Tool CaT
stdout:
MAYBE
Problem:
min(x,0()) -> 0()
min(0(),y) -> 0()
min(s(x),s(y)) -> s(min(x,y))
max(x,0()) -> x
max(0(),y) -> y
max(s(x),s(y)) -> s(max(x,y))
minus(x,0()) -> x
minus(s(x),s(y)) -> s(minus(x,y))
gcd(s(x),s(y)) -> gcd(minus(max(x,y),min(x,transform(y))),s(min(x,y)))
transform(x) -> s(s(x))
transform(cons(x,y)) -> cons(cons(x,x),x)
transform(cons(x,y)) -> y
transform(s(x)) -> s(s(transform(x)))
cons(x,y) -> y
cons(x,cons(y,s(z))) -> cons(y,x)
cons(cons(x,z),s(y)) -> transform(x)
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:
{ min(x, 0()) -> 0()
, min(0(), y) -> 0()
, min(s(x), s(y)) -> s(min(x, y))
, max(x, 0()) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> s(minus(x, y))
, gcd(s(x), s(y)) ->
gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))
, cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: min^#(x, 0()) -> c_0()
, 2: min^#(0(), y) -> c_1()
, 3: min^#(s(x), s(y)) -> c_2(min^#(x, y))
, 4: max^#(x, 0()) -> c_3()
, 5: max^#(0(), y) -> c_4()
, 6: max^#(s(x), s(y)) -> c_5(max^#(x, y))
, 7: minus^#(x, 0()) -> c_6()
, 8: minus^#(s(x), s(y)) -> c_7(minus^#(x, y))
, 9: gcd^#(s(x), s(y)) ->
c_8(gcd^#(minus(max(x, y), min(x, transform(y))), s(min(x, y))))
, 10: transform^#(x) -> c_9()
, 11: transform^#(cons(x, y)) -> c_10(cons^#(cons(x, x), x))
, 12: transform^#(cons(x, y)) -> c_11()
, 13: transform^#(s(x)) -> c_12(transform^#(x))
, 14: cons^#(x, y) -> c_13()
, 15: cons^#(x, cons(y, s(z))) -> c_14(cons^#(y, x))
, 16: cons^#(cons(x, z), s(y)) -> c_15(transform^#(x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11,16,15,13} [ YES(?,O(n^1)) ]
|
|->{10} [ YES(?,O(n^1)) ]
|
|->{12} [ YES(?,O(n^1)) ]
|
`->{14} [ YES(?,O(n^1)) ]
->{9} [ NA ]
->{8} [ YES(?,O(n^2)) ]
|
`->{7} [ YES(?,O(n^2)) ]
->{6} [ YES(?,O(n^2)) ]
|
|->{4} [ YES(?,O(n^2)) ]
|
`->{5} [ YES(?,O(n^2)) ]
->{3} [ YES(?,O(n^2)) ]
|
|->{1} [ YES(?,O(n^2)) ]
|
`->{2} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {1}, Uargs(max^#) = {},
Uargs(c_5) = {}, Uargs(minus^#) = {}, Uargs(c_7) = {},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_10) = {}, Uargs(cons^#) = {}, Uargs(c_12) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
transform(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
min^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
transform^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
cons^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11() = [0]
[0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(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^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
min^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_2(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {3}->{1}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {1}, Uargs(max^#) = {},
Uargs(c_5) = {}, Uargs(minus^#) = {}, Uargs(c_7) = {},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_10) = {}, Uargs(cons^#) = {}, Uargs(c_12) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
transform(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
transform^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
cons^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11() = [0]
[0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(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^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {min^#(x, 0()) -> c_0()}
Weak Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
min^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
[0 0] [4 1] [0]
c_0() = [1]
[0]
c_2(x1) = [1 0] x1 + [6]
[0 0] [7]
* Path {3}->{2}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {1}, Uargs(max^#) = {},
Uargs(c_5) = {}, Uargs(minus^#) = {}, Uargs(c_7) = {},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_10) = {}, Uargs(cons^#) = {}, Uargs(c_12) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
transform(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
transform^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
cons^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11() = [0]
[0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(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^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {min^#(0(), y) -> c_1()}
Weak Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
min^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
[4 1] [2 0] [0]
c_1() = [1]
[0]
c_2(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {6}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(max^#) = {},
Uargs(c_5) = {1}, Uargs(minus^#) = {}, Uargs(c_7) = {},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_10) = {}, Uargs(cons^#) = {}, Uargs(c_12) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
transform(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
max^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
transform^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
cons^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11() = [0]
[0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(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^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
max^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_5(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {6}->{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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(max^#) = {},
Uargs(c_5) = {1}, Uargs(minus^#) = {}, Uargs(c_7) = {},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_10) = {}, Uargs(cons^#) = {}, Uargs(c_12) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
transform(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
transform^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
cons^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11() = [0]
[0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(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^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {max^#(x, 0()) -> c_3()}
Weak Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
max^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
[0 0] [4 1] [0]
c_3() = [1]
[0]
c_5(x1) = [1 0] x1 + [6]
[0 0] [7]
* Path {6}->{5}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(max^#) = {},
Uargs(c_5) = {1}, Uargs(minus^#) = {}, Uargs(c_7) = {},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_10) = {}, Uargs(cons^#) = {}, Uargs(c_12) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
transform(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
transform^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
cons^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11() = [0]
[0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(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^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {max^#(0(), y) -> c_4()}
Weak Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
max^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
[4 1] [2 0] [0]
c_4() = [1]
[0]
c_5(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {8}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(max^#) = {},
Uargs(c_5) = {}, Uargs(minus^#) = {}, Uargs(c_7) = {1},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_10) = {}, Uargs(cons^#) = {}, Uargs(c_12) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
transform(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_6() = [0]
[0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
transform^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
cons^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11() = [0]
[0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(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^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {minus^#(s(x), s(y)) -> c_7(minus^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
minus^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_7(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {8}->{7}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(max^#) = {},
Uargs(c_5) = {}, Uargs(minus^#) = {}, Uargs(c_7) = {1},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_10) = {}, Uargs(cons^#) = {}, Uargs(c_12) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
transform(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
transform^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
cons^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11() = [0]
[0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(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^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {minus^#(x, 0()) -> c_6()}
Weak Rules: {minus^#(s(x), s(y)) -> c_7(minus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
minus^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
[0 0] [4 1] [0]
c_6() = [1]
[0]
c_7(x1) = [1 0] x1 + [6]
[0 0] [7]
* Path {9}: NA
------------
The usable rules for this path are:
{ min(x, 0()) -> 0()
, min(0(), y) -> 0()
, min(s(x), s(y)) -> s(min(x, y))
, max(x, 0()) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> s(minus(x, y))
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))
, cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)}
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 {11,16,15,13}: YES(?,O(n^1))
---------------------------------
The usable rules for this path are:
{ cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {1}, Uargs(max) = {},
Uargs(minus) = {}, Uargs(gcd) = {}, Uargs(transform) = {},
Uargs(cons) = {1}, Uargs(min^#) = {}, Uargs(c_2) = {},
Uargs(max^#) = {}, Uargs(c_5) = {}, Uargs(minus^#) = {},
Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_8) = {},
Uargs(transform^#) = {}, Uargs(c_10) = {1}, Uargs(cons^#) = {1},
Uargs(c_12) = {1}, Uargs(c_14) = {1}, Uargs(c_15) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 1] [1]
max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
transform(x1) = [2 2] x1 + [1]
[2 2] [2]
cons(x1, x2) = [1 1] x1 + [1 0] x2 + [2]
[1 1] [0 1] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
transform^#(x1) = [2 2] x1 + [0]
[3 3] [0]
c_9() = [0]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
cons^#(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_11() = [0]
[0]
c_12(x1) = [1 0] x1 + [0]
[0 1] [0]
c_13() = [0]
[0]
c_14(x1) = [1 0] x1 + [0]
[0 1] [0]
c_15(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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:
{ transform^#(cons(x, y)) -> c_10(cons^#(cons(x, x), x))
, cons^#(cons(x, z), s(y)) -> c_15(transform^#(x))
, cons^#(x, cons(y, s(z))) -> c_14(cons^#(y, x))
, transform^#(s(x)) -> c_12(transform^#(x))}
Weak Rules:
{ cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(transform^#) = {}, Uargs(c_10) = {1}, Uargs(cons^#) = {},
Uargs(c_12) = {1}, Uargs(c_14) = {1}, Uargs(c_15) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [1]
[0 1] [2]
transform(x1) = [1 6] x1 + [2]
[4 1] [6]
cons(x1, x2) = [1 3] x1 + [1 0] x2 + [2]
[2 1] [0 1] [0]
transform^#(x1) = [2 1] x1 + [0]
[0 0] [0]
c_10(x1) = [2 0] x1 + [3]
[0 0] [0]
cons^#(x1, x2) = [0 1] x1 + [0 1] x2 + [0]
[0 2] [0 0] [0]
c_12(x1) = [1 0] x1 + [1]
[0 0] [0]
c_14(x1) = [1 0] x1 + [1]
[0 0] [0]
c_15(x1) = [1 0] x1 + [1]
[0 0] [0]
* Path {11,16,15,13}->{10}: YES(?,O(n^1))
---------------------------------------
The usable rules for this path are:
{ cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {1}, Uargs(max) = {},
Uargs(minus) = {}, Uargs(gcd) = {}, Uargs(transform) = {},
Uargs(cons) = {1}, Uargs(min^#) = {}, Uargs(c_2) = {},
Uargs(max^#) = {}, Uargs(c_5) = {}, Uargs(minus^#) = {},
Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_8) = {},
Uargs(transform^#) = {}, Uargs(c_10) = {1}, Uargs(cons^#) = {1},
Uargs(c_12) = {1}, Uargs(c_14) = {1}, Uargs(c_15) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 1] [1]
max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
transform(x1) = [3 2] x1 + [2]
[2 2] [2]
cons(x1, x2) = [1 2] x1 + [1 1] x2 + [1]
[1 1] [0 1] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
transform^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
cons^#(x1, x2) = [3 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11() = [0]
[0]
c_12(x1) = [1 0] x1 + [0]
[0 1] [0]
c_13() = [0]
[0]
c_14(x1) = [1 0] x1 + [0]
[0 1] [0]
c_15(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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: {transform^#(x) -> c_9()}
Weak Rules:
{ transform^#(cons(x, y)) -> c_10(cons^#(cons(x, x), x))
, cons^#(cons(x, z), s(y)) -> c_15(transform^#(x))
, cons^#(x, cons(y, s(z))) -> c_14(cons^#(y, x))
, transform^#(s(x)) -> c_12(transform^#(x))
, cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(transform^#) = {}, Uargs(c_10) = {1}, Uargs(cons^#) = {},
Uargs(c_12) = {1}, Uargs(c_14) = {1}, Uargs(c_15) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 1] x1 + [0]
[0 0] [0]
transform(x1) = [4 0] x1 + [0]
[2 2] [0]
cons(x1, x2) = [2 1] x1 + [1 0] x2 + [0]
[1 1] [0 1] [0]
transform^#(x1) = [0 0] x1 + [1]
[0 0] [4]
c_9() = [0]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 0] [2]
cons^#(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[4 0] [0 0] [4]
c_12(x1) = [1 0] x1 + [0]
[0 0] [3]
c_14(x1) = [1 0] x1 + [0]
[0 0] [3]
c_15(x1) = [1 0] x1 + [0]
[0 0] [3]
* Path {11,16,15,13}->{12}: YES(?,O(n^1))
---------------------------------------
The usable rules for this path are:
{ cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {1}, Uargs(max) = {},
Uargs(minus) = {}, Uargs(gcd) = {}, Uargs(transform) = {},
Uargs(cons) = {1}, Uargs(min^#) = {}, Uargs(c_2) = {},
Uargs(max^#) = {}, Uargs(c_5) = {}, Uargs(minus^#) = {},
Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_8) = {},
Uargs(transform^#) = {}, Uargs(c_10) = {1}, Uargs(cons^#) = {1},
Uargs(c_12) = {1}, Uargs(c_14) = {1}, Uargs(c_15) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 1] [1]
max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
transform(x1) = [3 2] x1 + [2]
[2 2] [2]
cons(x1, x2) = [1 2] x1 + [1 1] x2 + [1]
[1 1] [0 1] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
transform^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
cons^#(x1, x2) = [3 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11() = [0]
[0]
c_12(x1) = [1 0] x1 + [0]
[0 1] [0]
c_13() = [0]
[0]
c_14(x1) = [1 0] x1 + [0]
[0 1] [0]
c_15(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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: {transform^#(cons(x, y)) -> c_11()}
Weak Rules:
{ transform^#(cons(x, y)) -> c_10(cons^#(cons(x, x), x))
, cons^#(cons(x, z), s(y)) -> c_15(transform^#(x))
, cons^#(x, cons(y, s(z))) -> c_14(cons^#(y, x))
, transform^#(s(x)) -> c_12(transform^#(x))
, cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(transform^#) = {}, Uargs(c_10) = {1}, Uargs(cons^#) = {},
Uargs(c_12) = {1}, Uargs(c_14) = {1}, Uargs(c_15) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [0]
transform(x1) = [4 2] x1 + [0]
[6 2] [0]
cons(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[4 1] [0 1] [1]
transform^#(x1) = [1 2] x1 + [0]
[2 0] [0]
c_10(x1) = [2 0] x1 + [2]
[0 0] [0]
cons^#(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [6 0] [0]
c_11() = [1]
[0]
c_12(x1) = [1 0] x1 + [0]
[0 0] [0]
c_14(x1) = [1 0] x1 + [0]
[0 0] [0]
c_15(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {11,16,15,13}->{14}: YES(?,O(n^1))
---------------------------------------
The usable rules for this path are:
{ cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {1}, Uargs(max) = {},
Uargs(minus) = {}, Uargs(gcd) = {}, Uargs(transform) = {},
Uargs(cons) = {1}, Uargs(min^#) = {}, Uargs(c_2) = {},
Uargs(max^#) = {}, Uargs(c_5) = {}, Uargs(minus^#) = {},
Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_8) = {},
Uargs(transform^#) = {}, Uargs(c_10) = {1}, Uargs(cons^#) = {1},
Uargs(c_12) = {1}, Uargs(c_14) = {1}, Uargs(c_15) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 1] [1]
max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
transform(x1) = [3 2] x1 + [2]
[2 2] [2]
cons(x1, x2) = [1 2] x1 + [1 1] x2 + [1]
[1 1] [0 1] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
transform^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
cons^#(x1, x2) = [3 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11() = [0]
[0]
c_12(x1) = [1 0] x1 + [0]
[0 1] [0]
c_13() = [0]
[0]
c_14(x1) = [1 0] x1 + [0]
[0 1] [0]
c_15(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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: {cons^#(x, y) -> c_13()}
Weak Rules:
{ transform^#(cons(x, y)) -> c_10(cons^#(cons(x, x), x))
, cons^#(cons(x, z), s(y)) -> c_15(transform^#(x))
, cons^#(x, cons(y, s(z))) -> c_14(cons^#(y, x))
, transform^#(s(x)) -> c_12(transform^#(x))
, cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(transform^#) = {}, Uargs(c_10) = {1}, Uargs(cons^#) = {},
Uargs(c_12) = {1}, Uargs(c_14) = {1}, Uargs(c_15) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 0] x1 + [0]
[0 0] [0]
transform(x1) = [1 4] x1 + [0]
[0 7] [0]
cons(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[2 2] [0 2] [0]
transform^#(x1) = [0 0] x1 + [1]
[0 0] [0]
c_10(x1) = [1 0] x1 + [0]
[0 0] [0]
cons^#(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [0]
c_12(x1) = [1 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [1 0] x1 + [0]
[0 0] [0]
c_15(x1) = [1 0] x1 + [0]
[0 0] [0]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: min^#(x, 0()) -> c_0()
, 2: min^#(0(), y) -> c_1()
, 3: min^#(s(x), s(y)) -> c_2(min^#(x, y))
, 4: max^#(x, 0()) -> c_3()
, 5: max^#(0(), y) -> c_4()
, 6: max^#(s(x), s(y)) -> c_5(max^#(x, y))
, 7: minus^#(x, 0()) -> c_6()
, 8: minus^#(s(x), s(y)) -> c_7(minus^#(x, y))
, 9: gcd^#(s(x), s(y)) ->
c_8(gcd^#(minus(max(x, y), min(x, transform(y))), s(min(x, y))))
, 10: transform^#(x) -> c_9()
, 11: transform^#(cons(x, y)) -> c_10(cons^#(cons(x, x), x))
, 12: transform^#(cons(x, y)) -> c_11()
, 13: transform^#(s(x)) -> c_12(transform^#(x))
, 14: cons^#(x, y) -> c_13()
, 15: cons^#(x, cons(y, s(z))) -> c_14(cons^#(y, x))
, 16: cons^#(cons(x, z), s(y)) -> c_15(transform^#(x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11,16,15,13} [ inherited ]
|
|->{10} [ YES(?,O(n^1)) ]
|
|->{12} [ YES(?,O(n^1)) ]
|
`->{14} [ YES(?,O(n^1)) ]
->{9} [ MAYBE ]
->{8} [ YES(?,O(n^1)) ]
|
`->{7} [ YES(?,O(n^1)) ]
->{6} [ YES(?,O(n^1)) ]
|
|->{4} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {1}, Uargs(max^#) = {},
Uargs(c_5) = {}, Uargs(minus^#) = {}, Uargs(c_7) = {},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_10) = {}, Uargs(cons^#) = {}, Uargs(c_12) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
gcd(x1, x2) = [0] x1 + [0] x2 + [0]
transform(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
min^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
transform^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11() = [0]
c_12(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [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: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
min^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_2(x1) = [1] x1 + [7]
* 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {1}, Uargs(max^#) = {},
Uargs(c_5) = {}, Uargs(minus^#) = {}, Uargs(c_7) = {},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_10) = {}, Uargs(cons^#) = {}, Uargs(c_12) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
gcd(x1, x2) = [0] x1 + [0] x2 + [0]
transform(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
transform^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11() = [0]
c_12(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [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: {min^#(x, 0()) -> c_0()}
Weak Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
min^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_0() = [1]
c_2(x1) = [1] x1 + [7]
* 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {1}, Uargs(max^#) = {},
Uargs(c_5) = {}, Uargs(minus^#) = {}, Uargs(c_7) = {},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_10) = {}, Uargs(cons^#) = {}, Uargs(c_12) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
gcd(x1, x2) = [0] x1 + [0] x2 + [0]
transform(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
transform^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11() = [0]
c_12(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [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: {min^#(0(), y) -> c_1()}
Weak Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
min^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_1() = [1]
c_2(x1) = [1] x1 + [2]
* Path {6}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(max^#) = {},
Uargs(c_5) = {1}, Uargs(minus^#) = {}, Uargs(c_7) = {},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_10) = {}, Uargs(cons^#) = {}, Uargs(c_12) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
gcd(x1, x2) = [0] x1 + [0] x2 + [0]
transform(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
max^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
transform^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11() = [0]
c_12(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [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: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
max^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_5(x1) = [1] x1 + [7]
* Path {6}->{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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(max^#) = {},
Uargs(c_5) = {1}, Uargs(minus^#) = {}, Uargs(c_7) = {},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_10) = {}, Uargs(cons^#) = {}, Uargs(c_12) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
gcd(x1, x2) = [0] x1 + [0] x2 + [0]
transform(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
transform^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11() = [0]
c_12(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [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: {max^#(x, 0()) -> c_3()}
Weak Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
max^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_3() = [1]
c_5(x1) = [1] x1 + [7]
* Path {6}->{5}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(max^#) = {},
Uargs(c_5) = {1}, Uargs(minus^#) = {}, Uargs(c_7) = {},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_10) = {}, Uargs(cons^#) = {}, Uargs(c_12) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
gcd(x1, x2) = [0] x1 + [0] x2 + [0]
transform(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
transform^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11() = [0]
c_12(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [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: {max^#(0(), y) -> c_4()}
Weak Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
max^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_4() = [1]
c_5(x1) = [1] x1 + [2]
* Path {8}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(max^#) = {},
Uargs(c_5) = {}, Uargs(minus^#) = {}, Uargs(c_7) = {1},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_10) = {}, Uargs(cons^#) = {}, Uargs(c_12) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
gcd(x1, x2) = [0] x1 + [0] x2 + [0]
transform(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
minus^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
transform^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11() = [0]
c_12(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [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: {minus^#(s(x), s(y)) -> c_7(minus^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
minus^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_7(x1) = [1] x1 + [7]
* Path {8}->{7}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(max^#) = {},
Uargs(c_5) = {}, Uargs(minus^#) = {}, Uargs(c_7) = {1},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_10) = {}, Uargs(cons^#) = {}, Uargs(c_12) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
gcd(x1, x2) = [0] x1 + [0] x2 + [0]
transform(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
transform^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11() = [0]
c_12(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [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: {minus^#(x, 0()) -> c_6()}
Weak Rules: {minus^#(s(x), s(y)) -> c_7(minus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
minus^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_6() = [1]
c_7(x1) = [1] x1 + [7]
* Path {9}: MAYBE
---------------
The usable rules for this path are:
{ min(x, 0()) -> 0()
, min(0(), y) -> 0()
, min(s(x), s(y)) -> s(min(x, y))
, max(x, 0()) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> s(minus(x, y))
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))
, cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)}
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:
{ gcd^#(s(x), s(y)) ->
c_8(gcd^#(minus(max(x, y), min(x, transform(y))), s(min(x, y))))
, min(x, 0()) -> 0()
, min(0(), y) -> 0()
, min(s(x), s(y)) -> s(min(x, y))
, max(x, 0()) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> s(minus(x, y))
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))
, cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)}
Proof Output:
The input cannot be shown compatible
* Path {11,16,15,13}: inherited
-----------------------------
This path is subsumed by the proof of path {11,16,15,13}->{10}.
* Path {11,16,15,13}->{10}: YES(?,O(n^1))
---------------------------------------
The usable rules for this path are:
{ cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
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: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ transform^#(cons(x, y)) -> c_10(cons^#(cons(x, x), x))
, cons^#(cons(x, z), s(y)) -> c_15(transform^#(x))
, cons^#(x, cons(y, s(z))) -> c_14(cons^#(y, x))
, transform^#(s(x)) -> c_12(transform^#(x))
, transform^#(x) -> c_9()
, cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {1}, Uargs(transform) = {}, Uargs(cons) = {1},
Uargs(transform^#) = {}, Uargs(c_10) = {1}, Uargs(cons^#) = {1},
Uargs(c_12) = {1}, Uargs(c_14) = {1}, Uargs(c_15) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [1]
transform(x1) = [4] x1 + [4]
cons(x1, x2) = [2] x1 + [1] x2 + [2]
transform^#(x1) = [2] x1 + [4]
c_9() = [1]
c_10(x1) = [1] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [5]
c_12(x1) = [1] x1 + [1]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [3]
* Path {11,16,15,13}->{12}: YES(?,O(n^1))
---------------------------------------
The usable rules for this path are:
{ cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
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: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ transform^#(cons(x, y)) -> c_10(cons^#(cons(x, x), x))
, cons^#(cons(x, z), s(y)) -> c_15(transform^#(x))
, cons^#(x, cons(y, s(z))) -> c_14(cons^#(y, x))
, transform^#(s(x)) -> c_12(transform^#(x))
, transform^#(cons(x, y)) -> c_11()
, cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {1}, Uargs(transform) = {}, Uargs(cons) = {1},
Uargs(transform^#) = {}, Uargs(c_10) = {1}, Uargs(cons^#) = {1},
Uargs(c_12) = {1}, Uargs(c_14) = {1}, Uargs(c_15) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [1]
transform(x1) = [5] x1 + [3]
cons(x1, x2) = [3] x1 + [1] x2 + [1]
transform^#(x1) = [2] x1 + [2]
c_10(x1) = [1] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [2]
c_11() = [1]
c_12(x1) = [1] x1 + [1]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [1]
* Path {11,16,15,13}->{14}: YES(?,O(n^1))
---------------------------------------
The usable rules for this path are:
{ cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
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: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ transform^#(cons(x, y)) -> c_10(cons^#(cons(x, x), x))
, cons^#(cons(x, z), s(y)) -> c_15(transform^#(x))
, cons^#(x, cons(y, s(z))) -> c_14(cons^#(y, x))
, transform^#(s(x)) -> c_12(transform^#(x))
, cons^#(x, y) -> c_13()
, cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {1}, Uargs(transform) = {}, Uargs(cons) = {1},
Uargs(transform^#) = {}, Uargs(c_10) = {1}, Uargs(cons^#) = {1},
Uargs(c_12) = {1}, Uargs(c_14) = {1}, Uargs(c_15) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
transform(x1) = [4] x1 + [6]
cons(x1, x2) = [2] x1 + [1] x2 + [2]
transform^#(x1) = [2] x1 + [0]
c_10(x1) = [1] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_12(x1) = [1] x1 + [3]
c_13() = [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [3]
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ min(x, 0()) -> 0()
, min(0(), y) -> 0()
, min(s(x), s(y)) -> s(min(x, y))
, max(x, 0()) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> s(minus(x, y))
, gcd(s(x), s(y)) ->
gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))
, cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: min^#(x, 0()) -> c_0()
, 2: min^#(0(), y) -> c_1()
, 3: min^#(s(x), s(y)) -> c_2(min^#(x, y))
, 4: max^#(x, 0()) -> c_3(x)
, 5: max^#(0(), y) -> c_4(y)
, 6: max^#(s(x), s(y)) -> c_5(max^#(x, y))
, 7: minus^#(x, 0()) -> c_6(x)
, 8: minus^#(s(x), s(y)) -> c_7(minus^#(x, y))
, 9: gcd^#(s(x), s(y)) ->
c_8(gcd^#(minus(max(x, y), min(x, transform(y))), s(min(x, y))))
, 10: transform^#(x) -> c_9(x)
, 11: transform^#(cons(x, y)) -> c_10(cons^#(cons(x, x), x))
, 12: transform^#(cons(x, y)) -> c_11(y)
, 13: transform^#(s(x)) -> c_12(transform^#(x))
, 14: cons^#(x, y) -> c_13(y)
, 15: cons^#(x, cons(y, s(z))) -> c_14(cons^#(y, x))
, 16: cons^#(cons(x, z), s(y)) -> c_15(transform^#(x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11,16,15,13} [ YES(?,O(n^1)) ]
|
|->{10} [ YES(?,O(n^1)) ]
|
|->{12} [ YES(?,O(n^1)) ]
|
`->{14} [ YES(?,O(n^1)) ]
->{9} [ NA ]
->{8} [ YES(?,O(n^2)) ]
|
`->{7} [ YES(?,O(n^2)) ]
->{6} [ YES(?,O(n^2)) ]
|
|->{4} [ YES(?,O(n^2)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{3} [ YES(?,O(n^2)) ]
|
|->{1} [ YES(?,O(n^2)) ]
|
`->{2} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {1}, Uargs(max^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(minus^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(cons^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
transform(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
min^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
transform^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
cons^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(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^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
min^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_2(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {3}->{1}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {1}, Uargs(max^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(minus^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(cons^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
transform(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
transform^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
cons^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(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^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {min^#(x, 0()) -> c_0()}
Weak Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
min^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
[0 0] [4 1] [0]
c_0() = [1]
[0]
c_2(x1) = [1 0] x1 + [6]
[0 0] [7]
* Path {3}->{2}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {1}, Uargs(max^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(minus^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(cons^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
transform(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
transform^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
cons^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(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^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {min^#(0(), y) -> c_1()}
Weak Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
min^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
[4 1] [2 0] [0]
c_1() = [1]
[0]
c_2(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {6}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(max^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1},
Uargs(minus^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(cons^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
transform(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
max^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[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]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
transform^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
cons^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(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^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
max^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_5(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {6}->{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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(max^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1},
Uargs(minus^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(cons^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
transform(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
max^#(x1, x2) = [3 3] x1 + [0 0] x2 + [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]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
transform^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
cons^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(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^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {max^#(x, 0()) -> c_3(x)}
Weak Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [1 2] x1 + [0]
[0 1] [0]
max^#(x1, x2) = [2 2] x1 + [0 2] x2 + [0]
[4 1] [3 2] [0]
c_3(x1) = [0 0] x1 + [1]
[0 1] [0]
c_5(x1) = [1 0] x1 + [0]
[2 0] [0]
* Path {6}->{5}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(max^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1},
Uargs(minus^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(cons^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
transform(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
max^#(x1, x2) = [0 0] x1 + [3 3] x2 + [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]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
transform^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
cons^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(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: {max^#(0(), y) -> c_4(y)}
Weak Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
max^#(x1, x2) = [0 2] x1 + [0 0] x2 + [4]
[0 2] [0 1] [0]
c_4(x1) = [0 0] x1 + [1]
[0 1] [0]
c_5(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {8}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(max^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(minus^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {1},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(cons^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
transform(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
transform^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
cons^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(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^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {minus^#(s(x), s(y)) -> c_7(minus^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
minus^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_7(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {8}->{7}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(max^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(minus^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {1},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(cons^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
transform(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [1 1] x1 + [0]
[0 0] [0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
transform^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
cons^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(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^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {minus^#(x, 0()) -> c_6(x)}
Weak Rules: {minus^#(s(x), s(y)) -> c_7(minus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [1 2] x1 + [0]
[0 1] [0]
minus^#(x1, x2) = [2 2] x1 + [0 2] x2 + [0]
[4 1] [3 2] [0]
c_6(x1) = [0 0] x1 + [1]
[0 1] [0]
c_7(x1) = [1 0] x1 + [0]
[2 0] [0]
* Path {9}: NA
------------
The usable rules for this path are:
{ min(x, 0()) -> 0()
, min(0(), y) -> 0()
, min(s(x), s(y)) -> s(min(x, y))
, max(x, 0()) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> s(minus(x, y))
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))
, cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)}
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 {11,16,15,13}: YES(?,O(n^1))
---------------------------------
The usable rules for this path are:
{ cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {1}, Uargs(max) = {},
Uargs(minus) = {}, Uargs(gcd) = {}, Uargs(transform) = {1},
Uargs(cons) = {1, 2}, Uargs(min^#) = {}, Uargs(c_2) = {},
Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(minus^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_8) = {},
Uargs(transform^#) = {1}, Uargs(c_9) = {}, Uargs(c_10) = {1},
Uargs(cons^#) = {1, 2}, Uargs(c_11) = {}, Uargs(c_12) = {1},
Uargs(c_13) = {}, Uargs(c_14) = {1}, Uargs(c_15) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 1] [1]
max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
transform(x1) = [2 2] x1 + [1]
[2 2] [2]
cons(x1, x2) = [1 1] x1 + [1 0] x2 + [1]
[1 1] [0 1] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
transform^#(x1) = [2 2] x1 + [0]
[3 3] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
cons^#(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [1 0] x1 + [0]
[0 1] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [1 0] x1 + [0]
[0 1] [0]
c_15(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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:
{ transform^#(cons(x, y)) -> c_10(cons^#(cons(x, x), x))
, cons^#(cons(x, z), s(y)) -> c_15(transform^#(x))
, cons^#(x, cons(y, s(z))) -> c_14(cons^#(y, x))
, transform^#(s(x)) -> c_12(transform^#(x))}
Weak Rules:
{ cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(transform^#) = {}, Uargs(c_10) = {1}, Uargs(cons^#) = {},
Uargs(c_12) = {1}, Uargs(c_14) = {1}, Uargs(c_15) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [1]
[0 1] [2]
transform(x1) = [1 6] x1 + [2]
[4 1] [6]
cons(x1, x2) = [1 3] x1 + [1 0] x2 + [2]
[2 1] [0 1] [0]
transform^#(x1) = [2 1] x1 + [0]
[0 0] [0]
c_10(x1) = [2 0] x1 + [3]
[0 0] [0]
cons^#(x1, x2) = [0 1] x1 + [0 1] x2 + [0]
[0 2] [0 0] [0]
c_12(x1) = [1 0] x1 + [1]
[0 0] [0]
c_14(x1) = [1 0] x1 + [1]
[0 0] [0]
c_15(x1) = [1 0] x1 + [1]
[0 0] [0]
* Path {11,16,15,13}->{10}: YES(?,O(n^1))
---------------------------------------
The usable rules for this path are:
{ cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {1}, Uargs(max) = {},
Uargs(minus) = {}, Uargs(gcd) = {}, Uargs(transform) = {1},
Uargs(cons) = {1, 2}, Uargs(min^#) = {}, Uargs(c_2) = {},
Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(minus^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_8) = {},
Uargs(transform^#) = {1}, Uargs(c_9) = {1}, Uargs(c_10) = {1},
Uargs(cons^#) = {1, 2}, Uargs(c_11) = {}, Uargs(c_12) = {1},
Uargs(c_13) = {}, Uargs(c_14) = {1}, Uargs(c_15) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 1] [1]
max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
transform(x1) = [2 2] x1 + [1]
[2 2] [2]
cons(x1, x2) = [1 1] x1 + [1 0] x2 + [2]
[1 1] [0 1] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
transform^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
cons^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [1 0] x1 + [0]
[0 1] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [1 0] x1 + [0]
[0 1] [0]
c_15(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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: {transform^#(x) -> c_9(x)}
Weak Rules:
{ transform^#(cons(x, y)) -> c_10(cons^#(cons(x, x), x))
, cons^#(cons(x, z), s(y)) -> c_15(transform^#(x))
, cons^#(x, cons(y, s(z))) -> c_14(cons^#(y, x))
, transform^#(s(x)) -> c_12(transform^#(x))
, cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(transform^#) = {}, Uargs(c_9) = {1}, Uargs(c_10) = {1},
Uargs(cons^#) = {}, Uargs(c_12) = {1}, Uargs(c_14) = {1},
Uargs(c_15) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [0]
transform(x1) = [2 5] x1 + [4]
[1 4] [0]
cons(x1, x2) = [1 4] x1 + [1 0] x2 + [2]
[1 1] [0 1] [0]
transform^#(x1) = [1 0] x1 + [4]
[0 0] [4]
c_9(x1) = [1 0] x1 + [1]
[0 0] [0]
c_10(x1) = [1 0] x1 + [2]
[0 0] [4]
cons^#(x1, x2) = [0 1] x1 + [0 1] x2 + [4]
[0 0] [0 0] [0]
c_12(x1) = [1 0] x1 + [0]
[0 0] [2]
c_14(x1) = [1 0] x1 + [0]
[0 0] [0]
c_15(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {11,16,15,13}->{12}: YES(?,O(n^1))
---------------------------------------
The usable rules for this path are:
{ cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {1}, Uargs(max) = {},
Uargs(minus) = {}, Uargs(gcd) = {}, Uargs(transform) = {1},
Uargs(cons) = {1, 2}, Uargs(min^#) = {}, Uargs(c_2) = {},
Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(minus^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_8) = {},
Uargs(transform^#) = {1}, Uargs(c_9) = {}, Uargs(c_10) = {1},
Uargs(cons^#) = {1, 2}, Uargs(c_11) = {1}, Uargs(c_12) = {1},
Uargs(c_13) = {}, Uargs(c_14) = {1}, Uargs(c_15) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 1] [1]
max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
transform(x1) = [3 1] x1 + [1]
[2 3] [2]
cons(x1, x2) = [1 1] x1 + [1 0] x2 + [1]
[2 1] [0 1] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
transform^#(x1) = [1 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
cons^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [1 0] x1 + [0]
[0 1] [0]
c_12(x1) = [1 0] x1 + [0]
[0 1] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [1 0] x1 + [0]
[0 1] [0]
c_15(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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: {transform^#(cons(x, y)) -> c_11(y)}
Weak Rules:
{ transform^#(cons(x, y)) -> c_10(cons^#(cons(x, x), x))
, cons^#(cons(x, z), s(y)) -> c_15(transform^#(x))
, cons^#(x, cons(y, s(z))) -> c_14(cons^#(y, x))
, transform^#(s(x)) -> c_12(transform^#(x))
, cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(transform^#) = {}, Uargs(c_10) = {1}, Uargs(cons^#) = {},
Uargs(c_11) = {1}, Uargs(c_12) = {1}, Uargs(c_14) = {1},
Uargs(c_15) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [0]
transform(x1) = [6 2] x1 + [0]
[3 2] [0]
cons(x1, x2) = [2 2] x1 + [2 2] x2 + [0]
[1 1] [1 1] [0]
transform^#(x1) = [2 1] x1 + [1]
[2 0] [0]
c_10(x1) = [1 0] x1 + [0]
[0 0] [0]
cons^#(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [0]
c_11(x1) = [1 1] x1 + [0]
[0 0] [0]
c_12(x1) = [1 0] x1 + [0]
[0 0] [0]
c_14(x1) = [1 0] x1 + [0]
[0 0] [0]
c_15(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {11,16,15,13}->{14}: YES(?,O(n^1))
---------------------------------------
The usable rules for this path are:
{ cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {1}, Uargs(max) = {},
Uargs(minus) = {}, Uargs(gcd) = {}, Uargs(transform) = {1},
Uargs(cons) = {1, 2}, Uargs(min^#) = {}, Uargs(c_2) = {},
Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(minus^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_8) = {},
Uargs(transform^#) = {1}, Uargs(c_9) = {}, Uargs(c_10) = {1},
Uargs(cons^#) = {1, 2}, Uargs(c_11) = {}, Uargs(c_12) = {1},
Uargs(c_13) = {1}, Uargs(c_14) = {1}, Uargs(c_15) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 1] [1]
max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
transform(x1) = [2 2] x1 + [1]
[2 2] [2]
cons(x1, x2) = [1 1] x1 + [1 0] x2 + [2]
[1 1] [0 1] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
transform^#(x1) = [3 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
cons^#(x1, x2) = [3 3] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [1 0] x1 + [0]
[0 1] [0]
c_13(x1) = [1 0] x1 + [0]
[0 1] [0]
c_14(x1) = [1 0] x1 + [0]
[0 1] [0]
c_15(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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: {cons^#(x, y) -> c_13(y)}
Weak Rules:
{ transform^#(cons(x, y)) -> c_10(cons^#(cons(x, x), x))
, cons^#(cons(x, z), s(y)) -> c_15(transform^#(x))
, cons^#(x, cons(y, s(z))) -> c_14(cons^#(y, x))
, transform^#(s(x)) -> c_12(transform^#(x))
, cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(transform^#) = {}, Uargs(c_10) = {1}, Uargs(cons^#) = {},
Uargs(c_12) = {1}, Uargs(c_13) = {1}, Uargs(c_14) = {1},
Uargs(c_15) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [2]
transform(x1) = [2 2] x1 + [0]
[3 5] [4]
cons(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[1 2] [0 2] [2]
transform^#(x1) = [1 3] x1 + [0]
[0 0] [0]
c_10(x1) = [1 0] x1 + [0]
[0 0] [0]
cons^#(x1, x2) = [1 1] x1 + [1 0] x2 + [1]
[7 1] [0 0] [0]
c_12(x1) = [1 0] x1 + [3]
[0 0] [0]
c_13(x1) = [1 0] x1 + [0]
[0 0] [0]
c_14(x1) = [1 0] x1 + [0]
[0 0] [0]
c_15(x1) = [1 0] x1 + [3]
[0 0] [2]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: min^#(x, 0()) -> c_0()
, 2: min^#(0(), y) -> c_1()
, 3: min^#(s(x), s(y)) -> c_2(min^#(x, y))
, 4: max^#(x, 0()) -> c_3(x)
, 5: max^#(0(), y) -> c_4(y)
, 6: max^#(s(x), s(y)) -> c_5(max^#(x, y))
, 7: minus^#(x, 0()) -> c_6(x)
, 8: minus^#(s(x), s(y)) -> c_7(minus^#(x, y))
, 9: gcd^#(s(x), s(y)) ->
c_8(gcd^#(minus(max(x, y), min(x, transform(y))), s(min(x, y))))
, 10: transform^#(x) -> c_9(x)
, 11: transform^#(cons(x, y)) -> c_10(cons^#(cons(x, x), x))
, 12: transform^#(cons(x, y)) -> c_11(y)
, 13: transform^#(s(x)) -> c_12(transform^#(x))
, 14: cons^#(x, y) -> c_13(y)
, 15: cons^#(x, cons(y, s(z))) -> c_14(cons^#(y, x))
, 16: cons^#(cons(x, z), s(y)) -> c_15(transform^#(x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11,16,15,13} [ inherited ]
|
|->{10} [ YES(?,O(n^1)) ]
|
|->{12} [ YES(?,O(n^1)) ]
|
`->{14} [ YES(?,O(n^1)) ]
->{9} [ MAYBE ]
->{8} [ YES(?,O(n^1)) ]
|
`->{7} [ YES(?,O(n^1)) ]
->{6} [ YES(?,O(n^1)) ]
|
|->{4} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {1}, Uargs(max^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(minus^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(cons^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
gcd(x1, x2) = [0] x1 + [0] x2 + [0]
transform(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
min^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
transform^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
min^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_2(x1) = [1] x1 + [7]
* 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {1}, Uargs(max^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(minus^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(cons^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
gcd(x1, x2) = [0] x1 + [0] x2 + [0]
transform(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
transform^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {min^#(x, 0()) -> c_0()}
Weak Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
min^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_0() = [1]
c_2(x1) = [1] x1 + [7]
* 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {1}, Uargs(max^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(minus^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(cons^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
gcd(x1, x2) = [0] x1 + [0] x2 + [0]
transform(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
transform^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {min^#(0(), y) -> c_1()}
Weak Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
min^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_1() = [1]
c_2(x1) = [1] x1 + [2]
* Path {6}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(max^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1},
Uargs(minus^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(cons^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
gcd(x1, x2) = [0] x1 + [0] x2 + [0]
transform(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
max^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
transform^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
max^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_5(x1) = [1] x1 + [7]
* Path {6}->{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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(max^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1},
Uargs(minus^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(cons^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
gcd(x1, x2) = [0] x1 + [0] x2 + [0]
transform(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
max^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
transform^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {max^#(x, 0()) -> c_3(x)}
Weak Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [1] x1 + [2]
max^#(x1, x2) = [2] x1 + [2] x2 + [2]
c_3(x1) = [0] x1 + [1]
c_5(x1) = [1] x1 + [5]
* Path {6}->{5}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(max^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1},
Uargs(minus^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(cons^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
gcd(x1, x2) = [0] x1 + [0] x2 + [0]
transform(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
max^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
transform^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {max^#(0(), y) -> c_4(y)}
Weak Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [1] x1 + [2]
max^#(x1, x2) = [2] x1 + [2] x2 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
* Path {8}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(max^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(minus^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {1},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(cons^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
gcd(x1, x2) = [0] x1 + [0] x2 + [0]
transform(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
minus^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
transform^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {minus^#(s(x), s(y)) -> c_7(minus^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
minus^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_7(x1) = [1] x1 + [7]
* Path {8}->{7}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(minus) = {},
Uargs(gcd) = {}, Uargs(transform) = {}, Uargs(cons) = {},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(max^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(minus^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {1},
Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(transform^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(cons^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
max(x1, x2) = [0] x1 + [0] x2 + [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
gcd(x1, x2) = [0] x1 + [0] x2 + [0]
transform(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
max^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
minus^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
transform^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {minus^#(x, 0()) -> c_6(x)}
Weak Rules: {minus^#(s(x), s(y)) -> c_7(minus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [1] x1 + [2]
minus^#(x1, x2) = [2] x1 + [2] x2 + [2]
c_6(x1) = [0] x1 + [1]
c_7(x1) = [1] x1 + [5]
* Path {9}: MAYBE
---------------
The usable rules for this path are:
{ min(x, 0()) -> 0()
, min(0(), y) -> 0()
, min(s(x), s(y)) -> s(min(x, y))
, max(x, 0()) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> s(minus(x, y))
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))
, cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)}
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:
{ gcd^#(s(x), s(y)) ->
c_8(gcd^#(minus(max(x, y), min(x, transform(y))), s(min(x, y))))
, min(x, 0()) -> 0()
, min(0(), y) -> 0()
, min(s(x), s(y)) -> s(min(x, y))
, max(x, 0()) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> s(minus(x, y))
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))
, cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)}
Proof Output:
The input cannot be shown compatible
* Path {11,16,15,13}: inherited
-----------------------------
This path is subsumed by the proof of path {11,16,15,13}->{10}.
* Path {11,16,15,13}->{10}: YES(?,O(n^1))
---------------------------------------
The usable rules for this path are:
{ cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
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: YES(?,O(n^1))
Input Problem: runtime-complexity with respect to
Rules:
{ transform^#(cons(x, y)) -> c_10(cons^#(cons(x, x), x))
, cons^#(cons(x, z), s(y)) -> c_15(transform^#(x))
, cons^#(x, cons(y, s(z))) -> c_14(cons^#(y, x))
, transform^#(s(x)) -> c_12(transform^#(x))
, transform^#(x) -> c_9(x)
, cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {1}, Uargs(transform) = {1}, Uargs(cons) = {1, 2},
Uargs(transform^#) = {1}, Uargs(c_9) = {1}, Uargs(c_10) = {1},
Uargs(cons^#) = {1, 2}, Uargs(c_12) = {1}, Uargs(c_14) = {1},
Uargs(c_15) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [1]
transform(x1) = [4] x1 + [3]
cons(x1, x2) = [2] x1 + [1] x2 + [3]
transform^#(x1) = [2] x1 + [1]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [1] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [2]
c_12(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [4]
* Path {11,16,15,13}->{12}: YES(?,O(n^1))
---------------------------------------
The usable rules for this path are:
{ cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
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: YES(?,O(n^1))
Input Problem: runtime-complexity with respect to
Rules:
{ transform^#(cons(x, y)) -> c_10(cons^#(cons(x, x), x))
, cons^#(cons(x, z), s(y)) -> c_15(transform^#(x))
, cons^#(x, cons(y, s(z))) -> c_14(cons^#(y, x))
, transform^#(s(x)) -> c_12(transform^#(x))
, transform^#(cons(x, y)) -> c_11(y)
, cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {1}, Uargs(transform) = {1}, Uargs(cons) = {1, 2},
Uargs(transform^#) = {1}, Uargs(c_10) = {1},
Uargs(cons^#) = {1, 2}, Uargs(c_11) = {1}, Uargs(c_12) = {1},
Uargs(c_14) = {1}, Uargs(c_15) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [1]
transform(x1) = [4] x1 + [3]
cons(x1, x2) = [2] x1 + [1] x2 + [1]
transform^#(x1) = [2] x1 + [0]
c_10(x1) = [1] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_11(x1) = [2] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [1]
* Path {11,16,15,13}->{14}: YES(?,O(n^1))
---------------------------------------
The usable rules for this path are:
{ cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
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: YES(?,O(n^1))
Input Problem: runtime-complexity with respect to
Rules:
{ transform^#(cons(x, y)) -> c_10(cons^#(cons(x, x), x))
, cons^#(cons(x, z), s(y)) -> c_15(transform^#(x))
, cons^#(x, cons(y, s(z))) -> c_14(cons^#(y, x))
, transform^#(s(x)) -> c_12(transform^#(x))
, cons^#(x, y) -> c_13(y)
, cons(x, y) -> y
, cons(x, cons(y, s(z))) -> cons(y, x)
, cons(cons(x, z), s(y)) -> transform(x)
, transform(x) -> s(s(x))
, transform(cons(x, y)) -> cons(cons(x, x), x)
, transform(cons(x, y)) -> y
, transform(s(x)) -> s(s(transform(x)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {1}, Uargs(transform) = {1}, Uargs(cons) = {1, 2},
Uargs(transform^#) = {1}, Uargs(c_10) = {1},
Uargs(cons^#) = {1, 2}, Uargs(c_12) = {1}, Uargs(c_13) = {1},
Uargs(c_14) = {1}, Uargs(c_15) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [1]
transform(x1) = [5] x1 + [4]
cons(x1, x2) = [3] x1 + [1] x2 + [2]
transform^#(x1) = [2] x1 + [2]
c_10(x1) = [1] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [1]
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.