Tool CaT
stdout:
MAYBE
Problem:
battle(H(0(),x),n) -> battle(x,s(n))
battle(H(H(0(),x),y),n) -> battle(f(x,y,n),s(n))
battle(H(H(H(0(),x),y),z),n) -> battle(H(f(x,y,n),z),s(n))
f(x,y,o()) -> y
f(x,y,s(n)) -> H(x,f(x,y,n))
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:
{ battle(H(0(), x), n) -> battle(x, s(n))
, battle(H(H(0(), x), y), n) -> battle(f(x, y, n), s(n))
, battle(H(H(H(0(), x), y), z), n) ->
battle(H(f(x, y, n), z), s(n))
, f(x, y, o()) -> y
, f(x, y, s(n)) -> H(x, f(x, y, n))}
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: battle^#(H(0(), x), n) -> c_0(battle^#(x, s(n)))
, 2: battle^#(H(H(0(), x), y), n) ->
c_1(battle^#(f(x, y, n), s(n)))
, 3: battle^#(H(H(H(0(), x), y), z), n) ->
c_2(battle^#(H(f(x, y, n), z), s(n)))
, 4: f^#(x, y, o()) -> c_3()
, 5: f^#(x, y, s(n)) -> c_4(f^#(x, y, n))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^2)) ]
->{1,3,2} [ NA ]
Sub-problems:
-------------
* Path {1,3,2}: NA
----------------
The usable rules for this path are:
{ f(x, y, o()) -> y
, f(x, y, s(n)) -> H(x, f(x, y, n))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {5}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(battle) = {}, Uargs(H) = {}, Uargs(s) = {}, Uargs(f) = {},
Uargs(battle^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(c_2) = {}, Uargs(f^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
battle(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]
H(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]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
f(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]
o() = [0]
[0]
[0]
battle^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f^#(x1, x2, x3) = [3 3 3] x1 + [3 3 3] 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_3() = [0]
[0]
[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^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(x, y, s(n)) -> c_4(f^#(x, y, n))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [0]
[0 0 0] [0]
[0 0 1] [4]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 1] x3 + [0]
[0 4 0] [0 0 0] [4 0 0] [0]
[4 4 0] [0 0 2] [0 0 0] [0]
c_4(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [0]
* Path {5}->{4}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(battle) = {}, Uargs(H) = {}, Uargs(s) = {}, Uargs(f) = {},
Uargs(battle^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(c_2) = {}, Uargs(f^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
battle(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]
H(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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f(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]
o() = [0]
[0]
[0]
battle^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f^#(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_3() = [0]
[0]
[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^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(x, y, o()) -> c_3()}
Weak Rules: {f^#(x, y, s(n)) -> c_4(f^#(x, y, n))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4 0] x1 + [0]
[0 1 2] [2]
[0 0 0] [0]
o() = [2]
[0]
[2]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [2 4 2] x3 + [0]
[4 0 0] [4 0 4] [2 4 0] [0]
[4 4 4] [0 0 4] [2 2 2] [4]
c_3() = [1]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: battle^#(H(0(), x), n) -> c_0(battle^#(x, s(n)))
, 2: battle^#(H(H(0(), x), y), n) ->
c_1(battle^#(f(x, y, n), s(n)))
, 3: battle^#(H(H(H(0(), x), y), z), n) ->
c_2(battle^#(H(f(x, y, n), z), s(n)))
, 4: f^#(x, y, o()) -> c_3()
, 5: f^#(x, y, s(n)) -> c_4(f^#(x, y, n))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{1,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,3,2}: MAYBE
-------------------
The usable rules for this path are:
{ f(x, y, o()) -> y
, f(x, y, s(n)) -> H(x, f(x, y, n))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ battle^#(H(0(), x), n) -> c_0(battle^#(x, s(n)))
, battle^#(H(H(H(0(), x), y), z), n) ->
c_2(battle^#(H(f(x, y, n), z), s(n)))
, battle^#(H(H(0(), x), y), n) -> c_1(battle^#(f(x, y, n), s(n)))
, f(x, y, o()) -> y
, f(x, y, s(n)) -> H(x, f(x, y, n))}
Proof Output:
The input cannot be shown compatible
* Path {5}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(battle) = {}, Uargs(H) = {}, Uargs(s) = {}, Uargs(f) = {},
Uargs(battle^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(c_2) = {}, Uargs(f^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
battle(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
H(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
o() = [0]
[0]
battle^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
f^#(x1, x2, x3) = [3 3] x1 + [3 3] x2 + [3 3] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_3() = [0]
[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: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(x, y, s(n)) -> c_4(f^#(x, y, n))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [2]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 5] x3 + [2]
[0 4] [4 4] [2 2] [0]
c_4(x1) = [1 0] x1 + [7]
[2 0] [3]
* Path {5}->{4}: 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(battle) = {}, Uargs(H) = {}, Uargs(s) = {}, Uargs(f) = {},
Uargs(battle^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(c_2) = {}, Uargs(f^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
battle(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
H(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
o() = [0]
[0]
battle^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3() = [0]
[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: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(x, y, o()) -> c_3()}
Weak Rules: {f^#(x, y, s(n)) -> c_4(f^#(x, y, n))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [2]
[0 1] [2]
o() = [2]
[2]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [2 2] x3 + [0]
[0 4] [4 4] [2 2] [0]
c_3() = [1]
[0]
c_4(x1) = [1 0] x1 + [3]
[0 0] [7]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: battle^#(H(0(), x), n) -> c_0(battle^#(x, s(n)))
, 2: battle^#(H(H(0(), x), y), n) ->
c_1(battle^#(f(x, y, n), s(n)))
, 3: battle^#(H(H(H(0(), x), y), z), n) ->
c_2(battle^#(H(f(x, y, n), z), s(n)))
, 4: f^#(x, y, o()) -> c_3()
, 5: f^#(x, y, s(n)) -> c_4(f^#(x, y, n))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{1,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,3,2}: MAYBE
-------------------
The usable rules for this path are:
{ f(x, y, o()) -> y
, f(x, y, s(n)) -> H(x, f(x, y, n))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ battle^#(H(0(), x), n) -> c_0(battle^#(x, s(n)))
, battle^#(H(H(H(0(), x), y), z), n) ->
c_2(battle^#(H(f(x, y, n), z), s(n)))
, battle^#(H(H(0(), x), y), n) -> c_1(battle^#(f(x, y, n), s(n)))
, f(x, y, o()) -> y
, f(x, y, s(n)) -> H(x, f(x, y, n))}
Proof Output:
The input cannot be shown compatible
* Path {5}: 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(battle) = {}, Uargs(H) = {}, Uargs(s) = {}, Uargs(f) = {},
Uargs(battle^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(c_2) = {}, Uargs(f^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
battle(x1, x2) = [0] x1 + [0] x2 + [0]
H(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
o() = [0]
battle^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
f^#(x1, x2, x3) = [3] x1 + [3] x2 + [3] x3 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(x, y, s(n)) -> c_4(f^#(x, y, n))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
f^#(x1, x2, x3) = [7] x1 + [7] x2 + [6] x3 + [0]
c_4(x1) = [1] x1 + [3]
* Path {5}->{4}: 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(battle) = {}, Uargs(H) = {}, Uargs(s) = {}, Uargs(f) = {},
Uargs(battle^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(c_2) = {}, Uargs(f^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
battle(x1, x2) = [0] x1 + [0] x2 + [0]
H(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
o() = [0]
battle^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(x, y, o()) -> c_3()}
Weak Rules: {f^#(x, y, s(n)) -> c_4(f^#(x, y, n))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
o() = [0]
f^#(x1, x2, x3) = [7] x1 + [7] x2 + [2] x3 + [1]
c_3() = [0]
c_4(x1) = [1] x1 + [4]
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:
{ battle(H(0(), x), n) -> battle(x, s(n))
, battle(H(H(0(), x), y), n) -> battle(f(x, y, n), s(n))
, battle(H(H(H(0(), x), y), z), n) ->
battle(H(f(x, y, n), z), s(n))
, f(x, y, o()) -> y
, f(x, y, s(n)) -> H(x, f(x, y, n))}
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: battle^#(H(0(), x), n) -> c_0(battle^#(x, s(n)))
, 2: battle^#(H(H(0(), x), y), n) ->
c_1(battle^#(f(x, y, n), s(n)))
, 3: battle^#(H(H(H(0(), x), y), z), n) ->
c_2(battle^#(H(f(x, y, n), z), s(n)))
, 4: f^#(x, y, o()) -> c_3(y)
, 5: f^#(x, y, s(n)) -> c_4(x, f^#(x, y, n))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^2)) ]
->{1,3,2} [ NA ]
Sub-problems:
-------------
* Path {1,3,2}: NA
----------------
The usable rules for this path are:
{ f(x, y, o()) -> y
, f(x, y, s(n)) -> H(x, f(x, y, n))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {5}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(battle) = {}, Uargs(H) = {}, Uargs(s) = {}, Uargs(f) = {},
Uargs(battle^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(c_2) = {}, Uargs(f^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
battle(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]
H(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]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
f(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]
o() = [0]
[0]
[0]
battle^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f^#(x1, x2, x3) = [2 3 3] x1 + [3 3 3] 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_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 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^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {f^#(x, y, s(n)) -> c_4(x, f^#(x, y, n))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_4) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 1] x1 + [0]
[0 1 2] [2]
[0 0 0] [1]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 5 0] x3 + [0]
[0 0 0] [0 0 0] [0 4 0] [2]
[0 0 0] [0 4 4] [4 2 4] [0]
c_4(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [7]
[0 0 0] [0 0 0] [6]
[0 0 0] [0 2 0] [3]
* Path {5}->{4}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(battle) = {}, Uargs(H) = {}, Uargs(s) = {}, Uargs(f) = {},
Uargs(battle^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(c_2) = {}, Uargs(f^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
battle(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]
H(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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f(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]
o() = [0]
[0]
[0]
battle^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f^#(x1, x2, x3) = [3 3 3] 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_3(x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 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^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {f^#(x, y, o()) -> c_3(y)}
Weak Rules: {f^#(x, y, s(n)) -> c_4(x, f^#(x, y, n))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0 2] x1 + [2]
[0 1 0] [2]
[0 0 0] [2]
o() = [2]
[2]
[2]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [2 2 0] x3 + [0]
[4 0 0] [4 0 0] [2 2 0] [0]
[4 4 0] [4 0 4] [2 2 2] [0]
c_3(x1) = [0 0 0] x1 + [1]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [7]
[3 0 0] [0 0 0] [7]
[3 3 0] [0 0 0] [7]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: battle^#(H(0(), x), n) -> c_0(battle^#(x, s(n)))
, 2: battle^#(H(H(0(), x), y), n) ->
c_1(battle^#(f(x, y, n), s(n)))
, 3: battle^#(H(H(H(0(), x), y), z), n) ->
c_2(battle^#(H(f(x, y, n), z), s(n)))
, 4: f^#(x, y, o()) -> c_3(y)
, 5: f^#(x, y, s(n)) -> c_4(x, f^#(x, y, n))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{1,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,3,2}: MAYBE
-------------------
The usable rules for this path are:
{ f(x, y, o()) -> y
, f(x, y, s(n)) -> H(x, f(x, y, n))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ battle^#(H(0(), x), n) -> c_0(battle^#(x, s(n)))
, battle^#(H(H(H(0(), x), y), z), n) ->
c_2(battle^#(H(f(x, y, n), z), s(n)))
, battle^#(H(H(0(), x), y), n) -> c_1(battle^#(f(x, y, n), s(n)))
, f(x, y, o()) -> y
, f(x, y, s(n)) -> H(x, f(x, y, n))}
Proof Output:
The input cannot be shown compatible
* Path {5}: 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(battle) = {}, Uargs(H) = {}, Uargs(s) = {}, Uargs(f) = {},
Uargs(battle^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(c_2) = {}, Uargs(f^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
battle(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
H(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
o() = [0]
[0]
battle^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
f^#(x1, x2, x3) = [3 3] x1 + [3 3] x2 + [3 3] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {f^#(x, y, s(n)) -> c_4(x, f^#(x, y, n))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_4) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 3] x1 + [2]
[0 0] [0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [2 0] x3 + [0]
[0 0] [0 4] [3 0] [4]
c_4(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [7]
* Path {5}->{4}: 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(battle) = {}, Uargs(H) = {}, Uargs(s) = {}, Uargs(f) = {},
Uargs(battle^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(c_2) = {}, Uargs(f^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
battle(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
H(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
o() = [0]
[0]
battle^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
f^#(x1, x2, x3) = [3 3] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3(x1) = [1 1] x1 + [0]
[0 0] [0]
c_4(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {f^#(x, y, o()) -> c_3(y)}
Weak Rules: {f^#(x, y, s(n)) -> c_4(x, f^#(x, y, n))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [2]
[0 1] [2]
o() = [2]
[2]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [2 2] x3 + [0]
[0 0] [4 0] [2 2] [0]
c_3(x1) = [0 0] x1 + [1]
[0 0] [0]
c_4(x1, x2) = [0 0] x1 + [1 0] x2 + [7]
[0 0] [0 0] [6]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: battle^#(H(0(), x), n) -> c_0(battle^#(x, s(n)))
, 2: battle^#(H(H(0(), x), y), n) ->
c_1(battle^#(f(x, y, n), s(n)))
, 3: battle^#(H(H(H(0(), x), y), z), n) ->
c_2(battle^#(H(f(x, y, n), z), s(n)))
, 4: f^#(x, y, o()) -> c_3(y)
, 5: f^#(x, y, s(n)) -> c_4(x, f^#(x, y, n))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{1,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,3,2}: MAYBE
-------------------
The usable rules for this path are:
{ f(x, y, o()) -> y
, f(x, y, s(n)) -> H(x, f(x, y, n))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ battle^#(H(0(), x), n) -> c_0(battle^#(x, s(n)))
, battle^#(H(H(H(0(), x), y), z), n) ->
c_2(battle^#(H(f(x, y, n), z), s(n)))
, battle^#(H(H(0(), x), y), n) -> c_1(battle^#(f(x, y, n), s(n)))
, f(x, y, o()) -> y
, f(x, y, s(n)) -> H(x, f(x, y, n))}
Proof Output:
The input cannot be shown compatible
* Path {5}: 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(battle) = {}, Uargs(H) = {}, Uargs(s) = {}, Uargs(f) = {},
Uargs(battle^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(c_2) = {}, Uargs(f^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
battle(x1, x2) = [0] x1 + [0] x2 + [0]
H(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
o() = [0]
battle^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
f^#(x1, x2, x3) = [3] x1 + [3] x2 + [3] x3 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1, x2) = [0] x1 + [1] x2 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {f^#(x, y, s(n)) -> c_4(x, f^#(x, y, n))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_4) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
f^#(x1, x2, x3) = [7] x1 + [7] x2 + [2] x3 + [0]
c_4(x1, x2) = [0] x1 + [1] x2 + [7]
* Path {5}->{4}: 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(battle) = {}, Uargs(H) = {}, Uargs(s) = {}, Uargs(f) = {},
Uargs(battle^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(c_2) = {}, Uargs(f^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
battle(x1, x2) = [0] x1 + [0] x2 + [0]
H(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
o() = [0]
battle^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
f^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1, x2) = [0] x1 + [1] x2 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {f^#(x, y, o()) -> c_3(y)}
Weak Rules: {f^#(x, y, s(n)) -> c_4(x, f^#(x, y, n))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
o() = [2]
f^#(x1, x2, x3) = [7] x1 + [7] x2 + [3] x3 + [2]
c_3(x1) = [1] x1 + [1]
c_4(x1, x2) = [0] x1 + [1] x2 + [0]
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 pair1rc
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ battle(H(0(), x), n) -> battle(x, s(n))
, battle(H(H(0(), x), y), n) -> battle(f(x, y, n), s(n))
, battle(H(H(H(0(), x), y), z), n) ->
battle(H(f(x, y, n), z), s(n))
, f(x, y, o()) -> y
, f(x, y, s(n)) -> H(x, f(x, y, n))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair1 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair2rc
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ battle(H(0(), x), n) -> battle(x, s(n))
, battle(H(H(0(), x), y), n) -> battle(f(x, y, n), s(n))
, battle(H(H(H(0(), x), y), z), n) ->
battle(H(f(x, y, n), z), s(n))
, f(x, y, o()) -> y
, f(x, y, s(n)) -> H(x, f(x, y, n))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair2 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair3irc
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ battle(H(0(), x), n) -> battle(x, s(n))
, battle(H(H(0(), x), y), n) -> battle(f(x, y, n), s(n))
, battle(H(H(H(0(), x), y), z), n) ->
battle(H(f(x, y, n), z), s(n))
, f(x, y, o()) -> y
, f(x, y, s(n)) -> H(x, f(x, y, n))}
StartTerms: basic terms
Strategy: innermost
Certificate: TIMEOUT
Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair3rc
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ battle(H(0(), x), n) -> battle(x, s(n))
, battle(H(H(0(), x), y), n) -> battle(f(x, y, n), s(n))
, battle(H(H(H(0(), x), y), z), n) ->
battle(H(f(x, y, n), z), s(n))
, f(x, y, o()) -> y
, f(x, y, s(n)) -> H(x, f(x, y, n))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool rc
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ battle(H(0(), x), n) -> battle(x, s(n))
, battle(H(H(0(), x), y), n) -> battle(f(x, y, n), s(n))
, battle(H(H(H(0(), x), y), z), n) ->
battle(H(f(x, y, n), z), s(n))
, f(x, y, o()) -> y
, f(x, y, s(n)) -> H(x, f(x, y, n))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'rc (timeout of 60.0 seconds)':
----------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool tup3irc
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ battle(H(0(), x), n) -> battle(x, s(n))
, battle(H(H(0(), x), y), n) -> battle(f(x, y, n), s(n))
, battle(H(H(H(0(), x), y), z), n) ->
battle(H(f(x, y, n), z), s(n))
, f(x, y, o()) -> y
, f(x, y, s(n)) -> H(x, f(x, y, n))}
StartTerms: basic terms
Strategy: innermost
Certificate: TIMEOUT
Application of 'tup3 (timeout of 60.0 seconds)':
------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..