The following DP Processors were used

Problem 1 was processed with processor DependencyGraph (0ms).
 | – Problem 2 remains open; application of the following processors failed [SubtermCriterion (0ms), DependencyGraph (1ms), PolynomialLinearRange4iUR (57ms), DependencyGraph (2ms), PolynomialLinearRange8NegiUR (37ms), DependencyGraph (1ms), ReductionPairSAT (16ms), DependencyGraph (1ms), SizeChangePrinciple (6ms)].
 | – Problem 3 was processed with processor SubtermCriterion (0ms).

The following open problems remain:

Open Dependency Pair Problem 2

Dependency Pairs

from^#(X)

→

from^#(s(X))

Rewrite Rules

from(X)	→	cons(X, from(s(X)))		after(0, XS)	→	XS
after(s(N), cons(X, XS))	→	after(N, XS)

Original Signature

Termination of terms over the following signature is verified: after, 0, s, from, cons

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

after^#(s(N), cons(X, XS))

→

after^#(N, XS)

from^#(X)

→

from^#(s(X))

Rewrite Rules

from(X)	→	cons(X, from(s(X)))		after(0, XS)	→	XS
after(s(N), cons(X, XS))	→	after(N, XS)

Original Signature

Termination of terms over the following signature is verified: after, 0, s, from, cons

Strategy

The following SCCs where found

after^#(s(N), cons(X, XS)) → after^#(N, XS)

from^#(X) → from^#(s(X))

Problem 3: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

after^#(s(N), cons(X, XS))

→

after^#(N, XS)

Rewrite Rules

from(X)	→	cons(X, from(s(X)))		after(0, XS)	→	XS
after(s(N), cons(X, XS))	→	after(N, XS)

Original Signature

Termination of terms over the following signature is verified: after, 0, s, from, cons

Strategy

Projection

The following projection was used:

π (after#): 1

Thus, the following dependency pairs are removed:

after^#(s(N), cons(X, XS))

→

after^#(N, XS)