```We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).

Strict Trs:
{ a__f(X1, X2) -> f(X1, X2)
, a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
, mark(g(X)) -> g(mark(X))
, mark(f(X1, X2)) -> a__f(mark(X1), X2) }
Obligation:
innermost runtime complexity
YES(O(1),O(n^2))

The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)

The following argument positions are usable:
Uargs(a__f) = {1}, Uargs(g) = {1}

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

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

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

[mark](x1) = [0]

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

The order satisfies the following ordering constraints:

[a__f(X1, X2)] =  [1] X1 + [1]
>  [1] X1 + [0]
=  [f(X1, X2)]

[a__f(g(X), Y)] =  [1] X + [1]
>= [1]
=  [a__f(mark(X), f(g(X), Y))]

[mark(g(X))] =  [0]
>= [0]
=  [g(mark(X))]

[mark(f(X1, X2))] =  [0]
?  [1]
=  [a__f(mark(X1), X2)]

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^2)).

Strict Trs:
{ a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
, mark(g(X)) -> g(mark(X))
, mark(f(X1, X2)) -> a__f(mark(X1), X2) }
Weak Trs: { a__f(X1, X2) -> f(X1, X2) }
Obligation:
innermost runtime complexity
YES(O(1),O(n^2))

The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)

The following argument positions are usable:
Uargs(a__f) = {1}, Uargs(g) = {1}

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

[a__f](x1, x2) = [1] x1 + [4]

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

[mark](x1) = [1]

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

The order satisfies the following ordering constraints:

[a__f(X1, X2)] = [1] X1 + [4]
> [1] X1 + [0]
= [f(X1, X2)]

[a__f(g(X), Y)] = [1] X + [8]
> [5]
= [a__f(mark(X), f(g(X), Y))]

[mark(g(X))] = [1]
? [5]
= [g(mark(X))]

[mark(f(X1, X2))] = [1]
? [5]
= [a__f(mark(X1), X2)]

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^2)).

Strict Trs:
{ mark(g(X)) -> g(mark(X))
, mark(f(X1, X2)) -> a__f(mark(X1), X2) }
Weak Trs:
{ a__f(X1, X2) -> f(X1, X2)
, a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y)) }
Obligation:
innermost runtime complexity
YES(O(1),O(n^2))

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

Trs: { mark(g(X)) -> g(mark(X)) }

The induced complexity on above rules (modulo remaining rules) is
YES(?,O(n^2)) . These rules are moved into the corresponding weak
component(s).

Sub-proof:
----------
The following argument positions are usable:
Uargs(a__f) = {1}, Uargs(g) = {1}

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

[a__f](x1, x2) = [1 1] x1 + [4]
[0 1]      [4]

[g](x1) = [1 1] x1 + [5]
[0 1]      [4]

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

[f](x1, x2) = [1 1] x1 + [3]
[0 1]      [4]

The order satisfies the following ordering constraints:

[a__f(X1, X2)] =  [1 1] X1 + [4]
[0 1]      [4]
>  [1 1] X1 + [3]
[0 1]      [4]
=  [f(X1, X2)]

[a__f(g(X), Y)] =  [1 2] X + [13]
[0 1]     [8]
>  [1 2] X + [9]
[0 1]     [7]
=  [a__f(mark(X), f(g(X), Y))]

[mark(g(X))] =  [1 2] X + [11]
[0 1]     [7]
>  [1 2] X + [10]
[0 1]     [7]
=  [g(mark(X))]

[mark(f(X1, X2))] =  [1 2] X1 + [9]
[0 1]      [7]
>= [1 2] X1 + [9]
[0 1]      [7]
=  [a__f(mark(X1), X2)]

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).

Strict Trs: { mark(f(X1, X2)) -> a__f(mark(X1), X2) }
Weak Trs:
{ a__f(X1, X2) -> f(X1, X2)
, a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
, mark(g(X)) -> g(mark(X)) }
Obligation:
innermost runtime complexity
YES(O(1),O(n^2))

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

Trs: { mark(f(X1, X2)) -> a__f(mark(X1), X2) }

The induced complexity on above rules (modulo remaining rules) is
YES(?,O(n^2)) . These rules are moved into the corresponding weak
component(s).

Sub-proof:
----------
The following argument positions are usable:
Uargs(a__f) = {1}, Uargs(g) = {1}

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

[a__f](x1, x2) = [1 4] x1 + [0]
[0 1]      [1]

[g](x1) = [1 4] x1 + [0]
[0 1]      [0]

[mark](x1) = [1 4] x1 + [0]
[0 1]      [0]

[f](x1, x2) = [1 4] x1 + [0]
[0 1]      [1]

The order satisfies the following ordering constraints:

[a__f(X1, X2)] =  [1 4] X1 + [0]
[0 1]      [1]
>= [1 4] X1 + [0]
[0 1]      [1]
=  [f(X1, X2)]

[a__f(g(X), Y)] =  [1 8] X + [0]
[0 1]     [1]
>= [1 8] X + [0]
[0 1]     [1]
=  [a__f(mark(X), f(g(X), Y))]

[mark(g(X))] =  [1 8] X + [0]
[0 1]     [0]
>= [1 8] X + [0]
[0 1]     [0]
=  [g(mark(X))]

[mark(f(X1, X2))] =  [1 8] X1 + [4]
[0 1]      [1]
>  [1 8] X1 + [0]
[0 1]      [1]
=  [a__f(mark(X1), X2)]

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

Weak Trs:
{ a__f(X1, X2) -> f(X1, X2)
, a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
, mark(g(X)) -> g(mark(X))
, mark(f(X1, X2)) -> a__f(mark(X1), X2) }
Obligation:
innermost runtime complexity