Tool CaT
stdout:
MAYBE
Problem:
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
lt(0(),s(y)) -> true()
lt(x,0()) -> false()
lt(s(x),s(y)) -> lt(x,y)
fib(x) -> fibiter(x,0(),0(),s(0()))
fibiter(b,c,x,y) -> if(lt(c,b),b,c,x,y)
if(false(),b,c,x,y) -> x
if(true(),b,c,x,y) -> fibiter(b,s(c),y,plus(x,y))
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:
{ plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, fib(x) -> fibiter(x, 0(), 0(), s(0()))
, fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y)
, if(false(), b, c, x, y) -> x
, if(true(), b, c, x, y) -> fibiter(b, s(c), y, plus(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: plus^#(0(), y) -> c_0()
, 2: plus^#(s(x), y) -> c_1(plus^#(x, y))
, 3: lt^#(0(), s(y)) -> c_2()
, 4: lt^#(x, 0()) -> c_3()
, 5: lt^#(s(x), s(y)) -> c_4(lt^#(x, y))
, 6: fib^#(x) -> c_5(fibiter^#(x, 0(), 0(), s(0())))
, 7: fibiter^#(b, c, x, y) -> c_6(if^#(lt(c, b), b, c, x, y))
, 8: if^#(false(), b, c, x, y) -> c_7()
, 9: if^#(true(), b, c, x, y) ->
c_8(fibiter^#(b, s(c), y, plus(x, y)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ inherited ]
|
`->{7,9} [ NA ]
|
`->{8} [ NA ]
->{5} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^2)) ]
->{2} [ YES(?,O(n^3)) ]
|
`->{1} [ YES(?,O(n^3)) ]
Sub-problems:
-------------
* Path {2}: YES(?,O(n^3))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {1}, Uargs(lt^#) = {}, Uargs(c_4) = {},
Uargs(fib^#) = {}, Uargs(c_5) = {}, Uargs(fibiter^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
0() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
lt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fibiter(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
plus^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
lt^#(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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [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_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fibiter^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(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^3))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {plus^#(s(x), y) -> c_1(plus^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4 4] x1 + [2]
[0 1 2] [2]
[0 0 1] [2]
plus^#(x1, x2) = [0 1 2] x1 + [0 0 0] x2 + [2]
[2 2 2] [4 4 4] [0]
[2 0 2] [4 4 4] [0]
c_1(x1) = [1 0 0] x1 + [3]
[0 0 0] [6]
[2 0 0] [2]
* Path {2}->{1}: YES(?,O(n^3))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {1}, Uargs(lt^#) = {}, Uargs(c_4) = {},
Uargs(fib^#) = {}, Uargs(c_5) = {}, Uargs(fibiter^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
lt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fibiter(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
lt^#(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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [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_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fibiter^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(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^3))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {plus^#(0(), y) -> c_0()}
Weak Rules: {plus^#(s(x), y) -> c_1(plus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 4 0] x1 + [0]
[0 1 3] [2]
[0 0 1] [2]
plus^#(x1, x2) = [0 3 2] x1 + [0 0 0] x2 + [0]
[2 2 2] [4 4 0] [0]
[2 2 2] [4 4 0] [0]
c_0() = [1]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1},
Uargs(fib^#) = {}, Uargs(c_5) = {}, Uargs(fibiter^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
lt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fibiter(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
lt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
fib^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fibiter^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(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: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 3] x1 + [0]
[0 0 1] [4]
[0 0 0] [2]
lt^#(x1, x2) = [1 1 0] x1 + [1 0 2] x2 + [0]
[0 0 2] [0 0 0] [2]
[1 1 0] [4 1 0] [0]
c_4(x1) = [1 2 0] x1 + [3]
[0 0 0] [2]
[0 2 0] [3]
* Path {5}->{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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1},
Uargs(fib^#) = {}, Uargs(c_5) = {}, Uargs(fibiter^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
lt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fibiter(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
lt^#(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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
fib^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fibiter^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(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^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {lt^#(0(), s(y)) -> c_2()}
Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[2]
s(x1) = [1 2 0] x1 + [2]
[0 1 2] [2]
[0 0 0] [2]
lt^#(x1, x2) = [2 1 2] x1 + [2 0 0] x2 + [0]
[0 0 2] [1 2 2] [0]
[0 0 0] [0 2 2] [0]
c_2() = [1]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [7]
[0 0 2] [2]
[0 0 0] [7]
* Path {5}->{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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1},
Uargs(fib^#) = {}, Uargs(c_5) = {}, Uargs(fibiter^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
lt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fibiter(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
lt^#(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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
fib^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fibiter^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(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^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {lt^#(x, 0()) -> c_3()}
Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 2 2] x1 + [2]
[0 1 2] [0]
[0 0 0] [3]
lt^#(x1, x2) = [4 0 0] x1 + [3 3 0] x2 + [0]
[2 0 2] [2 0 0] [0]
[0 4 0] [2 2 3] [0]
c_3() = [1]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [2]
[0 0 0] [2]
[0 0 0] [7]
* Path {6}: inherited
-------------------
This path is subsumed by the proof of path {6}->{7,9}.
* Path {6}->{7,9}: NA
-------------------
The usable rules for this path are:
{ plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(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 {6}->{7,9}->{8}: NA
------------------------
The usable rules for this path are:
{ plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {1}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {},
Uargs(fib^#) = {}, Uargs(c_5) = {1}, Uargs(fibiter^#) = {4},
Uargs(c_6) = {1}, Uargs(if^#) = {1}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [2 0 0] x1 + [3 3 3] x2 + [0]
[0 2 0] [3 3 3] [0]
[0 0 0] [3 3 3] [0]
0() = [2]
[0]
[0]
s(x1) = [1 0 0] x1 + [2]
[0 0 0] [2]
[0 0 0] [0]
lt(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [2]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [1]
[0]
[0]
false() = [1]
[0]
[0]
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fibiter(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
lt^#(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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [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_5(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
fibiter^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [3 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_6(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
if^#(x1, x2, x3, x4, x5) = [3 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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: plus^#(0(), y) -> c_0()
, 2: plus^#(s(x), y) -> c_1(plus^#(x, y))
, 3: lt^#(0(), s(y)) -> c_2()
, 4: lt^#(x, 0()) -> c_3()
, 5: lt^#(s(x), s(y)) -> c_4(lt^#(x, y))
, 6: fib^#(x) -> c_5(fibiter^#(x, 0(), 0(), s(0())))
, 7: fibiter^#(b, c, x, y) -> c_6(if^#(lt(c, b), b, c, x, y))
, 8: if^#(false(), b, c, x, y) -> c_7()
, 9: if^#(true(), b, c, x, y) ->
c_8(fibiter^#(b, s(c), y, plus(x, y)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ inherited ]
|
`->{7,9} [ MAYBE ]
|
`->{8} [ NA ]
->{5} [ YES(?,O(n^2)) ]
|
|->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^2)) ]
->{2} [ YES(?,O(n^1)) ]
|
`->{1} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {1}, Uargs(lt^#) = {}, Uargs(c_4) = {},
Uargs(fib^#) = {}, Uargs(c_5) = {}, Uargs(fibiter^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
fibiter(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
plus^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
fibiter^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {plus^#(s(x), y) -> c_1(plus^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [2]
plus^#(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
[4 5] [4 4] [0]
c_1(x1) = [1 0] x1 + [1]
[0 0] [7]
* Path {2}->{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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {1}, Uargs(lt^#) = {}, Uargs(c_4) = {},
Uargs(fib^#) = {}, Uargs(c_5) = {}, Uargs(fibiter^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
fibiter(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
fibiter^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {plus^#(0(), y) -> c_0()}
Weak Rules: {plus^#(s(x), y) -> c_1(plus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [1 3] x1 + [2]
[0 0] [2]
plus^#(x1, x2) = [3 0] x1 + [0 0] x2 + [4]
[2 2] [0 4] [0]
c_0() = [1]
[0]
c_1(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1},
Uargs(fib^#) = {}, Uargs(c_5) = {}, Uargs(fibiter^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
fibiter(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
lt^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
fibiter^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_7() = [0]
[0]
c_8(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: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [0]
lt^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
c_4(x1) = [1 0] x1 + [1]
[0 0] [3]
* Path {5}->{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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1},
Uargs(fib^#) = {}, Uargs(c_5) = {}, Uargs(fibiter^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
fibiter(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
fibiter^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {lt^#(0(), s(y)) -> c_2()}
Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 0] x1 + [0]
[0 1] [2]
lt^#(x1, x2) = [2 3] x1 + [0 2] x2 + [0]
[0 3] [0 1] [0]
c_2() = [1]
[0]
c_4(x1) = [1 0] x1 + [7]
[0 0] [7]
* Path {5}->{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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1},
Uargs(fib^#) = {}, Uargs(c_5) = {}, Uargs(fibiter^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
fibiter(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
fibiter^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_7() = [0]
[0]
c_8(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: {lt^#(x, 0()) -> c_3()}
Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [1 3] x1 + [2]
[0 1] [2]
lt^#(x1, x2) = [2 0] x1 + [2 3] x2 + [0]
[0 2] [3 2] [0]
c_3() = [1]
[0]
c_4(x1) = [1 0] x1 + [7]
[0 0] [7]
* Path {6}: inherited
-------------------
This path is subsumed by the proof of path {6}->{7,9}.
* Path {6}->{7,9}: MAYBE
----------------------
The usable rules for this path are:
{ plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(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:
{ fib^#(x) -> c_5(fibiter^#(x, 0(), 0(), s(0())))
, fibiter^#(b, c, x, y) -> c_6(if^#(lt(c, b), b, c, x, y))
, if^#(true(), b, c, x, y) ->
c_8(fibiter^#(b, s(c), y, plus(x, y)))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {6}->{7,9}->{8}: NA
------------------------
The usable rules for this path are:
{ plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {1}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {},
Uargs(fib^#) = {}, Uargs(c_5) = {1}, Uargs(fibiter^#) = {4},
Uargs(c_6) = {1}, Uargs(if^#) = {1}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [2 0] x1 + [3 3] x2 + [0]
[2 0] [3 3] [0]
0() = [2]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
lt(x1, x2) = [0 0] x1 + [3 0] x2 + [2]
[3 0] [0 0] [3]
true() = [1]
[1]
false() = [1]
[1]
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
fibiter(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
fibiter^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [3 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
if^#(x1, x2, x3, x4, x5) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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: plus^#(0(), y) -> c_0()
, 2: plus^#(s(x), y) -> c_1(plus^#(x, y))
, 3: lt^#(0(), s(y)) -> c_2()
, 4: lt^#(x, 0()) -> c_3()
, 5: lt^#(s(x), s(y)) -> c_4(lt^#(x, y))
, 6: fib^#(x) -> c_5(fibiter^#(x, 0(), 0(), s(0())))
, 7: fibiter^#(b, c, x, y) -> c_6(if^#(lt(c, b), b, c, x, y))
, 8: if^#(false(), b, c, x, y) -> c_7()
, 9: if^#(true(), b, c, x, y) ->
c_8(fibiter^#(b, s(c), y, plus(x, y)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ inherited ]
|
`->{7,9} [ MAYBE ]
|
`->{8} [ NA ]
->{5} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(n^1)) ]
|
`->{1} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {1}, Uargs(lt^#) = {}, Uargs(c_4) = {},
Uargs(fib^#) = {}, Uargs(c_5) = {}, Uargs(fibiter^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
fib(x1) = [0] x1 + [0]
fibiter(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
plus^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
fib^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
fibiter^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_6(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_7() = [0]
c_8(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: {plus^#(s(x), y) -> c_1(plus^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
plus^#(x1, x2) = [2] x1 + [7] x2 + [0]
c_1(x1) = [1] x1 + [7]
* Path {2}->{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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {1}, Uargs(lt^#) = {}, Uargs(c_4) = {},
Uargs(fib^#) = {}, Uargs(c_5) = {}, Uargs(fibiter^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
fib(x1) = [0] x1 + [0]
fibiter(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
fib^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
fibiter^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_6(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_7() = [0]
c_8(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: {plus^#(0(), y) -> c_0()}
Weak Rules: {plus^#(s(x), y) -> c_1(plus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
plus^#(x1, x2) = [2] x1 + [7] x2 + [4]
c_0() = [1]
c_1(x1) = [1] x1 + [4]
* Path {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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1},
Uargs(fib^#) = {}, Uargs(c_5) = {}, Uargs(fibiter^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
fib(x1) = [0] x1 + [0]
fibiter(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
lt^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
fib^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
fibiter^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_6(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_7() = [0]
c_8(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: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
lt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_4(x1) = [1] x1 + [7]
* Path {5}->{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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1},
Uargs(fib^#) = {}, Uargs(c_5) = {}, Uargs(fibiter^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
fib(x1) = [0] x1 + [0]
fibiter(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
fib^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
fibiter^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_6(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_7() = [0]
c_8(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: {lt^#(0(), s(y)) -> c_2()}
Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
lt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_2() = [1]
c_4(x1) = [1] x1 + [7]
* Path {5}->{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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1},
Uargs(fib^#) = {}, Uargs(c_5) = {}, Uargs(fibiter^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
fib(x1) = [0] x1 + [0]
fibiter(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
fib^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
fibiter^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_6(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_7() = [0]
c_8(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: {lt^#(x, 0()) -> c_3()}
Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
lt^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_3() = [1]
c_4(x1) = [1] x1 + [6]
* Path {6}: inherited
-------------------
This path is subsumed by the proof of path {6}->{7,9}.
* Path {6}->{7,9}: MAYBE
----------------------
The usable rules for this path are:
{ plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(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:
{ fib^#(x) -> c_5(fibiter^#(x, 0(), 0(), s(0())))
, fibiter^#(b, c, x, y) -> c_6(if^#(lt(c, b), b, c, x, y))
, if^#(true(), b, c, x, y) ->
c_8(fibiter^#(b, s(c), y, plus(x, y)))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {6}->{7,9}->{8}: NA
------------------------
The usable rules for this path are:
{ plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {1}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {},
Uargs(fib^#) = {}, Uargs(c_5) = {1}, Uargs(fibiter^#) = {4},
Uargs(c_6) = {1}, Uargs(if^#) = {1}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [2] x1 + [3] x2 + [0]
0() = [1]
s(x1) = [1] x1 + [1]
lt(x1, x2) = [0] x1 + [1] x2 + [0]
true() = [0]
false() = [0]
fib(x1) = [0] x1 + [0]
fibiter(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
fib^#(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
fibiter^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [3] x4 + [0]
c_6(x1) = [1] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [3] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, fib(x) -> fibiter(x, 0(), 0(), s(0()))
, fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y)
, if(false(), b, c, x, y) -> x
, if(true(), b, c, x, y) -> fibiter(b, s(c), y, plus(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: plus^#(0(), y) -> c_0(y)
, 2: plus^#(s(x), y) -> c_1(plus^#(x, y))
, 3: lt^#(0(), s(y)) -> c_2()
, 4: lt^#(x, 0()) -> c_3()
, 5: lt^#(s(x), s(y)) -> c_4(lt^#(x, y))
, 6: fib^#(x) -> c_5(fibiter^#(x, 0(), 0(), s(0())))
, 7: fibiter^#(b, c, x, y) -> c_6(if^#(lt(c, b), b, c, x, y))
, 8: if^#(false(), b, c, x, y) -> c_7(x)
, 9: if^#(true(), b, c, x, y) ->
c_8(fibiter^#(b, s(c), y, plus(x, y)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ inherited ]
|
`->{7,9} [ NA ]
|
`->{8} [ NA ]
->{5} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^2)) ]
->{2} [ YES(?,O(n^3)) ]
|
`->{1} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* Path {2}: YES(?,O(n^3))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {1}, Uargs(lt^#) = {},
Uargs(c_4) = {}, Uargs(fib^#) = {}, Uargs(c_5) = {},
Uargs(fibiter^#) = {}, Uargs(c_6) = {}, Uargs(if^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
0() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
lt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fibiter(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
plus^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
lt^#(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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [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_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fibiter^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 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]
c_8(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^3))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {plus^#(s(x), y) -> c_1(plus^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4 4] x1 + [2]
[0 1 2] [2]
[0 0 1] [2]
plus^#(x1, x2) = [0 1 2] x1 + [0 0 0] x2 + [2]
[2 2 2] [4 4 4] [0]
[2 0 2] [4 4 4] [0]
c_1(x1) = [1 0 0] x1 + [3]
[0 0 0] [6]
[2 0 0] [2]
* Path {2}->{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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {1}, Uargs(lt^#) = {},
Uargs(c_4) = {}, Uargs(fib^#) = {}, Uargs(c_5) = {},
Uargs(fibiter^#) = {}, Uargs(c_6) = {}, Uargs(if^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
lt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fibiter(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
plus^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
lt^#(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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [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_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fibiter^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 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]
c_8(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^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {plus^#(0(), y) -> c_0(y)}
Weak Rules: {plus^#(s(x), y) -> c_1(plus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
[0]
s(x1) = [1 2 3] x1 + [0]
[0 0 3] [2]
[0 0 1] [2]
plus^#(x1, x2) = [2 2 2] x1 + [0 0 0] x2 + [2]
[0 2 2] [0 0 0] [0]
[4 0 0] [4 0 4] [4]
c_0(x1) = [0 0 0] x1 + [1]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [1 0 0] x1 + [5]
[0 0 0] [7]
[2 2 0] [0]
* Path {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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
Uargs(c_4) = {1}, Uargs(fib^#) = {}, Uargs(c_5) = {},
Uargs(fibiter^#) = {}, Uargs(c_6) = {}, Uargs(if^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
lt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fibiter(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
lt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
fib^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fibiter^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 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]
c_8(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: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 3] x1 + [0]
[0 0 1] [4]
[0 0 0] [2]
lt^#(x1, x2) = [1 1 0] x1 + [1 0 2] x2 + [0]
[0 0 2] [0 0 0] [2]
[1 1 0] [4 1 0] [0]
c_4(x1) = [1 2 0] x1 + [3]
[0 0 0] [2]
[0 2 0] [3]
* Path {5}->{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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
Uargs(c_4) = {1}, Uargs(fib^#) = {}, Uargs(c_5) = {},
Uargs(fibiter^#) = {}, Uargs(c_6) = {}, Uargs(if^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
lt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fibiter(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
lt^#(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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
fib^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fibiter^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 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]
c_8(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^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {lt^#(0(), s(y)) -> c_2()}
Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[2]
s(x1) = [1 2 0] x1 + [2]
[0 1 2] [2]
[0 0 0] [2]
lt^#(x1, x2) = [2 1 2] x1 + [2 0 0] x2 + [0]
[0 0 2] [1 2 2] [0]
[0 0 0] [0 2 2] [0]
c_2() = [1]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [7]
[0 0 2] [2]
[0 0 0] [7]
* Path {5}->{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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
Uargs(c_4) = {1}, Uargs(fib^#) = {}, Uargs(c_5) = {},
Uargs(fibiter^#) = {}, Uargs(c_6) = {}, Uargs(if^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
lt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fibiter(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
lt^#(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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
fib^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fibiter^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 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]
c_8(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^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {lt^#(x, 0()) -> c_3()}
Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 2 2] x1 + [2]
[0 1 2] [0]
[0 0 0] [3]
lt^#(x1, x2) = [4 0 0] x1 + [3 3 0] x2 + [0]
[2 0 2] [2 0 0] [0]
[0 4 0] [2 2 3] [0]
c_3() = [1]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [2]
[0 0 0] [2]
[0 0 0] [7]
* Path {6}: inherited
-------------------
This path is subsumed by the proof of path {6}->{7,9}.
* Path {6}->{7,9}: NA
-------------------
The usable rules for this path are:
{ plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(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 {6}->{7,9}->{8}: NA
------------------------
The usable rules for this path are:
{ plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {1, 2}, Uargs(s) = {1}, Uargs(lt) = {},
Uargs(fib) = {}, Uargs(fibiter) = {}, Uargs(if) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(lt^#) = {}, Uargs(c_4) = {}, Uargs(fib^#) = {},
Uargs(c_5) = {1}, Uargs(fibiter^#) = {3, 4}, Uargs(c_6) = {1},
Uargs(if^#) = {1, 4, 5}, Uargs(c_7) = {1}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [2 0 0] x1 + [3 3 3] x2 + [0]
[0 0 1] [3 3 3] [2]
[2 0 0] [3 3 3] [0]
0() = [2]
[0]
[0]
s(x1) = [1 0 0] x1 + [2]
[0 1 0] [1]
[0 0 1] [2]
lt(x1, x2) = [0 0 0] x1 + [0 2 3] x2 + [2]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [1]
[0]
[0]
false() = [1]
[0]
[0]
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fibiter(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
lt^#(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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [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_5(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
fibiter^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [3 0 0] x3 + [3 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_6(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
if^#(x1, x2, x3, x4, x5) = [3 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [3 0 0] x4 + [3 3 3] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_7(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_8(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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: plus^#(0(), y) -> c_0(y)
, 2: plus^#(s(x), y) -> c_1(plus^#(x, y))
, 3: lt^#(0(), s(y)) -> c_2()
, 4: lt^#(x, 0()) -> c_3()
, 5: lt^#(s(x), s(y)) -> c_4(lt^#(x, y))
, 6: fib^#(x) -> c_5(fibiter^#(x, 0(), 0(), s(0())))
, 7: fibiter^#(b, c, x, y) -> c_6(if^#(lt(c, b), b, c, x, y))
, 8: if^#(false(), b, c, x, y) -> c_7(x)
, 9: if^#(true(), b, c, x, y) ->
c_8(fibiter^#(b, s(c), y, plus(x, y)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ inherited ]
|
`->{7,9} [ MAYBE ]
|
`->{8} [ NA ]
->{5} [ YES(?,O(n^2)) ]
|
|->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^2)) ]
->{2} [ YES(?,O(n^1)) ]
|
`->{1} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {1}, Uargs(lt^#) = {},
Uargs(c_4) = {}, Uargs(fib^#) = {}, Uargs(c_5) = {},
Uargs(fibiter^#) = {}, Uargs(c_6) = {}, Uargs(if^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
fibiter(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
plus^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
fibiter^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(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: {plus^#(s(x), y) -> c_1(plus^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [2]
plus^#(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
[4 5] [4 4] [0]
c_1(x1) = [1 0] x1 + [1]
[0 0] [7]
* Path {2}->{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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {1}, Uargs(lt^#) = {},
Uargs(c_4) = {}, Uargs(fib^#) = {}, Uargs(c_5) = {},
Uargs(fibiter^#) = {}, Uargs(c_6) = {}, Uargs(if^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
fibiter(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 1] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
fibiter^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(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: {plus^#(0(), y) -> c_0(y)}
Weak Rules: {plus^#(s(x), y) -> c_1(plus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
plus^#(x1, x2) = [2 0] x1 + [0 0] x2 + [4]
[2 2] [4 4] [0]
c_0(x1) = [0 0] x1 + [1]
[0 0] [0]
c_1(x1) = [1 0] x1 + [3]
[0 0] [4]
* Path {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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
Uargs(c_4) = {1}, Uargs(fib^#) = {}, Uargs(c_5) = {},
Uargs(fibiter^#) = {}, Uargs(c_6) = {}, Uargs(if^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
fibiter(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
lt^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
fibiter^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(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: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [0]
lt^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
c_4(x1) = [1 0] x1 + [1]
[0 0] [3]
* Path {5}->{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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
Uargs(c_4) = {1}, Uargs(fib^#) = {}, Uargs(c_5) = {},
Uargs(fibiter^#) = {}, Uargs(c_6) = {}, Uargs(if^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
fibiter(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
fibiter^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(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: {lt^#(0(), s(y)) -> c_2()}
Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 0] x1 + [0]
[0 1] [2]
lt^#(x1, x2) = [2 3] x1 + [0 2] x2 + [0]
[0 3] [0 1] [0]
c_2() = [1]
[0]
c_4(x1) = [1 0] x1 + [7]
[0 0] [7]
* Path {5}->{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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
Uargs(c_4) = {1}, Uargs(fib^#) = {}, Uargs(c_5) = {},
Uargs(fibiter^#) = {}, Uargs(c_6) = {}, Uargs(if^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
fibiter(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
fibiter^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(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: {lt^#(x, 0()) -> c_3()}
Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [1 3] x1 + [2]
[0 1] [2]
lt^#(x1, x2) = [2 0] x1 + [2 3] x2 + [0]
[0 2] [3 2] [0]
c_3() = [1]
[0]
c_4(x1) = [1 0] x1 + [7]
[0 0] [7]
* Path {6}: inherited
-------------------
This path is subsumed by the proof of path {6}->{7,9}.
* Path {6}->{7,9}: MAYBE
----------------------
The usable rules for this path are:
{ plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(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:
{ fib^#(x) -> c_5(fibiter^#(x, 0(), 0(), s(0())))
, fibiter^#(b, c, x, y) -> c_6(if^#(lt(c, b), b, c, x, y))
, if^#(true(), b, c, x, y) ->
c_8(fibiter^#(b, s(c), y, plus(x, y)))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {6}->{7,9}->{8}: NA
------------------------
The usable rules for this path are:
{ plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {1, 2}, Uargs(s) = {1}, Uargs(lt) = {},
Uargs(fib) = {}, Uargs(fibiter) = {}, Uargs(if) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(lt^#) = {}, Uargs(c_4) = {}, Uargs(fib^#) = {},
Uargs(c_5) = {1}, Uargs(fibiter^#) = {3, 4}, Uargs(c_6) = {1},
Uargs(if^#) = {1, 4, 5}, Uargs(c_7) = {1}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [2 0] x1 + [3 3] x2 + [0]
[0 0] [3 3] [0]
0() = [1]
[2]
s(x1) = [1 0] x1 + [2]
[0 1] [0]
lt(x1, x2) = [2 1] x1 + [0 0] x2 + [1]
[3 0] [0 0] [3]
true() = [0]
[0]
false() = [0]
[1]
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
fibiter(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
fibiter^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [3 0] x3 + [3 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
if^#(x1, x2, x3, x4, x5) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [3 0] x4 + [3 3] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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: plus^#(0(), y) -> c_0(y)
, 2: plus^#(s(x), y) -> c_1(plus^#(x, y))
, 3: lt^#(0(), s(y)) -> c_2()
, 4: lt^#(x, 0()) -> c_3()
, 5: lt^#(s(x), s(y)) -> c_4(lt^#(x, y))
, 6: fib^#(x) -> c_5(fibiter^#(x, 0(), 0(), s(0())))
, 7: fibiter^#(b, c, x, y) -> c_6(if^#(lt(c, b), b, c, x, y))
, 8: if^#(false(), b, c, x, y) -> c_7(x)
, 9: if^#(true(), b, c, x, y) ->
c_8(fibiter^#(b, s(c), y, plus(x, y)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ inherited ]
|
`->{7,9} [ MAYBE ]
|
`->{8} [ NA ]
->{5} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(n^1)) ]
|
`->{1} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {1}, Uargs(lt^#) = {},
Uargs(c_4) = {}, Uargs(fib^#) = {}, Uargs(c_5) = {},
Uargs(fibiter^#) = {}, Uargs(c_6) = {}, Uargs(if^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
fib(x1) = [0] x1 + [0]
fibiter(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
plus^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
fib^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
fibiter^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_6(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_7(x1) = [0] x1 + [0]
c_8(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: {plus^#(s(x), y) -> c_1(plus^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
plus^#(x1, x2) = [2] x1 + [7] x2 + [0]
c_1(x1) = [1] x1 + [7]
* Path {2}->{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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {1}, Uargs(lt^#) = {},
Uargs(c_4) = {}, Uargs(fib^#) = {}, Uargs(c_5) = {},
Uargs(fibiter^#) = {}, Uargs(c_6) = {}, Uargs(if^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
fib(x1) = [0] x1 + [0]
fibiter(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
plus^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
fib^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
fibiter^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_6(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_7(x1) = [0] x1 + [0]
c_8(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: {plus^#(0(), y) -> c_0(y)}
Weak Rules: {plus^#(s(x), y) -> c_1(plus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
plus^#(x1, x2) = [3] x1 + [7] x2 + [2]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [5]
* Path {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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
Uargs(c_4) = {1}, Uargs(fib^#) = {}, Uargs(c_5) = {},
Uargs(fibiter^#) = {}, Uargs(c_6) = {}, Uargs(if^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
fib(x1) = [0] x1 + [0]
fibiter(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
lt^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
fib^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
fibiter^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_6(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_7(x1) = [0] x1 + [0]
c_8(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: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
lt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_4(x1) = [1] x1 + [7]
* Path {5}->{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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
Uargs(c_4) = {1}, Uargs(fib^#) = {}, Uargs(c_5) = {},
Uargs(fibiter^#) = {}, Uargs(c_6) = {}, Uargs(if^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
fib(x1) = [0] x1 + [0]
fibiter(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
fib^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
fibiter^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_6(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_7(x1) = [0] x1 + [0]
c_8(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: {lt^#(0(), s(y)) -> c_2()}
Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
lt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_2() = [1]
c_4(x1) = [1] x1 + [7]
* Path {5}->{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(plus) = {}, Uargs(s) = {}, Uargs(lt) = {}, Uargs(fib) = {},
Uargs(fibiter) = {}, Uargs(if) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
Uargs(c_4) = {1}, Uargs(fib^#) = {}, Uargs(c_5) = {},
Uargs(fibiter^#) = {}, Uargs(c_6) = {}, Uargs(if^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
fib(x1) = [0] x1 + [0]
fibiter(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
fib^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
fibiter^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_6(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_7(x1) = [0] x1 + [0]
c_8(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: {lt^#(x, 0()) -> c_3()}
Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
lt^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_3() = [1]
c_4(x1) = [1] x1 + [6]
* Path {6}: inherited
-------------------
This path is subsumed by the proof of path {6}->{7,9}.
* Path {6}->{7,9}: MAYBE
----------------------
The usable rules for this path are:
{ plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(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:
{ fib^#(x) -> c_5(fibiter^#(x, 0(), 0(), s(0())))
, fibiter^#(b, c, x, y) -> c_6(if^#(lt(c, b), b, c, x, y))
, if^#(true(), b, c, x, y) ->
c_8(fibiter^#(b, s(c), y, plus(x, y)))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {6}->{7,9}->{8}: NA
------------------------
The usable rules for this path are:
{ plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {1, 2}, Uargs(s) = {1}, Uargs(lt) = {},
Uargs(fib) = {}, Uargs(fibiter) = {}, Uargs(if) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(lt^#) = {}, Uargs(c_4) = {}, Uargs(fib^#) = {},
Uargs(c_5) = {1}, Uargs(fibiter^#) = {3, 4}, Uargs(c_6) = {1},
Uargs(if^#) = {1, 4, 5}, Uargs(c_7) = {1}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [2] x1 + [3] x2 + [0]
0() = [2]
s(x1) = [1] x1 + [2]
lt(x1, x2) = [0] x1 + [2] x2 + [0]
true() = [1]
false() = [1]
fib(x1) = [0] x1 + [0]
fibiter(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
fib^#(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
fibiter^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [3] x3 + [3] x4 + [0]
c_6(x1) = [1] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [3] x1 + [0] x2 + [0] x3 + [3] x4 + [3] x5 + [0]
c_7(x1) = [1] x1 + [0]
c_8(x1) = [1] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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.