MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> cons(X,n__f(g(X))) f(X) -> n__f(X) g(0()) -> s(0()) g(s(X)) -> s(s(g(X))) sel(0(),cons(X,Y)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) - Signature: {activate/1,f/1,g/1,sel/2} / {0/0,cons/2,n__f/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,f,g,sel} and constructors {0,cons,n__f,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs activate#(X) -> c_1() activate#(n__f(X)) -> c_2(f#(X)) f#(X) -> c_3(g#(X)) f#(X) -> c_4() g#(0()) -> c_5() g#(s(X)) -> c_6(g#(X)) sel#(0(),cons(X,Y)) -> c_7() sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)) Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__f(X)) -> c_2(f#(X)) f#(X) -> c_3(g#(X)) f#(X) -> c_4() g#(0()) -> c_5() g#(s(X)) -> c_6(g#(X)) sel#(0(),cons(X,Y)) -> c_7() sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> cons(X,n__f(g(X))) f(X) -> n__f(X) g(0()) -> s(0()) g(s(X)) -> s(s(g(X))) sel(0(),cons(X,Y)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) - Signature: {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> cons(X,n__f(g(X))) f(X) -> n__f(X) g(0()) -> s(0()) g(s(X)) -> s(s(g(X))) activate#(X) -> c_1() activate#(n__f(X)) -> c_2(f#(X)) f#(X) -> c_3(g#(X)) f#(X) -> c_4() g#(0()) -> c_5() g#(s(X)) -> c_6(g#(X)) sel#(0(),cons(X,Y)) -> c_7() sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__f(X)) -> c_2(f#(X)) f#(X) -> c_3(g#(X)) f#(X) -> c_4() g#(0()) -> c_5() g#(s(X)) -> c_6(g#(X)) sel#(0(),cons(X,Y)) -> c_7() sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> cons(X,n__f(g(X))) f(X) -> n__f(X) g(0()) -> s(0()) g(s(X)) -> s(s(g(X))) - Signature: {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,4,5,7} by application of Pre({1,4,5,7}) = {2,3,6,8}. Here rules are labelled as follows: 1: activate#(X) -> c_1() 2: activate#(n__f(X)) -> c_2(f#(X)) 3: f#(X) -> c_3(g#(X)) 4: f#(X) -> c_4() 5: g#(0()) -> c_5() 6: g#(s(X)) -> c_6(g#(X)) 7: sel#(0(),cons(X,Y)) -> c_7() 8: sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)) * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: activate#(n__f(X)) -> c_2(f#(X)) f#(X) -> c_3(g#(X)) g#(s(X)) -> c_6(g#(X)) sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)) - Weak DPs: activate#(X) -> c_1() f#(X) -> c_4() g#(0()) -> c_5() sel#(0(),cons(X,Y)) -> c_7() - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> cons(X,n__f(g(X))) f(X) -> n__f(X) g(0()) -> s(0()) g(s(X)) -> s(s(g(X))) - Signature: {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__f(X)) -> c_2(f#(X)) -->_1 f#(X) -> c_3(g#(X)):2 -->_1 f#(X) -> c_4():6 2:S:f#(X) -> c_3(g#(X)) -->_1 g#(s(X)) -> c_6(g#(X)):3 -->_1 g#(0()) -> c_5():7 3:S:g#(s(X)) -> c_6(g#(X)) -->_1 g#(0()) -> c_5():7 -->_1 g#(s(X)) -> c_6(g#(X)):3 4:S:sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)) -->_1 sel#(0(),cons(X,Y)) -> c_7():8 -->_2 activate#(X) -> c_1():5 -->_1 sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)):4 -->_2 activate#(n__f(X)) -> c_2(f#(X)):1 5:W:activate#(X) -> c_1() 6:W:f#(X) -> c_4() 7:W:g#(0()) -> c_5() 8:W:sel#(0(),cons(X,Y)) -> c_7() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: activate#(X) -> c_1() 8: sel#(0(),cons(X,Y)) -> c_7() 6: f#(X) -> c_4() 7: g#(0()) -> c_5() * Step 5: Decompose MAYBE + Considered Problem: - Strict DPs: activate#(n__f(X)) -> c_2(f#(X)) f#(X) -> c_3(g#(X)) g#(s(X)) -> c_6(g#(X)) sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> cons(X,n__f(g(X))) f(X) -> n__f(X) g(0()) -> s(0()) g(s(X)) -> s(s(g(X))) - Signature: {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,s} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: activate#(n__f(X)) -> c_2(f#(X)) f#(X) -> c_3(g#(X)) g#(s(X)) -> c_6(g#(X)) - Weak DPs: sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> cons(X,n__f(g(X))) f(X) -> n__f(X) g(0()) -> s(0()) g(s(X)) -> s(s(g(X))) - Signature: {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,s} Problem (S) - Strict DPs: sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)) - Weak DPs: activate#(n__f(X)) -> c_2(f#(X)) f#(X) -> c_3(g#(X)) g#(s(X)) -> c_6(g#(X)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> cons(X,n__f(g(X))) f(X) -> n__f(X) g(0()) -> s(0()) g(s(X)) -> s(s(g(X))) - Signature: {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,s} ** Step 5.a:1: DecomposeDG MAYBE + Considered Problem: - Strict DPs: activate#(n__f(X)) -> c_2(f#(X)) f#(X) -> c_3(g#(X)) g#(s(X)) -> c_6(g#(X)) - Weak DPs: sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> cons(X,n__f(g(X))) f(X) -> n__f(X) g(0()) -> s(0()) g(s(X)) -> s(s(g(X))) - Signature: {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,s} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)) and a lower component activate#(n__f(X)) -> c_2(f#(X)) f#(X) -> c_3(g#(X)) g#(s(X)) -> c_6(g#(X)) Further, following extension rules are added to the lower component. sel#(s(X),cons(Y,Z)) -> activate#(Z) sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z)) *** Step 5.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> cons(X,n__f(g(X))) f(X) -> n__f(X) g(0()) -> s(0()) g(s(X)) -> s(s(g(X))) - Signature: {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__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: sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)) The strictly oriented rules are moved into the weak component. **** Step 5.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> cons(X,n__f(g(X))) f(X) -> n__f(X) g(0()) -> s(0()) g(s(X)) -> s(s(g(X))) - Signature: {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__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_8) = {1} Following symbols are considered usable: {activate#,f#,g#,sel#} TcT has computed the following interpretation: p(0) = [2] p(activate) = [0] p(cons) = [0] p(f) = [4] x1 + [0] p(g) = [4] x1 + [4] p(n__f) = [6] p(s) = [1] x1 + [5] p(sel) = [2] x1 + [1] x2 + [0] p(activate#) = [1] p(f#) = [1] p(g#) = [8] p(sel#) = [1] x1 + [1] p(c_1) = [1] p(c_2) = [4] p(c_3) = [2] x1 + [4] p(c_4) = [8] p(c_5) = [1] p(c_6) = [1] x1 + [1] p(c_7) = [1] p(c_8) = [1] x1 + [1] x2 + [2] Following rules are strictly oriented: sel#(s(X),cons(Y,Z)) = [1] X + [6] > [1] X + [4] = c_8(sel#(X,activate(Z)),activate#(Z)) Following rules are (at-least) weakly oriented: **** Step 5.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> cons(X,n__f(g(X))) f(X) -> n__f(X) g(0()) -> s(0()) g(s(X)) -> s(s(g(X))) - Signature: {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 5.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> cons(X,n__f(g(X))) f(X) -> n__f(X) g(0()) -> s(0()) g(s(X)) -> s(s(g(X))) - Signature: {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)) -->_1 sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)) **** Step 5.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> cons(X,n__f(g(X))) f(X) -> n__f(X) g(0()) -> s(0()) g(s(X)) -> s(s(g(X))) - Signature: {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 5.a:1.b:1: NaturalMI MAYBE + Considered Problem: - Strict DPs: activate#(n__f(X)) -> c_2(f#(X)) f#(X) -> c_3(g#(X)) g#(s(X)) -> c_6(g#(X)) - Weak DPs: sel#(s(X),cons(Y,Z)) -> activate#(Z) sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> cons(X,n__f(g(X))) f(X) -> n__f(X) g(0()) -> s(0()) g(s(X)) -> s(s(g(X))) - Signature: {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,s} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, 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}, uargs(c_6) = {1} Following symbols are considered usable: {activate#,f#,g#,sel#} TcT has computed the following interpretation: p(0) = [0] [0] p(activate) = [0] [0] p(cons) = [0 0] x1 + [0] [0 1] [0] p(f) = [0] [0] p(g) = [0] [0] p(n__f) = [0] [0] p(s) = [0] [0] p(sel) = [0] [0] p(activate#) = [4] [0] p(f#) = [0] [0] p(g#) = [0] [0] p(sel#) = [4] [0] p(c_1) = [0] [0] p(c_2) = [2 0] x1 + [0] [0 0] [0] p(c_3) = [1 0] x1 + [0] [0 0] [0] p(c_4) = [0] [0] p(c_5) = [0] [0] p(c_6) = [4 0] x1 + [0] [0 0] [0] p(c_7) = [0] [0] p(c_8) = [0] [0] Following rules are strictly oriented: activate#(n__f(X)) = [4] [0] > [0] [0] = c_2(f#(X)) Following rules are (at-least) weakly oriented: f#(X) = [0] [0] >= [0] [0] = c_3(g#(X)) g#(s(X)) = [0] [0] >= [0] [0] = c_6(g#(X)) sel#(s(X),cons(Y,Z)) = [4] [0] >= [4] [0] = activate#(Z) sel#(s(X),cons(Y,Z)) = [4] [0] >= [4] [0] = sel#(X,activate(Z)) *** Step 5.a:1.b:2: NaturalPI MAYBE + Considered Problem: - Strict DPs: f#(X) -> c_3(g#(X)) g#(s(X)) -> c_6(g#(X)) - Weak DPs: activate#(n__f(X)) -> c_2(f#(X)) sel#(s(X),cons(Y,Z)) -> activate#(Z) sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> cons(X,n__f(g(X))) f(X) -> n__f(X) g(0()) -> s(0()) g(s(X)) -> s(s(g(X))) - Signature: {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,s} + Applied Processor: NaturalPI {shape = Mixed 3, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(3)): The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_6) = {1} Following symbols are considered usable: {activate#,f#,g#,sel#} TcT has computed the following interpretation: p(0) = 0 p(activate) = 0 p(cons) = 0 p(f) = 2 + 2*x1 + 2*x1^2 p(g) = 2*x1 + 2*x1^2 + 2*x1^3 p(n__f) = x1 p(s) = 0 p(sel) = 2 + 2*x1*x2^2 + 2*x2^2 p(activate#) = 3 p(f#) = 3 p(g#) = 0 p(sel#) = 3 p(c_1) = 0 p(c_2) = x1 p(c_3) = 1 + x1 p(c_4) = 0 p(c_5) = 0 p(c_6) = x1 p(c_7) = 0 p(c_8) = 2 Following rules are strictly oriented: f#(X) = 3 > 1 = c_3(g#(X)) Following rules are (at-least) weakly oriented: activate#(n__f(X)) = 3 >= 3 = c_2(f#(X)) g#(s(X)) = 0 >= 0 = c_6(g#(X)) sel#(s(X),cons(Y,Z)) = 3 >= 3 = activate#(Z) sel#(s(X),cons(Y,Z)) = 3 >= 3 = sel#(X,activate(Z)) *** Step 5.a:1.b:3: Failure MAYBE + Considered Problem: - Strict DPs: g#(s(X)) -> c_6(g#(X)) - Weak DPs: activate#(n__f(X)) -> c_2(f#(X)) f#(X) -> c_3(g#(X)) sel#(s(X),cons(Y,Z)) -> activate#(Z) sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> cons(X,n__f(g(X))) f(X) -> n__f(X) g(0()) -> s(0()) g(s(X)) -> s(s(g(X))) - Signature: {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,s} + Applied Processor: EmptyProcessor + Details: The problem is still open. ** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)) - Weak DPs: activate#(n__f(X)) -> c_2(f#(X)) f#(X) -> c_3(g#(X)) g#(s(X)) -> c_6(g#(X)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> cons(X,n__f(g(X))) f(X) -> n__f(X) g(0()) -> s(0()) g(s(X)) -> s(s(g(X))) - Signature: {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)) -->_2 activate#(n__f(X)) -> c_2(f#(X)):2 -->_1 sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)):1 2:W:activate#(n__f(X)) -> c_2(f#(X)) -->_1 f#(X) -> c_3(g#(X)):3 3:W:f#(X) -> c_3(g#(X)) -->_1 g#(s(X)) -> c_6(g#(X)):4 4:W:g#(s(X)) -> c_6(g#(X)) -->_1 g#(s(X)) -> c_6(g#(X)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: activate#(n__f(X)) -> c_2(f#(X)) 3: f#(X) -> c_3(g#(X)) 4: g#(s(X)) -> c_6(g#(X)) ** Step 5.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> cons(X,n__f(g(X))) f(X) -> n__f(X) g(0()) -> s(0()) g(s(X)) -> s(s(g(X))) - Signature: {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)) -->_1 sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z))) ** Step 5.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z))) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> cons(X,n__f(g(X))) f(X) -> n__f(X) g(0()) -> s(0()) g(s(X)) -> s(s(g(X))) - Signature: {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__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: sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z))) The strictly oriented rules are moved into the weak component. *** Step 5.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z))) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> cons(X,n__f(g(X))) f(X) -> n__f(X) g(0()) -> s(0()) g(s(X)) -> s(s(g(X))) - Signature: {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__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_8) = {1} Following symbols are considered usable: {activate#,f#,g#,sel#} TcT has computed the following interpretation: p(0) = [7] p(activate) = [2] p(cons) = [1] x1 + [1] p(f) = [1] x1 + [2] p(g) = [10] p(n__f) = [2] p(s) = [1] x1 + [8] p(sel) = [1] x1 + [1] x2 + [0] p(activate#) = [8] p(f#) = [1] x1 + [1] p(g#) = [1] p(sel#) = [2] x1 + [1] p(c_1) = [2] p(c_2) = [0] p(c_3) = [4] x1 + [4] p(c_4) = [1] p(c_5) = [0] p(c_6) = [1] x1 + [2] p(c_7) = [0] p(c_8) = [1] x1 + [14] Following rules are strictly oriented: sel#(s(X),cons(Y,Z)) = [2] X + [17] > [2] X + [15] = c_8(sel#(X,activate(Z))) Following rules are (at-least) weakly oriented: *** Step 5.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z))) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> cons(X,n__f(g(X))) f(X) -> n__f(X) g(0()) -> s(0()) g(s(X)) -> s(s(g(X))) - Signature: {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 5.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z))) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> cons(X,n__f(g(X))) f(X) -> n__f(X) g(0()) -> s(0()) g(s(X)) -> s(s(g(X))) - Signature: {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z))) -->_1 sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z))) *** Step 5.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> cons(X,n__f(g(X))) f(X) -> n__f(X) g(0()) -> s(0()) g(s(X)) -> s(s(g(X))) - Signature: {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). MAYBE