Tool CaT
stdout:
MAYBE
Problem:
f(x,y) -> cond(and(isNat(x),isNat(y)),x,y)
cond(tt(),x,y) -> f(s(x),s(y))
isNat(s(x)) -> isNat(x)
isNat(0()) -> tt()
and(tt(),tt()) -> tt()
and(ff(),x) -> ff()
and(x,ff()) -> ff()
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:
{ f(x, y) -> cond(and(isNat(x), isNat(y)), x, y)
, cond(tt(), x, y) -> f(s(x), s(y))
, isNat(s(x)) -> isNat(x)
, isNat(0()) -> tt()
, and(tt(), tt()) -> tt()
, and(ff(), x) -> ff()
, and(x, ff()) -> ff()}
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: f^#(x, y) -> c_0(cond^#(and(isNat(x), isNat(y)), x, y))
, 2: cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
, 3: isNat^#(s(x)) -> c_2(isNat^#(x))
, 4: isNat^#(0()) -> c_3()
, 5: and^#(tt(), tt()) -> c_4()
, 6: and^#(ff(), x) -> c_5()
, 7: and^#(x, ff()) -> c_6()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ YES(?,O(1)) ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^2)) ]
->{1,2} [ NA ]
Sub-problems:
-------------
* Path {1,2}: NA
--------------
The usable rules for this path are:
{ isNat(s(x)) -> isNat(x)
, isNat(0()) -> tt()
, and(tt(), tt()) -> tt()
, and(ff(), x) -> ff()
, and(x, ff()) -> ff()}
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 {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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {1}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
cond(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]
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]
tt() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
ff() = [0]
[0]
[0]
f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(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_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isNat^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_3() = [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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6() = [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_2(isNat^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_2) = {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_2(x1) = [1 0 0] x1 + [1]
[2 0 2] [0]
[0 0 0] [0]
* Path {3}->{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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {1}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
cond(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]
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]
tt() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
ff() = [0]
[0]
[0]
f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(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_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isNat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_3() = [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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6() = [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_3()}
Weak Rules: {isNat^#(s(x)) -> c_2(isNat^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_2) = {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_2(x1) = [1 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [0]
c_3() = [1]
[0]
[0]
* Path {5}: 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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
cond(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]
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]
tt() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
ff() = [0]
[0]
[0]
f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(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_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isNat^#(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]
c_3() = [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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6() = [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^#(tt(), tt()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
tt() = [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_4() = [0]
[1]
[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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
cond(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]
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]
tt() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
ff() = [0]
[0]
[0]
f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(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_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isNat^#(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]
c_3() = [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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6() = [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^#(ff(), x) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ff() = [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_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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
cond(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]
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]
tt() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
ff() = [0]
[0]
[0]
f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(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_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isNat^#(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]
c_3() = [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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6() = [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, ff()) -> c_6()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ff() = [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_6() = [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: f^#(x, y) -> c_0(cond^#(and(isNat(x), isNat(y)), x, y))
, 2: cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
, 3: isNat^#(s(x)) -> c_2(isNat^#(x))
, 4: isNat^#(0()) -> c_3()
, 5: and^#(tt(), tt()) -> c_4()
, 6: and^#(ff(), x) -> c_5()
, 7: and^#(x, ff()) -> c_6()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ YES(?,O(1)) ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{1,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ isNat(s(x)) -> isNat(x)
, isNat(0()) -> tt()
, and(tt(), tt()) -> tt()
, and(ff(), x) -> ff()
, and(x, ff()) -> ff()}
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:
{ f^#(x, y) -> c_0(cond^#(and(isNat(x), isNat(y)), x, y))
, cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
, isNat(s(x)) -> isNat(x)
, isNat(0()) -> tt()
, and(tt(), tt()) -> tt()
, and(ff(), x) -> ff()
, and(x, ff()) -> ff()}
Proof Output:
The input cannot be shown compatible
* Path {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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {1}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 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]
tt() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
ff() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
c_3() = [0]
[0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6() = [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_2(isNat^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_2) = {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_2(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {3}->{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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {1}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 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]
tt() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
ff() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
c_3() = [0]
[0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6() = [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_3()}
Weak Rules: {isNat^#(s(x)) -> c_2(isNat^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_2) = {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_2(x1) = [1 0] x1 + [5]
[2 0] [3]
c_3() = [1]
[0]
* Path {5}: 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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 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]
tt() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
ff() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6() = [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^#(tt(), tt()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
tt() = [2]
[2]
and^#(x1, x2) = [2 2] x1 + [0 0] x2 + [7]
[2 2] [0 2] [3]
c_4() = [0]
[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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 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]
tt() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
ff() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6() = [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^#(ff(), x) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ff() = [2]
[2]
and^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
[2 2] [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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 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]
tt() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
ff() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6() = [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, ff()) -> c_6()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ff() = [2]
[2]
and^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
[0 0] [2 2] [7]
c_6() = [0]
[1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(x, y) -> c_0(cond^#(and(isNat(x), isNat(y)), x, y))
, 2: cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
, 3: isNat^#(s(x)) -> c_2(isNat^#(x))
, 4: isNat^#(0()) -> c_3()
, 5: and^#(tt(), tt()) -> c_4()
, 6: and^#(ff(), x) -> c_5()
, 7: and^#(x, ff()) -> c_6()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ YES(?,O(1)) ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{1,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ isNat(s(x)) -> isNat(x)
, isNat(0()) -> tt()
, and(tt(), tt()) -> tt()
, and(ff(), x) -> ff()
, and(x, ff()) -> ff()}
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:
{ f^#(x, y) -> c_0(cond^#(and(isNat(x), isNat(y)), x, y))
, cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
, isNat(s(x)) -> isNat(x)
, isNat(0()) -> tt()
, and(tt(), tt()) -> tt()
, and(ff(), x) -> ff()
, and(x, ff()) -> ff()}
Proof Output:
The input cannot be shown compatible
* Path {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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {1}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
tt() = [0]
s(x1) = [1] x1 + [0]
0() = [0]
ff() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
isNat^#(x1) = [3] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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_2(isNat^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
isNat^#(x1) = [2] x1 + [0]
c_2(x1) = [1] x1 + [7]
* Path {3}->{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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {1}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
tt() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
ff() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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_3()}
Weak Rules: {isNat^#(s(x)) -> c_2(isNat^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
0() = [2]
isNat^#(x1) = [2] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [1]
* Path {5}: 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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
tt() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
ff() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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^#(tt(), tt()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
tt() = [2]
and^#(x1, x2) = [2] x1 + [2] x2 + [7]
c_4() = [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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
tt() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
ff() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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^#(ff(), x) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ff() = [7]
and^#(x1, x2) = [1] x1 + [0] x2 + [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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
tt() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
ff() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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, ff()) -> c_6()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ff() = [7]
and^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_6() = [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:
{ f(x, y) -> cond(and(isNat(x), isNat(y)), x, y)
, cond(tt(), x, y) -> f(s(x), s(y))
, isNat(s(x)) -> isNat(x)
, isNat(0()) -> tt()
, and(tt(), tt()) -> tt()
, and(ff(), x) -> ff()
, and(x, ff()) -> ff()}
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: f^#(x, y) -> c_0(cond^#(and(isNat(x), isNat(y)), x, y))
, 2: cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
, 3: isNat^#(s(x)) -> c_2(isNat^#(x))
, 4: isNat^#(0()) -> c_3()
, 5: and^#(tt(), tt()) -> c_4()
, 6: and^#(ff(), x) -> c_5()
, 7: and^#(x, ff()) -> c_6()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ YES(?,O(1)) ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^2)) ]
->{1,2} [ NA ]
Sub-problems:
-------------
* Path {1,2}: NA
--------------
The usable rules for this path are:
{ isNat(s(x)) -> isNat(x)
, isNat(0()) -> tt()
, and(tt(), tt()) -> tt()
, and(ff(), x) -> ff()
, and(x, ff()) -> ff()}
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 {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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {1}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
cond(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]
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]
tt() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
ff() = [0]
[0]
[0]
f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(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_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isNat^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_3() = [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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6() = [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_2(isNat^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_2) = {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_2(x1) = [1 0 0] x1 + [1]
[2 0 2] [0]
[0 0 0] [0]
* Path {3}->{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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {1}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
cond(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]
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]
tt() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
ff() = [0]
[0]
[0]
f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(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_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isNat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_3() = [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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6() = [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_3()}
Weak Rules: {isNat^#(s(x)) -> c_2(isNat^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_2) = {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_2(x1) = [1 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [0]
c_3() = [1]
[0]
[0]
* Path {5}: 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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
cond(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]
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]
tt() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
ff() = [0]
[0]
[0]
f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(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_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isNat^#(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]
c_3() = [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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6() = [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^#(tt(), tt()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
tt() = [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_4() = [0]
[1]
[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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
cond(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]
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]
tt() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
ff() = [0]
[0]
[0]
f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(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_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isNat^#(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]
c_3() = [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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6() = [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^#(ff(), x) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ff() = [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_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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
cond(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]
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]
tt() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
ff() = [0]
[0]
[0]
f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(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_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isNat^#(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]
c_3() = [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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6() = [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, ff()) -> c_6()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ff() = [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_6() = [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: f^#(x, y) -> c_0(cond^#(and(isNat(x), isNat(y)), x, y))
, 2: cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
, 3: isNat^#(s(x)) -> c_2(isNat^#(x))
, 4: isNat^#(0()) -> c_3()
, 5: and^#(tt(), tt()) -> c_4()
, 6: and^#(ff(), x) -> c_5()
, 7: and^#(x, ff()) -> c_6()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ YES(?,O(1)) ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{1,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ isNat(s(x)) -> isNat(x)
, isNat(0()) -> tt()
, and(tt(), tt()) -> tt()
, and(ff(), x) -> ff()
, and(x, ff()) -> ff()}
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:
{ f^#(x, y) -> c_0(cond^#(and(isNat(x), isNat(y)), x, y))
, cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
, isNat(s(x)) -> isNat(x)
, isNat(0()) -> tt()
, and(tt(), tt()) -> tt()
, and(ff(), x) -> ff()
, and(x, ff()) -> ff()}
Proof Output:
The input cannot be shown compatible
* Path {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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {1}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 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]
tt() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
ff() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
c_3() = [0]
[0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6() = [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_2(isNat^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_2) = {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_2(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {3}->{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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {1}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 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]
tt() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
ff() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
c_3() = [0]
[0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6() = [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_3()}
Weak Rules: {isNat^#(s(x)) -> c_2(isNat^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_2) = {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_2(x1) = [1 0] x1 + [5]
[2 0] [3]
c_3() = [1]
[0]
* Path {5}: 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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 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]
tt() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
ff() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6() = [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^#(tt(), tt()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
tt() = [2]
[2]
and^#(x1, x2) = [2 2] x1 + [0 0] x2 + [7]
[2 2] [0 2] [3]
c_4() = [0]
[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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 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]
tt() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
ff() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6() = [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^#(ff(), x) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ff() = [2]
[2]
and^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
[2 2] [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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 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]
tt() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
ff() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
isNat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6() = [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, ff()) -> c_6()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ff() = [2]
[2]
and^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
[0 0] [2 2] [7]
c_6() = [0]
[1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(x, y) -> c_0(cond^#(and(isNat(x), isNat(y)), x, y))
, 2: cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
, 3: isNat^#(s(x)) -> c_2(isNat^#(x))
, 4: isNat^#(0()) -> c_3()
, 5: and^#(tt(), tt()) -> c_4()
, 6: and^#(ff(), x) -> c_5()
, 7: and^#(x, ff()) -> c_6()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ YES(?,O(1)) ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{1,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ isNat(s(x)) -> isNat(x)
, isNat(0()) -> tt()
, and(tt(), tt()) -> tt()
, and(ff(), x) -> ff()
, and(x, ff()) -> ff()}
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:
{ f^#(x, y) -> c_0(cond^#(and(isNat(x), isNat(y)), x, y))
, cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
, isNat(s(x)) -> isNat(x)
, isNat(0()) -> tt()
, and(tt(), tt()) -> tt()
, and(ff(), x) -> ff()
, and(x, ff()) -> ff()}
Proof Output:
The input cannot be shown compatible
* Path {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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {1}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
tt() = [0]
s(x1) = [1] x1 + [0]
0() = [0]
ff() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
isNat^#(x1) = [3] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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_2(isNat^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
isNat^#(x1) = [2] x1 + [0]
c_2(x1) = [1] x1 + [7]
* Path {3}->{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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {1}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
tt() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
ff() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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_3()}
Weak Rules: {isNat^#(s(x)) -> c_2(isNat^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
0() = [2]
isNat^#(x1) = [2] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [1]
* Path {5}: 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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
tt() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
ff() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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^#(tt(), tt()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
tt() = [2]
and^#(x1, x2) = [2] x1 + [2] x2 + [7]
c_4() = [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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
tt() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
ff() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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^#(ff(), x) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ff() = [7]
and^#(x1, x2) = [1] x1 + [0] x2 + [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(f) = {}, Uargs(cond) = {}, Uargs(and) = {},
Uargs(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(isNat^#) = {},
Uargs(c_2) = {}, Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
isNat(x1) = [0] x1 + [0]
tt() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
ff() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
isNat^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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, ff()) -> c_6()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ff() = [7]
and^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_6() = [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.