Tool CaT
stdout:
MAYBE
Problem:
min(0(),y) -> 0()
min(x,0()) -> 0()
min(s(x),s(y)) -> s(min(x,y))
max(0(),y) -> y
max(x,0()) -> x
max(s(x),s(y)) -> s(max(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
-(x,0()) -> x
-(s(x),s(y)) -> -(x,y)
*(x,0()) -> 0()
*(x,s(y)) -> +(x,*(x,y))
f(s(x)) ->
f(-(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0()))))))))
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:
{ min(0(), y) -> 0()
, min(x, 0()) -> 0()
, min(s(x), s(y)) -> s(min(x, y))
, max(0(), y) -> y
, max(x, 0()) -> x
, max(s(x), s(y)) -> s(max(x, y))
, +(0(), y) -> y
, +(s(x), y) -> s(+(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(x, *(x, y))
, f(s(x)) ->
f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0()))))),
max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0()))))))))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: min^#(0(), y) -> c_0()
, 2: min^#(x, 0()) -> c_1()
, 3: min^#(s(x), s(y)) -> c_2(min^#(x, y))
, 4: max^#(0(), y) -> c_3()
, 5: max^#(x, 0()) -> c_4()
, 6: max^#(s(x), s(y)) -> c_5(max^#(x, y))
, 7: +^#(0(), y) -> c_6()
, 8: +^#(s(x), y) -> c_7(+^#(x, y))
, 9: -^#(x, 0()) -> c_8()
, 10: -^#(s(x), s(y)) -> c_9(-^#(x, y))
, 11: *^#(x, 0()) -> c_10()
, 12: *^#(x, s(y)) -> c_11(+^#(x, *(x, y)))
, 13: f^#(s(x)) ->
c_12(f^#(-(max(*(s(x), s(x)), +(s(x), s(s(s(0()))))),
max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0())))))))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{13} [ NA ]
->{12} [ inherited ]
|
|->{7} [ MAYBE ]
|
`->{8} [ inherited ]
|
`->{7} [ NA ]
->{11} [ YES(?,O(1)) ]
->{10} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(n^1)) ]
->{6} [ YES(?,O(n^1)) ]
|
|->{4} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{3} [ YES(?,O(n^1)) ]
|
|->{1} [ YES(?,O(n^1)) ]
|
`->{2} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(+) = {},
Uargs(-) = {}, Uargs(*) = {}, Uargs(f) = {}, Uargs(min^#) = {},
Uargs(c_2) = {1}, Uargs(max^#) = {}, Uargs(c_5) = {},
Uargs(+^#) = {}, Uargs(c_7) = {}, Uargs(-^#) = {}, Uargs(c_9) = {},
Uargs(*^#) = {}, Uargs(c_11) = {}, Uargs(f^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
-(x1, x2) = [0] x1 + [0] x2 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1) = [0] x1 + [0]
min^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
-^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_12(x1) = [0] 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: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
min^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_2(x1) = [1] x1 + [7]
* Path {3}->{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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(+) = {},
Uargs(-) = {}, Uargs(*) = {}, Uargs(f) = {}, Uargs(min^#) = {},
Uargs(c_2) = {1}, Uargs(max^#) = {}, Uargs(c_5) = {},
Uargs(+^#) = {}, Uargs(c_7) = {}, Uargs(-^#) = {}, Uargs(c_9) = {},
Uargs(*^#) = {}, Uargs(c_11) = {}, Uargs(f^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
-(x1, x2) = [0] x1 + [0] x2 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1) = [0] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
-^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_12(x1) = [0] 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: {min^#(0(), y) -> c_0()}
Weak Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
min^#(x1, x2) = [3] x1 + [3] x2 + [2]
c_0() = [1]
c_2(x1) = [1] x1 + [5]
* Path {3}->{2}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(+) = {},
Uargs(-) = {}, Uargs(*) = {}, Uargs(f) = {}, Uargs(min^#) = {},
Uargs(c_2) = {1}, Uargs(max^#) = {}, Uargs(c_5) = {},
Uargs(+^#) = {}, Uargs(c_7) = {}, Uargs(-^#) = {}, Uargs(c_9) = {},
Uargs(*^#) = {}, Uargs(c_11) = {}, Uargs(f^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
-(x1, x2) = [0] x1 + [0] x2 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1) = [0] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
-^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_12(x1) = [0] 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: {min^#(x, 0()) -> c_1()}
Weak Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
min^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_1() = [1]
c_2(x1) = [1] x1 + [6]
* Path {6}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(+) = {},
Uargs(-) = {}, Uargs(*) = {}, Uargs(f) = {}, Uargs(min^#) = {},
Uargs(c_2) = {}, Uargs(max^#) = {}, Uargs(c_5) = {1},
Uargs(+^#) = {}, Uargs(c_7) = {}, Uargs(-^#) = {}, Uargs(c_9) = {},
Uargs(*^#) = {}, Uargs(c_11) = {}, Uargs(f^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
-(x1, x2) = [0] x1 + [0] x2 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1) = [0] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
max^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
-^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_12(x1) = [0] 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: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
max^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_5(x1) = [1] x1 + [7]
* Path {6}->{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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(+) = {},
Uargs(-) = {}, Uargs(*) = {}, Uargs(f) = {}, Uargs(min^#) = {},
Uargs(c_2) = {}, Uargs(max^#) = {}, Uargs(c_5) = {1},
Uargs(+^#) = {}, Uargs(c_7) = {}, Uargs(-^#) = {}, Uargs(c_9) = {},
Uargs(*^#) = {}, Uargs(c_11) = {}, Uargs(f^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
-(x1, x2) = [0] x1 + [0] x2 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1) = [0] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
-^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_12(x1) = [0] 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: {max^#(0(), y) -> c_3()}
Weak Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
max^#(x1, x2) = [3] x1 + [3] x2 + [2]
c_3() = [1]
c_5(x1) = [1] x1 + [5]
* Path {6}->{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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(+) = {},
Uargs(-) = {}, Uargs(*) = {}, Uargs(f) = {}, Uargs(min^#) = {},
Uargs(c_2) = {}, Uargs(max^#) = {}, Uargs(c_5) = {1},
Uargs(+^#) = {}, Uargs(c_7) = {}, Uargs(-^#) = {}, Uargs(c_9) = {},
Uargs(*^#) = {}, Uargs(c_11) = {}, Uargs(f^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
-(x1, x2) = [0] x1 + [0] x2 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1) = [0] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
-^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_12(x1) = [0] 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: {max^#(x, 0()) -> c_4()}
Weak Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
max^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_4() = [1]
c_5(x1) = [1] x1 + [6]
* Path {10}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(+) = {},
Uargs(-) = {}, Uargs(*) = {}, Uargs(f) = {}, Uargs(min^#) = {},
Uargs(c_2) = {}, Uargs(max^#) = {}, Uargs(c_5) = {},
Uargs(+^#) = {}, Uargs(c_7) = {}, Uargs(-^#) = {},
Uargs(c_9) = {1}, Uargs(*^#) = {}, Uargs(c_11) = {},
Uargs(f^#) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
-(x1, x2) = [0] x1 + [0] x2 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1) = [0] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
-^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_8() = [0]
c_9(x1) = [1] x1 + [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_12(x1) = [0] 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: {-^#(s(x), s(y)) -> c_9(-^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(-^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
-^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_9(x1) = [1] x1 + [7]
* Path {10}->{9}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(+) = {},
Uargs(-) = {}, Uargs(*) = {}, Uargs(f) = {}, Uargs(min^#) = {},
Uargs(c_2) = {}, Uargs(max^#) = {}, Uargs(c_5) = {},
Uargs(+^#) = {}, Uargs(c_7) = {}, Uargs(-^#) = {},
Uargs(c_9) = {1}, Uargs(*^#) = {}, Uargs(c_11) = {},
Uargs(f^#) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
-(x1, x2) = [0] x1 + [0] x2 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1) = [0] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
-^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9(x1) = [1] x1 + [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_12(x1) = [0] 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: {-^#(x, 0()) -> c_8()}
Weak Rules: {-^#(s(x), s(y)) -> c_9(-^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(-^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
-^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_8() = [1]
c_9(x1) = [1] x1 + [6]
* Path {11}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(+) = {},
Uargs(-) = {}, Uargs(*) = {}, Uargs(f) = {}, Uargs(min^#) = {},
Uargs(c_2) = {}, Uargs(max^#) = {}, Uargs(c_5) = {},
Uargs(+^#) = {}, Uargs(c_7) = {}, Uargs(-^#) = {}, Uargs(c_9) = {},
Uargs(*^#) = {}, Uargs(c_11) = {}, Uargs(f^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
-(x1, x2) = [0] x1 + [0] x2 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1) = [0] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
-^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [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: {*^#(x, 0()) -> c_10()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(*^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
*^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_10() = [1]
* Path {12}: inherited
--------------------
This path is subsumed by the proof of path {12}->{8}->{7}.
* Path {12}->{7}: MAYBE
---------------------
The usable rules for this path are:
{ *(x, 0()) -> 0()
, *(x, s(y)) -> +(x, *(x, y))
, +(0(), y) -> y
, +(s(x), y) -> s(+(x, y))}
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:
{ *^#(x, s(y)) -> c_11(+^#(x, *(x, y)))
, +^#(0(), y) -> c_6()
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(x, *(x, y))
, +(0(), y) -> y
, +(s(x), y) -> s(+(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {12}->{8}: inherited
-------------------------
This path is subsumed by the proof of path {12}->{8}->{7}.
* Path {12}->{8}->{7}: NA
-----------------------
The usable rules for this path are:
{ *(x, 0()) -> 0()
, *(x, s(y)) -> +(x, *(x, y))
, +(0(), y) -> y
, +(s(x), y) -> s(+(x, y))}
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 {13}: NA
-------------
The usable rules for this path are:
{ max(0(), y) -> y
, max(x, 0()) -> x
, max(s(x), s(y)) -> s(max(x, y))
, +(0(), y) -> y
, +(s(x), y) -> s(+(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(x, *(x, y))}
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.
3) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
4) '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:
{ min(0(), y) -> 0()
, min(x, 0()) -> 0()
, min(s(x), s(y)) -> s(min(x, y))
, max(0(), y) -> y
, max(x, 0()) -> x
, max(s(x), s(y)) -> s(max(x, y))
, +(0(), y) -> y
, +(s(x), y) -> s(+(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(x, *(x, y))
, f(s(x)) ->
f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0()))))),
max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0()))))))))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: min^#(0(), y) -> c_0()
, 2: min^#(x, 0()) -> c_1()
, 3: min^#(s(x), s(y)) -> c_2(min^#(x, y))
, 4: max^#(0(), y) -> c_3(y)
, 5: max^#(x, 0()) -> c_4(x)
, 6: max^#(s(x), s(y)) -> c_5(max^#(x, y))
, 7: +^#(0(), y) -> c_6(y)
, 8: +^#(s(x), y) -> c_7(+^#(x, y))
, 9: -^#(x, 0()) -> c_8(x)
, 10: -^#(s(x), s(y)) -> c_9(-^#(x, y))
, 11: *^#(x, 0()) -> c_10()
, 12: *^#(x, s(y)) -> c_11(+^#(x, *(x, y)))
, 13: f^#(s(x)) ->
c_12(f^#(-(max(*(s(x), s(x)), +(s(x), s(s(s(0()))))),
max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0())))))))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{13} [ NA ]
->{12} [ inherited ]
|
|->{7} [ MAYBE ]
|
`->{8} [ inherited ]
|
`->{7} [ NA ]
->{11} [ YES(?,O(1)) ]
->{10} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(n^1)) ]
->{6} [ YES(?,O(n^1)) ]
|
|->{4} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{3} [ YES(?,O(n^1)) ]
|
|->{1} [ YES(?,O(n^1)) ]
|
`->{2} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(+) = {},
Uargs(-) = {}, Uargs(*) = {}, Uargs(f) = {}, Uargs(min^#) = {},
Uargs(c_2) = {1}, Uargs(max^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(+^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(-^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(*^#) = {}, Uargs(c_11) = {}, Uargs(f^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
-(x1, x2) = [0] x1 + [0] x2 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1) = [0] x1 + [0]
min^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
-^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [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: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
min^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_2(x1) = [1] x1 + [7]
* Path {3}->{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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(+) = {},
Uargs(-) = {}, Uargs(*) = {}, Uargs(f) = {}, Uargs(min^#) = {},
Uargs(c_2) = {1}, Uargs(max^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(+^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(-^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(*^#) = {}, Uargs(c_11) = {}, Uargs(f^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
-(x1, x2) = [0] x1 + [0] x2 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1) = [0] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
-^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [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: {min^#(0(), y) -> c_0()}
Weak Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
min^#(x1, x2) = [3] x1 + [3] x2 + [2]
c_0() = [1]
c_2(x1) = [1] x1 + [5]
* Path {3}->{2}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(+) = {},
Uargs(-) = {}, Uargs(*) = {}, Uargs(f) = {}, Uargs(min^#) = {},
Uargs(c_2) = {1}, Uargs(max^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(+^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(-^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(*^#) = {}, Uargs(c_11) = {}, Uargs(f^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
-(x1, x2) = [0] x1 + [0] x2 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1) = [0] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
-^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [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: {min^#(x, 0()) -> c_1()}
Weak Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
min^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_1() = [1]
c_2(x1) = [1] x1 + [6]
* Path {6}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(+) = {},
Uargs(-) = {}, Uargs(*) = {}, Uargs(f) = {}, Uargs(min^#) = {},
Uargs(c_2) = {}, Uargs(max^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_5) = {1}, Uargs(+^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(-^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(*^#) = {}, Uargs(c_11) = {},
Uargs(f^#) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
-(x1, x2) = [0] x1 + [0] x2 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1) = [0] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
max^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
-^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [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: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
max^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_5(x1) = [1] x1 + [7]
* Path {6}->{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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(+) = {},
Uargs(-) = {}, Uargs(*) = {}, Uargs(f) = {}, Uargs(min^#) = {},
Uargs(c_2) = {}, Uargs(max^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_5) = {1}, Uargs(+^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(-^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(*^#) = {}, Uargs(c_11) = {},
Uargs(f^#) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
-(x1, x2) = [0] x1 + [0] x2 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1) = [0] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
max^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
-^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [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: {max^#(0(), y) -> c_3(y)}
Weak Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [1] x1 + [2]
max^#(x1, x2) = [2] x1 + [2] x2 + [2]
c_3(x1) = [0] x1 + [1]
c_5(x1) = [1] x1 + [5]
* Path {6}->{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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(+) = {},
Uargs(-) = {}, Uargs(*) = {}, Uargs(f) = {}, Uargs(min^#) = {},
Uargs(c_2) = {}, Uargs(max^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_5) = {1}, Uargs(+^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(-^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(*^#) = {}, Uargs(c_11) = {},
Uargs(f^#) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
-(x1, x2) = [0] x1 + [0] x2 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1) = [0] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
max^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
-^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [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: {max^#(x, 0()) -> c_4(x)}
Weak Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [1] x1 + [2]
max^#(x1, x2) = [2] x1 + [2] x2 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
* Path {10}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(+) = {},
Uargs(-) = {}, Uargs(*) = {}, Uargs(f) = {}, Uargs(min^#) = {},
Uargs(c_2) = {}, Uargs(max^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(+^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(-^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(*^#) = {}, Uargs(c_11) = {},
Uargs(f^#) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
-(x1, x2) = [0] x1 + [0] x2 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1) = [0] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
-^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [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: {-^#(s(x), s(y)) -> c_9(-^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(-^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
-^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_9(x1) = [1] x1 + [7]
* Path {10}->{9}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(+) = {},
Uargs(-) = {}, Uargs(*) = {}, Uargs(f) = {}, Uargs(min^#) = {},
Uargs(c_2) = {}, Uargs(max^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(+^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(-^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(*^#) = {}, Uargs(c_11) = {},
Uargs(f^#) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
-(x1, x2) = [0] x1 + [0] x2 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1) = [0] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
-^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [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: {-^#(x, 0()) -> c_8(x)}
Weak Rules: {-^#(s(x), s(y)) -> c_9(-^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(-^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [1] x1 + [2]
-^#(x1, x2) = [2] x1 + [2] x2 + [1]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
* Path {11}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(+) = {},
Uargs(-) = {}, Uargs(*) = {}, Uargs(f) = {}, Uargs(min^#) = {},
Uargs(c_2) = {}, Uargs(max^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(+^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(-^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(*^#) = {}, Uargs(c_11) = {}, Uargs(f^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
-(x1, x2) = [0] x1 + [0] x2 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1) = [0] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
-^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [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: {*^#(x, 0()) -> c_10()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(*^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
*^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_10() = [1]
* Path {12}: inherited
--------------------
This path is subsumed by the proof of path {12}->{8}->{7}.
* Path {12}->{7}: MAYBE
---------------------
The usable rules for this path are:
{ *(x, 0()) -> 0()
, *(x, s(y)) -> +(x, *(x, y))
, +(0(), y) -> y
, +(s(x), y) -> s(+(x, y))}
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:
{ *^#(x, s(y)) -> c_11(+^#(x, *(x, y)))
, +^#(0(), y) -> c_6(y)
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(x, *(x, y))
, +(0(), y) -> y
, +(s(x), y) -> s(+(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {12}->{8}: inherited
-------------------------
This path is subsumed by the proof of path {12}->{8}->{7}.
* Path {12}->{8}->{7}: NA
-----------------------
The usable rules for this path are:
{ *(x, 0()) -> 0()
, *(x, s(y)) -> +(x, *(x, y))
, +(0(), y) -> y
, +(s(x), y) -> s(+(x, y))}
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 {13}: NA
-------------
The usable rules for this path are:
{ max(0(), y) -> y
, max(x, 0()) -> x
, max(s(x), s(y)) -> s(max(x, y))
, +(0(), y) -> y
, +(s(x), y) -> s(+(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(x, *(x, y))}
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.
3) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
4) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.