```* Step 1: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
active(b()) -> mark(a())
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
proper(a()) -> ok(a())
proper(b()) -> ok(b())
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{active/1,f/2,proper/1,top/1} / {a/0,b/0,mark/1,ok/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,mark,ok}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(mark) = {1},
uargs(ok) = {1},
uargs(top) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(a) = [0]
p(active) = [1] x1 + [9]
p(b) = [0]
p(f) = [0]
p(mark) = [1] x1 + [0]
p(ok) = [1] x1 + [0]
p(proper) = [0]
p(top) = [1] x1 + [0]

Following rules are strictly oriented:
active(b()) = [9]
> [0]
= mark(a())

Following rules are (at-least) weakly oriented:
f(mark(X1),X2) =  [0]
>= [0]
=  mark(f(X1,X2))

f(ok(X1),ok(X2)) =  [0]
>= [0]
=  ok(f(X1,X2))

proper(a()) =  [0]
>= [0]
=  ok(a())

proper(b()) =  [0]
>= [0]
=  ok(b())

top(mark(X)) =  [1] X + [0]
>= [0]
=  top(proper(X))

top(ok(X)) =  [1] X + [0]
>= [1] X + [9]
=  top(active(X))

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
proper(a()) -> ok(a())
proper(b()) -> ok(b())
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Weak TRS:
active(b()) -> mark(a())
- Signature:
{active/1,f/2,proper/1,top/1} / {a/0,b/0,mark/1,ok/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,mark,ok}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(mark) = {1},
uargs(ok) = {1},
uargs(top) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(a) = [8]
p(active) = [8]
p(b) = [0]
p(f) = [9] x1 + [10]
p(mark) = [1] x1 + [0]
p(ok) = [1] x1 + [1]
p(proper) = [1] x1 + [9]
p(top) = [1] x1 + [0]

Following rules are strictly oriented:
f(ok(X1),ok(X2)) = [9] X1 + [19]
> [9] X1 + [11]
= ok(f(X1,X2))

proper(a()) = [17]
> [9]
= ok(a())

proper(b()) = [9]
> [1]
= ok(b())

Following rules are (at-least) weakly oriented:
active(b()) =  [8]
>= [8]
=  mark(a())

f(mark(X1),X2) =  [9] X1 + [10]
>= [9] X1 + [10]
=  mark(f(X1,X2))

top(mark(X)) =  [1] X + [0]
>= [1] X + [9]
=  top(proper(X))

top(ok(X)) =  [1] X + [1]
>= [8]
=  top(active(X))

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(mark(X1),X2) -> mark(f(X1,X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Weak TRS:
active(b()) -> mark(a())
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
proper(a()) -> ok(a())
proper(b()) -> ok(b())
- Signature:
{active/1,f/2,proper/1,top/1} / {a/0,b/0,mark/1,ok/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,mark,ok}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):

The following argument positions are considered usable:
uargs(mark) = {1},
uargs(ok) = {1},
uargs(top) = {1}

Following symbols are considered usable:
{active,f,proper,top}
TcT has computed the following interpretation:
p(a) = [0]
p(active) = [1] x_1 + [0]
p(b) = [8]
p(f) = [1] x_1 + [1] x_2 + [8]
p(mark) = [1] x_1 + [8]
p(ok) = [1] x_1 + [0]
p(proper) = [1] x_1 + [0]
p(top) = [1] x_1 + [4]

Following rules are strictly oriented:
top(mark(X)) = [1] X + [12]
> [1] X + [4]
= top(proper(X))

Following rules are (at-least) weakly oriented:
active(b()) =  [8]
>= [8]
=  mark(a())

f(mark(X1),X2) =  [1] X1 + [1] X2 + [16]
>= [1] X1 + [1] X2 + [16]
=  mark(f(X1,X2))

f(ok(X1),ok(X2)) =  [1] X1 + [1] X2 + [8]
>= [1] X1 + [1] X2 + [8]
=  ok(f(X1,X2))

proper(a()) =  [0]
>= [0]
=  ok(a())

proper(b()) =  [8]
>= [8]
=  ok(b())

top(ok(X)) =  [1] X + [4]
>= [1] X + [4]
=  top(active(X))

* Step 4: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(mark(X1),X2) -> mark(f(X1,X2))
top(ok(X)) -> top(active(X))
- Weak TRS:
active(b()) -> mark(a())
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
proper(a()) -> ok(a())
proper(b()) -> ok(b())
top(mark(X)) -> top(proper(X))
- Signature:
{active/1,f/2,proper/1,top/1} / {a/0,b/0,mark/1,ok/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,mark,ok}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(mark) = {1},
uargs(ok) = {1},
uargs(top) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(a) = [9]
p(active) = [1] x1 + [0]
p(b) = [12]
p(f) = [1] x1 + [14]
p(mark) = [1] x1 + [3]
p(ok) = [1] x1 + [3]
p(proper) = [1] x1 + [3]
p(top) = [1] x1 + [0]

Following rules are strictly oriented:
top(ok(X)) = [1] X + [3]
> [1] X + [0]
= top(active(X))

Following rules are (at-least) weakly oriented:
active(b()) =  [12]
>= [12]
=  mark(a())

f(mark(X1),X2) =  [1] X1 + [17]
>= [1] X1 + [17]
=  mark(f(X1,X2))

f(ok(X1),ok(X2)) =  [1] X1 + [17]
>= [1] X1 + [17]
=  ok(f(X1,X2))

proper(a()) =  [12]
>= [12]
=  ok(a())

proper(b()) =  [15]
>= [15]
=  ok(b())

top(mark(X)) =  [1] X + [3]
>= [1] X + [3]
=  top(proper(X))

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(mark(X1),X2) -> mark(f(X1,X2))
- Weak TRS:
active(b()) -> mark(a())
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
proper(a()) -> ok(a())
proper(b()) -> ok(b())
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{active/1,f/2,proper/1,top/1} / {a/0,b/0,mark/1,ok/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,mark,ok}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(mark) = {1},
uargs(ok) = {1},
uargs(top) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(a) = [2]
p(active) = [1] x1 + [1]
p(b) = [2]
p(f) = [12] x1 + [14] x2 + [3]
p(mark) = [1] x1 + [1]
p(ok) = [1] x1 + [1]
p(proper) = [1] x1 + [1]
p(top) = [1] x1 + [0]

Following rules are strictly oriented:
f(mark(X1),X2) = [12] X1 + [14] X2 + [15]
> [12] X1 + [14] X2 + [4]
= mark(f(X1,X2))

Following rules are (at-least) weakly oriented:
active(b()) =  [3]
>= [3]
=  mark(a())

f(ok(X1),ok(X2)) =  [12] X1 + [14] X2 + [29]
>= [12] X1 + [14] X2 + [4]
=  ok(f(X1,X2))

proper(a()) =  [3]
>= [3]
=  ok(a())

proper(b()) =  [3]
>= [3]
=  ok(b())

top(mark(X)) =  [1] X + [1]
>= [1] X + [1]
=  top(proper(X))

top(ok(X)) =  [1] X + [1]
>= [1] X + [1]
=  top(active(X))

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 6: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
active(b()) -> mark(a())
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
proper(a()) -> ok(a())
proper(b()) -> ok(b())
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{active/1,f/2,proper/1,top/1} / {a/0,b/0,mark/1,ok/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,mark,ok}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))
```