Tool CaT
stdout:
MAYBE
Problem:
ite(tt(),x,y) -> x
ite(ff(),x,y) -> y
lt(0(),s(y)) -> tt()
lt(x,0()) -> ff()
lt(s(x),s(y)) -> lt(x,y)
insert(a,nil()) -> cons(a,nil())
insert(a,cons(b,l)) -> ite(lt(a,b),cons(a,cons(b,l)),cons(b,insert(a,l)))
sort(nil()) -> nil()
sort(cons(a,l)) -> insert(a,sort(l))
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:
{ ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(x, y)
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))}
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: ite^#(tt(), x, y) -> c_0()
, 2: ite^#(ff(), x, y) -> c_1()
, 3: lt^#(0(), s(y)) -> c_2()
, 4: lt^#(x, 0()) -> c_3()
, 5: lt^#(s(x), s(y)) -> c_4(lt^#(x, y))
, 6: insert^#(a, nil()) -> c_5()
, 7: insert^#(a, cons(b, l)) ->
c_6(ite^#(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l))))
, 8: sort^#(nil()) -> c_7()
, 9: sort^#(cons(a, l)) -> c_8(insert^#(a, sort(l)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{9} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{7} [ inherited ]
|
|->{1} [ NA ]
|
`->{2} [ NA ]
->{8} [ YES(?,O(1)) ]
->{5} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^3)) ]
Sub-problems:
-------------
* Path {5}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
Uargs(lt^#) = {}, Uargs(c_4) = {1}, Uargs(insert^#) = {},
Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ite(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]
tt() = [0]
[0]
[0]
ff() = [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]
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
insert(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]
nil() = [0]
[0]
[0]
cons(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]
sort(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ite^#(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_0() = [0]
[0]
[0]
c_1() = [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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
insert^#(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]
sort^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(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: innermost DP runtime-complexity with respect to
Strict Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {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_4(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {5}->{3}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
Uargs(lt^#) = {}, Uargs(c_4) = {1}, Uargs(insert^#) = {},
Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ite(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]
tt() = [0]
[0]
[0]
ff() = [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]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
insert(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]
nil() = [0]
[0]
[0]
cons(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]
sort(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ite^#(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_0() = [0]
[0]
[0]
c_1() = [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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
insert^#(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]
sort^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(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: innermost DP runtime-complexity with respect to
Strict Rules: {lt^#(0(), s(y)) -> c_2()}
Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {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_2() = [1]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [3]
[2 0 0] [2]
[0 0 0] [2]
* Path {5}->{4}: 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(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
Uargs(lt^#) = {}, Uargs(c_4) = {1}, Uargs(insert^#) = {},
Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ite(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]
tt() = [0]
[0]
[0]
ff() = [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]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
insert(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]
nil() = [0]
[0]
[0]
cons(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]
sort(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ite^#(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_0() = [0]
[0]
[0]
c_1() = [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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
insert^#(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]
sort^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(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: innermost DP runtime-complexity with respect to
Strict Rules: {lt^#(x, 0()) -> c_3()}
Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {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_3() = [1]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [7]
* Path {8}: 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(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
Uargs(lt^#) = {}, Uargs(c_4) = {}, Uargs(insert^#) = {},
Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ite(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]
tt() = [0]
[0]
[0]
ff() = [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]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
insert(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]
nil() = [0]
[0]
[0]
cons(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]
sort(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ite^#(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_0() = [0]
[0]
[0]
c_1() = [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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
insert^#(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]
sort^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(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: innermost DP runtime-complexity with respect to
Strict Rules: {sort^#(nil()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(sort^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
[2]
sort^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_7() = [0]
[1]
[1]
* Path {9}: inherited
-------------------
This path is subsumed by the proof of path {9}->{7}->{1}.
* Path {9}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(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:
{ sort^#(cons(a, l)) -> c_8(insert^#(a, sort(l)))
, insert^#(a, nil()) -> c_5()
, sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {9}->{7}: inherited
------------------------
This path is subsumed by the proof of path {9}->{7}->{1}.
* Path {9}->{7}->{1}: NA
----------------------
The usable rules for this path are:
{ sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(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 {9}->{7}->{2}: NA
----------------------
The usable rules for this path are:
{ sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(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: ite^#(tt(), x, y) -> c_0()
, 2: ite^#(ff(), x, y) -> c_1()
, 3: lt^#(0(), s(y)) -> c_2()
, 4: lt^#(x, 0()) -> c_3()
, 5: lt^#(s(x), s(y)) -> c_4(lt^#(x, y))
, 6: insert^#(a, nil()) -> c_5()
, 7: insert^#(a, cons(b, l)) ->
c_6(ite^#(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l))))
, 8: sort^#(nil()) -> c_7()
, 9: sort^#(cons(a, l)) -> c_8(insert^#(a, sort(l)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{9} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{7} [ inherited ]
|
|->{1} [ NA ]
|
`->{2} [ NA ]
->{8} [ YES(?,O(1)) ]
->{5} [ YES(?,O(n^2)) ]
|
|->{3} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* Path {5}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
Uargs(lt^#) = {}, Uargs(c_4) = {1}, Uargs(insert^#) = {},
Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ite(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
tt() = [0]
[0]
ff() = [0]
[0]
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]
insert(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sort(x1) = [0 0] x1 + [0]
[0 0] [0]
ite^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
lt^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
insert^#(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]
sort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(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: innermost DP runtime-complexity with respect to
Strict Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {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_4(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {5}->{3}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
Uargs(lt^#) = {}, Uargs(c_4) = {1}, Uargs(insert^#) = {},
Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ite(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
tt() = [0]
[0]
ff() = [0]
[0]
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]
insert(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sort(x1) = [0 0] x1 + [0]
[0 0] [0]
ite^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
insert^#(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]
sort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(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: innermost DP runtime-complexity with respect to
Strict Rules: {lt^#(0(), s(y)) -> c_2()}
Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {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_2() = [1]
[0]
c_4(x1) = [1 0] x1 + [7]
[0 0] [7]
* Path {5}->{4}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
Uargs(lt^#) = {}, Uargs(c_4) = {1}, Uargs(insert^#) = {},
Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ite(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
tt() = [0]
[0]
ff() = [0]
[0]
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]
insert(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sort(x1) = [0 0] x1 + [0]
[0 0] [0]
ite^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
insert^#(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]
sort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(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: innermost DP runtime-complexity with respect to
Strict Rules: {lt^#(x, 0()) -> c_3()}
Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {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_3() = [1]
[0]
c_4(x1) = [1 0] x1 + [6]
[0 0] [7]
* Path {8}: 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(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
Uargs(lt^#) = {}, Uargs(c_4) = {}, Uargs(insert^#) = {},
Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ite(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
tt() = [0]
[0]
ff() = [0]
[0]
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]
insert(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sort(x1) = [0 0] x1 + [0]
[0 0] [0]
ite^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
insert^#(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]
sort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(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: innermost DP runtime-complexity with respect to
Strict Rules: {sort^#(nil()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(sort^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
sort^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_7() = [0]
[1]
* Path {9}: inherited
-------------------
This path is subsumed by the proof of path {9}->{7}->{1}.
* Path {9}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(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:
{ sort^#(cons(a, l)) -> c_8(insert^#(a, sort(l)))
, insert^#(a, nil()) -> c_5()
, sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {9}->{7}: inherited
------------------------
This path is subsumed by the proof of path {9}->{7}->{1}.
* Path {9}->{7}->{1}: NA
----------------------
The usable rules for this path are:
{ sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(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 {9}->{7}->{2}: NA
----------------------
The usable rules for this path are:
{ sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(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: ite^#(tt(), x, y) -> c_0()
, 2: ite^#(ff(), x, y) -> c_1()
, 3: lt^#(0(), s(y)) -> c_2()
, 4: lt^#(x, 0()) -> c_3()
, 5: lt^#(s(x), s(y)) -> c_4(lt^#(x, y))
, 6: insert^#(a, nil()) -> c_5()
, 7: insert^#(a, cons(b, l)) ->
c_6(ite^#(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l))))
, 8: sort^#(nil()) -> c_7()
, 9: sort^#(cons(a, l)) -> c_8(insert^#(a, sort(l)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{9} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{7} [ inherited ]
|
|->{1} [ NA ]
|
`->{2} [ NA ]
->{8} [ YES(?,O(1)) ]
->{5} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {5}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
Uargs(lt^#) = {}, Uargs(c_4) = {1}, Uargs(insert^#) = {},
Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ite(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
ff() = [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
insert(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sort(x1) = [0] x1 + [0]
ite^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0() = [0]
c_1() = [0]
lt^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
insert^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
sort^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(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: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {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_4(x1) = [1] x1 + [7]
* Path {5}->{3}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
Uargs(lt^#) = {}, Uargs(c_4) = {1}, Uargs(insert^#) = {},
Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ite(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
ff() = [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
insert(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sort(x1) = [0] x1 + [0]
ite^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0() = [0]
c_1() = [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
insert^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
sort^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(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: {lt^#(0(), s(y)) -> c_2()}
Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {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_2() = [1]
c_4(x1) = [1] x1 + [7]
* Path {5}->{4}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
Uargs(lt^#) = {}, Uargs(c_4) = {1}, Uargs(insert^#) = {},
Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ite(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
ff() = [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
insert(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sort(x1) = [0] x1 + [0]
ite^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0() = [0]
c_1() = [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
insert^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
sort^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(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: {lt^#(x, 0()) -> c_3()}
Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {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_3() = [1]
c_4(x1) = [1] x1 + [7]
* Path {8}: 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(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
Uargs(lt^#) = {}, Uargs(c_4) = {}, Uargs(insert^#) = {},
Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ite(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
ff() = [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
insert(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sort(x1) = [0] x1 + [0]
ite^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0() = [0]
c_1() = [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
insert^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
sort^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(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: {sort^#(nil()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(sort^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [7]
sort^#(x1) = [1] x1 + [7]
c_7() = [1]
* Path {9}: inherited
-------------------
This path is subsumed by the proof of path {9}->{7}->{1}.
* Path {9}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(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:
{ sort^#(cons(a, l)) -> c_8(insert^#(a, sort(l)))
, insert^#(a, nil()) -> c_5()
, sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {9}->{7}: inherited
------------------------
This path is subsumed by the proof of path {9}->{7}->{1}.
* Path {9}->{7}->{1}: NA
----------------------
The usable rules for this path are:
{ sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(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 {9}->{7}->{2}: NA
----------------------
The usable rules for this path are:
{ sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(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:
{ ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(x, y)
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))}
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: ite^#(tt(), x, y) -> c_0(x)
, 2: ite^#(ff(), x, y) -> c_1(y)
, 3: lt^#(0(), s(y)) -> c_2()
, 4: lt^#(x, 0()) -> c_3()
, 5: lt^#(s(x), s(y)) -> c_4(lt^#(x, y))
, 6: insert^#(a, nil()) -> c_5(a)
, 7: insert^#(a, cons(b, l)) ->
c_6(ite^#(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l))))
, 8: sort^#(nil()) -> c_7()
, 9: sort^#(cons(a, l)) -> c_8(insert^#(a, sort(l)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{9} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{7} [ inherited ]
|
|->{1} [ NA ]
|
`->{2} [ NA ]
->{8} [ YES(?,O(1)) ]
->{5} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^3)) ]
Sub-problems:
-------------
* Path {5}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
Uargs(c_4) = {1}, Uargs(insert^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ite(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]
tt() = [0]
[0]
[0]
ff() = [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]
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
insert(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]
nil() = [0]
[0]
[0]
cons(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]
sort(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ite^#(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_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
insert^#(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]
sort^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(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_4(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {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_4(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {5}->{3}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
Uargs(c_4) = {1}, Uargs(insert^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ite(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]
tt() = [0]
[0]
[0]
ff() = [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]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
insert(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]
nil() = [0]
[0]
[0]
cons(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]
sort(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ite^#(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_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
insert^#(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]
sort^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(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(y)) -> c_2()}
Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {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_2() = [1]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [3]
[2 0 0] [2]
[0 0 0] [2]
* Path {5}->{4}: 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(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
Uargs(c_4) = {1}, Uargs(insert^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ite(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]
tt() = [0]
[0]
[0]
ff() = [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]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
insert(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]
nil() = [0]
[0]
[0]
cons(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]
sort(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ite^#(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_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
insert^#(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]
sort^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(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_3()}
Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {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_3() = [1]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [7]
* Path {8}: 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(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
Uargs(c_4) = {}, Uargs(insert^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ite(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]
tt() = [0]
[0]
[0]
ff() = [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]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
insert(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]
nil() = [0]
[0]
[0]
cons(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]
sort(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ite^#(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_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
insert^#(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]
sort^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(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: {sort^#(nil()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(sort^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
[2]
sort^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_7() = [0]
[1]
[1]
* Path {9}: inherited
-------------------
This path is subsumed by the proof of path {9}->{7}->{1}.
* Path {9}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(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:
{ sort^#(cons(a, l)) -> c_8(insert^#(a, sort(l)))
, insert^#(a, nil()) -> c_5(a)
, sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {9}->{7}: inherited
------------------------
This path is subsumed by the proof of path {9}->{7}->{1}.
* Path {9}->{7}->{1}: NA
----------------------
The usable rules for this path are:
{ sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(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 {9}->{7}->{2}: NA
----------------------
The usable rules for this path are:
{ sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(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: ite^#(tt(), x, y) -> c_0(x)
, 2: ite^#(ff(), x, y) -> c_1(y)
, 3: lt^#(0(), s(y)) -> c_2()
, 4: lt^#(x, 0()) -> c_3()
, 5: lt^#(s(x), s(y)) -> c_4(lt^#(x, y))
, 6: insert^#(a, nil()) -> c_5(a)
, 7: insert^#(a, cons(b, l)) ->
c_6(ite^#(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l))))
, 8: sort^#(nil()) -> c_7()
, 9: sort^#(cons(a, l)) -> c_8(insert^#(a, sort(l)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{9} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{7} [ inherited ]
|
|->{1} [ NA ]
|
`->{2} [ NA ]
->{8} [ YES(?,O(1)) ]
->{5} [ YES(?,O(n^2)) ]
|
|->{3} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* Path {5}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
Uargs(c_4) = {1}, Uargs(insert^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ite(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
tt() = [0]
[0]
ff() = [0]
[0]
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]
insert(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sort(x1) = [0 0] x1 + [0]
[0 0] [0]
ite^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
lt^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
insert^#(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]
sort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(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_4(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {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_4(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {5}->{3}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
Uargs(c_4) = {1}, Uargs(insert^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ite(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
tt() = [0]
[0]
ff() = [0]
[0]
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]
insert(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sort(x1) = [0 0] x1 + [0]
[0 0] [0]
ite^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
insert^#(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]
sort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(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(y)) -> c_2()}
Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {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_2() = [1]
[0]
c_4(x1) = [1 0] x1 + [7]
[0 0] [7]
* Path {5}->{4}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
Uargs(c_4) = {1}, Uargs(insert^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ite(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
tt() = [0]
[0]
ff() = [0]
[0]
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]
insert(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sort(x1) = [0 0] x1 + [0]
[0 0] [0]
ite^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
insert^#(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]
sort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(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_3()}
Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {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_3() = [1]
[0]
c_4(x1) = [1 0] x1 + [6]
[0 0] [7]
* Path {8}: 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(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
Uargs(c_4) = {}, Uargs(insert^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ite(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
tt() = [0]
[0]
ff() = [0]
[0]
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]
insert(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sort(x1) = [0 0] x1 + [0]
[0 0] [0]
ite^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
insert^#(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]
sort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(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: {sort^#(nil()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(sort^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
sort^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_7() = [0]
[1]
* Path {9}: inherited
-------------------
This path is subsumed by the proof of path {9}->{7}->{1}.
* Path {9}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(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:
{ sort^#(cons(a, l)) -> c_8(insert^#(a, sort(l)))
, insert^#(a, nil()) -> c_5(a)
, sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {9}->{7}: inherited
------------------------
This path is subsumed by the proof of path {9}->{7}->{1}.
* Path {9}->{7}->{1}: NA
----------------------
The usable rules for this path are:
{ sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(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 {9}->{7}->{2}: NA
----------------------
The usable rules for this path are:
{ sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(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: ite^#(tt(), x, y) -> c_0(x)
, 2: ite^#(ff(), x, y) -> c_1(y)
, 3: lt^#(0(), s(y)) -> c_2()
, 4: lt^#(x, 0()) -> c_3()
, 5: lt^#(s(x), s(y)) -> c_4(lt^#(x, y))
, 6: insert^#(a, nil()) -> c_5(a)
, 7: insert^#(a, cons(b, l)) ->
c_6(ite^#(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l))))
, 8: sort^#(nil()) -> c_7()
, 9: sort^#(cons(a, l)) -> c_8(insert^#(a, sort(l)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{9} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{7} [ inherited ]
|
|->{1} [ NA ]
|
`->{2} [ NA ]
->{8} [ YES(?,O(1)) ]
->{5} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {5}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
Uargs(c_4) = {1}, Uargs(insert^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ite(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
ff() = [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
insert(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sort(x1) = [0] x1 + [0]
ite^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
lt^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
insert^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
sort^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(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_4(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {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_4(x1) = [1] x1 + [7]
* Path {5}->{3}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
Uargs(c_4) = {1}, Uargs(insert^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ite(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
ff() = [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
insert(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sort(x1) = [0] x1 + [0]
ite^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
insert^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
sort^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(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(y)) -> c_2()}
Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {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_2() = [1]
c_4(x1) = [1] x1 + [7]
* Path {5}->{4}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
Uargs(c_4) = {1}, Uargs(insert^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ite(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
ff() = [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
insert(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sort(x1) = [0] x1 + [0]
ite^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
insert^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
sort^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(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_3()}
Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {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_3() = [1]
c_4(x1) = [1] x1 + [7]
* Path {8}: 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(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
Uargs(c_4) = {}, Uargs(insert^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ite(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
ff() = [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
insert(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sort(x1) = [0] x1 + [0]
ite^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
insert^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
sort^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(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: {sort^#(nil()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(sort^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [7]
sort^#(x1) = [1] x1 + [7]
c_7() = [1]
* Path {9}: inherited
-------------------
This path is subsumed by the proof of path {9}->{7}->{1}.
* Path {9}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(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:
{ sort^#(cons(a, l)) -> c_8(insert^#(a, sort(l)))
, insert^#(a, nil()) -> c_5(a)
, sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {9}->{7}: inherited
------------------------
This path is subsumed by the proof of path {9}->{7}->{1}.
* Path {9}->{7}->{1}: NA
----------------------
The usable rules for this path are:
{ sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(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 {9}->{7}->{2}: NA
----------------------
The usable rules for this path are:
{ sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(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 pair2rc
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(x, y)
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair2 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair3irc
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(x, y)
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))}
StartTerms: basic terms
Strategy: innermost
Certificate: TIMEOUT
Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair3rc
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(x, y)
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool rc
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(x, y)
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'rc (timeout of 60.0 seconds)':
----------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool tup3irc
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ ite(tt(), x, y) -> x
, ite(ff(), x, y) -> y
, lt(0(), s(y)) -> tt()
, lt(x, 0()) -> ff()
, lt(s(x), s(y)) -> lt(x, y)
, insert(a, nil()) -> cons(a, nil())
, insert(a, cons(b, l)) ->
ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
, sort(nil()) -> nil()
, sort(cons(a, l)) -> insert(a, sort(l))}
StartTerms: basic terms
Strategy: innermost
Certificate: TIMEOUT
Application of 'tup3 (timeout of 60.0 seconds)':
------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..