Tool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ terms(N) -> cons(recip(sqr(N)), terms(s(N)))
, sqr(0()) -> 0()
, sqr(s(X)) -> s(add(sqr(X), dbl(X)))
, dbl(0()) -> 0()
, dbl(s(X)) -> s(s(dbl(X)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))
, first(0(), X) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))}
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: terms^#(N) -> c_0(sqr^#(N), terms^#(s(N)))
, 2: sqr^#(0()) -> c_1()
, 3: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
, 4: dbl^#(0()) -> c_3()
, 5: dbl^#(s(X)) -> c_4(dbl^#(X))
, 6: add^#(0(), X) -> c_5()
, 7: add^#(s(X), Y) -> c_6(add^#(X, Y))
, 8: first^#(0(), X) -> c_7()
, 9: first^#(s(X), cons(Y, Z)) -> c_8(first^#(X, Z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{9} [ YES(?,O(n^3)) ]
|
`->{8} [ YES(?,O(n^1)) ]
->{5} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^2)) ]
->{1} [ inherited ]
|
|->{2} [ YES(?,O(1)) ]
|
`->{3} [ inherited ]
|
|->{6} [ NA ]
|
`->{7} [ inherited ]
|
`->{6} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{3}->{7}->{6}.
* Path {1}->{2}: 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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1, 2},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
Uargs(first^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
recip(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sqr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
add(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]
dbl(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
first(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]
terms^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[0 1 0] [0 1 0] [0]
[0 0 1] [0 0 1] [0]
sqr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[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]
add^#(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]
dbl^#(x1) = [0 0 0] x1 + [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]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
first^#(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_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: {sqr^#(0()) -> c_1()}
Weak Rules: {terms^#(N) -> c_0(sqr^#(N), terms^#(s(N)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1, 2},
Uargs(sqr^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [2]
[0]
[0]
terms^#(x1) = [6 0 0] x1 + [0]
[2 2 2] [0]
[2 2 6] [0]
c_0(x1, x2) = [2 0 0] x1 + [2 2 2] x2 + [0]
[0 0 0] [2 2 0] [0]
[0 0 0] [2 2 2] [0]
sqr^#(x1) = [2 0 0] x1 + [0]
[0 0 0] [0]
[2 0 0] [0]
c_1() = [1]
[0]
[0]
* Path {1}->{3}: inherited
------------------------
This path is subsumed by the proof of path {1}->{3}->{7}->{6}.
* Path {1}->{3}->{6}: NA
----------------------
The usable rules for this path are:
{ sqr(0()) -> 0()
, sqr(s(X)) -> s(add(sqr(X), dbl(X)))
, dbl(0()) -> 0()
, dbl(s(X)) -> s(s(dbl(X)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(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 {1}->{3}->{7}: inherited
-----------------------------
This path is subsumed by the proof of path {1}->{3}->{7}->{6}.
* Path {1}->{3}->{7}->{6}: NA
---------------------------
The usable rules for this path are:
{ sqr(0()) -> 0()
, sqr(s(X)) -> s(add(sqr(X), dbl(X)))
, dbl(0()) -> 0()
, dbl(s(X)) -> s(s(dbl(X)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(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 {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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {1}, Uargs(c_6) = {},
Uargs(first^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
recip(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sqr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
add(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]
dbl(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
first(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]
terms^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(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]
sqr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[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]
add^#(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]
dbl^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
first^#(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_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: {dbl^#(s(X)) -> c_4(dbl^#(X))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 1 2] [2]
[0 0 0] [0]
dbl^#(x1) = [0 1 0] x1 + [2]
[6 0 0] [0]
[2 3 0] [2]
c_4(x1) = [1 0 0] x1 + [1]
[2 0 2] [0]
[0 0 0] [0]
* Path {5}->{4}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {1}, Uargs(c_6) = {},
Uargs(first^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
recip(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sqr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
add(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]
dbl(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
first(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]
terms^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(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]
sqr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[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]
add^#(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]
dbl^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
first^#(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_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: {dbl^#(0()) -> c_3()}
Weak Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [0]
[0 1 1] [1]
[0 0 0] [0]
0() = [2]
[2]
[2]
dbl^#(x1) = [2 2 2] x1 + [0]
[0 6 0] [0]
[0 0 2] [0]
c_3() = [1]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [0]
* Path {9}: 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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
Uargs(first^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [1 3 0] x1 + [0 0 0] x2 + [0]
[0 1 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
recip(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sqr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
add(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]
dbl(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
first(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]
terms^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(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]
sqr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[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]
add^#(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]
dbl^#(x1) = [0 0 0] x1 + [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]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
first^#(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_7() = [0]
[0]
[0]
c_8(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {first^#(s(X), cons(Y, Z)) -> c_8(first^#(X, Z))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(first^#) = {},
Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [1 4 4] x2 + [2]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 1] [2]
s(x1) = [1 4 4] x1 + [2]
[0 1 2] [2]
[0 0 0] [2]
first^#(x1, x2) = [0 2 0] x1 + [0 0 2] x2 + [0]
[2 0 2] [2 0 0] [0]
[0 2 2] [1 0 0] [0]
c_8(x1) = [1 0 0] x1 + [5]
[2 0 2] [3]
[0 0 0] [7]
* Path {9}->{8}: 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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
Uargs(first^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
recip(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sqr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
add(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]
dbl(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
first(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]
terms^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(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]
sqr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[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]
add^#(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]
dbl^#(x1) = [0 0 0] x1 + [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]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
first^#(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_7() = [0]
[0]
[0]
c_8(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {first^#(0(), X) -> c_7()}
Weak Rules: {first^#(s(X), cons(Y, Z)) -> c_8(first^#(X, Z))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(first^#) = {},
Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [1 4 4] x2 + [0]
[0 0 0] [0 0 4] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 2 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [2]
[0]
[0]
first^#(x1, x2) = [2 0 0] x1 + [2 0 0] x2 + [0]
[2 0 0] [2 0 0] [4]
[0 0 0] [0 2 0] [2]
c_7() = [1]
[0]
[0]
c_8(x1) = [1 0 0] x1 + [0]
[0 0 2] [0]
[0 0 0] [2]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: terms^#(N) -> c_0(sqr^#(N), terms^#(s(N)))
, 2: sqr^#(0()) -> c_1()
, 3: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
, 4: dbl^#(0()) -> c_3()
, 5: dbl^#(s(X)) -> c_4(dbl^#(X))
, 6: add^#(0(), X) -> c_5()
, 7: add^#(s(X), Y) -> c_6(add^#(X, Y))
, 8: first^#(0(), X) -> c_7()
, 9: first^#(s(X), cons(Y, Z)) -> c_8(first^#(X, Z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{9} [ YES(?,O(n^2)) ]
|
`->{8} [ YES(?,O(n^2)) ]
->{5} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{1} [ inherited ]
|
|->{2} [ YES(?,O(1)) ]
|
`->{3} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{7} [ inherited ]
|
`->{6} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{3}->{7}->{6}.
* Path {1}->{2}: 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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1, 2},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
Uargs(first^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
recip(x1) = [0 0] x1 + [0]
[0 0] [0]
sqr(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
dbl(x1) = [0 0] x1 + [0]
[0 0] [0]
first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
terms^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
sqr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
dbl^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 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: {sqr^#(0()) -> c_1()}
Weak Rules: {terms^#(N) -> c_0(sqr^#(N), terms^#(s(N)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1, 2},
Uargs(sqr^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [2]
[2]
terms^#(x1) = [2 4] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [1 0] x1 + [2 0] x2 + [0]
[0 0] [0 0] [0]
sqr^#(x1) = [2 2] x1 + [0]
[0 2] [0]
c_1() = [1]
[0]
* Path {1}->{3}: inherited
------------------------
This path is subsumed by the proof of path {1}->{3}->{7}->{6}.
* Path {1}->{3}->{6}: MAYBE
-------------------------
The usable rules for this path are:
{ sqr(0()) -> 0()
, sqr(s(X)) -> s(add(sqr(X), dbl(X)))
, dbl(0()) -> 0()
, dbl(s(X)) -> s(s(dbl(X)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(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:
{ sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
, terms^#(N) -> c_0(sqr^#(N), terms^#(s(N)))
, add^#(0(), X) -> c_5()
, sqr(0()) -> 0()
, sqr(s(X)) -> s(add(sqr(X), dbl(X)))
, dbl(0()) -> 0()
, dbl(s(X)) -> s(s(dbl(X)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
Proof Output:
The input cannot be shown compatible
* Path {1}->{3}->{7}: inherited
-----------------------------
This path is subsumed by the proof of path {1}->{3}->{7}->{6}.
* Path {1}->{3}->{7}->{6}: NA
---------------------------
The usable rules for this path are:
{ sqr(0()) -> 0()
, sqr(s(X)) -> s(add(sqr(X), dbl(X)))
, dbl(0()) -> 0()
, dbl(s(X)) -> s(s(dbl(X)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(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 {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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {1}, Uargs(c_6) = {},
Uargs(first^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
recip(x1) = [0 0] x1 + [0]
[0 0] [0]
sqr(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
dbl(x1) = [0 0] x1 + [0]
[0 0] [0]
first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
terms^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sqr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
dbl^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 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^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
dbl^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_4(x1) = [1 0] x1 + [0]
[0 0] [0]
* 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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {1}, Uargs(c_6) = {},
Uargs(first^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
recip(x1) = [0 0] x1 + [0]
[0 0] [0]
sqr(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
dbl(x1) = [0 0] x1 + [0]
[0 0] [0]
first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
terms^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sqr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
dbl^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 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^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {dbl^#(0()) -> c_3()}
Weak Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 0] [3]
0() = [2]
[2]
dbl^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_3() = [1]
[0]
c_4(x1) = [1 0] x1 + [5]
[2 0] [3]
* Path {9}: 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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
Uargs(first^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
recip(x1) = [0 0] x1 + [0]
[0 0] [0]
sqr(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
dbl(x1) = [0 0] x1 + [0]
[0 0] [0]
first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
terms^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sqr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
dbl^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [3 3] x1 + [1 3] x2 + [0]
[3 3] [3 3] [0]
c_7() = [0]
[0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {first^#(s(X), cons(Y, Z)) -> c_8(first^#(X, Z))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(first^#) = {},
Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [4]
s(x1) = [1 2] x1 + [2]
[0 0] [0]
first^#(x1, x2) = [0 0] x1 + [0 2] x2 + [0]
[4 0] [4 0] [0]
c_8(x1) = [1 0] x1 + [5]
[0 0] [7]
* Path {9}->{8}: 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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
Uargs(first^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
recip(x1) = [0 0] x1 + [0]
[0 0] [0]
sqr(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
dbl(x1) = [0 0] x1 + [0]
[0 0] [0]
first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
terms^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sqr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
dbl^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {first^#(0(), X) -> c_7()}
Weak Rules: {first^#(s(X), cons(Y, Z)) -> c_8(first^#(X, Z))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(first^#) = {},
Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [1 6] x2 + [0]
[0 0] [0 1] [2]
s(x1) = [1 6] x1 + [0]
[0 1] [2]
0() = [0]
[2]
first^#(x1, x2) = [1 2] x1 + [2 2] x2 + [0]
[0 2] [0 3] [0]
c_7() = [1]
[0]
c_8(x1) = [1 2] x1 + [6]
[0 0] [2]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: terms^#(N) -> c_0(sqr^#(N), terms^#(s(N)))
, 2: sqr^#(0()) -> c_1()
, 3: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
, 4: dbl^#(0()) -> c_3()
, 5: dbl^#(s(X)) -> c_4(dbl^#(X))
, 6: add^#(0(), X) -> c_5()
, 7: add^#(s(X), Y) -> c_6(add^#(X, Y))
, 8: first^#(0(), X) -> c_7()
, 9: first^#(s(X), cons(Y, Z)) -> c_8(first^#(X, Z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{9} [ YES(?,O(n^1)) ]
|
`->{8} [ YES(?,O(n^1)) ]
->{5} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{1} [ inherited ]
|
|->{2} [ YES(?,O(1)) ]
|
`->{3} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{7} [ inherited ]
|
`->{6} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{3}->{7}->{6}.
* Path {1}->{2}: 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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1, 2},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
Uargs(first^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
dbl(x1) = [0] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [1] x1 + [1] x2 + [0]
sqr^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
dbl^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [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: {sqr^#(0()) -> c_1()}
Weak Rules: {terms^#(N) -> c_0(sqr^#(N), terms^#(s(N)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1, 2},
Uargs(sqr^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0] x1 + [0]
0() = [1]
terms^#(x1) = [6] x1 + [0]
c_0(x1, x2) = [4] x1 + [2] x2 + [0]
sqr^#(x1) = [1] x1 + [0]
c_1() = [0]
* Path {1}->{3}: inherited
------------------------
This path is subsumed by the proof of path {1}->{3}->{7}->{6}.
* Path {1}->{3}->{6}: MAYBE
-------------------------
The usable rules for this path are:
{ sqr(0()) -> 0()
, sqr(s(X)) -> s(add(sqr(X), dbl(X)))
, dbl(0()) -> 0()
, dbl(s(X)) -> s(s(dbl(X)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(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:
{ sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
, terms^#(N) -> c_0(sqr^#(N), terms^#(s(N)))
, add^#(0(), X) -> c_5()
, sqr(0()) -> 0()
, sqr(s(X)) -> s(add(sqr(X), dbl(X)))
, dbl(0()) -> 0()
, dbl(s(X)) -> s(s(dbl(X)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
Proof Output:
The input cannot be shown compatible
* Path {1}->{3}->{7}: inherited
-----------------------------
This path is subsumed by the proof of path {1}->{3}->{7}->{6}.
* Path {1}->{3}->{7}->{6}: NA
---------------------------
The usable rules for this path are:
{ sqr(0()) -> 0()
, sqr(s(X)) -> s(add(sqr(X), dbl(X)))
, dbl(0()) -> 0()
, dbl(s(X)) -> s(s(dbl(X)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(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 {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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {1}, Uargs(c_6) = {},
Uargs(first^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
dbl(x1) = [0] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
sqr^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
dbl^#(x1) = [3] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [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: {dbl^#(s(X)) -> c_4(dbl^#(X))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
dbl^#(x1) = [2] x1 + [0]
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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {1}, Uargs(c_6) = {},
Uargs(first^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
dbl(x1) = [0] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
sqr^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
dbl^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [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: {dbl^#(0()) -> c_3()}
Weak Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
0() = [2]
dbl^#(x1) = [2] x1 + [0]
c_3() = [1]
c_4(x1) = [1] x1 + [0]
* Path {9}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
Uargs(first^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
dbl(x1) = [0] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
sqr^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
dbl^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
first^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {first^#(s(X), cons(Y, Z)) -> c_8(first^#(X, Z))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(first^#) = {},
Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [2]
first^#(x1, x2) = [3] x1 + [0] x2 + [2]
c_8(x1) = [1] x1 + [5]
* Path {9}->{8}: 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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
Uargs(first^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
dbl(x1) = [0] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
sqr^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
dbl^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {first^#(0(), X) -> c_7()}
Weak Rules: {first^#(s(X), cons(Y, Z)) -> c_8(first^#(X, Z))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(first^#) = {},
Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [1] x2 + [2]
s(x1) = [1] x1 + [0]
0() = [2]
first^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_7() = [1]
c_8(x1) = [1] x1 + [3]
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:
{ terms(N) -> cons(recip(sqr(N)), terms(s(N)))
, sqr(0()) -> 0()
, sqr(s(X)) -> s(add(sqr(X), dbl(X)))
, dbl(0()) -> 0()
, dbl(s(X)) -> s(s(dbl(X)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))
, first(0(), X) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))}
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: terms^#(N) -> c_0(sqr^#(N), terms^#(s(N)))
, 2: sqr^#(0()) -> c_1()
, 3: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
, 4: dbl^#(0()) -> c_3()
, 5: dbl^#(s(X)) -> c_4(dbl^#(X))
, 6: add^#(0(), X) -> c_5(X)
, 7: add^#(s(X), Y) -> c_6(add^#(X, Y))
, 8: first^#(0(), X) -> c_7()
, 9: first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{9} [ YES(?,O(n^3)) ]
|
`->{8} [ YES(?,O(n^2)) ]
->{5} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^2)) ]
->{1} [ inherited ]
|
|->{2} [ YES(?,O(1)) ]
|
`->{3} [ inherited ]
|
|->{6} [ NA ]
|
`->{7} [ inherited ]
|
`->{6} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{3}->{7}->{6}.
* Path {1}->{2}: 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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1, 2},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
recip(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sqr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
add(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]
dbl(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
first(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]
terms^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[0 1 0] [0 1 0] [0]
[0 0 1] [0 0 1] [0]
sqr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[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]
add^#(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]
dbl^#(x1) = [0 0 0] x1 + [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]
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]
first^#(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_7() = [0]
[0]
[0]
c_8(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]
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: {sqr^#(0()) -> c_1()}
Weak Rules: {terms^#(N) -> c_0(sqr^#(N), terms^#(s(N)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1, 2},
Uargs(sqr^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [2]
[0]
[0]
terms^#(x1) = [6 0 0] x1 + [0]
[2 2 2] [0]
[2 2 6] [0]
c_0(x1, x2) = [2 0 0] x1 + [2 2 2] x2 + [0]
[0 0 0] [2 2 0] [0]
[0 0 0] [2 2 2] [0]
sqr^#(x1) = [2 0 0] x1 + [0]
[0 0 0] [0]
[2 0 0] [0]
c_1() = [1]
[0]
[0]
* Path {1}->{3}: inherited
------------------------
This path is subsumed by the proof of path {1}->{3}->{7}->{6}.
* Path {1}->{3}->{6}: NA
----------------------
The usable rules for this path are:
{ sqr(0()) -> 0()
, sqr(s(X)) -> s(add(sqr(X), dbl(X)))
, dbl(0()) -> 0()
, dbl(s(X)) -> s(s(dbl(X)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(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 {1}->{3}->{7}: inherited
-----------------------------
This path is subsumed by the proof of path {1}->{3}->{7}->{6}.
* Path {1}->{3}->{7}->{6}: NA
---------------------------
The usable rules for this path are:
{ sqr(0()) -> 0()
, sqr(s(X)) -> s(add(sqr(X), dbl(X)))
, dbl(0()) -> 0()
, dbl(s(X)) -> s(s(dbl(X)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(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 {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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {1}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
recip(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sqr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
add(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]
dbl(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
first(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]
terms^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(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]
sqr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[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]
add^#(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]
dbl^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
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]
first^#(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_7() = [0]
[0]
[0]
c_8(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]
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: {dbl^#(s(X)) -> c_4(dbl^#(X))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 1 2] [2]
[0 0 0] [0]
dbl^#(x1) = [0 1 0] x1 + [2]
[6 0 0] [0]
[2 3 0] [2]
c_4(x1) = [1 0 0] x1 + [1]
[2 0 2] [0]
[0 0 0] [0]
* Path {5}->{4}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {1}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
recip(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sqr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
add(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]
dbl(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
first(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]
terms^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(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]
sqr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[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]
add^#(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]
dbl^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
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]
first^#(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_7() = [0]
[0]
[0]
c_8(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]
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: {dbl^#(0()) -> c_3()}
Weak Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [0]
[0 1 1] [1]
[0 0 0] [0]
0() = [2]
[2]
[2]
dbl^#(x1) = [2 2 2] x1 + [0]
[0 6 0] [0]
[0 0 2] [0]
c_3() = [1]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [0]
* Path {9}: 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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [1 3 3] x1 + [1 3 0] x2 + [0]
[0 1 3] [0 1 0] [0]
[0 0 1] [0 0 0] [0]
recip(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sqr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
add(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]
dbl(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
first(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]
terms^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(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]
sqr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[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]
add^#(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]
dbl^#(x1) = [0 0 0] x1 + [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]
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]
first^#(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_7() = [0]
[0]
[0]
c_8(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 0 0] [0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(first^#) = {},
Uargs(c_8) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 2 0] x1 + [1 0 2] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [6]
s(x1) = [1 1 1] x1 + [0]
[0 1 2] [3]
[0 0 1] [0]
first^#(x1, x2) = [1 1 1] x1 + [2 0 1] x2 + [0]
[4 2 0] [0 0 0] [2]
[0 4 0] [4 0 0] [0]
c_8(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [7]
[0 0 0] [0 0 0] [6]
[0 0 0] [0 0 0] [7]
* Path {9}->{8}: 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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
recip(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sqr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
add(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]
dbl(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
first(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]
terms^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(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]
sqr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[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]
add^#(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]
dbl^#(x1) = [0 0 0] x1 + [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]
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]
first^#(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_7() = [0]
[0]
[0]
c_8(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 0 0] [0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {first^#(0(), X) -> c_7()}
Weak Rules: {first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(first^#) = {},
Uargs(c_8) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 0 2] x1 + [1 3 0] x2 + [1]
[0 0 0] [0 1 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 2 0] x1 + [2]
[0 0 0] [0]
[0 0 0] [0]
0() = [2]
[2]
[2]
first^#(x1, x2) = [3 0 0] x1 + [3 0 0] x2 + [0]
[2 2 2] [0 5 0] [0]
[0 0 0] [1 1 0] [0]
c_7() = [1]
[0]
[0]
c_8(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [7]
[0 0 0] [0 0 0] [2]
[1 0 0] [0 0 0] [1]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: terms^#(N) -> c_0(sqr^#(N), terms^#(s(N)))
, 2: sqr^#(0()) -> c_1()
, 3: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
, 4: dbl^#(0()) -> c_3()
, 5: dbl^#(s(X)) -> c_4(dbl^#(X))
, 6: add^#(0(), X) -> c_5(X)
, 7: add^#(s(X), Y) -> c_6(add^#(X, Y))
, 8: first^#(0(), X) -> c_7()
, 9: first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{9} [ YES(?,O(n^2)) ]
|
`->{8} [ YES(?,O(n^1)) ]
->{5} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{1} [ inherited ]
|
|->{2} [ YES(?,O(1)) ]
|
`->{3} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{7} [ inherited ]
|
`->{6} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{3}->{7}->{6}.
* Path {1}->{2}: 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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1, 2},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
recip(x1) = [0 0] x1 + [0]
[0 0] [0]
sqr(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
dbl(x1) = [0 0] x1 + [0]
[0 0] [0]
first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
terms^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
sqr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
dbl^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 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: {sqr^#(0()) -> c_1()}
Weak Rules: {terms^#(N) -> c_0(sqr^#(N), terms^#(s(N)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1, 2},
Uargs(sqr^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [2]
[2]
terms^#(x1) = [2 4] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [1 0] x1 + [2 0] x2 + [0]
[0 0] [0 0] [0]
sqr^#(x1) = [2 2] x1 + [0]
[0 2] [0]
c_1() = [1]
[0]
* Path {1}->{3}: inherited
------------------------
This path is subsumed by the proof of path {1}->{3}->{7}->{6}.
* Path {1}->{3}->{6}: MAYBE
-------------------------
The usable rules for this path are:
{ sqr(0()) -> 0()
, sqr(s(X)) -> s(add(sqr(X), dbl(X)))
, dbl(0()) -> 0()
, dbl(s(X)) -> s(s(dbl(X)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(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:
{ sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
, terms^#(N) -> c_0(sqr^#(N), terms^#(s(N)))
, add^#(0(), X) -> c_5(X)
, sqr(0()) -> 0()
, sqr(s(X)) -> s(add(sqr(X), dbl(X)))
, dbl(0()) -> 0()
, dbl(s(X)) -> s(s(dbl(X)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
Proof Output:
The input cannot be shown compatible
* Path {1}->{3}->{7}: inherited
-----------------------------
This path is subsumed by the proof of path {1}->{3}->{7}->{6}.
* Path {1}->{3}->{7}->{6}: NA
---------------------------
The usable rules for this path are:
{ sqr(0()) -> 0()
, sqr(s(X)) -> s(add(sqr(X), dbl(X)))
, dbl(0()) -> 0()
, dbl(s(X)) -> s(s(dbl(X)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(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 {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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {1}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
recip(x1) = [0 0] x1 + [0]
[0 0] [0]
sqr(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
dbl(x1) = [0 0] x1 + [0]
[0 0] [0]
first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
terms^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sqr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
dbl^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
dbl^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_4(x1) = [1 0] x1 + [0]
[0 0] [0]
* 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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {1}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
recip(x1) = [0 0] x1 + [0]
[0 0] [0]
sqr(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
dbl(x1) = [0 0] x1 + [0]
[0 0] [0]
first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
terms^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sqr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
dbl^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {dbl^#(0()) -> c_3()}
Weak Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 0] [3]
0() = [2]
[2]
dbl^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_3() = [1]
[0]
c_4(x1) = [1 0] x1 + [5]
[2 0] [3]
* Path {9}: 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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 3] x1 + [1 3] x2 + [0]
[0 1] [0 1] [0]
recip(x1) = [0 0] x1 + [0]
[0 0] [0]
sqr(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
dbl(x1) = [0 0] x1 + [0]
[0 0] [0]
first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
terms^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sqr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
dbl^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [3 3] x1 + [1 3] x2 + [0]
[3 3] [3 3] [0]
c_7() = [0]
[0]
c_8(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(first^#) = {},
Uargs(c_8) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
[0 0] [0 1] [2]
s(x1) = [1 2] x1 + [0]
[0 1] [0]
first^#(x1, x2) = [0 2] x1 + [2 2] x2 + [2]
[2 5] [6 1] [0]
c_8(x1, x2) = [0 0] x1 + [1 0] x2 + [5]
[0 0] [2 0] [6]
* Path {9}->{8}: 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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
recip(x1) = [0 0] x1 + [0]
[0 0] [0]
sqr(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
dbl(x1) = [0 0] x1 + [0]
[0 0] [0]
first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
terms^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sqr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
dbl^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {first^#(0(), X) -> c_7()}
Weak Rules: {first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(first^#) = {},
Uargs(c_8) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 0] x1 + [1 4] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
0() = [2]
[2]
first^#(x1, x2) = [2 0] x1 + [2 0] x2 + [4]
[2 2] [1 0] [4]
c_7() = [1]
[0]
c_8(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[1 0] [0 0] [2]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: terms^#(N) -> c_0(sqr^#(N), terms^#(s(N)))
, 2: sqr^#(0()) -> c_1()
, 3: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
, 4: dbl^#(0()) -> c_3()
, 5: dbl^#(s(X)) -> c_4(dbl^#(X))
, 6: add^#(0(), X) -> c_5(X)
, 7: add^#(s(X), Y) -> c_6(add^#(X, Y))
, 8: first^#(0(), X) -> c_7()
, 9: first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{9} [ YES(?,O(n^1)) ]
|
`->{8} [ YES(?,O(n^1)) ]
->{5} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{1} [ inherited ]
|
|->{2} [ YES(?,O(1)) ]
|
`->{3} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{7} [ inherited ]
|
`->{6} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{3}->{7}->{6}.
* Path {1}->{2}: 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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1, 2},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
dbl(x1) = [0] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [1] x1 + [1] x2 + [0]
sqr^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
dbl^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8(x1, x2) = [0] x1 + [0] x2 + [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: {sqr^#(0()) -> c_1()}
Weak Rules: {terms^#(N) -> c_0(sqr^#(N), terms^#(s(N)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1, 2},
Uargs(sqr^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0] x1 + [0]
0() = [1]
terms^#(x1) = [6] x1 + [0]
c_0(x1, x2) = [4] x1 + [2] x2 + [0]
sqr^#(x1) = [1] x1 + [0]
c_1() = [0]
* Path {1}->{3}: inherited
------------------------
This path is subsumed by the proof of path {1}->{3}->{7}->{6}.
* Path {1}->{3}->{6}: MAYBE
-------------------------
The usable rules for this path are:
{ sqr(0()) -> 0()
, sqr(s(X)) -> s(add(sqr(X), dbl(X)))
, dbl(0()) -> 0()
, dbl(s(X)) -> s(s(dbl(X)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(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:
{ sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
, terms^#(N) -> c_0(sqr^#(N), terms^#(s(N)))
, add^#(0(), X) -> c_5(X)
, sqr(0()) -> 0()
, sqr(s(X)) -> s(add(sqr(X), dbl(X)))
, dbl(0()) -> 0()
, dbl(s(X)) -> s(s(dbl(X)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
Proof Output:
The input cannot be shown compatible
* Path {1}->{3}->{7}: inherited
-----------------------------
This path is subsumed by the proof of path {1}->{3}->{7}->{6}.
* Path {1}->{3}->{7}->{6}: NA
---------------------------
The usable rules for this path are:
{ sqr(0()) -> 0()
, sqr(s(X)) -> s(add(sqr(X), dbl(X)))
, dbl(0()) -> 0()
, dbl(s(X)) -> s(s(dbl(X)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(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 {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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {1}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
dbl(x1) = [0] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
sqr^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
dbl^#(x1) = [3] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8(x1, x2) = [0] x1 + [0] x2 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
dbl^#(x1) = [2] x1 + [0]
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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {1}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
dbl(x1) = [0] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
sqr^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
dbl^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8(x1, x2) = [0] x1 + [0] x2 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {dbl^#(0()) -> c_3()}
Weak Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
0() = [2]
dbl^#(x1) = [2] x1 + [0]
c_3() = [1]
c_4(x1) = [1] x1 + [0]
* Path {9}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
dbl(x1) = [0] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
sqr^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
dbl^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
first^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_7() = [0]
c_8(x1, x2) = [1] x1 + [1] x2 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(first^#) = {},
Uargs(c_8) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1] x1 + [1] x2 + [2]
s(x1) = [1] x1 + [0]
first^#(x1, x2) = [0] x1 + [2] x2 + [4]
c_8(x1, x2) = [0] x1 + [1] x2 + [3]
* Path {9}->{8}: 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(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
dbl(x1) = [0] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
sqr^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
dbl^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8(x1, x2) = [0] x1 + [1] x2 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {first^#(0(), X) -> c_7()}
Weak Rules: {first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(first^#) = {},
Uargs(c_8) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [2]
0() = [2]
first^#(x1, x2) = [3] x1 + [0] x2 + [2]
c_7() = [1]
c_8(x1, x2) = [0] x1 + [1] x2 + [5]
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.