WORST_CASE(?,O(n^1)) * Step 1: Sum. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 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} / {0/0,cons/2,n__f/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,f,p} and constructors {0,cons,n__f,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. MAYBE + Considered Problem: - Strict TRS: 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} / {0/0,cons/2,n__f/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,f,p} and constructors {0,cons,n__f,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: Ara. MAYBE + Considered Problem: - Strict TRS: 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} / {0/0,cons/2,n__f/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,f,p} and constructors {0,cons,n__f,s} + Applied Processor: Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False} + Details: Signatures used: ---------------- F (TrsFun "0") :: [] -(0)-> "A"(0, 0, 0) F (TrsFun "0") :: [] -(0)-> "A"(2, 2, 1) F (TrsFun "activate") :: ["A"(0, 0, 0)] -(1)-> "A"(0, 0, 0) F (TrsFun "cons") :: ["A"(0, 0, 0) x "A"(0, 0, 0)] -(0)-> "A"(0, 0, 0) F (TrsFun "f") :: ["A"(0, 0, 0)] -(1)-> "A"(0, 0, 0) F (TrsFun "main") :: ["A"(0, 0, 1)] -(1)-> "A"(0, 0, 0) F (TrsFun "n__f") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0) F (TrsFun "p") :: ["A"(1, 1, 1)] -(0)-> "A"(0, 0, 0) F (TrsFun "s") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0) F (TrsFun "s") :: ["A"(2, 2, 1)] -(1)-> "A"(1, 1, 1) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: 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() main(x1) -> p(x1) 2. Weak: ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 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} / {0/0,cons/2,n__f/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,f,p} and constructors {0,cons,n__f,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: 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. ** Step 1.b:2: PredecessorEstimation. WORST_CASE(?,O(n^1)) + Considered Problem: - 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 TRS: 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 runtime complexity wrt. defined symbols {activate#,f#,p#} and constructors {0,cons,n__f,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: 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() ** Step 1.b:3: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__f(X)) -> c_2(f#(X)) f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0()))) - Weak DPs: activate#(X) -> c_1() f#(X) -> c_3() f#(0()) -> c_4() p#(s(0())) -> c_6() - Weak TRS: 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 runtime complexity wrt. defined symbols {activate#,f#,p#} and constructors {0,cons,n__f,s} + Applied Processor: RemoveWeakSuffixes + Details: 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() ** Step 1.b:4: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__f(X)) -> c_2(f#(X)) f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0()))) - Weak TRS: 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 runtime complexity wrt. defined symbols {activate#,f#,p#} and constructors {0,cons,n__f,s} + Applied Processor: SimplifyRHS + Details: 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())))) ** Step 1.b:5: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__f(X)) -> c_2(f#(X)) f#(s(0())) -> c_5(f#(p(s(0())))) - Weak TRS: 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_5/1,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,p#} and constructors {0,cons,n__f,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: p(s(0())) -> 0() activate#(n__f(X)) -> c_2(f#(X)) f#(s(0())) -> c_5(f#(p(s(0())))) ** Step 1.b:6: RemoveHeads. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__f(X)) -> c_2(f#(X)) f#(s(0())) -> c_5(f#(p(s(0())))) - Weak TRS: 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 runtime complexity wrt. defined symbols {activate#,f#,p#} and constructors {0,cons,n__f,s} + Applied Processor: RemoveHeads + Details: 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)))] ** Step 1.b:7: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(s(0())) -> c_5(f#(p(s(0())))) - Weak TRS: 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 runtime complexity wrt. defined symbols {activate#,f#,p#} and constructors {0,cons,n__f,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_5) = {1} Following symbols are considered usable: {p,activate#,f#,p#} TcT has computed the following interpretation: p(0) = [3] p(activate) = [0] p(cons) = [1] x1 + [1] x2 + [0] p(f) = [0] p(n__f) = [0] p(p) = [3] p(s) = [8] p(activate#) = [2] 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) = [2] x1 + [0] p(c_6) = [0] Following rules are strictly oriented: f#(s(0())) = [8] > [6] = c_5(f#(p(s(0())))) Following rules are (at-least) weakly oriented: p(s(0())) = [3] >= [3] = 0() ** Step 1.b:8: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: f#(s(0())) -> c_5(f#(p(s(0())))) - Weak TRS: 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 runtime complexity wrt. defined symbols {activate#,f#,p#} and constructors {0,cons,n__f,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))