* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
+ Considered Problem:
- Strict TRS:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
times(x,0()) -> 0()
times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0()))),times(x,s(z)))
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {plus,times} 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:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
times(x,0()) -> 0()
times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0()))),times(x,s(z)))
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {plus,times} and constructors {0,s}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
plus(x,y){y -> s(y)} =
plus(x,s(y)) ->^+ s(plus(x,y))
= C[plus(x,y) = plus(x,y){}]
** Step 1.b:1: InnermostRuleRemoval WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
times(x,0()) -> 0()
times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0()))),times(x,s(z)))
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {plus,times} and constructors {0,s}
+ Applied Processor:
InnermostRuleRemoval
+ Details:
Arguments of following rules are not normal-forms.
times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0()))),times(x,s(z)))
All above mentioned rules can be savely removed.
** Step 1.b:2: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
times(x,0()) -> 0()
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {plus,times} 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(plus) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(plus) = [1] x1 + [0]
p(s) = [1] x1 + [1]
p(times) = [12] x1 + [8] x2 + [3]
Following rules are strictly oriented:
times(x,0()) = [12] x + [3]
> [0]
= 0()
times(x,s(y)) = [12] x + [8] y + [11]
> [12] x + [8] y + [3]
= plus(times(x,y),x)
Following rules are (at-least) weakly oriented:
plus(x,0()) = [1] x + [0]
>= [1] x + [0]
= x
plus(x,s(y)) = [1] x + [0]
>= [1] x + [1]
= s(plus(x,y))
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:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
- Weak TRS:
times(x,0()) -> 0()
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {plus,times} 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(plus) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [2]
p(plus) = [1] x1 + [2]
p(s) = [1] x1 + [1]
p(times) = [1] x1 + [4] x2 + [8]
Following rules are strictly oriented:
plus(x,0()) = [1] x + [2]
> [1] x + [0]
= x
Following rules are (at-least) weakly oriented:
plus(x,s(y)) = [1] x + [2]
>= [1] x + [3]
= s(plus(x,y))
times(x,0()) = [1] x + [16]
>= [2]
= 0()
times(x,s(y)) = [1] x + [4] y + [12]
>= [1] x + [4] y + [10]
= plus(times(x,y),x)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:4: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
plus(x,s(y)) -> s(plus(x,y))
- Weak TRS:
plus(x,0()) -> x
times(x,0()) -> 0()
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {plus,times} 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(plus) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{plus,times}
TcT has computed the following interpretation:
p(0) = 0
p(plus) = x1 + 3*x2
p(s) = 1 + x1
p(times) = 2*x1 + 3*x1*x2 + x1^2
Following rules are strictly oriented:
plus(x,s(y)) = 3 + x + 3*y
> 1 + x + 3*y
= s(plus(x,y))
Following rules are (at-least) weakly oriented:
plus(x,0()) = x
>= x
= x
times(x,0()) = 2*x + x^2
>= 0
= 0()
times(x,s(y)) = 5*x + 3*x*y + x^2
>= 5*x + 3*x*y + x^2
= plus(times(x,y),x)
** Step 1.b:5: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
times(x,0()) -> 0()
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {plus,times} 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))