WORST_CASE(?,O(n^2)) * Step 1: Ara WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: if_mod(false(),s(x),s(y)) -> s(x) if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) mod(0(),y) -> 0() mod(s(x),0()) -> 0() mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y)) - Signature: {if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if_mod,le,minus,mod} and constructors {0,false,s,true} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 1, maxDegree = 3, araTimeout = 60, araFindStrictRules = Nothing, araSmtSolver = MiniSMT} + Details: Signatures used: ---------------- 0 :: [] -(0)-> A(0, 0) 0 :: [] -(0)-> A(1, 0) 0 :: [] -(0)-> A(4, 3) 0 :: [] -(0)-> A(2, 0) 0 :: [] -(0)-> A(0, 1) false :: [] -(0)-> A(0, 0) false :: [] -(0)-> A(3, 2) if_mod :: [A(0, 0) x A(1, 3) x A(1, 0)] -(4)-> A(0, 0) le :: [A(0, 0) x A(1, 0)] -(1)-> A(2, 1) minus :: [A(4, 3) x A(0, 0)] -(1)-> A(4, 3) mod :: [A(4, 3) x A(1, 0)] -(3)-> A(0, 0) s :: [A(4, 3)] -(1)-> A(1, 3) s :: [A(1, 0)] -(1)-> A(1, 0) s :: [A(0, 0)] -(0)-> A(0, 0) s :: [A(7, 3)] -(4)-> A(4, 3) s :: [A(2, 1)] -(1)-> A(1, 1) true :: [] -(0)-> A(0, 0) true :: [] -(0)-> A(2, 1) Cost-free Signatures used: -------------------------- 0 :: [] -(0)-> A_cf(0, 0) 0 :: [] -(0)-> A_cf(1, 0) 0 :: [] -(0)-> A_cf(1, 2) false :: [] -(0)-> A_cf(0, 0) false :: [] -(0)-> A_cf(0, 1) false :: [] -(0)-> A_cf(4, 2) if_mod :: [A_cf(0, 0) x A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) if_mod :: [A_cf(0, 1) x A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) le :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) le :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(4, 2) minus :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) minus :: [A_cf(3, 0) x A_cf(0, 0)] -(3)-> A_cf(0, 0) mod :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) s :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) s :: [A_cf(3, 0)] -(3)-> A_cf(3, 0) true :: [] -(0)-> A_cf(0, 0) true :: [] -(0)-> A_cf(0, 1) true :: [] -(0)-> A_cf(4, 2) Base Constructor Signatures used: --------------------------------- 0_A :: [] -(0)-> A(1, 0) 0_A :: [] -(0)-> A(0, 1) false_A :: [] -(0)-> A(1, 0) false_A :: [] -(0)-> A(0, 1) s_A :: [A(1, 0)] -(1)-> A(1, 0) s_A :: [A(1, 1)] -(0)-> A(0, 1) true_A :: [] -(0)-> A(1, 0) true_A :: [] -(0)-> A(0, 1) * Step 2: Open MAYBE - Strict TRS: if_mod(false(),s(x),s(y)) -> s(x) if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) mod(0(),y) -> 0() mod(s(x),0()) -> 0() mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y)) - Signature: {if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if_mod,le,minus,mod} and constructors {0,false,s,true} Following problems could not be solved: - Strict TRS: if_mod(false(),s(x),s(y)) -> s(x) if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) mod(0(),y) -> 0() mod(s(x),0()) -> 0() mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y)) - Signature: {if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if_mod,le,minus,mod} and constructors {0,false,s,true} WORST_CASE(?,O(n^2))