TIMEOUT

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

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (165ms).
 | – Problem 2 was processed with processor SubtermCriterion (1ms).
 | – Problem 3 was processed with processor SubtermCriterion (1ms).
 | – Problem 4 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (4ms), PolynomialLinearRange4iUR (1119ms), DependencyGraph (6ms), PolynomialLinearRange8NegiUR (4925ms), DependencyGraph (4ms), ReductionPairSAT (timeout)].
 | – Problem 5 was processed with processor SubtermCriterion (1ms).

The following open problems remain:



Open Dependency Pair Problem 4

Dependency Pairs

helpb#(c, l, cons(y, ys), zs)helpa#(s(c), l, ys, zs)helpa#(c, l, ys, zs)if#(ge(c, l), c, l, ys, zs)
if#(false, c, l, ys, zs)helpb#(c, l, greater(ys, zs), smaller(ys, zs))

Rewrite Rules

app(x, y)helpa(0, plus(length(x), length(y)), x, y)plus(x, 0)x
plus(x, s(y))s(plus(x, y))length(nil)0
length(cons(x, y))s(length(y))helpa(c, l, ys, zs)if(ge(c, l), c, l, ys, zs)
ge(x, 0)truege(0, s(x))false
ge(s(x), s(y))ge(x, y)if(true, c, l, ys, zs)nil
if(false, c, l, ys, zs)helpb(c, l, greater(ys, zs), smaller(ys, zs))greater(ys, zs)helpc(ge(length(ys), length(zs)), ys, zs)
smaller(ys, zs)helpc(ge(length(ys), length(zs)), zs, ys)helpc(true, ys, zs)ys
helpc(false, ys, zs)zshelpb(c, l, cons(y, ys), zs)cons(y, helpa(s(c), l, ys, zs))

Original Signature

Termination of terms over the following signature is verified: plus, app, greater, true, ge, smaller, 0, s, if, helpa, length, helpb, false, helpc, nil, cons


Problem 1: DependencyGraph



Dependency Pair Problem

Dependency Pairs

smaller#(ys, zs)ge#(length(ys), length(zs))greater#(ys, zs)length#(zs)
greater#(ys, zs)ge#(length(ys), length(zs))helpa#(c, l, ys, zs)if#(ge(c, l), c, l, ys, zs)
if#(false, c, l, ys, zs)helpb#(c, l, greater(ys, zs), smaller(ys, zs))if#(false, c, l, ys, zs)smaller#(ys, zs)
helpb#(c, l, cons(y, ys), zs)helpa#(s(c), l, ys, zs)smaller#(ys, zs)length#(ys)
length#(cons(x, y))length#(y)app#(x, y)length#(y)
plus#(x, s(y))plus#(x, y)greater#(ys, zs)helpc#(ge(length(ys), length(zs)), ys, zs)
smaller#(ys, zs)helpc#(ge(length(ys), length(zs)), zs, ys)app#(x, y)length#(x)
helpa#(c, l, ys, zs)ge#(c, l)ge#(s(x), s(y))ge#(x, y)
if#(false, c, l, ys, zs)greater#(ys, zs)app#(x, y)plus#(length(x), length(y))
smaller#(ys, zs)length#(zs)app#(x, y)helpa#(0, plus(length(x), length(y)), x, y)
greater#(ys, zs)length#(ys)

Rewrite Rules

app(x, y)helpa(0, plus(length(x), length(y)), x, y)plus(x, 0)x
plus(x, s(y))s(plus(x, y))length(nil)0
length(cons(x, y))s(length(y))helpa(c, l, ys, zs)if(ge(c, l), c, l, ys, zs)
ge(x, 0)truege(0, s(x))false
ge(s(x), s(y))ge(x, y)if(true, c, l, ys, zs)nil
if(false, c, l, ys, zs)helpb(c, l, greater(ys, zs), smaller(ys, zs))greater(ys, zs)helpc(ge(length(ys), length(zs)), ys, zs)
smaller(ys, zs)helpc(ge(length(ys), length(zs)), zs, ys)helpc(true, ys, zs)ys
helpc(false, ys, zs)zshelpb(c, l, cons(y, ys), zs)cons(y, helpa(s(c), l, ys, zs))

Original Signature

Termination of terms over the following signature is verified: plus, app, greater, true, ge, smaller, 0, s, if, helpa, length, helpb, false, helpc, cons, nil

Strategy


The following SCCs where found

length#(cons(x, y)) → length#(y)

plus#(x, s(y)) → plus#(x, y)

helpb#(c, l, cons(y, ys), zs) → helpa#(s(c), l, ys, zs)helpa#(c, l, ys, zs) → if#(ge(c, l), c, l, ys, zs)
if#(false, c, l, ys, zs) → helpb#(c, l, greater(ys, zs), smaller(ys, zs))

ge#(s(x), s(y)) → ge#(x, y)

Problem 2: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

ge#(s(x), s(y))ge#(x, y)

Rewrite Rules

app(x, y)helpa(0, plus(length(x), length(y)), x, y)plus(x, 0)x
plus(x, s(y))s(plus(x, y))length(nil)0
length(cons(x, y))s(length(y))helpa(c, l, ys, zs)if(ge(c, l), c, l, ys, zs)
ge(x, 0)truege(0, s(x))false
ge(s(x), s(y))ge(x, y)if(true, c, l, ys, zs)nil
if(false, c, l, ys, zs)helpb(c, l, greater(ys, zs), smaller(ys, zs))greater(ys, zs)helpc(ge(length(ys), length(zs)), ys, zs)
smaller(ys, zs)helpc(ge(length(ys), length(zs)), zs, ys)helpc(true, ys, zs)ys
helpc(false, ys, zs)zshelpb(c, l, cons(y, ys), zs)cons(y, helpa(s(c), l, ys, zs))

Original Signature

Termination of terms over the following signature is verified: plus, app, greater, true, ge, smaller, 0, s, if, helpa, length, helpb, false, helpc, cons, nil

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

ge#(s(x), s(y))ge#(x, y)

Problem 3: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

plus#(x, s(y))plus#(x, y)

Rewrite Rules

app(x, y)helpa(0, plus(length(x), length(y)), x, y)plus(x, 0)x
plus(x, s(y))s(plus(x, y))length(nil)0
length(cons(x, y))s(length(y))helpa(c, l, ys, zs)if(ge(c, l), c, l, ys, zs)
ge(x, 0)truege(0, s(x))false
ge(s(x), s(y))ge(x, y)if(true, c, l, ys, zs)nil
if(false, c, l, ys, zs)helpb(c, l, greater(ys, zs), smaller(ys, zs))greater(ys, zs)helpc(ge(length(ys), length(zs)), ys, zs)
smaller(ys, zs)helpc(ge(length(ys), length(zs)), zs, ys)helpc(true, ys, zs)ys
helpc(false, ys, zs)zshelpb(c, l, cons(y, ys), zs)cons(y, helpa(s(c), l, ys, zs))

Original Signature

Termination of terms over the following signature is verified: plus, app, greater, true, ge, smaller, 0, s, if, helpa, length, helpb, false, helpc, cons, nil

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

plus#(x, s(y))plus#(x, y)

Problem 5: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

length#(cons(x, y))length#(y)

Rewrite Rules

app(x, y)helpa(0, plus(length(x), length(y)), x, y)plus(x, 0)x
plus(x, s(y))s(plus(x, y))length(nil)0
length(cons(x, y))s(length(y))helpa(c, l, ys, zs)if(ge(c, l), c, l, ys, zs)
ge(x, 0)truege(0, s(x))false
ge(s(x), s(y))ge(x, y)if(true, c, l, ys, zs)nil
if(false, c, l, ys, zs)helpb(c, l, greater(ys, zs), smaller(ys, zs))greater(ys, zs)helpc(ge(length(ys), length(zs)), ys, zs)
smaller(ys, zs)helpc(ge(length(ys), length(zs)), zs, ys)helpc(true, ys, zs)ys
helpc(false, ys, zs)zshelpb(c, l, cons(y, ys), zs)cons(y, helpa(s(c), l, ys, zs))

Original Signature

Termination of terms over the following signature is verified: plus, app, greater, true, ge, smaller, 0, s, if, helpa, length, helpb, false, helpc, cons, nil

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

length#(cons(x, y))length#(y)