*** 1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> n__f(X) f(0()) -> cons(0(),n__f(s(0()))) f(s(0())) -> f(p(s(0()))) p(s(0())) -> 0() Weak DP Rules: Weak TRS Rules: Signature: {activate/1,f/1,p/1} / {0/0,cons/2,n__f/1,s/1} Obligation: Full basic terms: {activate,f,p}/{0,cons,n__f,s} Applied Processor: ToInnermost Proof: switch to innermost, as the system is overlay and right linear and does not contain weak rules *** 1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> n__f(X) f(0()) -> cons(0(),n__f(s(0()))) f(s(0())) -> f(p(s(0()))) p(s(0())) -> 0() Weak DP Rules: Weak TRS Rules: Signature: {activate/1,f/1,p/1} / {0/0,cons/2,n__f/1,s/1} Obligation: Innermost basic terms: {activate,f,p}/{0,cons,n__f,s} Applied Processor: DependencyPairs {dpKind_ = DT} Proof: We add the following dependency tuples: Strict DPs activate#(X) -> c_1() activate#(n__f(X)) -> c_2(f#(X)) f#(X) -> c_3() f#(0()) -> c_4() f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0()))) p#(s(0())) -> c_6() Weak DPs and mark the set of starting terms. *** 1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: activate#(X) -> c_1() activate#(n__f(X)) -> c_2(f#(X)) f#(X) -> c_3() f#(0()) -> c_4() f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0()))) p#(s(0())) -> c_6() Strict TRS Rules: Weak DP Rules: Weak TRS Rules: activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> n__f(X) f(0()) -> cons(0(),n__f(s(0()))) f(s(0())) -> f(p(s(0()))) p(s(0())) -> 0() Signature: {activate/1,f/1,p/1,activate#/1,f#/1,p#/1} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2,c_6/0} Obligation: Innermost basic terms: {activate#,f#,p#}/{0,cons,n__f,s} Applied Processor: UsableRules Proof: We replace rewrite rules by usable rules: p(s(0())) -> 0() activate#(X) -> c_1() activate#(n__f(X)) -> c_2(f#(X)) f#(X) -> c_3() f#(0()) -> c_4() f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0()))) p#(s(0())) -> c_6() *** 1.1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: activate#(X) -> c_1() activate#(n__f(X)) -> c_2(f#(X)) f#(X) -> c_3() f#(0()) -> c_4() f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0()))) p#(s(0())) -> c_6() Strict TRS Rules: Weak DP Rules: Weak TRS Rules: p(s(0())) -> 0() Signature: {activate/1,f/1,p/1,activate#/1,f#/1,p#/1} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2,c_6/0} Obligation: Innermost basic terms: {activate#,f#,p#}/{0,cons,n__f,s} Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} Proof: We estimate the number of application of {1,3,4,6} by application of Pre({1,3,4,6}) = {2,5}. 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() 4: f#(0()) -> c_4() 5: f#(s(0())) -> c_5(f#(p(s(0()))) ,p#(s(0()))) 6: p#(s(0())) -> c_6() *** 1.1.1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: activate#(n__f(X)) -> c_2(f#(X)) f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0()))) Strict TRS Rules: Weak DP Rules: activate#(X) -> c_1() f#(X) -> c_3() f#(0()) -> c_4() p#(s(0())) -> c_6() Weak TRS Rules: p(s(0())) -> 0() Signature: {activate/1,f/1,p/1,activate#/1,f#/1,p#/1} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2,c_6/0} Obligation: Innermost basic terms: {activate#,f#,p#}/{0,cons,n__f,s} Applied Processor: RemoveWeakSuffixes Proof: Consider the dependency graph 1:S:activate#(n__f(X)) -> c_2(f#(X)) -->_1 f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0()))):2 -->_1 f#(0()) -> c_4():5 -->_1 f#(X) -> c_3():4 2:S:f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0()))) -->_2 p#(s(0())) -> c_6():6 -->_1 f#(0()) -> c_4():5 -->_1 f#(X) -> c_3():4 -->_1 f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0()))):2 3:W:activate#(X) -> c_1() 4:W:f#(X) -> c_3() 5:W:f#(0()) -> c_4() 6:W:p#(s(0())) -> c_6() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: activate#(X) -> c_1() 4: f#(X) -> c_3() 5: f#(0()) -> c_4() 6: p#(s(0())) -> c_6() *** 1.1.1.1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: activate#(n__f(X)) -> c_2(f#(X)) f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0()))) Strict TRS Rules: Weak DP Rules: Weak TRS Rules: p(s(0())) -> 0() Signature: {activate/1,f/1,p/1,activate#/1,f#/1,p#/1} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2,c_6/0} Obligation: Innermost basic terms: {activate#,f#,p#}/{0,cons,n__f,s} Applied Processor: SimplifyRHS Proof: Consider the dependency graph 1:S:activate#(n__f(X)) -> c_2(f#(X)) -->_1 f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0()))):2 2:S:f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0()))) -->_1 f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0()))):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: f#(s(0())) -> c_5(f#(p(s(0())))) *** 1.1.1.1.1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: activate#(n__f(X)) -> c_2(f#(X)) f#(s(0())) -> c_5(f#(p(s(0())))) Strict TRS Rules: Weak DP Rules: Weak TRS Rules: p(s(0())) -> 0() Signature: {activate/1,f/1,p/1,activate#/1,f#/1,p#/1} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0} Obligation: Innermost basic terms: {activate#,f#,p#}/{0,cons,n__f,s} Applied Processor: RemoveHeads Proof: Consider the dependency graph 1:S:activate#(n__f(X)) -> c_2(f#(X)) -->_1 f#(s(0())) -> c_5(f#(p(s(0())))):2 2:S:f#(s(0())) -> c_5(f#(p(s(0())))) -->_1 f#(s(0())) -> c_5(f#(p(s(0())))):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,activate#(n__f(X)) -> c_2(f#(X)))] *** 1.1.1.1.1.1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: f#(s(0())) -> c_5(f#(p(s(0())))) Strict TRS Rules: Weak DP Rules: Weak TRS Rules: p(s(0())) -> 0() Signature: {activate/1,f/1,p/1,activate#/1,f#/1,p#/1} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0} Obligation: Innermost basic terms: {activate#,f#,p#}/{0,cons,n__f,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following constant growth matrix-interpretation: 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(f#) = {1}, uargs(c_5) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [0] p(activate) = [0] p(cons) = [0] p(f) = [0] p(n__f) = [0] p(p) = [0] p(s) = [3] p(activate#) = [0] p(f#) = [1] x1 + [0] p(p#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [1] x1 + [0] p(c_6) = [0] Following rules are strictly oriented: f#(s(0())) = [3] > [0] = c_5(f#(p(s(0())))) Following rules are (at-least) weakly oriented: p(s(0())) = [0] >= [0] = 0() Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.1.1.1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: Weak DP Rules: f#(s(0())) -> c_5(f#(p(s(0())))) Weak TRS Rules: p(s(0())) -> 0() Signature: {activate/1,f/1,p/1,activate#/1,f#/1,p#/1} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0} Obligation: Innermost basic terms: {activate#,f#,p#}/{0,cons,n__f,s} Applied Processor: EmptyProcessor Proof: The problem is already closed. The intended complexity is O(1).