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