*** 1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) sel(0(),cons(X,Y)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) Weak DP Rules: Weak TRS Rules: Signature: {activate/1,from/1,sel/2} / {0/0,cons/2,n__from/1,s/1} Obligation: Innermost basic terms: {activate,from,sel}/{0,cons,n__from,s} Applied Processor: DependencyPairs {dpKind_ = WIDP} Proof: We add the following weak innermost dependency pairs: Strict DPs activate#(X) -> c_1() activate#(n__from(X)) -> c_2(from#(X)) from#(X) -> c_3() from#(X) -> c_4() sel#(0(),cons(X,Y)) -> c_5() sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) Weak DPs and mark the set of starting terms. *** 1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: activate#(X) -> c_1() activate#(n__from(X)) -> c_2(from#(X)) from#(X) -> c_3() from#(X) -> c_4() sel#(0(),cons(X,Y)) -> c_5() sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) Strict TRS Rules: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) sel(0(),cons(X,Y)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) Weak DP Rules: Weak TRS Rules: Signature: {activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0,c_6/1} Obligation: Innermost basic terms: {activate#,from#,sel#}/{0,cons,n__from,s} Applied Processor: UsableRules Proof: We replace rewrite rules by usable rules: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) activate#(X) -> c_1() activate#(n__from(X)) -> c_2(from#(X)) from#(X) -> c_3() from#(X) -> c_4() sel#(0(),cons(X,Y)) -> c_5() sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) *** 1.1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: activate#(X) -> c_1() activate#(n__from(X)) -> c_2(from#(X)) from#(X) -> c_3() from#(X) -> c_4() sel#(0(),cons(X,Y)) -> c_5() sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) Strict TRS Rules: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) Weak DP Rules: Weak TRS Rules: Signature: {activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0,c_6/1} Obligation: Innermost basic terms: {activate#,from#,sel#}/{0,cons,n__from,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs} Proof: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(sel#) = {2}, uargs(c_2) = {1}, uargs(c_6) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [7] p(cons) = [1] x2 + [0] p(from) = [1] x1 + [7] p(n__from) = [1] x1 + [1] p(s) = [1] x1 + [3] p(sel) = [0] p(activate#) = [3] x1 + [0] p(from#) = [3] x1 + [0] p(sel#) = [8] x1 + [1] x2 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] Following rules are strictly oriented: activate#(n__from(X)) = [3] X + [3] > [3] X + [0] = c_2(from#(X)) sel#(s(X),cons(Y,Z)) = [8] X + [1] Z + [24] > [8] X + [1] Z + [7] = c_6(sel#(X,activate(Z))) activate(X) = [1] X + [7] > [1] X + [0] = X activate(n__from(X)) = [1] X + [8] > [1] X + [7] = from(X) from(X) = [1] X + [7] > [1] X + [4] = cons(X,n__from(s(X))) from(X) = [1] X + [7] > [1] X + [1] = n__from(X) Following rules are (at-least) weakly oriented: activate#(X) = [3] X + [0] >= [0] = c_1() from#(X) = [3] X + [0] >= [0] = c_3() from#(X) = [3] X + [0] >= [0] = c_4() sel#(0(),cons(X,Y)) = [1] Y + [0] >= [0] = c_5() Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1 Progress [(?,O(1))] *** Considered Problem: Strict DP Rules: activate#(X) -> c_1() from#(X) -> c_3() from#(X) -> c_4() sel#(0(),cons(X,Y)) -> c_5() Strict TRS Rules: Weak DP Rules: activate#(n__from(X)) -> c_2(from#(X)) sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) Weak TRS Rules: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) Signature: {activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0,c_6/1} Obligation: Innermost basic terms: {activate#,from#,sel#}/{0,cons,n__from,s} Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} Proof: We estimate the number of application of {1} by application of Pre({1}) = {}. Here rules are labelled as follows: 1: activate#(X) -> c_1() 2: from#(X) -> c_3() 3: from#(X) -> c_4() 4: sel#(0(),cons(X,Y)) -> c_5() 5: activate#(n__from(X)) -> c_2(from#(X)) 6: sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) *** 1.1.1.1.1 Progress [(?,O(1))] *** Considered Problem: Strict DP Rules: from#(X) -> c_3() from#(X) -> c_4() sel#(0(),cons(X,Y)) -> c_5() Strict TRS Rules: Weak DP Rules: activate#(X) -> c_1() activate#(n__from(X)) -> c_2(from#(X)) sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) Weak TRS Rules: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) Signature: {activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0,c_6/1} Obligation: Innermost basic terms: {activate#,from#,sel#}/{0,cons,n__from,s} Applied Processor: RemoveWeakSuffixes Proof: Consider the dependency graph 1:S:from#(X) -> c_3() 2:S:from#(X) -> c_4() 3:S:sel#(0(),cons(X,Y)) -> c_5() 4:W:activate#(X) -> c_1() 5:W:activate#(n__from(X)) -> c_2(from#(X)) -->_1 from#(X) -> c_4():2 -->_1 from#(X) -> c_3():1 6:W:sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) -->_1 sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))):6 -->_1 sel#(0(),cons(X,Y)) -> c_5():3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: activate#(X) -> c_1() *** 1.1.1.1.1.1 Progress [(?,O(1))] *** Considered Problem: Strict DP Rules: from#(X) -> c_3() from#(X) -> c_4() sel#(0(),cons(X,Y)) -> c_5() Strict TRS Rules: Weak DP Rules: activate#(n__from(X)) -> c_2(from#(X)) sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) Weak TRS Rules: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) Signature: {activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0,c_6/1} Obligation: Innermost basic terms: {activate#,from#,sel#}/{0,cons,n__from,s} Applied Processor: RemoveHeads Proof: Consider the dependency graph 1:S:from#(X) -> c_3() 2:S:from#(X) -> c_4() 3:S:sel#(0(),cons(X,Y)) -> c_5() 5:W:activate#(n__from(X)) -> c_2(from#(X)) -->_1 from#(X) -> c_4():2 -->_1 from#(X) -> c_3():1 6:W:sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) -->_1 sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))):6 -->_1 sel#(0(),cons(X,Y)) -> c_5():3 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(5,activate#(n__from(X)) -> c_2(from#(X)))] *** 1.1.1.1.1.1.1 Progress [(?,O(1))] *** Considered Problem: Strict DP Rules: from#(X) -> c_3() from#(X) -> c_4() sel#(0(),cons(X,Y)) -> c_5() Strict TRS Rules: Weak DP Rules: sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) Weak TRS Rules: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) Signature: {activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0,c_6/1} Obligation: Innermost basic terms: {activate#,from#,sel#}/{0,cons,n__from,s} Applied Processor: RemoveHeads Proof: Consider the dependency graph 1:S:from#(X) -> c_3() 2:S:from#(X) -> c_4() 3:S:sel#(0(),cons(X,Y)) -> c_5() 6:W:sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) -->_1 sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))):6 -->_1 sel#(0(),cons(X,Y)) -> c_5():3 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(1,from#(X) -> c_3()),(2,from#(X) -> c_4())] *** 1.1.1.1.1.1.1.1 Progress [(?,O(1))] *** Considered Problem: Strict DP Rules: sel#(0(),cons(X,Y)) -> c_5() Strict TRS Rules: Weak DP Rules: sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) Weak TRS Rules: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) Signature: {activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0,c_6/1} Obligation: Innermost basic terms: {activate#,from#,sel#}/{0,cons,n__from,s} Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}} Proof: We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly: 3: sel#(0(),cons(X,Y)) -> c_5() The strictly oriented rules are moved into the weak component. *** 1.1.1.1.1.1.1.1.1 Progress [(?,O(1))] *** Considered Problem: Strict DP Rules: sel#(0(),cons(X,Y)) -> c_5() Strict TRS Rules: Weak DP Rules: sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) Weak TRS Rules: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) Signature: {activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0,c_6/1} Obligation: Innermost basic terms: {activate#,from#,sel#}/{0,cons,n__from,s} Applied Processor: NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy} Proof: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_6) = {1} Following symbols are considered usable: {activate#,from#,sel#} TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] p(cons) = [0] p(from) = [1] x1 + [1] p(n__from) = [2] p(s) = [0] p(sel) = [1] x1 + [1] p(activate#) = [1] p(from#) = [0] p(sel#) = [1] p(c_1) = [8] p(c_2) = [2] p(c_3) = [1] p(c_4) = [1] p(c_5) = [0] p(c_6) = [1] x1 + [0] Following rules are strictly oriented: sel#(0(),cons(X,Y)) = [1] > [0] = c_5() Following rules are (at-least) weakly oriented: sel#(s(X),cons(Y,Z)) = [1] >= [1] = c_6(sel#(X,activate(Z))) *** 1.1.1.1.1.1.1.1.1.1 Progress [(?,O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: Weak DP Rules: sel#(0(),cons(X,Y)) -> c_5() sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) Weak TRS Rules: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) Signature: {activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0,c_6/1} Obligation: Innermost basic terms: {activate#,from#,sel#}/{0,cons,n__from,s} Applied Processor: Assumption Proof: () *** 1.1.1.1.1.1.1.1.2 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: Weak DP Rules: sel#(0(),cons(X,Y)) -> c_5() sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) Weak TRS Rules: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) Signature: {activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0,c_6/1} Obligation: Innermost basic terms: {activate#,from#,sel#}/{0,cons,n__from,s} Applied Processor: RemoveWeakSuffixes Proof: Consider the dependency graph 1:W:sel#(0(),cons(X,Y)) -> c_5() 2:W:sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) -->_1 sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))):2 -->_1 sel#(0(),cons(X,Y)) -> c_5():1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) 1: sel#(0(),cons(X,Y)) -> c_5() *** 1.1.1.1.1.1.1.1.2.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: Weak DP Rules: Weak TRS Rules: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) Signature: {activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0,c_6/1} Obligation: Innermost basic terms: {activate#,from#,sel#}/{0,cons,n__from,s} Applied Processor: EmptyProcessor Proof: The problem is already closed. The intended complexity is O(1).