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