Tool CaT
stdout:
MAYBE
Problem:
f(true(),x,y,z) -> f(and(gt(x,y),gt(x,z)),x,s(y),z)
f(true(),x,y,z) -> f(and(gt(x,y),gt(x,z)),x,y,s(z))
gt(0(),v) -> false()
gt(s(u),0()) -> true()
gt(s(u),s(v)) -> gt(u,v)
and(x,true()) -> x
and(x,false()) -> false()
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:
{ f(true(), x, y, z) -> f(and(gt(x, y), gt(x, z)), x, s(y), z)
, f(true(), x, y, z) -> f(and(gt(x, y), gt(x, z)), x, y, s(z))
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, and(x, true()) -> x
, and(x, false()) -> false()}
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: f^#(true(), x, y, z) ->
c_0(f^#(and(gt(x, y), gt(x, z)), x, s(y), z))
, 2: f^#(true(), x, y, z) ->
c_1(f^#(and(gt(x, y), gt(x, z)), x, y, s(z)))
, 3: gt^#(0(), v) -> c_2()
, 4: gt^#(s(u), 0()) -> c_3()
, 5: gt^#(s(u), s(v)) -> c_4(gt^#(u, v))
, 6: and^#(x, true()) -> c_5()
, 7: and^#(x, false()) -> c_6()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ YES(?,O(1)) ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(n^2)) ]
|
|->{3} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{1,2} [ NA ]
Sub-problems:
-------------
* Path {1,2}: NA
--------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, and(x, true()) -> x
, and(x, false()) -> false()}
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(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(gt^#) = {}, Uargs(c_4) = {1}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
f^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 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]
gt^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[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: {gt^#(s(u), s(v)) -> c_4(gt^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [0]
gt^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
c_4(x1) = [1 0] x1 + [1]
[0 0] [3]
* Path {5}->{3}: 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(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(gt^#) = {}, Uargs(c_4) = {1}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
f^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 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]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[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: {gt^#(0(), v) -> c_2()}
Weak Rules: {gt^#(s(u), s(v)) -> c_4(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [0]
[0 1] [2]
0() = [2]
[2]
gt^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
[3 3] [0 1] [0]
c_2() = [1]
[0]
c_4(x1) = [1 0] x1 + [3]
[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(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(gt^#) = {}, Uargs(c_4) = {1}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
f^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 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]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[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: {gt^#(s(u), 0()) -> c_3()}
Weak Rules: {gt^#(s(u), s(v)) -> c_4(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [2]
[0 1] [2]
0() = [0]
[2]
gt^#(x1, x2) = [2 0] x1 + [0 2] x2 + [0]
[0 0] [0 0] [0]
c_3() = [1]
[0]
c_4(x1) = [1 0] x1 + [3]
[0 0] [0]
* Path {6}: YES(?,O(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(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(gt^#) = {}, Uargs(c_4) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
f^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 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]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {and^#(x, true()) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
[2]
and^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
[0 0] [2 2] [7]
c_5() = [0]
[1]
* Path {7}: YES(?,O(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(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(gt^#) = {}, Uargs(c_4) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
f^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 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]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {and^#(x, false()) -> c_6()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
and^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
[0 0] [2 2] [7]
c_6() = [0]
[1]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(true(), x, y, z) ->
c_0(f^#(and(gt(x, y), gt(x, z)), x, s(y), z))
, 2: f^#(true(), x, y, z) ->
c_1(f^#(and(gt(x, y), gt(x, z)), x, y, s(z)))
, 3: gt^#(0(), v) -> c_2()
, 4: gt^#(s(u), 0()) -> c_3()
, 5: gt^#(s(u), s(v)) -> c_4(gt^#(u, v))
, 6: and^#(x, true()) -> c_5()
, 7: and^#(x, false()) -> c_6()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ YES(?,O(1)) ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{1,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, and(x, true()) -> x
, and(x, false()) -> false()}
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:
{ f^#(true(), x, y, z) ->
c_0(f^#(and(gt(x, y), gt(x, z)), x, s(y), z))
, f^#(true(), x, y, z) ->
c_1(f^#(and(gt(x, y), gt(x, z)), x, y, s(z)))
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, and(x, true()) -> x
, and(x, false()) -> false()}
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(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(gt^#) = {}, Uargs(c_4) = {1}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
false() = [0]
f^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
gt^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [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: {gt^#(s(u), s(v)) -> c_4(gt^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_4(x1) = [1] x1 + [7]
* Path {5}->{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(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(gt^#) = {}, Uargs(c_4) = {1}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
false() = [0]
f^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [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: {gt^#(0(), v) -> c_2()}
Weak Rules: {gt^#(s(u), s(v)) -> c_4(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
gt^#(x1, x2) = [3] x1 + [3] x2 + [2]
c_2() = [1]
c_4(x1) = [1] x1 + [5]
* 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(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(gt^#) = {}, Uargs(c_4) = {1}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
false() = [0]
f^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [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: {gt^#(s(u), 0()) -> c_3()}
Weak Rules: {gt^#(s(u), s(v)) -> c_4(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_3() = [1]
c_4(x1) = [1] x1 + [7]
* Path {6}: YES(?,O(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(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(gt^#) = {}, Uargs(c_4) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
false() = [0]
f^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {and^#(x, true()) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [7]
and^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_5() = [1]
* Path {7}: YES(?,O(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(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(gt^#) = {}, Uargs(c_4) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
false() = [0]
f^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {and^#(x, false()) -> c_6()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [7]
and^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_6() = [1]
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) '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:
{ f(true(), x, y, z) -> f(and(gt(x, y), gt(x, z)), x, s(y), z)
, f(true(), x, y, z) -> f(and(gt(x, y), gt(x, z)), x, y, s(z))
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, and(x, true()) -> x
, and(x, false()) -> false()}
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: f^#(true(), x, y, z) ->
c_0(f^#(and(gt(x, y), gt(x, z)), x, s(y), z))
, 2: f^#(true(), x, y, z) ->
c_1(f^#(and(gt(x, y), gt(x, z)), x, y, s(z)))
, 3: gt^#(0(), v) -> c_2()
, 4: gt^#(s(u), 0()) -> c_3()
, 5: gt^#(s(u), s(v)) -> c_4(gt^#(u, v))
, 6: and^#(x, true()) -> c_5(x)
, 7: and^#(x, false()) -> c_6()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ YES(?,O(1)) ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(n^2)) ]
|
|->{3} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{1,2} [ NA ]
Sub-problems:
-------------
* Path {1,2}: NA
--------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, and(x, true()) -> x
, and(x, false()) -> false()}
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(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(gt^#) = {}, Uargs(c_4) = {1}, Uargs(and^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
f^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 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]
gt^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gt^#(s(u), s(v)) -> c_4(gt^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [0]
gt^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
c_4(x1) = [1 0] x1 + [1]
[0 0] [3]
* Path {5}->{3}: 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(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(gt^#) = {}, Uargs(c_4) = {1}, Uargs(and^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
f^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 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]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gt^#(0(), v) -> c_2()}
Weak Rules: {gt^#(s(u), s(v)) -> c_4(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [0]
[0 1] [2]
0() = [2]
[2]
gt^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
[3 3] [0 1] [0]
c_2() = [1]
[0]
c_4(x1) = [1 0] x1 + [3]
[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(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(gt^#) = {}, Uargs(c_4) = {1}, Uargs(and^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
f^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 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]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[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: {gt^#(s(u), 0()) -> c_3()}
Weak Rules: {gt^#(s(u), s(v)) -> c_4(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [2]
[0 1] [2]
0() = [0]
[2]
gt^#(x1, x2) = [2 0] x1 + [0 2] x2 + [0]
[0 0] [0 0] [0]
c_3() = [1]
[0]
c_4(x1) = [1 0] x1 + [3]
[0 0] [0]
* Path {6}: YES(?,O(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(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(gt^#) = {}, Uargs(c_4) = {}, Uargs(and^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
f^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 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]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [1 1] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {and^#(x, true()) -> c_5(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
[2]
and^#(x1, x2) = [7 7] x1 + [2 2] x2 + [7]
[7 7] [2 2] [3]
c_5(x1) = [1 3] x1 + [0]
[1 1] [1]
* Path {7}: YES(?,O(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(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(gt^#) = {}, Uargs(c_4) = {}, Uargs(and^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
f^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 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]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {and^#(x, false()) -> c_6()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
and^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
[0 0] [2 2] [7]
c_6() = [0]
[1]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(true(), x, y, z) ->
c_0(f^#(and(gt(x, y), gt(x, z)), x, s(y), z))
, 2: f^#(true(), x, y, z) ->
c_1(f^#(and(gt(x, y), gt(x, z)), x, y, s(z)))
, 3: gt^#(0(), v) -> c_2()
, 4: gt^#(s(u), 0()) -> c_3()
, 5: gt^#(s(u), s(v)) -> c_4(gt^#(u, v))
, 6: and^#(x, true()) -> c_5(x)
, 7: and^#(x, false()) -> c_6()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ YES(?,O(1)) ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{1,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, and(x, true()) -> x
, and(x, false()) -> false()}
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:
{ f^#(true(), x, y, z) ->
c_0(f^#(and(gt(x, y), gt(x, z)), x, s(y), z))
, f^#(true(), x, y, z) ->
c_1(f^#(and(gt(x, y), gt(x, z)), x, y, s(z)))
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, and(x, true()) -> x
, and(x, false()) -> false()}
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(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(gt^#) = {}, Uargs(c_4) = {1}, Uargs(and^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
false() = [0]
f^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
gt^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [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: {gt^#(s(u), s(v)) -> c_4(gt^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_4(x1) = [1] x1 + [7]
* Path {5}->{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(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(gt^#) = {}, Uargs(c_4) = {1}, Uargs(and^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
false() = [0]
f^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [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: {gt^#(0(), v) -> c_2()}
Weak Rules: {gt^#(s(u), s(v)) -> c_4(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
gt^#(x1, x2) = [3] x1 + [3] x2 + [2]
c_2() = [1]
c_4(x1) = [1] x1 + [5]
* 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(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(gt^#) = {}, Uargs(c_4) = {1}, Uargs(and^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
false() = [0]
f^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [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: {gt^#(s(u), 0()) -> c_3()}
Weak Rules: {gt^#(s(u), s(v)) -> c_4(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_3() = [1]
c_4(x1) = [1] x1 + [7]
* Path {6}: YES(?,O(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(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(gt^#) = {}, Uargs(c_4) = {}, Uargs(and^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
false() = [0]
f^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
and^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {and^#(x, true()) -> c_5(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [5]
and^#(x1, x2) = [7] x1 + [3] x2 + [0]
c_5(x1) = [1] x1 + [0]
* Path {7}: YES(?,O(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(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(gt^#) = {}, Uargs(c_4) = {}, Uargs(and^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
false() = [0]
f^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {and^#(x, false()) -> c_6()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [7]
and^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_6() = [1]
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.