Tool CaT
stdout:
MAYBE
Problem:
f(0()) -> true()
f(1()) -> false()
f(s(x)) -> f(x)
if(true(),x,y) -> x
if(false(),x,y) -> y
g(s(x),s(y)) -> if(f(x),s(x),s(y))
g(x,c(y)) -> g(x,g(s(c(y)),y))
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
YES(?,O(n^2))
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, g(s(x), s(y)) -> if(f(x), s(x), s(y))
, g(x, c(y)) -> g(x, g(s(c(y)), y))}
Proof Output:
'wdg' proved the best result:
Details:
--------
'wdg' succeeded with the following output:
'wdg'
-----
Answer: YES(?,O(n^2))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, g(s(x), s(y)) -> if(f(x), s(x), s(y))
, g(x, c(y)) -> g(x, g(s(c(y)), y))}
Proof Output:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(0()) -> c_0()
, 2: f^#(1()) -> c_1()
, 3: f^#(s(x)) -> c_2(f^#(x))
, 4: if^#(true(), x, y) -> c_3()
, 5: if^#(false(), x, y) -> c_4()
, 6: g^#(s(x), s(y)) -> c_5(if^#(f(x), s(x), s(y)))
, 7: g^#(x, c(y)) -> c_6(g^#(x, g(s(c(y)), y)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ YES(?,O(n^2)) ]
|
`->{6} [ YES(?,O(n^2)) ]
|
|->{4} [ YES(?,O(n^2)) ]
|
`->{5} [ YES(?,O(n^2)) ]
->{3} [ YES(?,O(n^1)) ]
|
|->{1} [ YES(?,O(n^1)) ]
|
`->{2} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* 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(s) = {}, Uargs(if) = {}, Uargs(g) = {},
Uargs(c) = {}, Uargs(f^#) = {}, Uargs(c_2) = {1}, Uargs(if^#) = {},
Uargs(g^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
1() = [0]
[0]
false() = [0]
[0]
s(x1) = [1 2] 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]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c(x1) = [0 0] x1 + [0]
[0 0] [0]
f^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
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: {f^#(s(x)) -> c_2(f^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
f^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_2(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {3}->{1}: 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(s) = {}, Uargs(if) = {}, Uargs(g) = {},
Uargs(c) = {}, Uargs(f^#) = {}, Uargs(c_2) = {1}, Uargs(if^#) = {},
Uargs(g^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
1() = [0]
[0]
false() = [0]
[0]
s(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]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c(x1) = [0 0] x1 + [0]
[0 0] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
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: {f^#(0()) -> c_0()}
Weak Rules: {f^#(s(x)) -> c_2(f^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [1]
[0 0] [3]
f^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_0() = [1]
[0]
c_2(x1) = [1 0] x1 + [5]
[2 0] [3]
* Path {3}->{2}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(g) = {},
Uargs(c) = {}, Uargs(f^#) = {}, Uargs(c_2) = {1}, Uargs(if^#) = {},
Uargs(g^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
1() = [0]
[0]
false() = [0]
[0]
s(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]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c(x1) = [0 0] x1 + [0]
[0 0] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
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: {f^#(1()) -> c_1()}
Weak Rules: {f^#(s(x)) -> c_2(f^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1() = [2]
[2]
s(x1) = [1 2] x1 + [1]
[0 0] [3]
f^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_1() = [1]
[0]
c_2(x1) = [1 0] x1 + [5]
[2 0] [3]
* Path {7}: YES(?,O(n^2))
-----------------------
The usable rules for this path are:
{ g(s(x), s(y)) -> if(f(x), s(x), s(y))
, g(x, c(y)) -> g(x, g(s(c(y)), y))
, f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(if) = {1}, Uargs(g) = {2},
Uargs(c) = {}, Uargs(f^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {},
Uargs(g^#) = {2}, Uargs(c_5) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 1] x1 + [0]
[0 1] [0]
0() = [0]
[2]
true() = [1]
[0]
1() = [0]
[2]
false() = [1]
[0]
s(x1) = [0 2] x1 + [0]
[0 1] [1]
if(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 1] [0 1] [1 2] [0]
g(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[1 2] [2 0] [1]
c(x1) = [1 2] x1 + [2]
[0 0] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(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]
c_3() = [0]
[0]
c_4() = [0]
[0]
g^#(x1, x2) = [3 3] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
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: {g^#(x, c(y)) -> c_6(g^#(x, g(s(c(y)), y)))}
Weak Rules:
{ g(s(x), s(y)) -> if(f(x), s(x), s(y))
, g(x, c(y)) -> g(x, g(s(c(y)), y))
, f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y}
Proof Output:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(g) = {},
Uargs(c) = {}, Uargs(g^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
1() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [4 0] x2 + [4 0] x3 + [0]
[0 0] [0 4] [0 2] [0]
g(x1, x2) = [2 0] x1 + [0 0] x2 + [0]
[2 2] [0 0] [0]
c(x1) = [0 0] x1 + [2]
[0 0] [0]
g^#(x1, x2) = [0 0] x1 + [2 2] x2 + [0]
[4 4] [0 0] [0]
c_6(x1) = [3 0] x1 + [3]
[0 0] [0]
* Path {7}->{6}: YES(?,O(n^2))
----------------------------
The usable rules for this path are:
{ g(s(x), s(y)) -> if(f(x), s(x), s(y))
, g(x, c(y)) -> g(x, g(s(c(y)), y))
, f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(if) = {1}, Uargs(g) = {2},
Uargs(c) = {}, Uargs(f^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {1},
Uargs(g^#) = {2}, Uargs(c_5) = {1}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 1] x1 + [0]
[0 0] [0]
0() = [0]
[2]
true() = [1]
[0]
1() = [0]
[2]
false() = [1]
[0]
s(x1) = [0 2] x1 + [0]
[0 1] [1]
if(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 0] [0 2] [0 1] [0]
g(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[1 2] [1 0] [1]
c(x1) = [1 0] x1 + [3]
[0 0] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3) = [1 0] x1 + [0 0] x2 + [0 1] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
g^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
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: {g^#(s(x), s(y)) -> c_5(if^#(f(x), s(x), s(y)))}
Weak Rules:
{ g^#(x, c(y)) -> c_6(g^#(x, g(s(c(y)), y)))
, g(s(x), s(y)) -> if(f(x), s(x), s(y))
, g(x, c(y)) -> g(x, g(s(c(y)), y))
, f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y}
Proof Output:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(g) = {},
Uargs(c) = {}, Uargs(if^#) = {}, Uargs(g^#) = {}, Uargs(c_5) = {1},
Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
1() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [4 0] x2 + [1 0] x3 + [0]
[0 0] [0 4] [0 4] [0]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c(x1) = [0 0] x1 + [0]
[0 0] [1]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
g^#(x1, x2) = [0 0] x1 + [0 4] x2 + [1]
[0 0] [0 2] [0]
c_5(x1) = [1 0] x1 + [0]
[0 0] [0]
c_6(x1) = [2 2] x1 + [1]
[0 0] [2]
* Path {7}->{6}->{4}: YES(?,O(n^2))
---------------------------------
The usable rules for this path are:
{ g(s(x), s(y)) -> if(f(x), s(x), s(y))
, g(x, c(y)) -> g(x, g(s(c(y)), y))
, f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(if) = {1}, Uargs(g) = {2},
Uargs(c) = {}, Uargs(f^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {1},
Uargs(g^#) = {2}, Uargs(c_5) = {1}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 2] x1 + [0]
[0 2] [0]
0() = [0]
[1]
true() = [0]
[0]
1() = [0]
[2]
false() = [0]
[0]
s(x1) = [0 2] x1 + [0]
[0 1] [1]
if(x1, x2, x3) = [2 1] x1 + [1 0] x2 + [1 0] x3 + [1]
[0 0] [0 1] [0 1] [0]
g(x1, x2) = [3 2] x1 + [1 0] x2 + [0]
[0 1] [2 0] [2]
c(x1) = [1 2] x1 + [3]
[0 0] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
g^#(x1, x2) = [0 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {if^#(true(), x, y) -> c_3()}
Weak Rules:
{ g^#(s(x), s(y)) -> c_5(if^#(f(x), s(x), s(y)))
, g^#(x, c(y)) -> c_6(g^#(x, g(s(c(y)), y)))
, g(s(x), s(y)) -> if(f(x), s(x), s(y))
, g(x, c(y)) -> g(x, g(s(c(y)), y))
, f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y}
Proof Output:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(g) = {},
Uargs(c) = {}, Uargs(if^#) = {}, Uargs(g^#) = {}, Uargs(c_5) = {1},
Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 1] x1 + [0]
[0 2] [0]
0() = [0]
[4]
true() = [2]
[2]
1() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 2] x1 + [0]
[0 1] [0]
if(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 0] [0 1] [0 1] [0]
g(x1, x2) = [0 2] x1 + [1 0] x2 + [0]
[4 0] [4 0] [0]
c(x1) = [1 2] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3) = [2 2] x1 + [0 0] x2 + [1 2] x3 + [0]
[2 0] [0 0] [0 0] [0]
c_3() = [1]
[0]
g^#(x1, x2) = [4 4] x1 + [4 0] x2 + [0]
[4 4] [2 5] [0]
c_5(x1) = [2 0] x1 + [0]
[2 0] [0]
c_6(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {7}->{6}->{5}: YES(?,O(n^2))
---------------------------------
The usable rules for this path are:
{ g(s(x), s(y)) -> if(f(x), s(x), s(y))
, g(x, c(y)) -> g(x, g(s(c(y)), y))
, f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(if) = {1}, Uargs(g) = {2},
Uargs(c) = {}, Uargs(f^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {1},
Uargs(g^#) = {2}, Uargs(c_5) = {1}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 2] x1 + [0]
[0 2] [0]
0() = [0]
[1]
true() = [0]
[0]
1() = [0]
[2]
false() = [0]
[0]
s(x1) = [0 2] x1 + [0]
[0 1] [1]
if(x1, x2, x3) = [2 1] x1 + [1 0] x2 + [1 0] x3 + [1]
[0 0] [0 1] [0 1] [0]
g(x1, x2) = [3 2] x1 + [1 0] x2 + [0]
[0 1] [2 0] [2]
c(x1) = [1 2] x1 + [3]
[0 0] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
g^#(x1, x2) = [0 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
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: {if^#(false(), x, y) -> c_4()}
Weak Rules:
{ g^#(s(x), s(y)) -> c_5(if^#(f(x), s(x), s(y)))
, g^#(x, c(y)) -> c_6(g^#(x, g(s(c(y)), y)))
, g(s(x), s(y)) -> if(f(x), s(x), s(y))
, g(x, c(y)) -> g(x, g(s(c(y)), y))
, f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y}
Proof Output:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(g) = {},
Uargs(c) = {}, Uargs(if^#) = {}, Uargs(g^#) = {}, Uargs(c_5) = {1},
Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
1() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [3]
[0 0] [1]
if(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [4 0] x3 + [0]
[0 0] [0 2] [0 1] [0]
g(x1, x2) = [0 4] x1 + [0 4] x2 + [7]
[0 0] [0 1] [2]
c(x1) = [0 0] x1 + [0]
[0 1] [2]
if^#(x1, x2, x3) = [2 0] x1 + [1 1] x2 + [0 0] x3 + [2]
[0 0] [0 0] [0 0] [0]
c_4() = [1]
[0]
g^#(x1, x2) = [1 6] x1 + [0 0] x2 + [4]
[0 4] [0 2] [4]
c_5(x1) = [2 0] x1 + [0]
[0 0] [6]
c_6(x1) = [1 0] x1 + [0]
[0 0] [7]Tool RC1
stdout:
MAYBE
Tool RC2
stdout:
YES(?,O(n^2))
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: YES(?,O(n^2))
Input Problem: runtime-complexity with respect to
Rules:
{ f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, g(s(x), s(y)) -> if(f(x), s(x), s(y))
, g(x, c(y)) -> g(x, g(s(c(y)), y))}
Proof Output:
'wdg' proved the best result:
Details:
--------
'wdg' succeeded with the following output:
'wdg'
-----
Answer: YES(?,O(n^2))
Input Problem: runtime-complexity with respect to
Rules:
{ f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, g(s(x), s(y)) -> if(f(x), s(x), s(y))
, g(x, c(y)) -> g(x, g(s(c(y)), y))}
Proof Output:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(0()) -> c_0()
, 2: f^#(1()) -> c_1()
, 3: f^#(s(x)) -> c_2(f^#(x))
, 4: if^#(true(), x, y) -> c_3(x)
, 5: if^#(false(), x, y) -> c_4(y)
, 6: g^#(s(x), s(y)) -> c_5(if^#(f(x), s(x), s(y)))
, 7: g^#(x, c(y)) -> c_6(g^#(x, g(s(c(y)), y)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ YES(?,O(n^2)) ]
|
`->{6} [ YES(?,O(n^2)) ]
|
|->{4} [ YES(?,O(n^2)) ]
|
`->{5} [ YES(?,O(n^2)) ]
->{3} [ YES(?,O(n^1)) ]
|
|->{1} [ YES(?,O(n^1)) ]
|
`->{2} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* 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(s) = {}, Uargs(if) = {}, Uargs(g) = {},
Uargs(c) = {}, Uargs(f^#) = {}, Uargs(c_2) = {1}, Uargs(if^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(g^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
1() = [0]
[0]
false() = [0]
[0]
s(x1) = [1 2] 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]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c(x1) = [0 0] x1 + [0]
[0 0] [0]
f^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
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: {f^#(s(x)) -> c_2(f^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
f^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_2(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {3}->{1}: 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(s) = {}, Uargs(if) = {}, Uargs(g) = {},
Uargs(c) = {}, Uargs(f^#) = {}, Uargs(c_2) = {1}, Uargs(if^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(g^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
1() = [0]
[0]
false() = [0]
[0]
s(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]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c(x1) = [0 0] x1 + [0]
[0 0] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
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: {f^#(0()) -> c_0()}
Weak Rules: {f^#(s(x)) -> c_2(f^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [1]
[0 0] [3]
f^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_0() = [1]
[0]
c_2(x1) = [1 0] x1 + [5]
[2 0] [3]
* Path {3}->{2}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(g) = {},
Uargs(c) = {}, Uargs(f^#) = {}, Uargs(c_2) = {1}, Uargs(if^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(g^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
1() = [0]
[0]
false() = [0]
[0]
s(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]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c(x1) = [0 0] x1 + [0]
[0 0] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
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: {f^#(1()) -> c_1()}
Weak Rules: {f^#(s(x)) -> c_2(f^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1() = [2]
[2]
s(x1) = [1 2] x1 + [1]
[0 0] [3]
f^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_1() = [1]
[0]
c_2(x1) = [1 0] x1 + [5]
[2 0] [3]
* Path {7}: YES(?,O(n^2))
-----------------------
The usable rules for this path are:
{ g(s(x), s(y)) -> if(f(x), s(x), s(y))
, g(x, c(y)) -> g(x, g(s(c(y)), y))
, f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(if) = {1}, Uargs(g) = {2},
Uargs(c) = {}, Uargs(f^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(g^#) = {2},
Uargs(c_5) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 1] x1 + [0]
[0 1] [0]
0() = [0]
[2]
true() = [1]
[0]
1() = [0]
[2]
false() = [1]
[0]
s(x1) = [0 2] x1 + [0]
[0 1] [1]
if(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 1] [0 1] [1 2] [0]
g(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[1 2] [2 0] [1]
c(x1) = [1 2] x1 + [2]
[0 0] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(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]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [3 3] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
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: {g^#(x, c(y)) -> c_6(g^#(x, g(s(c(y)), y)))}
Weak Rules:
{ g(s(x), s(y)) -> if(f(x), s(x), s(y))
, g(x, c(y)) -> g(x, g(s(c(y)), y))
, f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y}
Proof Output:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(g) = {},
Uargs(c) = {}, Uargs(g^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
1() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [4 0] x2 + [4 0] x3 + [0]
[0 0] [0 4] [0 2] [0]
g(x1, x2) = [2 0] x1 + [0 0] x2 + [0]
[2 2] [0 0] [0]
c(x1) = [0 0] x1 + [2]
[0 0] [0]
g^#(x1, x2) = [0 0] x1 + [2 2] x2 + [0]
[4 4] [0 0] [0]
c_6(x1) = [3 0] x1 + [3]
[0 0] [0]
* Path {7}->{6}: YES(?,O(n^2))
----------------------------
The usable rules for this path are:
{ g(s(x), s(y)) -> if(f(x), s(x), s(y))
, g(x, c(y)) -> g(x, g(s(c(y)), y))
, f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(if) = {1}, Uargs(g) = {2},
Uargs(c) = {}, Uargs(f^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {1},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(g^#) = {2},
Uargs(c_5) = {1}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 1] x1 + [0]
[0 0] [0]
0() = [0]
[2]
true() = [1]
[0]
1() = [0]
[2]
false() = [1]
[0]
s(x1) = [0 2] x1 + [0]
[0 1] [1]
if(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 0] [0 2] [0 1] [0]
g(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[1 2] [1 0] [1]
c(x1) = [1 0] x1 + [3]
[0 0] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3) = [1 0] x1 + [0 0] x2 + [0 1] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
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: {g^#(s(x), s(y)) -> c_5(if^#(f(x), s(x), s(y)))}
Weak Rules:
{ g^#(x, c(y)) -> c_6(g^#(x, g(s(c(y)), y)))
, g(s(x), s(y)) -> if(f(x), s(x), s(y))
, g(x, c(y)) -> g(x, g(s(c(y)), y))
, f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y}
Proof Output:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(g) = {},
Uargs(c) = {}, Uargs(if^#) = {}, Uargs(g^#) = {}, Uargs(c_5) = {1},
Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
1() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [4 0] x2 + [1 0] x3 + [0]
[0 0] [0 4] [0 4] [0]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c(x1) = [0 0] x1 + [0]
[0 0] [1]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
g^#(x1, x2) = [0 0] x1 + [0 4] x2 + [1]
[0 0] [0 2] [0]
c_5(x1) = [1 0] x1 + [0]
[0 0] [0]
c_6(x1) = [2 2] x1 + [1]
[0 0] [2]
* Path {7}->{6}->{4}: YES(?,O(n^2))
---------------------------------
The usable rules for this path are:
{ g(s(x), s(y)) -> if(f(x), s(x), s(y))
, g(x, c(y)) -> g(x, g(s(c(y)), y))
, f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(if) = {1}, Uargs(g) = {2},
Uargs(c) = {}, Uargs(f^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {1},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(g^#) = {2},
Uargs(c_5) = {1}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 1] x1 + [0]
[0 1] [0]
0() = [0]
[2]
true() = [0]
[1]
1() = [0]
[2]
false() = [0]
[1]
s(x1) = [0 2] x1 + [0]
[0 1] [2]
if(x1, x2, x3) = [1 1] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 0] [0 1] [0 2] [0]
g(x1, x2) = [2 0] x1 + [1 0] x2 + [2]
[0 2] [2 0] [2]
c(x1) = [1 2] x1 + [3]
[0 0] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3) = [3 0] x1 + [3 3] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3(x1) = [1 1] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [0 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {if^#(true(), x, y) -> c_3(x)}
Weak Rules:
{ g^#(s(x), s(y)) -> c_5(if^#(f(x), s(x), s(y)))
, g^#(x, c(y)) -> c_6(g^#(x, g(s(c(y)), y)))
, g(s(x), s(y)) -> if(f(x), s(x), s(y))
, g(x, c(y)) -> g(x, g(s(c(y)), y))
, f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y}
Proof Output:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(g) = {},
Uargs(c) = {}, Uargs(if^#) = {}, Uargs(c_3) = {}, Uargs(g^#) = {},
Uargs(c_5) = {1}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 1] x1 + [0]
[0 0] [1]
0() = [0]
[4]
true() = [3]
[1]
1() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [1]
[0 1] [1]
if(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[2 0] [0 1] [0 1] [0]
g(x1, x2) = [2 0] x1 + [0 0] x2 + [0]
[0 4] [1 1] [0]
c(x1) = [1 2] x1 + [6]
[0 0] [0]
if^#(x1, x2, x3) = [2 2] x1 + [0 1] x2 + [1 0] x3 + [0]
[2 4] [0 0] [0 0] [0]
c_3(x1) = [0 1] x1 + [1]
[0 0] [0]
g^#(x1, x2) = [2 6] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [2 0] x1 + [0]
[0 0] [0]
c_6(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {7}->{6}->{5}: YES(?,O(n^2))
---------------------------------
The usable rules for this path are:
{ g(s(x), s(y)) -> if(f(x), s(x), s(y))
, g(x, c(y)) -> g(x, g(s(c(y)), y))
, f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(if) = {1}, Uargs(g) = {2},
Uargs(c) = {}, Uargs(f^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {1},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(g^#) = {2},
Uargs(c_5) = {1}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 1] x1 + [0]
[0 1] [0]
0() = [0]
[2]
true() = [0]
[1]
1() = [0]
[2]
false() = [0]
[1]
s(x1) = [0 2] x1 + [0]
[0 1] [2]
if(x1, x2, x3) = [1 1] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 0] [0 1] [0 2] [0]
g(x1, x2) = [2 0] x1 + [1 0] x2 + [2]
[0 2] [2 0] [2]
c(x1) = [1 2] x1 + [3]
[0 0] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3) = [3 0] x1 + [3 3] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [1 1] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [0 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {if^#(false(), x, y) -> c_4(y)}
Weak Rules:
{ g^#(s(x), s(y)) -> c_5(if^#(f(x), s(x), s(y)))
, g^#(x, c(y)) -> c_6(g^#(x, g(s(c(y)), y)))
, g(s(x), s(y)) -> if(f(x), s(x), s(y))
, g(x, c(y)) -> g(x, g(s(c(y)), y))
, f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y}
Proof Output:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(g) = {},
Uargs(c) = {}, Uargs(if^#) = {}, Uargs(c_4) = {}, Uargs(g^#) = {},
Uargs(c_5) = {1}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 1] x1 + [0]
[0 0] [2]
0() = [0]
[0]
true() = [0]
[0]
1() = [0]
[2]
false() = [2]
[2]
s(x1) = [0 4] x1 + [0]
[0 1] [1]
if(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 0] [0 1] [0 1] [0]
g(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 1] [1 0] [3]
c(x1) = [1 4] x1 + [7]
[0 0] [1]
if^#(x1, x2, x3) = [2 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[2 2] [0 0] [0 0] [2]
c_4(x1) = [0 0] x1 + [1]
[0 0] [0]
g^#(x1, x2) = [2 0] x1 + [0 0] x2 + [0]
[1 1] [0 4] [7]
c_5(x1) = [2 0] x1 + [0]
[0 2] [0]
c_6(x1) = [1 0] x1 + [0]
[0 0] [7]