* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
+ Considered Problem:
- Strict TRS:
conv(0()) -> cons(nil(),0())
conv(s(x)) -> cons(conv(half(s(x))),lastbit(s(x)))
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
lastbit(0()) -> 0()
lastbit(s(0())) -> s(0())
lastbit(s(s(x))) -> lastbit(x)
- Signature:
{conv/1,half/1,lastbit/1} / {0/0,cons/2,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv,half,lastbit} and constructors {0,cons,nil,s}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
conv(0()) -> cons(nil(),0())
conv(s(x)) -> cons(conv(half(s(x))),lastbit(s(x)))
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
lastbit(0()) -> 0()
lastbit(s(0())) -> s(0())
lastbit(s(s(x))) -> lastbit(x)
- Signature:
{conv/1,half/1,lastbit/1} / {0/0,cons/2,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv,half,lastbit} and constructors {0,cons,nil,s}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
half(x){x -> s(s(x))} =
half(s(s(x))) ->^+ s(half(x))
= C[half(x) = half(x){}]
** Step 1.b:1: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
conv(0()) -> cons(nil(),0())
conv(s(x)) -> cons(conv(half(s(x))),lastbit(s(x)))
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
lastbit(0()) -> 0()
lastbit(s(0())) -> s(0())
lastbit(s(s(x))) -> lastbit(x)
- Signature:
{conv/1,half/1,lastbit/1} / {0/0,cons/2,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv,half,lastbit} and constructors {0,cons,nil,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(cons) = {1,2},
uargs(conv) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [8]
p(cons) = [1] x1 + [1] x2 + [10]
p(conv) = [1] x1 + [8]
p(half) = [1] x1 + [12]
p(lastbit) = [0]
p(nil) = [8]
p(s) = [1] x1 + [0]
Following rules are strictly oriented:
half(0()) = [20]
> [8]
= 0()
half(s(0())) = [20]
> [8]
= 0()
Following rules are (at-least) weakly oriented:
conv(0()) = [16]
>= [26]
= cons(nil(),0())
conv(s(x)) = [1] x + [8]
>= [1] x + [30]
= cons(conv(half(s(x))),lastbit(s(x)))
half(s(s(x))) = [1] x + [12]
>= [1] x + [12]
= s(half(x))
lastbit(0()) = [0]
>= [8]
= 0()
lastbit(s(0())) = [0]
>= [8]
= s(0())
lastbit(s(s(x))) = [0]
>= [0]
= lastbit(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:
conv(0()) -> cons(nil(),0())
conv(s(x)) -> cons(conv(half(s(x))),lastbit(s(x)))
half(s(s(x))) -> s(half(x))
lastbit(0()) -> 0()
lastbit(s(0())) -> s(0())
lastbit(s(s(x))) -> lastbit(x)
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
- Signature:
{conv/1,half/1,lastbit/1} / {0/0,cons/2,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv,half,lastbit} and constructors {0,cons,nil,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(cons) = {1,2},
uargs(conv) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(cons) = [1] x1 + [1] x2 + [0]
p(conv) = [1] x1 + [12]
p(half) = [0]
p(lastbit) = [1] x1 + [10]
p(nil) = [2]
p(s) = [1] x1 + [8]
Following rules are strictly oriented:
conv(0()) = [12]
> [2]
= cons(nil(),0())
lastbit(0()) = [10]
> [0]
= 0()
lastbit(s(0())) = [18]
> [8]
= s(0())
lastbit(s(s(x))) = [1] x + [26]
> [1] x + [10]
= lastbit(x)
Following rules are (at-least) weakly oriented:
conv(s(x)) = [1] x + [20]
>= [1] x + [30]
= cons(conv(half(s(x))),lastbit(s(x)))
half(0()) = [0]
>= [0]
= 0()
half(s(0())) = [0]
>= [0]
= 0()
half(s(s(x))) = [0]
>= [8]
= s(half(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:
conv(s(x)) -> cons(conv(half(s(x))),lastbit(s(x)))
half(s(s(x))) -> s(half(x))
- Weak TRS:
conv(0()) -> cons(nil(),0())
half(0()) -> 0()
half(s(0())) -> 0()
lastbit(0()) -> 0()
lastbit(s(0())) -> s(0())
lastbit(s(s(x))) -> lastbit(x)
- Signature:
{conv/1,half/1,lastbit/1} / {0/0,cons/2,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv,half,lastbit} and constructors {0,cons,nil,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(cons) = {1,2},
uargs(conv) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(cons) = [1] x1 + [1] x2 + [1]
p(conv) = [1] x1 + [5]
p(half) = [1] x1 + [0]
p(lastbit) = [9]
p(nil) = [3]
p(s) = [1] x1 + [8]
Following rules are strictly oriented:
half(s(s(x))) = [1] x + [16]
> [1] x + [8]
= s(half(x))
Following rules are (at-least) weakly oriented:
conv(0()) = [5]
>= [4]
= cons(nil(),0())
conv(s(x)) = [1] x + [13]
>= [1] x + [23]
= cons(conv(half(s(x))),lastbit(s(x)))
half(0()) = [0]
>= [0]
= 0()
half(s(0())) = [8]
>= [0]
= 0()
lastbit(0()) = [9]
>= [0]
= 0()
lastbit(s(0())) = [9]
>= [8]
= s(0())
lastbit(s(s(x))) = [9]
>= [9]
= lastbit(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:
conv(s(x)) -> cons(conv(half(s(x))),lastbit(s(x)))
- Weak TRS:
conv(0()) -> cons(nil(),0())
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
lastbit(0()) -> 0()
lastbit(s(0())) -> s(0())
lastbit(s(s(x))) -> lastbit(x)
- Signature:
{conv/1,half/1,lastbit/1} / {0/0,cons/2,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv,half,lastbit} and constructors {0,cons,nil,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(cons) = {1,2},
uargs(conv) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{conv,half,lastbit}
TcT has computed the following interpretation:
p(0) = [0]
[1]
[2]
p(cons) = [1 0 0] [1 0 0] [3]
[0 0 2] x_1 + [0 0 0] x_2 + [0]
[0 0 0] [0 0 0] [0]
p(conv) = [1 2 0] [6]
[6 0 0] x_1 + [2]
[5 0 0] [0]
p(half) = [0 0 0] [0]
[0 0 1] x_1 + [0]
[0 0 1] [0]
p(lastbit) = [0]
[6]
[4]
p(nil) = [5]
[0]
[1]
p(s) = [1 0 0] [0]
[0 0 1] x_1 + [4]
[0 0 1] [2]
Following rules are strictly oriented:
conv(s(x)) = [1 0 2] [14]
[6 0 0] x + [2]
[5 0 0] [0]
> [0 0 2] [13]
[0 0 0] x + [0]
[0 0 0] [0]
= cons(conv(half(s(x))),lastbit(s(x)))
Following rules are (at-least) weakly oriented:
conv(0()) = [8]
[2]
[0]
>= [8]
[2]
[0]
= cons(nil(),0())
half(0()) = [0]
[2]
[2]
>= [0]
[1]
[2]
= 0()
half(s(0())) = [0]
[4]
[4]
>= [0]
[1]
[2]
= 0()
half(s(s(x))) = [0 0 0] [0]
[0 0 1] x + [4]
[0 0 1] [4]
>= [0 0 0] [0]
[0 0 1] x + [4]
[0 0 1] [2]
= s(half(x))
lastbit(0()) = [0]
[6]
[4]
>= [0]
[1]
[2]
= 0()
lastbit(s(0())) = [0]
[6]
[4]
>= [0]
[6]
[4]
= s(0())
lastbit(s(s(x))) = [0]
[6]
[4]
>= [0]
[6]
[4]
= lastbit(x)
** Step 1.b:5: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
conv(0()) -> cons(nil(),0())
conv(s(x)) -> cons(conv(half(s(x))),lastbit(s(x)))
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
lastbit(0()) -> 0()
lastbit(s(0())) -> s(0())
lastbit(s(s(x))) -> lastbit(x)
- Signature:
{conv/1,half/1,lastbit/1} / {0/0,cons/2,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv,half,lastbit} and constructors {0,cons,nil,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^2))