(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:
Polynomial interpretation [POLO]:
POL(0) = 0
POL(cons(x1)) = 2 + x1
POL(first(x1, x2)) = x1 + 2·x2
POL(from(x1)) = 2 + 2·x1
POL(nil) = 0
POL(s(x1)) = x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
first(s(X), cons(Y)) → cons(Y)
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
first(0, X) → nil
from(X) → cons(X)
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Knuth-Bendix order [KBO] with precedence:
first2 > from1 > cons1 > nil > 0
and weight map:
0=1
nil=2
from_1=1
cons_1=1
first_2=0
The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
first(0, X) → nil
from(X) → cons(X)
(4) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(5) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(6) YES