We consider the following Problem: Strict Trs: { g(h(g(x))) -> g(x) , g(g(x)) -> g(h(g(x))) , h(h(x)) -> h(f(h(x), x))} StartTerms: basic terms Strategy: innermost Certificate: YES(?,O(n^1)) Proof: We consider the following Problem: Strict Trs: { g(h(g(x))) -> g(x) , g(g(x)) -> g(h(g(x))) , h(h(x)) -> h(f(h(x), x))} StartTerms: basic terms Strategy: innermost Certificate: YES(?,O(n^1)) Proof: The weightgap principle applies, where following rules are oriented strictly: TRS Component: {h(h(x)) -> h(f(h(x), x))} Interpretation of nonconstant growth: ------------------------------------- The following argument positions are usable: Uargs(g) = {}, Uargs(h) = {}, Uargs(f) = {} We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation: Interpretation Functions: g(x1) = [0 0] x1 + [0] [0 0] [1] h(x1) = [0 3] x1 + [2] [0 0] [2] f(x1, x2) = [0 0] x1 + [0 0] x2 + [0] [0 0] [0 0] [1] The strictly oriented rules are moved into the weak component. We consider the following Problem: Strict Trs: { g(h(g(x))) -> g(x) , g(g(x)) -> g(h(g(x)))} Weak Trs: {h(h(x)) -> h(f(h(x), x))} StartTerms: basic terms Strategy: innermost Certificate: YES(?,O(n^1)) Proof: The weightgap principle applies, where following rules are oriented strictly: TRS Component: {g(g(x)) -> g(h(g(x)))} Interpretation of nonconstant growth: ------------------------------------- The following argument positions are usable: Uargs(g) = {}, Uargs(h) = {}, Uargs(f) = {} We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation: Interpretation Functions: g(x1) = [1 1] x1 + [0] [0 0] [1] h(x1) = [1 0] x1 + [0] [0 0] [0] f(x1, x2) = [0 0] x1 + [0 0] x2 + [0] [0 0] [0 0] [0] The strictly oriented rules are moved into the weak component. We consider the following Problem: Strict Trs: {g(h(g(x))) -> g(x)} Weak Trs: { g(g(x)) -> g(h(g(x))) , h(h(x)) -> h(f(h(x), x))} StartTerms: basic terms Strategy: innermost Certificate: YES(?,O(n^1)) Proof: The weightgap principle applies, where following rules are oriented strictly: TRS Component: {g(h(g(x))) -> g(x)} Interpretation of nonconstant growth: ------------------------------------- The following argument positions are usable: Uargs(g) = {}, Uargs(h) = {}, Uargs(f) = {} We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation: Interpretation Functions: g(x1) = [1 1] x1 + [0] [0 0] [2] h(x1) = [1 0] x1 + [0] [0 0] [2] f(x1, x2) = [0 0] x1 + [0 0] x2 + [0] [0 0] [0 0] [0] The strictly oriented rules are moved into the weak component. We consider the following Problem: Weak Trs: { g(h(g(x))) -> g(x) , g(g(x)) -> g(h(g(x))) , h(h(x)) -> h(f(h(x), x))} StartTerms: basic terms Strategy: innermost Certificate: YES(O(1),O(1)) Proof: We consider the following Problem: Weak Trs: { g(h(g(x))) -> g(x) , g(g(x)) -> g(h(g(x))) , h(h(x)) -> h(f(h(x), x))} StartTerms: basic terms Strategy: innermost Certificate: YES(O(1),O(1)) Proof: Empty rules are trivially bounded Hurray, we answered YES(?,O(n^1))