(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
from(X) → cons(X, n__from(s(X)))
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0) = 1
POL(activate(x1)) = 2 + 2·x1
POL(cons(x1, x2)) = 1 + x1 + x2
POL(first(x1, x2)) = 1 + 2·x1 + 2·x2
POL(from(x1)) = 2 + 2·x1
POL(n__first(x1, x2)) = 2·x1 + x2
POL(n__from(x1)) = 1 + 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(0, X) → nil
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
from(X) → cons(X, n__from(s(X)))
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(activate(x1)) = x1
POL(cons(x1, x2)) = x1 + x2
POL(first(x1, x2)) = 2·x1 + x2
POL(from(x1)) = 1 + 2·x1
POL(n__first(x1, x2)) = x1 + x2
POL(n__from(x1)) = x1
POL(s(x1)) = x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
from(X) → cons(X, n__from(s(X)))
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Knuth-Bendix order [KBO] with precedence:
activate1 > nfirst2 > first2 > cons2 > s1
and weight map:
s_1=1
activate_1=1
first_2=0
n__first_2=0
cons_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(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
(6) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(7) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(8) YES