* Step 1: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: :(z,+(x,f(y))) -> :(g(z,y),+(x,a())) :(+(x,y),z) -> +(:(x,z),:(y,z)) - Signature: {:/2} / {+/2,a/0,f/1,g/2} - Obligation: innermost runtime complexity wrt. defined symbols {:} and constructors {+,a,f,g} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a()))) :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)) Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a()))) :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)) - Strict TRS: :(z,+(x,f(y))) -> :(g(z,y),+(x,a())) :(+(x,y),z) -> +(:(x,z),:(y,z)) - Signature: {:/2,:#/2} / {+/2,a/0,f/1,g/2,c_1/1,c_2/2} - Obligation: innermost runtime complexity wrt. defined symbols {:#} and constructors {+,a,f,g} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a()))) :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)) * Step 3: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a()))) :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)) - Signature: {:/2,:#/2} / {+/2,a/0,f/1,g/2,c_1/1,c_2/2} - Obligation: innermost runtime complexity wrt. defined symbols {:#} and constructors {+,a,f,g} + 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: :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a()))) 2: :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)) * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)) - Weak DPs: :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a()))) - Signature: {:/2,:#/2} / {+/2,a/0,f/1,g/2,c_1/1,c_2/2} - Obligation: innermost runtime complexity wrt. defined symbols {:#} and constructors {+,a,f,g} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S::#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)) -->_2 :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a()))):2 -->_1 :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a()))):2 -->_2 :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)):1 -->_1 :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)):1 2:W::#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a()))) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a()))) * Step 5: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)) - Signature: {:/2,:#/2} / {+/2,a/0,f/1,g/2,c_1/1,c_2/2} - Obligation: innermost runtime complexity wrt. defined symbols {:#} and constructors {+,a,f,g} + 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: :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)) The strictly oriented rules are moved into the weak component. ** Step 5.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)) - Signature: {:/2,:#/2} / {+/2,a/0,f/1,g/2,c_1/1,c_2/2} - Obligation: innermost runtime complexity wrt. defined symbols {:#} and constructors {+,a,f,g} + 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_2) = {1,2} Following symbols are considered usable: {:#} TcT has computed the following interpretation: p(+) = [1] x1 + [1] x2 + [9] p(:) = [1] x1 + [8] x2 + [4] p(a) = [0] p(f) = [1] p(g) = [1] x1 + [1] p(:#) = [2] x1 + [5] p(c_1) = [1] x1 + [1] p(c_2) = [1] x1 + [1] x2 + [1] Following rules are strictly oriented: :#(+(x,y),z) = [2] x + [2] y + [23] > [2] x + [2] y + [11] = c_2(:#(x,z),:#(y,z)) Following rules are (at-least) weakly oriented: ** Step 5.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)) - Signature: {:/2,:#/2} / {+/2,a/0,f/1,g/2,c_1/1,c_2/2} - Obligation: innermost runtime complexity wrt. defined symbols {:#} and constructors {+,a,f,g} + 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: :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)) - Signature: {:/2,:#/2} / {+/2,a/0,f/1,g/2,c_1/1,c_2/2} - Obligation: innermost runtime complexity wrt. defined symbols {:#} and constructors {+,a,f,g} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W::#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)) -->_2 :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)):1 -->_1 :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)) ** Step 5.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {:/2,:#/2} / {+/2,a/0,f/1,g/2,c_1/1,c_2/2} - Obligation: innermost runtime complexity wrt. defined symbols {:#} and constructors {+,a,f,g} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))