'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ U11(tt(), M, N) -> U12(tt(), activate(M), activate(N))
, U12(tt(), M, N) -> s(plus(activate(N), activate(M)))
, plus(N, 0()) -> N
, plus(N, s(M)) -> U11(tt(), M, N)
, activate(X) -> X}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ U11^#(tt(), M, N) -> c_0(U12^#(tt(), activate(M), activate(N)))
, U12^#(tt(), M, N) -> c_1(plus^#(activate(N), activate(M)))
, plus^#(N, 0()) -> c_2()
, plus^#(N, s(M)) -> c_3(U11^#(tt(), M, N))
, activate^#(X) -> c_4()}
The usable rules are:
{activate(X) -> X}
The estimated dependency graph contains the following edges:
{U11^#(tt(), M, N) -> c_0(U12^#(tt(), activate(M), activate(N)))}
==> {U12^#(tt(), M, N) -> c_1(plus^#(activate(N), activate(M)))}
{U12^#(tt(), M, N) -> c_1(plus^#(activate(N), activate(M)))}
==> {plus^#(N, s(M)) -> c_3(U11^#(tt(), M, N))}
{U12^#(tt(), M, N) -> c_1(plus^#(activate(N), activate(M)))}
==> {plus^#(N, 0()) -> c_2()}
{plus^#(N, s(M)) -> c_3(U11^#(tt(), M, N))}
==> {U11^#(tt(), M, N) ->
c_0(U12^#(tt(), activate(M), activate(N)))}
We consider the following path(s):
1) { U11^#(tt(), M, N) ->
c_0(U12^#(tt(), activate(M), activate(N)))
, plus^#(N, s(M)) -> c_3(U11^#(tt(), M, N))
, U12^#(tt(), M, N) -> c_1(plus^#(activate(N), activate(M)))}
The usable rules for this path are the following:
{activate(X) -> X}
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
U11(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
U12(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
activate(x1) = [1] x1 + [4]
s(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
c_4() = [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:
{ U11^#(tt(), M, N) -> c_0(U12^#(tt(), activate(M), activate(N)))
, plus^#(N, s(M)) -> c_3(U11^#(tt(), M, N))
, U12^#(tt(), M, N) -> c_1(plus^#(activate(N), activate(M)))}
Weak Rules: {activate(X) -> X}
Details:
We apply the weight gap principle, strictly orienting the rules
{plus^#(N, s(M)) -> c_3(U11^#(tt(), M, N))}
and weakly orienting the rules
{activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{plus^#(N, s(M)) -> c_3(U11^#(tt(), M, N))}
Details:
Interpretation Functions:
U11(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
U12(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
activate(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
c_0(x1) = [1] x1 + [1]
U12^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_1(x1) = [1] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [9]
c_2() = [0]
c_3(x1) = [1] x1 + [1]
activate^#(x1) = [0] x1 + [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{U11^#(tt(), M, N) -> c_0(U12^#(tt(), activate(M), activate(N)))}
and weakly orienting the rules
{ plus^#(N, s(M)) -> c_3(U11^#(tt(), M, N))
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{U11^#(tt(), M, N) -> c_0(U12^#(tt(), activate(M), activate(N)))}
Details:
Interpretation Functions:
U11(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
U12(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
activate(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [8]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
c_0(x1) = [1] x1 + [0]
U12^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_1(x1) = [1] x1 + [1]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [3]
activate^#(x1) = [0] x1 + [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{U12^#(tt(), M, N) -> c_1(plus^#(activate(N), activate(M)))}
and weakly orienting the rules
{ U11^#(tt(), M, N) -> c_0(U12^#(tt(), activate(M), activate(N)))
, plus^#(N, s(M)) -> c_3(U11^#(tt(), M, N))
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{U12^#(tt(), M, N) -> c_1(plus^#(activate(N), activate(M)))}
Details:
Interpretation Functions:
U11(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
U12(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
activate(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [8]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
c_0(x1) = [1] x1 + [0]
U12^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
c_1(x1) = [1] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [3]
activate^#(x1) = [0] x1 + [0]
c_4() = [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:
{ U12^#(tt(), M, N) -> c_1(plus^#(activate(N), activate(M)))
, U11^#(tt(), M, N) -> c_0(U12^#(tt(), activate(M), activate(N)))
, plus^#(N, s(M)) -> c_3(U11^#(tt(), M, N))
, activate(X) -> X}
Details:
The given problem does not contain any strict rules
2) { U11^#(tt(), M, N) ->
c_0(U12^#(tt(), activate(M), activate(N)))
, plus^#(N, s(M)) -> c_3(U11^#(tt(), M, N))
, U12^#(tt(), M, N) -> c_1(plus^#(activate(N), activate(M)))
, plus^#(N, 0()) -> c_2()}
The usable rules for this path are the following:
{activate(X) -> X}
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
U11(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
U12(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
activate(x1) = [1] x1 + [4]
s(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
c_4() = [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: {plus^#(N, 0()) -> c_2()}
Weak Rules:
{ activate(X) -> X
, U11^#(tt(), M, N) -> c_0(U12^#(tt(), activate(M), activate(N)))
, plus^#(N, s(M)) -> c_3(U11^#(tt(), M, N))
, U12^#(tt(), M, N) -> c_1(plus^#(activate(N), activate(M)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{plus^#(N, 0()) -> c_2()}
and weakly orienting the rules
{ activate(X) -> X
, U11^#(tt(), M, N) -> c_0(U12^#(tt(), activate(M), activate(N)))
, plus^#(N, s(M)) -> c_3(U11^#(tt(), M, N))
, U12^#(tt(), M, N) -> c_1(plus^#(activate(N), activate(M)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{plus^#(N, 0()) -> c_2()}
Details:
Interpretation Functions:
U11(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
U12(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
activate(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
c_0(x1) = [1] x1 + [0]
U12^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
c_1(x1) = [1] x1 + [0]
plus^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
activate^#(x1) = [0] x1 + [0]
c_4() = [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:
{ plus^#(N, 0()) -> c_2()
, activate(X) -> X
, U11^#(tt(), M, N) -> c_0(U12^#(tt(), activate(M), activate(N)))
, plus^#(N, s(M)) -> c_3(U11^#(tt(), M, N))
, U12^#(tt(), M, N) -> c_1(plus^#(activate(N), activate(M)))}
Details:
The given problem does not contain any strict rules
3) {activate^#(X) -> c_4()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
U11(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
U12(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
activate(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
c_4() = [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: {activate^#(X) -> c_4()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate^#(X) -> c_4()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(X) -> c_4()}
Details:
Interpretation Functions:
U11(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
U12(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
activate(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
activate^#(x1) = [1] x1 + [4]
c_4() = [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: {activate^#(X) -> c_4()}
Details:
The given problem does not contain any strict rules