```* Step 1: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
a__and(X1,X2) -> and(X1,X2)
a__and(false(),Y) -> false()
a__and(true(),X) -> mark(X)
a__first(X1,X2) -> first(X1,X2)
a__first(0(),X) -> nil()
a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
a__from(X) -> cons(X,from(s(X)))
a__from(X) -> from(X)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
mark(0()) -> 0()
mark(and(X1,X2)) -> a__and(mark(X1),X2)
mark(cons(X1,X2)) -> cons(X1,X2)
mark(false()) -> false()
mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
mark(from(X)) -> a__from(X)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(nil()) -> nil()
mark(s(X)) -> s(X)
mark(true()) -> true()
- Signature:
,if/3,nil/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if
+ 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(a__and) = {1},
uargs(a__first) = {1,2},
uargs(a__if) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [4]
p(a__add) = [1] x1 + [0]
p(a__and) = [1] x1 + [3]
p(a__first) = [1] x1 + [1] x2 + [0]
p(a__from) = [0]
p(a__if) = [1] x1 + [0]
p(and) = [0]
p(cons) = [0]
p(false) = [0]
p(first) = [0]
p(from) = [0]
p(if) = [0]
p(mark) = [4]
p(nil) = [0]
p(s) = [2]
p(true) = [5]

Following rules are strictly oriented:
a__and(X1,X2) = [1] X1 + [3]
> [0]
= and(X1,X2)

a__and(false(),Y) = [3]
> [0]
= false()

a__and(true(),X) = [8]
> [4]
= mark(X)

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

a__first(s(X),cons(Y,Z)) = [2]
> [0]
= cons(Y,first(X,Z))

a__if(true(),X,Y) = [5]
> [4]
= mark(X)

mark(cons(X1,X2)) = [4]
> [0]
= cons(X1,X2)

mark(false()) = [4]
> [0]
= false()

mark(from(X)) = [4]
> [0]
= a__from(X)

mark(nil()) = [4]
> [0]
= nil()

mark(s(X)) = [4]
> [2]
= s(X)

Following rules are (at-least) weakly oriented:
a__add(X1,X2) =  [1] X1 + [0]
>= [0]

>= [4]
=  mark(X)

>= [2]

a__first(X1,X2) =  [1] X1 + [1] X2 + [0]
>= [0]
=  first(X1,X2)

a__from(X) =  [0]
>= [0]
=  cons(X,from(s(X)))

a__from(X) =  [0]
>= [0]
=  from(X)

a__if(X1,X2,X3) =  [1] X1 + [0]
>= [0]
=  if(X1,X2,X3)

a__if(false(),X,Y) =  [0]
>= [4]
=  mark(Y)

mark(0()) =  [4]
>= [4]
=  0()

>= [4]

mark(and(X1,X2)) =  [4]
>= [7]
=  a__and(mark(X1),X2)

mark(first(X1,X2)) =  [4]
>= [8]
=  a__first(mark(X1),mark(X2))

mark(if(X1,X2,X3)) =  [4]
>= [4]
=  a__if(mark(X1),X2,X3)

mark(true()) =  [4]
>= [5]
=  true()

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:
a__first(X1,X2) -> first(X1,X2)
a__from(X) -> cons(X,from(s(X)))
a__from(X) -> from(X)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
mark(0()) -> 0()
mark(and(X1,X2)) -> a__and(mark(X1),X2)
mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(true()) -> true()
- Weak TRS:
a__and(X1,X2) -> and(X1,X2)
a__and(false(),Y) -> false()
a__and(true(),X) -> mark(X)
a__first(0(),X) -> nil()
a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
a__if(true(),X,Y) -> mark(X)
mark(cons(X1,X2)) -> cons(X1,X2)
mark(false()) -> false()
mark(from(X)) -> a__from(X)
mark(nil()) -> nil()
mark(s(X)) -> s(X)
- Signature:
,if/3,nil/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if
+ 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(a__and) = {1},
uargs(a__first) = {1,2},
uargs(a__if) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [5]
p(a__add) = [1] x1 + [1] x2 + [0]
p(a__and) = [1] x1 + [1] x2 + [4]
p(a__first) = [1] x1 + [1] x2 + [0]
p(a__from) = [0]
p(a__if) = [1] x1 + [1] x2 + [1] x3 + [5]
p(add) = [1] x1 + [1] x2 + [0]
p(and) = [1] x1 + [1] x2 + [0]
p(cons) = [1]
p(false) = [2]
p(first) = [1] x1 + [1] x2 + [2]
p(from) = [0]
p(if) = [1] x1 + [1] x2 + [1] x3 + [3]
p(mark) = [1] x1 + [1]
p(nil) = [4]
p(s) = [1] x1 + [0]
p(true) = [2]

Following rules are strictly oriented:
a__add(0(),X) = [1] X + [5]
> [1] X + [1]
= mark(X)

a__if(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [5]
> [1] X1 + [1] X2 + [1] X3 + [3]
= if(X1,X2,X3)

a__if(false(),X,Y) = [1] X + [1] Y + [7]
> [1] Y + [1]
= mark(Y)

mark(0()) = [6]
> [5]
= 0()

mark(first(X1,X2)) = [1] X1 + [1] X2 + [3]
> [1] X1 + [1] X2 + [2]
= a__first(mark(X1),mark(X2))

mark(true()) = [3]
> [2]
= true()

Following rules are (at-least) weakly oriented:
a__add(X1,X2) =  [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]

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

a__and(X1,X2) =  [1] X1 + [1] X2 + [4]
>= [1] X1 + [1] X2 + [0]
=  and(X1,X2)

a__and(false(),Y) =  [1] Y + [6]
>= [2]
=  false()

a__and(true(),X) =  [1] X + [6]
>= [1] X + [1]
=  mark(X)

a__first(X1,X2) =  [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [2]
=  first(X1,X2)

a__first(0(),X) =  [1] X + [5]
>= [4]
=  nil()

a__first(s(X),cons(Y,Z)) =  [1] X + [1]
>= [1]
=  cons(Y,first(X,Z))

a__from(X) =  [0]
>= [1]
=  cons(X,from(s(X)))

a__from(X) =  [0]
>= [0]
=  from(X)

a__if(true(),X,Y) =  [1] X + [1] Y + [7]
>= [1] X + [1]
=  mark(X)

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

mark(and(X1,X2)) =  [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [5]
=  a__and(mark(X1),X2)

mark(cons(X1,X2)) =  [2]
>= [1]
=  cons(X1,X2)

mark(false()) =  [3]
>= [2]
=  false()

mark(from(X)) =  [1]
>= [0]
=  a__from(X)

mark(if(X1,X2,X3)) =  [1] X1 + [1] X2 + [1] X3 + [4]
>= [1] X1 + [1] X2 + [1] X3 + [6]
=  a__if(mark(X1),X2,X3)

mark(nil()) =  [5]
>= [4]
=  nil()

mark(s(X)) =  [1] X + [1]
>= [1] X + [0]
=  s(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:
a__first(X1,X2) -> first(X1,X2)
a__from(X) -> cons(X,from(s(X)))
a__from(X) -> from(X)
mark(and(X1,X2)) -> a__and(mark(X1),X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
- Weak TRS:
a__and(X1,X2) -> and(X1,X2)
a__and(false(),Y) -> false()
a__and(true(),X) -> mark(X)
a__first(0(),X) -> nil()
a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(false()) -> false()
mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
mark(from(X)) -> a__from(X)
mark(nil()) -> nil()
mark(s(X)) -> s(X)
mark(true()) -> true()
- Signature:
,if/3,nil/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if
+ 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(a__and) = {1},
uargs(a__first) = {1,2},
uargs(a__if) = {1}

Following symbols are considered usable:
TcT has computed the following interpretation:
p(0) = [2]
p(a__add) = [1] x_1 + [8] x_2 + [0]
p(a__and) = [1] x_1 + [8] x_2 + [0]
p(a__first) = [1] x_1 + [1] x_2 + [15]
p(a__from) = [0]
p(a__if) = [1] x_1 + [8] x_2 + [8] x_3 + [0]
p(add) = [1] x_1 + [1] x_2 + [0]
p(and) = [1] x_1 + [1] x_2 + [0]
p(cons) = [0]
p(false) = [1]
p(first) = [1] x_1 + [1] x_2 + [2]
p(from) = [0]
p(if) = [1] x_1 + [1] x_2 + [1] x_3 + [0]
p(mark) = [8] x_1 + [1]
p(nil) = [0]
p(s) = [0]
p(true) = [2]

Following rules are strictly oriented:
a__first(X1,X2) = [1] X1 + [1] X2 + [15]
> [1] X1 + [1] X2 + [2]
= first(X1,X2)

Following rules are (at-least) weakly oriented:
a__add(X1,X2) =  [1] X1 + [8] X2 + [0]
>= [1] X1 + [1] X2 + [0]

a__add(0(),X) =  [8] X + [2]
>= [8] X + [1]
=  mark(X)

a__add(s(X),Y) =  [8] Y + [0]
>= [0]

a__and(X1,X2) =  [1] X1 + [8] X2 + [0]
>= [1] X1 + [1] X2 + [0]
=  and(X1,X2)

a__and(false(),Y) =  [8] Y + [1]
>= [1]
=  false()

a__and(true(),X) =  [8] X + [2]
>= [8] X + [1]
=  mark(X)

a__first(0(),X) =  [1] X + [17]
>= [0]
=  nil()

a__first(s(X),cons(Y,Z)) =  [15]
>= [0]
=  cons(Y,first(X,Z))

a__from(X) =  [0]
>= [0]
=  cons(X,from(s(X)))

a__from(X) =  [0]
>= [0]
=  from(X)

a__if(X1,X2,X3) =  [1] X1 + [8] X2 + [8] X3 + [0]
>= [1] X1 + [1] X2 + [1] X3 + [0]
=  if(X1,X2,X3)

a__if(false(),X,Y) =  [8] X + [8] Y + [1]
>= [8] Y + [1]
=  mark(Y)

a__if(true(),X,Y) =  [8] X + [8] Y + [2]
>= [8] X + [1]
=  mark(X)

mark(0()) =  [17]
>= [2]
=  0()

mark(add(X1,X2)) =  [8] X1 + [8] X2 + [1]
>= [8] X1 + [8] X2 + [1]

mark(and(X1,X2)) =  [8] X1 + [8] X2 + [1]
>= [8] X1 + [8] X2 + [1]
=  a__and(mark(X1),X2)

mark(cons(X1,X2)) =  [1]
>= [0]
=  cons(X1,X2)

mark(false()) =  [9]
>= [1]
=  false()

mark(first(X1,X2)) =  [8] X1 + [8] X2 + [17]
>= [8] X1 + [8] X2 + [17]
=  a__first(mark(X1),mark(X2))

mark(from(X)) =  [1]
>= [0]
=  a__from(X)

mark(if(X1,X2,X3)) =  [8] X1 + [8] X2 + [8] X3 + [1]
>= [8] X1 + [8] X2 + [8] X3 + [1]
=  a__if(mark(X1),X2,X3)

mark(nil()) =  [1]
>= [0]
=  nil()

mark(s(X)) =  [1]
>= [0]
=  s(X)

mark(true()) =  [17]
>= [2]
=  true()

* Step 4: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
a__from(X) -> cons(X,from(s(X)))
a__from(X) -> from(X)
mark(and(X1,X2)) -> a__and(mark(X1),X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
- Weak TRS:
a__and(X1,X2) -> and(X1,X2)
a__and(false(),Y) -> false()
a__and(true(),X) -> mark(X)
a__first(X1,X2) -> first(X1,X2)
a__first(0(),X) -> nil()
a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(false()) -> false()
mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
mark(from(X)) -> a__from(X)
mark(nil()) -> nil()
mark(s(X)) -> s(X)
mark(true()) -> true()
- Signature:
,if/3,nil/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if
+ 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(a__and) = {1},
uargs(a__first) = {1,2},
uargs(a__if) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(a__add) = [1] x1 + [4]
p(a__and) = [1] x1 + [2]
p(a__first) = [1] x1 + [1] x2 + [0]
p(a__from) = [0]
p(a__if) = [1] x1 + [4]
p(and) = [0]
p(cons) = [0]
p(false) = [0]
p(first) = [1] x2 + [0]
p(from) = [0]
p(if) = [0]
p(mark) = [0]
p(nil) = [0]
p(s) = [0]
p(true) = [0]

Following rules are strictly oriented:
a__add(X1,X2) = [1] X1 + [4]
> [0]

> [0]

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

a__and(X1,X2) =  [1] X1 + [2]
>= [0]
=  and(X1,X2)

a__and(false(),Y) =  [2]
>= [0]
=  false()

a__and(true(),X) =  [2]
>= [0]
=  mark(X)

a__first(X1,X2) =  [1] X1 + [1] X2 + [0]
>= [1] X2 + [0]
=  first(X1,X2)

a__first(0(),X) =  [1] X + [0]
>= [0]
=  nil()

a__first(s(X),cons(Y,Z)) =  [0]
>= [0]
=  cons(Y,first(X,Z))

a__from(X) =  [0]
>= [0]
=  cons(X,from(s(X)))

a__from(X) =  [0]
>= [0]
=  from(X)

a__if(X1,X2,X3) =  [1] X1 + [4]
>= [0]
=  if(X1,X2,X3)

a__if(false(),X,Y) =  [4]
>= [0]
=  mark(Y)

a__if(true(),X,Y) =  [4]
>= [0]
=  mark(X)

mark(0()) =  [0]
>= [0]
=  0()

>= [4]

mark(and(X1,X2)) =  [0]
>= [2]
=  a__and(mark(X1),X2)

mark(cons(X1,X2)) =  [0]
>= [0]
=  cons(X1,X2)

mark(false()) =  [0]
>= [0]
=  false()

mark(first(X1,X2)) =  [0]
>= [0]
=  a__first(mark(X1),mark(X2))

mark(from(X)) =  [0]
>= [0]
=  a__from(X)

mark(if(X1,X2,X3)) =  [0]
>= [4]
=  a__if(mark(X1),X2,X3)

mark(nil()) =  [0]
>= [0]
=  nil()

mark(s(X)) =  [0]
>= [0]
=  s(X)

mark(true()) =  [0]
>= [0]
=  true()

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:
a__from(X) -> cons(X,from(s(X)))
a__from(X) -> from(X)
mark(and(X1,X2)) -> a__and(mark(X1),X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
- Weak TRS:
a__and(X1,X2) -> and(X1,X2)
a__and(false(),Y) -> false()
a__and(true(),X) -> mark(X)
a__first(X1,X2) -> first(X1,X2)
a__first(0(),X) -> nil()
a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(false()) -> false()
mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
mark(from(X)) -> a__from(X)
mark(nil()) -> nil()
mark(s(X)) -> s(X)
mark(true()) -> true()
- Signature:
,if/3,nil/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if
+ 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(a__and) = {1},
uargs(a__first) = {1,2},
uargs(a__if) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [1]
p(a__add) = [1] x1 + [2] x2 + [4]
p(a__and) = [1] x1 + [2] x2 + [4]
p(a__first) = [1] x1 + [1] x2 + [0]
p(a__from) = [1] x1 + [4]
p(a__if) = [1] x1 + [2] x2 + [2] x3 + [1]
p(add) = [1] x1 + [1] x2 + [0]
p(and) = [1] x1 + [1] x2 + [0]
p(cons) = [1] x1 + [0]
p(false) = [0]
p(first) = [1] x1 + [1] x2 + [0]
p(from) = [1] x1 + [2]
p(if) = [1] x1 + [1] x2 + [1] x3 + [0]
p(mark) = [2] x1 + [0]
p(nil) = [0]
p(s) = [0]
p(true) = [4]

Following rules are strictly oriented:
a__from(X) = [1] X + [4]
> [1] X + [0]
= cons(X,from(s(X)))

a__from(X) = [1] X + [4]
> [1] X + [2]
= from(X)

Following rules are (at-least) weakly oriented:
a__add(X1,X2) =  [1] X1 + [2] X2 + [4]
>= [1] X1 + [1] X2 + [0]

a__add(0(),X) =  [2] X + [5]
>= [2] X + [0]
=  mark(X)

a__add(s(X),Y) =  [2] Y + [4]
>= [0]

a__and(X1,X2) =  [1] X1 + [2] X2 + [4]
>= [1] X1 + [1] X2 + [0]
=  and(X1,X2)

a__and(false(),Y) =  [2] Y + [4]
>= [0]
=  false()

a__and(true(),X) =  [2] X + [8]
>= [2] X + [0]
=  mark(X)

a__first(X1,X2) =  [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
=  first(X1,X2)

a__first(0(),X) =  [1] X + [1]
>= [0]
=  nil()

a__first(s(X),cons(Y,Z)) =  [1] Y + [0]
>= [1] Y + [0]
=  cons(Y,first(X,Z))

a__if(X1,X2,X3) =  [1] X1 + [2] X2 + [2] X3 + [1]
>= [1] X1 + [1] X2 + [1] X3 + [0]
=  if(X1,X2,X3)

a__if(false(),X,Y) =  [2] X + [2] Y + [1]
>= [2] Y + [0]
=  mark(Y)

a__if(true(),X,Y) =  [2] X + [2] Y + [5]
>= [2] X + [0]
=  mark(X)

mark(0()) =  [2]
>= [1]
=  0()

mark(add(X1,X2)) =  [2] X1 + [2] X2 + [0]
>= [2] X1 + [2] X2 + [4]

mark(and(X1,X2)) =  [2] X1 + [2] X2 + [0]
>= [2] X1 + [2] X2 + [4]
=  a__and(mark(X1),X2)

mark(cons(X1,X2)) =  [2] X1 + [0]
>= [1] X1 + [0]
=  cons(X1,X2)

mark(false()) =  [0]
>= [0]
=  false()

mark(first(X1,X2)) =  [2] X1 + [2] X2 + [0]
>= [2] X1 + [2] X2 + [0]
=  a__first(mark(X1),mark(X2))

mark(from(X)) =  [2] X + [4]
>= [1] X + [4]
=  a__from(X)

mark(if(X1,X2,X3)) =  [2] X1 + [2] X2 + [2] X3 + [0]
>= [2] X1 + [2] X2 + [2] X3 + [1]
=  a__if(mark(X1),X2,X3)

mark(nil()) =  [0]
>= [0]
=  nil()

mark(s(X)) =  [0]
>= [0]
=  s(X)

mark(true()) =  [8]
>= [4]
=  true()

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 6: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
mark(and(X1,X2)) -> a__and(mark(X1),X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
- Weak TRS:
a__and(X1,X2) -> and(X1,X2)
a__and(false(),Y) -> false()
a__and(true(),X) -> mark(X)
a__first(X1,X2) -> first(X1,X2)
a__first(0(),X) -> nil()
a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
a__from(X) -> cons(X,from(s(X)))
a__from(X) -> from(X)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(false()) -> false()
mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
mark(from(X)) -> a__from(X)
mark(nil()) -> nil()
mark(s(X)) -> s(X)
mark(true()) -> true()
- Signature:
,if/3,nil/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if
+ 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(a__and) = {1},
uargs(a__first) = {1,2},
uargs(a__if) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [1]
p(a__add) = [1] x1 + [4] x2 + [4]
p(a__and) = [1] x1 + [4] x2 + [1]
p(a__first) = [1] x1 + [1] x2 + [0]
p(a__from) = [6]
p(a__if) = [1] x1 + [4] x2 + [4] x3 + [2]
p(add) = [1] x1 + [1] x2 + [2]
p(and) = [1] x1 + [1] x2 + [0]
p(cons) = [0]
p(false) = [1]
p(first) = [1] x1 + [1] x2 + [0]
p(from) = [2]
p(if) = [1] x1 + [1] x2 + [1] x3 + [0]
p(mark) = [4] x1 + [0]
p(nil) = [0]
p(s) = [1]
p(true) = [0]

Following rules are strictly oriented:
mark(add(X1,X2)) = [4] X1 + [4] X2 + [8]
> [4] X1 + [4] X2 + [4]

Following rules are (at-least) weakly oriented:
a__add(X1,X2) =  [1] X1 + [4] X2 + [4]
>= [1] X1 + [1] X2 + [2]

a__add(0(),X) =  [4] X + [5]
>= [4] X + [0]
=  mark(X)

a__add(s(X),Y) =  [4] Y + [5]
>= [1]

a__and(X1,X2) =  [1] X1 + [4] X2 + [1]
>= [1] X1 + [1] X2 + [0]
=  and(X1,X2)

a__and(false(),Y) =  [4] Y + [2]
>= [1]
=  false()

a__and(true(),X) =  [4] X + [1]
>= [4] X + [0]
=  mark(X)

a__first(X1,X2) =  [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
=  first(X1,X2)

a__first(0(),X) =  [1] X + [1]
>= [0]
=  nil()

a__first(s(X),cons(Y,Z)) =  [1]
>= [0]
=  cons(Y,first(X,Z))

a__from(X) =  [6]
>= [0]
=  cons(X,from(s(X)))

a__from(X) =  [6]
>= [2]
=  from(X)

a__if(X1,X2,X3) =  [1] X1 + [4] X2 + [4] X3 + [2]
>= [1] X1 + [1] X2 + [1] X3 + [0]
=  if(X1,X2,X3)

a__if(false(),X,Y) =  [4] X + [4] Y + [3]
>= [4] Y + [0]
=  mark(Y)

a__if(true(),X,Y) =  [4] X + [4] Y + [2]
>= [4] X + [0]
=  mark(X)

mark(0()) =  [4]
>= [1]
=  0()

mark(and(X1,X2)) =  [4] X1 + [4] X2 + [0]
>= [4] X1 + [4] X2 + [1]
=  a__and(mark(X1),X2)

mark(cons(X1,X2)) =  [0]
>= [0]
=  cons(X1,X2)

mark(false()) =  [4]
>= [1]
=  false()

mark(first(X1,X2)) =  [4] X1 + [4] X2 + [0]
>= [4] X1 + [4] X2 + [0]
=  a__first(mark(X1),mark(X2))

mark(from(X)) =  [8]
>= [6]
=  a__from(X)

mark(if(X1,X2,X3)) =  [4] X1 + [4] X2 + [4] X3 + [0]
>= [4] X1 + [4] X2 + [4] X3 + [2]
=  a__if(mark(X1),X2,X3)

mark(nil()) =  [0]
>= [0]
=  nil()

mark(s(X)) =  [4]
>= [1]
=  s(X)

mark(true()) =  [0]
>= [0]
=  true()

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 7: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
mark(and(X1,X2)) -> a__and(mark(X1),X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
- Weak TRS:
a__and(X1,X2) -> and(X1,X2)
a__and(false(),Y) -> false()
a__and(true(),X) -> mark(X)
a__first(X1,X2) -> first(X1,X2)
a__first(0(),X) -> nil()
a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
a__from(X) -> cons(X,from(s(X)))
a__from(X) -> from(X)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(false()) -> false()
mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
mark(from(X)) -> a__from(X)
mark(nil()) -> nil()
mark(s(X)) -> s(X)
mark(true()) -> true()
- Signature:
,if/3,nil/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if
+ 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(a__and) = {1},
uargs(a__first) = {1,2},
uargs(a__if) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(a__add) = [1] x1 + [4] x2 + [6]
p(a__and) = [1] x1 + [4] x2 + [2]
p(a__first) = [1] x1 + [1] x2 + [4]
p(a__from) = [2]
p(a__if) = [1] x1 + [4] x2 + [4] x3 + [2]
p(add) = [1] x1 + [1] x2 + [3]
p(and) = [1] x1 + [1] x2 + [2]
p(cons) = [2]
p(false) = [0]
p(first) = [1] x1 + [1] x2 + [2]
p(from) = [1]
p(if) = [1] x1 + [1] x2 + [1] x3 + [0]
p(mark) = [4] x1 + [0]
p(nil) = [0]
p(s) = [2]
p(true) = [1]

Following rules are strictly oriented:
mark(and(X1,X2)) = [4] X1 + [4] X2 + [8]
> [4] X1 + [4] X2 + [2]
= a__and(mark(X1),X2)

Following rules are (at-least) weakly oriented:
a__add(X1,X2) =  [1] X1 + [4] X2 + [6]
>= [1] X1 + [1] X2 + [3]

a__add(0(),X) =  [4] X + [6]
>= [4] X + [0]
=  mark(X)

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

a__and(X1,X2) =  [1] X1 + [4] X2 + [2]
>= [1] X1 + [1] X2 + [2]
=  and(X1,X2)

a__and(false(),Y) =  [4] Y + [2]
>= [0]
=  false()

a__and(true(),X) =  [4] X + [3]
>= [4] X + [0]
=  mark(X)

a__first(X1,X2) =  [1] X1 + [1] X2 + [4]
>= [1] X1 + [1] X2 + [2]
=  first(X1,X2)

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

a__first(s(X),cons(Y,Z)) =  [8]
>= [2]
=  cons(Y,first(X,Z))

a__from(X) =  [2]
>= [2]
=  cons(X,from(s(X)))

a__from(X) =  [2]
>= [1]
=  from(X)

a__if(X1,X2,X3) =  [1] X1 + [4] X2 + [4] X3 + [2]
>= [1] X1 + [1] X2 + [1] X3 + [0]
=  if(X1,X2,X3)

a__if(false(),X,Y) =  [4] X + [4] Y + [2]
>= [4] Y + [0]
=  mark(Y)

a__if(true(),X,Y) =  [4] X + [4] Y + [3]
>= [4] X + [0]
=  mark(X)

mark(0()) =  [0]
>= [0]
=  0()

mark(add(X1,X2)) =  [4] X1 + [4] X2 + [12]
>= [4] X1 + [4] X2 + [6]

mark(cons(X1,X2)) =  [8]
>= [2]
=  cons(X1,X2)

mark(false()) =  [0]
>= [0]
=  false()

mark(first(X1,X2)) =  [4] X1 + [4] X2 + [8]
>= [4] X1 + [4] X2 + [4]
=  a__first(mark(X1),mark(X2))

mark(from(X)) =  [4]
>= [2]
=  a__from(X)

mark(if(X1,X2,X3)) =  [4] X1 + [4] X2 + [4] X3 + [0]
>= [4] X1 + [4] X2 + [4] X3 + [2]
=  a__if(mark(X1),X2,X3)

mark(nil()) =  [0]
>= [0]
=  nil()

mark(s(X)) =  [8]
>= [2]
=  s(X)

mark(true()) =  [4]
>= [1]
=  true()

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 8: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
- Weak TRS:
a__and(X1,X2) -> and(X1,X2)
a__and(false(),Y) -> false()
a__and(true(),X) -> mark(X)
a__first(X1,X2) -> first(X1,X2)
a__first(0(),X) -> nil()
a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
a__from(X) -> cons(X,from(s(X)))
a__from(X) -> from(X)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
mark(0()) -> 0()
mark(and(X1,X2)) -> a__and(mark(X1),X2)
mark(cons(X1,X2)) -> cons(X1,X2)
mark(false()) -> false()
mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
mark(from(X)) -> a__from(X)
mark(nil()) -> nil()
mark(s(X)) -> s(X)
mark(true()) -> true()
- Signature:
,if/3,nil/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if
+ 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(a__and) = {1},
uargs(a__first) = {1,2},
uargs(a__if) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [2]
p(a__add) = [1] x1 + [4] x2 + [4]
p(a__and) = [1] x1 + [4] x2 + [0]
p(a__first) = [1] x1 + [1] x2 + [7]
p(a__from) = [2] x1 + [1]
p(a__if) = [1] x1 + [4] x2 + [4] x3 + [1]
p(add) = [1] x1 + [1] x2 + [2]
p(and) = [1] x1 + [1] x2 + [0]
p(cons) = [1]
p(false) = [1]
p(first) = [1] x1 + [1] x2 + [3]
p(from) = [1] x1 + [1]
p(if) = [1] x1 + [1] x2 + [1] x3 + [1]
p(mark) = [4] x1 + [0]
p(nil) = [0]
p(s) = [1] x1 + [1]
p(true) = [0]

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

Following rules are (at-least) weakly oriented:
a__add(X1,X2) =  [1] X1 + [4] X2 + [4]
>= [1] X1 + [1] X2 + [2]

a__add(0(),X) =  [4] X + [6]
>= [4] X + [0]
=  mark(X)

a__add(s(X),Y) =  [1] X + [4] Y + [5]
>= [1] X + [1] Y + [3]

a__and(X1,X2) =  [1] X1 + [4] X2 + [0]
>= [1] X1 + [1] X2 + [0]
=  and(X1,X2)

a__and(false(),Y) =  [4] Y + [1]
>= [1]
=  false()

a__and(true(),X) =  [4] X + [0]
>= [4] X + [0]
=  mark(X)

a__first(X1,X2) =  [1] X1 + [1] X2 + [7]
>= [1] X1 + [1] X2 + [3]
=  first(X1,X2)

a__first(0(),X) =  [1] X + [9]
>= [0]
=  nil()

a__first(s(X),cons(Y,Z)) =  [1] X + [9]
>= [1]
=  cons(Y,first(X,Z))

a__from(X) =  [2] X + [1]
>= [1]
=  cons(X,from(s(X)))

a__from(X) =  [2] X + [1]
>= [1] X + [1]
=  from(X)

a__if(X1,X2,X3) =  [1] X1 + [4] X2 + [4] X3 + [1]
>= [1] X1 + [1] X2 + [1] X3 + [1]
=  if(X1,X2,X3)

a__if(false(),X,Y) =  [4] X + [4] Y + [2]
>= [4] Y + [0]
=  mark(Y)

a__if(true(),X,Y) =  [4] X + [4] Y + [1]
>= [4] X + [0]
=  mark(X)

mark(0()) =  [8]
>= [2]
=  0()

mark(add(X1,X2)) =  [4] X1 + [4] X2 + [8]
>= [4] X1 + [4] X2 + [4]

mark(and(X1,X2)) =  [4] X1 + [4] X2 + [0]
>= [4] X1 + [4] X2 + [0]
=  a__and(mark(X1),X2)

mark(cons(X1,X2)) =  [4]
>= [1]
=  cons(X1,X2)

mark(false()) =  [4]
>= [1]
=  false()

mark(first(X1,X2)) =  [4] X1 + [4] X2 + [12]
>= [4] X1 + [4] X2 + [7]
=  a__first(mark(X1),mark(X2))

mark(from(X)) =  [4] X + [4]
>= [2] X + [1]
=  a__from(X)

mark(nil()) =  [0]
>= [0]
=  nil()

mark(s(X)) =  [4] X + [4]
>= [1] X + [1]
=  s(X)

mark(true()) =  [0]
>= [0]
=  true()

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 9: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
a__and(X1,X2) -> and(X1,X2)
a__and(false(),Y) -> false()
a__and(true(),X) -> mark(X)
a__first(X1,X2) -> first(X1,X2)
a__first(0(),X) -> nil()
a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
a__from(X) -> cons(X,from(s(X)))
a__from(X) -> from(X)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
mark(0()) -> 0()
mark(and(X1,X2)) -> a__and(mark(X1),X2)
mark(cons(X1,X2)) -> cons(X1,X2)
mark(false()) -> false()
mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
mark(from(X)) -> a__from(X)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(nil()) -> nil()
mark(s(X)) -> s(X)
mark(true()) -> true()
- Signature: