* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: D(b(x,y)) -> b(D(x),D(y)) D(c(x,y)) -> b(c(y,D(x)),c(x,D(y))) D(constant()) -> h() D(div(x,y)) -> m(div(D(x),y),div(c(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(m(x,y)) -> m(D(x),D(y)) D(opp(x)) -> opp(D(x)) D(pow(x,y)) -> b(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))) D(t()) -> s(h()) b(x,h()) -> x b(b(x,y),z) -> b(x,b(y,z)) b(h(),x) -> x b(s(x),s(y)) -> s(s(b(x,y))) - Signature: {D/1,b/2} / {1/0,2/0,c/2,constant/0,div/2,h/0,ln/1,m/2,opp/1,pow/2,s/1,t/0} - Obligation: innermost runtime complexity wrt. defined symbols {D,b} and constructors {1,2,c,constant,div,h,ln,m,opp,pow ,s,t} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: D(b(x,y)) -> b(D(x),D(y)) D(c(x,y)) -> b(c(y,D(x)),c(x,D(y))) D(constant()) -> h() D(div(x,y)) -> m(div(D(x),y),div(c(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(m(x,y)) -> m(D(x),D(y)) D(opp(x)) -> opp(D(x)) D(pow(x,y)) -> b(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))) D(t()) -> s(h()) b(x,h()) -> x b(b(x,y),z) -> b(x,b(y,z)) b(h(),x) -> x b(s(x),s(y)) -> s(s(b(x,y))) - Signature: {D/1,b/2} / {1/0,2/0,c/2,constant/0,div/2,h/0,ln/1,m/2,opp/1,pow/2,s/1,t/0} - Obligation: innermost runtime complexity wrt. defined symbols {D,b} and constructors {1,2,c,constant,div,h,ln,m,opp,pow ,s,t} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: D(x){x -> c(x,y)} = D(c(x,y)) ->^+ b(c(y,D(x)),c(x,D(y))) = C[D(x) = D(x){}] WORST_CASE(Omega(n^1),?)