(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic path order with status [LPO].
Precedence:
first2 > nil
first2 > cons1
0 > nil
s1 > cons1
from1 > cons1

Status:
first2: [1,2]
0: []
nil: []
s1: [1]
cons1: [1]
from1: [1]
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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


(2) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(3) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(4) TRUE