(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
active(2nd(cons(X, cons(Y, Z)))) → mark(Y)
active(from(X)) → mark(cons(X, from(s(X))))
active(2nd(X)) → 2nd(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
2nd(mark(X)) → mark(2nd(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
proper(2nd(X)) → 2nd(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
2nd(ok(X)) → ok(2nd(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(2nd(x1)) = 2·x1
POL(active(x1)) = 2·x1
POL(cons(x1, x2)) = x1 + x2
POL(from(x1)) = 2·x1
POL(mark(x1)) = x1
POL(ok(x1)) = 1 + 2·x1
POL(proper(x1)) = x1
POL(s(x1)) = x1
POL(top(x1)) = 2·x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
2nd(ok(X)) → ok(2nd(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(ok(X)) → top(active(X))
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
active(2nd(cons(X, cons(Y, Z)))) → mark(Y)
active(from(X)) → mark(cons(X, from(s(X))))
active(2nd(X)) → 2nd(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
2nd(mark(X)) → mark(2nd(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
proper(2nd(X)) → 2nd(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
s(ok(X)) → ok(s(X))
top(mark(X)) → top(proper(X))
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(2nd(x1)) = 2·x1
POL(active(x1)) = 2·x1
POL(cons(x1, x2)) = 1 + x1 + x2
POL(from(x1)) = 2 + 2·x1
POL(mark(x1)) = 1 + x1
POL(ok(x1)) = x1
POL(proper(x1)) = x1
POL(s(x1)) = x1
POL(top(x1)) = x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
active(2nd(cons(X, cons(Y, Z)))) → mark(Y)
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
2nd(mark(X)) → mark(2nd(X))
from(mark(X)) → mark(from(X))
top(mark(X)) → top(proper(X))
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(2nd(X)) → 2nd(active(X))
active(s(X)) → s(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
proper(2nd(X)) → 2nd(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
s(ok(X)) → ok(s(X))
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:
active1 > from1
active1 > cons2 > mark1
active1 > s1 > mark1
active1 > s1 > ok1
active1 > 2nd1
proper1 > from1
proper1 > cons2 > mark1
proper1 > s1 > mark1
proper1 > s1 > ok1
proper1 > 2nd1
Status:
trivial
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
active(from(X)) → mark(cons(X, from(s(X))))
active(2nd(X)) → 2nd(active(X))
active(s(X)) → s(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
proper(2nd(X)) → 2nd(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
s(ok(X)) → ok(s(X))
(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) TRUE
(9) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(10) TRUE