* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: equal0(a,b) -> equal0[Ite](<(a,b),a,b) gcd(x,y) -> gcd[Ite](equal0(x,y),x,y) monus(S(x'),S(x)) -> monus(x',x) - Weak TRS: <(x,0()) -> False() <(0(),S(y)) -> True() <(S(x),S(y)) -> <(x,y) equal0[Ite](False(),a,b) -> False() equal0[Ite](True(),a,b) -> equal0[True][Ite](<(b,a),a,b) equal0[True][Ite](False(),a,b) -> False() equal0[True][Ite](True(),a,b) -> True() gcd[False][Ite](False(),x,y) -> gcd(y,monus(y,x)) gcd[False][Ite](True(),x,y) -> gcd(monus(x,y),y) gcd[Ite](False(),x,y) -> gcd[False][Ite](<(x,y),x,y) gcd[Ite](True(),x,y) -> x - Signature: {2,equal0/2,equal0[Ite]/3,equal0[True][Ite]/3,gcd/2,gcd[False][Ite]/3,gcd[Ite]/3,monus/2} / {0/0,False/0 ,S/1,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {<,equal0,equal0[Ite],equal0[True][Ite],gcd ,gcd[False][Ite],gcd[Ite],monus} and constructors {0,False,S,True} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: equal0(a,b) -> equal0[Ite](<(a,b),a,b) gcd(x,y) -> gcd[Ite](equal0(x,y),x,y) monus(S(x'),S(x)) -> monus(x',x) - Weak TRS: <(x,0()) -> False() <(0(),S(y)) -> True() <(S(x),S(y)) -> <(x,y) equal0[Ite](False(),a,b) -> False() equal0[Ite](True(),a,b) -> equal0[True][Ite](<(b,a),a,b) equal0[True][Ite](False(),a,b) -> False() equal0[True][Ite](True(),a,b) -> True() gcd[False][Ite](False(),x,y) -> gcd(y,monus(y,x)) gcd[False][Ite](True(),x,y) -> gcd(monus(x,y),y) gcd[Ite](False(),x,y) -> gcd[False][Ite](<(x,y),x,y) gcd[Ite](True(),x,y) -> x - Signature: {2,equal0/2,equal0[Ite]/3,equal0[True][Ite]/3,gcd/2,gcd[False][Ite]/3,gcd[Ite]/3,monus/2} / {0/0,False/0 ,S/1,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {<,equal0,equal0[Ite],equal0[True][Ite],gcd ,gcd[False][Ite],gcd[Ite],monus} and constructors {0,False,S,True} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: monus(x,y){x -> S(x),y -> S(y)} = monus(S(x),S(y)) ->^+ monus(x,y) = C[monus(x,y) = monus(x,y){}] WORST_CASE(Omega(n^1),?)