```* Step 1: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
active(c()) -> mark(a())
active(c()) -> mark(b())
f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3))
f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3))
proper(a()) -> ok(a())
proper(b()) -> ok(b())
proper(c()) -> ok(c())
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{active/1,f/3,proper/1,top/1} / {a/0,b/0,c/0,mark/1,ok/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,c,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 + [0]
p(b) = [0]
p(c) = [0]
p(f) = [15]
p(mark) = [1] x1 + [0]
p(ok) = [1] x1 + [11]
p(proper) = [1] x1 + [0]
p(top) = [1] x1 + [0]

Following rules are strictly oriented:
top(ok(X)) = [1] X + [11]
> [1] X + [0]
= top(active(X))

Following rules are (at-least) weakly oriented:
active(c()) =  [0]
>= [0]
=  mark(a())

active(c()) =  [0]
>= [0]
=  mark(b())

f(X1,X2,mark(X3)) =  [15]
>= [15]
=  mark(f(X1,X2,X3))

f(ok(X1),ok(X2),ok(X3)) =  [15]
>= [26]
=  ok(f(X1,X2,X3))

proper(a()) =  [0]
>= [11]
=  ok(a())

proper(b()) =  [0]
>= [11]
=  ok(b())

proper(c()) =  [0]
>= [11]
=  ok(c())

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

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
active(c()) -> mark(a())
active(c()) -> mark(b())
f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3))
f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3))
proper(a()) -> ok(a())
proper(b()) -> ok(b())
proper(c()) -> ok(c())
top(mark(X)) -> top(proper(X))
- Weak TRS:
top(ok(X)) -> top(active(X))
- Signature:
{active/1,f/3,proper/1,top/1} / {a/0,b/0,c/0,mark/1,ok/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,c,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) = [4]
p(active) = [1] x_1 + [0]
p(b) = [2]
p(c) = [5]
p(f) = [8] x_1 + [5]
p(mark) = [1] x_1 + [0]
p(ok) = [1] x_1 + [0]
p(proper) = [1] x_1 + [0]
p(top) = [2] x_1 + [0]

Following rules are strictly oriented:
active(c()) = [5]
> [4]
= mark(a())

active(c()) = [5]
> [2]
= mark(b())

Following rules are (at-least) weakly oriented:
f(X1,X2,mark(X3)) =  [8] X1 + [5]
>= [8] X1 + [5]
=  mark(f(X1,X2,X3))

f(ok(X1),ok(X2),ok(X3)) =  [8] X1 + [5]
>= [8] X1 + [5]
=  ok(f(X1,X2,X3))

proper(a()) =  [4]
>= [4]
=  ok(a())

proper(b()) =  [2]
>= [2]
=  ok(b())

proper(c()) =  [5]
>= [5]
=  ok(c())

top(mark(X)) =  [2] X + [0]
>= [2] X + [0]
=  top(proper(X))

top(ok(X)) =  [2] X + [0]
>= [2] X + [0]
=  top(active(X))

* Step 3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3))
f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3))
proper(a()) -> ok(a())
proper(b()) -> ok(b())
proper(c()) -> ok(c())
top(mark(X)) -> top(proper(X))
- Weak TRS:
active(c()) -> mark(a())
active(c()) -> mark(b())
top(ok(X)) -> top(active(X))
- Signature:
{active/1,f/3,proper/1,top/1} / {a/0,b/0,c/0,mark/1,ok/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,c,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) = [5]
p(active) = [1] x1 + [0]
p(b) = [5]
p(c) = [7]
p(f) = [1] x2 + [9] x3 + [0]
p(mark) = [1] x1 + [2]
p(ok) = [1] x1 + [0]
p(proper) = [1] x1 + [6]
p(top) = [1] x1 + [0]

Following rules are strictly oriented:
f(X1,X2,mark(X3)) = [1] X2 + [9] X3 + [18]
> [1] X2 + [9] X3 + [2]
= mark(f(X1,X2,X3))

proper(a()) = [11]
> [5]
= ok(a())

proper(b()) = [11]
> [5]
= ok(b())

proper(c()) = [13]
> [7]
= ok(c())

Following rules are (at-least) weakly oriented:
active(c()) =  [7]
>= [7]
=  mark(a())

active(c()) =  [7]
>= [7]
=  mark(b())

f(ok(X1),ok(X2),ok(X3)) =  [1] X2 + [9] X3 + [0]
>= [1] X2 + [9] X3 + [0]
=  ok(f(X1,X2,X3))

top(mark(X)) =  [1] X + [2]
>= [1] X + [6]
=  top(proper(X))

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

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3))
top(mark(X)) -> top(proper(X))
- Weak TRS:
active(c()) -> mark(a())
active(c()) -> mark(b())
f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3))
proper(a()) -> ok(a())
proper(b()) -> ok(b())
proper(c()) -> ok(c())
top(ok(X)) -> top(active(X))
- Signature:
{active/1,f/3,proper/1,top/1} / {a/0,b/0,c/0,mark/1,ok/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,c,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) = [1]
p(active) = [1] x1 + [0]
p(b) = [1]
p(c) = [4]
p(f) = [1] x3 + [12]
p(mark) = [1] x1 + [3]
p(ok) = [1] x1 + [0]
p(proper) = [1] x1 + [0]
p(top) = [1] x1 + [0]

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

Following rules are (at-least) weakly oriented:
active(c()) =  [4]
>= [4]
=  mark(a())

active(c()) =  [4]
>= [4]
=  mark(b())

f(X1,X2,mark(X3)) =  [1] X3 + [15]
>= [1] X3 + [15]
=  mark(f(X1,X2,X3))

f(ok(X1),ok(X2),ok(X3)) =  [1] X3 + [12]
>= [1] X3 + [12]
=  ok(f(X1,X2,X3))

proper(a()) =  [1]
>= [1]
=  ok(a())

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

proper(c()) =  [4]
>= [4]
=  ok(c())

top(ok(X)) =  [1] X + [0]
>= [1] X + [0]
=  top(active(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(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3))
- Weak TRS:
active(c()) -> mark(a())
active(c()) -> mark(b())
f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3))
proper(a()) -> ok(a())
proper(b()) -> ok(b())
proper(c()) -> ok(c())
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{active/1,f/3,proper/1,top/1} / {a/0,b/0,c/0,mark/1,ok/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,c,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 + [1]
p(b) = [6]
p(c) = [8]
p(f) = [2] x1 + [1] x3 + [0]
p(mark) = [1] x1 + [3]
p(ok) = [1] x1 + [1]
p(proper) = [1] x1 + [1]
p(top) = [1] x1 + [2]

Following rules are strictly oriented:
f(ok(X1),ok(X2),ok(X3)) = [2] X1 + [1] X3 + [3]
> [2] X1 + [1] X3 + [1]
= ok(f(X1,X2,X3))

Following rules are (at-least) weakly oriented:
active(c()) =  [9]
>= [3]
=  mark(a())

active(c()) =  [9]
>= [9]
=  mark(b())

f(X1,X2,mark(X3)) =  [2] X1 + [1] X3 + [3]
>= [2] X1 + [1] X3 + [3]
=  mark(f(X1,X2,X3))

proper(a()) =  [1]
>= [1]
=  ok(a())

proper(b()) =  [7]
>= [7]
=  ok(b())

proper(c()) =  [9]
>= [9]
=  ok(c())

top(mark(X)) =  [1] X + [5]
>= [1] X + [3]
=  top(proper(X))

top(ok(X)) =  [1] X + [3]
>= [1] X + [3]
=  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(c()) -> mark(a())
active(c()) -> mark(b())
f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3))
f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3))
proper(a()) -> ok(a())
proper(b()) -> ok(b())
proper(c()) -> ok(c())
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{active/1,f/3,proper/1,top/1} / {a/0,b/0,c/0,mark/1,ok/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,c,mark,ok}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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