```* 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(mark(X1),X2,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) = [11]
p(c) = [9]
p(f) = [1] x1 + [5] x2 + [0]
p(mark) = [1] x1 + [4]
p(ok) = [1] x1 + [5]
p(proper) = [1] x1 + [0]
p(top) = [1] x1 + [0]

Following rules are strictly oriented:
active(c()) = [9]
> [4]
= mark(a())

f(ok(X1),ok(X2),ok(X3)) = [1] X1 + [5] X2 + [30]
> [1] X1 + [5] X2 + [5]
= ok(f(X1,X2,X3))

top(mark(X)) = [1] X + [4]
> [1] X + [0]
= top(proper(X))

top(ok(X)) = [1] X + [5]
> [1] X + [0]
= top(active(X))

Following rules are (at-least) weakly oriented:
active(c()) =  [9]
>= [15]
=  mark(b())

f(X1,X2,mark(X3)) =  [1] X1 + [5] X2 + [0]
>= [1] X1 + [5] X2 + [4]
=  mark(f(X1,X2,X3))

f(mark(X1),X2,X3) =  [1] X1 + [5] X2 + [4]
>= [1] X1 + [5] X2 + [4]
=  mark(f(X1,X2,X3))

proper(a()) =  [0]
>= [5]
=  ok(a())

proper(b()) =  [11]
>= [16]
=  ok(b())

proper(c()) =  [9]
>= [14]
=  ok(c())

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:
active(c()) -> mark(b())
f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3))
f(mark(X1),X2,X3) -> mark(f(X1,X2,X3))
proper(a()) -> ok(a())
proper(b()) -> ok(b())
proper(c()) -> ok(c())
- Weak TRS:
active(c()) -> mark(a())
f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3))
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) = [10]
p(active) = [1] x1 + [9]
p(b) = [0]
p(c) = [11]
p(f) = [1] x1 + [0]
p(mark) = [1] x1 + [10]
p(ok) = [1] x1 + [9]
p(proper) = [1] x1 + [10]
p(top) = [1] x1 + [0]

Following rules are strictly oriented:
active(c()) = [20]
> [10]
= mark(b())

proper(a()) = [20]
> [19]
= ok(a())

proper(b()) = [10]
> [9]
= ok(b())

proper(c()) = [21]
> [20]
= ok(c())

Following rules are (at-least) weakly oriented:
active(c()) =  [20]
>= [20]
=  mark(a())

f(X1,X2,mark(X3)) =  [1] X1 + [0]
>= [1] X1 + [10]
=  mark(f(X1,X2,X3))

f(mark(X1),X2,X3) =  [1] X1 + [10]
>= [1] X1 + [10]
=  mark(f(X1,X2,X3))

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

top(mark(X)) =  [1] X + [10]
>= [1] X + [10]
=  top(proper(X))

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

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3))
f(mark(X1),X2,X3) -> mark(f(X1,X2,X3))
- Weak TRS:
active(c()) -> mark(a())
active(c()) -> mark(b())
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) = [1]
p(active) = [1] x1 + [0]
p(b) = [4]
p(c) = [7]
p(f) = [6] x1 + [9] x2 + [13]
p(mark) = [1] x1 + [3]
p(ok) = [1] x1 + [1]
p(proper) = [1] x1 + [1]
p(top) = [1] x1 + [0]

Following rules are strictly oriented:
f(mark(X1),X2,X3) = [6] X1 + [9] X2 + [31]
> [6] X1 + [9] X2 + [16]
= mark(f(X1,X2,X3))

Following rules are (at-least) weakly oriented:
active(c()) =  [7]
>= [4]
=  mark(a())

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

f(X1,X2,mark(X3)) =  [6] X1 + [9] X2 + [13]
>= [6] X1 + [9] X2 + [16]
=  mark(f(X1,X2,X3))

f(ok(X1),ok(X2),ok(X3)) =  [6] X1 + [9] X2 + [28]
>= [6] X1 + [9] X2 + [14]
=  ok(f(X1,X2,X3))

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

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

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

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

top(ok(X)) =  [1] X + [1]
>= [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(X1,X2,mark(X3)) -> mark(f(X1,X2,X3))
- Weak TRS:
active(c()) -> mark(a())
active(c()) -> mark(b())
f(mark(X1),X2,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) = [1]
p(f) = [2] x1 + [4] x3 + [0]
p(mark) = [1] x1 + [1]
p(ok) = [1] x1 + [0]
p(proper) = [1]
p(top) = [1] x1 + [8]

Following rules are strictly oriented:
f(X1,X2,mark(X3)) = [2] X1 + [4] X3 + [4]
> [2] X1 + [4] X3 + [1]
= mark(f(X1,X2,X3))

Following rules are (at-least) weakly oriented:
active(c()) =  [1]
>= [1]
=  mark(a())

active(c()) =  [1]
>= [1]
=  mark(b())

f(mark(X1),X2,X3) =  [2] X1 + [4] X3 + [2]
>= [2] X1 + [4] X3 + [1]
=  mark(f(X1,X2,X3))

f(ok(X1),ok(X2),ok(X3)) =  [2] X1 + [4] X3 + [0]
>= [2] X1 + [4] X3 + [0]
=  ok(f(X1,X2,X3))

proper(a()) =  [1]
>= [0]
=  ok(a())

proper(b()) =  [1]
>= [0]
=  ok(b())

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

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

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

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: 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(mark(X1),X2,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))
```