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(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
, zeros() -> cons(0(), zeros())
, take(0(), IL) -> uTake1(isNatIList(IL))
, uTake1(tt()) -> nil()
, take(s(M), cons(N, IL)) ->
uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)
, uTake2(tt(), M, N, IL) -> cons(N, take(M, IL))
, length(cons(N, L)) -> uLength(and(isNat(N), isNatList(L)), L)
, uLength(tt(), 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: and^#(tt(), T) -> c_0()
, 2: isNatIList^#(IL) -> c_1(isNatList^#(IL))
, 3: isNat^#(0()) -> c_2()
, 4: isNat^#(s(N)) -> c_3(isNat^#(N))
, 5: isNat^#(length(L)) -> c_4(isNatList^#(L))
, 6: isNatIList^#(zeros()) -> c_5()
, 7: isNatIList^#(cons(N, IL)) ->
c_6(and^#(isNat(N), isNatIList(IL)))
, 8: isNatList^#(nil()) -> c_7()
, 9: isNatList^#(cons(N, L)) -> c_8(and^#(isNat(N), isNatList(L)))
, 10: isNatList^#(take(N, IL)) ->
c_9(and^#(isNat(N), isNatIList(IL)))
, 11: zeros^#() -> c_10(zeros^#())
, 12: take^#(0(), IL) -> c_11(uTake1^#(isNatIList(IL)))
, 13: uTake1^#(tt()) -> c_12()
, 14: take^#(s(M), cons(N, IL)) ->
c_13(uTake2^#(and(isNat(M), and(isNat(N), isNatIList(IL))),
M,
N,
IL))
, 15: uTake2^#(tt(), M, N, IL) -> c_14(take^#(M, IL))
, 16: length^#(cons(N, L)) ->
c_15(uLength^#(and(isNat(N), isNatList(L)), L))
, 17: uLength^#(tt(), L) -> c_16(length^#(L))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{16,17} [ NA ]
->{14,15} [ NA ]
|
`->{12} [ NA ]
|
`->{13} [ NA ]
->{11} [ MAYBE ]
->{7} [ NA ]
|
`->{1} [ NA ]
->{6} [ NA ]
->{4} [ NA ]
|
|->{3} [ NA ]
|
`->{5} [ NA ]
|
|->{8} [ NA ]
|
|->{9} [ NA ]
| |
| `->{1} [ NA ]
|
`->{10} [ NA ]
|
`->{1} [ NA ]
->{2} [ NA ]
|
|->{8} [ NA ]
|
|->{9} [ NA ]
| |
| `->{1} [ NA ]
|
`->{10} [ NA ]
|
`->{1} [ 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(and) = {}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
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]
tt() = [0]
[0]
isNatIList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
isNatIList^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [1 1] x1 + [0]
[3 3] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {2}->{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(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {1},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
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]
tt() = [0]
[0]
isNatIList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {2}->{9}: NA
-----------------
The usable rules for this path are:
{ isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {1},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {1},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 2] [0]
tt() = [0]
[0]
isNatIList(x1) = [2 0] x1 + [1]
[0 0] [0]
isNatList(x1) = [2 0] x1 + [0]
[0 0] [0]
isNat(x1) = [2 0] x1 + [1]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
length(x1) = [1 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 1] [0 0] [0]
nil() = [2]
[0]
take(x1, x2) = [2 0] x1 + [1 2] x2 + [2]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
isNatList^#(x1) = [2 1] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {2}->{9}->{1}: NA
----------------------
The usable rules for this path are:
{ isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {1},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {1},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [0]
tt() = [0]
[0]
isNatIList(x1) = [2 0] x1 + [1]
[0 0] [0]
isNatList(x1) = [2 0] x1 + [0]
[0 0] [0]
isNat(x1) = [2 0] x1 + [1]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
length(x1) = [1 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 0] [0 0] [0]
nil() = [2]
[0]
take(x1, x2) = [2 0] x1 + [2 0] x2 + [2]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {2}->{10}: NA
------------------
The usable rules for this path are:
{ isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {1},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [0]
tt() = [0]
[0]
isNatIList(x1) = [2 2] x1 + [3]
[2 2] [0]
isNatList(x1) = [2 0] x1 + [0]
[1 0] [0]
isNat(x1) = [2 0] x1 + [1]
[2 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
length(x1) = [2 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [1 1] x1 + [1 1] x2 + [2]
[0 1] [0 1] [0]
nil() = [2]
[0]
take(x1, x2) = [1 0] x1 + [2 2] x2 + [3]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
isNatList^#(x1) = [2 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {2}->{10}->{1}: NA
-----------------------
The usable rules for this path are:
{ isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {1},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 2] [0 1] [0]
tt() = [0]
[1]
isNatIList(x1) = [3 2] x1 + [2]
[3 3] [2]
isNatList(x1) = [3 0] x1 + [0]
[3 0] [1]
isNat(x1) = [2 0] x1 + [1]
[1 0] [1]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
length(x1) = [3 0] x1 + [0]
[0 0] [0]
zeros() = [1]
[0]
cons(x1, x2) = [1 0] x1 + [1 3] x2 + [1]
[0 0] [0 0] [3]
nil() = [3]
[0]
take(x1, x2) = [1 0] x1 + [1 3] x2 + [2]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4}: 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(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {1}, Uargs(c_4) = {},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
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]
tt() = [0]
[0]
isNatIList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {4}->{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(and) = {}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {1}, Uargs(c_4) = {},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
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]
tt() = [0]
[0]
isNatIList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {4}->{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(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {1}, Uargs(c_4) = {},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
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]
tt() = [0]
[0]
isNatIList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {4}->{5}->{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(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {1}, Uargs(c_4) = {1},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
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]
tt() = [0]
[0]
isNatIList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {4}->{5}->{9}: NA
----------------------
The usable rules for this path are:
{ isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {1}, Uargs(c_4) = {1},
Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 2] [0]
tt() = [0]
[0]
isNatIList(x1) = [3 0] x1 + [1]
[0 0] [0]
isNatList(x1) = [2 0] x1 + [0]
[0 0] [0]
isNat(x1) = [2 0] x1 + [1]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
length(x1) = [1 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [3]
[0 0] [0 0] [0]
nil() = [2]
[0]
take(x1, x2) = [1 0] x1 + [2 0] x2 + [2]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [2 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4}->{5}->{9}->{1}: NA
---------------------------
The usable rules for this path are:
{ isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {1}, Uargs(c_4) = {1},
Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [0]
tt() = [0]
[0]
isNatIList(x1) = [2 0] x1 + [1]
[2 0] [1]
isNatList(x1) = [2 0] x1 + [0]
[1 0] [1]
isNat(x1) = [2 0] x1 + [1]
[2 0] [1]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
length(x1) = [1 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [1 1] x2 + [3]
[0 0] [0 0] [0]
nil() = [2]
[0]
take(x1, x2) = [1 0] x1 + [2 0] x2 + [2]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4}->{5}->{10}: NA
-----------------------
The usable rules for this path are:
{ isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {1}, Uargs(c_4) = {1},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 2] [0]
tt() = [0]
[0]
isNatIList(x1) = [3 0] x1 + [1]
[0 0] [0]
isNatList(x1) = [2 0] x1 + [0]
[0 0] [0]
isNat(x1) = [2 0] x1 + [1]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
length(x1) = [2 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [1 2] x1 + [1 0] x2 + [2]
[0 0] [0 0] [0]
nil() = [2]
[0]
take(x1, x2) = [1 0] x1 + [2 0] x2 + [2]
[3 0] [3 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [1 0] x1 + [3 0] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 3] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4}->{5}->{10}->{1}: NA
----------------------------
The usable rules for this path are:
{ isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {1}, Uargs(c_4) = {1},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [0]
tt() = [0]
[0]
isNatIList(x1) = [3 2] x1 + [3]
[1 0] [0]
isNatList(x1) = [3 2] x1 + [0]
[1 0] [0]
isNat(x1) = [1 0] x1 + [1]
[1 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
length(x1) = [3 2] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[0 0] [0 1] [2]
nil() = [0]
[2]
take(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[2 0] [0 1] [3]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* 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(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
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]
tt() = [0]
[0]
isNatIList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {7}: NA
------------
The usable rules for this path are:
{ isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 0] [0 2] [0]
tt() = [0]
[0]
isNatIList(x1) = [3 2] x1 + [2]
[0 0] [0]
isNatList(x1) = [3 1] x1 + [1]
[0 0] [0]
isNat(x1) = [1 0] x1 + [2]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
length(x1) = [3 1] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [1 3] x1 + [1 0] x2 + [2]
[0 0] [0 1] [2]
nil() = [0]
[0]
take(x1, x2) = [0 0] x1 + [1 3] x2 + [2]
[2 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
isNatIList^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {7}->{1}: NA
-----------------
The usable rules for this path are:
{ isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 2] [0]
tt() = [0]
[0]
isNatIList(x1) = [3 0] x1 + [1]
[0 0] [0]
isNatList(x1) = [2 0] x1 + [0]
[0 0] [0]
isNat(x1) = [2 0] x1 + [1]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
length(x1) = [1 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [1 3] x1 + [1 2] x2 + [2]
[0 0] [0 0] [0]
nil() = [2]
[0]
take(x1, x2) = [2 0] x1 + [2 0] x2 + [2]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {11}: 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(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {1}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
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]
tt() = [0]
[0]
isNatIList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(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: {zeros^#() -> c_10(zeros^#())}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {14,15}: NA
----------------
The usable rules for this path are:
{ and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {1}, Uargs(uTake2^#) = {1},
Uargs(c_14) = {1}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [2 0] x1 + [1 0] x2 + [0]
[0 0] [0 2] [0]
tt() = [1]
[0]
isNatIList(x1) = [0 2] x1 + [2]
[0 0] [0]
isNatList(x1) = [0 2] x1 + [0]
[0 0] [0]
isNat(x1) = [0 1] x1 + [0]
[0 0] [0]
0() = [0]
[2]
s(x1) = [1 2] x1 + [0]
[0 1] [1]
length(x1) = [0 0] x1 + [0]
[0 2] [2]
zeros() = [0]
[0]
cons(x1, x2) = [1 1] x1 + [1 3] x2 + [0]
[0 1] [0 1] [1]
nil() = [0]
[2]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 1] [0 2] [2]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[3 3] [3 3] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [1 0] x1 + [0]
[0 1] [0]
uTake2^#(x1, x2, x3, x4) = [1 0] x1 + [0 0] x2 + [1 0] x3 + [1 1] x4 + [0]
[3 3] [3 3] [3 3] [3 3] [0]
c_14(x1) = [1 0] x1 + [0]
[0 1] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {14,15}->{12}: NA
----------------------
The usable rules for this path are:
{ and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {1},
Uargs(uTake1^#) = {1}, Uargs(c_13) = {1}, Uargs(uTake2^#) = {1},
Uargs(c_14) = {1}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 0] [0 2] [0]
tt() = [0]
[0]
isNatIList(x1) = [3 0] x1 + [1]
[0 0] [0]
isNatList(x1) = [2 0] x1 + [0]
[0 0] [0]
isNat(x1) = [2 0] x1 + [2]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
length(x1) = [1 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [3]
[0 0] [0 0] [0]
nil() = [2]
[0]
take(x1, x2) = [2 2] x1 + [2 0] x2 + [3]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [1 0] x1 + [0]
[0 1] [0]
uTake1^#(x1) = [1 3] x1 + [0]
[3 3] [0]
c_12() = [0]
[0]
c_13(x1) = [1 0] x1 + [0]
[0 1] [0]
uTake2^#(x1, x2, x3, x4) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1) = [1 0] x1 + [0]
[0 1] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {14,15}->{12}->{13}: NA
----------------------------
The usable rules for this path are:
{ and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {1},
Uargs(uTake1^#) = {1}, Uargs(c_13) = {1}, Uargs(uTake2^#) = {1},
Uargs(c_14) = {1}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 2] [0]
tt() = [0]
[0]
isNatIList(x1) = [1 1] x1 + [3]
[0 0] [0]
isNatList(x1) = [1 1] x1 + [2]
[0 0] [0]
isNat(x1) = [1 0] x1 + [3]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
length(x1) = [1 1] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 0] [0 1] [3]
nil() = [0]
[0]
take(x1, x2) = [2 0] x1 + [1 0] x2 + [3]
[0 0] [0 2] [3]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [1 0] x1 + [0]
[0 1] [0]
uTake1^#(x1) = [3 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [1 0] x1 + [0]
[0 1] [0]
uTake2^#(x1, x2, x3, x4) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1) = [1 0] x1 + [0]
[0 1] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {16,17}: NA
----------------
The usable rules for this path are:
{ and(tt(), T) -> T
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
, isNatIList(IL) -> isNatList(IL)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {1},
Uargs(uLength^#) = {1}, Uargs(c_16) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
tt() = [1]
[0]
isNatIList(x1) = [0 1] x1 + [1]
[0 0] [0]
isNatList(x1) = [0 1] x1 + [0]
[0 0] [0]
isNat(x1) = [0 1] x1 + [0]
[3 1] [3]
0() = [0]
[2]
s(x1) = [1 0] x1 + [0]
[0 1] [2]
length(x1) = [0 0] x1 + [1]
[0 2] [2]
zeros() = [0]
[2]
cons(x1, x2) = [1 3] x1 + [1 2] x2 + [0]
[0 1] [0 1] [1]
nil() = [0]
[2]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 1] [0 1] [2]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
length^#(x1) = [1 1] x1 + [0]
[3 3] [0]
c_15(x1) = [1 0] x1 + [0]
[0 1] [0]
uLength^#(x1, x2) = [2 0] x1 + [1 1] x2 + [0]
[3 3] [3 3] [0]
c_16(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: and^#(tt(), T) -> c_0()
, 2: isNatIList^#(IL) -> c_1(isNatList^#(IL))
, 3: isNat^#(0()) -> c_2()
, 4: isNat^#(s(N)) -> c_3(isNat^#(N))
, 5: isNat^#(length(L)) -> c_4(isNatList^#(L))
, 6: isNatIList^#(zeros()) -> c_5()
, 7: isNatIList^#(cons(N, IL)) ->
c_6(and^#(isNat(N), isNatIList(IL)))
, 8: isNatList^#(nil()) -> c_7()
, 9: isNatList^#(cons(N, L)) -> c_8(and^#(isNat(N), isNatList(L)))
, 10: isNatList^#(take(N, IL)) ->
c_9(and^#(isNat(N), isNatIList(IL)))
, 11: zeros^#() -> c_10(zeros^#())
, 12: take^#(0(), IL) -> c_11(uTake1^#(isNatIList(IL)))
, 13: uTake1^#(tt()) -> c_12()
, 14: take^#(s(M), cons(N, IL)) ->
c_13(uTake2^#(and(isNat(M), and(isNat(N), isNatIList(IL))),
M,
N,
IL))
, 15: uTake2^#(tt(), M, N, IL) -> c_14(take^#(M, IL))
, 16: length^#(cons(N, L)) ->
c_15(uLength^#(and(isNat(N), isNatList(L)), L))
, 17: uLength^#(tt(), L) -> c_16(length^#(L))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{16,17} [ NA ]
->{14,15} [ NA ]
|
`->{12} [ NA ]
|
`->{13} [ NA ]
->{11} [ MAYBE ]
->{7} [ NA ]
|
`->{1} [ NA ]
->{6} [ NA ]
->{4} [ NA ]
|
|->{3} [ NA ]
|
`->{5} [ NA ]
|
|->{8} [ NA ]
|
|->{9} [ NA ]
| |
| `->{1} [ NA ]
|
`->{10} [ NA ]
|
`->{1} [ NA ]
->{2} [ NA ]
|
|->{8} [ NA ]
|
|->{9} [ NA ]
| |
| `->{1} [ NA ]
|
`->{10} [ NA ]
|
`->{1} [ 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(and) = {}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNatIList(x1) = [0] x1 + [0]
isNatList(x1) = [0] x1 + [0]
isNat(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
zeros() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
isNatIList^#(x1) = [3] x1 + [0]
c_1(x1) = [3] x1 + [0]
isNatList^#(x1) = [1] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {2}->{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(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {1},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNatIList(x1) = [0] x1 + [0]
isNatList(x1) = [0] x1 + [0]
isNat(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
zeros() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {2}->{9}: NA
-----------------
The usable rules for this path are:
{ isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {1},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {1},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [2]
isNatIList(x1) = [1] x1 + [1]
isNatList(x1) = [1] x1 + [0]
isNat(x1) = [1] x1 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
length(x1) = [3] x1 + [3]
zeros() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
nil() = [3]
take(x1, x2) = [2] x1 + [3] x2 + [2]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
isNatList^#(x1) = [3] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {2}->{9}->{1}: NA
----------------------
The usable rules for this path are:
{ isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {1},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {1},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [2] x1 + [1] x2 + [2]
tt() = [0]
isNatIList(x1) = [2] x1 + [2]
isNatList(x1) = [2] x1 + [1]
isNat(x1) = [1] x1 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
length(x1) = [3] x1 + [3]
zeros() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
nil() = [3]
take(x1, x2) = [3] x1 + [3] x2 + [2]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {2}->{10}: NA
------------------
The usable rules for this path are:
{ isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {1},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [2]
isNatIList(x1) = [2] x1 + [1]
isNatList(x1) = [2] x1 + [0]
isNat(x1) = [2] x1 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
length(x1) = [2] x1 + [3]
zeros() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
nil() = [3]
take(x1, x2) = [3] x1 + [3] x2 + [2]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [3] x1 + [2] x2 + [0]
c_0() = [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
isNatList^#(x1) = [3] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {2}->{10}->{1}: NA
-----------------------
The usable rules for this path are:
{ isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {1},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [1]
isNatIList(x1) = [1] x1 + [3]
isNatList(x1) = [1] x1 + [2]
isNat(x1) = [1] x1 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
length(x1) = [3] x1 + [3]
zeros() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
nil() = [3]
take(x1, x2) = [3] x1 + [1] x2 + [2]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4}: 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(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {1}, Uargs(c_4) = {},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNatIList(x1) = [0] x1 + [0]
isNatList(x1) = [0] x1 + [0]
isNat(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
length(x1) = [0] x1 + [0]
zeros() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [3] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {4}->{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(and) = {}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {1}, Uargs(c_4) = {},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNatIList(x1) = [0] x1 + [0]
isNatList(x1) = [0] x1 + [0]
isNat(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
zeros() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {4}->{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(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {1}, Uargs(c_4) = {},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNatIList(x1) = [0] x1 + [0]
isNatList(x1) = [0] x1 + [0]
isNat(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
zeros() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {4}->{5}->{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(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {1}, Uargs(c_4) = {1},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNatIList(x1) = [0] x1 + [0]
isNatList(x1) = [0] x1 + [0]
isNat(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
zeros() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {4}->{5}->{9}: NA
----------------------
The usable rules for this path are:
{ isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {1}, Uargs(c_4) = {1},
Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [2]
isNatIList(x1) = [1] x1 + [1]
isNatList(x1) = [1] x1 + [0]
isNat(x1) = [1] x1 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
length(x1) = [3] x1 + [3]
zeros() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [1]
nil() = [3]
take(x1, x2) = [3] x1 + [2] x2 + [2]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [3] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4}->{5}->{9}->{1}: NA
---------------------------
The usable rules for this path are:
{ isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {1}, Uargs(c_4) = {1},
Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [2]
isNatIList(x1) = [2] x1 + [1]
isNatList(x1) = [2] x1 + [0]
isNat(x1) = [2] x1 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
length(x1) = [2] x1 + [3]
zeros() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [1]
nil() = [3]
take(x1, x2) = [1] x1 + [1] x2 + [2]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4}->{5}->{10}: NA
-----------------------
The usable rules for this path are:
{ isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {1}, Uargs(c_4) = {1},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1] x1 + [1] x2 + [1]
tt() = [0]
isNatIList(x1) = [2] x1 + [3]
isNatList(x1) = [2] x1 + [2]
isNat(x1) = [2] x1 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
length(x1) = [2] x1 + [3]
zeros() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
nil() = [3]
take(x1, x2) = [2] x1 + [1] x2 + [3]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_0() = [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [2] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4}->{5}->{10}->{1}: NA
----------------------------
The usable rules for this path are:
{ isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {1}, Uargs(c_4) = {1},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [2] x1 + [1] x2 + [0]
tt() = [2]
isNatIList(x1) = [2] x1 + [1]
isNatList(x1) = [2] x1 + [0]
isNat(x1) = [1] x1 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
length(x1) = [3] x1 + [3]
zeros() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
nil() = [3]
take(x1, x2) = [1] x1 + [3] x2 + [2]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* 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(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNatIList(x1) = [0] x1 + [0]
isNatList(x1) = [0] x1 + [0]
isNat(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
zeros() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {7}: NA
------------
The usable rules for this path are:
{ isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [2]
isNatIList(x1) = [2] x1 + [1]
isNatList(x1) = [2] x1 + [0]
isNat(x1) = [2] x1 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
length(x1) = [2] x1 + [3]
zeros() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
nil() = [3]
take(x1, x2) = [3] x1 + [3] x2 + [1]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_0() = [0]
isNatIList^#(x1) = [3] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {7}->{1}: NA
-----------------
The usable rules for this path are:
{ isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [1]
isNatIList(x1) = [2] x1 + [1]
isNatList(x1) = [2] x1 + [0]
isNat(x1) = [1] x1 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
length(x1) = [3] x1 + [3]
zeros() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
nil() = [3]
take(x1, x2) = [2] x1 + [1] x2 + [1]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {11}: 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(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {1}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNatIList(x1) = [0] x1 + [0]
isNatList(x1) = [0] x1 + [0]
isNat(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
zeros() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [1] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(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: {zeros^#() -> c_10(zeros^#())}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {14,15}: NA
----------------
The usable rules for this path are:
{ and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {14,15}->{12}: NA
----------------------
The usable rules for this path are:
{ and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {1},
Uargs(uTake1^#) = {1}, Uargs(c_13) = {1}, Uargs(uTake2^#) = {1},
Uargs(c_14) = {1}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [2] x1 + [1] x2 + [0]
tt() = [2]
isNatIList(x1) = [2] x1 + [1]
isNatList(x1) = [2] x1 + [0]
isNat(x1) = [1] x1 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
length(x1) = [3] x1 + [3]
zeros() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
nil() = [3]
take(x1, x2) = [3] x1 + [3] x2 + [2]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_11(x1) = [1] x1 + [0]
uTake1^#(x1) = [1] x1 + [0]
c_12() = [0]
c_13(x1) = [1] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [3] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1) = [1] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {14,15}->{12}->{13}: NA
----------------------------
The usable rules for this path are:
{ and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(isNatIList^#) = {}, Uargs(c_1) = {}, Uargs(isNatList^#) = {},
Uargs(isNat^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {1},
Uargs(uTake1^#) = {1}, Uargs(c_13) = {1}, Uargs(uTake2^#) = {1},
Uargs(c_14) = {1}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [2]
isNatIList(x1) = [2] x1 + [1]
isNatList(x1) = [2] x1 + [0]
isNat(x1) = [1] x1 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
length(x1) = [3] x1 + [3]
zeros() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
nil() = [3]
take(x1, x2) = [1] x1 + [3] x2 + [2]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [1] x1 + [0]
uTake1^#(x1) = [3] x1 + [0]
c_12() = [0]
c_13(x1) = [1] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [3] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1) = [1] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {16,17}: NA
----------------
The usable rules for this path are:
{ and(tt(), T) -> T
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
, isNatIList(IL) -> isNatList(IL)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) '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(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
, zeros() -> cons(0(), zeros())
, take(0(), IL) -> uTake1(isNatIList(IL))
, uTake1(tt()) -> nil()
, take(s(M), cons(N, IL)) ->
uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)
, uTake2(tt(), M, N, IL) -> cons(N, take(M, IL))
, length(cons(N, L)) -> uLength(and(isNat(N), isNatList(L)), L)
, uLength(tt(), 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: and^#(tt(), T) -> c_0(T)
, 2: isNatIList^#(IL) -> c_1(isNatList^#(IL))
, 3: isNat^#(0()) -> c_2()
, 4: isNat^#(s(N)) -> c_3(isNat^#(N))
, 5: isNat^#(length(L)) -> c_4(isNatList^#(L))
, 6: isNatIList^#(zeros()) -> c_5()
, 7: isNatIList^#(cons(N, IL)) ->
c_6(and^#(isNat(N), isNatIList(IL)))
, 8: isNatList^#(nil()) -> c_7()
, 9: isNatList^#(cons(N, L)) -> c_8(and^#(isNat(N), isNatList(L)))
, 10: isNatList^#(take(N, IL)) ->
c_9(and^#(isNat(N), isNatIList(IL)))
, 11: zeros^#() -> c_10(zeros^#())
, 12: take^#(0(), IL) -> c_11(uTake1^#(isNatIList(IL)))
, 13: uTake1^#(tt()) -> c_12()
, 14: take^#(s(M), cons(N, IL)) ->
c_13(uTake2^#(and(isNat(M), and(isNat(N), isNatIList(IL))),
M,
N,
IL))
, 15: uTake2^#(tt(), M, N, IL) -> c_14(N, take^#(M, IL))
, 16: length^#(cons(N, L)) ->
c_15(uLength^#(and(isNat(N), isNatList(L)), L))
, 17: uLength^#(tt(), L) -> c_16(length^#(L))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{16,17} [ NA ]
->{14,15} [ NA ]
|
`->{12} [ NA ]
|
`->{13} [ NA ]
->{11} [ MAYBE ]
->{7} [ NA ]
|
`->{1} [ NA ]
->{6} [ NA ]
->{4} [ NA ]
|
|->{3} [ NA ]
|
`->{5} [ NA ]
|
|->{8} [ NA ]
|
|->{9} [ NA ]
| |
| `->{1} [ NA ]
|
`->{10} [ NA ]
|
`->{1} [ NA ]
->{2} [ NA ]
|
|->{8} [ NA ]
|
|->{9} [ NA ]
| |
| `->{1} [ NA ]
|
`->{10} [ NA ]
|
`->{1} [ 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(and) = {}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
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]
tt() = [0]
[0]
isNatIList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 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]
isNatIList^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [1 1] x1 + [0]
[3 3] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {2}->{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(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {1},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
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]
tt() = [0]
[0]
isNatIList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 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]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {2}->{9}: NA
-----------------
The usable rules for this path are:
{ isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {1},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {1},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 2] [0]
tt() = [0]
[0]
isNatIList(x1) = [2 0] x1 + [1]
[0 0] [0]
isNatList(x1) = [2 0] x1 + [0]
[0 0] [0]
isNat(x1) = [2 0] x1 + [1]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
length(x1) = [1 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 1] [0 0] [0]
nil() = [2]
[0]
take(x1, x2) = [2 0] x1 + [1 2] x2 + [2]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
isNatList^#(x1) = [2 1] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {2}->{9}->{1}: NA
----------------------
The usable rules for this path are:
{ isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(c_0) = {1}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {1},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {1},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 2] [0]
tt() = [0]
[0]
isNatIList(x1) = [1 2] x1 + [2]
[0 0] [0]
isNatList(x1) = [1 1] x1 + [1]
[0 0] [0]
isNat(x1) = [1 0] x1 + [2]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
length(x1) = [1 1] x1 + [0]
[0 0] [0]
zeros() = [2]
[0]
cons(x1, x2) = [1 0] x1 + [1 2] x2 + [2]
[0 0] [0 1] [2]
nil() = [0]
[0]
take(x1, x2) = [2 0] x1 + [0 3] x2 + [3]
[2 0] [1 1] [3]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [3 0] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {2}->{10}: NA
------------------
The usable rules for this path are:
{ isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {1},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [0]
tt() = [0]
[0]
isNatIList(x1) = [2 2] x1 + [3]
[2 2] [0]
isNatList(x1) = [2 0] x1 + [0]
[1 0] [0]
isNat(x1) = [2 0] x1 + [1]
[2 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
length(x1) = [2 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [1 1] x1 + [1 1] x2 + [2]
[0 1] [0 1] [0]
nil() = [2]
[0]
take(x1, x2) = [1 0] x1 + [2 2] x2 + [3]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[3 3] [3 3] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
isNatList^#(x1) = [2 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {2}->{10}->{1}: NA
-----------------------
The usable rules for this path are:
{ isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(c_0) = {1}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {1},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 0] [0 1] [0]
tt() = [0]
[0]
isNatIList(x1) = [3 2] x1 + [2]
[0 1] [0]
isNatList(x1) = [3 1] x1 + [0]
[0 1] [0]
isNat(x1) = [1 0] x1 + [1]
[2 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
length(x1) = [3 1] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 0] [0 1] [2]
nil() = [0]
[2]
take(x1, x2) = [0 0] x1 + [1 2] x2 + [2]
[2 0] [0 1] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [3 0] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4}: 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(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
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]
tt() = [0]
[0]
isNatIList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 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]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {4}->{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(and) = {}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
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]
tt() = [0]
[0]
isNatIList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 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]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {4}->{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(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
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]
tt() = [0]
[0]
isNatIList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 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]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {4}->{5}->{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(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1},
Uargs(c_4) = {1}, Uargs(c_6) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
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]
tt() = [0]
[0]
isNatIList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 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]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {4}->{5}->{9}: NA
----------------------
The usable rules for this path are:
{ isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1},
Uargs(c_4) = {1}, Uargs(c_6) = {}, Uargs(c_8) = {1},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 2] [0]
tt() = [0]
[0]
isNatIList(x1) = [3 0] x1 + [1]
[0 0] [0]
isNatList(x1) = [2 0] x1 + [0]
[0 0] [0]
isNat(x1) = [2 0] x1 + [1]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
length(x1) = [1 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [3]
[0 0] [0 0] [0]
nil() = [2]
[0]
take(x1, x2) = [1 0] x1 + [2 0] x2 + [2]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [2 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4}->{5}->{9}->{1}: NA
---------------------------
The usable rules for this path are:
{ isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(c_0) = {1}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1},
Uargs(c_4) = {1}, Uargs(c_6) = {}, Uargs(c_8) = {1},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [0]
tt() = [0]
[0]
isNatIList(x1) = [1 2] x1 + [1]
[2 0] [0]
isNatList(x1) = [1 2] x1 + [0]
[2 0] [0]
isNat(x1) = [1 0] x1 + [1]
[2 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
length(x1) = [1 2] x1 + [1]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [1 2] x1 + [1 0] x2 + [2]
[0 1] [0 1] [3]
nil() = [0]
[2]
take(x1, x2) = [0 0] x1 + [1 0] x2 + [2]
[1 0] [0 2] [3]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [3 0] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {4}->{5}->{10}: NA
-----------------------
The usable rules for this path are:
{ isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1},
Uargs(c_4) = {1}, Uargs(c_6) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 2] [0]
tt() = [0]
[0]
isNatIList(x1) = [3 0] x1 + [1]
[0 0] [0]
isNatList(x1) = [2 0] x1 + [0]
[0 0] [0]
isNat(x1) = [2 0] x1 + [1]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
length(x1) = [2 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [1 2] x1 + [1 0] x2 + [2]
[0 0] [0 0] [0]
nil() = [2]
[0]
take(x1, x2) = [1 0] x1 + [2 0] x2 + [2]
[3 0] [3 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [1 0] x1 + [3 0] x2 + [0]
[3 3] [3 3] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 3] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4}->{5}->{10}->{1}: NA
----------------------------
The usable rules for this path are:
{ isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(c_0) = {1}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1},
Uargs(c_4) = {1}, Uargs(c_6) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [3 2] x1 + [1 3] x2 + [0]
[0 0] [0 1] [0]
tt() = [0]
[1]
isNatIList(x1) = [3 3] x1 + [3]
[0 0] [1]
isNatList(x1) = [3 3] x1 + [0]
[0 0] [1]
isNat(x1) = [1 0] x1 + [1]
[0 0] [1]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
length(x1) = [3 3] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [1 3] x2 + [3]
[0 0] [0 0] [3]
nil() = [3]
[0]
take(x1, x2) = [3 3] x1 + [0 0] x2 + [1]
[0 0] [3 3] [3]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [3 0] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* 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(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
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]
tt() = [0]
[0]
isNatIList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 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]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {7}: NA
------------
The usable rules for this path are:
{ isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {1}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 0] [0 2] [0]
tt() = [0]
[0]
isNatIList(x1) = [3 2] x1 + [2]
[0 0] [0]
isNatList(x1) = [3 1] x1 + [1]
[0 0] [0]
isNat(x1) = [1 0] x1 + [2]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
length(x1) = [3 1] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [1 3] x1 + [1 0] x2 + [2]
[0 0] [0 1] [2]
nil() = [0]
[0]
take(x1, x2) = [0 0] x1 + [1 3] x2 + [2]
[2 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatIList^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {7}->{1}: NA
-----------------
The usable rules for this path are:
{ isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(c_0) = {1}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {1}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [2 0] x1 + [1 0] x2 + [2]
[0 0] [0 1] [0]
tt() = [0]
[0]
isNatIList(x1) = [3 2] x1 + [1]
[2 2] [0]
isNatList(x1) = [3 1] x1 + [0]
[2 2] [0]
isNat(x1) = [1 0] x1 + [1]
[3 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
length(x1) = [3 2] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [1 2] x2 + [2]
[0 0] [0 0] [2]
nil() = [0]
[2]
take(x1, x2) = [2 0] x1 + [0 3] x2 + [2]
[0 0] [3 2] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [3 0] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {11}: 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(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {1}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
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]
tt() = [0]
[0]
isNatIList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 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]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(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: {zeros^#() -> c_10(zeros^#())}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {14,15}: NA
----------------
The usable rules for this path are:
{ and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {1}, Uargs(uTake2^#) = {1},
Uargs(c_14) = {2}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 2] [0]
tt() = [0]
[0]
isNatIList(x1) = [0 1] x1 + [1]
[0 0] [0]
isNatList(x1) = [0 1] x1 + [0]
[0 0] [0]
isNat(x1) = [0 1] x1 + [0]
[2 0] [0]
0() = [0]
[2]
s(x1) = [1 2] x1 + [0]
[0 1] [1]
length(x1) = [0 0] x1 + [2]
[0 2] [1]
zeros() = [0]
[0]
cons(x1, x2) = [1 3] x1 + [1 3] x2 + [0]
[0 1] [0 1] [2]
nil() = [0]
[2]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 1] [0 1] [3]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 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]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[3 3] [3 3] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [1 0] x1 + [0]
[0 1] [0]
uTake2^#(x1, x2, x3, x4) = [2 0] x1 + [1 0] x2 + [0 0] x3 + [1 2] x4 + [0]
[3 3] [3 3] [3 3] [3 3] [0]
c_14(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {14,15}->{12}: NA
----------------------
The usable rules for this path are:
{ and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {1},
Uargs(uTake1^#) = {1}, Uargs(c_13) = {1}, Uargs(uTake2^#) = {1},
Uargs(c_14) = {2}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 0] [0 2] [0]
tt() = [0]
[0]
isNatIList(x1) = [3 0] x1 + [1]
[0 0] [0]
isNatList(x1) = [2 0] x1 + [0]
[0 0] [0]
isNat(x1) = [2 0] x1 + [2]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
length(x1) = [1 0] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [3]
[0 0] [0 0] [0]
nil() = [2]
[0]
take(x1, x2) = [2 2] x1 + [2 0] x2 + [3]
[0 0] [0 0] [0]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 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]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [1 0] x1 + [0]
[0 1] [0]
uTake1^#(x1) = [1 3] x1 + [0]
[3 3] [0]
c_12() = [0]
[0]
c_13(x1) = [1 0] x1 + [0]
[0 1] [0]
uTake2^#(x1, x2, x3, x4) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {14,15}->{12}->{13}: NA
----------------------------
The usable rules for this path are:
{ and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {1},
Uargs(uTake1^#) = {1}, Uargs(c_13) = {1}, Uargs(uTake2^#) = {1},
Uargs(c_14) = {2}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 2] [0]
tt() = [0]
[0]
isNatIList(x1) = [1 1] x1 + [3]
[0 0] [0]
isNatList(x1) = [1 1] x1 + [2]
[0 0] [0]
isNat(x1) = [1 0] x1 + [3]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
length(x1) = [1 1] x1 + [0]
[0 0] [0]
zeros() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 0] [0 1] [3]
nil() = [0]
[0]
take(x1, x2) = [2 0] x1 + [1 0] x2 + [3]
[0 0] [0 2] [3]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 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]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [1 0] x1 + [0]
[0 1] [0]
uTake1^#(x1) = [3 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [1 0] x1 + [0]
[0 1] [0]
uTake2^#(x1, x2, x3, x4) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
uLength^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {16,17}: NA
----------------
The usable rules for this path are:
{ and(tt(), T) -> T
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
, isNatIList(IL) -> isNatList(IL)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {1},
Uargs(uLength^#) = {1}, Uargs(c_16) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
tt() = [1]
[0]
isNatIList(x1) = [0 1] x1 + [1]
[0 0] [0]
isNatList(x1) = [0 1] x1 + [0]
[0 0] [0]
isNat(x1) = [0 1] x1 + [0]
[3 1] [3]
0() = [0]
[2]
s(x1) = [1 0] x1 + [0]
[0 1] [2]
length(x1) = [0 0] x1 + [1]
[0 2] [2]
zeros() = [0]
[2]
cons(x1, x2) = [1 3] x1 + [1 2] x2 + [0]
[0 1] [0 1] [1]
nil() = [0]
[2]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 1] [0 1] [2]
uTake1(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
uLength(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 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]
isNatIList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNatList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
zeros^#() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
uTake2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_14(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [1 1] x1 + [0]
[3 3] [0]
c_15(x1) = [1 0] x1 + [0]
[0 1] [0]
uLength^#(x1, x2) = [2 0] x1 + [1 1] x2 + [0]
[3 3] [3 3] [0]
c_16(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: and^#(tt(), T) -> c_0(T)
, 2: isNatIList^#(IL) -> c_1(isNatList^#(IL))
, 3: isNat^#(0()) -> c_2()
, 4: isNat^#(s(N)) -> c_3(isNat^#(N))
, 5: isNat^#(length(L)) -> c_4(isNatList^#(L))
, 6: isNatIList^#(zeros()) -> c_5()
, 7: isNatIList^#(cons(N, IL)) ->
c_6(and^#(isNat(N), isNatIList(IL)))
, 8: isNatList^#(nil()) -> c_7()
, 9: isNatList^#(cons(N, L)) -> c_8(and^#(isNat(N), isNatList(L)))
, 10: isNatList^#(take(N, IL)) ->
c_9(and^#(isNat(N), isNatIList(IL)))
, 11: zeros^#() -> c_10(zeros^#())
, 12: take^#(0(), IL) -> c_11(uTake1^#(isNatIList(IL)))
, 13: uTake1^#(tt()) -> c_12()
, 14: take^#(s(M), cons(N, IL)) ->
c_13(uTake2^#(and(isNat(M), and(isNat(N), isNatIList(IL))),
M,
N,
IL))
, 15: uTake2^#(tt(), M, N, IL) -> c_14(N, take^#(M, IL))
, 16: length^#(cons(N, L)) ->
c_15(uLength^#(and(isNat(N), isNatList(L)), L))
, 17: uLength^#(tt(), L) -> c_16(length^#(L))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{16,17} [ NA ]
->{14,15} [ NA ]
|
`->{12} [ NA ]
|
`->{13} [ NA ]
->{11} [ MAYBE ]
->{7} [ NA ]
|
`->{1} [ NA ]
->{6} [ NA ]
->{4} [ NA ]
|
|->{3} [ NA ]
|
`->{5} [ NA ]
|
|->{8} [ NA ]
|
|->{9} [ NA ]
| |
| `->{1} [ NA ]
|
`->{10} [ NA ]
|
`->{1} [ NA ]
->{2} [ NA ]
|
|->{8} [ NA ]
|
|->{9} [ NA ]
| |
| `->{1} [ NA ]
|
`->{10} [ NA ]
|
`->{1} [ 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(and) = {}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNatIList(x1) = [0] x1 + [0]
isNatList(x1) = [0] x1 + [0]
isNat(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
zeros() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
isNatIList^#(x1) = [3] x1 + [0]
c_1(x1) = [3] x1 + [0]
isNatList^#(x1) = [1] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {2}->{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(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {1},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNatIList(x1) = [0] x1 + [0]
isNatList(x1) = [0] x1 + [0]
isNat(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
zeros() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {2}->{9}: NA
-----------------
The usable rules for this path are:
{ isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {1},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {1},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [2]
isNatIList(x1) = [1] x1 + [1]
isNatList(x1) = [1] x1 + [0]
isNat(x1) = [1] x1 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
length(x1) = [3] x1 + [3]
zeros() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
nil() = [3]
take(x1, x2) = [2] x1 + [3] x2 + [2]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0(x1) = [0] x1 + [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
isNatList^#(x1) = [3] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {2}->{9}->{1}: NA
----------------------
The usable rules for this path are:
{ isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(c_0) = {1}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {1},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {1},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1] x1 + [1] x2 + [1]
tt() = [0]
isNatIList(x1) = [2] x1 + [2]
isNatList(x1) = [2] x1 + [0]
isNat(x1) = [2] x1 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
length(x1) = [2] x1 + [3]
zeros() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [1]
nil() = [3]
take(x1, x2) = [1] x1 + [1] x2 + [2]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0(x1) = [1] x1 + [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {2}->{10}: NA
------------------
The usable rules for this path are:
{ isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {1},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [2]
isNatIList(x1) = [2] x1 + [1]
isNatList(x1) = [2] x1 + [0]
isNat(x1) = [2] x1 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
length(x1) = [2] x1 + [3]
zeros() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
nil() = [3]
take(x1, x2) = [3] x1 + [3] x2 + [2]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [3] x1 + [2] x2 + [0]
c_0(x1) = [0] x1 + [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
isNatList^#(x1) = [3] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {2}->{10}->{1}: NA
-----------------------
The usable rules for this path are:
{ isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(c_0) = {1}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {1},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1] x1 + [1] x2 + [1]
tt() = [0]
isNatIList(x1) = [2] x1 + [2]
isNatList(x1) = [2] x1 + [0]
isNat(x1) = [2] x1 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
length(x1) = [2] x1 + [3]
zeros() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [1]
nil() = [3]
take(x1, x2) = [3] x1 + [3] x2 + [2]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0(x1) = [1] x1 + [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4}: 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(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNatIList(x1) = [0] x1 + [0]
isNatList(x1) = [0] x1 + [0]
isNat(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
length(x1) = [0] x1 + [0]
zeros() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [3] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {4}->{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(and) = {}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNatIList(x1) = [0] x1 + [0]
isNatList(x1) = [0] x1 + [0]
isNat(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
zeros() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {4}->{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(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNatIList(x1) = [0] x1 + [0]
isNatList(x1) = [0] x1 + [0]
isNat(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
zeros() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {4}->{5}->{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(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1},
Uargs(c_4) = {1}, Uargs(c_6) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNatIList(x1) = [0] x1 + [0]
isNatList(x1) = [0] x1 + [0]
isNat(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
zeros() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {4}->{5}->{9}: NA
----------------------
The usable rules for this path are:
{ isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1},
Uargs(c_4) = {1}, Uargs(c_6) = {}, Uargs(c_8) = {1},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [2]
isNatIList(x1) = [1] x1 + [1]
isNatList(x1) = [1] x1 + [0]
isNat(x1) = [1] x1 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
length(x1) = [3] x1 + [3]
zeros() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [1]
nil() = [3]
take(x1, x2) = [3] x1 + [2] x2 + [2]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0(x1) = [0] x1 + [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [3] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4}->{5}->{9}->{1}: NA
---------------------------
The usable rules for this path are:
{ isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(c_0) = {1}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1},
Uargs(c_4) = {1}, Uargs(c_6) = {}, Uargs(c_8) = {1},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1] x1 + [1] x2 + [2]
tt() = [2]
isNatIList(x1) = [2] x1 + [2]
isNatList(x1) = [2] x1 + [1]
isNat(x1) = [2] x1 + [1]
0() = [3]
s(x1) = [1] x1 + [3]
length(x1) = [2] x1 + [3]
zeros() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [3]
nil() = [3]
take(x1, x2) = [3] x1 + [3] x2 + [3]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0(x1) = [1] x1 + [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4}->{5}->{10}: NA
-----------------------
The usable rules for this path are:
{ isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1},
Uargs(c_4) = {1}, Uargs(c_6) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1] x1 + [1] x2 + [1]
tt() = [0]
isNatIList(x1) = [2] x1 + [3]
isNatList(x1) = [2] x1 + [2]
isNat(x1) = [2] x1 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
length(x1) = [2] x1 + [3]
zeros() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
nil() = [3]
take(x1, x2) = [2] x1 + [1] x2 + [3]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_0(x1) = [0] x1 + [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [2] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4}->{5}->{10}->{1}: NA
----------------------------
The usable rules for this path are:
{ isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(c_0) = {1}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1},
Uargs(c_4) = {1}, Uargs(c_6) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [2]
isNatIList(x1) = [2] x1 + [1]
isNatList(x1) = [2] x1 + [0]
isNat(x1) = [2] x1 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
length(x1) = [2] x1 + [3]
zeros() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [1]
nil() = [3]
take(x1, x2) = [2] x1 + [2] x2 + [2]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0(x1) = [1] x1 + [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* 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(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNatIList(x1) = [0] x1 + [0]
isNatList(x1) = [0] x1 + [0]
isNat(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
zeros() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {7}: NA
------------
The usable rules for this path are:
{ isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {1}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [2]
isNatIList(x1) = [2] x1 + [1]
isNatList(x1) = [2] x1 + [0]
isNat(x1) = [2] x1 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
length(x1) = [2] x1 + [3]
zeros() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
nil() = [3]
take(x1, x2) = [3] x1 + [3] x2 + [1]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_0(x1) = [0] x1 + [0]
isNatIList^#(x1) = [3] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {7}->{1}: NA
-----------------
The usable rules for this path are:
{ isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, and(tt(), T) -> T
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {1, 2},
Uargs(c_0) = {1}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {1}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(take^#) = {},
Uargs(c_11) = {}, Uargs(uTake1^#) = {}, Uargs(c_13) = {},
Uargs(uTake2^#) = {}, Uargs(c_14) = {}, Uargs(length^#) = {},
Uargs(c_15) = {}, Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [1]
isNatIList(x1) = [2] x1 + [1]
isNatList(x1) = [2] x1 + [0]
isNat(x1) = [1] x1 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
length(x1) = [3] x1 + [3]
zeros() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [1]
nil() = [3]
take(x1, x2) = [1] x1 + [1] x2 + [2]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0(x1) = [1] x1 + [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {11}: 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(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {1}, Uargs(take^#) = {}, Uargs(c_11) = {},
Uargs(uTake1^#) = {}, Uargs(c_13) = {}, Uargs(uTake2^#) = {},
Uargs(c_14) = {}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNatIList(x1) = [0] x1 + [0]
isNatList(x1) = [0] x1 + [0]
isNat(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
zeros() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [1] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
uTake1^#(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(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: {zeros^#() -> c_10(zeros^#())}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {14,15}: NA
----------------
The usable rules for this path are:
{ and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {14,15}->{12}: NA
----------------------
The usable rules for this path are:
{ and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {1},
Uargs(uTake1^#) = {1}, Uargs(c_13) = {1}, Uargs(uTake2^#) = {1},
Uargs(c_14) = {2}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [2] x1 + [1] x2 + [0]
tt() = [2]
isNatIList(x1) = [2] x1 + [1]
isNatList(x1) = [2] x1 + [0]
isNat(x1) = [1] x1 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
length(x1) = [3] x1 + [3]
zeros() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
nil() = [3]
take(x1, x2) = [3] x1 + [3] x2 + [2]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_11(x1) = [1] x1 + [0]
uTake1^#(x1) = [1] x1 + [0]
c_12() = [0]
c_13(x1) = [1] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [3] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1, x2) = [0] x1 + [1] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {14,15}->{12}->{13}: NA
----------------------------
The usable rules for this path are:
{ and(tt(), T) -> T
, isNatIList(IL) -> isNatList(IL)
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(isNatIList) = {}, Uargs(isNatList) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(length) = {},
Uargs(cons) = {}, Uargs(take) = {}, Uargs(uTake1) = {},
Uargs(uTake2) = {}, Uargs(uLength) = {}, Uargs(and^#) = {},
Uargs(c_0) = {}, Uargs(isNatIList^#) = {}, Uargs(c_1) = {},
Uargs(isNatList^#) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(take^#) = {}, Uargs(c_11) = {1},
Uargs(uTake1^#) = {1}, Uargs(c_13) = {1}, Uargs(uTake2^#) = {1},
Uargs(c_14) = {2}, Uargs(length^#) = {}, Uargs(c_15) = {},
Uargs(uLength^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [2]
isNatIList(x1) = [2] x1 + [1]
isNatList(x1) = [2] x1 + [0]
isNat(x1) = [1] x1 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
length(x1) = [3] x1 + [3]
zeros() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
nil() = [3]
take(x1, x2) = [1] x1 + [3] x2 + [2]
uTake1(x1) = [0] x1 + [0]
uTake2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
uLength(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
isNatIList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
isNatList^#(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
zeros^#() = [0]
c_10(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [1] x1 + [0]
uTake1^#(x1) = [3] x1 + [0]
c_12() = [0]
c_13(x1) = [1] x1 + [0]
uTake2^#(x1, x2, x3, x4) = [3] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_14(x1, x2) = [0] x1 + [1] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
uLength^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_16(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {16,17}: NA
----------------
The usable rules for this path are:
{ and(tt(), T) -> T
, isNat(0()) -> tt()
, isNat(s(N)) -> isNat(N)
, isNat(length(L)) -> isNatList(L)
, isNatList(nil()) -> tt()
, isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
, isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
, isNatIList(IL) -> isNatList(IL)
, isNatIList(zeros()) -> tt()
, isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.