YES(?,ELEMENTARY) 'epo* (timeout of 60.0 seconds)' -------------------------------- Answer: YES(?,ELEMENTARY) Input Problem: innermost runtime-complexity with respect to Rules: { f(x, y) -> g1(x, x, y) , f(x, y) -> g1(y, x, x) , f(x, y) -> g2(x, y, y) , f(x, y) -> g2(y, y, x) , g1(x, x, y) -> h(x, y) , g1(y, x, x) -> h(x, y) , g2(x, y, y) -> h(x, y) , g2(y, y, x) -> h(x, y) , h(x, x) -> x} Proof Output: Strict Rules in Predicative Notation: { f(x, y;) -> g1(; x, x, y) , f(x, y;) -> g1(; y, x, x) , f(x, y;) -> g2(x, y; y) , f(x, y;) -> g2(y, x; y) , g1(; x, x, y) -> h(; x, y) , g1(; y, x, x) -> h(; x, y) , g2(x, y; y) -> h(; x, y) , g2(y, x; y) -> h(; x, y) , h(; x, x) -> x} Safe Mapping: safe(f) = {}, safe(g1) = {1, 2, 3}, safe(g2) = {2}, safe(h) = {1, 2} Argument Permutation: mu(f) = [1, 2], mu(g1) = [1, 2, 3], mu(g2) = [2, 3, 1], mu(h) = [2, 1] Precedence: h ~ h, g2 > h, g2 ~ g2, g2 > g1, g1 > h, g1 ~ g1, f > h, f > g2, f > g1, f ~ f