*** 1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: 2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1))) 2nd(cons1(X,cons(Y,Z))) -> Y 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: {2nd/1,activate/1,from/1} / {cons/2,cons1/2,n__from/1,s/1} Obligation: Full basic terms: {2nd,activate,from}/{cons,cons1,n__from,s} Applied Processor: DependencyPairs {dpKind_ = WIDP} Proof: We add the following weak dependency pairs: Strict DPs 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1)))) 2nd#(cons1(X,cons(Y,Z))) -> c_2(Y) activate#(X) -> c_3(X) activate#(n__from(X)) -> c_4(from#(X)) from#(X) -> c_5(X,X) from#(X) -> c_6(X) Weak DPs and mark the set of starting terms. *** 1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1)))) 2nd#(cons1(X,cons(Y,Z))) -> c_2(Y) activate#(X) -> c_3(X) activate#(n__from(X)) -> c_4(from#(X)) from#(X) -> c_5(X,X) from#(X) -> c_6(X) Strict TRS Rules: 2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1))) 2nd(cons1(X,cons(Y,Z))) -> Y 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: {2nd/1,activate/1,from/1,2nd#/1,activate#/1,from#/1} / {cons/2,cons1/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1} Obligation: Full basic terms: {2nd#,activate#,from#}/{cons,cons1,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) 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1)))) 2nd#(cons1(X,cons(Y,Z))) -> c_2(Y) activate#(X) -> c_3(X) activate#(n__from(X)) -> c_4(from#(X)) from#(X) -> c_5(X,X) from#(X) -> c_6(X) *** 1.1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1)))) 2nd#(cons1(X,cons(Y,Z))) -> c_2(Y) activate#(X) -> c_3(X) activate#(n__from(X)) -> c_4(from#(X)) from#(X) -> c_5(X,X) from#(X) -> c_6(X) 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: {2nd/1,activate/1,from/1,2nd#/1,activate#/1,from#/1} / {cons/2,cons1/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1} Obligation: Full basic terms: {2nd#,activate#,from#}/{cons,cons1,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(cons1) = {2}, uargs(2nd#) = {1}, uargs(c_1) = {1}, uargs(c_4) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(2nd) = [0] p(activate) = [1] x1 + [15] p(cons) = [1] x2 + [5] p(cons1) = [1] x2 + [0] p(from) = [1] x1 + [6] p(n__from) = [1] x1 + [0] p(s) = [1] x1 + [0] p(2nd#) = [1] x1 + [0] p(activate#) = [0] p(from#) = [0] p(c_1) = [1] x1 + [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [0] Following rules are strictly oriented: 2nd#(cons1(X,cons(Y,Z))) = [1] Z + [5] > [0] = c_2(Y) activate(X) = [1] X + [15] > [1] X + [0] = X activate(n__from(X)) = [1] X + [15] > [1] X + [6] = from(X) from(X) = [1] X + [6] > [1] X + [5] = cons(X,n__from(s(X))) from(X) = [1] X + [6] > [1] X + [0] = n__from(X) Following rules are (at-least) weakly oriented: 2nd#(cons(X,X1)) = [1] X1 + [5] >= [1] X1 + [15] = c_1(2nd#(cons1(X,activate(X1)))) activate#(X) = [0] >= [0] = c_3(X) activate#(n__from(X)) = [0] >= [0] = c_4(from#(X)) from#(X) = [0] >= [0] = c_5(X,X) from#(X) = [0] >= [0] = c_6(X) 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: 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1)))) activate#(X) -> c_3(X) activate#(n__from(X)) -> c_4(from#(X)) from#(X) -> c_5(X,X) from#(X) -> c_6(X) Strict TRS Rules: Weak DP Rules: 2nd#(cons1(X,cons(Y,Z))) -> c_2(Y) 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: {2nd/1,activate/1,from/1,2nd#/1,activate#/1,from#/1} / {cons/2,cons1/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1} Obligation: Full basic terms: {2nd#,activate#,from#}/{cons,cons1,n__from,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, greedy = NoGreedy}} Proof: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly: 1: 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1)))) 2: activate#(X) -> c_3(X) 3: activate#(n__from(X)) -> c_4(from#(X)) 5: from#(X) -> c_6(X) The strictly oriented rules are moved into the weak component. *** 1.1.1.1.1 Progress [(?,O(1))] *** Considered Problem: Strict DP Rules: 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1)))) activate#(X) -> c_3(X) activate#(n__from(X)) -> c_4(from#(X)) from#(X) -> c_5(X,X) from#(X) -> c_6(X) Strict TRS Rules: Weak DP Rules: 2nd#(cons1(X,cons(Y,Z))) -> c_2(Y) 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: {2nd/1,activate/1,from/1,2nd#/1,activate#/1,from#/1} / {cons/2,cons1/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1} Obligation: Full basic terms: {2nd#,activate#,from#}/{cons,cons1,n__from,s} Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, 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: The following argument positions are considered usable: uargs(c_1) = {1}, uargs(c_4) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(2nd) = [8] p(activate) = [4] x1 + [1] p(cons) = [4] p(cons1) = [0] p(from) = [4] p(n__from) = [4] p(s) = [1] p(2nd#) = [4] x1 + [0] p(activate#) = [9] p(from#) = [3] p(c_1) = [4] x1 + [8] p(c_2) = [0] p(c_3) = [5] p(c_4) = [1] x1 + [3] p(c_5) = [3] p(c_6) = [0] Following rules are strictly oriented: 2nd#(cons(X,X1)) = [16] > [8] = c_1(2nd#(cons1(X,activate(X1)))) activate#(X) = [9] > [5] = c_3(X) activate#(n__from(X)) = [9] > [6] = c_4(from#(X)) from#(X) = [3] > [0] = c_6(X) Following rules are (at-least) weakly oriented: 2nd#(cons1(X,cons(Y,Z))) = [0] >= [0] = c_2(Y) from#(X) = [3] >= [3] = c_5(X,X) activate(X) = [4] X + [1] >= [1] X + [0] = X activate(n__from(X)) = [17] >= [4] = from(X) from(X) = [4] >= [4] = cons(X,n__from(s(X))) from(X) = [4] >= [4] = n__from(X) *** 1.1.1.1.1.1 Progress [(?,O(1))] *** Considered Problem: Strict DP Rules: from#(X) -> c_5(X,X) Strict TRS Rules: Weak DP Rules: 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1)))) 2nd#(cons1(X,cons(Y,Z))) -> c_2(Y) activate#(X) -> c_3(X) activate#(n__from(X)) -> c_4(from#(X)) from#(X) -> c_6(X) 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: {2nd/1,activate/1,from/1,2nd#/1,activate#/1,from#/1} / {cons/2,cons1/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1} Obligation: Full basic terms: {2nd#,activate#,from#}/{cons,cons1,n__from,s} Applied Processor: Assumption Proof: () *** 1.1.1.1.2 Progress [(?,O(1))] *** Considered Problem: Strict DP Rules: from#(X) -> c_5(X,X) Strict TRS Rules: Weak DP Rules: 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1)))) 2nd#(cons1(X,cons(Y,Z))) -> c_2(Y) activate#(X) -> c_3(X) activate#(n__from(X)) -> c_4(from#(X)) from#(X) -> c_6(X) 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: {2nd/1,activate/1,from/1,2nd#/1,activate#/1,from#/1} / {cons/2,cons1/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1} Obligation: Full basic terms: {2nd#,activate#,from#}/{cons,cons1,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: 1: from#(X) -> c_5(X,X) The strictly oriented rules are moved into the weak component. *** 1.1.1.1.2.1 Progress [(?,O(1))] *** Considered Problem: Strict DP Rules: from#(X) -> c_5(X,X) Strict TRS Rules: Weak DP Rules: 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1)))) 2nd#(cons1(X,cons(Y,Z))) -> c_2(Y) activate#(X) -> c_3(X) activate#(n__from(X)) -> c_4(from#(X)) from#(X) -> c_6(X) 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: {2nd/1,activate/1,from/1,2nd#/1,activate#/1,from#/1} / {cons/2,cons1/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1} Obligation: Full basic terms: {2nd#,activate#,from#}/{cons,cons1,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_1) = {1}, uargs(c_4) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(2nd) = [8] x1 + [8] p(activate) = [1] x1 + [8] p(cons) = [10] p(cons1) = [0] p(from) = [12] p(n__from) = [9] p(s) = [0] p(2nd#) = [2] x1 + [1] p(activate#) = [1] x1 + [0] p(from#) = [9] p(c_1) = [8] x1 + [0] p(c_2) = [1] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [5] p(c_6) = [9] Following rules are strictly oriented: from#(X) = [9] > [5] = c_5(X,X) Following rules are (at-least) weakly oriented: 2nd#(cons(X,X1)) = [21] >= [8] = c_1(2nd#(cons1(X,activate(X1)))) 2nd#(cons1(X,cons(Y,Z))) = [1] >= [1] = c_2(Y) activate#(X) = [1] X + [0] >= [1] X + [0] = c_3(X) activate#(n__from(X)) = [9] >= [9] = c_4(from#(X)) from#(X) = [9] >= [9] = c_6(X) activate(X) = [1] X + [8] >= [1] X + [0] = X activate(n__from(X)) = [17] >= [12] = from(X) from(X) = [12] >= [10] = cons(X,n__from(s(X))) from(X) = [12] >= [9] = n__from(X) *** 1.1.1.1.2.1.1 Progress [(?,O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: Weak DP Rules: 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1)))) 2nd#(cons1(X,cons(Y,Z))) -> c_2(Y) activate#(X) -> c_3(X) activate#(n__from(X)) -> c_4(from#(X)) from#(X) -> c_5(X,X) from#(X) -> c_6(X) 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: {2nd/1,activate/1,from/1,2nd#/1,activate#/1,from#/1} / {cons/2,cons1/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1} Obligation: Full basic terms: {2nd#,activate#,from#}/{cons,cons1,n__from,s} Applied Processor: Assumption Proof: () *** 1.1.1.1.2.2 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: Weak DP Rules: 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1)))) 2nd#(cons1(X,cons(Y,Z))) -> c_2(Y) activate#(X) -> c_3(X) activate#(n__from(X)) -> c_4(from#(X)) from#(X) -> c_5(X,X) from#(X) -> c_6(X) 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: {2nd/1,activate/1,from/1,2nd#/1,activate#/1,from#/1} / {cons/2,cons1/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1} Obligation: Full basic terms: {2nd#,activate#,from#}/{cons,cons1,n__from,s} Applied Processor: RemoveWeakSuffixes Proof: Consider the dependency graph 1:W:2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1)))) -->_1 2nd#(cons1(X,cons(Y,Z))) -> c_2(Y):2 2:W:2nd#(cons1(X,cons(Y,Z))) -> c_2(Y) -->_1 from#(X) -> c_6(X):6 -->_1 from#(X) -> c_5(X,X):5 -->_1 activate#(n__from(X)) -> c_4(from#(X)):4 -->_1 activate#(X) -> c_3(X):3 -->_1 2nd#(cons1(X,cons(Y,Z))) -> c_2(Y):2 -->_1 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1)))):1 3:W:activate#(X) -> c_3(X) -->_1 from#(X) -> c_6(X):6 -->_1 from#(X) -> c_5(X,X):5 -->_1 activate#(n__from(X)) -> c_4(from#(X)):4 -->_1 activate#(X) -> c_3(X):3 -->_1 2nd#(cons1(X,cons(Y,Z))) -> c_2(Y):2 -->_1 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1)))):1 4:W:activate#(n__from(X)) -> c_4(from#(X)) -->_1 from#(X) -> c_6(X):6 -->_1 from#(X) -> c_5(X,X):5 5:W:from#(X) -> c_5(X,X) -->_2 from#(X) -> c_6(X):6 -->_1 from#(X) -> c_6(X):6 -->_2 from#(X) -> c_5(X,X):5 -->_1 from#(X) -> c_5(X,X):5 -->_2 activate#(n__from(X)) -> c_4(from#(X)):4 -->_1 activate#(n__from(X)) -> c_4(from#(X)):4 -->_2 activate#(X) -> c_3(X):3 -->_1 activate#(X) -> c_3(X):3 -->_2 2nd#(cons1(X,cons(Y,Z))) -> c_2(Y):2 -->_1 2nd#(cons1(X,cons(Y,Z))) -> c_2(Y):2 -->_2 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1)))):1 -->_1 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1)))):1 6:W:from#(X) -> c_6(X) -->_1 from#(X) -> c_6(X):6 -->_1 from#(X) -> c_5(X,X):5 -->_1 activate#(n__from(X)) -> c_4(from#(X)):4 -->_1 activate#(X) -> c_3(X):3 -->_1 2nd#(cons1(X,cons(Y,Z))) -> c_2(Y):2 -->_1 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1)))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1)))) 6: from#(X) -> c_6(X) 5: from#(X) -> c_5(X,X) 4: activate#(n__from(X)) -> c_4(from#(X)) 3: activate#(X) -> c_3(X) 2: 2nd#(cons1(X,cons(Y,Z))) -> c_2(Y) *** 1.1.1.1.2.2.1 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: {2nd/1,activate/1,from/1,2nd#/1,activate#/1,from#/1} / {cons/2,cons1/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1} Obligation: Full basic terms: {2nd#,activate#,from#}/{cons,cons1,n__from,s} Applied Processor: EmptyProcessor Proof: The problem is already closed. The intended complexity is O(1).