MAYBE

The TRS could not be proven terminating. The proof attempt took 267 ms.

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)nilfirst(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)nilfirst(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)nilfirst(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:

Thus, the following dependency pairs are removed:

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