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