* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
+ Considered Problem:
- Strict TRS:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
-(x,0()) -> x
-(0(),y) -> 0()
-(s(x),s(y)) -> -(x,y)
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2} / {+/2,0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*,-,exp} 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:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
-(x,0()) -> x
-(0(),y) -> 0()
-(s(x),s(y)) -> -(x,y)
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2} / {+/2,0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*,-,exp} and constructors {+,0,s}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
*(x,y){x -> s(x)} =
*(s(x),y) ->^+ +(y,*(x,y))
= C[*(x,y) = *(x,y){}]
** Step 1.b:1: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
-(x,0()) -> x
-(0(),y) -> 0()
-(s(x),s(y)) -> -(x,y)
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2} / {+/2,0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*,-,exp} 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(*) = {2},
uargs(+) = {2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(*) = [1] x2 + [2]
p(+) = [1] x2 + [0]
p(-) = [4] x1 + [1] x2 + [6]
p(0) = [0]
p(exp) = [10] x2 + [1]
p(s) = [1] x1 + [1]
Following rules are strictly oriented:
*(0(),y) = [1] y + [2]
> [0]
= 0()
-(x,0()) = [4] x + [6]
> [1] x + [0]
= x
-(0(),y) = [1] y + [6]
> [0]
= 0()
-(s(x),s(y)) = [4] x + [1] y + [11]
> [4] x + [1] y + [6]
= -(x,y)
exp(x,s(y)) = [10] y + [11]
> [10] y + [3]
= *(x,exp(x,y))
Following rules are (at-least) weakly oriented:
*(s(x),y) = [1] y + [2]
>= [1] y + [2]
= +(y,*(x,y))
exp(x,0()) = [1]
>= [1]
= s(0())
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:
*(s(x),y) -> +(y,*(x,y))
exp(x,0()) -> s(0())
- Weak TRS:
*(0(),y) -> 0()
-(x,0()) -> x
-(0(),y) -> 0()
-(s(x),s(y)) -> -(x,y)
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2} / {+/2,0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*,-,exp} 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(*) = {2},
uargs(+) = {2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(*) = [1] x2 + [0]
p(+) = [1] x2 + [0]
p(-) = [2] x1 + [0]
p(0) = [0]
p(exp) = [7]
p(s) = [1] x1 + [0]
Following rules are strictly oriented:
exp(x,0()) = [7]
> [0]
= s(0())
Following rules are (at-least) weakly oriented:
*(0(),y) = [1] y + [0]
>= [0]
= 0()
*(s(x),y) = [1] y + [0]
>= [1] y + [0]
= +(y,*(x,y))
-(x,0()) = [2] x + [0]
>= [1] x + [0]
= x
-(0(),y) = [0]
>= [0]
= 0()
-(s(x),s(y)) = [2] x + [0]
>= [2] x + [0]
= -(x,y)
exp(x,s(y)) = [7]
>= [7]
= *(x,exp(x,y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
*(s(x),y) -> +(y,*(x,y))
- Weak TRS:
*(0(),y) -> 0()
-(x,0()) -> x
-(0(),y) -> 0()
-(s(x),s(y)) -> -(x,y)
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2} / {+/2,0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*,-,exp} and constructors {+,0,s}
+ Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(*) = {2},
uargs(+) = {2}
Following symbols are considered usable:
{*,-,exp}
TcT has computed the following interpretation:
p(*) = 2*x1 + x2
p(+) = x2
p(-) = 4 + 4*x1 + 4*x1*x2
p(0) = 1
p(exp) = 2*x1*x2 + 2*x2
p(s) = 1 + x1
Following rules are strictly oriented:
*(s(x),y) = 2 + 2*x + y
> 2*x + y
= +(y,*(x,y))
Following rules are (at-least) weakly oriented:
*(0(),y) = 2 + y
>= 1
= 0()
-(x,0()) = 4 + 8*x
>= x
= x
-(0(),y) = 8 + 4*y
>= 1
= 0()
-(s(x),s(y)) = 12 + 8*x + 4*x*y + 4*y
>= 4 + 4*x + 4*x*y
= -(x,y)
exp(x,0()) = 2 + 2*x
>= 2
= s(0())
exp(x,s(y)) = 2 + 2*x + 2*x*y + 2*y
>= 2*x + 2*x*y + 2*y
= *(x,exp(x,y))
** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
-(x,0()) -> x
-(0(),y) -> 0()
-(s(x),s(y)) -> -(x,y)
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2} / {+/2,0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*,-,exp} 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^2))