WORST_CASE(?,O(n^2)) * Step 1: Ara WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: if(false(),x,y) -> y if(true(),x,y) -> x le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y))))) p(0()) -> 0() p(s(x)) -> x - Signature: {if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if,le,minus,p} and constructors {0,false,s,true} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 1, maxDegree = 3, araTimeout = 60, araFindStrictRules = Nothing, araSmtSolver = Z3} + Details: Signatures used: ---------------- 0 :: [] -(0)-> A(0, 0) 0 :: [] -(0)-> A(1, 0) 0 :: [] -(0)-> A(7, 7) 0 :: [] -(0)-> A(0, 7) 0 :: [] -(0)-> A(15, 15) 0 :: [] -(0)-> A(13, 13) false :: [] -(0)-> A(0, 0) false :: [] -(0)-> A(14, 14) if :: [A(0, 0) x A(5, 7) x A(6, 7)] -(2)-> A(5, 7) le :: [A(0, 0) x A(1, 0)] -(1)-> A(14, 14) minus :: [A(14, 13) x A(7, 7)] -(8)-> A(4, 7) p :: [A(0, 7)] -(1)-> A(7, 7) s :: [A(0, 0)] -(0)-> A(0, 0) s :: [A(1, 0)] -(1)-> A(1, 0) s :: [A(14, 7)] -(7)-> A(7, 7) s :: [A(7, 7)] -(0)-> A(0, 7) true :: [] -(0)-> A(0, 0) true :: [] -(0)-> A(14, 14) Cost-free Signatures used: -------------------------- 0 :: [] -(0)-> A_cf(0, 0) false :: [] -(0)-> A_cf(0, 0) if :: [A_cf(0, 0) 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) minus :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) p :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) s :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) true :: [] -(0)-> A_cf(0, 0) 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(false(),x,y) -> y if(true(),x,y) -> x le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y))))) p(0()) -> 0() p(s(x)) -> x - Signature: {if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if,le,minus,p} and constructors {0,false,s,true} Following problems could not be solved: - Strict TRS: if(false(),x,y) -> y if(true(),x,y) -> x le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y))))) p(0()) -> 0() p(s(x)) -> x - Signature: {if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if,le,minus,p} and constructors {0,false,s,true} WORST_CASE(?,O(n^2))