* Step 1: ToInnermost WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: g(f(x),y) -> f(h(x,y)) h(x,y) -> g(x,f(y)) - Signature: {g/2,h/2} / {f/1} - Obligation: runtime complexity wrt. defined symbols {g,h} and constructors {f} + Applied Processor: ToInnermost + Details: switch to innermost, as the system is overlay and right linear and does not contain weak rules * Step 2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: g(f(x),y) -> f(h(x,y)) h(x,y) -> g(x,f(y)) - Signature: {g/2,h/2} / {f/1} - Obligation: innermost runtime complexity wrt. defined symbols {g,h} and constructors {f} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(f) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(f) = [1] x1 + [5] p(g) = [4] x1 + [1] x2 + [0] p(h) = [4] x1 + [1] x2 + [14] Following rules are strictly oriented: g(f(x),y) = [4] x + [1] y + [20] > [4] x + [1] y + [19] = f(h(x,y)) h(x,y) = [4] x + [1] y + [14] > [4] x + [1] y + [5] = g(x,f(y)) Following rules are (at-least) weakly oriented: Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: g(f(x),y) -> f(h(x,y)) h(x,y) -> g(x,f(y)) - Signature: {g/2,h/2} / {f/1} - Obligation: innermost runtime complexity wrt. defined symbols {g,h} and constructors {f} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))