* 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),?)