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

Strict Trs:
{ minus(X, 0()) -> X
, minus(s(X), s(Y)) -> p(minus(X, Y))
, p(s(X)) -> X
, div(0(), s(Y)) -> 0()
, div(s(X), s(Y)) -> s(div(minus(X, Y), 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(s) = {1}, Uargs(p) = {1}, Uargs(div) = {1}

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

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

[0] = [5]

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

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

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

The order satisfies the following ordering constraints:

[minus(X, 0())] =  [1] X + [0]
>= [1] X + [0]
=  [X]

[minus(s(X), s(Y))] =  [1] X + [1]
>  [1] X + [0]
=  [p(minus(X, Y))]

[p(s(X))] =  [1] X + [1]
>  [1] X + [0]
=  [X]

[div(0(), s(Y))] =  [5]
>= [5]
=  [0()]

[div(s(X), s(Y))] =  [1] X + [1]
>= [1] X + [1]
=  [s(div(minus(X, Y), 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:
{ minus(X, 0()) -> X
, div(0(), s(Y)) -> 0()
, div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) }
Weak Trs:
{ minus(s(X), s(Y)) -> p(minus(X, Y))
, p(s(X)) -> 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(p) = {1}, Uargs(div) = {1}

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

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

[0] = [4]

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

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

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

The order satisfies the following ordering constraints:

[minus(X, 0())] =  [1] X + [1]
>  [1] X + [0]
=  [X]

[minus(s(X), s(Y))] =  [1] X + [1]
>= [1] X + [1]
=  [p(minus(X, Y))]

[p(s(X))] =  [1] X + [0]
>= [1] X + [0]
=  [X]

[div(0(), s(Y))] =  [4]
>= [4]
=  [0()]

[div(s(X), s(Y))] =  [1] X + [0]
?  [1] X + [1]
=  [s(div(minus(X, Y), 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:
{ div(0(), s(Y)) -> 0()
, div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) }
Weak Trs:
{ minus(X, 0()) -> X
, minus(s(X), s(Y)) -> p(minus(X, Y))
, p(s(X)) -> 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(p) = {1}, Uargs(div) = {1}

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

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

[0] = [0]

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

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

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

The order satisfies the following ordering constraints:

[minus(X, 0())] =  [1] X + [4]
>  [1] X + [0]
=  [X]

[minus(s(X), s(Y))] =  [1] X + [8]
>= [1] X + [8]
=  [p(minus(X, Y))]

[p(s(X))] =  [1] X + [8]
>  [1] X + [0]
=  [X]

[div(0(), s(Y))] =  [1] Y + [4]
>  [0]
=  [0()]

[div(s(X), s(Y))] =  [1] X + [1] Y + [8]
?  [1] X + [1] Y + [12]
=  [s(div(minus(X, Y), 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: { div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) }
Weak Trs:
{ minus(X, 0()) -> X
, minus(s(X), s(Y)) -> p(minus(X, Y))
, p(s(X)) -> X
, div(0(), s(Y)) -> 0() }
Obligation:
innermost runtime complexity
YES(O(1),O(n^1))

We use the processor 'matrix interpretation of dimension 1' to
orient following rules strictly.

Trs: { div(s(X), s(Y)) -> s(div(minus(X, Y), s(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(p) = {1}, Uargs(div) = {1}

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

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

[0] = [0]

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

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

[div](x1, x2) = [2] x1 + [0]

The order satisfies the following ordering constraints:

[minus(X, 0())] =  [1] X + [0]
>= [1] X + [0]
=  [X]

[minus(s(X), s(Y))] =  [1] X + [4]
>  [1] X + [0]
=  [p(minus(X, Y))]

[p(s(X))] =  [1] X + [4]
>  [1] X + [0]
=  [X]

[div(0(), s(Y))] =  [0]
>= [0]
=  [0()]

[div(s(X), s(Y))] =  [2] X + [8]
>  [2] X + [4]
=  [s(div(minus(X, Y), s(Y)))]

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

Weak Trs:
{ minus(X, 0()) -> X
, minus(s(X), s(Y)) -> p(minus(X, Y))
, p(s(X)) -> X
, div(0(), s(Y)) -> 0()
, div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) }
Obligation:
innermost runtime complexity