WORST_CASE(?,O(n^1)) * Step 1: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: dx(X) -> one() dx(a()) -> zero() dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two())))) dx(exp(ALPHA,BETA)) -> plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))) ,times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA)))) dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA) dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA)) dx(neg(ALPHA)) -> neg(dx(ALPHA)) dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA)) dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA))) - Signature: {dx/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0} - Obligation: innermost runtime complexity wrt. defined symbols {dx} and constructors {a,div,exp,ln,minus,neg,one,plus ,times,two,zero} + Applied Processor: Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 5, araRuleShifting = Nothing} + Details: Signatures used: ---------------- a :: [] -(0)-> "A"(15) div :: ["A"(15) x "A"(15)] -(15)-> "A"(15) div :: ["A"(0) x "A"(0)] -(0)-> "A"(0) dx :: ["A"(15)] -(8)-> "A"(0) exp :: ["A"(15) x "A"(15)] -(15)-> "A"(15) exp :: ["A"(0) x "A"(0)] -(0)-> "A"(0) ln :: ["A"(15)] -(15)-> "A"(15) ln :: ["A"(0)] -(0)-> "A"(0) minus :: ["A"(15) x "A"(15)] -(15)-> "A"(15) minus :: ["A"(0) x "A"(0)] -(0)-> "A"(0) neg :: ["A"(15)] -(15)-> "A"(15) neg :: ["A"(0)] -(0)-> "A"(0) one :: [] -(0)-> "A"(6) one :: [] -(0)-> "A"(12) plus :: ["A"(15) x "A"(15)] -(15)-> "A"(15) plus :: ["A"(0) x "A"(0)] -(0)-> "A"(0) times :: ["A"(15) x "A"(15)] -(15)-> "A"(15) times :: ["A"(0) x "A"(0)] -(0)-> "A"(0) two :: [] -(0)-> "A"(12) zero :: [] -(0)-> "A"(6) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "a_A" :: [] -(0)-> "A"(1) "div_A" :: ["A"(1) x "A"(1)] -(1)-> "A"(1) "exp_A" :: ["A"(1) x "A"(1)] -(1)-> "A"(1) "ln_A" :: ["A"(1)] -(1)-> "A"(1) "minus_A" :: ["A"(1) x "A"(1)] -(1)-> "A"(1) "neg_A" :: ["A"(1)] -(1)-> "A"(1) "one_A" :: [] -(0)-> "A"(1) "plus_A" :: ["A"(1) x "A"(1)] -(1)-> "A"(1) "times_A" :: ["A"(1) x "A"(1)] -(1)-> "A"(1) "two_A" :: [] -(0)-> "A"(1) "zero_A" :: [] -(0)-> "A"(1) WORST_CASE(?,O(n^1))