* Step 1: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f(g(x),s(0()),y) -> f(y,y,g(x)) g(0()) -> 0() g(s(x)) -> s(g(x)) - Signature: {f/3,g/1} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {f,g} and constructors {0,s} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak dependency pairs: Strict DPs f#(g(x),s(0()),y) -> c_1(f#(y,y,g(x))) g#(0()) -> c_2() g#(s(x)) -> c_3(g#(x)) Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(g(x),s(0()),y) -> c_1(f#(y,y,g(x))) g#(0()) -> c_2() g#(s(x)) -> c_3(g#(x)) - Strict TRS: f(g(x),s(0()),y) -> f(y,y,g(x)) g(0()) -> 0() g(s(x)) -> s(g(x)) - Signature: {f/3,g/1,f#/3,g#/1} / {0/0,s/1,c_1/1,c_2/0,c_3/1} - Obligation: runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: g#(0()) -> c_2() g#(s(x)) -> c_3(g#(x)) * Step 3: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: g#(0()) -> c_2() g#(s(x)) -> c_3(g#(x)) - Signature: {f/3,g/1,f#/3,g#/1} / {0/0,s/1,c_1/1,c_2/0,c_3/1} - Obligation: runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1} by application of Pre({1}) = {2}. Here rules are labelled as follows: 1: g#(0()) -> c_2() 2: g#(s(x)) -> c_3(g#(x)) * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: g#(s(x)) -> c_3(g#(x)) - Weak DPs: g#(0()) -> c_2() - Signature: {f/3,g/1,f#/3,g#/1} / {0/0,s/1,c_1/1,c_2/0,c_3/1} - Obligation: runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:g#(s(x)) -> c_3(g#(x)) -->_1 g#(0()) -> c_2():2 -->_1 g#(s(x)) -> c_3(g#(x)):1 2:W:g#(0()) -> c_2() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: g#(0()) -> c_2() * Step 5: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: g#(s(x)) -> c_3(g#(x)) - Signature: {f/3,g/1,f#/3,g#/1} / {0/0,s/1,c_1/1,c_2/0,c_3/1} - Obligation: runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: g#(s(x)) -> c_3(g#(x)) The strictly oriented rules are moved into the weak component. ** Step 5.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: g#(s(x)) -> c_3(g#(x)) - Signature: {f/3,g/1,f#/3,g#/1} / {0/0,s/1,c_1/1,c_2/0,c_3/1} - Obligation: runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_3) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(f) = [1] x1 + [4] x2 + [0] p(g) = [2] x1 + [1] p(s) = [1] x1 + [2] p(f#) = [1] x2 + [2] x3 + [1] p(g#) = [4] x1 + [7] p(c_1) = [2] x1 + [8] p(c_2) = [2] p(c_3) = [1] x1 + [0] Following rules are strictly oriented: g#(s(x)) = [4] x + [15] > [4] x + [7] = c_3(g#(x)) Following rules are (at-least) weakly oriented: ** Step 5.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: g#(s(x)) -> c_3(g#(x)) - Signature: {f/3,g/1,f#/3,g#/1} / {0/0,s/1,c_1/1,c_2/0,c_3/1} - Obligation: runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: g#(s(x)) -> c_3(g#(x)) - Signature: {f/3,g/1,f#/3,g#/1} / {0/0,s/1,c_1/1,c_2/0,c_3/1} - Obligation: runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:g#(s(x)) -> c_3(g#(x)) -->_1 g#(s(x)) -> c_3(g#(x)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: g#(s(x)) -> c_3(g#(x)) ** Step 5.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {f/3,g/1,f#/3,g#/1} / {0/0,s/1,c_1/1,c_2/0,c_3/1} - Obligation: runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))