```* 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) = 
p(active) =  x1 + 
p(b) = 
p(c) = 
p(f) =  x1 +  x2 + 
p(mark) =  x1 + 
p(ok) =  x1 + 
p(proper) =  x1 + 
p(top) =  x1 + 

Following rules are strictly oriented:
active(c()) = 
> 
= mark(a())

f(ok(X1),ok(X2),ok(X3)) =  X1 +  X2 + 
>  X1 +  X2 + 
= ok(f(X1,X2,X3))

top(mark(X)) =  X + 
>  X + 
= top(proper(X))

top(ok(X)) =  X + 
>  X + 
= top(active(X))

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

f(X1,X2,mark(X3)) =   X1 +  X2 + 
>=  X1 +  X2 + 
=  mark(f(X1,X2,X3))

f(mark(X1),X2,X3) =   X1 +  X2 + 
>=  X1 +  X2 + 
=  mark(f(X1,X2,X3))

proper(a()) =  
>= 
=  ok(a())

proper(b()) =  
>= 
=  ok(b())

proper(c()) =  
>= 
=  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) = 
p(active) =  x1 + 
p(b) = 
p(c) = 
p(f) =  x1 + 
p(mark) =  x1 + 
p(ok) =  x1 + 
p(proper) =  x1 + 
p(top) =  x1 + 

Following rules are strictly oriented:
active(c()) = 
> 
= mark(b())

proper(a()) = 
> 
= ok(a())

proper(b()) = 
> 
= ok(b())

proper(c()) = 
> 
= ok(c())

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

f(X1,X2,mark(X3)) =   X1 + 
>=  X1 + 
=  mark(f(X1,X2,X3))

f(mark(X1),X2,X3) =   X1 + 
>=  X1 + 
=  mark(f(X1,X2,X3))

f(ok(X1),ok(X2),ok(X3)) =   X1 + 
>=  X1 + 
=  ok(f(X1,X2,X3))

top(mark(X)) =   X + 
>=  X + 
=  top(proper(X))

top(ok(X)) =   X + 
>=  X + 
=  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) = 
p(active) =  x1 + 
p(b) = 
p(c) = 
p(f) =  x1 +  x2 + 
p(mark) =  x1 + 
p(ok) =  x1 + 
p(proper) =  x1 + 
p(top) =  x1 + 

Following rules are strictly oriented:
f(mark(X1),X2,X3) =  X1 +  X2 + 
>  X1 +  X2 + 
= mark(f(X1,X2,X3))

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

active(c()) =  
>= 
=  mark(b())

f(X1,X2,mark(X3)) =   X1 +  X2 + 
>=  X1 +  X2 + 
=  mark(f(X1,X2,X3))

f(ok(X1),ok(X2),ok(X3)) =   X1 +  X2 + 
>=  X1 +  X2 + 
=  ok(f(X1,X2,X3))

proper(a()) =  
>= 
=  ok(a())

proper(b()) =  
>= 
=  ok(b())

proper(c()) =  
>= 
=  ok(c())

top(mark(X)) =   X + 
>=  X + 
=  top(proper(X))

top(ok(X)) =   X + 
>=  X + 
=  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) = 
p(active) =  x1 + 
p(b) = 
p(c) = 
p(f) =  x1 +  x3 + 
p(mark) =  x1 + 
p(ok) =  x1 + 
p(proper) = 
p(top) =  x1 + 

Following rules are strictly oriented:
f(X1,X2,mark(X3)) =  X1 +  X3 + 
>  X1 +  X3 + 
= mark(f(X1,X2,X3))

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

active(c()) =  
>= 
=  mark(b())

f(mark(X1),X2,X3) =   X1 +  X3 + 
>=  X1 +  X3 + 
=  mark(f(X1,X2,X3))

f(ok(X1),ok(X2),ok(X3)) =   X1 +  X3 + 
>=  X1 +  X3 + 
=  ok(f(X1,X2,X3))

proper(a()) =  
>= 
=  ok(a())

proper(b()) =  
>= 
=  ok(b())

proper(c()) =  
>= 
=  ok(c())

top(mark(X)) =   X + 
>= 
=  top(proper(X))

top(ok(X)) =   X + 
>=  X + 
=  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))
```