Tool CaT
stdout:
MAYBE
Problem:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
minus(minus(x,y),z) -> minus(x,plus(y,z))
app(nil(),k) -> k
app(l,nil()) -> l
app(cons(x,l),k) -> cons(x,app(l,k))
sum(cons(x,nil())) -> cons(x,nil())
sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
plus(zero(),y) -> y
plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
id(x) -> x
if(true(),x,y) -> x
if(false(),x,y) -> y
not(x) -> if(x,false(),true())
gt(s(x),zero()) -> true()
gt(zero(),y) -> false()
gt(s(x),s(y)) -> gt(x,y)
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, minus(minus(x, y), z) -> minus(x, plus(y, z))
, app(nil(), k) -> k
, app(l, nil()) -> l
, app(cons(x, l), k) -> cons(x, app(l, k))
, sum(cons(x, nil())) -> cons(x, nil())
, sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
, sum(app(l, cons(x, cons(y, k)))) ->
sum(app(l, sum(cons(x, cons(y, k)))))
, plus(s(x), s(y)) ->
s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
, plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
, plus(zero(), y) -> y
, plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
, id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(x, y)}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: minus^#(x, 0()) -> c_0()
, 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
, 3: quot^#(0(), s(y)) -> c_2()
, 4: quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))
, 5: plus^#(0(), y) -> c_4()
, 6: plus^#(s(x), y) -> c_5(plus^#(x, y))
, 7: minus^#(minus(x, y), z) -> c_6(minus^#(x, plus(y, z)))
, 8: app^#(nil(), k) -> c_7()
, 9: app^#(l, nil()) -> c_8()
, 10: app^#(cons(x, l), k) -> c_9(app^#(l, k))
, 11: sum^#(cons(x, nil())) -> c_10()
, 12: sum^#(cons(x, cons(y, l))) ->
c_11(sum^#(cons(plus(x, y), l)))
, 13: sum^#(app(l, cons(x, cons(y, k)))) ->
c_12(sum^#(app(l, sum(cons(x, cons(y, k))))))
, 14: plus^#(s(x), s(y)) ->
c_13(plus^#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))
, 15: plus^#(s(x), x) ->
c_14(plus^#(if(gt(x, x), id(x), id(x)), s(x)))
, 16: plus^#(zero(), y) -> c_15()
, 17: plus^#(id(x), s(y)) ->
c_16(plus^#(x, if(gt(s(y), y), y, s(y))))
, 18: id^#(x) -> c_17()
, 19: if^#(true(), x, y) -> c_18()
, 20: if^#(false(), x, y) -> c_19()
, 21: not^#(x) -> c_20(if^#(x, false(), true()))
, 22: gt^#(s(x), zero()) -> c_21()
, 23: gt^#(zero(), y) -> c_22()
, 24: gt^#(s(x), s(y)) -> c_23(gt^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{24} [ YES(?,O(n^1)) ]
|
|->{22} [ YES(?,O(n^1)) ]
|
`->{23} [ YES(?,O(n^1)) ]
->{21} [ YES(?,O(1)) ]
|
|->{19} [ YES(?,O(1)) ]
|
`->{20} [ YES(?,O(1)) ]
->{18} [ YES(?,O(1)) ]
->{13} [ inherited ]
|
|->{11} [ NA ]
|
`->{12} [ inherited ]
|
`->{11} [ NA ]
->{10} [ YES(?,O(n^1)) ]
|
|->{8} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(n^1)) ]
->{6,17,15,14} [ NA ]
|
|->{5} [ NA ]
|
`->{16} [ NA ]
->{4} [ inherited ]
|
`->{3} [ NA ]
->{2,7} [ inherited ]
|
`->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {2,7}: inherited
---------------------
This path is subsumed by the proof of path {2,7}->{1}.
* Path {2,7}->{1}: MAYBE
----------------------
The usable rules for this path are:
{ plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, plus(s(x), s(y)) ->
s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
, plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
, plus(zero(), y) -> y
, plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
, id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(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:
{ minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
, minus^#(minus(x, y), z) -> c_6(minus^#(x, plus(y, z)))
, minus^#(x, 0()) -> c_0()
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, plus(s(x), s(y)) ->
s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
, plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
, plus(zero(), y) -> y
, plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
, id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{3}.
* Path {4}->{3}: NA
-----------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, minus(minus(x, y), z) -> minus(x, plus(y, z))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, plus(s(x), s(y)) ->
s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
, plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
, plus(zero(), y) -> y
, plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
, id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {6,17,15,14}: NA
---------------------
The usable rules for this path are:
{ id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {6,17,15,14}->{5}: NA
--------------------------
The usable rules for this path are:
{ id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(if) = {1, 2, 3}, Uargs(gt) = {},
Uargs(not) = {1}, Uargs(id) = {}, Uargs(minus^#) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(plus^#) = {1, 2}, Uargs(c_5) = {1}, Uargs(c_6) = {},
Uargs(app^#) = {}, Uargs(c_9) = {}, Uargs(sum^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(c_13) = {1},
Uargs(c_14) = {1}, Uargs(c_16) = {1}, Uargs(id^#) = {},
Uargs(if^#) = {}, Uargs(not^#) = {}, Uargs(c_20) = {},
Uargs(gt^#) = {}, Uargs(c_23) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [1]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
if(x1, x2, x3) = [1] x1 + [3] x2 + [3] x3 + [1]
gt(x1, x2) = [0] x1 + [1] x2 + [2]
not(x1) = [3] x1 + [3]
id(x1) = [3] x1 + [3]
zero() = [0]
true() = [0]
false() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [0]
c_15() = [0]
c_16(x1) = [1] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_17() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_18() = [0]
c_19() = [0]
not^#(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_21() = [0]
c_22() = [0]
c_23(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {6,17,15,14}->{16}: NA
---------------------------
The usable rules for this path are:
{ id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(if) = {1, 2, 3}, Uargs(gt) = {},
Uargs(not) = {1}, Uargs(id) = {}, Uargs(minus^#) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(plus^#) = {1, 2}, Uargs(c_5) = {1}, Uargs(c_6) = {},
Uargs(app^#) = {}, Uargs(c_9) = {}, Uargs(sum^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(c_13) = {1},
Uargs(c_14) = {1}, Uargs(c_16) = {1}, Uargs(id^#) = {},
Uargs(if^#) = {}, Uargs(not^#) = {}, Uargs(c_20) = {},
Uargs(gt^#) = {}, Uargs(c_23) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [1]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
if(x1, x2, x3) = [1] x1 + [3] x2 + [3] x3 + [1]
gt(x1, x2) = [0] x1 + [1] x2 + [2]
not(x1) = [3] x1 + [3]
id(x1) = [3] x1 + [3]
zero() = [0]
true() = [0]
false() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [0]
c_15() = [0]
c_16(x1) = [1] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_17() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_18() = [0]
c_19() = [0]
not^#(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_21() = [0]
c_22() = [0]
c_23(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {10}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(if) = {}, Uargs(gt) = {}, Uargs(not) = {},
Uargs(id) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_9) = {1}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_16) = {}, Uargs(id^#) = {}, Uargs(if^#) = {},
Uargs(not^#) = {}, Uargs(c_20) = {}, Uargs(gt^#) = {},
Uargs(c_23) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [1] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_17() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_18() = [0]
c_19() = [0]
not^#(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_21() = [0]
c_22() = [0]
c_23(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: {app^#(cons(x, l), k) -> c_9(app^#(l, k))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(app^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [1] x2 + [4]
app^#(x1, x2) = [2] x1 + [7] x2 + [0]
c_9(x1) = [1] x1 + [7]
* Path {10}->{8}: YES(?,O(n^1))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(if) = {}, Uargs(gt) = {}, Uargs(not) = {},
Uargs(id) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_9) = {1}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_16) = {}, Uargs(id^#) = {}, Uargs(if^#) = {},
Uargs(not^#) = {}, Uargs(c_20) = {}, Uargs(gt^#) = {},
Uargs(c_23) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [1] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_17() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_18() = [0]
c_19() = [0]
not^#(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_21() = [0]
c_22() = [0]
c_23(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: {app^#(nil(), k) -> c_7()}
Weak Rules: {app^#(cons(x, l), k) -> c_9(app^#(l, k))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(app^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
cons(x1, x2) = [0] x1 + [1] x2 + [2]
app^#(x1, x2) = [2] x1 + [7] x2 + [4]
c_7() = [1]
c_9(x1) = [1] x1 + [4]
* Path {10}->{9}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(if) = {}, Uargs(gt) = {}, Uargs(not) = {},
Uargs(id) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_9) = {1}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_16) = {}, Uargs(id^#) = {}, Uargs(if^#) = {},
Uargs(not^#) = {}, Uargs(c_20) = {}, Uargs(gt^#) = {},
Uargs(c_23) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [1] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_17() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_18() = [0]
c_19() = [0]
not^#(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_21() = [0]
c_22() = [0]
c_23(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: {app^#(l, nil()) -> c_8()}
Weak Rules: {app^#(cons(x, l), k) -> c_9(app^#(l, k))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(app^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
cons(x1, x2) = [0] x1 + [1] x2 + [2]
app^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_8() = [1]
c_9(x1) = [1] x1 + [4]
* Path {13}: inherited
--------------------
This path is subsumed by the proof of path {13}->{12}->{11}.
* Path {13}->{11}: NA
-------------------
The usable rules for this path are:
{ app(nil(), k) -> k
, app(l, nil()) -> l
, app(cons(x, l), k) -> cons(x, app(l, k))
, sum(cons(x, nil())) -> cons(x, nil())
, sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
, sum(app(l, cons(x, cons(y, k)))) ->
sum(app(l, sum(cons(x, cons(y, k)))))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, plus(s(x), s(y)) ->
s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
, plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
, plus(zero(), y) -> y
, plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
, id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(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 {13}->{12}: inherited
--------------------------
This path is subsumed by the proof of path {13}->{12}->{11}.
* Path {13}->{12}->{11}: NA
-------------------------
The usable rules for this path are:
{ app(nil(), k) -> k
, app(l, nil()) -> l
, app(cons(x, l), k) -> cons(x, app(l, k))
, sum(cons(x, nil())) -> cons(x, nil())
, sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
, sum(app(l, cons(x, cons(y, k)))) ->
sum(app(l, sum(cons(x, cons(y, k)))))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, plus(s(x), s(y)) ->
s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
, plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
, plus(zero(), y) -> y
, plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
, id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(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 {18}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(if) = {}, Uargs(gt) = {}, Uargs(not) = {},
Uargs(id) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_9) = {}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_16) = {}, Uargs(id^#) = {}, Uargs(if^#) = {},
Uargs(not^#) = {}, Uargs(c_20) = {}, Uargs(gt^#) = {},
Uargs(c_23) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_17() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_18() = [0]
c_19() = [0]
not^#(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_21() = [0]
c_22() = [0]
c_23(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: {id^#(x) -> c_17()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(id^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
id^#(x1) = [0] x1 + [7]
c_17() = [0]
* Path {21}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(if) = {}, Uargs(gt) = {}, Uargs(not) = {},
Uargs(id) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_9) = {}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_16) = {}, Uargs(id^#) = {}, Uargs(if^#) = {},
Uargs(not^#) = {}, Uargs(c_20) = {}, Uargs(gt^#) = {},
Uargs(c_23) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_17() = [0]
if^#(x1, x2, x3) = [1] x1 + [0] x2 + [0] x3 + [0]
c_18() = [0]
c_19() = [0]
not^#(x1) = [3] x1 + [0]
c_20(x1) = [3] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_21() = [0]
c_22() = [0]
c_23(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: {not^#(x) -> c_20(if^#(x, false(), true()))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(not^#) = {}, Uargs(c_20) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [0]
false() = [0]
if^#(x1, x2, x3) = [7] x1 + [0] x2 + [0] x3 + [4]
not^#(x1) = [7] x1 + [7]
c_20(x1) = [0] x1 + [3]
* Path {21}->{19}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(if) = {}, Uargs(gt) = {}, Uargs(not) = {},
Uargs(id) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_9) = {}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_16) = {}, Uargs(id^#) = {}, Uargs(if^#) = {},
Uargs(not^#) = {}, Uargs(c_20) = {1}, Uargs(gt^#) = {},
Uargs(c_23) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_17() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_18() = [0]
c_19() = [0]
not^#(x1) = [0] x1 + [0]
c_20(x1) = [1] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_21() = [0]
c_22() = [0]
c_23(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: {if^#(true(), x, y) -> c_18()}
Weak Rules: {not^#(x) -> c_20(if^#(x, false(), true()))}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(not^#) = {}, Uargs(c_20) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
false() = [0]
if^#(x1, x2, x3) = [2] x1 + [0] x2 + [0] x3 + [0]
c_18() = [1]
not^#(x1) = [7] x1 + [7]
c_20(x1) = [2] x1 + [7]
* Path {21}->{20}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(if) = {}, Uargs(gt) = {}, Uargs(not) = {},
Uargs(id) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_9) = {}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_16) = {}, Uargs(id^#) = {}, Uargs(if^#) = {},
Uargs(not^#) = {}, Uargs(c_20) = {1}, Uargs(gt^#) = {},
Uargs(c_23) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_17() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_18() = [0]
c_19() = [0]
not^#(x1) = [0] x1 + [0]
c_20(x1) = [1] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_21() = [0]
c_22() = [0]
c_23(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: {if^#(false(), x, y) -> c_19()}
Weak Rules: {not^#(x) -> c_20(if^#(x, false(), true()))}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(not^#) = {}, Uargs(c_20) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [0]
false() = [2]
if^#(x1, x2, x3) = [2] x1 + [0] x2 + [0] x3 + [4]
c_19() = [1]
not^#(x1) = [7] x1 + [7]
c_20(x1) = [1] x1 + [3]
* Path {24}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(if) = {}, Uargs(gt) = {}, Uargs(not) = {},
Uargs(id) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_9) = {}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_16) = {}, Uargs(id^#) = {}, Uargs(if^#) = {},
Uargs(not^#) = {}, Uargs(c_20) = {}, Uargs(gt^#) = {},
Uargs(c_23) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_17() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_18() = [0]
c_19() = [0]
not^#(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
gt^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_21() = [0]
c_22() = [0]
c_23(x1) = [1] 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: {gt^#(s(x), s(y)) -> c_23(gt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_23) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_23(x1) = [1] x1 + [7]
* Path {24}->{22}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(if) = {}, Uargs(gt) = {}, Uargs(not) = {},
Uargs(id) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_9) = {}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_16) = {}, Uargs(id^#) = {}, Uargs(if^#) = {},
Uargs(not^#) = {}, Uargs(c_20) = {}, Uargs(gt^#) = {},
Uargs(c_23) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_17() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_18() = [0]
c_19() = [0]
not^#(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_21() = [0]
c_22() = [0]
c_23(x1) = [1] 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: {gt^#(s(x), zero()) -> c_21()}
Weak Rules: {gt^#(s(x), s(y)) -> c_23(gt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_23) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
zero() = [2]
gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_21() = [1]
c_23(x1) = [1] x1 + [7]
* Path {24}->{23}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(if) = {}, Uargs(gt) = {}, Uargs(not) = {},
Uargs(id) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_9) = {}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_16) = {}, Uargs(id^#) = {}, Uargs(if^#) = {},
Uargs(not^#) = {}, Uargs(c_20) = {}, Uargs(gt^#) = {},
Uargs(c_23) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_17() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_18() = [0]
c_19() = [0]
not^#(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_21() = [0]
c_22() = [0]
c_23(x1) = [1] 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: {gt^#(zero(), y) -> c_22()}
Weak Rules: {gt^#(s(x), s(y)) -> c_23(gt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_23) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
zero() = [2]
gt^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_22() = [1]
c_23(x1) = [1] x1 + [2]
2) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
3) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
4) '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:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, minus(minus(x, y), z) -> minus(x, plus(y, z))
, app(nil(), k) -> k
, app(l, nil()) -> l
, app(cons(x, l), k) -> cons(x, app(l, k))
, sum(cons(x, nil())) -> cons(x, nil())
, sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
, sum(app(l, cons(x, cons(y, k)))) ->
sum(app(l, sum(cons(x, cons(y, k)))))
, plus(s(x), s(y)) ->
s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
, plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
, plus(zero(), y) -> y
, plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
, id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(x, y)}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: minus^#(x, 0()) -> c_0(x)
, 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
, 3: quot^#(0(), s(y)) -> c_2()
, 4: quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))
, 5: plus^#(0(), y) -> c_4(y)
, 6: plus^#(s(x), y) -> c_5(plus^#(x, y))
, 7: minus^#(minus(x, y), z) -> c_6(minus^#(x, plus(y, z)))
, 8: app^#(nil(), k) -> c_7(k)
, 9: app^#(l, nil()) -> c_8(l)
, 10: app^#(cons(x, l), k) -> c_9(x, app^#(l, k))
, 11: sum^#(cons(x, nil())) -> c_10(x)
, 12: sum^#(cons(x, cons(y, l))) ->
c_11(sum^#(cons(plus(x, y), l)))
, 13: sum^#(app(l, cons(x, cons(y, k)))) ->
c_12(sum^#(app(l, sum(cons(x, cons(y, k))))))
, 14: plus^#(s(x), s(y)) ->
c_13(plus^#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))
, 15: plus^#(s(x), x) ->
c_14(plus^#(if(gt(x, x), id(x), id(x)), s(x)))
, 16: plus^#(zero(), y) -> c_15(y)
, 17: plus^#(id(x), s(y)) ->
c_16(plus^#(x, if(gt(s(y), y), y, s(y))))
, 18: id^#(x) -> c_17(x)
, 19: if^#(true(), x, y) -> c_18(x)
, 20: if^#(false(), x, y) -> c_19(y)
, 21: not^#(x) -> c_20(if^#(x, false(), true()))
, 22: gt^#(s(x), zero()) -> c_21()
, 23: gt^#(zero(), y) -> c_22()
, 24: gt^#(s(x), s(y)) -> c_23(gt^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{24} [ YES(?,O(n^1)) ]
|
|->{22} [ YES(?,O(n^1)) ]
|
`->{23} [ YES(?,O(n^1)) ]
->{21} [ YES(?,O(1)) ]
|
|->{19} [ YES(?,O(1)) ]
|
`->{20} [ YES(?,O(1)) ]
->{18} [ YES(?,O(1)) ]
->{13} [ inherited ]
|
|->{11} [ NA ]
|
`->{12} [ inherited ]
|
`->{11} [ NA ]
->{10} [ YES(?,O(n^1)) ]
|
|->{8} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(n^1)) ]
->{6,17,15,14} [ NA ]
|
|->{5} [ NA ]
|
`->{16} [ NA ]
->{4} [ inherited ]
|
`->{3} [ NA ]
->{2,7} [ inherited ]
|
`->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {2,7}: inherited
---------------------
This path is subsumed by the proof of path {2,7}->{1}.
* Path {2,7}->{1}: MAYBE
----------------------
The usable rules for this path are:
{ plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, plus(s(x), s(y)) ->
s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
, plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
, plus(zero(), y) -> y
, plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
, id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(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:
{ minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
, minus^#(minus(x, y), z) -> c_6(minus^#(x, plus(y, z)))
, minus^#(x, 0()) -> c_0(x)
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, plus(s(x), s(y)) ->
s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
, plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
, plus(zero(), y) -> y
, plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
, id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{3}.
* Path {4}->{3}: NA
-----------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, minus(minus(x, y), z) -> minus(x, plus(y, z))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, plus(s(x), s(y)) ->
s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
, plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
, plus(zero(), y) -> y
, plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
, id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {6,17,15,14}: NA
---------------------
The usable rules for this path are:
{ id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {6,17,15,14}->{5}: NA
--------------------------
The usable rules for this path are:
{ id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(if) = {1, 2, 3}, Uargs(gt) = {},
Uargs(not) = {1}, Uargs(id) = {}, Uargs(minus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(quot^#) = {},
Uargs(c_3) = {}, Uargs(plus^#) = {1, 2}, Uargs(c_4) = {1},
Uargs(c_5) = {1}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(sum^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(c_13) = {1}, Uargs(c_14) = {1},
Uargs(c_15) = {}, Uargs(c_16) = {1}, Uargs(id^#) = {},
Uargs(c_17) = {}, Uargs(if^#) = {}, Uargs(c_18) = {},
Uargs(c_19) = {}, Uargs(not^#) = {}, Uargs(c_20) = {},
Uargs(gt^#) = {}, Uargs(c_23) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [3]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
if(x1, x2, x3) = [3] x1 + [3] x2 + [3] x3 + [2]
gt(x1, x2) = [1] x1 + [1] x2 + [3]
not(x1) = [3] x1 + [3]
id(x1) = [3] x1 + [3]
zero() = [3]
true() = [0]
false() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1, x2) = [0] x1 + [0] x2 + [0]
sum^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [1] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
not^#(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_21() = [0]
c_22() = [0]
c_23(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {6,17,15,14}->{16}: NA
---------------------------
The usable rules for this path are:
{ id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(if) = {1, 2, 3}, Uargs(gt) = {},
Uargs(not) = {1}, Uargs(id) = {}, Uargs(minus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(quot^#) = {},
Uargs(c_3) = {}, Uargs(plus^#) = {1, 2}, Uargs(c_4) = {},
Uargs(c_5) = {1}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(sum^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(c_13) = {1}, Uargs(c_14) = {1},
Uargs(c_15) = {1}, Uargs(c_16) = {1}, Uargs(id^#) = {},
Uargs(c_17) = {}, Uargs(if^#) = {}, Uargs(c_18) = {},
Uargs(c_19) = {}, Uargs(not^#) = {}, Uargs(c_20) = {},
Uargs(gt^#) = {}, Uargs(c_23) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [3]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
if(x1, x2, x3) = [3] x1 + [3] x2 + [3] x3 + [2]
gt(x1, x2) = [1] x1 + [1] x2 + [3]
not(x1) = [3] x1 + [3]
id(x1) = [3] x1 + [3]
zero() = [3]
true() = [0]
false() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1, x2) = [0] x1 + [0] x2 + [0]
sum^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [1] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
not^#(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_21() = [0]
c_22() = [0]
c_23(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {10}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(if) = {}, Uargs(gt) = {}, Uargs(not) = {},
Uargs(id) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(c_9) = {2}, Uargs(sum^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}, Uargs(c_12) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(id^#) = {}, Uargs(c_17) = {},
Uargs(if^#) = {}, Uargs(c_18) = {}, Uargs(c_19) = {},
Uargs(not^#) = {}, Uargs(c_20) = {}, Uargs(gt^#) = {},
Uargs(c_23) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
sum(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1, x2) = [1] x1 + [1] x2 + [0]
sum^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
not^#(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_21() = [0]
c_22() = [0]
c_23(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: {app^#(cons(x, l), k) -> c_9(x, app^#(l, k))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(app^#) = {}, Uargs(c_9) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1] x1 + [1] x2 + [2]
app^#(x1, x2) = [6] x1 + [7] x2 + [0]
c_9(x1, x2) = [0] x1 + [1] x2 + [7]
* Path {10}->{8}: YES(?,O(n^1))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(if) = {}, Uargs(gt) = {}, Uargs(not) = {},
Uargs(id) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(c_9) = {2}, Uargs(sum^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}, Uargs(c_12) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(id^#) = {}, Uargs(c_17) = {},
Uargs(if^#) = {}, Uargs(c_18) = {}, Uargs(c_19) = {},
Uargs(not^#) = {}, Uargs(c_20) = {}, Uargs(gt^#) = {},
Uargs(c_23) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_7(x1) = [1] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1, x2) = [0] x1 + [1] x2 + [0]
sum^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
not^#(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_21() = [0]
c_22() = [0]
c_23(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: {app^#(nil(), k) -> c_7(k)}
Weak Rules: {app^#(cons(x, l), k) -> c_9(x, app^#(l, k))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(app^#) = {}, Uargs(c_7) = {},
Uargs(c_9) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
app^#(x1, x2) = [3] x1 + [7] x2 + [2]
c_7(x1) = [1] x1 + [1]
c_9(x1, x2) = [1] x1 + [1] x2 + [5]
* Path {10}->{9}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(if) = {}, Uargs(gt) = {}, Uargs(not) = {},
Uargs(id) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(c_9) = {2}, Uargs(sum^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}, Uargs(c_12) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(id^#) = {}, Uargs(c_17) = {},
Uargs(if^#) = {}, Uargs(c_18) = {}, Uargs(c_19) = {},
Uargs(not^#) = {}, Uargs(c_20) = {}, Uargs(gt^#) = {},
Uargs(c_23) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [1] x1 + [0]
c_9(x1, x2) = [0] x1 + [1] x2 + [0]
sum^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
not^#(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_21() = [0]
c_22() = [0]
c_23(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: {app^#(l, nil()) -> c_8(l)}
Weak Rules: {app^#(cons(x, l), k) -> c_9(x, app^#(l, k))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(app^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
app^#(x1, x2) = [3] x1 + [3] x2 + [2]
c_8(x1) = [1] x1 + [1]
c_9(x1, x2) = [1] x1 + [1] x2 + [5]
* Path {13}: inherited
--------------------
This path is subsumed by the proof of path {13}->{12}->{11}.
* Path {13}->{11}: NA
-------------------
The usable rules for this path are:
{ app(nil(), k) -> k
, app(l, nil()) -> l
, app(cons(x, l), k) -> cons(x, app(l, k))
, sum(cons(x, nil())) -> cons(x, nil())
, sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
, sum(app(l, cons(x, cons(y, k)))) ->
sum(app(l, sum(cons(x, cons(y, k)))))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, plus(s(x), s(y)) ->
s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
, plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
, plus(zero(), y) -> y
, plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
, id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(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 {13}->{12}: inherited
--------------------------
This path is subsumed by the proof of path {13}->{12}->{11}.
* Path {13}->{12}->{11}: NA
-------------------------
The usable rules for this path are:
{ app(nil(), k) -> k
, app(l, nil()) -> l
, app(cons(x, l), k) -> cons(x, app(l, k))
, sum(cons(x, nil())) -> cons(x, nil())
, sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
, sum(app(l, cons(x, cons(y, k)))) ->
sum(app(l, sum(cons(x, cons(y, k)))))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, plus(s(x), s(y)) ->
s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
, plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
, plus(zero(), y) -> y
, plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
, id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(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 {18}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(if) = {}, Uargs(gt) = {}, Uargs(not) = {},
Uargs(id) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(sum^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}, Uargs(c_12) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(id^#) = {}, Uargs(c_17) = {},
Uargs(if^#) = {}, Uargs(c_18) = {}, Uargs(c_19) = {},
Uargs(not^#) = {}, Uargs(c_20) = {}, Uargs(gt^#) = {},
Uargs(c_23) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1, x2) = [0] x1 + [0] x2 + [0]
sum^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
id^#(x1) = [3] x1 + [0]
c_17(x1) = [1] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
not^#(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_21() = [0]
c_22() = [0]
c_23(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: {id^#(x) -> c_17(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(id^#) = {}, Uargs(c_17) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
id^#(x1) = [7] x1 + [7]
c_17(x1) = [1] x1 + [0]
* Path {21}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(if) = {}, Uargs(gt) = {}, Uargs(not) = {},
Uargs(id) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(sum^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}, Uargs(c_12) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(id^#) = {}, Uargs(c_17) = {},
Uargs(if^#) = {}, Uargs(c_18) = {}, Uargs(c_19) = {},
Uargs(not^#) = {}, Uargs(c_20) = {}, Uargs(gt^#) = {},
Uargs(c_23) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1, x2) = [0] x1 + [0] x2 + [0]
sum^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [1] x1 + [0] x2 + [0] x3 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
not^#(x1) = [3] x1 + [0]
c_20(x1) = [3] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_21() = [0]
c_22() = [0]
c_23(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: {not^#(x) -> c_20(if^#(x, false(), true()))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(not^#) = {}, Uargs(c_20) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [0]
false() = [0]
if^#(x1, x2, x3) = [7] x1 + [0] x2 + [0] x3 + [4]
not^#(x1) = [7] x1 + [7]
c_20(x1) = [0] x1 + [3]
* Path {21}->{19}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(if) = {}, Uargs(gt) = {}, Uargs(not) = {},
Uargs(id) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(sum^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}, Uargs(c_12) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(id^#) = {}, Uargs(c_17) = {},
Uargs(if^#) = {}, Uargs(c_18) = {}, Uargs(c_19) = {},
Uargs(not^#) = {}, Uargs(c_20) = {1}, Uargs(gt^#) = {},
Uargs(c_23) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1, x2) = [0] x1 + [0] x2 + [0]
sum^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [3] x2 + [0] x3 + [0]
c_18(x1) = [1] x1 + [0]
c_19(x1) = [0] x1 + [0]
not^#(x1) = [0] x1 + [0]
c_20(x1) = [1] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_21() = [0]
c_22() = [0]
c_23(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: {if^#(true(), x, y) -> c_18(x)}
Weak Rules: {not^#(x) -> c_20(if^#(x, false(), true()))}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(c_18) = {}, Uargs(not^#) = {},
Uargs(c_20) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
false() = [0]
if^#(x1, x2, x3) = [2] x1 + [2] x2 + [0] x3 + [0]
c_18(x1) = [0] x1 + [1]
not^#(x1) = [7] x1 + [7]
c_20(x1) = [2] x1 + [7]
* Path {21}->{20}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(if) = {}, Uargs(gt) = {}, Uargs(not) = {},
Uargs(id) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(sum^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}, Uargs(c_12) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(id^#) = {}, Uargs(c_17) = {},
Uargs(if^#) = {}, Uargs(c_18) = {}, Uargs(c_19) = {},
Uargs(not^#) = {}, Uargs(c_20) = {1}, Uargs(gt^#) = {},
Uargs(c_23) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1, x2) = [0] x1 + [0] x2 + [0]
sum^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [3] x2 + [0] x3 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [1] x1 + [0]
not^#(x1) = [0] x1 + [0]
c_20(x1) = [1] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_21() = [0]
c_22() = [0]
c_23(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: {if^#(false(), x, y) -> c_19(y)}
Weak Rules: {not^#(x) -> c_20(if^#(x, false(), true()))}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(c_19) = {}, Uargs(not^#) = {},
Uargs(c_20) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [0]
false() = [2]
if^#(x1, x2, x3) = [2] x1 + [0] x2 + [0] x3 + [4]
c_19(x1) = [0] x1 + [1]
not^#(x1) = [7] x1 + [7]
c_20(x1) = [1] x1 + [3]
* Path {24}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(if) = {}, Uargs(gt) = {}, Uargs(not) = {},
Uargs(id) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(sum^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}, Uargs(c_12) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(id^#) = {}, Uargs(c_17) = {},
Uargs(if^#) = {}, Uargs(c_18) = {}, Uargs(c_19) = {},
Uargs(not^#) = {}, Uargs(c_20) = {}, Uargs(gt^#) = {},
Uargs(c_23) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1, x2) = [0] x1 + [0] x2 + [0]
sum^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
not^#(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
gt^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_21() = [0]
c_22() = [0]
c_23(x1) = [1] 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: {gt^#(s(x), s(y)) -> c_23(gt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_23) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_23(x1) = [1] x1 + [7]
* Path {24}->{22}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(if) = {}, Uargs(gt) = {}, Uargs(not) = {},
Uargs(id) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(sum^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}, Uargs(c_12) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(id^#) = {}, Uargs(c_17) = {},
Uargs(if^#) = {}, Uargs(c_18) = {}, Uargs(c_19) = {},
Uargs(not^#) = {}, Uargs(c_20) = {}, Uargs(gt^#) = {},
Uargs(c_23) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1, x2) = [0] x1 + [0] x2 + [0]
sum^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
not^#(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_21() = [0]
c_22() = [0]
c_23(x1) = [1] 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: {gt^#(s(x), zero()) -> c_21()}
Weak Rules: {gt^#(s(x), s(y)) -> c_23(gt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_23) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
zero() = [2]
gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_21() = [1]
c_23(x1) = [1] x1 + [7]
* Path {24}->{23}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(if) = {}, Uargs(gt) = {}, Uargs(not) = {},
Uargs(id) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(sum^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}, Uargs(c_12) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(id^#) = {}, Uargs(c_17) = {},
Uargs(if^#) = {}, Uargs(c_18) = {}, Uargs(c_19) = {},
Uargs(not^#) = {}, Uargs(c_20) = {}, Uargs(gt^#) = {},
Uargs(c_23) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1, x2) = [0] x1 + [0] x2 + [0]
sum^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
not^#(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_21() = [0]
c_22() = [0]
c_23(x1) = [1] 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: {gt^#(zero(), y) -> c_22()}
Weak Rules: {gt^#(s(x), s(y)) -> c_23(gt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_23) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
zero() = [2]
gt^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_22() = [1]
c_23(x1) = [1] x1 + [2]
2) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
3) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
4) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.