*** 1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
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)
Weak DP Rules:
Weak TRS Rules:
Signature:
{conv/1,half/1,lastbit/1} / {0/0,cons/2,nil/0,s/1}
Obligation:
Innermost
basic terms: {conv,half,lastbit}/{0,cons,nil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
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:
{}
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.
*** 1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
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 DP Rules:
Weak TRS Rules:
half(0()) -> 0()
half(s(0())) -> 0()
Signature:
{conv/1,half/1,lastbit/1} / {0/0,cons/2,nil/0,s/1}
Obligation:
Innermost
basic terms: {conv,half,lastbit}/{0,cons,nil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
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:
{}
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.
*** 1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
conv(s(x)) -> cons(conv(half(s(x))),lastbit(s(x)))
half(s(s(x))) -> s(half(x))
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {conv,half,lastbit}/{0,cons,nil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
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:
{}
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.
*** 1.1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
conv(s(x)) -> cons(conv(half(s(x))),lastbit(s(x)))
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {conv,half,lastbit}/{0,cons,nil,s}
Applied Processor:
NaturalMI {miDimension = 4, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 3 non-zero interpretation-entries in the diagonal of the component-wise maxima):
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]
[0]
[0]
[0]
p(cons) = [1 0 0 0] [1 0 0
0] [0]
[0 0 0 1] x1 + [0 0 1
0] x2 + [0]
[0 0 0 0] [0 0 0
0] [0]
[0 0 0 0] [0 0 0
0] [0]
p(conv) = [1 1 0 0] [0]
[1 0 1 0] x1 + [1]
[0 0 0 0] [0]
[0 0 1 0] [0]
p(half) = [0 0 0 0] [0]
[1 0 1 0] x1 + [0]
[0 0 0 1] [0]
[0 0 0 1] [0]
p(lastbit) = [0 0 0 0] [0]
[0 0 0 0] x1 + [1]
[1 0 0 0] [0]
[0 0 0 0] [1]
p(nil) = [0]
[0]
[0]
[1]
p(s) = [1 0 0 0] [0]
[0 0 0 1] x1 + [1]
[0 0 0 1] [0]
[0 0 0 1] [1]
Following rules are strictly oriented:
conv(s(x)) = [1 0 0 1] [1]
[1 0 0 1] x + [1]
[0 0 0 0] [0]
[0 0 0 1] [0]
> [1 0 0 1] [0]
[1 0 0 1] x + [1]
[0 0 0 0] [0]
[0 0 0 0] [0]
= cons(conv(half(s(x)))
,lastbit(s(x)))
Following rules are (at-least) weakly oriented:
conv(0()) = [0]
[1]
[0]
[0]
>= [0]
[1]
[0]
[0]
= cons(nil(),0())
half(0()) = [0]
[0]
[0]
[0]
>= [0]
[0]
[0]
[0]
= 0()
half(s(0())) = [0]
[0]
[1]
[1]
>= [0]
[0]
[0]
[0]
= 0()
half(s(s(x))) = [0 0 0 0] [0]
[1 0 0 1] x + [1]
[0 0 0 1] [2]
[0 0 0 1] [2]
>= [0 0 0 0] [0]
[0 0 0 1] x + [1]
[0 0 0 1] [0]
[0 0 0 1] [1]
= s(half(x))
lastbit(0()) = [0]
[1]
[0]
[1]
>= [0]
[0]
[0]
[0]
= 0()
lastbit(s(0())) = [0]
[1]
[0]
[1]
>= [0]
[1]
[0]
[1]
= s(0())
lastbit(s(s(x))) = [0 0 0 0] [0]
[0 0 0 0] x + [1]
[1 0 0 0] [0]
[0 0 0 0] [1]
>= [0 0 0 0] [0]
[0 0 0 0] x + [1]
[1 0 0 0] [0]
[0 0 0 0] [1]
= lastbit(x)
*** 1.1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {conv,half,lastbit}/{0,cons,nil,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).