We consider the following Problem: Strict Trs: { a(f(), a(f(), x)) -> a(x, g()) , a(x, g()) -> a(f(), a(g(), a(f(), x)))} StartTerms: basic terms Strategy: innermost Certificate: YES(?,O(n^1)) Proof: We consider the following Problem: Strict Trs: { a(f(), a(f(), x)) -> a(x, g()) , a(x, g()) -> a(f(), a(g(), a(f(), x)))} StartTerms: basic terms Strategy: innermost Certificate: YES(?,O(n^1)) Proof: The weightgap principle applies, where following rules are oriented strictly: TRS Component: {a(f(), a(f(), x)) -> a(x, g())} Interpretation of nonconstant growth: ------------------------------------- The following argument positions are usable: Uargs(a) = {2} We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation: Interpretation Functions: a(x1, x2) = [1 0] x1 + [1 0] x2 + [2] [0 0] [0 1] [1] f() = [0] [0] g() = [0] [0] The strictly oriented rules are moved into the weak component. We consider the following Problem: Strict Trs: {a(x, g()) -> a(f(), a(g(), a(f(), x)))} Weak Trs: {a(f(), a(f(), x)) -> a(x, g())} StartTerms: basic terms Strategy: innermost Certificate: YES(?,O(n^1)) Proof: We consider the following Problem: Strict Trs: {a(x, g()) -> a(f(), a(g(), a(f(), x)))} Weak Trs: {a(f(), a(f(), x)) -> a(x, g())} StartTerms: basic terms Strategy: innermost Certificate: YES(?,O(n^1)) Proof: The problem is match-bounded by 1. The enriched problem is compatible with the following automaton: { a_0(2, 2) -> 1 , a_1(3, 4) -> 1 , a_1(5, 6) -> 4 , a_1(5, 8) -> 2 , a_1(7, 2) -> 6 , a_1(7, 7) -> 8 , f_0() -> 2 , f_1() -> 3 , f_1() -> 7 , g_0() -> 2 , g_1() -> 5} Hurray, we answered YES(?,O(n^1))