MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> cons(X,n__f(n__g(X))) f(X) -> n__f(X) g(X) -> n__g(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,n__g/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,f,g,sel} and constructors {0,cons,n__f,n__g,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#(activate(X)),activate#(X)) activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) f#(X) -> c_4() f#(X) -> c_5() g#(X) -> c_6() g#(0()) -> c_7() g#(s(X)) -> c_8(g#(X)) sel#(0(),cons(X,Y)) -> c_9() sel#(s(X),cons(Y,Z)) -> c_10(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#(activate(X)),activate#(X)) activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) f#(X) -> c_4() f#(X) -> c_5() g#(X) -> c_6() g#(0()) -> c_7() g#(s(X)) -> c_8(g#(X)) sel#(0(),cons(X,Y)) -> c_9() sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> cons(X,n__f(n__g(X))) f(X) -> n__f(X) g(X) -> n__g(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,n__g/1,s/1,c_1/0,c_2/2,c_3/2 ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,n__g ,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> cons(X,n__f(n__g(X))) f(X) -> n__f(X) g(X) -> n__g(X) g(0()) -> s(0()) g(s(X)) -> s(s(g(X))) activate#(X) -> c_1() activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X)) activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) f#(X) -> c_4() f#(X) -> c_5() g#(X) -> c_6() g#(0()) -> c_7() g#(s(X)) -> c_8(g#(X)) sel#(0(),cons(X,Y)) -> c_9() sel#(s(X),cons(Y,Z)) -> c_10(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#(activate(X)),activate#(X)) activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) f#(X) -> c_4() f#(X) -> c_5() g#(X) -> c_6() g#(0()) -> c_7() g#(s(X)) -> c_8(g#(X)) sel#(0(),cons(X,Y)) -> c_9() sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> cons(X,n__f(n__g(X))) f(X) -> n__f(X) g(X) -> n__g(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,n__g/1,s/1,c_1/0,c_2/2,c_3/2 ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,n__g ,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,4,5,6,7,9} by application of Pre({1,4,5,6,7,9}) = {2,3,8,10}. Here rules are labelled as follows: 1: activate#(X) -> c_1() 2: activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X)) 3: activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) 4: f#(X) -> c_4() 5: f#(X) -> c_5() 6: g#(X) -> c_6() 7: g#(0()) -> c_7() 8: g#(s(X)) -> c_8(g#(X)) 9: sel#(0(),cons(X,Y)) -> c_9() 10: sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X)) activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) g#(s(X)) -> c_8(g#(X)) sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) - Weak DPs: activate#(X) -> c_1() f#(X) -> c_4() f#(X) -> c_5() g#(X) -> c_6() g#(0()) -> c_7() sel#(0(),cons(X,Y)) -> c_9() - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> cons(X,n__f(n__g(X))) f(X) -> n__f(X) g(X) -> n__g(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,n__g/1,s/1,c_1/0,c_2/2,c_3/2 ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,n__g ,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X)) -->_2 activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)):2 -->_1 f#(X) -> c_5():7 -->_1 f#(X) -> c_4():6 -->_2 activate#(X) -> c_1():5 -->_2 activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X)):1 2:S:activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) -->_1 g#(s(X)) -> c_8(g#(X)):3 -->_1 g#(0()) -> c_7():9 -->_1 g#(X) -> c_6():8 -->_2 activate#(X) -> c_1():5 -->_2 activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)):2 -->_2 activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X)):1 3:S:g#(s(X)) -> c_8(g#(X)) -->_1 g#(0()) -> c_7():9 -->_1 g#(X) -> c_6():8 -->_1 g#(s(X)) -> c_8(g#(X)):3 4:S:sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) -->_1 sel#(0(),cons(X,Y)) -> c_9():10 -->_2 activate#(X) -> c_1():5 -->_1 sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)):4 -->_2 activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)):2 -->_2 activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X)):1 5:W:activate#(X) -> c_1() 6:W:f#(X) -> c_4() 7:W:f#(X) -> c_5() 8:W:g#(X) -> c_6() 9:W:g#(0()) -> c_7() 10:W:sel#(0(),cons(X,Y)) -> c_9() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 10: sel#(0(),cons(X,Y)) -> c_9() 6: f#(X) -> c_4() 7: f#(X) -> c_5() 5: activate#(X) -> c_1() 8: g#(X) -> c_6() 9: g#(0()) -> c_7() * Step 5: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X)) activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) g#(s(X)) -> c_8(g#(X)) sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> cons(X,n__f(n__g(X))) f(X) -> n__f(X) g(X) -> n__g(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,n__g/1,s/1,c_1/0,c_2/2,c_3/2 ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,n__g ,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X)) -->_2 activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)):2 -->_2 activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X)):1 2:S:activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) -->_1 g#(s(X)) -> c_8(g#(X)):3 -->_2 activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)):2 -->_2 activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X)):1 3:S:g#(s(X)) -> c_8(g#(X)) -->_1 g#(s(X)) -> c_8(g#(X)):3 4:S:sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) -->_1 sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)):4 -->_2 activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)):2 -->_2 activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: activate#(n__f(X)) -> c_2(activate#(X)) * Step 6: Decompose MAYBE + Considered Problem: - Strict DPs: activate#(n__f(X)) -> c_2(activate#(X)) activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) g#(s(X)) -> c_8(g#(X)) sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> cons(X,n__f(n__g(X))) f(X) -> n__f(X) g(X) -> n__g(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,n__g/1,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,n__g ,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(activate#(X)) activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) g#(s(X)) -> c_8(g#(X)) - Weak DPs: sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> cons(X,n__f(n__g(X))) f(X) -> n__f(X) g(X) -> n__g(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,n__g/1,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f ,n__g,s} Problem (S) - Strict DPs: sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) - Weak DPs: activate#(n__f(X)) -> c_2(activate#(X)) activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) g#(s(X)) -> c_8(g#(X)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> cons(X,n__f(n__g(X))) f(X) -> n__f(X) g(X) -> n__g(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,n__g/1,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f ,n__g,s} ** Step 6.a:1: DecomposeDG MAYBE + Considered Problem: - Strict DPs: activate#(n__f(X)) -> c_2(activate#(X)) activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) g#(s(X)) -> c_8(g#(X)) - Weak DPs: sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> cons(X,n__f(n__g(X))) f(X) -> n__f(X) g(X) -> n__g(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,n__g/1,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,n__g ,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_10(sel#(X,activate(Z)),activate#(Z)) and a lower component activate#(n__f(X)) -> c_2(activate#(X)) activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) g#(s(X)) -> c_8(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 6.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> cons(X,n__f(n__g(X))) f(X) -> n__f(X) g(X) -> n__g(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,n__g/1,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,n__g ,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_10(sel#(X,activate(Z)),activate#(Z)) The strictly oriented rules are moved into the weak component. **** Step 6.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> cons(X,n__f(n__g(X))) f(X) -> n__f(X) g(X) -> n__g(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,n__g/1,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,n__g ,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_10) = {1} Following symbols are considered usable: {activate#,f#,g#,sel#} TcT has computed the following interpretation: p(0) = [3] p(activate) = [0] p(cons) = [1] x1 + [0] p(f) = [1] x1 + [0] p(g) = [0] p(n__f) = [1] x1 + [4] p(n__g) = [15] p(s) = [1] x1 + [5] p(sel) = [1] x2 + [1] p(activate#) = [4] p(f#) = [1] p(g#) = [0] p(sel#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [8] x1 + [1] p(c_3) = [1] x1 + [0] p(c_4) = [1] p(c_5) = [1] p(c_6) = [8] p(c_7) = [2] p(c_8) = [1] p(c_9) = [0] p(c_10) = [1] x1 + [1] x2 + [0] Following rules are strictly oriented: sel#(s(X),cons(Y,Z)) = [1] X + [5] > [1] X + [4] = c_10(sel#(X,activate(Z)),activate#(Z)) Following rules are (at-least) weakly oriented: **** Step 6.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> cons(X,n__f(n__g(X))) f(X) -> n__f(X) g(X) -> n__g(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,n__g/1,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,n__g ,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 6.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> cons(X,n__f(n__g(X))) f(X) -> n__f(X) g(X) -> n__g(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,n__g/1,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,n__g ,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) -->_1 sel#(s(X),cons(Y,Z)) -> c_10(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_10(sel#(X,activate(Z)),activate#(Z)) **** Step 6.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> cons(X,n__f(n__g(X))) f(X) -> n__f(X) g(X) -> n__g(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,n__g/1,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,n__g ,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 6.a:1.b:1: NaturalPI MAYBE + Considered Problem: - Strict DPs: activate#(n__f(X)) -> c_2(activate#(X)) activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) g#(s(X)) -> c_8(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(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> cons(X,n__f(n__g(X))) f(X) -> n__f(X) g(X) -> n__g(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,n__g/1,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,n__g ,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) = {1,2}, uargs(c_8) = {1} Following symbols are considered usable: {activate,f,g,activate#,f#,g#,sel#} TcT has computed the following interpretation: p(0) = 0 p(activate) = x1 p(cons) = x2 p(f) = 1 + x1 p(g) = x1 p(n__f) = 1 + x1 p(n__g) = x1 p(s) = x1 p(sel) = 2*x1 + x1*x2^2 + 2*x1^2*x2 + x1^3 + x2^2 + x2^3 p(activate#) = x1^3 p(f#) = 1 + 2*x1^2 p(g#) = 0 p(sel#) = 3*x1 + 2*x2^3 p(c_1) = 2 p(c_2) = x1 p(c_3) = x1 + x2 p(c_4) = 0 p(c_5) = 0 p(c_6) = 0 p(c_7) = 0 p(c_8) = x1 p(c_9) = 1 p(c_10) = 0 Following rules are strictly oriented: activate#(n__f(X)) = 1 + 3*X + 3*X^2 + X^3 > X^3 = c_2(activate#(X)) Following rules are (at-least) weakly oriented: activate#(n__g(X)) = X^3 >= X^3 = c_3(g#(activate(X)),activate#(X)) g#(s(X)) = 0 >= 0 = c_8(g#(X)) sel#(s(X),cons(Y,Z)) = 3*X + 2*Z^3 >= Z^3 = activate#(Z) sel#(s(X),cons(Y,Z)) = 3*X + 2*Z^3 >= 3*X + 2*Z^3 = sel#(X,activate(Z)) activate(X) = X >= X = X activate(n__f(X)) = 1 + X >= 1 + X = f(activate(X)) activate(n__g(X)) = X >= X = g(activate(X)) f(X) = 1 + X >= 1 + X = cons(X,n__f(n__g(X))) f(X) = 1 + X >= 1 + X = n__f(X) g(X) = X >= X = n__g(X) g(0()) = 0 >= 0 = s(0()) g(s(X)) = X >= X = s(s(g(X))) *** Step 6.a:1.b:2: Failure MAYBE + Considered Problem: - Strict DPs: activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) g#(s(X)) -> c_8(g#(X)) - Weak DPs: activate#(n__f(X)) -> c_2(activate#(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(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> cons(X,n__f(n__g(X))) f(X) -> n__f(X) g(X) -> n__g(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,n__g/1,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,n__g ,s} + Applied Processor: EmptyProcessor + Details: The problem is still open. ** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) - Weak DPs: activate#(n__f(X)) -> c_2(activate#(X)) activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) g#(s(X)) -> c_8(g#(X)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> cons(X,n__f(n__g(X))) f(X) -> n__f(X) g(X) -> n__g(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,n__g/1,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,n__g ,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) -->_2 activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)):3 -->_2 activate#(n__f(X)) -> c_2(activate#(X)):2 -->_1 sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)):1 2:W:activate#(n__f(X)) -> c_2(activate#(X)) -->_1 activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)):3 -->_1 activate#(n__f(X)) -> c_2(activate#(X)):2 3:W:activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) -->_1 g#(s(X)) -> c_8(g#(X)):4 -->_2 activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)):3 -->_2 activate#(n__f(X)) -> c_2(activate#(X)):2 4:W:g#(s(X)) -> c_8(g#(X)) -->_1 g#(s(X)) -> c_8(g#(X)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)) 2: activate#(n__f(X)) -> c_2(activate#(X)) 4: g#(s(X)) -> c_8(g#(X)) ** Step 6.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> cons(X,n__f(n__g(X))) f(X) -> n__f(X) g(X) -> n__g(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,n__g/1,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,n__g ,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) -->_1 sel#(s(X),cons(Y,Z)) -> c_10(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_10(sel#(X,activate(Z))) ** Step 6.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z))) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> cons(X,n__f(n__g(X))) f(X) -> n__f(X) g(X) -> n__g(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,n__g/1,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,n__g ,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_10(sel#(X,activate(Z))) The strictly oriented rules are moved into the weak component. *** Step 6.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z))) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> cons(X,n__f(n__g(X))) f(X) -> n__f(X) g(X) -> n__g(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,n__g/1,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,n__g ,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_10) = {1} Following symbols are considered usable: {activate#,f#,g#,sel#} TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] p(cons) = [0] p(f) = [1] x1 + [0] p(g) = [8] p(n__f) = [14] p(n__g) = [1] x1 + [0] p(s) = [1] x1 + [4] p(sel) = [2] x1 + [2] x2 + [4] p(activate#) = [1] p(f#) = [1] x1 + [0] p(g#) = [1] x1 + [8] p(sel#) = [1] x1 + [14] p(c_1) = [0] p(c_2) = [8] x1 + [1] p(c_3) = [1] x1 + [1] x2 + [1] p(c_4) = [2] p(c_5) = [1] p(c_6) = [2] p(c_7) = [1] p(c_8) = [2] x1 + [8] p(c_9) = [1] p(c_10) = [1] x1 + [0] Following rules are strictly oriented: sel#(s(X),cons(Y,Z)) = [1] X + [18] > [1] X + [14] = c_10(sel#(X,activate(Z))) Following rules are (at-least) weakly oriented: *** Step 6.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z))) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> cons(X,n__f(n__g(X))) f(X) -> n__f(X) g(X) -> n__g(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,n__g/1,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,n__g ,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 6.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z))) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> cons(X,n__f(n__g(X))) f(X) -> n__f(X) g(X) -> n__g(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,n__g/1,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,n__g ,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z))) -->_1 sel#(s(X),cons(Y,Z)) -> c_10(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_10(sel#(X,activate(Z))) *** Step 6.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> cons(X,n__f(n__g(X))) f(X) -> n__f(X) g(X) -> n__g(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,n__g/1,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,n__g ,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). MAYBE