'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ concat(leaf(), Y) -> Y
, concat(cons(U, V), Y) -> cons(U, concat(V, Y))
, lessleaves(X, leaf()) -> false()
, lessleaves(leaf(), cons(W, Z)) -> true()
, lessleaves(cons(U, V), cons(W, Z)) ->
lessleaves(concat(U, V), concat(W, Z))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ concat^#(leaf(), Y) -> c_0()
, concat^#(cons(U, V), Y) -> c_1(concat^#(V, Y))
, lessleaves^#(X, leaf()) -> c_2()
, lessleaves^#(leaf(), cons(W, Z)) -> c_3()
, lessleaves^#(cons(U, V), cons(W, Z)) ->
c_4(lessleaves^#(concat(U, V), concat(W, Z)))}
The usable rules are:
{ concat(leaf(), Y) -> Y
, concat(cons(U, V), Y) -> cons(U, concat(V, Y))}
The estimated dependency graph contains the following edges:
{concat^#(cons(U, V), Y) -> c_1(concat^#(V, Y))}
==> {concat^#(cons(U, V), Y) -> c_1(concat^#(V, Y))}
{concat^#(cons(U, V), Y) -> c_1(concat^#(V, Y))}
==> {concat^#(leaf(), Y) -> c_0()}
{lessleaves^#(cons(U, V), cons(W, Z)) ->
c_4(lessleaves^#(concat(U, V), concat(W, Z)))}
==> {lessleaves^#(cons(U, V), cons(W, Z)) ->
c_4(lessleaves^#(concat(U, V), concat(W, Z)))}
{lessleaves^#(cons(U, V), cons(W, Z)) ->
c_4(lessleaves^#(concat(U, V), concat(W, Z)))}
==> {lessleaves^#(leaf(), cons(W, Z)) -> c_3()}
{lessleaves^#(cons(U, V), cons(W, Z)) ->
c_4(lessleaves^#(concat(U, V), concat(W, Z)))}
==> {lessleaves^#(X, leaf()) -> c_2()}
We consider the following path(s):
1) { lessleaves^#(cons(U, V), cons(W, Z)) ->
c_4(lessleaves^#(concat(U, V), concat(W, Z)))
, lessleaves^#(leaf(), cons(W, Z)) -> c_3()}
The usable rules for this path are the following:
{ concat(leaf(), Y) -> Y
, concat(cons(U, V), Y) -> cons(U, concat(V, Y))}
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:
{ concat(leaf(), Y) -> Y
, concat(cons(U, V), Y) -> cons(U, concat(V, Y))
, lessleaves^#(cons(U, V), cons(W, Z)) ->
c_4(lessleaves^#(concat(U, V), concat(W, Z)))
, lessleaves^#(leaf(), cons(W, Z)) -> c_3()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ concat(leaf(), Y) -> Y
, lessleaves^#(leaf(), cons(W, Z)) -> c_3()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ concat(leaf(), Y) -> Y
, lessleaves^#(leaf(), cons(W, Z)) -> c_3()}
Details:
Interpretation Functions:
concat(x1, x2) = [1] x1 + [1] x2 + [1]
leaf() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
lessleaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
concat^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
lessleaves^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{lessleaves^#(cons(U, V), cons(W, Z)) ->
c_4(lessleaves^#(concat(U, V), concat(W, Z)))}
and weakly orienting the rules
{ concat(leaf(), Y) -> Y
, lessleaves^#(leaf(), cons(W, Z)) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{lessleaves^#(cons(U, V), cons(W, Z)) ->
c_4(lessleaves^#(concat(U, V), concat(W, Z)))}
Details:
Interpretation Functions:
concat(x1, x2) = [1] x1 + [1] x2 + [1]
leaf() = [9]
cons(x1, x2) = [1] x1 + [1] x2 + [8]
lessleaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
concat^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
lessleaves^#(x1, x2) = [1] x1 + [1] x2 + [14]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [8]
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: {concat(cons(U, V), Y) -> cons(U, concat(V, Y))}
Weak Rules:
{ lessleaves^#(cons(U, V), cons(W, Z)) ->
c_4(lessleaves^#(concat(U, V), concat(W, Z)))
, concat(leaf(), Y) -> Y
, lessleaves^#(leaf(), cons(W, Z)) -> c_3()}
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: {concat(cons(U, V), Y) -> cons(U, concat(V, Y))}
Weak Rules:
{ lessleaves^#(cons(U, V), cons(W, Z)) ->
c_4(lessleaves^#(concat(U, V), concat(W, Z)))
, concat(leaf(), Y) -> Y
, lessleaves^#(leaf(), cons(W, Z)) -> c_3()}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ concat_0(2, 2) -> 4
, concat_0(2, 2) -> 5
, concat_1(2, 2) -> 6
, concat_1(2, 6) -> 6
, leaf_0() -> 2
, leaf_0() -> 4
, leaf_0() -> 5
, leaf_0() -> 6
, cons_0(2, 2) -> 2
, cons_0(2, 2) -> 4
, cons_0(2, 2) -> 5
, cons_0(2, 2) -> 6
, cons_1(2, 6) -> 4
, cons_1(2, 6) -> 5
, cons_1(2, 6) -> 6
, lessleaves^#_0(2, 2) -> 1
, lessleaves^#_0(4, 5) -> 3
, lessleaves^#_1(6, 6) -> 7
, c_3_0() -> 1
, c_3_0() -> 3
, c_3_1() -> 3
, c_3_1() -> 7
, c_4_0(3) -> 1
, c_4_1(7) -> 1
, c_4_1(7) -> 3
, c_4_1(7) -> 7}
2) { lessleaves^#(cons(U, V), cons(W, Z)) ->
c_4(lessleaves^#(concat(U, V), concat(W, Z)))
, lessleaves^#(X, leaf()) -> c_2()}
The usable rules for this path are the following:
{ concat(leaf(), Y) -> Y
, concat(cons(U, V), Y) -> cons(U, concat(V, Y))}
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:
{ concat(leaf(), Y) -> Y
, concat(cons(U, V), Y) -> cons(U, concat(V, Y))
, lessleaves^#(cons(U, V), cons(W, Z)) ->
c_4(lessleaves^#(concat(U, V), concat(W, Z)))
, lessleaves^#(X, leaf()) -> c_2()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ concat(leaf(), Y) -> Y
, lessleaves^#(X, leaf()) -> c_2()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ concat(leaf(), Y) -> Y
, lessleaves^#(X, leaf()) -> c_2()}
Details:
Interpretation Functions:
concat(x1, x2) = [1] x1 + [1] x2 + [1]
leaf() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
lessleaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
concat^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
lessleaves^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{lessleaves^#(cons(U, V), cons(W, Z)) ->
c_4(lessleaves^#(concat(U, V), concat(W, Z)))}
and weakly orienting the rules
{ concat(leaf(), Y) -> Y
, lessleaves^#(X, leaf()) -> c_2()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{lessleaves^#(cons(U, V), cons(W, Z)) ->
c_4(lessleaves^#(concat(U, V), concat(W, Z)))}
Details:
Interpretation Functions:
concat(x1, x2) = [1] x1 + [1] x2 + [1]
leaf() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [8]
lessleaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
concat^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
lessleaves^#(x1, x2) = [1] x1 + [1] x2 + [7]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [9]
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: {concat(cons(U, V), Y) -> cons(U, concat(V, Y))}
Weak Rules:
{ lessleaves^#(cons(U, V), cons(W, Z)) ->
c_4(lessleaves^#(concat(U, V), concat(W, Z)))
, concat(leaf(), Y) -> Y
, lessleaves^#(X, leaf()) -> c_2()}
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: {concat(cons(U, V), Y) -> cons(U, concat(V, Y))}
Weak Rules:
{ lessleaves^#(cons(U, V), cons(W, Z)) ->
c_4(lessleaves^#(concat(U, V), concat(W, Z)))
, concat(leaf(), Y) -> Y
, lessleaves^#(X, leaf()) -> c_2()}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ concat_0(2, 2) -> 4
, concat_0(2, 2) -> 5
, concat_1(2, 2) -> 6
, concat_1(2, 6) -> 6
, leaf_0() -> 2
, leaf_0() -> 4
, leaf_0() -> 5
, leaf_0() -> 6
, cons_0(2, 2) -> 2
, cons_0(2, 2) -> 4
, cons_0(2, 2) -> 5
, cons_0(2, 2) -> 6
, cons_1(2, 6) -> 4
, cons_1(2, 6) -> 5
, cons_1(2, 6) -> 6
, lessleaves^#_0(2, 2) -> 1
, lessleaves^#_0(4, 5) -> 3
, lessleaves^#_1(6, 6) -> 7
, c_2_0() -> 1
, c_2_0() -> 3
, c_2_1() -> 7
, c_4_0(3) -> 1
, c_4_1(7) -> 1
, c_4_1(7) -> 3
, c_4_1(7) -> 7}
3) {lessleaves^#(cons(U, V), cons(W, Z)) ->
c_4(lessleaves^#(concat(U, V), concat(W, Z)))}
The usable rules for this path are the following:
{ concat(leaf(), Y) -> Y
, concat(cons(U, V), Y) -> cons(U, concat(V, Y))}
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:
{ concat(leaf(), Y) -> Y
, concat(cons(U, V), Y) -> cons(U, concat(V, Y))
, lessleaves^#(cons(U, V), cons(W, Z)) ->
c_4(lessleaves^#(concat(U, V), concat(W, Z)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{concat(leaf(), Y) -> Y}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{concat(leaf(), Y) -> Y}
Details:
Interpretation Functions:
concat(x1, x2) = [1] x1 + [1] x2 + [1]
leaf() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
lessleaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
concat^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
lessleaves^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{lessleaves^#(cons(U, V), cons(W, Z)) ->
c_4(lessleaves^#(concat(U, V), concat(W, Z)))}
and weakly orienting the rules
{concat(leaf(), Y) -> Y}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{lessleaves^#(cons(U, V), cons(W, Z)) ->
c_4(lessleaves^#(concat(U, V), concat(W, Z)))}
Details:
Interpretation Functions:
concat(x1, x2) = [1] x1 + [1] x2 + [1]
leaf() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [8]
lessleaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
concat^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
lessleaves^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(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: {concat(cons(U, V), Y) -> cons(U, concat(V, Y))}
Weak Rules:
{ lessleaves^#(cons(U, V), cons(W, Z)) ->
c_4(lessleaves^#(concat(U, V), concat(W, Z)))
, concat(leaf(), Y) -> Y}
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: {concat(cons(U, V), Y) -> cons(U, concat(V, Y))}
Weak Rules:
{ lessleaves^#(cons(U, V), cons(W, Z)) ->
c_4(lessleaves^#(concat(U, V), concat(W, Z)))
, concat(leaf(), Y) -> Y}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ concat_0(2, 2) -> 4
, concat_0(2, 2) -> 5
, concat_1(2, 2) -> 6
, concat_1(2, 6) -> 6
, leaf_0() -> 2
, leaf_0() -> 4
, leaf_0() -> 5
, leaf_0() -> 6
, cons_0(2, 2) -> 2
, cons_0(2, 2) -> 4
, cons_0(2, 2) -> 5
, cons_0(2, 2) -> 6
, cons_1(2, 6) -> 4
, cons_1(2, 6) -> 5
, cons_1(2, 6) -> 6
, lessleaves^#_0(2, 2) -> 1
, lessleaves^#_0(4, 5) -> 3
, lessleaves^#_1(6, 6) -> 7
, c_4_0(3) -> 1
, c_4_1(7) -> 1
, c_4_1(7) -> 3
, c_4_1(7) -> 7}
4) { concat^#(cons(U, V), Y) -> c_1(concat^#(V, Y))
, concat^#(leaf(), Y) -> c_0()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
concat(x1, x2) = [0] x1 + [0] x2 + [0]
leaf() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
lessleaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
concat^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
lessleaves^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(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: {concat^#(leaf(), Y) -> c_0()}
Weak Rules: {concat^#(cons(U, V), Y) -> c_1(concat^#(V, Y))}
Details:
We apply the weight gap principle, strictly orienting the rules
{concat^#(leaf(), Y) -> c_0()}
and weakly orienting the rules
{concat^#(cons(U, V), Y) -> c_1(concat^#(V, Y))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{concat^#(leaf(), Y) -> c_0()}
Details:
Interpretation Functions:
concat(x1, x2) = [0] x1 + [0] x2 + [0]
leaf() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
lessleaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
concat^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
lessleaves^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(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:
{ concat^#(leaf(), Y) -> c_0()
, concat^#(cons(U, V), Y) -> c_1(concat^#(V, Y))}
Details:
The given problem does not contain any strict rules
5) {concat^#(cons(U, V), Y) -> c_1(concat^#(V, Y))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
concat(x1, x2) = [0] x1 + [0] x2 + [0]
leaf() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
lessleaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
concat^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
lessleaves^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(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: {concat^#(cons(U, V), Y) -> c_1(concat^#(V, Y))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{concat^#(cons(U, V), Y) -> c_1(concat^#(V, Y))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{concat^#(cons(U, V), Y) -> c_1(concat^#(V, Y))}
Details:
Interpretation Functions:
concat(x1, x2) = [0] x1 + [0] x2 + [0]
leaf() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [8]
lessleaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
concat^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [3]
lessleaves^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(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: {concat^#(cons(U, V), Y) -> c_1(concat^#(V, Y))}
Details:
The given problem does not contain any strict rules