WORST_CASE(?,O(n^1)) * Step 1: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: divp(x,y) -> =(rem(x,y),0()) prime(0()) -> false() prime(s(0())) -> false() prime(s(s(x))) -> prime1(s(s(x)),s(x)) prime1(x,0()) -> false() prime1(x,s(0())) -> true() prime1(x,s(s(y))) -> and(not(divp(s(s(y)),x)),prime1(x,s(y))) - Signature: {divp/2,prime/1,prime1/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {divp,prime,prime1} and constructors {0,=,and,false,not ,rem,s,true} + Applied Processor: NaturalPI {shape = StronglyLinear, restrict = NoRestrict, uargs = UArgs, urules = URules, selector = Nothing} + Details: We apply a polynomial interpretation of kind constructor-based(stronglyLinear): The following argument positions are considered usable: uargs(and) = {1,2}, uargs(not) = {1} Following symbols are considered usable: {divp,prime,prime1} TcT has computed the following interpretation: p(0) = 4 p(=) = x1 p(and) = x1 + x2 p(divp) = 1 p(false) = 4 p(not) = x1 p(prime) = 11 + x1 p(prime1) = 10 + x2 p(rem) = 0 p(s) = 3 + x1 p(true) = 2 Following rules are strictly oriented: divp(x,y) = 1 > 0 = =(rem(x,y),0()) prime(0()) = 15 > 4 = false() prime(s(0())) = 18 > 4 = false() prime(s(s(x))) = 17 + x > 13 + x = prime1(s(s(x)),s(x)) prime1(x,0()) = 14 > 4 = false() prime1(x,s(0())) = 17 > 2 = true() prime1(x,s(s(y))) = 16 + y > 14 + y = and(not(divp(s(s(y)),x)),prime1(x,s(y))) Following rules are (at-least) weakly oriented: WORST_CASE(?,O(n^1))