* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
*(#(),x) -> #()
*(*(x,y),z) -> *(x,*(y,z))
*(0(x),y) -> 0(*(x,y))
*(1(x),y) -> +(0(*(x,y)),y)
*(j(x),y) -> -(0(*(x,y)),y)
+(x,#()) -> x
+(#(),x) -> x
+(+(x,y),z) -> +(x,+(y,z))
+(0(x),0(y)) -> 0(+(x,y))
+(0(x),1(y)) -> 1(+(x,y))
+(0(x),j(y)) -> j(+(x,y))
+(1(x),0(y)) -> 1(+(x,y))
+(1(x),1(y)) -> j(+(+(x,y),1(#())))
+(1(x),j(y)) -> 0(+(x,y))
+(j(x),0(y)) -> j(+(x,y))
+(j(x),1(y)) -> 0(+(x,y))
+(j(x),j(y)) -> 1(+(+(x,y),j(#())))
-(x,y) -> +(x,opp(y))
0(#()) -> #()
opp(#()) -> #()
opp(0(x)) -> 0(opp(x))
opp(1(x)) -> j(opp(x))
opp(j(x)) -> 1(opp(x))
- Signature:
{*/2,+/2,-/2,0/1,opp/1} / {#/0,1/1,j/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*,+,-,0,opp} and constructors {#,1,j}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
*(#(),x) -> #()
*(*(x,y),z) -> *(x,*(y,z))
*(0(x),y) -> 0(*(x,y))
*(1(x),y) -> +(0(*(x,y)),y)
*(j(x),y) -> -(0(*(x,y)),y)
+(x,#()) -> x
+(#(),x) -> x
+(+(x,y),z) -> +(x,+(y,z))
+(0(x),0(y)) -> 0(+(x,y))
+(0(x),1(y)) -> 1(+(x,y))
+(0(x),j(y)) -> j(+(x,y))
+(1(x),0(y)) -> 1(+(x,y))
+(1(x),1(y)) -> j(+(+(x,y),1(#())))
+(1(x),j(y)) -> 0(+(x,y))
+(j(x),0(y)) -> j(+(x,y))
+(j(x),1(y)) -> 0(+(x,y))
+(j(x),j(y)) -> 1(+(+(x,y),j(#())))
-(x,y) -> +(x,opp(y))
0(#()) -> #()
opp(#()) -> #()
opp(0(x)) -> 0(opp(x))
opp(1(x)) -> j(opp(x))
opp(j(x)) -> 1(opp(x))
- Signature:
{*/2,+/2,-/2,0/1,opp/1} / {#/0,1/1,j/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*,+,-,0,opp} and constructors {#,1,j}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
*(x,y){x -> 1(x)} =
*(1(x),y) ->^+ +(0(*(x,y)),y)
= C[*(x,y) = *(x,y){}]
WORST_CASE(Omega(n^1),?)