* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: cadr(cons(x,cons(y,l))) -> y car(cons(x,l)) -> x cddr(cons(x,cons(y,l))) -> l cddr(cons(x,nil())) -> nil() cddr(nil()) -> nil() if(false(),b,l) -> if2(b,l) if(true(),b,l) -> s(0()) if2(false(),l) -> prod(cons(times(car(l),cadr(l)),cddr(l))) if2(true(),l) -> car(l) ifplus(false(),x,y) -> s(plus(p(x),y)) ifplus(true(),x,y) -> y iftimes(false(),x,y) -> plus(y,times(p(x),y)) iftimes(true(),x,y) -> 0() isZero(0()) -> true() isZero(s(x)) -> false() p(0()) -> 0() p(s(x)) -> x plus(x,y) -> ifplus(isZero(x),x,y) prod(l) -> if(shorter(l,0()),shorter(l,s(0())),l) shorter(cons(x,l),0()) -> false() shorter(cons(x,l),s(y)) -> shorter(l,y) shorter(nil(),y) -> true() times(x,y) -> iftimes(isZero(x),x,y) - Signature: {cadr/1,car/1,cddr/1,if/3,if2/2,ifplus/3,iftimes/3,isZero/1,p/1,plus/2,prod/1,shorter/2,times/2} / {0/0 ,cons/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {cadr,car,cddr,if,if2,ifplus,iftimes,isZero,p,plus,prod ,shorter,times} and constructors {0,cons,false,nil,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: cadr(cons(x,cons(y,l))) -> y car(cons(x,l)) -> x cddr(cons(x,cons(y,l))) -> l cddr(cons(x,nil())) -> nil() cddr(nil()) -> nil() if(false(),b,l) -> if2(b,l) if(true(),b,l) -> s(0()) if2(false(),l) -> prod(cons(times(car(l),cadr(l)),cddr(l))) if2(true(),l) -> car(l) ifplus(false(),x,y) -> s(plus(p(x),y)) ifplus(true(),x,y) -> y iftimes(false(),x,y) -> plus(y,times(p(x),y)) iftimes(true(),x,y) -> 0() isZero(0()) -> true() isZero(s(x)) -> false() p(0()) -> 0() p(s(x)) -> x plus(x,y) -> ifplus(isZero(x),x,y) prod(l) -> if(shorter(l,0()),shorter(l,s(0())),l) shorter(cons(x,l),0()) -> false() shorter(cons(x,l),s(y)) -> shorter(l,y) shorter(nil(),y) -> true() times(x,y) -> iftimes(isZero(x),x,y) - Signature: {cadr/1,car/1,cddr/1,if/3,if2/2,ifplus/3,iftimes/3,isZero/1,p/1,plus/2,prod/1,shorter/2,times/2} / {0/0 ,cons/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {cadr,car,cddr,if,if2,ifplus,iftimes,isZero,p,plus,prod ,shorter,times} and constructors {0,cons,false,nil,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: shorter(y,z){y -> cons(x,y),z -> s(z)} = shorter(cons(x,y),s(z)) ->^+ shorter(y,z) = C[shorter(y,z) = shorter(y,z){}] WORST_CASE(Omega(n^1),?)