YES Termination w.r.t. Q proof of Transformed_CSR_04_Ex6_Luc98_L.ari

(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