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

Strict Trs:
{ f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, g(x, c(y)) -> g(x, g(s(c(y)), y))
, g(s(x), s(y)) -> if(f(x), s(x), s(y)) }
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(if) = {1}, Uargs(g) = {2}

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

[f](x1) = [0]

[0] = [0]

[true] = [0]

[1] = [0]

[false] = [4]

[s](x1) = [0]

[if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]

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

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

The order satisfies the following ordering constraints:

[f(0())] =  [0]
>= [0]
=  [true()]

[f(1())] =  [0]
?  [4]
=  [false()]

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

[if(true(), x, y)] =  [1] x + [1] y + [1]
>  [1] x + [0]
=  [x]

[if(false(), x, y)] =  [1] x + [1] y + [5]
>  [1] y + [0]
=  [y]

[g(x, c(y))] =  [1] x + [1] y + [0]
>= [1] x + [1] y + [0]
=  [g(x, g(s(c(y)), y))]

[g(s(x), s(y))] =  [0]
?  [1]
=  [if(f(x), s(x), s(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:
{ f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, g(x, c(y)) -> g(x, g(s(c(y)), y))
, g(s(x), s(y)) -> if(f(x), s(x), s(y)) }
Weak Trs:
{ if(true(), x, y) -> x
, if(false(), x, y) -> y }
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(if) = {1}, Uargs(g) = {2}

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

[f](x1) = [1]

[0] = [0]

[true] = [0]

[1] = [0]

[false] = [0]

[s](x1) = [0]

[if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]

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

[c](x1) = [1] x1 + [2]

The order satisfies the following ordering constraints:

[f(0())] =  [1]
>  [0]
=  [true()]

[f(1())] =  [1]
>  [0]
=  [false()]

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

[if(true(), x, y)] =  [1] x + [1] y + [0]
>= [1] x + [0]
=  [x]

[if(false(), x, y)] =  [1] x + [1] y + [0]
>= [1] y + [0]
=  [y]

[g(x, c(y))] =  [1] y + [2]
>  [1] y + [0]
=  [g(x, g(s(c(y)), y))]

[g(s(x), s(y))] =  [0]
?  [1]
=  [if(f(x), s(x), s(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:
{ f(s(x)) -> f(x)
, g(s(x), s(y)) -> if(f(x), s(x), s(y)) }
Weak Trs:
{ f(0()) -> true()
, f(1()) -> false()
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, g(x, c(y)) -> g(x, g(s(c(y)), y)) }
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(if) = {1}, Uargs(g) = {2}

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

[f](x1) = [1]

[0] = [0]

[true] = [0]

[1] = [0]

[false] = [0]

[s](x1) = [0]

[if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]

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

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

The order satisfies the following ordering constraints:

[f(0())] =  [1]
>  [0]
=  [true()]

[f(1())] =  [1]
>  [0]
=  [false()]

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

[if(true(), x, y)] =  [1] x + [1] y + [0]
>= [1] x + [0]
=  [x]

[if(false(), x, y)] =  [1] x + [1] y + [0]
>= [1] y + [0]
=  [y]

[g(x, c(y))] =  [1] y + [11]
>  [1] y + [8]
=  [g(x, g(s(c(y)), y))]

[g(s(x), s(y))] =  [4]
>  [1]
=  [if(f(x), s(x), s(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: { f(s(x)) -> f(x) }
Weak Trs:
{ f(0()) -> true()
, f(1()) -> false()
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, g(x, c(y)) -> g(x, g(s(c(y)), y))
, g(s(x), s(y)) -> if(f(x), s(x), s(y)) }
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: { f(s(x)) -> f(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(if) = {1}, Uargs(g) = {2}

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

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

[0] = [0]
[0]

[true] = [0]
[0]

[1] = [0]
[0]

[false] = [0]
[0]

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

[if](x1, x2, x3) = [2 0] x1 + [2 0] x2 + [1 0] x3 + [0]
[0 2]      [0 1]      [0 4]      [0]

[g](x1, x2) = [3 1] x1 + [1 0] x2 + [0]
[1 7]      [1 0]      [6]

[c](x1) = [1 0] x1 + [7]
[0 0]      [0]

The order satisfies the following ordering constraints:

[f(0())] =  [0]
[4]
>= [0]
[0]
=  [true()]

[f(1())] =  [0]
[4]
>= [0]
[0]
=  [false()]

[f(s(x))] =  [0 1] x + [1]
[0 5]     [9]
>  [0 1] x + [0]
[0 5]     [4]
=  [f(x)]

[if(true(), x, y)] =  [2 0] x + [1 0] y + [0]
[0 1]     [0 4]     [0]
>= [1 0] x + [0]
[0 1]     [0]
=  [x]

[if(false(), x, y)] =  [2 0] x + [1 0] y + [0]
[0 1]     [0 4]     [0]
>= [1 0] y + [0]
[0 1]     [0]
=  [y]

[g(x, c(y))] =  [3 1] x + [1 0] y + [7]
[1 7]     [1 0]     [13]
>  [3 1] x + [1 0] y + [1]
[1 7]     [1 0]     [7]
=  [g(x, g(s(c(y)), y))]

[g(s(x), s(y))] =  [0 13] x + [0 4] y + [1]
[0 11]     [0 4]     [13]
>  [0 10] x + [0 4] y + [0]
[0 11]     [0 4]     [13]
=  [if(f(x), s(x), s(y))]

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

Weak Trs:
{ f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, g(x, c(y)) -> g(x, g(s(c(y)), y))
, g(s(x), s(y)) -> if(f(x), s(x), s(y)) }
Obligation:
innermost runtime complexity