Tool CaT
stdout:
MAYBE
Problem:
add(true(),x,xs) -> add(and(isNat(x),isList(xs)),x,Cons(x,xs))
isList(Cons(x,xs)) -> isList(xs)
isList(nil()) -> true()
isNat(s(x)) -> isNat(x)
isNat(0()) -> true()
if(true(),x,y) -> x
if(false(),x,y) -> y
and(true(),true()) -> true()
and(false(),x) -> false()
and(x,false()) -> false()
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ add(true(), x, xs) ->
add(and(isNat(x), isList(xs)), x, Cons(x, xs))
, isList(Cons(x, xs)) -> isList(xs)
, isList(nil()) -> true()
, isNat(s(x)) -> isNat(x)
, isNat(0()) -> true()
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, and(true(), true()) -> true()
, and(false(), x) -> false()
, and(x, false()) -> false()}
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: add^#(true(), x, xs) ->
c_0(add^#(and(isNat(x), isList(xs)), x, Cons(x, xs)))
, 2: isList^#(Cons(x, xs)) -> c_1(isList^#(xs))
, 3: isList^#(nil()) -> c_2()
, 4: isNat^#(s(x)) -> c_3(isNat^#(x))
, 5: isNat^#(0()) -> c_4()
, 6: if^#(true(), x, y) -> c_5()
, 7: if^#(false(), x, y) -> c_6()
, 8: and^#(true(), true()) -> c_7()
, 9: and^#(false(), x) -> c_8()
, 10: and^#(x, false()) -> c_9()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ YES(?,O(1)) ]
->{9} [ YES(?,O(1)) ]
->{8} [ YES(?,O(1)) ]
->{7} [ YES(?,O(1)) ]
->{6} [ YES(?,O(1)) ]
->{4} [ YES(?,O(n^2)) ]
|
`->{5} [ YES(?,O(n^2)) ]
->{2} [ YES(?,O(n^2)) ]
|
`->{3} [ YES(?,O(n^2)) ]
->{1} [ NA ]
Sub-problems:
-------------
* Path {1}: NA
------------
The usable rules for this path are:
{ isList(Cons(x, xs)) -> isList(xs)
, isList(nil()) -> true()
, isNat(s(x)) -> isNat(x)
, isNat(0()) -> true()
, and(true(), true()) -> true()
, and(false(), x) -> false()
, and(x, false()) -> false()}
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 {2}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {1}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
isNat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Cons(x1, x2) = [1 3 0] x1 + [0 0 0] x2 + [0]
[0 1 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
nil() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
add^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_2() = [0]
[0]
[0]
isNat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isList^#(Cons(x, xs)) -> c_1(isList^#(xs))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(Cons) = {}, Uargs(isList^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
Cons(x1, x2) = [0 0 0] x1 + [1 2 2] x2 + [2]
[0 0 0] [0 1 2] [2]
[0 0 0] [0 0 0] [0]
isList^#(x1) = [0 1 0] x1 + [2]
[6 0 0] [0]
[2 3 0] [2]
c_1(x1) = [1 0 0] x1 + [1]
[2 0 2] [0]
[0 0 0] [0]
* Path {2}->{3}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {1}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
isNat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
nil() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
add^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_2() = [0]
[0]
[0]
isNat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isList^#(nil()) -> c_2()}
Weak Rules: {isList^#(Cons(x, xs)) -> c_1(isList^#(xs))}
Proof Output:
The following argument positions are usable:
Uargs(Cons) = {}, Uargs(isList^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
Cons(x1, x2) = [0 0 0] x1 + [1 1 0] x2 + [0]
[0 0 0] [0 1 1] [1]
[0 0 0] [0 0 0] [0]
nil() = [2]
[2]
[2]
isList^#(x1) = [2 2 2] x1 + [0]
[0 6 0] [0]
[0 0 2] [0]
c_1(x1) = [1 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [0]
c_2() = [1]
[0]
[0]
* Path {4}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {1}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
isNat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
nil() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
add^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
isNat^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_4() = [0]
[0]
[0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isNat^#(s(x)) -> c_3(isNat^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 1 2] [2]
[0 0 0] [0]
isNat^#(x1) = [0 1 0] x1 + [2]
[6 0 0] [0]
[2 3 0] [2]
c_3(x1) = [1 0 0] x1 + [1]
[2 0 2] [0]
[0 0 0] [0]
* Path {4}->{5}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {1}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
isNat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
nil() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
add^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
isNat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_4() = [0]
[0]
[0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isNat^#(0()) -> c_4()}
Weak Rules: {isNat^#(s(x)) -> c_3(isNat^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [0]
[0 1 1] [1]
[0 0 0] [0]
0() = [2]
[2]
[2]
isNat^#(x1) = [2 2 2] x1 + [0]
[0 6 0] [0]
[0 0 2] [0]
c_3(x1) = [1 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [0]
c_4() = [1]
[0]
[0]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
isNat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
nil() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
add^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
isNat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {if^#(true(), x, y) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
[2]
[2]
if^#(x1, x2, x3) = [0 2 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [7]
[2 2 0] [0 0 0] [0 0 0] [3]
[2 2 2] [0 0 0] [0 0 0] [3]
c_5() = [0]
[1]
[1]
* Path {7}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
isNat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
nil() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
add^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
isNat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {if^#(false(), x, y) -> c_6()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
[2]
if^#(x1, x2, x3) = [0 2 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [7]
[2 2 0] [0 0 0] [0 0 0] [3]
[2 2 2] [0 0 0] [0 0 0] [3]
c_6() = [0]
[1]
[1]
* Path {8}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
isNat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
nil() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
add^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
isNat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {and^#(true(), true()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
[2]
[3]
and^#(x1, x2) = [2 0 2] x1 + [1 0 0] x2 + [3]
[0 0 0] [0 2 2] [3]
[2 0 0] [0 2 0] [7]
c_7() = [0]
[1]
[1]
* Path {9}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
isNat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
nil() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
add^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
isNat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {and^#(false(), x) -> c_8()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
[2]
and^#(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [7]
[2 2 0] [0 0 0] [3]
[2 2 2] [0 0 0] [3]
c_8() = [0]
[1]
[1]
* Path {10}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
isNat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
nil() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
add^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
isNat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {and^#(x, false()) -> c_9()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
[2]
and^#(x1, x2) = [0 0 0] x1 + [0 2 0] x2 + [7]
[0 0 0] [2 2 0] [3]
[0 0 0] [2 2 2] [3]
c_9() = [0]
[1]
[1]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: add^#(true(), x, xs) ->
c_0(add^#(and(isNat(x), isList(xs)), x, Cons(x, xs)))
, 2: isList^#(Cons(x, xs)) -> c_1(isList^#(xs))
, 3: isList^#(nil()) -> c_2()
, 4: isNat^#(s(x)) -> c_3(isNat^#(x))
, 5: isNat^#(0()) -> c_4()
, 6: if^#(true(), x, y) -> c_5()
, 7: if^#(false(), x, y) -> c_6()
, 8: and^#(true(), true()) -> c_7()
, 9: and^#(false(), x) -> c_8()
, 10: and^#(x, false()) -> c_9()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ YES(?,O(1)) ]
->{9} [ YES(?,O(1)) ]
->{8} [ YES(?,O(1)) ]
->{7} [ YES(?,O(1)) ]
->{6} [ YES(?,O(1)) ]
->{4} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ isList(Cons(x, xs)) -> isList(xs)
, isList(nil()) -> true()
, isNat(s(x)) -> isNat(x)
, isNat(0()) -> true()
, and(true(), true()) -> true()
, and(false(), x) -> false()
, and(x, false()) -> false()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ add^#(true(), x, xs) ->
c_0(add^#(and(isNat(x), isList(xs)), x, Cons(x, xs)))
, isList(Cons(x, xs)) -> isList(xs)
, isList(nil()) -> true()
, isNat(s(x)) -> isNat(x)
, isNat(0()) -> true()
, and(true(), true()) -> true()
, and(false(), x) -> false()
, and(x, false()) -> false()}
Proof Output:
The input cannot be shown compatible
* Path {2}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {1}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
Cons(x1, x2) = [1 2] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
false() = [0]
[0]
add^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isList^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [0]
[0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isList^#(Cons(x, xs)) -> c_1(isList^#(xs))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(Cons) = {}, Uargs(isList^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
Cons(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [1]
isList^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {2}->{3}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {1}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
false() = [0]
[0]
add^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [0]
[0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isList^#(nil()) -> c_2()}
Weak Rules: {isList^#(Cons(x, xs)) -> c_1(isList^#(xs))}
Proof Output:
The following argument positions are usable:
Uargs(Cons) = {}, Uargs(isList^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
Cons(x1, x2) = [0 0] x1 + [1 2] x2 + [1]
[0 0] [0 0] [3]
nil() = [2]
[2]
isList^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_1(x1) = [1 0] x1 + [5]
[2 0] [3]
c_2() = [1]
[0]
* Path {4}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {1}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
false() = [0]
[0]
add^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
isNat^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isNat^#(s(x)) -> c_3(isNat^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
isNat^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_3(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {4}->{5}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {1}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
false() = [0]
[0]
add^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isNat^#(0()) -> c_4()}
Weak Rules: {isNat^#(s(x)) -> c_3(isNat^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 0] [3]
0() = [2]
[2]
isNat^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_3(x1) = [1 0] x1 + [5]
[2 0] [3]
c_4() = [1]
[0]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
false() = [0]
[0]
add^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {if^#(true(), x, y) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
[2]
if^#(x1, x2, x3) = [2 0] x1 + [0 0] x2 + [0 0] x3 + [7]
[2 2] [0 0] [0 0] [7]
c_5() = [0]
[1]
* Path {7}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
false() = [0]
[0]
add^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {if^#(false(), x, y) -> c_6()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
if^#(x1, x2, x3) = [2 0] x1 + [0 0] x2 + [0 0] x3 + [7]
[2 2] [0 0] [0 0] [7]
c_6() = [0]
[1]
* Path {8}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
false() = [0]
[0]
add^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {and^#(true(), true()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
[2]
and^#(x1, x2) = [2 2] x1 + [0 0] x2 + [7]
[2 2] [0 2] [3]
c_7() = [0]
[1]
* Path {9}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
false() = [0]
[0]
add^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {and^#(false(), x) -> c_8()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
and^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
[2 2] [0 0] [7]
c_8() = [0]
[1]
* Path {10}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
false() = [0]
[0]
add^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {and^#(x, false()) -> c_9()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
and^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
[0 0] [2 2] [7]
c_9() = [0]
[1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: add^#(true(), x, xs) ->
c_0(add^#(and(isNat(x), isList(xs)), x, Cons(x, xs)))
, 2: isList^#(Cons(x, xs)) -> c_1(isList^#(xs))
, 3: isList^#(nil()) -> c_2()
, 4: isNat^#(s(x)) -> c_3(isNat^#(x))
, 5: isNat^#(0()) -> c_4()
, 6: if^#(true(), x, y) -> c_5()
, 7: if^#(false(), x, y) -> c_6()
, 8: and^#(true(), true()) -> c_7()
, 9: and^#(false(), x) -> c_8()
, 10: and^#(x, false()) -> c_9()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ YES(?,O(1)) ]
->{9} [ YES(?,O(1)) ]
->{8} [ YES(?,O(1)) ]
->{7} [ YES(?,O(1)) ]
->{6} [ YES(?,O(1)) ]
->{4} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ isList(Cons(x, xs)) -> isList(xs)
, isList(nil()) -> true()
, isNat(s(x)) -> isNat(x)
, isNat(0()) -> true()
, and(true(), true()) -> true()
, and(false(), x) -> false()
, and(x, false()) -> false()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ add^#(true(), x, xs) ->
c_0(add^#(and(isNat(x), isList(xs)), x, Cons(x, xs)))
, isList(Cons(x, xs)) -> isList(xs)
, isList(nil()) -> true()
, isNat(s(x)) -> isNat(x)
, isNat(0()) -> true()
, and(true(), true()) -> true()
, and(false(), x) -> false()
, and(x, false()) -> false()}
Proof Output:
The input cannot be shown compatible
* Path {2}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {1}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
isList(x1) = [0] x1 + [0]
Cons(x1, x2) = [1] x1 + [0] x2 + [0]
nil() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
false() = [0]
add^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
isList^#(x1) = [3] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
isNat^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_5() = [0]
c_6() = [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isList^#(Cons(x, xs)) -> c_1(isList^#(xs))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(Cons) = {}, Uargs(isList^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
Cons(x1, x2) = [0] x1 + [1] x2 + [4]
isList^#(x1) = [2] x1 + [0]
c_1(x1) = [1] x1 + [7]
* Path {2}->{3}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {1}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
isList(x1) = [0] x1 + [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
false() = [0]
add^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
isList^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
isNat^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_5() = [0]
c_6() = [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isList^#(nil()) -> c_2()}
Weak Rules: {isList^#(Cons(x, xs)) -> c_1(isList^#(xs))}
Proof Output:
The following argument positions are usable:
Uargs(Cons) = {}, Uargs(isList^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
Cons(x1, x2) = [0] x1 + [1] x2 + [0]
nil() = [2]
isList^#(x1) = [2] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [1]
* Path {4}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {1}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
isList(x1) = [0] x1 + [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
s(x1) = [1] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
false() = [0]
add^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
isList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
isNat^#(x1) = [3] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_5() = [0]
c_6() = [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isNat^#(s(x)) -> c_3(isNat^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
isNat^#(x1) = [2] x1 + [0]
c_3(x1) = [1] x1 + [7]
* Path {4}->{5}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {1}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
isList(x1) = [0] x1 + [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
false() = [0]
add^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
isList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
isNat^#(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_5() = [0]
c_6() = [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isNat^#(0()) -> c_4()}
Weak Rules: {isNat^#(s(x)) -> c_3(isNat^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
0() = [2]
isNat^#(x1) = [2] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [1]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
isList(x1) = [0] x1 + [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
false() = [0]
add^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
isList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
isNat^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_5() = [0]
c_6() = [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {if^#(true(), x, y) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [7]
if^#(x1, x2, x3) = [1] x1 + [0] x2 + [0] x3 + [7]
c_5() = [1]
* Path {7}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
isList(x1) = [0] x1 + [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
false() = [0]
add^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
isList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
isNat^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_5() = [0]
c_6() = [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {if^#(false(), x, y) -> c_6()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [7]
if^#(x1, x2, x3) = [1] x1 + [0] x2 + [0] x3 + [7]
c_6() = [1]
* Path {8}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
isList(x1) = [0] x1 + [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
false() = [0]
add^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
isList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
isNat^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_5() = [0]
c_6() = [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {and^#(true(), true()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
and^#(x1, x2) = [2] x1 + [2] x2 + [7]
c_7() = [0]
* Path {9}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
isList(x1) = [0] x1 + [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
false() = [0]
add^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
isList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
isNat^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_5() = [0]
c_6() = [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {and^#(false(), x) -> c_8()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [7]
and^#(x1, x2) = [1] x1 + [0] x2 + [7]
c_8() = [1]
* Path {10}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
isList(x1) = [0] x1 + [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
false() = [0]
add^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
isList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
isNat^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_5() = [0]
c_6() = [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {and^#(x, false()) -> c_9()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [7]
and^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_9() = [1]
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ add(true(), x, xs) ->
add(and(isNat(x), isList(xs)), x, Cons(x, xs))
, isList(Cons(x, xs)) -> isList(xs)
, isList(nil()) -> true()
, isNat(s(x)) -> isNat(x)
, isNat(0()) -> true()
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, and(true(), true()) -> true()
, and(false(), x) -> false()
, and(x, false()) -> false()}
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: add^#(true(), x, xs) ->
c_0(add^#(and(isNat(x), isList(xs)), x, Cons(x, xs)))
, 2: isList^#(Cons(x, xs)) -> c_1(isList^#(xs))
, 3: isList^#(nil()) -> c_2()
, 4: isNat^#(s(x)) -> c_3(isNat^#(x))
, 5: isNat^#(0()) -> c_4()
, 6: if^#(true(), x, y) -> c_5(x)
, 7: if^#(false(), x, y) -> c_6(y)
, 8: and^#(true(), true()) -> c_7()
, 9: and^#(false(), x) -> c_8()
, 10: and^#(x, false()) -> c_9()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ YES(?,O(1)) ]
->{9} [ YES(?,O(1)) ]
->{8} [ YES(?,O(1)) ]
->{7} [ YES(?,O(1)) ]
->{6} [ YES(?,O(1)) ]
->{4} [ YES(?,O(n^2)) ]
|
`->{5} [ YES(?,O(n^2)) ]
->{2} [ YES(?,O(n^2)) ]
|
`->{3} [ YES(?,O(n^2)) ]
->{1} [ NA ]
Sub-problems:
-------------
* Path {1}: NA
------------
The usable rules for this path are:
{ isList(Cons(x, xs)) -> isList(xs)
, isList(nil()) -> true()
, isNat(s(x)) -> isNat(x)
, isNat(0()) -> true()
, and(true(), true()) -> true()
, and(false(), x) -> false()
, and(x, false()) -> false()}
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 {2}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {1}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
isNat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Cons(x1, x2) = [1 3 0] x1 + [0 0 0] x2 + [0]
[0 1 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
nil() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
add^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_2() = [0]
[0]
[0]
isNat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isList^#(Cons(x, xs)) -> c_1(isList^#(xs))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(Cons) = {}, Uargs(isList^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
Cons(x1, x2) = [0 0 0] x1 + [1 2 2] x2 + [2]
[0 0 0] [0 1 2] [2]
[0 0 0] [0 0 0] [0]
isList^#(x1) = [0 1 0] x1 + [2]
[6 0 0] [0]
[2 3 0] [2]
c_1(x1) = [1 0 0] x1 + [1]
[2 0 2] [0]
[0 0 0] [0]
* Path {2}->{3}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {1}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
isNat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
nil() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
add^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_2() = [0]
[0]
[0]
isNat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isList^#(nil()) -> c_2()}
Weak Rules: {isList^#(Cons(x, xs)) -> c_1(isList^#(xs))}
Proof Output:
The following argument positions are usable:
Uargs(Cons) = {}, Uargs(isList^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
Cons(x1, x2) = [0 0 0] x1 + [1 1 0] x2 + [0]
[0 0 0] [0 1 1] [1]
[0 0 0] [0 0 0] [0]
nil() = [2]
[2]
[2]
isList^#(x1) = [2 2 2] x1 + [0]
[0 6 0] [0]
[0 0 2] [0]
c_1(x1) = [1 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [0]
c_2() = [1]
[0]
[0]
* Path {4}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {1}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
isNat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
nil() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
add^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
isNat^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_4() = [0]
[0]
[0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isNat^#(s(x)) -> c_3(isNat^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 1 2] [2]
[0 0 0] [0]
isNat^#(x1) = [0 1 0] x1 + [2]
[6 0 0] [0]
[2 3 0] [2]
c_3(x1) = [1 0 0] x1 + [1]
[2 0 2] [0]
[0 0 0] [0]
* Path {4}->{5}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {1}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
isNat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
nil() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
add^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
isNat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_4() = [0]
[0]
[0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isNat^#(0()) -> c_4()}
Weak Rules: {isNat^#(s(x)) -> c_3(isNat^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [0]
[0 1 1] [1]
[0 0 0] [0]
0() = [2]
[2]
[2]
isNat^#(x1) = [2 2 2] x1 + [0]
[0 6 0] [0]
[0 0 2] [0]
c_3(x1) = [1 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [0]
c_4() = [1]
[0]
[0]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
isNat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
nil() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
add^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
isNat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
if^#(x1, x2, x3) = [0 0 0] x1 + [3 3 3] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_5(x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {if^#(true(), x, y) -> c_5(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
[0]
[2]
if^#(x1, x2, x3) = [2 0 2] x1 + [7 7 7] x2 + [0 0 0] x3 + [7]
[2 0 2] [7 7 7] [0 0 0] [7]
[2 0 2] [7 7 7] [0 0 0] [7]
c_5(x1) = [1 3 3] x1 + [0]
[1 1 1] [1]
[1 1 1] [1]
* Path {7}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
isNat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
nil() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
add^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
isNat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
if^#(x1, x2, x3) = [0 0 0] x1 + [3 3 3] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {if^#(false(), x, y) -> c_6(y)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[0]
[2]
if^#(x1, x2, x3) = [2 0 2] x1 + [0 0 0] x2 + [7 7 7] x3 + [7]
[2 0 2] [0 0 0] [7 7 7] [7]
[2 0 2] [0 0 0] [7 7 7] [7]
c_6(x1) = [1 3 3] x1 + [0]
[1 1 1] [1]
[1 1 1] [1]
* Path {8}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
isNat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
nil() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
add^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
isNat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {and^#(true(), true()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
[2]
[3]
and^#(x1, x2) = [2 0 2] x1 + [1 0 0] x2 + [3]
[0 0 0] [0 2 2] [3]
[2 0 0] [0 2 0] [7]
c_7() = [0]
[1]
[1]
* Path {9}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
isNat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
nil() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
add^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
isNat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {and^#(false(), x) -> c_8()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
[2]
and^#(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [7]
[2 2 0] [0 0 0] [3]
[2 2 2] [0 0 0] [3]
c_8() = [0]
[1]
[1]
* Path {10}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
isNat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
nil() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
add^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
isNat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {and^#(x, false()) -> c_9()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
[2]
and^#(x1, x2) = [0 0 0] x1 + [0 2 0] x2 + [7]
[0 0 0] [2 2 0] [3]
[0 0 0] [2 2 2] [3]
c_9() = [0]
[1]
[1]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: add^#(true(), x, xs) ->
c_0(add^#(and(isNat(x), isList(xs)), x, Cons(x, xs)))
, 2: isList^#(Cons(x, xs)) -> c_1(isList^#(xs))
, 3: isList^#(nil()) -> c_2()
, 4: isNat^#(s(x)) -> c_3(isNat^#(x))
, 5: isNat^#(0()) -> c_4()
, 6: if^#(true(), x, y) -> c_5(x)
, 7: if^#(false(), x, y) -> c_6(y)
, 8: and^#(true(), true()) -> c_7()
, 9: and^#(false(), x) -> c_8()
, 10: and^#(x, false()) -> c_9()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ YES(?,O(1)) ]
->{9} [ YES(?,O(1)) ]
->{8} [ YES(?,O(1)) ]
->{7} [ YES(?,O(1)) ]
->{6} [ YES(?,O(1)) ]
->{4} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ isList(Cons(x, xs)) -> isList(xs)
, isList(nil()) -> true()
, isNat(s(x)) -> isNat(x)
, isNat(0()) -> true()
, and(true(), true()) -> true()
, and(false(), x) -> false()
, and(x, false()) -> false()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ add^#(true(), x, xs) ->
c_0(add^#(and(isNat(x), isList(xs)), x, Cons(x, xs)))
, isList(Cons(x, xs)) -> isList(xs)
, isList(nil()) -> true()
, isNat(s(x)) -> isNat(x)
, isNat(0()) -> true()
, and(true(), true()) -> true()
, and(false(), x) -> false()
, and(x, false()) -> false()}
Proof Output:
The input cannot be shown compatible
* Path {2}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {1}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
Cons(x1, x2) = [1 2] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
false() = [0]
[0]
add^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isList^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [0]
[0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isList^#(Cons(x, xs)) -> c_1(isList^#(xs))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(Cons) = {}, Uargs(isList^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
Cons(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [1]
isList^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {2}->{3}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {1}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
false() = [0]
[0]
add^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [0]
[0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isList^#(nil()) -> c_2()}
Weak Rules: {isList^#(Cons(x, xs)) -> c_1(isList^#(xs))}
Proof Output:
The following argument positions are usable:
Uargs(Cons) = {}, Uargs(isList^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
Cons(x1, x2) = [0 0] x1 + [1 2] x2 + [1]
[0 0] [0 0] [3]
nil() = [2]
[2]
isList^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_1(x1) = [1 0] x1 + [5]
[2 0] [3]
c_2() = [1]
[0]
* Path {4}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {1}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
false() = [0]
[0]
add^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
isNat^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isNat^#(s(x)) -> c_3(isNat^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
isNat^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_3(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {4}->{5}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {1}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
false() = [0]
[0]
add^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isNat^#(0()) -> c_4()}
Weak Rules: {isNat^#(s(x)) -> c_3(isNat^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 0] [3]
0() = [2]
[2]
isNat^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_3(x1) = [1 0] x1 + [5]
[2 0] [3]
c_4() = [1]
[0]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
false() = [0]
[0]
add^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [3 3] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_5(x1) = [1 1] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {if^#(true(), x, y) -> c_5(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
[2]
if^#(x1, x2, x3) = [2 2] x1 + [7 7] x2 + [0 0] x3 + [7]
[2 2] [7 7] [0 0] [3]
c_5(x1) = [1 3] x1 + [0]
[1 1] [1]
* Path {7}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
false() = [0]
[0]
add^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [3 3] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [1 1] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {if^#(false(), x, y) -> c_6(y)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
if^#(x1, x2, x3) = [2 2] x1 + [0 0] x2 + [7 7] x3 + [7]
[2 2] [0 0] [7 7] [3]
c_6(x1) = [1 3] x1 + [0]
[1 1] [1]
* Path {8}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
false() = [0]
[0]
add^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {and^#(true(), true()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
[2]
and^#(x1, x2) = [2 2] x1 + [0 0] x2 + [7]
[2 2] [0 2] [3]
c_7() = [0]
[1]
* Path {9}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
false() = [0]
[0]
add^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {and^#(false(), x) -> c_8()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
and^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
[2 2] [0 0] [7]
c_8() = [0]
[1]
* Path {10}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isNat(x1) = [0 0] x1 + [0]
[0 0] [0]
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
false() = [0]
[0]
add^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {and^#(x, false()) -> c_9()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
and^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
[0 0] [2 2] [7]
c_9() = [0]
[1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: add^#(true(), x, xs) ->
c_0(add^#(and(isNat(x), isList(xs)), x, Cons(x, xs)))
, 2: isList^#(Cons(x, xs)) -> c_1(isList^#(xs))
, 3: isList^#(nil()) -> c_2()
, 4: isNat^#(s(x)) -> c_3(isNat^#(x))
, 5: isNat^#(0()) -> c_4()
, 6: if^#(true(), x, y) -> c_5(x)
, 7: if^#(false(), x, y) -> c_6(y)
, 8: and^#(true(), true()) -> c_7()
, 9: and^#(false(), x) -> c_8()
, 10: and^#(x, false()) -> c_9()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ YES(?,O(1)) ]
->{9} [ YES(?,O(1)) ]
->{8} [ YES(?,O(1)) ]
->{7} [ YES(?,O(1)) ]
->{6} [ YES(?,O(1)) ]
->{4} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ isList(Cons(x, xs)) -> isList(xs)
, isList(nil()) -> true()
, isNat(s(x)) -> isNat(x)
, isNat(0()) -> true()
, and(true(), true()) -> true()
, and(false(), x) -> false()
, and(x, false()) -> false()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ add^#(true(), x, xs) ->
c_0(add^#(and(isNat(x), isList(xs)), x, Cons(x, xs)))
, isList(Cons(x, xs)) -> isList(xs)
, isList(nil()) -> true()
, isNat(s(x)) -> isNat(x)
, isNat(0()) -> true()
, and(true(), true()) -> true()
, and(false(), x) -> false()
, and(x, false()) -> false()}
Proof Output:
The input cannot be shown compatible
* Path {2}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {1}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
isList(x1) = [0] x1 + [0]
Cons(x1, x2) = [1] x1 + [0] x2 + [0]
nil() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
false() = [0]
add^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
isList^#(x1) = [3] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
isNat^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isList^#(Cons(x, xs)) -> c_1(isList^#(xs))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(Cons) = {}, Uargs(isList^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
Cons(x1, x2) = [0] x1 + [1] x2 + [4]
isList^#(x1) = [2] x1 + [0]
c_1(x1) = [1] x1 + [7]
* Path {2}->{3}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {1}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
isList(x1) = [0] x1 + [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
false() = [0]
add^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
isList^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
isNat^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isList^#(nil()) -> c_2()}
Weak Rules: {isList^#(Cons(x, xs)) -> c_1(isList^#(xs))}
Proof Output:
The following argument positions are usable:
Uargs(Cons) = {}, Uargs(isList^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
Cons(x1, x2) = [0] x1 + [1] x2 + [0]
nil() = [2]
isList^#(x1) = [2] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [1]
* Path {4}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {1}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
isList(x1) = [0] x1 + [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
s(x1) = [1] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
false() = [0]
add^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
isList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
isNat^#(x1) = [3] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isNat^#(s(x)) -> c_3(isNat^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
isNat^#(x1) = [2] x1 + [0]
c_3(x1) = [1] x1 + [7]
* Path {4}->{5}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {1}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
isList(x1) = [0] x1 + [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
false() = [0]
add^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
isList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
isNat^#(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isNat^#(0()) -> c_4()}
Weak Rules: {isNat^#(s(x)) -> c_3(isNat^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
0() = [2]
isNat^#(x1) = [2] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [1]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
isList(x1) = [0] x1 + [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
false() = [0]
add^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
isList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
isNat^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
if^#(x1, x2, x3) = [0] x1 + [3] x2 + [0] x3 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {if^#(true(), x, y) -> c_5(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [5]
if^#(x1, x2, x3) = [3] x1 + [7] x2 + [0] x3 + [0]
c_5(x1) = [1] x1 + [0]
* Path {7}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
isList(x1) = [0] x1 + [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
false() = [0]
add^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
isList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
isNat^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
if^#(x1, x2, x3) = [0] x1 + [3] x2 + [0] x3 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {if^#(false(), x, y) -> c_6(y)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [5]
if^#(x1, x2, x3) = [3] x1 + [0] x2 + [7] x3 + [0]
c_6(x1) = [1] x1 + [0]
* Path {8}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
isList(x1) = [0] x1 + [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
false() = [0]
add^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
isList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
isNat^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {and^#(true(), true()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
and^#(x1, x2) = [2] x1 + [2] x2 + [7]
c_7() = [0]
* Path {9}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
isList(x1) = [0] x1 + [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
false() = [0]
add^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
isList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
isNat^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {and^#(false(), x) -> c_8()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [7]
and^#(x1, x2) = [1] x1 + [0] x2 + [7]
c_8() = [1]
* Path {10}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(add) = {}, Uargs(and) = {}, Uargs(isNat) = {},
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(s) = {},
Uargs(if) = {}, Uargs(add^#) = {}, Uargs(c_0) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_3) = {}, Uargs(if^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
isList(x1) = [0] x1 + [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
false() = [0]
add^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
isList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
isNat^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {and^#(x, false()) -> c_9()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [7]
and^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_9() = [1]
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.