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