*** 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(activate(X)) activate(n__s(X)) -> s(activate(X)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) Weak DP Rules: Weak TRS Rules: Signature: {2nd/1,activate/1,from/1,s/1} / {cons/2,cons1/2,n__from/1,n__s/1} Obligation: Innermost basic terms: {2nd,activate,from,s}/{cons,cons1,n__from,n__s} Applied Processor: DependencyPairs {dpKind_ = DT} Proof: We add the following dependency tuples: Strict DPs 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1)) 2nd#(cons1(X,cons(Y,Z))) -> c_2() activate#(X) -> c_3() activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) from#(X) -> c_6() from#(X) -> c_7() s#(X) -> c_8() 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))),activate#(X1)) 2nd#(cons1(X,cons(Y,Z))) -> c_2() activate#(X) -> c_3() activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) from#(X) -> c_6() from#(X) -> c_7() s#(X) -> c_8() Strict TRS Rules: Weak DP Rules: Weak 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(activate(X)) activate(n__s(X)) -> s(activate(X)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) Signature: {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/2,c_2/0,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0,c_8/0} Obligation: Innermost basic terms: {2nd#,activate#,from#,s#}/{cons,cons1,n__from,n__s} Applied Processor: UsableRules Proof: We replace rewrite rules by usable rules: activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1)) 2nd#(cons1(X,cons(Y,Z))) -> c_2() activate#(X) -> c_3() activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) from#(X) -> c_6() from#(X) -> c_7() s#(X) -> c_8() *** 1.1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1)) 2nd#(cons1(X,cons(Y,Z))) -> c_2() activate#(X) -> c_3() activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) from#(X) -> c_6() from#(X) -> c_7() s#(X) -> c_8() Strict TRS Rules: Weak DP Rules: Weak TRS Rules: activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) Signature: {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/2,c_2/0,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0,c_8/0} Obligation: Innermost basic terms: {2nd#,activate#,from#,s#}/{cons,cons1,n__from,n__s} Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} Proof: We estimate the number of application of {2,3,6,7,8} by application of Pre({2,3,6,7,8}) = {1,4,5}. Here rules are labelled as follows: 1: 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))) ,activate#(X1)) 2: 2nd#(cons1(X,cons(Y,Z))) -> c_2() 3: activate#(X) -> c_3() 4: activate#(n__from(X)) -> c_4(from#(activate(X)) ,activate#(X)) 5: activate#(n__s(X)) -> c_5(s#(activate(X)) ,activate#(X)) 6: from#(X) -> c_6() 7: from#(X) -> c_7() 8: s#(X) -> c_8() *** 1.1.1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1)) activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) Strict TRS Rules: Weak DP Rules: 2nd#(cons1(X,cons(Y,Z))) -> c_2() activate#(X) -> c_3() from#(X) -> c_6() from#(X) -> c_7() s#(X) -> c_8() Weak TRS Rules: activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) Signature: {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/2,c_2/0,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0,c_8/0} Obligation: Innermost basic terms: {2nd#,activate#,from#,s#}/{cons,cons1,n__from,n__s} Applied Processor: RemoveWeakSuffixes Proof: Consider the dependency graph 1:S:2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1)) -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):2 -->_2 activate#(X) -> c_3():5 -->_1 2nd#(cons1(X,cons(Y,Z))) -> c_2():4 2:S:activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3 -->_1 from#(X) -> c_7():7 -->_1 from#(X) -> c_6():6 -->_2 activate#(X) -> c_3():5 -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):2 3:S:activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) -->_1 s#(X) -> c_8():8 -->_2 activate#(X) -> c_3():5 -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):2 4:W:2nd#(cons1(X,cons(Y,Z))) -> c_2() 5:W:activate#(X) -> c_3() 6:W:from#(X) -> c_6() 7:W:from#(X) -> c_7() 8:W:s#(X) -> c_8() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: 2nd#(cons1(X,cons(Y,Z))) -> c_2() 6: from#(X) -> c_6() 7: from#(X) -> c_7() 5: activate#(X) -> c_3() 8: s#(X) -> c_8() *** 1.1.1.1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1)) activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) Strict TRS Rules: Weak DP Rules: Weak TRS Rules: activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) Signature: {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/2,c_2/0,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0,c_8/0} Obligation: Innermost basic terms: {2nd#,activate#,from#,s#}/{cons,cons1,n__from,n__s} Applied Processor: SimplifyRHS Proof: Consider the dependency graph 1:S:2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1)) -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):2 2:S:activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):2 3:S:activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: 2nd#(cons(X,X1)) -> c_1(activate#(X1)) activate#(n__from(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) *** 1.1.1.1.1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: 2nd#(cons(X,X1)) -> c_1(activate#(X1)) activate#(n__from(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) Strict TRS Rules: Weak DP Rules: Weak TRS Rules: activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) Signature: {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0} Obligation: Innermost basic terms: {2nd#,activate#,from#,s#}/{cons,cons1,n__from,n__s} Applied Processor: UsableRules Proof: We replace rewrite rules by usable rules: 2nd#(cons(X,X1)) -> c_1(activate#(X1)) activate#(n__from(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) *** 1.1.1.1.1.1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: 2nd#(cons(X,X1)) -> c_1(activate#(X1)) activate#(n__from(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) Strict TRS Rules: Weak DP Rules: Weak TRS Rules: Signature: {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0} Obligation: Innermost basic terms: {2nd#,activate#,from#,s#}/{cons,cons1,n__from,n__s} Applied Processor: RemoveHeads Proof: Consider the dependency graph 1:S:2nd#(cons(X,X1)) -> c_1(activate#(X1)) -->_1 activate#(n__s(X)) -> c_5(activate#(X)):3 -->_1 activate#(n__from(X)) -> c_4(activate#(X)):2 2:S:activate#(n__from(X)) -> c_4(activate#(X)) -->_1 activate#(n__s(X)) -> c_5(activate#(X)):3 -->_1 activate#(n__from(X)) -> c_4(activate#(X)):2 3:S:activate#(n__s(X)) -> c_5(activate#(X)) -->_1 activate#(n__s(X)) -> c_5(activate#(X)):3 -->_1 activate#(n__from(X)) -> c_4(activate#(X)):2 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). [(1,2nd#(cons(X,X1)) -> c_1(activate#(X1)))] *** 1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: activate#(n__from(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) Strict TRS Rules: Weak DP Rules: Weak TRS Rules: Signature: {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0} Obligation: Innermost basic terms: {2nd#,activate#,from#,s#}/{cons,cons1,n__from,n__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: 3: activate#(n__s(X)) -> c_5(activate#(X)) The strictly oriented rules are moved into the weak component. *** 1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: activate#(n__from(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) Strict TRS Rules: Weak DP Rules: Weak TRS Rules: Signature: {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0} Obligation: Innermost basic terms: {2nd#,activate#,from#,s#}/{cons,cons1,n__from,n__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_4) = {1}, uargs(c_5) = {1} Following symbols are considered usable: {2nd#,activate#,from#,s#} TcT has computed the following interpretation: p(2nd) = [1] x1 + [1] p(activate) = [1] p(cons) = [1] x1 + [0] p(cons1) = [1] x2 + [1] p(from) = [1] x1 + [8] p(n__from) = [1] x1 + [0] p(n__s) = [1] x1 + [2] p(s) = [1] x1 + [1] p(2nd#) = [1] x1 + [1] p(activate#) = [8] x1 + [0] p(from#) = [1] p(s#) = [0] p(c_1) = [1] x1 + [0] p(c_2) = [1] p(c_3) = [1] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [6] p(c_6) = [0] p(c_7) = [8] p(c_8) = [2] Following rules are strictly oriented: activate#(n__s(X)) = [8] X + [16] > [8] X + [6] = c_5(activate#(X)) Following rules are (at-least) weakly oriented: activate#(n__from(X)) = [8] X + [0] >= [8] X + [0] = c_4(activate#(X)) *** 1.1.1.1.1.1.1.1.1.1 Progress [(?,O(1))] *** Considered Problem: Strict DP Rules: activate#(n__from(X)) -> c_4(activate#(X)) Strict TRS Rules: Weak DP Rules: activate#(n__s(X)) -> c_5(activate#(X)) Weak TRS Rules: Signature: {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0} Obligation: Innermost basic terms: {2nd#,activate#,from#,s#}/{cons,cons1,n__from,n__s} Applied Processor: Assumption Proof: () *** 1.1.1.1.1.1.1.1.2 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: activate#(n__from(X)) -> c_4(activate#(X)) Strict TRS Rules: Weak DP Rules: activate#(n__s(X)) -> c_5(activate#(X)) Weak TRS Rules: Signature: {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0} Obligation: Innermost basic terms: {2nd#,activate#,from#,s#}/{cons,cons1,n__from,n__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: activate#(n__from(X)) -> c_4(activate#(X)) The strictly oriented rules are moved into the weak component. *** 1.1.1.1.1.1.1.1.2.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: activate#(n__from(X)) -> c_4(activate#(X)) Strict TRS Rules: Weak DP Rules: activate#(n__s(X)) -> c_5(activate#(X)) Weak TRS Rules: Signature: {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0} Obligation: Innermost basic terms: {2nd#,activate#,from#,s#}/{cons,cons1,n__from,n__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_4) = {1}, uargs(c_5) = {1} Following symbols are considered usable: {2nd#,activate#,from#,s#} TcT has computed the following interpretation: p(2nd) = [2] x1 + [1] p(activate) = [1] p(cons) = [0] p(cons1) = [1] x2 + [1] p(from) = [1] p(n__from) = [1] x1 + [8] p(n__s) = [1] x1 + [6] p(s) = [8] x1 + [0] p(2nd#) = [2] x1 + [2] p(activate#) = [1] x1 + [0] p(from#) = [4] p(s#) = [0] p(c_1) = [2] x1 + [1] p(c_2) = [1] p(c_3) = [1] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [0] p(c_6) = [2] p(c_7) = [1] p(c_8) = [0] Following rules are strictly oriented: activate#(n__from(X)) = [1] X + [8] > [1] X + [0] = c_4(activate#(X)) Following rules are (at-least) weakly oriented: activate#(n__s(X)) = [1] X + [6] >= [1] X + [0] = c_5(activate#(X)) *** 1.1.1.1.1.1.1.1.2.1.1 Progress [(?,O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: Weak DP Rules: activate#(n__from(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) Weak TRS Rules: Signature: {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0} Obligation: Innermost basic terms: {2nd#,activate#,from#,s#}/{cons,cons1,n__from,n__s} Applied Processor: Assumption Proof: () *** 1.1.1.1.1.1.1.1.2.2 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: Weak DP Rules: activate#(n__from(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) Weak TRS Rules: Signature: {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0} Obligation: Innermost basic terms: {2nd#,activate#,from#,s#}/{cons,cons1,n__from,n__s} Applied Processor: RemoveWeakSuffixes Proof: Consider the dependency graph 1:W:activate#(n__from(X)) -> c_4(activate#(X)) -->_1 activate#(n__s(X)) -> c_5(activate#(X)):2 -->_1 activate#(n__from(X)) -> c_4(activate#(X)):1 2:W:activate#(n__s(X)) -> c_5(activate#(X)) -->_1 activate#(n__s(X)) -> c_5(activate#(X)):2 -->_1 activate#(n__from(X)) -> c_4(activate#(X)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: activate#(n__from(X)) -> c_4(activate#(X)) 2: activate#(n__s(X)) -> c_5(activate#(X)) *** 1.1.1.1.1.1.1.1.2.2.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: Weak DP Rules: Weak TRS Rules: Signature: {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0} Obligation: Innermost basic terms: {2nd#,activate#,from#,s#}/{cons,cons1,n__from,n__s} Applied Processor: EmptyProcessor Proof: The problem is already closed. The intended complexity is O(1).