Tool CaT
stdout:
MAYBE
Problem:
lt(0(),s(x)) -> true()
lt(x,0()) -> false()
lt(s(x),s(y)) -> lt(x,y)
times(0(),y) -> 0()
times(s(x),y) -> plus(y,times(x,y))
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
fac(x) -> loop(x,s(0()),s(0()))
loop(x,c,y) -> if(lt(x,c),x,c,y)
if(false(),x,c,y) -> loop(x,s(c),times(y,s(c)))
if(true(),x,c,y) -> y
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:
{ lt(0(), s(x)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, fac(x) -> loop(x, s(0()), s(0()))
, loop(x, c, y) -> if(lt(x, c), x, c, y)
, if(false(), x, c, y) -> loop(x, s(c), times(y, s(c)))
, if(true(), x, c, y) -> y}
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: lt^#(0(), s(x)) -> c_0()
, 2: lt^#(x, 0()) -> c_1()
, 3: lt^#(s(x), s(y)) -> c_2(lt^#(x, y))
, 4: times^#(0(), y) -> c_3()
, 5: times^#(s(x), y) -> c_4(plus^#(y, times(x, y)))
, 6: plus^#(0(), y) -> c_5()
, 7: plus^#(s(x), y) -> c_6(plus^#(x, y))
, 8: fac^#(x) -> c_7(loop^#(x, s(0()), s(0())))
, 9: loop^#(x, c, y) -> c_8(if^#(lt(x, c), x, c, y))
, 10: if^#(false(), x, c, y) ->
c_9(loop^#(x, s(c), times(y, s(c))))
, 11: if^#(true(), x, c, y) -> c_10()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8} [ inherited ]
|
`->{9,10} [ inherited ]
|
`->{11} [ NA ]
->{5} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{7} [ inherited ]
|
`->{6} [ NA ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^1)) ]
|
|->{1} [ YES(?,O(n^2)) ]
|
`->{2} [ YES(?,O(n^3)) ]
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(lt) = {}, Uargs(s) = {}, Uargs(times) = {}, Uargs(plus) = {},
Uargs(fac) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(lt^#) = {}, Uargs(c_2) = {1}, Uargs(times^#) = {},
Uargs(c_4) = {}, Uargs(plus^#) = {}, Uargs(c_6) = {},
Uargs(fac^#) = {}, Uargs(c_7) = {}, Uargs(loop^#) = {},
Uargs(c_8) = {}, Uargs(if^#) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
lt(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 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
times(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]
plus(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]
fac(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop(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]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
lt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
times^#(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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plus^#(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_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fac^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop^#(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_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {lt^#(s(x), s(y)) -> c_2(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [2]
[0 0 0] [0]
lt^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
[2 2 0] [0 2 0] [0]
[4 0 0] [0 2 0] [0]
c_2(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {3}->{1}: 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(lt) = {}, Uargs(s) = {}, Uargs(times) = {}, Uargs(plus) = {},
Uargs(fac) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(lt^#) = {}, Uargs(c_2) = {1}, Uargs(times^#) = {},
Uargs(c_4) = {}, Uargs(plus^#) = {}, Uargs(c_6) = {},
Uargs(fac^#) = {}, Uargs(c_7) = {}, Uargs(loop^#) = {},
Uargs(c_8) = {}, Uargs(if^#) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
lt(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
times(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]
plus(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]
fac(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop(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]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
lt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
times^#(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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plus^#(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_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fac^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop^#(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_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[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: {lt^#(0(), s(x)) -> c_0()}
Weak Rules: {lt^#(s(x), s(y)) -> c_2(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
[0]
s(x1) = [1 1 2] x1 + [0]
[0 1 0] [2]
[0 0 0] [0]
lt^#(x1, x2) = [0 2 0] x1 + [1 0 2] x2 + [0]
[7 1 0] [4 0 0] [0]
[4 2 0] [4 0 0] [0]
c_0() = [1]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [3]
[2 0 0] [2]
[0 0 0] [2]
* Path {3}->{2}: YES(?,O(n^3))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(lt) = {}, Uargs(s) = {}, Uargs(times) = {}, Uargs(plus) = {},
Uargs(fac) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(lt^#) = {}, Uargs(c_2) = {1}, Uargs(times^#) = {},
Uargs(c_4) = {}, Uargs(plus^#) = {}, Uargs(c_6) = {},
Uargs(fac^#) = {}, Uargs(c_7) = {}, Uargs(loop^#) = {},
Uargs(c_8) = {}, Uargs(if^#) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
lt(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
times(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]
plus(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]
fac(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop(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]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
lt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
times^#(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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plus^#(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_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fac^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop^#(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_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {lt^#(x, 0()) -> c_1()}
Weak Rules: {lt^#(s(x), s(y)) -> c_2(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 2 0] x1 + [2]
[0 1 3] [2]
[0 0 1] [2]
lt^#(x1, x2) = [0 0 0] x1 + [0 2 2] x2 + [0]
[0 0 2] [2 2 0] [0]
[0 0 0] [0 2 2] [0]
c_1() = [1]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [7]
* Path {4}: 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(lt) = {}, Uargs(s) = {}, Uargs(times) = {}, Uargs(plus) = {},
Uargs(fac) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(lt^#) = {}, Uargs(c_2) = {}, Uargs(times^#) = {},
Uargs(c_4) = {}, Uargs(plus^#) = {}, Uargs(c_6) = {},
Uargs(fac^#) = {}, Uargs(c_7) = {}, Uargs(loop^#) = {},
Uargs(c_8) = {}, Uargs(if^#) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
lt(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
times(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]
plus(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]
fac(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop(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]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
lt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plus^#(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_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fac^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop^#(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_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {times^#(0(), y) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
times^#(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [7]
[2 2 0] [0 0 0] [3]
[2 2 2] [0 0 0] [3]
c_3() = [0]
[1]
[1]
* Path {5}: inherited
-------------------
This path is subsumed by the proof of path {5}->{7}->{6}.
* Path {5}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ times^#(s(x), y) -> c_4(plus^#(y, times(x, y)))
, plus^#(0(), y) -> c_5()
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {5}->{7}: inherited
------------------------
This path is subsumed by the proof of path {5}->{7}->{6}.
* Path {5}->{7}->{6}: NA
----------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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 {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{9,10}->{11}.
* Path {8}->{9,10}: inherited
---------------------------
This path is subsumed by the proof of path {8}->{9,10}->{11}.
* Path {8}->{9,10}->{11}: NA
--------------------------
The usable rules for this path are:
{ lt(0(), s(x)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: lt^#(0(), s(x)) -> c_0()
, 2: lt^#(x, 0()) -> c_1()
, 3: lt^#(s(x), s(y)) -> c_2(lt^#(x, y))
, 4: times^#(0(), y) -> c_3()
, 5: times^#(s(x), y) -> c_4(plus^#(y, times(x, y)))
, 6: plus^#(0(), y) -> c_5()
, 7: plus^#(s(x), y) -> c_6(plus^#(x, y))
, 8: fac^#(x) -> c_7(loop^#(x, s(0()), s(0())))
, 9: loop^#(x, c, y) -> c_8(if^#(lt(x, c), x, c, y))
, 10: if^#(false(), x, c, y) ->
c_9(loop^#(x, s(c), times(y, s(c))))
, 11: if^#(true(), x, c, y) -> c_10()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8} [ inherited ]
|
`->{9,10} [ inherited ]
|
`->{11} [ NA ]
->{5} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{7} [ inherited ]
|
`->{6} [ NA ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^2)) ]
|
|->{1} [ YES(?,O(n^2)) ]
|
`->{2} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* Path {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(lt) = {}, Uargs(s) = {}, Uargs(times) = {}, Uargs(plus) = {},
Uargs(fac) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(lt^#) = {}, Uargs(c_2) = {1}, Uargs(times^#) = {},
Uargs(c_4) = {}, Uargs(plus^#) = {}, Uargs(c_6) = {},
Uargs(fac^#) = {}, Uargs(c_7) = {}, Uargs(loop^#) = {},
Uargs(c_8) = {}, Uargs(if^#) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
lt(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]
true() = [0]
[0]
false() = [0]
[0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fac(x1) = [0 0] x1 + [0]
[0 0] [0]
loop(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
if(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]
lt^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
fac^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
loop^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(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_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [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: {lt^#(s(x), s(y)) -> c_2(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
lt^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_2(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {3}->{1}: 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(lt) = {}, Uargs(s) = {}, Uargs(times) = {}, Uargs(plus) = {},
Uargs(fac) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(lt^#) = {}, Uargs(c_2) = {1}, Uargs(times^#) = {},
Uargs(c_4) = {}, Uargs(plus^#) = {}, Uargs(c_6) = {},
Uargs(fac^#) = {}, Uargs(c_7) = {}, Uargs(loop^#) = {},
Uargs(c_8) = {}, Uargs(if^#) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
lt(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]
true() = [0]
[0]
false() = [0]
[0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fac(x1) = [0 0] x1 + [0]
[0 0] [0]
loop(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
if(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]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
fac^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
loop^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(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_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [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: {lt^#(0(), s(x)) -> c_0()}
Weak Rules: {lt^#(s(x), s(y)) -> c_2(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 6] x1 + [2]
[0 1] [2]
lt^#(x1, x2) = [2 1] x1 + [0 1] x2 + [0]
[1 2] [2 0] [0]
c_0() = [1]
[0]
c_2(x1) = [1 0] x1 + [7]
[0 0] [7]
* Path {3}->{2}: 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(lt) = {}, Uargs(s) = {}, Uargs(times) = {}, Uargs(plus) = {},
Uargs(fac) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(lt^#) = {}, Uargs(c_2) = {1}, Uargs(times^#) = {},
Uargs(c_4) = {}, Uargs(plus^#) = {}, Uargs(c_6) = {},
Uargs(fac^#) = {}, Uargs(c_7) = {}, Uargs(loop^#) = {},
Uargs(c_8) = {}, Uargs(if^#) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
lt(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]
true() = [0]
[0]
false() = [0]
[0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fac(x1) = [0 0] x1 + [0]
[0 0] [0]
loop(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
if(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]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
fac^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
loop^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(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_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [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: {lt^#(x, 0()) -> c_1()}
Weak Rules: {lt^#(s(x), s(y)) -> c_2(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
lt^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
[0 0] [4 1] [0]
c_1() = [1]
[0]
c_2(x1) = [1 0] x1 + [6]
[0 0] [7]
* Path {4}: 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(lt) = {}, Uargs(s) = {}, Uargs(times) = {}, Uargs(plus) = {},
Uargs(fac) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(lt^#) = {}, Uargs(c_2) = {}, Uargs(times^#) = {},
Uargs(c_4) = {}, Uargs(plus^#) = {}, Uargs(c_6) = {},
Uargs(fac^#) = {}, Uargs(c_7) = {}, Uargs(loop^#) = {},
Uargs(c_8) = {}, Uargs(if^#) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
lt(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]
true() = [0]
[0]
false() = [0]
[0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fac(x1) = [0 0] x1 + [0]
[0 0] [0]
loop(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
if(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]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
fac^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
loop^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(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_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [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: {times^#(0(), y) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
times^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
[2 2] [0 0] [7]
c_3() = [0]
[1]
* Path {5}: inherited
-------------------
This path is subsumed by the proof of path {5}->{7}->{6}.
* Path {5}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ times^#(s(x), y) -> c_4(plus^#(y, times(x, y)))
, plus^#(0(), y) -> c_5()
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {5}->{7}: inherited
------------------------
This path is subsumed by the proof of path {5}->{7}->{6}.
* Path {5}->{7}->{6}: NA
----------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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 {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{9,10}->{11}.
* Path {8}->{9,10}: inherited
---------------------------
This path is subsumed by the proof of path {8}->{9,10}->{11}.
* Path {8}->{9,10}->{11}: NA
--------------------------
The usable rules for this path are:
{ lt(0(), s(x)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: lt^#(0(), s(x)) -> c_0()
, 2: lt^#(x, 0()) -> c_1()
, 3: lt^#(s(x), s(y)) -> c_2(lt^#(x, y))
, 4: times^#(0(), y) -> c_3()
, 5: times^#(s(x), y) -> c_4(plus^#(y, times(x, y)))
, 6: plus^#(0(), y) -> c_5()
, 7: plus^#(s(x), y) -> c_6(plus^#(x, y))
, 8: fac^#(x) -> c_7(loop^#(x, s(0()), s(0())))
, 9: loop^#(x, c, y) -> c_8(if^#(lt(x, c), x, c, y))
, 10: if^#(false(), x, c, y) ->
c_9(loop^#(x, s(c), times(y, s(c))))
, 11: if^#(true(), x, c, y) -> c_10()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8} [ inherited ]
|
`->{9,10} [ inherited ]
|
`->{11} [ NA ]
->{5} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{7} [ inherited ]
|
`->{6} [ NA ]
->{4} [ YES(?,O(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(lt) = {}, Uargs(s) = {}, Uargs(times) = {}, Uargs(plus) = {},
Uargs(fac) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(lt^#) = {}, Uargs(c_2) = {1}, Uargs(times^#) = {},
Uargs(c_4) = {}, Uargs(plus^#) = {}, Uargs(c_6) = {},
Uargs(fac^#) = {}, Uargs(c_7) = {}, Uargs(loop^#) = {},
Uargs(c_8) = {}, Uargs(if^#) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
lt(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
true() = [0]
false() = [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
fac(x1) = [0] x1 + [0]
loop(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
lt^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
fac^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
loop^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9(x1) = [0] x1 + [0]
c_10() = [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: {lt^#(s(x), s(y)) -> c_2(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
lt^#(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(lt) = {}, Uargs(s) = {}, Uargs(times) = {}, Uargs(plus) = {},
Uargs(fac) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(lt^#) = {}, Uargs(c_2) = {1}, Uargs(times^#) = {},
Uargs(c_4) = {}, Uargs(plus^#) = {}, Uargs(c_6) = {},
Uargs(fac^#) = {}, Uargs(c_7) = {}, Uargs(loop^#) = {},
Uargs(c_8) = {}, Uargs(if^#) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
lt(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
true() = [0]
false() = [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
fac(x1) = [0] x1 + [0]
loop(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
fac^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
loop^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9(x1) = [0] x1 + [0]
c_10() = [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: {lt^#(0(), s(x)) -> c_0()}
Weak Rules: {lt^#(s(x), s(y)) -> c_2(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
lt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_0() = [1]
c_2(x1) = [1] x1 + [7]
* 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(lt) = {}, Uargs(s) = {}, Uargs(times) = {}, Uargs(plus) = {},
Uargs(fac) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(lt^#) = {}, Uargs(c_2) = {1}, Uargs(times^#) = {},
Uargs(c_4) = {}, Uargs(plus^#) = {}, Uargs(c_6) = {},
Uargs(fac^#) = {}, Uargs(c_7) = {}, Uargs(loop^#) = {},
Uargs(c_8) = {}, Uargs(if^#) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
lt(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
true() = [0]
false() = [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
fac(x1) = [0] x1 + [0]
loop(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
fac^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
loop^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9(x1) = [0] x1 + [0]
c_10() = [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: {lt^#(x, 0()) -> c_1()}
Weak Rules: {lt^#(s(x), s(y)) -> c_2(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
lt^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_1() = [1]
c_2(x1) = [1] x1 + [7]
* Path {4}: 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(lt) = {}, Uargs(s) = {}, Uargs(times) = {}, Uargs(plus) = {},
Uargs(fac) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(lt^#) = {}, Uargs(c_2) = {}, Uargs(times^#) = {},
Uargs(c_4) = {}, Uargs(plus^#) = {}, Uargs(c_6) = {},
Uargs(fac^#) = {}, Uargs(c_7) = {}, Uargs(loop^#) = {},
Uargs(c_8) = {}, Uargs(if^#) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
lt(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
true() = [0]
false() = [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
fac(x1) = [0] x1 + [0]
loop(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
fac^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
loop^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9(x1) = [0] x1 + [0]
c_10() = [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: {times^#(0(), y) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
times^#(x1, x2) = [1] x1 + [0] x2 + [7]
c_3() = [1]
* Path {5}: inherited
-------------------
This path is subsumed by the proof of path {5}->{7}->{6}.
* Path {5}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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:
{ times^#(s(x), y) -> c_4(plus^#(y, times(x, y)))
, plus^#(0(), y) -> c_5()
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {5}->{7}: inherited
------------------------
This path is subsumed by the proof of path {5}->{7}->{6}.
* Path {5}->{7}->{6}: NA
----------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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 {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{9,10}->{11}.
* Path {8}->{9,10}: inherited
---------------------------
This path is subsumed by the proof of path {8}->{9,10}->{11}.
* Path {8}->{9,10}->{11}: NA
--------------------------
The usable rules for this path are:
{ lt(0(), s(x)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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.
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:
{ lt(0(), s(x)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, fac(x) -> loop(x, s(0()), s(0()))
, loop(x, c, y) -> if(lt(x, c), x, c, y)
, if(false(), x, c, y) -> loop(x, s(c), times(y, s(c)))
, if(true(), x, c, y) -> y}
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: lt^#(0(), s(x)) -> c_0()
, 2: lt^#(x, 0()) -> c_1()
, 3: lt^#(s(x), s(y)) -> c_2(lt^#(x, y))
, 4: times^#(0(), y) -> c_3()
, 5: times^#(s(x), y) -> c_4(plus^#(y, times(x, y)))
, 6: plus^#(0(), y) -> c_5(y)
, 7: plus^#(s(x), y) -> c_6(plus^#(x, y))
, 8: fac^#(x) -> c_7(loop^#(x, s(0()), s(0())))
, 9: loop^#(x, c, y) -> c_8(if^#(lt(x, c), x, c, y))
, 10: if^#(false(), x, c, y) ->
c_9(loop^#(x, s(c), times(y, s(c))))
, 11: if^#(true(), x, c, y) -> c_10(y)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8} [ inherited ]
|
`->{9,10} [ inherited ]
|
`->{11} [ NA ]
->{5} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{7} [ inherited ]
|
`->{6} [ NA ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^1)) ]
|
|->{1} [ YES(?,O(n^2)) ]
|
`->{2} [ YES(?,O(n^3)) ]
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(lt) = {}, Uargs(s) = {}, Uargs(times) = {}, Uargs(plus) = {},
Uargs(fac) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(lt^#) = {}, Uargs(c_2) = {1}, Uargs(times^#) = {},
Uargs(c_4) = {}, Uargs(plus^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(fac^#) = {}, Uargs(c_7) = {},
Uargs(loop^#) = {}, Uargs(c_8) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
lt(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 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
times(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]
plus(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]
fac(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop(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]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
lt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
times^#(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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plus^#(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_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fac^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop^#(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_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {lt^#(s(x), s(y)) -> c_2(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [2]
[0 0 0] [0]
lt^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
[2 2 0] [0 2 0] [0]
[4 0 0] [0 2 0] [0]
c_2(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {3}->{1}: 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(lt) = {}, Uargs(s) = {}, Uargs(times) = {}, Uargs(plus) = {},
Uargs(fac) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(lt^#) = {}, Uargs(c_2) = {1}, Uargs(times^#) = {},
Uargs(c_4) = {}, Uargs(plus^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(fac^#) = {}, Uargs(c_7) = {},
Uargs(loop^#) = {}, Uargs(c_8) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
lt(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
times(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]
plus(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]
fac(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop(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]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
lt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
times^#(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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plus^#(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_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fac^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop^#(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_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [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: {lt^#(0(), s(x)) -> c_0()}
Weak Rules: {lt^#(s(x), s(y)) -> c_2(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
[0]
s(x1) = [1 1 2] x1 + [0]
[0 1 0] [2]
[0 0 0] [0]
lt^#(x1, x2) = [0 2 0] x1 + [1 0 2] x2 + [0]
[7 1 0] [4 0 0] [0]
[4 2 0] [4 0 0] [0]
c_0() = [1]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [3]
[2 0 0] [2]
[0 0 0] [2]
* Path {3}->{2}: YES(?,O(n^3))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(lt) = {}, Uargs(s) = {}, Uargs(times) = {}, Uargs(plus) = {},
Uargs(fac) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(lt^#) = {}, Uargs(c_2) = {1}, Uargs(times^#) = {},
Uargs(c_4) = {}, Uargs(plus^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(fac^#) = {}, Uargs(c_7) = {},
Uargs(loop^#) = {}, Uargs(c_8) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
lt(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
times(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]
plus(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]
fac(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop(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]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
lt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
times^#(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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plus^#(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_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fac^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop^#(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_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {lt^#(x, 0()) -> c_1()}
Weak Rules: {lt^#(s(x), s(y)) -> c_2(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 2 0] x1 + [2]
[0 1 3] [2]
[0 0 1] [2]
lt^#(x1, x2) = [0 0 0] x1 + [0 2 2] x2 + [0]
[0 0 2] [2 2 0] [0]
[0 0 0] [0 2 2] [0]
c_1() = [1]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [7]
* Path {4}: 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(lt) = {}, Uargs(s) = {}, Uargs(times) = {}, Uargs(plus) = {},
Uargs(fac) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(lt^#) = {}, Uargs(c_2) = {}, Uargs(times^#) = {},
Uargs(c_4) = {}, Uargs(plus^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(fac^#) = {}, Uargs(c_7) = {},
Uargs(loop^#) = {}, Uargs(c_8) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
lt(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
times(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]
plus(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]
fac(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop(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]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
lt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plus^#(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_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fac^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop^#(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_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {times^#(0(), y) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
times^#(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [7]
[2 2 0] [0 0 0] [3]
[2 2 2] [0 0 0] [3]
c_3() = [0]
[1]
[1]
* Path {5}: inherited
-------------------
This path is subsumed by the proof of path {5}->{7}->{6}.
* Path {5}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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 3'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ times^#(s(x), y) -> c_4(plus^#(y, times(x, y)))
, plus^#(0(), y) -> c_5(y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {5}->{7}: inherited
------------------------
This path is subsumed by the proof of path {5}->{7}->{6}.
* Path {5}->{7}->{6}: NA
----------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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 {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{9,10}->{11}.
* Path {8}->{9,10}: inherited
---------------------------
This path is subsumed by the proof of path {8}->{9,10}->{11}.
* Path {8}->{9,10}->{11}: NA
--------------------------
The usable rules for this path are:
{ lt(0(), s(x)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: lt^#(0(), s(x)) -> c_0()
, 2: lt^#(x, 0()) -> c_1()
, 3: lt^#(s(x), s(y)) -> c_2(lt^#(x, y))
, 4: times^#(0(), y) -> c_3()
, 5: times^#(s(x), y) -> c_4(plus^#(y, times(x, y)))
, 6: plus^#(0(), y) -> c_5(y)
, 7: plus^#(s(x), y) -> c_6(plus^#(x, y))
, 8: fac^#(x) -> c_7(loop^#(x, s(0()), s(0())))
, 9: loop^#(x, c, y) -> c_8(if^#(lt(x, c), x, c, y))
, 10: if^#(false(), x, c, y) ->
c_9(loop^#(x, s(c), times(y, s(c))))
, 11: if^#(true(), x, c, y) -> c_10(y)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8} [ inherited ]
|
`->{9,10} [ inherited ]
|
`->{11} [ NA ]
->{5} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{7} [ inherited ]
|
`->{6} [ NA ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^2)) ]
|
|->{1} [ YES(?,O(n^2)) ]
|
`->{2} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* Path {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(lt) = {}, Uargs(s) = {}, Uargs(times) = {}, Uargs(plus) = {},
Uargs(fac) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(lt^#) = {}, Uargs(c_2) = {1}, Uargs(times^#) = {},
Uargs(c_4) = {}, Uargs(plus^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(fac^#) = {}, Uargs(c_7) = {},
Uargs(loop^#) = {}, Uargs(c_8) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
lt(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]
true() = [0]
[0]
false() = [0]
[0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fac(x1) = [0 0] x1 + [0]
[0 0] [0]
loop(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
if(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]
lt^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(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(x1) = [0 0] x1 + [0]
[0 0] [0]
fac^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
loop^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(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_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 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: {lt^#(s(x), s(y)) -> c_2(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
lt^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_2(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {3}->{1}: 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(lt) = {}, Uargs(s) = {}, Uargs(times) = {}, Uargs(plus) = {},
Uargs(fac) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(lt^#) = {}, Uargs(c_2) = {1}, Uargs(times^#) = {},
Uargs(c_4) = {}, Uargs(plus^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(fac^#) = {}, Uargs(c_7) = {},
Uargs(loop^#) = {}, Uargs(c_8) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
lt(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]
true() = [0]
[0]
false() = [0]
[0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fac(x1) = [0 0] x1 + [0]
[0 0] [0]
loop(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
if(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]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(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(x1) = [0 0] x1 + [0]
[0 0] [0]
fac^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
loop^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(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_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 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: {lt^#(0(), s(x)) -> c_0()}
Weak Rules: {lt^#(s(x), s(y)) -> c_2(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 6] x1 + [2]
[0 1] [2]
lt^#(x1, x2) = [2 1] x1 + [0 1] x2 + [0]
[1 2] [2 0] [0]
c_0() = [1]
[0]
c_2(x1) = [1 0] x1 + [7]
[0 0] [7]
* Path {3}->{2}: 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(lt) = {}, Uargs(s) = {}, Uargs(times) = {}, Uargs(plus) = {},
Uargs(fac) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(lt^#) = {}, Uargs(c_2) = {1}, Uargs(times^#) = {},
Uargs(c_4) = {}, Uargs(plus^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(fac^#) = {}, Uargs(c_7) = {},
Uargs(loop^#) = {}, Uargs(c_8) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
lt(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]
true() = [0]
[0]
false() = [0]
[0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fac(x1) = [0 0] x1 + [0]
[0 0] [0]
loop(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
if(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]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(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(x1) = [0 0] x1 + [0]
[0 0] [0]
fac^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
loop^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(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_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 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: {lt^#(x, 0()) -> c_1()}
Weak Rules: {lt^#(s(x), s(y)) -> c_2(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
lt^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
[0 0] [4 1] [0]
c_1() = [1]
[0]
c_2(x1) = [1 0] x1 + [6]
[0 0] [7]
* Path {4}: 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(lt) = {}, Uargs(s) = {}, Uargs(times) = {}, Uargs(plus) = {},
Uargs(fac) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(lt^#) = {}, Uargs(c_2) = {}, Uargs(times^#) = {},
Uargs(c_4) = {}, Uargs(plus^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(fac^#) = {}, Uargs(c_7) = {},
Uargs(loop^#) = {}, Uargs(c_8) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
lt(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]
true() = [0]
[0]
false() = [0]
[0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fac(x1) = [0 0] x1 + [0]
[0 0] [0]
loop(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
if(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]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(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(x1) = [0 0] x1 + [0]
[0 0] [0]
fac^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
loop^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(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_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 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: {times^#(0(), y) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
times^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
[2 2] [0 0] [7]
c_3() = [0]
[1]
* Path {5}: inherited
-------------------
This path is subsumed by the proof of path {5}->{7}->{6}.
* Path {5}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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 2'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ times^#(s(x), y) -> c_4(plus^#(y, times(x, y)))
, plus^#(0(), y) -> c_5(y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {5}->{7}: inherited
------------------------
This path is subsumed by the proof of path {5}->{7}->{6}.
* Path {5}->{7}->{6}: NA
----------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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 {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{9,10}->{11}.
* Path {8}->{9,10}: inherited
---------------------------
This path is subsumed by the proof of path {8}->{9,10}->{11}.
* Path {8}->{9,10}->{11}: NA
--------------------------
The usable rules for this path are:
{ lt(0(), s(x)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: lt^#(0(), s(x)) -> c_0()
, 2: lt^#(x, 0()) -> c_1()
, 3: lt^#(s(x), s(y)) -> c_2(lt^#(x, y))
, 4: times^#(0(), y) -> c_3()
, 5: times^#(s(x), y) -> c_4(plus^#(y, times(x, y)))
, 6: plus^#(0(), y) -> c_5(y)
, 7: plus^#(s(x), y) -> c_6(plus^#(x, y))
, 8: fac^#(x) -> c_7(loop^#(x, s(0()), s(0())))
, 9: loop^#(x, c, y) -> c_8(if^#(lt(x, c), x, c, y))
, 10: if^#(false(), x, c, y) ->
c_9(loop^#(x, s(c), times(y, s(c))))
, 11: if^#(true(), x, c, y) -> c_10(y)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8} [ inherited ]
|
`->{9,10} [ inherited ]
|
`->{11} [ NA ]
->{5} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{7} [ inherited ]
|
`->{6} [ NA ]
->{4} [ YES(?,O(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(lt) = {}, Uargs(s) = {}, Uargs(times) = {}, Uargs(plus) = {},
Uargs(fac) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(lt^#) = {}, Uargs(c_2) = {1}, Uargs(times^#) = {},
Uargs(c_4) = {}, Uargs(plus^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(fac^#) = {}, Uargs(c_7) = {},
Uargs(loop^#) = {}, Uargs(c_8) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
lt(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
true() = [0]
false() = [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
fac(x1) = [0] x1 + [0]
loop(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
lt^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
fac^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
loop^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9(x1) = [0] x1 + [0]
c_10(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: {lt^#(s(x), s(y)) -> c_2(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
lt^#(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(lt) = {}, Uargs(s) = {}, Uargs(times) = {}, Uargs(plus) = {},
Uargs(fac) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(lt^#) = {}, Uargs(c_2) = {1}, Uargs(times^#) = {},
Uargs(c_4) = {}, Uargs(plus^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(fac^#) = {}, Uargs(c_7) = {},
Uargs(loop^#) = {}, Uargs(c_8) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
lt(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
true() = [0]
false() = [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
fac(x1) = [0] x1 + [0]
loop(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
fac^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
loop^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9(x1) = [0] x1 + [0]
c_10(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: {lt^#(0(), s(x)) -> c_0()}
Weak Rules: {lt^#(s(x), s(y)) -> c_2(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
lt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_0() = [1]
c_2(x1) = [1] x1 + [7]
* 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(lt) = {}, Uargs(s) = {}, Uargs(times) = {}, Uargs(plus) = {},
Uargs(fac) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(lt^#) = {}, Uargs(c_2) = {1}, Uargs(times^#) = {},
Uargs(c_4) = {}, Uargs(plus^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(fac^#) = {}, Uargs(c_7) = {},
Uargs(loop^#) = {}, Uargs(c_8) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
lt(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
true() = [0]
false() = [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
fac(x1) = [0] x1 + [0]
loop(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
fac^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
loop^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9(x1) = [0] x1 + [0]
c_10(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: {lt^#(x, 0()) -> c_1()}
Weak Rules: {lt^#(s(x), s(y)) -> c_2(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
lt^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_1() = [1]
c_2(x1) = [1] x1 + [7]
* Path {4}: 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(lt) = {}, Uargs(s) = {}, Uargs(times) = {}, Uargs(plus) = {},
Uargs(fac) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(lt^#) = {}, Uargs(c_2) = {}, Uargs(times^#) = {},
Uargs(c_4) = {}, Uargs(plus^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(fac^#) = {}, Uargs(c_7) = {},
Uargs(loop^#) = {}, Uargs(c_8) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
lt(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
true() = [0]
false() = [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
fac(x1) = [0] x1 + [0]
loop(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
fac^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
loop^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9(x1) = [0] x1 + [0]
c_10(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: {times^#(0(), y) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
times^#(x1, x2) = [1] x1 + [0] x2 + [7]
c_3() = [1]
* Path {5}: inherited
-------------------
This path is subsumed by the proof of path {5}->{7}->{6}.
* Path {5}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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:
{ times^#(s(x), y) -> c_4(plus^#(y, times(x, y)))
, plus^#(0(), y) -> c_5(y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {5}->{7}: inherited
------------------------
This path is subsumed by the proof of path {5}->{7}->{6}.
* Path {5}->{7}->{6}: NA
----------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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 {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{9,10}->{11}.
* Path {8}->{9,10}: inherited
---------------------------
This path is subsumed by the proof of path {8}->{9,10}->{11}.
* Path {8}->{9,10}->{11}: NA
--------------------------
The usable rules for this path are:
{ lt(0(), s(x)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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.
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.