'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ ++(nil(), y) -> y
, ++(x, nil()) -> x
, ++(.(x, y), z) -> .(x, ++(y, z))
, ++(++(x, y), z) -> ++(x, ++(y, z))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ ++^#(nil(), y) -> c_0()
, ++^#(x, nil()) -> c_1()
, ++^#(.(x, y), z) -> c_2(++^#(y, z))
, ++^#(++(x, y), z) -> c_3(++^#(x, ++(y, z)))}
The usable rules are:
{ ++(nil(), y) -> y
, ++(x, nil()) -> x
, ++(.(x, y), z) -> .(x, ++(y, z))
, ++(++(x, y), z) -> ++(x, ++(y, z))}
The estimated dependency graph contains the following edges:
{++^#(.(x, y), z) -> c_2(++^#(y, z))}
==> {++^#(++(x, y), z) -> c_3(++^#(x, ++(y, z)))}
{++^#(.(x, y), z) -> c_2(++^#(y, z))}
==> {++^#(.(x, y), z) -> c_2(++^#(y, z))}
{++^#(.(x, y), z) -> c_2(++^#(y, z))}
==> {++^#(x, nil()) -> c_1()}
{++^#(.(x, y), z) -> c_2(++^#(y, z))}
==> {++^#(nil(), y) -> c_0()}
{++^#(++(x, y), z) -> c_3(++^#(x, ++(y, z)))}
==> {++^#(++(x, y), z) -> c_3(++^#(x, ++(y, z)))}
{++^#(++(x, y), z) -> c_3(++^#(x, ++(y, z)))}
==> {++^#(.(x, y), z) -> c_2(++^#(y, z))}
{++^#(++(x, y), z) -> c_3(++^#(x, ++(y, z)))}
==> {++^#(x, nil()) -> c_1()}
{++^#(++(x, y), z) -> c_3(++^#(x, ++(y, z)))}
==> {++^#(nil(), y) -> c_0()}
We consider the following path(s):
1) { ++^#(.(x, y), z) -> c_2(++^#(y, z))
, ++^#(++(x, y), z) -> c_3(++^#(x, ++(y, z)))
, ++^#(x, nil()) -> c_1()}
The usable rules for this path are the following:
{ ++(nil(), y) -> y
, ++(x, nil()) -> x
, ++(.(x, y), z) -> .(x, ++(y, z))
, ++(++(x, y), z) -> ++(x, ++(y, z))}
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:
{ ++(nil(), y) -> y
, ++(x, nil()) -> x
, ++(.(x, y), z) -> .(x, ++(y, z))
, ++(++(x, y), z) -> ++(x, ++(y, z))
, ++^#(.(x, y), z) -> c_2(++^#(y, z))
, ++^#(++(x, y), z) -> c_3(++^#(x, ++(y, z)))
, ++^#(x, nil()) -> c_1()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ ++(nil(), y) -> y
, ++(x, nil()) -> x
, ++^#(x, nil()) -> c_1()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ ++(nil(), y) -> y
, ++(x, nil()) -> x
, ++^#(x, nil()) -> c_1()}
Details:
Interpretation Functions:
++(x1, x2) = [1] x1 + [1] x2 + [1]
nil() = [0]
.(x1, x2) = [1] x1 + [1] x2 + [0]
++^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [15]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{++^#(.(x, y), z) -> c_2(++^#(y, z))}
and weakly orienting the rules
{ ++(nil(), y) -> y
, ++(x, nil()) -> x
, ++^#(x, nil()) -> c_1()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{++^#(.(x, y), z) -> c_2(++^#(y, z))}
Details:
Interpretation Functions:
++(x1, x2) = [1] x1 + [1] x2 + [1]
nil() = [3]
.(x1, x2) = [1] x1 + [1] x2 + [8]
++^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [7]
c_3(x1) = [1] x1 + [1]
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:
{ ++(.(x, y), z) -> .(x, ++(y, z))
, ++(++(x, y), z) -> ++(x, ++(y, z))
, ++^#(++(x, y), z) -> c_3(++^#(x, ++(y, z)))}
Weak Rules:
{ ++^#(.(x, y), z) -> c_2(++^#(y, z))
, ++(nil(), y) -> y
, ++(x, nil()) -> x
, ++^#(x, nil()) -> c_1()}
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:
{ ++(.(x, y), z) -> .(x, ++(y, z))
, ++(++(x, y), z) -> ++(x, ++(y, z))
, ++^#(++(x, y), z) -> c_3(++^#(x, ++(y, z)))}
Weak Rules:
{ ++^#(.(x, y), z) -> c_2(++^#(y, z))
, ++(nil(), y) -> y
, ++(x, nil()) -> x
, ++^#(x, nil()) -> c_1()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ nil_0() -> 2
, ._0(2, 2) -> 3
, ._0(2, 3) -> 3
, ._0(3, 2) -> 3
, ._0(3, 3) -> 3
, ++^#_0(2, 2) -> 4
, ++^#_0(2, 3) -> 4
, ++^#_0(3, 2) -> 4
, ++^#_0(3, 3) -> 4
, c_1_0() -> 4
, c_2_0(4) -> 4}
2) { ++^#(.(x, y), z) -> c_2(++^#(y, z))
, ++^#(++(x, y), z) -> c_3(++^#(x, ++(y, z)))}
The usable rules for this path are the following:
{ ++(nil(), y) -> y
, ++(x, nil()) -> x
, ++(.(x, y), z) -> .(x, ++(y, z))
, ++(++(x, y), z) -> ++(x, ++(y, z))}
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:
{ ++(nil(), y) -> y
, ++(x, nil()) -> x
, ++(.(x, y), z) -> .(x, ++(y, z))
, ++(++(x, y), z) -> ++(x, ++(y, z))
, ++^#(.(x, y), z) -> c_2(++^#(y, z))
, ++^#(++(x, y), z) -> c_3(++^#(x, ++(y, z)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ ++(nil(), y) -> y
, ++(x, nil()) -> x}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ ++(nil(), y) -> y
, ++(x, nil()) -> x}
Details:
Interpretation Functions:
++(x1, x2) = [1] x1 + [1] x2 + [1]
nil() = [0]
.(x1, x2) = [1] x1 + [1] x2 + [0]
++^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{++^#(.(x, y), z) -> c_2(++^#(y, z))}
and weakly orienting the rules
{ ++(nil(), y) -> y
, ++(x, nil()) -> x}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{++^#(.(x, y), z) -> c_2(++^#(y, z))}
Details:
Interpretation Functions:
++(x1, x2) = [1] x1 + [1] x2 + [1]
nil() = [7]
.(x1, x2) = [1] x1 + [1] x2 + [8]
++^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
c_3(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:
{ ++(.(x, y), z) -> .(x, ++(y, z))
, ++(++(x, y), z) -> ++(x, ++(y, z))
, ++^#(++(x, y), z) -> c_3(++^#(x, ++(y, z)))}
Weak Rules:
{ ++^#(.(x, y), z) -> c_2(++^#(y, z))
, ++(nil(), y) -> y
, ++(x, nil()) -> 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:
{ ++(.(x, y), z) -> .(x, ++(y, z))
, ++(++(x, y), z) -> ++(x, ++(y, z))
, ++^#(++(x, y), z) -> c_3(++^#(x, ++(y, z)))}
Weak Rules:
{ ++^#(.(x, y), z) -> c_2(++^#(y, z))
, ++(nil(), y) -> y
, ++(x, nil()) -> x}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ nil_0() -> 2
, ._0(2, 2) -> 3
, ._0(2, 3) -> 3
, ._0(3, 2) -> 3
, ._0(3, 3) -> 3
, ++^#_0(2, 2) -> 4
, ++^#_0(2, 3) -> 4
, ++^#_0(3, 2) -> 4
, ++^#_0(3, 3) -> 4
, c_2_0(4) -> 4}
3) { ++^#(.(x, y), z) -> c_2(++^#(y, z))
, ++^#(++(x, y), z) -> c_3(++^#(x, ++(y, z)))
, ++^#(nil(), y) -> c_0()}
The usable rules for this path are the following:
{ ++(nil(), y) -> y
, ++(x, nil()) -> x
, ++(.(x, y), z) -> .(x, ++(y, z))
, ++(++(x, y), z) -> ++(x, ++(y, z))}
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:
{ ++(nil(), y) -> y
, ++(x, nil()) -> x
, ++(.(x, y), z) -> .(x, ++(y, z))
, ++(++(x, y), z) -> ++(x, ++(y, z))
, ++^#(.(x, y), z) -> c_2(++^#(y, z))
, ++^#(++(x, y), z) -> c_3(++^#(x, ++(y, z)))
, ++^#(nil(), y) -> c_0()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ ++(nil(), y) -> y
, ++(x, nil()) -> x
, ++^#(nil(), y) -> c_0()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ ++(nil(), y) -> y
, ++(x, nil()) -> x
, ++^#(nil(), y) -> c_0()}
Details:
Interpretation Functions:
++(x1, x2) = [1] x1 + [1] x2 + [1]
nil() = [0]
.(x1, x2) = [1] x1 + [1] x2 + [0]
++^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [15]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{++^#(.(x, y), z) -> c_2(++^#(y, z))}
and weakly orienting the rules
{ ++(nil(), y) -> y
, ++(x, nil()) -> x
, ++^#(nil(), y) -> c_0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{++^#(.(x, y), z) -> c_2(++^#(y, z))}
Details:
Interpretation Functions:
++(x1, x2) = [1] x1 + [1] x2 + [1]
nil() = [3]
.(x1, x2) = [1] x1 + [1] x2 + [8]
++^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [7]
c_3(x1) = [1] x1 + [1]
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:
{ ++(.(x, y), z) -> .(x, ++(y, z))
, ++(++(x, y), z) -> ++(x, ++(y, z))
, ++^#(++(x, y), z) -> c_3(++^#(x, ++(y, z)))}
Weak Rules:
{ ++^#(.(x, y), z) -> c_2(++^#(y, z))
, ++(nil(), y) -> y
, ++(x, nil()) -> x
, ++^#(nil(), y) -> c_0()}
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:
{ ++(.(x, y), z) -> .(x, ++(y, z))
, ++(++(x, y), z) -> ++(x, ++(y, z))
, ++^#(++(x, y), z) -> c_3(++^#(x, ++(y, z)))}
Weak Rules:
{ ++^#(.(x, y), z) -> c_2(++^#(y, z))
, ++(nil(), y) -> y
, ++(x, nil()) -> x
, ++^#(nil(), y) -> c_0()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ nil_0() -> 2
, ._0(2, 2) -> 3
, ._0(2, 3) -> 3
, ._0(3, 2) -> 3
, ._0(3, 3) -> 3
, ++^#_0(2, 2) -> 4
, ++^#_0(2, 3) -> 4
, ++^#_0(3, 2) -> 4
, ++^#_0(3, 3) -> 4
, c_0_0() -> 4
, c_2_0(4) -> 4}