* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: +(x,#()) -> x +(#(),x) -> x +(+(x,y),z) -> +(x,+(y,z)) +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() ge(x,#()) -> true() ge(#(),0(x)) -> ge(#(),x) ge(#(),1(x)) -> false() ge(0(x),0(y)) -> ge(x,y) ge(0(x),1(y)) -> not(ge(y,x)) ge(1(x),0(y)) -> ge(x,y) ge(1(x),1(y)) -> ge(x,y) if(false(),x,y) -> y if(true(),x,y) -> x log(x) -> -(log'(x),1(#())) log'(#()) -> #() log'(0(x)) -> if(ge(x,1(#())),+(log'(x),1(#())),#()) log'(1(x)) -> +(log'(x),1(#())) not(false()) -> true() not(true()) -> false() - Signature: {+/2,-/2,0/1,ge/2,if/3,log/1,log'/1,not/1} / {#/0,1/1,false/0,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {+,-,0,ge,if,log,log',not} and constructors {#,1,false ,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: +(x,#()) -> x +(#(),x) -> x +(+(x,y),z) -> +(x,+(y,z)) +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() ge(x,#()) -> true() ge(#(),0(x)) -> ge(#(),x) ge(#(),1(x)) -> false() ge(0(x),0(y)) -> ge(x,y) ge(0(x),1(y)) -> not(ge(y,x)) ge(1(x),0(y)) -> ge(x,y) ge(1(x),1(y)) -> ge(x,y) if(false(),x,y) -> y if(true(),x,y) -> x log(x) -> -(log'(x),1(#())) log'(#()) -> #() log'(0(x)) -> if(ge(x,1(#())),+(log'(x),1(#())),#()) log'(1(x)) -> +(log'(x),1(#())) not(false()) -> true() not(true()) -> false() - Signature: {+/2,-/2,0/1,ge/2,if/3,log/1,log'/1,not/1} / {#/0,1/1,false/0,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {+,-,0,ge,if,log,log',not} and constructors {#,1,false ,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: +(x,y){x -> 1(x),y -> 1(y)} = +(1(x),1(y)) ->^+ 0(+(+(x,y),1(#()))) = C[+(x,y) = +(x,y){}] WORST_CASE(Omega(n^1),?)