* Step 1: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: g(X,s(Y)) -> g(X,Y) g(0(),Y) -> 0() h(X,Z) -> f(X,s(X),Z) - Signature: {g/2,h/2} / {0/0,f/3,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {g,h} and constructors {0,f,s} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs g#(X,s(Y)) -> c_1(g#(X,Y)) g#(0(),Y) -> c_2() h#(X,Z) -> c_3() Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: g#(X,s(Y)) -> c_1(g#(X,Y)) g#(0(),Y) -> c_2() h#(X,Z) -> c_3() - Strict TRS: g(X,s(Y)) -> g(X,Y) g(0(),Y) -> 0() h(X,Z) -> f(X,s(X),Z) - Signature: {g/2,h/2,g#/2,h#/2} / {0/0,f/3,s/1,c_1/1,c_2/0,c_3/0} - Obligation: innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {0,f,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: g#(X,s(Y)) -> c_1(g#(X,Y)) g#(0(),Y) -> c_2() h#(X,Z) -> c_3() * Step 3: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: g#(X,s(Y)) -> c_1(g#(X,Y)) g#(0(),Y) -> c_2() h#(X,Z) -> c_3() - Signature: {g/2,h/2,g#/2,h#/2} / {0/0,f/3,s/1,c_1/1,c_2/0,c_3/0} - Obligation: innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {0,f,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,3} by application of Pre({2,3}) = {1}. Here rules are labelled as follows: 1: g#(X,s(Y)) -> c_1(g#(X,Y)) 2: g#(0(),Y) -> c_2() 3: h#(X,Z) -> c_3() * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: g#(X,s(Y)) -> c_1(g#(X,Y)) - Weak DPs: g#(0(),Y) -> c_2() h#(X,Z) -> c_3() - Signature: {g/2,h/2,g#/2,h#/2} / {0/0,f/3,s/1,c_1/1,c_2/0,c_3/0} - Obligation: innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {0,f,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:g#(X,s(Y)) -> c_1(g#(X,Y)) -->_1 g#(0(),Y) -> c_2():2 -->_1 g#(X,s(Y)) -> c_1(g#(X,Y)):1 2:W:g#(0(),Y) -> c_2() 3:W:h#(X,Z) -> c_3() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: h#(X,Z) -> c_3() 2: g#(0(),Y) -> c_2() * Step 5: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: g#(X,s(Y)) -> c_1(g#(X,Y)) - Signature: {g/2,h/2,g#/2,h#/2} / {0/0,f/3,s/1,c_1/1,c_2/0,c_3/0} - Obligation: innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {0,f,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#(X,s(Y)) -> c_1(g#(X,Y)) 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#(X,s(Y)) -> c_1(g#(X,Y)) - Signature: {g/2,h/2,g#/2,h#/2} / {0/0,f/3,s/1,c_1/1,c_2/0,c_3/0} - Obligation: innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {0,f,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_1) = {1} Following symbols are considered usable: {g#,h#} TcT has computed the following interpretation: p(0) = [0] p(f) = [1] x3 + [0] p(g) = [2] x2 + [1] p(h) = [2] x1 + [4] x2 + [1] p(s) = [1] x1 + [6] p(g#) = [2] x2 + [12] p(h#) = [4] x2 + [0] p(c_1) = [1] x1 + [6] p(c_2) = [0] p(c_3) = [1] Following rules are strictly oriented: g#(X,s(Y)) = [2] Y + [24] > [2] Y + [18] = c_1(g#(X,Y)) Following rules are (at-least) weakly oriented: ** Step 5.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: g#(X,s(Y)) -> c_1(g#(X,Y)) - Signature: {g/2,h/2,g#/2,h#/2} / {0/0,f/3,s/1,c_1/1,c_2/0,c_3/0} - Obligation: innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {0,f,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#(X,s(Y)) -> c_1(g#(X,Y)) - Signature: {g/2,h/2,g#/2,h#/2} / {0/0,f/3,s/1,c_1/1,c_2/0,c_3/0} - Obligation: innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {0,f,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:g#(X,s(Y)) -> c_1(g#(X,Y)) -->_1 g#(X,s(Y)) -> c_1(g#(X,Y)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: g#(X,s(Y)) -> c_1(g#(X,Y)) ** Step 5.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {g/2,h/2,g#/2,h#/2} / {0/0,f/3,s/1,c_1/1,c_2/0,c_3/0} - Obligation: innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {0,f,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))