* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
average(x,s(s(s(y)))) -> s(average(s(x),y))
average(0(),0()) -> 0()
average(0(),s(0())) -> 0()
average(0(),s(s(0()))) -> s(0())
average(s(x),y) -> average(x,s(y))
- Signature:
{average/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {average} and constructors {0,s}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
average(x,s(s(s(y)))) -> s(average(s(x),y))
average(0(),0()) -> 0()
average(0(),s(0())) -> 0()
average(0(),s(s(0()))) -> s(0())
average(s(x),y) -> average(x,s(y))
- Signature:
{average/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {average} and constructors {0,s}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
average(x,y){y -> s(s(s(y)))} =
average(x,s(s(s(y)))) ->^+ s(average(s(x),y))
= C[average(s(x),y) = average(x,y){x -> s(x)}]
** Step 1.b:1: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
average(x,s(s(s(y)))) -> s(average(s(x),y))
average(0(),0()) -> 0()
average(0(),s(0())) -> 0()
average(0(),s(s(0()))) -> s(0())
average(s(x),y) -> average(x,s(y))
- Signature:
{average/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {average} and constructors {0,s}
+ 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(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [12]
p(average) = [1] x1 + [13]
p(s) = [1] x1 + [0]
Following rules are strictly oriented:
average(0(),0()) = [25]
> [12]
= 0()
average(0(),s(0())) = [25]
> [12]
= 0()
average(0(),s(s(0()))) = [25]
> [12]
= s(0())
Following rules are (at-least) weakly oriented:
average(x,s(s(s(y)))) = [1] x + [13]
>= [1] x + [13]
= s(average(s(x),y))
average(s(x),y) = [1] x + [13]
>= [1] x + [13]
= average(x,s(y))
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^1))
+ Considered Problem:
- Strict TRS:
average(x,s(s(s(y)))) -> s(average(s(x),y))
average(s(x),y) -> average(x,s(y))
- Weak TRS:
average(0(),0()) -> 0()
average(0(),s(0())) -> 0()
average(0(),s(s(0()))) -> s(0())
- Signature:
{average/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {average} and constructors {0,s}
+ 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(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [1]
p(average) = [10] x1 + [0]
p(s) = [1] x1 + [2]
Following rules are strictly oriented:
average(s(x),y) = [10] x + [20]
> [10] x + [0]
= average(x,s(y))
Following rules are (at-least) weakly oriented:
average(x,s(s(s(y)))) = [10] x + [0]
>= [10] x + [22]
= s(average(s(x),y))
average(0(),0()) = [10]
>= [1]
= 0()
average(0(),s(0())) = [10]
>= [1]
= 0()
average(0(),s(s(0()))) = [10]
>= [3]
= s(0())
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^1))
+ Considered Problem:
- Strict TRS:
average(x,s(s(s(y)))) -> s(average(s(x),y))
- Weak TRS:
average(0(),0()) -> 0()
average(0(),s(0())) -> 0()
average(0(),s(s(0()))) -> s(0())
average(s(x),y) -> average(x,s(y))
- Signature:
{average/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {average} and constructors {0,s}
+ 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(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [1]
p(average) = [9] x1 + [6] x2 + [0]
p(s) = [1] x1 + [1]
Following rules are strictly oriented:
average(x,s(s(s(y)))) = [9] x + [6] y + [18]
> [9] x + [6] y + [10]
= s(average(s(x),y))
Following rules are (at-least) weakly oriented:
average(0(),0()) = [15]
>= [1]
= 0()
average(0(),s(0())) = [21]
>= [1]
= 0()
average(0(),s(s(0()))) = [27]
>= [2]
= s(0())
average(s(x),y) = [9] x + [6] y + [9]
>= [9] x + [6] y + [6]
= average(x,s(y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
average(x,s(s(s(y)))) -> s(average(s(x),y))
average(0(),0()) -> 0()
average(0(),s(0())) -> 0()
average(0(),s(s(0()))) -> s(0())
average(s(x),y) -> average(x,s(y))
- Signature:
{average/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {average} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^1))