WORST_CASE(?,O(n^1)) * Step 1: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if ,mark} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 1, maxDegree = 3, araTimeout = 60, araFindStrictRules = Nothing, araSmtSolver = Z3} + Details: Signatures used: ---------------- 0 :: [] -(0)-> A(5) 0 :: [] -(0)-> A(15) 0 :: [] -(0)-> A(11) a__add :: [A(5) x A(15)] -(8)-> A(5) a__and :: [A(5) x A(15)] -(9)-> A(5) a__first :: [A(5) x A(5)] -(6)-> A(5) a__from :: [A(11)] -(15)-> A(8) a__if :: [A(5) x A(15) x A(15)] -(6)-> A(5) add :: [A(15) x A(15)] -(15)-> A(15) add :: [A(5) x A(5)] -(5)-> A(5) add :: [A(0) x A(0)] -(0)-> A(0) and :: [A(15) x A(15)] -(15)-> A(15) and :: [A(5) x A(5)] -(5)-> A(5) cons :: [A(5) x A(0)] -(5)-> A(5) cons :: [A(15) x A(0)] -(15)-> A(15) cons :: [A(8) x A(0)] -(8)-> A(8) false :: [] -(0)-> A(5) false :: [] -(0)-> A(15) false :: [] -(0)-> A(11) first :: [A(15) x A(15)] -(15)-> A(15) first :: [A(5) x A(5)] -(5)-> A(5) first :: [A(0) x A(0)] -(0)-> A(0) from :: [A(15)] -(15)-> A(15) from :: [A(0)] -(0)-> A(0) from :: [A(11)] -(11)-> A(11) if :: [A(15) x A(15) x A(15)] -(15)-> A(15) if :: [A(5) x A(5) x A(5)] -(5)-> A(5) mark :: [A(15)] -(1)-> A(5) nil :: [] -(0)-> A(15) nil :: [] -(0)-> A(11) s :: [A(0)] -(5)-> A(5) s :: [A(0)] -(15)-> A(15) s :: [A(0)] -(6)-> A(6) s :: [A(0)] -(0)-> A(0) true :: [] -(0)-> A(5) true :: [] -(0)-> A(15) true :: [] -(0)-> A(11) Cost-free Signatures used: -------------------------- 0 :: [] -(0)-> A_cf(0) a__add :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) a__and :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) a__first :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) a__from :: [A_cf(0)] -(0)-> A_cf(0) a__if :: [A_cf(0) x A_cf(0) x A_cf(0)] -(0)-> A_cf(0) add :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) and :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) cons :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) false :: [] -(0)-> A_cf(0) first :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) from :: [A_cf(0)] -(0)-> A_cf(0) if :: [A_cf(0) x A_cf(0) x A_cf(0)] -(0)-> A_cf(0) mark :: [A_cf(0)] -(0)-> A_cf(0) nil :: [] -(0)-> A_cf(0) s :: [A_cf(0)] -(0)-> A_cf(0) true :: [] -(0)-> A_cf(0) Base Constructor Signatures used: --------------------------------- 0_A :: [] -(0)-> A(1) add_A :: [A(1) x A(1)] -(1)-> A(1) and_A :: [A(1) x A(1)] -(1)-> A(1) cons_A :: [A(1) x A(0)] -(1)-> A(1) false_A :: [] -(0)-> A(1) first_A :: [A(1) x A(1)] -(1)-> A(1) from_A :: [A(1)] -(1)-> A(1) if_A :: [A(1) x A(1) x A(1)] -(1)-> A(1) nil_A :: [] -(0)-> A(1) s_A :: [A(0)] -(1)-> A(1) true_A :: [] -(0)-> A(1) * Step 2: Open MAYBE - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if ,mark} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} Following problems could not be solved: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if ,mark} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} WORST_CASE(?,O(n^1))