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:
{ eq(0(), 0()) -> true()
, eq(s(X), s(Y)) -> eq(X, Y)
, eq(X, Y) -> false()
, inf(X) -> cons(X, inf(s(X)))
, take(0(), X) -> nil()
, take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
, length(nil()) -> 0()
, length(cons(X, L)) -> s(length(L))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: eq^#(0(), 0()) -> c_0()
, 2: eq^#(s(X), s(Y)) -> c_1(eq^#(X, Y))
, 3: eq^#(X, Y) -> c_2()
, 4: inf^#(X) -> c_3(inf^#(s(X)))
, 5: take^#(0(), X) -> c_4()
, 6: take^#(s(X), cons(Y, L)) -> c_5(take^#(X, L))
, 7: length^#(nil()) -> c_6()
, 8: length^#(cons(X, L)) -> c_7(length^#(L))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8} [ YES(?,O(n^2)) ]
|
`->{7} [ YES(?,O(n^2)) ]
->{6} [ YES(?,O(n^3)) ]
|
`->{5} [ NA ]
->{4} [ MAYBE ]
->{2} [ YES(?,O(n^1)) ]
|
|->{1} [ NA ]
|
`->{3} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {2}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, 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]
true() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
inf(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
take(x1, 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]
length(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
eq^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_2() = [0]
[0]
[0]
inf^#(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]
take^#(x1, 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]
length^#(x1) = [0 0 0] x1 + [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]
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: {eq^#(s(X), s(Y)) -> c_1(eq^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [2]
[0 0 0] [0]
eq^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
[2 2 0] [0 2 0] [0]
[4 0 0] [0 2 0] [0]
c_1(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {2}->{1}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, 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]
true() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
inf(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
take(x1, 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]
length(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
eq^#(x1, 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(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_2() = [0]
[0]
[0]
inf^#(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]
take^#(x1, 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]
length^#(x1) = [0 0 0] x1 + [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]
We have not generated a proof for the resulting sub-problem.
* Path {2}->{3}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, 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]
true() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
inf(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
take(x1, 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]
length(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
eq^#(x1, 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(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_2() = [0]
[0]
[0]
inf^#(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]
take^#(x1, 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]
length^#(x1) = [0 0 0] x1 + [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]
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: {eq^#(X, Y) -> c_2()}
Weak Rules: {eq^#(s(X), s(Y)) -> c_1(eq^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 0] x1 + [0]
[0 0 2] [0]
[0 0 0] [0]
eq^#(x1, x2) = [0 0 0] x1 + [4 0 0] x2 + [1]
[2 0 0] [0 2 0] [0]
[0 4 0] [2 0 0] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
* Path {4}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {1},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, 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]
true() = [0]
[0]
[0]
s(x1) = [1 1 0] x1 + [0]
[0 0 1] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
inf(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
take(x1, 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]
length(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
eq^#(x1, 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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
inf^#(x1) = [3 3 3] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
take^#(x1, 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]
length^#(x1) = [0 0 0] x1 + [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]
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: {inf^#(X) -> c_3(inf^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {6}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {1}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, 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]
true() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
inf(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [1 3 0] x1 + [0 0 0] x2 + [0]
[0 1 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
take(x1, 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]
length(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
eq^#(x1, 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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
inf^#(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]
take^#(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_4() = [0]
[0]
[0]
c_5(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
length^#(x1) = [0 0 0] x1 + [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]
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: {take^#(s(X), cons(Y, L)) -> c_5(take^#(X, L))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(cons) = {}, Uargs(take^#) = {},
Uargs(c_5) = {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]
take^#(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_5(x1) = [1 0 0] x1 + [5]
[2 0 2] [3]
[0 0 0] [7]
* 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {1}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, 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]
true() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
inf(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
take(x1, 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]
length(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
eq^#(x1, 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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
inf^#(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]
take^#(x1, 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]
length^#(x1) = [0 0 0] x1 + [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]
We have not generated a proof for the resulting sub-problem.
* Path {8}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, 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]
true() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
inf(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [1 3 0] x1 + [0 0 0] x2 + [0]
[0 1 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
take(x1, 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]
length(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
eq^#(x1, 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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
inf^#(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]
take^#(x1, 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]
length^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[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]
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: {length^#(cons(X, L)) -> c_7(length^#(L))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(length^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [1 2 2] x2 + [2]
[0 0 0] [0 1 2] [2]
[0 0 0] [0 0 0] [0]
length^#(x1) = [0 1 0] x1 + [2]
[6 0 0] [0]
[2 3 0] [2]
c_7(x1) = [1 0 0] x1 + [1]
[2 0 2] [0]
[0 0 0] [0]
* Path {8}->{7}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, 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]
true() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
inf(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
take(x1, 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]
length(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
eq^#(x1, 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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
inf^#(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]
take^#(x1, 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]
length^#(x1) = [0 0 0] x1 + [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]
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: {length^#(nil()) -> c_6()}
Weak Rules: {length^#(cons(X, L)) -> c_7(length^#(L))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(length^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [1 1 0] x2 + [0]
[0 0 0] [0 1 1] [1]
[0 0 0] [0 0 0] [0]
nil() = [2]
[2]
[2]
length^#(x1) = [2 2 2] x1 + [0]
[0 6 0] [0]
[0 0 2] [0]
c_6() = [1]
[0]
[0]
c_7(x1) = [1 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [0]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: eq^#(0(), 0()) -> c_0()
, 2: eq^#(s(X), s(Y)) -> c_1(eq^#(X, Y))
, 3: eq^#(X, Y) -> c_2()
, 4: inf^#(X) -> c_3(inf^#(s(X)))
, 5: take^#(0(), X) -> c_4()
, 6: take^#(s(X), cons(Y, L)) -> c_5(take^#(X, L))
, 7: length^#(nil()) -> c_6()
, 8: length^#(cons(X, L)) -> c_7(length^#(L))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8} [ YES(?,O(n^1)) ]
|
`->{7} [ YES(?,O(n^1)) ]
->{6} [ NA ]
|
`->{5} [ NA ]
->{4} [ MAYBE ]
->{2} [ NA ]
|
|->{1} [ NA ]
|
`->{3} [ NA ]
Sub-problems:
-------------
* Path {2}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
false() = [0]
[0]
inf(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [0]
[0]
inf^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(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]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {2}->{1}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
inf(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [0]
[0]
inf^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(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]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {2}->{3}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
inf(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [0]
[0]
inf^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(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]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {4}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {1},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
false() = [0]
[0]
inf(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
inf^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
take^#(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]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [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: {inf^#(X) -> c_3(inf^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {1}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
false() = [0]
[0]
inf(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
inf^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [3 3] x1 + [1 3] x2 + [0]
[3 3] [3 3] [0]
c_4() = [0]
[0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {1}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
inf(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
inf^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(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]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {8}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
inf(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 2] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
inf^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(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]
length^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_6() = [0]
[0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {length^#(cons(X, L)) -> c_7(length^#(L))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(length^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [1]
length^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_7(x1) = [1 0] x1 + [0]
[0 0] [0]
* 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
inf(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
inf^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(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]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {length^#(nil()) -> c_6()}
Weak Rules: {length^#(cons(X, L)) -> c_7(length^#(L))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(length^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [1 2] x2 + [1]
[0 0] [0 0] [3]
nil() = [2]
[2]
length^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_6() = [1]
[0]
c_7(x1) = [1 0] x1 + [5]
[2 0] [3]
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: eq^#(0(), 0()) -> c_0()
, 2: eq^#(s(X), s(Y)) -> c_1(eq^#(X, Y))
, 3: eq^#(X, Y) -> c_2()
, 4: inf^#(X) -> c_3(inf^#(s(X)))
, 5: take^#(0(), X) -> c_4()
, 6: take^#(s(X), cons(Y, L)) -> c_5(take^#(X, L))
, 7: length^#(nil()) -> c_6()
, 8: length^#(cons(X, L)) -> c_7(length^#(L))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8} [ NA ]
|
`->{7} [ NA ]
->{6} [ NA ]
|
`->{5} [ NA ]
->{4} [ MAYBE ]
->{2} [ NA ]
|
|->{1} [ NA ]
|
`->{3} [ NA ]
Sub-problems:
-------------
* Path {2}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
true() = [0]
s(x1) = [1] x1 + [0]
false() = [0]
inf(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
length(x1) = [0] x1 + [0]
eq^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
inf^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {2}->{1}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
true() = [0]
s(x1) = [0] x1 + [0]
false() = [0]
inf(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
length(x1) = [0] x1 + [0]
eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
inf^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {2}->{3}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
true() = [0]
s(x1) = [0] x1 + [0]
false() = [0]
inf(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
length(x1) = [0] x1 + [0]
eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
inf^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {4}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {1},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
true() = [0]
s(x1) = [0] x1 + [0]
false() = [0]
inf(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
length(x1) = [0] x1 + [0]
eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
inf^#(x1) = [3] x1 + [0]
c_3(x1) = [1] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] 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: {inf^#(X) -> c_3(inf^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {1}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
true() = [0]
s(x1) = [1] x1 + [0]
false() = [0]
inf(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
length(x1) = [0] x1 + [0]
eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
inf^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
take^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {1}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
true() = [0]
s(x1) = [0] x1 + [0]
false() = [0]
inf(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
length(x1) = [0] x1 + [0]
eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
inf^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
true() = [0]
s(x1) = [0] x1 + [0]
false() = [0]
inf(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
length(x1) = [0] x1 + [0]
eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
inf^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
length^#(x1) = [3] x1 + [0]
c_6() = [0]
c_7(x1) = [1] 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
true() = [0]
s(x1) = [0] x1 + [0]
false() = [0]
inf(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
length(x1) = [0] x1 + [0]
eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
inf^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
We have not generated a proof for the resulting sub-problem.
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:
{ eq(0(), 0()) -> true()
, eq(s(X), s(Y)) -> eq(X, Y)
, eq(X, Y) -> false()
, inf(X) -> cons(X, inf(s(X)))
, take(0(), X) -> nil()
, take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
, length(nil()) -> 0()
, length(cons(X, L)) -> s(length(L))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: eq^#(0(), 0()) -> c_0()
, 2: eq^#(s(X), s(Y)) -> c_1(eq^#(X, Y))
, 3: eq^#(X, Y) -> c_2()
, 4: inf^#(X) -> c_3(X, inf^#(s(X)))
, 5: take^#(0(), X) -> c_4()
, 6: take^#(s(X), cons(Y, L)) -> c_5(Y, take^#(X, L))
, 7: length^#(nil()) -> c_6()
, 8: length^#(cons(X, L)) -> c_7(length^#(L))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8} [ YES(?,O(n^2)) ]
|
`->{7} [ YES(?,O(n^2)) ]
->{6} [ NA ]
|
`->{5} [ NA ]
->{4} [ MAYBE ]
->{2} [ YES(?,O(n^1)) ]
|
|->{1} [ NA ]
|
`->{3} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {2}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, 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]
true() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
inf(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
take(x1, 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]
length(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
eq^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_2() = [0]
[0]
[0]
inf^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
take^#(x1, 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, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
length^#(x1) = [0 0 0] x1 + [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]
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: {eq^#(s(X), s(Y)) -> c_1(eq^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [2]
[0 0 0] [0]
eq^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
[2 2 0] [0 2 0] [0]
[4 0 0] [0 2 0] [0]
c_1(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {2}->{1}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, 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]
true() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
inf(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
take(x1, 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]
length(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
eq^#(x1, 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(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_2() = [0]
[0]
[0]
inf^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
take^#(x1, 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, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
length^#(x1) = [0 0 0] x1 + [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]
We have not generated a proof for the resulting sub-problem.
* Path {2}->{3}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, 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]
true() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
inf(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
take(x1, 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]
length(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
eq^#(x1, 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(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_2() = [0]
[0]
[0]
inf^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
take^#(x1, 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, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
length^#(x1) = [0 0 0] x1 + [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]
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: {eq^#(X, Y) -> c_2()}
Weak Rules: {eq^#(s(X), s(Y)) -> c_1(eq^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 0] x1 + [0]
[0 0 2] [0]
[0 0 0] [0]
eq^#(x1, x2) = [0 0 0] x1 + [4 0 0] x2 + [1]
[2 0 0] [0 2 0] [0]
[0 4 0] [2 0 0] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
* Path {4}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {2},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, 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]
true() = [0]
[0]
[0]
s(x1) = [0 1 0] x1 + [0]
[0 0 0] [0]
[0 0 1] [0]
false() = [0]
[0]
[0]
inf(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
take(x1, 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]
length(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
eq^#(x1, 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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
inf^#(x1) = [1 3 3] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_3(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]
take^#(x1, 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, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
length^#(x1) = [0 0 0] x1 + [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]
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: {inf^#(X) -> c_3(X, inf^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {2}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, 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]
true() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
inf(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [1 3 3] x1 + [1 3 0] x2 + [0]
[0 1 3] [0 1 0] [0]
[0 0 1] [0 0 0] [0]
take(x1, 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]
length(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
eq^#(x1, 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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
inf^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
take^#(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_4() = [0]
[0]
[0]
c_5(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]
length^#(x1) = [0 0 0] x1 + [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]
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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {2}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, 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]
true() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
inf(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
take(x1, 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]
length(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
eq^#(x1, 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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
inf^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
take^#(x1, 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, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 0 0] [0 0 1] [0]
length^#(x1) = [0 0 0] x1 + [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]
We have not generated a proof for the resulting sub-problem.
* Path {8}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, 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]
true() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
inf(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [1 3 0] x1 + [0 0 0] x2 + [0]
[0 1 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
take(x1, 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]
length(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
eq^#(x1, 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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
inf^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
take^#(x1, 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, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
length^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[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]
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: {length^#(cons(X, L)) -> c_7(length^#(L))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(length^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [1 2 2] x2 + [2]
[0 0 0] [0 1 2] [2]
[0 0 0] [0 0 0] [0]
length^#(x1) = [0 1 0] x1 + [2]
[6 0 0] [0]
[2 3 0] [2]
c_7(x1) = [1 0 0] x1 + [1]
[2 0 2] [0]
[0 0 0] [0]
* Path {8}->{7}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, 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]
true() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
inf(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
take(x1, 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]
length(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
eq^#(x1, 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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
inf^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
take^#(x1, 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, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
length^#(x1) = [0 0 0] x1 + [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]
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: {length^#(nil()) -> c_6()}
Weak Rules: {length^#(cons(X, L)) -> c_7(length^#(L))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(length^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [1 1 0] x2 + [0]
[0 0 0] [0 1 1] [1]
[0 0 0] [0 0 0] [0]
nil() = [2]
[2]
[2]
length^#(x1) = [2 2 2] x1 + [0]
[0 6 0] [0]
[0 0 2] [0]
c_6() = [1]
[0]
[0]
c_7(x1) = [1 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [0]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: eq^#(0(), 0()) -> c_0()
, 2: eq^#(s(X), s(Y)) -> c_1(eq^#(X, Y))
, 3: eq^#(X, Y) -> c_2()
, 4: inf^#(X) -> c_3(X, inf^#(s(X)))
, 5: take^#(0(), X) -> c_4()
, 6: take^#(s(X), cons(Y, L)) -> c_5(Y, take^#(X, L))
, 7: length^#(nil()) -> c_6()
, 8: length^#(cons(X, L)) -> c_7(length^#(L))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8} [ YES(?,O(n^1)) ]
|
`->{7} [ YES(?,O(n^1)) ]
->{6} [ NA ]
|
`->{5} [ NA ]
->{4} [ MAYBE ]
->{2} [ NA ]
|
|->{1} [ NA ]
|
`->{3} [ NA ]
Sub-problems:
-------------
* Path {2}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
false() = [0]
[0]
inf(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [0]
[0]
inf^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {2}->{1}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
inf(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [0]
[0]
inf^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {2}->{3}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
inf(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [0]
[0]
inf^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {4}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {2},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [1 1] x1 + [0]
[0 0] [0]
false() = [0]
[0]
inf(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
inf^#(x1) = [1 3] x1 + [0]
[3 3] [0]
c_3(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [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: {inf^#(X) -> c_3(X, inf^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {2}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
false() = [0]
[0]
inf(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 3] x1 + [1 3] x2 + [0]
[0 1] [0 1] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
inf^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take^#(x1, x2) = [3 3] x1 + [1 3] x2 + [0]
[3 3] [3 3] [0]
c_4() = [0]
[0]
c_5(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {2}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
inf(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
inf^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {8}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
inf(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 2] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
inf^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_6() = [0]
[0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {length^#(cons(X, L)) -> c_7(length^#(L))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(length^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [1]
length^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_7(x1) = [1 0] x1 + [0]
[0 0] [0]
* 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
inf(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
inf^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {length^#(nil()) -> c_6()}
Weak Rules: {length^#(cons(X, L)) -> c_7(length^#(L))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(length^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [1 2] x2 + [1]
[0 0] [0 0] [3]
nil() = [2]
[2]
length^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_6() = [1]
[0]
c_7(x1) = [1 0] x1 + [5]
[2 0] [3]
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: eq^#(0(), 0()) -> c_0()
, 2: eq^#(s(X), s(Y)) -> c_1(eq^#(X, Y))
, 3: eq^#(X, Y) -> c_2()
, 4: inf^#(X) -> c_3(X, inf^#(s(X)))
, 5: take^#(0(), X) -> c_4()
, 6: take^#(s(X), cons(Y, L)) -> c_5(Y, take^#(X, L))
, 7: length^#(nil()) -> c_6()
, 8: length^#(cons(X, L)) -> c_7(length^#(L))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8} [ NA ]
|
`->{7} [ NA ]
->{6} [ NA ]
|
`->{5} [ NA ]
->{4} [ MAYBE ]
->{2} [ NA ]
|
|->{1} [ NA ]
|
`->{3} [ NA ]
Sub-problems:
-------------
* Path {2}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
true() = [0]
s(x1) = [1] x1 + [0]
false() = [0]
inf(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
length(x1) = [0] x1 + [0]
eq^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
inf^#(x1) = [0] x1 + [0]
c_3(x1, x2) = [0] x1 + [0] x2 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {2}->{1}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
true() = [0]
s(x1) = [0] x1 + [0]
false() = [0]
inf(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
length(x1) = [0] x1 + [0]
eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
inf^#(x1) = [0] x1 + [0]
c_3(x1, x2) = [0] x1 + [0] x2 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {2}->{3}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
true() = [0]
s(x1) = [0] x1 + [0]
false() = [0]
inf(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
length(x1) = [0] x1 + [0]
eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
inf^#(x1) = [0] x1 + [0]
c_3(x1, x2) = [0] x1 + [0] x2 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {4}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {2},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
true() = [0]
s(x1) = [0] x1 + [0]
false() = [0]
inf(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
length(x1) = [0] x1 + [0]
eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
inf^#(x1) = [3] x1 + [0]
c_3(x1, x2) = [2] x1 + [1] x2 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [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: {inf^#(X) -> c_3(X, inf^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {2}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
true() = [0]
s(x1) = [1] x1 + [0]
false() = [0]
inf(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
length(x1) = [0] x1 + [0]
eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
inf^#(x1) = [0] x1 + [0]
c_3(x1, x2) = [0] x1 + [0] x2 + [0]
take^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_4() = [0]
c_5(x1, x2) = [1] x1 + [1] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {2}, Uargs(length^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
true() = [0]
s(x1) = [0] x1 + [0]
false() = [0]
inf(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
length(x1) = [0] x1 + [0]
eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
inf^#(x1) = [0] x1 + [0]
c_3(x1, x2) = [0] x1 + [0] x2 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1, x2) = [0] x1 + [1] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
true() = [0]
s(x1) = [0] x1 + [0]
false() = [0]
inf(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
length(x1) = [0] x1 + [0]
eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
inf^#(x1) = [0] x1 + [0]
c_3(x1, x2) = [0] x1 + [0] x2 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [3] x1 + [0]
c_6() = [0]
c_7(x1) = [1] 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
eq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
true() = [0]
s(x1) = [0] x1 + [0]
false() = [0]
inf(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
length(x1) = [0] x1 + [0]
eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
inf^#(x1) = [0] x1 + [0]
c_3(x1, x2) = [0] x1 + [0] x2 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
We have not generated a proof for the resulting sub-problem.
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.