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 (2133ms).
 | – Problem 2 was processed with processor SubtermCriterion (2ms).
 | – Problem 3 was processed with processor SubtermCriterion (3ms).
 |    | – Problem 9 was processed with processor ReductionPairSAT (84ms).
 |    |    | – Problem 14 was processed with processor ReductionPairSAT (58ms).
 | – Problem 4 was processed with processor ReductionPairSAT (4701ms).
 |    | – Problem 13 was processed with processor ReductionPairSAT (4353ms).
 |    |    | – Problem 18 was processed with processor ReductionPairSAT (7559ms).
 |    |    |    | – Problem 19 remains open; application of the following processors failed [DependencyGraph (203ms), ReductionPairSAT (9498ms), DependencyGraph (195ms), SizeChangePrinciple (timeout)].
 | – Problem 5 was processed with processor SubtermCriterion (3ms).
 |    | – Problem 10 was processed with processor ReductionPairSAT (62ms).
 |    |    | – Problem 15 was processed with processor ReductionPairSAT (24ms).
 | – Problem 6 was processed with processor SubtermCriterion (1ms).
 | – Problem 7 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 11 was processed with processor ReductionPairSAT (37ms).
 |    |    | – Problem 16 was processed with processor ReductionPairSAT (66ms).
 | – Problem 8 was processed with processor SubtermCriterion (0ms).
 |    | – Problem 12 was processed with processor ReductionPairSAT (73ms).
 |    |    | – Problem 17 was processed with processor ReductionPairSAT (91ms).

The following open problems remain:

Open Dependency Pair Problem 19

Dependency Pairs

mark#(fib1(X1, X2)) → active#(fib1(mark(X1), mark(X2)))mark#(cons(X1, X2)) → mark#(X1)
active#(fib(N)) → mark#(sel(N, fib1(s(0), s(0))))active#(add(s(X), Y)) → mark#(s(add(X, Y)))
active#(add(0, X)) → mark#(X)mark#(fib1(X1, X2)) → mark#(X1)
mark#(sel(X1, X2)) → active#(sel(mark(X1), mark(X2)))mark#(fib1(X1, X2)) → mark#(X2)
mark#(sel(X1, X2)) → mark#(X2)mark#(fib(X)) → mark#(X)
mark#(add(X1, X2)) → mark#(X2)mark#(fib(X)) → active#(fib(mark(X)))
mark#(s(X)) → mark#(X)active#(sel(0, cons(X, XS))) → mark#(X)
mark#(add(X1, X2)) → active#(add(mark(X1), mark(X2)))mark#(sel(X1, X2)) → mark#(X1)
active#(sel(s(N), cons(X, XS))) → mark#(sel(N, XS))mark#(add(X1, X2)) → mark#(X1)
active#(fib1(X, Y)) → mark#(cons(X, fib1(Y, add(X, Y))))

Rewrite Rules

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Original Signature

Termination of terms over the following signature is verified: 0, s, active, mark, fib1, add, sel, fib, cons

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

mark#(cons(X1, X2)) → active#(cons(mark(X1), X2))add#(X1, mark(X2)) → add#(X1, X2)
active#(add(s(X), Y)) → mark#(s(add(X, Y)))mark#(s(X)) → s#(mark(X))
fib#(mark(X)) → fib#(X)active#(fib1(X, Y)) → add#(X, Y)
mark#(fib1(X1, X2)) → fib1#(mark(X1), mark(X2))mark#(add(X1, X2)) → mark#(X2)
fib1#(active(X1), X2) → fib1#(X1, X2)mark#(s(X)) → mark#(X)
mark#(add(X1, X2)) → active#(add(mark(X1), mark(X2)))mark#(sel(X1, X2)) → mark#(X1)
mark#(add(X1, X2)) → mark#(X1)mark#(sel(X1, X2)) → sel#(mark(X1), mark(X2))
mark#(fib(X)) → fib#(mark(X))active#(fib1(X, Y)) → cons#(X, fib1(Y, add(X, Y)))
mark#(cons(X1, X2)) → mark#(X1)active#(add(0, X)) → mark#(X)
add#(mark(X1), X2) → add#(X1, X2)mark#(sel(X1, X2)) → active#(sel(mark(X1), mark(X2)))
sel#(X1, mark(X2)) → sel#(X1, X2)fib1#(mark(X1), X2) → fib1#(X1, X2)
sel#(active(X1), X2) → sel#(X1, X2)cons#(X1, active(X2)) → cons#(X1, X2)
active#(add(s(X), Y)) → add#(X, Y)sel#(X1, active(X2)) → sel#(X1, X2)
mark#(add(X1, X2)) → add#(mark(X1), mark(X2))add#(active(X1), X2) → add#(X1, X2)
mark#(fib(X)) → mark#(X)active#(sel(s(N), cons(X, XS))) → mark#(sel(N, XS))
active#(fib1(X, Y)) → fib1#(Y, add(X, Y))active#(fib1(X, Y)) → mark#(cons(X, fib1(Y, add(X, Y))))
sel#(mark(X1), X2) → sel#(X1, X2)active#(fib(N)) → fib1#(s(0), s(0))
cons#(mark(X1), X2) → cons#(X1, X2)fib1#(X1, active(X2)) → fib1#(X1, X2)
active#(fib(N)) → mark#(sel(N, fib1(s(0), s(0))))active#(add(s(X), Y)) → s#(add(X, Y))
mark#(sel(X1, X2)) → mark#(X2)active#(sel(s(N), cons(X, XS))) → sel#(N, XS)
add#(X1, active(X2)) → add#(X1, X2)mark#(fib(X)) → active#(fib(mark(X)))
cons#(X1, mark(X2)) → cons#(X1, X2)active#(sel(0, cons(X, XS))) → mark#(X)
fib1#(X1, mark(X2)) → fib1#(X1, X2)mark#(cons(X1, X2)) → cons#(mark(X1), X2)
mark#(0) → active#(0)mark#(s(X)) → active#(s(mark(X)))
mark#(fib1(X1, X2)) → active#(fib1(mark(X1), mark(X2)))active#(fib(N)) → sel#(N, fib1(s(0), s(0)))
mark#(fib1(X1, X2)) → mark#(X1)cons#(active(X1), X2) → cons#(X1, X2)
active#(fib(N)) → s#(0)s#(mark(X)) → s#(X)
mark#(fib1(X1, X2)) → mark#(X2)fib#(active(X)) → fib#(X)
s#(active(X)) → s#(X)

Rewrite Rules

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Original Signature

Termination of terms over the following signature is verified: 0, s, active, mark, fib1, add, sel, fib, cons

Strategy

The following SCCs where found

mark#(cons(X1, X2)) → active#(cons(mark(X1), X2))mark#(0) → active#(0)
mark#(s(X)) → active#(s(mark(X)))mark#(fib1(X1, X2)) → active#(fib1(mark(X1), mark(X2)))
mark#(cons(X1, X2)) → mark#(X1)active#(add(s(X), Y)) → mark#(s(add(X, Y)))
active#(fib(N)) → mark#(sel(N, fib1(s(0), s(0))))active#(add(0, X)) → mark#(X)
mark#(fib1(X1, X2)) → mark#(X1)mark#(sel(X1, X2)) → active#(sel(mark(X1), mark(X2)))
mark#(fib1(X1, X2)) → mark#(X2)mark#(sel(X1, X2)) → mark#(X2)
mark#(fib(X)) → mark#(X)mark#(add(X1, X2)) → mark#(X2)
mark#(fib(X)) → active#(fib(mark(X)))mark#(s(X)) → mark#(X)
active#(sel(0, cons(X, XS))) → mark#(X)mark#(add(X1, X2)) → active#(add(mark(X1), mark(X2)))
mark#(sel(X1, X2)) → mark#(X1)active#(sel(s(N), cons(X, XS))) → mark#(sel(N, XS))
mark#(add(X1, X2)) → mark#(X1)active#(fib1(X, Y)) → mark#(cons(X, fib1(Y, add(X, Y))))
fib1#(X1, active(X2)) → fib1#(X1, X2)fib1#(active(X1), X2) → fib1#(X1, X2)
fib1#(X1, mark(X2)) → fib1#(X1, X2)fib1#(mark(X1), X2) → fib1#(X1, X2)
s#(mark(X)) → s#(X)s#(active(X)) → s#(X)
cons#(X1, active(X2)) → cons#(X1, X2)cons#(mark(X1), X2) → cons#(X1, X2)
cons#(X1, mark(X2)) → cons#(X1, X2)cons#(active(X1), X2) → cons#(X1, X2)
fib#(active(X)) → fib#(X)fib#(mark(X)) → fib#(X)
sel#(mark(X1), X2) → sel#(X1, X2)sel#(active(X1), X2) → sel#(X1, X2)
sel#(X1, active(X2)) → sel#(X1, X2)sel#(X1, mark(X2)) → sel#(X1, X2)
add#(X1, mark(X2)) → add#(X1, X2)add#(active(X1), X2) → add#(X1, X2)
add#(X1, active(X2)) → add#(X1, X2)add#(mark(X1), X2) → add#(X1, X2)

Problem 2: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

fib#(active(X)) → fib#(X)fib#(mark(X)) → fib#(X)

Rewrite Rules

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Original Signature

Termination of terms over the following signature is verified: 0, s, active, mark, fib1, add, sel, fib, cons

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

fib#(active(X)) → fib#(X)fib#(mark(X)) → fib#(X)

Problem 3: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

fib1#(X1, active(X2)) → fib1#(X1, X2)fib1#(active(X1), X2) → fib1#(X1, X2)
fib1#(X1, mark(X2)) → fib1#(X1, X2)fib1#(mark(X1), X2) → fib1#(X1, X2)

Rewrite Rules

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Original Signature

Termination of terms over the following signature is verified: 0, s, active, mark, fib1, add, sel, fib, cons

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

fib1#(active(X1), X2) → fib1#(X1, X2)fib1#(mark(X1), X2) → fib1#(X1, X2)

Problem 9: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

fib1#(X1, active(X2)) → fib1#(X1, X2)fib1#(X1, mark(X2)) → fib1#(X1, X2)

Rewrite Rules

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Original Signature

Termination of terms over the following signature is verified: 0, s, active, mark, add, fib1, sel, fib, cons

Strategy

Function Precedence

active < fib1# < 0 = s = mark = add = fib1 = sel = fib = cons

Argument Filtering

0: all arguments are removed from 0
s: all arguments are removed from s
active: collapses to 1
mark: 1
add: all arguments are removed from add
fib1: 1 2
sel: all arguments are removed from sel
fib: collapses to 1
fib1#: 2
cons: all arguments are removed from cons

Status

0: multiset
s: multiset
mark: multiset
add: multiset
fib1: lexicographic with permutation 1 → 1 2 → 2
sel: multiset
fib1#: lexicographic with permutation 2 → 1
cons: 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.

fib1#(X1, mark(X2)) → fib1#(X1, X2)

Problem 14: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

fib1#(X1, active(X2)) → fib1#(X1, X2)

Rewrite Rules

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Original Signature

Termination of terms over the following signature is verified: 0, s, active, mark, fib1, add, sel, fib, cons

Strategy

Function Precedence

active < 0 = s = mark = add = fib1 = sel = fib = fib1# = cons

Argument Filtering

0: all arguments are removed from 0
s: all arguments are removed from s
active: 1
mark: all arguments are removed from mark
add: all arguments are removed from add
fib1: all arguments are removed from fib1
sel: all arguments are removed from sel
fib: all arguments are removed from fib
fib1#: collapses to 2
cons: collapses to 1

Status

0: multiset
s: multiset
active: multiset
mark: multiset
add: multiset
fib1: multiset
sel: multiset
fib: 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.

fib1#(X1, active(X2)) → fib1#(X1, X2)

Problem 4: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

mark#(0) → active#(0)mark#(cons(X1, X2)) → active#(cons(mark(X1), X2))
mark#(s(X)) → active#(s(mark(X)))mark#(fib1(X1, X2)) → active#(fib1(mark(X1), mark(X2)))
mark#(cons(X1, X2)) → mark#(X1)active#(fib(N)) → mark#(sel(N, fib1(s(0), s(0))))
active#(add(s(X), Y)) → mark#(s(add(X, Y)))active#(add(0, X)) → mark#(X)
mark#(fib1(X1, X2)) → mark#(X1)mark#(sel(X1, X2)) → active#(sel(mark(X1), mark(X2)))
mark#(fib1(X1, X2)) → mark#(X2)mark#(sel(X1, X2)) → mark#(X2)
mark#(fib(X)) → mark#(X)mark#(add(X1, X2)) → mark#(X2)
mark#(fib(X)) → active#(fib(mark(X)))mark#(s(X)) → mark#(X)
active#(sel(0, cons(X, XS))) → mark#(X)mark#(add(X1, X2)) → active#(add(mark(X1), mark(X2)))
mark#(sel(X1, X2)) → mark#(X1)active#(sel(s(N), cons(X, XS))) → mark#(sel(N, XS))
mark#(add(X1, X2)) → mark#(X1)active#(fib1(X, Y)) → mark#(cons(X, fib1(Y, add(X, Y))))

Rewrite Rules

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Original Signature

Termination of terms over the following signature is verified: 0, s, active, mark, fib1, add, sel, fib, cons

Strategy

Function Precedence

0 = s = active = mark = active# = add = fib1 = sel = mark# = fib = cons

Argument Filtering

0: all arguments are removed from 0
s: all arguments are removed from s
active: collapses to 1
mark: all arguments are removed from mark
active#: collapses to 1
add: all arguments are removed from add
fib1: all arguments are removed from fib1
sel: all arguments are removed from sel
mark#: all arguments are removed from mark#
fib: all arguments are removed from fib
cons: all arguments are removed from cons

Status

0: multiset
s: multiset
mark: multiset
add: multiset
fib1: multiset
sel: multiset
mark#: multiset
fib: multiset
cons: multiset

Usable Rules

active(sel(0, cons(X, XS))) → mark(X)cons(active(X1), X2) → cons(X1, X2)
fib(mark(X)) → fib(X)sel(X1, mark(X2)) → sel(X1, X2)
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))mark(fib(X)) → active(fib(mark(X)))
fib1(X1, active(X2)) → fib1(X1, X2)add(X1, mark(X2)) → add(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)add(X1, active(X2)) → add(X1, X2)
add(mark(X1), X2) → add(X1, X2)mark(0) → active(0)
s(active(X)) → s(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
cons(X1, active(X2)) → cons(X1, X2)add(active(X1), X2) → add(X1, X2)
active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))active(add(s(X), Y)) → mark(s(add(X, Y)))
active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))sel(X1, active(X2)) → sel(X1, X2)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))fib1(active(X1), X2) → fib1(X1, X2)
fib(active(X)) → fib(X)cons(mark(X1), X2) → cons(X1, X2)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))active(add(0, X)) → mark(X)
fib1(mark(X1), X2) → fib1(X1, X2)fib1(X1, mark(X2)) → fib1(X1, X2)
s(mark(X)) → s(X)sel(active(X1), X2) → sel(X1, X2)
sel(mark(X1), X2) → sel(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#(0) → active#(0)

Problem 13: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

mark#(cons(X1, X2)) → active#(cons(mark(X1), X2))mark#(s(X)) → active#(s(mark(X)))
mark#(fib1(X1, X2)) → active#(fib1(mark(X1), mark(X2)))mark#(cons(X1, X2)) → mark#(X1)
active#(fib(N)) → mark#(sel(N, fib1(s(0), s(0))))active#(add(s(X), Y)) → mark#(s(add(X, Y)))
active#(add(0, X)) → mark#(X)mark#(fib1(X1, X2)) → mark#(X1)
mark#(sel(X1, X2)) → active#(sel(mark(X1), mark(X2)))mark#(fib1(X1, X2)) → mark#(X2)
mark#(sel(X1, X2)) → mark#(X2)mark#(fib(X)) → mark#(X)
mark#(add(X1, X2)) → mark#(X2)mark#(fib(X)) → active#(fib(mark(X)))
mark#(add(X1, X2)) → active#(add(mark(X1), mark(X2)))active#(sel(0, cons(X, XS))) → mark#(X)
mark#(s(X)) → mark#(X)active#(sel(s(N), cons(X, XS))) → mark#(sel(N, XS))
mark#(sel(X1, X2)) → mark#(X1)mark#(add(X1, X2)) → mark#(X1)
active#(fib1(X, Y)) → mark#(cons(X, fib1(Y, add(X, Y))))

Rewrite Rules

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Original Signature

Termination of terms over the following signature is verified: 0, s, active, mark, add, fib1, sel, fib, cons

Strategy

Function Precedence

0 < active# = add = fib1 = sel = mark# = fib < s = cons < active = mark

Argument Filtering

0: all arguments are removed from 0
s: all arguments are removed from s
active: all arguments are removed from active
mark: all arguments are removed from mark
active#: collapses to 1
add: all arguments are removed from add
fib1: all arguments are removed from fib1
sel: all arguments are removed from sel
mark#: all arguments are removed from mark#
fib: all arguments are removed from fib
cons: all arguments are removed from cons

Status

0: multiset
s: multiset
active: multiset
mark: multiset
add: multiset
fib1: multiset
sel: multiset
mark#: multiset
fib: multiset
cons: multiset

Usable Rules

active(sel(0, cons(X, XS))) → mark(X)cons(active(X1), X2) → cons(X1, X2)
fib(mark(X)) → fib(X)sel(X1, mark(X2)) → sel(X1, X2)
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))mark(fib(X)) → active(fib(mark(X)))
fib1(X1, active(X2)) → fib1(X1, X2)add(X1, mark(X2)) → add(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)add(X1, active(X2)) → add(X1, X2)
add(mark(X1), X2) → add(X1, X2)mark(0) → active(0)
s(active(X)) → s(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
cons(X1, active(X2)) → cons(X1, X2)add(active(X1), X2) → add(X1, X2)
active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))active(add(s(X), Y)) → mark(s(add(X, Y)))
active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))sel(X1, active(X2)) → sel(X1, X2)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))fib1(active(X1), X2) → fib1(X1, X2)
fib(active(X)) → fib(X)cons(mark(X1), X2) → cons(X1, X2)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))active(add(0, X)) → mark(X)
fib1(mark(X1), X2) → fib1(X1, X2)fib1(X1, mark(X2)) → fib1(X1, X2)
s(mark(X)) → s(X)sel(active(X1), X2) → sel(X1, X2)
sel(mark(X1), X2) → sel(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#(s(X)) → active#(s(mark(X)))

Problem 18: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

mark#(cons(X1, X2)) → active#(cons(mark(X1), X2))mark#(fib1(X1, X2)) → active#(fib1(mark(X1), mark(X2)))
mark#(cons(X1, X2)) → mark#(X1)active#(fib(N)) → mark#(sel(N, fib1(s(0), s(0))))
active#(add(s(X), Y)) → mark#(s(add(X, Y)))active#(add(0, X)) → mark#(X)
mark#(fib1(X1, X2)) → mark#(X1)mark#(sel(X1, X2)) → active#(sel(mark(X1), mark(X2)))
mark#(fib1(X1, X2)) → mark#(X2)mark#(sel(X1, X2)) → mark#(X2)
mark#(fib(X)) → mark#(X)mark#(add(X1, X2)) → mark#(X2)
mark#(fib(X)) → active#(fib(mark(X)))mark#(s(X)) → mark#(X)
active#(sel(0, cons(X, XS))) → mark#(X)mark#(add(X1, X2)) → active#(add(mark(X1), mark(X2)))
mark#(sel(X1, X2)) → mark#(X1)active#(sel(s(N), cons(X, XS))) → mark#(sel(N, XS))
mark#(add(X1, X2)) → mark#(X1)active#(fib1(X, Y)) → mark#(cons(X, fib1(Y, add(X, Y))))

Rewrite Rules

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Original Signature

Termination of terms over the following signature is verified: 0, s, active, mark, fib1, add, sel, fib, cons

Strategy

Function Precedence

0 < active# < add = fib1 = sel = mark# = fib < s = active = mark = cons

Argument Filtering

0: all arguments are removed from 0
s: all arguments are removed from s
active: all arguments are removed from active
mark: all arguments are removed from mark
active#: collapses to 1
add: all arguments are removed from add
fib1: all arguments are removed from fib1
sel: all arguments are removed from sel
mark#: all arguments are removed from mark#
fib: all arguments are removed from fib
cons: all arguments are removed from cons

Status

0: multiset
s: multiset
active: multiset
mark: multiset
add: multiset
fib1: multiset
sel: multiset
mark#: multiset
fib: multiset
cons: multiset

Usable Rules

active(sel(0, cons(X, XS))) → mark(X)cons(active(X1), X2) → cons(X1, X2)
fib(mark(X)) → fib(X)sel(X1, mark(X2)) → sel(X1, X2)
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))mark(fib(X)) → active(fib(mark(X)))
fib1(X1, active(X2)) → fib1(X1, X2)add(X1, mark(X2)) → add(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)add(X1, active(X2)) → add(X1, X2)
add(mark(X1), X2) → add(X1, X2)mark(0) → active(0)
s(active(X)) → s(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
cons(X1, active(X2)) → cons(X1, X2)add(active(X1), X2) → add(X1, X2)
active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))active(add(s(X), Y)) → mark(s(add(X, Y)))
active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))sel(X1, active(X2)) → sel(X1, X2)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))fib1(active(X1), X2) → fib1(X1, X2)
fib(active(X)) → fib(X)cons(mark(X1), X2) → cons(X1, X2)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))active(add(0, X)) → mark(X)
fib1(mark(X1), X2) → fib1(X1, X2)fib1(X1, mark(X2)) → fib1(X1, X2)
s(mark(X)) → s(X)sel(active(X1), X2) → sel(X1, X2)
sel(mark(X1), X2) → sel(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#(cons(X1, X2)) → active#(cons(mark(X1), X2))

Problem 5: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Original Signature

Termination of terms over the following signature is verified: 0, s, active, mark, fib1, add, sel, fib, cons

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 10: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Original Signature

Termination of terms over the following signature is verified: 0, s, active, mark, add, fib1, sel, fib, cons

Strategy

Function Precedence

active < mark < cons# = 0 = s = add = fib1 = sel = fib = cons

Argument Filtering

cons#: collapses to 2
0: all arguments are removed from 0
s: all arguments are removed from s
active: collapses to 1
mark: 1
add: all arguments are removed from add
fib1: all arguments are removed from fib1
sel: all arguments are removed from sel
fib: collapses to 1
cons: all arguments are removed from cons

Status

0: multiset
s: multiset
mark: multiset
add: multiset
fib1: multiset
sel: multiset
cons: 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.

cons#(X1, mark(X2)) → cons#(X1, X2)

Problem 15: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

cons#(X1, active(X2)) → cons#(X1, X2)

Rewrite Rules

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Original Signature

Termination of terms over the following signature is verified: 0, s, active, mark, fib1, add, sel, fib, cons

Strategy

Function Precedence

active < cons# < 0 = s = mark = add = fib1 = sel = fib = cons

Argument Filtering

cons#: collapses to 2
0: all arguments are removed from 0
s: all arguments are removed from s
active: 1
mark: all arguments are removed from mark
add: all arguments are removed from add
fib1: all arguments are removed from fib1
sel: all arguments are removed from sel
fib: all arguments are removed from fib
cons: 1 2

Status

0: multiset
s: multiset
active: multiset
mark: multiset
add: multiset
fib1: multiset
sel: multiset
fib: multiset
cons: lexicographic with permutation 1 → 1 2 → 2

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.

cons#(X1, active(X2)) → cons#(X1, X2)

Problem 6: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Original Signature

Termination of terms over the following signature is verified: 0, s, active, mark, fib1, add, sel, fib, cons

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 7: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Original Signature

Termination of terms over the following signature is verified: 0, s, active, mark, fib1, add, sel, fib, cons

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 11: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Original Signature

Termination of terms over the following signature is verified: 0, s, active, mark, add, fib1, sel, fib, cons

Strategy

Function Precedence

active = mark < add# < 0 = s = add = fib1 = sel = fib = cons

Argument Filtering

0: all arguments are removed from 0
s: all arguments are removed from s
active: 1
mark: collapses to 1
add#: collapses to 2
add: all arguments are removed from add
fib1: 1 2
sel: all arguments are removed from sel
fib: all arguments are removed from fib
cons: 1 2

Status

0: multiset
s: multiset
active: multiset
add: multiset
fib1: lexicographic with permutation 1 → 2 2 → 1
sel: multiset
fib: multiset
cons: lexicographic with permutation 1 → 1 2 → 2

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.

add#(X1, active(X2)) → add#(X1, X2)

Problem 16: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

add#(X1, mark(X2)) → add#(X1, X2)

Rewrite Rules

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Original Signature

Termination of terms over the following signature is verified: 0, s, active, mark, fib1, add, sel, fib, cons

Strategy

Function Precedence

add# < mark < 0 = s = active = add = fib1 = sel = fib = cons

Argument Filtering

0: all arguments are removed from 0
s: all arguments are removed from s
active: all arguments are removed from active
mark: 1
add#: collapses to 2
add: 1 2
fib1: all arguments are removed from fib1
sel: all arguments are removed from sel
fib: all arguments are removed from fib
cons: all arguments are removed from cons

Status

0: multiset
s: multiset
active: multiset
mark: multiset
add: lexicographic with permutation 1 → 2 2 → 1
fib1: multiset
sel: multiset
fib: multiset
cons: 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.

add#(X1, mark(X2)) → add#(X1, X2)

Problem 8: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Original Signature

Termination of terms over the following signature is verified: 0, s, active, mark, fib1, add, sel, fib, cons

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 12: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Original Signature

Termination of terms over the following signature is verified: 0, s, active, mark, add, fib1, sel, fib, cons

Strategy

Function Precedence

active < 0 = s = mark = sel# = add = fib1 = sel = fib = cons

Argument Filtering

0: all arguments are removed from 0
s: all arguments are removed from s
active: 1
mark: collapses to 1
sel#: collapses to 2
add: 1
fib1: 1 2
sel: 1 2
fib: all arguments are removed from fib
cons: all arguments are removed from cons

Status

0: multiset
s: multiset
active: multiset
add: lexicographic with permutation 1 → 1
fib1: lexicographic with permutation 1 → 2 2 → 1
sel: lexicographic with permutation 1 → 2 2 → 1
fib: multiset
cons: 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.

sel#(X1, active(X2)) → sel#(X1, X2)

Problem 17: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

sel#(X1, mark(X2)) → sel#(X1, X2)

Rewrite Rules

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)s(mark(X)) → s(X)
s(active(X)) → s(X)cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Original Signature

Termination of terms over the following signature is verified: 0, s, active, mark, fib1, add, sel, fib, cons

Strategy

Function Precedence

sel# < mark < 0 = s = active = add = fib1 = sel = fib = cons

Argument Filtering

0: all arguments are removed from 0
s: all arguments are removed from s
active: all arguments are removed from active
mark: 1
sel#: 2
add: all arguments are removed from add
fib1: all arguments are removed from fib1
sel: all arguments are removed from sel
fib: all arguments are removed from fib
cons: all arguments are removed from cons

Status

0: multiset
s: multiset
active: multiset
mark: multiset
sel#: lexicographic with permutation 2 → 1
add: multiset
fib1: multiset
sel: multiset
fib: multiset
cons: 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.

sel#(X1, mark(X2)) → sel#(X1, X2)