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 (1ms), DependencyGraph (1ms), PolynomialLinearRange4iUR (57ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (37ms), DependencyGraph (1ms), ReductionPairSAT (19ms), DependencyGraph (1ms), SizeChangePrinciple (4ms)].
 | – 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

first(0, X)	→	nil		first(s(X), cons(Y, Z))	→	cons(Y, first(X, Z))
from(X)	→	cons(X, from(s(X)))

Original Signature

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

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

first^#(s(X), cons(Y, Z))

→

first^#(X, Z)

from^#(X)

→

from^#(s(X))

Rewrite Rules

first(0, X)	→	nil		first(s(X), cons(Y, Z))	→	cons(Y, first(X, Z))
from(X)	→	cons(X, from(s(X)))

Original Signature

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

Strategy

The following SCCs where found

first^#(s(X), cons(Y, Z)) → first^#(X, Z)

from^#(X) → from^#(s(X))

Problem 3: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

first^#(s(X), cons(Y, Z))

→

first^#(X, Z)

Rewrite Rules

first(0, X)	→	nil		first(s(X), cons(Y, Z))	→	cons(Y, first(X, Z))
from(X)	→	cons(X, from(s(X)))

Original Signature

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

Strategy

Projection

The following projection was used:

π (first#): 1

Thus, the following dependency pairs are removed:

first^#(s(X), cons(Y, Z))

→

first^#(X, Z)