'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, length(mark(X)) -> mark(length(X))
, proper(zeros()) -> ok(zeros())
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, proper(nil()) -> ok(nil())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))
, length(ok(X)) -> ok(length(X))
, top(mark(X)) -> top(proper(X))
, top(ok(X)) -> top(active(X))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ active^#(zeros()) -> c_0(cons^#(0(), zeros()))
, active^#(U11(tt(), L)) -> c_1(U12^#(tt(), L))
, active^#(U12(tt(), L)) -> c_2(s^#(length(L)))
, active^#(length(nil())) -> c_3()
, active^#(length(cons(N, L))) -> c_4(U11^#(tt(), L))
, active^#(cons(X1, X2)) -> c_5(cons^#(active(X1), X2))
, active^#(U11(X1, X2)) -> c_6(U11^#(active(X1), X2))
, active^#(U12(X1, X2)) -> c_7(U12^#(active(X1), X2))
, active^#(s(X)) -> c_8(s^#(active(X)))
, active^#(length(X)) -> c_9(length^#(active(X)))
, cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))
, U11^#(mark(X1), X2) -> c_11(U11^#(X1, X2))
, U12^#(mark(X1), X2) -> c_12(U12^#(X1, X2))
, s^#(mark(X)) -> c_13(s^#(X))
, length^#(mark(X)) -> c_14(length^#(X))
, proper^#(zeros()) -> c_15()
, proper^#(cons(X1, X2)) -> c_16(cons^#(proper(X1), proper(X2)))
, proper^#(0()) -> c_17()
, proper^#(U11(X1, X2)) -> c_18(U11^#(proper(X1), proper(X2)))
, proper^#(tt()) -> c_19()
, proper^#(U12(X1, X2)) -> c_20(U12^#(proper(X1), proper(X2)))
, proper^#(s(X)) -> c_21(s^#(proper(X)))
, proper^#(length(X)) -> c_22(length^#(proper(X)))
, proper^#(nil()) -> c_23()
, cons^#(ok(X1), ok(X2)) -> c_24(cons^#(X1, X2))
, U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))
, U12^#(ok(X1), ok(X2)) -> c_26(U12^#(X1, X2))
, s^#(ok(X)) -> c_27(s^#(X))
, length^#(ok(X)) -> c_28(length^#(X))
, top^#(mark(X)) -> c_29(top^#(proper(X)))
, top^#(ok(X)) -> c_30(top^#(active(X)))}
The usable rules are:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, proper(zeros()) -> ok(zeros())
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, proper(nil()) -> ok(nil())
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))}
The estimated dependency graph contains the following edges:
{active^#(U12(tt(), L)) -> c_2(s^#(length(L)))}
==> {s^#(ok(X)) -> c_27(s^#(X))}
{active^#(U12(tt(), L)) -> c_2(s^#(length(L)))}
==> {s^#(mark(X)) -> c_13(s^#(X))}
{active^#(cons(X1, X2)) -> c_5(cons^#(active(X1), X2))}
==> {cons^#(ok(X1), ok(X2)) -> c_24(cons^#(X1, X2))}
{active^#(cons(X1, X2)) -> c_5(cons^#(active(X1), X2))}
==> {cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))}
{active^#(U11(X1, X2)) -> c_6(U11^#(active(X1), X2))}
==> {U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))}
{active^#(U11(X1, X2)) -> c_6(U11^#(active(X1), X2))}
==> {U11^#(mark(X1), X2) -> c_11(U11^#(X1, X2))}
{active^#(U12(X1, X2)) -> c_7(U12^#(active(X1), X2))}
==> {U12^#(ok(X1), ok(X2)) -> c_26(U12^#(X1, X2))}
{active^#(U12(X1, X2)) -> c_7(U12^#(active(X1), X2))}
==> {U12^#(mark(X1), X2) -> c_12(U12^#(X1, X2))}
{active^#(s(X)) -> c_8(s^#(active(X)))}
==> {s^#(ok(X)) -> c_27(s^#(X))}
{active^#(s(X)) -> c_8(s^#(active(X)))}
==> {s^#(mark(X)) -> c_13(s^#(X))}
{active^#(length(X)) -> c_9(length^#(active(X)))}
==> {length^#(ok(X)) -> c_28(length^#(X))}
{active^#(length(X)) -> c_9(length^#(active(X)))}
==> {length^#(mark(X)) -> c_14(length^#(X))}
{cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))}
==> {cons^#(ok(X1), ok(X2)) -> c_24(cons^#(X1, X2))}
{cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))}
==> {cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))}
{U11^#(mark(X1), X2) -> c_11(U11^#(X1, X2))}
==> {U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))}
{U11^#(mark(X1), X2) -> c_11(U11^#(X1, X2))}
==> {U11^#(mark(X1), X2) -> c_11(U11^#(X1, X2))}
{U12^#(mark(X1), X2) -> c_12(U12^#(X1, X2))}
==> {U12^#(ok(X1), ok(X2)) -> c_26(U12^#(X1, X2))}
{U12^#(mark(X1), X2) -> c_12(U12^#(X1, X2))}
==> {U12^#(mark(X1), X2) -> c_12(U12^#(X1, X2))}
{s^#(mark(X)) -> c_13(s^#(X))}
==> {s^#(ok(X)) -> c_27(s^#(X))}
{s^#(mark(X)) -> c_13(s^#(X))}
==> {s^#(mark(X)) -> c_13(s^#(X))}
{length^#(mark(X)) -> c_14(length^#(X))}
==> {length^#(ok(X)) -> c_28(length^#(X))}
{length^#(mark(X)) -> c_14(length^#(X))}
==> {length^#(mark(X)) -> c_14(length^#(X))}
{proper^#(cons(X1, X2)) -> c_16(cons^#(proper(X1), proper(X2)))}
==> {cons^#(ok(X1), ok(X2)) -> c_24(cons^#(X1, X2))}
{proper^#(cons(X1, X2)) -> c_16(cons^#(proper(X1), proper(X2)))}
==> {cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))}
{proper^#(U11(X1, X2)) -> c_18(U11^#(proper(X1), proper(X2)))}
==> {U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))}
{proper^#(U11(X1, X2)) -> c_18(U11^#(proper(X1), proper(X2)))}
==> {U11^#(mark(X1), X2) -> c_11(U11^#(X1, X2))}
{proper^#(U12(X1, X2)) -> c_20(U12^#(proper(X1), proper(X2)))}
==> {U12^#(ok(X1), ok(X2)) -> c_26(U12^#(X1, X2))}
{proper^#(U12(X1, X2)) -> c_20(U12^#(proper(X1), proper(X2)))}
==> {U12^#(mark(X1), X2) -> c_12(U12^#(X1, X2))}
{proper^#(s(X)) -> c_21(s^#(proper(X)))}
==> {s^#(ok(X)) -> c_27(s^#(X))}
{proper^#(s(X)) -> c_21(s^#(proper(X)))}
==> {s^#(mark(X)) -> c_13(s^#(X))}
{proper^#(length(X)) -> c_22(length^#(proper(X)))}
==> {length^#(ok(X)) -> c_28(length^#(X))}
{proper^#(length(X)) -> c_22(length^#(proper(X)))}
==> {length^#(mark(X)) -> c_14(length^#(X))}
{cons^#(ok(X1), ok(X2)) -> c_24(cons^#(X1, X2))}
==> {cons^#(ok(X1), ok(X2)) -> c_24(cons^#(X1, X2))}
{cons^#(ok(X1), ok(X2)) -> c_24(cons^#(X1, X2))}
==> {cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))}
{U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))}
==> {U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))}
{U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))}
==> {U11^#(mark(X1), X2) -> c_11(U11^#(X1, X2))}
{U12^#(ok(X1), ok(X2)) -> c_26(U12^#(X1, X2))}
==> {U12^#(ok(X1), ok(X2)) -> c_26(U12^#(X1, X2))}
{U12^#(ok(X1), ok(X2)) -> c_26(U12^#(X1, X2))}
==> {U12^#(mark(X1), X2) -> c_12(U12^#(X1, X2))}
{s^#(ok(X)) -> c_27(s^#(X))}
==> {s^#(ok(X)) -> c_27(s^#(X))}
{s^#(ok(X)) -> c_27(s^#(X))}
==> {s^#(mark(X)) -> c_13(s^#(X))}
{length^#(ok(X)) -> c_28(length^#(X))}
==> {length^#(ok(X)) -> c_28(length^#(X))}
{length^#(ok(X)) -> c_28(length^#(X))}
==> {length^#(mark(X)) -> c_14(length^#(X))}
{top^#(mark(X)) -> c_29(top^#(proper(X)))}
==> {top^#(ok(X)) -> c_30(top^#(active(X)))}
{top^#(mark(X)) -> c_29(top^#(proper(X)))}
==> {top^#(mark(X)) -> c_29(top^#(proper(X)))}
{top^#(ok(X)) -> c_30(top^#(active(X)))}
==> {top^#(ok(X)) -> c_30(top^#(active(X)))}
{top^#(ok(X)) -> c_30(top^#(active(X)))}
==> {top^#(mark(X)) -> c_29(top^#(proper(X)))}
We consider the following path(s):
1) { top^#(mark(X)) -> c_29(top^#(proper(X)))
, top^#(ok(X)) -> c_30(top^#(active(X)))}
The usable rules for this path are the following:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, proper(zeros()) -> ok(zeros())
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, proper(nil()) -> ok(nil())
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, proper(zeros()) -> ok(zeros())
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, proper(nil()) -> ok(nil())
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))
, top^#(mark(X)) -> c_29(top^#(proper(X)))
, top^#(ok(X)) -> c_30(top^#(active(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [0]
c_29(x1) = [1] x1 + [0]
c_30(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, top^#(ok(X)) -> c_30(top^#(active(X)))}
and weakly orienting the rules
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, top^#(ok(X)) -> c_30(top^#(active(X)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [9]
ok(x1) = [1] x1 + [2]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [0]
c_29(x1) = [1] x1 + [0]
c_30(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(length(nil())) -> mark(0())}
and weakly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, top^#(ok(X)) -> c_30(top^#(active(X)))
, proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(length(nil())) -> mark(0())}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [12]
length(x1) = [1] x1 + [0]
nil() = [6]
proper(x1) = [1] x1 + [4]
ok(x1) = [1] x1 + [2]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [0]
c_29(x1) = [1] x1 + [13]
c_30(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U12(tt(), L)) -> mark(s(length(L)))}
and weakly orienting the rules
{ active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, top^#(ok(X)) -> c_30(top^#(active(X)))
, proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U12(tt(), L)) -> mark(s(length(L)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [3]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [7]
c_29(x1) = [1] x1 + [0]
c_30(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(length(cons(N, L))) -> mark(U11(tt(), L))}
and weakly orienting the rules
{ active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, top^#(ok(X)) -> c_30(top^#(active(X)))
, proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(length(cons(N, L))) -> mark(U11(tt(), L))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [4]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [2]
nil() = [1]
proper(x1) = [1] x1 + [7]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [1]
c_29(x1) = [1] x1 + [0]
c_30(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U11(tt(), L)) -> mark(U12(tt(), L))}
and weakly orienting the rules
{ active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, top^#(ok(X)) -> c_30(top^#(active(X)))
, proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U11(tt(), L)) -> mark(U12(tt(), L))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [4]
zeros() = [6]
mark(x1) = [1] x1 + [3]
cons(x1, x2) = [1] x1 + [1] x2 + [5]
0() = [9]
U11(x1, x2) = [1] x1 + [1] x2 + [5]
tt() = [1]
U12(x1, x2) = [1] x1 + [1] x2 + [5]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [12]
proper(x1) = [1] x1 + [13]
ok(x1) = [1] x1 + [4]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [1]
c_29(x1) = [1] x1 + [0]
c_30(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{top^#(mark(X)) -> c_29(top^#(proper(X)))}
and weakly orienting the rules
{ active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, top^#(ok(X)) -> c_30(top^#(active(X)))
, proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{top^#(mark(X)) -> c_29(top^#(proper(X)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [2]
cons(x1, x2) = [1] x1 + [1] x2 + [8]
0() = [6]
U11(x1, x2) = [1] x1 + [1] x2 + [2]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [1]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [15]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [1]
c_29(x1) = [1] x1 + [0]
c_30(x1) = [1] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ top^#(mark(X)) -> c_29(top^#(proper(X)))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, top^#(ok(X)) -> c_30(top^#(active(X)))
, proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ top^#(mark(X)) -> c_29(top^#(proper(X)))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, top^#(ok(X)) -> c_30(top^#(active(X)))
, proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
Details:
The problem is Match-bounded by 4.
The enriched problem is compatible with the following automaton:
{ active_0(2) -> 6
, active_1(2) -> 15
, active_2(8) -> 21
, active_2(9) -> 21
, active_2(24) -> 28
, active_3(23) -> 29
, active_3(24) -> 35
, active_3(38) -> 40
, active_4(37) -> 41
, active_4(38) -> 43
, zeros_0() -> 2
, zeros_1() -> 9
, zeros_2() -> 22
, zeros_3() -> 36
, mark_0(2) -> 2
, mark_1(7) -> 6
, mark_1(7) -> 15
, mark_2(24) -> 21
, cons_1(8, 9) -> 7
, cons_2(16, 17) -> 13
, cons_2(16, 17) -> 19
, cons_2(23, 22) -> 24
, cons_3(29, 22) -> 28
, cons_3(29, 22) -> 35
, cons_3(30, 31) -> 26
, cons_3(30, 31) -> 33
, cons_3(37, 36) -> 38
, cons_4(41, 36) -> 40
, cons_4(41, 36) -> 43
, 0_0() -> 2
, 0_1() -> 8
, 0_2() -> 23
, 0_3() -> 37
, tt_0() -> 2
, tt_1() -> 8
, tt_2() -> 23
, tt_3() -> 37
, nil_0() -> 2
, nil_1() -> 8
, nil_2() -> 23
, nil_3() -> 37
, proper_0(2) -> 4
, proper_1(2) -> 11
, proper_1(7) -> 13
, proper_2(7) -> 19
, proper_2(8) -> 16
, proper_2(9) -> 17
, proper_2(24) -> 26
, proper_3(22) -> 31
, proper_3(23) -> 30
, proper_3(24) -> 33
, ok_0(2) -> 2
, ok_0(2) -> 4
, ok_1(8) -> 11
, ok_1(9) -> 11
, ok_2(22) -> 17
, ok_2(23) -> 16
, ok_2(24) -> 13
, ok_2(24) -> 19
, ok_3(36) -> 31
, ok_3(37) -> 30
, ok_3(38) -> 26
, ok_3(38) -> 33
, top^#_0(2) -> 1
, top^#_0(4) -> 3
, top^#_0(6) -> 5
, top^#_1(11) -> 10
, top^#_1(13) -> 12
, top^#_1(15) -> 14
, top^#_2(19) -> 18
, top^#_2(21) -> 20
, top^#_2(26) -> 25
, top^#_2(28) -> 27
, top^#_3(33) -> 32
, top^#_3(35) -> 34
, top^#_3(40) -> 39
, top^#_4(43) -> 42
, c_29_0(3) -> 1
, c_29_1(10) -> 1
, c_29_1(12) -> 5
, c_29_2(18) -> 14
, c_29_2(25) -> 20
, c_29_3(32) -> 20
, c_30_0(5) -> 1
, c_30_1(14) -> 1
, c_30_1(14) -> 3
, c_30_2(20) -> 10
, c_30_2(27) -> 12
, c_30_2(27) -> 18
, c_30_3(34) -> 18
, c_30_3(39) -> 25
, c_30_3(39) -> 32
, c_30_4(42) -> 32}
2) { active^#(length(X)) -> c_9(length^#(active(X)))
, length^#(ok(X)) -> c_28(length^#(X))
, length^#(mark(X)) -> c_14(length^#(X))}
The usable rules for this path are the following:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))
, active^#(length(X)) -> c_9(length^#(active(X)))
, length^#(ok(X)) -> c_28(length^#(X))
, length^#(mark(X)) -> c_14(length^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{length^#(mark(X)) -> c_14(length^#(X))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{length^#(mark(X)) -> c_14(length^#(X))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [1]
length^#(x1) = [1] x1 + [12]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [1] x1 + [5]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U12(tt(), L)) -> mark(s(length(L)))}
and weakly orienting the rules
{length^#(mark(X)) -> c_14(length^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U12(tt(), L)) -> mark(s(length(L)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [4]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
length^#(x1) = [1] x1 + [8]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [1]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [1] x1 + [1]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active^#(length(X)) -> c_9(length^#(active(X)))}
and weakly orienting the rules
{ active(U12(tt(), L)) -> mark(s(length(L)))
, length^#(mark(X)) -> c_14(length^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(length(X)) -> c_9(length^#(active(X)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [3]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
length^#(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [1]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [1] x1 + [1]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, length^#(ok(X)) -> c_28(length^#(X))}
and weakly orienting the rules
{ active^#(length(X)) -> c_9(length^#(active(X)))
, active(U12(tt(), L)) -> mark(s(length(L)))
, length^#(mark(X)) -> c_14(length^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, length^#(ok(X)) -> c_28(length^#(X))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [4]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [7]
length^#(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(length(nil())) -> mark(0())}
and weakly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, length^#(ok(X)) -> c_28(length^#(X))
, active^#(length(X)) -> c_9(length^#(active(X)))
, active(U12(tt(), L)) -> mark(s(length(L)))
, length^#(mark(X)) -> c_14(length^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(length(nil())) -> mark(0())}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [4]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [11]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [1]
length^#(x1) = [1] x1 + [7]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(length(cons(N, L))) -> mark(U11(tt(), L))}
and weakly orienting the rules
{ active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, length^#(ok(X)) -> c_28(length^#(X))
, active^#(length(X)) -> c_9(length^#(active(X)))
, active(U12(tt(), L)) -> mark(s(length(L)))
, length^#(mark(X)) -> c_14(length^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(length(cons(N, L))) -> mark(U11(tt(), L))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [8]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [3]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [5]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
length^#(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [1]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [1] x1 + [1]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U11(tt(), L)) -> mark(U12(tt(), L))}
and weakly orienting the rules
{ active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, length^#(ok(X)) -> c_28(length^#(X))
, active^#(length(X)) -> c_9(length^#(active(X)))
, active(U12(tt(), L)) -> mark(s(length(L)))
, length^#(mark(X)) -> c_14(length^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U11(tt(), L)) -> mark(U12(tt(), L))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [0]
zeros() = [2]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [1]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [10]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [5]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
length^#(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(zeros()) -> mark(cons(0(), zeros()))}
and weakly orienting the rules
{ active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, length^#(ok(X)) -> c_28(length^#(X))
, active^#(length(X)) -> c_9(length^#(active(X)))
, active(U12(tt(), L)) -> mark(s(length(L)))
, length^#(mark(X)) -> c_14(length^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(zeros()) -> mark(cons(0(), zeros()))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [4]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [2]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [9]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [1]
length^#(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [1] x1 + [1]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, length^#(ok(X)) -> c_28(length^#(X))
, active^#(length(X)) -> c_9(length^#(active(X)))
, active(U12(tt(), L)) -> mark(s(length(L)))
, length^#(mark(X)) -> c_14(length^#(X))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, length^#(ok(X)) -> c_28(length^#(X))
, active^#(length(X)) -> c_9(length^#(active(X)))
, active(U12(tt(), L)) -> mark(s(length(L)))
, length^#(mark(X)) -> c_14(length^#(X))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ zeros_0() -> 2
, mark_0(2) -> 3
, mark_0(3) -> 3
, mark_0(5) -> 3
, mark_0(7) -> 3
, mark_0(11) -> 3
, mark_0(13) -> 3
, 0_0() -> 5
, tt_0() -> 7
, nil_0() -> 11
, ok_0(2) -> 13
, ok_0(3) -> 13
, ok_0(5) -> 13
, ok_0(7) -> 13
, ok_0(11) -> 13
, ok_0(13) -> 13
, active^#_0(2) -> 15
, active^#_0(3) -> 15
, active^#_0(5) -> 15
, active^#_0(7) -> 15
, active^#_0(11) -> 15
, active^#_0(13) -> 15
, length^#_0(2) -> 30
, length^#_0(3) -> 30
, length^#_0(5) -> 30
, length^#_0(7) -> 30
, length^#_0(11) -> 30
, length^#_0(13) -> 30
, c_14_0(30) -> 30
, c_28_0(30) -> 30}
3) { active^#(U12(X1, X2)) -> c_7(U12^#(active(X1), X2))
, U12^#(ok(X1), ok(X2)) -> c_26(U12^#(X1, X2))
, U12^#(mark(X1), X2) -> c_12(U12^#(X1, X2))}
The usable rules for this path are the following:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))
, active^#(U12(X1, X2)) -> c_7(U12^#(active(X1), X2))
, U12^#(ok(X1), ok(X2)) -> c_26(U12^#(X1, X2))
, U12^#(mark(X1), X2) -> c_12(U12^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{U12^#(mark(X1), X2) -> c_12(U12^#(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{U12^#(mark(X1), X2) -> c_12(U12^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [1] x1 + [1] x2 + [8]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [1] x1 + [1]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ active(U12(tt(), L)) -> mark(s(length(L)))
, active^#(U12(X1, X2)) -> c_7(U12^#(active(X1), X2))}
and weakly orienting the rules
{U12^#(mark(X1), X2) -> c_12(U12^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(U12(tt(), L)) -> mark(s(length(L)))
, active^#(U12(X1, X2)) -> c_7(U12^#(active(X1), X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [8]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [1] x1 + [1]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, U12^#(ok(X1), ok(X2)) -> c_26(U12^#(X1, X2))}
and weakly orienting the rules
{ active(U12(tt(), L)) -> mark(s(length(L)))
, active^#(U12(X1, X2)) -> c_7(U12^#(active(X1), X2))
, U12^#(mark(X1), X2) -> c_12(U12^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, U12^#(ok(X1), ok(X2)) -> c_26(U12^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [3]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [14]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [1] x1 + [4]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(length(nil())) -> mark(0())}
and weakly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, U12^#(ok(X1), ok(X2)) -> c_26(U12^#(X1, X2))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active^#(U12(X1, X2)) -> c_7(U12^#(active(X1), X2))
, U12^#(mark(X1), X2) -> c_12(U12^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(length(nil())) -> mark(0())}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [12]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [1] x1 + [1] x2 + [4]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [1] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(length(cons(N, L))) -> mark(U11(tt(), L))}
and weakly orienting the rules
{ active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, U12^#(ok(X1), ok(X2)) -> c_26(U12^#(X1, X2))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active^#(U12(X1, X2)) -> c_7(U12^#(active(X1), X2))
, U12^#(mark(X1), X2) -> c_12(U12^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(length(cons(N, L))) -> mark(U11(tt(), L))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [4]
s(x1) = [1] x1 + [2]
length(x1) = [1] x1 + [2]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [13]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [1] x1 + [1] x2 + [15]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [1] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U11(tt(), L)) -> mark(U12(tt(), L))}
and weakly orienting the rules
{ active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, U12^#(ok(X1), ok(X2)) -> c_26(U12^#(X1, X2))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active^#(U12(X1, X2)) -> c_7(U12^#(active(X1), X2))
, U12^#(mark(X1), X2) -> c_12(U12^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U11(tt(), L)) -> mark(U12(tt(), L))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [8]
zeros() = [8]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [8]
0() = [9]
U11(x1, x2) = [1] x1 + [1] x2 + [4]
tt() = [4]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [8]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [12]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [1] x1 + [1] x2 + [2]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [1]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [1] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(zeros()) -> mark(cons(0(), zeros()))}
and weakly orienting the rules
{ active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, U12^#(ok(X1), ok(X2)) -> c_26(U12^#(X1, X2))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active^#(U12(X1, X2)) -> c_7(U12^#(active(X1), X2))
, U12^#(mark(X1), X2) -> c_12(U12^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(zeros()) -> mark(cons(0(), zeros()))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [1]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [1]
s(x1) = [1] x1 + [1]
length(x1) = [1] x1 + [0]
nil() = [3]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [15]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [1] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, U12^#(ok(X1), ok(X2)) -> c_26(U12^#(X1, X2))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active^#(U12(X1, X2)) -> c_7(U12^#(active(X1), X2))
, U12^#(mark(X1), X2) -> c_12(U12^#(X1, X2))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, U12^#(ok(X1), ok(X2)) -> c_26(U12^#(X1, X2))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active^#(U12(X1, X2)) -> c_7(U12^#(active(X1), X2))
, U12^#(mark(X1), X2) -> c_12(U12^#(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ zeros_0() -> 2
, mark_0(2) -> 2
, 0_0() -> 2
, tt_0() -> 2
, nil_0() -> 2
, ok_0(2) -> 2
, active^#_0(2) -> 1
, U12^#_0(2, 2) -> 1
, c_12_0(1) -> 1
, c_26_0(1) -> 1}
4) { active^#(s(X)) -> c_8(s^#(active(X)))
, s^#(ok(X)) -> c_27(s^#(X))
, s^#(mark(X)) -> c_13(s^#(X))}
The usable rules for this path are the following:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))
, active^#(s(X)) -> c_8(s^#(active(X)))
, s^#(ok(X)) -> c_27(s^#(X))
, s^#(mark(X)) -> c_13(s^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s^#(ok(X)) -> c_27(s^#(X))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s^#(ok(X)) -> c_27(s^#(X))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [10]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [1] x1 + [1]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{s^#(mark(X)) -> c_13(s^#(X))}
and weakly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s^#(ok(X)) -> c_27(s^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{s^#(mark(X)) -> c_13(s^#(X))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [1] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active^#(s(X)) -> c_8(s^#(active(X)))}
and weakly orienting the rules
{ s^#(mark(X)) -> c_13(s^#(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s^#(ok(X)) -> c_27(s^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(s(X)) -> c_8(s^#(active(X)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [4]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [1] x1 + [1]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [1] x1 + [1]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(length(nil())) -> mark(0())}
and weakly orienting the rules
{ active^#(s(X)) -> c_8(s^#(active(X)))
, s^#(mark(X)) -> c_13(s^#(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s^#(ok(X)) -> c_27(s^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(length(nil())) -> mark(0())}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [8]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [1] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(length(cons(N, L))) -> mark(U11(tt(), L))}
and weakly orienting the rules
{ active(length(nil())) -> mark(0())
, active^#(s(X)) -> c_8(s^#(active(X)))
, s^#(mark(X)) -> c_13(s^#(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s^#(ok(X)) -> c_27(s^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(length(cons(N, L))) -> mark(U11(tt(), L))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [4]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [4]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [8]
s(x1) = [1] x1 + [12]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [4]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [12]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [12]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [1] x1 + [1]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [1] x1 + [4]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U11(tt(), L)) -> mark(U12(tt(), L))}
and weakly orienting the rules
{ active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, active^#(s(X)) -> c_8(s^#(active(X)))
, s^#(mark(X)) -> c_13(s^#(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s^#(ok(X)) -> c_27(s^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U11(tt(), L)) -> mark(U12(tt(), L))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [2]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [15]
0() = [8]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [3]
nil() = [12]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [8]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [3]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [1] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U12(tt(), L)) -> mark(s(length(L)))}
and weakly orienting the rules
{ active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, active^#(s(X)) -> c_8(s^#(active(X)))
, s^#(mark(X)) -> c_13(s^#(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s^#(ok(X)) -> c_27(s^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U12(tt(), L)) -> mark(s(length(L)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [9]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [3]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [2]
U12(x1, x2) = [1] x1 + [1] x2 + [5]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [4]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [1] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, active^#(s(X)) -> c_8(s^#(active(X)))
, s^#(mark(X)) -> c_13(s^#(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s^#(ok(X)) -> c_27(s^#(X))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, active^#(s(X)) -> c_8(s^#(active(X)))
, s^#(mark(X)) -> c_13(s^#(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s^#(ok(X)) -> c_27(s^#(X))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ zeros_0() -> 2
, mark_0(2) -> 3
, mark_0(3) -> 3
, mark_0(5) -> 3
, mark_0(7) -> 3
, mark_0(11) -> 3
, mark_0(13) -> 3
, 0_0() -> 5
, tt_0() -> 7
, nil_0() -> 11
, ok_0(2) -> 13
, ok_0(3) -> 13
, ok_0(5) -> 13
, ok_0(7) -> 13
, ok_0(11) -> 13
, ok_0(13) -> 13
, active^#_0(2) -> 15
, active^#_0(3) -> 15
, active^#_0(5) -> 15
, active^#_0(7) -> 15
, active^#_0(11) -> 15
, active^#_0(13) -> 15
, s^#_0(2) -> 21
, s^#_0(3) -> 21
, s^#_0(5) -> 21
, s^#_0(7) -> 21
, s^#_0(11) -> 21
, s^#_0(13) -> 21
, c_13_0(21) -> 21
, c_27_0(21) -> 21}
5) { active^#(cons(X1, X2)) -> c_5(cons^#(active(X1), X2))
, cons^#(ok(X1), ok(X2)) -> c_24(cons^#(X1, X2))
, cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))}
The usable rules for this path are the following:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))
, active^#(cons(X1, X2)) -> c_5(cons^#(active(X1), X2))
, cons^#(ok(X1), ok(X2)) -> c_24(cons^#(X1, X2))
, cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [1] x1 + [1]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U12(tt(), L)) -> mark(s(length(L)))}
and weakly orienting the rules
{cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U12(tt(), L)) -> mark(s(length(L)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [8]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [3]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [1] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, cons^#(ok(X1), ok(X2)) -> c_24(cons^#(X1, X2))}
and weakly orienting the rules
{ active(U12(tt(), L)) -> mark(s(length(L)))
, cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, cons^#(ok(X1), ok(X2)) -> c_24(cons^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [9]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [5]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [4]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [2]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [1] x1 + [1]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active^#(cons(X1, X2)) -> c_5(cons^#(active(X1), X2))}
and weakly orienting the rules
{ active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, cons^#(ok(X1), ok(X2)) -> c_24(cons^#(X1, X2))
, active(U12(tt(), L)) -> mark(s(length(L)))
, cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(cons(X1, X2)) -> c_5(cons^#(active(X1), X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [1] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U11(tt(), L)) -> mark(U12(tt(), L))}
and weakly orienting the rules
{ active^#(cons(X1, X2)) -> c_5(cons^#(active(X1), X2))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, cons^#(ok(X1), ok(X2)) -> c_24(cons^#(X1, X2))
, active(U12(tt(), L)) -> mark(s(length(L)))
, cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U11(tt(), L)) -> mark(U12(tt(), L))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [8]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [2]
tt() = [6]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [3]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [1] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(zeros()) -> mark(cons(0(), zeros()))}
and weakly orienting the rules
{ active(U11(tt(), L)) -> mark(U12(tt(), L))
, active^#(cons(X1, X2)) -> c_5(cons^#(active(X1), X2))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, cons^#(ok(X1), ok(X2)) -> c_24(cons^#(X1, X2))
, active(U12(tt(), L)) -> mark(s(length(L)))
, cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(zeros()) -> mark(cons(0(), zeros()))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [4]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [1] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active^#(cons(X1, X2)) -> c_5(cons^#(active(X1), X2))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, cons^#(ok(X1), ok(X2)) -> c_24(cons^#(X1, X2))
, active(U12(tt(), L)) -> mark(s(length(L)))
, cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active^#(cons(X1, X2)) -> c_5(cons^#(active(X1), X2))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, cons^#(ok(X1), ok(X2)) -> c_24(cons^#(X1, X2))
, active(U12(tt(), L)) -> mark(s(length(L)))
, cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ zeros_0() -> 2
, mark_0(2) -> 2
, 0_0() -> 2
, tt_0() -> 2
, nil_0() -> 2
, ok_0(2) -> 2
, active^#_0(2) -> 1
, cons^#_0(2, 2) -> 1
, c_10_0(1) -> 1
, c_24_0(1) -> 1}
6) {active^#(U11(X1, X2)) -> c_6(U11^#(active(X1), X2))}
The usable rules for this path are the following:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))
, active^#(U11(X1, X2)) -> c_6(U11^#(active(X1), X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(U11(X1, X2)) -> c_6(U11^#(active(X1), X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(U11(X1, X2)) -> c_6(U11^#(active(X1), X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
and weakly orienting the rules
{active^#(U11(X1, X2)) -> c_6(U11^#(active(X1), X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [12]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [8]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(length(nil())) -> mark(0())}
and weakly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, active^#(U11(X1, X2)) -> c_6(U11^#(active(X1), X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(length(nil())) -> mark(0())}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [4]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [7]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(length(cons(N, L))) -> mark(U11(tt(), L))}
and weakly orienting the rules
{ active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, active^#(U11(X1, X2)) -> c_6(U11^#(active(X1), X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(length(cons(N, L))) -> mark(U11(tt(), L))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [2]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [13]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [1] x1 + [1] x2 + [7]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U12(tt(), L)) -> mark(s(length(L)))}
and weakly orienting the rules
{ active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, active^#(U11(X1, X2)) -> c_6(U11^#(active(X1), X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U12(tt(), L)) -> mark(s(length(L)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [7]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [9]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(zeros()) -> mark(cons(0(), zeros()))}
and weakly orienting the rules
{ active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, active^#(U11(X1, X2)) -> c_6(U11^#(active(X1), X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(zeros()) -> mark(cons(0(), zeros()))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [4]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [15]
s(x1) = [1] x1 + [13]
length(x1) = [1] x1 + [1]
nil() = [9]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U11(tt(), L)) -> mark(U12(tt(), L))}
and weakly orienting the rules
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, active^#(U11(X1, X2)) -> c_6(U11^#(active(X1), X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U11(tt(), L)) -> mark(U12(tt(), L))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [15]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(zeros()) -> mark(cons(0(), zeros()))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, active^#(U11(X1, X2)) -> c_6(U11^#(active(X1), X2))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(zeros()) -> mark(cons(0(), zeros()))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, active^#(U11(X1, X2)) -> c_6(U11^#(active(X1), X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ zeros_0() -> 2
, mark_0(2) -> 3
, mark_0(3) -> 3
, mark_0(5) -> 3
, mark_0(7) -> 3
, mark_0(11) -> 3
, mark_0(13) -> 3
, 0_0() -> 5
, tt_0() -> 7
, nil_0() -> 11
, ok_0(2) -> 13
, ok_0(3) -> 13
, ok_0(5) -> 13
, ok_0(7) -> 13
, ok_0(11) -> 13
, ok_0(13) -> 13
, active^#_0(2) -> 15
, active^#_0(3) -> 15
, active^#_0(5) -> 15
, active^#_0(7) -> 15
, active^#_0(11) -> 15
, active^#_0(13) -> 15
, U11^#_0(2, 2) -> 24
, U11^#_0(2, 3) -> 24
, U11^#_0(2, 5) -> 24
, U11^#_0(2, 7) -> 24
, U11^#_0(2, 11) -> 24
, U11^#_0(2, 13) -> 24
, U11^#_0(3, 2) -> 24
, U11^#_0(3, 3) -> 24
, U11^#_0(3, 5) -> 24
, U11^#_0(3, 7) -> 24
, U11^#_0(3, 11) -> 24
, U11^#_0(3, 13) -> 24
, U11^#_0(5, 2) -> 24
, U11^#_0(5, 3) -> 24
, U11^#_0(5, 5) -> 24
, U11^#_0(5, 7) -> 24
, U11^#_0(5, 11) -> 24
, U11^#_0(5, 13) -> 24
, U11^#_0(7, 2) -> 24
, U11^#_0(7, 3) -> 24
, U11^#_0(7, 5) -> 24
, U11^#_0(7, 7) -> 24
, U11^#_0(7, 11) -> 24
, U11^#_0(7, 13) -> 24
, U11^#_0(11, 2) -> 24
, U11^#_0(11, 3) -> 24
, U11^#_0(11, 5) -> 24
, U11^#_0(11, 7) -> 24
, U11^#_0(11, 11) -> 24
, U11^#_0(11, 13) -> 24
, U11^#_0(13, 2) -> 24
, U11^#_0(13, 3) -> 24
, U11^#_0(13, 5) -> 24
, U11^#_0(13, 7) -> 24
, U11^#_0(13, 11) -> 24
, U11^#_0(13, 13) -> 24}
7) {active^#(cons(X1, X2)) -> c_5(cons^#(active(X1), X2))}
The usable rules for this path are the following:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))
, active^#(cons(X1, X2)) -> c_5(cons^#(active(X1), X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(cons(X1, X2)) -> c_5(cons^#(active(X1), X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(cons(X1, X2)) -> c_5(cons^#(active(X1), X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [5]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
and weakly orienting the rules
{active^#(cons(X1, X2)) -> c_5(cons^#(active(X1), X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(length(nil())) -> mark(0())}
and weakly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, active^#(cons(X1, X2)) -> c_5(cons^#(active(X1), X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(length(nil())) -> mark(0())}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [4]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(length(cons(N, L))) -> mark(U11(tt(), L))}
and weakly orienting the rules
{ active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, active^#(cons(X1, X2)) -> c_5(cons^#(active(X1), X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(length(cons(N, L))) -> mark(U11(tt(), L))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [2]
nil() = [14]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [5]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U12(tt(), L)) -> mark(s(length(L)))}
and weakly orienting the rules
{ active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, active^#(cons(X1, X2)) -> c_5(cons^#(active(X1), X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U12(tt(), L)) -> mark(s(length(L)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [7]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [5]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U11(tt(), L)) -> mark(U12(tt(), L))}
and weakly orienting the rules
{ active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, active^#(cons(X1, X2)) -> c_5(cons^#(active(X1), X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U11(tt(), L)) -> mark(U12(tt(), L))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [1]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [3]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [4]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [15]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [8]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [1] x1 + [8]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(zeros()) -> mark(cons(0(), zeros()))}
and weakly orienting the rules
{ active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, active^#(cons(X1, X2)) -> c_5(cons^#(active(X1), X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(zeros()) -> mark(cons(0(), zeros()))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [8]
zeros() = [0]
mark(x1) = [1] x1 + [4]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [8]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [13]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, active^#(cons(X1, X2)) -> c_5(cons^#(active(X1), X2))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, active^#(cons(X1, X2)) -> c_5(cons^#(active(X1), X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ zeros_0() -> 2
, mark_0(2) -> 2
, 0_0() -> 2
, tt_0() -> 2
, nil_0() -> 2
, ok_0(2) -> 2
, active^#_0(2) -> 1
, cons^#_0(2, 2) -> 1}
8) { active^#(U11(X1, X2)) -> c_6(U11^#(active(X1), X2))
, U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))
, U11^#(mark(X1), X2) -> c_11(U11^#(X1, X2))}
The usable rules for this path are the following:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))
, active^#(U11(X1, X2)) -> c_6(U11^#(active(X1), X2))
, U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))
, U11^#(mark(X1), X2) -> c_11(U11^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{U11^#(mark(X1), X2) -> c_11(U11^#(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{U11^#(mark(X1), X2) -> c_11(U11^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [1] x1 + [1] x2 + [8]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [1] x1 + [1]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U12(tt(), L)) -> mark(s(length(L)))}
and weakly orienting the rules
{U11^#(mark(X1), X2) -> c_11(U11^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U12(tt(), L)) -> mark(s(length(L)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [8]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [1] x1 + [1]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))}
and weakly orienting the rules
{ active(U12(tt(), L)) -> mark(s(length(L)))
, U11^#(mark(X1), X2) -> c_11(U11^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [9]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [5]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [4]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [1] x1 + [1]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active^#(U11(X1, X2)) -> c_6(U11^#(active(X1), X2))}
and weakly orienting the rules
{ active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))
, active(U12(tt(), L)) -> mark(s(length(L)))
, U11^#(mark(X1), X2) -> c_11(U11^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(U11(X1, X2)) -> c_6(U11^#(active(X1), X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [1] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))}
and weakly orienting the rules
{ active^#(U11(X1, X2)) -> c_6(U11^#(active(X1), X2))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))
, active(U12(tt(), L)) -> mark(s(length(L)))
, U11^#(mark(X1), X2) -> c_11(U11^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [1]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [1]
nil() = [8]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [4]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [1] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active^#(U11(X1, X2)) -> c_6(U11^#(active(X1), X2))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))
, active(U12(tt(), L)) -> mark(s(length(L)))
, U11^#(mark(X1), X2) -> c_11(U11^#(X1, X2))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active^#(U11(X1, X2)) -> c_6(U11^#(active(X1), X2))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))
, active(U12(tt(), L)) -> mark(s(length(L)))
, U11^#(mark(X1), X2) -> c_11(U11^#(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ zeros_0() -> 2
, mark_0(2) -> 2
, 0_0() -> 2
, tt_0() -> 2
, nil_0() -> 2
, ok_0(2) -> 2
, active^#_0(2) -> 1
, U11^#_0(2, 2) -> 1
, c_11_0(1) -> 1
, c_25_0(1) -> 1}
9) {active^#(length(X)) -> c_9(length^#(active(X)))}
The usable rules for this path are the following:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))
, active^#(length(X)) -> c_9(length^#(active(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
length^#(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active^#(length(X)) -> c_9(length^#(active(X)))}
and weakly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(length(X)) -> c_9(length^#(active(X)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
length^#(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(length(nil())) -> mark(0())}
and weakly orienting the rules
{ active^#(length(X)) -> c_9(length^#(active(X)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(length(nil())) -> mark(0())}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [4]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [3]
length^#(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(length(cons(N, L))) -> mark(U11(tt(), L))}
and weakly orienting the rules
{ active(length(nil())) -> mark(0())
, active^#(length(X)) -> c_9(length^#(active(X)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(length(cons(N, L))) -> mark(U11(tt(), L))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [2]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
length^#(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U12(tt(), L)) -> mark(s(length(L)))}
and weakly orienting the rules
{ active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, active^#(length(X)) -> c_9(length^#(active(X)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U12(tt(), L)) -> mark(s(length(L)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [7]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
length^#(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))}
and weakly orienting the rules
{ active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, active^#(length(X)) -> c_9(length^#(active(X)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [2]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
length^#(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, active^#(length(X)) -> c_9(length^#(active(X)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, active^#(length(X)) -> c_9(length^#(active(X)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ zeros_0() -> 2
, mark_0(2) -> 3
, mark_0(3) -> 3
, mark_0(5) -> 3
, mark_0(7) -> 3
, mark_0(11) -> 3
, mark_0(13) -> 3
, 0_0() -> 5
, tt_0() -> 7
, nil_0() -> 11
, ok_0(2) -> 13
, ok_0(3) -> 13
, ok_0(5) -> 13
, ok_0(7) -> 13
, ok_0(11) -> 13
, ok_0(13) -> 13
, active^#_0(2) -> 15
, active^#_0(3) -> 15
, active^#_0(5) -> 15
, active^#_0(7) -> 15
, active^#_0(11) -> 15
, active^#_0(13) -> 15
, length^#_0(2) -> 30
, length^#_0(3) -> 30
, length^#_0(5) -> 30
, length^#_0(7) -> 30
, length^#_0(11) -> 30
, length^#_0(13) -> 30}
10)
{active^#(s(X)) -> c_8(s^#(active(X)))}
The usable rules for this path are the following:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))
, active^#(s(X)) -> c_8(s^#(active(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active^#(s(X)) -> c_8(s^#(active(X)))}
and weakly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(s(X)) -> c_8(s^#(active(X)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(length(nil())) -> mark(0())}
and weakly orienting the rules
{ active^#(s(X)) -> c_8(s^#(active(X)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(length(nil())) -> mark(0())}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [1]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(length(cons(N, L))) -> mark(U11(tt(), L))}
and weakly orienting the rules
{ active(length(nil())) -> mark(0())
, active^#(s(X)) -> c_8(s^#(active(X)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(length(cons(N, L))) -> mark(U11(tt(), L))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [14]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [2]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [1] x1 + [1]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U12(tt(), L)) -> mark(s(length(L)))}
and weakly orienting the rules
{ active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, active^#(s(X)) -> c_8(s^#(active(X)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U12(tt(), L)) -> mark(s(length(L)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [7]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))}
and weakly orienting the rules
{ active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, active^#(s(X)) -> c_8(s^#(active(X)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [4]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, active^#(s(X)) -> c_8(s^#(active(X)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, active^#(s(X)) -> c_8(s^#(active(X)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ zeros_0() -> 2
, mark_0(2) -> 3
, mark_0(3) -> 3
, mark_0(5) -> 3
, mark_0(7) -> 3
, mark_0(11) -> 3
, mark_0(13) -> 3
, 0_0() -> 5
, tt_0() -> 7
, nil_0() -> 11
, ok_0(2) -> 13
, ok_0(3) -> 13
, ok_0(5) -> 13
, ok_0(7) -> 13
, ok_0(11) -> 13
, ok_0(13) -> 13
, active^#_0(2) -> 15
, active^#_0(3) -> 15
, active^#_0(5) -> 15
, active^#_0(7) -> 15
, active^#_0(11) -> 15
, active^#_0(13) -> 15
, s^#_0(2) -> 21
, s^#_0(3) -> 21
, s^#_0(5) -> 21
, s^#_0(7) -> 21
, s^#_0(11) -> 21
, s^#_0(13) -> 21}
11)
{active^#(U12(X1, X2)) -> c_7(U12^#(active(X1), X2))}
The usable rules for this path are the following:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(nil())) -> mark(0())
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))
, active^#(U12(X1, X2)) -> c_7(U12^#(active(X1), X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, active^#(U12(X1, X2)) -> c_7(U12^#(active(X1), X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, active^#(U12(X1, X2)) -> c_7(U12^#(active(X1), X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [2]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [3]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(length(nil())) -> mark(0())}
and weakly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, active^#(U12(X1, X2)) -> c_7(U12^#(active(X1), X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(length(nil())) -> mark(0())}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [4]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(length(cons(N, L))) -> mark(U11(tt(), L))}
and weakly orienting the rules
{ active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, active^#(U12(X1, X2)) -> c_7(U12^#(active(X1), X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(length(cons(N, L))) -> mark(U11(tt(), L))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [2]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [13]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [1] x1 + [1] x2 + [7]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U12(tt(), L)) -> mark(s(length(L)))}
and weakly orienting the rules
{ active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, active^#(U12(X1, X2)) -> c_7(U12^#(active(X1), X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U12(tt(), L)) -> mark(s(length(L)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [7]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [9]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(zeros()) -> mark(cons(0(), zeros()))}
and weakly orienting the rules
{ active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, active^#(U12(X1, X2)) -> c_7(U12^#(active(X1), X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(zeros()) -> mark(cons(0(), zeros()))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [1]
U12(x1, x2) = [1] x1 + [1] x2 + [12]
s(x1) = [1] x1 + [13]
length(x1) = [1] x1 + [1]
nil() = [9]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U11(tt(), L)) -> mark(U12(tt(), L))}
and weakly orienting the rules
{ active(zeros()) -> mark(cons(0(), zeros()))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, active^#(U12(X1, X2)) -> c_7(U12^#(active(X1), X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U11(tt(), L)) -> mark(U12(tt(), L))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [15]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(zeros()) -> mark(cons(0(), zeros()))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, active^#(U12(X1, X2)) -> c_7(U12^#(active(X1), X2))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ active(cons(X1, X2)) -> cons(active(X1), X2)
, active(U11(X1, X2)) -> U11(active(X1), X2)
, active(U12(X1, X2)) -> U12(active(X1), X2)
, active(s(X)) -> s(active(X))
, active(length(X)) -> length(active(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(U11(tt(), L)) -> mark(U12(tt(), L))
, active(zeros()) -> mark(cons(0(), zeros()))
, active(U12(tt(), L)) -> mark(s(length(L)))
, active(length(cons(N, L))) -> mark(U11(tt(), L))
, active(length(nil())) -> mark(0())
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, active^#(U12(X1, X2)) -> c_7(U12^#(active(X1), X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ zeros_0() -> 2
, mark_0(2) -> 3
, mark_0(3) -> 3
, mark_0(5) -> 3
, mark_0(7) -> 3
, mark_0(11) -> 3
, mark_0(13) -> 3
, 0_0() -> 5
, tt_0() -> 7
, nil_0() -> 11
, ok_0(2) -> 13
, ok_0(3) -> 13
, ok_0(5) -> 13
, ok_0(7) -> 13
, ok_0(11) -> 13
, ok_0(13) -> 13
, active^#_0(2) -> 15
, active^#_0(3) -> 15
, active^#_0(5) -> 15
, active^#_0(7) -> 15
, active^#_0(11) -> 15
, active^#_0(13) -> 15
, U12^#_0(2, 2) -> 19
, U12^#_0(2, 3) -> 19
, U12^#_0(2, 5) -> 19
, U12^#_0(2, 7) -> 19
, U12^#_0(2, 11) -> 19
, U12^#_0(2, 13) -> 19
, U12^#_0(3, 2) -> 19
, U12^#_0(3, 3) -> 19
, U12^#_0(3, 5) -> 19
, U12^#_0(3, 7) -> 19
, U12^#_0(3, 11) -> 19
, U12^#_0(3, 13) -> 19
, U12^#_0(5, 2) -> 19
, U12^#_0(5, 3) -> 19
, U12^#_0(5, 5) -> 19
, U12^#_0(5, 7) -> 19
, U12^#_0(5, 11) -> 19
, U12^#_0(5, 13) -> 19
, U12^#_0(7, 2) -> 19
, U12^#_0(7, 3) -> 19
, U12^#_0(7, 5) -> 19
, U12^#_0(7, 7) -> 19
, U12^#_0(7, 11) -> 19
, U12^#_0(7, 13) -> 19
, U12^#_0(11, 2) -> 19
, U12^#_0(11, 3) -> 19
, U12^#_0(11, 5) -> 19
, U12^#_0(11, 7) -> 19
, U12^#_0(11, 11) -> 19
, U12^#_0(11, 13) -> 19
, U12^#_0(13, 2) -> 19
, U12^#_0(13, 3) -> 19
, U12^#_0(13, 5) -> 19
, U12^#_0(13, 7) -> 19
, U12^#_0(13, 11) -> 19
, U12^#_0(13, 13) -> 19}
12)
{ proper^#(s(X)) -> c_21(s^#(proper(X)))
, s^#(ok(X)) -> c_27(s^#(X))
, s^#(mark(X)) -> c_13(s^#(X))}
The usable rules for this path are the following:
{ proper(zeros()) -> ok(zeros())
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, proper(nil()) -> ok(nil())
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ proper(zeros()) -> ok(zeros())
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, proper(nil()) -> ok(nil())
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))
, proper^#(s(X)) -> c_21(s^#(proper(X)))
, s^#(ok(X)) -> c_27(s^#(X))
, s^#(mark(X)) -> c_13(s^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [8]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [1] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [1] x1 + [1]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{s^#(ok(X)) -> c_27(s^#(X))}
and weakly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{s^#(ok(X)) -> c_27(s^#(X))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [1]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [1] x1 + [3]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [1] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(s(X)) -> c_21(s^#(proper(X)))}
and weakly orienting the rules
{ s^#(ok(X)) -> c_27(s^#(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(s(X)) -> c_21(s^#(proper(X)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [1]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [1] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [1] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{s^#(mark(X)) -> c_13(s^#(X))}
and weakly orienting the rules
{ proper^#(s(X)) -> c_21(s^#(proper(X)))
, s^#(ok(X)) -> c_27(s^#(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{s^#(mark(X)) -> c_13(s^#(X))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [8]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [8]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [1] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [1] x1 + [1]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
and weakly orienting the rules
{ s^#(mark(X)) -> c_13(s^#(X))
, proper^#(s(X)) -> c_21(s^#(proper(X)))
, s^#(ok(X)) -> c_27(s^#(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [4]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [4]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [13]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [1] x1 + [7]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [1] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())
, s^#(mark(X)) -> c_13(s^#(X))
, proper^#(s(X)) -> c_21(s^#(proper(X)))
, s^#(ok(X)) -> c_27(s^#(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())
, s^#(mark(X)) -> c_13(s^#(X))
, proper^#(s(X)) -> c_21(s^#(proper(X)))
, s^#(ok(X)) -> c_27(s^#(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ zeros_0() -> 2
, mark_0(2) -> 3
, mark_0(3) -> 3
, mark_0(5) -> 3
, mark_0(7) -> 3
, mark_0(11) -> 3
, mark_0(13) -> 3
, 0_0() -> 5
, tt_0() -> 7
, nil_0() -> 11
, ok_0(2) -> 13
, ok_0(3) -> 13
, ok_0(5) -> 13
, ok_0(7) -> 13
, ok_0(11) -> 13
, ok_0(13) -> 13
, s^#_0(2) -> 21
, s^#_0(3) -> 21
, s^#_0(5) -> 21
, s^#_0(7) -> 21
, s^#_0(11) -> 21
, s^#_0(13) -> 21
, c_13_0(21) -> 21
, proper^#_0(2) -> 36
, proper^#_0(3) -> 36
, proper^#_0(5) -> 36
, proper^#_0(7) -> 36
, proper^#_0(11) -> 36
, proper^#_0(13) -> 36
, c_27_0(21) -> 21}
13)
{ proper^#(U11(X1, X2)) -> c_18(U11^#(proper(X1), proper(X2)))
, U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))
, U11^#(mark(X1), X2) -> c_11(U11^#(X1, X2))}
The usable rules for this path are the following:
{ proper(zeros()) -> ok(zeros())
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, proper(nil()) -> ok(nil())
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ proper(zeros()) -> ok(zeros())
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, proper(nil()) -> ok(nil())
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))
, proper^#(U11(X1, X2)) -> c_18(U11^#(proper(X1), proper(X2)))
, U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))
, U11^#(mark(X1), X2) -> c_11(U11^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [4]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [1] x1 + [1]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [1] x1 + [1]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{U11^#(mark(X1), X2) -> c_11(U11^#(X1, X2))}
and weakly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{U11^#(mark(X1), X2) -> c_11(U11^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [4]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [8]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [1] x1 + [1] x2 + [4]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [1] x1 + [1]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [1] x1 + [1]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(U11(X1, X2)) -> c_18(U11^#(proper(X1), proper(X2)))}
and weakly orienting the rules
{ U11^#(mark(X1), X2) -> c_11(U11^#(X1, X2))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(U11(X1, X2)) -> c_18(U11^#(proper(X1), proper(X2)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [1] x1 + [1]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [1] x1 + [1]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
and weakly orienting the rules
{ proper^#(U11(X1, X2)) -> c_18(U11^#(proper(X1), proper(X2)))
, U11^#(mark(X1), X2) -> c_11(U11^#(X1, X2))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [3]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [8]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [1]
length(x1) = [1] x1 + [8]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [5]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [1] x1 + [2]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [1] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())
, proper^#(U11(X1, X2)) -> c_18(U11^#(proper(X1), proper(X2)))
, U11^#(mark(X1), X2) -> c_11(U11^#(X1, X2))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())
, proper^#(U11(X1, X2)) -> c_18(U11^#(proper(X1), proper(X2)))
, U11^#(mark(X1), X2) -> c_11(U11^#(X1, X2))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ zeros_0() -> 2
, mark_0(2) -> 2
, 0_0() -> 2
, tt_0() -> 2
, nil_0() -> 2
, ok_0(2) -> 2
, U11^#_0(2, 2) -> 1
, c_11_0(1) -> 1
, proper^#_0(2) -> 1
, c_25_0(1) -> 1}
14)
{ proper^#(cons(X1, X2)) -> c_16(cons^#(proper(X1), proper(X2)))
, cons^#(ok(X1), ok(X2)) -> c_24(cons^#(X1, X2))
, cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))}
The usable rules for this path are the following:
{ proper(zeros()) -> ok(zeros())
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, proper(nil()) -> ok(nil())
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ proper(zeros()) -> ok(zeros())
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, proper(nil()) -> ok(nil())
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))
, proper^#(cons(X1, X2)) -> c_16(cons^#(proper(X1), proper(X2)))
, cons^#(ok(X1), ok(X2)) -> c_24(cons^#(X1, X2))
, cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, cons^#(ok(X1), ok(X2)) -> c_24(cons^#(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, cons^#(ok(X1), ok(X2)) -> c_24(cons^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [4]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_15() = [0]
c_16(x1) = [1] x1 + [1]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [1] x1 + [1]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))}
and weakly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, cons^#(ok(X1), ok(X2)) -> c_24(cons^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [4]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [8]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [4]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_15() = [0]
c_16(x1) = [1] x1 + [1]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [1] x1 + [1]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(cons(X1, X2)) -> c_16(cons^#(proper(X1), proper(X2)))}
and weakly orienting the rules
{ cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, cons^#(ok(X1), ok(X2)) -> c_24(cons^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(cons(X1, X2)) -> c_16(cons^#(proper(X1), proper(X2)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_15() = [0]
c_16(x1) = [1] x1 + [1]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [1] x1 + [1]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
and weakly orienting the rules
{ proper^#(cons(X1, X2)) -> c_16(cons^#(proper(X1), proper(X2)))
, cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, cons^#(ok(X1), ok(X2)) -> c_24(cons^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [2]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [2]
s(x1) = [1] x1 + [4]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [4]
c_15() = [0]
c_16(x1) = [1] x1 + [1]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [1] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())
, proper^#(cons(X1, X2)) -> c_16(cons^#(proper(X1), proper(X2)))
, cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, cons^#(ok(X1), ok(X2)) -> c_24(cons^#(X1, X2))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())
, proper^#(cons(X1, X2)) -> c_16(cons^#(proper(X1), proper(X2)))
, cons^#(mark(X1), X2) -> c_10(cons^#(X1, X2))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, cons^#(ok(X1), ok(X2)) -> c_24(cons^#(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ zeros_0() -> 2
, mark_0(2) -> 2
, 0_0() -> 2
, tt_0() -> 2
, nil_0() -> 2
, ok_0(2) -> 2
, cons^#_0(2, 2) -> 1
, c_10_0(1) -> 1
, proper^#_0(2) -> 1
, c_24_0(1) -> 1}
15)
{ proper^#(length(X)) -> c_22(length^#(proper(X)))
, length^#(ok(X)) -> c_28(length^#(X))
, length^#(mark(X)) -> c_14(length^#(X))}
The usable rules for this path are the following:
{ proper(zeros()) -> ok(zeros())
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, proper(nil()) -> ok(nil())
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ proper(zeros()) -> ok(zeros())
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, proper(nil()) -> ok(nil())
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))
, proper^#(length(X)) -> c_22(length^#(proper(X)))
, length^#(ok(X)) -> c_28(length^#(X))
, length^#(mark(X)) -> c_14(length^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [1]
proper^#(x1) = [1] x1 + [1]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [1] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [1] x1 + [1]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ length^#(ok(X)) -> c_28(length^#(X))
, length^#(mark(X)) -> c_14(length^#(X))}
and weakly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ length^#(ok(X)) -> c_28(length^#(X))
, length^#(mark(X)) -> c_14(length^#(X))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [9]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [1] x1 + [1]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(length(X)) -> c_22(length^#(proper(X)))}
and weakly orienting the rules
{ length^#(ok(X)) -> c_28(length^#(X))
, length^#(mark(X)) -> c_14(length^#(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(length(X)) -> c_22(length^#(proper(X)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [4]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [1] x1 + [3]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
and weakly orienting the rules
{ proper^#(length(X)) -> c_22(length^#(proper(X)))
, length^#(ok(X)) -> c_28(length^#(X))
, length^#(mark(X)) -> c_14(length^#(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [1]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [0]
proper^#(x1) = [1] x1 + [2]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [1] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())
, proper^#(length(X)) -> c_22(length^#(proper(X)))
, length^#(ok(X)) -> c_28(length^#(X))
, length^#(mark(X)) -> c_14(length^#(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())
, proper^#(length(X)) -> c_22(length^#(proper(X)))
, length^#(ok(X)) -> c_28(length^#(X))
, length^#(mark(X)) -> c_14(length^#(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ zeros_0() -> 2
, mark_0(2) -> 3
, mark_0(3) -> 3
, mark_0(5) -> 3
, mark_0(7) -> 3
, mark_0(11) -> 3
, mark_0(13) -> 3
, 0_0() -> 5
, tt_0() -> 7
, nil_0() -> 11
, ok_0(2) -> 13
, ok_0(3) -> 13
, ok_0(5) -> 13
, ok_0(7) -> 13
, ok_0(11) -> 13
, ok_0(13) -> 13
, length^#_0(2) -> 30
, length^#_0(3) -> 30
, length^#_0(5) -> 30
, length^#_0(7) -> 30
, length^#_0(11) -> 30
, length^#_0(13) -> 30
, c_14_0(30) -> 30
, proper^#_0(2) -> 36
, proper^#_0(3) -> 36
, proper^#_0(5) -> 36
, proper^#_0(7) -> 36
, proper^#_0(11) -> 36
, proper^#_0(13) -> 36
, c_28_0(30) -> 30}
16)
{ proper^#(U12(X1, X2)) -> c_20(U12^#(proper(X1), proper(X2)))
, U12^#(ok(X1), ok(X2)) -> c_26(U12^#(X1, X2))
, U12^#(mark(X1), X2) -> c_12(U12^#(X1, X2))}
The usable rules for this path are the following:
{ proper(zeros()) -> ok(zeros())
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, proper(nil()) -> ok(nil())
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ proper(zeros()) -> ok(zeros())
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, proper(nil()) -> ok(nil())
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))
, proper^#(U12(X1, X2)) -> c_20(U12^#(proper(X1), proper(X2)))
, U12^#(ok(X1), ok(X2)) -> c_26(U12^#(X1, X2))
, U12^#(mark(X1), X2) -> c_12(U12^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, proper^#(U12(X1, X2)) -> c_20(U12^#(proper(X1), proper(X2)))
, U12^#(ok(X1), ok(X2)) -> c_26(U12^#(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, proper^#(U12(X1, X2)) -> c_20(U12^#(proper(X1), proper(X2)))
, U12^#(ok(X1), ok(X2)) -> c_26(U12^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [4]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [1] x1 + [1]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [1] x1 + [1]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())
, U12^#(mark(X1), X2) -> c_12(U12^#(X1, X2))}
and weakly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, proper^#(U12(X1, X2)) -> c_20(U12^#(proper(X1), proper(X2)))
, U12^#(ok(X1), ok(X2)) -> c_26(U12^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())
, U12^#(mark(X1), X2) -> c_12(U12^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [4]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [15]
s(x1) = [1] x1 + [1]
length(x1) = [1] x1 + [4]
nil() = [8]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [1] x1 + [1] x2 + [4]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [13]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [1] x1 + [12]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [1] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())
, U12^#(mark(X1), X2) -> c_12(U12^#(X1, X2))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, proper^#(U12(X1, X2)) -> c_20(U12^#(proper(X1), proper(X2)))
, U12^#(ok(X1), ok(X2)) -> c_26(U12^#(X1, X2))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())
, U12^#(mark(X1), X2) -> c_12(U12^#(X1, X2))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, proper^#(U12(X1, X2)) -> c_20(U12^#(proper(X1), proper(X2)))
, U12^#(ok(X1), ok(X2)) -> c_26(U12^#(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ zeros_0() -> 2
, mark_0(2) -> 2
, 0_0() -> 2
, tt_0() -> 2
, nil_0() -> 2
, ok_0(2) -> 2
, U12^#_0(2, 2) -> 1
, c_12_0(1) -> 1
, proper^#_0(2) -> 1
, c_26_0(1) -> 1}
17)
{proper^#(U11(X1, X2)) -> c_18(U11^#(proper(X1), proper(X2)))}
The usable rules for this path are the following:
{ proper(zeros()) -> ok(zeros())
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, proper(nil()) -> ok(nil())
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ proper(zeros()) -> ok(zeros())
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, proper(nil()) -> ok(nil())
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))
, proper^#(U11(X1, X2)) -> c_18(U11^#(proper(X1), proper(X2)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [1] x1 + [1]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(U11(X1, X2)) -> c_18(U11^#(proper(X1), proper(X2)))}
and weakly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(U11(X1, X2)) -> c_18(U11^#(proper(X1), proper(X2)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [1] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
and weakly orienting the rules
{ proper^#(U11(X1, X2)) -> c_18(U11^#(proper(X1), proper(X2)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [2]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [1] x1 + [1] x2 + [3]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [1] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())
, proper^#(U11(X1, X2)) -> c_18(U11^#(proper(X1), proper(X2)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())
, proper^#(U11(X1, X2)) -> c_18(U11^#(proper(X1), proper(X2)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ zeros_0() -> 2
, mark_0(2) -> 3
, mark_0(3) -> 3
, mark_0(5) -> 3
, mark_0(7) -> 3
, mark_0(11) -> 3
, mark_0(13) -> 3
, 0_0() -> 5
, tt_0() -> 7
, nil_0() -> 11
, ok_0(2) -> 13
, ok_0(3) -> 13
, ok_0(5) -> 13
, ok_0(7) -> 13
, ok_0(11) -> 13
, ok_0(13) -> 13
, U11^#_0(2, 2) -> 24
, U11^#_0(2, 3) -> 24
, U11^#_0(2, 5) -> 24
, U11^#_0(2, 7) -> 24
, U11^#_0(2, 11) -> 24
, U11^#_0(2, 13) -> 24
, U11^#_0(3, 2) -> 24
, U11^#_0(3, 3) -> 24
, U11^#_0(3, 5) -> 24
, U11^#_0(3, 7) -> 24
, U11^#_0(3, 11) -> 24
, U11^#_0(3, 13) -> 24
, U11^#_0(5, 2) -> 24
, U11^#_0(5, 3) -> 24
, U11^#_0(5, 5) -> 24
, U11^#_0(5, 7) -> 24
, U11^#_0(5, 11) -> 24
, U11^#_0(5, 13) -> 24
, U11^#_0(7, 2) -> 24
, U11^#_0(7, 3) -> 24
, U11^#_0(7, 5) -> 24
, U11^#_0(7, 7) -> 24
, U11^#_0(7, 11) -> 24
, U11^#_0(7, 13) -> 24
, U11^#_0(11, 2) -> 24
, U11^#_0(11, 3) -> 24
, U11^#_0(11, 5) -> 24
, U11^#_0(11, 7) -> 24
, U11^#_0(11, 11) -> 24
, U11^#_0(11, 13) -> 24
, U11^#_0(13, 2) -> 24
, U11^#_0(13, 3) -> 24
, U11^#_0(13, 5) -> 24
, U11^#_0(13, 7) -> 24
, U11^#_0(13, 11) -> 24
, U11^#_0(13, 13) -> 24
, proper^#_0(2) -> 36
, proper^#_0(3) -> 36
, proper^#_0(5) -> 36
, proper^#_0(7) -> 36
, proper^#_0(11) -> 36
, proper^#_0(13) -> 36}
18)
{proper^#(cons(X1, X2)) -> c_16(cons^#(proper(X1), proper(X2)))}
The usable rules for this path are the following:
{ proper(zeros()) -> ok(zeros())
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, proper(nil()) -> ok(nil())
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ proper(zeros()) -> ok(zeros())
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, proper(nil()) -> ok(nil())
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))
, proper^#(cons(X1, X2)) -> c_16(cons^#(proper(X1), proper(X2)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_15() = [0]
c_16(x1) = [1] x1 + [1]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(cons(X1, X2)) -> c_16(cons^#(proper(X1), proper(X2)))}
and weakly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(cons(X1, X2)) -> c_16(cons^#(proper(X1), proper(X2)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_15() = [0]
c_16(x1) = [1] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
and weakly orienting the rules
{ proper^#(cons(X1, X2)) -> c_16(cons^#(proper(X1), proper(X2)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [8]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [8]
length(x1) = [1] x1 + [2]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [11]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [13]
c_15() = [0]
c_16(x1) = [1] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())
, proper^#(cons(X1, X2)) -> c_16(cons^#(proper(X1), proper(X2)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())
, proper^#(cons(X1, X2)) -> c_16(cons^#(proper(X1), proper(X2)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ zeros_0() -> 2
, mark_0(2) -> 3
, mark_0(3) -> 3
, mark_0(5) -> 3
, mark_0(7) -> 3
, mark_0(11) -> 3
, mark_0(13) -> 3
, 0_0() -> 5
, tt_0() -> 7
, nil_0() -> 11
, ok_0(2) -> 13
, ok_0(3) -> 13
, ok_0(5) -> 13
, ok_0(7) -> 13
, ok_0(11) -> 13
, ok_0(13) -> 13
, cons^#_0(2, 2) -> 17
, cons^#_0(2, 3) -> 17
, cons^#_0(2, 5) -> 17
, cons^#_0(2, 7) -> 17
, cons^#_0(2, 11) -> 17
, cons^#_0(2, 13) -> 17
, cons^#_0(3, 2) -> 17
, cons^#_0(3, 3) -> 17
, cons^#_0(3, 5) -> 17
, cons^#_0(3, 7) -> 17
, cons^#_0(3, 11) -> 17
, cons^#_0(3, 13) -> 17
, cons^#_0(5, 2) -> 17
, cons^#_0(5, 3) -> 17
, cons^#_0(5, 5) -> 17
, cons^#_0(5, 7) -> 17
, cons^#_0(5, 11) -> 17
, cons^#_0(5, 13) -> 17
, cons^#_0(7, 2) -> 17
, cons^#_0(7, 3) -> 17
, cons^#_0(7, 5) -> 17
, cons^#_0(7, 7) -> 17
, cons^#_0(7, 11) -> 17
, cons^#_0(7, 13) -> 17
, cons^#_0(11, 2) -> 17
, cons^#_0(11, 3) -> 17
, cons^#_0(11, 5) -> 17
, cons^#_0(11, 7) -> 17
, cons^#_0(11, 11) -> 17
, cons^#_0(11, 13) -> 17
, cons^#_0(13, 2) -> 17
, cons^#_0(13, 3) -> 17
, cons^#_0(13, 5) -> 17
, cons^#_0(13, 7) -> 17
, cons^#_0(13, 11) -> 17
, cons^#_0(13, 13) -> 17
, proper^#_0(2) -> 36
, proper^#_0(3) -> 36
, proper^#_0(5) -> 36
, proper^#_0(7) -> 36
, proper^#_0(11) -> 36
, proper^#_0(13) -> 36}
19)
{proper^#(U12(X1, X2)) -> c_20(U12^#(proper(X1), proper(X2)))}
The usable rules for this path are the following:
{ proper(zeros()) -> ok(zeros())
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, proper(nil()) -> ok(nil())
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ proper(zeros()) -> ok(zeros())
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, proper(nil()) -> ok(nil())
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))
, proper^#(U12(X1, X2)) -> c_20(U12^#(proper(X1), proper(X2)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [1] x1 + [1]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(U12(X1, X2)) -> c_20(U12^#(proper(X1), proper(X2)))}
and weakly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(U12(X1, X2)) -> c_20(U12^#(proper(X1), proper(X2)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [1] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
and weakly orienting the rules
{ proper^#(U12(X1, X2)) -> c_20(U12^#(proper(X1), proper(X2)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [8]
cons(x1, x2) = [1] x1 + [1] x2 + [8]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [2]
s(x1) = [1] x1 + [8]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [1] x1 + [1] x2 + [3]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [1] x1 + [4]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())
, proper^#(U12(X1, X2)) -> c_20(U12^#(proper(X1), proper(X2)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())
, proper^#(U12(X1, X2)) -> c_20(U12^#(proper(X1), proper(X2)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ zeros_0() -> 2
, mark_0(2) -> 3
, mark_0(3) -> 3
, mark_0(5) -> 3
, mark_0(7) -> 3
, mark_0(11) -> 3
, mark_0(13) -> 3
, 0_0() -> 5
, tt_0() -> 7
, nil_0() -> 11
, ok_0(2) -> 13
, ok_0(3) -> 13
, ok_0(5) -> 13
, ok_0(7) -> 13
, ok_0(11) -> 13
, ok_0(13) -> 13
, U12^#_0(2, 2) -> 19
, U12^#_0(2, 3) -> 19
, U12^#_0(2, 5) -> 19
, U12^#_0(2, 7) -> 19
, U12^#_0(2, 11) -> 19
, U12^#_0(2, 13) -> 19
, U12^#_0(3, 2) -> 19
, U12^#_0(3, 3) -> 19
, U12^#_0(3, 5) -> 19
, U12^#_0(3, 7) -> 19
, U12^#_0(3, 11) -> 19
, U12^#_0(3, 13) -> 19
, U12^#_0(5, 2) -> 19
, U12^#_0(5, 3) -> 19
, U12^#_0(5, 5) -> 19
, U12^#_0(5, 7) -> 19
, U12^#_0(5, 11) -> 19
, U12^#_0(5, 13) -> 19
, U12^#_0(7, 2) -> 19
, U12^#_0(7, 3) -> 19
, U12^#_0(7, 5) -> 19
, U12^#_0(7, 7) -> 19
, U12^#_0(7, 11) -> 19
, U12^#_0(7, 13) -> 19
, U12^#_0(11, 2) -> 19
, U12^#_0(11, 3) -> 19
, U12^#_0(11, 5) -> 19
, U12^#_0(11, 7) -> 19
, U12^#_0(11, 11) -> 19
, U12^#_0(11, 13) -> 19
, U12^#_0(13, 2) -> 19
, U12^#_0(13, 3) -> 19
, U12^#_0(13, 5) -> 19
, U12^#_0(13, 7) -> 19
, U12^#_0(13, 11) -> 19
, U12^#_0(13, 13) -> 19
, proper^#_0(2) -> 36
, proper^#_0(3) -> 36
, proper^#_0(5) -> 36
, proper^#_0(7) -> 36
, proper^#_0(11) -> 36
, proper^#_0(13) -> 36}
20)
{proper^#(length(X)) -> c_22(length^#(proper(X)))}
The usable rules for this path are the following:
{ proper(zeros()) -> ok(zeros())
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, proper(nil()) -> ok(nil())
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ proper(zeros()) -> ok(zeros())
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, proper(nil()) -> ok(nil())
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))
, proper^#(length(X)) -> c_22(length^#(proper(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [4]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [1] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(length(X)) -> c_22(length^#(proper(X)))}
and weakly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(length(X)) -> c_22(length^#(proper(X)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [1] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
and weakly orienting the rules
{ proper^#(length(X)) -> c_22(length^#(proper(X)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [1]
mark(x1) = [1] x1 + [8]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [8]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [1] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())
, proper^#(length(X)) -> c_22(length^#(proper(X)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())
, proper^#(length(X)) -> c_22(length^#(proper(X)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ zeros_0() -> 2
, mark_0(2) -> 3
, mark_0(3) -> 3
, mark_0(5) -> 3
, mark_0(7) -> 3
, mark_0(11) -> 3
, mark_0(13) -> 3
, 0_0() -> 5
, tt_0() -> 7
, nil_0() -> 11
, ok_0(2) -> 13
, ok_0(3) -> 13
, ok_0(5) -> 13
, ok_0(7) -> 13
, ok_0(11) -> 13
, ok_0(13) -> 13
, length^#_0(2) -> 30
, length^#_0(3) -> 30
, length^#_0(5) -> 30
, length^#_0(7) -> 30
, length^#_0(11) -> 30
, length^#_0(13) -> 30
, proper^#_0(2) -> 36
, proper^#_0(3) -> 36
, proper^#_0(5) -> 36
, proper^#_0(7) -> 36
, proper^#_0(11) -> 36
, proper^#_0(13) -> 36}
21)
{proper^#(s(X)) -> c_21(s^#(proper(X)))}
The usable rules for this path are the following:
{ proper(zeros()) -> ok(zeros())
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, proper(nil()) -> ok(nil())
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ proper(zeros()) -> ok(zeros())
, proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, proper(nil()) -> ok(nil())
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
, s(ok(X)) -> ok(s(X))
, proper^#(s(X)) -> c_21(s^#(proper(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [4]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [1] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(s(X)) -> c_21(s^#(proper(X)))}
and weakly orienting the rules
{ cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(s(X)) -> c_21(s^#(proper(X)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [1] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
and weakly orienting the rules
{ proper^#(s(X)) -> c_21(s^#(proper(X)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [1]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [2]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [12]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [1] x1 + [6]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())
, proper^#(s(X)) -> c_21(s^#(proper(X)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
, proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
, proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
, proper(s(X)) -> s(proper(X))
, proper(length(X)) -> length(proper(X))
, length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, cons(mark(X1), X2) -> mark(cons(X1, X2))
, U11(mark(X1), X2) -> mark(U11(X1, X2))
, U12(mark(X1), X2) -> mark(U12(X1, X2))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(zeros()) -> ok(zeros())
, proper(0()) -> ok(0())
, proper(tt()) -> ok(tt())
, proper(nil()) -> ok(nil())
, proper^#(s(X)) -> c_21(s^#(proper(X)))
, cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
, U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
, U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ zeros_0() -> 2
, mark_0(2) -> 3
, mark_0(3) -> 3
, mark_0(5) -> 3
, mark_0(7) -> 3
, mark_0(11) -> 3
, mark_0(13) -> 3
, 0_0() -> 5
, tt_0() -> 7
, nil_0() -> 11
, ok_0(2) -> 13
, ok_0(3) -> 13
, ok_0(5) -> 13
, ok_0(7) -> 13
, ok_0(11) -> 13
, ok_0(13) -> 13
, s^#_0(2) -> 21
, s^#_0(3) -> 21
, s^#_0(5) -> 21
, s^#_0(7) -> 21
, s^#_0(11) -> 21
, s^#_0(13) -> 21
, proper^#_0(2) -> 36
, proper^#_0(3) -> 36
, proper^#_0(5) -> 36
, proper^#_0(7) -> 36
, proper^#_0(11) -> 36
, proper^#_0(13) -> 36}
22)
{ active^#(U12(tt(), L)) -> c_2(s^#(length(L)))
, s^#(ok(X)) -> c_27(s^#(X))
, s^#(mark(X)) -> c_13(s^#(X))}
The usable rules for this path are the following:
{ length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, active^#(U12(tt(), L)) -> c_2(s^#(length(L)))
, s^#(ok(X)) -> c_27(s^#(X))
, s^#(mark(X)) -> c_13(s^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(U12(tt(), L)) -> c_2(s^#(length(L)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(U12(tt(), L)) -> c_2(s^#(length(L)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [0] x1 + [0]
length(x1) = [1] x1 + [1]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [1] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [1] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{s^#(ok(X)) -> c_27(s^#(X))}
and weakly orienting the rules
{active^#(U12(tt(), L)) -> c_2(s^#(length(L)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{s^#(ok(X)) -> c_27(s^#(X))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [8]
U12(x1, x2) = [1] x1 + [1] x2 + [10]
s(x1) = [0] x1 + [0]
length(x1) = [1] x1 + [1]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [1] x1 + [7]
s^#(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [1] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{s^#(mark(X)) -> c_13(s^#(X))}
and weakly orienting the rules
{ s^#(ok(X)) -> c_27(s^#(X))
, active^#(U12(tt(), L)) -> c_2(s^#(length(L)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{s^#(mark(X)) -> c_13(s^#(X))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [8]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [0] x1 + [0]
length(x1) = [1] x1 + [1]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [1] x1 + [0]
s^#(x1) = [1] x1 + [8]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [1] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))}
Weak Rules:
{ s^#(mark(X)) -> c_13(s^#(X))
, s^#(ok(X)) -> c_27(s^#(X))
, active^#(U12(tt(), L)) -> c_2(s^#(length(L)))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))}
Weak Rules:
{ s^#(mark(X)) -> c_13(s^#(X))
, s^#(ok(X)) -> c_27(s^#(X))
, active^#(U12(tt(), L)) -> c_2(s^#(length(L)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(3) -> 3
, mark_0(7) -> 3
, mark_0(13) -> 3
, tt_0() -> 7
, ok_0(3) -> 13
, ok_0(7) -> 13
, ok_0(13) -> 13
, active^#_0(3) -> 15
, active^#_0(7) -> 15
, active^#_0(13) -> 15
, s^#_0(3) -> 21
, s^#_0(7) -> 21
, s^#_0(13) -> 21
, c_13_0(21) -> 21
, c_27_0(21) -> 21}
23)
{active^#(U12(tt(), L)) -> c_2(s^#(length(L)))}
The usable rules for this path are the following:
{ length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))
, active^#(U12(tt(), L)) -> c_2(s^#(length(L)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(U12(tt(), L)) -> c_2(s^#(length(L)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(U12(tt(), L)) -> c_2(s^#(length(L)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [1] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
U12(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [0] x1 + [0]
length(x1) = [1] x1 + [1]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [1] x1 + [1]
s^#(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))}
Weak Rules: {active^#(U12(tt(), L)) -> c_2(s^#(length(L)))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ length(mark(X)) -> mark(length(X))
, length(ok(X)) -> ok(length(X))}
Weak Rules: {active^#(U12(tt(), L)) -> c_2(s^#(length(L)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(3) -> 3
, mark_0(7) -> 3
, mark_0(13) -> 3
, tt_0() -> 7
, ok_0(3) -> 13
, ok_0(7) -> 13
, ok_0(13) -> 13
, active^#_0(3) -> 15
, active^#_0(7) -> 15
, active^#_0(13) -> 15
, s^#_0(3) -> 21
, s^#_0(7) -> 21
, s^#_0(13) -> 21}
24)
{active^#(U11(tt(), L)) -> c_1(U12^#(tt(), L))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
U12(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {active^#(U11(tt(), L)) -> c_1(U12^#(tt(), L))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(U11(tt(), L)) -> c_1(U12^#(tt(), L))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(U11(tt(), L)) -> c_1(U12^#(tt(), L))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
U12(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [1] x1 + [0]
U12^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {active^#(U11(tt(), L)) -> c_1(U12^#(tt(), L))}
Details:
The given problem does not contain any strict rules
25)
{active^#(zeros()) -> c_0(cons^#(0(), zeros()))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
U12(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {active^#(zeros()) -> c_0(cons^#(0(), zeros()))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(zeros()) -> c_0(cons^#(0(), zeros()))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(zeros()) -> c_0(cons^#(0(), zeros()))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
U12(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {active^#(zeros()) -> c_0(cons^#(0(), zeros()))}
Details:
The given problem does not contain any strict rules
26)
{active^#(length(cons(N, L))) -> c_4(U11^#(tt(), L))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
U12(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {active^#(length(cons(N, L))) -> c_4(U11^#(tt(), L))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(length(cons(N, L))) -> c_4(U11^#(tt(), L))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(length(cons(N, L))) -> c_4(U11^#(tt(), L))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
U11(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
U12(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
U11^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {active^#(length(cons(N, L))) -> c_4(U11^#(tt(), L))}
Details:
The given problem does not contain any strict rules
27)
{active^#(length(nil())) -> c_3()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
U12(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {active^#(length(nil())) -> c_3()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(length(nil())) -> c_3()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(length(nil())) -> c_3()}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
U12(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
length(x1) = [1] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {active^#(length(nil())) -> c_3()}
Details:
The given problem does not contain any strict rules
28)
{proper^#(tt()) -> c_19()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
U12(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {proper^#(tt()) -> c_19()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{proper^#(tt()) -> c_19()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(tt()) -> c_19()}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
U12(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {proper^#(tt()) -> c_19()}
Details:
The given problem does not contain any strict rules
29)
{proper^#(zeros()) -> c_15()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
U12(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {proper^#(zeros()) -> c_15()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{proper^#(zeros()) -> c_15()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(zeros()) -> c_15()}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
U12(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {proper^#(zeros()) -> c_15()}
Details:
The given problem does not contain any strict rules
30)
{proper^#(0()) -> c_17()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
U12(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {proper^#(0()) -> c_17()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{proper^#(0()) -> c_17()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(0()) -> c_17()}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
U12(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {proper^#(0()) -> c_17()}
Details:
The given problem does not contain any strict rules
31)
{proper^#(nil()) -> c_23()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
U12(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {proper^#(nil()) -> c_23()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{proper^#(nil()) -> c_23()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(nil()) -> c_23()}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
zeros() = [0]
mark(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
U12(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
nil() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23() = [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
c_26(x1) = [0] x1 + [0]
c_27(x1) = [0] x1 + [0]
c_28(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_29(x1) = [0] x1 + [0]
c_30(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {proper^#(nil()) -> c_23()}
Details:
The given problem does not contain any strict rules