We consider the following Problem: Strict Trs: { g(x, y) -> x , g(x, y) -> y , f(s(x), y, y) -> f(y, x, s(x))} StartTerms: basic terms Strategy: innermost Certificate: YES(?,O(n^1)) Proof: We consider the following Problem: Strict Trs: { g(x, y) -> x , g(x, y) -> y , f(s(x), y, y) -> f(y, x, s(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(x, y) -> x} Interpretation of nonconstant growth: ------------------------------------- The following argument positions are usable: Uargs(g) = {}, Uargs(f) = {}, Uargs(s) = {} We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation: Interpretation Functions: g(x1, x2) = [1 0] x1 + [0 0] x2 + [2] [0 1] [1 1] [0] f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 1] x3 + [1] [0 0] [1 1] [0 0] [1] s(x1) = [0 0] x1 + [0] [0 0] [0] The strictly oriented rules are moved into the weak component. We consider the following Problem: Strict Trs: { g(x, y) -> y , f(s(x), y, y) -> f(y, x, s(x))} Weak Trs: {g(x, y) -> 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(x, y) -> y} Interpretation of nonconstant growth: ------------------------------------- The following argument positions are usable: Uargs(g) = {}, Uargs(f) = {}, Uargs(s) = {} We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation: Interpretation Functions: g(x1, x2) = [1 0] x1 + [1 0] x2 + [2] [0 1] [0 1] [0] f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 1] x3 + [1] [1 1] [1 1] [0 0] [1] s(x1) = [0 0] x1 + [0] [0 0] [0] The strictly oriented rules are moved into the weak component. We consider the following Problem: Strict Trs: {f(s(x), y, y) -> f(y, x, s(x))} Weak Trs: { g(x, y) -> y , g(x, y) -> x} StartTerms: basic terms Strategy: innermost Certificate: YES(?,O(n^1)) Proof: The weightgap principle applies, where following rules are oriented strictly: TRS Component: {f(s(x), y, y) -> f(y, x, s(x))} Interpretation of nonconstant growth: ------------------------------------- The following argument positions are usable: Uargs(g) = {}, Uargs(f) = {}, Uargs(s) = {} We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation: Interpretation Functions: g(x1, x2) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 1] [0] f(x1, x2, x3) = [0 3] x1 + [0 3] x2 + [1 0] x3 + [0] [0 0] [0 0] [0 0] [1] s(x1) = [0 0] x1 + [0] [0 1] [3] The strictly oriented rules are moved into the weak component. We consider the following Problem: Weak Trs: { f(s(x), y, y) -> f(y, x, s(x)) , g(x, y) -> y , g(x, y) -> x} StartTerms: basic terms Strategy: innermost Certificate: YES(O(1),O(1)) Proof: We consider the following Problem: Weak Trs: { f(s(x), y, y) -> f(y, x, s(x)) , g(x, y) -> y , g(x, y) -> 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))