* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a() -> b() a() -> c() gcd(x,y) -> gcd2(x,y,0()) gcd2(x,y,i) -> if1(le(x,0()),le(y,0()),le(x,y),le(y,x),x,y,inc(i)) if1(false(),b1,b2,b3,x,y,i) -> if2(b1,b2,b3,x,y,i) if1(true(),b1,b2,b3,x,y,i) -> pair(result(y),neededIterations(i)) if2(false(),b2,b3,x,y,i) -> if3(b2,b3,x,y,i) if2(true(),b2,b3,x,y,i) -> pair(result(x),neededIterations(i)) if3(false(),b3,x,y,i) -> gcd2(minus(x,y),y,i) if3(true(),b3,x,y,i) -> if4(b3,x,y,i) if4(false(),x,y,i) -> gcd2(x,minus(y,x),i) if4(true(),x,y,i) -> pair(result(x),neededIterations(i)) inc(0()) -> 0() inc(s(i)) -> s(inc(i)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(0(),y) -> 0() minus(s(x),s(y)) -> minus(x,y) - Signature: {a/0,gcd/2,gcd2/3,if1/7,if2/6,if3/5,if4/4,inc/1,le/2,minus/2} / {0/0,b/0,c/0,false/0,neededIterations/1 ,pair/2,result/1,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a,gcd,gcd2,if1,if2,if3,if4,inc,le ,minus} and constructors {0,b,c,false,neededIterations,pair,result,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a() -> b() a() -> c() gcd(x,y) -> gcd2(x,y,0()) gcd2(x,y,i) -> if1(le(x,0()),le(y,0()),le(x,y),le(y,x),x,y,inc(i)) if1(false(),b1,b2,b3,x,y,i) -> if2(b1,b2,b3,x,y,i) if1(true(),b1,b2,b3,x,y,i) -> pair(result(y),neededIterations(i)) if2(false(),b2,b3,x,y,i) -> if3(b2,b3,x,y,i) if2(true(),b2,b3,x,y,i) -> pair(result(x),neededIterations(i)) if3(false(),b3,x,y,i) -> gcd2(minus(x,y),y,i) if3(true(),b3,x,y,i) -> if4(b3,x,y,i) if4(false(),x,y,i) -> gcd2(x,minus(y,x),i) if4(true(),x,y,i) -> pair(result(x),neededIterations(i)) inc(0()) -> 0() inc(s(i)) -> s(inc(i)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(0(),y) -> 0() minus(s(x),s(y)) -> minus(x,y) - Signature: {a/0,gcd/2,gcd2/3,if1/7,if2/6,if3/5,if4/4,inc/1,le/2,minus/2} / {0/0,b/0,c/0,false/0,neededIterations/1 ,pair/2,result/1,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a,gcd,gcd2,if1,if2,if3,if4,inc,le ,minus} and constructors {0,b,c,false,neededIterations,pair,result,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: inc(x){x -> s(x)} = inc(s(x)) ->^+ s(inc(x)) = C[inc(x) = inc(x){}] WORST_CASE(Omega(n^1),?)