* Step 1: DependencyPairs WORST_CASE(?,O(1)) + Considered Problem: - Strict TRS: c(b(a(),a()),b(y,z),x) -> b(a(),b(z,z)) f(b(b(x,f(y)),z)) -> c(z,x,f(b(b(f(a()),y),y))) f(c(c(a(),y,a()),b(x,z),a())) -> b(y,f(c(f(a()),z,z))) - Signature: {c/3,f/1} / {a/0,b/2} - Obligation: runtime complexity wrt. defined symbols {c,f} and constructors {a,b} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following weak dependency pairs: Strict DPs c#(b(a(),a()),b(y,z),x) -> c_1(z,z) f#(b(b(x,f(y)),z)) -> c_2(c#(z,x,f(b(b(f(a()),y),y)))) f#(c(c(a(),y,a()),b(x,z),a())) -> c_3(y,f#(c(f(a()),z,z))) Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: c#(b(a(),a()),b(y,z),x) -> c_1(z,z) f#(b(b(x,f(y)),z)) -> c_2(c#(z,x,f(b(b(f(a()),y),y)))) f#(c(c(a(),y,a()),b(x,z),a())) -> c_3(y,f#(c(f(a()),z,z))) - Strict TRS: c(b(a(),a()),b(y,z),x) -> b(a(),b(z,z)) f(b(b(x,f(y)),z)) -> c(z,x,f(b(b(f(a()),y),y))) f(c(c(a(),y,a()),b(x,z),a())) -> b(y,f(c(f(a()),z,z))) - Signature: {c/3,f/1,c#/3,f#/1} / {a/0,b/2,c_1/2,c_2/1,c_3/2} - Obligation: runtime complexity wrt. defined symbols {c#,f#} and constructors {a,b} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: c#(b(a(),a()),b(y,z),x) -> c_1(z,z) * Step 3: WeightGap WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: c#(b(a(),a()),b(y,z),x) -> c_1(z,z) - Signature: {c/3,f/1,c#/3,f#/1} / {a/0,b/2,c_1/2,c_2/1,c_3/2} - Obligation: runtime complexity wrt. defined symbols {c#,f#} and constructors {a,b} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: none Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [0] p(b) = [0] p(c) = [0] p(f) = [0] p(c#) = [11] p(f#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] Following rules are strictly oriented: c#(b(a(),a()),b(y,z),x) = [11] > [0] = c_1(z,z) Following rules are (at-least) weakly oriented: Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: c#(b(a(),a()),b(y,z),x) -> c_1(z,z) - Signature: {c/3,f/1,c#/3,f#/1} / {a/0,b/2,c_1/2,c_2/1,c_3/2} - Obligation: runtime complexity wrt. defined symbols {c#,f#} and constructors {a,b} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(1))