Tool CaT
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | SK90 2.27 |
|---|
stdout:
MAYBE
Problem:
fib(0()) -> 0()
fib(s(0())) -> s(0())
fib(s(s(0()))) -> s(0())
fib(s(s(x))) -> sp(g(x))
g(0()) -> pair(s(0()),0())
g(s(0())) -> pair(s(0()),s(0()))
g(s(x)) -> np(g(x))
sp(pair(x,y)) -> +(x,y)
np(pair(x,y)) -> pair(+(x,y),x)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
Proof:
OpenTool IRC1
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | SK90 2.27 |
|---|
stdout:
MAYBE
Tool IRC2
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | SK90 2.27 |
|---|
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(0()))) -> s(0())
, fib(s(s(x))) -> sp(g(x))
, g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, sp(pair(x, y)) -> +(x, y)
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
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: fib^#(0()) -> c_0()
, 2: fib^#(s(0())) -> c_1()
, 3: fib^#(s(s(0()))) -> c_2()
, 4: fib^#(s(s(x))) -> c_3(sp^#(g(x)))
, 5: g^#(0()) -> c_4()
, 6: g^#(s(0())) -> c_5()
, 7: g^#(s(x)) -> c_6(np^#(g(x)))
, 8: sp^#(pair(x, y)) -> c_7(+^#(x, y))
, 9: np^#(pair(x, y)) -> c_8(+^#(x, y))
, 10: +^#(x, 0()) -> c_9()
, 11: +^#(x, s(y)) -> c_10(+^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ inherited ]
|
`->{9} [ inherited ]
|
|->{10} [ MAYBE ]
|
`->{11} [ inherited ]
|
`->{10} [ NA ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ inherited ]
|
`->{8} [ inherited ]
|
|->{10} [ NA ]
|
`->{11} [ inherited ]
|
`->{10} [ NA ]
->{3} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sp(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
np(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
+(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
fib^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sp^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
np^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
+^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {fib^#(0()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
fib^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_0() = [0]
[1]
[1]
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sp(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
np(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
+(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
fib^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sp^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
np^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
+^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {fib^#(s(0())) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
[0]
s(x1) = [0 3 2] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fib^#(x1) = [2 0 0] x1 + [3]
[0 0 0] [7]
[0 0 0] [7]
c_1() = [0]
[1]
[1]
* 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(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sp(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
np(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
+(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
fib^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sp^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
np^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
+^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {fib^#(s(s(0()))) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
[0]
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [0]
[0 0 0] [0]
fib^#(x1) = [2 0 0] x1 + [3]
[0 0 0] [7]
[2 2 0] [3]
c_2() = [0]
[1]
[1]
* Path {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{8}->{11}->{10}.
* Path {4}->{8}: inherited
------------------------
This path is subsumed by the proof of path {4}->{8}->{11}->{10}.
* Path {4}->{8}->{10}: NA
-----------------------
The usable rules for this path are:
{ g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {4}->{8}->{11}: inherited
------------------------------
This path is subsumed by the proof of path {4}->{8}->{11}->{10}.
* Path {4}->{8}->{11}->{10}: NA
-----------------------------
The usable rules for this path are:
{ g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {5}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sp(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
np(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
+(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
fib^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sp^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
np^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
+^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(0()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
g^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_4() = [0]
[1]
[1]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sp(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
np(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
+(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
fib^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sp^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
np^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
+^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(s(0())) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
[0]
s(x1) = [0 3 2] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1) = [2 0 0] x1 + [3]
[0 0 0] [7]
[0 0 0] [7]
c_5() = [0]
[1]
[1]
* Path {7}: inherited
-------------------
This path is subsumed by the proof of path {7}->{9}->{11}->{10}.
* Path {7}->{9}: inherited
------------------------
This path is subsumed by the proof of path {7}->{9}->{11}->{10}.
* Path {7}->{9}->{10}: MAYBE
--------------------------
The usable rules for this path are:
{ g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ np^#(pair(x, y)) -> c_8(+^#(x, y))
, g^#(s(x)) -> c_6(np^#(g(x)))
, +^#(x, 0()) -> c_9()
, g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {7}->{9}->{11}: inherited
------------------------------
This path is subsumed by the proof of path {7}->{9}->{11}->{10}.
* Path {7}->{9}->{11}->{10}: NA
-----------------------------
The usable rules for this path are:
{ g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: fib^#(0()) -> c_0()
, 2: fib^#(s(0())) -> c_1()
, 3: fib^#(s(s(0()))) -> c_2()
, 4: fib^#(s(s(x))) -> c_3(sp^#(g(x)))
, 5: g^#(0()) -> c_4()
, 6: g^#(s(0())) -> c_5()
, 7: g^#(s(x)) -> c_6(np^#(g(x)))
, 8: sp^#(pair(x, y)) -> c_7(+^#(x, y))
, 9: np^#(pair(x, y)) -> c_8(+^#(x, y))
, 10: +^#(x, 0()) -> c_9()
, 11: +^#(x, s(y)) -> c_10(+^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ inherited ]
|
`->{9} [ inherited ]
|
|->{10} [ MAYBE ]
|
`->{11} [ inherited ]
|
`->{10} [ NA ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ inherited ]
|
`->{8} [ inherited ]
|
|->{10} [ NA ]
|
`->{11} [ inherited ]
|
`->{10} [ NA ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sp(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
np(x1) = [0 0] x1 + [0]
[0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
sp^#(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
np^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
+^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {fib^#(0()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
fib^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_0() = [0]
[1]
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sp(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
np(x1) = [0 0] x1 + [0]
[0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
sp^#(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
np^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
+^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {fib^#(s(0())) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [0 1] x1 + [2]
[0 0] [2]
fib^#(x1) = [2 2] x1 + [3]
[2 2] [3]
c_1() = [0]
[1]
* Path {3}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sp(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
np(x1) = [0 0] x1 + [0]
[0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
sp^#(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
np^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
+^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {fib^#(s(s(0()))) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
s(x1) = [0 3] x1 + [0]
[0 0] [2]
fib^#(x1) = [2 0] x1 + [3]
[2 0] [3]
c_2() = [0]
[1]
* Path {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{8}->{11}->{10}.
* Path {4}->{8}: inherited
------------------------
This path is subsumed by the proof of path {4}->{8}->{11}->{10}.
* Path {4}->{8}->{10}: NA
-----------------------
The usable rules for this path are:
{ g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {4}->{8}->{11}: inherited
------------------------------
This path is subsumed by the proof of path {4}->{8}->{11}->{10}.
* Path {4}->{8}->{11}->{10}: NA
-----------------------------
The usable rules for this path are:
{ g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {5}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sp(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
np(x1) = [0 0] x1 + [0]
[0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
sp^#(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
np^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
+^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(0()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
g^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_4() = [0]
[1]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sp(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
np(x1) = [0 0] x1 + [0]
[0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
sp^#(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
np^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
+^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(s(0())) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [0 1] x1 + [2]
[0 0] [2]
g^#(x1) = [2 2] x1 + [3]
[2 2] [3]
c_5() = [0]
[1]
* Path {7}: inherited
-------------------
This path is subsumed by the proof of path {7}->{9}->{11}->{10}.
* Path {7}->{9}: inherited
------------------------
This path is subsumed by the proof of path {7}->{9}->{11}->{10}.
* Path {7}->{9}->{10}: MAYBE
--------------------------
The usable rules for this path are:
{ g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ np^#(pair(x, y)) -> c_8(+^#(x, y))
, g^#(s(x)) -> c_6(np^#(g(x)))
, +^#(x, 0()) -> c_9()
, g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {7}->{9}->{11}: inherited
------------------------------
This path is subsumed by the proof of path {7}->{9}->{11}->{10}.
* Path {7}->{9}->{11}->{10}: NA
-----------------------------
The usable rules for this path are:
{ g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: fib^#(0()) -> c_0()
, 2: fib^#(s(0())) -> c_1()
, 3: fib^#(s(s(0()))) -> c_2()
, 4: fib^#(s(s(x))) -> c_3(sp^#(g(x)))
, 5: g^#(0()) -> c_4()
, 6: g^#(s(0())) -> c_5()
, 7: g^#(s(x)) -> c_6(np^#(g(x)))
, 8: sp^#(pair(x, y)) -> c_7(+^#(x, y))
, 9: np^#(pair(x, y)) -> c_8(+^#(x, y))
, 10: +^#(x, 0()) -> c_9()
, 11: +^#(x, s(y)) -> c_10(+^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ inherited ]
|
`->{9} [ inherited ]
|
|->{10} [ MAYBE ]
|
`->{11} [ inherited ]
|
`->{10} [ NA ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ inherited ]
|
`->{8} [ inherited ]
|
|->{10} [ NA ]
|
`->{11} [ inherited ]
|
`->{10} [ NA ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
sp(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
np(x1) = [0] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
fib^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
sp^#(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
np^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {fib^#(0()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
fib^#(x1) = [1] x1 + [7]
c_0() = [1]
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
sp(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
np(x1) = [0] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
fib^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
sp^#(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
np^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {fib^#(s(0())) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [0] x1 + [2]
fib^#(x1) = [2] x1 + [7]
c_1() = [0]
* Path {3}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
sp(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
np(x1) = [0] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
fib^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
sp^#(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
np^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {fib^#(s(s(0()))) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [0] x1 + [0]
fib^#(x1) = [0] x1 + [7]
c_2() = [0]
* Path {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{8}->{11}->{10}.
* Path {4}->{8}: inherited
------------------------
This path is subsumed by the proof of path {4}->{8}->{11}->{10}.
* Path {4}->{8}->{10}: NA
-----------------------
The usable rules for this path are:
{ g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {4}->{8}->{11}: inherited
------------------------------
This path is subsumed by the proof of path {4}->{8}->{11}->{10}.
* Path {4}->{8}->{11}->{10}: NA
-----------------------------
The usable rules for this path are:
{ g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {5}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
sp(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
np(x1) = [0] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
fib^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
sp^#(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
np^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(0()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
g^#(x1) = [1] x1 + [7]
c_4() = [1]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
sp(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
np(x1) = [0] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
fib^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
sp^#(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
np^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(s(0())) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [0] x1 + [2]
g^#(x1) = [2] x1 + [7]
c_5() = [0]
* Path {7}: inherited
-------------------
This path is subsumed by the proof of path {7}->{9}->{11}->{10}.
* Path {7}->{9}: inherited
------------------------
This path is subsumed by the proof of path {7}->{9}->{11}->{10}.
* Path {7}->{9}->{10}: MAYBE
--------------------------
The usable rules for this path are:
{ g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ np^#(pair(x, y)) -> c_8(+^#(x, y))
, g^#(s(x)) -> c_6(np^#(g(x)))
, +^#(x, 0()) -> c_9()
, g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {7}->{9}->{11}: inherited
------------------------------
This path is subsumed by the proof of path {7}->{9}->{11}->{10}.
* Path {7}->{9}->{11}->{10}: NA
-----------------------------
The usable rules for this path are:
{ g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | SK90 2.27 |
|---|
stdout:
MAYBE
Tool RC2
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | SK90 2.27 |
|---|
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(0()))) -> s(0())
, fib(s(s(x))) -> sp(g(x))
, g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, sp(pair(x, y)) -> +(x, y)
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
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: fib^#(0()) -> c_0()
, 2: fib^#(s(0())) -> c_1()
, 3: fib^#(s(s(0()))) -> c_2()
, 4: fib^#(s(s(x))) -> c_3(sp^#(g(x)))
, 5: g^#(0()) -> c_4()
, 6: g^#(s(0())) -> c_5()
, 7: g^#(s(x)) -> c_6(np^#(g(x)))
, 8: sp^#(pair(x, y)) -> c_7(+^#(x, y))
, 9: np^#(pair(x, y)) -> c_8(+^#(x, y), x)
, 10: +^#(x, 0()) -> c_9(x)
, 11: +^#(x, s(y)) -> c_10(+^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ inherited ]
|
`->{9} [ inherited ]
|
|->{10} [ MAYBE ]
|
`->{11} [ inherited ]
|
`->{10} [ NA ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ inherited ]
|
`->{8} [ inherited ]
|
|->{10} [ NA ]
|
`->{11} [ inherited ]
|
`->{10} [ NA ]
->{3} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sp(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
np(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
+(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
fib^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sp^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
np^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
+^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {fib^#(0()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
fib^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_0() = [0]
[1]
[1]
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sp(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
np(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
+(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
fib^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sp^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
np^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
+^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {fib^#(s(0())) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
[0]
s(x1) = [0 3 2] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fib^#(x1) = [2 0 0] x1 + [3]
[0 0 0] [7]
[0 0 0] [7]
c_1() = [0]
[1]
[1]
* 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(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sp(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
np(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
+(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
fib^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sp^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
np^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
+^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {fib^#(s(s(0()))) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
[0]
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [0]
[0 0 0] [0]
fib^#(x1) = [2 0 0] x1 + [3]
[0 0 0] [7]
[2 2 0] [3]
c_2() = [0]
[1]
[1]
* Path {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{8}->{11}->{10}.
* Path {4}->{8}: inherited
------------------------
This path is subsumed by the proof of path {4}->{8}->{11}->{10}.
* Path {4}->{8}->{10}: NA
-----------------------
The usable rules for this path are:
{ g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {4}->{8}->{11}: inherited
------------------------------
This path is subsumed by the proof of path {4}->{8}->{11}->{10}.
* Path {4}->{8}->{11}->{10}: NA
-----------------------------
The usable rules for this path are:
{ g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {5}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sp(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
np(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
+(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
fib^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sp^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
np^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
+^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {g^#(0()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
g^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_4() = [0]
[1]
[1]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sp(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
np(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
+(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
fib^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sp^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
np^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
+^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {g^#(s(0())) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
[0]
s(x1) = [0 3 2] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1) = [2 0 0] x1 + [3]
[0 0 0] [7]
[0 0 0] [7]
c_5() = [0]
[1]
[1]
* Path {7}: inherited
-------------------
This path is subsumed by the proof of path {7}->{9}->{11}->{10}.
* Path {7}->{9}: inherited
------------------------
This path is subsumed by the proof of path {7}->{9}->{11}->{10}.
* Path {7}->{9}->{10}: MAYBE
--------------------------
The usable rules for this path are:
{ g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ np^#(pair(x, y)) -> c_8(+^#(x, y), x)
, g^#(s(x)) -> c_6(np^#(g(x)))
, +^#(x, 0()) -> c_9(x)
, g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {7}->{9}->{11}: inherited
------------------------------
This path is subsumed by the proof of path {7}->{9}->{11}->{10}.
* Path {7}->{9}->{11}->{10}: NA
-----------------------------
The usable rules for this path are:
{ g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: fib^#(0()) -> c_0()
, 2: fib^#(s(0())) -> c_1()
, 3: fib^#(s(s(0()))) -> c_2()
, 4: fib^#(s(s(x))) -> c_3(sp^#(g(x)))
, 5: g^#(0()) -> c_4()
, 6: g^#(s(0())) -> c_5()
, 7: g^#(s(x)) -> c_6(np^#(g(x)))
, 8: sp^#(pair(x, y)) -> c_7(+^#(x, y))
, 9: np^#(pair(x, y)) -> c_8(+^#(x, y), x)
, 10: +^#(x, 0()) -> c_9(x)
, 11: +^#(x, s(y)) -> c_10(+^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ inherited ]
|
`->{9} [ inherited ]
|
|->{10} [ MAYBE ]
|
`->{11} [ inherited ]
|
`->{10} [ NA ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ inherited ]
|
`->{8} [ inherited ]
|
|->{10} [ NA ]
|
`->{11} [ inherited ]
|
`->{10} [ NA ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sp(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
np(x1) = [0 0] x1 + [0]
[0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
sp^#(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
np^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
+^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {fib^#(0()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
fib^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_0() = [0]
[1]
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sp(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
np(x1) = [0 0] x1 + [0]
[0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
sp^#(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
np^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
+^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {fib^#(s(0())) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [0 1] x1 + [2]
[0 0] [2]
fib^#(x1) = [2 2] x1 + [3]
[2 2] [3]
c_1() = [0]
[1]
* Path {3}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sp(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
np(x1) = [0 0] x1 + [0]
[0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
sp^#(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
np^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
+^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {fib^#(s(s(0()))) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
s(x1) = [0 3] x1 + [0]
[0 0] [2]
fib^#(x1) = [2 0] x1 + [3]
[2 0] [3]
c_2() = [0]
[1]
* Path {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{8}->{11}->{10}.
* Path {4}->{8}: inherited
------------------------
This path is subsumed by the proof of path {4}->{8}->{11}->{10}.
* Path {4}->{8}->{10}: NA
-----------------------
The usable rules for this path are:
{ g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {4}->{8}->{11}: inherited
------------------------------
This path is subsumed by the proof of path {4}->{8}->{11}->{10}.
* Path {4}->{8}->{11}->{10}: NA
-----------------------------
The usable rules for this path are:
{ g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {5}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sp(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
np(x1) = [0 0] x1 + [0]
[0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
sp^#(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
np^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
+^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {g^#(0()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
g^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_4() = [0]
[1]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sp(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
np(x1) = [0 0] x1 + [0]
[0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
sp^#(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
np^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
+^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {g^#(s(0())) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [0 1] x1 + [2]
[0 0] [2]
g^#(x1) = [2 2] x1 + [3]
[2 2] [3]
c_5() = [0]
[1]
* Path {7}: inherited
-------------------
This path is subsumed by the proof of path {7}->{9}->{11}->{10}.
* Path {7}->{9}: inherited
------------------------
This path is subsumed by the proof of path {7}->{9}->{11}->{10}.
* Path {7}->{9}->{10}: MAYBE
--------------------------
The usable rules for this path are:
{ g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ np^#(pair(x, y)) -> c_8(+^#(x, y), x)
, g^#(s(x)) -> c_6(np^#(g(x)))
, +^#(x, 0()) -> c_9(x)
, g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {7}->{9}->{11}: inherited
------------------------------
This path is subsumed by the proof of path {7}->{9}->{11}->{10}.
* Path {7}->{9}->{11}->{10}: NA
-----------------------------
The usable rules for this path are:
{ g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: fib^#(0()) -> c_0()
, 2: fib^#(s(0())) -> c_1()
, 3: fib^#(s(s(0()))) -> c_2()
, 4: fib^#(s(s(x))) -> c_3(sp^#(g(x)))
, 5: g^#(0()) -> c_4()
, 6: g^#(s(0())) -> c_5()
, 7: g^#(s(x)) -> c_6(np^#(g(x)))
, 8: sp^#(pair(x, y)) -> c_7(+^#(x, y))
, 9: np^#(pair(x, y)) -> c_8(+^#(x, y), x)
, 10: +^#(x, 0()) -> c_9(x)
, 11: +^#(x, s(y)) -> c_10(+^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ inherited ]
|
`->{9} [ inherited ]
|
|->{10} [ MAYBE ]
|
`->{11} [ inherited ]
|
`->{10} [ NA ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ inherited ]
|
`->{8} [ inherited ]
|
|->{10} [ NA ]
|
`->{11} [ inherited ]
|
`->{10} [ NA ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
sp(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
np(x1) = [0] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
fib^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
sp^#(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
np^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {fib^#(0()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
fib^#(x1) = [1] x1 + [7]
c_0() = [1]
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
sp(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
np(x1) = [0] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
fib^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
sp^#(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
np^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {fib^#(s(0())) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [0] x1 + [2]
fib^#(x1) = [2] x1 + [7]
c_1() = [0]
* Path {3}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
sp(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
np(x1) = [0] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
fib^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
sp^#(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
np^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {fib^#(s(s(0()))) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [0] x1 + [0]
fib^#(x1) = [0] x1 + [7]
c_2() = [0]
* Path {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{8}->{11}->{10}.
* Path {4}->{8}: inherited
------------------------
This path is subsumed by the proof of path {4}->{8}->{11}->{10}.
* Path {4}->{8}->{10}: NA
-----------------------
The usable rules for this path are:
{ g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {4}->{8}->{11}: inherited
------------------------------
This path is subsumed by the proof of path {4}->{8}->{11}->{10}.
* Path {4}->{8}->{11}->{10}: NA
-----------------------------
The usable rules for this path are:
{ g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {5}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
sp(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
np(x1) = [0] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
fib^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
sp^#(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
np^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {g^#(0()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
g^#(x1) = [1] x1 + [7]
c_4() = [1]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(fib) = {}, Uargs(s) = {}, Uargs(sp) = {}, Uargs(g) = {},
Uargs(pair) = {}, Uargs(np) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_3) = {}, Uargs(sp^#) = {}, Uargs(g^#) = {},
Uargs(c_6) = {}, Uargs(np^#) = {}, Uargs(c_7) = {},
Uargs(+^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
sp(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
np(x1) = [0] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
fib^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
sp^#(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
np^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {g^#(s(0())) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [0] x1 + [2]
g^#(x1) = [2] x1 + [7]
c_5() = [0]
* Path {7}: inherited
-------------------
This path is subsumed by the proof of path {7}->{9}->{11}->{10}.
* Path {7}->{9}: inherited
------------------------
This path is subsumed by the proof of path {7}->{9}->{11}->{10}.
* Path {7}->{9}->{10}: MAYBE
--------------------------
The usable rules for this path are:
{ g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ np^#(pair(x, y)) -> c_8(+^#(x, y), x)
, g^#(s(x)) -> c_6(np^#(g(x)))
, +^#(x, 0()) -> c_9(x)
, g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {7}->{9}->{11}: inherited
------------------------------
This path is subsumed by the proof of path {7}->{9}->{11}->{10}.
* Path {7}->{9}->{11}->{10}: NA
-----------------------------
The usable rules for this path are:
{ g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool pair1rc
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | SK90 2.27 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(0()))) -> s(0())
, fib(s(s(x))) -> sp(g(x))
, g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, sp(pair(x, y)) -> +(x, y)
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair1 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair2rc
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | SK90 2.27 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(0()))) -> s(0())
, fib(s(s(x))) -> sp(g(x))
, g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, sp(pair(x, y)) -> +(x, y)
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair2 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair3irc
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | SK90 2.27 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(0()))) -> s(0())
, fib(s(s(x))) -> sp(g(x))
, g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, sp(pair(x, y)) -> +(x, y)
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
StartTerms: basic terms
Strategy: innermost
Certificate: TIMEOUT
Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair3rc
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | SK90 2.27 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(0()))) -> s(0())
, fib(s(s(x))) -> sp(g(x))
, g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, sp(pair(x, y)) -> +(x, y)
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool rc
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | SK90 2.27 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(0()))) -> s(0())
, fib(s(s(x))) -> sp(g(x))
, g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, sp(pair(x, y)) -> +(x, y)
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'rc (timeout of 60.0 seconds)':
----------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool tup3irc
| Execution Time | 60.115536ms |
|---|
| Answer | TIMEOUT |
|---|
| Input | SK90 2.27 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(0()))) -> s(0())
, fib(s(s(x))) -> sp(g(x))
, g(0()) -> pair(s(0()), 0())
, g(s(0())) -> pair(s(0()), s(0()))
, g(s(x)) -> np(g(x))
, sp(pair(x, y)) -> +(x, y)
, np(pair(x, y)) -> pair(+(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
StartTerms: basic terms
Strategy: innermost
Certificate: TIMEOUT
Application of 'tup3 (timeout of 60.0 seconds)':
------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..