We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ minus_active(x, y) -> minus(x, y)
, minus_active(0(), y) -> 0()
, minus_active(s(x), s(y)) -> minus_active(x, y)
, mark(0()) -> 0()
, mark(s(x)) -> s(mark(x))
, mark(minus(x, y)) -> minus_active(x, y)
, mark(ge(x, y)) -> ge_active(x, y)
, mark(div(x, y)) -> div_active(mark(x), y)
, mark(if(x, y, z)) -> if_active(mark(x), y, z)
, ge_active(x, y) -> ge(x, y)
, ge_active(x, 0()) -> true()
, ge_active(0(), s(y)) -> false()
, ge_active(s(x), s(y)) -> ge_active(x, y)
, div_active(x, y) -> div(x, y)
, div_active(0(), s(y)) -> 0()
, div_active(s(x), s(y)) ->
if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())
, if_active(x, y, z) -> if(x, y, z)
, if_active(true(), x, y) -> mark(x)
, if_active(false(), x, y) -> mark(y) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[minus_active](x1, x2) = [4]
[0] = [0]
[mark](x1) = [0]
[s](x1) = [1] x1 + [0]
[ge_active](x1, x2) = [0]
[true] = [0]
[minus](x1, x2) = [0]
[false] = [0]
[ge](x1, x2) = [0]
[div](x1, x2) = [0]
[div_active](x1, x2) = [1] x1 + [0]
[if](x1, x2, x3) = [1] x1 + [0]
[if_active](x1, x2, x3) = [1] x1 + [0]
The order satisfies the following ordering constraints:
[minus_active(x, y)] = [4]
> [0]
= [minus(x, y)]
[minus_active(0(), y)] = [4]
> [0]
= [0()]
[minus_active(s(x), s(y))] = [4]
>= [4]
= [minus_active(x, y)]
[mark(0())] = [0]
>= [0]
= [0()]
[mark(s(x))] = [0]
>= [0]
= [s(mark(x))]
[mark(minus(x, y))] = [0]
? [4]
= [minus_active(x, y)]
[mark(ge(x, y))] = [0]
>= [0]
= [ge_active(x, y)]
[mark(div(x, y))] = [0]
>= [0]
= [div_active(mark(x), y)]
[mark(if(x, y, z))] = [0]
>= [0]
= [if_active(mark(x), y, z)]
[ge_active(x, y)] = [0]
>= [0]
= [ge(x, y)]
[ge_active(x, 0())] = [0]
>= [0]
= [true()]
[ge_active(0(), s(y))] = [0]
>= [0]
= [false()]
[ge_active(s(x), s(y))] = [0]
>= [0]
= [ge_active(x, y)]
[div_active(x, y)] = [1] x + [0]
>= [0]
= [div(x, y)]
[div_active(0(), s(y))] = [0]
>= [0]
= [0()]
[div_active(s(x), s(y))] = [1] x + [0]
>= [0]
= [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())]
[if_active(x, y, z)] = [1] x + [0]
>= [1] x + [0]
= [if(x, y, z)]
[if_active(true(), x, y)] = [0]
>= [0]
= [mark(x)]
[if_active(false(), x, y)] = [0]
>= [0]
= [mark(y)]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ minus_active(s(x), s(y)) -> minus_active(x, y)
, mark(0()) -> 0()
, mark(s(x)) -> s(mark(x))
, mark(minus(x, y)) -> minus_active(x, y)
, mark(ge(x, y)) -> ge_active(x, y)
, mark(div(x, y)) -> div_active(mark(x), y)
, mark(if(x, y, z)) -> if_active(mark(x), y, z)
, ge_active(x, y) -> ge(x, y)
, ge_active(x, 0()) -> true()
, ge_active(0(), s(y)) -> false()
, ge_active(s(x), s(y)) -> ge_active(x, y)
, div_active(x, y) -> div(x, y)
, div_active(0(), s(y)) -> 0()
, div_active(s(x), s(y)) ->
if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())
, if_active(x, y, z) -> if(x, y, z)
, if_active(true(), x, y) -> mark(x)
, if_active(false(), x, y) -> mark(y) }
Weak Trs:
{ minus_active(x, y) -> minus(x, y)
, minus_active(0(), y) -> 0() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[minus_active](x1, x2) = [4]
[0] = [1]
[mark](x1) = [1] x1 + [0]
[s](x1) = [1] x1 + [0]
[ge_active](x1, x2) = [1] x1 + [0]
[true] = [0]
[minus](x1, x2) = [0]
[false] = [0]
[ge](x1, x2) = [1] x1 + [4]
[div](x1, x2) = [1] x1 + [1] x2 + [0]
[div_active](x1, x2) = [1] x1 + [1] x2 + [0]
[if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
[if_active](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
The order satisfies the following ordering constraints:
[minus_active(x, y)] = [4]
> [0]
= [minus(x, y)]
[minus_active(0(), y)] = [4]
> [1]
= [0()]
[minus_active(s(x), s(y))] = [4]
>= [4]
= [minus_active(x, y)]
[mark(0())] = [1]
>= [1]
= [0()]
[mark(s(x))] = [1] x + [0]
>= [1] x + [0]
= [s(mark(x))]
[mark(minus(x, y))] = [0]
? [4]
= [minus_active(x, y)]
[mark(ge(x, y))] = [1] x + [4]
> [1] x + [0]
= [ge_active(x, y)]
[mark(div(x, y))] = [1] y + [1] x + [0]
>= [1] y + [1] x + [0]
= [div_active(mark(x), y)]
[mark(if(x, y, z))] = [1] y + [1] x + [1] z + [0]
>= [1] y + [1] x + [1] z + [0]
= [if_active(mark(x), y, z)]
[ge_active(x, y)] = [1] x + [0]
? [1] x + [4]
= [ge(x, y)]
[ge_active(x, 0())] = [1] x + [0]
>= [0]
= [true()]
[ge_active(0(), s(y))] = [1]
> [0]
= [false()]
[ge_active(s(x), s(y))] = [1] x + [0]
>= [1] x + [0]
= [ge_active(x, y)]
[div_active(x, y)] = [1] y + [1] x + [0]
>= [1] y + [1] x + [0]
= [div(x, y)]
[div_active(0(), s(y))] = [1] y + [1]
>= [1]
= [0()]
[div_active(s(x), s(y))] = [1] y + [1] x + [0]
? [1] y + [1] x + [1]
= [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())]
[if_active(x, y, z)] = [1] y + [1] x + [1] z + [0]
>= [1] y + [1] x + [1] z + [0]
= [if(x, y, z)]
[if_active(true(), x, y)] = [1] y + [1] x + [0]
>= [1] x + [0]
= [mark(x)]
[if_active(false(), x, y)] = [1] y + [1] x + [0]
>= [1] y + [0]
= [mark(y)]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ minus_active(s(x), s(y)) -> minus_active(x, y)
, mark(0()) -> 0()
, mark(s(x)) -> s(mark(x))
, mark(minus(x, y)) -> minus_active(x, y)
, mark(div(x, y)) -> div_active(mark(x), y)
, mark(if(x, y, z)) -> if_active(mark(x), y, z)
, ge_active(x, y) -> ge(x, y)
, ge_active(x, 0()) -> true()
, ge_active(s(x), s(y)) -> ge_active(x, y)
, div_active(x, y) -> div(x, y)
, div_active(0(), s(y)) -> 0()
, div_active(s(x), s(y)) ->
if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())
, if_active(x, y, z) -> if(x, y, z)
, if_active(true(), x, y) -> mark(x)
, if_active(false(), x, y) -> mark(y) }
Weak Trs:
{ minus_active(x, y) -> minus(x, y)
, minus_active(0(), y) -> 0()
, mark(ge(x, y)) -> ge_active(x, y)
, ge_active(0(), s(y)) -> false() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[minus_active](x1, x2) = [4]
[0] = [0]
[mark](x1) = [1] x1 + [1]
[s](x1) = [1] x1 + [0]
[ge_active](x1, x2) = [1] x1 + [5]
[true] = [0]
[minus](x1, x2) = [3]
[false] = [0]
[ge](x1, x2) = [1] x1 + [7]
[div](x1, x2) = [1] x1 + [0]
[div_active](x1, x2) = [1] x1 + [0]
[if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
[if_active](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
The order satisfies the following ordering constraints:
[minus_active(x, y)] = [4]
> [3]
= [minus(x, y)]
[minus_active(0(), y)] = [4]
> [0]
= [0()]
[minus_active(s(x), s(y))] = [4]
>= [4]
= [minus_active(x, y)]
[mark(0())] = [1]
> [0]
= [0()]
[mark(s(x))] = [1] x + [1]
>= [1] x + [1]
= [s(mark(x))]
[mark(minus(x, y))] = [4]
>= [4]
= [minus_active(x, y)]
[mark(ge(x, y))] = [1] x + [8]
> [1] x + [5]
= [ge_active(x, y)]
[mark(div(x, y))] = [1] x + [1]
>= [1] x + [1]
= [div_active(mark(x), y)]
[mark(if(x, y, z))] = [1] y + [1] x + [1] z + [1]
>= [1] y + [1] x + [1] z + [1]
= [if_active(mark(x), y, z)]
[ge_active(x, y)] = [1] x + [5]
? [1] x + [7]
= [ge(x, y)]
[ge_active(x, 0())] = [1] x + [5]
> [0]
= [true()]
[ge_active(0(), s(y))] = [5]
> [0]
= [false()]
[ge_active(s(x), s(y))] = [1] x + [5]
>= [1] x + [5]
= [ge_active(x, y)]
[div_active(x, y)] = [1] x + [0]
>= [1] x + [0]
= [div(x, y)]
[div_active(0(), s(y))] = [0]
>= [0]
= [0()]
[div_active(s(x), s(y))] = [1] x + [0]
? [1] x + [8]
= [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())]
[if_active(x, y, z)] = [1] y + [1] x + [1] z + [0]
>= [1] y + [1] x + [1] z + [0]
= [if(x, y, z)]
[if_active(true(), x, y)] = [1] y + [1] x + [0]
? [1] x + [1]
= [mark(x)]
[if_active(false(), x, y)] = [1] y + [1] x + [0]
? [1] y + [1]
= [mark(y)]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ minus_active(s(x), s(y)) -> minus_active(x, y)
, mark(s(x)) -> s(mark(x))
, mark(minus(x, y)) -> minus_active(x, y)
, mark(div(x, y)) -> div_active(mark(x), y)
, mark(if(x, y, z)) -> if_active(mark(x), y, z)
, ge_active(x, y) -> ge(x, y)
, ge_active(s(x), s(y)) -> ge_active(x, y)
, div_active(x, y) -> div(x, y)
, div_active(0(), s(y)) -> 0()
, div_active(s(x), s(y)) ->
if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())
, if_active(x, y, z) -> if(x, y, z)
, if_active(true(), x, y) -> mark(x)
, if_active(false(), x, y) -> mark(y) }
Weak Trs:
{ minus_active(x, y) -> minus(x, y)
, minus_active(0(), y) -> 0()
, mark(0()) -> 0()
, mark(ge(x, y)) -> ge_active(x, y)
, ge_active(x, 0()) -> true()
, ge_active(0(), s(y)) -> false() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[minus_active](x1, x2) = [4]
[0] = [1]
[mark](x1) = [1] x1 + [1]
[s](x1) = [1] x1 + [0]
[ge_active](x1, x2) = [0]
[true] = [0]
[minus](x1, x2) = [0]
[false] = [0]
[ge](x1, x2) = [0]
[div](x1, x2) = [1] x1 + [0]
[div_active](x1, x2) = [1] x1 + [0]
[if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [7]
[if_active](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
The order satisfies the following ordering constraints:
[minus_active(x, y)] = [4]
> [0]
= [minus(x, y)]
[minus_active(0(), y)] = [4]
> [1]
= [0()]
[minus_active(s(x), s(y))] = [4]
>= [4]
= [minus_active(x, y)]
[mark(0())] = [2]
> [1]
= [0()]
[mark(s(x))] = [1] x + [1]
>= [1] x + [1]
= [s(mark(x))]
[mark(minus(x, y))] = [1]
? [4]
= [minus_active(x, y)]
[mark(ge(x, y))] = [1]
> [0]
= [ge_active(x, y)]
[mark(div(x, y))] = [1] x + [1]
>= [1] x + [1]
= [div_active(mark(x), y)]
[mark(if(x, y, z))] = [1] y + [1] x + [1] z + [8]
> [1] y + [1] x + [1] z + [1]
= [if_active(mark(x), y, z)]
[ge_active(x, y)] = [0]
>= [0]
= [ge(x, y)]
[ge_active(x, 0())] = [0]
>= [0]
= [true()]
[ge_active(0(), s(y))] = [0]
>= [0]
= [false()]
[ge_active(s(x), s(y))] = [0]
>= [0]
= [ge_active(x, y)]
[div_active(x, y)] = [1] x + [0]
>= [1] x + [0]
= [div(x, y)]
[div_active(0(), s(y))] = [1]
>= [1]
= [0()]
[div_active(s(x), s(y))] = [1] x + [0]
? [1]
= [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())]
[if_active(x, y, z)] = [1] y + [1] x + [1] z + [0]
? [1] y + [1] x + [1] z + [7]
= [if(x, y, z)]
[if_active(true(), x, y)] = [1] y + [1] x + [0]
? [1] x + [1]
= [mark(x)]
[if_active(false(), x, y)] = [1] y + [1] x + [0]
? [1] y + [1]
= [mark(y)]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ minus_active(s(x), s(y)) -> minus_active(x, y)
, mark(s(x)) -> s(mark(x))
, mark(minus(x, y)) -> minus_active(x, y)
, mark(div(x, y)) -> div_active(mark(x), y)
, ge_active(x, y) -> ge(x, y)
, ge_active(s(x), s(y)) -> ge_active(x, y)
, div_active(x, y) -> div(x, y)
, div_active(0(), s(y)) -> 0()
, div_active(s(x), s(y)) ->
if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())
, if_active(x, y, z) -> if(x, y, z)
, if_active(true(), x, y) -> mark(x)
, if_active(false(), x, y) -> mark(y) }
Weak Trs:
{ minus_active(x, y) -> minus(x, y)
, minus_active(0(), y) -> 0()
, mark(0()) -> 0()
, mark(ge(x, y)) -> ge_active(x, y)
, mark(if(x, y, z)) -> if_active(mark(x), y, z)
, ge_active(x, 0()) -> true()
, ge_active(0(), s(y)) -> false() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[minus_active](x1, x2) = [4]
[0] = [0]
[mark](x1) = [1] x1 + [1]
[s](x1) = [1] x1 + [0]
[ge_active](x1, x2) = [5]
[true] = [0]
[minus](x1, x2) = [0]
[false] = [0]
[ge](x1, x2) = [7]
[div](x1, x2) = [1] x1 + [7]
[div_active](x1, x2) = [1] x1 + [0]
[if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [7]
[if_active](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
The order satisfies the following ordering constraints:
[minus_active(x, y)] = [4]
> [0]
= [minus(x, y)]
[minus_active(0(), y)] = [4]
> [0]
= [0()]
[minus_active(s(x), s(y))] = [4]
>= [4]
= [minus_active(x, y)]
[mark(0())] = [1]
> [0]
= [0()]
[mark(s(x))] = [1] x + [1]
>= [1] x + [1]
= [s(mark(x))]
[mark(minus(x, y))] = [1]
? [4]
= [minus_active(x, y)]
[mark(ge(x, y))] = [8]
> [5]
= [ge_active(x, y)]
[mark(div(x, y))] = [1] x + [8]
> [1] x + [1]
= [div_active(mark(x), y)]
[mark(if(x, y, z))] = [1] y + [1] x + [1] z + [8]
> [1] y + [1] x + [1] z + [1]
= [if_active(mark(x), y, z)]
[ge_active(x, y)] = [5]
? [7]
= [ge(x, y)]
[ge_active(x, 0())] = [5]
> [0]
= [true()]
[ge_active(0(), s(y))] = [5]
> [0]
= [false()]
[ge_active(s(x), s(y))] = [5]
>= [5]
= [ge_active(x, y)]
[div_active(x, y)] = [1] x + [0]
? [1] x + [7]
= [div(x, y)]
[div_active(0(), s(y))] = [0]
>= [0]
= [0()]
[div_active(s(x), s(y))] = [1] x + [0]
? [12]
= [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())]
[if_active(x, y, z)] = [1] y + [1] x + [1] z + [0]
? [1] y + [1] x + [1] z + [7]
= [if(x, y, z)]
[if_active(true(), x, y)] = [1] y + [1] x + [0]
? [1] x + [1]
= [mark(x)]
[if_active(false(), x, y)] = [1] y + [1] x + [0]
? [1] y + [1]
= [mark(y)]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ minus_active(s(x), s(y)) -> minus_active(x, y)
, mark(s(x)) -> s(mark(x))
, mark(minus(x, y)) -> minus_active(x, y)
, ge_active(x, y) -> ge(x, y)
, ge_active(s(x), s(y)) -> ge_active(x, y)
, div_active(x, y) -> div(x, y)
, div_active(0(), s(y)) -> 0()
, div_active(s(x), s(y)) ->
if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())
, if_active(x, y, z) -> if(x, y, z)
, if_active(true(), x, y) -> mark(x)
, if_active(false(), x, y) -> mark(y) }
Weak Trs:
{ minus_active(x, y) -> minus(x, y)
, minus_active(0(), y) -> 0()
, mark(0()) -> 0()
, mark(ge(x, y)) -> ge_active(x, y)
, mark(div(x, y)) -> div_active(mark(x), y)
, mark(if(x, y, z)) -> if_active(mark(x), y, z)
, ge_active(x, 0()) -> true()
, ge_active(0(), s(y)) -> false() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[minus_active](x1, x2) = [0]
[0] = [0]
[mark](x1) = [1] x1 + [1]
[s](x1) = [1] x1 + [0]
[ge_active](x1, x2) = [5]
[true] = [0]
[minus](x1, x2) = [0]
[false] = [0]
[ge](x1, x2) = [7]
[div](x1, x2) = [1] x1 + [7]
[div_active](x1, x2) = [1] x1 + [0]
[if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [7]
[if_active](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
The order satisfies the following ordering constraints:
[minus_active(x, y)] = [0]
>= [0]
= [minus(x, y)]
[minus_active(0(), y)] = [0]
>= [0]
= [0()]
[minus_active(s(x), s(y))] = [0]
>= [0]
= [minus_active(x, y)]
[mark(0())] = [1]
> [0]
= [0()]
[mark(s(x))] = [1] x + [1]
>= [1] x + [1]
= [s(mark(x))]
[mark(minus(x, y))] = [1]
> [0]
= [minus_active(x, y)]
[mark(ge(x, y))] = [8]
> [5]
= [ge_active(x, y)]
[mark(div(x, y))] = [1] x + [8]
> [1] x + [1]
= [div_active(mark(x), y)]
[mark(if(x, y, z))] = [1] y + [1] x + [1] z + [8]
> [1] y + [1] x + [1] z + [1]
= [if_active(mark(x), y, z)]
[ge_active(x, y)] = [5]
? [7]
= [ge(x, y)]
[ge_active(x, 0())] = [5]
> [0]
= [true()]
[ge_active(0(), s(y))] = [5]
> [0]
= [false()]
[ge_active(s(x), s(y))] = [5]
>= [5]
= [ge_active(x, y)]
[div_active(x, y)] = [1] x + [0]
? [1] x + [7]
= [div(x, y)]
[div_active(0(), s(y))] = [0]
>= [0]
= [0()]
[div_active(s(x), s(y))] = [1] x + [0]
? [12]
= [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())]
[if_active(x, y, z)] = [1] y + [1] x + [1] z + [0]
? [1] y + [1] x + [1] z + [7]
= [if(x, y, z)]
[if_active(true(), x, y)] = [1] y + [1] x + [0]
? [1] x + [1]
= [mark(x)]
[if_active(false(), x, y)] = [1] y + [1] x + [0]
? [1] y + [1]
= [mark(y)]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ minus_active(s(x), s(y)) -> minus_active(x, y)
, mark(s(x)) -> s(mark(x))
, ge_active(x, y) -> ge(x, y)
, ge_active(s(x), s(y)) -> ge_active(x, y)
, div_active(x, y) -> div(x, y)
, div_active(0(), s(y)) -> 0()
, div_active(s(x), s(y)) ->
if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())
, if_active(x, y, z) -> if(x, y, z)
, if_active(true(), x, y) -> mark(x)
, if_active(false(), x, y) -> mark(y) }
Weak Trs:
{ minus_active(x, y) -> minus(x, y)
, minus_active(0(), y) -> 0()
, mark(0()) -> 0()
, mark(minus(x, y)) -> minus_active(x, y)
, mark(ge(x, y)) -> ge_active(x, y)
, mark(div(x, y)) -> div_active(mark(x), y)
, mark(if(x, y, z)) -> if_active(mark(x), y, z)
, ge_active(x, 0()) -> true()
, ge_active(0(), s(y)) -> false() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[minus_active](x1, x2) = [1] x1 + [0]
[0] = [4]
[mark](x1) = [1] x1 + [0]
[s](x1) = [1] x1 + [0]
[ge_active](x1, x2) = [4]
[true] = [3]
[minus](x1, x2) = [1] x1 + [0]
[false] = [0]
[ge](x1, x2) = [4]
[div](x1, x2) = [1] x1 + [1] x2 + [0]
[div_active](x1, x2) = [1] x1 + [1] x2 + [0]
[if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [4]
[if_active](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
The order satisfies the following ordering constraints:
[minus_active(x, y)] = [1] x + [0]
>= [1] x + [0]
= [minus(x, y)]
[minus_active(0(), y)] = [4]
>= [4]
= [0()]
[minus_active(s(x), s(y))] = [1] x + [0]
>= [1] x + [0]
= [minus_active(x, y)]
[mark(0())] = [4]
>= [4]
= [0()]
[mark(s(x))] = [1] x + [0]
>= [1] x + [0]
= [s(mark(x))]
[mark(minus(x, y))] = [1] x + [0]
>= [1] x + [0]
= [minus_active(x, y)]
[mark(ge(x, y))] = [4]
>= [4]
= [ge_active(x, y)]
[mark(div(x, y))] = [1] y + [1] x + [0]
>= [1] y + [1] x + [0]
= [div_active(mark(x), y)]
[mark(if(x, y, z))] = [1] y + [1] x + [1] z + [4]
> [1] y + [1] x + [1] z + [1]
= [if_active(mark(x), y, z)]
[ge_active(x, y)] = [4]
>= [4]
= [ge(x, y)]
[ge_active(x, 0())] = [4]
> [3]
= [true()]
[ge_active(0(), s(y))] = [4]
> [0]
= [false()]
[ge_active(s(x), s(y))] = [4]
>= [4]
= [ge_active(x, y)]
[div_active(x, y)] = [1] y + [1] x + [0]
>= [1] y + [1] x + [0]
= [div(x, y)]
[div_active(0(), s(y))] = [1] y + [4]
>= [4]
= [0()]
[div_active(s(x), s(y))] = [1] y + [1] x + [0]
? [1] y + [1] x + [9]
= [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())]
[if_active(x, y, z)] = [1] y + [1] x + [1] z + [1]
? [1] y + [1] x + [1] z + [4]
= [if(x, y, z)]
[if_active(true(), x, y)] = [1] y + [1] x + [4]
> [1] x + [0]
= [mark(x)]
[if_active(false(), x, y)] = [1] y + [1] x + [1]
> [1] y + [0]
= [mark(y)]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ minus_active(s(x), s(y)) -> minus_active(x, y)
, mark(s(x)) -> s(mark(x))
, ge_active(x, y) -> ge(x, y)
, ge_active(s(x), s(y)) -> ge_active(x, y)
, div_active(x, y) -> div(x, y)
, div_active(0(), s(y)) -> 0()
, div_active(s(x), s(y)) ->
if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())
, if_active(x, y, z) -> if(x, y, z) }
Weak Trs:
{ minus_active(x, y) -> minus(x, y)
, minus_active(0(), y) -> 0()
, mark(0()) -> 0()
, mark(minus(x, y)) -> minus_active(x, y)
, mark(ge(x, y)) -> ge_active(x, y)
, mark(div(x, y)) -> div_active(mark(x), y)
, mark(if(x, y, z)) -> if_active(mark(x), y, z)
, ge_active(x, 0()) -> true()
, ge_active(0(), s(y)) -> false()
, if_active(true(), x, y) -> mark(x)
, if_active(false(), x, y) -> mark(y) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[minus_active](x1, x2) = [4]
[0] = [0]
[mark](x1) = [1] x1 + [0]
[s](x1) = [1] x1 + [0]
[ge_active](x1, x2) = [1] x1 + [4]
[true] = [0]
[minus](x1, x2) = [4]
[false] = [4]
[ge](x1, x2) = [1] x1 + [5]
[div](x1, x2) = [1] x1 + [1]
[div_active](x1, x2) = [1] x1 + [1]
[if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
[if_active](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
The order satisfies the following ordering constraints:
[minus_active(x, y)] = [4]
>= [4]
= [minus(x, y)]
[minus_active(0(), y)] = [4]
> [0]
= [0()]
[minus_active(s(x), s(y))] = [4]
>= [4]
= [minus_active(x, y)]
[mark(0())] = [0]
>= [0]
= [0()]
[mark(s(x))] = [1] x + [0]
>= [1] x + [0]
= [s(mark(x))]
[mark(minus(x, y))] = [4]
>= [4]
= [minus_active(x, y)]
[mark(ge(x, y))] = [1] x + [5]
> [1] x + [4]
= [ge_active(x, y)]
[mark(div(x, y))] = [1] x + [1]
>= [1] x + [1]
= [div_active(mark(x), y)]
[mark(if(x, y, z))] = [1] y + [1] x + [1] z + [0]
>= [1] y + [1] x + [1] z + [0]
= [if_active(mark(x), y, z)]
[ge_active(x, y)] = [1] x + [4]
? [1] x + [5]
= [ge(x, y)]
[ge_active(x, 0())] = [1] x + [4]
> [0]
= [true()]
[ge_active(0(), s(y))] = [4]
>= [4]
= [false()]
[ge_active(s(x), s(y))] = [1] x + [4]
>= [1] x + [4]
= [ge_active(x, y)]
[div_active(x, y)] = [1] x + [1]
>= [1] x + [1]
= [div(x, y)]
[div_active(0(), s(y))] = [1]
> [0]
= [0()]
[div_active(s(x), s(y))] = [1] x + [1]
? [1] x + [9]
= [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())]
[if_active(x, y, z)] = [1] y + [1] x + [1] z + [0]
>= [1] y + [1] x + [1] z + [0]
= [if(x, y, z)]
[if_active(true(), x, y)] = [1] y + [1] x + [0]
>= [1] x + [0]
= [mark(x)]
[if_active(false(), x, y)] = [1] y + [1] x + [4]
> [1] y + [0]
= [mark(y)]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ minus_active(s(x), s(y)) -> minus_active(x, y)
, mark(s(x)) -> s(mark(x))
, ge_active(x, y) -> ge(x, y)
, ge_active(s(x), s(y)) -> ge_active(x, y)
, div_active(x, y) -> div(x, y)
, div_active(s(x), s(y)) ->
if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())
, if_active(x, y, z) -> if(x, y, z) }
Weak Trs:
{ minus_active(x, y) -> minus(x, y)
, minus_active(0(), y) -> 0()
, mark(0()) -> 0()
, mark(minus(x, y)) -> minus_active(x, y)
, mark(ge(x, y)) -> ge_active(x, y)
, mark(div(x, y)) -> div_active(mark(x), y)
, mark(if(x, y, z)) -> if_active(mark(x), y, z)
, ge_active(x, 0()) -> true()
, ge_active(0(), s(y)) -> false()
, div_active(0(), s(y)) -> 0()
, if_active(true(), x, y) -> mark(x)
, if_active(false(), x, y) -> mark(y) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[minus_active](x1, x2) = [4]
[0] = [4]
[mark](x1) = [1] x1 + [1]
[s](x1) = [1] x1 + [0]
[ge_active](x1, x2) = [1] x1 + [1]
[true] = [1]
[minus](x1, x2) = [3]
[false] = [4]
[ge](x1, x2) = [1] x1 + [0]
[div](x1, x2) = [1] x1 + [0]
[div_active](x1, x2) = [1] x1 + [0]
[if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
[if_active](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
The order satisfies the following ordering constraints:
[minus_active(x, y)] = [4]
> [3]
= [minus(x, y)]
[minus_active(0(), y)] = [4]
>= [4]
= [0()]
[minus_active(s(x), s(y))] = [4]
>= [4]
= [minus_active(x, y)]
[mark(0())] = [5]
> [4]
= [0()]
[mark(s(x))] = [1] x + [1]
>= [1] x + [1]
= [s(mark(x))]
[mark(minus(x, y))] = [4]
>= [4]
= [minus_active(x, y)]
[mark(ge(x, y))] = [1] x + [1]
>= [1] x + [1]
= [ge_active(x, y)]
[mark(div(x, y))] = [1] x + [1]
>= [1] x + [1]
= [div_active(mark(x), y)]
[mark(if(x, y, z))] = [1] y + [1] x + [1] z + [1]
>= [1] y + [1] x + [1] z + [1]
= [if_active(mark(x), y, z)]
[ge_active(x, y)] = [1] x + [1]
> [1] x + [0]
= [ge(x, y)]
[ge_active(x, 0())] = [1] x + [1]
>= [1]
= [true()]
[ge_active(0(), s(y))] = [5]
> [4]
= [false()]
[ge_active(s(x), s(y))] = [1] x + [1]
>= [1] x + [1]
= [ge_active(x, y)]
[div_active(x, y)] = [1] x + [0]
>= [1] x + [0]
= [div(x, y)]
[div_active(0(), s(y))] = [4]
>= [4]
= [0()]
[div_active(s(x), s(y))] = [1] x + [0]
? [1] x + [8]
= [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())]
[if_active(x, y, z)] = [1] y + [1] x + [1] z + [0]
>= [1] y + [1] x + [1] z + [0]
= [if(x, y, z)]
[if_active(true(), x, y)] = [1] y + [1] x + [1]
>= [1] x + [1]
= [mark(x)]
[if_active(false(), x, y)] = [1] y + [1] x + [4]
> [1] y + [1]
= [mark(y)]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ minus_active(s(x), s(y)) -> minus_active(x, y)
, mark(s(x)) -> s(mark(x))
, ge_active(s(x), s(y)) -> ge_active(x, y)
, div_active(x, y) -> div(x, y)
, div_active(s(x), s(y)) ->
if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())
, if_active(x, y, z) -> if(x, y, z) }
Weak Trs:
{ minus_active(x, y) -> minus(x, y)
, minus_active(0(), y) -> 0()
, mark(0()) -> 0()
, mark(minus(x, y)) -> minus_active(x, y)
, mark(ge(x, y)) -> ge_active(x, y)
, mark(div(x, y)) -> div_active(mark(x), y)
, mark(if(x, y, z)) -> if_active(mark(x), y, z)
, ge_active(x, y) -> ge(x, y)
, ge_active(x, 0()) -> true()
, ge_active(0(), s(y)) -> false()
, div_active(0(), s(y)) -> 0()
, if_active(true(), x, y) -> mark(x)
, if_active(false(), x, y) -> mark(y) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[minus_active](x1, x2) = [0]
[0] = [0]
[mark](x1) = [0]
[s](x1) = [1] x1 + [7]
[ge_active](x1, x2) = [0]
[true] = [0]
[minus](x1, x2) = [0]
[false] = [0]
[ge](x1, x2) = [0]
[div](x1, x2) = [1] x1 + [0]
[div_active](x1, x2) = [1] x1 + [0]
[if](x1, x2, x3) = [1] x1 + [0]
[if_active](x1, x2, x3) = [1] x1 + [0]
The order satisfies the following ordering constraints:
[minus_active(x, y)] = [0]
>= [0]
= [minus(x, y)]
[minus_active(0(), y)] = [0]
>= [0]
= [0()]
[minus_active(s(x), s(y))] = [0]
>= [0]
= [minus_active(x, y)]
[mark(0())] = [0]
>= [0]
= [0()]
[mark(s(x))] = [0]
? [7]
= [s(mark(x))]
[mark(minus(x, y))] = [0]
>= [0]
= [minus_active(x, y)]
[mark(ge(x, y))] = [0]
>= [0]
= [ge_active(x, y)]
[mark(div(x, y))] = [0]
>= [0]
= [div_active(mark(x), y)]
[mark(if(x, y, z))] = [0]
>= [0]
= [if_active(mark(x), y, z)]
[ge_active(x, y)] = [0]
>= [0]
= [ge(x, y)]
[ge_active(x, 0())] = [0]
>= [0]
= [true()]
[ge_active(0(), s(y))] = [0]
>= [0]
= [false()]
[ge_active(s(x), s(y))] = [0]
>= [0]
= [ge_active(x, y)]
[div_active(x, y)] = [1] x + [0]
>= [1] x + [0]
= [div(x, y)]
[div_active(0(), s(y))] = [0]
>= [0]
= [0()]
[div_active(s(x), s(y))] = [1] x + [7]
> [0]
= [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())]
[if_active(x, y, z)] = [1] x + [0]
>= [1] x + [0]
= [if(x, y, z)]
[if_active(true(), x, y)] = [0]
>= [0]
= [mark(x)]
[if_active(false(), x, y)] = [0]
>= [0]
= [mark(y)]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ minus_active(s(x), s(y)) -> minus_active(x, y)
, mark(s(x)) -> s(mark(x))
, ge_active(s(x), s(y)) -> ge_active(x, y)
, div_active(x, y) -> div(x, y)
, if_active(x, y, z) -> if(x, y, z) }
Weak Trs:
{ minus_active(x, y) -> minus(x, y)
, minus_active(0(), y) -> 0()
, mark(0()) -> 0()
, mark(minus(x, y)) -> minus_active(x, y)
, mark(ge(x, y)) -> ge_active(x, y)
, mark(div(x, y)) -> div_active(mark(x), y)
, mark(if(x, y, z)) -> if_active(mark(x), y, z)
, ge_active(x, y) -> ge(x, y)
, ge_active(x, 0()) -> true()
, ge_active(0(), s(y)) -> false()
, div_active(0(), s(y)) -> 0()
, div_active(s(x), s(y)) ->
if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())
, if_active(true(), x, y) -> mark(x)
, if_active(false(), x, y) -> mark(y) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[minus_active](x1, x2) = [0]
[0] = [0]
[mark](x1) = [1] x1 + [0]
[s](x1) = [1] x1 + [1]
[ge_active](x1, x2) = [1] x1 + [0]
[true] = [0]
[minus](x1, x2) = [0]
[false] = [0]
[ge](x1, x2) = [1] x1 + [0]
[div](x1, x2) = [1] x1 + [0]
[div_active](x1, x2) = [1] x1 + [0]
[if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
[if_active](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
The order satisfies the following ordering constraints:
[minus_active(x, y)] = [0]
>= [0]
= [minus(x, y)]
[minus_active(0(), y)] = [0]
>= [0]
= [0()]
[minus_active(s(x), s(y))] = [0]
>= [0]
= [minus_active(x, y)]
[mark(0())] = [0]
>= [0]
= [0()]
[mark(s(x))] = [1] x + [1]
>= [1] x + [1]
= [s(mark(x))]
[mark(minus(x, y))] = [0]
>= [0]
= [minus_active(x, y)]
[mark(ge(x, y))] = [1] x + [0]
>= [1] x + [0]
= [ge_active(x, y)]
[mark(div(x, y))] = [1] x + [0]
>= [1] x + [0]
= [div_active(mark(x), y)]
[mark(if(x, y, z))] = [1] y + [1] x + [1] z + [0]
>= [1] y + [1] x + [1] z + [0]
= [if_active(mark(x), y, z)]
[ge_active(x, y)] = [1] x + [0]
>= [1] x + [0]
= [ge(x, y)]
[ge_active(x, 0())] = [1] x + [0]
>= [0]
= [true()]
[ge_active(0(), s(y))] = [0]
>= [0]
= [false()]
[ge_active(s(x), s(y))] = [1] x + [1]
> [1] x + [0]
= [ge_active(x, y)]
[div_active(x, y)] = [1] x + [0]
>= [1] x + [0]
= [div(x, y)]
[div_active(0(), s(y))] = [0]
>= [0]
= [0()]
[div_active(s(x), s(y))] = [1] x + [1]
>= [1] x + [1]
= [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())]
[if_active(x, y, z)] = [1] y + [1] x + [1] z + [0]
>= [1] y + [1] x + [1] z + [0]
= [if(x, y, z)]
[if_active(true(), x, y)] = [1] y + [1] x + [0]
>= [1] x + [0]
= [mark(x)]
[if_active(false(), x, y)] = [1] y + [1] x + [0]
>= [1] y + [0]
= [mark(y)]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ minus_active(s(x), s(y)) -> minus_active(x, y)
, mark(s(x)) -> s(mark(x))
, div_active(x, y) -> div(x, y)
, if_active(x, y, z) -> if(x, y, z) }
Weak Trs:
{ minus_active(x, y) -> minus(x, y)
, minus_active(0(), y) -> 0()
, mark(0()) -> 0()
, mark(minus(x, y)) -> minus_active(x, y)
, mark(ge(x, y)) -> ge_active(x, y)
, mark(div(x, y)) -> div_active(mark(x), y)
, mark(if(x, y, z)) -> if_active(mark(x), y, z)
, ge_active(x, y) -> ge(x, y)
, ge_active(x, 0()) -> true()
, ge_active(0(), s(y)) -> false()
, ge_active(s(x), s(y)) -> ge_active(x, y)
, div_active(0(), s(y)) -> 0()
, div_active(s(x), s(y)) ->
if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())
, if_active(true(), x, y) -> mark(x)
, if_active(false(), x, y) -> mark(y) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[minus_active](x1, x2) = [1] x1 + [0]
[0] = [0]
[mark](x1) = [1] x1 + [0]
[s](x1) = [1] x1 + [3]
[ge_active](x1, x2) = [0]
[true] = [0]
[minus](x1, x2) = [1] x1 + [0]
[false] = [0]
[ge](x1, x2) = [0]
[div](x1, x2) = [1] x1 + [1] x2 + [6]
[div_active](x1, x2) = [1] x1 + [1] x2 + [6]
[if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [4]
[if_active](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
The order satisfies the following ordering constraints:
[minus_active(x, y)] = [1] x + [0]
>= [1] x + [0]
= [minus(x, y)]
[minus_active(0(), y)] = [0]
>= [0]
= [0()]
[minus_active(s(x), s(y))] = [1] x + [3]
> [1] x + [0]
= [minus_active(x, y)]
[mark(0())] = [0]
>= [0]
= [0()]
[mark(s(x))] = [1] x + [3]
>= [1] x + [3]
= [s(mark(x))]
[mark(minus(x, y))] = [1] x + [0]
>= [1] x + [0]
= [minus_active(x, y)]
[mark(ge(x, y))] = [0]
>= [0]
= [ge_active(x, y)]
[mark(div(x, y))] = [1] y + [1] x + [6]
>= [1] y + [1] x + [6]
= [div_active(mark(x), y)]
[mark(if(x, y, z))] = [1] y + [1] x + [1] z + [4]
> [1] y + [1] x + [1] z + [0]
= [if_active(mark(x), y, z)]
[ge_active(x, y)] = [0]
>= [0]
= [ge(x, y)]
[ge_active(x, 0())] = [0]
>= [0]
= [true()]
[ge_active(0(), s(y))] = [0]
>= [0]
= [false()]
[ge_active(s(x), s(y))] = [0]
>= [0]
= [ge_active(x, y)]
[div_active(x, y)] = [1] y + [1] x + [6]
>= [1] y + [1] x + [6]
= [div(x, y)]
[div_active(0(), s(y))] = [1] y + [9]
> [0]
= [0()]
[div_active(s(x), s(y))] = [1] y + [1] x + [12]
>= [1] y + [1] x + [12]
= [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())]
[if_active(x, y, z)] = [1] y + [1] x + [1] z + [0]
? [1] y + [1] x + [1] z + [4]
= [if(x, y, z)]
[if_active(true(), x, y)] = [1] y + [1] x + [0]
>= [1] x + [0]
= [mark(x)]
[if_active(false(), x, y)] = [1] y + [1] x + [0]
>= [1] y + [0]
= [mark(y)]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ mark(s(x)) -> s(mark(x))
, div_active(x, y) -> div(x, y)
, if_active(x, y, z) -> if(x, y, z) }
Weak Trs:
{ minus_active(x, y) -> minus(x, y)
, minus_active(0(), y) -> 0()
, minus_active(s(x), s(y)) -> minus_active(x, y)
, mark(0()) -> 0()
, mark(minus(x, y)) -> minus_active(x, y)
, mark(ge(x, y)) -> ge_active(x, y)
, mark(div(x, y)) -> div_active(mark(x), y)
, mark(if(x, y, z)) -> if_active(mark(x), y, z)
, ge_active(x, y) -> ge(x, y)
, ge_active(x, 0()) -> true()
, ge_active(0(), s(y)) -> false()
, ge_active(s(x), s(y)) -> ge_active(x, y)
, div_active(0(), s(y)) -> 0()
, div_active(s(x), s(y)) ->
if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())
, if_active(true(), x, y) -> mark(x)
, if_active(false(), x, y) -> mark(y) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We use the processor 'matrix interpretation of dimension 1' to
orient following rules strictly.
Trs: { div_active(x, y) -> div(x, y) }
The induced complexity on above rules (modulo remaining rules) is
YES(?,O(n^1)) . These rules are moved into the corresponding weak
component(s).
Sub-proof:
----------
The following argument positions are usable:
Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[minus_active](x1, x2) = [0]
[0] = [0]
[mark](x1) = [5] x1 + [0]
[s](x1) = [1] x1 + [0]
[ge_active](x1, x2) = [0]
[true] = [0]
[minus](x1, x2) = [0]
[false] = [0]
[ge](x1, x2) = [0]
[div](x1, x2) = [1] x1 + [1] x2 + [1]
[div_active](x1, x2) = [1] x1 + [5] x2 + [5]
[if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
[if_active](x1, x2, x3) = [1] x1 + [5] x2 + [5] x3 + [0]
The order satisfies the following ordering constraints:
[minus_active(x, y)] = [0]
>= [0]
= [minus(x, y)]
[minus_active(0(), y)] = [0]
>= [0]
= [0()]
[minus_active(s(x), s(y))] = [0]
>= [0]
= [minus_active(x, y)]
[mark(0())] = [0]
>= [0]
= [0()]
[mark(s(x))] = [5] x + [0]
>= [5] x + [0]
= [s(mark(x))]
[mark(minus(x, y))] = [0]
>= [0]
= [minus_active(x, y)]
[mark(ge(x, y))] = [0]
>= [0]
= [ge_active(x, y)]
[mark(div(x, y))] = [5] y + [5] x + [5]
>= [5] y + [5] x + [5]
= [div_active(mark(x), y)]
[mark(if(x, y, z))] = [5] y + [5] x + [5] z + [0]
>= [5] y + [5] x + [5] z + [0]
= [if_active(mark(x), y, z)]
[ge_active(x, y)] = [0]
>= [0]
= [ge(x, y)]
[ge_active(x, 0())] = [0]
>= [0]
= [true()]
[ge_active(0(), s(y))] = [0]
>= [0]
= [false()]
[ge_active(s(x), s(y))] = [0]
>= [0]
= [ge_active(x, y)]
[div_active(x, y)] = [5] y + [1] x + [5]
> [1] y + [1] x + [1]
= [div(x, y)]
[div_active(0(), s(y))] = [5] y + [5]
> [0]
= [0()]
[div_active(s(x), s(y))] = [5] y + [1] x + [5]
>= [5] y + [5]
= [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())]
[if_active(x, y, z)] = [5] y + [1] x + [5] z + [0]
>= [1] y + [1] x + [1] z + [0]
= [if(x, y, z)]
[if_active(true(), x, y)] = [5] y + [5] x + [0]
>= [5] x + [0]
= [mark(x)]
[if_active(false(), x, y)] = [5] y + [5] x + [0]
>= [5] y + [0]
= [mark(y)]
We return to the main proof.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ mark(s(x)) -> s(mark(x))
, if_active(x, y, z) -> if(x, y, z) }
Weak Trs:
{ minus_active(x, y) -> minus(x, y)
, minus_active(0(), y) -> 0()
, minus_active(s(x), s(y)) -> minus_active(x, y)
, mark(0()) -> 0()
, mark(minus(x, y)) -> minus_active(x, y)
, mark(ge(x, y)) -> ge_active(x, y)
, mark(div(x, y)) -> div_active(mark(x), y)
, mark(if(x, y, z)) -> if_active(mark(x), y, z)
, ge_active(x, y) -> ge(x, y)
, ge_active(x, 0()) -> true()
, ge_active(0(), s(y)) -> false()
, ge_active(s(x), s(y)) -> ge_active(x, y)
, div_active(x, y) -> div(x, y)
, div_active(0(), s(y)) -> 0()
, div_active(s(x), s(y)) ->
if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())
, if_active(true(), x, y) -> mark(x)
, if_active(false(), x, y) -> mark(y) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We use the processor 'matrix interpretation of dimension 2' to
orient following rules strictly.
Trs:
{ mark(s(x)) -> s(mark(x))
, if_active(x, y, z) -> if(x, y, z) }
The induced complexity on above rules (modulo remaining rules) is
YES(?,O(n^1)) . These rules are moved into the corresponding weak
component(s).
Sub-proof:
----------
The following argument positions are usable:
Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA) and not(IDA(1)).
[minus_active](x1, x2) = [0]
[0]
[0] = [0]
[0]
[mark](x1) = [2 0] x1 + [0]
[0 2] [0]
[s](x1) = [1 0] x1 + [3]
[0 0] [4]
[ge_active](x1, x2) = [1]
[0]
[true] = [1]
[0]
[minus](x1, x2) = [0]
[0]
[false] = [0]
[0]
[ge](x1, x2) = [1]
[0]
[div](x1, x2) = [1 3] x1 + [0 0] x2 + [0]
[0 0] [0 1] [0]
[div_active](x1, x2) = [1 3] x1 + [0 0] x2 + [0]
[0 0] [0 2] [0]
[if](x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [3]
[0 1] [0 1] [0 1] [0]
[if_active](x1, x2, x3) = [1 0] x1 + [2 0] x2 + [2 0] x3 + [5]
[0 1] [0 2] [0 2] [0]
The order satisfies the following ordering constraints:
[minus_active(x, y)] = [0]
[0]
>= [0]
[0]
= [minus(x, y)]
[minus_active(0(), y)] = [0]
[0]
>= [0]
[0]
= [0()]
[minus_active(s(x), s(y))] = [0]
[0]
>= [0]
[0]
= [minus_active(x, y)]
[mark(0())] = [0]
[0]
>= [0]
[0]
= [0()]
[mark(s(x))] = [2 0] x + [6]
[0 0] [8]
> [2 0] x + [3]
[0 0] [4]
= [s(mark(x))]
[mark(minus(x, y))] = [0]
[0]
>= [0]
[0]
= [minus_active(x, y)]
[mark(ge(x, y))] = [2]
[0]
> [1]
[0]
= [ge_active(x, y)]
[mark(div(x, y))] = [0 0] y + [2 6] x + [0]
[0 2] [0 0] [0]
>= [0 0] y + [2 6] x + [0]
[0 2] [0 0] [0]
= [div_active(mark(x), y)]
[mark(if(x, y, z))] = [2 0] y + [2 0] x + [2 0] z + [6]
[0 2] [0 2] [0 2] [0]
> [2 0] y + [2 0] x + [2 0] z + [5]
[0 2] [0 2] [0 2] [0]
= [if_active(mark(x), y, z)]
[ge_active(x, y)] = [1]
[0]
>= [1]
[0]
= [ge(x, y)]
[ge_active(x, 0())] = [1]
[0]
>= [1]
[0]
= [true()]
[ge_active(0(), s(y))] = [1]
[0]
> [0]
[0]
= [false()]
[ge_active(s(x), s(y))] = [1]
[0]
>= [1]
[0]
= [ge_active(x, y)]
[div_active(x, y)] = [0 0] y + [1 3] x + [0]
[0 2] [0 0] [0]
>= [0 0] y + [1 3] x + [0]
[0 1] [0 0] [0]
= [div(x, y)]
[div_active(0(), s(y))] = [0]
[8]
>= [0]
[0]
= [0()]
[div_active(s(x), s(y))] = [1 0] x + [15]
[0 0] [8]
> [12]
[8]
= [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())]
[if_active(x, y, z)] = [2 0] y + [1 0] x + [2 0] z + [5]
[0 2] [0 1] [0 2] [0]
> [1 0] y + [1 0] x + [1 0] z + [3]
[0 1] [0 1] [0 1] [0]
= [if(x, y, z)]
[if_active(true(), x, y)] = [2 0] y + [2 0] x + [6]
[0 2] [0 2] [0]
> [2 0] x + [0]
[0 2] [0]
= [mark(x)]
[if_active(false(), x, y)] = [2 0] y + [2 0] x + [5]
[0 2] [0 2] [0]
> [2 0] y + [0]
[0 2] [0]
= [mark(y)]
We return to the main proof.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Weak Trs:
{ minus_active(x, y) -> minus(x, y)
, minus_active(0(), y) -> 0()
, minus_active(s(x), s(y)) -> minus_active(x, y)
, mark(0()) -> 0()
, mark(s(x)) -> s(mark(x))
, mark(minus(x, y)) -> minus_active(x, y)
, mark(ge(x, y)) -> ge_active(x, y)
, mark(div(x, y)) -> div_active(mark(x), y)
, mark(if(x, y, z)) -> if_active(mark(x), y, z)
, ge_active(x, y) -> ge(x, y)
, ge_active(x, 0()) -> true()
, ge_active(0(), s(y)) -> false()
, ge_active(s(x), s(y)) -> ge_active(x, y)
, div_active(x, y) -> div(x, y)
, div_active(0(), s(y)) -> 0()
, div_active(s(x), s(y)) ->
if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())
, if_active(x, y, z) -> if(x, y, z)
, if_active(true(), x, y) -> mark(x)
, if_active(false(), x, y) -> mark(y) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
Empty rules are trivially bounded
Hurray, we answered YES(O(1),O(n^1))