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:
{ and(true(), X) -> X
, and(false(), Y) -> false()
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y
, 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))
, from(X) -> cons(X, from(s(X)))}
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: and^#(true(), X) -> c_0()
, 2: and^#(false(), Y) -> c_1()
, 3: if^#(true(), X, Y) -> c_2()
, 4: if^#(false(), X, Y) -> c_3()
, 5: add^#(0(), X) -> c_4()
, 6: add^#(s(X), Y) -> c_5(add^#(X, Y))
, 7: first^#(0(), X) -> c_6()
, 8: first^#(s(X), cons(Y, Z)) -> c_7(first^#(X, Z))
, 9: from^#(X) -> c_8(from^#(s(X)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{9} [ MAYBE ]
->{8} [ YES(?,O(n^3)) ]
|
`->{7} [ YES(?,O(n^1)) ]
->{6} [ YES(?,O(n^2)) ]
|
`->{5} [ YES(?,O(n^2)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
if(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]
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]
0() = [0]
[0]
[0]
s(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]
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]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
if^#(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_2() = [0]
[0]
[0]
c_3() = [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]
c_4() = [0]
[0]
[0]
c_5(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_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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: {and^#(true(), X) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
[2]
[2]
and^#(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [7]
[2 2 0] [0 0 0] [3]
[2 2 2] [0 0 0] [3]
c_0() = [0]
[1]
[1]
* Path {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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
if(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]
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]
0() = [0]
[0]
[0]
s(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]
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]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
if^#(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_2() = [0]
[0]
[0]
c_3() = [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]
c_4() = [0]
[0]
[0]
c_5(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_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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: {and^#(false(), Y) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
[2]
and^#(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [7]
[2 2 0] [0 0 0] [3]
[2 2 2] [0 0 0] [3]
c_1() = [0]
[1]
[1]
* Path {3}: 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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
if(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]
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]
0() = [0]
[0]
[0]
s(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]
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]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
if^#(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_2() = [0]
[0]
[0]
c_3() = [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]
c_4() = [0]
[0]
[0]
c_5(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_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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: {if^#(true(), X, Y) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
[2]
[2]
if^#(x1, x2, x3) = [0 2 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [7]
[2 2 0] [0 0 0] [0 0 0] [3]
[2 2 2] [0 0 0] [0 0 0] [3]
c_2() = [0]
[1]
[1]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
if(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]
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]
0() = [0]
[0]
[0]
s(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]
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]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
if^#(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_2() = [0]
[0]
[0]
c_3() = [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]
c_4() = [0]
[0]
[0]
c_5(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_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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: {if^#(false(), X, Y) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
[2]
if^#(x1, x2, x3) = [0 2 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [7]
[2 2 0] [0 0 0] [0 0 0] [3]
[2 2 2] [0 0 0] [0 0 0] [3]
c_3() = [0]
[1]
[1]
* Path {6}: 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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {1}, Uargs(first^#) = {}, Uargs(c_7) = {},
Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
if(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]
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]
0() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 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]
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]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
if^#(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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
add^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_4() = [0]
[0]
[0]
c_5(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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: {add^#(s(X), Y) -> c_5(add^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 0 2] [3]
[0 0 1] [2]
add^#(x1, x2) = [0 0 1] x1 + [0 0 0] x2 + [2]
[0 2 1] [0 0 0] [2]
[4 0 2] [0 0 4] [0]
c_5(x1) = [1 0 0] x1 + [1]
[0 0 0] [2]
[2 2 0] [3]
* Path {6}->{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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {1}, Uargs(first^#) = {}, Uargs(c_7) = {},
Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
if(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]
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]
0() = [0]
[0]
[0]
s(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]
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]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
if^#(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_2() = [0]
[0]
[0]
c_3() = [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]
c_4() = [0]
[0]
[0]
c_5(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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: {add^#(0(), X) -> c_4()}
Weak Rules: {add^#(s(X), Y) -> c_5(add^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 2 0] x1 + [2]
[0 1 4] [0]
[0 0 0] [2]
add^#(x1, x2) = [1 3 1] x1 + [0 0 0] x2 + [0]
[3 2 2] [0 0 4] [0]
[0 2 2] [0 0 2] [2]
c_4() = [1]
[0]
[0]
c_5(x1) = [1 0 0] x1 + [0]
[0 0 2] [3]
[0 0 0] [2]
* Path {8}: 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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {1},
Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
if(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]
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]
0() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 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]
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]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
if^#(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_2() = [0]
[0]
[0]
c_3() = [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]
c_4() = [0]
[0]
[0]
c_5(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_6() = [0]
[0]
[0]
c_7(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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_7(first^#(X, Z))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(cons) = {}, Uargs(first^#) = {},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4 4] x1 + [2]
[0 1 2] [2]
[0 0 0] [2]
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]
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_7(x1) = [1 0 0] x1 + [5]
[2 0 2] [3]
[0 0 0] [7]
* Path {8}->{7}: 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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {1},
Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
if(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]
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]
0() = [0]
[0]
[0]
s(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]
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]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
if^#(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_2() = [0]
[0]
[0]
c_3() = [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]
c_4() = [0]
[0]
[0]
c_5(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_6() = [0]
[0]
[0]
c_7(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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_6()}
Weak Rules: {first^#(s(X), cons(Y, Z)) -> c_7(first^#(X, Z))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(cons) = {}, Uargs(first^#) = {},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
[0]
s(x1) = [1 2 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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]
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_6() = [1]
[0]
[0]
c_7(x1) = [1 0 0] x1 + [0]
[0 0 2] [0]
[0 0 0] [2]
* Path {9}: MAYBE
---------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
Uargs(from^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
if(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]
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]
0() = [0]
[0]
[0]
s(x1) = [1 1 0] x1 + [0]
[0 0 1] [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]
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]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
if^#(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_2() = [0]
[0]
[0]
c_3() = [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]
c_4() = [0]
[0]
[0]
c_5(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_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
from^#(x1) = [3 3 3] x1 + [0]
[3 3 3] [0]
[3 3 3] [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: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {from^#(X) -> c_8(from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: and^#(true(), X) -> c_0()
, 2: and^#(false(), Y) -> c_1()
, 3: if^#(true(), X, Y) -> c_2()
, 4: if^#(false(), X, Y) -> c_3()
, 5: add^#(0(), X) -> c_4()
, 6: add^#(s(X), Y) -> c_5(add^#(X, Y))
, 7: first^#(0(), X) -> c_6()
, 8: first^#(s(X), cons(Y, Z)) -> c_7(first^#(X, Z))
, 9: from^#(X) -> c_8(from^#(s(X)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{9} [ MAYBE ]
->{8} [ NA ]
|
`->{7} [ NA ]
->{6} [ YES(?,O(n^1)) ]
|
`->{5} [ NA ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
add(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]
first(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]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 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: {and^#(true(), X) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
[2]
and^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
[2 2] [0 0] [7]
c_0() = [0]
[1]
* Path {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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
add(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]
first(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]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 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: {and^#(false(), Y) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
and^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
[2 2] [0 0] [7]
c_1() = [0]
[1]
* Path {3}: 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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
add(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]
first(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]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 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: {if^#(true(), X, Y) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
[2]
if^#(x1, x2, x3) = [2 0] x1 + [0 0] x2 + [0 0] x3 + [7]
[2 2] [0 0] [0 0] [7]
c_2() = [0]
[1]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
add(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]
first(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]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 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: {if^#(false(), X, Y) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
if^#(x1, x2, x3) = [2 0] x1 + [0 0] x2 + [0 0] x3 + [7]
[2 2] [0 0] [0 0] [7]
c_3() = [0]
[1]
* Path {6}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {1}, Uargs(first^#) = {}, Uargs(c_7) = {},
Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
add(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]
first(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]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
add^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_4() = [0]
[0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 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: {add^#(s(X), Y) -> c_5(add^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
add^#(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
[0 0] [0 4] [4]
c_5(x1) = [1 0] x1 + [0]
[0 0] [3]
* Path {6}->{5}: NA
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {1}, Uargs(first^#) = {}, Uargs(c_7) = {},
Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
add(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]
first(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]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {8}: NA
------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {1},
Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
add(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]
first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [3 3] x1 + [1 3] x2 + [0]
[3 3] [3 3] [0]
c_6() = [0]
[0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {8}->{7}: NA
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {1},
Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
add(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]
first(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]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {9}: MAYBE
---------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
Uargs(from^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
first(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]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [3 3] x1 + [0]
[3 3] [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: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {from^#(X) -> c_8(from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: and^#(true(), X) -> c_0()
, 2: and^#(false(), Y) -> c_1()
, 3: if^#(true(), X, Y) -> c_2()
, 4: if^#(false(), X, Y) -> c_3()
, 5: add^#(0(), X) -> c_4()
, 6: add^#(s(X), Y) -> c_5(add^#(X, Y))
, 7: first^#(0(), X) -> c_6()
, 8: first^#(s(X), cons(Y, Z)) -> c_7(first^#(X, Z))
, 9: from^#(X) -> c_8(from^#(s(X)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{9} [ MAYBE ]
->{8} [ NA ]
|
`->{7} [ NA ]
->{6} [ NA ]
|
`->{5} [ NA ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2() = [0]
c_3() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [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: {and^#(true(), X) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [7]
and^#(x1, x2) = [1] x1 + [0] x2 + [7]
c_0() = [1]
* Path {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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2() = [0]
c_3() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [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: {and^#(false(), Y) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [7]
and^#(x1, x2) = [1] x1 + [0] x2 + [7]
c_1() = [1]
* Path {3}: 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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2() = [0]
c_3() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [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: {if^#(true(), X, Y) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [7]
if^#(x1, x2, x3) = [1] x1 + [0] x2 + [0] x3 + [7]
c_2() = [1]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2() = [0]
c_3() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [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: {if^#(false(), X, Y) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [7]
if^#(x1, x2, x3) = [1] x1 + [0] x2 + [0] x3 + [7]
c_3() = [1]
* Path {6}: NA
------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {1}, Uargs(first^#) = {}, Uargs(c_7) = {},
Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2() = [0]
c_3() = [0]
add^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {6}->{5}: NA
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {1}, Uargs(first^#) = {}, Uargs(c_7) = {},
Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2() = [0]
c_3() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {8}: NA
------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {1},
Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2() = [0]
c_3() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
first^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {8}->{7}: NA
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {1},
Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2() = [0]
c_3() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {9}: MAYBE
---------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
Uargs(from^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2() = [0]
c_3() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
from^#(x1) = [3] x1 + [0]
c_8(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {from^#(X) -> c_8(from^#(s(X)))}
Weak Rules: {}
Proof Output:
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:
{ and(true(), X) -> X
, and(false(), Y) -> false()
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y
, 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))
, from(X) -> cons(X, from(s(X)))}
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: and^#(true(), X) -> c_0(X)
, 2: and^#(false(), Y) -> c_1()
, 3: if^#(true(), X, Y) -> c_2(X)
, 4: if^#(false(), X, Y) -> c_3(Y)
, 5: add^#(0(), X) -> c_4(X)
, 6: add^#(s(X), Y) -> c_5(add^#(X, Y))
, 7: first^#(0(), X) -> c_6()
, 8: first^#(s(X), cons(Y, Z)) -> c_7(Y, first^#(X, Z))
, 9: from^#(X) -> c_8(X, from^#(s(X)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{9} [ MAYBE ]
->{8} [ YES(?,O(n^3)) ]
|
`->{7} [ NA ]
->{6} [ YES(?,O(n^2)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
if(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]
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]
0() = [0]
[0]
[0]
s(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]
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]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
if^#(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_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(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]
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]
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_6() = [0]
[0]
[0]
c_7(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]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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: {and^#(true(), X) -> c_0(X)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
[0]
[2]
and^#(x1, x2) = [2 0 2] x1 + [7 7 7] x2 + [7]
[2 0 2] [7 7 7] [7]
[2 0 2] [7 7 7] [7]
c_0(x1) = [1 3 3] x1 + [0]
[1 1 1] [1]
[1 1 1] [1]
* Path {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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
if(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]
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]
0() = [0]
[0]
[0]
s(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]
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]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
if^#(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_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(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]
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]
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_6() = [0]
[0]
[0]
c_7(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]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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: {and^#(false(), Y) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
[2]
and^#(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [7]
[2 2 0] [0 0 0] [3]
[2 2 2] [0 0 0] [3]
c_1() = [0]
[1]
[1]
* Path {3}: 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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
if(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]
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]
0() = [0]
[0]
[0]
s(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]
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]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
if^#(x1, x2, x3) = [0 0 0] x1 + [3 3 3] 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_2(x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(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]
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]
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_6() = [0]
[0]
[0]
c_7(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]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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: {if^#(true(), X, Y) -> c_2(X)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
[0]
[2]
if^#(x1, x2, x3) = [2 0 2] x1 + [7 7 7] x2 + [0 0 0] x3 + [7]
[2 0 2] [7 7 7] [0 0 0] [7]
[2 0 2] [7 7 7] [0 0 0] [7]
c_2(x1) = [1 3 3] x1 + [0]
[1 1 1] [1]
[1 1 1] [1]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
if(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]
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]
0() = [0]
[0]
[0]
s(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]
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]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
if^#(x1, x2, x3) = [0 0 0] x1 + [3 3 3] 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_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [1 1 1] 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]
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]
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_6() = [0]
[0]
[0]
c_7(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]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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: {if^#(false(), X, Y) -> c_3(Y)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[0]
[2]
if^#(x1, x2, x3) = [2 0 2] x1 + [0 0 0] x2 + [7 7 7] x3 + [7]
[2 0 2] [0 0 0] [7 7 7] [7]
[2 0 2] [0 0 0] [7 7 7] [7]
c_3(x1) = [1 3 3] x1 + [0]
[1 1 1] [1]
[1 1 1] [1]
* Path {6}: 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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {1}, Uargs(first^#) = {},
Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
if(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]
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]
0() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 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]
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]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
if^#(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_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
add^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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_6() = [0]
[0]
[0]
c_7(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]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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: {add^#(s(X), Y) -> c_5(add^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 0 2] [3]
[0 0 1] [2]
add^#(x1, x2) = [0 0 1] x1 + [0 0 0] x2 + [2]
[0 2 1] [0 0 0] [2]
[4 0 2] [0 0 4] [0]
c_5(x1) = [1 0 0] x1 + [1]
[0 0 0] [2]
[2 2 0] [3]
* Path {6}->{5}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {1}, Uargs(first^#) = {},
Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
if(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]
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]
0() = [0]
[0]
[0]
s(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]
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]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
if^#(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_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
add^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_4(x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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_6() = [0]
[0]
[0]
c_7(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]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {add^#(0(), X) -> c_4(X)}
Weak Rules: {add^#(s(X), Y) -> c_5(add^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 2 2] x1 + [2]
[0 0 2] [2]
[0 0 0] [2]
add^#(x1, x2) = [2 2 2] x1 + [0 0 0] x2 + [0]
[2 2 2] [4 4 4] [0]
[2 2 2] [4 0 4] [0]
c_4(x1) = [0 0 0] x1 + [1]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [2]
* Path {8}: 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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
Uargs(c_7) = {2}, Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
if(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]
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]
0() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 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]
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]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
if^#(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_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(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]
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]
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_6() = [0]
[0]
[0]
c_7(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]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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^3))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {first^#(s(X), cons(Y, Z)) -> c_7(Y, first^#(X, Z))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(cons) = {}, Uargs(first^#) = {},
Uargs(c_7) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 1] x1 + [0]
[0 1 2] [3]
[0 0 1] [0]
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]
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_7(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 {8}->{7}: NA
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
Uargs(c_7) = {2}, Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
if(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]
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]
0() = [0]
[0]
[0]
s(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]
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]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
if^#(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_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(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]
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]
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_6() = [0]
[0]
[0]
c_7(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]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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 have not generated a proof for the resulting sub-problem.
* Path {9}: MAYBE
---------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
if(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]
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]
0() = [0]
[0]
[0]
s(x1) = [0 1 0] x1 + [0]
[0 0 0] [0]
[0 0 1] [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]
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]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
if^#(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_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(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]
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]
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_6() = [0]
[0]
[0]
c_7(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]
from^#(x1) = [1 3 3] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_8(x1, x2) = [0 1 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: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules: {from^#(X) -> c_8(X, from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: and^#(true(), X) -> c_0(X)
, 2: and^#(false(), Y) -> c_1()
, 3: if^#(true(), X, Y) -> c_2(X)
, 4: if^#(false(), X, Y) -> c_3(Y)
, 5: add^#(0(), X) -> c_4(X)
, 6: add^#(s(X), Y) -> c_5(add^#(X, Y))
, 7: first^#(0(), X) -> c_6()
, 8: first^#(s(X), cons(Y, Z)) -> c_7(Y, first^#(X, Z))
, 9: from^#(X) -> c_8(X, from^#(s(X)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{9} [ MAYBE ]
->{8} [ NA ]
|
`->{7} [ NA ]
->{6} [ YES(?,O(n^1)) ]
|
`->{5} [ NA ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
add(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]
first(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]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 1] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 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: {and^#(true(), X) -> c_0(X)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
[2]
and^#(x1, x2) = [2 2] x1 + [7 7] x2 + [7]
[2 2] [7 7] [3]
c_0(x1) = [1 3] x1 + [0]
[1 1] [1]
* Path {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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
add(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]
first(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]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 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: {and^#(false(), Y) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
and^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
[2 2] [0 0] [7]
c_1() = [0]
[1]
* Path {3}: 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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
add(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]
first(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]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [3 3] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2(x1) = [1 1] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 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: {if^#(true(), X, Y) -> c_2(X)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
[2]
if^#(x1, x2, x3) = [2 2] x1 + [7 7] x2 + [0 0] x3 + [7]
[2 2] [7 7] [0 0] [3]
c_2(x1) = [1 3] x1 + [0]
[1 1] [1]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
add(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]
first(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]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [3 3] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [1 1] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 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: {if^#(false(), X, Y) -> c_3(Y)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
if^#(x1, x2, x3) = [2 2] x1 + [0 0] x2 + [7 7] x3 + [7]
[2 2] [0 0] [7 7] [3]
c_3(x1) = [1 3] x1 + [0]
[1 1] [1]
* Path {6}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {1}, Uargs(first^#) = {},
Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
add(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]
first(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]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 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: {add^#(s(X), Y) -> c_5(add^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
add^#(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
[0 0] [0 4] [4]
c_5(x1) = [1 0] x1 + [0]
[0 0] [3]
* Path {6}->{5}: NA
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {1}, Uargs(first^#) = {},
Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
add(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]
first(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]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [1 1] x1 + [0]
[0 0] [0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {8}: NA
------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
Uargs(c_7) = {2}, Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
add(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]
first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [1 3] x1 + [1 3] x2 + [0]
[0 1] [0 1] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [3 3] x1 + [1 3] x2 + [0]
[3 3] [3 3] [0]
c_6() = [0]
[0]
c_7(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {8}->{7}: NA
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
Uargs(c_7) = {2}, Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
add(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]
first(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]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {9}: MAYBE
---------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 1] x1 + [0]
[0 0] [0]
first(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]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
from^#(x1) = [1 3] x1 + [0]
[3 3] [0]
c_8(x1, x2) = [0 1] 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: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules: {from^#(X) -> c_8(X, from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: and^#(true(), X) -> c_0(X)
, 2: and^#(false(), Y) -> c_1()
, 3: if^#(true(), X, Y) -> c_2(X)
, 4: if^#(false(), X, Y) -> c_3(Y)
, 5: add^#(0(), X) -> c_4(X)
, 6: add^#(s(X), Y) -> c_5(add^#(X, Y))
, 7: first^#(0(), X) -> c_6()
, 8: first^#(s(X), cons(Y, Z)) -> c_7(Y, first^#(X, Z))
, 9: from^#(X) -> c_8(X, from^#(s(X)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{9} [ MAYBE ]
->{8} [ NA ]
|
`->{7} [ NA ]
->{6} [ NA ]
|
`->{5} [ NA ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_0(x1) = [1] x1 + [0]
c_1() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1, x2) = [0] x1 + [0] x2 + [0]
from^#(x1) = [0] x1 + [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: {and^#(true(), X) -> c_0(X)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [5]
and^#(x1, x2) = [3] x1 + [7] x2 + [0]
c_0(x1) = [1] x1 + [0]
* Path {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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1, x2) = [0] x1 + [0] x2 + [0]
from^#(x1) = [0] x1 + [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: {and^#(false(), Y) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [7]
and^#(x1, x2) = [1] x1 + [0] x2 + [7]
c_1() = [1]
* Path {3}: 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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
if^#(x1, x2, x3) = [0] x1 + [3] x2 + [0] x3 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1, x2) = [0] x1 + [0] x2 + [0]
from^#(x1) = [0] x1 + [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: {if^#(true(), X, Y) -> c_2(X)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [5]
if^#(x1, x2, x3) = [3] x1 + [7] x2 + [0] x3 + [0]
c_2(x1) = [1] x1 + [0]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
if^#(x1, x2, x3) = [0] x1 + [3] x2 + [0] x3 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1, x2) = [0] x1 + [0] x2 + [0]
from^#(x1) = [0] x1 + [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: {if^#(false(), X, Y) -> c_3(Y)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [5]
if^#(x1, x2, x3) = [3] x1 + [0] x2 + [7] x3 + [0]
c_3(x1) = [1] x1 + [0]
* Path {6}: NA
------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {1}, Uargs(first^#) = {},
Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
add^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1, x2) = [0] x1 + [0] x2 + [0]
from^#(x1) = [0] x1 + [0]
c_8(x1, x2) = [0] x1 + [0] x2 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {6}->{5}: NA
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {1}, Uargs(first^#) = {},
Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
add^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1, x2) = [0] x1 + [0] x2 + [0]
from^#(x1) = [0] x1 + [0]
c_8(x1, x2) = [0] x1 + [0] x2 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {8}: NA
------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
Uargs(c_7) = {2}, Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
from(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
first^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_6() = [0]
c_7(x1, x2) = [1] x1 + [1] x2 + [0]
from^#(x1) = [0] x1 + [0]
c_8(x1, x2) = [0] x1 + [0] x2 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {8}->{7}: NA
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
Uargs(c_7) = {2}, Uargs(from^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1, x2) = [0] x1 + [1] x2 + [0]
from^#(x1) = [0] x1 + [0]
c_8(x1, x2) = [0] x1 + [0] x2 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {9}: MAYBE
---------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1, x2) = [0] x1 + [0] x2 + [0]
from^#(x1) = [3] x1 + [0]
c_8(x1, x2) = [2] x1 + [1] x2 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules: {from^#(X) -> c_8(X, from^#(s(X)))}
Weak Rules: {}
Proof Output:
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.