WORST_CASE(?,O(n^1)) * Step 1: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1))) 2nd(cons1(X,cons(Y,Z))) -> Y activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2nd/1,activate/1,from/1,s/1} / {cons/2,cons1/2,n__from/1,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,from,s} and constructors {cons,cons1,n__from ,n__s} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 1, maxDegree = 1, araTimeout = 15, araFindStrictRules = Just 1} + Details: Signatures used: ---------------- 2nd :: [A(14)] -(14)-> A(0) activate :: [A(8)] -(5)-> A(0) cons :: [A(14) x A(14)] -(14)-> A(14) cons :: [A(0) x A(0)] -(0)-> A(0) cons1 :: [A(14) x A(0)] -(0)-> A(14) from :: [A(0)] -(3)-> A(0) n__from :: [A(8)] -(8)-> A(8) n__from :: [A(0)] -(0)-> A(0) n__s :: [A(8)] -(8)-> A(8) n__s :: [A(0)] -(0)-> A(0) s :: [A(0)] -(0)-> A(0) Cost-free Signatures used: -------------------------- 2nd :: [A_cf(0)] -(0)-> A_cf(0) activate :: [A_cf(0)] -(0)-> A_cf(0) cons :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) cons1 :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) from :: [A_cf(0)] -(0)-> A_cf(0) n__from :: [A_cf(0)] -(0)-> A_cf(0) n__s :: [A_cf(0)] -(0)-> A_cf(0) s :: [A_cf(0)] -(0)-> A_cf(0) Base Constructor Signatures used: --------------------------------- cons1_A :: [A(1) x A(0)] -(0)-> A(1) cons_A :: [A(1) x A(1)] -(1)-> A(1) n__from_A :: [A(0)] -(1)-> A(1) n__s_A :: [A(0)] -(1)-> A(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: activate(n__s(X)) -> s(activate(X)) from(X) -> n__from(X) 2. Weak: 2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1))) 2nd(cons1(X,cons(Y,Z))) -> Y activate(X) -> X activate(n__from(X)) -> from(activate(X)) from(X) -> cons(X,n__from(n__s(X))) s(X) -> n__s(X) * Step 2: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1))) 2nd(cons1(X,cons(Y,Z))) -> Y activate(X) -> X activate(n__from(X)) -> from(activate(X)) from(X) -> cons(X,n__from(n__s(X))) s(X) -> n__s(X) - Weak TRS: activate(n__s(X)) -> s(activate(X)) from(X) -> n__from(X) - Signature: {2nd/1,activate/1,from/1,s/1} / {cons/2,cons1/2,n__from/1,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,from,s} and constructors {cons,cons1,n__from ,n__s} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 1, maxDegree = 1, araTimeout = 15, araFindStrictRules = Just 1} + Details: Signatures used: ---------------- 2nd :: [A(13)] -(7)-> A(0) activate :: [A(0)] -(1)-> A(0) cons :: [A(13) x A(0)] -(13)-> A(13) cons :: [A(0) x A(0)] -(0)-> A(0) cons1 :: [A(0) x A(0)] -(0)-> A(13) cons1 :: [A(0) x A(0)] -(0)-> A(15) from :: [A(0)] -(0)-> A(0) n__from :: [A(0)] -(0)-> A(0) n__from :: [A(0)] -(0)-> A(9) n__from :: [A(0)] -(0)-> A(1) n__s :: [A(0)] -(0)-> A(0) n__s :: [A(0)] -(0)-> A(5) n__s :: [A(0)] -(0)-> A(14) s :: [A(0)] -(0)-> A(13) Cost-free Signatures used: -------------------------- 2nd :: [A_cf(0)] -(0)-> A_cf(0) activate :: [A_cf(0)] -(0)-> A_cf(0) cons :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) cons1 :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) from :: [A_cf(0)] -(0)-> A_cf(0) n__from :: [A_cf(0)] -(0)-> A_cf(0) n__s :: [A_cf(0)] -(0)-> A_cf(0) s :: [A_cf(0)] -(0)-> A_cf(0) Base Constructor Signatures used: --------------------------------- cons1_A :: [A(0) x A(0)] -(0)-> A(1) cons_A :: [A(1) x A(0)] -(1)-> A(1) n__from_A :: [A(0)] -(0)-> A(1) n__s_A :: [A(0)] -(0)-> A(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: 2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1))) 2nd(cons1(X,cons(Y,Z))) -> Y 2. Weak: activate(X) -> X activate(n__from(X)) -> from(activate(X)) from(X) -> cons(X,n__from(n__s(X))) s(X) -> n__s(X) * Step 3: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__from(X)) -> from(activate(X)) from(X) -> cons(X,n__from(n__s(X))) s(X) -> n__s(X) - Weak TRS: 2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1))) 2nd(cons1(X,cons(Y,Z))) -> Y activate(n__s(X)) -> s(activate(X)) from(X) -> n__from(X) - Signature: {2nd/1,activate/1,from/1,s/1} / {cons/2,cons1/2,n__from/1,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,from,s} and constructors {cons,cons1,n__from ,n__s} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 1, maxDegree = 1, araTimeout = 15, araFindStrictRules = Just 1} + Details: Signatures used: ---------------- 2nd :: [A(15)] -(7)-> A(0) activate :: [A(15)] -(8)-> A(0) cons :: [A(0) x A(15)] -(15)-> A(15) cons :: [A(0) x A(0)] -(0)-> A(0) cons1 :: [A(0) x A(0)] -(0)-> A(15) from :: [A(0)] -(15)-> A(0) n__from :: [A(15)] -(15)-> A(15) n__from :: [A(0)] -(0)-> A(0) n__s :: [A(15)] -(15)-> A(15) n__s :: [A(0)] -(0)-> A(0) s :: [A(0)] -(0)-> A(0) Cost-free Signatures used: -------------------------- 2nd :: [A_cf(0)] -(0)-> A_cf(0) activate :: [A_cf(0)] -(0)-> A_cf(0) cons :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) cons1 :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) from :: [A_cf(0)] -(0)-> A_cf(0) n__from :: [A_cf(0)] -(0)-> A_cf(0) n__s :: [A_cf(0)] -(0)-> A_cf(0) s :: [A_cf(0)] -(0)-> A_cf(0) Base Constructor Signatures used: --------------------------------- cons1_A :: [A(0) x A(0)] -(0)-> A(1) cons_A :: [A(0) x A(1)] -(1)-> A(1) n__from_A :: [A(0)] -(1)-> A(1) n__s_A :: [A(0)] -(1)-> A(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: activate(X) -> X 2. Weak: activate(n__from(X)) -> from(activate(X)) from(X) -> cons(X,n__from(n__s(X))) s(X) -> n__s(X) * Step 4: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(n__from(X)) -> from(activate(X)) from(X) -> cons(X,n__from(n__s(X))) s(X) -> n__s(X) - Weak TRS: 2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1))) 2nd(cons1(X,cons(Y,Z))) -> Y activate(X) -> X activate(n__s(X)) -> s(activate(X)) from(X) -> n__from(X) - Signature: {2nd/1,activate/1,from/1,s/1} / {cons/2,cons1/2,n__from/1,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,from,s} and constructors {cons,cons1,n__from ,n__s} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 1, maxDegree = 1, araTimeout = 15, araFindStrictRules = Just 1} + Details: Signatures used: ---------------- 2nd :: [A(15)] -(9)-> A(0) activate :: [A(11)] -(15)-> A(0) cons :: [A(0) x A(15)] -(15)-> A(15) cons :: [A(0) x A(0)] -(0)-> A(0) cons1 :: [A(0) x A(0)] -(0)-> A(15) from :: [A(0)] -(11)-> A(0) n__from :: [A(11)] -(11)-> A(11) n__from :: [A(0)] -(0)-> A(0) n__s :: [A(11)] -(11)-> A(11) n__s :: [A(0)] -(0)-> A(0) s :: [A(0)] -(4)-> A(0) Cost-free Signatures used: -------------------------- 2nd :: [A_cf(0)] -(0)-> A_cf(0) activate :: [A_cf(0)] -(0)-> A_cf(0) cons :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) cons1 :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) from :: [A_cf(0)] -(0)-> A_cf(0) n__from :: [A_cf(0)] -(0)-> A_cf(0) n__s :: [A_cf(0)] -(0)-> A_cf(0) s :: [A_cf(0)] -(0)-> A_cf(0) Base Constructor Signatures used: --------------------------------- cons1_A :: [A(0) x A(0)] -(0)-> A(1) cons_A :: [A(0) x A(1)] -(1)-> A(1) n__from_A :: [A(0)] -(1)-> A(1) n__s_A :: [A(0)] -(1)-> A(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: s(X) -> n__s(X) 2. Weak: activate(n__from(X)) -> from(activate(X)) from(X) -> cons(X,n__from(n__s(X))) * Step 5: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(n__from(X)) -> from(activate(X)) from(X) -> cons(X,n__from(n__s(X))) - Weak TRS: 2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1))) 2nd(cons1(X,cons(Y,Z))) -> Y activate(X) -> X activate(n__s(X)) -> s(activate(X)) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2nd/1,activate/1,from/1,s/1} / {cons/2,cons1/2,n__from/1,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,from,s} and constructors {cons,cons1,n__from ,n__s} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 1, maxDegree = 1, araTimeout = 15, araFindStrictRules = Just 1} + Details: Signatures used: ---------------- 2nd :: [A(11)] -(13)-> A(0) activate :: [A(11)] -(2)-> A(0) cons :: [A(0) x A(11)] -(11)-> A(11) cons :: [A(0) x A(0)] -(0)-> A(0) cons1 :: [A(0) x A(0)] -(0)-> A(11) cons1 :: [A(0) x A(0)] -(0)-> A(13) from :: [A(0)] -(7)-> A(0) n__from :: [A(11)] -(11)-> A(11) n__from :: [A(0)] -(0)-> A(0) n__s :: [A(11)] -(11)-> A(11) n__s :: [A(0)] -(0)-> A(0) s :: [A(0)] -(1)-> A(0) Cost-free Signatures used: -------------------------- 2nd :: [A_cf(0)] -(0)-> A_cf(0) activate :: [A_cf(0)] -(0)-> A_cf(0) cons :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) cons1 :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) from :: [A_cf(0)] -(0)-> A_cf(0) n__from :: [A_cf(0)] -(0)-> A_cf(0) n__s :: [A_cf(0)] -(0)-> A_cf(0) s :: [A_cf(0)] -(0)-> A_cf(0) Base Constructor Signatures used: --------------------------------- cons1_A :: [A(0) x A(0)] -(0)-> A(1) cons_A :: [A(0) x A(1)] -(1)-> A(1) n__from_A :: [A(0)] -(1)-> A(1) n__s_A :: [A(0)] -(1)-> A(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: from(X) -> cons(X,n__from(n__s(X))) 2. Weak: activate(n__from(X)) -> from(activate(X)) * Step 6: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(n__from(X)) -> from(activate(X)) - Weak TRS: 2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1))) 2nd(cons1(X,cons(Y,Z))) -> Y activate(X) -> X activate(n__s(X)) -> s(activate(X)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2nd/1,activate/1,from/1,s/1} / {cons/2,cons1/2,n__from/1,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,from,s} and constructors {cons,cons1,n__from ,n__s} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 1, maxDegree = 1, araTimeout = 15, araFindStrictRules = Just 1} + Details: Signatures used: ---------------- 2nd :: [A(15)] -(4)-> A(0) activate :: [A(15)] -(0)-> A(8) cons :: [A(0) x A(15)] -(0)-> A(15) cons :: [A(0) x A(0)] -(0)-> A(0) cons :: [A(0) x A(8)] -(0)-> A(8) cons1 :: [A(0) x A(0)] -(0)-> A(15) from :: [A(8)] -(10)-> A(8) n__from :: [A(15)] -(15)-> A(15) n__from :: [A(8)] -(8)-> A(8) n__s :: [A(15)] -(0)-> A(15) n__s :: [A(8)] -(0)-> A(8) s :: [A(8)] -(0)-> A(8) Cost-free Signatures used: -------------------------- 2nd :: [A_cf(0)] -(0)-> A_cf(0) activate :: [A_cf(0)] -(0)-> A_cf(0) cons :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) cons1 :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) from :: [A_cf(0)] -(0)-> A_cf(0) n__from :: [A_cf(0)] -(0)-> A_cf(0) n__s :: [A_cf(0)] -(0)-> A_cf(0) s :: [A_cf(0)] -(0)-> A_cf(0) Base Constructor Signatures used: --------------------------------- cons1_A :: [A(0) x A(0)] -(0)-> A(1) cons_A :: [A(0) x A(1)] -(0)-> A(1) n__from_A :: [A(0)] -(1)-> A(1) n__s_A :: [A(0)] -(0)-> A(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: activate(n__from(X)) -> from(activate(X)) 2. Weak: * Step 7: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1))) 2nd(cons1(X,cons(Y,Z))) -> Y activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2nd/1,activate/1,from/1,s/1} / {cons/2,cons1/2,n__from/1,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,from,s} and constructors {cons,cons1,n__from ,n__s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))