Tool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ fact(X) -> if(zero(X), s(0()), prod(X, fact(p(X))))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))
, prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y
, zero(0()) -> true()
, zero(s(X)) -> false()
, p(s(X)) -> X}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: fact^#(X) -> c_0(if^#(zero(X), s(0()), prod(X, fact(p(X)))))
, 2: add^#(0(), X) -> c_1()
, 3: add^#(s(X), Y) -> c_2(add^#(X, Y))
, 4: prod^#(0(), X) -> c_3()
, 5: prod^#(s(X), Y) -> c_4(add^#(Y, prod(X, Y)))
, 6: if^#(true(), X, Y) -> c_5()
, 7: if^#(false(), X, Y) -> c_6()
, 8: zero^#(0()) -> c_7()
, 9: zero^#(s(X)) -> c_8()
, 10: p^#(s(X)) -> 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)) ]
->{5} [ inherited ]
|
|->{2} [ MAYBE ]
|
`->{3} [ inherited ]
|
`->{2} [ NA ]
->{4} [ YES(?,O(1)) ]
->{1} [ inherited ]
|
|->{6} [ NA ]
|
`->{7} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{7}.
* Path {1}->{6}: NA
-----------------
The usable rules for this path are:
{ fact(X) -> if(zero(X), s(0()), prod(X, fact(p(X))))
, prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, zero(0()) -> true()
, zero(s(X)) -> false()
, p(s(X)) -> X
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {1}->{7}: NA
-----------------
The usable rules for this path are:
{ fact(X) -> if(zero(X), s(0()), prod(X, fact(p(X))))
, prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, zero(0()) -> true()
, zero(s(X)) -> false()
, p(s(X)) -> X
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fact) = {}, Uargs(if) = {}, Uargs(zero) = {}, Uargs(s) = {},
Uargs(prod) = {}, Uargs(p) = {}, Uargs(add) = {},
Uargs(fact^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(add^#) = {}, Uargs(c_2) = {}, Uargs(prod^#) = {},
Uargs(c_4) = {}, Uargs(zero^#) = {}, Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fact(x1) = [0 0 0] x1 + [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]
zero(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
prod(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
fact^#(x1) = [0 0 0] x1 + [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]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
prod^#(x1, 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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
zero^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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: {prod^#(0(), X) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(prod^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
prod^#(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_3() = [0]
[1]
[1]
* Path {5}: inherited
-------------------
This path is subsumed by the proof of path {5}->{3}->{2}.
* Path {5}->{2}: MAYBE
--------------------
The usable rules for this path are:
{ prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ prod^#(s(X), Y) -> c_4(add^#(Y, prod(X, Y)))
, add^#(0(), X) -> c_1()
, prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
Proof Output:
The input cannot be shown compatible
* Path {5}->{3}: inherited
------------------------
This path is subsumed by the proof of path {5}->{3}->{2}.
* Path {5}->{3}->{2}: NA
----------------------
The usable rules for this path are:
{ prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {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(fact) = {}, Uargs(if) = {}, Uargs(zero) = {}, Uargs(s) = {},
Uargs(prod) = {}, Uargs(p) = {}, Uargs(add) = {},
Uargs(fact^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(add^#) = {}, Uargs(c_2) = {}, Uargs(prod^#) = {},
Uargs(c_4) = {}, Uargs(zero^#) = {}, Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fact(x1) = [0 0 0] x1 + [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]
zero(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
prod(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
fact^#(x1) = [0 0 0] x1 + [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]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
prod^#(x1, 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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
zero^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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: {zero^#(0()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(zero^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
zero^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
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(fact) = {}, Uargs(if) = {}, Uargs(zero) = {}, Uargs(s) = {},
Uargs(prod) = {}, Uargs(p) = {}, Uargs(add) = {},
Uargs(fact^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(add^#) = {}, Uargs(c_2) = {}, Uargs(prod^#) = {},
Uargs(c_4) = {}, Uargs(zero^#) = {}, Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fact(x1) = [0 0 0] x1 + [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]
zero(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
prod(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
fact^#(x1) = [0 0 0] x1 + [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]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
prod^#(x1, 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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
zero^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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: {zero^#(s(X)) -> c_8()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(zero^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 0 0] x1 + [2]
[0 0 0] [2]
[0 0 0] [2]
zero^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [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(fact) = {}, Uargs(if) = {}, Uargs(zero) = {}, Uargs(s) = {},
Uargs(prod) = {}, Uargs(p) = {}, Uargs(add) = {},
Uargs(fact^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(add^#) = {}, Uargs(c_2) = {}, Uargs(prod^#) = {},
Uargs(c_4) = {}, Uargs(zero^#) = {}, Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fact(x1) = [0 0 0] x1 + [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]
zero(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
prod(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
fact^#(x1) = [0 0 0] x1 + [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]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
prod^#(x1, 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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
zero^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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: {p^#(s(X)) -> c_9()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 0 0] x1 + [2]
[0 0 0] [2]
[0 0 0] [2]
p^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[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: fact^#(X) -> c_0(if^#(zero(X), s(0()), prod(X, fact(p(X)))))
, 2: add^#(0(), X) -> c_1()
, 3: add^#(s(X), Y) -> c_2(add^#(X, Y))
, 4: prod^#(0(), X) -> c_3()
, 5: prod^#(s(X), Y) -> c_4(add^#(Y, prod(X, Y)))
, 6: if^#(true(), X, Y) -> c_5()
, 7: if^#(false(), X, Y) -> c_6()
, 8: zero^#(0()) -> c_7()
, 9: zero^#(s(X)) -> c_8()
, 10: p^#(s(X)) -> 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)) ]
->{5} [ inherited ]
|
|->{2} [ MAYBE ]
|
`->{3} [ inherited ]
|
`->{2} [ NA ]
->{4} [ YES(?,O(1)) ]
->{1} [ inherited ]
|
|->{6} [ NA ]
|
`->{7} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{7}.
* Path {1}->{6}: NA
-----------------
The usable rules for this path are:
{ fact(X) -> if(zero(X), s(0()), prod(X, fact(p(X))))
, prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, zero(0()) -> true()
, zero(s(X)) -> false()
, p(s(X)) -> X
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {1}->{7}: NA
-----------------
The usable rules for this path are:
{ fact(X) -> if(zero(X), s(0()), prod(X, fact(p(X))))
, prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, zero(0()) -> true()
, zero(s(X)) -> false()
, p(s(X)) -> X
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fact) = {}, Uargs(if) = {}, Uargs(zero) = {}, Uargs(s) = {},
Uargs(prod) = {}, Uargs(p) = {}, Uargs(add) = {},
Uargs(fact^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(add^#) = {}, Uargs(c_2) = {}, Uargs(prod^#) = {},
Uargs(c_4) = {}, Uargs(zero^#) = {}, Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fact(x1) = [0 0] x1 + [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]
zero(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
prod(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
fact^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [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]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
prod^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
zero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 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: {prod^#(0(), X) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(prod^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
prod^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
[2 2] [0 0] [7]
c_3() = [0]
[1]
* Path {5}: inherited
-------------------
This path is subsumed by the proof of path {5}->{3}->{2}.
* Path {5}->{2}: MAYBE
--------------------
The usable rules for this path are:
{ prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ prod^#(s(X), Y) -> c_4(add^#(Y, prod(X, Y)))
, add^#(0(), X) -> c_1()
, prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
Proof Output:
The input cannot be shown compatible
* Path {5}->{3}: inherited
------------------------
This path is subsumed by the proof of path {5}->{3}->{2}.
* Path {5}->{3}->{2}: NA
----------------------
The usable rules for this path are:
{ prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {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(fact) = {}, Uargs(if) = {}, Uargs(zero) = {}, Uargs(s) = {},
Uargs(prod) = {}, Uargs(p) = {}, Uargs(add) = {},
Uargs(fact^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(add^#) = {}, Uargs(c_2) = {}, Uargs(prod^#) = {},
Uargs(c_4) = {}, Uargs(zero^#) = {}, Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fact(x1) = [0 0] x1 + [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]
zero(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
prod(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
fact^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [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]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
prod^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
zero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 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: {zero^#(0()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(zero^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
zero^#(x1) = [2 0] x1 + [7]
[2 2] [7]
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(fact) = {}, Uargs(if) = {}, Uargs(zero) = {}, Uargs(s) = {},
Uargs(prod) = {}, Uargs(p) = {}, Uargs(add) = {},
Uargs(fact^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(add^#) = {}, Uargs(c_2) = {}, Uargs(prod^#) = {},
Uargs(c_4) = {}, Uargs(zero^#) = {}, Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fact(x1) = [0 0] x1 + [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]
zero(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
prod(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
fact^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [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]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
prod^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
zero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 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: {zero^#(s(X)) -> c_8()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(zero^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 0] x1 + [2]
[0 0] [2]
zero^#(x1) = [2 0] x1 + [7]
[2 2] [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(fact) = {}, Uargs(if) = {}, Uargs(zero) = {}, Uargs(s) = {},
Uargs(prod) = {}, Uargs(p) = {}, Uargs(add) = {},
Uargs(fact^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(add^#) = {}, Uargs(c_2) = {}, Uargs(prod^#) = {},
Uargs(c_4) = {}, Uargs(zero^#) = {}, Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fact(x1) = [0 0] x1 + [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]
zero(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
prod(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
fact^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [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]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
prod^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
zero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 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: {p^#(s(X)) -> c_9()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 0] x1 + [2]
[0 0] [2]
p^#(x1) = [2 0] x1 + [7]
[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: fact^#(X) -> c_0(if^#(zero(X), s(0()), prod(X, fact(p(X)))))
, 2: add^#(0(), X) -> c_1()
, 3: add^#(s(X), Y) -> c_2(add^#(X, Y))
, 4: prod^#(0(), X) -> c_3()
, 5: prod^#(s(X), Y) -> c_4(add^#(Y, prod(X, Y)))
, 6: if^#(true(), X, Y) -> c_5()
, 7: if^#(false(), X, Y) -> c_6()
, 8: zero^#(0()) -> c_7()
, 9: zero^#(s(X)) -> c_8()
, 10: p^#(s(X)) -> 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)) ]
->{5} [ inherited ]
|
|->{2} [ MAYBE ]
|
`->{3} [ inherited ]
|
`->{2} [ NA ]
->{4} [ YES(?,O(1)) ]
->{1} [ inherited ]
|
|->{6} [ NA ]
|
`->{7} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{7}.
* Path {1}->{6}: NA
-----------------
The usable rules for this path are:
{ fact(X) -> if(zero(X), s(0()), prod(X, fact(p(X))))
, prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, zero(0()) -> true()
, zero(s(X)) -> false()
, p(s(X)) -> X
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {1}->{7}: NA
-----------------
The usable rules for this path are:
{ fact(X) -> if(zero(X), s(0()), prod(X, fact(p(X))))
, prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, zero(0()) -> true()
, zero(s(X)) -> false()
, p(s(X)) -> X
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fact) = {}, Uargs(if) = {}, Uargs(zero) = {}, Uargs(s) = {},
Uargs(prod) = {}, Uargs(p) = {}, Uargs(add) = {},
Uargs(fact^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(add^#) = {}, Uargs(c_2) = {}, Uargs(prod^#) = {},
Uargs(c_4) = {}, Uargs(zero^#) = {}, Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fact(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
zero(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
prod(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
fact^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
prod^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6() = [0]
zero^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
p^#(x1) = [0] x1 + [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: {prod^#(0(), X) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(prod^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
prod^#(x1, x2) = [1] x1 + [0] x2 + [7]
c_3() = [1]
* Path {5}: inherited
-------------------
This path is subsumed by the proof of path {5}->{3}->{2}.
* Path {5}->{2}: MAYBE
--------------------
The usable rules for this path are:
{ prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ prod^#(s(X), Y) -> c_4(add^#(Y, prod(X, Y)))
, add^#(0(), X) -> c_1()
, prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
Proof Output:
The input cannot be shown compatible
* Path {5}->{3}: inherited
------------------------
This path is subsumed by the proof of path {5}->{3}->{2}.
* Path {5}->{3}->{2}: NA
----------------------
The usable rules for this path are:
{ prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {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(fact) = {}, Uargs(if) = {}, Uargs(zero) = {}, Uargs(s) = {},
Uargs(prod) = {}, Uargs(p) = {}, Uargs(add) = {},
Uargs(fact^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(add^#) = {}, Uargs(c_2) = {}, Uargs(prod^#) = {},
Uargs(c_4) = {}, Uargs(zero^#) = {}, Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fact(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
zero(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
prod(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
fact^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
prod^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6() = [0]
zero^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
p^#(x1) = [0] x1 + [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: {zero^#(0()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(zero^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
zero^#(x1) = [1] x1 + [7]
c_7() = [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(fact) = {}, Uargs(if) = {}, Uargs(zero) = {}, Uargs(s) = {},
Uargs(prod) = {}, Uargs(p) = {}, Uargs(add) = {},
Uargs(fact^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(add^#) = {}, Uargs(c_2) = {}, Uargs(prod^#) = {},
Uargs(c_4) = {}, Uargs(zero^#) = {}, Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fact(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
zero(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
prod(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
fact^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
prod^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6() = [0]
zero^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
p^#(x1) = [0] x1 + [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: {zero^#(s(X)) -> c_8()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(zero^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0] x1 + [7]
zero^#(x1) = [1] x1 + [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(fact) = {}, Uargs(if) = {}, Uargs(zero) = {}, Uargs(s) = {},
Uargs(prod) = {}, Uargs(p) = {}, Uargs(add) = {},
Uargs(fact^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(add^#) = {}, Uargs(c_2) = {}, Uargs(prod^#) = {},
Uargs(c_4) = {}, Uargs(zero^#) = {}, Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fact(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
zero(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
prod(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
fact^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
prod^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6() = [0]
zero^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
p^#(x1) = [0] x1 + [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: {p^#(s(X)) -> c_9()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0] x1 + [7]
p^#(x1) = [1] x1 + [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:
{ fact(X) -> if(zero(X), s(0()), prod(X, fact(p(X))))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))
, prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y
, zero(0()) -> true()
, zero(s(X)) -> false()
, p(s(X)) -> X}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: fact^#(X) -> c_0(if^#(zero(X), s(0()), prod(X, fact(p(X)))))
, 2: add^#(0(), X) -> c_1(X)
, 3: add^#(s(X), Y) -> c_2(add^#(X, Y))
, 4: prod^#(0(), X) -> c_3()
, 5: prod^#(s(X), Y) -> c_4(add^#(Y, prod(X, Y)))
, 6: if^#(true(), X, Y) -> c_5(X)
, 7: if^#(false(), X, Y) -> c_6(Y)
, 8: zero^#(0()) -> c_7()
, 9: zero^#(s(X)) -> c_8()
, 10: p^#(s(X)) -> c_9(X)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ YES(?,O(n^3)) ]
->{9} [ YES(?,O(1)) ]
->{8} [ YES(?,O(1)) ]
->{5} [ inherited ]
|
|->{2} [ MAYBE ]
|
`->{3} [ inherited ]
|
`->{2} [ NA ]
->{4} [ YES(?,O(1)) ]
->{1} [ inherited ]
|
|->{6} [ NA ]
|
`->{7} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{7}.
* Path {1}->{6}: NA
-----------------
The usable rules for this path are:
{ fact(X) -> if(zero(X), s(0()), prod(X, fact(p(X))))
, prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, zero(0()) -> true()
, zero(s(X)) -> false()
, p(s(X)) -> X
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {1}->{7}: NA
-----------------
The usable rules for this path are:
{ fact(X) -> if(zero(X), s(0()), prod(X, fact(p(X))))
, prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, zero(0()) -> true()
, zero(s(X)) -> false()
, p(s(X)) -> X
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fact) = {}, Uargs(if) = {}, Uargs(zero) = {}, Uargs(s) = {},
Uargs(prod) = {}, Uargs(p) = {}, Uargs(add) = {},
Uargs(fact^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(add^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(prod^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(zero^#) = {}, Uargs(p^#) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fact(x1) = [0 0 0] x1 + [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]
zero(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
prod(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
fact^#(x1) = [0 0 0] x1 + [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]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
prod^#(x1, 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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
zero^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {prod^#(0(), X) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(prod^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
prod^#(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_3() = [0]
[1]
[1]
* Path {5}: inherited
-------------------
This path is subsumed by the proof of path {5}->{3}->{2}.
* Path {5}->{2}: MAYBE
--------------------
The usable rules for this path are:
{ prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ prod^#(s(X), Y) -> c_4(add^#(Y, prod(X, Y)))
, add^#(0(), X) -> c_1(X)
, prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
Proof Output:
The input cannot be shown compatible
* Path {5}->{3}: inherited
------------------------
This path is subsumed by the proof of path {5}->{3}->{2}.
* Path {5}->{3}->{2}: NA
----------------------
The usable rules for this path are:
{ prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {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(fact) = {}, Uargs(if) = {}, Uargs(zero) = {}, Uargs(s) = {},
Uargs(prod) = {}, Uargs(p) = {}, Uargs(add) = {},
Uargs(fact^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(add^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(prod^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(zero^#) = {}, Uargs(p^#) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fact(x1) = [0 0 0] x1 + [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]
zero(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
prod(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
fact^#(x1) = [0 0 0] x1 + [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]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
prod^#(x1, 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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
zero^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {zero^#(0()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(zero^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
zero^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
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(fact) = {}, Uargs(if) = {}, Uargs(zero) = {}, Uargs(s) = {},
Uargs(prod) = {}, Uargs(p) = {}, Uargs(add) = {},
Uargs(fact^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(add^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(prod^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(zero^#) = {}, Uargs(p^#) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fact(x1) = [0 0 0] x1 + [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]
zero(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
prod(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
fact^#(x1) = [0 0 0] x1 + [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]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
prod^#(x1, 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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
zero^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {zero^#(s(X)) -> c_8()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(zero^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 0 0] x1 + [2]
[0 0 0] [2]
[0 0 0] [2]
zero^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_8() = [0]
[1]
[1]
* Path {10}: YES(?,O(n^3))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fact) = {}, Uargs(if) = {}, Uargs(zero) = {}, Uargs(s) = {},
Uargs(prod) = {}, Uargs(p) = {}, Uargs(add) = {},
Uargs(fact^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(add^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(prod^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(zero^#) = {}, Uargs(p^#) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fact(x1) = [0 0 0] x1 + [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]
zero(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [1 3 3] x1 + [0]
[0 1 1] [0]
[0 0 1] [0]
0() = [0]
[0]
[0]
prod(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
fact^#(x1) = [0 0 0] x1 + [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]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
prod^#(x1, 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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
zero^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
p^#(x1) = [1 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9(x1) = [1 0 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {p^#(s(X)) -> c_9(X)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 0 2] [2]
[0 0 0] [2]
p^#(x1) = [2 2 2] x1 + [3]
[2 2 2] [3]
[2 2 2] [3]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [1]
[0 0 0] [1]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: fact^#(X) -> c_0(if^#(zero(X), s(0()), prod(X, fact(p(X)))))
, 2: add^#(0(), X) -> c_1(X)
, 3: add^#(s(X), Y) -> c_2(add^#(X, Y))
, 4: prod^#(0(), X) -> c_3()
, 5: prod^#(s(X), Y) -> c_4(add^#(Y, prod(X, Y)))
, 6: if^#(true(), X, Y) -> c_5(X)
, 7: if^#(false(), X, Y) -> c_6(Y)
, 8: zero^#(0()) -> c_7()
, 9: zero^#(s(X)) -> c_8()
, 10: p^#(s(X)) -> c_9(X)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ YES(?,O(n^2)) ]
->{9} [ YES(?,O(1)) ]
->{8} [ YES(?,O(1)) ]
->{5} [ inherited ]
|
|->{2} [ MAYBE ]
|
`->{3} [ inherited ]
|
`->{2} [ NA ]
->{4} [ YES(?,O(1)) ]
->{1} [ inherited ]
|
|->{6} [ NA ]
|
`->{7} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{7}.
* Path {1}->{6}: NA
-----------------
The usable rules for this path are:
{ fact(X) -> if(zero(X), s(0()), prod(X, fact(p(X))))
, prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, zero(0()) -> true()
, zero(s(X)) -> false()
, p(s(X)) -> X
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {1}->{7}: NA
-----------------
The usable rules for this path are:
{ fact(X) -> if(zero(X), s(0()), prod(X, fact(p(X))))
, prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, zero(0()) -> true()
, zero(s(X)) -> false()
, p(s(X)) -> X
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fact) = {}, Uargs(if) = {}, Uargs(zero) = {}, Uargs(s) = {},
Uargs(prod) = {}, Uargs(p) = {}, Uargs(add) = {},
Uargs(fact^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(add^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(prod^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(zero^#) = {}, Uargs(p^#) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fact(x1) = [0 0] x1 + [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]
zero(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
prod(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
fact^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [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]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
prod^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
zero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {prod^#(0(), X) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(prod^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
prod^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
[2 2] [0 0] [7]
c_3() = [0]
[1]
* Path {5}: inherited
-------------------
This path is subsumed by the proof of path {5}->{3}->{2}.
* Path {5}->{2}: MAYBE
--------------------
The usable rules for this path are:
{ prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ prod^#(s(X), Y) -> c_4(add^#(Y, prod(X, Y)))
, add^#(0(), X) -> c_1(X)
, prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
Proof Output:
The input cannot be shown compatible
* Path {5}->{3}: inherited
------------------------
This path is subsumed by the proof of path {5}->{3}->{2}.
* Path {5}->{3}->{2}: NA
----------------------
The usable rules for this path are:
{ prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {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(fact) = {}, Uargs(if) = {}, Uargs(zero) = {}, Uargs(s) = {},
Uargs(prod) = {}, Uargs(p) = {}, Uargs(add) = {},
Uargs(fact^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(add^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(prod^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(zero^#) = {}, Uargs(p^#) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fact(x1) = [0 0] x1 + [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]
zero(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
prod(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
fact^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [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]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
prod^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
zero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {zero^#(0()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(zero^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
zero^#(x1) = [2 0] x1 + [7]
[2 2] [7]
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(fact) = {}, Uargs(if) = {}, Uargs(zero) = {}, Uargs(s) = {},
Uargs(prod) = {}, Uargs(p) = {}, Uargs(add) = {},
Uargs(fact^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(add^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(prod^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(zero^#) = {}, Uargs(p^#) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fact(x1) = [0 0] x1 + [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]
zero(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
prod(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
fact^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [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]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
prod^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
zero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {zero^#(s(X)) -> c_8()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(zero^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 0] x1 + [2]
[0 0] [2]
zero^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_8() = [0]
[1]
* Path {10}: 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(fact) = {}, Uargs(if) = {}, Uargs(zero) = {}, Uargs(s) = {},
Uargs(prod) = {}, Uargs(p) = {}, Uargs(add) = {},
Uargs(fact^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(add^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(prod^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(zero^#) = {}, Uargs(p^#) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fact(x1) = [0 0] x1 + [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]
zero(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 1] x1 + [0]
[0 1] [0]
0() = [0]
[0]
prod(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
fact^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [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]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
prod^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
zero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
p^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_9(x1) = [1 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {p^#(s(X)) -> c_9(X)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [2]
[0 0] [2]
p^#(x1) = [2 2] x1 + [7]
[2 0] [7]
c_9(x1) = [0 0] x1 + [0]
[0 0] [1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: fact^#(X) -> c_0(if^#(zero(X), s(0()), prod(X, fact(p(X)))))
, 2: add^#(0(), X) -> c_1(X)
, 3: add^#(s(X), Y) -> c_2(add^#(X, Y))
, 4: prod^#(0(), X) -> c_3()
, 5: prod^#(s(X), Y) -> c_4(add^#(Y, prod(X, Y)))
, 6: if^#(true(), X, Y) -> c_5(X)
, 7: if^#(false(), X, Y) -> c_6(Y)
, 8: zero^#(0()) -> c_7()
, 9: zero^#(s(X)) -> c_8()
, 10: p^#(s(X)) -> c_9(X)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ YES(?,O(n^1)) ]
->{9} [ YES(?,O(1)) ]
->{8} [ YES(?,O(1)) ]
->{5} [ inherited ]
|
|->{2} [ MAYBE ]
|
`->{3} [ inherited ]
|
`->{2} [ NA ]
->{4} [ YES(?,O(1)) ]
->{1} [ inherited ]
|
|->{6} [ NA ]
|
`->{7} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{7}.
* Path {1}->{6}: NA
-----------------
The usable rules for this path are:
{ fact(X) -> if(zero(X), s(0()), prod(X, fact(p(X))))
, prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, zero(0()) -> true()
, zero(s(X)) -> false()
, p(s(X)) -> X
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {1}->{7}: NA
-----------------
The usable rules for this path are:
{ fact(X) -> if(zero(X), s(0()), prod(X, fact(p(X))))
, prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, zero(0()) -> true()
, zero(s(X)) -> false()
, p(s(X)) -> X
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fact) = {}, Uargs(if) = {}, Uargs(zero) = {}, Uargs(s) = {},
Uargs(prod) = {}, Uargs(p) = {}, Uargs(add) = {},
Uargs(fact^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(add^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(prod^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(zero^#) = {}, Uargs(p^#) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fact(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
zero(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
prod(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
fact^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
prod^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
zero^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
p^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {prod^#(0(), X) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(prod^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
prod^#(x1, x2) = [1] x1 + [0] x2 + [7]
c_3() = [1]
* Path {5}: inherited
-------------------
This path is subsumed by the proof of path {5}->{3}->{2}.
* Path {5}->{2}: MAYBE
--------------------
The usable rules for this path are:
{ prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ prod^#(s(X), Y) -> c_4(add^#(Y, prod(X, Y)))
, add^#(0(), X) -> c_1(X)
, prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
Proof Output:
The input cannot be shown compatible
* Path {5}->{3}: inherited
------------------------
This path is subsumed by the proof of path {5}->{3}->{2}.
* Path {5}->{3}->{2}: NA
----------------------
The usable rules for this path are:
{ prod(0(), X) -> 0()
, prod(s(X), Y) -> add(Y, prod(X, Y))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {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(fact) = {}, Uargs(if) = {}, Uargs(zero) = {}, Uargs(s) = {},
Uargs(prod) = {}, Uargs(p) = {}, Uargs(add) = {},
Uargs(fact^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(add^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(prod^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(zero^#) = {}, Uargs(p^#) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fact(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
zero(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
prod(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
fact^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
prod^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
zero^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
p^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {zero^#(0()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(zero^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
zero^#(x1) = [1] x1 + [7]
c_7() = [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(fact) = {}, Uargs(if) = {}, Uargs(zero) = {}, Uargs(s) = {},
Uargs(prod) = {}, Uargs(p) = {}, Uargs(add) = {},
Uargs(fact^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(add^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(prod^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(zero^#) = {}, Uargs(p^#) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fact(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
zero(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
prod(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
fact^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
prod^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
zero^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
p^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {zero^#(s(X)) -> c_8()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(zero^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0] x1 + [7]
zero^#(x1) = [1] x1 + [7]
c_8() = [1]
* Path {10}: 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(fact) = {}, Uargs(if) = {}, Uargs(zero) = {}, Uargs(s) = {},
Uargs(prod) = {}, Uargs(p) = {}, Uargs(add) = {},
Uargs(fact^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
Uargs(add^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(prod^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(zero^#) = {}, Uargs(p^#) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fact(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
zero(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
prod(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
fact^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
prod^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
zero^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
p^#(x1) = [3] x1 + [0]
c_9(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {p^#(s(X)) -> c_9(X)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [7]
p^#(x1) = [1] x1 + [7]
c_9(x1) = [1] x1 + [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.