* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{minus/2,quot/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus,quot} 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:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{minus/2,quot/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus,quot} and constructors {0,s}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
minus(x,y){x -> s(x),y -> s(y)} =
minus(s(x),s(y)) ->^+ minus(x,y)
= C[minus(x,y) = minus(x,y){}]
** Step 1.b:1: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{minus/2,quot/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus,quot} 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(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [8]
p(minus) = [1] x1 + [0]
p(quot) = [1] x1 + [1] x2 + [0]
p(s) = [1] x1 + [8]
Following rules are strictly oriented:
minus(s(x),s(y)) = [1] x + [8]
> [1] x + [0]
= minus(x,y)
quot(0(),s(y)) = [1] y + [16]
> [8]
= 0()
Following rules are (at-least) weakly oriented:
minus(x,0()) = [1] x + [0]
>= [1] x + [0]
= x
quot(s(x),s(y)) = [1] x + [1] y + [16]
>= [1] x + [1] y + [16]
= s(quot(minus(x,y),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:
minus(x,0()) -> x
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Weak TRS:
minus(s(x),s(y)) -> minus(x,y)
quot(0(),s(y)) -> 0()
- Signature:
{minus/2,quot/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus,quot} 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(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [2]
p(minus) = [1] x1 + [1]
p(quot) = [1] x1 + [1]
p(s) = [1] x1 + [1]
Following rules are strictly oriented:
minus(x,0()) = [1] x + [1]
> [1] x + [0]
= x
Following rules are (at-least) weakly oriented:
minus(s(x),s(y)) = [1] x + [2]
>= [1] x + [1]
= minus(x,y)
quot(0(),s(y)) = [3]
>= [2]
= 0()
quot(s(x),s(y)) = [1] x + [2]
>= [1] x + [3]
= s(quot(minus(x,y),s(y)))
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^1))
+ Considered Problem:
- Strict TRS:
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Weak TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
quot(0(),s(y)) -> 0()
- Signature:
{minus/2,quot/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus,quot} and constructors {0,s}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
The following argument positions are considered usable:
uargs(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{minus,quot}
TcT has computed the following interpretation:
p(0) = [0]
p(minus) = [1] x_1 + [0]
p(quot) = [2] x_1 + [0]
p(s) = [1] x_1 + [8]
Following rules are strictly oriented:
quot(s(x),s(y)) = [2] x + [16]
> [2] x + [8]
= s(quot(minus(x,y),s(y)))
Following rules are (at-least) weakly oriented:
minus(x,0()) = [1] x + [0]
>= [1] x + [0]
= x
minus(s(x),s(y)) = [1] x + [8]
>= [1] x + [0]
= minus(x,y)
quot(0(),s(y)) = [0]
>= [0]
= 0()
** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{minus/2,quot/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus,quot} 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))