* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
+ Considered Problem:
- Strict TRS:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
top(sent(x)) -> top(check(rest(x)))
- Signature:
{check/1,rest/1,top/1} / {cons/2,nil/0,sent/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {check,rest,top} and constructors {cons,nil,sent}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
top(sent(x)) -> top(check(rest(x)))
- Signature:
{check/1,rest/1,top/1} / {cons/2,nil/0,sent/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {check,rest,top} and constructors {cons,nil,sent}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
check(y){y -> cons(x,y)} =
check(cons(x,y)) ->^+ cons(x,check(y))
= C[check(y) = check(y){}]
** Step 1.b:1: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
top(sent(x)) -> top(check(rest(x)))
- Signature:
{check/1,rest/1,top/1} / {cons/2,nil/0,sent/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {check,rest,top} and constructors {cons,nil,sent}
+ 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:
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(check) = [1] x1 + [0]
p(cons) = [1] x1 + [1] x2 + [5]
p(nil) = [0]
p(rest) = [1] x1 + [0]
p(sent) = [1] x1 + [5]
p(top) = [1] x1 + [0]
Following rules are strictly oriented:
top(sent(x)) = [1] x + [5]
> [1] x + [0]
= top(check(rest(x)))
Following rules are (at-least) weakly oriented:
check(cons(x,y)) = [1] x + [1] y + [5]
>= [1] x + [1] y + [5]
= cons(x,y)
check(cons(x,y)) = [1] x + [1] y + [5]
>= [1] x + [1] y + [5]
= cons(x,check(y))
check(cons(x,y)) = [1] x + [1] y + [5]
>= [1] x + [1] y + [5]
= cons(check(x),y)
check(rest(x)) = [1] x + [0]
>= [1] x + [0]
= rest(check(x))
check(sent(x)) = [1] x + [5]
>= [1] x + [5]
= sent(check(x))
rest(cons(x,y)) = [1] x + [1] y + [5]
>= [1] y + [5]
= sent(y)
rest(nil()) = [0]
>= [5]
= sent(nil())
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:2: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
- Weak TRS:
top(sent(x)) -> top(check(rest(x)))
- Signature:
{check/1,rest/1,top/1} / {cons/2,nil/0,sent/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {check,rest,top} and constructors {cons,nil,sent}
+ 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:
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(check) = [1] x1 + [0]
p(cons) = [1] x1 + [1] x2 + [1]
p(nil) = [9]
p(rest) = [1] x1 + [0]
p(sent) = [1] x1 + [0]
p(top) = [1] x1 + [0]
Following rules are strictly oriented:
rest(cons(x,y)) = [1] x + [1] y + [1]
> [1] y + [0]
= sent(y)
Following rules are (at-least) weakly oriented:
check(cons(x,y)) = [1] x + [1] y + [1]
>= [1] x + [1] y + [1]
= cons(x,y)
check(cons(x,y)) = [1] x + [1] y + [1]
>= [1] x + [1] y + [1]
= cons(x,check(y))
check(cons(x,y)) = [1] x + [1] y + [1]
>= [1] x + [1] y + [1]
= cons(check(x),y)
check(rest(x)) = [1] x + [0]
>= [1] x + [0]
= rest(check(x))
check(sent(x)) = [1] x + [0]
>= [1] x + [0]
= sent(check(x))
rest(nil()) = [9]
>= [9]
= sent(nil())
top(sent(x)) = [1] x + [0]
>= [1] x + [0]
= top(check(rest(x)))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:3: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(nil()) -> sent(nil())
- Weak TRS:
rest(cons(x,y)) -> sent(y)
top(sent(x)) -> top(check(rest(x)))
- Signature:
{check/1,rest/1,top/1} / {cons/2,nil/0,sent/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {check,rest,top} and constructors {cons,nil,sent}
+ 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:
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(check) = [1] x1 + [8]
p(cons) = [1] x1 + [1] x2 + [8]
p(nil) = [1]
p(rest) = [1] x1 + [0]
p(sent) = [1] x1 + [8]
p(top) = [1] x1 + [8]
Following rules are strictly oriented:
check(cons(x,y)) = [1] x + [1] y + [16]
> [1] x + [1] y + [8]
= cons(x,y)
Following rules are (at-least) weakly oriented:
check(cons(x,y)) = [1] x + [1] y + [16]
>= [1] x + [1] y + [16]
= cons(x,check(y))
check(cons(x,y)) = [1] x + [1] y + [16]
>= [1] x + [1] y + [16]
= cons(check(x),y)
check(rest(x)) = [1] x + [8]
>= [1] x + [8]
= rest(check(x))
check(sent(x)) = [1] x + [16]
>= [1] x + [16]
= sent(check(x))
rest(cons(x,y)) = [1] x + [1] y + [8]
>= [1] y + [8]
= sent(y)
rest(nil()) = [1]
>= [9]
= sent(nil())
top(sent(x)) = [1] x + [16]
>= [1] x + [16]
= top(check(rest(x)))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:4: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(nil()) -> sent(nil())
- Weak TRS:
check(cons(x,y)) -> cons(x,y)
rest(cons(x,y)) -> sent(y)
top(sent(x)) -> top(check(rest(x)))
- Signature:
{check/1,rest/1,top/1} / {cons/2,nil/0,sent/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {check,rest,top} and constructors {cons,nil,sent}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
Following symbols are considered usable:
{check,rest,top}
TcT has computed the following interpretation:
p(check) = [1 1] x_1 + [0]
[0 1] [0]
p(cons) = [1 0] x_1 + [1 3] x_2 + [0]
[0 1] [0 1] [1]
p(nil) = [0]
[0]
p(rest) = [1 0] x_1 + [0]
[0 2] [0]
p(sent) = [1 2] x_1 + [0]
[0 1] [0]
p(top) = [4 0] x_1 + [6]
[0 0] [0]
Following rules are strictly oriented:
check(cons(x,y)) = [1 1] x + [1 4] y + [1]
[0 1] [0 1] [1]
> [1 0] x + [1 4] y + [0]
[0 1] [0 1] [1]
= cons(x,check(y))
check(cons(x,y)) = [1 1] x + [1 4] y + [1]
[0 1] [0 1] [1]
> [1 1] x + [1 3] y + [0]
[0 1] [0 1] [1]
= cons(check(x),y)
Following rules are (at-least) weakly oriented:
check(cons(x,y)) = [1 1] x + [1 4] y + [1]
[0 1] [0 1] [1]
>= [1 0] x + [1 3] y + [0]
[0 1] [0 1] [1]
= cons(x,y)
check(rest(x)) = [1 2] x + [0]
[0 2] [0]
>= [1 1] x + [0]
[0 2] [0]
= rest(check(x))
check(sent(x)) = [1 3] x + [0]
[0 1] [0]
>= [1 3] x + [0]
[0 1] [0]
= sent(check(x))
rest(cons(x,y)) = [1 0] x + [1 3] y + [0]
[0 2] [0 2] [2]
>= [1 2] y + [0]
[0 1] [0]
= sent(y)
rest(nil()) = [0]
[0]
>= [0]
[0]
= sent(nil())
top(sent(x)) = [4 8] x + [6]
[0 0] [0]
>= [4 8] x + [6]
[0 0] [0]
= top(check(rest(x)))
** Step 1.b:5: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(nil()) -> sent(nil())
- Weak TRS:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
rest(cons(x,y)) -> sent(y)
top(sent(x)) -> top(check(rest(x)))
- Signature:
{check/1,rest/1,top/1} / {cons/2,nil/0,sent/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {check,rest,top} and constructors {cons,nil,sent}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 3))), miDimension = 4, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 3))):
Following symbols are considered usable:
{check,rest,top}
TcT has computed the following interpretation:
p(check) = [1 0 0 0] [0]
[0 1 0 0] x_1 + [0]
[0 1 0 0] [0]
[0 0 0 0] [0]
p(cons) = [1 2 0 1] [1 2 0 0] [0]
[0 1 0 0] x_1 + [0 0 0 1] x_2 + [0]
[0 0 0 0] [0 0 0 1] [0]
[0 0 0 0] [0 0 0 0] [0]
p(nil) = [2]
[0]
[2]
[2]
p(rest) = [1 0 0 2] [0]
[0 2 0 0] x_1 + [0]
[0 0 2 0] [0]
[0 0 0 0] [0]
p(sent) = [1 2 0 0] [0]
[0 0 0 0] x_1 + [0]
[0 0 0 2] [0]
[0 0 0 0] [0]
p(top) = [2 0 2 0] [0]
[0 0 0 0] x_1 + [0]
[2 0 2 1] [0]
[1 0 1 0] [0]
Following rules are strictly oriented:
rest(nil()) = [6]
[0]
[4]
[0]
> [2]
[0]
[4]
[0]
= sent(nil())
Following rules are (at-least) weakly oriented:
check(cons(x,y)) = [1 2 0 1] [1 2 0 0] [0]
[0 1 0 0] x + [0 0 0 1] y + [0]
[0 1 0 0] [0 0 0 1] [0]
[0 0 0 0] [0 0 0 0] [0]
>= [1 2 0 1] [1 2 0 0] [0]
[0 1 0 0] x + [0 0 0 1] y + [0]
[0 0 0 0] [0 0 0 1] [0]
[0 0 0 0] [0 0 0 0] [0]
= cons(x,y)
check(cons(x,y)) = [1 2 0 1] [1 2 0 0] [0]
[0 1 0 0] x + [0 0 0 1] y + [0]
[0 1 0 0] [0 0 0 1] [0]
[0 0 0 0] [0 0 0 0] [0]
>= [1 2 0 1] [1 2 0 0] [0]
[0 1 0 0] x + [0 0 0 0] y + [0]
[0 0 0 0] [0 0 0 0] [0]
[0 0 0 0] [0 0 0 0] [0]
= cons(x,check(y))
check(cons(x,y)) = [1 2 0 1] [1 2 0 0] [0]
[0 1 0 0] x + [0 0 0 1] y + [0]
[0 1 0 0] [0 0 0 1] [0]
[0 0 0 0] [0 0 0 0] [0]
>= [1 2 0 0] [1 2 0 0] [0]
[0 1 0 0] x + [0 0 0 1] y + [0]
[0 0 0 0] [0 0 0 1] [0]
[0 0 0 0] [0 0 0 0] [0]
= cons(check(x),y)
check(rest(x)) = [1 0 0 2] [0]
[0 2 0 0] x + [0]
[0 2 0 0] [0]
[0 0 0 0] [0]
>= [1 0 0 0] [0]
[0 2 0 0] x + [0]
[0 2 0 0] [0]
[0 0 0 0] [0]
= rest(check(x))
check(sent(x)) = [1 2 0 0] [0]
[0 0 0 0] x + [0]
[0 0 0 0] [0]
[0 0 0 0] [0]
>= [1 2 0 0] [0]
[0 0 0 0] x + [0]
[0 0 0 0] [0]
[0 0 0 0] [0]
= sent(check(x))
rest(cons(x,y)) = [1 2 0 1] [1 2 0 0] [0]
[0 2 0 0] x + [0 0 0 2] y + [0]
[0 0 0 0] [0 0 0 2] [0]
[0 0 0 0] [0 0 0 0] [0]
>= [1 2 0 0] [0]
[0 0 0 0] y + [0]
[0 0 0 2] [0]
[0 0 0 0] [0]
= sent(y)
top(sent(x)) = [2 4 0 4] [0]
[0 0 0 0] x + [0]
[2 4 0 4] [0]
[1 2 0 2] [0]
>= [2 4 0 4] [0]
[0 0 0 0] x + [0]
[2 4 0 4] [0]
[1 2 0 2] [0]
= top(check(rest(x)))
** Step 1.b:6: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
- Weak TRS:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
top(sent(x)) -> top(check(rest(x)))
- Signature:
{check/1,rest/1,top/1} / {cons/2,nil/0,sent/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {check,rest,top} and constructors {cons,nil,sent}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 3))), miDimension = 4, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 3))):
Following symbols are considered usable:
{check,rest,top}
TcT has computed the following interpretation:
p(check) = [1 2 0 0] [0]
[0 1 0 0] x_1 + [0]
[0 1 0 0] [0]
[0 0 0 0] [0]
p(cons) = [1 1 0 3] [1 3 0 0] [0]
[0 1 0 1] x_1 + [0 1 0 3] x_2 + [1]
[0 0 0 0] [0 0 0 3] [1]
[0 0 0 0] [0 0 0 0] [0]
p(nil) = [2]
[0]
[0]
[2]
p(rest) = [1 0 0 0] [1]
[0 1 0 1] x_1 + [0]
[0 0 1 3] [0]
[0 0 0 0] [0]
p(sent) = [1 3 0 0] [1]
[0 1 0 0] x_1 + [1]
[0 0 0 3] [0]
[0 0 0 0] [0]
p(top) = [2 0 2 0] [0]
[0 0 0 0] x_1 + [2]
[0 0 0 0] [1]
[0 0 0 0] [0]
Following rules are strictly oriented:
check(sent(x)) = [1 5 0 0] [3]
[0 1 0 0] x + [1]
[0 1 0 0] [1]
[0 0 0 0] [0]
> [1 5 0 0] [1]
[0 1 0 0] x + [1]
[0 0 0 0] [0]
[0 0 0 0] [0]
= sent(check(x))
Following rules are (at-least) weakly oriented:
check(cons(x,y)) = [1 3 0 5] [1 5 0 6] [2]
[0 1 0 1] x + [0 1 0 3] y + [1]
[0 1 0 1] [0 1 0 3] [1]
[0 0 0 0] [0 0 0 0] [0]
>= [1 1 0 3] [1 3 0 0] [0]
[0 1 0 1] x + [0 1 0 3] y + [1]
[0 0 0 0] [0 0 0 3] [1]
[0 0 0 0] [0 0 0 0] [0]
= cons(x,y)
check(cons(x,y)) = [1 3 0 5] [1 5 0 6] [2]
[0 1 0 1] x + [0 1 0 3] y + [1]
[0 1 0 1] [0 1 0 3] [1]
[0 0 0 0] [0 0 0 0] [0]
>= [1 1 0 3] [1 5 0 0] [0]
[0 1 0 1] x + [0 1 0 0] y + [1]
[0 0 0 0] [0 0 0 0] [1]
[0 0 0 0] [0 0 0 0] [0]
= cons(x,check(y))
check(cons(x,y)) = [1 3 0 5] [1 5 0 6] [2]
[0 1 0 1] x + [0 1 0 3] y + [1]
[0 1 0 1] [0 1 0 3] [1]
[0 0 0 0] [0 0 0 0] [0]
>= [1 3 0 0] [1 3 0 0] [0]
[0 1 0 0] x + [0 1 0 3] y + [1]
[0 0 0 0] [0 0 0 3] [1]
[0 0 0 0] [0 0 0 0] [0]
= cons(check(x),y)
check(rest(x)) = [1 2 0 2] [1]
[0 1 0 1] x + [0]
[0 1 0 1] [0]
[0 0 0 0] [0]
>= [1 2 0 0] [1]
[0 1 0 0] x + [0]
[0 1 0 0] [0]
[0 0 0 0] [0]
= rest(check(x))
rest(cons(x,y)) = [1 1 0 3] [1 3 0 0] [1]
[0 1 0 1] x + [0 1 0 3] y + [1]
[0 0 0 0] [0 0 0 3] [1]
[0 0 0 0] [0 0 0 0] [0]
>= [1 3 0 0] [1]
[0 1 0 0] y + [1]
[0 0 0 3] [0]
[0 0 0 0] [0]
= sent(y)
rest(nil()) = [3]
[2]
[6]
[0]
>= [3]
[1]
[6]
[0]
= sent(nil())
top(sent(x)) = [2 6 0 6] [2]
[0 0 0 0] x + [2]
[0 0 0 0] [1]
[0 0 0 0] [0]
>= [2 6 0 6] [2]
[0 0 0 0] x + [2]
[0 0 0 0] [1]
[0 0 0 0] [0]
= top(check(rest(x)))
** Step 1.b:7: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
check(rest(x)) -> rest(check(x))
- Weak TRS:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
top(sent(x)) -> top(check(rest(x)))
- Signature:
{check/1,rest/1,top/1} / {cons/2,nil/0,sent/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {check,rest,top} and constructors {cons,nil,sent}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 3))), miDimension = 4, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 3))):
Following symbols are considered usable:
{check,rest,top}
TcT has computed the following interpretation:
p(check) = [1 0 2 0] [0]
[0 0 2 0] x_1 + [0]
[0 0 1 0] [0]
[0 0 0 0] [0]
p(cons) = [1 0 2 0] [1 0 3 2] [3]
[0 0 0 0] x_1 + [0 0 2 2] x_2 + [2]
[0 0 1 0] [0 0 1 2] [2]
[0 0 0 0] [0 0 0 0] [0]
p(nil) = [2]
[0]
[0]
[2]
p(rest) = [1 0 0 2] [0]
[0 1 0 2] x_1 + [2]
[0 0 1 0] [1]
[0 0 0 0] [0]
p(sent) = [1 0 3 1] [2]
[0 0 0 2] x_1 + [2]
[0 0 1 0] [1]
[0 0 0 0] [0]
p(top) = [2 1 0 0] [1]
[2 1 1 2] x_1 + [0]
[2 1 1 0] [0]
[0 0 2 0] [2]
Following rules are strictly oriented:
check(rest(x)) = [1 0 2 2] [2]
[0 0 2 0] x + [2]
[0 0 1 0] [1]
[0 0 0 0] [0]
> [1 0 2 0] [0]
[0 0 2 0] x + [2]
[0 0 1 0] [1]
[0 0 0 0] [0]
= rest(check(x))
Following rules are (at-least) weakly oriented:
check(cons(x,y)) = [1 0 4 0] [1 0 5 6] [7]
[0 0 2 0] x + [0 0 2 4] y + [4]
[0 0 1 0] [0 0 1 2] [2]
[0 0 0 0] [0 0 0 0] [0]
>= [1 0 2 0] [1 0 3 2] [3]
[0 0 0 0] x + [0 0 2 2] y + [2]
[0 0 1 0] [0 0 1 2] [2]
[0 0 0 0] [0 0 0 0] [0]
= cons(x,y)
check(cons(x,y)) = [1 0 4 0] [1 0 5 6] [7]
[0 0 2 0] x + [0 0 2 4] y + [4]
[0 0 1 0] [0 0 1 2] [2]
[0 0 0 0] [0 0 0 0] [0]
>= [1 0 2 0] [1 0 5 0] [3]
[0 0 0 0] x + [0 0 2 0] y + [2]
[0 0 1 0] [0 0 1 0] [2]
[0 0 0 0] [0 0 0 0] [0]
= cons(x,check(y))
check(cons(x,y)) = [1 0 4 0] [1 0 5 6] [7]
[0 0 2 0] x + [0 0 2 4] y + [4]
[0 0 1 0] [0 0 1 2] [2]
[0 0 0 0] [0 0 0 0] [0]
>= [1 0 4 0] [1 0 3 2] [3]
[0 0 0 0] x + [0 0 2 2] y + [2]
[0 0 1 0] [0 0 1 2] [2]
[0 0 0 0] [0 0 0 0] [0]
= cons(check(x),y)
check(sent(x)) = [1 0 5 1] [4]
[0 0 2 0] x + [2]
[0 0 1 0] [1]
[0 0 0 0] [0]
>= [1 0 5 0] [2]
[0 0 0 0] x + [2]
[0 0 1 0] [1]
[0 0 0 0] [0]
= sent(check(x))
rest(cons(x,y)) = [1 0 2 0] [1 0 3 2] [3]
[0 0 0 0] x + [0 0 2 2] y + [4]
[0 0 1 0] [0 0 1 2] [3]
[0 0 0 0] [0 0 0 0] [0]
>= [1 0 3 1] [2]
[0 0 0 2] y + [2]
[0 0 1 0] [1]
[0 0 0 0] [0]
= sent(y)
rest(nil()) = [6]
[6]
[1]
[0]
>= [6]
[6]
[1]
[0]
= sent(nil())
top(sent(x)) = [2 0 6 4] [7]
[2 0 7 4] x + [7]
[2 0 7 4] [7]
[0 0 2 0] [4]
>= [2 0 6 4] [7]
[2 0 7 4] x + [7]
[2 0 7 4] [7]
[0 0 2 0] [4]
= top(check(rest(x)))
** Step 1.b:8: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
top(sent(x)) -> top(check(rest(x)))
- Signature:
{check/1,rest/1,top/1} / {cons/2,nil/0,sent/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {check,rest,top} and constructors {cons,nil,sent}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^2))