* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: active(diff(X,Y)) -> mark(if(leq(X,Y),0(),s(diff(p(X),Y)))) active(diff(X1,X2)) -> diff(X1,active(X2)) active(diff(X1,X2)) -> diff(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(false(),X,Y)) -> mark(Y) active(if(true(),X,Y)) -> mark(X) active(leq(X1,X2)) -> leq(X1,active(X2)) active(leq(X1,X2)) -> leq(active(X1),X2) active(leq(0(),Y)) -> mark(true()) active(leq(s(X),0())) -> mark(false()) active(leq(s(X),s(Y))) -> mark(leq(X,Y)) active(p(X)) -> p(active(X)) active(p(0())) -> mark(0()) active(p(s(X))) -> mark(X) active(s(X)) -> s(active(X)) diff(X1,mark(X2)) -> mark(diff(X1,X2)) diff(mark(X1),X2) -> mark(diff(X1,X2)) diff(ok(X1),ok(X2)) -> ok(diff(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) leq(X1,mark(X2)) -> mark(leq(X1,X2)) leq(mark(X1),X2) -> mark(leq(X1,X2)) leq(ok(X1),ok(X2)) -> ok(leq(X1,X2)) p(mark(X)) -> mark(p(X)) p(ok(X)) -> ok(p(X)) proper(0()) -> ok(0()) proper(diff(X1,X2)) -> diff(proper(X1),proper(X2)) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(leq(X1,X2)) -> leq(proper(X1),proper(X2)) proper(p(X)) -> p(proper(X)) proper(s(X)) -> s(proper(X)) proper(true()) -> ok(true()) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,diff/2,if/3,leq/2,p/1,proper/1,s/1,top/1} / {0/0,false/0,mark/1,ok/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {active,diff,if,leq,p,proper,s,top} and constructors {0 ,false,mark,ok,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: active(diff(X,Y)) -> mark(if(leq(X,Y),0(),s(diff(p(X),Y)))) active(diff(X1,X2)) -> diff(X1,active(X2)) active(diff(X1,X2)) -> diff(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(false(),X,Y)) -> mark(Y) active(if(true(),X,Y)) -> mark(X) active(leq(X1,X2)) -> leq(X1,active(X2)) active(leq(X1,X2)) -> leq(active(X1),X2) active(leq(0(),Y)) -> mark(true()) active(leq(s(X),0())) -> mark(false()) active(leq(s(X),s(Y))) -> mark(leq(X,Y)) active(p(X)) -> p(active(X)) active(p(0())) -> mark(0()) active(p(s(X))) -> mark(X) active(s(X)) -> s(active(X)) diff(X1,mark(X2)) -> mark(diff(X1,X2)) diff(mark(X1),X2) -> mark(diff(X1,X2)) diff(ok(X1),ok(X2)) -> ok(diff(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) leq(X1,mark(X2)) -> mark(leq(X1,X2)) leq(mark(X1),X2) -> mark(leq(X1,X2)) leq(ok(X1),ok(X2)) -> ok(leq(X1,X2)) p(mark(X)) -> mark(p(X)) p(ok(X)) -> ok(p(X)) proper(0()) -> ok(0()) proper(diff(X1,X2)) -> diff(proper(X1),proper(X2)) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(leq(X1,X2)) -> leq(proper(X1),proper(X2)) proper(p(X)) -> p(proper(X)) proper(s(X)) -> s(proper(X)) proper(true()) -> ok(true()) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,diff/2,if/3,leq/2,p/1,proper/1,s/1,top/1} / {0/0,false/0,mark/1,ok/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {active,diff,if,leq,p,proper,s,top} and constructors {0 ,false,mark,ok,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: diff(x,y){y -> mark(y)} = diff(x,mark(y)) ->^+ mark(diff(x,y)) = C[diff(x,y) = diff(x,y){}] WORST_CASE(Omega(n^1),?)