Tool CaT
stdout:
MAYBE
Problem:
div(0(),y) -> 0()
div(x,y) -> quot(x,y,y)
quot(0(),s(y),z) -> 0()
quot(s(x),s(y),z) -> quot(x,y,z)
quot(x,0(),s(z)) -> s(div(x,s(z)))
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ div(0(), y) -> 0()
, div(x, y) -> quot(x, y, y)
, quot(0(), s(y), z) -> 0()
, quot(s(x), s(y), z) -> quot(x, y, z)
, quot(x, 0(), s(z)) -> s(div(x, s(z)))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: div^#(0(), y) -> c_0()
, 2: div^#(x, y) -> c_1(quot^#(x, y, y))
, 3: quot^#(0(), s(y), z) -> c_2()
, 4: quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))
, 5: quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,5,4} [ MAYBE ]
|
|->{1} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {2,5,4}: MAYBE
-------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
quot(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
div^#(x1, x2) = [0 0 0] x1 + [0 0 3] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
quot^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[3 3 3] [3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [3 3 3] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ div^#(x, y) -> c_1(quot^#(x, y, y))
, quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))
, quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {2,5,4}->{1}: YES(?,O(n^1))
--------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
quot(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
div^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
quot^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {div^#(0(), y) -> c_0()}
Weak Rules:
{ div^#(x, y) -> c_1(quot^#(x, y, y))
, quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))
, quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(div^#) = {}, Uargs(c_1) = {1},
Uargs(quot^#) = {}, Uargs(c_3) = {1}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[2]
s(x1) = [0 0 1] x1 + [0]
[0 0 1] [0]
[0 0 1] [0]
div^#(x1, x2) = [0 0 2] x1 + [0 0 0] x2 + [0]
[4 4 4] [0 0 4] [4]
[0 0 0] [2 0 0] [0]
c_0() = [1]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 0 0] [3]
[0 0 0] [0]
quot^#(x1, x2, x3) = [0 0 2] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[4 3 1] [0 0 0] [1 3 4] [0]
[4 0 0] [0 0 0] [0 0 0] [0]
c_3(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [1 0 0] x1 + [0]
[0 0 2] [0]
[0 0 0] [0]
* Path {2,5,4}->{3}: YES(?,O(n^1))
--------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
quot(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
div^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
quot^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {quot^#(0(), s(y), z) -> c_2()}
Weak Rules:
{ div^#(x, y) -> c_1(quot^#(x, y, y))
, quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))
, quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(div^#) = {}, Uargs(c_1) = {1},
Uargs(quot^#) = {}, Uargs(c_3) = {1}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
[0]
s(x1) = [1 0 4] x1 + [0]
[0 0 1] [0]
[0 0 0] [0]
div^#(x1, x2) = [2 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 2 0] [0]
[4 4 4] [0 0 0] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2, x3) = [2 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[2 0 4] [0 0 0] [2 0 0] [0]
[0 4 0] [0 0 0] [2 0 0] [0]
c_2() = [1]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 2 0] [0]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: div^#(0(), y) -> c_0()
, 2: div^#(x, y) -> c_1(quot^#(x, y, y))
, 3: quot^#(0(), s(y), z) -> c_2()
, 4: quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))
, 5: quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,5,4} [ MAYBE ]
|
|->{1} [ NA ]
|
`->{3} [ NA ]
Sub-problems:
-------------
* Path {2,5,4}: MAYBE
-------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
div^#(x1, x2) = [3 3] x1 + [0 1] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
quot^#(x1, x2, x3) = [3 3] x1 + [0 0] x2 + [0 1] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ div^#(x, y) -> c_1(quot^#(x, y, y))
, quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))
, quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {2,5,4}->{1}: NA
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
quot^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
We have not generated a proof for the resulting sub-problem.
* Path {2,5,4}->{3}: NA
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
quot^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
We have not generated a proof for the resulting sub-problem.
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: div^#(0(), y) -> c_0()
, 2: div^#(x, y) -> c_1(quot^#(x, y, y))
, 3: quot^#(0(), s(y), z) -> c_2()
, 4: quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))
, 5: quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,5,4} [ MAYBE ]
|
|->{1} [ NA ]
|
`->{3} [ NA ]
Sub-problems:
-------------
* Path {2,5,4}: MAYBE
-------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [1] x1 + [0]
div^#(x1, x2) = [3] x1 + [2] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
quot^#(x1, x2, x3) = [3] x1 + [0] x2 + [2] x3 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ div^#(x, y) -> c_1(quot^#(x, y, y))
, quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))
, quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {2,5,4}->{1}: NA
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [0] x1 + [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
quot^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {2,5,4}->{3}: NA
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [0] x1 + [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
quot^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
We have not generated a proof for the resulting sub-problem.
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ div(0(), y) -> 0()
, div(x, y) -> quot(x, y, y)
, quot(0(), s(y), z) -> 0()
, quot(s(x), s(y), z) -> quot(x, y, z)
, quot(x, 0(), s(z)) -> s(div(x, s(z)))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: div^#(0(), y) -> c_0()
, 2: div^#(x, y) -> c_1(quot^#(x, y, y))
, 3: quot^#(0(), s(y), z) -> c_2()
, 4: quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))
, 5: quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,5,4} [ MAYBE ]
|
|->{1} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {2,5,4}: MAYBE
-------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
quot(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
div^#(x1, x2) = [0 0 0] x1 + [0 0 3] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
quot^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[3 3 3] [3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [3 3 3] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{ div^#(x, y) -> c_1(quot^#(x, y, y))
, quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))
, quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {2,5,4}->{1}: YES(?,O(n^1))
--------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
quot(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
div^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
quot^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {div^#(0(), y) -> c_0()}
Weak Rules:
{ div^#(x, y) -> c_1(quot^#(x, y, y))
, quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))
, quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(div^#) = {}, Uargs(c_1) = {1},
Uargs(quot^#) = {}, Uargs(c_3) = {1}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[2]
s(x1) = [0 0 1] x1 + [0]
[0 0 1] [0]
[0 0 1] [0]
div^#(x1, x2) = [0 0 2] x1 + [0 0 0] x2 + [0]
[4 4 4] [0 0 4] [4]
[0 0 0] [2 0 0] [0]
c_0() = [1]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 0 0] [3]
[0 0 0] [0]
quot^#(x1, x2, x3) = [0 0 2] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[4 3 1] [0 0 0] [1 3 4] [0]
[4 0 0] [0 0 0] [0 0 0] [0]
c_3(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [1 0 0] x1 + [0]
[0 0 2] [0]
[0 0 0] [0]
* Path {2,5,4}->{3}: YES(?,O(n^1))
--------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
quot(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
div^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
quot^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {quot^#(0(), s(y), z) -> c_2()}
Weak Rules:
{ div^#(x, y) -> c_1(quot^#(x, y, y))
, quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))
, quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(div^#) = {}, Uargs(c_1) = {1},
Uargs(quot^#) = {}, Uargs(c_3) = {1}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
[0]
s(x1) = [1 0 4] x1 + [0]
[0 0 1] [0]
[0 0 0] [0]
div^#(x1, x2) = [2 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 2 0] [0]
[4 4 4] [0 0 0] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2, x3) = [2 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[2 0 4] [0 0 0] [2 0 0] [0]
[0 4 0] [0 0 0] [2 0 0] [0]
c_2() = [1]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 2 0] [0]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: div^#(0(), y) -> c_0()
, 2: div^#(x, y) -> c_1(quot^#(x, y, y))
, 3: quot^#(0(), s(y), z) -> c_2()
, 4: quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))
, 5: quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,5,4} [ MAYBE ]
|
|->{1} [ NA ]
|
`->{3} [ NA ]
Sub-problems:
-------------
* Path {2,5,4}: MAYBE
-------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
div^#(x1, x2) = [3 3] x1 + [0 1] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
quot^#(x1, x2, x3) = [3 3] x1 + [0 0] x2 + [0 1] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{ div^#(x, y) -> c_1(quot^#(x, y, y))
, quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))
, quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {2,5,4}->{1}: NA
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
quot^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
We have not generated a proof for the resulting sub-problem.
* Path {2,5,4}->{3}: NA
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
quot^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
We have not generated a proof for the resulting sub-problem.
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: div^#(0(), y) -> c_0()
, 2: div^#(x, y) -> c_1(quot^#(x, y, y))
, 3: quot^#(0(), s(y), z) -> c_2()
, 4: quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))
, 5: quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,5,4} [ MAYBE ]
|
|->{1} [ NA ]
|
`->{3} [ NA ]
Sub-problems:
-------------
* Path {2,5,4}: MAYBE
-------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [1] x1 + [0]
div^#(x1, x2) = [3] x1 + [2] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
quot^#(x1, x2, x3) = [3] x1 + [0] x2 + [2] x3 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{ div^#(x, y) -> c_1(quot^#(x, y, y))
, quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))
, quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {2,5,4}->{1}: NA
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [0] x1 + [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
quot^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {2,5,4}->{3}: NA
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [0] x1 + [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
quot^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
We have not generated a proof for the resulting sub-problem.
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.