```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))
, bits(0()) -> 0()
, bits(s(x)) -> s(bits(half(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(bits) = {1}

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

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

[0] = [0]

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

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

The order satisfies the following ordering constraints:

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

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

[half(s(s(x)))] =  [1] x + [8]
>  [1] x + [4]
=  [s(half(x))]

[bits(0())] =  [0]
>= [0]
=  [0()]

[bits(s(x))] =  [1] x + [4]
?  [1] x + [8]
=  [s(bits(half(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()
, bits(0()) -> 0()
, bits(s(x)) -> s(bits(half(s(x)))) }
Weak Trs:
{ half(s(0())) -> 0()
, half(s(s(x))) -> s(half(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(bits) = {1}

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

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

[0] = [4]

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

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

The order satisfies the following ordering constraints:

[half(0())] =  [8]
>  [4]
=  [0()]

[half(s(0()))] =  [12]
>  [4]
=  [0()]

[half(s(s(x)))] =  [1] x + [12]
>  [1] x + [8]
=  [s(half(x))]

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

[bits(s(x))] =  [1] x + [4]
?  [1] x + [12]
=  [s(bits(half(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:
{ bits(0()) -> 0()
, bits(s(x)) -> s(bits(half(s(x)))) }
Weak Trs:
{ half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(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(bits) = {1}

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

[half](x1) = [7]

[0] = [0]

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

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

The order satisfies the following ordering constraints:

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

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

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

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

[bits(s(x))] =  [1] x + [1]
?  [8]
=  [s(bits(half(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: { bits(s(x)) -> s(bits(half(s(x)))) }
Weak Trs:
{ half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))
, bits(0()) -> 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: { bits(s(x)) -> s(bits(half(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(bits) = {1}

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

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

[0] = [0]
[0]

[s](x1) = [1 0] x1 + [2]
[1 0]      [4]

[bits](x1) = [1 3] x1 + [1]
[3 1]      [3]

The order satisfies the following ordering constraints:

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

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

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

[bits(0())] =  [1]
[3]
>  [0]
[0]
=  [0()]

[bits(s(x))] =  [4 0] x + [15]
[4 0]     [13]
>  [4 0] x + [11]
[4 0]     [13]
=  [s(bits(half(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))
, bits(0()) -> 0()
, bits(s(x)) -> s(bits(half(s(x)))) }
Obligation:
innermost runtime complexity