'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, proper(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s(ok(X)) -> ok(s(X))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, 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^#(U11(tt(), M, N)) -> c_0(U12^#(tt(), M, N))
, active^#(U12(tt(), M, N)) -> c_1(s^#(plus(N, M)))
, active^#(plus(N, 0())) -> c_2()
, active^#(plus(N, s(M))) -> c_3(U11^#(tt(), M, N))
, active^#(U11(X1, X2, X3)) -> c_4(U11^#(active(X1), X2, X3))
, active^#(U12(X1, X2, X3)) -> c_5(U12^#(active(X1), X2, X3))
, active^#(s(X)) -> c_6(s^#(active(X)))
, active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2))
, active^#(plus(X1, X2)) -> c_8(plus^#(X1, active(X2)))
, U11^#(mark(X1), X2, X3) -> c_9(U11^#(X1, X2, X3))
, U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))
, s^#(mark(X)) -> c_11(s^#(X))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))
, plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))
, proper^#(U11(X1, X2, X3)) ->
c_14(U11^#(proper(X1), proper(X2), proper(X3)))
, proper^#(tt()) -> c_15()
, proper^#(U12(X1, X2, X3)) ->
c_16(U12^#(proper(X1), proper(X2), proper(X3)))
, proper^#(s(X)) -> c_17(s^#(proper(X)))
, proper^#(plus(X1, X2)) -> c_18(plus^#(proper(X1), proper(X2)))
, proper^#(0()) -> c_19()
, U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))
, U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))
, s^#(ok(X)) -> c_22(s^#(X))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))
, top^#(mark(X)) -> c_24(top^#(proper(X)))
, top^#(ok(X)) -> c_25(top^#(active(X)))}
The usable rules are:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, proper(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s(ok(X)) -> ok(s(X))}
The estimated dependency graph contains the following edges:
{active^#(U12(tt(), M, N)) -> c_1(s^#(plus(N, M)))}
==> {s^#(ok(X)) -> c_22(s^#(X))}
{active^#(U12(tt(), M, N)) -> c_1(s^#(plus(N, M)))}
==> {s^#(mark(X)) -> c_11(s^#(X))}
{active^#(U11(X1, X2, X3)) -> c_4(U11^#(active(X1), X2, X3))}
==> {U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))}
{active^#(U11(X1, X2, X3)) -> c_4(U11^#(active(X1), X2, X3))}
==> {U11^#(mark(X1), X2, X3) -> c_9(U11^#(X1, X2, X3))}
{active^#(U12(X1, X2, X3)) -> c_5(U12^#(active(X1), X2, X3))}
==> {U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))}
{active^#(U12(X1, X2, X3)) -> c_5(U12^#(active(X1), X2, X3))}
==> {U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))}
{active^#(s(X)) -> c_6(s^#(active(X)))}
==> {s^#(ok(X)) -> c_22(s^#(X))}
{active^#(s(X)) -> c_6(s^#(active(X)))}
==> {s^#(mark(X)) -> c_11(s^#(X))}
{active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2))}
==> {plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))}
{active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2))}
==> {plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))}
{active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2))}
==> {plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
{active^#(plus(X1, X2)) -> c_8(plus^#(X1, active(X2)))}
==> {plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))}
{active^#(plus(X1, X2)) -> c_8(plus^#(X1, active(X2)))}
==> {plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))}
{active^#(plus(X1, X2)) -> c_8(plus^#(X1, active(X2)))}
==> {plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
{U11^#(mark(X1), X2, X3) -> c_9(U11^#(X1, X2, X3))}
==> {U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))}
{U11^#(mark(X1), X2, X3) -> c_9(U11^#(X1, X2, X3))}
==> {U11^#(mark(X1), X2, X3) -> c_9(U11^#(X1, X2, X3))}
{U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))}
==> {U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))}
{U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))}
==> {U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))}
{s^#(mark(X)) -> c_11(s^#(X))}
==> {s^#(ok(X)) -> c_22(s^#(X))}
{s^#(mark(X)) -> c_11(s^#(X))}
==> {s^#(mark(X)) -> c_11(s^#(X))}
{plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
==> {plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))}
{plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
==> {plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))}
{plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
==> {plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
{plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))}
==> {plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))}
{plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))}
==> {plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))}
{plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))}
==> {plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
{proper^#(U11(X1, X2, X3)) ->
c_14(U11^#(proper(X1), proper(X2), proper(X3)))}
==> {U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))}
{proper^#(U11(X1, X2, X3)) ->
c_14(U11^#(proper(X1), proper(X2), proper(X3)))}
==> {U11^#(mark(X1), X2, X3) -> c_9(U11^#(X1, X2, X3))}
{proper^#(U12(X1, X2, X3)) ->
c_16(U12^#(proper(X1), proper(X2), proper(X3)))}
==> {U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))}
{proper^#(U12(X1, X2, X3)) ->
c_16(U12^#(proper(X1), proper(X2), proper(X3)))}
==> {U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))}
{proper^#(s(X)) -> c_17(s^#(proper(X)))}
==> {s^#(ok(X)) -> c_22(s^#(X))}
{proper^#(s(X)) -> c_17(s^#(proper(X)))}
==> {s^#(mark(X)) -> c_11(s^#(X))}
{proper^#(plus(X1, X2)) -> c_18(plus^#(proper(X1), proper(X2)))}
==> {plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))}
{proper^#(plus(X1, X2)) -> c_18(plus^#(proper(X1), proper(X2)))}
==> {plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))}
{proper^#(plus(X1, X2)) -> c_18(plus^#(proper(X1), proper(X2)))}
==> {plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
{U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))}
==> {U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))}
{U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))}
==> {U11^#(mark(X1), X2, X3) -> c_9(U11^#(X1, X2, X3))}
{U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))}
==> {U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))}
{U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))}
==> {U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))}
{s^#(ok(X)) -> c_22(s^#(X))}
==> {s^#(ok(X)) -> c_22(s^#(X))}
{s^#(ok(X)) -> c_22(s^#(X))}
==> {s^#(mark(X)) -> c_11(s^#(X))}
{plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))}
==> {plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))}
{plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))}
==> {plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))}
{plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))}
==> {plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
{top^#(mark(X)) -> c_24(top^#(proper(X)))}
==> {top^#(ok(X)) -> c_25(top^#(active(X)))}
{top^#(mark(X)) -> c_24(top^#(proper(X)))}
==> {top^#(mark(X)) -> c_24(top^#(proper(X)))}
{top^#(ok(X)) -> c_25(top^#(active(X)))}
==> {top^#(ok(X)) -> c_25(top^#(active(X)))}
{top^#(ok(X)) -> c_25(top^#(active(X)))}
==> {top^#(mark(X)) -> c_24(top^#(proper(X)))}
We consider the following path(s):
1) { top^#(mark(X)) -> c_24(top^#(proper(X)))
, top^#(ok(X)) -> c_25(top^#(active(X)))}
The usable rules for this path are the following:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, proper(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, 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(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, proper(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s(ok(X)) -> ok(s(X))
, top^#(mark(X)) -> c_24(top^#(proper(X)))
, top^#(ok(X)) -> c_25(top^#(active(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [0]
c_24(x1) = [1] x1 + [0]
c_25(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{top^#(mark(X)) -> c_24(top^#(proper(X)))}
and weakly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{top^#(mark(X)) -> c_24(top^#(proper(X)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
proper(x1) = [1] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [0]
c_24(x1) = [1] x1 + [0]
c_25(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())}
and weakly orienting the rules
{ top^#(mark(X)) -> c_24(top^#(proper(X)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [1]
c_24(x1) = [1] x1 + [0]
c_25(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(plus(N, 0())) -> mark(N)}
and weakly orienting the rules
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())
, top^#(mark(X)) -> c_24(top^#(proper(X)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(plus(N, 0())) -> mark(N)}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [14]
tt() = [14]
mark(x1) = [1] x1 + [15]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [14]
s(x1) = [1] x1 + [4]
plus(x1, x2) = [1] x1 + [1] x2 + [14]
0() = [8]
proper(x1) = [1] x1 + [8]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [1]
c_24(x1) = [1] x1 + [0]
c_25(x1) = [1] x1 + [7]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(plus(N, s(M))) -> mark(U11(tt(), M, N))}
and weakly orienting the rules
{ active(plus(N, 0())) -> mark(N)
, proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())
, top^#(mark(X)) -> c_24(top^#(proper(X)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(plus(N, s(M))) -> mark(U11(tt(), M, N))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [8]
plus(x1, x2) = [1] x1 + [1] x2 + [10]
0() = [0]
proper(x1) = [1] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [0]
c_24(x1) = [1] x1 + [0]
c_25(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U12(tt(), M, N)) -> mark(s(plus(N, M)))}
and weakly orienting the rules
{ active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(plus(N, 0())) -> mark(N)
, proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())
, top^#(mark(X)) -> c_24(top^#(proper(X)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U12(tt(), M, N)) -> mark(s(plus(N, M)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [4]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [5]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [8]
proper(x1) = [1] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [1]
c_24(x1) = [1] x1 + [0]
c_25(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{top^#(ok(X)) -> c_25(top^#(active(X)))}
and weakly orienting the rules
{ active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(plus(N, 0())) -> mark(N)
, proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())
, top^#(mark(X)) -> c_24(top^#(proper(X)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{top^#(ok(X)) -> c_25(top^#(active(X)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [8]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [14]
s(x1) = [1] x1 + [2]
plus(x1, x2) = [1] x1 + [1] x2 + [5]
0() = [9]
proper(x1) = [1] x1 + [8]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [0]
c_24(x1) = [1] x1 + [0]
c_25(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(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, proper(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ top^#(ok(X)) -> c_25(top^#(active(X)))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(plus(N, 0())) -> mark(N)
, proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())
, top^#(mark(X)) -> c_24(top^#(proper(X)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
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(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, proper(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ top^#(ok(X)) -> c_25(top^#(active(X)))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(plus(N, 0())) -> mark(N)
, proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())
, top^#(mark(X)) -> c_24(top^#(proper(X)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ active_0(2) -> 4
, tt_0() -> 2
, mark_0(2) -> 2
, 0_0() -> 2
, proper_0(2) -> 6
, ok_0(2) -> 2
, ok_0(2) -> 6
, top^#_0(2) -> 1
, top^#_0(4) -> 3
, top^#_0(6) -> 5
, c_24_0(5) -> 1
, c_25_0(3) -> 1
, c_25_0(3) -> 5}
2) { active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))
, plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
The usable rules for this path are the following:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, 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(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s(ok(X)) -> ok(s(X))
, active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))
, plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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) = [1] x1 + [1]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [1] x1 + [1]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))}
and weakly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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) = [1] x1 + [1]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))}
and weakly orienting the rules
{ active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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) = [1] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2))}
and weakly orienting the rules
{ plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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) = [1] x1 + [1]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U12(tt(), M, N)) -> mark(s(plus(N, M)))}
and weakly orienting the rules
{ active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2))
, plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U12(tt(), M, N)) -> mark(s(plus(N, M)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [6]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [11]
s(x1) = [1] x1 + [3]
plus(x1, x2) = [1] x1 + [1] x2 + [2]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [1] x1 + [6]
c_13(x1) = [1] x1 + [1]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))}
and weakly orienting the rules
{ active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2))
, plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [8]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [5]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [4]
c_8(x1) = [0] x1 + [0]
c_9(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) = [1] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2))
, plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(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(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2))
, plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ tt_0() -> 2
, mark_0(2) -> 2
, 0_0() -> 2
, ok_0(2) -> 2
, active^#_0(2) -> 1
, plus^#_0(2, 2) -> 1
, c_12_0(1) -> 1
, c_13_0(1) -> 1
, c_23_0(1) -> 1}
3) { active^#(plus(X1, X2)) -> c_8(plus^#(X1, active(X2)))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))
, plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
The usable rules for this path are the following:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, 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(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s(ok(X)) -> ok(s(X))
, active^#(plus(X1, X2)) -> c_8(plus^#(X1, active(X2)))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))
, plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [1] x1 + [0]
c_9(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) = [1] x1 + [1]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [1] x1 + [1]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))}
and weakly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [1] x1 + [0]
c_9(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) = [1] x1 + [1]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))}
and weakly orienting the rules
{ active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [1] x1 + [0]
c_9(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) = [1] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active^#(plus(X1, X2)) -> c_8(plus^#(X1, active(X2)))}
and weakly orienting the rules
{ plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(plus(X1, X2)) -> c_8(plus^#(X1, active(X2)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [1] x1 + [0]
c_9(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) = [1] x1 + [1]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U12(tt(), M, N)) -> mark(s(plus(N, M)))}
and weakly orienting the rules
{ active^#(plus(X1, X2)) -> c_8(plus^#(X1, active(X2)))
, plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U12(tt(), M, N)) -> mark(s(plus(N, M)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [6]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [11]
s(x1) = [1] x1 + [3]
plus(x1, x2) = [1] x1 + [1] x2 + [2]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [1] x1 + [6]
c_13(x1) = [1] x1 + [1]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))}
and weakly orienting the rules
{ active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active^#(plus(X1, X2)) -> c_8(plus^#(X1, active(X2)))
, plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [8]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [5]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [4]
c_8(x1) = [1] x1 + [0]
c_9(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) = [1] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active^#(plus(X1, X2)) -> c_8(plus^#(X1, active(X2)))
, plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(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(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active^#(plus(X1, X2)) -> c_8(plus^#(X1, active(X2)))
, plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ tt_0() -> 2
, mark_0(2) -> 2
, 0_0() -> 2
, ok_0(2) -> 2
, active^#_0(2) -> 1
, plus^#_0(2, 2) -> 1
, c_12_0(1) -> 1
, c_13_0(1) -> 1
, c_23_0(1) -> 1}
4) { active^#(U12(X1, X2, X3)) -> c_5(U12^#(active(X1), X2, X3))
, U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))
, U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))}
The usable rules for this path are the following:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, 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(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s(ok(X)) -> ok(s(X))
, active^#(U12(X1, X2, X3)) -> c_5(U12^#(active(X1), X2, X3))
, U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))
, U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))}
Details:
We apply the weight gap principle, strictly orienting the rules
{U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [1] x1 + [1]
c_22(x1) = [0] x1 + [0]
c_23(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))}
and weakly orienting the rules
{U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [12]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [2]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active^#(U12(X1, X2, X3)) -> c_5(U12^#(active(X1), X2, X3))}
and weakly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))
, U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(U12(X1, X2, X3)) -> c_5(U12^#(active(X1), X2, X3))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [9]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [7]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [5]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(plus(N, 0())) -> mark(N)}
and weakly orienting the rules
{ active^#(U12(X1, X2, X3)) -> c_5(U12^#(active(X1), X2, X3))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))
, U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(plus(N, 0())) -> mark(N)}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [1]
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]
U12^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [1] x1 + [1]
c_22(x1) = [0] x1 + [0]
c_23(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(plus(N, s(M))) -> mark(U11(tt(), M, N))}
and weakly orienting the rules
{ active(plus(N, 0())) -> mark(N)
, active^#(U12(X1, X2, X3)) -> c_5(U12^#(active(X1), X2, X3))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))
, U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(plus(N, s(M))) -> mark(U11(tt(), M, N))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [1]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [8]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [2]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [13]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [12]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [1] x1 + [1]
c_22(x1) = [0] x1 + [0]
c_23(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U12(tt(), M, N)) -> mark(s(plus(N, M)))}
and weakly orienting the rules
{ active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(plus(N, 0())) -> mark(N)
, active^#(U12(X1, X2, X3)) -> c_5(U12^#(active(X1), X2, X3))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))
, U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U12(tt(), M, N)) -> mark(s(plus(N, M)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [8]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [9]
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]
U12^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [4]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))}
and weakly orienting the rules
{ active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(plus(N, 0())) -> mark(N)
, active^#(U12(X1, X2, X3)) -> c_5(U12^#(active(X1), X2, X3))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))
, U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [8]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(plus(N, 0())) -> mark(N)
, active^#(U12(X1, X2, X3)) -> c_5(U12^#(active(X1), X2, X3))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))
, U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))}
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(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(plus(N, 0())) -> mark(N)
, active^#(U12(X1, X2, X3)) -> c_5(U12^#(active(X1), X2, X3))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))
, U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ tt_0() -> 2
, mark_0(2) -> 2
, 0_0() -> 2
, ok_0(2) -> 2
, active^#_0(2) -> 1
, U12^#_0(2, 2, 2) -> 1
, c_10_0(1) -> 1
, c_21_0(1) -> 1}
5) { active^#(U11(X1, X2, X3)) -> c_4(U11^#(active(X1), X2, X3))
, U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))
, U11^#(mark(X1), X2, X3) -> c_9(U11^#(X1, X2, X3))}
The usable rules for this path are the following:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, 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(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s(ok(X)) -> ok(s(X))
, active^#(U11(X1, X2, X3)) -> c_4(U11^#(active(X1), X2, X3))
, U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))
, U11^#(mark(X1), X2, X3) -> c_9(U11^#(X1, X2, X3))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{U11^#(mark(X1), X2, X3) -> c_9(U11^#(X1, X2, X3))}
and weakly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{U11^#(mark(X1), X2, X3) -> c_9(U11^#(X1, X2, X3))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active^#(U11(X1, X2, X3)) -> c_4(U11^#(active(X1), X2, X3))}
and weakly orienting the rules
{ U11^#(mark(X1), X2, X3) -> c_9(U11^#(X1, X2, X3))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(U11(X1, X2, X3)) -> c_4(U11^#(active(X1), X2, X3))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(plus(N, s(M))) -> mark(U11(tt(), M, N))}
and weakly orienting the rules
{ active^#(U11(X1, X2, X3)) -> c_4(U11^#(active(X1), X2, X3))
, U11^#(mark(X1), X2, X3) -> c_9(U11^#(X1, X2, X3))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(plus(N, s(M))) -> mark(U11(tt(), M, N))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [8]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(plus(N, 0())) -> mark(N)}
and weakly orienting the rules
{ active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active^#(U11(X1, X2, X3)) -> c_4(U11^#(active(X1), X2, X3))
, U11^#(mark(X1), X2, X3) -> c_9(U11^#(X1, X2, X3))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(plus(N, 0())) -> mark(N)}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [8]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))}
and weakly orienting the rules
{ active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active^#(U11(X1, X2, X3)) -> c_4(U11^#(active(X1), X2, X3))
, U11^#(mark(X1), X2, X3) -> c_9(U11^#(X1, X2, X3))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [4]
0() = [0]
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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
c_4(x1) = [1] x1 + [5]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U12(tt(), M, N)) -> mark(s(plus(N, M)))}
and weakly orienting the rules
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active^#(U11(X1, X2, X3)) -> c_4(U11^#(active(X1), X2, X3))
, U11^#(mark(X1), X2, X3) -> c_9(U11^#(X1, X2, X3))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U12(tt(), M, N)) -> mark(s(plus(N, M)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [6]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active^#(U11(X1, X2, X3)) -> c_4(U11^#(active(X1), X2, X3))
, U11^#(mark(X1), X2, X3) -> c_9(U11^#(X1, X2, X3))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))}
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(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active^#(U11(X1, X2, X3)) -> c_4(U11^#(active(X1), X2, X3))
, U11^#(mark(X1), X2, X3) -> c_9(U11^#(X1, X2, X3))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ tt_0() -> 2
, mark_0(2) -> 2
, 0_0() -> 2
, ok_0(2) -> 2
, active^#_0(2) -> 1
, U11^#_0(2, 2, 2) -> 1
, c_9_0(1) -> 1
, c_20_0(1) -> 1}
6) { active^#(s(X)) -> c_6(s^#(active(X)))
, s^#(ok(X)) -> c_22(s^#(X))
, s^#(mark(X)) -> c_11(s^#(X))}
The usable rules for this path are the following:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, 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(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s(ok(X)) -> ok(s(X))
, active^#(s(X)) -> c_6(s^#(active(X)))
, s^#(ok(X)) -> c_22(s^#(X))
, s^#(mark(X)) -> c_11(s^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s^#(ok(X)) -> c_22(s^#(X))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s^#(ok(X)) -> c_22(s^#(X))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [12]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{s^#(mark(X)) -> c_11(s^#(X))}
and weakly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s^#(ok(X)) -> c_22(s^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{s^#(mark(X)) -> c_11(s^#(X))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [12]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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 + [4]
c_23(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active^#(s(X)) -> c_6(s^#(active(X)))}
and weakly orienting the rules
{ s^#(mark(X)) -> c_11(s^#(X))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s^#(ok(X)) -> c_22(s^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(s(X)) -> c_6(s^#(active(X)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [9]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [7]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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 + [7]
c_23(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(plus(N, s(M))) -> mark(U11(tt(), M, N))}
and weakly orienting the rules
{ active^#(s(X)) -> c_6(s^#(active(X)))
, s^#(mark(X)) -> c_11(s^#(X))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s^#(ok(X)) -> c_22(s^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(plus(N, s(M))) -> mark(U11(tt(), M, N))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [1]
mark(x1) = [1] x1 + [4]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [6]
plus(x1, x2) = [1] x1 + [1] x2 + [2]
0() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [2]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(plus(N, 0())) -> mark(N)}
and weakly orienting the rules
{ active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active^#(s(X)) -> c_6(s^#(active(X)))
, s^#(mark(X)) -> c_11(s^#(X))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s^#(ok(X)) -> c_22(s^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(plus(N, 0())) -> mark(N)}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [8]
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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [2]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))}
and weakly orienting the rules
{ active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active^#(s(X)) -> c_6(s^#(active(X)))
, s^#(mark(X)) -> c_11(s^#(X))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s^#(ok(X)) -> c_22(s^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [1]
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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active^#(s(X)) -> c_6(s^#(active(X)))
, s^#(mark(X)) -> c_11(s^#(X))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s^#(ok(X)) -> c_22(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(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active^#(s(X)) -> c_6(s^#(active(X)))
, s^#(mark(X)) -> c_11(s^#(X))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s^#(ok(X)) -> c_22(s^#(X))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ tt_0() -> 3
, mark_0(3) -> 4
, mark_0(4) -> 4
, mark_0(8) -> 4
, mark_0(10) -> 4
, 0_0() -> 8
, ok_0(3) -> 10
, ok_0(4) -> 10
, ok_0(8) -> 10
, ok_0(10) -> 10
, active^#_0(3) -> 12
, active^#_0(4) -> 12
, active^#_0(8) -> 12
, active^#_0(10) -> 12
, s^#_0(3) -> 16
, s^#_0(4) -> 16
, s^#_0(8) -> 16
, s^#_0(10) -> 16
, c_11_0(16) -> 16
, c_22_0(16) -> 16}
7) {active^#(U12(X1, X2, X3)) -> c_5(U12^#(active(X1), X2, X3))}
The usable rules for this path are the following:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, 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(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s(ok(X)) -> ok(s(X))
, active^#(U12(X1, X2, X3)) -> c_5(U12^#(active(X1), X2, X3))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [2]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [8]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [4]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active^#(U12(X1, X2, X3)) -> c_5(U12^#(active(X1), X2, X3))}
and weakly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(U12(X1, X2, X3)) -> c_5(U12^#(active(X1), X2, X3))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [4]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [2]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(plus(N, s(M))) -> mark(U11(tt(), M, N))}
and weakly orienting the rules
{ active^#(U12(X1, X2, X3)) -> c_5(U12^#(active(X1), X2, X3))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(plus(N, s(M))) -> mark(U11(tt(), M, N))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [8]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(plus(N, 0())) -> mark(N)}
and weakly orienting the rules
{ active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active^#(U12(X1, X2, X3)) -> c_5(U12^#(active(X1), X2, X3))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(plus(N, 0())) -> mark(N)}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [12]
plus(x1, x2) = [1] x1 + [1] x2 + [2]
0() = [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]
U12^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))}
and weakly orienting the rules
{ active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active^#(U12(X1, X2, X3)) -> c_5(U12^#(active(X1), X2, X3))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [7]
0() = [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]
U12^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [2]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U12(tt(), M, N)) -> mark(s(plus(N, M)))}
and weakly orienting the rules
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active^#(U12(X1, X2, X3)) -> c_5(U12^#(active(X1), X2, X3))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U12(tt(), M, N)) -> mark(s(plus(N, M)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active^#(U12(X1, X2, X3)) -> c_5(U12^#(active(X1), X2, X3))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
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(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active^#(U12(X1, X2, X3)) -> c_5(U12^#(active(X1), X2, X3))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ tt_0() -> 3
, mark_0(3) -> 4
, mark_0(4) -> 4
, mark_0(8) -> 4
, mark_0(10) -> 4
, 0_0() -> 8
, ok_0(3) -> 10
, ok_0(4) -> 10
, ok_0(8) -> 10
, ok_0(10) -> 10
, active^#_0(3) -> 12
, active^#_0(4) -> 12
, active^#_0(8) -> 12
, active^#_0(10) -> 12
, U12^#_0(3, 3, 3) -> 14
, U12^#_0(3, 3, 4) -> 14
, U12^#_0(3, 3, 8) -> 14
, U12^#_0(3, 3, 10) -> 14
, U12^#_0(3, 4, 3) -> 14
, U12^#_0(3, 4, 4) -> 14
, U12^#_0(3, 4, 8) -> 14
, U12^#_0(3, 4, 10) -> 14
, U12^#_0(3, 8, 3) -> 14
, U12^#_0(3, 8, 4) -> 14
, U12^#_0(3, 8, 8) -> 14
, U12^#_0(3, 8, 10) -> 14
, U12^#_0(3, 10, 3) -> 14
, U12^#_0(3, 10, 4) -> 14
, U12^#_0(3, 10, 8) -> 14
, U12^#_0(3, 10, 10) -> 14
, U12^#_0(4, 3, 3) -> 14
, U12^#_0(4, 3, 4) -> 14
, U12^#_0(4, 3, 8) -> 14
, U12^#_0(4, 3, 10) -> 14
, U12^#_0(4, 4, 3) -> 14
, U12^#_0(4, 4, 4) -> 14
, U12^#_0(4, 4, 8) -> 14
, U12^#_0(4, 4, 10) -> 14
, U12^#_0(4, 8, 3) -> 14
, U12^#_0(4, 8, 4) -> 14
, U12^#_0(4, 8, 8) -> 14
, U12^#_0(4, 8, 10) -> 14
, U12^#_0(4, 10, 3) -> 14
, U12^#_0(4, 10, 4) -> 14
, U12^#_0(4, 10, 8) -> 14
, U12^#_0(4, 10, 10) -> 14
, U12^#_0(8, 3, 3) -> 14
, U12^#_0(8, 3, 4) -> 14
, U12^#_0(8, 3, 8) -> 14
, U12^#_0(8, 3, 10) -> 14
, U12^#_0(8, 4, 3) -> 14
, U12^#_0(8, 4, 4) -> 14
, U12^#_0(8, 4, 8) -> 14
, U12^#_0(8, 4, 10) -> 14
, U12^#_0(8, 8, 3) -> 14
, U12^#_0(8, 8, 4) -> 14
, U12^#_0(8, 8, 8) -> 14
, U12^#_0(8, 8, 10) -> 14
, U12^#_0(8, 10, 3) -> 14
, U12^#_0(8, 10, 4) -> 14
, U12^#_0(8, 10, 8) -> 14
, U12^#_0(8, 10, 10) -> 14
, U12^#_0(10, 3, 3) -> 14
, U12^#_0(10, 3, 4) -> 14
, U12^#_0(10, 3, 8) -> 14
, U12^#_0(10, 3, 10) -> 14
, U12^#_0(10, 4, 3) -> 14
, U12^#_0(10, 4, 4) -> 14
, U12^#_0(10, 4, 8) -> 14
, U12^#_0(10, 4, 10) -> 14
, U12^#_0(10, 8, 3) -> 14
, U12^#_0(10, 8, 4) -> 14
, U12^#_0(10, 8, 8) -> 14
, U12^#_0(10, 8, 10) -> 14
, U12^#_0(10, 10, 3) -> 14
, U12^#_0(10, 10, 4) -> 14
, U12^#_0(10, 10, 8) -> 14
, U12^#_0(10, 10, 10) -> 14}
8) {active^#(U11(X1, X2, X3)) -> c_4(U11^#(active(X1), X2, X3))}
The usable rules for this path are the following:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, 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(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s(ok(X)) -> ok(s(X))
, active^#(U11(X1, X2, X3)) -> c_4(U11^#(active(X1), X2, X3))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [2]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active^#(U11(X1, X2, X3)) -> c_4(U11^#(active(X1), X2, X3))}
and weakly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(U11(X1, X2, X3)) -> c_4(U11^#(active(X1), X2, X3))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [4]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [2]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(plus(N, s(M))) -> mark(U11(tt(), M, N))}
and weakly orienting the rules
{ active^#(U11(X1, X2, X3)) -> c_4(U11^#(active(X1), X2, X3))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(plus(N, s(M))) -> mark(U11(tt(), M, N))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [8]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(plus(N, 0())) -> mark(N)}
and weakly orienting the rules
{ active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active^#(U11(X1, X2, X3)) -> c_4(U11^#(active(X1), X2, X3))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(plus(N, 0())) -> mark(N)}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [12]
plus(x1, x2) = [1] x1 + [1] x2 + [2]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U12(tt(), M, N)) -> mark(s(plus(N, M)))}
and weakly orienting the rules
{ active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active^#(U11(X1, X2, X3)) -> c_4(U11^#(active(X1), X2, X3))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U12(tt(), M, N)) -> mark(s(plus(N, M)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [2]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [5]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [2]
0() = [2]
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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))}
and weakly orienting the rules
{ active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active^#(U11(X1, X2, X3)) -> c_4(U11^#(active(X1), X2, X3))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active^#(U11(X1, X2, X3)) -> c_4(U11^#(active(X1), X2, X3))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
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(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active^#(U11(X1, X2, X3)) -> c_4(U11^#(active(X1), X2, X3))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ tt_0() -> 3
, mark_0(3) -> 4
, mark_0(4) -> 4
, mark_0(8) -> 4
, mark_0(10) -> 4
, 0_0() -> 8
, ok_0(3) -> 10
, ok_0(4) -> 10
, ok_0(8) -> 10
, ok_0(10) -> 10
, active^#_0(3) -> 12
, active^#_0(4) -> 12
, active^#_0(8) -> 12
, active^#_0(10) -> 12
, U11^#_0(3, 3, 3) -> 19
, U11^#_0(3, 3, 4) -> 19
, U11^#_0(3, 3, 8) -> 19
, U11^#_0(3, 3, 10) -> 19
, U11^#_0(3, 4, 3) -> 19
, U11^#_0(3, 4, 4) -> 19
, U11^#_0(3, 4, 8) -> 19
, U11^#_0(3, 4, 10) -> 19
, U11^#_0(3, 8, 3) -> 19
, U11^#_0(3, 8, 4) -> 19
, U11^#_0(3, 8, 8) -> 19
, U11^#_0(3, 8, 10) -> 19
, U11^#_0(3, 10, 3) -> 19
, U11^#_0(3, 10, 4) -> 19
, U11^#_0(3, 10, 8) -> 19
, U11^#_0(3, 10, 10) -> 19
, U11^#_0(4, 3, 3) -> 19
, U11^#_0(4, 3, 4) -> 19
, U11^#_0(4, 3, 8) -> 19
, U11^#_0(4, 3, 10) -> 19
, U11^#_0(4, 4, 3) -> 19
, U11^#_0(4, 4, 4) -> 19
, U11^#_0(4, 4, 8) -> 19
, U11^#_0(4, 4, 10) -> 19
, U11^#_0(4, 8, 3) -> 19
, U11^#_0(4, 8, 4) -> 19
, U11^#_0(4, 8, 8) -> 19
, U11^#_0(4, 8, 10) -> 19
, U11^#_0(4, 10, 3) -> 19
, U11^#_0(4, 10, 4) -> 19
, U11^#_0(4, 10, 8) -> 19
, U11^#_0(4, 10, 10) -> 19
, U11^#_0(8, 3, 3) -> 19
, U11^#_0(8, 3, 4) -> 19
, U11^#_0(8, 3, 8) -> 19
, U11^#_0(8, 3, 10) -> 19
, U11^#_0(8, 4, 3) -> 19
, U11^#_0(8, 4, 4) -> 19
, U11^#_0(8, 4, 8) -> 19
, U11^#_0(8, 4, 10) -> 19
, U11^#_0(8, 8, 3) -> 19
, U11^#_0(8, 8, 4) -> 19
, U11^#_0(8, 8, 8) -> 19
, U11^#_0(8, 8, 10) -> 19
, U11^#_0(8, 10, 3) -> 19
, U11^#_0(8, 10, 4) -> 19
, U11^#_0(8, 10, 8) -> 19
, U11^#_0(8, 10, 10) -> 19
, U11^#_0(10, 3, 3) -> 19
, U11^#_0(10, 3, 4) -> 19
, U11^#_0(10, 3, 8) -> 19
, U11^#_0(10, 3, 10) -> 19
, U11^#_0(10, 4, 3) -> 19
, U11^#_0(10, 4, 4) -> 19
, U11^#_0(10, 4, 8) -> 19
, U11^#_0(10, 4, 10) -> 19
, U11^#_0(10, 8, 3) -> 19
, U11^#_0(10, 8, 4) -> 19
, U11^#_0(10, 8, 8) -> 19
, U11^#_0(10, 8, 10) -> 19
, U11^#_0(10, 10, 3) -> 19
, U11^#_0(10, 10, 4) -> 19
, U11^#_0(10, 10, 8) -> 19
, U11^#_0(10, 10, 10) -> 19}
9) {active^#(plus(X1, X2)) -> c_8(plus^#(X1, active(X2)))}
The usable rules for this path are the following:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, 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(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s(ok(X)) -> ok(s(X))
, active^#(plus(X1, X2)) -> c_8(plus^#(X1, active(X2)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [2]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [2]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [1] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active^#(plus(X1, X2)) -> c_8(plus^#(X1, active(X2)))}
and weakly orienting the rules
{ active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(plus(X1, X2)) -> c_8(plus^#(X1, active(X2)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [1] x1 + [3]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(plus(N, 0())) -> mark(N)}
and weakly orienting the rules
{ active^#(plus(X1, X2)) -> c_8(plus^#(X1, active(X2)))
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(plus(N, 0())) -> mark(N)}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [8]
plus(x1, x2) = [1] x1 + [1] x2 + [2]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [1] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))}
and weakly orienting the rules
{ active(plus(N, 0())) -> mark(N)
, active^#(plus(X1, X2)) -> c_8(plus^#(X1, active(X2)))
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [9]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [8]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [8]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [1] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U12(tt(), M, N)) -> mark(s(plus(N, M)))}
and weakly orienting the rules
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(plus(N, 0())) -> mark(N)
, active^#(plus(X1, X2)) -> c_8(plus^#(X1, active(X2)))
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U12(tt(), M, N)) -> mark(s(plus(N, M)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [1] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(plus(N, 0())) -> mark(N)
, active^#(plus(X1, X2)) -> c_8(plus^#(X1, active(X2)))
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
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(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(plus(N, 0())) -> mark(N)
, active^#(plus(X1, X2)) -> c_8(plus^#(X1, active(X2)))
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ tt_0() -> 3
, mark_0(3) -> 4
, mark_0(4) -> 4
, mark_0(8) -> 4
, mark_0(10) -> 4
, 0_0() -> 8
, ok_0(3) -> 10
, ok_0(4) -> 10
, ok_0(8) -> 10
, ok_0(10) -> 10
, active^#_0(3) -> 12
, active^#_0(4) -> 12
, active^#_0(8) -> 12
, active^#_0(10) -> 12
, plus^#_0(3, 3) -> 24
, plus^#_0(3, 4) -> 24
, plus^#_0(3, 8) -> 24
, plus^#_0(3, 10) -> 24
, plus^#_0(4, 3) -> 24
, plus^#_0(4, 4) -> 24
, plus^#_0(4, 8) -> 24
, plus^#_0(4, 10) -> 24
, plus^#_0(8, 3) -> 24
, plus^#_0(8, 4) -> 24
, plus^#_0(8, 8) -> 24
, plus^#_0(8, 10) -> 24
, plus^#_0(10, 3) -> 24
, plus^#_0(10, 4) -> 24
, plus^#_0(10, 8) -> 24
, plus^#_0(10, 10) -> 24}
10)
{active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2))}
The usable rules for this path are the following:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, 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(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s(ok(X)) -> ok(s(X))
, active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [2]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [2]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2))}
and weakly orienting the rules
{ active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [3]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(plus(N, 0())) -> mark(N)}
and weakly orienting the rules
{ active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2))
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(plus(N, 0())) -> mark(N)}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [8]
plus(x1, x2) = [1] x1 + [1] x2 + [2]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))}
and weakly orienting the rules
{ active(plus(N, 0())) -> mark(N)
, active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2))
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [9]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [8]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [8]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(U12(tt(), M, N)) -> mark(s(plus(N, M)))}
and weakly orienting the rules
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(plus(N, 0())) -> mark(N)
, active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2))
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(U12(tt(), M, N)) -> mark(s(plus(N, M)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(plus(N, 0())) -> mark(N)
, active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2))
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
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(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(plus(N, 0())) -> mark(N)
, active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2))
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ tt_0() -> 3
, mark_0(3) -> 4
, mark_0(4) -> 4
, mark_0(8) -> 4
, mark_0(10) -> 4
, 0_0() -> 8
, ok_0(3) -> 10
, ok_0(4) -> 10
, ok_0(8) -> 10
, ok_0(10) -> 10
, active^#_0(3) -> 12
, active^#_0(4) -> 12
, active^#_0(8) -> 12
, active^#_0(10) -> 12
, plus^#_0(3, 3) -> 24
, plus^#_0(3, 4) -> 24
, plus^#_0(3, 8) -> 24
, plus^#_0(3, 10) -> 24
, plus^#_0(4, 3) -> 24
, plus^#_0(4, 4) -> 24
, plus^#_0(4, 8) -> 24
, plus^#_0(4, 10) -> 24
, plus^#_0(8, 3) -> 24
, plus^#_0(8, 4) -> 24
, plus^#_0(8, 8) -> 24
, plus^#_0(8, 10) -> 24
, plus^#_0(10, 3) -> 24
, plus^#_0(10, 4) -> 24
, plus^#_0(10, 8) -> 24
, plus^#_0(10, 10) -> 24}
11)
{ proper^#(U11(X1, X2, X3)) ->
c_14(U11^#(proper(X1), proper(X2), proper(X3)))
, U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))
, U11^#(mark(X1), X2, X3) -> c_9(U11^#(X1, X2, X3))}
The usable rules for this path are the following:
{ proper(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, 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(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s(ok(X)) -> ok(s(X))
, proper^#(U11(X1, X2, X3)) ->
c_14(U11^#(proper(X1), proper(X2), proper(X3)))
, U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))
, U11^#(mark(X1), X2, X3) -> c_9(U11^#(X1, X2, X3))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [4]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [5]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_14(x1) = [1] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(U11(X1, X2, X3)) ->
c_14(U11^#(proper(X1), proper(X2), proper(X3)))}
and weakly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(U11(X1, X2, X3)) ->
c_14(U11^#(proper(X1), proper(X2), proper(X3)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [4]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_14(x1) = [1] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())}
and weakly orienting the rules
{ proper^#(U11(X1, X2, X3)) ->
c_14(U11^#(proper(X1), proper(X2), proper(X3)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [1] x1 + [9]
c_14(x1) = [1] x1 + [5]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{U11^#(mark(X1), X2, X3) -> c_9(U11^#(X1, X2, X3))}
and weakly orienting the rules
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())
, proper^#(U11(X1, X2, X3)) ->
c_14(U11^#(proper(X1), proper(X2), proper(X3)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{U11^#(mark(X1), X2, X3) -> c_9(U11^#(X1, X2, X3))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [4]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [4]
proper(x1) = [1] x1 + [2]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [5]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [3]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [11]
c_14(x1) = [1] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ U11^#(mark(X1), X2, X3) -> c_9(U11^#(X1, X2, X3))
, proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())
, proper^#(U11(X1, X2, X3)) ->
c_14(U11^#(proper(X1), proper(X2), proper(X3)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))}
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(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ U11^#(mark(X1), X2, X3) -> c_9(U11^#(X1, X2, X3))
, proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())
, proper^#(U11(X1, X2, X3)) ->
c_14(U11^#(proper(X1), proper(X2), proper(X3)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U11^#(ok(X1), ok(X2), ok(X3)) -> c_20(U11^#(X1, X2, X3))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ tt_0() -> 2
, mark_0(2) -> 2
, 0_0() -> 2
, ok_0(2) -> 2
, U11^#_0(2, 2, 2) -> 1
, c_9_0(1) -> 1
, proper^#_0(2) -> 1
, c_20_0(1) -> 1}
12)
{ proper^#(plus(X1, X2)) -> c_18(plus^#(proper(X1), proper(X2)))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))
, plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
The usable rules for this path are the following:
{ proper(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, 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(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s(ok(X)) -> ok(s(X))
, proper^#(plus(X1, X2)) -> c_18(plus^#(proper(X1), proper(X2)))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))
, plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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) = [1] x1 + [1]
proper^#(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [1] x1 + [1]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(plus(X1, X2)) -> c_18(plus^#(proper(X1), proper(X2)))}
and weakly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(plus(X1, X2)) -> c_18(plus^#(proper(X1), proper(X2)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [1] x1 + [3]
c_13(x1) = [1] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [1] x1 + [1]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
and weakly orienting the rules
{ proper^#(plus(X1, X2)) -> c_18(plus^#(proper(X1), proper(X2)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [8]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [1] x1 + [2]
c_13(x1) = [1] x1 + [1]
proper^#(x1) = [1] x1 + [9]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [1] x1 + [1]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())}
and weakly orienting the rules
{ plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))
, proper^#(plus(X1, X2)) -> c_18(plus^#(proper(X1), proper(X2)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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) = [1] x1 + [0]
proper^#(x1) = [1] x1 + [15]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [1] x1 + [13]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())
, plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))
, proper^#(plus(X1, X2)) -> c_18(plus^#(proper(X1), proper(X2)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(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(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())
, plus^#(X1, mark(X2)) -> c_13(plus^#(X1, X2))
, plus^#(mark(X1), X2) -> c_12(plus^#(X1, X2))
, proper^#(plus(X1, X2)) -> c_18(plus^#(proper(X1), proper(X2)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ tt_0() -> 2
, mark_0(2) -> 2
, 0_0() -> 2
, ok_0(2) -> 2
, plus^#_0(2, 2) -> 1
, c_12_0(1) -> 1
, c_13_0(1) -> 1
, proper^#_0(2) -> 1
, c_23_0(1) -> 1}
13)
{ proper^#(U12(X1, X2, X3)) ->
c_16(U12^#(proper(X1), proper(X2), proper(X3)))
, U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))
, U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))}
The usable rules for this path are the following:
{ proper(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, 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(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s(ok(X)) -> ok(s(X))
, proper^#(U12(X1, X2, X3)) ->
c_16(U12^#(proper(X1), proper(X2), proper(X3)))
, U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))
, U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [1] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [1] x1 + [1]
c_22(x1) = [0] x1 + [0]
c_23(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(U12(X1, X2, X3)) ->
c_16(U12^#(proper(X1), proper(X2), proper(X3)))}
and weakly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(U12(X1, X2, X3)) ->
c_16(U12^#(proper(X1), proper(X2), proper(X3)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [4]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [1] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [1] x1 + [1]
c_22(x1) = [0] x1 + [0]
c_23(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))}
and weakly orienting the rules
{ proper^#(U12(X1, X2, X3)) ->
c_16(U12^#(proper(X1), proper(X2), proper(X3)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [8]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [5]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [4]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [1] x1 + [1]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())}
and weakly orienting the rules
{ U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))
, proper^#(U12(X1, X2, X3)) ->
c_16(U12^#(proper(X1), proper(X2), proper(X3)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [3]
mark(x1) = [1] x1 + [8]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [4]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [1] x1 + [9]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [1] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())
, U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))
, proper^#(U12(X1, X2, X3)) ->
c_16(U12^#(proper(X1), proper(X2), proper(X3)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))}
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(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())
, U12^#(mark(X1), X2, X3) -> c_10(U12^#(X1, X2, X3))
, proper^#(U12(X1, X2, X3)) ->
c_16(U12^#(proper(X1), proper(X2), proper(X3)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, U12^#(ok(X1), ok(X2), ok(X3)) -> c_21(U12^#(X1, X2, X3))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ tt_0() -> 2
, mark_0(2) -> 2
, 0_0() -> 2
, ok_0(2) -> 2
, U12^#_0(2, 2, 2) -> 1
, c_10_0(1) -> 1
, proper^#_0(2) -> 1
, c_21_0(1) -> 1}
14)
{active^#(s(X)) -> c_6(s^#(active(X)))}
The usable rules for this path are the following:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, 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(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s(ok(X)) -> ok(s(X))
, active^#(s(X)) -> c_6(s^#(active(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active^#(s(X)) -> c_6(s^#(active(X)))}
and weakly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(s(X)) -> c_6(s^#(active(X)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [9]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [3]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))}
and weakly orienting the rules
{ active^#(s(X)) -> c_6(s^#(active(X)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [8]
plus(x1, x2) = [1] x1 + [1] x2 + [2]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))}
and weakly orienting the rules
{ active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active^#(s(X)) -> c_6(s^#(active(X)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [8]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [8]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [5]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active^#(s(X)) -> c_6(s^#(active(X)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
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(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
, active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
, active(s(X)) -> s(active(X))
, active(plus(X1, X2)) -> plus(active(X1), X2)
, active(plus(X1, X2)) -> plus(X1, active(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ active(U11(tt(), M, N)) -> mark(U12(tt(), M, N))
, active(U12(tt(), M, N)) -> mark(s(plus(N, M)))
, active(plus(N, 0())) -> mark(N)
, active(plus(N, s(M))) -> mark(U11(tt(), M, N))
, active^#(s(X)) -> c_6(s^#(active(X)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ tt_0() -> 3
, mark_0(3) -> 4
, mark_0(4) -> 4
, mark_0(8) -> 4
, mark_0(10) -> 4
, 0_0() -> 8
, ok_0(3) -> 10
, ok_0(4) -> 10
, ok_0(8) -> 10
, ok_0(10) -> 10
, active^#_0(3) -> 12
, active^#_0(4) -> 12
, active^#_0(8) -> 12
, active^#_0(10) -> 12
, s^#_0(3) -> 16
, s^#_0(4) -> 16
, s^#_0(8) -> 16
, s^#_0(10) -> 16}
15)
{ proper^#(s(X)) -> c_17(s^#(proper(X)))
, s^#(ok(X)) -> c_22(s^#(X))
, s^#(mark(X)) -> c_11(s^#(X))}
The usable rules for this path are the following:
{ proper(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, 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(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s(ok(X)) -> ok(s(X))
, proper^#(s(X)) -> c_17(s^#(proper(X)))
, s^#(ok(X)) -> c_22(s^#(X))
, s^#(mark(X)) -> c_11(s^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s^#(ok(X)) -> c_22(s^#(X))
, s^#(mark(X)) -> c_11(s^#(X))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s^#(ok(X)) -> c_22(s^#(X))
, s^#(mark(X)) -> c_11(s^#(X))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [1]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [1]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [4]
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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [1]
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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(s(X)) -> c_17(s^#(proper(X)))}
and weakly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s^#(ok(X)) -> c_22(s^#(X))
, s^#(mark(X)) -> c_11(s^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(s(X)) -> c_17(s^#(proper(X)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [2]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [1] x1 + [9]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())}
and weakly orienting the rules
{ proper^#(s(X)) -> c_17(s^#(proper(X)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s^#(ok(X)) -> c_22(s^#(X))
, s^#(mark(X)) -> c_11(s^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [1] x1 + [5]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())
, proper^#(s(X)) -> c_17(s^#(proper(X)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s^#(ok(X)) -> c_22(s^#(X))
, s^#(mark(X)) -> c_11(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:
{ proper(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())
, proper^#(s(X)) -> c_17(s^#(proper(X)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s^#(ok(X)) -> c_22(s^#(X))
, s^#(mark(X)) -> c_11(s^#(X))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ tt_0() -> 3
, mark_0(3) -> 4
, mark_0(4) -> 4
, mark_0(8) -> 4
, mark_0(10) -> 4
, 0_0() -> 8
, ok_0(3) -> 10
, ok_0(4) -> 10
, ok_0(8) -> 10
, ok_0(10) -> 10
, s^#_0(3) -> 16
, s^#_0(4) -> 16
, s^#_0(8) -> 16
, s^#_0(10) -> 16
, c_11_0(16) -> 16
, proper^#_0(3) -> 31
, proper^#_0(4) -> 31
, proper^#_0(8) -> 31
, proper^#_0(10) -> 31
, c_22_0(16) -> 16}
16)
{proper^#(U12(X1, X2, X3)) ->
c_16(U12^#(proper(X1), proper(X2), proper(X3)))}
The usable rules for this path are the following:
{ proper(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, 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(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s(ok(X)) -> ok(s(X))
, proper^#(U12(X1, X2, X3)) ->
c_16(U12^#(proper(X1), proper(X2), proper(X3)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [1] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(U12(X1, X2, X3)) ->
c_16(U12^#(proper(X1), proper(X2), proper(X3)))}
and weakly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(U12(X1, X2, X3)) ->
c_16(U12^#(proper(X1), proper(X2), proper(X3)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [1] x1 + [9]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [1] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())}
and weakly orienting the rules
{ proper^#(U12(X1, X2, X3)) ->
c_16(U12^#(proper(X1), proper(X2), proper(X3)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [1]
mark(x1) = [1] x1 + [8]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [1] x1 + [8]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [1] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())
, proper^#(U12(X1, X2, X3)) ->
c_16(U12^#(proper(X1), proper(X2), proper(X3)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
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(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())
, proper^#(U12(X1, X2, X3)) ->
c_16(U12^#(proper(X1), proper(X2), proper(X3)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ tt_0() -> 3
, mark_0(3) -> 4
, mark_0(4) -> 4
, mark_0(8) -> 4
, mark_0(10) -> 4
, 0_0() -> 8
, ok_0(3) -> 10
, ok_0(4) -> 10
, ok_0(8) -> 10
, ok_0(10) -> 10
, U12^#_0(3, 3, 3) -> 14
, U12^#_0(3, 3, 4) -> 14
, U12^#_0(3, 3, 8) -> 14
, U12^#_0(3, 3, 10) -> 14
, U12^#_0(3, 4, 3) -> 14
, U12^#_0(3, 4, 4) -> 14
, U12^#_0(3, 4, 8) -> 14
, U12^#_0(3, 4, 10) -> 14
, U12^#_0(3, 8, 3) -> 14
, U12^#_0(3, 8, 4) -> 14
, U12^#_0(3, 8, 8) -> 14
, U12^#_0(3, 8, 10) -> 14
, U12^#_0(3, 10, 3) -> 14
, U12^#_0(3, 10, 4) -> 14
, U12^#_0(3, 10, 8) -> 14
, U12^#_0(3, 10, 10) -> 14
, U12^#_0(4, 3, 3) -> 14
, U12^#_0(4, 3, 4) -> 14
, U12^#_0(4, 3, 8) -> 14
, U12^#_0(4, 3, 10) -> 14
, U12^#_0(4, 4, 3) -> 14
, U12^#_0(4, 4, 4) -> 14
, U12^#_0(4, 4, 8) -> 14
, U12^#_0(4, 4, 10) -> 14
, U12^#_0(4, 8, 3) -> 14
, U12^#_0(4, 8, 4) -> 14
, U12^#_0(4, 8, 8) -> 14
, U12^#_0(4, 8, 10) -> 14
, U12^#_0(4, 10, 3) -> 14
, U12^#_0(4, 10, 4) -> 14
, U12^#_0(4, 10, 8) -> 14
, U12^#_0(4, 10, 10) -> 14
, U12^#_0(8, 3, 3) -> 14
, U12^#_0(8, 3, 4) -> 14
, U12^#_0(8, 3, 8) -> 14
, U12^#_0(8, 3, 10) -> 14
, U12^#_0(8, 4, 3) -> 14
, U12^#_0(8, 4, 4) -> 14
, U12^#_0(8, 4, 8) -> 14
, U12^#_0(8, 4, 10) -> 14
, U12^#_0(8, 8, 3) -> 14
, U12^#_0(8, 8, 4) -> 14
, U12^#_0(8, 8, 8) -> 14
, U12^#_0(8, 8, 10) -> 14
, U12^#_0(8, 10, 3) -> 14
, U12^#_0(8, 10, 4) -> 14
, U12^#_0(8, 10, 8) -> 14
, U12^#_0(8, 10, 10) -> 14
, U12^#_0(10, 3, 3) -> 14
, U12^#_0(10, 3, 4) -> 14
, U12^#_0(10, 3, 8) -> 14
, U12^#_0(10, 3, 10) -> 14
, U12^#_0(10, 4, 3) -> 14
, U12^#_0(10, 4, 4) -> 14
, U12^#_0(10, 4, 8) -> 14
, U12^#_0(10, 4, 10) -> 14
, U12^#_0(10, 8, 3) -> 14
, U12^#_0(10, 8, 4) -> 14
, U12^#_0(10, 8, 8) -> 14
, U12^#_0(10, 8, 10) -> 14
, U12^#_0(10, 10, 3) -> 14
, U12^#_0(10, 10, 4) -> 14
, U12^#_0(10, 10, 8) -> 14
, U12^#_0(10, 10, 10) -> 14
, proper^#_0(3) -> 31
, proper^#_0(4) -> 31
, proper^#_0(8) -> 31
, proper^#_0(10) -> 31}
17)
{proper^#(U11(X1, X2, X3)) ->
c_14(U11^#(proper(X1), proper(X2), proper(X3)))}
The usable rules for this path are the following:
{ proper(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, 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(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s(ok(X)) -> ok(s(X))
, proper^#(U11(X1, X2, X3)) ->
c_14(U11^#(proper(X1), proper(X2), proper(X3)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [1] x1 + [1]
c_14(x1) = [1] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(U11(X1, X2, X3)) ->
c_14(U11^#(proper(X1), proper(X2), proper(X3)))}
and weakly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(U11(X1, X2, X3)) ->
c_14(U11^#(proper(X1), proper(X2), proper(X3)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [1] x1 + [9]
c_14(x1) = [1] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())}
and weakly orienting the rules
{ proper^#(U11(X1, X2, X3)) ->
c_14(U11^#(proper(X1), proper(X2), proper(X3)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [1]
mark(x1) = [1] x1 + [8]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [1] x1 + [8]
c_14(x1) = [1] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())
, proper^#(U11(X1, X2, X3)) ->
c_14(U11^#(proper(X1), proper(X2), proper(X3)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
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(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())
, proper^#(U11(X1, X2, X3)) ->
c_14(U11^#(proper(X1), proper(X2), proper(X3)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ tt_0() -> 3
, mark_0(3) -> 4
, mark_0(4) -> 4
, mark_0(8) -> 4
, mark_0(10) -> 4
, 0_0() -> 8
, ok_0(3) -> 10
, ok_0(4) -> 10
, ok_0(8) -> 10
, ok_0(10) -> 10
, U11^#_0(3, 3, 3) -> 19
, U11^#_0(3, 3, 4) -> 19
, U11^#_0(3, 3, 8) -> 19
, U11^#_0(3, 3, 10) -> 19
, U11^#_0(3, 4, 3) -> 19
, U11^#_0(3, 4, 4) -> 19
, U11^#_0(3, 4, 8) -> 19
, U11^#_0(3, 4, 10) -> 19
, U11^#_0(3, 8, 3) -> 19
, U11^#_0(3, 8, 4) -> 19
, U11^#_0(3, 8, 8) -> 19
, U11^#_0(3, 8, 10) -> 19
, U11^#_0(3, 10, 3) -> 19
, U11^#_0(3, 10, 4) -> 19
, U11^#_0(3, 10, 8) -> 19
, U11^#_0(3, 10, 10) -> 19
, U11^#_0(4, 3, 3) -> 19
, U11^#_0(4, 3, 4) -> 19
, U11^#_0(4, 3, 8) -> 19
, U11^#_0(4, 3, 10) -> 19
, U11^#_0(4, 4, 3) -> 19
, U11^#_0(4, 4, 4) -> 19
, U11^#_0(4, 4, 8) -> 19
, U11^#_0(4, 4, 10) -> 19
, U11^#_0(4, 8, 3) -> 19
, U11^#_0(4, 8, 4) -> 19
, U11^#_0(4, 8, 8) -> 19
, U11^#_0(4, 8, 10) -> 19
, U11^#_0(4, 10, 3) -> 19
, U11^#_0(4, 10, 4) -> 19
, U11^#_0(4, 10, 8) -> 19
, U11^#_0(4, 10, 10) -> 19
, U11^#_0(8, 3, 3) -> 19
, U11^#_0(8, 3, 4) -> 19
, U11^#_0(8, 3, 8) -> 19
, U11^#_0(8, 3, 10) -> 19
, U11^#_0(8, 4, 3) -> 19
, U11^#_0(8, 4, 4) -> 19
, U11^#_0(8, 4, 8) -> 19
, U11^#_0(8, 4, 10) -> 19
, U11^#_0(8, 8, 3) -> 19
, U11^#_0(8, 8, 4) -> 19
, U11^#_0(8, 8, 8) -> 19
, U11^#_0(8, 8, 10) -> 19
, U11^#_0(8, 10, 3) -> 19
, U11^#_0(8, 10, 4) -> 19
, U11^#_0(8, 10, 8) -> 19
, U11^#_0(8, 10, 10) -> 19
, U11^#_0(10, 3, 3) -> 19
, U11^#_0(10, 3, 4) -> 19
, U11^#_0(10, 3, 8) -> 19
, U11^#_0(10, 3, 10) -> 19
, U11^#_0(10, 4, 3) -> 19
, U11^#_0(10, 4, 4) -> 19
, U11^#_0(10, 4, 8) -> 19
, U11^#_0(10, 4, 10) -> 19
, U11^#_0(10, 8, 3) -> 19
, U11^#_0(10, 8, 4) -> 19
, U11^#_0(10, 8, 8) -> 19
, U11^#_0(10, 8, 10) -> 19
, U11^#_0(10, 10, 3) -> 19
, U11^#_0(10, 10, 4) -> 19
, U11^#_0(10, 10, 8) -> 19
, U11^#_0(10, 10, 10) -> 19
, proper^#_0(3) -> 31
, proper^#_0(4) -> 31
, proper^#_0(8) -> 31
, proper^#_0(10) -> 31}
18)
{proper^#(plus(X1, X2)) -> c_18(plus^#(proper(X1), proper(X2)))}
The usable rules for this path are the following:
{ proper(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, 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(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s(ok(X)) -> ok(s(X))
, proper^#(plus(X1, X2)) -> c_18(plus^#(proper(X1), proper(X2)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(plus(X1, X2)) -> c_18(plus^#(proper(X1), proper(X2)))}
and weakly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(plus(X1, X2)) -> c_18(plus^#(proper(X1), proper(X2)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [2]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [8]
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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [1] x1 + [9]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())}
and weakly orienting the rules
{ proper^#(plus(X1, X2)) -> c_18(plus^#(proper(X1), proper(X2)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [2]
tt() = [0]
mark(x1) = [1] x1 + [14]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [2]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [1] x1 + [8]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [1] x1 + [3]
c_19() = [0]
c_20(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
c_23(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())
, proper^#(plus(X1, X2)) -> c_18(plus^#(proper(X1), proper(X2)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
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(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())
, proper^#(plus(X1, X2)) -> c_18(plus^#(proper(X1), proper(X2)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ tt_0() -> 3
, mark_0(3) -> 4
, mark_0(4) -> 4
, mark_0(8) -> 4
, mark_0(10) -> 4
, 0_0() -> 8
, ok_0(3) -> 10
, ok_0(4) -> 10
, ok_0(8) -> 10
, ok_0(10) -> 10
, plus^#_0(3, 3) -> 24
, plus^#_0(3, 4) -> 24
, plus^#_0(3, 8) -> 24
, plus^#_0(3, 10) -> 24
, plus^#_0(4, 3) -> 24
, plus^#_0(4, 4) -> 24
, plus^#_0(4, 8) -> 24
, plus^#_0(4, 10) -> 24
, plus^#_0(8, 3) -> 24
, plus^#_0(8, 4) -> 24
, plus^#_0(8, 8) -> 24
, plus^#_0(8, 10) -> 24
, plus^#_0(10, 3) -> 24
, plus^#_0(10, 4) -> 24
, plus^#_0(10, 8) -> 24
, plus^#_0(10, 10) -> 24
, proper^#_0(3) -> 31
, proper^#_0(4) -> 31
, proper^#_0(8) -> 31
, proper^#_0(10) -> 31}
19)
{proper^#(s(X)) -> c_17(s^#(proper(X)))}
The usable rules for this path are the following:
{ proper(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, 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(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(tt()) -> ok(tt())
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, proper(0()) -> ok(0())
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
, s(ok(X)) -> ok(s(X))
, proper^#(s(X)) -> c_17(s^#(proper(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(s(X)) -> c_17(s^#(proper(X)))}
and weakly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(s(X)) -> c_17(s^#(proper(X)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [8]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [3]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [1] x1 + [9]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())}
and weakly orienting the rules
{ proper^#(s(X)) -> c_17(s^#(proper(X)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [2]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [1] x1 + [5]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())
, proper^#(s(X)) -> c_17(s^#(proper(X)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
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(U11(X1, X2, X3)) ->
U11(proper(X1), proper(X2), proper(X3))
, proper(U12(X1, X2, X3)) ->
U12(proper(X1), proper(X2), proper(X3))
, proper(s(X)) -> s(proper(X))
, proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
, plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
, U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
, s(mark(X)) -> mark(s(X))
, s(ok(X)) -> ok(s(X))}
Weak Rules:
{ proper(tt()) -> ok(tt())
, proper(0()) -> ok(0())
, proper^#(s(X)) -> c_17(s^#(proper(X)))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
, U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ tt_0() -> 3
, mark_0(3) -> 4
, mark_0(4) -> 4
, mark_0(8) -> 4
, mark_0(10) -> 4
, 0_0() -> 8
, ok_0(3) -> 10
, ok_0(4) -> 10
, ok_0(8) -> 10
, ok_0(10) -> 10
, s^#_0(3) -> 16
, s^#_0(4) -> 16
, s^#_0(8) -> 16
, s^#_0(10) -> 16
, proper^#_0(3) -> 31
, proper^#_0(4) -> 31
, proper^#_0(8) -> 31
, proper^#_0(10) -> 31}
20)
{ active^#(U12(tt(), M, N)) -> c_1(s^#(plus(N, M)))
, s^#(ok(X)) -> c_22(s^#(X))
, s^#(mark(X)) -> c_11(s^#(X))}
The usable rules for this path are the following:
{ plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))}
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:
{ plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, active^#(U12(tt(), M, N)) -> c_1(s^#(plus(N, M)))
, s^#(ok(X)) -> c_22(s^#(X))
, s^#(mark(X)) -> c_11(s^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(U12(tt(), M, N)) -> c_1(s^#(plus(N, M)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(U12(tt(), M, N)) -> c_1(s^#(plus(N, M)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [0] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [1] x1 + [1]
s^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [4]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, s^#(ok(X)) -> c_22(s^#(X))}
and weakly orienting the rules
{active^#(U12(tt(), M, N)) -> c_1(s^#(plus(N, M)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, s^#(ok(X)) -> c_22(s^#(X))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [0] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [3]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [1] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{s^#(mark(X)) -> c_11(s^#(X))}
and weakly orienting the rules
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, s^#(ok(X)) -> c_22(s^#(X))
, active^#(U12(tt(), M, N)) -> c_1(s^#(plus(N, M)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{s^#(mark(X)) -> c_11(s^#(X))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [8]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [3]
s(x1) = [0] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [1] x1 + [1]
s^#(x1) = [1] x1 + [8]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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:
{ plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))}
Weak Rules:
{ s^#(mark(X)) -> c_11(s^#(X))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, s^#(ok(X)) -> c_22(s^#(X))
, active^#(U12(tt(), M, N)) -> c_1(s^#(plus(N, M)))}
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:
{ plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))}
Weak Rules:
{ s^#(mark(X)) -> c_11(s^#(X))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, s^#(ok(X)) -> c_22(s^#(X))
, active^#(U12(tt(), M, N)) -> c_1(s^#(plus(N, M)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ tt_0() -> 3
, mark_0(3) -> 4
, mark_0(4) -> 4
, mark_0(10) -> 4
, ok_0(3) -> 10
, ok_0(4) -> 10
, ok_0(10) -> 10
, active^#_0(3) -> 12
, active^#_0(4) -> 12
, active^#_0(10) -> 12
, s^#_0(3) -> 16
, s^#_0(4) -> 16
, s^#_0(10) -> 16
, c_11_0(16) -> 16
, c_22_0(16) -> 16}
21)
{active^#(U12(tt(), M, N)) -> c_1(s^#(plus(N, M)))}
The usable rules for this path are the following:
{ plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))}
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:
{ plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))
, plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, active^#(U12(tt(), M, N)) -> c_1(s^#(plus(N, M)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(U12(tt(), M, N)) -> c_1(s^#(plus(N, M)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(U12(tt(), M, N)) -> c_1(s^#(plus(N, M)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [0] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [1] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))}
and weakly orienting the rules
{active^#(U12(tt(), M, N)) -> c_1(s^#(plus(N, M)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
mark(x1) = [1] x1 + [0]
U12(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
s(x1) = [0] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [1] x1 + [0]
s^#(x1) = [1] x1 + [2]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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:
{ plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))}
Weak Rules:
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, active^#(U12(tt(), M, N)) -> c_1(s^#(plus(N, M)))}
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:
{ plus(mark(X1), X2) -> mark(plus(X1, X2))
, plus(X1, mark(X2)) -> mark(plus(X1, X2))}
Weak Rules:
{ plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
, active^#(U12(tt(), M, N)) -> c_1(s^#(plus(N, M)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ tt_0() -> 3
, mark_0(3) -> 4
, mark_0(4) -> 4
, mark_0(10) -> 4
, ok_0(3) -> 10
, ok_0(4) -> 10
, ok_0(10) -> 10
, active^#_0(3) -> 12
, active^#_0(4) -> 12
, active^#_0(10) -> 12
, s^#_0(3) -> 16
, s^#_0(4) -> 16
, s^#_0(10) -> 16}
22)
{active^#(plus(N, s(M))) -> c_3(U11^#(tt(), M, N))}
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]
U11(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
mark(x1) = [0] x1 + [0]
U12(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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^#(plus(N, s(M))) -> c_3(U11^#(tt(), M, N))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(plus(N, s(M))) -> c_3(U11^#(tt(), M, N))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(plus(N, s(M))) -> c_3(U11^#(tt(), M, N))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
mark(x1) = [0] x1 + [0]
U12(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
U11^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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^#(plus(N, s(M))) -> c_3(U11^#(tt(), M, N))}
Details:
The given problem does not contain any strict rules
23)
{active^#(U11(tt(), M, N)) -> c_0(U12^#(tt(), M, N))}
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]
U11(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
mark(x1) = [0] x1 + [0]
U12(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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(), M, N)) -> c_0(U12^#(tt(), M, N))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(U11(tt(), M, N)) -> c_0(U12^#(tt(), M, N))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(U11(tt(), M, N)) -> c_0(U12^#(tt(), M, N))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tt() = [0]
mark(x1) = [0] x1 + [0]
U12(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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(), M, N)) -> c_0(U12^#(tt(), M, N))}
Details:
The given problem does not contain any strict rules
24)
{active^#(plus(N, 0())) -> c_2()}
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]
U11(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
mark(x1) = [0] x1 + [0]
U12(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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^#(plus(N, 0())) -> c_2()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(plus(N, 0())) -> c_2()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(plus(N, 0())) -> c_2()}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
mark(x1) = [0] x1 + [0]
U12(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [0] x1 + [0]
plus(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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^#(plus(N, 0())) -> c_2()}
Details:
The given problem does not contain any strict rules
25)
{proper^#(tt()) -> 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]
U11(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
mark(x1) = [0] x1 + [0]
U12(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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_15()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{proper^#(tt()) -> c_15()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(tt()) -> c_15()}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
mark(x1) = [0] x1 + [0]
U12(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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_15()}
Details:
The given problem does not contain any strict rules
26)
{proper^#(0()) -> 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]
U11(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
mark(x1) = [0] x1 + [0]
U12(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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_19()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{proper^#(0()) -> c_19()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(0()) -> c_19()}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
U11(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
mark(x1) = [0] x1 + [0]
U12(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [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]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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]
proper^#(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [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(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_24(x1) = [0] x1 + [0]
c_25(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_19()}
Details:
The given problem does not contain any strict rules