Tool CaT
stdout:
MAYBE
Problem:
le(s(x),0()) -> false()
le(0(),y) -> true()
le(s(x),s(y)) -> le(x,y)
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
times(0(),y) -> 0()
times(s(x),y) -> plus(y,times(x,y))
log(x,0()) -> baseError()
log(x,s(0())) -> baseError()
log(0(),s(s(b))) -> logZeroError()
log(s(x),s(s(b))) -> loop(s(x),s(s(b)),s(0()),0())
loop(x,s(s(b)),s(y),z) -> if(le(x,s(y)),x,s(s(b)),s(y),z)
if(true(),x,b,y,z) -> z
if(false(),x,b,y,z) -> loop(x,b,times(b,y),s(z))
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:
{ le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, log(x, 0()) -> baseError()
, log(x, s(0())) -> baseError()
, log(0(), s(s(b))) -> logZeroError()
, log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0()), 0())
, loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z)
, if(true(), x, b, y, z) -> z
, if(false(), x, b, y, z) -> loop(x, b, times(b, y), s(z))}
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: le^#(s(x), 0()) -> c_0()
, 2: le^#(0(), y) -> c_1()
, 3: le^#(s(x), s(y)) -> c_2(le^#(x, y))
, 4: plus^#(0(), y) -> c_3()
, 5: plus^#(s(x), y) -> c_4(plus^#(x, y))
, 6: times^#(0(), y) -> c_5()
, 7: times^#(s(x), y) -> c_6(plus^#(y, times(x, y)))
, 8: log^#(x, 0()) -> c_7()
, 9: log^#(x, s(0())) -> c_8()
, 10: log^#(0(), s(s(b))) -> c_9()
, 11: log^#(s(x), s(s(b))) ->
c_10(loop^#(s(x), s(s(b)), s(0()), 0()))
, 12: loop^#(x, s(s(b)), s(y), z) ->
c_11(if^#(le(x, s(y)), x, s(s(b)), s(y), z))
, 13: if^#(true(), x, b, y, z) -> c_12()
, 14: if^#(false(), x, b, y, z) ->
c_13(loop^#(x, b, times(b, y), s(z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ inherited ]
|
`->{12,14} [ inherited ]
|
`->{13} [ NA ]
->{10} [ YES(?,O(1)) ]
->{9} [ YES(?,O(1)) ]
->{8} [ YES(?,O(1)) ]
->{7} [ inherited ]
|
|->{4} [ MAYBE ]
|
`->{5} [ inherited ]
|
`->{4} [ NA ]
->{6} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^1)) ]
|
|->{1} [ YES(?,O(n^3)) ]
|
`->{2} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* Path {3}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(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]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
true() = [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]
times(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]
log(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]
baseError() = [0]
[0]
[0]
logZeroError() = [0]
[0]
[0]
loop(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]
le^#(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_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(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_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
log^#(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_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop^#(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_11(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_12() = [0]
[0]
[0]
c_13(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: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [2]
[0 0 0] [0]
le^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
[2 2 0] [0 2 0] [0]
[4 0 0] [0 2 0] [0]
c_2(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {3}->{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(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
true() = [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]
times(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]
log(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]
baseError() = [0]
[0]
[0]
logZeroError() = [0]
[0]
[0]
loop(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]
le^#(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() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(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_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
log^#(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_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop^#(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_11(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_12() = [0]
[0]
[0]
c_13(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: {le^#(s(x), 0()) -> c_0()}
Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 3 2] x1 + [2]
[0 1 2] [0]
[0 0 1] [2]
0() = [0]
[0]
[0]
le^#(x1, x2) = [2 2 0] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 2] [0]
[0 1 0] [0 2 4] [0]
c_0() = [1]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [6]
* Path {3}->{2}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
true() = [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]
times(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]
log(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]
baseError() = [0]
[0]
[0]
logZeroError() = [0]
[0]
[0]
loop(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]
le^#(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() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(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_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
log^#(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_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop^#(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_11(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_12() = [0]
[0]
[0]
c_13(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: {le^#(0(), y) -> c_1()}
Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4 2] x1 + [0]
[0 0 2] [0]
[0 0 1] [0]
0() = [2]
[2]
[2]
le^#(x1, x2) = [2 2 2] x1 + [2 0 0] x2 + [0]
[2 2 2] [0 0 4] [0]
[2 2 2] [0 0 0] [0]
c_1() = [1]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
true() = [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]
times(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]
log(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]
baseError() = [0]
[0]
[0]
logZeroError() = [0]
[0]
[0]
loop(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]
le^#(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() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(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_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
log^#(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_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop^#(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_11(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_12() = [0]
[0]
[0]
c_13(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {times^#(0(), y) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
times^#(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [7]
[2 2 0] [0 0 0] [3]
[2 2 2] [0 0 0] [3]
c_5() = [0]
[1]
[1]
* Path {7}: inherited
-------------------
This path is subsumed by the proof of path {7}->{5}->{4}.
* Path {7}->{4}: MAYBE
--------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ times^#(s(x), y) -> c_6(plus^#(y, times(x, y)))
, plus^#(0(), y) -> c_3()
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {7}->{5}: inherited
------------------------
This path is subsumed by the proof of path {7}->{5}->{4}.
* Path {7}->{5}->{4}: NA
----------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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 {8}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
true() = [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]
times(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]
log(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]
baseError() = [0]
[0]
[0]
logZeroError() = [0]
[0]
[0]
loop(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]
le^#(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() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(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_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
log^#(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_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop^#(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_11(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_12() = [0]
[0]
[0]
c_13(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {log^#(x, 0()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(log^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
log^#(x1, x2) = [0 0 0] x1 + [0 2 0] x2 + [7]
[0 0 0] [2 2 0] [3]
[0 0 0] [2 2 2] [3]
c_7() = [0]
[1]
[1]
* Path {9}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
true() = [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]
times(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]
log(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]
baseError() = [0]
[0]
[0]
logZeroError() = [0]
[0]
[0]
loop(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]
le^#(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() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(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_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
log^#(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_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop^#(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_11(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_12() = [0]
[0]
[0]
c_13(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {log^#(x, s(0())) -> c_8()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(log^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 3 2] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[2]
[0]
log^#(x1, x2) = [0 0 0] x1 + [2 0 0] x2 + [3]
[0 0 0] [0 0 0] [7]
[0 0 0] [0 0 0] [7]
c_8() = [0]
[1]
[1]
* Path {10}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
true() = [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]
times(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]
log(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]
baseError() = [0]
[0]
[0]
logZeroError() = [0]
[0]
[0]
loop(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]
le^#(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() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(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_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
log^#(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_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop^#(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_11(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_12() = [0]
[0]
[0]
c_13(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {log^#(0(), s(s(b))) -> c_9()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(log^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 2 2] x1 + [2]
[0 0 2] [0]
[0 0 0] [0]
0() = [2]
[2]
[2]
log^#(x1, x2) = [0 2 2] x1 + [2 2 0] x2 + [3]
[0 0 0] [2 0 0] [7]
[2 2 2] [0 0 0] [3]
c_9() = [0]
[1]
[1]
* Path {11}: inherited
--------------------
This path is subsumed by the proof of path {11}->{12,14}->{13}.
* Path {11}->{12,14}: inherited
-----------------------------
This path is subsumed by the proof of path {11}->{12,14}->{13}.
* Path {11}->{12,14}->{13}: NA
----------------------------
The usable rules for this path are:
{ le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: le^#(s(x), 0()) -> c_0()
, 2: le^#(0(), y) -> c_1()
, 3: le^#(s(x), s(y)) -> c_2(le^#(x, y))
, 4: plus^#(0(), y) -> c_3()
, 5: plus^#(s(x), y) -> c_4(plus^#(x, y))
, 6: times^#(0(), y) -> c_5()
, 7: times^#(s(x), y) -> c_6(plus^#(y, times(x, y)))
, 8: log^#(x, 0()) -> c_7()
, 9: log^#(x, s(0())) -> c_8()
, 10: log^#(0(), s(s(b))) -> c_9()
, 11: log^#(s(x), s(s(b))) ->
c_10(loop^#(s(x), s(s(b)), s(0()), 0()))
, 12: loop^#(x, s(s(b)), s(y), z) ->
c_11(if^#(le(x, s(y)), x, s(s(b)), s(y), z))
, 13: if^#(true(), x, b, y, z) -> c_12()
, 14: if^#(false(), x, b, y, z) ->
c_13(loop^#(x, b, times(b, y), s(z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ inherited ]
|
`->{12,14} [ inherited ]
|
`->{13} [ NA ]
->{10} [ YES(?,O(1)) ]
->{9} [ YES(?,O(1)) ]
->{8} [ YES(?,O(1)) ]
->{7} [ inherited ]
|
|->{4} [ MAYBE ]
|
`->{5} [ inherited ]
|
`->{4} [ NA ]
->{6} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^2)) ]
|
|->{1} [ YES(?,O(n^1)) ]
|
`->{2} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* Path {3}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
true() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
baseError() = [0]
[0]
logZeroError() = [0]
[0]
loop(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]
le^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
loop^#(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_11(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_12() = [0]
[0]
c_13(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: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
le^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_2(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {3}->{1}: YES(?,O(n^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(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
true() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
baseError() = [0]
[0]
logZeroError() = [0]
[0]
loop(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
loop^#(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_11(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_12() = [0]
[0]
c_13(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: {le^#(s(x), 0()) -> c_0()}
Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4] x1 + [2]
[0 0] [0]
0() = [0]
[0]
le^#(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
[0 0] [0 2] [0]
c_0() = [1]
[0]
c_2(x1) = [1 2] x1 + [3]
[0 0] [0]
* Path {3}->{2}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
true() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
baseError() = [0]
[0]
logZeroError() = [0]
[0]
loop(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
loop^#(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_11(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_12() = [0]
[0]
c_13(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: {le^#(0(), y) -> c_1()}
Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [2]
[0 1] [0]
0() = [2]
[2]
le^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
[4 1] [2 0] [0]
c_1() = [1]
[0]
c_2(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {6}: YES(?,O(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(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
true() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
baseError() = [0]
[0]
logZeroError() = [0]
[0]
loop(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
loop^#(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_11(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_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {times^#(0(), y) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
times^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
[2 2] [0 0] [7]
c_5() = [0]
[1]
* Path {7}: inherited
-------------------
This path is subsumed by the proof of path {7}->{5}->{4}.
* Path {7}->{4}: MAYBE
--------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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:
{ times^#(s(x), y) -> c_6(plus^#(y, times(x, y)))
, plus^#(0(), y) -> c_3()
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {7}->{5}: inherited
------------------------
This path is subsumed by the proof of path {7}->{5}->{4}.
* Path {7}->{5}->{4}: NA
----------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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 {8}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
true() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
baseError() = [0]
[0]
logZeroError() = [0]
[0]
loop(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
loop^#(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_11(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_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {log^#(x, 0()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(log^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
log^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
[0 0] [2 2] [7]
c_7() = [0]
[1]
* Path {9}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
true() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
baseError() = [0]
[0]
logZeroError() = [0]
[0]
loop(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
loop^#(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_11(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_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {log^#(x, s(0())) -> c_8()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(log^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 1] x1 + [2]
[0 0] [2]
0() = [0]
[2]
log^#(x1, x2) = [0 0] x1 + [2 2] x2 + [3]
[0 0] [2 2] [3]
c_8() = [0]
[1]
* Path {10}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
true() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
baseError() = [0]
[0]
logZeroError() = [0]
[0]
loop(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
loop^#(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_11(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_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {log^#(0(), s(s(b))) -> c_9()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(log^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 2] x1 + [2]
[0 0] [0]
0() = [0]
[2]
log^#(x1, x2) = [0 2] x1 + [2 0] x2 + [7]
[0 0] [0 0] [7]
c_9() = [0]
[1]
* Path {11}: inherited
--------------------
This path is subsumed by the proof of path {11}->{12,14}->{13}.
* Path {11}->{12,14}: inherited
-----------------------------
This path is subsumed by the proof of path {11}->{12,14}->{13}.
* Path {11}->{12,14}->{13}: NA
----------------------------
The usable rules for this path are:
{ le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: le^#(s(x), 0()) -> c_0()
, 2: le^#(0(), y) -> c_1()
, 3: le^#(s(x), s(y)) -> c_2(le^#(x, y))
, 4: plus^#(0(), y) -> c_3()
, 5: plus^#(s(x), y) -> c_4(plus^#(x, y))
, 6: times^#(0(), y) -> c_5()
, 7: times^#(s(x), y) -> c_6(plus^#(y, times(x, y)))
, 8: log^#(x, 0()) -> c_7()
, 9: log^#(x, s(0())) -> c_8()
, 10: log^#(0(), s(s(b))) -> c_9()
, 11: log^#(s(x), s(s(b))) ->
c_10(loop^#(s(x), s(s(b)), s(0()), 0()))
, 12: loop^#(x, s(s(b)), s(y), z) ->
c_11(if^#(le(x, s(y)), x, s(s(b)), s(y), z))
, 13: if^#(true(), x, b, y, z) -> c_12()
, 14: if^#(false(), x, b, y, z) ->
c_13(loop^#(x, b, times(b, y), s(z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ inherited ]
|
`->{12,14} [ inherited ]
|
`->{13} [ NA ]
->{10} [ YES(?,O(1)) ]
->{9} [ YES(?,O(1)) ]
->{8} [ YES(?,O(1)) ]
->{7} [ inherited ]
|
|->{4} [ MAYBE ]
|
`->{5} [ inherited ]
|
`->{4} [ NA ]
->{6} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^1)) ]
|
|->{1} [ YES(?,O(n^1)) ]
|
`->{2} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {3}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
false() = [0]
true() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
log(x1, x2) = [0] x1 + [0] x2 + [0]
baseError() = [0]
logZeroError() = [0]
loop(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]
le^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
log^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_12() = [0]
c_13(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: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
le^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_2(x1) = [1] x1 + [7]
* Path {3}->{1}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
false() = [0]
true() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
log(x1, x2) = [0] x1 + [0] x2 + [0]
baseError() = [0]
logZeroError() = [0]
loop(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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
log^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_12() = [0]
c_13(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: {le^#(s(x), 0()) -> c_0()}
Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
le^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_0() = [1]
c_2(x1) = [1] x1 + [7]
* Path {3}->{2}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
false() = [0]
true() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
log(x1, x2) = [0] x1 + [0] x2 + [0]
baseError() = [0]
logZeroError() = [0]
loop(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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
log^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_12() = [0]
c_13(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: {le^#(0(), y) -> c_1()}
Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
le^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_1() = [1]
c_2(x1) = [1] x1 + [2]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
false() = [0]
true() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
log(x1, x2) = [0] x1 + [0] x2 + [0]
baseError() = [0]
logZeroError() = [0]
loop(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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
log^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {times^#(0(), y) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
times^#(x1, x2) = [1] x1 + [0] x2 + [7]
c_5() = [1]
* Path {7}: inherited
-------------------
This path is subsumed by the proof of path {7}->{5}->{4}.
* Path {7}->{4}: MAYBE
--------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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:
{ times^#(s(x), y) -> c_6(plus^#(y, times(x, y)))
, plus^#(0(), y) -> c_3()
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {7}->{5}: inherited
------------------------
This path is subsumed by the proof of path {7}->{5}->{4}.
* Path {7}->{5}->{4}: NA
----------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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 {8}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
false() = [0]
true() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
log(x1, x2) = [0] x1 + [0] x2 + [0]
baseError() = [0]
logZeroError() = [0]
loop(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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
log^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {log^#(x, 0()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(log^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
log^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_7() = [1]
* Path {9}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
false() = [0]
true() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
log(x1, x2) = [0] x1 + [0] x2 + [0]
baseError() = [0]
logZeroError() = [0]
loop(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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
log^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {log^#(x, s(0())) -> c_8()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(log^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0] x1 + [2]
0() = [0]
log^#(x1, x2) = [0] x1 + [2] x2 + [7]
c_8() = [0]
* Path {10}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
false() = [0]
true() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
log(x1, x2) = [0] x1 + [0] x2 + [0]
baseError() = [0]
logZeroError() = [0]
loop(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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
log^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {log^#(0(), s(s(b))) -> c_9()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(log^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0] x1 + [2]
0() = [2]
log^#(x1, x2) = [2] x1 + [2] x2 + [7]
c_9() = [0]
* Path {11}: inherited
--------------------
This path is subsumed by the proof of path {11}->{12,14}->{13}.
* Path {11}->{12,14}: inherited
-----------------------------
This path is subsumed by the proof of path {11}->{12,14}->{13}.
* Path {11}->{12,14}->{13}: NA
----------------------------
The usable rules for this path are:
{ le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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.
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:
{ le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, log(x, 0()) -> baseError()
, log(x, s(0())) -> baseError()
, log(0(), s(s(b))) -> logZeroError()
, log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0()), 0())
, loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z)
, if(true(), x, b, y, z) -> z
, if(false(), x, b, y, z) -> loop(x, b, times(b, y), s(z))}
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: le^#(s(x), 0()) -> c_0()
, 2: le^#(0(), y) -> c_1()
, 3: le^#(s(x), s(y)) -> c_2(le^#(x, y))
, 4: plus^#(0(), y) -> c_3(y)
, 5: plus^#(s(x), y) -> c_4(plus^#(x, y))
, 6: times^#(0(), y) -> c_5()
, 7: times^#(s(x), y) -> c_6(plus^#(y, times(x, y)))
, 8: log^#(x, 0()) -> c_7()
, 9: log^#(x, s(0())) -> c_8()
, 10: log^#(0(), s(s(b))) -> c_9()
, 11: log^#(s(x), s(s(b))) ->
c_10(loop^#(s(x), s(s(b)), s(0()), 0()))
, 12: loop^#(x, s(s(b)), s(y), z) ->
c_11(if^#(le(x, s(y)), x, s(s(b)), s(y), z))
, 13: if^#(true(), x, b, y, z) -> c_12(z)
, 14: if^#(false(), x, b, y, z) ->
c_13(loop^#(x, b, times(b, y), s(z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ inherited ]
|
`->{12,14} [ inherited ]
|
`->{13} [ NA ]
->{10} [ YES(?,O(1)) ]
->{9} [ YES(?,O(1)) ]
->{8} [ YES(?,O(1)) ]
->{7} [ inherited ]
|
|->{4} [ MAYBE ]
|
`->{5} [ inherited ]
|
`->{4} [ NA ]
->{6} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^1)) ]
|
|->{1} [ YES(?,O(n^3)) ]
|
`->{2} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* Path {3}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(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]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
true() = [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]
times(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]
log(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]
baseError() = [0]
[0]
[0]
logZeroError() = [0]
[0]
[0]
loop(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]
le^#(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_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(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_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
log^#(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_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop^#(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_11(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_12(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13(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: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [2]
[0 0 0] [0]
le^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
[2 2 0] [0 2 0] [0]
[4 0 0] [0 2 0] [0]
c_2(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {3}->{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(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
true() = [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]
times(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]
log(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]
baseError() = [0]
[0]
[0]
logZeroError() = [0]
[0]
[0]
loop(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]
le^#(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() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(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_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
log^#(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_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop^#(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_11(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_12(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13(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: {le^#(s(x), 0()) -> c_0()}
Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 3 2] x1 + [2]
[0 1 2] [0]
[0 0 1] [2]
0() = [0]
[0]
[0]
le^#(x1, x2) = [2 2 0] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 2] [0]
[0 1 0] [0 2 4] [0]
c_0() = [1]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [6]
* Path {3}->{2}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
true() = [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]
times(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]
log(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]
baseError() = [0]
[0]
[0]
logZeroError() = [0]
[0]
[0]
loop(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]
le^#(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() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(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_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
log^#(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_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop^#(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_11(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_12(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13(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: {le^#(0(), y) -> c_1()}
Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4 2] x1 + [0]
[0 0 2] [0]
[0 0 1] [0]
0() = [2]
[2]
[2]
le^#(x1, x2) = [2 2 2] x1 + [2 0 0] x2 + [0]
[2 2 2] [0 0 4] [0]
[2 2 2] [0 0 0] [0]
c_1() = [1]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
true() = [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]
times(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]
log(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]
baseError() = [0]
[0]
[0]
logZeroError() = [0]
[0]
[0]
loop(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]
le^#(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() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [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_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(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_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
log^#(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_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop^#(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_11(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_12(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {times^#(0(), y) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
times^#(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [7]
[2 2 0] [0 0 0] [3]
[2 2 2] [0 0 0] [3]
c_5() = [0]
[1]
[1]
* Path {7}: inherited
-------------------
This path is subsumed by the proof of path {7}->{5}->{4}.
* Path {7}->{4}: MAYBE
--------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ times^#(s(x), y) -> c_6(plus^#(y, times(x, y)))
, plus^#(0(), y) -> c_3(y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {7}->{5}: inherited
------------------------
This path is subsumed by the proof of path {7}->{5}->{4}.
* Path {7}->{5}->{4}: NA
----------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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 {8}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
true() = [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]
times(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]
log(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]
baseError() = [0]
[0]
[0]
logZeroError() = [0]
[0]
[0]
loop(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]
le^#(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() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [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_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(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_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
log^#(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_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop^#(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_11(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_12(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {log^#(x, 0()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(log^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
log^#(x1, x2) = [0 0 0] x1 + [0 2 0] x2 + [7]
[0 0 0] [2 2 0] [3]
[0 0 0] [2 2 2] [3]
c_7() = [0]
[1]
[1]
* Path {9}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
true() = [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]
times(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]
log(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]
baseError() = [0]
[0]
[0]
logZeroError() = [0]
[0]
[0]
loop(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]
le^#(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() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [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_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(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_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
log^#(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_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop^#(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_11(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_12(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {log^#(x, s(0())) -> c_8()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(log^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 3 2] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[2]
[0]
log^#(x1, x2) = [0 0 0] x1 + [2 0 0] x2 + [3]
[0 0 0] [0 0 0] [7]
[0 0 0] [0 0 0] [7]
c_8() = [0]
[1]
[1]
* Path {10}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
true() = [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]
times(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]
log(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]
baseError() = [0]
[0]
[0]
logZeroError() = [0]
[0]
[0]
loop(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]
le^#(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() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [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_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(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_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
log^#(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_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
loop^#(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_11(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_12(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {log^#(0(), s(s(b))) -> c_9()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(log^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 2 2] x1 + [2]
[0 0 2] [0]
[0 0 0] [0]
0() = [2]
[2]
[2]
log^#(x1, x2) = [0 2 2] x1 + [2 2 0] x2 + [3]
[0 0 0] [2 0 0] [7]
[2 2 2] [0 0 0] [3]
c_9() = [0]
[1]
[1]
* Path {11}: inherited
--------------------
This path is subsumed by the proof of path {11}->{12,14}->{13}.
* Path {11}->{12,14}: inherited
-----------------------------
This path is subsumed by the proof of path {11}->{12,14}->{13}.
* Path {11}->{12,14}->{13}: NA
----------------------------
The usable rules for this path are:
{ le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: le^#(s(x), 0()) -> c_0()
, 2: le^#(0(), y) -> c_1()
, 3: le^#(s(x), s(y)) -> c_2(le^#(x, y))
, 4: plus^#(0(), y) -> c_3(y)
, 5: plus^#(s(x), y) -> c_4(plus^#(x, y))
, 6: times^#(0(), y) -> c_5()
, 7: times^#(s(x), y) -> c_6(plus^#(y, times(x, y)))
, 8: log^#(x, 0()) -> c_7()
, 9: log^#(x, s(0())) -> c_8()
, 10: log^#(0(), s(s(b))) -> c_9()
, 11: log^#(s(x), s(s(b))) ->
c_10(loop^#(s(x), s(s(b)), s(0()), 0()))
, 12: loop^#(x, s(s(b)), s(y), z) ->
c_11(if^#(le(x, s(y)), x, s(s(b)), s(y), z))
, 13: if^#(true(), x, b, y, z) -> c_12(z)
, 14: if^#(false(), x, b, y, z) ->
c_13(loop^#(x, b, times(b, y), s(z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ inherited ]
|
`->{12,14} [ inherited ]
|
`->{13} [ NA ]
->{10} [ YES(?,O(1)) ]
->{9} [ YES(?,O(1)) ]
->{8} [ YES(?,O(1)) ]
->{7} [ inherited ]
|
|->{4} [ MAYBE ]
|
`->{5} [ inherited ]
|
`->{4} [ NA ]
->{6} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^2)) ]
|
|->{1} [ YES(?,O(n^1)) ]
|
`->{2} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* Path {3}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
true() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
baseError() = [0]
[0]
logZeroError() = [0]
[0]
loop(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]
le^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
loop^#(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_11(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_12(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13(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: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
le^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_2(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {3}->{1}: YES(?,O(n^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(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
true() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
baseError() = [0]
[0]
logZeroError() = [0]
[0]
loop(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
loop^#(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_11(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_12(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13(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: {le^#(s(x), 0()) -> c_0()}
Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4] x1 + [2]
[0 0] [0]
0() = [0]
[0]
le^#(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
[0 0] [0 2] [0]
c_0() = [1]
[0]
c_2(x1) = [1 2] x1 + [3]
[0 0] [0]
* Path {3}->{2}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
true() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
baseError() = [0]
[0]
logZeroError() = [0]
[0]
loop(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
loop^#(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_11(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_12(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13(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: {le^#(0(), y) -> c_1()}
Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [2]
[0 1] [0]
0() = [2]
[2]
le^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
[4 1] [2 0] [0]
c_1() = [1]
[0]
c_2(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {6}: YES(?,O(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(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
true() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
baseError() = [0]
[0]
logZeroError() = [0]
[0]
loop(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
loop^#(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_11(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_12(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {times^#(0(), y) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
times^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
[2 2] [0 0] [7]
c_5() = [0]
[1]
* Path {7}: inherited
-------------------
This path is subsumed by the proof of path {7}->{5}->{4}.
* Path {7}->{4}: MAYBE
--------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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:
{ times^#(s(x), y) -> c_6(plus^#(y, times(x, y)))
, plus^#(0(), y) -> c_3(y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {7}->{5}: inherited
------------------------
This path is subsumed by the proof of path {7}->{5}->{4}.
* Path {7}->{5}->{4}: NA
----------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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 {8}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
true() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
baseError() = [0]
[0]
logZeroError() = [0]
[0]
loop(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
loop^#(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_11(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_12(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {log^#(x, 0()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(log^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
log^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
[0 0] [2 2] [7]
c_7() = [0]
[1]
* Path {9}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
true() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
baseError() = [0]
[0]
logZeroError() = [0]
[0]
loop(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
loop^#(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_11(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_12(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {log^#(x, s(0())) -> c_8()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(log^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 1] x1 + [2]
[0 0] [2]
0() = [0]
[2]
log^#(x1, x2) = [0 0] x1 + [2 2] x2 + [3]
[0 0] [2 2] [3]
c_8() = [0]
[1]
* Path {10}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
true() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
baseError() = [0]
[0]
logZeroError() = [0]
[0]
loop(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
loop^#(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_11(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_12(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {log^#(0(), s(s(b))) -> c_9()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(log^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 2] x1 + [2]
[0 0] [0]
0() = [0]
[2]
log^#(x1, x2) = [0 2] x1 + [2 0] x2 + [7]
[0 0] [0 0] [7]
c_9() = [0]
[1]
* Path {11}: inherited
--------------------
This path is subsumed by the proof of path {11}->{12,14}->{13}.
* Path {11}->{12,14}: inherited
-----------------------------
This path is subsumed by the proof of path {11}->{12,14}->{13}.
* Path {11}->{12,14}->{13}: NA
----------------------------
The usable rules for this path are:
{ le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: le^#(s(x), 0()) -> c_0()
, 2: le^#(0(), y) -> c_1()
, 3: le^#(s(x), s(y)) -> c_2(le^#(x, y))
, 4: plus^#(0(), y) -> c_3(y)
, 5: plus^#(s(x), y) -> c_4(plus^#(x, y))
, 6: times^#(0(), y) -> c_5()
, 7: times^#(s(x), y) -> c_6(plus^#(y, times(x, y)))
, 8: log^#(x, 0()) -> c_7()
, 9: log^#(x, s(0())) -> c_8()
, 10: log^#(0(), s(s(b))) -> c_9()
, 11: log^#(s(x), s(s(b))) ->
c_10(loop^#(s(x), s(s(b)), s(0()), 0()))
, 12: loop^#(x, s(s(b)), s(y), z) ->
c_11(if^#(le(x, s(y)), x, s(s(b)), s(y), z))
, 13: if^#(true(), x, b, y, z) -> c_12(z)
, 14: if^#(false(), x, b, y, z) ->
c_13(loop^#(x, b, times(b, y), s(z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ inherited ]
|
`->{12,14} [ inherited ]
|
`->{13} [ NA ]
->{10} [ YES(?,O(1)) ]
->{9} [ YES(?,O(1)) ]
->{8} [ YES(?,O(1)) ]
->{7} [ inherited ]
|
|->{4} [ MAYBE ]
|
`->{5} [ inherited ]
|
`->{4} [ NA ]
->{6} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^1)) ]
|
|->{1} [ YES(?,O(n^1)) ]
|
`->{2} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {3}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
false() = [0]
true() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
log(x1, x2) = [0] x1 + [0] x2 + [0]
baseError() = [0]
logZeroError() = [0]
loop(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]
le^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
log^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_12(x1) = [0] x1 + [0]
c_13(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: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
le^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_2(x1) = [1] x1 + [7]
* Path {3}->{1}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
false() = [0]
true() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
log(x1, x2) = [0] x1 + [0] x2 + [0]
baseError() = [0]
logZeroError() = [0]
loop(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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
log^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_12(x1) = [0] x1 + [0]
c_13(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: {le^#(s(x), 0()) -> c_0()}
Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
le^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_0() = [1]
c_2(x1) = [1] x1 + [7]
* Path {3}->{2}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
false() = [0]
true() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
log(x1, x2) = [0] x1 + [0] x2 + [0]
baseError() = [0]
logZeroError() = [0]
loop(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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
log^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_12(x1) = [0] x1 + [0]
c_13(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: {le^#(0(), y) -> c_1()}
Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
le^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_1() = [1]
c_2(x1) = [1] x1 + [2]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
false() = [0]
true() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
log(x1, x2) = [0] x1 + [0] x2 + [0]
baseError() = [0]
logZeroError() = [0]
loop(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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
log^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {times^#(0(), y) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
times^#(x1, x2) = [1] x1 + [0] x2 + [7]
c_5() = [1]
* Path {7}: inherited
-------------------
This path is subsumed by the proof of path {7}->{5}->{4}.
* Path {7}->{4}: MAYBE
--------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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:
{ times^#(s(x), y) -> c_6(plus^#(y, times(x, y)))
, plus^#(0(), y) -> c_3(y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {7}->{5}: inherited
------------------------
This path is subsumed by the proof of path {7}->{5}->{4}.
* Path {7}->{5}->{4}: NA
----------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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 {8}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
false() = [0]
true() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
log(x1, x2) = [0] x1 + [0] x2 + [0]
baseError() = [0]
logZeroError() = [0]
loop(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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
log^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {log^#(x, 0()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(log^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
log^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_7() = [1]
* Path {9}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
false() = [0]
true() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
log(x1, x2) = [0] x1 + [0] x2 + [0]
baseError() = [0]
logZeroError() = [0]
loop(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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
log^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {log^#(x, s(0())) -> c_8()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(log^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0] x1 + [2]
0() = [0]
log^#(x1, x2) = [0] x1 + [2] x2 + [7]
c_8() = [0]
* Path {10}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
false() = [0]
true() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
log(x1, x2) = [0] x1 + [0] x2 + [0]
baseError() = [0]
logZeroError() = [0]
loop(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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
log^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {log^#(0(), s(s(b))) -> c_9()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(log^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0] x1 + [2]
0() = [2]
log^#(x1, x2) = [2] x1 + [2] x2 + [7]
c_9() = [0]
* Path {11}: inherited
--------------------
This path is subsumed by the proof of path {11}->{12,14}->{13}.
* Path {11}->{12,14}: inherited
-----------------------------
This path is subsumed by the proof of path {11}->{12,14}->{13}.
* Path {11}->{12,14}->{13}: NA
----------------------------
The usable rules for this path are:
{ le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(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.
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.