*** 1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: f(0()) -> cons(0()) f(s(0())) -> f(p(s(0()))) p(s(0())) -> 0() Weak DP Rules: Weak TRS Rules: Signature: {f/1,p/1} / {0/0,cons/1,s/1} Obligation: Full basic terms: {f,p}/{0,cons,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: f(0()) -> cons(0()) f(s(0())) -> f(p(s(0()))) p(s(0())) -> 0() Weak DP Rules: Weak TRS Rules: Signature: {f/1,p/1} / {0/0,cons/1,s/1} Obligation: Innermost basic terms: {f,p}/{0,cons,s} Applied Processor: DependencyPairs {dpKind_ = DT} Proof: We add the following dependency tuples: Strict DPs f#(0()) -> c_1() f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0()))) p#(s(0())) -> c_3() Weak DPs and mark the set of starting terms. *** 1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: f#(0()) -> c_1() f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0()))) p#(s(0())) -> c_3() Strict TRS Rules: Weak DP Rules: Weak TRS Rules: f(0()) -> cons(0()) f(s(0())) -> f(p(s(0()))) p(s(0())) -> 0() Signature: {f/1,p/1,f#/1,p#/1} / {0/0,cons/1,s/1,c_1/0,c_2/2,c_3/0} Obligation: Innermost basic terms: {f#,p#}/{0,cons,s} Applied Processor: UsableRules Proof: We replace rewrite rules by usable rules: p(s(0())) -> 0() f#(0()) -> c_1() f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0()))) p#(s(0())) -> c_3() *** 1.1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: f#(0()) -> c_1() f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0()))) p#(s(0())) -> c_3() Strict TRS Rules: Weak DP Rules: Weak TRS Rules: p(s(0())) -> 0() Signature: {f/1,p/1,f#/1,p#/1} / {0/0,cons/1,s/1,c_1/0,c_2/2,c_3/0} Obligation: Innermost basic terms: {f#,p#}/{0,cons,s} Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} Proof: We estimate the number of application of {1,3} by application of Pre({1,3}) = {2}. Here rules are labelled as follows: 1: f#(0()) -> c_1() 2: f#(s(0())) -> c_2(f#(p(s(0()))) ,p#(s(0()))) 3: p#(s(0())) -> c_3() *** 1.1.1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0()))) Strict TRS Rules: Weak DP Rules: f#(0()) -> c_1() p#(s(0())) -> c_3() Weak TRS Rules: p(s(0())) -> 0() Signature: {f/1,p/1,f#/1,p#/1} / {0/0,cons/1,s/1,c_1/0,c_2/2,c_3/0} Obligation: Innermost basic terms: {f#,p#}/{0,cons,s} Applied Processor: RemoveWeakSuffixes Proof: Consider the dependency graph 1:S:f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0()))) -->_2 p#(s(0())) -> c_3():3 -->_1 f#(0()) -> c_1():2 -->_1 f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0()))):1 2:W:f#(0()) -> c_1() 3:W:p#(s(0())) -> c_3() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: f#(0()) -> c_1() 3: p#(s(0())) -> c_3() *** 1.1.1.1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0()))) Strict TRS Rules: Weak DP Rules: Weak TRS Rules: p(s(0())) -> 0() Signature: {f/1,p/1,f#/1,p#/1} / {0/0,cons/1,s/1,c_1/0,c_2/2,c_3/0} Obligation: Innermost basic terms: {f#,p#}/{0,cons,s} Applied Processor: SimplifyRHS Proof: Consider the dependency graph 1:S:f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0()))) -->_1 f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0()))):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: f#(s(0())) -> c_2(f#(p(s(0())))) *** 1.1.1.1.1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: f#(s(0())) -> c_2(f#(p(s(0())))) Strict TRS Rules: Weak DP Rules: Weak TRS Rules: p(s(0())) -> 0() Signature: {f/1,p/1,f#/1,p#/1} / {0/0,cons/1,s/1,c_1/0,c_2/1,c_3/0} Obligation: Innermost basic terms: {f#,p#}/{0,cons,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_2) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [0] p(cons) = [0] p(f) = [0] p(p) = [0] p(s) = [1] p(f#) = [1] x1 + [0] p(p#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [0] Following rules are strictly oriented: f#(s(0())) = [1] > [0] = c_2(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 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: Weak DP Rules: f#(s(0())) -> c_2(f#(p(s(0())))) Weak TRS Rules: p(s(0())) -> 0() Signature: {f/1,p/1,f#/1,p#/1} / {0/0,cons/1,s/1,c_1/0,c_2/1,c_3/0} Obligation: Innermost basic terms: {f#,p#}/{0,cons,s} Applied Processor: EmptyProcessor Proof: The problem is already closed. The intended complexity is O(1).