```We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict Trs:
{ half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))
, lastbit(0()) -> 0()
, lastbit(s(0())) -> s(0())
, lastbit(s(s(x))) -> lastbit(x)
, conv(0()) -> cons(nil(), 0())
, conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) }
Obligation:
innermost runtime complexity
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(conv) = {1}, Uargs(cons) = {1, 2}

TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).

[half](x1) = [0]

[0] = [0]

[s](x1) = [1] x1 + [0]

[lastbit](x1) = [4]

[conv](x1) = [1] x1 + [0]

[cons](x1, x2) = [1] x1 + [1] x2 + [0]

[nil] = [0]

The order satisfies the following ordering constraints:

[half(0())] =  [0]
>= [0]
=  [0()]

[half(s(0()))] =  [0]
>= [0]
=  [0()]

[half(s(s(x)))] =  [0]
>= [0]
=  [s(half(x))]

[lastbit(0())] =  [4]
>  [0]
=  [0()]

[lastbit(s(0()))] =  [4]
>  [0]
=  [s(0())]

[lastbit(s(s(x)))] =  [4]
>= [4]
=  [lastbit(x)]

[conv(0())] =  [0]
>= [0]
=  [cons(nil(), 0())]

[conv(s(x))] =  [1] x + [0]
?  [4]
=  [cons(conv(half(s(x))), lastbit(s(x)))]

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:
{ half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))
, lastbit(s(s(x))) -> lastbit(x)
, conv(0()) -> cons(nil(), 0())
, conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) }
Weak Trs:
{ lastbit(0()) -> 0()
, lastbit(s(0())) -> s(0()) }
Obligation:
innermost runtime complexity
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(conv) = {1}, Uargs(cons) = {1, 2}

TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).

[half](x1) = [0]

[0] = [1]

[s](x1) = [1] x1 + [1]

[lastbit](x1) = [1] x1 + [7]

[conv](x1) = [1] x1 + [0]

[cons](x1, x2) = [1] x1 + [1] x2 + [0]

[nil] = [7]

The order satisfies the following ordering constraints:

[half(0())] = [0]
? [1]
= [0()]

[half(s(0()))] = [0]
? [1]
= [0()]

[half(s(s(x)))] = [0]
? [1]
= [s(half(x))]

[lastbit(0())] = [8]
> [1]
= [0()]

[lastbit(s(0()))] = [9]
> [2]
= [s(0())]

[lastbit(s(s(x)))] = [1] x + [9]
> [1] x + [7]
= [lastbit(x)]

[conv(0())] = [1]
? [8]
= [cons(nil(), 0())]

[conv(s(x))] = [1] x + [1]
? [1] x + [8]
= [cons(conv(half(s(x))), lastbit(s(x)))]

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:
{ half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))
, conv(0()) -> cons(nil(), 0())
, conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) }
Weak Trs:
{ lastbit(0()) -> 0()
, lastbit(s(0())) -> s(0())
, lastbit(s(s(x))) -> lastbit(x) }
Obligation:
innermost runtime complexity
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(conv) = {1}, Uargs(cons) = {1, 2}

TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).

[half](x1) = [1]

[0] = [0]

[s](x1) = [1] x1 + [0]

[lastbit](x1) = [7]

[conv](x1) = [1] x1 + [0]

[cons](x1, x2) = [1] x1 + [1] x2 + [1]

[nil] = [3]

The order satisfies the following ordering constraints:

[half(0())] =  [1]
>  [0]
=  [0()]

[half(s(0()))] =  [1]
>  [0]
=  [0()]

[half(s(s(x)))] =  [1]
>= [1]
=  [s(half(x))]

[lastbit(0())] =  [7]
>  [0]
=  [0()]

[lastbit(s(0()))] =  [7]
>  [0]
=  [s(0())]

[lastbit(s(s(x)))] =  [7]
>= [7]
=  [lastbit(x)]

[conv(0())] =  [0]
?  [4]
=  [cons(nil(), 0())]

[conv(s(x))] =  [1] x + [0]
?  [9]
=  [cons(conv(half(s(x))), lastbit(s(x)))]

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:
{ half(s(s(x))) -> s(half(x))
, conv(0()) -> cons(nil(), 0())
, conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) }
Weak Trs:
{ half(0()) -> 0()
, half(s(0())) -> 0()
, lastbit(0()) -> 0()
, lastbit(s(0())) -> s(0())
, lastbit(s(s(x))) -> lastbit(x) }
Obligation:
innermost runtime complexity
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(conv) = {1}, Uargs(cons) = {1, 2}

TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).

[half](x1) = [1] x1 + [1]

[0] = [0]

[s](x1) = [1] x1 + [0]

[lastbit](x1) = [6]

[conv](x1) = [1] x1 + [1]

[cons](x1, x2) = [1] x1 + [1] x2 + [0]

[nil] = [0]

The order satisfies the following ordering constraints:

[half(0())] =  [1]
>  [0]
=  [0()]

[half(s(0()))] =  [1]
>  [0]
=  [0()]

[half(s(s(x)))] =  [1] x + [1]
>= [1] x + [1]
=  [s(half(x))]

[lastbit(0())] =  [6]
>  [0]
=  [0()]

[lastbit(s(0()))] =  [6]
>  [0]
=  [s(0())]

[lastbit(s(s(x)))] =  [6]
>= [6]
=  [lastbit(x)]

[conv(0())] =  [1]
>  [0]
=  [cons(nil(), 0())]

[conv(s(x))] =  [1] x + [1]
?  [1] x + [8]
=  [cons(conv(half(s(x))), lastbit(s(x)))]

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:
{ half(s(s(x))) -> s(half(x))
, conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) }
Weak Trs:
{ half(0()) -> 0()
, half(s(0())) -> 0()
, lastbit(0()) -> 0()
, lastbit(s(0())) -> s(0())
, lastbit(s(s(x))) -> lastbit(x)
, conv(0()) -> cons(nil(), 0()) }
Obligation:
innermost runtime complexity
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(conv) = {1}, Uargs(cons) = {1, 2}

TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).

[half](x1) = [1] x1 + [1]

[0] = [0]

[s](x1) = [1] x1 + [4]

[lastbit](x1) = [6]

[conv](x1) = [1] x1 + [1]

[cons](x1, x2) = [1] x1 + [1] x2 + [0]

[nil] = [0]

The order satisfies the following ordering constraints:

[half(0())] =  [1]
>  [0]
=  [0()]

[half(s(0()))] =  [5]
>  [0]
=  [0()]

[half(s(s(x)))] =  [1] x + [9]
>  [1] x + [5]
=  [s(half(x))]

[lastbit(0())] =  [6]
>  [0]
=  [0()]

[lastbit(s(0()))] =  [6]
>  [4]
=  [s(0())]

[lastbit(s(s(x)))] =  [6]
>= [6]
=  [lastbit(x)]

[conv(0())] =  [1]
>  [0]
=  [cons(nil(), 0())]

[conv(s(x))] =  [1] x + [5]
?  [1] x + [12]
=  [cons(conv(half(s(x))), lastbit(s(x)))]

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: { conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) }
Weak Trs:
{ half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))
, lastbit(0()) -> 0()
, lastbit(s(0())) -> s(0())
, lastbit(s(s(x))) -> lastbit(x)
, conv(0()) -> cons(nil(), 0()) }
Obligation:
innermost runtime complexity
YES(O(1),O(n^1))

We use the processor 'matrix interpretation of dimension 2' to
orient following rules strictly.

Trs: { conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) }

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(conv) = {1}, Uargs(cons) = {1, 2}

TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA) and not(IDA(1)).

[half](x1) = [1 0] x1 + [0]
[1 0]      [1]

[0] = [0]
[1]

[s](x1) = [1 0] x1 + [2]
[1 0]      [5]

[lastbit](x1) = [0 0] x1 + [2]
[1 0]      [7]

[conv](x1) = [1 2] x1 + [0]
[1 0]      [3]

[cons](x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0]      [0 0]      [0]

[nil] = [1]
[0]

The order satisfies the following ordering constraints:

[half(0())] =  [0]
[1]
>= [0]
[1]
=  [0()]

[half(s(0()))] =  [2]
[3]
>  [0]
[1]
=  [0()]

[half(s(s(x)))] =  [1 0] x + [4]
[1 0]     [5]
>  [1 0] x + [2]
[1 0]     [5]
=  [s(half(x))]

[lastbit(0())] =  [2]
[7]
>  [0]
[1]
=  [0()]

[lastbit(s(0()))] =  [2]
[9]
>= [2]
[5]
=  [s(0())]

[lastbit(s(s(x)))] =  [0 0] x + [2]
[1 0]     [11]
>= [0 0] x + [2]
[1 0]     [7]
=  [lastbit(x)]

[conv(0())] =  [2]
[3]
>= [2]
[0]
=  [cons(nil(), 0())]

[conv(s(x))] =  [3 0] x + [12]
[1 0]     [5]
>  [3 0] x + [11]
[0 0]     [0]
=  [cons(conv(half(s(x))), lastbit(s(x)))]

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).

Weak Trs:
{ half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))
, lastbit(0()) -> 0()
, lastbit(s(0())) -> s(0())
, lastbit(s(s(x))) -> lastbit(x)
, conv(0()) -> cons(nil(), 0())
, conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) }
Obligation:
innermost runtime complexity