WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__f(X1,X2)) -> f(activate(X1),X2) activate(n__g(X)) -> g(activate(X)) f(X1,X2) -> n__f(X1,X2) f(g(X),Y) -> f(X,n__f(n__g(X),activate(Y))) g(X) -> n__g(X) - Signature: {activate/1,f/2,g/1} / {n__f/2,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,f,g} and constructors {n__f,n__g} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__f(X1,X2)) -> f(activate(X1),X2) activate(n__g(X)) -> g(activate(X)) f(X1,X2) -> n__f(X1,X2) f(g(X),Y) -> f(X,n__f(n__g(X),activate(Y))) g(X) -> n__g(X) - Signature: {activate/1,f/2,g/1} / {n__f/2,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,f,g} and constructors {n__f,n__g} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: activate(x){x -> n__f(x,y)} = activate(n__f(x,y)) ->^+ f(activate(x),y) = C[activate(x) = activate(x){}] ** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__f(X1,X2)) -> f(activate(X1),X2) activate(n__g(X)) -> g(activate(X)) f(X1,X2) -> n__f(X1,X2) f(g(X),Y) -> f(X,n__f(n__g(X),activate(Y))) g(X) -> n__g(X) - Signature: {activate/1,f/2,g/1} / {n__f/2,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,f,g} and constructors {n__f,n__g} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs activate#(X) -> c_1() activate#(n__f(X1,X2)) -> c_2(f#(activate(X1),X2),activate#(X1)) activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) f#(X1,X2) -> c_4() f#(g(X),Y) -> c_5(f#(X,n__f(n__g(X),activate(Y))),activate#(Y)) g#(X) -> c_6() Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__f(X1,X2)) -> c_2(f#(activate(X1),X2),activate#(X1)) activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) f#(X1,X2) -> c_4() f#(g(X),Y) -> c_5(f#(X,n__f(n__g(X),activate(Y))),activate#(Y)) g#(X) -> c_6() - Weak TRS: activate(X) -> X activate(n__f(X1,X2)) -> f(activate(X1),X2) activate(n__g(X)) -> g(activate(X)) f(X1,X2) -> n__f(X1,X2) f(g(X),Y) -> f(X,n__f(n__g(X),activate(Y))) g(X) -> n__g(X) - Signature: {activate/1,f/2,g/1,activate#/1,f#/2,g#/1} / {n__f/2,n__g/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/2,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#} and constructors {n__f,n__g} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,4,5,6} by application of Pre({1,4,5,6}) = {2,3}. Here rules are labelled as follows: 1: activate#(X) -> c_1() 2: activate#(n__f(X1,X2)) -> c_2(f#(activate(X1),X2),activate#(X1)) 3: activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) 4: f#(X1,X2) -> c_4() 5: f#(g(X),Y) -> c_5(f#(X,n__f(n__g(X),activate(Y))),activate#(Y)) 6: g#(X) -> c_6() ** Step 1.b:3: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__f(X1,X2)) -> c_2(f#(activate(X1),X2),activate#(X1)) activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) - Weak DPs: activate#(X) -> c_1() f#(X1,X2) -> c_4() f#(g(X),Y) -> c_5(f#(X,n__f(n__g(X),activate(Y))),activate#(Y)) g#(X) -> c_6() - Weak TRS: activate(X) -> X activate(n__f(X1,X2)) -> f(activate(X1),X2) activate(n__g(X)) -> g(activate(X)) f(X1,X2) -> n__f(X1,X2) f(g(X),Y) -> f(X,n__f(n__g(X),activate(Y))) g(X) -> n__g(X) - Signature: {activate/1,f/2,g/1,activate#/1,f#/2,g#/1} / {n__f/2,n__g/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/2,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#} and constructors {n__f,n__g} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__f(X1,X2)) -> c_2(f#(activate(X1),X2),activate#(X1)) -->_2 activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)):2 -->_1 f#(X1,X2) -> c_4():4 -->_2 activate#(X) -> c_1():3 -->_2 activate#(n__f(X1,X2)) -> c_2(f#(activate(X1),X2),activate#(X1)):1 2:S:activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) -->_1 g#(X) -> c_6():6 -->_2 activate#(X) -> c_1():3 -->_2 activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)):2 -->_2 activate#(n__f(X1,X2)) -> c_2(f#(activate(X1),X2),activate#(X1)):1 3:W:activate#(X) -> c_1() 4:W:f#(X1,X2) -> c_4() 5:W:f#(g(X),Y) -> c_5(f#(X,n__f(n__g(X),activate(Y))),activate#(Y)) 6:W:g#(X) -> c_6() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: f#(g(X),Y) -> c_5(f#(X,n__f(n__g(X),activate(Y))),activate#(Y)) 4: f#(X1,X2) -> c_4() 3: activate#(X) -> c_1() 6: g#(X) -> c_6() ** Step 1.b:4: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__f(X1,X2)) -> c_2(f#(activate(X1),X2),activate#(X1)) activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) - Weak TRS: activate(X) -> X activate(n__f(X1,X2)) -> f(activate(X1),X2) activate(n__g(X)) -> g(activate(X)) f(X1,X2) -> n__f(X1,X2) f(g(X),Y) -> f(X,n__f(n__g(X),activate(Y))) g(X) -> n__g(X) - Signature: {activate/1,f/2,g/1,activate#/1,f#/2,g#/1} / {n__f/2,n__g/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/2,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#} and constructors {n__f,n__g} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:activate#(n__f(X1,X2)) -> c_2(f#(activate(X1),X2),activate#(X1)) -->_2 activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)):2 -->_2 activate#(n__f(X1,X2)) -> c_2(f#(activate(X1),X2),activate#(X1)):1 2:S:activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) -->_2 activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)):2 -->_2 activate#(n__f(X1,X2)) -> c_2(f#(activate(X1),X2),activate#(X1)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: activate#(n__f(X1,X2)) -> c_2(activate#(X1)) activate#(n__g(X)) -> c_3(activate#(X)) ** Step 1.b:5: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__f(X1,X2)) -> c_2(activate#(X1)) activate#(n__g(X)) -> c_3(activate#(X)) - Weak TRS: activate(X) -> X activate(n__f(X1,X2)) -> f(activate(X1),X2) activate(n__g(X)) -> g(activate(X)) f(X1,X2) -> n__f(X1,X2) f(g(X),Y) -> f(X,n__f(n__g(X),activate(Y))) g(X) -> n__g(X) - Signature: {activate/1,f/2,g/1,activate#/1,f#/2,g#/1} / {n__f/2,n__g/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/2,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#} and constructors {n__f,n__g} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate#(n__f(X1,X2)) -> c_2(activate#(X1)) activate#(n__g(X)) -> c_3(activate#(X)) ** Step 1.b:6: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__f(X1,X2)) -> c_2(activate#(X1)) activate#(n__g(X)) -> c_3(activate#(X)) - Signature: {activate/1,f/2,g/1,activate#/1,f#/2,g#/1} / {n__f/2,n__g/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/2,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#} and constructors {n__f,n__g} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any 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}, uargs(c_3) = {1} Following symbols are considered usable: {activate#,f#,g#} TcT has computed the following interpretation: p(activate) = [2] p(f) = [1] x1 + [0] p(g) = [1] x1 + [2] p(n__f) = [1] x1 + [1] p(n__g) = [1] x1 + [2] p(activate#) = [8] x1 + [2] p(f#) = [1] x1 + [1] x2 + [2] p(g#) = [1] x1 + [0] p(c_1) = [1] p(c_2) = [1] x1 + [8] p(c_3) = [1] x1 + [3] p(c_4) = [1] p(c_5) = [1] p(c_6) = [1] Following rules are strictly oriented: activate#(n__g(X)) = [8] X + [18] > [8] X + [5] = c_3(activate#(X)) Following rules are (at-least) weakly oriented: activate#(n__f(X1,X2)) = [8] X1 + [10] >= [8] X1 + [10] = c_2(activate#(X1)) ** Step 1.b:7: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__f(X1,X2)) -> c_2(activate#(X1)) - Weak DPs: activate#(n__g(X)) -> c_3(activate#(X)) - Signature: {activate/1,f/2,g/1,activate#/1,f#/2,g#/1} / {n__f/2,n__g/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/2,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#} and constructors {n__f,n__g} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any 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}, uargs(c_3) = {1} Following symbols are considered usable: {activate#,f#,g#} TcT has computed the following interpretation: p(activate) = [1] x1 + [1] p(f) = [1] p(g) = [1] x1 + [0] p(n__f) = [1] x1 + [1] p(n__g) = [1] x1 + [11] p(activate#) = [1] x1 + [2] p(f#) = [0] p(g#) = [2] x1 + [1] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [4] p(c_4) = [4] p(c_5) = [1] x1 + [0] p(c_6) = [1] Following rules are strictly oriented: activate#(n__f(X1,X2)) = [1] X1 + [3] > [1] X1 + [2] = c_2(activate#(X1)) Following rules are (at-least) weakly oriented: activate#(n__g(X)) = [1] X + [13] >= [1] X + [6] = c_3(activate#(X)) ** Step 1.b:8: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(n__f(X1,X2)) -> c_2(activate#(X1)) activate#(n__g(X)) -> c_3(activate#(X)) - Signature: {activate/1,f/2,g/1,activate#/1,f#/2,g#/1} / {n__f/2,n__g/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/2,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#} and constructors {n__f,n__g} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))