, X2)
s(active(X)) → s(X)mark(0) → active(0)
plus(X1, active(X2)) → plus(X1, X2)plus(X1, mark(X2)) → plus(X1, X2)
mark(tt) → active(tt)active(plus(N, s(M))) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)plus(active(X1), X2) → plus(X1, X2)
and(mark(X1), X2) → and(X1, X2)x(X1, mark(X2)) → x(X1, X2)
x(mark(X1), X2) → x(X1, X2)active(plus(N, 0)) → mark(N)
s(mark(X)) → s(X)mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
and(X1, active(X2)) → and(X1, X2)

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

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

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

Problem 16: DependencyGraph

Dependency Pair Problem

Dependency Pairs

mark#(and(X1, X2)) → mark#(X1)mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2)))
mark#(x(X1, X2)) → mark#(X2)mark#(plus(X1, X2)) → mark#(X2)
mark#(s(X)) → mark#(X)mark#(x(X1, X2)) → mark#(X1)
mark#(plus(X1, X2)) → mark#(X1)

Rewrite Rules

active(and(tt, X)) → mark(X)active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(tt) → active(tt)mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)mark(s(X)) → active(s(mark(X)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

Original Signature

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

Strategy

The following SCCs where found

mark#(and(X1, X2)) → mark#(X1)mark#(x(X1, X2)) → mark#(X2)
mark#(plus(X1, X2)) → mark#(X2)mark#(s(X)) → mark#(X)
mark#(x(X1, X2)) → mark#(X1)mark#(plus(X1, X2)) → mark#(X1)

Problem 17: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

mark#(and(X1, X2)) → mark#(X1)mark#(x(X1, X2)) → mark#(X2)
mark#(plus(X1, X2)) → mark#(X2)mark#(s(X)) → mark#(X)
mark#(x(X1, X2)) → mark#(X1)mark#(plus(X1, X2)) → mark#(X1)

Rewrite Rules

active(and(tt, X)) → mark(X)active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(tt) → active(tt)mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)mark(s(X)) → active(s(mark(X)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

Original Signature

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

Strategy

Function Precedence

and < s < plus = 0 = tt = active = mark = mark# = x

Argument Filtering

plus: 1 2
0: all arguments are removed from 0
s: collapses to 1
tt: all arguments are removed from tt
active: all arguments are removed from active
mark: all arguments are removed from mark
mark#: collapses to 1
x: 1 2
and: collapses to 1

Status

plus: multiset
0: multiset
tt: multiset
active: multiset
mark: multiset
x: multiset

Usable Rules

There are no usable rules.

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

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

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

Problem 18: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

mark#(and(X1, X2)) → mark#(X1)mark#(s(X)) → mark#(X)

Rewrite Rules

active(and(tt, X)) → mark(X)active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(tt) → active(tt)mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)mark(s(X)) → active(s(mark(X)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

Original Signature

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

Strategy

Function Precedence

s < plus = 0 = tt = active = mark = mark# = x = and

Argument Filtering

plus: all arguments are removed from plus
0: all arguments are removed from 0
s: collapses to 1
tt: all arguments are removed from tt
active: collapses to 1
mark: all arguments are removed from mark
mark#: 1
x: all arguments are removed from x
and: 1 2

Status

plus: multiset
0: multiset
tt: multiset
mark: multiset
mark#: lexicographic with permutation 1 → 1
x: multiset
and: multiset

Usable Rules

There are no usable rules.

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

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

mark#(and(X1, X2)) → mark#(X1)

Problem 19: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

mark#(s(X)) → mark#(X)

Rewrite Rules

active(and(tt, X)) → mark(X)active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(tt) → active(tt)mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)mark(s(X)) → active(s(mark(X)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

Original Signature

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

Strategy

Function Precedence

plus = 0 = s = tt = active = mark = mark# = x = and

Argument Filtering

plus: all arguments are removed from plus
0: all arguments are removed from 0
s: 1
tt: all arguments are removed from tt
active: all arguments are removed from active
mark: all arguments are removed from mark
mark#: 1
x: all arguments are removed from x
and: all arguments are removed from and

Status

plus: multiset
0: multiset
s: multiset
tt: multiset
active: multiset
mark: multiset
mark#: multiset
x: multiset
and: multiset

Usable Rules

There are no usable rules.

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

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

mark#(s(X)) → mark#(X)

Problem 4: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(and(tt, X)) → mark(X)active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(tt) → active(tt)mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)mark(s(X)) → active(s(mark(X)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 5: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(and(tt, X)) → mark(X)active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(tt) → active(tt)mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)mark(s(X)) → active(s(mark(X)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 8: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(and(tt, X)) → mark(X)active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(tt) → active(tt)mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)mark(s(X)) → active(s(mark(X)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 6: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(and(tt, X)) → mark(X)active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(tt) → active(tt)mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)mark(s(X)) → active(s(mark(X)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 9: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(and(tt, X)) → mark(X)active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(plus(x(N, M), N))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(tt) → active(tt)mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)mark(s(X)) → active(s(mark(X)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

Original Signature

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

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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