* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
+ Considered Problem:
- Strict TRS:
fac(s(x)) -> *(fac(p(s(x))),s(x))
p(s(0())) -> 0()
p(s(s(x))) -> s(p(s(x)))
- Signature:
{fac/1,p/1} / {*/2,0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {fac,p} 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:
fac(s(x)) -> *(fac(p(s(x))),s(x))
p(s(0())) -> 0()
p(s(s(x))) -> s(p(s(x)))
- Signature:
{fac/1,p/1} / {*/2,0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {fac,p} and constructors {*,0,s}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
p(s(x)){x -> s(x)} =
p(s(s(x))) ->^+ s(p(s(x)))
= C[p(s(x)) = p(s(x)){}]
** Step 1.b:1: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
fac(s(x)) -> *(fac(p(s(x))),s(x))
p(s(0())) -> 0()
p(s(s(x))) -> s(p(s(x)))
- Signature:
{fac/1,p/1} / {*/2,0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {fac,p} 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(*) = {1},
uargs(fac) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(*) = [1] x1 + [0]
p(0) = [0]
p(fac) = [1] x1 + [7]
p(p) = [4]
p(s) = [1] x1 + [1]
Following rules are strictly oriented:
p(s(0())) = [4]
> [0]
= 0()
Following rules are (at-least) weakly oriented:
fac(s(x)) = [1] x + [8]
>= [11]
= *(fac(p(s(x))),s(x))
p(s(s(x))) = [4]
>= [5]
= s(p(s(x)))
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:
fac(s(x)) -> *(fac(p(s(x))),s(x))
p(s(s(x))) -> s(p(s(x)))
- Weak TRS:
p(s(0())) -> 0()
- Signature:
{fac/1,p/1} / {*/2,0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {fac,p} 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(*) = {1},
uargs(fac) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(*) = [1] x1 + [0]
p(0) = [0]
p(fac) = [1] x1 + [0]
p(p) = [0]
p(s) = [1] x1 + [1]
Following rules are strictly oriented:
fac(s(x)) = [1] x + [1]
> [0]
= *(fac(p(s(x))),s(x))
Following rules are (at-least) weakly oriented:
p(s(0())) = [0]
>= [0]
= 0()
p(s(s(x))) = [0]
>= [1]
= s(p(s(x)))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:3: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
p(s(s(x))) -> s(p(s(x)))
- Weak TRS:
fac(s(x)) -> *(fac(p(s(x))),s(x))
p(s(0())) -> 0()
- Signature:
{fac/1,p/1} / {*/2,0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {fac,p} and constructors {*,0,s}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 2))), miDimension = 3, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 2))):
The following argument positions are considered usable:
uargs(*) = {1},
uargs(fac) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{fac,p}
TcT has computed the following interpretation:
p(*) = [1 0 0] [0]
[0 0 0] x_1 + [3]
[0 0 0] [0]
p(0) = [4]
[0]
[0]
p(fac) = [2 0 0] [0]
[0 2 0] x_1 + [3]
[0 0 0] [3]
p(p) = [1 0 0] [0]
[0 1 0] x_1 + [2]
[0 1 0] [0]
p(s) = [1 4 3] [0]
[0 0 1] x_1 + [0]
[0 0 1] [4]
Following rules are strictly oriented:
p(s(s(x))) = [1 4 10] [12]
[0 0 1] x + [6]
[0 0 1] [4]
> [1 4 10] [8]
[0 0 1] x + [0]
[0 0 1] [4]
= s(p(s(x)))
Following rules are (at-least) weakly oriented:
fac(s(x)) = [2 8 6] [0]
[0 0 2] x + [3]
[0 0 0] [3]
>= [2 8 6] [0]
[0 0 0] x + [3]
[0 0 0] [0]
= *(fac(p(s(x))),s(x))
p(s(0())) = [4]
[2]
[0]
>= [4]
[0]
[0]
= 0()
** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
fac(s(x)) -> *(fac(p(s(x))),s(x))
p(s(0())) -> 0()
p(s(s(x))) -> s(p(s(x)))
- Signature:
{fac/1,p/1} / {*/2,0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {fac,p} 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))